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Removing DESpecs directory which deserted to git
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Wed, 29 Nov 2017 16:55:37 +0100 |
| parents | 22d6a63640c6 |
| children |
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<?xml version="1.0" encoding="utf-8"?><echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" xmlns:dcterms="http://purl.org/dc/terms" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.0RC"> <metadata> <dcterms:identifier>ECHO:H8QSKVP5.xml</dcterms:identifier> <dcterms:creator identifier="GND:118648160">Ibn-al-Haitam, al-Hasan Ibn-al-Hasan</dcterms:creator> <dcterms:creator identifier="GND:118975765">Witelo</dcterms:creator> <dcterms:creator identifier="GND:124534112">Risner, Friedrich</dcterms:creator> <dcterms:alternative xml:lang="la">Opticae thesaurus Alhazeni Arabis libri septem, nunc primùm editi. Eiusdem liber De Crepusculis & Nubium ascensionibus. Item Vitellonis Thuringopoloni Libri X. Omnes instaurati, figuris illustrati & aucti, adiectis etiam in Alhazenum commentariis, a Federico Risnero</dcterms:alternative> <dcterms:title xml:lang="la">Opticae thesavrvs Alhazeni Arabis libri septem, nunc primùm editi. Eivsdem liber De Crepvscvlis & Nubium ascensionibus. Item Vitellonis Thuvringopoloni Libri X. Omnes instaurati, figuris illustrati & aucti, adiectis etiam in Alhazenum commentarijs, a Federico Risnero</dcterms:title> <dcterms:date xsi:type="dcterms:W3CDTF">1572</dcterms:date> <dcterms:language xsi:type="dcterms:ISO639-3">lat</dcterms:language> <dcterms:rights>open access</dcterms:rights> <dcterms:license>http://echo.mpiwg-berlin.mpg.de/policy/oa_basics/declaration</dcterms:license> <dcterms:accessRights>free</dcterms:accessRights> <echodir>/permanent/library/H8QSKVP5</echodir> <log> http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/mpiwg/online/permanent/library/H8QSKVP5/ </log> </metadata> <text xml:lang="lat" type="free"> <div xml:id="echoid-div1" type="section" level="0" n="0"><pb file="0001" n="1"/> </div> <div xml:id="echoid-div2" type="section" level="0" n="0"> <head xml:id="echoid-head1" xml:space="preserve"><emph style="sc">Optic ae</emph></head> <head xml:id="echoid-head2" xml:space="preserve">THE SAVRVS.</head> <head xml:id="echoid-head3" xml:space="preserve">ALHAZENI</head> <head xml:id="echoid-head4" xml:space="preserve">ARABIS</head> <head xml:id="echoid-head5" xml:space="preserve">libri ſeptem, nuncprimùm <lb/>editi.</head> <head xml:id="echoid-head6" xml:space="preserve" style="it">EIVSDEM liber DE CREPVSCVLIS <lb/>& Nubium aſcenſionibus.</head> <head xml:id="echoid-head7" xml:space="preserve">ITEM</head> <head xml:id="echoid-head8" xml:space="preserve">VITELLONIS</head> <head xml:id="echoid-head9" xml:space="preserve">THVRINGOPOLONI</head> <head xml:id="echoid-head10" xml:space="preserve">LIBRI X.</head> <head xml:id="echoid-head11" xml:space="preserve">Omnes inſtaurati, figuris illuſtrati & aucti, adiectis etiam in <lb/>Alhazenum commentarijs,</head> <head xml:id="echoid-head12" xml:space="preserve">A'</head> <head xml:id="echoid-head13" xml:space="preserve"><emph style="sc">Federico</emph> <emph style="sc">Risnero</emph>.</head> <figure> <image file="0001-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0001-01"/> </figure> </div> <div xml:id="echoid-div3" type="section" level="0" n="0"> <head xml:id="echoid-head14" xml:space="preserve" style="it">Cum priuilegio Cæſareo & Regis Galliæ ad ſexennium<gap/></head> <head xml:id="echoid-head15" xml:space="preserve">BASILE AE, <lb/>PER EPISCOPIOS. M D LXXII.</head> <figure> <description xml:id="echoid-description1" xml:space="preserve">CIN EMATH EQUE FRANCAISE</description> <description xml:id="echoid-description2" xml:space="preserve">BIBLIOTHEQUE MUSEE</description> </figure> <pb file="0002" n="2"/> </div> <div xml:id="echoid-div4" type="section" level="0" n="0"> <head xml:id="echoid-head16" xml:space="preserve">Triplicis uiſus, directi, reflexi & refracti, de <lb/>quo optica diſputat, ar-<lb/>gumenta.</head> <figure> <image file="0002-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0002-01"/> </figure> <figure> <image file="0002-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0002-02"/> </figure> <pb file="0003" n="3"/> </div> <div xml:id="echoid-div5" type="section" level="0" n="0"> <head xml:id="echoid-head17" xml:space="preserve">FEDERICI RISNE-<lb/>RI IN ALHAZENI ARABIS</head> <head xml:id="echoid-head18" xml:space="preserve">OPTICAM PRAEFATIO</head> <head xml:id="echoid-head19" xml:space="preserve">A D</head> <head xml:id="echoid-head20" xml:space="preserve" style="it">IL LVSTRISSIMAM REGINAM CA-<lb/>tharinam Mediceam, matrem regis Galliæ <lb/>Caroli noni.</head> <p> <s xml:id="echoid-s1" xml:space="preserve">PLVRES adorant orientem ſolem, Regina illu-<lb/>ſtriſsima, quàm occidentem, utuulgò fertur:</s> <s xml:id="echoid-s2" xml:space="preserve"> at <lb/>contrà fieri meo iudicio debuit, cum ſolis oc-<lb/>caſus ditiſsimas & opulentiſsimas orbis partes <lb/>complexus ſit:</s> <s xml:id="echoid-s3" xml:space="preserve"> utſub æquatore eſt America, & <lb/>ſupra omnes inſulas fortunatiſsimæ Moluccæ.</s> <s xml:id="echoid-s4" xml:space="preserve"> Etin uita ho <lb/>minis, ſi qua ætas pręcipuè laudabilis ſit, ipſa ſenectus eſt, re-<lb/>liquarum ætatum gubernatrix & magiſtra.</s> <s xml:id="echoid-s5" xml:space="preserve"> Ideoq́ hoc mihi <lb/>à P.</s> <s xml:id="echoid-s6" xml:space="preserve"> Ramo conſilium maieſtatis tuæ colendæ ſuccurrit:</s> <s xml:id="echoid-s7" xml:space="preserve"> qui <lb/>cum tuo no miniliberales artes gallico ſermone conſecra-<lb/>rit, multos profectò mortales exemplo ſuo inuitauit ad ea-<lb/>dem uota concipiendum atq;</s> <s xml:id="echoid-s8" xml:space="preserve"> nun cup an dum:</s> <s xml:id="echoid-s9" xml:space="preserve"> tanquam iu-<lb/>dicaritunam tein Gallia non ſolùm tanto rege matrem di-<lb/>gniſsimam, ſed omnium regiarum uirtutum laudumq́ pa-<lb/>tronam am antiſsim am.</s> <s xml:id="echoid-s10" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s11" xml:space="preserve"> tanquam clientem Alhazenũ <lb/>tibi dico, nun cupoq́, Arabem opticæ ſcriptorem, utèno-<lb/>mine Alhazen arabico, (quod Latinè bonum uirum ſonat) <lb/>intelligitur:</s> <s xml:id="echoid-s12" xml:space="preserve"> & inſcriptio operis ipſum arabepatre Alhayzen <lb/>natum indicat.</s> <s xml:id="echoid-s13" xml:space="preserve"> Quatuor autẽ philoſophos arabes hoc no-<lb/>mine fuiſſe è peritis Arabicæ linguæ hominibus didici:</s> <s xml:id="echoid-s14" xml:space="preserve"> quo <lb/>tamen tempore Alhazenus noſter floruerit, nondum neque <lb/>legendo neq;</s> <s xml:id="echoid-s15" xml:space="preserve"> percontando certò cognoſcere potui.</s> <s xml:id="echoid-s16" xml:space="preserve"> Equi-<lb/>dem à præſtantibus mathematicis in uetuſtiſsim orum Ara-<lb/>bum numero ipſum haberi animaduerto:</s> <s xml:id="echoid-s17" xml:space="preserve"> quanquam tem-<lb/>poris, quo uixerit, mentio nulla fiat.</s> <s xml:id="echoid-s18" xml:space="preserve"> Coniectura quædam <lb/>eſt anno Chriſti milleſimo ac circiter centeſimo ipſum ui-<lb/>xiſſe:</s> <s xml:id="echoid-s19" xml:space="preserve"> ætatenimirum Auicennæ, Auerrois, Zoaræ, aliorumq́;</s> <s xml:id="echoid-s20" xml:space="preserve"> <lb/>excellentium Arabum:</s> <s xml:id="echoid-s21" xml:space="preserve"> quo fæculo apud Arabes & Sarrace-<lb/>nos, tum omniuming enuarum artium ſtudia, tum uerò ma <lb/>thematicas diſciplinas inprimis floruiſſe ex hiſtoriarũ com-<lb/>mentarijs ſatis conſtat.</s> <s xml:id="echoid-s22" xml:space="preserve"> Huncigitur authorem (cuius editio <lb/>nem abhinc amplius annis triginta à clariſsimis matchema-<lb/> <pb file="0004" n="4" rhead="FEDERICI RISNERI"/> ticis expectauimus) cum P.</s> <s xml:id="echoid-s23" xml:space="preserve"> Ramus diu multumq́;</s> <s xml:id="echoid-s24" xml:space="preserve"> peruarias <lb/>bibliothecas requiſitum, ueſtigijsq́;</s> <s xml:id="echoid-s25" xml:space="preserve"> omnibus indagatũ tan <lb/>dem in auctione publica proſtitutu, & tanquam pro deſer-<lb/>to habitum coemiſſet:</s> <s xml:id="echoid-s26" xml:space="preserve"> alterũ poſtea etiam exemplar nactus <lb/>eſſet:</s> <s xml:id="echoid-s27" xml:space="preserve"> utrunq;</s> <s xml:id="echoid-s28" xml:space="preserve"> mihi (quem aliquot antè annos mathematicę <lb/>exercitationis conſortem & adiutorem habuiſſet) conferen <lb/>dum tradidit:</s> <s xml:id="echoid-s29" xml:space="preserve"> poſteaq́;</s> <s xml:id="echoid-s30" xml:space="preserve">, cum è medio renaſcentium bellorũ <lb/>ardore mathemata quædam ſibi chariora, & hunc Alhaze-<lb/>nũ inprimis è bibliotheca ſua ſub duxiſſet, Baſileam me cum <lb/>his tanquam penatibus ſecum abduxit, & annum integrum <lb/>in authore iſto reſtituẽdo & conformando occupauit.</s> <s xml:id="echoid-s31" xml:space="preserve"> Dili-<lb/>gentiam ſanè & doctrinam in arabe homine mirabilem de-<lb/>prehendi, nec admodum, quod animaduertere potuerim, à <lb/>ueteribus Græciæ opticis adiutã.</s> <s xml:id="echoid-s32" xml:space="preserve"> Euclideũ hic uel Ptolemai <lb/>cum nihil ferè eſt.</s> <s xml:id="echoid-s33" xml:space="preserve"> Aliquid fortaſſe ſumpſerat ex Archimede, <lb/>Apollonio & Auenello, à quibus optica quædam conſcri-<lb/>pta eſſe monimẽtis literarũ teſtatum extat:</s> <s xml:id="echoid-s34" xml:space="preserve"> item à Damiano <lb/>& alijs opticis, quorũ libri in manus meas nondum incide-<lb/>runt.</s> <s xml:id="echoid-s35" xml:space="preserve"> Veterum tamen opticorum lectionem Alhazenus ipſe <lb/>confitetur lib.</s> <s xml:id="echoid-s36" xml:space="preserve"> 6.</s> <s xml:id="echoid-s37" xml:space="preserve"> cap.</s> <s xml:id="echoid-s38" xml:space="preserve"> 4.</s> <s xml:id="echoid-s39" xml:space="preserve"> de errore, qui accidit in ſpeculis ſphę <lb/>ricis conuexis.</s> <s xml:id="echoid-s40" xml:space="preserve"> & lib.</s> <s xml:id="echoid-s41" xml:space="preserve"> 7.</s> <s xml:id="echoid-s42" xml:space="preserve"> cap.</s> <s xml:id="echoid-s43" xml:space="preserve"> 6.</s> <s xml:id="echoid-s44" xml:space="preserve"> quomodo uiſus cõprehendat <lb/>uiſibilia ſecundum refractionẽ.</s> <s xml:id="echoid-s45" xml:space="preserve"> Quam obrem cum luculen <lb/>tum quidem ſcriptorem & copioſum opticũ, ſed ualde con <lb/>fuſum perſpexiſſem:</s> <s xml:id="echoid-s46" xml:space="preserve"> id mihi P.</s> <s xml:id="echoid-s47" xml:space="preserve"> Ramo ſuaſore & authore con.</s> <s xml:id="echoid-s48" xml:space="preserve"> <lb/>ſilium ſumpſi:</s> <s xml:id="echoid-s49" xml:space="preserve"> quod eſt eiuſmodi.</s> <s xml:id="echoid-s50" xml:space="preserve"> Primò quia totũ opus in <lb/>pauca & prolixa capita continuo perpetuoq́;</s> <s xml:id="echoid-s51" xml:space="preserve"> ſermone diur <lb/>ſum fuit:</s> <s xml:id="echoid-s52" xml:space="preserve"> ſingulos libros & capita in propoſitiones diſtinxi:</s> <s xml:id="echoid-s53" xml:space="preserve"> <lb/>& quæ Vitellonis theoremata his reſponderent, annotaui:</s> <s xml:id="echoid-s54" xml:space="preserve"> <lb/>ut collatione theorematũ utriuſq;</s> <s xml:id="echoid-s55" xml:space="preserve"> optica materies rudi ac <lb/>nouitio lectori difficilior atq;</s> <s xml:id="echoid-s56" xml:space="preserve"> obſcurior lucem aliquam & <lb/>perſpicuitatem acciperet.</s> <s xml:id="echoid-s57" xml:space="preserve"> Deinde demonſtrationes omnes <lb/>emendaui & reſtitui:</s> <s xml:id="echoid-s58" xml:space="preserve"> earum firm amenta acrobora (quæ ple <lb/>riſq;</s> <s xml:id="echoid-s59" xml:space="preserve"> locis omnibus deerant) ex Euclide, Theodoſio, Apollo <lb/>nio, Sereno, alijsq́;</s> <s xml:id="echoid-s60" xml:space="preserve"> geometris addidi:</s> <s xml:id="echoid-s61" xml:space="preserve"> pręcipuè uerò quintũ <lb/>& ſextum libros, quibus catoptrica cõprehenduntur, & ſe-<lb/>ptimũ, qui refractionem interpretatur, propter demonſtra-<lb/>tionum obſcuritatem breuitatemq́;</s> <s xml:id="echoid-s62" xml:space="preserve"> cõmentariolis quibuſ-<lb/>dam illuſtrare conatus ſum.</s> <s xml:id="echoid-s63" xml:space="preserve"> Deniq;</s> <s xml:id="echoid-s64" xml:space="preserve"> figuras omnium propo <lb/>ſitionũ de integro conformaui.</s> <s xml:id="echoid-s65" xml:space="preserve"> Atq;</s> <s xml:id="echoid-s66" xml:space="preserve"> hic in rebus ſentẽtijsq́;</s> <s xml:id="echoid-s67" xml:space="preserve"> <lb/> <pb file="0005" n="5" rhead="PR AE FATIO."/> labornobis fuit:</s> <s xml:id="echoid-s68" xml:space="preserve"> qui nequaquam par aut ſimilis fuit in uer-<lb/>bis.</s> <s xml:id="echoid-s69" xml:space="preserve"> In his enim nihil admodum mihi immutandum eſſe exi <lb/>ſtimaui, niſi ut pro ijs, quæ obſcuritatem rebus & ambigui-<lb/>tatem allatura uidebantur, alia reponerẽ:</s> <s xml:id="echoid-s70" xml:space="preserve"> & utinſcriptionẽ <lb/>operis (quæ authori eſt de aſpectibus) gręco, concinniore & <lb/>breuiore nomine opticam nominarẽ.</s> <s xml:id="echoid-s71" xml:space="preserve"> Verùm, illuſtriſsima <lb/>regina, uiderer pleriſq;</s> <s xml:id="echoid-s72" xml:space="preserve"> urbanis hominibus inſtituti fortaſſe <lb/>mei nõ immemor, ſed certè nõ ſatis memor maieſtatis tuæ, <lb/>qui præfatione iſta apud te tam multa & ſcholaſtica, & tam <lb/>ſcholaſticè diſputem:</s> <s xml:id="echoid-s73" xml:space="preserve"> niſi tuo nomine ſcholis omnibus & <lb/>ſcholaſticis beneficiũ tam ſingulare, tamq́;</s> <s xml:id="echoid-s74" xml:space="preserve"> populare ac re-<lb/>gale munus afferrem.</s> <s xml:id="echoid-s75" xml:space="preserve"> Etenim cum præſtantium regum regi-<lb/>narumq́;</s> <s xml:id="echoid-s76" xml:space="preserve"> in populos beneficia muneraq́;</s> <s xml:id="echoid-s77" xml:space="preserve"> multa queãt eſſe:</s> <s xml:id="echoid-s78" xml:space="preserve"> <lb/>certè uirtute doctrinaq́;</s> <s xml:id="echoid-s79" xml:space="preserve"> magnificentius regaliús ue dari ni-<lb/>hil poteſt.</s> <s xml:id="echoid-s80" xml:space="preserve"> Et quidem ſi de optices uſu declamandi locus the <lb/>tori cuipiam hic eſſet, marathonius quidam campus ei me-<lb/>ritò uideretur, ſiue ſupera illa mundi ſiue hæc infera ſpecten <lb/>tur.</s> <s xml:id="echoid-s81" xml:space="preserve"> Etenim quæcunq;</s> <s xml:id="echoid-s82" xml:space="preserve"> hominibus de corporum cœleſtium <lb/>materia, numero, ordine, deq́;</s> <s xml:id="echoid-s83" xml:space="preserve"> motuũ cœleſtium infinita ua <lb/>rietate aperta ac patefacta ſunt, optica ferè aperuit & patefe <lb/>cit:</s> <s xml:id="echoid-s84" xml:space="preserve"> meteora:</s> <s xml:id="echoid-s85" xml:space="preserve"> miracula in iride una præſertim opticis radijs <lb/>diſtincta ſunt:</s> <s xml:id="echoid-s86" xml:space="preserve"> falſas opiniones de numero, motu, atq;</s> <s xml:id="echoid-s87" xml:space="preserve"> loco <lb/>elementorũ optica ſolertia deprehendit & conuicit.</s> <s xml:id="echoid-s88" xml:space="preserve"> In uita <lb/>uerò hominum pleraq;</s> <s xml:id="echoid-s89" xml:space="preserve"> dæmonum præſtigijs attributa:</s> <s xml:id="echoid-s90" xml:space="preserve"> ut <lb/>imagines in aëre quocũq;</s> <s xml:id="echoid-s91" xml:space="preserve"> mobiles repræſentare:</s> <s xml:id="echoid-s92" xml:space="preserve"> utlongin-<lb/>quo ſpatio diſiunctum exercitũ uelut ante oculos intueri:</s> <s xml:id="echoid-s93" xml:space="preserve"> <lb/>ut claſſem hoſtium incendio conſumere, opticæ artis ui ac <lb/>facultate omnia efficiuntur:</s> <s xml:id="echoid-s94" xml:space="preserve"> ut picturam, architecturam, me <lb/>chanicam interea taceam nihil admodum niſi opticã eſſe.</s> <s xml:id="echoid-s95" xml:space="preserve"> <lb/>Quamobrem quòd Alhazenus uetuſtiſsimus & copioſiſsi-<lb/>mus opticæ doctrinæ ſcriptorè tenebris tam diuturnis eru-<lb/>tus, ſqualore, ſitu, puluere abſterſo in publicam lucem pro-<lb/>deat quòd in mathematicas ſcholas ingrediatur:</s> <s xml:id="echoid-s96" xml:space="preserve"> quòd in-<lb/>uenta ſua publicis ſtudijs communicet, Catharinæ <lb/>Mediceæ in poſteritatem fama gloriaq́;</s> <s xml:id="echoid-s97" xml:space="preserve"> <lb/>immortalis eſto.</s> <s xml:id="echoid-s98" xml:space="preserve"/> </p> <pb file="0006" n="6"/> </div> <div xml:id="echoid-div6" type="section" level="0" n="0"> <head xml:id="echoid-head21" xml:space="preserve">CANDIDO LECTORI</head> <p style="it"> <s xml:id="echoid-s99" xml:space="preserve">DVo ſunt, candide Lector, de quib{us} te paucis admonendum hic duxim{us}.</s> <s xml:id="echoid-s100" xml:space="preserve"> Primum ect, quòd ea, quæ commentario-<lb/>rum uice in Alhazeno paßim inſeruim{us}, duab{us} clauſulis (ut uocant) ſic [ ] incluſimus.</s> <s xml:id="echoid-s101" xml:space="preserve"> Alterum, quòd in fir-<lb/>mamentis demonſtrationum è uarijs géometris allegandis γραφι\~Ν<unsure/> quadam breuitate uſi ſum{us}:</s> <s xml:id="echoid-s102" xml:space="preserve"> & Euclidis <lb/>quidem propoſitiones omiſſo nomine hoc modo notauim{us}:</s> <s xml:id="echoid-s103" xml:space="preserve"> per 29 p 1.</s> <s xml:id="echoid-s104" xml:space="preserve">1 αχ.</s> <s xml:id="echoid-s105" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s106" xml:space="preserve"> 1 d 6:</s> <s xml:id="echoid-s107" xml:space="preserve"> id ect per 29 propoſitionem 1 li-<lb/>bri:</s> <s xml:id="echoid-s108" xml:space="preserve"> 1 axioma.</s> <s xml:id="echoid-s109" xml:space="preserve"> 4 propoſitionem.</s> <s xml:id="echoid-s110" xml:space="preserve"> 1 definitionem 6 libri elementorum Euclidis, &c.</s> <s xml:id="echoid-s111" xml:space="preserve"> In reliquorum uerò geometrarum al-<lb/>legationib{us} ſimiliter numeros theorematis & libri literis th (theorema ſignificantib{us}) inter utrũq;</s> <s xml:id="echoid-s112" xml:space="preserve"> intermedijs unà cun<gap/> <lb/>nomine adiecim{us}.</s> <s xml:id="echoid-s113" xml:space="preserve"> Alhazeni autem propoſitiones (in qu{as} à nobis diſtinctus ect) differentiæ breuitatisq́;</s> <s xml:id="echoid-s114" xml:space="preserve"> cauſſa in citandis <lb/>Euclidis & Alhazeni propoſitionib{us} numeros appellauim{us}:</s> <s xml:id="echoid-s115" xml:space="preserve"> ij q́;</s> <s xml:id="echoid-s116" xml:space="preserve"> it a nobis notati ſunt:</s> <s xml:id="echoid-s117" xml:space="preserve"> per 19 n 1.</s> <s xml:id="echoid-s118" xml:space="preserve">14 n 2.</s> <s xml:id="echoid-s119" xml:space="preserve">13 n.</s> <s xml:id="echoid-s120" xml:space="preserve"> id ect per <lb/>19 numerum 1 libri optices Alhazeni.</s> <s xml:id="echoid-s121" xml:space="preserve"> 14 numerum 2 libri.</s> <s xml:id="echoid-s122" xml:space="preserve"> 13 numerum:</s> <s xml:id="echoid-s123" xml:space="preserve"> ei{us} nimirum libri, in quo fit iſta allegatio:</s> <s xml:id="echoid-s124" xml:space="preserve"> id <lb/>ect per 13 numerum primi, ſecundi, tertij, &c.</s> <s xml:id="echoid-s125" xml:space="preserve"> libri optices Albazeni, ſi ille numer{us} in primo, ſecundo, tertio, &c.</s> <s xml:id="echoid-s126" xml:space="preserve"> libro <lb/>optices Albazeni allegat{us} ſit.</s> <s xml:id="echoid-s127" xml:space="preserve"> Quod ad errata attinet, placuit ea in ueſtibulo libri adſcribere, quò emendare uolenti in <lb/>promptu eſſent:</s> <s xml:id="echoid-s128" xml:space="preserve"> ubi prim{us} numer{us} paginam, ſecund{us} lineam indicat.</s> <s xml:id="echoid-s129" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s130" xml:space="preserve">Pagina 4.</s> <s xml:id="echoid-s131" xml:space="preserve"> linea 55 centrum.</s> <s xml:id="echoid-s132" xml:space="preserve"> 7.</s> <s xml:id="echoid-s133" xml:space="preserve"> ult.</s> <s xml:id="echoid-s134" xml:space="preserve"> unaquaq;</s> <s xml:id="echoid-s135" xml:space="preserve">. 16.</s> <s xml:id="echoid-s136" xml:space="preserve">35 ultimum.</s> <s xml:id="echoid-s137" xml:space="preserve"> 26.</s> <s xml:id="echoid-s138" xml:space="preserve">22 earum.</s> <s xml:id="echoid-s139" xml:space="preserve"> 34.</s> <s xml:id="echoid-s140" xml:space="preserve">23 & 24 humiditas.</s> <s xml:id="echoid-s141" xml:space="preserve"> 41.</s> <s xml:id="echoid-s142" xml:space="preserve">46 <lb/>certitudinẽ.</s> <s xml:id="echoid-s143" xml:space="preserve"> 61.</s> <s xml:id="echoid-s144" xml:space="preserve">1 pro 711,111.</s> <s xml:id="echoid-s145" xml:space="preserve"> 67.</s> <s xml:id="echoid-s146" xml:space="preserve">16 formarũ.</s> <s xml:id="echoid-s147" xml:space="preserve"> 80.</s> <s xml:id="echoid-s148" xml:space="preserve">60 poſitio.</s> <s xml:id="echoid-s149" xml:space="preserve"> .</s> <s xml:id="echoid-s150" xml:space="preserve">81.</s> <s xml:id="echoid-s151" xml:space="preserve">82 figurarũlloca permutata ſunt.</s> <s xml:id="echoid-s152" xml:space="preserve"> 83.</s> <s xml:id="echoid-s153" xml:space="preserve">56 poſi <lb/>tio.</s> <s xml:id="echoid-s154" xml:space="preserve"> 93.</s> <s xml:id="echoid-s155" xml:space="preserve">28 poſt iterũ, adde in.</s> <s xml:id="echoid-s156" xml:space="preserve"> 106.</s> <s xml:id="echoid-s157" xml:space="preserve">53 utroq;</s> <s xml:id="echoid-s158" xml:space="preserve">. 111.</s> <s xml:id="echoid-s159" xml:space="preserve">10 lucis.</s> <s xml:id="echoid-s160" xml:space="preserve"> 134.</s> <s xml:id="echoid-s161" xml:space="preserve">60 angulus age.</s> <s xml:id="echoid-s162" xml:space="preserve"> 135.</s> <s xml:id="echoid-s163" xml:space="preserve">14 poſt lineis ponè colon, & <lb/>poſt reflexionis, comma.</s> <s xml:id="echoid-s164" xml:space="preserve"> 142.</s> <s xml:id="echoid-s165" xml:space="preserve">11 uiſu & uiſibili.</s> <s xml:id="echoid-s166" xml:space="preserve"> 146.</s> <s xml:id="echoid-s167" xml:space="preserve">31 concurret.</s> <s xml:id="echoid-s168" xml:space="preserve"> 156.</s> <s xml:id="echoid-s169" xml:space="preserve">49 pro 44 poſt exteriore, repone 45-<lb/>159.</s> <s xml:id="echoid-s170" xml:space="preserve">26 æquidiſtans.</s> <s xml:id="echoid-s171" xml:space="preserve"> 181.</s> <s xml:id="echoid-s172" xml:space="preserve">36 & 37 pro ibi, ubi.</s> <s xml:id="echoid-s173" xml:space="preserve"> 185.</s> <s xml:id="echoid-s174" xml:space="preserve">12 pro t, b.</s> <s xml:id="echoid-s175" xml:space="preserve"> 189.</s> <s xml:id="echoid-s176" xml:space="preserve">6 fit.</s> <s xml:id="echoid-s177" xml:space="preserve"> 194.</s> <s xml:id="echoid-s178" xml:space="preserve">53 & 54 pro œ, œ d.</s> <s xml:id="echoid-s179" xml:space="preserve"> ibid.</s> <s xml:id="echoid-s180" xml:space="preserve"> 62 pro i <lb/>poſt a g, repone &.</s> <s xml:id="echoid-s181" xml:space="preserve"> 197.</s> <s xml:id="echoid-s182" xml:space="preserve">15 pro 4,45.</s> <s xml:id="echoid-s183" xml:space="preserve"> Pag.</s> <s xml:id="echoid-s184" xml:space="preserve"> 201 figura prior 15 numeri, quæ ad ſiniſtram eſt, pertinet ad 14 nume <lb/>rum præcedentem:</s> <s xml:id="echoid-s185" xml:space="preserve"> neq;</s> <s xml:id="echoid-s186" xml:space="preserve"> enim ſuo quamq;</s> <s xml:id="echoid-s187" xml:space="preserve"> loco collocari compoſitionis, ut uocant, ratio permiſit.</s> <s xml:id="echoid-s188" xml:space="preserve"> 205.</s> <s xml:id="echoid-s189" xml:space="preserve">1 z q l.</s> <s xml:id="echoid-s190" xml:space="preserve"> <lb/>211.</s> <s xml:id="echoid-s191" xml:space="preserve">21 poſt le tolle comma.</s> <s xml:id="echoid-s192" xml:space="preserve"> 213.</s> <s xml:id="echoid-s193" xml:space="preserve">43 ſpeculιs.</s> <s xml:id="echoid-s194" xml:space="preserve"> 217.</s> <s xml:id="echoid-s195" xml:space="preserve">39 pro 4, 41.</s> <s xml:id="echoid-s196" xml:space="preserve"> 219.</s> <s xml:id="echoid-s197" xml:space="preserve">39 pro erit, reflectetur, & pro in, ex.</s> <s xml:id="echoid-s198" xml:space="preserve"> Pagina <lb/>221 in figura litera proximè infra rin linea gr propter linearum concurſum obſcurior, eſt k.</s> <s xml:id="echoid-s199" xml:space="preserve"> 223.</s> <s xml:id="echoid-s200" xml:space="preserve">17 peripheria <lb/>a z.</s> <s xml:id="echoid-s201" xml:space="preserve"> 226.</s> <s xml:id="echoid-s202" xml:space="preserve">28 perueniat ergo.</s> <s xml:id="echoid-s203" xml:space="preserve"> 229.</s> <s xml:id="echoid-s204" xml:space="preserve">58 tranſeunte.</s> <s xml:id="echoid-s205" xml:space="preserve"> 237.</s> <s xml:id="echoid-s206" xml:space="preserve">21 poſt laminæ adde, & inter ſuperficiem laminæ.</s> <s xml:id="echoid-s207" xml:space="preserve"> Ibid.</s> <s xml:id="echoid-s208" xml:space="preserve"> 35 <lb/>dele fit.</s> <s xml:id="echoid-s209" xml:space="preserve"> 240.</s> <s xml:id="echoid-s210" xml:space="preserve">46 exiſtunt.</s> <s xml:id="echoid-s211" xml:space="preserve"> 250.</s> <s xml:id="echoid-s212" xml:space="preserve">55 lineam.</s> <s xml:id="echoid-s213" xml:space="preserve"> 252.</s> <s xml:id="echoid-s214" xml:space="preserve">7 poſt eleuatio, pone comma.</s> <s xml:id="echoid-s215" xml:space="preserve"> Ibid.</s> <s xml:id="echoid-s216" xml:space="preserve"> 19 k b g.</s> <s xml:id="echoid-s217" xml:space="preserve"> Ibid.</s> <s xml:id="echoid-s218" xml:space="preserve"> in figura po-<lb/>natur litera a è regione e centri mundi ad peripheriam meridiani b g.</s> <s xml:id="echoid-s219" xml:space="preserve"> Ibid.</s> <s xml:id="echoid-s220" xml:space="preserve"> 49 exiſtunt.</s> <s xml:id="echoid-s221" xml:space="preserve"> 255.</s> <s xml:id="echoid-s222" xml:space="preserve">57 pro in, à.</s> <s xml:id="echoid-s223" xml:space="preserve"> 258 <lb/>in ſecunda figura ad terminum lineæ fo continuatæ, pone literam q.</s> <s xml:id="echoid-s224" xml:space="preserve"> 266.</s> <s xml:id="echoid-s225" xml:space="preserve">48 tolle &.</s> <s xml:id="echoid-s226" xml:space="preserve"> Pag.</s> <s xml:id="echoid-s227" xml:space="preserve"> 273 in prima figu-<lb/>ra ad terminum lineæ d m continuatæ pone literam e.</s> <s xml:id="echoid-s228" xml:space="preserve"> Pagina 275 ſub numero 45 duæ figuræ cõiunctæ ſunt, <lb/>quarum poſterior, quæ ad dextram eſt, pertinet ad 46 numerum:</s> <s xml:id="echoid-s229" xml:space="preserve"> ad quem prior quoq;</s> <s xml:id="echoid-s230" xml:space="preserve"> figura 47 numeri refer-<lb/>ri debet.</s> <s xml:id="echoid-s231" xml:space="preserve"> 281.</s> <s xml:id="echoid-s232" xml:space="preserve">38 poſt diuerſæ dele comma.</s> <s xml:id="echoid-s233" xml:space="preserve"> 285.</s> <s xml:id="echoid-s234" xml:space="preserve">23 pro e, ei.</s> <s xml:id="echoid-s235" xml:space="preserve"> 286.</s> <s xml:id="echoid-s236" xml:space="preserve">51 reſtant.</s> <s xml:id="echoid-s237" xml:space="preserve"> 288.</s> <s xml:id="echoid-s238" xml:space="preserve">58 pro 48,40.</s> <s xml:id="echoid-s239" xml:space="preserve"> Deniq;</s> <s xml:id="echoid-s240" xml:space="preserve"> illud e-<lb/>tiam in erratis reponere placuit, quod in commentarijs ad quartam & tertiam figuras 64 n 5 prætermiſſum eſt:</s> <s xml:id="echoid-s241" xml:space="preserve"> <lb/>nempe.</s> <s xml:id="echoid-s242" xml:space="preserve"> [Quando imago uidetur in puncto a:</s> <s xml:id="echoid-s243" xml:space="preserve"> & b q, quæ parallela ducitur ipſi a l, cadit extra triangulum at <lb/>g:</s> <s xml:id="echoid-s244" xml:space="preserve"> tum omiſſa b q & oſtenſa æqualitate linearum a g, a t, ut prius, breuius per 3 p 6 propoſitum concludetur.</s> <s xml:id="echoid-s245" xml:space="preserve"> <lb/>Erit enim ut b e ad e a, ſic b g ad a g, id eſt per 7 p 5 ad a l:</s> <s xml:id="echoid-s246" xml:space="preserve"> ſed ut b g ad a l, ſic b t ad t a propter ſimilitudinem <lb/>triangulorum b g t, a l t.</s> <s xml:id="echoid-s247" xml:space="preserve"> Ergo per 11 p 5 ut b e ad e a, fic b t ad t a.</s> <s xml:id="echoid-s248" xml:space="preserve"> Eodem modo cadente b q intra triangulum a <lb/>g t:</s> <s xml:id="echoid-s249" xml:space="preserve"> breuior & facilior abſq;</s> <s xml:id="echoid-s250" xml:space="preserve"> linea a l erit demonſtratio.</s> <s xml:id="echoid-s251" xml:space="preserve">]</s> </p> <pb o="1" file="0007" n="7"/> </div> <div xml:id="echoid-div7" type="section" level="0" n="0"> <head xml:id="echoid-head22" xml:space="preserve">ALHAZEN FILII</head> <head xml:id="echoid-head23" xml:space="preserve">ALHAYZEN OPTICAE</head> <head xml:id="echoid-head24" xml:space="preserve">LIBER PRIMVS.</head> <p style="it"> <s xml:id="echoid-s252" xml:space="preserve">PRIMVS Liber in ſeptem capita diuiditur.</s> <s xml:id="echoid-s253" xml:space="preserve"> Primum eſt quòd lux per ſe <lb/>& colores illuminati operentur in uiſum aliquam operationem.</s> <s xml:id="echoid-s254" xml:space="preserve"> Secun-<lb/>dum quòd lux uehemens occultat quædam uiſibilia, quæ lux debilis ma-<lb/>nifeſtat, & contrà.</s> <s xml:id="echoid-s255" xml:space="preserve"> Tertium quòd colores corporum diuerſificantur a-<lb/>pud uiſum ſecundum dιuerſitatem lucium orientium ſuper ipſos.</s> <s xml:id="echoid-s256" xml:space="preserve"> Quartum ect de com <lb/><gap/>oſitione oculi, forma, & ſitu.</s> <s xml:id="echoid-s257" xml:space="preserve"> Quintum declarat qualitatem uiſionis, & dependen-<lb/>tia ab illa.</s> <s xml:id="echoid-s258" xml:space="preserve"> Sextum ect de officio & utilitate inſtrumentorum uiſus.</s> <s xml:id="echoid-s259" xml:space="preserve"> Septimum de ijs, <lb/>ſine quib{us} uiſio non potect compleri.</s> <s xml:id="echoid-s260" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div8" type="section" level="0" n="0"> <head xml:id="echoid-head25" xml:space="preserve">QVOD LVX PER SE, ET COLORES ILLVMINATI OPE-<lb/>renturin uiſum aliquam operationem. Cap. 1.</head> <head xml:id="echoid-head26" xml:space="preserve" style="it">1. Lux per ſe, & color illuminat{us} feriunt oculos. Vitell. in hypotheſ. 6. 16 p 3.</head> <p> <s xml:id="echoid-s261" xml:space="preserve">INuenimus quòd uiſus, quãdo inſpexerit luces ualde fortes, fortiter dolebit ex eis, & habebit no <lb/>cumentum:</s> <s xml:id="echoid-s262" xml:space="preserve"> aſpiciens enim quando aſpexerit corpus ſolis, non poteſt bene aſpicere ipſum, quo-<lb/>niam uiſus eius dolebit propter ipſius lucem.</s> <s xml:id="echoid-s263" xml:space="preserve"> Et ſimiliter quando inſpexerit ſpeculum terſum, <lb/>ſuper quod aſcendebat lux ſolis, & fuerit uiſus eius in loco, ad quem reflectitur lux ab illo ſpecu-<lb/>lo:</s> <s xml:id="echoid-s264" xml:space="preserve"> dolebit iterum propter lumen reflexum, perueniens ad ſuum uiſum à ſpeculo, & non poterit a-<lb/>perire oculum ad inſpicien dum lumen illud.</s> <s xml:id="echoid-s265" xml:space="preserve"> Et inuenimus iterum quando aſpiciens intuetur cor-<lb/>pus mundum album, ſuper quod aſcen debat lux ſolis, & moretur in aſpectu ipſius:</s> <s xml:id="echoid-s266" xml:space="preserve"> deinde conuer-<lb/>tat uiſum ſuum ab eo ad locum obſcurum, debilis lucis:</s> <s xml:id="echoid-s267" xml:space="preserve"> quòd fere<unsure/> nõ poterit comprehendere res <lb/>uiſibiles illius loci comprehenſione uera:</s> <s xml:id="echoid-s268" xml:space="preserve"> & inueniet coopertorium quaſi inter uiſum & ipſas:</s> <s xml:id="echoid-s269" xml:space="preserve"> de-<lb/>in de paulatim diſcooperietur, & reuertetur uiſus in ſuam diſpoſitionem.</s> <s xml:id="echoid-s270" xml:space="preserve"> Et iterum, quando inſpi-<lb/>ciens inſpexerit ignem fortem:</s> <s xml:id="echoid-s271" xml:space="preserve"> & fuerit intuitus ipſum:</s> <s xml:id="echoid-s272" xml:space="preserve"> & moretur in aſpiciendo longo tempore:</s> <s xml:id="echoid-s273" xml:space="preserve"> <lb/>deinde declinet uiſum ſuum ad locum obſcurum, debilis lucis:</s> <s xml:id="echoid-s274" xml:space="preserve"> inueniet iterum idem in uiſu ſuo.</s> <s xml:id="echoid-s275" xml:space="preserve"> <lb/>Et iterum inuenimus, quando inſpiciens inſpexerit corpus mũdum album, ſuper quod oriebatur <lb/>lux diei:</s> <s xml:id="echoid-s276" xml:space="preserve"> & fuerit illa lux fortis, quamuis non ſit lux ſolis:</s> <s xml:id="echoid-s277" xml:space="preserve"> & moretur in aſpectu diu deinde conuer-<lb/>tat uiſum ſuum ad locum obſcurum:</s> <s xml:id="echoid-s278" xml:space="preserve"> inueniet formam lucis illius in loco illo, & inueniet cum hoc <lb/>figuram eius:</s> <s xml:id="echoid-s279" xml:space="preserve"> deinde ſi clauſerit uiſum:</s> <s xml:id="echoid-s280" xml:space="preserve"> inueniet in ipſo formam illius lucis:</s> <s xml:id="echoid-s281" xml:space="preserve"> deinde auferetur hoc, <lb/>& reuertetur oculus in ſuam diſpoſitionem.</s> <s xml:id="echoid-s282" xml:space="preserve"> Et ſimiliter erit diſpoſitus uiſus, quãdo inſpexerit cor-<lb/>pus, ſuper quod oriebatur lux ſolis.</s> <s xml:id="echoid-s283" xml:space="preserve"> Et ſimiliter quando inſpexerit corpus clarè album, ſuper quod <lb/>oriebatur lux ignis, quando luxignis fuerit fortis, & moretur in aſpiciendo ipſum:</s> <s xml:id="echoid-s284" xml:space="preserve"> deinde receſſe-<lb/>rit ad locum obſcurum:</s> <s xml:id="echoid-s285" xml:space="preserve"> inueniet iterum in eo idem hoc in fuo uiſu.</s> <s xml:id="echoid-s286" xml:space="preserve"> Et ſimiliter quando aſpiciens <lb/>fuerit in domo, in qua fuerit foramen amplum diſcoopertum ad cœlum:</s> <s xml:id="echoid-s287" xml:space="preserve"> & aſpexerit ex illo loco <lb/>cœlum in luce diei:</s> <s xml:id="echoid-s288" xml:space="preserve"> & moretur in aſpiciendo ipſum:</s> <s xml:id="echoid-s289" xml:space="preserve"> deinde reuertatur uiſus eius ad locum obſcu-<lb/>rum in domo:</s> <s xml:id="echoid-s290" xml:space="preserve"> inueniet formam lucis, quam comprehendebat ex foramine cum figura foraminis <lb/>in loco obſcuro:</s> <s xml:id="echoid-s291" xml:space="preserve"> & ſi clauſerit oculum ſuum:</s> <s xml:id="echoid-s292" xml:space="preserve"> inuenier iterum in eo formam illam.</s> <s xml:id="echoid-s293" xml:space="preserve"> Omnia ergo iſta <lb/>ſignificant, quòd lux operetur in uiſum aliquam operationẽ.</s> <s xml:id="echoid-s294" xml:space="preserve"> Et inuenimus iterũ quod, quãdo aſpi <lb/>ciẽs inſp exerit uiridarium multæ ſpiſsitudinis herbarum, ſuper quod oriebatur lux ſolis:</s> <s xml:id="echoid-s295" xml:space="preserve"> & more-<lb/>turin aſpiciendo ipſum:</s> <s xml:id="echoid-s296" xml:space="preserve"> deinde conuertat ſuum uiſum ad locum obſcurum:</s> <s xml:id="echoid-s297" xml:space="preserve"> inueniet in illo loco <lb/>obſcuro formam coloratam à uirore illarum herbarum:</s> <s xml:id="echoid-s298" xml:space="preserve"> deinde ſi aſpexerit in iſta diſpoſitione uiſi-<lb/>bilia alba:</s> <s xml:id="echoid-s299" xml:space="preserve"> & fuerint illa uiſibilia in umbra, & loco debilis lucis:</s> <s xml:id="echoid-s300" xml:space="preserve"> inueniet colores iſtos admixtos cũ <lb/>uirore:</s> <s xml:id="echoid-s301" xml:space="preserve"> & ſi clauſerit oculum ſuum:</s> <s xml:id="echoid-s302" xml:space="preserve"> iterum inueniet in ipſo formam lucis & formam uiroris:</s> <s xml:id="echoid-s303" xml:space="preserve"> deinde <lb/>diſcooperietur illud, & auferetur.</s> <s xml:id="echoid-s304" xml:space="preserve"> Et ſimiliter ſi aſpexerit corpus coloratum colore cæruleo uel ru-<lb/>beo, uel alio colore forti ſcintillante, ſuper quod oriebatur lux ſolis:</s> <s xml:id="echoid-s305" xml:space="preserve"> & moretu:</s> <s xml:id="echoid-s306" xml:space="preserve"> in <gap/>ſpiciẽdo ipſum:</s> <s xml:id="echoid-s307" xml:space="preserve"> <lb/>deinde auferat uiſum ſuum ad uiſibilia alba in loco debilis lucis:</s> <s xml:id="echoid-s308" xml:space="preserve"> inueniet colores illos admixtos cũ <lb/>illo colore.</s> <s xml:id="echoid-s309" xml:space="preserve"> Iſta ergo ſignificant quòd colores illuminati operentur in uiſum.</s> <s xml:id="echoid-s310" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div9" type="section" level="0" n="0"> <head xml:id="echoid-head27" xml:space="preserve">QVOD LVX VEHEMENS OCCVLTAT QVAEDAM VI-<lb/>ſibilia quæ lux debilis manifeſtat: & contrà. Cap. 2.</head> <head xml:id="echoid-head28" xml:space="preserve" style="it">2. Lux uehemens obſcur at quædam uiſibilia, quæ lux debilis illuſtrat: & <lb/>contrà. 28. 97. 109. 150. 155. 156 p 4.</head> <p> <s xml:id="echoid-s311" xml:space="preserve">ET iterum uidemus ſtellas in nocte, & non uidemus ipſas in luce diei:</s> <s xml:id="echoid-s312" xml:space="preserve"> & nulla eſt differentia in-<lb/>ter tempora, niſi quòd aer medius inter uiſum noſtrum & cœlum eſt in die illuminatus, & in <lb/> <pb o="2" file="0008" n="8" rhead="ALHAZEN"/> nocte obſcurus:</s> <s xml:id="echoid-s313" xml:space="preserve"> cum ergo aer fuerit obſcurus, nos uidemus ſtellas:</s> <s xml:id="echoid-s314" xml:space="preserve"> cum autem illuminatus fue-<lb/>rit aer medius inter uiſum noſtrum & ſtellas, latebunt nos ſtellæ.</s> <s xml:id="echoid-s315" xml:space="preserve"> Et ſimiliter ſi aſpiciens nocte <lb/>aſpexerit in loco luminoſo lumine ignis:</s> <s xml:id="echoid-s316" xml:space="preserve"> & fuerit lumen ignis extenſum ſuper terram:</s> <s xml:id="echoid-s317" xml:space="preserve"> & fuerint in <lb/>illo loco uiſibilia ſubtilia, aut uiſibilia, in quibus ſunt res ſubtiles:</s> <s xml:id="echoid-s318" xml:space="preserve"> & fuerint in aliqua umbra, ſed nõ <lb/>forti:</s> <s xml:id="echoid-s319" xml:space="preserve"> & non fuerit ignis medius inter illa uiſibilia & uiſum:</s> <s xml:id="echoid-s320" xml:space="preserve"> & comprehenderit tunc aſprciens illa ui <lb/>ſibilia, & res ſubtiles, quæ ſuntin eis:</s> <s xml:id="echoid-s321" xml:space="preserve"> deinde moueatur à ſuo loco, donec ſit ignis medius inter illa <lb/>uiſibilia & ſuum uiſum:</s> <s xml:id="echoid-s322" xml:space="preserve"> tunc illa uiſibilia latebunt ipſum, ſi fuerint ſubtilia, uel ſubtilia, quæ in eis <lb/>ſunt, & ferè non comprehendet ipſa, cum ignis fuerit medius inter uiſum ſuum & ipſa uiſibilia:</s> <s xml:id="echoid-s323" xml:space="preserve"> & ſi <lb/>cooperiatur ignis a uiſu ſuo:</s> <s xml:id="echoid-s324" xml:space="preserve"> comprehendet ſtatim uiſibilia illa, quæ latebant ipſum:</s> <s xml:id="echoid-s325" xml:space="preserve"> & ſi auferatur <lb/>coopertorium inter uiſum ſuum & ignem:</s> <s xml:id="echoid-s326" xml:space="preserve"> latebunt ipſum iterum illa uiſibilia.</s> <s xml:id="echoid-s327" xml:space="preserve"> Iſtæ ergo diſpoſitio-<lb/>nes ſignificant, quòd luces fortes orientes ſuper uiſum & ſuper aerem inter oculum & rem uiſam, <lb/>prohibent uiſum à comprehenſione quorundam uiſibilium, quorum luces ſunt debiles.</s> <s xml:id="echoid-s328" xml:space="preserve"> Et iterum, <lb/>quando aſpiciens aſpexerit corpus terſum:</s> <s xml:id="echoid-s329" xml:space="preserve"> & fuerint in illo corpore ſculpturæ ſubtiles:</s> <s xml:id="echoid-s330" xml:space="preserve"> & non fue-<lb/>rint illæ ſculpturæ diuerſorum colorum à colore corporis:</s> <s xml:id="echoid-s331" xml:space="preserve"> & fuerit aſpiciens in loco temperatæ lu-<lb/>cis:</s> <s xml:id="echoid-s332" xml:space="preserve"> deinde oppoſitum fuerit corpus illud ſoli aut parieti illuminato lumine forti:</s> <s xml:id="echoid-s333" xml:space="preserve"> reflectetur ab eo <lb/>aliqua lux ad uiſum:</s> <s xml:id="echoid-s334" xml:space="preserve"> & inueniet aſpiciens lucem apparentem in ſuperficie corporis, & in loco, à que <lb/>reflectitur lux, fortiorem, & magis ſcintillantem:</s> <s xml:id="echoid-s335" xml:space="preserve"> & in iſta diſpoſitione ſi inſpiciens fuerit intuitus <lb/>illud corpus terſum:</s> <s xml:id="echoid-s336" xml:space="preserve"> non uidebit in eo aliquam ſculpturam ex ſculpturis, quæ funt in loco lucis for <lb/>tis, & ſcintillátis:</s> <s xml:id="echoid-s337" xml:space="preserve"> deinde ſi inſpiciens declinauerit illud corpus ab illo loco, ita ut reflexio ſiat ad a-<lb/>lium locũ, extra locum uiſus ſui, & fuerit præterea ſuper corpus illud lux temperata:</s> <s xml:id="echoid-s338" xml:space="preserve"> tunc inſpiciẽs <lb/>cõprehendet ſculpturas, quæ ſuntin eo, quas prius non comprehendebat in reflexione lucis à cor-<lb/>pore ad ſuum uiſum.</s> <s xml:id="echoid-s339" xml:space="preserve"> Et ſimiliter, quando lux reflectetur à paginaterſa, in qua ſunt ſculpturæ ſubti-<lb/>les ad uiſum:</s> <s xml:id="echoid-s340" xml:space="preserve"> non diſtinguet uiſus illas ſculpturas necuerificabit, donec ſit lux reflexa ad uiſum ab <lb/>illa pagina:</s> <s xml:id="echoid-s341" xml:space="preserve"> & ſi declinetur ſuperficies paginæ, ita ut ſitus eius mutetur, & non reflectatur ab ea lux <lb/>ad uiſum:</s> <s xml:id="echoid-s342" xml:space="preserve"> cõprehendet tunc uiſus illas ſculpturas, & diſtinguet.</s> <s xml:id="echoid-s343" xml:space="preserve"> Et iterum quando ignis debilis fue <lb/>rit in lumine debili:</s> <s xml:id="echoid-s344" xml:space="preserve"> apparebit & comprehẽdetur à uiſu:</s> <s xml:id="echoid-s345" xml:space="preserve"> & cum fuerit in lumine ſolis:</s> <s xml:id="echoid-s346" xml:space="preserve"> apparebit cor-<lb/>pus, in quo eſt, denſum, coloratũ colore ſcintillante forti:</s> <s xml:id="echoid-s347" xml:space="preserve"> & tũ ſi fuerit prope illud corpus, aliquod <lb/>corpus albũ claræ albedinis:</s> <s xml:id="echoid-s348" xml:space="preserve"> & fuerit corpus illud in umbra & luce debili:</s> <s xml:id="echoid-s349" xml:space="preserve"> apparebit ſuper ipſum co <lb/>lor corporis illius, ſicut narrauimus ſuperius:</s> <s xml:id="echoid-s350" xml:space="preserve"> deinde ſi moueatur illud corpus album, donec ſit in <lb/>lumine ſolis, latebit iam ille color, qui eſt in eo:</s> <s xml:id="echoid-s351" xml:space="preserve"> & ſi reducatur ad umbram:</s> <s xml:id="echoid-s352" xml:space="preserve"> apparebit color ille ful-<lb/>gens, qui eſt in ipſo:</s> <s xml:id="echoid-s353" xml:space="preserve"> & apparebit color ille ſuper ipſum in luce forti, & apud latitationẽ coloris, qui <lb/>eſt ſuper ipſum, ſi obumbretur corpore denſo:</s> <s xml:id="echoid-s354" xml:space="preserve"> & ſi maneat in ſuo loco, donec debilitetur lux, quæ <lb/>eſt ſuper ipſum:</s> <s xml:id="echoid-s355" xml:space="preserve"> apparebit color, qui eſt ſuper ipſum:</s> <s xml:id="echoid-s356" xml:space="preserve"> & ſi auferatur corpus obũbrãs, donec uigore<gap/>-<lb/>cat lux ſuper corpus album:</s> <s xml:id="echoid-s357" xml:space="preserve"> latebit color, qui eſt ſuper ipſum.</s> <s xml:id="echoid-s358" xml:space="preserve"> Et ſimiliter quãdo admouerimus cor-<lb/>pus diaphanum coloratum colore ſcintillante, igni uehementer forti:</s> <s xml:id="echoid-s359" xml:space="preserve"> & admouerimus umbræ il-<lb/>lius corporis pannũ album:</s> <s xml:id="echoid-s360" xml:space="preserve"> apparebit color illius corporis diaphani ſuper illum pannum, ſicut nar-<lb/>rauimus prius:</s> <s xml:id="echoid-s361" xml:space="preserve"> deinde ſi admouerimus illi panno alium ignem, ita ut lux eius oriatur ſuper illũ pan <lb/>num:</s> <s xml:id="echoid-s362" xml:space="preserve"> latebit ille color, qui apparebat ſuper pannum, & non apparebit niſi albedo panni tantum:</s> <s xml:id="echoid-s363" xml:space="preserve"> & <lb/>ſi auferamus illum ignem, ſecundus apparebit color ſuper pannum.</s> <s xml:id="echoid-s364" xml:space="preserve"> Et iterum quædam animalia <lb/>marina habent conchas & teſtas, & cum fuerint in loco obſcuro, in quo non eſtlux, apparebuntillę <lb/>conchæ quaſi ignis:</s> <s xml:id="echoid-s365" xml:space="preserve"> & ſi aſpiciens inſpexerit illas in luce diei uelignis, comprehendet eas, & non <lb/>uidebit in eis lumen uel aliquem ignem.</s> <s xml:id="echoid-s366" xml:space="preserve"> Et ſimiliter, quando animal, quod dicitur noctiluca, uo-<lb/>lat de nocte, apparet quaſi lampas, & cum aſpiciens inſpexerit illud in luce diei uel in luce ignis, ap <lb/>parebit animal ſine igne.</s> <s xml:id="echoid-s367" xml:space="preserve"> Significant ergo omnes iſtæ diſpoſitiones, quas declarauimus, quòd lu-<lb/>ces fortes uifibilium, aliquando occultantres, quæ ſuntin quibuſdam uiſibilibus:</s> <s xml:id="echoid-s368" xml:space="preserve"> & quòd luces de-<lb/>biles aliquando manifeſtant quafdam res, quæ ſunt in quibuſdam uiſibilibus.</s> <s xml:id="echoid-s369" xml:space="preserve"> Et iterum uiſum mul <lb/>toties latent quædam res, quę funt inuiſibiles ex ſculpturis & ſcripturis ſubtilibus, quando fuerint <lb/>in locis obſcuris uel in lucibus debilibus:</s> <s xml:id="echoid-s370" xml:space="preserve"> & ſi extrahantur ad loca luminoſa fortis luminis, uel po-<lb/>nantur in luce ſolis:</s> <s xml:id="echoid-s371" xml:space="preserve"> appare buntres, quæ ſunt in eis, quæ latebant in locis & lucibus debilibus.</s> <s xml:id="echoid-s372" xml:space="preserve"> Et <lb/>ſimiliter ſculpturarum ſubtilium comprehenſiones nequit uiſus comprehendere in locis obſcuris <lb/>& lucibus debilibus:</s> <s xml:id="echoid-s373" xml:space="preserve"> & cum extrahuntur ad luces fortes, comprehenduntur à uiſu.</s> <s xml:id="echoid-s374" xml:space="preserve"> Significatur er-<lb/>go per hanc diſputationem, quòd luces fortes maniſeſtant multas res uiſibiles, & quòd luces debi-<lb/>les occultant multas res uiſibiles.</s> <s xml:id="echoid-s375" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div10" type="section" level="0" n="0"> <head xml:id="echoid-head29" xml:space="preserve">QVOD COLORES CORPORVM DIVERSIFICENTVR APVD VI-<lb/>ſum ſecundum diuerſitatem lucium ordentium ſuper ipſos. Cap. 3.</head> <head xml:id="echoid-head30" xml:space="preserve" style="it">3. Color uariatur pro lucis qualitate. 1 p 3.</head> <p> <s xml:id="echoid-s376" xml:space="preserve">ET iterum inuenimus, quod colores corporum denſorum coloratorum coloribus ſcintillan-<lb/>tibus, ſicut Lazuleis, uinoſis & cœleſtib.</s> <s xml:id="echoid-s377" xml:space="preserve"> quádo ipſa fuerint in locis obſcuris & lucibus debili <lb/>bus, apparent turbidi, & cum fuerint in luce forti, apparent colores eorum ſcintillantes & cla <lb/>ri:</s> <s xml:id="echoid-s378" xml:space="preserve"> & quanto magis augmentabitur lux ſuperipſa, tanto magis augmentabitur ſuper ipſa ſcintil-<lb/>lantis coloris claritas.</s> <s xml:id="echoid-s379" xml:space="preserve"> Et ſi fuerit aliquod iſtorum corporum in loco obſcuro:</s> <s xml:id="echoid-s380" xml:space="preserve"> & non fuerit in eo niſi <lb/>lux parua ualde:</s> <s xml:id="echoid-s381" xml:space="preserve"> apparebit illud corpus obſcurum, & non diſtinguet uiſus colorem eius, & uidebi <lb/>tur quaſi niger:</s> <s xml:id="echoid-s382" xml:space="preserve"> & cum extrahitur ad loca luminoſa lumine forti:</s> <s xml:id="echoid-s383" xml:space="preserve"> apparebit color eius, & diſtingue-<lb/> <pb o="3" file="0009" n="9" rhead="OPTICAE LIBER I."/> tur à uiſu.</s> <s xml:id="echoid-s384" xml:space="preserve"> Et inuenimus iterum quòd colores corporum ferrei coloris, quando lux oritur ſuper ipſa <lb/>fortis, clareſcunt.</s> <s xml:id="echoid-s385" xml:space="preserve"> Et inuenimus etiam quòd, quando lux fortis oritur ſuper corpora denſa alba, au-<lb/>gmentantur in albedine & ſcintillatione apud ſenſum.</s> <s xml:id="echoid-s386" xml:space="preserve"> Et inuenimus iterum, quòd, corpora diapha <lb/>na colorata coloribus fortibus, ſicut uina fortia fortis ruboris, quæ ſunt in uaſis diaphanis, quando <lb/>fuerint in locis obſcuris & lucibus debilibus, apparent nigra & obſcura, & quaſi non diaphana, & <lb/>cum fuerint in lucibus fortibus, & orta fuerit ſuper ipſa lux ſolis, clareſcunt colores eorum, & appa-<lb/>ret in eis diaphanitas.</s> <s xml:id="echoid-s387" xml:space="preserve"> Et ſimiliter colores lapidum diaphanorum coloratorum, quando fuerint in <lb/>locis obſcuris, apparent turbidi & obſcuri:</s> <s xml:id="echoid-s388" xml:space="preserve"> & cum ſuper ipſos oritur lux fortis, uel ponuntur in op-<lb/>poſitione lucis, ita quòd lux per ipſos pertranſeat, apparent colores eorum clari, & apparet in eis <lb/>diaphanitas propter penetrationem lucis.</s> <s xml:id="echoid-s389" xml:space="preserve"> Et iterum quando corpora diaphana colorata ponuntur <lb/>in oppoſitione lucis:</s> <s xml:id="echoid-s390" xml:space="preserve"> & fuerit poſitum ex parte contraria parti lucis, corpus album, ſicut diximus <lb/>ſuperius:</s> <s xml:id="echoid-s391" xml:space="preserve"> & ſi lux fuerit fortis:</s> <s xml:id="echoid-s392" xml:space="preserve"> apparebit form a illius coloris in umbra eius ſuper corpus album op-<lb/>poſitum ei:</s> <s xml:id="echoid-s393" xml:space="preserve"> & ſi lux oriens ſuper ipſum, fuerit debilis, apparebit ſuper corpus album oppoſitum ei <lb/>umbra tantùm, & non apparebit color.</s> <s xml:id="echoid-s394" xml:space="preserve"> Et iterum inuenimus quòd pennæ pauonis, & pannus, qui <lb/>dicitur amilialmon, id eſt ſericus uiridis mixtus cum fuſco roſeo, diuerſificantur in colore apud ui-<lb/>ſum in diuerſis temporibus diei, ſecundum diuerſitatem lucis orientis ſuper ipſa.</s> <s xml:id="echoid-s395" xml:space="preserve"> Significant ergo <lb/>iſtæ diſpoſitiones apparentes in coloribus, quòd colores corporum coloratorum non comprehen-<lb/>duntur à uiſu, niſi ſecundum luces orientes ſuper ipſa.</s> <s xml:id="echoid-s396" xml:space="preserve"> Et cum luces fortes uiſibilium occultent <lb/>quaſdam res, quæ ſunt in quibuſdam uiſibilibus aliquando, & aliquando manifeſtent nobis res <lb/>quaſdam, quæ ſunt in quibuſdam uiſibilibus:</s> <s xml:id="echoid-s397" xml:space="preserve"> & luces debiles uiſibilium aliquando manifeſtent <lb/>quaſdam res, quæ ſunt in quibuſdam uiſibilibus, & aliquando occultent quaſdam res, quæ ſunt in <lb/>quibuſdam uiſibilibus:</s> <s xml:id="echoid-s398" xml:space="preserve"> & corporum coloratorum colores aliquando alterentur ſecundum diuerſi-<lb/>tatem lucis, quæ oritur ſuper ipſa:</s> <s xml:id="echoid-s399" xml:space="preserve"> & luces fortes orientes ſuper ipſum uiſum, aliquando prohibeãt <lb/>ipſum uiſum à comprehenſione quorundam uiſibilium:</s> <s xml:id="echoid-s400" xml:space="preserve"> & uiſus tamen in omnibus iſtis nihil com-<lb/>prehendat ex uiſibilibus, niſi ſit illuminata forma.</s> <s xml:id="echoid-s401" xml:space="preserve"> Ergo quod comprehendit uiſus ex re uiſa, non <lb/>eſt niſi ſecundum lucem, quæ eſt in illa re uiſa, & ſecundum luces, quæ oriuntur ſuper ipſum uiſum <lb/>in comprehenſione illius rei uiſibilis, & ſuper aerem medium inter uiſum & rem uiſam.</s> <s xml:id="echoid-s402" xml:space="preserve"> Quare ue-<lb/>rò luces fortes prohibeant uiſum à comprehenſione uiſibilium quorũdam, declarabitur à nobis in <lb/>ſermone noſtro de qualitate uiſionis.</s> <s xml:id="echoid-s403" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div11" type="section" level="0" n="0"> <head xml:id="echoid-head31" xml:space="preserve">DE COMPOSITIONE OCVLI, FORMA ET SI-<lb/>tu. Caput quartum.</head> <head xml:id="echoid-head32" xml:space="preserve" style="it">4. Ortus & principium oculi exiſtit è cerebro: & conſtat è tribus humori-<lb/>bus & quatuor tunicis. 4 p 3.</head> <p> <s xml:id="echoid-s404" xml:space="preserve">OCulus eſt compoſitus ex telis & corporibus diuerſis:</s> <s xml:id="echoid-s405" xml:space="preserve"> & principium & incrementum eius <lb/>eſt ex anteriore parte cerebri:</s> <s xml:id="echoid-s406" xml:space="preserve"> quoniam ex anteriore parte creſcunt duo nerui optici conſi-<lb/>miles, & incipiunt oriri ex duobus locis à duabus partib.</s> <s xml:id="echoid-s407" xml:space="preserve"> anterioris cerebri:</s> <s xml:id="echoid-s408" xml:space="preserve"> & dicitur quòd <lb/>uterque illorum habet duas tunicas, & quòd creſcunt à duabus telis cerebri, & perueniunt ad me-<lb/>dium exterioris partis cerebri & anterioris cerebri, deinde concurrunt & efficiunt unum neruum <lb/>opticum:</s> <s xml:id="echoid-s409" xml:space="preserve"> deinde iſte neruus diuiditur, & efficiuntur iterum duo nerui optici ęquales & conſimiles:</s> <s xml:id="echoid-s410" xml:space="preserve"> <lb/>deinde extenduntur iſti duo nerui, donec perueniant ad duo conuexa duorum oculorum oſsium <lb/>concauorum continentium duos oculos:</s> <s xml:id="echoid-s411" xml:space="preserve"> & in duobus medijs iſtorum duorum concauorum oſsiũ <lb/>ſunt duo foramina æqualiter perforata:</s> <s xml:id="echoid-s412" xml:space="preserve"> & ſitus eorum in neruo communi, eſt ſitus conſimilis illi.</s> <s xml:id="echoid-s413" xml:space="preserve"> <lb/>Nerui ergo intrãt iſta duo foramina, & exeunt ad concaua duorum oſsium, & illic dilatantur & am-<lb/>pliantur, & efficitur extremitas utriuſq;</s> <s xml:id="echoid-s414" xml:space="preserve"> eorum quaſi inſtrum entum ponendi uinum in dolijs:</s> <s xml:id="echoid-s415" xml:space="preserve"> & u-<lb/>terq;</s> <s xml:id="echoid-s416" xml:space="preserve"> oculorum eſt compoſitus ſuper iſtam extremitatem nerui, quæ eſt prædictum inſtrumentum, <lb/>& conſolidatur cum ipſo:</s> <s xml:id="echoid-s417" xml:space="preserve"> & ſitus utriuſque oculorum ex neruo communi eſt ſitus conſimilis.</s> <s xml:id="echoid-s418" xml:space="preserve"> Etto <lb/>tus uterq;</s> <s xml:id="echoid-s419" xml:space="preserve"> oculus eſt compoſitus extunicis multis.</s> <s xml:id="echoid-s420" xml:space="preserve"> Prima ergo illarum eſt pinguedo alba, quæ im-<lb/>pler concauum oſsis:</s> <s xml:id="echoid-s421" xml:space="preserve"> & eſt maxima pars oculi:</s> <s xml:id="echoid-s422" xml:space="preserve"> & dicitur conſolidatiua.</s> <s xml:id="echoid-s423" xml:space="preserve"> Et intra iſtam pinguedinem <lb/>eſt ſphæra rotunda, concaua, nιgra ut plurimum, & uiridis, & glauca in quibuſdã oculis:</s> <s xml:id="echoid-s424" xml:space="preserve"> & corpus <lb/>iſtius ſphæræ eſt tenue, & inſuper denſum & non rarum:</s> <s xml:id="echoid-s425" xml:space="preserve"> & manifeſtũ eius eſt applicatum cũ conſo-<lb/>lidatiua:</s> <s xml:id="echoid-s426" xml:space="preserve"> & interius eius eſt concauum:</s> <s xml:id="echoid-s427" xml:space="preserve"> & in parte concauitatis eſt quaſi quædã attritio:</s> <s xml:id="echoid-s428" xml:space="preserve"> & quaſi con <lb/>ſolidatiua continet iſtam ſphæram, præterquàm ſuum anterius:</s> <s xml:id="echoid-s429" xml:space="preserve"> quoniam conſolidatiua non coope <lb/>rit anterius iſtius ſphæræ, ſed circulatur ſuper anterius eius:</s> <s xml:id="echoid-s430" xml:space="preserve"> & iſta tunica dicitur uuea, quia aſsimi-<lb/>latur uuæ Et in medio anterioris uueæ eſt foramẽ rotundum perforatũ uſq;</s> <s xml:id="echoid-s431" xml:space="preserve"> ad eius concauũ:</s> <s xml:id="echoid-s432" xml:space="preserve"> & eſt <lb/>oppoſitum extremitati concauitatis nerui, ſuper quem componitur oculus.</s> <s xml:id="echoid-s433" xml:space="preserve"> Et cooperit iſtud fora-<lb/>men, & omne anterius uueæ, in cuius circuitu circulatur conſolidatiua, extrinſecus tunica fortis, al <lb/>ba, diaphana:</s> <s xml:id="echoid-s434" xml:space="preserve"> & dicitur cornea, quia aſsimilatur cornu albo & claro.</s> <s xml:id="echoid-s435" xml:space="preserve"> Et intra concauũ uueæ eſt ſphæ <lb/>ra alba, parua, humida, receptibilis humiditatis formarũ uiſibilium:</s> <s xml:id="echoid-s436" xml:space="preserve"> & in ea eſt diaphanitas non in-<lb/>tenſa ualde, ſed aliqua ſpiſsitudo:</s> <s xml:id="echoid-s437" xml:space="preserve"> & diaphanitas eius aſsimilatur diaphanitati glaciei:</s> <s xml:id="echoid-s438" xml:space="preserve"> & ideo dici-<lb/>tur glacialis:</s> <s xml:id="echoid-s439" xml:space="preserve"> & eſt cõpoſita ſuper extremitatẽ cõcauitatis nerui:</s> <s xml:id="echoid-s440" xml:space="preserve"> & in anteriori iſtius ſphæræ eſt cõ-<lb/>preſsio ſuperficialis parua, & aſsimilatur compreſsioni ſuperficiei lenticulæ:</s> <s xml:id="echoid-s441" xml:space="preserve"> ſuperficies ergo ante-<lb/>rioris eius eſt portio ſuperficiei ſphæræ, maioris ſuperſicie ſphęrica, continente duo eius foramina:</s> <s xml:id="echoid-s442" xml:space="preserve"> <lb/> <pb o="4" file="0010" n="10" rhead="ALHAZEN"/> & iſta compreſsio eſt oppoſita foramini, quod eſt in anteriori uueæ:</s> <s xml:id="echoid-s443" xml:space="preserve"> & ſitus eius conſimilis eſt cum <lb/>eo.</s> <s xml:id="echoid-s444" xml:space="preserve"> Et iſte humor diuiditur in partes duas diuerſæ diaphanitatis, & altera illarum ſequitur anterius <lb/>eius, & altera ſequitur eius poſterius:</s> <s xml:id="echoid-s445" xml:space="preserve"> & diaphanitas partis poſterioris eius aſsimilatur diaphanita <lb/>ti uitri quaſi fruſtati:</s> <s xml:id="echoid-s446" xml:space="preserve"> & iſta pars dicitur humor uitreus.</s> <s xml:id="echoid-s447" xml:space="preserve"> Et continet duas has partes cõgregatas tela <lb/>ualde tenuis, & dicitur aranea, quoniam aſsimilatur texturæ araneæ.</s> <s xml:id="echoid-s448" xml:space="preserve"> Et in poſteriore parte conca-<lb/>uitatis ſphæræ uueæ dicitur, quod eſt foramen rotundum, & eſt ſuper extremitatem concauitatis <lb/>nerui:</s> <s xml:id="echoid-s449" xml:space="preserve"> & ſphæra glacialis eſt compoſita in iſto foramine:</s> <s xml:id="echoid-s450" xml:space="preserve"> & rotunditas iſtius foraminis (& eſt extre-<lb/>mitas nerui) continet medium ſphæræ glacialis:</s> <s xml:id="echoid-s451" xml:space="preserve"> & conſolidatur uuea cum glaciali in circulo cõti-<lb/>nente iſtud foramen.</s> <s xml:id="echoid-s452" xml:space="preserve"> Et dicitur quòd ortus uueæ eſt ex tunica interiore duarum tunicarũ duorum <lb/>neruorum opticorum:</s> <s xml:id="echoid-s453" xml:space="preserve"> & quòd ortus corneæ eſt ex tunica exteriore duarum tunicarũ iſtius nerui.</s> <s xml:id="echoid-s454" xml:space="preserve"> <lb/>& implet concauitatem uueæ humor albus, tenuis, clarus, diaphanus:</s> <s xml:id="echoid-s455" xml:space="preserve"> & dicitur albugineus, quoniã <lb/>aſsimilatur albumini oui in tenuitate, & albedine, & diaphanitate eius:</s> <s xml:id="echoid-s456" xml:space="preserve"> & ipſe implet concauitatem <lb/>uueæ, & contingit anterius glacialis, & implet foramen, quod eſt in anteriori uueæ, & contingit cõ-<lb/>cauum corneæ.</s> <s xml:id="echoid-s457" xml:space="preserve"> Et ſphæra glacialis eſt cõpoſita ſuper concauitatem nerui:</s> <s xml:id="echoid-s458" xml:space="preserve"> & ſequitur cõcauitatem <lb/>nerui humor uitreus.</s> <s xml:id="echoid-s459" xml:space="preserve"> Erunt ergo cornea, & humor albugineus, & glacialis, & uitreus ſe conſequen-<lb/>tes.</s> <s xml:id="echoid-s460" xml:space="preserve"> Et omnes iſtæ tunicæ ſunt diaphanæ:</s> <s xml:id="echoid-s461" xml:space="preserve"> Et foramen, quod eſt in anteriori uueæ, eſt oppoſitum fo-<lb/>ramini concauitatis nerui.</s> <s xml:id="echoid-s462" xml:space="preserve"> Et dicitur, quòd ſpiritus uiſibilis emittitur ex anteriori parte cerebri, & <lb/>implet duas concauitates duorum neruorũ primorum coniunctorum cum cerebro, & peruenit ad <lb/>neruum communem, & implet concauitatem eius, & uenit ad duos neruos ſecũdos opticos, & im-<lb/>plet ipſos, & peruenit ad glacialem, & dat ei uirtutem uiſibilem.</s> <s xml:id="echoid-s463" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div12" type="section" level="0" n="0"> <head xml:id="echoid-head33" xml:space="preserve" style="it">5. In toti{us} oculi ſeu motu ſeu quiete, ſit{us} partium ſtabilis permanet. 25 p 3.</head> <p> <s xml:id="echoid-s464" xml:space="preserve">ET inter circumferentiam glacialis coniunctam cum uuea, & foramen, quod eſt in cõcauo oſ-<lb/>ſis, ex quo exit neruus, eſt ſpatium aliquantulum:</s> <s xml:id="echoid-s465" xml:space="preserve"> & neruus extẽditur in iſto ſpatio ex fine fo-<lb/>raminis uſq;</s> <s xml:id="echoid-s466" xml:space="preserve"> ad circumferentiam glacialis ſecundũ pyramid alitatẽ & amplificationẽ:</s> <s xml:id="echoid-s467" xml:space="preserve"> & quã-<lb/>tò magis elongatur à foramine oſsis, tantò m agis amplificatur, quouſq;</s> <s xml:id="echoid-s468" xml:space="preserve"> perueniat ad circumferen-<lb/>tiam ſphæræ glacialis, & conſolidetur cũ circumferentia eius.</s> <s xml:id="echoid-s469" xml:space="preserve"> Et corpus conſolidatiuæ continet i-<lb/>ſtam partem pyramidalem nerui, & continet ſphæram uueã:</s> <s xml:id="echoid-s470" xml:space="preserve"> & ſphæra uueæ antecedit medium cõ-<lb/>ſolidatiuæ ad partem manifeſtam oculi.</s> <s xml:id="echoid-s471" xml:space="preserve"> Et corpus conſolidatiuæ eſt conſolidatum cum ſphæra u-<lb/>uea, & cum extremitate pyramidali, & cuſtodit ſitum eius.</s> <s xml:id="echoid-s472" xml:space="preserve"> Cum ergo mouetur oculus, mouebitur <lb/>ſecundum ſe totum, & ſic declinabit neruus, ſuper quem componitur oculus, apud motum eius, & <lb/>erit declinatio apud foramen, quod eſt in concauitate oſsis:</s> <s xml:id="echoid-s473" xml:space="preserve"> quoniam concauitas oſsis continet to-<lb/>tum oculum, & oculus mouetur ſecundum ſe totum in iſta cõcauitate:</s> <s xml:id="echoid-s474" xml:space="preserve"> & cõſolidatiua conſolidatur <lb/>cum eo quod eſt in anteriori oculi ex neruo, & ex tunicis reſiduis, & cuſtodit ſemper ſitũ eius.</s> <s xml:id="echoid-s475" xml:space="preserve"> De-<lb/>clinatio ergo nerui apud motum oculi, non eſt, niſi à poſteriori totius oculi:</s> <s xml:id="echoid-s476" xml:space="preserve"> eſt ergo apud foramen, <lb/>quod eſt in concauitate totius oſsis.</s> <s xml:id="echoid-s477" xml:space="preserve"> Similiter quando oculus quieuerit, & neruus declinauerit, nõ <lb/>erit declinatio niſi apud foramen, quod eſt in concauitate oſsis.</s> <s xml:id="echoid-s478" xml:space="preserve"> Nã non mutatur ſitus partiũ totius <lb/>oculi inter ſe, neque apud motum neque apud quietem.</s> <s xml:id="echoid-s479" xml:space="preserve"> Declinatio ergo nerui, ſuper quem cõponi <lb/>tur oculus, nõ eſt, niſi apud foramẽ, quod eſt in cõcauitate oſsis, ſiue moueatur oculus, ſiue quieſcat.</s> <s xml:id="echoid-s480" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div13" type="section" level="0" n="0"> <head xml:id="echoid-head34" xml:space="preserve" style="it">6. Ocul{us} tot{us} & ſpher a uuea centris differunt: & oculi centrum <lb/>ect alti{us}. 8 p 3.</head> <p> <s xml:id="echoid-s481" xml:space="preserve">SVperficies autem manifeſta corneæ eſt ſuperficies ſphærica, & eſt continuata cum ſuperfi ie <lb/>totius oculi & cum toto oculo:</s> <s xml:id="echoid-s482" xml:space="preserve"> & totus oculus eſt maior ſphæra uuea, quæ eſt quiddam eius:</s> <s xml:id="echoid-s483" xml:space="preserve"> <lb/>Superficies autem manifeſta corneæ eſt cũ ſuperficie totius oculi, & eſt maior ſuperficie ſphę-<lb/>ræ uueæ:</s> <s xml:id="echoid-s484" xml:space="preserve"> ſemidiameter ergo eius eſt maior ſemidiametro uueæ.</s> <s xml:id="echoid-s485" xml:space="preserve"> Et quia ſuperficies intrinſeca cor-<lb/>neæ ſuperpoſita foramini uueæ eſt ſuperficies ſphærica concaua, æquidiſtans ſuperficiei manifeſtæ <lb/>ipſius corneæ, quoniam tota cornea eſt æqualis ſpiſsitudinis, propterea quòd centrum ſuperficiei <lb/>concauæ corneæ eſt idẽ cũ centro manifeſtæ ſuperficiei ſuæ conuexæ:</s> <s xml:id="echoid-s486" xml:space="preserve"> ſed ſuperficies concaua cor <lb/>neæ ſecat ſuperficiem ſpheræ uueę ſuper circumferentiam foraminis, quod eſt in anteriori parte u-<lb/>ueæ:</s> <s xml:id="echoid-s487" xml:space="preserve"> centrum ergo eius eſt remotius in profundo, quã centrum uueæ, quoniam hoc eſt in proprie-<lb/>tatibus centrorũ ſphęrarũ ſe interſecantium.</s> <s xml:id="echoid-s488" xml:space="preserve"> Et etiã quia ſphęra uuea non eſt in medio conſolidati-<lb/>uę, ſed antecedit ad partẽ ſuperficiei manifeſtæ oculi, & ſuperficies manifeſta oculi eſt ex ſphęra ma <lb/>iore ſphæra uueæ, erit centrum ſuperficiei manifeſtæ oculi remotius in profundo centro uueæ.</s> <s xml:id="echoid-s489" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div14" type="section" level="0" n="0"> <head xml:id="echoid-head35" xml:space="preserve" style="it">7. Rect a connectens centra ſphær arum corneæ & uueæ, continuata tranſit per <lb/>centrum for aminis uueæ, & medium caui nerui optici. 9 p 3.</head> <p> <s xml:id="echoid-s490" xml:space="preserve">ET recta linea, quæ continuat duo centra, ſcilicet ctrum ſuperficiei corneæ, & centrum uueę, <lb/>quando extrahitur<gap/>ct<gap/>, peruenit ad centrum foraminis, quod eſt in anteriori uueę, & ad duo <lb/>media duarum ſuperficierũ corneæ æquidiſtantium:</s> <s xml:id="echoid-s491" xml:space="preserve"> ſuperficies enim concaua corneę & con <lb/>uexa uueæ ſunt ſuperficies ſphæricę ſecantes ſe:</s> <s xml:id="echoid-s492" xml:space="preserve"> & linea quæ continuat centra earũ, tranſit per cen <lb/>trum circuliſectionis, & eſt perpendicularis ſuper ſuperficiem eius:</s> <s xml:id="echoid-s493" xml:space="preserve"> quia linea, quę exit à cen-<lb/>tro circuli ſectionis, & eft perpendicularis ſuper ſuperficiem eius, tranſit <lb/>per centra duarum ſphærarum.</s> <s xml:id="echoid-s494" xml:space="preserve"/> </p> <pb o="5" file="0011" n="11" rhead="OPTICAE LIBER I."/> </div> <div xml:id="echoid-div15" type="section" level="0" n="0"> <head xml:id="echoid-head36" xml:space="preserve" style="it">8. Centrum ſphæræ uueæ eſt inferi{us} centris reliquarum oculi partium. 8 p 3.</head> <p> <s xml:id="echoid-s495" xml:space="preserve">ET quia ſuperficies concaua corneæ contingit ſuperficiem humoris albuginei, qui eſt in ante-<lb/>riori foramine uueę, & ſuperponitur ipſi.</s> <s xml:id="echoid-s496" xml:space="preserve"> Superficies ergo humoris albuginei conuexa etiam <lb/>eſt ſuperficies ſphærica, cuius centrum eſt centrũ ſuperficiei ipſi ſuperpoſitæ.</s> <s xml:id="echoid-s497" xml:space="preserve"> Superficies er-<lb/>go manifeſta corneæ, & ſuperficies intrinſeca ipſius, & ſuperficies humoris albuginei cõuexa, quæ <lb/>contingit concauum corneæ, ſunt ſuperficies ſphæricę æ quidiſtantes.</s> <s xml:id="echoid-s498" xml:space="preserve"> Centrum igitur earum eſt u-<lb/>num punctum cõmune, & eſt remotius in profundo centro uueæ:</s> <s xml:id="echoid-s499" xml:space="preserve"> & linea, quæ tranſit per centrum <lb/>uueæ, & per centrum corneæ, & per centrũ foraminis, quod eſt in anteriori uueæ, quando extendi-<lb/>tur rectè, tranſibit per medium concauitatis nerui, ſuper quem cõponitur oculus:</s> <s xml:id="echoid-s500" xml:space="preserve"> quoniã foramen <lb/>quod eſt in anteriori uueæ, eſt oppoſitũ foramini, quod eſt in poſteriore parte uueæ, quod eſt extre <lb/>mitas concauitatis nerui [per 4 n.</s> <s xml:id="echoid-s501" xml:space="preserve">]</s> </p> </div> <div xml:id="echoid-div16" type="section" level="0" n="0"> <head xml:id="echoid-head37" xml:space="preserve" style="it">9. Recta connectẽs centra ſphærarũ cryſtallinæ & uueæ, cõtinuata cadit in centrũ circuli <lb/>cõglutinãtis cryſtallinã & uitreã ſphær {as} cũ uuea: & eſt ad ipſum perpendicularis. 10 p 3.</head> <p> <s xml:id="echoid-s502" xml:space="preserve">ET ſuperficies anterioris glacialis etiã eſt ſphęrica ſuperficies, & ipſa ſecat ſphęrã uueę:</s> <s xml:id="echoid-s503" xml:space="preserve"> centrũ <lb/>ergo eius eſt remotius in profundo cẽtro uueæ.</s> <s xml:id="echoid-s504" xml:space="preserve"> Et linea recta, quę cõtinuat cẽtra earũ, tranſit <lb/>per centrũ circuli ſectionis, & eſt perpendicularis ſuper ipſum.</s> <s xml:id="echoid-s505" xml:space="preserve"> Et circulus ſectionis inter ſu-<lb/>perficiẽ anterioris glacialis, & ſuperficiẽ ſphęrę uueæ, eſt aut circulus diſtinguẽs finẽ conſolidatio-<lb/>nis inter glacialẽ & uueã, aut æquidiſtans ei:</s> <s xml:id="echoid-s506" xml:space="preserve"> quoniã ſuperficies quę eſt in anteriori glacialis, eſt op-<lb/>poſita foramini, qđ eſt in anteriori uueę, & ſitus eius eſt cõſimilis cũ eo.</s> <s xml:id="echoid-s507" xml:space="preserve"> Finis ergo iſtius ſuperficiei <lb/>(& eſt circulus ſectionis inter duas ſuperficies glacialis & uueę) aut eſt ipſe circulus conſolidatio-<lb/>nis, aut ęquidiſtãs ei.</s> <s xml:id="echoid-s508" xml:space="preserve"> Si ergo circulus ſectionis inter duas ſuperficies glacialis, fuerit circulus cõſoli <lb/>dationis, iſte circulus eſt circulus ſectionis inter ſuperficiẽ anterioris glacialis, & inter ſuperficiem <lb/>uueę.</s> <s xml:id="echoid-s509" xml:space="preserve"> Et ſi circulus ſectionis inter duas ſuperficies glacialis fuerit ęquidiſtãs circulo cõſolidationis <lb/>ſphęrę glacialis cũ uuea:</s> <s xml:id="echoid-s510" xml:space="preserve"> (quod quidẽ accidit, ſi fuerit cõſolidatio in poſteriori parte glacialis) tune <lb/>ſuperficies anterioris partis glacialis, quando fuerit mẽte extenſa ſuper illud, ſuper quod eſt ex ſua <lb/>ſphęra, ſecabit ſphęrã uueæ ſuper circulum æquidiſtantẽ iſti circulo, ſcilicet circulo ſectionis inter <lb/>duas ſuperficies glacialis propter ſimilitudinẽ ſitus iſtius circuli ad circumferentiam ſphæræ uueę.</s> <s xml:id="echoid-s511" xml:space="preserve"> <lb/>Et quia iſte circulus eſt æquidiſtans circulo cõſolidationis, erit ergo circulus ſectionis inter ſuper-<lb/>ficiẽ anterioris glacialis, & inter ſphæram uueã, aut ipſe circulus cõſolidationis, aut ęquidiſtãs ei.</s> <s xml:id="echoid-s512" xml:space="preserve"> Si <lb/>ergo iſte circulus fuerit ipſe circulus cõſolidationis, linea recta, quæ trãfit per centrũ anterioris gla-<lb/>cialis, & per centrũ uueę, tranſibit per centrũ ipſius circuli:</s> <s xml:id="echoid-s513" xml:space="preserve"> & erit perpendicularis ſuper ipſum:</s> <s xml:id="echoid-s514" xml:space="preserve"> quo <lb/>niã iſte circulus erit circulus ſectionis inter duas illas ſphęricas ſuperficies.</s> <s xml:id="echoid-s515" xml:space="preserve"> Sed ſi iſte circulus fue-<lb/>rit ęquidiſtans circulo conſolidationis, & eſt ęquidiſtãs circulo ſectionis inter duas ſuperficies gla-<lb/>cialis:</s> <s xml:id="echoid-s516" xml:space="preserve"> eſt ergo cũ circulo ſectionis inter duas ſuperficies glacialis:</s> <s xml:id="echoid-s517" xml:space="preserve"> in ſuperficie una ſphęrica:</s> <s xml:id="echoid-s518" xml:space="preserve"> quæ <lb/>eſt ſuperficies anterioris glacialis, & eſt ęquidiſtãs circulo ſectiõis.</s> <s xml:id="echoid-s519" xml:space="preserve"> Linea ergo quę trãſit per centrũ <lb/>uueę, & per centrũ ſuperficiei anterioris glacialis, tranſit per centrũ circuli cõſolidationis ſecundũ <lb/>oẽs diſpoſitiones, & eſt perpendicularis ſuper ipſum, ſiue ſit circulus conſolidationis ipſe circulus <lb/>ſectionis inter ſuperficiem anterioris glacialis & inter ſphærã uueę, ſiue ſit ęquidiſtans iſti circulo.</s> <s xml:id="echoid-s520" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div17" type="section" level="0" n="0"> <head xml:id="echoid-head38" xml:space="preserve" style="it">10. Centrum ſphæræ cryſtallinæ alti{us} eſt centro ſphæræ uitreæ. 11 p 3.</head> <p> <s xml:id="echoid-s521" xml:space="preserve">ET iterũ ſuperficies anterioris glacialis, & ſuperficies reſidui glacialis, ſunt duę ſuperficies ſphę <lb/>ricæ ſecantes ſe:</s> <s xml:id="echoid-s522" xml:space="preserve"> centrum ergo ſuperficiei anterioris, eſt remotius in profundo centro ſuper-<lb/>ficiei poſterioris.</s> <s xml:id="echoid-s523" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div18" type="section" level="0" n="0"> <head xml:id="echoid-head39" xml:space="preserve" style="it">11. Rect a connectens centra ſphær arum & uueæ, continuata cadit in centrum ui-<lb/>treæ, & medium cauinerui optici. 12 p 3.</head> <p> <s xml:id="echoid-s524" xml:space="preserve">ET linea recta, quæ continuat iſta duo cẽtra, tranſit per cen trũ circuli ſectionis, & eſt perpẽdi-<lb/>cularis ſuper ipſum:</s> <s xml:id="echoid-s525" xml:space="preserve"> & iam declaratũ eſt [9 n] quod tranſit per centrũ circuli conſolidatiõis, <lb/>& eſt perpẽdicularis ſuper ipſum:</s> <s xml:id="echoid-s526" xml:space="preserve"> hie uerò circulus aut eſt circulus ſectionis, aut ęquidiſtans <lb/>ei.</s> <s xml:id="echoid-s527" xml:space="preserve"> Linea ergo quę tranſit per centrũ uueæ, & per centrum anterioris glacialis, & per centrũ circuli <lb/>eõſolidationis, & eſt perpendicularis ſuper iſtũ circulũ, tranſit per centrũ reſidui glacialis.</s> <s xml:id="echoid-s528" xml:space="preserve"> Et cum <lb/>linea iſta tranſeat per cẽtrum reſidui glacialis, & per centrum circuli cõſolidationis, & ſit erecta ſu-<lb/>per circulum cõſolidationis ſecundum angulos rectos:</s> <s xml:id="echoid-s529" xml:space="preserve"> extenditur ergo in medio concauitatis ner <lb/>ui, ſuper quẽ cõponitur oculus:</s> <s xml:id="echoid-s530" xml:space="preserve"> quoniã circulus cõſolidationis eſt extremitas cõcauitatis nerui.</s> <s xml:id="echoid-s531" xml:space="preserve"> Et <lb/>iam declaratum eſt [7 n] quòd linea trãſiens per centrum uueæ, & per centrum corneę, & per cen-<lb/>trum foraminis, quod eſt in exteriori ſiue anteriori uueę, extẽditur in medio cõcauitatis nerui.</s> <s xml:id="echoid-s532" xml:space="preserve"> Iſta <lb/>ergo linea, quę tranſit per duo centra ſuperficiei glacialis, & per cẽtrum uueę, eſt ipſa linea, quę trã <lb/>ſit per centrum corneę, & per cẽtrum foraminis, quod eſt in anteriori uueę.</s> <s xml:id="echoid-s533" xml:space="preserve"> Iſta ergo linea trãſit per <lb/>cẽtrum corneę, & per cẽtrum uueę, & per duo cẽtra ſuperficiei glacial<gap/>s & per centrum foraminis, <lb/>quod eſt in anteriore uueæ, & per <gap/>cẽtrum circuli cõſolidationis, & trãſit per duo media tunicarum <lb/>omniũ oppoſitarũ foramini uueę:</s> <s xml:id="echoid-s534" xml:space="preserve"> Et eſt perpẽdicularis ſuper ſuքficies omniũ tunicarũ oppoſitarũ <lb/>foramini uueę, & eſt perpẽdicularis ſuper ſuքficiẽ foraminis uueę, & eſt perpẽdicularis ſuper ſuքfi <lb/>ciẽ circuli conſolidationis, & extenditur in medio cõcauitatis nerui, ſuper quẽ cõponitur oculus.</s> <s xml:id="echoid-s535" xml:space="preserve"/> </p> <pb o="6" file="0012" n="12" rhead="ALHAZEN"/> </div> <div xml:id="echoid-div19" type="section" level="0" n="0"> <head xml:id="echoid-head40" xml:space="preserve" style="it">12. Centra ſphær arum toti{us} oculi, cryſtallinæ, utriuſ ſuperficiei corneæ, & con-<lb/>uexæ humoris albuginei, eſt unum punctum. 7 p 3.</head> <p> <s xml:id="echoid-s536" xml:space="preserve">ET cum declaratũ ſit, [6.</s> <s xml:id="echoid-s537" xml:space="preserve">8 n] quòd centrũ corneę, & centrũ ſuperficiei anterioris glacialis, am-<lb/>bo ſint ſuper iſtam lineã, & ambo ſint remotiora in profundo centro uueæ, melius eſt, ut cen-<lb/>trũ ſuperficiei anterioris glacialis ſit ipſum centrũ corneæ, ita ut centra omniũ ſuperficierum <lb/>oppoſitarũ foramini uueæ, ſint unũ punctũ cõmune:</s> <s xml:id="echoid-s538" xml:space="preserve"> & ſic erunt omnes lineæ exeuntes à centro ad <lb/>ſuperficiem oculi perpendiculares ſuper oẽs ſuperficies oppoſitas foramini:</s> <s xml:id="echoid-s539" xml:space="preserve"> & hinc poſterius decla <lb/>rabitur, apud noſtrum ſermonem de qualitate uiſionis, quòd centrum ſuperficiei corneæ & centrũ <lb/>ſuperficiei anterioris glacialis, eſt unum centrũ cõmune.</s> <s xml:id="echoid-s540" xml:space="preserve"> Superficies ergo tunicarũ uiſus, oppoſita-<lb/>rum foramini uueæ, ſunt ſuperficies ſphæricæ, quarum centrum eſt unum punctum commune.</s> <s xml:id="echoid-s541" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div20" type="section" level="0" n="0"> <head xml:id="echoid-head41" xml:space="preserve" style="it">13. In toti{us} oculi ſeu motu ſeu quiete ſit{us} partium ſtabilis permanet. 25 p 3. Idem 9 n.</head> <p> <s xml:id="echoid-s542" xml:space="preserve">ET iterum quia iſtud centrũ eſt centrum ſuperficiei manifeſtæ oculi, cõtinuatę cũ ſuperficie cõ <lb/>tinente totum oculum, & totus oculus eſt rotundus, niſi quantũ deficit de cõpletione ſphęrę <lb/>pinguedinis cõſolidatiuæ à parte anteriore ipſius oculi, & iſte defectus nõ operatur diuerſita <lb/>tem in motu oculi, quoniã non tangit cõcauum oſsis.</s> <s xml:id="echoid-s543" xml:space="preserve"> Iſtud ergo centrum erit centrum totius oculi:</s> <s xml:id="echoid-s544" xml:space="preserve"> <lb/>ergo eſt intra totũ oculum.</s> <s xml:id="echoid-s545" xml:space="preserve"> Centrũ ergo ſuperficierum tunicarum uiſus, oppoſitarũ foramini uueæ, <lb/>eſt intra totum oculum.</s> <s xml:id="echoid-s546" xml:space="preserve"> Cum ergo mouetur oculus, non mutabitur punctum oculi, quod eſt cẽtrũ <lb/>ſuperficierũ tunicarum uiſus, nec mutabitur ſitus eius ab iſtis ſuperficieb.</s> <s xml:id="echoid-s547" xml:space="preserve"> ſed cuſtodit ſitum ſuum.</s> <s xml:id="echoid-s548" xml:space="preserve"> <lb/>Nam oculus qñ mouetur, nõ mouetur niſi ſecundum ſe totũ, & ſitus partium totius inter ſe nõ mu-<lb/>tatur apud motum:</s> <s xml:id="echoid-s549" xml:space="preserve"> & iſtud centrũ eſt intra.</s> <s xml:id="echoid-s550" xml:space="preserve"> Situs ergo eius nõ mutatur apud ſuũ motum.</s> <s xml:id="echoid-s551" xml:space="preserve"> Et ſimili-<lb/>ter tunicarum ſitus nõ mutatur apud totum oculum, id eſt apud motũ ipſius uiſus.</s> <s xml:id="echoid-s552" xml:space="preserve"> Situs ergo iſtius <lb/>centri apud ſuperficiem tunicarum uiſus non mutatur neq;</s> <s xml:id="echoid-s553" xml:space="preserve"> in motu, neq;</s> <s xml:id="echoid-s554" xml:space="preserve"> in quiete.</s> <s xml:id="echoid-s555" xml:space="preserve"> Et iam declara-<lb/>tum eſt [5 n] quòd declinatio nerui apud motum uiſus, & apud quietem non eſt, niſi apud foramen <lb/>oculi, quod eſt in concauitate oſsis:</s> <s xml:id="echoid-s556" xml:space="preserve"> quoniam non eſt niſi à poſteriori totius oculi.</s> <s xml:id="echoid-s557" xml:space="preserve"> Declinatio ue-<lb/>rò nerui apud motum uiſus & quietem, non eſt niſi à poſteriori centri eius, & non mutatur ſitus <lb/>partium totius oculi inter ſe neque in motu, neque in quiete.</s> <s xml:id="echoid-s558" xml:space="preserve"> Situs ergo centrorum tunicarum o-<lb/>culi apud totum oculum non mutatur, neque in motu uiſus, neque in quiete.</s> <s xml:id="echoid-s559" xml:space="preserve"> Linea ergo tranſiens <lb/>per cẽtrum non mutat ſuum locum uel ſitum apud totum oculum, neque apud partes eius, ſcilicet <lb/>neq;</s> <s xml:id="echoid-s560" xml:space="preserve"> in motu, neque in quiete.</s> <s xml:id="echoid-s561" xml:space="preserve"> Et cum ſitus iſtius lineæ non mutetur apud totum oculum, neque <lb/>apud partes eius:</s> <s xml:id="echoid-s562" xml:space="preserve"> Situs erg<gap/> iſtius lineæ non <lb/> <anchor type="figure" xlink:label="fig-0012-01a" xlink:href="fig-0012-01"/> mutatur apud ſuperficiem circuli conſolida-<lb/>tionis, neque apud ſuam circumferentiam:</s> <s xml:id="echoid-s563" xml:space="preserve"> Et <lb/>iſte circulus eſt extremitas concauitatis ner-<lb/>ui.</s> <s xml:id="echoid-s564" xml:space="preserve"> Situs ergo ſuperficiei eius à ſuperficie con-<lb/>cauitatis nerui, eſt ſitus conſimilis.</s> <s xml:id="echoid-s565" xml:space="preserve"> Et decli-<lb/>natio partis pyramidalis nerui ſuper ſuperfi-<lb/>ciem iſtius circuli, eſt declinatio cõſimilis:</s> <s xml:id="echoid-s566" xml:space="preserve"> quo <lb/>niam, ſitus glacialis ab iſto neruo eſt ſitus con-<lb/>ſimilis, & ſitus partium oculi non mutatur in-<lb/>ter ſe.</s> <s xml:id="echoid-s567" xml:space="preserve"> Superficies ergo concauitatis nerui à <lb/>loco circũferentiæ circuli cõſolidationis uſq;</s> <s xml:id="echoid-s568" xml:space="preserve"> <lb/>ad locũ declinationis nerui, qui eſt pars pyra-<lb/>midalis, nõ mutat ſitum ſuũ apud totũ oculũ, <lb/>neq;</s> <s xml:id="echoid-s569" xml:space="preserve"> apud circulũ cõſolidationis.</s> <s xml:id="echoid-s570" xml:space="preserve"> Et iam decla <lb/>ratũ eſt [5 n] quòd ſitus lineæ, quę trãſit per cẽ <lb/>tra oĩa, nõ mutatur apud circulũ cõſolidatiõis, <lb/>& quòd ipſa extẽditur in medio concauitatis <lb/>nerui.</s> <s xml:id="echoid-s571" xml:space="preserve"> Et cũ ſitus iſtius lineæ nõ mutetur apud <lb/>circulũ cõſolidationis, neq;</s> <s xml:id="echoid-s572" xml:space="preserve"> ſuperficies conca-<lb/>uitatis nerui, quę eſt à loco circũferentię circu <lb/>li cõſolidationis uſq;</s> <s xml:id="echoid-s573" xml:space="preserve"> ad locũ declinatiõis, mu-<lb/>tet ſuũ ſitũ apud circulũ cõſolidationis:</s> <s xml:id="echoid-s574" xml:space="preserve"> iſta er-<lb/>go linea nõ mutat ſuũ ſitũ apud cõcauitatẽ ner <lb/>ui, quouſq;</s> <s xml:id="echoid-s575" xml:space="preserve"> perueniat ad locũ declinationis.</s> <s xml:id="echoid-s576" xml:space="preserve"> Li <lb/>nea ergo quæ trãſit per centra tunicarũ, tranſit <lb/>per centrum cõſolidationis:</s> <s xml:id="echoid-s577" xml:space="preserve"> & eſt erecta ſuper <lb/>ipſum ſecundũ angulos rectos, & extẽditur in <lb/>medio cõcauitatis nerui pyramidalis, quouſq;</s> <s xml:id="echoid-s578" xml:space="preserve"> <lb/>perueniat ad locũ declinatiõis nerui:</s> <s xml:id="echoid-s579" xml:space="preserve"> & erit ſi-<lb/>tus ſuus ſemper à ſuperficie cõcauitatis nerui, <lb/>quę eſt intra totũ oculũ, & ab omnib.</s> <s xml:id="echoid-s580" xml:space="preserve"> partib.</s> <s xml:id="echoid-s581" xml:space="preserve"> o-<lb/>culi, & ab omnib.</s> <s xml:id="echoid-s582" xml:space="preserve"> ſuքficieb.</s> <s xml:id="echoid-s583" xml:space="preserve"> tunicarũ uiſus, idẽ <lb/>ſitus, & nõ mutatur neq;</s> <s xml:id="echoid-s584" xml:space="preserve"> in motu uiſus, neq;</s> <s xml:id="echoid-s585" xml:space="preserve"> in <lb/> <pb o="7" file="0013" n="13" rhead="OPTIC AE LIBER I."/> motu eius.</s> <s xml:id="echoid-s586" xml:space="preserve"> Iſti ergo ſunt ſitus tunicarũ uiſus, & ſitus centrorũ earũ, & ſitus lineæ rectæ trãſeũtis per <lb/>centra eorum.</s> <s xml:id="echoid-s587" xml:space="preserve"> Oculi autem ambo ſunt conſimiles in omnibus ſuis diſpoſitionibus, & in ſuis tuni-<lb/>cis, & figuris ſuarum tunicarum, & in ſitu cuiuslibet tunicæ, reſpectu totius oculi.</s> <s xml:id="echoid-s588" xml:space="preserve"> Et cum ita ſit, ſi-<lb/>tus ergo cuiuslibet centrorum, quorum diſtinctio declarata fuit, apud totum oculum, & apud par-<lb/>tes eius, eſt ſicut ſitus centri reſpondentis illi centro in alio oculo apud totum oculnm illum, & a-<lb/>pud partes eius.</s> <s xml:id="echoid-s589" xml:space="preserve"> Et cum ſitus centrorum in utroq;</s> <s xml:id="echoid-s590" xml:space="preserve"> oculo ſit ſimilis ſitus, erit ſitus lineæ tranſeun-<lb/>tis per centrũ in uno oculo apud totũ oculũ, & apud partes eius, & apud fuas tunicas, ſimilis ſitui <lb/>lineæ tranſeuntis per centrum alterius oculi apud totum oculum, & apud partes eius, & apud ſuas <lb/>tunicas.</s> <s xml:id="echoid-s591" xml:space="preserve"> Situs ergo duarum linearum tranſeuntium per cẽtra tunicarum uiſus ab utroq;</s> <s xml:id="echoid-s592" xml:space="preserve"> oculo, eſt <lb/>ſitus conſimilis in omnibus ſuis diſpoſitionibus.</s> <s xml:id="echoid-s593" xml:space="preserve"> Et utraq;</s> <s xml:id="echoid-s594" xml:space="preserve"> conſolidatiuarum cõſolidatur cum eis:</s> <s xml:id="echoid-s595" xml:space="preserve"> <lb/>cum ex eis exeant duo lacerti paruuli, quorum unus eſt in parte lachrymarum oculi, & alius in par <lb/>te poſteriore.</s> <s xml:id="echoid-s596" xml:space="preserve"> Et continent utrunq;</s> <s xml:id="echoid-s597" xml:space="preserve"> oculum palpebrę & cilia.</s> <s xml:id="echoid-s598" xml:space="preserve"> Hoc ergo quod declarauimus, eſt diſ-<lb/>poſitio compoſitionis oculi, & forma eius, & forma ſuarũ tunicarum.</s> <s xml:id="echoid-s599" xml:space="preserve"> Et omne, quod diximus de <lb/>tunicis oculi, & compoſitione earum, iam declaratum eſt ab anatomicis in libris anatomiæ.</s> <s xml:id="echoid-s600" xml:space="preserve"/> </p> <div xml:id="echoid-div20" type="float" level="0" n="0"> <figure xlink:label="fig-0012-01" xlink:href="fig-0012-01a"> <image file="0012-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0012-01"/> </figure> </div> </div> <div xml:id="echoid-div22" type="section" level="0" n="0"> <head xml:id="echoid-head42" xml:space="preserve">DE QVALITATE VISIONIS, ET AB ILLA DE-<lb/>pendentibus. Cap. 5.</head> <head xml:id="echoid-head43" xml:space="preserve" style="it">14. Viſio fit radijs à uiſibili extrinſec{us} ad uiſum manantib{us}. 6 p 3.</head> <p> <s xml:id="echoid-s601" xml:space="preserve">IAm declaratum eſt ſuperius [1 n] quòd ex corpore quolibet illuminato cum quolibet lumine <lb/>exit lux ad quamlibet partem oppoſitam ei.</s> <s xml:id="echoid-s602" xml:space="preserve"> Cum ergo uiſus opponitur alicui rei uiſæ, & fuerit <lb/>res illa illuminata cũ quolibet lumine, exlumine rei uiſæ ueniet lumẽ ad ſuperficiẽ uiſus.</s> <s xml:id="echoid-s603" xml:space="preserve"> Et de-<lb/>claratũ fuit quòd ex proprietate lucis eſt operari in uiſum, & quòd natura uiſus eſt pati ex luce.</s> <s xml:id="echoid-s604" xml:space="preserve"> Di <lb/>gnum eſt ergo, ut non ſentiat uiſus lumen rei uiſae;</s> <s xml:id="echoid-s605" xml:space="preserve">, niſi ex lumine ueniente ex ea ad uiſum.</s> <s xml:id="echoid-s606" xml:space="preserve"> Et decla <lb/>ratũ fuit iã, quòd forma coloris cuiuslibet corporis colorati & illuminaticũ quolibet lumine, aſſo-<lb/>ciatur ſemper lumini uenienti ab illo corpore ad quamlibet partem oppoſitã illi corpori, & erit lu-<lb/>men & forma coloris ſemper ſimul.</s> <s xml:id="echoid-s607" xml:space="preserve"> Ergo cũ lumine ueniente ad uiſum ex lumine corporis uiſi, e-<lb/>rit ſemper forma coloris corporis uiſi.</s> <s xml:id="echoid-s608" xml:space="preserve"> Et cũ lumẽ & color ueniant ſimul ad ſuperficiem uiſus, uiſus <lb/>ſentit colorẽ, qui eſt in re uiſa ex lumine ueniente ad ſe ex re uiſa.</s> <s xml:id="echoid-s609" xml:space="preserve"> Dignius ergo eſt, ut non ſit ſenſus <lb/>uiſus coloris rei uiſæ, niſi ex forma coloris uenientis ad ipſum uiſum cum lumine, & forma coloris <lb/>ſemper eſt admixta cũ forma lucis, & nõ eſt diſtincta ab ea.</s> <s xml:id="echoid-s610" xml:space="preserve"> Viſus ergo nõ ſentit lumẽ, niſi admixtũ <lb/>cũ colore.</s> <s xml:id="echoid-s611" xml:space="preserve"> Dignius ergo eſt, ut nõ ſit ſenſus uiſus coloris rei uiſæ & luminis, qđ eſt in ea, niſi ex for-<lb/>ma admixta cũ lumine & colore ueniente ad ipſum ex ſuperficie rei uiſæ.</s> <s xml:id="echoid-s612" xml:space="preserve"> Et iterũ tunicæ uiſus quę <lb/>ſituantur ad mediũ anterioris uiſus, ſunt diaphanæ cõtingẽtes ſe, [per 4 n] & prima illarũ, ſcilicet <lb/>cornea tãgit aerẽ, in quo primo uenit forma.</s> <s xml:id="echoid-s613" xml:space="preserve"> Et ex proprietate lucis eſt pertrãſire in quodlibet cor <lb/>pus diaphanũ:</s> <s xml:id="echoid-s614" xml:space="preserve"> & ſimiliter eſt proprietas formæ coloris, quæ aſſociatur lumini, pertrãſire in corpus <lb/>diaphanũ, & ideo extẽditur in aere diaphano, ſicut extẽditur lumẽ.</s> <s xml:id="echoid-s615" xml:space="preserve"> Et ex natura corporũ diaphano <lb/>rũ eſt, recipere formas lucis & coloris, & reddere ipſas partibus ſibi oppoſitis.</s> <s xml:id="echoid-s616" xml:space="preserve"> Forma ergo ueniẽs <lb/>ex re uiſa ad ſuperficiẽ uiſus, trãſibit per diaphanitatẽ tunicarũ uiſus, per foramẽ quod eſt in anteri <lb/>ore uueæ, perueniet ergo ad humorẽ glacialẽ, & pertrãſibit in eo, ſecundũ diaphanitatẽ ſuã.</s> <s xml:id="echoid-s617" xml:space="preserve"> Digni-<lb/>us ergo eſt, ut tunicę uiſus nõ ſint diaphanæ, niſi ut pertrãſeant in eis formæ lucis & colorũ, uenien <lb/>tiũ ad ipſum.</s> <s xml:id="echoid-s618" xml:space="preserve"> Aggregemus ergo modò qđ cõponitur ex omnibus iſtis, & dicamus, quòd uiſus ſen-<lb/>tit lumẽ & colores, qui ſunt in ſuperficie rei uiſæ, & quòd pertranſeunt per diaphanitatẽ tunicarum <lb/>uiſus.</s> <s xml:id="echoid-s619" xml:space="preserve"> Et hoc eſt illud in quo quieſcebat phyſicorũ opinio de qualitate uiſionis.</s> <s xml:id="echoid-s620" xml:space="preserve"> Dicemus ergo mo-<lb/>dò, quòd qualitas uiſionis nõ aſſeritur huiuſmodi eſſe tãtùm, quoniã iſte modus deſtruitur, niſi ad-<lb/>datur ei aliud.</s> <s xml:id="echoid-s621" xml:space="preserve"> Quoniã enim forma lucis & coloris cuiuslibet colorati & illuminati extẽditur in ae-<lb/>re diaphano, cõtinuato cũ eo ad oẽs partes oppoſitas, uiſus aũt opponitur eodẽ tẽpore multis reb.</s> <s xml:id="echoid-s622" xml:space="preserve"> <lb/>uiſis diuerſi coloris, & inter quãlibet earũ & uiſum ſunt in aere lineæ rectæ cõtinuato medio inter <lb/>eas:</s> <s xml:id="echoid-s623" xml:space="preserve"> & cũ formæ lucis & coloris, quæ ſunt in re uiſa oppoſita uiſui ueniant ad ſuperficiẽ uiſus:</s> <s xml:id="echoid-s624" xml:space="preserve"> for-<lb/>mę ergo lucis & coloris cuiuslibet rerũ uiſibiliũ, oppoſitarũ uiſui, in eodẽ tẽpore ueniẽt ad ſuperfi <lb/>ciẽ uiſus.</s> <s xml:id="echoid-s625" xml:space="preserve"> Et cũ formę extẽdãtur ex re uiſa ad quãlibet partẽ oppoſitã, & nõ perueniãt ad uiſum, niſi <lb/>ꝓpter oppoſitionẽ:</s> <s xml:id="echoid-s626" xml:space="preserve"> forma, quę peruenit ex re uiſa ad uiſum, peruenit ad totã ſuperficiẽ uiſus.</s> <s xml:id="echoid-s627" xml:space="preserve"> Et cũ <lb/>ita ſit, quãdo uiſus opponitur alicui ſuperficiei rei uiſę, & peruenit forma coloris eius & lucis ad ſu <lb/>perficiẽ uiſus, & uiderit in illo tẽpore aſpiciẽs alia uiſibilia diuerſi coloris oppoſita uiſui:</s> <s xml:id="echoid-s628" xml:space="preserve"> tũc forma <lb/>lucis & coloris cuiuslibet illorũ uiſibiliũ uen<gap/>et ad ſuperficiẽ uiſus, & forma omniũ illorũ uiſibiliũ <lb/>perueniet ad totã ſuperficiẽ uiſus.</s> <s xml:id="echoid-s629" xml:space="preserve"> Perueniẽt ergo ad totã ſuperficiẽ uiſus multa lumina diuerſa, & <lb/>multi colores diuerſi, & quilibet illorũ implet ſuperficiẽ uiſus:</s> <s xml:id="echoid-s630" xml:space="preserve"> perueniet ergo in ſuperficiem uiſus <lb/>forma admixta ex colorib.</s> <s xml:id="echoid-s631" xml:space="preserve"> diuerſis, & luminibus diuerſis.</s> <s xml:id="echoid-s632" xml:space="preserve"> Si ergo ſenſerit uiſus illã formã admixtã, <lb/>ſentiet colorẽ diuerſum à colore cuiuslibet illarũ rerũ, & nõ diſtinguẽtur ab eo uiſibilia.</s> <s xml:id="echoid-s633" xml:space="preserve"> Et ſi ſenſe <lb/>rit unã illarũ rerũ uiſibiliũ, & nõ ſenſerit reſiduas:</s> <s xml:id="echoid-s634" xml:space="preserve"> cõprehẽdet unã rẽ uiſibilẽ, & nõ alias:</s> <s xml:id="echoid-s635" xml:space="preserve"> ſed ipſe cõ-<lb/>prehẽdit omnia illa uiſibilia in eodẽ tẽpore, & cõprehẽdit ipſa diſtincta.</s> <s xml:id="echoid-s636" xml:space="preserve"> Et ſi non ſenſerit unã illarũ <lb/>formarũ, nihil ſentiet ex ipſis, uel ex alijs uiſibilibus oppoſitis illi:</s> <s xml:id="echoid-s637" xml:space="preserve"> ſed ipſe ſentit omnia.</s> <s xml:id="echoid-s638" xml:space="preserve"> Et iterũ poſ <lb/>ſunt eſſe in eodẽ uiſo diuerſi colores, & à qualibet parte eius exit lumẽ & color ſecũdum oẽs lineas <lb/>rectas, quæ extẽdũtur in aere cõtinuo.</s> <s xml:id="echoid-s639" xml:space="preserve"> Cũ ergo fuerint partes unius rei uiſę diuerſi coloris:</s> <s xml:id="echoid-s640" xml:space="preserve"> ueniet <lb/>ad totã ſuperficiẽ uiſus ex unoquoq;</s> <s xml:id="echoid-s641" xml:space="preserve"> illarũ forma coloris & lucis, & ſic քmiſcebũtur colores illarũ <lb/> <pb o="8" file="0014" n="14" rhead="ALHAZEN"/> partiũ in<gap/>ſuperficie uiſus.</s> <s xml:id="echoid-s642" xml:space="preserve"> Quare cõprehendet uiſus ipſos admixtos, aut nihil cõprehẽdet ex eis.</s> <s xml:id="echoid-s643" xml:space="preserve"> Si <lb/>uerò cõprehẽdet eos permixtos, nõ diſtinguũtur, nec ordinabũtur ab eo partes ſiue colores parti-<lb/>um.</s> <s xml:id="echoid-s644" xml:space="preserve"> Et ſi nihil cõprehendit ex iſtis formis, nihil cõprehẽdet ex iſtis partibus:</s> <s xml:id="echoid-s645" xml:space="preserve"> & ſi nihil cõprehendit <lb/>ex partibus, nihil cõprehendet ex re uiſa:</s> <s xml:id="echoid-s646" xml:space="preserve"> ſed uiſus cõprehẽdit rẽ uiſam ſibi oppoſitã illuminatã, & <lb/>comprehendit partes eius diuerſi coloris ordinatas, & diſtιnctas.</s> <s xml:id="echoid-s647" xml:space="preserve"> Et cum ita ſit, conſtat quòd aut <lb/>qualitas uiſionis erit alio modo, aut erit iſte modus pars propoſiti modi uidendi.</s> <s xml:id="echoid-s648" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div23" type="section" level="0" n="0"> <head xml:id="echoid-head44" xml:space="preserve" style="it">15. Viſ{us} è ſingulis ſuæ ſuperficiei punctis ſingula uiſibilis punct a uidet. 17. 18 p 3.</head> <p> <s xml:id="echoid-s649" xml:space="preserve">DEbemus ergo cõſiderare utrũ iſte modus poſsit cõuenire cõditionibus, per quas diſtinguã-<lb/>tur colores rerũ uiſibiliũ, & ordinãtur partes earũ apud uiſum, & cõueniunt ad eorũ eſſe in <lb/>corpore.</s> <s xml:id="echoid-s650" xml:space="preserve"> Dicimus ergo quòd, quãdo uiſus fuerit oppoſitus alicui rei uiſibili, ueniet ex quo-<lb/>libet puncto ſuperficiei rei uiſæ forma & coloris & lucis, quæ ſunt in ea, ad totã ſuperficiẽ uiſus, & <lb/>ex quolibet puncto cuiuslibet rerũ uiſibilium oppoſitarũ uiſui in illa diſpoſitione, etiam uenient <lb/>formę coloris & lucis, quę ſunt in illis, ad totã ſuperficiẽ uiſus.</s> <s xml:id="echoid-s651" xml:space="preserve"> Si ergo uiſus ſenſerit ex tota eius ſu-<lb/>perficie formas coloris & lucis, quæ ueniunt ex aliquo puncto ſuperficiei rei uiſæ, ſentiet ex tota e-<lb/>ius ſuperficie, formã cuiuslibet puncti ſuperficiei rei uiſæ, & formã cuiuslibet puncti ſuperficierũ <lb/>omniũ uiſibilium rerũ oppoſitarũ illi in illa diſpoſitione:</s> <s xml:id="echoid-s652" xml:space="preserve"> & ſic non ordinabuntur ab eo partes uni-<lb/>us rei uiſæ, neq;</s> <s xml:id="echoid-s653" xml:space="preserve"> diſtinguentur ab eo.</s> <s xml:id="echoid-s654" xml:space="preserve"> Et ſi ſenſerit formã uenientẽ ex uno puncto ſuperficiei rei ui-<lb/>ſæ ad totã ſuperficiẽ uiſus, ex uno puncto tantùm ſuperficiei ipſius uiſus, & nõ ſenſerit formã illius <lb/>puncti tota eius ſuperficie:</s> <s xml:id="echoid-s655" xml:space="preserve"> ordinabuntur ab eo partes rei uiſæ, & diſtinguẽtur omnia uiſibilia op-<lb/>poſita.</s> <s xml:id="echoid-s656" xml:space="preserve"> Quoniã quando cõprehenderit colorem puncti unius ex uno puncto tantùm ſuperficiei e-<lb/>ius, cõprehendet colorẽ unius partis rei uiſæ ex una parte ſuperficiei ſuæ, & cõprehendet colorem <lb/>alterius partis ex alia parte ſuperficiei ſuæ, & cõprehendet unam quamq;</s> <s xml:id="echoid-s657" xml:space="preserve"> partẽ uiſibiliũ ex loco ſu-<lb/>perficiei ſuæ diuerſo & oppoſito ei, per quem cõprehendit aliã rem uiſibilem.</s> <s xml:id="echoid-s658" xml:space="preserve"> Quare uiſibilia erunt <lb/>ab eo ordinata & diſtincta, & ſimiliter partes cuiuslibet illorum.</s> <s xml:id="echoid-s659" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div24" type="section" level="0" n="0"> <head xml:id="echoid-head45" xml:space="preserve" style="it">16. Humor cryſtallin{us} eſt præcipuum organum facult atis opticæ. 4. 18 p 3.</head> <p> <s xml:id="echoid-s660" xml:space="preserve">MOdò ergo cõſideremus utrũ hoc ſit cõueniens, & poſsibile ad eſſe.</s> <s xml:id="echoid-s661" xml:space="preserve"> Et dicamus prius, quòd <lb/>uiſio nõ eſt niſi per glacialẽ, ſiue fiat uiſio per formas uenientes ex re uiſa ad uiſum, ſiue ſe-<lb/>cundum alium modũ.</s> <s xml:id="echoid-s662" xml:space="preserve"> Viſio autem nõ eſt per unã aliarũ tunicarum antecedentiũ ſe, quo-<lb/>niam illæ tunicæ non ſunt niſi inſtrumentũ glacialis.</s> <s xml:id="echoid-s663" xml:space="preserve"> Quoniã ſi contigerit humori glaciali læſio cũ <lb/>ſalute aliarum tunicarũ, deſtruitur uiſio, & ſi acciderit reſiduis tunicis corruptio, remanente ipſa-<lb/>rum diaphanitate cum ſalute glacialis, non corrumpetur uiſus:</s> <s xml:id="echoid-s664" xml:space="preserve"> Et etiam ſi in foramine uueæ fuerit <lb/>oppilatio, & deſtruatur diaphanitas humoris eius, deſtruetur uiſus cum ſalute corneæ, & ſi aufera-<lb/>tur oppilatio, reuertetur uiſus.</s> <s xml:id="echoid-s665" xml:space="preserve"> Et ſimiliter ſi peruenerit intra humorẽ albugineum pars craſſa, non <lb/>diaphana, & fuerit in facie humoris glacialis, & media inter ipſum & foramen uueæ, deſtruetur ui-<lb/>ſio, & quando auferetur illud craſſum, uel declinabitur auerſione lineæ rectæ, quæ eſt inter glacia-<lb/>lẽ & foramẽ uueæ ad aliquã partẽ, reuertetur uiſus.</s> <s xml:id="echoid-s666" xml:space="preserve"> Et omnibus iſtis atteſtatur medicina.</s> <s xml:id="echoid-s667" xml:space="preserve"> Deſtru-<lb/>ctio ergo ſenſus uiſus eſt apud corruptionẽ glacialis cũ ſalute tunicarũ antecedentiũ illũ.</s> <s xml:id="echoid-s668" xml:space="preserve"> Et illud <lb/>eſt argumentũ, quòd ſenſus uiſus nõ eſt, niſi per iſtũ humorẽ, nõ per tunicas reſiduas antecedentes <lb/>illũ.</s> <s xml:id="echoid-s669" xml:space="preserve"> Et deſtructio ſenſus apud deſtructionem diaphanitatis, quæ eſt inter glacialẽ & ſuperficiem ui <lb/>ſus per corpus denſum non translucẽs, ſignificat quòd diaphanitas iſtarum tunicarum non eſt, ni-<lb/>ſiut continuetur diaphanitas tunicarum uiſus cum diaphanitate aeris, & efficiantur corpora, quæ <lb/>ſunt inter glacialem & rem uiſam, diaphana continuitate diaphanitatis.</s> <s xml:id="echoid-s670" xml:space="preserve"> Et deſtructio ſenſus apud <lb/>deſtructionem linearum, quæ ſunt inter glacialem & ſuperficiem uiſus:</s> <s xml:id="echoid-s671" xml:space="preserve"> ſignificat, quòd ſenſus gla-<lb/>cialis non erit, niſi ex lineis rectis, quæ ſunt inter ipſum & ſuperficiem uiſus.</s> <s xml:id="echoid-s672" xml:space="preserve"> Dicemus ergo ſi ſen-<lb/>ſus uiſus eſt ex colore rei uiſæ & lucis, quę ſunt in eo, & ex forma ueniente ex rebus uiſis ad ſuper-<lb/>ficiem uiſus, & ſenſus non eſt niſi per glacialem.</s> <s xml:id="echoid-s673" xml:space="preserve"> Ergo non per ſuperſiciem uiſus ſentiet uiſus iſtam <lb/>formam, ſed poſtquã tranſierit ſuperficiem uiſus, & peruenerit ad glacialẽ.</s> <s xml:id="echoid-s674" xml:space="preserve"> Et forma quæ uenit ex <lb/>re uiſa ad ſuperficiem uiſus, pertranſit in diaphanitate tunicarum uiſus:</s> <s xml:id="echoid-s675" xml:space="preserve"> quoniam ex proprietate <lb/>diaphanitatis eſt, ut tranſeant in ea formæ lucis & coloris, & extendantur rectè.</s> <s xml:id="echoid-s676" xml:space="preserve"> Et iam declaraui-<lb/>mus hoc in aere [14 n.</s> <s xml:id="echoid-s677" xml:space="preserve">] Et cum fuerint experimẽtata omnia corpora diaphana, inuenietur quòd <lb/>lux extenditur in eis ſecundum lineas rectas:</s> <s xml:id="echoid-s678" xml:space="preserve"> & nos declarabimus pòſt apud noſtrum ſermonem <lb/>de obliquatione, quomodo hoc experiendum ſit.</s> <s xml:id="echoid-s679" xml:space="preserve"> Si ergo ſenſus uiſus lucis & coloris, quæ ſunt in <lb/>re uiſa, eſt ex forma ueniente ad uiſum ex re uiſa:</s> <s xml:id="echoid-s680" xml:space="preserve"> apud peruentionem ipſius formæ ad glacialem e-<lb/>rit ſenſus.</s> <s xml:id="echoid-s681" xml:space="preserve"> Et iam declaratum eſt antea [15 n] quòd non eſt poſsibile, ut uiſus comprehendat rem <lb/>uiſam ſecundum ſuum eſſe, niſi quando comprehenderit formam unius puncti rei uiſæ ex uno <lb/>puncto tantùm ſuæ ſuperficiei.</s> <s xml:id="echoid-s682" xml:space="preserve"> Non eſt ergo poſsibile, ut glacialis comprehendat rem uiſam ſecun <lb/>dum ſuum eſſe, niſi quando comprehenderit colorem unius puncti rei uiſæ ex forma ueniente ad <lb/>ipſum ex uno puncto tantùm ſuperficiei uiſus:</s> <s xml:id="echoid-s683" xml:space="preserve"> forma autem uenit ex quolibet puncto ſuperficiei <lb/>rei uiſæ, & pertranſit totam uiſus ſuperficiem uſque ad interius.</s> <s xml:id="echoid-s684" xml:space="preserve"> Si uerò ex eo, quod uenit ex u-<lb/>no puncto rei uiſæ ad totam ſuperficiem uiſus, & pertranſit tunicas uiſus, & peruenit ad glacia-<lb/>lem, non comprehendit glacialis niſi quod uenit ad ipſum ex uno puncto tantùm ſuperficiei ui-<lb/>ſus, & ſentit colorem illius puncti tantùm ex ſuperficie uiſus, & peruenit ad unum punctum <lb/> <pb o="9" file="0015" n="15" rhead="OPTIC AE LIBER I."/> tantùm ſuperficiei eius, & non comprehendit illud punctum rei uiſæ ex reſidua forma perue-<lb/>niente ad ſuperficiem eius ex reſidua ſuperficie uiſus:</s> <s xml:id="echoid-s685" xml:space="preserve"> complebitur uiſio, & ordinabuntur partes <lb/>rei uiſæ, & diſtinguentur res in ſe apud uiſum, & non complebitur uiſio, niſi ſecundum iſtum mo-<lb/>dum.</s> <s xml:id="echoid-s686" xml:space="preserve"> Et hoc non poteſt eſſe ita, niſi quando fuerit unum punctorũ, quę ſuntin ſuperficle uiſus, per <lb/>quam tranſit forma unius puncti ſuperficiei rei uiſæ, diſtinctuш à punctis reſiduis, quæ ſunt in ſu-<lb/>perficie uiſus, &<gap/>fuerit linea, ſuper quam uenit forma ad illud punctũ ſuperficiei niſus, diſtincta à <lb/>reſiduis lineis, ſuper quas uenit forma.</s> <s xml:id="echoid-s687" xml:space="preserve"> Et propter hoc poteſt glacialis cõprehendere formã ueni-<lb/>entem ſuper illã lineam, & ex puncto ſuperficiei uiſus, quod eſt ſuper illam lineã, & nõ poteſt com-<lb/>prehendere ipſam per aliam.</s> <s xml:id="echoid-s688" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div25" type="section" level="0" n="0"> <head xml:id="echoid-head46" xml:space="preserve" style="it">17. Lux perpendicularis penetr at per qualibet diuerſa media: obliqua refringitur. 42. 43. <lb/>44. 45. 47 p 2.</head> <p> <s xml:id="echoid-s689" xml:space="preserve">ET cum inducuntur luces, & experimẽtatur qualitas tranſitus earum, & extenſionis earũ in <lb/>corporibus diaphanis, inuenitur quòd lux extenditur per corpus diaphanum ſecundum li-<lb/>neas rectas, dum corpus diaphanũ fuerit cõſimilis diaphanitatis:</s> <s xml:id="echoid-s690" xml:space="preserve"> & cũ occurrerit corpus ali-<lb/>ud diuerſæ diaphanitatis à diaphanitate corporis præcedèntis, in quo extendebatur, non pertran-<lb/>ſibit ſecũdum rectitudinem linearũ, ſuper quas extendebatur antè, niſi quando illæ lineæ fuerint <lb/>perpendiculares ſuper ſuperficiẽ ſecũdi corporis diaphani:</s> <s xml:id="echoid-s691" xml:space="preserve"> & ſi illæ lineæ fuerint obliquatæ ſuper <lb/>ſuperficiẽ ſecũdi corporis, & nõ perpendiculares, obliquabitur lux apud ſuperficiẽ ſecũdi corpo-<lb/>ris, & non extendetur rectè:</s> <s xml:id="echoid-s692" xml:space="preserve"> & cum obliquatur, extẽdetur in ſecundo corpore ſecundũ illas lineas <lb/>rectas, ſuper quas obliquabatur:</s> <s xml:id="echoid-s693" xml:space="preserve"> & erũt lineæ ſuper quas obliquabatur lux in ſecũdo corpore, etiã <lb/>declinantes ſuper ſuperficiẽ ſecundi corporis, & nõ perpendiculares.</s> <s xml:id="echoid-s694" xml:space="preserve"> Et ſi fuerint quædã lineæ ſu-<lb/>per quas uenit lux in primo corpore, perpendiculares ſuper ſuperficiẽ ſecundi corporis, & quædã <lb/>declinantes:</s> <s xml:id="echoid-s695" xml:space="preserve"> extendetur lux, quæ erat ſuper lineas perpendiculares in ſecundo corpore ſecundum <lb/>rectitudinẽ, & quę erat ſuper lineas declinantes, obliquabitur apud ſuperficiẽ ſecundi corporis ſe-<lb/>cundum lineas declinantes, & extendetur in eo ſecundũ rectitudinẽ illarũ linearum declinantiũ, <lb/>ſuper quas obliquabatur.</s> <s xml:id="echoid-s696" xml:space="preserve"> Et hoc nos declarabimus in ſermone de refractione, & oſtendemus uiã, <lb/>per quã poterit quis experiri iſtã diſpoſitionẽ:</s> <s xml:id="echoid-s697" xml:space="preserve"> & apparebit ſenſui, & cadet ſuper ipſam certitudo.</s> <s xml:id="echoid-s698" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div26" type="section" level="0" n="0"> <head xml:id="echoid-head47" xml:space="preserve" style="it">18. Viſio diſtincta fit rectis lineis à uiſibili ad ſuperficiem uiſ{us} perpẽdicularibus. Ita ſin-<lb/>gula uiſibilis punct a eundem obtinent ſitum in ſuperficie uiſ{us}, quem in uiſibili. 17 p 3.</head> <p> <s xml:id="echoid-s699" xml:space="preserve">ET cum ita ſit, ex forma ergo lucis & coloris, quæ ueniunt ex quolibet puncto rei uiſæ ad ſu-<lb/>perficiem uiſus, quando peruenerit ad ſuperficiẽ uiſus, nihil pertranſibit per diaphanitatem <lb/>tunicarũ uiſus ſecũdũ rectitudinẽ, niſi illud, quod erit ſuper lineã rectã eleuatã ſuper ſuperfi-<lb/>ciẽ uiſus ſecundũ angulos rectos, & illud, quod fuerit ſuper aliã, refringetur, & non pertranſibit re-<lb/>ctè:</s> <s xml:id="echoid-s700" xml:space="preserve"> quoniam diaphanitas tunicarum uiſus nõ eſt, ficut diaphanitas aeris contingentis fuperficiẽ <lb/>uiſus.</s> <s xml:id="echoid-s701" xml:space="preserve"> Et illud, quod refringitur ex iſtis formis, refringetur etiam ſuper lineas declinantes, non ſu-<lb/>per lineas perpendiculares extenſas ex loco refractionis:</s> <s xml:id="echoid-s702" xml:space="preserve"> & una linea recta tantùm exit ad punctũ <lb/>ſuperficiei uiſus ab uno puncto ſuperficiei rei uiſæ, ita ut ſit perpendicularis ad ſuperficiem uiſus:</s> <s xml:id="echoid-s703" xml:space="preserve"> <lb/>[per 13 p 11] & exeunt ad eã lineæ infinitæ declinãtes ſuper ſuperficiẽ uiſus.</s> <s xml:id="echoid-s704" xml:space="preserve"> Et forma ueniẽs ſecun <lb/>dũ rectitudinẽ perpendicularis, pertranfit tunicas uiſus ſecun dũ rectitudinem perpendicularis:</s> <s xml:id="echoid-s705" xml:space="preserve"> & <lb/>omnes formæ uenientes ſecundum lineas declinantes ad illud punctum, refringuntur apud illud <lb/>punctum, & tranſeunt in tunicis uiſus ſecundum lineas declinantes:</s> <s xml:id="echoid-s706" xml:space="preserve"> & nihil ex eis tranſit ſecundũ <lb/>extenſionem linearũ, ſuper quas uenerũt, neq;</s> <s xml:id="echoid-s707" xml:space="preserve"> etiam ſecundũ rectitudinem linearũ perpendicula-<lb/>riter erectarũ ſuper illud punctũ.</s> <s xml:id="echoid-s708" xml:space="preserve"> Et ad quodlibet punctũ ſuperficiei uiſus ueniunt in eodem tem-<lb/>pore formæ omniũ punctorũ, quæ ſunt in ſuperficiebus omniũ uiſibiliũ & illuminatorũ oppoſito-<lb/>rũ illi in illo tempore:</s> <s xml:id="echoid-s709" xml:space="preserve"> quoniam inter ipſum & quodlibet punctũ oppoſitum illi eſt linea recta:</s> <s xml:id="echoid-s710" xml:space="preserve"> & à <lb/>quolibet punctorum, quæ ſunt in ſuperficiebus uiſibilium illuminatorum, extenduntur formæ ſu-<lb/>per quamlibet lineam rectam, quæ poteſt extendi ex illo puncto, & forma unius puncti tantùm de <lb/>numero omnium punctorum oppoſitorum uiſui, quæ uenit ad illud punctũ ſuperficiei uiſus in il-<lb/>lo tempore, uenit ſuper perpendicularem eleuatam ſuper illud punctũ ſuperficiei uiſus:</s> <s xml:id="echoid-s711" xml:space="preserve"> & formæ <lb/>omniũ punctorũ reſiduorũ ueniũt ad illud punctũ ſuperficiei uiſus ſuper lineas declinantes:</s> <s xml:id="echoid-s712" xml:space="preserve"> & in <lb/>quolibet puncto ſuperficiei uiſus tranſeunt in eo dẽ tempore formæ omniũ punctorũ, quæ ſunt in <lb/>ſuperficiebus omniũ uiſibiliũ oppoſitorũ in illo tẽpore:</s> <s xml:id="echoid-s713" xml:space="preserve"> & forma unius puncti tantùm trãſit rectè <lb/>per diaphanitatẽ tunicarũ uiſus:</s> <s xml:id="echoid-s714" xml:space="preserve"> & eſt punctũ, quod eſt apud extremitatẽ perpẽdicularis exeuntis <lb/>ab illo puncto ſuperficiei uiſus:</s> <s xml:id="echoid-s715" xml:space="preserve"> & formæ omniũ punctorũ reliquorũ refringuntur apud illud pun <lb/>etũ ſuperficiei uiſus, & trãſeũt per diaphanitatẽ tunicarũ uiſus ſecundũ lineas declínãtes ad ſuperfi <lb/>ciẽ uiſus.</s> <s xml:id="echoid-s716" xml:space="preserve"> Et ex quolibet pũcto ſuperficiei glacialis exit una linea tãtũ perpẽdicularis ſuper ſuperfi-<lb/>ciẽ uiſus:</s> <s xml:id="echoid-s717" xml:space="preserve"> & ab eodẽ exeunt lineę infinitæ ad ſuperficiẽ uiſus, & ſunt declinãtes ſuper ipſam.</s> <s xml:id="echoid-s718" xml:space="preserve"> A pun <lb/>cto ergo ſuperficiei glacialis, ex quo exit perpendicularis ſuper ſuperficiẽ uiſus, & pertrãſit foramẽ <lb/>uueæ, exeunt lineæ infinitæ, quæ trãſeunt in foramẽ uueæ, & perueniũt ad ſuperficiẽ uiſus, pręter <lb/>illã perpẽdicularẽ:</s> <s xml:id="echoid-s719" xml:space="preserve"> & extrem itates omniũ linearũ exeuntiũ à pũcto aliquo ſuperficiei glacialis, & <lb/>trãſeuntiũ ք foramẽ uueæ, & perueniẽtiũ ad ſuperficiẽ uiſus, & declinãtiũ ſuper illã, quãdo fuerint <lb/> <anchor type="figure" xlink:label="fig-0015-01a" xlink:href="fig-0015-01"/> <pb o="10" file="0016" n="16" rhead="ALHAZEN"/> intellectæ refringi ſecundum modum, quem affirmat diuerſitas diaphanitatis, quæ eſt inter diapha <lb/>nitatem corporis corneæ & corporis aeris, perueniunt ad diuerſa loca, & ad puncta diuerſa denu-<lb/>mero punctorum, quæ ſunt in ſuperficiebus uiſibilium oppoſitorum uiſui in uno tempore:</s> <s xml:id="echoid-s720" xml:space="preserve"> & nul-<lb/>la iſtarum linearum occurrit puncto, quod eſt apud extremitatem perpendicularis.</s> <s xml:id="echoid-s721" xml:space="preserve"> Et formæ pun-<lb/>ctorũ, quæ ſunt apud extremitates omnium iſtarum linearum ſuperficierum uiſibilium, extendun <lb/>tur ſecundum rectitudinem iſtarum linearum, & perueniunt ad ſuperficiem uiſus, & refringuntur <lb/>ad idem punctum ſuperficiei glacialis, præter formam puncti, quod eſt apud extremitatem perpen <lb/>dicularis:</s> <s xml:id="echoid-s722" xml:space="preserve"> quoniam ipſa extenditur ſecundum rectitudinem perpendicularis, & pertranſit ad illud <lb/>punctum glacialis.</s> <s xml:id="echoid-s723" xml:space="preserve"> Si ergo glacialis ſentit ex uno puncto omnes formas uenientes ad ipſum ex o-<lb/>mnibus uerticationibus, ſentiet ex omni puncto formas admixtas ex multis formis diuerſis, & co-<lb/>loribus multis uiſibilium oppoſitorum uiſui in illo tempore:</s> <s xml:id="echoid-s724" xml:space="preserve"> & ſic nihil diftinguetur ab eo ex pun-<lb/>ctis, quæ ſunt in ſuperficiebus uiſibilium, neque ordinabuntur formæ punctorum uenientes ad il-<lb/>lud punctum:</s> <s xml:id="echoid-s725" xml:space="preserve"> at ſi glacialis ſenſerit ex uno ſui puncto illud, quod uenit ad ipſum ex una uerticatio-<lb/>ne tantùm, diſtinguentur ab eo puncta, quæ ſunt in ſuperficiebus uiſibilium.</s> <s xml:id="echoid-s726" xml:space="preserve"> Et nullum punctorũ, <lb/>quorum formæ perueniunt ad glacialem ſuper lineas refractas, eſt dignius alio ex formis refractis, <lb/>neque ulla refracta uerticatio eſt dignior alia:</s> <s xml:id="echoid-s727" xml:space="preserve"> & formæ refractę ad unum punctum glacialis in uno <lb/>tempore, ſunt multæ non determinatæ.</s> <s xml:id="echoid-s728" xml:space="preserve"> Et punctum, cuius forma uenit ſecundum rectitudinem <lb/>perpendicularis ad unum punctum glacialis, eſt unum punctum tantùm, & nulla alia forma uenit <lb/>cum ea ſecundum rectitudinem perpendicularis:</s> <s xml:id="echoid-s729" xml:space="preserve"> quoniam omnes formæ refractæ non refringun-<lb/>tur niſi ſecundum lineas declinantes.</s> <s xml:id="echoid-s730" xml:space="preserve"> Et cum centrum ſuperficiei uiſus ſit idem cum centro ſuper-<lb/>ficiei glacialis [per 12 n] linea, quæ eſt perpendicularis ſuper ſuperficiem uiſus, eft perpendicula-<lb/>ris ſuper ſuperficiem glacialis.</s> <s xml:id="echoid-s731" xml:space="preserve"> Form a ergo, quæ uenit ſuper perpendicularem, diſtinguitur ab alijs <lb/>formis duabus diſpoſitionibus:</s> <s xml:id="echoid-s732" xml:space="preserve"> quarum altera eſt, quòd ipſa extenditur à ſuperficie rei uiſę ad pun <lb/>ctum glacialis ſuper lineam rectam, & reſiduæ ueniunt ſuper lineas refractas:</s> <s xml:id="echoid-s733" xml:space="preserve"> altera autem eſt, <lb/>quòd ipſa perpendicularis erecta ſuper ſuperficiem uiſus, eſt etiam perpendicularis ſuper ſuperfi-<lb/>ciem glacialis:</s> <s xml:id="echoid-s734" xml:space="preserve"> & lineæ reſiduæ, ſuper quas ueniunt formæ reſiduæ refractæ, ſunt declinantes ſu-<lb/>per ſuperficiem uiſus.</s> <s xml:id="echoid-s735" xml:space="preserve"> Et operatio lucis uenientis ſuper perpendiculares, eſt fortior operatione lu <lb/>cis uenientis ſuper lineas inclinatas.</s> <s xml:id="echoid-s736" xml:space="preserve"> Dignius ergo eſt, ut glacialis non ſentiat ex quolibet puncto, <lb/>niſi formam uenientem ad ipſum punctum ſuper rectitudinem perpendicularis tantùm, & non <lb/>ſentiat ex illo puncto illud, quod uenit ad illud punctum ſecundum uerticationes refractas.</s> <s xml:id="echoid-s737" xml:space="preserve"> Et ite-<lb/>rum cum centrum ſuperficiei uiſus, & centrum ſuperficiei glacialis, fit idem punctum, omnes per-<lb/>pendiculares eleuatæ ſuper ſuperficiem glacialis & ſuperficiem uiſus, concurrent ſuper centrum <lb/>commune, & erunt diametri in ſuperficiebus tunicarum uiſus, perpẽdiculares ſuper ipſas tunicas <lb/>uiſus:</s> <s xml:id="echoid-s738" xml:space="preserve"> & erit quælibet perpendicularis occurrens ſuperficiei corneæ in uno puncto, & occurrens <lb/>ſuperficiei glacialis in uno puncto:</s> <s xml:id="echoid-s739" xml:space="preserve"> & non exit ad illud punctum corneæ, niſi una perpendicularis, <lb/>neque exit ad illud punctũ glacialis, niſi una perpendicularis tantùm.</s> <s xml:id="echoid-s740" xml:space="preserve"> Forma ergo, quæ exit à quo-<lb/>libet puncto ſuperficiei rei uiſæ ſuper perpendicularem, quæ extenditur ab eo ad ſuperficiem ui-<lb/>ſus, occurrit ſuperficiei uiſus ſuper unum punctum, ſuper quod ei non occurrit aliqua alia forma, <lb/>non uenientium ſuper perpendiculares.</s> <s xml:id="echoid-s741" xml:space="preserve"/> </p> <div xml:id="echoid-div26" type="float" level="0" n="0"> <figure xlink:label="fig-0015-01" xlink:href="fig-0015-01a"> <image file="0015-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0015-01"/> </figure> </div> </div> <div xml:id="echoid-div28" type="section" level="0" n="0"> <head xml:id="echoid-head48" xml:space="preserve" style="it">19. Viſio fit per pyramidem, cui{us} uertex eſt in uiſu, baſis in uiſibili. 18. 21. 22 p 3.</head> <p> <s xml:id="echoid-s742" xml:space="preserve">ET iterum iam determinatũ eſt, [14.</s> <s xml:id="echoid-s743" xml:space="preserve"> 18 n] quòd ex quolibet puncto cuiuslibet corporis co-<lb/>lorati & illuminati cum quolibet lumine, exeunt lux & color ſuper quamlibet lineam rectã, <lb/>quæ poterit extendi ab illo puncto:</s> <s xml:id="echoid-s744" xml:space="preserve"> ergo inter quodlibet punctum uiſus, & quodlibet pun-<lb/>ctum oppoſitum alicui ſuperficiei, & quodlibet punctum illius ſuperficiei, eſt linea recta imagina-<lb/>bilis, & inter illud punctum, & illam ſuperficiem eſt pyramis imaginabilis, cuius uertex eſt illud <lb/>punctum, & cuius baſis eſt illa ſuperficies:</s> <s xml:id="echoid-s745" xml:space="preserve"> & illa pyramis continet omnes lineas rectas intellectas, <lb/>quæ ſunt inter illud punctum & omnia puncta illius ſuperficiei.</s> <s xml:id="echoid-s746" xml:space="preserve"> Cum ergo forma lucis & coloris <lb/>exierint à quolibet puncto ſuperficiei corporis colorati, illuminati, ſuper quamlibet lineam rectã, <lb/>quę poterit extendi ab illo puncto, ad quodlibet punctum oppoſitum corpori illuminato & colo-<lb/>rato:</s> <s xml:id="echoid-s747" xml:space="preserve"> forma lucis & coloris, quæ ſunt in ſuperficie illius corporis, extendetur à quolibet puncto ſu <lb/>perficiei illius corporis, ad illud punctum, oppoſitũ illi ſuper lineam rectam extenſam inter ipſum <lb/>corpus & illud punctum.</s> <s xml:id="echoid-s748" xml:space="preserve"> Form a ergo lucis & coloris cuiuslibet corporis colorati & illuminati cũ <lb/>quolibet lumine, extẽditur à ſua ſuperficie ad quodlibet punctum oppoſitum illi ſuperficiei ſecun <lb/>dum uerticationem pyramidis, quæ formatur inter illud punctum & illam ſuperficiem:</s> <s xml:id="echoid-s749" xml:space="preserve"> & erit for-<lb/>ma ordinata in illa pyramide per lineas illas concurrentes ad illud punctum, quod eſt uertex pyra-<lb/>midis, ſicut eſt ordinatio in partibus coloris, qui eſt in ſuperficie illius corporis.</s> <s xml:id="echoid-s750" xml:space="preserve"> Cũ ergo uiſus fue-<lb/>rit oppoſitus alicui rei uiſibili, formabitur inter punctum, quod eſt centrum uiſus, & ſuperficiem <lb/>illius rei uiſæ, pyramis imaginabilis, cuius uertex erit centrum uiſus, & baſis erit ſuperficies illius <lb/>rei uiſæ:</s> <s xml:id="echoid-s751" xml:space="preserve"> & cum aer medius inter illam rem uiſam & uiſum fuerit continuus, & non fuerit medium <lb/>inter rem uiſam & uiſum, corpus denſum, & fuerit illa res uiſa illuminata cum quolibet lumine:</s> <s xml:id="echoid-s752" xml:space="preserve"> <lb/>extendetur forma lucis & coloris, quæ ſunt in ſuperficie illius rei uiſæ, ad uiſum ſecundum uertica <lb/>tionem illius pyramidis, & extendetur forma cuiuslibet puncti ſuperficiei illius rei uiſæ ſecundum <lb/>rectitu dinem lιneæ, quę eſt inter illud punctum, & uerticem illius pyramidis, qui eſt cẽtrum uiſus.</s> <s xml:id="echoid-s753" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0016-01a" xlink:href="fig-0016-01"/> <pb o="11" file="0017" n="17" rhead="OPTIC AE LIBER I."/> Et quia centrum uiſus idem eſt cum centro ſuperficiei glacialis, [per 12 n] erunt omnes iſtæ lineæ <lb/>perpendicularesſuper ſuperficiem oculi, & ſuperficiẽ glacialis, & ſuper oẽs ſuperficies uiſus æqui-<lb/>diſtantes:</s> <s xml:id="echoid-s754" xml:space="preserve"> & erit pyramis continua ſuper omnes iſtas perpendiculares, continens omnes iſtas per-<lb/>pendiculares, & aerem, in quo extenditur forma à tota ſuperficie illius rei uiſæ oppoſitæ uiſui, ſe-<lb/>cundum uerticationes perpendicularium:</s> <s xml:id="echoid-s755" xml:space="preserve"> & ſuperficies glacialis ſecabit iſtam pyramidem:</s> <s xml:id="echoid-s756" xml:space="preserve"> & ſic <lb/>peruenit forma lucis & coloris, quæ ſunt in ſuperficie illius rei uiſæ, in partem ſuperficiei, quã com <lb/>prehendit pyramis.</s> <s xml:id="echoid-s757" xml:space="preserve"> Et ad quodlibet punctum iſtius ſuperficiei glacialis ueniet forma puncti oppo <lb/>ſiti ſuperficiei rei uiſæ, ſecundum rectitudinem perpendicularis exeuntis ab iſto puncto ſuperfici-<lb/>ei rei uiſæ ſuper ſuperficiem tunicarum uiſus, & ſuper ſuperficiem glacialis, & pertranſibit diapha-<lb/>nitatem tunicarum uiſus ſecundum rectitudinem illius perpendicularis, & non pertranſibit cum <lb/>illa forma ſecundum rectitudinem illius perpendicularis alia forma.</s> <s xml:id="echoid-s758" xml:space="preserve"> Et iſta forma perueniet ad iſtã <lb/>partem glacialis ordinata in ea ſecũdum lineas rectas, ſuper quas peruenit ad ipſam, quæ ſunt per-<lb/>pendiculares ad ipſam, & cõcurrẽtes apud centrũ uiſus, ſicut ordinatio partiũ ſuperficiei rei uiſæ.</s> <s xml:id="echoid-s759" xml:space="preserve"> <lb/>Præterea ueniunt in illa diſpoſitione ad quodlibet punctum huius partis ſuperficiei glacialis mul <lb/>tæ formæ à multis punctis ſuperficierum uiſarum in eodem tempore.</s> <s xml:id="echoid-s760" xml:space="preserve"> Perueniunt ergo in iſtã par-<lb/>tem ſuperficiei glacialis, quæ diſtinguebatur à pyramide, multæ formæ ex multis coloribus diuer-<lb/>ſis.</s> <s xml:id="echoid-s761" xml:space="preserve"> Si ergo glacialis ſenſerit ex parte diſtincta per pyramidem, formam uenientem ad ſe ex uertica-<lb/>tione illius pyramidis tantùm, neque ſenſerit ex illa parte ſuæ ſuperficiei aliam formam, niſi formã <lb/>uenientem ſuper illam uerticationem:</s> <s xml:id="echoid-s762" xml:space="preserve"> ſentiet formam illius rei ſecundũ ſuũ eſſe, & ſentiet ordina-<lb/>tam ſecundum ſuam ordinationem.</s> <s xml:id="echoid-s763" xml:space="preserve"> Et poterit etiam ſentire in illa diſpoſitione formas aliarum re-<lb/>rum uiſarum, præter illam rem uiſam ex pyramidibus diſtinguentibus ex ſua ſuperficie alias par-<lb/>tes ab illa parte:</s> <s xml:id="echoid-s764" xml:space="preserve"> & poterit ſentire formam cuiuslibet rerum uiſarum ſecũdum ſuum eſſe, & ſentire <lb/>ſitus earum inter ſe ſecundum ſuũ eſſe.</s> <s xml:id="echoid-s765" xml:space="preserve"> Et ſi glacialis ſenſerit formas uenientes ad ſe ex uerticatio-<lb/>nibus refractis, ſentiet ex eadem parte, quæ diſtinguebatur ex ſua ſuperficie per illam pyramidem, <lb/>formas admixtas ex formis partium illius rei uiſæ, & ex formis multarum rerum uiſarum diuerſa-<lb/>rum, & erunt admixtæ ex multis coloribus diuerſis, & ſentiet ex qualibet parte ſuæ ſuperficiei, <lb/>præter illam partem, formam permixtam exformis multarum rerum diuerſarum:</s> <s xml:id="echoid-s766" xml:space="preserve"> & ſic non ſenti-<lb/>et formam uenientem ſecundum pyramidis uerticationem ſecundum ſuũ eſſe, neque aliquam for <lb/>mam uenientem ſuper perpendicularem ſecũdum ſuũ eſſe, neque aliquam formam uenientem ex <lb/>uerticationibus refractis.</s> <s xml:id="echoid-s767" xml:space="preserve"> Non ſentiet ergo formam unius rei uiſæ ſecundum ſuũ eſſe, neq;</s> <s xml:id="echoid-s768" xml:space="preserve"> diſtin-<lb/>guentur ab ea res uiſæ oppoſitæ illi in eodem tempore:</s> <s xml:id="echoid-s769" xml:space="preserve"> ſed uiſus comprehẽdit res uiſas diſtinctas, <lb/>& comprehendit partes unius rei uiſæ ordinatas ſecundum ſuum eſſe in ſuperficie rei uiſæ, & com <lb/>prehendit res uiſas multas ſimul in eodem tempore.</s> <s xml:id="echoid-s770" xml:space="preserve"> Et cum uiſio ſit ex formis uenientibus ex re-<lb/>bus uiſis ad uiſum [per 14 n] nihil ſentiet glacialis ex formis rerum uiſarum ex uerticationibus <lb/>refractis:</s> <s xml:id="echoid-s771" xml:space="preserve"> & ſic nulla formarũ peruenientiũ ad ſuperficiẽ glacialis ex formis rerum uiſarũ, ordinabi <lb/>tur in ſuperficie glacialis ſecundũ ſuũ eſſe:</s> <s xml:id="echoid-s772" xml:space="preserve"> neq;</s> <s xml:id="echoid-s773" xml:space="preserve"> ulla formarũ partium unius rei uiſæ peruenientiũ <lb/>ad ſuperficiem glacialis, ordinabitur in ſuperficie glacialis ſecundum ſuũ eſſe in ſuperficie rei uiſæ, <lb/>niſi formæ peruenientes ad eam ſecundum rectitudinem perpendicularium eleuatarum ſuper ſu-<lb/>perficiem uiſus tantùm.</s> <s xml:id="echoid-s774" xml:space="preserve"> Situs autem formarum refractarum apud ſuperficiem uiſus etiam perue-<lb/>niuntin ſuperficiem glacialis conuerſi, & peruenit inſuper forma unius puncti in portionem ſu-<lb/>perficiei glacialis, nõ in unum punctum.</s> <s xml:id="echoid-s775" xml:space="preserve"> Et illud eſt, quòd forma puncti dextri apud uiſum, quan-<lb/>do extendetur ad punctum ſuperficiei uiſus, & linea, ſuper quam extenditur forma, obliqua ſuper <lb/>ſuperficiem uiſus, refringetur ad partem ſiniſtram à perpendiculari, quæ extendetur à centro ui-<lb/>ſus ad illud punctum ſuæ ſuperficiei:</s> <s xml:id="echoid-s776" xml:space="preserve"> & peruenit forma, quæ refringitur ab extremitate perpendi-<lb/>cularis, ſecundum hunc modum ad punctum ſiniſtrum à puncto glacialis ſuperficiei, ſuper quod <lb/>abſcindit illã illa perpendicularis:</s> <s xml:id="echoid-s777" xml:space="preserve"> Et ſimiliter forma puncti ſiniſtri à uiſu, quæ extendetur ad illud <lb/>idem punctũ ſuperficiei uiſus, & declinat ſuper ipſam, refringetur ad punctũ dextrũ à perpendicu-<lb/>lari, & à puncto ſuperficiei glacialis, quod eſt ſuper illã perpendicularẽ:</s> <s xml:id="echoid-s778" xml:space="preserve"> quoniã formæ refractæ nõ <lb/>appropinquant poſt refractionẽ perpendiculari exeunti à loco refractionis, & non perueniunt per <lb/>applicationẽ formæ ad perpendicularẽ neq;</s> <s xml:id="echoid-s779" xml:space="preserve"> poſt refractionẽ pertranſeunt ipſam, neq;</s> <s xml:id="echoid-s780" xml:space="preserve"> pręcedunt:</s> <s xml:id="echoid-s781" xml:space="preserve"> <lb/>quoniã hæc eſt proprietas formarũ refractarũ.</s> <s xml:id="echoid-s782" xml:space="preserve"> Et ſimiliter formæ duorũ punctorũ, quæ ſuntin ea-<lb/>dẽ parte à uiſu, quæ exeunt ad unũ punctũ ſuperficiei uiſus, & declinant ſuper ipſam in eadẽ parte, <lb/>perueniuntin ſuperficiẽ glacialis conuerſæ:</s> <s xml:id="echoid-s783" xml:space="preserve"> quoniã duæ lineæ, ſuper quas extendũtur duæ formæ <lb/>punctorũ, ſecant ſe ad punctũ ſuperficiei uiſus, ſuper quod cõcurrunt duæ formæ, & occurrũt per-<lb/>pẽdiculari exeunti ad illud punctũ ſuperficiei uiſus, ſuper illud punctũ.</s> <s xml:id="echoid-s784" xml:space="preserve"> Cũ ergo iſtæ duæ lineę fue <lb/>rint declinantes à ſuperficie uiſus in eadẽ parte à perpẽdiculari exeunte à cẽtro uiſus ad illud pun-<lb/>ctum, refringũtur formæ duorũ punctorũ ad punctũ oppoſitũ illi parti.</s> <s xml:id="echoid-s785" xml:space="preserve"> Et etiã quia duæ lineæ, ſu-<lb/>per quas extenduntur duę formę ad unum punctũ ſuperficiei uiſus, ſecant ſe ſuper illud punctum:</s> <s xml:id="echoid-s786" xml:space="preserve"> <lb/>oportet, quando extenduntur ſecundũ ſuam rectitudinẽ poſt ſectionẽ, ut appareat ſitus eorũ con-<lb/>uerſus in reſpectu eius, qui eſt in re uiſa, & reſpectu etiã perpendicularis, & efficitur linea, quæ erat <lb/>dextra ante ſuã peruentionẽ ad ſuperficiẽ uiſus ex illis duabus lineis, ſiniſtra poſt ſuũ pertranſitũ <lb/>in ſuperficiẽ uiſus, & ſiniſtra, dextra.</s> <s xml:id="echoid-s787" xml:space="preserve"> Et ſimiliter erit ſitus duarũ linearũ, ſuper quasrefringẽtur duę <lb/>formæ ex uno puncto ſuperficiei uiſus:</s> <s xml:id="echoid-s788" xml:space="preserve"> quoniã duæ formæ, quæ refringuntur ex uno puncto, ap-<lb/>propinquant ambo perpẽdiculari, & extenditur forma, quæ erat ſuper lineã remotiorẽ à perpendi <lb/> <pb o="12" file="0018" n="18" rhead="ALHAZEN"/> culari, poſt ſectionẽ ſuper lineá remotiorẽ etiá à perpédiculari, ſed minoris remotióis quàm linea, <lb/>ſuper quã erat:</s> <s xml:id="echoid-s789" xml:space="preserve"> & extẽditur forma, quҫ erat ſuper lineã propinquioré perpẽdiculari, etiá poſt ſectio <lb/>né ſuper lineá propinquiorem etiam perpendiculari, ſed maioris propinquitatis, quàm linea, ſuper <lb/>quá erat.</s> <s xml:id="echoid-s790" xml:space="preserve"> Et ſimiliter omnes formæ, quæ extenduntur ab uno puncto.</s> <s xml:id="echoid-s791" xml:space="preserve"> Et cúſuerit experimẽtatum <lb/>experimétatione ſubtili, inuenietur, ſecúdú quod diximus.</s> <s xml:id="echoid-s792" xml:space="preserve"> Et nos oſten demus uiam, per quá expe <lb/>rimentabitur hoc experimentatione uera apud noſtrum fermonẽ de refractione, & tũc diſcoope-<lb/>rientur omnia depẽdentia à refractione:</s> <s xml:id="echoid-s793" xml:space="preserve"> & nos nó utemurillic in demóſtratione rebus, quibus uſi <lb/>fuimus in iſto tractatu.</s> <s xml:id="echoid-s794" xml:space="preserve"> Duo ergo puncta declinantia ad uná partetn à re uiſa, quando formæ eorũ <lb/>extenduntur ad unũ punctũ ſuperficiei uiſus, ſecabũt ſe ſuper duas lineas, quarũ ſitus erit apud ui-<lb/>ſum in reſpectu rei uiſę contrarius ſitui duarũ linearum primarũ, ſuper quas extêdebãtur duæ for-<lb/>mæ ad ſuperficiẽ uiſus.</s> <s xml:id="echoid-s795" xml:space="preserve"> Erit ergo ſitus duorum punctorum ſuperficiei glacialis, ad quæ perueniunt <lb/>duælformæ contrarius ſitui, duorũ punctorũ, ex quibus ueniunt duæ formæ.</s> <s xml:id="echoid-s796" xml:space="preserve"> Omnes ergo ſormæ, <lb/>quæ refringũtur ab uno pũcto ſuperſiciei uiſus, perueniũt in ſuperficiẽ glacialis cõuerſæ.</s> <s xml:id="echoid-s797" xml:space="preserve"> Etiterũ <lb/>forma cuiuslibet pũcti oppoſiti uiſui uenit ad totá ſuperficiẽ uiſus:</s> <s xml:id="echoid-s798" xml:space="preserve"> ergo refringetur à tota ſuperfi-<lb/>cie uiſus:</s> <s xml:id="echoid-s799" xml:space="preserve"> & forma, quæ refringitur à tota ſuperficie uiſus, refringitur ad partẽ alicuius quãtitatis ſu <lb/>perficiei glacialis, nó ad unũ pũctũ.</s> <s xml:id="echoid-s800" xml:space="preserve"> Quoniá formę refractionis ſi cõcurrerẽt poſt refractionẽ ſuper <lb/>unũ punctũ, ſecarẽt perpẽdiculares, apud quarũ extremitates reſringebãtur, aut pertrãſirent ipſas, <lb/>aut exiret forma à ſuperſicie, in qua refringebatur:</s> <s xml:id="echoid-s801" xml:space="preserve"> ſed nulla forma refracta occurrit perpẽdiculari, <lb/>apud cuius extremitatẽ fuerit refracta poſt refractionẽ, neq;</s> <s xml:id="echoid-s802" xml:space="preserve"> pertrãſit illã, neq, exit à ſuperſicie, in <lb/>qua fuit reſracta.</s> <s xml:id="echoid-s803" xml:space="preserve"> Et omnia iſta maniſeſtãtur per experimẽtationẽ.</s> <s xml:id="echoid-s804" xml:space="preserve"> Forma ergo unius pũcti rei uiſæ, <lb/>quæ peruenit in ſuperficiẽ glacialis, poſt refractionẽ nõ erit in uno pũcto, ſed in parte alicuius quã-<lb/>titatis ſuperficiei glacialis, & nõ erit ſitus formarũ rerũ diuerſarũ uel pũctorũ diuerſorũ ſuperficiei <lb/>rei uiſæ, quæ perueniũt in ſuperficiẽ glacialis ք refractionẽ inter ſe, ſicut ſitus earũ ſecũdũ ſuũ eſſe <lb/>in ſuperficiebus rerũ uiſarũ, ſed cótrarius.</s> <s xml:id="echoid-s805" xml:space="preserve"> Nulla ergo formarũ refractarũ rerũ uiſarũ peruenientiũ <lb/>ad ſuperficiẽ glacialis eſt ſecũ dũ ſuũ eſſe in ſuperficieb.</s> <s xml:id="echoid-s806" xml:space="preserve"> uiſarũ rerũ.</s> <s xml:id="echoid-s807" xml:space="preserve"> Et iã declaratũ eſt [18 n] quòd <lb/>formæ uenientes ſuper perpẽdiculares, ordinãtur in ſuperficie glacialis ſecundũ ſuũ eſſe, quoniã <lb/>extẽduntur rectè à ſuperficiebus rerũ uiſarũ ad ſuperficiẽ glacialis.</s> <s xml:id="echoid-s808" xml:space="preserve"> Nulla ergo formarũ rerũ uiſarũ <lb/>uenientiũ ad ſuperficiẽ glacialis ordinatur in ſuperficie glacialis ſecundũ ſuũ eſſe, quod habentin <lb/>ſuperficiebus rerũ uiſarum, niſi formæ extenſæ ſuper uerticationes perpédiculariũ tãtùm.</s> <s xml:id="echoid-s809" xml:space="preserve"> Si ergo <lb/>ſenſus uiſus rerũ uiſarũ ſit ex form is uenientib.</s> <s xml:id="echoid-s810" xml:space="preserve"> ad ipſum ex ſuperſiciebus rerũ uiſarũ, nihil cõpre-<lb/>hendet uiſus ex formis rerũ uiſarũ peruenientibus ad ipſum, niſi ex uerticationibus, quarũ extre-<lb/>mitates cõcurrunt apud centrũ uiſus tátùm:</s> <s xml:id="echoid-s811" xml:space="preserve"> quoniã uiſus nihil cõprehendit ex ſormis rerum uiſa-<lb/>rum, niſi ordinatum ſecundum ſuum eſſe in ſuperficiebus rerum uiſarum.</s> <s xml:id="echoid-s812" xml:space="preserve"/> </p> <div xml:id="echoid-div28" type="float" level="0" n="0"> <figure xlink:label="fig-0016-01" xlink:href="fig-0016-01a"> <image file="0016-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0016-01"/> </figure> </div> </div> <div xml:id="echoid-div30" type="section" level="0" n="0"> <head xml:id="echoid-head49" xml:space="preserve" style="it">20. Oculus & ſphæra cryſtallina habent idem centrum. 7 p 3. Idem 12 n.</head> <p> <s xml:id="echoid-s813" xml:space="preserve">ET iterũ ſi centrũ uiſus nõ eſt centrũ ſuperficiei glacialis:</s> <s xml:id="echoid-s814" xml:space="preserve"> lineæ rectæ, quę exeunt à cẽtro ſu-<lb/>perficiei uiſus, & extẽduntur in foramine uueæ, & perueniũt ad res uiſas, nõ erũt perpẽdicu-<lb/>lares ſuper ſuperficiẽ glacialis, ſed declinãtes ſuper ipſam:</s> <s xml:id="echoid-s815" xml:space="preserve"> neq;</s> <s xml:id="echoid-s816" xml:space="preserve"> ſitus earũ ſuper ſuperficiẽ gla <lb/>cialis erũt ſitus cõſimiles, niſi una linea tantũ, ſcilicet, quæ trãſit per duo cẽtra.</s> <s xml:id="echoid-s817" xml:space="preserve"> Formas ergo uenien <lb/>tes à ſuperficieb.</s> <s xml:id="echoid-s818" xml:space="preserve"> rerũ uiſarũ ad ſuperficiẽ glacialis, nõ poteſt ſentire glacialis, niſi ex uerticationib.</s> <s xml:id="echoid-s819" xml:space="preserve"> <lb/>iſtarũ linearũ tãtùm, ſcilicet quę ſunt perpẽdiculares ſuper ſuperficiẽ uiſus, quę eſt ſuperficies cor-<lb/>neę:</s> <s xml:id="echoid-s820" xml:space="preserve"> quoniã formę, quę ſunt ſuper iſtas քpẽdiculares, tãtùm ſunt ordinatę in ſuperficie glacialis ſe-<lb/>cundũ ordinationẽ earũ in ſuperficieb, rerũ uiſarũ.</s> <s xml:id="echoid-s821" xml:space="preserve"> Si ergo glacialis cõprehẽditres uiſas ex formis <lb/>uenientib.</s> <s xml:id="echoid-s822" xml:space="preserve"> ad ſe, & nõ cõprehendit formã, niſi ex uerticationibus iſtarũ linearũ, & iſtę lineę nõ ſunt <lb/>perpendiculares ſuper ſuperficiẽ eius:</s> <s xml:id="echoid-s823" xml:space="preserve"> cõprehendet tũc formas ex uerticationibus, quarũ ſitus à ſu <lb/>perficie ſua ſunt diuerſi ſitus, & declinãtes ſuper ſuã ſuperficiẽ, & cõprehẽdet formas ex uerticatio-<lb/>nibus dιuerſorũ ſituũ declinãtibus, & cõprehẽdet oẽs formas refractas ex uerticationib.</s> <s xml:id="echoid-s824" xml:space="preserve"> diuerſorũ <lb/>ſituũ apud ſuã ſuperficiẽ.</s> <s xml:id="echoid-s825" xml:space="preserve"> Et ſi cõprehendit oẽs formas refractas ex uerticationib.</s> <s xml:id="echoid-s826" xml:space="preserve"> diuerſorũ ſituũ, <lb/>nihil diſtinguetur ab eo ex rebus uiſis, propter hoc, quod declaratũ fuit ſuperius.</s> <s xml:id="echoid-s827" xml:space="preserve"> Et cũ nõ ſit poſsi <lb/>bile, ut cõprehẽdat formas refractas ex uerticationib.</s> <s xml:id="echoid-s828" xml:space="preserve"> diuerſorũ ſituũ, nõ eſt poſsibile, ut cõprehen <lb/>dat formas rerũ uiſarũ ex uerticationib.</s> <s xml:id="echoid-s829" xml:space="preserve"> linearũ, quę ſunt perpẽdiculares ſuper ſuperficiẽ uiſus, niſi <lb/>quãdo lineæ fuerint perpẽdiculares ſuper ſuperficiẽ eius, & fuerint ſitus eorũ in ſuperficie cõſimi-<lb/>les:</s> <s xml:id="echoid-s830" xml:space="preserve"> & iſtę lineę nõ erũt perpẽdiculares ſuper ſuperficiẽ ſuã, niſi quãdo cẽtrũ ſuę ſuperficiei, & cẽtrũ <lb/>ſuperficiei uiſus fuerint idẽ pũctũ.</s> <s xml:id="echoid-s831" xml:space="preserve"> Si ergo ſenſus uiſus rerũ uiſarũ eſt ex formis ueniẽtib.</s> <s xml:id="echoid-s832" xml:space="preserve"> ad ipſum <lb/>ex coloribus rerũ uiſarũ, & lucibus earũ, & hoc diſtinctè:</s> <s xml:id="echoid-s833" xml:space="preserve"> oportet, ut centrũ ſuperficiei uiſus & cen <lb/>trũ ſuperficiei glacialis ſit unũ punctũ cõmune, & nihil cõprehẽdat uiſus ex formis rerũ uifarũ, niſi <lb/>ex uerticationib.</s> <s xml:id="echoid-s834" xml:space="preserve"> rectarũ linearũ, quarũ extremitates cõcurrũt apud unũ & idẽ pũctũ tãtùm.</s> <s xml:id="echoid-s835" xml:space="preserve"> Et nõ <lb/>eſt impoſsibile, ut duo cẽtra ſint idẽ:</s> <s xml:id="echoid-s836" xml:space="preserve"> quoniã declaratũ eſt, [6.</s> <s xml:id="echoid-s837" xml:space="preserve">8 n] quòd duo cẽtra ſunt ex poſte-<lb/>riori cẽtro uueę, & ſuper unã lineã rectã trãſeuntẽ per omnia cẽtra.</s> <s xml:id="echoid-s838" xml:space="preserve"> Et quoniã nõ eſt impoſsibile, ut <lb/>duo cẽtra ſint idẽ, & ut lineæ rectæ, quæ exeunt à cẽtris, ſint perpẽdiculares ſuper duas ſuperficies, <lb/>ſcilicet ſuperficiẽ glacialis, & ſuperficiẽ uiſus:</s> <s xml:id="echoid-s839" xml:space="preserve"> nõ eſt etiã impoſsibile, ut ſit cõprehẽſio uiſus rerũ ui-<lb/>ſarũ ex formis uenientib.</s> <s xml:id="echoid-s840" xml:space="preserve"> ad ipſum, lucis & coloris, quæ ſunt in ſuperficie rerũ uiſarũ, cũ cõprehen-<lb/>ſio formarum iſtarũ ſit ex uerticationibus perpendiculariũ tantùm.</s> <s xml:id="echoid-s841" xml:space="preserve"> Et illud eſt, utnatura uiſus re-<lb/>cipiatea, quæ ueniunt ad ſe, ex formis rerum uiſarũ:</s> <s xml:id="echoid-s842" xml:space="preserve"> & etiam ut ſit natura uiſus infuper appropria-<lb/>ta, ut non recipiat ea, quæ ueniunt ad ſe ex formis, niſi ex proprijs uerticationibus, non ex omni-<lb/> <pb o="13" file="0019" n="19" rhead="OPTICAE LIBER I."/> bus uerticationibus:</s> <s xml:id="echoid-s843" xml:space="preserve"> & ſunt uerticationes linearum rectarum, quarum extremitates concurrunt a-<lb/>pud centrum uiſus tantùm.</s> <s xml:id="echoid-s844" xml:space="preserve"> Et iſtæ lineæ appropinquantur in centro, quia ſunt diametri eius uiſus <lb/>ſcilicet, & perpendiculares ſuper ſuperficiem uiſus ſentientis.</s> <s xml:id="echoid-s845" xml:space="preserve"> Et ſic erit ſenſus ex formis uenienti-<lb/>bus ex rebus uiſis, & erunt iſtæ lineæ quaſi inſtrumentũ uiſus, per quod diſtinguẽtur à uiſu res uiſæ, <lb/>& per quod ordinabuntur à uiſu partes cuiuslibet rerum uiſarum.</s> <s xml:id="echoid-s846" xml:space="preserve"> Et quòd eſſe uiſus appropriatur <lb/>aliquibus uerticationibus tantùm, habet ſimilia in rebus naturalibus.</s> <s xml:id="echoid-s847" xml:space="preserve"> Quoniam lux oritur ex cor-<lb/>poribus luminoſis, & extenditur ſuper uerticationes rectas tantùm, & non extenditur ſuper lineas <lb/>arcuales aut tortuoſas.</s> <s xml:id="echoid-s848" xml:space="preserve"> Et corpora ponderoſa mouentur ad inferius motu naturali ſuper lineas re-<lb/>ctas:</s> <s xml:id="echoid-s849" xml:space="preserve"> non ſuper lineas curuas, aut arcuales, aut tortuoſas:</s> <s xml:id="echoid-s850" xml:space="preserve"> nec tamen mouebuntur ſuper omnes li-<lb/>neas rectas, quæ ſunt inter ea & ſuperficiem terræ, ſed ſuper lineas rectas proprias, quę ſunt perpen <lb/>diculares ſuper ſuperficiem terræ & diametrũ eius.</s> <s xml:id="echoid-s851" xml:space="preserve"> Et corpora cœleſtia mouẽtur ſuper lineas ſphæ.</s> <s xml:id="echoid-s852" xml:space="preserve"> <lb/>ricas, & non ſuper lineas rectas, neq;</s> <s xml:id="echoid-s853" xml:space="preserve"> ſuper lineas diuerſi ordinis.</s> <s xml:id="echoid-s854" xml:space="preserve"> Et cum fuerimus intuiti motus na <lb/>turales, inueniemus, quòd quilibet eorum eſt appropriatus aliquibus uerticationibus tantùm.</s> <s xml:id="echoid-s855" xml:space="preserve"> Nõ <lb/>eſt ergo impoſsibile, ut ſit uiſus appropriatus in receptione operationum lucis & coloris aliquibus <lb/>uerticationibus rectis, quæ concurrunt apud eius centrum tantùm, & ſunt perpendiculares ſuper <lb/>ſuperficiem eius.</s> <s xml:id="echoid-s856" xml:space="preserve"> Comprehenſio autem uiſus de rebus uiſis ex uerticationibus linearum rectarum, <lb/>quarum extremitates concurrunt apud centrum uiſus, eſt conceſſa à mathematicis, & nulla diuerſi <lb/>tas eſt inter eos in hoc:</s> <s xml:id="echoid-s857" xml:space="preserve"> & iſtæ lineæ uocantur ab eis lineæ radiales.</s> <s xml:id="echoid-s858" xml:space="preserve"> Et cum hoc ſit poſsibile, & for-<lb/>mæ lucis & coloris ueniant ad uiſum, & pertranſeant per diaphanitatem tunicarum uiſus, & uiſio <lb/>non compleatur ex receptione iſtarum formarum, niſi quando uiſus receperit ipſas ex uerticatio-<lb/>nibus tantùm:</s> <s xml:id="echoid-s859" xml:space="preserve"> uiſus ergo non comprehendit luces & colores rerum uiſarum, niſi ex formis uenien-<lb/>tibus ad ipſum ex ſuperficiebus rerum uiſarum, & non comprehendit iſtas formas, niſi ex uertica-<lb/>tionibus linearum rectarum, quarum extremitates concurrunt apud centrum uiſus tantùm.</s> <s xml:id="echoid-s860" xml:space="preserve"> Aggre <lb/>gemus ergo modò ea, quæ poſſunt aggregari ex omni, quod diximus, & dicamus:</s> <s xml:id="echoid-s861" xml:space="preserve"> quòd uiſus ſentit <lb/>lucem & colores, qui ſunt in ſuperficie rei uiſę, ex forma extenſa, & ex luce, & colore, qui ſunt in ſu-<lb/>perficie rei uiſę per corpus diaphanũ, quod eſt medium inter uiſum & rem uiſam.</s> <s xml:id="echoid-s862" xml:space="preserve"> Et nihil compre-<lb/>hendit uiſus ex formis rerum uiſarum, niſi ex uerticationibus linearum extenſarũ inter rem uiſam <lb/>& centrum uiſus tantùm.</s> <s xml:id="echoid-s863" xml:space="preserve"> Et declaratum eſt, quòd hoc ſit poſsibile.</s> <s xml:id="echoid-s864" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div31" type="section" level="0" n="0"> <head xml:id="echoid-head50" xml:space="preserve" style="it">21. Viſibile uiſui oppoſitum uidetur. 2 p 3.</head> <p> <s xml:id="echoid-s865" xml:space="preserve">NOs uerò modò exponemus quęſtionem, quare fiat uiſio ſecundum modum hunc, dicendo, <lb/>quòd uiſio nõ poteſt eſſe niſi ſecundum hunc modum.</s> <s xml:id="echoid-s866" xml:space="preserve"> Quoniam uiſus quando ſenſerit rem <lb/>uiſam, poſtquam non ſentiebat ipſam, aliquid accidit ei, quod nõ erat prius:</s> <s xml:id="echoid-s867" xml:space="preserve"> & nihil accidet, <lb/>poſt quam non erat prius, niſi per aliquam cauſam.</s> <s xml:id="echoid-s868" xml:space="preserve"> Et inuenimus, quòd uiſus quando fuerit oppoſi-<lb/>tus rei uiſæ, ſentiet ipſam, & cũ auferetur ab eius oppoſitione, non ſentiet ipſam, & cum reuertetur <lb/>ad oppoſititionem, reuertetur uiſus.</s> <s xml:id="echoid-s869" xml:space="preserve"> Et ſimiliter inuenimus, quando uiſus ſenſerit rem uiſam, dein-<lb/>de clauſerit palpebras, quòd ſenſus deſtruitur, & cum aperit palpebras, & res uiſa fuerit in oppoſi-<lb/>tione, reuertitur ſenſus.</s> <s xml:id="echoid-s870" xml:space="preserve"> Sed cauſſa eſt illud, quòd quando deſtruitur cauſſa, deſtruitur cauſſatum, & <lb/>quando reuertitur cauſſa, reuertitur cauſſatum.</s> <s xml:id="echoid-s871" xml:space="preserve"> Cauſſa ergo, quę facit contingere rem illam in uiſu, <lb/>eſt res uiſa, quando opponitur uiſui.</s> <s xml:id="echoid-s872" xml:space="preserve"> Viſus ergo non ſentit rem uiſam, niſi propter illud, quod facit <lb/>res uiſas contingere in uiſu, quando ſcilicet opponuntur uiſui.</s> <s xml:id="echoid-s873" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div32" type="section" level="0" n="0"> <head xml:id="echoid-head51" xml:space="preserve" style="it">22. Viſibile per medium perſpicuum uidetur. 13 p 3.</head> <p> <s xml:id="echoid-s874" xml:space="preserve">ET iterum uiſus non comprehendit rem uiſam, niſi quando corpus, quod eſt medium inter ea, <lb/>fuerit diaphanũ.</s> <s xml:id="echoid-s875" xml:space="preserve"> Nam comprehenſio uiſus de re uiſa ex poſteriori aeris, qui eſt medius inter <lb/>eos, non eſt propter humiditatem aeris, ſed propter diaphanitatẽ eius.</s> <s xml:id="echoid-s876" xml:space="preserve"> Quoniam ſi medium <lb/>fuerit inter uiſum & rem uiſam aliquis lapis, aut aliud corpus diaphanũ quodcunq;</s> <s xml:id="echoid-s877" xml:space="preserve">: comprehendet <lb/>tunc uiſus rem uiſam, & erit comprehenſio ſecundum diaphanitatẽ corporis mediantis:</s> <s xml:id="echoid-s878" xml:space="preserve"> & quantò <lb/>corpus mediũ fuerit magis diaphanũ, tantò erit ſenſus uiſus de re illa manifeſtior.</s> <s xml:id="echoid-s879" xml:space="preserve"> Et ſimiliter quan <lb/>do fuerit inter uiſum & rem uiſam aqua clara diaphana, comprehendet uiſus rem uiſam à poſteriori <lb/>aquæ:</s> <s xml:id="echoid-s880" xml:space="preserve"> & ſi illa aqua fuerit tincta aliqua tinctura forti, ita ut deſtruatur diaphanitas, quamuis rema-<lb/>neat in ea humiditas, tunc uiſus non comprehendet illam rem uiſam, quæ eſt in aqua.</s> <s xml:id="echoid-s881" xml:space="preserve"> Declarabitur <lb/>ergo ex iſtis diſpoſitionibus, quòd uiſio non completur, niſi per diaphanitatem corporis medij, & <lb/>non per humiditatẽ.</s> <s xml:id="echoid-s882" xml:space="preserve"> Illud ergo quòd res uiſa operatur in uiſum apud ſuam oppoſitionẽ contra illũ, <lb/>ex quo eſt ſenſus, non cõpletur niſi per diaphanitatẽ corporis medij inter uiſum & rem uiſam.</s> <s xml:id="echoid-s883" xml:space="preserve"> Lux <lb/>ergo & color rei uiſæ non cõprehendetur à uiſu, niſi ex aliquo, quod ſit ex illa luce & colore in uiſu:</s> <s xml:id="echoid-s884" xml:space="preserve"> <lb/>& illud non accidit ex luce & colore in uiſu, niſi quando corpus medium inter uiſum & rem uiſam <lb/>fuerit diaphanum.</s> <s xml:id="echoid-s885" xml:space="preserve"> Diaphanitas aũt non appropriatur alicui ex eis, quæ pendent ex luce & colore, <lb/>quo diuerſiſicetur à nõ diaphanitate, niſi quia forma lucis & coloris pertranſit per diaphanũ, & non <lb/>pertranſit per non diaphanũ:</s> <s xml:id="echoid-s886" xml:space="preserve"> & quia corpus diaphanum recipit formam lucis & coloris, & reddit <lb/>ipſam partibus oppoſitis luci & colori, corpus aũt nõ diaphanũ nõ habet iſtam proprietatẽ.</s> <s xml:id="echoid-s887" xml:space="preserve"> Et quia <lb/>uiſus non ſentit lucem & colorem, quæ ſuntin re uiſa, niſi ex aliquo cõtingente ex luce & colore in <lb/>uiſu, & illud non contingit in uiſu, niſi quando corpus medium inter uiſum & rem uiſam fuerit dia-<lb/>phanum:</s> <s xml:id="echoid-s888" xml:space="preserve"> & corpus diaphanum nulli appropriatur, quo diſtinguatur à corpore nõ diaphano ex eis, <lb/> <pb o="14" file="0020" n="20" rhead="ALHAZEN"/> quę pendent à luce & colore, niſi per receptionẽ formarum colorum, & redditionẽ eorum ad partes <lb/>oppoſitas:</s> <s xml:id="echoid-s889" xml:space="preserve"> & declaratũ eſt, [19 n] quòd quando uiſus fuerit oppoſitus rei uiſę, formę lucis & coloris, <lb/>quæ ſuntin re uiſa, reddentur uiſui, & peruenientin ſuperficiem ſentientis:</s> <s xml:id="echoid-s890" xml:space="preserve"> uiſus ergo non ſentit l<gap/> <lb/>cem & colorem rei uiſę, niſi ex forma extenſa per corpus diaphanum inter rem uiſam & uiſum, & ex <lb/>re, quam facit contingere res uiſa in uiſu, dum opponitur illi, mediante corpore diaphano.</s> <s xml:id="echoid-s891" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div33" type="section" level="0" n="0"> <head xml:id="echoid-head52" xml:space="preserve" style="it">23. Viſio non fit radijs à uiſu emißis. s p 3.</head> <p> <s xml:id="echoid-s892" xml:space="preserve">ET licet nobis dicere, quòd corpus diaphanũ recipit à uiſu aliquid, & red dit ipſum rei uiſæ, & <lb/>per continuationẽ iſtius rei uiſæ inter uiſum & rem uiſam, euenit ſenſus.</s> <s xml:id="echoid-s893" xml:space="preserve"> Et hæc eſt opinio po <lb/>nentiũ radios exire à uiſu.</s> <s xml:id="echoid-s894" xml:space="preserve"> Ponatur ergo quòd ita ſit, quòd radij exeant à uiſu, & pertranſeant <lb/>per corpus diaphanum peruenientes ad rem uiſam, & per iſtos radios fiat ſenſus.</s> <s xml:id="echoid-s895" xml:space="preserve"> Et cum ita fiat ſen <lb/>ſus, quæro an per iſtos radios reddatur uiſui aliquid, aut nõ reddatur?</s> <s xml:id="echoid-s896" xml:space="preserve"> Si uero ſenſus fiat per radios, <lb/>& non reddant uiſui aliquid, uiſus non ſentiet:</s> <s xml:id="echoid-s897" xml:space="preserve"> ſed uiſus ſentit rem uiſam, & non ſentit, niſi median-<lb/>tibus radijs:</s> <s xml:id="echoid-s898" xml:space="preserve"> iſti ergo radij, qui ſentiũt rem uiſam, reddunt uiſui aliquid, per quod uiſus ſentit rem ui <lb/>ſam.</s> <s xml:id="echoid-s899" xml:space="preserve"> Et cum radij reddant uiſui aliquid, per quod ſentit rem uiſam, uiſus non ſentiet lucem & colo-<lb/>rem, quæ ſunt in re uiſa, niſi ex aliquo ueniente à luce & colore, quæ ſunt in re uiſa ad uiſum, & radij <lb/>reddunt illa.</s> <s xml:id="echoid-s900" xml:space="preserve"> Secundum ergo omnes diſpoſitiones non erit uiſus, niſi per aduentum alicuius rei ui-<lb/>ſæ à re uiſa, ſiue exierint radij, ſiue non.</s> <s xml:id="echoid-s901" xml:space="preserve"> Et iam declaratum eſt, [22 n] quòd uiſio non completur, niſi <lb/>per diaphanitatem corporis medij inter uiſum & rem uiſam, & nõ completur, quando fuerit mediũ <lb/>inter ea corpus non diaphanũ.</s> <s xml:id="echoid-s902" xml:space="preserve"> Et eſt manifeſtum, quòd corpus diaphanum à nõ diaphano in nullo <lb/>diſtinguitur, niſi ſecundum modum prædictum.</s> <s xml:id="echoid-s903" xml:space="preserve"> Et cum ita ſit, ut diximus, & ſit declaratum, quòd <lb/>formæ lucis & coloris, quæ ſunt in re uiſa, perueniãt ad uiſum, quando fuerint oppoſitæ uiſui.</s> <s xml:id="echoid-s904" xml:space="preserve"> Illud <lb/>ergo, quod uenit ex re uiſa ad uiſum, per quod uiſus comprehendit lucem & colores, quæ ſunt in re <lb/>uiſa ſecundum omnem diſpoſitionẽ, non eſt niſi iſta forma, ſiue exeant radij, ſiue non.</s> <s xml:id="echoid-s905" xml:space="preserve"> Et iam decla-<lb/>ratũ eſt [14.</s> <s xml:id="echoid-s906" xml:space="preserve">18 n] quòd formæ lucis & coloris ſemper generentur in aere, & in omnibus corporibus <lb/>diaphanis, & ſemper extendantur in aere, & in corporibus diaphanis ad partes oppoſitas, ſiue ocu-<lb/>lus fuerit præſens, ſiue non.</s> <s xml:id="echoid-s907" xml:space="preserve"> Exitus ergo radiorũ eſt ſuperfluus & otioſus.</s> <s xml:id="echoid-s908" xml:space="preserve"> Viſus ergo non ſentit lucẽ <lb/>& colorẽ rei uiſæ, niſi ex forma ueniente à luce & colore, quæ ſunt in re uiſa.</s> <s xml:id="echoid-s909" xml:space="preserve"> Et declaratũ eſt [19 n] <lb/>quòd forma cuiuslibet puncti rei uiſæ, oppoſiti uiſui, peruenit ad uiſum ſecun dũ uerticationes mul <lb/>tas diuerſas, & quòd uiſus non poteſt apprehendere formã rei uiſæ ſecundũ ſuam ordinationẽ in ſu <lb/>perficie rei uiſæ, niſi quando receptio formarũ fuerit ex uerticationibus linearũ rectarum, quæ ſunt <lb/>perpendiculares ſuper ſuperficiem uiſus, & ſuper ſuperficiem mẽbri ſentientis, & quòd lineæ rectæ <lb/>perpendiculares nõ erunt ſuper iſtas ſuperficies, niſi quando centrũ iſtarum ſuperficierũ fuerit unũ <lb/>punctũ.</s> <s xml:id="echoid-s910" xml:space="preserve"> Et cum hoc totum ſit, ſicut dictũ eſt:</s> <s xml:id="echoid-s911" xml:space="preserve"> oportet, ut centrũ ſuperficiei glacialis & centrũ ſuperfi <lb/>ciei uiſus ſint unum punctũ.</s> <s xml:id="echoid-s912" xml:space="preserve"> Viſus ergo nihil poteſt cõprehendere ex formis rerũ uiſarũ, niſi ex uer <lb/>ticationibus linearũ rectarum, quarũ extremitates concurrunt apud hoc centrũ tantùm.</s> <s xml:id="echoid-s913" xml:space="preserve"> Et hoc eſt, <lb/>quod promiſimus antè declarare in hoc capitulo in præcedente ſermone [12 n] de forma uiſus:</s> <s xml:id="echoid-s914" xml:space="preserve"> ſcili <lb/>cet quòd centrũ glacialis & centrũ ſuperficiei uiſus ſunt idem punctũ commune.</s> <s xml:id="echoid-s915" xml:space="preserve"> Et cum hoc decla <lb/>ratũ ſit, remanet ergo modò conſiderare opinionem ponentiũ radios exire à uiſu, & declarare quid <lb/>in ea falſum, & quid uerũ.</s> <s xml:id="echoid-s916" xml:space="preserve"> Dicamus ergo, ſi uiſio ſit ex re exeunte ex uiſu ad rem uiſam:</s> <s xml:id="echoid-s917" xml:space="preserve"> iſta res aut <lb/>eſt corpus, aut nõ corpus:</s> <s xml:id="echoid-s918" xml:space="preserve"> Si eſt corpus quando aſpexerimus cœlũ, & uiderimus ſtellas, quę ſunt in <lb/>eo, oportet, quòd in illa hora exeat à uiſu noſtro corpus, & impleat illud, quod eſt inter cœlũ & terrã, <lb/>& quòd nihil diminuatur à uiſu:</s> <s xml:id="echoid-s919" xml:space="preserve"> & hoc eſt falſum.</s> <s xml:id="echoid-s920" xml:space="preserve"> Viſio ergo nõ eſt per corpus exiens à uiſu ad rem <lb/>uiſam.</s> <s xml:id="echoid-s921" xml:space="preserve"> Et ſi illud, quod exit à uiſu, nõ eſt corpus, illud nõ ſentiet rem uiſam:</s> <s xml:id="echoid-s922" xml:space="preserve"> ſenſus enim nõ eſt, niſi in <lb/>corporibus.</s> <s xml:id="echoid-s923" xml:space="preserve"> Nihil ergo exit à uiſu ad rem uiſam, ſentiens rem illã.</s> <s xml:id="echoid-s924" xml:space="preserve"> Et manifeſtũ eſt, quòd uiſio eſt per <lb/>uiſum:</s> <s xml:id="echoid-s925" xml:space="preserve"> & cũ hoc ſit, & uiſus nõ cõprehendat rem uiſam, niſi quando exit ab eo ad rem uiſam, & illud <lb/>quod exit, nõ ſentit rem illam uiſam.</s> <s xml:id="echoid-s926" xml:space="preserve"> Illud ergo quod exit à uiſu ad rem uiſam, nõ redditad uiſum ali <lb/>quid, quo uiſus cõprehendat rem uiſam.</s> <s xml:id="echoid-s927" xml:space="preserve"> Et hoc quod exit à uiſu, nõ eſt ſenſibile, ſed opinabile, & ni <lb/>hil debet putari niſi per rationẽ.</s> <s xml:id="echoid-s928" xml:space="preserve"> Ponentes aũt radios exire à uiſu, opinãtur hoc, quod illi inuenerũt:</s> <s xml:id="echoid-s929" xml:space="preserve"> <lb/>quòd uiſus cõprehendit rem uiſam, & inter illa eſt ſpatiũ, & magnũ eſt hominibus, quòd ſenſus non <lb/>eſt, niſi per contactũ.</s> <s xml:id="echoid-s930" xml:space="preserve"> Quare illi opinati ſunt, quòd uiſio nõ ſit, niſi per aliquid exiens à uiſu ad rem ui <lb/>ſam, ita ut illud exiens ſentiat rem in ſuo loco, aut accipiat aliquid à re uiſa, & reddat ipſum uiſui, & <lb/>tunc ſentiat illud uiſus.</s> <s xml:id="echoid-s931" xml:space="preserve"> Et quia non poteſt exire à uiſu corpus ſentiẽs rem uiſam, & nihil ſentit rem <lb/>uiſam niſi corpus:</s> <s xml:id="echoid-s932" xml:space="preserve"> nõ remanſit opinari, niſi ut illud, quod à uiſu exit ad rem uiſam, recipiat à uiſo ali-<lb/>quid, & reddat ipſum uiſui.</s> <s xml:id="echoid-s933" xml:space="preserve"> Et quia declaratũ eſt [14.</s> <s xml:id="echoid-s934" xml:space="preserve">18.</s> <s xml:id="echoid-s935" xml:space="preserve">19 n] quòd aer, & corpora diaphana recipiũt <lb/>formã rei uiſæ, & reddunt ipſam uiſui, & omni corpori oppoſito rei uiſæ:</s> <s xml:id="echoid-s936" xml:space="preserve"> tunc illud, quod opinãtur, <lb/>quod reddit uiſui aliquid ex re uiſa, nõ eſt, niſi aer & corpora diaphana inter uiſum & rem uiſam.</s> <s xml:id="echoid-s937" xml:space="preserve"> Et <lb/>cum aer & corpora diaphana reddũt uiſui aliquid ex re uiſa, in quolibet tẽpore reddunt, & ſecundũ <lb/>omnes diſpoſitiones, quando uiſus fuerit oppoſitus rei uiſæ, ſine indigẽtia alicuius rei exeuntis à ui <lb/>ſu.</s> <s xml:id="echoid-s938" xml:space="preserve"> Ratio ergo quæ induxit ponentes radios ad dicendũ radios eſſe, eſt ſuperflua:</s> <s xml:id="echoid-s939" xml:space="preserve"> quoniã illud, quod <lb/>induxit eos ad dicendũ, quòd radij eſſent, eſt illorũ opinio:</s> <s xml:id="echoid-s940" xml:space="preserve"> quia uiſio non poteſt copleri, niſi per ali-<lb/>quid extenſum inter uiſum & rem uiſam, quod reddat uiſui aliquid ex re uiſa.</s> <s xml:id="echoid-s941" xml:space="preserve"> Et cum aer & corpora <lb/>diaphana faciant hoc ſine indigentia alicuius rei exeuntis à uiſu, & ſint inſuper extenſa inter uiſum <lb/>& rem uiſam ſine indigentia:</s> <s xml:id="echoid-s942" xml:space="preserve"> tunc ad ponendũ aliam rem reddentẽ aliquid uiſui de re uiſa, nulla eſt <lb/> <pb o="15" file="0021" n="21" rhead="OPTICAE LIBER I."/> opinio.</s> <s xml:id="echoid-s943" xml:space="preserve"> Dicere ergo eſſe radios, eſt nihil.</s> <s xml:id="echoid-s944" xml:space="preserve"> Et etiã omnes mathematici dicẽtes eſſe radios, nõ utuntur <lb/>in demonſtrationibus, niſi lineis imaginarijs tantùm, & uocant ipſas lineas radiales.</s> <s xml:id="echoid-s945" xml:space="preserve"> Et iam declara <lb/>uimus nos, quòd uiſus nihil cõprehendit ex rebus uiſis, niſi ex uerticatiõibus iſtarũ linearũ tantùm.</s> <s xml:id="echoid-s946" xml:space="preserve"> <lb/>Opinio ergo opinantiũ, quòd lineæ radiales ſint imaginariæ, eſt opinio uera:</s> <s xml:id="echoid-s947" xml:space="preserve"> & opinio opinantium, <lb/>quòd aliquid exit à uiſu, eſt opinio falſa.</s> <s xml:id="echoid-s948" xml:space="preserve"> Iam ergo declaratũ eſt ex omnibus, quę diximus, quòd uiſus <lb/>nõ ſentit lucẽ & colorẽ, quę ſunt in ſuperficie rei uiſæ, niſi per formã extenſam à ſuperficie rei uiſę ad <lb/>uiſum per corpus diaphanũ mediũ inter uiſum & rem uiſam:</s> <s xml:id="echoid-s949" xml:space="preserve"> & quòd uiſus nihil cõprehendit ex for <lb/>mis, niſi ex uerticationibus linearũ rectarũ, quæ intelliguntur extenſæ inter rem uiſam & centrũ ui-<lb/>ſus tantùm, quę ſunt perpendiculares ſuper omnes ſuperficies tunicarũ uiſus.</s> <s xml:id="echoid-s950" xml:space="preserve"> Et hoc eſt quod uolui <lb/>mus declarare.</s> <s xml:id="echoid-s951" xml:space="preserve"> Iſta eſt ergo qualitas uiſionis generaliter, quòd uiſus nõ cõprehendit ex re uiſa, ſenſu <lb/>ſpoliato, niſi lucẽ & colorẽ, quę ſunt in re uiſa, tantũ.</s> <s xml:id="echoid-s952" xml:space="preserve"> Res aũt reſiduę, quas cõprehẽdit uiſus ex rebus <lb/>uiſis, ſicut figurã, & magnitudinẽ, & ſimilia, nõ cõprehenduntur à uiſu, ſenſu ſpoliato, ſed per rationẽ <lb/>& ſigna.</s> <s xml:id="echoid-s953" xml:space="preserve"> Et hoc declarabimus nos pòſt in ſecundo tractatu poſt declarationẽ completam apud ſer-<lb/>monem noſtrum de diſtinctione rerum uiſibilium, quas comprehendit uiſus.</s> <s xml:id="echoid-s954" xml:space="preserve"> Et hoc, quod declara-<lb/>uimus ſcilicet qualitatem uiſionis, eſt conueniens opinioni uerificantium matheſin & naturam.</s> <s xml:id="echoid-s955" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div34" type="section" level="0" n="0"> <head xml:id="echoid-head53" xml:space="preserve" style="it">24. Viſio uidetur fieri per σ {υν}{άν}γ{δι}αμ, id eſt receptos ſimul & emiſſos radios.</head> <p> <s xml:id="echoid-s956" xml:space="preserve">ET declaratũ eſt ex hoc, quòd duæ ſectæ dicant uerũ:</s> <s xml:id="echoid-s957" xml:space="preserve"> & quòd duæ opiniones ſint rectæ & cõue <lb/>nιentes:</s> <s xml:id="echoid-s958" xml:space="preserve"> ſed non completur altera earũ, niſi per alterã, neq;</s> <s xml:id="echoid-s959" xml:space="preserve"> poteſt eſſe uiſio, niſi per illud, quod <lb/>aggregatur ex duabus ſectis.</s> <s xml:id="echoid-s960" xml:space="preserve"> Senſus ergo nõ eſt, niſi ex forma & ex operatiõe formę in uiſum, <lb/>& ex paſsione uiſus à forma:</s> <s xml:id="echoid-s961" xml:space="preserve"> & uiſus eſt paratus ad patiendũ ex iſta forma ſecundũ ſitum proprium, <lb/>ſcilicet ſitum uerticationũ perpendiculariũ ſuper ſuam ſuperficiem.</s> <s xml:id="echoid-s962" xml:space="preserve"> Natura aũt uiſus non congruit <lb/>iſti proprietati, niſi quia nõ diſtinguuntur uiſibilia, neq;</s> <s xml:id="echoid-s963" xml:space="preserve"> ordinantur partes cuiuslibet eorũ apud ui-<lb/>ſum, niſi quãdo ſenſus eius fuerit ex uerticationibus iſtis tãtùm.</s> <s xml:id="echoid-s964" xml:space="preserve"> Lineæ ergo radiales ſunt lineæ ima <lb/>ginabiles, & figuratur per eas qualitas ſitus, ſuper quã patitur uiſus ex forma.</s> <s xml:id="echoid-s965" xml:space="preserve"> Et iam declaratum eſt <lb/>[19 n] quòd quando uiſus oppoſitus fuerit rei uiſæ, figurabitur inter rem uiſam & centrũ uiſus pyra <lb/>mis, cuius uertex erit centrũ uiſus, & baſis eius ſuperficies rei uiſæ, & erit inter quodlibet punctũ ſu <lb/>perficiei rei uiſæ, & inter centrũ uiſus linea recta, intellecta perpendiculariter ſuper ſuperficies tuni <lb/>carũ uiſus:</s> <s xml:id="echoid-s966" xml:space="preserve"> & ſic pyramis cõtinebit omnes iſtas lineas, & ſuperficies glacialis ſecabit iſtã pyramidẽ:</s> <s xml:id="echoid-s967" xml:space="preserve"> <lb/>quoniã centrũ uiſus, quod eſt uertex pyramidis, eſt à poſteriori ſuperficiei glacialis.</s> <s xml:id="echoid-s968" xml:space="preserve"> Et cũ aer, qui eſt <lb/>inter uiſum & rem uiſam, fuerit continuus, erit forma extenſa ab illa re uiſa ſecundũ uerticationẽ il-<lb/>lius pyramidis in aere, quã diſtinguit ipſa pyramis, & in tunicis uiſus diaphanis uſq;</s> <s xml:id="echoid-s969" xml:space="preserve"> ad partẽ ſuperfi <lb/>ciei glacialis, quę diſtinguitur per iſtã pyramidẽ:</s> <s xml:id="echoid-s970" xml:space="preserve"> & iſta pyramis cõtinebit omnes uerticationes, quę <lb/>ſunt inter uiſum & rem uiſam, ex quib cõprehendit uiſus formã rei uiſę:</s> <s xml:id="echoid-s971" xml:space="preserve"> & erit forma ordinata, ſicut <lb/>eſt ordinata in ſuperficie rei uiſæ, & in parte iſta ſuperficiei glacialis:</s> <s xml:id="echoid-s972" xml:space="preserve"> & ιam declaratũ eſt, [16 n] quòd <lb/>ſenſus nõ eſt, niſi per glacialẽ.</s> <s xml:id="echoid-s973" xml:space="preserve"> Senſus ergo uiſus ex luce & colore, quę ſunt in ſuperficie rei uiſæ, non <lb/>eſt niſi ex parte glacialis, quam diſtinguit pyramis figurata inter illam rem uiſam & centrum uiſus.</s> <s xml:id="echoid-s974" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div35" type="section" level="0" n="0"> <head xml:id="echoid-head54" xml:space="preserve" style="it">25. Viſio perſicitur, cŭ forma uiſibilis cryſtallino humore recepta, in neruũ opticum <lb/>peruenerit. 20 p 3.</head> <p> <s xml:id="echoid-s975" xml:space="preserve">ET iam declaratũ eſt, [4 n] quòd in iſto humore eſt aliquãtula diaphanitas, & aliquantula ſpiſsi <lb/>tudo:</s> <s xml:id="echoid-s976" xml:space="preserve"> & propter hoc aſsimιlatur glaciei.</s> <s xml:id="echoid-s977" xml:space="preserve"> Quia ergo in eo eſt aliquantulũ diaphanitatis, recipit <lb/>formas:</s> <s xml:id="echoid-s978" xml:space="preserve"> & hæ pertranſeũt in eo, cũ eo, quod eſt ex eo de diaphanitate:</s> <s xml:id="echoid-s979" xml:space="preserve"> & quia in eo eſt aliquan-<lb/>tulum ſpiſsitudinis, prohibet form as à trãſitu in eo, cũ eo, quod eſt ex eo de ſpiſsitudine:</s> <s xml:id="echoid-s980" xml:space="preserve"> & figuntur <lb/>formæ in eius ſuperficie & corpore, ſed debiliter.</s> <s xml:id="echoid-s981" xml:space="preserve"> Et ſimiliter eſt quodlibet corpus diaphanũ, in quo <lb/>eſt aliquid ſpiſsitudinis, quando ſuper ipſum oritur lux, pertranſibit in eo ſecundũ quod eſt in eo de <lb/>diaphanitate, & figetur lux in ſuperficie eius ſecundũ id, quod eſt in eo de ſpiſsitudine.</s> <s xml:id="echoid-s982" xml:space="preserve"> Et etiã glacia <lb/>lis eſt præparatus ad recipiendũ ιſtas formas, & ad ſentiendũipſas.</s> <s xml:id="echoid-s983" xml:space="preserve"> Formæ ergo pertŕanſeunt in eo <lb/>propter uirtutẽ ſenſibilem recipientem.</s> <s xml:id="echoid-s984" xml:space="preserve"> Et cum forma peruenerit in ſuperficiem glacialis, operatur <lb/>in ea, & glacialis patitur ex ea:</s> <s xml:id="echoid-s985" xml:space="preserve"> quoniã ex proprietate lucis eſt, ut operetur in uiſum, & ex proprieta-<lb/>te uiſus, ut patiatur à luce.</s> <s xml:id="echoid-s986" xml:space="preserve"> Et iſta operatio, quam operatur lux in glaciali, pertranſit corpus glacialis <lb/>ſecundum rectitudinem linearũ radialium tantùm:</s> <s xml:id="echoid-s987" xml:space="preserve"> quoniam glacialis eſt præparatus ad recipiendũ <lb/>formas lucis ex uerticationibus linearum radialiũ.</s> <s xml:id="echoid-s988" xml:space="preserve"> Et cum lux pertranſit in corpus glacialis, color <lb/>pertranſit cum ea:</s> <s xml:id="echoid-s989" xml:space="preserve"> color enim eſt permixtus luci, & glacialis recipit iſtam operationẽ, & iſtum per-<lb/>tranſitum:</s> <s xml:id="echoid-s990" xml:space="preserve"> & ex iſta operatione & paſsione erit ſenſus glacialis ex formis rerum uiſibiliũ, quæ ſunt <lb/>in ſuperficie ſua, & pertranſeunt per totum ſuum corpus:</s> <s xml:id="echoid-s991" xml:space="preserve"> & ex ordinatione partium formæ in ſua ſu <lb/>perſicie & ſuo toto corpore, erit ſen ſus eius ex ordinatione partium operantis.</s> <s xml:id="echoid-s992" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div36" type="section" level="0" n="0"> <head xml:id="echoid-head55" xml:space="preserve" style="it">26. Viſio eſt ex eorum numero, quæ dolorem faciunt. 16 p 3.</head> <p> <s xml:id="echoid-s993" xml:space="preserve">ET iſta operatio, quã operatur lux in glacialem, eſt ex genere doloris, cum quidam dolores ſint <lb/>paſsibiles, & nõ læ ditur membrũ propter eos:</s> <s xml:id="echoid-s994" xml:space="preserve"> & tales dolores nõ manifeſtantur ſenſui, neq;</s> <s xml:id="echoid-s995" xml:space="preserve"> iu <lb/>dicat dolens, quòd ſit dolor.</s> <s xml:id="echoid-s996" xml:space="preserve"> Et ſignificatio ſuper hoc eſt, quòd lux inducit dolorẽ:</s> <s xml:id="echoid-s997" xml:space="preserve"> quιa luces <lb/>fortes offendũt uiſum, & lædunt manifeſtè, ſicut lux ſolis, quando aſpiciẽs aſpexerit corpus ipſius, <lb/>& ſicut lux ſolis reflexa à corporibus terſis ad uiſum, quoniã iſtæ luces inducũt dolores manifeſtos <lb/>in uiſũ.</s> <s xml:id="echoid-s998" xml:space="preserve"> Et operatio omnis lucis in uiſum eſt ex eode<unsure/> genere, & nõ diuerſificatur, niſi ſecundũ magis <lb/> <pb o="16" file="0022" n="22" rhead="ALHAZEN"/> & minus:</s> <s xml:id="echoid-s999" xml:space="preserve"> & cum omnes ſint ex uno genere, & operatio fortiorũ luciũ eſt ex genere doloris:</s> <s xml:id="echoid-s1000" xml:space="preserve"> omnes <lb/>ergo operationes luciũ ſunt ex genere doloris:</s> <s xml:id="echoid-s1001" xml:space="preserve"> & non diuerſificantur, niſi ſecundũ magis & minus:</s> <s xml:id="echoid-s1002" xml:space="preserve"> <lb/>& propter leuitatẽ operationũ luciũ debiliũ temperatarũ in uiſum, latet ſenſum eas in ducere dolo-<lb/>rem.</s> <s xml:id="echoid-s1003" xml:space="preserve"> Senſus ergo glacialis ex operatione lucis eſt de genere ſenſibilis doloroſi.</s> <s xml:id="echoid-s1004" xml:space="preserve"> Deinde iſte ſenſus, <lb/>qui cadit in glacialẽ, extenditur in neruo optico, & uenit ad anterius cerebri, & illic eſt ultimus ſen-<lb/>ſus, & ſentiens ultimũ, quod eſt uirtus ſenſitiua, quę eſt in anteriore cerebri.</s> <s xml:id="echoid-s1005" xml:space="preserve"> Et iſta uirtus cõprehen <lb/>dit ſenſibilia:</s> <s xml:id="echoid-s1006" xml:space="preserve"> uiſus aũt nõ eſt, niſi quoddam inſtrumentũ iſtius uirtutis:</s> <s xml:id="echoid-s1007" xml:space="preserve"> quoniã uiſus recipit formas <lb/>rerum uiſarum, & reddit eas ſentienti ultimo, & ſentiens ultimũ comprehendit iſtas formas, & com <lb/>prehendit ex eis res uiſibiles, quę ſunt in eis.</s> <s xml:id="echoid-s1008" xml:space="preserve"> Et illa forma in ſuperficie glacialis extenditur in corpo <lb/>re glacialis:</s> <s xml:id="echoid-s1009" xml:space="preserve"> deinde in corpus ſubtile, quod eſt in concauo nerui, quouſq;</s> <s xml:id="echoid-s1010" xml:space="preserve"> perueniat ad neruum com <lb/>munem, & apud peruentum formæ apud neruum communem completur uiſio, & ex forma uenien <lb/>te in neruum communem, comprehendet ultimum ſentiens formas rerum uiſarum.</s> <s xml:id="echoid-s1011" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div37" type="section" level="0" n="0"> <head xml:id="echoid-head56" xml:space="preserve" style="it">27. Vtro uiſu una uiſibilis forma plerun uidetur. 28 p 3.</head> <p> <s xml:id="echoid-s1012" xml:space="preserve">ET aſpiciens cõprehendet res uiſas duobus oculis, & ſic oportet, ut forma rei uiſę perueniat ad <lb/>utrunq;</s> <s xml:id="echoid-s1013" xml:space="preserve"> uiſum:</s> <s xml:id="echoid-s1014" xml:space="preserve"> quare peruenient ad uiſum ab una re uiſa duæ formę, cũ aſpiciens comprehen <lb/>dat unam rem uiſam.</s> <s xml:id="echoid-s1015" xml:space="preserve"> Et hoc eſt, quia duæ formæ, quæ perueniunt ad duos uiſus ex uno uiſo, <lb/>quando perueniunt ad neruum cõmunem, concurrunt, & ſuperponitur una alij, & efficitur una for <lb/>ma, & ex illa forma adunata ex duabus formis comprehendit ultimũ ſentiens formam illius uιſi.</s> <s xml:id="echoid-s1016" xml:space="preserve"> Et <lb/>ſignificatio ſuper hoc eſt, quod duæ formæ, quæ perueniũt ad duos oculos ab uno uiſo, ordinantur <lb/>& efficiuntur una forma, antequam cõprehendat ipſas ultimum ſentiens.</s> <s xml:id="echoid-s1017" xml:space="preserve"> Quòd aũt ultimũ ſentiens <lb/>non cõprehendat formã, nιſi poſt adunationẽ duarum formarũ:</s> <s xml:id="echoid-s1018" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s1019" xml:space="preserve"> quòd quando aſpicιens mutaue <lb/>rit ſitũ oculi unius, & alius fuerit immotus, & motus unius oculi mutati ſecundũ ſitum, fuerit ad an <lb/>terius, uidebit de re una oppoſita duas, & ſi aperuerit unũ oculum, & cooperuerit alterũ, nõ uidebit <lb/>niſi unum.</s> <s xml:id="echoid-s1020" xml:space="preserve"> Si ergo ſentiens comprehendιſſet unũ, quia unum, deberet ipſum cõprehendere ſemper <lb/>unum:</s> <s xml:id="echoid-s1021" xml:space="preserve"> & ſi ueniſſent ad ipſum ſemper duæ formæ ab uno uiſo, cõprehenderet ſemper unum uiſum, <lb/>duo.</s> <s xml:id="echoid-s1022" xml:space="preserve"> Et cum ulti nũ ſentiens non comprehendat uiſum, niſi ex forma ueniente ad ipſum, & aliquan <lb/>do comprehendat unã rem uiſam, duas, & aliquando unam:</s> <s xml:id="echoid-s1023" xml:space="preserve"> eſt ſignũ, quòd id, quod uenit ad ipſum, <lb/>quando comprehendit ipſum duo, eſt forma duplex:</s> <s xml:id="echoid-s1024" xml:space="preserve"> & quando cõprehendit unã rem uiſam, unam, <lb/>quod uenitad ipſum, eſt forma una.</s> <s xml:id="echoid-s1025" xml:space="preserve"> Et cum in utraq;</s> <s xml:id="echoid-s1026" xml:space="preserve"> diſpoſitione perueniunt ab uno uiſo ad duos <lb/>oculos duæ formæ:</s> <s xml:id="echoid-s1027" xml:space="preserve"> & illud, quod redditur ultimo ſentienti, aliquando eſt duplex forma, aliquando <lb/>una:</s> <s xml:id="echoid-s1028" xml:space="preserve"> & forma, quæ redditur ultimo ſentienti, non redditur niſi à uiſu:</s> <s xml:id="echoid-s1029" xml:space="preserve"> tunc illud, quod redditur ulti-<lb/>mo ſentienti ex duabus formis, quæ perueniunt ad duos oculos ab uno uiſo, quando cõprehende-<lb/>rit ipſum unum, eſt una forma.</s> <s xml:id="echoid-s1030" xml:space="preserve"> Et cum ita ſit, duæ ergo formæ prædictæ exten duntur à duobus ocu <lb/>lis, & concurrunt, antequam comprehendatipſas ultimũ ſentiens, & poſt cõcurſum interſe, com-<lb/>prehendet ſentiens ultimã formam adunatam ex eis.</s> <s xml:id="echoid-s1031" xml:space="preserve"> Et duæ formæ, quæ perueniũt ad duos oculos <lb/>ab uno uiſo, quando ultimũ ſentiens comprehendit ipſum duo, extenduntur à duobus oculis, & nõ <lb/>concurrunt, & perueniunt ad ultimũ ſentiens, & ſunt duæ formæ.</s> <s xml:id="echoid-s1032" xml:space="preserve"> Et comprehenſio unius uiſi, quod <lb/>apparet aliquando unũ, aliquando duo, ſignificat quòd uiſio non eſt per oculum ſolummodo:</s> <s xml:id="echoid-s1033" xml:space="preserve"> quo-<lb/>niam ſi ita eſſet apud comprehenſionẽ uiſi, quod unũ apparet, comprehenderent duo oculi ex dua-<lb/>bus formis peruenientibus ad eos, unam & eandem formã:</s> <s xml:id="echoid-s1034" xml:space="preserve"> & ſi ita eſſet, cõprehenderent ſemper ex <lb/>duabus formis unam formã.</s> <s xml:id="echoid-s1035" xml:space="preserve"> Et cum unum uiſum cõprehendatur aliquando unum, alιquando duo, <lb/>& in utraq;</s> <s xml:id="echoid-s1036" xml:space="preserve"> diſpoſitione ſint in duobus oculis duæ formę:</s> <s xml:id="echoid-s1037" xml:space="preserve"> ſignificatur, quòd ιllic eſt aliud ſentiens, <lb/>præter duos oculos, ad quod perueniunt ab uno uiſo, quando cõprehenduntur per unũ, duę formę <lb/>unũ, & apud quod cõprehenduntur duæ formæ, quando cõprehenduntur, duę:</s> <s xml:id="echoid-s1038" xml:space="preserve"> & quòd ſenſus non <lb/>cõpletur, niſi per illud ſentiens tantùm, non per oculũ tantùm.</s> <s xml:id="echoid-s1039" xml:space="preserve"> Et etiã ſenſus non ext<gap/>nditur à mem <lb/>bris ad ultimũ ſentiens, niſi in neruis continuatis membris & cerebro.</s> <s xml:id="echoid-s1040" xml:space="preserve"> Duæ ergo formæ extendun <lb/>tur ab oculo in neruo extenſo inter oculum & cerebrũ, quouſq;</s> <s xml:id="echoid-s1041" xml:space="preserve"> perueniant ad ultimũ ſentiens.</s> <s xml:id="echoid-s1042" xml:space="preserve"> Iſtæ <lb/>ergo duæ formæ extenduntur à duobus oculis, & concurrunt in loco concurſus duorũ neruorũ.</s> <s xml:id="echoid-s1043" xml:space="preserve"> Et <lb/>ſignificatio manifeſta, quòd formæ rerum uiſarũ extenduntur in concauo nerui, & perueniũt ad ul <lb/>timum ſentiens, & poſt peruentũ compleatur uiſio:</s> <s xml:id="echoid-s1044" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s1045" xml:space="preserve"> quòd quando fuerit oppilatio in ιſto neruo, <lb/>deſtruitur uiſio, & quãdo deſtruitur oppilatio, reuertitur uiſio.</s> <s xml:id="echoid-s1046" xml:space="preserve"> Et ars medicinalis teſtatur hoc.</s> <s xml:id="echoid-s1047" xml:space="preserve"> Qua <lb/>re uerò aliquando concurrant duæ formæ, aliquando non:</s> <s xml:id="echoid-s1048" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s1049" xml:space="preserve"> quia quando ſitus duorũ oculorum <lb/>fuerit naturalis, erit ſitus eorũ ab uno uιſo ſitus conſimilis:</s> <s xml:id="echoid-s1050" xml:space="preserve"> & ſic perueniet forma unius uiſi in duo <lb/>loca conſimilis ſitus:</s> <s xml:id="echoid-s1051" xml:space="preserve"> & cum fuerit declinãs ſitus unius oculi, diuerſabitur ſitus oculorũ ab illo uiſo:</s> <s xml:id="echoid-s1052" xml:space="preserve"> <lb/>& ſic peruenient duæ formæ illius uiſi diuerſi ſitus.</s> <s xml:id="echoid-s1053" xml:space="preserve"> Et iam prædictũ eſt in forma oculi, [4 n] quòd <lb/>ſitus nerui cõmunis à duobus oculis, eſt ſitus conſimilis:</s> <s xml:id="echoid-s1054" xml:space="preserve"> & ſic erit ſitus duorum locorũ conſimilis, <lb/>ſitus à duobus oculis ab eodem loco nerui cõmunis ſitus conſimilis, & ex duobus neruis concauis <lb/>fit unus, in quo uniuntur duæ formæ uiſus.</s> <s xml:id="echoid-s1055" xml:space="preserve"> Et licet dicere, quòd formę uenientes ad oculũ, non per-<lb/>ueniunt ad neruũ cõmunem, ſed ſenſus extenditur ab oculo ad neruũ cõmunẽ, ſicut extenditur ſen <lb/>ſus doloris & tactus, & tũc cõprehendit ultimũ ſentιẽs illud ſenſibile.</s> <s xml:id="echoid-s1056" xml:space="preserve"> Et nos dicemus, quòd ſenſus <lb/>ipſe ueniens ad oculum, peruenit ad neruum cõmunem omnino, tamen ſenſus, qui peruenit ad ocu <lb/>lum, non eſt ſenſus doloris tantùm, ſed eſt ſenſus operationis de genere doloris, & eſt ſenſus lucis & <lb/>coloris, & ſen<gap/>us ordinationis partiũ uiſi.</s> <s xml:id="echoid-s1057" xml:space="preserve"> Senſus aũt diuerſitatis coloris & ordinationis partiũ uiſi <lb/> <pb o="17" file="0023" n="23" rhead="OPTICAE LIBER I."/> non eſt in genere doloris.</s> <s xml:id="echoid-s1058" xml:space="preserve"> Et nos declarabimus pòſt, quomodo erit ſenſus uiſus ex omnibus rebus <lb/>iſtis.</s> <s xml:id="echoid-s1059" xml:space="preserve"> Senſus ergo perueniens in neruum communem eſt ſenſus lucis, & coloris & ordinationis, & <lb/>illud, à quo comprehendit ſentiens ultimum lucem & colorem, eſt aliqua forma.</s> <s xml:id="echoid-s1060" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div38" type="section" level="0" n="0"> <head xml:id="echoid-head57" xml:space="preserve" style="it">28. Corpora perſpicua nata at apta ſunt ad recipiendum reddenduḿ obiectis corporibus <lb/>lucem & colorem, abſ ulla ſui mutatione. 4 p 2.</head> <p> <s xml:id="echoid-s1061" xml:space="preserve">ET remanet modò explicare quæſtionẽ, quæ eſt.</s> <s xml:id="echoid-s1062" xml:space="preserve"> Quoniam formę lucis & coloris extenduntur <lb/>in aere, & in corporibus diaphanis;</s> <s xml:id="echoid-s1063" xml:space="preserve"> & perueniunt ad uiſum:</s> <s xml:id="echoid-s1064" xml:space="preserve"> & aer & corpora diaphana reci-<lb/>piunt omnes colores:</s> <s xml:id="echoid-s1065" xml:space="preserve"> & formæ cuiuslibet lucis, quæ ſunt præſentes in eodẽ tempore, exten-<lb/>duntur in eodem tẽpore, & in eodẽ aere, & perueniunt ad uiſum, & pertranſeunt diaphanitatẽ tuni <lb/>carum uiſus:</s> <s xml:id="echoid-s1066" xml:space="preserve"> quare oportet, ut admiſceantur iſti colores & lux in aere, & in corporibus diaphanis, <lb/>& perueniant ad uiſum mixta omnia.</s> <s xml:id="echoid-s1067" xml:space="preserve"> Et ſic non diſtinguuntur à uiſu colores rerũ uiſarũ.</s> <s xml:id="echoid-s1068" xml:space="preserve"> Et ſi ita eſt:</s> <s xml:id="echoid-s1069" xml:space="preserve"> <lb/>Senſus ergo uiſus non poteſt eſſe ex iſtis formis.</s> <s xml:id="echoid-s1070" xml:space="preserve"> Dicamus ergo quòd corpora diaphana non immu <lb/>tantur à coloribus, neq;</s> <s xml:id="echoid-s1071" xml:space="preserve"> alterantur ab eis alteratiõe fixa, ſed proprietas coloris & lucis eſt, ut formæ <lb/>eorum extendantur ſecundum uerticationes rectas:</s> <s xml:id="echoid-s1072" xml:space="preserve"> & ex proprietate eſt corporis diaphani, ut non <lb/>prohibeat formas lucis & coloris tranſire per ſuam diaphanitatem:</s> <s xml:id="echoid-s1073" xml:space="preserve"> & illud non recipit formas, niſi <lb/>receptione ad reddendum, non receptione ad alterandum.</s> <s xml:id="echoid-s1074" xml:space="preserve"> Et declaratum eſt [14.</s> <s xml:id="echoid-s1075" xml:space="preserve">18 n] quòd formæ <lb/>lucis & coloris non extenduntur in aere, niſi ſecundum lineas rectas.</s> <s xml:id="echoid-s1076" xml:space="preserve"> Formæ ergo lucis & coloris, <lb/>quæ ſuntin corporibus præſentibus ſimul in eodem aere, extenduntur ſecundum lineas rectas, & <lb/>erunt illæ lineæ, ſuper quas extenduntur formæ diuerſæ, quæ dam æquidiſtantes, & quædam ſecan <lb/>tes ſe, & quædam diuerſi ſitus:</s> <s xml:id="echoid-s1077" xml:space="preserve"> & quælibet uerticatio earum eſt diſtincta per corpus, à quo extendi-<lb/>tur forma ſuper illam uerticationem.</s> <s xml:id="echoid-s1078" xml:space="preserve"> Formarum ergo extenſarum à corporibus diuerſis in eodem <lb/>aere, quælibet extenditur ſuper ſuam uerticationem, & pertranſit ad formas oppoſitas.</s> <s xml:id="echoid-s1079" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div39" type="section" level="0" n="0"> <head xml:id="echoid-head58" xml:space="preserve" style="it">29. Lux & color per corpor a perſpicua diſtinctè penetrant. s p 2.</head> <p> <s xml:id="echoid-s1080" xml:space="preserve">ET ſignificatio, quòd luces & colores nõ permiſceantur in aere, neq;</s> <s xml:id="echoid-s1081" xml:space="preserve"> in corporibus diaphanis:</s> <s xml:id="echoid-s1082" xml:space="preserve"> <lb/>eſt, quòd quando in uno loco fuerint multę candelæ in locis diuerſis & diſtinctis, & fuerint o-<lb/>mnes oppoſitæ uni formaini pertranſeunti ad locum obſcurum, & fuerit in oppoſitione illius <lb/>foraminis in obſcuro loco paries, aut corpus non diaphanum:</s> <s xml:id="echoid-s1083" xml:space="preserve"> luces illarum candelarum apparent <lb/>ſuper corpus uel ſuper illũ parietẽ, diſtinctę ſecundum numerũ candelarum illarũ:</s> <s xml:id="echoid-s1084" xml:space="preserve"> & quælibet illa-<lb/>rum apparet oppoſita uni candelę ſecundũ lineam tranſeuntẽ per foramẽ:</s> <s xml:id="echoid-s1085" xml:space="preserve"> & ſi cooperiatur una can <lb/>dela, deſtruetur lux oppoſita uni candelę tantùm:</s> <s xml:id="echoid-s1086" xml:space="preserve"> & ſi auferatur coopertoriũ, reuertetur lux.</s> <s xml:id="echoid-s1087" xml:space="preserve"> Et hoc <lb/>poterit omni hora probari:</s> <s xml:id="echoid-s1088" xml:space="preserve"> quòd ſi luces admiſcerentur cũ aere, admiſcerentur cũ aere foraminis, & <lb/>deberent tranſire admixtæ, & nõ diſtinguerentur poſtea.</s> <s xml:id="echoid-s1089" xml:space="preserve"> Et nos nõ inuenimus ita.</s> <s xml:id="echoid-s1090" xml:space="preserve"> Luces ergo non <lb/>admiſcẽtur in aere, ſed quælibet illarũ extenditur ſuper uerticationes rectas:</s> <s xml:id="echoid-s1091" xml:space="preserve"> & illę uerticatiões ſunt <lb/>æquidiſtãtes, & ſecãtes ſe, & diuerſi ſirus.</s> <s xml:id="echoid-s1092" xml:space="preserve"> Et forma cuiuslibet lucis extẽditur ſuper oẽs uerticatiões, <lb/>quę poſſunt extendi in illo aere ab illa hora:</s> <s xml:id="echoid-s1093" xml:space="preserve"> neq;</s> <s xml:id="echoid-s1094" xml:space="preserve"> tamẽ admiſcentur in aere, nec aer tingitur per eas, <lb/>ſed pertranſeunt per ipſius diaphanitatẽ tantùm, & aer non amittit ſuã formã.</s> <s xml:id="echoid-s1095" xml:space="preserve"> Et quod diximus de <lb/>luce, & colore, & aere, intelligendũ eſt de omnib.</s> <s xml:id="echoid-s1096" xml:space="preserve"> corporibus diaphanis, & tunicis uiſus diaphanis.</s> <s xml:id="echoid-s1097" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div40" type="section" level="0" n="0"> <head xml:id="echoid-head59" xml:space="preserve" style="it">30. Humor cryſtallin{us} lucem & colorẽ aliter recipit, quàm cætera perſpicua corpora. 22 p 3.</head> <p> <s xml:id="echoid-s1098" xml:space="preserve">MEmbrum uerò ſentiens, ſcilicet glacialis, nõ recipit formã lucis & coloris, ſicut recipit aer, & <lb/>alia diaphana nõ ſentientia, ſed ſecundũ modũ diuerſum ab illo modo:</s> <s xml:id="echoid-s1099" xml:space="preserve"> Quoniã iſtud mem-<lb/>brum eſt præparatũ ad recipiendũ iſtam formã:</s> <s xml:id="echoid-s1100" xml:space="preserve"> recipit ergo iſtam, quatenus eſt ſentiens, & <lb/>quatenus eſt diaphanũ.</s> <s xml:id="echoid-s1101" xml:space="preserve"> Et iam declaratũ eſt, [26 n] quòd paſsio eius ex iſta forma, eſt ex genere do <lb/>loris.</s> <s xml:id="echoid-s1102" xml:space="preserve"> Qualitas ergo receptiõis eius ab iſta forma, eſt diuerſa à qualitate receptionis corporũ diapha <lb/>norum nõ ſentientiũ:</s> <s xml:id="echoid-s1103" xml:space="preserve"> Sed tamen iſtud membrũ cum ſua receptione ab iſta forma, quatenus eſt ſen-<lb/>tiens, & cum ſua alteratione uel mutatione, nõ tingitur per iſtam formã illius tinctura, neq;</s> <s xml:id="echoid-s1104" xml:space="preserve"> remanẽt <lb/>formæ coloris & lucis poſt receſſum eius à ſua oppoſitione, uel receſſu earũ.</s> <s xml:id="echoid-s1105" xml:space="preserve"> Et poteſt cõtradici huic <lb/>ſermoni dicendo.</s> <s xml:id="echoid-s1106" xml:space="preserve"> Quoniã iam prædictũ eſt, [1 n] quòd colores fortes ſcintillantes, ſuper quos oriun <lb/>tur luces fortes, operantur in oculũ, & remanent illarũ alterationes in uiſu poſt receſſum, & rema-<lb/>nent formæ coloris in oculo tẽpore aliquanto, & quodcunq;</s> <s xml:id="echoid-s1107" xml:space="preserve"> cõprehenderit uiſus poſt hoc, erit ad-<lb/>mixtum cũ illis coloribus.</s> <s xml:id="echoid-s1108" xml:space="preserve"> Et hoc eſt manifeſtũ;</s> <s xml:id="echoid-s1109" xml:space="preserve"> & non dubitatur.</s> <s xml:id="echoid-s1110" xml:space="preserve"> Quod cum ita ſit:</s> <s xml:id="echoid-s1111" xml:space="preserve"> uiſus ergo tingi-<lb/>tur à colore & luce:</s> <s xml:id="echoid-s1112" xml:space="preserve"> & ſequitur quòd corpora diaphana tingantur à lucibus & coloribus.</s> <s xml:id="echoid-s1113" xml:space="preserve"> Et nos di-<lb/>cemus, reſpondendo ad hoc:</s> <s xml:id="echoid-s1114" xml:space="preserve"> quòd hoc ipſum ſignificat, quòd uiſus nõ tingitur à colore & luce, neq;</s> <s xml:id="echoid-s1115" xml:space="preserve"> <lb/>remanent in eo alterationes coloris & lucis:</s> <s xml:id="echoid-s1116" xml:space="preserve"> quoniã lſtæ alterationes, quas diximus, nõ accidũt niſi <lb/>extranea fortitudine, ſcilicet fortitudine lucis & coloris.</s> <s xml:id="echoid-s1117" xml:space="preserve"> Et manifeſtũ eſt, quòd iſtę alterationes nõ <lb/>remanent in uiſu, niſi modico tẽpore, & pòſt auferuntur, & tunc debiles ſunt immutationes, nec re-<lb/>manet aliquid:</s> <s xml:id="echoid-s1118" xml:space="preserve"> Tunc ergo uiſus nõ tingitur ab iſtis alterationibus alteratione fixa, nec remanent in <lb/>eo poſt receſſum.</s> <s xml:id="echoid-s1119" xml:space="preserve"> Et ex hoc declarabitur, quòd luces & colores operantur in uiſum, nec remanent <lb/>eorũ alterationes poſt receſſum, niſi paruo tẽpore.</s> <s xml:id="echoid-s1120" xml:space="preserve"> Glacialis ergo alteratur à luce & coloribus tan-<lb/>tùm, ut ſentiat, & deinde aufertur immutatio poſt receſſum.</s> <s xml:id="echoid-s1121" xml:space="preserve"> Alteratio ergo eius à luce & colore eſt <lb/>neceſſaria, ſed natura nõ fixa.</s> <s xml:id="echoid-s1122" xml:space="preserve"> Et etiã uiſus eſt præparatus ad patiendũ colores & luces, & ad ſentien <lb/>dum eos:</s> <s xml:id="echoid-s1123" xml:space="preserve"> neq;</s> <s xml:id="echoid-s1124" xml:space="preserve"> tamẽ remanet in eo alteratio.</s> <s xml:id="echoid-s1125" xml:space="preserve"> Et aer & corpora diaphana, & tunicæ uiſus diaphanæ <lb/>anterioris glacialis nõ ſunt præparatæ ad patiendũ lucẽ & colorẽ, & ſentiendũ ea, ſed ad reddendũ <lb/> <pb o="18" file="0024" n="24" rhead="ALHAZEN"/> luces & colores tantùm.</s> <s xml:id="echoid-s1126" xml:space="preserve"> Iam ergo declaratũ eſt, quòd uiſus nõ tingitur ex coloribus & formis lucis <lb/>tinctura fixa:</s> <s xml:id="echoid-s1127" xml:space="preserve"> & declaratũ eſt, quòd formę lucis & coloris non admiſcẽtur in aere & corporibus dia-<lb/>phanis:</s> <s xml:id="echoid-s1128" xml:space="preserve"> & quòd uiſus multi cõprehendunt ipſos in aere, & in eodem tempore, & quilibet eorũ com <lb/>prehendit ipſos ſecundum pyramidem, quæ diſtinguitur inter ipſos & centrum uiſus.</s> <s xml:id="echoid-s1129" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div41" type="section" level="0" n="0"> <head xml:id="echoid-head60" xml:space="preserve" style="it">31. Colores uiſibilium in obiect is corporib{us} illuminantur, & obſcur antur præcipuè, pro lucis <lb/>qualitate & obiectorum corporum colorib{us}. Vide 3 n.</head> <p> <s xml:id="echoid-s1130" xml:space="preserve">QVare uerò nõ appareant omnes formę omniũ corporũ uel colorũ ſuper omnia corpora op-<lb/>poſita, ſed quædam appareant, quædã non:</s> <s xml:id="echoid-s1131" xml:space="preserve"> nõ eſt:</s> <s xml:id="echoid-s1132" xml:space="preserve"> niſi quando color fuerit fortis, & lux, quæ <lb/>eſt in corpore, fuerit fortis, & lux, quę eſt in corpore, ſuper quod apparet forma coloris, debi <lb/>lis:</s> <s xml:id="echoid-s1133" xml:space="preserve"> & hoc pertinet ad uiſum:</s> <s xml:id="echoid-s1134" xml:space="preserve"> quoniã iſtę formę colorũ nõ oriuntur ſuper corpora oppoſita illis, niſi il <lb/>luminẽtur, ſed ſuper corpora illuminata cũ quolibet lumine:</s> <s xml:id="echoid-s1135" xml:space="preserve"> quoniã formę lucis & coloris eius ſem <lb/>per oriũtur ſuper omnia corpora oppoſita illis, quorũ remotio nõ eſt extranea, multa, fortis, lõga.</s> <s xml:id="echoid-s1136" xml:space="preserve"> In <lb/>lucibus uerò hoc manifeſtatur.</s> <s xml:id="echoid-s1137" xml:space="preserve"> Quoniã, quãdo fuerit experimẽtatũ omne corpus illuminatũ quoli <lb/>bet lumine, ita quòd fuerit lux ualde debilis, & fuerit experimẽtatũ ſecũdũ modos, quos declaraui <lb/>mus, [2 n] ſcilicet, ut ſit poſitũ in ſua oppoſitione corpus albũ, & illud corpus ſit in loco obſcuro, & <lb/>fuerit inter corpus illuminatũ, & locũ illũ obſcurũ foramen ſtrictũ:</s> <s xml:id="echoid-s1138" xml:space="preserve"> inuenietur quòd ſuper illud cor <lb/>pus tũc apparebit lux, colores aũt nõ apparebũt, niſi ſecundũ modũ prędictũ:</s> <s xml:id="echoid-s1139" xml:space="preserve"> quoniã declaratũ eſt <lb/>per inductionẽ, quòd formæ colorũ ſemper ſunt debiliores ipſis coloribus, & quãtò formæ fuerint <lb/>remotiores à ſuo principio, tantò erũt debiliores.</s> <s xml:id="echoid-s1140" xml:space="preserve"> Et declaratũ eſt [2 n] per inductionẽ, quòd fortes <lb/>colores, quando fuerint in locis obſcuris, & fuerint luces, quæ ſunt ſuper ipſos, ualde debiles:</s> <s xml:id="echoid-s1141" xml:space="preserve"> iſti co <lb/>lores apparebunt obſcuri, & nõ diſtinguentur à uiſu:</s> <s xml:id="echoid-s1142" xml:space="preserve"> & quando fuerint in locis illuminatis, & fuerit <lb/>lux, quę eſt ſuper ipſos, fortis:</s> <s xml:id="echoid-s1143" xml:space="preserve"> apparebunt colores, & diſtinguentur à uiſu.</s> <s xml:id="echoid-s1144" xml:space="preserve"> Et declaratũ eſt etiam per <lb/>inductionẽ, quòd, quando lux fortis fuerit ſuper formas colorũ apparentes ſuper corpora oppoſita <lb/>illis, latebunt uiſum, & nõ apparebunt, niſi quando lux fuerit remota.</s> <s xml:id="echoid-s1145" xml:space="preserve"> Et etiã declaratum eſt, quòd, <lb/>quãdo lux fuerit fortis, & peruenerit ad uiſum, prohibebit ipſum ab apprehenſione rerũ uiſarũ non <lb/>apparentium in ſe multùm, & oppoſitarum illi tunc.</s> <s xml:id="echoid-s1146" xml:space="preserve"> Et etiam declaratum eſt, [3 n] quòd uiſus non <lb/>comprehendit colores, niſi ex forma ueniente ad ipſum ex illo colore, & quòd comprehenſio ipſius <lb/>erit ſecundum uerticationes proprias.</s> <s xml:id="echoid-s1147" xml:space="preserve"> Quando ergo inſpiciens aſpexerit corpus denſum, ſuper <lb/>quod oriebatur forma coloris:</s> <s xml:id="echoid-s1148" xml:space="preserve"> non comprehendit illam formam, niſi ex ſecunda forma ueniente <lb/>ad ipſum ex illa forma:</s> <s xml:id="echoid-s1149" xml:space="preserve"> & iſta forma ſecunda eſt debilior forma prima, quæ eſt ſuper illud corpus, <lb/>& prima forma eſt debilior ipſo colore.</s> <s xml:id="echoid-s1150" xml:space="preserve"> Et uiſus non comprehendit illud corpus denſum, ſuper <lb/>quod apparet forma, niſi quando in eo apparuerit aliqua lux, ſiue lux ueniens cum forma coloris <lb/>ſuper ipſum orientis, ſiue illa lux cum alia.</s> <s xml:id="echoid-s1151" xml:space="preserve"> Forma ergo ſecunda, quæ uenit ad uiſum ex prima for-<lb/>ma coloris:</s> <s xml:id="echoid-s1152" xml:space="preserve"> uenit ad ipſum cum forma lucis, quæ eſt in illo corpore denſo, & color illius corporis <lb/>denſi, ſuper quod eſt iſta forma, comprehendetur à uiſu etiam in illa diſpoſitione.</s> <s xml:id="echoid-s1153" xml:space="preserve"> Forma ergo co-<lb/>loris uenit ad uiſum cum forma ſecunda ueniente ad ipſum ex forma coloris, quæ eſt ſuper ipſum:</s> <s xml:id="echoid-s1154" xml:space="preserve"> <lb/>& forma coloris iſtius corporis, quæ uenit ad ipſum uiſum in illa diſpoſitione, eſt prima forma:</s> <s xml:id="echoid-s1155" xml:space="preserve"> ui-<lb/>ſus autem non comprehendit illud, quod comprehendit, niſi ex uerticationibus proprijs:</s> <s xml:id="echoid-s1156" xml:space="preserve"> & uerti-<lb/>catio propria, quæ eſt inter ipſum & corpus denſum, ſecundum quod comprehendit formam il-<lb/>lius corporis denſi, eſt eadem cum uerticatione ſua, ſecundum quam comprehendit formam ſe-<lb/>cundam uenientem ex forma coloris orientis ſuper illud corpus:</s> <s xml:id="echoid-s1157" xml:space="preserve"> quoniam illa forma eſt in ſuper-<lb/>ficie illius corporis.</s> <s xml:id="echoid-s1158" xml:space="preserve"> Viſus ergo cõprehendit ipſam ex uerticationibus, quæ ſunt inter ipſum & illud <lb/>corpus:</s> <s xml:id="echoid-s1159" xml:space="preserve"> & ipſe comprehendit colorẽ ipſius ex uerticationibus, quę ſunt inter ipſum & illud corpus.</s> <s xml:id="echoid-s1160" xml:space="preserve"> <lb/>Et ſimiliter comprehendit uiſus lucem, quæ eſt in illo corpore, ex illis eiſdem uerticationibus.</s> <s xml:id="echoid-s1161" xml:space="preserve"> Tres <lb/>ergo formæ uenientes ex illo corpore ad uiſum, comprehenduntur à uiſu ex eadem uerticatione:</s> <s xml:id="echoid-s1162" xml:space="preserve"> & <lb/>quidem admixtæ.</s> <s xml:id="echoid-s1163" xml:space="preserve"> Et ſormæ ſecundæ, quæ ueniunt ad uiſum ex forma coloris, quæ eſt ſuper cor-<lb/>pus oppoſitum illi, comprehenduntur à uiſu ſemper admixtæ cum forma coloris illius corporis, & <lb/>cum forma lucis eius.</s> <s xml:id="echoid-s1164" xml:space="preserve"> Viſus ergo comprehendit ex congregatione duorum colorum formam diuer <lb/>ſam à forma cuiuslibet eorum.</s> <s xml:id="echoid-s1165" xml:space="preserve"> Si ergo corpus, ſuper quod eſt forma, habuerit fortem colorem, e-<lb/>rit forma eius, quæ uenit ad uiſum, fortis:</s> <s xml:id="echoid-s1166" xml:space="preserve"> & eſt prima forma, & eſt admixta cum ſecunda forma, <lb/>quæ uenit ad ipſum ex forma coloris uenientis ſuper illud corpus:</s> <s xml:id="echoid-s1167" xml:space="preserve"> & iſta forma eſt debilis:</s> <s xml:id="echoid-s1168" xml:space="preserve"> quare <lb/>nõ apparet uiſui.</s> <s xml:id="echoid-s1169" xml:space="preserve"> quoniam quando cum colore debili fuerit admixtus color fortis, ipſe ſcilicet color <lb/>fortis uincet debilem:</s> <s xml:id="echoid-s1170" xml:space="preserve"> & ſimiliter inueniuntur ſemper colores & tincturæ, quando admiſcentur <lb/>inter ſe.</s> <s xml:id="echoid-s1171" xml:space="preserve"> Forma uerò coloris non latet, quando lux eſt ſuper ipſam fortis, & cum albedine corporis.</s> <s xml:id="echoid-s1172" xml:space="preserve"> <lb/>Et iam declaratum eſt, [2 n] quòd lux fortis, quando uenit ad uiſum, prohibet uiſum à compre-<lb/>henſione formarum debilium.</s> <s xml:id="echoid-s1173" xml:space="preserve"> Quando ergo ueniet ad uiſum lux fortis cum albedine corporis, ſu-<lb/>per quod cadit, prohibet ipſum à comprehenſione ſecundæ formæ debilis, quæ uenit ad ipſum <lb/>cum ea:</s> <s xml:id="echoid-s1174" xml:space="preserve"> & ſi corpus, ſuper quod eſt forma coloris, fuerit album, & lux, quæ eſt ſuper ipſum, fuerit <lb/>debilis, & forma coloris, quæ eſt ſuper ipſum, fuerit debilis:</s> <s xml:id="echoid-s1175" xml:space="preserve"> tunc forma lucis, quæ eſt in illo cor-<lb/>pore, quamuis ſit debilis cum albedine corporis, fortè uincet formam coloris, quę eſt ualde debilis, <lb/>& cum uenerit ad uiſum, non diſtinguetur forma illa à uiſu.</s> <s xml:id="echoid-s1176" xml:space="preserve"> Et ſi corpus, ſuper quod eſt lux, fuerit al <lb/>bum, & color cum forma, quæ oritur ſuper ipſum, fuerit niger, aut obſcurus, non obſcurabitur illa <lb/>forma, niſi albedine illius corporis tantùm, & erit quaſi umbra, & comprehendet uiſus illud corpus <lb/> <pb o="19" file="0025" n="25" rhead="OPTICAE LIBER I."/> non ualde album, ſicut comprehendit corpus album in umbra.</s> <s xml:id="echoid-s1177" xml:space="preserve"> Quare non diſtinguetur ab eo for-<lb/>ma.</s> <s xml:id="echoid-s1178" xml:space="preserve"> Et omne hoc erit ita, quando lux, quæ eſt in corpore colorato, fuerit fortis, & forma, quæ oritur <lb/>ab eo ſuper corpus oppoſitum, fuerit albedinis debilis.</s> <s xml:id="echoid-s1179" xml:space="preserve"> Si autem lux, quæ eſt in corpore colorato, <lb/>fuerit debilis:</s> <s xml:id="echoid-s1180" xml:space="preserve"> tunc forma, quæ exit ab eo ſuper corpus oppoſitum, erit obſcura, & erit apud uiſum, <lb/>ſicut colores, quos comprehendit in locis obſcuris, in quibus eſt lux ualde debilis, & quaſi colores <lb/>corporum diaphanorum, ſuper quæ oritur lux debilis.</s> <s xml:id="echoid-s1181" xml:space="preserve"> Formæ ergo colorum, quæ ſunt in corpori-<lb/>bus coloratis, quando lux, quæ eſt ſuper ipſa, fuerit debilis, quando oriuntur ſuper corpora oppoſi-<lb/>ta ſibi, non erunt, niſi umbræ tantùm quo ad ſenſum uiſus.</s> <s xml:id="echoid-s1182" xml:space="preserve"> Et ſi corpus oppoſitum colori fuerit in <lb/>loco obſcuro, nihil apparebit ſuper ipſum propter ſuam obſcuritatẽ, & obſcuritatem formæ uenien <lb/>tis ad ipſum.</s> <s xml:id="echoid-s1183" xml:space="preserve"> Et ſi corpus oppoſitũ illi colori fuerit in illuminato loco, & fuerit ſuper ipſum lux, præ <lb/>ter lucẽ illius formæ, & fuerit illud corpus illuminatum:</s> <s xml:id="echoid-s1184" xml:space="preserve"> apparebit color eius ſuper iſtam formam, <lb/>& apparebit uiſui color iſtius corporis, & nõ apparebit forma:</s> <s xml:id="echoid-s1185" xml:space="preserve"> quoniam eſt ſicut umbra, & non di-<lb/>ſtinguetur à uiſu iſta diminutio.</s> <s xml:id="echoid-s1186" xml:space="preserve"> Et ſi iſtud corpus, ſuper quod eſt forma, fuerit album, & præterea <lb/>fuerit illuminatum cum alio lumine, præter lumen formæ:</s> <s xml:id="echoid-s1187" xml:space="preserve"> tunc forma obſcurabit albedinem iſtius <lb/>corporis, & lucem eíus tantùm propter ſuam obſcuritatem, ſicut faciũt umbræ in corporibus alijs:</s> <s xml:id="echoid-s1188" xml:space="preserve"> <lb/>& formæ, quæ ſunt huiuſmodi, tantùm comprehẽduntur à uiſu ſuper corpora oppoſita coloribus.</s> <s xml:id="echoid-s1189" xml:space="preserve"> <lb/>Viſus ergo non comprehendit formam coloris ſuper corpus oppoſitum colori, niſi quando forma <lb/>ſecũda ueniens ad ipſum ex forma coloris, fuerit fortior, & potẽtior prima forma ueniẽte ad ipſam <lb/>cum ea, ex luce & colore, quæ ſunt in corpore, ſuper quod eſt forma.</s> <s xml:id="echoid-s1190" xml:space="preserve"> Et iſte modus eſt ualde rarus:</s> <s xml:id="echoid-s1191" xml:space="preserve"> <lb/>& propter hoc rarò apparet huiuſmodi forma, & non apparet ex ea, niſi illud, quod eſt ex coloribus <lb/>fortibus ſcintillantibus.</s> <s xml:id="echoid-s1192" xml:space="preserve"> Et ſimiliter quòd lux debilis, nõ apparet ſuper corpus oppoſitum ſibi:</s> <s xml:id="echoid-s1193" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s1194" xml:space="preserve"> <lb/>quia corpus oppoſitũ luci debili, quando fuerit illuminatũ ab alio lumine, admiſcebuntur duæ lu-<lb/>ces, & ſic non diſtinguetur lux debilis à uiſu.</s> <s xml:id="echoid-s1195" xml:space="preserve"> Et cũ corpus oppoſitum luci debili fuerit obſcurum, <lb/>non apparebit forma lucis debilis ſuper ipſum:</s> <s xml:id="echoid-s1196" xml:space="preserve"> quoniam forma lucis debilior eſt ipſa luce:</s> <s xml:id="echoid-s1197" xml:space="preserve"> & forma <lb/>ſecunda ueniens ad oculũ ab iſta forma, ex qua oportet uiſum comprehẽdere iſtam formam ſuper <lb/>corpus oppoſitũ luci, eſt debilior iſta forma.</s> <s xml:id="echoid-s1198" xml:space="preserve"> Cum ergo lux fuerit debilis, & corpus oppoſitũ fuerit <lb/>obſcurum:</s> <s xml:id="echoid-s1199" xml:space="preserve"> erit forma;</s> <s xml:id="echoid-s1200" xml:space="preserve"> quæ eſt ſuper corpus oppoſitũ, ualde debilis, & erit forma ſecũda, quæ uenit <lb/>exilla, in fine debilitatis.</s> <s xml:id="echoid-s1201" xml:space="preserve"> Viſus autẽ non comprehendit lucẽ, quæ eſt in fine debilitatis.</s> <s xml:id="echoid-s1202" xml:space="preserve"> Formæ ergo <lb/>omnium colorũ illuminatorum, & formæ omnis lucis oriuntur ſuper corpora oppoſita, & nõ appa <lb/>rent pleræq;</s> <s xml:id="echoid-s1203" xml:space="preserve"> illarũ propter cauſſas, quas diximus:</s> <s xml:id="echoid-s1204" xml:space="preserve"> & quædã apparent, quando fuerint ſecundũ mo-<lb/>dum, quem narrauimus.</s> <s xml:id="echoid-s1205" xml:space="preserve"> Iam ergo declarata eſt cauſſa, propter quã non comprehẽdit uiſus formas <lb/>omnium colorũ, quæ ſunt in corporibus coloratis ſuper omnia corpora oppoſita illi, & cõprehen-<lb/>dit quaſdã:</s> <s xml:id="echoid-s1206" xml:space="preserve"> & cum hoc comprehẽdit omnes colores, qui ſunt in corporibus coloratis.</s> <s xml:id="echoid-s1207" xml:space="preserve"> Et cauſſa eſt, <lb/>quia comprehendit colores, qui ſunt in corporibus coloratis ex propria forma ueniente ad ipſum <lb/>ex eis, quæ eſt fortior forma ſecunda ueniente ad ipſum ex formis colorum, qui ſunt ſuper corpora <lb/>oppoſita illi.</s> <s xml:id="echoid-s1208" xml:space="preserve"> Et comprehendit etiã formam colorum ſingularẽ non admixtam cum alia:</s> <s xml:id="echoid-s1209" xml:space="preserve"> & compre-<lb/>hendit ſecundã formam uenientẽ ad ipſum ex formis colorum admixtam cum alia.</s> <s xml:id="echoid-s1210" xml:space="preserve"> Et hoc eſt quod <lb/>promiſimus declarare in fine capitis tertij.</s> <s xml:id="echoid-s1211" xml:space="preserve"> Et declaratũ eſt modò, quòd colores, quos comprehen-<lb/>dit uiſus ex rebus uiſis, non comprehendit, niſi admixtos cũ formis lucis, quæ ſunt in eis, & admix-<lb/>tos cum omnibus formis orientibus ſuper ipſos ex coloribus corporum oppoſitorum.</s> <s xml:id="echoid-s1212" xml:space="preserve"> Et ſi in cor-<lb/>pore diaphano, quod eſt medium inter ipſos & uiſum, fuerit aliqua ſpiſsitudo, admiſcebitur color <lb/>eius etiam cum eis, & uiſus non comprehendit illum colorem ſingularem:</s> <s xml:id="echoid-s1213" xml:space="preserve"> ſed tamen formæ, quæ <lb/>oriuntur ſuper corpora colorata, ſunt in maiori parte ualde debiles:</s> <s xml:id="echoid-s1214" xml:space="preserve"> & formæ ſecũdæ, quæ ueniunt <lb/>ex eis ad uiſum, ſunt in fine debilitatis:</s> <s xml:id="echoid-s1215" xml:space="preserve"> & propter hoc erunt colores ipſorum corporum plerunque <lb/>fortiores formis orientibus ſuper ipſa.</s> <s xml:id="echoid-s1216" xml:space="preserve"> Et ſimiliter ſi in corpore diaphano, quod eſt inter uiſuin & <lb/>rem uiſam, fuerit modica ſpiſsitudo, non diſtinguetur à uiſu color eius à colore uiſi uenientis cum <lb/>eo, quando color uiſi uenientis cum eo, fuerit fortior colore illius.</s> <s xml:id="echoid-s1217" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div42" type="section" level="0" n="0"> <head xml:id="echoid-head61" xml:space="preserve" style="it">32. Lux uehemens trib{us} potißimùm de caußis uiſibilia quædam obſcur at. Vide 2 n.</head> <p> <s xml:id="echoid-s1218" xml:space="preserve">QVare uerò lux fortis prohibeat uiſum à comprehẽſione quarundam rerũ uiſarum:</s> <s xml:id="echoid-s1219" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s1220" xml:space="preserve"> quia <lb/>formæ, quæ ueniunt ad uiſum ſuper unam uerticationem, non comprehẽduntur à uiſu, niſi <lb/>admixtæ.</s> <s xml:id="echoid-s1221" xml:space="preserve"> Et cum quæ dam formæ admixtæ fuerint fortis ſcintillationis, & quædam debi-<lb/>lis, ſuperabit forma fortis formam debilem, & ſic nõ comprehendetur forma debilis à uiſu.</s> <s xml:id="echoid-s1222" xml:space="preserve"> Et cum <lb/>formæ admixtæ fuerint propinquæ in fortitudine, comprehendentur à uiſu, & erit comprehenſio <lb/>cuiuslibet illarum ſecundum illud, quod permiſcebitur cum eis ex formis admixtis cum eis:</s> <s xml:id="echoid-s1223" xml:space="preserve"> quo-<lb/>niam formæ admixtæ non comprehenduntur à uiſu ſingulariter, ſed admixtæ.</s> <s xml:id="echoid-s1224" xml:space="preserve"> Stellæ ergo non <lb/>comprehenduntur à uiſu in luce diei, quia lux, quæ peruẽnit in aerem, eſt fortior luce ſtellarum.</s> <s xml:id="echoid-s1225" xml:space="preserve"> <lb/>Cum ergo inſpiciens aſpexerit cœlum in luce diei:</s> <s xml:id="echoid-s1226" xml:space="preserve"> erit aer, qui eſt inter ipſum & cœlum, illu-<lb/>minatus à luce ſolis, & continuatur cum uiſu, & erunt ſtellæ ex poſteriori illius lucis.</s> <s xml:id="echoid-s1227" xml:space="preserve"> Venient <lb/>ergo forma ſtellæ, & forma lucis, quæ eſt in aere medio inter uiſum & ſtellam, ad uiſum <lb/>ſuper unam uerticationem:</s> <s xml:id="echoid-s1228" xml:space="preserve"> & ſic comprehendentur admixtæ.</s> <s xml:id="echoid-s1229" xml:space="preserve"> Sed forma lucis diei eſt in ae-<lb/>re fortior multò forma lucis ſtellæ:</s> <s xml:id="echoid-s1230" xml:space="preserve"> quare ſuperabit lux aeris lucem ſtellæ, & ſic non diſtin-<lb/>guetur forma ſtellæ.</s> <s xml:id="echoid-s1231" xml:space="preserve"> Et ſimiliter eſt lux debilis, quæ eſt in medio lucis fortis, ſicut ignis de-<lb/>bilis in luce ſolis, & ſicut noctiluca in luce diei, & ſimilibus:</s> <s xml:id="echoid-s1232" xml:space="preserve"> iſta enim uiſibilia quando fuerint in <lb/> <pb o="20" file="0026" n="26" rhead="ALHAZEN"/> luce ſolis aut diei, uenient formæ eorum ad uiſum admixtæ cum forma lucis fortis orientis ſuper <lb/>ipſas, & comprehendet uiſus formam huiuſmodi rerũ uiſarum admixtam cum forma lucis fortis.</s> <s xml:id="echoid-s1233" xml:space="preserve"> <lb/>Quare ſuperabit forma lucis fortis formam debilem.</s> <s xml:id="echoid-s1234" xml:space="preserve"> Et multoties latet lux debilis, & forma rei ui-<lb/>ſæ debilis, quando peruenerit in uiſum lux fortis, quamuis nõ ſit peruentus duarum formarum ad <lb/>uiſum ex una uerticatione.</s> <s xml:id="echoid-s1235" xml:space="preserve"> Et hoc erit, quãdo peruentus duarum formarum fuerit ex duabus uer-<lb/>ticationibus uicinantibus.</s> <s xml:id="echoid-s1236" xml:space="preserve"> Et hoc apparet nocte, & in luce ignis.</s> <s xml:id="echoid-s1237" xml:space="preserve"> Quoniam uiſus quando compre-<lb/>henderit lucem ignis, & fuerit ignis propinquus uiſui, & fuerit lux eius fortis, & fuerit in oppoſi-<lb/>tione uiſus in illa diſpoſitione aliquod uiſibile, in quo eſt lux debilis accidentalis, & fuerit illud ui-<lb/>ſibile remotius à uiſu quàm ignis, & fuerit ſuper uerticationem uicinãtem uerticationi ignis:</s> <s xml:id="echoid-s1238" xml:space="preserve"> tunc <lb/>uiſus non comprehendet uiſibile illud comprehenſione uera.</s> <s xml:id="echoid-s1239" xml:space="preserve"> Et ſi aſpiciens cooperuerit ignem à <lb/>ſuo uiſu, aut remouerit ſe à uerticatione ignis, ita ut ſit uerticatio, à qua comprehendet illud uiſibi-<lb/>le, remota à uerticatìone ignis:</s> <s xml:id="echoid-s1240" xml:space="preserve"> tunc comprehendet illud uiſibile comprehenſione manifeſtiore.</s> <s xml:id="echoid-s1241" xml:space="preserve"><gap/>Et <lb/>cauſſa illius eſt, quòd uiſibile, in quo eſt lux debilis accidentalis, habet formam obſcuram, & cum <lb/>ipſam comprehenderit uiſus, non autem comprehenderit cum ea lucem fortem:</s> <s xml:id="echoid-s1242" xml:space="preserve"> ſentiet lucem de-<lb/>bilem, in qua eſt aliquid obſcuritatis inter uiſum, aut priuationem lucis fortis à parte eius, in quam <lb/>peruenit lux debilis.</s> <s xml:id="echoid-s1243" xml:space="preserve"> Et cum uiſus comprehenderit formam lucis debilis, & comprehenderit cum <lb/>ea lucem fortem:</s> <s xml:id="echoid-s1244" xml:space="preserve"> tunc etiam comprehendet lucem fortem in parte contingente partem uiſus, qua <lb/>comprehendebat formam obſcuram:</s> <s xml:id="echoid-s1245" xml:space="preserve"> non comprehendet autem uiſus lucem debilem, quæ eſt in <lb/>forma obſcura propter duo:</s> <s xml:id="echoid-s1246" xml:space="preserve"> quorum unum eſt, quòd lux fortis, quando peruenerit ad uiſum, illu-<lb/>minatur totus uiſus, & cum totus uiſus fuerit illuminatus, non apparebit in eo lux debilis, & maxi-<lb/>mè quando lux debilis füerit proportionis minimæ reſpectu lucis fortis.</s> <s xml:id="echoid-s1247" xml:space="preserve"> Alterum eſt coniunctio <lb/>lucis debilis cum luce forti in duabus partibus uiſus uicinãtibus, & quia lux debilis reſpectu lucis <lb/>fortis eſt ferè obſcuritas.</s> <s xml:id="echoid-s1248" xml:space="preserve"> Et cum lux appropin quabit ad formam obſcuram debilem, & forma lucis <lb/>fortis fuerit in uiſu:</s> <s xml:id="echoid-s1249" xml:space="preserve"> non comprehendet uiſus formam, quæ eſt in luce obſcura, neque comprehen-<lb/>det etiam formam obſcuram, niſi obſcuritatem tantùm:</s> <s xml:id="echoid-s1250" xml:space="preserve"> & ſic non diſtinguetur ab eo forma, neque <lb/>comprehendet eam comprehenſione uera.</s> <s xml:id="echoid-s1251" xml:space="preserve"> Et occultatio formarum debilis lucis propter uicinita-<lb/>tem lucis fortis, habet ſimile in coloribus:</s> <s xml:id="echoid-s1252" xml:space="preserve"> quoniam color fuſcus ſi intingatur cũ corpore albo pun-<lb/>ctatim, apparebunt ipſa puncta nigra propter fortitudinem albedinis:</s> <s xml:id="echoid-s1253" xml:space="preserve"> & ſi eadẽ puncta fuerint po-<lb/>ſita ſupra corpora ualde nigra, apparebunt ferè alba, & non apparebit obſcuritas, quæ eſt in eis.</s> <s xml:id="echoid-s1254" xml:space="preserve"> Et <lb/>quando illa tinctura fuerit in corporibus, quæ non ſunt multùm alba, neque multùm nigra:</s> <s xml:id="echoid-s1255" xml:space="preserve"> appa-<lb/>rebit color ſecundum ſuum eſſe.</s> <s xml:id="echoid-s1256" xml:space="preserve"> Et ſimiliter quando color uiridis ſegetalis fuerit ſuper corpus ci-<lb/>trinum, apparebit illa tinctura obſcura:</s> <s xml:id="echoid-s1257" xml:space="preserve"> & quando fuerit in corpore nigro, apparebit illa tinctura <lb/>ſimilis colori origani.</s> <s xml:id="echoid-s1258" xml:space="preserve"> Et ſimiliter eſt omnis tinctura media inter duas extremitates.</s> <s xml:id="echoid-s1259" xml:space="preserve"> Viſibilia ergo <lb/>uicinantia, quando fuerint remota in fortitudine & debilitate coloris:</s> <s xml:id="echoid-s1260" xml:space="preserve"> quod eſt debilis coloris, la-<lb/>tebit uiſum:</s> <s xml:id="echoid-s1261" xml:space="preserve"> quoniam qualitates lucis & coloris non comprehenduntur à uiſu, niſi reſpectu eorum <lb/>inter ſe.</s> <s xml:id="echoid-s1262" xml:space="preserve"> Et lux fortis non prohibebit uiſum à comprehenſione uiſibilium lucis debilis, niſi pro-<lb/>pter admixtionem formæ lucis debilis cum formis eorum, & propter uictoriam formarum lucis <lb/>fortis ſuper formas lucis debilis, & debilitatem ſenſus ad comprehendendum illud, quod eſt mini-<lb/>mæ proportionis, reſpectu fortis.</s> <s xml:id="echoid-s1263" xml:space="preserve"> Iam ergo compleuimus declarationem omnium rerum depen-<lb/>dentium ab illo capitulo.</s> <s xml:id="echoid-s1264" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div43" type="section" level="0" n="0"> <head xml:id="echoid-head62" xml:space="preserve">DE OFFICIO ET VTILITATE INSTRVMEN-<lb/>torum uiſus. Caput ſextum.</head> <head xml:id="echoid-head63" xml:space="preserve" style="it">33. Multiplex & uaria eſt partium uiſ{us} utilit{as}: diuerſá ſunt ipſarum inter <lb/>ipſas officìa. 4 p 3.</head> <p> <s xml:id="echoid-s1265" xml:space="preserve">TVnicæ, quas diximus in declaratione formæ uiſus, ſunt inſtrumenta, per quæ completur ui-<lb/>ſio.</s> <s xml:id="echoid-s1266" xml:space="preserve"> Tunica uerò prima, quæ dicitur cornea, eſt tunica diaphana, & nõnihil fortis, & eſt ſuper-<lb/>poſita foramini, quod eſt in anteriori uueæ.</s> <s xml:id="echoid-s1267" xml:space="preserve"> Et prima utilitas eius eſt, quòd cooperit foramen <lb/>uueæ:</s> <s xml:id="echoid-s1268" xml:space="preserve"> quare retinet humorem albugineum, qui eſt in anteriori uueæ:</s> <s xml:id="echoid-s1269" xml:space="preserve"> & eſt diaphana, ut tranſeant <lb/>in ea formæ lucis & coloris ad interius uiſus:</s> <s xml:id="echoid-s1270" xml:space="preserve"> quoniam non tranſeunt, niſi per diaphana.</s> <s xml:id="echoid-s1271" xml:space="preserve"> Fortitudo <lb/>autem eius eſt, ut non corrumpatur citò:</s> <s xml:id="echoid-s1272" xml:space="preserve"> quoniam eſt expoſita aeri, & poteſt citò corrumpi ex fu-<lb/>mo, & puluere, & ſimilibus.</s> <s xml:id="echoid-s1273" xml:space="preserve"> Humor autem albugineus eſt diaphanus, & eſt humidus & fluxibilis.</s> <s xml:id="echoid-s1274" xml:space="preserve"> <lb/>Diaphanus autem eſt, ut pertranſeant in eo formæ, & perueniant in eo ad humorem glacialem:</s> <s xml:id="echoid-s1275" xml:space="preserve"> <lb/>humiditas autem eius eſt, ut ſemper humefaciat humorem glacialem, ita ut eius natura ſit cu-<lb/>ſtodita:</s> <s xml:id="echoid-s1276" xml:space="preserve"> quoniam tela, quæ eſt ſuper glacialem, eſt ualde tenuis, & nimia ſiccitate poteſt cor-<lb/>rumpi.</s> <s xml:id="echoid-s1277" xml:space="preserve"> Tunica autem nigra continens humorem albugineum, quæ eſt uuea, eſt nigra, & fortis, <lb/>ſpiſſa, & ſphærica:</s> <s xml:id="echoid-s1278" xml:space="preserve"> & in anteriori eius eſt foramen rotundum, ſicut narrauimus.</s> <s xml:id="echoid-s1279" xml:space="preserve"> Nigredo ue-<lb/>rò eius eſt, ut obſcuretur humor albugineus & glacialis, ita ut appareant in eis formæ lucis debi-<lb/>lis:</s> <s xml:id="echoid-s1280" xml:space="preserve"> quoniam lux debilis üalde apparet in locis obſcuris, & latet in locis luminoſis.</s> <s xml:id="echoid-s1281" xml:space="preserve"> Et eſt aliquan-<lb/>tulum fortis, ut retineat humorem albugineum, & ut non reſudet ex eo aliquid foras.</s> <s xml:id="echoid-s1282" xml:space="preserve"> Et eſt ſpiſſa, <lb/>ut ſit obſcura:</s> <s xml:id="echoid-s1283" xml:space="preserve"> quoniam ſi eſſet rara, eſſet diàphana:</s> <s xml:id="echoid-s1284" xml:space="preserve"> ſed cum fuerit ſpiſſa, obſcurabitur anterior pars <lb/>eius.</s> <s xml:id="echoid-s1285" xml:space="preserve"> Et eſt ſphærica, quia magis temperata figurarum eſt ſphærica, & eſt magis remota ab offen-<lb/>ſionibus:</s> <s xml:id="echoid-s1286" xml:space="preserve"> habens enim angulos, citius alteratur per angulos.</s> <s xml:id="echoid-s1287" xml:space="preserve"> Foramen autem, quod eſt in a<gap/>te-<lb/> <pb o="21" file="0027" n="27" rhead="OPTICAE LIBER I."/> riori iſtius tunicæ, eſt, ut pertrãſeant ipſum formæ ad interius uiſus:</s> <s xml:id="echoid-s1288" xml:space="preserve"> & eſt rotundum, quia rotundi-<lb/>tas eſt ſimpliciſsima figurarũ, & ampliſsima iſoperimetrarum.</s> <s xml:id="echoid-s1289" xml:space="preserve"> Humor autẽ glacialis habet multas <lb/>proprietates, per quas completur ſenſus:</s> <s xml:id="echoid-s1290" xml:space="preserve"> quoniam eſt humidus & ſubtilis:</s> <s xml:id="echoid-s1291" xml:space="preserve"> & eſt in eo aliquid dia-<lb/>phanitatis & ſpiſsitudinis:</s> <s xml:id="echoid-s1292" xml:space="preserve"> & ſuper ipſum eſt tela ualde rara:</s> <s xml:id="echoid-s1293" xml:space="preserve"> & figura ſuperficiei eius eſt cõpoſita ex <lb/>duabus ſuperficiebus ſphæricis diuerſis:</s> <s xml:id="echoid-s1294" xml:space="preserve"> & anterior illarũ eſt maioris ſphæricitatis altera.</s> <s xml:id="echoid-s1295" xml:space="preserve"> Eſt au-<lb/>tem humidus, ut citius patiatur à luce:</s> <s xml:id="echoid-s1296" xml:space="preserve"> & eſt ſubtilis, quia talia corpora ſunt ſubtilis ſenſus:</s> <s xml:id="echoid-s1297" xml:space="preserve"> & eſt ali <lb/>quantulum diaphanus, ut recipiat formas lucis & coloris, & ut pertrãſeant per ipſum lux & color:</s> <s xml:id="echoid-s1298" xml:space="preserve"> <lb/>& eſt aliquantulũ ſpiſſus, ut remaneant in eo diu formæ lucis & coloris, ita ut appareat uirtuti ſen-<lb/>ſibili forma lucis & coloris, quæ figebantur in eo.</s> <s xml:id="echoid-s1299" xml:space="preserve"> Nam ſi eſſet diaphanus in fine diaphanitatis, per-<lb/>tranſirent formæ in eo, & non pateretur à formis paſsione, quæ eſt ex genere doloris:</s> <s xml:id="echoid-s1300" xml:space="preserve"> & ſic nõ com-<lb/>prehenderet formas.</s> <s xml:id="echoid-s1301" xml:space="preserve"> Tela autẽ quæ eſt ſuper iſtum humorẽ, eſt, ut retineat ipſum, ne fluat:</s> <s xml:id="echoid-s1302" xml:space="preserve"> quoniã <lb/>humores nõ retinerentur, ſed aliquò fluerent, & nõ remanerent ſecundũ unam figuram.</s> <s xml:id="echoid-s1303" xml:space="preserve"> Et iſta tela <lb/>eſt ualde rara, ut nõ occultet formas uenientes:</s> <s xml:id="echoid-s1304" xml:space="preserve"> & eſt ſphærica propter cauſſam, quam diximus.</s> <s xml:id="echoid-s1305" xml:space="preserve"> Et <lb/>ſuperficies anterioris eius eſt ex ſphæra maiori, ut ſit æquidiſtans ſuperficiei anteriori uiſus, ita ut <lb/>centrum illarũ ſit unumpunctum.</s> <s xml:id="echoid-s1306" xml:space="preserve"> Neruus autẽ opticus, ſuper quem componitur oculus totus, eſt <lb/>cauus, ut currat per ipſum ſpiritus uiſibilis à cerebro, & perueniat ad glacialem, & det ipſi uirtutem <lb/>ſenſibilem ſucceſsiuè, & ut pertranſeant etiã formæ in corpore ſubtili currẽte in ſuo concauo, quo-<lb/>uſq;</s> <s xml:id="echoid-s1307" xml:space="preserve"> perueniant ad ultimum ſentiens, quod eſt in anteriori cerebri.</s> <s xml:id="echoid-s1308" xml:space="preserve"> Et principia duorũ neruorum, <lb/>ſuper quos componũtur oculi duo, ſunt in duabus partibus anterioris cerebri, ut ſitus duorũ ocu-<lb/>lorum à ſuis principijs ſit ſitus conſimilis:</s> <s xml:id="echoid-s1309" xml:space="preserve"> & nõ fuit principium eorum à medio anterioris cerebri, <lb/>quia iſte locus eſt proprius ſenſui ordinatus.</s> <s xml:id="echoid-s1310" xml:space="preserve"> Quare autẽ ſint duo oculi, eſt benignitas operatoris, <lb/>ut ſi uni illorum accideret interitus, remaneret alter, & ut forma faciei eſſet pulchrior.</s> <s xml:id="echoid-s1311" xml:space="preserve"> Cauſſa autẽ, <lb/>propter quam concurrant iſti duo nerui, iam fuit dicta in qualitate uiſionis.</s> <s xml:id="echoid-s1312" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div44" type="section" level="0" n="0"> <head xml:id="echoid-head64" xml:space="preserve" style="it">34. Superſicies tunicarum uiſ{us} ſunt globoſæ. 3. 4 p 3.</head> <p> <s xml:id="echoid-s1313" xml:space="preserve">SVperficies uerò tunicarum oculi ſunt ſphæricæ & æquidiſtantes, & centrum illarũ eſt unum <lb/>punctum:</s> <s xml:id="echoid-s1314" xml:space="preserve"> ita ut perpendicularis, quæ eſt ſuper primam illarũ, ſit pe<gap/>pendicularis etiam ſuper <lb/>omnes:</s> <s xml:id="echoid-s1315" xml:space="preserve"> & ſunt ſphæricæ, nt exeant omnes ab uno puncto, quod eſt centrum illarũ:</s> <s xml:id="echoid-s1316" xml:space="preserve"> deinde di-<lb/>ſtent apud extremitates ſecundũ remotionem à centro:</s> <s xml:id="echoid-s1317" xml:space="preserve"> ita ut pyramis extenſa à centro, contineat <lb/>omnes perpendiculares exeuntes ab illa re uiſa, & diſtinguat ex ſuperficie uiſus & mẽbri ſentientis <lb/>partem, licet paruam, continentem tamen totam formam uenientem à re uiſa ad uiſum.</s> <s xml:id="echoid-s1318" xml:space="preserve"> Et ſi ſuper-<lb/>ficies tunicarum uiſus eſſent planæ, nõ ueniret forma uiſi ad uiſum ſuper perpendiculares, niſi eſſet <lb/>uiſus æqualis uiſo.</s> <s xml:id="echoid-s1319" xml:space="preserve"> Et nulla figura eſt, in qua adunantur perpendiculares, & cõcurrunt in unũ pun-<lb/>ctum, niſi figura ſphærica:</s> <s xml:id="echoid-s1320" xml:space="preserve"> & cum iſta diſpoſitione poſſunt exire à centro uiſus multæ pyramides ad <lb/>multa uiſa in eodem tempore:</s> <s xml:id="echoid-s1321" xml:space="preserve"> & quælibet illarũ diſtinguet partem paruam membri ſentientis, con <lb/>tinentem formam illius uiſi.</s> <s xml:id="echoid-s1322" xml:space="preserve"> Et omnes tunicæ habentidẽ centrum propter illud, quod diximus:</s> <s xml:id="echoid-s1323" xml:space="preserve"> & <lb/>eſt, ut perpendiculares exeuntes à re uiſa ad unam iſtarum, ſint perpendiculares ſuper omnes, & ut <lb/>pertranſeant etiã formæ omnes ſecundũ unam uerticationem.</s> <s xml:id="echoid-s1324" xml:space="preserve"> Quare uerò nihil comprehẽdat ui-<lb/>ſus ex rebus uiſibilibus, niſi ex uerticationibus iſtarũ perpendicularium tantùm:</s> <s xml:id="echoid-s1325" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s1326" xml:space="preserve"> quia per iſtas <lb/>perpendiculares tantùm ordinãtur partes rei uiſæ in ſuperficie mẽbri ſentiẽtis.</s> <s xml:id="echoid-s1327" xml:space="preserve"> Et hoc fuit iam ma-<lb/>niſeſtum antea [18 n] quoniã non poteſt ordinari forma rei uiſæ in ſuperficie membri ſentiẽtis, niſi <lb/>ſit receptio eius ad formã ex iſtis uerticationibus tãtùm.</s> <s xml:id="echoid-s1328" xml:space="preserve"> Et propter hoc appropriatur natura uiſus <lb/>iſta proprietate, & naturatur, ut non recipiat aliquã formam, niſi ſecundũ ſitum iſtarum uerticatio-<lb/>num tantùm.</s> <s xml:id="echoid-s1329" xml:space="preserve"> Et appropriatio uiſus, habita hac proprietate, eſt una rerũ ex quibus apparet maxima <lb/>diſcretio operatoris, & bonitas præparationis naturæ, præparãdo inſtrumẽta uiſus, & formam, per <lb/>quã eõpletur ſenſus, & per quã diſtinguũtur uiſibilia.</s> <s xml:id="echoid-s1330" xml:space="preserve"> Conſolidatiua autẽ cõtinet omnes iſtas tuni-<lb/>cas:</s> <s xml:id="echoid-s1331" xml:space="preserve"> & in ea eſt aliquid humiditatis, & pręterea habet aliquid retẽtionis, & eſt aliquãtulùm fortis.</s> <s xml:id="echoid-s1332" xml:space="preserve"> Et <lb/>cõtinet iſtas tunicas, ut cõgreget & cõſeruet illas:</s> <s xml:id="echoid-s1333" xml:space="preserve"> & eſt aliquãtulũ humida, ut præparẽtur loca tuni <lb/>carũ ex ea, & ut nõ accidat ſiccitas uelociter illis tunicis:</s> <s xml:id="echoid-s1334" xml:space="preserve"> & eſt aliquãtulùm retẽtiua & fortis, ut cõ-<lb/>ſeruet ſitus & figuras tunicarũ, ut nõ alterẽtur citò:</s> <s xml:id="echoid-s1335" xml:space="preserve"> & eſt alba, ut ſit per ipſam forma faciei pulchra.</s> <s xml:id="echoid-s1336" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div45" type="section" level="0" n="0"> <head xml:id="echoid-head65" xml:space="preserve" style="it">35. Ocul{us} eſt globoſ{us}. 3 p 3.</head> <p> <s xml:id="echoid-s1337" xml:space="preserve">ET totus oculus eſt rotũdus, quoniã rotunditas eſt melior figuris, & maior, & leuioris motus.</s> <s xml:id="echoid-s1338" xml:space="preserve"> <lb/>Oculus autẽ indiget motu, & uelocitate motus, ita, ut ſit oppoſitus per motũ multis uiſibili-<lb/>bus in eodẽ tẽpore, & ut ſit oppoſitus propter motũ omnibus partibus rei uiſæ, mediũ aſpi-<lb/>ciens, ita ut cõprehendat ipſum comprehenſione uera, & conſimili:</s> <s xml:id="echoid-s1339" xml:space="preserve"> quoniã ſenſus per mediũ mem-<lb/>bri ſentientis eſt manifeſtior.</s> <s xml:id="echoid-s1340" xml:space="preserve"> Et hoc declarabimus pòſt in loco cõueniente.</s> <s xml:id="echoid-s1341" xml:space="preserve"> Velocitas autẽ motus <lb/>uiſus eſt, ut aſpiciat omnes partes rei uiſæ, & uiſibilia ſibi oppoſita in modico tõpore.</s> <s xml:id="echoid-s1342" xml:space="preserve"> Palpebræ au-<lb/>tem ſunt, ut conſeruent oculum in ſomno, & ut faciant oculũ quieſcere, quãdo fatigatur à lumine, <lb/>quoniam luces fortes nocent oculis:</s> <s xml:id="echoid-s1343" xml:space="preserve"> & ſi continuè aperirentur oculi, ſupra modũ debilitarentur:</s> <s xml:id="echoid-s1344" xml:space="preserve"> & <lb/>hoc apparet, quando oculi aſpiciunt lucẽ fortem longo tempore.</s> <s xml:id="echoid-s1345" xml:space="preserve"> Et ſimiliter nocet uiſui aer, quan-<lb/>do in eo fuerit fumus, aut puluis.</s> <s xml:id="echoid-s1346" xml:space="preserve"> Palpebræ ergo cooperiunt oculũ à luce, quãdo indiget, & conſer-<lb/>uant ipſos ab aere, & abſtergunt ab eis multa nocumẽta:</s> <s xml:id="echoid-s1347" xml:space="preserve"> deinde quãdo fatigantur, ſuperponuntur <lb/>palpebræ eis, ita ut compleatur in eis ſua requies:</s> <s xml:id="echoid-s1348" xml:space="preserve"> & ſunt uelocis motus, ut citius ſuperponantur <lb/> <pb o="22" file="0028" n="28" rhead="ALHAZEN"/> oculis, dum appropinquant nocumenta oculis.</s> <s xml:id="echoid-s1349" xml:space="preserve"> Cilia autem ſunt ad temperandam quandam par-<lb/>tem lucis, quando dolebit uiſus propter fortitudinem lucis:</s> <s xml:id="echoid-s1350" xml:space="preserve"> & propter hoc adunat aſpiciens ocu-<lb/>lum, & conſtringit, ita ut poſsit aſpicere ab anguſto, quando lux fortis nocuerit ei.</s> <s xml:id="echoid-s1351" xml:space="preserve"> Iſta ergo, quæ di-<lb/>ximus, ſunt utilitates inſtrumentorum uiſus:</s> <s xml:id="echoid-s1352" xml:space="preserve"> ex quibus manifeſtatur magna diſcretio operatoris.</s> <s xml:id="echoid-s1353" xml:space="preserve"> <lb/>Sit ergo nomen eius benedictum, & bonitas præparationis naturæ.</s> <s xml:id="echoid-s1354" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div46" type="section" level="0" n="0"> <head xml:id="echoid-head66" xml:space="preserve">DE IIS SINE QVIBVS VISIO NON PO-<lb/>teſt compleri. Caput ſeptimum.</head> <head xml:id="echoid-head67" xml:space="preserve" style="it">36. Ad uiſionem perſiciendam ſex inprimis neceſſaria ſunt.</head> <p> <s xml:id="echoid-s1355" xml:space="preserve">IAm ergo declaratum eſt ſuperius, quòd uiſus nihil comprehendit ex rebus uiſis, quæ ſunt cum <lb/>eo in eodem aere, ita ut comprehenſio earum ab eo non ſit ſecundum refractionem, niſi quãdo <lb/>aggregatæ fuerint iſtæ res:</s> <s xml:id="echoid-s1356" xml:space="preserve"> & ſunt, ut ſit inter ea aliquid ſpatij:</s> <s xml:id="echoid-s1357" xml:space="preserve"> & ſit oppoſita uiſui illa res, ita ut <lb/>ſit inter quodlibet punctum eius ſuperficiei, quam comprehendit uiſus, & inter aliud punctum ſu-<lb/>perficiei uiſus, linea recta imaginabilis:</s> <s xml:id="echoid-s1358" xml:space="preserve"> & ut ſit in ea lux:</s> <s xml:id="echoid-s1359" xml:space="preserve"> & ut ſit corpus eius aliquãtulum, in reſpe-<lb/>ctu uirtutis ſenſus uiſus:</s> <s xml:id="echoid-s1360" xml:space="preserve"> & ut ſit aer medius diaphanus, cõtinuæ diaphanitatis, & nõ ſit in eo alιud <lb/>corpus non diaphanum:</s> <s xml:id="echoid-s1361" xml:space="preserve"> & ut ſit res uiſa reſiſtens uiſui, ſcilicet ut non ſit in ea diaphanitas, aut ſi ſit, <lb/>ſit ſpiſsior diaphanitate aeris medij inter ipſam & uiſum.</s> <s xml:id="echoid-s1362" xml:space="preserve"> Viſus autem non comprehendet rem ui-<lb/>ſam, niſi quando aggregabuntur iſtæ ſex intentiones:</s> <s xml:id="echoid-s1363" xml:space="preserve"> & ſi res uiſa caruerit una iſtarum intentionũ, <lb/>non comprehendetur à uiſu.</s> <s xml:id="echoid-s1364" xml:space="preserve"> Indigentia autem uiſus ab unaquaque iſtarum intentionum, non eſt <lb/>niſi propter aliquam cauſſam.</s> <s xml:id="echoid-s1365" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div47" type="section" level="0" n="0"> <head xml:id="echoid-head68" xml:space="preserve" style="it">37. Diſt antia inter uiſum & uiſibile. 15 p 3.</head> <p> <s xml:id="echoid-s1366" xml:space="preserve">QVare ergo non comprehendat uiſus rem uiſam, niſi quando inter ea fuerit diſtantia aliqua, <lb/>& non comprehendatipſam, quando applicatur ei, eſt propter duas cauſſas.</s> <s xml:id="echoid-s1367" xml:space="preserve"> Quarum una <lb/>eſt, quia uiſus non comprehendit rem uiſam, niſi quando in ea fuerit lux aliqua, [per 3 n] & <lb/>quando fuerit applicata uiſui, & non fuerit illuminata per ſe, non erit in ſua ſuperficie uicinante ui-<lb/>ſui lux:</s> <s xml:id="echoid-s1368" xml:space="preserve"> quoniã corpus oculi ſecundũ ſitum ſuum tunc prohibetur à uiſu.</s> <s xml:id="echoid-s1369" xml:space="preserve"> Res autẽ luminoſæ per ſe <lb/>non poſſunt applicari ſuperſiciei uiſus:</s> <s xml:id="echoid-s1370" xml:space="preserve"> quoniam res illumιnatæ per ſe ſunt ſtellæ, & ignis, quæ non <lb/>poſſunt applicari ſuperficiei uiſus.</s> <s xml:id="echoid-s1371" xml:space="preserve"> Cauſſa autem ſecunda eſt, quia uiſio nõ fit, niſi ex parte oppoſita <lb/>foramini uueæ ex medio ſuperficiei uiſus [per 4 n] & ſi res uiſa applicetur uiſui, nõ ſuperponetur <lb/>iſti parti uiſui, niſi pars æqualis illi tantùm ex re uiſa:</s> <s xml:id="echoid-s1372" xml:space="preserve"> & ſi uiſus comprehenderet rem uiſam per ap-<lb/>plicationem, non comprehenderet, niſi partem applicatam parti oppoſitæ foramini tantùm, & non <lb/>comprehenderet reſiduum rei uiſæ.</s> <s xml:id="echoid-s1373" xml:space="preserve"> Et ſi moueatur res uiſa ſuper ſuperficiem uiſus, quouſq;</s> <s xml:id="echoid-s1374" xml:space="preserve"> cõtin-<lb/>gat totam ſuperficiem rei uiſæ ſecundũ partem mediam uiſus, comprehendet partem poſt partem <lb/>aliam, & dum comprehendet partem ſecundam non comprehendet partem primam:</s> <s xml:id="echoid-s1375" xml:space="preserve"> & ſic non po <lb/>terit comprehendere totam rem uiſam ſimul.</s> <s xml:id="echoid-s1376" xml:space="preserve"> Et cum ita ſit, non figurabitur in eo forma rei uiſæ:</s> <s xml:id="echoid-s1377" xml:space="preserve"> ita <lb/>ut ſi aliqua res uiſa eſſet ſuper corpus denſum, & eſſet in illo corpore denſo foramen minoris quan-<lb/>titatis re uiſa, & res uiſa eſſet applicata foramini, non comprehẽderet ex ea, niſi partem ſuppoſitam <lb/>foramini tantùm:</s> <s xml:id="echoid-s1378" xml:space="preserve"> deinde ſi res uiſa moueatur ſuper foramen, quouſq;</s> <s xml:id="echoid-s1379" xml:space="preserve"> comprehendatur à uiſu pars <lb/>poſt aliam, non figuratur in uiſu tota forma eius.</s> <s xml:id="echoid-s1380" xml:space="preserve"> Si ergo uiſio eſſet per tactum, nõ comprehenderet <lb/>uiſus totam rem uiſam, neq;</s> <s xml:id="echoid-s1381" xml:space="preserve"> figuram & formam eius, niſi eſſet res uiſa æqualis parti mediæ ſuperfi-<lb/>ciei uiſus, per quam erit uiſio:</s> <s xml:id="echoid-s1382" xml:space="preserve"> neq;</s> <s xml:id="echoid-s1383" xml:space="preserve"> etiam ſic poteſt comprehẽdere multas res uiſas in eodem tem-<lb/>pore.</s> <s xml:id="echoid-s1384" xml:space="preserve"> Et cum inter uiſum & rem uiſam fuerit aliquod ſpatium, poterit rem uiſam comprehẽdere in <lb/>eodem tempore totam ex parte parua, quamuis ſit res uiſa magna:</s> <s xml:id="echoid-s1385" xml:space="preserve"> & poteſt comprehẽdere res ui-<lb/>ſas multas ſimul in eodem tempore:</s> <s xml:id="echoid-s1386" xml:space="preserve"> & cum res uiſa fuerit remota à uiſu, erit poſsibile oriri lucem <lb/>ſuper ſuperficiem uiſus oppoſitam uiſui.</s> <s xml:id="echoid-s1387" xml:space="preserve"> Propter iſtas igitur duas cauſſas non comprehendit uiſus <lb/>quicquam ex rebus uiſibilibus, niſi ſit inter ea aliquod ſpatium.</s> <s xml:id="echoid-s1388" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div48" type="section" level="0" n="0"> <head xml:id="echoid-head69" xml:space="preserve" style="it">38. Collocatio uiſibilis ante uiſum directa. 2 p 3.</head> <p> <s xml:id="echoid-s1389" xml:space="preserve">QVare uerò non comprehendat uiſus rem uiſam, quæ eſt cum eo in eodem aere, & in parte <lb/>oppoſita illi, niſi ſit inter quodlibet punctum eius, & aliquod punctũ partis ſuperficiei ui-<lb/>ſus, per quam erit uiſio, linea recta:</s> <s xml:id="echoid-s1390" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s1391" xml:space="preserve"> quia declaratum eſt, quòd uiſio non ſit, niſi ex formis <lb/>uenientibus à re uiſa ad uiſum, & quòd formæ non comprehendantur, niſi ſecundum lineas rectas:</s> <s xml:id="echoid-s1392" xml:space="preserve"> <lb/>[per 14 n] & propter hanc cauſſam non comprehendit uiſus rem, niſi ſit inter ea linea recta.</s> <s xml:id="echoid-s1393" xml:space="preserve"> Et ſi ſe-<lb/>cuerint corpora denſa media omnes lineas, quæ ſunt inter ea, latebunt res uiſæ uiſum:</s> <s xml:id="echoid-s1394" xml:space="preserve"> & ſi ſecuerit <lb/>illud corpus quaſdam illarum linearum rectarum, latebit uiſum quædam pars, quæ eſt apud extre-<lb/>mitatem linearum reſectarum per corpus denſum.</s> <s xml:id="echoid-s1395" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div49" type="section" level="0" n="0"> <head xml:id="echoid-head70" xml:space="preserve" style="it">39. Lux. 1 p 3.</head> <p> <s xml:id="echoid-s1396" xml:space="preserve">QVare uerò uiſus nõ cõprehendat rẽ uiſam, niſi ſit in ea lux, eſt ꝓpter duas cauſſas:</s> <s xml:id="echoid-s1397" xml:space="preserve"> aut ꝗafor <lb/>mæ uiſæ nõ extẽdũtur in aere, niſi ſit lux cũ colore, [ք 3 n] aut ꝗa forma coloris extẽditur in <lb/>aere, quamuis nõ ſit cũ ea lux:</s> <s xml:id="echoid-s1398" xml:space="preserve"> ſed nõ operatur in uiſum operatione ſenſibili, niſi per lucẽ.</s> <s xml:id="echoid-s1399" xml:space="preserve"> Et <lb/>manifeſtũ eſt, quod forma lucis manifeſtior eſt, forma coloris, & quòd lux oքatur oքatione manife <lb/>ſtiore:</s> <s xml:id="echoid-s1400" xml:space="preserve"> & quòd forma coloris, ꝗa eſt debilis, nõ poteſt operari in uiſum, ſicut operatur lux.</s> <s xml:id="echoid-s1401" xml:space="preserve"> Et forma <lb/> <pb o="23" file="0029" n="29" rhead="OPTICAE LIBER I."/> coloris, quæ eſt in corpore illuminato, ſemper eſt admixta cum forma lucis, & cum peruenerit ad <lb/>uiſum, ſemper operatur in ipſum per ſuam fortitudinem & præparationem uiſus, ut patiatur ex ea:</s> <s xml:id="echoid-s1402" xml:space="preserve"> <lb/>& quia admiſcetur cum forma coloris, & non diſtinguitur ab ea, non ſentit uiſus formam lucis, niſi <lb/>admixtam cum forma coloris.</s> <s xml:id="echoid-s1403" xml:space="preserve"> Viſus ergo non ſentit colorem rei uiſæ, niſi ex colore admixto cum <lb/>forma lucis ueniente ad<gap/>ipſum ex re uiſa:</s> <s xml:id="echoid-s1404" xml:space="preserve"> & propter hoc alterantur colores multarum rerũ uiſarum <lb/>apud uiſum per alterationem lucis ſuper ipſas.</s> <s xml:id="echoid-s1405" xml:space="preserve"> Quia ergo forma coloris non operatur in uiſum, niſi <lb/>admixta ſit cum lumine, & non eſt ex colore forma niſi ſit in ea lux:</s> <s xml:id="echoid-s1406" xml:space="preserve"> nihil comprehendit uiſus exre-<lb/>bus uiſibilibus, niſi quando in eis fuerit aliqua lux.</s> <s xml:id="echoid-s1407" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div50" type="section" level="0" n="0"> <head xml:id="echoid-head71" xml:space="preserve" style="it">40. Magnitudo rei uiſibilis. 19 p 3.</head> <p> <s xml:id="echoid-s1408" xml:space="preserve">QVare uerò non comprehendat uiſus rem uiſam, niſi ſit corpus eius in aliqua quantitate:</s> <s xml:id="echoid-s1409" xml:space="preserve"> eſt;</s> <s xml:id="echoid-s1410" xml:space="preserve"> <lb/>quia declaratũ eſt, [19 n] quòd forma rei uiſæ non perueniat ad uiſum, niſi ex pyramidibus, <lb/>quarum caput eſt cẽtrum uiſus, & baſis ſuperficies rei uiſæ, & quòd iſta pyramis diſtinguat <lb/>ex ſuperficie mẽbri ſentientis paruam partẽ, in qua ordinatur forma rei uiſæ:</s> <s xml:id="echoid-s1411" xml:space="preserve"> & ſi res uiſa fuerit ual-<lb/>de parua, erit pyramis, quæ eſt inter ipſam & centrum uiſus, ualde parua.</s> <s xml:id="echoid-s1412" xml:space="preserve"> Erit ergo pars diſtincta ex <lb/>mẽbro ſentiente, quaſi punctũ, ualde parua:</s> <s xml:id="echoid-s1413" xml:space="preserve"> ſed ſentiens nõ ſentit formã, niſi quãdo pars ſuæ ſuper-<lb/>ficiei, ad quã peruenit forma, fuerit quantitatis ſenſibilis, reſpectu totius apud totũ membrũ.</s> <s xml:id="echoid-s1414" xml:space="preserve"> Et uir <lb/>tutes ſenſus etiã ſunt finitæ.</s> <s xml:id="echoid-s1415" xml:space="preserve"> Et cum pars membri ſentientis, ad quam peruenit forma, non eſt quan <lb/>titatis ſenſibilis apud totũ membrum ſentiens, non ſentiet paſsionem, quæ ìlli accidit, propter par-<lb/>uitatem ipſius.</s> <s xml:id="echoid-s1416" xml:space="preserve"> Quare non comprehendit formam.</s> <s xml:id="echoid-s1417" xml:space="preserve"> Res ergo uiſa, quæ poteſt comprehẽdi à uiſu, eſt <lb/>illa, in qua pyramis, quæ figuratur inter rem uiſam & cẽtrum uiſus, diſtinguet ex ſuperficie glacialis <lb/>partem quãtitatis ſenſibilis reſpectu totius ſuperficiei glacialis.</s> <s xml:id="echoid-s1418" xml:space="preserve"> Et iſte ſenſus erit ſecundũ tantum, <lb/>ad quantum peruenit uirtus ſenſitiua, & non extenditur ad infinitum, & diuerſatur ſecundũ diuer-<lb/>ſitatem uirtutis oculi.</s> <s xml:id="echoid-s1419" xml:space="preserve"> Et cũ pyramis, quæ figuratur inter rem uiſam & centrum uiſus, diſtin xerit ex <lb/>ſuperficie glacialis partem quantitatis inſenſibilis, reſpectu totius ſuperficiei glacialis, non poteſt <lb/>uiſus comprehendere illam rem.</s> <s xml:id="echoid-s1420" xml:space="preserve"> Et propter hoc non comprehendet uiſus rem ualde paruam.</s> <s xml:id="echoid-s1421" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div51" type="section" level="0" n="0"> <head xml:id="echoid-head72" xml:space="preserve" style="it">41. Perſpicuit{as} corporis inter uiſum & uiſibile interiecti. 13 p 3.</head> <p> <s xml:id="echoid-s1422" xml:space="preserve">QVare uerò uiſus non comprehendat rem uiſam, niſi quando corpus medium inter ipſum <lb/>uiſum & rem uiſam fuerit diaphanum:</s> <s xml:id="echoid-s1423" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s1424" xml:space="preserve"> quia uiſio non eſt niſi ex forma ueniente ex re ui-<lb/>ſa ad uiſum [per 14 n] formæ autem non extenduntur niſi in corporibus diaphanis, & ui-<lb/>ſio non completur, quando res uiſa fuerit cum uiſu in eodem aere, & fuerit comprehenſio non ſe-<lb/>cundum refractionem, niſi quando aer fuerit continuus inter rem uiſam, & nõ abſciderit rectas li-<lb/>neas, quæ ſunt inter ea, corpus denſum:</s> <s xml:id="echoid-s1425" xml:space="preserve"> quoniam forma non extenditur in aere conſimilis diapha-<lb/>nitatis, niſi ſecundum lineas rectas.</s> <s xml:id="echoid-s1426" xml:space="preserve"> Et propter hoc uiſus nõ comprehendit rem uiſam, quæ eſt cum <lb/>eo in eodem aere, & in parte oppoſita uiſui, niſi quando aer medιus inter ea fuerit diaphanus, con-<lb/>ſimilis diaphanitatis.</s> <s xml:id="echoid-s1427" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div52" type="section" level="0" n="0"> <head xml:id="echoid-head73" xml:space="preserve" style="it">42. Denſit{as} ac ſolidit {as} uiſibilis. 14 p 3.</head> <p> <s xml:id="echoid-s1428" xml:space="preserve">QVare uerò uiſus non comprehendat uiſam rem, niſi quãdo in ea fuerit denſitas, aut aliquid <lb/>denſitatis:</s> <s xml:id="echoid-s1429" xml:space="preserve"> eſt propter duas cauſſas:</s> <s xml:id="echoid-s1430" xml:space="preserve"> quarũ altera eſt, quia quod eſt denſum, eſt coloratum, <lb/>& ex colore uenit forma ad uiſum, ex qua comprehendit uiſus colorẽ rei uiſæ:</s> <s xml:id="echoid-s1431" xml:space="preserve"> quod autem <lb/>eſt in fine diaphanitatis, caret colore:</s> <s xml:id="echoid-s1432" xml:space="preserve"> quare non comprehendetur à uiſu.</s> <s xml:id="echoid-s1433" xml:space="preserve"> Et cauſſa ſecũda eſt, quo-<lb/>niam uiſus non comprehendit rem uiſam, niſi ſit illuminata, & ueniat ex luce, quæ eſt in ea, forma <lb/>fecunda ad uiſum cum forma coloris, & non erit forma ſecunda ex luce oriente ſuper aliquod cor-<lb/>pus, niſi figatur lux in illo corpore, ſuper quod oritur:</s> <s xml:id="echoid-s1434" xml:space="preserve"> ergo cũ lux fuerit fixa in corpore illo, erit ex <lb/>eo forma ſecunda:</s> <s xml:id="echoid-s1435" xml:space="preserve"> & quando lux orietur ſuper corpus diaphanũ ualde, non figetur in eo, ſed exten <lb/>detur in ſua diaphanitate.</s> <s xml:id="echoid-s1436" xml:space="preserve"> Cum ergo corpus diaphanum fuerit oppoſitũ uiſui, & ſuper ipſum oritur <lb/>lux ex parte, in qua eſt uiſus, in eo extendetur, & non figetur in ſua ſuperficie:</s> <s xml:id="echoid-s1437" xml:space="preserve"> & ſic nõ erit in ſuper-<lb/>ficie oppoſita uiſui iſtius corporis lux, ex qua uenit forma ad uiſum.</s> <s xml:id="echoid-s1438" xml:space="preserve"> Et ſi fuerit illud illuminatum, <lb/>cuius lux oritur ſuper illud corpus diaphanum, oppoſitum uiſui, pertrãſibit lux eius in corpus dia-<lb/>phanum, & perueniet ad uiſum, & nihil deferet ſecũ ad uiſum ex colore corporis diaphani:</s> <s xml:id="echoid-s1439" xml:space="preserve"> quoniã <lb/>corpus diaphanum, quod eſt in fine diaphanitatis, nõ habet colorem.</s> <s xml:id="echoid-s1440" xml:space="preserve"> Viſus ergo comprehendet ex <lb/>illo loco corpus illuminatum, cuius lux oritur ſuper corpus diaphanum, poſt corpus diaphanum:</s> <s xml:id="echoid-s1441" xml:space="preserve"> <lb/>& non comprehendet corpus diaphanũ propter hoc:</s> <s xml:id="echoid-s1442" xml:space="preserve"> quia non comprehẽdit uiſus rem uiſam, quæ <lb/>eſt in fine diaphanitatis.</s> <s xml:id="echoid-s1443" xml:space="preserve"> Et cũ diaphanitas corporis fuerit ſimilis diaphanitati aeris, erit eius diſpo <lb/>ſitio, ſicut diſpoſitio aeris, & nõ comprehẽdetur à uiſu, ſicut nec aer.</s> <s xml:id="echoid-s1444" xml:space="preserve"> Et corpora diaphana, quorum <lb/>diaphanitas non eſt ſpiſsior diaphanitate aeris, non comprehendẽtur à uiſu:</s> <s xml:id="echoid-s1445" xml:space="preserve"> quoniam nulla forma <lb/>uenit ex eis ad uiſum, quæ poſsit operari in uiſum.</s> <s xml:id="echoid-s1446" xml:space="preserve"> Et ſimiliter accidit ſi inter uiſum & rem uiſam <lb/>fuerit medium corpus diaphanum præter aerem, & fuerit diaphanitas rei uiſæ nõ ſpiſsior diapha-<lb/>nitate corporis medij.</s> <s xml:id="echoid-s1447" xml:space="preserve"> Et cum res uiſa fuerit denſa, erit colorata, & cum ſuper ipſam oritur lux, fige-<lb/>tur in ſua ſuperficie, & erit ex colore eius, & ex luce, quæ oritur ſuper ipſam, forma, quæ extendi-<lb/>tur in aere, & in corporibus diaphanis:</s> <s xml:id="echoid-s1448" xml:space="preserve"> & cum iſta forma peruenerit ad ipſum uiſum, operabitur <lb/>in eo, & ex ea ſentiet uiſus rem uiſam.</s> <s xml:id="echoid-s1449" xml:space="preserve"> Et cum res uiſa fuerit diaphana, ſed minus quàm aer:</s> <s xml:id="echoid-s1450" xml:space="preserve"> habe-<lb/>bit colorem ſecundum ſuam ſpiſsitudinem:</s> <s xml:id="echoid-s1451" xml:space="preserve"> & cum aer ſuper ipſam oritur, lux figetur in éa aliqua <lb/> <pb o="24" file="0030" n="30" rhead="ALHAZEN"/> fixione, ſecundum illud, quod eſt in ea de ſpiſsitudine, & pertranſibit in ea ſecũdum ſuam diapha-<lb/>nitatem, & erit ex ea, forma in aere ſecundum colorem & lucem, quæ ſunt in ſua ſuperficie:</s> <s xml:id="echoid-s1452" xml:space="preserve"> & cum <lb/>illa forma peruenerit ad uiſum, operabitur in uiſum, & ſentiet uiſus illã rem uiſam.</s> <s xml:id="echoid-s1453" xml:space="preserve"> Et propter iſtam <lb/>cauſſam non comprehendit uiſus ex rebus uiſibilibus, niſi quãdo ipſum uiſibile fuerit denſum, aut <lb/>fuerit in eo aliquid denſitatis.</s> <s xml:id="echoid-s1454" xml:space="preserve"> Iam ergo declaratę ſunt cauſſæ, propter quas nihil comprehendit ui-<lb/>ſus, niſi quãdo fuerint aggregatæ intentiones prædictæ.</s> <s xml:id="echoid-s1455" xml:space="preserve"> Et hoc, quod declarauimus, eſt illud, quod <lb/>intendimus declarare in iſto tractatu.</s> <s xml:id="echoid-s1456" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div53" type="section" level="0" n="0"> <head xml:id="echoid-head74" xml:space="preserve">ALHAZEN FILII</head> <head xml:id="echoid-head75" xml:space="preserve">ALHAYZEN OPTICAE</head> <head xml:id="echoid-head76" xml:space="preserve">LIBER SECVNDVS.</head> <p style="it"> <s xml:id="echoid-s1457" xml:space="preserve">DECLARATVM eſt qualiter fiat uiſio:</s> <s xml:id="echoid-s1458" xml:space="preserve"> & eſt qualit{as} ſenſus uiſ{us} à forma <lb/>lucis & coloris, quæ ſunt in re uiſa, ordinatorum ita, ſicut ſunt in ſuperficie <lb/>rei uiſæ.</s> <s xml:id="echoid-s1459" xml:space="preserve"> Viſ{us} autem comprehendit ex reb{us} uiſibilib{us} mult{as} intentio-<lb/>nes præter lucem & colorem.</s> <s xml:id="echoid-s1460" xml:space="preserve"> Et etiam declaratum eſt in primo tractatu <lb/>[18 n] quòd uiſio non ſit, niſi ex uerticationib{us} linearum radialium:</s> <s xml:id="echoid-s1461" xml:space="preserve"> & lineæ radia-<lb/>les diuerſentur in ſuis diſpoſitionib{us}:</s> <s xml:id="echoid-s1462" xml:space="preserve"> & ſimiliter diuerſantur diſpoſitiones formarum <lb/>uenientium ſuper ipſas ad uiſum.</s> <s xml:id="echoid-s1463" xml:space="preserve"> Et etiam comprehenſio uiſ{us} à re uiſa nõ eſt in omni-<lb/>b{us} corporib{us}, & in omnib{us} uiſibilib{us}:</s> <s xml:id="echoid-s1464" xml:space="preserve"> ſed diuerſatur qualit{as} ſenſ{us} uiſ{us} à reb{us} <lb/>uiſibilib{us}:</s> <s xml:id="echoid-s1465" xml:space="preserve"> & diuerſatur qualit{as} ſenſ{us} uiſ{us} ab una re uiſa ſecundum ſitum unum, <lb/>& ſecundum eandem dιſtantiam.</s> <s xml:id="echoid-s1466" xml:space="preserve"> Et nos diuidem{us} iſtum tractatum in tria capita.</s> <s xml:id="echoid-s1467" xml:space="preserve"> <lb/>In primo declarabim{us} diuerſitatem diſpoſitionum linearũ radialium, & diſtingue-<lb/>m{us} proprietates earum.</s> <s xml:id="echoid-s1468" xml:space="preserve"> In ſecundo declarabim{us} omnes intentiones comprehenſas à <lb/>uiſu, & qualiter comprehendat uiſ{us} quamlibet illarum.</s> <s xml:id="echoid-s1469" xml:space="preserve"> In tertio declarabim{us} diuer-<lb/>ſitatem comprehenſionis uιſ{us} ab eis.</s> <s xml:id="echoid-s1470" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div54" type="section" level="0" n="0"> <head xml:id="echoid-head77" xml:space="preserve">DE DIVERSITATE DISPOSITIONVM LINEARVM <lb/>radialium, & diſtinctione proprietatum ipſarum. <lb/>Caput primum.</head> <head xml:id="echoid-head78" xml:space="preserve" style="it">1. Recta connectens centra partium uiſ{us}, eſt axis pyramidis opticæ. 18 p 3.</head> <p> <s xml:id="echoid-s1471" xml:space="preserve">IAm declaratum eſt in primo tractatu [18.</s> <s xml:id="echoid-s1472" xml:space="preserve">20 n] quòd lineæ radiales, ex quarum uerticationibus <lb/>comprehendit uiſus uiſibilia, ſunt lineæ rectæ, quarum extremitates concurrunt apud centrum <lb/>uiſus.</s> <s xml:id="echoid-s1473" xml:space="preserve"> Et iam declaratum eſt in forma uiſus [4 n 1] quòd membrum ſentiẽs, quod eſt membrum <lb/>glacialis, eſt compoſitum ſuper extremitatem concauitatis nerui, ſuper quem compoſitus eſt ocu-<lb/>lus totus:</s> <s xml:id="echoid-s1474" xml:space="preserve"> & quòd iſte neruus non gyratur niſi à poſteriori centri uiſus, & à poſteriori totius oculi, <lb/>& apud foramen, quod eſt in concauo oſsis.</s> <s xml:id="echoid-s1475" xml:space="preserve"> Et iam declaratum eſt [7.</s> <s xml:id="echoid-s1476" xml:space="preserve">9 n 1] quòd linea recta tran-<lb/>ſiens per omnia centra tunicarum uiſus, extenditur in medio concaui nerui, & tranſit per medium <lb/>foraminis, quod eſt in anteriori uueæ.</s> <s xml:id="echoid-s1477" xml:space="preserve"> Et iam declaratum eſt [5.</s> <s xml:id="echoid-s1478" xml:space="preserve">13 n 1] quòd centrũ iſtius lineæ non <lb/>diuerſatur reſpectu totius uiſus, neq;</s> <s xml:id="echoid-s1479" xml:space="preserve"> reſpectu ſuperficierum tunicarum uiſus, neque reſpectu par-<lb/>tium uiſus.</s> <s xml:id="echoid-s1480" xml:space="preserve"> Linea ergo recta tranſiens per omnia centra tunicarum uiſus, ſemper extenditur rectè <lb/>ad locum gyrationis concaui nerui, ſuper quem componitur oculus, in omnibus diſpoſitionibus, <lb/>ſiue ſit uiſus in motu, ſiue in quiete.</s> <s xml:id="echoid-s1481" xml:space="preserve"> Et quia iſta linea tranſit per centrum uiſus, & per centrum fo-<lb/>raminis, quod eſt in anteriori uueæ, & per centrũ uueæ extenditur in medio pyramidis, cuius cen-<lb/>trum eſt uiſus:</s> <s xml:id="echoid-s1482" xml:space="preserve"> & continet ipſam circumferentia foraminis, quod eſt in anteriori uueæ:</s> <s xml:id="echoid-s1483" xml:space="preserve"> appellemus <lb/>ergo iſtam lineam axem pyramidis.</s> <s xml:id="echoid-s1484" xml:space="preserve"> Et declaratum eſt etiam in ipſo tractatu primo [19 n] quòd py-<lb/>ramis figurata inter rem uiſam & centrum uiſus, diſtinguit ex ſuperficie glacialis partem continen <lb/>tem totam formam rei uiſæ, quæ eſt apud baſim illius pyramidis:</s> <s xml:id="echoid-s1485" xml:space="preserve"> & erit forma ordinata in iſta parte <lb/>ſuperficiei glacialis per uerticationem linearum radialium extenſarum inter rem uiſam & uiſum, <lb/>ſecundum ordinationem partium ſuperficiei rei uiſæ.</s> <s xml:id="echoid-s1486" xml:space="preserve"> Cum ergo uiſus comprehenderit aliquã rem <lb/>uiſam, & peruenerit eius forma in partem ſuperficiei glacialis, quam diſtinguit pyramis prædicta:</s> <s xml:id="echoid-s1487" xml:space="preserve"> <lb/>quodlibet punctum formæ prædictæ eſt ſuper lineam radialem extenſam inter illud punctum, & <lb/>punctum oppoſitum illi in ſuperficie rei uiſæ, ſuper quam uenit forma ad illud punctum in ſuperfi-<lb/>ciem glacialis rectè.</s> <s xml:id="echoid-s1488" xml:space="preserve"> Cum ergo forma rei uiſæ fuerit in medio ſuperficiei glacialis, erit axis prædi-<lb/>ctus una linearum, ſuper quas ueniunt formæ punctorum, quæ ſunt in ſuperficie rei uiſæ:</s> <s xml:id="echoid-s1489" xml:space="preserve"> & erit <lb/> <pb o="25" file="0031" n="31" rhead="OPTICAE LIBER II."/> punctum ſuperficiei rei uiſæ, quod eſt apud extremitatem iſtius axis, illud, ſuper quod uenit forma <lb/>eιus ſuper iſtum axem.</s> <s xml:id="echoid-s1490" xml:space="preserve"> Et declaratũ eſt in primo tractatu [26 n] quòd formæ, quæ comprehendun <lb/>tur per uiſum, extenduntur in corpore glacialis, & in concauo nerui, ſuper quem componitur ocu-<lb/>lus, & perueniunt ad neruum communem, qui eſt apud medium interioris cerebri, & illic eſt com-<lb/>prehenſio ſentientis ultimò à formis rerum uiſibilium:</s> <s xml:id="echoid-s1491" xml:space="preserve"> & quòd uiſio non completur, niſi per aduen <lb/>tum formæ ad neruum communem:</s> <s xml:id="echoid-s1492" xml:space="preserve"> & quòd extenſio formarum à ſuperficie glacialis intra corpus <lb/>glacialis, eſt ſecundum rectitudinem linearum rectarum radialium tantum:</s> <s xml:id="echoid-s1493" xml:space="preserve"> quoniam glacialis non <lb/>recipit iſtas formas, niſi ſecun dum uerticationem linearum radialium tantùm.</s> <s xml:id="echoid-s1494" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div55" type="section" level="0" n="0"> <head xml:id="echoid-head79" xml:space="preserve" style="it">2. Cryſtallin{us} & uitre{us} humores perſpicuitate differunt. Ita forma uiſibilis <lb/>refringitur in ſuperſicie uitrei humoris. 21 p 3.</head> <p> <s xml:id="echoid-s1495" xml:space="preserve">ET ultimum ſentiens non comprehendit ſitus partium rei uiſæ, niſi ſecundum ſuum ſitum in <lb/>ſuperficie rei uiſæ.</s> <s xml:id="echoid-s1496" xml:space="preserve"> Et cum ſitus partium formæ inter ſe ſcilicet formæ peruenientis ad ſuper-<lb/>ficiem glacialis, ſint ſitus partium ſuperficiei rei uiſæ inter ſe [per 18 n 1] & iſtæ formæ exten <lb/>dantur, ſicut prædictum eſt:</s> <s xml:id="echoid-s1497" xml:space="preserve"> & cum omnia iſta ita ſint:</s> <s xml:id="echoid-s1498" xml:space="preserve"> uiſio ergo non complebitur, niſi poſt aduen-<lb/>tum formæ, quæ eſt in ſuperficie glacialis, ad neruum communem, & ſitus partium eius ſecundum <lb/>ſuum eſſe in ſuperficiem glacialis ſine aliqua admixtione.</s> <s xml:id="echoid-s1499" xml:space="preserve"> Forma autem non peruenit à ſuperficie <lb/>glacialis ad neruum communem, niſi per extenſionem eius in concauo nerui, ſuper quem compo-<lb/>nitur oculus ſiue humor glacialis.</s> <s xml:id="echoid-s1500" xml:space="preserve"> Si ergo forma nõ perueniat in cõcauum iſtius nerui ſecundũ ſuũ <lb/>eſſe in glaciali, neq;</s> <s xml:id="echoid-s1501" xml:space="preserve"> etiã perueniet ad neruum communẽ ſecundum ſuũ eſſe.</s> <s xml:id="echoid-s1502" xml:space="preserve"> Forma autẽ nõ poteſt <lb/>extendi à ſuperficie glacialis ad concauum nerui ſecundum rectitudinem linearũ rectarum, & con <lb/>ſeruare ſitus partium ſecundũ eſſe ſuum:</s> <s xml:id="echoid-s1503" xml:space="preserve"> quoniam omnes illæ lineæ concurrunt apud centrum ui-<lb/>ſus, & quando fuerint extenſæ ſecundum rectitudinem, poſt centrum conuertetur ſitus earum, & <lb/>quod eſt dextrũ, efficietur ſiniſtrum, & è contrario, & ſuperius inferius, & inferius ſuperius.</s> <s xml:id="echoid-s1504" xml:space="preserve"> Si ergo <lb/>forma fuerit extenſa ſecundum rectitudinem linearum radialium, cõgregabitur apud centrum ui-<lb/>ſus, & efficietur quaſi unum punctum.</s> <s xml:id="echoid-s1505" xml:space="preserve"> Et quia centrum uiſus eſt in medio totius oculi, & ante locũ <lb/>gyrationis concaui nerui:</s> <s xml:id="echoid-s1506" xml:space="preserve"> ſi forma fuerit extenſa à centro oculi, & ipſius unũ punctum ſuper unam <lb/>lineam:</s> <s xml:id="echoid-s1507" xml:space="preserve"> perueniet ad locum gyrationis, & ipſius unum punctum:</s> <s xml:id="echoid-s1508" xml:space="preserve"> & ſic non perueniet formatota ad <lb/>locum gyrationis:</s> <s xml:id="echoid-s1509" xml:space="preserve"> quia non niſi unum punctum, ſcilicet, quod eſt in extremitate axis pyramidis.</s> <s xml:id="echoid-s1510" xml:space="preserve"> Et <lb/>ſi fuerit extenſa ſecundum rectitudinem linearum radialium, & pertranſierit per centrum:</s> <s xml:id="echoid-s1511" xml:space="preserve"> erit con <lb/>uerſa ſecundum conuerſionem linearum ſe ſecantium, ſuper quas extendebatur.</s> <s xml:id="echoid-s1512" xml:space="preserve"> Non poteſt ergo <lb/>forma peruenire à ſuperficie glacialis ad cõcauum nerui, ita ut ſitus partium ſit ſecundũ ſuum eſſe:</s> <s xml:id="echoid-s1513" xml:space="preserve"> <lb/>non poteſt ergo forma peruenire à ſuperficie glacialis ad concauum nerui, niſi ſecundum lineas re-<lb/>fractas, ſecantes lineas radiales.</s> <s xml:id="echoid-s1514" xml:space="preserve"> Et cũ ita ſit, uiſio ergo non complebitur, niſi poſtquam refracta fue-<lb/>rit forma, quæ peruenit à ſuperficie glacialis, & extenditur ſuper lineas ſecantes lineas radiales.</s> <s xml:id="echoid-s1515" xml:space="preserve"> Iſta <lb/>ergo refractio debet eſſe ante peruentum ad centrum:</s> <s xml:id="echoid-s1516" xml:space="preserve"> quoniam ſi fuerint refractæ poſt tranſitũ cen <lb/>tri, erunt conuerſæ.</s> <s xml:id="echoid-s1517" xml:space="preserve"> Et iam declaratum eſt [18 n 1] quòd iſta forma pertranſeat in corpore glacialis <lb/>ſecundum rectitudinem linearum radialium:</s> <s xml:id="echoid-s1518" xml:space="preserve"> & cum non poſsit peruenire ad concauum nerui, niſi <lb/>poſtquam refracta fuerit ſuper lineas ſecantes lineas radiales:</s> <s xml:id="echoid-s1519" xml:space="preserve"> forma ergo non refringitur, niſi per <lb/>tranſitum eius in corpore glacialis.</s> <s xml:id="echoid-s1520" xml:space="preserve"> Et iam prædictum eſt [4 n 1] in forma uiſus, quòd corpus gla-<lb/>cialis eſt diuerſæ diaphanitatis, & quòd pars poſterior eius, quæ dicitur humor uitreus, eſt diuerſæ <lb/>diaphanitatis à parte anteriore:</s> <s xml:id="echoid-s1521" xml:space="preserve"> & nullum corpus eſt in glaciali diuerſæ formæ à forma corporis an <lb/>terioris, præter corpus uitreum:</s> <s xml:id="echoid-s1522" xml:space="preserve"> & ex proprietate formarũ lucis & coloris eſt, ut refringantur, quã-<lb/>do occurrerint alij corpori diuerſæ diaphanitatis à corpore primo.</s> <s xml:id="echoid-s1523" xml:space="preserve"> Formæ ergo non refringuntur, <lb/>niſi apud peruentum earum ad humorem uitreum.</s> <s xml:id="echoid-s1524" xml:space="preserve"> Et iſtud corpus non fuit diuerſæ diaphanitatis <lb/>à corpore anterioris glacialis, niſi ut refringerentur formæ in ipſo.</s> <s xml:id="echoid-s1525" xml:space="preserve"> Et debet ſuperficies iſtius corpo <lb/>ris antecedere centrum, ut refringantur formæ apud ipſum, antequam pertranſeant cẽtrum:</s> <s xml:id="echoid-s1526" xml:space="preserve"> & de-<lb/>bet iſta ſuperficies eſſe conſimilis ordinationis:</s> <s xml:id="echoid-s1527" xml:space="preserve"> quoniam ſi non fuerit conſimilis ordinationis, ap-<lb/>parebit forma monſtruoſa propter refractionem.</s> <s xml:id="echoid-s1528" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div56" type="section" level="0" n="0"> <head xml:id="echoid-head80" xml:space="preserve" style="it">3. Communis ſectio cryſtallinæ & uitreæ ſphærarum aut eſt plana: aut eſt pars <lb/>ſphæræ maioris cryſtallina ſphæra. Et habet centrum diuer-<lb/>ſum ab oculi centro. 23 p 3.</head> <p> <s xml:id="echoid-s1529" xml:space="preserve">SV perficies autẽ cõſimilis ordinationis aut eſt plana, aut ſphærica.</s> <s xml:id="echoid-s1530" xml:space="preserve"> Et nõ poteſt iſta ſuperficies <lb/>eſſe ex ſphæra, cuius centrũ eſt centrum uiſus:</s> <s xml:id="echoid-s1531" xml:space="preserve"> quoniã ſi ita eſſet, eſſent lineæ radiales ſemper <lb/>perpendiculares ſuper ipſam:</s> <s xml:id="echoid-s1532" xml:space="preserve"> & ſic extenderetur forma ſecundũ rectitudinem earũ, & non re-<lb/>fringeretur.</s> <s xml:id="echoid-s1533" xml:space="preserve"> Neq;</s> <s xml:id="echoid-s1534" xml:space="preserve"> poteſt eſſe ex ſphæra parua:</s> <s xml:id="echoid-s1535" xml:space="preserve"> quoniã, ſi fuerit ex ſphæra parua, quãdo forma refrin-<lb/>getur ab ea, & elongabitur ab ea, fiet monſtruoſa.</s> <s xml:id="echoid-s1536" xml:space="preserve"> Iſta ergo ſuperficies aut eſt plana, aut ſphærica è <lb/>ſphæra alicuius bonæ quantitatis:</s> <s xml:id="echoid-s1537" xml:space="preserve"> ita quòd ſphæricitas eius nõ operabitur in ordinatione formæ.</s> <s xml:id="echoid-s1538" xml:space="preserve"> <lb/>Superficies ergo humoris glacialis, quæ eſt differentia cõmunis inter iſtud corpus uitrei & corpus <lb/>anterius glacialis, eſt ſuperficies cõſimilis ordinationis antecedẽs centrum uiſus.</s> <s xml:id="echoid-s1539" xml:space="preserve"> Et omnes formæ <lb/>perueniẽtes in ſuperficiẽ glacialis, extenduntur in corpore glacialis ſecundũ rectitudinem linearũ <lb/>radialiũ, quouſq;</s> <s xml:id="echoid-s1540" xml:space="preserve"> perueniãt ad iſtã ſuperficiem, & cũ peruenerint ad ſuperficiẽ iſtam:</s> <s xml:id="echoid-s1541" xml:space="preserve"> refringuntur <lb/>apud ipſam ſecundũ lineas cõſimilis ordinationis, ſecantes lineas radiales.</s> <s xml:id="echoid-s1542" xml:space="preserve"> Lineæ ergo radiales nõ <lb/> <pb o="26" file="0032" n="32" rhead="ALHAZEN"/> iuuant ad ordinationem formarum rerum uiſibilium, niſi apud glacialem tantùm:</s> <s xml:id="echoid-s1543" xml:space="preserve"> quoniam apud <lb/>membrum iſtud principium eſt ſenſus.</s> <s xml:id="echoid-s1544" xml:space="preserve"> Et declaratum eſt in primo tractatu etiam [15.</s> <s xml:id="echoid-s1545" xml:space="preserve">16.</s> <s xml:id="echoid-s1546" xml:space="preserve">18 n] quod <lb/>impoſsibile eſt, ut forma rei uiſæ ſit ordinata in ſuperficie uiſus cũ imagine rei uiſæ & paruitate rei <lb/>ſentientis, niſi per iſtas lineas.</s> <s xml:id="echoid-s1547" xml:space="preserve"> Iſtæ ergo lineæ non ſunt, niſi inſtrumentũ uiſus, per quas completur <lb/>comprehenſio rerum uiſarum ſecundum ſuum eſſe.</s> <s xml:id="echoid-s1548" xml:space="preserve"> Peruentus autem formarum ad ultimum ſen-<lb/>tiens, non indiget extenſione ſecundum rectitudinem iſtarum linearum.</s> <s xml:id="echoid-s1549" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div57" type="section" level="0" n="0"> <head xml:id="echoid-head81" xml:space="preserve" style="it">4. Humor cryſtallin{us} lucem & colorem aliter recipit, quàm cæter a perſpicua corpora. <lb/>22 p 3. Idem 30 n 1.</head> <p> <s xml:id="echoid-s1550" xml:space="preserve">ET receptio formarum in membro ſentiente non eſt, ſicut receptio formarum in corporibus <lb/>diaphanis:</s> <s xml:id="echoid-s1551" xml:space="preserve"> quoniam membrũ ſentiens recipit iſtas formas, & ſentit eas, & pertranſeunt in eo <lb/>propter ſuam diaphanitatem & uirtutem ſenſibilem, quæ eſt in eo.</s> <s xml:id="echoid-s1552" xml:space="preserve"> Recipit ergo iſtas formas <lb/>ſecundum receptionem ſenſus.</s> <s xml:id="echoid-s1553" xml:space="preserve"> Corpora autem diaphana non recipiunt iſtas formas, niſi receptio-<lb/>ne, qua recipiunt ad reddendum, & non ſentiunt ipſas.</s> <s xml:id="echoid-s1554" xml:space="preserve"> Et cum receptio corporis ſentientis ab iſtis <lb/>formis non ſit ſicut receptio corporum diaphanorum, non ſentientium:</s> <s xml:id="echoid-s1555" xml:space="preserve"> extenſio formarum in cor-<lb/>pore ſentiente non debet eſſe ſecundum uerticationes, quas corpora diaphana exigũt.</s> <s xml:id="echoid-s1556" xml:space="preserve"> Viſus ergo <lb/>non eſt appropriatus receptioni formarum ex uerticationibus linearũ radialium tantùm:</s> <s xml:id="echoid-s1557" xml:space="preserve"> niſi quia <lb/>proprietas formarũ eſt, ut extendãtur in corporibus diaphanis ſuper omnes uerticationes rectas.</s> <s xml:id="echoid-s1558" xml:space="preserve"> <lb/>Et cum iſtæ formæ peruenerint ad membrum ſentiens ordinatæ, & comprehendantur à membro <lb/>ſentiente ordinatæ:</s> <s xml:id="echoid-s1559" xml:space="preserve"> nihil remanebit pòſt, indigẽs iſtarum uerticationibus.</s> <s xml:id="echoid-s1560" xml:space="preserve"> Pars ergo anterior tan-<lb/>tùm glacialis eſt appropriata receptioni formarum ex uerticationibus linearum radialium:</s> <s xml:id="echoid-s1561" xml:space="preserve"> poſte-<lb/>rior autẽ pars, quæ eſt humor uitreus:</s> <s xml:id="echoid-s1562" xml:space="preserve"> & uirtus recipiens, quæ eſt in illo corpore, nõ eſt appropriata <lb/>cum ſuo ſenſu iſtarum formarum, niſi ad cuſtodiendum eorum ordinationem tantùm.</s> <s xml:id="echoid-s1563" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div58" type="section" level="0" n="0"> <head xml:id="echoid-head82" xml:space="preserve" style="it">5. Cryſtallin{us} & uitre{us} humores dißimiliter lucem & colorem recipiunt. 22 p 3.</head> <p> <s xml:id="echoid-s1564" xml:space="preserve">ET cũ ita ſit, qualitas ergo receptionis uitrei à formis non eſt ſicut receptio corporis ſiue qua-<lb/>litas corporis anterioris glacialis:</s> <s xml:id="echoid-s1565" xml:space="preserve"> & uirtus recipiens, quæ eſt in uitreo, nõ eſt uirtus recipiẽs, <lb/>quæ eſt in parte anteriori.</s> <s xml:id="echoid-s1566" xml:space="preserve"> Et cum qualitas receptionis uitrei à formis, non ſit qualitas partis <lb/>anterioris glacialis:</s> <s xml:id="echoid-s1567" xml:space="preserve"> refractio ergo formarum apud ſuperficiem uitrei, non eſt niſi propter diuerſi-<lb/>tatem qualitatis receptionis ſenſus inter iſta duo corpora.</s> <s xml:id="echoid-s1568" xml:space="preserve"> Formæ ergo refringuntur apud uitreum <lb/>duabus de cauſsis:</s> <s xml:id="echoid-s1569" xml:space="preserve"> quarum altera eſt diuerſitas diaphanitatis duorum corporum:</s> <s xml:id="echoid-s1570" xml:space="preserve"> & altera diuerſi-<lb/>tas qualitatis receptionis ſenſus inter iſta duo corpora.</s> <s xml:id="echoid-s1571" xml:space="preserve"> Et ſi diaphanitas iſta duorum corporum eſ-<lb/>ſet conſimilis:</s> <s xml:id="echoid-s1572" xml:space="preserve"> eſſet forma extenſa in corpore uitreo ſecundum rectitudinem linearum radialium, <lb/>propter conſimilιtudinem diaphanitatis:</s> <s xml:id="echoid-s1573" xml:space="preserve"> & eſſet refracta propter diuerſitatem qualitatis ſenſus:</s> <s xml:id="echoid-s1574" xml:space="preserve"> & <lb/>ſic eſſet forma propter refractionem monſtruoſa, aut duæ formæ eſſent propter iſtã diſpoſitionem.</s> <s xml:id="echoid-s1575" xml:space="preserve"> <lb/>Et cum diuerſitas diaphanitatis affirmet refractionem, & diuerſitas qualitatis ſenſus affirmet illam <lb/>refractionem aut obliquationem:</s> <s xml:id="echoid-s1576" xml:space="preserve"> erit forma poſt refractionem una forma.</s> <s xml:id="echoid-s1577" xml:space="preserve"> Et propter hoc diuerſa-<lb/>tur diaphanitas corporis uitrei, & diaphanitas corporis anterioris glacialis.</s> <s xml:id="echoid-s1578" xml:space="preserve"> Formę ergo perueniũt <lb/>ad uitreum ordinatæ, ſecũdum ordinationem earum in ſuperficie uiſi:</s> <s xml:id="echoid-s1579" xml:space="preserve"> & recipit ipſas iſtud corpus, <lb/>& ſentit ipſas:</s> <s xml:id="echoid-s1580" xml:space="preserve"> deinde refringitur forma propter diuerſitatem diaphanitatis, & diuerſitatem ſenſus <lb/>iſtius corporis, & ſic peruenit forma ſecũdum diſpoſitionem ſuam:</s> <s xml:id="echoid-s1581" xml:space="preserve"> deinde extenditur iſte ſenſus, <lb/>& iſtæ formæ per hoc corpus, quouſq;</s> <s xml:id="echoid-s1582" xml:space="preserve"> perueniat iſte ſenſus, & iſtæ formæ ad ultimum ſentiens:</s> <s xml:id="echoid-s1583" xml:space="preserve"> & <lb/>erit extenſio ſenſus & extenſio formæ in corpore uitreo, & in corpore ſentiente extẽſo in concauo <lb/>nerui, ad ultimum ſentiens, ſicut extenſio ſenſus tactus & ſenſus doloris ad ultimum ſentiens:</s> <s xml:id="echoid-s1584" xml:space="preserve"> ſen-<lb/>ſus autem tactus & ſenſus doloris non extenduntur à membris, niſi in filis neruorum, & in ſpiritu <lb/>extenſo ſecundum iſta fila.</s> <s xml:id="echoid-s1585" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div59" type="section" level="0" n="0"> <head xml:id="echoid-head83" xml:space="preserve" style="it">6. Humor uitre{us} & ſpirit{us} uiſibilis eadem ferè perſpicuitate præditi ſunt. 22 p 3.</head> <p> <s xml:id="echoid-s1586" xml:space="preserve">ET formæ rerum uiſibilium quãdo peruenerint in corpus humoris uitrei, extendetur ſenſus <lb/>ab illo membro in corpus ſentiens, extenſum in concauo nerui continuati inter uiſum & an-<lb/>terius cerebri:</s> <s xml:id="echoid-s1587" xml:space="preserve"> & ſecundum extenſionem ſenſus, extenduntur formæ ordinatæ ſecundum <lb/>ſuam diſpoſitionem:</s> <s xml:id="echoid-s1588" xml:space="preserve"> quoniam corpus ſentiẽs naturaliter ſeruat ordinationem iſtarum formarum.</s> <s xml:id="echoid-s1589" xml:space="preserve"> <lb/>Et iſta ordinatio conſeruatur in corpore ſentiente:</s> <s xml:id="echoid-s1590" xml:space="preserve"> quoniam ordinatio partium corporis ſentien-<lb/>tis, recipientium partes formarum, & ordinatio uirtutis recipientis, quæ eſt in partibus corporis <lb/>recipientis, eſt in corpore uitrei, & in omni corpore ſubtili extenſo in cõcauo nerui, ordinatio con-<lb/>ſimilis.</s> <s xml:id="echoid-s1591" xml:space="preserve"> Et cum ita ſit, quãdo forma peruenit ad quodlibet punctum ſuperficiei uitrei, curret in uer-<lb/>ticatione continua, & non alterabitur eius ſitus in concauitate nerui, in quo extenditur corpus ſen <lb/>tiens:</s> <s xml:id="echoid-s1592" xml:space="preserve"> & erunt omnes uerticationes iſtæ, per quas currunt omnia puncta, quæ ſunt in forma, conſi-<lb/>milis ordinationis interſe:</s> <s xml:id="echoid-s1593" xml:space="preserve"> & erunt omnes iſtæ uerticationes gyrantes apud gyrationem nerui:</s> <s xml:id="echoid-s1594" xml:space="preserve"> & <lb/>erunt apud gyrationem ordinatæ ſecundum ſuam ordinationem ante gyrationem, & poſt, propter <lb/>qualitatem ſenſus iſtius corporis:</s> <s xml:id="echoid-s1595" xml:space="preserve"> & ſic perueniet forma ad neruum communem ſecundum ſuam <lb/>diſpoſitionem.</s> <s xml:id="echoid-s1596" xml:space="preserve"> Et non eſt poſsibile, ut ſit extenſio formarum uiſibilium uſq;</s> <s xml:id="echoid-s1597" xml:space="preserve"> ad ultimum ſentiens, <lb/>niſi ſecundum hunc modum:</s> <s xml:id="echoid-s1598" xml:space="preserve"> quoniam nõ eſt poſsibile, ut formæ perueniant ad neruum commu-<lb/>nem ſecundum ſuum eſſe, niſi ſit extenſio earum ſecundum hunc modum.</s> <s xml:id="echoid-s1599" xml:space="preserve"> Et cum formæ exten-<lb/> <pb o="27" file="0033" n="33" rhead="OPTICAE LIBER II."/> duntur ſecundum iſtam ordinationem, oportet, ut forma perueniens ad quodlibet punctum ſuper <lb/>ficiei glacialis, ſemper extendatur ſuper eandem uerticationem ad idem punctum loci nerui com-<lb/>munis, ad quod peruenit forma:</s> <s xml:id="echoid-s1600" xml:space="preserve"> ſed tamẽ forma perueniens ad quodlibet punctum ſuperficiei gla-<lb/>cialis, peruenit ſemper ad idem punctum ſuperficiei uitrei.</s> <s xml:id="echoid-s1601" xml:space="preserve"> Et ſequitur ex hoc, ut ex omnibus duo-<lb/>bus punctis conſimilis ſitus in reſpectu duorum oculorum, extendantur duæ formæ ad idem pun-<lb/>ctum in neruo communi:</s> <s xml:id="echoid-s1602" xml:space="preserve"> & etiam ſequitur ex hoc, ut corpus ſentiens, quod eſt in cõcauo nerui, ſit <lb/>aliquantulum diaphanum, ut appareant in eo formæ lucis & coloris.</s> <s xml:id="echoid-s1603" xml:space="preserve"> Et etiam ſequitur, ut ſit eius <lb/>diaphanitas ſimilis diaphanitati humoris uitrei, ut nõ refringantur formæ apud peruentum earum <lb/>ad ultimam ſuperficiem uitrei, uicinantem concauo nerui:</s> <s xml:id="echoid-s1604" xml:space="preserve"> quoniam quando diaphanitas duorum <lb/>corporum fuerit conſimilis, non refringentur formæ.</s> <s xml:id="echoid-s1605" xml:space="preserve"> Et non eſt poſsibile, ut formæ refringantur <lb/>apud iſtam ſuperficiem:</s> <s xml:id="echoid-s1606" xml:space="preserve"> quoniam iſta ſuperficies eſt ſphærica.</s> <s xml:id="echoid-s1607" xml:space="preserve"> Si autem formæ refringerẽtur ab iſta <lb/>ſuperficie, non elongarentur ab ea, niſi modicùm, & fierent ſtatim monſtruoſæ.</s> <s xml:id="echoid-s1608" xml:space="preserve"> Refractio ergo for-<lb/>marum non poteſt eſſe apud iſtam ſuperficiem.</s> <s xml:id="echoid-s1609" xml:space="preserve"> Et cum diaphanitas corporis ſentientis, quod eſt in <lb/>concauo nerui, non ſit diuerſa à diaphanitate humoris uitrei:</s> <s xml:id="echoid-s1610" xml:space="preserve"> non faciet contingere iſta diuerſitas <lb/>aliquam diuerſitatem in forma.</s> <s xml:id="echoid-s1611" xml:space="preserve"> Et quamuis forma extendatur cum extenſione ſenſus:</s> <s xml:id="echoid-s1612" xml:space="preserve"> diaphanitas <lb/>tamen corporis ſentientis, quod eſt in concauo nerui, nõ eſt diuerſa à diaphanitate corporis uitrei.</s> <s xml:id="echoid-s1613" xml:space="preserve"> <lb/>Diaphanitas autem iſta iſtius corporis non eſt, niſi ut extendantur formæ in eo ſecundum uertica-<lb/>tiones, quas exigit diaphanitas, & ut recipiat formas lucis & coloris, & ut appareãt in eo:</s> <s xml:id="echoid-s1614" xml:space="preserve"> quoniam <lb/>corpus non recipit lucem & colorem, neque pertranſeuntin eo formæ lucis & coloris, niſi ſit dia-<lb/>phanum, aut fuerit in eo aliquid diaphanitatis.</s> <s xml:id="echoid-s1615" xml:space="preserve"> Et nõ apparet lux & color in corpore diaphano, niſi <lb/>ſit in eius diaphanitate aliquid ſpiſsitudinis:</s> <s xml:id="echoid-s1616" xml:space="preserve"> & propter hoc non eſt glacialis in fine diaphanitatis, <lb/>neque in ſine ſpiſsitudinis.</s> <s xml:id="echoid-s1617" xml:space="preserve"> Corpus ergo ſentiens, quod eſt in concauo nerui, eſt diaphanum, & in <lb/>eo eſt inſuper aliquid ſpiſsitudinis.</s> <s xml:id="echoid-s1618" xml:space="preserve"> Forma autem pertranſit in iſto corpore cũ eo, quod eſt in eo de <lb/>diaphanitate:</s> <s xml:id="echoid-s1619" xml:space="preserve"> & apparent in eo formæ uirtuti ſenſitiuæ cũ eo, quod eſt in eo de ſpiſsitudine.</s> <s xml:id="echoid-s1620" xml:space="preserve"> Et ſen-<lb/>tiens ultimum non comprehendit formas lucis & coloris, niſi ex formis peruenientibus ad iſtud <lb/>corpu<gap/> apud peruentum earum ad neruum communem:</s> <s xml:id="echoid-s1621" xml:space="preserve"> & comprehẽdit lucem ex illuminatione <lb/>iſtius corporis, & colorem ex coloratione.</s> <s xml:id="echoid-s1622" xml:space="preserve"> Secũdum ergo hunc modum erit peruentus formarum <lb/>ad ultimum ſentiens, & comprehenſio ultimi ſentientis quò ad illas.</s> <s xml:id="echoid-s1623" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div60" type="section" level="0" n="0"> <head xml:id="echoid-head84" xml:space="preserve" style="it">7. Axis pyramidis opticæ ſol{us} ad perpendiculum eſt cõmuni ſectioni cryſtallinæ & uitreæ <lb/>ſphærarum. 24 p 3.</head> <p> <s xml:id="echoid-s1624" xml:space="preserve">ET poſtquam declaratum eſt, quòd formæ refringãtur apud ſuperficiem uitrei:</s> <s xml:id="echoid-s1625" xml:space="preserve"> dicamus quòd <lb/>axis pyramidis radialis nõ poteſt eſſe declinans ſuper iſtam ſuperficiem, neq;</s> <s xml:id="echoid-s1626" xml:space="preserve"> poteſt eſſe alia <lb/>linea perpendicularis ſuper ipſam.</s> <s xml:id="echoid-s1627" xml:space="preserve"> Quoniam ſi axis fuerit declinans ſuper iſtam ſuperficiem, <lb/>quando formæ peruenirent ad iſtam ſuperficiem, diuerſificarentur in ordinatione, & mutarentur <lb/>ipſarum diſpoſitiones.</s> <s xml:id="echoid-s1628" xml:space="preserve"> Formæ autem non poſſunt peruenire in ſuperficiem uitrei ſecundum ſuum <lb/>eſſe, niſi fuerit axis pyramidis ſuper iſtam ſuperficiem perpendicularis.</s> <s xml:id="echoid-s1629" xml:space="preserve"> Quoniam quãdo uiſus fue-<lb/>rit oppoſitus alicui rei uiſæ, & peruenerit axis radialis ſuper iſtam ſuperficiem iſtius rei uiſæ:</s> <s xml:id="echoid-s1630" xml:space="preserve"> per-<lb/>ueniet forma illius rei uiſæ in ſuperficiem glacialis ordinata ſecundum ordinationem partium ſu-<lb/>perſiciei rei uiſæ, & perueniet forma puncti, quod eſt apud extremitatem axis ſuperficiei rei uiſæ, <lb/>ad punctum, quod eſt ſuper axem in ſuperficie glacialis [per 18 n 1] & peruenient formæ omnium <lb/>punctorum ſuperficiei rei uiſæ, quorũ remotio à puncto, quod eſt apud extremitatẽ axis, eſt æqua-<lb/>lis, ad puncta formarum, quæ ſunt in ſuperficie glacialis, quorum remotio à puncto, quod eſt ſuper <lb/>axem, æqualis eſt:</s> <s xml:id="echoid-s1631" xml:space="preserve"> quoniam omnia puncta peruenientia ad ſuperficiem glacialis, ſunt ſuper lineas <lb/>radiales extenſas à centro uiſus ad ſuperficiem uiſus, & axis radialis eſt perpendicularis ſuper ſu-<lb/>perficiem glacialis.</s> <s xml:id="echoid-s1632" xml:space="preserve"> Omnes ergo ſuperficies planæ exeuntes ab axe, & ſecantes ſuperficiem glacia-<lb/>lis, erunt [per 18 p 11] perpendiculares ſuper iſtam ſuperficiem.</s> <s xml:id="echoid-s1633" xml:space="preserve"> Et iam declaratum eſt [3 n] quòd <lb/>ſuperficies humoris uitrei, aut eſt plana, aut eſt ſphærica, & centrum eius non eſt centrum uiſus.</s> <s xml:id="echoid-s1634" xml:space="preserve"> Si <lb/>ergo axis radialis eſt declinans ſuper iſtam ſuperficiem, & nõ eſt perpendicularis ſuper ipſam:</s> <s xml:id="echoid-s1635" xml:space="preserve"> non <lb/>exibit ab axe ſuperficies plana perpendicularis ſuper iſtam ſuperficiẽ, niſi una ſuperficies tantùm, <lb/>& omnes ſuperficies reſiduæ exeuntes ab axe erunt declinãtes ſuper ipſam:</s> <s xml:id="echoid-s1636" xml:space="preserve"> quoniam hæc eſt pro-<lb/>prietas linearũ declinantium ſuper ſuperficies planas & ſphæricas.</s> <s xml:id="echoid-s1637" xml:space="preserve"> Imaginemur igitur ſuperficiem <lb/>a b c d, exeuntem ab axe a c, & perpendicula-<lb/> <anchor type="figure" xlink:label="fig-0033-01a" xlink:href="fig-0033-01"/> riter ſuper ſuperficiem uitrei f g e extendi:</s> <s xml:id="echoid-s1638" xml:space="preserve"> ſe-<lb/>cabit ergo ſuperficiem uitrei & ſuperficiẽ gla-<lb/>cialis, & ſignabit in eis duas differentias com-<lb/>munes:</s> <s xml:id="echoid-s1639" xml:space="preserve"> in glaciali quidem b d, in uitreo uerò <lb/>e f:</s> <s xml:id="echoid-s1640" xml:space="preserve"> & imaginemur ſuper differẽtiam commu-<lb/>nem, quæ eſt communis huic ſuperficiei & ſu-<lb/>perficiei glacialis, duo puncta b, d:</s> <s xml:id="echoid-s1641" xml:space="preserve"> & ſint re-<lb/>mota à puncto a, quod eſt ſuper axem, æquali-<lb/>ter:</s> <s xml:id="echoid-s1642" xml:space="preserve"> & imaginemur duas lineas exeuntes à cen <lb/>tro glacialis, quod eſt c, uſq;</s> <s xml:id="echoid-s1643" xml:space="preserve"> ad iſta duo pũcta <lb/>b, d, & ſint c b, c d.</s> <s xml:id="echoid-s1644" xml:space="preserve"> Erũt ergo [per 1 p 11] hę duæ <lb/> <pb o="28" file="0034" n="34" rhead="ALHAZEN"/> lineæ cũ axe a c in ſuperficie communi a b c d perpendiculari ſuper ſuperficiem uitrei e g f:</s> <s xml:id="echoid-s1645" xml:space="preserve"> quoniã <lb/>duo puncta b, d, & punctũ centri c ſunt in iſta ſuperficie:</s> <s xml:id="echoid-s1646" xml:space="preserve"> & erunt [per 8 p 1 ductis rectis a b, a d] duo <lb/>anguli, qui fient ex iſtis duabus lineis & axe, ſcilicet anguli a c b, a c d, æquales:</s> <s xml:id="echoid-s1647" xml:space="preserve"> & ſint iſtæ duę lineæ <lb/>c b, c d ſecantes differentiã communẽ, quæ eſt in ſuperficie uitrei, ſuper duobus punctis e, f:</s> <s xml:id="echoid-s1648" xml:space="preserve"> & ſimi-<lb/>liter axis ſecet differentiam iſtam communẽ ſuper punctum g, interiectum inter illa duo puncta e, f.</s> <s xml:id="echoid-s1649" xml:space="preserve"> <lb/>Si ergo ſuperficies uitrei eſt plana, erit [per 3 p 11] differentia cõmunis linea recta:</s> <s xml:id="echoid-s1650" xml:space="preserve"> Et ſi axis a c fuerit <lb/>declinans ſuper ſuperficiem uitrei, & fuerit ſuperficies, quæ fecit differentiã communẽ, perpendi-<lb/>cularis ſuper iſtam ſuperficiem:</s> <s xml:id="echoid-s1651" xml:space="preserve"> erit etiã axis a c declinans ſuper cõmunem differentiã, ſuper lineam <lb/>e f:</s> <s xml:id="echoid-s1652" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s1653" xml:space="preserve"> duo anguli e g c, f g c inæquales:</s> <s xml:id="echoid-s1654" xml:space="preserve"> quoniã ſi axis a c eſſet perpendicularis ſuper communẽ <lb/>differentiã e f, eſſet perpendicularis ſuper ſuperficiẽ uitrei [per 4 d 11] & duo anguli e g c, f g c æqua <lb/>les.</s> <s xml:id="echoid-s1655" xml:space="preserve"> Sed cũ hi duo prædicti anguli ſint inæquales, & duo anguli e c g, f c g, qui ſunt apud centrũ gla-<lb/>cialis c, quod eſt extremitas axis a c, ſint æquales:</s> <s xml:id="echoid-s1656" xml:space="preserve"> erũt e g & g f duæ partes lineæ e f, quæ eſt differẽ-<lb/>tia<gap/>communis, inæquales [Quia enim trianguli c e f latera c e, c f ſunt inæqualia (ſecus axis a c eſſet <lb/>perpendicularis ad f e per 4 p.</s> <s xml:id="echoid-s1657" xml:space="preserve"> 10 d 1, cõtra hypotheſim) eſto maius c e:</s> <s xml:id="echoid-s1658" xml:space="preserve"> factoq́ ipſi c e æquali c h, du-<lb/>catur g h recta, quę per conſtructionẽ & 4 p 1 erit æqualis ipſi g e:</s> <s xml:id="echoid-s1659" xml:space="preserve"> ductaq́;</s> <s xml:id="echoid-s1660" xml:space="preserve"> ex g perpendiculari g i ſu-<lb/>per h c:</s> <s xml:id="echoid-s1661" xml:space="preserve"> erit per 16 p 1 angulus g f h obtuſus:</s> <s xml:id="echoid-s1662" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s1663" xml:space="preserve"> ք 19 p 1 latus h g, id eſt e g, erit maius latere f g] Ergo <lb/>erunt duo puncta e, f extremitatũ ipſius, diuerſæ diſtantiæ à puncto g exiſtẽte ſuper axem in illa li-<lb/>nea.</s> <s xml:id="echoid-s1664" xml:space="preserve"> Et iſta duo puncta ſunt illa, ad quæ perueniunt formæ duorũ punctorum ſuperficiei glacialis, <lb/>quę ſunt æqualiter diſtãtia ab axe a c:</s> <s xml:id="echoid-s1665" xml:space="preserve"> quoniã ſunt apud duas extremitates duarũ linearũ radialium <lb/>tranſeuntiũ per iſta duo puncta.</s> <s xml:id="echoid-s1666" xml:space="preserve"> Et punctũ g, quod eſt ſuper axẽ a c ex ſuperficie uitrei, eſt illud, ad <lb/>quod peruenit forma puncti a, quod eſt ſuper axem ex ſuperficie glacialis.</s> <s xml:id="echoid-s1667" xml:space="preserve"> Et cũ axis a c fuerit decli <lb/>nans ſuper ſuperficiẽ uitrei, & ſuperficies uitrei fuerit plana:</s> <s xml:id="echoid-s1668" xml:space="preserve"> tunc quando duo puncta, (quorũ for-<lb/>mæ perueniunt in ſuperficiẽ glacialis, & quorũ diſtantia à puncto a, quod eſt ſuper axem, eſt æqua-<lb/>lis, & quę ſunt in ſuperficie perpendiculari ſuper ſuperficiẽ uitrei) peruenerint ad ſuperficiẽ uitrei, <lb/>erit diſtantia eorũ à puncto g ueniente ſuper axem, diſtantia inæqualis.</s> <s xml:id="echoid-s1669" xml:space="preserve"> Et auãdo axis fuerit decli-<lb/>nans ſuper ſuperficiẽ uitrei, & fuerit ſuperficies uitrei plana:</s> <s xml:id="echoid-s1670" xml:space="preserve"> tũc differentia cõmunis, quæ fit à qua-<lb/>libet ſuperficie exeũte ab axe, & ſecante ſuperficiẽ uitrei, continebit cũ axe duos angulos inæqua-<lb/>les, præter unã ſuperficiem tantùm:</s> <s xml:id="echoid-s1671" xml:space="preserve"> & eſt illa, quæ ſecat ſuperficiẽ perpendicularem ſuper uitreum:</s> <s xml:id="echoid-s1672" xml:space="preserve"> <lb/>quoniam differentia cõmunis eius continebit cum axe duos angulos rectos, & erit axis declinans <lb/>ſuper differentias communes omniũ ſuperficierum reſiduarum.</s> <s xml:id="echoid-s1673" xml:space="preserve"> Et cũ duo anguli prædicti fuerint <lb/>inæquales, & fuerint duo anguli, reſpicientes duas partes differentiæ cõmunis, ſcilicet anguli, qui <lb/>ſunt apud centrum ſuperficiei glacialis, æquales:</s> <s xml:id="echoid-s1674" xml:space="preserve"> erunt duæ partes differentiæ cõmunis, quæ eſt in <lb/>ſuperficie uitrei, inæquales:</s> <s xml:id="echoid-s1675" xml:space="preserve"> & erunt duo puncta quę ſunt extremitates iſtius differentiæ cõmunis, <lb/>diuerſæ diſtantiæ à puncto quod eſt ſuper axem:</s> <s xml:id="echoid-s1676" xml:space="preserve"> duæ autẽ partes differentiæ cõmunis, quæ ſunt in <lb/>ſuperficie glacialis, erũt æquales:</s> <s xml:id="echoid-s1677" xml:space="preserve"> & erunt duo puncta quæ ſunt in extremitate iſtius differẽtiæ com <lb/>munis, æqualis diſtantiæ à puncto, quod eſt ſuper axem in ſuperficie glacialis.</s> <s xml:id="echoid-s1678" xml:space="preserve"> Et cum ita ſit, quãdo <lb/>forma peruenerit à ſuperficie glacialis ad ſuperficiem uitrei, erit ordinatio eius non ſecundũ ſuum <lb/>eſſe in ſuperficie glacialis, neq;</s> <s xml:id="echoid-s1679" xml:space="preserve"> ſecũdum ſuũ eſſe in ſuperficie rei uiſæ.</s> <s xml:id="echoid-s1680" xml:space="preserve"> Et ſimiliter declarabitur etiã <lb/>quando ſuperficies uitrei fuerit ſphærica, & fuerit axis declinans ſuper ipſam:</s> <s xml:id="echoid-s1681" xml:space="preserve"> quoniã puncta, quæ <lb/>ſunt in ſuperficie glacialis, quorũ diſtantia ab axe eſt æqualis, quando peruenerint ad ſuperficiẽ ui-<lb/>trei, diſta bunt inæqualiter à puncto axis.</s> <s xml:id="echoid-s1682" xml:space="preserve"> Quoniam quando axis non fuerit perpendicularis ſuper <lb/>ſuperficiem uitrei, & ſuperficies uitrea fuerit <lb/> <anchor type="figure" xlink:label="fig-0034-01a" xlink:href="fig-0034-01"/> ſphęrica, non pertranſibit axis iſte per centrũ <lb/>uitrei, & pertranſibit per centrum ſuperficiei <lb/>glacialis.</s> <s xml:id="echoid-s1683" xml:space="preserve"> Lineæ ergo, quæ exeunt à cẽtro gla-<lb/>cialis ad puncta, quorũ diſtãtia à puncto axis <lb/>in ſuperficie glacialis eſt æqualis, continent <lb/>cum axe apud centrũ glacialis angulos æqua <lb/>les.</s> <s xml:id="echoid-s1684" xml:space="preserve"> Et cum ita ſit, & centrum glacialis non ſit <lb/>centrum uitrei [per 10 n 1] iſtæ lineæ diſtin-<lb/>guent ex ſuperficie uitrei arcus inæquales:</s> <s xml:id="echoid-s1685" xml:space="preserve"> & <lb/>nullæ lineæ cõtinentes cum axe angulos re-<lb/>ctos, & exiſtentes cum axe in eadẽ ſuperficie, <lb/>diſtinguent ex ſuperficie uitrei arcus æquales, niſi duæ lineæ tantũ:</s> <s xml:id="echoid-s1686" xml:space="preserve"> & ſunt illæ, quæ ſunt in ſuperfi-<lb/>cie ſecante ſuperficiẽ perpendicularem ſuper ſuperficiem uitrei.</s> <s xml:id="echoid-s1687" xml:space="preserve"> Cum ergo axis fuerit declinãs ſu-<lb/>per ſuperficiem uitrei:</s> <s xml:id="echoid-s1688" xml:space="preserve"> formæ peruenientes in ſuperficiem uitrei, erũt diuerſæ ordinationis, ſiue ſit <lb/>iſta ſuperficies plana, ſiue ſphærica:</s> <s xml:id="echoid-s1689" xml:space="preserve"> & cum axis fuerit perpendicularis ſuper ſuperficiem uitrei, erit <lb/>perpendicularis ſuper omnes differentias cõmunes:</s> <s xml:id="echoid-s1690" xml:space="preserve"> & quælibet duæ lineæ exeuntes à centro gla-<lb/>cialis, quod eſt punctum in axe, continebunt angulos rectos, & diſtinguent ex differentia cõmuni, <lb/>quæ eſt in ſuperficie<gap/>uitrei, duas partes æquales:</s> <s xml:id="echoid-s1691" xml:space="preserve"> & erit diſtantia duorum punctorum, quę ſunt ex-<lb/>tremitates duarum partium æqualium à puncto, quod eſt ſuper axem in ſuperficie uitrei, æqualis, <lb/>ſiue ſit ſuperficies uitrei plana, ſiue ſphærica.</s> <s xml:id="echoid-s1692" xml:space="preserve"> Secundum ergo diſpoſitiones omnes non peruenit <lb/>forma ad ſuperficiem uitrei, & ſitus partium eius ſecundum eſſe ſuum in ſuperficie uiſus, niſi axis <lb/>perpendicularis ſit ſuper ſuperficiem uitrei, & ſentiens nõ ſentit formam, niſi ſecundum eſſe ſuum <lb/> <pb o="29" file="0035" n="35" rhead="OPTICAE LIBER II."/> apud eius peruentum ad ſe, & ſentiens comprehendit ordinationem partium rei uiſæ ſecundum <lb/>ſuum eſſe in ſuperficie rei uiſæ.</s> <s xml:id="echoid-s1693" xml:space="preserve"> Non eſt ergo poſsibile, ut formæ perueniãt in ſuperficiem uitrei, niſi <lb/>ſit ordinatio partium ſuarum ſecundum ſuum eſſe.</s> <s xml:id="echoid-s1694" xml:space="preserve"> Non eſt ergo poſsibile, ut axis radialis ſit decli-<lb/>nans ſuper ſuperficiem uitrei:</s> <s xml:id="echoid-s1695" xml:space="preserve"> erit ergo perpendicularis.</s> <s xml:id="echoid-s1696" xml:space="preserve"> Omnes ergo lineæ radiales reſiduæ erunt <lb/>obliquatæ ſuper ſuperficiem iſtam, ſiue ſit plana, ſiue ſit ſphærica, quoniam ſecant axem ſuper cen-<lb/>trum glacialis.</s> <s xml:id="echoid-s1697" xml:space="preserve"> Nulla ergo iſtarum linearum tranſit per centrum ſuperficiei uitrei, ſi fuerit ſphærica, <lb/>niſi axis tantùm, quoniam eſt perpendicularis ſuper ipſam, & quia cẽtrum ſuperficiei glacialis non <lb/>eſt cen trum ſuperficiei uitrei.</s> <s xml:id="echoid-s1698" xml:space="preserve"/> </p> <div xml:id="echoid-div60" type="float" level="0" n="0"> <figure xlink:label="fig-0033-01" xlink:href="fig-0033-01a"> <variables xml:id="echoid-variables1" xml:space="preserve">a b d h g e f i c</variables> </figure> <figure xlink:label="fig-0034-01" xlink:href="fig-0034-01a"> <variables xml:id="echoid-variables2" xml:space="preserve">a b d h g e f i c</variables> </figure> </div> </div> <div xml:id="echoid-div62" type="section" level="0" n="0"> <head xml:id="echoid-head85" xml:space="preserve" style="it">8. Viſio per axem pyramidis opticæ certißima eſt: per aliam lineam tantò certior, quantò <lb/>ipſa axi propinquior fuerit. 43 p 3.</head> <p> <s xml:id="echoid-s1699" xml:space="preserve">ET quoniam declaratum eſt [2 n] quòd formæ peruenientes in ſuperficiem glacialis, nõ per-<lb/>ueniunt ad concauum nerui, niſi poſtquam fuerint refractæ, & non eſt refractio earum, niſi <lb/>apud ſuperficiem uitrei, & axis eſt perpendicularis ſuper iſtam ſuperficiem, & omnes lineæ <lb/>radiales reſiduæ ſunt obliquatæ ſuper iſtam ſuperficiem:</s> <s xml:id="echoid-s1700" xml:space="preserve"> quãdo formę peruenerint ad ſuperficiem <lb/>uitrei, refringẽtur omnia puncta, quæ ſunt in ea, pręter punctum axis:</s> <s xml:id="echoid-s1701" xml:space="preserve"> quoniam iſte punctus exten-<lb/>ditur ſecundum rectitudinem axis, quouſq;</s> <s xml:id="echoid-s1702" xml:space="preserve"> perueniat ad locum gyrationis concaui nerui [per 17 <lb/>n 1] Nulla ergo forma perueniens ad ſuperficiem glacialis extenditur ad concauum nerui ſecundũ <lb/>rectitudinem, niſi punctum axis tantùm, & omnia puncta reſidua perueniunt ad concauum nerui <lb/>ſecundum lineas refractas.</s> <s xml:id="echoid-s1703" xml:space="preserve"> Cum ergo uiſus comprehendit rem uiſam, & illa res uiſa fuerit oppoſi-<lb/>ta medio uiſus, & fuerit axis intra pyramidem radialem continentem illam rem uiſam:</s> <s xml:id="echoid-s1704" xml:space="preserve"> forma illius <lb/>rei uiſæ perueniet ad ſuperficiem glacialis ſecũdum rectitudinem linearum radialium:</s> <s xml:id="echoid-s1705" xml:space="preserve"> deinde ex-<lb/>tenduntur formæ ab iſta ſuperficie ſecundum rectitudinem linearum radialium etiam, quouſque <lb/>perueniant ad ſuperficiem uitrei:</s> <s xml:id="echoid-s1706" xml:space="preserve"> deinde punctum axis extendetur ab iſta ſuperficie ſecundũ recti-<lb/>tudinem axis, quouſq;</s> <s xml:id="echoid-s1707" xml:space="preserve"> perueniat ad locũ gyrationis concaui nerui, & omnia puncta reſidua refrin-<lb/>guntur ſuper lineas ſecantes lineas radiales, & conſimilis ordinationis, quouſq;</s> <s xml:id="echoid-s1708" xml:space="preserve"> perueniant ad lo-<lb/>cum gyrationis concaui nerui.</s> <s xml:id="echoid-s1709" xml:space="preserve"> Perueniet ergo forma in illum locum ordinata ſecundum ſuum or-<lb/>dinem in ſuperficie glacialis, & ordinata ſecundum ſuam ordinationem in ſuperficie rei uiſæ.</s> <s xml:id="echoid-s1710" xml:space="preserve"> Sed <lb/>diſpoſitio formarum obliquatarum non eſt ſicut diſpoſitio formarum extenſarum rectè, quoniam <lb/>obliquatio alterabit ipſas aliqua alteratione neceſſariò.</s> <s xml:id="echoid-s1711" xml:space="preserve"> Sequitur ergo de iſta diſpoſitione, ut pun-<lb/>ctum perueniens ad locum gyrationis concaui nerui, quod extendebatur ſecundum rectitudinem <lb/>axis, ſit magis uerificatum omnibus punctis formarum.</s> <s xml:id="echoid-s1712" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div63" type="section" level="0" n="0"> <head xml:id="echoid-head86" xml:space="preserve" style="it">9. Radi{us} pyramidis opticæ obliqu{us}, axi propior ad minores angulos refringitur, remotior <lb/>ad maiores: & duo æqualiter remoti, ad æquales. 36 p 3.</head> <p> <s xml:id="echoid-s1713" xml:space="preserve">ET etiam refractio punctorũ peruenientium in ſuperficiem refractionis propinquiorum pun-<lb/>cto axis, eſt minor, & remotiorum, maior:</s> <s xml:id="echoid-s1714" xml:space="preserve"> quoniam refractio non eſt, niſi ſecundum angulos, <lb/>qui fiunt ex lineis, ſuper quas formæ ueniunt, & ex perpendicularibus, quæ ſunt ſuper ſuper-<lb/>ficiem refractionis:</s> <s xml:id="echoid-s1715" xml:space="preserve"> & linearum continentium cum perpendicularibus angulos minores, erit refra-<lb/>ctio ſecundum angulos minores:</s> <s xml:id="echoid-s1716" xml:space="preserve"> & linearum continentium cum perpendicularibus angulos ma-<lb/>iores, erit refractio ſecundum angulos maiores.</s> <s xml:id="echoid-s1717" xml:space="preserve"> Et lineæ propinquiores axi minus declinant ſuper <lb/>ſuperficiem refractionis, & ſic continent cum perpendicularibus, quę ſunt ſuper ſuperficiem refra-<lb/>ctionis, angulos minores:</s> <s xml:id="echoid-s1718" xml:space="preserve"> & illæ, quæ ſunt remotiores ab axe, magis declinãt ſuper ſuperficiem re-<lb/>fractionis, & ſic continent cum perpendicularibus angulos maiores.</s> <s xml:id="echoid-s1719" xml:space="preserve"> Et formæ, quorum refractio <lb/>eſt minor, magis manifeſtantur, & quarum refractio eſt maior, minus.</s> <s xml:id="echoid-s1720" xml:space="preserve"> Punctum ergo, quod eſt ſu-<lb/>per axem, perueniens ad locum gyrationis nerui cõcaui, eſt manifeſtius omnibus alijs punctis re-<lb/>ſiduis, & quod eſt propinquum illi, eſt manifeſtius remotiore ab illo.</s> <s xml:id="echoid-s1721" xml:space="preserve"> Et iſtæ formę ſunt, quę exten-<lb/>duntur ad neruum communem, & ex illis comprehẽdit ultimum ſentiens formam rei uiſæ.</s> <s xml:id="echoid-s1722" xml:space="preserve"> Et cum <lb/>iſta forma perueniens ad locũ gyrationis cõcaui nerui:</s> <s xml:id="echoid-s1723" xml:space="preserve"> ſit diuerſæ diſpoſitionis, ſcilicet quòd pun-<lb/>ctum axis eſt manifeſtius omnibus punctis reſiduis, & quod eſt propinquius illi, eſt remotiore ma-<lb/>nifeſtius:</s> <s xml:id="echoid-s1724" xml:space="preserve"> forma ergo perueniens in neruo communi, ex qua comprehẽdit uirtus ſenſitiua formam <lb/>rei uiſæ, erit diuerſæ diſpoſitionis, & punctum eius reſpondens puncto axis in ſuperficie rei uiſæ, <lb/>erit manifeſtius omnibus punctis reſiduis formæ, & huιc propinquius, manifeſtius remotiore.</s> <s xml:id="echoid-s1725" xml:space="preserve"> Et <lb/>ſi in ducantur diſpoſitiones rerum uiſarum, & diſtinguatur qualitas comprehenſionis uiſus à rebus <lb/>uiſis, quas comprehenderit uiſus ſimul, & qualitas comprehẽſionis uiſus à partibus unius rei uiſæ:</s> <s xml:id="echoid-s1726" xml:space="preserve"> <lb/>inuenientur conuenientes omnino in hoc, quod declarauimus.</s> <s xml:id="echoid-s1727" xml:space="preserve"> Quoniam aſpiciens quando in eo-<lb/>dem tempore fuerit oppoſitus multis rebus uiſibilibus, & uiſus eius fuerit quietus, & non mouerit <lb/>ipſum:</s> <s xml:id="echoid-s1728" xml:space="preserve"> inueniet rem uiſam oppoſitam medio ſui uiſus manifeſtiorem illis, quę ſunt à parte laterum <lb/>illius medij, & quę eſt propin quior medio, erit manifeſtior.</s> <s xml:id="echoid-s1729" xml:space="preserve"> Et ſimiliter quando inſpiciens inſpexe-<lb/>rit rem uiſam magnam, & uiſus eius fuerit oppoſitus medio illius rei uiſæ, & fuerit quietus, compre <lb/>hendet medium illius rei uiſæ manifeſtius iſtius rei extremitatibus.</s> <s xml:id="echoid-s1730" xml:space="preserve"> Et hoc manifeſtabitur bene, <lb/>quando fuerint multa uiſibilia ſibi propinqua, & aſpicien<gap/> fuerit oppoſitus uni illorum;</s> <s xml:id="echoid-s1731" xml:space="preserve"> quod erit <lb/>medium inter illa uiſibilia quieto uiſu:</s> <s xml:id="echoid-s1732" xml:space="preserve"> quoniam tunc comprehendet comprehenſione manifeſta <lb/>illud medium, & ſimul etiam comprehendet illa, quæ ſunt in lateribus illius, ſed non manifeſtè.</s> <s xml:id="echoid-s1733" xml:space="preserve"> Et <lb/> <pb o="30" file="0036" n="36" rhead="ALHAZEN"/> hoc manifeſtatur magis, quando ſpatium, ſuper quod ſunt illa uiſibilia, fuerit longum, quoniã tun<gap/> <lb/>erit inter comprehenſionem medij, & comprehenſionem extremitatum magna diuerſitas.</s> <s xml:id="echoid-s1734" xml:space="preserve"> Deinde <lb/>ſi hęc ſpecies motus mouerit uiſum in aſpiciente, & fuerit oppoſitus alij rei uiſæ, præter illam rem <lb/>uiſam, quæ antè erat oppoſita:</s> <s xml:id="echoid-s1735" xml:space="preserve"> comprehendet iſtam ſecundam rem uiſam comprehenſione mani-<lb/>feſta, primam autem comprehendet comprehenſione debili:</s> <s xml:id="echoid-s1736" xml:space="preserve"> & ſi fuerit oppoſitus extremitati, & <lb/>intueatur ipſam:</s> <s xml:id="echoid-s1737" xml:space="preserve"> comprehendet ipſam comprehenſione manifeſtiore, quàm in comprehenſione <lb/>primæ diſpoſitionis ſecundum eius remotionem ab eo, & ſimul comprehendet medium compre-<lb/>henſione debili, quamuis ſit propinquius:</s> <s xml:id="echoid-s1738" xml:space="preserve"> & erit inter comprehenſionem medij, dum aſpiciens op-<lb/>ponitur extremitati, & inter comprehenſionem medij, dum opponitur medio, diuerſitas ſenſibilis.</s> <s xml:id="echoid-s1739" xml:space="preserve"> <lb/>Manifeſtabitur ergo ex hac experimẽtatione, quòd uiſio per medium uiſus, & per axem, quem de-<lb/>finiuimus, eſt manifeſtior uiſione per extremitates, & per lineas continentes axem.</s> <s xml:id="echoid-s1740" xml:space="preserve"> Declaratum <lb/>eſt ergo, quòd uiſio erit per axem pyramidis radialis manifeſtior, quàm uiſio per omnes lineas ra-<lb/>diales, & quòd uiſio per propinquiores axi, eſt manifeſtior, quàm per remotiores.</s> <s xml:id="echoid-s1741" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div64" type="section" level="0" n="0"> <head xml:id="echoid-head87" xml:space="preserve" style="it">10. Viſibile percipitur aut ſolo uiſu: aut uiſu & ſyllogiſmo: aut uiſu & anticipata notione. In <lb/>hypothe. 3 lib. inpræfa. 4 lib. 59. 60 p 3.</head> <p> <s xml:id="echoid-s1742" xml:space="preserve">SEnſus autẽ uiſus nihil comprehendit de rebus uiſibilibus niſi in corpore:</s> <s xml:id="echoid-s1743" xml:space="preserve"> in corpore uerò res <lb/>multæ congregantur, & accidunt ei multæ res, & uiſus comprehendit de corporibus multas <lb/>res, quæ ſunt in eis, & quæ accidunt illis.</s> <s xml:id="echoid-s1744" xml:space="preserve"> Et color eſt unum eorum, quæ accidunt corporibus, <lb/>& ſimiliter lux, & ſenſus uiſus comprehendit utrunque iſtorum in corporibus:</s> <s xml:id="echoid-s1745" xml:space="preserve"> & comprehendit <lb/>etiam alias res præter iſtas duas, ſicut figuram, & ſitum, & magnitudinem, & motum, & alia, quæ <lb/>nos diſtinguemus pòſt:</s> <s xml:id="echoid-s1746" xml:space="preserve"> & comprehendit etiam ſimilitudinem colorum, & diuerſitatem eorum, & <lb/>ſimilitudinem lucis, & diuerſitatem eius:</s> <s xml:id="echoid-s1747" xml:space="preserve"> & ſimiliter etiam comprehendit conſimilitudinem figu-<lb/>rarum, & ſituum, & motuum.</s> <s xml:id="echoid-s1748" xml:space="preserve"> Et comprehenſio omnium iſtorum nõ eſt ſecundum unum modum, <lb/>neq;</s> <s xml:id="echoid-s1749" xml:space="preserve"> comprehenſio cuiuslibet iſtorum eſt ſolo ſenſu.</s> <s xml:id="echoid-s1750" xml:space="preserve"> Quoniam uiſus quando comprehendit duo <lb/>indiuidua in eodem tempore, & fuerint conſimilia in forma, comprehendet indiuidua, & compre-<lb/>hendet ſimilia.</s> <s xml:id="echoid-s1751" xml:space="preserve"> Sed ſimilitudo duarum formarum in duobus indiuiduis non ſunt ipſæ formæ am-<lb/>bæ, nec<gap/>una illarum.</s> <s xml:id="echoid-s1752" xml:space="preserve"> Et cum uiſus comprehendit indiuidua ex formis peruenientibus ad ipſum ex <lb/>duobus indiuiduis, ipſe comprehendit conſimilitudinem duorum indiuiduorum ex ſimilitudine <lb/>duarum formarum peruenientium à forma ad uiſum:</s> <s xml:id="echoid-s1753" xml:space="preserve"> & conſimilitudo duarum formarum nõ ſunt <lb/>ipſæ formæ, neque tertia forma propria conſimilitudini:</s> <s xml:id="echoid-s1754" xml:space="preserve"> ſed conſimilitudo duarum formarum eſt <lb/>conuenientia illarum in aliquo.</s> <s xml:id="echoid-s1755" xml:space="preserve"> Non ergo comprehendetur duarum formarum ſimilitudo, niſi ex <lb/>comparatione unius ad alteram, & ex comprehenſione iſtius, in quo ſunt conſimiles.</s> <s xml:id="echoid-s1756" xml:space="preserve"> Et quia uiſus <lb/>comprehendit ſimilitudinem, & non eſt in eo tertia forma, ex qua comprehendit ſimilitudinem:</s> <s xml:id="echoid-s1757" xml:space="preserve"> <lb/>uiſus ergo non comprehendit ſimilitudinem duarum formarum, niſi ex comparatione unius ad al-<lb/>teram.</s> <s xml:id="echoid-s1758" xml:space="preserve"> Et cum ita ſit, comprehenſio ergo ſenſus uiſus à conſimilitudine formarum, & diuerſitate il-<lb/>larum, non eſt per ſolum ſenſum, ſed per comparationem formarum inter ſe.</s> <s xml:id="echoid-s1759" xml:space="preserve"> Et etiam quando ui-<lb/>ſus comprehendit duos colores unius generis, & fuerit unus illorum fortior altero, ſicut uiridem <lb/>myrti & uiridem leuiſtici:</s> <s xml:id="echoid-s1760" xml:space="preserve"> comprehendet, quòd ſunt uirides, & comprehendet etiam quòd alter il-<lb/>lorum eſt fortioris uiriditatis, & diſtinguet inter duas uiriditates, & comprehendet conſimilitudi-<lb/>nem illorum in uiriditate, & diuerſitatem illorum in fortitudine & debilitate:</s> <s xml:id="echoid-s1761" xml:space="preserve"> ſed diſtinctio inter <lb/>duas uiriditates non eſt ipſe ſenſus uiriditatis, quoniam ſenſus uiriditatis eſt ex uiridificatione ui-<lb/>ſus ab utraq;</s> <s xml:id="echoid-s1762" xml:space="preserve"> uiriditate:</s> <s xml:id="echoid-s1763" xml:space="preserve"> & comprehẽdet, quòd ſunt unius generis.</s> <s xml:id="echoid-s1764" xml:space="preserve"> Comprehenſio ergo uiſus, quòd <lb/>altera uiriditas eſt fortior altera, & quòd duæ ſunt unius generis, eſt diſtinctio colorationis, quę eſt <lb/>in uiſu, non ipſe ſenſus coloris.</s> <s xml:id="echoid-s1765" xml:space="preserve"> Et ſimiliter, quãdo duo colores ſimiles in fortitudine ſuerint unius <lb/>generis, uiſus comprehendit duos colores, & comprehendit quòd unius generis ſunt, & quòd ſunt <lb/>conſimiles in fortitudine.</s> <s xml:id="echoid-s1766" xml:space="preserve"> Et ſimiliter eſt diſpoſitio lucis apud uiſum, quoniam uiſus comprehen-<lb/>dit lucem, & diſtinguit inter lucem fortem & debilem.</s> <s xml:id="echoid-s1767" xml:space="preserve"> Comprehenſio ergo uiſus quò ad conſimi-<lb/>litudinem colorum, & diuerſitatem eorum, & conſimilitudinem lucis & diuerſitatem eius, & con-<lb/>ſimilitudinem lineationum formarum rerum uiſibilium, & figuræ, & ſitus earum, & diuerſitates <lb/>earum, non eſt, niſi ex comparatione illarum inter ſe, non ſolo ſenſu.</s> <s xml:id="echoid-s1768" xml:space="preserve"> Et etiam ſenſus uiſus com-<lb/>prehendit diaphanitatem corporum diaphanorum, & diaphanitatem corporum, quæ ſunt in fine <lb/>diaphanitatis:</s> <s xml:id="echoid-s1769" xml:space="preserve"> ſed non comprehendit diaphanitatem talem alia ratione, niſi per comparationem:</s> <s xml:id="echoid-s1770" xml:space="preserve"> <lb/>quoniam lapides diaphani, quorum diaphanitas eſt modica, nõ comprehenduntur à uiſu eſſe dia-<lb/>phani, niſi poſtquam fuerint oppoſiti luci, & comprehendatur lux à poſteriori eorum:</s> <s xml:id="echoid-s1771" xml:space="preserve"> & com-<lb/>prehendentur, quòd ſunt diaphani.</s> <s xml:id="echoid-s1772" xml:space="preserve"> Et ſimiliter diaphanitas cuiuslibet corporis diaphani, non <lb/>comprehenditur à uiſu, niſi poſtquam comprehenſum fuerit corpus aut lux, quæ eſt à poſterio-<lb/>ri eius, & comprehendatur inſuper per diſtinctionem, quòd illud, quod appareat à poſteriori, eſt <lb/>diuerſum à corpore diaphano:</s> <s xml:id="echoid-s1773" xml:space="preserve"> comprehenſio autem eius, quòd illud, quod eſt à poſteriori cor-<lb/>poris diaphani, eſt diuerſum ab illo corpore, non eſt comprehenſio ſolo ſenſu, ſed eſt compre-<lb/>henſio per rationem.</s> <s xml:id="echoid-s1774" xml:space="preserve"> Et cum diaphanitas non comprehendatur niſi per ſignationem, ergo non <lb/>comprehendetur, niſi diſtinctione & ratione.</s> <s xml:id="echoid-s1775" xml:space="preserve"> Et etiam ſcriptura non comprehenditur, niſi ex <lb/>diſtinctione formarum literarum, & compoſitione illarum, & comparatione illarum ex ſibi ſimi-<lb/>libus, quæ ſunt notæ ſcriptori antè.</s> <s xml:id="echoid-s1776" xml:space="preserve"> Et ſimiliter multæ res uiſibiles, quando conſiderabitur quali-<lb/> <pb o="31" file="0037" n="37" rhead="OPTICAE LIBER II."/> tas comprehenſionis illarum, non comprehenduntur ſolo ſenſu, ſed ratione & diſtinctione.</s> <s xml:id="echoid-s1777" xml:space="preserve"> Et cum <lb/>ita ſit, non ergo omne, quod comprehenditur à uiſu, comprehenditur ſolo ſenſu:</s> <s xml:id="echoid-s1778" xml:space="preserve"> ſed multæ in-<lb/>tentiones uiſibiles comprehenduntur per rationem & diſtinctionem cum ſenſu formæ uiſæ.</s> <s xml:id="echoid-s1779" xml:space="preserve"> Viſus <lb/>autem non habet uirtutem diſtinguendi, ſed uirtus diſtinctiua diſtinguit iſtas res:</s> <s xml:id="echoid-s1780" xml:space="preserve"> attamen diſtin-<lb/>ctio uirtutis diſtinctiuæ in iſtis rebus uiſibilibus non eſt, niſi mediante uiſu.</s> <s xml:id="echoid-s1781" xml:space="preserve"> Et etiam uiſus compre-<lb/>hendit multas res uiſas per cognitionem, & cognoſcit hominem eſſe hominem, & equum equum, <lb/>& Socratem eſſe Socratem, quando uiderit illum prius:</s> <s xml:id="echoid-s1782" xml:space="preserve"> & cognoſcit animalia ſibi aſſueta, & arbo-<lb/>res, & plantas, & lapides, quando prius uiderit ipſa, & conſimilia.</s> <s xml:id="echoid-s1783" xml:space="preserve"> Et cognoſcit omnes intentiones <lb/>in rebus uιſibilibus ſibi aſſuetas.</s> <s xml:id="echoid-s1784" xml:space="preserve"> Et non cõprehendit uiſus quidditatẽ alicuius rei, niſi per cognitio <lb/>nem.</s> <s xml:id="echoid-s1785" xml:space="preserve"> Cognitio autẽ non eſt comprehenſio ſolo ſenſu, quoniã uiſus nõ cognoſcit omne, quod uidit <lb/>prius.</s> <s xml:id="echoid-s1786" xml:space="preserve"> Et cum uiſus comprehenderit aliquod indiuiduum, & poſtea ſeparabitur ab illo longo tem-<lb/>pore, & pòſt uiderit ipſum:</s> <s xml:id="echoid-s1787" xml:space="preserve"> & non fuerit memor ipſius:</s> <s xml:id="echoid-s1788" xml:space="preserve"> non cognoſcet ipſum:</s> <s xml:id="echoid-s1789" xml:space="preserve"> quoniam non cogno-<lb/>ſcit illud, quod cognouit, niſi quando fuerit memor.</s> <s xml:id="echoid-s1790" xml:space="preserve"> Si ergo cognitio eſſet comprehenſio ſolo ſenſu:</s> <s xml:id="echoid-s1791" xml:space="preserve"> <lb/>oporteret, quãdo uideret uiſus aliquod indiuiduũ, quod prius uidit, quòd ſtatim cognoſceret ipſum <lb/>in ſecunda uiſione ſecundum omnes diſpoſitiones:</s> <s xml:id="echoid-s1792" xml:space="preserve"> ſed non eſt ita.</s> <s xml:id="echoid-s1793" xml:space="preserve"> Et cum cognitio nõ ſit niſi per re <lb/>memorationem:</s> <s xml:id="echoid-s1794" xml:space="preserve"> cognitio ergo non eſt comprehenſio ſolo ſenſu.</s> <s xml:id="echoid-s1795" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div65" type="section" level="0" n="0"> <head xml:id="echoid-head88" xml:space="preserve" style="it">11. Viſio per anticipatam notionem fit quodammodo per ſyllogiſmum. 63 p 3.</head> <p> <s xml:id="echoid-s1796" xml:space="preserve">COmprehenſio autem per cognitionem eſt comprehenſio per aliquem modorum ratiocina-<lb/>tionis, quoniam cognitio eſt comprehenſio conſimilitudinis duarum formarum, ſcilicet for <lb/>mæ quam comprehendit uiſus apud cognitionem, & formę illius rei uiſæ uel ſibi ſimilis, quã <lb/>comprehendebat in prima uice:</s> <s xml:id="echoid-s1797" xml:space="preserve"> & propter hoc nõ erit cognitio niſi per rememorationẽ.</s> <s xml:id="echoid-s1798" xml:space="preserve"> Quoniam <lb/>ſi prima forma non fuerit præſens memorię, non comprehendet ui<gap/>us ſimιlitudinem duarum for-<lb/>marum, & ſic non cognoſcit rem uiſam.</s> <s xml:id="echoid-s1799" xml:space="preserve"> Cognitio autem eſt formæ alicuius rei indiuiduæ, & formæ <lb/>ſpeciei.</s> <s xml:id="echoid-s1800" xml:space="preserve"> Cognitio ergo indiuidui eſt ex aſsimilatione formæ indiuidui, quam comprehendit uiſus a-<lb/>pud cognitionem indiuidui, alij formæ, quam prius comprehendebat.</s> <s xml:id="echoid-s1801" xml:space="preserve"> Et cognitio ſpeciei eſt ex aſsi <lb/>milatione formæ rei uiſæ ad alias formas ſimiles indiuiduis ſuæ ſpeciei, quæ prius cõprehendebat.</s> <s xml:id="echoid-s1802" xml:space="preserve"> <lb/>Et comprehenſio ſimilitudinis eſt comprehenſio per rationem, quoniam non eſt, niſi ex compara-<lb/>tione unius formę ad alteram.</s> <s xml:id="echoid-s1803" xml:space="preserve"> Cognitio ergo non eſt, niſi modus rationis.</s> <s xml:id="echoid-s1804" xml:space="preserve"> Sed iſta ratio diſtinguitur <lb/>ab omnibus rationibus:</s> <s xml:id="echoid-s1805" xml:space="preserve"> quoniam cognitio non erit per inductionem omnium intentionum, quæ <lb/>ſunt in forma, ſed per ſigna.</s> <s xml:id="echoid-s1806" xml:space="preserve"> Cum ergo uiſus comprehendit aliquam intentionum, quæ ſunt in for-<lb/>ma, & fuerit memor primæ formæ, ſtatim cognoſcet formam, & non eſt ita omne, quod comprehen <lb/>dit per rationem:</s> <s xml:id="echoid-s1807" xml:space="preserve"> quonιam plura eorum, quę comprehenduntur per rationem, non comprehendun <lb/>tur, niſi poſt inductionem omnium intentionum, quæ ſunt in eis.</s> <s xml:id="echoid-s1808" xml:space="preserve"> Quoniã ſcriptor quando momen <lb/>to aſpexerit formam a b c, ſtatim comprehendet, quod eſt a b c.</s> <s xml:id="echoid-s1809" xml:space="preserve"> ex apprehenſione ergo eius, quòd a <lb/>eſt præcedens, & c eſt ultimum, comprehendet, quod eſt a b c.</s> <s xml:id="echoid-s1810" xml:space="preserve"> Et ſimiliter ſi uiderit (<emph style="sc">DOMINVS</emph>) <lb/>ſcriptum, ſtatim comprehendet ipſum per cognitionem & conſuetudinem:</s> <s xml:id="echoid-s1811" xml:space="preserve"> & ſimiliter omnes diſ-<lb/>poſitiones ſibi aſſuetas, quando ſcriptor uiderit ipſas, ſtatim comprehendet ſine indigentia diſtin-<lb/>ctionis unius ab altera:</s> <s xml:id="echoid-s1812" xml:space="preserve"> & non eſt ita, ſi ſcriptor inſpexerit dictionem extraneam ſcriptam, quam <lb/>antè non uidit, quoniam ſcriptor non comprehendet iſtam dictionem, niſi poſtquam diſtinxerit e-<lb/>ius literas, & pòſt comprehendet dictionem.</s> <s xml:id="echoid-s1813" xml:space="preserve"> Omnis ergo forma, quam prius non uidit uiſus, neq;</s> <s xml:id="echoid-s1814" xml:space="preserve"> <lb/>ſimilem illi, quando comprehendetur à uiſu, non comprehendet uiſus, quòd eſt illa forma, niſi poſt-<lb/>quam diſtinxerit omnes illas intentiones illius formæ, aut plures illarum.</s> <s xml:id="echoid-s1815" xml:space="preserve"> Forma autem conſue-<lb/>ta comprehendetur à uiſu ſtatim comprehenſione quarundam intentionum, quæ ſunt in illa for-<lb/>ma.</s> <s xml:id="echoid-s1816" xml:space="preserve"> Illud ergo quod comprehenditur per cognitionem, comprehendetur per ſignum:</s> <s xml:id="echoid-s1817" xml:space="preserve"> & non o-<lb/>mne quod comprehenditur per rationem, comprehenditur per ſignum.</s> <s xml:id="echoid-s1818" xml:space="preserve"> Et plures intentiones uiſi-<lb/>bilium non comprehenduntur niſi per cognitionem.</s> <s xml:id="echoid-s1819" xml:space="preserve"> Et non comprehendetur quidditas alicuius <lb/>rei uiſæ, neque alicuius rei ſenſibilis alio ſenſu, niſi per cognitionem.</s> <s xml:id="echoid-s1820" xml:space="preserve"> Et uirtus cognitionis eſt con-<lb/>iuncta uirtuti ſenſus:</s> <s xml:id="echoid-s1821" xml:space="preserve"> & non completur comprehenſio uiſibilium, niſi per cognitionem.</s> <s xml:id="echoid-s1822" xml:space="preserve"> Cogni-<lb/>tio autem non eſt ſolo ſenſu.</s> <s xml:id="echoid-s1823" xml:space="preserve"> Intentiones ergo quæ comprehenduntur à ſenſu uiſu quædam com-<lb/>prehenduntur ſolo ſenſu, quædam per cognitionem, quædam per rationem & diſtinctionem.</s> <s xml:id="echoid-s1824" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div66" type="section" level="0" n="0"> <head xml:id="echoid-head89" xml:space="preserve" style="it">12. Viſio per ſyllogiſmum, fit plerun breui tempore. 69 p 3.</head> <p> <s xml:id="echoid-s1825" xml:space="preserve">ET plures intentiones uiſibilium, quæ comprehenduntur per rationem & diſtinctionem, <lb/>comprehenduntur in tempore ualde paruo, & non apparet, quòd comprehenſio earum ſit <lb/>per rationem & diſtinctionem, propter uelocitatem rationis, per quam comprehenduntur <lb/>iſtæ intentiones.</s> <s xml:id="echoid-s1826" xml:space="preserve"> Quoniam figura, & magnitudo, & diaphanitas corporis, & ſimilia, ex intentioni-<lb/>bus, quæ ſunt in rebus uiſibilibus, comprehenduntur in maiori parte comprehenſione ualde ue-<lb/>loci, & nõ comprehenditur tunc, quòd comprehenſio earum ſit per rationẽ.</s> <s xml:id="echoid-s1827" xml:space="preserve"> Et cum comprehenſio <lb/>iſtarum intentionum eſt per rationem, non eſt, niſi per manifeſtationem poſitionum illarum, & per <lb/>conſuetudinem uirtutis diſtinctiuæ ad iſtas intentiones.</s> <s xml:id="echoid-s1828" xml:space="preserve"> A pud peruentum ergo illius formæ com-<lb/>prehendit omnes intentiones, quæ ſunt in ea, & ſic diſtinguentur ab eo apud comprehenſio <lb/> <pb o="32" file="0038" n="38" rhead="ALHAZEN"/> nem.</s> <s xml:id="echoid-s1829" xml:space="preserve"> Et ſimiliter in argumentatione & omnibus rationibus, quarũ propoſitiones ſunt uniuerſale<gap/> <lb/>& manifeſtæ, non indiget uirtus diſtinctiua aliquanto tempore in comprehendendo illarum con-<lb/>cluſiones, ſed apud intellectum ſtatim propoſitionis intelligetur concluſio.</s> <s xml:id="echoid-s1830" xml:space="preserve"> Et cauſa in hoc eſt, <lb/>quòd uirtus diſtinctiua non arguit per compoſitionem & ordinationem propoſitionis, ſicut com-<lb/>ponitur argumentatio per uocabula.</s> <s xml:id="echoid-s1831" xml:space="preserve"> Quoniam argumentum, quod concludit, erit ſecundum uer-<lb/>bum, & ſecundum ordinationem propoſitionum:</s> <s xml:id="echoid-s1832" xml:space="preserve"> argumentum autem uirtutis diſtinctiuæ non eſt <lb/>ita, quoniam uirtus diſtinctiua comprehendit concluſionem ſine indigentia in uerbis, & ſine in-<lb/>digentia ordinationis propoſitionum, & ordinationis uerborum:</s> <s xml:id="echoid-s1833" xml:space="preserve"> quoniam ordinatio uerborum <lb/>argumenti non eſt, niſi modus qualitatis comprehenſionis uirtutis diſtinctiuæ à concluſione:</s> <s xml:id="echoid-s1834" xml:space="preserve"> Sed <lb/>comprehenſio uirtutis diſtinctiuæ ad concluſionem non indiget modo qualitatis, nec ordine qua-<lb/>litatis comprehenſionis.</s> <s xml:id="echoid-s1835" xml:space="preserve"> Intentiones ergo uiſibiles, quæ comprehenduntur à ratione, compre-<lb/>henduntur utplurimùm, ualde uelociter, & non apparet in maiori parte, ſi comprehenſio earum <lb/>ſit in ratione.</s> <s xml:id="echoid-s1836" xml:space="preserve"> Et etiam intentiones uiſibiles, quæ comprehenduntur per rationem & diſtinctio-<lb/>nem, quoniam multoties comprehenduntur per rationem, & intelligit uirtus diſtinctiua intentio-<lb/>nes earum:</s> <s xml:id="echoid-s1837" xml:space="preserve"> ſi pòſt uiderit ipſas, comprehendet eas per cognitionem ſine indigentia diſtinctionis <lb/>omnium intentionum, quæ ſunt in ſecundis, ſed per ſigna tantùm, & diſtinguet illam concluſio-<lb/>nem per cognitionem ſine indigentia argumentationis alicuius iterandæ:</s> <s xml:id="echoid-s1838" xml:space="preserve"> & eſt exemplum in eo <lb/>ſcriptore, qui primo uidet uerbum extraneum.</s> <s xml:id="echoid-s1839" xml:space="preserve"> Et ſimiliter ſunt omnes intentiones, quæ compre-<lb/>henduntur per rationem, quando propoſitiones earum fuerint manifeſtæ, & concluſiones fuerint <lb/>ueræ.</s> <s xml:id="echoid-s1840" xml:space="preserve"> Quoniam quando anim a intellexerit concluſionem eſſe ueram, deinde multoties uenerit in <lb/>animam:</s> <s xml:id="echoid-s1841" xml:space="preserve"> erit concluſio quaſi propoſitio manifeſta:</s> <s xml:id="echoid-s1842" xml:space="preserve"> & ſic, quando anima uiderit propoſitionem, ſta-<lb/>tim intelliget concluſionem ſine indigentia argumentationis iterandæ.</s> <s xml:id="echoid-s1843" xml:space="preserve"> Et plures intentiones, quas <lb/>non comprehendit uirtus diſtinctiua, quòd ſint ueræ, niſi per rationem, putantur quòd ſint propo-<lb/>ſitiones primæ, & quòd non comprehendantur, niſi per naturam & intellectum, non per rationem:</s> <s xml:id="echoid-s1844" xml:space="preserve"> <lb/>uerbi gratia, quòd totum ſit maius ſua parte, putatur quòd natura intellectus iudicet quòd ſit ue-<lb/>rum, & quòd comprehenſio ueritatis ipſius non eſt per rationem.</s> <s xml:id="echoid-s1845" xml:space="preserve"> Sed totum eſt maius ſua parte, <lb/>non comprehendet prius, niſi per rationem, quoniam diſtinctio non habet uiam ad comprehenden <lb/>dum, quòd totum ſit maius ſua parte, niſi poſtquã intellexerit intentiones totius & partis, & inten-<lb/>tionem maioritatis & minoritatis:</s> <s xml:id="echoid-s1846" xml:space="preserve"> quoniam ſi non intellexerit intentionem partium, non intelli-<lb/>get intentionem totius.</s> <s xml:id="echoid-s1847" xml:space="preserve"> Intentio autem totius non eſt niſi communitas, & intentio partis, niſi ali-<lb/>quiditas, & maioritas eſt relatio ad alterum, & intentio maioris eſt illud, quod eſt æquale alij, & <lb/>plus.</s> <s xml:id="echoid-s1848" xml:space="preserve"> Et probatio quòd omne totum eſt maius ſua parte, eſt quod confertur ei cum quadam æqui-<lb/>ualentia, & addit ſuper ipſam cum reſiduo, quod eſt plus ſcilicet:</s> <s xml:id="echoid-s1849" xml:space="preserve"> & ex conuenientia intentionis <lb/>maioris cum intentione totius:</s> <s xml:id="echoid-s1850" xml:space="preserve"> & argumentatione apparet, quòd totum ſit maius ſua parte.</s> <s xml:id="echoid-s1851" xml:space="preserve"> Et <lb/>cum comprehenſio huius propoſitionis, quòd totum ſit maius ſua parte, non ſit niſi per iſtam <lb/>uiam:</s> <s xml:id="echoid-s1852" xml:space="preserve"> comprehenſio ergo eius non eſt, niſi per rationem, non per naturam intellectus:</s> <s xml:id="echoid-s1853" xml:space="preserve"> & illud, <lb/>quod eſt in natura intellectus, non eſt niſi comprehenſio conuenientiæ intentionis totius, & inten-<lb/>tionis maioris, & in augmentatione tantùm.</s> <s xml:id="echoid-s1854" xml:space="preserve"> Et ordinatio iſtius ſyllogiſmi eſt ita:</s> <s xml:id="echoid-s1855" xml:space="preserve"> omne totum ad-<lb/>dit ſuper partem:</s> <s xml:id="echoid-s1856" xml:space="preserve"> & omne addens ſuper aliud, eſt maius ipſo:</s> <s xml:id="echoid-s1857" xml:space="preserve"> ergo omne totum eſt maius ſua par-<lb/>te.</s> <s xml:id="echoid-s1858" xml:space="preserve"> Et uelocitas comprehenſionis uirtutis diſtinctiuæ circa concluſionem, non eſt, niſi quia pro-<lb/>poſitio uniuerſalis eſt manifeſta ex comprehenſione uirtutis diſtinctiuæ:</s> <s xml:id="echoid-s1859" xml:space="preserve"> ſed comprehenſio, quòd <lb/>totum eſt maius parte, eſt per rationem.</s> <s xml:id="echoid-s1860" xml:space="preserve"> Et quia propoſitio uniuerſalis eſt ei manifeſta, compre-<lb/>hendet concluſionem apud euentum propoſitionis minoris particularis, & propoſitio particula-<lb/>ris eſt additio intentionis totiu<gap/> ſuper partem.</s> <s xml:id="echoid-s1861" xml:space="preserve"> Et quia ueritas concluſionis iſtius ſyllogiſmi eſt cer-<lb/>tiſsima in anima, & præſens in memoria:</s> <s xml:id="echoid-s1862" xml:space="preserve"> quando ueniet propoſitio ad ipſum, recipit ipſam intelle-<lb/>ctus ſine indigentia argumentationis iterandæ, ſed per cognitionem tantùm.</s> <s xml:id="echoid-s1863" xml:space="preserve"> Et omne, quod eſt i-<lb/>ſtius generis, uocatur ab hominibus propoſitio prima:</s> <s xml:id="echoid-s1864" xml:space="preserve"> & putatur, quòd comprehendatur ſolo in-<lb/>tellectu, & quòd non indigeatur in comprehenſione ueritatis circa ipſum, niſi ſolo intellectu.</s> <s xml:id="echoid-s1865" xml:space="preserve"> Et <lb/>cauſſa illius eſt, quòd comprehenditur ſtatim.</s> <s xml:id="echoid-s1866" xml:space="preserve"> Syllogiſmi ergo, quorum propoſitiones ſunt uniuer <lb/>ſales & manifeſtæ, comprehenduntur in tempore inſenſibili:</s> <s xml:id="echoid-s1867" xml:space="preserve"> deinde quando ſyllogizatur toties, <lb/>ut ueritas concluſionis certificetur in anima, tunc efficietur concluſio quaſi propoſitio manifeſta.</s> <s xml:id="echoid-s1868" xml:space="preserve"> <lb/>Et ſecundum hunc modum erit comprehenſio uirtutis diſtinctiuæ ad plures intentiones, quæ com <lb/>prehenduntur ratione in tempore inſenſibili, ſine indigentia argumentationis iterandæ.</s> <s xml:id="echoid-s1869" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div67" type="section" level="0" n="0"> <head xml:id="echoid-head90" xml:space="preserve" style="it">13. Viſio per anticipatam notionem fit in tempore: & qualitas ei{us} plerunque ignoratur. <lb/>64. 69 p 3.</head> <p> <s xml:id="echoid-s1870" xml:space="preserve">ET etiam multoties non apparet qualitas comprehenſionis intentionum uiſibilium, quæ com <lb/>prehenduntur ratione & cognitione, quoniam comprehenſio earum non fit ualde uelociter, <lb/>& quia comprehenſio qualitatis comprehenſionis non eſt, niſi per ſecundum argumentum <lb/>poſt primum argumentum, per quod fuit uiſio.</s> <s xml:id="echoid-s1871" xml:space="preserve"> Virtus àutem diſtinctiua non utitur iſto ſecun-<lb/>do argumento, in tempore, in quo comprehendit aliquam intentionem uiſibilem, neque diſtin-<lb/>guit qualiter comprehendit illam intentionem, neq;</s> <s xml:id="echoid-s1872" xml:space="preserve"> poteſt, propter uelocitatem comprehenſionis <lb/><gap/>ius ad intentiones cõprehenſas per cognitionẽ & per argumentũ, cuius propoſitiones ſunt mani-<lb/> <pb o="33" file="0039" n="39" rhead="OPTIC AE LIBER II."/> feſtæ & certæ in anima.</s> <s xml:id="echoid-s1873" xml:space="preserve"> Et propter hoc non ſentitur qualitas comprehenſionis ueritatis plurium <lb/>propoſitionum uerarum, quæ comprehenduntur per cognitionem:</s> <s xml:id="echoid-s1874" xml:space="preserve"> Et radix affirmationis ueritatis <lb/>earum eſt per rationem apud earum euentum.</s> <s xml:id="echoid-s1875" xml:space="preserve"> Quoniam quando iſtæ propoſitiones eueniunt <lb/>uirtuti diſtinctiuæ, ſtatim iudicat, quòd ſint ueræ per cognitionem:</s> <s xml:id="echoid-s1876" xml:space="preserve"> ſed apud cognitionem non in-<lb/>quirit qualiter affirmata fuerit prius ueritas, neque inquirit, qualiter comprehendit, quòd ueræ ſint <lb/>apud euentum earum.</s> <s xml:id="echoid-s1877" xml:space="preserve"> Et etiam pari modo argumentum, per quod comprehendit uirtus diſtincti-<lb/>ua qualitatem comprehenſionis eius ad illud, quod comprehendit, non eſt argumentum in fine ue-<lb/>locitatis, ſed indiget conſideratione, quoniam comprehenſiones diuerſantur, & quædam ſunt per <lb/>naturam intellectus, & quædam per cognitionem, & quædam per conſiderationem & diſtinctio-<lb/>nem.</s> <s xml:id="echoid-s1878" xml:space="preserve"> Comprehenſio ergo qualitatis comprehenſionis, & quæ coprehenſio eiuſmodi cõprehenſio-<lb/>nis eſt, non eſt, niſi per argumentum & diſtinctionem non uelocem.</s> <s xml:id="echoid-s1879" xml:space="preserve"> Et propter hoc non apparet <lb/>multoties qualitas comprehenſionis rerum uiſibilium, quæ comprehenduntur ratione apud com-<lb/>prehenſionem.</s> <s xml:id="echoid-s1880" xml:space="preserve"> Et etiam eſt homo natus ad diſtinguendum ſine difficultate, & arguendum ſine la-<lb/>bore, & non percipit, quod arguit, niſi quando arguit cum difficultate, quando uerò non utitur dif-<lb/>ficultate, & cognitione, non percipit, quod arguit.</s> <s xml:id="echoid-s1881" xml:space="preserve"> Argumenta ergo aſſueta, quorum propoſitiones <lb/>ſunt manifeſtæ, & non indigent difficultate, ſunt in homine naturaliter:</s> <s xml:id="echoid-s1882" xml:space="preserve"> & propter hoc percipit, <lb/>quando comprehendit concluſiones eorum, quòd comprehendat ipſas per argumentum.</s> <s xml:id="echoid-s1883" xml:space="preserve"> Et ſigni-<lb/>ficatio eſt, quòd homo natus eſt ad arguendum, quòd ipſe arguit, & non percipit quòd arguit, quod <lb/>apparet in pueris in primo incremento:</s> <s xml:id="echoid-s1884" xml:space="preserve"> quoniam ipſi comprehendunt plures res, ſicut homo per-<lb/>fectus, & diſtinguens, & utuntur multis operationibus per diſtinctionem:</s> <s xml:id="echoid-s1885" xml:space="preserve"> uerbi gratia:</s> <s xml:id="echoid-s1886" xml:space="preserve"> Puer quan-<lb/>do ei demonſtrantur duo ex eodem genere, ſicut duo poma, & fuerit unum pulchrius alio, accipiet <lb/>pulchrius, & dimittet alterum, ſed electio rei pulchrioris non eſt, niſi per comparationem alterius <lb/>ad alterum:</s> <s xml:id="echoid-s1887" xml:space="preserve"> & comprehenſio pulchri, quòd ſit pulchrum, & fœdi, quòd ſit fœdum:</s> <s xml:id="echoid-s1888" xml:space="preserve"> & ſimiliter quan <lb/>do elegerit pulchrius alio pulchro minoris pulchritudinis, ſignificat quòd non elegit ipſum, niſi <lb/>poſt comparationem unius ad alterum, & comprehenſionem formæ cuiuslibet eorum, & compre-<lb/>henſionem argumenti pulchritudinis pulchrioris ſuper minus pulchrum:</s> <s xml:id="echoid-s1889" xml:space="preserve"> & electio pulchrioris <lb/>non eſt, niſi per propoſitionem uniuerſalem dicentem:</s> <s xml:id="echoid-s1890" xml:space="preserve"> Quòd pulchrius eſt, melius eſt:</s> <s xml:id="echoid-s1891" xml:space="preserve"> & quod eſt <lb/>melius, dignius eſt ad eligendum:</s> <s xml:id="echoid-s1892" xml:space="preserve"> ergo ipſe utitur hac propoſitione, & non percipit, quòd utatur ea.</s> <s xml:id="echoid-s1893" xml:space="preserve"> <lb/>Et cum ita ſit:</s> <s xml:id="echoid-s1894" xml:space="preserve"> puer ergo arguit & diſtinguit:</s> <s xml:id="echoid-s1895" xml:space="preserve"> & non eſt dubium, quòd puer neſcit, quod eſt argumen <lb/>tum, neque percipit quando arguit, utrum arguat, aut non:</s> <s xml:id="echoid-s1896" xml:space="preserve"> & ſi quis etiam intenderet ipſum inſtrue <lb/>re, quid ſit argumentum, uel arguere, non intelligeret.</s> <s xml:id="echoid-s1897" xml:space="preserve"> Et quia puer arguit, & neſcit, quid ſit argu-<lb/>mentum, anima ergo humana nata eſt ad arguendum ſine difficultate & labore, & non percipit ho-<lb/>mo apud comprehenſionem rei, quòd ſit huiuſmodi, quòd ſit per argumentum.</s> <s xml:id="echoid-s1898" xml:space="preserve"> Sed intentiones, <lb/>quæ comprehenduntur ratione, non ſunt, niſi intentiones manifeſtæ, quarum propoſitiones ſunt <lb/>ualde manifeſtæ:</s> <s xml:id="echoid-s1899" xml:space="preserve"> intentiones uerò, quarum propoſitiones non ſunt ualde manifeſtæ, & quarum ar-<lb/>gumenta indigent difficultate, quando comprehenduntur ab homine, fortè percipit, quòd compre <lb/>hendit ipſas per rationem, quando fuerint illæ ueræ diſtinctionis.</s> <s xml:id="echoid-s1900" xml:space="preserve"> Iam ergo declaratum eſt ex omni <lb/>quod diximus, quòd quædam intentiones, quæ comprehenduntur per uiſum, comprehenduntur <lb/>ſolo ſenſu, & quædam per diſtinctionem, & quædam per cognitionem, & argumentum, & rationem <lb/>& poſitionem:</s> <s xml:id="echoid-s1901" xml:space="preserve"> & quòd qualitas comprehenſionis intentionum particularium per uiſum, non appa <lb/>ret in maiori parte propter uelocitatem iſtius, quod comprehenditur per cognitionem, & propter <lb/>uelocitatem argumenti, per quod comprehenduntur intentiones uiſibiles:</s> <s xml:id="echoid-s1902" xml:space="preserve"> & quòd uirtus diſtincti <lb/>ua eſt nata ad arguendum ſine labore & difficultate, ſed natura & conſuetudine, & non indiget ar-<lb/>gumentatione iteranda illa uirtus in comprehenſione alicuius intentionum particulariũ, quę mu<gap/> <lb/>toties fuerint uiſæ.</s> <s xml:id="echoid-s1903" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div68" type="section" level="0" n="0"> <head xml:id="echoid-head91" xml:space="preserve" style="it">14. È uiſibili ſæpi{us} uiſoremanet in animo gener alis notio, qua quodlibet uiſibile ſimile per-<lb/>cipitur & cognoſcitur. 61 p 3.</head> <p> <s xml:id="echoid-s1904" xml:space="preserve">ET comprehenduntur etiam intentiones, quæ multoties fuerint uiſæ, ratione & diſtinctione, <lb/>quæ ſunt in anima, ita quòd homo non percipit quietem illarum, neque quies illarum habe<gap/> <lb/>principium ſenſibile, quoniam habet experientia, quòd comprehendit uiſibilia:</s> <s xml:id="echoid-s1905" xml:space="preserve"> & experien-<lb/>tia eſt in eo quædam diſtinctio, & præcipuè diſtinctio, per quam comprehenduntur intentiones <lb/>ſenſib iles:</s> <s xml:id="echoid-s1906" xml:space="preserve"> Ipſe ergo comprehendit intentiones ſenſibiles ratione & diſtinctione, & acquirit inten-<lb/>tiones ſenſibilium.</s> <s xml:id="echoid-s1907" xml:space="preserve"> Et multoties redduntur ipſæ intentiones ſenſibiles illi ſucceſsiuè, quouſq;</s> <s xml:id="echoid-s1908" xml:space="preserve"> quie-<lb/>ſcant in eius anima:</s> <s xml:id="echoid-s1909" xml:space="preserve"> ita etiam ut non percipiat quietem earum:</s> <s xml:id="echoid-s1910" xml:space="preserve"> & ſic quando uenerit ipſa intentio <lb/>particularis, quæ quieuerit in anima eius, cõprehendet eam apud eius euentũ per cognitionẽ, neq;</s> <s xml:id="echoid-s1911" xml:space="preserve"> <lb/>tamẽ percipit qualitatẽ comprehenſionis, neq;</s> <s xml:id="echoid-s1912" xml:space="preserve"> qualitatẽ cognitionis, neq;</s> <s xml:id="echoid-s1913" xml:space="preserve"> qualiter quieuerit in ani <lb/>ma eius, cognitio ipſius intentiõis.</s> <s xml:id="echoid-s1914" xml:space="preserve"> Oẽs ergo intentiões particulares, quę cõprehenduntur ratione, <lb/>& diſtinctiõe, & multoties redduntur, iam cõprehenſæ ſunt ab homine in præterito tẽpore, & quie <lb/>uerũt in anima, & facta eſt forma uniuerſalis quieſcẽs ex qualibet intentione particulariũ.</s> <s xml:id="echoid-s1915" xml:space="preserve"> Compre <lb/>henduntur ergo intentiones iſtæ ſine argumentatione iteranda, quã primò fecit, & ſine ratione, per <lb/>quã cõprehenſa eſt ueritas illius intentionis, & ſine cõprehenſione qualitatis cõprehenſionis ipſius <lb/>apud comprehenſionẽ, & ſine cõprehenſione qualitatis cognitionis apud comprehenſionem, & ni-<lb/>hil remanet argumentatione iteranda indigens, niſi conſiderare intentiones particulares, quæ ſunt <lb/> <pb o="34" file="0040" n="40" rhead="ALHAZEN"/> in ipſis indiuiduis particularibus, ſicut figura in re indiuidua, ſcilicet in re uiſa ſignata, aut ſitus rei <lb/>uiſæ indiuiduæ, aut magnitudo rei uiſæ indiuiduæ, aut comparatio coloris alicuius rei uiſæ indiui-<lb/>duæ cum colore alterius rei uiſæ & illi ſimilis.</s> <s xml:id="echoid-s1916" xml:space="preserve"> Et ſecundum iſtos modos erit comprehenſio omniũ <lb/>intentionum particularium, quæ ſunt in rebus uiſibilibus.</s> <s xml:id="echoid-s1917" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div69" type="section" level="0" n="0"> <head xml:id="echoid-head92" xml:space="preserve">DE OMNIBVS INTENTIONIBVS COMPREHENSIS À<unsure/> VISV: <lb/>& qualiter comprehendat uiſus quamlib et illarum. Cap. XI.</head> <head xml:id="echoid-head93" xml:space="preserve" style="it">15. Species uiſibiles principes ſunt uigintiduæ: adquas reliquæ omnes referuntur. In hypo. <lb/>3 lib. in præfa. 4 libr.</head> <p> <s xml:id="echoid-s1918" xml:space="preserve">ET cum declarata ſint omnia iſta, incipiemus modò ad declarandum qualitates comprehenſio <lb/>nis cuiuslibet intentionum particularium, quæ comprehenduntur per uiſum, & qualitates <lb/>argumentorum, per quæ acquirit uirtus diſtinctiua intentiones comprehenſas ſenſu uiſus.</s> <s xml:id="echoid-s1919" xml:space="preserve"> <lb/>Intentiones particulares, quæ comprehenduntur ſenſu uiſu, ſunt multæ, ſed generaliter diuiduntur <lb/>in 22:</s> <s xml:id="echoid-s1920" xml:space="preserve"> & ſunt lux, color, remotio, ſitus, corporeitas, figura, magnitudo, continuum, diſcretio & ſepa-<lb/>ratio, numerus, motus, quies, aſperitas, leuitas, diaphanitas, ſpiſsitudo, umbra, obſcuritas, pulchri-<lb/>tudo, turpitudo, conſimilitudo, & diuerſitas in omnibus intentionibus particularibus, & in omni-<lb/>bus formis compoſitis ex omnibus intentionibus particularibus.</s> <s xml:id="echoid-s1921" xml:space="preserve"> Iſta ergo ſunt omnia quæ com-<lb/>prehenduntur per ſenſum uiſus:</s> <s xml:id="echoid-s1922" xml:space="preserve"> & ſi aliqua intentio uiſibilis eſt pręter iſtas, collocabitur ſub aliqua <lb/>iſtarum, ſicut ordinatio, quæ collocabitur ſub ſitu, & ſcriptura, & pictura, quæ collocabuntur ſub fi-<lb/>gura & ordine:</s> <s xml:id="echoid-s1923" xml:space="preserve"> & ſicut rectitudo, & curuitas, & concauitas, & conuexitas, quæ collocantur ſub figu <lb/>ra:</s> <s xml:id="echoid-s1924" xml:space="preserve"> & multitudo & paucitas, quæ collocantur ſub numero:</s> <s xml:id="echoid-s1925" xml:space="preserve"> & ſicut æqualitas & augmentum, quæ <lb/>collocantur ſub ſimilitudine & diuerſitate:</s> <s xml:id="echoid-s1926" xml:space="preserve"> & alacritas, & riſus, & triſtitia, quę comprehenduntur ex <lb/>figura formæ faciei:</s> <s xml:id="echoid-s1927" xml:space="preserve"> collocantur ergo ſub figura:</s> <s xml:id="echoid-s1928" xml:space="preserve"> & ſicut fletus, qui continetur ſub figura faciei cum <lb/>motu lachrymarum, collocatur ergo ſub figura & motu:</s> <s xml:id="echoid-s1929" xml:space="preserve"> & ſicut humilitas & ſiccitas, quæ collocan-<lb/>tur ſub motu & quiete, quoniam humilitas comprehenditur ſenſu uiſu, ſed non ſenſu uiſu compre-<lb/>henditur, niſi ex liquiditate corporis humidi, & ex motu unius partis illius ante aliã, & ſiccitas com <lb/>prehenditur ſenſu uiſus, ſed nõ comprehenditur, niſi ex retentione partium corporis ſicci, & ex pri <lb/>uatione motus liquiditatis:</s> <s xml:id="echoid-s1930" xml:space="preserve"> & ſimiliter quælibet intentio particularis comprehenſa à uiſu, colloca-<lb/>tur ſub partibus, quas diximus prius.</s> <s xml:id="echoid-s1931" xml:space="preserve"> Et omnes intentiones uiſibiles ſunt, ſicut ſuperius diximus.</s> <s xml:id="echoid-s1932" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div70" type="section" level="0" n="0"> <head xml:id="echoid-head94" xml:space="preserve" style="it">16. Viſio perficitur, cum forma uiſibilis cryſtallino humore recepta, in neruum opticum per-<lb/>uenerit. 20 p 3. Idem 25 n 1.</head> <p> <s xml:id="echoid-s1933" xml:space="preserve">ET cum ita ſit, diſtinctio & argumentatio uirtutis diſtinctiuæ, & cognitio formarum & ſigno-<lb/>rum eorum non erunt, niſi ex cognitione uel diſtinctione uirtutis diſtinctiuæ ex formis per-<lb/>uenientibus intra concauum nerui communis, apud comprehenſionem ultimi ſentientis il-<lb/>las, & ex cognitione ſignorum formarum iſtarum.</s> <s xml:id="echoid-s1934" xml:space="preserve"> Et ita corpus ſentiẽs extenſum à ſuperficie mem <lb/>bri ſentientis uſq;</s> <s xml:id="echoid-s1935" xml:space="preserve"> ad concauum nerui communis, ſcilicet ſpiritus uiſibilis eſt ſentiens per totum, <lb/>quoniã uirtus ſenſitiua eſt per totum iſtius corporis.</s> <s xml:id="echoid-s1936" xml:space="preserve"> Cum ergo forma extenditur à ſuperficie mem <lb/>bri ſentientis uſq;</s> <s xml:id="echoid-s1937" xml:space="preserve"> ad concauum nerui communis, quælibet pars corporis ſentientis ſentiet formã:</s> <s xml:id="echoid-s1938" xml:space="preserve"> <lb/>& cum peruenerit forma in concauum nerui communis, comprehendetur ab ultimo ſentiente, & <lb/>tunc erit diſtinctio & argumentatio.</s> <s xml:id="echoid-s1939" xml:space="preserve"> Virtus autem ſenſitiua ſentit formam rei uiſæ ex toto corpore <lb/>ſentiente extenſam à ſuperficie membri ſentientis uſque ad concauum nerui communis:</s> <s xml:id="echoid-s1940" xml:space="preserve"> & uirtus <lb/>diſtinctiua diſtinguit intentiones, quæ ſunt in forma apud comprehenſionem ultimi ſentientis cir-<lb/>ca formam.</s> <s xml:id="echoid-s1941" xml:space="preserve"> Secundum ergo hunc modum erit comprehenſio formarum rerum uiſibilium à uir-<lb/>tute ſenſitiua, & ab ultimo ſentiente, & à uirtute diſtinctiua.</s> <s xml:id="echoid-s1942" xml:space="preserve"> Et declarabitur ex iſta diſpoſitiõe, quòd <lb/>uirtus ſenſitiua ſentit locũ membri ſentientis, in quem peruenit forma, quoniã non ſentit formam, <lb/>niſi ex loco, in quem peruenit forma.</s> <s xml:id="echoid-s1943" xml:space="preserve"> Et declaratũ eſt etiam [25 n 1] quòd à quolibet puncto ſuper-<lb/>ficiei glacialis extenditur forma ſecundum unam uerticationem continuam, cum eo, quod eſt in <lb/>eadem de obliquatione & incuruatione, quouſque perueniat ad unum punctum loci, in quem per-<lb/>uenit forma in concauo nerui communis.</s> <s xml:id="echoid-s1944" xml:space="preserve"> Et cum ita ſit, forma ergo perueniens in partem ſuperfi-<lb/>ciei glacialis, extenditur ab illa parte ad aliam partem concaui nerui communis.</s> <s xml:id="echoid-s1945" xml:space="preserve"> Et forma cuiusli-<lb/>bet uiſarum rerum diuerſarum, quæ comprehenduntur ſimul in eodem tempore:</s> <s xml:id="echoid-s1946" xml:space="preserve"> extenditur ad lo-<lb/>cum certum in concauo nerui communis:</s> <s xml:id="echoid-s1947" xml:space="preserve"> & perueniunt formæ omnium illarum rerum uiſarum <lb/>ad concauum nerui communis:</s> <s xml:id="echoid-s1948" xml:space="preserve"> & erit ordinatio formarum illarum inter ſe in concauo nerui cõmu <lb/>nis, ſicut ordinatio ipſarum rerum inter ſe uiſarum.</s> <s xml:id="echoid-s1949" xml:space="preserve"> Cum ergo uiſus fuerit oppoſitus alicui rei uiſæ, <lb/>formæ lucis & coloris iſtius rei uiſæ perueniunt ad ſuperficiem uiſus, & perueniunt in ſuperficiem <lb/>glacialis, & extenduntur ſecundum uerticationes determinatas, quas diximus ſecundũ ſuam ordi-<lb/>nationem, & figurã, & formã, quouſq;</s> <s xml:id="echoid-s1950" xml:space="preserve"> perueniant ad concauũ nerui cõmunis, & comprehendentur <lb/>à uirtute ſenſitiua apud peruentũ earũ in corpore glacialis, & apud peruentũ earũ in toto corpore <lb/>ſentiente, & uirtus diſtinctiua diſtinguit omnes intentiones, quæ ſunt in eis:</s> <s xml:id="echoid-s1951" xml:space="preserve"> & forma lucis & forma <lb/>coloris nõ perueniũt ad cõcauũ nerui, niſi quia corpus ſentiẽs extẽſum in cõcauo nerui, coloratur à <lb/>forma lucis & coloris, & illuminatur à forma lucis, & peruenit forma ad cõcauũ nerui cõmunis:</s> <s xml:id="echoid-s1952" xml:space="preserve"> & <lb/> <pb o="35" file="0041" n="41" rhead="OPTIC AE LIBER II."/> erit pars corporis ſentientis, quod eſt in concauo nerui cõmunis, ad quam peruenit forma rei uiſæ, <lb/>colorata colore illius rei uiſæ, & illuminata luce, quæ eſt in illa re uiſa:</s> <s xml:id="echoid-s1953" xml:space="preserve"> & ſi res uiſa habuerit unũ co-<lb/>lorem, erit illa pars corporis ſentientis unius coloris, & ſi partes rei uiſæ fuerint diuerſi coloris, erũt <lb/>partes illius corporis partis ſentientis, quod eſt in concauo nerui cõmunis, diuerſi coloris:</s> <s xml:id="echoid-s1954" xml:space="preserve"> & ulti-<lb/>mum <gap/>entiens ſentit colorem rei uiſæ ex coloratione, quam inuenit in illa parte, & cõprehendit lu-<lb/>cem rei uiſæ ex illuminationè, quam inuenit in illa parte.</s> <s xml:id="echoid-s1955" xml:space="preserve"> Et uirtus diſtinctiua comprehendit plures <lb/>intentiones particulares, quæ ſunt in re uiſa, ex diſtinctione intentionum, quæ ſunt in illa forma ab <lb/>ea, ſcilicet ex ordinatione partium formæ, & ex figuratione illius, quod continet formam, & ex figu <lb/>ratione partium eius, & diuerſitate colorum, & ſituum & ordinationum, quæ ſunt in partibus illius <lb/>formæ, & ex conſimilitudine & diuerſitate earum.</s> <s xml:id="echoid-s1956" xml:space="preserve"> Et etiam lux ueniens à re uiſa, colorata ad uiſum, <lb/>non uenit per ſe ſine colore, & forma coloris ueniens à re uiſa, colorata ad uiſum, non uenit ſine lu-<lb/>ce, & non uenit forma lucis & coloris, quæ ſunt in re uiſa, niſi admixtæ, neq;</s> <s xml:id="echoid-s1957" xml:space="preserve"> comprehendit eas ulti-<lb/>mum ſentiens, niſi admixtas:</s> <s xml:id="echoid-s1958" xml:space="preserve"> tamen etiam ſentiens comprehendit rem uiſam illuminatã, & compre <lb/>hendit, quòd lux apparens in re uiſa, eſt diuerſa à colore:</s> <s xml:id="echoid-s1959" xml:space="preserve"> & iſta comprehenſio eſt diſtincti<gap/>.</s> <s xml:id="echoid-s1960" xml:space="preserve"> Diſt n-<lb/>ctio autem non eſt, niſi uirtutis diſtinctiuæ, non ſenſitiuæ<gap/>tamen cum comprehenſione iſtius inten <lb/>tionis à uirtute diſtinctiua, iſta intentio quieſcit in anima, & non indiget argumentatione iteranda <lb/>apud euentũ cuiuslibet formę.</s> <s xml:id="echoid-s1961" xml:space="preserve"> Sed quod lux, quæ eſt in ea, eſt diuerſa à colore, quæ eſt in ea:</s> <s xml:id="echoid-s1962" xml:space="preserve"> & com <lb/>prehenſio uirtutis diſtinctiuæ, quòd lux accidentalis, quæ eſt in re uiſa colorata, eſt diuerſa à colo-<lb/>re, qui eſt in ea:</s> <s xml:id="echoid-s1963" xml:space="preserve"> eſt, quia ſuper unam rem uiſam diuerſatur lux, & aliquando augmentatur, & aliquan <lb/>do diminuitur.</s> <s xml:id="echoid-s1964" xml:space="preserve"> Et cum hoc eſt, remanet color eius idem, quamuis diuerſetur ſcintillatio coloris ſe-<lb/>cundum diuerſitatem lucis, tamen genus coloris nõ diuerſatur.</s> <s xml:id="echoid-s1965" xml:space="preserve"> Et etiam lux accidentalis fortè per-<lb/>uenit ad rem uiſam ex foramine, & cum fuerit obſtructum illud foramen, obſcurabitur illa res uiſa.</s> <s xml:id="echoid-s1966" xml:space="preserve"> <lb/>Ex comprehenſione ergo uirtutis dιſtinctiuæ circa diuerſitatem lucis ſuper res uiſas, & ex compre <lb/>henſione eius circa illuminationem rei uιſæ, aliquando etiam priuationem lucis ab ea, comprehen-<lb/>dit uiſus, quòd colores, qui ſunt in rebus uiſis, ſunt diuerſi à luce, quæ accidit in eis.</s> <s xml:id="echoid-s1967" xml:space="preserve"> Forma ergo, <lb/>quam comprehendit ſentiens ex re uiſa colorata, eſt forma admixta ex forma lucis & forma coloris, <lb/>quæ ſunt in re uiſa:</s> <s xml:id="echoid-s1968" xml:space="preserve"> Et uirtus diſtιnctiua comprehendit, quòd color, qui eſt in eo, eſt diuerſus à luce, <lb/>quę eſt in ea.</s> <s xml:id="echoid-s1969" xml:space="preserve"> Et iſta cõprehenſio, eſt comprehenſio ſecundum cognit<gap/>onẽ apud euentũ formæ, quæ <lb/>eſt in ſentiente:</s> <s xml:id="echoid-s1970" xml:space="preserve"> quoniam iam quieſcit in anima, quòd lux cuiuslibet formæ admixtæ ex luce & colo <lb/>re, eſt diuerſa à colore, qui eſt in ea.</s> <s xml:id="echoid-s1971" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div71" type="section" level="0" n="0"> <head xml:id="echoid-head95" xml:space="preserve" style="it">17. È<unsure/> ſpecieb{us} uiſibilib{us} primùm percipitur eſſentia lucis & coloris. 67 p 3.</head> <p> <s xml:id="echoid-s1972" xml:space="preserve">ET primum, quod comprehendit uirtus diſtinctiua ex intentionibus, quæ appropriantur for-<lb/>mę, eſt quidditas coloris:</s> <s xml:id="echoid-s1973" xml:space="preserve"> quidditas autem coloris non comprehendetur à uirtute diſtinctiua, <lb/>niſi per cognitionẽ, quando color rei uiſę fuerit ex coloribus aſſuetis:</s> <s xml:id="echoid-s1974" xml:space="preserve"> & comprehenſio quid-<lb/>dιtatis coloris à uirtute diſtinctiua ſecundum cognitionem non eſt, niſi ex comparatione formæ co <lb/>loris ad formas, quas comprehendebat antè, ex formis ſcilicet ſimilibus illi colori.</s> <s xml:id="echoid-s1975" xml:space="preserve"> Quoniam quan-<lb/>do uiſus comprehendit colorem rubeum, & comprehendit, quòd ſit rubeus, non comprehendit, <lb/>quòd ſit rubeus, niſi quia cognoſcit ιpſum:</s> <s xml:id="echoid-s1976" xml:space="preserve"> & iſta cognitio non eſt, niſi ex aſsimilatione formæ eius <lb/>ad res, quas comprehendebat prius.</s> <s xml:id="echoid-s1977" xml:space="preserve"> Si autem uiſus nunquam comprehendiſſet rubeum colorẽ, niſi <lb/>inodo, neſciret apud cõprehenſionem rubei, quòd eſſet rubeus.</s> <s xml:id="echoid-s1978" xml:space="preserve"> Cum ergo color fuerit ex coloribus <lb/>aſſuetis, cognoſcetur à uiſu ſecundum cognitionẽ, & ſi fuerit ex coloribus extraneis, ita quòd uiſus <lb/>nunquã comprehendit talem antè;</s> <s xml:id="echoid-s1979" xml:space="preserve"> non comprehendetur à uiſu, ut cognoſcat ipſum, ſed aſsimilabit <lb/>ipſum coloribus propinquis, ſcilicet quos cognoſcebat.</s> <s xml:id="echoid-s1980" xml:space="preserve"> Radix ergo comprehenſionis coloris eſt à <lb/>ſenſu ſolo, deinde quando ſuper uiſum multoties redierit, per cognitionẽ comprehendetur, ſcilicet <lb/>cuiuſmodifuerit coloris.</s> <s xml:id="echoid-s1981" xml:space="preserve"> Et quidditas lucis etiã non comprehendetur à uiſu, niſi per cognitionem:</s> <s xml:id="echoid-s1982" xml:space="preserve"> <lb/>quoniam uiſus cognoſcit lumen ignis & lumen ſolis, & diſtinguit inter ipſum lumẽ lunæ & ignis:</s> <s xml:id="echoid-s1983" xml:space="preserve"> & <lb/>fic cognoſcit lucem lunæ, & lucem ignis.</s> <s xml:id="echoid-s1984" xml:space="preserve"> Comprehenſio ergo quidditatis iſtarũ lucium à uiſu, non <lb/>eſt, niſi per cognitionẽ.</s> <s xml:id="echoid-s1985" xml:space="preserve"> Deinde omne, quod comprehenditur per ſenſum uiſum, poſt lucem & colo-<lb/>rem, non comprehenditur ſolo ſenſu, ſed comprehenditur per diſtinctionem & argumentationem <lb/>cum ſenſu, quoniã omne, quod cõprehenditur per diſtinctionẽ & argumentationẽ, non cõprehen-<lb/>ditur niſi ex diſtinctione intentionũ, quæ ſunt in forma ſenſibili.</s> <s xml:id="echoid-s1986" xml:space="preserve"> Et intentiones, quæ cõprehendun <lb/>tur per diſtinctionẽ, & argumentationẽ, & cognitionẽ, non cõprehenduntur, niſi cum ſenſu formæ.</s> <s xml:id="echoid-s1987" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div72" type="section" level="0" n="0"> <head xml:id="echoid-head96" xml:space="preserve" style="it">18. Lux & color ex ſeſe, ſolo uiſu percipiuntur. 59 p 3.</head> <p> <s xml:id="echoid-s1988" xml:space="preserve">LVx autem, quæ eſt in corpore illuminato, per ſe comprehenditur à uiſu ſecundum ſuum eſſe, <lb/>& per ſe & ex ipſo ſenſu:</s> <s xml:id="echoid-s1989" xml:space="preserve"> & lux & color, quæ ſunt in corpore colorato, illuminato lumine acci-<lb/>dentali, comprehenduntur à uiſu ſimul & admixta, & ſolo ſenſu.</s> <s xml:id="echoid-s1990" xml:space="preserve"> Lux ergo eſſentialis compre-<lb/>henditur à ſentiente ex illuminatione corporis ſentientis, & color comprehenditur à ſentiente ex <lb/>ãlteratione formæ corporis ſentientis, & ex eius coloratione, & cum huiuſmodi comprehenſione <lb/>lucis à corpore ſentiente per lumen accidentale admixtum cum illo colore.</s> <s xml:id="echoid-s1991" xml:space="preserve"> Sentiens ergo compre-<lb/>hendit ex corpore apud peruentum formæ coloris ad ſe lucem coloratam, & comprehendit ex eo <lb/>apud peruentum formæ lucis eſſentialis lucem ſolam.</s> <s xml:id="echoid-s1992" xml:space="preserve"> Iſta ergo duo tantum comprehenduntur <lb/>ã<unsure/> uiſu ſolo ſenſu.</s> <s xml:id="echoid-s1993" xml:space="preserve"/> </p> <pb o="36" file="0042" n="42" rhead="ALHAZEN"/> </div> <div xml:id="echoid-div73" type="section" level="0" n="0"> <head xml:id="echoid-head97" xml:space="preserve" style="it">19. Color ex ſeſe, pri{us} percipitur, quàm ipſi{us} eſſentia. Ita uiſibile quodlibet ex <lb/>ſeſe pri{us} percipitur, quàm ipſi{us} eſſentia. 68 p 3.</head> <p> <s xml:id="echoid-s1994" xml:space="preserve">ET iterum dicemus, quòd comprehenſio coloris in eo, quod eſt color, eſt ante comprehenſio-<lb/>nem quid ditatis coloris, ſcilicet, quòd uiſus comprehendit colorem, & ſentit, quòd eſt color, <lb/>antequam ſentiat cuiuſmodi ſit coloris:</s> <s xml:id="echoid-s1995" xml:space="preserve"> quoniam apud peruentũ formæ in uiſu, coloratur ui-<lb/>ſus, & cum uiſus coloratur, ſentit, quòd ſit coloratus, & ſic ſentit colorem:</s> <s xml:id="echoid-s1996" xml:space="preserve"> deinde ex diſtinctione co <lb/>loris, & comparatione ipſius ad colores notos uiſui, comprehendit quidditatẽ coloris.</s> <s xml:id="echoid-s1997" xml:space="preserve"> Comprehen <lb/>ſio ergo coloris in eo, quòd eſt color, eſt ante comprehenſionẽ quidditatis coloris, & erit cõprehen-<lb/>ſio quidditatis coloris per cognitionẽ.</s> <s xml:id="echoid-s1998" xml:space="preserve"> Et ſignificatio, quòd uiſus comprehendit colorẽ in eo, quòd <lb/>eſt color, antequam comprehendat cuiuſmodi ſit ratio coloris:</s> <s xml:id="echoid-s1999" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s2000" xml:space="preserve"> quia uiſibilia, quorum colores <lb/>ſunt fortes, ſicut uiriditas profunda, & fuſcitas, & ſimiles, quãdo fuerint in obſcuro ualde loco, non <lb/>comprehenduntur à uiſu in illo loco, niſi quaſi colores tantùm:</s> <s xml:id="echoid-s2001" xml:space="preserve"> tamen ſentit quòd ſint colores, & <lb/>non diſtinguit cuiuſinodi ſint colores in principio comprehenſionis.</s> <s xml:id="echoid-s2002" xml:space="preserve"> Et quando locus non fuerit <lb/>ualde obſcurus, & uiſus multũ intueatur, comprehendit uiſus, cuiuſmodi ſint coloris:</s> <s xml:id="echoid-s2003" xml:space="preserve"> aut ſi lux au-<lb/>gmentetur & intendatur in illo loco.</s> <s xml:id="echoid-s2004" xml:space="preserve"> Declarabitur ergo ex iſta experimentatione, quod uiſus com-<lb/>prehendit colorem in eo, quòd eſt color, antequam comprehendat cuiuſmodi ſit coloris:</s> <s xml:id="echoid-s2005" xml:space="preserve"> & illud, <lb/>quod comprehendit uiſus ex colore in principio ſui peruentus ad uiſum, eſt coloratio, & coloratio <lb/>eſt quaſi obſcuritas aut umbra, quando color fuerit ſubtilis.</s> <s xml:id="echoid-s2006" xml:space="preserve"> Et ſi res uiſa fuerit diuerſorũ colorum, <lb/>comprehendet uiſus in principio ex forma illius rei uiſæ obſcuritatem partium diuerſæ qualitatis, <lb/>ſecundum fortitudinẽ & debilitatem, aut quaſi umbras diuerſas in fortitudine & debilitate.</s> <s xml:id="echoid-s2007" xml:space="preserve"> Primũ <lb/>ergo, quod comprehendit uiſus ex forma coloris, eſt mutatio membri ſentientis, & coloratio eius, <lb/>quæ eſt obſcuritas aut ſimilitudo obſcuritatis:</s> <s xml:id="echoid-s2008" xml:space="preserve"> deinde ſentiens diſtinguet illam colorationem:</s> <s xml:id="echoid-s2009" xml:space="preserve"> & ſi <lb/>res uiſa fuerit illuminata, diſtinguetur ille color à uiſu, & comprehendetur eius quidditas, quando <lb/>fuerit ex coloribus, quos multoties comprehendebat prius:</s> <s xml:id="echoid-s2010" xml:space="preserve"> & ſi fuerit ex coloribus, quos ferè ſem-<lb/>per antè comprehendebat, comprehendetur in minore tempore, & in inſtanti ſecundo, inter quod <lb/>& primum, in quo comprehendit colorem, quatenus eſt color, non eſt ſenſibile tempus:</s> <s xml:id="echoid-s2011" xml:space="preserve"> ſi autẽ fue-<lb/>rit ex coloribus non manifeſtis, quos uiſus non comprehendit antè, niſi rarò, aut fuerit in loco ob-<lb/>ſcuro & debilis lucis, nõ comprehendetur à uiſu quidditas eius, niſi in tempore ſenſibili:</s> <s xml:id="echoid-s2012" xml:space="preserve"> & ſi res ui-<lb/>ſa fuerit obſcura, & fuerit in ea, niſi modica lux, ſicut illud, quod comprehenditur nocte, & in locis <lb/>ualde obſcuris, non diſtinguetur à ſentiente, niſi obſcuritas tantùm.</s> <s xml:id="echoid-s2013" xml:space="preserve"> Declaratum eſt ergo ex com <lb/>prehenſione colorũ in locis obſcuris, quòd comprehenſio coloris in eo, quòd eſt color, eſt ante com <lb/>prehenſionẽ quidditatis eius.</s> <s xml:id="echoid-s2014" xml:space="preserve"> Et etiam ſignificatio quòd uiſus comprehendit colorem in eo, quòd <lb/>eſt color, antequam comprehendat cuiuſmodi ſit coloris:</s> <s xml:id="echoid-s2015" xml:space="preserve"> eſt, quia uiſus cum cõprehendit colorem <lb/>extraneum, quem nunquam uidit antè, comprehendit quòd eſt color, & tamen neſcit, cuiuſmodi ſit <lb/>coloris:</s> <s xml:id="echoid-s2016" xml:space="preserve"> & cum fuerit multùm circa ipſum, aſsimilabit ipſum propinquiori colori ſimili illi.</s> <s xml:id="echoid-s2017" xml:space="preserve"> Ex iſtis <lb/>ergo experimẽtationibus declaratur declaratione manifeſta, quòd cõprehenſio coloris in eo, quòd <lb/>eſt color, erit ante comprehenſionem quidditatis coloris:</s> <s xml:id="echoid-s2018" xml:space="preserve"> & declaratũ eſt etiam ex iſtis experimen-<lb/>tationibus, quòd comprehenſio quidditatis coloris nõ erit niſi per diſtinctionem.</s> <s xml:id="echoid-s2019" xml:space="preserve"> Illud ergo quod <lb/>comprehendit uiſus ſolo ſenſu, non eſt, niſi color in eo, quòd eſt color, & lux in eo, quòd eſt lux:</s> <s xml:id="echoid-s2020" xml:space="preserve"> & <lb/>præter iſta nihil comprehendit ſolo ſenſu, ſed per diſtinctionem, & argumentationẽ & cognitionẽ.</s> <s xml:id="echoid-s2021" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div74" type="section" level="0" n="0"> <head xml:id="echoid-head98" xml:space="preserve" style="it">20. Eſſentia coloris percipitur in tempore. Ita eſſentia cui{us}libet uiſibilis percipi-<lb/>tur in tempore. 70 p 3.</head> <p> <s xml:id="echoid-s2022" xml:space="preserve">ET etiã dicamus, quòd comprehenſio quidditatis coloris nõ eſt, niſi in tempore.</s> <s xml:id="echoid-s2023" xml:space="preserve"> Quoniã enim <lb/>comprehenſio quidditatis coloris non eſt, niſi per diſtinctionẽ & aſsimilationẽ, ſed diſtinctio <lb/>non eſt, niſi in tempore:</s> <s xml:id="echoid-s2024" xml:space="preserve"> ergo comprehenſio quidditatis coloris non eſt, niſi in tempore.</s> <s xml:id="echoid-s2025" xml:space="preserve"> Signi <lb/>ficationem aũt manifeſtam, quòd comprehenſio quidditatis coloris nõ eſt, niſi in tempore, præbet <lb/>illud, quod apparet in trocho apud motum eius:</s> <s xml:id="echoid-s2026" xml:space="preserve"> quoniã quando in trocho fuerint tincturæ diuerſę, <lb/>& illæ tincturæ fuerint lineæ extenſæ ex medio ſuperficiei eius manifeſtæ, & ex parte colli eius uſq:</s> <s xml:id="echoid-s2027" xml:space="preserve"> <lb/>ad finẽ ſuæ circumferentię, & trochus fuerit circumgyratus motu forti, & aſpexerit ipſum quis, com <lb/>prehendet omnes colores eius quaſi unũ, diuerſum ab omnibus coloribus eius, qui ſunt in eo, quaſi <lb/>eſſet color cõpoſitus ex omnibus coloribus illarum linearum, & non comprehendet lineationẽ, nec <lb/>diuerſitatem colorum:</s> <s xml:id="echoid-s2028" xml:space="preserve"> & ſimul comprehendet ipſum quaſi quietum, quando motus eius fuerit ual-<lb/>de fortis, quoniam quodlibet punctum nõ figitur in eodem loco, tempore ſenſibili, ſed in quantum <lb/>minimo tempore gyrat circumferentiã totam, ſuper quam reuoluitur.</s> <s xml:id="echoid-s2029" xml:space="preserve"> Peruenit ergo forma puncti <lb/>in uiſum ſuper circumferentiam circuli in uiſu, & uiſus non comprehendit colorem illius puncti in <lb/>minimo tempore, niſi ex tota circumferentia circuli peruenientis in uiſum:</s> <s xml:id="echoid-s2030" xml:space="preserve"> cõprehendit ergo colo-<lb/>rem illius puncti in minimo tempore circumgyratũ.</s> <s xml:id="echoid-s2031" xml:space="preserve"> Et ſimiliter omnia puncta, quæ ſunt in ſuperfi-<lb/>cie trochi, ſignificant quòd uiſus comprehendit colorem cuiuslibet illorum ſuper totam circumſe-<lb/>rentiam circuli, ſuper quam mouetur illud punctum in minimo tempore.</s> <s xml:id="echoid-s2032" xml:space="preserve"> Et omnia puncta, quorũ <lb/>remotio à centro eſt æqualis, mouentur apud circumgyrationẽ trochi ſuper eandem circuli unius <lb/>circumferentiã.</s> <s xml:id="echoid-s2033" xml:space="preserve"> Accidit ergo ex hoc, ut appareat color cuiuslibet puncti illorum punctorũ, quorũ <lb/>remotio à centro eſt æqualis, ſuper circumferentiam eiuſdem circuli in minimo tempore, quod erit <lb/> <pb o="37" file="0043" n="43" rhead="OPTICAE LIBER II."/> tempus reuolutionis.</s> <s xml:id="echoid-s2034" xml:space="preserve"> Quare apparebũt colores omniũ punctorũ in tota circũferẽtia illius circuli <lb/>admixti:</s> <s xml:id="echoid-s2035" xml:space="preserve"> & propter hoc cõprehẽditur color ſuperficiei trochi, quaſi color unus admixtus ex omni-<lb/>bus coloribus, qui ſunt in ſua ſuperficie.</s> <s xml:id="echoid-s2036" xml:space="preserve"> Si ergo uiſus cõprehendiſſet quidditatẽ coloris in uno in-<lb/>ſtanti, & nõ indiguiſſet ad cõprehẽdendũ quidditatẽ eius, tẽpore:</s> <s xml:id="echoid-s2037" xml:space="preserve"> cõprehendiſſet in uno inſtãti, & <lb/>in quolibet inſtãti tẽporis, in quo mouetur trochus, quidditates omniũ colorũ, qui ſunt in trocho, <lb/>diſtinctæ eſſent apud motum.</s> <s xml:id="echoid-s2038" xml:space="preserve"> Quoniam quando indiguerit tempore ad comprehẽdendũ quiddi-<lb/>tates eornm:</s> <s xml:id="echoid-s2039" xml:space="preserve"> comprehendet illos in parte temporis reuolutionis, & in quolibet inſtanti temporis <lb/>reuolutionis apud motum eorum, ſicut comprehẽdet quidditatem eorum, apud eorum quietem:</s> <s xml:id="echoid-s2040" xml:space="preserve"> <lb/>Quoniam quidditates omnium colorum uiſibilium aſſuetorũ in quiete & in motu, ſunt uniuſmo-<lb/>di, non mutatæ:</s> <s xml:id="echoid-s2041" xml:space="preserve"> In quolibet ergo inſtanti, in quo mouetur res uiſa, non mutatur color eius.</s> <s xml:id="echoid-s2042" xml:space="preserve"> Et quia <lb/>uiſus non comprehendit quidditatem colorum, qui ſunt in ſuperficie trochi, quando trochus mo-<lb/>uebitur motu ueloci, & comprehenditipſam, quando trochus quieuerit uel fuerit in motu tardo:</s> <s xml:id="echoid-s2043" xml:space="preserve"> <lb/>uiſus ergo non comprehendit quidditatem coloris, niſi ſit color fixus in eodem loco, tempore ſen <lb/>ſibili, uel fuerit in motu, tempore ſenſibili in ſpatio, cuius quantitas non operatur in ſitu coloris <lb/>iſtius à uiſu operatione extranea.</s> <s xml:id="echoid-s2044" xml:space="preserve"> Declarabitur ergo ex iſta diſpoſitione, quòd comprehẽſio quid-<lb/>ditatis coloris non erit, niſi in tempore:</s> <s xml:id="echoid-s2045" xml:space="preserve"> & declarabitur ex iſta diſpoſitione, quòd comprehenſio <lb/>quidditatis omnium uiſibilium non eſt, niſi in tempore.</s> <s xml:id="echoid-s2046" xml:space="preserve"> Quoniam quando uiſus non cõprehendit <lb/>quidditatem coloris, qui comprehenditur ſolo ſenſu, niſi in tempore:</s> <s xml:id="echoid-s2047" xml:space="preserve"> maximè igitur indiget tem-<lb/>pore in comprehenſione intentionũ uiſibiliũ, quæ cõprehenduntur per diſtinctionem & argumen <lb/>tationẽ.</s> <s xml:id="echoid-s2048" xml:space="preserve"> Cõprehenſio ergo quidditatis uiſibiliũ, & cõprehenſio, per cognitionẽ, & cõprehenſio per <lb/>diſtinctionem & argumentationem, nõ erit, niſi in tẽpore:</s> <s xml:id="echoid-s2049" xml:space="preserve"> fed multoties erit in minimo tẽpore.</s> <s xml:id="echoid-s2050" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div75" type="section" level="0" n="0"> <head xml:id="echoid-head99" xml:space="preserve" style="it">21. Lux & color exſeſe, percipiuntur in tempore.</head> <p> <s xml:id="echoid-s2051" xml:space="preserve">ET dicemus, quòd color in eo, quòd eſt color, & lux in eo, quòd eſt lux, non comprehendetur <lb/>à uiſu, niſi in tempore, ſcilicet, quòd inſtans, apud quod erit comprehẽſio coloris in eo, quòd <lb/>eſt color, & comprehenſio lucis in eo, quòd eſt lux, eſt diuerſum ab inſtãti, quod eſt primum <lb/>inſtans, in quo conting it ſuperficiẽ uiſus aer deferens formã.</s> <s xml:id="echoid-s2052" xml:space="preserve"> Quoniã color in eo, quòd eſt color, & <lb/>lux in eo, quòd eſt lux, non comprehenduntur à ſentiente, niſi poſt peruentum formæ in corpore <lb/>ſenſibili, & non comprehenduntur ab ultimo ſentiente, niſi poſt peruentum formæ ad concauum <lb/>nerui communis, & peruentus formę ad concauum nerui communis, eſt ſicut peruentus lucis à fo <lb/>raminibus, per quæ intrat lux ad corpora oppoſita illis foraminibus:</s> <s xml:id="echoid-s2053" xml:space="preserve"> peruentus igitur lucis à fora <lb/>mine ad corpus oppoſitum foramini, non erit, niſi in tempore, quamuis lateat ſenſum.</s> <s xml:id="echoid-s2054" xml:space="preserve"> Quoniam <lb/>enim peruẽtus lucis à foramine ad corpus oppoſitum foramini non poteſt euadere ab altero duo-<lb/>rum modorum, ſcilicet, quòd aut lux ueniet in partem aeris uicinantis foramini, antequam perue-<lb/>niat in partẽ aliam ſequentem, deinde perueniet ad aliam partem, deinde ad aliam, quouſque per-<lb/>ueniat ad corpus oppoſitũ foramini:</s> <s xml:id="echoid-s2055" xml:space="preserve"> aut quòd lux perueniet in totum aerem medium, qui eſt in-<lb/>ter foramen & corpus oppoſitum foramini, & in ipſum corpus oppoſitum foramini ſimul.</s> <s xml:id="echoid-s2056" xml:space="preserve"> Siergo <lb/>aer reciperet lucem ſucceſsiuè, nõ perueniret lux ad corpus oppoſitum foramini, niſi per motum:</s> <s xml:id="echoid-s2057" xml:space="preserve"> <lb/>ſed non eſt motus, niſi in tempore:</s> <s xml:id="echoid-s2058" xml:space="preserve"> ſi autem totus aer recipit lucem ſimul, peruentus lucis etiam in <lb/>aerem, poſtquam non erat in eo, non erit, niſi in tempore, quamuis lateat ſenſum.</s> <s xml:id="echoid-s2059" xml:space="preserve"> Quoniam quan-<lb/>do foramen, per quod intrat lux, fuerit obturatum, & deinde fuerit ablatum obturans:</s> <s xml:id="echoid-s2060" xml:space="preserve"> inſtans, in <lb/>quo fuerit ablatum obturans à prima parte foraminis, & in quo fuerit difcoopertus aer, qui eſt in <lb/>foramine ad partem lucis, eſt diuerſum ab inſtanti, in quo peruenit lux in aerem contingentem il-<lb/>lam partem, quæ eſt intra foramen, & in aerem continuatum cum illo aere ſecundum omnes diſpo <lb/>ſitiones:</s> <s xml:id="echoid-s2061" xml:space="preserve"> quoniam lux non peruenit in aliquam partem aeris, qui eſt intra foramen, quod eſt coo-<lb/>pertum contralucem, niſi poſtquam fuerit diſcooperta aliqua pars foraminis contra lucem, & nul-<lb/>la pars foraminis diſcooperitur in minori, uno inſtanti:</s> <s xml:id="echoid-s2062" xml:space="preserve"> ſed inſtans non diuiditur:</s> <s xml:id="echoid-s2063" xml:space="preserve"> nihil ergo exlu-<lb/>ce peruenit in interius foraminis in illo inſtanti, in quo fuerit diſcooperta pars foraminis:</s> <s xml:id="echoid-s2064" xml:space="preserve"> quoniã <lb/>illud, quod eſt diſcoopertum ex foramine in uno inſtanti, non diſcooperitur ſucceſsiuè, neque il-<lb/>lud, quod diſcooperitur ex foramine in uno inſtanti, eſt pars alicuius quantitatis, quoniam non <lb/>diſcooperitur in uno inſtanti, niſi punctum carens quantitate, aut linea carẽs latitudine, quoniam <lb/>non auferetur cooperiens ab habente longitudinem & latitudinem, niſi ſucceſsiuè.</s> <s xml:id="echoid-s2065" xml:space="preserve"> Igitur per mo-<lb/>tum:</s> <s xml:id="echoid-s2066" xml:space="preserve"> ſed motus non erit, niſi in tempore:</s> <s xml:id="echoid-s2067" xml:space="preserve"> & illud quod diſcooperitur à foramine in uno inſtanti, ca <lb/>ret latitudine:</s> <s xml:id="echoid-s2068" xml:space="preserve"> eſt ergo punctum aut linea:</s> <s xml:id="echoid-s2069" xml:space="preserve"> ſed punctum carens quantitate, & linea carens latitudi-<lb/>ne, non eſt pars aeris:</s> <s xml:id="echoid-s2070" xml:space="preserve"> Punctum ergo carens quantitate, & linea carens latitudine, quod eſt punctũ, <lb/>quod diſcooperitur exforamine in inſtanti, non eſt, niſi finis alicuius partium aeris, qui eſt intra fo <lb/>ramen, non pars aeris.</s> <s xml:id="echoid-s2071" xml:space="preserve"> Et punctum carens quantitate, non recipit lucem, neq;</s> <s xml:id="echoid-s2072" xml:space="preserve"> linea carens latitudi <lb/>ne, quoniam non recipit lucem, niſi corpus.</s> <s xml:id="echoid-s2073" xml:space="preserve"> Et cum ita ſit, nihil peruenit exluce in aerem, qui eſt <lb/>intra foramen, in inſtanti, in quo diſcooperitur primùm, quod diſcooperitur ex foramine.</s> <s xml:id="echoid-s2074" xml:space="preserve"> Inſtans <lb/>ergo, quod eſt punctum uel primum inſtans, in quo peruenit lux in aerem, qui eſt intra foramẽ, aut <lb/>in partem eius, eſt diuerſum ab inſtanti, in quo diſcooperitur primùm, quod diſcooperitur ex fo-<lb/>ramine:</s> <s xml:id="echoid-s2075" xml:space="preserve"> ſed inter quælibet duo inſtantia eſt tempus.</s> <s xml:id="echoid-s2076" xml:space="preserve"> Lux ergo non peruenit ex aere, qui eſt extra <lb/>foramen, ad aerem, qui eſt intra foramẽ, niſi in tempore:</s> <s xml:id="echoid-s2077" xml:space="preserve"> ſed id tempus ualde latet ſenſum, propter <lb/>uelocitatem receptionis formarũ lucis ab aere.</s> <s xml:id="echoid-s2078" xml:space="preserve"> Et ſimiliter accidit in uiſu, quãdo fuerit oppoſitus <lb/> <pb o="38" file="0044" n="44" rhead="ALHAZEN"/> rei uiſæ, poſtquam non erat ita, & aer deferens formam rei uiſæ, contigerit ſuperficiem uiſus, poſt-<lb/>quam non contingebatipſam prius:</s> <s xml:id="echoid-s2079" xml:space="preserve"> non peruenit forma ex aere deferente formã ad interius con-<lb/>caui nerui communis, niſi in tempore:</s> <s xml:id="echoid-s2080" xml:space="preserve"> ſed ſenſus caret uia cõprehẽſionis iſtius temporis propter <lb/>paruitatẽ eius, & errorẽ eius, & debilitatẽ eius ad cõprehendendum id, quod eſt in fine paruitatis.</s> <s xml:id="echoid-s2081" xml:space="preserve"> <lb/>Iſtud ergo tẽpus reſpectu ſenſus eſt ſicut inſtans.</s> <s xml:id="echoid-s2082" xml:space="preserve"> Et etiã mẽbrũ ſentiens non ſentit formas uenien <lb/>tes ad ipſum, niſi poſtquã patitur ab illis:</s> <s xml:id="echoid-s2083" xml:space="preserve"> non ſentit ergo colorẽ in eo, quòd eſt color, neq;</s> <s xml:id="echoid-s2084" xml:space="preserve"> lucẽ in <lb/>eo, quòd eſt lux, niſi poſtquã patitur à forma lucis & coloris:</s> <s xml:id="echoid-s2085" xml:space="preserve"> ſed paſsio mẽbri ſentientis à forma co <lb/>loris & forma lucis, eſt aliqua alteratio:</s> <s xml:id="echoid-s2086" xml:space="preserve"> ſed nulla alteratio eſt, niſi in tẽpore:</s> <s xml:id="echoid-s2087" xml:space="preserve"> uiſus ergo non cõpre-<lb/>hẽdit colorẽ in eo, quòd eſt color, neq;</s> <s xml:id="echoid-s2088" xml:space="preserve"> lucem in eo, quòd eſt lux, niſi in tẽpore.</s> <s xml:id="echoid-s2089" xml:space="preserve"> Et in tẽpore, in quo <lb/>extenditur forma à ſuperficie mẽbri ſentientis ad concauũ nerui cõmunis, erit cõprehẽſio coloris <lb/>in eo, quòd eſt color, & lucis in eo, quòd eſt lux, à uirtute ſentiente, quæ eſt in toto corpore ſentien <lb/>te, & apud peruentum formę in concauum nerui cõmunis, erit cõprehenſio coloris, in eo quòd eſt <lb/>color, & lucis in eo, quòd eſt lux, ab ultimo ſentiente.</s> <s xml:id="echoid-s2090" xml:space="preserve"> Comprehenſio ergo coloris in eo, quòd eſt <lb/>color, & lucis in eo, quòd eſt lux, eſt in tempore ſequente tempus, in quo peruenit forma à ſuperfi-<lb/>cie membri ſentientis ad concauũ nerui communis.</s> <s xml:id="echoid-s2091" xml:space="preserve"> Et etiam inſtãs, quod eſt primum, in quo per-<lb/>uenit forma in ſuperficiem uiſus, diuerſum eſt ab inſtanti, quod eſt primum inſtans, in quo aer de-<lb/>ferens formam, contingit primum punctum ſuperficiei uiſus, quando uiſus fuerit oppoſitus rei ui-<lb/>ſæ, poſtquam non fuerat ita, & poſtquã oculus aperuerit palpebras, poſtquã fuerunt clauſæ.</s> <s xml:id="echoid-s2092" xml:space="preserve"> Quo-<lb/>niam quando ita fuerit, primum, quod contingit ſuperficiem uiſus exaere deferente formam illi-<lb/>us rei uiſæ, eſt unum punctum, aut linea carens latitudine, deinde pars poſt aliam, quouſq;</s> <s xml:id="echoid-s2093" xml:space="preserve"> aer de-<lb/>ferens formam, contingat partem ſuperficiei uiſus, in quam peruenit forma:</s> <s xml:id="echoid-s2094" xml:space="preserve"> & apud contactum il-<lb/>lius puncti carentis quantitate, aut lineæ carentis latitudine ſuperficiei uiſus, ad punctum carens <lb/>quantitate, aut ad lineam carentem quantitate ſuperficiei aeris deferentis formam, nihil peruenit <lb/>ex forma lucis & coloris in ſuperficiem uiſus:</s> <s xml:id="echoid-s2095" xml:space="preserve"> quoniam minimum ex ſuperficie, in quod peruenit <lb/>lux, aut forma coloris, non erit, niſi ſuperficies.</s> <s xml:id="echoid-s2096" xml:space="preserve"> In inſtanti ergo, in quo contingit punctum ſuperfi-<lb/>ciei uiſus primum punctum aeris deferentis formam;</s> <s xml:id="echoid-s2097" xml:space="preserve"> nihil peruenit in ſuperficiem uiſus.</s> <s xml:id="echoid-s2098" xml:space="preserve"> Inſtans er <lb/>go, quod eſt primum inſtans, in quo peruenit forma in ſuperficiem uiſus, eſt diuerſum ab inſtanti, <lb/>quod eſt primum inſtans, in quo contingit aer deferens formam, ſuperficiem uiſus, quando fuerit <lb/>uiſus oppoſitus rei uiſæ, & aperuerit palpebras eius, poſtquam fuerunt clauſæ.</s> <s xml:id="echoid-s2099" xml:space="preserve"> Et cum ita ſit, non <lb/>peruenit forma lucis aut coloris in aliquam partem membri ſentientis, neq;</s> <s xml:id="echoid-s2100" xml:space="preserve"> in ſuperficiem uiſus, <lb/>niſi in tempore.</s> <s xml:id="echoid-s2101" xml:space="preserve"> Non comprehendit ergo ſentiens colorem in eo, quòd eſt color, neq;</s> <s xml:id="echoid-s2102" xml:space="preserve"> lucem in eo, <lb/>quòd eſt lux, niſi in tempore, ſcilicet quòd inſtans, in quo cadit ſenſus coloris in eo, quòd eſt color, <lb/>& lucis in eo, quòd eſt lux, eſt diuerſum ab inſtanti, quod eſt inſtans primum, in quo contingit aer <lb/>deferens formam, ſuperficiem uiſus.</s> <s xml:id="echoid-s2103" xml:space="preserve"> Iam ergo declaratum eſt ex omnibus, quæ diximus, quo-<lb/>modo comprehendat uiſus lucem in eo, quòd eſt lux, & quomodo comprehendat colorem in eo, <lb/>quòd eſt color, & quomodo comprehendat quidditatem lucis & coloris, & quomodo compre-<lb/>hendat qualitatem lucis.</s> <s xml:id="echoid-s2104" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div76" type="section" level="0" n="0"> <head xml:id="echoid-head100" xml:space="preserve" style="it">22. Perceptio diſtantiæ uiſibilis differt à perceptionibus loci uiſibilis, & uiſibilis in ſuo lo-<lb/>60. 14 p 4.</head> <p> <s xml:id="echoid-s2105" xml:space="preserve">SEd remotio rei uiſæ à uiſu nõ comprehenditur à uiſu ſolo ſenſu, neq;</s> <s xml:id="echoid-s2106" xml:space="preserve"> comprehenſio remotio-<lb/>nis rei uiſæ, eſt comprehenſio loci rei uiſæ, neq;</s> <s xml:id="echoid-s2107" xml:space="preserve"> comprehenſio reiuiſæ in loco ſuo eſt ex com-<lb/>prehenſione remotionis eius tantùm.</s> <s xml:id="echoid-s2108" xml:space="preserve"> Quoniam locus rei uiſæ fit ex tribus intentionibus, ſci-<lb/>licet, ex remotione, & ex parte uniuerſi, & ex quantitate remotionis.</s> <s xml:id="echoid-s2109" xml:space="preserve"> Quantitas ergo remotionis <lb/>eſt diuerſa ab intentione remotionis in eo, quòd eſt remotio, quoniam intentio remotionis inter <lb/>duo corpora eſt priuatio contactus, & priuatio contactus eſt, eſſe aliquod ſpatiũ inter illa duo cor-<lb/>pora, & quantitas remotionis eſt quantitas illius ſpatij.</s> <s xml:id="echoid-s2110" xml:space="preserve"> Intentio ergo remotionis in eo, quòd eſt re <lb/>motio, eſt ex ſitu:</s> <s xml:id="echoid-s2111" xml:space="preserve"> Non eſt ergo quantitas remotionis.</s> <s xml:id="echoid-s2112" xml:space="preserve"> Comprehẽſio ergo intentionis remotionis, <lb/>quæ eſt priuatio contactus, eſt diuerſa à comprehenſione quantitatis ſpatij, quæ eſt menſura remo <lb/>tionis.</s> <s xml:id="echoid-s2113" xml:space="preserve"> Et comprehenſio quantitatis remotionis eſt ex comprehenſione magnitudinis:</s> <s xml:id="echoid-s2114" xml:space="preserve"> & compre-<lb/>henſio remotionis rei uiſæ, & comprehenſio partis eius ſunt ex comprehẽſione ſitus loci.</s> <s xml:id="echoid-s2115" xml:space="preserve"> Et qua-<lb/>litas comprehenſionis utriuſq;</s> <s xml:id="echoid-s2116" xml:space="preserve"> iſtorum, eſt diuerſa à qualitate comprehenſionis remotionis alteri-<lb/>us illorum, quoniam priuatio contactus eſt diuerſa à parte.</s> <s xml:id="echoid-s2117" xml:space="preserve"> Comprehenſio ergo loci rei uiſæ, non <lb/>eſt comprehenſio remotionis reiuiſæ.</s> <s xml:id="echoid-s2118" xml:space="preserve"> Et comprehenſio rei uiſæ in ſuo loco, cõſiſtit in comprehen <lb/>ſione quinq;</s> <s xml:id="echoid-s2119" xml:space="preserve"> rerum, ſcilicet in comprehenſione lucis, quæ eſt in ea, & comprehenſione coloris e-<lb/>ius, & comprehenſione remotionis eius, & comprehenſione partis eius, & comprehenſione quan-<lb/>titatis remotionis eius:</s> <s xml:id="echoid-s2120" xml:space="preserve"> & nullum iſtorum comprehenditur per ſe ſolum, neq;</s> <s xml:id="echoid-s2121" xml:space="preserve"> comprehenditur u-<lb/>num poſt aliud, ſed omnia comprehenduntur ſimul, quando comprehenduntur per cognitionem, <lb/>non per argumentationem iterandam.</s> <s xml:id="echoid-s2122" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div77" type="section" level="0" n="0"> <head xml:id="echoid-head101" xml:space="preserve" style="it">23. Viſio non fit radijs ab oculo emißis. 5 p 3. Vide 23 n 1.</head> <p> <s xml:id="echoid-s2123" xml:space="preserve">ET ex comprehenſione rei uiſæ in ſuo loco, opinati ſunt ponentes radios:</s> <s xml:id="echoid-s2124" xml:space="preserve"> quòd uiſio eſſet per <lb/>radios exeuntes à uiſu, & peruenientes ad rem uiſam, & quòd uiſio eſſet per extremitatem ra <lb/>dij, & ratiocinati ſunt contra phyſicos, dicentes.</s> <s xml:id="echoid-s2125" xml:space="preserve"> Cum uiſio fuerit per formam uenientem à <lb/> <pb o="39" file="0045" n="45" rhead="OPTICAE LIBER II."/> re uiſa ad uiſum, & illa forma peruenit ad interius uiſus:</s> <s xml:id="echoid-s2126" xml:space="preserve"> quare comprehẽditur res uiſa in ſuo loco, <lb/>qui eſt extra uiſum, & forma eius iam peruenit ad interius uiſus?</s> <s xml:id="echoid-s2127" xml:space="preserve"> Et non ſciuerunt iſti, quòd uiſio <lb/>non completur ſolo ſenſu tantùm, & quòd uiſio non completur, niſi per cognitionem & diſtinctio-<lb/>nem antecedentem, & ſi cognitio & diſtinctio antecedens non eſſet, nõ compleretur in uiſu uiſio.</s> <s xml:id="echoid-s2128" xml:space="preserve"> <lb/>Et non comprehendit uiſus quid eſt res uiſa apud uiſionem:</s> <s xml:id="echoid-s2129" xml:space="preserve"> quoniam quid eſt res uiſa non com-<lb/>prehenditur ſolo ſenſu, niſi per diſtinctionem, aut cognitionem, aut argumentationem iterandam <lb/>apud uiſionem.</s> <s xml:id="echoid-s2130" xml:space="preserve"> Si ergo uiſio eſſet ſolo ſenſu tantùm, & omnia, quæ comprehen duntur ex intentio <lb/>nibus, quæ ſunt in rebus uiſibilibus, comprehenderentur ſolo ſenſu, non comprehenderetur res ui <lb/>ſa in ſuo loco, niſi poſtquam perueniſſet aliquid ad ipſam, quod contingeret & ſentiret eam.</s> <s xml:id="echoid-s2131" xml:space="preserve"> Cum <lb/>autem uiſio non compleatur ſolo ſenſu, ſed per diſtinctionem, & argumentationem, & cognitionẽ:</s> <s xml:id="echoid-s2132" xml:space="preserve"> <lb/>non indiget in cõprehenſione rei in ſuo loco, ſentiente extenſo ad ipſam, & contingente ipſam.</s> <s xml:id="echoid-s2133" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div78" type="section" level="0" n="0"> <head xml:id="echoid-head102" xml:space="preserve" style="it">24. Remotio uiſibilis percipitur diſtinctione & anticipata notione. 9 p 4.</head> <p> <s xml:id="echoid-s2134" xml:space="preserve">REdeamus ergo ad narrandum qualitatem comprehenſionis uiſionis, & dicamus:</s> <s xml:id="echoid-s2135" xml:space="preserve"> Remotio <lb/>rei uiſæ non comprehenditur per ſe, niſi per diſtinctionem:</s> <s xml:id="echoid-s2136" xml:space="preserve"> & iſta intentio eſt ex intentioni-<lb/>bus, quæ quieſcunt in anima ſecundum tempora pertranſita, ita quòd percepta non recedit <lb/>ab anima, propter nimiam frequẽtationem & iterationem eius ſuper uirtutem diſtinctiuam.</s> <s xml:id="echoid-s2137" xml:space="preserve"> Qua-<lb/>re non opus eſt in comprehenſione eius argumentatione iteranda apud comprehenſionẽ cuiusli-<lb/>bet rei uiſæ;</s> <s xml:id="echoid-s2138" xml:space="preserve"> neque quærit etiam uirtus diſtinctiua apud comprehenſionem cuiuslibet rei uiſæ, <lb/>quomodo quieuit intentio rei uiſæ in ea:</s> <s xml:id="echoid-s2139" xml:space="preserve"> quoniam non diſtinguit qualitatem comprehenſionis a-<lb/>pud comprehenſionem cuiuslibet rei uiſæ, & non comprehendit remotionem, niſi cum alijs inten <lb/>tionibus, quæ ſunt in re uiſa:</s> <s xml:id="echoid-s2140" xml:space="preserve"> & comprehendit illam apud comprehenſionem rei uiſæ per cognitio <lb/>nem antecedentem.</s> <s xml:id="echoid-s2141" xml:space="preserve"> Quomodo autem uirtus diſtinctiua comprehendat remotionem per diſtin-<lb/>ctionem, eſt, ſecundum quod narrabo.</s> <s xml:id="echoid-s2142" xml:space="preserve"> Quando uiſus fuerit oppoſitus rei uiſæ, poſtquam non fue-<lb/>rat oppoſitus:</s> <s xml:id="echoid-s2143" xml:space="preserve"> comprehendit rem uiſam, & quando aufertur ab oppoſitione, deſtruitur compre-<lb/>henſio.</s> <s xml:id="echoid-s2144" xml:space="preserve"> Et ſimiliter, quando uiſus aperuerit palpebras, poſtquam fuerunt clauſę, & fuerit oppoſitus <lb/>alicui rei uiſæ:</s> <s xml:id="echoid-s2145" xml:space="preserve"> comprehendet illam rem uiſam, & cum clauſerit palpebras, deſtruetur comprehen-<lb/>ſio.</s> <s xml:id="echoid-s2146" xml:space="preserve"> Et in natura intellectus eſt, quòd illud, quod accidit in uiſis apud aliquem ſitum, & deſtruitur a-<lb/>pud eius ablationem, non eſt fixum intra uiſum, neque faciens ipſum accidere, eſt intra uiſum.</s> <s xml:id="echoid-s2147" xml:space="preserve"> Et <lb/>in natura intellectus eſt etiam, quòd id, quod apparet apud apertionem palpebrarum, & deſtruitur <lb/>apud clauſionem earum, non eſt fixum intra uiſum, neque faciens ipſum accidere, eſt intra uiſum.</s> <s xml:id="echoid-s2148" xml:space="preserve"> <lb/>Et cum uirtus diſtinctiua comprehendit, quòd id, quod accidit in uiſu, ex quo uiſus comprehendit <lb/>rem uiſam, neque eſt res fixa intra uiſum, neque operans ipſum eſt intra uiſum:</s> <s xml:id="echoid-s2149" xml:space="preserve"> ſtatim comprehen-<lb/>dit, quòd id, quod accidit in uiſu, aduenit extrinſecus, & operans ipſum eſt extra uiſum.</s> <s xml:id="echoid-s2150" xml:space="preserve"> Et cum ui-<lb/>ſio deſtruitur apud clauſionem palpebrarum, & apud ablationem ab oppoſitione, & fit apud aper-<lb/>tionem palpebrarum, & apud oppoſitionem:</s> <s xml:id="echoid-s2151" xml:space="preserve"> uirtus diſtinctiua comprehendit, quòd id, quod ui-<lb/>detur in uiſu, non eſt applicatum cum uiſu.</s> <s xml:id="echoid-s2152" xml:space="preserve"> Et cum uirtus diſtinctiua comprehendit, quòd illud, <lb/>quod uidetur, non eſt intra uiſum, neque eſt applicatum cum uiſu, ſtatim comprehẽdit, quòd inter <lb/>ipſum & uiſum eſt remotio:</s> <s xml:id="echoid-s2153" xml:space="preserve"> quoniam in natura intellectús eſt, aut in fine manifeſtationis diſtincti-<lb/>onis, quòd omne, quod non eſt in corpore, neque eſt applicatum cum ipſo, ſit remotum ab eo.</s> <s xml:id="echoid-s2154" xml:space="preserve"> Et <lb/>hæc eſt qualitas comprehenſionis remotionis rei uiſæ in eo, quòd eſt remotio.</s> <s xml:id="echoid-s2155" xml:space="preserve"> Sed uirtus diſtincti <lb/>ua non indiget in comprehenſione remotionis rei uiſæ ad diuidendum ea, quæ diuiſimus, quoniã <lb/>non fecimus hoc, niſi gratia declarandi.</s> <s xml:id="echoid-s2156" xml:space="preserve"> Et uirtus diſtinctiua comprehendit concluſionem iſtius <lb/>diſtinctionis apud uiſionem ſine indigentia illius diuiſionis.</s> <s xml:id="echoid-s2157" xml:space="preserve"> Ex comprehenſione ergo rei apud op <lb/>poſitionem, & apertionem palpebrarum, & ex deſtructione eius apud ablationem oppoſitionis, & <lb/>apud clauſionem palpebrarum comprehendit uirtus diſtinctiua, quòd res uiſa eſt extra uiſum, & <lb/>quòd non eſt applicata cum uiſu.</s> <s xml:id="echoid-s2158" xml:space="preserve"> Et ſecundum iſtum modum comprehendit uirtus diſtinctiua, <lb/>quòd inter uiſum & rem uiſam ſit remotio:</s> <s xml:id="echoid-s2159" xml:space="preserve"> deinde propter frequentationem iſtius intentionis, & <lb/>iterationem eius, quieuit in anima, ita quòd non percipit quietem eius, neque qualitatem quietis <lb/>eius, ſcilicet quòd omnia uiſibilia ſunt extra uiſum, & quòd inter quamlibet rem uiſam & uiſum eſt <lb/>remotio.</s> <s xml:id="echoid-s2160" xml:space="preserve"> Remotio ergo rei uiſæ à uiſu non comprehenditur, niſi per modicam diſtinctionem, ſcili-<lb/>cet quòd uirtus diſtinctiua comprehendit, quòd uiſio eſt propter intentionem extrinſecam à uiſu:</s> <s xml:id="echoid-s2161" xml:space="preserve"> <lb/>& cum hoc, quando fuerit quieſcens in anima, intelliget uirtus diſtinctiua, quòd quælibet res uiſa <lb/>comprehenſa à uiſu, eſt extra uiſum, & inter ipſam & uiſum eſt remotio:</s> <s xml:id="echoid-s2162" xml:space="preserve"> & etiam ſicut diximus ſu-<lb/>perius, non comprehenditur remotio niſi cum alijs:</s> <s xml:id="echoid-s2163" xml:space="preserve"> & apud noſtrum ſermonem de qualitate com-<lb/>prehenſionis ſitus declarabitur, quomodo comprehendatur remotio cum ſitu, & quomodo com-<lb/>prehendatur res uiſa in loco ſuo.</s> <s xml:id="echoid-s2164" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div79" type="section" level="0" n="0"> <head xml:id="echoid-head103" xml:space="preserve" style="it">25. Magnitudo diſtantiæ percipitur è corporibus communibus inter uiſum & uiſibile in-<lb/>teriectis. 10 p 4.</head> <p> <s xml:id="echoid-s2165" xml:space="preserve">COmprehenſio uerò quantitatis remotionis à uiſu, diuerſatur.</s> <s xml:id="echoid-s2166" xml:space="preserve"> Quoniam quædam compre-<lb/>henduntur per ſenſum uiſus, & certificatur eorum quantitas:</s> <s xml:id="echoid-s2167" xml:space="preserve"> & quædam comprehendun-<lb/>tur, quorum quantitas non certificatur.</s> <s xml:id="echoid-s2168" xml:space="preserve"> Remotio rei uiſæ à uiſu comprehenditur in quali-<lb/> <pb o="40" file="0046" n="46" rhead="ALHAZEN"/> betre uiſa, & certificatur in qualibet re uiſa:</s> <s xml:id="echoid-s2169" xml:space="preserve"> quantitas autem remotionis non certificatur uiſui i<gap/> <lb/>qualibet re uiſa:</s> <s xml:id="echoid-s2170" xml:space="preserve"> quoniam inter quædam uiſibilia & uiſum ſunt corpora ordinata continuata:</s> <s xml:id="echoid-s2171" xml:space="preserve"> inter <lb/>quædam uerò & uiſum non ſunt corpora ordinata continuata, neque remotio eorum reſpicit cor-<lb/>pora ordinata continuata.</s> <s xml:id="echoid-s2172" xml:space="preserve"> Illa ergo, quorum remotio reſpicit corpora ordinata continuata, quãdo <lb/>uiſus comprehen derit corpora ordinata, quæ reſpiciunt remotionem eorum uiſibiliũ, quãdo com <lb/>prehendet ſcilicet quantitates illorum corporum, & cum comprehenderit menſuras illorum cor-<lb/>porum, comprehendet quãtitates ſpatiorum, quæ ſunt inter extremitates illorum.</s> <s xml:id="echoid-s2173" xml:space="preserve"> Et ſpatiũ, quod <lb/>eſt inter duas extremitates corporis uiſi, quod reſpicit remotionem, quæ eſt inter uiſum & rem ui-<lb/>ſam, quarum altera eſt in parte rei uiſę, & altera in parte aſpicientis:</s> <s xml:id="echoid-s2174" xml:space="preserve"> eſt remotio rei uiſæ à uiſu, quo-<lb/>niam reſpicit ſpatium, quod eſt inter uiſum & rem uiſam.</s> <s xml:id="echoid-s2175" xml:space="preserve"> Cum ergo uiſus comprehendet menſu-<lb/>rã iſtius ſpatij:</s> <s xml:id="echoid-s2176" xml:space="preserve"> comprehendet menſurã remotionis rei uiſæ.</s> <s xml:id="echoid-s2177" xml:space="preserve"> Viſus ergo comprehendit quantitatẽ <lb/>remotionis rerũ uiſibiliũ (quarũ remotio reſpicit corpora ordinata cõtinuata) ex cõprehenſione <lb/>mẽſurarum corporũ ordinatorũ reſpicientium remotiones earũ.</s> <s xml:id="echoid-s2178" xml:space="preserve"> Et remotio quarundã rerũ iſtarũ <lb/>uiſibilium eſt mediocris:</s> <s xml:id="echoid-s2179" xml:space="preserve"> & remotio quarundã eſt extra medio critatem.</s> <s xml:id="echoid-s2180" xml:space="preserve"> Remotio ergo uiſibilium, <lb/>quorũ remotio eſt mediocris:</s> <s xml:id="echoid-s2181" xml:space="preserve"> comprehenditur à uiſu comprehenſione uera certificata:</s> <s xml:id="echoid-s2182" xml:space="preserve"> quoniam <lb/>uiſibilia, quorũ remotio eſt mediocris, & inter quæ, & uiſum ſunt corpora ordinata cõtinuata, cõ-<lb/>prehenduntur à uiſu uera comprehenſione:</s> <s xml:id="echoid-s2183" xml:space="preserve"> Et cum uiſus cõprehendit iſta uiſibilia uera cõprehen <lb/>ſione:</s> <s xml:id="echoid-s2184" xml:space="preserve"> cõprehendit corpora ordinata interiacẽtia inter ipſum & ipſa uiſibilia uera cõprehenſione:</s> <s xml:id="echoid-s2185" xml:space="preserve"> <lb/>& cũ cõprehendit iſta corpora uera cõprehenſione:</s> <s xml:id="echoid-s2186" xml:space="preserve"> cõprehendit ſpatia interiacentia inter extremi <lb/>tates eorum uera comprehenſione:</s> <s xml:id="echoid-s2187" xml:space="preserve"> & cum comprehendit ſpatia uera comprehenſione:</s> <s xml:id="echoid-s2188" xml:space="preserve"> cõprehen-<lb/>det menſuras remotionum uiſibilium, reſpicientium iſta ſpatia uera comprehenſione & certifica-<lb/>ta.</s> <s xml:id="echoid-s2189" xml:space="preserve"> Viſibilium ergo, quorum remotio reſpicit corpora ordinata continuata, & quorum remotio à <lb/>uiſu eſt mediocris, menſuras remotionum comprehendit uiſus uera comprehenſione & certa:</s> <s xml:id="echoid-s2190" xml:space="preserve"> & <lb/>eſt dicere, certa, in ultimitate, in qua poterit ſenſus comprehendere.</s> <s xml:id="echoid-s2191" xml:space="preserve"> Menſuræ uerò remotionum ui <lb/>ſibilium, quorum remotio eſt extra mediocritatem, & quorum remotio reſpicit corpora ordinata <lb/>continuata, ſi comprehenduntur à uiſu:</s> <s xml:id="echoid-s2192" xml:space="preserve"> non comprehenduntur uera comprehenſione & certifica-<lb/>ta:</s> <s xml:id="echoid-s2193" xml:space="preserve"> quoniam uiſibilia, quorum remotio eſt extra mediocritatem, non comprehenduntur à uiſu ue-<lb/>ra comprehenſione.</s> <s xml:id="echoid-s2194" xml:space="preserve"> Et cum inter uiſum & iſta uiſibilia fuerint corpora ordinata continuata:</s> <s xml:id="echoid-s2195" xml:space="preserve"> non <lb/>comprehenduntur à uiſu omnia iſta uiſibilia uera comprehenſione propter extraneitatem remo-<lb/>tionum extremitatum ſuarum, & exitus eorum à mediocritate, per quam uiſus certificat uiſibilia.</s> <s xml:id="echoid-s2196" xml:space="preserve"> <lb/>Et cum uiſus non comprehendat iſta corpora uera comprehenſione:</s> <s xml:id="echoid-s2197" xml:space="preserve"> non comprehendet ſpatia in-<lb/>teriacentia inter extremitates uera comprehenſione.</s> <s xml:id="echoid-s2198" xml:space="preserve"> Non comprehendet ergo remotiones, quæ <lb/>ſuntinteriacentes inter ipſum & uiſibilia, quæ ſunt apud extremitates iſtorum corporũ, uera com-<lb/>prehenſione.</s> <s xml:id="echoid-s2199" xml:space="preserve"> Quãtitates ergo remotionum uiſibilium, quorum remotio eſt extra mediocritatem, <lb/>& inter quam & uiſum ſunt corpora ordinata continuata, non comprehenduntur à uiſu uera com <lb/>prehenſione.</s> <s xml:id="echoid-s2200" xml:space="preserve"> Similiter remotiones uiſibilium, quorum remotio nõ reſpicit corpora ordinata con-<lb/>tinuata, non comprehenduntur à uiſu uera comprehenſione.</s> <s xml:id="echoid-s2201" xml:space="preserve"> Quare uiſus, quando comprehende-<lb/>rit nubes in plano & in locis carentibus montibus:</s> <s xml:id="echoid-s2202" xml:space="preserve"> exiſtimabit, quòd ſint magnæ remotionis in re-<lb/>ſpectu corporum cœleſtium, & cum nubes fuerint inter montes, & fuerint continuatæ:</s> <s xml:id="echoid-s2203" xml:space="preserve"> fortè coo-<lb/>perientur cacumina montium à nubibus:</s> <s xml:id="echoid-s2204" xml:space="preserve"> & cum nubes diſtiterint, una ab altera:</s> <s xml:id="echoid-s2205" xml:space="preserve"> fortè apparebunt <lb/>cacumina montium ſuperiora nubibus:</s> <s xml:id="echoid-s2206" xml:space="preserve"> & fortè comprehendet uiſus partes nubium applicatas <lb/>cum uertice montium, & fortè erit hoc in montibus non ualde altis.</s> <s xml:id="echoid-s2207" xml:space="preserve"> Ex iſta ergo experimentatio-<lb/>ne apparet, quod remotio nubium non eſt extranea:</s> <s xml:id="echoid-s2208" xml:space="preserve"> & quòd plures illarum ſunt propinquiores <lb/>terrę cacuminibus montium:</s> <s xml:id="echoid-s2209" xml:space="preserve"> & quòd illud, quod exiſtimatur de extraneitate remotionis illarum, <lb/>error eſt.</s> <s xml:id="echoid-s2210" xml:space="preserve"> Et declarabitur inde, quòd uiſus non comprehendit menſuram remotionis nubium in <lb/>plano:</s> <s xml:id="echoid-s2211" xml:space="preserve"> & quòd menſura remotionis nubium comprehendetur à uiſu, quando fuerint inter mon-<lb/>tes, & apparuerint cacumina montium ſuperiora.</s> <s xml:id="echoid-s2212" xml:space="preserve"> Et hoc inuenitur etiam in pluribus uiſibilibus, <lb/>quæ ſunt ſuper faciem terræ, ſcilicet, quòd menſuræ remotionum non reſpicientes corpora or-<lb/>dinata continuata, non comprehenduntur à uiſu.</s> <s xml:id="echoid-s2213" xml:space="preserve"> Ex illis ergo, ex quibus manifeſtatur hoc, ſci-<lb/>licet quòd uiſus non comprehendat quantitatem remotionis rei uiſæ, niſi quando remotio eius <lb/>reſpexerit corpora ordinata continuata, & comprehenderit uiſus illa corpora interpoſita, & cer-<lb/>tificauerit menſuras eorum:</s> <s xml:id="echoid-s2214" xml:space="preserve"> eſt experimentatio ſequens.</s> <s xml:id="echoid-s2215" xml:space="preserve"> Sit domus, in quam experimentator <lb/>non intrauerit ante horam experimentationis:</s> <s xml:id="echoid-s2216" xml:space="preserve"> & ſit in quodam pariete illius domus ſtrictum fo-<lb/>ramen:</s> <s xml:id="echoid-s2217" xml:space="preserve"> & ſit poſt illud foramen uacuitas, quam ante illam horam non uidit:</s> <s xml:id="echoid-s2218" xml:space="preserve"> & ſint in illa uacui-<lb/>tate duo parietes, quorum unus ſit propinquior foramini quàm alius:</s> <s xml:id="echoid-s2219" xml:space="preserve"> & ſit inter illos duos pari-<lb/>etes diſtantia alicuius quantitatis:</s> <s xml:id="echoid-s2220" xml:space="preserve"> & ſit paries propinquior cooperiens quandam partem parie-<lb/>tis remotioris:</s> <s xml:id="echoid-s2221" xml:space="preserve"> & ſit quædam pars parietis remotioris apparens:</s> <s xml:id="echoid-s2222" xml:space="preserve"> & ſit foramen eleuatum à ter-<lb/>ra, ita ut quando aſpiciens aſpexerit per ipſum, non uideat faciem terræ, quæ eſt poſt parietem, <lb/>in quo foramen eſt.</s> <s xml:id="echoid-s2223" xml:space="preserve"> Experimentator igitur quando acceſſerit ad iſtum locum:</s> <s xml:id="echoid-s2224" xml:space="preserve"> & inſpexerit per <lb/>iſtud foramen:</s> <s xml:id="echoid-s2225" xml:space="preserve"> uidebit duos parietes ſimul, & non comprehendet remotionem, quæ eſt inter <lb/>ipſos.</s> <s xml:id="echoid-s2226" xml:space="preserve"> Siuerò remotio primi parietis fuerit magna, remotio extranea à foramine, comprehendet <lb/>duos parietes quaſi ſe contingentes, & fortè exiſtimabit quòd ſit unus continuus, quando color <lb/>eorum fuerit unus.</s> <s xml:id="echoid-s2227" xml:space="preserve"> Et ſi paries primus fuerit remotus à foramine mediocriter, & percipiatur, <lb/>quòd ſint duo parietes:</s> <s xml:id="echoid-s2228" xml:space="preserve"> exiſtimabitur, quòd ſint propinqui ſibi, aut ſe contingentes, & non cer-<lb/> <pb o="41" file="0047" n="47" rhead="OPTICAE LIBER II."/> tificabitur remotio, quæ eſt inter ipſos.</s> <s xml:id="echoid-s2229" xml:space="preserve"> Et cum comprehenderit primum parietem uiſus, quan-<lb/>do remotio eius fuerit mediocris, quaſi eſſet propinquus alteri:</s> <s xml:id="echoid-s2230" xml:space="preserve"> non certificabit remotionem e-<lb/>ius, & non certificabitur remotio, quæ eſt inter iſta duo corpora huiuſmodi per ſenſum uiſus:</s> <s xml:id="echoid-s2231" xml:space="preserve"> quo-<lb/>niam ante illam horam non uiderat iſtum locum, neque illos duos parietes:</s> <s xml:id="echoid-s2232" xml:space="preserve"> & fortè comprehen-<lb/>det uiſus illa duo corpora quaſi ſe contingentia, quamuis antè ſciuerit diſtantiam, quæ eſt inter <lb/>ea.</s> <s xml:id="echoid-s2233" xml:space="preserve"> Et cum uiſus non comprehendat remotionem, quæ eſt inter duo corpora huiuſmodi:</s> <s xml:id="echoid-s2234" xml:space="preserve"> non <lb/>comprehendit quantitatem remotionis ultimi corporis:</s> <s xml:id="echoid-s2235" xml:space="preserve"> & tamen comprehendit formam eius cor <lb/>poris.</s> <s xml:id="echoid-s2236" xml:space="preserve"> Et cum non comprehendat quantitatem remotionis iſtius corporis, quamuis comprehen-<lb/>dat illud corpus:</s> <s xml:id="echoid-s2237" xml:space="preserve"> non comprehendet corpora continuata reſpicientia remotionem eius:</s> <s xml:id="echoid-s2238" xml:space="preserve"> & non <lb/>comprehendet uiſus quantitatem remotionis rei uiſæ certè ex comprehenſione formæ rei uiſæ:</s> <s xml:id="echoid-s2239" xml:space="preserve"> & <lb/>non comprehendet uiſus quantitatem remotionis rei uiſæ, niſi per argumentationem.</s> <s xml:id="echoid-s2240" xml:space="preserve"> Viſus au-<lb/>tem non arguit ſuper aliquam menſuram, niſi per argumentum comparationis, ſiue per compa-<lb/>rationem illius menſuræ ad aliam menſuram iam comprehenſam à uiſu, uel ad menſuram tunc <lb/>comprehenſam cum ea.</s> <s xml:id="echoid-s2241" xml:space="preserve"> Et nihil eſt, per quod uiſus poteſt menſurare remotionem rei uiſæ, & com <lb/>parare ad ipſam, ita ut comprehendat menſuram eius uerè, niſi per corpora ordinata reſpicientia <lb/>remotionem rei uiſæ:</s> <s xml:id="echoid-s2242" xml:space="preserve"> ſi autem menſurauerit uiſus remotionem per alia quàm per iſta corpora, e-<lb/>rit menſuratio qualiſcunque, non certa.</s> <s xml:id="echoid-s2243" xml:space="preserve"> Non igitur comprehenditur quantitas remotionis rei ui-<lb/>ſæ à ſenſu uiſus, niſi remotio eius reſpexerit corpora ordinata continuata:</s> <s xml:id="echoid-s2244" xml:space="preserve"> comprehendit enim <lb/>uiſus illa corpora, & menſuras illorum.</s> <s xml:id="echoid-s2245" xml:space="preserve"> Et iſta experimentatio, quam diximus, habet multa ſimi-<lb/>lia in uiſibilibus, ſicut ex duabus arboribus erectis ſecũdum modum, quem diximus in parietibus, <lb/>aut in ligno tranſuerſim poſito ſuper foramen, ſecundum modum, quem diximus de pariete <lb/>primo.</s> <s xml:id="echoid-s2246" xml:space="preserve"> Remotiones autem uiſibilium diſtantium abinuicem comprehenduntur à uiſu ex com-<lb/>prehenſione diuiſionis, quæ eſt inter uiſibilia.</s> <s xml:id="echoid-s2247" xml:space="preserve"> Diſpoſitiones autem quantitatis remotionis uiſi-<lb/>bilium inter ſe ſunt apud uiſum, ſicut diſpoſitiones remotionum uiſibilium à uiſu.</s> <s xml:id="echoid-s2248" xml:space="preserve"> Quoniam ſi in-<lb/>ter duas res uiſas diſtinctas fuerint corpora ordinata continuata, & comprehenderit uiſus illa cor-<lb/>pora & menſuras eorum:</s> <s xml:id="echoid-s2249" xml:space="preserve"> comprehendet quantitatem remotionis, quæ eſt inter res uiſas:</s> <s xml:id="echoid-s2250" xml:space="preserve"> ſi au-<lb/>tem non:</s> <s xml:id="echoid-s2251" xml:space="preserve"> non comprehendet quantitatem diſtantiæ, quæ eſt inter illas res, uerè.</s> <s xml:id="echoid-s2252" xml:space="preserve"> Et ſimiliter, ſi in-<lb/>ter iſtas duas res uiſas fuerint corpora ordinata continuata:</s> <s xml:id="echoid-s2253" xml:space="preserve"> & fuerint ualde extraneæ remotio-<lb/>nis, ita ut uiſus non poſsit certificare menſuras illorum corporum:</s> <s xml:id="echoid-s2254" xml:space="preserve"> non certificabitur menſura, <lb/>quæ eſt inter illas duas res uiſas.</s> <s xml:id="echoid-s2255" xml:space="preserve"> Remotiones ergo uiſibilium à uiſu non comprehenduntur, niſi <lb/>ex comprehenſione uirtutis diſtinctiuæ:</s> <s xml:id="echoid-s2256" xml:space="preserve"> quoniam illud, quod accidit in uiſu apud uiſionem, non <lb/>accidit, niſi per aliquid extrinſecum.</s> <s xml:id="echoid-s2257" xml:space="preserve"> Et nulla quantitas remotionis uiſibilium comprehenditur <lb/>per ſenſum uiſum uera comprehenſione, niſi remotiones uiſibilium, quorum remotio reſpicit cor-<lb/>pora ordinata & continuata, quorum remotio ſimul eſt mediocris.</s> <s xml:id="echoid-s2258" xml:space="preserve"> Et uiſus unà etiam compre-<lb/>hendit corpora ordinata reſpicientia remotiones eorum, & certificat menſuras illorum corpo-<lb/>rum, ut ſe conſequuntur.</s> <s xml:id="echoid-s2259" xml:space="preserve"> Menſuræ autem remotionum, præter huiuſmodi, non certificantur à <lb/>uiſu.</s> <s xml:id="echoid-s2260" xml:space="preserve"> Viſibilium autem, quorum remotionum menſuræ non certificantur à uiſu, quædam remo-<lb/>tiones reſpiciunt corpora ordinata continuata, & uiſus comprehendit illa corpora cum hoc:</s> <s xml:id="echoid-s2261" xml:space="preserve"> & <lb/>ſunt illa corpora, quorum extremitatum remotio eſt extranea:</s> <s xml:id="echoid-s2262" xml:space="preserve"> & quædam remotiones eorum <lb/>reſpiciunt corpora ordinata continuata, ſed uiſus non cõprehendit illa corpora, ſiue ſint remotio-<lb/>nes eorum extraneæ, ſiue ſint mediocres:</s> <s xml:id="echoid-s2263" xml:space="preserve"> & quædam remotiones eorum non reſpiciunt corpora <lb/>ordinata continuata, & ſuntilla uiſibilia, quæ ſunt ualde eleuata à terra, quæ ſunt extraneæ remo-<lb/>tionis:</s> <s xml:id="echoid-s2264" xml:space="preserve"> & quæ non habent propè ipſam remotionem, neq;</s> <s xml:id="echoid-s2265" xml:space="preserve"> parietem reſpicientem remotionem eo-<lb/>rum.</s> <s xml:id="echoid-s2266" xml:space="preserve"> Et omnia uiſibilia diuiduntur in iſtas partes.</s> <s xml:id="echoid-s2267" xml:space="preserve"> Et quando uiſus comprehendit uiſibilia, quo-<lb/>rum remotionum quantitates non certificantur à uiſu:</s> <s xml:id="echoid-s2268" xml:space="preserve"> uirtus diſtinctiua ſtatim cognoſcit menſu-<lb/>ras remotionum eorum ſecundum æſtimationem, non ſecundum rectitudinem, & comparat re-<lb/>motionem eorum adremotionem ſibi ſimilium ex uiſibilibus comprehenſis à uiſu antè, & ſuſten-<lb/>tat ſe in argumentatione ſuper formam rei uiſæ, & comparat formam rei uiſæ ad formam uiſibili-<lb/>um ſimilium, quæ uiſus comprehendit antè, & in quibus quantitates remotionum iam certifican-<lb/>tur à uirtute diſtinctiua:</s> <s xml:id="echoid-s2269" xml:space="preserve"> & ſic comparat remotionem rei uiſæ, cuius quantitatem remotionis non <lb/>certificat, ad remotionem uiſibilium ſibi ſimilium, quæ comprehendιtuiſus antè, & quorum remo <lb/>tionum menſuræ iam certificantur à uirtute diſtinctiua.</s> <s xml:id="echoid-s2270" xml:space="preserve"> Cum ergo uirtus diſtinctiua non certifi-<lb/>cauerit lineationes formæ rei uiſæ:</s> <s xml:id="echoid-s2271" xml:space="preserve"> comparabit quantitatem totius formæ ad menſuras formarum <lb/>uiſibilium, æqualium illis formis in menſura, quarum remotionum quãtitates iam certificatę ſunt <lb/>in uirtute diſtinctiua, & aſsimilabit remotionem rei uiſæ, cuius quantitas remotionis non certifi-<lb/>cabitur ab eo, ad remotionem uiſibilium in menſura, quorum remotiones iam ſunt certificatæ.</s> <s xml:id="echoid-s2272" xml:space="preserve"> Et <lb/>eſt hoc maximum, ſuper quod poteſt uirtus diſtinctiua in comprehendendo menſuras remotionũ <lb/>uiſibilium.</s> <s xml:id="echoid-s2273" xml:space="preserve"> Fortè ergo inueniet per iſtam argumentationem certitudinem in comprehendendo re <lb/>motionem illius, quod eſt huiuſmodi:</s> <s xml:id="echoid-s2274" xml:space="preserve"> & fortè errabit:</s> <s xml:id="echoid-s2275" xml:space="preserve"> & in illis, in quibus inueniet certitudinem, <lb/>non certificatur, utrum inuenit certitudinem, an non.</s> <s xml:id="echoid-s2276" xml:space="preserve"> Et iſta argumentatio erit argumentatio in <lb/>fine uelocitatis propter aſſuetudinem uirtutis diſtinctiuæ, in comprehen dendo remotionem uiſi-<lb/>bilium per argumentationem & certificationem.</s> <s xml:id="echoid-s2277" xml:space="preserve"> Et fortè æſtimabit uirtus diſtinctiua menſuras re <lb/>motionis rei uiſæ, ſiremotio eius reſpexerit corpora ordinata, & fuerit ex remotionibus mediocri <lb/>bus, propter aſſuetudinem ſuam in æſtimando uel arguendo remotiones uiſibiliũ, & propter ue-<lb/> <pb o="42" file="0048" n="48" rhead="ALHAZEN"/> locitatem cum ſuæ æſtimationis argumentatione.</s> <s xml:id="echoid-s2278" xml:space="preserve"> Et cum remotio rei uiſæ fuerit mediocris, non <lb/>erit inter æſtimationem remotionis, & inter ueram remotionem magna diuerſitas.</s> <s xml:id="echoid-s2279" xml:space="preserve"> Cum ergo ui-<lb/>ſus comprehenderit aliquam rem uiſam, ſtatim uirtus diſtinctiua comprehendet remotionem e-<lb/>ius, & menſuram remotionis eius, ſecundum quod poterit comprehendere, ſcilicet aut per certi-<lb/>tudinem, aut per æſtimationem, & ſtatim remotio eius habebit in anima menſuram conceptam.</s> <s xml:id="echoid-s2280" xml:space="preserve"> <lb/>Menſura ergo remotionis rei uiſæ comprehenſa à uiſu, cuius forma eſt concepta in anima, quan-<lb/>do illa remotio reſpexerit corpora ordinata continuata, & ſimul fuerit mediocris, & comprehen-<lb/>derit uiſus illa corpora ordinata reſpicientia eius remotionem, & etiam iam uirtus diſtinctiua co-<lb/>gnouerit ipſam, & certificauerit menſuras corporum ordinatorum, certificata eſt.</s> <s xml:id="echoid-s2281" xml:space="preserve"> Si autem eius re <lb/>motio non reſpexerit corpora ordinata continuata:</s> <s xml:id="echoid-s2282" xml:space="preserve"> aut reſpexerit corpora ordinata continuata, <lb/>& comprehenderit uiſus illa corpora:</s> <s xml:id="echoid-s2283" xml:space="preserve"> & ſimul fuerit remotio extranea, ita ut uiſus non poſsit cer-<lb/>tificare menſuras illorum corporum:</s> <s xml:id="echoid-s2284" xml:space="preserve"> aut reſpexerit corpora ordinata continuata, & non com-<lb/>prehenderit uiſus illa corpora, neque certificauerit menſuras eorum:</s> <s xml:id="echoid-s2285" xml:space="preserve"> aut poſsit comprehendere <lb/>illa corpora, ſed non aſpexerit illa tunc, nec menſurauerit quantitates eorum, ſiue ſint remotio-<lb/>nes illorum uiſibilium extraneæ, ſiue mediocres:</s> <s xml:id="echoid-s2286" xml:space="preserve"> erit tunc menſura eius remotionis, quæ eſt con-<lb/>cepta in anima, neque certificata, neque uerificata.</s> <s xml:id="echoid-s2287" xml:space="preserve"> Etremotiones, quæ ſunt inter uiſibilia di-<lb/>ſtincta, non comprehenduntur, niſi ex comprehenſione diuiſionis, quæ eſt inter illa uiſibilia:</s> <s xml:id="echoid-s2288" xml:space="preserve"> & <lb/>quædam quantitates remotionum, quæ ſunt inter uiſibilia diſtincta, comprehenduntur uera com <lb/>prehenſione, & quædam comprehenduntur per æſtimationem.</s> <s xml:id="echoid-s2289" xml:space="preserve"> Menſura ergo remotionis, quæ eſt <lb/>inter duo uiſibilia, inter quæ ſunt corpora ordinata continuata, quæ uiſus comprehendit, & quo-<lb/>rum certificat menſuras:</s> <s xml:id="echoid-s2290" xml:space="preserve"> eſt menſura certificata:</s> <s xml:id="echoid-s2291" xml:space="preserve"> menſura autem remotionis, quæ eſt inter duo <lb/>uiſibilia, inter quæ non ſunt corpora ordinata continuata:</s> <s xml:id="echoid-s2292" xml:space="preserve"> aut inter quæ ſunt corpora ordinata <lb/>continuata, ſed uiſus non certificat menſuras illorum corporum:</s> <s xml:id="echoid-s2293" xml:space="preserve"> aut non comprehendit illa, eſt <lb/>menſura non certificata.</s> <s xml:id="echoid-s2294" xml:space="preserve"> Secundum ergo iſtos modos erit comprehenſio remotionum uiſibilium <lb/>per ſenſum uiſus.</s> <s xml:id="echoid-s2295" xml:space="preserve"> Et etiam corpora reſpicientia remotiones uiſibilium aſſuetorum, quæ ſunt in re-<lb/>motionibus aſſuetis, quæ aſſuetæ comprehenduntur à uiſu, comprehenduntur à uiſu, & certifi-<lb/>cantur menſuræ eorum propter frequentationem eorum, ita ut uiſus propter hoc comprehendat <lb/>menſuras remotionum eorum per cognitionem.</s> <s xml:id="echoid-s2296" xml:space="preserve"> Quoniam uiſus quando comprehendit aliquod <lb/>uiſibile aſſuetum, & fuerit in remotione aſſueta:</s> <s xml:id="echoid-s2297" xml:space="preserve"> cognoſcetipſum, & cognoſcet eius remotionem, <lb/>& æſtimabit quantitatem remotion is eius.</s> <s xml:id="echoid-s2298" xml:space="preserve"> Quando ergo æſtimabit quantitatem remotionis hu-<lb/>iuſmodi uiſibilium:</s> <s xml:id="echoid-s2299" xml:space="preserve"> erit æſtimatio eorum propè uera, & non erit inter æſtimationem eius, & ueri-<lb/>tatem magna diuerſitas.</s> <s xml:id="echoid-s2300" xml:space="preserve"> Quantitas ergo remotionum uiſibilium aſſuetorum, quæ ſunt in remotio, <lb/>nibus aſſuetis, comprehenduntur à uiſu per cognitionem ex æſtimatione quantitatum eorum:</s> <s xml:id="echoid-s2301" xml:space="preserve"> & <lb/>plures remotiones uiſibilium comprehenduntur ſecundum huiuſmodi modum.</s> <s xml:id="echoid-s2302" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div80" type="section" level="0" n="0"> <head xml:id="echoid-head104" xml:space="preserve" style="it">26. Situs percipitur è uiſibilis ſiti moderata diſt antia. 29 p 4.</head> <p> <s xml:id="echoid-s2303" xml:space="preserve">SItus uerò, quem uiſus comprehẽdit ex uiſibilibus, diuiditur in tres modos:</s> <s xml:id="echoid-s2304" xml:space="preserve"> quorum unus eſt <lb/>ſitus totius rei uiſæ apud uiſum, aut ſitus cuiuſdam partis rei uiſæ apud uiſum:</s> <s xml:id="echoid-s2305" xml:space="preserve"> & iſte modus <lb/>eſt oppoſitio.</s> <s xml:id="echoid-s2306" xml:space="preserve"> Secundus eſt ſitus ſuperficiei rei uiſæ oppoſitæ uiſui apud uiſum:</s> <s xml:id="echoid-s2307" xml:space="preserve"> & ſitus ſuperfi <lb/>cierum rei uiſæ oppoſitarum uiſui apud uiſum, quando res uiſa fuerit multarum ſuperficierum, & <lb/>fuerit illud, quod apparet uiſui ex eis, multæ ſuperficies:</s> <s xml:id="echoid-s2308" xml:space="preserve"> & ſitus terminorum ſuperficierum uiſibi-<lb/>lium apud uiſum, & ſitus linearũ, & ſpatiorum, quæ ſunt inter quælibet duo puncta, aut inter quæ-<lb/>libet duo uiſibilia, quæ ſimul comprehenduntur à uiſu.</s> <s xml:id="echoid-s2309" xml:space="preserve"> Modus tertius eſt ſitus partium rei uiſę in-<lb/>ter ſe, & ſitus terminorum ſuperficiei rei uiſæ inter ſe, & ſitus partium terminorum ſuperficiei rei <lb/>uiſæ inter ſe, & ſitus partium ſuperficiei terminorum rei uiſæ inter ſe:</s> <s xml:id="echoid-s2310" xml:space="preserve"> & iſte modus eſt ordinatio:</s> <s xml:id="echoid-s2311" xml:space="preserve"> <lb/>& conſimiliter ſitus uiſibilium diuerſorum inter ſe, collocatur ſub hoc modo.</s> <s xml:id="echoid-s2312" xml:space="preserve"> Omnes ergo ſitus, <lb/>qui comprehenduntur à uiſu, diuiduntur in iſtos tres modos.</s> <s xml:id="echoid-s2313" xml:space="preserve"> Et ſitus cuiuslibet habentis ſitum a-<lb/>pud aliud, componitur ex remotione illius habentis ſitum ab illo alio, & ex ſitu illius habentis ſitũ <lb/>reſpectu illius alterius.</s> <s xml:id="echoid-s2314" xml:space="preserve"> Oppoſitio ergo rei uiſæ ad uiſum componitur ex remotione rei uiſæ à uiſu, <lb/>& ex parte, in qua eſt res uiſa, reſpectu uiſus.</s> <s xml:id="echoid-s2315" xml:space="preserve"> Comprehenſio autem remotionis rei uiſæ iam decla-<lb/>rata eſt, quòd eſt intentio quieſcens in anima.</s> <s xml:id="echoid-s2316" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div81" type="section" level="0" n="0"> <head xml:id="echoid-head105" xml:space="preserve" style="it">27. Locus & oppoſitio uiſibilis percipiuntur è ſitu, quem obtinent in ſuperficie uiſus. 30 p 4. <lb/>Vide 22 n.</head> <p> <s xml:id="echoid-s2317" xml:space="preserve">VErus autem locus rei uiſæ comprehenditur ex ſitu rei uiſæ apud uiſionem, quoniam uiſus <lb/>non comprehenditrem uiſam, niſi ex oppoſitione:</s> <s xml:id="echoid-s2318" xml:space="preserve"> & loca, quæ comprehenduntur à ſenſu, <lb/>comprehenduntur à diſtinctione:</s> <s xml:id="echoid-s2319" xml:space="preserve"> & ſenſus & diſtinctio diſtinguunt inter loca, quamuis in <lb/>eis nihil ſit ex uiſibilibus:</s> <s xml:id="echoid-s2320" xml:space="preserve"> & diſtinguit diſtinctio inter locum obiectum uiſui, & locum propinquũ <lb/>ei:</s> <s xml:id="echoid-s2321" xml:space="preserve"> Et uirtus diſtinctiua comprehendit omnia loca per imaginationem.</s> <s xml:id="echoid-s2322" xml:space="preserve"> Cum ergo uiſus fuerit op-<lb/>poſitus alicuiloco, & comprehenderit aliquod uiſibile:</s> <s xml:id="echoid-s2323" xml:space="preserve"> & uiſus poſtea fuerit ablatus ab illo loco:</s> <s xml:id="echoid-s2324" xml:space="preserve"> <lb/>& fuerit oppoſitus alij loco:</s> <s xml:id="echoid-s2325" xml:space="preserve"> deſtruetur uiſio illius rei uiſæ:</s> <s xml:id="echoid-s2326" xml:space="preserve"> & cum reuertetur iterum ad oppoſitio-<lb/>nem illius loci, reuertetur iterum uiſio illius rei uiſæ.</s> <s xml:id="echoid-s2327" xml:space="preserve"> Et cum uiſus comprehenderit rem uiſam a-<lb/>pud oppoſitionem illius in loco, in quo eſt res uiſa:</s> <s xml:id="echoid-s2328" xml:space="preserve"> & comprehenderit uirtus diſtinctiua lo-<lb/> <pb o="43" file="0049" n="49" rhead="OPTICAE LIBER II."/> cum oppoſitum uiſui apud comprehenſionem illius rei uiſæ:</s> <s xml:id="echoid-s2329" xml:space="preserve"> & cum uiſus fuerit ablatus ab op-<lb/>poſitione illius loci:</s> <s xml:id="echoid-s2330" xml:space="preserve"> deſtruitur uiſio illius rei uiſæ:</s> <s xml:id="echoid-s2331" xml:space="preserve"> tunc ergo uirtus diſtinctiua comprehender, <lb/>quòd res uiſa non eſt, niſi in parte oppoſita uiſui apud uiſionem illius rei uiſæ.</s> <s xml:id="echoid-s2332" xml:space="preserve"> Et etiam declara-<lb/>tum eſt [18 n 1] quòd uiſus recipit formas propriè ex uerticationibus linearum radialium, & <lb/>quòd ipſe non patitur à formis, niſi ex uerticationibus iſtarum linearum tantùm.</s> <s xml:id="echoid-s2333" xml:space="preserve"> Et etiam decla-<lb/>ratum eſt, [19 n 1] quòd forma extenditur in corpore uiſus ſecundum rectitudinem linearum <lb/>radialium.</s> <s xml:id="echoid-s2334" xml:space="preserve"> Cum ergo forma rei uiſæ peruenerit in uiſum:</s> <s xml:id="echoid-s2335" xml:space="preserve"> ſtatim ſentiens ſentiet formam, & ſen-<lb/>tiet partem uiſus, in quam peruenit forma, & ſentiet uerticationem, per quam extenditur forma in <lb/>corpore membri ſentientis.</s> <s xml:id="echoid-s2336" xml:space="preserve"> Cum ergo comprehenderit uiſus locum formæ in uiſu, & compre-<lb/>henderit uerticationem, per quam extendebatur illa forma:</s> <s xml:id="echoid-s2337" xml:space="preserve"> ſtatim uirtus diſtinctiua compre-<lb/>hendet locum, in quem, ex quo, & per quem extendebatur illa uerticatio.</s> <s xml:id="echoid-s2338" xml:space="preserve"> Locus autem per <lb/>quem & ex quo extendetur illa uerticatio, eſt locus, in quo eſt illa res uiſa.</s> <s xml:id="echoid-s2339" xml:space="preserve"> Ex comprehenſione <lb/>ergo partis uiſus, in quam peruenit forma rei uiſæ, & ex comprehenſione uerticationis, per quam <lb/>extendebatur forma, & ex qua patitur uiſus à forma, comprehendit uirtus diſtinctiua uerticatio-<lb/>nem, per quam extendebatur forma rei uiſæ ſecundum ueritatem.</s> <s xml:id="echoid-s2340" xml:space="preserve"> Et ſecundum hunc modum <lb/>diſtinguuntur loca uiſibilium:</s> <s xml:id="echoid-s2341" xml:space="preserve"> quoniam uiſibilia diſtincta non diſtinguuntur à uiſu, niſi ex diſtin-<lb/>ctione locorum diſtinctorum in ſuperficie membri ſentientis, ad quę perueniunt formæ uiſibilium <lb/>diſtinctorum.</s> <s xml:id="echoid-s2342" xml:space="preserve"> Et comprehenſio loci rei uiſæ ſecundum hunc modum habet ſimile in auditu:</s> <s xml:id="echoid-s2343" xml:space="preserve"> quo-<lb/>niam ſentiens comprehendit uocem per ſenſum auditus, & comprehendit locum, à quo uenit uox, <lb/>& diſtinguit inter uocem uenientem à dextra, & uocem uenientem à ſiniſtra, & antè, & retro:</s> <s xml:id="echoid-s2344" xml:space="preserve"> Imò <lb/>diſtinguit etiam inter loca uocum diſtinctione ſubtiliori iſta:</s> <s xml:id="echoid-s2345" xml:space="preserve"> & diſtinguit inter locum uocis ue-<lb/>nientis à loco ſibi oppoſito facialiter, & locum uocis uenientis à loco obliquo à uerticatione oppo-<lb/>ſitionis:</s> <s xml:id="echoid-s2346" xml:space="preserve"> & non diſtinguuntur loca, à quibus ueniunt uoces reſpectu auditus, niſi per uertica-<lb/>tiones, ſuper quas ueniunt uoces ad auditum.</s> <s xml:id="echoid-s2347" xml:space="preserve"> Senſus ergo auditus comprehendit uoces, & <lb/>comprehendit uerticationes, ex quibus ueniunt uoces:</s> <s xml:id="echoid-s2348" xml:space="preserve"> & ex comprehenſione uerticationum, <lb/>ſuper quas ueniunt uoces ad auditum, & ſuper quarum rectitudinem percutit uox auditum, com-<lb/>prehendit uirtus diſtinctiua locum, à quo uenit uox.</s> <s xml:id="echoid-s2349" xml:space="preserve"> Sicut ergo loca uocum comprehendun-<lb/>tur à ſenſu auditus:</s> <s xml:id="echoid-s2350" xml:space="preserve"> deinde à uirtute diſtinctiua mediante auditu:</s> <s xml:id="echoid-s2351" xml:space="preserve"> ita loca uiſibilium comprehen-<lb/>duntur à uirtute diſtinctiua per ſenſum uiſus.</s> <s xml:id="echoid-s2352" xml:space="preserve"> Et ex illis, ex quibus declaratur, quòd ſentiens <lb/>comprehendit uerticationem, ſecundum quod patitur uiſus à forma rei uiſæ, eſt illud, quod com-<lb/>prehenditur in ſpeculis ſecundum reflexionem.</s> <s xml:id="echoid-s2353" xml:space="preserve"> Quoniam res uiſa, quam comprehendit uiſus <lb/>ſecundum reflexionem, non comprehenditur à uiſu, niſi in oppoſitione, & cum eſt oppoſita illi:</s> <s xml:id="echoid-s2354" xml:space="preserve"> <lb/>ſed forma eius peruenit ad uiſum ſecundum linearum rectarum uerticationes, quæ ſunt lineæ <lb/>radiales extenſæ à uiſu in partem oppoſitionis.</s> <s xml:id="echoid-s2355" xml:space="preserve"> Cum ergo uiſus ſenſerit formam ex uertica-<lb/>tionibus linearum radialium:</s> <s xml:id="echoid-s2356" xml:space="preserve"> æſtimabit rem uiſam eſſe apud extremitates illarum linearum:</s> <s xml:id="echoid-s2357" xml:space="preserve"> <lb/>quoniam nihil comprehendit ex uiſibilibus aſſuetis, quæ ſemper comprehendit, niſi apud ex-<lb/>tremitates linearum imaginatarum inter uiſum & rem uiſam, quæ ſunt lineæ radiales.</s> <s xml:id="echoid-s2358" xml:space="preserve"> Ex com-<lb/>prehenſione ergo rei uiſæ à uiſu ſecundum reflexionem uiſus ad oppoſitionem, & ſecundum <lb/>rectitudinem uerticationum, ſuper quas formæ reflexæ perueniunt ad uiſum, uidebitur quòd <lb/>ſentiens ſentit uerticationem, per quam uenit forma, & ex qua patitur uiſus à forma.</s> <s xml:id="echoid-s2359" xml:space="preserve"> Et cum <lb/>ſentiens ſentit uerticationem uiſus, ex qua patitur uiſus à forma, comprehendit uirtus diſtin-<lb/>ctiua locum, in quo extenditur illa uerticatio, & comprehendet locum rei uiſæ.</s> <s xml:id="echoid-s2360" xml:space="preserve"> Locus ergo rei <lb/>uiſæ comprehendetur à ſentiente, comprehenſione larga, & ex comprehenſione ſitus apud <lb/>uiſionem:</s> <s xml:id="echoid-s2361" xml:space="preserve"> & comprehendetur à uirtute diſtinctiua, comprehenſione larga, ex comprehenſione <lb/>ſitus rei uiſæ apud uiſionem:</s> <s xml:id="echoid-s2362" xml:space="preserve"> & comprehendetur uera comprehenſione, & certificata, ex com-<lb/>prehenſione uerticationis, ex qua patitur uiſus à forma rei uiſæ.</s> <s xml:id="echoid-s2363" xml:space="preserve"> Remotio autem rei uiſæ eſt <lb/>intentio, quæ iam quieuit in anima.</s> <s xml:id="echoid-s2364" xml:space="preserve"> Igitur apud peruentum rei uiſæ ad uiſum, comprehendit <lb/>uirtus diſtinctiua locum rei uiſæ cum quiete intentionis remotionis apud ipſam:</s> <s xml:id="echoid-s2365" xml:space="preserve"> & adiunctio <lb/>remotionis & loci eſt oppoſitio.</s> <s xml:id="echoid-s2366" xml:space="preserve"> Cum ergo uirtus diſtinctiua comprehenderit locum rei ui-<lb/>ſæ, & ſuam remotionem, ſimul comprehendet eius oppoſitionem.</s> <s xml:id="echoid-s2367" xml:space="preserve"> Comprehenſio ergo oppo-<lb/>ſitionis eſt ex comprehenſione loci rei uiſæ, & ex comprehenſione remotionis rei uiſæ in ſi-<lb/>mul.</s> <s xml:id="echoid-s2368" xml:space="preserve"> Et comprehenſio loci erit ſecundum modum, quem diximus.</s> <s xml:id="echoid-s2369" xml:space="preserve"> Cum ergo forma rei ui-<lb/>ſæ peruenerit in uiſum, ſentiet ſentiens locum membri ſentientis, in quem peruenit forma, & <lb/>comprehendet uirtus diſtinctiua locum rei uiſæ ex uerticatione, per quam extenditur forma:</s> <s xml:id="echoid-s2370" xml:space="preserve"> <lb/>& intentio remotionis iam quieta eſt apud ipſam.</s> <s xml:id="echoid-s2371" xml:space="preserve"> Ipſa ergo comprehendet locum, & remotio-<lb/>nem ſimul apud comprehenſionem formæ à ſentiente.</s> <s xml:id="echoid-s2372" xml:space="preserve"> Igitur apud comprehenſionem formæ à <lb/>ſentiente comprehendet uirtus diſtinctiua oppoſitionem.</s> <s xml:id="echoid-s2373" xml:space="preserve"> Secundum ergo hunc modum dictum, <lb/>erit comprehenſio oppoſitionis.</s> <s xml:id="echoid-s2374" xml:space="preserve"> Et iam declaratum eſt, [10 n] quo modo uiſus comprehendat <lb/>formam rei uiſæ ſolo ſenſu.</s> <s xml:id="echoid-s2375" xml:space="preserve"> A pud peruentum ergo formæ rei uiſæ in uiſum comprehendet ſentiens <lb/>colorem rei uiſæ, & lucem eius, & locum uiſus, qui colorabatur & illuminabatur ab illa forma:</s> <s xml:id="echoid-s2376" xml:space="preserve"> & <lb/>comprehendet uirtus diſtinctiua locum eius, & remotionem apud comprehenſionem lucis & co-<lb/>loris eius à ſentiente.</s> <s xml:id="echoid-s2377" xml:space="preserve"> Et ſic comprehenduntur lux & color, locus & remotio ſimul in minimo tem-<lb/>pore.</s> <s xml:id="echoid-s2378" xml:space="preserve"> Sed locus & remotio ſunt oppoſita, & lux & color ſunt forma rei uiſæ, & ex comprehenſione <lb/>formæ & comprehenſione oppoſitionis ſuſtentatur cõprehenſio rei uiſę in oppoſitione uiſus Ergo <lb/> <pb o="44" file="0050" n="50" rhead="ALHAZEN"/> comprehenſio rei uiſæ in oppoſitione uiſus non eſt, niſi quia forma & oppoſitio comprehenduntur <lb/>ſimul:</s> <s xml:id="echoid-s2379" xml:space="preserve"> deinde propter frequentationem iſtius intentionis, & multitudinem iterationis eius eſt facta <lb/>forma ſignum ſenſui, & uirtuti diſtinctiuę.</s> <s xml:id="echoid-s2380" xml:space="preserve"> Apud peruentum ergo formæ in uiſum comprehenditur <lb/>à ſentiente, & comprehendit uirtus diſtinctiua oppoſitionem, & efficitur ex hoc ab ipſo ſentiente <lb/>comprehenſio rei uiſæ in ſuo loco:</s> <s xml:id="echoid-s2381" xml:space="preserve"> & ſimiliter de qualibet parte rei uiſæ.</s> <s xml:id="echoid-s2382" xml:space="preserve"> Secundum ergo hunc <lb/>modum erit comprehenſio rei uiſæ in loco ſuo:</s> <s xml:id="echoid-s2383" xml:space="preserve"> & ſimiliter de qualibet parte rei uiſæ.</s> <s xml:id="echoid-s2384" xml:space="preserve"> Cum ergo re-<lb/>motio rei uiſæ fuerit ex remotionibus mediocribus certificatæ quantitatis:</s> <s xml:id="echoid-s2385" xml:space="preserve"> erit locus rei uiſæ, in <lb/>quo comprehenditur à uiſu, locus uerus:</s> <s xml:id="echoid-s2386" xml:space="preserve"> & ſi remotio rei uiſæ non fuerit ex remotionibus certifi-<lb/>catæ menſuræ:</s> <s xml:id="echoid-s2387" xml:space="preserve"> erit comprehenſio rei uiſæ in oppoſitione certificata ſecundum oppoſitiones:</s> <s xml:id="echoid-s2388" xml:space="preserve"> quo-<lb/>niam oppoſitio componitur exubitate & remotione in eo, quod eſt remotio.</s> <s xml:id="echoid-s2389" xml:space="preserve"> Sed locus rei uiſæ, in <lb/>quo comprehenditur à uiſu, eſt æſtimatus, non certificatus:</s> <s xml:id="echoid-s2390" xml:space="preserve"> quoniam locus certificatus non com-<lb/>prehenditur, niſi ex certificatione quantitatis remotionis.</s> <s xml:id="echoid-s2391" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div82" type="section" level="0" n="0"> <head xml:id="echoid-head106" xml:space="preserve" style="it">28. Situs directus & obliquus lineæ, ſuperficiei, & ſpatij percipitur ex æquabili & inæqua-<lb/>bili terminorum diſtantia. 31 p 4.</head> <p> <s xml:id="echoid-s2392" xml:space="preserve">SItus uerò ſuperficierum uiſibilium apud uiſum diuiditur in duo, ſcilicet in directam oppoſi-<lb/>tionem, & obliquationem.</s> <s xml:id="echoid-s2393" xml:space="preserve"> Superficies autem directa oppoſita uiſui eſt illa, cuius axis ra-<lb/>dialis, (quando ſuperficies comprehenditur à uiſu apud rectam oppoſitionem) occurrit ali-<lb/>cui puncto ex ea, & eſt ſimul eleuatus ſuper ſuperficiem eleuatione æquali.</s> <s xml:id="echoid-s2394" xml:space="preserve"> Et ſuperficies obli-<lb/>quata eſt illa, cuius axis radialis, (quando ipſa comprehenditur à uiſu apud obliquationem) oc-<lb/>currit alicui puncto ex ea, & eſt obliquatus ſuper ſuperficiem, non eleuatus ſuper ipſam eleuatio-<lb/>ne æquali ſecundum omnes diuerſitates modorum obliquationis.</s> <s xml:id="echoid-s2395" xml:space="preserve"> Termini uerò ſuperficierum <lb/>uiſibilium, & lineæ, quæ ſunt in rebus, & ſpatia quæ ſunt inter uiſibilia, & inter partes uiſibilium, <lb/>diuiduntur in duo:</s> <s xml:id="echoid-s2396" xml:space="preserve"> quorum alterum ſuntlineæ, & ſpatia ſecantia lineas radiales:</s> <s xml:id="echoid-s2397" xml:space="preserve"> & alterum ſunt li-<lb/>neæ & ſpatia æquidiſtantia lineis radialibus, & reſpicientia ipſas.</s> <s xml:id="echoid-s2398" xml:space="preserve"> Et lineæ & ſpatia ſecantia lineas <lb/>radiales diuiduntur ſecundum ſitum in duo:</s> <s xml:id="echoid-s2399" xml:space="preserve"> in obliquationem & directionem, ſecundum diuiſio-<lb/>nem ſituum & ſuperficierum in iſta duo.</s> <s xml:id="echoid-s2400" xml:space="preserve"> Linea autem directa eſt illa, ad cuius aliquod punctum <lb/>perueniet axis radialis:</s> <s xml:id="echoid-s2401" xml:space="preserve"> & erit perpendicularis ſuper ipſam:</s> <s xml:id="echoid-s2402" xml:space="preserve"> & linea obliquata eſt illa, cuius axis ra <lb/>dialis, quando peruenerit ad aliquod punctum eius, erit obliquatus ſuper ipſam, non perpendicu-<lb/>laris.</s> <s xml:id="echoid-s2403" xml:space="preserve"> Viſus autem comprehendit directionem & obliquationem ſuperficierum, & linearum, & di-<lb/>ſtinctionem earum ex comprehenſione diuerſitatis remotionum extremitatum ſuperficierum & <lb/>linearum, & æqualitatis earum.</s> <s xml:id="echoid-s2404" xml:space="preserve"> Quoniam quando uiſus comprehenderit ſuperficiem rei uiſæ:</s> <s xml:id="echoid-s2405" xml:space="preserve"> & <lb/>comprehenderit remotiones extremitatum eius:</s> <s xml:id="echoid-s2406" xml:space="preserve"> & ſenſerit æqualitatem remotionum termino-<lb/>rum ſuperficiei ab eo, aut æqualitatem duorum locorum oppoſitorum æqualis remotionis à loco <lb/>ſuperficiei, ad quam intuetur quis:</s> <s xml:id="echoid-s2407" xml:space="preserve"> comprehendet ſuperficiem eſſe directè oppoſitam, & iudica-<lb/>bit uirtus diſtinctiua, quòd ſit directa.</s> <s xml:id="echoid-s2408" xml:space="preserve"> Et cum uiſus comprehenderit ſuperficiem rei uiſæ, & com-<lb/>prehenderit remotionem extremitatum eius & diuerſitatem, & non inuenerit in ſuperficie duo lo-<lb/>ca æqualis remotionis à loco ſuperficiei, ad quam intuetur, quorum remotio ab eo fuerit æqualis:</s> <s xml:id="echoid-s2409" xml:space="preserve"> <lb/>comprehendet ſuperficiem obliquatam in reſpectu ſui, & iudicabit uirtus diſtinctiua, quòd ſit ob-<lb/>liquata.</s> <s xml:id="echoid-s2410" xml:space="preserve"> Et ſimiliter de ſitibus linearum, & ſpatiorum directorum & obliquorum:</s> <s xml:id="echoid-s2411" xml:space="preserve"> ſcilicet, quòd <lb/>uiſus comprehendat directionem lineæ & ſpatij, quando ſenſerit, quòd duæ remotiones duarum <lb/>extremitatum lineæ aut ſpatij ſunt æquales ab eo:</s> <s xml:id="echoid-s2412" xml:space="preserve"> aut quòd duæ remotiones duorum punctorum <lb/>lineæ aut ſpatij, quorum remotio à puncto, ad quod intuetur quis, puncto ſcilicet lineæ, aut ſpa-<lb/>tij eſt æqualis:</s> <s xml:id="echoid-s2413" xml:space="preserve"> & comprehendit uiſus obliquationem lineæ aut ſpatij, quando ſenſerit, quòd duæ <lb/>remotiones duarum extremitatum lineæ aut ſpatij ab eo ſuntinæquales:</s> <s xml:id="echoid-s2414" xml:space="preserve"> aut quòd duæ remotio-<lb/>nes duorum punctorum, & æqualis remotionis à puncto, ad quod intuetur quis, lineæ aut ſpa-<lb/>tij, ſunt diuerſæ.</s> <s xml:id="echoid-s2415" xml:space="preserve"> Et iſta æqualitas & diuerſitas multoties comprehenduntur à ſentiente per æ-<lb/>ſtimationem & ſigna.</s> <s xml:id="echoid-s2416" xml:space="preserve"> Secundum ergo hunc modum erit obliquationis comprehenſio, & dire-<lb/>ctionis à uiſu.</s> <s xml:id="echoid-s2417" xml:space="preserve"> Et cum ſuperficies tota, aut linea tota fuerit directa uiſui, non erit quælibet pars <lb/>eius per ſe directè oppoſita uiſui:</s> <s xml:id="echoid-s2418" xml:space="preserve"> imò nulla pars eius eſt directè oppoſita uiſui per ſe, niſi pars, <lb/>ſupra quam eſt axis apud directam oppoſitionem.</s> <s xml:id="echoid-s2419" xml:space="preserve"> Cum ergo mouetur axis radialis ſuper ſuperfi-<lb/>ciem directam, aut ſuper lineam directam, erit obliquatus ſuper quamlibet ipſius partem, ſu-<lb/>pra quam tranſit, præter primam partem, in qua eſt punctum, ſuper quod fuerit perpendicularis:</s> <s xml:id="echoid-s2420" xml:space="preserve"> <lb/>& ſic erit quælibet pars ſuperficiei directè oppoſitæ, & lineæ directè oppoſitæ, quando fuerit ſum-<lb/>pta perſe, obliquata, præter partem prædictam:</s> <s xml:id="echoid-s2421" xml:space="preserve"> & quando accipietur tota linea, aut ſuperficies, <lb/>erit directa.</s> <s xml:id="echoid-s2422" xml:space="preserve"> Et cum punctum, apud quod erit axis perpendicularis ſuper ſuperficiem aut li-<lb/>neam, fuerit in medio ſuperficiei aut lineæ:</s> <s xml:id="echoid-s2423" xml:space="preserve"> erit ſuperficies aut linea in fine directæ oppoſitionis <lb/>ad uiſum.</s> <s xml:id="echoid-s2424" xml:space="preserve"> Si autem punctum non fuerit in medio:</s> <s xml:id="echoid-s2425" xml:space="preserve"> erit ſuperficies aut linea directa, ſed non in fi-<lb/>ne directionis:</s> <s xml:id="echoid-s2426" xml:space="preserve"> & quantò fuerit punctum, apud quod axis fuerit perpendicularis ſuper ſuperfi-<lb/>ciem aut lineam, medio ſuperficiei aut lineæ propinquius, tantò erit ſuperficies autlinea directio-<lb/>ris oppoſitionis.</s> <s xml:id="echoid-s2427" xml:space="preserve"> Situs autem linearum & ſpatiorum æquidiſtantium lineis radialibus, compre-<lb/>henduntur à uiſu ex comprehenſione oppoſitionis.</s> <s xml:id="echoid-s2428" xml:space="preserve"> Quoniam, quando uiſus comprehende-<lb/>rit extremitates linearum aut ſpatiorum, quæ ſequuntur uiſibilia oppoſita uiſui illi, & extre-<lb/>mitates eorum propinquas, quæ ſequuntur eundem uiſum, comprehendet ſitus eorum, & com-<lb/> <pb o="45" file="0051" n="51" rhead="OPTICAE LIBER II."/> prehendet extenſionem eorum in uerticatione oppoſitionis.</s> <s xml:id="echoid-s2429" xml:space="preserve"> Secundum ergo iſtos modos erit <lb/>comprehenſio ſituum, ſuperficierum, linearum, & ſpatiorum à uiſu, reſpectu illius.</s> <s xml:id="echoid-s2430" xml:space="preserve"> Quædam au-<lb/>tem ſuperficies, & lineæ, & ſpatia ſecantia lineas radiales ſunt obliquationis ualde magnæ ſuper <lb/>radiales lineas, & quædam ſunt modicæ, & quædam ſunt perpendiculares ſuper lineas radiales:</s> <s xml:id="echoid-s2431" xml:space="preserve"> <lb/>& ſunt ſuperficies, & lineæ, & ſpatia directè oppoſita uiſui.</s> <s xml:id="echoid-s2432" xml:space="preserve"> Extremitas autem remotior cuiusli-<lb/>bet ſuperficiei, & lineæ, & ſpatij ſequitur partem remotam à uiſu, ſcilicet partem ſequentem ex-<lb/>tremitates linearum radialium, & extremitas propinquior ſequitur partem propinquam uiſui, ſci-<lb/>licet partem ſequentem uiſum.</s> <s xml:id="echoid-s2433" xml:space="preserve"> Et quando uiſus comprehenderit aliquam lineam, uel aliquod <lb/>ſpatium, ſtatim comprehendet duas ubitates ſequentes extremitates lineæ illius, aut illius ſpatij:</s> <s xml:id="echoid-s2434" xml:space="preserve"> <lb/>& ſimiliter quando uiſus comprehenderit aliquam ſuperficiem:</s> <s xml:id="echoid-s2435" xml:space="preserve"> comprehendet ubitates ſequentes <lb/>extremitates illius ſuperficiei ex comprehenſione extenſionis illius ſuperficiei, in longitudine, & <lb/>latitudine.</s> <s xml:id="echoid-s2436" xml:space="preserve"> Cum ergo uiſus comprehenderit ſuperficiem obliquam ſuper lineas radiales, & fue-<lb/>rit illa ſuperficies maximæ declinationis:</s> <s xml:id="echoid-s2437" xml:space="preserve"> comprehendet uiſus ubitatem ſequentem extremitatem <lb/>remotiorem apud comprehenſionem ſuperficiei, & comprehendet ipſam eſſe ſequentem extre-<lb/>mitates linearum radialium, & comprehendet ubitatem ſequentem extremitatem propinquio-<lb/>rem, & comprehendet ipſam eſſe ſequentem illud, quod eſt prope uiſum.</s> <s xml:id="echoid-s2438" xml:space="preserve"> Et ſimiliter de linea, & <lb/>ſpatio maximæ obliquationis.</s> <s xml:id="echoid-s2439" xml:space="preserve"> Et cum uiſus perceperit, quòd una duarum extremitatum ſuper-<lb/>ficiei, aut lineæ, aut ſpatij ſequantur ubitatem remotam à uiſu, & quòd altera extremitas fequatur <lb/>ubitatem propinquam uiſui:</s> <s xml:id="echoid-s2440" xml:space="preserve"> ſtatim percipiet remotionem unius duarum extremitatum, aut lineę, <lb/>aut ſpatij, aut ſuperficiei, & appropinquationem alterius.</s> <s xml:id="echoid-s2441" xml:space="preserve"> Et cum perceperit remotionem unius <lb/>duarum extremitatum, aut lineæ, aut ſpatij, aut ſuperficiei, & appropinquationem alterius:</s> <s xml:id="echoid-s2442" xml:space="preserve"> ſtatim <lb/>percipiet obliquationem ſitus illius ſuperficiei, aut lineæ, aut ſpatij.</s> <s xml:id="echoid-s2443" xml:space="preserve"> Obliquatio ergo ſuperficie-<lb/>rum, & linearum, & ſpatiorum obliquatorum ſuper lineas radiales extraneæ obliquationis, com-<lb/>prehenditur à uiſu ex comprehenſione duarum ubitatum extremitatum eorum.</s> <s xml:id="echoid-s2444" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div83" type="section" level="0" n="0"> <head xml:id="echoid-head107" xml:space="preserve" style="it">29. Situs uiſibilis obliquus ex immoderata diſtantia uidetur direct{us}. 34 p 4.</head> <p> <s xml:id="echoid-s2445" xml:space="preserve">DEclinatio autem & directa oppoſitio linearum, & ſuperficierum, & ſpatiorum modicæ obli-<lb/>quationis, & directionis, non comprehenduntur à uiſu uera comprehenſione certificata, <lb/>niſi remotio eorum ſit mediocris, & reſpiciat corpora ordinata comprehenſa à uiſu, & com-<lb/>prehenderit ex menſuris eorum corporum menſuras remotionum extremitatum illarum ſuper-<lb/>ficierum, & linearum, & ſpatiorum, & comprehenderit æqualitatem duarum remotionum <lb/>duarum extremitatum ſuperficiei, aut lineæ, aut ſpatij:</s> <s xml:id="echoid-s2446" xml:space="preserve"> aut inæqualitatem earum:</s> <s xml:id="echoid-s2447" xml:space="preserve"> quoniam nul-<lb/>la ubitatum ſequentium extremitates ſuperficierum, & linearum, & ſpatiorum directè oppoſi-<lb/>torum, aut declinantium modica declinatione, ſequitur uiſum:</s> <s xml:id="echoid-s2448" xml:space="preserve"> Sed extremitates eorum oppo-<lb/>ſitæ ſequuntur ubitates dextras, aut ſiniſtras, aut ſuperiores, aut inferiores.</s> <s xml:id="echoid-s2449" xml:space="preserve"> Si ergo uiſus non <lb/>comprehenderit menſuras remotionum eorum, quæ ſunt huiuſmodi à uiſu, non comprehendet <lb/>æqualitatem remotionum extremitatum eorum, aut inæqualitatem:</s> <s xml:id="echoid-s2450" xml:space="preserve"> & ſi hæc non comprehen-<lb/>derit, non comprehendet obliquationem eorum, neque directionem.</s> <s xml:id="echoid-s2451" xml:space="preserve"> Cum ergo ſuperficies, & <lb/>lineæ, & ſpatia fuerint maximæ remotionis, & fuerit obliquatio eorum modica:</s> <s xml:id="echoid-s2452" xml:space="preserve"> non poterit ui-<lb/>ſus comprehendere obliquationem eorum, neque poteſt diſtinguere inter obliquum, & re-<lb/>ctum:</s> <s xml:id="echoid-s2453" xml:space="preserve"> quoniam quantitates remotionum ſuperficierum, & linearum, & ſpatiorum, quorum re-<lb/>motio eſt magna, non certificantur à uiſu, ſed æſtimantur.</s> <s xml:id="echoid-s2454" xml:space="preserve"> Et cum remotio eorum fuerit magna, <lb/>& fuerint ipſa modicæ obliquationis:</s> <s xml:id="echoid-s2455" xml:space="preserve"> erit differentia, quæ eſt in ter remotas extremitates eorum <lb/>oppoſitorum, ualde modica, ferè carens quantitate reſpectu quantitatum remotionum eorum.</s> <s xml:id="echoid-s2456" xml:space="preserve"> <lb/>Et cum uiſus non certificauerit quantitates remotionum extremitatum eorum, non comprehen-<lb/>det diuerſitatem remotionum, quæ eſt inter extremitates eorum.</s> <s xml:id="echoid-s2457" xml:space="preserve"> Et cum non comprehende-<lb/>rit diuerſitatem, quæ eſt inter remotiones extremitatum ſuperficiei, lineæ, & ſpatij, æſtima-<lb/>bitremotiones illas eſſe æquales, & non comprehendet obliquationem illius ſuperficiei, aut li-<lb/>neæ, aut ſpatij:</s> <s xml:id="echoid-s2458" xml:space="preserve"> & cum non comprehenderit obliquationem illius ſuperficiei, aut lineæ, aut ſpa-<lb/>tij, æſtimabit ipſum eſſe directum.</s> <s xml:id="echoid-s2459" xml:space="preserve"> Et obliquatio modica ſuperficierum, & linearum, & ſpatio-<lb/>rum, quorum remotio eſt maxima, non comprehenditur à uiſu.</s> <s xml:id="echoid-s2460" xml:space="preserve"> Viſus ergo comprehendit o-<lb/>mnes ſuperficies, & lineas, & ſpatia, quæ ſunt maximæ remotionis, & minimæ obliquationis, <lb/>quaſi directè oppoſita, & non certificat ſitus eorum, neque diſtinguit inter obliquum, & directè <lb/>oppoſitum, ſed comprehendit obliquum, & rectum ſecundum unum modum.</s> <s xml:id="echoid-s2461" xml:space="preserve"> Et ſimiliter ſitus <lb/>ſuperficierum, & linearum, & ſpatiorum, quorum remotio eſt mediocris, quando non reſpexe-<lb/>rint corpora ordinata, aut uiſus non comprehenderit corpora reſpicientia remotiones eorum, & <lb/>non certificauerit quantitates remotionum eorum, non certificatur à uiſu, nec diſtinguit ui-<lb/>ſus inter obliquum eorum & directum, ſed accipit ſitum eorum æſtimatione:</s> <s xml:id="echoid-s2462" xml:space="preserve"> & fortaſſe æſtimabit <lb/>illud, quod eſt huiuſmodi, eſſe directum, quamuis ſit obliquum.</s> <s xml:id="echoid-s2463" xml:space="preserve"> Et cum ſuperficies, & lineæ, & ſpa <lb/>tia fuerint in remotione mediocri, & remotiones eorum reſpexerint corpora ordinata, & compre-<lb/>henderit uiſus illa corpora ordinata, & quantitates eorum, comprehendet quantitates remo-<lb/>tionum extremitatum ſuperficierum illarum, & linearum, & ſpatiorum, & comprehendet æqua-<lb/>litatem remotionum extremitatum eorum oppoſitorum, ſi fuerint extremitates illæ æquales, & <lb/> <pb o="46" file="0052" n="52" rhead="ALHAZEN"/> inæqualitatem eorum, ſi fuerint inæquales.</s> <s xml:id="echoid-s2464" xml:space="preserve"> Et cum comprehenderit æqualitatem remotionum <lb/>extremitatum ſuperficierum, aut linearum, aut ſpatiorum, aut inæqualitatem eorum:</s> <s xml:id="echoid-s2465" xml:space="preserve"> comprehen-<lb/>det directionem illius ſuperficiei, autlineæ, aut ſpatij, aut eorum obliquationem certificata com-<lb/>prehenſione.</s> <s xml:id="echoid-s2466" xml:space="preserve"> Et ſimiliter obliquatio linearum, aut ſuperficierum, aut ſpatiorum, quæ ſunt maximę <lb/>obliquationis, non comprehenditur à uiſu, niſi ipſa ſint in remotione mediocri, reſpectu magnitu-<lb/>dinis eorum.</s> <s xml:id="echoid-s2467" xml:space="preserve"> Nam uiſus non comprehendit ubitates ſequentes extremitates ſuperficiei, aut lineę, <lb/>aut ſpatij:</s> <s xml:id="echoid-s2468" xml:space="preserve"> niſi quando comprehenderit quantitatem extenſionis illius ſuperficiei, aut lineę, aut ſpa <lb/>tij:</s> <s xml:id="echoid-s2469" xml:space="preserve"> ſed uiſus non comprehendit quantitatem extenſionis ſuperficiei, aut lineæ, aut ſpatij, niſi quã-<lb/>do fuerit in remotione mediocri reſpectu quantitatis illius ſuperficiei, aut lineæ, aut ſpatij.</s> <s xml:id="echoid-s2470" xml:space="preserve"> Decli-<lb/>natio ergo ſuperficiei, aut lineæ, aut ſpatij ſecantium lineas radiales, quando fuerit maxima, com-<lb/>prehendetur à uiſu comprehenſione ubitatum extremitatum eius:</s> <s xml:id="echoid-s2471" xml:space="preserve"> & ſi fuerit modicæ obliquatio-<lb/>nis, aut directæ oppoſitionis:</s> <s xml:id="echoid-s2472" xml:space="preserve"> comprehendetur à uiſu eſſe obliquum, aut eſſe directum uiſibile ex <lb/>comprehenſione quantitatum remotionum extremitatum eorum.</s> <s xml:id="echoid-s2473" xml:space="preserve"> Et uiſus non certificat quali-<lb/>tatem ſituum ſuperficierum, & linearum, & ſpatiorum, quæ ſunt maximæ obliquationis, niſi quan-<lb/>do certificauerit qualitatem extenſionis eorum.</s> <s xml:id="echoid-s2474" xml:space="preserve"> Et non certificat ſitum ſuperficierum, & linea-<lb/>rum, & ſpatiorum, quæ ſunt modicæ obliquationis, aut directæ oppoſitionis, niſi quando certifi-<lb/>cauerit quantitates remotionum extremitatum eorum, & comprehenderit inæqualitatem remo-<lb/>tionum extremitatum eorum oppoſitorum, aut æqualitatem.</s> <s xml:id="echoid-s2475" xml:space="preserve"> Sed uiſus rarò certificat ſitus uiſi-<lb/>bilium, & plura, quæ comprehendit uiſus ex ſitibus uiſibilium, non comprehendit, niſi per æſti-<lb/>mationem.</s> <s xml:id="echoid-s2476" xml:space="preserve"> Suſtentatio ergo uiſus in comprehenſione ſituum uiſibilium non eſt, niſi per æſtimatio <lb/>nem.</s> <s xml:id="echoid-s2477" xml:space="preserve"> Cum ergo aſpiciens aſpexerit, & uoluerit certificare ſitum alicuius ſuperficiei, aut ſitum a-<lb/>licuius lineæ, quæ ſunt in uiſibilibus, aut ſitum alicuius ſpatij, in ſuperficiebus uiſibilium:</s> <s xml:id="echoid-s2478" xml:space="preserve"> intue-<lb/>bitur formam illius rei uiſæ, & qualitatem extenſionis illius ſuperficiei, autlineæ, aut ſpatij.</s> <s xml:id="echoid-s2479" xml:space="preserve"> Si er-<lb/>go forma illius rei uiſæ, in qua eſt illa ſuperficies, aut linea, aut ſpatium, fuerit manifeſta, & certifi-<lb/>cata, & fuerit obliquatio iſtius ſuperficiei, aut lineæ, aut ſpatij maxima:</s> <s xml:id="echoid-s2480" xml:space="preserve"> comprehendet uiſus obli-<lb/>quationem eius uerè ex comprehenſione qualitatis extenſionis eius, & ex comprehenſione dua-<lb/>rum ubitatum extremitatum eius.</s> <s xml:id="echoid-s2481" xml:space="preserve"> Et ſi forma illius rei uiſæ fuerit manifeſta, & non fuerit maxi-<lb/>mæ obliquationis, & remotio eius reſpexerit corpora ordinata:</s> <s xml:id="echoid-s2482" xml:space="preserve"> uidebit corpora reſpicientia re-<lb/>motiones extremitatum eius, & conſiderabit quantitatem eorum:</s> <s xml:id="echoid-s2483" xml:space="preserve"> & comprehendet remotionem <lb/>illius ſuperficiei, aut lineæ, aut ſpatij, & quantitatem obliquationis eius, aut directionem eius ex <lb/>comprehenſione quantitatum remotionum extremitatum eius.</s> <s xml:id="echoid-s2484" xml:space="preserve"> Et ſi forma rei uiſæ non fuerit <lb/>manifeſta, aut fuerit manifeſta, ſed obliquatio fuerit maxima, & remotio non reſpexerit corpo-<lb/>ra ordinata:</s> <s xml:id="echoid-s2485" xml:space="preserve"> non comprehendet uiſus certitudinem ſitus huiuſmodi ſuperficiei, aut lineæ, aut <lb/>ſpatij.</s> <s xml:id="echoid-s2486" xml:space="preserve"> Et quando uiſus comprehenderit formam non manifeſtam, & inuenerit remotiones eius <lb/>reſpicere corpora ordinata:</s> <s xml:id="echoid-s2487" xml:space="preserve"> ſtatim percipiet, quòd ſitus illius ſuperficiei, aut lineæ, aut ſpatij <lb/>non certificatur.</s> <s xml:id="echoid-s2488" xml:space="preserve"> Secundum ergo iſtos modos comprehendit uiſus ſitus ſuperficierum uiſibilium, <lb/>& ſitus linearum, & ſpatiorum, quæ ſunt in ſuperficiebus uiſibilium, ſcilicet quæ omnes ſecant li-<lb/>neas radiales.</s> <s xml:id="echoid-s2489" xml:space="preserve"> Quod uerò eſt ex ſpatijs, quæ ſunt inter uiſibilia diſtincta in rebus remotioribus <lb/>maximis, ſcilicet, quando fuerit remotio utriuſque uiſibilium, quæ ſunt apud duas extremitates <lb/>ſpatij, maxima remotio, comprehenditur à uiſu tunc quaſi directè oppoſitum, quamuis ſit obli-<lb/>quum:</s> <s xml:id="echoid-s2490" xml:space="preserve"> quoniam non comprehendit diuerſitatem, quæ eſt inter remotiones extremitatum eius.</s> <s xml:id="echoid-s2491" xml:space="preserve"> <lb/>Et ſi alterum duorum uiſibilium, quæ ſunt apud duas extremitates ſpatij, fuerit propinquius al-<lb/>tero, & ſenſerit uiſus appropinquationem eius:</s> <s xml:id="echoid-s2492" xml:space="preserve"> comprehendet ſpatium, quod eſt inter ea, eſſe <lb/>obliquum, ſecundum quod comprehendit ex appropinquatione propinquioris illorum duorum <lb/>uiſibilium, & ex remotione remotioris illorum.</s> <s xml:id="echoid-s2493" xml:space="preserve"> Et ſi alterum duorum uiſibilium fuerit propin-<lb/>quius, ſed uiſus non comprehenderit appropinquationem eius:</s> <s xml:id="echoid-s2494" xml:space="preserve"> non ſentiet obliquationem ſpa-<lb/>tij, quod eſt inter ea.</s> <s xml:id="echoid-s2495" xml:space="preserve"> Situs ergo ſuperficierum, & linearum, & ſpatiorum ſecantium lineas radia-<lb/>les, non certificatur à uiſu, niſi ſit remotio eorum mediocris:</s> <s xml:id="echoid-s2496" xml:space="preserve"> & ſimul non certificat uiſus æquali-<lb/>tatem aut inęqualitatẽ remotionum extremitatũ eorum.</s> <s xml:id="echoid-s2497" xml:space="preserve"> Si autem uiſus nõ certificauerit æqualita <lb/>tẽ remotionis extremitatum eorũ, aut inæqualitatẽ, non poterit certificare ſitum illorum.</s> <s xml:id="echoid-s2498" xml:space="preserve"> Et plura <lb/>illorum, quæ comprehenduntur à uiſu ex ſitibus uiſibilium, non comprehenduntur niſi per æſti-<lb/>mationem.</s> <s xml:id="echoid-s2499" xml:space="preserve"> Si ergo ipſa fuerint in remotione mediocri, non erit magna diuerſitas inter ſitum com-<lb/>prehenſum à uiſu per æſtimationem, & uerum ſitum:</s> <s xml:id="echoid-s2500" xml:space="preserve"> & ſi fuerint in remotione maxima, non di-<lb/>ſtinguet inter obliquum & directum.</s> <s xml:id="echoid-s2501" xml:space="preserve"> Quoniam uiſus quando non comprehenderit inæqualita-<lb/>tem duarum remotionum duarum extremitatum rei uiſæ:</s> <s xml:id="echoid-s2502" xml:space="preserve"> comprehendet ipſas eſſe æquales, & ſic <lb/>iudicabit ipſam rem uiſam eſſe directam.</s> <s xml:id="echoid-s2503" xml:space="preserve"> Secundum ergo iſtos modos erit comprehenſio ſituum <lb/>ſuperficierum, & linearum, & ſpatiorum per ſenſum uiſus.</s> <s xml:id="echoid-s2504" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div84" type="section" level="0" n="0"> <head xml:id="echoid-head108" xml:space="preserve" style="it">30. Situs partium & terminorum rei uiſibilis, & ſitus uiſibilium diſtinctorum per-<lb/>cipiuntur ex æquabili & inæquabili diſtantia, ordinéque formarum ad uiſum manantium. <lb/>32 p 4.</head> <p> <s xml:id="echoid-s2505" xml:space="preserve">SItus uerò partium rei uiſæ inter ſe, & ſitus terminorum ſuperficiei rei uiſæ, aut ſuperficiei eius <lb/>inter ſe, & ſitus uiſibilium diſtinctorum inter ſe, quæ collocantur ſub ordinatione, compreheri-<lb/> <pb o="47" file="0053" n="53" rhead="OPTICAE LIBER II."/> duntur à uiſu ex comprehenſione locorum uiſus, ad quæ perueniunt formæ partium, & ex compre <lb/>henſione ordinationis partium formę peruenientis ad uiſum, per uirtutem diſtinctiuam.</s> <s xml:id="echoid-s2506" xml:space="preserve"> Quoniam <lb/>enim forma cuiuslibet partium ſuperficiei rei uiſæ peruenit in aliquam partem ſuperficiei membri <lb/>ſentientis, in quam peruenit forma totius:</s> <s xml:id="echoid-s2507" xml:space="preserve"> unde cum ſuperficies rei uiſæ fuerit diuerſorum colo-<lb/>rum, & fuerint inter partes eius differentiæ, per quas diſtinguantur partes inter ſe:</s> <s xml:id="echoid-s2508" xml:space="preserve"> erit forma per-<lb/>ueniens ad uiſum diuerſorum colorum, & erunt partes eius diſtinctæ ſecundum partium diſtin-<lb/>ctionem ſuperficiei rei uiſæ.</s> <s xml:id="echoid-s2509" xml:space="preserve"> Et ſentiens ſentit formam, & ſentit quamlibet partium formæ ex ſen-<lb/>ſu colorum illarum partium, & lucis quæ eſt in eis:</s> <s xml:id="echoid-s2510" xml:space="preserve"> & ſentit loca formarum partium in uiſu ex ſen-<lb/>ſu colorum partium illarum, & lucis illarum.</s> <s xml:id="echoid-s2511" xml:space="preserve"> Et uirtus diſtinctiua comprehendit ordinem illorum <lb/>locorum ex comprehenſione diuerſitatis colorum partium formæ, & ex comprehenſione differen <lb/>tiarum partium.</s> <s xml:id="echoid-s2512" xml:space="preserve"> Et ſic comprehendit dextrum & ſiniſtrum, ſuperius & inferius ex comparatione <lb/>illorum inter ſe:</s> <s xml:id="echoid-s2513" xml:space="preserve"> & ſic comprehendit etiam contiguum & ſeparatum.</s> <s xml:id="echoid-s2514" xml:space="preserve"> Situs uerò partium rei uiſæ, <lb/>inter ſe ſecundum acceſsionem & remotionem, ſcilicet ſecundum pręeminentiam & profundatio-<lb/>nem, comprehenduntnr à uiſu, ex comprehenſione quantitatis remotionum partium à uiſu, & <lb/>comprehenſione diuerſitatis remotionum partium ſecundum magis & minus.</s> <s xml:id="echoid-s2515" xml:space="preserve"> Situs uerò partium <lb/>rei uiſæ quando fuerint in remotione mediocri inter ſe, ſecundum acceſsionem & remotionem <lb/>comprehenduntur à uiſu:</s> <s xml:id="echoid-s2516" xml:space="preserve"> & hoc, cum uiſus comprehenderit quantitatem illius remotionis, & cõ-<lb/>prehenderit inæqualitatem, quæ eſt inter remotiones partium à uiſu, & æqualitatem.</s> <s xml:id="echoid-s2517" xml:space="preserve"> Si autem ui-<lb/>ſus non certificauerit quantitates rem otionis eius, & quantitates remotionum partium eius:</s> <s xml:id="echoid-s2518" xml:space="preserve"> non <lb/>comprehendet uiſus ordinationem partium eius ſecundum acceſsionem & remotionem apud ui-<lb/>ſionem.</s> <s xml:id="echoid-s2519" xml:space="preserve"> Si autem fuerit aliquid ex uiſibilibus aſſuetis, quæ cognoſcuntur à uiſu, comprehendet <lb/>ordinationem partium eius ſecundum præeminentiam & profunditatem, & figuram ſuperficiei e-<lb/>ius per cognitionem, non ſola uiſione:</s> <s xml:id="echoid-s2520" xml:space="preserve"> & ſi fuerit ex uiſibilibus extraneis, quæ uiſus non cognoſcit:</s> <s xml:id="echoid-s2521" xml:space="preserve"> <lb/>comprehendet ſuperficiem eius quaſi planam, quando non certificauerit quantitates remotio-<lb/>num partium eius.</s> <s xml:id="echoid-s2522" xml:space="preserve"> Et iſta intentio apparet, quando uiſus inſpexerit aliquod corpus conuexum, <lb/>aut concauum, & fuerit in remotione maxima:</s> <s xml:id="echoid-s2523" xml:space="preserve"> quoniam uiſus tunc non comprehendet conuexi-<lb/>tatem aut concauitatem, ſed comprehendet ipſum quaſi planum.</s> <s xml:id="echoid-s2524" xml:space="preserve"> Et ſitus partium ſuperficiei rei <lb/>uiſæ inter ſe in diuerſitate ubitatum, & in ſeparatione, & in continuatione non comprehendun-<lb/>tur à uiſu, niſi ex comprehenſione partium formæ peruenientis in uiſum, & comprehenſione di-<lb/>uerſitatis colorum & differentiarum, per quas diſtinguuntur partes, & ex comprehenſione ordi-<lb/>nationis partium formæ per uirtutem diſtinctiuam.</s> <s xml:id="echoid-s2525" xml:space="preserve"> Et ſitus partium ſuperficiei rei uiſæ inter ſe in <lb/>acceſsione, & etiam ſecundum remotionem reſpectu uiſus, non comprehenduntur à uiſu, niſi ex <lb/>comprehenſione quantitatis remotionis partium, & ex comprehenſione æqualitatis & inæquali-<lb/>tatis quantitatum remotionum earum.</s> <s xml:id="echoid-s2526" xml:space="preserve"> Ordinatio ergo partium rei uiſæ ſecundum acceſsionem & <lb/>remotionem illius, cuius quantitates remotionum partium certificantur à uiſu, comprehendi-<lb/>tur à uiſu:</s> <s xml:id="echoid-s2527" xml:space="preserve"> ordinatio uerò partium illius remotionum partium, cuius quantitates non certifican-<lb/>tur à uiſu, non comprehenditur à uiſu.</s> <s xml:id="echoid-s2528" xml:space="preserve"> Ordinatio autem partium rei uiſæ diſtinctarum com-<lb/>prehenditur à uiſu ex comprehenſione locorum uiſus, in quæ perueniunt formæ illarum par-<lb/>tium, & ex comprehenſione diſtinctionis in uiſu per uirtutem diſtinctiuam.</s> <s xml:id="echoid-s2529" xml:space="preserve"> Et ſimiliter eſt de ui-<lb/>ſibilibus diſtinctis.</s> <s xml:id="echoid-s2530" xml:space="preserve"> Termini autem ſuperficiei rei uiſæ, aut ſuperficierum eius, & ordinatio eo-<lb/>rum comprehenduntur à uiſu ex comprehenſione partis ſuperficiei eius, in quam peruenit color <lb/>illius ſuperficiei, & lux eius à uiſu, & ex comprehenſione terminorum illius partis, & ordinatio-<lb/>nis circumferentiæ illius partis per uirtutem diſtinctiuam.</s> <s xml:id="echoid-s2531" xml:space="preserve"> Secundum ergo iſtos modos compre-<lb/>hendit uiſus ſitus partium uiſibilium, & ſitus partium ſuperficierum uiſibilium inter ſe, & ſitus ter-<lb/>minorum ſuperficierum, & ſitus partium diſtinctarum uiſibilium inter ſe, & ſitus uiſibilium di-<lb/>ſtinctorum inter ſe.</s> <s xml:id="echoid-s2532" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div85" type="section" level="0" n="0"> <head xml:id="echoid-head109" xml:space="preserve" style="it">31. Solidit{as} quorundam corporum ſolo uiſu percipitur: quorundam uiſu & ſyllo-<lb/>giſmo ſimul. 63 p 4.</head> <p> <s xml:id="echoid-s2533" xml:space="preserve">COrporeitas uerò, quæ eſt extenſio ſecundum trinam dimenſionem, comprehenditur à uiſu <lb/>in quibuſdam corporibus, & in quibuſdam non.</s> <s xml:id="echoid-s2534" xml:space="preserve"> Tamen apud hominem diſtinguentem iam <lb/>quietum eſt principium, quod non comprehenditur ſenſu uiſu, niſi corpus:</s> <s xml:id="echoid-s2535" xml:space="preserve"> & ſic quando <lb/>ipſe comprehendet uiſibile:</s> <s xml:id="echoid-s2536" xml:space="preserve"> ſciet ſtatim quod eſt corpus, quamuis non comprehendat extenſio-<lb/>nem ſecundum trinam dimenſionem.</s> <s xml:id="echoid-s2537" xml:space="preserve"> Et uiſus comprehendit in corporibus extenſionem eorum <lb/>ſecundum longitudinem & latitudinem ex comprehenſione ſuperficierum corporum oppoſito-<lb/>rum illi.</s> <s xml:id="echoid-s2538" xml:space="preserve"> Cum ergo comprehenderit ſuperficiem corporis, ſciendo quòd illud uiſibile eſt corpus:</s> <s xml:id="echoid-s2539" xml:space="preserve"> <lb/>comprehendet ſtatim extenſionem illius corporis ſecundum longitudinem & latitudinem, & non <lb/>remanet niſi dimenſio tertia.</s> <s xml:id="echoid-s2540" xml:space="preserve"> Et quædam corpora continentur à ſuperficiebus planis ſecantibus ſe <lb/>obliquè:</s> <s xml:id="echoid-s2541" xml:space="preserve"> & quædam continentur à ſuperficiebus concauis, aut conuexis:</s> <s xml:id="echoid-s2542" xml:space="preserve"> & quædam continentur à <lb/>ſuperficiebus diuerſarum figurarum ſecantibus ſe obliquè:</s> <s xml:id="echoid-s2543" xml:space="preserve"> & quædam continentur ab una ſuperfi-<lb/>cie rotunda.</s> <s xml:id="echoid-s2544" xml:space="preserve"> Corpus autem, quod continetur à ſuperficiebus ſecãtibus ſe, cuius una ſuperficies eſt <lb/>plana, quando comprehenditur à uiſu, & fuerit ſuperficies eius plana oppoſita uiſui & directa, & ſu <lb/>perficies reſiduę ſecuerint ſuperficiẽ directè oppoſitam, aut perpẽdiculares ſuper ſuperficiẽ directè <lb/> <pb o="48" file="0054" n="54" rhead="ALHAZEN"/> oppoſitam, aut obliquæ ſuper ipſam ad partem ſtrictam ex parte poſteriori ſuperficiei directè op-<lb/>poſitæ:</s> <s xml:id="echoid-s2545" xml:space="preserve"> non apparebit uiſui ex eo, niſi ſuperficies directè oppoſita tantùm.</s> <s xml:id="echoid-s2546" xml:space="preserve"> Ergo ex huiuſmodi cor-<lb/>poribus non comprehendit uiſus, niſi longitudinem & latitudinem tantùm:</s> <s xml:id="echoid-s2547" xml:space="preserve"> ergo non ſentit corpo-<lb/>reitatem huiuſmodi.</s> <s xml:id="echoid-s2548" xml:space="preserve"> Corpus autem, quod continetur à ſuperficiebus ſecantibus ſe, quando ſuper-<lb/>ficies eius fuerit oppoſita uiſui, ſed non ſecundum directam oppoſitionem, & fuerit ſectio iſtius ſu-<lb/>perficiei cum alia ſuperficie illius corporis, comprehenſa à uiſu, ita ut poſsit comprehendere duas <lb/>ſuperficies ſimul:</s> <s xml:id="echoid-s2549" xml:space="preserve"> comprehendetur à uiſu tunc eius corporeitas:</s> <s xml:id="echoid-s2550" xml:space="preserve"> quoniam comprehendet obliqua-<lb/>tionem ſuperficiei corporis ad eius profunditatem:</s> <s xml:id="echoid-s2551" xml:space="preserve"> quare comprehendet extenſionem corporis ſe-<lb/>cundum profunditatem, cum comprehenderit ex ſuperficie obliqua extenſionem in longum & la-<lb/>tum.</s> <s xml:id="echoid-s2552" xml:space="preserve"> Et ſic comprehendet corporeitatem huiuſmodi corporum.</s> <s xml:id="echoid-s2553" xml:space="preserve"> Et ſimiliter erit, quando una ſu-<lb/>perficierum corporis directè fuerit oppoſita uiſui, & fuerint ſuperficies ſecantes illam ſuperficiem, <lb/>aut una illarum obliqua ſuper ſuperficiem directè oppoſitam ad partem amplam ex parte poſterio-<lb/>ri ſuperficiei directè oppoſitæ:</s> <s xml:id="echoid-s2554" xml:space="preserve"> quoniam uiſus comprehendet in tali corpore ſuperficiem directè op <lb/>poſitam, & ſuperficiem obliquè ſecantem ſuperficiem directè oppoſitam, & comprehendet etiam <lb/>ſectionem iſtarum ſuperficierum:</s> <s xml:id="echoid-s2555" xml:space="preserve"> & ſic, ſicut diximus, comprehendet corporeitatem illius corpo-<lb/>ris.</s> <s xml:id="echoid-s2556" xml:space="preserve"> Et generaliter dico, quòd omne corpus, in quo poteſt uiſus comprehendere duas ſuperficies <lb/>ſecantes ſe, comprehendetur in ſua corporeitate à uiſu.</s> <s xml:id="echoid-s2557" xml:space="preserve"> Corporum autem, in quibus eſt ſuperficies <lb/>conuexa comprehenſa à uiſu, & illud, quod continet ipſa, eſt aut una ſuperficies, aut multæ ſuperfi-<lb/>cies, corporeitatem uiſus comprehendere poterit ex comprehenſione ueritatis eius.</s> <s xml:id="echoid-s2558" xml:space="preserve"> Quoniam ſi <lb/>ſuperficies conuexa fuerit oppoſita uiſui:</s> <s xml:id="echoid-s2559" xml:space="preserve"> erunt remotiones partium eius à uiſu inæquales, & erit <lb/>medium eius propinquius extremitatibus uiſus:</s> <s xml:id="echoid-s2560" xml:space="preserve"> & cum uiſus comprehenderit conuexitatem eius, <lb/>comprehendet quòd medium eius eſt ſibi propinquius extremitatibus:</s> <s xml:id="echoid-s2561" xml:space="preserve"> & cum ſenſerit quòd me-<lb/>dium eius eſt propinquius illi, & quòd extremitates eius ſunt remotiores:</s> <s xml:id="echoid-s2562" xml:space="preserve"> ſentiet ſtatim, quòd ſu-<lb/>perficies exit ad ipſum ab ultimis tendentibus ad poſterius:</s> <s xml:id="echoid-s2563" xml:space="preserve"> & ſic ſentiet extenſionem corporis in <lb/>profunditate, reſpectu ſuperficiei directè oppoſitæ.</s> <s xml:id="echoid-s2564" xml:space="preserve"> Et ipſe comprehendet extenſionem corporis <lb/>illius ſecundum longitudinem & latitudinem, ex comprehenſione extenſionis ſuperficiei conuexę <lb/>ſecundum longitudinem & latitudinem.</s> <s xml:id="echoid-s2565" xml:space="preserve"> Et ſimiliter ſi alia ſuperficies corporis præter ſuperficiem <lb/>directè oppoſitam, fuerit conuexa:</s> <s xml:id="echoid-s2566" xml:space="preserve"> & comprehenderit uiſus conuexitatem eius:</s> <s xml:id="echoid-s2567" xml:space="preserve"> comprehendet e-<lb/>tiam extenſionem eius ſecundum trinam dimenſionem.</s> <s xml:id="echoid-s2568" xml:space="preserve"> Si uerò corporis, in quo eſt ſuperficies con <lb/>caua comprehenſa à uiſu, aliam ſuperficiem ſenſerit uiſus, & ſenſerit ſectionem eius cum ſuperficie <lb/>concaua:</s> <s xml:id="echoid-s2569" xml:space="preserve"> tunc ſentiet obliquationem ſuperficiei corporis illius, & cum ſenſerit obliquationem il-<lb/>lius ſuperficiei, ſtatim ſentiet corporeitatem eius.</s> <s xml:id="echoid-s2570" xml:space="preserve"> Si autem ſuperficies fuerit concaua, comprehen <lb/>ſa à uiſu, & non apparuerit uiſui alia ſuperficierum reſiduarum:</s> <s xml:id="echoid-s2571" xml:space="preserve"> non comprehendet uiſus corporei-<lb/>tatem illius corporis:</s> <s xml:id="echoid-s2572" xml:space="preserve"> neque uiſus comprehendet ex huiuſmodi corporibus, niſi extenſiones eius <lb/>ſecundum duas dimenſiones tantùm, & non ſentiet corporeitatem huiuſmodi corporum, niſi per <lb/>ſcientiam præcedentem tantùm, non per ſenſum trium dimenſionum illius corporis.</s> <s xml:id="echoid-s2573" xml:space="preserve"> Et ſuperfi-<lb/>cies concaua etiam extenditur in profunditate propter propinquitatem extremitatum eius ad ui-<lb/>ſum & remotionem medij:</s> <s xml:id="echoid-s2574" xml:space="preserve"> Sed non comprehenditur ex extenſione profunditatis, niſi extenſio ua-<lb/>cuitatis, non extenſio corporis uiſi, cuius ſuperficies eſt illa ſuperficies concaua.</s> <s xml:id="echoid-s2575" xml:space="preserve"> Comprehenſio <lb/>ergo corporeitatis à uiſu, non eſt, niſi ex comprehenſione obliquationis ſuperficierum corporum:</s> <s xml:id="echoid-s2576" xml:space="preserve"> <lb/>& obliquitates ſuperficierum corporũ, per quas ſignificatur uiſui, quòd corpora ſint corpora, non <lb/>comprehenduntur à uiſu, niſi in corporibus, quorum remotio eſt mediocris.</s> <s xml:id="echoid-s2577" xml:space="preserve"> In corporibus au-<lb/>tem maximę remotionis, quorum remotio non certificatur à uiſu, non comprehendit uiſus obli-<lb/>quationes ſuperficierum:</s> <s xml:id="echoid-s2578" xml:space="preserve"> & ſic non comprehendit corporeitatem eius per ſenſum uiſus.</s> <s xml:id="echoid-s2579" xml:space="preserve"> Quoniam <lb/>in talibus corporibus non comprehendit uiſus ſitus partium ſuperficierum eorum inter ſe, neque <lb/>comprehendit ipſas niſi planas, & ſic non comprehendit obliquationes ſuperficierum, & ſic deni-<lb/>que non comprehendit corporeitatem.</s> <s xml:id="echoid-s2580" xml:space="preserve"> Viſus ergo non comprehẽdit corporeitatem corporis ma-<lb/>ximæ remotionis, cuius remotio non certificatur illi.</s> <s xml:id="echoid-s2581" xml:space="preserve"> Et ipſe comprehendit corporeitatem cor-<lb/>porum ex comprehenſione obliquationum ſuperficierum corporum:</s> <s xml:id="echoid-s2582" xml:space="preserve"> & obliquationes ſuperficie-<lb/>rum corporum non comprehenduntur à uiſu, niſi in uiſibilibus mediocris remotionis, quorum ſi-<lb/>tus partium ſuperficierum inter ſe comprehenduntur à uiſu.</s> <s xml:id="echoid-s2583" xml:space="preserve"> Et præter iſtorum uiſibilium corpo-<lb/>reitatem, non comprehendit corporeitatem uiſus, niſi per ſcientiam antecedentem tantùm.</s> <s xml:id="echoid-s2584" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div86" type="section" level="0" n="0"> <head xml:id="echoid-head110" xml:space="preserve" style="it">32. Circulus percipitur è ſitu, quem obtinet in ſuperficie uiſus. 45 p 4.</head> <p> <s xml:id="echoid-s2585" xml:space="preserve">FIgura autem rei uiſæ diuiditur in duo:</s> <s xml:id="echoid-s2586" xml:space="preserve"> quorum alterum eſt figura circumferentiæ ſuperficiei <lb/>rei uiſæ, aut circumferentiæ alicuius partis rei uiſæ:</s> <s xml:id="echoid-s2587" xml:space="preserve"> ſecundum autem eſt figura corporeita-<lb/>tis rei uiſæ, aut figura corporeitatis alicuius partis rei uiſæ.</s> <s xml:id="echoid-s2588" xml:space="preserve"> Et iſte modus eſt forma ſuperfi-<lb/>ciei rei uiſæ, cuius corporeitas comprehenditur per ſenſum uiſus, aut forma partis ſuperficiei rei <lb/>uiſæ, cuius corporeitas comprehenditur.</s> <s xml:id="echoid-s2589" xml:space="preserve"> Et omne, quod uiſus comprehendit ex figuris uiſi-<lb/>bilium, diuiditur in iſtos modos.</s> <s xml:id="echoid-s2590" xml:space="preserve"> Figura uerò circumferentiæ ſuperficiei rei uiſæ comprehendi-<lb/>tur à ſentiente, ex comprehenſione circumferentiæ formæ, quæ peruenit in concauum nerui com-<lb/>munιs, & ex comprehenſione circumferentiæ partis ſuperficiei membri ſentientis, in quam per-<lb/>uenit forma rei uiſæ:</s> <s xml:id="echoid-s2591" xml:space="preserve"> quoniam in utroque iſtorum locorum figuratur circumferentia ſuperficiei rei <lb/>uiſæ.</s> <s xml:id="echoid-s2592" xml:space="preserve"> Quemcunque ergo iſtorum locorum animaduerterit ſentiens, poterit comprehendere in eo <lb/> <pb o="49" file="0055" n="55" rhead="OPTICAE LIBER II."/> figuram circumferentiæ rei uiſæ.</s> <s xml:id="echoid-s2593" xml:space="preserve"> Et ſimiliter figura circumferentiæ cuiuslibet partium ſuperficiei <lb/>rei uiſæ comprehenditur à ſentiente ex ſenſu ordinationis partium terminorum partis formæ.</s> <s xml:id="echoid-s2594" xml:space="preserve"> Et <lb/>cum ſentiens uoluerit certificare figuram circumferentiæ ſuperficiei rei uiſæ, aut figuram circum-<lb/>ferentiæ partis rei uiſæ, mouebit axem radialem ſuper circumferentiam rei uiſæ:</s> <s xml:id="echoid-s2595" xml:space="preserve"> & ſic per motum <lb/>certificabit ſitum partium terminorum formæ ſuperficiei, quæ eſt in ſuperficie membri ſentientis, <lb/>& in concauo nerui communis.</s> <s xml:id="echoid-s2596" xml:space="preserve"> Quare comprehendet ex certificatione ſituum terminorum for-<lb/>mæ, figuram circumferentiæ ſuperficiei rei uiſæ.</s> <s xml:id="echoid-s2597" xml:space="preserve"> Secundum ergo hunc modum erit comprehen-<lb/>ſio figuræ circumferentiæ rei uiſæ, & figuræ circumferentiæ cuiuslibet partis ſuperficiei rei uiſæ <lb/>per ſenſum uiſus.</s> <s xml:id="echoid-s2598" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div87" type="section" level="0" n="0"> <head xml:id="echoid-head111" xml:space="preserve" style="it">33. Superficies globoſa percipitur è propinquitate partium mediarum, & æquabi-<lb/>li longinquitate extremarum. 48 p 4.</head> <p> <s xml:id="echoid-s2599" xml:space="preserve">FOrma autem ſuperficiei rei uiſæ non comprehenditur à uiſu, niſi ex comprehenſione ſituum <lb/>partium ſuperficiei rei uiſæ, & ex conſimilitudine & diſsimilitudine eorundem ſituum.</s> <s xml:id="echoid-s2600" xml:space="preserve"> Et <lb/>certificatur forma ſuperficiei ex comprehenſione diuerſitatis inæqualitatis remotionum par-<lb/>tium ſuperficiei rei uiſę, & æqualitatis earum, aut inæqualitatis eleuationum partium ſuperficiei <lb/>& æqualitatis earum.</s> <s xml:id="echoid-s2601" xml:space="preserve"> Quoniam conuexitas ſuperficiei non comprehenditur à uiſu, niſi aut ex <lb/>comprehenſione propinquitatis partium mediarum in ſuperficie, & remotionis partium in termi-<lb/>nis:</s> <s xml:id="echoid-s2602" xml:space="preserve"> aut ex inęqualitate eleuationum partium eius, quando ſuperficies ſuperior corporis fuerit cõ-<lb/>uexa.</s> <s xml:id="echoid-s2603" xml:space="preserve"> Et ſimiliter conuexitas termini ſuperficiei non comprehenditur à uiſu, niſi aut ex compre-<lb/>henſione propinquitatis medij, & remotionis extremitatum, quando conuexitas eius opponitur <lb/>uiſui:</s> <s xml:id="echoid-s2604" xml:space="preserve"> aut ex inęqualitate eleuationum partium eius, quãdo gibboſitas eius fuerit deorſum, aut ſur-<lb/>ſum:</s> <s xml:id="echoid-s2605" xml:space="preserve"> aut ex inęqualitate partium eius, quod in eo dextrum eſt, aut ſiniſtrum, quando gibboſitas e-<lb/>ius fuerit dextra aut ſiniſtra.</s> <s xml:id="echoid-s2606" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div88" type="section" level="0" n="0"> <head xml:id="echoid-head112" xml:space="preserve" style="it">34. Superficies caua percipit ur è longinquit ate partium mediarum, & æquabilipro-<lb/>pinquitate extremarum. 49 p 4.</head> <p> <s xml:id="echoid-s2607" xml:space="preserve">COncauitas autem ſuperficiei, quando opponitur uiſui, comprehenditur à uiſu ex compre-<lb/>henſione remotionis partium mediarum, & appropinquatione extremitatum terminorum.</s> <s xml:id="echoid-s2608" xml:space="preserve"> <lb/>Similiter eſt de concauitate terminorum ſuperficiei, quando opponitur uiſui:</s> <s xml:id="echoid-s2609" xml:space="preserve"> & uiſus non <lb/>comprehendit concauitatem ſuperficiei, quando concauitas fuerit oppoſita ſurſum, aut deorſum, <lb/>aut ad latus, niſi quando ſuperficies concaua fuerit in parte abſciſſa, & apparuerit arcualitas termi-<lb/>ni eius, quę eſt uerſus uiſum.</s> <s xml:id="echoid-s2610" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div89" type="section" level="0" n="0"> <head xml:id="echoid-head113" xml:space="preserve" style="it">35. Planities in diſtantia moderata directè oppoſita uiſui: percipitur ex æquabili <lb/>partium longinquitate, & ſimilitudine collocationis atque ordinis ipſarum inter i-<lb/>pſas. 47 p 4.</head> <p> <s xml:id="echoid-s2611" xml:space="preserve">PLanities autem ſuperficierum comprehenditur à uiſu ex comprehenſione æqualitatis remo-<lb/>tionum partium & conſimilitudinis ordinationis earum.</s> <s xml:id="echoid-s2612" xml:space="preserve"> Et ſimiliter comprehenditur recti-<lb/>tudo termini ſuperficiei, quando terminus opponetur uiſui.</s> <s xml:id="echoid-s2613" xml:space="preserve"> Rectitudo enim termini ſuper-<lb/>ficiei, & arcualitas, aut curuitas eius, quando ſuperficies fuerit oppoſita uiſui, & termini continue-<lb/>rint ipſam, comprehenditur à uiſu ex ordinatione partium eius inter ſe.</s> <s xml:id="echoid-s2614" xml:space="preserve"> Conuexitas ergo ſuperfi-<lb/>ciei rei uiſæ, quæ opponitur uiſui, & concauitas eius, & planities comprehenduntur à uiſu ex com <lb/>prehenſione diuerſitatis remotionis partium ſuperficiei, aut eleuationum earum, aut latitudinum <lb/>earum, & ex quantitatibus exceſſus remotionis partium, aut eleuationum, aut latitudinum ea-<lb/>rum interſe.</s> <s xml:id="echoid-s2615" xml:space="preserve"> Et ſimiliter conuexitas, & concauitas, & planities cuiuslibet partis rei uiſæ compre-<lb/>henditur à uiſu ex comprehenſione exceſſus remotionum partium illius partis, aut exceſſus eleua-<lb/>tionum, aut latitudinum earum, aut æqualitatis earum.</s> <s xml:id="echoid-s2616" xml:space="preserve"> Et propter iſtam cauſſam non comprehen <lb/>dit uiſus concauitatem & conuexitatem, niſi in uiſibilibus, quorum remotio eſt mediocris.</s> <s xml:id="echoid-s2617" xml:space="preserve"> Vi-<lb/>ſus autem comprehendit propinquitatem quarundam partium ſuperficiei, & remotionem qua-<lb/>rundam per quædam corpora interuenientia inter ipſum, & ſuperficiem, & per corpora reſpicien-<lb/>tia remotiones partium, quarum appropinquatio & remotio certificatur à uiſu.</s> <s xml:id="echoid-s2618" xml:space="preserve"> Et cum quædam <lb/>partes ſuperficiei fuerint prominentes, & quædam profundæ:</s> <s xml:id="echoid-s2619" xml:space="preserve"> comprehendet uiſus prominen-<lb/>tiam & profunditatem illarum per obliquationem ſuperficierum partium, & ſectiones partium, & <lb/>curuitates earum in locis profunditatis, & per ſitus ſuperficierum partium inter ſe.</s> <s xml:id="echoid-s2620" xml:space="preserve"> Et hoc erit, <lb/>quando uiſus non comprehenderit illam ſuperficiem antè, neque aliquam huius generis.</s> <s xml:id="echoid-s2621" xml:space="preserve"> Si autem <lb/>illa res uiſa fuerit ex uiſibilibus aſſuetis, comprehendet uiſus formam eius, & formam ſuperſiciei <lb/>per cognitionem antecedentem.</s> <s xml:id="echoid-s2622" xml:space="preserve"> Forma a<gap/>tem rei uiſæ, quæ continetur ex ſuperficiebus ſecanti-<lb/>bus ſe, & diuerſorum ſituum, comprehenditur à uiſu ex comprehenſione ſectionis ſuperficiei e-<lb/>ius, & ex comprehenſione ſitus cuiuslibet ſuperficierum eius, & ex comprehenſione ſuperficie-<lb/>rum earum inter ſe.</s> <s xml:id="echoid-s2623" xml:space="preserve"> Formæ igitur figurarum rerum uiſarum, quarum corporeitas comprehendi-<lb/>tur à uiſu, comprehenduntur ex comprehenſione formarum ſuperficierum earum, & ex compre-<lb/> <pb o="50" file="0056" n="56" rhead="ALHAZEN"/> henſione ſituum ſuperficierum earum inter ſe.</s> <s xml:id="echoid-s2624" xml:space="preserve"> Et ſormæ ſuperficierum uiſibilium, quarum par-<lb/>tes ſunt diuerſi ſitus:</s> <s xml:id="echoid-s2625" xml:space="preserve"> comprehenduntur à uiſu ex comprehenſione conuexitatis & concauitatis, & <lb/>planitiei partium ſuperficierum in uiſibilibus, & prominentiæ, & profunditatis partium ſuperfi-<lb/>ciei.</s> <s xml:id="echoid-s2626" xml:space="preserve"> Secundum ergo hunc modum erit comprehenſio ſuperficierum formarum uiſibilium, & fi-<lb/>gurarum earum.</s> <s xml:id="echoid-s2627" xml:space="preserve"> Et cum ſentiens uoluerit certificare formam ſuperficiei rei uiſæ, aut formam ali-<lb/>cuius partis rei uiſæ, mouebit uiſum in oppoſitionem eius, & faciet tranſire axem radialem ſuper <lb/>omnes partes eius, donec ſentiat remotiones partium eius, & ſitus cuiuslibet illarum a pud uiſum, <lb/>& ſitum earum inter ſe.</s> <s xml:id="echoid-s2628" xml:space="preserve"> Et cum ſentiens comprehenderit remotionem partium ſuperficierum, & <lb/>ſitus earum, & comprehenderit prominentiam & profunditatem:</s> <s xml:id="echoid-s2629" xml:space="preserve"> comprehendet formam illius <lb/>ſuperficiei rei uiſæ, & certificabit figuram eius.</s> <s xml:id="echoid-s2630" xml:space="preserve"> Et multoties errat uiſus in eo, quod comprehen-<lb/>dit ex formis ſuperficierum uiſibilium, & formis figurarum uiſibilium, & non percipit errorem.</s> <s xml:id="echoid-s2631" xml:space="preserve"> <lb/>Quoniam conuexitas parua & concauitas parua, & prominentia, & profunditas parua non com-<lb/>prehenduntur ſecundum acceſſum ad uiſum, quamuis earum remotio ſit mediocris, niſi ſit propin-<lb/>qua ualde uiſui.</s> <s xml:id="echoid-s2632" xml:space="preserve"> Viſibilia ergo, quorum formæ comprehenduntur à uiſu, ſunt illa, quorum quan-<lb/>titates partium ſuperficierum comprehenduntur à uiſu, & quorum exceſſus & æqualitates remo-<lb/>tionum partium comprehenduntur à uiſu.</s> <s xml:id="echoid-s2633" xml:space="preserve"> Et uiſibilia, quorum formæ certificantur à uiſu, ſunt il-<lb/>la, quorum quantitates remotionum partium, & quorum quantitates exceſſus remotionis par-<lb/>tium certificantur à uiſu.</s> <s xml:id="echoid-s2634" xml:space="preserve"> Et ſimiliter figuræ circumferentiarum ſuperficierum uiſibilium, & figu-<lb/>ræ circumferentiarum partium ſuperficierum uiſibilium non certificantur à uiſu, niſi ſint in re-<lb/>motionibus mediocribus, & certificauerit uiſus ordinationem terminorum earum, & ſitum par-<lb/>tium terminorum earum inter ſe, & certificauerit angulos earum.</s> <s xml:id="echoid-s2635" xml:space="preserve"> Et in quibus ſitus terminorum <lb/>non certificãtur à uiſu, neque anguli, ſi habuerint angulos:</s> <s xml:id="echoid-s2636" xml:space="preserve"> in ijs non certificabit uiſus figuras.</s> <s xml:id="echoid-s2637" xml:space="preserve"> O-<lb/>mnes ergo figuræ uiſibilium comprehenduntur à uiſu, ſecundum modos, quos declarauimus.</s> <s xml:id="echoid-s2638" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div90" type="section" level="0" n="0"> <head xml:id="echoid-head114" xml:space="preserve" style="it">36. Magnitudo nec ex angulo pyramidis opticæ tantum: nec ex anguli & diſtantiæ compa-<lb/>ratione percipitur. 27 p 4.</head> <p> <s xml:id="echoid-s2639" xml:space="preserve">MAgnitudo uerò & quantitas rei uiſæ comprehenduntur à uiſu:</s> <s xml:id="echoid-s2640" xml:space="preserve"> ſed qualitas comprehenſio-<lb/>nis eius eſt ex intentionibus dubitabilibus.</s> <s xml:id="echoid-s2641" xml:space="preserve"> Et plures opinantur, quòd quantitas magni-<lb/>tudinis rei uiſæ non comprehenditur à uiſu, niſi ex quantitate anguli, qui fit apud centrum <lb/>uiſus, quem continet ſuperficies pyramidis radialis, cuius baſis continet rem uiſam:</s> <s xml:id="echoid-s2642" xml:space="preserve"> & quòd uiſus <lb/>comparat quantitates rerum uiſarum ad quantitates angulorum, qui fiunt à radijs, qui continent <lb/>res uiſas apud centrum uiſus, & non ſuſtentatur in comprehenſione magnitudinis, niſi ſuper an-<lb/>gulos tantùm.</s> <s xml:id="echoid-s2643" xml:space="preserve"> Et quidam illorum opinantur, quòd comprehenſio magnitudinis non completur <lb/>in comparatione ad angulos tantùm, ſed per conſiderationem remotionis rei uiſæ, & ſitus eius <lb/>cum comparatione ad angulos.</s> <s xml:id="echoid-s2644" xml:space="preserve"> Et ueritas eſt, quòd non eſt poſsibile, ut ſit comprehenſio quanti-<lb/>tatum rerum uiſarum à uiſu ex comparatione ad angulos, quos res uiſæ reſpiciunt apud centrum <lb/>uiſus tantùm.</s> <s xml:id="echoid-s2645" xml:space="preserve"> Quoniam eadem res uiſa non diuerſatur in quantitate apud uiſum, quamuis remo-<lb/>tiones eius diuerſentur diuerſitate non magna.</s> <s xml:id="echoid-s2646" xml:space="preserve"> Quoniam quãdo res fuerit prope uiſum, & ipſe cõ-<lb/>prehenderit quantitatem eius:</s> <s xml:id="echoid-s2647" xml:space="preserve"> & poſtea fuerit elongata à uiſu non multum:</s> <s xml:id="echoid-s2648" xml:space="preserve"> non diminuetur eius <lb/>quantitas apud uiſum, quando eius remotio fuerit mediocris.</s> <s xml:id="echoid-s2649" xml:space="preserve"> Et nunquam diuerſatur quantitas <lb/>alicuius rei uiſæ aſſuetæ apud uiſum, quando remotiones eius diuerſantur, & fuerint ex remotioni <lb/>bus mediocribus.</s> <s xml:id="echoid-s2650" xml:space="preserve"> Et ſimiliter corpora æqualia diuerſarum remotionum, quando remotio illo-<lb/>rum fuerit mediocris, cõ-<lb/> <anchor type="figure" xlink:label="fig-0056-01a" xlink:href="fig-0056-01"/> <anchor type="figure" xlink:label="fig-0056-02a" xlink:href="fig-0056-02"/> prehenduntur à uiſu æ-<lb/>qualia:</s> <s xml:id="echoid-s2651" xml:space="preserve"> Sed anguli, quos <lb/>reſpicit una & eadem res <lb/>uiſa in remotionibus di-<lb/>uerſis mediocribus, diuer <lb/>ſantur diuerſitate alicu-<lb/>ius quantitatis, [ut patet <lb/>per 21 p 1] Quoniam <lb/>quando res uiſa fuerit re-<lb/>mota à uiſu per unum cu-<lb/>bitum, deinde ſi elonge-<lb/>tur à uiſu, donec fuerit e-<lb/>ius remotio per duos cu-<lb/>bitos:</s> <s xml:id="echoid-s2652" xml:space="preserve"> erit inter duos an-<lb/>gulos, qui fiunt apud ui-<lb/>ſum ab illa re uiſa, ma-<lb/>gnus exceſſus:</s> <s xml:id="echoid-s2653" xml:space="preserve"> & tamen <lb/>non comprehendit uiſus <lb/>rem uiſam in remotione <lb/>duorum cubitorum, mi-<lb/> <pb o="51" file="0057" n="57" rhead="OPTICAE LIBER II."/> norem, quàm in remotione unius cubiti.</s> <s xml:id="echoid-s2654" xml:space="preserve"> Et ſimiliter ſi elongetur à uiſu pertres cubitos aut qua-<lb/>tuor, non uidebitur minor, quamuis anguli, qui fiunt apud uiſum, diuerſentur diuerſitate extra-<lb/>nea.</s> <s xml:id="echoid-s2655" xml:space="preserve"> Et etiam ſi in ſuperficie alicuius corporis ſignetur figura quadrata ęqualium laterum, & recto-<lb/>rum angulorum:</s> <s xml:id="echoid-s2656" xml:space="preserve"> & eleuetur illud corpus, donec ſuperficies eius, in qua eſt quadratio, ſit prope <lb/>æquidiſtantiam uiſus, & ita ut uiſus comprehendat figuram quadratam:</s> <s xml:id="echoid-s2657" xml:space="preserve"> comprehendet uiſus fi-<lb/>guram quadrilateram æqualium laterum:</s> <s xml:id="echoid-s2658" xml:space="preserve"> & tamen anguli, quos reſpiciunt latera quadrati apud <lb/>centrum uiſus, quando centrum uiſus fuerit prope ſuperficiem, in qua eſt quadratio, erunt diuer-<lb/>ſi:</s> <s xml:id="echoid-s2659" xml:space="preserve"> cum nihilominus uiſus comprehendat latera quadrati ęqualia.</s> <s xml:id="echoid-s2660" xml:space="preserve"> Et ſimiliter quando in circulo ex-<lb/>trahuntur diametri diuerſorum ſituum, deinde eleuatur ſuperficies, in qua eſt circulus, donec ſit <lb/>prope ęquidiſtantiam uiſus:</s> <s xml:id="echoid-s2661" xml:space="preserve"> erunt anguli, quos reſpiciunt diametri circuli apud centrum uiſus, di-<lb/>uerſi diuerſitate magna ſecundum diuerſitatem ſitus diametrorum:</s> <s xml:id="echoid-s2662" xml:space="preserve"> & tamen uiſus non compre-<lb/>hendit diametros circuli, niſi æquales, quando remotio circulorum fuerit mediocris.</s> <s xml:id="echoid-s2663" xml:space="preserve"> Si ergo com-<lb/>prehenſio rerum uiſarum eſſet ex comparatione ad angulos tantùm, qui fiunt ex uiſibilibus a-<lb/>pud centrum uiſus:</s> <s xml:id="echoid-s2664" xml:space="preserve"> non comprehenderentur quadrati latera æqualia, neque comprehenderen-<lb/>tur diametri circuli æquales, neque comprehenderetur circulus rotun dus, neque comprehende-<lb/>retur una res uiſa in rebus diuerſis unius quantitatis.</s> <s xml:id="echoid-s2665" xml:space="preserve"> Experimentatione igitur iſtarum intentio-<lb/>num patet, quòd comprehenſio quantitatum rerum uiſarum non eſt ex comparatione ad angu-<lb/>los tantùm.</s> <s xml:id="echoid-s2666" xml:space="preserve"/> </p> <div xml:id="echoid-div90" type="float" level="0" n="0"> <figure xlink:label="fig-0056-01" xlink:href="fig-0056-01a"> <variables xml:id="echoid-variables3" xml:space="preserve">e b a d c</variables> </figure> <figure xlink:label="fig-0056-02" xlink:href="fig-0056-02a"> <variables xml:id="echoid-variables4" xml:space="preserve">b e g a h d k f z</variables> </figure> </div> </div> <div xml:id="echoid-div92" type="section" level="0" n="0"> <head xml:id="echoid-head115" xml:space="preserve" style="it">37. Magnitudo rei uiſibilis percipitur è magnitudine partis ſuperficiei uiſ{us} (in quam per-<lb/>uenit forma) & angulo pyramidis opticæ. 17 p 4.</head> <p> <s xml:id="echoid-s2667" xml:space="preserve">ET quia hoc declaratum eſt, quomodo certificemus qualitatem comprehenſionis magnitu-<lb/>dinis:</s> <s xml:id="echoid-s2668" xml:space="preserve"> & iam declaratum eſt, quòd ſuſtentatio in comprehenſione plurium ſenſibilium non <lb/>eſt, niſi per argumentationem & diſtinctionem:</s> <s xml:id="echoid-s2669" xml:space="preserve"> magnitudo autem eſt una intentionum, quę <lb/>comprehenduntur ratione & argumentatione:</s> <s xml:id="echoid-s2670" xml:space="preserve"> & radix, ſuper quam ſuſtentatur uirtus diſtincti-<lb/>ua in diſtinctione quantitatis magnitudinis rei uiſæ, eſt quantitas partis uiſus, in quam peruenit <lb/>forma rei uiſæ:</s> <s xml:id="echoid-s2671" xml:space="preserve"> & pars, in quam peruenit forma rei uiſæ, determinatur, & menſuratur per angulum, <lb/>qui eſt apud centrum uiſus, quem continet pyramis radialis, continens rem uiſam, & partem ui-<lb/>ſus, in quam peruenit forma rei uiſæ.</s> <s xml:id="echoid-s2672" xml:space="preserve"> Pars ergo uiſus, in quam peruenit forma rei uiſæ, & angulus, <lb/>quem continet pyramis radialis, continens illam partem, ſunt radix, quam non poteſt ſenſus & <lb/>diſtinctio uitare in comprehenſione magnitudinis rei uiſæ.</s> <s xml:id="echoid-s2673" xml:space="preserve"> Sed tamen non ſufficit uirtuti diſtin-<lb/>ctiuæ in comprehenſione magnitudinis conſideratio anguli tantùm, aut conſideratio partis uiſus <lb/>reſpicientis angulum tantùm.</s> <s xml:id="echoid-s2674" xml:space="preserve"> Quoniam una res uiſa quando comprehenditur à uiſu, & eſt prope i-<lb/>pſum:</s> <s xml:id="echoid-s2675" xml:space="preserve"> comprehendet ſentiẽs locũ uiſus, in quem peruenit forma rei uiſæ, & comprehendet quan-<lb/>titatem illius loci:</s> <s xml:id="echoid-s2676" xml:space="preserve"> deinde quando illa res uiſa elongabitur à uiſu:</s> <s xml:id="echoid-s2677" xml:space="preserve"> comprehendetur etiam à uiſu, <lb/>& comprehendet ſentiens locum uiſus, in quem peruenit forma eius ſecundò, & comprehendet <lb/>quantitatem loci.</s> <s xml:id="echoid-s2678" xml:space="preserve"> Et manifeſtum eſt, quòd locus uiſus, in quem peruenit forma eius primò, & lo-<lb/>cus uiſus, in quem peruenit forma eius ſecundò, diuerſantur ſecundum quantitatem:</s> <s xml:id="echoid-s2679" xml:space="preserve"> quoniam lo-<lb/>cus formę in uiſu erit ſecundum quantitatem anguli, quem reſpicit illa res uiſa apud centrum uiſus.</s> <s xml:id="echoid-s2680" xml:space="preserve"> <lb/>Et quãtò magis elongabitur res uiſa, tantò magis anguſtabitur pyramis cõtinens ipſam, & eius an-<lb/>gulus, & locus uiſus, in quem peruenit forma.</s> <s xml:id="echoid-s2681" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div93" type="section" level="0" n="0"> <head xml:id="echoid-head116" xml:space="preserve" style="it">38. Magnitudo uera uiſibilis percipitur è comparatione baſis anguli, & longitu-<lb/>dine pyramidis opticæ. 27 p 4.</head> <p> <s xml:id="echoid-s2682" xml:space="preserve">ET cum ſentiens comprehenderit locum, in quem peruenit forma rei uiſæ, & comprehende-<lb/>rit quantitatem loci:</s> <s xml:id="echoid-s2683" xml:space="preserve"> comprehendet diminutionem loci apud remotionem rei uiſæ à uiſu.</s> <s xml:id="echoid-s2684" xml:space="preserve"> Et <lb/>iſta intentio ſæpe reuertitur ad uiſum:</s> <s xml:id="echoid-s2685" xml:space="preserve"> ſcilicet quòd uiſibilia ſæ pe elongantur à uiſu, & uiſus <lb/>ab eis, & appropin quant uiſui, & uiſus illis:</s> <s xml:id="echoid-s2686" xml:space="preserve"> & uiſus comprehendit ipſa, & comprehendit diminu-<lb/>tionem locorum formarum illarum in uiſu apud remotionem, & comprehendit augmentationem <lb/>locorum formarum illarum in uiſu apud appropinquationem.</s> <s xml:id="echoid-s2687" xml:space="preserve"> Quare ad comprehenſionem quan-<lb/>titatis rei uiſæ adiungit uirtus diſtinctiua remotionem rei uiſæ ad angulum pyramidis radialis, qui <lb/>eſt in centro oculi.</s> <s xml:id="echoid-s2688" xml:space="preserve"> Ex frequentia ergo iſtius intentionis quieuit in anima apud uirtutem diſtincti-<lb/>uam, quòd quantò magis elongatur res uiſa à uiſu, tantò magis diminuitur locus formæ eius in ui-<lb/>ſu, & angulus, quem reſpicit res uiſa apud centrum uiſus.</s> <s xml:id="echoid-s2689" xml:space="preserve"> Et cum hoc eſt:</s> <s xml:id="echoid-s2690" xml:space="preserve"> eſt quietum in anima, <lb/>quòd locus, in quem peruenit ſorma rei uiſæ, & angulus, quem reſpicit res uiſa apud centrum ui-<lb/>ſus, non erit niſi ſecundum remotionem rei uiſæ à uiſu.</s> <s xml:id="echoid-s2691" xml:space="preserve"> Et cum hoc quietum eſt in anima, quando <lb/>uirtus diſtinctiua diſtinguet quantitatem rei uiſæ, non conſiderabit angulum tantùm, ſed conſi-<lb/>derabit angulum & remotionem ſimul:</s> <s xml:id="echoid-s2692" xml:space="preserve"> quoniam quietum eſt apud ipſam, quòd angulus non erit, <lb/>nιſi ſecundum remotionem.</s> <s xml:id="echoid-s2693" xml:space="preserve"> Quantitates ergo uiſibilium non comprehenduntur, niſi per diſtin-<lb/>ctionem & comparationem.</s> <s xml:id="echoid-s2694" xml:space="preserve"> Comparatio autem, per quam comprehenditur quantitas rei uiſæ, <lb/>eſt comparatio baſis pyramidis radialis, quæ eſt ſuperficies rei uiſæ, ad angulum pyramidis, & ad <lb/>quantitatem longitudinis pyramidis, quæ eſt remotio rei uiſæ à uiſu.</s> <s xml:id="echoid-s2695" xml:space="preserve"> Et conſideratio uirtutis di-<lb/> <pb o="52" file="0058" n="58" rhead="ALHAZEN"/> ſtinctiuæ non eſt, niſi in parte ſuperficiei membri ſentientis, in quam peruenit forma rei uiſæ, cum <lb/>conſideratione remotionis rei uiſæ à ſuperficie uiſus.</s> <s xml:id="echoid-s2696" xml:space="preserve"> Quoniam quantitas partis, in quam perue-<lb/>nit forma, nunquam erit niſi ſecundum quantitatem anguli, quem reſpicit illa pars apud centrum <lb/>uiſus.</s> <s xml:id="echoid-s2697" xml:space="preserve"> Et non eſt inter remotionem rei uiſæ à ſuperficie uiſus, & remotionem eius à centro uiſus <lb/>in maiori parte diuerſitas operans in remotionem.</s> <s xml:id="echoid-s2698" xml:space="preserve"> Et etiam iam declaratum eſt, [18 n 1.</s> <s xml:id="echoid-s2699" xml:space="preserve"> & 24.</s> <s xml:id="echoid-s2700" xml:space="preserve"> <lb/>25 n] quòd ſentiens comprehendit uerticationes, quæ ſunt inter centrum uiſus & rem uiſam, quæ <lb/>ſunt uerticationes linearum radialium, & comprehendit uerticationum ordinationem, & ordina-<lb/>tionem uiſibilium, & ordinationem partium rei uiſæ.</s> <s xml:id="echoid-s2701" xml:space="preserve"> Et cum ſentiens comprehendit hæc:</s> <s xml:id="echoid-s2702" xml:space="preserve"> uirtus <lb/>diſtinctiua comprehendit, quòd iſtæ uerticationes quantò magis elongantur à uiſu, tantò magis <lb/>ampliantur ſpatia, quæ ſunt inter earum extremitates.</s> <s xml:id="echoid-s2703" xml:space="preserve"> Et iſta intentio iam etiam quieta eſt in ani-<lb/>ma:</s> <s xml:id="echoid-s2704" xml:space="preserve"> & præterea, quietum eſt etiam in anima, quòd lineæ radiales quantò magis elongabuntur à <lb/>uiſu, tantò erit res uiſa apud earum extremitates minor.</s> <s xml:id="echoid-s2705" xml:space="preserve"> Cum ergo uiſus comprehenderit ali-<lb/>quam rem uiſam, & comprehenderit terminos eius:</s> <s xml:id="echoid-s2706" xml:space="preserve"> comprehendet uerticationes, ex quibus com-<lb/>prehendet terminos rei uiſæ:</s> <s xml:id="echoid-s2707" xml:space="preserve"> & uerticationes, ex quibus comprehendet terminos rei uiſæ, ſunt <lb/>lineæ continentes angulum, qui eſt apud centrum uiſus, quem reſpicit illa res uiſa, & ſunt lineæ <lb/>continentes locum uiſus, in quem peruenit forma rei uiſæ.</s> <s xml:id="echoid-s2708" xml:space="preserve"> Cum ergo uiſus comprehenderit uer-<lb/>ticationes:</s> <s xml:id="echoid-s2709" xml:space="preserve"> imaginabitur uirtus diſtinctiua extenſionem iſtarum linearum à centro uiſus uſque ad <lb/>terminos rei uiſæ:</s> <s xml:id="echoid-s2710" xml:space="preserve"> & quando ſimul comprehenderit quantitatem remotionis rei uiſæ:</s> <s xml:id="echoid-s2711" xml:space="preserve"> imagina-<lb/>bitur quantitatem longitudinum iſtarum linearum, & quantitatem ſpatij, quod eſt inter extremi-<lb/>tates earum:</s> <s xml:id="echoid-s2712" xml:space="preserve"> & ſpatia, quæ ſunt inter extremitates iſtarum linearum, ſunt diametri rei uiſæ.</s> <s xml:id="echoid-s2713" xml:space="preserve"> Et <lb/>quando uirtus diſtinctiua imaginabitur quantitatem anguli, & quantitatem longitudinum linea-<lb/>rum radialium, & quantitatem ſpatiorum, quæ ſunt inter extremitates earum:</s> <s xml:id="echoid-s2714" xml:space="preserve"> comprehendet <lb/>quantitatem rei uiſæ ſecundum ſuum eſſe.</s> <s xml:id="echoid-s2715" xml:space="preserve"> Verticationes autem, quæ extenduntur inter centrum <lb/>uiſus & terminos cuiuslibet rei uiſæ comprehenſæ à uiſu, comprehenduntur à ſentiente, & à uir-<lb/>tute diſtinctiua:</s> <s xml:id="echoid-s2716" xml:space="preserve"> & ſentiens & uirtus diſtinctiua comprehendunt quantitatem partis uiſus, in <lb/>quam peruenit forma illius rei uiſæ.</s> <s xml:id="echoid-s2717" xml:space="preserve"> Et cum uirtus diſtinctiua comprehenderit uerticationes li-<lb/>nearum radialium:</s> <s xml:id="echoid-s2718" xml:space="preserve"> comprehendet ſitus earum inter ſe, & comprehendet appropinquationem ea-<lb/>rum inter ſe, & comprehendet qualitatem extenſionis earum:</s> <s xml:id="echoid-s2719" xml:space="preserve"> & nihil remanet, quo completur <lb/>comprehenſio magnitudinis rei uiſæ, niſi quantitas remotionis rei uiſæ.</s> <s xml:id="echoid-s2720" xml:space="preserve"> Et iam declaratum eſt in <lb/>qualitate comprehenſionis remotionis rei uiſæ, [24 n] quòd cuiuslibet rei uiſæ remotio com-<lb/>prehenditur à uiſu, aut certè, aut æſtimatione.</s> <s xml:id="echoid-s2721" xml:space="preserve"> Et cum uirtus diſtinctiua comprehenderit ſitus li-<lb/>nearum radialium continentium terminos rei uiſæ, & quantitatem partis, quæ eſt inter ipſas li-<lb/>neas radiales, & ſuperficiem membri ſentientis, quæ eſt quantitas anguli, & imaginata fuerit ſi-<lb/>mul quantitatem remotionis rei uiſæ:</s> <s xml:id="echoid-s2722" xml:space="preserve"> ſtatim imaginabitur quantitatem anguli, & remotionis ſi-<lb/>mul.</s> <s xml:id="echoid-s2723" xml:space="preserve"> Et cum imaginata fuerit quantitatem anguli & remotionis ſimul, comprehendet quantita-<lb/>tem rei uiſæ ſecundum quantitatem anguli, & ſecundum quantitatem remotionis ſimul.</s> <s xml:id="echoid-s2724" xml:space="preserve"> Et uir-<lb/>tus diſtinctiua imaginatur quantitatem remotionis cuiuslibet rei uiſæ comprehenſæ à uiſu, & i-<lb/>maginatur uerticationes continentes terminos illius, & per imaginationem iſtam perueniet ad i-<lb/>pſam forma pyramidis continentis rem uiſam, & quantitas baſis eius, quæ eſt res uiſa:</s> <s xml:id="echoid-s2725" xml:space="preserve"> & ſic per-<lb/>ueniet ad illam quantitas rei uiſæ.</s> <s xml:id="echoid-s2726" xml:space="preserve"> Et ſignificatio, quòd comprehenſio magnitudinis rei uiſæ ſit <lb/>per comparationem magnitudinis ad remotionem rei uiſæ, eſt:</s> <s xml:id="echoid-s2727" xml:space="preserve"> Quia uiſus quando comprehen-<lb/>derit duo uiſibilia diuerſæ remotionis, & reſpicientia eundem angulum apud centrum uiſus:</s> <s xml:id="echoid-s2728" xml:space="preserve"> ſci-<lb/>licet, ut radij tranſeuntes per extrema primi illorum, perueniant ad extrema ſecundi, & primum <lb/>illorum non cooperuerit totum ſecundum, & comprehenderit uiſus remotionem cuiuslibet illo-<lb/>rum comprehenſione certificata:</s> <s xml:id="echoid-s2729" xml:space="preserve"> ſemper uiſibile remotius comprehendetur à uiſu uiſibili pro-<lb/>pinquiore maius.</s> <s xml:id="echoid-s2730" xml:space="preserve"> Et quantò remotius uiſibile magis elongabitur, & uiſus certificauerit quanti-<lb/>tatem remotionis eius, tantò comprehendetur maius.</s> <s xml:id="echoid-s2731" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s2732" xml:space="preserve"> quando aliquis aſpexerit pa-<lb/>rietem remotum à uiſu remotione mediocri:</s> <s xml:id="echoid-s2733" xml:space="preserve"> & certificauerit uiſus remotionem illius parietis, <lb/>& quantitatem eius:</s> <s xml:id="echoid-s2734" xml:space="preserve"> & certificauerit quantitatem latitudinis eius:</s> <s xml:id="echoid-s2735" xml:space="preserve"> deinde appoſuerit manum u-<lb/>ni uiſui inter uiſum & parietem:</s> <s xml:id="echoid-s2736" xml:space="preserve"> & clauſerit alterum oculum:</s> <s xml:id="echoid-s2737" xml:space="preserve"> inueniet tunc, quòd manus eius co-<lb/>operiet portionem magnam illius parietis, & comprehendet quantitatem manus eius in iſta diſ-<lb/>poſitione, & comprehendet quòd quantitas cooperta à manu ex pariete, eſt multò maior quan-<lb/>titate manus eius:</s> <s xml:id="echoid-s2738" xml:space="preserve"> & uiſus ſimul comprehendet uerticationes linearum radialium, & compre-<lb/>hendet angulum, quem continent lineæ radiales.</s> <s xml:id="echoid-s2739" xml:space="preserve"> Tunc ergo uiſus comprehendet, quòd angu-<lb/>lus, quem reſpiciunt manus & paries, eſt idem angulus:</s> <s xml:id="echoid-s2740" xml:space="preserve"> & tunc etiam comprehendet, quòd <lb/>pars parietis cooperta manu eius, eſt multò maior manu.</s> <s xml:id="echoid-s2741" xml:space="preserve"> Et cum ita ſit:</s> <s xml:id="echoid-s2742" xml:space="preserve"> uirtus diſtinctiua in illa <lb/>comprehenſione comprehendit, quòd remotius duorum uiſibilium diuerſæ remotionis, reſpici-<lb/>entium unum angulum, eſt maioris quantitatis.</s> <s xml:id="echoid-s2743" xml:space="preserve"> Deinde quando quis in illa diſpoſitione uiſum <lb/>ſuum auerterit:</s> <s xml:id="echoid-s2744" xml:space="preserve"> & aſpexerit alium parietem remotiorem illo pariete:</s> <s xml:id="echoid-s2745" xml:space="preserve"> & appoſuerit manum ſu-<lb/>am inter uiſum & illum parietem:</s> <s xml:id="echoid-s2746" xml:space="preserve"> inueniet, quòd illud, quod cooperitur ex ſecundo pariete, eſt <lb/>maius illo, quod cooperitur ex primo.</s> <s xml:id="echoid-s2747" xml:space="preserve"> Et ſi tunc afpexerit cœlum:</s> <s xml:id="echoid-s2748" xml:space="preserve"> inueniet quòd manus eius <lb/>cooperiet medium illius, quod apparet de cœlo, aut magnam portionem eius:</s> <s xml:id="echoid-s2749" xml:space="preserve"> tamen aſpiciens <lb/>non dubitabit, quin manus eius nihil ſit reſpectu illius, quod cooperuerit de cœlo ſecundum ſen-<lb/>ſum.</s> <s xml:id="echoid-s2750" xml:space="preserve"> Determinabitur ergo ex iſta experimentatione, quòd uiſus non comprehendit quantita-<lb/> <pb o="53" file="0059" n="59" rhead="OPTICAE LIBER II."/> tem magnitudinis rei uiſæ, niſi ex comparatione magnitudinis rei uiſæ ad quantitatem remotionis <lb/>eius cum comparatione ad angulum, non ex comparatione ad angulum tantùm.</s> <s xml:id="echoid-s2751" xml:space="preserve"> Et ſi comprehen-<lb/>ſio quantitatis magnitudinis eſſet ſecundum angulum tantùm:</s> <s xml:id="echoid-s2752" xml:space="preserve"> oporteret ut duo uiſibilia diuerſæ <lb/>remotionis, reſpicientia unum angulum apud centrum uiſus, uiderentur æqualia.</s> <s xml:id="echoid-s2753" xml:space="preserve"> Et non eſt ita.</s> <s xml:id="echoid-s2754" xml:space="preserve"> <lb/>Quantitas ergo magnitudinis rei uiſæ non comprehenditur per diſtinctionem, niſi ex imaginatio-<lb/>ne pyramidis continentis rem uiſam à uirtute diſtinctiua, & eximaginatione quantitatis anguli py <lb/>ramidis, & ex comparatione baſis pyramidis ad quantitatem anguli eius, & ad quantitatem lon-<lb/>gitudinis eius ſimul.</s> <s xml:id="echoid-s2755" xml:space="preserve"> Et hæc eſt qualitas comprehenſionis magnitudinis.</s> <s xml:id="echoid-s2756" xml:space="preserve"> Et propter multitudi-<lb/>nem conſuetudinis uiſus in diſtinctione remotionum uiſibilium, quando ſenſerit formam & remo <lb/>tionem rei uiſæ:</s> <s xml:id="echoid-s2757" xml:space="preserve"> ſtatim imaginabitur quantitatem loci formæ, & quantitatem remotionis, & com-<lb/>prehendet ex congregatione iſtarum duarum intentionum magnitudinem rei uiſæ:</s> <s xml:id="echoid-s2758" xml:space="preserve"> Sed tamen <lb/>quantitates remotionum uiſibilium ſunt collocatæ ſub magnitudinibus, quæ comprehenduntur <lb/>à uiſu.</s> <s xml:id="echoid-s2759" xml:space="preserve"> Et iam prædictum eſt, [24.</s> <s xml:id="echoid-s2760" xml:space="preserve"> 25 n] quòd quædam quantitates remotionum uiſibilium com-<lb/>prehenduntur certè, & quædam æſtimatiuè.</s> <s xml:id="echoid-s2761" xml:space="preserve"> & quòd illæ, quæ comprehenduntur æſtimatiuè, com <lb/>prehenduntur à ſimilitudine remotionis rei uiſæ ad remotiones ſibi ſimilium ex uiſibilibus certifi-<lb/>catæ remotionis:</s> <s xml:id="echoid-s2762" xml:space="preserve"> & quòd remotiones certificatæ quantitatis ſunt illæ, quæ reſpiciunt corpora or-<lb/>dinata & continuata.</s> <s xml:id="echoid-s2763" xml:space="preserve"> Et ex comprehenſione corporum ordinatorum continuatorum reſpicien-<lb/>tium ipſas à uiſu, & ex certificatione quantitatum illorum corporum, erit certificatio quantitatum <lb/>remotionum uiſibilium, quæ ſunt apud extremitates eorum.</s> <s xml:id="echoid-s2764" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div94" type="section" level="0" n="0"> <head xml:id="echoid-head117" xml:space="preserve" style="it">39. Magnitudo diſt antiæ percipiturè corporib{us} communib{us}, inter uiſum & ui-<lb/>ſibile interiectis. 10 p 4. Idem 25 n.</head> <p> <s xml:id="echoid-s2765" xml:space="preserve">REmanet ergo declarandum, quomodo uiſus comprehendat quantitates remotionum uiſibi-<lb/>lium refpicientium corpora ordinata continuata, & quomodo certificet quantitates corpo-<lb/>rum ordinatorum continuatorum reſpicientium remotiones uiſibilium.</s> <s xml:id="echoid-s2766" xml:space="preserve"> Corpora ergo ordi-<lb/>nata continuata reſpicientia remotiones uiſibilium, ſunt in maiori parte partes terræ, & uiſibilia <lb/>aſſueta, quæ ſemper comprehenduntur à uiſu, & frequentius ſunt ſuperficies terræ, & corpus ter-<lb/>ræ interiacet inter ipſa, & corpus hominis aſpicientis.</s> <s xml:id="echoid-s2767" xml:space="preserve"> Et quantitates partium terræ interiacen-<lb/>tium inter aſpicientem & uiſibilia, quæ ſunt ſuper faciem terræ reſpicientes remotionem iſtorum <lb/>uiſibilium à uiſu, ſemper comprehenduntur à uiſu.</s> <s xml:id="echoid-s2768" xml:space="preserve"> Et comprehenſio quantitatum partium terræ <lb/>interiacentium inter aſpicientem, & uiſibilia, quæ ſunt ſuper faciem terræ, non eſt niſi ex menſu-<lb/>ratione illarum inter ſe à uiſu, & ex menſuratione partium terræ remotarum ab eo, ad partes ter-<lb/>ræ propinquas illi, quarum quantitates ſunt certificatæ.</s> <s xml:id="echoid-s2769" xml:space="preserve"> Deinde ex frequentatione comprehen-<lb/>ſionis partium terræ à uiſu, & ex frequentatione menſurationis illarum à uiſu, comprehendet ui-<lb/>ſus quantitatem partium terræ, quæ ſunt apud pedes per cognitionem & aſsimilationem illarum <lb/>per ſimiles iam prius comprehenſas.</s> <s xml:id="echoid-s2770" xml:space="preserve"> Viſus ergo quando aſpexerit partem terræ interiacentem in-<lb/>ter ipſum & rem uiſam:</s> <s xml:id="echoid-s2771" xml:space="preserve"> cognoſcet quantitatem eius propter frequentationem comprehenſionis <lb/>ſimilium illi parti terræ.</s> <s xml:id="echoid-s2772" xml:space="preserve"> Et iſta intentio eſt ex illis intentionibus, quas ſentiens acquirie à prin-<lb/>cipio quieſcentiæ.</s> <s xml:id="echoid-s2773" xml:space="preserve"> Et ſic peruenient quantitates remotionum uiſibilium aſſuetorum figuratæ in i-<lb/>maginationem & quietem in anima, ita ut homo non percipiat qualitatem quieſcentiæ earum.</s> <s xml:id="echoid-s2774" xml:space="preserve"> Vn <lb/>de uerò ſit principium comprehenſionis partium terræ interiacentium inter uiſum & uiſibilia, eſt, <lb/>ſecundum quod narrabo.</s> <s xml:id="echoid-s2775" xml:space="preserve"> Principium eius, cuius quantitas certificatur à uiſu, eſt illud, quod eſt a-<lb/>pud pedes:</s> <s xml:id="echoid-s2776" xml:space="preserve"> quoniam quantitas illius, quod eſt apud pedes, comprehenditur à uiſu & à uirtute di-<lb/>ſtinctiua, & uiſus certificat ipſam per menſuram corporis hominis.</s> <s xml:id="echoid-s2777" xml:space="preserve"> Quoniam illud, quod eſt ex ter <lb/>ra apud pedes, ſemper menſuratur ab homine ſine intentione per pedes eius, quando ambulat ſu-<lb/>per ipſum, & per brachium eius, quando extendit manus ad ipſum.</s> <s xml:id="echoid-s2778" xml:space="preserve"> Et omne, quod eſt prope ho-<lb/>minem ex terra, ſemper menſuratur per corpus hominis ſine intentione, & uiſus comprehendit il <lb/>lam menſuram, & ſentit ipſam:</s> <s xml:id="echoid-s2779" xml:space="preserve"> & uirtus diſtinctiua comprehendit iſtam menſuram, & intelligit i-<lb/>pſam, & certificat ex ea quantitates partium terræ continuatarum cum corpore hominis.</s> <s xml:id="echoid-s2780" xml:space="preserve"> Quanti-<lb/>tates ergo partium terræ propinquarum homini iam ſunt intellectæ apud ſentientem, & apud uir-<lb/>tutem diſtinctiuam.</s> <s xml:id="echoid-s2781" xml:space="preserve"> Et iam formæ earum ſunt conceptæ apud uirtutem diſtinctiuam, & quietæ <lb/>in anima:</s> <s xml:id="echoid-s2782" xml:space="preserve"> & uiſus comprehendit iſtas partes terræ ſemper, & ſentiens ſentit uerticationes, quę ex-<lb/>tenduntur à uiſu ad extremitates iſtarum partium apud comprehenſionem illarum à uiſu, & apud <lb/>conſiderationem corporis terræ à uiſu, & comprehendit partes ſuperficiei membri ſentientis, in <lb/>quas perueniunt formæ iſtarum partium terræ, & comprehendit quantitates partium, & quantita-<lb/>tes angulorum, quos reſpiciunt iſtæ partes uiſus.</s> <s xml:id="echoid-s2783" xml:space="preserve"> Anguli uerò, quos reſpiciunt partes terræ pro-<lb/>pinquæ homini, intelliguntur apud membrum ſentiens ſecundum tranſitum temporis, & formæ <lb/>eorum ſunt conceptæ in anima.</s> <s xml:id="echoid-s2784" xml:space="preserve"> Et quantitates longitudinum linearum radialium, quæ extendun <lb/>tur à centro uiſus ad extremitates partium terræ propinquarum homini, comprehenduntur à ſen-<lb/>tiente, & à uirtute diſtinctiua, & certificantur ab ea.</s> <s xml:id="echoid-s2785" xml:space="preserve"> Quoniam uerò longitudines iſtarum uertica-<lb/>tionum ſemper menſurantur per corpus hominis ſine intentione:</s> <s xml:id="echoid-s2786" xml:space="preserve"> ſi ergo homo fuerit erectus, & <lb/>aſpexerit terram apud pedes eius, erunt longitudines linearum radialium ſecundum quantitatem <lb/>erectionis hominis:</s> <s xml:id="echoid-s2787" xml:space="preserve"> & uirtus diſtinctiua intelliget certè, quòd remotio interiacens inter uiſum & <lb/> <pb o="54" file="0060" n="60" rhead="ALHAZEN"/> partem terræ, eſt quantitas erectionis hominis:</s> <s xml:id="echoid-s2788" xml:space="preserve"> & longitudines locorum terræ continuatorum <lb/>cum corpore hominis, ſunt intellectæ & perceptæ quantitates apud uirtutem diſtinctiuam, & for-<lb/>mæ eorum ſunt quietæ in anima.</s> <s xml:id="echoid-s2789" xml:space="preserve"> Cum ergo uiſus aſpexerit partem, quæ eſt apud pedes:</s> <s xml:id="echoid-s2790" xml:space="preserve"> ſtatim ſen <lb/>tiens comprehendet uerticationes peruenientes ad extremitates illius partis:</s> <s xml:id="echoid-s2791" xml:space="preserve"> & imaginabitur uir-<lb/>tus diſtinctiua quantitates longitudinum uerticationum peruenientium ad extremitates earum, <lb/>& quantitates angulorum, quos continent illæ uerticationes.</s> <s xml:id="echoid-s2792" xml:space="preserve"> Et cum uirtus diſtinctiua imaginata <lb/>fuerit quantitates longitudinum uerticationnm, & quantitates angulorum, quos continent uerti-<lb/>cationes:</s> <s xml:id="echoid-s2793" xml:space="preserve"> comprehendet quantitatem ſpatij, quæ eſt inter extremitates illarum uerticationum, cer <lb/>ta comprehenſione.</s> <s xml:id="echoid-s2794" xml:space="preserve"> Secundum ergo hunc modum certificantur quantitates partium terræ per ſen <lb/>ſum uiſus.</s> <s xml:id="echoid-s2795" xml:space="preserve"> Deinde quantitates partium ſequentium iſtas partes in remotione, comprehenduntur <lb/>â uiſu ex comparatione quantitatum partium linearum radialium, quæ extenduntur ad extremita-<lb/>tes earum, ad quantitates radialium, quæ extenduntur ad primas partes, quæ ſequuntur homi-<lb/>nem:</s> <s xml:id="echoid-s2796" xml:space="preserve"> & ſic comparat uirtus diſtinctiua lineas radiales tertio loco uenientes ad radios ſecundo ue-<lb/>nientes, communes primæ parti & ſecundæ, & percipit quantitatem augmentationis tertij radij ſu-<lb/>pra ſecundum, & cum ſenſerit, ſentiet quantitatem tertij radij:</s> <s xml:id="echoid-s2797" xml:space="preserve"> & ipſe comprehendet quantitatem <lb/>ſecundi radij certa comprehenſione.</s> <s xml:id="echoid-s2798" xml:space="preserve"> Erunt ergo duo radij continentes ſecundam partem terræ no <lb/>tæ quantitatis apud uirtutem diſtinctiuam:</s> <s xml:id="echoid-s2799" xml:space="preserve"> & ſimiliter erit ſitus eorum notus apud ipſam.</s> <s xml:id="echoid-s2800" xml:space="preserve"> Et cum <lb/>comprehenderit longitudinem duorum radiorum, & ſitum eorum:</s> <s xml:id="echoid-s2801" xml:space="preserve"> comprehendet ſpatium, quod <lb/>eſt inter extremitates eorum certa comprehenſione.</s> <s xml:id="echoid-s2802" xml:space="preserve"> Secundum ergo hunc modum comprehen-<lb/>det uirtus diſtinctiua etiam quantitates partium terræ, ſequentium partes continentes pedes.</s> <s xml:id="echoid-s2803" xml:space="preserve"> Et <lb/>etiam partes ſequentes partes continentes pedes, ſemper menſurantur per corpus hominis:</s> <s xml:id="echoid-s2804" xml:space="preserve"> quo-<lb/>niam quando homo ambulauerit ſuper terram:</s> <s xml:id="echoid-s2805" xml:space="preserve"> menſurabitur terra, ſuper quam ambulat, per pedes <lb/>eius & paſſus, & comprehendet uirtus diſtinctiua quantitatem eius.</s> <s xml:id="echoid-s2806" xml:space="preserve"> Et cum homo pertranſierit lo <lb/>cum, in quo fuerit, & partes continuatas cum pedibus eius:</s> <s xml:id="echoid-s2807" xml:space="preserve"> & peruenerit ad illas partesſequen-<lb/>tes:</s> <s xml:id="echoid-s2808" xml:space="preserve"> menſurabuntur etiam illæ partes ſequentes, ſicut menſurabantur etiam priores, & compre-<lb/>hendet etiam ſequentes, ſicut comprehendebat priores.</s> <s xml:id="echoid-s2809" xml:space="preserve"> Et iſta comprehenſio erit certificata ſine <lb/>dubio:</s> <s xml:id="echoid-s2810" xml:space="preserve"> & ſic certificabitur ab eo per comprehenſionem iſtam ſecundam prima comprehenſio.</s> <s xml:id="echoid-s2811" xml:space="preserve"> Si er <lb/>go quantitas eius non fuerit certificata primò, certificabitur ſecundò.</s> <s xml:id="echoid-s2812" xml:space="preserve"> Et iſta commenſuratio com-<lb/>prehenditur à ſentiente ſemper, & utitur ipſa ſine intentione ſolicita.</s> <s xml:id="echoid-s2813" xml:space="preserve"> Sed aſpecta aliqua partiũ ter-<lb/>ræ à uiſu:</s> <s xml:id="echoid-s2814" xml:space="preserve"> comprehendit ſentiens & uirtus diſtinctiua iſtam menſurationem per uiam accidentalẽ <lb/>ſine intentione:</s> <s xml:id="echoid-s2815" xml:space="preserve"> deinde propter frequentationem iſtius intentionis ſunt iam certificatæ quantita-<lb/>tes partium terræ ſequentium pedes, & quantitates earum, quæ ſequuntur ipſas.</s> <s xml:id="echoid-s2816" xml:space="preserve"> Secundum ergo <lb/>hunc modum acquirit ſentiens & uirtus diſtinctiua quantitates partium terræ continentium homi <lb/>nem, interiacentium inter uiſum & uiſibilia.</s> <s xml:id="echoid-s2817" xml:space="preserve"> Et iſta acquiſitio eſt in principio quieſcentiæ hominis:</s> <s xml:id="echoid-s2818" xml:space="preserve"> <lb/>deinde acquieſcunt quantitates remotionum uiſibilium aſſuetorum, quæ ſunt ſuper faciem terræ <lb/>apud ſentientem & apud uirtutem diſtinctiuam.</s> <s xml:id="echoid-s2819" xml:space="preserve"> Erit ergo comprehenſio remotionum uiſibilium <lb/>aſſuetorum, quæ ſunt ſuper faciem terræ, per cognitionem & aſsimilationem eorum adinuicem:</s> <s xml:id="echoid-s2820" xml:space="preserve"> & <lb/>eſt dicere, comprehenſionem quantitatum remotionũ uiſibilium eſſe per acquiſitionem à ſentien-<lb/>te, & à uirtute diſtinctiua:</s> <s xml:id="echoid-s2821" xml:space="preserve"> non quòd per iſta comprehendat aſpiciens quot cubiti ſint in qualibet re <lb/>motione, ſed acquirit ex qualibet remotione, & ex qualibet parte terræ quantitatem determinatã, <lb/>& ad illas quantitates determinatas comparat quantitates remotionum uiſibilium, quas compre-<lb/>hendit poſt.</s> <s xml:id="echoid-s2822" xml:space="preserve"> Et ſimiliter acquirit ex cubito & palmo, & à qualibet quantitate menſurata quantita-<lb/>tem determinatam.</s> <s xml:id="echoid-s2823" xml:space="preserve"> Quando ergo aſpiciens comprehenderit aliquod ſpatium, & uoluerit ſcire, <lb/>quot cubiti fuerint in eo:</s> <s xml:id="echoid-s2824" xml:space="preserve"> comparabit formam acquiſitam ex imaginatione ex illo ſpatio, ad formã <lb/>acquiſitam in imaginatione ex cubito, & comprehendet per iſtam comparationem ſpatij quantita-<lb/>tem reſpectu cubiti.</s> <s xml:id="echoid-s2825" xml:space="preserve"> Et etiam ex aſſuetudine hominis eſt, quòd quãdo uoluerit certificare aliquam <lb/>intentionem:</s> <s xml:id="echoid-s2826" xml:space="preserve"> iterabit aſpectum ſuum:</s> <s xml:id="echoid-s2827" xml:space="preserve"> & diſtinguet intentiones eius:</s> <s xml:id="echoid-s2828" xml:space="preserve"> & conſiderabit tempus:</s> <s xml:id="echoid-s2829" xml:space="preserve"> & per <lb/>illud comprehendet illam intentionem ſecun dum ueritatem.</s> <s xml:id="echoid-s2830" xml:space="preserve"> Aſpiciens ergo quando comprehen-<lb/>derit aliquam rem uiſam ſuper faciem terræ, & uoluerit certificare remotionem eius:</s> <s xml:id="echoid-s2831" xml:space="preserve"> intuebitur <lb/>partem terræ continuatam, interiacentem inter ipſum & rem uiſam, & mouebitur uiſus in longitu-<lb/>dine ipſius, & ſic mouebitur axis radialis ſuper illam partem, & menſurabit ipſam, & comprehen-<lb/>det ipſam ſecundum ſingulas partes, & ſentiet partes eius paruas, quando remotio illius ultimi ſpa <lb/>tij fuerit mediocris.</s> <s xml:id="echoid-s2832" xml:space="preserve"> Et quando uiſus comprehenderit partes terræ, & comprehenderit partes par-<lb/>uas:</s> <s xml:id="echoid-s2833" xml:space="preserve"> comprehendet uirtus diſtinctiua quantitatem totius ſpatij:</s> <s xml:id="echoid-s2834" xml:space="preserve"> quoniam per motum axis ra-<lb/>dialis ſuper ſpatium, certificabit uirtus diſtinctiua quantitatem partis uiſus, in quam peruenit for-<lb/>ma illius ſpatij, & quantitatem anguli, quem reſpicit illud ſpatium, & quantitatem longitudi-<lb/>nis radij, qui extenditur ad ultimum ſpatij:</s> <s xml:id="echoid-s2835" xml:space="preserve"> & cum iſtæ duæ intentiones certificabuntur à uirtu-<lb/>te diſtinctiua, certificabitur quantitas partis terræ uiſæ.</s> <s xml:id="echoid-s2836" xml:space="preserve"> Et ſimiliter quantitates longitudinum <lb/>corporum eleuatorum à terra extenſorum in parte remota (ſicut parietum & montium) com-<lb/>prehenduntur à uiſu, ſicut comprehenduntur quantitates partium terræ, & comprehendun-<lb/>tur remotiones uiſibilium reſpicientium ipſas, ex comprehenſione quantitatum longitudinum <lb/>earum.</s> <s xml:id="echoid-s2837" xml:space="preserve"> Secundum ergo hunc modum certificat uiſus quantitates remotionum uiſibilium, quan-<lb/>do fuerint ex remotionibus mediocribus, & fuerint reſpicientia corpora ordinata continua-<lb/>ta.</s> <s xml:id="echoid-s2838" xml:space="preserve"> Quædam autem uiſibilia, quæ ſunt ſuper faciem terræ, habent remotionem mediocrem, <lb/> <pb o="55" file="0061" n="61" rhead="OPTICAE LIBER II."/> & quantitates partium terræ interiacentium inter uiſum & ipſa, ſunt quantitates mediocres:</s> <s xml:id="echoid-s2839" xml:space="preserve"> & <lb/>quædam ſunt, quorum remotio eſt maxima & extra mediocritatem, & quantitates partium terræ <lb/>interiacentium inter uiſum & ipſa, ſunt extraneæ magnitudinis.</s> <s xml:id="echoid-s2840" xml:space="preserve"> Et quantitates partium terrę com <lb/>prehenduntur à uiſu ſecundum modos, quos narrauimus.</s> <s xml:id="echoid-s2841" xml:space="preserve"> Illud ergo, quod eſt propinquum & me-<lb/>diocris quantitatis, comprehenditur, & certificatur à uiſu, & quantitas eius, quòd eſt extraneæ re-<lb/>motionis, non certificatur à uiſu:</s> <s xml:id="echoid-s2842" xml:space="preserve"> quoniam uiſus quando comprehenderit ſpatia:</s> <s xml:id="echoid-s2843" xml:space="preserve"> comprehen-<lb/>det quantitates eorundem, dum ſenſerit augmentationem longitudinis radij:</s> <s xml:id="echoid-s2844" xml:space="preserve"> & dum ſenſerit an-<lb/>gulos, quos reſpiciunt partes paruæ partium ſpatij apud motum axis ſuper ſpatium:</s> <s xml:id="echoid-s2845" xml:space="preserve"> & certifica-<lb/>bit quantitatem ſpatij, dum ſenſerit paruam augmentationem in longitudine radij, & augmen-<lb/>tationem paruam in angulo, quem reſpicit ſpatium.</s> <s xml:id="echoid-s2846" xml:space="preserve"> Et cum remotio fuerit maxima, non ſentiet <lb/>augmentationem paruam in longitudine radij, nec ſentiet motum radij propter paruam partem <lb/>ſpatij, cuius remotio eſt maxima, nec ſentiet angulum, quem reſpicit parua pars remotionis ma-<lb/>ximæ, nec certificabit longitudinem radij peruenientis ad extremum ſpatij, nec certificabit quan-<lb/>titatem anguli, quem reſpicit ſpatium illud.</s> <s xml:id="echoid-s2847" xml:space="preserve"> Et cum non certificauerit longitudinem radij perue-<lb/>nientis ad extremum ſpatij, & non certificauerit quantitatem anguli, quem reſpicit ſpatium:</s> <s xml:id="echoid-s2848" xml:space="preserve"> non <lb/>certificabit quantitatem ſpatij.</s> <s xml:id="echoid-s2849" xml:space="preserve"> Et etiam, quando remotio fuerit maxima, partes paruæ ſpatij, quæ <lb/>ſunt in ultimo ſpatij, non comprehenduntur à uiſu, nec diſtinguuntur ab eo:</s> <s xml:id="echoid-s2850" xml:space="preserve"> quoniam parua <lb/>quantitas in remotione maxima latet uiſum.</s> <s xml:id="echoid-s2851" xml:space="preserve"> Cum ergo axis radialis mouebitur ſuper ſpatium re-<lb/>motum maximè, & perueniet ad remotionem maximam, tranſibit partem paruam ſpatij, & <lb/>non ſentiet ſentiens motum eius:</s> <s xml:id="echoid-s2852" xml:space="preserve"> quoniam parua pars in remotione maxima non facit angulum <lb/>ſenſibilem apud centrum uiſus.</s> <s xml:id="echoid-s2853" xml:space="preserve"> Cum ergo axis radialis mouebitur ſuper ſpatium remotum, & ſen-<lb/>ſerit uiſus, quòd ipſe iam tranſierit aliquam partem ſpatij:</s> <s xml:id="echoid-s2854" xml:space="preserve"> quantitas illius partis ſpatij, quam tran-<lb/>ſiuit, non erit quantitas, quam comprehendit per ſenſum, ſed erit maior:</s> <s xml:id="echoid-s2855" xml:space="preserve"> & quantò magis augmen-<lb/>tabitur remotio ſpatij, tantò magis partes latebunt uiſum apud ultimum ſpatij, & ſuper quas la-<lb/>tet motus radij uiſus, erunt maiores.</s> <s xml:id="echoid-s2856" xml:space="preserve"> Quantitates ergo remotionum maximarum, quæ ſunt ſu-<lb/>per faciem terræ, non certificantur à uiſu;</s> <s xml:id="echoid-s2857" xml:space="preserve"> quonram non certificat quantitatem longitudinis radij <lb/>peruenientis ad ultimum earum, nec quantitatem anguli, quem reſpicit illud ſpatium.</s> <s xml:id="echoid-s2858" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div95" type="section" level="0" n="0"> <head xml:id="echoid-head118" xml:space="preserve" style="it">40. Viſibile propinquum uiſui accur ati{us} uidetur. 15 p 4.</head> <p> <s xml:id="echoid-s2859" xml:space="preserve">ET etiam ſentiens ſentit certificationem quantitatis ſpatij:</s> <s xml:id="echoid-s2860" xml:space="preserve"> quoniam uiſibile propinquum <lb/>uiſui eſt certioris uiſionis:</s> <s xml:id="echoid-s2861" xml:space="preserve"> ſcilicet quia formæ uiſibilium propinquorum ſunt manifeſtio-<lb/>res, & comprehenduntur à uiſu manifeſtiore comprehenſione, & color & lux eorum ſunt <lb/>manifeſtiores, & ſitus ſuperficierum eorum apud uiſum, & ſitus partium eorum, & forma parti-<lb/>um eorum, & partium ſuperficierum ſunt manifeſtiores uiſui:</s> <s xml:id="echoid-s2862" xml:space="preserve"> & ſi in eis fuerit lineatio, aut pictura, <lb/>aut partes paruæ, apparebunt uiſui manifeſtius:</s> <s xml:id="echoid-s2863" xml:space="preserve"> & non eſt ita de uiſibilibus remotionis maximæ:</s> <s xml:id="echoid-s2864" xml:space="preserve"> <lb/>quoniã formã rei uiſę, quę fuerit in remotione maxima, non certificabit uiſus ſecundum ſuum eſſe, <lb/>& dubitabit in colore, luce, & forma ſuperficierum eius, & nihil apparebit in ea ex ſubtilibus in-<lb/>tentionibus & ex partibus paruis.</s> <s xml:id="echoid-s2865" xml:space="preserve"> Et iſta intentio manifeſta eſt ſenſui.</s> <s xml:id="echoid-s2866" xml:space="preserve"> Cum ergo uiſus compre-<lb/>henderit aliquod ſpatium ſuper faciem terræ, ſtatim ſentiet, priuſquam uiderit ultimum eius, & <lb/>quædam uiſibilia in ultimo eius, quòd illud ſpatium eſt ex ſpatijs mediocribus, aut ex ſpatijs ma-<lb/>ximæ remotionis.</s> <s xml:id="echoid-s2867" xml:space="preserve"> Si uerò certificauerit formam ultimi eius, aut formam rei uiſæ, quæ eſt apud ul-<lb/>tim<gap/><gap/> eius, manifeſtè, & diſtinxerit etiam quantitatẽ illius ſpatij ſecundum modum prædictum:</s> <s xml:id="echoid-s2868" xml:space="preserve"> <lb/>tunc uirtus diſtinctiua etiam comprehendet, quòd quantitas illius ſpatij eſt certificata ex compre-<lb/>henſione manifeſtationis formæ ultimi eius, aut formæ rei uiſæ, quæ eſt apud ultimum eius.</s> <s xml:id="echoid-s2869" xml:space="preserve"> Si au-<lb/>tem non certificauerit formam ultimi eius, aut formam rei uiſæ, quæ eſt apud ultimum eius, non <lb/>certificabit quantitatem illius ſpatij.</s> <s xml:id="echoid-s2870" xml:space="preserve"> Et uirtus diſtinctiua apud conſiderationem iſtius ſpatij ſimul <lb/>comprehendet, quòd iſtud ſpatium non eſt certificatæ quantitatis, propter latentiam formæ ulti-<lb/>mi eius, aut formæ rei uiſæ, quæ eſt apud ultimum eius.</s> <s xml:id="echoid-s2871" xml:space="preserve"> Quantitates ergo remotionum uiſibili-<lb/>um diſtinguuntur à uiſu, & qualitas comprehenſionis quantitatum earum certificatur apud intui-<lb/>tionem.</s> <s xml:id="echoid-s2872" xml:space="preserve"> Et quando aſpiciens uoluerit certificare quantitatem rei uiſæ, & certificare quantitatem <lb/>remotionis rei uiſæ:</s> <s xml:id="echoid-s2873" xml:space="preserve"> intuebitur remotionem, & diſtinguet ipſam, & ſic diſtinguetur ab eo remotio <lb/>certificata à remotione non certificata.</s> <s xml:id="echoid-s2874" xml:space="preserve"> Nihil ergo eſt ex intentionibus uiſibilium, cuius quantitas <lb/>ſit certificata, niſi remotiones reſpicientes corpora ordinata continuata, & cuius etiam remotio-<lb/>nes ſunt mediocres.</s> <s xml:id="echoid-s2875" xml:space="preserve"> Quantitates ergo huiuſmodi remotionum comprehenduntur à uiſu ſecun-<lb/>dum modum, quem declarauimus.</s> <s xml:id="echoid-s2876" xml:space="preserve"> Et præter iſta non certificantur à uiſu, ſed æſtimantur & aſsi-<lb/>milantur:</s> <s xml:id="echoid-s2877" xml:space="preserve"> ſcilicet quòd uiſus aſsimilat remotionem rei uiſæ remotioni ſibi ſimilium ex uiſibilibus <lb/>aſſuetis, quorum quantitas remotionis eſt certificata iam ab eo.</s> <s xml:id="echoid-s2878" xml:space="preserve"> Et cum uiſus ſenſerit iam laten-<lb/>tiam formæ rei uiſæ propter remotionem, dubitabit de quantitate remotionis eius.</s> <s xml:id="echoid-s2879" xml:space="preserve"> Et remotio <lb/>mediocris, cuius quantitas certificatur à uiſu, eſt remotio, apud cuiu<gap/> ultimum non latet uiſum <lb/>pars, habens proportionem ſenſibilem ad totam remotionem:</s> <s xml:id="echoid-s2880" xml:space="preserve"> & remotio mediocris reſpectu rei <lb/>uiſæ, in qua uiſus comprehendit ueram quantitatem rei uiſæ, eſt remotio mediocris, apud cu-<lb/>ius ultimum non latet pars illius rei uiſæ, habens proportionem ſenſibilem ad quantitatem rei <lb/>uiſæ, quando uiſus intuebitur illam partem per ſe.</s> <s xml:id="echoid-s2881" xml:space="preserve"> Omne ergo ſpatium, in quo cuiuslibet partis <lb/> <pb o="56" file="0062" n="62" rhead="ALHAZEN"/> longitudo habet proportionem ſenſibilem ad quantitatem longitudinis ſpatij, comprehenditur à <lb/>uiſu, & non latet uiſum ex partibus ſpatij, quæ ſunt apud ultimum eius, nιſi illud, quod caret pro-<lb/>portione ſenſibili ad longitudinem illius ſpatij:</s> <s xml:id="echoid-s2882" xml:space="preserve"> & omne tale ſpatium eſt ex remotionibus medio-<lb/>cribus.</s> <s xml:id="echoid-s2883" xml:space="preserve"> Remotio autem, quæ eſt extra mediocritatem in longitudine, eſt illa, apud cuius ultimum <lb/>latet quantitas habens proportionem ſenſibilem ad totam illam remotionem:</s> <s xml:id="echoid-s2884" xml:space="preserve"> & remotio, quæ eſt <lb/>extra mediocritatem reſpectu uiſus, eſt illa, in qua latet quantitas aliqua ex illa re uiſa, habens <lb/>proportionem ſenſibilem ad totam illam rem uiſam:</s> <s xml:id="echoid-s2885" xml:space="preserve"> aut latet aliqua intentio illius rei uiſæ, cuius <lb/>latentia operatur in latentiam quidditatis illius rei uiſæ.</s> <s xml:id="echoid-s2886" xml:space="preserve"> Et etiam ſentiens comprehendit quanti-<lb/>tatem remotionis rei uiſæ ex quantitate anguli, quem reſpicit illa res uiſa.</s> <s xml:id="echoid-s2887" xml:space="preserve"> Quoniam quando ui-<lb/>ſus comprehendit uiſibilia aſſueta, quæ ſunt in remotionibus aſſuetis, ſtatim apud comprehen-<lb/>ſionem cognoſcet ipſa uiſus:</s> <s xml:id="echoid-s2888" xml:space="preserve"> & quando uiſus cognouerit ipſa:</s> <s xml:id="echoid-s2889" xml:space="preserve"> cognoſcet ipſas quantitates magni-<lb/>tudinum eorum:</s> <s xml:id="echoid-s2890" xml:space="preserve"> quoniam quantitates magnitudinum eorum iam fuerunt certificatæ propter <lb/>frequentationem cuiuslibet comprehenſionis uiſibilium aſſuetorum, & iam ſunt quietæ in ima-<lb/>ginatione.</s> <s xml:id="echoid-s2891" xml:space="preserve"> Et uiſus, cum comprehenderit rem uiſam aſſuetam, ſtatim comprehendit partem uiſus, <lb/>in quam peruenit forma illius rei uiſæ, quam reſpicit illa pars.</s> <s xml:id="echoid-s2892" xml:space="preserve"> Et cum ſentiens comprehenderit <lb/>quantitatem magnitudinis rei uiſæ per cognitionem, & comprehenderit angulum, quem tunc re-<lb/>ſpicit illa res uiſa:</s> <s xml:id="echoid-s2893" xml:space="preserve"> comprehendet quantitatem remotionis illius rei uiſæ in illa diſpoſitione:</s> <s xml:id="echoid-s2894" xml:space="preserve"> quo-<lb/>niam angulus, quem reſpicit illa res uiſa, non erit, niſi ſecundum quantitatem remotionis.</s> <s xml:id="echoid-s2895" xml:space="preserve"> Et ſi-<lb/>cut ſentiens recipit ſignificationem ſuper quantitatem magnitudinis ex remotione cum illo angu-<lb/>lo:</s> <s xml:id="echoid-s2896" xml:space="preserve"> ita accipit ſignificationem ſuper quantitatem remotionis ex quantitate magnitudinis cognitæ <lb/>apud ipſam cum illo angulo:</s> <s xml:id="echoid-s2897" xml:space="preserve"> quoniam illa magnitudo non reſpicit illum angulum, niſi ex illa ea-<lb/>dem remotione, aut ex remotione æquali illi, non ex omnibus remotionibus.</s> <s xml:id="echoid-s2898" xml:space="preserve"> Et cum ſentiens <lb/>comprehenderit quantitatem remotionis illius rei uiſæ aſſuetæ multoties & frequenter in horis, <lb/>in quibus illa res uiſa reſpicit apud centrum uiſus ſimilem illi angulo, & multoties acceperit ſigni-<lb/>ficationem ſuper quantitatem magnitudinis illius rei uiſæ ex quantitate remotionis illius rei uiſæ <lb/>cum quantitate anguli, qui eſt æqualis illi angulo:</s> <s xml:id="echoid-s2899" xml:space="preserve"> uirtus diſtinctiua intelliget quantitatem remo-<lb/>tionis, in qua comprehendit magnitudinem illius rei uiſæ, reſpectu illius anguli.</s> <s xml:id="echoid-s2900" xml:space="preserve"> Et cum uirtus di-<lb/>ſtinctiua intellexerit quantitatem illius rei uiſæ, reſpectu illius anguli, & comprehenderit in iſta re-<lb/>motione magnitudinem rei uiſæ, reſpectu illius eiuſdem anguli, & cognouerit illam rem uiſam, & <lb/>cognouerit quantitatem magnitudinis eius, quam antè comprehendit, & comprehenderit quan-<lb/>titatem illius anguli, quem tunc reſpicit illa res uiſa:</s> <s xml:id="echoid-s2901" xml:space="preserve"> cognoſcet quantitatem remotionis, ſecun-<lb/>dum quam illa remotio reſpicit illum angulum.</s> <s xml:id="echoid-s2902" xml:space="preserve"> Sentiens ergo comprehendit quantitatem remo-<lb/>tionum uiſibilium aſſuetorum ex comparatione anguli ad magnitudinem rei uiſæ:</s> <s xml:id="echoid-s2903" xml:space="preserve"> deinde pro-<lb/>pter frequentationem comprehendit ſentiens remotionem rei uiſæ aſſuetæ per cognitionem.</s> <s xml:id="echoid-s2904" xml:space="preserve"> Et <lb/>erit quantitas anguli, quem reſpicit res uiſa aſſueta apud comprehenſionem anguli eius, cum co-<lb/>gnitione illius rei uiſæ, ſignum ſuper quantitatem remotionis illius rei uiſæ.</s> <s xml:id="echoid-s2905" xml:space="preserve"> Et plures remotio-<lb/>nes uiſibilium aſſuetorum comprehenduntur ſecundum hunc modum.</s> <s xml:id="echoid-s2906" xml:space="preserve"> Et iſta comprehenſio non <lb/>eſt in fine certitudinis:</s> <s xml:id="echoid-s2907" xml:space="preserve"> T amen inter remotionem iſtam & remotionem certificatam non eſt maxi-<lb/>ma diuerſitas.</s> <s xml:id="echoid-s2908" xml:space="preserve"> Et ex iſta comprehenſione opinati ſunt mathematici, quòd magnitudo rei compre-<lb/>hendatur per angulum.</s> <s xml:id="echoid-s2909" xml:space="preserve"> Quando ergo uiſus comprehenderit uiſibilia aſſueta, քuæ ſunt in remo-<lb/>tionibus aſſuetis, & cognouerit quantitates remotionum illorum ſecundum iſtam uiam:</s> <s xml:id="echoid-s2910" xml:space="preserve"> inueniet <lb/>ueritatem in maiori parte in quantitatibus remotionum ipſorum, aut non erit inter illud, quod <lb/>comprehendit ex quantitatibus remotionum eorum, & inter remotiones ueras magna diuerſi-<lb/>tas.</s> <s xml:id="echoid-s2911" xml:space="preserve"> In illo autem, quod uiſus comprehendit ex quantitatibus remotionum uiſibilium extraneo-<lb/>rum, quæ non frequenter comprehendit uiſus, errat in maiori parte:</s> <s xml:id="echoid-s2912" xml:space="preserve"> & eum hoc fortè inueniet a-<lb/>liquid in eo, quod comprehendit ex quantitatibus eorum ſecundum hunc modum.</s> <s xml:id="echoid-s2913" xml:space="preserve"> Secundum <lb/>ergo iſtos modos, quos declarauimus, comprehenduntur quantitates remotionum uiſibilium <lb/>per ſenſum uiſus.</s> <s xml:id="echoid-s2914" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div96" type="section" level="0" n="0"> <head xml:id="echoid-head119" xml:space="preserve" style="it">41. Magnitudines uiſibiles ſunt ſuperficies, earum partes, termini, & ſpatia, quæinter di-<lb/>ſtincta uiſibilia interijciuntur. 18 p 4.</head> <p> <s xml:id="echoid-s2915" xml:space="preserve">ET poſtquam declarata eſt qualitas comprehenſionis quantitatum remotionum uiſibilium, <lb/>& diſtinctæ ſunt remotiones uiſibilium:</s> <s xml:id="echoid-s2916" xml:space="preserve"> diſtinguemus modò magnitudines uiſibilium, quæ <lb/>comprehenduntur à uiſu, & diſtinguemus comprehenſionem illarum à uiſu.</s> <s xml:id="echoid-s2917" xml:space="preserve"> Dicamus ergo, <lb/>quòd magnitudines, quas comprehendit uiſus apud oppoſitionem, ſunt quantitates ſuperficie-<lb/>rum uiſibilium, & quantitates partium ſuperficierum uiſibilium, & quantitates terminorum ſu-<lb/>perficierum uiſibilium, & quantitates ſpatiorum, quæ ſunt inter terminos partium ſuperficierum <lb/>uiſibilium, & quantitates ſpatiorum, quæ ſunt inter uiſibilia diſtincta.</s> <s xml:id="echoid-s2918" xml:space="preserve"> Et iſti ſunt omnes modi <lb/>quantitatum, quas comprehendit uiſus apud oppoſitionem rei uiſæ.</s> <s xml:id="echoid-s2919" xml:space="preserve"> Quantitas autem corporis <lb/>rei uiſæ non comprehenditur à uiſu apud oppoſitionem quoniam uiſus non comprehendit totã <lb/>ſuperficiẽ corporis apud oppoſitionẽ, & non cõprehendit niſi illud, quod ſibi opponitur ex ſuperfi <lb/>cie corporis, aut ex ſuperficiebus eius, quãuis corpus ſit paruũ.</s> <s xml:id="echoid-s2920" xml:space="preserve"> Et ſi uiſus cõprehẽderit corporeita <lb/>tẽ corporis, nõ cõprehẽdet quantitatẽ corporis eius, ſed figurã corporeitatis tãtùm.</s> <s xml:id="echoid-s2921" xml:space="preserve"> Si ergo corpus <lb/>fuerit motum, aut uiſus moueatur, ita ut comprehendat uiſus totam ſuperficiem corporis per ſen-<lb/> <pb o="57" file="0063" n="63" rhead="OPTICAE LIBER II."/> ſum, aut per ſignificationem:</s> <s xml:id="echoid-s2922" xml:space="preserve"> tunc uirtus diſtinctiua comprehendet quantitates corporeitatis e-<lb/>ius per ſecundam argumentationem, præter argumentationem, qua uſa eſt apud uiſionem.</s> <s xml:id="echoid-s2923" xml:space="preserve"> Et ſi-<lb/>militer ſi uirtus diſtinctiua comprehendet quantitatem corporeitatis cuiuslibet partium corpo-<lb/>ris, non comprehendet ipſam, niſi per argumentationem ſecundam, præter argumentationem, <lb/>quæ eſt apud uiſionem.</s> <s xml:id="echoid-s2924" xml:space="preserve"> Quantitates ergo, quas uiſus comprehendit apud oppoſitionem, non ſunt <lb/>niſi quantitates ſuperficierum, & linearum quas determinauimus tantùm.</s> <s xml:id="echoid-s2925" xml:space="preserve"> Et iam declaratum eſt, <lb/>[38 n] quòd comprehenſio magnitudinis non eſt, niſi ex comparatione baſis pyramidis radialis <lb/>continentis magnitudinem, ad angulum pyramidis, qui eſt apud centrum uiſus, & longitudinem <lb/>pyramidis, quæ eſt remotio magnitudinis rei uiſæ:</s> <s xml:id="echoid-s2926" xml:space="preserve"> & iam declaratum eſt, [24.</s> <s xml:id="echoid-s2927" xml:space="preserve"> 25 n] quòd quæ-<lb/>dam remotiones uiſibilium ſunt certificatæ, & quædam æſtimatæ:</s> <s xml:id="echoid-s2928" xml:space="preserve"> magnitudines autem uiſibili-<lb/>um, quorum remotio eſt certificata, comprehenduntur à uiſu ex comparatione magnitudinum <lb/>earum ad angulos, quos reſpiciunt illæ magnitudines apud centrum uiſus, & ad remotiones eo-<lb/>rum certificatas.</s> <s xml:id="echoid-s2929" xml:space="preserve"> Comprehenſio ergo quantitatum remotionum huiuſmodi uiſibilium erit com-<lb/>prehenſio certificata.</s> <s xml:id="echoid-s2930" xml:space="preserve"> Quantitates autem remotionum uiſibilium, quorum remotio eſt æſtimata, <lb/>& non certificata:</s> <s xml:id="echoid-s2931" xml:space="preserve"> comprehenduntur à uiſu ex comparatione magnitudinis eorum ad angulos, <lb/>quos reſpiciunt illæ magnitudines apud centrum uiſus:</s> <s xml:id="echoid-s2932" xml:space="preserve"> & ad remotiones earum æſtimatas & non <lb/>certificatas.</s> <s xml:id="echoid-s2933" xml:space="preserve"> Comprehenſio ergo quantitatum remotionum uiſibilium huiuſmodi, erit compre-<lb/>henſio non certificata.</s> <s xml:id="echoid-s2934" xml:space="preserve"> Cum ergo ſentiens uoluerit certificare quantitatem magnitudinis alicuius <lb/>rei niſæ, mouebit uiſum ſuper illius diametros, & ſic mouebitur axis radialis ſuper omnes partes <lb/>rei uiſæ.</s> <s xml:id="echoid-s2935" xml:space="preserve"> Si ergo remotio rei uiſæ fuerit ex remotionibus maximis:</s> <s xml:id="echoid-s2936" xml:space="preserve"> ſtatim apparebit ſenſui laten-<lb/>tia formæ eius, & manifeſtabitur ſentienti, quòd quantitas eius non eſt certificata:</s> <s xml:id="echoid-s2937" xml:space="preserve"> ſi uerò re-<lb/>motio rei uiſæ fuerit ex remotionibus mediocribus:</s> <s xml:id="echoid-s2938" xml:space="preserve"> ſtatim apparebit ſenſui uerificatio uiſionis e-<lb/>ius.</s> <s xml:id="echoid-s2939" xml:space="preserve"> Si ergo axis radialis moueatur ſuper illud, quod eſt in huiuſmodi uiſibilibus:</s> <s xml:id="echoid-s2940" xml:space="preserve"> menſurabit <lb/>ipſum uera menſuratione, & comprehendet partes eius, & certificabit quantitates partium eius, <lb/>& per motum certificabit quantitatem partium ſuperficiei membri ſentientis, in quam peruenit <lb/>forma illius rei uiſæ, & quantitatem anguli pyramidis, quem reſpicit illa pars.</s> <s xml:id="echoid-s2941" xml:space="preserve"> Et cum ſentiens uo-<lb/>luerit certificare remotionem ſuper corpus reſpiciens remotionem eius, per motum certificabit <lb/>quantitatem corporis reſpicientis remotionem eius, quæ eſt æqualis ſecundum ſenſum longitu-<lb/>dinibus linearum radialium.</s> <s xml:id="echoid-s2942" xml:space="preserve"> Et cum ſentiens certificauerit quantitatem remotionis rei uiſæ, & <lb/>quantitatem anguli, quem continet pyramis, continens rem uiſam:</s> <s xml:id="echoid-s2943" xml:space="preserve"> certificabit quantitatem il-<lb/>lius rei uiſæ.</s> <s xml:id="echoid-s2944" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div97" type="section" level="0" n="0"> <head xml:id="echoid-head120" xml:space="preserve" style="it">42. Axis opticæpyramidis, oculo moto immut abilis permanet. 53 p 3.</head> <p> <s xml:id="echoid-s2945" xml:space="preserve">MOtus autem axis ſuper partes rei uiſæ non erit per gyrationem axis à loco centri, & per <lb/>motum eius per ſe ſuper partes rei uiſæ:</s> <s xml:id="echoid-s2946" xml:space="preserve"> quoniam iam declaratum eſt, [11 n 1 & 7 n] quòd <lb/>iſta linea ſemper eſt extenſa rectè uſque ad locum gyrationis nerui, ſuper quem componi-<lb/>tur oculus, & quòd ſitus eius à uiſu non mutatur, & totus oculus mouetur in oppoſitione rei ui-<lb/>ſæ, & medium loci, qui eſt locus ſenſus uiſus, opponitur cuilibet parti partium rei uiſæ.</s> <s xml:id="echoid-s2947" xml:space="preserve"> Ergo cum <lb/>totus uiſus mouebitur in oppoſitione rei uiſæ:</s> <s xml:id="echoid-s2948" xml:space="preserve"> axis tranſibit per quamlibet partium rei uiſæ:</s> <s xml:id="echoid-s2949" xml:space="preserve"> & <lb/>tunc forma cuiuslibet partium rei uiſæ extendetur ad uiſum apud peruentum axis ad ipſam ſuper <lb/>rectitudinem axis:</s> <s xml:id="echoid-s2950" xml:space="preserve"> & erit axis fixus in ſuo loco, & non mutabitur à ſuo loco reſpectu omnium par-<lb/>tium totius oculi:</s> <s xml:id="echoid-s2951" xml:space="preserve"> & erit gyratio eius in ſua diſpoſitione apud motum totius uiſus in loco nerui, <lb/>qui eſt apud concauum oſsis tantùm.</s> <s xml:id="echoid-s2952" xml:space="preserve"> Et cum uiſus uoluerit intueri rem uiſam, & incœperit intue-<lb/>ri in extremitatem rei uiſæ:</s> <s xml:id="echoid-s2953" xml:space="preserve"> erit tunc extremum axis ſuper partem extremam rei uiſæ.</s> <s xml:id="echoid-s2954" xml:space="preserve"> Erit ergo in <lb/>iſta diſpoſitione maior pars totius rei uiſæ in parte ſuperficiei uiſus declinante, aut obliqua ab axe <lb/>ad aliquam partem, præter partem, ſuper quam eſt axis:</s> <s xml:id="echoid-s2955" xml:space="preserve"> quoniam forma eius erit in medio eius <lb/>& in loco axis in uiſu, & erit reſiduum formæ obliquum aut declinans ad aliam partem ab axe.</s> <s xml:id="echoid-s2956" xml:space="preserve"> De-<lb/>inde quando uiſus mouebitur poſt illam diſpoſitionem ſuper aliam diametrum rei uiſæ:</s> <s xml:id="echoid-s2957" xml:space="preserve"> transfe-<lb/>retur axis ad partem ſequentem illam partem, & forma primæ partis declinabit ſuper alteram u-<lb/>bitatem oppoſitam ubitati, ad quam mouetur axis:</s> <s xml:id="echoid-s2958" xml:space="preserve"> iam deinde non ceſſabit forma declinare, dum <lb/>axis mouetur ſuper illam diametrum, quouſque axis perueniat ad ultimum illius diametri rei ui-<lb/>fæ, & ad partem extremam rei uiſæ oppoſitam primæ parti.</s> <s xml:id="echoid-s2959" xml:space="preserve"> Erit ergo forma totius rei uiſæ in iſta <lb/>diſpoſitione obliqua ad ubitatem oppoſitam ubitati, ad quam prius fuit obliqua, præterquam ul-<lb/>tima pars, quæ erat ſuper axem, & in medio uiſus.</s> <s xml:id="echoid-s2960" xml:space="preserve"> Et axis in toto iſto motu erit fixus in ſuo ſitu, & <lb/>erit iſte motus ualde uelox, & in maiori parte eſt inſenſibilis propter uelocitatem.</s> <s xml:id="echoid-s2961" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div98" type="section" level="0" n="0"> <head xml:id="echoid-head121" xml:space="preserve" style="it">43. Axis optic{us} in ſuo motu nunquã fit baſis anguli à ſuperficie uiſibilis ſubtenſi: nec ſem-<lb/>per ſet at angulum ab aliqua uiſibilis diametro ſubtenſum. 54 p 3.</head> <p> <s xml:id="echoid-s2962" xml:space="preserve">AXis autem nõ ſupponitur in ſuo motu terminus anguli, quem reſpicit illa res uiſa apud cen-<lb/>trum uiſus, neq;</s> <s xml:id="echoid-s2963" xml:space="preserve"> ſecat latitudinem anguli, quem reſpicit aliqua diametrorum rei uiſę:</s> <s xml:id="echoid-s2964" xml:space="preserve"> quo-<lb/>niam hoc non erit, niſi quando axis fuerit motus per ſe, & totus oculus quieuerit, quod <lb/>eſt impoſsibile;</s> <s xml:id="echoid-s2965" xml:space="preserve"> totus enim oculus mouetur apud intuitionem, & axis <lb/>mouetur per motum eius.</s> <s xml:id="echoid-s2966" xml:space="preserve"/> </p> <pb o="58" file="0064" n="64" rhead="ALHAZEN"/> </div> <div xml:id="echoid-div99" type="section" level="0" n="0"> <head xml:id="echoid-head122" xml:space="preserve" style="it">44. Viſ{us} percipit magnitudinem anguli optici è parte ſuperficiei uiſ{us}, in qua formatur <lb/>rei uiſibilis forma. 73 p 3.</head> <p> <s xml:id="echoid-s2967" xml:space="preserve">SEntiens autem non comprehendit quantitatem anguli, quem reſpicit res uiſa apud centrum <lb/>uiſus, niſi ex comprehenſione quantitatis partis ſuperficiei uiſus, in qua figuratur forma rei <lb/>uiſæ, & ex imaginatione anguli, quem reſpicit illa pars apud centrum uiſus.</s> <s xml:id="echoid-s2968" xml:space="preserve"> Nam ſenſus uiſus <lb/>comprehendit naturaliter quantitates partium uiſus, in quibus figurantur formæ, & naturaliter i-<lb/>maginatur angulos, quos reſpiciunt iſtæ partes.</s> <s xml:id="echoid-s2969" xml:space="preserve"> Sentiens autem non certificat formam rei uiſæ, & <lb/>quantitatem magnitudinis rei uiſæ per motum uiſus, niſi quia per iſtum motum comprehendit <lb/>quamlibet partium rei uiſæ per eius medium & per locum axis in uiſu:</s> <s xml:id="echoid-s2970" xml:space="preserve"> & per iſtum motum moue-<lb/>tur forma rei uiſæ ſuper ſuperficiem uiſus, & ſic mutabitur pars ſuperficiei uiſus, in qua fuit forma:</s> <s xml:id="echoid-s2971" xml:space="preserve"> <lb/>quoniam forma rei uiſæ apud motum, erit in parte poſt aliam partem in ſuperficie uiſus.</s> <s xml:id="echoid-s2972" xml:space="preserve"> Et quo-<lb/>ties comprehenderit ſentiens partem rei uiſæ, quæ eſt apud extremum axis:</s> <s xml:id="echoid-s2973" xml:space="preserve"> comprehendet ſimul <lb/>totam rem uiſam, & comprehendet totam partem ſuperficiei uiſus, in quam peruenit forma toti-<lb/>us rei uiſæ, & comprehendet quantitatem illius partis, & comprehendet quantitatem anguli, <lb/>quem reſpicit illa pars, apud centrum uiſus.</s> <s xml:id="echoid-s2974" xml:space="preserve"> Et ſic multoties comprehendet ſentiens quantita-<lb/>tem anguli, quem reſpicit illa res uiſa.</s> <s xml:id="echoid-s2975" xml:space="preserve"> Quare erit ab eo certificata:</s> <s xml:id="echoid-s2976" xml:space="preserve"> quare etiam uirtus diſtinctiua <lb/>intelliget quantitatem anguli, & quantitatem remotionis, ex quibus comprehendet quantitatem <lb/>magnitudinis rei uiſæ ſecundum ueritatem.</s> <s xml:id="echoid-s2977" xml:space="preserve"> Secundum ergo hunc modum erit intuitio uiſibilium <lb/>à uiſu, & certificatio quantitatis magnitudinum rerum uiſarum per intuitionem.</s> <s xml:id="echoid-s2978" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div100" type="section" level="0" n="0"> <head xml:id="echoid-head123" xml:space="preserve" style="it">45. Sit{us} direct{us} & obliqu{us} lineæ, ſuperficiei, & ſpatij percipitur ex æquabili & inæqua-<lb/>bili terminorum diſtantia. 12 p 4. Idem 28 n.</head> <p> <s xml:id="echoid-s2979" xml:space="preserve">ET etiam quando uiſus comprehendet quantitates longitudinum linearum radialium, quæ <lb/>ſunt inter uiſum & terminos rei uiſæ, aut partes ſuperficiei rei uiſæ, ſentiet æqualitatem & <lb/>inæqualitatem earum quantitatum.</s> <s xml:id="echoid-s2980" xml:space="preserve"> Si ſuperficies rei uiſæ, quam uiſus comprehendit, fuerit <lb/>obliqua:</s> <s xml:id="echoid-s2981" xml:space="preserve"> ſentiet obliquationem eius ex ſenſu inæqualitatis quantitatum remotionum extremo-<lb/>rum eius.</s> <s xml:id="echoid-s2982" xml:space="preserve"> Et ſi ſuperficies fuerit directè oppoſita, ſentiet directionem ex ſenſu æqualitatis remo-<lb/>tionum:</s> <s xml:id="echoid-s2983" xml:space="preserve"> & ſic non latebit quantitas magnitudinis eius uirtutem diſtinctiuam:</s> <s xml:id="echoid-s2984" xml:space="preserve"> quoniam uirtus di-<lb/>ſtinctiua comprehendit ex inæqualitate remotionum diametrorum extremorum ſpatij obliqui, <lb/>obliquationẽ pyramidis continentis ipſum.</s> <s xml:id="echoid-s2985" xml:space="preserve"> Quare ſentiet exceſſum magnitudinis eius baſis pro-<lb/>pter obliquationem.</s> <s xml:id="echoid-s2986" xml:space="preserve"> Et non admiſcetur ſecundum aſsimilationem quantitas magnitudinis obli-<lb/>quæ magnitudini directè oppoſitæ, niſi quando comparatio fuerit ad angulum tantùm:</s> <s xml:id="echoid-s2987" xml:space="preserve"> ſi autem <lb/>comparatio fuerit ad angulum & ad longitudines linearum radialium interiacentium inter uiſum <lb/>& extrema rei uiſæ:</s> <s xml:id="echoid-s2988" xml:space="preserve"> non dubitabit uirtus diſtinctiua in quantitate magnitudinis.</s> <s xml:id="echoid-s2989" xml:space="preserve"> Quantitates er-<lb/>go magnitudinum, linearum & ſpatiorum comprehenduntur à uiſu ex comprehenſione quanti-<lb/>tatum remotionum extremorum in illis, & ex comprehenſione inęqualitatis & ęqualitatis eorum.</s> <s xml:id="echoid-s2990" xml:space="preserve"> <lb/>Sed remotio remotiſsima remotionum mediocrium, reſpectu rei uiſæ, quando res uiſa fuerit obli <lb/>qua, eſt minor remotiſsima remotionum mediocriumr, eſpectu illius eiuſdem rei uiſæ, quando res <lb/>uiſa fuerit directè oppoſita:</s> <s xml:id="echoid-s2991" xml:space="preserve"> quoniam remotio mediocris reſpectu rei uiſæ eſt, in qua non latet ui-<lb/>ſum pars rei uiſæ habens proportionem ſenſibilem ad totam rem uiſam.</s> <s xml:id="echoid-s2992" xml:space="preserve"> Et cum res uiſa fuerit ob-<lb/>liqua, angulus, quem continent duo radij exeuntes à uiſu ad aliquam partem rei uiſæ obliquæ, e-<lb/>rit minor angulo, quem continent duo radij exeuntes à uiſu ad il-<lb/> <anchor type="figure" xlink:label="fig-0064-01a" xlink:href="fig-0064-01"/> lam eandem partem & ad illam eandem remotionem, quando res <lb/>uiſa fuerit directè oppoſita uiſui.</s> <s xml:id="echoid-s2993" xml:space="preserve"> Et pars habens ſenſibilem pro-<lb/>portionem ad totam rem uiſam, quando res uiſa fuerit obliqua:</s> <s xml:id="echoid-s2994" xml:space="preserve"> la-<lb/>tet in remotione minori quàm eſt remotio, in qua latet eadem illa <lb/>pars, quando illa res uiſa fuerit directè oppoſita.</s> <s xml:id="echoid-s2995" xml:space="preserve"> Remotiſsima er-<lb/>go remotionum mediocrium reſpectu rei uiſæ obliquæ, eſt minor <lb/>remotiſsima remotionum mediocrium reſpectu illius eiuſdem rei <lb/>uiſæ, quando illa res uiſa fuerit directè oppoſita:</s> <s xml:id="echoid-s2996" xml:space="preserve"> & tota res uiſa ob-<lb/>liqua latet in remotione minori quàm eſt remotio, in qua latet illa <lb/>res uiſa, quando fuerit directè oppoſita:</s> <s xml:id="echoid-s2997" xml:space="preserve"> & diminuitur quantitas <lb/>eius in remotione minore remotione, in qua diminuitur quanti-<lb/>tas eius, quando fuerit directè oppoſita.</s> <s xml:id="echoid-s2998" xml:space="preserve"> Magnitudines ergo re-<lb/>rum uiſarum, quarum quantitates certificantur à uiſu, ſunt illæ, <lb/>quarum remotio eſt mediocris, & quarum remotio reſpicit corpo-<lb/>ra ordinata continuata:</s> <s xml:id="echoid-s2999" xml:space="preserve"> & comprehenduntur à uiſu ex comparati <lb/>one illarum ad angulos pyramidum radialium continentium ipſas, <lb/>& ad longitudines linearum radialium.</s> <s xml:id="echoid-s3000" xml:space="preserve"> Remotiones autem me-<lb/>diocres reſpectu rei uiſæ ſunt ſecundum ſitum illius rei uiſæ in ob-<lb/>liquatione, aut in directa oppoſitione.</s> <s xml:id="echoid-s3001" xml:space="preserve"> Et anguli nõ certificãtur, niſi <lb/>per motũ uiſus reſpicientis ſuper diametros ſuperficiei rei uiſæ, aut <lb/> <pb o="59" file="0065" n="65" rhead="OPTICAE LIBER II."/> ſuper ſpatiũ, cuius magnitudinẽ uoluerit ſcire.</s> <s xml:id="echoid-s3002" xml:space="preserve"> Et certificatur remotio per motum uiſus ſuper cor-<lb/>pus reſpiciẽs remotiones extremorũ illius ſuperficiei, aut illius ſpatij.</s> <s xml:id="echoid-s3003" xml:space="preserve"> Et generaliter formareiuiſę, <lb/>& forma remotionis rei uiſæ, cuius remotio eſt mediocris, & refpicit corpora ordinata continuata, <lb/>perueniunt cõmuniter in imaginationem ſimul apud intuitionem rei uiſæ:</s> <s xml:id="echoid-s3004" xml:space="preserve"> quoniam uiſus cõpre-<lb/>hendit corpus reſpiciens remotionem rei uiſæ apud comprehenſionem rei uiſæ:</s> <s xml:id="echoid-s3005" xml:space="preserve"> & ſic uirtus diſtin <lb/>ctiua comprehendet magnitudinem rei uiſæ ſecundum quantitatem formæ remotionis eius cer-<lb/>tificatæ, & coniunctę cum forma eius.</s> <s xml:id="echoid-s3006" xml:space="preserve"> Quantitates ergo huiuſmodi uiſibilium tantùm comprehen <lb/>duntur à uiſu uera comprehenſione.</s> <s xml:id="echoid-s3007" xml:space="preserve"> Secundum ergo hunc modum, quem declarauimus, compre-<lb/>henduntur magnitudines rerum uiſarum per ſenſum uiſus.</s> <s xml:id="echoid-s3008" xml:space="preserve"> Quare uerò res uiſa comprehendatur <lb/>in maxima remotione minoris quantitatis ſua uera quantitate:</s> <s xml:id="echoid-s3009" xml:space="preserve"> & quare comprehendatur quan-<lb/>titas rei uiſæ in propinquiſsima remotione maior quantitate ſua uera, declarabimus in noſtro ſer-<lb/>mone de erroribus uiſus.</s> <s xml:id="echoid-s3010" xml:space="preserve"/> </p> <div xml:id="echoid-div100" type="float" level="0" n="0"> <figure xlink:label="fig-0064-01" xlink:href="fig-0064-01a"> <variables xml:id="echoid-variables5" xml:space="preserve">d a a b c <gap/></variables> </figure> </div> </div> <div xml:id="echoid-div102" type="section" level="0" n="0"> <head xml:id="echoid-head124" xml:space="preserve" style="it">46. Diſtinctio uiſibilium percipitur è diſtinctione formarum, quæ in diuerſis ſuperficiei ui-<lb/>ſ{us} partib{us} ſunt impreſſæ. 99 p 4.</head> <p> <s xml:id="echoid-s3011" xml:space="preserve">DIſtinctio uerò, quæ eſt inter uiſibilia, comprehẽditur à uiſu ex diſtinctione formarum duo-<lb/>rum corporum ſiue duorum uiſibilium diſtinctorum peruenientium in uiſum.</s> <s xml:id="echoid-s3012" xml:space="preserve"> Sed in diſtin <lb/>ctione, quæ eſt inter quælibet duo corpora diſtincta, aut eſt lux:</s> <s xml:id="echoid-s3013" xml:space="preserve"> aut eſt corpus coloratum il <lb/>luminatum:</s> <s xml:id="echoid-s3014" xml:space="preserve"> aut eſt obſcuritas.</s> <s xml:id="echoid-s3015" xml:space="preserve"> Cum ergo uiſus comprehẽderit duo corpora diſtincta:</s> <s xml:id="echoid-s3016" xml:space="preserve"> forma lucis, <lb/>aut forma coloris corporis, aut forma obſcuritatis, quæ eſt in loco diſtinctionis, peruenit in partẽ <lb/>uiſus interiacentem inter duas formas duorum corporum diſtinctorum peruenientium in uiſum.</s> <s xml:id="echoid-s3017" xml:space="preserve"> <lb/>Lux uerò, aut color, aut obſcuritas aliquando erit in corpore medio interiacente inter duo corpo-<lb/>ra continuata cum utroque corporum.</s> <s xml:id="echoid-s3018" xml:space="preserve"> Si ergo uiſus non ſenſerit, quòd lux, color, aut obſcuritas, <lb/>quæ eſt in loco diſtinctionis, non eſt in corpore continuato cum utroq;</s> <s xml:id="echoid-s3019" xml:space="preserve"> corporum, quę ſunt in eius <lb/>lateribus, non ſentiet diſtinctionem duorum corporum.</s> <s xml:id="echoid-s3020" xml:space="preserve"> Et etiam ſuperficies cuiuslibet illorũ duo <lb/>rum corporum eſt obliqua ad locum remotionis.</s> <s xml:id="echoid-s3021" xml:space="preserve"> In loco ergo diſtinctionis fortè erit obliquatio <lb/>duarum ſuperficierum duorum corporum, aut ſuperficiei alterius duorum corporum manifeſta ui <lb/>ſui, & fortè non.</s> <s xml:id="echoid-s3022" xml:space="preserve"> Cum ergo obliquatio duarum ſuperficierum duorum corporum, aut ſuperficiei <lb/>alterius duorum corporum fuerit manifeſta uiſui:</s> <s xml:id="echoid-s3023" xml:space="preserve"> tunc ſentiet uiſus diſtinctionem duorum corpo <lb/>rum.</s> <s xml:id="echoid-s3024" xml:space="preserve"> Viſus ergo comprehendit diſtinctionem corporum ex comprehenſione intentionum, quas <lb/>diximus, aut ex comprehenſione lucis in loco diſtinctionis, ſentiendo, quòd illa lux eſt ex poſte-<lb/>riori duarum ſuperficierum duorum corporum diſtinctorum:</s> <s xml:id="echoid-s3025" xml:space="preserve"> aut ex comprehenſione corporis co <lb/>lorati in loco diſtinctionis, ſentiendo, quòd illud corpus eſt diuerſum ab utroque corporum diſtin <lb/>ctorum:</s> <s xml:id="echoid-s3026" xml:space="preserve"> aut ex comprehenſione obſcurationis loci diſtinctionis, comprehendendo, quòd iſtud eſt <lb/>obſcuritas, & non eſt corpus continuatum cum duobus corporibus:</s> <s xml:id="echoid-s3027" xml:space="preserve"> aut ex comprehenſione obli-<lb/>quationis utriuſque ſuperficiei duorum corporum in loco diſtinctionis, aut obliquationis ſuper-<lb/>ficiei alterius duorum corporum.</s> <s xml:id="echoid-s3028" xml:space="preserve"> Omne ergo, quod uiſus comprehendit ex diſtinctione corpo-<lb/>rum:</s> <s xml:id="echoid-s3029" xml:space="preserve"> non comprehendit, niſi ſecundum aliquam iſtarum intentionum.</s> <s xml:id="echoid-s3030" xml:space="preserve"> Diſtinctio autem fortè erit <lb/>inter duo corpora diſtincta:</s> <s xml:id="echoid-s3031" xml:space="preserve"> & fortè inter duo corpora non diuerſa, ſcilicet quòd duo corpora <lb/>ſunt continuata ſecundum quaſdam partes, & diuerſa ſecũdum quaſdam inter ſe, ut digiti, & mem-<lb/>bra animalis, & rami arborum:</s> <s xml:id="echoid-s3032" xml:space="preserve"> & ſecundum utramlibet diſpoſitionum uiſus non comprehendit di <lb/>ſtinctionem, niſi ſecundum modos, quos declarauimus.</s> <s xml:id="echoid-s3033" xml:space="preserve"> Et fortè comprehenditur diſtinctio corpo <lb/>rum per cognitionem & per ſcientiam antecedentem:</s> <s xml:id="echoid-s3034" xml:space="preserve"> ſed illa comprehenſio non eſt per ſenſum ui-<lb/>ſus.</s> <s xml:id="echoid-s3035" xml:space="preserve"> Et quædam diſtinctio corporum eſt ampla, & quædam ſtricta.</s> <s xml:id="echoid-s3036" xml:space="preserve"> Diſtinctio uerò ampla non late<gap/> <lb/>uiſum in maiori parte, propter apparentiam corporis reſpicientis diſtantiam diſtinctam, & pro-<lb/>pter hoc, quòd illud corpus apparet diuerſum ab utro que corporum diſtinctorum, & propter com <lb/>prehenſionem lucis & uacuitatis illuminati reſpicientis diſtantiam.</s> <s xml:id="echoid-s3037" xml:space="preserve"> Diſtinctio autem modica & <lb/>ſtricta non comprehenditur à uiſu, niſi in remotione, in qua non latet uiſum corpus, cuius quanti-<lb/>tas eſt æqualis quantitati amplitudinis diſtantię.</s> <s xml:id="echoid-s3038" xml:space="preserve"> Si autem diſtantia inter duo corpora fuerit ſtricta <lb/>& occulta:</s> <s xml:id="echoid-s3039" xml:space="preserve"> & fuerit remotio illius à uiſu ſimilis illi, in qua lateant corpora, quorum quantitas eſt, <lb/>ſicut quantitas amplitudinis diſtantiæ:</s> <s xml:id="echoid-s3040" xml:space="preserve"> nõ comprehendet uiſus illam diſtãtiam.</s> <s xml:id="echoid-s3041" xml:space="preserve"> Et ſi remotio duo-<lb/>rum corporum à uiſu ſit ex remotionibus mediocribus, & uiſus comprehenderit duo corpora ue-<lb/>ra comprehenſione:</s> <s xml:id="echoid-s3042" xml:space="preserve"> (mediocris autem remotio eſt illa, in qua non latet omnino quantitas ſenſibi-<lb/>lis reſpectu quantitatis totius remotionis:</s> <s xml:id="echoid-s3043" xml:space="preserve"> & uera comprehẽſio eſt illa, inter quam & ueritatem rei <lb/>uiſæ non eſt diuerſitas ſenſibilis omnino reſpectu totius rei uiſæ) amplitudo autem diſtantiæ for-<lb/>tè ſit quantitatis carentis proportione ſenſibili ad remotionem rei uiſæ, & carentis quantitate ſen-<lb/>ſibili reſpectu duorum corporum diſtinctorum:</s> <s xml:id="echoid-s3044" xml:space="preserve"> (quoniam diſtinctio fortè erit in quantitate unius <lb/>capilli:</s> <s xml:id="echoid-s3045" xml:space="preserve">) tum illud diminutum non aufert diſtantiam ſenſibilem in uiſu.</s> <s xml:id="echoid-s3046" xml:space="preserve"> Diſtãtia igitur inter uiſibi <lb/>lia comprehenditur à uiſu fecundum modos, quos declarauimus.</s> <s xml:id="echoid-s3047" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div103" type="section" level="0" n="0"> <head xml:id="echoid-head125" xml:space="preserve" style="it">47. Continuatio uiſibilis percipitur è diſtantiæ priuatione. 100 p 4.</head> <p> <s xml:id="echoid-s3048" xml:space="preserve">COntinuatio aũt cõprehẽditur à uiſu ex priuatione diſtantiæ.</s> <s xml:id="echoid-s3049" xml:space="preserve"> Cũ ergo uiſus nõ ſenſerit in ali-<lb/>quo corpore diſtãtiã:</s> <s xml:id="echoid-s3050" xml:space="preserve"> cõprehẽdet ipſum eſſe continuũ.</s> <s xml:id="echoid-s3051" xml:space="preserve"> Et ſi in corpore fuerit diſtãtia occul-<lb/>ta, nõ cõprehẽſa à uiſu:</s> <s xml:id="echoid-s3052" xml:space="preserve"> cõprehẽdet uiſus illud corpus eſſe cõtinuũ, quãuis in eo ſit diſcretio.</s> <s xml:id="echoid-s3053" xml:space="preserve"> <lb/> <pb o="60" file="0066" n="66" rhead="ALHAZEN"/> Et uiſus cõprehendit continuationem, & diſcernit inter cõtinuationẽ & contiguationẽ ex cõpre-<lb/>henſione aggregationis duorum terminorum duorum corporum.</s> <s xml:id="echoid-s3054" xml:space="preserve"> Et uiſus non iudicat contigua-<lb/>tionem, niſi poſtquam ſciuerit, quòd utrumque duorum corporum contiguorum eſt diuerſum ab <lb/>altero:</s> <s xml:id="echoid-s3055" xml:space="preserve"> quoniam differentia, quæ eſt inter duo cõtigua, fortè inuenitur in duobus corporibus con-<lb/>tinuis.</s> <s xml:id="echoid-s3056" xml:space="preserve"> Si ergo ſentiens non ſenſerit, quòd utrumq;</s> <s xml:id="echoid-s3057" xml:space="preserve"> duorum corporum contiguorum eſt diuerſum <lb/>ab altero, & diſtinctum ab eo:</s> <s xml:id="echoid-s3058" xml:space="preserve">non ſentiet contiguationem, & iudicabit continuationem.</s> <s xml:id="echoid-s3059" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div104" type="section" level="0" n="0"> <head xml:id="echoid-head126" xml:space="preserve" style="it">48. Numerus percipitur è uiſibilium diſtinctione. 101 p 4.</head> <p> <s xml:id="echoid-s3060" xml:space="preserve">NVmerus uerò comprehenditur à uiſu, & numeri medietas.</s> <s xml:id="echoid-s3061" xml:space="preserve"> Quoniam uiſus comprehendit <lb/>in una hora multa uiſibilia ſimul:</s> <s xml:id="echoid-s3062" xml:space="preserve"> & cum uiſus comprehenderit diſtinctionem illorũ, com-<lb/>prehẽdet quodlibet illorũ eſſe diuerſum ab alio:</s> <s xml:id="echoid-s3063" xml:space="preserve"> & ſic comprehendit multitudinem.</s> <s xml:id="echoid-s3064" xml:space="preserve"> Et uir-<lb/>tus diſtinctiua comprehendit numerum ex multitudine.</s> <s xml:id="echoid-s3065" xml:space="preserve"> Numerus ergo comprehenditur per ſen-<lb/>ſum uiſus ex cõprehenſione multorũ uiſibilium diſtinctorũ, quando uiſus cõprehendit ipſa ſimul:</s> <s xml:id="echoid-s3066" xml:space="preserve"> <lb/>& comprehenderit diſtinctionem illorum:</s> <s xml:id="echoid-s3067" xml:space="preserve"> & comprehenderit quòd quodlibet illorum eſt diuer-<lb/>ſum ab alio.</s> <s xml:id="echoid-s3068" xml:space="preserve"> Secundum ergo iſtum modum comprehenditur numerus per ſenſum uiſus.</s> <s xml:id="echoid-s3069" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div105" type="section" level="0" n="0"> <head xml:id="echoid-head127" xml:space="preserve" style="it">49. Motus uiſibilis percipitur è mutatione ſitus eius in ſenſilitempore. 110 p 4.</head> <p> <s xml:id="echoid-s3070" xml:space="preserve">MOtus autem comprehenditur à uiſu ex comparatione rei motæ ad aliud uiſibile.</s> <s xml:id="echoid-s3071" xml:space="preserve"> Quoniam <lb/>quando uiſus comprehenderit uiſibile motum, & cum ipſo comprehenderit aliud uiſibi-<lb/>le, comprehendet ſitum eius reſpectu illius uiſibilis moti.</s> <s xml:id="echoid-s3072" xml:space="preserve"> Et cum illud uiſibile fuerit motũ, <lb/>& illud aliud uiſibile fuerit non motum:</s> <s xml:id="echoid-s3073" xml:space="preserve"> per motum illius uiſibilis moti, ſitus illius uiſibilis moti di <lb/>uerſabitur reſpectu illius uiſibilis non moti.</s> <s xml:id="echoid-s3074" xml:space="preserve"> Et cum uiſus comprehenderit ipſum, & cum eo com-<lb/>prehenderit aliud uiſibile:</s> <s xml:id="echoid-s3075" xml:space="preserve"> comprehendet ſitum eius reſpectu illius uiſibilis, & comprehendet mo <lb/>tum eius.</s> <s xml:id="echoid-s3076" xml:space="preserve"> Motus ergo comprehenditur à uiſu ex comprehenſione diuerſitatis ſitus rei uiſæ motæ <lb/>reſpectu alterius.</s> <s xml:id="echoid-s3077" xml:space="preserve"> Et motus cõprehenditur à uiſu ſecũdum aliquem trium modorũ:</s> <s xml:id="echoid-s3078" xml:space="preserve"> aut ex reſpectu <lb/>rei uiſæ motæ ad multa uiſibilia:</s> <s xml:id="echoid-s3079" xml:space="preserve"> aut ex reſpectu rei uiſæ motæ ad unum uiſibile:</s> <s xml:id="echoid-s3080" xml:space="preserve"> aut ex reſpectu rei <lb/>uiſæ motæ ad ipſum uiſum.</s> <s xml:id="echoid-s3081" xml:space="preserve"> Primum autẽ quando uiſus comprehenderit rẽ uiſam & eius motũ, & <lb/>comprehenderit ipſam reſpicientẽ aliquod uiſibile:</s> <s xml:id="echoid-s3082" xml:space="preserve"> deinde comprehenderitipſam reſpicientem a-<lb/>liquod aliud uiſibile diuerſum à primo, exiſtente uiſu in ſuo loco:</s> <s xml:id="echoid-s3083" xml:space="preserve"> ſentiet motũ illius rei uiſæ.</s> <s xml:id="echoid-s3084" xml:space="preserve"> Reſpe <lb/>ctus autem rei uiſæ motæ ad unum ſolum uiſibile eſt, quando uiſus comprehẽderit rẽ uiſam motã, <lb/>& comprehenderit ſitum eius reſpectu alterius uiſibilis:</s> <s xml:id="echoid-s3085" xml:space="preserve"> deinde cõprehenderit ſitũ eius, qui muta-<lb/>tus eſt reſpectu illius alterius uiſibilis:</s> <s xml:id="echoid-s3086" xml:space="preserve"> aut quòd eſt remotius:</s> <s xml:id="echoid-s3087" xml:space="preserve"> aut quòd propinquius:</s> <s xml:id="echoid-s3088" xml:space="preserve"> aut quòd eſt <lb/>in parte altera, uiſu exiſtente in ſuo loco:</s> <s xml:id="echoid-s3089" xml:space="preserve"> aut per mutationem ſitus alicuius partis rei uiſæ motæ, <lb/>reſpectu illius uiſibilis immoti:</s> <s xml:id="echoid-s3090" xml:space="preserve"> aut per mutationem ſitus partiũ eius reſpectu uiſibilis illius:</s> <s xml:id="echoid-s3091" xml:space="preserve"> & ſe-<lb/>cundum iſtum ultimũ modum comprehendit uiſus motum uiſibilis moti circulariter, quando ho-<lb/>mo comparauerit ipſum ad aliud uiſibile.</s> <s xml:id="echoid-s3092" xml:space="preserve"> Cum ergo uiſus comprehenderit ſitũ rei uiſæ motæ, aut <lb/>ſitum partium eius, aut ſitũ alicuius partis eius:</s> <s xml:id="echoid-s3093" xml:space="preserve"> cõprehendet motũ rei uiſę motæ.</s> <s xml:id="echoid-s3094" xml:space="preserve"> Reſpectus autem <lb/>rei uiſæ motæ ad ipſum uiſum eſt, quando uiſus comprehendit rem uiſam motã, cõprehendetubr-<lb/>tatem eius & remotionẽ eius à uiſu:</s> <s xml:id="echoid-s3095" xml:space="preserve"> & cum uiſus fuerit quietus, & res uiſa fuerit mota:</s> <s xml:id="echoid-s3096" xml:space="preserve"> tunc muta-<lb/>bitur ſitus rei uiſæ motæ reſpectu uiſus.</s> <s xml:id="echoid-s3097" xml:space="preserve"> Si ergo motus rei uiſæ fuerit ſecundum ſpatium latũ:</s> <s xml:id="echoid-s3098" xml:space="preserve"> mu-<lb/>tabitur ubitas eius, & ſentiet uiſus mutationem ubitatis.</s> <s xml:id="echoid-s3099" xml:space="preserve"> Et cum uiſus ſenſerit mutationem ubita-<lb/>tis eius, uiſu quieſcente, ſentiet motum eius.</s> <s xml:id="echoid-s3100" xml:space="preserve"> Et ſi motus rei uiſæ fuerit in longitudine extenſa inter <lb/>ipſum & uiſum:</s> <s xml:id="echoid-s3101" xml:space="preserve"> tuncres uiſa aut elongabitur à uiſu per motũ, aut appropinquabit.</s> <s xml:id="echoid-s3102" xml:space="preserve"> Et cũ uiſus ſen-<lb/>ſerit elongationem aut appropinquationem eius, uiſu exiſtẽte in ſuo loco:</s> <s xml:id="echoid-s3103" xml:space="preserve"> uiſus ſentiet motum e-<lb/>ius.</s> <s xml:id="echoid-s3104" xml:space="preserve"> Et ſi motus rei uiſæ fuerit circularis, neceſſariò mutabitur pars rei uiſæ eius, quę opponitur ui-<lb/>ſui:</s> <s xml:id="echoid-s3105" xml:space="preserve"> & cum illa pars rei uiſæ fuerit mutata, & ſenſerit uiſus mutation em eius, uiſu exiſtente in ſuo lo <lb/>co:</s> <s xml:id="echoid-s3106" xml:space="preserve"> ſentiet motum rei uiſæ.</s> <s xml:id="echoid-s3107" xml:space="preserve"> Secundum ergo iſtos modos comprehendit uiſus motum, quando ui-<lb/>ſus fuerit fixus in ſuo loco.</s> <s xml:id="echoid-s3108" xml:space="preserve"> Et uiſus comprehendet etiam motum ſecundum quemlibet iſtorũ mo-<lb/>dorum, quamuis uiſus etiam moueatur.</s> <s xml:id="echoid-s3109" xml:space="preserve"> Et hoc erit quando uiſus ſenſerit diuerſitatem ſitus rei ui-<lb/>ſæ motæ, ſentiendo, quòd illa diuerſitas non eſt propter motum eius, & diſtinguendo inter diuerſi <lb/>tatem ſitus, quæ accidit illi rei propter motum illius rei uiſæ, & inter diuerſitatem ſitus, quę accidit <lb/>ei propter motum uiſus.</s> <s xml:id="echoid-s3110" xml:space="preserve"> Cum ergo uiſus ſenſerit diuerſitatem ſitus rei uiſæ, & ſenſerit, quòd diuer-<lb/>ſitas eius ſitus non eſt propter motum uiſus:</s> <s xml:id="echoid-s3111" xml:space="preserve"> ſentiet motum rei uiſæ.</s> <s xml:id="echoid-s3112" xml:space="preserve"> Et forma rei uiſæ motæ moue <lb/>tur etiam in uiſu propter motum eius:</s> <s xml:id="echoid-s3113" xml:space="preserve"> ſed uiſus non cõprehendit motum rei uiſæ ex motu ſuæ for-<lb/>mæ in uiſu tantùm:</s> <s xml:id="echoid-s3114" xml:space="preserve"> imo uiſus non comprehendit motum rei uiſæ, niſi ex comparatione reiuiſæ ad <lb/>aliam ſecundum modos, quos declarauimus:</s> <s xml:id="echoid-s3115" xml:space="preserve"> quoniã forma rei uiſæ quieſcentis aliquando moue-<lb/>turin uiſu cum quiete rei uiſæ, & inde uiſus non comprehenditipſam motam.</s> <s xml:id="echoid-s3116" xml:space="preserve"> Quoniã uiſus quan-<lb/>do mouebitur ſuper oppoſitionem rerum uiſarum:</s> <s xml:id="echoid-s3117" xml:space="preserve"> mouebitur forma cuiuslιbet rei uiſæ oppoſitæ <lb/>uiſui in ſuperficie uiſus apud motũ eius, ſiue quieſcat, ſiue moueatur.</s> <s xml:id="echoid-s3118" xml:space="preserve"> Et quia uiſus iam aſſuefactus <lb/>eſt ad motum formarum rerum uiſarum in ſuperficie eius cum quiete illarum rerum uiſarum:</s> <s xml:id="echoid-s3119" xml:space="preserve"> non <lb/>iudicabit motum rei uiſæ propter motum formæ eius, niſi quando in uiſum peruenerit forma ali-<lb/>cuius rei uiſæ, & comprehenderit uiſus diuerſitatẽ ſitus formæ rei uiſæ motæ, reſpectu alterius for-<lb/>mæ rei uiſæ:</s> <s xml:id="echoid-s3120" xml:space="preserve"> aut ex mutatione formarum in eodem loco uiſus, qui erιt in loco circulari.</s> <s xml:id="echoid-s3121" xml:space="preserve"> Motus er-<lb/>go non comprehenditur à uiſu, niſi ſecundum modos, quos diſtinximus.</s> <s xml:id="echoid-s3122" xml:space="preserve"/> </p> <pb o="61" file="0067" n="67" rhead="OPTICAE LIBER II."/> </div> <div xml:id="echoid-div106" type="section" level="0" n="0"> <head xml:id="echoid-head128" xml:space="preserve" style="it">50. Qualitas motus percipitur è ſpatio, per quoduiſibile mouetur. 711 p 4.</head> <p> <s xml:id="echoid-s3123" xml:space="preserve">COmprehenſio aũt qualitatis motus eſt ex cõprehenſione ſpatij, ſuper quod mouetur res ui-<lb/>ſa, quando res uiſa mouebitur ſecundum ſe totam.</s> <s xml:id="echoid-s3124" xml:space="preserve"> Et uiſus certificat qualitatẽ motus, quan-<lb/>do certificauerit figuram ſpatij, ſuper quod mouetur res uiſa mota.</s> <s xml:id="echoid-s3125" xml:space="preserve"> Et cum res uiſa mouebi-<lb/>tur circulariter:</s> <s xml:id="echoid-s3126" xml:space="preserve"> uiſus comprehendet motum eius eſſe circularem ex comprehenſione mutationis <lb/>partium eius ſequentium uiſum apud aliquam rem uiſam:</s> <s xml:id="echoid-s3127" xml:space="preserve"> aut ex reſpicientia alicuius partis illius <lb/>ad diuerſa uiſibilia, unum poſt alterum:</s> <s xml:id="echoid-s3128" xml:space="preserve"> aut ad partes unius rei uiſæ unam partem poſt aliam, cum <lb/>quiete totalitatis rei uiſæ in ſuo loco.</s> <s xml:id="echoid-s3129" xml:space="preserve"> Et ſi motus rei uiſæ fuerit compoſitus ex motu circulari & lo <lb/>cali, uiſus comprehendet illum eſſe compoſitum ex comprehenſione mutationis partium rei uiſæ <lb/>motæ reſpectu uiſus, aut reſpectu alterius rei uiſæ cum comprehenſione motus totalitatis rei uiſæ <lb/>à ſuo loco.</s> <s xml:id="echoid-s3130" xml:space="preserve"> Secundum ergo iſtos modos uiſus comprehendit qualitates motus uiſibilium.</s> <s xml:id="echoid-s3131" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div107" type="section" level="0" n="0"> <head xml:id="echoid-head129" xml:space="preserve" style="it">51. Motus uiſibilis percipitur in tempore ſenſili.</head> <p> <s xml:id="echoid-s3132" xml:space="preserve">ET uiſus non comprehendit motum, niſi in tempore:</s> <s xml:id="echoid-s3133" xml:space="preserve"> quoniam motus non eſt, niſi in tempore, <lb/>& omnis pars motus non eſt, niſi in tempore.</s> <s xml:id="echoid-s3134" xml:space="preserve"> Et uiſus non comprehendit motum rei uiſæ, ni-<lb/>ſi ex comprehenſione rei uiſæ in duobus locis diuerſis, aut ſecundum duos ſitus.</s> <s xml:id="echoid-s3135" xml:space="preserve"> Locus au-<lb/>tem & ſitus rei uiſæ non diuerſantur, niſi in temporibus.</s> <s xml:id="echoid-s3136" xml:space="preserve"> Cum ergo uiſus comprehenderit rem ui-<lb/>ſam in duobus locis diuerſis, aut in duobus ſitibus diuerſis, nõ eſt, niſi in duabus horis diuerſis.</s> <s xml:id="echoid-s3137" xml:space="preserve"> Sed <lb/>inter quaslibet horas duas diuerſas eſt tempus mediũ.</s> <s xml:id="echoid-s3138" xml:space="preserve"> Viſus ergo nõ comprehendit motum, niſi in <lb/>tempore.</s> <s xml:id="echoid-s3139" xml:space="preserve"> Et etiam dicemus quòd tempus, in quo uiſus comprehendit motũ, non erit, niſi ſenſibile:</s> <s xml:id="echoid-s3140" xml:space="preserve"> <lb/>quoniam uiſus nõ comprehendit motũ, niſi ex comprehenſione rei uiſæ in duobus locis diuerſis in <lb/>uno loco poſt aliũ:</s> <s xml:id="echoid-s3141" xml:space="preserve"> aut ſecundũ duos ſitus diuerſos unũ ſitum poſtaliũ.</s> <s xml:id="echoid-s3142" xml:space="preserve"> Cum ergo uiſus cõprehen-<lb/>derit rem uiſam motã in ſuo loco ſecundo, & nõ comprehenderit tunc ipſam in primo loco, in quo <lb/>cõprehendit antè ipſam:</s> <s xml:id="echoid-s3143" xml:space="preserve"> ſtatim ſentiet ſentiens, quòd hora, in qua cõprehendit ipſam in ſecundo lo <lb/>co, eſt diuerſa ab hora, in qua comprehendit ipſam in primo loco.</s> <s xml:id="echoid-s3144" xml:space="preserve"> Quare ſentiet diuerſitatẽ duarum <lb/>horarum.</s> <s xml:id="echoid-s3145" xml:space="preserve"> Et ſimiliter quando comprehenderit motũ ex diuerſitate ſitus rei uiſæ.</s> <s xml:id="echoid-s3146" xml:space="preserve"> Quoniam ſi com-<lb/>prehenderit rem uiſam motam ſecundum ſitum, & non comprehenderit ipſam tunc ſecundum pri-<lb/>mum ſitum, ſecundum quem comprehendit ipſam antè:</s> <s xml:id="echoid-s3147" xml:space="preserve"> ſtatim ſentiet diuerſitatem duarum hora-<lb/>rum.</s> <s xml:id="echoid-s3148" xml:space="preserve"> Quare ſentiet tempus quod eſt inter ipſas.</s> <s xml:id="echoid-s3149" xml:space="preserve"> Tempus ergo, in quo uiſus comprehendit motum, <lb/>eſt ſenſibile neceſſariò.</s> <s xml:id="echoid-s3150" xml:space="preserve"> Et cum omnes iſtæ intentiones ſint declaratę, narremus modò quod coacer <lb/>uatur ex eis.</s> <s xml:id="echoid-s3151" xml:space="preserve"> Dicemus ergo, quòd uiſus comprehendit motum ex comprehenſione rei uiſę motę ſe-<lb/>cundum duos ſitus diuerſos, in duabus horis diuerſis, inter quas eſt tẽpus ſenſibile:</s> <s xml:id="echoid-s3152" xml:space="preserve"> & hæc eſt quali <lb/>tas comprehenſionis motus à uiſu.</s> <s xml:id="echoid-s3153" xml:space="preserve"> Et uiſus comprehendit diuerſitatẽ motuum ſecundum ueloci-<lb/>tatem & tarditatem, & æqualitatem motuum ex comprehenſione ſpatiorum, ſuper quæ mouentur <lb/>uiſibilia mota.</s> <s xml:id="echoid-s3154" xml:space="preserve"> Cum ergo uiſus comprehenderit duo uiſibilia mota, & cõprehenderit ſpatia, ſuper <lb/>quæ mouentur illa duo uiſibilia, & ſenſerit quòd alterum duorum ſpatiorum, quæ à duobus uiſibili <lb/>bus motis pertranſeunturin eodem tempore, eſt maius altero, ſentiet uelocitatem rei uiſæ motæ <lb/>tranſeuntis ſuper maius ſpatium.</s> <s xml:id="echoid-s3155" xml:space="preserve"> Et cum duo ſpatia, ſuper quæ mouentur uiſibilia, ſunt pertranſita <lb/>in duobus temporibus æqualibus, & ſenſerit uiſus æqualitatem illorum ſpatiorum, ſentiet æquali-<lb/>tatem duarum rerum motarũ.</s> <s xml:id="echoid-s3156" xml:space="preserve"> Et ſimiliter, ſi uiſus ſenſerit æqualitatem duorum ſpatiorũ cum inæ-<lb/>qualitate duorum temporum duorum motuũ:</s> <s xml:id="echoid-s3157" xml:space="preserve"> ſentiet uelocitatem motus rei motæ tranſeuntis per <lb/>ſpatium in minore tempore.</s> <s xml:id="echoid-s3158" xml:space="preserve"> Et ſimiliter quando duo mota tranſierint in duobus temporibus æqua <lb/>libus per duo ſpatia æqualia, & ſenſerit uiſus æqualitatẽ temporis & æqualitatem ſpatiorũ:</s> <s xml:id="echoid-s3159" xml:space="preserve"> ſentiet <lb/>æqualitatem duorum motuũ.</s> <s xml:id="echoid-s3160" xml:space="preserve"> Iam diximus, qualiter uiſus comprehendat motum, & diſtinguat mo-<lb/>tum, & qualitatem eius, & æqualitatem & inæqualitatem eius.</s> <s xml:id="echoid-s3161" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div108" type="section" level="0" n="0"> <head xml:id="echoid-head130" xml:space="preserve" style="it">52. Quies percipitur è uiſibili, eundem ſitum locuḿ tempore ſenſili occupante. 112 p 4.</head> <p> <s xml:id="echoid-s3162" xml:space="preserve">QVies autem comprehenditur à uiſu ex comprehenſione rei uiſę in tempore ſenſibili in eo-<lb/>dem loco & in eodem ſitu.</s> <s xml:id="echoid-s3163" xml:space="preserve"> Cum ergo uiſus comprehenderit uiſum in eodẽ loco, & ſecundũ <lb/>eundem ſitũ in duabus horis diuerſis, inter quas eſt tẽpus ſenſibile:</s> <s xml:id="echoid-s3164" xml:space="preserve"> cõprehendet rem uiſam <lb/>in illo tempore quieſcentem.</s> <s xml:id="echoid-s3165" xml:space="preserve"> Et uiſus comprehendit ſitum rei uiſæ quieſcentis reſpectu alterius rei <lb/>uiſę, & reſpectu ipſius uiſus.</s> <s xml:id="echoid-s3166" xml:space="preserve"> Secundũ ergo hunc modum erit comprehenſio quietis uiſibiliũ à uiſu.</s> <s xml:id="echoid-s3167" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div109" type="section" level="0" n="0"> <head xml:id="echoid-head131" xml:space="preserve" style="it">53. Aſperitas percipitur è luce aſper am ſuperficiem illuminante. 139 p 4.</head> <p> <s xml:id="echoid-s3168" xml:space="preserve">ASperitas uerò comprehenditur à uiſu in maiori parte ex forma lucis apparẽtis in ſuperſicie <lb/>corporis aſperi:</s> <s xml:id="echoid-s3169" xml:space="preserve"> quoniã aſperitas eſt diuerſitas ſitus partium ſuperficiei corporis.</s> <s xml:id="echoid-s3170" xml:space="preserve"> Quare lux <lb/>quando oritur ſuper ſuperficiem illius corporis, partes prominentes facient umbrã in maio-<lb/>ri parte.</s> <s xml:id="echoid-s3171" xml:space="preserve"> Et cum lux peruenerit in partes profundas, erunt cum eo etiam umbrę, & partes prominen <lb/>tes erunt manifeſtæ luce, & diſcoopertæ luce.</s> <s xml:id="echoid-s3172" xml:space="preserve"> Et cum in partes profundas ueniunt umbræ, & ſuper <lb/>prominentes non fuerit aliqua umbra:</s> <s xml:id="echoid-s3173" xml:space="preserve"> diuerſabitur forma lucis in ſuperficie illius corporis.</s> <s xml:id="echoid-s3174" xml:space="preserve"> In ſu-<lb/>perficie autem plana non eſt ita:</s> <s xml:id="echoid-s3175" xml:space="preserve"> quoniam ſuperficiei planæ partes ſunt conſimilis ſitus:</s> <s xml:id="echoid-s3176" xml:space="preserve"> & cum lux <lb/>orietur ſuperipſas, erit forma lucis in tota ſuperficie conſimilis.</s> <s xml:id="echoid-s3177" xml:space="preserve"> Forma ergo lucis in ſuperficie cor-<lb/>poris aſperi eſt diuerſa à forma lucis in ſuperficie plana.</s> <s xml:id="echoid-s3178" xml:space="preserve"> Et uiſus cognoſcit formam lucis, quæ eſt in <lb/> <pb o="62" file="0068" n="68" rhead="ALHAZEN"/> ſuperficiebus aſperis, & formam lucis, quæ eſt in ſuperficiebus planis propter frequentationem ui-<lb/>ſionis ſuperficierũ aſperarum & planarum.</s> <s xml:id="echoid-s3179" xml:space="preserve"> Cum ergo uiſus ſenſerit lucem, quæ eſt in ſuperficiebus <lb/>corporis ſecundum modum, quẽ aſſueuit in ſuperficiebus aſperis:</s> <s xml:id="echoid-s3180" xml:space="preserve"> iudicabit aſperitatẽ illius corpo-<lb/>ris:</s> <s xml:id="echoid-s3181" xml:space="preserve"> & cum ſenſerit lucem in ſuperficie corporis ſecundum modum, quem aſſueuit in ſuperficiebus <lb/>planis:</s> <s xml:id="echoid-s3182" xml:space="preserve"> iudicabit planitiem in ſuperficiebus illius corporis.</s> <s xml:id="echoid-s3183" xml:space="preserve"> Et cum aſperitas fuerit extranea:</s> <s xml:id="echoid-s3184" xml:space="preserve"> erunt <lb/>partes prominentes alicuius quantitatis:</s> <s xml:id="echoid-s3185" xml:space="preserve"> & ſic uiſus comprehendet prominentiã illarum partium:</s> <s xml:id="echoid-s3186" xml:space="preserve"> <lb/>& comprehendet ſitum ſuperficiei corporis ex comprehenſione diſtantiæ, quæ eſt inter partes.</s> <s xml:id="echoid-s3187" xml:space="preserve"> Et <lb/>cum uiſus comprehenderit diuerſitatẽ ſituum partium ſuperficiei corporis:</s> <s xml:id="echoid-s3188" xml:space="preserve"> comprehendet aſperi-<lb/>tatem eius ſine indigentia ad conſiderandum lucem.</s> <s xml:id="echoid-s3189" xml:space="preserve"> Et etiam quando aſperitas corporis fuerit ex-<lb/>tranea, & oritur ſuper ipſam lux:</s> <s xml:id="echoid-s3190" xml:space="preserve"> erit forma lucis in ſuperficie eius diuerſa maxima diuerſitate.</s> <s xml:id="echoid-s3191" xml:space="preserve"> Vi-<lb/>debitur ergo ex diuerſitate lucis diſtantia partium & diuerſitas ſitus earum:</s> <s xml:id="echoid-s3192" xml:space="preserve"> & ex hoc apparebit a-<lb/>ſperitas corporis.</s> <s xml:id="echoid-s3193" xml:space="preserve"> Si ergo lux oriens ſuper corpus aſperum, fuerit ex parte oppoſita ſuperficiei aſpe-<lb/>ræ, & fuerit lux fortis:</s> <s xml:id="echoid-s3194" xml:space="preserve"> non comprehendet uiſus aſperitatem huius corporis, niſi quando compre-<lb/>henderit prominentiam quarundam partium & profunditatem quarundam.</s> <s xml:id="echoid-s3195" xml:space="preserve"> Si ergo aſperitas hu-<lb/>ius corporis fuerit extranea, id eſt, maxima:</s> <s xml:id="echoid-s3196" xml:space="preserve"> comprehendet uiſus diſtantiam partium & diuerſita-<lb/>tem ſitus earum, & comprehendet aſperitatem corporis in maiori parte.</s> <s xml:id="echoid-s3197" xml:space="preserve"> Si autem aſperitas fuerit <lb/>modica, & partes fuerint profundæ, & pori illius corporis in ultimitate paruitatis:</s> <s xml:id="echoid-s3198" xml:space="preserve"> latebit uiſum in <lb/>maiori parte, & nunquam uiſus comprehendet aſperitatem huius corporis, niſi in magna appropin <lb/>quatione cum intuitu partium ſuperficiei corporis.</s> <s xml:id="echoid-s3199" xml:space="preserve"> Cum ergo uiſus diſtinxerit diſtantiam partium <lb/>huiuſmodi corporis, & prominentiam & profunditatem illarum:</s> <s xml:id="echoid-s3200" xml:space="preserve"> comprehender aſperitatem eius.</s> <s xml:id="echoid-s3201" xml:space="preserve"> <lb/>Si autem uiſus non diſtinxerit diſtantiam partium eius, nec prominentiam & profunditatem par-<lb/>tium eius:</s> <s xml:id="echoid-s3202" xml:space="preserve"> non comprehendet aſperitatem eius.</s> <s xml:id="echoid-s3203" xml:space="preserve"> Aſperitas ergo comprehenditur à uiſu ex compre-<lb/>henſione diuerſitatis ſituũ partium ſuperficiei corporis, aut ex forma lucis, quam uiſus aſſueuit ui-<lb/>dere in ſuperficiebus corporum aſperorum.</s> <s xml:id="echoid-s3204" xml:space="preserve"> Et uiſus cognoſcit etiam aſperitatem ex prænotione <lb/>conſimilitudinis.</s> <s xml:id="echoid-s3205" xml:space="preserve"> Cum ergo uiſus nihil ſenſerit in corpore, ex conſimilitudine, iudicabit eius aſpe-<lb/>ritatem.</s> <s xml:id="echoid-s3206" xml:space="preserve"> Sed multoties errat uiſus in aſperitate, quando uoluerit cognoſcere ipſam per iſtam inten-<lb/>tionem:</s> <s xml:id="echoid-s3207" xml:space="preserve"> quoniam erit ſuperficies terſa, & non apparet eius terſitudo:</s> <s xml:id="echoid-s3208" xml:space="preserve"> quoniam terſitudo non appa-<lb/>ret, niſi in ſitu proprio.</s> <s xml:id="echoid-s3209" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div110" type="section" level="0" n="0"> <head xml:id="echoid-head132" xml:space="preserve" style="it">54. Lenit as percipitur è luce lenem ſuperficiem illuminante. 140 p 4.</head> <p> <s xml:id="echoid-s3210" xml:space="preserve">PLanities autem & æqualitas ſuperficiei corporis comprehenditur à uiſu in maiori parte ex <lb/>forma lucis apparentis in ſuperficie corporis plani, quam aſſueuit uidere in ſuperficiebus pla-<lb/>nis.</s> <s xml:id="echoid-s3211" xml:space="preserve"> Et cum lux, quæ eſt in ſuperficiebus corporis, fuerit conſimilis formæ:</s> <s xml:id="echoid-s3212" xml:space="preserve"> cognoſcet peri-<lb/>pſam planitiem ſuperficiei.</s> <s xml:id="echoid-s3213" xml:space="preserve"> Et uiſus comprehendit aliquando planitiem per intuitum etiam.</s> <s xml:id="echoid-s3214" xml:space="preserve"> Cum <lb/>ergo uiſus intuebitur ſuperficiem corporis plani, comprehendet æqualitatem partium eius:</s> <s xml:id="echoid-s3215" xml:space="preserve"> & ſic <lb/>comprehendet planitiem.</s> <s xml:id="echoid-s3216" xml:space="preserve"> Terſitudo autem (& eſt fortis planities) comprehenditur à uiſu ex ſcin-<lb/>tillatione lucis in ſuperficie corporis ſui.</s> <s xml:id="echoid-s3217" xml:space="preserve"> Planities ergo comprehenditur à uiſu ex comprehenſio-<lb/>ne æqualitatis ſuperficiei.</s> <s xml:id="echoid-s3218" xml:space="preserve"> Aequalitas autem ſuperficiei comprehenditur à uiſu in maiori parte ex <lb/>ſimilitudine formæ lucis in ſuperficie corporis.</s> <s xml:id="echoid-s3219" xml:space="preserve"> Et terſitudo comprehenditur à uiſu ex ſcintilla-<lb/>tione lucis in ſuperficie corporis, & ex ſitu, ſecundum quem reflectitur lux.</s> <s xml:id="echoid-s3220" xml:space="preserve"> Et fortè ſimul aggre-<lb/>gatur aſperitas & planities in eadem ſuperficie, ſcilicet quòd ſint in ſuperficie alicuius corporis par <lb/>tes diuerſi ſitus, profundæ & prominentes, & ſint partes cuiuslibet partium diuerſiſitus prominen <lb/>tium & profundarum ad quaſdam partes, uel ad partes quarundam conſimilis ſitus, ita ut tota ſu-<lb/>perficies ſit aſpera, & partes eius, aut quædam ſint planæ.</s> <s xml:id="echoid-s3221" xml:space="preserve"> Et aſperitas huiuſmodi ſuperficiei com-<lb/>prehenditur à uiſu ex comprehenſione diuerſitatis ſitus partium prominentium & profundarum.</s> <s xml:id="echoid-s3222" xml:space="preserve"> <lb/>Et planities partium comprehenditur à uiſu in ſuperficie bus partium.</s> <s xml:id="echoid-s3223" xml:space="preserve"> Et aliquando uiſus compre-<lb/>hendit planitiem huiuſmodi partium per intuitionem, & ex comprehenſione conſimilitudinis ſu-<lb/>perficiei cuiuslibet iHarum.</s> <s xml:id="echoid-s3224" xml:space="preserve"> Et ſecundum iſtos modos comprehendit uiſus planitiem & terſitudi-<lb/>nem & aſperitatem.</s> <s xml:id="echoid-s3225" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div111" type="section" level="0" n="0"> <head xml:id="echoid-head133" xml:space="preserve" style="it">55. Perſpicuit{as} percipitur è perceptione corporis denſi ultra corp{us} perſpicuum poſiti. <lb/>142 p 4.</head> <p> <s xml:id="echoid-s3226" xml:space="preserve">DIaphanitas autem comprehenditur à uiſu per argumentationem ex comprehenſione illius, <lb/>quod eſt ultra corpus diaphanum.</s> <s xml:id="echoid-s3227" xml:space="preserve"> Et diaphanitas corporis diaphani non comprehenditur <lb/>à uiſu, niſi quando fuerit in eo ſpiſsitudo quædam, & fuerit diaphanitas eius ſpiſsior, dia-<lb/>phanitate aeris interiacentis inter uiſum & ipſum.</s> <s xml:id="echoid-s3228" xml:space="preserve"> Si autem fuerit in fine diaphanitatis, non com-<lb/>prehendet uiſus diaphanitatem eius, & non comprehendet, niſi illud, quod eſt ultra ipſum tantùm.</s> <s xml:id="echoid-s3229" xml:space="preserve"> <lb/>Et cum in eo fuerit quædam diaphanitas:</s> <s xml:id="echoid-s3230" xml:space="preserve"> comprehendetur à uiſu propter illud, quod eſt de ſpiſsi-<lb/>tudine in eo, & diaphanitas eius comprehendetur ex comprehenſione illius, quod eſt ultra ipſum.</s> <s xml:id="echoid-s3231" xml:space="preserve"> <lb/>Quoniam quando ultra corpus diaphanum fuerit lux aut corpus coloratum illuminatum, uidebi-<lb/>tur ultra corpus diaphanum.</s> <s xml:id="echoid-s3232" xml:space="preserve"> Et uiſus non ſentit diaphanitatem corporis, quando ſenſerit illud, <lb/>quod eſt ultra ipſum, niſi cum ſenſerit quòd color & lux, quæ comprehenduntur ultra corpus dia-<lb/>phanum, eſt lux & color ultra corpus diaphanum, & non eſt color & lux ipſius corporis:</s> <s xml:id="echoid-s3233" xml:space="preserve"> ſi autem <lb/> <pb o="63" file="0069" n="69" rhead="OPTICAE LIBER II."/> non:</s> <s xml:id="echoid-s3234" xml:space="preserve"> nõ ſentiet diaphanitatem corporis diaphani.</s> <s xml:id="echoid-s3235" xml:space="preserve"> Si ergo ultra corpus diaphanum non fuerit lux, <lb/>nec corpus illuminatum, nec in circuitu eius, & non apparuerit ultra ipſum, neq;</s> <s xml:id="echoid-s3236" xml:space="preserve"> in aliqua alia par-<lb/>te lux aut color:</s> <s xml:id="echoid-s3237" xml:space="preserve"> diaphanitas illius corporis non comprehenditur.</s> <s xml:id="echoid-s3238" xml:space="preserve"> Et hoc erit quando corpus dia-<lb/>phanum fuerit applicatum cum aliquo corpore ſpiſſo, & illud corpus ſpiſſum continuerit ipſum, <lb/>aut reſpexerit ipſum, & fuerit quoq;</s> <s xml:id="echoid-s3239" xml:space="preserve"> corpus diaphanum obſcuri coloris:</s> <s xml:id="echoid-s3240" xml:space="preserve"> quoniam tunc uiſus non <lb/>ſentiet diaphanitatem huius corporis.</s> <s xml:id="echoid-s3241" xml:space="preserve"> Et ſimiliter quàndo ultra corpus diaphanũ fuerit locus ob-<lb/>ſcurus, & non apparuerit ultra ipſum aliqua lux:</s> <s xml:id="echoid-s3242" xml:space="preserve"> non comprehendetur diaphanitas eius.</s> <s xml:id="echoid-s3243" xml:space="preserve"> Cum ergo <lb/>uiſus ſenſerit, quòd color, qui comprehẽditur ultra corpus diaphanum, eſt color corporis ultra cor-<lb/>pus diaphanum, ſentiet diaphanitatem corporis diaphani.</s> <s xml:id="echoid-s3244" xml:space="preserve"> Et ſimiliter quando corpus diaphanum <lb/>fuerit debilis diaphanitatis, & fuerit corpus, quod eſt ultra ipſum, & corpora quæ ſunt in circuitu e-<lb/>ius, debilis lucis:</s> <s xml:id="echoid-s3245" xml:space="preserve"> tunc diaphanitas eius non comprehenditur à uiſu, niſi apponatur forti luci.</s> <s xml:id="echoid-s3246" xml:space="preserve"> Cum <lb/>autem cognoſcet lucem ultra ipſum:</s> <s xml:id="echoid-s3247" xml:space="preserve"> comprehendet diaphanitatem.</s> <s xml:id="echoid-s3248" xml:space="preserve"> Secundum ergo iſtos inodos <lb/>comprehendet uiſus diaphanitatem corporum diaphanorum.</s> <s xml:id="echoid-s3249" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div112" type="section" level="0" n="0"> <head xml:id="echoid-head134" xml:space="preserve" style="it">56. Denſitas percipitur è perſpicuitatis priuatione. 143 p 4.</head> <p> <s xml:id="echoid-s3250" xml:space="preserve">SPiſsitudo comprehenditur à uiſu ex priuatione diaphanitatis.</s> <s xml:id="echoid-s3251" xml:space="preserve"> Cum ergo uiſus comprehenderit <lb/>corpus, & non ſenſerit in ipſo aliquam diaphanitatem, arguet eius ſpiſsitudinem.</s> <s xml:id="echoid-s3252" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div113" type="section" level="0" n="0"> <head xml:id="echoid-head135" xml:space="preserve" style="it">57. Vmbra percipitur è lucis unius abſentia, alterius præſentia. 145 p 4.</head> <p> <s xml:id="echoid-s3253" xml:space="preserve">VMbra uerò comprehenditur à uiſu reſpectu lucis illuminantis, aut partis lucis illuminantis.</s> <s xml:id="echoid-s3254" xml:space="preserve"> <lb/>Quoniã enim umbra eſt priuatio quarundam lucium cum illuminatione loci umbrę ab ex-<lb/>tranea luce priuata à loco umbræ:</s> <s xml:id="echoid-s3255" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s3256" xml:space="preserve"> cum ſenſerit uiſus illud, quod eſt uicinum ipſi, & fue-<lb/>rit ſuper illud corpus uicinum lux fortiorluce, quæ eſt in loco umbræ, ſentiet umbrationem illius <lb/>loci, & priuationẽ à luce oriente ſuper corpus uicinũ illi.</s> <s xml:id="echoid-s3257" xml:space="preserve"> Quoniã quando uiſus ſenſerit aliquã lucẽ <lb/>in aliquo loco:</s> <s xml:id="echoid-s3258" xml:space="preserve"> & caruerit ille locus luce ſolis, aut aliqua luce forti:</s> <s xml:id="echoid-s3259" xml:space="preserve"> ſentiet obumbrationẽ illius loci <lb/>& priuationẽ à luce ſolis, aut ab alia luce forti.</s> <s xml:id="echoid-s3260" xml:space="preserve"> Et fortè uiſus ſentiet corpus faciẽs umbram, & fortè <lb/>non diſtinguetur ab eo ſtatim corpus obumbrans, ſed tandem, quando uiſus comprehenderit lo-<lb/>cũ, in quo eſt luxdebilis, & cõprehenderit ultima corporà in loco lucis debilis eſſe fortioris lucis il-<lb/>la luce debili:</s> <s xml:id="echoid-s3261" xml:space="preserve"> ſentiet ſtatim umbrã illius loci.</s> <s xml:id="echoid-s3262" xml:space="preserve"> Secundũ ergo hunc modũ uiſus cõprehendit umbrã.</s> <s xml:id="echoid-s3263" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div114" type="section" level="0" n="0"> <head xml:id="echoid-head136" xml:space="preserve" style="it">58. Obſcurit{as} percipitur è lucis priuatione & abſentia. 146 p 4.</head> <p> <s xml:id="echoid-s3264" xml:space="preserve">OBſcuritas uerò comprehenditur à uiſu per argumentationem expriuatione lucis.</s> <s xml:id="echoid-s3265" xml:space="preserve"> Cum er-<lb/>go uiſus comprehenderit aliquem locum, & non comprehenderit in eo aliquam lucem, ſen <lb/>tiet obſcuritatem eius.</s> <s xml:id="echoid-s3266" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div115" type="section" level="0" n="0"> <head xml:id="echoid-head137" xml:space="preserve" style="it">59. Pulchritudo percipitur tum è ſingulis uiſibilibus ſpeciebus, tum è pluribus ſimul coniun <lb/>ctis, ſymmetris inter ſe. 148 p 4.</head> <p> <s xml:id="echoid-s3267" xml:space="preserve">PVlchritudo autem comprehenditur à uiſu ex comprehenſione intentionum particularium, <lb/>quarum comprehenſionis qualitas eſt declarata antè.</s> <s xml:id="echoid-s3268" xml:space="preserve"> Nam unaquæque intentionum particu-<lb/>larium præ dictarum faciet per ſe aliquem modum pulchritudinis, & coniugationes illaruin <lb/>faoiunt etiam alios modos pulchritudinis.</s> <s xml:id="echoid-s3269" xml:space="preserve"> Et uiſus non comprehendit pulchritudinem, niſi in for-<lb/>mis uiſibiliũ, quæ comprehenduntur per ſenſum uiſus.</s> <s xml:id="echoid-s3270" xml:space="preserve"> Et formæ uiſibiliũ ſunt compoſitæ ex inten <lb/>tionibus particularibus, quarũ diſtinctio iam eſt declarata.</s> <s xml:id="echoid-s3271" xml:space="preserve"> Et uiſus comprehendit formas ex com-<lb/>prehenſione iſtarum intentionum.</s> <s xml:id="echoid-s3272" xml:space="preserve"> Ipſe ergo comprehendit pulchritudinem ex comprehenſione <lb/>iſtarum intentionum.</s> <s xml:id="echoid-s3273" xml:space="preserve"> Modi autem pulchritudinis, qui comprehenduntur à uiſu in formis uiſibi-<lb/>lium, ſunt multi.</s> <s xml:id="echoid-s3274" xml:space="preserve"> Quædam ergo uiſibilia habent unam cauſſam ex intentionibus particularibus, quę <lb/>ſunt in forma:</s> <s xml:id="echoid-s3275" xml:space="preserve"> & cauſſa quorundam non eſt, niſi intentionum inter ſe coniunctio, non ipſæ inten-<lb/>tiones:</s> <s xml:id="echoid-s3276" xml:space="preserve"> & cauſſa quorundam eſt compoſita ex intentionibus & ex compoſitione illarum.</s> <s xml:id="echoid-s3277" xml:space="preserve"> Et uiſus <lb/>comprehendit quamlibet intentionum, quæ ſunt in qualibet forma perſe:</s> <s xml:id="echoid-s3278" xml:space="preserve"> & comprehendit ipſas <lb/>compoſitas:</s> <s xml:id="echoid-s3279" xml:space="preserve"> & comprehendit compoſitionem & coniugationem illarum.</s> <s xml:id="echoid-s3280" xml:space="preserve"> Viſus ergo comprehen-<lb/>dit pulchritudinem ſecundum diuerſos modos.</s> <s xml:id="echoid-s3281" xml:space="preserve"> Et omnes modi, ex quibus uiſus comprehendit <lb/>pulchritudinem, reuertuntur ad comprehenſionem intentionum particularium.</s> <s xml:id="echoid-s3282" xml:space="preserve"> Si uerò iſtæ in-<lb/>tentiones particulares faciunt pulchritudinem:</s> <s xml:id="echoid-s3283" xml:space="preserve"> etiam compoſitæ ſimiliter.</s> <s xml:id="echoid-s3284" xml:space="preserve"> Et eſt dicere:</s> <s xml:id="echoid-s3285" xml:space="preserve"> facere pul <lb/>chritudinem, eſt inducere diſpoſitionem in anima, qua uidebiture ei, quòd ſit res pulchra, quæ ui-<lb/>detur.</s> <s xml:id="echoid-s3286" xml:space="preserve"> Et hoc apparebit per modicam inſpectionem:</s> <s xml:id="echoid-s3287" xml:space="preserve"> quoniam lux facit pulchritudinem:</s> <s xml:id="echoid-s3288" xml:space="preserve"> & pro-<lb/>pter hoc apparebunt pulchra ſol, luna & ſteliæ:</s> <s xml:id="echoid-s3289" xml:space="preserve"> & non eſt in ſole, luna & ſtellis cauſa, propter <lb/>quam apparebunt decora, niſi luxearum.</s> <s xml:id="echoid-s3290" xml:space="preserve"> Lux ergo per ſe facit pulchritudinem.</s> <s xml:id="echoid-s3291" xml:space="preserve"> Et color etiam <lb/>facit pulchritudinem.</s> <s xml:id="echoid-s3292" xml:space="preserve"> Quoniam quilibet color ſcintillans ſicut uiridis & roſeus, & his ſimiles ap-<lb/>parebunt pulchri uiſui, & delectatur uiſus eis.</s> <s xml:id="echoid-s3293" xml:space="preserve"> Et propter hoc apparebunt pulchri panni tincti, & <lb/>flores, & uiridia.</s> <s xml:id="echoid-s3294" xml:space="preserve"> Color ergo per ſe facit pulchritudinem.</s> <s xml:id="echoid-s3295" xml:space="preserve"> Et remotio etiam aliquando facit pulchri-<lb/>tudinem accidentaliter.</s> <s xml:id="echoid-s3296" xml:space="preserve"> Quoniam in quibuſdam formis pulchris ſunt maculę & rugæ, quæ faciunt <lb/>turpitudinem in formis:</s> <s xml:id="echoid-s3297" xml:space="preserve"> & cum elongabuntur à uiſu, latent illæ intentiones ſubtiles, quæ faciunt <lb/> <pb o="64" file="0070" n="70" rhead="ALHAZEN"/> turpitudinem in illis formis, & apud latentiam illarum intentionum apparebit pulchritudo illius <lb/>formæ.</s> <s xml:id="echoid-s3298" xml:space="preserve"> Et ſimiliter etiam in multis formis pulchris ſunt intentiones ſubtiles, per quas forma eſt <lb/>pulchra, ſicut lineatio & ordinatio, & multæ iſtarum intentionum latent uiſum in multis remotio-<lb/>nibus mediocribus:</s> <s xml:id="echoid-s3299" xml:space="preserve"> & quando ſunt prope uiſum, apparebunt illę intentiones ſubtiles uiſui, & appa <lb/>rebit pulchritudo formæ.</s> <s xml:id="echoid-s3300" xml:space="preserve"> Remotio ergo & appropinquatio faciunt pulchritudinẽ.</s> <s xml:id="echoid-s3301" xml:space="preserve"> Et ſitus aliquan-<lb/>do facit pulchritudinem:</s> <s xml:id="echoid-s3302" xml:space="preserve"> & plures intentiones pulchræ non apparent pulchræ, niſi propter ordi-<lb/>nem & ſitum tantùm.</s> <s xml:id="echoid-s3303" xml:space="preserve"> Quoniam omnes diſtinctiones ordinatæ quaſi punctatæ non apparent pul-<lb/>chræ, niſi propter ordinem.</s> <s xml:id="echoid-s3304" xml:space="preserve"> Et ſcriptura non apparet pulchra, niſi propter ordinationem:</s> <s xml:id="echoid-s3305" xml:space="preserve"> quoniam <lb/>pulchritudo non eſt, niſi ex directione figurarum literarum, & ex compoſitione earum inter ſe.</s> <s xml:id="echoid-s3306" xml:space="preserve"> Si <lb/>autem compoſitio literarum & ordinatio non fuerit ſecundum unam proportionem, ſcilicet, ut u-<lb/>na magna, alia parua:</s> <s xml:id="echoid-s3307" xml:space="preserve"> tunc non erit pulchra ſcriptura, quamuis figuræ literarum per ſe ſint benè po <lb/>ſitæ.</s> <s xml:id="echoid-s3308" xml:space="preserve"> Et aliquando apparet ſcriptura pulchra, quando compoſitio eius fuerit proportionalis, quam-<lb/>uis literæ non ſint in fine bonæ diſpoſitionis.</s> <s xml:id="echoid-s3309" xml:space="preserve"> Et ſimiliter plures formæ uiſibilium non apparent <lb/>pulchræ, niſi propter diſpoſitionem & ordinationem partium inter ſe.</s> <s xml:id="echoid-s3310" xml:space="preserve"> Et corporeitas etiam facit <lb/>pulchritudinem:</s> <s xml:id="echoid-s3311" xml:space="preserve"> & propter hoc apparent pulchra corpora hominum & multorum animalium.</s> <s xml:id="echoid-s3312" xml:space="preserve"> Et <lb/>figura facit pulchritudinem:</s> <s xml:id="echoid-s3313" xml:space="preserve"> & propter hoc luna, & formæ pulchræ hominum & multorum anima-<lb/>lium, & arborum, & plantarum non apparent pulchræ, niſi propter formas eorum, aut propter fi-<lb/>guras partium eorum, aut propter eorum figuras, aut propter figuras partium formæ.</s> <s xml:id="echoid-s3314" xml:space="preserve"> Et magnitu-<lb/>do facit pulchritudinem:</s> <s xml:id="echoid-s3315" xml:space="preserve"> & propter hoc apparet luna pulchrior ſtellis, & ſtellæ magnæ pulchrio-<lb/>res ſtellis paruis.</s> <s xml:id="echoid-s3316" xml:space="preserve"> Et diuiſio facit pulchritudinem:</s> <s xml:id="echoid-s3317" xml:space="preserve"> & propter hoc ſtellæ ſeparatæ ſunt pulchriores <lb/>ſtellis extenſis, & pulchriores ſtellis galaxiæ:</s> <s xml:id="echoid-s3318" xml:space="preserve"> & propter hoc candelæ diſtinctæ ſunt pulchriores i-<lb/>gne.</s> <s xml:id="echoid-s3319" xml:space="preserve"> Et continuatio etiam facit pulchritudinem:</s> <s xml:id="echoid-s3320" xml:space="preserve"> & propter hoc uiridale continuum, & plantæ con-<lb/>tinuæ & ſpiſſæ ſunt pulchriores diſtinctis.</s> <s xml:id="echoid-s3321" xml:space="preserve"> Et numerus facit pulchritudinem:</s> <s xml:id="echoid-s3322" xml:space="preserve"> & propter hoc loca <lb/>cœli multarum ſtellarum ſunt pulchriora locis paucarum ſtellarum:</s> <s xml:id="echoid-s3323" xml:space="preserve"> & propter hoc candelæ multæ <lb/>in eodem loco faciunt pulchritudinem.</s> <s xml:id="echoid-s3324" xml:space="preserve"> Et etiam motus hominis in ſerm one facit pulchritudinem.</s> <s xml:id="echoid-s3325" xml:space="preserve"> <lb/>Et quies eius facit pulchritudinem:</s> <s xml:id="echoid-s3326" xml:space="preserve"> & propter hoc apparet pulchra grauitas & taciturnitas.</s> <s xml:id="echoid-s3327" xml:space="preserve"> Et aſpe-<lb/>ritas facit pulchritudinem:</s> <s xml:id="echoid-s3328" xml:space="preserve"> & propter hocapparet uilloſitas pulchra, ut uilloſitas in multis pannis.</s> <s xml:id="echoid-s3329" xml:space="preserve"> <lb/>Et planities facit pulchritudinem:</s> <s xml:id="echoid-s3330" xml:space="preserve"> & propter hoc apparet pulchrum in pannis.</s> <s xml:id="echoid-s3331" xml:space="preserve"> Et diaphanitas facit <lb/>pulchritudinem:</s> <s xml:id="echoid-s3332" xml:space="preserve"> & propter hoc apparent de nocte micantes diaphani.</s> <s xml:id="echoid-s3333" xml:space="preserve"> Et ſpiſsitudo facit pulchri-<lb/>tudinem:</s> <s xml:id="echoid-s3334" xml:space="preserve"> quoniam color & lux, & figura, & lineatio, & omnes intentiones pulchræ apparentes in <lb/>formis uiſibilium non comprehenduntur ſimiliter à uiſu, niſi propter ſpiſsitudinem & umbram.</s> <s xml:id="echoid-s3335" xml:space="preserve"> <lb/>Et umbra facit apparere pulchritudinem:</s> <s xml:id="echoid-s3336" xml:space="preserve"> quoniam in multis formis uiſibilium ſunt maculæ, & <lb/>pori ſubtiles reddentes eas turpes:</s> <s xml:id="echoid-s3337" xml:space="preserve"> & cum fuerint in luce ſolis, apparebunt maculæ in eis:</s> <s xml:id="echoid-s3338" xml:space="preserve"> quare la-<lb/>tebit pulchritudo earum:</s> <s xml:id="echoid-s3339" xml:space="preserve"> & cum fuerint in umbra aut luce debili, latebunt illæ maculæ & rugæ:</s> <s xml:id="echoid-s3340" xml:space="preserve"> <lb/>quare comprehendetur pulchritudo earum.</s> <s xml:id="echoid-s3341" xml:space="preserve"> Et etiam tortuoſitates, quæ apparent in plumis a-<lb/>uium, & in panno, qui dicitur amilialmon, in umbra non apparent & in luce debili.</s> <s xml:id="echoid-s3342" xml:space="preserve"> Et obſcuritas <lb/>facit pulchritudinem apparere:</s> <s xml:id="echoid-s3343" xml:space="preserve"> quoniam ſtellæ non apparent, niſi in obſcuro:</s> <s xml:id="echoid-s3344" xml:space="preserve"> & ſimiliter non ap-<lb/>paret pulchritudo earum, niſi in nigredine noctis, & in locis obſcuris, & latet in luce diei:</s> <s xml:id="echoid-s3345" xml:space="preserve"> & ſtellæ <lb/>in noctibus obſcuris ſunt pulchriores, quàm in noctibus lunæ.</s> <s xml:id="echoid-s3346" xml:space="preserve"> Et conſimilitudo facit pulchritudi-<lb/>nem:</s> <s xml:id="echoid-s3347" xml:space="preserve"> quoniam membra animalis eiuſdem ſpeciei, ut oculus oculo, non apparent pulchra, niſi quan <lb/>do fuerint conſimilia:</s> <s xml:id="echoid-s3348" xml:space="preserve"> quoniam oculi, quando fuerint diuerſæ figuræ, ſcilicet quod unus ſit rotun-<lb/>dus, & alter longus, erunt in fine turpitudinis:</s> <s xml:id="echoid-s3349" xml:space="preserve"> & etiã ſi unus fuerit niger & alter uiridis, erũt etiã tur <lb/>pes:</s> <s xml:id="echoid-s3350" xml:space="preserve"> & ſimiliter ſi unus fuerit maior altero.</s> <s xml:id="echoid-s3351" xml:space="preserve"> Et ſimiliter ſi una gena fuerit profunda, & altera pro-<lb/>minens, erunt in fine turpitudinis.</s> <s xml:id="echoid-s3352" xml:space="preserve"> Et ſimiliter quando unum ſuperciliorum fuerit groſſum, & al-<lb/>terum ſubtile, aut unum illorum longum, & alterum breue, erunt turpia.</s> <s xml:id="echoid-s3353" xml:space="preserve"> Omnia ergo membra ani-<lb/>malium uniuſmodi non erunt pulchra, niſi cum fuerint conſimilia.</s> <s xml:id="echoid-s3354" xml:space="preserve"> Et ſimiliter literæ & picturæ <lb/>non apparent pulchræ, niſi quando literæ fuerint, quæ ſunt uniuſmodi:</s> <s xml:id="echoid-s3355" xml:space="preserve"> & partes illarum, quæ ſunt <lb/>uniuſmodi, conſimiles.</s> <s xml:id="echoid-s3356" xml:space="preserve"> Et diuerſitas facit pulchritudinem:</s> <s xml:id="echoid-s3357" xml:space="preserve"> quoniam figuræ membrorum anima-<lb/>lis ſunt diuerſarum partium, & non ſunt pulchræ, niſi propter illam diuerſitatem.</s> <s xml:id="echoid-s3358" xml:space="preserve"> Quoniam ſina-<lb/>ſus totus eſſet eiuſdem groſsitudinis, eſſet in fine turpitudinis:</s> <s xml:id="echoid-s3359" xml:space="preserve"> & pulchritudo eius non eſt, niſi pro-<lb/>pter diuerſitatem duorum extremorum eius, & eius pyramidalitatem.</s> <s xml:id="echoid-s3360" xml:space="preserve"> Et ſimiliter pulchritudo ſu-<lb/>perciliorum non eſt, niſi quando extrema eorum fuerint ſubtiliora reſiduis anterioribus.</s> <s xml:id="echoid-s3361" xml:space="preserve"> Et ſimi-<lb/>liter omnia membra animalium quando aſpiciuntur:</s> <s xml:id="echoid-s3362" xml:space="preserve"> inuenitur quòd pulchritudo eorum non <lb/>eſt, niſi ex diuerſitate figurarum partium eorum.</s> <s xml:id="echoid-s3363" xml:space="preserve"> Et ſimiliter ſcripturæ:</s> <s xml:id="echoid-s3364" xml:space="preserve"> quoniam ſi partes ſcriptu-<lb/>ræ eſſent æqualis groſsitudinis, non apparerent pulchræ:</s> <s xml:id="echoid-s3365" xml:space="preserve"> quoniam extrema literarum non appa-<lb/>rent pulchra, niſi quando fuerint ſubtiliora reſiduo.</s> <s xml:id="echoid-s3366" xml:space="preserve"> Quoniam ſi extrema literarum & media ea-<lb/>rum, & continuatio earum eſſent unius ſpiſsitudinis:</s> <s xml:id="echoid-s3367" xml:space="preserve"> eſſet ſcriptura in fine turpitudinis.</s> <s xml:id="echoid-s3368" xml:space="preserve"> Diuerſi-<lb/>tas ergo facit pulchritudinem in multis formis uiſibilium.</s> <s xml:id="echoid-s3369" xml:space="preserve"> Iam ergo declaratum eſt ex eo, quod di-<lb/>ximus, quòd unaquæque intentionum particularium, quando comprehenditur per ſenſum uiſus, <lb/>aliquando facit pulchritudinem per ſe.</s> <s xml:id="echoid-s3370" xml:space="preserve"> Et cum ſermo fuerit factus de multis corporibus inductiuè <lb/>per ſe:</s> <s xml:id="echoid-s3371" xml:space="preserve"> cum inducentur omnia corpora:</s> <s xml:id="echoid-s3372" xml:space="preserve"> inuenietur, quòd quælibet iſtarum intentionum facit pul-<lb/>chritudinem in multis locis.</s> <s xml:id="echoid-s3373" xml:space="preserve"> Et non diximus, ea quæ diximus, niſi gratia exempli, & ut poſſent <lb/>acquiri alia exempla per iſta.</s> <s xml:id="echoid-s3374" xml:space="preserve"> Sed tamen iſtæ intentiones non faciunt pulchritudinem in omnibus <lb/>locis, neque unaiſtarum intentionum facit pulchritudinem in qualibet forma, in quam peruenit <lb/> <pb o="65" file="0071" n="71" rhead="OPTICAE LIBER II."/> illa intentio, ſed in quibuſdam formis, & in quibuſdam non.</s> <s xml:id="echoid-s3375" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s3376" xml:space="preserve"> non quælibet magni-<lb/>tudo facit pulchritudinem in quolibet corpore alicuius magnitudinis:</s> <s xml:id="echoid-s3377" xml:space="preserve"> & ſimiliter non quili-<lb/>bet color facit pulchritudinem:</s> <s xml:id="echoid-s3378" xml:space="preserve"> neque uiridis color facit pulchritudinem in quolibet corpore, in <lb/>quod peruenit ille color:</s> <s xml:id="echoid-s3379" xml:space="preserve"> & ſimiliter non quælibet figura facit pulchritudinem.</s> <s xml:id="echoid-s3380" xml:space="preserve"> Et quælibet illa-<lb/>rum intentionum, quas diximus, facit pulchritudinem per ſe, ſed in quibuſdam locis, & in quibuſ-<lb/>dam non, & ſecundum quoſdam modos, & ſecundum alios non.</s> <s xml:id="echoid-s3381" xml:space="preserve"> Et etiam iſtæ intentiones faciunt <lb/>pulchritudinem per coniunctionem illarum interſe:</s> <s xml:id="echoid-s3382" xml:space="preserve"> quoniam ſcriptura pulchra eſt illa, cum figu-<lb/>ræ literarum ſunt pulchræ, & compoſitio illarum inter ſe eſt compoſitio pulchra:</s> <s xml:id="echoid-s3383" xml:space="preserve"> quoniam ſcri-<lb/>ptura, in qua adunantur iſtæ duæ intentiones, eſt pulchrior ſcriptura, in qua eſt una iſtarum dua-<lb/>rum intentionum tantùm.</s> <s xml:id="echoid-s3384" xml:space="preserve"> Finis ergo pulchritudinis ſcripturæ non eſt, niſi ex coniugatione figu-<lb/>ræ & ſitus.</s> <s xml:id="echoid-s3385" xml:space="preserve"> Et ſimiliter quando colores ſcintillantes & picturæ fuerint ordinatæ ordinatione con-<lb/>ſimili, ſunt pulchriores coloribus & picturis carentibus ordinatione conſimili.</s> <s xml:id="echoid-s3386" xml:space="preserve"> Et ſimiliter pul-<lb/>chritudo apparet in forma hominum & animalium ex coniugatione intentionum particularium, <lb/>quæ ſunt in eis.</s> <s xml:id="echoid-s3387" xml:space="preserve"> Quoniam magnitudo oculorum mediocris cum figura eius amygdalata eſt pul-<lb/>chrior oculo, qui non habet, niſi magnitudinem tantùm aut figuram amygdalatam tantùm.</s> <s xml:id="echoid-s3388" xml:space="preserve"> Et ſi-<lb/>militer rotunditas faciei cum tenuitate & ſubtilitate cutis & coloris, eſt pulchrior quàm unum ſi-<lb/>ne altero.</s> <s xml:id="echoid-s3389" xml:space="preserve"> Et ſimiliter paruitas oris cum ſubtilitate labiorum & mediocritate, eſt pulchrior paruita-<lb/>te oris cum groſsitudine labiorũ:</s> <s xml:id="echoid-s3390" xml:space="preserve"> & pulchrior gracilitate labiorũ cũ amplitudine oris.</s> <s xml:id="echoid-s3391" xml:space="preserve"> Et iſta inten-<lb/>tio eſt multæ diuerſitatis, & multorũ modorũ.</s> <s xml:id="echoid-s3392" xml:space="preserve"> Et cum feceris inductionẽ in formis pulchris omniũ <lb/>modorũ uiſibiliũ:</s> <s xml:id="echoid-s3393" xml:space="preserve"> inuenies quòd coniunctio intentionũ particulariũ, quæ ſunt in formis, facit in eis <lb/>modos pulchritudinis, quosnõ facit una intentionũ per ſe.</s> <s xml:id="echoid-s3394" xml:space="preserve"> Et pulchritudo in maiori parte nõ fit, niſi <lb/>ex coniunctione iſtarum intentionũ inter ſe.</s> <s xml:id="echoid-s3395" xml:space="preserve"> Quoniã intentiones particulares, quas diximus, faciũt <lb/>pulchritudinẽ per ſe, & faciunt pulchritudinem per coniunctionẽ earum inter ſe.</s> <s xml:id="echoid-s3396" xml:space="preserve"> Et etiã pulchritu-<lb/>do fit ex alia intentione præter iſtas duas intentiones, quas prædiximus:</s> <s xml:id="echoid-s3397" xml:space="preserve"> & eſt proportionalitas & <lb/>conſonoritas.</s> <s xml:id="echoid-s3398" xml:space="preserve"> Quoniam formæ compoſitæ ex membris diuerſis, & partibus diuerſis, habent fi-<lb/>guras diuerſas, & magnitudines diuerſas, & ſitus diuerſos, & continuationem & coniunctionem, <lb/>& perueniunt in quamlibet illarum multæ intentiones particulares, tamen omnes non ſunt pro-<lb/>portionales.</s> <s xml:id="echoid-s3399" xml:space="preserve"> Quoniam non quælibet figura eſt pulchra cum qualibet figura:</s> <s xml:id="echoid-s3400" xml:space="preserve"> nec quælibet magni-<lb/>tudo eſt pulchra cum qualibet magnitudine:</s> <s xml:id="echoid-s3401" xml:space="preserve"> neque quilibet ſitus eſt pulcher cum quolibet ſitu:</s> <s xml:id="echoid-s3402" xml:space="preserve"> <lb/>neque quælibet figura cum qualibet magnitudine:</s> <s xml:id="echoid-s3403" xml:space="preserve"> neque quælibet magnitudo cum quolibet ſi-<lb/>tu.</s> <s xml:id="echoid-s3404" xml:space="preserve"> Sed quælibet intentionum particularium habet proportionem cum quibuſdam intentionibus, <lb/>& eſt aſymmetra quibuſdam.</s> <s xml:id="echoid-s3405" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s3406" xml:space="preserve"> Simitas naſi cum profunditate oculorum non eſt pul-<lb/>chra:</s> <s xml:id="echoid-s3407" xml:space="preserve"> & ſimiliter magnitudo naſi cum magnitudine oculorum non eſt pulchra:</s> <s xml:id="echoid-s3408" xml:space="preserve"> & ſimiliter pro-<lb/>minentia frontis cum profunditate oculorum non eſt pulchra:</s> <s xml:id="echoid-s3409" xml:space="preserve"> & ſimiliter frontis planities cum <lb/>prominentia oculorum non eſt pulchra.</s> <s xml:id="echoid-s3410" xml:space="preserve"> Quodlibet ergo membrorum habet figuram, quæ facit <lb/>formam eius pulchram:</s> <s xml:id="echoid-s3411" xml:space="preserve"> & etiam quælibet figura cuiuslibet membri non habet proportionem, ni-<lb/>ſi cum quibuſdam figuris reſiduorum membrorum, & cum alijs non.</s> <s xml:id="echoid-s3412" xml:space="preserve"> Et forma fit pulchra per con-<lb/>gregationem figurarum proportionalium:</s> <s xml:id="echoid-s3413" xml:space="preserve"> & ſimiliter magnitudines & ſitus, & ordinatio eorum.</s> <s xml:id="echoid-s3414" xml:space="preserve"> <lb/>Quoniam magnitudo oculorum cum pulchritudine figuræ eorum, & cum mediocritate ſimita-<lb/>tis naſi, & cum magnitudine proportionali ad magnitudinem oculorum, eſt pulchra.</s> <s xml:id="echoid-s3415" xml:space="preserve"> Et ſimili-<lb/>ter amygdalitas oculorum, & dulcẽdo, & tenuitas figuræ eius:</s> <s xml:id="echoid-s3416" xml:space="preserve"> & ſi fuerint parui cum ſubtilitate <lb/>naſi & mediocritate figuræ quantitatis eius, erunt pulchri.</s> <s xml:id="echoid-s3417" xml:space="preserve"> Et ſimiliter gracilitas labiorum cum <lb/>ſubtilitate oris eſt pulchra, quando gracilitas oris fuerit proportionalis ad gracilitatem labiorum:</s> <s xml:id="echoid-s3418" xml:space="preserve"> <lb/>ſcilicet quòd labia non ſint in fine gracilitatis, & os non in fine paruitatis, ſed erit paruitas oris me-<lb/>diocris, & labia gracilia, & præterea proportionalia ad quantitatem oris.</s> <s xml:id="echoid-s3419" xml:space="preserve"> Et ſimiliter amplitudo <lb/>faciei, quando fuerit proportionalis ad quantitates membrorum faciei, erit pulchra:</s> <s xml:id="echoid-s3420" xml:space="preserve"> ſcilicet, <lb/>quòd facies non ſit in fine amplitudinis, & membra faciei ſint proportionalia ad quantitatem to-<lb/>tius faciei.</s> <s xml:id="echoid-s3421" xml:space="preserve"> Quoniam quando facies fuerit ampla maximæ amplitudinis, & membra, quæ ſunt in <lb/>ea, ſunt parua, non proportionalia ad quantitatem eius:</s> <s xml:id="echoid-s3422" xml:space="preserve"> non erit facies pulchra, quamuis quan-<lb/>titates membrorum ſint proportionales, & figuræ eorum ſint pulchræ.</s> <s xml:id="echoid-s3423" xml:space="preserve"> Et ſimiliter quando fue-<lb/>rit parua facies, & ſtricta, & membra eius fuerint magna, membra dico faciei:</s> <s xml:id="echoid-s3424" xml:space="preserve"> erit facies turpis:</s> <s xml:id="echoid-s3425" xml:space="preserve"> <lb/>& cum membra fuerint proportionalia inter ſe, & proportionalia ad quantitatem amplitudinis <lb/>faciei:</s> <s xml:id="echoid-s3426" xml:space="preserve"> erit facies pulchra, quamuis membra per ſe non ſint pulchra:</s> <s xml:id="echoid-s3427" xml:space="preserve"> ſed proportionalitas tantum-<lb/>modò facit pulchritudinẽ.</s> <s xml:id="echoid-s3428" xml:space="preserve"> Cum ergo in forma congregabitur pulchritudo figuræ cuiuslibet partis <lb/>eius, erit pulchritudo quantitatis & compoſitionis, & proportionalitas membrorum ſecundum fi-<lb/>guras, & magnitudines, & ſitus:</s> <s xml:id="echoid-s3429" xml:space="preserve"> & fuerint præterea proportionalia ad totam figuram faciei & <lb/>quantitatem eius, erit in fine pulchritudinis.</s> <s xml:id="echoid-s3430" xml:space="preserve"> Et ſimiliter ſcriptura non erit pulchra, niſi quando <lb/>fuerint literæ eius proportionales in figura, & quantitate, & ſitu, & ordine.</s> <s xml:id="echoid-s3431" xml:space="preserve"> Et ſimiliter eſt cum o-<lb/>mnibus modis uiſibilium, cum quibus congregantur partes diuerſæ.</s> <s xml:id="echoid-s3432" xml:space="preserve"> Et cum conſideraueris for-<lb/>mas pulchras de omnibus modis uiſibilium:</s> <s xml:id="echoid-s3433" xml:space="preserve"> inuenies quòd proportionalitas facit pulchritudi-<lb/>nem magis, quàm aliqua alia intentio, uel etiam aliquæ coniunctæ per ſe.</s> <s xml:id="echoid-s3434" xml:space="preserve"> Et cum conſiderabun-<lb/>tur intentiones pulchræ, quas faciunt intentiones particulares per coniunctionem earum inter ſe:</s> <s xml:id="echoid-s3435" xml:space="preserve"> <lb/>inuenietur, quòd pulchritudo, quæ apparet ex coniunctione illarum inter ſe, non apparet, niſi <lb/> <pb o="66" file="0072" n="72" rhead="ALHAZEN"/> propter proportionalitatem illarum intentionum coniunctarum inter ſe.</s> <s xml:id="echoid-s3436" xml:space="preserve"> Quoniam non, quan-<lb/>docunque adunabuntur illæ intentiones, fit pulchritudo, ſed in quibuſdam formis fit, & in alijs <lb/>non.</s> <s xml:id="echoid-s3437" xml:space="preserve"> Et hoc eſt propter proportionalitatem, quæ contingit inter illas intentiones.</s> <s xml:id="echoid-s3438" xml:space="preserve"> Pulchritudo <lb/>ergo non eſt, niſi ex intentionibus particularibus, & perfectio eius non eſt, niſi ex proportiona-<lb/>litate & conſonantia, quæ fit inter intentiones particulares.</s> <s xml:id="echoid-s3439" xml:space="preserve"> Iam ergo declaratum eſt ex omni, <lb/>quod diximus, quòd formæ pulchræ comprehenſæ à uiſu non ſunt pulchræ, niſi ex intentionibus <lb/>particularibus, quæ comprehenduntur per ſenſum uiſus, & ex coniunctione earum inter ſe, & ex <lb/>proportionalitate earum inter ſe.</s> <s xml:id="echoid-s3440" xml:space="preserve"> Et uiſus comprehendit intentiones particulares prædictas ſim-<lb/>plices & compoſitas.</s> <s xml:id="echoid-s3441" xml:space="preserve"> Cum ergo uiſus comprehenderit aliquam rem uiſam, & fuerit aliqua inten-<lb/>tio in illa re uiſa particularis, faciens pulchritudinem per ſe aliquam:</s> <s xml:id="echoid-s3442" xml:space="preserve"> & intueatur uiſus illam in-<lb/>tentionem per ſe:</s> <s xml:id="echoid-s3443" xml:space="preserve"> perueniet forma illius intentionis poſt intuitum apud membrum ſentiens, & <lb/>comprehendet uirtus diſtinctiua pulchritudinem rei uiſæ, in qua eſt illa intentio.</s> <s xml:id="echoid-s3444" xml:space="preserve"> Quoniam ue-<lb/>ro forma cuiuslibet rei uiſę eſt compoſita ex multis intentionibus earum intentionum, quarum di-<lb/>uiſionem prædiximus:</s> <s xml:id="echoid-s3445" xml:space="preserve"> cum ergo uiſus comprehenderit rem uiſam, & non diſtinxerit intentio-<lb/>nes, quæ ſuntin ea:</s> <s xml:id="echoid-s3446" xml:space="preserve"> non comprehendet pulchritudinem eius:</s> <s xml:id="echoid-s3447" xml:space="preserve"> & cum diſtinxerit intentiones, quæ <lb/>ſunt in ea, & fuerit aliqua intentio earum, quæ ſunt in ea, ſecundum modum facientem pulchri-<lb/>tudinem in anima:</s> <s xml:id="echoid-s3448" xml:space="preserve"> ſtatim uiſus apud intuitionem illius intention is comprehendet illam intentio-<lb/>nem per ſe.</s> <s xml:id="echoid-s3449" xml:space="preserve"> Et cum comprehenderit illam intentionem perſe:</s> <s xml:id="echoid-s3450" xml:space="preserve"> perueniet illa comprehenſio apud <lb/>membrum ſentiens:</s> <s xml:id="echoid-s3451" xml:space="preserve"> & ſic uirtus diſtinctiua comprehendet pulchritudinem, quæ eſt in ea:</s> <s xml:id="echoid-s3452" xml:space="preserve"> & per <lb/>iſtam comprehenſionem comprehendet pulchritudinem illius rei uiſæ.</s> <s xml:id="echoid-s3453" xml:space="preserve"> Cum ergo uiſus compre-<lb/>henderit aliquam rem uiſam, & in illa re uiſa fuerit pulchritudo compoſita ex intentionibus coniun <lb/>ctis:</s> <s xml:id="echoid-s3454" xml:space="preserve"> & fuerit uiſus intuitus illam rem uiſam:</s> <s xml:id="echoid-s3455" xml:space="preserve"> & diſtinxerit intentiones, quæ ſunt in ea:</s> <s xml:id="echoid-s3456" xml:space="preserve"> & compre-<lb/>henderit intentiones, quæ faciunt pulchritudinem per coniunctionem earum inter ſe, aut propor-<lb/>tionalitatem earum inter ſe:</s> <s xml:id="echoid-s3457" xml:space="preserve"> & peruenerit illa comprehenſio apud membrum ſentiens:</s> <s xml:id="echoid-s3458" xml:space="preserve"> & compa-<lb/>rauerit uirtus diſtinctiua illas intentiones inter ſe:</s> <s xml:id="echoid-s3459" xml:space="preserve"> comprehendet pulchritudinem illius rei uiſæ <lb/>compoſitam ex coniunctione intentionum, quæ ſunt in ea.</s> <s xml:id="echoid-s3460" xml:space="preserve"> Viſus ergo comprehendet pulchritu-<lb/>dinem, quæ eſt in uiſibilibus ex compoſitione illarum intentionum inter ſe ſecundum modum, <lb/>quem declarauimus.</s> <s xml:id="echoid-s3461" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div116" type="section" level="0" n="0"> <head xml:id="echoid-head138" xml:space="preserve" style="it">60. Deformitas percipitur tum è ſingulis uiſibilibus ſpeciebus, tum è pluribus ſimul coniun-<lb/>ctis, aſymmetris inter ſe. 149 p 4.</head> <p> <s xml:id="echoid-s3462" xml:space="preserve">TVrpitudo uerò eſt forma carens intentione qualibet pulchra.</s> <s xml:id="echoid-s3463" xml:space="preserve"> Quoniam enim iam prædictum <lb/>eſt, quòd intentiones particulares faciunt pulchritudinem, ſed non in omnibus locis, ne<gap/> <lb/>in omnibus formis, ſed in alijs, & in alijs non:</s> <s xml:id="echoid-s3464" xml:space="preserve"> & ſimiliter proportionalitas non eſt in omni-<lb/>bus formis, ſed in quibuſdam formis, & in quibuſdam non.</s> <s xml:id="echoid-s3465" xml:space="preserve"> Formæ ergo, in quibus non faciunt in-<lb/>tentiones particulares pulchritudinem aliquam perſe, nec per ſuam coniunctionem, & in quibus <lb/>non eſt aliqua proportionalitas inter partes earum, carent omni pulchritudine:</s> <s xml:id="echoid-s3466" xml:space="preserve"> & ſic ſunt turpes:</s> <s xml:id="echoid-s3467" xml:space="preserve"> <lb/>quoniam turpitudo formarum eſt priuatio pulchritudinis in eis.</s> <s xml:id="echoid-s3468" xml:space="preserve"> Et fortè aggregantur in eadem <lb/>forma intentiones pulchræ & turpes:</s> <s xml:id="echoid-s3469" xml:space="preserve"> ſed uiſus comprehendit pulchritudinem ex pulchro, & tur-<lb/>pitudinem exturpi, quando diſtinxerit, & fuerit intuitus intentiones quæ ſunt in ea.</s> <s xml:id="echoid-s3470" xml:space="preserve"> Turpitudo er-<lb/>go comprehenditur à uiſu in formis carentibus omnibus pulchritudinibus, ex priuatione pulchri-<lb/>tudinis apud comprehenſionem.</s> <s xml:id="echoid-s3471" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div117" type="section" level="0" n="0"> <head xml:id="echoid-head139" xml:space="preserve" style="it">61. Similitudo percipitur è uiſibilium inter ſe conuenientia. 151 p 4.</head> <p> <s xml:id="echoid-s3472" xml:space="preserve">COnſimilitudo autem eſt æqualitas duarum formarum aut duarum intentionum in re, in qua <lb/>ſunt conſimiles.</s> <s xml:id="echoid-s3473" xml:space="preserve"> Cum ergo uiſus comprehenderit duas formas, aut duas intentiones conſi-<lb/>miles:</s> <s xml:id="echoid-s3474" xml:space="preserve"> ſimul comprehendet conſimilitudinem ex illarum comprehenſione cuiuslibet dua-<lb/>rum formarum, uel intentionum, & ex compcratione alterius illarum ad alteram.</s> <s xml:id="echoid-s3475" xml:space="preserve"> Viſus ergo com-<lb/>prehendit conſimilitudinem in formis, uel intentionibus conſimilibus ex comprehenſione cuiusli <lb/>bet formarum uel intentionum ſecundum ſuum eſſe, & ex comparatione illarum inter ſe.</s> <s xml:id="echoid-s3476" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div118" type="section" level="0" n="0"> <head xml:id="echoid-head140" xml:space="preserve" style="it">62. Dißimilitudo percipitur è priuatione ſimilitudinis & conuenientiæ uiſibilium inter <lb/>ſe. 152 p 4.</head> <p> <s xml:id="echoid-s3477" xml:space="preserve">DIuerſitas autem comprehenditur à uiſu in formis diuerſis ex comprehenſione cuiuslibet for <lb/>marum diuerſarum, & ex comparatione alterius illarum ad alterã, & ex comprehenſione pri <lb/>uationis æqualitatis, id eſt conſimilitudinis in eis.</s> <s xml:id="echoid-s3478" xml:space="preserve"> Diuerſitas ergo comprehenditur per ſen-<lb/>ſum uiſus ex comprehenſione cuiuslibet formarum & intentionum per ſe, & ex comparatione ea-<lb/>rum inter ſe, & ex ſenſu priuationis æqualitatis à ſentiente.</s> <s xml:id="echoid-s3479" xml:space="preserve"> Iam ergo cõpleuimus, & declarauimus <lb/>declarationẽ qualitatis cõprehenſionis cuiuslibet intentionũ particulariũ, quæ comprehenduntur <lb/>per ſenſum uiſus.</s> <s xml:id="echoid-s3480" xml:space="preserve"> Et declaratũ eſt ex omnibus his, quòd quædã intentiones particulares cõprehen-<lb/>duntur ſolo ſenſu:</s> <s xml:id="echoid-s3481" xml:space="preserve"> & quædã cõprehenduntur per cognitionem:</s> <s xml:id="echoid-s3482" xml:space="preserve"> & quædam per argumentationẽ & <lb/> <pb o="67" file="0073" n="73" rhead="OPTICAE LIBER II."/> ſignificationem ſecundum uias, quarum declarationem prædiximus.</s> <s xml:id="echoid-s3483" xml:space="preserve"> Et iſtæ ſunt intentiones, qua-<lb/>rum declarationem intendimus in hoc opere.</s> <s xml:id="echoid-s3484" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div119" type="section" level="0" n="0"> <head xml:id="echoid-head141" xml:space="preserve">DE DIVERSITATE COMPREHENSIONIS VISVS AB</head> <head xml:id="echoid-head142" xml:space="preserve">intentionibus particularibus. Cap. III.</head> <head xml:id="echoid-head143" xml:space="preserve" style="it">63. Viſus plures uiſibiles ſpecies ſimul percipit. 2 p 4.</head> <p> <s xml:id="echoid-s3485" xml:space="preserve">IAm declaratum eſt, quomodo uiſus comprehẽdat quamlibet intentionum particularium, quæ <lb/>comprehenduntur per ſenſum uiſus.</s> <s xml:id="echoid-s3486" xml:space="preserve"> Et uiſus non comprehendit niſi formas uiſibilium, quæ <lb/>ſunt corpora:</s> <s xml:id="echoid-s3487" xml:space="preserve"> ſed formæ uiſibilium ſunt compoſitæ exintentionibus particularibus prædictis, <lb/>ſicut, figura, & magnitudine, & colore, & ſitu, & ordine, & ſimilibus.</s> <s xml:id="echoid-s3488" xml:space="preserve"> Viſus ergo non comprehendit <lb/>quamlibet intentionum, niſi ex comprehenſione formarum uiſibilium compoſitarum ex intentio-<lb/>nibus particularibus:</s> <s xml:id="echoid-s3489" xml:space="preserve"> & uiſus comprehendit quamlibet formarum uiſibilium ſecundum intentio-<lb/>nes particulares, quæ ſunt in formis uiſibilibus:</s> <s xml:id="echoid-s3490" xml:space="preserve"> & nihil comprehendit uiſus ex intentionibus par-<lb/>ticularibus per ſe:</s> <s xml:id="echoid-s3491" xml:space="preserve"> quoniam nulla intentionum prædictarum eſt ſola per ſe.</s> <s xml:id="echoid-s3492" xml:space="preserve"> Nam omnes iſtæ parti-<lb/>culares intentiones non inueniũtur, niſi in corporibus, & nullum corpus eſt, in quo eſt aliqua iſta-<lb/>rum intentionum ſola ſine alia.</s> <s xml:id="echoid-s3493" xml:space="preserve"> Viſus ergo non comprehendit niſi formas uiſibilium:</s> <s xml:id="echoid-s3494" xml:space="preserve"> ſed quælibet <lb/>forma uiſibilium eſt compoſita ex multis intentionibus particularibus.</s> <s xml:id="echoid-s3495" xml:space="preserve"> Ergo uiſus comprehendit <lb/>in qualibet formarum uiſibilium multas intentiones particulares, quæ diſtinguuntur in imagina-<lb/>tione.</s> <s xml:id="echoid-s3496" xml:space="preserve"> Viſus ergo comprehendit quamlibet intentionũ particularium apud uiſionem rei uiſæ, con-<lb/>iunctam cum intentione aliqua particulari:</s> <s xml:id="echoid-s3497" xml:space="preserve"> deinde ex diſtinctione eius inter intentiones, quæ ſunt <lb/>in forma, comprehendit quamlibet intentionum per ſe.</s> <s xml:id="echoid-s3498" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div120" type="section" level="0" n="0"> <head xml:id="echoid-head144" xml:space="preserve" style="it">64. Viſio fit aſpectu, aut obtutu. 51 p 3.</head> <p> <s xml:id="echoid-s3499" xml:space="preserve">ET iam declaratum eſt, & determinatum, qualiter uiſus comprehẽdat formas uiſibilium, quæ <lb/>componuntur ex intentionibus particularibus.</s> <s xml:id="echoid-s3500" xml:space="preserve"> Et quędam intentiones particulares, ex qui-<lb/>bus componuntur formæ uiſibilium, apparent apud aſpectum rei uiſæ:</s> <s xml:id="echoid-s3501" xml:space="preserve"> & quædam non ap-<lb/>parent, niſi poſt intuitionem & conſiderationem ſubtilem:</s> <s xml:id="echoid-s3502" xml:space="preserve"> ſicut ſcriptura ſubtilis, & lineatio ſubti-<lb/>lis, & diuerſitas colorum conſimilium ferè.</s> <s xml:id="echoid-s3503" xml:space="preserve"> Et genéraliter omnes intentiones ſubtiles non appa-<lb/>rent uiſui apud aſpectum rei uiſæ, ſed poſt intuitionem & conſiderationem.</s> <s xml:id="echoid-s3504" xml:space="preserve"> Et forma rei uiſæ com-<lb/>prehenſa per ſenſum uiſus, eſt illa quæ componitur ex omnibus intentionibus particularibus, quæ <lb/>ſunt ex forma rei uiſæ, quas poſsibile eſt uiſum comprehendere.</s> <s xml:id="echoid-s3505" xml:space="preserve"> Et uiſus non comprehendit ue-<lb/>ram formam rei uiſæ, niſi per comprehenſionem omnium intentionum particularium, quæ ſunt in <lb/>forma rei uiſæ.</s> <s xml:id="echoid-s3506" xml:space="preserve"> Et cum ita ſit, forma ergo uera rei uiſæ, in qua ſunt intentiones ſubtiles, non com-<lb/>prehenditura à uiſu, niſi poſt intuitionem.</s> <s xml:id="echoid-s3507" xml:space="preserve"> Et cum uiſus nõ comprehendat ſubtiles intentiones, niſi <lb/>per intuitionem, & non appareant intentiones ſubtiles uiſui apud aſpectum rei uiſæ:</s> <s xml:id="echoid-s3508" xml:space="preserve"> quando igi-<lb/>tur uiſus comprehenderit aliquam rem uiſam, & comprehenderit formam eius, & fuerint in illa re <lb/>uiſa ſubtiles intentiones, non apparentillæ per aſpectũ, ſed per intuitionem.</s> <s xml:id="echoid-s3509" xml:space="preserve"> Cum ergo uiſus com-<lb/>prehenderit aliquam rem uiſam, & non fuerit in ea aliqua intentio ſubtilis:</s> <s xml:id="echoid-s3510" xml:space="preserve"> comprehendet ueram <lb/>eius formam:</s> <s xml:id="echoid-s3511" xml:space="preserve"> quamuis non certificabit, quod illa forma eſt uera, niſi poſtquam habuerit fortem in-<lb/>tuitionem ſuper quamlibet partem illius rei uiſæ, & certificauerit, quòd nulla intentio ſubtilis eſt <lb/>in ea, & tunc certificabit quod forma, quam comprehendit, eſt uera forma.</s> <s xml:id="echoid-s3512" xml:space="preserve"> Secundum ergo omnes <lb/>diſpoſitiones non certificat uiſus formam rei uiſæ, niſi per conſiderationem omnium partium rei <lb/>uiſæ, & per intuitionem omnium partium, quæ poſſunt apparere in re uiſa.</s> <s xml:id="echoid-s3513" xml:space="preserve"> Et quia hoc eſt decla-<lb/>ratum, dicamus quòd comprehenſio uiſibilium erit ſecundum duos modos, quiſunt comprehen-<lb/>ſio ſuperficialis, & comprehenſio per intuitionem, quæ profundum aſpicit.</s> <s xml:id="echoid-s3514" xml:space="preserve"> Quoniam quando ui-<lb/>ſus aſpicit rem uiſam, comprehendit intentiones manifeſtas, quæ ſunt in ea apud aſpectum:</s> <s xml:id="echoid-s3515" xml:space="preserve"> dein-<lb/>de ſi præter illud inſpexerit ipſam, & conſiderauerit omnes partes eius, certificabit formam eius:</s> <s xml:id="echoid-s3516" xml:space="preserve"> ſi <lb/>autem non intuetur partes eius, non comprehendet formam certificatam.</s> <s xml:id="echoid-s3517" xml:space="preserve"> Et illa forma, quæ eſt in <lb/>uiſu, aut erit uera eius forma, ſed uiſus non certificat, quòd ſit uera eius forma:</s> <s xml:id="echoid-s3518" xml:space="preserve"> aut non erit forma <lb/>eius uera.</s> <s xml:id="echoid-s3519" xml:space="preserve"> Et cum ita ſit, comprehenſio ergo uiſibilium erit ſecundum duos modos:</s> <s xml:id="echoid-s3520" xml:space="preserve"> & eſt compre-<lb/>henſio ſuperficialis, quæ eſt in primo aſpectu, & comprehenſio, quæ eſt per intuitionem.</s> <s xml:id="echoid-s3521" xml:space="preserve"> Compre-<lb/>henſio autem per primum aſpectum, eſt comprehenſio non certificata:</s> <s xml:id="echoid-s3522" xml:space="preserve"> & comprehenſio per intui-<lb/>tionem, eſt comprehenſio, per quam certificantur formæ uiſibilium.</s> <s xml:id="echoid-s3523" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div121" type="section" level="0" n="0"> <head xml:id="echoid-head145" xml:space="preserve" style="it">65. Viſio per aſpectum, fit per quemlibet pyramidis opticæ radium: per obtutum uerò fit per <lb/>ſolum axem. 52 p 3.</head> <p> <s xml:id="echoid-s3524" xml:space="preserve">ET cum hoc declaratum ſit, dicamus quòd intuitio, per quam comprehenduntur ueræ formæ <lb/>uiſibilium, erit per ipſum uiſum, & erit per diſtinctionẽ.</s> <s xml:id="echoid-s3525" xml:space="preserve"> Quoniam iam declaratũ eſt in diſtin-<lb/>ctione linearum radialium [8 n] quòd formæ, quæ à uiſu comprehenduntur ex axe radiali, <lb/>& exillo, qui eſt prope axem, ſunt manifeſtiores, & maioris certificationis, formis, quæ compre-<lb/> <pb o="68" file="0074" n="74" rhead="ALHAZEN"/> henduntur ex reſiduis uerticationibus.</s> <s xml:id="echoid-s3526" xml:space="preserve"> Cum ergo uiſus fuerit oppoſitus alicui rei uiſæ:</s> <s xml:id="echoid-s3527" xml:space="preserve"> & illa res-<lb/>uiſa non fuerit in fine paruitatis, ſed alicuius quantitatis:</s> <s xml:id="echoid-s3528" xml:space="preserve"> & uiſus fuerit fixus in oppoſitione eius <lb/>apud aſpectum:</s> <s xml:id="echoid-s3529" xml:space="preserve"> illud, quod opponitur medio uiſus ex illa re uiſa, & fuerit ſuper axem aut prope <lb/>axem:</s> <s xml:id="echoid-s3530" xml:space="preserve"> erit manifeſtius partibus reſiduis rei uiſæ:</s> <s xml:id="echoid-s3531" xml:space="preserve"> & uiſus percipit iſtam diſpoſitionem.</s> <s xml:id="echoid-s3532" xml:space="preserve"> Quoniam <lb/>quando comprehenderit rem uiſam totam:</s> <s xml:id="echoid-s3533" xml:space="preserve"> inueniet locum oppoſitum medio eius, cuius forma <lb/>peruenit in medium uiſus, eſſe manifeſtiorem partibus reſiduis.</s> <s xml:id="echoid-s3534" xml:space="preserve"> Et ſuperius declaratum eſt, quòd <lb/>iſta intentio apparet ſenſui, quando res uiſa fuerit magnæ quantitatis.</s> <s xml:id="echoid-s3535" xml:space="preserve"> Cum ergo uiſus comprehen <lb/>derit totam rem uiſam:</s> <s xml:id="echoid-s3536" xml:space="preserve"> inueniet, quòd forma partis oppoſitæ medio eius, eſt manifeſtior omni-<lb/>bus partibus reſiduis.</s> <s xml:id="echoid-s3537" xml:space="preserve"> Et cum uoluerit certificare formam rei uiſæ, mouebitur, ita ut medium eius <lb/>ſit oppoſitum cuilibet parti partium rei uiſæ:</s> <s xml:id="echoid-s3538" xml:space="preserve"> & ſic comprehendet formam cuiuslibet partis par-<lb/>tium rei uiſæ, comprehenſione manifeſta & certificata, ſicut comprehendit partem oppoſitam me-<lb/>dio eius apud aſpectum rei uiſæ.</s> <s xml:id="echoid-s3539" xml:space="preserve"> Cum igitur ſentiens uoluerit certificare rem uiſam:</s> <s xml:id="echoid-s3540" xml:space="preserve"> mouebitur ui-<lb/>ſus ita, ut ſit medium eius oppoſitum cuilibet parti partium rei uiſæ.</s> <s xml:id="echoid-s3541" xml:space="preserve"> Et per iſtum modum compre-<lb/>hendet formam cuiuslibet partium rei uiſæ ualde manifeſtè:</s> <s xml:id="echoid-s3542" xml:space="preserve"> & uirtus diſtinctiua diſtinguet omnes <lb/>formas uenientes ad ipſam, & diſtinguet colores partium, & diuerſitatem colorum, & ordinatio-<lb/>nem partium interſe.</s> <s xml:id="echoid-s3543" xml:space="preserve"> Et generaliter diſtinguet omnes intentiones rei uiſæ, quæ apparent per in-<lb/>tuitum, & formam totius rei uiſæ comp oſitam ex illis intentionibus.</s> <s xml:id="echoid-s3544" xml:space="preserve"> Secundum ergo hunc mo-<lb/>dum erit certificatio cuiuslibet partium rei uiſæ ſecundum ſuum eſſe, & certificatio omnium in-<lb/>tentionum rei uiſæ.</s> <s xml:id="echoid-s3545" xml:space="preserve"> Et non certificatur forma cuiuslibet partium rei uiſæ, niſi poſt motum uiſus <lb/>ſuper omnes partes.</s> <s xml:id="echoid-s3546" xml:space="preserve"> Et præterea natus eſt uiſus ad motum intuitionis, & ad faciendum axem ra-<lb/>dialem tranſire ſuper omnes partes rei uiſæ.</s> <s xml:id="echoid-s3547" xml:space="preserve"> Cum ergo uirtus diſtinctiua quæſierit intueri rem ui-<lb/>ſam:</s> <s xml:id="echoid-s3548" xml:space="preserve"> mouebitur axis radialis ſuper omnes partes rei uiſæ.</s> <s xml:id="echoid-s3549" xml:space="preserve"> Et cum intentiones ſubtiles, quæ ſunt in <lb/>illa re uiſa, non appareant, niſi per motum uiſus, & per tranſitum axis, aut linearum radialium, <lb/>quæ ſunt prope ipſum ſuper quamlibet partium rei uiſæ:</s> <s xml:id="echoid-s3550" xml:space="preserve"> non perueniet forma rei uiſæ certifica-<lb/>ta ad ſentientem, quando corpus eius fuerit alicuius quantitatis, niſi per motum uiſus, & per op-<lb/>poſitionem cuiuslibet partium rei uiſæ, medio uiſus.</s> <s xml:id="echoid-s3551" xml:space="preserve"> Et etiam quando res uiſa fuerit in fine par-<lb/>uitatis, & non fuerit oppoſita medio uiſus:</s> <s xml:id="echoid-s3552" xml:space="preserve"> etiam nõ complebitur intuitio eius, niſi poſtquam mo-<lb/>tus fuerit uiſus, donec axis trãſeat in illam rem uiſam, & perueniat forma illius rei uiſæ in medium <lb/>uiſus, & appareat forma rei uiſæ.</s> <s xml:id="echoid-s3553" xml:space="preserve"> Et cum ita ſit, intuitio, per quam uiſus comprehendit ueras for-<lb/>mas uiſibilium, fortè erit per ipſum uiſum & fortè per diſtinctionem ſimul.</s> <s xml:id="echoid-s3554" xml:space="preserve"> Comprehẽſio ergo for-<lb/>mæ ueræ rei uiſæ non erit, niſi per intuitionem:</s> <s xml:id="echoid-s3555" xml:space="preserve"> & intuitio, per quam certificabitur forma rei uiſæ, <lb/>non complebitur, niſi per motum uiſus.</s> <s xml:id="echoid-s3556" xml:space="preserve"> Et cum corpus rei uiſæ fuerit alicuius quantitatis, non <lb/>complebitur intuitio eius, niſi per motum axis radialis in omnes diametros rei uiſæ.</s> <s xml:id="echoid-s3557" xml:space="preserve"> Et iſtam in-<lb/>tentionem uoluit dicere ille, qui opinabatur, quòd uiſio non fieret niſi per motum:</s> <s xml:id="echoid-s3558" xml:space="preserve"> & quòd nulla <lb/>res uiſa uideretur tota ſimul.</s> <s xml:id="echoid-s3559" xml:space="preserve"> Quoniam ipſe intendebat dicere uiſionem certificatam, quæ non po-<lb/>teſt eſſe, niſi per intuitionem, & per motum uiſus, & per motum axis radialis ſuper omnes diame-<lb/>tros rei uiſæ.</s> <s xml:id="echoid-s3560" xml:space="preserve"> Quomodo uerò ſentiens certificet per intuitionem & per motum, formam rei uiſæ, <lb/>eſt:</s> <s xml:id="echoid-s3561" xml:space="preserve"> quia quando uiſus fuerit oppoſitus rei uiſæ, ſentiens comprehendet totam formam apud op-<lb/>poſitionem comprehenſione qualicunque, & comprehendet partem, quæ eſt apud extremum axis <lb/>uera comprehenſione in fine ueritatis:</s> <s xml:id="echoid-s3562" xml:space="preserve"> & etiam tunc quamlibet partem reſiduarum partium for-<lb/>mæ aliqua comprehenſione.</s> <s xml:id="echoid-s3563" xml:space="preserve"> Deinde quando uiſus mouebitur, & mutabitur axis à parte, in qua <lb/>erat, ad aliam partem:</s> <s xml:id="echoid-s3564" xml:space="preserve"> comprehendet ſentiens in iſta diſpoſitione formam totius rei uiſæ ſecunda <lb/>comprehenſione, & comprehendet partem, quæ eſt apud extremum axis ſecunda comprehenſio-<lb/>ne etiam.</s> <s xml:id="echoid-s3565" xml:space="preserve"> Et erit comprehenſio iſtius partis, quæ eſt apud extremum axis, in ſecunda diſpoſitio-<lb/>ne, manifeſtior comprehenſione eius in prima diſpoſitione.</s> <s xml:id="echoid-s3566" xml:space="preserve"> Et in iſta diſpoſitione etiam ſentiens <lb/>comprehendet partes reſiduas aliqua comprehenſione.</s> <s xml:id="echoid-s3567" xml:space="preserve"> Et ſimiliter, quando axis mutabitur per <lb/>motum ad tertiam partem, comprehendet ſentiens in tertia diſpoſitione totam rem uiſam tertia <lb/>comprehenſione, & comprehendet partem, quæ eſt apud extremitatem axis tertia comprehenſio-<lb/>ne etiam.</s> <s xml:id="echoid-s3568" xml:space="preserve"> Et erit comprehenſio iſtius partis ab eo in iſta diſpoſitione manifeſtior comprehenſio-<lb/>ne in duabus primis diſpoſitionibus:</s> <s xml:id="echoid-s3569" xml:space="preserve"> & tunc ſentiens comprehendet in iſta diſpoſitione etiam <lb/>quamlibet partium reſiduarum aliqua comprehenſione.</s> <s xml:id="echoid-s3570" xml:space="preserve"> Per motum ergo uiſus ſuper partes rei ui-<lb/>ſæ acquirit ſentiens duas diſpoſitiones:</s> <s xml:id="echoid-s3571" xml:space="preserve"> quarum altera eſt frequentatio comprehenſionis totius <lb/>rei uiſæ, & ſecunda eſt, quæ comprehendit quamlibet partium rei uiſæ per axem radialem, aut per <lb/>illud, quod eſt prope axem radialem, manifeſta comprehenſione.</s> <s xml:id="echoid-s3572" xml:space="preserve"> Apparet ergo ſenſui omne, quod <lb/>eſt poſsibile apparere exillis partibus.</s> <s xml:id="echoid-s3573" xml:space="preserve"> Et cum ſentiens ſæpe comprehenderit rem uiſam totam, & <lb/>quamlibet partium rei uiſæ:</s> <s xml:id="echoid-s3574" xml:space="preserve"> comprehendet per iſtam diſpoſitionem omne, quod eſt poſsibile com <lb/>prehendi ab illa re uiſa.</s> <s xml:id="echoid-s3575" xml:space="preserve"> Et cum hac comprehenſione multoties iterata in duplicationibus & itera-<lb/>tionibus comprehenſionis totius rei uiſæ, diſtinguit uirtus diſtinctiua illud, quod apparet ex colo-<lb/>ribus partium, & luce, & magnitudine, & remotione, & figura, & ſitu earum, & æqualitate illarum, <lb/>quæ ſunt cõſimiles in iſtis diſtinctionibus, & diuerſitate earum, quæ ſunt diuerſæ in omnibus iſtis <lb/>intentionibus aut in quibuſdá, & ex ordine partiũ inter ſe:</s> <s xml:id="echoid-s3576" xml:space="preserve"> & comprehendit ex diſtinctione omniũ <lb/>iſtarum intentionũ ad ea, quæ cognoſcuntur ex ſimilibus earũ, formam compoſitam ex omnibus:</s> <s xml:id="echoid-s3577" xml:space="preserve"> <lb/>& ſic ſignatur in imaginatione forma compoſita ex omnibus iſtis intentionibus:</s> <s xml:id="echoid-s3578" xml:space="preserve"> & ſic certificatur <lb/> <pb o="69" file="0075" n="75" rhead="OPTICAE LIBER II."/> forma rei uiſæ, per quam appropriatur illa res uiſa apud ſentientem.</s> <s xml:id="echoid-s3579" xml:space="preserve"> Secundum ergo hũc modum <lb/>certificat ſentiens per intuitionem formas uiſibilium.</s> <s xml:id="echoid-s3580" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div122" type="section" level="0" n="0"> <head xml:id="echoid-head146" xml:space="preserve" style="it">66. Obtut{us} iteratio alti{us} imprimit formas uiſibiles animo, certioreś efficit. 58 p 3.</head> <p> <s xml:id="echoid-s3581" xml:space="preserve">ET etiam dicamus, quòd quando uiſus comprehenderit aliquam rem uiſam, & fuerit certifi-<lb/>cata forma eius apud ſentientem:</s> <s xml:id="echoid-s3582" xml:space="preserve"> forma illius rei uiſæ remanet in anima:</s> <s xml:id="echoid-s3583" xml:space="preserve"> & figuratur in ima-<lb/>ginatione, & iteratur comprehẽſio rei uiſæ, & erit forma eius magis fixa in anima, quàm for-<lb/>ma rei uiſæ, quam uiſus non comprehendit, niſi ſemel autrarò.</s> <s xml:id="echoid-s3584" xml:space="preserve"> Et quòd uiſus quando comprehen-<lb/>derit aliquod indiuiduũ:</s> <s xml:id="echoid-s3585" xml:space="preserve"> deinde comprehenderit alia indiuidua eiuſmodi indiuidui, & iterata fue-<lb/>rit comprehenſio in diuiduorum frequenter:</s> <s xml:id="echoid-s3586" xml:space="preserve"> quieſcet forma illiuſmodi in anima, & perueniet for-<lb/>ma uniuerſaliter figurata in imaginatione.</s> <s xml:id="echoid-s3587" xml:space="preserve"> Et ſignificatio ſuper hoc, quòd formæ uiſibilium rema-<lb/>neant in anima & in imaginatione, eſt:</s> <s xml:id="echoid-s3588" xml:space="preserve"> Quia homo, quãdo meminerit de aliquo homine, quem co-<lb/>gnouit antè, & certificauerit formam eius, & meminerit tempus, in quo uidit illum hominẽ, & lo-<lb/>cum uera memoratione:</s> <s xml:id="echoid-s3589" xml:space="preserve"> ſtatim imaginabitur formã illius hominis, & figuram faciei eius, & ſitum <lb/>illius, in quo erat in illo tempore, & imaginabitur locum, in quo uidit ipſum:</s> <s xml:id="echoid-s3590" xml:space="preserve"> & fortè imaginabitur <lb/>alia uiſibilia, quæ fuerunt præſentia in illo loco, quando uidit ipſum.</s> <s xml:id="echoid-s3591" xml:space="preserve"> Et hæc eſt ſignificatio manife-<lb/>ſta, quòd forma illius hominis & forma illius loci ſunt fixæ in anima, & remanent in imaginatione.</s> <s xml:id="echoid-s3592" xml:space="preserve"> <lb/>Et propterhoc, quando homo meminerit de aliqua ciuitate, quam uidit, imaginabitur formam il-<lb/>lius ciuitatis, & formam locorum, in quibus fuit in illa ciuitate, & formas indiuiduorum, quæ co-<lb/>gnouit in illa ciuitate.</s> <s xml:id="echoid-s3593" xml:space="preserve"> Et ſimiliter omnium, quæ uidit ex uiſibilibus, quando ei occurrunt ad me-<lb/>moriam:</s> <s xml:id="echoid-s3594" xml:space="preserve"> imag<gap/>nabitur formas ſecundum modum & eſſe, ut percepit ea antea.</s> <s xml:id="echoid-s3595" xml:space="preserve"> Imaginatio ergo for-<lb/>marum uiſibilium, quas homo antè uidit, & modò ſciuerit, cum ſunt abſentes:</s> <s xml:id="echoid-s3596" xml:space="preserve"> eſt ſignificatio, quòd <lb/>formæ uiſibilium, quas comprehendit, perueniunt in animam, & figurãtur in imaginatione.</s> <s xml:id="echoid-s3597" xml:space="preserve"> Quòd <lb/>uerò forma rei, cuius comprehenſio iterabitur à uiſu, ſit magis fixa in anima & in imaginatione, <lb/>quàm forma rei uiſæ, cuius comprehenſio non iterabitur, eſt:</s> <s xml:id="echoid-s3598" xml:space="preserve"> quia, quando ad animam peruenit ali-<lb/>qua intentio, ſtatim perueniet forma illius intentionis in animam.</s> <s xml:id="echoid-s3599" xml:space="preserve"> Et cum tempus pertranſierit, & <lb/>intra multum tempus nõ redierit iterum ad animam:</s> <s xml:id="echoid-s3600" xml:space="preserve"> fortè tradetur illa intentio obliuioni, aut ali-<lb/>qua intentionum, quæ ſunt in illa intentione:</s> <s xml:id="echoid-s3601" xml:space="preserve"> & ſi redierit ad animam ante obliuionem, renouatur <lb/>forma illius in anima, & rememorabit anima per formam ſecũdam, formam primam.</s> <s xml:id="echoid-s3602" xml:space="preserve"> Et cum mul-<lb/>toties iterabitur euentus illius intentionis ſuper animam, anima magis meminerit de illa intentio-<lb/>ne:</s> <s xml:id="echoid-s3603" xml:space="preserve"> & ſic erit illa intentio magis fixa in anima.</s> <s xml:id="echoid-s3604" xml:space="preserve"> Et etiã prima uice, in qua intentio uenit ad animam, <lb/>aut in qua forma rei uiſæ uenit ad animam, fortè anima non comprehẽdet omnes intentiones, quæ <lb/>ſunt in illa forma, neque certificabit ipſas, ſed comprehendet tantùm quaſdã intentiones, quæ ſunt <lb/>in ea.</s> <s xml:id="echoid-s3605" xml:space="preserve"> Et cum forma redierit ſecundò, comprehendet anima ex ea aliquid, quod in prima uice non <lb/>comprehendit:</s> <s xml:id="echoid-s3606" xml:space="preserve"> & quantò magis iterabitur forma ſuper animam, tantò magis manifeſtabitur ex ea, <lb/>quod prius non apparebat.</s> <s xml:id="echoid-s3607" xml:space="preserve"> Et cum anima comprehenderit ex forma intentiones ſubtiles eius, & <lb/>certificauerit formam eius:</s> <s xml:id="echoid-s3608" xml:space="preserve"> erit magis fixa in anima, & in imaginatione, quàm forma, ex qua non <lb/>uerè comprehendit mens omnes intentiones, quæ ſunt in ea.</s> <s xml:id="echoid-s3609" xml:space="preserve"> Et cum anima comprehenderit ex <lb/>forma omnes intentiones, quæ ſunt in ea prima uice:</s> <s xml:id="echoid-s3610" xml:space="preserve"> deinde iterabitur peruentus formæ ſuper <lb/>ipſam, & comprehenderit in ea ſecundò intentiones:</s> <s xml:id="echoid-s3611" xml:space="preserve"> plus certificabit:</s> <s xml:id="echoid-s3612" xml:space="preserve"> quod illud quod in prima ui-<lb/>ce comprehendit, eſt uera forma illius.</s> <s xml:id="echoid-s3613" xml:space="preserve"> Forma autem, uera uerificata & certificata eſt magis fixa in <lb/>anima & in imaginatione, quàm forma non certificata.</s> <s xml:id="echoid-s3614" xml:space="preserve"> Forma ergo rei uiſæ, quando multoties <lb/>iterabitur comprehenſio eius, erit magis certificata apud animam, & in imaginatione, & per fixio-<lb/>nem formæ in anima, & per fixionem formæ in imaginatione erit memoratio illarum ab anima.</s> <s xml:id="echoid-s3615" xml:space="preserve"> Et <lb/>ſignificatio ſuper hoc manifeſta, quòd intentiones & formæ quando iterabuntur in anima, erunt <lb/>magis fixæ, quàm intentiones & formæ non iteratæ, eſt:</s> <s xml:id="echoid-s3616" xml:space="preserve"> Quia quando homo uoluerit corde tene-<lb/>re aliquem ſermonem, uel uerſum aliquem, iterabit ſermonem illius intentionis multoties:</s> <s xml:id="echoid-s3617" xml:space="preserve"> & ſic <lb/>figetur in ſua anima.</s> <s xml:id="echoid-s3618" xml:space="preserve"> Et quantò magis iterabit lectionem eius, tantò magis erit fixa in anima, & re-<lb/>motioris obliuionis:</s> <s xml:id="echoid-s3619" xml:space="preserve"> & ſi ſemel legeritipſam uel ipſum uerſum, non remanebit uerſus ille fixus in <lb/>anima:</s> <s xml:id="echoid-s3620" xml:space="preserve"> & ſimiliter, ſi bis legerit ipſum, fortè non figetur in anima eius:</s> <s xml:id="echoid-s3621" xml:space="preserve"> & ſi figatur, ſtatim tradetur <lb/>obliuioni.</s> <s xml:id="echoid-s3622" xml:space="preserve"> Experimentatione ergo iſtius intentionis patet, quòd formæ uenientes ad animã, quan-<lb/>tò magis iterabuntur, tantò magis erunt fixæ in anima & in imaginatione.</s> <s xml:id="echoid-s3623" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div123" type="section" level="0" n="0"> <head xml:id="echoid-head147" xml:space="preserve" style="it">67. E uiſibili ſæpi{us} uiſo remanet in animo generalis notio: qua quodlibet uiſibile ſimile per <lb/>cipitur & cognoſcitur. 61 p 3. Idem 14 n.</head> <p> <s xml:id="echoid-s3624" xml:space="preserve">PEruentus autem formarum uniuerſalium modorum uiſibilium in anima, & figuratio eorum <lb/>in imaginatione, eſt.</s> <s xml:id="echoid-s3625" xml:space="preserve"> Quia quodlibet indiuiduorum uiſibilium habet formam & figuram, in <lb/>quibus æquabuntur omnia indiuidua illiuſmodi:</s> <s xml:id="echoid-s3626" xml:space="preserve"> & illa indiuidua diuerſantur tantùm inten-<lb/>tionibus particularibus comprehenſis per ſenſum uiſus:</s> <s xml:id="echoid-s3627" xml:space="preserve"> & forte erit color in omnibus indiuiduis <lb/>illiuſmodi unus.</s> <s xml:id="echoid-s3628" xml:space="preserve"> Et forma, & figura, & color, & omnes intentiones, ex quibus cõponitur forma cu-<lb/>iuslibet indiuidui ſpeciei, eſt forma uniuerſalis illiuſmodi:</s> <s xml:id="echoid-s3629" xml:space="preserve"> & uiſus cõprehendit illam formã & uni-<lb/>uerſalẽ illã figurã, & cõprehendit omnẽ intentionẽ, in qua æquabuntur omnia indiuidua ſpeciei in <lb/>omnib.</s> <s xml:id="echoid-s3630" xml:space="preserve"> indiuiduis, quæ cõprehendũtur ex indiuiduis omnib.</s> <s xml:id="echoid-s3631" xml:space="preserve"> illi<emph style="sub">9</emph> ſpeciei:</s> <s xml:id="echoid-s3632" xml:space="preserve"> & cõprehendũtur etiã in-<lb/>tẽtiones particulares, p quas diuerſantur illa indiuidua.</s> <s xml:id="echoid-s3633" xml:space="preserve"> Per intuitionẽ ergo cõprehẽſionis indiui-<lb/> <pb o="70" file="0076" n="76" rhead="ALHAZEN"/> duorum omnium uniuſmodi à uiſu, iteratur forma uniuerſalis, quæ eſt in illa ſpecie, cum diuerſita <lb/>te formarum particularium illorum indiuiduorum.</s> <s xml:id="echoid-s3634" xml:space="preserve"> Et cum forma uniuerſalis iterabitur in anima, <lb/>figetur in anima, & quieſcet:</s> <s xml:id="echoid-s3635" xml:space="preserve"> & ex diuerſitate formarum particularium uenientium ad uiſum cum <lb/>formis uniuerſalibus apud intuitionem, comprehendet anima, quòd forma, in qua æquabuntur o-<lb/>mnia indiuidua illiuſmodi;</s> <s xml:id="echoid-s3636" xml:space="preserve"> eſt forma uniuerſalis illiuſmodi.</s> <s xml:id="echoid-s3637" xml:space="preserve"> Secundum ergo hũc modum erit per-<lb/>uentus formarum uniuerſalium, quas uiſus comprehendit ex modis uiſibilium in anima & in ima-<lb/>ginatione.</s> <s xml:id="echoid-s3638" xml:space="preserve"> Formæ ergo indiuiduorum uiſibilium, quas uiſus comprehendit, remanent in anima, & <lb/>figurantur in imaginatione:</s> <s xml:id="echoid-s3639" xml:space="preserve"> & quantò magis iterabitur comprehenſio eorum â uiſu, tantò magis <lb/>erunt fixæ in anima & in imaginatione.</s> <s xml:id="echoid-s3640" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div124" type="section" level="0" n="0"> <head xml:id="echoid-head148" xml:space="preserve" style="it">68. Eſſentia uiſibilis percipitur è ſpecieb{us} uifibilib{us}, beneficio formæ in animo reſiden-<lb/>tis. 66 p 3.</head> <p> <s xml:id="echoid-s3641" xml:space="preserve">ET ſuſtentatio ſentientis in comprehenſione quidditatis uiſibilium non eſt, niſi ſuper formas <lb/>peruenientes in animam:</s> <s xml:id="echoid-s3642" xml:space="preserve"> quoniam comprehenſio quidditatis uiſibilium nõ erit, niſi per co-<lb/>gnitionem:</s> <s xml:id="echoid-s3643" xml:space="preserve"> & cognitio non eſt, niſi ex comprehenſione formæ, quã uiſus comprehendit mo-<lb/>dò ad formam ſecundam, quæ eſt in imaginatione ex formis uiſibilium, quas uiſus comprehendit <lb/>antè:</s> <s xml:id="echoid-s3644" xml:space="preserve"> & ex comprehenſione conſiderationis formæ comprehenſæ modò ad aliam formarum per-<lb/>uenientium in imaginationem.</s> <s xml:id="echoid-s3645" xml:space="preserve"> Comprehenſio ergo quidditatis rei uiſæ nõ eſt, oiſi ex comprehen-<lb/>ſione aſsimilationis formæ rei uiſæ alicuius formarum quieſcẽtium in anima, fixarũ in imaginatio-<lb/>ne.</s> <s xml:id="echoid-s3646" xml:space="preserve"> Suſtentatio ergo ſentientis in comprehenſione quidditatis uiſibilium nõ eſt, niſi ſuper formam <lb/>uniuerſalem peruenientem in animam:</s> <s xml:id="echoid-s3647" xml:space="preserve"> & ſuſtentatio eius in cognitione indiuiduorum uiſibilium <lb/>non eſt, niſi ſuper formas indiuiduorum perueniẽtes in animam cuiuslibet indiuiduorum, quæ ui-<lb/>ſus comprehendit antè, & quorum formæ ſunt cõceptæ imaginatione antè & intellectæ.</s> <s xml:id="echoid-s3648" xml:space="preserve"> Et uirtus <lb/>diſtinctiua naturaliter aſsimilat formas uiſibilium apud uiſionẽ, formis uiſis fixis in imaginatione, <lb/>quas anima acquirit ex formis uiſibilium.</s> <s xml:id="echoid-s3649" xml:space="preserve"> Cum ergo uiſus comprehẽderit aliquam rem uiſam, ſta-<lb/>tim uirtus diſtinctiua quærit eius ſimile in formis exiſtentibus in imaginatione:</s> <s xml:id="echoid-s3650" xml:space="preserve"> & cũ inuenerit in <lb/>imaginatione aliquam ſimilem formæ illius rei uiſæ:</s> <s xml:id="echoid-s3651" xml:space="preserve"> cognoſcet illam rem uiſam, & comprehendet <lb/>quidditatem eius:</s> <s xml:id="echoid-s3652" xml:space="preserve"> & ſi non inuenerit ex formis exiſtentibus in imaginatione formam ſimilem for-<lb/>mæ illius rei uiſæ:</s> <s xml:id="echoid-s3653" xml:space="preserve"> non cognoſcet illam rem uiſam, neq;</s> <s xml:id="echoid-s3654" xml:space="preserve"> comprehẽdet quidditatem eius.</s> <s xml:id="echoid-s3655" xml:space="preserve"> Et propter <lb/>uelocitatem aſsimilationis formæ reiuiſæ apud uiſionem à uirtute diſtinctiua, fortè accidet ei er-<lb/>ror, ita quòd aſsimilabit rem uiſam alij rei uiſæ, quando in re uiſa fuerit aliqua intentio, quæ eſt in <lb/>illa alia re:</s> <s xml:id="echoid-s3656" xml:space="preserve"> deinde ſi conſiderauerit cum iteratione illam rem uiſam poſt iſtam diſpoſitionem, & cer <lb/>tificauerit formam eius:</s> <s xml:id="echoid-s3657" xml:space="preserve"> aſsimilabit ipſam formæ ſimili ei in rei ueritate, & manifeſtabitur illi ſe-<lb/>cundò, quòd errauerat in prima aſsimilatione.</s> <s xml:id="echoid-s3658" xml:space="preserve"> Secundum ergo hunc modum comprehenduntur <lb/>quidditates uiſibilium per ſenſum uiſus.</s> <s xml:id="echoid-s3659" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div125" type="section" level="0" n="0"> <head xml:id="echoid-head149" xml:space="preserve" style="it">69. Diſtinctauiſio fit aut obtutu ſolo: aut obtutu & anticipata notione ſimul. 62 p 3.</head> <p> <s xml:id="echoid-s3660" xml:space="preserve">ET cum omnes iſtæ intentiones ſint declaratæ, dicamus modò:</s> <s xml:id="echoid-s3661" xml:space="preserve"> quòd comprehenſio uiſibi-<lb/>lium per intuitionem erit duobus modis:</s> <s xml:id="echoid-s3662" xml:space="preserve"> comprehenſio ſola intuitione:</s> <s xml:id="echoid-s3663" xml:space="preserve"> & comprehenſio <lb/>per intuitionem cum ſcientia præcedente.</s> <s xml:id="echoid-s3664" xml:space="preserve"> Comprehenſio uerò, quæ eſt ſola intuitione, eſt <lb/>comprehenſio uiſibilium extraneorum, quæ uiſus non uidit antè:</s> <s xml:id="echoid-s3665" xml:space="preserve"> aut uiſibilium, quæ uiſus com-<lb/>prehendit antè, ſed non meminit uiſionis illorum.</s> <s xml:id="echoid-s3666" xml:space="preserve"> Quoniam uiſus quando comprehenderit ali-<lb/>quam rem uiſam, quam antè non percepit uidendo, necrem uiſam huius ſpeciei, & uoluerit aſpi-<lb/>ciens certificare formam huius rei uiſæ:</s> <s xml:id="echoid-s3667" xml:space="preserve"> intuebitur ipſam, & conſiderabit per intuitionem omnes <lb/>intentiones, quæ ſunt in ea, & comprehendet per intuitionem formam eius ueram.</s> <s xml:id="echoid-s3668" xml:space="preserve"> Et cum antè <lb/>non perceperitillam rem uiſam, neq;</s> <s xml:id="echoid-s3669" xml:space="preserve"> aliquam rem huius ſpeciei:</s> <s xml:id="echoid-s3670" xml:space="preserve"> non cognoſcet formam eius apud <lb/>eius comprehenſionem:</s> <s xml:id="echoid-s3671" xml:space="preserve"> & in talibus indiget uiſus intuitione ad formam propriam.</s> <s xml:id="echoid-s3672" xml:space="preserve"> Erit ergo cer-<lb/>tificatio formæ huiuſinodi uiſibilium non niſi per ſolam intuitionem tantùm.</s> <s xml:id="echoid-s3673" xml:space="preserve"> Et ſimiliter quando <lb/>uiſus comprehenderit aliquam rem uiſam, quam antè percepit, & non meminit ipſius:</s> <s xml:id="echoid-s3674" xml:space="preserve"> non cogno-<lb/>ſcet formam eius niſi per intuitionem.</s> <s xml:id="echoid-s3675" xml:space="preserve"> Erit ergo comprehenſio huiuſmodi uiſibilium per ſolam in-<lb/>tuitionem.</s> <s xml:id="echoid-s3676" xml:space="preserve"> Comprehenſio uerò, quæ eſt per intuitionem cum ſcientia pręcedente, eſt comprehen-<lb/>ſio omnium uiſibilium, quæ uiſus comprehendit antè, aut de quorum ſpecie aliquid comprehen-<lb/>dit uiſus antè, & peruenerunt formæ ſpecierum eorum & indiuiduorum eorum in animam.</s> <s xml:id="echoid-s3677" xml:space="preserve"> Cum <lb/>ergo uiſus comprehendit aliquam rem uiſam, quam antè comprehendit, aut cuius ſpeciei àliquam <lb/>rem antè comprehendit:</s> <s xml:id="echoid-s3678" xml:space="preserve"> ſtatim apud aſpectum illius rei uiſæ comprehendet totam formaim eius:</s> <s xml:id="echoid-s3679" xml:space="preserve"> <lb/>deinde modica intuitione comprehendet totam formam eius, quæ eſt uniuerſalis forma ſpeciei.</s> <s xml:id="echoid-s3680" xml:space="preserve"> <lb/>Cum ergo antè comprehendit uiſibilia illiuſmodi rei uiſæ, & peruenerit forma ſpeciei illius rei ul-<lb/>ſæ in ſuam animam, & fuerit memor ex forma uniuerſali illiuſmodi rei uiſæ:</s> <s xml:id="echoid-s3681" xml:space="preserve"> cognoſcet formam <lb/>uniuerſalem, quam comprehendit in illa re uiſa apud comprehenſionem eius, & apud cognitio-<lb/>nem formæ uniuerſalis, quam comprehendit in illa re uiſa, ſtatim cognoſcet illam rem uiſam ſpe-<lb/>cialiter:</s> <s xml:id="echoid-s3682" xml:space="preserve"> deinde quando intuitus fuerit intentiones reſiduas, quæ ſunt in illa re uiſa, certifica-<lb/>bit formam eius particularem.</s> <s xml:id="echoid-s3683" xml:space="preserve"> Si autem non percepit antè illam rem uiſam, aut fortè percepit il-<lb/>lam, ſed non meminit de perceptione illius:</s> <s xml:id="echoid-s3684" xml:space="preserve"> non cognoſcet formam particularem:</s> <s xml:id="echoid-s3685" xml:space="preserve"> & cum non <lb/> <pb o="71" file="0077" n="77" rhead="OPTICAE LIBER II."/> cognouerit formam particularem, non cognoſcet illam rem uiſam:</s> <s xml:id="echoid-s3686" xml:space="preserve"> & ſic erit cognitio illius rei uiſæ <lb/>ab eo ſecundum ſpeciem tantùm, & acquiret ex intuitione & certificatione formæ eius, formã eius <lb/>particularem, quæ appropriatur ſuo indiuiduo.</s> <s xml:id="echoid-s3687" xml:space="preserve"> Et ſi antè perceperit illam rem uiſam, & non perce-<lb/>perit alia indiuidua huiuſmodi ſpeciei, & fuerit memor illius formæ, quam antè comprehendit ex <lb/>illa re uiſa:</s> <s xml:id="echoid-s3688" xml:space="preserve"> quando comprehenderit formam eius particularem, cognoſcet per cognitionẽ formam <lb/>particularem, & apud cognitionem formæ particularis comprehendet rem uiſam:</s> <s xml:id="echoid-s3689" xml:space="preserve"> & ſic per com-<lb/>prehenſionem formæ eius particularis certificabit formam rei uiſæ, & ſimul cognoſcet ipſam rem <lb/>uiſam:</s> <s xml:id="echoid-s3690" xml:space="preserve"> & erit cognitio rei uiſæ ab eo ſpecialiter & ſecundum indiuiduum ſimul.</s> <s xml:id="echoid-s3691" xml:space="preserve"> Et ſi antè percepe-<lb/>rit illam rem uiſam, ſed non perceperit ex modo illius rei uiſæ, niſi illud indiuiduum tantùm, & <lb/>non diſtinguatur ab eo forma uniuerſalis illius modi rei uiſæ:</s> <s xml:id="echoid-s3692" xml:space="preserve"> quando comprehenderit illam rem <lb/>uiſam, & comprehenderit intentiones uniuerſales, quæ ſunt in illa re uiſa, & in omnibus rebus il-<lb/>lius ſpeciei, non cognoſcet illam rem uiſam, neq, comprehendet quidditatem eius ex comprehen-<lb/>ſione formæ uniuerſalis.</s> <s xml:id="echoid-s3693" xml:space="preserve"> Cum ergo comprehenderit intentiones reſiduas, quæ ſunt in illa re uiſa, <lb/>& comprehenderit formam particularem eius, & fuerit memor formæ particularis, quam compre-<lb/>hendit in illa re uiſa:</s> <s xml:id="echoid-s3694" xml:space="preserve"> cognoſcet formam particularem apud comprehenſionem eius:</s> <s xml:id="echoid-s3695" xml:space="preserve"> & cum cogno-<lb/>uerit formam particularem, cognoſcet eandem rem uiſam:</s> <s xml:id="echoid-s3696" xml:space="preserve"> & erit cognitio illius rei uiſæ ab eo in-<lb/>diuidualiter.</s> <s xml:id="echoid-s3697" xml:space="preserve"> Et nulla res uiſa comprehendetur per intuitionem, niſi ſecundũ aliquem iſtorum mo-<lb/>dorum.</s> <s xml:id="echoid-s3698" xml:space="preserve"> Comprehenſio ergo omniũ uiſibilium ſecundum intuitionem erit duobus modis:</s> <s xml:id="echoid-s3699" xml:space="preserve"> ſola in-<lb/>tuitione, & comprehenſione per intuitionem cũ ſcientia præcedẽte.</s> <s xml:id="echoid-s3700" xml:space="preserve"> Cognitio autem talis & ſcien-<lb/>tia quandoq;</s> <s xml:id="echoid-s3701" xml:space="preserve"> erit ſecundum ſpeciem tantùm, quandoq;</s> <s xml:id="echoid-s3702" xml:space="preserve"> ſecundum ſpeciem & indiuiduum ſimul.</s> <s xml:id="echoid-s3703" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div126" type="section" level="0" n="0"> <head xml:id="echoid-head150" xml:space="preserve" style="it">70. Obtut{us} fit in tempore. 56 p 3.</head> <p> <s xml:id="echoid-s3704" xml:space="preserve">ET etiam comprehenſio per intuitionem non erit, niſi in tempore:</s> <s xml:id="echoid-s3705" xml:space="preserve"> quoniam intuitio non erit <lb/>niſi per diſtinctionem & motum uiſus:</s> <s xml:id="echoid-s3706" xml:space="preserve"> ſed diſtinctio & motus non erunt niſi in tempore.</s> <s xml:id="echoid-s3707" xml:space="preserve"> In-<lb/>tuitio ergo non erit niſi in tempore.</s> <s xml:id="echoid-s3708" xml:space="preserve"> Et ſuperius declaratum eſt [12.</s> <s xml:id="echoid-s3709" xml:space="preserve">13 n] quòd comprehenſio <lb/>per cognitionem & comprehenſio per diſtinctionem non eſt niſi in tempore.</s> <s xml:id="echoid-s3710" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div127" type="section" level="0" n="0"> <head xml:id="echoid-head151" xml:space="preserve" style="it">71. Viſibile obtutu & antegreſſa cognitione ſimul, minore tempore percipitur, quàm ſolo ob-<lb/>tutu. 64 p 3.</head> <p> <s xml:id="echoid-s3711" xml:space="preserve">ET quia declaratum eſt [69 n] quòd comprehẽſio uiſibilium per intuitionem, erit quandoq;</s> <s xml:id="echoid-s3712" xml:space="preserve"> <lb/>ſola intuitione & quandoq;</s> <s xml:id="echoid-s3713" xml:space="preserve"> per intuitionem cum cognitione præcedẽte:</s> <s xml:id="echoid-s3714" xml:space="preserve"> & quòd illud, quod <lb/>comprehenditur per intuitionem & quod comprehenditur per cognitionem, non compre-<lb/>henditur, niſi in tempore:</s> <s xml:id="echoid-s3715" xml:space="preserve"> dicemus quòd comprehenſio, quæ erit per intuitionem cum cognitione <lb/>uel ſcientia præcedente, erit in maiori parte in minori tempore, quàm ſit tempus, in quo erit com-<lb/>prehenſio per ſolam intuitionem.</s> <s xml:id="echoid-s3716" xml:space="preserve"> Quoniã enim formæ exiſtentes in anima & pręſentes memoriæ, <lb/>non indigent, ut cognoſcantur omnes intentiones, quæ ſunt in eis, ex quibus componuntur in rei <lb/>ueritate:</s> <s xml:id="echoid-s3717" xml:space="preserve"> ſed ſufficit in comprehenſione earum cõprehenſio alicuius intentionis proprię illis.</s> <s xml:id="echoid-s3718" xml:space="preserve"> Cum <lb/>ergo uirtus diſtinctiua comprehenderit in forma ueniente ad ipſam, aliquã intentionem propriam <lb/>illi formę, & fuerit memor primę formæ:</s> <s xml:id="echoid-s3719" xml:space="preserve"> cognoſcet omnes formas uenientes ad ipſam:</s> <s xml:id="echoid-s3720" xml:space="preserve"> quoniam o-<lb/>mnis intentio, quę appropriatur alicui formæ, eſt ſignum ſignans ſuper illas formas.</s> <s xml:id="echoid-s3721" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s3722" xml:space="preserve"> <lb/>Quia quando uiſus comprehenderit indiuiduum hominis, & comprehẽderit lineationem ſuæ ma-<lb/>nus tantùm:</s> <s xml:id="echoid-s3723" xml:space="preserve"> ſtatim comprehendet, quòd ſit homo antequam comprehẽdat lineationem faciei ſuæ, <lb/>& antequam comprehendat lineationem partium reſiduarum eius.</s> <s xml:id="echoid-s3724" xml:space="preserve"> Et ſimiliter ſi comprehenderit <lb/>lineationem faciei ſuæ, antequã comprehendat partes reſiduas eius.</s> <s xml:id="echoid-s3725" xml:space="preserve"> Ex comprehẽſione ergo qua-<lb/>rundam intentionũ, quę appropriantur formæ hominis, comprehẽdit, quòd illud uiſibile ſit homo <lb/>ſine indigentia comprehenſionis partium reſiduatum:</s> <s xml:id="echoid-s3726" xml:space="preserve"> quoniam comprehẽdet partes reſiduas per <lb/>cognitionem præcedentẽ ex formis reſidentibus in anima, formis d<gap/><gap/>o hominũ.</s> <s xml:id="echoid-s3727" xml:space="preserve"> Et ſimiliter quan-<lb/>do uiſus comprehenderit aliquas intentiones, quæ appropriantur formæ particulari alicuius indi-<lb/>uidui, quod antè uiſus percepit, ſicut ſimitatem in naſo, aut uiriditatem in oculo, aut arcualitatẽ in <lb/>ſupercilijs:</s> <s xml:id="echoid-s3728" xml:space="preserve"> comprehendet comprehenſione totius ſuæ formę illud in diuiduũ, & cognoſcet ipſum.</s> <s xml:id="echoid-s3729" xml:space="preserve"> <lb/>Et ſimiliter cognoſcet equũ per aliquã maculam in fronte eius, aut per diuerſitatẽ coloris.</s> <s xml:id="echoid-s3730" xml:space="preserve"> Et ſimi-<lb/>liter ſcriptor quãdo cõprehenderit formã alicuius dictionis, ſuperficialiter cognoſcet eam, antequã <lb/>cõſideret literas particulares.</s> <s xml:id="echoid-s3731" xml:space="preserve"> Et ſimiliter omnes partes, quas ſcriptor frequẽter & continuè uidet, <lb/>cognoſcentur ab eo ex comprehenſione quarundã literarum.</s> <s xml:id="echoid-s3732" xml:space="preserve"> Viſibilia ergo, quæ uiſus antè cõpre-<lb/>henderit, & modò cognoſcit formas illorũ, & eſt memor illorũ:</s> <s xml:id="echoid-s3733" xml:space="preserve"> comprehenduntur à uiſu per ſigna.</s> <s xml:id="echoid-s3734" xml:space="preserve"> <lb/>Viſibilia autẽ extranea, quæ uiſus antè nõ percepit, aut uiſibilia, quę antè percepit, ſed non eſt me-<lb/>mor illorũ, non ſunt ita.</s> <s xml:id="echoid-s3735" xml:space="preserve"> Quoniã quando uiſus comprehẽderit aliquã rem uiſam, quã antè nõ uidit, <lb/>& comprehenderit lineationem quarundã partium:</s> <s xml:id="echoid-s3736" xml:space="preserve"> non comprehẽdet ex eo quidditatem illius reĩ <lb/>uiſæ:</s> <s xml:id="echoid-s3737" xml:space="preserve"> quoniam apud ipſum non quieſcit forma partium reſiduarum.</s> <s xml:id="echoid-s3738" xml:space="preserve"> Viſus ergo non comprehendit <lb/>certitudinem rei uiſæ, quam antè non uidit, niſi per conſiderationem omnium ſuarum partium, & <lb/>omniũ intentionum, quę ſunt in ea.</s> <s xml:id="echoid-s3739" xml:space="preserve"> Et ſimiliter forma rei uiſæ, quã uiſus antè percepit, ſed non me-<lb/>minit eius:</s> <s xml:id="echoid-s3740" xml:space="preserve"> non certificatur ab eo, niſi poſt cõſiderationem omnium intentionũ, quę ſunt in ea.</s> <s xml:id="echoid-s3741" xml:space="preserve"> Sed <lb/>comprehenſio quarundã intentionum, quę ſunt in forma, erit in minoritempore illo, in quo cõpre-<lb/>hendit omnes intentiones, quæ ſunt in forma.</s> <s xml:id="echoid-s3742" xml:space="preserve"> Viſio ergo, quæ eſt per intuitionem cum cognitione <lb/> <pb o="72" file="0078" n="78" rhead="ALHAZEN"/> præcedente, erit in maiori parte in breuiore tempore, illo tempore, in quo erit uiſio ſola intuitione.</s> <s xml:id="echoid-s3743" xml:space="preserve"> <lb/>Et propter hoc uiſus comprehendit uiſibilia cõſueta comprehenſione ualde ueloci in tempore la-<lb/>tente ſenſum:</s> <s xml:id="echoid-s3744" xml:space="preserve"> & non eritinter oppoſitionem uiſus & rem uiſam, & inter comprehenſionem quid-<lb/>ditatis rei uiſæ aſſuetæ tempus ſenſibile in maiori parte.</s> <s xml:id="echoid-s3745" xml:space="preserve"> Quoniã homo ex pueritia & ex principio <lb/>incrementi comprehẽdit uiſibilia, & iterantur ſuper eius aſpectum indiuidua uiſibilium, & formæ <lb/>uniuerſales uiſibilium.</s> <s xml:id="echoid-s3746" xml:space="preserve"> Et etiam declaratum eſt [14.</s> <s xml:id="echoid-s3747" xml:space="preserve">67 n] quòd formæ uiſibilium, quas uiſus com-<lb/>prehendit, perueniunt in animam, & figurantur in imaginatione:</s> <s xml:id="echoid-s3748" xml:space="preserve"> & quòd formæ, quæ iterantur ui-<lb/>ſui, figurantur in anima:</s> <s xml:id="echoid-s3749" xml:space="preserve"> & quas uiſus comprehẽdit, perueniunt in animam, & quieſcit figuratio ea-<lb/>rum in imaginatione.</s> <s xml:id="echoid-s3750" xml:space="preserve"> Omnia ergo uiſibilia aſſueta, & omnes modi aſſueti exiſtũt in anima, & quie-<lb/>ſcunt figurati in imaginatione & præſentes memoriæ.</s> <s xml:id="echoid-s3751" xml:space="preserve"> Cum ergo uiſus comprehenderit aliquam <lb/>rem uiſam aſſuetam, & comprehẽderit totam formam ſuam, & poſt illud comprehenderit aliquod <lb/>ſignum proprium illius rei uiſæ:</s> <s xml:id="echoid-s3752" xml:space="preserve"> comprehendet quidditatem rei uiſæ apud compreheſionem illius <lb/>ſigni:</s> <s xml:id="echoid-s3753" xml:space="preserve"> & erit comprehenſio rei uiſæ ab eo per cognitionem præcedentem & modicam intuitionem.</s> <s xml:id="echoid-s3754" xml:space="preserve"> <lb/>Viſibilia ergo aſſueta comprehenduntur à uiſu per ſigna & per cognitionem præcedentem.</s> <s xml:id="echoid-s3755" xml:space="preserve"> Quare <lb/>erit comprehenſio quidditatum eorum in maiori parte in tempore ſenſibili.</s> <s xml:id="echoid-s3756" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div128" type="section" level="0" n="0"> <head xml:id="echoid-head152" xml:space="preserve" style="it">72. Generales uiſibilis ſpecies citi{us} percipiuntur ſingularib{us}. 71 p 3.</head> <p> <s xml:id="echoid-s3757" xml:space="preserve">ET etiam comprehenſio ſpeciei rei uiſæ eſt in maiori parte in minore tempore, quàm compre-<lb/>hendatur indiuiduitas rei uiſæ.</s> <s xml:id="echoid-s3758" xml:space="preserve"> Et eſt:</s> <s xml:id="echoid-s3759" xml:space="preserve"> quoniam quando uiſus comprehenderit aliquod indi-<lb/>uiduum hominis, primò comprehendet ipſum eſſe hominem, antequam comprehendat for-<lb/>mam eius particularem:</s> <s xml:id="echoid-s3760" xml:space="preserve"> & fortè comprehendet ipſum eſſe hominem, quamuis nõ comprehendat <lb/>lineationem faciei, ſed ex erectione ſui corporis, & ordinatione membrorum corporis eius com-<lb/>prehendet ipſum eſſe hominem, quamuis non uiderit faciem eius.</s> <s xml:id="echoid-s3761" xml:space="preserve"> Et ſimiliter uiſus fortè compre-<lb/>hendet quandoq;</s> <s xml:id="echoid-s3762" xml:space="preserve"> ſpecialitatem modorum alicuius uiſibilium aſſuetorum per quædam ſigna, quæ <lb/>appropriantur illi ſpeciei.</s> <s xml:id="echoid-s3763" xml:space="preserve"> Et non eſt ſic comprehenſio indiuiduitatis rei uiſæ.</s> <s xml:id="echoid-s3764" xml:space="preserve"> Indiuidualitas enim <lb/>rei uiſæ non comprehenditur, niſi ex comprehenſione intentionũ particularium, quæ approprian-<lb/>tur illi indiuiduo, aut ex comprehenſione quarundam:</s> <s xml:id="echoid-s3765" xml:space="preserve"> ſed comprehẽſio quarundam intentionum <lb/>particularium, quæ appropriantur indiuiduo, non comprehenduntur, niſi poſt comprehenſionem <lb/>intentionum uniuerſalium, quę ſunt in illo indiuiduo, aut poſt comprehenſionem quarundam:</s> <s xml:id="echoid-s3766" xml:space="preserve"> aut <lb/>generaliter, intentiones, quæ ſunt in formis uniuerſalibus illiuſmodi indiuidui, ſunt ante intentio-<lb/>nes, quæ ſunt in forma eius indiuiduali:</s> <s xml:id="echoid-s3767" xml:space="preserve"> ſed comprehenſio partis eſt in minori tempore, quàm tem-<lb/>pus, in quo comprehenditur totum.</s> <s xml:id="echoid-s3768" xml:space="preserve"> Comprehenſio ergo ſpecialitatis rei uiſæ à uiſu eſt in minori <lb/>tempore, quàm tempus, in quo comprehenditur in diuidualitas illius rei uiſæ.</s> <s xml:id="echoid-s3769" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div129" type="section" level="0" n="0"> <head xml:id="echoid-head153" xml:space="preserve" style="it">73. E uiſibilib{us} communib{us} alia alijs citi{us} percipiuntur. 72 p 3.</head> <p> <s xml:id="echoid-s3770" xml:space="preserve">ET etiam tempus comprehenſionis ſpecialitatis uiſibiliũ ſcilicet aſſuetorum diuerſatur.</s> <s xml:id="echoid-s3771" xml:space="preserve"> Quo <lb/>niam quædam ſpecierum uiſibilium aſſuetorum aſsimilantur alijs ſpeciebus:</s> <s xml:id="echoid-s3772" xml:space="preserve"> & quædã non, <lb/>ut ſpecies hominis & ſpecies equi:</s> <s xml:id="echoid-s3773" xml:space="preserve"> quoniam forma ſpeciei hominis non aſsimilatur alij ſpe-<lb/>ciei animalium:</s> <s xml:id="echoid-s3774" xml:space="preserve"> & nõ eſt ita in equis.</s> <s xml:id="echoid-s3775" xml:space="preserve"> Quoniam equus aliquis aſsimilatur multis animalibus in to-<lb/>ta forma.</s> <s xml:id="echoid-s3776" xml:space="preserve"> Tempus ergo, in quo uiſus comprehendit ſpeciem indiuidui hominis, & comprehendit <lb/>ipſum eſſe hominem, non eſt ſicut tempus, in quo comprehendit ſpeciem equi, & comprehendit <lb/>ipſum eſſe equum:</s> <s xml:id="echoid-s3777" xml:space="preserve"> & maximè quando comprehendit utrumq;</s> <s xml:id="echoid-s3778" xml:space="preserve"> in remotione alicuius quantitatis.</s> <s xml:id="echoid-s3779" xml:space="preserve"> <lb/>Quoniam quando uiſus comprehenderit indiuiduum hominis alicuius motum localiter:</s> <s xml:id="echoid-s3780" xml:space="preserve"> ſtatim <lb/>comprehendet ipſum eſſe animal ex motu, & ex erectione corporis comprehendet ipſum eſſe ho-<lb/>minem:</s> <s xml:id="echoid-s3781" xml:space="preserve"> & non eſt ita, quando comprehenderit equum.</s> <s xml:id="echoid-s3782" xml:space="preserve"> Quoniam quando uiſus comprehenderit <lb/>indiuiduum equi mouens ſe, & comprehenderit ſimul motum eius, & numerum pedum, non com <lb/>prehendet ex hocipſum eſſe equum:</s> <s xml:id="echoid-s3783" xml:space="preserve"> quoniã illæ intentiones ſunt in pluribus quadrupedibus, quę <lb/>aſsimilãtur equo in pluribus intentionibus, & maximè in mulo:</s> <s xml:id="echoid-s3784" xml:space="preserve"> quoniam mulus aſsimilatur equo <lb/>in multis diſpoſitionibus:</s> <s xml:id="echoid-s3785" xml:space="preserve"> quoniam mulus non diſtinguitur ab equo, niſi per intentiones ferè non <lb/>manifeſtas, ſicut lineationem faciei, & extenſionem colli, & uelocitatem motus, & amplitudinem <lb/>paſſuum.</s> <s xml:id="echoid-s3786" xml:space="preserve"> Si autem uiſus non comprehenderit aliquam intentionum iſtarum, per quas comprehen <lb/>ditur equus cũ comprehenſione totius ſuæ formæ, non comprehendet ipſum eſſe equum.</s> <s xml:id="echoid-s3787" xml:space="preserve"> Et tem-<lb/>pus, in quo uiſus comprehendit erectionem corporis hominis, non eſt ſicut tempus, in quo com-<lb/>prehendit formam equi cum intentionibus particularibus, per quas diſtinguitur equus ab alio.</s> <s xml:id="echoid-s3788" xml:space="preserve"> <lb/>Comprehenſio ergo ſpeciei hominis eſt in minore tempore, quàm tempus, in quo comprehendi-<lb/>tur ſpecies equi:</s> <s xml:id="echoid-s3789" xml:space="preserve"> quamuis duo tempora ſint parua:</s> <s xml:id="echoid-s3790" xml:space="preserve"> tamen unum eorum ſecundum omnes diſpoſi-<lb/>tiones eſt maius altero.</s> <s xml:id="echoid-s3791" xml:space="preserve"> Et ſimiliter quando uiſus comprehenderit colorem roſeum in floribus cu-<lb/>iuſdam horti:</s> <s xml:id="echoid-s3792" xml:space="preserve"> ſtatim comprehendet quòd ſubſtantiæ illorum colorum ſunt roſæ propter colorem <lb/>proprium roſarum:</s> <s xml:id="echoid-s3793" xml:space="preserve"> & cum hoc, quòd ille color eſt in rebus exiſtentibus in horto:</s> <s xml:id="echoid-s3794" xml:space="preserve"> comprehenditur <lb/>ante comprehenſionem rotunditatis, & ante rotunditatem foliorum eius, & applicationum folio-<lb/>rum eius, unius ſuper alterum, & ante comprehenſionem omnium intentionum eius, ex quibus <lb/>componitur forma roſæ:</s> <s xml:id="echoid-s3795" xml:space="preserve"> & non eſt ita, quando comprehenderit uiriditatem myrti in horto:</s> <s xml:id="echoid-s3796" xml:space="preserve"> quo-<lb/>niam quando uiſus comprehenderit tantùm uiriditatem myrti in horto:</s> <s xml:id="echoid-s3797" xml:space="preserve"> non comprehendet ipſam <lb/>eſſe myrtum ex comprehenſione uiriditatis tantùm:</s> <s xml:id="echoid-s3798" xml:space="preserve"> quoniam plures plantæ ſunt uirides, & plures <lb/> <pb o="73" file="0079" n="79" rhead="OPTICAE LIBER II."/> plantæ aſsimilantur myrto in uiriditate & figura.</s> <s xml:id="echoid-s3799" xml:space="preserve"> Si ergo non comprehenderit figuram foliorum <lb/>eius, & ſpiſsitudinem eorum, & intentionem propriam myrti:</s> <s xml:id="echoid-s3800" xml:space="preserve"> non comprehendet ipſam eſſe myr-<lb/>tum.</s> <s xml:id="echoid-s3801" xml:space="preserve"> Et tempus, in quo comprehendit figuram foliorum myrti & intentiones, ſecundum quas ap-<lb/>propriatur myrtus cum comprehenſione uiriditatis, non eſt ſicut tempus, in quo comprehẽdit co-<lb/>lorem roſaceum tantùm.</s> <s xml:id="echoid-s3802" xml:space="preserve"> Et ſimiliter quidditates omnium ſpecierum, quæ poſſunt aſsimilari alijs, <lb/>non comprehenduntur à uiſu, niſi per magnam intuitionem:</s> <s xml:id="echoid-s3803" xml:space="preserve"> quidditas autem paucæ aſsimilatio-<lb/>nis ad alia, comprehenditur à uiſu pauca intuitione.</s> <s xml:id="echoid-s3804" xml:space="preserve"> Et ſimiliter de indiuiduis:</s> <s xml:id="echoid-s3805" xml:space="preserve"> quoniã indiuiduum, <lb/>quod uiſu non aſsimilatur alij indiuiduo, comprehenditur à uiſu per modicam intuitionem, & per <lb/>ſigna:</s> <s xml:id="echoid-s3806" xml:space="preserve"> & indiuiduum, quòd uiſus cognoſcit, & quod aſsimilatur alij indiuiduo, quamuis cognoſcit, <lb/>tamen comprehenditur à uiſu per magnam intuitionem.</s> <s xml:id="echoid-s3807" xml:space="preserve"> Species ergo & indiuiduum omnium ui-<lb/>ſibilium aſſuetorum comprehenditur à uiſu per modicam intuitionem cũ cognitione præcedente.</s> <s xml:id="echoid-s3808" xml:space="preserve"> <lb/>Et erit comprehenſio eorum in maiori parte in tempore ſenſibili:</s> <s xml:id="echoid-s3809" xml:space="preserve"> tamen diuerſatur tempus com-<lb/>prehenſionis eorum ſecundum diuerſitatem ſpecierum & indiuiduorum eorum:</s> <s xml:id="echoid-s3810" xml:space="preserve"> & erit compre-<lb/>henſio ſpeciei uelocior comprehenſione indiuidui:</s> <s xml:id="echoid-s3811" xml:space="preserve"> & erit comprehenſio ſpeciei paucæ aſsimila-<lb/>tionis ad alia, uelocior comprehenſione ſpeciei multæ aſsimilationis.</s> <s xml:id="echoid-s3812" xml:space="preserve"> Et ſimiliter comprehenſio <lb/>indiuidui paucæ aſsimilationis, erit uelocior comprehenſione indiuidui multæ aſsimilationis.</s> <s xml:id="echoid-s3813" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div130" type="section" level="0" n="0"> <head xml:id="echoid-head154" xml:space="preserve" style="it">74. Temp{us} obtut{us} pro ſpecierum uiſibilium uarietate uariat. 56 p 3.</head> <p> <s xml:id="echoid-s3814" xml:space="preserve">ET tempus intuitionis diuerſatur ſecundum intentiones, quas quiſque intuetur in uiſibili-<lb/>bus.</s> <s xml:id="echoid-s3815" xml:space="preserve"> Verbi gratia.</s> <s xml:id="echoid-s3816" xml:space="preserve"> Quia quando uiſus comprehenderit animal multipes paruorum pedum, <lb/>& illud animal fuerit in motu:</s> <s xml:id="echoid-s3817" xml:space="preserve"> per modicam intuitionem comprehendet motum eius, & cum <lb/>comprehenderit motum eius, comprehendet ipſum eſſe animal:</s> <s xml:id="echoid-s3818" xml:space="preserve"> deinde per modicam intuitionem <lb/>in pedibus comprehendet ipſum eſſe multipes ex comprehenſione diſtantiæ inter pedes:</s> <s xml:id="echoid-s3819" xml:space="preserve"> & ſic <lb/>non cognoſcet ſtatim numerum pedum:</s> <s xml:id="echoid-s3820" xml:space="preserve"> & ſi uoluerit cognoſcere numerum pedum, indigebit lon-<lb/>giore intuitione, & maiore tempore.</s> <s xml:id="echoid-s3821" xml:space="preserve"> Comprehenſio ergo animalitatis eius erit in tempore paruo:</s> <s xml:id="echoid-s3822" xml:space="preserve"> <lb/>deinde comprehenſio multitudinis pedum erit in tempore paruo:</s> <s xml:id="echoid-s3823" xml:space="preserve"> ſed numerus pedum non com-<lb/>prehẽdetur, niſi poſtquam fuerit uiſus intuitus quemlibet pedem, & numeraueritipſos, quod non <lb/>poteſt eſſe, niſi in tempore alicuius quantitatis:</s> <s xml:id="echoid-s3824" xml:space="preserve"> & erit quantitas temporis ſecũdum multitudinem <lb/>pedum & paucitatem eorum.</s> <s xml:id="echoid-s3825" xml:space="preserve"> Et ſimiliter quando uiſus comprehenderit figuram rotundam, intra <lb/>quam eſt figura multorum laterum, & fuerint latera illius figuræ parua, & cum hoc fuerit diuerſo-<lb/>rum laterum non maxima diuerſitas:</s> <s xml:id="echoid-s3826" xml:space="preserve"> apud comprehenſionem totalis figuræ comprehendet ipſam <lb/>eſſe rotundam, & non comprehendet ſtatim, quòd intra ipſam ſit laterata figura:</s> <s xml:id="echoid-s3827" xml:space="preserve"> quoniam latera <lb/>eius fuerunt in fine paruitatis.</s> <s xml:id="echoid-s3828" xml:space="preserve"> Et cum intuitus fuerit figuram rotundam profundiore intuitione, <lb/>apparebit figura laterata, quæ eſt intra rotundam.</s> <s xml:id="echoid-s3829" xml:space="preserve"> Erit ergo comprehenſio rotunditatis figuræ ue-<lb/>locior comprehenſione figuræ lateratæ, quæ eſt intra:</s> <s xml:id="echoid-s3830" xml:space="preserve"> deinde apud comprehenſionem iſtius non <lb/>apparebit diuerſitas laterum iſtius figuræ, nec diſtinguetur à uiſu an ſint æqualia, an non:</s> <s xml:id="echoid-s3831" xml:space="preserve"> & non <lb/>apparebit inæqualitas laterum figuræ lateratæ, niſi poſt magnam intuitionem & in tempore ali-<lb/>cuius quantitatis.</s> <s xml:id="echoid-s3832" xml:space="preserve"> Et etiam ſentiens quando uoluerit intueri figuram totius rei uiſæ, ſufficit ei, ut <lb/>tranſeat uiſus ſuper ſuperficiem rei uiſæ tantùm.</s> <s xml:id="echoid-s3833" xml:space="preserve"> Et ſimiliter quando uoluerit intueri colorem rei <lb/>uiſæ, ſufficit ei tranſire uiſum ſuper ipſum tantùm.</s> <s xml:id="echoid-s3834" xml:space="preserve"> Et ſimiliter quando uoluerit intueri aſperitatem <lb/>ſuperficiei rei uiſæ, aut planitiem, aut diaphanitatem, aut ſpiſsitudinem:</s> <s xml:id="echoid-s3835" xml:space="preserve"> & non ſuntita intentiones <lb/>occultæ ſubtiles, quæ ſunt in uiſibilibus, ſicut figuræ, quæ ſunt in quibuslibet partibus uiſibilium:</s> <s xml:id="echoid-s3836" xml:space="preserve"> <lb/>& conſimilitudo figurarum & quantitatis partium, & diuerſitas quantitatum, & colorum, & con-<lb/>ſimilitudo eorum, & ordinatio partium paruarum inter ſe:</s> <s xml:id="echoid-s3837" xml:space="preserve"> quoniam iſtæ intentiones non compre-<lb/>henduntur per intuitionem, niſi poſtquam fuerit uiſus fixus ſuper quamlibet partium, & conſide-<lb/>rauerit figuras illarum partium, & comparauerit unam ad alteram:</s> <s xml:id="echoid-s3838" xml:space="preserve"> & hoc non complebitur in tem-<lb/>pore paruo, & per motum uelocem, ſed in tempore alicuius quantitatis.</s> <s xml:id="echoid-s3839" xml:space="preserve"> Tempus ergo intuitionis <lb/>intentionum uiſibilium diuerſatur ſecundum diuerſitatem intentionum intuitarum.</s> <s xml:id="echoid-s3840" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div131" type="section" level="0" n="0"> <head xml:id="echoid-head155" xml:space="preserve" style="it">75. Viſio per anticipatam notionem & breuem obtutum, eſt incerta. 65 p 3.</head> <p> <s xml:id="echoid-s3841" xml:space="preserve">ET cum hoc ſit declaratum, dicamus:</s> <s xml:id="echoid-s3842" xml:space="preserve"> quòd uiſio, quæ eſt per cognitionem præcedentem, & <lb/>per ſigna & per modicam intuitionem, non eſt comprehenſio certificata.</s> <s xml:id="echoid-s3843" xml:space="preserve"> Quoniam compre-<lb/>henſio rei uiſæ per cognitιonem præcedentem & per ſigna non eſt, niſi circa totalitatem & <lb/>uniuerſalitatem rei uiſæ in groſſo:</s> <s xml:id="echoid-s3844" xml:space="preserve"> & uirtus diſtinctiua comprehẽdit intentiones particulares, quæ <lb/>ſunt in illa re uiſa, ſecundum modum, quo cognouit illas res uiſas ex prima forma illius rei uiſę exi-<lb/>ſtente in anima:</s> <s xml:id="echoid-s3845" xml:space="preserve"> ſed iſtæ intentiones particulares, quæ ſunt in uiſibilibus, mutantur ſecundum tran <lb/>ſitum temporis:</s> <s xml:id="echoid-s3846" xml:space="preserve"> & ſic uiſus non comprehendit intentiones, quæ ſunt mutatæ in illa re uiſa per co-<lb/>gnitionem præcedentem.</s> <s xml:id="echoid-s3847" xml:space="preserve"> Et cum mutatio fuerit occulta & non bene manifeſta, non comprehen-<lb/>ditur à uiſ<gap/> primo aſpectu, & non comprehenditur, quando non fuerit ualde manifeſta, niſi per in-<lb/>tuitionem.</s> <s xml:id="echoid-s3848" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s3849" xml:space="preserve"> quando uiſus cognoſcit aliquem hominem, & fuerit facies illius hominis <lb/>munda, & certificauerit uiſus formam eius:</s> <s xml:id="echoid-s3850" xml:space="preserve"> deinde receſſerit ille homo à uiſu longo tempore:</s> <s xml:id="echoid-s3851" xml:space="preserve"> & <lb/>contingat in facie eius macula:</s> <s xml:id="echoid-s3852" xml:space="preserve"> & fuerit occulta illa macula:</s> <s xml:id="echoid-s3853" xml:space="preserve"> & comprehenderit ipſum poſt iſtam <lb/>diſpoſitionem:</s> <s xml:id="echoid-s3854" xml:space="preserve"> cognoſcet ipſum apud comprehenſionem:</s> <s xml:id="echoid-s3855" xml:space="preserve"> ſed tamen non propter comprehenſio-<lb/> <pb o="74" file="0080" n="80" rhead="ALHAZEN"/> nem & cognitionem illius hominis, comprehẽdet maculam in facie eius, niſi ſit manifeſta:</s> <s xml:id="echoid-s3856" xml:space="preserve"> & ſi non <lb/>fuerit intuitus ipſam, non comprehendet ipſam ſecundum ſuum eſſe:</s> <s xml:id="echoid-s3857" xml:space="preserve"> & ſi intuitus fuerit ipſam pu-<lb/>riore intuitione:</s> <s xml:id="echoid-s3858" xml:space="preserve"> apparebit ei macula, quæ eſt in facie eius:</s> <s xml:id="echoid-s3859" xml:space="preserve"> & tunc comprehendet formam eius ſe-<lb/>cundum ſuum eſſe.</s> <s xml:id="echoid-s3860" xml:space="preserve"> Et ſimiliter quando uiſus comprehenderit aliquam arborem, & intuitus fuerit <lb/>ipſam, & certificauerit formá eius:</s> <s xml:id="echoid-s3861" xml:space="preserve"> deinde receſſerit ab eadem diu, dum creuerit illa arbor, & aucta <lb/>fuerit:</s> <s xml:id="echoid-s3862" xml:space="preserve"> & mutata figura eius:</s> <s xml:id="echoid-s3863" xml:space="preserve"> & facta ſit in ea aliqua mutatio:</s> <s xml:id="echoid-s3864" xml:space="preserve"> & illa mutatio, quæ fuerit in arbore, <lb/>fuerit modica:</s> <s xml:id="echoid-s3865" xml:space="preserve"> deinde ſi reuertatur uiſus ad illam arborem, & cognoſcat eam:</s> <s xml:id="echoid-s3866" xml:space="preserve"> non comprehendet <lb/>apud comprehenſionem per cognitionem illam modicam, mutationem, quæ contigit in ea:</s> <s xml:id="echoid-s3867" xml:space="preserve"> ſi au-<lb/>tem intuitus fuerit ipſam ſecundò, & ſimul fuerit memor ueræ formæ eius, quam habebat prima <lb/>uice:</s> <s xml:id="echoid-s3868" xml:space="preserve"> comprehendet mutationẽ, quæ contigit in ea, & certificabit formam eius ſecundò:</s> <s xml:id="echoid-s3869" xml:space="preserve"> & ſi non <lb/>fuerit intuitus ipſam, non erit illa forma, quam comprehendit ex illa arbore per cognitionem ante-<lb/>cedentem, ipſa forma uera, quam habet ſecunda comprehenſione.</s> <s xml:id="echoid-s3870" xml:space="preserve"> Et ſimiliter, quando uiſus com-<lb/>prehẽderit parietem in quibuſdam locis:</s> <s xml:id="echoid-s3871" xml:space="preserve"> & ille paries fuerit planus:</s> <s xml:id="echoid-s3872" xml:space="preserve"> & fuerint in eo picturæ & ſcul-<lb/>pturæ:</s> <s xml:id="echoid-s3873" xml:space="preserve"> & intuitus fuerit uiſus illum parietem:</s> <s xml:id="echoid-s3874" xml:space="preserve"> & certificauerit formã eius:</s> <s xml:id="echoid-s3875" xml:space="preserve"> deinde receſſerit ab illo <lb/>loco diu:</s> <s xml:id="echoid-s3876" xml:space="preserve"> & contingat pòſt mutatio in illo pariete ex aſperitate ſuperficiei, aut ex intentione qua-<lb/>rundam picturarum:</s> <s xml:id="echoid-s3877" xml:space="preserve"> & non fuerit illa mutatio ualde manifeſta:</s> <s xml:id="echoid-s3878" xml:space="preserve"> deinde ſi reuertatur uiſus ad illum <lb/>locum:</s> <s xml:id="echoid-s3879" xml:space="preserve"> & aſpexerit illum parietem:</s> <s xml:id="echoid-s3880" xml:space="preserve"> & fuerit memor formæ primæ:</s> <s xml:id="echoid-s3881" xml:space="preserve"> comprehendet ipſam apud pri-<lb/>mam uiſionem:</s> <s xml:id="echoid-s3882" xml:space="preserve"> ſed apud comprehenſionem per cognitionem nõ comprehendet mutationem oc-<lb/>cultam, quæ in eo contigit:</s> <s xml:id="echoid-s3883" xml:space="preserve"> & ipſe cognoſcet formam eius ſine aliqua mutatione.</s> <s xml:id="echoid-s3884" xml:space="preserve"> Si ergo in eo con-<lb/>tigit aliqua aſperitas, æſtimabit ipſam eſſe læuem, ſicut conſueuit eſſe:</s> <s xml:id="echoid-s3885" xml:space="preserve"> & ſi picturæ primò fuerint <lb/>certificatæ uerè, & fuerint mutatæ, æſtimabit eas eſſe quaſi certificatas.</s> <s xml:id="echoid-s3886" xml:space="preserve"> Et omnia uiſibilia, quę ſunt <lb/>apud nos, ſunt recipientia mutationem ſecundum colorem, & figuram, & magnitudinem, & ſitum, <lb/>& aſperitatem, & læuitatem, & ordinationem partium, & ſecundum multas intentiones particula <lb/>res:</s> <s xml:id="echoid-s3887" xml:space="preserve"> quoniam naturæ earum ſunt mutabiles & præparatæ paſsioni ab eo, quod accidit eis extrinſe-<lb/>cus.</s> <s xml:id="echoid-s3888" xml:space="preserve"> Et quia mutatio eſt poſsibilis in eis, poſsibile eſtipſam comprehendi à uiſu in omnibus illis.</s> <s xml:id="echoid-s3889" xml:space="preserve"> <lb/>Et quamuis ſit in eis aliqua mutatio, quæ non poteſt apparere uiſui:</s> <s xml:id="echoid-s3890" xml:space="preserve"> nihil eſt tamen ex eis, in quo <lb/>non accidat extrinſecus mutatio, quæ poſsit apparere uiſui.</s> <s xml:id="echoid-s3891" xml:space="preserve"> Et cum omnia uiſibilia ſint præparata <lb/>mutationi, quæ poſsit comprehendi à uiſu:</s> <s xml:id="echoid-s3892" xml:space="preserve"> nullum ergo uiſibile, quod uiſus comprehendit modò, <lb/>& erat prius comprehenſum:</s> <s xml:id="echoid-s3893" xml:space="preserve"> certificatum eſt apud comprehenſionem ſecundam â uiſu, ſcilicet, <lb/>quòd uiſus ſit ſecurus ſecundò, quòd non fuerit mutatum, cum mutatio ſit poſsibilis in omnibus <lb/>uiſibilibus.</s> <s xml:id="echoid-s3894" xml:space="preserve"> Cum ergo uiſus comprehenderit aliquam rem uiſam, quam antè comprehendit:</s> <s xml:id="echoid-s3895" xml:space="preserve"> & in-<lb/>tuitus fuerit ipſam:</s> <s xml:id="echoid-s3896" xml:space="preserve"> & certificauerit formam eius:</s> <s xml:id="echoid-s3897" xml:space="preserve"> & fuerit memor ſuæ formæ apud comprehenſio-<lb/>nem, cognoſcet ipſam.</s> <s xml:id="echoid-s3898" xml:space="preserve"> Et ſi in illa re uiſa contigit mutatio manifeſta, comprehẽdet illam mutatio <lb/>nem apud uiſionem:</s> <s xml:id="echoid-s3899" xml:space="preserve"> ſi autem nõ fuerit manifeſta:</s> <s xml:id="echoid-s3900" xml:space="preserve"> cognoſcet illam rem, & æſtimabit illam eſſe apud <lb/>cognitionem ſecundũ modum primum:</s> <s xml:id="echoid-s3901" xml:space="preserve"> & ſic, ſi non iterauerit intuitionem, non erit ſecúrus, quòd <lb/>forma, quam antè cognoſcebat, remaneat ſecundum ſuum eſſe, cum ſit poſsibile, quòd in ea conti-<lb/>gerit mutatio occulta, quæ non poteſt apparere, niſi per intuitionem.</s> <s xml:id="echoid-s3902" xml:space="preserve"> Si ergo iterauerit intuitio-<lb/>nem, certificabit formam eius:</s> <s xml:id="echoid-s3903" xml:space="preserve"> & ſi non iterauerit intuitionem, non erit comprehenſio illius rei ui-<lb/>ſæ certificata.</s> <s xml:id="echoid-s3904" xml:space="preserve"> Comprehenſio ergo uiſibilium per cognitionem præcedentem, & per ſigna, & per <lb/>modicam intuitionem, non eſt uera comprehenſio.</s> <s xml:id="echoid-s3905" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div132" type="section" level="0" n="0"> <head xml:id="echoid-head156" xml:space="preserve" style="it">76. Vera uiſibilis forma percipitur obtutu: accurata conſideratione: & dilig enti omnium <lb/>uiſibilium ſpecierum diſtinctione. 57 p 3.</head> <p> <s xml:id="echoid-s3906" xml:space="preserve">ET uiſus non comprehendit rem uiſam uera comprehenſione, niſi per intuitionem rei uiſæ <lb/>apud comprehenſionem eius, & per conſiderationem omnium intentionum, quę ſunt in illa <lb/>re uiſa, & per diſtinctionem omnium apud comprehenſionem illius rei uiſæ.</s> <s xml:id="echoid-s3907" xml:space="preserve"> Viſio ergo erit <lb/>ſecundum duos modos:</s> <s xml:id="echoid-s3908" xml:space="preserve"> uiſio in primo aſpectu, & uiſio quæ eſt per intuitionem.</s> <s xml:id="echoid-s3909" xml:space="preserve"> Et per uiſionem, <lb/>quæ eſt in primo aſpectu, comprehendet uiſus intentiones rei uiſæ manifeſtas tantùm, & non cer-<lb/>tificatur per huiuſmodi aſpectum forma rei uiſæ.</s> <s xml:id="echoid-s3910" xml:space="preserve"> Et uiſio, quæ eſt in primo aſpectu:</s> <s xml:id="echoid-s3911" xml:space="preserve"> quandoque eſt <lb/>ſolùm phantaſtica:</s> <s xml:id="echoid-s3912" xml:space="preserve"> & quandoq;</s> <s xml:id="echoid-s3913" xml:space="preserve"> cum cognitione præcedente:</s> <s xml:id="echoid-s3914" xml:space="preserve"> & uiſio talis, quæ eſt ſecundum phan <lb/>taſiam, eſt uiſio uiſibilium, quæ uiſus non cognouit apud aſpectum:</s> <s xml:id="echoid-s3915" xml:space="preserve"> & cum hoc intuetur ipſa.</s> <s xml:id="echoid-s3916" xml:space="preserve"> Et ui-<lb/>ſio, quæ eſt ſecundum phantaſiam cum cognitione præcedente, eſt uiſio uiſibilium, quæ uiſus co-<lb/>gnouit antè:</s> <s xml:id="echoid-s3917" xml:space="preserve"> & cum hoc non intuetur intentiones eorum.</s> <s xml:id="echoid-s3918" xml:space="preserve"> Et ſecundum diſpoſitionem utriuſque <lb/>earum non comprehendit uiſus per phantaſiam ueritatem rei uiſæ, ſiue præcognouerit illam rem, <lb/>ſiue non.</s> <s xml:id="echoid-s3919" xml:space="preserve"> Et uiſio per intuitionem erit ſecundum duos modos, ſcilicet uiſio ſola intuitione, & uiſio <lb/>per intuitionem cum præcedente cognitione.</s> <s xml:id="echoid-s3920" xml:space="preserve"> Viſio autem, quæ eſt ſola intuitione, eſt uiſibilium, <lb/>quæ uiſus antè non comprehendit, aut non eſt memor comprehenſionis eorum, quando intuetur <lb/>modò ipſa.</s> <s xml:id="echoid-s3921" xml:space="preserve"> Et uiſio per intuitionem cum ſcientia præcedente, eſt uiſio omnium uiſibilium, quæ ui-<lb/>ſus comprehendit:</s> <s xml:id="echoid-s3922" xml:space="preserve"> & eſt memor comprehenſionis eorum, quando intuitus fuerit eorum intentio-<lb/>nes, & conſyderauerit intentiones omnes, quæ ſunt in eis.</s> <s xml:id="echoid-s3923" xml:space="preserve"> Et iſta uiſio diuiditur in duos modos:</s> <s xml:id="echoid-s3924" xml:space="preserve"> <lb/>quorum unus, eſt uiſio aſſueta uiſibilium aſſuetorum:</s> <s xml:id="echoid-s3925" xml:space="preserve"> & iſta pars erit per ſigna, quæ comprehen-<lb/>duntur modica intuitione, & per cõſyderationem quarundam intentionum, quæ ſunt in illa re ui-<lb/>ſa cum cognitione præcedente.</s> <s xml:id="echoid-s3926" xml:space="preserve"> Et illa uiſio eſt in maiore parte in tempore inſenſibili;</s> <s xml:id="echoid-s3927" xml:space="preserve"> & compre-<lb/> <pb o="75" file="0081" n="81" rhead="OPTICAE LIBER III."/> henſio illius, quod comprehenditur ſecundum hunc modum, non eſt comprehenſio in fine certi-<lb/>tudinis.</s> <s xml:id="echoid-s3928" xml:space="preserve"> Pars autem ſecunda eſt per finem intuitionis, & per conſyderationem omnium intentio-<lb/>num, quæ ſunt in re uiſa apud comprehenſionem illius rei uiſæ, & cum cognitione præcedente:</s> <s xml:id="echoid-s3929" xml:space="preserve"> & <lb/>erit in maiori parte in tempore ſenſibili:</s> <s xml:id="echoid-s3930" xml:space="preserve"> & diuerſatur tempus ſecundum intentiones, quæ ſunt in <lb/>re uiſa.</s> <s xml:id="echoid-s3931" xml:space="preserve"> Et uiſio, quæ eſt ſecũdum hunc modum, per quem uiſibilia aſſueta comprehenduntur com-<lb/>prehenſione in fine certitudinis, non eſt niſi per intuitionem omnium intentionum, quæ ſunt in re <lb/>uiſa, & per conſyderationem omnium partium rei uiſæ, & per diſtinctionem omniũ intentionum, <lb/>quæ ſunt in re uiſa apud comprehenſionem rei uiſæ, ſiue præcognouerit illam rem ſiue non.</s> <s xml:id="echoid-s3932" xml:space="preserve"> Et iſta <lb/>certificatio, quæ eſt reſpectu ſenſus, eſt intentio certificata:</s> <s xml:id="echoid-s3933" xml:space="preserve"> & eſt dicere finem certificationis in iſtis <lb/>locis, finem illius, quod poteſt comprehendi à ſenſu.</s> <s xml:id="echoid-s3934" xml:space="preserve"> Et cum omnibus iſtis comprehẽſio uiſibilium <lb/>à uiſu eſt ſecundum fortitudinem uiſus:</s> <s xml:id="echoid-s3935" xml:space="preserve"> quoniam ſenſus uiſus oculorum diuerſatur ſecundum ui-<lb/>gorem & debilitatem.</s> <s xml:id="echoid-s3936" xml:space="preserve"> Secundum ergo iſtos modos erit comprehenſio uiſibilium à uiſu, & iſti ſunt <lb/>omnes modi uiſibiliũ.</s> <s xml:id="echoid-s3937" xml:space="preserve"> Et hoc eſt illud, quod intendebamus declarare in iſto capitulo.</s> <s xml:id="echoid-s3938" xml:space="preserve"> Et iam com-<lb/>pleuimus diuiſionem omnium uiſibilium, & diuiſionem omnium intentionum uiſibilium, & de-<lb/>clarauimus omnes intentiones, per quas uenit uiſus ad comprehenſionem uiſibilium & intentio-<lb/>num uiſibilium, & diſtinximus omnes partes, in quas diuiduntur omnes modi uiſionum.</s> <s xml:id="echoid-s3939" xml:space="preserve"> Et iſtæ <lb/>ſunt intentiones, quas intendebamus declarare in iſto tractatu.</s> <s xml:id="echoid-s3940" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div133" type="section" level="0" n="0"> <head xml:id="echoid-head157" xml:space="preserve">ALHAZEN FILII</head> <head xml:id="echoid-head158" xml:space="preserve">ALHAYZEN OPTICAE</head> <head xml:id="echoid-head159" xml:space="preserve">LIBER TERTIVS.</head> <p style="it"> <s xml:id="echoid-s3941" xml:space="preserve">TERTIVS tractat{us} eſt ex ſeptem capitulis.</s> <s xml:id="echoid-s3942" xml:space="preserve"> Primum capitulum eſt proœ-<lb/>mium.</s> <s xml:id="echoid-s3943" xml:space="preserve"> Secundum de ijs, quæ debent præponi ſermoni in deceptionib{us} ui-<lb/>ſ{us}.</s> <s xml:id="echoid-s3944" xml:space="preserve"> Tertium de caußis, quib{us} deceptio accidit uiſui.</s> <s xml:id="echoid-s3945" xml:space="preserve"> Quartum in diſtin-<lb/>guendo deceptiones uiſ{us}.</s> <s xml:id="echoid-s3946" xml:space="preserve"> Quintum de qualitatib{us} deceptionum uiſ{us}, <lb/>quæ fiunt ſolo ſenſu.</s> <s xml:id="echoid-s3947" xml:space="preserve"> Sextum de qualitatib{us} deceptionum uiſ{us}, quæ fiunt in cognitio-<lb/>ne.</s> <s xml:id="echoid-s3948" xml:space="preserve"> Septimum de qualitatib{us} deceptionum uiſ{us}, quæ fiunt in ratione.</s> <s xml:id="echoid-s3949" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div134" type="section" level="0" n="0"> <head xml:id="echoid-head160" xml:space="preserve">PROOEMIVM LIBRI. CAP. 1.</head> <head xml:id="echoid-head161" xml:space="preserve" style="it">1. Viſ{us} in perceptione uiſibilium aliquando allucinatur. 1 p 4.</head> <p> <s xml:id="echoid-s3950" xml:space="preserve">DEclaratum eſt in primo tractatu & ſecundo, quomodo uiſus comprehendat uiſibilia ſecun-<lb/>dum quod ſunt, ſi comprehenſio eius fuerit rectè:</s> <s xml:id="echoid-s3951" xml:space="preserve"> & quomodo certificet formam uiſi:</s> <s xml:id="echoid-s3952" xml:space="preserve"> & quo <lb/>modo comprehendat unamquamque intentionum particularium, ſecundum quod eſt:</s> <s xml:id="echoid-s3953" xml:space="preserve"> & <lb/>quomodo certificet illam.</s> <s xml:id="echoid-s3954" xml:space="preserve"> Sed non omne comprehenſibile à uiſu, comprehẽditur ab eo ſecundum <lb/>quod eſt, neq;</s> <s xml:id="echoid-s3955" xml:space="preserve"> omne, quod uidetur ab aſpiciente comprehendi in rei ueritate, eſt rectè comprehen-<lb/>ſum.</s> <s xml:id="echoid-s3956" xml:space="preserve"> Sed multoties decipitur uiſus in multis eorum, quæ comprehendit ex uiſibilibus, & compre-<lb/>hendit illa alio modo ab eo, quo ſunt:</s> <s xml:id="echoid-s3957" xml:space="preserve"> & fortè percipit ſuam deceptionem etiam cum decipitur, & <lb/>fortè non, ſed reputat ſe benè comprehendere.</s> <s xml:id="echoid-s3958" xml:space="preserve"> Cum enim uiſus comprehenderit aliquod uiſum <lb/>per ſpatium remotum:</s> <s xml:id="echoid-s3959" xml:space="preserve"> tunc menſura eius uidebitur minor, quàm uera menſura:</s> <s xml:id="echoid-s3960" xml:space="preserve"> & quando illud <lb/>uiſum fuerit fortè propinquum uiſui:</s> <s xml:id="echoid-s3961" xml:space="preserve"> comprehendet menſuram eius maiorem uera.</s> <s xml:id="echoid-s3962" xml:space="preserve"> Et amplius, <lb/>quando uiſus comprehenderit quadratum, aut polygonum à remoto:</s> <s xml:id="echoid-s3963" xml:space="preserve"> comprehendet illud rotun-<lb/>dum, ſifuerit æqualium diametrorum:</s> <s xml:id="echoid-s3964" xml:space="preserve"> aut longum, ſi fuerit inæqualium diametrorum.</s> <s xml:id="echoid-s3965" xml:space="preserve"> Et ſi com-<lb/>prehenderit ſphæram à remotiſsimo, comprehendet eam planam.</s> <s xml:id="echoid-s3966" xml:space="preserve"> Et talia ſunt multa & multimo-<lb/>da:</s> <s xml:id="echoid-s3967" xml:space="preserve"> & omnia quæ ſunt comprehenſa à uiſu tali modo, ſunt fallibilia.</s> <s xml:id="echoid-s3968" xml:space="preserve"> Amplius, quando uiſus inſpe-<lb/>xerit aliquam ſtellam, comprehendet eam quieſcentem, licet ſtella tunc moueatur:</s> <s xml:id="echoid-s3969" xml:space="preserve"> & cũ inſpiciens <lb/>reuertetur ad ſcientiam:</s> <s xml:id="echoid-s3970" xml:space="preserve"> ſciet illam ſtellam moueri apud aſpectum:</s> <s xml:id="echoid-s3971" xml:space="preserve"> & cum inſpiciens diſtinxerit il-<lb/>lud:</s> <s xml:id="echoid-s3972" xml:space="preserve"> ſtatim comprehendet ſe decipi in hoc, quod comprehenderit de quiete ſtellæ.</s> <s xml:id="echoid-s3973" xml:space="preserve"> Et cum aliquis <lb/>inſpexerit aliquod indiuiduum ſuper faciem terræ à remotiſsimo interuallo, & illud indiuiduum <lb/>fuerit motum motu tardiſsimo, & non diu durauerit aſpectus:</s> <s xml:id="echoid-s3974" xml:space="preserve"> tunc in tali ſtatu aſpiciens compre-<lb/>hendet ipſum quieſcens:</s> <s xml:id="echoid-s3975" xml:space="preserve"> & ſi aſpiciens non perceperit antè motum illius indiuidui, & nõ diu dura-<lb/>uerit in eius oppoſitione:</s> <s xml:id="echoid-s3976" xml:space="preserve"> tunc non percipiet ſe eſſe deceptum in hoc, quod comprehẽdit de quiete <lb/>illius indiuidui:</s> <s xml:id="echoid-s3977" xml:space="preserve"> & in comprehenſione huius erit deceptus, & tamen non percipiet ſe decipi.</s> <s xml:id="echoid-s3978" xml:space="preserve"> Acci-<lb/>dit igitur uiſui deceptio in multis eorum, quæ comprehẽdit:</s> <s xml:id="echoid-s3979" xml:space="preserve"> & fortè percipitur ab eo, & fortè non.</s> <s xml:id="echoid-s3980" xml:space="preserve"> <lb/>Et cum in duobus libris pręcedentibus ſit declaratum, quomodo uiſus comprehendat uiſibilia, ſe-<lb/>cundum quod ſunt:</s> <s xml:id="echoid-s3981" xml:space="preserve"> In hoc autem capitulo declaratum eſt ex eis, quæ diximus, quòd multoties ac-<lb/>cidit uiſui deceptio in multis eorum, quæ comprehendit:</s> <s xml:id="echoid-s3982" xml:space="preserve"> remanet declarãdum, quare deceptio ac-<lb/>cidat uiſui, & quando, & quomodo.</s> <s xml:id="echoid-s3983" xml:space="preserve"> Nos autem in hoc tractatu contenti ſumus de deceptionibus <lb/>uiſus in eis, quæ comprehẽdit rectè:</s> <s xml:id="echoid-s3984" xml:space="preserve"> & declarabimus cauſſam in hoc, & diuerſitates deceptionum, <lb/>& quomodo accidat unaquæq;</s> <s xml:id="echoid-s3985" xml:space="preserve"> deceptio.</s> <s xml:id="echoid-s3986" xml:space="preserve"/> </p> <pb o="76" file="0082" n="82" rhead="ALHAZEN"/> </div> <div xml:id="echoid-div135" type="section" level="0" n="0"> <head xml:id="echoid-head162" xml:space="preserve">DE IIS QVAE DEBENT PRAEPONI SERMONI</head> <head xml:id="echoid-head163" xml:space="preserve">in deceptionibus uiſus. Cap. II.</head> <head xml:id="echoid-head164" xml:space="preserve" style="it">2. Axes pyramidum opticarum utriuſ uiſ{us} per centrum foraminis uueæ tranſeuntes, <lb/>in uno uiſibilis puncto ſemper concurrunt: & ſunt perpendiculares ſuperficiei uiſ{us}. 32. 35 p 3.</head> <p> <s xml:id="echoid-s3987" xml:space="preserve">DEclaratum eſt in primo tractatu [18 n] quòd uiſus nihil comprehendat ex uiſibilibus, niſi <lb/>ſecundum uerticationes refractas linearum radialium:</s> <s xml:id="echoid-s3988" xml:space="preserve"> & quòd ordo uiſibilium & partium <lb/>eorum non comprehenditur, niſi ex ordinatione linearum radialium.</s> <s xml:id="echoid-s3989" xml:space="preserve"> Et dictum eſt etiam <lb/>[27 n 1] quòd unum uiſum, quod comprehenditur duobus oculis ſimul, non comprehenditur <lb/>unum, niſi quando poſitio eius in reſpectu duorum oculorum fuerit poſitio conſimilis:</s> <s xml:id="echoid-s3990" xml:space="preserve"> & quòd ſi <lb/>poſitio fuerit diuerſa:</s> <s xml:id="echoid-s3991" xml:space="preserve"> tunc comprehendetur unum duo.</s> <s xml:id="echoid-s3992" xml:space="preserve"> Sed unumquodq;</s> <s xml:id="echoid-s3993" xml:space="preserve"> uiſibilium aſſuetorum, <lb/>quæ ſemper comprehenduntur à duobus uiſibus, ſemper comprehendetur unum.</s> <s xml:id="echoid-s3994" xml:space="preserve"> Vnde oportet <lb/>nos declarare, quomodo unum uiſum comprehendatur à duobus uiſibus unum in maiore parte <lb/>temporis & in pluribus poſitionibus:</s> <s xml:id="echoid-s3995" xml:space="preserve"> & quomodo poſitio unius uiſi ab ambobus oculis in maiore <lb/>parte temporis, & in pluribus erit conſimilis.</s> <s xml:id="echoid-s3996" xml:space="preserve"> Et declarabimus etiã <lb/> <anchor type="figure" xlink:label="fig-0082-01a" xlink:href="fig-0082-01"/> quomodo poſitio unius uiſi ab ambobus uiſibus erit poſitio diuer-<lb/>ſa, & quomodo accidat hoc.</s> <s xml:id="echoid-s3997" xml:space="preserve"> Et iam diximus hoc in primo tractatu <lb/>[27 n] & declarauimus ipſum uniuerſaliter, non determinatè.</s> <s xml:id="echoid-s3998" xml:space="preserve"> Dica <lb/>mus ergo quòd cum inſpiciẽs inſpexerit aliquod uiſum, tunc uterq;</s> <s xml:id="echoid-s3999" xml:space="preserve"> <lb/>uiſus erit in oppoſitione illius uiſi:</s> <s xml:id="echoid-s4000" xml:space="preserve"> & cum inſpiciens direxerit pu-<lb/>pillam ad illud uiſum:</s> <s xml:id="echoid-s4001" xml:space="preserve"> tunc uterq;</s> <s xml:id="echoid-s4002" xml:space="preserve"> uiſus diriget pupillam ad illud ui-<lb/>ſum directione æquali.</s> <s xml:id="echoid-s4003" xml:space="preserve"> Et cum uiſus fuerit motus ſuper rem uiſam:</s> <s xml:id="echoid-s4004" xml:space="preserve"> <lb/>tunc uterq;</s> <s xml:id="echoid-s4005" xml:space="preserve"> uiſus mouebitur ſuper illud.</s> <s xml:id="echoid-s4006" xml:space="preserve"> Et cum uiſus direxerit pu-<lb/>pillam ad rem uiſam:</s> <s xml:id="echoid-s4007" xml:space="preserve"> tunc axes duorum uiſuum congregabuntur in <lb/>illa re uiſa, & coniungentur in aliquo puncto illius ſuperficiei.</s> <s xml:id="echoid-s4008" xml:space="preserve"> Et ſi <lb/>inſpiciens mouerit uiſum per illam rem uiſam:</s> <s xml:id="echoid-s4009" xml:space="preserve"> tũc illi duo axes mo-<lb/>uebuntur ſimul ſuper ſuperficiẽ illius uiſi, & per omnes partes eius.</s> <s xml:id="echoid-s4010" xml:space="preserve"> <lb/>Et uniuerſaliter duo oculi ſunt æquales in omnibus ſuis diſpoſitio-<lb/>nibus:</s> <s xml:id="echoid-s4011" xml:space="preserve"> & uirtus ſenſibilis, quæ eſt in eis, eſt eadem, & actio & paſsio <lb/>eorum ſemper eſt æqualis & omnino cõſimilis.</s> <s xml:id="echoid-s4012" xml:space="preserve"> Et ſi alter uiſus fue-<lb/>rit motus ad uidendum, ſtatim reliquus mouebitur ad illud uiſum <lb/>illo eodem motu:</s> <s xml:id="echoid-s4013" xml:space="preserve"> & ſi alter uiſus quieuerit, reliquus quieſcit.</s> <s xml:id="echoid-s4014" xml:space="preserve"> Et im-<lb/>poſsibile eſt, ut alter uiſus moueatur ad uidẽdum, & reliquus quie-<lb/>ſcat, niſi impediatur.</s> <s xml:id="echoid-s4015" xml:space="preserve"> Et declaratũ eſt in præteritis [19 n 1] quòd in-<lb/>ter quodlibet uiſum & cẽtrum uiſus eſt pyramis imaginabilis apud <lb/>uiſionem, cuius uertex eſt centrum uiſus, & baſis ſuperficies uiſi, <lb/>quod uiſus comprehendit:</s> <s xml:id="echoid-s4016" xml:space="preserve"> & iſta pyramis continet omnes uertica-<lb/>tiones, ex quibus comprehendit illã rem uiſam.</s> <s xml:id="echoid-s4017" xml:space="preserve"> Cum ergo duo axes <lb/>amborum uiſuum fuerint cõiuncti in aliquo puncto ſuperficiei uiſi:</s> <s xml:id="echoid-s4018" xml:space="preserve"> <lb/>tunc ſuperficies uiſi erit baſis communis ambabus pyramidibus ra-<lb/>dialibus, figuratis inter duo cẽtra amborum uiſuum & illud uiſum:</s> <s xml:id="echoid-s4019" xml:space="preserve"> <lb/>& tunc poſitio puncti, in quo axes ſunt cõiuncti apud ambos uiſus, <lb/>eſt poſitio cõſimilis:</s> <s xml:id="echoid-s4020" xml:space="preserve"> quia eſt oppoſitũ duobus medijs amborum ui-<lb/>ſuum, & duo axes, qui ſunt inter illud & duos uiſus, ſunt perpendi-<lb/>culares ſuper ſuperficiem duorum uiſuum.</s> <s xml:id="echoid-s4021" xml:space="preserve"/> </p> <div xml:id="echoid-div135" type="float" level="0" n="0"> <figure xlink:label="fig-0082-01" xlink:href="fig-0082-01a"> <variables xml:id="echoid-variables6" xml:space="preserve">a e g b f z q x c u d</variables> </figure> </div> </div> <div xml:id="echoid-div137" type="section" level="0" n="0"> <head xml:id="echoid-head165" xml:space="preserve" style="it">3. Sit{us} uiſibilis erga utrun uiſum eſt plerun ſit{us} ſimilis. Ita axes pyramidum optica-<lb/>rum & lineæ ab utro uiſu ductæ ad cõcurſum duorum axιum, factũ in recta linea adutrun <lb/>axem perpendiculari, ſunt æquales. 40. 42 p 3.</head> <p> <s xml:id="echoid-s4022" xml:space="preserve">QVod autẽ remanet de ſuperficie uiſi, inter quodlibet punctũ eius, & inter duo cẽtra ambo-<lb/>rum uiſuũ, ſunt duæ lineæ, quarũ poſitio in reſpectu duorũ axiũ, erit poſitio cõſimilis in par <lb/>te ſcilicet:</s> <s xml:id="echoid-s4023" xml:space="preserve"> quoniã omnes duæ lineæ imaginabiles inter duo cẽtra duorũ uiſuum & punctũ <lb/>ſuperficiei uiſæ, in quo coniungũtur duo axes duorũ uiſuũ:</s> <s xml:id="echoid-s4024" xml:space="preserve"> erunt declinabiles à duobus axibus ad <lb/>unã partẽ.</s> <s xml:id="echoid-s4025" xml:space="preserve"> Nã omne punctũ ſuperficiei uiſi, in quo duo axes coniungũtur, declinabit à puncto con-<lb/>iunctionis ad eandẽ partẽ:</s> <s xml:id="echoid-s4026" xml:space="preserve"> punctũ uerò cõiunctionis eſt ſuper utrumq;</s> <s xml:id="echoid-s4027" xml:space="preserve"> axem.</s> <s xml:id="echoid-s4028" xml:space="preserve"> Remotiones autem <lb/>iſtarũ linearum à duobus axibus ſunt æquales:</s> <s xml:id="echoid-s4029" xml:space="preserve"> quoniã omnes duæ lineæ exeuntes à duobus cẽtris <lb/>duorũ uiſuum ad quodlibet punctum punctorũ ualde propinquorũ puncto cõiunctionis, æquali-<lb/>ter diſtant à duobus axibus, quantũ ad ſenſum.</s> <s xml:id="echoid-s4030" xml:space="preserve"> Duo enim axes exeuntes ad punctũ cõiunctionis, <lb/>erũt æquales, aut nõ erit inter eos diuerſitas ſenſibilis, quãdo res uiſa nõ fuerit ualde propinqua ui-<lb/>ſui, & diſtãtia eius à uiſu fuerit mediocris.</s> <s xml:id="echoid-s4031" xml:space="preserve"> Et ſimiliter eſt diſpoſitio cuiuslibet pũcti multũ propin-<lb/>qui pũcto cõiunctionis, ſcilicet, quòd omnes duæ lineæ exeũtes à duobus cẽtris duorũ uiſuum ad <lb/>quodlibet punctũ eorũ, ferè nõ differũt in longitudine quantùm ad ſenſum, ſed ferè erũt æquales.</s> <s xml:id="echoid-s4032" xml:space="preserve"/> </p> <pb o="77" file="0083" n="83" rhead="OPTICAE LIBER III."/> </div> <div xml:id="echoid-div138" type="section" level="0" n="0"> <head xml:id="echoid-head166" xml:space="preserve" style="it">4. Duærectæ lineæ ab utro uiſu ductæad concurſum duorum axium, factum in recta linea <lb/>ad utrun axem obliqua, ſunt ferè inæquales. 41 p 3.</head> <p> <s xml:id="echoid-s4033" xml:space="preserve">QVando uero lineæ duæ declinantes, fuerint coniunctæ in ſuperficie, in qua ſunt duo axes, <lb/>erunt inæ quales.</s> <s xml:id="echoid-s4034" xml:space="preserve"> Nãlinea, quæ exit ex puncto, in quo duo axes coniunguntur, ad punctum <lb/>declinans ab illo, continet cũ duobus axibus angulos inæquales, & duo axes ſunt æquales, <lb/>& linea copulans duo puncta, eſt cõmunis.</s> <s xml:id="echoid-s4035" xml:space="preserve"> Quapropter duæ lineæ declinãtes erunt inæquales:</s> <s xml:id="echoid-s4036" xml:space="preserve"> ſed <lb/>iſta inæqualitas nõ operatur in ſenſum, ſi punctũ declinans fuerit propinquum puncto cõiunctio-<lb/> <anchor type="figure" xlink:label="fig-0083-01a" xlink:href="fig-0083-01"/> nis.</s> <s xml:id="echoid-s4037" xml:space="preserve"> Si autem duæ lineæ declinãtes fuerint ſub axi <lb/>bus, aut ſuper illos, poſſunt eſſe æquales.</s> <s xml:id="echoid-s4038" xml:space="preserve"> Duo enim <lb/>anguli, quos cõtinent duo axes cũ linea cõtinuante <lb/>duo pũcta, poſſunt eſſe æquales, ſi punctũ fuerit ſub <lb/>axibus, aut ſuper eos.</s> <s xml:id="echoid-s4039" xml:space="preserve"> Et in poſitionibus, quę ſunt in <lb/>ter has duas poſitiones, erit diuerſitas, quæ eſt inter <lb/>duas declinãtes, minor quàm diuerſitas, quæ eſt in-<lb/>ter duas lineas primas declinãtes:</s> <s xml:id="echoid-s4040" xml:space="preserve"> & ſic nõ erit inter <lb/>eas differẽtia operãs in ſenſum.</s> <s xml:id="echoid-s4041" xml:space="preserve"> Ergo duæ lineæ ex-<lb/>euntes à duobus cẽtris duorũ uiſuum ad pũcta pro <lb/>pinqua puncto, in quo coniungũtur duo axes, non <lb/>differũt ferè in longitudine, quantùm ad ſenſum:</s> <s xml:id="echoid-s4042" xml:space="preserve"> & <lb/>axes ſunt æquales:</s> <s xml:id="echoid-s4043" xml:space="preserve"> & linea quæ copulat punctũ c on <lb/>iunctionis cũ puncto declinãte, ad quod exeũt duæ <lb/>lineæ à duobus centris, eſt cõmunis duobus trian-<lb/>gulis factis ex iſtis lineis.</s> <s xml:id="echoid-s4044" xml:space="preserve"> Ergo duo anguli, qui ſunt <lb/>apud duo centra duorũ uiſuum, quibus ſubtẽditur <lb/>apud ſuperficiem uiſi linea cõmunis, erũt æquales:</s> <s xml:id="echoid-s4045" xml:space="preserve"> <lb/>aut ferè inter eos nõ eſt diuerſitas ſenſibilis:</s> <s xml:id="echoid-s4046" xml:space="preserve"> & iſti duo anguli ſemper erũt minimi, quãdo punctum <lb/>fuerit ualde propinquũ cõiunctioni duorũ axium.</s> <s xml:id="echoid-s4047" xml:space="preserve"> Et cũ duæ lineæ, quæ exeunt ad quodlibet pun-<lb/>ctum propinquũ puncto cõiunctionis, continent cũ duobus axibus angulos æquales:</s> <s xml:id="echoid-s4048" xml:space="preserve"> tũc remotio <lb/>quarumlibet duarũ linearum, exeuntium ad idẽ punctum punctorũ propinquorum puncto cõiun-<lb/>ctionis à duobus axibus duorũ uiſuum, erit remotio æqualis.</s> <s xml:id="echoid-s4049" xml:space="preserve"> Ergo poſitio cuiuslibet puncti ſuper-<lb/>ficiei uiſi, in quo coniunguntur duo axes uiſuum, ſi fuerit propinquum puncto cõiunctionis, in re-<lb/>ſpectu duorũ uiſuũ, eſt poſitio cõſimilis in parte & in remotione à duobus axibus.</s> <s xml:id="echoid-s4050" xml:space="preserve"> Diſpoſitio autẽ <lb/>in punctis remotis à puncto cõiunctionis, declinãtibus ad unã partẽ ab ambobus axibus, eſt talis.</s> <s xml:id="echoid-s4051" xml:space="preserve"> <lb/>Anguli, qui ſuntinter duas lineas exeũtes ad aliquod punctum eorũ & inter duos axes, fortaſſe dif-<lb/>ferunt diuerſitate aliquantula:</s> <s xml:id="echoid-s4052" xml:space="preserve"> & poſitio omniũ huiuſmodi punctorũ remotorum à puncto cõiun-<lb/>ctionis in reſpectu duorũ uiſuum, eſt poſitio cõſimilis in parte tantùm:</s> <s xml:id="echoid-s4053" xml:space="preserve"> ſed nõ in remotione à duo-<lb/>bus axibus.</s> <s xml:id="echoid-s4054" xml:space="preserve"> Poſitio igitur cuiuslibet pũcti uiſi cõprehenſi ambobus uiſibus, cũ fuerit alicuius quan <lb/>titatis & propinquarum diametrorũ, apud duos uiſus eſt poſitio conſimilis in parte, & in remotio-<lb/>ne.</s> <s xml:id="echoid-s4055" xml:space="preserve"> Quapropter forma eius ſtatuetur in duobus locis cõſimilis poſitionis à duobus uiſibus:</s> <s xml:id="echoid-s4056" xml:space="preserve"> & cum <lb/>uiſum cõprehenſum ambobus uiſibus, fuerit maximarũ diametrorũ:</s> <s xml:id="echoid-s4057" xml:space="preserve"> tũc poſitio eius puncti, in quo <lb/>coniungũtur duo axes, erit poſitio cõſimilis apud duos uiſus Et quantò magis appropinquauerint <lb/>illi duo pũcta, quæ ſunt in ſuperficie illius uiſi, tantò magis poſitio illorũ apud duos uiſus erit cõſi-<lb/>milis in parte & in remotione ſimul.</s> <s xml:id="echoid-s4058" xml:space="preserve"> Puncta autẽ, quæ ſunt in ſuperficie illius uiſi, remota à puncto <lb/>cõiunctionis, & declinãtia ab ambobus axibus ad unã partẽ, habent poſitionẽ conſimilem in parte <lb/>apud duos uiſus, & in remotione fortè conſimilẽ, & fortè nõ.</s> <s xml:id="echoid-s4059" xml:space="preserve"> Forma igitur partis, quę eſt apud pun-<lb/>ctum cõiunctionis huius uiſi, & eius, quod cõtinet punctũ coniunctionis, & eius, quod eſt illi pro-<lb/>pinquum, inſtituitur in duobus locis duorũ uiſuũ cõſimilis poſitionis in omnibus diſpoſitionibus.</s> <s xml:id="echoid-s4060" xml:space="preserve"> <lb/>Et inſtituentur formæ partiũ reſiduarũ remotarũ à puncto cõiunctionis circundantiũ partem cõ-<lb/>ſimilis poſitionis cõtinuæ cũ forma partis cõſimilis poſitionis:</s> <s xml:id="echoid-s4061" xml:space="preserve"> & ſic uniuerſum duarũ formarũ in-<lb/>ſtituitur in duob locis duorũ uiſuũ, inter quæ nõ eſt maxima differẽtia in poſitione:</s> <s xml:id="echoid-s4062" xml:space="preserve"> & ſi fuerit, erit <lb/>extrema tantùm, & erit modica propter cõtinuationem duorũ extremorũ cũ duobus medijs, quæ <lb/>ſunt, cõſimilis poſitionis.</s> <s xml:id="echoid-s4063" xml:space="preserve"> Et hocerit, cũ duo uiſus fixi fuerint in oppoſitione uiſi, & duo axes fuerint <lb/>fixi in uno puncto eius.</s> <s xml:id="echoid-s4064" xml:space="preserve"> Cũ autem duo uiſus fuerint moti ſuper rem uiſam:</s> <s xml:id="echoid-s4065" xml:space="preserve"> & duo axes fuerint trãs-<lb/>lati ab illo pũcto:</s> <s xml:id="echoid-s4066" xml:space="preserve"> & fuerint moti ſimul per ſuperficiẽ uiſi:</s> <s xml:id="echoid-s4067" xml:space="preserve"> tũc poſitio cuiuslibet puncti illius uiſi, & <lb/>poſitio punctorũ propinquorũ illi, in reſpectu duorũ uiſuum apud coniunctionẽ duorum axiũ in <lb/>ipſo, erit poſitio cõſimilis ualde.</s> <s xml:id="echoid-s4068" xml:space="preserve"> Et forma cuiuslibet partis uiſi apud motum duorũ axium per ſu-<lb/>perficiẽ, erit in duob.</s> <s xml:id="echoid-s4069" xml:space="preserve"> locis poſitionis conſimilis apud duos uiſus:</s> <s xml:id="echoid-s4070" xml:space="preserve"> & ſic forma omnium partium uiſi <lb/>apud motum & intuitionem, erit conſimilis diſpoſitionis apud ambos uiſus.</s> <s xml:id="echoid-s4071" xml:space="preserve"/> </p> <div xml:id="echoid-div138" type="float" level="0" n="0"> <figure xlink:label="fig-0083-01" xlink:href="fig-0083-01a"> <variables xml:id="echoid-variables7" xml:space="preserve">e r g b z f k m a n l c u d</variables> </figure> </div> </div> <div xml:id="echoid-div140" type="section" level="0" n="0"> <head xml:id="echoid-head167" xml:space="preserve" style="it">5. E plurib. uiſibilib. ordinatim intraopticos axes diſpoſitis: remotiora incertè uidẽtur. 50 p 3.</head> <p> <s xml:id="echoid-s4072" xml:space="preserve">ET ſimiliter etiam quando uiſus comprehẽderit uiſibilia ſeparata in eadem hora ſimul:</s> <s xml:id="echoid-s4073" xml:space="preserve"> & duo <lb/>axes fuerint cõiuncti in aliquo eorũ:</s> <s xml:id="echoid-s4074" xml:space="preserve"> & illud uiſum, in quo ſunt cõluncti duo axes, fuerit pro-<lb/>pinquarum diametrorũ:</s> <s xml:id="echoid-s4075" xml:space="preserve"> tunc forma illius uiſi inſtituetur in duobus locis duorũ uiſuum cõ-<lb/>ſimilis poſitionis.</s> <s xml:id="echoid-s4076" xml:space="preserve"> Et etiã forma eius, quod propinquum eſt illi uiſo, ſi fuerit paruæ quãtitatis:</s> <s xml:id="echoid-s4077" xml:space="preserve"> inſti-<lb/> <pb o="78" file="0084" n="84" rhead="ALHAZEN"/> tuetur in duobus locis duorum uiſuum, inter quorum poſitiones non erit differentia ſenſibilis.</s> <s xml:id="echoid-s4078" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0084-01a" xlink:href="fig-0084-01"/> Forma autem uiſi remoti à uiſo, in quo duo axes coniungun-<lb/>tur, quando ambo uiſus comprehendunt illud uiſum, dum duo <lb/>axes ſunt fixi in illo uiſo:</s> <s xml:id="echoid-s4079" xml:space="preserve"> inſtituetur in duobus locis duorũ ui-<lb/>ſuum conſimilis poſitionis in parte tantùm, & non inremotio-<lb/>ne:</s> <s xml:id="echoid-s4080" xml:space="preserve"> aut non omnes partes eorum erunt conſimilis poſitionis <lb/>in remotione à duobus axibus:</s> <s xml:id="echoid-s4081" xml:space="preserve"> nec forma erit certificata.</s> <s xml:id="echoid-s4082" xml:space="preserve"> Dein-<lb/>de ſi duo uiſus fuerint moti, & duo axes:</s> <s xml:id="echoid-s4083" xml:space="preserve"> & fuerint coniuncti in <lb/>unoquoq;</s> <s xml:id="echoid-s4084" xml:space="preserve"> uiſibilium comprehenſorũ ſimul:</s> <s xml:id="echoid-s4085" xml:space="preserve"> tunc forma utriuſq;</s> <s xml:id="echoid-s4086" xml:space="preserve"> <lb/>eorum inſtituetur in duobus locis cõſimilis poſitionis in reſpe-<lb/>ctu duorum uiſuum in parte & in remotione:</s> <s xml:id="echoid-s4087" xml:space="preserve"> & tunc certificabi-<lb/>tur forma uniuſcuiuſq;</s> <s xml:id="echoid-s4088" xml:space="preserve"> illorum uiſibilium.</s> <s xml:id="echoid-s4089" xml:space="preserve"> Et multoties coniun-<lb/>guntur duo axes amborũ uiſuum in aliquo uiſo:</s> <s xml:id="echoid-s4090" xml:space="preserve"> & cum hoc duo <lb/>uiſus comprehẽdent aliam rem uiſam, cuius poſitio in reſpectu <lb/>amborum uiſuum erit diuerſa in parte.</s> <s xml:id="echoid-s4091" xml:space="preserve"> Et hoc erit, quando illud <lb/>aliud uiſum fuerit propin quius ambobus uiſibus uiſo, in quo di <lb/>ſtinguuntur duo axes:</s> <s xml:id="echoid-s4092" xml:space="preserve"> & fuerit ſimul inter duos axes:</s> <s xml:id="echoid-s4093" xml:space="preserve"> aut fuerit <lb/>remotius ab ambobus uiſibus uiſo, in quo coniunguntur duo <lb/>axes, & fuerit etia inter duos axes, cũ fuerimus imaginati eos ex <lb/>tenſospoſt cõiunctionẽ:</s> <s xml:id="echoid-s4094" xml:space="preserve"> & uiſum, in quo cõiungũtur duo axes, <lb/>nõ cooperiet uiſum, qđ eſt remotius ipſo, aut cooperiet quiddã <lb/>illius.</s> <s xml:id="echoid-s4095" xml:space="preserve"> His ergo modιs fit cõprehenſio uiſibiliũ ambobus uiſibus.</s> <s xml:id="echoid-s4096" xml:space="preserve"/> </p> <div xml:id="echoid-div140" type="float" level="0" n="0"> <figure xlink:label="fig-0084-01" xlink:href="fig-0084-01a"> <variables xml:id="echoid-variables8" xml:space="preserve">n m a b k c e d f g p h q ſ r o</variables> </figure> </div> </div> <div xml:id="echoid-div142" type="section" level="0" n="0"> <head xml:id="echoid-head168" xml:space="preserve" style="it">6. Si duæ rectæ lineæ à medio nerui cõmunis ſint contermi-<lb/>nærectæ cõnectenti centra for aminum gyrineruorum cauo-<lb/>rum: conſtituent triangulum æquicrurum. 30 p 3.</head> <p> <s xml:id="echoid-s4097" xml:space="preserve">ET etiam declaratum eſt in ſecundo tractatu [1.</s> <s xml:id="echoid-s4098" xml:space="preserve">42 n] quòd <lb/>axis radialis in utroq;</s> <s xml:id="echoid-s4099" xml:space="preserve"> uiſu eſt eadẽ linea, quę nõ tranſmu-<lb/>tatur:</s> <s xml:id="echoid-s4100" xml:space="preserve"> & quòd pertranſit centra omniũ tunicarum uiſus, & <lb/>extenditur rectè per centra omniũ tunicarum ad mediũ loci in <lb/>curuationis ex cõcauo nerui, ſuper quem cõponitur oculus, qui eſt apud foramen, quod eſt in con-<lb/>cauo oſsis:</s> <s xml:id="echoid-s4101" xml:space="preserve"> & quòd eſt inſeparabilis ab omnibus cẽtris:</s> <s xml:id="echoid-s4102" xml:space="preserve"> & quòd poſitio eius apud omnes partes ui-<lb/> <anchor type="figure" xlink:label="fig-0084-02a" xlink:href="fig-0084-02"/> ſus, eſt poſitio ſemper eadẽ, nõ tranſmutabilis apud <lb/>motũ uiſus, nec apud quietem eius:</s> <s xml:id="echoid-s4103" xml:space="preserve"> & quòd poſitio <lb/>duorũ axium apud duos uiſus eſt poſit<unsure/>io conſimilis <lb/>in reſpectu amborũ uiſuum, apud cõcauitatem ner-<lb/>ui cõmunis, ex quo ultimum ſentiens comprehẽdit <lb/>formas uiſibilium.</s> <s xml:id="echoid-s4104" xml:space="preserve"> Imaginemur ergo lineam rectam <lb/>copulãtem duo centra duorũ foraminum, quæ ſunt <lb/>in duabus concauitatibus duorum oſsium cõtinen-<lb/>tiũ duos oculos:</s> <s xml:id="echoid-s4105" xml:space="preserve"> & imaginemur duas lineas exeun-<lb/>tes à duobus centris duorũ foraminum, extenſas in <lb/>duobus medijs duarũ concauitatum neruorum.</s> <s xml:id="echoid-s4106" xml:space="preserve"> Hæ <lb/>ergo lineæ cõiunguntur in medio concauitatis ner-<lb/>uι communis:</s> <s xml:id="echoid-s4107" xml:space="preserve"> quia poſitio duorum neruorum in re-<lb/>ſpectu communis nerui, eſt poſitio cõſimilis [per 4 <lb/>n 1] & poſitio duarum harum linearum apud lineam <lb/>copulãtem duo centra duorum foraminum, erit po-<lb/>ſitio conſimilis:</s> <s xml:id="echoid-s4108" xml:space="preserve"> quia duorum neruorum poſitiones, <lb/>in reſpectu duorum foraminum, eſt poſitio cõſimi-<lb/>lis [per 4 n 1] & ſic duo anguli, qui ſunt inter has duas lineas & lineam copulatem duo centra duo-<lb/>rum foraminum, erunt æquales [ſecus diſsimilis eſſet poſitio neruorum.</s> <s xml:id="echoid-s4109" xml:space="preserve">]</s> </p> <div xml:id="echoid-div142" type="float" level="0" n="0"> <figure xlink:label="fig-0084-02" xlink:href="fig-0084-02a"> <variables xml:id="echoid-variables9" xml:space="preserve">a r t</variables> </figure> </div> </div> <div xml:id="echoid-div144" type="section" level="0" n="0"> <head xml:id="echoid-head169" xml:space="preserve" style="it">7. Si recta linea ſit à medio nerui communis admedium rectæ lineæ connectentis centra fo-<lb/>raminum gyrineruorum cauorum: erit ad ipſam perpendicularis. 33 p 3.</head> <p> <s xml:id="echoid-s4110" xml:space="preserve">ET imaginemur etiã lineã copulantem duo cẽtra duorũ foraminũ, diuiſam in duo æqualia:</s> <s xml:id="echoid-s4111" xml:space="preserve"> & <lb/>imaginemur lineã exeuntẽ à puncto, qđ eſt in medio cõcauitatis nerui cõmunis, in quo duæ <lb/>lineæ extenſæ in cõcauitatibus duorũ neruorũ ſunt cõiunctæ, extẽſam ad punctũ diuidẽs li-<lb/>neã copulantẽ duo cẽtra duorũ foraminã in duo æqualia.</s> <s xml:id="echoid-s4112" xml:space="preserve"> Hęcigitur linea erit perpẽdicularis ſuper <lb/>lineã copulantẽ duo cẽtra duorũ foraminũ [Nã recta cõnectens cẽtra duorũ foraminũ, fit baſis tri-<lb/>anguli æquicruri, cuius latera, ſunt rectæ à medio nerui cõmunis:</s> <s xml:id="echoid-s4113" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s4114" xml:space="preserve"> ſi recta ſit à uertice in mediũ <lb/>baſis, erit քpẽdicularis ad baſim, ք 8 p.</s> <s xml:id="echoid-s4115" xml:space="preserve"> 10 d 1.</s> <s xml:id="echoid-s4116" xml:space="preserve">] Et imaginemur iſtã քpẽdicularẽ extẽſam rectè in par-<lb/>tẽ oppoſitã uiſui:</s> <s xml:id="echoid-s4117" xml:space="preserve"> & ſic iſta linea erit fixa in eodẽ ſtatu, & poſitio ei<emph style="sub">9</emph> nõ trãſmutabitur:</s> <s xml:id="echoid-s4118" xml:space="preserve"> ꝗa pũctũ, qđ <lb/>eſt in medio cõcauitatis nerui cõmunis, in quo duę lineæ extẽſæ in duob.</s> <s xml:id="echoid-s4119" xml:space="preserve"> medιjs concauitatũ duo-<lb/>rũ neruorũ ſunt cõiunctæ, eſt unũ nõ tranſmutabile:</s> <s xml:id="echoid-s4120" xml:space="preserve"> & punctũ etiã, qđ diuidit lineã copulantẽ duo <lb/> <pb o="79" file="0085" n="85" rhead="OPTICAE LIBER III."/> centra duorum foraminum, eſt unum nõ tranſmutabile.</s> <s xml:id="echoid-s4121" xml:space="preserve"> Quapropter poſitio lineæ tranſeuntis per <lb/>illa, eſt una poſitio, non tranſmutabilis.</s> <s xml:id="echoid-s4122" xml:space="preserve"> Hæc igitur linea uocetur axis communis.</s> <s xml:id="echoid-s4123" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div145" type="section" level="0" n="0"> <head xml:id="echoid-head170" xml:space="preserve" style="it">8. Si axes, communis & duo optici, in uno uiſibilis puncto concurrant: erunt in eodem plano <lb/>cum rectis, connectente centra foraminum gyrineruorum cauorum, & duab{us} à medio nerui <lb/>communis connectenti conterminis. 34 p 3.</head> <p> <s xml:id="echoid-s4124" xml:space="preserve">ET imaginemur apud punctũ aliquod iſtius lineę, in parte oppoſita uiſui aliquod uiſum, & ima <lb/>ginemur duos uiſus inſpicere illud uiſum, & duos axes ſimul cõiungi in puncto ſuperficiei ui <lb/>ſi, in quo axis communis occurrerit ſuperficiei illius uiſi:</s> <s xml:id="echoid-s4125" xml:space="preserve"> & hoc quidem poſsibile eſt in omni <lb/>uiſo, cuius ſitus ex duobus uiſibus eſt ſitus cõſimilis.</s> <s xml:id="echoid-s4126" xml:space="preserve"> Cum ergo duo axes fuerint cõiuncti in aliquo <lb/>puncto axis cõmunis, tunc duo axes & axis cõmunis, & linea, quæ copulat duo centra foraminum <lb/>duorum oſsium, & duę lineæ extenſę in concauitatibus duorũ neruorũ, omnia erunt in una ſuperſi <lb/>cie.</s> <s xml:id="echoid-s4127" xml:space="preserve"> Duo enim axes tranſeunt per centra duorum foraminũ:</s> <s xml:id="echoid-s4128" xml:space="preserve"> tranſeunt enim per duo media concaui <lb/>tatum duorũ neruorũ, in loco pyramidationis duorum neruorũ.</s> <s xml:id="echoid-s4129" xml:space="preserve"> Cum igitur duo axes fuerint con-<lb/>iuncti in axe cõmuni, erunt omnes in ſuperficie, in qua eſt axis cõmunis, [per 2 p 11] & ſimiliter linea <lb/>ſecans ipſam, quę copulat centra foraminũ duorũ oſsiũ, & duæ lineę extẽſę in cõcauitatibus duorũ <lb/>neruorũ:</s> <s xml:id="echoid-s4130" xml:space="preserve"> & duo axes de loco centrorum duorũ foraminũ, uſq;</s> <s xml:id="echoid-s4131" xml:space="preserve"> ad punctum cõiunctiõis, quod eſt in <lb/>axe cõmuni, erunt æquales:</s> <s xml:id="echoid-s4132" xml:space="preserve"> & poſitio eorũ apud axem communẽ, erit poſitio conſimilis:</s> <s xml:id="echoid-s4133" xml:space="preserve"> & duæ par <lb/>tes duorum axiũ, quę ſunt de centris duorũ uiſuum uſq;</s> <s xml:id="echoid-s4134" xml:space="preserve"> ad punctũ coniunctionis, erunt æquales:</s> <s xml:id="echoid-s4135" xml:space="preserve"> & <lb/>remotιo duorum centrorũ uiſuum à foraminibus duorum oſsium, & à centris duorum foraminũ, <lb/>eſt remotio æqualis:</s> <s xml:id="echoid-s4136" xml:space="preserve"> & etiam duæ partes duorum axium, quæ ſunt de ſuperficiebus duorũ uiſuum <lb/>uſq;</s> <s xml:id="echoid-s4137" xml:space="preserve"> ad punctum coniunctionis, erunt æquales:</s> <s xml:id="echoid-s4138" xml:space="preserve"> nam duæ medietates diametrorũ ſphærarum duo-<lb/>rum uiſuum ſunt æquales.</s> <s xml:id="echoid-s4139" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div146" type="section" level="0" n="0"> <head xml:id="echoid-head171" xml:space="preserve" style="it">9. Vtro uiſu uiſibile unum plerun uidetur. 28 p 3. Idem 27 n 1.</head> <p> <s xml:id="echoid-s4140" xml:space="preserve">ET quia ita eſt:</s> <s xml:id="echoid-s4141" xml:space="preserve"> poſitio puncti ſuperficiei uiſi, in quo coniuncti ſunt duo axes, apud duo puncta, <lb/>per quæ tranſeunt duo axes, erit poſitio conſimilis:</s> <s xml:id="echoid-s4142" xml:space="preserve"> & remotio eius ab eis erit æqualis.</s> <s xml:id="echoid-s4143" xml:space="preserve"> Et hæc <lb/>duo puncta ſuperficierum uiſuum ſuntilla, in quibus infigitur forma puncti, in quo coniuncti <lb/>ſunt duo axes.</s> <s xml:id="echoid-s4144" xml:space="preserve"> Et etiam poſitio utriuſq;</s> <s xml:id="echoid-s4145" xml:space="preserve"> duorum punctorũ, quæ ſunt in duobus axibus ſuperficierũ <lb/>duorum uiſuum, apud concauitatẽ nerui cõmunis, erit poſitio conſimilis.</s> <s xml:id="echoid-s4146" xml:space="preserve"> Et poſitio iſtorũ duorum <lb/>punctorum apud quodlibet punctum in axe communi, eſt poſitio conſimilis.</s> <s xml:id="echoid-s4147" xml:space="preserve"> Ergo poſitio duorum <lb/>punctorum, quę ſunt in duobus axibus ſuperficierũ duorum uiſuum, apud punctum axis cõmunis, <lb/>qui eſt in medio concauitatis nerui cõmunis, in quo ſunt coniunctæ duæ lineæ exeuntes à centris <lb/>duorum foraminũ, eſt poſitio ualde cõſimilis & æqualis.</s> <s xml:id="echoid-s4148" xml:space="preserve"> Et ambæ formæ, quæ inſtituuntur in duo-<lb/>bus punctis ſuperficierum duorum uiſuum, quæ ſunt in duobus axibus, cum peruenerint ad conca <lb/>uitatem communis nerui, infigentur in puncto, quod eſt in axe communi, quod eſt in medio conca <lb/>uitatis communis nerui, in quo lineæ ſunt coniunctæ, & efficietur una forma.</s> <s xml:id="echoid-s4149" xml:space="preserve"> Et cum duæ formæ, <lb/>quæ ſunt in duobus punctis, quæ ſunt in duobus axibus ſuperficierum duorum uiſuum, figuntur <lb/>in puncto, quod eſt in axe cõmuni, quod eſt in medio concauitatis nerui cõmunis:</s> <s xml:id="echoid-s4150" xml:space="preserve"> formæ, quæ ſunt <lb/>in punctis circundantibus utrunq;</s> <s xml:id="echoid-s4151" xml:space="preserve"> duorũ punctorũ, quæ ſunt in duobus axibus ſuperficierũ duorũ <lb/>uiſuũ, infiguntur in concauitate cõmunis nerui, in punctis circundantibus punctũ, quod eſt in axe <lb/>cõmuni.</s> <s xml:id="echoid-s4152" xml:space="preserve"> Et poſitio quorumlibet duorũ punctorum ſuperficierũ duorum uiſuũ, quorũ poſitio apud <lb/>duo puncta, in medio in duobus axibus duorũ uiſuum eſt poſitio cõſimilis in parte & in remotiõe:</s> <s xml:id="echoid-s4153" xml:space="preserve"> <lb/>apud idem punctũ concauitatis nerui cõmunis eſt poſitio conſimilis.</s> <s xml:id="echoid-s4154" xml:space="preserve"> Et puncta, quorũ poſitio apud <lb/>ipſum eſt poſitio conſimilis, declinabunt à puncto, quod eſt in axe cõmuni, quod eſt in loco cõiun-<lb/>ctionis linearum ex cõcauitate nerui cõmunis in partem, ad quã ambo puncta, quæ ſunt in ſuperfi <lb/>ciebus duorũ uiſuũ, declinant:</s> <s xml:id="echoid-s4155" xml:space="preserve"> & remotio eorũ ab ipſo erit ſecundũ remotiones eorũ à duobus axi-<lb/>bus:</s> <s xml:id="echoid-s4156" xml:space="preserve"> & duæ formæ, quę infiguntur in duobus punctis, quę ſunt cõſimilis poſitionis apud ſuperficies <lb/>duorum uiſuũ, peruenient ad illud idem punctũ concauitatis cõmunis ipſius nerui, & ſuperponen <lb/>tur illi apud illud punctũ, & efficietur una forma.</s> <s xml:id="echoid-s4157" xml:space="preserve"> Et poſitio uniuſcuiuſq;</s> <s xml:id="echoid-s4158" xml:space="preserve"> punctorũ ſuperficiei uiſi, <lb/>quæ ſunt in circuitu puncti, quod eſt in axe cõmuni, apud duos axes duorum uiſuũ eſt poſitio con-<lb/>ſimilis.</s> <s xml:id="echoid-s4159" xml:space="preserve"> Ergo forma cuiuslibet puncti eorũ infigetur in duobus uiſibus in duobus locis cõſimilis po <lb/>ſitionis, in reſpectu duorũ punctorũ, quæ ſunt in duobus axibus ſuperficierũ duorũ uiſuũ.</s> <s xml:id="echoid-s4160" xml:space="preserve"> Duæ er-<lb/>go formę uiſi, in quo cõiuncti ſunt tres axes, infiguntur in duobus medijs duarũ ſuperficierũ duorũ <lb/>uiſuũ.</s> <s xml:id="echoid-s4161" xml:space="preserve"> Et duę formę puncti, in quo ſunt cõiuncti tres axes, infigentur in duobus punctis, quę ſunt in <lb/>duobus axibus ſuperficierũ duorum uiſuũ.</s> <s xml:id="echoid-s4162" xml:space="preserve"> Et quodlibet punctũ duarum formarũ infigetur in duo-<lb/>bus locis cõſimilis poſitionis de duobus uiſibus:</s> <s xml:id="echoid-s4163" xml:space="preserve"> deinde duæ formæ uiſæ perueniẽt ad concauitatẽ <lb/>nerui cõmunis:</s> <s xml:id="echoid-s4164" xml:space="preserve"> & perueniẽt duæ formæ, quę ſunt in puncto, quod eſt in duobus axibus, ad punctũ, <lb/>quod eſt in cõmuni axe, & efficietur una forma.</s> <s xml:id="echoid-s4165" xml:space="preserve"> Et quælibet quę formę, quę ſunt in duobus punctis <lb/>conſimilis poſitionis à duobus uiſibus, peruenient ad idem punctũ punctorũ circundantiũ punctũ, <lb/>quod eſt in axe cõmuni:</s> <s xml:id="echoid-s4166" xml:space="preserve"> & ſic duę formę totius uiſi ſuperponentur ſibi, & efficietur una forma, & ſic <lb/>unũ cõprehendetur unũ.</s> <s xml:id="echoid-s4167" xml:space="preserve"> Secundũ ergo hũc modũ duę formę, quę infigẽtur duobus uiſibus ab uno <lb/>uiſo, cuius poſitio in reſpectu duorũ uiſuũ eſt conſimilis:</s> <s xml:id="echoid-s4168" xml:space="preserve"> efficiuntur una forma:</s> <s xml:id="echoid-s4169" xml:space="preserve"> & ſic ſentiẽs cõpre-<lb/>hendit unũ uiſum, licet duæ formę infigãtur ab eo in duobus uiſibus.</s> <s xml:id="echoid-s4170" xml:space="preserve"> Et cũ duæ formę, quę ſunt in <lb/>duob.</s> <s xml:id="echoid-s4171" xml:space="preserve"> pũctis, quę ſunt in duob.</s> <s xml:id="echoid-s4172" xml:space="preserve"> medijs ſuperficierũ duorũ uiſuũ, quę ſunt in duob.</s> <s xml:id="echoid-s4173" xml:space="preserve"> axibus, peruene <lb/>rint ad punctũ, qđ eſt in axe cõmuni:</s> <s xml:id="echoid-s4174" xml:space="preserve"> tũc quælibet duæ formæ infixæ in duab.</s> <s xml:id="echoid-s4175" xml:space="preserve"> ſuperficiebus duorũ <lb/> <pb o="80" file="0086" n="86" rhead="ALHAZEN"/> uiſuum in duobus punctis, quæ ſunt in duobus axibus, peruenient ſemper ad illud idem punctum <lb/>concauitatis nerui cõmunis, quod eſt in cõmuni axe.</s> <s xml:id="echoid-s4176" xml:space="preserve"> Nam duo puncta, per quæ tranſeunt duo axes <lb/>duorũ uiſuum nõ mutantur:</s> <s xml:id="echoid-s4177" xml:space="preserve"> quoniã poſitio duorũ axium apud duos uiſus ſemper eſt eadẽ poſitio, <lb/>non tranſmutabilis.</s> <s xml:id="echoid-s4178" xml:space="preserve"> Ergo punctũ concauitatis cõmunis nerui, ad quod perueniunt duæ formę, quę <lb/>infiguntur in duobus punctis, quę ſunt in duobus axibus ſuperficierũ duorum uiſuũ, ſemper eſt idẽ <lb/>punctũ:</s> <s xml:id="echoid-s4179" xml:space="preserve"> & eſt punctũ, quod eſt in cõmuni axe, in quo cõcurrunt duæ lineę exeuntes à duobus cen-<lb/>tris foraminũ duorum oſsium extenſorũ in duobus medijs concauitatũ duorum neruorũ.</s> <s xml:id="echoid-s4180" xml:space="preserve"> Iſtud igi <lb/>tur punctum, quod eſt in concauitate communis nerui, quod eſt in cõmuni axe, uocetur centrum.</s> <s xml:id="echoid-s4181" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div147" type="section" level="0" n="0"> <head xml:id="echoid-head172" xml:space="preserve" style="it">10. Concurſiis axium opticorum in axe communifacit uiſionem certißimam: extrà, tantò <lb/>certiorem, quantò axi propinquior fuerit. 44 p 3.</head> <p> <s xml:id="echoid-s4182" xml:space="preserve">HOc igitur declarato, declaratũ eſt, quòd forma cuiuslibet comprehenſi, quod cõprehenditur <lb/>ambobus uiſibus, in cuius ſuperficiei puncto concurrunt axes duorũ uiſuũ, infigitur in duo-<lb/>bus locis ſuperficierũ duorum uiſuum, quæ ſunt duo media ſuperficierũ duorum uiſuũ:</s> <s xml:id="echoid-s4183" xml:space="preserve"> dein <lb/>de iſtæ duæ formæ perueniunt à duobus uiſibus ad concauitatem cõmunis neruiad eundem locũ, <lb/>& ſuperponuntur ſibi, & efficitur una forma.</s> <s xml:id="echoid-s4184" xml:space="preserve"> Et duæ formæ puncti, in quo concurrunt duo axes ex <lb/>uiſo, infigentur in duobus punctis, quæ ſunt in duobus axibus ſuperficierũ duorum uiſuũ, & ibunt <lb/>ab iſtis duobus punctis ad punctũ centri concauitatis cõmunis nerui, & indifferenter, ſiue punctũ, <lb/>in quo concurrunt duo axes, fuerit in axe cõmuni, ſiue extrà.</s> <s xml:id="echoid-s4185" xml:space="preserve"> Sed tamẽ cum uiſum fuerit in axe com <lb/>muni, & duo axes cõcurrerint in puncto ipſius, quod eſt in axe cõmuni, tunc duæ formæ iſtius pun <lb/>cti erunt magis cõſimiles.</s> <s xml:id="echoid-s4186" xml:space="preserve"> Remotiones enim iſtius puncti à duobus punctis, in quibus figuntur duę <lb/>formę iſtius puncti ſuperficierũ duorum uiſuũ (& ſunt illa, quæ ſunt ſuper axes) erũt æquales:</s> <s xml:id="echoid-s4187" xml:space="preserve"> quo-<lb/>niam duo axes in hac diſpoſitione erunt æquales in longitudine.</s> <s xml:id="echoid-s4188" xml:space="preserve"> Et ſimiliter formæ cuiuslibet pun-<lb/>cti propinqui iſti puncto, cuius remotiones à duobus punctis, in quibus infiguntur formæ ſuæ, ſunt <lb/>æquales, quantùm ad ſenſum, erunt magis conſimiles, quàm duæ formæ uiſi, quod eſt extra cõmu-<lb/>nem axem.</s> <s xml:id="echoid-s4189" xml:space="preserve"> Quapropter forma uiſi, quod eſt in cõmuni axe, cum fuerit infixa in concauitate cõmu-<lb/>nis nerui, erit magis certificata.</s> <s xml:id="echoid-s4190" xml:space="preserve"> Sed cum uiſum fuerit extra cõmunem axem, & remotio non fuerit <lb/>maxima:</s> <s xml:id="echoid-s4191" xml:space="preserve"> tunc ſuæ duę formæ, quę infiguntur in duobus uiſibus, nõ maximè different.</s> <s xml:id="echoid-s4192" xml:space="preserve"> Quapropter <lb/>formæ eius, quæ infiguntur in concauitate nerui cõmunis, non erunt duæ.</s> <s xml:id="echoid-s4193" xml:space="preserve"> Cum uerò uiſum fuerit <lb/>extra cõmunem axem, & maximè fuerit remotũ ab ipſo:</s> <s xml:id="echoid-s4194" xml:space="preserve"> & axes duorũ uiſuũ cõcurrerint in aliquo <lb/>puncto ipſius:</s> <s xml:id="echoid-s4195" xml:space="preserve"> tũc forma eius infigetur in cõcauitate cõmunis nerui una forma:</s> <s xml:id="echoid-s4196" xml:space="preserve"> & forma pũcti eius, <lb/>in quo duo axes concurrunt, infigetur in puncto cõmunis centri:</s> <s xml:id="echoid-s4197" xml:space="preserve"> ſed tamen forma eius non erit ue-<lb/>rificata, ſed dubitabilis.</s> <s xml:id="echoid-s4198" xml:space="preserve"> Forma igitur puncti uiſi, in quo duo axes concurrunt, infigetur in omnibus <lb/>diſpoſitionibus, in puncto centri concauitatis cõmunis nerui, ſiue punctũ concurſus fuerit in com-<lb/>muni axe, ſiue extra illum:</s> <s xml:id="echoid-s4199" xml:space="preserve"> quod aũt remanet de forma uiſi, infigetur in circuitu puncti centri.</s> <s xml:id="echoid-s4200" xml:space="preserve"> Si aũt <lb/>uiſum fuerit minimi corporis, & propinquarũ diametrorum, & fuerit in cõmuni axe, uel prope:</s> <s xml:id="echoid-s4201" xml:space="preserve"> tũc <lb/>forma eius infigetur in cõcauitate cõmunis nerui una forma, & uerificata:</s> <s xml:id="echoid-s4202" xml:space="preserve"> & poſitio cuiuslibet pun <lb/>cti eius apud duos uiſus, eſt poſitio cõſimilis, ut prius declarauimus.</s> <s xml:id="echoid-s4203" xml:space="preserve"> Si uerò uiſum fuerit magni cor <lb/>poris & remotarũ diametrorum, & etiam fuerit in cõmuni axe:</s> <s xml:id="echoid-s4204" xml:space="preserve"> tunc forma illius partis, quæ eſt a-<lb/>pud locum coniunctionis duorum axium, quæ circundat punctum coniunctionis, infigetur in com <lb/>muni neruo una forma & uerificata, & forma reſiduarum partium infigetur continua cum forma i-<lb/>ſtius partis.</s> <s xml:id="echoid-s4205" xml:space="preserve"> Quapropter forma totius uiſi infigetur una in omnibus diſpoſitionibus:</s> <s xml:id="echoid-s4206" xml:space="preserve"> ſed tamen for <lb/>ma extremorum, & illorum, quæ remota ſunt à puncto concurſus, erit non certificata.</s> <s xml:id="echoid-s4207" xml:space="preserve"> Quoniam o-<lb/>mnis puncti remoti à puncto concurſus, figentur duæ formæ in duobus punctis conſimilis poſitio-<lb/>nis, in reſpectu amborum uiſuum in fine conſimilitudinis:</s> <s xml:id="echoid-s4208" xml:space="preserve"> ſed forma cuiuslibet puncti remoti à pun <lb/>cto concurſus, figetur in duobus punctis amborum uiſuum, quorum poſitio apud duos uiſus eſt po <lb/>ſitio conſimilis in parte, & fortè cõſimilis in remotione à duobus axibus, & fortè non conſimilis in <lb/>remotione à duobus axibus.</s> <s xml:id="echoid-s4209" xml:space="preserve"> Formę aũt eorum, quorũ remotio non eſt conſimilis, figentur in conca <lb/>uitate communis nerui, in duobus punctis obliquis à centro in una parte:</s> <s xml:id="echoid-s4210" xml:space="preserve"> & erunt duæ.</s> <s xml:id="echoid-s4211" xml:space="preserve"> Et ſi uiſum <lb/>fuerit unius coloris, tunc iſtud ferè nihil operabitur in ipſum, propter conſimilitudinem coloris & <lb/>identitatẽ formæ:</s> <s xml:id="echoid-s4212" xml:space="preserve"> Si autẽ uiſum habuerit diuerſos colores, aut fuerit in eo lineatio, aut pictura, aut <lb/>ſubtiles intentiones:</s> <s xml:id="echoid-s4213" xml:space="preserve"> tũc iſtud operatur in ipſum.</s> <s xml:id="echoid-s4214" xml:space="preserve"> Quapropter extremorũ forma erit dubitabilis, nõ <lb/>certificata.</s> <s xml:id="echoid-s4215" xml:space="preserve"> Et cum uiſum fuerit magni corporis & remotarum diametrorum, & axes amborum ui-<lb/>ſuum fuerint fixi in aliquo puncto eius, & immobiles:</s> <s xml:id="echoid-s4216" xml:space="preserve"> tunc forma eius apparet una, & locus concur <lb/>ſus eius, & illud, quod ei propinquum eſt, erunt certificata & indubitabilia:</s> <s xml:id="echoid-s4217" xml:space="preserve"> extrema autem, & <lb/>illa, quæ uicina ſunt eis, erunt non certificata propter duas cauſſas:</s> <s xml:id="echoid-s4218" xml:space="preserve"> quarum una eſt, quòd extre-<lb/>ma comprehendantur per radios remotos ab axe:</s> <s xml:id="echoid-s4219" xml:space="preserve"> quapropter non bene erunt manifeſta.</s> <s xml:id="echoid-s4220" xml:space="preserve"> Secun-<lb/>da eſt, quia non forma cuiuslibet puncti eius inſtituitur in concauitate communis nerui in uno <lb/>puncto, ſed quæ dam ſunt, quorum forma inſtituitur in duobus punctis, non in uno.</s> <s xml:id="echoid-s4221" xml:space="preserve"> Cum ergo <lb/>duo axes fuerint moti ſuper omnes partes huius uiſi:</s> <s xml:id="echoid-s4222" xml:space="preserve"> tunc certificabitur forma eius.</s> <s xml:id="echoid-s4223" xml:space="preserve"> Si autem <lb/>uiſum fuerit extra axem communem, & remotum ab ipſo:</s> <s xml:id="echoid-s4224" xml:space="preserve"> tunc forma eius non erit certificata.</s> <s xml:id="echoid-s4225" xml:space="preserve"> Porſi <lb/>tio enim cuiuslibet puncti illius apud ambos uiſus, non eſt poſitio conſimilis propter inæqua-<lb/>litatem remotionum puncti huius uiſi à duobus punctis ſuperficierum duorum uiſuum, in qui-<lb/>bus inſtituuntur duæ formæ eius, & à duobus axibus.</s> <s xml:id="echoid-s4226" xml:space="preserve"> Cum igitur ambo uiſus obliquabun-<lb/> <pb o="81" file="0087" n="87" rhead="OPTICAE LIBER III."/> tur ad huiuſinodi uiſum, adeò ut axis communis ueniat ad iſtud uiſum, aut prope, tunc certificabi-<lb/>tur forma eius.</s> <s xml:id="echoid-s4227" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div148" type="section" level="0" n="0"> <head xml:id="echoid-head173" xml:space="preserve" style="it">11. Viſibile intra axes opticos ſitum: ueluni uiſui rectè, reliquo obliquè oppoſitum: uidetur <lb/>geminum. 104.103 p 4.</head> <p> <s xml:id="echoid-s4228" xml:space="preserve">ET ſimiliter cum ambo uiſus comprehenderint multa uiſa ſimul:</s> <s xml:id="echoid-s4229" xml:space="preserve"> & axes amborum uiſuum ſi-<lb/>mul concurrerint in aliquod unum uiſorum illorum:</s> <s xml:id="echoid-s4230" xml:space="preserve"> & fuerint fixi in illo:</s> <s xml:id="echoid-s4231" xml:space="preserve"> reſidua autem uiſa <lb/>fuerint extra duos axes:</s> <s xml:id="echoid-s4232" xml:space="preserve"> & uiſum, in quo concurrunt duo axes, fuerit minimi corporis:</s> <s xml:id="echoid-s4233" xml:space="preserve"> tunc <lb/>forma uiſi, in quo concurrunt duo axes, in concauitate nerui communis, erit una forma & certifica <lb/>ta.</s> <s xml:id="echoid-s4234" xml:space="preserve"> Et ſi uiſum fuerit ſuper axem communem:</s> <s xml:id="echoid-s4235" xml:space="preserve"> tunc forma eius erit magis certificata, quàm forma ui-<lb/>ſi, quæ eſt extra axem communem, & ſi in ipſo concurrunt duo axes.</s> <s xml:id="echoid-s4236" xml:space="preserve"> Viſorum autem, quæ compre-<lb/>henduntur à uiſu in illo ſtatu, quæ ſunt propinqua uiſo, in quo duo axes concurrunt, ſi etiam fue-<lb/>rint ipſa minimi corporis:</s> <s xml:id="echoid-s4237" xml:space="preserve"> forma inſtituitur in concauitate communis nerui una, in qua non erit du <lb/>bitatio maxima:</s> <s xml:id="echoid-s4238" xml:space="preserve"> nam forma eius erit propinqua centro.</s> <s xml:id="echoid-s4239" xml:space="preserve"> Ex illis autem uiſibilibus, quæ compre-<lb/>henduntur à uiſu in iſto ſtatu, quod fuerit remotum à uiſo, in quo concurrunt duo axes:</s> <s xml:id="echoid-s4240" xml:space="preserve"> eius forma <lb/>inſtituetur in concauitate iſtius nerui, dubitabilis:</s> <s xml:id="echoid-s4241" xml:space="preserve"> & tunc aut erunt duæ formæ ſe mutuò pene-<lb/>trantes, quia ſunt in una parte:</s> <s xml:id="echoid-s4242" xml:space="preserve"> quapropter inæqualitas, quæ eſt inter ſuas poſitiones in remotione, <lb/>non erit maxima:</s> <s xml:id="echoid-s4243" xml:space="preserve"> unde duæ formæ ſe mutuò penetrabunt:</s> <s xml:id="echoid-s4244" xml:space="preserve"> aut forma quarundam partium erit du-<lb/>plex, & forma quarundam erit una:</s> <s xml:id="echoid-s4245" xml:space="preserve"> & ſic forma huiuſmodi uiſibilium erit dubitabilis in omnibus <lb/>diſpoſitionibus, propter diuerſitatem poſitionis radiorum exeuntium ad illa, & quia radij exeun-<lb/>tes ad illa, erunt remoti à duobus axibus.</s> <s xml:id="echoid-s4246" xml:space="preserve"> Forma autem obliqui uiſi à duobus axibus, remoti à loco <lb/>concurſus duorum axium, erit non certificata, dum fuerit remota à concurſu duorum axium.</s> <s xml:id="echoid-s4247" xml:space="preserve"> Cum <lb/>autem duo axes fuerint remoti, & concurrerint in ipſo:</s> <s xml:id="echoid-s4248" xml:space="preserve"> tunc uerificabitur forma eius.</s> <s xml:id="echoid-s4249" xml:space="preserve"> Cum autem <lb/>duo axes duorum uiſuum concurrerint in aliquo uiſo, & hi duo uiſus comprehenderint aliud ui-<lb/>ſum propinquius duobus uiſibus, uiſo, in quo concurrunt duo axes:</s> <s xml:id="echoid-s4250" xml:space="preserve"> aut remotius:</s> <s xml:id="echoid-s4251" xml:space="preserve"> & fuerit etiam <lb/>inter duos axes:</s> <s xml:id="echoid-s4252" xml:space="preserve"> tunc poſitio eius apud duos uiſus erit diuerſa in parte.</s> <s xml:id="echoid-s4253" xml:space="preserve"> Nam cum fuerit inter duos <lb/>axes, erit dextrum unius axis, ſiniſtrum alterius, & radij exeuntes ad ipſum ab altero uiſo, erunt de-<lb/>xtri ab axe, & qui exeũt ad ipſum à reliquo uiſo, erunt ſiniſtri:</s> <s xml:id="echoid-s4254" xml:space="preserve"> & ſic poſitio eius apud duos uiſus erit <lb/>diuerſa in parte.</s> <s xml:id="echoid-s4255" xml:space="preserve"> Et forma huiuſmodi uiſorũ inſtituitur in duobus uiſibus, in duobus locis diuerſæ <lb/>poſitionis:</s> <s xml:id="echoid-s4256" xml:space="preserve"> & duæ formæ, quæ inſtituuntur in duobus uiſibus, perueniẽt ad duo loca diuerſa conca <lb/>uitatum communis nerui, & erunt à duobus lateribus centri.</s> <s xml:id="echoid-s4257" xml:space="preserve"> Quapropter erunt duæ formę, & non <lb/>ſuperponentur ſibi.</s> <s xml:id="echoid-s4258" xml:space="preserve"> Et ſimiliter cum fuerit uiſum in altero axe, & extra reliquum, forma eius inſti-<lb/>tuetur in concauitate communis nerui, in duobus locis, una ſcilicet in centro, & alia obliqua à cen-<lb/>tro, & non ſuperponentur ſibi.</s> <s xml:id="echoid-s4259" xml:space="preserve"> Secundum ergo hos modos inſtituetur forma uiſibilium in duobus <lb/>uiſibus, & in concauitate communis nerui.</s> <s xml:id="echoid-s4260" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div149" type="section" level="0" n="0"> <head xml:id="echoid-head174" xml:space="preserve" style="it">12. Viſibile aliàs unum: aliàs geminum uideri organo ostenditur. 108 p 4.</head> <p> <s xml:id="echoid-s4261" xml:space="preserve">OMnia autẽ, quę diximus, ſic poſſunt experimentari experimẽto:</s> <s xml:id="echoid-s4262" xml:space="preserve"> cum quo ueniet certifica-<lb/>tio.</s> <s xml:id="echoid-s4263" xml:space="preserve"> Accipiatur tabula leuis ligni:</s> <s xml:id="echoid-s4264" xml:space="preserve"> cuius longitudo ſit unius cubiti:</s> <s xml:id="echoid-s4265" xml:space="preserve"> & cuius latitudo ſit qua-<lb/> <anchor type="figure" xlink:label="fig-0087-01a" xlink:href="fig-0087-01"/> tuor dígitorũ:</s> <s xml:id="echoid-s4266" xml:space="preserve"> & ſit bene plana & æqualis <lb/>& læuis:</s> <s xml:id="echoid-s4267" xml:space="preserve"> & ſint fines ſuæ longitudinis æquidiſtan <lb/>tes, & ſuæ latitudines æquidιſtantes:</s> <s xml:id="echoid-s4268" xml:space="preserve"> & ſint in ipſa <lb/>duæ diametrι ſe ſecantes:</s> <s xml:id="echoid-s4269" xml:space="preserve"> à quarũ loco ſectionis <lb/>extrahatur linea recta æquidiſtans duobus fini-<lb/>bus longitudinis [per 31 p 1.</s> <s xml:id="echoid-s4270" xml:space="preserve">] Et extrahatur etiam <lb/>à loco ſectionis linea recta perpendicularis ſuper <lb/>lineam primam poſitam in medio:</s> <s xml:id="echoid-s4271" xml:space="preserve"> [per 11 p 1] & <lb/>intingantur iſtæ lineæ tincturis lucidis dιuerſo-<lb/>rum colorum, ut bene appareant:</s> <s xml:id="echoid-s4272" xml:space="preserve"> ſed tamen duæ <lb/>diametri ſint unius coloris.</s> <s xml:id="echoid-s4273" xml:space="preserve"> Et fiat cauatura in me <lb/>dio latitudιnis tabulæ, apud extremum lineæ re-<lb/>ctæ poſitę in medio, & inter duas diametros con-<lb/>cauιtate rotũda, & quaſi pyramidaliter, ſic ut poſ-<lb/>ſit intrare cornu naſi, quando tabula ſuperpone-<lb/>tur ei, quouſq;</s> <s xml:id="echoid-s4274" xml:space="preserve"> tangãt duo anguli tabulę ferè duo <lb/>media ſuperficierum duorum uiſuum, quamuis <lb/>non tangent.</s> <s xml:id="echoid-s4275" xml:space="preserve"> Sit igitur tabula in figura a b c d:</s> <s xml:id="echoid-s4276" xml:space="preserve"> & <lb/>diametrι a d, b c:</s> <s xml:id="echoid-s4277" xml:space="preserve"> & punctus ſectionιs ſit q:</s> <s xml:id="echoid-s4278" xml:space="preserve"> & linea <lb/>extenſa in medio longitudinis ſit h q z:</s> <s xml:id="echoid-s4279" xml:space="preserve"> & linea ſe-<lb/>cans hanc lineam ſecundum angulos rectos ſit k <lb/>q t:</s> <s xml:id="echoid-s4280" xml:space="preserve"> & concauitas, quæ eſt in medio latitudinis ta-<lb/>bulæ, ſit illa, quæ continetur à linea m h n.</s> <s xml:id="echoid-s4281" xml:space="preserve"> Hac <lb/>igitur tabula facta hoc modo:</s> <s xml:id="echoid-s4282" xml:space="preserve"> accipiatur cera al-<lb/>ba, ex qua fiant tria indiuidua parua columna-<lb/> <pb o="82" file="0088" n="88" rhead="ALHAZEN"/> ta:</s> <s xml:id="echoid-s4283" xml:space="preserve"> & intingantur diuerſis coloribus, & erigatur unum indiuiduorum in medio tabulæ in puncto q, <lb/>& applicetur tabulæ adeò, ut non poſsit auferri à ſuo loco:</s> <s xml:id="echoid-s4284" xml:space="preserve"> & ſit ſtans ſuper tabulam ſtatu æquali:</s> <s xml:id="echoid-s4285" xml:space="preserve"> <lb/>duo aũt indiuidua reliqua erigantur ſuper extrema lineæ latę in duobus punctis k, t:</s> <s xml:id="echoid-s4286" xml:space="preserve"> & ſic tria indi-<lb/>uidua erunt in una uerticatione.</s> <s xml:id="echoid-s4287" xml:space="preserve"> Et hoc quidem facto:</s> <s xml:id="echoid-s4288" xml:space="preserve"> eleuet experimentator hanc tabulam, & ſu-<lb/>perponat concauitatẽ, quæ eſt in medio longitudinis, cornu naſi, & inter oculos adeò, ut cornu naſi <lb/>intret concauitatẽ, & applicetur cum tabula, & fientduo anguli tabulę apud duo media ſuperficierũ <lb/>duorum uiſuum, & propinqui, ut tangãt ipſa ferè.</s> <s xml:id="echoid-s4289" xml:space="preserve"> Deinde experimentator debet inſpicere indiui-<lb/>duum poſitum in medio tabulę, & pupillam ſuper ipſum tenere fortiter.</s> <s xml:id="echoid-s4290" xml:space="preserve"> Cum igitur experimẽtator <lb/>inſpexerit indiuiduum poſitum in medio hoc modo:</s> <s xml:id="echoid-s4291" xml:space="preserve"> axes duorum uiſuum concurrent in hoc indi-<lb/>uiduo, & ſuperponentur duabus diametris, aut erũt æquidiſtantes illis:</s> <s xml:id="echoid-s4292" xml:space="preserve"> & erit axis cõmunis, quem <lb/>prius determinauimus, ſuperpoſitus lineæ extẽſę in medio lõgitudinis tabulę, quę eſt linea h z.</s> <s xml:id="echoid-s4293" xml:space="preserve"> De-<lb/>inde experimẽtator in hac diſpoſitione debet intueri omnia, quę ſunt in ſuperficie tabulę:</s> <s xml:id="echoid-s4294" xml:space="preserve"> tunc aũt <lb/>inueniet unum quodq;</s> <s xml:id="echoid-s4295" xml:space="preserve"> triũ indiuiduorũ, quę ſunt in punctis k, q, t unum:</s> <s xml:id="echoid-s4296" xml:space="preserve"> & inueniet lineã k q t etiã <lb/>unam:</s> <s xml:id="echoid-s4297" xml:space="preserve"> linea aũt h z extenſa in longitudine tabulę, inuenientur duę, ſe ſecantes apud indiuiduũ poſi <lb/>tum in medio.</s> <s xml:id="echoid-s4298" xml:space="preserve"> Et ſimiliter duæ diametri etiã, cum experimentator intuetur eas in hoc ſtatu, appare <lb/>bunt quatuor:</s> <s xml:id="echoid-s4299" xml:space="preserve"> utraq;</s> <s xml:id="echoid-s4300" xml:space="preserve"> earũ ſcilicet duplex.</s> <s xml:id="echoid-s4301" xml:space="preserve"> Deinde experimentator debet ponere pupillã circa alte-<lb/>rum indiuiduorũ, quæ ſunt in duobus punctis k, t, ut duo axes concurrãt in indiuiduo poſito in ex-<lb/>tremo:</s> <s xml:id="echoid-s4302" xml:space="preserve"> deinde intueatur etiã in hac diſpoſitione:</s> <s xml:id="echoid-s4303" xml:space="preserve"> & inueniet triũ indiuiduorũ unumquodq;</s> <s xml:id="echoid-s4304" xml:space="preserve"> unum:</s> <s xml:id="echoid-s4305" xml:space="preserve"> <lb/>& lineam poſitã in latitudine etiã unam:</s> <s xml:id="echoid-s4306" xml:space="preserve"> & inueniet lineã mediam extenſam in longitudine tabulæ <lb/>duas:</s> <s xml:id="echoid-s4307" xml:space="preserve"> & utrãq;</s> <s xml:id="echoid-s4308" xml:space="preserve"> diametrorũ duas.</s> <s xml:id="echoid-s4309" xml:space="preserve"> Cum igitur experimentator cõprehenderit has lineas & indiuidua <lb/>poſita ſuper tabulã:</s> <s xml:id="echoid-s4310" xml:space="preserve"> auferat duo indiuidua, quę ſunt in duob.</s> <s xml:id="echoid-s4311" xml:space="preserve"> punctis k, t:</s> <s xml:id="echoid-s4312" xml:space="preserve"> & ponat ea ſuper lineã h z, <lb/>extẽſam in lõgitudine, unũ ſcilicet in puncto l, quod ſequitur uiſum, & reliquũ in puncto s, quod eſt <lb/>ultra indiuiduũ poſitum in medio:</s> <s xml:id="echoid-s4313" xml:space="preserve"> deinde uertat tabulam ad ſuam primã poſitionem, & dirigat pu <lb/>pillam ad indiuiduũ poſitũ in medio:</s> <s xml:id="echoid-s4314" xml:space="preserve"> tunc aũt inueniet duo indiuidua, quatuor, & obliqua à medio, <lb/>duo ſcilicet in dextro, & duo in ſiniſtro:</s> <s xml:id="echoid-s4315" xml:space="preserve"> & inueniet ea ſuper duas lineas, quæ in rei ueritate ſunt una <lb/>linea in medio, ſed apparent duę:</s> <s xml:id="echoid-s4316" xml:space="preserve"> & inueniet quælibet duo horũ quatuor ſuper alterã duarũ linearũ.</s> <s xml:id="echoid-s4317" xml:space="preserve"> <lb/>Et ſimiliter ſi abſtulerit duo indiuidua ab hac linea, & poſuerit ea ſuper alterã diametrorũ duarũ, u-<lb/>num in parte uiſus, & reliquũ ultra indiuiduũ poſitũ in medio inueniet illa quatuor:</s> <s xml:id="echoid-s4318" xml:space="preserve"> nam utraq;</s> <s xml:id="echoid-s4319" xml:space="preserve"> dia <lb/>metrorũ apparebit duplex.</s> <s xml:id="echoid-s4320" xml:space="preserve"> Quaproter apparebunt ſuper utrãq;</s> <s xml:id="echoid-s4321" xml:space="preserve"> linearũ, quæ ſunt unius diametri, in <lb/>rei ueritate duo indiuidua, unum in parte uiſus, & aliud ultra indiuiduũ poſitum in medio.</s> <s xml:id="echoid-s4322" xml:space="preserve"> Et ſimi-<lb/>liter ſi poſuerit duo indiuidua ſuper ambas diametros, utrumq;</s> <s xml:id="echoid-s4323" xml:space="preserve"> ſuper alterã diametrum, & poſuerit <lb/>in ea parte uiſus:</s> <s xml:id="echoid-s4324" xml:space="preserve"> inueniet illa quatuor:</s> <s xml:id="echoid-s4325" xml:space="preserve"> duo propinqua, & duo remota.</s> <s xml:id="echoid-s4326" xml:space="preserve"> Deinde experimentator de-<lb/>bet auferre duo indiuidua à tabula, & ponere alterum eorum ſuper marginẽ tabulæ, ultra punctum <lb/>k, & prope ipſum ualde, utſuper punctum r, & reuertatur tabula ad ſuam primam poſitionem, & di-<lb/>rigat pupillam ad indiuiduũ poſitũ in medio:</s> <s xml:id="echoid-s4327" xml:space="preserve"> tunc inueniet indiuiduũ poſitum in puncto r, unum.</s> <s xml:id="echoid-s4328" xml:space="preserve"> <lb/>Deinde auferat indiuiduũ à puncto r, & ponat ipſum in margine tabulæ etiam ultra punctum k, ſu-<lb/>per punctum remotum à puncto k, ut ſuper punctũ f, & dirigat pupillam ad indiuiduum poſitum in <lb/>medio:</s> <s xml:id="echoid-s4329" xml:space="preserve"> quoniã tunc inueniet indiuiduum poſitum in puncto f, duo.</s> <s xml:id="echoid-s4330" xml:space="preserve"> Experimentator aũt inueniet <lb/>omnia, quæ diximus, cum direxerit pupillam ad indiuiduũ poſitũ in medio, aut ad indiuiduũ poſitũ <lb/> <anchor type="figure" xlink:label="fig-0088-01a" xlink:href="fig-0088-01"/> in linea recta in latitudine, aut ad punctũ unius li-<lb/>neę, quodcunq;</s> <s xml:id="echoid-s4331" xml:space="preserve"> ſit, & dum duo axes cõcurrunt in <lb/>indiuiduo poſito in medio, aut in aliquo puncto li <lb/>neæ poſitæ in latitudine.</s> <s xml:id="echoid-s4332" xml:space="preserve"> Si ergo experimentator <lb/>direxerit pupillã in illo ſitu ad indiuiduũ, poſitũ <lb/>extra lineam poſitam in latitudine, aut ad pun-<lb/>ctum poſitum extra lineam illam, & concurrerint <lb/>duo axes in aliquo puncto extra lineam poſitam <lb/>in latitudine:</s> <s xml:id="echoid-s4333" xml:space="preserve"> tunc indiuiduum poſitum in medio <lb/>uidebitur duo:</s> <s xml:id="echoid-s4334" xml:space="preserve"> & ſi reliqua indiuidua fuerint in <lb/>duobus punctis k, t:</s> <s xml:id="echoid-s4335" xml:space="preserve"> tũc utrumq;</s> <s xml:id="echoid-s4336" xml:space="preserve"> eorum etiã uide <lb/>bitur duo.</s> <s xml:id="echoid-s4337" xml:space="preserve"> Deinde cũ experimẽtator direxerit pu <lb/>pillam ad mediũ indiuiduum, aut ad aliquẽ locũ <lb/>lineæ poſitæ in latitudine:</s> <s xml:id="echoid-s4338" xml:space="preserve"> ſtatim diſpoſitio reuer-<lb/>tetur ut in prima figura.</s> <s xml:id="echoid-s4339" xml:space="preserve"> Igitur à puncto b extra-<lb/>hantur lineæ b k, b r, b f, linea igitur k b eſt maior <lb/>linea b t, [per theſin & 19 p 1] & linea k q eſt æ-<lb/>qualis q t[ex theſi.</s> <s xml:id="echoid-s4340" xml:space="preserve">] Sic igitur angulus t b q, eſt ma <lb/>ior angulo q b k [per 4 p geometriæ Iordani.</s> <s xml:id="echoid-s4341" xml:space="preserve"> In <lb/>triangulo enim b t k ab angulo t b k, inæqualibus <lb/>lateribus b t, b k comprehenſo, recta b q eſt in me-<lb/>diũ baſis t k:</s> <s xml:id="echoid-s4342" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s4343" xml:space="preserve"> angulus q b k ab ipſa b q & ma-<lb/>iore latere b k coprehẽſus, minor eſt angulo t b q, <lb/>ab eadẽ b q & minore latere b t comprehenſo] & <lb/>angulus t b q eſt æqualis angulo k a q [per 8 p 1] <lb/> <pb o="83" file="0089" n="89" rhead="OPTICAE LIBER III."/> ergo àngulus k a q eſt maior angulo k b q.</s> <s xml:id="echoid-s4344" xml:space="preserve"> Ergo remotio lineę a k ab axe a q, eſt maior quã rem otio <lb/>lineæ b k ab axe b q:</s> <s xml:id="echoid-s4345" xml:space="preserve">ſed differẽtia inter has duas remotiones eſt modica:</s> <s xml:id="echoid-s4346" xml:space="preserve"> differẽtia enim inter duos <lb/>angulos k a q, k b q eſt parua, & indiuiduum, quod eſt apud punctum k, uidetur ambobus uiſibus u-<lb/>num, quando axes concurrerint in indiuiduo, quod eſt a pud punctum q.</s> <s xml:id="echoid-s4347" xml:space="preserve"> Et duæ lineæ a k, b k, ſunt <lb/>æ quidiſtantes duobus radijs exeuntibus ad indiuiduũ, quod eſt a pud punctũ k, cum duo axes con-<lb/>currerint in indiuiduo, quod eſt apud q.</s> <s xml:id="echoid-s4348" xml:space="preserve"> Similiter diſpoſitio indiuidui, quod eſt apud punctum r, ſci-<lb/>tur:</s> <s xml:id="echoid-s4349" xml:space="preserve"> quoniam radij exeuntes ad ipſum, erũt in uerticatione duarum linearum a r, b r, & uidebitur u-<lb/>num:</s> <s xml:id="echoid-s4350" xml:space="preserve"> & duo anguli r a q, r b q non maxim è differunt:</s> <s xml:id="echoid-s4351" xml:space="preserve"> & angulus k b r non habet ſenſibilem quantita <lb/>tem, quando punctum r fuerit ualde propinquũ puncto k.</s> <s xml:id="echoid-s4352" xml:space="preserve"> Declarabitur igitur ex hac diſpoſitione:</s> <s xml:id="echoid-s4353" xml:space="preserve"> <lb/>quòd uiſum, cuius diſpoſitio apud duos axes eſt una poſitio in parte, & remotio radiorum exeun-<lb/>tium ad ipſum à duobus uiſibus, non eſt maximè differens:</s> <s xml:id="echoid-s4354" xml:space="preserve"> illud uiſum uidebitur duobus uiſibus <lb/>unum.</s> <s xml:id="echoid-s4355" xml:space="preserve"> Anguli autem f a q, f b q ſunt diuerſi diuerſitate maxima:</s> <s xml:id="echoid-s4356" xml:space="preserve"> & indiuiduum, quod eſt apud pun-<lb/>ctum f, uidebitur duo:</s> <s xml:id="echoid-s4357" xml:space="preserve">quoniã duo axes concurrent in indiuiduo, quod eſt apud punctum q.</s> <s xml:id="echoid-s4358" xml:space="preserve"> Decla-<lb/>rabitur igitur ex hac diſpoſitione, quòd uiſum, ad quod poſitio radiorum exeuntium à duobus uiſi-<lb/>bus eſt diuerſa in remotione à duobus axibus maxima diuerſitate, uidetur duo:</s> <s xml:id="echoid-s4359" xml:space="preserve"> licet poſitio eius in <lb/>reſpectu duorum axium eadem eſt poſitio in parte.</s> <s xml:id="echoid-s4360" xml:space="preserve"> Poſitio autem lineæ h q z in reſpectu axium <lb/>duorum uiſuum, eſt poſitio diuerſa in parte:</s> <s xml:id="echoid-s4361" xml:space="preserve"> radij etenim exeuntes ad partem h q à dextro uiſu, <lb/>ſunt ſiniſtri ab axe a q:</s> <s xml:id="echoid-s4362" xml:space="preserve"> radij autem exeuntes ad hanc partem à ſiniſtro uiſu, ſunt dextri ab axe b q:</s> <s xml:id="echoid-s4363" xml:space="preserve"> <lb/>radij uerò exeuntes ad partem q z à dextro uiſu, ſunt dextri ab axe a q:</s> <s xml:id="echoid-s4364" xml:space="preserve"> & radij exeuntes ad ipſam à <lb/>ſiniſtro uiſu, ſunt ſiniſtri ab axe b q:</s> <s xml:id="echoid-s4365" xml:space="preserve"> & radij qui exeunt ad ipſum, ſunt diuerſæ poſitionis in parte:</s> <s xml:id="echoid-s4366" xml:space="preserve"> & <lb/>remotio duorum radiorum exeuntium ad quodlibet punctum illius lineæ à duobus uiſibus, à duo-<lb/>bus axibus eſt æ qualis:</s> <s xml:id="echoid-s4367" xml:space="preserve"> & iſta linea, & omnia poſita ſuper ipſam, pręter indiuiduum poſitum in me-<lb/>dio, ſemper uidentur duo, cum duo axes concurrerint in indiuiduo poſito in medio.</s> <s xml:id="echoid-s4368" xml:space="preserve"> Declaratum <lb/>igitur eſt ex hac diſpoſitione, quòd uiſum, cuius poſitio in reſpectu duorum axium eſt diuerſa in <lb/>parte, ſemper uidetur duo:</s> <s xml:id="echoid-s4369" xml:space="preserve"> quamuis remotiones radiorum exeuntium ad ipſum à duobus uiſibus, <lb/>à duobus axibus ſint æquales.</s> <s xml:id="echoid-s4370" xml:space="preserve"> Remotiones enim quorumlibet duorum radiorum exeuntium à <lb/>duobus uiſibus ad aliquod punctum eius, erunt in duabus partibus diuerſis.</s> <s xml:id="echoid-s4371" xml:space="preserve"> Quapropter duæ for-<lb/>mæ cuiuslibet puncti eius inſtituentur in duobus punctis concauitatis communis nerui à duobus <lb/>lateribus centri.</s> <s xml:id="echoid-s4372" xml:space="preserve"> Et ſimiliter etiam eſt diſpoſitio utriuſque diametrorum.</s> <s xml:id="echoid-s4373" xml:space="preserve"> Quoniam radij exeuntes <lb/>ad utramlibet earum à uiſu ſequente ipſam, erunt à medio uiſus, & propinqui axi, & ſub axe, & ſu-<lb/>pra axem:</s> <s xml:id="echoid-s4374" xml:space="preserve"> & radij exeuntes ad ipſam à reliquo uiſu, erunt declinantes à reliquo axe:</s> <s xml:id="echoid-s4375" xml:space="preserve"> qui uerò à de-<lb/>xtro uiſu ad ſiniſtram diametrum, erunt ſiniſtri ab axe:</s> <s xml:id="echoid-s4376" xml:space="preserve"> qui autem exeunt à ſiniſtro uiſu ad dextram, <lb/>erunt dextri ab axe.</s> <s xml:id="echoid-s4377" xml:space="preserve"> Et formæ diametrorum iſtarum, & omnia puncta, & omnia poſita ſuper i-<lb/>pſas, uidentur duo, præter indiuiduum poſitum in medio, quando duo axes concurrerint in me-<lb/>dio indiuiduo.</s> <s xml:id="echoid-s4378" xml:space="preserve"/> </p> <div xml:id="echoid-div149" type="float" level="0" n="0"> <figure xlink:label="fig-0087-01" xlink:href="fig-0087-01a"> <variables xml:id="echoid-variables10" xml:space="preserve">d z c s f r t q k l h b n m a</variables> </figure> <figure xlink:label="fig-0088-01" xlink:href="fig-0088-01a"> <variables xml:id="echoid-variables11" xml:space="preserve">d z c s f r t q k l h b n m a</variables> </figure> </div> </div> <div xml:id="echoid-div151" type="section" level="0" n="0"> <head xml:id="echoid-head175" xml:space="preserve" style="it">13. Viſibile medio unius uiſus rectè, reliquo obliquè oppoſitum, uidetur geminum. 103 p 4. <lb/>Idem II n.</head> <p> <s xml:id="echoid-s4379" xml:space="preserve">DEclarabitur igitur exhoc, quòd uiſum, quod in reſpectu alterius uiſus eſt oppoſitum medio <lb/>eius, in reſpectu autem reliqui eſt obliquum à medio, uidetur duo.</s> <s xml:id="echoid-s4380" xml:space="preserve"> Nam formæ puncti, quæ <lb/>inſtituitur in medio alterius uiſi, ueniet ad centrum:</s> <s xml:id="echoid-s4381" xml:space="preserve"> forma uerò puncti obliqui à medio re-<lb/>liqui uiſus, ueniet ad punctum aliud à centro, & obliquum à centro, ſecundum obliquationem pun <lb/>cti ſuperficiei uiſus.</s> <s xml:id="echoid-s4382" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div152" type="section" level="0" n="0"> <head xml:id="echoid-head176" xml:space="preserve" style="it">14. Viſibile, in quo concurrunt axes optici, aut radij his propinqui: uidetur unum. 46 p 3.</head> <p> <s xml:id="echoid-s4383" xml:space="preserve">EX hac igitur experimentatione & expoſitione declaratur bene, quòd uiſum, in quo concur-<lb/>runt duo axes, ſemper uidetur unum:</s> <s xml:id="echoid-s4384" xml:space="preserve"> & quòd unum quod que uiſorum, etiam in quibus con-<lb/>currunt radij, qui ſunt conſimilis poſitionis in parte, inter quos non eſt maxima diuerſitas in <lb/>remotione à duobus axibus, uidetur etiam unum:</s> <s xml:id="echoid-s4385" xml:space="preserve"> & quòd uiſum, in quo concurrunt radij conſimi-<lb/>lis poſitionis in parte, & diuerſæ poſitionis in remotione à duobus axibus maxima diuerſitate, uide <lb/>tur duo:</s> <s xml:id="echoid-s4386" xml:space="preserve"> & quòd uiſum, quod comprehen ditur per radios diuerſæ poſitionis in parte, uidetur duo:</s> <s xml:id="echoid-s4387" xml:space="preserve"> <lb/>quamuis remotiones radiorum exeuntium ad ipſum à duobus axibus, ſunt ęquales:</s> <s xml:id="echoid-s4388" xml:space="preserve"> & quòd omnia <lb/>iſta erunt ſic:</s> <s xml:id="echoid-s4389" xml:space="preserve"> dum duo axes concurrent in uno uiſo.</s> <s xml:id="echoid-s4390" xml:space="preserve"> Et omnia uiſa aſſueta ſunt oppoſita ambo-<lb/>bus uiſibus, & ambo uiſus inſpiciunt ad quodlibet eorum.</s> <s xml:id="echoid-s4391" xml:space="preserve"> Ergo duo axes duorum uiſuum ſem-<lb/>per concurrunt in eis, & poſitio radiorum reſiduorum, qui concurrũt in communi puncto eorum, <lb/>eſt poſitio conſimilis in parte, & non differt in remotione à duobus axibus maxima differentia.</s> <s xml:id="echoid-s4392" xml:space="preserve"> Et <lb/>ideo quodlibet uiſibilium aſſuetorum uidetur ambobus uiſibus unum:</s> <s xml:id="echoid-s4393" xml:space="preserve"> & nullum uiſibilium uide-<lb/>tur duo, niſi rarò.</s> <s xml:id="echoid-s4394" xml:space="preserve"> Nullum enium uiſibiliũ uidetur duo, niſi cum cõpoſitio eius in reſpectu amborũ ui <lb/>fuũ fuerit diuerſa maxima diuerſitate, aut in parte, aut in remotione, aut in utroq;</s> <s xml:id="echoid-s4395" xml:space="preserve">. Et poſitio unius <lb/>uiſi apud duos uiſus non diuerſatur quidẽ maxima diuerſitate, niſi rarò.</s> <s xml:id="echoid-s4396" xml:space="preserve"> Cauſſa igitur propter quã <lb/>unũquodq;</s> <s xml:id="echoid-s4397" xml:space="preserve"> uiſorũ aſſuetorũ uidetur unũ ambobus uiſibus, declarata eſt ratiõe & experientia.</s> <s xml:id="echoid-s4398" xml:space="preserve"> Et e-<lb/>tiã cũ experimẽtator abſtulerit indiuiduũ, quod eſt in medio tabulę, & inſpexerit mediũ ſectionis, <lb/>quę eſt in medio tabulę:</s> <s xml:id="echoid-s4399" xml:space="preserve"> & intuitus fuerit tũc lineas ſcriptas in tabula:</s> <s xml:id="echoid-s4400" xml:space="preserve"> inueniet duas diametros qua <lb/>tuor:</s> <s xml:id="echoid-s4401" xml:space="preserve"> & inueniet ſimul duas illarũ quatuor ꝓpinquas ſibi, & duas à ſe remotas:</s> <s xml:id="echoid-s4402" xml:space="preserve"> & etiã oẽs ſe ſecãtes <lb/>ſuperpunctũ mediũ, qđ eſt punctũ ſectiõis duarũ diametrorũ, qđ eſt ſuper axẽ cõmunẽ:</s> <s xml:id="echoid-s4403" xml:space="preserve"> & inueniet <lb/> <pb o="84" file="0090" n="90" rhead="ALHAZEN"/> utramque illarum remotarum, magis remotam à medio, quàm ſit in rei ueritate.</s> <s xml:id="echoid-s4404" xml:space="preserve"> Deinde cum ex-<lb/>perimentator cooperuerit alterum uiſum:</s> <s xml:id="echoid-s4405" xml:space="preserve"> uidebit duas diametros, & uidebit ſpatium inter eas ma-<lb/>ius, quàm in rei ueritate ſecundum ſuam pyramidationem:</s> <s xml:id="echoid-s4406" xml:space="preserve"> quod autem eſt magis amplum de ipſo, <lb/>eſt latitudo tabulæ:</s> <s xml:id="echoid-s4407" xml:space="preserve"> & apparebit, quòd diameter remota à medio, eſt diameter, quæ ſequitur uiſum <lb/>coopertum.</s> <s xml:id="echoid-s4408" xml:space="preserve"> Ex quo declaratur, quòd duæ diametri, quæ uidentur propinquæ, cum uiſio fuerit in <lb/>utroque uiſu:</s> <s xml:id="echoid-s4409" xml:space="preserve"> ſunt illæ, quarum utraque uidetur uiſu ſequente:</s> <s xml:id="echoid-s4410" xml:space="preserve"> & quòd duæ diametri remotæ ſunt <lb/>illæ, quarum utraque uidetur uiſu obliquo.</s> <s xml:id="echoid-s4411" xml:space="preserve"> Propinquitas autem duarum è quatuor eſt:</s> <s xml:id="echoid-s4412" xml:space="preserve"> quia cum <lb/>duo axes concurrerint in indiuiduo poſito in medio:</s> <s xml:id="echoid-s4413" xml:space="preserve"> tunc utraque diametrorum comprehende-<lb/>tur à uiſu ſequente per radios ualde propinquos axi.</s> <s xml:id="echoid-s4414" xml:space="preserve"> Quapropter formæ eorum propter hoc e-<lb/>runt in concauitate communis nerui ualde propinquæ centro, & erit punctus ſectionis eorum in <lb/>ipſo centro:</s> <s xml:id="echoid-s4415" xml:space="preserve"> unde uidentur propinquæ ſibi, & medio.</s> <s xml:id="echoid-s4416" xml:space="preserve"> Remotio autem duarum è quatuor eſt:</s> <s xml:id="echoid-s4417" xml:space="preserve"> quia <lb/>utraque diametrorum comprehenditur etiam alio uiſu obliquo ab ipſo.</s> <s xml:id="echoid-s4418" xml:space="preserve"> Quapropter comprehen-<lb/>ditur per radios remotos ab axe:</s> <s xml:id="echoid-s4419" xml:space="preserve"> & altera comprehenditur per radios dextros ab axe, & reliqua per <lb/>radios ſiniſtros ab axe alio.</s> <s xml:id="echoid-s4420" xml:space="preserve"> Quapropter formæ earum inſtituentur in concauitate communis nerui <lb/>remotæ.</s> <s xml:id="echoid-s4421" xml:space="preserve"> Infigentur enim in duabus partibus contrarijs in reſpectu centri, & etiam remotis à cen-<lb/>tro:</s> <s xml:id="echoid-s4422" xml:space="preserve"> unde duę diametri habent duas formas propinquas ſibi, & duas formas remotas à ſe.</s> <s xml:id="echoid-s4423" xml:space="preserve"> Quare ue <lb/>rò comprehendatur remotio utriuſq;</s> <s xml:id="echoid-s4424" xml:space="preserve"> remotarũ à medio, maior quàm ſit ſua remotio uera:</s> <s xml:id="echoid-s4425" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s4426" xml:space="preserve"> quia <lb/>remotio, quę eſt inter duas diametros, cõprehenditur ab utroq;</s> <s xml:id="echoid-s4427" xml:space="preserve"> uiſu maior, quàm ſit in rei ueritate:</s> <s xml:id="echoid-s4428" xml:space="preserve"> <lb/>& hoc apparet, quando experimentator cooperuerit alterũ uiſum, & inſpexerit per reliquũ.</s> <s xml:id="echoid-s4429" xml:space="preserve"> Quare <lb/>uerò, quando experimentator cooperuerit alterum uiſum, & inſpexerit per reliquum tantùm:</s> <s xml:id="echoid-s4430" xml:space="preserve"> inue <lb/>niat ſpatium inter duas diametros magis amplum, quàm in rei ueritate:</s> <s xml:id="echoid-s4431" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s4432" xml:space="preserve"> quia ſpatium, quod eſt <lb/>inter duas diametros, comprehenditur ab utroq;</s> <s xml:id="echoid-s4433" xml:space="preserve"> uiſu ualde propinquũ uiſui:</s> <s xml:id="echoid-s4434" xml:space="preserve"> & omne, quod eſt ual <lb/>de propinquum uiſui, uidetur maius, quàm ſit in rei ueritate.</s> <s xml:id="echoid-s4435" xml:space="preserve"> Et cauſa huius declarabitur pòſt, cum <lb/>loquemur de deceptionibus uiſus.</s> <s xml:id="echoid-s4436" xml:space="preserve"> Ex conſideratione igitur diſpoſitionum diametrorum, quæ ſunt <lb/>in tabula, & indiuiduorum poſitorum ſuper eas, non in medio:</s> <s xml:id="echoid-s4437" xml:space="preserve"> apparet, quòd omne uiſum poſitum <lb/>ſuper axem communem, & comprehenſum à uiſu per axem radialem, comprehenditur in ſuo loco, <lb/>ſiue comprehendatur uno uiſu, & per unũ axem axiũ duorũ uiſuum, ſiue cõprehendatur per duos <lb/>uiſus & ambos axes.</s> <s xml:id="echoid-s4438" xml:space="preserve"> Et declaratur, quòd omne uiſum comprehenſum per unum uiſum & per axem <lb/>radialem, quod uiſum non eſt ſuper axem cõmunem, comprehenditur in loco propinquiore cõmu-<lb/>muni axi quàm ſuo loco uero:</s> <s xml:id="echoid-s4439" xml:space="preserve"> & hoc etiã ſequitur in eis, quæ cõprehenduntur per reſiduos radios, <lb/>præter axem.</s> <s xml:id="echoid-s4440" xml:space="preserve"> Quoniã cũ uiſus comprehenderit rem uiſam ſecundũ quod eſt:</s> <s xml:id="echoid-s4441" xml:space="preserve"> & inſtituta fuerit for-<lb/>ma in cõcauitate cõmunis nerui in uno loco:</s> <s xml:id="echoid-s4442" xml:space="preserve"> & continua ſibi inuicem ſecundum continuationẽ rei <lb/>uiſæ:</s> <s xml:id="echoid-s4443" xml:space="preserve"> & punctũ uiſi, quod eſt ſuper axem radialem, cum nõ fuerit ſuper axem cõmunem, uideatur in <lb/>loco propinquiore cõmuni axi, quàm ſuo loco uero:</s> <s xml:id="echoid-s4444" xml:space="preserve"> tunc puncta ſua reſidua etiam uidentur in loco <lb/>propinquiore cõmuni axi, ſuo loco uero, quia ſunt continuata cum parte, quæ eſt apud extremum <lb/>axis.</s> <s xml:id="echoid-s4445" xml:space="preserve"> Et ſi axes duorũ uiſuum concurrerẽt in aliquo uiſo extra axem cõmunem, ſequeretur etiã iſta <lb/>diſpoſitio:</s> <s xml:id="echoid-s4446" xml:space="preserve"> ſcilicet quòd uideretur in loco propinquiore cõmuni axi, quàm ſuo loco uero.</s> <s xml:id="echoid-s4447" xml:space="preserve"> Sed iſta po <lb/>ſitio rarò accidit.</s> <s xml:id="echoid-s4448" xml:space="preserve"> Cum enim illi axes duorũ uiſuum cõcurrerint in aliquo uiſo:</s> <s xml:id="echoid-s4449" xml:space="preserve"> tunc in pluribus diſ-<lb/>poſitionibus axis cõmunis tranſibit per illud uiſum, & nunquã axes duorum uiſuũ concurrentin <lb/>aliquo uiſo extra axem cõmunem, niſi per laborem aut per impedimentũ cogens uiſum ad hoc.</s> <s xml:id="echoid-s4450" xml:space="preserve"> Et <lb/>hæc diſpoſitio nõ apparetin uiſis aſſuetis.</s> <s xml:id="echoid-s4451" xml:space="preserve"> Nam cum acciderit hoc in aliquo uiſo aſſueto:</s> <s xml:id="echoid-s4452" xml:space="preserve"> continget <lb/>in omnibus uiſis continuis cum illo uiſo:</s> <s xml:id="echoid-s4453" xml:space="preserve"> unde poſitio uiſorum inter ſe inuicem nõ tranſmutabitur <lb/>propter hoc.</s> <s xml:id="echoid-s4454" xml:space="preserve"> Et cum poſitio illius uiſi in reſpectu uiſorũ uicinantium non fuerit tranſmutata:</s> <s xml:id="echoid-s4455" xml:space="preserve"> tunc <lb/>non apparebit tranſmutatio ſuiloci, cum acciderit in uiſis aſſuetis.</s> <s xml:id="echoid-s4456" xml:space="preserve"> Quando igitur conſideratur hæc <lb/>uia prædicta:</s> <s xml:id="echoid-s4457" xml:space="preserve"> declarabitur ex illa experientia, quòd hoc ſequitur in omnibus uiſis, in quibus cõcur-<lb/>runt axes duorũ uiſuum, quæ ſunt extra axem cõmunem.</s> <s xml:id="echoid-s4458" xml:space="preserve"> Et etiam oportet experimentatorẽ acci-<lb/>pere tres ſchedulas pergameni, paruas, æquales:</s> <s xml:id="echoid-s4459" xml:space="preserve"> & ſcribat in una uerbum aliquod ſcriptura mani-<lb/>feſta:</s> <s xml:id="echoid-s4460" xml:space="preserve"> & in reſiduis ſcribat illam eandem partem:</s> <s xml:id="echoid-s4461" xml:space="preserve"> & in illa quantitate & in illa figura:</s> <s xml:id="echoid-s4462" xml:space="preserve"> & ponat in diui-<lb/>duum unum in medio tabulæ, ut prius:</s> <s xml:id="echoid-s4463" xml:space="preserve"> & ponat etiam alterum indiuiduum ſuper punctum k.</s> <s xml:id="echoid-s4464" xml:space="preserve"> Dein <lb/>de applicet unam ſchedulam cum indiuiduo, quod eſt in medio tabulæ, & aliã in puncto k:</s> <s xml:id="echoid-s4465" xml:space="preserve"> & obſer <lb/>uet, ut poſitio eius ſit, ſicut poſitio primæ ſchedulæ:</s> <s xml:id="echoid-s4466" xml:space="preserve"> & ponat tabulam, ut prius fecit:</s> <s xml:id="echoid-s4467" xml:space="preserve"> & dirigat pupil <lb/>lam ad ſchedulam, quę eſt in medio in diuiduo:</s> <s xml:id="echoid-s4468" xml:space="preserve"> & intueatur illam:</s> <s xml:id="echoid-s4469" xml:space="preserve"> tunc cõprehendet partẽ ſcriptam <lb/>ſuper illam certa comprehenſione:</s> <s xml:id="echoid-s4470" xml:space="preserve"> & comprehendet ſimul in illa diſpoſitione aliam ſchedulã, & par <lb/>tem ſcriptã in ea, ſed non bene declaratã, ſicut eſt pars ſimilis illi, quæ eſt ſcripta in media ſchedula, <lb/>licet ſint cõſimiles in figura, forma & quãtitate.</s> <s xml:id="echoid-s4471" xml:space="preserve"> Deinde in hac diſpoſitiõe oportet experimentatorẽ <lb/>accipere tertiam ſchedulam manu ſequente punctum k:</s> <s xml:id="echoid-s4472" xml:space="preserve"> & ponat illam in uerticatione duarum ſche <lb/>dularum, quę ſunt in tabula, & in rectitudine extenſionis lineæ, quę eſt in latitudine tabulæ, quæ eſt <lb/>in ſuperficie tabulæ, quantũ ad ſenſum:</s> <s xml:id="echoid-s4473" xml:space="preserve"> ſed tamen ſit remota à tabula:</s> <s xml:id="echoid-s4474" xml:space="preserve"> Et huius uerticatio uocetur <lb/>uerticatio facialis.</s> <s xml:id="echoid-s4475" xml:space="preserve"> Et obſeruet experimentator, ut poſitio tertiæ ſchedulæ, & poſitio partis, quæ eſt <lb/>in illa, quando ponit ſchedulã, ſit ſimilis poſitioni duarũ ſchedularũ, quæ ſunt in tabula:</s> <s xml:id="echoid-s4476" xml:space="preserve"> & tunc figat <lb/>ambos uiſus in ſchedulam poſitã in medio, & dirigat pupillam ad ipſam:</s> <s xml:id="echoid-s4477" xml:space="preserve"> & tunc quidẽ cõprehendet <lb/>tertiã ſchedulam, ſi non fuerit multũ remota à tabula:</s> <s xml:id="echoid-s4478" xml:space="preserve"> ſed comprehendet formã partis, quæ eſt in ea, <lb/>dubitabilẽ, non intelligibilẽ, & nõ inueniet eam, ſicut inuenit formã partis ſimilis illi, quæ eſt in me-<lb/>dio tabulę:</s> <s xml:id="echoid-s4479" xml:space="preserve">nec ſicut inuenit formã partis, quę eſt apud punctũ k, dum ambo uiſus direxerint pupillã <lb/> <pb o="85" file="0091" n="91" rhead="OPTICAE LIBER III."/> ad ſchedulam, quæ eſt in medio.</s> <s xml:id="echoid-s4480" xml:space="preserve"> Deinde auferat experimẽtator indiuiduum, quod eſt apud punctũ <lb/>k, & ſchedulam, quæ eſt in illo:</s> <s xml:id="echoid-s4481" xml:space="preserve"> & appropinquet ſchedulam, quam tenet in manu, quouſq;</s> <s xml:id="echoid-s4482" xml:space="preserve"> applicet <lb/>eam ad latus ſchedulæ, applicatæ cũ indiuiduo poſito in medio:</s> <s xml:id="echoid-s4483" xml:space="preserve"> & præſeruet ſe, ut ſchedula ſit per-<lb/>pendicularis ſuper lineam poſitam in latitudine:</s> <s xml:id="echoid-s4484" xml:space="preserve"> & dirigat pupillam, ſicut prius, ad ſchedulam po-<lb/>ſitam in medio:</s> <s xml:id="echoid-s4485" xml:space="preserve"> tunc quidem in medio comprehendet ambas partes, quæ ſuntin duabus ſchedu-<lb/>lis comprehenſione manifeſta & certificata, & non erit inter duas formas duarum partium in de-<lb/>claratione & certificatione differentia ſenſibilis.</s> <s xml:id="echoid-s4486" xml:space="preserve"> Dein de experimẽtator moueat ſchedulam, quam <lb/>tenet in manu motu ſubtili ſuper lineam poſitam in latitudine:</s> <s xml:id="echoid-s4487" xml:space="preserve"> & præſeruet ſe, ut ſitus eius ſit, ſicut <lb/>erat prius:</s> <s xml:id="echoid-s4488" xml:space="preserve"> & intendat certificare ſchedulam, quæ eſt in medio, & intueatur bene duas ſchedulas in <lb/>hoc ſtatu:</s> <s xml:id="echoid-s4489" xml:space="preserve"> tunc quidem uidebit, quòd quantò magis ſchedula mota remouetur à medio, tantò ma-<lb/>gis diminuitur declaratio partis, quæ eſt in ea.</s> <s xml:id="echoid-s4490" xml:space="preserve"> Cum igitur uenerit apud punctum k:</s> <s xml:id="echoid-s4491" xml:space="preserve"> tunc inueniet <lb/>formam partis intelligibilem, ſed non tantùm, quantum, cum eſſet apud ſuam applicationem cum <lb/>ſchedula, quæ eſt in medio.</s> <s xml:id="echoid-s4492" xml:space="preserve"> Deinde experimẽtator moueat ſchedulam etiam:</s> <s xml:id="echoid-s4493" xml:space="preserve"> & extrahat illam à ra <lb/>bula:</s> <s xml:id="echoid-s4494" xml:space="preserve"> & rem oueat illã paulatim & paulatim in uerticatione lineæ poſitæ in latitudine:</s> <s xml:id="echoid-s4495" xml:space="preserve"> & intueatur <lb/>cõſiderans optimè;</s> <s xml:id="echoid-s4496" xml:space="preserve"> & dirigat pupillam ad ſchedulã poſitam in medio:</s> <s xml:id="echoid-s4497" xml:space="preserve"> quoniã tunc inueniet, quòd <lb/>ſchedula mota, quantò magis remouetur à medio, tantò minus apparebit pars ſcripta in ea, adeò <lb/>quòd erit nõ intelligibilis omnino.</s> <s xml:id="echoid-s4498" xml:space="preserve"> Deinde cum mouerit illam poſt hoc:</s> <s xml:id="echoid-s4499" xml:space="preserve"> uidebit, quòd quantò ma-<lb/>gis illa remouetur à medio, tantò magis latebit forma illius partis ſcriptæ in ea.</s> <s xml:id="echoid-s4500" xml:space="preserve"> Et etiam cooperiat <lb/>experimentator uiſum, qui ſequitur punctumt:</s> <s xml:id="echoid-s4501" xml:space="preserve"> & figat tabulam in eadẽ diſpoſitione:</s> <s xml:id="echoid-s4502" xml:space="preserve"> & dirigat pu-<lb/>pillam unius uiſus, qui ſequitur punctum k, ad ſchedulam poſitam in medio:</s> <s xml:id="echoid-s4503" xml:space="preserve"> & applicet aliam ſche-<lb/>dulam ad latus ſchedulæ poſitæ in medio, ſicut fecit prius:</s> <s xml:id="echoid-s4504" xml:space="preserve"> tũc quidem inueniet partem, quæ eſt in <lb/>alia ſchedula, manifeſtam, inter quam & ſchedulam poſitam in medio, non eſt differentia ſenſibilis.</s> <s xml:id="echoid-s4505" xml:space="preserve"> <lb/>Deinde moueat ſecundam ſchedulam, ut primò fecit:</s> <s xml:id="echoid-s4506" xml:space="preserve"> & intendat ſchedulam poſitam in medio:</s> <s xml:id="echoid-s4507" xml:space="preserve"> & <lb/>dirigat pupillam ad ipſam:</s> <s xml:id="echoid-s4508" xml:space="preserve"> tunc quidem inueniet partem, quæ eſt in ſecunda ſchedula apud motũ <lb/>latére.</s> <s xml:id="echoid-s4509" xml:space="preserve"> Et cum peruenerit ad punctum k:</s> <s xml:id="echoid-s4510" xml:space="preserve"> tunc erit inter ſuam certificationem in hoc ſtatu, & ſuam <lb/>certificationem apud applicationem ſuam cum ea, quæ eſt in medio:</s> <s xml:id="echoid-s4511" xml:space="preserve"> differentia ſenſibilis.</s> <s xml:id="echoid-s4512" xml:space="preserve"> Deinde <lb/>moueat hãc ſchedulam, & extrahat illam à tabula, ut primò fecit:</s> <s xml:id="echoid-s4513" xml:space="preserve"> & intueatur ſchedulam in medio <lb/>poſitam:</s> <s xml:id="echoid-s4514" xml:space="preserve"> tunc quidem inueniet, quòd ſchedula mota, quantò minus remouetur à medio, tantò mi-<lb/>nus diminuitur declaratio, quæ eſt in ea:</s> <s xml:id="echoid-s4515" xml:space="preserve"> adeò quòd forma eius omnino erit intelligibilis:</s> <s xml:id="echoid-s4516" xml:space="preserve"> & quan-<lb/>tò magis remouetur à medio, tantò magis latebit.</s> <s xml:id="echoid-s4517" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div153" type="section" level="0" n="0"> <head xml:id="echoid-head177" xml:space="preserve" style="it">15. Viſibile in axium opticorum concurſu certißimè uidetur: extratantò certius, quantò <lb/>concurſui fuerit propinquius. 45 p 3.</head> <p> <s xml:id="echoid-s4518" xml:space="preserve">APparet ergo ex hac conſideratione, quòd manifeſtiſsimum uiſibilium facialium uiſui, quæ <lb/>comprehenduntur ambobus uiſibus:</s> <s xml:id="echoid-s4519" xml:space="preserve"> eſt illud, quod eſt apud concurſum duorum axium:</s> <s xml:id="echoid-s4520" xml:space="preserve"> & <lb/>quod eſt propin quius concurſui duorum axium, eſt manifeſtius remotiore:</s> <s xml:id="echoid-s4521" xml:space="preserve"> & quòd forma <lb/>remoti uiſi ad concurſum duorum axium eſt non certificata, licet comprehendatur utroque uiſu.</s> <s xml:id="echoid-s4522" xml:space="preserve"> <lb/>Amplius apparet ex hac conſideratione, quòd manifeſtiſsimum uiſibilium facialium, quę compre-<lb/>henduntur uno uiſu:</s> <s xml:id="echoid-s4523" xml:space="preserve"> eſt illud, quod uidetur per axem radialem:</s> <s xml:id="echoid-s4524" xml:space="preserve"> & illud, quod eſt propinquius illi, <lb/>eſt manifeſtius, quàm illud, quod eſt remotius:</s> <s xml:id="echoid-s4525" xml:space="preserve"> & quod remotum uiſum à radiali axe habet formã <lb/>dubitabilem, non certificatam.</s> <s xml:id="echoid-s4526" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div154" type="section" level="0" n="0"> <head xml:id="echoid-head178" xml:space="preserve" style="it">16. Viſibile magnum ſimul totum æquabiliter non uidetur. 48 p 3.</head> <p> <s xml:id="echoid-s4527" xml:space="preserve">AMplius apparet, quòd uiſus non comprehendit rem uiſam, quæ eſt remotarum diametrorũ, <lb/>uera comprehenſione, niſi moueat radialem axem ſuper omnes eius diametros, & ſuper o-<lb/>mnes eius partes, ſiue comprehenſio ſit ambobus uiſibus, ſiue uno.</s> <s xml:id="echoid-s4528" xml:space="preserve"> Viſus enim cum fuerit <lb/>ſixus in oppoſitione uiſi, quod eſt maximarum diametrorum, non comprehendet totum uera com <lb/>prehenſione:</s> <s xml:id="echoid-s4529" xml:space="preserve"> ſed ſolùm illud, quod eſt ſuper axem & prope, certificata ſcilicet cõprehenſione:</s> <s xml:id="echoid-s4530" xml:space="preserve"> reſi-<lb/>duæ uerò partes eius, & illud, quod remotum eſt ab axe ſcilicet, comprehendetur, ſed non certè, li-<lb/>cet uiſum ſit faciale, & indifferenter, ſiue comprehenſio ſit utroq;</s> <s xml:id="echoid-s4531" xml:space="preserve"> uiſu, ſiue uno tantùm.</s> <s xml:id="echoid-s4532" xml:space="preserve"> Poſtea o-<lb/>portet experimentatorem accipere pergamenum quatuor digitorũ in omni diuiſione, in quo ſcri-<lb/>bat lineas ſcriptura ſubtili, tamen manifeſta & intelligibili.</s> <s xml:id="echoid-s4533" xml:space="preserve"> Deinde auferat indiuiduum poſitum <lb/>ſuper tabulam:</s> <s xml:id="echoid-s4534" xml:space="preserve"> & ſuperponat tabulam prope uiſum, ut prius fecit:</s> <s xml:id="echoid-s4535" xml:space="preserve"> & erigat pergamenum ſuper li-<lb/>neam poſitam in latitudine, quæ eſt in medio tabulæ:</s> <s xml:id="echoid-s4536" xml:space="preserve"> & dirigat pupillam utroque uiſu ad medium <lb/>pergameni, & intueatur ipſum:</s> <s xml:id="echoid-s4537" xml:space="preserve"> quoniam tunc inueniet ſcripturam, quæ eſt in pergameno, apertã <lb/>& intelligibilem:</s> <s xml:id="echoid-s4538" xml:space="preserve"> Sed tamen ſcriptura, quæ eſt in medio pergameni, eſt manifeſtior, quàm quæ eſt <lb/>in extremis:</s> <s xml:id="echoid-s4539" xml:space="preserve"> quando uiſus direxerit pupillam ad medium pergameni, & non fuerit motus ſuper o-<lb/>mnes eius diametros.</s> <s xml:id="echoid-s4540" xml:space="preserve"> Dein de obliquet pergamenum adeò, ut ſecet lineam poſitam in latitudine, <lb/>in puncto poſito in medio tabulæ, quod eſt punctum ſectionis (obliquatio autem pergameni ſu-<lb/>per lineam poſitam in latitudine ſit parua) & inſpiciat ambobus uiſibus medium pergameni:</s> <s xml:id="echoid-s4541" xml:space="preserve"> quo-<lb/>niam tunc inueniet ſcripturam legibilem, ſed non tantùm, quantùm cum pergamenum erat facia-<lb/>le.</s> <s xml:id="echoid-s4542" xml:space="preserve"> Deinde experimentator debet obliquare pergamenum obliquatione maiore prima, ita ut me-<lb/>dium eius ſit ſuper punctum ſectionis:</s> <s xml:id="echoid-s4543" xml:space="preserve"> & dirigat pupillam utroq;</s> <s xml:id="echoid-s4544" xml:space="preserve"> uiſu ad medium eius:</s> <s xml:id="echoid-s4545" xml:space="preserve"> tunc quidẽ <lb/> <pb o="86" file="0092" n="92" rhead="ALHAZEN"/> uidebit ſcripturam latentiorem prima.</s> <s xml:id="echoid-s4546" xml:space="preserve"> Deinde etiam obliquet pergamenum paulatim, ita ut me-<lb/>dium eius ſemper ſit in puncto ſectionis, & intueatur ſucceſsiuè:</s> <s xml:id="echoid-s4547" xml:space="preserve"> & tunc inueniet ſcripturam latẽ-<lb/>re apud obl quationes pergameni:</s> <s xml:id="echoid-s4548" xml:space="preserve"> & quantò magis pergamenum fuerit obliquum, tantò magis <lb/>latebit ſcriptura, adeò ut pergamenum appropinquet lineæ extenſæ in medio longitudinis tabulę:</s> <s xml:id="echoid-s4549" xml:space="preserve"> <lb/>& tunc ſcriptura, quæ eſt in pergameno:</s> <s xml:id="echoid-s4550" xml:space="preserve"> uidebitur multum dubitabilis, & ferè non intelligibilis, <lb/>& non certificata.</s> <s xml:id="echoid-s4551" xml:space="preserve"> Deinde oportet experimentatorem uertere pergamenum ad primam poſitio-<lb/>nem:</s> <s xml:id="echoid-s4552" xml:space="preserve"> & erigere ipſum ſuper lineam poſitam in latitudine:</s> <s xml:id="echoid-s4553" xml:space="preserve"> & cooperire alterum uiſum:</s> <s xml:id="echoid-s4554" xml:space="preserve"> & inſpicere <lb/>pergamenum reliquo uiſu:</s> <s xml:id="echoid-s4555" xml:space="preserve"> & tunc inueniet ſcripturam manifeſtam, & legibilem.</s> <s xml:id="echoid-s4556" xml:space="preserve"> Deinde obliquer <lb/>pergamenum, ut prius fecit:</s> <s xml:id="echoid-s4557" xml:space="preserve"> & inſpiciat ipſum uno uiſu:</s> <s xml:id="echoid-s4558" xml:space="preserve"> & tunc inueniet ſcripturam latentiorem, <lb/>quàm cum eratapud oppoſitionem facialem.</s> <s xml:id="echoid-s4559" xml:space="preserve"> Deinde obliquet pergamenum plus paulatim & pau <lb/>latim:</s> <s xml:id="echoid-s4560" xml:space="preserve"> & intueatur ipſum multoties:</s> <s xml:id="echoid-s4561" xml:space="preserve"> & tunc inueniet, quòd quanto magis obliquatur, tanto ma-<lb/>gis latet pars ſcripta, adeò ut pergamenum appropinquet diametro, quæ ſequitur uiſum apertum.</s> <s xml:id="echoid-s4562" xml:space="preserve"> <lb/>Declarabitur ergo ex hac conſideratione, quòd manifeſtiſsimum uiſibilium, quæ ſunt ſuper axem <lb/>radialem:</s> <s xml:id="echoid-s4563" xml:space="preserve"> eſt illud, quod eſt faciale uiſui:</s> <s xml:id="echoid-s4564" xml:space="preserve"> & quòd illud, cuius poſitio eſt magis facialis, eſt manife-<lb/>ſtius illo, cuius poſitio eſt minus facialis:</s> <s xml:id="echoid-s4565" xml:space="preserve"> & quòd illud, quod eſt obliquum ab axe radiali obliqua-<lb/>tione maxima, eſt dubitabile, non intelligibile, ſiue uiſio ſit utroque uiſu, ſiue uno.</s> <s xml:id="echoid-s4566" xml:space="preserve"> Deinde oportet <lb/>experimentatorem uertere indiuiduum, quod erat ſuper tabulam:</s> <s xml:id="echoid-s4567" xml:space="preserve"> & ponereipſum in medio tabu-<lb/>læ:</s> <s xml:id="echoid-s4568" xml:space="preserve"> & applicare ipſum ad punctum ſectionis, ut in prima conſideratione.</s> <s xml:id="echoid-s4569" xml:space="preserve"> Deinde erigat pergame-<lb/>num ſuper alteram partem lineæ poſitæ in latitudine ſuper uerticationem facialem:</s> <s xml:id="echoid-s4570" xml:space="preserve"> & dirigat pu-<lb/>pillam utroq;</s> <s xml:id="echoid-s4571" xml:space="preserve"> uiſu ad indiuiduum poſitum in medio:</s> <s xml:id="echoid-s4572" xml:space="preserve"> In hac quidem diſpoſitione comprehendet <lb/>pergamenum, & ſcripturam, quæ eſt in ipſo:</s> <s xml:id="echoid-s4573" xml:space="preserve"> ſed illud, quod propinquum eſt indiuiduo poſito in <lb/>medio:</s> <s xml:id="echoid-s4574" xml:space="preserve"> erit manifeſtum, & quod remotum eſt ab illo, eſt dubitabile & latens:</s> <s xml:id="echoid-s4575" xml:space="preserve"> & quanto magis re <lb/>mouetur ab indiuiduo, tantò magis latet.</s> <s xml:id="echoid-s4576" xml:space="preserve"> Etiterum oportet experimentatorem obliquare perga-<lb/>menum in hoc ſtatu, ita ut ſecet lineam poſitam in latitudine ſuper aliquod punctum alterms eius <lb/>partis:</s> <s xml:id="echoid-s4577" xml:space="preserve"> & ſit parua obliquatio:</s> <s xml:id="echoid-s4578" xml:space="preserve"> & dirigat pupillam ad indiuiduum poſitum in medio:</s> <s xml:id="echoid-s4579" xml:space="preserve"> tunc quidem <lb/>uidebit ſcripturam, quæ eſt in pergameno latentiorem, quàm cum erat facialis.</s> <s xml:id="echoid-s4580" xml:space="preserve"> Deinde obliquet <lb/>plus pergamenum:</s> <s xml:id="echoid-s4581" xml:space="preserve"> & dirigat pupillam ad indiuiduum poſitum in medio:</s> <s xml:id="echoid-s4582" xml:space="preserve"> tunc quidem uidebit <lb/>dcripturam dubitabilem, non manifeſtam, nec legibilem.</s> <s xml:id="echoid-s4583" xml:space="preserve"> Deinde oportet experimentatorem coo-<lb/>perire alterum uiſum, & inſpicere uno uiſu:</s> <s xml:id="echoid-s4584" xml:space="preserve"> & uertat pergamenum in ſua prima poſitione:</s> <s xml:id="echoid-s4585" xml:space="preserve"> & erigat <lb/>ipſum ſuper partem lineæ poſitæ in latitudine, quæ ſequitur uiſum inſpicientem:</s> <s xml:id="echoid-s4586" xml:space="preserve"> & dirigat pupil-<lb/>lam unius uiſus ad indiuiduum poſitum in medio:</s> <s xml:id="echoid-s4587" xml:space="preserve"> tunc quidem comprehendet etiam ſcripturam, <lb/>quæ eſt in pergameno, & uidebit illam, quæ eſt prope indiuiduum, manifeſtiorem remota, & uide-<lb/>bit illam, quæ eſt remotiſsima ab indiuiduo, dubitabilem, & non legibilem.</s> <s xml:id="echoid-s4588" xml:space="preserve"> Deinde obliquet per-<lb/>gamenum ita, ut ſecet lineam poſitam in latitudine ſuper punctum partis, ſuper quam erat erectũ, <lb/>& inſpiciat indiuiduum poſitum in medio, illo eodem uiſu:</s> <s xml:id="echoid-s4589" xml:space="preserve"> tunc quidem uidebit ſcripturam, quæ <lb/>eſt in pergameno, dubitabilem, & illegibilem magis, quàm cum pergamenum erat faciale.</s> <s xml:id="echoid-s4590" xml:space="preserve"> Dein-<lb/>de obliquet pergamenum magis paulatim ac paulatim, & uidebit, quòd quantò magis obliquatur <lb/>pergamenum, tantò magis latebit ſcriptura.</s> <s xml:id="echoid-s4591" xml:space="preserve"> Apparet ergo exhac conſideratione, quòd uiſum, <lb/>quod eſt faciale, eſt manifeſtius uiſo obliquo:</s> <s xml:id="echoid-s4592" xml:space="preserve"> quamuis uiſum non fuerit ſuper axem radialem, fed <lb/>extra ipſum.</s> <s xml:id="echoid-s4593" xml:space="preserve"> Viſum enim quando multùm eſt obliquum, latet multùm, licet non ſit ſuper axem ra-<lb/>dialem, ſiue uiſio ſit utroque uiſu, ſiue uno tantùm.</s> <s xml:id="echoid-s4594" xml:space="preserve"> Et iterum oportet experimentatorem auſerre <lb/>indiuiduum à tabula:</s> <s xml:id="echoid-s4595" xml:space="preserve"> & erigere pergamenum ſuper extremum tabulæ:</s> <s xml:id="echoid-s4596" xml:space="preserve"> & ſuperponere finem erus <lb/>fini latitudinis tabulæ, qui eſt c d:</s> <s xml:id="echoid-s4597" xml:space="preserve"> & dirigat pupillam utroq;</s> <s xml:id="echoid-s4598" xml:space="preserve"> uiſu ad medium pergameni:</s> <s xml:id="echoid-s4599" xml:space="preserve"> quoniam <lb/>tunc inueniet ſcripturam manifeſtam & legibilem.</s> <s xml:id="echoid-s4600" xml:space="preserve"> Deinde obliquet pergamenum ita, ut ſecetla-<lb/>titudinem tabulę ſuper punctum z, quod eſt in medio latitudinis tabulę, & dirigat pupillam utro-<lb/>que uiſu ad medium pergameni:</s> <s xml:id="echoid-s4601" xml:space="preserve"> tunc quidem uidebit ſcripturam latentiorem, quàm prius.</s> <s xml:id="echoid-s4602" xml:space="preserve"> Dein-<lb/>de addat in obliquatione pergameni paulatim & paulatim:</s> <s xml:id="echoid-s4603" xml:space="preserve"> & uidebit ſcripturam latére paulatim <lb/>& paulatim, adeò, utſi obliquatio pergameni fuerit maxima:</s> <s xml:id="echoid-s4604" xml:space="preserve"> uideat ſcripturam ualde latentem in <lb/>eadem diſpoſitione, in qua erat, quando conſiderabatur in medio tabulæ.</s> <s xml:id="echoid-s4605" xml:space="preserve"> Et ſimiliter ſi conſidera-<lb/>uerit ipſum in hoc loco uno uiſu.</s> <s xml:id="echoid-s4606" xml:space="preserve"> Deinde oportet experimentatorem ponere indiuiduum ſuper-<lb/>punctum z, & erigere pergamenum ſuper alteram partem latitudinis, apud extremum tabulæ, ſi-<lb/>cut fecit in medio tabulæ:</s> <s xml:id="echoid-s4607" xml:space="preserve"> & dirigat pupillam ad indiuiduum poſitum in medio, & intueatur per-<lb/>gamenum, & conſideret ſcripturam:</s> <s xml:id="echoid-s4608" xml:space="preserve"> tunc enim uidebit diſpoſitionem, ſicut uidebateam, quan-<lb/>do erat in medio tabulæ, ſiue conſideretur utroque uiſu, ſiue uno.</s> <s xml:id="echoid-s4609" xml:space="preserve"> Deinde oportet experimenta-<lb/>torem etiam experiri ſchedulas paruas, quas præ diximus, apud extremum tabulæ, & uidebit diſ-<lb/>poſitionem in eis, ſicut cum erant in medio, ſcilicet, quòd pars, quæ eſt in media ſchedula, eſt <lb/>manifeſtior parte, quæ eſt in ſchedula remota à medio:</s> <s xml:id="echoid-s4610" xml:space="preserve"> & quantò ſchedula magis eſt remota à me-<lb/>dio, tantò magis latebit pars.</s> <s xml:id="echoid-s4611" xml:space="preserve"> Sed tamen uidebit, quòd remotio à medio, apud quam latet pars po <lb/>ſita in extremo, quando conſideratio fuerit apud extre mum tabulæ, eſt proportionalis ad remo-<lb/>tionem à medio, apud quam latet pars poſita in extremo, quando conſideratio fuerit in medio ta-<lb/>bulæ:</s> <s xml:id="echoid-s4612" xml:space="preserve"> eſt enim ſecundum remotionem radiorum exeuntium ad extremum ab axe.</s> <s xml:id="echoid-s4613" xml:space="preserve"> Proportio igi-<lb/>tur remotionis, apud quam latet forma poſita in extremo, à forma poſita in medio, ad remotionem <lb/>formæ poſitæ in medio, eſt eadẽ proportio in conſideratione apud mediũ tabulæ, & in conſidera-<lb/>tiõe apud extremũ eius.</s> <s xml:id="echoid-s4614" xml:space="preserve"> Et ſimiliter etiã ſi experimẽtator abſtulerit tabulã;</s> <s xml:id="echoid-s4615" xml:space="preserve"> & poſuerit pergamenũ, <lb/> <pb o="87" file="0093" n="93" rhead="OPTICAE LIBER III."/> in quo eſt ſcriptura in maiore diſtantia, quàm longitudo tabulæ ſit, & ubi poſsit legere ſcripturam:</s> <s xml:id="echoid-s4616" xml:space="preserve"> <lb/>& fuerit faciale uiſui:</s> <s xml:id="echoid-s4617" xml:space="preserve"> & intueatur ipſum:</s> <s xml:id="echoid-s4618" xml:space="preserve"> deinde obliquauerit ipſum in ſuo loco:</s> <s xml:id="echoid-s4619" xml:space="preserve"> inueniet ſcriptu-<lb/>ram latêre:</s> <s xml:id="echoid-s4620" xml:space="preserve"> & ſi magis obliquauerit, magis latebit, ita quòd ſi multùm obliquauerit ipſum, adeò ut <lb/>poſitio eius ſit propin qua poſitioni radiorum exeuntium ad medium eius:</s> <s xml:id="echoid-s4621" xml:space="preserve"> tunc uidebit ſcripturã <lb/>in pergameno latentem ualde, adeò, ut non poſsit legi:</s> <s xml:id="echoid-s4622" xml:space="preserve"> & hoc uidebit, ſiue conſideretur utroque <lb/>uiſu, ſiue uno tantùm.</s> <s xml:id="echoid-s4623" xml:space="preserve"> Et ſimiliter cum fixerit aliquam ſchedularum paruarum in loco oppoſito ui <lb/>ſui remotiore, quàm ſit longitudo tabulæ:</s> <s xml:id="echoid-s4624" xml:space="preserve"> & poſuerit ipſam facialem uiſui:</s> <s xml:id="echoid-s4625" xml:space="preserve"> & direxerit pupillam ad <lb/>ipſam utroque uiſu:</s> <s xml:id="echoid-s4626" xml:space="preserve"> & poſuerit aliam ſchedulam obliquam ſuper illam, aut dextrorſum aut ſini-<lb/>ſtrorſum:</s> <s xml:id="echoid-s4627" xml:space="preserve"> & erexerit eam ita, ut ſit facialis:</s> <s xml:id="echoid-s4628" xml:space="preserve"> inueniet eam latentiorem.</s> <s xml:id="echoid-s4629" xml:space="preserve"> Deinde ſi aliquis mouerit <lb/>ſecundam ſchedulam, & remouerit eam paulatim & paulatim à ſchedula, ad quam dirigit pupil-<lb/>lam:</s> <s xml:id="echoid-s4630" xml:space="preserve"> inueniet, quòd forma partis, quæ eſt in ſchedula, quæ eſt in extremo, quantò magis illa re-<lb/>motior eſt à ſecunda ſchedula, tantò magis latet, adeò ut fiat illegibilis omnino.</s> <s xml:id="echoid-s4631" xml:space="preserve"> Et ſimiliter ſi con-<lb/>ſiderauerit has duas ſchedulas, uno uiſu:</s> <s xml:id="echoid-s4632" xml:space="preserve"> inueniet talem diſpoſitionem.</s> <s xml:id="echoid-s4633" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div155" type="section" level="0" n="0"> <head xml:id="echoid-head179" xml:space="preserve" style="it">17. Viſibile uiſui directũ, certißimè uidetur: obliquũ tantò minus, quantò obliquius. 33 p 4.</head> <p> <s xml:id="echoid-s4634" xml:space="preserve">DEclaratur ergo exiſtis conſiderationibus omnibus, quòd manifeſtiſsimum uiſibilium in o-<lb/>muibus remotionibus eſt illud, quod eſt ſuper axem radialem:</s> <s xml:id="echoid-s4635" xml:space="preserve"> & quòd illud, quod eſt pro-<lb/>pinquius axi, eſt manifeſtius remotiore ab ipſo:</s> <s xml:id="echoid-s4636" xml:space="preserve"> & quòd uiſum remotum ab axe maxima re <lb/>motione, eſt dubitabilis formę, nõ certificabilis, & indifferenter, ſiue uiſio ſit uno uiſu, ſiue utroq;</s> <s xml:id="echoid-s4637" xml:space="preserve">. <lb/>Amplius etiam quòd uiſum faciale eſt in omnibus remotionibus manifeſtius uiſo obliquo:</s> <s xml:id="echoid-s4638" xml:space="preserve"> & <lb/>quòd quantò magis poſitio uiſi appropinquat poſitioni faciali, tantò erit manifeſtius:</s> <s xml:id="echoid-s4639" xml:space="preserve"> & quòd ui-<lb/>ſum obliquum ſuper lineas radiales obliquatione maxima, habet formam multùm dubitabilem, & <lb/>nõ certificatam à uiſu, ſiue uiſio ſit uno uiſu, ſiue utroq;</s> <s xml:id="echoid-s4640" xml:space="preserve">, & ſiue uiſum ſit ſuper axem, ſiue extra axẽ.</s> <s xml:id="echoid-s4641" xml:space="preserve"> <lb/>Quare uerò uiſum multùm obliquum ſit dubitabilis formę, licet remotio eius ſit mediocris, & licet <lb/>magnitudo ſit comprehenſa, ſecundum quod eſt:</s> <s xml:id="echoid-s4642" xml:space="preserve"> & quare uiſum faciale ſit manifeſtius obliquo, <lb/>hæc eſt:</s> <s xml:id="echoid-s4643" xml:space="preserve"> quia forma uiſi multùm obliqui inſtituitur in ſuperficie uiſus cõgregata propter ſuam ob-<lb/>liquationem.</s> <s xml:id="echoid-s4644" xml:space="preserve"> Quoniam cum uiſus fuerit multùm obliquus, tunc angulus, quem ſubtendit uiſum <lb/>ſuper centrum uiſus, erit paruus, & pars uiſus, in qua inſtituitur forma illius uiſi, erit minor multò <lb/>parte, in qua inſtituitur forma illius, ſi fuerit faciale uiſui, & partes eius paruæ ſuſtentantur apud ui <lb/>ſum angulis inſenſibilibus, propter maximã obliquationem.</s> <s xml:id="echoid-s4645" xml:space="preserve"> Pars enim parua cum multùm fuerit <lb/>obliqua:</s> <s xml:id="echoid-s4646" xml:space="preserve"> tunc duæ lineæ exeuntes à centro uiſus ad extrema illius partis, fient quaſi una linea.</s> <s xml:id="echoid-s4647" xml:space="preserve"> <lb/>Quapropter ſentiens non comprehendit angulum contentum inter eas, neque partem, quam di-<lb/>ſtinguit ex ſuperficie uiſus.</s> <s xml:id="echoid-s4648" xml:space="preserve"> Et uiſum multùm obliquum erit dubitabile, quia forma eius, quæ infi-<lb/>gitur in uiſu, erit congregata maxima congregatione, & partes eius paruæ erunt inſenſibiles, & i-<lb/>deo forma eius erit dubitabilis.</s> <s xml:id="echoid-s4649" xml:space="preserve"> Et ideo ſi in huiuſmodi uiſo fuerint ſubtiles intentiones, non com <lb/>prehendentur à uiſu propter latentiam ſuarum partium paruarum, & propter congregationem <lb/>formæ.</s> <s xml:id="echoid-s4650" xml:space="preserve"> Viſum autem faciale eſt è contrario.</s> <s xml:id="echoid-s4651" xml:space="preserve"> Nam forma eius, quæ inſtituitur in uiſu, erit ordinata <lb/>ſecundum quod eſt in ſuperficie uiſi, & partes eius paruæ, quæ poſſunt com prehendi à uiſu, erunt <lb/>manifeſtæ & ordinatæ in ſuperficie uiſus ſecundum ſuam ordinationem in ſuperficie uiſi:</s> <s xml:id="echoid-s4652" xml:space="preserve"> & tunc <lb/>forma erit manifeſta, & non dubitabilis.</s> <s xml:id="echoid-s4653" xml:space="preserve"> Et uniuerſaliter intentiones ſubtiles, & partes ſubtiles, & <lb/>ordinatio partium uiſi non comprehenduntur à uiſu uera comprehenſione, niſi cum forma impri-<lb/>mitur in ſuperficie membri ſentientis, & inſtituitur quælibet pars eius in parte ſenſibili ſuperficiei <lb/>membri ſentientis.</s> <s xml:id="echoid-s4654" xml:space="preserve"> Et cum uiſum fuerit multùm obliquum:</s> <s xml:id="echoid-s4655" xml:space="preserve"> tunc forma eius non imprimetur in <lb/>uiſu, neque formæ aliquarum partium paruarum infigentur in parte ſenſibili uiſus.</s> <s xml:id="echoid-s4656" xml:space="preserve"> Hoc enim non <lb/>fit, niſi quando uiſum fuerit faciale, aut quando obliquatio eius fuerit parua, & fuerit remotio eius <lb/>ſimul ex remotionibus mediocribus, in reſpectu remotionum, quæ ſunt in illo uiſo.</s> <s xml:id="echoid-s4657" xml:space="preserve"> Comprehen-<lb/>ſio uerò magnitudinis uiſi obliqui multùm, ſecundum quod eſt, cum fuerit in remotione medio-<lb/>cri, licet obliquatio eius ſit maxima:</s> <s xml:id="echoid-s4658" xml:space="preserve"> non eſt ex ipſa forma uiſi, quæ inſtituitur in uiſu, tantùm, ſed <lb/>ex ratione extra formam, ſcilicet ex hoc, quòd comprehendens comprehendit diuerſitatem dua-<lb/>rum remotionum extremorum eius, cum hoc, quòd comprehendit menſuram formæ.</s> <s xml:id="echoid-s4659" xml:space="preserve"> Et cum ui-<lb/>ſus comprehenderit diuerſitatem remotionis duorum extremorum uiſi multùm obliqui, & com-<lb/>prehenderit differentiam maximam inter eas:</s> <s xml:id="echoid-s4660" xml:space="preserve"> ſtatim uirtus diſtinctiua imaginabitur poſitionem il <lb/>lius uiſi, & comprehendet menſuram eius ſecundum diuerſitatem remotionum duorum extremo-<lb/>rum eius:</s> <s xml:id="echoid-s4661" xml:space="preserve"> & ſecundum menſuram partis, in qua inſtituitur forma:</s> <s xml:id="echoid-s4662" xml:space="preserve"> & ſecundum menſuram anguli, <lb/>quem ſubtenditilla pars apud centrũ uiſus, non ſolum modò exipſa forma.</s> <s xml:id="echoid-s4663" xml:space="preserve"> Et cũ uirtus diſtinctiua <lb/>cõprehenderit diuerſitatẽ duorũ extremorũ uiſi multũ obliqui, & cõprehẽderit obliquationẽ eius:</s> <s xml:id="echoid-s4664" xml:space="preserve"> <lb/>ſtatim percipiet congregationẽ formę.</s> <s xml:id="echoid-s4665" xml:space="preserve"> Cõprehendit ergo menſurã eius, cũ ſenſerit quantitatẽ obli <lb/>quationis eius non ſecundum menſurã formæ, ſed ſecundũ poſitionẽ eius.</s> <s xml:id="echoid-s4666" xml:space="preserve"> Et partes paruæ & ſub-<lb/>tiles intentiones, quæ ſunt in uiſo, non poſſunt comprehendi ratione, ſi uiſus non ſenſerit illas par <lb/>tes, aut illas intentiones.</s> <s xml:id="echoid-s4667" xml:space="preserve"> Latentia igitur formæ uiſi accidit ex congregatione formæ eius in uiſu, & <lb/>ex latẽtia partiũ eius paruarũ.</s> <s xml:id="echoid-s4668" xml:space="preserve"> Et apparẽtia formæ uiſi cũ fuerit in remotione mediocri, eſt propter <lb/>impreſsionẽ formę in uiſu, ſecũdũ qđ eſt, & propter hoc, quòd ſentit uiſus partes eius paruas.</s> <s xml:id="echoid-s4669" xml:space="preserve"> Qua-<lb/>re igitur forma uiſi maximè obliqui ſit dubitabilis, forma aũt uiſi facialis ſit manifeſta, declaratum <lb/>eſt.</s> <s xml:id="echoid-s4670" xml:space="preserve"> His aũt declaratis, incipiẽdũ eſt à ſermõe de deceptiõe uiſus, & declarãdę cauſſę & ſpecies earũ.</s> <s xml:id="echoid-s4671" xml:space="preserve"/> </p> <pb o="88" file="0094" n="94" rhead="ALHAZEN"/> </div> <div xml:id="echoid-div156" type="section" level="0" n="0"> <head xml:id="echoid-head180" xml:space="preserve">DE CAVSSIS, QVIBVS VISVI ACCIDIT DE-<lb/>ceptio. Cap. III.</head> <head xml:id="echoid-head181" xml:space="preserve" style="it">18. Ad uiſionem perficiendam octo neceſſaria ſunt: quorum quodlibet ad uitandum allu-<lb/>cinationes, uiſibili ſymmetrum eſſe oportet. 1. 2. 13. 14. 15. 16. 19. 56 p 3. 1 p 4. Vide 36 n 1.</head> <p> <s xml:id="echoid-s4672" xml:space="preserve">DEclaratum eſt in libro primo [36 n] quòd ad hoc, ut formas corporis uiſi directè uiſus com <lb/>prehendat, neceſſaria eſt quorundam aggregatio, quæ ſunt Longitudo:</s> <s xml:id="echoid-s4673" xml:space="preserve"> Oppoſitio:</s> <s xml:id="echoid-s4674" xml:space="preserve"> Lux non <lb/>multùm debilis:</s> <s xml:id="echoid-s4675" xml:space="preserve"> Soliditas corporis:</s> <s xml:id="echoid-s4676" xml:space="preserve"> Magnitudo eiuſdem:</s> <s xml:id="echoid-s4677" xml:space="preserve"> Raritas intermedij aeris:</s> <s xml:id="echoid-s4678" xml:space="preserve"> ſi enim <lb/>adfuerit alicuius horum defectus, non erit uiſus.</s> <s xml:id="echoid-s4679" xml:space="preserve"> Planum eſt etiam exlibro ſecundo [12.</s> <s xml:id="echoid-s4680" xml:space="preserve"> 13.</s> <s xml:id="echoid-s4681" xml:space="preserve"> 20 n] <lb/>quòd nihil poteſt uiſus comprehendere ex corporibus, niſi in tempore.</s> <s xml:id="echoid-s4682" xml:space="preserve"> Tẽpus igitur eſt unum eo-<lb/>rum, quæ neceſſaria ſunt ad hoc, ut fiat uiſus.</s> <s xml:id="echoid-s4683" xml:space="preserve"> Similiter infirmitas oculi impedit uiſum:</s> <s xml:id="echoid-s4684" xml:space="preserve"> quare ſani-<lb/>tas erit unum neceſſariorum.</s> <s xml:id="echoid-s4685" xml:space="preserve"> Amplius iam explanatum eſt in parte præcedente [15.</s> <s xml:id="echoid-s4686" xml:space="preserve"> 17n] quòd <lb/>corpus multùm elongatum ab axe, occultatur uiſui:</s> <s xml:id="echoid-s4687" xml:space="preserve"> & ſi multùm tunc fuerit declinatum, non ple-<lb/>nè comprehendetur.</s> <s xml:id="echoid-s4688" xml:space="preserve"> Neceſſarius ergo eſt ſitus ad complementum uiſus, cum non plena fiat com-<lb/>prehenſio, niſi in ſitu determinato.</s> <s xml:id="echoid-s4689" xml:space="preserve"> Sunt ergo octo neceſſaria ad operationem uiſus, Longitudo:</s> <s xml:id="echoid-s4690" xml:space="preserve"> <lb/>Situs:</s> <s xml:id="echoid-s4691" xml:space="preserve"> Lux:</s> <s xml:id="echoid-s4692" xml:space="preserve"> Magnitudo corporis:</s> <s xml:id="echoid-s4693" xml:space="preserve"> Soliditas:</s> <s xml:id="echoid-s4694" xml:space="preserve"> Raritas aeris:</s> <s xml:id="echoid-s4695" xml:space="preserve"> Tẽpus:</s> <s xml:id="echoid-s4696" xml:space="preserve"> Sanitas uiſus.</s> <s xml:id="echoid-s4697" xml:space="preserve"> Et quodlibet iſto-<lb/>rum latitudinem habet proportionatam ad rem uiſam.</s> <s xml:id="echoid-s4698" xml:space="preserve"> Verbi gratia, corpus aliquod ab aliqua di-<lb/>ſtantia plenè comprehenditur, ab alia non plenè:</s> <s xml:id="echoid-s4699" xml:space="preserve"> & inter illas diſtantias eſt latitudo magna, in qua <lb/>fit plena comprehenſio illius corporis, quæ eſt latitudo longitudinis, reſpectu tanti corporis, & ſe-<lb/>cundum quod maius fuerit corpus, maior erit latitudo diſtantiæ eius.</s> <s xml:id="echoid-s4700" xml:space="preserve"> Pari modo cum magna fue-<lb/>rit corporis alicuius declinatio:</s> <s xml:id="echoid-s4701" xml:space="preserve">non comprehendentur notæ, uel particulæ, quæ ſunt in eo:</s> <s xml:id="echoid-s4702" xml:space="preserve"> ſi autẽ <lb/>in eadem declinatione uideatur corpus, in quo maioris quantitatis notæ, uel partes minus minutę <lb/>fuerint:</s> <s xml:id="echoid-s4703" xml:space="preserve"> comprehendentur:</s> <s xml:id="echoid-s4704" xml:space="preserve"> in minore autem declinatione corporis primi, uidebuntur eius minu-<lb/>tiæ:</s> <s xml:id="echoid-s4705" xml:space="preserve"> & eſt inter has declinationes latitudo.</s> <s xml:id="echoid-s4706" xml:space="preserve"> Similiter corpus paruum circa axem ſitum uidetur:</s> <s xml:id="echoid-s4707" xml:space="preserve"> mul <lb/>tùm elongatum, occultatur:</s> <s xml:id="echoid-s4708" xml:space="preserve"> & in eadem elongatione corpus maius uidebitur.</s> <s xml:id="echoid-s4709" xml:space="preserve"> Palàm ergo, quòd <lb/>ſitus habet latitudinem proportionatam ad corporis magnitudinem & minutias eius.</s> <s xml:id="echoid-s4710" xml:space="preserve"> Lucem pla-<lb/>num eſt habere latitudinem:</s> <s xml:id="echoid-s4711" xml:space="preserve"> fortitu do enim lucis cum magna fuerit, obfuſcat apparentiam corpo-<lb/>ris:</s> <s xml:id="echoid-s4712" xml:space="preserve"> & ſimiliter etiam eiuſdem debilitas:</s> <s xml:id="echoid-s4713" xml:space="preserve"> ſed erit corporum apparentia in lucibus intermedijs.</s> <s xml:id="echoid-s4714" xml:space="preserve"> Præ-<lb/>terea in luce aliqua quædã partes corporis cõprehenduntur, & in eadẽluce aliæ minutiſsimæ ab-<lb/>ſconduntur, quæ in luce maiore uiderẽtur.</s> <s xml:id="echoid-s4715" xml:space="preserve"> Eſt ergo latitudo lucis proportionata ad magnitudinẽ <lb/>corporis.</s> <s xml:id="echoid-s4716" xml:space="preserve"> Magnitudo corporis habet latitudinẽ:</s> <s xml:id="echoid-s4717" xml:space="preserve"> Si enim partes rei uiſæ nõ fuerint proportionales <lb/>totali:</s> <s xml:id="echoid-s4718" xml:space="preserve"> occultabũtur uiſui:</s> <s xml:id="echoid-s4719" xml:space="preserve"> ſi uerò fuerint ꝓportionales, & corpus totale fuerit modicũ, adhuc ab-<lb/>ſcondentur.</s> <s xml:id="echoid-s4720" xml:space="preserve"> Vnde in auibus & animalibus minutis particulas aliquas nõ percipimus, licet ſint pro-<lb/>portionales eis:</s> <s xml:id="echoid-s4721" xml:space="preserve"> Si aũt magnũ fuerit corpus uiſum, & partes eius ꝓportionales:</s> <s xml:id="echoid-s4722" xml:space="preserve"> nõ latebũtuſque-<lb/>adeò.</s> <s xml:id="echoid-s4723" xml:space="preserve"> Eſt igitur latitudo magnitudinis rei uiſæ proportionata ad totale corpus, cuius pars fuerit.</s> <s xml:id="echoid-s4724" xml:space="preserve"> <lb/>Soliditas aũt habet latitudinẽ ꝓportionatã ad rem uiſam.</s> <s xml:id="echoid-s4725" xml:space="preserve"> Si enim in corpore aliquo coloracutus <lb/>fuerit:</s> <s xml:id="echoid-s4726" xml:space="preserve"> licet paucæ ſoliditatis:</s> <s xml:id="echoid-s4727" xml:space="preserve"> uideri poterit, quòd eadẽ ſoliditate manẽte nõ accideret, ſi color eſſet <lb/>obtuſus.</s> <s xml:id="echoid-s4728" xml:space="preserve"> Raritas aeris habet latitudinẽ.</s> <s xml:id="echoid-s4729" xml:space="preserve"> Si enim uiſui & ſcripturæ interponatur aer parũ ſolidus, ut <lb/>flãma uel fumus, ſcriptura nõ diſcernetur, pergamenũ tamẽ uidebitur:</s> <s xml:id="echoid-s4730" xml:space="preserve"> & ſic in huiuſmodi alijs.</s> <s xml:id="echoid-s4731" xml:space="preserve"> Eſt <lb/>ergo proportionata hæc latitudo ſecũdũ uiſa.</s> <s xml:id="echoid-s4732" xml:space="preserve"> Tempus habet latitudinẽ.</s> <s xml:id="echoid-s4733" xml:space="preserve"> Si quis enim per foramen <lb/>inſpiciat corpus, quod ſtatim tranſeat, non percipietur.</s> <s xml:id="echoid-s4734" xml:space="preserve"> Similiter motus trochi (quia uelociſsi-<lb/>mus) in tempore multùm paruo non attenditur.</s> <s xml:id="echoid-s4735" xml:space="preserve"> Similiter accidit in motu multùm paruo.</s> <s xml:id="echoid-s4736" xml:space="preserve"> Sani-<lb/>tas habet latitudinem.</s> <s xml:id="echoid-s4737" xml:space="preserve"> In quadam enim infirmitate minutiæ corporis uiſi ab ſconduntur, in mino-<lb/>re percipiuntur.</s> <s xml:id="echoid-s4738" xml:space="preserve"> Et generaliter quilibet ſitus, in quo non uerificatur forma rei uiſæ, ſicut eſt in ueri-<lb/>tate, eſt ſitus egreſſus à remperantia ad rem uiſam illam proportionata.</s> <s xml:id="echoid-s4739" xml:space="preserve"> Egreditur autem ſitus rei <lb/>uiſæ à temperamento in longitudine:</s> <s xml:id="echoid-s4740" xml:space="preserve"> uel propter maximum longitudinis excrementum:</s> <s xml:id="echoid-s4741" xml:space="preserve"> uel maxi <lb/>mam eius diminutionem.</s> <s xml:id="echoid-s4742" xml:space="preserve"> In ſitu ſit egreſsio à temperantia per maximam ab axe elongationem:</s> <s xml:id="echoid-s4743" xml:space="preserve"> <lb/>per ſitus corporis reſpectu duorum uiſuum diuerſitatem:</s> <s xml:id="echoid-s4744" xml:space="preserve"> per maximam eius decliñationem.</s> <s xml:id="echoid-s4745" xml:space="preserve"> In lu-<lb/>ce egreſſum à temperantia efficit fortitudo maxima eius, uel debilitas nimia.</s> <s xml:id="echoid-s4746" xml:space="preserve"> In magnitudine di-<lb/>minutio quantitatis rei uiſæ.</s> <s xml:id="echoid-s4747" xml:space="preserve"> In ſoliditate raritatis intenſio.</s> <s xml:id="echoid-s4748" xml:space="preserve"> In aere nimia eius ſpiſsitudo.</s> <s xml:id="echoid-s4749" xml:space="preserve"> In tèmpo-<lb/>re minima eius duratio.</s> <s xml:id="echoid-s4750" xml:space="preserve"> In ſanitate debilitas uiſus maxima, uel eius immutatio ſecundum ægritu-<lb/>dinem.</s> <s xml:id="echoid-s4751" xml:space="preserve"> Habet autem temperamentum latitudinem, quæ ſic patebit.</s> <s xml:id="echoid-s4752" xml:space="preserve"> Viſo aliquo corpore, & pau-<lb/>lulum à uiſu elongato uel adducto:</s> <s xml:id="echoid-s4753" xml:space="preserve"> dum uidetur diſtans à ueritate inſenſibili proportione, adhuc <lb/>eſt de temperamento:</s> <s xml:id="echoid-s4754" xml:space="preserve"> & ita donec proportionalis ſit, & ſenſibilis apparentiæ mutatio.</s> <s xml:id="echoid-s4755" xml:space="preserve"> Menſura-<lb/>tur etiam temperamenti latitudo in quolibet iſtorum ſecundum proportionem eius ad alia ſeptẽ:</s> <s xml:id="echoid-s4756" xml:space="preserve"> <lb/>& ſecundum colorem & partium corporis paruitatem.</s> <s xml:id="echoid-s4757" xml:space="preserve"> Igitur latitudo temperamenti longitudinis <lb/>attenditur, & ſecundum colorem & ſecundum minutias, quæ in corpore fuerint, & ſecundum lu-<lb/>cem, & ſexalia, quæ dicta ſunt.</s> <s xml:id="echoid-s4758" xml:space="preserve"> Secundum coloris uarietatem:</s> <s xml:id="echoid-s4759" xml:space="preserve"> quoniam corpus fortis & acuti co-<lb/>loris, à maiore longitudine percipitur, quàm obſcuri & debilis.</s> <s xml:id="echoid-s4760" xml:space="preserve"> Vnde latitudo temperamenti <lb/>longitudinis maior, eſt proportionata magis ad colorem fortem, quàm ad debilem.</s> <s xml:id="echoid-s4761" xml:space="preserve"> Similiter ſi <lb/>fuerint in corpore uiſo notæ notabiles, à maiore longitudine comprehendentur, quàm ſi multùm <lb/>paruæ.</s> <s xml:id="echoid-s4762" xml:space="preserve"> Vnde maior longitudinis temperantia, reſpectu partium corporis notabilium, quàm reſpe <lb/>ctu minutarum.</s> <s xml:id="echoid-s4763" xml:space="preserve"> Pari modo maius eſt temperamẽtum longitudinis ad rectam corporis oppoſitio-<lb/> <pb o="89" file="0095" n="95" rhead="OPTICAE LIBER III."/> nem proportionatum, quàm ad eius declinationem.</s> <s xml:id="echoid-s4764" xml:space="preserve"> Similiter erit maius ſecundum propinquita-<lb/>tem corporis ab axe, quàm elongationem.</s> <s xml:id="echoid-s4765" xml:space="preserve"> Eodem modo maior eſt temperamenti longitudinis la-<lb/>titudo in fortiluce, quàm in debili.</s> <s xml:id="echoid-s4766" xml:space="preserve"> Et maior, ſi corpus uiſum fuerit magnum, quàm ſi paruum.</s> <s xml:id="echoid-s4767" xml:space="preserve"> <lb/>Similiter corpus multùm ſolidum à maiore longitudine percipitur, quàm minus ſolidum.</s> <s xml:id="echoid-s4768" xml:space="preserve"> Vnde <lb/>ſoliditati corporis proportionatur longitudinis temperamentum.</s> <s xml:id="echoid-s4769" xml:space="preserve"> Ad qualitatem aeris propor-<lb/>tionatur temperamentum longitudinis:</s> <s xml:id="echoid-s4770" xml:space="preserve"> quoniam ſpiſsitu do aeris ab aliqua longitudine corpora <lb/>uiſui abſcondit, quæ ab eadem, uel à maiore longitudine, claritas exponit.</s> <s xml:id="echoid-s4771" xml:space="preserve"> Temporis quantita-<lb/>ti proportionatur temperamentum longitudinis.</s> <s xml:id="echoid-s4772" xml:space="preserve"> Quoniam in tempore aliquo motus corporis <lb/>percipitur ab aliqua longitudine, & à maiore percipietur in maiore tempore.</s> <s xml:id="echoid-s4773" xml:space="preserve"> Pari modo in aliquo <lb/>ſtatu ſanitatis uiſus, in maiore longitudine uidebitur corpus, quàm in minore.</s> <s xml:id="echoid-s4774" xml:space="preserve"> Similiter menſura-<lb/>tur temperamentum ſitus, ſecundum proportionem factam ad longitudinem, ad colorem, ad mi-<lb/>nutias corporis, ad lucem, & ad alia, quæ enumerauimus.</s> <s xml:id="echoid-s4775" xml:space="preserve"> Et tu conſidera, & ſingulis adapta, & ui-<lb/>dere poteris facile:</s> <s xml:id="echoid-s4776" xml:space="preserve"> & eodem modo proportionabis temperamentum cuiuslibet iſtorum ad omnia <lb/>alia, & uidebis, quod dictum eſt per ſingula.</s> <s xml:id="echoid-s4777" xml:space="preserve"> Quando ergo ſingula eorum, quæ enumerata ſunt, <lb/>fuerint in latitudine temperamenti ſui:</s> <s xml:id="echoid-s4778" xml:space="preserve"> apparebit ueritas formæ rei uiſæ, ſicut eſt in re:</s> <s xml:id="echoid-s4779" xml:space="preserve"> quando au-<lb/>tem non apparet forma, ſicut eſt in ueritate, egreſſum eſt uel aliquod prædictorum à temperamen-<lb/>to, aut plura eorum.</s> <s xml:id="echoid-s4780" xml:space="preserve"> Igitur cauſſa, quare erret uiſus in comprehenſione formarum, nõ eſt, niſi egreſ-<lb/>ſus alicuius prædictorum à temperamento, aut plurium.</s> <s xml:id="echoid-s4781" xml:space="preserve"> Et hæc dicenda in hac erant parte.</s> <s xml:id="echoid-s4782" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div157" type="section" level="0" n="0"> <head xml:id="echoid-head182" xml:space="preserve">DE DISTINGVENDIS ERRORIBVS VI-<lb/>ſus. Cap. IIII.</head> <head xml:id="echoid-head183" xml:space="preserve" style="it">19. In uiſione erratur aut ſolo uiſu: aut anticipata notione: aut ſyllogiſmo.</head> <p> <s xml:id="echoid-s4783" xml:space="preserve">PLanum eſt ex libro ſecundo [10 n] quòd comprehenſio rerum fit per ſenſum, ſcientiam, ſyl-<lb/>logiſmum.</s> <s xml:id="echoid-s4784" xml:space="preserve"> Cum autem accidit error ιn his, quorum fit comprehenſio perſolum ſenſum:</s> <s xml:id="echoid-s4785" xml:space="preserve"> ſci-<lb/>mus quòd eſt error ſenſus tantùm.</s> <s xml:id="echoid-s4786" xml:space="preserve"> Cum uerò in ijs, quæ per ſcientiam comprehendit, quis <lb/>errauerit:</s> <s xml:id="echoid-s4787" xml:space="preserve"> in ſcientia tantùm erit error.</s> <s xml:id="echoid-s4788" xml:space="preserve"> Si uerò in his, quæ per ſyllogiſmum comprehenduntur, er <lb/>ret quis:</s> <s xml:id="echoid-s4789" xml:space="preserve"> erit error in ſyllogiſmo tantùm.</s> <s xml:id="echoid-s4790" xml:space="preserve"> Senſus acquirit lucem & colorem tantùm, ſicut dictum <lb/>eſt [17 n 2.</s> <s xml:id="echoid-s4791" xml:space="preserve">] Scientia uero prætendit ea, quæ prius ſunt uiſa & in uiſu habita, ut lux ſolis cogno-<lb/>ſcitur, quòd plurimùm uiſa ſit, & inter lucem ſolis & lunæ diſcernitur:</s> <s xml:id="echoid-s4792" xml:space="preserve"> & licet, fiat comprehenſio <lb/>lucis per ſenſum tantùm:</s> <s xml:id="echoid-s4793" xml:space="preserve"> tamen per ſcientiam accidit diſtinctio lucis.</s> <s xml:id="echoid-s4794" xml:space="preserve"> Similiter accidit per ſcien-<lb/>tiam notitia figurarum, ut trianguli, quadrati, circuli, & aliarum ſimilium.</s> <s xml:id="echoid-s4795" xml:space="preserve"> Similiter notitia aſperi-<lb/>tatis, læuitatis, umbræ, decoris, & ſimilium.</s> <s xml:id="echoid-s4796" xml:space="preserve"> Per ſyllogiſmum fit comprehenſio eorum, quæ ſu-<lb/>prà explanauimus, licet ea non plurimùm nouerit ſenſus.</s> <s xml:id="echoid-s4797" xml:space="preserve"> Omnis autem comprehenſio rerum con <lb/>tinetur ſub aliquo horum trium modorum:</s> <s xml:id="echoid-s4798" xml:space="preserve"> & cum error accidit in comprehenſione formarum, <lb/>non accidit, niſi in aliquo iſtorum.</s> <s xml:id="echoid-s4799" xml:space="preserve"> Accidit error ſenſui, ſi corpus, in quo ſit multa colorum parti-<lb/>cularium diuerſitas, occurrat uiſui ſub luce multùm debili, ut ueſtis aliqua diuerſis coloribus & mi <lb/>nutis picturata, apparebit unius coloris.</s> <s xml:id="echoid-s4800" xml:space="preserve"> Et erit error in ſenſu propter lucem à temperamento ſuo <lb/>egreſſam, cæteris a temperantia non egreſsis.</s> <s xml:id="echoid-s4801" xml:space="preserve"> In ſcientia error accidit, cum in magna longitudine <lb/>uidetur aliquando homo notus, æſtimatur eſſe alius, ſimiliter cognitus:</s> <s xml:id="echoid-s4802" xml:space="preserve"> unde ab aliqua longitu-<lb/>dine uidens fratrem, putat ſe uidere patrem, uel aliquem in hunc modum.</s> <s xml:id="echoid-s4803" xml:space="preserve"> Et eſt error in ſcientia, <lb/>propter egreſſum ſolius longitudinis à temperamento.</s> <s xml:id="echoid-s4804" xml:space="preserve"> In ſyllogiſmo accidit error, ut quando mo-<lb/>tis nubibus, æſtimatur eſſe lunæ motus.</s> <s xml:id="echoid-s4805" xml:space="preserve"> Et accidit error iſte ex intemperata longitudine.</s> <s xml:id="echoid-s4806" xml:space="preserve"> Quo-<lb/>niam quando uiſilongitudinis eſt temperantia, non euenit ita:</s> <s xml:id="echoid-s4807" xml:space="preserve"> ut baculum fundoaquæ infixum, <lb/>& aquam ſupereminentem, in motu etiam immotum uidemus, & motum tranſeuntis aquæ per-<lb/>cipimus.</s> <s xml:id="echoid-s4808" xml:space="preserve"> Accidit autem error prædictus in motu lunæ, cum nubes fuerint multæ & continuæ.</s> <s xml:id="echoid-s4809" xml:space="preserve"> Et <lb/>cauſſa eius eſt:</s> <s xml:id="echoid-s4810" xml:space="preserve"> quoniam ſicut patuit ſuperius, [49 n 2] non comprehenditur motus, niſi per ac-<lb/>ceſſum alicuius ad aliquid, uel receſſum conſideratum.</s> <s xml:id="echoid-s4811" xml:space="preserve"> Cum ergo paucitas fuerit nubium:</s> <s xml:id="echoid-s4812" xml:space="preserve"> poſſu-<lb/>mus diſcernere motus earum propter uniuſcuiuſq;</s> <s xml:id="echoid-s4813" xml:space="preserve"> ad ſtellam aliquam acceſſum apparentem, aut <lb/>receſſum:</s> <s xml:id="echoid-s4814" xml:space="preserve"> cum uerò cœlum nubibus fuerit coopertum, propter continuitatem earum non diſcer-<lb/>nimus motum, ueruntamen lunam modò in una parte uidemus, modò in alia:</s> <s xml:id="echoid-s4815" xml:space="preserve"> unde ipſam mo-<lb/>tu celerrimo moueri concludimus.</s> <s xml:id="echoid-s4816" xml:space="preserve"> Eodem modo erit error per ſitum à temperamento egreſſum, <lb/>& per unum quodque octo ſuprà dictorum in comprehenſione per ſenſum, per ſcientiam, & per <lb/>ſyllogiſmum.</s> <s xml:id="echoid-s4817" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div158" type="section" level="0" n="0"> <head xml:id="echoid-head184" xml:space="preserve">DE QVALITATIBVS DECEPTIONVM VISVS, QVAE <lb/>fiunt ſolo ſenſu. Cap. v.</head> <head xml:id="echoid-head185" xml:space="preserve" style="it">20. Erratur ſolo uiſu in luce & colore, propter ſingulorum uiſionem perficientium aſymme-<lb/>triam. 156 p 4.</head> <p> <s xml:id="echoid-s4818" xml:space="preserve">EX prædictis palàm, quòd non fit comprehenſio per ſenſum, niſi lucis & coloris tantùm.</s> <s xml:id="echoid-s4819" xml:space="preserve"> Non <lb/>ergo error accidit ſenſui, niſi in luce & coloretãtùm.</s> <s xml:id="echoid-s4820" xml:space="preserve"> Nec accidit perlucem aut colorem, ni-<lb/>ſi propter intemperatam debilitatẽ eius aut fortitudinem:</s> <s xml:id="echoid-s4821" xml:space="preserve"> uel propter colorum minutorũ & <lb/> <pb o="90" file="0096" n="96" rhead="ALHAZEN"/> debilium diuerſitatem.</s> <s xml:id="echoid-s4822" xml:space="preserve"> Et hæc colorum diuerſitas in luce debili uenitad oculum, tanquam aliquid <lb/>obſcurum aut tenebroſum:</s> <s xml:id="echoid-s4823" xml:space="preserve"> & etiam in luce forti, quando ſubſtantia colorum fuerit ualde parua.</s> <s xml:id="echoid-s4824" xml:space="preserve"> <lb/>Longitudo inducit errorem ſenſus, cum temperata fuerit elongatio corporis à uiſu, & fuerint in <lb/>corpore partes minutę in coloribus diuerſę, ad quas prop ortionata partiũ elongatio fit intempera <lb/>ta:</s> <s xml:id="echoid-s4825" xml:space="preserve"> apparebit enim corpus illud unius coloris tantùm:</s> <s xml:id="echoid-s4826" xml:space="preserve"> quoniam extra temperantiam eſt longitu-<lb/>do, reſpectu particularium, licet omnia alia conueniant in temperantia.</s> <s xml:id="echoid-s4827" xml:space="preserve"> Et eſt error iſte ſenſualis, <lb/>cum ſenſus ſit comprehenſiuus coloris.</s> <s xml:id="echoid-s4828" xml:space="preserve"> Situs ſenſum errare facit, cum maxima fuerit corporis ui-<lb/>ſi declinatio:</s> <s xml:id="echoid-s4829" xml:space="preserve"> occultabuntur uiſui minutæ eius particulæ.</s> <s xml:id="echoid-s4830" xml:space="preserve"> Et ſi in partibus minutis fuerit colorum <lb/>diuerſitas:</s> <s xml:id="echoid-s4831" xml:space="preserve"> apparebit in totali corpore colorum unitas.</s> <s xml:id="echoid-s4832" xml:space="preserve"> Et accidit error propter ſitũ tantùm.</s> <s xml:id="echoid-s4833" xml:space="preserve"> Quia <lb/>oppoſito corpore uiſui, in ſitu recto, alijs (ſicut ſunt) immotis, percipientur etiam partes corpo-<lb/>ris & coloris, cum ſolus ſitus egreſſus ſit à temperamento.</s> <s xml:id="echoid-s4834" xml:space="preserve"> Idem error accidit ex ſitus intemperan-<lb/>tia, cum elongatio partium minutarum ab axe fuerit magna.</s> <s xml:id="echoid-s4835" xml:space="preserve"> Lux multùm debilis errorem facit, ab-<lb/>ſcondit enim uiſui particulas corporis, & prætendit unitatẽ tenebroſi coloris:</s> <s xml:id="echoid-s4836" xml:space="preserve"> & ſi lux ad temperã-<lb/>tiam reduceretur:</s> <s xml:id="echoid-s4837" xml:space="preserve"> diuerſitas colorum aut diminutio partium non occultaretur:</s> <s xml:id="echoid-s4838" xml:space="preserve"> quoniam lux ſola <lb/>extra temperantiam eſt ſita.</s> <s xml:id="echoid-s4839" xml:space="preserve"> Magnitudo errorem inuehit.</s> <s xml:id="echoid-s4840" xml:space="preserve"> Cum enim partes corporis minutiſsimæ <lb/>diſsimiles fuerint in totali colore:</s> <s xml:id="echoid-s4841" xml:space="preserve"> latebunt uiſum partes illæ propter ſuam paruitatem, & ſimiliter <lb/>corum colores:</s> <s xml:id="echoid-s4842" xml:space="preserve"> & apparebit color unicus in corpore, magnitudine ſola extra temperantiam ſita:</s> <s xml:id="echoid-s4843" xml:space="preserve"> <lb/>quod nõ appareret, ſi paruitas partiũ extra tẽperamentũ non exiret.</s> <s xml:id="echoid-s4844" xml:space="preserve"> Soliditas cauſſa eſt erroris ſem <lb/>ſualis, ſi remiſſa fuerit ſoliditas, ut in cryſtallo:</s> <s xml:id="echoid-s4845" xml:space="preserve">unde cum ei ſupponitur corpus coloratum, uidetur <lb/>cryſtallus colore illo affecta, propter ſoliditatis paruitatem à temperamento egreſſam:</s> <s xml:id="echoid-s4846" xml:space="preserve"> quod non <lb/>accideret, ſi cryſtallus magis ſolida eſſet.</s> <s xml:id="echoid-s4847" xml:space="preserve"> Ex raritate aeris procedit error ſenſualis:</s> <s xml:id="echoid-s4848" xml:space="preserve"> cum intercidit <lb/>inter uiſum & corpus oppoſitum, flamma, licet fortis coloris ſit corpus uiſum:</s> <s xml:id="echoid-s4849" xml:space="preserve"> uidebitur tenebro-<lb/>ſum.</s> <s xml:id="echoid-s4850" xml:space="preserve"> Et ſola raritas aeris egreſſa eſt temperamentum.</s> <s xml:id="echoid-s4851" xml:space="preserve"> Tempus eſt cauſſa erroris:</s> <s xml:id="echoid-s4852" xml:space="preserve"> quoniam ſi ſubitò <lb/>ſuper corpus diuerſorum colorum fiat uiſus directio:</s> <s xml:id="echoid-s4853" xml:space="preserve"> apparebit color ſingularis, donec prolonge-<lb/>tur inſpectionis duratio:</s> <s xml:id="echoid-s4854" xml:space="preserve"> luce dico, ſub qua comprehenditur corpus, non forti.</s> <s xml:id="echoid-s4855" xml:space="preserve"> In luce enim debi-<lb/>li non ſtatim immutatur uiſus ſecundum quemlibet colorum particularium:</s> <s xml:id="echoid-s4856" xml:space="preserve"> quod accideret in lu-<lb/>ce forti.</s> <s xml:id="echoid-s4857" xml:space="preserve"> Viſus aliquan do errorem prætendit:</s> <s xml:id="echoid-s4858" xml:space="preserve"> Luce enim forti in uiſum cadente:</s> <s xml:id="echoid-s4859" xml:space="preserve"> læditur uiſus, & <lb/>ſtatim ad colorem alicuius corporis conuerſus, ipſum tenebroſum recipit, donec paululum ſtete-<lb/>rit, & læſio receſſerit.</s> <s xml:id="echoid-s4860" xml:space="preserve"> Pari modo cum aderit oculi infirmitas:</s> <s xml:id="echoid-s4861" xml:space="preserve"> occultabitur uiſui colorum ueritas.</s> <s xml:id="echoid-s4862" xml:space="preserve"> <lb/>Vnde error eſt ex ſola uiſus qualitate à temperamento recedente.</s> <s xml:id="echoid-s4863" xml:space="preserve"> Patet ergo, quòd accidanterro-<lb/>res uiſui ſecundum quo dlibet prædictorum conſiderati.</s> <s xml:id="echoid-s4864" xml:space="preserve"> Et acciduntin ſenſu tantùm:</s> <s xml:id="echoid-s4865" xml:space="preserve"> cum ex ſolo <lb/>ſenſu fiat comprehenſio colorum.</s> <s xml:id="echoid-s4866" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div159" type="section" level="0" n="0"> <head xml:id="echoid-head186" xml:space="preserve">DE QVALITATIBVS DECEPTIONVM VISVS, QVAE <lb/>fiuntin ſcientia & cognitione. Cap. VI.</head> <head xml:id="echoid-head187" xml:space="preserve" style="it">21. Erratur anticipata nõtione: cum forma anticipata, obiecto uiſibili perperam aßimila-<lb/>tur, propter ſingulorum uiſionem perficientium aſymmetriam. 155 p 4.</head> <p> <s xml:id="echoid-s4867" xml:space="preserve">DIctum eſt in ſecundo libro, [14.</s> <s xml:id="echoid-s4868" xml:space="preserve"> 67 n] quòd non niſi per ſcientiam fit definitionis rei acqui-<lb/>ſitio.</s> <s xml:id="echoid-s4869" xml:space="preserve"> Peruenit enim definitio ex ſimilitudine uel diſsimilitudine alicuius rei cũ alia, in com-<lb/>muni forma.</s> <s xml:id="echoid-s4870" xml:space="preserve"> Et proprium eſt ſcientiæ communicare rem uiſui præſentem cum re prius ui-<lb/>ſa in ſorma recepta:</s> <s xml:id="echoid-s4871" xml:space="preserve"> & ex hac communicatione acquiritur definitio rei cuiufcunque.</s> <s xml:id="echoid-s4872" xml:space="preserve"> Diuerſifica-<lb/>tur autem ſcientia in ſcientiam ideæ uniuerſalis, aut ſingularis, aut utriuſque.</s> <s xml:id="echoid-s4873" xml:space="preserve"> Et omnis error ſci-<lb/>entiæ erit error in aliquo iſtorum, aut in utroque.</s> <s xml:id="echoid-s4874" xml:space="preserve"> Cum ergo res aliqua, aut alia, aut alterius ſpe-<lb/>ciei apparet, quàm ſit in rei ueritate:</s> <s xml:id="echoid-s4875" xml:space="preserve"> erit error in definitionis aſsignatione.</s> <s xml:id="echoid-s4876" xml:space="preserve"> Nec accidit error iſte, <lb/>niſi aliquod prædictorum fuerit extra temperamentum.</s> <s xml:id="echoid-s4877" xml:space="preserve"> Error ſcientiæ in longitudine erit:</s> <s xml:id="echoid-s4878" xml:space="preserve"> ſi à lon <lb/>gitudine magna uideatur homo notus:</s> <s xml:id="echoid-s4879" xml:space="preserve"> apparebit forſitan eſſe alius uidenti notus.</s> <s xml:id="echoid-s4880" xml:space="preserve"> Vnde aliquan-<lb/>do uidens Petrum, uiſum dicit eſſe Martinum, cum conſtet utrumq;</s> <s xml:id="echoid-s4881" xml:space="preserve"> ei eſſe notum.</s> <s xml:id="echoid-s4882" xml:space="preserve"> In forma com-<lb/>muni erit error:</s> <s xml:id="echoid-s4883" xml:space="preserve"> ſi quis ab aliqua longitudine uideat equum, & puter ſe uidere aſinum.</s> <s xml:id="echoid-s4884" xml:space="preserve"> In utraque <lb/>formarum, ſcilicet ſingularis & cõmunis, eſt error:</s> <s xml:id="echoid-s4885" xml:space="preserve"> ut ſi quis à longitudine maxima uideat equum <lb/>ſibi notum, & æſtimet ſe uidere aſinum ſibi cognitum.</s> <s xml:id="echoid-s4886" xml:space="preserve"> Pari modo accidit error in arboribus tri-<lb/>plex:</s> <s xml:id="echoid-s4887" xml:space="preserve"> in indiuiduis:</s> <s xml:id="echoid-s4888" xml:space="preserve"> in communibus formis:</s> <s xml:id="echoid-s4889" xml:space="preserve"> in utriſque.</s> <s xml:id="echoid-s4890" xml:space="preserve"> Vnde aliquando una amygdalus æſtima-<lb/>tur alia:</s> <s xml:id="echoid-s4891" xml:space="preserve"> aliquando à longitudine magna pyrus æſtimatur amygdalus:</s> <s xml:id="echoid-s4892" xml:space="preserve"> aliquando pyrus Petri, cre-<lb/>ditur amygdalus Martini.</s> <s xml:id="echoid-s4893" xml:space="preserve"> Eadem triplicitas erroris ex longitudine accidit plurimum in ueſtibus, <lb/>lapidibus, & alijs.</s> <s xml:id="echoid-s4894" xml:space="preserve"> Aliquando uidetur res ignota, & contingit error in ſcientia:</s> <s xml:id="echoid-s4895" xml:space="preserve"> ſicut ſi aliquis ui-<lb/>derit ignem longè remotum in aere, æſtimat forſitan ſe ſtellam uidere.</s> <s xml:id="echoid-s4896" xml:space="preserve"> Planum autem, quemlibet <lb/>errorem prædictum cadere in ſcientiam, cum in eo fiat aſsignatio definitionis rei uiſæ, quæ non <lb/>eſt in ea, in ueritate.</s> <s xml:id="echoid-s4897" xml:space="preserve"> Palàm etiam, quòd accidit error præfatus exlongitudine extra temperamen-<lb/>tum exeunte.</s> <s xml:id="echoid-s4898" xml:space="preserve"> Ea enim ad temperamentum reducta, alijs erroris cauſsis (ſicut ſunt,) manenti-<lb/>bus, non accidit error in ſcientia prædictus.</s> <s xml:id="echoid-s4899" xml:space="preserve"> Situs errorem infert ſcientiæ, cum corpus aliquod <lb/>multùm fuerit elongatũ ab axe:</s> <s xml:id="echoid-s4900" xml:space="preserve"> non erit certa formæ cõprehenſio.</s> <s xml:id="echoid-s4901" xml:space="preserve"> Vnde ali quando in hoc ſitu Pe-<lb/>trus æſtimabitur Martinus, aliquãdo equus æſtimabitur eſſe aſinus.</s> <s xml:id="echoid-s4902" xml:space="preserve"> Et in hac incertitudine forſan <lb/>eligetur ueritas, forſan falſitas.</s> <s xml:id="echoid-s4903" xml:space="preserve"> Cum enim in hoc ſtatu incertum ſit iudiciũ:</s> <s xml:id="echoid-s4904" xml:space="preserve"> caſualis electio erit.</s> <s xml:id="echoid-s4905" xml:space="preserve"> Ac-<lb/> <pb o="91" file="0097" n="97" rhead="OPTICAE LIBER III."/> cidit autem error ex intemperamento ſitus:</s> <s xml:id="echoid-s4906" xml:space="preserve"> quoniam ipſo ad tẽperantiam reducto, non errabit iu-<lb/>dicium ex ſcientia ſumptum.</s> <s xml:id="echoid-s4907" xml:space="preserve"> Pari modo in magna corporis declinatione non uerificantur particu-<lb/>læ minutæ.</s> <s xml:id="echoid-s4908" xml:space="preserve"> Vnde accidit in hoc ſitu error figuræ, coloris, magnitudinis.</s> <s xml:id="echoid-s4909" xml:space="preserve"> Forſan enim quadratum ui-<lb/>detur circulare:</s> <s xml:id="echoid-s4910" xml:space="preserve"> & ita error in quantitate & colore.</s> <s xml:id="echoid-s4911" xml:space="preserve"> Egreſsio lucis à temperamẽto errorẽ inducit ſci <lb/>entiæ.</s> <s xml:id="echoid-s4912" xml:space="preserve"> Debilitas enim lucis nimia errorem infert formę.</s> <s xml:id="echoid-s4913" xml:space="preserve"> Vnde accidit error in crepuſculis, in anima <lb/>libus, ueſtib.</s> <s xml:id="echoid-s4914" xml:space="preserve"> arboribus, ſcilicet triplex, uel in indiuiduo, uel in ſpecie, uel in utroq;</s> <s xml:id="echoid-s4915" xml:space="preserve">:<unsure/> quod non acci-<lb/>deret in temperata luce.</s> <s xml:id="echoid-s4916" xml:space="preserve"> Amplius ſi fuerit egreſsio lucis à temperamento proportionato uiſo, oppo <lb/>ſito uiſui:</s> <s xml:id="echoid-s4917" xml:space="preserve"> accidet error prædictus, licet non ſit intemperata in ſe lux:</s> <s xml:id="echoid-s4918" xml:space="preserve"> ſicut euenit in quadam aue ara <lb/>bicè aluerach dicta:</s> <s xml:id="echoid-s4919" xml:space="preserve"> non enim uideri poteſt, niſi de nocte:</s> <s xml:id="echoid-s4920" xml:space="preserve"> egreditur enim lux à temperamẽto, reſpe <lb/>ctu illius:</s> <s xml:id="echoid-s4921" xml:space="preserve"> percipitur aũt de nocte, ſicut ignis:</s> <s xml:id="echoid-s4922" xml:space="preserve"> de die uerò cũ nõ plenè diſcernatur, forſan papilio (cui <lb/>eſt ſimilis) putabitur.</s> <s xml:id="echoid-s4923" xml:space="preserve"> Etaccidit error in definitione rei ex intemperata luce.</s> <s xml:id="echoid-s4924" xml:space="preserve"> Quantitas extra tem-<lb/>perantiam ſita errare facit ſcientiam.</s> <s xml:id="echoid-s4925" xml:space="preserve"> Vn de aliquan do formica præ ſui paruitate æſtimatur muſca <lb/>tritico innata:</s> <s xml:id="echoid-s4926" xml:space="preserve"> & aliquando eadẽ de cauſſa ſinapis granũ reputatur naſturtium.</s> <s xml:id="echoid-s4927" xml:space="preserve"> Soliditas à tẽpera <lb/>mento egreſſa errorẽ efficit, ut cũ cryſtallo cõtinuatur corpus rubeum, alia cryſtalli facie uiſui oppo <lb/>ſita:</s> <s xml:id="echoid-s4928" xml:space="preserve"> æſtimabit uidens colorẽ cryſtalli, eſſe rubedinẽ:</s> <s xml:id="echoid-s4929" xml:space="preserve"> unde error eſt ſcientiæ, quia in coloris defini-<lb/>tione.</s> <s xml:id="echoid-s4930" xml:space="preserve"> Raritas aeris nimis diminuta, erroris eſt cauſſa:</s> <s xml:id="echoid-s4931" xml:space="preserve">unde in eius ſpiſsitudine fit error in rei defini <lb/>tione.</s> <s xml:id="echoid-s4932" xml:space="preserve"> Similiter ſi oculo & corpori uiſo interponatur corpus, cuius raritas extra temperantiam eſt, <lb/>reſpectu aeris tẽperatæ raritatis, ſicut eſt uitrum:</s> <s xml:id="echoid-s4933" xml:space="preserve"> æſtimabitur color corporis oppoſiti mixtus ex co <lb/>lore proprio & colore uitri.</s> <s xml:id="echoid-s4934" xml:space="preserve"> Et ita eſt error in coloris definitiõe.</s> <s xml:id="echoid-s4935" xml:space="preserve"> Pari modo ſi anreponatur oculo pã <lb/>nus multũ rarus, & poſt illũ uideatur corpus:</s> <s xml:id="echoid-s4936" xml:space="preserve"> apparebit color corporis mixtus.</s> <s xml:id="echoid-s4937" xml:space="preserve"> Sed oritur quæſtio, <lb/>quomodo poſt pãni oppoſitionẽ appareat coloris corporis mixtura, cũ partiales corporis colores <lb/>accedãt ad oculũ non niſi per pãni foramina:</s> <s xml:id="echoid-s4938" xml:space="preserve"> & ex pãno nõ accedat ad oculũ color, niſi ex filis eius, <lb/>per quę non tranſeunt colores corporis.</s> <s xml:id="echoid-s4939" xml:space="preserve"> Et huius rei ueritas eſt Quod licet partiales corporis colo <lb/>res ſigillatim ueniant, & in ſual loca cadãt, nec commiſceantur filorum coloribus, ſed filorum colo-<lb/>res ſint ab eis ſeparati intra uiſum & extra, nec ſit ibi aliqua confuſio:</s> <s xml:id="echoid-s4940" xml:space="preserve"> tñ quia ualde propinqua ſunt <lb/>puncta, in quæ incidunt color corporis ſuperficialis & color fili (cum non ſit diſtantia ſenſibilis in-<lb/>ter ea) uidentur quaſi punctum:</s> <s xml:id="echoid-s4941" xml:space="preserve"> unde colores ibi apparent unus ex eis mixtus.</s> <s xml:id="echoid-s4942" xml:space="preserve"> Si uerò magna fue-<lb/>rint panni foramina, diſcernetur & panni & coloris corporis ueritas ſine mixtura.</s> <s xml:id="echoid-s4943" xml:space="preserve"> Et quantò com-<lb/>preſsior fuerit foraminum ſtrictura, tantò uerior apparebit mixtura.</s> <s xml:id="echoid-s4944" xml:space="preserve"> Vnde uiſo corpore poſt pan-<lb/>num lanæ, uidebitur mixtura colorum plurimùm conſonans colori filorum.</s> <s xml:id="echoid-s4945" xml:space="preserve"> Foramina enim panni <lb/>lanei in ſe ſunt ſtricta, & quoniam pilis teguntur, efficiuntur ſtrictiora.</s> <s xml:id="echoid-s4946" xml:space="preserve"> Similiter cum aliquis iocula-<lb/>tor facit imagines ligneas moueri, umbræ earum inſpicienti per pannum, (ſicut ſolet fieri) lineum <lb/>ſubtilem, apparebunt aues, aut animalia form is imaginum conſona.</s> <s xml:id="echoid-s4947" xml:space="preserve"> Nec accidit error iſte in defini-<lb/>tionis aſsignatione, niſi ex raritatis aeris diminutione.</s> <s xml:id="echoid-s4948" xml:space="preserve"> Temporis diſtantia extra temperamentum <lb/>erroris ſcientiæ eſt cauſſa.</s> <s xml:id="echoid-s4949" xml:space="preserve"> Si quis enim per foramen inſpiciat corpus tranſiens ueloci motu, non <lb/>plenè acquirit formam corporis.</s> <s xml:id="echoid-s4950" xml:space="preserve"> Vnde accidit error in indiuiduo, in ſpecie, in utroque, ut in equis, <lb/>hominibus & arboribus.</s> <s xml:id="echoid-s4951" xml:space="preserve"> Similiter etiã accidit ſine foramine, ut ſi quis ſubitò aliquid uideat, quod <lb/>ſtatim à uiſu recedat, errabit in comprehenſione illius formæ:</s> <s xml:id="echoid-s4952" xml:space="preserve"> unde forſan erit error in ſpecie, in <lb/>indiuiduo, uel in utroque.</s> <s xml:id="echoid-s4953" xml:space="preserve"> Et erit error iſte ex ſolo tempore.</s> <s xml:id="echoid-s4954" xml:space="preserve"> Viſus ſolus errorem facit:</s> <s xml:id="echoid-s4955" xml:space="preserve"> ſi lux ſolis <lb/>fortiter deſcendat ſuper colorem uiridem fortem, uel intenſam rubedinem, adhibito uiſu lædetur:</s> <s xml:id="echoid-s4956" xml:space="preserve"> <lb/>& cum aliquid deinde inſpexerit:</s> <s xml:id="echoid-s4957" xml:space="preserve"> aliud ei, quàm ſit in ueritate, apparebit, aut alterius coloris, pro-<lb/>pter præſentiam læſionis.</s> <s xml:id="echoid-s4958" xml:space="preserve"> Et modo ſimili accidunt errores plurimi.</s> <s xml:id="echoid-s4959" xml:space="preserve"> Pari modo in oculorum ægri-<lb/>tudine aliquando equus apparet aſinus.</s> <s xml:id="echoid-s4960" xml:space="preserve"> Et accidit error triplex prædictus & in pluribus.</s> <s xml:id="echoid-s4961" xml:space="preserve"> Et planũ <lb/>eſt, errorẽ eſſe in ſcientia, exſola immoderatione uiſus.</s> <s xml:id="echoid-s4962" xml:space="preserve"> Plani ergo ſunt errores, qui in uiſu ſcienti<gap/> <lb/>accidunt ſecundum ſingulas erroris uiſus cauſſas.</s> <s xml:id="echoid-s4963" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div160" type="section" level="0" n="0"> <head xml:id="echoid-head188" xml:space="preserve">DE QVALITATIBVS DECEPTIONVM VISVS, QVAE AC-<lb/>cidunt in ſyllogiſmo & ratione. Cap. VII.</head> <head xml:id="echoid-head189" xml:space="preserve" style="it">22. Erratur ſyllogiſmo propter ſingulorum uiſionem perficientium aſymmetriam.</head> <p> <s xml:id="echoid-s4964" xml:space="preserve">PLurima eorum, quorum in uiſu fit comprehenſio, acquiruntur ex ſyllogiſmo, ſicut patuit ex <lb/>præcedente libro:</s> <s xml:id="echoid-s4965" xml:space="preserve"> & præceſsit explanatio eorum, quorum per ſyllogiſmum fit comprehen-<lb/>ſio:</s> <s xml:id="echoid-s4966" xml:space="preserve"> & quòd exeis occurrat ſenſui compoſitio in ſingulis formis.</s> <s xml:id="echoid-s4967" xml:space="preserve"> Cum ergo acciderit error in <lb/>aliquo illorum:</s> <s xml:id="echoid-s4968" xml:space="preserve"> erit error in comprehenſione facta per ſyllogiſmum.</s> <s xml:id="echoid-s4969" xml:space="preserve"> Bipertita eſt autem partitio <lb/>erroris in ſyllogiſmo:</s> <s xml:id="echoid-s4970" xml:space="preserve"> aut enim erit in propoſitionibus:</s> <s xml:id="echoid-s4971" xml:space="preserve"> aut in earum congregatione.</s> <s xml:id="echoid-s4972" xml:space="preserve"> In propoſitio-<lb/>nibus triplex:</s> <s xml:id="echoid-s4973" xml:space="preserve"> aut enim falſa loco ueræ ſumitur:</s> <s xml:id="echoid-s4974" xml:space="preserve"> aut particularis loco uniuerſalis:</s> <s xml:id="echoid-s4975" xml:space="preserve"> aut in compara-<lb/>tione propoſitionum erratur.</s> <s xml:id="echoid-s4976" xml:space="preserve"> Verbi gratia.</s> <s xml:id="echoid-s4977" xml:space="preserve"> Sifuerint in re uiſa partes, quæ appareant, & partes, <lb/>quæ lateant, quæ tamen comprehenſibiles ſint uiſui:</s> <s xml:id="echoid-s4978" xml:space="preserve"> Sim illam figatur uiſus intentio, cum uiden-<lb/>tem partes illæ præcedant:</s> <s xml:id="echoid-s4979" xml:space="preserve"> ex eis tantùm, quæ in re uiſa acquirit, concludit.</s> <s xml:id="echoid-s4980" xml:space="preserve"> Cum etiam con-<lb/>cluſiones aliquas, quas rei illi accidentes conſiderat:</s> <s xml:id="echoid-s4981" xml:space="preserve"> æſtimat eas accidere ei expartibus eius ap-<lb/>parentibus:</s> <s xml:id="echoid-s4982" xml:space="preserve"> quoniam non mſi eas computat.</s> <s xml:id="echoid-s4983" xml:space="preserve"> Cum uero intuitus diligentiam in re uiſa figit, par-<lb/>tes prius latentes percipit, & errorem cognoſcit.</s> <s xml:id="echoid-s4984" xml:space="preserve"> Enumerabo igitur errores eorum, quæ compre-<lb/>bendit uiſus per ſyllogiſmum, quæ numero ſunt uiginti duo, ut ſic pateant errores in ſyllogiſmo.</s> <s xml:id="echoid-s4985" xml:space="preserve"> <lb/> <pb o="92" file="0098" n="98" rhead="ALHAZEN"/> Et hæc erit enumeratio ſecundum unam quam que octo cauſſarum prius dictarum, & primò ſecun-<lb/>dum longitudinem.</s> <s xml:id="echoid-s4986" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div161" type="section" level="0" n="0"> <head xml:id="echoid-head190" xml:space="preserve" style="it">23. Diſtantia immoder ata cre at errores in ſingulis uiſibilibus ſpeciebus. In remotione. 16 p 4.</head> <p> <s xml:id="echoid-s4987" xml:space="preserve">DIco ergo, quòd longitudo egreſſa à temperamento errare facit uidentem in longitudine:</s> <s xml:id="echoid-s4988" xml:space="preserve"> ſi-<lb/>cut accidit, cum quis arbores ualde remotas inſpexerit, licet plurimùm diſtent inter ſe, uide-<lb/>buntur tamen quaſi coniunctæ, aut ſaltem æſtimabuntur ſibi propinquæ.</s> <s xml:id="echoid-s4989" xml:space="preserve"> Ob eandem cauſ-<lb/>ſam euenit, ut ſtellæ aliquando reputentur quaſi coniunctæ, licet plurimùm diſtent in ueritate.</s> <s xml:id="echoid-s4990" xml:space="preserve"> Ob <lb/>hoc ſtellæ erraticæ æſtimantur ab hominib.</s> <s xml:id="echoid-s4991" xml:space="preserve"> in eadem ſuperficie cum fixis, licet plurimùm elongatę <lb/>ſint ab eis.</s> <s xml:id="echoid-s4992" xml:space="preserve"> Eſt ergo error in longitudine propter egreſſum longitudinis à temperantia.</s> <s xml:id="echoid-s4993" xml:space="preserve"> Et eſt error <lb/>iſte in ſyllogiſmo, cum longitudin is tantùm per ſyllogiſmum fiat comprehenſio.</s> <s xml:id="echoid-s4994" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div162" type="section" level="0" n="0"> <head xml:id="echoid-head191" xml:space="preserve" style="it">24. In ſitu. 44. 59. 61. 62. 97 p 4.</head> <p> <s xml:id="echoid-s4995" xml:space="preserve">LOngitudo extra temperantiam, ſitus errorem inducit:</s> <s xml:id="echoid-s4996" xml:space="preserve"> quoniam à tali longitudine corpus de-<lb/>clinatum apparebit rectum:</s> <s xml:id="echoid-s4997" xml:space="preserve"> & ob hoc corpus quadratũ in hac lõgitudine declinatũ, uidebitur <lb/>oblongum.</s> <s xml:id="echoid-s4998" xml:space="preserve"> Eodẽ modo oblõga apparebit circularis forma in hac longitudine declinata.</s> <s xml:id="echoid-s4999" xml:space="preserve"> Nec <lb/>accidit error iſte, niſi ex declinationis occultatione, quę latet in tanta lõgitudine.</s> <s xml:id="echoid-s5000" xml:space="preserve"> Si enim appareret <lb/>declinatio, nõ eſſet aſsignare, quare occultaretur ueritas corporalis formę.</s> <s xml:id="echoid-s5001" xml:space="preserve"> Eſt igitur error in ſolo ſi-<lb/>tu ex lõgitudinis immoderatione.</s> <s xml:id="echoid-s5002" xml:space="preserve"> Et quare ignoretur ſitus, eſt hęc ratio.</s> <s xml:id="echoid-s5003" xml:space="preserve"> Exceſſus unius radiorũ in <lb/>latus quadrati cadentium ſuper longitudinem alterius, nõ eſt proportionalis, reſpectu totalis remo <lb/>tionis corporis à uiſu:</s> <s xml:id="echoid-s5004" xml:space="preserve"> proportione dico ſenſibili:</s> <s xml:id="echoid-s5005" xml:space="preserve"> unde propter inſenſibilitatẽ exceſſus nõ ęſtimatur <lb/>maior aliquo aliquis radius.</s> <s xml:id="echoid-s5006" xml:space="preserve"> Reputatur uerò oblõga quadrati forma, qñ unũ eius latus nõ declina-<lb/>tũ, reſpectu uiſus, cadit in partem oculi, & in minorem incidit forma lateris declinati, quoniã ſub mi <lb/>nore angulo.</s> <s xml:id="echoid-s5007" xml:space="preserve"> Et erit huiuſmodi minoritatis perceptio, ſecundum quod fuerit quadrati declinatio.</s> <s xml:id="echoid-s5008" xml:space="preserve"> <lb/>Et quoniã non attenditur declinatio, æſtimabitur unũ latus maius alio:</s> <s xml:id="echoid-s5009" xml:space="preserve"> quoniã ſub maiore angulo.</s> <s xml:id="echoid-s5010" xml:space="preserve"> <lb/>Proinde forma apparebit oblonga.</s> <s xml:id="echoid-s5011" xml:space="preserve"> Pari ratione in circulari forma, una diameter maior apparet alia:</s> <s xml:id="echoid-s5012" xml:space="preserve"> <lb/>unde reputatur oblõga.</s> <s xml:id="echoid-s5013" xml:space="preserve"> Et eſt error iſte exintẽperantia longitudinis:</s> <s xml:id="echoid-s5014" xml:space="preserve"> quod nõ accideret, ſi tẽperata <lb/>eſſet.</s> <s xml:id="echoid-s5015" xml:space="preserve"> Si uerò lõgitudo, licet intẽperata, non fuerit multũ magna, ſed ualida ſit illius corporis declina <lb/>tio:</s> <s xml:id="echoid-s5016" xml:space="preserve"> perpendet fortaſſe uidens declinationem, ſed non declinationis ueritatẽ:</s> <s xml:id="echoid-s5017" xml:space="preserve"> imò minorem æſtima <lb/>bit quàm ſit, & conferet declinationẽ lateris cum angulo, ſub quo cõprehenditur:</s> <s xml:id="echoid-s5018" xml:space="preserve"> unde minor appa <lb/>rebit quantitas talis quã ſit:</s> <s xml:id="echoid-s5019" xml:space="preserve"> unde & ſic reputabitur quadrati forma oblõga, ſed minus quàm prius.</s> <s xml:id="echoid-s5020" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div163" type="section" level="0" n="0"> <head xml:id="echoid-head192" xml:space="preserve" style="it">25. In ſoliditate & figura. 98. 97. 95. 50. 65 p 4.</head> <p> <s xml:id="echoid-s5021" xml:space="preserve">SVperfluitas longitudinis errorem generat corporeitatis.</s> <s xml:id="echoid-s5022" xml:space="preserve"> Corporeitas aũt eſt ex diſpoſitione <lb/>ſpeciei, & cõprehenditur notitia corporeitatis ex notitia huiuſmodi diſpoſitionis.</s> <s xml:id="echoid-s5023" xml:space="preserve"> Cum ergo <lb/>error acciditin corporeitate, erit in ſpeciei uel ſpecierum diſp oſitione:</s> <s xml:id="echoid-s5024" xml:space="preserve">uelut ſi ſpecies corporis <lb/>incuruata ex aliqua lõgitudine uideatur plana, aut plana æſtimetur curua.</s> <s xml:id="echoid-s5025" xml:space="preserve"> Et hęc apparentia erit in <lb/>figura.</s> <s xml:id="echoid-s5026" xml:space="preserve"> Eſt igitur figura ſpecierũ corporis diſpoſitio.</s> <s xml:id="echoid-s5027" xml:space="preserve"> Recipit etiã ſitũ ſpecierũ diſpoſitio:</s> <s xml:id="echoid-s5028" xml:space="preserve"> unde corpo <lb/>reitas includitur ſub figura & ſitu:</s> <s xml:id="echoid-s5029" xml:space="preserve"> unde errorem corporeitatis gerit in ſe error ſitus & figuræ.</s> <s xml:id="echoid-s5030" xml:space="preserve"> Acci-<lb/>dit aũt error figuræ abſque ſitus errore ex longitudinis immoderatione.</s> <s xml:id="echoid-s5031" xml:space="preserve"> Verbi gratia, figura multo <lb/>rum laterum ęqualium, directè oppoſita uiſui in longitudine intemperata, circularis apparet:</s> <s xml:id="echoid-s5032" xml:space="preserve"> nõ ob <lb/>aliud quidẽ, niſi quia anguli figuræ ſunt imperceptibiles uiſui.</s> <s xml:id="echoid-s5033" xml:space="preserve"> Longitudo enim illa abſcondit uiſui <lb/>etiam proportionalia toti, quamuis nõ totum.</s> <s xml:id="echoid-s5034" xml:space="preserve"> Eodem erroris tenore ab hac longitudine linea cur-<lb/>ua æſtimatur recta.</s> <s xml:id="echoid-s5035" xml:space="preserve"> Non enim perceptibilis eſt maioritas acceſſus unius lineæ partis incuruatæ ad <lb/>uiſum, ſuper partis eiuſdem remotioris acceſſum:</s> <s xml:id="echoid-s5036" xml:space="preserve"> quia occultatur incuruatio partiũ, licet error non <lb/>accidit in ſitu lineę illius.</s> <s xml:id="echoid-s5037" xml:space="preserve"> Similiter uiſa ſphæra ab hac longitudine æſtimabitur ſpecies plana.</s> <s xml:id="echoid-s5038" xml:space="preserve"> Quo-<lb/>niam propin quitas tumoris eius imperceptibiliter propinquitatẽ extremitatũ ab hac longitudine <lb/>excedit:</s> <s xml:id="echoid-s5039" xml:space="preserve"> unde ęſtimatur æqualis partium propinquitas:</s> <s xml:id="echoid-s5040" xml:space="preserve"> unde ſpeciei planitudo.</s> <s xml:id="echoid-s5041" xml:space="preserve"> Inde eſt, quòd ſol & <lb/>luna ſuperficiales uidentibus reputantur;</s> <s xml:id="echoid-s5042" xml:space="preserve"> quę erronea excluderetur figuræ reputatio, ſi temperata <lb/>eſſet longitudo.</s> <s xml:id="echoid-s5043" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div164" type="section" level="0" n="0"> <head xml:id="echoid-head193" xml:space="preserve" style="it">26. In magnitudine. 28 p 4.</head> <p> <s xml:id="echoid-s5044" xml:space="preserve">IN magnitudine corporis erit error ex intemperata lõgitudine:</s> <s xml:id="echoid-s5045" xml:space="preserve"> quoniã uidebitur multò minus, <lb/>quàm ſit in ueritate.</s> <s xml:id="echoid-s5046" xml:space="preserve"> Ethuius rei ratio eſt.</s> <s xml:id="echoid-s5047" xml:space="preserve"> Quoniã, ut diximus, longitudo intẽperata eſt, quæ par <lb/>tes proportionales toti proportione etiam ſenſibili abſcondit uiſui.</s> <s xml:id="echoid-s5048" xml:space="preserve"> Et cum fuerit occultatio par <lb/>tium ſenſui perceptibilium:</s> <s xml:id="echoid-s5049" xml:space="preserve"> anguli, in quos cadunt, non ſentientur, licet totali angulo proportiona-<lb/>les ſint.</s> <s xml:id="echoid-s5050" xml:space="preserve"> Vnde cum diſcurrit axis rem uiſam, abſconduntur ei lineæ multę ex ea, & partes multæ.</s> <s xml:id="echoid-s5051" xml:space="preserve"> Vn <lb/>de minor efficitur totalis apparentia.</s> <s xml:id="echoid-s5052" xml:space="preserve"> Amplius magnitudo partis alicuius corporis non conſidera-<lb/>tur, niſi ſecundum magnitudinem anguli, in quẽ cadit:</s> <s xml:id="echoid-s5053" xml:space="preserve"> & magnitudo anguli attenditur ſecundũ par <lb/>tem in uiſu ſectam:</s> <s xml:id="echoid-s5054" xml:space="preserve"> & partis ſectę quantitas æſtimatur ſecundũ duo puncta illius partis terminalia:</s> <s xml:id="echoid-s5055" xml:space="preserve"> <lb/>& puncta illa ſenſibilia ſunt, & parti ſectæ proportionalia.</s> <s xml:id="echoid-s5056" xml:space="preserve"> Quoniã à lõgitudine tanta æſtimatur res <lb/>uiſa ſecundũ fines toti uiſo proportionales:</s> <s xml:id="echoid-s5057" xml:space="preserve"> aliter enim non eſſent fines illi ſenſibiles:</s> <s xml:id="echoid-s5058" xml:space="preserve"> & fines partis <lb/>ſectę directè opponuntur finib.</s> <s xml:id="echoid-s5059" xml:space="preserve"> partis uiſę proportionalib.</s> <s xml:id="echoid-s5060" xml:space="preserve"> Puncta ergo illa partis ſe ctæ terminalia <lb/>abſcondunt ex re uiſa partes ſenſibiles.</s> <s xml:id="echoid-s5061" xml:space="preserve"> Cum ergo incedit axis ſuperſingulas rei partes:</s> <s xml:id="echoid-s5062" xml:space="preserve"> ex ſingulis <lb/>partib.</s> <s xml:id="echoid-s5063" xml:space="preserve"> abſconduntur partes ſenſibiles:</s> <s xml:id="echoid-s5064" xml:space="preserve"> & ita minor apparet tota rei quantitas.</s> <s xml:id="echoid-s5065" xml:space="preserve"> Cum aũt uidetur cor <lb/>pus à tẽperata long tudine, puncta terminalia partis ſectę ualde ſunt parua, & quaſi inſenſibilia ad <lb/> <pb o="93" file="0099" n="99" rhead="OPTICAE LIBER III."/> ipſam collata.</s> <s xml:id="echoid-s5066" xml:space="preserve"> Fines enim rebus uiſis inſenſibiles eligit in longitudine tẽperata exiſtimatio uiden-<lb/>tis:</s> <s xml:id="echoid-s5067" xml:space="preserve"> unde non abſconduntur partes toti proportionales.</s> <s xml:id="echoid-s5068" xml:space="preserve"> Quare corpus nõ apparet minus, quàm ha-<lb/>beat ueritas eius.</s> <s xml:id="echoid-s5069" xml:space="preserve"> Amplius, ſicut dictũ eſt in ſuperioribus, [38 n 2] magnitudo nõ acquiritur in cor <lb/>pore, niſi ex lõgitudinis & anguli collatione:</s> <s xml:id="echoid-s5070" xml:space="preserve"> & iã dictũ eſt, quòd eximmoderata lõgitudine apparet <lb/>minor angulus:</s> <s xml:id="echoid-s5071" xml:space="preserve"> quia minor eſt in ueritate.</s> <s xml:id="echoid-s5072" xml:space="preserve"> Sed remotionis nõ fit diſcretio.</s> <s xml:id="echoid-s5073" xml:space="preserve"> Iã enim ſuprà patuit [39 n <lb/>2] quòd remotio moderata cõprehenditur per corpora interpoſita:</s> <s xml:id="echoid-s5074" xml:space="preserve"> immoderata uerò minimè.</s> <s xml:id="echoid-s5075" xml:space="preserve"> Cũ <lb/>ergo remotio rei uiſæ ſit ignota:</s> <s xml:id="echoid-s5076" xml:space="preserve"> fiet fortaſsis collatio ipſius ad longitudinẽ notã, & æſtimabit eã mi <lb/>norem.</s> <s xml:id="echoid-s5077" xml:space="preserve"> Quare putabitur minor & in angulo minoritas, & in longitudine, quã ſit in ueritate:</s> <s xml:id="echoid-s5078" xml:space="preserve"> unde er <lb/>ror in corporis quãtitate.</s> <s xml:id="echoid-s5079" xml:space="preserve"> Et quãtò augmẽtabitur longitudo, tantò inualeſcet error.</s> <s xml:id="echoid-s5080" xml:space="preserve"> Et adeò poterit <lb/>augmentari lõgitudo, ut æſtimetur quãtitas corporis quaſi punctualis.</s> <s xml:id="echoid-s5081" xml:space="preserve"> Et ſi ultrà creuerit lõgitudo, <lb/>occultabitur uiſui corpus illud.</s> <s xml:id="echoid-s5082" xml:space="preserve"> Simili modo accidit corporis occultatio in lõgitudine tẽperata, nõ <lb/>ex ipſa remotione, ſed ex coloris corporis debilitate.</s> <s xml:id="echoid-s5083" xml:space="preserve"> Et patet occultationem fieri ex debili colore.</s> <s xml:id="echoid-s5084" xml:space="preserve"> <lb/>Quoniã ſi loco huius corporis in eadem longitudine ſtatuatur corpus eiuſdẽ quantitatis, in quo ſit <lb/>fortitudo coloris, nõ latebit uiſum, ſicut corpus, in quo fuerat coloris debilitas.</s> <s xml:id="echoid-s5085" xml:space="preserve"> Quare aliquãdo oc-<lb/>cultat corpus uiſui, non elongatio, nõ diminuta quãtitas, ſed ſola coloris debilitas.</s> <s xml:id="echoid-s5086" xml:space="preserve"> Amplius, aliquã-<lb/>do euenit corporis occultatio ex coloris eius ſimilitudine, cũ interpoſitorum ipſi & uiſui corporũ <lb/>colore:</s> <s xml:id="echoid-s5087" xml:space="preserve"> & hoc in tẽperata longitudine.</s> <s xml:id="echoid-s5088" xml:space="preserve"> Vnde corpus albũ à longè poſitũ, effuſa niue ſuper ſupficiem <lb/>interiacẽtis terrę, non diſcernitur:</s> <s xml:id="echoid-s5089" xml:space="preserve"> niue uerò remota percipitur.</s> <s xml:id="echoid-s5090" xml:space="preserve"> Et palã, quòd erat occultatio ex hac <lb/>colorũ identitate.</s> <s xml:id="echoid-s5091" xml:space="preserve"> Quoniã ſi loco illius corporis opponatur uiſui ab eadem remotione corpus ęqua <lb/>le alterius coloris, non occultabitur.</s> <s xml:id="echoid-s5092" xml:space="preserve"> Cũigitur aliqua res oppoſita uiſui nõ քcipitur, poterit eſſe cau <lb/>ſa abſconſionis ſuperfluitas elõgationis, ad partem uiſus inſenſibilẽ formã dirigentis, uel quaſi pun <lb/>ctualẽ.</s> <s xml:id="echoid-s5093" xml:space="preserve"> Quòd ſi in partẽ uiſus ſenſibilẽ forma inciderit:</s> <s xml:id="echoid-s5094" xml:space="preserve"> poterit iterũ pręterire uiſum, uel propter co-<lb/>loris remiſsionẽ, uel colorũ rei uiſę & corporũ interiacentiũ conformitatẽ.</s> <s xml:id="echoid-s5095" xml:space="preserve"> A mplius accidit error in <lb/>rei uiſę quantitate, etiã in longitudine tẽperata.</s> <s xml:id="echoid-s5096" xml:space="preserve"> Quoniã corpore aliquo ſecondũ moderationẽ elon <lb/>gato & uiſo:</s> <s xml:id="echoid-s5097" xml:space="preserve"> occultabuntur uiſui partes eius minutę:</s> <s xml:id="echoid-s5098" xml:space="preserve"> quę quidẽ in minore elongatione apparent, li-<lb/>cet fortaſsis non plenè:</s> <s xml:id="echoid-s5099" xml:space="preserve"> & paululũ amplius elongatæ iterum, minus plenè.</s> <s xml:id="echoid-s5100" xml:space="preserve"> Et minuetur cõprehẽſio <lb/>nis plenitudo, inualeſcente remotionis augmento:</s> <s xml:id="echoid-s5101" xml:space="preserve"> donec occurrat partiũ occultatio:</s> <s xml:id="echoid-s5102" xml:space="preserve"> licet nõ egre-<lb/>diatur tẽperantiam illa elongatio.</s> <s xml:id="echoid-s5103" xml:space="preserve"> Iterũ immoderata remotione pars aliqua plenè cõprehenditur, <lb/>aliqua minimarum eius partium occultatur.</s> <s xml:id="echoid-s5104" xml:space="preserve"> Quoniã elongatio rei egreſſa eſt à temperamento pro-<lb/>portionato ad partes illas, licet non reſpectu totius corporis, aut cõprehenſę partis.</s> <s xml:id="echoid-s5105" xml:space="preserve"> Et licet nota ſit <lb/>hominihęclongitudo:</s> <s xml:id="echoid-s5106" xml:space="preserve"> tñ accidit error in cõprehenſione quãtitatis partium:</s> <s xml:id="echoid-s5107" xml:space="preserve"> & hoc propter angulũ, <lb/>ſub quo pars cõprehenditur, cuius capacitas minor æſtimatur, ꝗ̃ habeat ueritas.</s> <s xml:id="echoid-s5108" xml:space="preserve"> Et cauſſa apparen-<lb/>tiæ minoritatis eius, eſt ex punctis terminalib.</s> <s xml:id="echoid-s5109" xml:space="preserve"> ſectę partis, in uiſu partẽ occultantib.</s> <s xml:id="echoid-s5110" xml:space="preserve"> & anguli capa-<lb/>citatem conſtringentib.</s> <s xml:id="echoid-s5111" xml:space="preserve"> Igitur cũ immoderata ſuerit rei uiſæ ab aliquo diſtantia:</s> <s xml:id="echoid-s5112" xml:space="preserve"> proueniet error in <lb/>eius quantitate dupliciter:</s> <s xml:id="echoid-s5113" xml:space="preserve"> & ex anguli minoritate:</s> <s xml:id="echoid-s5114" xml:space="preserve"> & exlõgitudinis incertitudine.</s> <s xml:id="echoid-s5115" xml:space="preserve"> In moderata ue-<lb/>rò longitudine erit error in quantitate minutarũ partium ex errore anguli tantùm.</s> <s xml:id="echoid-s5116" xml:space="preserve"> Et hæ ſunt cauſ-<lb/>ſæ, quare corpus æſtimetur minus, quàm ſit in longitudine temperata.</s> <s xml:id="echoid-s5117" xml:space="preserve"> Immoderatio longitudinis <lb/>aliquando errorem inducit maioritatis.</s> <s xml:id="echoid-s5118" xml:space="preserve"> Vnde in longitudine immoderata, minima ſcilicet, qñ cor-<lb/>pus uiſum fuerit multùm uicinum uiſui, uidebitur corpus maioris quantitatis, quàm in longitudi-<lb/>ne temperata, uel quàm ſit reuera:</s> <s xml:id="echoid-s5119" xml:space="preserve"> & hoc duplici de cauſſa.</s> <s xml:id="echoid-s5120" xml:space="preserve"> Quoniã, ut dictũ eſt [38 n 2] intellectus <lb/>longitudinem & angulum conſiderat, & inde quantitatem corporis ſyllogizat.</s> <s xml:id="echoid-s5121" xml:space="preserve"> Et in hac elongatio-<lb/>ne angulus pyramidis eſt ualde magnus:</s> <s xml:id="echoid-s5122" xml:space="preserve"> & elongatio corporis nõ æſtimatur, niſi à uiſus ſuperficie <lb/>ad ſuperficiem corporis.</s> <s xml:id="echoid-s5123" xml:space="preserve"> Non enim poteſt cadere in uiſus æſtimationẽ lõgitudo, ad interiora uiſus <lb/>penetrans à corpore uiſo:</s> <s xml:id="echoid-s5124" xml:space="preserve"> cũ pars eius interior radijs non ſubiaceat, nec menſurari à uiſu queat.</s> <s xml:id="echoid-s5125" xml:space="preserve"> Syl <lb/>logizat igitur uiſus ex anguli capacitate & tota longitudine.</s> <s xml:id="echoid-s5126" xml:space="preserve"> Vera aũt remotio corporis attenditur <lb/>ſecundum lineam à centro oculi ad corpus procedentem:</s> <s xml:id="echoid-s5127" xml:space="preserve"> cum reſpectu centri fiat conſideratio an-<lb/>guli.</s> <s xml:id="echoid-s5128" xml:space="preserve"> Et in temperata corporis diſtantia ſemidiameter oculi, qua uera corporis elongatio excedit <lb/>apparentem, inſenſibilis eſt, reſpectu totalis diſtantiæ corporis.</s> <s xml:id="echoid-s5129" xml:space="preserve"> Vnde non facit errorem in longitu <lb/>dinis æſtimatione:</s> <s xml:id="echoid-s5130" xml:space="preserve"> ſed corpore circa oculum exiſtente, erit magnitudo ſemidiametri proportiona-<lb/>lis diſtantiæ corporis proportione ſenſibili.</s> <s xml:id="echoid-s5131" xml:space="preserve"> Erit enim aliquando maior, aliquando ęqualis, aliquan <lb/>do minor, ſed proportione modica, uelut ſubdupla, uel huiuſmodi.</s> <s xml:id="echoid-s5132" xml:space="preserve"> Vnde in propinquitate rei uiſæ <lb/>excrementum anguli pyramidalis, & ſenſibilis minoritas longitudinis æſtimatæ, reſpectu ueræ, in-<lb/>ducunt apparentiam maioritatis in corpore.</s> <s xml:id="echoid-s5133" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div165" type="section" level="0" n="0"> <head xml:id="echoid-head194" xml:space="preserve" style="it">27. In diuiſione, & continuatione & numero 109 p 4.</head> <p> <s xml:id="echoid-s5134" xml:space="preserve">IMmoderata extenſio remotionis errorẽ inuehit diſtinctionis.</s> <s xml:id="echoid-s5135" xml:space="preserve"> Pariete enim aliquo à lõgè uiſo, ſi <lb/>in parte eius fuerit color tenebroſus:</s> <s xml:id="echoid-s5136" xml:space="preserve"> fiet uidenti fides, colorẽ illũ eſſe diſtinctionem partiũ:</s> <s xml:id="echoid-s5137" xml:space="preserve"> un-<lb/>de continuum ex hoc errore reputabitur diſcretum.</s> <s xml:id="echoid-s5138" xml:space="preserve"> Similiter ſi prope parietem illum creſcat al <lb/>titudo herbarum, uidebitur forſan diſtinctio partium, inter quas fuerit pars occulta ab omni oppo <lb/>ſitione herbarum:</s> <s xml:id="echoid-s5139" xml:space="preserve"> Vnde non reputabitur paries aliquid continuum.</s> <s xml:id="echoid-s5140" xml:space="preserve"> Pari modo luce ſolis in parie-<lb/>tem deſcendente non multùm forti:</s> <s xml:id="echoid-s5141" xml:space="preserve"> ſi corpus aliquod umbram iaciat, quæ umbra in parietem ca-<lb/>dat:</s> <s xml:id="echoid-s5142" xml:space="preserve"> accidet error idem in partium, ſine intermedio, ſeparatione.</s> <s xml:id="echoid-s5143" xml:space="preserve"> Palàm ergo, quòd error diſtinctio <lb/>nis in ſyllogiſmo eſt ex immoderatione remotionis.</s> <s xml:id="echoid-s5144" xml:space="preserve"> Longitudo à moderatione egreſſa erroris con <lb/>tinuitatis eſt cauſſa.</s> <s xml:id="echoid-s5145" xml:space="preserve"> Corpora enim à longè uiſa in colore ſimilia ſibi, propinqua credũtur cõtinua.</s> <s xml:id="echoid-s5146" xml:space="preserve"> <lb/> <pb o="94" file="0100" n="100" rhead="ALHAZEN"/> Hinc accidit quòd tabulę parietis uel ſcamni apparent ali quãdo cõtinuæ:</s> <s xml:id="echoid-s5147" xml:space="preserve"> licet abinuicẽ ſint diuiſæ, <lb/>modica, dico, diſtinctione.</s> <s xml:id="echoid-s5148" xml:space="preserve"> Et accidet hoc etiã in intẽperata remotione rei uiſæ, ſcilicet immodera-<lb/>ta, quantũ ad comprehenſionẽ remotionis diſtinctionis tam paruæ.</s> <s xml:id="echoid-s5149" xml:space="preserve"> Et ita ex hoc remotionis errore <lb/>diſcretum creditur continuum.</s> <s xml:id="echoid-s5150" xml:space="preserve"> Et quoniam ſecundum conſiderationem continuitatis & diſcretio-<lb/>nis attenditur numeri comprehenſio:</s> <s xml:id="echoid-s5151" xml:space="preserve"> accidi terror in numero, cum in rebus diſcretis apparebit u-<lb/>nitas, aut in re una prætendetur pluralitas.</s> <s xml:id="echoid-s5152" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div166" type="section" level="0" n="0"> <head xml:id="echoid-head195" xml:space="preserve" style="it">28. In motu & quiete. 138 p 4.</head> <p> <s xml:id="echoid-s5153" xml:space="preserve">EGreſſus remotionis à moderamine errorem efficit motus.</s> <s xml:id="echoid-s5154" xml:space="preserve"> Si quis ad partem, in qua lunã, aut <lb/>ſolem, aut ſtellam aliquam uiderit, moueatur, cum plurimùm motus, lunam ante ſe uiderit e-<lb/>longatã, non minus, quàm in principio motus:</s> <s xml:id="echoid-s5155" xml:space="preserve"> concludit ipſam in eandem partem moueri, & <lb/>ab eo recedere:</s> <s xml:id="echoid-s5156" xml:space="preserve"> & ob hoc elõgationes durare.</s> <s xml:id="echoid-s5157" xml:space="preserve"> Et accidit hoc, luna etiã ad partem contrariam prope-<lb/>rante.</s> <s xml:id="echoid-s5158" xml:space="preserve"> Ethuius erroris ratio eſt:</s> <s xml:id="echoid-s5159" xml:space="preserve"> Quia notum eſt uidenti, quòd in his inferiorib.</s> <s xml:id="echoid-s5160" xml:space="preserve"> Naturis, ſtatutis duo <lb/>bus corporib.</s> <s xml:id="echoid-s5161" xml:space="preserve"> quorum unũ moueatur in partem aliquã, ſi permanſerit idẽtitas ſitus unius reſpectu <lb/>alterius:</s> <s xml:id="echoid-s5162" xml:space="preserve"> neceſſe eſt aliud moueri in eandem partẽ, & motu æquali.</s> <s xml:id="echoid-s5163" xml:space="preserve"> Verũ hoc nõ oportet exiſtimare <lb/>in luna & ſtellis.</s> <s xml:id="echoid-s5164" xml:space="preserve"> Cum enim in his non percipiatur ſitus motus mouentis ad ſtellã motam:</s> <s xml:id="echoid-s5165" xml:space="preserve"> occultè <lb/>ex propoſitionib.</s> <s xml:id="echoid-s5166" xml:space="preserve"> iam dudum animo notis infertur ſyllogiſticè motio, & occultatur immutatio ſitus <lb/>mouentis ad ſtellã.</s> <s xml:id="echoid-s5167" xml:space="preserve"> Quoniam uia, quã quis peragit motu ſuo, nõ eſt proportionalis ipſius ſtellę ma-<lb/>gnitudini:</s> <s xml:id="echoid-s5168" xml:space="preserve"> multò magis igitur exceſſus poſtremę propin quitatis eius ad ſtellã ſuper primam propin <lb/>quitatẽ, nõ eſt ſenſibilis reſpectu totalis remotionis.</s> <s xml:id="echoid-s5169" xml:space="preserve"> Idem error accidit in motu nubiũ:</s> <s xml:id="echoid-s5170" xml:space="preserve"> creditur e-<lb/>nim uelociſsimus eſſe lunę motus, licet non ſit, ut nos ſuprà [19 n] explanauimus.</s> <s xml:id="echoid-s5171" xml:space="preserve"> Euagatio remo-<lb/>tionis à tẽperamento, errorẽ infert quietis.</s> <s xml:id="echoid-s5172" xml:space="preserve"> Si quis à longè uiſus motu non ueloci moueatur:</s> <s xml:id="echoid-s5173" xml:space="preserve"> putabi <lb/>tur quieſcere:</s> <s xml:id="echoid-s5174" xml:space="preserve"> unde ſtellas errantes credimus immotas:</s> <s xml:id="echoid-s5175" xml:space="preserve"> licet inſit eis motus uelocitas.</s> <s xml:id="echoid-s5176" xml:space="preserve"> Et eſt hęc ꝗe-<lb/>tis ſtellarum æſtimatio.</s> <s xml:id="echoid-s5177" xml:space="preserve"> Quoniã uiæ, quas incedunt etiã in tẽpore magno, nõ ſunt perceptibiles ui-<lb/>ſui à tanta remotione.</s> <s xml:id="echoid-s5178" xml:space="preserve"> Vnde durante ſitu earũ, reſpectu uidentis, identitate æſtimãtur quieſcere.</s> <s xml:id="echoid-s5179" xml:space="preserve"> Pa <lb/>ri modo ſi corpus aliquod à lõgitudine moueatur ſuper radios uiſus:</s> <s xml:id="echoid-s5180" xml:space="preserve"> & accedat ad ipſum uiſum, uel <lb/>recedat ab eo:</s> <s xml:id="echoid-s5181" xml:space="preserve"> putabitur immotũ, niſi morus eius fuerit ualde fortis.</s> <s xml:id="echoid-s5182" xml:space="preserve"> Et accidit iſte error, quoniã, ut <lb/>ſuprà [49 n 2] patuit, motus non cõprehenditur in corpore, niſi quia modò uidetur cũ aliquo cor-<lb/>pore, modò cum alio.</s> <s xml:id="echoid-s5183" xml:space="preserve"> Hic autem excluditur hæc perceptio:</s> <s xml:id="echoid-s5184" xml:space="preserve"> quoniam uia, quam incedit mouens ſu-<lb/>per radios, imperceptibilis eſt à tanta longitudine.</s> <s xml:id="echoid-s5185" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div167" type="section" level="0" n="0"> <head xml:id="echoid-head196" xml:space="preserve" style="it">29. In aſperitate & lenitate. 141 p 4.</head> <p> <s xml:id="echoid-s5186" xml:space="preserve">SVperflua longitudo errorẽ ingerit aſperitatis.</s> <s xml:id="echoid-s5187" xml:space="preserve"> Vnde in capillis alicuius pictę imaginis à lõgitu <lb/>dine intẽperata æſtimatur aſperitas, cũ expreſſa fuerit pictura.</s> <s xml:id="echoid-s5188" xml:space="preserve"> Quia enim notum eſt aſperita-<lb/>tem eſſe in ueris capillis:</s> <s xml:id="echoid-s5189" xml:space="preserve"> concludit eã animus illis ſimiliter ineſſe propter expreſsionẽ formæ.</s> <s xml:id="echoid-s5190" xml:space="preserve"> <lb/>Idem error accidit in ueſtib.</s> <s xml:id="echoid-s5191" xml:space="preserve"> depictis, & animalium pilis expreſsè depictorum.</s> <s xml:id="echoid-s5192" xml:space="preserve"> In his aũt omnib.</s> <s xml:id="echoid-s5193" xml:space="preserve"> nõ <lb/>eſt aſperitas, ſed immenſa læuitas.</s> <s xml:id="echoid-s5194" xml:space="preserve"> Etlicet à corporib.</s> <s xml:id="echoid-s5195" xml:space="preserve"> læuibus fiat reflexio lucis, nõ ab aſperis:</s> <s xml:id="echoid-s5196" xml:space="preserve"> rñ in <lb/>pictura aliquando uidetur reflexio lucis, nec ob hoc excluditur opinio aſperitatis.</s> <s xml:id="echoid-s5197" xml:space="preserve"> Quoniam opinã <lb/>ti eſt certũ aliquando in eodẽ corpore aſperitatis & reflexionis fieri concurſum, ſicut accidit in ca-<lb/>pillis hominis nigerrimis & benè lotis:</s> <s xml:id="echoid-s5198" xml:space="preserve"> reflectitur enim lux in eis, licet aſperis.</s> <s xml:id="echoid-s5199" xml:space="preserve"> Vnde ex hac ſimilitu <lb/>dine accidit error in æſtimatione aſperitatis picturæ per immoderatã remotionẽ, ad corpus pictum <lb/>proportionatum.</s> <s xml:id="echoid-s5200" xml:space="preserve"> Non enim poterit cõprehendi lęuitas in pictura, niſi cum multùm fuerit certa.</s> <s xml:id="echoid-s5201" xml:space="preserve"> Vn <lb/>de diſtantia reſpectu aliarũ rerũ extra temperantiã, eſt ad acquiſitionẽ læuitatis comparata.</s> <s xml:id="echoid-s5202" xml:space="preserve"> Ex eua-<lb/>gata remotione accidit error in læuitate.</s> <s xml:id="echoid-s5203" xml:space="preserve"> Si enim à magna longitudine opponatur uiſui corpus, in <lb/>quo modica eſt aſperitas, putabitur læue.</s> <s xml:id="echoid-s5204" xml:space="preserve"> Aſperitas enim nõ acquiritur in corpore, niſi ex diuerſita-<lb/>te ſitus partiũ inter ſe, uel luce eminẽtiũ, uel umbra depreſſarũ, ſicut explanatũ eſt ſuperius [53 n 2:</s> <s xml:id="echoid-s5205" xml:space="preserve">] <lb/>& à tali longitudine non percipitur diuerſitas ſitus partium eminentium ſuper depreſſas, aut proie-<lb/>ctio umbræ.</s> <s xml:id="echoid-s5206" xml:space="preserve"> Vnde iudicatur in eo læuitas.</s> <s xml:id="echoid-s5207" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div168" type="section" level="0" n="0"> <head xml:id="echoid-head197" xml:space="preserve" style="it">30. In raritate & denſitate. 144 p 4.</head> <p> <s xml:id="echoid-s5208" xml:space="preserve">EX immoderatione elongationis oritur error raritatis.</s> <s xml:id="echoid-s5209" xml:space="preserve"> Cum enim circa oculũ erigitur acus, <lb/>aut aliquid ſubtile multum:</s> <s xml:id="echoid-s5210" xml:space="preserve"> licet appareat uiſui maius, quàm ſit:</s> <s xml:id="echoid-s5211" xml:space="preserve"> tñ nihil occultat ei de oppoſi-<lb/>to pariete, aut alio oppoſito corpore.</s> <s xml:id="echoid-s5212" xml:space="preserve"> Vnde cum fiat raritatis comprehenſio in corpore, ex eo, <lb/>quòd poſtipſum poſſum us aliquid uidere:</s> <s xml:id="echoid-s5213" xml:space="preserve"> in acu erecta, aut in aliquo cõſimili, raritas æſtimabitur, <lb/>cum poſt ipſam totus paries uideatur.</s> <s xml:id="echoid-s5214" xml:space="preserve"> Quare aũt acus prope uiſum ſita maior appareat, patet ex ſu-<lb/>perioribus.</s> <s xml:id="echoid-s5215" xml:space="preserve"> Quare autem in tanta propin quitate nihil abſcondat uiſui ex pariete oppoſito:</s> <s xml:id="echoid-s5216" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s5217" xml:space="preserve"> quia <lb/>remotio tam modica, reſpectu occultationis acus, immoderata eſt.</s> <s xml:id="echoid-s5218" xml:space="preserve"> Si enim paululũ elongetur ab o-<lb/>culo acus illa:</s> <s xml:id="echoid-s5219" xml:space="preserve"> occultabitur pars parietis maior acuipſa.</s> <s xml:id="echoid-s5220" xml:space="preserve"> Et huius rei cauſſa plenius explanabitur.</s> <s xml:id="echoid-s5221" xml:space="preserve"> <lb/>Ex ſuperabundantia longitudinis accidit error ſoliditatis.</s> <s xml:id="echoid-s5222" xml:space="preserve"> Si quis enim à lõgè intueatur corpus ra-<lb/>rum, & ſtatuatur poſt ipſum corpus coloratum, aut quid tenebroſum:</s> <s xml:id="echoid-s5223" xml:space="preserve"> non reputabitur corpus il-<lb/>lud rarum, ſed ſolidum.</s> <s xml:id="echoid-s5224" xml:space="preserve"> Et eſt error:</s> <s xml:id="echoid-s5225" xml:space="preserve"> quoniã poſt corpus illud non percipit aliud, cũ natura rari ſit, ut <lb/>poſtipſum poſsit uideri ſolidum:</s> <s xml:id="echoid-s5226" xml:space="preserve"> concludetur corpus illud non eſſe rarum, ſed ſolidum.</s> <s xml:id="echoid-s5227" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div169" type="section" level="0" n="0"> <head xml:id="echoid-head198" xml:space="preserve" style="it">31. In umbra & tenebris. 147 p 4.</head> <p> <s xml:id="echoid-s5228" xml:space="preserve">EX ſuperfluitate remotionis oritur error in umbra.</s> <s xml:id="echoid-s5229" xml:space="preserve"> Si enim à tali lõgitudine opponatur uiſui cor <lb/>pus albũ, in quo ſit pars tenebroſa, luce ſolis ſuper corpus illud deſcẽdente:</s> <s xml:id="echoid-s5230" xml:space="preserve"> apparebit umbra in <lb/> <pb o="95" file="0101" n="101" rhead="OPTICAE LIBER III."/> parte corporis tenebroſa:</s> <s xml:id="echoid-s5231" xml:space="preserve"> & ſi circa corpus illud uideatur aliud:</s> <s xml:id="echoid-s5232" xml:space="preserve"> fiet conclu ſio, quòd umbra apparẽs <lb/>proijciatur ab illo alio.</s> <s xml:id="echoid-s5233" xml:space="preserve"> Et palàm, quòd accidit error iſte ex nimia remotione.</s> <s xml:id="echoid-s5234" xml:space="preserve"> Propter diſtantiæ ex-<lb/>ceſſum ſe ingerit error tenebrarum.</s> <s xml:id="echoid-s5235" xml:space="preserve"> Si enim procul uideatur corpus album, in quo pars nigra mul-<lb/>tùm ſit:</s> <s xml:id="echoid-s5236" xml:space="preserve"> æſtimabuntur fortaſsis in parte illa tenebræ:</s> <s xml:id="echoid-s5237" xml:space="preserve"> unde fiet concluſio, quòd in directo illius par-<lb/>tis ſit foramen corporis, per quod appareat tenebrarum egreſsio poſt corpus illud exiſtentium.</s> <s xml:id="echoid-s5238" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div170" type="section" level="0" n="0"> <head xml:id="echoid-head199" xml:space="preserve" style="it">32. In pulchritudine & deformitate. 150 p 4.</head> <p> <s xml:id="echoid-s5239" xml:space="preserve">REmotio excedens modum cauſſa eſt erroris pulchritudinis & deformitatis.</s> <s xml:id="echoid-s5240" xml:space="preserve"> Cũ enim procul <lb/>inſpicitur res aliqua, ſi fuerint in ea maculæ paruę, eã deformantes, quia occultantur exlõgi <lb/>tudine, iudicatur formoſa:</s> <s xml:id="echoid-s5241" xml:space="preserve"> quoniam ex ſolis apparentib.</s> <s xml:id="echoid-s5242" xml:space="preserve"> fit concluſio, & latent maculæ, appa <lb/>rent uerò partes formoſę.</s> <s xml:id="echoid-s5243" xml:space="preserve"> Similiter ſi à tanta lõgitudine uideatur res, in qua ſunt picturę, ſed minu-<lb/>tæ, rei totali decorem conferentes:</s> <s xml:id="echoid-s5244" xml:space="preserve"> cum lateant uiſum cauſſæ decoris:</s> <s xml:id="echoid-s5245" xml:space="preserve"> iudicabitur res illa deformis, <lb/>cum ex apparentibus tantùm ſumat iudex iudicium.</s> <s xml:id="echoid-s5246" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div171" type="section" level="0" n="0"> <head xml:id="echoid-head200" xml:space="preserve" style="it">33. In ſimilitudine & dißimilitudine. 153 p 4.</head> <p> <s xml:id="echoid-s5247" xml:space="preserve">EX ſuperflua elõgatione accidit error in ſimilitudine corporum & diſsimilitudine.</s> <s xml:id="echoid-s5248" xml:space="preserve"> Si enim di-<lb/>rigantur uiſus in corpora lõgè remota in colore ſimilia, & ſi fuerint in eis notæ uel protractio <lb/>nes minutæ ſibi diſsimiles & diuerſę, quę cũ uiſus prætereant:</s> <s xml:id="echoid-s5249" xml:space="preserve"> iudicabuntur corpora ex toto <lb/>ſimilia.</s> <s xml:id="echoid-s5250" xml:space="preserve"> È<unsure/>contrario ſi diuerſitas fuerit in totalibus corporum coloribus, ſed in eis ſint notæ minutę, <lb/>inter quas ſit ſimilitudo:</s> <s xml:id="echoid-s5251" xml:space="preserve"> iudicabuntur diſsimilia ex toto.</s> <s xml:id="echoid-s5252" xml:space="preserve"> Et accidet error:</s> <s xml:id="echoid-s5253" xml:space="preserve"> quoniam ex ſolùm appa-<lb/>rentibus fiet concluſio.</s> <s xml:id="echoid-s5254" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div172" type="section" level="0" n="0"> <head xml:id="echoid-head201" xml:space="preserve" style="it">34. Situs immoderatus creat errores in ſingulis uiſibilibus ſpeciebus. In diſtantia. 16 p 4.</head> <p> <s xml:id="echoid-s5255" xml:space="preserve">SItus egreditur à temperamento, & errorem inducit in quòlibet eorum, quorum fit cõprehen-<lb/>ſio perſyllogiſmum.</s> <s xml:id="echoid-s5256" xml:space="preserve"> In longitudine, ut ſi uideãtur duo corpora, quorũ unum ſit poſt aliud dire <lb/>cte, ita ut unũ cooperiat partẽ alterius, & pars poſterioris emineat:</s> <s xml:id="echoid-s5257" xml:space="preserve"> & hoc in lõgitudine tẽpera <lb/>ta, non tñ multùm certa, nec inter ea fuerint alia corpora:</s> <s xml:id="echoid-s5258" xml:space="preserve"> nõ plenè æſtimabitur longitudinis unius <lb/>ad aliud menſura.</s> <s xml:id="echoid-s5259" xml:space="preserve"> Et forſitã iudicabit uidens ea ualde ſibi eſſe propinqua.</s> <s xml:id="echoid-s5260" xml:space="preserve"> Et eſt error in ſyllogiſmo, <lb/>cum perſyllogiſmũ tantùm comprehendatur longitudo:</s> <s xml:id="echoid-s5261" xml:space="preserve"> per ſitum uerò, quoniam, ſi unũ nõ occul-<lb/>taret alterius partem, ſed utrunq;</s> <s xml:id="echoid-s5262" xml:space="preserve"> totum exponeretur uiſui, utuia inter ipſa in diuerſos, nõ in eun-<lb/>dem incideret radios:</s> <s xml:id="echoid-s5263" xml:space="preserve"> diſcerneretur diſtantia unius ab alio Et eſt error exſola ſitus intẽperãtia:</s> <s xml:id="echoid-s5264" xml:space="preserve"> quo <lb/>niam ſitu ad temperantiam reducto (cæteris partibus non mutatis) non accidit error talis.</s> <s xml:id="echoid-s5265" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div173" type="section" level="0" n="0"> <head xml:id="echoid-head202" xml:space="preserve" style="it">35. In ſitu. 44 p 4.</head> <p> <s xml:id="echoid-s5266" xml:space="preserve">SItus extra temperantiam, in ſitu errorem inuehit:</s> <s xml:id="echoid-s5267" xml:space="preserve"> cadente enim axe uiſuali in corpus à tempe <lb/>rata longitudine oppoſitum uiſui, ſumpto alio corpore multũ elõgato ab axe, & declinato mo <lb/>dicũ ſuper lineam intellectualẽ, ſuper quã cadit axis perpendiculariter:</s> <s xml:id="echoid-s5268" xml:space="preserve"> nõ cõprehendetuidẽs <lb/>corporis illius declinationẽ propter ſitum à temperamento egreſſum:</s> <s xml:id="echoid-s5269" xml:space="preserve"> quoniã non plena fit cõpre-<lb/>henſio corporum ab axe longè poſitorũ [per 15 n.</s> <s xml:id="echoid-s5270" xml:space="preserve">] Et in hoc errore declinatũ iudicabit uiſus rectũ.</s> <s xml:id="echoid-s5271" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div174" type="section" level="0" n="0"> <head xml:id="echoid-head203" xml:space="preserve" style="it">36. In figura. 97. 96. 61. 62 p 4.</head> <p> <s xml:id="echoid-s5272" xml:space="preserve">IN figura autem error eſt per ſitum.</s> <s xml:id="echoid-s5273" xml:space="preserve"> Si enim corpus circulare, ut ſchyphus uel ſcutella ab axe elõ-<lb/>getur, & modicum ſuper lineã intellectualẽ, quã diximus, declinetur:</s> <s xml:id="echoid-s5274" xml:space="preserve"> occultabitur eius declina-<lb/>tio, & una eius diameter ſub maiore angulo comprehendetur, quàm alia.</s> <s xml:id="echoid-s5275" xml:space="preserve"> Quæ enim apparetre-<lb/>cta, maiorẽ reſpicit angulũ, quã declinata.</s> <s xml:id="echoid-s5276" xml:space="preserve"> Et quia notabilis eſt unius anguli ad aliũ exceſſus:</s> <s xml:id="echoid-s5277" xml:space="preserve"> iudica <lb/>tur diameter recta maior declinata:</s> <s xml:id="echoid-s5278" xml:space="preserve"> unde circularis figura corporis, iudicabitur oblonga.</s> <s xml:id="echoid-s5279" xml:space="preserve"> Pari erro-<lb/>re figura quadrãgula æſtimabitur oblonga, cũ latus eius directè oppoſitũ oculo, maius appareat la-<lb/>tere declinato.</s> <s xml:id="echoid-s5280" xml:space="preserve"> Et eſt error in ſyllogiſmo.</s> <s xml:id="echoid-s5281" xml:space="preserve"> Pręmittit enim propoſiiones, in quibus eſt falſitas, ſcilicet:</s> <s xml:id="echoid-s5282" xml:space="preserve"> <lb/>neutrum laterum eſſe declinatum:</s> <s xml:id="echoid-s5283" xml:space="preserve"> & uiſa ab eadem longitudine ſub eodem & inæqualib.</s> <s xml:id="echoid-s5284" xml:space="preserve"> angulis, <lb/>eſſe inæqualia:</s> <s xml:id="echoid-s5285" xml:space="preserve"> & oblongam eſſe formam, cuius unum latus eſt inæquale alij.</s> <s xml:id="echoid-s5286" xml:space="preserve"> Inde concluditur er-<lb/>ror, non ueritas figuræ.</s> <s xml:id="echoid-s5287" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div175" type="section" level="0" n="0"> <head xml:id="echoid-head204" xml:space="preserve" style="it">37. In magnitudine. 28 p 4.</head> <p> <s xml:id="echoid-s5288" xml:space="preserve">EX eadem cauſſa palàm, errorem eſſe in quantitate, cũ diameter circularis corporis maior uide <lb/>tur alia eiuſdem diametro, cui eſt æqualis.</s> <s xml:id="echoid-s5289" xml:space="preserve"> Amplius alio modo accidit error in magnitudine, <lb/>ex ſitu intẽperato & ſolo:</s> <s xml:id="echoid-s5290" xml:space="preserve"> cũ aliquis in alto poſitus intuetur ſub altitudine illa incedẽtes & in-<lb/>ter ſe æquales, eis in ordine uno poſt aliũ diſpoſitis, radius cadens ſuper primũ abſq;</s> <s xml:id="echoid-s5291" xml:space="preserve"> dubio demil-<lb/>ſior erit radio cadente ſuper ſecundũ:</s> <s xml:id="echoid-s5292" xml:space="preserve"> & ſecundum, quod augmentabitur elõgatio alicuius eorum à <lb/>primo, ſecundum illud maior erit radij ſuper ipſum cadentis altitudo.</s> <s xml:id="echoid-s5293" xml:space="preserve"> Vnde altior erit radius cadẽs <lb/>in poſtremũ, quàm in aliquẽ aliũ.</s> <s xml:id="echoid-s5294" xml:space="preserve"> Iudicabitur ergo à uidente poſtremus maior omnib.</s> <s xml:id="echoid-s5295" xml:space="preserve"> Ita dico, ſi ter <lb/>ræ ſpatium inter quoslibet duos ſitum lateat uiſum, ne in collatione ad terram apparẽtem facta, cõ-<lb/>prehendi poſsit altitudinis hominũ menſura:</s> <s xml:id="echoid-s5296" xml:space="preserve"> erit error in ſyllogiſmo:</s> <s xml:id="echoid-s5297" xml:space="preserve"> quoniã errat in antecedentib.</s> <s xml:id="echoid-s5298" xml:space="preserve"> <lb/>quorum unum eſt:</s> <s xml:id="echoid-s5299" xml:space="preserve"> Quæcunq;</s> <s xml:id="echoid-s5300" xml:space="preserve"> apparent altiora, ſunt maiora:</s> <s xml:id="echoid-s5301" xml:space="preserve"> & hoc non inuenitur in omnibus, ſed <lb/>in pluribus.</s> <s xml:id="echoid-s5302" xml:space="preserve"> Et eſt error ex ſitus immoderatione, reſpectu cõprehẽſionis magnitudinis rei ſic diſpo-<lb/>ſitæ.</s> <s xml:id="echoid-s5303" xml:space="preserve"> Si enim radius cadens in primum ſit æquidiſtans terræ, & idẽ radius cadat in quemlibet alium <lb/>proceſſu ſuo:</s> <s xml:id="echoid-s5304" xml:space="preserve"> non habebit locum error iſte.</s> <s xml:id="echoid-s5305" xml:space="preserve"/> </p> <pb o="96" file="0102" n="102" rhead="ALHAZEN"/> </div> <div xml:id="echoid-div176" type="section" level="0" n="0"> <head xml:id="echoid-head205" xml:space="preserve" style="it">38. In diuiſione, continuatione, & numero. 109 p 4.</head> <p> <s xml:id="echoid-s5306" xml:space="preserve">IN diſtinctione prouenit error ex exceſſu ſitus:</s> <s xml:id="echoid-s5307" xml:space="preserve"> ſi enim magna fuerit corporis alicuius ſuper radi <lb/>os declinatio, & fuerint in eo puncta ſenſibilia nigra, uel ualde tenebroſa:</s> <s xml:id="echoid-s5308" xml:space="preserve"> putabũtur forſitã eſſe <lb/>foramina:</s> <s xml:id="echoid-s5309" xml:space="preserve"> & ita inter partes huic tenebricoſitati affines iudicabitur diuiſio, licet ibi ſit cõtinuita <lb/>tis unio.</s> <s xml:id="echoid-s5310" xml:space="preserve"> Si uerò in hoc corpore fuerint lineæ ſenſibiles tenebroſæ:</s> <s xml:id="echoid-s5311" xml:space="preserve"> iudicabuntur contermin ales di-<lb/>uiſę, cũ ſint continuę.</s> <s xml:id="echoid-s5312" xml:space="preserve"> Et ita error accidit ex corporis declinatione.</s> <s xml:id="echoid-s5313" xml:space="preserve"> In cõtinuitate erit error ex ſitu:</s> <s xml:id="echoid-s5314" xml:space="preserve"> ſi <lb/>opponatur uiſui plurium parietum diſpoſitio, quorum unus ſit ordinatim poſt aliũ, modicũ diſtans <lb/>ab eo, & omnes cadant ſuper eundem radium:</s> <s xml:id="echoid-s5315" xml:space="preserve"> occultabitur forſitã uidenti ſpatium, quod inter eos <lb/>fuerit:</s> <s xml:id="echoid-s5316" xml:space="preserve"> unde putabuntur cõtinui, cum ſint diuiſi:</s> <s xml:id="echoid-s5317" xml:space="preserve"> quod non accidet, ſitu parietũ immutato, utnon cõ <lb/>prehendantur ſub eodem radio.</s> <s xml:id="echoid-s5318" xml:space="preserve"> Error inducitur in numero ex ſitu immoderato, quando corpus ali <lb/>quod uidetur duo:</s> <s xml:id="echoid-s5319" xml:space="preserve"> & hoc accidit, cum reſpectu duorum uiſuum, corporis diuerſus fuerit ſitus.</s> <s xml:id="echoid-s5320" xml:space="preserve"> Pari <lb/>modo & in corpore uno iudicatur pluralitas, cũ inter duos axes corpus uiſum ceciderit, ſicut ſuprà <lb/>patuit [11 n.</s> <s xml:id="echoid-s5321" xml:space="preserve">] Et eſt error in ſyllogiſmo:</s> <s xml:id="echoid-s5322" xml:space="preserve"> præmittit enim uidens eſſe diuerſa corpora exterius uiſa, cũ <lb/>forma interius in diuerſa uiſus ceciderit loca:</s> <s xml:id="echoid-s5323" xml:space="preserve"> Inde diuerſitatem, ubi identitas eſt, concludit.</s> <s xml:id="echoid-s5324" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div177" type="section" level="0" n="0"> <head xml:id="echoid-head206" xml:space="preserve" style="it">39. In motu & quiete. 138 p 4</head> <p> <s xml:id="echoid-s5325" xml:space="preserve">IN motu oritur error ex ſitu, ut nauim currentẽ in flumine, aliquo inſpiciente, ſi fuerint in littore <lb/>fluminis arbores ab axe multùm elongatę, putabuntur moueri:</s> <s xml:id="echoid-s5326" xml:space="preserve"> & ſi fiat directio axium ſuper eas, <lb/>uidebuntur immotæ.</s> <s xml:id="echoid-s5327" xml:space="preserve"> In quiete error ex ſitu ſe ingerit:</s> <s xml:id="echoid-s5328" xml:space="preserve"> uiſa re aliqua, ut rota, quæ motu citiſsimo <lb/>uoluatur ab axe elongata:</s> <s xml:id="echoid-s5329" xml:space="preserve"> apparebit immota.</s> <s xml:id="echoid-s5330" xml:space="preserve"> Et planum eſt per ſitum eſſe errorem:</s> <s xml:id="echoid-s5331" xml:space="preserve"> quoniã ſitu mu-<lb/>tato percipietur eius motio:</s> <s xml:id="echoid-s5332" xml:space="preserve"> unde error eſt ex ſitu ſolo intemperato.</s> <s xml:id="echoid-s5333" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div178" type="section" level="0" n="0"> <head xml:id="echoid-head207" xml:space="preserve" style="it">40. In aſperitate & lenitate. 141 p 4.</head> <p> <s xml:id="echoid-s5334" xml:space="preserve">IN aſperitate ſitus errorem facit.</s> <s xml:id="echoid-s5335" xml:space="preserve"> Si enim à capillis expreſsè depictis, fiat reflexio lucis, nec fuerit <lb/>uiſus in loco reflexionis:</s> <s xml:id="echoid-s5336" xml:space="preserve"> fiet in eis comprehenſio aſperitatis, cum ſola ſit in eis læuitas.</s> <s xml:id="echoid-s5337" xml:space="preserve"> Et ex ſitu <lb/>ſolo eſt error:</s> <s xml:id="echoid-s5338" xml:space="preserve"> quoniam uiſu ſub luce reflexa fixo, non cõprehenditur aſperitas in corpore uiſo.</s> <s xml:id="echoid-s5339" xml:space="preserve"> In <lb/>læuitate erit error exſitu:</s> <s xml:id="echoid-s5340" xml:space="preserve"> cũ aliquid fuerit elongatum ab axe, & modica fuerit in eo aſperitas:</s> <s xml:id="echoid-s5341" xml:space="preserve"> appa-<lb/>rebit lęue:</s> <s xml:id="echoid-s5342" xml:space="preserve"> cuius quidẽ aſperitatẽ (ſitu ad temperantiam reducto) poſſet uidens comprehendere.</s> <s xml:id="echoid-s5343" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div179" type="section" level="0" n="0"> <head xml:id="echoid-head208" xml:space="preserve" style="it">41. In raritate & denſitate. 144 p 4.</head> <p> <s xml:id="echoid-s5344" xml:space="preserve">IN raritate & ſoliditate fiet error ex ſitus immoderamine.</s> <s xml:id="echoid-s5345" xml:space="preserve"> Si enim deſcenderit lux declinata in ui <lb/>trũ uino plenum, & lateat uiſum tranſitus lucis per uitrũ, & magna ſit declinatio illius lucis à ra-<lb/>dijs incidentibus, & uidentem lateat uinum eſſe in uaſe uitreo:</s> <s xml:id="echoid-s5346" xml:space="preserve"> æſtimabitur à uidente uinum ſo-<lb/>lidum corpus unum cum uaſe.</s> <s xml:id="echoid-s5347" xml:space="preserve"> Et non accidit error iſte tranſitu luci per uas uitreum patente.</s> <s xml:id="echoid-s5348" xml:space="preserve"> Vn-<lb/>de error in ſitu ex raritate & ſoliditate.</s> <s xml:id="echoid-s5349" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div180" type="section" level="0" n="0"> <head xml:id="echoid-head209" xml:space="preserve" style="it">42. In umbra & tenebris. 147 p 4.</head> <p> <s xml:id="echoid-s5350" xml:space="preserve">IN umbra & tenebris.</s> <s xml:id="echoid-s5351" xml:space="preserve"> Corpore enim aliquo ab axe elongato, ſi fuerit in eo pars tenebroſa:</s> <s xml:id="echoid-s5352" xml:space="preserve"> putabi <lb/>tur fortaſsis umbra:</s> <s xml:id="echoid-s5353" xml:space="preserve"> & corpore aliquo circumpoſito:</s> <s xml:id="echoid-s5354" xml:space="preserve"> ęſtimabitur procedere ab illo.</s> <s xml:id="echoid-s5355" xml:space="preserve"> Si aũt in cor-<lb/>pore illo fuerit pars multum nigra:</s> <s xml:id="echoid-s5356" xml:space="preserve"> æſtimabitur forſitan in loco nigredinis perforatio, per quam <lb/>egrediantur tenebrę.</s> <s xml:id="echoid-s5357" xml:space="preserve"> Quod non accideret in corpore ſtatuto in ſitus temperantia.</s> <s xml:id="echoid-s5358" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div181" type="section" level="0" n="0"> <head xml:id="echoid-head210" xml:space="preserve" style="it">43. In pulchritudine & deformitate. 150 p 4.</head> <p> <s xml:id="echoid-s5359" xml:space="preserve">IN ſpecie & deformitate aũt error accidit ex ſitu:</s> <s xml:id="echoid-s5360" xml:space="preserve"> cum corpus aliquod remotum fuerit ab axe, & <lb/>fuerintin eo multæ minutæ maculæ, ipſum deturpantes:</s> <s xml:id="echoid-s5361" xml:space="preserve"> occultabuntur, & iudicabitur in corpo <lb/>re ſpecies.</s> <s xml:id="echoid-s5362" xml:space="preserve"> Vnde facies lentiginoſa in hoc ſitu uidetur ſpecioſa.</s> <s xml:id="echoid-s5363" xml:space="preserve"> Similiter in hoc ſitu obliquo latẽt <lb/>uidentẽ lunę adhęrentes maculæ:</s> <s xml:id="echoid-s5364" xml:space="preserve"> unde adſcribitur decor lunæ ſic inſpectæ.</s> <s xml:id="echoid-s5365" xml:space="preserve"> Si autẽ in corpore uiſo <lb/>fuerint picturę, ei ſpeciem reddentes, nec ſit corpus decorum, niſi ex prætentu earum, cum ipſæ in <lb/>hoc ſtatu lateant uiſum:</s> <s xml:id="echoid-s5366" xml:space="preserve"> iudicabitur corpus deforme.</s> <s xml:id="echoid-s5367" xml:space="preserve"> Et eſt error in ſyllogiſmo:</s> <s xml:id="echoid-s5368" xml:space="preserve"> quia per apparen-<lb/>tiam tantùm fiet deformitatis uel decoris concluſio.</s> <s xml:id="echoid-s5369" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div182" type="section" level="0" n="0"> <head xml:id="echoid-head211" xml:space="preserve" style="it">44. In ſimilitudine & dißimilitudine. 153 p 4.</head> <p> <s xml:id="echoid-s5370" xml:space="preserve">IN ſimilitudine & diſsimilitudine ex ſitu error oritur.</s> <s xml:id="echoid-s5371" xml:space="preserve"> Si enim longè ab axe ſtatuãtur duo cõcor-<lb/>dantia in figura, ſpecie & colore, ſed in eis ſint modicæ & diſsimiles notę:</s> <s xml:id="echoid-s5372" xml:space="preserve"> iudicabitur in eis ſimi-<lb/>litudo omnimoda:</s> <s xml:id="echoid-s5373" xml:space="preserve"> cum notæ illæ uidenti ſint ignotę.</s> <s xml:id="echoid-s5374" xml:space="preserve"> Si aũt fuerit diuerſitas inter ea, in ſpecie, fi-<lb/>gura & colore, ſed in eis ſint notę ſimiles:</s> <s xml:id="echoid-s5375" xml:space="preserve"> putabuntur ex toto diſsimilia, cũ aliqua diſsimilitudo ſit <lb/>inter ea.</s> <s xml:id="echoid-s5376" xml:space="preserve"> Et ita eſt error in ſimilitudine & diſsimilitudine, propter concluſionẽ ex apparentib.</s> <s xml:id="echoid-s5377" xml:space="preserve"> tantũ <lb/>factam.</s> <s xml:id="echoid-s5378" xml:space="preserve"> Et in omnib.</s> <s xml:id="echoid-s5379" xml:space="preserve"> prædictis procreatur error ex ſolo ſitu intẽperato:</s> <s xml:id="echoid-s5380" xml:space="preserve"> quoniam eo intra tempera-<lb/>mentum ſito, alijs (ſicut ſunt) manentibus, non accidit erronea æſtimatio.</s> <s xml:id="echoid-s5381" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div183" type="section" level="0" n="0"> <head xml:id="echoid-head212" xml:space="preserve" style="it">45. Lux immoderata creat errores in ſingulis uiſibilibus ſpeciebus. In diſtantia. 16 p 4.</head> <p> <s xml:id="echoid-s5382" xml:space="preserve">LVx à temperantiæ finibus egreditur, & ob hoc ſolùm in omnib.</s> <s xml:id="echoid-s5383" xml:space="preserve"> quorum fit acquiſitio perſyl-<lb/>logiſmum, error procreatur in longitudine exlucis paruitate.</s> <s xml:id="echoid-s5384" xml:space="preserve"> Si enim in longitudine tẽpera-<lb/>ta non multũ certa, fiat hominũ diſpoſitio, utſit unus poſtaliũ, & uiſu huic diſpoſitioni de no-<lb/>cte adhibito:</s> <s xml:id="echoid-s5385" xml:space="preserve"> uidebuntur ſibi cohęrere, & incõprehenſa inter eos diſtantia, propter debilitatẽ lucis, <lb/> <pb o="97" file="0103" n="103" rhead="OPTICAE LIBER III."/> quæ pateret, ſi lux eſſet fortis:</s> <s xml:id="echoid-s5386" xml:space="preserve"> qui homines, ſi in eandem partem moueantur, æquali motu ſimul ſem <lb/>per moueri putabuntur.</s> <s xml:id="echoid-s5387" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div184" type="section" level="0" n="0"> <head xml:id="echoid-head213" xml:space="preserve" style="it">46. In ſitu. 44 p 4.</head> <p> <s xml:id="echoid-s5388" xml:space="preserve">IN ſitu.</s> <s xml:id="echoid-s5389" xml:space="preserve"> Vt ſi in nocte non obſcura aliquid modicè à uiſu declinatum, opponatur uiſui:</s> <s xml:id="echoid-s5390" xml:space="preserve">ęſtimabitur <lb/>in eo ſitus rectitudo, propter debilitatem lucis egreſſæ à temperamento.</s> <s xml:id="echoid-s5391" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div185" type="section" level="0" n="0"> <head xml:id="echoid-head214" xml:space="preserve" style="it">47. In figura & magnitudine. 97. 28 p 4.</head> <p> <s xml:id="echoid-s5392" xml:space="preserve">SImiliter figura multorum laterum æqualium, circularis apparebit de nocte aſpecta:</s> <s xml:id="echoid-s5393" xml:space="preserve"> quoniã oc-<lb/>cultat angulos lux nimium debilis.</s> <s xml:id="echoid-s5394" xml:space="preserve"> Pari modo ſphæra ſic uiſa reputatur ſuperficies plana:</s> <s xml:id="echoid-s5395" xml:space="preserve"> quia <lb/>occultatur uiſui partium eminentia.</s> <s xml:id="echoid-s5396" xml:space="preserve"> In ma gnitudine.</s> <s xml:id="echoid-s5397" xml:space="preserve"> Vt nocte inſpecto homine & uiſo nemo-<lb/>re, aut remoto ab eo, pariete, uidebitur propin quitas hominis ad nemus uel parietem, cum lateat ui <lb/>ſum diſtantia eorum, licet ſit plurima.</s> <s xml:id="echoid-s5398" xml:space="preserve"> Et forſan exibit idem radius ſuper caput hominis & altitudi-<lb/>nem nemoris, ſecundum quantitatem diſtantiæ à nemore:</s> <s xml:id="echoid-s5399" xml:space="preserve"> & in hoc ſitu uidebuntur eſſe eiuſdem al <lb/>titudinis:</s> <s xml:id="echoid-s5400" xml:space="preserve"> aut forſitan homo uidebitur eſſe maioris:</s> <s xml:id="echoid-s5401" xml:space="preserve"> quod non accideret, ſi lux in temperamento eſ-<lb/>ſet:</s> <s xml:id="echoid-s5402" xml:space="preserve"> quoniam diſtantia hominis ad nemus diſcerneretur, & altitudo uniuſcuiuſque ſecundum ter-<lb/>ram apparentem menſuraretur.</s> <s xml:id="echoid-s5403" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div186" type="section" level="0" n="0"> <head xml:id="echoid-head215" xml:space="preserve" style="it">48. In diuiſione, continuatione & numero. 109 p 4.</head> <p> <s xml:id="echoid-s5404" xml:space="preserve">IN diſtinctione, numero, continuitate erit error ex lucis debilitate.</s> <s xml:id="echoid-s5405" xml:space="preserve"> Vt ſi de nocte uideatur tabu-<lb/>la, in qua ſit linearum obſcurarum protractio:</s> <s xml:id="echoid-s5406" xml:space="preserve"> putabit forſan uidẽs diuiſiones eſſe uel fiſſuras.</s> <s xml:id="echoid-s5407" xml:space="preserve"> Et <lb/>ita error eſt in diſtinctione, quia continuũ apparet diuiſum.</s> <s xml:id="echoid-s5408" xml:space="preserve"> Et in numero, quia pluralitas in uno.</s> <s xml:id="echoid-s5409" xml:space="preserve"> <lb/>Similiter exiſtente uiſu in lucis fortis reflexione:</s> <s xml:id="echoid-s5410" xml:space="preserve"> ſi adhibeantur corpora modicũ diſtantia:</s> <s xml:id="echoid-s5411" xml:space="preserve"> appare-<lb/>bunt continua.</s> <s xml:id="echoid-s5412" xml:space="preserve"> Et ita error eſt in cõtinuitate, propter lucem nimiùm aut fortem aut debilem.</s> <s xml:id="echoid-s5413" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div187" type="section" level="0" n="0"> <head xml:id="echoid-head216" xml:space="preserve" style="it">49. In motu & quiete. 138 p 4.</head> <p> <s xml:id="echoid-s5414" xml:space="preserve">IN motu & quiete accidit error ex luce.</s> <s xml:id="echoid-s5415" xml:space="preserve"> Si enim nocte cõprehenderit uiſus hominem, & remotũ <lb/>ab eo nemus:</s> <s xml:id="echoid-s5416" xml:space="preserve"> occultabitur diſtantia hominis ad nemus:</s> <s xml:id="echoid-s5417" xml:space="preserve"> & ſi moueatur uidens ad hominem illũ, <lb/>quantò magis ad illum acceſſerit, tantò diſtantiam illam certius uidebit.</s> <s xml:id="echoid-s5418" xml:space="preserve"> Vnde cum prius ſimul <lb/>cum nemore appareret ei homo uiſus, quando ad eum accedit, plus uidetur à nemore remotus:</s> <s xml:id="echoid-s5419" xml:space="preserve"> & <lb/>cum certum ſit ei, nemus immotum manere:</s> <s xml:id="echoid-s5420" xml:space="preserve"> ſyllogizabit hominem uiſum à parte nemoris incede-<lb/>re, licet ueritas habeat ipſum immotum eſſe:</s> <s xml:id="echoid-s5421" xml:space="preserve"> qui error nõ accideret in temperata luce.</s> <s xml:id="echoid-s5422" xml:space="preserve"> In quiete.</s> <s xml:id="echoid-s5423" xml:space="preserve"> Vt <lb/>homo de nocte uiſus non plenè comprehenditur:</s> <s xml:id="echoid-s5424" xml:space="preserve"> unde ſi modicum uideatur, nõ diſcernitur, & mo-<lb/>tus putabitur quieſcere.</s> <s xml:id="echoid-s5425" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div188" type="section" level="0" n="0"> <head xml:id="echoid-head217" xml:space="preserve" style="it">50. In aſperitate & lenitate: raritate & denſitate: umbra & tenebris. 141. 144 p 4.</head> <p> <s xml:id="echoid-s5426" xml:space="preserve">IN aſperitate & læuitate erit error.</s> <s xml:id="echoid-s5427" xml:space="preserve"> De nocte enim uiſa aſperitas iudicatur forſan læuitas:</s> <s xml:id="echoid-s5428" xml:space="preserve"> aut è cõ <lb/>trario, ſecundum quod fuerit rei uiſæ qualitas.</s> <s xml:id="echoid-s5429" xml:space="preserve"> In raritate & denſitate.</s> <s xml:id="echoid-s5430" xml:space="preserve"> De nocte enim remiſſa iu-<lb/>dicabitur in corpore multũ raro raritas:</s> <s xml:id="echoid-s5431" xml:space="preserve"> quia cũ poſt ipſum non plena fiat comprehenſio ſolidi:</s> <s xml:id="echoid-s5432" xml:space="preserve"> <lb/>æſtimabitur remiſsio raritatis eius uiam negare uiſui:</s> <s xml:id="echoid-s5433" xml:space="preserve"> Corpus uerò modicè rarũ uidebitur ſolidũ.</s> <s xml:id="echoid-s5434" xml:space="preserve"> <lb/>In umbra & tenebris.</s> <s xml:id="echoid-s5435" xml:space="preserve"> Si enim in pariete albo fuerint partes obſcurę, & cadat ſuper parietẽ illum lux <lb/>candelæ:</s> <s xml:id="echoid-s5436" xml:space="preserve"> iudicabit forſitan uidens obſcuritatẽ illam eſſe umbrã:</s> <s xml:id="echoid-s5437" xml:space="preserve"> & uidebiture ei forſitan, quòd proce <lb/>dat apparens umbra à uicino pariete:</s> <s xml:id="echoid-s5438" xml:space="preserve"> & ita error eſt in umbrę æſtimatione.</s> <s xml:id="echoid-s5439" xml:space="preserve"> Similiter ſi fuerit in par-<lb/>te parietis nigredo multùm:</s> <s xml:id="echoid-s5440" xml:space="preserve"> æſtimabitur forſitan uacuitas foraminis iter prębens egredientib.</s> <s xml:id="echoid-s5441" xml:space="preserve"> tene <lb/>bris.</s> <s xml:id="echoid-s5442" xml:space="preserve"> Et ſi tota parietis ſuperficies afficiatur intenſa nigredine:</s> <s xml:id="echoid-s5443" xml:space="preserve"> totus forſitan putabitur tenebrę, ut <lb/>accidit in pariete cooperto ignis fuligine, & uiſo in debili luce.</s> <s xml:id="echoid-s5444" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div189" type="section" level="0" n="0"> <head xml:id="echoid-head218" xml:space="preserve" style="it">51. In pulchritudine & deformitate: ſimilitudine & dißimilitudine. 150. 153 p 4.</head> <p> <s xml:id="echoid-s5445" xml:space="preserve">IN ſpecie & deformitate.</s> <s xml:id="echoid-s5446" xml:space="preserve"> Palàm enim, quòd de nocte uidetur facies formoſa, licet in ea ſint macu-<lb/>læ, ſicut in lentiginoſa.</s> <s xml:id="echoid-s5447" xml:space="preserve"> Et ſi fuerint in re uiſa picturę ſubtiles, totalis ſpeciei cauſſæ, cum in nocte <lb/>uiſum lateant:</s> <s xml:id="echoid-s5448" xml:space="preserve"> uidebitur res deformis.</s> <s xml:id="echoid-s5449" xml:space="preserve"> In ſimilitudine & diſsimilitudine.</s> <s xml:id="echoid-s5450" xml:space="preserve"> In corporib.</s> <s xml:id="echoid-s5451" xml:space="preserve"> enim eiuſ-<lb/>dem ſpeciei, coloris, & figurę, in quibus eſt partialis diuerſitas per latẽtes notas:</s> <s xml:id="echoid-s5452" xml:space="preserve"> in debili luce omni <lb/>moda ſimilitudo iudicabitur.</s> <s xml:id="echoid-s5453" xml:space="preserve"> Et ſi diuerſa fuerint corpora, in ſpecie, colore, & figura, ſed ex aliquib.</s> <s xml:id="echoid-s5454" xml:space="preserve"> <lb/>notis conformitas eſt partialis:</s> <s xml:id="echoid-s5455" xml:space="preserve"> propter occultationem notarum exremiſsione lucis, iudicabitur o-<lb/>mnimoda diuerſitas corporum.</s> <s xml:id="echoid-s5456" xml:space="preserve"> Et palàm, in omnibus prædictis errorem accidere exſola debilita-<lb/>te lucis, cum ipſa intra terminos temperantiæ ſita, error non accidat, alijs immotis.</s> <s xml:id="echoid-s5457" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div190" type="section" level="0" n="0"> <head xml:id="echoid-head219" xml:space="preserve" style="it">52. Magnitudo immoderata creat errores in ſingulis uiſibilib. ſpeciebus. In diſtantia. 16 p 4.</head> <p> <s xml:id="echoid-s5458" xml:space="preserve">QVantitas egreditur à temperantia, & ille egreſſus cauſſa eſt erroris in omnibus, quibus fidẽ <lb/>facit ſyllogiſmus.</s> <s xml:id="echoid-s5459" xml:space="preserve"> Error erit in longitudine ex cauſſa prædicta:</s> <s xml:id="echoid-s5460" xml:space="preserve"> ut ſi uideantur duo homines <lb/>à longitudine temperata, & in ſuo genere maxima, & unus paululum fuerit ante alium:</s> <s xml:id="echoid-s5461" xml:space="preserve"> non <lb/>diſcernetur uia inter cos ſita:</s> <s xml:id="echoid-s5462" xml:space="preserve"> unde unus eorum apparebit circa alium.</s> <s xml:id="echoid-s5463" xml:space="preserve"> Et accidit error:</s> <s xml:id="echoid-s5464" xml:space="preserve"> quoniam di <lb/>ſtantia eorum cum multùm ſit parua, non eſt proportionalis totali eorũ à uiſu elongationi, licet elõ <lb/>gatio ſit temperata.</s> <s xml:id="echoid-s5465" xml:space="preserve"> Eſt aũt error in longitudine, quoniam homines illi iudicabuntur ab oculo ęquè <lb/>remoti:</s> <s xml:id="echoid-s5466" xml:space="preserve"> & ita quantitas unius longitudinis maior, quàm ſit in ueritate.</s> <s xml:id="echoid-s5467" xml:space="preserve"> Vnde error in longitudine.</s> <s xml:id="echoid-s5468" xml:space="preserve"/> </p> <pb o="98" file="0104" n="104" rhead="ALHAZEN"/> </div> <div xml:id="echoid-div191" type="section" level="0" n="0"> <head xml:id="echoid-head220" xml:space="preserve" style="it">53. In ſitu. 44 p 4.</head> <p> <s xml:id="echoid-s5469" xml:space="preserve">IN ſitu propter quantitatis paruitatem eſt error.</s> <s xml:id="echoid-s5470" xml:space="preserve"> Quoniam granum ſinapis ſi fuerit ab oculo de-<lb/>clinatum, tamen uidetur rectum:</s> <s xml:id="echoid-s5471" xml:space="preserve"> quoniam pro paruitate nimia non poteſt deprehendi declina-<lb/>tio huius grani ſuper lineam intellectualem, in quam axis communis cadit orthogonaliter:</s> <s xml:id="echoid-s5472" xml:space="preserve"> quo-<lb/>niam non plenè diſcernitur longitudo inter hanc lineam & extremitates grani, cum ſit minima.</s> <s xml:id="echoid-s5473" xml:space="preserve"> Et <lb/>ſecundum hanc longitudinem conſideratur declinatio eius ſuper lineam illam.</s> <s xml:id="echoid-s5474" xml:space="preserve"> Et ſecundum hanc <lb/>lineam conſideratur ſemper declinatio rei uiſæ, reſpectu uiſus utriuſque.</s> <s xml:id="echoid-s5475" xml:space="preserve"> Et ita error eſt in ſitu, ex <lb/>quantitate immoderata.</s> <s xml:id="echoid-s5476" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div192" type="section" level="0" n="0"> <head xml:id="echoid-head221" xml:space="preserve" style="it">54. In figura & magnitudine. 97. 28 p 4.</head> <p> <s xml:id="echoid-s5477" xml:space="preserve">IN figura.</s> <s xml:id="echoid-s5478" xml:space="preserve"> Cum enim res uiſa fuerit multùm parua, & fuerint in ea anguli:</s> <s xml:id="echoid-s5479" xml:space="preserve"> anguli occultab untur <lb/>uiſui:</s> <s xml:id="echoid-s5480" xml:space="preserve"> unde fortaſſe eius forma, cum non ſit, æſtimabitur rotunda aut longa:</s> <s xml:id="echoid-s5481" xml:space="preserve"> & ſi fuerit in ea incur <lb/>uatio modica, latebit uiſum, & æſtimabitur ſuperficies eius plana:</s> <s xml:id="echoid-s5482" xml:space="preserve"> unde palàm, quòd error eſt in <lb/>figura.</s> <s xml:id="echoid-s5483" xml:space="preserve"> In quantitate.</s> <s xml:id="echoid-s5484" xml:space="preserve"> Quantitas intemperata errorem inuehit:</s> <s xml:id="echoid-s5485" xml:space="preserve"> Propoſitis enim uiſui duobus corpo <lb/>ribus, quorum unum modicè excedat aliud in longitudine ſola, aut in latitudine:</s> <s xml:id="echoid-s5486" xml:space="preserve"> forſitã iudicabun-<lb/>tur æqualia omni dimenſione.</s> <s xml:id="echoid-s5487" xml:space="preserve"> Et eſt error iſte:</s> <s xml:id="echoid-s5488" xml:space="preserve"> quoniam excrementum unius dimenſionis ſuper a-<lb/>liam, euaſit fines temperantiæ, reſpectu uiſus, cum ſit ei inſenſibile præ nimia ſua diminutione:</s> <s xml:id="echoid-s5489" xml:space="preserve"> Ob <lb/>hoc neceſſariæ ſunt menſuræ, ut uerificentur quantitates corporum:</s> <s xml:id="echoid-s5490" xml:space="preserve"> cum non acquiratur certitu-<lb/>do per uiſum.</s> <s xml:id="echoid-s5491" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div193" type="section" level="0" n="0"> <head xml:id="echoid-head222" xml:space="preserve" style="it">55. In diuiſione, continuatione, & numero: motu & quiete. 109. 138 p 4.</head> <p> <s xml:id="echoid-s5492" xml:space="preserve">IN diuiſione error accidit.</s> <s xml:id="echoid-s5493" xml:space="preserve"> Capillo enim adhęrente uitro:</s> <s xml:id="echoid-s5494" xml:space="preserve"> apparebit diuiſio eſſe in uitro & fiſſura, <lb/>cum ibi ſit continuitas uera:</s> <s xml:id="echoid-s5495" xml:space="preserve"> & prouenit hoc ex capilli tenuitate, quoniam ſi adhęſerit uitro quan <lb/>titas corpulenta:</s> <s xml:id="echoid-s5496" xml:space="preserve"> non ęſtimabitur in eo fiſſura.</s> <s xml:id="echoid-s5497" xml:space="preserve"> In continuitate.</s> <s xml:id="echoid-s5498" xml:space="preserve"> Si enim prætendantur uiſui folia <lb/>pergameni tenuia, æqualis latitudinis bene compreſſa, & ignoret uidẽs eſſe folia:</s> <s xml:id="echoid-s5499" xml:space="preserve"> iudicabit ipſa eſſe <lb/>cõtinua, & unũ corpus efficere.</s> <s xml:id="echoid-s5500" xml:space="preserve"> Et eſt erroris cauſſa quãtitas uię interiacẽtis inter folia, quę præ ſua <lb/>paruitate non percipitur à uidente.</s> <s xml:id="echoid-s5501" xml:space="preserve"> Et eadẽ erit cauſſa erroris numeri, quæ cõtinuitatis.</s> <s xml:id="echoid-s5502" xml:space="preserve"> In motu.</s> <s xml:id="echoid-s5503" xml:space="preserve"> Si <lb/>enim moueantur duo, quorum unum moueatur paulò uelocius alio:</s> <s xml:id="echoid-s5504" xml:space="preserve"> putabit uidẽs æqualẽ eſſe mo <lb/>tum corum:</s> <s xml:id="echoid-s5505" xml:space="preserve"> quia inſenſibile eſt unius ſuper aliud excrementum uidenti.</s> <s xml:id="echoid-s5506" xml:space="preserve"> Similiter quantitas exceſ-<lb/>ſus uiæ, quã incedit unus ſuper eã, quã incedit alius, imperceptibilis eſt uiſui.</s> <s xml:id="echoid-s5507" xml:space="preserve"> Vnde iudicatur ęqua-<lb/>litas uiarum & motuũ.</s> <s xml:id="echoid-s5508" xml:space="preserve"> In quiete.</s> <s xml:id="echoid-s5509" xml:space="preserve"> Cũ enim offertur uiſui aliquid multũ paruum, forſitan mouebitur <lb/>pars eius aliqua, & ipſum iudicabitur immotum, cum motus partis lateat uiſum.</s> <s xml:id="echoid-s5510" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div194" type="section" level="0" n="0"> <head xml:id="echoid-head223" xml:space="preserve" style="it">56. In aſperitate & lenitate: raritate & denſitate: umbra & tenebris. 141. 144. 147 p 4.</head> <p> <s xml:id="echoid-s5511" xml:space="preserve">IN aſperitate & læuitate.</s> <s xml:id="echoid-s5512" xml:space="preserve"> Cũ enim occurrerit uiſui res multũ parua:</s> <s xml:id="echoid-s5513" xml:space="preserve"> iudicabitur forſan lęuitas, ubi <lb/>fuerit aſperitas, & è cõtrario.</s> <s xml:id="echoid-s5514" xml:space="preserve"> Quoniã, ut dictũ eſt, [53 n 2] aſperitas nõ cõprehẽditur in corpore, <lb/>niſi ex umbra quarun dam partiũ ſuper alias, uel eminentia earum, & depreſsione aliarũ:</s> <s xml:id="echoid-s5515" xml:space="preserve"> quod to <lb/>tum occultabitur iudicio uidentis, præ nimia paruitate corporis.</s> <s xml:id="echoid-s5516" xml:space="preserve"> In raritate & ſoliditate.</s> <s xml:id="echoid-s5517" xml:space="preserve"> Si quis e-<lb/>nim intueatur corpus ualde paruum politũ, ut ab eo lux poſsit reflecti, ſicut eſt margaritæ ſimile:</s> <s xml:id="echoid-s5518" xml:space="preserve"> ra <lb/>rum eſſe iudicabitur, cum non ſit.</s> <s xml:id="echoid-s5519" xml:space="preserve"> Similiter uiſo corpore raro multũ paruo, quòd poſtipſum non ſit <lb/>corporis ſolidi comprehenſio:</s> <s xml:id="echoid-s5520" xml:space="preserve"> exiſtimatur eſſe ſolidum.</s> <s xml:id="echoid-s5521" xml:space="preserve"> In umbra & tenebris.</s> <s xml:id="echoid-s5522" xml:space="preserve"> Si enim in pariete al-<lb/>bo uiſui oppoſito fuerit punctorum ualde nigrorum diſtinctio, adhibita ſolis luce, ſed directè in pa-<lb/>rietem cadente uel prope:</s> <s xml:id="echoid-s5523" xml:space="preserve"> æſtimabuntur à uidente ſingula puncta ſingula eſſe foramina, poſt quę e-<lb/>rumpant tenebræ.</s> <s xml:id="echoid-s5524" xml:space="preserve"> unde error cum tenebrarum æſtimatione ex ſola punctorum paruitate:</s> <s xml:id="echoid-s5525" xml:space="preserve"> qui non <lb/>accideret, ſi nigredo quantumcunq;</s> <s xml:id="echoid-s5526" xml:space="preserve"> intenſa magnam partem parietis inficeret.</s> <s xml:id="echoid-s5527" xml:space="preserve"> Si aũt fuerit in pun <lb/>ctis illis nigredo non adec̀ intenſa:</s> <s xml:id="echoid-s5528" xml:space="preserve"> reputabũtur quidem puncta illa, foramina, in quibus ſit umbra:</s> <s xml:id="echoid-s5529" xml:space="preserve"> <lb/>cum lux nõ penetret ea, ſicut ſolet accidere luce ſuper multorum foraminum ſpeciem cadente.</s> <s xml:id="echoid-s5530" xml:space="preserve"> Vn <lb/>de error um bræ ex ſo a punctorum diminutione.</s> <s xml:id="echoid-s5531" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div195" type="section" level="0" n="0"> <head xml:id="echoid-head224" xml:space="preserve" style="it">57. In pulchritudine & deformitate: ſimilitudine & dißimilitudine. 150. 153 p 4.</head> <p> <s xml:id="echoid-s5532" xml:space="preserve">IN ſpecie & deformitate:</s> <s xml:id="echoid-s5533" xml:space="preserve"> Cum præ ſua paruitate occultantur uiſui deturpantes corpus uiſum ma <lb/>culę, accidit erroneum de ſpecie iudicium:</s> <s xml:id="echoid-s5534" xml:space="preserve"> quia ſumitur ex apparentibus tantùm:</s> <s xml:id="echoid-s5535" xml:space="preserve"> Sicut eſt error <lb/>in deformitate, cum propter paruitatem lateant picturæ decorem ingerentes rei uiſæ.</s> <s xml:id="echoid-s5536" xml:space="preserve"> In ſimilitu <lb/>dine & diſsimilitudine.</s> <s xml:id="echoid-s5537" xml:space="preserve"> Cum enim notæ minutiſsimæ inter aliqua corpora, ſimilitudinis aut diſsi-<lb/>militudinis fuerint cauſſæ:</s> <s xml:id="echoid-s5538" xml:space="preserve"> quia prætereunt uiſum præ paruitate ſua, iudicabitur ſimilitudo aut diſ-<lb/>ſimilitudo omnimoda:</s> <s xml:id="echoid-s5539" xml:space="preserve"> & ſumetur iudicium ex apparentibus tãtùm.</s> <s xml:id="echoid-s5540" xml:space="preserve"> In omnibus prędictis eſt error <lb/>in ſyllogiſmo ex paruitate corporis:</s> <s xml:id="echoid-s5541" xml:space="preserve"> cum ea exiſtente tẽperata non accidat error, alijs immotis.</s> <s xml:id="echoid-s5542" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div196" type="section" level="0" n="0"> <head xml:id="echoid-head225" xml:space="preserve" style="it">58. Solidit {as} immoderata creat errores in ſingulis uiſibilibus ſpeciebus. In diſtantia & <lb/>ſitu. 16. 44 p 4.</head> <p> <s xml:id="echoid-s5543" xml:space="preserve">SOliditas aliquando egreditur temperamentum, & errorem inducit in quolibet eorum, quæ cõ <lb/>prehenduntur per ſyllogiſmum.</s> <s xml:id="echoid-s5544" xml:space="preserve"> In lõgitudine.</s> <s xml:id="echoid-s5545" xml:space="preserve"> Si enim minima fuerit corporis ſoliditas:</s> <s xml:id="echoid-s5546" xml:space="preserve"> & eſt:</s> <s xml:id="echoid-s5547" xml:space="preserve"> <lb/>ut ſit ualde rarum, ſicut eſt cryſtallus pura, & ſit poſt ipſam corpus lucidũ luce forti:</s> <s xml:id="echoid-s5548" xml:space="preserve"> non cõpre-<lb/>hendetur cryſtallus, ſed quaſi nullum eſſet intermedium, cõprehẽdetur corpus per ipſam:</s> <s xml:id="echoid-s5549" xml:space="preserve"> unde, cũ <lb/>quaſi non ſit, fiat rari acquiſitio:</s> <s xml:id="echoid-s5550" xml:space="preserve"> non plena erit longitudinis eius ab eo comprehẽſio.</s> <s xml:id="echoid-s5551" xml:space="preserve"> Vnde error in <lb/>longitudine.</s> <s xml:id="echoid-s5552" xml:space="preserve"> Quare ſi corporis rari ſitus fuerit declinatus, occultabitur uidẽti declinatio, & iudica-<lb/> <pb o="99" file="0105" n="105" rhead="OPTICAE LIBER III."/> bitur ſorſitan rectitudo.</s> <s xml:id="echoid-s5553" xml:space="preserve"> Vnde error in ſitu, & etiam error in longitudine:</s> <s xml:id="echoid-s5554" xml:space="preserve"> quoniam una eius extre-<lb/>mitas eiuſdem longitudinis reputabitur cum alia, cum ſint diuerſæ.</s> <s xml:id="echoid-s5555" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div197" type="section" level="0" n="0"> <head xml:id="echoid-head226" xml:space="preserve" style="it">59. In magnitudine & figura: diuiſione, continuatione & numero. 28. 97. 109 p 4.</head> <p> <s xml:id="echoid-s5556" xml:space="preserve">DEinde quoniam quantitas corporis comprehenditur ex longitudine, & anguli, ſub quo ui-<lb/>detur, capacitate:</s> <s xml:id="echoid-s5557" xml:space="preserve"> ignorata longitudine:</s> <s xml:id="echoid-s5558" xml:space="preserve"> accidit error in quãtitate.</s> <s xml:id="echoid-s5559" xml:space="preserve"> Modo conſimili erit error <lb/>in figura.</s> <s xml:id="echoid-s5560" xml:space="preserve"> Si enim in corpore fuerint anguli:</s> <s xml:id="echoid-s5561" xml:space="preserve"> occultabũtur uidenti:</s> <s xml:id="echoid-s5562" xml:space="preserve"> unde ſexangula forma pu-<lb/>tabitur ſphærica.</s> <s xml:id="echoid-s5563" xml:space="preserve"> Si uerò modica fuerit incuruatio in corpore, latebit, & iudicabitur corpus planũ <lb/>eſſe.</s> <s xml:id="echoid-s5564" xml:space="preserve"> In diſtinctione erit error.</s> <s xml:id="echoid-s5565" xml:space="preserve"> Si enim fuerit per corpus magnæ raritatis linea nigra, apparebit cor-<lb/>pus diuiſum in loco, in quẽ cadit linea.</s> <s xml:id="echoid-s5566" xml:space="preserve"> Si uerò fuerint duo corpora talia modicũ à ſe diſtãtia:</s> <s xml:id="echoid-s5567" xml:space="preserve"> repu-<lb/>tabuntur continua.</s> <s xml:id="echoid-s5568" xml:space="preserve"> Vnde error in continuitate.</s> <s xml:id="echoid-s5569" xml:space="preserve"> Et palàm, quòd ex his erit error in numeri compre-<lb/>henſione:</s> <s xml:id="echoid-s5570" xml:space="preserve"> cum uel unum plura, uel plura unum apparebunt.</s> <s xml:id="echoid-s5571" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div198" type="section" level="0" n="0"> <head xml:id="echoid-head227" xml:space="preserve" style="it">60. In motu & quiete. 138 p 4.</head> <p> <s xml:id="echoid-s5572" xml:space="preserve">IN motu erit error ex immoderatione raritatis:</s> <s xml:id="echoid-s5573" xml:space="preserve"> ſi opponatur foramini corpus ualde rarum, ut cry <lb/>ſtallus:</s> <s xml:id="echoid-s5574" xml:space="preserve"> & huius corporis extremitates lateant uiſum:</s> <s xml:id="echoid-s5575" xml:space="preserve"> & poſt corpus hoc moueatur aliud:</s> <s xml:id="echoid-s5576" xml:space="preserve"> puta-<lb/>bit uidens corpus rarum moueri, cum ſit immotum:</s> <s xml:id="echoid-s5577" xml:space="preserve"> quod non accideret ipſo temperatè ſolido.</s> <s xml:id="echoid-s5578" xml:space="preserve"> <lb/>In quiete accidit error ex eadem intemperantia.</s> <s xml:id="echoid-s5579" xml:space="preserve"> Si enim corpus ualde rarum includatur in manu, <lb/>coniunctum manui, & ab ea recedat, & moueatur intra manum reuolutionis motu, immota manu:</s> <s xml:id="echoid-s5580" xml:space="preserve"> <lb/>ita tamen, ut appareat diuiſio aliqua inter ipſum & manum:</s> <s xml:id="echoid-s5581" xml:space="preserve"> iudicabitur corpus illud immotũ:</s> <s xml:id="echoid-s5582" xml:space="preserve"> quo-<lb/>niã non poteſt in eo cõprehendi motus, niſi mutatione ſitus partiũ partis alicuius, reſpectu manus, <lb/>uel partis eius.</s> <s xml:id="echoid-s5583" xml:space="preserve"> Et quia omnimoda eſt ſimilitudo in partibus, uel prætenditur:</s> <s xml:id="echoid-s5584" xml:space="preserve"> propter raritatẽ non <lb/>poteſt diſcerni alicuius partium ſitus:</s> <s xml:id="echoid-s5585" xml:space="preserve"> quare nec motus.</s> <s xml:id="echoid-s5586" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div199" type="section" level="0" n="0"> <head xml:id="echoid-head228" xml:space="preserve" style="it">61. In aſperitate & lenitate: raritate & denſitate. 141. 144 p 4.</head> <p> <s xml:id="echoid-s5587" xml:space="preserve">IN aſperitate & lęuitate.</s> <s xml:id="echoid-s5588" xml:space="preserve"> Si enim in corpore multùm raro fuerit aſperitas non magna, putabitur <lb/>forſitan læue.</s> <s xml:id="echoid-s5589" xml:space="preserve"> Si uerò fuerit læue, & ſtatuatur poſt ipſum corpus aſperum, aut corpus diuerſorũ <lb/>colorum, æſtimabitur hoc corpus rarum & lęue, aſperũ.</s> <s xml:id="echoid-s5590" xml:space="preserve"> Vnde error in læuitate.</s> <s xml:id="echoid-s5591" xml:space="preserve"> In raritate.</s> <s xml:id="echoid-s5592" xml:space="preserve"> Si e-<lb/>nim poſt corpus ualde rarũ ſit aliud corpus rarũ non multũ, & colore forti coloratũ:</s> <s xml:id="echoid-s5593" xml:space="preserve"> apparebit pri-<lb/>mum non multũ rarum, ſed æſtimabitur eius raritas ſecundum raritatem poſtpoſiti.</s> <s xml:id="echoid-s5594" xml:space="preserve"> Vnde uitrum <lb/>alij uitro ſuperpoſitum non apparet ita rarum, ſicut appareret eo ſolo uiſui adhibito.</s> <s xml:id="echoid-s5595" xml:space="preserve"> Vnde error in <lb/>raritate.</s> <s xml:id="echoid-s5596" xml:space="preserve"> Si aũt poſt poſt primũ rarum ſtatuatur corpus ſolidum:</s> <s xml:id="echoid-s5597" xml:space="preserve"> iudicabitur primũ ſolidũ:</s> <s xml:id="echoid-s5598" xml:space="preserve"> unde er-<lb/>ror in ſoliditate.</s> <s xml:id="echoid-s5599" xml:space="preserve"> Pari modo ſiuas ualde rarum contineat uinũ, cum poſtillud nõ percipiatur luxaut <lb/>corpus aliud:</s> <s xml:id="echoid-s5600" xml:space="preserve"> iudicabitur forſan totum cum uino uitrum eſſe unum corpus ſolidum.</s> <s xml:id="echoid-s5601" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div200" type="section" level="0" n="0"> <head xml:id="echoid-head229" xml:space="preserve" style="it">62. In umbra & tenebris. 147 p 4. 67 p 10.</head> <p> <s xml:id="echoid-s5602" xml:space="preserve">IN umbra erit error ex raritate.</s> <s xml:id="echoid-s5603" xml:space="preserve"> Luce enim ſolis in domũ aliquam per foramen aliquod deſcendẽ <lb/>te, & ſuper feneſtram uitream cadente, cum domus illa ſit umbroſa:</s> <s xml:id="echoid-s5604" xml:space="preserve"> apparebit ſuper feneſtram il <lb/>lam umbra, licet in ueritate lux in ipſam incidat:</s> <s xml:id="echoid-s5605" xml:space="preserve"> quæ quidem <gap/><gap/>x comprehenderetur, ſi ſolidum <lb/>eſſet feneſtræ corpus:</s> <s xml:id="echoid-s5606" xml:space="preserve"> quoniam non tranſiret, & ita ſuper ſolidum appareret.</s> <s xml:id="echoid-s5607" xml:space="preserve"> Vnde error in umbra.</s> <s xml:id="echoid-s5608" xml:space="preserve"> <lb/>In tenebris.</s> <s xml:id="echoid-s5609" xml:space="preserve"> Luce enim ſolis in aquam fluminis nõ deſcendente, aut in mare, ſicut accidit hora ma-<lb/>tutina & ueſpertina:</s> <s xml:id="echoid-s5610" xml:space="preserve"> & ſi fuerit claritas in aqua:</s> <s xml:id="echoid-s5611" xml:space="preserve"> apparebit tenebroſa:</s> <s xml:id="echoid-s5612" xml:space="preserve"> & quantò fuerit clarior, tantò <lb/>putabitur tenebroſior.</s> <s xml:id="echoid-s5613" xml:space="preserve"> Et accidit hoc:</s> <s xml:id="echoid-s5614" xml:space="preserve"> quoniam pars aquæ ſuperior umbrã iacit ſuper proximã par <lb/>tem inferiorem, & illa proxima ſuper aliam inferiorem propinquam:</s> <s xml:id="echoid-s5615" xml:space="preserve"> & ita per ſingulas uſq;</s> <s xml:id="echoid-s5616" xml:space="preserve"> ad fun-<lb/>dum.</s> <s xml:id="echoid-s5617" xml:space="preserve"> Et licet ſingularum partium umbra in ſe ſit modica:</s> <s xml:id="echoid-s5618" xml:space="preserve"> tamen coniunctæ unam efficiunt maxi-<lb/>mam, ſicut palàm eſt in colore uini accidere:</s> <s xml:id="echoid-s5619" xml:space="preserve"> In modica enim quantitate uini color eſt debilis:</s> <s xml:id="echoid-s5620" xml:space="preserve"> & in <lb/>multa, licet eiuſdem modi, fortis.</s> <s xml:id="echoid-s5621" xml:space="preserve"> Cauſſa autem quare in mari umbram iaciente uideantur eſſe tene <lb/>bræ in maris claritate, eſt:</s> <s xml:id="echoid-s5622" xml:space="preserve"> quoniam intenſa claritas intenſam reddit raritatem:</s> <s xml:id="echoid-s5623" xml:space="preserve"> unde uiſui maiorem <lb/>reddit penetrationem:</s> <s xml:id="echoid-s5624" xml:space="preserve"> Vnde fit acquiſitio plurium maris partium umbram facientium:</s> <s xml:id="echoid-s5625" xml:space="preserve"> quoniã um <lb/>brarum aggregatarum perceptio inducit fidem tenebrarum.</s> <s xml:id="echoid-s5626" xml:space="preserve"> Si uerò mare fuerit turbulentum, pro-<lb/>pter diminutam raritatem penetrabit uiſus paululum, & comprehendet modicam aquæ partem:</s> <s xml:id="echoid-s5627" xml:space="preserve"> & <lb/>licet faciat umbram, cum ipſa ſit remiſſa, color illius partis uincit umbram.</s> <s xml:id="echoid-s5628" xml:space="preserve"> In turbida enim aqua co <lb/>lor apparet, in clara nullus:</s> <s xml:id="echoid-s5629" xml:space="preserve"> unde & propter turbidæ aquæ colorem & propter umbræ partis appa-<lb/>rentis remiſsionem non comprehenduntur in aqua tenebrę:</s> <s xml:id="echoid-s5630" xml:space="preserve"> unde ipſa turbida, apparebit colora-<lb/>ta, & clara tenebroſa.</s> <s xml:id="echoid-s5631" xml:space="preserve"> Solis autem radio cadente ſuper faciem maris, cum ei per raritatem ipſius pa-<lb/>teat tranſitus:</s> <s xml:id="echoid-s5632" xml:space="preserve"> abijcietur omnis tenebrarum & umbræ apparentia.</s> <s xml:id="echoid-s5633" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div201" type="section" level="0" n="0"> <head xml:id="echoid-head230" xml:space="preserve" style="it">63. In pulchritudine & deformitate: ſimilitudine & dißimilitudine. 150. 153 p 4.</head> <p> <s xml:id="echoid-s5634" xml:space="preserve">IN decore & deformitate.</s> <s xml:id="echoid-s5635" xml:space="preserve"> Si enim in uaſe multùm raro ſint particulæ uel inciſuræ ipſi decorem <lb/>afferentes:</s> <s xml:id="echoid-s5636" xml:space="preserve"> & imponatur uaſi illi uinum turbidum & turpe:</s> <s xml:id="echoid-s5637" xml:space="preserve"> occultabuntur decoris cauſſæ:</s> <s xml:id="echoid-s5638" xml:space="preserve"> & iu-<lb/>dicabitur uas deforme, ut aliquando accidit in uitreo uaſe.</s> <s xml:id="echoid-s5639" xml:space="preserve"> Econtrariò ſi uas tale deformente-<lb/>ius aliquæ particulæ, & imponatur ei uinum clarum lucidum, & in colore formoſum:</s> <s xml:id="echoid-s5640" xml:space="preserve"> occultabũtur <lb/>deformitatis cauſſæ, & reputabitur uas ſpecioſum, cum ſit deforme.</s> <s xml:id="echoid-s5641" xml:space="preserve"> In ſimilitudine & diſsimilitudi <lb/>ne.</s> <s xml:id="echoid-s5642" xml:space="preserve"> Si duo uaſa multũ rara conueniant in forma, ſpecie, raritate:</s> <s xml:id="echoid-s5643" xml:space="preserve"> ſed diſcrepent in aliquarum partiũ <lb/> <pb o="100" file="0106" n="106" rhead="ALHAZEN"/> diſpoſitione, & uino eiuſdem coloris, eiuſdẽ claritatis impleantur:</s> <s xml:id="echoid-s5644" xml:space="preserve"> latebunt cauſſę diuerſitatis, & re <lb/>putabuntur omnino ſimilia.</s> <s xml:id="echoid-s5645" xml:space="preserve"> Si uerò inter ea fuerit diuerſitas in ſpecie & forma:</s> <s xml:id="echoid-s5646" xml:space="preserve"> ſed in aliquibus par <lb/>tialibus conuenientia, & uino ſimili impleantur:</s> <s xml:id="echoid-s5647" xml:space="preserve"> putabuntur omnino diſsimilia.</s> <s xml:id="echoid-s5648" xml:space="preserve"> Vnde error in ſimi <lb/>litudine & diſsimilitudine:</s> <s xml:id="echoid-s5649" xml:space="preserve"> quia ſumitur iudicium ex apparẽtib.</s> <s xml:id="echoid-s5650" xml:space="preserve"> tantùm.</s> <s xml:id="echoid-s5651" xml:space="preserve"> Et in omnib, prædictis ac-<lb/>cidit error ex ſola ſoliditatis intemperantia:</s> <s xml:id="echoid-s5652" xml:space="preserve"> quoniã alijs in ſuo eſſe manentibus, non accidit error, <lb/>ea ad temperantiam reuocata.</s> <s xml:id="echoid-s5653" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div202" type="section" level="0" n="0"> <head xml:id="echoid-head231" xml:space="preserve" style="it">64. Perſpicuitas medij immoder ata creat errores in ſingulis uiſibilibus ſpeciebus. In diſtãtia: <lb/>ſitu: figura: magnitudine: diuiſione: continuatione & numero. 16. 44. 97. 28. 109 p 4.</head> <p> <s xml:id="echoid-s5654" xml:space="preserve">RAritas aeris inter uiſum & rem uiſam intercidentis egreditur temperamenti proprij metas, & <lb/>errorem generat in omnibus, quorum fidem uiſus efficit per ſyllogiſmum.</s> <s xml:id="echoid-s5655" xml:space="preserve"> In longitudine.</s> <s xml:id="echoid-s5656" xml:space="preserve"> Si <lb/>enim fueritaer pruinoſus & obſcurus, ſicut in horis matutinis ſolet accidere:</s> <s xml:id="echoid-s5657" xml:space="preserve"> turri aliqua ui-<lb/>ſui oppoſita in longitudine temperata:</s> <s xml:id="echoid-s5658" xml:space="preserve"> æſtimabitur plus à uiſu elongata, quàm habeat ueritas.</s> <s xml:id="echoid-s5659" xml:space="preserve"> Vn-<lb/>de error in longitudine eſt.</s> <s xml:id="echoid-s5660" xml:space="preserve"> Et cauſſa eſt:</s> <s xml:id="echoid-s5661" xml:space="preserve"> quoniã non comprehenditur longitudo inferioris terræ, ſu <lb/>per quã elongationis turris ſumitur menſura:</s> <s xml:id="echoid-s5662" xml:space="preserve"> & occultatur terra ex raritate aeris diminuta.</s> <s xml:id="echoid-s5663" xml:space="preserve"> Vnde <lb/>raritas eſt cauſſa erroris.</s> <s xml:id="echoid-s5664" xml:space="preserve"> Si aũt in hoc aere declinetur modicè corpus uiſum:</s> <s xml:id="echoid-s5665" xml:space="preserve"> occultabitur declina-<lb/>tio, quæ pateret in aere claro.</s> <s xml:id="echoid-s5666" xml:space="preserve"> Vnde error in ſitu.</s> <s xml:id="echoid-s5667" xml:space="preserve"> Et ſi fuerit in corpore gibboſitas modica:</s> <s xml:id="echoid-s5668" xml:space="preserve"> appare-<lb/>bit planum in tali aere:</s> <s xml:id="echoid-s5669" xml:space="preserve"> & ſi fuerint in corpore anguli, latebunt.</s> <s xml:id="echoid-s5670" xml:space="preserve"> Vnde erroneum erit figuræ iudiciũ.</s> <s xml:id="echoid-s5671" xml:space="preserve"> <lb/>In quantitate erit error extali aere:</s> <s xml:id="echoid-s5672" xml:space="preserve"> quoniã uiſum maius apparebit, quã in temperato aere:</s> <s xml:id="echoid-s5673" xml:space="preserve"> Sicut ac-<lb/>cidit in corporib poſt aquę raritatem cõprehenſis.</s> <s xml:id="echoid-s5674" xml:space="preserve"> Et ſi fuerit in corpore quaſi linea nigra:</s> <s xml:id="echoid-s5675" xml:space="preserve"> putabi-<lb/>tur eſſe partium diuiſio.</s> <s xml:id="echoid-s5676" xml:space="preserve"> Vnde error in diuiſione.</s> <s xml:id="echoid-s5677" xml:space="preserve"> Et ſi fuerint duo corpora modicũ à ſe diſiuncta:</s> <s xml:id="echoid-s5678" xml:space="preserve"> ap <lb/>parebunt in hoc aere cõtinua.</s> <s xml:id="echoid-s5679" xml:space="preserve"> Vnde error in cõtinuitate.</s> <s xml:id="echoid-s5680" xml:space="preserve"> Et ex his palã, quòd error eſt in numero.</s> <s xml:id="echoid-s5681" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div203" type="section" level="0" n="0"> <head xml:id="echoid-head232" xml:space="preserve" style="it">65. In motu: quiete: aſperitate: lenitate: raritate: denſitate: umbra: tenebris: pulchritudine: <lb/>deformitate: ſimilitudine & dißimilitudine. 138. 141. 144. 147. 150. 153 p 4.</head> <p> <s xml:id="echoid-s5682" xml:space="preserve">IN motu.</s> <s xml:id="echoid-s5683" xml:space="preserve"> Si enim in aere duo uideantur, quorum unum alio paulò uelocius moueatur:</s> <s xml:id="echoid-s5684" xml:space="preserve"> iudicabũ <lb/>tur fortaſſe æquales eſſe eorum motus:</s> <s xml:id="echoid-s5685" xml:space="preserve"> cũ in temperato aere diſcerni poſſet unius ad alium ex-<lb/>ceſſus.</s> <s xml:id="echoid-s5686" xml:space="preserve"> Et eſt error propter latẽs excrementũ uię unius ſuper uiã alterius.</s> <s xml:id="echoid-s5687" xml:space="preserve"> In quiete.</s> <s xml:id="echoid-s5688" xml:space="preserve"> Si quis enim <lb/>per talem aerem à longitudine tẽperata non tñ parua uideat aquã fluentem:</s> <s xml:id="echoid-s5689" xml:space="preserve"> aut iudicabit eam im-<lb/>motam:</s> <s xml:id="echoid-s5690" xml:space="preserve"> aut ſi fuerit fortis eius fluxus:</s> <s xml:id="echoid-s5691" xml:space="preserve"> minus, ꝗ̃ moueatur, motam.</s> <s xml:id="echoid-s5692" xml:space="preserve"> In aſperitate & læuitate.</s> <s xml:id="echoid-s5693" xml:space="preserve"> Quia in <lb/>hoc aere uidebitur aſperum læue, propter latentes aſperitatis cauſſas.</s> <s xml:id="echoid-s5694" xml:space="preserve"> Et uiſa re polita, cũ nõ diſcer <lb/>natur in ea reflexio:</s> <s xml:id="echoid-s5695" xml:space="preserve"> æſtimabitur aſpera.</s> <s xml:id="echoid-s5696" xml:space="preserve"> In umbra.</s> <s xml:id="echoid-s5697" xml:space="preserve"> Si enim poſt hunc aerem uideatur corpus album, <lb/>in quo ſint particulæ rotundæ nigræ, luce ignis in corpus illud cadente, ita tñ, ut fit interpoſitio hu-<lb/>ius aeris:</s> <s xml:id="echoid-s5698" xml:space="preserve"> apparebit in locis illis umbra, aut forſitan reputabuntur foramina uiã tenebris erũpentib.</s> <s xml:id="echoid-s5699" xml:space="preserve"> <lb/>pręſtantia.</s> <s xml:id="echoid-s5700" xml:space="preserve"> Vnde error in umbra & tenebris.</s> <s xml:id="echoid-s5701" xml:space="preserve"> Quare poſt hunc aerem corpus rarũ apparebit minus <lb/>rarũ:</s> <s xml:id="echoid-s5702" xml:space="preserve"> & forſan putabitur ſolidum.</s> <s xml:id="echoid-s5703" xml:space="preserve"> Et ita error in ſoliditate & raritate.</s> <s xml:id="echoid-s5704" xml:space="preserve"> In ſpecie & deformitate, ꝓpter <lb/>cauſſas particulares corpus decorantes, uel deformantes, in hoc aere latentes.</s> <s xml:id="echoid-s5705" xml:space="preserve"> In ſimilitudine & diſ <lb/>ſimilitudine propter particulares diuerſitatis, aut conuenientiæ cauſſas, inter duo corpora non ap-<lb/>parentes.</s> <s xml:id="echoid-s5706" xml:space="preserve"> Et in his omnibus prouenit error ex raritate aeris ſola immoderata, cum alijs immotis, in <lb/>aere temperato non accideret.</s> <s xml:id="echoid-s5707" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div204" type="section" level="0" n="0"> <head xml:id="echoid-head233" xml:space="preserve" style="it">66. Tempus immoderatum creat errores in ſingulis uiſibilibus ſpeciebus. In diſtantia: ſitu: <lb/>figura: magnitudine. 16. 44. 97. 28 p 4.</head> <p> <s xml:id="echoid-s5708" xml:space="preserve">TEmpus extra temperamenti ſui fines locatũ cauſſa eſt erroris per ſingula, quorũ fides in uiſu <lb/>ſumitur ex ſyllogiſmo.</s> <s xml:id="echoid-s5709" xml:space="preserve"> In lõgitudine.</s> <s xml:id="echoid-s5710" xml:space="preserve"> Si enim ſubitò intueatur quis aliquod remotum à turri, <lb/>quod ſtatim uiſui ſurripiatur:</s> <s xml:id="echoid-s5711" xml:space="preserve"> nõ poterit plenè diſcernere lõgitudinẽ inter illud & turrim:</s> <s xml:id="echoid-s5712" xml:space="preserve"> & iu <lb/>dicabitur forſan aut minus remotũ à turri, ꝗ̃ ſit in ueritate, aut magis.</s> <s xml:id="echoid-s5713" xml:space="preserve"> Et eſt cauſſa:</s> <s xml:id="echoid-s5714" xml:space="preserve"> quoniã in illa tẽ-<lb/>poris inſtantia nõ percipitur à uidente terra intermedia inter turrim & rem uiſam, ſecundũ quã ſu-<lb/>mitur diſtantiæ menſura:</s> <s xml:id="echoid-s5715" xml:space="preserve"> aut quoniã in breui tẽpore, nõ poterit axis uiã intermediã diſcernere.</s> <s xml:id="echoid-s5716" xml:space="preserve"> Vn-<lb/>de nec plenè cõprehendere.</s> <s xml:id="echoid-s5717" xml:space="preserve"> Et ita error in lõgitudine.</s> <s xml:id="echoid-s5718" xml:space="preserve"> In fitu.</s> <s xml:id="echoid-s5719" xml:space="preserve"> Cũ aliquid ſubit ò occurrit uiſui, & ſta <lb/>tim recedit:</s> <s xml:id="echoid-s5720" xml:space="preserve"> reputabitur forſitan rectum, declinatum, aut econtrariò.</s> <s xml:id="echoid-s5721" xml:space="preserve"> In ſigura.</s> <s xml:id="echoid-s5722" xml:space="preserve"> Si fuerit modica gib-<lb/>boſitas in re ſubitò uiſa:</s> <s xml:id="echoid-s5723" xml:space="preserve"> latebit, & putabitur res plana, aut latebunt anguli, ſi fuerint in ea.</s> <s xml:id="echoid-s5724" xml:space="preserve"> In quãti-<lb/>rate.</s> <s xml:id="echoid-s5725" xml:space="preserve"> Si quis enim titionem ardentem moueat motu citiſsimo, & intra uiam modicam, ut ſæpe ua-<lb/>dat & reuertatur per eam:</s> <s xml:id="echoid-s5726" xml:space="preserve"> apparebit uia motus ignea:</s> <s xml:id="echoid-s5727" xml:space="preserve"> quoniam motus titionis ab uno uiæ termino <lb/>ad alium fit quaſi in inſtanti.</s> <s xml:id="echoid-s5728" xml:space="preserve"> Vnde propter breuιtatem temporis non poteſt diſcerni uel quantitas <lb/>uel motus titionis.</s> <s xml:id="echoid-s5729" xml:space="preserve"> Vnde & hic error in motu.</s> <s xml:id="echoid-s5730" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div205" type="section" level="0" n="0"> <head xml:id="echoid-head234" xml:space="preserve" style="it">67. In diuiſione: continuatione: numero: quiete & motu. 109. 138 p 4.</head> <p> <s xml:id="echoid-s5731" xml:space="preserve">IN diuiſione.</s> <s xml:id="echoid-s5732" xml:space="preserve"> Si quid enim ſubitò uiſum à uiſu diuertatur, & fuerit in eo linea nigra:</s> <s xml:id="echoid-s5733" xml:space="preserve"> putabitur eſ-<lb/>ſe diuiſio partiũ, illa nigredo.</s> <s xml:id="echoid-s5734" xml:space="preserve"> Et ſi corpora contigua uel ualde propinqua ſubitò uideantur:</s> <s xml:id="echoid-s5735" xml:space="preserve"> ęſti-<lb/>mabuntur continua:</s> <s xml:id="echoid-s5736" xml:space="preserve"> ſicut accidit in ſcamnorum tabulis ſubitò inſpectis.</s> <s xml:id="echoid-s5737" xml:space="preserve"> Vnde error in cõtinui-<lb/>tate.</s> <s xml:id="echoid-s5738" xml:space="preserve"> In motu.</s> <s xml:id="echoid-s5739" xml:space="preserve"> Cum duorũ unum paulò uelocius alio mouetur:</s> <s xml:id="echoid-s5740" xml:space="preserve"> motus in tẽpore modico cõprehenſi <lb/>æquales iudicabuntur, cum non tam ſubitò cõprehenſibilis ſit exceſſus.</s> <s xml:id="echoid-s5741" xml:space="preserve"> In quiete.</s> <s xml:id="echoid-s5742" xml:space="preserve"> Si enim aliquid <lb/>modicè moueatur:</s> <s xml:id="echoid-s5743" xml:space="preserve"> ſubitò uiſum moueri nõ uidebitur:</s> <s xml:id="echoid-s5744" xml:space="preserve"> quoniã uia, quã percurrit in tẽpore ſuę perce <lb/>ptionis, imperceptibilis eſt uiſui p̃ ſui paruitate.</s> <s xml:id="echoid-s5745" xml:space="preserve"> Superius aũt explanatũ eſt, [51 n 2] quòd non com <lb/> <pb o="101" file="0107" n="107" rhead="OPTICAE LIBER III."/> prehenditur motus in corpore, niſi in tempore ſenſibili.</s> <s xml:id="echoid-s5746" xml:space="preserve"> Similis error accidit in rota modica:</s> <s xml:id="echoid-s5747" xml:space="preserve"> cum <lb/>citiſsimè uoluitur, apparet immota:</s> <s xml:id="echoid-s5748" xml:space="preserve"> cum non poſsit fieri comprehenſio reuolutionis eius in tempo <lb/>re tam paruo, quàm paruum eſt, in quo fit una eius reuolutio.</s> <s xml:id="echoid-s5749" xml:space="preserve"> Idem error accidit in trocho.</s> <s xml:id="echoid-s5750" xml:space="preserve"> Vnde er <lb/>ror in quiete:</s> <s xml:id="echoid-s5751" xml:space="preserve"> quoniam non poteſt diſcerni mutatio ſitus partium trochi:</s> <s xml:id="echoid-s5752" xml:space="preserve"> quare nec motus eius.</s> <s xml:id="echoid-s5753" xml:space="preserve"> Et ſi <lb/>unius coloris fuerit trochus:</s> <s xml:id="echoid-s5754" xml:space="preserve"> palàm, quòd non comprehenditur motus.</s> <s xml:id="echoid-s5755" xml:space="preserve"> Si uerò plurium & diuerſo-<lb/>rum colorum:</s> <s xml:id="echoid-s5756" xml:space="preserve"> nec ſic etiam apparebit motus:</s> <s xml:id="echoid-s5757" xml:space="preserve"> cum lateat colorum diuerſitas, & prætendatur ex ni-<lb/>mia feſtinatione, confuſa quædam colorum unitas.</s> <s xml:id="echoid-s5758" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div206" type="section" level="0" n="0"> <head xml:id="echoid-head235" xml:space="preserve" style="it">68. In aſperitate: lenitate: raritate: denſitate: umbra: tenebris: pulchritudine: deformitate: <lb/>ſimilitudine: dißimilitudine. 141. 144. 147. 150. 153 p 4.</head> <p> <s xml:id="echoid-s5759" xml:space="preserve">IN aſperitate.</s> <s xml:id="echoid-s5760" xml:space="preserve"> Cum enim ſubitò uidetur aſperum:</s> <s xml:id="echoid-s5761" xml:space="preserve"> putabitur forſitan læue:</s> <s xml:id="echoid-s5762" xml:space="preserve"> & ſi hoc modo uidea-<lb/>tur, non poterit in eo diſcerni læuitas aut aſperitas.</s> <s xml:id="echoid-s5763" xml:space="preserve"> Vnde dubitatio & error.</s> <s xml:id="echoid-s5764" xml:space="preserve"> In raritate.</s> <s xml:id="echoid-s5765" xml:space="preserve"> Luce e-<lb/>nim declinata ſuper corpus remiſsè rarũ deſcendente, ſubitò uiſum, cũ non percipiatur declina-<lb/>tio lucis:</s> <s xml:id="echoid-s5766" xml:space="preserve"> putabitur forſitan, quod in fine raritatis ſit apparẽs raritas corporis.</s> <s xml:id="echoid-s5767" xml:space="preserve"> Quòd ſi in tẽpore pau <lb/>lo maiore adhibeatur uiſus:</s> <s xml:id="echoid-s5768" xml:space="preserve"> percipietur declinatio cauſſa apparentiæ raritatis remiſſę.</s> <s xml:id="echoid-s5769" xml:space="preserve"> In ſoliditate.</s> <s xml:id="echoid-s5770" xml:space="preserve"> <lb/>Si quis enim inſtanter uideat corpus rarũ, & poſt ipſum nõ diſcernat lucis tranſitũ, putabit illud eſ-<lb/>ſe ſolidũ.</s> <s xml:id="echoid-s5771" xml:space="preserve"> In umbra.</s> <s xml:id="echoid-s5772" xml:space="preserve"> Si in albo pariete ſint partes ſubnigrę, deſcendẽte ſuper ipſum ignis luce, ſubitò <lb/>uiſæ putabuntur eſſe umbræ.</s> <s xml:id="echoid-s5773" xml:space="preserve"> Si uerò nigredo earum uiſa fuerit intenſa:</s> <s xml:id="echoid-s5774" xml:space="preserve"> æſtimabuntur foramma te-<lb/>nebris plena.</s> <s xml:id="echoid-s5775" xml:space="preserve"> In ſpecie & deformitate.</s> <s xml:id="echoid-s5776" xml:space="preserve"> Quia in tã paruo tẽpore non ſunt cõprehenſibiles minutę de-<lb/>coris & deformitatis cauſſæ:</s> <s xml:id="echoid-s5777" xml:space="preserve"> ſicut accidit cũ aliquis inſpiciens per foramen intuetur faciem, iudicat <lb/>aliquando fœdam:</s> <s xml:id="echoid-s5778" xml:space="preserve"> formoſam:</s> <s xml:id="echoid-s5779" xml:space="preserve"> uel econtrariò.</s> <s xml:id="echoid-s5780" xml:space="preserve"> Et idẽ error accidit mota re uiſa, oculo immoto.</s> <s xml:id="echoid-s5781" xml:space="preserve"> In ſi-<lb/>militudine & diſsimilitudine:</s> <s xml:id="echoid-s5782" xml:space="preserve"> Quoniam latent particulares ſimilitudinis & diſsimilitudinis cauſſę.</s> <s xml:id="echoid-s5783" xml:space="preserve"> <lb/>Et in his omnibus ex ſolo tempore non moderato accidit error:</s> <s xml:id="echoid-s5784" xml:space="preserve"> cum in prædictis nullus accideret, <lb/>eo ad temperantiam reducto.</s> <s xml:id="echoid-s5785" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div207" type="section" level="0" n="0"> <head xml:id="echoid-head236" xml:space="preserve" style="it">69. Imbecillit{as} uiſus creat errores in ſingulis uiſibilibus ſpeciebus. In diſtantia: ſitu: magni-<lb/>tudine: figura: diuiſione: continuatione: numero. 16. 44. 28. 97. 109 p 4.</head> <p> <s xml:id="echoid-s5786" xml:space="preserve">VIſus debilitas & immoderatio errorẽ inuehit ſingulis per ſyllo giſmũ in uiſu comprehẽſis.</s> <s xml:id="echoid-s5787" xml:space="preserve"> In <lb/>lõgitudine.</s> <s xml:id="echoid-s5788" xml:space="preserve"> Si enim opponãtur uiſui duo corpora, quorum unũ ſit coloris fortis, & remotius:</s> <s xml:id="echoid-s5789" xml:space="preserve"> <lb/>aliud coloris debilis, & oculo propinquius:</s> <s xml:id="echoid-s5790" xml:space="preserve"> cũ nõ fiat cõprehenſio longirudinis, niſi facta col <lb/>latione ad aliqua corpora interiecta:</s> <s xml:id="echoid-s5791" xml:space="preserve"> [per 25.</s> <s xml:id="echoid-s5792" xml:space="preserve"> 39 n 2] faciet incertã collationẽ debilitas uiſus.</s> <s xml:id="echoid-s5793" xml:space="preserve"> Et quia <lb/>certum eſt homini, quòd ex locis propin quiorib.</s> <s xml:id="echoid-s5794" xml:space="preserve"> certior fit fides uiſui, ꝗ̃ ex remotiorib.</s> <s xml:id="echoid-s5795" xml:space="preserve"> concludit il-<lb/>lud, quod apparet ei certius ex his corporib eſſe propinquius.</s> <s xml:id="echoid-s5796" xml:space="preserve"> Et planum, quòd uiſui debili certior <lb/>fit fides coloris fortis, quàm debilis:</s> <s xml:id="echoid-s5797" xml:space="preserve"> licet paulò plus elon gati.</s> <s xml:id="echoid-s5798" xml:space="preserve"> Idem error accidit etiã in tẽperantia <lb/>uiſus:</s> <s xml:id="echoid-s5799" xml:space="preserve"> quoniã à longitudine magna propinquius iudicatur corpus, cuius color fortis, quã cuius co-<lb/>lor debilis:</s> <s xml:id="echoid-s5800" xml:space="preserve"> licet nõ ſit multò remotius.</s> <s xml:id="echoid-s5801" xml:space="preserve"> In ſitu errat uiſus debilitas.</s> <s xml:id="echoid-s5802" xml:space="preserve"> Si enim ab aliquãta longitudine, <lb/>licet temperata declinetur corpus, & ſit modica declinatio:</s> <s xml:id="echoid-s5803" xml:space="preserve"> ignorabitur, cũ plenè comprehenditur <lb/>lõgitudo.</s> <s xml:id="echoid-s5804" xml:space="preserve"> Et incertitudo longitudinis quãtitatis, errorẽ etiã ſitus ingerit.</s> <s xml:id="echoid-s5805" xml:space="preserve"> In figura.</s> <s xml:id="echoid-s5806" xml:space="preserve"> Quia gibbus mo <lb/>dicus, & multiplex angulus latent debilitatẽ uiſus.</s> <s xml:id="echoid-s5807" xml:space="preserve"> Et ſi in corpore linea nigra fuerit:</s> <s xml:id="echoid-s5808" xml:space="preserve"> æſtimabitur di <lb/>uiſio uel fiſſura:</s> <s xml:id="echoid-s5809" xml:space="preserve"> & æſtimabuntur corpora contigua, unũ continuum.</s> <s xml:id="echoid-s5810" xml:space="preserve"> Vnde error in diuiſione:</s> <s xml:id="echoid-s5811" xml:space="preserve"> conti-<lb/>nuitate:</s> <s xml:id="echoid-s5812" xml:space="preserve"> numero.</s> <s xml:id="echoid-s5813" xml:space="preserve"> Eadem erroris cauſſa ſtrabo unum iudicat duo:</s> <s xml:id="echoid-s5814" xml:space="preserve">ſi fuerit deformitas in uno tantùm <lb/>oculo.</s> <s xml:id="echoid-s5815" xml:space="preserve"> Quoniam habet res uiſa diuerſitatem ſitus, reſpectu duorum oculorum eius.</s> <s xml:id="echoid-s5816" xml:space="preserve"> Si aũt in duo-<lb/>bus oculis eius ſit deformatio:</s> <s xml:id="echoid-s5817" xml:space="preserve"> cum acciditeos moueri forſitan accidet eis diuerſitas ſitus, reſpectu <lb/>rei uiſæ:</s> <s xml:id="echoid-s5818" xml:space="preserve"> & ita in uno pluralitas.</s> <s xml:id="echoid-s5819" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div208" type="section" level="0" n="0"> <head xml:id="echoid-head237" xml:space="preserve" style="it">70. In motu & quiete. 138 p 4.</head> <p> <s xml:id="echoid-s5820" xml:space="preserve">IN motu.</s> <s xml:id="echoid-s5821" xml:space="preserve"> Si quis enim ſæpius in circuitũ uoluitur, cũ quieſcit:</s> <s xml:id="echoid-s5822" xml:space="preserve"> putat, quòd parietes moueãtur.</s> <s xml:id="echoid-s5823" xml:space="preserve"> Et <lb/>eſt, quoniam moto uidente, mouetur intrinſecus uis uiſibilis:</s> <s xml:id="echoid-s5824" xml:space="preserve"> & licet uidẽs ſteterit, nõ ſtatim uis <lb/>uiſibilis ſtabit:</s> <s xml:id="echoid-s5825" xml:space="preserve"> ſed motus eius in uidentis quiete durabit:</s> <s xml:id="echoid-s5826" xml:space="preserve"> & ob hoc motus uiſarum rerũ æſtima-<lb/>tio inſurgit.</s> <s xml:id="echoid-s5827" xml:space="preserve"> Et huius motus exemplum in trocho uidemus:</s> <s xml:id="echoid-s5828" xml:space="preserve"> quoniã diu poſt manus mouentis quie <lb/>tem uoluitur trochus.</s> <s xml:id="echoid-s5829" xml:space="preserve"> Eſt etiam infirmitas, in qua uidentur patienti omnia uolui.</s> <s xml:id="echoid-s5830" xml:space="preserve"> In quiete.</s> <s xml:id="echoid-s5831" xml:space="preserve"> Quan-<lb/>do corpus ſimilium partium, ut ſunt quædã rotæ horologiorum, reuoluitur reuolutione pauca:</s> <s xml:id="echoid-s5832" xml:space="preserve"> ui-<lb/>ſus debilis non percipit eius motum, quẽ quidem perciperet uiſus temperatus.</s> <s xml:id="echoid-s5833" xml:space="preserve"> Si aũt multa ſit reuo <lb/>lutio, non percipitur etiam à temperato.</s> <s xml:id="echoid-s5834" xml:space="preserve"> Si uerò ſit diſsimiliũ partium corpus motũ, ut in rota mo-<lb/>letrinæ:</s> <s xml:id="echoid-s5835" xml:space="preserve"> tunc uiſus debilis comprehendet motum:</s> <s xml:id="echoid-s5836" xml:space="preserve"> Si autem feſtina fueritrotæreuolutio:</s> <s xml:id="echoid-s5837" xml:space="preserve"> occultabi <lb/>tur uiſui debili motus.</s> <s xml:id="echoid-s5838" xml:space="preserve"> Quoniã partes rotæ multũ diſsimiles ſunt:</s> <s xml:id="echoid-s5839" xml:space="preserve"> non plenè comprehendetur diſsi <lb/>militudo in feſtinatione:</s> <s xml:id="echoid-s5840" xml:space="preserve"> & per diſs imilitudinem partium fit comprehenſio motus earum.</s> <s xml:id="echoid-s5841" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div209" type="section" level="0" n="0"> <head xml:id="echoid-head238" xml:space="preserve" style="it">71. In aſperitate: lenitate: raritate: denſitate: umbra: tenebris: pulchritudine: deformitate: <lb/>ſimilitudine: dißimilitudine. 141. 144. 147. 150. 153 p 4.</head> <p> <s xml:id="echoid-s5842" xml:space="preserve">IN aſperitate & læuitate.</s> <s xml:id="echoid-s5843" xml:space="preserve"> Quia forſan reputabit modicè lęue, aſperũ:</s> <s xml:id="echoid-s5844" xml:space="preserve"> uel ecõtrariò, ſi inter formas <lb/>aſperi & lęuis fuerit diſsimilitudo.</s> <s xml:id="echoid-s5845" xml:space="preserve"> In raritate.</s> <s xml:id="echoid-s5846" xml:space="preserve"> Cũ fuerit in corpore raro ſoliditas pauca:</s> <s xml:id="echoid-s5847" xml:space="preserve"> æſtimabi <lb/>tur à uiſu debili maior uera.</s> <s xml:id="echoid-s5848" xml:space="preserve"> In ſoliditate.</s> <s xml:id="echoid-s5849" xml:space="preserve"> Si fuerit in corpore raro color fortis, aut poſt ipſum, & <lb/>raritas nõ maxima:</s> <s xml:id="echoid-s5850" xml:space="preserve"> putabit illud eſſe ſolidũ.</s> <s xml:id="echoid-s5851" xml:space="preserve"> In umbra.</s> <s xml:id="echoid-s5852" xml:space="preserve"> Notæ parietis ſubnigræ, deſcẽdẽte ſuք ipſum <lb/>luce, apparẽt huic uiſui umbrę:</s> <s xml:id="echoid-s5853" xml:space="preserve"> & ſi fuerintmultũ nigrę:</s> <s xml:id="echoid-s5854" xml:space="preserve"> apparebũt foramina, in quibus tenebrę.</s> <s xml:id="echoid-s5855" xml:space="preserve"> In <lb/> <pb o="102" file="0108" n="108" rhead="ALHAZEN"/> deformitate & decore:</s> <s xml:id="echoid-s5856" xml:space="preserve"> ſimilitudine & diſsimilitudine ꝓpter particulares decoris uel fœditatis, ſimi <lb/>litudinis & diſsimilitudinis cauſſas uiſũ latẽtes.</s> <s xml:id="echoid-s5857" xml:space="preserve"> Et eſt error in p̃ dictis omnib.</s> <s xml:id="echoid-s5858" xml:space="preserve"> ex ſola debilitate uiſ 9.</s> <s xml:id="echoid-s5859" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div210" type="section" level="0" n="0"> <head xml:id="echoid-head239" xml:space="preserve" style="it">72. In uiſione errores creantur aliàs quidem à ſingulis uiſionẽ perficientibus: aliàs uerò à plu-<lb/>ribus ſimul, quorum nullum per ſe errorem crearet. 154 p 4.</head> <p> <s xml:id="echoid-s5860" xml:space="preserve">IAm diximus, quomodo accidat error in ſyllogiſmo, ſecundum unamquãq;</s> <s xml:id="echoid-s5861" xml:space="preserve"> cauſſarum erroris ui <lb/>ſus in qualibet partium, quæ acquiruntur per ſyllogiſmum, & inceſsimus ſuper quemlibet erro-<lb/>ris modum, & cuiuslibet ſuppoſuimus exemplum.</s> <s xml:id="echoid-s5862" xml:space="preserve"> Et licet in errorib.</s> <s xml:id="echoid-s5863" xml:space="preserve"> uiſus ſit copioſa multitu-<lb/>do:</s> <s xml:id="echoid-s5864" xml:space="preserve"> tñ omniũ ad modos dictos fiet reductio, & ad exẽpla ordinatim propoſita:</s> <s xml:id="echoid-s5865" xml:space="preserve"> aſsignauimus quoq;</s> <s xml:id="echoid-s5866" xml:space="preserve"> <lb/>errores, ſecundum quod ſinguli eorum accidunt ab unica tantũ cauſſa.</s> <s xml:id="echoid-s5867" xml:space="preserve"> Et aliquan do error infertur <lb/>non ab una tantùm, ſed à duabus cauſsis uel plurib.</s> <s xml:id="echoid-s5868" xml:space="preserve"> Verbi gratia.</s> <s xml:id="echoid-s5869" xml:space="preserve"> Simoueatur aliquid à longitudi-<lb/>ne magna motulento:</s> <s xml:id="echoid-s5870" xml:space="preserve"> ſubitò uiſum uidebitur immotũ:</s> <s xml:id="echoid-s5871" xml:space="preserve"> & percipi poſſet motus ille in diſtantia tem <lb/>perata etiam celeri uiſu, uel etiã in illa longitudine intemperata non occultaretur motus:</s> <s xml:id="echoid-s5872" xml:space="preserve"> ſi tẽpera-<lb/>tum eſſet in ſpectionis tẽpus.</s> <s xml:id="echoid-s5873" xml:space="preserve"> Prouenit igitur error ex duabus intemperantijs, quarum neutra per ſe <lb/>ſufficit:</s> <s xml:id="echoid-s5874" xml:space="preserve"> triũ aggregatio errorem efficit.</s> <s xml:id="echoid-s5875" xml:space="preserve"> Si à magna lõgitudine, ſub debili luce, in modico tẽpore, op-<lb/>ponatur uiſui corporis diuerſorum colorũ reuolutio non cita:</s> <s xml:id="echoid-s5876" xml:space="preserve"> æſtimabitur corpus ſtare:</s> <s xml:id="echoid-s5877" xml:space="preserve"> Et ſi ab ea-<lb/>dem longitudine, ſub eadem luce, tempore temperato, adhibeatur intuitus:</s> <s xml:id="echoid-s5878" xml:space="preserve"> comprehendetur mo-<lb/>tus:</s> <s xml:id="echoid-s5879" xml:space="preserve"> qui ſimiliter non latebit in tẽperata longitudine, ſub eadem luce & modico tẽpore:</s> <s xml:id="echoid-s5880" xml:space="preserve"> & etiã perci <lb/>pi poterit in eadẽ longitudine ſub fortiluce.</s> <s xml:id="echoid-s5881" xml:space="preserve"> Et generaliter ex omnib.</s> <s xml:id="echoid-s5882" xml:space="preserve"> errorib.</s> <s xml:id="echoid-s5883" xml:space="preserve"> uiſui accidentib.</s> <s xml:id="echoid-s5884" xml:space="preserve"> nec <lb/>unus nec plures congregati euadũt cauſſas, quas diximus.</s> <s xml:id="echoid-s5885" xml:space="preserve"> Quælibet aũt forma rei uiſæ, ex ijs, quæ <lb/>enumerauimus, eſt cõpoſita.</s> <s xml:id="echoid-s5886" xml:space="preserve"> Et cum uiſus non ac quirat ex rebus uiſis, niſi aliquas iſtarũ:</s> <s xml:id="echoid-s5887" xml:space="preserve"> non acci-<lb/>dit error in uiſu, niſi in aliqua iſtarũ.</s> <s xml:id="echoid-s5888" xml:space="preserve"> Et omnis error, qui accidit in ſcientia, eſt, quoniam intellectus <lb/>ſimilia efficit, quæ percipit, cũ ijs, quę percepit in modo aliquo, aut diſsimilia.</s> <s xml:id="echoid-s5889" xml:space="preserve"> Et omnis error in par <lb/>tialibus erit, aut in ſenſu, aut in ſcientia, aut in ſyllogiſmo:</s> <s xml:id="echoid-s5890" xml:space="preserve"> & non poteſt eſſe, quin ſit in aliquo iſtorũ, <lb/>aut duobus, autipſis tribus.</s> <s xml:id="echoid-s5891" xml:space="preserve"> Et quicunque error accidit in huiuſmodi tribus, non erit, niſi per erro-<lb/>rem uiſus in partibus.</s> <s xml:id="echoid-s5892" xml:space="preserve"> Et iam patuit, quòd error uiſus in partialib.</s> <s xml:id="echoid-s5893" xml:space="preserve"> non erit, niſi propter cauſſas, quas <lb/>aſsignauimus, aut ex una earum tantùm, aut ex pluribus.</s> <s xml:id="echoid-s5894" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div211" type="section" level="0" n="0"> <head xml:id="echoid-head240" xml:space="preserve">ALHAZEN FILII</head> <head xml:id="echoid-head241" xml:space="preserve">ALHAYZEN OPTICAE</head> <head xml:id="echoid-head242" xml:space="preserve">LIBER QVARTVS.</head> <p style="it"> <s xml:id="echoid-s5895" xml:space="preserve">LIBER iſte diuiditur in quinque partes.</s> <s xml:id="echoid-s5896" xml:space="preserve"> Pars prima eſt proæmium libri.</s> <s xml:id="echoid-s5897" xml:space="preserve"> Se-<lb/>cundaest in declaratione, quòd luci accidat reflexio à politis corporib.</s> <s xml:id="echoid-s5898" xml:space="preserve"> Tertia <lb/>est in modo reflexionis formæ.</s> <s xml:id="echoid-s5899" xml:space="preserve"> Quarta in oſtenſione, quòd comprehenſio for <lb/>mæ ex corporibus politis non est, niſi ex reflexione.</s> <s xml:id="echoid-s5900" xml:space="preserve"> Quinta est in modo comprehenſio-<lb/>nis formarum per reflexionem.</s> <s xml:id="echoid-s5901" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div212" type="section" level="0" n="0"> <head xml:id="echoid-head243" xml:space="preserve">PROOEMIVM LIBRI. CAP. I.</head> <head xml:id="echoid-head244" xml:space="preserve" style="it">1. Viſio fit trifariam: rectè: reflexè: & refr actè. In præf. 1. 3. 10 Libr.</head> <p> <s xml:id="echoid-s5902" xml:space="preserve">IAm explanauimus in libris tribus modũ comprehenſionis formarũ in uiſu, cum fuerit directus:</s> <s xml:id="echoid-s5903" xml:space="preserve"> <lb/>& enumerauimus ſingula, quæ in rebus uiſis cõprehendit uiſus.</s> <s xml:id="echoid-s5904" xml:space="preserve"> Sed diuerſificatur acquiſitio ui-<lb/>ſus tripliciter:</s> <s xml:id="echoid-s5905" xml:space="preserve"> Aut enim directè, ſicut diximus:</s> <s xml:id="echoid-s5906" xml:space="preserve"> aut per reflexionem in politis corporibus:</s> <s xml:id="echoid-s5907" xml:space="preserve"> aut per <lb/>penetrationem, ut in raris, quorum non eſtraritas, ſicut raritas aeris.</s> <s xml:id="echoid-s5908" xml:space="preserve"> Et non poteſt diuerſificari ui-<lb/>ſus, niſi in his modis tribus.</s> <s xml:id="echoid-s5909" xml:space="preserve"> Et his duobus modis poſterioribus comprehendit uiſus in rebus uiſis, <lb/>quæ ſuprà expoſuimus, & quorum acquiſitionẽ in uiſu directo patefecimus.</s> <s xml:id="echoid-s5910" xml:space="preserve"> Et forſitan uiſus in his <lb/>incurrit in errorem, aut conſequitur ueritatem.</s> <s xml:id="echoid-s5911" xml:space="preserve"> Et nos aſsign abimus in hoc libro, quando per refle-<lb/>xionem fiat formarum acquiſitio:</s> <s xml:id="echoid-s5912" xml:space="preserve"> & quomodo erit reflexio:</s> <s xml:id="echoid-s5913" xml:space="preserve"> & quis linearũ reflexarum ſitus:</s> <s xml:id="echoid-s5914" xml:space="preserve"> Et præ <lb/>ponemus quædam accidentia præponenda.</s> <s xml:id="echoid-s5915" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div213" type="section" level="0" n="0"> <head xml:id="echoid-head245" xml:space="preserve">QVOD LVCI ACCIDAT REFLEXIO À<unsure/> POLITIS <lb/>corporibus. Cap. II.</head> <head xml:id="echoid-head246" xml:space="preserve" style="it">2. Lux & color reflectuntur à quolibet politæ ſuperficiei puncto, lineis rectis. 1 p 5.</head> <p> <s xml:id="echoid-s5916" xml:space="preserve">PLanum eſt ex libro primo [1.</s> <s xml:id="echoid-s5917" xml:space="preserve"> 2.</s> <s xml:id="echoid-s5918" xml:space="preserve"> 3.</s> <s xml:id="echoid-s5919" xml:space="preserve"> 14.</s> <s xml:id="echoid-s5920" xml:space="preserve"> 18.</s> <s xml:id="echoid-s5921" xml:space="preserve"> 19 n] quòd lux à corpore lucido luce propria uel accidẽ-<lb/>tali dirigatur in omne corpus ei oppoſitum:</s> <s xml:id="echoid-s5922" xml:space="preserve"> & eodem modo color, cũ in eo lux fuerit, mittitur-<lb/>Itaq;</s> <s xml:id="echoid-s5923" xml:space="preserve"> corpore polito oppoſito corpori lucido, mittitur ad ipſum lux mixtim cũ colore, & refle-<lb/>ctitur lux cũ colore, ſiue fuerit fortis, ſiue debilis, ſiue prima, ſiue ſecundaria.</s> <s xml:id="echoid-s5924" xml:space="preserve"> Et quòd fiat in luce for <lb/>ti reflexio, poteſt patére:</s> <s xml:id="echoid-s5925" xml:space="preserve"> oppoſito luci forti ſpeculo ferreo, ſi oppoſitus fuerit paries ſpeculo, & de-<lb/>ſcẽderit ſuք ipſum lux declinata, nõ recta:</s> <s xml:id="echoid-s5926" xml:space="preserve"> uidebitur in pariete lux fortis reflexa:</s> <s xml:id="echoid-s5927" xml:space="preserve"> quę ꝗdẽ nõ uidebi-<lb/>tur ſuper eun dem locum, ſi ſpeculũ auferatur uel moueatur:</s> <s xml:id="echoid-s5928" xml:space="preserve"> imò ſecũdum motum ſpeculi mutabi-<lb/> <pb o="103" file="0109" n="109" rhead="OPTICAE LIBER IIII."/> tur locus lucis reflexæ in pariete.</s> <s xml:id="echoid-s5929" xml:space="preserve"> Quare palàm, reflexionem fieri in luce forti.</s> <s xml:id="echoid-s5930" xml:space="preserve"> In luce debili patére <lb/>poteſt facile.</s> <s xml:id="echoid-s5931" xml:space="preserve"> Si intra domũ aliquã, per foramen unicũ à terra elongatũ, ſed non multùm, deſcendat <lb/>lux diei, non ſolis, ſuper aliquod corpus:</s> <s xml:id="echoid-s5932" xml:space="preserve"> & circa corpus ſtatuatur ſpeculũ ferreum:</s> <s xml:id="echoid-s5933" xml:space="preserve"> & circa ſpeculũ <lb/>corpus aliquod albũ:</s> <s xml:id="echoid-s5934" xml:space="preserve"> apparebit in ſecun do corpore albo lux maior quàm ſine ſpeculo:</s> <s xml:id="echoid-s5935" xml:space="preserve"> & augmen-<lb/>tũ illius nõ eſt, niſi ex ſpeculi reflexione, quoniã ablato ſpeculo ſola lux ſecundaria debilis appare-<lb/>bit in corpore albo.</s> <s xml:id="echoid-s5936" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s5937" xml:space="preserve"> ſi diligens figatur intuitus in lineis, per quas à corpore primo lux in <lb/>ſpeculũ mittitur:</s> <s xml:id="echoid-s5938" xml:space="preserve"> perpendetur quidẽ linearũ illarũ declinatio ſuper ſpeculũ, & ſuper idẽ linearum <lb/>punctũ, reflexionis declinatio eadẽ.</s> <s xml:id="echoid-s5939" xml:space="preserve"> Et eſt propriũ reflexionis, ut ſit eadẽ declinatio, & idẽ angulus <lb/>linearũ uenientiũ & reflexarũ.</s> <s xml:id="echoid-s5940" xml:space="preserve"> Quòd ſi moueatur corpus albũ à loco reflexionis in aliũ locũ:</s> <s xml:id="echoid-s5941" xml:space="preserve"> tamẽ <lb/>circa ſpeculũ:</s> <s xml:id="echoid-s5942" xml:space="preserve"> nõ uidebitur in eo lucis augmentũ:</s> <s xml:id="echoid-s5943" xml:space="preserve"> nec uideri poterit, niſi in illo ſitu tantùm.</s> <s xml:id="echoid-s5944" xml:space="preserve"> Quare <lb/>planũ eſt propriũ eſſe reflexionis hunc ſitũ.</s> <s xml:id="echoid-s5945" xml:space="preserve"> Hoc idẽ poterit uideri in ſecundaria luce:</s> <s xml:id="echoid-s5946" xml:space="preserve"> ſi prædictum <lb/>ſpeculũ ſit argenteum, & corpus tertiũ albũ ſit ex alia parte ſpeculi:</s> <s xml:id="echoid-s5947" xml:space="preserve"> apparebit quidẽ ſuper corpus <lb/>tertium lux ſecundaria, & ſuper corpus ſecundũ lux maior illa:</s> <s xml:id="echoid-s5948" xml:space="preserve"> Et palàm, huius maioritatis cauſſam <lb/>ſolã eſſe reflexionẽ.</s> <s xml:id="echoid-s5949" xml:space="preserve"> Patebit aũt in omni loco lucis reflexio:</s> <s xml:id="echoid-s5950" xml:space="preserve"> ubi ſuper corpus deſcendit per foramẽ <lb/>aliquod lux fortis, adhibito luci ſpeculo, & ei corpore albo oppoſito, modo ſuprà dicto.</s> <s xml:id="echoid-s5951" xml:space="preserve"> Verùm lo-<lb/>cũ reflexionis & linearũ ſitũ explanabimus.</s> <s xml:id="echoid-s5952" xml:space="preserve"> Iã patuit in libro primo, [1.</s> <s xml:id="echoid-s5953" xml:space="preserve"> 2.</s> <s xml:id="echoid-s5954" xml:space="preserve"> 3.</s> <s xml:id="echoid-s5955" xml:space="preserve"> 14.</s> <s xml:id="echoid-s5956" xml:space="preserve"> 18.</s> <s xml:id="echoid-s5957" xml:space="preserve"> 19 n] quòd lux re-<lb/>flexa ſequitur rectitudinem linearum:</s> <s xml:id="echoid-s5958" xml:space="preserve"> quare ex corporibus politis fit reflexio ſecũdum proceſſum <lb/>rectitudinis in ſitu proprio.</s> <s xml:id="echoid-s5959" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div214" type="section" level="0" n="0"> <head xml:id="echoid-head247" xml:space="preserve" style="it">3. Lux & color à quolibet ſuperficiei coloratæ puncto ad quodlibet ſuperficiei politæ oppoſitæ <lb/>punctum permixti confluunt. 2 p 5.</head> <p> <s xml:id="echoid-s5960" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s5961" xml:space="preserve"> Planum eſt ex ſuperioribus, quòd lux ſecunda à corpore illuminato, accidentali lu-<lb/>ce procèdẽs, ſecũ fert colorẽ corporis.</s> <s xml:id="echoid-s5962" xml:space="preserve"> Ab omni igitur corpore illuminato ſeu lucido color <lb/>mixtim cũ luce ad corpora oppoſita polita mittitur, & mixtim in partẽ debitã reflectitur.</s> <s xml:id="echoid-s5963" xml:space="preserve"> Et <lb/>huic rei fides poterit fieri, ſi intra domũ unius foraminis tãtùm, deſcendat lux ſuper corpus forti & <lb/>ſpecioſo colore:</s> <s xml:id="echoid-s5964" xml:space="preserve"> & ſtatuatur circa ipſum ſpeculũ ferreũ, & circa ſpeculũ corpus concauũ ad ſcyphi <lb/>modũ, intra quod ſit corpus albũ, & aptetur hoc uas in loco reflexionis, ut lux reflexa incidat in cor <lb/>pus albũ:</s> <s xml:id="echoid-s5965" xml:space="preserve"> apparebit quidẽ ſuper faciẽ albi corporis color illius, in qđ fit prim ò deſcenſus lucis:</s> <s xml:id="echoid-s5966" xml:space="preserve"> qđ <lb/>quidẽ nõ accidet, ſi extra propriũ ſitũ reflexiõis ſtatuatur corpus albũ.</s> <s xml:id="echoid-s5967" xml:space="preserve"> Et ſecundũ diuerſas colorũ <lb/>ſpecies hoc ꝓbatũ inuenies, uelut in colore cœleſti, rubore, uiriditate, & huiuſmodi.</s> <s xml:id="echoid-s5968" xml:space="preserve"> Quare planũ, <lb/>colorẽ mixtũ cũ luce remitti, & certiorẽ eſſe coloris reflexi apparẽtiã, ſi ſpeculũ fuerit argenteum.</s> <s xml:id="echoid-s5969" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div215" type="section" level="0" n="0"> <head xml:id="echoid-head248" xml:space="preserve" style="it">4. Reflexio debilit at lucem & colorem: & omnino totam uiſibilis ſpeciem. 3 p 5.</head> <p> <s xml:id="echoid-s5970" xml:space="preserve">QVare autẽ nõ appareat hæc probatio, ſcilicet, quòd cõprehẽdatur color reflexus, cuicunq;</s> <s xml:id="echoid-s5971" xml:space="preserve"> <lb/>corpori opponatur ſpeculũ, ſed ei adhibeatur albũ, hæc eſt ratio:</s> <s xml:id="echoid-s5972" xml:space="preserve"> ſicut ſuprà dictũ eſt:</s> <s xml:id="echoid-s5973" xml:space="preserve"> colo-<lb/>res debiles (licet ſimul cũ luce mittãtur) nõ ſentiuntur.</s> <s xml:id="echoid-s5974" xml:space="preserve"> Formæ enim, quæ reflectũtur, de-<lb/>biliores ſunt formis, à quibus reflexio oritur.</s> <s xml:id="echoid-s5975" xml:space="preserve"> Et hoc in hac luce poteſt patére.</s> <s xml:id="echoid-s5976" xml:space="preserve"> Quoniã luce forti in <lb/>ſpeculum cadẽte, & reflexa in pariete:</s> <s xml:id="echoid-s5977" xml:space="preserve"> debilior uidebitur lux parietis, quàm ſpeculi, & notabilis eſt <lb/>inter eas proportio.</s> <s xml:id="echoid-s5978" xml:space="preserve"> Idẽ patebit in luce debili pari modo, ut in domo in prima diſpoſitiõe:</s> <s xml:id="echoid-s5979" xml:space="preserve"> ſi corpus <lb/>terſum tertiũ albũ ponamus loco ſpeculi ferrei, uel circa ipſum:</s> <s xml:id="echoid-s5980" xml:space="preserve"> maior apparebit lux ſuper hoc cor-<lb/>pus, quàm ſuper ſecũdũ:</s> <s xml:id="echoid-s5981" xml:space="preserve"> quod nõ accideret, niſi reflexio lucẽ debilitaret.</s> <s xml:id="echoid-s5982" xml:space="preserve"> Sed dicet aliquis, cauſſam <lb/>huius rei eſſe nigredinẽ ſpeculi ferrei, quæ admixta luci, in ſpeculũ cadenti, ipſam obumbrat, & re-<lb/>flexa in corpus ſecũdũ debilis & fuſca apparet:</s> <s xml:id="echoid-s5983" xml:space="preserve"> ſed in corpus tertiũ loco ſpeculi poſitũ nõ deſcẽdit <lb/>lux, niſi à corpore primo nulli admixta nigredini.</s> <s xml:id="echoid-s5984" xml:space="preserve"> Verùm quòd hoc nõ ſit in cauſſa, palàm ex eo eſt:</s> <s xml:id="echoid-s5985" xml:space="preserve"> <lb/>quòd loco ſpeculi ferrei, argenteo poſito, eadẽ accidit probatio.</s> <s xml:id="echoid-s5986" xml:space="preserve"> Pari modo reflexus color debilior <lb/>erit colore, à quo fit reflexio:</s> <s xml:id="echoid-s5987" xml:space="preserve"> quod in domo & uaſe, ut antea, patére poteſt:</s> <s xml:id="echoid-s5988" xml:space="preserve"> ſi corpus albũ loco ſpe-<lb/>culi ponatur, uel circa:</s> <s xml:id="echoid-s5989" xml:space="preserve"> fortior apparebit in ipſo color, quã in corpore albo intra uas poſito.</s> <s xml:id="echoid-s5990" xml:space="preserve"> Et idem <lb/>patebit, ſi loco ferrei ſpeculi argenteũ ponatur ſpeculũ.</s> <s xml:id="echoid-s5991" xml:space="preserve"> Igitur reflexio debilitat & luces & colores:</s> <s xml:id="echoid-s5992" xml:space="preserve"> <lb/>ſed colores amplius, quàm luces, ſecun dum utrumq;</s> <s xml:id="echoid-s5993" xml:space="preserve"> ſpeculum.</s> <s xml:id="echoid-s5994" xml:space="preserve"> Et eſt:</s> <s xml:id="echoid-s5995" xml:space="preserve"> quoniam colores accedunt <lb/>debiliores, quàm luces:</s> <s xml:id="echoid-s5996" xml:space="preserve"> unde facile efficiũtur in reflexione debiliores.</s> <s xml:id="echoid-s5997" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s5998" xml:space="preserve"> color debilis cum <lb/>peruenerit ad ſpeculũ, miſcetur colori eius:</s> <s xml:id="echoid-s5999" xml:space="preserve"> quare reflexus apparebit debilis & tenebroſus.</s> <s xml:id="echoid-s6000" xml:space="preserve"> Et for-<lb/>mæ debiliores ſunt reflexæ, quàm in loco reflexionis:</s> <s xml:id="echoid-s6001" xml:space="preserve"> & reflexio cauſa eſt debilitatis.</s> <s xml:id="echoid-s6002" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div216" type="section" level="0" n="0"> <head xml:id="echoid-head249" xml:space="preserve" style="it">5. Lux & color reflexi ſunt debiliores luce & colore primis: fortiores autem ſecundis, cum <lb/>quibus ab eodem ortu æquabiliter diſtant. 4 p 5.</head> <p> <s xml:id="echoid-s6003" xml:space="preserve">POterit aliquis dicere, nõ eſſe debilitatẽ formarũ in reflexione, niſi ex elongatione earũ à ſua <lb/>origine.</s> <s xml:id="echoid-s6004" xml:space="preserve"> Sed explanabitur, quòd licet ab ortu æqualiter elongentur lux directa & lux reflexa:</s> <s xml:id="echoid-s6005" xml:space="preserve"> <lb/>tamen debilior erit reflexa.</s> <s xml:id="echoid-s6006" xml:space="preserve"> Intret radius ſolis in domum aliquã per foramen:</s> <s xml:id="echoid-s6007" xml:space="preserve"> & opponatur ſo <lb/>ramini in aere ſpeculũ ferreũ minus foramine:</s> <s xml:id="echoid-s6008" xml:space="preserve"> & lux foraminis reſidua cadat in terrã ſuper corpus <lb/>albũ:</s> <s xml:id="echoid-s6009" xml:space="preserve"> & lux à ſpeculo reflexa cadat in corpus albũ eleuatum:</s> <s xml:id="echoid-s6010" xml:space="preserve"> hoc obſeruato:</s> <s xml:id="echoid-s6011" xml:space="preserve"> ut eadẽ fit eleuati & ia-<lb/>centis à foramine longitudo:</s> <s xml:id="echoid-s6012" xml:space="preserve"> uidebitur quidẽ ſuper eleuatum lux minor, quàm ſuper iacens.</s> <s xml:id="echoid-s6013" xml:space="preserve"> Et hu <lb/>ius minoritatis nõ poteſt aſsignari cauſſa, niſi reflexio ſola.</s> <s xml:id="echoid-s6014" xml:space="preserve"> Idẽ accidit, ſi ſpeculum ſit argenteũ.</s> <s xml:id="echoid-s6015" xml:space="preserve"> Idẽ <lb/>in colore poteſt patére:</s> <s xml:id="echoid-s6016" xml:space="preserve"> lux enim ſolis in domũ aliquã per foramẽ deſcẽdat ſuper corpus coloris for <lb/>tis, cui adhibeatur ſpeculũ, & aliud corpus concauũ:</s> <s xml:id="echoid-s6017" xml:space="preserve"> intra quod ſit corpus albũ, in quod cadit refle <lb/>xio:</s> <s xml:id="echoid-s6018" xml:space="preserve"> & ſtatuatur in domo aliud corpus albũ eiuſdẽ modi, cũ eo, quod eſt in cõcauo:</s> <s xml:id="echoid-s6019" xml:space="preserve"> & ſit elongatio <lb/> <pb o="104" file="0110" n="110" rhead="ALHAZEN"/> huius albi à corpore colorato, in quod cadit lux foraminis, eadem cum elongatione albi, quod eſt <lb/>in concauo ab eodem, & cum elongatione ſpeculi ab codem:</s> <s xml:id="echoid-s6020" xml:space="preserve"> tunc comprehendetur color debilior <lb/>in albo, quod eſt intra concauum, quàm in eo, quod eſt extra:</s> <s xml:id="echoid-s6021" xml:space="preserve"> licet æquidiſtent ab ortu ſuo, id eſt à <lb/>corpore colorato.</s> <s xml:id="echoid-s6022" xml:space="preserve"> Et in cauſſa eſt reflexio colorem debilitans.</s> <s xml:id="echoid-s6023" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s6024" xml:space="preserve"> lux reflexa fortior eſt lu-<lb/>ceſecundaria:</s> <s xml:id="echoid-s6025" xml:space="preserve"> licet eiuſdem ſint elongationis ab origine ſua.</s> <s xml:id="echoid-s6026" xml:space="preserve"> Luce enim reflexa cadente in cor-<lb/>pus aliquod:</s> <s xml:id="echoid-s6027" xml:space="preserve"> ſi aliud eiuſmodi corpus ponatur extra locum reflexionis:</s> <s xml:id="echoid-s6028" xml:space="preserve"> & ſit cum eo eiuſdem e-<lb/>longationis à ſpeculo:</s> <s xml:id="echoid-s6029" xml:space="preserve"> uidebitur ſuper ipſum lux minor, quàm in illo.</s> <s xml:id="echoid-s6030" xml:space="preserve"> Idem etiam planum erit in <lb/>domo:</s> <s xml:id="echoid-s6031" xml:space="preserve"> ſi deponatur in terram, in directo foraminis ſpeculum, quod accipiat totam foramin is lu-<lb/>cem:</s> <s xml:id="echoid-s6032" xml:space="preserve"> erit lux fortior ſuper corpus in loco reflexionis poſitum, quàm ſuper aliud eiuſdem modi <lb/>extra hunc locum, tantundem elongatum à ſpeculo.</s> <s xml:id="echoid-s6033" xml:space="preserve"> Eodem modo ſi excedat lux foraminis quan-<lb/>titatem ſpeculi:</s> <s xml:id="echoid-s6034" xml:space="preserve"> & cadat circa ſpeculum lux in terram, aut corpus album, à quo aliud corpus tan-<lb/>tùm elongatur, quantùm corpus reflexionis à ſpeculo:</s> <s xml:id="echoid-s6035" xml:space="preserve"> debilior apparebit in eo lux, quàm ſuper <lb/>reflexionis corpus.</s> <s xml:id="echoid-s6036" xml:space="preserve"> Similiter accidit in colore, ſi corpus aliquod tantùm diſtet à ſpeculo extra ſi-<lb/>tum reflexionis, quantùm aliud ei ſimile, quod eſt in ſitu reflexionis:</s> <s xml:id="echoid-s6037" xml:space="preserve"> apparebit quidem ſuper cor-<lb/>pus, quod eſt in ſitu reflexionis, color reflexus:</s> <s xml:id="echoid-s6038" xml:space="preserve"> ſuper aliud forſitan nullus.</s> <s xml:id="echoid-s6039" xml:space="preserve"> Si enim ferreum fuerit <lb/>ſpeculum:</s> <s xml:id="echoid-s6040" xml:space="preserve"> aut modicus uidebitur, aut omnino nullus.</s> <s xml:id="echoid-s6041" xml:space="preserve"> Si uerò argenteum fuerit ſpeculum:</s> <s xml:id="echoid-s6042" xml:space="preserve"> appa-<lb/>rebit ſuper ipſum color aliquis, ſed ualde debilis, & longè debilior, quàm in corpore, quod eſt in <lb/>ſitu reflexionis.</s> <s xml:id="echoid-s6043" xml:space="preserve"> Et iã igitur planũ, quòd formæ luciũ & colorũ ex corporibus politis reflectuntur, <lb/>& in reflexione debilitantur:</s> <s xml:id="echoid-s6044" xml:space="preserve"> & erit forma directa fortior reflexa, cũ eadẽ fuerit earũ origo, & æqua <lb/>lis ab ea origine elongatio:</s> <s xml:id="echoid-s6045" xml:space="preserve"> & reflexa fortior ſecũdaria, cũ eſt idẽ uel æqualis ortus, & par elõgatio.</s> <s xml:id="echoid-s6046" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div217" type="section" level="0" n="0"> <head xml:id="echoid-head250" xml:space="preserve">DE MODO REFLEXIONIS FORMARVM À<unsure/> POLI-<lb/>tis corporibus. Cap. III.</head> <head xml:id="echoid-head251" xml:space="preserve" style="it">6. Lenitatis: politæ ſuperficiei: & perpendicularis incidentiæ definitiones. In def. 5 libr.</head> <p> <s xml:id="echoid-s6047" xml:space="preserve">POlitum eſt læue multùm in ſuperficie:</s> <s xml:id="echoid-s6048" xml:space="preserve"> Et læuitas eſt, ut ſint partes ſuperficiei continuæ, ſine <lb/>pororum multitudine.</s> <s xml:id="echoid-s6049" xml:space="preserve"> Læuitas intenſa eſt, ubi eſt multa partium ſuperficiei continuitas, & <lb/>pororum paruitas & paucitas:</s> <s xml:id="echoid-s6050" xml:space="preserve"> & finis læuitatis eſt priuatio pororũ, & priuatio diuiſionis par <lb/>tium.</s> <s xml:id="echoid-s6051" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s6052" xml:space="preserve"> politio eſt politiua cõtinuitas partium ſuperficiei, cum poris raris & exiguis:</s> <s xml:id="echoid-s6053" xml:space="preserve"> & finis po-<lb/>litionis eſt uera continuitas partium, & priuatio pororum.</s> <s xml:id="echoid-s6054" xml:space="preserve"> In omnibus politis ſuperficiebus, licet <lb/>diuerſis ſubiaceant figuris, accidet reflexio:</s> <s xml:id="echoid-s6055" xml:space="preserve"> & idem reflexionis modus & eadem proprietas eſt.</s> <s xml:id="echoid-s6056" xml:space="preserve"> Et <lb/>eſt, quòd ab omni politi ſuperficie & quolibet eius puncto fit reflexio.</s> <s xml:id="echoid-s6057" xml:space="preserve"> Et ſumpto quocunq;</s> <s xml:id="echoid-s6058" xml:space="preserve"> pũcto <lb/>in ſuperficie, à qua fit reflexio:</s> <s xml:id="echoid-s6059" xml:space="preserve"> linea acceſſus formæ ad illud punctum, & linea reflexionis in eadẽ <lb/>ſuperficie erunt cum linea perpendiculari ſuper illud punctum erecta:</s> <s xml:id="echoid-s6060" xml:space="preserve"> & tenebunthę lineæ eundẽ <lb/>ſitum reſpectu perpendicularis, & æqualitatem angulorum.</s> <s xml:id="echoid-s6061" xml:space="preserve"> Et uolo dicere perpendicularem:</s> <s xml:id="echoid-s6062" xml:space="preserve"> quæ <lb/>ſit perpendicularis ſuper ſuperficiẽ, tangentem corpus politum in illo puncto.</s> <s xml:id="echoid-s6063" xml:space="preserve"> Et duę lineę cũ per-<lb/>pendiculari ſunt in eadem ſuperficie orthogonaliter cadente ſuper ſuperficiem, corpus politum in <lb/>puncto, à quo fit reflexio, tangentem.</s> <s xml:id="echoid-s6064" xml:space="preserve"> Si autem linea, per quam accedit ad ſpeculũ forma, cadat per-<lb/>pendiculariter ſuper illud:</s> <s xml:id="echoid-s6065" xml:space="preserve"> fiet reflexio formæ per ipſam, non per aliam.</s> <s xml:id="echoid-s6066" xml:space="preserve"> Et hoc eſt propriũ in omni <lb/>reflexione, in omni polito corpore.</s> <s xml:id="echoid-s6067" xml:space="preserve"> Si igitur corpus politum fuerit planum:</s> <s xml:id="echoid-s6068" xml:space="preserve"> ſuperficies tangẽs pun <lb/>ctũ reflexionis, erit una & eadem cũ ſuperficie corporis.</s> <s xml:id="echoid-s6069" xml:space="preserve"> Si uerò fuerit columnare ſpeculũ interius <lb/>aut exterius politũ:</s> <s xml:id="echoid-s6070" xml:space="preserve"> erunt contactus ſuperficiei ſpeculi & ſuperficiei contingentis linea tantùm, ſe-<lb/>cundum longitudinem ſpeculi intellecta.</s> <s xml:id="echoid-s6071" xml:space="preserve"> Idẽ in ſpeculo pyramidali intus uel extrà polito.</s> <s xml:id="echoid-s6072" xml:space="preserve"> In ſphæ-<lb/>rico ſiue interius ſiue exterius polito, contingens ſuperficies tangit in ſolo puncto.</s> <s xml:id="echoid-s6073" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div218" type="section" level="0" n="0"> <head xml:id="echoid-head252" xml:space="preserve" style="it">7. Fabricatio & uſus organi reflexionis. 9 p 5.</head> <p> <s xml:id="echoid-s6074" xml:space="preserve">QVomodo autem etiam ad oculum pateat hic modus reflexionis in ſpeculis omnibus, expla <lb/>nabimus.</s> <s xml:id="echoid-s6075" xml:space="preserve"> Accipe tabulam æneam ſpiſſam, ut firmior ſit:</s> <s xml:id="echoid-s6076" xml:space="preserve"> eius longitudo ſit non minor duo-<lb/>decim digitis:</s> <s xml:id="echoid-s6077" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s6078" xml:space="preserve"> latitudo ſex digitorum, & fiat linea æquidiſtans extremitati longitudi-<lb/>nis:</s> <s xml:id="echoid-s6079" xml:space="preserve"> & circa illam extremitatẽ, & ſuper pũctũ <lb/> <anchor type="figure" xlink:label="fig-0110-01a" xlink:href="fig-0110-01"/> huius lineæ mediũ ponatur pes circini, & fiat <lb/>ſemicirculus, cuius ſemidiameter ſit latitudo <lb/>tabulæ & [per 11 p 1] extrahatur à puncto, <lb/>quod eſt centrũ, linea orthogonaliter ſuper <lb/>diametrũ iã factã:</s> <s xml:id="echoid-s6080" xml:space="preserve"> & erit linea illa ſemidiame-<lb/>ter diuidens ſemicirculum per æqualia [per <lb/>33 p 6.</s> <s xml:id="echoid-s6081" xml:space="preserve">] Et in hac ſemidiametro ſumatur men <lb/>ſura unius digiti, & poſito pede circini ſuper <lb/>centrũ, fiat ſemicirculus ſecundũ quantitatẽ <lb/>partis reſiduæ ſemidiametri, ſecundum ſemi-<lb/>diametrũ quinq;</s> <s xml:id="echoid-s6082" xml:space="preserve"> digitorũ.</s> <s xml:id="echoid-s6083" xml:space="preserve"> Et diuidiantur ſe-<lb/>micirculi primi medietates, in quot libuerit, partes, ita ut ſibi reſpondeant in æ qualitate, prima ſcili <lb/>cet primæ, ſecũda ſecundæ, & ſic de alijs:</s> <s xml:id="echoid-s6084" xml:space="preserve"> & protrahantur lineę à centro ad pũcta diuiſionum.</s> <s xml:id="echoid-s6085" xml:space="preserve"> De-<lb/>inceps in ſemidiametro menſura digiti ſignetur:</s> <s xml:id="echoid-s6086" xml:space="preserve"> & ex parte centri & ſuper punctum ſignatum pro-<lb/>trahatur linea, æquidiſtans diametro ſemicirculi, ſiue tabulæ extremitati [per 31 p 1:</s> <s xml:id="echoid-s6087" xml:space="preserve">] & ſecetur è <lb/>tabula, quod interiacet hanc lineam & ſemidiametrũ, uſq;</s> <s xml:id="echoid-s6088" xml:space="preserve"> ad centrum & lineas primas, ad diuiſio-<lb/> <pb o="105" file="0111" n="111" rhead="OPTICAE LIBER IIII."/> nes ſemicirculi protractas, id eſt ad lineas tales ſemidiametro propinquiores.</s> <s xml:id="echoid-s6089" xml:space="preserve"> Pòſt ſecetur tabula <lb/>circa ſemicirculum maiorem, ut ſolum remaneat ſemicirculus:</s> <s xml:id="echoid-s6090" xml:space="preserve"> & ſecetur tabula ſub centro, ut cen-<lb/>tri locus acuatur quaſi punctum:</s> <s xml:id="echoid-s6091" xml:space="preserve"> hoc tamen modo, ut in eadem ſuperficie remaneat cum ſemicir-<lb/>culo & alijs lineis.</s> <s xml:id="echoid-s6092" xml:space="preserve"> Pòſt ſumatur tabula lignea plana excedens æneam in longitudine duobus digi-<lb/>tis:</s> <s xml:id="echoid-s6093" xml:space="preserve"> & ſit quadrata:</s> <s xml:id="echoid-s6094" xml:space="preserve"> & eius altitudo fiue ſpiſsitudo ſeptem digitorum.</s> <s xml:id="echoid-s6095" xml:space="preserve"> Signetur ergo in hac tabula <lb/>punctum medium:</s> <s xml:id="echoid-s6096" xml:space="preserve"> & ſuper ipſum fiat circulus excedens maiorem circulum tabulę æneæ, quanti-<lb/>tate digiti magni:</s> <s xml:id="echoid-s6097" xml:space="preserve"> & fiat ſuper idem centrum circulus, æqualis circulo minori tabulę æneę:</s> <s xml:id="echoid-s6098" xml:space="preserve"> & diui-<lb/>datur circulus maior in partes, in æqualitate reſpondentes partibus ſemicirculi tabulæ æneę:</s> <s xml:id="echoid-s6099" xml:space="preserve"> ut <lb/>ſcilicet prima reſpondeat primæ, ſecunda ſecundæ, & ſic de alijs:</s> <s xml:id="echoid-s6100" xml:space="preserve"> & circumquaque ſecetur ta-<lb/>bula lignea, ut ſolum remaneat maior circulus:</s> <s xml:id="echoid-s6101" xml:space="preserve"> & fiet hæc ſectio uſitato ſecandi modo.</s> <s xml:id="echoid-s6102" xml:space="preserve"> Secetur e-<lb/>tiam pars tabulæ minore circulo contenta:</s> <s xml:id="echoid-s6103" xml:space="preserve"> & modus ſectionis erit:</s> <s xml:id="echoid-s6104" xml:space="preserve"> uthuic tabulæ aſſocietur alia <lb/>tabula, ita ut linea à centro huius ad centrum illius tranſiens, ſit perpendicularis ſuper illam:</s> <s xml:id="echoid-s6105" xml:space="preserve"> & ad-<lb/>hibito tornatili inſtrumento centris earum, fiat ſectio partis circularis iam dictæ:</s> <s xml:id="echoid-s6106" xml:space="preserve"> (eſt autem alte-<lb/>rius tabulæ aſſociatio, ut fixa ſtet in ſectione) igitur reſtabit tabula quaſi annulus circularis, cuius <lb/>latitudo erit duorum digitorum:</s> <s xml:id="echoid-s6107" xml:space="preserve"> longitudo quatuordecim:</s> <s xml:id="echoid-s6108" xml:space="preserve"> altitudo ſeptem.</s> <s xml:id="echoid-s6109" xml:space="preserve"> Et ſit hæc altitudo <lb/> <anchor type="figure" xlink:label="fig-0111-01a" xlink:href="fig-0111-01"/> optimè circula-<lb/>ta ad modum, co <lb/>lumnę:</s> <s xml:id="echoid-s6110" xml:space="preserve"> remanẽt <lb/>autẽ in latitudi <lb/>ne huius annuli <lb/>lineę diuidentes <lb/>circulũ eius ſe <lb/>cundum diuiſio <lb/>nẽ ſemicirculi ta <lb/>bulæ æneę.</s> <s xml:id="echoid-s6111" xml:space="preserve"> À<unsure/> ca <lb/>pitibus autem li <lb/>nearum harũ ꝓ-<lb/>ducantur lineæ <lb/>in ſuperficie al <lb/>titudinis exteri <lb/>oris, perpẽdicu <lb/>lares ſuper ſu-<lb/>perficiem latitu <lb/>dinis:</s> <s xml:id="echoid-s6112" xml:space="preserve"> & poterit <lb/>hoc modo fieri.</s> <s xml:id="echoid-s6113" xml:space="preserve"> <lb/>Quæratur regu-<lb/>la bene aeuta, cu <lb/>ius capiti linéæ <lb/>adhibeantur, & <lb/>regula mouea-<lb/>tur, donec tran <lb/>ſeat ſuperficiẽ al <lb/>titudinis, in qua <lb/>libet parte acu-<lb/>minis:</s> <s xml:id="echoid-s6114" xml:space="preserve"> Signa e-<lb/>ius capita, & fac <lb/>lineam, quoniam illa erit perpendicularis, quam quæris.</s> <s xml:id="echoid-s6115" xml:space="preserve"> Aliter poterit hoc idem fieri.</s> <s xml:id="echoid-s6116" xml:space="preserve"> Ponatur pes <lb/>circini ſuper terminũ lineæ diuidentis circulũ, & fiat ſemicirculus ſecũdũ altitudinẽ annuli, qui di <lb/>uidatur per æqualia, & protrahatur à puncto in punctũ linea, & ita de ſingulis.</s> <s xml:id="echoid-s6117" xml:space="preserve"> Pari modo à termi-<lb/>nis illarum diuidentium protrahantur perpẽdiculares ex parte interioris altitudinis.</s> <s xml:id="echoid-s6118" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s6119" xml:space="preserve"> ſu <lb/>matur in altitudine interiori ex parte faciei non diuiſę, altitudo duorum digitorum:</s> <s xml:id="echoid-s6120" xml:space="preserve"> & in perpen-<lb/>dicularibus fiat ſignum, & in ſignis illis fiat circulus, æquidiſtans faciei annuli hoc modo.</s> <s xml:id="echoid-s6121" xml:space="preserve"> Tabula <lb/>aliqua plana fiat circularis, æqualis circulo minori tabulę æneę:</s> <s xml:id="echoid-s6122" xml:space="preserve"> & ſecetur ex ea pars aliqua uſque <lb/>ad centrum, quaſitriangulum ex duabus ſemidiametris & arcu circuli, ſecundum quod libuerit, <lb/>ut poſsis tabulam cum manu imponere, & locis aſsignatis aptare.</s> <s xml:id="echoid-s6123" xml:space="preserve"> Apta ergo locis illis, ut ſit æqui-<lb/>diſtans faciei annuli, & fac circulum ſecundum ipſam.</s> <s xml:id="echoid-s6124" xml:space="preserve"> Sumatur etiam infra hunc circulum altitu-<lb/>do medietatis grani hordei, & fiant ſigna, & in punctis aſsignatis fiat circulus per aptationem ta-<lb/>bulę.</s> <s xml:id="echoid-s6125" xml:space="preserve"> Et in hoc poſtremo circulo fiat circularis concauitas, & ſit unius digiti eius profunditas, & <lb/>altitudo tanquam altitudo tabulę æneę:</s> <s xml:id="echoid-s6126" xml:space="preserve"> & ſit hęc altitudo intra altitudinem duorum digitorum, ut <lb/>eadem ſit poſtremi circuli & cõcauitatis ſpecies.</s> <s xml:id="echoid-s6127" xml:space="preserve"> Aptetur autem huic concauitati tabula ęnea, quę <lb/>quidem intret concauitatem uſq;</s> <s xml:id="echoid-s6128" xml:space="preserve"> ad circulum minorem.</s> <s xml:id="echoid-s6129" xml:space="preserve"> Et cum diſtantia minoris à maiori ſit uni-<lb/>us digiti, & concauitas ſimiliter:</s> <s xml:id="echoid-s6130" xml:space="preserve"> igitur circulo poſtremo & tabulę ęneę communis erit ſuperficies:</s> <s xml:id="echoid-s6131" xml:space="preserve"> <lb/>& line æ perpendiculares in altitudine annuli, tangent lineas diuiſionis tabulæ æneæ, & cadent <lb/>perpendiculariter ſuper tabulam ęneam.</s> <s xml:id="echoid-s6132" xml:space="preserve"> Sit autem ſuperficies tabulę ęneę diuiſa ex parte faciei <lb/> <pb o="106" file="0112" n="112" rhead="ALHAZEN"/> annuli diuiſæ.</s> <s xml:id="echoid-s6133" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s6134" xml:space="preserve"> in exteriore planitudine annuli ſignetur punctus, à longitudine duorum <lb/>digitorum:</s> <s xml:id="echoid-s6135" xml:space="preserve"> & poſito pede circini ſuper punctum ſignatum, fiat circulus, ſecundum quantitatem u-<lb/>nius grani hordei, & inſtrumento ferreo, cuius ſimiliter latitudo ſit quantitas unius grani hordei, <lb/>perforetur foramine columnari:</s> <s xml:id="echoid-s6136" xml:space="preserve"> & baculus ligneus foramini aptetur:</s> <s xml:id="echoid-s6137" xml:space="preserve"> qui quidem cum tranſierit ad <lb/>interiorem concauitatem, tanget tabulæ æneę ſuperficiem.</s> <s xml:id="echoid-s6138" xml:space="preserve"> Pari modo ſuper ſingulas exterioris al-<lb/>titudinis perpendiculares ſimilia & æqualia efficiantur foramina, in quantitate & altitudine.</s> <s xml:id="echoid-s6139" xml:space="preserve"> De-<lb/>inde ſumatur tabula lignea quadrata, cuius latus ſit æquale diametro annuli, & protrahatur in eius <lb/>ſuperficie linea diuidens per medium quadratum, æquidiſtans lateribus.</s> <s xml:id="echoid-s6140" xml:space="preserve"> Et ab una parte ſumatur <lb/>longitudo duorum digitorum, & fiat ſignum:</s> <s xml:id="echoid-s6141" xml:space="preserve"> & pòſt ſumatur longitudo ſemidiametri minoris cir-<lb/>culi tabulæ æneæ, & poſito pede circini, fiat circulus tranſiens per ſignum:</s> <s xml:id="echoid-s6142" xml:space="preserve"> qui quidem circulus e-<lb/>rit æqualis minori circulo tabulæ æneæ & concauitati annuli.</s> <s xml:id="echoid-s6143" xml:space="preserve"> Deinde ſupra centrum huius circu-<lb/>li ſumatur longitudo duorum digitorum, & infra centrum ſimiliter:</s> <s xml:id="echoid-s6144" xml:space="preserve"> & ſignentur puncta ab utro-<lb/>que in utranque partem:</s> <s xml:id="echoid-s6145" xml:space="preserve"> & protrahatur linea æquidiſtans lateribus quadrati:</s> <s xml:id="echoid-s6146" xml:space="preserve"> & in utraq;</s> <s xml:id="echoid-s6147" xml:space="preserve"> harum <lb/>linearum ſignetur longitudo duorum digitorum, ex utraq;</s> <s xml:id="echoid-s6148" xml:space="preserve"> parte puncti ſignati:</s> <s xml:id="echoid-s6149" xml:space="preserve"> & à punctis unius <lb/>lineæ ſignatis protrahantur lineę æquidiſtantes ad puncta alterius lineę ſignata:</s> <s xml:id="echoid-s6150" xml:space="preserve"> & fiat quadratum <lb/>quatuor digitorum.</s> <s xml:id="echoid-s6151" xml:space="preserve"> Cauetur poſtea hoc quadratum, ſecundum altitudinem unius digiti, & conca-<lb/>uationis latera efficiantur plana, & orthogonalia, & fundus ſimiliter planus.</s> <s xml:id="echoid-s6152" xml:space="preserve"> Deinde aptetur hæc <lb/>tabula faciei annuli, ita ut circulus minor applicetur foramini eius, & extremitas eius extremitati:</s> <s xml:id="echoid-s6153" xml:space="preserve"> <lb/>& firmetur hæc applicatio cum clauis, ut immota maneat tabula.</s> <s xml:id="echoid-s6154" xml:space="preserve"> Notandum uerò, quòd in omni-<lb/>bus prædictis, dictorum digitorum menſura certa debet eſſe & determinata:</s> <s xml:id="echoid-s6155" xml:space="preserve"> & ob hoc in linea ali-<lb/>qua fiat immutabili, ne ex mutatione menſurę error accidat.</s> <s xml:id="echoid-s6156" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s6157" xml:space="preserve"> fiat columna ferrea conca-<lb/>ua, plana, aliquantulum ſpiſſa, ut ſtatim intret, nec immutari queat:</s> <s xml:id="echoid-s6158" xml:space="preserve"> & ſit quantitas diametri circuli <lb/>eius unius grani hordei:</s> <s xml:id="echoid-s6159" xml:space="preserve"> & ponatur columna in foraminibus:</s> <s xml:id="echoid-s6160" xml:space="preserve"> quę quidem cum ad interiora annuli <lb/>peruenerit:</s> <s xml:id="echoid-s6161" xml:space="preserve"> continget lineas in tabula ænea factas.</s> <s xml:id="echoid-s6162" xml:space="preserve"> Et erit operis eius complementum, ſi linea ta-<lb/>bulę æneę contingat circulum columnę in puncto lineæ altitudinis annuli, perpendicularis ſuper <lb/>tabulam æneam, & tranſeuntis per centrum circuli columnę.</s> <s xml:id="echoid-s6163" xml:space="preserve"> Fiat autem in capite columnæ annu-<lb/>lus aut repagulum, quod non permittat columnam intrare, niſi ad locum determinatum.</s> <s xml:id="echoid-s6164" xml:space="preserve"> Sit autem <lb/>huius longitudinis columna, ut procedens ſuper tabulam æneam, attingat lineam æquidiſtantem <lb/>diametro tabulæ, intra quam facta eſt ſectio.</s> <s xml:id="echoid-s6165" xml:space="preserve"> Et hæc eſt linea illa æquidiſtans baſi trianguli ta-<lb/>bulæ æneæ.</s> <s xml:id="echoid-s6166" xml:space="preserve"/> </p> <div xml:id="echoid-div218" type="float" level="0" n="0"> <figure xlink:label="fig-0110-01" xlink:href="fig-0110-01a"> <variables xml:id="echoid-variables12" xml:space="preserve">n m l b h i k e p t r o s u q a f d g c</variables> </figure> <figure xlink:label="fig-0111-01" xlink:href="fig-0111-01a"> <image file="0111-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0111-01"/> </figure> </div> </div> <div xml:id="echoid-div220" type="section" level="0" n="0"> <head xml:id="echoid-head253" xml:space="preserve" style="it">8. Fabricatio ſeptem ſpeculorum regularium. 8 p 5.</head> <p> <s xml:id="echoid-s6167" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s6168" xml:space="preserve"> fabricentur ſeptem ſpecula ferrea, quorum unum planum:</s> <s xml:id="echoid-s6169" xml:space="preserve"> duo ſphærica, unum con <lb/>cauum intrà politum, aliud extrà:</s> <s xml:id="echoid-s6170" xml:space="preserve"> duo pyramidalia, unum politum in facie, aliud in conca-<lb/>uitate:</s> <s xml:id="echoid-s6171" xml:space="preserve"> duo columnaria, unum concauum, aliud in ſuperficie politum.</s> <s xml:id="echoid-s6172" xml:space="preserve"> Speculum autem pla <lb/>num ſit circulare:</s> <s xml:id="echoid-s6173" xml:space="preserve"> & ſit eius diameter trium digitorum.</s> <s xml:id="echoid-s6174" xml:space="preserve"> Speculum columnare politum in ſuperficie <lb/>ſit lucidum, & perfectè politum:</s> <s xml:id="echoid-s6175" xml:space="preserve"> & ſit diameter circuli longitudinis ſex digitorum, qui circulus eſt <lb/>baſis eius.</s> <s xml:id="echoid-s6176" xml:space="preserve"> Longitudo autem columnę ſit trium digitorum.</s> <s xml:id="echoid-s6177" xml:space="preserve"> In baſi columnę ſumatur chorda longi-<lb/>tudinis trium digitorum:</s> <s xml:id="echoid-s6178" xml:space="preserve"> ſimiliter in baſi eiuſdem columnę oppoſita ſumatur huic æqualis chor-<lb/>da, & ei oppoſita, ut lineę à capitibus unius chordę ad capita alteri-<lb/> <anchor type="figure" xlink:label="fig-0112-01a" xlink:href="fig-0112-01"/> us productę, ſintrectę.</s> <s xml:id="echoid-s6179" xml:space="preserve"> Et ſecetur hæc columna ſecundum harum li-<lb/>nearum proceſſum, ut reſtet nobis pars columnę, cuius capita ſint <lb/>portiones chordarum:</s> <s xml:id="echoid-s6180" xml:space="preserve"> altitudo autem axis remanentis portionis <lb/>minor, quàm altitudo dimidij digiti.</s> <s xml:id="echoid-s6181" xml:space="preserve"> Axem autem dico lineã à me-<lb/>dio puncto arcus, ad mediũ chordę punctũ productã.</s> <s xml:id="echoid-s6182" xml:space="preserve"> Columnę aũt <lb/>cõcauę longitudo ſit triũ digitorũ, & diameter baſis eius ſex digito-<lb/>rum:</s> <s xml:id="echoid-s6183" xml:space="preserve"> & in ea ſumatur chorda trium digitorum, & ſiat ſectio, ſicut in <lb/>prima:</s> <s xml:id="echoid-s6184" xml:space="preserve"> & erit altitudo axis partis remanentis minor, quàm altitudo <lb/>dimidij digiti.</s> <s xml:id="echoid-s6185" xml:space="preserve"> Sit autem in his omnibus politura exquiſita, & æqua-<lb/>litas omnimoda.</s> <s xml:id="echoid-s6186" xml:space="preserve"> In ſpeculo pyramidali quęratur diameter baſis:</s> <s xml:id="echoid-s6187" xml:space="preserve"> cu-<lb/>ius quantitas ſit ſex digitorũ, & chorda trium:</s> <s xml:id="echoid-s6188" xml:space="preserve"> & longitudo quatuor <lb/>digitorum & dimidij:</s> <s xml:id="echoid-s6189" xml:space="preserve"> & fiat ſectio ſecũdum lineas rectas:</s> <s xml:id="echoid-s6190" xml:space="preserve"> & axis por-<lb/>tionis altitudo ſit minor quàm altitudo dimidij digiti.</s> <s xml:id="echoid-s6191" xml:space="preserve"> Et hæc in <lb/>utraque pyramidali intellige.</s> <s xml:id="echoid-s6192" xml:space="preserve"> Speculum ſphęricum ſit portio ſphę-<lb/>rę, cuius diameter ſit ſex digitorum, & diameter baſis huius ſpecu-<lb/>li trium digitorum:</s> <s xml:id="echoid-s6193" xml:space="preserve"> & erit axis altitudo minor quàm altitudo dimi-<lb/>dij digiti.</s> <s xml:id="echoid-s6194" xml:space="preserve"> Idem operare in ſpeculo ſphęrico concauo.</s> <s xml:id="echoid-s6195" xml:space="preserve"> Deinde fa-<lb/>cias ſeptem regulas ligneas planas, quarum latera ſint æquidiſtan-<lb/>tia & orthogonalia, ſuper capita æquidiſtantia in fine poſsibilita-<lb/>tis:</s> <s xml:id="echoid-s6196" xml:space="preserve"> & ſit longitudo regularum ſex digitorum, latitudo quatuor.</s> <s xml:id="echoid-s6197" xml:space="preserve"> Poſtea quadrato adaptetur ali-<lb/>qua regularum, ita ut orthogonaliter cadatſuper inferiorem concaui quadrati ſuperficiem, & ui-<lb/>de, ut facile intret quadratum:</s> <s xml:id="echoid-s6198" xml:space="preserve"> ne compreſſa immutetur.</s> <s xml:id="echoid-s6199" xml:space="preserve"> Cadat igitur ſuper faciem lateris re-<lb/>gulę acumen tabulę ęneę, & ubi continuabitur ei, fiat ſignum:</s> <s xml:id="echoid-s6200" xml:space="preserve"> & à puncto aſsignato produca-<lb/>tur in extremitates regulę, linea æquidiſtans lateribus regulæ, ut ſit linea illa, linea longitudinis <lb/> <pb o="107" file="0113" n="113" rhead="OPTICAE LIBER IIII."/> regulæ.</s> <s xml:id="echoid-s6201" xml:space="preserve"> Deinceps in longiore parte illius lineæ circa punctum ſumptum, ſumatur altitudo me-<lb/>dij grani hordei, & fiat punctum.</s> <s xml:id="echoid-s6202" xml:space="preserve"> Dico quod illud eſt punctum medium regulæ, quod etiam cen-<lb/>tris foraminum opponitur rectè.</s> <s xml:id="echoid-s6203" xml:space="preserve"> Quoniam enim centra foraminum elongantur ſuper ſuperfi-<lb/>ciem tabulæ æneæ, in medij grani quantitate, & diſtant à ſuperficie annuli per duos digitos:</s> <s xml:id="echoid-s6204" xml:space="preserve"> lgi-<lb/>tur punctum illud diſtat ab eadem per duos digitos, & in quadrato concauo per digitum unum.</s> <s xml:id="echoid-s6205" xml:space="preserve"> <lb/>Quare ab extremitatibus regulæ ad punctũ ſunt tres digiti.</s> <s xml:id="echoid-s6206" xml:space="preserve"> Quare punctũ illud erit mediũ.</s> <s xml:id="echoid-s6207" xml:space="preserve"> Super <lb/>hoc mediũ punctum producatur in utrãq;</s> <s xml:id="echoid-s6208" xml:space="preserve"> partẽ linea, ſecundũ latitudinẽ æquidiſtans extremιtati-<lb/>bus:</s> <s xml:id="echoid-s6209" xml:space="preserve"> & medietates lineæ longitudinis (ſuper quam eſt hæc perpendicularis) diuidãtur per æqua-<lb/>lia, per lineas latitudinis perpendiculares extremitatibus æquidiſtantes.</s> <s xml:id="echoid-s6210" xml:space="preserve"> Et ita diuiſa erit regula in <lb/>quatuor æquales partes.</s> <s xml:id="echoid-s6211" xml:space="preserve"> Similis fiat in alijs regulis operatio.</s> <s xml:id="echoid-s6212" xml:space="preserve"/> </p> <div xml:id="echoid-div220" type="float" level="0" n="0"> <figure xlink:label="fig-0112-01" xlink:href="fig-0112-01a"> <variables xml:id="echoid-variables13" xml:space="preserve">p k c z q x y b</variables> </figure> </div> </div> <div xml:id="echoid-div222" type="section" level="0" n="0"> <head xml:id="echoid-head254" xml:space="preserve" style="it">9. Sit{us} & collocatio ſpeculorum regulariũ in reflexionis organo.10.12.13.14.15.16.17 p 5.</head> <p> <s xml:id="echoid-s6213" xml:space="preserve">HIs completis, adaptetur ſpeculum planũ uni regularum:</s> <s xml:id="echoid-s6214" xml:space="preserve"> & eſt:</s> <s xml:id="echoid-s6215" xml:space="preserve"> ut ſit regula cauata ſecundũ <lb/>altitudinem ſpeculi, ita ut ſuperficies ſpeculi ſit in eadem ſuperficie cũ ſuperficie regulæ:</s> <s xml:id="echoid-s6216" xml:space="preserve"> & <lb/>ita, ut medium ſuperficiei ſpeculi punctũ, directè ſupponatur medio ſuperficiei regulæ pun <lb/>cto:</s> <s xml:id="echoid-s6217" xml:space="preserve"> & ita, ut linea diuidens ſuperficiẽ regulę in duo æqualia:</s> <s xml:id="echoid-s6218" xml:space="preserve"> diuidat etiã ſuperficiem ſpeculiper ę-<lb/>qualia, & ut cõtinuentur partes ſpeculi cũ linea diuidente:</s> <s xml:id="echoid-s6219" xml:space="preserve"> & hoc obſeruetur in poſsibilitatis fine.</s> <s xml:id="echoid-s6220" xml:space="preserve"> <lb/>Deinde ſpeculum columnare politũ in facie applicetur alicui regulæ ita, ut mediũ punctũ eius ca-<lb/>dat ſuper mediũ regulæ punctũ, & ita, ut linea in longitudine ſpeculi ſumpta, diuidens ipſum per <lb/>æqualia, cõtinuetur cũ partibus lineæ lõgitudinis ſuperficiei regulæ æ què diuidenti, & ut media <lb/>longitudinis ſpeculi linea ſit in ſuperficie regulę.</s> <s xml:id="echoid-s6221" xml:space="preserve"> Et hoc ſic fieri poterit.</s> <s xml:id="echoid-s6222" xml:space="preserve"> Vtriuſq;</s> <s xml:id="echoid-s6223" xml:space="preserve"> baſis ſpeculi arcus <lb/>per æqualia diuidantur, & à puncto diuiſionis ſignato ad oppoſitũ ſignatum punctũ linea produca <lb/>tur, & lineę mediæ longitudinis aptetur & cõtinuetur.</s> <s xml:id="echoid-s6224" xml:space="preserve"> Speculũ columnare concauũ aptetur regu-<lb/>læ, ut media lõgitudinis eius linea ſecundũ æ qualẽ baſium arcuum diuiſionẽ ſumpta, æquidiſtãs <lb/>ſit line æ mediæ longitudinis regulæ:</s> <s xml:id="echoid-s6225" xml:space="preserve"> & etiã ut utriuſq, arcus chordę cũ lineæ lõgitudinis extremis <lb/>ſint in ſuperficie regulæ.</s> <s xml:id="echoid-s6226" xml:space="preserve"> Pyramidale ſpeculũ extrà politũ applicetur regulæ, ut acumen eius ſit in <lb/>termino line æ mediæ lõgitudinis regulæ, & linea diuidens portionẽ pyramidalis per æ qua, quę ſci <lb/>licet à uertice ad medium arcus baſis punctũ producitur, ſit in ſuperficie continuata cum parte re-<lb/>ſtante lineę mediæ, longitudinis regulæ.</s> <s xml:id="echoid-s6227" xml:space="preserve"> Speculum pyramidale concauum applicetur regulę ita, ut <lb/>acumen eius ſit in directo mediæ lineæ longitudinis regulæ, chorda uerò arcus baſis ſit in ſuperfi-<lb/>cie regulæ, & linea à uertice ad medium arcus baſis punctum ducta, ſit æ quidiſtans mediæ lineæ <lb/>longitudinis regulæ.</s> <s xml:id="echoid-s6228" xml:space="preserve"> Cum autem longitudo pyramidis ſit quatuor digitorum & dimidij:</s> <s xml:id="echoid-s6229" xml:space="preserve"> reſtabũt <lb/>ex longitudine regulæ digitus & dimidius.</s> <s xml:id="echoid-s6230" xml:space="preserve"> Ad aptandum regulæ ſpeculum ſphæricum extrà poli-<lb/>tum:</s> <s xml:id="echoid-s6231" xml:space="preserve"> fiat in regula circulus ſecundum quantitatem trium digitorum:</s> <s xml:id="echoid-s6232" xml:space="preserve"> eius centrum ſit medium re-<lb/>gulæ punctum:</s> <s xml:id="echoid-s6233" xml:space="preserve"> & aptetur ſpeculum, ut medium ſuperficiei eius punctum ſit in ſuperficie regulæ, <lb/>& in medio puncto mediæ lineæ longitudinis regulæ:</s> <s xml:id="echoid-s6234" xml:space="preserve"> quod quidẽ ſciri poterit per application em <lb/>alterius regulæ acutæ, æ qualis huic in longitudine, & diuiſæ per æ qualitatem, & applicatę mediæ <lb/>lineæ longitudinis regulæ, ita ut medium huius regulæ acutæ punctum, tangat medium ſpeculi <lb/>ſphærici punctum.</s> <s xml:id="echoid-s6235" xml:space="preserve"> Sphęricum concauum aptatur:</s> <s xml:id="echoid-s6236" xml:space="preserve"> facto in regula circulo ſecundum quantitatem <lb/>trium digitorum, cuius centrum medium regulæ punctum:</s> <s xml:id="echoid-s6237" xml:space="preserve"> Cauato circulo imponatur ita, ut circu <lb/>lus baſis ſpeculi ſit in ſuperficie regulæ, & punctum medium concauitatis ſpeculi, ſit directè oppo-<lb/>ſitũ medio regulæ puncto, & diameter baſis ſpeculi continuetur mediæ lineæ regulę:</s> <s xml:id="echoid-s6238" xml:space="preserve"> Quæita per-<lb/>pendetur.</s> <s xml:id="echoid-s6239" xml:space="preserve"> In regula acuta punctũ ſignetur:</s> <s xml:id="echoid-s6240" xml:space="preserve"> & ab illo puncto lõgitudo ſemidiametri baſis ſpeculi no <lb/>retur ex utra que parte, & ita hæc acuta regula mediæ lineæ regulæ applicetur, ut punctum ſignatũ <lb/>in ea, directè opponatur medio cõcauitatis ſpeculi puncto, & diameter in ea facta ſimul ſit cum ba-<lb/>fis diametro.</s> <s xml:id="echoid-s6241" xml:space="preserve"> His peractis in ſemidiametro tabulæ æneæ triangulum per æqualia diuidente:</s> <s xml:id="echoid-s6242" xml:space="preserve"> ſigne-<lb/>tur ab acumine eius longitudo, æqualis axi huius ſpeculi concaui, & fiat punctum.</s> <s xml:id="echoid-s6243" xml:space="preserve"> Axis autem ſic <lb/>dignoſcitur.</s> <s xml:id="echoid-s6244" xml:space="preserve"> Regula acuta ſuperficiei ſpeculi applicetur, ut acuitas directè ſit ſuper mediam longi-<lb/>tudinis lineam, puncto eius ſuper medium concaui ſpeculi punctum directè ſtatuto:</s> <s xml:id="echoid-s6245" xml:space="preserve"> deinde acus <lb/>recta & ſubtilis ſuper illud regulæ acutæ punctum perpendiculariter cadat in ſpeculum:</s> <s xml:id="echoid-s6246" xml:space="preserve"> deſcen-<lb/>det quidem ſuper medium concaui punctum:</s> <s xml:id="echoid-s6247" xml:space="preserve"> ſignetur autem in acu punctum, quod poſt ſuum <lb/>deſcẽſum tangat concauitas regulæ:</s> <s xml:id="echoid-s6248" xml:space="preserve"> & ſit modicum declinata regula, ut certius poſsit fieri in acu <lb/>ſignum.</s> <s xml:id="echoid-s6249" xml:space="preserve"> Poſtea ſecun dum longitudinem acus à puncto ſignato in ea, metire ab acumine tabulæ æ-<lb/>neæ in linea triangulum diuidente, & fac punctum.</s> <s xml:id="echoid-s6250" xml:space="preserve"> Deinceps hanc regulam facias intrare quadra-<lb/>tum concauum, ita ut acumen tabulæ æneæ deſcendat ſupra ſpeculum, & adhibeatur regula acu-<lb/>ta, ut ſignetur punctum in linea diuidente triangulum, quod tetigerit ex ea regula acuta, cum a-<lb/>cumen trianguli deſeenderit uſque ad ſuperficiem concaui:</s> <s xml:id="echoid-s6251" xml:space="preserve"> Signa igitur punctum:</s> <s xml:id="echoid-s6252" xml:space="preserve"> hoc uerò ſe-<lb/>cundum punctum minus diſtabit ab acumine quàm primum.</s> <s xml:id="echoid-s6253" xml:space="preserve"> Superficies enim tabulæ æneæ di-<lb/>ſtat à ſuperficie annuli ſiue tabulæ, in qua eſt quadratum concauum, per duos digitos minus me-<lb/>dietate grani hordei:</s> <s xml:id="echoid-s6254" xml:space="preserve"> punctum autem medium regulæ directè eſt oppoſitum medio ſpeculi conca-<lb/>ui puncto:</s> <s xml:id="echoid-s6255" xml:space="preserve"> quod quidem diſtat ab eadem ſuperficie tabulæ per duos digitos.</s> <s xml:id="echoid-s6256" xml:space="preserve"> Cum ergo acumen ta-<lb/>bulę orthogonaliter deſcendat:</s> <s xml:id="echoid-s6257" xml:space="preserve"> nõ cadet ſuper mediũ cõcaui punctũ, quod eſt terminus axis, ſed in <lb/>punctũ altius.</s> <s xml:id="echoid-s6258" xml:space="preserve"> Quare patet propoſitũ.</s> <s xml:id="echoid-s6259" xml:space="preserve"> Signetur uerò in ſpeculo cõcauo pũctũ, in qđ incidit acumẽ <lb/>tabulæ æneæ, & extracto in pũcto illo foramine, orthogonaliter deſcẽdẽte & modico, ad hãc quidẽ <lb/> <pb o="108" file="0114" n="114" rhead="ALHAZEN"/> menſuram, ut in eo deſcendat acumen:</s> <s xml:id="echoid-s6260" xml:space="preserve"> donec acuitas lineæ adhibitæ, cõtingat punctũ lineæ diui-<lb/>dentis triangulum primò ſignatũ:</s> <s xml:id="echoid-s6261" xml:space="preserve"> quod cum fuerit:</s> <s xml:id="echoid-s6262" xml:space="preserve"> erit quidem acumen tabulæ æneæ in eadem ſu <lb/>perficie cũ termino axis ſpeculi:</s> <s xml:id="echoid-s6263" xml:space="preserve"> quæ ſuperficies eſt æquidiſtãs ſuperficiei regulę:</s> <s xml:id="echoid-s6264" xml:space="preserve"> & erit linea à ter-<lb/>mino axis ad acumen ducta, perpendicularis ſuper ſuperficiẽ tabulæ æneæ.</s> <s xml:id="echoid-s6265" xml:space="preserve"> Axis aũt ſpeculi in ea-<lb/>dem erit ſuperficie cũ centris foraminũ:</s> <s xml:id="echoid-s6266" xml:space="preserve"> quoniam diſtantia eorũ à ſuperficie annuli duorum eſt di-<lb/>gitorum, & terminus axis ſimiliter.</s> <s xml:id="echoid-s6267" xml:space="preserve"> His cum diligentia præparatis, poterit uĩderi, quod ꝓmiſimus.</s> <s xml:id="echoid-s6268" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div223" type="section" level="0" n="0"> <head xml:id="echoid-head255" xml:space="preserve" style="it">10. Radi{us} ſpeculo plano obliqu{us}, in oppoſitam partem reflectitur: & æquat angulos inci-<lb/>dentiæ & reflexionis. 10 p 5.</head> <p> <s xml:id="echoid-s6269" xml:space="preserve">IMmittatur annulo regula, ſuper quã eſt ſpeculum planũ, donec acumen tabulæ æneæ cadat ſu-<lb/>per ſpeculũ, & ſit infixa regula quadrato concauo:</s> <s xml:id="echoid-s6270" xml:space="preserve"> & in eo ſubtus regulã aliquid appoǹatur, qđ <lb/>ei cõferat firmitatẽ, ne uacillet:</s> <s xml:id="echoid-s6271" xml:space="preserve"> deinde opponatur pergamenũ foraminibus, & cũ digito fiat im <lb/>preſsio, ut obturentur, & impreſsionẽ percipere poſsis, & ſignum foraminis fiat in pergameno cũ <lb/>in cauſto uel aliquo alio:</s> <s xml:id="echoid-s6272" xml:space="preserve"> Vnum autem foraminum relinquatur apertum, declinatũ non ſuper me-<lb/>diam regulam, & adhibeatur radio ſolis foramen apertum:</s> <s xml:id="echoid-s6273" xml:space="preserve"> certior autem erit huius rei comprehen <lb/>ſio, ſi adhibeatur radio ſolis per foramen domus intranti.</s> <s xml:id="echoid-s6274" xml:space="preserve"> Cum igitur radius foramẽ intrans ad ſpe <lb/>culum peruenerit, uidebis ipſum reflecti ad foramen illud, reſpiciens ſuper lineam tabulæ æneę æ-<lb/>qualem angulum continentem cum linea triangulum per æqua diuidente, ei angulo, quem tenet li <lb/>nea à foramine diſcooperto cum eadem tabulæ ſemidiametro.</s> <s xml:id="echoid-s6275" xml:space="preserve"> Siuerò foramẽ, in quod fit reflexio, <lb/>diſcoopertum opponas radio, priore cooperto:</s> <s xml:id="echoid-s6276" xml:space="preserve"> uidebis radium reflecti in coopertum.</s> <s xml:id="echoid-s6277" xml:space="preserve"> Si uerò fora <lb/>mini imponatur columna ferrea concaua, quam ad quantitatem foraminum fieri pręcepimus:</s> <s xml:id="echoid-s6278" xml:space="preserve"> quę <lb/>ut firmius ſtet, modicum ceræ circa eam apponatur:</s> <s xml:id="echoid-s6279" xml:space="preserve"> deſcendet lux per columnæ concauitatem, ſi-<lb/>cut deſcendit per foramen, & reflectetur in foramen ſibi reſpondens, & ſuper lineas tabulæ æneæ <lb/>erit deſcenſus & reflexio pari modo, ut prius.</s> <s xml:id="echoid-s6280" xml:space="preserve"> Et ſi ad ſecundum foramen columnã tranſtulerimus:</s> <s xml:id="echoid-s6281" xml:space="preserve"> <lb/>in primum lucem reflexam uidebimus.</s> <s xml:id="echoid-s6282" xml:space="preserve"> Erit autem debilior lux per columnam deſcendens, quàm <lb/>ſine columna per foramen.</s> <s xml:id="echoid-s6283" xml:space="preserve"> Erit autem uιdere eundem reflectendi modum in debiliore luce.</s> <s xml:id="echoid-s6284" xml:space="preserve"> Obtu <lb/>retur foramen cum cera, ut modicum circa centrum ei reſtet uacuum:</s> <s xml:id="echoid-s6285" xml:space="preserve"> & uidebitur lucis reflexio in <lb/>foramine ſimili circa centrum.</s> <s xml:id="echoid-s6286" xml:space="preserve"> Pari modo, ſi concauitatem columnæ cum cera obturaueris, ut re-<lb/>maneat quaſi terminus ſolius axis:</s> <s xml:id="echoid-s6287" xml:space="preserve"> deſcen det lux ſuper axem columnæ, & reflectetur ad centrum <lb/>foraminis ſimilis.</s> <s xml:id="echoid-s6288" xml:space="preserve"> Eodem modo altera columna impoſita, cũ deſcen derit lux ſuper axem unius fo-<lb/>raminis:</s> <s xml:id="echoid-s6289" xml:space="preserve"> reflectetur ſuper axem ſimilis.</s> <s xml:id="echoid-s6290" xml:space="preserve"> Centrum enim foraminis directè axi opponitur:</s> <s xml:id="echoid-s6291" xml:space="preserve"> & cũ lucis <lb/>reflexio cadat in centrũ, nec moueatur, niſi per lineam rectã, oportet, ut procedat ſecundum axem.</s> <s xml:id="echoid-s6292" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div224" type="section" level="0" n="0"> <head xml:id="echoid-head256" xml:space="preserve" style="it">11. Radi{us} ſpeculo perpendicularis, reflectitur in ſeipſum. 11.12 p 5.</head> <p> <s xml:id="echoid-s6293" xml:space="preserve">OBturatis autem foraminibus ſingulis, præter medium, quod directè ſuper tabulam æneam <lb/>incidit:</s> <s xml:id="echoid-s6294" xml:space="preserve"> fiat baculus columnaris ad quantitatẽ foraminis, & extremitas eius acuatur, ut re-<lb/>maneat ſolus terminus axis eius, & deſcẽdat per foramen ad ſpeculũ, & ſignetur punctum, <lb/>in quod ceciderit:</s> <s xml:id="echoid-s6295" xml:space="preserve"> deinde deſcen dat radius ſolis per foramen illud:</s> <s xml:id="echoid-s6296" xml:space="preserve"> cadet quidẽ ſuper punctũ ſigna <lb/>tum, & circa ipſum efficiet circulum.</s> <s xml:id="echoid-s6297" xml:space="preserve"> Signetur igitur in fine huius lucis circularis punctũ, & ſecun-<lb/>dum quantitatẽ lineæ interiacentis puncta ſignata, fiat circulus:</s> <s xml:id="echoid-s6298" xml:space="preserve"> erit quidẽ circulus iſte maior circu <lb/>lo foraminis:</s> <s xml:id="echoid-s6299" xml:space="preserve"> quoniam proceſſus lucis perforamen ingredientis, eſt per modum pyramidis.</s> <s xml:id="echoid-s6300" xml:space="preserve"> Vnde <lb/>palàm, quòd lux deſcendens per axem, reflectitur ſuper eundem.</s> <s xml:id="echoid-s6301" xml:space="preserve"> Veruntamen apparebit lux circu <lb/>laris circa baſim interiorem foraminis, maioris quidem capacitatis luce incidente uel radio, & ma-<lb/>ioris etiã lucis, interioris lucis circulo:</s> <s xml:id="echoid-s6302" xml:space="preserve"> & palã eſt, hãc lucẽ eſſe per reflexionẽ:</s> <s xml:id="echoid-s6303" xml:space="preserve"> uerùm nõ per reflexi <lb/>onem lucis ſuper axem deſcendentis:</s> <s xml:id="echoid-s6304" xml:space="preserve"> quod ex hoc poterit patére.</s> <s xml:id="echoid-s6305" xml:space="preserve"> Obturata utraq;</s> <s xml:id="echoid-s6306" xml:space="preserve"> foraminis baſi, <lb/>ut quaſi ſola remaneat axis uia, & radio ſolis per uiam axis deſcendente:</s> <s xml:id="echoid-s6307" xml:space="preserve"> nõ apparebit lux illa circu <lb/>laris, circa interiorẽ baſim foraminis.</s> <s xml:id="echoid-s6308" xml:space="preserve"> Quare nõ procedebat ex reflexa lúcis axe.</s> <s xml:id="echoid-s6309" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s6310" xml:space="preserve"> ut ſuprà <lb/>quidẽ ſuppoſuimus, ut regula orthogonaliter caderet in quadratũ concauũ;</s> <s xml:id="echoid-s6311" xml:space="preserve"><unsure/> ſi aliquantulũ inde au <lb/>feratur, ut regula declinetur, ita, ut extremitas à quadrato remotior, ſit demiſsior radio deſcen-<lb/>dente ſuper foramen mediũ:</s> <s xml:id="echoid-s6312" xml:space="preserve"> non cadet perpen diculariter ſuper ſpeculũ:</s> <s xml:id="echoid-s6313" xml:space="preserve"> & apparebit lux reflexa à <lb/>foramine medio remota:</s> <s xml:id="echoid-s6314" xml:space="preserve"> & quantò maior erit declinatio, tantò maior erit lucis reflexæ à foramine <lb/>remotio.</s> <s xml:id="echoid-s6315" xml:space="preserve"> Si uerò ad rectitudinẽ regula reducatur, lux reflexa circa interiorẽ foraminis baſim, ut pri <lb/>us, uidebitur.</s> <s xml:id="echoid-s6316" xml:space="preserve"> Palã igitur, quòd luce ſuper ſpeculũ perpendiculariter cadẽte, regreditur ad foramẽ, <lb/>per quod ingreſſa eſt.</s> <s xml:id="echoid-s6317" xml:space="preserve"> Cũ uerò lux axis declinata ceciderit, nõ reflectetur ad foramen, per quod in-<lb/>greſſa eſt, ſed tamẽ apparebit centrum lucis ſemper ſuper lineam ſuperficiei concauæ annuli, per-<lb/>pendicularem ſuper tabulam æneam, & deſcendentem per centrum foraminis medij.</s> <s xml:id="echoid-s6318" xml:space="preserve"> Quæcun-<lb/>que autem dicta ſunt in duobus foraminibus primis declinatis:</s> <s xml:id="echoid-s6319" xml:space="preserve"> intellige in ſingulis:</s> <s xml:id="echoid-s6320" xml:space="preserve"> & quod dictũ <lb/>eſt in ſpeculo plano, de luce per foramen ſeu declinatum ſeu medium deſcendente:</s> <s xml:id="echoid-s6321" xml:space="preserve"> regula ſeu recta <lb/>ſeu declanata:</s> <s xml:id="echoid-s6322" xml:space="preserve"> in alijs ſpeculis intellige.</s> <s xml:id="echoid-s6323" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div225" type="section" level="0" n="0"> <head xml:id="echoid-head257" xml:space="preserve" style="it">12. In ſpeculis, conuexis, cauis: ſphærico, conico cylindraceo, anguli incidentiæ & reflexio-<lb/>nis æquantur. 12.13.14.15.16.17.20 p 5.</head> <p> <s xml:id="echoid-s6324" xml:space="preserve">SIautẽ regula, in qua fuerit ſpeculũ columnare extrà politũ, declinetur in quadrato, ita ut non <lb/>orthogonaliter cadat ſuper quadratum, ſed declinetur ſuper partem dextram uel ſiniſtram:</s> <s xml:id="echoid-s6325" xml:space="preserve"> <lb/> <pb o="109" file="0115" n="115" rhead="OPTICAE LIBER IIII."/> <gap/>pparebit tamen lux reflecti ſuper foramen, ſimile eius deſcenſui, & medium lucis ſuper medium <lb/>foraminis, ſicut uiſum eſt in regula non declinata.</s> <s xml:id="echoid-s6326" xml:space="preserve"> Regulam, in qua ſitum eſt columnare concauum, <lb/>impones, ut deſcendat acumen tabulæ æneæ, donec tangat ſuperficiem ſpeculi:</s> <s xml:id="echoid-s6327" xml:space="preserve"> & declinabis hoc <lb/>ſpeculum ſecundum latus ſuum, ſicut declinaſti extrà politum.</s> <s xml:id="echoid-s6328" xml:space="preserve"> Idem in ſpeculis pyramidalibus con <lb/>cauis operaberis.</s> <s xml:id="echoid-s6329" xml:space="preserve"> Sphæricum concauũ aptetur, donec deſcendat acumen tabulæ æneæ in foramen <lb/>ſpeculi, factum ſecundum acuminis deſcenſum.</s> <s xml:id="echoid-s6330" xml:space="preserve"> Sphæricum extrà politum ſic imponatur, ut acu-<lb/>men tabulæ æneæ ſit in ſuperficie regulæ, & in eadem ſuperficie cum medio ſpeculi puncto:</s> <s xml:id="echoid-s6331" xml:space="preserve"> quod <lb/>ſic fieri poterit.</s> <s xml:id="echoid-s6332" xml:space="preserve"> Adhibeatur regula acuta regulę, & puncto ſpeculi medio, & deſcendat acumen tabu <lb/>læ æneæ, quouſq;</s> <s xml:id="echoid-s6333" xml:space="preserve"> ſit in directo acuitatis regulæ:</s> <s xml:id="echoid-s6334" xml:space="preserve"> & tunc cogatur ſiſtere.</s> <s xml:id="echoid-s6335" xml:space="preserve"> In ſpeculis columnaribus ui <lb/>debis reflexionem hoc modo.</s> <s xml:id="echoid-s6336" xml:space="preserve"> Aptetur ſpeculum, ſicut dictum eſt:</s> <s xml:id="echoid-s6337" xml:space="preserve"> & per foramen medium deſcen-<lb/>dat baculus columnaris, ſicut factũ eſt in ſpeculis planis:</s> <s xml:id="echoid-s6338" xml:space="preserve"> Cadet quidem baculus ſuper mediam lon <lb/>gitudinis ſpeculi lineam, & erit eius terminus in ſuperficie regulę.</s> <s xml:id="echoid-s6339" xml:space="preserve"> Super mediam igitur lineã ſigne-<lb/>tur punctum, in quod cadit:</s> <s xml:id="echoid-s6340" xml:space="preserve"> & ab hoc puncto in ſuperficie regulæ ſumatur longitudo ſemidiametri <lb/>circuli facti in regula, ad diſcernendum circularem lucis caſum:</s> <s xml:id="echoid-s6341" xml:space="preserve"> & ex alia parte puncti ſumatur lon-<lb/>gitudo eadem, & habebitur linea æqualis diametro prædicti circuli.</s> <s xml:id="echoid-s6342" xml:space="preserve"> Videbitur autem lux cadens, <lb/>extendi ſuper præd ctam lineam tantùm, & reflectetur ad foramen medium:</s> <s xml:id="echoid-s6343" xml:space="preserve"> & circa eius baſim in-<lb/>teriorem uidebitur lux circularis maior circulo interiori, ſicut in ſpeculis planis uiſum eſt.</s> <s xml:id="echoid-s6344" xml:space="preserve"> Idem in <lb/>ſpeculis pyramidalibus uidere poteris.</s> <s xml:id="echoid-s6345" xml:space="preserve"> Pari modo in ſpeculis ſphæricis, luce per foramen mediũ <lb/>deſcendente:</s> <s xml:id="echoid-s6346" xml:space="preserve"> fiat circulus in ſuperficie regulæ ad quantitatem circuli iam dicti:</s> <s xml:id="echoid-s6347" xml:space="preserve"> & uidebitur lux ex-<lb/>tendi ſuper hunc circulum, & reflecti ad foramen medium modo iam dicto.</s> <s xml:id="echoid-s6348" xml:space="preserve"> Et apparebit in his o-<lb/>mnibus rectis reflexionibus, linea perpendicularis in interiore ſuperficie annuli ſecare lucẽ circu-<lb/>larem reflexam, & diuidere circulum eius per medium.</s> <s xml:id="echoid-s6349" xml:space="preserve"> Quod autem dictum eſt de luce naturali:</s> <s xml:id="echoid-s6350" xml:space="preserve"> ui-<lb/>deri poterit in luce accidentali.</s> <s xml:id="echoid-s6351" xml:space="preserve"> Domus unici foraminis opponatur parieti, in quem deſcendit ra-<lb/>dius ſolis, & applicetur inſtrumentum foramini.</s> <s xml:id="echoid-s6352" xml:space="preserve"> Cum ergo intrauerit lux accidentalis per foramen <lb/>non medium, uidebitur reflecti per eius oppoſitum:</s> <s xml:id="echoid-s6353" xml:space="preserve"> & ſi aptetur inſtrumentum, utintret per duo <lb/>foramina, reflectetur per duo ſimilia.</s> <s xml:id="echoid-s6354" xml:space="preserve"> Verùm ut poſsis perpendere lucem, cum intrauerit directè:</s> <s xml:id="echoid-s6355" xml:space="preserve"> <lb/>appone ſuperius pergamenum album, & inclina inſtrumentum, donec uideas locem cadentem ſu-<lb/>per pergamenum:</s> <s xml:id="echoid-s6356" xml:space="preserve"> in ſpeculis enim non plenè comprehenditur lucis accidentalis caſus, propter de-<lb/>bilitatẽ eius.</s> <s xml:id="echoid-s6357" xml:space="preserve"> Idem autem in hac luce patebit, quod in naturali patuit:</s> <s xml:id="echoid-s6358" xml:space="preserve"> non enim eſt diuerſitas in ea-<lb/>rum natura, niſi quòd una'<unsure/>fortis eſt, & alia debilis.</s> <s xml:id="echoid-s6359" xml:space="preserve"> Palàm ergo, quòd luces per diuerſas lineas ad ſpe <lb/>cula accedentes, per diuerſas reflectuntur lineas:</s> <s xml:id="echoid-s6360" xml:space="preserve"> & quòd ſecundum rectam perpendicularem in-<lb/>cidentes, ſecundum eandem regrediuntur:</s> <s xml:id="echoid-s6361" xml:space="preserve"> & quòd declinatio linearũ reflexionis, eſt æqualis decli <lb/>nationi linearum acceſſus.</s> <s xml:id="echoid-s6362" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div226" type="section" level="0" n="0"> <head xml:id="echoid-head258" xml:space="preserve" style="it">13. Superficies reflexionis eſt perpendicularis plano ſpeculum in reflexionis puncto tan-<lb/>genti. 25 p 5.</head> <p> <s xml:id="echoid-s6363" xml:space="preserve">ET planũ, quòd lineæ lucis reflexæ & aduenientis, ſunt in eadem ſuperficie orthogonali ſuper <lb/>ſuperficiem politi, aut ſuperficiem contingentem punctum politi, a quo fit reflexio:</s> <s xml:id="echoid-s6364" xml:space="preserve"> & ſi lux ſu <lb/>per perpendicularem uenerit, reflectitur ſuper perpendicularem:</s> <s xml:id="echoid-s6365" xml:space="preserve"> & in quodcunq;</s> <s xml:id="echoid-s6366" xml:space="preserve"> punctum <lb/>ceciderit, reflectitur in ſuperficie perpendiculari, ſuper ſuperficiem tangentem illud punctum:</s> <s xml:id="echoid-s6367" xml:space="preserve"> & <lb/>ſemper linea reflexa cum perpendiculari ſuper illud punctum, æqualem tenet angulum, angulo, <lb/>quem includit linea ueniens cum eadem perpendiculari.</s> <s xml:id="echoid-s6368" xml:space="preserve"> Et huius rei probatio eſt.</s> <s xml:id="echoid-s6369" xml:space="preserve"> Quia palàm [per <lb/>10.</s> <s xml:id="echoid-s6370" xml:space="preserve">11.</s> <s xml:id="echoid-s6371" xml:space="preserve">12.</s> <s xml:id="echoid-s6372" xml:space="preserve">n] quòd ſi deſcendat lux quæcunq;</s> <s xml:id="echoid-s6373" xml:space="preserve"> per foramen aliquod:</s> <s xml:id="echoid-s6374" xml:space="preserve"> reflectitur per aliud ipſum reſpi-<lb/>ciens:</s> <s xml:id="echoid-s6375" xml:space="preserve"> & ſi conſtrin gatur foramen, ut reſtet quaſi ſolus axis:</s> <s xml:id="echoid-s6376" xml:space="preserve"> reflectitur per axem reſpicientis:</s> <s xml:id="echoid-s6377" xml:space="preserve"> & ſi <lb/>fiat alteratio deſcenſus lucis:</s> <s xml:id="echoid-s6378" xml:space="preserve"> reflectitur per lineas, per quas prius deſcenderat.</s> <s xml:id="echoid-s6379" xml:space="preserve"> Et palàm [ex inſtru-<lb/>menti reflexionis cõſtructione] quòd foramina ſe reſpicientia eundem habent ſitum, reſpectu me-<lb/>dij.</s> <s xml:id="echoid-s6380" xml:space="preserve"> Et cum non procedat lux, niſi per rectas lineas:</s> <s xml:id="echoid-s6381" xml:space="preserve"> palam, quòd reflectitur per lineas eiuſdem ſitus, <lb/>reſpectu medij foraminis, cum lineis deſcenſus:</s> <s xml:id="echoid-s6382" xml:space="preserve"> Vnde cũ accedit per orthogonalẽ, per eam reflecti-<lb/>tur ſolam.</s> <s xml:id="echoid-s6383" xml:space="preserve"> Quare ſemper lineæ reflexionis eundem ſeruant ſitum cum lineis deſcenſus, reſpectu ſu-<lb/>perficiei contingentis punctum reflexionis.</s> <s xml:id="echoid-s6384" xml:space="preserve"> Et hoc uerum eſt ſiue in ſubſtantiali ſiue in accidenta-<lb/>li luce, ſiue forti ſiue debili:</s> <s xml:id="echoid-s6385" xml:space="preserve"> & generaliter in omni.</s> <s xml:id="echoid-s6386" xml:space="preserve"> Et nos oſtendemus identitatẽ ſitus.</s> <s xml:id="echoid-s6387" xml:space="preserve"> Iam ſcimus, <lb/>quòd ſuperficies regulæ cadit ſuper tabulam, in qua quadratum fecimus, orthogonaliter.</s> <s xml:id="echoid-s6388" xml:space="preserve"> Igitur li-<lb/>nea media tabulæ quadrati orthogonalis eſt ſuper lineam communem ſectioni ipſius & regulæ, & <lb/>ſuper lineam latitudinis regulæ:</s> <s xml:id="echoid-s6389" xml:space="preserve"> Et tabula quadrati æquidiſtat æneæ tabulæ, & linea eius, id eſt, ta-<lb/>bulæ quadratæ concauæ media, æquidiſtat lineæ mediæ tabulæ æneæ, quę eſt linea à centro tabu-<lb/>læ æneæ producta, & diuidens ſemicirculum per æqualia.</s> <s xml:id="echoid-s6390" xml:space="preserve"> Linea autem comunis ſuperficιei tabulę <lb/>æneæ & ſuperficiei regulæ, in qua eſt linea latitudinis, eſt æquidiſtãs lineæ communi concauę tabu <lb/>læ & regulæ [per 28 p 1:</s> <s xml:id="echoid-s6391" xml:space="preserve"> linea enim longitudinis regulæ rectè ſecat latitudinis lineas.</s> <s xml:id="echoid-s6392" xml:space="preserve">] Quare linea <lb/>media tabulæ æneæ cadet perpendiculariter ſuper lineam cõmunem regulæ & tabulę æneæ.</s> <s xml:id="echoid-s6393" xml:space="preserve"> Et re-<lb/>gula perpendicularis eſt ſuper ſuperficiem quadrati, & ſuperficies quadrati æquidiſtans ſuperficiei <lb/>tabulæ æneę Quare ſuperficies tabulę æneę orthogonalis eſt ſuper ſuperficiem regulę:</s> <s xml:id="echoid-s6394" xml:space="preserve"> & linea me-<lb/>dia latitudinis regulę, eſt perpendicularis ſuper mediam longitudinis regulæ lineam:</s> <s xml:id="echoid-s6395" xml:space="preserve"> & ſimiliter li-<lb/>nea media tabulę æneę, eſt perpendicularis ſuper eandẽ.</s> <s xml:id="echoid-s6396" xml:space="preserve"> Et ita media linea tabulę æneæ eſt perpen <lb/>dicularis ſuper ſuperficiem regulæ, & ſuper mediam longitudinis eius lineam:</s> <s xml:id="echoid-s6397" xml:space="preserve"> Eſt ergo perpendicu <lb/> <anchor type="figure" xlink:label="fig-0115-01a" xlink:href="fig-0115-01"/> <pb o="110" file="0116" n="116" rhead="ALHAZEN"/> laris ſuper ſuperficiẽ ſpeculi plani, & ſuper mediã longitudinis eius lineã.</s> <s xml:id="echoid-s6398" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s6399" xml:space="preserve"> ſuperficies tabu <lb/>læ æneę eſt æquidiſtans ſuperficiei deſcendenti per centra foraminũ.</s> <s xml:id="echoid-s6400" xml:space="preserve"> Nam longitudo centrorũ à ſu <lb/>perficie tabulę æneę eſt eadem, id eſt medietatis unius grani hordei, & diameter foraminis eſt unius <lb/>grani hordei:</s> <s xml:id="echoid-s6401" xml:space="preserve"> ſimiliter latitudo ſuperficiei columnæ eſt unius grani:</s> <s xml:id="echoid-s6402" xml:space="preserve"> & ſuperficies deſcendens per <lb/>centra foraminum, ſecat columnã per medium:</s> <s xml:id="echoid-s6403" xml:space="preserve"> & ita axis columnæ eſt in ſuperficie illa.</s> <s xml:id="echoid-s6404" xml:space="preserve"> Et columna <lb/>deſcenſu ſuo tangit lineã in tabula ænea, cui quidẽ æquidiſtat axis:</s> <s xml:id="echoid-s6405" xml:space="preserve"> quoniã axis eſt æquidiſtans cuili <lb/>bet lineę ſuperficiei columnæ.</s> <s xml:id="echoid-s6406" xml:space="preserve"> Et axis colũnæ cadit in punctũ ſuperficiei regulæ, à quo puncto linea <lb/>ducta ad centrum tabulę æneę, eſt perpendicularis ſuper tabulam æneam:</s> <s xml:id="echoid-s6407" xml:space="preserve"> quoniam per quodcunq;</s> <s xml:id="echoid-s6408" xml:space="preserve"> <lb/>foramen deſcendat columna:</s> <s xml:id="echoid-s6409" xml:space="preserve"> axis eius cadit ſuper mediã longitudinis regulæ lineam:</s> <s xml:id="echoid-s6410" xml:space="preserve"> & linea pro-<lb/>tracta à puncto regulæ, in quod cadit axis per centra foraminum, eſt æ quidiſtans lineę protractæ à <lb/>centro tabulę æneę ad terminum diametri foraminis:</s> <s xml:id="echoid-s6411" xml:space="preserve"> [per 33 p 1] quoniam linea inter punctum il-<lb/>lud & centrum eſt orthogonalis ſuper ſuperficiem tabulę æneę, cum ſit pars lineę medię longitudi-<lb/>nis regulę:</s> <s xml:id="echoid-s6412" xml:space="preserve"> & huic lineę interiacenti centrum tabulę æneę & punctum, eſt æquidιſtans linea annuli, <lb/>tranſiẽs per centra foraminũ, & perpendiculariter cadẽs ſuper ſuperficiem tabulę æneę [per 6 p 11:</s> <s xml:id="echoid-s6413" xml:space="preserve"> <lb/>Vtraq;</s> <s xml:id="echoid-s6414" xml:space="preserve"> enim linea ad perpendiculum eſt tabulę æneę.</s> <s xml:id="echoid-s6415" xml:space="preserve">] Quare æquidiſtantes erunt lineę cadentes à <lb/>puncto regulę ad centra foraminũ, lineis à tabulæ æneę centro, ad terminos diametrorũ eorundem <lb/>foraminũ in ſuperficie tabulę ductis.</s> <s xml:id="echoid-s6416" xml:space="preserve"> Pari modo in ſingulis foraminibus.</s> <s xml:id="echoid-s6417" xml:space="preserve"> Quare lineę à puncto regu <lb/>læ, in quod cadιt axis, productę ad centrũ duorum foraminũ ſe reſpicientiũ, æquidiſtantes duabus <lb/>lineis, à centro tabulę æneę ad extremitates diametrorũ eorundem foraminũ protractis, æqualem <lb/>cum his lineis tenent angulũ [per 10 p 11.</s> <s xml:id="echoid-s6418" xml:space="preserve">] Et ſi à termino axis erigatur linea ad centrũ foraminis:</s> <s xml:id="echoid-s6419" xml:space="preserve"> <lb/>erit in ſuperficie per centrum deſcendente, & erit æquidiſtans medię lineę tabulę æneę:</s> <s xml:id="echoid-s6420" xml:space="preserve"> [per 6 p 11] <lb/>quoniã lιnea inferior interiacens capita earũ, eſt perpendicularis ſuper tabulam æneam, & æqualis <lb/>ſuperiori eadẽ capita interiacenti, & ſuper tabulã æneã perpendiculari:</s> <s xml:id="echoid-s6421" xml:space="preserve"> & eſt æquidiſtans ei.</s> <s xml:id="echoid-s6422" xml:space="preserve"> Et ſimi <lb/>liter linea à centro foraminis medij ad terminũ axis colũnę, eſt æquidiſtãs medię lineę tabulę æneę:</s> <s xml:id="echoid-s6423" xml:space="preserve"> <lb/>& eſt illa perpendicularis ſuper regulã:</s> <s xml:id="echoid-s6424" xml:space="preserve"> quare & iſta [per 8 p 11.</s> <s xml:id="echoid-s6425" xml:space="preserve">] Igitur hęc linea & altera an gulũ cõti <lb/>nentes, æquidiſtant medię lineę tabulę æneę, & alteri lineæ in tabula ænea reliquũ angulũ cõtinen-<lb/>ti.</s> <s xml:id="echoid-s6426" xml:space="preserve"> Quare anguli partiales ſibi oppoſiti ſunt æquales [per 10 p 11.</s> <s xml:id="echoid-s6427" xml:space="preserve">] Igitur linea tabulæ æneę media di-<lb/>uidit angulum ſuum per æqualia.</s> <s xml:id="echoid-s6428" xml:space="preserve"> Quare linea à centro foraminis medij, diuidit angulum ſuum per <lb/>æqualia.</s> <s xml:id="echoid-s6429" xml:space="preserve"> Et cum certum ſit, quòd lux foramen declinatum intrans per illas lineas angulũ continen-<lb/>tes moueatur:</s> <s xml:id="echoid-s6430" xml:space="preserve"> planũ, quòd lux omnis reflectitur per lineas, quę cum lineis deſcenſus ſunt in ſuper-<lb/>ficie orthogonali ſuper ſuperficiem reflexionis, & angulum æqualem facientes cum linea perpen-<lb/>diculari, angulo, quẽ continet perpendicularis cum lineis deſcẽſus:</s> <s xml:id="echoid-s6431" xml:space="preserve"> & quòd lux perpendiculariter <lb/>deſcendens:</s> <s xml:id="echoid-s6432" xml:space="preserve"> reflectitur per perpendicularem Et hoc generale eſt in omni luce.</s> <s xml:id="echoid-s6433" xml:space="preserve"> Si aũt declinetur re <lb/>gula, non in latus ſuũ, ſed in caput, ut axis foraminis medij non ſit perpendicularis ſuper regulã:</s> <s xml:id="echoid-s6434" xml:space="preserve"> re-<lb/>flectetur lux, & uidebitur ſuper lineam altitudinis annuli perpendicularẽ, & per centrum foraminis <lb/>tranſeuntẽ:</s> <s xml:id="echoid-s6435" xml:space="preserve"> & quantò maior fuerit declinatio, tantò maior erit lucis reflexæ à foramine uel axe elon <lb/>gatio:</s> <s xml:id="echoid-s6436" xml:space="preserve"> & ſi diminuatur declinatio, diminuetur elongatiò:</s> <s xml:id="echoid-s6437" xml:space="preserve"> & ita, donec ſitus regulæ ad rectitudinẽ re-<lb/>grediatur, & ſuper perpendicularẽ illã reflectatur lux.</s> <s xml:id="echoid-s6438" xml:space="preserve"> Quòd aũt in hac declinatione axis foraminis <lb/>medij & linea reflexionis, ſint in eadem ſuperficie orthogonali ſuper ſuperficiem reflexionis, planũ <lb/>per hoc.</s> <s xml:id="echoid-s6439" xml:space="preserve"> Quoniã enim axis foraminis medij eſt perpendicularis ſuper latitudinẽ regulæ, id eſt ſuper <lb/>lineã communẽ ſuperficiei regulę, & ſuperficiei per centra foraminũ deſcendentis, & media linea ta <lb/>bulę, ſcilicet annuli, eſt æquidiſtãs huic axi, & æquidiſtans medię lineæ tabulæ æneę, & media linea <lb/>tabulę æneę eſt perpendicularis ſuper latitudinẽ regulę, & ſuper lineã cõmunem ſuperficiei regulę, <lb/>& ſuperficiei tabulę æneę.</s> <s xml:id="echoid-s6440" xml:space="preserve"> Quare ſuperficies, in qua ſunt, media linea tabulę æneę, & axis foraminis <lb/>medij, etiã orthogonalis eſt ſuper ſuperficiem regulę:</s> <s xml:id="echoid-s6441" xml:space="preserve"> & in hac ſuperficie eſt linea perpendicularis <lb/>in altitudine annuli:</s> <s xml:id="echoid-s6442" xml:space="preserve"> [per 7 p 11] quoniã tranſit per terminos æquidiſtantium, ſcilicet medię tabulę <lb/>æneę & axis foraminis medij.</s> <s xml:id="echoid-s6443" xml:space="preserve"> Palàm igitur, quòd lux reflexa, quę apparet in perpendiculari altitudi <lb/>nis annuli, reflectitur per lineam, quę cum axe, per quem fit deſcenſus, eſt in ſuperficie orthogonali <lb/>ſuper ſuperficiem regulę.</s> <s xml:id="echoid-s6444" xml:space="preserve"> Luce ergo deſcendente in ſpeculum planum, fit reflexio ſecundũ lineas, <lb/>quarũ eadem declinatio ſuper ſuperficiem ſpeculi:</s> <s xml:id="echoid-s6445" xml:space="preserve"> & ipſę ſunt cum perpendiculari in ſuperficie or-<lb/>thogonali ſuper ſpeculi ſuperficiem.</s> <s xml:id="echoid-s6446" xml:space="preserve"> In ſpeculo columnari exteriori eadẽ penitus probatio, quę eſt <lb/>in plano:</s> <s xml:id="echoid-s6447" xml:space="preserve"> ſcilicet quòd acumẽ tabulę æneę cadat ſuper lineam longitudinis ſpeculi orthogonaliter:</s> <s xml:id="echoid-s6448" xml:space="preserve"> <lb/>& ſimiliter colũna deſcen dẽs ſuper eandẽ:</s> <s xml:id="echoid-s6449" xml:space="preserve"> & pars illius lineę inter hos caſus eſt orthogonalis ſuper <lb/>tabulã æneam.</s> <s xml:id="echoid-s6450" xml:space="preserve"> Et ſemper, ſiue per foramẽ mediũ, ſiue per declinatũ deſcen derit lux:</s> <s xml:id="echoid-s6451" xml:space="preserve"> reflexio eius cũ <lb/>deſcenſu erit in eadem ſuperficie, orthogonali ſuper ſuperficiem contingentẽ lineam longitudinis <lb/>ſpeculi.</s> <s xml:id="echoid-s6452" xml:space="preserve"> In pyramidali uero exteriori, cum ſuperficies regulæ ſit in eadem ſuperficie cum linea longi <lb/>tudinis pyramidis, ſicut in columnari:</s> <s xml:id="echoid-s6453" xml:space="preserve"> erit idem ſitus linearũ ſuperficiei, & idem reflexionis modus, <lb/>ſicut in plano ſpeculo, & eadem penitus probatio.</s> <s xml:id="echoid-s6454" xml:space="preserve"> In ſpeculo columnari concauo deſcẽdit acumen <lb/>tabulæ æneę uſq;</s> <s xml:id="echoid-s6455" xml:space="preserve"> ad lineam longitudinis eius mediam, & ſuper eandẽ cadit axis cuiuſq;</s> <s xml:id="echoid-s6456" xml:space="preserve"> foraminis:</s> <s xml:id="echoid-s6457" xml:space="preserve"> <lb/>& pars illius lineę inter hos caſus eſt orthogonalis ſuper ſuperficiẽ tabulę æneę:</s> <s xml:id="echoid-s6458" xml:space="preserve"> & axis foraminis, <lb/>& media linea tabulę æneæ, ſunt orthogonales ſuper ſuperficiem, tangentẽ ſpeculum illud in linea <lb/>longitudinis (quę eſt locus reflexionis) & æquidiſtantẽ ſuperficiei regulę.</s> <s xml:id="echoid-s6459" xml:space="preserve"> Et ita idẽ modus proban <lb/>di, qui prius:</s> <s xml:id="echoid-s6460" xml:space="preserve"> quòd ſcilicet reflexio & deſcenſus ſint in eadẽ ſuperficie, orthogonali ſuper ſuperficiẽ <lb/>loci reflexionis:</s> <s xml:id="echoid-s6461" xml:space="preserve"> & quòd eiuſdẽ ſint declinationis:</s> <s xml:id="echoid-s6462" xml:space="preserve"> & quòd deſcenſus per mediũ, efficit reflexionem <lb/> <anchor type="figure" xlink:label="fig-0116-01a" xlink:href="fig-0116-01"/> <pb o="111" file="0117" n="117" rhead="OPTICAE LIBER IIII."/> per ipſum:</s> <s xml:id="echoid-s6463" xml:space="preserve"> & declinato capite regulæ:</s> <s xml:id="echoid-s6464" xml:space="preserve"> erit reflexio ſuper perpendicularẽ annuli, ſicut dictũ eſt in pla <lb/>no.</s> <s xml:id="echoid-s6465" xml:space="preserve"> In ſpeculo pyramidali concauo eadẽ in omnibus probatio.</s> <s xml:id="echoid-s6466" xml:space="preserve"> In ſpeculo ſphęrico exteriori palàm, <lb/>quòd mediũ eius punctũ eſt in ſuperficie regulæ, & axis cadit in punctũ illud:</s> <s xml:id="echoid-s6467" xml:space="preserve"> & erit in eo idẽ ſitus li <lb/>nearum & aliorũ penitus, qui in plano:</s> <s xml:id="echoid-s6468" xml:space="preserve"> & eadem demonſtratio.</s> <s xml:id="echoid-s6469" xml:space="preserve"> In ſpeculo ſphærico cõcauo iam de-<lb/>claratum eſt, [9 n] quòd axis foraminis deſcendat ad punctum eius mediũ, & acumen tabulæ æneę <lb/>tranſeat per foramẽ in ſpeculo iam factũ, uſq;</s> <s xml:id="echoid-s6470" xml:space="preserve"> dum ſit in eadem ſuperficie cum puncto illo medio:</s> <s xml:id="echoid-s6471" xml:space="preserve"> & <lb/>linea à puncto illo ad acumen protracta, eſt æquidiſtans mediæ lineę longitudinis regulæ.</s> <s xml:id="echoid-s6472" xml:space="preserve"> Et ita de-<lb/>ſcenſus & reflexio ſunt in eadẽ ſuperficie, orthogonali ſuper ſuperficiem contingentẽ ſpeculũ in illo <lb/>puncto mediò, & æquidiſtantẽ ſuperficiei regulę.</s> <s xml:id="echoid-s6473" xml:space="preserve"> Et eadem probatio penitus, quę in alijs.</s> <s xml:id="echoid-s6474" xml:space="preserve"> Planũ er-<lb/>go, quòd omnis lux, in quodcunq;</s> <s xml:id="echoid-s6475" xml:space="preserve"> horum ſpeculorũ ceciderit, reflexio & deſcenſus ſunt in eadẽ ſu-<lb/>perficie orthogonali.</s> <s xml:id="echoid-s6476" xml:space="preserve"> Hic aũt modus reflexionis non accidit ex proprietate axis uel puncti, in quod <lb/>cadit:</s> <s xml:id="echoid-s6477" xml:space="preserve"> uel foraminis, per quod intrat:</s> <s xml:id="echoid-s6478" xml:space="preserve"> uel ꝓprietate ſpeculi.</s> <s xml:id="echoid-s6479" xml:space="preserve"> Accidit enim in quolibet foramine, quæ-<lb/>cunq;</s> <s xml:id="echoid-s6480" xml:space="preserve"> ſit lux, & per quamcunq;</s> <s xml:id="echoid-s6481" xml:space="preserve"> lineã deſcendat, & in quodcunq;</s> <s xml:id="echoid-s6482" xml:space="preserve"> ſpeculi punctũ cadat.</s> <s xml:id="echoid-s6483" xml:space="preserve"> Quoniã quo-<lb/>cunq;</s> <s xml:id="echoid-s6484" xml:space="preserve"> puncto ſpeculi ſumpto, ſi lux in ipſum deſcendat, cũ idem ſit ei ſitus, reſpectu longitudinis ſpe <lb/>culi, & cuicunq;</s> <s xml:id="echoid-s6485" xml:space="preserve"> alij:</s> <s xml:id="echoid-s6486" xml:space="preserve"> erunt ſimiliter ijdem reſpectu linearũ ab eo protractarũ, quæ eiuſdẽ ſunt decli-<lb/>nationis cũ lineis à puncto priore intellectis, ſicut puncto priori uel cuicunq;</s> <s xml:id="echoid-s6487" xml:space="preserve"> alij.</s> <s xml:id="echoid-s6488" xml:space="preserve"> Et generaliter idẽ <lb/>eſt ſitus cuilibet puncto, in quod cadit lux, qui & in priore ſumpto, & reſpectu axis & reſpectu acu-<lb/>minis tabulæ æneę:</s> <s xml:id="echoid-s6489" xml:space="preserve"> & eadem in omnibus probatio, & ſimilis demonſtratio.</s> <s xml:id="echoid-s6490" xml:space="preserve"> Vnde eſt certũ, non eſſe <lb/>hoc ex proprietate lucis uel figura alicuius ſpeculi, ſed ex proprietate quadam communi rei politæ <lb/>& cuilibet luci.</s> <s xml:id="echoid-s6491" xml:space="preserve"> Si autem per diuerſa in quodcunq;</s> <s xml:id="echoid-s6492" xml:space="preserve"> punctum deſcenderit lux foramina, uidebitur re <lb/>flexio diuerſa, & angulorum diuerſitas ſuo deſcenſui conſona:</s> <s xml:id="echoid-s6493" xml:space="preserve"> & ſic in omnibus.</s> <s xml:id="echoid-s6494" xml:space="preserve"/> </p> <div xml:id="echoid-div226" type="float" level="0" n="0"> <figure xlink:label="fig-0115-01" xlink:href="fig-0115-01a"> <image file="0115-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0115-01"/> </figure> <figure xlink:label="fig-0116-01" xlink:href="fig-0116-01a"> <image file="0116-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0116-01"/> </figure> </div> </div> <div xml:id="echoid-div228" type="section" level="0" n="0"> <head xml:id="echoid-head259" xml:space="preserve" style="it">14. Inter uiſibile & ſpeculũ innumer abiles pyramides fiũt alternis baſib. & uerticib{us}. 22 p 5.</head> <p> <s xml:id="echoid-s6495" xml:space="preserve">MAnifeſtũ aũt ex ſuperioribus [2.</s> <s xml:id="echoid-s6496" xml:space="preserve">3 n] quòd ſi corpus politum opponatur corpori luminoſo:</s> <s xml:id="echoid-s6497" xml:space="preserve"> <lb/>cadet in quodlibet punctũ eius lux à quolibet puncto luminoſi:</s> <s xml:id="echoid-s6498" xml:space="preserve"> unde ſuper quodlibet politi <lb/>punctũ cadit pyramis, cuius acumẽ in eo, & ſuperficies luminoſi eſt baſis:</s> <s xml:id="echoid-s6499" xml:space="preserve"> & à quolibet pun <lb/>cto luminoſi procedit pyramis, cuius acumẽ in eo, & baſis ſuperficies politi.</s> <s xml:id="echoid-s6500" xml:space="preserve"> Si aũt inter luminoſum <lb/>& politũ intelligatur punctũ aliquod:</s> <s xml:id="echoid-s6501" xml:space="preserve"> ueniet quidẽ ad illud punctũ lux luminoſi, in modum pyrami <lb/>dis, cuius acumen in puncto, & latera huius pyramidis procedentia, uſq;</s> <s xml:id="echoid-s6502" xml:space="preserve"> dum cadant in ſuperficiem <lb/>politi, pyramidẽ efficiunt.</s> <s xml:id="echoid-s6503" xml:space="preserve"> Vnde in puncto intellecto erunt acumina duarũ pyramidũ, quarũ baſes <lb/>ſunt ſuperficies luminoſi & politi.</s> <s xml:id="echoid-s6504" xml:space="preserve"> Et ſi ad punctũ quodcũq;</s> <s xml:id="echoid-s6505" xml:space="preserve"> intermediũ intelligatur pyramis, cuius <lb/>baſis ſuperficies politi, & procedant huius pyramidis lineę:</s> <s xml:id="echoid-s6506" xml:space="preserve"> illud, quod occupabunt ex ſuperficie lu <lb/>minoſi, hoc eſt, à quo procedebat lux ad politũ:</s> <s xml:id="echoid-s6507" xml:space="preserve"> erit ſecun dũ duas pyramides, quarũ acumina ſunt in <lb/>puncto intellecto:</s> <s xml:id="echoid-s6508" xml:space="preserve"> & quicquid procedit lucis in his duabus pyramidibus, procedit & includitur in <lb/>duabus primis pyramidibus.</s> <s xml:id="echoid-s6509" xml:space="preserve"> Et à luminoſo ſecundũ lineas æquidiſtantes procedit lux ad ſpeculũ:</s> <s xml:id="echoid-s6510" xml:space="preserve"> <lb/>ſed hæ lineę includuntur in duabus primis pyramidibus:</s> <s xml:id="echoid-s6511" xml:space="preserve"> & per quaſcũq;</s> <s xml:id="echoid-s6512" xml:space="preserve"> lineas mouetur lux ad ſpe <lb/>culũ:</s> <s xml:id="echoid-s6513" xml:space="preserve"> obſeruant lineę reflexionis eundẽ penitus ſitum, quẽ habebant lineæ motus lucis.</s> <s xml:id="echoid-s6514" xml:space="preserve"> Vnde ſi mo <lb/>ueatur lux per æquidiſtantes, reflectitur per æquidiſtantes:</s> <s xml:id="echoid-s6515" xml:space="preserve"> & lux cadẽs in politũ, ad modũ pyrami-<lb/>dis reflectitur, obſeruãs modũ eiuſdem pyramidis.</s> <s xml:id="echoid-s6516" xml:space="preserve"> Et cũ deſcendit lux à corpore luminoſo per fora <lb/>men aliquod ad corpus politũ:</s> <s xml:id="echoid-s6517" xml:space="preserve"> ſi in ſuperficie foraminis ex parte luminoſi intelligatur pũctũ, à quo <lb/>puncto intelligãtur duę pyramides, baſis unius in luminoſo, alterius in polito:</s> <s xml:id="echoid-s6518" xml:space="preserve"> à ſola baſi pyramidis, <lb/>cuius luminoſum baſis:</s> <s xml:id="echoid-s6519" xml:space="preserve"> uenit lux ad politũ ſuper illud punctũ.</s> <s xml:id="echoid-s6520" xml:space="preserve"> Similiter ſi in ſuperficie foraminis <lb/>ex parte politi intelligatur punctũ, in quo acumina duarũ pyramidum, unius ad ſpeculũ, alterius ad <lb/>luminoſum:</s> <s xml:id="echoid-s6521" xml:space="preserve"> à ſola baſi pyramidis, quę baſis eſt in luminoſo, accedit lux ad ſpeculum ſuper hoc pun-<lb/>ctum:</s> <s xml:id="echoid-s6522" xml:space="preserve"> & à parte luminoſi his duabus pyramidibus cõmunis, accedit lux ad partẽ ſpeculi communẽ <lb/>duabus pyramidibus.</s> <s xml:id="echoid-s6523" xml:space="preserve"> Venit etiã lux à luminoſo ad ſpeculũ per lineas ęquidiſtãtes:</s> <s xml:id="echoid-s6524" xml:space="preserve"> ſed per quaſcũq;</s> <s xml:id="echoid-s6525" xml:space="preserve"> <lb/>accedat:</s> <s xml:id="echoid-s6526" xml:space="preserve"> fit reflexio modo prædicto:</s> <s xml:id="echoid-s6527" xml:space="preserve"> & quælibet lineę reflexionis obſeruant ſitum linearum deſcen <lb/>ſus lucis eas reſpicientium:</s> <s xml:id="echoid-s6528" xml:space="preserve"> & in omni reflexione obſeruatur identitas formę lucis, quę fuerit in po-<lb/>lito corpore:</s> <s xml:id="echoid-s6529" xml:space="preserve"> & hæc deinceps explanabimus explanatione euidenti.</s> <s xml:id="echoid-s6530" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div229" type="section" level="0" n="0"> <head xml:id="echoid-head260" xml:space="preserve" style="it">15. Lux à ſuperficie polita longinquiore reflexa, trifariam debilitatur.</head> <p> <s xml:id="echoid-s6531" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s6532" xml:space="preserve"> Patuit [4.</s> <s xml:id="echoid-s6533" xml:space="preserve">5 n] quòd lux quanto plus ab ortu ſuo elongatur, tantò plus debilitatur:</s> <s xml:id="echoid-s6534" xml:space="preserve"> <lb/>patuit etiã, quòd lux cõtinua fortior eſt diſgregata.</s> <s xml:id="echoid-s6535" xml:space="preserve"> Cũ igitur ab aliquo puncto luminoſi pro <lb/>cedit lux ad ſuperficiẽ ſpeculi in modũ pyramidis, quãto magis elongatur ab illo puncto:</s> <s xml:id="echoid-s6536" xml:space="preserve"> tan <lb/>tò maior erit eius debilitas duplici de cauſſa:</s> <s xml:id="echoid-s6537" xml:space="preserve"> & propter elongationẽ ab ortu ſuo, & propter diſgre-<lb/>gationẽ.</s> <s xml:id="echoid-s6538" xml:space="preserve"> Cum aũt ab aliquo ſpeculi puncto reflectitur lux iſta, fit debilior tripliciter:</s> <s xml:id="echoid-s6539" xml:space="preserve"> & propter refle <lb/>xionẽ, quæ debilitat, & propter elongationẽ à loco reflexionis, & propter diſgregationẽ.</s> <s xml:id="echoid-s6540" xml:space="preserve"> Si uerò lux <lb/>reflexa à ſpeculo aggregetur in punctũ aliquod:</s> <s xml:id="echoid-s6541" xml:space="preserve"> fiet quidẽ fortior propter aggregationẽ, ſed debilita <lb/>bitur propter reflexionẽ & elongationẽ.</s> <s xml:id="echoid-s6542" xml:space="preserve"> Si igitur aggregatio lucis tantũ reddit ei fortitudinis, quan <lb/>tum ſubtrahunt reflexio & elongatio:</s> <s xml:id="echoid-s6543" xml:space="preserve"> erit lux reflexa aggregata eiuſdẽ fortitudinis, cuius eſt in ſu-<lb/>perficie ſpeculi:</s> <s xml:id="echoid-s6544" xml:space="preserve"> ſi uerò aggre gatio minus addat fortitudinis, quàm diminuũt illa duo:</s> <s xml:id="echoid-s6545" xml:space="preserve"> erit debilior:</s> <s xml:id="echoid-s6546" xml:space="preserve"> <lb/>& ſi plus addat, erit fortior.</s> <s xml:id="echoid-s6547" xml:space="preserve"> Sumiliter ſi à ſuperficie luminoſi procedat pyramis ad aliquod punctum <lb/>ſpeculi:</s> <s xml:id="echoid-s6548" xml:space="preserve"> erit lux procedẽs ſecundum hanc pyramidalitatẽ debilior propter elongationẽ, ſed fortior <lb/>propter aggregationẽ.</s> <s xml:id="echoid-s6549" xml:space="preserve"> Si aũt aggregatio poteſt ſuper elongationẽ:</s> <s xml:id="echoid-s6550" xml:space="preserve"> erit lux in pũcto ſpeculi aggrega <lb/>ta fortior luce unica à luminoſo ueniente per lineã unã:</s> <s xml:id="echoid-s6551" xml:space="preserve"> unica dico:</s> <s xml:id="echoid-s6552" xml:space="preserve"> quia ad quodlibet punctũ lineæ <lb/> <pb o="112" file="0118" n="118" rhead="ALHAZEN"/> ex illis ſumptæ uenit etiã pyramis à luminoſo, quæ quidẽ pyramis cum ſimilibus excluditur in ha<gap/> <lb/>conſideratione.</s> <s xml:id="echoid-s6553" xml:space="preserve"> Si uerò elongatio ponderet ſuper aggregationem:</s> <s xml:id="echoid-s6554" xml:space="preserve"> erit lux puncti politi minor luce <lb/>ſola unius lineæ ſumpta:</s> <s xml:id="echoid-s6555" xml:space="preserve"> & ſi aggre gatio plus ponderet elongatione:</s> <s xml:id="echoid-s6556" xml:space="preserve"> erit fortior.</s> <s xml:id="echoid-s6557" xml:space="preserve"> Luces aũt, quę à lu <lb/>minoſo ad ſpeculũ accedunt ſuper lineas æquidiſtantes:</s> <s xml:id="echoid-s6558" xml:space="preserve"> erunt debiliores, quàm modo alio acceden <lb/>tes:</s> <s xml:id="echoid-s6559" xml:space="preserve"> quoniã debilitatę propter elon gationẽ non aggregantur in ſpeculũ, ſed in reflexione per lineas <lb/>æquidiſtantes mouentur:</s> <s xml:id="echoid-s6560" xml:space="preserve"> unde per reflexionẽ & elongationẽ debilitantur.</s> <s xml:id="echoid-s6561" xml:space="preserve"> Et ſi aggregentur in re-<lb/>flexione:</s> <s xml:id="echoid-s6562" xml:space="preserve"> conferetur eis fortitudo comparata ad fortitudinem, quam habuerunt in ſpeculo, ſecun-<lb/>dum poſſe aggregans ſuper reflexionem & elongationem.</s> <s xml:id="echoid-s6563" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div230" type="section" level="0" n="0"> <head xml:id="echoid-head261" xml:space="preserve" style="it">16. Lux & color reflectuntur per line{as} phyſic{as}, latitudine quadam prędit{as}. 3 p 2.6 p 5.</head> <p> <s xml:id="echoid-s6564" xml:space="preserve">AMplius.</s> <s xml:id="echoid-s6565" xml:space="preserve"> Omnis linea, per quã mouetur lux à corpore luminoſo ad corpus oppoſitũ, eſt linea <lb/>ſenſualis, non ſine latitudine.</s> <s xml:id="echoid-s6566" xml:space="preserve"> Lux enim nõ procedit, niſi à corpore, quoniã non eſt, niſi in cor <lb/>pore:</s> <s xml:id="echoid-s6567" xml:space="preserve"> ſed in minore luce, quæ ſumi poteſt, eſt latitudo, & in linea proceſſus eius eſt latitudo:</s> <s xml:id="echoid-s6568" xml:space="preserve"> <lb/>& in medio illius lineæ ſenſualis, eſt linea intellectualis, & aliæ eius lineæ ſunt æquidιſtantes illi.</s> <s xml:id="echoid-s6569" xml:space="preserve"> Et <lb/>ſi diuidatur minor ex lucibus:</s> <s xml:id="echoid-s6570" xml:space="preserve"> neutra eius pars erit lux, ſed utraq, extinguetur, nec apparebit.</s> <s xml:id="echoid-s6571" xml:space="preserve"> Si aũt <lb/>lux minima doplicetur, aut amplius multiplicetur, & cõpacta per æqualιa diuidatur:</s> <s xml:id="echoid-s6572" xml:space="preserve"> erit lux utraq;</s> <s xml:id="echoid-s6573" xml:space="preserve"> <lb/>pars eius:</s> <s xml:id="echoid-s6574" xml:space="preserve"> Si uerò per inæqualia fiat diuiſio:</s> <s xml:id="echoid-s6575" xml:space="preserve"> erit altera pars eius lux, altera minime.</s> <s xml:id="echoid-s6576" xml:space="preserve"> Lux aũt minima <lb/>procedit in minimã corporis partem, quã lux occupare poſsit, & proceſſus eius eſt ſecundum lineã <lb/>intellectualem, lineæ ſenſualis mediam, & extremitates ei æquidiſtantes.</s> <s xml:id="echoid-s6577" xml:space="preserve"> Et cadit lux minima non <lb/>in punctũ corporis intelligibile, ſed ſenſibile, & reflectitur per lineam ſenſibilem, cuius latitudo eſt <lb/>æqualis latitudini lineę ſenſibilis ueniẽtis.</s> <s xml:id="echoid-s6578" xml:space="preserve"> Et ſi intelligatur in linea ſinſibili, linea reflexa intellectua <lb/>lis media:</s> <s xml:id="echoid-s6579" xml:space="preserve"> eundẽ habebit ſitũ ſuper reflexiõis locũ, quẽ habet linea intelligιbιlis media, lineę ſenſibi-<lb/>lis ueniẽtis.</s> <s xml:id="echoid-s6580" xml:space="preserve"> Et quęlibet linea intellectualis in linea reflexa ſenſibili eundẽ penitus obſeruat ſitũ cũ li <lb/>nea intell gibili alterius lineę ſenſibilis, ipſã reſpiciẽte.</s> <s xml:id="echoid-s6581" xml:space="preserve"> Obſeruatur ergo in omniluce ratio linearũ <lb/>& punctorũ intellectorũ, licet ab eis aut per ipſas nõ procedat lux:</s> <s xml:id="echoid-s6582" xml:space="preserve"> & in hũc modũ erit reflexio lucis.</s> <s xml:id="echoid-s6583" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div231" type="section" level="0" n="0"> <head xml:id="echoid-head262" xml:space="preserve" style="it">17. Reflexio lucis & coloris à ſuperficie aſper a facta, plerun fugit uiſum. 1 p 5.</head> <p> <s xml:id="echoid-s6584" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s6585" xml:space="preserve"> quare ex politis corporibus, non ex aſperis fiat reflexio:</s> <s xml:id="echoid-s6586" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s6587" xml:space="preserve"> Quoniã lux, ut diximus, <lb/>non accedit ad corpus, niſi per motum citiſsimũ, & cum peruenerit ad politũ:</s> <s xml:id="echoid-s6588" xml:space="preserve"> eijcit eam po-<lb/>litum à ſe:</s> <s xml:id="echoid-s6589" xml:space="preserve"> corpus uerò aſperum non poteſt eam eijcere:</s> <s xml:id="echoid-s6590" xml:space="preserve"> quoniã in corpore aſpero ſunt pori, <lb/>quos lux ſubintrat:</s> <s xml:id="echoid-s6591" xml:space="preserve"> in polιtis aũt non inuenit poros, nec accidit eiectio hæc, propter corporis forti-<lb/>tudinẽ uel duriciem, quia uidemus in aqua reflexionẽ:</s> <s xml:id="echoid-s6592" xml:space="preserve"> ſed eſt hæc repulſio propria politurę:</s> <s xml:id="echoid-s6593" xml:space="preserve"> ſicut de <lb/>natura accidit, quod aliquid poroſum cadens ab alto ſuper lapidẽ durum, reuertitur in altũ:</s> <s xml:id="echoid-s6594" xml:space="preserve"> & quan <lb/>tò minor fuerit duricies lapidis, in quẽ ceciderit, tantò regreſsio cadentis debilior erit:</s> <s xml:id="echoid-s6595" xml:space="preserve"> & ſemper re <lb/>greditur cadens uerſus partẽ, à qua proceſsit:</s> <s xml:id="echoid-s6596" xml:space="preserve"> Verũ in arena propter eius mollitiẽ non fit regreſsio, <lb/>quæ accidit in corpore duro.</s> <s xml:id="echoid-s6597" xml:space="preserve"> Si aũt in poris aſperi corporis ſit politio:</s> <s xml:id="echoid-s6598" xml:space="preserve"> tamen lux intrans poros nõ <lb/>reflectitur:</s> <s xml:id="echoid-s6599" xml:space="preserve"> & ſi eam reflecti acciderit, diſpergitur, & propter diſperſionem à uiſu nõ percιpitur.</s> <s xml:id="echoid-s6600" xml:space="preserve"> Pari <lb/>modo ſi in aſpero corpore partes elatiores fuerint politæ:</s> <s xml:id="echoid-s6601" xml:space="preserve"> fiet reflexa diſperſio:</s> <s xml:id="echoid-s6602" xml:space="preserve"> & ob hoc occultabi-<lb/>tur uiſui.</s> <s xml:id="echoid-s6603" xml:space="preserve"> Si uerò eminentia partiũ adeò ſit modica, ut eius quaſi ſit idem ſitus cũ depreſsis:</s> <s xml:id="echoid-s6604" xml:space="preserve"> compre-<lb/>hendetur eius reflexio tan quam in polito, non aſpero, licet minus perfectè.</s> <s xml:id="echoid-s6605" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div232" type="section" level="0" n="0"> <head xml:id="echoid-head263" xml:space="preserve" style="it">18. Radij incidentiæ & reflexionis, ſit{us} ſimilitudine conueniunt. Ita anguli incidentiæ & <lb/>reflexionis æquantur. 20 p 5.</head> <p> <s xml:id="echoid-s6606" xml:space="preserve">QVare aũt fiat reflexio lucis ſecundum lineam eiuſdem ſitus cum linea, per quã accedit ad ſpe <lb/>culũ ipſa lux:</s> <s xml:id="echoid-s6607" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s6608" xml:space="preserve"> Quoniã lux motu citiſsimo mouetur:</s> <s xml:id="echoid-s6609" xml:space="preserve"> & quando cadit in ſpeculũ, nõ recipi <lb/>tur:</s> <s xml:id="echoid-s6610" xml:space="preserve"> ſed ei fixio in corpore illo negatur:</s> <s xml:id="echoid-s6611" xml:space="preserve"> & cũ in ea perſeueret adhuc prioris motus uis & na-<lb/>tura, reflectitur ad partẽ, à qua proceſsit, & ſecundũ lineas eundẽ ſitũ cũ prioribus habentes.</s> <s xml:id="echoid-s6612" xml:space="preserve"> Huius <lb/>aũt rei ſimile in naturalibus motibus uidere poſſumus, & etiã in accidentalibus.</s> <s xml:id="echoid-s6613" xml:space="preserve"> Si corpus ſphæricũ <lb/>ponderoſum ab aliqua altitudine deſcendere permittamus perpendiculariter ſuper politũ corpus:</s> <s xml:id="echoid-s6614" xml:space="preserve"> <lb/>uide bimus ipſum ſuper perpendicularẽ reflecti, per quã deſcenderat.</s> <s xml:id="echoid-s6615" xml:space="preserve"> In accidentali motu.</s> <s xml:id="echoid-s6616" xml:space="preserve"> Si eleue-<lb/>tur aliquod ſpeculum ſecundum aliquã altitudinẽ hominis, & firmiter in pariete figatur:</s> <s xml:id="echoid-s6617" xml:space="preserve"> & in acu-<lb/>mine ſagittæ cõſolidetur corpus ſphæricũ:</s> <s xml:id="echoid-s6618" xml:space="preserve"> & proijciatur ſagitta per arcum in ſpeculũ hoc modo, ut <lb/>eleuatio ſagittę ſit æqualis eleuationi ſpeculi, & ſit ſagitta æquidiſtans horizonti:</s> <s xml:id="echoid-s6619" xml:space="preserve"> planũ, quòd ſuper <lb/>perpendicularẽ accedit ſagitta ad ſpeculũ, & uidebitur ſuper eandẽ perpendicularẽ eius regreſſus.</s> <s xml:id="echoid-s6620" xml:space="preserve"> <lb/>Si uerò motus ſagittæ fuerit ſuper lineam declinatã in ipſum, uidebitur reflecti non per lineam, per <lb/>quam uenerat, ſed per lineam æquidiſtantẽ horizonti, ſicut & alia erat, & eiuſdẽ ſitus, reſpectu ſpe-<lb/>culi cum ea, & reſpectu' perpendicularis in ſpeculo.</s> <s xml:id="echoid-s6621" xml:space="preserve"> Quòd aũt ex prohibitione corporis politi acci-<lb/>dat luci motus reflexionis, palàm:</s> <s xml:id="echoid-s6622" xml:space="preserve"> quia cum fortior fuerit repulſio uel prohibitio, fortior erit lucis re <lb/>flexio.</s> <s xml:id="echoid-s6623" xml:space="preserve"> Quare aũt accidat idem motus reflexionis & eius acceſſus, hæc eſt ratio.</s> <s xml:id="echoid-s6624" xml:space="preserve"> Cum deſcendit cor <lb/>pus ponderoſum ſuper perpendicularẽ:</s> <s xml:id="echoid-s6625" xml:space="preserve"> repulſio corporis politi, & motus deſcendentis ponderoſi <lb/>directè ſibi ſunt oppoſiti, nec eſt ibi motus, niſi perpendicularis:</s> <s xml:id="echoid-s6626" xml:space="preserve"> & prohibitio fit per perpendicula-<lb/>rem:</s> <s xml:id="echoid-s6627" xml:space="preserve"> quare repellitur corpus ſecundum perpendicularẽ.</s> <s xml:id="echoid-s6628" xml:space="preserve"> Vnde perpendiculariter regreditur.</s> <s xml:id="echoid-s6629" xml:space="preserve"> Cum <lb/>uerò deſcenderit corpus ſuper lineam declinatã:</s> <s xml:id="echoid-s6630" xml:space="preserve"> cadit quidem linea deſcenſus inter perpendicula-<lb/>rem ſuperficiei politi, per ipſum politum tranſeuntẽ, & lineam ſuperficiei eius orthogonalem ſuper <lb/> <pb o="113" file="0119" n="119" rhead="OPTICAE LIBER IIII."/> hanc perpendicularẽ:</s> <s xml:id="echoid-s6631" xml:space="preserve"> & ſi penetraret motus ultra punctum, in quod cadit, ut liberũ inueniret tranſi <lb/>tum:</s> <s xml:id="echoid-s6632" xml:space="preserve"> caderet quidem hæclinea inter perpendicularem, tranſeuntem per politum, & lineam ſuperfi-<lb/>ciei orthogonalem ſuper perpendicularem, & obſeruaret menſuram ſitus, reſpectu perpendicularis <lb/>tranſeuntis, & reſpectu lineæ alterius, quæ orthogonalis eſt ſuper perpen dicularem illam.</s> <s xml:id="echoid-s6633" xml:space="preserve"> Compa-<lb/>cta eſt enim menſura ſitus huius motus ex ſitu ad perpendicularem & ſitu ad orthogonalem.</s> <s xml:id="echoid-s6634" xml:space="preserve"> Re-<lb/>pulſio uerò per perpendicularem incedens, cum no poſsit repellere motum ſecundum menſuram, <lb/>quam habet ad perpen dicularem tranſeuntem per politum:</s> <s xml:id="echoid-s6635" xml:space="preserve"> quia nec modicum intrat:</s> <s xml:id="echoid-s6636" xml:space="preserve"> repellit ergo <lb/>ſecundum menſuram ſitus ad perpendicularem, quam habet ad orthogonalem.</s> <s xml:id="echoid-s6637" xml:space="preserve"> Et quando motus <lb/>regreſsio eadem fuerit menſura ſitus ad orthogonalem, quæ fuit prius ad eandem ex alia parte:</s> <s xml:id="echoid-s6638" xml:space="preserve"> erit <lb/>ſimiliter ei eadem menſura ſitus ad perpendicularem tranſeuntem, quę fuit prius.</s> <s xml:id="echoid-s6639" xml:space="preserve"> Sed ponderoſum <lb/>corpus in regreſſu, cum finitur repulſionis motus, ex natura ſua deſcendit, & ad centrũ tendit.</s> <s xml:id="echoid-s6640" xml:space="preserve"> Lux <lb/>autem eandem habens reflectendi naturam, cum ei naturale non ſit aſcendere aut deſcendere, mo-<lb/>uetur in reflexione ſecundum lineam incœptam uſq;</s> <s xml:id="echoid-s6641" xml:space="preserve"> ad obſtaculum, quod ſiſtere faciat motum.</s> <s xml:id="echoid-s6642" xml:space="preserve"> Et <lb/>hæc eſt cauſſa reflexionis.</s> <s xml:id="echoid-s6643" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div233" type="section" level="0" n="0"> <head xml:id="echoid-head264" xml:space="preserve" style="it">19. Colorem luci permiſtum reflecti, reflexionis organo ostenditur. 3 p 5.</head> <p> <s xml:id="echoid-s6644" xml:space="preserve">PAtet etiam ex ſuperioribus, quòd colores ſimul mouentur cum lucibus:</s> <s xml:id="echoid-s6645" xml:space="preserve"> unde erit reflexio co-<lb/>loris, ſicut & lucis.</s> <s xml:id="echoid-s6646" xml:space="preserve"> Et ſi probationẽ eius uidere uolueris ſecundum modum in parte ſecunda <lb/>aſsignatũ, poteris:</s> <s xml:id="echoid-s6647" xml:space="preserve"> Verùm per inſtrumentũ ad hanc denotandam reflexionẽ factum, nõ plenè <lb/>uidebis propter debilitatẽ coloris.</s> <s xml:id="echoid-s6648" xml:space="preserve"> Debilitatur enim color per elongationẽ:</s> <s xml:id="echoid-s6649" xml:space="preserve"> per reflexionẽ:</s> <s xml:id="echoid-s6650" xml:space="preserve"> per fora <lb/>men, per quod intrat.</s> <s xml:id="echoid-s6651" xml:space="preserve"> Quòd aũt foramen debilitet, planũ per hoc:</s> <s xml:id="echoid-s6652" xml:space="preserve"> Quia lux apparet maior poſt fora <lb/>men magnũ, quàm poſt paruũ.</s> <s xml:id="echoid-s6653" xml:space="preserve"> Pari modo cũ foramina ſtricta ſunt:</s> <s xml:id="echoid-s6654" xml:space="preserve"> color poſt reflexionẽ aut nullus <lb/>apparebit, aut ualde modicus.</s> <s xml:id="echoid-s6655" xml:space="preserve"> Tamen ſi in prædicto inſtrumento uidere uolueris:</s> <s xml:id="echoid-s6656" xml:space="preserve"> facias ſpeculũ ar-<lb/>genteum:</s> <s xml:id="echoid-s6657" xml:space="preserve"> in ferreo enim ſpeculo color apparet debilior:</s> <s xml:id="echoid-s6658" xml:space="preserve"> quoniã in reflexione miſcetur cum luce re-<lb/>flexa, mixta exluce deſcendente & luce ſpeculi ferrei modica, & color ferreus colori reflexo mixtus <lb/>debilitat ipſum.</s> <s xml:id="echoid-s6659" xml:space="preserve"> Verùm in domo unici foraminis tantùm, habeatur inſtrumentũ prædictum, cui do <lb/>mui paries opponatur albus;</s> <s xml:id="echoid-s6660" xml:space="preserve"> & inſtrumentũ foramini domus aptetur:</s> <s xml:id="echoid-s6661" xml:space="preserve"> cuius foraminis latitudo ſit, <lb/>ut duo inſtrumenti foramina occupare poſsit, per quorũ alterũ inſpiciatur paries albus, domui op-<lb/>poſitus, & parti comprehenſæ parietis apponatur corpus coloris fortis, & per aliud inſtrumenti fo-<lb/>ramen uideatur pars parietis.</s> <s xml:id="echoid-s6662" xml:space="preserve"> Cum ergo lux intrauerit per foramina inſtrumẽti:</s> <s xml:id="echoid-s6663" xml:space="preserve"> uidebitur color re-<lb/>flecti per foramen illud reſpiciẽs, quod oppoſitum eſt colorato corpori, per aliud minimè:</s> <s xml:id="echoid-s6664" xml:space="preserve"> & ita ac-<lb/>cidet quocunq;</s> <s xml:id="echoid-s6665" xml:space="preserve"> oppoſito corpori foramine.</s> <s xml:id="echoid-s6666" xml:space="preserve"> Et quæ dicta ſunt in reflexione lucis, conſiderari po-<lb/>terunt in reflexione coloris.</s> <s xml:id="echoid-s6667" xml:space="preserve"> Occupauit autem latitudo foraminis parietis duo inſtrumenti fora-<lb/>mina ei adhibita, ut maior deſcendat in ſpeculum lux, & melius appareat color reflexus.</s> <s xml:id="echoid-s6668" xml:space="preserve"> Et quoniã <lb/>color debilitatur per foramen directus, & ſimiliter reflexus, cum in corpus ceciderit uiſui oppoſitũ, <lb/>percipietur ſecundus:</s> <s xml:id="echoid-s6669" xml:space="preserve"> unde ſi poſt reflexionem cadat in corpus album foramini colorationis adhi-<lb/>bitum:</s> <s xml:id="echoid-s6670" xml:space="preserve"> forſan propter debilitatem non comprehendet eum uiſus:</s> <s xml:id="echoid-s6671" xml:space="preserve"> adhibito autem ſecundo uiſu fo-<lb/>ramini colorationis, forſan comprehendetur:</s> <s xml:id="echoid-s6672" xml:space="preserve"> quoniam primus, non ſecundus uidebitur.</s> <s xml:id="echoid-s6673" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div234" type="section" level="0" n="0"> <head xml:id="echoid-head265" xml:space="preserve">QVOÒ<unsure/>D COMPREHENSIO FORMARVM È<unsure/> CORPORIBVS <lb/>politis fiat reflexione. Cap. 1111.</head> <head xml:id="echoid-head266" xml:space="preserve" style="it">20. Falſa eſt utra opinio: & radios à uiſu ad ſpeculum miſſos, indé ad uiſibile reflexos, ima <lb/>ginem percipere: & imaginẽ in ſpeculo iam antè impreſſam inde ad uiſum manare. 23. 24 p 5.</head> <p> <s xml:id="echoid-s6674" xml:space="preserve">SVper modum cõprehenſionis formæ in politis corporibus diſſentiunt plurimi.</s> <s xml:id="echoid-s6675" xml:space="preserve"> Vnde quidam <lb/>eorum radios à uiſu exire ad ſpeculũ, & à ſpeculo redire, & formã rei in reditu comprehendere <lb/>exiſtimant:</s> <s xml:id="echoid-s6676" xml:space="preserve"> alij affirmant formã corporis ſpeculo ei oppoſito imprimi:</s> <s xml:id="echoid-s6677" xml:space="preserve"> & proin de in eo uideri, <lb/>ſicut in corporibus fit cõprehenſio formarũ naturalium eius.</s> <s xml:id="echoid-s6678" xml:space="preserve"> Verùm quòd aliter ſit, palàm per hoc.</s> <s xml:id="echoid-s6679" xml:space="preserve"> <lb/>Quoniã ſi quis ſe uiderit in aliqua ſpeculi parte, motus in partẽ aliam, non uidebit ſe in parte prima, <lb/>ſed in ſecunda:</s> <s xml:id="echoid-s6680" xml:space="preserve"> quod non accideret, ſi in parte prima infixa eſſet eius forma:</s> <s xml:id="echoid-s6681" xml:space="preserve"> pari modo ſi ad tertiam <lb/>mutetur partem, mutabitur locus apparentiæ formæ, nec apparebit in prima uel in ſecunda parte.</s> <s xml:id="echoid-s6682" xml:space="preserve"> <lb/>Amplius:</s> <s xml:id="echoid-s6683" xml:space="preserve">uiſo corpore aliquo, & uidente ab eo ſitu remoto:</s> <s xml:id="echoid-s6684" xml:space="preserve"> poterit accidere, ut non uideat corpus <lb/>illud in ſpeculo illo, licet uideat totam ſpeculi ſuperficiem:</s> <s xml:id="echoid-s6685" xml:space="preserve"> quod quidem non eſſet, ſi imprimeretur <lb/>forma in ſpeculo, cum uideatur ſpeculum, & non mutet locum, & corpus ſimiliter ſit immotum, & <lb/>forma eius inficiat ſpeculum, ſicut & prius.</s> <s xml:id="echoid-s6686" xml:space="preserve"> Et ut planè appareat, non accidere hoc ex comprehen-<lb/>ſione formæ:</s> <s xml:id="echoid-s6687" xml:space="preserve"> obturetur medietas foraminum inſtrumenti, & in aliquo obturatorum ſit ſcriptura <lb/>aliqua, ſi inſpiciatur ſpeculum regulæ per foramen ſcripturã reſpiciens:</s> <s xml:id="echoid-s6688" xml:space="preserve"> comprehendet<gap/>r in ſpecu-<lb/>lo ſcriptura:</s> <s xml:id="echoid-s6689" xml:space="preserve"> per quodcũq;</s> <s xml:id="echoid-s6690" xml:space="preserve"> aliud minimè:</s> <s xml:id="echoid-s6691" xml:space="preserve"> quod ſi ſcripturæ forma ſpeculo eſſet impreſſa, per quod-<lb/>cunq;</s> <s xml:id="echoid-s6692" xml:space="preserve"> foramẽ inſtrumẽti poſſet percipi.</s> <s xml:id="echoid-s6693" xml:space="preserve"> Simili modo in ſpeculis columnaribus perforamen, reſpi-<lb/>ciens tantùm foramẽ obturatũ, in quo eſt ſcriptura, cõprehendetur ſcripturę ſitus.</s> <s xml:id="echoid-s6694" xml:space="preserve"> Verùm in ſpecu-<lb/>lis pyramidalibus & ſphæricis ſitus & magnitudo ſcripturæ mutabitur.</s> <s xml:id="echoid-s6695" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s6696" xml:space="preserve"> ſpeculo columna <lb/>ri extracto, regula ſuper baſes ſuas directè ſita apparebit facies hominis in eo directa.</s> <s xml:id="echoid-s6697" xml:space="preserve"> Si uerò eriga-<lb/>tur regula, aut multum inclinetur, uidebitur diſtorta.</s> <s xml:id="echoid-s6698" xml:space="preserve"> Palàm ergo, quòd non accidit comprehen-<lb/>ſio ex forma fixa in ſpeculo, cum non comprehendatur res uiſa in ſpeculis, niſi fuerit uiſus in ſitu re-<lb/> <pb o="114" file="0120" n="120" rhead="ALHAZEN"/> flexionis.</s> <s xml:id="echoid-s6699" xml:space="preserve"> Palàm etiã, quòd diſtortio faciei apparentis non eſt ex forma rei, ſed diſpoſitione ſpeculi.</s> <s xml:id="echoid-s6700" xml:space="preserve"> <lb/>Amplius:</s> <s xml:id="echoid-s6701" xml:space="preserve"> uiſo corpore in ſpeculo, & pòſt elongato:</s> <s xml:id="echoid-s6702" xml:space="preserve"> comprehendetur corpus magis intra ſpeculum, <lb/>quàm prius:</s> <s xml:id="echoid-s6703" xml:space="preserve"> quod non eſſet, ſi forma corporis in ſuperficie ſpeculi eſſet, & ibi comprehenderetur.</s> <s xml:id="echoid-s6704" xml:space="preserve"> <lb/>Comprehenſionem igitur formæ in ſpeculo efficit reflexio.</s> <s xml:id="echoid-s6705" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div235" type="section" level="0" n="0"> <head xml:id="echoid-head267" xml:space="preserve">DE MODO COMPREHENSIONIS FORMARVM È<unsure/> COR-<lb/>poribus politis. Cap. V.</head> <head xml:id="echoid-head268" xml:space="preserve" style="it">21. Imago uiſibilis percipitur è reflexione formæ uiſibilis à ſpeculo ad uiſum facta. 24 p 5.</head> <p> <s xml:id="echoid-s6706" xml:space="preserve">IAm patuit in parte ſuperiori, [3 n] quòd ſi opponatur ſpeculo corpus coloratũ lucidum:</s> <s xml:id="echoid-s6707" xml:space="preserve"> à quo-<lb/>libet eius puncto procedit lux cum colore ad totam ſpeculi ſuperficiem, & reflectitur per lineas <lb/>reflexionis proprias.</s> <s xml:id="echoid-s6708" xml:space="preserve"> Igitur à puncto ſumpto in corpore, oppoſito ſpeculo procedit lux cum co-<lb/>lore ad ſpeculum, in modum pyramidis continuæ, cuius baſis eſt ſuperficies ſpeculi.</s> <s xml:id="echoid-s6709" xml:space="preserve"> Et forma illa re <lb/>flectitur per lineas eiuſdem ſitus cum lineis acceſſus, & erit poſt reflexionem continuitas, ſicut in <lb/>acceſſu.</s> <s xml:id="echoid-s6710" xml:space="preserve"> Et ſi lineis reflexis occurrat ſuperficies corporis, propter continuitatem earum tota occu-<lb/>pabitur, ut nihil interſit uacuum.</s> <s xml:id="echoid-s6711" xml:space="preserve"> Si ergo forma illius corporis moueatur ad ſpeculum per lineas il-<lb/>las reflexas, & ad baſim pyramidis peruenerit:</s> <s xml:id="echoid-s6712" xml:space="preserve"> quoniã lineę pyramidis eiuſdẽ ſunt ſitus cũ lineis re-<lb/>flexis:</s> <s xml:id="echoid-s6713" xml:space="preserve"> reflectetur forma per lineas pyramidis, & aggregabitur tota in puncto ſumpto.</s> <s xml:id="echoid-s6714" xml:space="preserve"> Quoties ergo <lb/>forma alιcuius corporis ad ſpeculũ uenerit per aliquas lineas:</s> <s xml:id="echoid-s6715" xml:space="preserve"> ſi lineæ iſtæ eiuſdẽ ſint ſitus cũ lineis <lb/>pyramidis, à puncto ſumpto ad ſpeculum (intellige tamẽ) eas reſpicientibus:</s> <s xml:id="echoid-s6716" xml:space="preserve"> mouebitur forma per <lb/>pyramidem illam ad punctum ſumptum:</s> <s xml:id="echoid-s6717" xml:space="preserve"> & ſi in puncto ſumpto fuerit uiſus:</s> <s xml:id="echoid-s6718" xml:space="preserve"> uidebit corpus, cuius <lb/>eſt forma illa.</s> <s xml:id="echoid-s6719" xml:space="preserve"> Et ſuperius declaratum eſt [2.</s> <s xml:id="echoid-s6720" xml:space="preserve">17.</s> <s xml:id="echoid-s6721" xml:space="preserve">18 n] quòd in ſitu determinato fiat acquiſitio formæ <lb/>in ſpeculo.</s> <s xml:id="echoid-s6722" xml:space="preserve"> Situs igitur proprius & naturalis acquiſitionis uiſus per reflexionem hic eſt:</s> <s xml:id="echoid-s6723" xml:space="preserve"> ut lineę ac-<lb/>ceſſus formæ ad ſpeculum, eundem habeant ſitum cum lineis pyramidis à centro uiſus ad capita il-<lb/>larum linearum, ſcilicet unaquæq;</s> <s xml:id="echoid-s6724" xml:space="preserve"> cum ſua reſpiciente:</s> <s xml:id="echoid-s6725" xml:space="preserve"> nec accidit formę reflexę comprehenſio, ni-<lb/>ſi in iſto ſitu.</s> <s xml:id="echoid-s6726" xml:space="preserve"> Palàm ergo, quòd ſecundum hanc diſpoſitionem linearum tantùm fiat comprehenſio <lb/>formarum.</s> <s xml:id="echoid-s6727" xml:space="preserve"> Et palàm, quòd ex corpore colorato luminoſo procedat lux cum colore ad ſpeculum, & <lb/>reflectatur, nec procedat aliquid ex corpore, præter lucem & colorẽ.</s> <s xml:id="echoid-s6728" xml:space="preserve"> Patet ergo, quòd ex luce & co-<lb/>lore cantùm huiuſmodi forma comprehenditur.</s> <s xml:id="echoid-s6729" xml:space="preserve"> Et cum moueatur forma ex colore & luce compa-<lb/>cta ſecundum prædictam ſitus obſeruationem:</s> <s xml:id="echoid-s6730" xml:space="preserve"> ſuperfluum eſt dicere, quòd ab oculo exeant radij <lb/>ad ſpeculum, & reflectantur ſecundum ſitum prædictum, ſicut à pluribus dictum eſt.</s> <s xml:id="echoid-s6731" xml:space="preserve"> Hic eſt igitur <lb/>reflexionis modus geometrarum doctrinæ non aduerſus, ſed conſonus:</s> <s xml:id="echoid-s6732" xml:space="preserve"> cum in eo geometricè ra-<lb/>diorum exeuntiũ opinione obſeruetur ſitus.</s> <s xml:id="echoid-s6733" xml:space="preserve"> Et hic modus mihi ſoli uſq;</s> <s xml:id="echoid-s6734" xml:space="preserve"> nunc patuit.</s> <s xml:id="echoid-s6735" xml:space="preserve"> Verùm cum à <lb/>corpore luminoſo procedat forma ad ſpeculum ſecundum uarietatẽ ſituum, propter lineas à quoli-<lb/>bet puncto corporis ad totam ſpeculi ſuperficiem intellectas:</s> <s xml:id="echoid-s6736" xml:space="preserve"> erit formæ eiuſdem reflexio per diuer <lb/>ſas pyramides, quarum capita ſunt diuerſa puncta, & baſes ſpeculi ſuperficies, ſitum linearũ motus <lb/>formæ obſeruantes.</s> <s xml:id="echoid-s6737" xml:space="preserve"> Ob hoc accidit, ut eadem hora fixo ſpeculo, eadem percipiatur corporis forma <lb/>à diuerſis, ſuper quorũ intuitus cadunt capita pyramidum reflexarũ.</s> <s xml:id="echoid-s6738" xml:space="preserve"> Similiter ſi idem uiſus mouea-<lb/>tur ſuper illa pyramidum capita:</s> <s xml:id="echoid-s6739" xml:space="preserve"> apparebit ei, ſpeculo immoto, à locis diuerſis eadẽ ſorma.</s> <s xml:id="echoid-s6740" xml:space="preserve"> Sed di-<lb/>uerſis in ſpeculo eandem formã comprehendentibus, in diuerſa ſpeculi loca cadunt eorũ intuitus.</s> <s xml:id="echoid-s6741" xml:space="preserve"> <lb/>Quoniã ab eodem ſpeculi puncto diuerſorũ punctorum formas comprehendere eaſdẽ nõ poſſunt.</s> <s xml:id="echoid-s6742" xml:space="preserve"> <lb/>Et iam dictũ eſt [3 n] quòd à quolibet puncto corporis procedit lux ad quodlibet punctum ſpeculi.</s> <s xml:id="echoid-s6743" xml:space="preserve"> <lb/>Vnde ſuper quodlibet corporis punctũ eſt acumen pyramidis, cuius ſuperficies ſpeculi eſt baſis:</s> <s xml:id="echoid-s6744" xml:space="preserve"> & <lb/>quodlibet ſuperficiei ſpeculi punctũ, eſt acumẽ pyramidis, cuius baſis ſuperficies corporis tota.</s> <s xml:id="echoid-s6745" xml:space="preserve"> Er-<lb/>go forma corporis erit in quolibet puncto ſpeculi, per lineas procedẽtes in partes diuerſas, nec con <lb/>currere poſsibiles.</s> <s xml:id="echoid-s6746" xml:space="preserve"> Et forma à corpore ad quodcunq;</s> <s xml:id="echoid-s6747" xml:space="preserve"> ſpeculi punctum accedens per pyramidem:</s> <s xml:id="echoid-s6748" xml:space="preserve"> <lb/>reflectetur per pyramidem.</s> <s xml:id="echoid-s6749" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div236" type="section" level="0" n="0"> <head xml:id="echoid-head269" xml:space="preserve" style="it">22. Si uiſibile & ſpeculum figuræ ſit{us}́ ſimilitudine conueniant: uera & distincta imago <lb/>uidetur. 35 p 5.</head> <p> <s xml:id="echoid-s6750" xml:space="preserve">ET licet in ſpeculi ſuperficie ſuper numerũ multiplicetur eadẽ iteratio formæ, cũ concurrat for <lb/>matotalis cũ qualibet parte & in quolibet puncto, & non ſit in formis illis diſcretio, ſed conti-<lb/>nuitas inſeparabilis in reflexione:</s> <s xml:id="echoid-s6751" xml:space="preserve"> tamẽ, quia forma totalis nõ cadit in diuerſas ſpeculi partes, <lb/>ſecundũ identitatẽ ſitus dirigitur ad loca diuerſa, in quibus eam cõprehẽdit uiſus.</s> <s xml:id="echoid-s6752" xml:space="preserve"> Cũ igitur ſimilis <lb/>fuerit forma ſpeculi formę corporis:</s> <s xml:id="echoid-s6753" xml:space="preserve"> erit in ſpeculo complementũ formę corporis & figurę:</s> <s xml:id="echoid-s6754" xml:space="preserve"> quoniã <lb/>in ſpeculo eiuſdẽ figurę cũ corpore, forma primi puncti dirigitur ad primũ punctũ ſpeculi, ſecundi <lb/>ad ſecundum, & ſic in omnibus ſe reſpicientibus:</s> <s xml:id="echoid-s6755" xml:space="preserve"> & ita erit in ſpeculi ſuperficie figura totalis figurę:</s> <s xml:id="echoid-s6756" xml:space="preserve"> <lb/>quod nõ accidit in ſpeculo alterius figurę.</s> <s xml:id="echoid-s6757" xml:space="preserve"> Similiter ſumpta quacunq;</s> <s xml:id="echoid-s6758" xml:space="preserve"> ſpeculi parte, cui eadẽ cũ cor-<lb/>pore figura, erit complementũ figurę corporis in ea.</s> <s xml:id="echoid-s6759" xml:space="preserve"> Et cũ infinitę ſint tales ſpeculi partes, infinitæ <lb/>erũt formę corporis reflexionis, ſed ad puncta diuerſa procedentes, à quibus formã cõprehendit ui <lb/>ſus.</s> <s xml:id="echoid-s6760" xml:space="preserve"> Cũ igitur ſecundũ hanc linearũ diſpoſitionẽ fiat formæ cõprehenſio, non erit formę proceden-<lb/>tis à corpore in ſpeculi ſuperficie fixio.</s> <s xml:id="echoid-s6761" xml:space="preserve"> Et in hũc modũ accidit in omnibus ſpeculis, ſed in planis cer <lb/>tinus:</s> <s xml:id="echoid-s6762" xml:space="preserve"> in alijs aũt accidit quædã diuerſitas ex errore uiſus, ſecundũ modum prædictũ.</s> <s xml:id="echoid-s6763" xml:space="preserve"> Et quilibet ui-<lb/>ſus ſecundum modum prędictum ab uno ſpeculi puncto non percipit, niſi unũ corporis punctum:</s> <s xml:id="echoid-s6764" xml:space="preserve"> <lb/>nec à uiſibus duobus percipitur in eodem ſpeculi puncto idem corporis punctum.</s> <s xml:id="echoid-s6765" xml:space="preserve"/> </p> <pb o="115" file="0121" n="121" rhead="OPTICAE LIBER IIII."/> </div> <div xml:id="echoid-div237" type="section" level="0" n="0"> <head xml:id="echoid-head270" xml:space="preserve" style="it">23. Superficies reflexionis quatuor habet puncta: uiſibilis: reflexionis: uiſ{us}: & terminũ per-<lb/>pendicularis ductæ à puncto reflexionis ſuper planum in eodem puncto ſpeculum tangens. Ita <lb/>perpendicularis hæc cõmunis eſt omnib{us} reflexionis ſuperficieb{us}. 27 p 5.6 p 6.24 p 7.3 p 8.3 p 9.</head> <p> <s xml:id="echoid-s6766" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s6767" xml:space="preserve"> ſi opponatur ſpeculum uiſui:</s> <s xml:id="echoid-s6768" xml:space="preserve"> & intelligatur à cẽtro uiſus ad ſuperficiem ſpeculi py-<lb/>ramis & baſis illius pyramidis:</s> <s xml:id="echoid-s6769" xml:space="preserve"> & ſumatur punctum:</s> <s xml:id="echoid-s6770" xml:space="preserve"> & intelligatur linea pyramidis à centro <lb/>uiſus ad illud punctum:</s> <s xml:id="echoid-s6771" xml:space="preserve"> cum à puncto illo infinitæ poſsint produci lineæ:</s> <s xml:id="echoid-s6772" xml:space="preserve"> ſi aliqua earũ cum <lb/>latere pyramidis eundem habeat ſitum, & æqualem cum perpendiculari teneat angulum, & ita ac-<lb/>cidat quolibet puncto ſpeculi ſumpto:</s> <s xml:id="echoid-s6773" xml:space="preserve"> planũ, quòd à quolibet puncto ſpeculi poteſt fieri reflexio.</s> <s xml:id="echoid-s6774" xml:space="preserve"> <lb/>Dico igitur, quòd inter lineas à puncto ſumpto productas, eſt linea, quæ eundẽ habet ſitum cum la-<lb/>tere pyramidis, & æqualem tenet angulum cum perpẽdiculari ſuper illud punctum:</s> <s xml:id="echoid-s6775" xml:space="preserve"> & illa linea eſt <lb/>latus pyramidis intellectæ à puncto illo ſuperficiei rei occurrẽtis:</s> <s xml:id="echoid-s6776" xml:space="preserve"> & quod ſuper terminum illius li-<lb/>neæ ceciderit, cum per eam ad punctũ ſumptum uenerit:</s> <s xml:id="echoid-s6777" xml:space="preserve"> reflectetur ad uiſum, per latus pyramidis <lb/>iam dictũ.</s> <s xml:id="echoid-s6778" xml:space="preserve"> Et huius pyramidis latus cum linea à puncto illo producta erit in eadẽ ſuperficie, ortho-<lb/>gonali ſuper ſuperficiẽ tãgentẽ ſpeculũ in illo pũcto.</s> <s xml:id="echoid-s6779" xml:space="preserve"> Et hoc dico, cũ lateris pyramidis ſuper punctũ <lb/>ſumptũ fuerit declinatio.</s> <s xml:id="echoid-s6780" xml:space="preserve"> Si enim orthogonaliter cadat ſuper ſuperficiẽ tangentẽ ſpeculũ in pũcto <lb/>ſumpto, latus pyramidis productum à cẽtro uiſus reflectetur in ſe, & redibit in uiſum ad originem <lb/>ſui motus [per 11 n.</s> <s xml:id="echoid-s6781" xml:space="preserve">] In ſpeculo plano planũ eſt:</s> <s xml:id="echoid-s6782" xml:space="preserve"> quod diximus.</s> <s xml:id="echoid-s6783" xml:space="preserve"> Quo <lb/> <anchor type="figure" xlink:label="fig-0121-01a" xlink:href="fig-0121-01"/> niã in quodcunq;</s> <s xml:id="echoid-s6784" xml:space="preserve"> punctũ ſuperficiei planæ ceciderit radius:</s> <s xml:id="echoid-s6785" xml:space="preserve"> à pũcto <lb/>illo poteſt erigi linea orthogonalis ſuper ſuperficiẽ illã:</s> <s xml:id="echoid-s6786" xml:space="preserve"> & à cẽtro ui <lb/>ſus poteſt intelligi linea perpendiculariter cadẽs in ſuperficiẽ planã <lb/>prædictæ continuam, aut in eandẽ:</s> <s xml:id="echoid-s6787" xml:space="preserve"> & [per 35 d 1] hæ duæ perpendi-<lb/>culares erũt in eadẽ ſuperficie:</s> <s xml:id="echoid-s6788" xml:space="preserve"> quoniã ſunt æquidiſtãtes [per 6 p 11] <lb/>& linea à termino unius uſq;</s> <s xml:id="echoid-s6789" xml:space="preserve"> ad terminũ alterius protracta in ſuper-<lb/>ficie plana tenebit angulũ cum utraq;</s> <s xml:id="echoid-s6790" xml:space="preserve">: & erit in eadẽ ſuperficie cum <lb/>utraq;</s> <s xml:id="echoid-s6791" xml:space="preserve"> [per 2 p 11] & radius, qui à linea illa eleuatur:</s> <s xml:id="echoid-s6792" xml:space="preserve"> tenebit acutum <lb/>angulum cũ perpendiculari ſpeculi, & ſimiliter cum perpendiculari <lb/>uiſus [angulus enim d c e acutus eſt:</s> <s xml:id="echoid-s6793" xml:space="preserve"> quia pars recti d c a:</s> <s xml:id="echoid-s6794" xml:space="preserve"> & huic æ-<lb/>quatur a e c per 29 p 1:</s> <s xml:id="echoid-s6795" xml:space="preserve"> quia a e, d c ſunt parallelæ.</s> <s xml:id="echoid-s6796" xml:space="preserve">] Et ſi intelligatur <lb/>in partem alteram produci linea ſuperficiei planæ, tranſiens ortho-<lb/>gonaliter ſuper terminos perpendicularium:</s> <s xml:id="echoid-s6797" xml:space="preserve"> tenebit ex parte alte-<lb/>ra cum perpendiculari ſpeculi angulum rectum [per 29 p 1:</s> <s xml:id="echoid-s6798" xml:space="preserve">] unde <lb/>ex illo recto poterit abſcindi angulus acutus, æqualis angulo acu-<lb/>to, quem cum eadem perpendiculari tenet radius.</s> <s xml:id="echoid-s6799" xml:space="preserve"> Et hi duo anguli <lb/>ſunt in eadem ſuperficie.</s> <s xml:id="echoid-s6800" xml:space="preserve"> Quare radius exiens & reflexus in eadem <lb/>ſunt ſuperficie, & in ſuperficie perpendicularium dictarum.</s> <s xml:id="echoid-s6801" xml:space="preserve"> Inſpe-<lb/>cto autem alio puncto, idem ſitus accidet radiorum cum perpendi-<lb/>cularibus:</s> <s xml:id="echoid-s6802" xml:space="preserve"> quarum una à centro uiſus:</s> <s xml:id="echoid-s6803" xml:space="preserve"> alia à puncto uiſo.</s> <s xml:id="echoid-s6804" xml:space="preserve"> In omni ergo ſuperficie reflexionis accidit <lb/>quatuor punctorũ concurſus, quæ ſunt:</s> <s xml:id="echoid-s6805" xml:space="preserve"> centrũ uiſus:</s> <s xml:id="echoid-s6806" xml:space="preserve"> & punctũ apprehenſum:</s> <s xml:id="echoid-s6807" xml:space="preserve"> & terminus perpen-<lb/>dicularis à cẽtro uiſus ductæ:</s> <s xml:id="echoid-s6808" xml:space="preserve"> & punctũ reflexionis.</s> <s xml:id="echoid-s6809" xml:space="preserve"> Et oẽs reflexionis ſuքficies ſecãt ſe in քpẽdicu-<lb/>lari, à pũcto reflexionis intellecta:</s> <s xml:id="echoid-s6810" xml:space="preserve"> & eſt ipſa cõmunis omnib.</s> <s xml:id="echoid-s6811" xml:space="preserve"> ſuperficieb.</s> <s xml:id="echoid-s6812" xml:space="preserve"> reflexionis.</s> <s xml:id="echoid-s6813" xml:space="preserve"> Et cũ idẽ ac-<lb/>cidat, quolibet pũcto ſuքficiei planæ inſpecto:</s> <s xml:id="echoid-s6814" xml:space="preserve"> erit ex omnib.</s> <s xml:id="echoid-s6815" xml:space="preserve"> pũctis ſimilis reflexio & eodẽ modo.</s> <s xml:id="echoid-s6816" xml:space="preserve"/> </p> <div xml:id="echoid-div237" type="float" level="0" n="0"> <figure xlink:label="fig-0121-01" xlink:href="fig-0121-01a"> <variables xml:id="echoid-variables14" xml:space="preserve">e d f a c b</variables> </figure> </div> </div> <div xml:id="echoid-div239" type="section" level="0" n="0"> <head xml:id="echoid-head271" xml:space="preserve" style="it">24. Si uiſ{us} ſit extra ſuperficiem ſpeculi ſphærici conuexi, uelipſi continuam: communis ſe-<lb/>ctio baſis pyramidis opticæ & ſuperficiei ſpeculi, erit peripheria <lb/>minimi in ſphæra circuli. 3 p 6.</head> <p> <s xml:id="echoid-s6817" xml:space="preserve">IN ſpeculis autem ſphęricis palàm erit, quod diximus:</s> <s xml:id="echoid-s6818" xml:space="preserve"> oppoſito <lb/> <anchor type="figure" xlink:label="fig-0121-02a" xlink:href="fig-0121-02"/> uiſui ſpeculo ſphærico:</s> <s xml:id="echoid-s6819" xml:space="preserve"> (& eſt oppoſitio, ut uiſus nõ ſit in ſuper-<lb/>ficie illius ſpeculi:</s> <s xml:id="echoid-s6820" xml:space="preserve"> aut in ſuperficie ei continua) & inſpecto hoc <lb/>ſpeculo:</s> <s xml:id="echoid-s6821" xml:space="preserve"> pars eius à uiſu comprehenſa, erit pars ſphæræ circulo mi-<lb/>nore incluſa, quem efficit motu ſuo radius, tangẽs ſuperficiẽ ſphæ-<lb/>ræ, ſi per gyrum moueatur contingendo ſphæram, donec redeat ad <lb/>punctum primum, à quo ſumpſit motus principium:</s> <s xml:id="echoid-s6822" xml:space="preserve"> quia ſi intelli-<lb/>gantur ſuperficies ſe ſecantes ſuper diametrum ſphæræ, à polo cir-<lb/>culi prædicti intellectam:</s> <s xml:id="echoid-s6823" xml:space="preserve"> quilibet arcuum ſuperficiei ſphęræ, & his <lb/>ſuperficiebus communium, à polo circuli ad ipſum circulum intel-<lb/>lectorum, erit minor quarta circuli magni.</s> <s xml:id="echoid-s6824" xml:space="preserve"> Quoniam linea à centro <lb/>ſphæræ ad terminum radij, ſphæram contingentis protracta (quæ <lb/>eſt ad circulum prædictum) tenet cum radio angulum rectum ra-<lb/>tione contingentiæ [per 18 p 3.</s> <s xml:id="echoid-s6825" xml:space="preserve">] Tenet ergo angulum acutum cum <lb/>ſemidiametro à polo circuli producta [per 17 p 1] & hunc angulum <lb/>reſpicit arcus interiacens<gap/>polum circuli & circulum [Quare per 33 <lb/>p 6 peripheria c s minor eſt quadrãte peripheriæ maximi in ſphæ-<lb/>ra circuli.</s> <s xml:id="echoid-s6826" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s6827" xml:space="preserve"> cum per 16 th.</s> <s xml:id="echoid-s6828" xml:space="preserve"> 1 ſphęr.</s> <s xml:id="echoid-s6829" xml:space="preserve"> Theodoſij peripheria maximi <lb/> <pb o="116" file="0122" n="122" rhead="ALHAZEN"/> circuli diſtet à ſuo polo quadrante peripheriæ maximi circuli:</s> <s xml:id="echoid-s6830" xml:space="preserve"> erit peripheria, conuerſione radij ab <lb/>uno uiſu ſphæram tangentis, in ſphærica ſuperficie deſcripta, minor maximi circuli peripheria.</s> <s xml:id="echoid-s6831" xml:space="preserve">]</s> </p> <div xml:id="echoid-div239" type="float" level="0" n="0"> <figure xlink:label="fig-0121-02" xlink:href="fig-0121-02a"> <variables xml:id="echoid-variables15" xml:space="preserve">a s b c</variables> </figure> </div> </div> <div xml:id="echoid-div241" type="section" level="0" n="0"> <head xml:id="echoid-head272" xml:space="preserve" style="it">25. Si duarum rectarum linearum à uiſu, alter a ſpeculum ſphæricum conuexum tangat, re-<lb/>liqua per centrum ſecet: tangens circa ſecantem fixam cõuerſa, definiet ſegmentum ſuperficiei <lb/>ſpeculι: à cui{us} puncto quolibet poteſt ad uiſum fieri reflexio. Et centra uiſ{us} & ſpeculi, puncta <lb/>reflexionis & uiſibilis ſunt in reflexionis ſuperficie. 2.5.6 p 6.</head> <p> <s xml:id="echoid-s6832" xml:space="preserve">DIco igitur, quòd à quolibet puncto huius portionis poterit fieri reflexio.</s> <s xml:id="echoid-s6833" xml:space="preserve"> Quoniã ſumpto ali-<lb/>quo eius puncto:</s> <s xml:id="echoid-s6834" xml:space="preserve"> diameter ſphæræ ab illo puncto intellecta, erit perpẽdicularis ſuper ſuper-<lb/>ficiem planam tangentem ſphæram in puncto illo [per 4 th.</s> <s xml:id="echoid-s6835" xml:space="preserve"> 1 ſphæ.</s> <s xml:id="echoid-s6836" xml:space="preserve">] Et huius rei probatio <lb/>eſt.</s> <s xml:id="echoid-s6837" xml:space="preserve"> Intellectis duabus ſuperficiebus ſphæram ſuper diametrum à puncto ſumptam, intellectam ſe-<lb/>cantibus:</s> <s xml:id="echoid-s6838" xml:space="preserve"> lineæ communes ſuperficiei ſphæræ & his ſuperficiebus ſunt circuli ſphæræ tranſeuntes <lb/>per punctum ſumptum [per 1 th.</s> <s xml:id="echoid-s6839" xml:space="preserve"> 1 ſphæ:</s> <s xml:id="echoid-s6840" xml:space="preserve">] & intellectis duabus lineis, tangentibus hos circulos in <lb/>puncto ſumpto:</s> <s xml:id="echoid-s6841" xml:space="preserve"> erit diameter perpendicularis ſuper utramq;</s> <s xml:id="echoid-s6842" xml:space="preserve"> lineam [per 18 p 3.</s> <s xml:id="echoid-s6843" xml:space="preserve">] Quare ſuper ſu-<lb/>perficiem, in qua ſunt illæ lineæ [per 4 p 11.</s> <s xml:id="echoid-s6844" xml:space="preserve">] Et cum deſcenderit radius ſuper punctum<gap/> ſumptum:</s> <s xml:id="echoid-s6845" xml:space="preserve"> <lb/>eritin eadem ſuperficie cũ diametro ſphæræ, cuius terminus punctum eſt ſumptum [per 2 p 11] & <lb/>linea à centro uiſus ad centrũ ſphæræ intellecta:</s> <s xml:id="echoid-s6846" xml:space="preserve"> quæ quidẽ tranſit per polum circuli (& eſt radius <lb/>orthogonaliter cadens ſuper ſuperficiem ſphęræ) [quia per 4 th.</s> <s xml:id="echoid-s6847" xml:space="preserve"> 1 ſphær.</s> <s xml:id="echoid-s6848" xml:space="preserve"> eſt perpendicularis plano <lb/>ſphæram in puncto d tangenti] eſt ſimiliter in eadem ſuperficie [per 2 p 11:</s> <s xml:id="echoid-s6849" xml:space="preserve">] & exhis tribus lineis <lb/>erit triangulum:</s> <s xml:id="echoid-s6850" xml:space="preserve"> & radius ſuper punctũ ſumptũ incidẽs;</s> <s xml:id="echoid-s6851" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0122-01a" xlink:href="fig-0122-01"/> tenet acutũ angulũ cũ diametro ſphæræ ab exteriori par <lb/>te:</s> <s xml:id="echoid-s6852" xml:space="preserve"> quoniã cũ elatior ſit iſte radius radio ſphæram cõtin-<lb/>gente:</s> <s xml:id="echoid-s6853" xml:space="preserve"> ſecabit ſphęram cũ producta intelligitur:</s> <s xml:id="echoid-s6854" xml:space="preserve"> & ſuper-<lb/>ficies tangẽs ſphærã in pũcto ſumpto demiſsior erit hoe <lb/>radio:</s> <s xml:id="echoid-s6855" xml:space="preserve"> & ſecabit inter ſphærã & uiſum, uiſam diametrũ, <lb/>id eſt lineã à cẽtro uiſus ad centrũ ſphæræ intellectã, per <lb/>polum circuli tranſeuntem:</s> <s xml:id="echoid-s6856" xml:space="preserve"> unde cũ diameter ſphęræ ſit <lb/>orthogonalis in ſuperficie punctũ tangente:</s> <s xml:id="echoid-s6857" xml:space="preserve"> tenebit an-<lb/>gulũ recto maiorẽ ex parte interiori cũ radio in punctũ <lb/>deſcendente:</s> <s xml:id="echoid-s6858" xml:space="preserve"> unde [per 13 p 1] in exteriori parte tenebit <lb/>cum eo angulũ minorẽ recto:</s> <s xml:id="echoid-s6859" xml:space="preserve"> & producta, orthogonalis <lb/>erit ſuper ſuperficiẽ cõtingentẽ exterius [ք 4 th.</s> <s xml:id="echoid-s6860" xml:space="preserve"> 1 ſphæ.</s> <s xml:id="echoid-s6861" xml:space="preserve">] <lb/>Quare ex angulo recto, quẽ tenebit cũ ſuperficie ex alia <lb/>radij parte, poterit abſcindi acutus æqualis ei, quẽ inclu-<lb/>dit radius cũ illa diametro:</s> <s xml:id="echoid-s6862" xml:space="preserve"> & erũt lineę tres hos angulos <lb/>duos includêtes in eadẽ ſuperficie [per 6.</s> <s xml:id="echoid-s6863" xml:space="preserve"> 13 n.</s> <s xml:id="echoid-s6864" xml:space="preserve">] Quare à <lb/>puncto portionis ſumpto poteſt produci linea in eadem <lb/>ſuperficie cum radio, in punctũ illud cadẽte, & linea or-<lb/>thogonali in ſuperficie punctũ contingẽte, & ad parita-<lb/>tem angulorum cũ perpẽdiculari illa:</s> <s xml:id="echoid-s6865" xml:space="preserve"> & illi lineæ occur-<lb/>rer forma puncti mota ad ſuperficiẽ ſpeculi per radium <lb/>illum.</s> <s xml:id="echoid-s6866" xml:space="preserve"> Igitur eiuſdem eſt ſitus cum linea, quæ poterit re-<lb/>flecti [per 12 uel 18 n.</s> <s xml:id="echoid-s6867" xml:space="preserve">] Et erit ſuperficies, in qua ſunt hæ <lb/>lineæ, orthogonalis ſuper ſuperficiem, ſphærã in puncto <lb/>contingentẽ [per 13 n.</s> <s xml:id="echoid-s6868" xml:space="preserve">] Et ita in quolibet portionis pun-<lb/>cto intelligendum.</s> <s xml:id="echoid-s6869" xml:space="preserve"> Ergo in omni ſuperficie reflexionis <lb/>erũt centrũ uiſus:</s> <s xml:id="echoid-s6870" xml:space="preserve"> centrũ ſphæræ:</s> <s xml:id="echoid-s6871" xml:space="preserve"> punctũ reflexionis:</s> <s xml:id="echoid-s6872" xml:space="preserve"> & punctũ reflexũ.</s> <s xml:id="echoid-s6873" xml:space="preserve"> Et oẽs hæ ſuքficies ſecabũt <lb/>ſe ſuք lineã à cẽtro uiſus ad cẽtrũ ſphęræ ptractã:</s> <s xml:id="echoid-s6874" xml:space="preserve"> & cuilibet reflexiõis ſuքficiei & ſuքficiei ſphæræ, <lb/>cõmunis linea erit circulus ſphęræ [ք 1th.</s> <s xml:id="echoid-s6875" xml:space="preserve"> 1 ſphæ:</s> <s xml:id="echoid-s6876" xml:space="preserve">] & oẽs circuli ſecabũt ſe ſuք pũctũ ſphęræ, in qđ <lb/>cadit diameter uiſus:</s> <s xml:id="echoid-s6877" xml:space="preserve"> & eſt ſuք circuli portiõis polũ.</s> <s xml:id="echoid-s6878" xml:space="preserve"> Cũ aũt radius ceciderit in ſpeculũ orthogona-<lb/>liter ſuք ſuքficiẽ, in pũcto, in qđ radius cadit, ſphærã tãgentẽ (& eſt radius ille, diameter uiſus ք po-<lb/>lũ circuli portiõis ad cẽtrũ ſphęræ) fiet reflexio ad uiſum ք eũdẽ radiũ ad motus radij ortũ [ք 11 n.</s> <s xml:id="echoid-s6879" xml:space="preserve">]</s> </p> <div xml:id="echoid-div241" type="float" level="0" n="0"> <figure xlink:label="fig-0122-01" xlink:href="fig-0122-01a"> <variables xml:id="echoid-variables16" xml:space="preserve">a k f s d m b g c h</variables> </figure> </div> </div> <div xml:id="echoid-div243" type="section" level="0" n="0"> <head xml:id="echoid-head273" xml:space="preserve" style="it">26. Siduo plana à cẽtro uiſiis, ducãtur ք later a cõſpicuam ſpeculi cylindracei cõuexi ſuperficiẽ <lb/>terminãtia: tangẽt ſpeculũ: & facient in uiſu cõmunem ſectionẽ par allelã axiſpeculi. 2.3 p 7.</head> <p> <s xml:id="echoid-s6880" xml:space="preserve">IN ſpeculis autẽ columnaribus patebit, quod diximus.</s> <s xml:id="echoid-s6881" xml:space="preserve"> Opponatur ſpeculũ columnare exterius <lb/>politum oculo:</s> <s xml:id="echoid-s6882" xml:space="preserve"> (& eſt oppoſitio, ut non ſit uiſus in ſuperficie columnæ, aut ſuperficie ei conti-<lb/>nua) & intelligamus ſuperficiem à centro uiſus ad columnæ ſuperficiem, ſecantem columnam <lb/>ſuper circulum æquidiſtantẽ baſibus columnæ:</s> <s xml:id="echoid-s6883" xml:space="preserve"> & in hac ſuperficie ſumantur duæ lineæ, tangentes <lb/>circulũ ſectionis in duobus punctis oppoſitis:</s> <s xml:id="echoid-s6884" xml:space="preserve"> ab utroq;</s> <s xml:id="echoid-s6885" xml:space="preserve"> illorũ punctorum producatur linea ſecun-<lb/>dum longitudinem columnæ:</s> <s xml:id="echoid-s6886" xml:space="preserve"> & intelligãtur duæ ſuperficies, in quibus ſint hæ duæ lineæ longitu-<lb/>dinis, & duæ lineæ à centro uiſus ductæ, contin gentes circulũ ſectionis.</s> <s xml:id="echoid-s6887" xml:space="preserve"> Dico, quòd hæ ſuperficies <lb/>tangent columnã.</s> <s xml:id="echoid-s6888" xml:space="preserve"> Si enim dicatur, quòd altera ſecat illã:</s> <s xml:id="echoid-s6889" xml:space="preserve"> planũ eſt, quòd ſectio eſt ſuper lineã longi-<lb/>tudinis colũnæ, in quã ſuperficies cadit:</s> <s xml:id="echoid-s6890" xml:space="preserve"> [per 21 def.</s> <s xml:id="echoid-s6891" xml:space="preserve"> 11] & ſimiliter erit ſectio ſuper lineã lõgitudinis <lb/> <pb o="117" file="0123" n="123" rhead="OPTICAE LIBER IIII."/> columnæ huic oppoſitam:</s> <s xml:id="echoid-s6892" xml:space="preserve"> & circulus ſectionis trãſit per has duas lineas longitudinis:</s> <s xml:id="echoid-s6893" xml:space="preserve"> & linea con-<lb/>tingens circulum ſectionis:</s> <s xml:id="echoid-s6894" xml:space="preserve"> cum ſit in ſuperficie aliqua:</s> <s xml:id="echoid-s6895" xml:space="preserve"> ſecat columnam ſuper aliquas longitudinis <lb/>lineas, ſibi inuicem æquidiſtantes:</s> <s xml:id="echoid-s6896" xml:space="preserve"> & ſi tranſit per unam earum, tran-<lb/> <anchor type="figure" xlink:label="fig-0123-01a" xlink:href="fig-0123-01"/> ſibit per alteram, & ad paritatem angulorum.</s> <s xml:id="echoid-s6897" xml:space="preserve"> Cum ergo tranſeat per <lb/>punctum, in quo circulus ſectionis ſecat primã longιtudinis lineam:</s> <s xml:id="echoid-s6898" xml:space="preserve"> <lb/>tranſibit etiam per punctũ, in quo alia longitudinis linea tangit hunc <lb/>circulum:</s> <s xml:id="echoid-s6899" xml:space="preserve"> & ita ſecat circulum.</s> <s xml:id="echoid-s6900" xml:space="preserve"> Quare non contingit, quod eſt contra <lb/>hypotheſin.</s> <s xml:id="echoid-s6901" xml:space="preserve"> Palàm ergo, quòd duæ illæ ſuperficies cõtingunt ſpecu-<lb/>lum, & quod inter illas cadit ex ſuperficie ſpeculi, eſt, quod apparet <lb/>uiſui.</s> <s xml:id="echoid-s6902" xml:space="preserve"> Cum autem duarum illarum ſuperficierum ſit concurſus in cen <lb/>tro uiſus, ſecabunt ſe, & linea ſectionis communis tranſibit per cen-<lb/>trum uiſus:</s> <s xml:id="echoid-s6903" xml:space="preserve"> & erit æquidiftans axi columnæ.</s> <s xml:id="echoid-s6904" xml:space="preserve"> Quoniam enim axis co-<lb/>lumnæ orthogonalis eſt ſuper circulum ſectionis [per conuerſam 14 <lb/>p 11] & lineæ longitudinis columnæ orthogonales ſuper eundẽ cir-<lb/>culum [per 8 p 11:</s> <s xml:id="echoid-s6905" xml:space="preserve"> latera enim cylindri parallela ſunt axi, perpendicu-<lb/>lari ad circulum ſectionis per 21 d 11] etiam ſuperficies tangẽtes co-<lb/>lumnam ſecũdum lineas has:</s> <s xml:id="echoid-s6906" xml:space="preserve"> orthogonales erunt ſuper circulũ eun-<lb/>dem [per 18 p 11:</s> <s xml:id="echoid-s6907" xml:space="preserve">] ergo & ſuper ſuperficiem ſecantẽ columnam in illo <lb/>circulo.</s> <s xml:id="echoid-s6908" xml:space="preserve"> Quare linea communis harum ſuperficierum eſt orthogona <lb/>lis ſuper eandem ſuperficiem [per 19 p 11] quare æquidiſtans axi co-<lb/>lumnæ [per 6 p 11.</s> <s xml:id="echoid-s6909" xml:space="preserve">]</s> </p> <div xml:id="echoid-div243" type="float" level="0" n="0"> <figure xlink:label="fig-0123-01" xlink:href="fig-0123-01a"> <variables xml:id="echoid-variables17" xml:space="preserve">a e g c b d h f</variables> </figure> </div> </div> <div xml:id="echoid-div245" type="section" level="0" n="0"> <head xml:id="echoid-head274" xml:space="preserve" style="it">27. Si linea recta à cẽtro uiſ{us}, ducta ad punctũ cõſpicuæ ſuper-<lb/>ficiei ſpeculi cylindr acei cõuexi, cõtinuetur: ſecabit ſpeculũ. 4.5 p 7.</head> <p> <s xml:id="echoid-s6910" xml:space="preserve">DIco ergo, quòd quocunq;</s> <s xml:id="echoid-s6911" xml:space="preserve"> puncto in ſectione ſpeculi apparen <lb/>te ſumpto:</s> <s xml:id="echoid-s6912" xml:space="preserve"> linea à centro uiſus ad punctum producta, ſecabit <lb/>ſpeculum.</s> <s xml:id="echoid-s6913" xml:space="preserve"> Quoniam intellecta linea longitudinis columnæ à <lb/>puncto ſumpto, tranſibit per circulum ſectionis, & tanget ipſum in <lb/>puncto:</s> <s xml:id="echoid-s6914" xml:space="preserve"> ad quod punctum ſi ducatur linea à centro uiſus:</s> <s xml:id="echoid-s6915" xml:space="preserve"> ſecabit ſpe-<lb/>culum:</s> <s xml:id="echoid-s6916" xml:space="preserve"> quia cadit inter lineas contingẽtes hunc circulum:</s> <s xml:id="echoid-s6917" xml:space="preserve"> ergo & ſu-<lb/>perficies à centro uiſus procedens, in qua fuerit hæc linea, ſecabit ſpe <lb/>culum.</s> <s xml:id="echoid-s6918" xml:space="preserve"> Cum ergo in eadem ſuperficie ſit linea à centro uiſus, ad pun-<lb/>ctum ſumptum ducta:</s> <s xml:id="echoid-s6919" xml:space="preserve"> ſecabit linea illa ſpeculum:</s> <s xml:id="echoid-s6920" xml:space="preserve"> & ita quælibet linea à centro uiſus, ad portionem <lb/>ſpeculi intellecta, ſecat ſpeculum.</s> <s xml:id="echoid-s6921" xml:space="preserve"> Eodẽ modo quælibet linea à linea cõmuni, per centrum uiſus in-<lb/>tellecta, ad hãc portionẽ ducta, ſecat ſpeculũ.</s> <s xml:id="echoid-s6922" xml:space="preserve"> Vnde quælibet ſuperficies tangens ſpeculũ in aliqua <lb/>portionis apparentis linea, ſecat ſuperficies, quę contingũt portionis extremitates:</s> <s xml:id="echoid-s6923" xml:space="preserve"> & nulla omniũ <lb/>ſuperficierum portionẽ tangentiũ, peruenit ad uiſus centrũ, ſed inter uiſum extẽditur & ſpeculum.</s> <s xml:id="echoid-s6924" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div246" type="section" level="0" n="0"> <head xml:id="echoid-head275" xml:space="preserve" style="it">28. In ſpeculo cylindraceo conuexo, à quolibet conſpicuæ ſuperficiei puncto poteſt ad uiſum <lb/>reflexio fieri. 25 p 7.</head> <p> <s xml:id="echoid-s6925" xml:space="preserve">DIco ergo, quòd à quolibet puncto portionis huius poteſt fieri reflexio lucis.</s> <s xml:id="echoid-s6926" xml:space="preserve"> Dato enim pun-<lb/>cto, fiat ſuper ipſum circulus æquidiſtans columnæ baſibus:</s> <s xml:id="echoid-s6927" xml:space="preserve"> ſi ergo ſuperficies à cẽtro uiſus <lb/>procedens, & columnę ſuperficiem æquidiſtanter baſi ſecans, ſecet eam ſuper hunc circulũ:</s> <s xml:id="echoid-s6928" xml:space="preserve"> <lb/>& linea à centro uiſus ad circuli centrũ ducta, tranſeat per punctum datũ:</s> <s xml:id="echoid-s6929" xml:space="preserve"> fiet reflexio ſormæ illius <lb/>puncti per eandem lineam ad lineæ ortum [per 11 n] quia linea illa eſt axis uiſus ſuper axem colu-<lb/>mnæ perpendicularis [per 21 d 11, 29 p 1.</s> <s xml:id="echoid-s6930" xml:space="preserve">] Sumpto autem puncto quocunq;</s> <s xml:id="echoid-s6931" xml:space="preserve"> per quod tranſeat axis, <lb/>perpendicularis ſuper axem columnæ:</s> <s xml:id="echoid-s6932" xml:space="preserve"> fiet reflexio illius puncti per eundẽ axem [per 11 n.</s> <s xml:id="echoid-s6933" xml:space="preserve">] Si ueró <lb/>prætereat axem punctum ſumptum, quæcunq;</s> <s xml:id="echoid-s6934" xml:space="preserve"> ſit linea à centro circuli, æquidiſtantis baſibus per <lb/>ipſum punctum ducti, ad ſuperficiem in linea longitudinis columnæ per punctũ illud tranſeuntis, <lb/>contingentem:</s> <s xml:id="echoid-s6935" xml:space="preserve"> erit ſuper axem orthogonalis [per 21 d 11, & conuerſam 14 p 11.</s> <s xml:id="echoid-s6936" xml:space="preserve">] Quare ſuper lineam <lb/>longitudinis per punctum illud trãſeuntem [per 29 p 1.</s> <s xml:id="echoid-s6937" xml:space="preserve">] Et quoniã uiſus eſt altior ſuperficie pun-<lb/>ctum contingẽte:</s> <s xml:id="echoid-s6938" xml:space="preserve"> linea à cẽtro uiſus ad punctum ſumptũ ducta, tenebit acutum angulũ cũ perpen-<lb/>diculari illa, à pũcto ad centrũ circuli ducta:</s> <s xml:id="echoid-s6939" xml:space="preserve"> & hic eſt ex parte exteriore, quia obtuſum habet ex in-<lb/>teriore:</s> <s xml:id="echoid-s6940" xml:space="preserve"> & ex angulo recto, quem illa perpendicularis tenet cum linea ſuperficiei contingentis cir-<lb/>culum [per 18 p 3] poterit abſcindi acutus huic æqualis:</s> <s xml:id="echoid-s6941" xml:space="preserve"> & perpendicularis illa cum cẽtro uiſus eſt <lb/>in eadem ſuperficie:</s> <s xml:id="echoid-s6942" xml:space="preserve"> quare etiam cum linea à cẽtro ad punctum ducta:</s> <s xml:id="echoid-s6943" xml:space="preserve"> & erit linea reflexa in eadem <lb/>ſuperficie:</s> <s xml:id="echoid-s6944" xml:space="preserve"> quare cum linea à centro ad punctum ducta.</s> <s xml:id="echoid-s6945" xml:space="preserve"> Et erit hæc ſuperficies orthogonalis ſuper <lb/>ſuperficiem, contingentem ſpeculum in puncto illo:</s> <s xml:id="echoid-s6946" xml:space="preserve"> quoniam perpendicularis orthogonaliter ca-<lb/>dit ſuper hanc ſuperficiem:</s> <s xml:id="echoid-s6947" xml:space="preserve"> & huiuſinodi erit reflexionis ſuperficies.</s> <s xml:id="echoid-s6948" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div247" type="section" level="0" n="0"> <head xml:id="echoid-head276" xml:space="preserve" style="it">29. Si uiſ{us} ſit extra ſuperficiem ſpeculi cylindr acei conuexi, in plano uiſibilis per axem du-<lb/>cto: cõm unis ſectio ſuperficier um reflexionis & ſpeculi, erit lat{us} cylindri: & unicum tantùm <lb/>eſt in eadem conſpicua ſuperficie planum, à quo ad eundem uiſum reflexio fieri poteſt. 7.16 p 7.</head> <p> <s xml:id="echoid-s6949" xml:space="preserve">ESt autẽ diuerſitas inter lineas ſuperficiebus reflexionis & ſuperficiei columnæ cõmunes.</s> <s xml:id="echoid-s6950" xml:space="preserve"> Cũ <lb/>enim reflexio erit per eundẽ radium:</s> <s xml:id="echoid-s6951" xml:space="preserve"> cadet idẽ radius ille orthogonaliter ſuper axem, & linea <lb/> <pb o="118" file="0124" n="124" rhead="ALHAZEN"/> cõmunis ſuperficiei columnæ & ſuperficiei reflexionis, erit linea recta, ſcilicet latus columnæ:</s> <s xml:id="echoid-s6952" xml:space="preserve"> cum <lb/>in ſuperficie reflexionis ſit diameter columnæ.</s> <s xml:id="echoid-s6953" xml:space="preserve"> Et planum hoc eſt, quoniam columnæ compoſitio <lb/>eſt ex motu ſuperficiei æquidiſtantium laterum ſuper unum latus immotum [per 21 d 11.</s> <s xml:id="echoid-s6954" xml:space="preserve">] Vnde ſu-<lb/>perficiei columnam ſecanti, in qua ſit axis, id eſt latus immotum, & ſuperficiei columnę communis <lb/>linea, erit latus motum.</s> <s xml:id="echoid-s6955" xml:space="preserve"> Et dico, quòd ex omnibus reflexionis ſuperficiebus una ſola eſt, cui & co-<lb/>lumnæ ſuperficiei ſit linea communis recta.</s> <s xml:id="echoid-s6956" xml:space="preserve"> Quoniã unica poteſt intelligi ſuperficies, in qua ſit axis <lb/>columnæ & centrum uiſus:</s> <s xml:id="echoid-s6957" xml:space="preserve"> & non plures.</s> <s xml:id="echoid-s6958" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div248" type="section" level="0" n="0"> <head xml:id="echoid-head277" xml:space="preserve" style="it">30. Si uiſ{us} ſit extrá ſuperficiem ſpeculi cylindracei cõuexi, in planò uiſibilis ad axem recto: <lb/>communis ſectio ſuperficierum reflexionis & ſpeculi, erit circul{us}: & unic{us} tantùm eſt in ea-<lb/>dem conſpicuà ſuperficie, à quo ad uiſum reflexio fieri poteſt. 9.17 p 7.</head> <p> <s xml:id="echoid-s6959" xml:space="preserve">SI uerò ſuperficies reflexionis ſit æquidiſtans baſibus columnæ:</s> <s xml:id="echoid-s6960" xml:space="preserve"> erit linea communis circulus <lb/>[per 5 th Sereni de ſectione cylindri] & hæc ſola eſt ſuperficies, quæ cum columnæ ſuperficie <lb/>lineam communem habeat circularem.</s> <s xml:id="echoid-s6961" xml:space="preserve"> Quoniam in omni reflexione, perpendicularis ſuper <lb/>ſuperficiem, contingẽtem punctum reflexionis, eſt diameter circuli, baſibus columnæ æquidiſtan-<lb/>tis:</s> <s xml:id="echoid-s6962" xml:space="preserve"> & non poteſt eſſe in columnæ ſuperficie, niſi unus circulus æquidiſtans baſibus, qui cum cen-<lb/>tro uiſus ſit in eadem ſuperficie.</s> <s xml:id="echoid-s6963" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div249" type="section" level="0" n="0"> <head xml:id="echoid-head278" xml:space="preserve" style="it">31. Si uiſ{us} ſit extra ſuperficiem ſpeculi cylindracei conuexi, in plano uiſibilis ad axem obli-<lb/>quo: communis ſectio ſuperficierum reflexionis & ſpeculi erit ellipſis: & plures in eadem conſpi-<lb/>cua ſuperficie eſſe poſſunt, à quib{us} ad eundem uiſum reflexio fiat. 10. 18 p 7.</head> <p> <s xml:id="echoid-s6964" xml:space="preserve">OMnes autẽ aliæ ſuperficies reflexionis, ſecant columnã & axẽ columnæ:</s> <s xml:id="echoid-s6965" xml:space="preserve"> quoniã perpendi-<lb/>cularis ducta à pũcto reflexionis ſecat axẽ columnæ:</s> <s xml:id="echoid-s6966" xml:space="preserve"> & lineæ cõmunes his ſuperficiebus & <lb/>ſuperficiebus columnę, ſunt ſectiones, quas in colũnis & pyramidibus aſsignãt geometræ.</s> <s xml:id="echoid-s6967" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div250" type="section" level="0" n="0"> <head xml:id="echoid-head279" xml:space="preserve" style="it">32. Si communis ſectio ſuperficierum reflexionis & ſpeculi cylindr acei conuexi, fuerit lat{us} <lb/>cylindri, uel cιrcul{us}: reflexio à quocun communis ſectionis puncto facta, in eadem ſuperficie <lb/>ſemper fiet. 19. 20 p 7.</head> <p> <s xml:id="echoid-s6968" xml:space="preserve">CVm ſuperficiebus columnæ & reflexionis linea recta fuerit cõmunis, quodcunq;</s> <s xml:id="echoid-s6969" xml:space="preserve"> punctum <lb/>illius lineæ intueatur uiſus:</s> <s xml:id="echoid-s6970" xml:space="preserve"> fiet reflexio in ſuperficie eadem, in qua eſt axis.</s> <s xml:id="echoid-s6971" xml:space="preserve"> Quoniam eſt ſu-<lb/>perficies unica, contingens columnam in linea illa longitudinis:</s> <s xml:id="echoid-s6972" xml:space="preserve"> & quocunq;</s> <s xml:id="echoid-s6973" xml:space="preserve"> puncto huius <lb/>lineæ ſumpto:</s> <s xml:id="echoid-s6974" xml:space="preserve"> perpendicularis ab eo ad axem ducta, erit in eadem ſuperficie cum axe:</s> <s xml:id="echoid-s6975" xml:space="preserve"> & hæc linea <lb/>erit orthogonalis ſuper ſuperficiem, contingentem ſuperficiem columnæ [Nam quia per 21 d 11 la-<lb/>tus cylindri eſt parallelum axi:</s> <s xml:id="echoid-s6976" xml:space="preserve"> erit recta linea perpendicularis axi:</s> <s xml:id="echoid-s6977" xml:space="preserve"> perpendicularis tum lateri per <lb/>29 p 1, tum rectæ circulum per idem lateris punctum deſcriptum, tangenti, per 18 p 3.</s> <s xml:id="echoid-s6978" xml:space="preserve"> Quare per <lb/>4 p 11 erit perpendicularis plano ſpeculum tangenti.</s> <s xml:id="echoid-s6979" xml:space="preserve">] Sed centrum uiſus eſt in ſuperficie orthogo-<lb/>nali ſuper eandem ſuperficiem:</s> <s xml:id="echoid-s6980" xml:space="preserve"> quia in una ſuperficie eſt centrum uiſus & linea communis & axis <lb/>columnæ [per 6.</s> <s xml:id="echoid-s6981" xml:space="preserve"> 13 n] & una ſola eſt ſuperficies orthogonalis ſuper illam ſuperficiem [per 13 p 11.</s> <s xml:id="echoid-s6982" xml:space="preserve">] <lb/>Quare omnes reflexiones à punctis huius lineæ factæ, ſunt in eadem reflexionis ſuperficie.</s> <s xml:id="echoid-s6983" xml:space="preserve"> Verùm <lb/>cum linea cõmunis ſuperficiei reflexionis & columnæ fuerit circulus, quo cunq;</s> <s xml:id="echoid-s6984" xml:space="preserve"> puncto illius cir-<lb/>culi uiſo:</s> <s xml:id="echoid-s6985" xml:space="preserve"> fiet in una & eadem ſuperficie reflexio.</s> <s xml:id="echoid-s6986" xml:space="preserve"> Quoniam quæcunq;</s> <s xml:id="echoid-s6987" xml:space="preserve"> perpendicularis à puncto re-<lb/>flexionis ducta:</s> <s xml:id="echoid-s6988" xml:space="preserve"> erit diameter huius circuli:</s> <s xml:id="echoid-s6989" xml:space="preserve"> quare in ſuperficie huius circuli eſt:</s> <s xml:id="echoid-s6990" xml:space="preserve"> & punctum uiſus <lb/>ſimiliter:</s> <s xml:id="echoid-s6991" xml:space="preserve"> & ſuperficies hæc orthogonalis eſt ſuper ſuperficiẽ, quodcunq;</s> <s xml:id="echoid-s6992" xml:space="preserve"> punctũ huius circuli ſum-<lb/>ptum contingentem.</s> <s xml:id="echoid-s6993" xml:space="preserve"> Quare in hac ſola ſuperficie erit cuiuslibet puncti, prædicti circuli reflexio.</s> <s xml:id="echoid-s6994" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div251" type="section" level="0" n="0"> <head xml:id="echoid-head280" xml:space="preserve" style="it">33. Ab uno cõmunis ſectionis ſuperficierum reflexionis & ſpeculi cylindr acei conuexi pun-<lb/>cto, unum uiſibilis punctum ad unum uiſum in eadem ſuperficie reflectitur. 22 p 7.</head> <p> <s xml:id="echoid-s6995" xml:space="preserve">QVacunq;</s> <s xml:id="echoid-s6996" xml:space="preserve"> uerò alia linea communi ſumpta:</s> <s xml:id="echoid-s6997" xml:space="preserve"> nõ fiet in eadem reflexionis ſuperficie reflexio, <lb/>niſi ex uno tantùm huius lineæ puncto.</s> <s xml:id="echoid-s6998" xml:space="preserve"> Quoniam perpẽdicularis ducta à puncto reflexio-<lb/>nis, orthogonalis eſt ſuper lineam longitudinis columnæ per punctũ illud tranſeuntis [per <lb/>3 d 11] quare & ſuper axem [per 29 p 1] & perpendicularis illa, eſt diameter circuli, æquidiſtantis <lb/>baſibus columnæ:</s> <s xml:id="echoid-s6999" xml:space="preserve"> & ſuperficies reflexionis & circulus ille ſecant ſe:</s> <s xml:id="echoid-s7000" xml:space="preserve"> & linea ijs communis, eſt dia-<lb/>meter illius circuli:</s> <s xml:id="echoid-s7001" xml:space="preserve"> & eſt illa diameter perpendicularis ſuper ſuperficiem, columnam in illo puncto <lb/>contingentem, & ſuperficies reflexionis ſecat illam lineam longitudinis columnæ ſuper quam fit <lb/>contingentia, & eſt declinata ſuper ipſam:</s> <s xml:id="echoid-s7002" xml:space="preserve"> ergo & ſuper axem erit illa ſuperficies reflexionis decli-<lb/>nata:</s> <s xml:id="echoid-s7003" xml:space="preserve"> & in ſuperficie plana ſuper lineam aliquam declinata nõ poteſt intelligi, niſi una linea ortho-<lb/>gonaliter cadens in illam.</s> <s xml:id="echoid-s7004" xml:space="preserve"> Sed ſi à duobus ſuperficiei reflexionis punctis fieret reflexio in eadem <lb/>ſuperficie:</s> <s xml:id="echoid-s7005" xml:space="preserve"> eſſent duæ lineæ illius ſuperficiei orthogonales ſuper axem:</s> <s xml:id="echoid-s7006" xml:space="preserve"> quod eſſe non poteſt, cum <lb/>ſuperficies illa ſit declinata ſuper eum.</s> <s xml:id="echoid-s7007" xml:space="preserve"> Nam perpendicularis à puncto reflexionis cadit in circu-<lb/>lum, æquidiſtantem baſibus columnæ, & in punctum axis, & eſt ſectio cõmunis ſuperficiei circuli <lb/>& ſuperficiei reflexionis.</s> <s xml:id="echoid-s7008" xml:space="preserve"> Si ergo ab alio lineæ communis puncto, in eadem ſuperficie fieret refle-<lb/>xio:</s> <s xml:id="echoid-s7009" xml:space="preserve"> alia perpendicularis ab alio puncto ducta:</s> <s xml:id="echoid-s7010" xml:space="preserve"> eſſet diameter alte<gap/>ius circuli columnæ, huic æqui-<lb/>diſtãtis, & caderet in punctũ axis, in quod nõ cadit ſuperficies reflexionis.</s> <s xml:id="echoid-s7011" xml:space="preserve"> Et ita in omnibus ſuper-<lb/>ficiebus reflexionis eſt intelligendũ:</s> <s xml:id="echoid-s7012" xml:space="preserve"> quòd ab uno puncto tantùm lineæ communis fiat reflexio in <lb/> <pb o="119" file="0125" n="125" rhead="OPTICAE LIBER IIII."/> cadem ſuperficie, reſpectu eiuſdem uiſus:</s> <s xml:id="echoid-s7013" xml:space="preserve"> quoniam reſpectu duorum uiſuum poteſt reflexio fieri à <lb/>duobus pũctis ſuperficiei ſpeculi, ut circuli diametri terminis, quæ eſt perpendicularis ſuper ipſam <lb/>ſectionem:</s> <s xml:id="echoid-s7014" xml:space="preserve"> reſpectu uerò unius uiſus non accidit:</s> <s xml:id="echoid-s7015" xml:space="preserve"> quoniam illa duo puncta nõ ſimul ab eodem uiſu <lb/>poſſunt comprehendi:</s> <s xml:id="echoid-s7016" xml:space="preserve"> ſemper enim neceſſe eſt partem columnæ medietate minorem uideri.</s> <s xml:id="echoid-s7017" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div252" type="section" level="0" n="0"> <head xml:id="echoid-head281" xml:space="preserve" style="it">34. Si rect a line à reflexionis puncto, ſit perpendicularis ſpeculo cylindraceo conuexo: in-<lb/>t{us} continuata, tranſibit per centrum circuli baſib{us} par alleli: & contrà. 21 p 7.</head> <p> <s xml:id="echoid-s7018" xml:space="preserve">PAlàm ex prædictis, perpendicularẽ ſuper punctum reflexionis intellectam extrà & intrà pro-<lb/>duci, diametrum circuli efficere.</s> <s xml:id="echoid-s7019" xml:space="preserve"> Quia ſi non:</s> <s xml:id="echoid-s7020" xml:space="preserve"> cum conſtet diametrum circuli ſuper punctum <lb/>illud tranſeuntem, perpendicularem eſſe ſuper ſuperficiem contingentem columnam in illo <lb/>puncto [ut oſtenſum eſt 32 n] & perpendicularem extrà ſimiliter:</s> <s xml:id="echoid-s7021" xml:space="preserve"> erit [per 14 p 1] cõtinuitas inter <lb/>has perpendiculares, & unam efficient lineam.</s> <s xml:id="echoid-s7022" xml:space="preserve"> Quia ſi non eſt, quòd diameter extrà producta, per-<lb/>pendicularis ſit ſuper illã ſuperficiem:</s> <s xml:id="echoid-s7023" xml:space="preserve"> accidet ex eodẽ ſuperficiei puncto duas erigi perpendicula-<lb/>res [cõtra 13 p 11] In omni ergo ſuperficie reflexionis patet quatuor punctorũ cõcurſus:</s> <s xml:id="echoid-s7024" xml:space="preserve"> cẽtri uiſus:</s> <s xml:id="echoid-s7025" xml:space="preserve"> <lb/>pũcti axis, in qđ cadit քpẽdicularis:</s> <s xml:id="echoid-s7026" xml:space="preserve"> pũcti reflexiõis in ſpeculo:</s> <s xml:id="echoid-s7027" xml:space="preserve"> pũcti, à quo forma corporis ꝓcedit.</s> <s xml:id="echoid-s7028" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div253" type="section" level="0" n="0"> <head xml:id="echoid-head282" xml:space="preserve" style="it">35. Si à uiſu extra ſpeculi conici conuexirecti ſuperficiem, uel ipſi continuam ſito, recta li-<lb/>nea cum uertice axis acutum angulũ faciat: duo plana educta per rect{as} à uiſu, ſpeculum tan-<lb/>gentes & conica latera, per tact{us} puncta tranſeuntia, tangent ſpeculum, & cõſpicuam ſuper-<lb/>ficiem dimidiat a minorem, à qua ad uiſum reflexio fiat, terminabunt. 1. 2 p 7.</head> <p> <s xml:id="echoid-s7029" xml:space="preserve">IN ſpeculis pyramidalibus ſuper baſes ſuas orthogonalibus politis exterius eſt oppoſitio uiſus:</s> <s xml:id="echoid-s7030" xml:space="preserve"> <lb/>ut non ſit uiſus in ſuperficie ſpeculi, aut in continua ei:</s> <s xml:id="echoid-s7031" xml:space="preserve"> & ſecũdum uiſus ſitum, reſpectu ſpeculi <lb/>pyramidalis erit quantitas comprehẽſæ in eo partis.</s> <s xml:id="echoid-s7032" xml:space="preserve"> Igitur ſi radius ab oculi centro ad terminũ <lb/>axis pyramidis, id eſt ad acumen intellectus, faciat cum axe angulũ <lb/> <anchor type="figure" xlink:label="fig-0125-01a" xlink:href="fig-0125-01"/> acutum ex parte pyramidis:</s> <s xml:id="echoid-s7033" xml:space="preserve"> intelligemus à centro uiſus ſuperficiem <lb/>ſecantem pyramidem ſuper circulũ æquidiſtantem baſi pyramidis:</s> <s xml:id="echoid-s7034" xml:space="preserve"> <lb/>& intelligemus duas lineas à centro quidẽ uiſus, tangẽtes illum cir-<lb/>culum in punctis oppoſitis, à quibus protrahemus lineas ſecundum <lb/>longitudinẽ pyramidis.</s> <s xml:id="echoid-s7035" xml:space="preserve"> Superficies ergo ex una harum linearũ lon-<lb/>gitudinis & altera contingentium circulum, continget pyramidem.</s> <s xml:id="echoid-s7036" xml:space="preserve"> <lb/>Si enim ſecuerit:</s> <s xml:id="echoid-s7037" xml:space="preserve"> continget aliud punctum, quàm punctum contin-<lb/>gentiæ circuli:</s> <s xml:id="echoid-s7038" xml:space="preserve"> ſuper illud punctum producatur linea longitudinis, <lb/>& illud punctum & acumen pyramidis ſimul ſunt in hac ſuperficie.</s> <s xml:id="echoid-s7039" xml:space="preserve"> <lb/>Quare illa linea erit in hac ſuperficie, & tranſibit per aliquod punctũ <lb/>circuli:</s> <s xml:id="echoid-s7040" xml:space="preserve"> illud igitur punctum in hac ſuperficie eſt, & in circulo:</s> <s xml:id="echoid-s7041" xml:space="preserve"> quare <lb/>eſt in linea cõmuni circulo & ſuperficiei:</s> <s xml:id="echoid-s7042" xml:space="preserve"> ſed illa contingit circulum:</s> <s xml:id="echoid-s7043" xml:space="preserve"> <lb/>quare cõtingens tranſit per duo puncta circuli, quẽ contingit, quod <lb/>eſt impoſsibile [& contra 2 d 3.</s> <s xml:id="echoid-s7044" xml:space="preserve">] Reſtat igitur, ut illa ſuperficies tan-<lb/>gat pyramidem.</s> <s xml:id="echoid-s7045" xml:space="preserve"> Et generaliter omnis ſuperficies, in qua cõcurrunt <lb/>linea, tangẽs aliquod punctum pyramidis, & longitudinis linea, per <lb/>punctum illud tranſiens, tangit pyramidem ſuper lineam longitudi-<lb/>nis.</s> <s xml:id="echoid-s7046" xml:space="preserve"> Habemus ergo duas ſuperficies ab oculi centro procedẽtes, py-<lb/>ramidem contingentes, inter quas eſt portio pyramidis apparentis <lb/>uiſui in hoc ſitu:</s> <s xml:id="echoid-s7047" xml:space="preserve"> & eſt minor medietate pyramidis:</s> <s xml:id="echoid-s7048" xml:space="preserve"> quoniam lineæ tangentes circulum, includun<gap/> <lb/>eius partem medietate minorem.</s> <s xml:id="echoid-s7049" xml:space="preserve"/> </p> <div xml:id="echoid-div253" type="float" level="0" n="0"> <figure xlink:label="fig-0125-01" xlink:href="fig-0125-01a"> <variables xml:id="echoid-variables18" xml:space="preserve">a b f g c d n</variables> </figure> </div> </div> <div xml:id="echoid-div255" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables19" xml:space="preserve">b a f l g e k h n d c</variables> </figure> <head xml:id="echoid-head283" xml:space="preserve" style="it">36. Si à uiſu recta linea, ſit perpendicularis uertici axis ſpecu-<lb/> li conici cõuexi recti: duo plana educta per rect{as} ſpeculum in ter- minis diametricirculi, ad baſim paralleli tangentes, & later a co- nica per tact{us} puncta tranſeuntia: tangent ſpeculum: & dimi- diatam ſuperficiem conſpicuam, à qua ad uiſum reflexio fiat, ter- minabunt. 89 p 4.</head> <p> <s xml:id="echoid-s7050" xml:space="preserve">SI uerò linea à centro uiſus ad acumen pyramidis ducta, teneat <lb/>angulum rectum cum axe, & intelligatur circulus ſecans pyra-<lb/>midem æquidiſtanter baſi:</s> <s xml:id="echoid-s7051" xml:space="preserve"> linea communis huic circulo, & ſu-<lb/>perficiei, in qua ſunt axis pyramidis, & centrũ uiſus:</s> <s xml:id="echoid-s7052" xml:space="preserve"> erit orthogona-<lb/>lis ſuper axem pyramidis:</s> <s xml:id="echoid-s7053" xml:space="preserve"> quoniã axis eſt orthogonalis ſuper ſuper-<lb/>ficiem circuli [per cõuerſam 14 p 11:</s> <s xml:id="echoid-s7054" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s7055" xml:space="preserve"> per 3 d 11 axis coni eſt ad per <lb/>pendiculum omnibus lineis, à quibus in plano circuli tangitur.</s> <s xml:id="echoid-s7056" xml:space="preserve">] Et <lb/>ſuper lineam communem protrahatur per cẽtrum circuli diameter <lb/>orthogonalis ſuper hãc lineam:</s> <s xml:id="echoid-s7057" xml:space="preserve"> & à terminis huius diametri ortho-<lb/>gonalis protrahãtur duæ cõtingentes circulum:</s> <s xml:id="echoid-s7058" xml:space="preserve"> & etiam duæ lineæ <lb/>uſq;</s> <s xml:id="echoid-s7059" xml:space="preserve"> ad acumen pyramidis.</s> <s xml:id="echoid-s7060" xml:space="preserve"> Duæ ſuperficies, in quibus erũt hæ duæ <lb/>lineæ cũ contingẽtibus, cõtingẽt pyramidẽ ſecũdũ modũ prædictũ.</s> <s xml:id="echoid-s7061" xml:space="preserve"> <lb/> <pb o="120" file="0126" n="126" rhead="ALHAZEN"/> Et quoniam linea communis circulo & ſuperficiei, in qua ſunt centrum uiſus, & axis pyramidis:</s> <s xml:id="echoid-s7062" xml:space="preserve"> eſt <lb/>æquidiſtans lineæ, à centro illius uiſus ad terminum axis productæ [per 28 p 1:</s> <s xml:id="echoid-s7063" xml:space="preserve"> quia axis ad perpen <lb/>diculum eſt utriq;</s> <s xml:id="echoid-s7064" xml:space="preserve">] & huic lineæ communi ſunt æquidiſtantes lineæ, circulum in prædictis pun-<lb/>ctis contingentes [per 28 p 1:</s> <s xml:id="echoid-s7065" xml:space="preserve"> quia per 18 p 3 diameter ipſis ad perpendiculum eſt] erunt illæ lineæ <lb/>æquidiſtantes lineæ à centro uiſus ad terminum axis ductæ [per 9 p 11.</s> <s xml:id="echoid-s7066" xml:space="preserve">] Quare erunt in eadem ſu-<lb/>perficie cum illa [per 35 d 1.</s> <s xml:id="echoid-s7067" xml:space="preserve">] Igitur utraq;</s> <s xml:id="echoid-s7068" xml:space="preserve"> ſuperficierum circulum contingentium, tranſit per cen-<lb/>tra uiſus:</s> <s xml:id="echoid-s7069" xml:space="preserve"> & communis illarum ſuperficierum ſectio, eſt linea à cẽtro uiſus ad terminum axis ducta:</s> <s xml:id="echoid-s7070" xml:space="preserve"> <lb/>& quod inter illas ſuperficies cadit ex pyramide, apparet uiſui:</s> <s xml:id="echoid-s7071" xml:space="preserve"> & eſt medietas pyramidis:</s> <s xml:id="echoid-s7072" xml:space="preserve"> quoniam <lb/>lineas has contingentes circulum interiacet medietas circuli.</s> <s xml:id="echoid-s7073" xml:space="preserve"> Et ita palàm, quòd in hoc ſitu appa-<lb/>ret medietas pyramidalis ſpeculi.</s> <s xml:id="echoid-s7074" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div256" type="section" level="0" n="0"> <head xml:id="echoid-head284" xml:space="preserve" style="it">37. Si recta linea à centro uiſ{us}, cum uertice ſpeculi conici conuexi recti angulum obtuſum <lb/>faciens, continuata concurr at extra ſpeculum, cum diametro circuli ad baſim par alleli conti-<lb/>nuata: duo plana educta per rect{as} à concurſu ſpeculum in dicto circulo tangentes, & later a <lb/>conica per tact{us} puncta tranſeuntia, tangent ſpeculum: & ſuperficiem conſpicuam dimidiata <lb/>maiorem, à qua ad uiſum reflexio fiat: terminabunt. 90 p 4.</head> <p> <s xml:id="echoid-s7075" xml:space="preserve">VErùm ſi linea à centro uiſus ducta ad terminum axis pyramidis, teneat cũ axe angulum ob-<lb/>tuſum ex parte ſuperiori apparente:</s> <s xml:id="echoid-s7076" xml:space="preserve"> & fiat circulus ſecans pyramidem æquidiſtanter baſi-<lb/>linea communis huic circulo & ſuperficiei, in qua eſt centrum uiſus & axis, eſt perpendicu-<lb/>laris ſuper axem pyramidis [per demonſtrata numero præcedente] <lb/> <anchor type="figure" xlink:label="fig-0126-01a" xlink:href="fig-0126-01"/> Et hæc linea communis extra producta, concurret cum linea à cen-<lb/>tro uiſus ad terminum axis ducta [per 11 ax] propter angulum acu-<lb/>tum, quem facit hæc linea cum axe ex inferiori parte [per theſin & <lb/>13 p 1:</s> <s xml:id="echoid-s7077" xml:space="preserve"> & propter angulum b c g rectum.</s> <s xml:id="echoid-s7078" xml:space="preserve">] A puncto igitur concurſus <lb/>linearum protrahantur duæ lineæ, contingêtes circulum in duobus <lb/>punctis oppoſitis:</s> <s xml:id="echoid-s7079" xml:space="preserve"> & producantur lineæ ab his punctis ad acumen <lb/>pyramidis:</s> <s xml:id="echoid-s7080" xml:space="preserve"> ſuperficies, in quibus ſunt lineæ contingentes cum his <lb/>longitudinis lineis, contingunt pyramidem:</s> <s xml:id="echoid-s7081" xml:space="preserve"> & in utraq;</s> <s xml:id="echoid-s7082" xml:space="preserve"> harum ſu-<lb/>perficierum ſunt duo puncta lineæ à centro uiſus ad terminum axis <lb/>ductæ, ſcilicet terminus axis & terminus perpendicularis, in quo ſci <lb/>licet concurrunt linea illa & perpendicularis.</s> <s xml:id="echoid-s7083" xml:space="preserve"> Quare linea illa, quæ <lb/>ducitur à cẽtro uiſus per terminum axis, eſt in utraq;</s> <s xml:id="echoid-s7084" xml:space="preserve"> ſuperficie [per <lb/>1 p 11.</s> <s xml:id="echoid-s7085" xml:space="preserve">] Igitur utraq;</s> <s xml:id="echoid-s7086" xml:space="preserve"> ſuperficies tranſit per cẽtrum uiſus.</s> <s xml:id="echoid-s7087" xml:space="preserve"> Et includunt <lb/>hæ ſuperficies ex inferiori parte minorẽ partem pyramidis medie-<lb/>tate:</s> <s xml:id="echoid-s7088" xml:space="preserve"> quia lineæ contingentes circulum, includunt partem eius mi-<lb/>norem medietate.</s> <s xml:id="echoid-s7089" xml:space="preserve"> Vnde ex parte ſuperiori interiacet ſuperficies py-<lb/>ramidem contingentes pars medietate maior:</s> <s xml:id="echoid-s7090" xml:space="preserve"> & illa eſt, quæ appa-<lb/>ret uiſui.</s> <s xml:id="echoid-s7091" xml:space="preserve"> Quare in hoc ſitu comprehendit uiſus partem pyramidis <lb/>medietate maiorem.</s> <s xml:id="echoid-s7092" xml:space="preserve"/> </p> <div xml:id="echoid-div256" type="float" level="0" n="0"> <figure xlink:label="fig-0126-01" xlink:href="fig-0126-01a"> <variables xml:id="echoid-variables20" xml:space="preserve">a b e c f h g r i d m</variables> </figure> </div> </div> <div xml:id="echoid-div258" type="section" level="0" n="0"> <head xml:id="echoid-head285" xml:space="preserve" style="it">38. Sirecta linea à uiſu per uerticem ſpeculi conici conuexi recti, continuetur cum conico <lb/>latere: tota ſuperficies, præter dictum lat{us}, uidebitur. 91 p 4.</head> <p> <s xml:id="echoid-s7093" xml:space="preserve">SI autem linea à centro uiſus ad terminum axis producta, cadit ſuper latus pyramidis, ut ex ea <lb/>& latere unum efficiatur continuum latus:</s> <s xml:id="echoid-s7094" xml:space="preserve"> Dico quòd non la-<lb/> <anchor type="figure" xlink:label="fig-0126-02a" xlink:href="fig-0126-02"/> tebit uiſum ex hac pyramide, præter lineam quandam intelle-<lb/>ctualem.</s> <s xml:id="echoid-s7095" xml:space="preserve"> Quoniam omnis ſuperficies, in qua eſt linea à centro uiſus <lb/>ad terminum axis ducta, & ſecundum lateris longitudinem prolon-<lb/>gata, ſecat pyramidem, una tantùm excepta, quæ contingit pyrami-<lb/>dem in latere, quod eſt pars lineæ:</s> <s xml:id="echoid-s7096" xml:space="preserve"> & hoc ſolùm latus intellectuale, <lb/>in tota pyramidis ſuperficie ſub hoc ſitu uiſum præterit.</s> <s xml:id="echoid-s7097" xml:space="preserve"> Et huius rei <lb/>ueritas patet ex hoc.</s> <s xml:id="echoid-s7098" xml:space="preserve"> Quòd quocunq;</s> <s xml:id="echoid-s7099" xml:space="preserve"> pyramidis puncto ſumpto ex-<lb/>tra latus intellectuale, ſi ad ipſum ducatur linea à centro uiſus, & ab <lb/>eo linea longitudinis pyramidis ad terminum axis, efficient hæ duæ <lb/>lineę triangulum cum linea lateri applicata:</s> <s xml:id="echoid-s7100" xml:space="preserve"> & erit triangulum in ſu-<lb/>perficie â centro uiſus intellecta, pyramidem ſecante.</s> <s xml:id="echoid-s7101" xml:space="preserve"> [Nam ſi conus <lb/>ſecetur plano per axem:</s> <s xml:id="echoid-s7102" xml:space="preserve"> cõmunis ſectio eſt triangulum per 3 th 1 co-<lb/>nico.</s> <s xml:id="echoid-s7103" xml:space="preserve"> Apollonij] Et ex his lineis huius ſuperficiei nõ niſi duæ cadunt <lb/>in ſuperficiem pyramidis, ſcilicet linea longitudinis, à punct<gap/>ſum-<lb/>pto ad acumen pyramidis, & linea oppoſita huic ex altera parte.</s> <s xml:id="echoid-s7104" xml:space="preserve"> Et <lb/>linea à centro uiſus ad punctum ſumptum ducta, ſecat lineam longi-<lb/>tudinis in puncto ſumpto, & lineam lateris continuati cum uiſu in <lb/>centro uiſus.</s> <s xml:id="echoid-s7105" xml:space="preserve"> Quare huic lineæ à centro uiſus non accidet concurſus <lb/>cum aliqua line arum, niſi in ipſo centro uiſus.</s> <s xml:id="echoid-s7106" xml:space="preserve"> Cum igitur non poſsit <lb/> <pb o="121" file="0127" n="127" rhead="OPTICAE LIBER IIII."/> ſumi punctum aliud, ad quod linea à centro uiſus accedat, & in hoc punctum tranſeat:</s> <s xml:id="echoid-s7107" xml:space="preserve"> nõ occulta-<lb/>tur punctum iſtud ab alio puncto, quòd non perueniat ad centrum uiſus:</s> <s xml:id="echoid-s7108" xml:space="preserve"> quare apparet uiſui, cum <lb/>inter ipſum & uiſum non intercidat corporis ſolidi obiectio.</s> <s xml:id="echoid-s7109" xml:space="preserve"> Et eadem probatio eſt de quolibet ſu-<lb/>perficiei pyramidis puncto.</s> <s xml:id="echoid-s7110" xml:space="preserve"/> </p> <div xml:id="echoid-div258" type="float" level="0" n="0"> <figure xlink:label="fig-0126-02" xlink:href="fig-0126-02a"> <variables xml:id="echoid-variables21" xml:space="preserve">a b h c</variables> </figure> </div> </div> <div xml:id="echoid-div260" type="section" level="0" n="0"> <head xml:id="echoid-head286" xml:space="preserve" style="it">39. Si recta linea à uiſu in uerticem ſpeculi conici conuexi recti, continuetur cum axe: tota <lb/>ſuperficies conica uidebitur. 92 p 4.</head> <p> <s xml:id="echoid-s7111" xml:space="preserve">ET ſi linea à centro uiſus in terminum axis cadens, intret pyramidem:</s> <s xml:id="echoid-s7112" xml:space="preserve"> dico quòd nullũ occul-<lb/>tatur uiſui punctũ in tota pyramidis ſuperficie.</s> <s xml:id="echoid-s7113" xml:space="preserve"> Sumpto enim quocunq;</s> <s xml:id="echoid-s7114" xml:space="preserve"> pũcto in pyramidis <lb/>ſuperficie:</s> <s xml:id="echoid-s7115" xml:space="preserve"> intelligatur ad ipſum linea à centro uiſus, & alia ab <lb/> <anchor type="figure" xlink:label="fig-0127-01a" xlink:href="fig-0127-01"/> <gap/>o uſq;</s> <s xml:id="echoid-s7116" xml:space="preserve"> ad acumen pyramidis:</s> <s xml:id="echoid-s7117" xml:space="preserve"> hæ duæ lineæ includunt ſuperficiem <lb/>triangularem cũ linea à centro uiſus ad terminũ axis ducta, pyrami-<lb/>dem intrãte:</s> <s xml:id="echoid-s7118" xml:space="preserve"> & eſt iſtud triangulũ in ſuperficie pyramidem ſecante:</s> <s xml:id="echoid-s7119" xml:space="preserve"> <lb/>cum omnis ſuperficies, in qua fuerit linea intrans pyramidem, ſecet <lb/>eam.</s> <s xml:id="echoid-s7120" xml:space="preserve"> Linea uerò à centro uiſus ad punctũ ſumptũ ducta, ſecat in illo <lb/>puncto lineã longitudinis ab eo ad acumẽ pyramidis ductã.</s> <s xml:id="echoid-s7121" xml:space="preserve"> Et ex li-<lb/>neis ſuperficièi, in qua ſunt hæ duæ lineæ, non ſunt, niſi duæ lineæ in <lb/>ſuperficie pyramidis, ſcilicet hæc linea longitudinis, à pũcto ad acu-<lb/>men ducta, & alia oppoſita, ſecans angulũ, quem includit hæc cũ li-<lb/>nea pyramidẽ intrante.</s> <s xml:id="echoid-s7122" xml:space="preserve"> Igitur linea illa oppoſita, extra pyramidẽ pro <lb/>ducta, ſecat lineam à centro ad punctũ ſumptum ductã.</s> <s xml:id="echoid-s7123" xml:space="preserve"> Quare linea <lb/>hæc ſecat duas lineas, quę ſolæ ex lineis huius ſuperficiei ſunt in py-<lb/>ramidis ſuperficie:</s> <s xml:id="echoid-s7124" xml:space="preserve"> unam extra pyramidem, aliã in puncto ſumpto.</s> <s xml:id="echoid-s7125" xml:space="preserve"> <lb/>Quare producta in infinitum nõ concurret cũ aliqua illarum linea-<lb/>rum:</s> <s xml:id="echoid-s7126" xml:space="preserve"> unde nõ occultatur uiſui ſumptum punctum, ſecundũ modum <lb/>ſuprà dictum.</s> <s xml:id="echoid-s7127" xml:space="preserve"> In hoc ſitu ergo nulla ſuperficιerũ pyramidem tangen <lb/>tium tranſibit per centrũ uiſus, ſed quęlibet ſecabit lineam à uiſu ſu-<lb/>per terminum axis pyramidem intrãtis, inter uiſum & pyramidem:</s> <s xml:id="echoid-s7128" xml:space="preserve"> <lb/>& eſt in termino axis.</s> <s xml:id="echoid-s7129" xml:space="preserve"> Cum uero linea uiſus lineæ longitudinis pyra <lb/>midis applicatur:</s> <s xml:id="echoid-s7130" xml:space="preserve"> nulla ſuperficierum pyramidem tangentium pertinetad centrum uiſus præter il-<lb/>lam, quæ in prædicta linea contingit pyramidem:</s> <s xml:id="echoid-s7131" xml:space="preserve"> & omnes ſuperficies contingẽtes, ſecabunt lineã <lb/>illam inter uiſum & uerticem pyramιdis.</s> <s xml:id="echoid-s7132" xml:space="preserve"> Similiter in ſitu, in quo duæ ſuperficies contingentes py-<lb/>ramidẽ per centrũ uiſus tranſeunt:</s> <s xml:id="echoid-s7133" xml:space="preserve"> quælibet ſuperficies tangens pyramidẽ in portione pyramidis <lb/>apparẽte, quę duas contingẽtes interiacet, à centro uiſus diuertit:</s> <s xml:id="echoid-s7134" xml:space="preserve"> & ſuper quodcunq;</s> <s xml:id="echoid-s7135" xml:space="preserve"> punctũ illius <lb/>portionis cadat linea uiſualis:</s> <s xml:id="echoid-s7136" xml:space="preserve"> ſecabit pyramidẽ, cũ intercidat inter duas cõtingẽtes uiſuales:</s> <s xml:id="echoid-s7137" xml:space="preserve"> & ſu-<lb/>perficies, in qua fuerit linea hæc uiſualis, & linea longitudinis pyramidis, ſecabit pyramidẽ:</s> <s xml:id="echoid-s7138" xml:space="preserve"> & erit <lb/>hæc uiſualis ſuperficies cuicunq;</s> <s xml:id="echoid-s7139" xml:space="preserve"> ſuperficiei pyramidis in hac portione, continua:</s> <s xml:id="echoid-s7140" xml:space="preserve"> quare & uiſus.</s> <s xml:id="echoid-s7141" xml:space="preserve"/> </p> <div xml:id="echoid-div260" type="float" level="0" n="0"> <figure xlink:label="fig-0127-01" xlink:href="fig-0127-01a"> <variables xml:id="echoid-variables22" xml:space="preserve">a d b k ſ c</variables> </figure> </div> </div> <div xml:id="echoid-div262" type="section" level="0" n="0"> <head xml:id="echoid-head287" xml:space="preserve" style="it">40. Si communis ſectio ſuperficierum, reflexionis & ſpeculi conici conuexi fuerit lat{us} coni-<lb/>cum: à quolιbet conſpicuæ ſuperficiei puncto ad uiſum reflexio fieri poteſt. 31 p 7.</head> <p> <s xml:id="echoid-s7142" xml:space="preserve">DIco ergo, quòd in quolibet ſitu, à quolibet puncto poteſt fieri reflexio.</s> <s xml:id="echoid-s7143" xml:space="preserve"> Sumatur enim pun-<lb/>ctum, & intelligatur circulus per punctum tranſiens, baſi py-<lb/> <anchor type="figure" xlink:label="fig-0127-02a" xlink:href="fig-0127-02"/> ramidi æquidiſtãs:</s> <s xml:id="echoid-s7144" xml:space="preserve"> diameter igitur huius circuli ab hoc pun-<lb/>cto incipiens, erit perpendicularis ſuper axem [per 3 d 11] cũ axis ſit <lb/>perpendicularis ſuper circuli ſuperficiẽ [per 18 d 11, & conuerſam 14 <lb/>p 11.</s> <s xml:id="echoid-s7145" xml:space="preserve">] Quare linea longitudinis à puncto ad acumẽ pyramidis ducta, <lb/>tenet angulum acutum cum diametro, & acutum cum axis termino <lb/>in eadem ſuperficie [per 32 p 1, quia angulus ab axe & ſemidiametro <lb/>g d comprehenſus, eſt rectus.</s> <s xml:id="echoid-s7146" xml:space="preserve">] Sit linea uiſualis ſuper punctũ cadens <lb/>in ſuperficie, in qua eſt linea lõgitudinis & axis, in qua ſuperficie de-<lb/>ducatur perpendicularis ſuper lineã longitudinis in puncto illo:</s> <s xml:id="echoid-s7147" xml:space="preserve"> con <lb/>curret hæc quidem perpendicularis cum axe:</s> <s xml:id="echoid-s7148" xml:space="preserve"> [per 11 ax] & ex ea, & <lb/>axe, & linea longitudinis efficietur triangulum.</s> <s xml:id="echoid-s7149" xml:space="preserve"> Super punctũ illud <lb/>intelligatur linea contingens, & ſuper diametrum circuli, quem feci <lb/>mus, intelligatur diameter alia orthogonalis ſuper ipſam:</s> <s xml:id="echoid-s7150" xml:space="preserve"> quæ erit <lb/>orthogonalis ſuper ipſum axem:</s> <s xml:id="echoid-s7151" xml:space="preserve"> & ſuper ſuperficiem, in qua eſt axis, <lb/>& diameter prima [per 4 p 11] & hæc diameter ſecunda eſt æquidi-<lb/>ſtans contingenti [per 28 p 1] quoniam contingens perpendicularis <lb/>eſt ſuper diametrum primam [per 18 p 3] & ita linea contingens or-<lb/>thogonalis eſt ſuper ſuperficiem, in qua ſunt axis & diameter prima <lb/>[per 8 p 11.</s> <s xml:id="echoid-s7152" xml:space="preserve">] Quare erit perpendicularis ſuper perpendicularẽ, quam <lb/>primo fecimus [per 3 d 11] & ita illa prima perpendicularis orthogonaliter cadit ſuper ſuperficiem, <lb/>contingẽtem pyramidem, in qua punctum eſt ſumptum.</s> <s xml:id="echoid-s7153" xml:space="preserve"> Igitur ſi linea uiſualis, cadens in punctum <lb/>ſumptum, trãſeat ſecundũ proceſſum perpendicularis:</s> <s xml:id="echoid-s7154" xml:space="preserve"> erit quidẽ orthogonalis ſuper ſuperficiem, <lb/> <pb o="122" file="0128" n="128" rhead="ALHAZEN"/> pyramidem illã in puncto contingentẽ, & fiet reflexio formæ per eandẽ lineam [per 11 n.</s> <s xml:id="echoid-s7155" xml:space="preserve">] Si aute<gap/> <lb/>deuiet à proceſſu perpẽdicularis:</s> <s xml:id="echoid-s7156" xml:space="preserve"> faciet quidẽ angulum cum perpendiculari acutũ in puncto ſum-<lb/>pto:</s> <s xml:id="echoid-s7157" xml:space="preserve"> & poterit produci in ſuperficie eius lineæ uiſualis, alia linea à puncto illo, quæ æqualẽ angulũ <lb/>huic teneat cum perpendiculari:</s> <s xml:id="echoid-s7158" xml:space="preserve"> cum perpendicularis orthogonalis ſit ſuper ſuperficiẽ contingen <lb/>tem.</s> <s xml:id="echoid-s7159" xml:space="preserve"> Linea autẽ quæcunq;</s> <s xml:id="echoid-s7160" xml:space="preserve"> ſuper ſuperficiem, contingentem in puncto ſumpto orthogonaliter ca-<lb/>dens, tranſit ad axem [per 11 a x:</s> <s xml:id="echoid-s7161" xml:space="preserve"> eſt enim perpẽdicularis conico lateri:</s> <s xml:id="echoid-s7162" xml:space="preserve"> quia, cum ex theſi ſit perpen-<lb/>dicularis plano conum tangenti in latere per 6 uel 35 n:</s> <s xml:id="echoid-s7163" xml:space="preserve"> erit per 3 d 11 ipſi lateri perpendicularis] & <lb/>ſi ab axe ducatur orthogonalis ad hanc ſuperficiem, efficient perpendiculares, interior & exterior, <lb/>lineã unam [per 14 p 1:</s> <s xml:id="echoid-s7164" xml:space="preserve">] quòd ſi non:</s> <s xml:id="echoid-s7165" xml:space="preserve"> cũ perpendicularis interior, extrà producta, ſit etiã perpendi-<lb/>cularis ſuper ſuperficiem:</s> <s xml:id="echoid-s7166" xml:space="preserve"> accidet ab eodẽ puncto ſuper aliquam ſuperficiem, erigi duas perpendi-<lb/>culares in eandẽ partem [contra 13 p 11.</s> <s xml:id="echoid-s7167" xml:space="preserve">] Palàm igitur, quòd à quocunq;</s> <s xml:id="echoid-s7168" xml:space="preserve"> puncto ſuperficiei pyrami-<lb/>dis uiſo, poteſt fieri reflexio ad paritatem angulorum.</s> <s xml:id="echoid-s7169" xml:space="preserve"> Et cum linea declinata occurrerit:</s> <s xml:id="echoid-s7170" xml:space="preserve"> forma ue-<lb/>niet ad ſpeculum ſuper lineam hanc, & reflectetur ad uiſum ſuper aliam:</s> <s xml:id="echoid-s7171" xml:space="preserve"> & ſunt hæ lineæ in eadem <lb/>ſuperficie orthogonali, ſuper ſuperficiem contingentem pyramidem in puncto reflexionis [per 6.</s> <s xml:id="echoid-s7172" xml:space="preserve"> <lb/>13 n.</s> <s xml:id="echoid-s7173" xml:space="preserve">] Et hæc eſt ſuperficies reflexionis, in qua ſemper fit comprehenſio quatuor punctorũ, ſcilicet, <lb/>centri uiſus, puncti uiſi, puncti reflexionis, termini perpendicularis.</s> <s xml:id="echoid-s7174" xml:space="preserve"/> </p> <div xml:id="echoid-div262" type="float" level="0" n="0"> <figure xlink:label="fig-0127-02" xlink:href="fig-0127-02a"> <variables xml:id="echoid-variables23" xml:space="preserve">b ſ a u f d c h n g r k s x q p</variables> </figure> </div> </div> <div xml:id="echoid-div264" type="section" level="0" n="0"> <head xml:id="echoid-head288" xml:space="preserve" style="it">41. Communis ſectio ſuperficierum reflexionis & ſpeculi conici cõuexi eſt lat{us} conicum uel <lb/>ellipſis: nunquam uerò circul{us}. 12 p 7.</head> <p> <s xml:id="echoid-s7175" xml:space="preserve">DIuerſificantur autẽ lineæ cõmunes ſuperficiei reflexionis, & ſuperficiei pyramidis.</s> <s xml:id="echoid-s7176" xml:space="preserve"> Cũ enim <lb/>radius uiſualis continuus fuerit axi pyramidis, ſcilicet, cũ in ſuperficie reflexionis fuerit to-<lb/>tus axis, & perpendicularis ad axem tranſiens:</s> <s xml:id="echoid-s7177" xml:space="preserve"> erit ſuperficici reflexionis & ſuperficiei py-<lb/>ramidis cõmunis linea, linea longitudinis in hoc ſitu.</s> <s xml:id="echoid-s7178" xml:space="preserve"> Quoniã quælibet ſuperficies, in qua eſt totus <lb/>axis, hanchabet lineam communem cum ſuperficie pyramidis [ut patet è 18 d 11.</s> <s xml:id="echoid-s7179" xml:space="preserve">] Et in omni alio <lb/>ſitu unica longitudinis pyramidis linea erit communis, illa ſcilicet, quę fuerit in ſuperficie uiſus cẽ-<lb/>trum & axẽ continẽte.</s> <s xml:id="echoid-s7180" xml:space="preserve"> Et quãdo centrũ uiſus nõ erit in directo axis, una tantùm erit ſuperficies ta-<lb/>lis:</s> <s xml:id="echoid-s7181" xml:space="preserve"> & omnis alia cõmunis linea, erit ſectio pyramidalis, nõ circulus.</s> <s xml:id="echoid-s7182" xml:space="preserve"> Si enim fuerit circul<emph style="sub">9</emph>:</s> <s xml:id="echoid-s7183" xml:space="preserve"> erit ſuք-<lb/>ficies illi<emph style="sub">9</emph> circuli in ſuքficie reflexiõis.</s> <s xml:id="echoid-s7184" xml:space="preserve"> Et ꝗa axis orthogonalis eſt ſuք illũ circulũ [ք 18 d 11, & cõuer-<lb/>ſam 14 p 11] cũ quilibet circulus pyramidis ſit æquidiſtãs baſi [ք 4 th.</s> <s xml:id="echoid-s7185" xml:space="preserve"> 1 conicorũ Apollonij] erũt la-<lb/>tera pyramidis declinata ſuք circulũ:</s> <s xml:id="echoid-s7186" xml:space="preserve"> & ita ſuք ſuքficiẽ reflexiõis.</s> <s xml:id="echoid-s7187" xml:space="preserve"> Quare in ſuperficie illa nõ poteſt <lb/>duci քpẽdicularis ſuք lineã lõgitudinis pyramidis:</s> <s xml:id="echoid-s7188" xml:space="preserve"> ſed [ք 6.</s> <s xml:id="echoid-s7189" xml:space="preserve">13 n] քpendicularis ducta ſuք ſuքficiẽ, <lb/>cõtingẽtẽ locũ reflexiõis, eſt in ſuքficie reflexiõis, & քpẽdicularis ſuք lineã lõgitudinis [ut oſtẽſum <lb/>eſt ꝓximo numero] cũ q̃libet ſuքficies tãgẽs tãgat in linea lõgitudinis [ք 6.</s> <s xml:id="echoid-s7190" xml:space="preserve"> 35 n.</s> <s xml:id="echoid-s7191" xml:space="preserve">] Accidit igitur im <lb/>poſsibile [cõtra 13 n.</s> <s xml:id="echoid-s7192" xml:space="preserve">] Quare reſtat oẽs alias cõmunes reflexiõis lineas, ſectiões pyramidales eſſe.</s> <s xml:id="echoid-s7193" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div265" type="section" level="0" n="0"> <head xml:id="echoid-head289" xml:space="preserve" style="it">42. Si communis ſectio ſuperficierum reflexionis & ſpeculi conici conuexi, fuerit lat{us} co-<lb/>nicum: reflexio à quocun ipſi{us} puncto facta, in eadem ſuperficie ſemper fiet. 19 p 7.</head> <p> <s xml:id="echoid-s7194" xml:space="preserve">ET cũ fuerit linea cõmunis, linea lõgitudinis, ex quocũ q;</s> <s xml:id="echoid-s7195" xml:space="preserve"> pũcto illius lineæ fiat reflexio:</s> <s xml:id="echoid-s7196" xml:space="preserve"> erit in <lb/>eadẽ ſuperficie cũ cuiuſcunq;</s> <s xml:id="echoid-s7197" xml:space="preserve"> alterius pũcti reflexione.</s> <s xml:id="echoid-s7198" xml:space="preserve"> Quoniã à quolibet huius lineæ pũcto <lb/>ducta perpẽdicularis cõtin get axẽ [ut oſtẽſum eſt 40 n:</s> <s xml:id="echoid-s7199" xml:space="preserve">] & erũt in ſuքficie reflexiõis cẽtrum <lb/>uiſus:</s> <s xml:id="echoid-s7200" xml:space="preserve"> & punctũ reflexionis:</s> <s xml:id="echoid-s7201" xml:space="preserve"> & punctũ axis.</s> <s xml:id="echoid-s7202" xml:space="preserve"> Quare in hac ſuperficie fit reflexio à quocunq;</s> <s xml:id="echoid-s7203" xml:space="preserve"> puncto.</s> <s xml:id="echoid-s7204" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div266" type="section" level="0" n="0"> <head xml:id="echoid-head290" xml:space="preserve" style="it">43. Si cõmunis ſectio ſuperficierũ, reflexionis & ſpeculi conici cõuexi fuerit ellipſis: ab uno uel <lb/>duob. cõſpicuæ ſuperficiei pũctis quib{us}libet, in eadẽ ſuքficie ad uiſum reflexio fieri poteſt. 34 p 7.</head> <p> <s xml:id="echoid-s7205" xml:space="preserve">SI uerò cõmunis linea nõ fuerit linea lõgitudinis:</s> <s xml:id="echoid-s7206" xml:space="preserve"> dico quòd uel <lb/> <anchor type="figure" xlink:label="fig-0128-01a" xlink:href="fig-0128-01"/> ab uno cõmunis lineæ pũcto, in eadẽ ſuperficie fiat reflexio, uel <lb/>à duobus tantùm.</s> <s xml:id="echoid-s7207" xml:space="preserve"> Quoniã ducta perpendiculari à puncto refle-<lb/>xionis:</s> <s xml:id="echoid-s7208" xml:space="preserve"> perueniet ad axẽ, & cadet in aliquod punctũ eius [ut patuit <lb/>40 n:</s> <s xml:id="echoid-s7209" xml:space="preserve">] & intellecto circulo ſuper punctũ reflexionis, orthogonaliter <lb/>ſecabit circulus axem [Quia enim circulus parallelus eſt baſi per 4 <lb/>th 1 conico.</s> <s xml:id="echoid-s7210" xml:space="preserve"> Apollonij:</s> <s xml:id="echoid-s7211" xml:space="preserve"> erit axis ad ipſum perpendicularis per 18 d, & <lb/>conuerſam 14 p 11.</s> <s xml:id="echoid-s7212" xml:space="preserve">] Et quia perpendicularis tenet angulum acutum <lb/>cum axe:</s> <s xml:id="echoid-s7213" xml:space="preserve"> erit perpẽdicularis declinata ſuper circulũ;</s> <s xml:id="echoid-s7214" xml:space="preserve"> & circumquaq;</s> <s xml:id="echoid-s7215" xml:space="preserve"> <lb/>ducta, ſemper erit æqualis.</s> <s xml:id="echoid-s7216" xml:space="preserve"> Vnde fiet pyramis, cuius baſis circulus, <lb/>acumen punctum axis, in quod cadit perpendicularis.</s> <s xml:id="echoid-s7217" xml:space="preserve"> Igitur ſuper-<lb/>ficies reflexionis aut tanget hanc pyramidẽ, aut ſecabit.</s> <s xml:id="echoid-s7218" xml:space="preserve"> Si tangat:</s> <s xml:id="echoid-s7219" xml:space="preserve"> di-<lb/>co quòd à puncto reflexionis ſumpto poſsit tantùm fieri in eadẽ ſu-<lb/>perficie reflexio.</s> <s xml:id="echoid-s7220" xml:space="preserve"> Planũ enim, quòd ſuperficies reflexionis continget <lb/>hanc pyramidẽ ſuper perpendicularẽ, quæ eſt linea orthogonalis in <lb/>ſuperficie reflexionis [per 6.</s> <s xml:id="echoid-s7221" xml:space="preserve"> 35 n.</s> <s xml:id="echoid-s7222" xml:space="preserve">] Et ſi ab acumine totalis pyrami-<lb/>dis ducantur lineæ ad ſectionem communẽ ſuperficiei reflexionis & <lb/>pyramidis totalis, prius cadent in circulũ, qui eſt baſis pyramidιs in-<lb/>tellectæ, quàm in ſectionẽ:</s> <s xml:id="echoid-s7223" xml:space="preserve"> præter unã, quæ in punctũ reflexionis ca-<lb/>dit.</s> <s xml:id="echoid-s7224" xml:space="preserve"> Si ergo ab alio puncto cõmunis ſectionis fieret reflexio:</s> <s xml:id="echoid-s7225" xml:space="preserve"> linea ab <lb/>illo puncto ad acumen intellectæ pyramidis ducta:</s> <s xml:id="echoid-s7226" xml:space="preserve"> erit perpendicularis ſuper lineam longitudinis <lb/> <pb o="123" file="0129" n="129" rhead="OPTICAE LIBER IIII."/> pyramidis, per punctum illud tranſeuntem [ut antè patuit:</s> <s xml:id="echoid-s7227" xml:space="preserve">] ſed linea ab acumine pyramidis intel-<lb/>lectæ ad punctũ circuli, per quod tranſit illa linea longitudinis, abſq;</s> <s xml:id="echoid-s7228" xml:space="preserve"> dubio eſt perpendicularis ſu-<lb/>per eam.</s> <s xml:id="echoid-s7229" xml:space="preserve"> Quare alia angulum tenet acutum cum hac linea, non rectum.</s> <s xml:id="echoid-s7230" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0129-01a" xlink:href="fig-0129-01"/> [ſecus tres anguli trianguli rectilinei maiores eſſent duobus rectis cõ-<lb/>tra 32 p 1:</s> <s xml:id="echoid-s7231" xml:space="preserve"> quod tamen abſurdum ex angulis c r i, cir rectis concluſis <lb/>ſequitur.</s> <s xml:id="echoid-s7232" xml:space="preserve">] Si uerò ſuperficies reflexionis ſecet intellectualem pyrami-<lb/>dem:</s> <s xml:id="echoid-s7233" xml:space="preserve"> ſecabit circulum, qui eſt baſis, in duobus punctis.</s> <s xml:id="echoid-s7234" xml:space="preserve"> [Quia enim cõ-<lb/>munis ſectio ellipſis (quæ ex theſi eſt reflexionis ſuperficies) & circuli <lb/>(qui eſt fictæ pyramidis baſis) eſt linea recta per 3 p 11, duobus punctis <lb/>terminata:</s> <s xml:id="echoid-s7235" xml:space="preserve"> ellipſis igitur ſecat circulũ in duobus punctis, nempe lineæ <lb/>rectæ terminis.</s> <s xml:id="echoid-s7236" xml:space="preserve">] Dico, quòd hæc ſola ſunt puncta in tota ſectione com-<lb/>muni, à quibus fieri poſsit reflexio in eadẽ ſuperficie.</s> <s xml:id="echoid-s7237" xml:space="preserve"> Quoniã ab utroq;</s> <s xml:id="echoid-s7238" xml:space="preserve"> <lb/>iſtorum punctorũ linea ducta ad acumen intellectæ pyramidis, eſt per-<lb/>pendicularis ſuper lineam longitudinis ſuper punctum ſuũ tranſeun-<lb/>tem.</s> <s xml:id="echoid-s7239" xml:space="preserve"> À<unsure/>quocunq;</s> <s xml:id="echoid-s7240" xml:space="preserve"> enim ſectionis puncto alio ducatur linea ad acumen <lb/>illius pyramidis:</s> <s xml:id="echoid-s7241" xml:space="preserve"> tenebit angulum acutum cum linea longitudinis per <lb/>ipſum tranſeunte, cũ perpendicularis cum eadẽ longitudinis linea an-<lb/>gulum rectum teneat in circulo.</s> <s xml:id="echoid-s7242" xml:space="preserve"> Et lineæ ductæ ab acumine pyramidis <lb/>intellectæ ad puncta ſectionis, quæ intercidunt inter ſpeculi acumen & <lb/>circulum:</s> <s xml:id="echoid-s7243" xml:space="preserve"> facient angulos obtuſos cum lineis longitudinis uerſus par-<lb/>tem acuminis pyramιdis totalis:</s> <s xml:id="echoid-s7244" xml:space="preserve"> & quæ ducuntur ad puncta inter cir-<lb/>culum & baſim ſpeculi interiacentia, faciẽt cum linea longitudinis an-<lb/>gulos acutos ex parte acuminis ſpeculi, obtuſos ex parte baſis.</s> <s xml:id="echoid-s7245" xml:space="preserve"> Ergo à nullo iſtorũ punctorum po-<lb/>teſt fieri reflexio.</s> <s xml:id="echoid-s7246" xml:space="preserve"/> </p> <div xml:id="echoid-div266" type="float" level="0" n="0"> <figure xlink:label="fig-0128-01" xlink:href="fig-0128-01a"> <variables xml:id="echoid-variables24" xml:space="preserve">f d d e r b g c h i p ſ q s n k</variables> </figure> <figure xlink:label="fig-0129-01" xlink:href="fig-0129-01a"> <variables xml:id="echoid-variables25" xml:space="preserve">f a r d e b g c h p ſ s n k</variables> </figure> </div> </div> <div xml:id="echoid-div268" type="section" level="0" n="0"> <head xml:id="echoid-head291" xml:space="preserve" style="it">44. Si uiſ{us} fuerit in caua ſpeculi ſphærici ſuperficie: uidebit totam: ſi intra uel extra: aliâs <lb/>hemiſp hærium, aliâs pl{us}, aliâs min{us}: ſi in centro: ſe ipſum tantùm uidebit. 71. 72 p 4. 4 p 8.</head> <p> <s xml:id="echoid-s7247" xml:space="preserve">IN ſpeculis ſphæricis concauis ſi uiſus fuerit intra concauitatem ſpeculi:</s> <s xml:id="echoid-s7248" xml:space="preserve"> tota ſpeculi ſuperficies <lb/>apparebit ei:</s> <s xml:id="echoid-s7249" xml:space="preserve"> quod ſi extra fuerit:</s> <s xml:id="echoid-s7250" xml:space="preserve"> poterit comprehendere portionem eius maiorem medietate, <lb/>quam ſcilicet fecit circulus ſphæræ, quem contingunt duo radij à centro uiſus ducti:</s> <s xml:id="echoid-s7251" xml:space="preserve"> uiſu autem <lb/>in centro huius ſpeculi exiſtente, non fiet ab aliquo puncto ſpeculi reflexio, niſi in ſe.</s> <s xml:id="echoid-s7252" xml:space="preserve"> Quoniã enim <lb/>quælibet linea à centro ſphæræ ad ſphæram ducta perpendicularis eſt ſuper ſuperficiem, ſphæram <lb/>in puncto illo tangentem [per 25 n uel 4 th.</s> <s xml:id="echoid-s7253" xml:space="preserve"> 1 ſphæricorum:</s> <s xml:id="echoid-s7254" xml:space="preserve">] ergo in hoc ſitu non comprehendet <lb/>uiſus per reflexionem, niſi ſe tantùm [per 11 n.</s> <s xml:id="echoid-s7255" xml:space="preserve">]</s> </p> </div> <div xml:id="echoid-div269" type="section" level="0" n="0"> <head xml:id="echoid-head292" xml:space="preserve" style="it">45. Si uiſ{us} ſit extra centrum ſpeculi ſphærici caui: uiſibile à quolibet ei{us} puncto ad uiſum <lb/>reflecti poteſt: excepto eo, in quod recta à uiſu per centrum ſpeculi ducta, cadit. 6. 3 p 8.</head> <p> <s xml:id="echoid-s7256" xml:space="preserve">SI uerò ſtatuatur uiſus extra centrum ſphæræ:</s> <s xml:id="echoid-s7257" xml:space="preserve"> poterit fieri reflexio alterius rei uiſibilis à quo-<lb/>cunq;</s> <s xml:id="echoid-s7258" xml:space="preserve"> ſpeculi puncto:</s> <s xml:id="echoid-s7259" xml:space="preserve"> præterquam ab eo, in quod cadit diameter, à centro uiſus ad ſphæram <lb/>per centrum ſphæræ ducta:</s> <s xml:id="echoid-s7260" xml:space="preserve"> quoniam diameter cadit ſuper ſuperficiem contingentem ſphæ-<lb/>ram, orthogonaliter [per 25 n, ideoq́;</s> <s xml:id="echoid-s7261" xml:space="preserve"> reflectitur in ſeipſam per 11 n.</s> <s xml:id="echoid-s7262" xml:space="preserve">] Sumpto autẽ alio puncto, du-<lb/>catur ad ipſum diameter à centro ſphæræ, & linea à <lb/> <anchor type="figure" xlink:label="fig-0129-02a" xlink:href="fig-0129-02"/> centro uiſus.</s> <s xml:id="echoid-s7263" xml:space="preserve"> Ex his ergo lineis acutus includetur <lb/>angulus:</s> <s xml:id="echoid-s7264" xml:space="preserve"> quoniam linea uiſualis cadit inter diame-<lb/>trum & ſuperficiem contingentem punctum, quæ <lb/>ſcilicet eſt extra ſphæram:</s> <s xml:id="echoid-s7265" xml:space="preserve"> & ſiue ſit oculus intra ſpe <lb/>culum, ſiue extra, cadit uiſualis linea intra ſpecu-<lb/>lum:</s> <s xml:id="echoid-s7266" xml:space="preserve"> quia cadit inter lineas uiſuales contingentes <lb/>circulum portionis ſphæræ.</s> <s xml:id="echoid-s7267" xml:space="preserve"> [Itaq;</s> <s xml:id="echoid-s7268" xml:space="preserve"> ſi diameter g b & <lb/>linea reflexionis g a in peripheriam cõtinuatæ, con-<lb/>nectantur:</s> <s xml:id="echoid-s7269" xml:space="preserve"> erit angulus a g b acutus per 31 p 3.</s> <s xml:id="echoid-s7270" xml:space="preserve">32 p 1.</s> <s xml:id="echoid-s7271" xml:space="preserve">] <lb/>Cum igitur diameter angulum rectum teneat cum <lb/>contingẽte [per 18 p 3:</s> <s xml:id="echoid-s7272" xml:space="preserve">] ſecetur ex eo acutus, æqua-<lb/>lis prædicto in eadem ſuperficie:</s> <s xml:id="echoid-s7273" xml:space="preserve"> dico ergo, quòd li-<lb/>nea reflexionis cadit intra ſpeculum:</s> <s xml:id="echoid-s7274" xml:space="preserve"> quoniam com <lb/>munis linea ſpeculi & ſuperficiei reflexionis, eſt cir-<lb/>culus, tenens cum diametro angulum acutum ma-<lb/>iorem omni rectilineo acuto [per 31 p 3.</s> <s xml:id="echoid-s7275" xml:space="preserve">] Et in ſin-<lb/>gulis punctis erit hic modus reflexionis.</s> <s xml:id="echoid-s7276" xml:space="preserve"> Palàm ex <lb/>his, quòd in omni ſuperficie reflexionis erunt centrum uiſus:</s> <s xml:id="echoid-s7277" xml:space="preserve"> centrum ſpeculi:</s> <s xml:id="echoid-s7278" xml:space="preserve"> punctum reflexio-<lb/>nis:</s> <s xml:id="echoid-s7279" xml:space="preserve"> punctum uiſum:</s> <s xml:id="echoid-s7280" xml:space="preserve"> terminus diametri à centro uiſus per centrum ſphæræ ductæ:</s> <s xml:id="echoid-s7281" xml:space="preserve"> & quòd com-<lb/>munis omnium ſuperficierum reflexionis linea cum ſuperficie ſpeculi, eſt circulus:</s> <s xml:id="echoid-s7282" xml:space="preserve"> & quòd à quo-<lb/>libet lineæ communis puncto poteſt fieri in eadem ſuperficie reflexio.</s> <s xml:id="echoid-s7283" xml:space="preserve"/> </p> <div xml:id="echoid-div269" type="float" level="0" n="0"> <figure xlink:label="fig-0129-02" xlink:href="fig-0129-02a"> <variables xml:id="echoid-variables26" xml:space="preserve">ſ g d f h b a <gap/></variables> </figure> </div> <pb o="124" file="0130" n="130" rhead="ALHAZEN"/> </div> <div xml:id="echoid-div271" type="section" level="0" n="0"> <head xml:id="echoid-head293" xml:space="preserve" style="it">46. In ſpeculo cylindraceo cauo ſuperficies reflexionis quatuor habet puncta: uiſ{us}, uiſibilis, <lb/>reflexionis, & axis, in quod perpendicularis à reflexionis puncto ducta, cadit. 3 p 9.83 p 4.</head> <p> <s xml:id="echoid-s7284" xml:space="preserve">IN ſpeculis columnaribus concauis poteſt comprehendi totum ſpeculum:</s> <s xml:id="echoid-s7285" xml:space="preserve"> ſi fuerit uiſus intra <lb/>ipſum:</s> <s xml:id="echoid-s7286" xml:space="preserve"> ſed eo extrà ſito, uidebitur maior medietate ſpeculi portio, quæ ſcilicet interiacet duas <lb/>ſuperficies à centro uiſus procedentes, columnam contingentes.</s> <s xml:id="echoid-s7287" xml:space="preserve"> Intelligemus autem ſuperfi-<lb/>ciem à centro uiſus procedẽtem, baſibus columnæ æquidiſtantem:</s> <s xml:id="echoid-s7288" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0130-01a" xlink:href="fig-0130-01"/> hæc ſuperficies aut cadet in columnã, aut nõ:</s> <s xml:id="echoid-s7289" xml:space="preserve"> ſi ceciderit, linea com-<lb/>munis huic ſuperficiei & columnæ erit circulus [per 5th.</s> <s xml:id="echoid-s7290" xml:space="preserve"> Sereni de <lb/>ſectione cylindri:</s> <s xml:id="echoid-s7291" xml:space="preserve">] & linea uiſualis, tranſiens per centrum huius cir-<lb/>culi, cadet orthogonaliter ſuper ſuperficiem, contingentem colu-<lb/>mnam in puncto, in quod cadit linea [ut dem õſtratũ eſt 32 n] & fiet <lb/>reflexio per eandem lineam ad eius originem [per 11 n.</s> <s xml:id="echoid-s7292" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s7293" xml:space="preserve"> cum li-<lb/>nea recta (quæ per 1 p 11 in uno eſt plano) tranſeat per puncta uiſus, <lb/>uiſibilis, reflexionis, & axis, in quod perpendicularis à reflexionis <lb/>puncto ducta, cadit:</s> <s xml:id="echoid-s7294" xml:space="preserve"> erũt ipĩa in uno reflexionis plano.</s> <s xml:id="echoid-s7295" xml:space="preserve">] Quodcunq;</s> <s xml:id="echoid-s7296" xml:space="preserve"> <lb/>aliud ſumatur punctum, linea perpendiculariter ab hoc puncto du-<lb/>cta, cadet in axem [ut patuit 40 n:</s> <s xml:id="echoid-s7297" xml:space="preserve">] & linea uiſualis in punctũ illud <lb/>cadens, faciet angulum acutum cum linea perpendiculari [ut oſten-<lb/>ſum eſt ſuperiore numero] cũ ſit inter perpendicularẽ & contingen <lb/>tem.</s> <s xml:id="echoid-s7298" xml:space="preserve"> Et quòd hęc linea cadat intra ſpeculũ, planum eſt ex hoc:</s> <s xml:id="echoid-s7299" xml:space="preserve"> quòd <lb/>cadit inter ſuperficies portionem contingentes.</s> <s xml:id="echoid-s7300" xml:space="preserve"> Poterimus igitur in <lb/>eadem reflexionis ſuperficie ex angulo, quem facit perpendicularis <lb/>cum contingente, excipere angulum acutum, æqualem angulo acu <lb/>to prædicto:</s> <s xml:id="echoid-s7301" xml:space="preserve"> & cadet linea reflexionis, hunc angulum continens, in-<lb/>tra columnam:</s> <s xml:id="echoid-s7302" xml:space="preserve"> quoniam cadet inter perpendicularem & lineam lõ-<lb/>gitudinis, per terminum perpendicularis tranſeuntem.</s> <s xml:id="echoid-s7303" xml:space="preserve"> Erunt igitur in ſuperficie reflexionis cen-<lb/>trum uiſus, punctum reflexionis, punctum uiſum, punctum axis, in quod cadit perpendicularis.</s> <s xml:id="echoid-s7304" xml:space="preserve"/> </p> <div xml:id="echoid-div271" type="float" level="0" n="0"> <figure xlink:label="fig-0130-01" xlink:href="fig-0130-01a"> <variables xml:id="echoid-variables27" xml:space="preserve">a d f <gap/> t e b</variables> </figure> </div> </div> <div xml:id="echoid-div273" type="section" level="0" n="0"> <head xml:id="echoid-head294" xml:space="preserve" style="it">47. Si communis ſectio ſuperficierum, reflexionis & ſpeculi cylindracei caui, fuerit lat{us} cy-<lb/>lindr aceum, aut circul{us}: reflexio à quocun ſectionis puncto facta, in eadem ſuperficie fiet.</head> <p> <s xml:id="echoid-s7305" xml:space="preserve">ET ſi hoc modo ſtatuatur uiſus, ut communis linea ſuperficiei reflexionis & ſuperficiei colu-<lb/>mnæ ſit linea longitudinis;</s> <s xml:id="echoid-s7306" xml:space="preserve"> à quo cunq;</s> <s xml:id="echoid-s7307" xml:space="preserve"> puncto cõmunis lineæ fiat reflexio:</s> <s xml:id="echoid-s7308" xml:space="preserve"> in una determi-<lb/>nata erit ſuperficie, omnibus his reflexionibus communi, ea ſcilicet, in qua centrum uiſus, & <lb/>axis columnæ totus, ſicut dictum eſt ſuperius in columnari ſpeculo non concauo [32 n.</s> <s xml:id="echoid-s7309" xml:space="preserve">] Similiter <lb/>ſi linea communis fuerit circulus, omnes reflexiones à punctis illius circuli factæ, procedent in ea-<lb/>dem ſuperficie, ſicut in alijs circulis patuit.</s> <s xml:id="echoid-s7310" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div274" type="section" level="0" n="0"> <head xml:id="echoid-head295" xml:space="preserve" style="it">48. Si communis ſectio ſuperficierum, reflexionis & ſpeculi cylindracei caui fuerit elli-<lb/>pſis: à plurib{us} ei{us} punctis idem uiſibile ad eundem uiſum, in eadem ſuperficie reflecti po-<lb/>teſt. 9 p 9.</head> <p> <s xml:id="echoid-s7311" xml:space="preserve">ET ſi ſectio columnaris, fuerit linea communis:</s> <s xml:id="echoid-s7312" xml:space="preserve"> à duobus quidem eius punctis tantùm fiet re-<lb/>flexio in eadẽ ſuperficie, licet in ſuperioribus columnis [33 n] tantùm ab uno puncto in uni-<lb/>ca ſuperficie fieret reflexio, unico uiſu adhibito:</s> <s xml:id="echoid-s7313" xml:space="preserve"> quoniam illic latebant uiſum puncta ſectio-<lb/>nis ſe reſpicientia, per quæ ſcilicet tranſit circulus columnæ baſibus æquidiſtans:</s> <s xml:id="echoid-s7314" xml:space="preserve"> uiſo enim uno il-<lb/>lorum punctorum, latebat aliud, propter minoris columnæ portionis apparentiam:</s> <s xml:id="echoid-s7315" xml:space="preserve"> ſed in his appa <lb/>ret maior columnæ portio:</s> <s xml:id="echoid-s7316" xml:space="preserve"> unde ab uno uiſu percipiuntur puncta terminantia diametrum circuli, <lb/>æquidiſtantis baſibus columnæ.</s> <s xml:id="echoid-s7317" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div275" type="section" level="0" n="0"> <head xml:id="echoid-head296" xml:space="preserve" style="it">49. Si uiſ{us} fuerit intra ſpeculum conicum cauum: tota ei{us} ſuperficies uidebitur: ſi extra & <lb/>recta à uiſu continuetur cum axe, uel conico latere: tot a occultabitur. 5. 2. 9. 3 p 9.</head> <p> <s xml:id="echoid-s7318" xml:space="preserve">IN ſpeculis pyramidalibus concauis, ſi fuerit uiſus intra ſpeculum:</s> <s xml:id="echoid-s7319" xml:space="preserve"> uidebit ipſum totum:</s> <s xml:id="echoid-s7320" xml:space="preserve"> ſi uerò <lb/>extra, & linea à cẽtro uiſus ad acumen pyramidis ducta, intret pyramidem, aut applicetur lineæ <lb/>longitudinis pyramidis, nihil uidebitur ex ſpeculò.</s> <s xml:id="echoid-s7321" xml:space="preserve"> Quohiam quæcunq;</s> <s xml:id="echoid-s7322" xml:space="preserve"> alia linea ab oculo ad <lb/>pyramidem ducta, cadet in pyramidis ſuperficiem exteriorem:</s> <s xml:id="echoid-s7323" xml:space="preserve"> unde occultabitur interior ſuperfi-<lb/>cies.</s> <s xml:id="echoid-s7324" xml:space="preserve"> Si autem auferatur portio à pyramide, poterit uideri pars pyramidis, cadens inter contingen-<lb/>tes ſuperficies à centro ductas, ſcilicet maior.</s> <s xml:id="echoid-s7325" xml:space="preserve"> Et ſi linea à centro uiſus, ſit perpendicularis ſuper ſu-<lb/>perficiem contingentem pyramidem, & continuetur axi:</s> <s xml:id="echoid-s7326" xml:space="preserve"> erunt lineæ communes (ſicut dictum eſt <lb/>in alijs pyramidalibus) aut lineæ longitudinis pyramidum, aut ſectiones.</s> <s xml:id="echoid-s7327" xml:space="preserve"> Et in his à duobus pun-<lb/>ctis ſectionis poterit fieri reflexio, in eadem ſuperficie, reſpectu eiuſdem uiſus.</s> <s xml:id="echoid-s7328" xml:space="preserve"> Et in ſuperficie re-<lb/>flexionis erunt, centrum uiſus, punctum uiſum, punctum reflexionis, punctum axis, in quod cadit <lb/>perpendicularis.</s> <s xml:id="echoid-s7329" xml:space="preserve"/> </p> <pb o="125" file="0131" n="131" rhead="OPTICAE LIBER IIII."/> </div> <div xml:id="echoid-div276" type="section" level="0" n="0"> <head xml:id="echoid-head297" xml:space="preserve" style="it">50. Si uiſ{us} opponatur baſi ſpeculi conici caui: uiſibile intra ſpeculum poſitum, tantùm uide-<lb/>bitur. 6 p 9.</head> <p> <s xml:id="echoid-s7330" xml:space="preserve">SEd ſpeculum pyramidale integrum ſi opponatur uiſui, & ſit uiſus ex parte baſis, non percipiet <lb/>niſi hoc, quod fuerit intra ſpeculum:</s> <s xml:id="echoid-s7331" xml:space="preserve"> quoniam perpendicularis tenet angulum acutum cum <lb/>linea ab oculo ad ipſam ducta, ex parte baſis:</s> <s xml:id="echoid-s7332" xml:space="preserve"> unde fit reflexio ex parte acuminis [radius enim <lb/>reflexus declinat ad partem oppoſitam radio, obliquè ſpeculo incidẽti per 10 n:</s> <s xml:id="echoid-s7333" xml:space="preserve">] & cadent omnes <lb/>lineæ reflexæ intra pyramidem, & uideri poterit, quod intra pyramidem poſitum eſt.</s> <s xml:id="echoid-s7334" xml:space="preserve"> Si autem au-<lb/>feratur ex eo portio ſecundum longitudinem:</s> <s xml:id="echoid-s7335" xml:space="preserve"> poterunt quidem comprehendi exteriora, cum pa-<lb/>teat exitus lineis reflexionis.</s> <s xml:id="echoid-s7336" xml:space="preserve"> Similiter ſi ſecetur pyramis ad modum annuli, ut auferatur uertex:</s> <s xml:id="echoid-s7337" xml:space="preserve"> li-<lb/>berum habebunt lineæ ingreſſum, & exteriora apparebũt:</s> <s xml:id="echoid-s7338" xml:space="preserve"> & ſi fuerit uiſus ex parte ſuperficiei con-<lb/>cauitatis ſpeculi:</s> <s xml:id="echoid-s7339" xml:space="preserve"> plura poterit comprehendere exteriora, quàm ex parte baſis:</s> <s xml:id="echoid-s7340" xml:space="preserve"> quia latior inciden-<lb/>tibus datur lineis uia.</s> <s xml:id="echoid-s7341" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div277" type="section" level="0" n="0"> <head xml:id="echoid-head298" xml:space="preserve" style="it">51. Ab uno cui{us}libet ſpeculi puncto, unum uiſibilis punctum ad unum uiſum reflectitur. <lb/>29. 30. 31 p 5. Item 37 p 5: item in præfat. 1. 5. & 10 librorum.</head> <p> <s xml:id="echoid-s7342" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s7343" xml:space="preserve"> ſumpto uniuſcuiuſq;</s> <s xml:id="echoid-s7344" xml:space="preserve"> ſpeculi puncto, nõ eſt poſsibile in eo percipi formam, niſi for-<lb/>mam unius puncti ab eodem uiſu.</s> <s xml:id="echoid-s7345" xml:space="preserve"> Quoniam enim per perpendicularem & centrum uiſus <lb/>unica tranſit ſuperficies:</s> <s xml:id="echoid-s7346" xml:space="preserve"> & una ſola eſt linea à centro uiſus ad punctum:</s> <s xml:id="echoid-s7347" xml:space="preserve"> & unicus angulus <lb/>ex linea perpendiculari acutus, & unicus angulus in eadem ſuperficie acutus æqualis huic [ſecus <lb/>pars æquaretur toti contra 9 ax.</s> <s xml:id="echoid-s7348" xml:space="preserve">] ergo eſt unica linea, quæ angulum æqualem huic cum perpendi-<lb/>culari facit:</s> <s xml:id="echoid-s7349" xml:space="preserve"> & cum linea peruenerit ad partem corporis, nõ poteſt forma alterius puncti per ipſam <lb/>uehi, cum punctum præcedens occultet poſtpoſitum.</s> <s xml:id="echoid-s7350" xml:space="preserve"> Sed duobus uiſibus poſſunt in eodem ſpe-<lb/>culi puncto comprehendi duæ punctuales formæ:</s> <s xml:id="echoid-s7351" xml:space="preserve"> quoniam infinitæ poſſunt ſumi ſuperficies, ſu-<lb/>per perpendicularem ſe ſecantes, in quarum qualibet circa perpendicularem ſumi poterunt duo <lb/>anguli æquales acuti.</s> <s xml:id="echoid-s7352" xml:space="preserve"> Iam ergo proprietatem reflexionis declarauimus, & ſimiliter cuiuslibet ſpe-<lb/>culi proprium.</s> <s xml:id="echoid-s7353" xml:space="preserve"> Viſus autem cum per reflexionem formas comprehendit, non animaduertit quòd <lb/>hæc acquiſitio per reflexionem ſit.</s> <s xml:id="echoid-s7354" xml:space="preserve"> Non enim accidit ex proprietate uiſus reflexio:</s> <s xml:id="echoid-s7355" xml:space="preserve"> quoniam uiſu <lb/>remoto, procedit non minus forma à corpore ad ſpeculum, & reflectitur ſecundum modum prędi-<lb/>ctum:</s> <s xml:id="echoid-s7356" xml:space="preserve"> & ſi accidat uiſum eſſe in loco, in quem linearum reflexarum fit aggregatio:</s> <s xml:id="echoid-s7357" xml:space="preserve"> comprehendet <lb/>uiſus formam illam in capitibus harum linearum:</s> <s xml:id="echoid-s7358" xml:space="preserve"> & eſt in ſpeculo tanquam non adueniens, ſed na-<lb/>turalis eſſet forma ſpeculo.</s> <s xml:id="echoid-s7359" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s7360" xml:space="preserve"> aliquando acquirit uiſus formas in ſpeculis in ſola ſuperficie, <lb/>aliquando intra ſpeculum, aliquando ultra.</s> <s xml:id="echoid-s7361" xml:space="preserve"> Et erit apparens locus formæ ſecundum figuram ſpe-<lb/>culi & ſitum rei uiſæ:</s> <s xml:id="echoid-s7362" xml:space="preserve"> & ſemper comprehendetur forma in loco proprio, mutato ſitu uiſus & ſpecu-<lb/>li:</s> <s xml:id="echoid-s7363" xml:space="preserve"> & erit diuerſitas elongationis loci formæ ad ſpeculi ſuperficiem, ſecundum diuerſitatem figuræ <lb/>ſpeculi.</s> <s xml:id="echoid-s7364" xml:space="preserve"> Et locus formæ dicitur locus imaginis.</s> <s xml:id="echoid-s7365" xml:space="preserve"> Et forma dicitur imago.</s> <s xml:id="echoid-s7366" xml:space="preserve"> Viſus autem comprehen-<lb/>dit rem uiſam in loco imaginis.</s> <s xml:id="echoid-s7367" xml:space="preserve"> Et nos dicemus illum locum, & eius proprium in quolibet ſpecu-<lb/>lorum, quæ enumerauimus:</s> <s xml:id="echoid-s7368" xml:space="preserve"> & aſsignabimus cauſas, propter quas comprehendantur res uiſæ in <lb/>loco illo:</s> <s xml:id="echoid-s7369" xml:space="preserve"> & hoc in ſequente libro, ſi deus uoluerit.</s> <s xml:id="echoid-s7370" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div278" type="section" level="0" n="0"> <head xml:id="echoid-head299" xml:space="preserve">ALHAZEN FILII</head> <head xml:id="echoid-head300" xml:space="preserve">ALHAYZEN OPTICAE</head> <head xml:id="echoid-head301" xml:space="preserve">LIBER QVINTVS.</head> <p style="it"> <s xml:id="echoid-s7371" xml:space="preserve">LIBER iſte in du{as} partes diuiſ{us} est.</s> <s xml:id="echoid-s7372" xml:space="preserve"> Prima pars eſt proœmium libri.</s> <s xml:id="echoid-s7373" xml:space="preserve"> Secunda <lb/>de imaginib{us}.</s> <s xml:id="echoid-s7374" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div279" type="section" level="0" n="0"> <head xml:id="echoid-head302" xml:space="preserve">PROOEMIVM LIBRI. CAP. I.</head> <head xml:id="echoid-head303" xml:space="preserve" style="it">1. Imago eſt form a uiſibilis, à polit a ſuperficie reflexa. In def. 5 libri.</head> <p> <s xml:id="echoid-s7375" xml:space="preserve">LIquet ex quarto libro [2 n] quòd formæ rerum uiſarum reflectuntur ex corporibus politis, & <lb/>uiſus comprehendit eas in corporibus politis propter reflexionem:</s> <s xml:id="echoid-s7376" xml:space="preserve"> & patuit [20.</s> <s xml:id="echoid-s7377" xml:space="preserve"> 21 n 4] quo-<lb/>modo fieret acquiſitio rerum ex reflexione formarum.</s> <s xml:id="echoid-s7378" xml:space="preserve"> Et uiſus comprehendit rem uiſam in loco <lb/>determinato:</s> <s xml:id="echoid-s7379" xml:space="preserve"> & primò, cum non fuerit ſitus rei uiſæ ad uiſum mutatio.</s> <s xml:id="echoid-s7380" xml:space="preserve"> Et forma comprehenſa in <lb/>corpore polito nominatur imago.</s> <s xml:id="echoid-s7381" xml:space="preserve"> Et nos explanabimus in hoc libro loca imaginũ ex corporibus <lb/>politis:</s> <s xml:id="echoid-s7382" xml:space="preserve"> & dicemus quomodo acquiratur horũ locorũ ſcientia, & quomodo inueniatur ſyllogiſticè.</s> <s xml:id="echoid-s7383" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div280" type="section" level="0" n="0"> <head xml:id="echoid-head304" xml:space="preserve">DE LOCIS IMAGINVM. CAP. II.</head> <head xml:id="echoid-head305" xml:space="preserve" style="it">2. In ſpeculo plano imago uidetur in concurſu perpendicularis incidentiæ & lineæ reflexio-<lb/>nis. 37 p 5.</head> <p> <s xml:id="echoid-s7384" xml:space="preserve">IMaginis cuiuſcunq;</s> <s xml:id="echoid-s7385" xml:space="preserve"> puncti locus, eſt punctum in quõ linea reflexionis ſecat perpendicularem <lb/>à puncto rei uiſæ intellectam ſuper lineam contingentem lineam cõmunem ſuperficiei ſpeculi, <lb/> <pb o="126" file="0132" n="132" rhead="ALHAZEN"/> uel ſuperficiei ſpeculo continuæ, & ſuperficiei reflexionis.</s> <s xml:id="echoid-s7386" xml:space="preserve"> Et nos hæc declarabímus.</s> <s xml:id="echoid-s7387" xml:space="preserve"> Sumatur ſpe-<lb/>culum planum, & ſtatuatur æquidiſtans horizonti:</s> <s xml:id="echoid-s7388" xml:space="preserve"> & lignum directũ & politum erigatur ſuper ſpe <lb/>culum:</s> <s xml:id="echoid-s7389" xml:space="preserve"> & ſit ſpeculi quantitas, ut totũ poſsit uideri lignum:</s> <s xml:id="echoid-s7390" xml:space="preserve"> niſi enim totum appareat, error inerit:</s> <s xml:id="echoid-s7391" xml:space="preserve"> <lb/>& ſignetur in ligno punctum aliquod nigrum:</s> <s xml:id="echoid-s7392" xml:space="preserve"> apparebit quidem uiſui lignũ æquale huic ultra ſpe-<lb/>culum, huic ligno continuum, & orthogonale ſupra ſpeculum, & in ligno apparẽte apparebit pun-<lb/>ctum ſignatum, tantùm diſtans à ſuperficie ſpeculi, quantùm ab eadem diſtat in ligno ſuperiore.</s> <s xml:id="echoid-s7393" xml:space="preserve"> Et <lb/>ſi declinetur lignum ſupra ſpeculum:</s> <s xml:id="echoid-s7394" xml:space="preserve"> apparebit apparens eadem declinatione declinatum:</s> <s xml:id="echoid-s7395" xml:space="preserve"> & pun-<lb/>ctum ſignatum in apparente apparebit æquè remotum à ſuperficie ſpeculi.</s> <s xml:id="echoid-s7396" xml:space="preserve"> Et ſi à puncto ſignato <lb/>lignum aliquod erigatur orthogonaliter ſupra ſpeculum:</s> <s xml:id="echoid-s7397" xml:space="preserve"> uidebitur etiam hoc lignum à puncto ap-<lb/>parente orthogonaliter ſupra ſpeculum, & huic orthogonali continuum.</s> <s xml:id="echoid-s7398" xml:space="preserve"> Idem accidet pluribus <lb/>punctis in ligno ſignatis.</s> <s xml:id="echoid-s7399" xml:space="preserve"> Idemq́ue penitus accidet eleuato aut depreſſo ſpeculo.</s> <s xml:id="echoid-s7400" xml:space="preserve"> Planum ergo per <lb/>hoc, quòd imago puncti uiſi apparet in perpendiculari, ducta à puncto uiſo ad ſuperficiem ſpeculi.</s> <s xml:id="echoid-s7401" xml:space="preserve"> <lb/>Et in hoc ſpeculo, quæ perpendicularis eſt ſuper ſuperficiem ſpeculi, eſt perpendicularis ſuper li-<lb/>neam communem ſuperficiei ſpeculi & ſuperficiei reflexionis.</s> <s xml:id="echoid-s7402" xml:space="preserve"> Idem patére poteſt in pyramide ſu-<lb/>per baſim orthogonali, cuius baſis plana ſpeculo plano ſit orthogonaliter adhibita:</s> <s xml:id="echoid-s7403" xml:space="preserve"> apparebit enim <lb/>huic pyramis alia continua, & harum pyramidum baſis eadem, & acumina ipſarum æqualiter à ſpe <lb/>culo diſtantia.</s> <s xml:id="echoid-s7404" xml:space="preserve"> Et planum, quòd ſi ab acumine ad acumen ducatur linea recta, erit perpendicularis <lb/>ſuper baſim:</s> <s xml:id="echoid-s7405" xml:space="preserve"> & ita ſuper ſpeculum, cum eadem ſit ſuperficies ſpeculi & baſis.</s> <s xml:id="echoid-s7406" xml:space="preserve"> Quare uertex pyra-<lb/>midis in perpendiculari uidebitur ab eo ad ſpeculum ducta.</s> <s xml:id="echoid-s7407" xml:space="preserve"> Similiter à quocunq;</s> <s xml:id="echoid-s7408" xml:space="preserve"> puncto pyrami-<lb/>dis ducatur linea æquidiſtans axi, cadet ad punctum reſpiciens ipſum in apparente pyramide:</s> <s xml:id="echoid-s7409" xml:space="preserve"> & <lb/>erit linea illa perpendicularis ſuper baſim & ſuper ſpeculi ſuperficiem [per 8 p 11.</s> <s xml:id="echoid-s7410" xml:space="preserve">] Quare imago <lb/>cuiuſq;</s> <s xml:id="echoid-s7411" xml:space="preserve"> puncti pyramidis cadit in perpẽdicularem, intellectam à puncto illo in ſpeculi ſuperficiem.</s> <s xml:id="echoid-s7412" xml:space="preserve"> <lb/>Sed quodcunq;</s> <s xml:id="echoid-s7413" xml:space="preserve"> punctum opponatur ſpeculo plano, eſt intelligere pyramidem, cuius punctum il-<lb/>lud uertex:</s> <s xml:id="echoid-s7414" xml:space="preserve"> [per 14 n 4] quæ quidem pyramis ſuper baſim orthogonalis eſt, & etiam ſuper ſpeculi <lb/>ſuperficiem, aut ei continuam:</s> <s xml:id="echoid-s7415" xml:space="preserve"> & eſt intelligere aliam huic pyramidi oppoſitam, quarum baſis ea-<lb/>dem, & perpendicularis à uertice ad uerticẽ orthogonalis erit ſuper ſpeculum.</s> <s xml:id="echoid-s7416" xml:space="preserve"> Quare imago cuiuſ-<lb/>cunq;</s> <s xml:id="echoid-s7417" xml:space="preserve"> puncti ſpeculo oppoſiti, cadit in perpendicularem ductam à puncto ad ſpeculi ſuperficiem, <lb/>aut ei continuam.</s> <s xml:id="echoid-s7418" xml:space="preserve"> Sed [per 21 n 4] planum eſt, quòd in ſpeculis non accidit comprehenſio forma-<lb/>rum, niſi per lineas reflexionum.</s> <s xml:id="echoid-s7419" xml:space="preserve"> Quare imago puncti uiſi cadit in lineam reflexionis:</s> <s xml:id="echoid-s7420" xml:space="preserve"> & quælibet <lb/>talis linea eſt recta.</s> <s xml:id="echoid-s7421" xml:space="preserve"> Quare imago cuiuſcunq;</s> <s xml:id="echoid-s7422" xml:space="preserve"> puncti cadit in punctum ſectionis perpẽdicularis, du-<lb/>ctæ ab illo puncto ad ſuperficiem ſpeculi, & lineæ reflexionis.</s> <s xml:id="echoid-s7423" xml:space="preserve"> Et in ſpeculis planis linea communis <lb/>ſuperficiei ſpeculι & ſuperficiei reflexionis eſt una linea cum linea contingente locum reflexionis.</s> <s xml:id="echoid-s7424" xml:space="preserve"> <lb/>Quare planum, quòd in ſpeculis planis imaginis locus, eſt punctũ ſectionis perpendicularis à pun-<lb/>cto uiſo ſuper lineam, contingentem communem lineam ſuperficiei ſpeculi & ſuperficiei reflexio-<lb/>nis, & lineæ reflexionis.</s> <s xml:id="echoid-s7425" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div281" type="section" level="0" n="0"> <head xml:id="echoid-head306" xml:space="preserve" style="it">3. In ſpeculo ſphærico conuexo, imago uidetur in concurſu perpendicularis incidentiæ & li-<lb/>neæ reflexionis. 11 p 6.</head> <p> <s xml:id="echoid-s7426" xml:space="preserve">IN ſpeculis ſphæricis extrà politis patebit quod diximus.</s> <s xml:id="echoid-s7427" xml:space="preserve"> Quęratur ſuperficies ſpeculi talis ma-<lb/>gna, in qua appareat forma baculi gracilis, perpendiculariter erecti ſuper ipſum:</s> <s xml:id="echoid-s7428" xml:space="preserve"> apparebit qui-<lb/>dem forma baculi baculo continua:</s> <s xml:id="echoid-s7429" xml:space="preserve"> & apparebit in forma baculi punctum ſignatum, diſtans à <lb/>ſuperficie ſpeculi ſecundum diſtantiam eius ab eodem, in baculo:</s> <s xml:id="echoid-s7430" xml:space="preserve"> & ſi fuerit baculus gracilior ex <lb/>parte unius capitis, quàm ex parte alterius:</s> <s xml:id="echoid-s7431" xml:space="preserve"> apparebit quidem in hoc ſpeculo forma eius pyrami-<lb/>dalis:</s> <s xml:id="echoid-s7432" xml:space="preserve"> & eſt error uiſus, quem poſtea aſsignabimus.</s> <s xml:id="echoid-s7433" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s7434" xml:space="preserve"> fiat pyramis orthogonalis ſuper baſim <lb/>circularem circulatione perfecta:</s> <s xml:id="echoid-s7435" xml:space="preserve"> & applicetur etiam huic ſpeculo:</s> <s xml:id="echoid-s7436" xml:space="preserve"> uidebitur quidem pyramis huic <lb/>cõtinua ſuper eandem baſim erecta, ſed minoriſta.</s> <s xml:id="echoid-s7437" xml:space="preserve"> Quòd autem appareat pyramis, planum eſt per <lb/>hoc, quòd omnes lineæ ab apparehte imagine uerticis ad circulum baſis, uideantur æquales.</s> <s xml:id="echoid-s7438" xml:space="preserve"> Et ſi <lb/>declinetur pyramis modicùm ſupra ſpeculum a ſitu, in quo tota uidetur, ut ſcilicet aliquid ex ea ab-<lb/>ſcondatur, dum tamen locus reflexionis in ſpeculo uiſui exponatur:</s> <s xml:id="echoid-s7439" xml:space="preserve"> apparebit etiam inde imago <lb/>pyramidis.</s> <s xml:id="echoid-s7440" xml:space="preserve"> Et ſi elongetur uiſus à ſpeculo, aut accedat, dum tamẽ ſuper lineam à loco ad ipſum pro-<lb/>tractam cadat:</s> <s xml:id="echoid-s7441" xml:space="preserve"> comprehendetur imago pyramidis.</s> <s xml:id="echoid-s7442" xml:space="preserve"> Sed acceſſus uel receſſus ſecundum hanc li-<lb/>neam erit, ut notetur locus reflexionis, & à nota ad locum uiſus ducatur linea, ſecundum quam <lb/>fiat proceſſus.</s> <s xml:id="echoid-s7443" xml:space="preserve"> Verùm quoniam imago pyramidis orthogonalis eſt ſuper baſim pyramidis, & ba-<lb/>ſis eſt circulus ex circulis in ſphæra:</s> <s xml:id="echoid-s7444" xml:space="preserve"> erit linea à uertice pyramidis ad uerticem imaginis ducta, or-<lb/>thogonalis ſuper circulum illum, & tranſibit per centrum eius [per 6.</s> <s xml:id="echoid-s7445" xml:space="preserve">8 d 1 conicorum] & erit or-<lb/>thogonalis ſuper ſphæram, & tranſibit per centrum ſphæræ, & erit perpendicularis ſuper ſuperfi-<lb/>ficiem, ſphæram contingentem in puncto, per quod tranſit hæc linea [per 4 th.</s> <s xml:id="echoid-s7446" xml:space="preserve"> 1 ſphær.</s> <s xml:id="echoid-s7447" xml:space="preserve"> uel 25 <lb/>n 4] & erit ſimiliter orthogonalis ſuper lineam, contingentem circulum ſphæræ per punctum <lb/>illud tranſeuntem [per 3 d 11.</s> <s xml:id="echoid-s7448" xml:space="preserve">] Et hæc contingens eſt linea, communis ſuperficiei reflexionis & <lb/>ſuperficiei contingenti ſphæram in puncto illo:</s> <s xml:id="echoid-s7449" xml:space="preserve"> & hæc linea contingit circulum ſphæræ, commu-<lb/>nem ſuperficiei ſphæræ & ſuperficiei reflexionis.</s> <s xml:id="echoid-s7450" xml:space="preserve"> Linea ergo à uertice pyramidis ad uerticem <lb/>imaginis ducta, eſt perpendicularis ſuper lineam contingentem, lineam communem ſuperficiei <lb/>reflexionis & ſuperficiei ſpeculi:</s> <s xml:id="echoid-s7451" xml:space="preserve"> quæ quidem eſt circulus.</s> <s xml:id="echoid-s7452" xml:space="preserve"> In hac igitur perpendiculari uide-<lb/>tur imago uerticis.</s> <s xml:id="echoid-s7453" xml:space="preserve"> Et planum [per 21 n 4] quòd imago uerticis eſt in linea reflexionis.</s> <s xml:id="echoid-s7454" xml:space="preserve"> Quare <lb/> <lb/> <pb o="127" file="0133" n="133" rhead="OPTICAE LIBER V."/> comprehendetur imago uerticis in cõcurſu lineæ reflexionis, & perpendicularis à uertice ad ſphæ-<lb/>ram ductæ, ſiue ad contingentem, circulum communem ſuperficiei ſphæræ & ſuperficiei reflexio-<lb/>nis.</s> <s xml:id="echoid-s7455" xml:space="preserve"> Sumpto autem quocunque puncto huic ſpeculo oppoſito, eſt intelligere pyramidem ſuper <lb/>ſuperficiem ſpeculi orthogonalem, aut ſuper continuam ei, cuius uertex ſit punctũ ſumptum:</s> <s xml:id="echoid-s7456" xml:space="preserve"> [per <lb/>14 n 4] & linea ab illo puncto ad imaginẽ puncti illius, erit in ſuperficie reflexionis, & perpendicu-<lb/>laris ſuper ſuperficiem ſpeculi, uel ei continuam modo prædicto:</s> <s xml:id="echoid-s7457" xml:space="preserve"> quoniam punctum uiſum & ima-<lb/>go ſemper ſunt ſimul in ſuperficie reflexionis [per 23 n 4.</s> <s xml:id="echoid-s7458" xml:space="preserve">] Quare & linea à puncto uiſo ad eius <lb/>imaginem ducta.</s> <s xml:id="echoid-s7459" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div282" type="section" level="0" n="0"> <head xml:id="echoid-head307" xml:space="preserve" style="it">4. In ſpeculis conuexis cylindraceo, conico, imago uidetur in concurſu perpendicularis inci-<lb/>dentiæ & lineæ reflexionis. 37 p 5.</head> <p> <s xml:id="echoid-s7460" xml:space="preserve">IN ſpeculis columnaribus exterius politis non apparent, quæ in ligno & pyramide diximus:</s> <s xml:id="echoid-s7461" xml:space="preserve"> <lb/>quoniam recta in his ſpeculis uidetur non recta:</s> <s xml:id="echoid-s7462" xml:space="preserve"> & eſt error uiſus communis, cuius poſtea cauſ-<lb/>ſam aſsignabimus.</s> <s xml:id="echoid-s7463" xml:space="preserve"> Accidit tamen in ſolo corporis puncto uidere locum imaginis prædictum, <lb/>hoc modo.</s> <s xml:id="echoid-s7464" xml:space="preserve"> Adhibito præcedentis libri inſtrumento, immittatur regula, cui ſit infixum columnare <lb/>ſpeculum, ut media portionis ſpeculi linea ſit in ſuperficie regulæ, & non tranſeat hæc regula tabu-<lb/>lam æneam, ſed ſuper ipſam cadat orthogonaliter, ita ut altitudo regulæ ſit ſuper lineam, diuidentẽ <lb/>triangulum tabulæ æneæ.</s> <s xml:id="echoid-s7465" xml:space="preserve"> Erectione facta in hac tabula, impleatur cera, & inducatur ei planities, ut <lb/>ſit in eadem ſuperficie cum tabula:</s> <s xml:id="echoid-s7466" xml:space="preserve"> & eſt;</s> <s xml:id="echoid-s7467" xml:space="preserve"> ut certior fiat orthogonalis regulę directio ſuper tabulam.</s> <s xml:id="echoid-s7468" xml:space="preserve"> <lb/>Deinde quæratur regula acuta, & acuatur extremitas, & applicetur huius regulæ acuitas mediæ ſu <lb/>perficiei annuli lineæ, & deſcendat ſecundum lineam hanc, & ubi ceciderit ſuper regulam, fiat ſi-<lb/>gnum.</s> <s xml:id="echoid-s7469" xml:space="preserve"> Poſtea acus deſcendat, in qua infixum ſit modicũ corpus album:</s> <s xml:id="echoid-s7470" xml:space="preserve"> & hoc in termino, ne de-<lb/>icendat acus uſq;</s> <s xml:id="echoid-s7471" xml:space="preserve"> ad regulam.</s> <s xml:id="echoid-s7472" xml:space="preserve"> Adhibeatur autem uiſus, ut ſit in ſuperficie regulæ, & claudatur unus <lb/>uiſuum:</s> <s xml:id="echoid-s7473" xml:space="preserve"> uidebitur quidem imago corporis ſuper lineam, à puncto ſignato ad acumen acus protra-<lb/>ctam:</s> <s xml:id="echoid-s7474" xml:space="preserve"> quæ quidem linea perpendicularis eſt ſuper ſuperficiem regulæ;</s> <s xml:id="echoid-s7475" xml:space="preserve"> quæ ſuperficies tangit colu-<lb/>mnam in linea longitudinis:</s> <s xml:id="echoid-s7476" xml:space="preserve"> & eſt perpendicularis ſuper lineam longitudinis columnæ, quæ eſt in <lb/>ſuperficie regulæ:</s> <s xml:id="echoid-s7477" xml:space="preserve"> & eſt linea cõmunis ſuperficiei regulæ & ſuperficiei reflexionis:</s> <s xml:id="echoid-s7478" xml:space="preserve"> & in ſuperficie re <lb/>flexionis ſunt linea longitudinis & linea perpendicularis.</s> <s xml:id="echoid-s7479" xml:space="preserve"> Et ſi ſitus uiſus mutetur, & circa annuli ſu <lb/>perficiem uiſus uoluatur:</s> <s xml:id="echoid-s7480" xml:space="preserve"> apparebunt ſicut prius, & in eadem linea corpus, & imago corporis, & a-<lb/>cus.</s> <s xml:id="echoid-s7481" xml:space="preserve"> Et eſt linea illa perpendicularis ſuper mediam longitudinis columnæ lineam:</s> <s xml:id="echoid-s7482" xml:space="preserve"> & hæc eſt per-<lb/>pendicularis in ſuperficie reflexionis:</s> <s xml:id="echoid-s7483" xml:space="preserve"> quoniam ſuperficies annuli ſecat columnam ſuper circulum, <lb/>æquidiſtantem baſi columnæ:</s> <s xml:id="echoid-s7484" xml:space="preserve"> & in hac ſuperficie eſt uiſus.</s> <s xml:id="echoid-s7485" xml:space="preserve"> Et nos probabimus poſtea, quòd quan-<lb/>do uiſus, & uiſum corpus fuerint in ſuperficie, æquidiſtante baſi columnæ, illa eſt ſuperficies refle-<lb/>xionis.</s> <s xml:id="echoid-s7486" xml:space="preserve"> In hoc autem ſitu, linea communis ſuperficiei columnæ, & ſuperficiei reflexionis, eſt circu-<lb/>lus:</s> <s xml:id="echoid-s7487" xml:space="preserve"> & perpendicularis, in qua uidetur imago & corpus, orthogonaliter cadunt ſuper lineam, hunc <lb/>circulum contingentem.</s> <s xml:id="echoid-s7488" xml:space="preserve"> His peractis auferatur acus à loco ſuo, & ponatur regula acuta ſuper li-<lb/>neam annuli mediam, ita ut cadat ſuper mediam longitudinis regulæ lineam, & adhibeatur regula <lb/>acuta ſuperficiei annuli cera firmiter.</s> <s xml:id="echoid-s7489" xml:space="preserve"> Poſtea auferatur regula, in qua eſt ſpeculum, & accipiatur <lb/>regula acuta, & applicetur eius acuitas mediæ longitudinis regulæ lineæ, & ſecundum proceſſum <lb/>acuitatis fiat cum incanſto ſuper ſpeculum protractio.</s> <s xml:id="echoid-s7490" xml:space="preserve"> Pòſt ſumatur triangulum cereum modi-<lb/>cum, cuius unum latus ſit æquale altitudini regulæ, in qua eſt ſpeculum, & ſit ſpiſsitudo huius tri-<lb/>anguli moderata, & ſuperficies huius trianguli ſint planæ pro poſſe:</s> <s xml:id="echoid-s7491" xml:space="preserve"> & adhibeatur columnæ re-<lb/>gulæ triangulum firmiter ſub baſi regulæ, & latus eius æquale altitudini regulæ ponatur ſuper la-<lb/>tus baſis regulæ.</s> <s xml:id="echoid-s7492" xml:space="preserve"> Cum ita fuerit, erit huius trianguli altitudo ſuper baſim columnæ æqualem regu-<lb/>læ.</s> <s xml:id="echoid-s7493" xml:space="preserve"> Et ut efficiatur ſuperficies plana ad modum ſuperficiei regulæ, includatur triangulum inter <lb/>regulam & ſuperficiem planam, & comprimatur, donec ſit bene complanatum, & ſuper ſuperfi-<lb/>ciem huius trianguli ponatur regula acuta, & ſecetur finis huius trianguli cum acuitate regulæ, & <lb/>erit finis eius linea recta, & erit linea hæc baſis regulæ, in qua eſt ſpeculum.</s> <s xml:id="echoid-s7494" xml:space="preserve"> Poſtea ponatur regu-<lb/>la ſuper ſuperficiem tabulæ, quæ eſt in inſtrumento, & ponatur finis eius baſis, quæ eſt in longitu-<lb/>dine, quæ eſt latus trianguli cerei, ſuper lineam, quę eſt in longitudine tabulę, ſicut factum eſt prius:</s> <s xml:id="echoid-s7495" xml:space="preserve"> <lb/>& erit ſuperficies regulæ, in qua eſt ſpeculum, orthogonalis ſuper tabulam æneam:</s> <s xml:id="echoid-s7496" xml:space="preserve"> & hæc ſuperfi-<lb/>cies ſecat tabulam æneam ſuper lineam, quæ eſt in longitudine eius:</s> <s xml:id="echoid-s7497" xml:space="preserve"> & hæc ſuperficies tangit ſu-<lb/>perficiem ſpeculi ſuper lineam, quæ eſt in ſuperficie ſpeculi:</s> <s xml:id="echoid-s7498" xml:space="preserve"> & hæc eſt ſuperficies regulæ, in qua eſt <lb/>ſpeculum:</s> <s xml:id="echoid-s7499" xml:space="preserve"> & erit angulus regulæ acutæ, adhærentis in media linea ſuperficiei annuli, in qua ſuper-<lb/>ficie erit ſpeculum, declinatus in partem, in qua eſt caput trianguli:</s> <s xml:id="echoid-s7500" xml:space="preserve"> quia regula exaltauit unam <lb/>partem eius cum corpore trianguli, & alia pars, quæ eſt poſt caput trianguli, eſt ſuperficies tabulæ <lb/>æneæ:</s> <s xml:id="echoid-s7501" xml:space="preserve"> & erit linea, quæ eſt in medietate ſpeculi, declinata.</s> <s xml:id="echoid-s7502" xml:space="preserve"> Et quando fuerit latus trianguli cerei ſu-<lb/>per lineam, quæ eſt in longitudine æneæ tabulæ:</s> <s xml:id="echoid-s7503" xml:space="preserve"> mouebitur regula, in qua eſt ſpeculum:</s> <s xml:id="echoid-s7504" xml:space="preserve"> & latus <lb/>trianguli in hoc motu, ſi ſit ſuper lineam longitudinis tabulę æneæ, & procedat uel retrocedat, do-<lb/>nec concurrat angulus regulæ acutę cum puncto aliquo lineę ſuperficiei ſpeculi, donec firmetur re <lb/>gula acuta, & auferatur linea in ſpeculo cum incauſto facta:</s> <s xml:id="echoid-s7505" xml:space="preserve"> & fiat punctum in ſuperficie ſpecu-<lb/>li in directo capitis regulæ acutę, & auferatur regula acuta, & apponatur acus, & ſit acus ſuper li-<lb/>neam mediam ſuperficiei annuli, & adhærere cogatur cum cera:</s> <s xml:id="echoid-s7506" xml:space="preserve"> erit linea intellectualis ab acu in <lb/>punctum ſignatum in ſuperficie ſpeculi, perpendicularis ſuper ſuperficiem regulæ, quæ tangit ſu-<lb/> <pb o="128" file="0134" n="134" rhead="ALHAZEN"/> perficiem ſpeculi ſuper punctum ſignatum, & perpendicularis ſuper quamlibet lineam ab illo pun-<lb/>cto protractam, in ſuperficiem contingentem ſpeculum.</s> <s xml:id="echoid-s7507" xml:space="preserve"> Erit ergo perpen dicularis ſuper lineam re-<lb/>ctam, contingentem lineam communem ſuperficiei altæ annuli & ſuperficiei ſpeculi.</s> <s xml:id="echoid-s7508" xml:space="preserve"> Ponatur au-<lb/>tem uiſus in ſuperficie annuli, in capite eius, & uidebit in ſpeculo, donec comprehendat formam <lb/>corporis parui, quod eſt in acu:</s> <s xml:id="echoid-s7509" xml:space="preserve"> & tunc percipiet corpus illud, & punctum in ſpeculo ſignatum, & <lb/>imaginem illius corporis.</s> <s xml:id="echoid-s7510" xml:space="preserve"> Et linea tranſiens per corpus paruum, & per punctum in ſuperficie ſigna-<lb/>tum, eſt perpendicularis ſuper ſuperficiem, contingentem ſpeculi ſuperficiem ſuper punctũ ſigna-<lb/>tum:</s> <s xml:id="echoid-s7511" xml:space="preserve"> & hæc ſuperficies annuli, eſt ex ſuperficiebus reflexionis:</s> <s xml:id="echoid-s7512" xml:space="preserve"> & corpus paruum, & centrum uiſus <lb/>ſunt in hac ſuperficie, & punctus reflexionis eſt in hac ſuperficie:</s> <s xml:id="echoid-s7513" xml:space="preserve"> & hæc deinceps probabimus.</s> <s xml:id="echoid-s7514" xml:space="preserve"> <lb/>Et imago corporis parui in hoc ſitu, erit ſuper lineam rectam, à corpore paruo protràctam ſuper ſu-<lb/>perficiem, contingentem ſuperficiem ſpeculi:</s> <s xml:id="echoid-s7515" xml:space="preserve"> & eſt hæc linea perpendicularis ſuper lineam rectam, <lb/>contingentem lineam communem ſuperficiei ſpeculi, & ſuperficiei reflexionis, quæ eſt ſuperficies <lb/>annuli.</s> <s xml:id="echoid-s7516" xml:space="preserve"> Et ſuperficies reflexionis eſt ex ſuperficiebus declinantibus, ſecantibus columnam inter li-<lb/>neas longitudinis columnæ, & circulos eius æquidiſtantes baſibus:</s> <s xml:id="echoid-s7517" xml:space="preserve"> quia regula & ſpeculum, quod <lb/>eſt in ea, ſunt declinata.</s> <s xml:id="echoid-s7518" xml:space="preserve"> Linea ergo communis huic ſuperficiei & ſuperficiei ſpeculi, eſt ex ſectio-<lb/>nibus columnaribus.</s> <s xml:id="echoid-s7519" xml:space="preserve"> Et ita explanabimus locum imaginis, ut mutetur ſitus regulæ, in qua eſt ſpe-<lb/>culum & declinetur ſuper ſuperficiem eius aliqua declinatione maiore uel minore.</s> <s xml:id="echoid-s7520" xml:space="preserve"> Palàm ergo ex <lb/>his, quòd imago percipitur, ubi perpendicularis à uiſo puncto ad ſpeculi ſuperficiem ducta, concur <lb/>rit cum linea reflexionis.</s> <s xml:id="echoid-s7521" xml:space="preserve"> Et hic eſt ſitus prædictus.</s> <s xml:id="echoid-s7522" xml:space="preserve"> Eadem poterit adhiberi operatio in ſpeculo py-<lb/>ramidali exteriore:</s> <s xml:id="echoid-s7523" xml:space="preserve"> & idem patebit ſiue ſintimagines rerum uiſarum in ſectionibus pyramidalibus, <lb/>ſiue in ijs, quæ fiunt ſecundum lineas longitudinis.</s> <s xml:id="echoid-s7524" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div283" type="section" level="0" n="0"> <head xml:id="echoid-head308" xml:space="preserve" style="it">5. Rectarum linearum ab eodem uiſibilis puncto in ſpecula planum uel conuexum caden-<lb/>tium: minima eſt perpendicularis. 21 p 1.</head> <p> <s xml:id="echoid-s7525" xml:space="preserve">SI à puncto uiſo ad ſpeculi ſuperficiem ducantur lineę:</s> <s xml:id="echoid-s7526" xml:space="preserve"> quæ perpendicularis eſt, minor eſt quali <lb/>bet alia.</s> <s xml:id="echoid-s7527" xml:space="preserve"> Quoniã quælibet alia prius ſecat communẽ lineã ſuperficiei cõtingentis ſpeculum, in <lb/>quam orthogonaliter cadit perpendicularis, & ſuperficiei reflexionis, antequã ueniat ad ſpe-<lb/>culum:</s> <s xml:id="echoid-s7528" xml:space="preserve"> & quælibet linea à puncto uiſo in hac ſuperfi-<lb/> <anchor type="figure" xlink:label="fig-0134-01a" xlink:href="fig-0134-01"/> cie, ad hanc lineã cõmunẽ ducta, eſt maior perpendi <lb/>culari [per 19 p 1] quia maiorẽ reſpicit angulũ [rectũ <lb/>nẽpe a e f in triangulo a e f.</s> <s xml:id="echoid-s7529" xml:space="preserve">] Quare patet propoſitũ.</s> <s xml:id="echoid-s7530" xml:space="preserve"/> </p> <div xml:id="echoid-div283" type="float" level="0" n="0"> <figure xlink:label="fig-0134-01" xlink:href="fig-0134-01a"> <variables xml:id="echoid-variables28" xml:space="preserve">d b c e f g b d <gap/></variables> </figure> </div> </div> <div xml:id="echoid-div285" type="section" level="0" n="0"> <head xml:id="echoid-head309" xml:space="preserve" style="it">6. In ſpeculo ſpbærico cauo, imago uidetur in <lb/>concurſu perpendicularis incidentiæ & lineæ refle <lb/>xionis. 37 p 5.</head> <p> <s xml:id="echoid-s7531" xml:space="preserve">IN ſpeculis ſphæricis concauis comprehendun-<lb/>tur imagines quædam ultra ſpeculum:</s> <s xml:id="echoid-s7532" xml:space="preserve"> quædam <lb/>in ſuperficie:</s> <s xml:id="echoid-s7533" xml:space="preserve"> quædam citra ſuperficiem.</s> <s xml:id="echoid-s7534" xml:space="preserve"> Et harũ <lb/>quædam comprehenduntur in ueritate, quædam <lb/>præter ueritatem.</s> <s xml:id="echoid-s7535" xml:space="preserve"> Omnes, quarum comprehenditur <lb/>ueritas, apparent in loco ſectionis perpendicularis <lb/>& lineæ reflexionis:</s> <s xml:id="echoid-s7536" xml:space="preserve"> quod ſic patebit.</s> <s xml:id="echoid-s7537" xml:space="preserve"> Fiat pyramis, <lb/>& eius axis ſit orthogonalis ſuper baſim:</s> <s xml:id="echoid-s7538" xml:space="preserve"> & diame-<lb/>ter baſis ſit minor medietate diametri ſphæræ:</s> <s xml:id="echoid-s7539" xml:space="preserve"> & li-<lb/>nea longitudinis pyramidis, ſit maior eadẽ ſemidia-<lb/>metro:</s> <s xml:id="echoid-s7540" xml:space="preserve"> & ſecetur ex parte baſis, ad quantitatẽ eius, ſcilicet ſemidiametri:</s> <s xml:id="echoid-s7541" xml:space="preserve"> & fiat ſuper ſectionẽ circu <lb/>lus:</s> <s xml:id="echoid-s7542" xml:space="preserve"> & ſecetur pyramis ſuper hũc circulũ.</s> <s xml:id="echoid-s7543" xml:space="preserve"> Poſtea in medio ſpeculi fiat circulus ad quantitatẽ baſis py <lb/>ramidis remanentis:</s> <s xml:id="echoid-s7544" xml:space="preserve"> & aptetur huic circulo pyramis, & firmetur cum cera.</s> <s xml:id="echoid-s7545" xml:space="preserve"> Deinde ſtatuatur uiſus <lb/>in ſitu, in quo imaginem pyramidis poſsit comprehendere:</s> <s xml:id="echoid-s7546" xml:space="preserve"> & adhibeatur lux, ut certior fiat com-<lb/>prehenſio:</s> <s xml:id="echoid-s7547" xml:space="preserve"> non uidebis quidem pyramidem huic coniumctam, ſed comprehendes hanc ultra ſpecu-<lb/>lum extenſam:</s> <s xml:id="echoid-s7548" xml:space="preserve"> unde apparebit pyramis quædam continua, cuius baſis ultra ſpeculum eſt, & pars <lb/>cius pyramis cerea.</s> <s xml:id="echoid-s7549" xml:space="preserve"> Et ſi in hac pyramide ſignetur linea longitudinis cum incauſto:</s> <s xml:id="echoid-s7550" xml:space="preserve"> uidebitur hæc <lb/>linea protendi ſuper ſuperficiẽ pyramidis apparentis.</s> <s xml:id="echoid-s7551" xml:space="preserve"> Et quoniã uertex pyramidis eſt centrũ ſphæ-<lb/>ræ:</s> <s xml:id="echoid-s7552" xml:space="preserve"> linea à uertice ſecundum longitudinem pyramidis ducta, erit perpendicularis ſuper lineam, con <lb/>tingentem quemlibet circulum ſphæræ, per caput lineæ tranſeuntem[quodlibet enim conilatus æ-<lb/>quatur ſemidiametro ſphæræ per fabricam:</s> <s xml:id="echoid-s7553" xml:space="preserve"> uertex igitur coni eſt centrum maximi in ſphæra circu-<lb/>li:</s> <s xml:id="echoid-s7554" xml:space="preserve"> cuius ſemidiameter eſt latus:</s> <s xml:id="echoid-s7555" xml:space="preserve"> itaque per 18 p 3 ad lineam tan gentem eſt perpendiculare.</s> <s xml:id="echoid-s7556" xml:space="preserve">] Quare <lb/>quælibet linea longitudinis pyramidis apparentis, eſt perpendicularis ſuper lineam, contingen-<lb/>tem lineam cõmunem ſuperficiei reflexionis & ſuperficiei ſphæræ:</s> <s xml:id="echoid-s7557" xml:space="preserve"> quę quidem linea cõmunis eſt <lb/>circulus [per 1 th 1 ſphæ.</s> <s xml:id="echoid-s7558" xml:space="preserve">] & quodlibet punctum pyramidis in hac uidetur perpendiculari:</s> <s xml:id="echoid-s7559" xml:space="preserve"> & quæ-<lb/>libet perpendicularis eſt in ſuperficie reflexionis [per 23 n 4:</s> <s xml:id="echoid-s7560" xml:space="preserve">] quoniam punctum uiſum & ima-<lb/>go eius ſunt in perpendiculari, & in hac ſuperficie:</s> <s xml:id="echoid-s7561" xml:space="preserve"> & omnis imago comprehenditur in linea re-<lb/>flexionis [per 21 n 4.</s> <s xml:id="echoid-s7562" xml:space="preserve">] Quare imago cuiuſcũq;</s> <s xml:id="echoid-s7563" xml:space="preserve"> puncti pyramidis, erit in puncto ſectionis perpendi-<lb/> <pb o="129" file="0135" n="135" rhead="OPTICAE LIBER V."/> cularis & lineæ reflexionis.</s> <s xml:id="echoid-s7564" xml:space="preserve"> Puncta autem, quorum imagines citra ſpeculum eomprehenduntur, <lb/>hoc eſt inter uiſum & ſpeculum, ſunt, cum à quolibet eorum linea ducta ad centrum ſpeculi, ſecat la <lb/>titudinem uiæ inter uiſum & ſpeculum interiacentis.</s> <s xml:id="echoid-s7565" xml:space="preserve"> Et ut uideatur hoc:</s> <s xml:id="echoid-s7566" xml:space="preserve"> auferatur pyramis à me-<lb/>dio ſpeculi:</s> <s xml:id="echoid-s7567" xml:space="preserve"> & collocetur in parte, erit uertex centrum ſpeculi:</s> <s xml:id="echoid-s7568" xml:space="preserve"> & remotio uiſus ſit maior ſemidiame <lb/>tro ſphæræ.</s> <s xml:id="echoid-s7569" xml:space="preserve"> Deinde ſumatur lignum gracile album, & ſtatuatur in ſpeculo, ut ſit centrum ſpeculi <lb/>directè medium inter caput ligni & centrum uiſus, & dirigatur intuitus in punctum ſpeculi, à quo <lb/>linea ad uerticem pyramidis ducta, ſit inter caput ligni & uiſum:</s> <s xml:id="echoid-s7570" xml:space="preserve"> & apparebit forma capitis ligni ci-<lb/>tra ſpeculum, & propin quior uiſui uertice pyramidis:</s> <s xml:id="echoid-s7571" xml:space="preserve"> & erunt in eadem linea recta, uertex pyrami-<lb/>dis, & caput ligni, & imago capitis.</s> <s xml:id="echoid-s7572" xml:space="preserve"> Et hæc linea eſt perpendicularis ſuper lineam, contingentem <lb/>lineam communem ſuperficiei ſpeculi & ſuperficiei reflexionis [per 25 n 4:</s> <s xml:id="echoid-s7573" xml:space="preserve">] quoniam ſuperficies <lb/>reflexionis tranſit per centrum & punctum uiſus.</s> <s xml:id="echoid-s7574" xml:space="preserve"> Et linea tranſiens per hæc duo puncta, eſt in <lb/>ſuperficie reflexionis.</s> <s xml:id="echoid-s7575" xml:space="preserve"> Et linea cõmunis eſt circulus:</s> <s xml:id="echoid-s7576" xml:space="preserve"> & hæc linea huic circulo erit diameter:</s> <s xml:id="echoid-s7577" xml:space="preserve"> quoniã <lb/>centrum illius circuli, eſt centrum ſphæræ.</s> <s xml:id="echoid-s7578" xml:space="preserve"> Quare erit hæc linea perpendicularis ſuper lineam, <lb/>contingentem circulum in capite huius lineæ [per 18 p 3:</s> <s xml:id="echoid-s7579" xml:space="preserve">] & hæc linea tranſit per punctum uiſum, <lb/>& eius imaginem.</s> <s xml:id="echoid-s7580" xml:space="preserve"> Et ita quodlibet punctum citra ſpeculum uiſum, comprehenditur in eadem li-<lb/>nea cum centro & cum imagine eius:</s> <s xml:id="echoid-s7581" xml:space="preserve"> & quodlibet punctum uidetur in linea reflexionis [per 21 n <lb/>4.</s> <s xml:id="echoid-s7582" xml:space="preserve">] Quare in loco ſectionis perpendicularis & lineæ reflexionis.</s> <s xml:id="echoid-s7583" xml:space="preserve"> Et ea, quorum ueritas in his ſpe-<lb/>culis comprehenditur, ſunt, quorum imagines apparent ultra ſpeculum uel citra ſuperficiem eius:</s> <s xml:id="echoid-s7584" xml:space="preserve"> <lb/>& præter hæc, nulla ſunt, quæ in hoc ſpeculo in ueritate comprehendat uiſus, ipſa enim prohibent <lb/>imagines ſuas ueras apparere.</s> <s xml:id="echoid-s7585" xml:space="preserve"> Imagines, quæ apparent in ſuperficie huius ſpeculi, ſunt ex ultima <lb/>partitione:</s> <s xml:id="echoid-s7586" xml:space="preserve"> & hæc explanabimus, cum erit ſermo de erroribus uiſus.</s> <s xml:id="echoid-s7587" xml:space="preserve"> Quodlibet ë<unsure/>rgo punctum in <lb/>ueritate in hoc ſpeculo comprehenſum, apparet in concurſu perpendicularis & lineæ reflexionis:</s> <s xml:id="echoid-s7588" xml:space="preserve"> <lb/>quæ quidem perpendicularis tranſit à puncto uiſo ad centrum ſphæræ, & cadit orthogonaliter in <lb/>contingentem, lineam communem.</s> <s xml:id="echoid-s7589" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div286" type="section" level="0" n="0"> <head xml:id="echoid-head310" xml:space="preserve" style="it">7. In ſpeculis cauis cylindraceo, conico, imago uidetur in concurſu perpendicularis inciden-<lb/>tiæ & lineæ reflexionis. 37 p 5.</head> <p> <s xml:id="echoid-s7590" xml:space="preserve">IN ſpeculis columnaribus concauis diuerſificatur imago:</s> <s xml:id="echoid-s7591" xml:space="preserve"> aliquando enim erit locus eius in ſu-<lb/>perficie ſpeculi:</s> <s xml:id="echoid-s7592" xml:space="preserve"> aliquando ultra:</s> <s xml:id="echoid-s7593" xml:space="preserve"> & in his omnibus aliquando in ueritate comprehendetur:</s> <s xml:id="echoid-s7594" xml:space="preserve"> ali-<lb/>quando non.</s> <s xml:id="echoid-s7595" xml:space="preserve"> Cum uolueris in his locum imaginis percipere:</s> <s xml:id="echoid-s7596" xml:space="preserve"> facias, ſicut feciſti in columnari-<lb/>bus exterioribus.</s> <s xml:id="echoid-s7597" xml:space="preserve"> Adhibeatur enim regula, in qua ſit columna concaua, ſicut adhibita eſt ſuperius, <lb/>& acus ſimiliter, & corpus modicum, in ſummitate acus:</s> <s xml:id="echoid-s7598" xml:space="preserve"> & ponatur uiſus oppoſitus in medio cir-<lb/>culi, & in medio ſuperficiei annuli:</s> <s xml:id="echoid-s7599" xml:space="preserve"> & ſubleuetur uiſus modicum à ſuperficie annuli:</s> <s xml:id="echoid-s7600" xml:space="preserve"> & inſpiciat, <lb/>donec imaginem corporis uideat, & comprehendat formam corporis, & corpus, & punctum in ſpe <lb/>culo, ſignatum in eadem linea perpendiculari, ſuper ſuperficiem ſpeculi:</s> <s xml:id="echoid-s7601" xml:space="preserve"> & hoc per ſyllogiſmum ſen <lb/>ſualem.</s> <s xml:id="echoid-s7602" xml:space="preserve"> Et erit imago ultra ſpeculum, & erit reflexio ex puncto lineæ rectæ, quæ eſt in medio ſpe-<lb/>culi.</s> <s xml:id="echoid-s7603" xml:space="preserve"> Deinde ſtatuatur uiſus in ſuperficie annuli, ſed extra medium, donec uideat imaginem cor-<lb/>poris parui:</s> <s xml:id="echoid-s7604" xml:space="preserve"> uidebit quidem eam citra ſpeculum:</s> <s xml:id="echoid-s7605" xml:space="preserve"> & uidebit corpus, & eius imaginem, & punctum <lb/>in ſpeculo ſignatum, in una linea recta perpendiculari, ſuper lineam rectam contingentem circu-<lb/>lum æquidiſtantem baſi ſpeculi, ſuper punctum ſignatum in ſpeculi ſuperficie:</s> <s xml:id="echoid-s7606" xml:space="preserve"> & ſuperficies huius, <lb/>eſt ſuperficies reflexionis in hocſitu:</s> <s xml:id="echoid-s7607" xml:space="preserve"> & eſt ſuperficies faciei annuli:</s> <s xml:id="echoid-s7608" xml:space="preserve"> & punctum reflexionis eſt pun-<lb/>ctum illius circuli.</s> <s xml:id="echoid-s7609" xml:space="preserve"> Poſtea adhibeatur cum manu alia acus, in cuius ſummitate ſit corpus modicum:</s> <s xml:id="echoid-s7610" xml:space="preserve"> <lb/>& ſtatuaturin ſuperficiem & axem, hoc modo, ut corpus, & punctum ſignatum ſint in eadem li-<lb/>nea, ſecundum ſenſualem ſyllogiſmum:</s> <s xml:id="echoid-s7611" xml:space="preserve"> & ſit uiſus in ſuperficie annuli, inter caput eius & medium:</s> <s xml:id="echoid-s7612" xml:space="preserve"> <lb/>uidebit quidem imaginem corporis, & uidebit hanc imaginem & corpus eius, & punctum ſigna-<lb/>tum in ſuperficie ſpeculi, in eadem linea recta.</s> <s xml:id="echoid-s7613" xml:space="preserve"> Si autem declinetur linea recta cum triangulo par-<lb/>uo, quod fecimus, & ſit uiſus in medio annuli:</s> <s xml:id="echoid-s7614" xml:space="preserve"> uidebit imaginem citra ſpeculum, ſed in eadem linea <lb/>recta cum corpore, & puncto ſignato.</s> <s xml:id="echoid-s7615" xml:space="preserve"> Et hæc reflexio erit ex columnaribus ſectionibus:</s> <s xml:id="echoid-s7616" xml:space="preserve"> quoniam <lb/>ſpeculum eſt declinatum:</s> <s xml:id="echoid-s7617" xml:space="preserve"> & ſcimus [è 21 n 4] quòd non percipitur imago, niſi in linea reflexio-<lb/>nis.</s> <s xml:id="echoid-s7618" xml:space="preserve"> Palàm ergo, quòd locus imaginis eſt, ubi ſecat perpendicularis prædictam lineam reflexio-<lb/>nis, cum comprehenditur ueritas.</s> <s xml:id="echoid-s7619" xml:space="preserve"> Et licet non comprehendatur certitudo imaginis, tamen erit <lb/>modus harum imaginum cum ueritatis imaginibus.</s> <s xml:id="echoid-s7620" xml:space="preserve"> Pari modo uidere poteris imaginem in py-<lb/>ramidalibus concauis in concurſu perpendicularis cum linea reflexionis.</s> <s xml:id="echoid-s7621" xml:space="preserve"> Palàm ergo, quòd in o-<lb/>mnibus ſpeculis comprehenduntur imagines in loco prædicto:</s> <s xml:id="echoid-s7622" xml:space="preserve"> qui quidem locus ſimiliter dicitur <lb/>imaginis locus.</s> <s xml:id="echoid-s7623" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div287" type="section" level="0" n="0"> <head xml:id="echoid-head311" xml:space="preserve" style="it">8. Imago in quocun ſpeculo, uidetur in concurſu perpendicularis incidentiæ & lineæ refle-<lb/>scionis. 37 p 5.</head> <p> <s xml:id="echoid-s7624" xml:space="preserve">QVare autem comprehendantur res uiſæ per reflexionem in locis imaginum:</s> <s xml:id="echoid-s7625" xml:space="preserve"> & quare ima-<lb/>go ſit ſuper perpendicularem à re uiſa in ſpeculi ſuperficiem, declarabimus cauſſam.</s> <s xml:id="echoid-s7626" xml:space="preserve"> Viſus <lb/>cum acquirit form am per reflexionẽ, acquirit eam ſtatim ſine certitudine, & acquirit longi-<lb/>tudinẽ per æſtimationẽ, & hanc longitudinẽ cõprehendet forſitan in ueritate, per diligentiã intui-<lb/>tus adhibitã, forſitan nõ.</s> <s xml:id="echoid-s7627" xml:space="preserve"> Et iſtud explanauimus in libro ſecũdo [24.</s> <s xml:id="echoid-s7628" xml:space="preserve"> 25.</s> <s xml:id="echoid-s7629" xml:space="preserve"> 38.</s> <s xml:id="echoid-s7630" xml:space="preserve"> 39 n:</s> <s xml:id="echoid-s7631" xml:space="preserve">] & ibi dictũ eſt, quòd <lb/> <pb o="130" file="0136" n="136" rhead="ALHAZEN"/> uiſus acquirit longitudinem per ſyllogiſmum ex magnitudine corporis, & angulo aliquo, ſub quo <lb/>comprehenditur magnitudo.</s> <s xml:id="echoid-s7632" xml:space="preserve"> Et acquiſitio rei uiſæ notæ manifeſta eſt in hunc modum.</s> <s xml:id="echoid-s7633" xml:space="preserve"> Res etiam <lb/>ignotæ comprehenduntur in hunc modum:</s> <s xml:id="echoid-s7634" xml:space="preserve"> conferuntur enim rebus cognitis & magnitudinibus <lb/>uel longitudinibus notis.</s> <s xml:id="echoid-s7635" xml:space="preserve"> Cum uiſus comprehendit rem aliquam per reflexionem:</s> <s xml:id="echoid-s7636" xml:space="preserve"> non compre-<lb/>hendit longitudinem imaginis, niſi per æſtimationem:</s> <s xml:id="echoid-s7637" xml:space="preserve"> dein de adhibita diligentia, acquirit longitu-<lb/>dinem, & uerificat per ſyllogiſmum ex magnitudine rei uiſæ & angulo pyramidis, ſuper quam for-<lb/>ma reflectitur ad uiſum.</s> <s xml:id="echoid-s7638" xml:space="preserve"> Cum ergo res uiſa ex rebus notis fuerit, uiſus acquirit eius longitudinem <lb/>per iam notam longitudinem angulum æqualem huic tenentem, & huic longitudini ſimilem.</s> <s xml:id="echoid-s7639" xml:space="preserve"> Simi-<lb/>liter res uiſa cum fuerit ignota, confertur magnitudo eius alij magnitudini rerum uiſarum nota-<lb/>rum, & acquiritur longitudo eius imaginis per ſyllogiſmum menſuræ anguli, quem tenet imago in <lb/>centro uiſus, in hora reflexionis.</s> <s xml:id="echoid-s7640" xml:space="preserve"> Et à loco, in quo eſt forma rei uiſæ comprehenſa per reflexionem, <lb/>forma directè ueniens ad angulum circa oculum, accedit ſuper pyramidem ipſam, per quam for-<lb/>ma reflectitur ad uiſum:</s> <s xml:id="echoid-s7641" xml:space="preserve"> & eadem pyramis occupabit totam formam, quæ fuerit in loco imaginis.</s> <s xml:id="echoid-s7642" xml:space="preserve"> <lb/>Viſus ergo cum acquirit rem uiſam per reflexionem:</s> <s xml:id="echoid-s7643" xml:space="preserve"> acquirit eam in loco imaginis:</s> <s xml:id="echoid-s7644" xml:space="preserve"> quoniam for-<lb/>ma comprehenſa eſt in loco imaginis per reflexionem.</s> <s xml:id="echoid-s7645" xml:space="preserve"> Quare ſimilis eſt formæ directè comprehen <lb/>ſæ, occupatæ ab illa pyramide.</s> <s xml:id="echoid-s7646" xml:space="preserve"> Et hæc eſt cauſa, quare comprehendatur in loco imaginis.</s> <s xml:id="echoid-s7647" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div288" type="section" level="0" n="0"> <head xml:id="echoid-head312" xml:space="preserve" style="it">9. Imago in ſpeculo plano uidetur in perpendiculari incidentiæ. 36 p 5.</head> <p> <s xml:id="echoid-s7648" xml:space="preserve">QVare autem comprehendatur imago in perpendiculari, dicemus.</s> <s xml:id="echoid-s7649" xml:space="preserve"> Scimus [per16 n 4] quòd <lb/>punctum uiſui perceptibile, non eſt intellectuale, ſed ſenſuale, & forma eius ſenſualis.</s> <s xml:id="echoid-s7650" xml:space="preserve"> Dico <lb/>igitur in ſpeculis planis, quòd cum imago non appareat in ſuperficie ſpeculi, ſed ultra:</s> <s xml:id="echoid-s7651" xml:space="preserve"> com-<lb/>petentius eſt, & rationabilius, ut appareat ſupra perpendicularem, quàm extra eam.</s> <s xml:id="echoid-s7652" xml:space="preserve"> Cum enim in <lb/>loco perpendicularis aſsignata fuerit diſtantia eius à puncto refle-<lb/> <anchor type="figure" xlink:label="fig-0136-01a" xlink:href="fig-0136-01"/> xionis ſpeculi, quæ ſcilicet eſt pars lineæ reflexionis, à loco imaginis <lb/>ad punctum reflexionis ductæ:</s> <s xml:id="echoid-s7653" xml:space="preserve"> erit æqualis diſtantiæ puncti uiſi à <lb/>puncto reflexionis.</s> <s xml:id="echoid-s7654" xml:space="preserve"> Quia enim ſuperficies ſpeculi eſt orthogonalis <lb/>ſuper perpendicularem, [per theſin] & linea à puncto reflexionis <lb/>ad perpendicularem ducta eſt latus duobus triangulis commune, & <lb/>angulus lineæ acceſſus eſt æqualis angulo reflexionis [per 10 n 4, & <lb/>angulus f c d æquatur angulo e c b per 15 p 1:</s> <s xml:id="echoid-s7655" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s7656" xml:space="preserve"> angulo a c b] <lb/>quare duo anguli unius trianguli ſuntæquales duobus angulis al-<lb/>terius trianguli [anguli enim ad b recti ſunt per theſin & 3 d 11] & u-<lb/>num latus commune eſt:</s> <s xml:id="echoid-s7657" xml:space="preserve"> quare [per 26 p 1] reliqua latera æqualia <lb/>ſunt reliquis lateribus.</s> <s xml:id="echoid-s7658" xml:space="preserve"> Si ergo imago in perpendiculari apparuerit:</s> <s xml:id="echoid-s7659" xml:space="preserve"> <lb/>æqualiter à ſpeculo diſtabit cum corpore, à quo procedit:</s> <s xml:id="echoid-s7660" xml:space="preserve"> & erit ima <lb/>gini idem ſitus, reſpectu puncti reflexionis, qui eſt in puncto uiſo, <lb/>reſpectu puncti eiuſdẽ:</s> <s xml:id="echoid-s7661" xml:space="preserve"> & idem eſt ſitus, reſpectu uiſus.</s> <s xml:id="echoid-s7662" xml:space="preserve"> Vnde in hoc <lb/>ſitu apparebit ueritas & puncti uiſi, & imaginis.</s> <s xml:id="echoid-s7663" xml:space="preserve"> Si uerò imago fuerit <lb/>extra perpendicularem, cum fuerit neceſſe eam in linea reflexionis <lb/>eſſe, [per 2 n 4] aut erit ultra perpendicularem, aut citra, reſpectu <lb/>uiſus.</s> <s xml:id="echoid-s7664" xml:space="preserve"> Si fuerit ultra:</s> <s xml:id="echoid-s7665" xml:space="preserve"> erit quidem remotior à puncto reflexionis, & à <lb/>uiſu, quàm punctum uiſum, unde tenebit minorem angulum in ocu <lb/>lo, quàm punctum uiſum, & minorem occupabit uiſus partem:</s> <s xml:id="echoid-s7666" xml:space="preserve"> unde cum ſit æqualis, uidebitur mi-<lb/>nor eo.</s> <s xml:id="echoid-s7667" xml:space="preserve"> Si autem fuerit citra perpendicularem, uidebitur maior, cum ſit propinquior.</s> <s xml:id="echoid-s7668" xml:space="preserve"/> </p> <div xml:id="echoid-div288" type="float" level="0" n="0"> <figure xlink:label="fig-0136-01" xlink:href="fig-0136-01a"> <variables xml:id="echoid-variables29" xml:space="preserve">a f b c d e</variables> </figure> </div> </div> <div xml:id="echoid-div290" type="section" level="0" n="0"> <head xml:id="echoid-head313" xml:space="preserve" style="it">10. Imago in ſpeculis conuexis, cauis: ſphærico, cylindraceo, conico uidetur in perpendiculari <lb/>incidentiæ. 36 p 5.</head> <p> <s xml:id="echoid-s7669" xml:space="preserve">IN ſpeculo ſphærico extrà polito uidetur imago ſuper perpendicularem.</s> <s xml:id="echoid-s7670" xml:space="preserve"> Aut enim uidetur ima-<lb/>go centri uiſus:</s> <s xml:id="echoid-s7671" xml:space="preserve"> aut alterius puncti.</s> <s xml:id="echoid-s7672" xml:space="preserve"> Si imago centri uiſus:</s> <s xml:id="echoid-s7673" xml:space="preserve"> dico, quòd dignior eſt perpendicula-<lb/>ris ab oculo ad centrum ſphæræ ducta, ut ſuper eam appareat imago centri uiſus, quàm alia.</s> <s xml:id="echoid-s7674" xml:space="preserve"> Si <lb/>enim forma directè procedat ſecundum hanc perpendicularem uſque ad centrum ſphæræ, eun-<lb/>dem ſemper ſeruabit ſitum, reſpectu uiſus:</s> <s xml:id="echoid-s7675" xml:space="preserve"> & ita cuicunque puncto ſphæræ opponatur forma:</s> <s xml:id="echoid-s7676" xml:space="preserve"> per-<lb/>pendicularis ad centrum mota, identitatem ſitus tenebit, reſpectu uiſus:</s> <s xml:id="echoid-s7677" xml:space="preserve"> & idem erit ſitus for-<lb/>mæ in una perpendiculari, quæ & in alia:</s> <s xml:id="echoid-s7678" xml:space="preserve"> quoniam centrum ſphæræ eundem habet ſitum, reſpe-<lb/>ctu cuiuslibet puncti ſphæræ, & omnes huiuſmodi perpendiculares eiuſdem ſunt ſitus.</s> <s xml:id="echoid-s7679" xml:space="preserve"> Si autem <lb/>extra perpendicularem imago moueatur, ad quodcunque punctum ſphæræ mutabitur ſitus e-<lb/>ius, reſpectu uiſus:</s> <s xml:id="echoid-s7680" xml:space="preserve"> quoniam alium habebit ſitum extra perpendicularem, quàm in perpendicu-<lb/>lari, & extra ſpeculum mouebitur perpendicularis, & non intra:</s> <s xml:id="echoid-s7681" xml:space="preserve"> & ſi extra ſpeculum appareat, <lb/>non ſeruabit ſitum.</s> <s xml:id="echoid-s7682" xml:space="preserve"> Et conuenientius fuit, ut ſeruaret ſitum imago, quàm ut mutaret, ut uiſus rem <lb/>uiſam certius comprehenderet.</s> <s xml:id="echoid-s7683" xml:space="preserve"> Ob hoc imago centri uiſus ſuper perpendicularem apparet.</s> <s xml:id="echoid-s7684" xml:space="preserve"> Et <lb/>huic imagini non poſſumus certum aſsignare in perpendiculari punctum:</s> <s xml:id="echoid-s7685" xml:space="preserve"> quoniam non inueni-<lb/>tur dignitas in uno perpendicularis puncto maior, quàm in alio, ut hæc imago determinatè appa-<lb/>reatin eo:</s> <s xml:id="echoid-s7686" xml:space="preserve"> ſed ſcimus, quòd in quocunque puncto huius perpendicularis appareat, ſemper appa-<lb/> <pb o="131" file="0137" n="137" rhead="OPTICAE LIBER V"/> ret continua cum apparente oculo:</s> <s xml:id="echoid-s7687" xml:space="preserve"> & ſemper in totali forma apparente eundem tenet locum & ſi-<lb/>tum.</s> <s xml:id="echoid-s7688" xml:space="preserve"> Cuiuſcunq;</s> <s xml:id="echoid-s7689" xml:space="preserve"> uerò puncti imago, præter centrũ uiſus, ad ſpeculum accedit, mouetur declinatè:</s> <s xml:id="echoid-s7690" xml:space="preserve"> <lb/>quare nõ durat ei ſimilitudo ſitus, reſpectu uiſus:</s> <s xml:id="echoid-s7691" xml:space="preserve"> & perpendicularis à puncto uiſo ad ſpeculũ ducta, <lb/>cadit ſuper centrũ ſphęrę:</s> <s xml:id="echoid-s7692" xml:space="preserve"> in qua quidẽ perpẽdiculari obſeruat imago ſimilitu dinẽ ſitus.</s> <s xml:id="echoid-s7693" xml:space="preserve"> Nõ eſt ergo <lb/>punctum, in quo cõprehenſa imago ſeruet ſimilitudinẽ ſitus, niſi in perpendiculari illa.</s> <s xml:id="echoid-s7694" xml:space="preserve"> Et cũ opor-<lb/>teat ipſam comprehendi in linea reflexionis, [per 21 n 4] comprehendetur in concurſu huius lineæ <lb/>cum hac perpendiculari.</s> <s xml:id="echoid-s7695" xml:space="preserve"> Iam ergo aſsignauimus cauſſam huius rei.</s> <s xml:id="echoid-s7696" xml:space="preserve"> Verùm rerum naturaliũ ſtatus <lb/>reſpicit ſitus ſuorum principiorũ, & principia rerum naturaliũ ſunt occulta.</s> <s xml:id="echoid-s7697" xml:space="preserve"> Idem erit modus proba <lb/>tionis in ſpeculo ſphærico concauo.</s> <s xml:id="echoid-s7698" xml:space="preserve"> Similiter in pyramidali concauo, uel extrà polito.</s> <s xml:id="echoid-s7699" xml:space="preserve"> Et uniuerſali <lb/>ter erit locus imaginis in perpendiculari in quocunq;</s> <s xml:id="echoid-s7700" xml:space="preserve"> ſpeculo:</s> <s xml:id="echoid-s7701" xml:space="preserve"> quoniam non eſt locus extra perpen <lb/>pendicularem, in quo forma obſeruet ſimilitudinem ſitus & identitatem.</s> <s xml:id="echoid-s7702" xml:space="preserve"> His explanatis reſtat de-<lb/>monſtratiuè declarare locum imaginis, in qualibet ſpeculorum ſpecie.</s> <s xml:id="echoid-s7703" xml:space="preserve"> Dicimus ergo, quod linea, <lb/>per quam reflectitur forma puncti cuiuslibet comprehenſi à uiſu in ſpeculo plano, quando ipſum e-<lb/>greſſum eſt à perpendiculari, quæ à centro uiſus cadit in ſuperficiem ſpeculi plani:</s> <s xml:id="echoid-s7704" xml:space="preserve"> concurret cum <lb/>perpendiculari, producta ab illo puncto ad ſuperficiem ſpeculi:</s> <s xml:id="echoid-s7705" xml:space="preserve"> & erit punctum concurſus (qui eſt <lb/>locus imaginis) ultra ſpeculum:</s> <s xml:id="echoid-s7706" xml:space="preserve"> & erit longitudo illius à ſuperficie ſpeculi, æqualis longitudini pun <lb/>cti uiſi à ſuperficie ſpeculi:</s> <s xml:id="echoid-s7707" xml:space="preserve"> & uiſus non acquirit imaginem puncti uiſi, niſi in loco illo.</s> <s xml:id="echoid-s7708" xml:space="preserve"> Et quodcunq;</s> <s xml:id="echoid-s7709" xml:space="preserve"> <lb/>punctum acquirit uiſus in hoc ſpeculo:</s> <s xml:id="echoid-s7710" xml:space="preserve"> non apparebit ex eo, niſi unica imago.</s> <s xml:id="echoid-s7711" xml:space="preserve"> Quodcunq;</s> <s xml:id="echoid-s7712" xml:space="preserve"> autẽ pun <lb/>ctum comprehendit uiſus in ſpeculo ſphærico extrà polito, quando egreditur forma à perpendicu-<lb/>lari, ducta à centro uiſus ad centrum ſpeculi:</s> <s xml:id="echoid-s7713" xml:space="preserve"> linea, per quã reflectitur imago ad oculum, concurret <lb/>cum linea producta à puncto illo ad centrum ſpeculi:</s> <s xml:id="echoid-s7714" xml:space="preserve"> quæ linea eſt perpendicularis, ducta à puncto <lb/>illo orthogonaliter ſuper lineã, contingentẽ lineam cõmunem ſuperficiei reflexionis, & ſuperficiei <lb/>ſpeculi.</s> <s xml:id="echoid-s7715" xml:space="preserve"> Et ſitus puncti concurſus, qui eſt locus imaginis, à ſuperficie ſpeculi erit ſecundũ ſitum ui-<lb/>ſus à ſuperficie ſpeculi.</s> <s xml:id="echoid-s7716" xml:space="preserve"> Et forſitan erit punctum concurſus ultra ſpeculum, forſitan in ſuperficie ſpe <lb/>culi, forſitan intra ſpeculum.</s> <s xml:id="echoid-s7717" xml:space="preserve"> Et uiſus comprehendit imagines omnes ultra ſpeculum, licet diuerſa <lb/>ſint earum loca:</s> <s xml:id="echoid-s7718" xml:space="preserve"> & non comprehendit locum cuiuslibet imaginis, niſi ſyllogiſticè in ſuperficie ſpe-<lb/>culi.</s> <s xml:id="echoid-s7719" xml:space="preserve"> Et quodlibet punctum comprehenſum in hoc ſpeculo, non prætendit, niſi unam imaginem.</s> <s xml:id="echoid-s7720" xml:space="preserve"> In <lb/>ſpeculo columnari extrà polito, & pyramidali extrà polito, quodcunq;</s> <s xml:id="echoid-s7721" xml:space="preserve"> punctum comprehendit ui-<lb/>ſus, cum fuerit extra perpendicularem, ductam à centro uiſus, orthogonalem ſuper ſuperficiem con <lb/>tingentem ſuperficiem ſpeculi:</s> <s xml:id="echoid-s7722" xml:space="preserve"> linea, per quam reflectitur forma ad uiſum, concurret cum perpendi <lb/>culari, ducta ab illo puncto ſuper rectam lineam, contingentem lineam communem ſuperficiei re-<lb/>flexionis, & ſpeculi.</s> <s xml:id="echoid-s7723" xml:space="preserve"> Et loca imagihum horum ſpeculorum quædam ſunt ultra ſuperficiem ſpeculi:</s> <s xml:id="echoid-s7724" xml:space="preserve"> <lb/>quædam in ſuperficie:</s> <s xml:id="echoid-s7725" xml:space="preserve"> quædam citra.</s> <s xml:id="echoid-s7726" xml:space="preserve"> Et uiſus acquirit omnes imagines horum ſpeculorum ultra ſu <lb/>perficiem ſpeculi.</s> <s xml:id="echoid-s7727" xml:space="preserve"> Et quodcunq;</s> <s xml:id="echoid-s7728" xml:space="preserve"> punctum comprehendit uiſus in his ſpeculis, non efficit, niſi unam <lb/>imaginem tantùm.</s> <s xml:id="echoid-s7729" xml:space="preserve"> In ſpeculo ſphærico concauo lineæ, per quas reflectuntur formæ punctorũ uiſo-<lb/>rum:</s> <s xml:id="echoid-s7730" xml:space="preserve"> quædam concurrunt cum perpendicularibus, ductis à punctis illis ſuper lineas, contingentes <lb/>lineas communes ſuperficiei ſpeculi & ſuperficiei reflexionis:</s> <s xml:id="echoid-s7731" xml:space="preserve"> quædam ſunt æquidiſtantes his per-<lb/>pendicularibus.</s> <s xml:id="echoid-s7732" xml:space="preserve"> Et earum, quæ concurrunt cum perpendicularibus, quædam habent locum con-<lb/>curſus (qui eſt locus imaginis) ultra ſpeculum:</s> <s xml:id="echoid-s7733" xml:space="preserve"> quædam citra ſpeculum.</s> <s xml:id="echoid-s7734" xml:space="preserve"> Et quæ citra ſpeculum ha-<lb/>bent:</s> <s xml:id="echoid-s7735" xml:space="preserve"> quædam inter uiſum & ſpeculum:</s> <s xml:id="echoid-s7736" xml:space="preserve"> quædam ſuper ipſum centrũ uiſus:</s> <s xml:id="echoid-s7737" xml:space="preserve"> quædam ultra centrum <lb/>uiſus.</s> <s xml:id="echoid-s7738" xml:space="preserve"> Et uiſus quaſdam formarum rerum uiſarum, quas acquirit in his ſpeculis, comprehendit in lo <lb/>co imaginis, qui eſt punctum concurſus:</s> <s xml:id="echoid-s7739" xml:space="preserve"> & hæ ſunt, quas uiſus certò comprehendit:</s> <s xml:id="echoid-s7740" xml:space="preserve"> quaſdam com-<lb/>prehendit extra locum concurſus:</s> <s xml:id="echoid-s7741" xml:space="preserve"> & eſt comprehenſio ſine certitudine.</s> <s xml:id="echoid-s7742" xml:space="preserve"> Et res uiſæ, quas acquirit ui <lb/>ſus in hoc ſpeculo, quædam unam præ ſe ferunt imaginem tantùm:</s> <s xml:id="echoid-s7743" xml:space="preserve"> quædam duas:</s> <s xml:id="echoid-s7744" xml:space="preserve"> quædam tres:</s> <s xml:id="echoid-s7745" xml:space="preserve"> <lb/>quædã quatuor.</s> <s xml:id="echoid-s7746" xml:space="preserve"> Nec poteſt eſſe, quod una res prætendat plures.</s> <s xml:id="echoid-s7747" xml:space="preserve"> In ſpeculo pyramidali cõcauo & co <lb/>lumnari concauo lineæ, per quas reflectuntur formæ ad uiſum:</s> <s xml:id="echoid-s7748" xml:space="preserve"> quædam concurrunt cum perpendi <lb/>cularibus, ductis à punctis uiſis ſuper lineas, contingentes lineas communes:</s> <s xml:id="echoid-s7749" xml:space="preserve"> & quædam ſunt æqui <lb/>diſtantes perpendιcularibus.</s> <s xml:id="echoid-s7750" xml:space="preserve"> Quæ concurrunt cum perpendicularibus:</s> <s xml:id="echoid-s7751" xml:space="preserve"> quædam habent concur-<lb/>ſum ultra ſpeculum:</s> <s xml:id="echoid-s7752" xml:space="preserve"> quædam citra.</s> <s xml:id="echoid-s7753" xml:space="preserve"> Quæ autem citra:</s> <s xml:id="echoid-s7754" xml:space="preserve"> quædam inter ſpeculum & uiſum:</s> <s xml:id="echoid-s7755" xml:space="preserve"> quædam ſu <lb/>per centrum uiſus:</s> <s xml:id="echoid-s7756" xml:space="preserve"> quædam ultra centrum uiſus.</s> <s xml:id="echoid-s7757" xml:space="preserve"> Et comprehenſio rerum uiſarum in hoc ſpeculo <lb/>per uiſum, quædam fit in loco imaginis (qui eſt locus concurſus) quædam extra locum concurſus.</s> <s xml:id="echoid-s7758" xml:space="preserve"> <lb/>Et eorum, quæ comprehenduntur, aliud prætendit unam imaginem tantùm:</s> <s xml:id="echoid-s7759" xml:space="preserve"> aliud duas:</s> <s xml:id="echoid-s7760" xml:space="preserve"> aliud tres:</s> <s xml:id="echoid-s7761" xml:space="preserve"> <lb/>alind quatuor.</s> <s xml:id="echoid-s7762" xml:space="preserve"> Nec aliquod eſt, quod poſsit prætendere plures, quàm quatuor.</s> <s xml:id="echoid-s7763" xml:space="preserve"> Et nos declarabi-<lb/>mus hæc omnia demonſtratiuè.</s> <s xml:id="echoid-s7764" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div291" type="section" level="0" n="0"> <head xml:id="echoid-head314" xml:space="preserve" style="it">11. Viſibile & imago à ſpeculi plani ſuperficie in oppoſit {as} partes æquabiliter distant. 49 p 5.</head> <p> <s xml:id="echoid-s7765" xml:space="preserve">SIt a punctum uiſum:</s> <s xml:id="echoid-s7766" xml:space="preserve"> b centrum uiſus:</s> <s xml:id="echoid-s7767" xml:space="preserve"> c d e ſpeculum planum:</s> <s xml:id="echoid-s7768" xml:space="preserve"> & ſit d punctum reflexionis:</s> <s xml:id="echoid-s7769" xml:space="preserve"> c d e <lb/>linea communis ſuperficiei reflexionis & ſuperficiei ſpeculi.</s> <s xml:id="echoid-s7770" xml:space="preserve"> A puncto d ducatur d f perpendi-<lb/>cularis ſuper lineã cõmunẽ:</s> <s xml:id="echoid-s7771" xml:space="preserve"> [per 11 p 1] & à puncto a ducatur perpendicularis ſuper ſpeculi ſu <lb/>perficiẽ, [per 11 p 11] quæ ſit a c, & producatur ultra ſpeculũ:</s> <s xml:id="echoid-s7772" xml:space="preserve"> & a d ſit linea, per quã forma accedit ad <lb/>ſpeculũ:</s> <s xml:id="echoid-s7773" xml:space="preserve"> b d, per quã reflectitur ad uiſum.</s> <s xml:id="echoid-s7774" xml:space="preserve"> Igitur b d, f d, a d, ſunt in ſuperficie reflexionis [per 23 n 4.</s> <s xml:id="echoid-s7775" xml:space="preserve">] <lb/>Et cũ f d ſit æquidiſtãs a c [per 28 p 1:</s> <s xml:id="echoid-s7776" xml:space="preserve"> quia cũ a c ſit քpẽdicularis ſuperficiei ſpeculi per fabricatiõem:</s> <s xml:id="echoid-s7777" xml:space="preserve"> <lb/>erit perpẽdicularis lineę c d e per 3 d 11] & [per 13 p 11] d b declinata ſit ſuper f d, cõcurret [per lemma <lb/>Procli ad 29 p 1] b d cũ a c.</s> <s xml:id="echoid-s7778" xml:space="preserve"> Cõcurrat ergo in puncto g.</s> <s xml:id="echoid-s7779" xml:space="preserve"> Dico, quòd g c eſt æqualis c a.</s> <s xml:id="echoid-s7780" xml:space="preserve"> Quoniã enim <lb/> <pb o="132" file="0138" n="138" rhead="ALHAZEN"/> angulus b d e æqualis eſt angulo a d c [per 10 n 4, & per 15 p 1 angulus b d e æqualis angulo g d c:</s> <s xml:id="echoid-s7781" xml:space="preserve"> er-<lb/>go per 1 ax:</s> <s xml:id="echoid-s7782" xml:space="preserve"> angulus a d c æquatur angulo g d c] & angulus a c d æ-<lb/> <anchor type="figure" xlink:label="fig-0138-01a" xlink:href="fig-0138-01"/> qualis angulo g c d [per 10 ax:</s> <s xml:id="echoid-s7783" xml:space="preserve">] & latus c d commune.</s> <s xml:id="echoid-s7784" xml:space="preserve"> Quare [per 26 <lb/>p 1] triangulum æquale triangulo.</s> <s xml:id="echoid-s7785" xml:space="preserve"> Quare g c æqualis a c.</s> <s xml:id="echoid-s7786" xml:space="preserve"/> </p> <div xml:id="echoid-div291" type="float" level="0" n="0"> <figure xlink:label="fig-0138-01" xlink:href="fig-0138-01a"> <variables xml:id="echoid-variables30" xml:space="preserve">a f b c d e g</variables> </figure> </div> </div> <div xml:id="echoid-div293" type="section" level="0" n="0"> <head xml:id="echoid-head315" xml:space="preserve" style="it">12. Viſu & uiſibili datis, in ſpeculo plano punctum reflexionis <lb/>inuenire. 46 p 5.</head> <p> <s xml:id="echoid-s7787" xml:space="preserve">ET ſi uoluerimus per perpendicularem inuenire locum reflexio <lb/>nis:</s> <s xml:id="echoid-s7788" xml:space="preserve"> ſecetur ex perpendiculari ultra ſpeculum pars, æqualis par <lb/>ti eius uſq;</s> <s xml:id="echoid-s7789" xml:space="preserve"> ad ſpeculum:</s> <s xml:id="echoid-s7790" xml:space="preserve"> & eſt, ut ſit g c æqualis a c:</s> <s xml:id="echoid-s7791" xml:space="preserve"> & ducatur li <lb/>nea à centro uiſus ad punctum g, quæ ſit b d g.</s> <s xml:id="echoid-s7792" xml:space="preserve"> Dico, quòd d, eſt pun-<lb/>ctum reflexionis.</s> <s xml:id="echoid-s7793" xml:space="preserve"> Quoniam enim [per fabricationem & 2 ax:</s> <s xml:id="echoid-s7794" xml:space="preserve">] a c & <lb/>c d ſunt æqualia c g & c d, & angulus angulo [a c d ipſi g c d per theſin <lb/>& 10 ax.</s> <s xml:id="echoid-s7795" xml:space="preserve">] Ergo [per 4 p 1] triangulum triangulo.</s> <s xml:id="echoid-s7796" xml:space="preserve"> Igitur angulus g d c <lb/>eſt æqualis angulo a d c:</s> <s xml:id="echoid-s7797" xml:space="preserve"> Sed g d c eſt æqualis angulo b d e [per 15 p 1] <lb/>reſtat ergo [per 1 ax] ut angulus b d e ſit æqualis angulo a d c.</s> <s xml:id="echoid-s7798" xml:space="preserve"> Et ita <lb/>[per 10 n 4] d eſt punctum reflexionis:</s> <s xml:id="echoid-s7799" xml:space="preserve"> & ita patet propoſitum.</s> <s xml:id="echoid-s7800" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div294" type="section" level="0" n="0"> <head xml:id="echoid-head316" xml:space="preserve" style="it">13. Si recta linea ab uno uiſu ſit perpendicularis ſpeculo plano, <lb/>unum ipſi{us} punctũ; in quo uiſ{us} ſuperficiem ſecat, ab uno ſpeculi <lb/>puncto, in quod cadit, ad eundem uiſum reflectetur. 32 p 5.</head> <p> <s xml:id="echoid-s7801" xml:space="preserve">SIt a centrum uiſus:</s> <s xml:id="echoid-s7802" xml:space="preserve"> & a g perpendicularis ſuper ſpeculũ planũ:</s> <s xml:id="echoid-s7803" xml:space="preserve"> & d ſecet hanc perpendicularẽ in <lb/>ſuperficie oculi.</s> <s xml:id="echoid-s7804" xml:space="preserve"> Dico, quòd in hac perpendiculari non eſt punctũ, quod reflectatur ab hoc ſpe-<lb/>culo ad uiſum, præter d.</s> <s xml:id="echoid-s7805" xml:space="preserve"> Sin autem:</s> <s xml:id="echoid-s7806" xml:space="preserve"> ſumatur ultra uiſum punctum in hac perpendiculari:</s> <s xml:id="echoid-s7807" xml:space="preserve"> & ſit <lb/>h:</s> <s xml:id="echoid-s7808" xml:space="preserve"> Non iam perueniet forma eius ad ſpeculũ ſuper <lb/> <anchor type="figure" xlink:label="fig-0138-02a" xlink:href="fig-0138-02"/> perpendicularẽ a h, propter ſolidi corporis inter-<lb/>poſitionem:</s> <s xml:id="echoid-s7809" xml:space="preserve"> & ita nõ reflectetur forma eius ſuper <lb/>perpendicularẽ.</s> <s xml:id="echoid-s7810" xml:space="preserve"> Et ſi dicatur, quòd ab alio puncto <lb/>ſpeculi poſsit reflecti:</s> <s xml:id="echoid-s7811" xml:space="preserve"> ſit illud b.</s> <s xml:id="echoid-s7812" xml:space="preserve"> Mouebitur quidẽ <lb/>forma eius ad punctũ b per lineã h b:</s> <s xml:id="echoid-s7813" xml:space="preserve"> & reflectetur <lb/>per lineam b a.</s> <s xml:id="echoid-s7814" xml:space="preserve"> Diuidatur angulus h b a [per 9 p 1] <lb/>per ęqualia, per lineã t b.</s> <s xml:id="echoid-s7815" xml:space="preserve"> Igitur erit perpẽdicularis <lb/>ſuper ſuperficiẽ ſpeculi.</s> <s xml:id="echoid-s7816" xml:space="preserve"> [Quia enim angulus h b c <lb/>æquatur angulo a b g ք theſin & 10 n 4, & h b t ipſi <lb/>a b t per fabricationẽ:</s> <s xml:id="echoid-s7817" xml:space="preserve"> totus t b c æquabitur toti t b <lb/>g.</s> <s xml:id="echoid-s7818" xml:space="preserve"> quare per 10 d 1 t b eſt perpendicularis ipſi g c cõ <lb/>muni ſectioni ſuperficierũ reflexionis & ſpeculi.</s> <s xml:id="echoid-s7819" xml:space="preserve"> <lb/>Itaq;</s> <s xml:id="echoid-s7820" xml:space="preserve"> cũ reflexiõis ſuperficies, in qua eſt t b, ſit per <lb/>pendicularis ſuperficiei ſpeculi per 13 n 4:</s> <s xml:id="echoid-s7821" xml:space="preserve"> erit t b <lb/>քpẽdicularis ſuperficiei ſpeculi per cõuerſam 4 d <lb/>11] ſed [per hypotheſin] t g eſt perpẽdicularis ſuper <lb/>eandẽ.</s> <s xml:id="echoid-s7822" xml:space="preserve"> Quare ab eodẽ puncto eſt ducere duas per <lb/>pendiculares ad ſuperficiem ſpeculi, quod eſt im-<lb/>poſsibile:</s> <s xml:id="echoid-s7823" xml:space="preserve"> [ſic enim tres interiores anguli triangu-<lb/>li eſſent maiores duobus rectis, cõtra 32 p 1.</s> <s xml:id="echoid-s7824" xml:space="preserve">] Eadẽ <lb/>erit probatio, quòd forma puncti d nõ poteſt refle <lb/>cti ab alio ſpeculi puncto, quam à puncto g.</s> <s xml:id="echoid-s7825" xml:space="preserve"> Quare <lb/>non reflectitur, niſi ſuper perpendicularẽ d g.</s> <s xml:id="echoid-s7826" xml:space="preserve"> Pun <lb/>ctum aũt in hac perpendiculari ſumptum inter g & d:</s> <s xml:id="echoid-s7827" xml:space="preserve"> ſi dicatur formã per reflexionẽ ad uiſum mit-<lb/>tere:</s> <s xml:id="echoid-s7828" xml:space="preserve"> improbo.</s> <s xml:id="echoid-s7829" xml:space="preserve"> Quoniã aut erit corpus ſolidum, aut rarũ.</s> <s xml:id="echoid-s7830" xml:space="preserve"> Si ſolidum, procedet ſecundum perpendi-<lb/>cularem forma eius ad ſpeculum, & regredietur ſecundũ eandem uſq;</s> <s xml:id="echoid-s7831" xml:space="preserve"> ad ipſum, [per 11 n 4] & pro-<lb/>pter ſoliditatẽ non poterit tranſire, & ad uiſum peruenire.</s> <s xml:id="echoid-s7832" xml:space="preserve"> Si aũt punctum illud fuerit rarum:</s> <s xml:id="echoid-s7833" xml:space="preserve"> forma <lb/>eius regrediẽs à ſpeculo ſuper perpendicularẽ miſcebitur ei, & adhærebit, nec reflectetur ad uiſum.</s> <s xml:id="echoid-s7834" xml:space="preserve"> <lb/>Quòd autem forma cuiuſcunq;</s> <s xml:id="echoid-s7835" xml:space="preserve"> puncti in hac perpendiculari inter g & d ſumpti non poſsit ab alio <lb/>puncto ſpeculi ad uiſum reflecti, modo ſuprà dicto poteſt probari.</s> <s xml:id="echoid-s7836" xml:space="preserve"> Similiter forma puncti inter a & <lb/>d ſumpti non reflectitur ad uiſum per perpendicularem, nec per aliam.</s> <s xml:id="echoid-s7837" xml:space="preserve"> Quoniã puncta inter centrũ <lb/>uiſus & ſuperficiem eius interpoſita ſunt ualde rara.</s> <s xml:id="echoid-s7838" xml:space="preserve"> Vnde nec mittitur eorum forma, nec reflecti-<lb/>tur, ut ſentiatur.</s> <s xml:id="echoid-s7839" xml:space="preserve"> Et quoniám quodlibet punctum, præter d in ſuperficie uiſus ſumptum:</s> <s xml:id="echoid-s7840" xml:space="preserve"> opponitur <lb/>ſpeculo, non ad rectum angulum, uidebitur quodlibet ſuper perpendicularem ab eo ad ſpeculum <lb/>ductam, & imago eius ultra ſpeculum æquè diſtans à ſuperficie, ſicut ipſum punctum [per 11 n.</s> <s xml:id="echoid-s7841" xml:space="preserve">] Et <lb/>quoniam d uidetur continuum cum alijs ſuperficiei uiſus punctis, & imago eius cõtinua cum alijs <lb/>imaginibus:</s> <s xml:id="echoid-s7842" xml:space="preserve"> uidebitur imago d tantùm diſtans à ſuperficiei ſpeculi, quantùm diſtat d ab eadem.</s> <s xml:id="echoid-s7843" xml:space="preserve"> Pa-<lb/>làm ergo, quòd cuiuſcunq;</s> <s xml:id="echoid-s7844" xml:space="preserve"> puncti in ſpeculo uiſi imago uidebitur ſuper perpendicularem:</s> <s xml:id="echoid-s7845" xml:space="preserve"> & elon-<lb/>gatio imaginis, & uiſi corporis à ſuperficie ſpeculi eſt eadem.</s> <s xml:id="echoid-s7846" xml:space="preserve"/> </p> <div xml:id="echoid-div294" type="float" level="0" n="0"> <figure xlink:label="fig-0138-02" xlink:href="fig-0138-02a"> <variables xml:id="echoid-variables31" xml:space="preserve">h t a d ſ s g k b e</variables> </figure> </div> <pb o="133" file="0139" n="139" rhead="OPTICAE LIBER V."/> </div> <div xml:id="echoid-div296" type="section" level="0" n="0"> <head xml:id="echoid-head317" xml:space="preserve" style="it">14. Ab uno ſpeculi plani puncto, unum uiſibilis punctũ ad unũ uiſum reflectitur. 45 p 5.</head> <p> <s xml:id="echoid-s7847" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s7848" xml:space="preserve"> forma puncti uiſi in ſpeculo plano non reflectitur ad eundẽ uiſum, niſi ab uno pun-<lb/>cto tantùm.</s> <s xml:id="echoid-s7849" xml:space="preserve"> Sit enim a centrum uiſus:</s> <s xml:id="echoid-s7850" xml:space="preserve"> b pun <lb/> <anchor type="figure" xlink:label="fig-0139-01a" xlink:href="fig-0139-01"/> ctum uiſum:</s> <s xml:id="echoid-s7851" xml:space="preserve"> z h ſpeculum.</s> <s xml:id="echoid-s7852" xml:space="preserve"> Si ergo dicatur, <lb/>quod à duobus punctis ſpeculi reflectatur forma b <lb/>ad uiſum a:</s> <s xml:id="echoid-s7853" xml:space="preserve"> ſit unum punctũ d, aliud e:</s> <s xml:id="echoid-s7854" xml:space="preserve"> & ducatur <lb/>linea à puncto uiſo ad uiſum, ſcilicet b a:</s> <s xml:id="echoid-s7855" xml:space="preserve"> quæ quidẽ <lb/>linea aut erit perpendicularis ſupra ſpeculũ:</s> <s xml:id="echoid-s7856" xml:space="preserve"> aut nõ.</s> <s xml:id="echoid-s7857" xml:space="preserve"> <lb/>[Siquidẽ cum ſpeculi ſuperficie concurrit.</s> <s xml:id="echoid-s7858" xml:space="preserve"> Nã cum <lb/>ſit in plano lineæ h z per 23 n 4:</s> <s xml:id="echoid-s7859" xml:space="preserve"> h neceſſariò uel ad <lb/>ipſam parallela eſt, uel concurrit.</s> <s xml:id="echoid-s7860" xml:space="preserve">] Si non fuerit per <lb/>pendicularis, ſcimus, quòd illa linea eſt in ſuperficie <lb/>reflexionis orthogonali ſuper ſuperficiem ſpeculi <lb/>[quia cõnectit duo pũcta a & b, quæ per 23 n 4 ſunt <lb/>in reflexionis ſuperficie, perpẽdiculari ad ſpeculi ſu-<lb/>perficiẽ, per 13 n 4:</s> <s xml:id="echoid-s7861" xml:space="preserve">] & in una ſola tali.</s> <s xml:id="echoid-s7862" xml:space="preserve"> Quoniam ſi in <lb/>duabus:</s> <s xml:id="echoid-s7863" xml:space="preserve"> erit communis duabus ſuperficiebus ortho <lb/>gonalibus:</s> <s xml:id="echoid-s7864" xml:space="preserve"> & ſumpto in ea puncto, & ducta ab illo <lb/>linea in alteram ſuperficierum, ſuper lineam, com-<lb/>munem huic ſuperficiei & ſuperficiei ſpeculi, erit <lb/>[per 19 p 11] hæc linea orthogonalis ſuper ſpeculum.</s> <s xml:id="echoid-s7865" xml:space="preserve"> Similiter ab eodem puncto ducatur linea in <lb/>alia ſuperficie ſuper lineam, communem huic ſuperficiei & ſuperficiei ſpeculi:</s> <s xml:id="echoid-s7866" xml:space="preserve"> erit hęc linea ortho-<lb/>gonalis ſuper ſpeculum.</s> <s xml:id="echoid-s7867" xml:space="preserve"> Quare ab eodem puncto erit ducere duas perpendiculares ad ſuperficiem <lb/>ſpeculi [& ſic connexis per rectam lineam perpendicularium duarũ terminis:</s> <s xml:id="echoid-s7868" xml:space="preserve"> erunt ipſæ ad con-<lb/>nectentem perpendiculares, per 3 d 11:</s> <s xml:id="echoid-s7869" xml:space="preserve"> itaque in triangulo rectilineo erunt duo anguli recti, co n-<lb/>tra 32 p 1.</s> <s xml:id="echoid-s7870" xml:space="preserve">] Cum ergo b a ſit in una ſola ſuperficie orthogonali:</s> <s xml:id="echoid-s7871" xml:space="preserve"> & tria puncta a, b, e ſint in eadem ſu-<lb/>perficie orthogonali [per 23 n 4] erunt a e, e b in illa ſuperficie orthogonali:</s> <s xml:id="echoid-s7872" xml:space="preserve"> ſimiliter [per 2 p 11] <lb/>e d, d b, d a.</s> <s xml:id="echoid-s7873" xml:space="preserve"> Quare e a, e b ſunt in eadem ſuperficie cum d a, d b:</s> <s xml:id="echoid-s7874" xml:space="preserve"> ſed angulus a e h eſt æqualis angu-<lb/>lo b e d, [per 10 n 4] & angulus a e h maior angulo a d e, [per 16 p 1] quia exterior.</s> <s xml:id="echoid-s7875" xml:space="preserve"> Quare b ed ma <lb/>ior a d e.</s> <s xml:id="echoid-s7876" xml:space="preserve"> Sed b d z æqualis a d e [per 10 n 4, & per 16 p 1 b d z maior b e d.</s> <s xml:id="echoid-s7877" xml:space="preserve">] Quare a d e maior b e d:</s> <s xml:id="echoid-s7878" xml:space="preserve"> <lb/>& dictum eſt, quod minor.</s> <s xml:id="echoid-s7879" xml:space="preserve"> Reſtat ergo, ut à ſolo puncto fiat reflexio.</s> <s xml:id="echoid-s7880" xml:space="preserve"> Si uerò a b ſit perpendicularis <lb/>ſuper ſpeculum:</s> <s xml:id="echoid-s7881" xml:space="preserve"> iam dictum eſt, [13 n] quò d unicum eſt punctum in linea, à centro uiſus ad ſpecu <lb/>lum orthogonaliter ducta, cuius forma reflectitur à ſpeculo ad uiſum.</s> <s xml:id="echoid-s7882" xml:space="preserve"> Et iam probatum eſt, quòd <lb/>imago illius puncti ab uno ſolo reflectitur puncto.</s> <s xml:id="echoid-s7883" xml:space="preserve"> Quare patet propoſitum.</s> <s xml:id="echoid-s7884" xml:space="preserve"/> </p> <div xml:id="echoid-div296" type="float" level="0" n="0"> <figure xlink:label="fig-0139-01" xlink:href="fig-0139-01a"> <variables xml:id="echoid-variables32" xml:space="preserve">a b h e d z</variables> </figure> </div> </div> <div xml:id="echoid-div298" type="section" level="0" n="0"> <head xml:id="echoid-head318" xml:space="preserve" style="it">15. In ſpeculo plano, imagouni{us} puncti, una, & uno eodeḿ in loco ab utroque uiſu uide-<lb/>tur. 51 p 5.</head> <p> <s xml:id="echoid-s7885" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s7886" xml:space="preserve"> inſpecto aliquo puncto ab utroque uiſu:</s> <s xml:id="echoid-s7887" xml:space="preserve"> una tantùm & eadem imago apparet u-<lb/>trique uiſui & in loco prædicto.</s> <s xml:id="echoid-s7888" xml:space="preserve"> Vnde planum eſt, quòd forma puncti non reflectitur ad u-<lb/>trumque uiſum ab eodem puncto ſpeculi.</s> <s xml:id="echoid-s7889" xml:space="preserve"> Quia enim linea reflexionis ad unum uiſum pro-<lb/>cedens, angulum tenet cum perpendiculari erecta ſuper ſuperficiem ſpeculi, æqualem angulo, quẽ <lb/>tenet linea acceſſus formæ a d ſpeculum cum eadem perpendiculari [per 10 n 4:</s> <s xml:id="echoid-s7890" xml:space="preserve">] non poterit in <lb/>eadem ſuperficie ſumi alia linea, quæ æqualem angulum huic efficiat cum perpendiculari [ſecus <lb/> <anchor type="figure" xlink:label="fig-0139-02a" xlink:href="fig-0139-02"/> <anchor type="figure" xlink:label="fig-0139-03a" xlink:href="fig-0139-03"/> pars æquaretur toti, contra 9 ax:</s> <s xml:id="echoid-s7891" xml:space="preserve">] Vnde ab hoc puncto non reflectetur linea aliqua ad alterũ ui-<lb/>ſum.</s> <s xml:id="echoid-s7892" xml:space="preserve"> Oportet ergo ut à diuerſis punctis ſpeculi fiat reflexio.</s> <s xml:id="echoid-s7893" xml:space="preserve"> Sint illa puncta t, z:</s> <s xml:id="echoid-s7894" xml:space="preserve"> & ſit ſpeculũ pla-<lb/>num q e:</s> <s xml:id="echoid-s7895" xml:space="preserve"> punctum uiſum a:</s> <s xml:id="echoid-s7896" xml:space="preserve"> duo uiſus b, g:</s> <s xml:id="echoid-s7897" xml:space="preserve"> perpendicularis a d.</s> <s xml:id="echoid-s7898" xml:space="preserve"> Palàm ergo [per 23 n 4] quòd b t, <lb/> <pb o="134" file="0140" n="140" rhead="ALHAZEN"/> at, ad ſunt in eadem ſuperficie orthogonali ſuper ſuperficiem ſpeculi.</s> <s xml:id="echoid-s7899" xml:space="preserve"> Similiter g z, a z, a d ſunt in <lb/>eadem ſuperficie orthogonali:</s> <s xml:id="echoid-s7900" xml:space="preserve"> & linea d t communis ſuperficiei a d t b & ſuperficiei ſpeculi:</s> <s xml:id="echoid-s7901" xml:space="preserve"> & d z <lb/>linea communis ſuperficiei a d z g & ſuperficiei ſpeculi.</s> <s xml:id="echoid-s7902" xml:space="preserve"> Si iam b t, g z fuerint in eadem ſuperficie <lb/>orthogonali, erit [per 3 p 11] t d z linea una recta:</s> <s xml:id="echoid-s7903" xml:space="preserve"> & perpendicula-<lb/> <anchor type="figure" xlink:label="fig-0140-01a" xlink:href="fig-0140-01"/> ris a d aut erit inter duas perpendiculares productas ad ſuperficiem <lb/>ſpeculi à duobus uiſibus:</s> <s xml:id="echoid-s7904" xml:space="preserve"> aut extra.</s> <s xml:id="echoid-s7905" xml:space="preserve"> Vtrumlibet ſit:</s> <s xml:id="echoid-s7906" xml:space="preserve"> linea b t ſecabit <lb/>ex perpendiculari a d ultra ſpeculum partem, æqualem parti, quæ eſt <lb/>a d [per 11 n.</s> <s xml:id="echoid-s7907" xml:space="preserve">] Similiter g z ſecabit ex eadem perpendiculari partem <lb/>ultra ſpeculum, æqualem illi parti.</s> <s xml:id="echoid-s7908" xml:space="preserve"> Illæ igitur duæ lineæ reflexionis <lb/>ſecabunt perpendicularem ultra ſpeculum in eodem puncto.</s> <s xml:id="echoid-s7909" xml:space="preserve"> Ergo <lb/>imago puncti a in eodem perpendicularis puncto percipietur ab u-<lb/>troque uiſu.</s> <s xml:id="echoid-s7910" xml:space="preserve"> Quare unica tantùm erit imago & eadem:</s> <s xml:id="echoid-s7911" xml:space="preserve"> & in eodem <lb/>loco:</s> <s xml:id="echoid-s7912" xml:space="preserve"> quæ eſſet uno tantùm uiſu adhibito.</s> <s xml:id="echoid-s7913" xml:space="preserve"> Si uerò puncta t, z non <lb/>fuerint in eadem ſuperficie reflexionis orthogonali ſuper ſpeculum:</s> <s xml:id="echoid-s7914" xml:space="preserve"> <lb/>eadem tamen erit probatio:</s> <s xml:id="echoid-s7915" xml:space="preserve"> quòd utraque linea reflexionis ſecet ex <lb/>perpendiculari partem, æqualẽ parti ſuperiori:</s> <s xml:id="echoid-s7916" xml:space="preserve"> & erit ſectio linearũ <lb/>reflexionis cum perpendiculari in eodem puncto.</s> <s xml:id="echoid-s7917" xml:space="preserve"> Quare patet pro-<lb/>poſitum.</s> <s xml:id="echoid-s7918" xml:space="preserve"> Si uerò fuerit punctum a in perpendiculari ducta ab uno <lb/>uiſu ad ſuperficiem ſpeculi tantùm, ſecundum eundem uiſum com-<lb/>prehendetur [per 11 n 4] ultra ſpeculum in puncto perpẽdicularis, <lb/>tãtùm elõgato à ſuperficie ſpeculi, quantũ diſtat a ab eadẽ [per 11 n.</s> <s xml:id="echoid-s7919" xml:space="preserve">] <lb/>Quia forma a uidetur continua cum formis aliorum punctorũ, quæ <lb/>quidem uidentur in locis ſimilibus:</s> <s xml:id="echoid-s7920" xml:space="preserve"> & ab alio uiſu comprehendetur <lb/>imago a in eodem perpendicularis puncto.</s> <s xml:id="echoid-s7921" xml:space="preserve"> Quare & ſic utriq;</s> <s xml:id="echoid-s7922" xml:space="preserve"> uiſui unica tantùm apparet image <lb/>puncti a, & in eodem eiuſdem perpendicularis puncto.</s> <s xml:id="echoid-s7923" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s7924" xml:space="preserve"/> </p> <div xml:id="echoid-div298" type="float" level="0" n="0"> <figure xlink:label="fig-0139-02" xlink:href="fig-0139-02a"> <variables xml:id="echoid-variables33" xml:space="preserve">b a g q t d z e h</variables> </figure> <figure xlink:label="fig-0139-03" xlink:href="fig-0139-03a"> <variables xml:id="echoid-variables34" xml:space="preserve">a g b e d z t q h</variables> </figure> <figure xlink:label="fig-0140-01" xlink:href="fig-0140-01a"> <variables xml:id="echoid-variables35" xml:space="preserve">b g a t z d h</variables> </figure> </div> </div> <div xml:id="echoid-div300" type="section" level="0" n="0"> <head xml:id="echoid-head319" xml:space="preserve" style="it">16. In ſpeculo ſphærico conuexo linea reflexionis & perpendicularis incidentiæ concurrunt: <lb/>& imago uidetur in ipſarum concurſu. 9. 11 p 6. Idem 3 n.</head> <p> <s xml:id="echoid-s7925" xml:space="preserve">IN ſpeculis ſphæricis extrà politis patebit, quod diximus.</s> <s xml:id="echoid-s7926" xml:space="preserve"> Sit a punctum uiſum:</s> <s xml:id="echoid-s7927" xml:space="preserve"> b cẽtrum uiſus:</s> <s xml:id="echoid-s7928" xml:space="preserve"> <lb/>g punctũ reflexionis.</s> <s xml:id="echoid-s7929" xml:space="preserve"> Palàm [per 23.</s> <s xml:id="echoid-s7930" xml:space="preserve"> 13 n 4] quòd b g, a g ſunt in eadem ſuperficie orthogona-<lb/>lι ſuper ſuperficiem ſphæram contingentẽ in puncto g:</s> <s xml:id="echoid-s7931" xml:space="preserve"> linea com <lb/> <anchor type="figure" xlink:label="fig-0140-02a" xlink:href="fig-0140-02"/> munis ſuperficiei reflexionis & fuperficiei ſphæræ eſt circumferen-<lb/>tia [per 1 th 1 ſphær:</s> <s xml:id="echoid-s7932" xml:space="preserve"> uel 25.</s> <s xml:id="echoid-s7933" xml:space="preserve"> uel 45 n 4] & ſit z g q.</s> <s xml:id="echoid-s7934" xml:space="preserve"> Linea contingẽs <lb/>hunc circulum in puncto reflexionis ſit p g e:</s> <s xml:id="echoid-s7935" xml:space="preserve"> perpendicularis ſuper <lb/>hanc lineam ſit h g:</s> <s xml:id="echoid-s7936" xml:space="preserve"> planum, quòd h g perueniet ad centrum ſphæræ.</s> <s xml:id="echoid-s7937" xml:space="preserve"> <lb/>Quod ſi non:</s> <s xml:id="echoid-s7938" xml:space="preserve"> cum linea à centro ſphæræ ducta ad punctum g, ſit e-<lb/>tiam perpendicularis ſuper lineam p g e [per 25 n 4 & 3 d 11:</s> <s xml:id="echoid-s7939" xml:space="preserve">] erit ab <lb/>eodem puncto in eandem partem ducere duas lineas perpendicula-<lb/>res ſuper unam lineam [& ſic pars æquaretur toti, contra 9 ax.</s> <s xml:id="echoid-s7940" xml:space="preserve">] Sit <lb/>autem centrum ſphæræ n:</s> <s xml:id="echoid-s7941" xml:space="preserve"> & ducatur linea à puncto uiſo ad centrum <lb/>ſphæræ, ſcilicet a n:</s> <s xml:id="echoid-s7942" xml:space="preserve"> quæ quidem erit perpendicularis ſuper ſuperfi-<lb/>ciem, contingentem ſphæram in puncto ſphæræ, per quod tranſit <lb/>[per 25 n 4.</s> <s xml:id="echoid-s7943" xml:space="preserve">] Et quoniam planum eſt, quòd b g ſecat ſphęram:</s> <s xml:id="echoid-s7944" xml:space="preserve"> cum <lb/>ſit inter h g, g p, quæ continent rectum angulum:</s> <s xml:id="echoid-s7945" xml:space="preserve"> concurret cum li-<lb/>nea a n:</s> <s xml:id="echoid-s7946" xml:space="preserve"> Et cũ perpẽdicularis h g ſit in ſuքficie reflexiõis [per 23 n 4] <lb/>erit centrum ſphæræ in eadem [per 1 p 11:</s> <s xml:id="echoid-s7947" xml:space="preserve"> quia h g continuata cadit <lb/>in n centrum ſphæræ, ut patuit] & ita a n in eadem ſuperficie cum <lb/>h g.</s> <s xml:id="echoid-s7948" xml:space="preserve"> Sit ergo concurſus b g cum a n, punctum d.</s> <s xml:id="echoid-s7949" xml:space="preserve"> Planum [per 3 n] <lb/>quòd d erit locus imaginis.</s> <s xml:id="echoid-s7950" xml:space="preserve"> Et hæc quidem intelligenda ſunt, quan-<lb/>do linea ducta à puncto uiſo ad centrum uiſus, non fuerit perpendi-<lb/>cularis ſuper ſpeculum [uiſu enim & uiſibili in recta linea perpendiculari ſuper ſpeculum colloca-<lb/>tis, reflexio fit per eandem perpendicularem, per 11 n 4.</s> <s xml:id="echoid-s7951" xml:space="preserve">]</s> </p> <div xml:id="echoid-div300" type="float" level="0" n="0"> <figure xlink:label="fig-0140-02" xlink:href="fig-0140-02a"> <variables xml:id="echoid-variables36" xml:space="preserve">a h b e g p d z n q</variables> </figure> </div> </div> <div xml:id="echoid-div302" type="section" level="0" n="0"> <head xml:id="echoid-head320" xml:space="preserve" style="it">17. Finis contingentiæ in ſpeculo ſphærico, eſt concurſ{us} rectæ ſpeculum in reflexionis puncto <lb/>tangentis, cum perpendiculari incidentiæ uel reflexionis. Et rect a à centro ſpeculi ſphærici <lb/>conuexi ad imaginem, maior est recta ab imagine ad reflexionis punctum ducta. In def. 13 p 6.</head> <p> <s xml:id="echoid-s7952" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s7953" xml:space="preserve"> linea p g e ſecat lineam a n:</s> <s xml:id="echoid-s7954" xml:space="preserve"> ſit punctum ſectionis e:</s> <s xml:id="echoid-s7955" xml:space="preserve"> & dicitur punctum iſtud finis <lb/>contingentiæ.</s> <s xml:id="echoid-s7956" xml:space="preserve"> Dico, quòd in hoc ſitu linea à centro ſphæræ a d locum imaginis ducta, ma-<lb/>ior eſt linea, à loco imaginis ducta ad locum reflexionis, id eſt d n maior d g.</s> <s xml:id="echoid-s7957" xml:space="preserve"> Quoniam e-<lb/>nim angulus b g h eſt æqualis angulo h g a [ut demonſtratum eſt 13 n] ſed [per 15 p 1] angulus <lb/>b g h æqualis eſt angulo n g d:</s> <s xml:id="echoid-s7958" xml:space="preserve"> ergo [per 1 ax] angulus h g a æqualis eſt eidem:</s> <s xml:id="echoid-s7959" xml:space="preserve"> & e g perpendicu-<lb/>laris ſuper h g n [per fabricationem.</s> <s xml:id="echoid-s7960" xml:space="preserve">] Quare [per 3 ax] angulus a g æqualis eſt angulo e g d.</s> <s xml:id="echoid-s7961" xml:space="preserve"> Igi-<lb/>tur [per 3 p 6] proportio a g ad g d, ſicut a e ad e d.</s> <s xml:id="echoid-s7962" xml:space="preserve"> Protrahatur à puncto a æquidiſtans ipſi d g <lb/> <pb o="135" file="0141" n="141" rhead="OPTICAE LIBER V."/> [per 31 p 1] & concurrat cum linea h n in puncto h [cõcurret autem per lemma Procli ad 29 p 1.</s> <s xml:id="echoid-s7963" xml:space="preserve">] <lb/>Erit igitur [per 29 p 1] angulus n g d æqualis angulo g h a:</s> <s xml:id="echoid-s7964" xml:space="preserve"> ſed an-<lb/> <anchor type="figure" xlink:label="fig-0141-01a" xlink:href="fig-0141-01"/> gulus n g d æqualis eſt angulo a g h [ergo per 1 ax angulus g h a <lb/>æqualis eſt angulo a g h.</s> <s xml:id="echoid-s7965" xml:space="preserve">] Quare [per 6 p 1] duo latera a g, h a <lb/>ſunt æqualia.</s> <s xml:id="echoid-s7966" xml:space="preserve"> Igitur [per 7 p 5] proportio a h ad g d, ſicut a g ad <lb/>eandem.</s> <s xml:id="echoid-s7967" xml:space="preserve"> Sed proportio a h ad g d, ſicut a n ad d n [per 4 p 6:</s> <s xml:id="echoid-s7968" xml:space="preserve"> <lb/>ſunt enim triangula a h n, d g n æquiangula per 29 p 1, & quia an-<lb/>gulus ad n communis eſt utrique triangulo.</s> <s xml:id="echoid-s7969" xml:space="preserve">] Quare [per 11 p 5] <lb/>a n ad d n, ſicut a g ad g d:</s> <s xml:id="echoid-s7970" xml:space="preserve"> Igitur [per 16 p 5] proportio a n ad <lb/>a g:</s> <s xml:id="echoid-s7971" xml:space="preserve"> ſicut d n ad d g:</s> <s xml:id="echoid-s7972" xml:space="preserve"> Sed a n eſt maior a g:</s> <s xml:id="echoid-s7973" xml:space="preserve"> [per 19 p 1] quia reſpicit <lb/>angulum maiorem recto in triangulo a g n [rectus enim eſt, ut pa-<lb/>tuit, e g n.</s> <s xml:id="echoid-s7974" xml:space="preserve">] Igitur d n maior d g:</s> <s xml:id="echoid-s7975" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s7976" xml:space="preserve"/> </p> <div xml:id="echoid-div302" type="float" level="0" n="0"> <figure xlink:label="fig-0141-01" xlink:href="fig-0141-01a"> <variables xml:id="echoid-variables37" xml:space="preserve">h a b e g p d z n q</variables> </figure> </div> </div> <div xml:id="echoid-div304" type="section" level="0" n="0"> <head xml:id="echoid-head321" xml:space="preserve" style="it">18. Si in ſpeculo ſphærico conuexo perpendicularis incidentiæ <lb/>ſecetur à lineis reflexionis: & ſpeculum in reflexionis puncto tan-<lb/>gente: erit, ut tota perpendicularis ad inferum ſegmentum: ſic ſu-<lb/>perum ad intermedium. Et pars perpendicularis inter punctum <lb/>contingentiæ, & peripheriam, communem ſectionem ſuperficie-<lb/>rum reflexionis, & ſpeculi, erit minor eiuſdem peripheriæ ſemidia <lb/>metro. 12. 14 p 6.</head> <p> <s xml:id="echoid-s7977" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s7978" xml:space="preserve"> dico quòd linea ducta à fine contingentiæ, qui eſt e, uſque ad ſphæram perpendicu <lb/>lariter, id eſt e f, pars lineæ e n minor eſt ſemidiametro.</s> <s xml:id="echoid-s7979" xml:space="preserve"> Sit f punctum, in quo a n ſecat ſu-<lb/>perficiem ſphæræ.</s> <s xml:id="echoid-s7980" xml:space="preserve"> Dico ergo, quòd e f minor eſt n f.</s> <s xml:id="echoid-s7981" xml:space="preserve"> Quo <lb/> <anchor type="figure" xlink:label="fig-0141-02a" xlink:href="fig-0141-02"/> niam ut dictum eſt [proximo numero] proportio a g ad g d, ſicut <lb/>a e ad e d:</s> <s xml:id="echoid-s7982" xml:space="preserve"> ſed a n ad d n, ſicut a g ad g d:</s> <s xml:id="echoid-s7983" xml:space="preserve"> Igitur [per 11 p 5] a n <lb/>ad d n, ſicut a e ad e d:</s> <s xml:id="echoid-s7984" xml:space="preserve"> Igitur [per 16 p 5] a n ad a e, ſicut d n ad <lb/>d e:</s> <s xml:id="echoid-s7985" xml:space="preserve"> ſed [per 9 ax] a n maior a e.</s> <s xml:id="echoid-s7986" xml:space="preserve"> Quare d n maior d e:</s> <s xml:id="echoid-s7987" xml:space="preserve"> quare <lb/>d n maior d f:</s> <s xml:id="echoid-s7988" xml:space="preserve"> quare n f maior e f:</s> <s xml:id="echoid-s7989" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s7990" xml:space="preserve"/> </p> <div xml:id="echoid-div304" type="float" level="0" n="0"> <figure xlink:label="fig-0141-02" xlink:href="fig-0141-02a"> <variables xml:id="echoid-variables38" xml:space="preserve">a h b e g p f d z n q</variables> </figure> </div> </div> <div xml:id="echoid-div306" type="section" level="0" n="0"> <head xml:id="echoid-head322" xml:space="preserve" style="it">19. Sirecta linea ab uno uiſu ſit perpendicularis ſpeculo ſphæ-<lb/>rico conuexo: unum ipſi{us} punctum, in quo uiſ{us} ſuperficiem ſe-<lb/>cat, ab uno ſpeculi puncto, in quod cadit, ad eundem uiſum refle-<lb/>ctetur. 10 p 6.</head> <p> <s xml:id="echoid-s7991" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s7992" xml:space="preserve"> ſit g centrum uiſus:</s> <s xml:id="echoid-s7993" xml:space="preserve"> d centrum ſphæræ:</s> <s xml:id="echoid-s7994" xml:space="preserve"> d z g per-<lb/>pendicularis à centro uiſus a d ſphæram.</s> <s xml:id="echoid-s7995" xml:space="preserve"> Dico, quòd nullius <lb/>puncti forma reflectitur per hãc perpendicularem, niſi pun-<lb/>cti eius, quod eſt in ſuperficie uiſus.</s> <s xml:id="echoid-s7996" xml:space="preserve"> Punctorum enim formæ poſt <lb/>centrum uiſus ſum ptorum non reflectuntur per eam, propter cauſ-<lb/>ſam ſupradictam [13 n.</s> <s xml:id="echoid-s7997" xml:space="preserve">] Similiter nec puncta inter ſuperficiem ui-<lb/>ſus & ſpeculum ſumpta.</s> <s xml:id="echoid-s7998" xml:space="preserve"> Dico etiam, quòd nullum punctum huius <lb/>perpendicularis reflectitur ab alio puncto ſpeculi.</s> <s xml:id="echoid-s7999" xml:space="preserve"> Si enim dicatur, <lb/>quòd ab alio puncto:</s> <s xml:id="echoid-s8000" xml:space="preserve"> ſit illud punctum a:</s> <s xml:id="echoid-s8001" xml:space="preserve"> erit <lb/> <anchor type="figure" xlink:label="fig-0141-03a" xlink:href="fig-0141-03"/> linea g a linea reflexionis:</s> <s xml:id="echoid-s8002" xml:space="preserve"> & à puncto illo in-<lb/>telligamus lineam ad a, quæ eſt linea, per quã <lb/>mouetur forma:</s> <s xml:id="echoid-s8003" xml:space="preserve"> & includunt hæ duæ lineæ <lb/>angulum ſuper a:</s> <s xml:id="echoid-s8004" xml:space="preserve"> quem quidem angulum ne-<lb/>ceſſariò diuidet per æqualia diameter d a, cum <lb/>ſit perpẽdicularis ſuper punctum a.</s> <s xml:id="echoid-s8005" xml:space="preserve"> Quia per-<lb/>pendicularis diuidit angulum ex linea motus <lb/>formę & linea reflexiõis, per ęqua [per 13 n 4.</s> <s xml:id="echoid-s8006" xml:space="preserve">] <lb/>Etita diameter d a concurret cum perpendicu <lb/>lari g d, inter punctum ſumptum & g.</s> <s xml:id="echoid-s8007" xml:space="preserve"> Et ita <lb/>duæ lineæ rectæ in duobus punctis concur-<lb/>rent, & ſuperficiem includent [contra 12 ax:</s> <s xml:id="echoid-s8008" xml:space="preserve">] Reſtat ergo, ut ſolius puncti, quod eſt in ſuperficie <lb/>uiſus, forma reflectatur à ſpeculo per perpendicularem, & uideatur in proprio imaginis loco, pro-<lb/>pter eius cum alijs punctis continuitatem.</s> <s xml:id="echoid-s8009" xml:space="preserve"/> </p> <div xml:id="echoid-div306" type="float" level="0" n="0"> <figure xlink:label="fig-0141-03" xlink:href="fig-0141-03a"> <variables xml:id="echoid-variables39" xml:space="preserve">x e g k z a d</variables> </figure> </div> </div> <div xml:id="echoid-div308" type="section" level="0" n="0"> <head xml:id="echoid-head323" xml:space="preserve" style="it">20. Sipars lineæ reflexionis, intra peripheriam circuli (qui eſt communis ſectio ſuperficie-<lb/>rum reflexionis & ſpeculi ſphærici conuexi) continuatæ, æquetur ſemidiametro eiuſdem peri-<lb/>pheriæ: imago intra ſpeculum uidebitur. 24 p 6.</head> <p> <s xml:id="echoid-s8010" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s8011" xml:space="preserve"> g a, g b ſint lineæ à centro uiſus ductæ, contingentes ſphæram:</s> <s xml:id="echoid-s8012" xml:space="preserve"> & ſignetur circulus, ſu <lb/> <pb o="136" file="0142" n="142" rhead="ALHAZEN"/> per quem ſuperficies his lineis incluſa ſecat ſphæram:</s> <s xml:id="echoid-s8013" xml:space="preserve"> erit [per 25 n 4] a b portio apparens ex <lb/>hoc circulo.</s> <s xml:id="echoid-s8014" xml:space="preserve"> Dico ergo, quòd loca imaginum, quæ per reflexiones ab hac portione factas compre-<lb/>henduntur:</s> <s xml:id="echoid-s8015" xml:space="preserve"> quædam ſunt intra ſpeculum:</s> <s xml:id="echoid-s8016" xml:space="preserve"> quędam in ſu <lb/> <anchor type="figure" xlink:label="fig-0142-01a" xlink:href="fig-0142-01"/> perficie ſpeculi:</s> <s xml:id="echoid-s8017" xml:space="preserve"> quędam extra ſpeculũ.</s> <s xml:id="echoid-s8018" xml:space="preserve"> Et unumquod-<lb/>que horum eſt determinandum.</s> <s xml:id="echoid-s8019" xml:space="preserve"> Ducatur à puncto g li-<lb/>nea ſecans circulum, & pars eius, quæ eſt chorda arcus <lb/>circuli, ſit æqualis ſemidiametro circuli [id quod per 1 <lb/>p 4 fieri poteſt:</s> <s xml:id="echoid-s8020" xml:space="preserve">] ſit linea illa g h k:</s> <s xml:id="echoid-s8021" xml:space="preserve"> & chorda æqualis <lb/>ſemidiametro ſit h k:</s> <s xml:id="echoid-s8022" xml:space="preserve"> & producatur à puncto h perpen-<lb/>dicularis, quæ ſit d h m.</s> <s xml:id="echoid-s8023" xml:space="preserve"> Dico, quòd formæ reflexę à pun <lb/>cto h locus eſt intra ſphęram.</s> <s xml:id="echoid-s8024" xml:space="preserve"> Ducatur [per 23 p 1] à pun <lb/>cto h linea æ qualem renens angulum cum m h, angulo <lb/>m h g:</s> <s xml:id="echoid-s8025" xml:space="preserve"> & ſit p h:</s> <s xml:id="echoid-s8026" xml:space="preserve"> reflectentur quidem puncta huius lineæ <lb/>à puncto h ad uiſum g, & nõ alterius [per 12 n 4.</s> <s xml:id="echoid-s8027" xml:space="preserve">] Suma <lb/>tur ergo aliquod eius punctum:</s> <s xml:id="echoid-s8028" xml:space="preserve"> & ſit p:</s> <s xml:id="echoid-s8029" xml:space="preserve"> & ducatur ab eo <lb/>linea ad centrum ſphærę quę ſit p d:</s> <s xml:id="echoid-s8030" xml:space="preserve"> erit [ut demonſtra <lb/>tum eſt 25 n 4] p d perpendicularis ſuper ſuperficiem, <lb/>contingentem ſphæram ſuper punctum eius, per quod <lb/>tranſit p d:</s> <s xml:id="echoid-s8031" xml:space="preserve"> & coniungatur d k.</s> <s xml:id="echoid-s8032" xml:space="preserve"> Verùm angulus p h m eſt <lb/>æqualis angulo m h g [ex fabricatione.</s> <s xml:id="echoid-s8033" xml:space="preserve">] Quare [per 15 <lb/>p 1] ſimiliter æqualis eſt angulo contrapoſito k h d:</s> <s xml:id="echoid-s8034" xml:space="preserve"> ſed <lb/>[per hypotheſim & 5 p 1] k h d eſt æqualis k d h:</s> <s xml:id="echoid-s8035" xml:space="preserve"> quoni-<lb/>am reſpiciunt æqualia latera:</s> <s xml:id="echoid-s8036" xml:space="preserve"> Igitur [per 1 ax:</s> <s xml:id="echoid-s8037" xml:space="preserve">] angulus <lb/>p h m æ qualis eſt angulo k d m.</s> <s xml:id="echoid-s8038" xml:space="preserve"> Quare [per 28 p 1] lineę <lb/>k d, p h ſunt ęquidiſtantes:</s> <s xml:id="echoid-s8039" xml:space="preserve"> ergo [per 35 def 1] in infi-<lb/>nitum productę nun quam concurrent:</s> <s xml:id="echoid-s8040" xml:space="preserve"> & linea p d ſeca-<lb/>bit lineam, interiacentem inter k d, & p h [quia ſecat an-<lb/>gulum h d k ipſi h k ſubtenſum.</s> <s xml:id="echoid-s8041" xml:space="preserve">] Et ita quodcunq;</s> <s xml:id="echoid-s8042" xml:space="preserve"> pun-<lb/>ctum ſumatur in linea p h:</s> <s xml:id="echoid-s8043" xml:space="preserve"> linea ducta ab illo puncto, ad <lb/>punctum d, ſecabit lineam reflexionis intra ſphęram:</s> <s xml:id="echoid-s8044" xml:space="preserve"> quę quidem linea perpendicularis erit ſuper <lb/>ſphęram [per 25 n 4] ſicut eſt p d.</s> <s xml:id="echoid-s8045" xml:space="preserve"> Quare imago cuiuſcunque puncti lineę p h apparebit intra ſphę <lb/>ram [per 3 n.</s> <s xml:id="echoid-s8046" xml:space="preserve">]</s> </p> <div xml:id="echoid-div308" type="float" level="0" n="0"> <figure xlink:label="fig-0142-01" xlink:href="fig-0142-01a"> <variables xml:id="echoid-variables40" xml:space="preserve">g m h z p b d a k</variables> </figure> </div> </div> <div xml:id="echoid-div310" type="section" level="0" n="0"> <head xml:id="echoid-head324" xml:space="preserve" style="it">21. Si reflexio fiat à peripheria circuli (qui eſt communis ſectio ſuperficierum, reflexionis & <lb/>ſpeculi ſphærici conuexi) inter rectam à uiſu ad ſpeculi centrum ductam, & lineam reflexionis, <lb/>æquantem partem ſuam intra peripheriam, eiuſdem ſemidiametro: imago intra ſpeculum ui-<lb/>debitur. 25 p 6.</head> <p> <s xml:id="echoid-s8047" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s8048" xml:space="preserve"> arcus circuli interiacens inter punctum h, & punctum, per quod tranſit perpendi-<lb/>cularis à centro uiſus ducta:</s> <s xml:id="echoid-s8049" xml:space="preserve"> eſto h z.</s> <s xml:id="echoid-s8050" xml:space="preserve"> Dico, quod <lb/> <anchor type="figure" xlink:label="fig-0142-02a" xlink:href="fig-0142-02"/> à quocunq, puncto huius arcus fiat reflexio:</s> <s xml:id="echoid-s8051" xml:space="preserve"> lo-<lb/>cus imaginis erit intra ſphæram.</s> <s xml:id="echoid-s8052" xml:space="preserve"> Sit i punctum ſumptũ:</s> <s xml:id="echoid-s8053" xml:space="preserve"> <lb/>& ducatur linea à centro uiſus ſecans cιrculũ ſuper pun-<lb/>ctum illud, quę ſit g is:</s> <s xml:id="echoid-s8054" xml:space="preserve"> & ducatur perpendicularis per <lb/>punctum hoc, quę ſit d i t:</s> <s xml:id="echoid-s8055" xml:space="preserve"> & [per 23 p 1] fiat linea p i, æ-<lb/>qualem tenens angulum cum it angulo tig.</s> <s xml:id="echoid-s8056" xml:space="preserve"> Palàm [per <lb/>12 n 4] quòd ſola puncta lineę p i reflectuntur à puncto <lb/>iad uiſum.</s> <s xml:id="echoid-s8057" xml:space="preserve"> Palàm etiam [per 15 p 3] quòd linea i s ma-<lb/>ior eſt linea k h.</s> <s xml:id="echoid-s8058" xml:space="preserve"> Quare maior s d [eſt enim h k ex prima <lb/>hypotheſi ęqualis ſemidiametro s d.</s> <s xml:id="echoid-s8059" xml:space="preserve">] Igitur [per 18 p 1] <lb/>angulus s d i maior eſt angulo s i d:</s> <s xml:id="echoid-s8060" xml:space="preserve"> quare [per 15 p 1] <lb/>eſt maior angulo g i t:</s> <s xml:id="echoid-s8061" xml:space="preserve"> quare eſt maior angulo tip.</s> <s xml:id="echoid-s8062" xml:space="preserve"> Igitur <lb/>lineę p i & s d nunquam concurrent [ad partes p & s:</s> <s xml:id="echoid-s8063" xml:space="preserve"> <lb/>ſecus ſpatium comprehenderent contra 12 ax.</s> <s xml:id="echoid-s8064" xml:space="preserve"> quia con-<lb/>currunt ad partes i & d per 11 ax.</s> <s xml:id="echoid-s8065" xml:space="preserve">] Et linea ducta à pun-<lb/>cto quocunque p i lineę, ad punctum d, ſecat lineam s i <lb/>intra ſphęram:</s> <s xml:id="echoid-s8066" xml:space="preserve"> quę s i eſt linea reflexionis:</s> <s xml:id="echoid-s8067" xml:space="preserve"> & omnis <lb/>linea ducta à quocunq;</s> <s xml:id="echoid-s8068" xml:space="preserve"> puncto p i lineę, ad punctum d:</s> <s xml:id="echoid-s8069" xml:space="preserve"> <lb/>erit perpendicularis ſuper ſphęram [ut oſtenſum eſt 25 <lb/>n 4,] ſicut eſt p d.</s> <s xml:id="echoid-s8070" xml:space="preserve"> Et cum locus imaginis ſit in concur-<lb/>ſu perpendicularis à puncto uiſo & lineę reflexionis:</s> <s xml:id="echoid-s8071" xml:space="preserve"> <lb/>[per 3n] erit imago cuiuslibet puncti lineę p i intra <lb/>ſphę<gap/>a n.</s> <s xml:id="echoid-s8072" xml:space="preserve"> Palàm ergo, quòd omnium imaginum arcus <lb/>hz, locus proprius erit intra ſpeculum:</s> <s xml:id="echoid-s8073" xml:space="preserve"> Quod <lb/>eſt propoſitum.</s> <s xml:id="echoid-s8074" xml:space="preserve"/> </p> <div xml:id="echoid-div310" type="float" level="0" n="0"> <figure xlink:label="fig-0142-02" xlink:href="fig-0142-02a"> <variables xml:id="echoid-variables41" xml:space="preserve">t g p b h i z d a k s</variables> </figure> </div> <pb o="137" file="0143" n="143" rhead="OPTICAE LIBER V."/> </div> <div xml:id="echoid-div312" type="section" level="0" n="0"> <head xml:id="echoid-head325" xml:space="preserve" style="it">22. Si reflexio fiat à peripheria circuli (qui eſt communis ſectio ſuperficierum reflexionis & <lb/>ſpeculi ſphærici conuexi) inter rectam à uiſu ſpeculum tangentem, reflexionis puncto proxi-<lb/>mam, & lineam reflexionis æquãtem partem ſuam intra peripheriam eiuſdem ſemidiametro: <lb/>imago aliàs intra ſpeculum: aliàs in ſuperficie: aliàs extra uidebitur. 26 p 6. Item 27. 7 p 6.</head> <p> <s xml:id="echoid-s8075" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s8076" xml:space="preserve"> ſumpto quocunque puncto arcus h b:</s> <s xml:id="echoid-s8077" xml:space="preserve"> dico, quòd quędam eius imago erit intra ſpe <lb/>culum:</s> <s xml:id="echoid-s8078" xml:space="preserve"> quædam in ſuperficie ſpeculi:</s> <s xml:id="echoid-s8079" xml:space="preserve"> quædam extra ſpeculum.</s> <s xml:id="echoid-s8080" xml:space="preserve"> Sumatur aliquod eius pun-<lb/>ctum:</s> <s xml:id="echoid-s8081" xml:space="preserve"> & ſit n:</s> <s xml:id="echoid-s8082" xml:space="preserve"> & ducatur linea à pũcto g ſe-<lb/> <anchor type="figure" xlink:label="fig-0143-01a" xlink:href="fig-0143-01"/> cans circulum, quæ ſit g n q:</s> <s xml:id="echoid-s8083" xml:space="preserve"> & ducatur perpen-<lb/>dicularis d n f:</s> <s xml:id="echoid-s8084" xml:space="preserve"> & [per 23 p 1] protrahatur linea, <lb/>æqualem angulum tenens cum perpendiculari, <lb/>angulo f n g:</s> <s xml:id="echoid-s8085" xml:space="preserve"> & ſit e n.</s> <s xml:id="echoid-s8086" xml:space="preserve"> Quoniam linea n q minor <lb/>eſt k h [per 15 p 3] eſt etiam minor linea q d [nã <lb/>h k poſita eſt æqualis ſemidiametro ſphæræ] & <lb/>ita [per 18 p 1] q d n angulus minor eſt angu-<lb/>lo d n q:</s> <s xml:id="echoid-s8087" xml:space="preserve"> quare [per 15 p 1] minor angulo g <lb/>n f:</s> <s xml:id="echoid-s8088" xml:space="preserve"> quare etiam minor angulo e n f:</s> <s xml:id="echoid-s8089" xml:space="preserve"> Igitur e n <lb/>& d q concurrent [ad partes e & q per 11 ax.</s> <s xml:id="echoid-s8090" xml:space="preserve">] <lb/>Sit ergo concurſus in puncto e.</s> <s xml:id="echoid-s8091" xml:space="preserve"> Palàm [per 25 <lb/>n 4] quòd linea e q d eſt perpendicularis ſuper <lb/>ſphæram:</s> <s xml:id="echoid-s8092" xml:space="preserve"> & ſecat lineam g n q, quæ eſt linea re-<lb/>flexionis, in puncto q, quod eſt punctum ſphæ-<lb/>ræ.</s> <s xml:id="echoid-s8093" xml:space="preserve"> Quare imago puncti e, cum fuerit reflexio ſu-<lb/>per punctum n, apparebit in puncto q:</s> <s xml:id="echoid-s8094" xml:space="preserve"> [per 3 n] <lb/>& eſt in ſuperficie ſphæræ.</s> <s xml:id="echoid-s8095" xml:space="preserve"> Si uerò in linea n e ſu-<lb/>matur punctum ultra e, utpoter:</s> <s xml:id="echoid-s8096" xml:space="preserve"> perpẽdicularis <lb/>ducta ab eo ad centrum ſphæræ, quæ ſit r d, ſeca <lb/>bit lineam g n q reflexionis, ultra punctum q:</s> <s xml:id="echoid-s8097" xml:space="preserve"> & <lb/>eſt extra ſphæram.</s> <s xml:id="echoid-s8098" xml:space="preserve"> Quare imago cuiuslibet pun-<lb/>cti lineæ e n ultra e ſumpti, erit extra ſuperficiem <lb/>ſpeculi.</s> <s xml:id="echoid-s8099" xml:space="preserve"> Si uerò in linea e n, citra punctum e ſu-<lb/>matur aliquod punctum:</s> <s xml:id="echoid-s8100" xml:space="preserve"> perpendicularis ab eo <lb/>ducta ad ſpeculum, ſecabit lineam g n q intra ſphæram:</s> <s xml:id="echoid-s8101" xml:space="preserve"> quoniam in puncto, quod eſt inter n & q.</s> <s xml:id="echoid-s8102" xml:space="preserve"> <lb/>Quare imago cuiuslibet puncti lineæ e n inter e & n ſumpti, apparebit intra ſphæram.</s> <s xml:id="echoid-s8103" xml:space="preserve"> Eadem peni <lb/>tus erit probatio, ſumpto quocunque alio arcus b h puncto:</s> <s xml:id="echoid-s8104" xml:space="preserve"> & ita imago cuiuslibet puncti arcus <lb/>b h una ſola eſt imago in ſuperficie ſpeculi:</s> <s xml:id="echoid-s8105" xml:space="preserve"> aliarum quædam in ſpeculo:</s> <s xml:id="echoid-s8106" xml:space="preserve"> quædam extra.</s> <s xml:id="echoid-s8107" xml:space="preserve"> Et quod <lb/>demonſtratum eſt in arcu z b, eodem modo poteſt patére in arcu z a:</s> <s xml:id="echoid-s8108" xml:space="preserve"> & eadem penitus erit demon-<lb/>ſtratio, cuiuſcunque circuli ſphæræ ſumatur portio, uiſui oppoſita, à perpendiculari g d æqualiter <lb/>diuiſa.</s> <s xml:id="echoid-s8109" xml:space="preserve"> Vnde uiſu immoto, & perpendiculari g z d manente, ſi moueatur æquidiſtanter perpendi-<lb/>culari uiſus linea g h, ſecabit ex ſphæra motu ſuo portionem circularem:</s> <s xml:id="echoid-s8110" xml:space="preserve"> & cuiuslibet puncti hu-<lb/>ius portionis imago apparebit intra ſphæram.</s> <s xml:id="echoid-s8111" xml:space="preserve"> Si uerò linea g b contingens, moueatur æquidiſtan-<lb/>ter perpendiculari uiſus, ſecabit ex ſphæra portionem prædicta maiorem:</s> <s xml:id="echoid-s8112" xml:space="preserve"> & à quolibet puncto <lb/>excrementi unius portionis ſuper aliam reflectitur imago, cuius locus erit in ſuperficie ſphæ-<lb/>ræ:</s> <s xml:id="echoid-s8113" xml:space="preserve"> & aliarum quædam intra ſphæram:</s> <s xml:id="echoid-s8114" xml:space="preserve"> quædam extra.</s> <s xml:id="echoid-s8115" xml:space="preserve"> Scimus ex his, quòd in hoc ſpeculo quæli-<lb/>bet imago apparet in diametro ſphæræ:</s> <s xml:id="echoid-s8116" xml:space="preserve"> aut intra ſphærã:</s> <s xml:id="echoid-s8117" xml:space="preserve"> aut extra:</s> <s xml:id="echoid-s8118" xml:space="preserve"> aut in ſuperficie.</s> <s xml:id="echoid-s8119" xml:space="preserve"> Et omnis dia-<lb/>meter, in qua apparet imago aliqua in ſuperficie ſphæræ, aut extra, demiſsior eſt puncto ſphæræ, <lb/>quod tangit linea contingens à centro uiſus, ducta in ultimum punctum portionis apparentis.</s> <s xml:id="echoid-s8120" xml:space="preserve"> Sci <lb/>mus etiam, quòd quælibet linea reflexionis ſecat ſphæram in duobus punctis, in puncto reflexio-<lb/>nis, & in alio.</s> <s xml:id="echoid-s8121" xml:space="preserve"> Reſtatiam, ut loca imaginum certius determinemus.</s> <s xml:id="echoid-s8122" xml:space="preserve"/> </p> <div xml:id="echoid-div312" type="float" level="0" n="0"> <figure xlink:label="fig-0143-01" xlink:href="fig-0143-01a"> <variables xml:id="echoid-variables42" xml:space="preserve">g z f h a b d c q e k ſ r</variables> </figure> </div> </div> <div xml:id="echoid-div314" type="section" level="0" n="0"> <head xml:id="echoid-head326" xml:space="preserve" style="it">23. Si linea reflexionis ſecans diametrum ſpeculi ſphærici conuexi: æquet ſegmentum ſuum <lb/>inter ſpeculi ſuperficiem & dictam diametrum, ſegmento eiuſdem diametri contermino centro <lb/>ſpeculi: erit hoc ſegmentum imaginum expers. 28 p 6.</head> <p> <s xml:id="echoid-s8123" xml:space="preserve">DIco, quòd ſumpta diametro, ſi ad ipſam ducatur linea ſecans ſphæram à centro uiſus, cuius <lb/>pars interiacens punctum ſectionis ſphæræ & punctum diametri, quam attingit, eſt æqua-<lb/>lis parti diametri, interiacenti inter punctum illud & centrum:</s> <s xml:id="echoid-s8124" xml:space="preserve"> punctum illud non eſt locus <lb/>alicuius imaginis.</s> <s xml:id="echoid-s8125" xml:space="preserve"> Verbi gratia.</s> <s xml:id="echoid-s8126" xml:space="preserve"> ſit a g circulus ſphæræ:</s> <s xml:id="echoid-s8127" xml:space="preserve"> h uiſus:</s> <s xml:id="echoid-s8128" xml:space="preserve"> e d ſemidiameter ſphæræ, ſiue per-<lb/>pendicularis:</s> <s xml:id="echoid-s8129" xml:space="preserve"> & h z ſit linea ſecans ſphæram ſuper punctum f, & concurrens cum e d in puncto z:</s> <s xml:id="echoid-s8130" xml:space="preserve"> & <lb/>ſit z f æqualis z d.</s> <s xml:id="echoid-s8131" xml:space="preserve"> Dico, quòd z non eſt locus alicuius imaginis.</s> <s xml:id="echoid-s8132" xml:space="preserve"> Palàm enim, quòd nõ eſt locus i-<lb/>maginis alterius, quàm alicuius puncti lineæ e d:</s> <s xml:id="echoid-s8133" xml:space="preserve"> quoniam imago cuiuslibet puncti eſt ſuper dia-<lb/>metrum, ab eo ad centrum ſphæræ ductam [per 10 n.</s> <s xml:id="echoid-s8134" xml:space="preserve">] Et quòd locus imaginis alicuius puncti e d <lb/>non ſit in z:</s> <s xml:id="echoid-s8135" xml:space="preserve"> ſic conſtabit.</s> <s xml:id="echoid-s8136" xml:space="preserve"> Ducatur perpendicularis à puncto d ſuper punctum f:</s> <s xml:id="echoid-s8137" xml:space="preserve"> & ſit d f n:</s> <s xml:id="echoid-s8138" xml:space="preserve"> & [per <lb/>23 p 1] ſuper punctum f fiat angulus æqualis angulo n fh:</s> <s xml:id="echoid-s8139" xml:space="preserve"> & ſit q f n.</s> <s xml:id="echoid-s8140" xml:space="preserve"> Palàm ergo [per 15 p 1.</s> <s xml:id="echoid-s8141" xml:space="preserve"> 1 ax] <lb/> <pb o="138" file="0144" n="144" rhead="ALHAZEN"/> quòd angulus q fn æqualis eſt angulo z f d:</s> <s xml:id="echoid-s8142" xml:space="preserve"> ſed [per hypotheſim, & 5 p 1] z f d eſt æqualis angulo <lb/>z d f:</s> <s xml:id="echoid-s8143" xml:space="preserve"> Igitur [per 1 ax] q f n eſt æqualis angulo z d n.</s> <s xml:id="echoid-s8144" xml:space="preserve"> Quare [per 28 p 1] linea f q eſt æqui-<lb/>diſtans lineę e d.</s> <s xml:id="echoid-s8145" xml:space="preserve"> Igitur [per 35 d 1] in infinitum productæ nun-<lb/> <anchor type="figure" xlink:label="fig-0144-01a" xlink:href="fig-0144-01"/> quam concurrent.</s> <s xml:id="echoid-s8146" xml:space="preserve"> Igitur nullius puncti e d forma mouebitur ad <lb/>punctum f per q f:</s> <s xml:id="echoid-s8147" xml:space="preserve"> non poteſt autem eſſe locus imaginis alicuiu <lb/>puncti in puncto z, niſi forma eius moueatur ad f per lineam q f <lb/>[quia h ex theſi eſt uiſus, & h f linea reflexionis.</s> <s xml:id="echoid-s8148" xml:space="preserve">] Eadem erit pro-<lb/>batio ſumpta quacunque diametro.</s> <s xml:id="echoid-s8149" xml:space="preserve"> Quare patet propoſitum.</s> <s xml:id="echoid-s8150" xml:space="preserve"> Am-<lb/>plius:</s> <s xml:id="echoid-s8151" xml:space="preserve"> dico quòd nullum punctum lineæ z d poteſt eſſe locus ali-<lb/>cuius imaginis.</s> <s xml:id="echoid-s8152" xml:space="preserve"> Sumatur enim punctum p:</s> <s xml:id="echoid-s8153" xml:space="preserve"> & ducatur linea h p, ſe-<lb/>cans ſphæram in puncto b:</s> <s xml:id="echoid-s8154" xml:space="preserve"> & ducatur perpendicularis d b m:</s> <s xml:id="echoid-s8155" xml:space="preserve"> & <lb/>[per 23 p 1] angulo m b h fiat angulus æqualis, qui ſit t b m.</s> <s xml:id="echoid-s8156" xml:space="preserve"> Palàm <lb/>[per 15 p 1.</s> <s xml:id="echoid-s8157" xml:space="preserve"> 1 ax] quòd t.</s> <s xml:id="echoid-s8158" xml:space="preserve"> b m eſt æqualis p b d:</s> <s xml:id="echoid-s8159" xml:space="preserve"> & palàm [per 16 <lb/>p 1] quòd angulus d p h eſt maior angulo p z f:</s> <s xml:id="echoid-s8160" xml:space="preserve"> quia exterior.</s> <s xml:id="echoid-s8161" xml:space="preserve"> Igi-<lb/>tur duo alij anguli trianguli d p b ſunt minores duobus alijs angu-<lb/>lis trianguli z d f [per 32 p 1.</s> <s xml:id="echoid-s8162" xml:space="preserve">] Sed [per 9 ax.</s> <s xml:id="echoid-s8163" xml:space="preserve">] p d b eſt maior an-<lb/>gulo z d f:</s> <s xml:id="echoid-s8164" xml:space="preserve"> reſtat ergo ut angulus d p b ſit minor angulo d f z:</s> <s xml:id="echoid-s8165" xml:space="preserve"> ſed <lb/>angulus d f z eſt æqualis angulo z d f:</s> <s xml:id="echoid-s8166" xml:space="preserve"> [utiam patuit per theſin <lb/>& 5 p 1:</s> <s xml:id="echoid-s8167" xml:space="preserve">] quare angulus d b p minor eſt angulo z d f:</s> <s xml:id="echoid-s8168" xml:space="preserve"> Igitur mul-<lb/>to minor angulo p d b:</s> <s xml:id="echoid-s8169" xml:space="preserve"> ergo t b m minor eſt p d b.</s> <s xml:id="echoid-s8170" xml:space="preserve"> Ergo t b, e d <lb/>nunquam concurrent [ad partes t, e:</s> <s xml:id="echoid-s8171" xml:space="preserve"> & ita nulla forma à puncto <lb/>b reflectetur ad punctum h, ut p ſitlocus imaginis.</s> <s xml:id="echoid-s8172" xml:space="preserve"> ſimiliter neci-<lb/>mago alterius puncti.</s> <s xml:id="echoid-s8173" xml:space="preserve"> Et ſimiliter de quolibet puncto lineæ z d.</s> <s xml:id="echoid-s8174" xml:space="preserve"> Reſtat ergo, ut tota linea z d ſit ua-<lb/>cua à locis imaginum.</s> <s xml:id="echoid-s8175" xml:space="preserve"/> </p> <div xml:id="echoid-div314" type="float" level="0" n="0"> <figure xlink:label="fig-0144-01" xlink:href="fig-0144-01a"> <variables xml:id="echoid-variables43" xml:space="preserve">m t n q h b f e z p d a g</variables> </figure> </div> </div> <div xml:id="echoid-div316" type="section" level="0" n="0"> <head xml:id="echoid-head327" xml:space="preserve" style="it">24. Si in diametro ſpeculi ſphærici conuexi extra uiſ{us} centrum ducta, iń apparentem <lb/>ſuperficiem continuata, imaginum meta notetur: Imagines dictæ diametri uidebuntur inter <lb/>metam & ſpeculi ſuperficiem. 29 p 6.</head> <p> <s xml:id="echoid-s8176" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s8177" xml:space="preserve"> ſumpta quacunque diametro inter lineas contingentiæ à uiſu ad ſphæram ductas, <lb/>præter diametrum à centro uiſus ad centrum ſphæræ intellectam, & determinato in ea pun <lb/>cto, quod diximus, quod eſt meta locorum imaginum.</s> <s xml:id="echoid-s8178" xml:space="preserve"> Dico, quòd in punctis tantùm illius <lb/>diametri, quæ ſunt inter ſuperficiem ſphæræ, & metam prædictam, ſuntloca imaginum, puncto-<lb/>rum illius diametri.</s> <s xml:id="echoid-s8179" xml:space="preserve"> Verbi gratia, ſint b z, b e lineæ contingentes:</s> <s xml:id="echoid-s8180" xml:space="preserve"> b centrum uiſus:</s> <s xml:id="echoid-s8181" xml:space="preserve"> a centrum ſphæ-<lb/>ræ:</s> <s xml:id="echoid-s8182" xml:space="preserve"> b h a diameter uiſualis:</s> <s xml:id="echoid-s8183" xml:space="preserve"> d a diameter ſumpta, cuius meta ſit t:</s> <s xml:id="echoid-s8184" xml:space="preserve"> g punctum ſphærę, in quo dia-<lb/>meter ſecat ſphæram.</s> <s xml:id="echoid-s8185" xml:space="preserve"> Dico, quòd in ſola puncta inter g, t interiacentia, cadunt imagines puncto-<lb/>rum rectæ d a.</s> <s xml:id="echoid-s8186" xml:space="preserve"> Quòd enim non cadant in punctum g, uel extra ſu-<lb/> <anchor type="figure" xlink:label="fig-0144-02a" xlink:href="fig-0144-02"/> perficiem ſphæræ:</s> <s xml:id="echoid-s8187" xml:space="preserve"> palàm per hoc, quod ſuprà dictum eſt [22 n,] <lb/>diametrum, in qua eſt locus imaginis in ſuperficie ſphæræ aut ex-<lb/>tra, demiſsiorem eſſe puncto contingentiæ:</s> <s xml:id="echoid-s8188" xml:space="preserve"> & cum diameter d a ſit <lb/>inter lineas contingentes:</s> <s xml:id="echoid-s8189" xml:space="preserve"> non erit in ea locus imaginis, aut in ſu-<lb/>perficie ſphæræ, aut extra.</s> <s xml:id="echoid-s8190" xml:space="preserve"> Quòd autem in quodlibet punctum in-<lb/>ter g & t ſumptum, cadat imago:</s> <s xml:id="echoid-s8191" xml:space="preserve"> ſic conſtabit.</s> <s xml:id="echoid-s8192" xml:space="preserve"> Sumatur punctum:</s> <s xml:id="echoid-s8193" xml:space="preserve"> <lb/>& ſit q:</s> <s xml:id="echoid-s8194" xml:space="preserve"> & ducatur linea b q, ſecans ſphæram in puncto p:</s> <s xml:id="echoid-s8195" xml:space="preserve"> & duca-<lb/>tur perpendicularis a p l:</s> <s xml:id="echoid-s8196" xml:space="preserve"> & [per 23 p 1] angulo l p b fiat æqualis <lb/>angulus d p l:</s> <s xml:id="echoid-s8197" xml:space="preserve"> & educatur linea b t, ſecans ſphæram in puncto f:</s> <s xml:id="echoid-s8198" xml:space="preserve"> & <lb/>ducatur perpendicularis a f.</s> <s xml:id="echoid-s8199" xml:space="preserve"> Igitur triangulum a p b continet tri-<lb/>angulum a f b:</s> <s xml:id="echoid-s8200" xml:space="preserve"> quare [per 21 p 1] angulas a f b maior eſt angu-<lb/>lo a p b:</s> <s xml:id="echoid-s8201" xml:space="preserve"> reſtat ergo [per 13 p 1] ut angulus a f t ſit minor a p q:</s> <s xml:id="echoid-s8202" xml:space="preserve"> <lb/>ſed angulus a f t eſt æqualis angulo f a t, quia æqualia latera reſpi-<lb/>ciunt:</s> <s xml:id="echoid-s8203" xml:space="preserve"> [per hypotheſin & pręcedentem numerum.</s> <s xml:id="echoid-s8204" xml:space="preserve">] Igitur a p q e-<lb/>rit maior angulo f a t:</s> <s xml:id="echoid-s8205" xml:space="preserve"> ergo & angulo p a q, [per 9 ax.</s> <s xml:id="echoid-s8206" xml:space="preserve">] Quare [per <lb/>15 p 1.</s> <s xml:id="echoid-s8207" xml:space="preserve"> 1 ax.</s> <s xml:id="echoid-s8208" xml:space="preserve">] l p b maior eſt p a q.</s> <s xml:id="echoid-s8209" xml:space="preserve"> Vnde d p l maior p a q:</s> <s xml:id="echoid-s8210" xml:space="preserve"> Igitur p <lb/>d, a q concurrent [per 11 ax.</s> <s xml:id="echoid-s8211" xml:space="preserve">] ſit d concurſus.</s> <s xml:id="echoid-s8212" xml:space="preserve"> Forma igitur pun-<lb/>cti d reflectetur à puncto p per lineam p b:</s> <s xml:id="echoid-s8213" xml:space="preserve"> & locus imaginis eius <lb/>eſt q [per 3 n.</s> <s xml:id="echoid-s8214" xml:space="preserve">] Et eadem eſt probatio, ſumpto quocunque pun-<lb/>cto inter g & t.</s> <s xml:id="echoid-s8215" xml:space="preserve"> Reſtat, utaſsignemus loca imaginum in ſectione ſphæræ occulta uiſui.</s> <s xml:id="echoid-s8216" xml:space="preserve"/> </p> <div xml:id="echoid-div316" type="float" level="0" n="0"> <figure xlink:label="fig-0144-02" xlink:href="fig-0144-02a"> <variables xml:id="echoid-variables44" xml:space="preserve">b ſ d h f r g z q t e a</variables> </figure> </div> </div> <div xml:id="echoid-div318" type="section" level="0" n="0"> <head xml:id="echoid-head328" xml:space="preserve" style="it">25. Si linea reflexionis ſecans ſpeculum ſphæricum conuexum, æquet ſegmentum intra ipſi-<lb/>{us} ſuperficiem, eiuſdem ſemidiametro: & ſemidiameter per terminum lineæ reflexionis con-<lb/>currat cum rect a à uiſu ſpeculum tangente: Imagines concurrentis ſemidiametri, inter concur <lb/>ſum & ſpeculι ſuperficiem uidebuntur. 30 p 6.</head> <p> <s xml:id="echoid-s8217" xml:space="preserve">SInt ergo a c, a g lineæ contin gentes portionem apparentem:</s> <s xml:id="echoid-s8218" xml:space="preserve"> a centrum uiſus:</s> <s xml:id="echoid-s8219" xml:space="preserve"> b centrum ſphæ-<lb/> <pb o="139" file="0145" n="145" rhead="OPTICAE LIBER V."/> ræ:</s> <s xml:id="echoid-s8220" xml:space="preserve"> a d b z diameteruiſualis:</s> <s xml:id="echoid-s8221" xml:space="preserve"> z c g circulus ſphæræ in ſuperficie linearũ contingẽtiæ:</s> <s xml:id="echoid-s8222" xml:space="preserve"> & protrahatur <lb/>à centro ad punctũ contingentiæ diameter b g.</s> <s xml:id="echoid-s8223" xml:space="preserve"> Palàm, quòd angulus z b g eſt maior recto.</s> <s xml:id="echoid-s8224" xml:space="preserve"> Cũ enim <lb/>in triãgulo b a g angulus b g a [per 18 p 3] <lb/> <anchor type="figure" xlink:label="fig-0145-01a" xlink:href="fig-0145-01"/> ſit rectus, erit [per 17 p 1] angulus g b a mi <lb/>nor recto:</s> <s xml:id="echoid-s8225" xml:space="preserve"> quare [per 13 p 1] z b g maior.</s> <s xml:id="echoid-s8226" xml:space="preserve"> <lb/>Sit ergo [per 23 p 1] h b g rectus:</s> <s xml:id="echoid-s8227" xml:space="preserve"> erit ergo <lb/>[per 28 p 1] h b æquidiſtans lineę cõtingẽ <lb/>tię a g:</s> <s xml:id="echoid-s8228" xml:space="preserve"> Igitur [per 35 d 1] productæ nunꝗ̃ <lb/>concurrent:</s> <s xml:id="echoid-s8229" xml:space="preserve"> & quęlibet diameter inter h <lb/>& g concurret cũ linea a g [per lẽma Pro-<lb/>cli ad 29 p 1.</s> <s xml:id="echoid-s8230" xml:space="preserve">] Ducatur à pũcto a linea ſe-<lb/>cans ſphęrã:</s> <s xml:id="echoid-s8231" xml:space="preserve"> quæ ſit a m o:</s> <s xml:id="echoid-s8232" xml:space="preserve"> ita quod chor-<lb/>da, quę eſt m o, ſit ęqualis ſemidiametro <lb/>o b:</s> <s xml:id="echoid-s8233" xml:space="preserve"> & cõcurrat ſemidiameter b o cum li-<lb/>nea a g, in puncto t.</s> <s xml:id="echoid-s8234" xml:space="preserve"> Dico, quòd in quoli-<lb/>bet pũcto t o eſt locus imaginis:</s> <s xml:id="echoid-s8235" xml:space="preserve"> & in nul <lb/>lo alio puncto diametri t b eſt locus ima-<lb/>ginis:</s> <s xml:id="echoid-s8236" xml:space="preserve"> & ſunt o, t termini locorũ imaginũ <lb/>[per 23 n.</s> <s xml:id="echoid-s8237" xml:space="preserve">] Sumatur enim punctũ:</s> <s xml:id="echoid-s8238" xml:space="preserve"> & ſit k:</s> <s xml:id="echoid-s8239" xml:space="preserve"> <lb/>& a n k ducatur ſecans ſphærã in puncto <lb/>n:</s> <s xml:id="echoid-s8240" xml:space="preserve"> & ducatur perpendicularis b n x:</s> <s xml:id="echoid-s8241" xml:space="preserve"> & [ք <lb/>23 p 1] angulo x n a fiat angulus ęqualis per lineam f n.</s> <s xml:id="echoid-s8242" xml:space="preserve"> Palàm, quò d n f nõ cadet inter b, g.</s> <s xml:id="echoid-s8243" xml:space="preserve"> Quoniã ſic <lb/>aut ſecaret ſphæram, aut ſecaret contingentẽ a g in duobus punctis [& ſic duę lineę rectę ſpatiũ cõ-<lb/>prehenderent contra 12 ax.</s> <s xml:id="echoid-s8244" xml:space="preserve">] Igitur forma puncti f mouebitur per f n ad punctum n, & reflectetur ad <lb/>a per lineam a n:</s> <s xml:id="echoid-s8245" xml:space="preserve"> & apparebit imago eius in puncto k [per 3 n.</s> <s xml:id="echoid-s8246" xml:space="preserve">] Et eadem probatio eſt, ſumpto <lb/>quocunque alio puncto.</s> <s xml:id="echoid-s8247" xml:space="preserve"/> </p> <div xml:id="echoid-div318" type="float" level="0" n="0"> <figure xlink:label="fig-0145-01" xlink:href="fig-0145-01a"> <variables xml:id="echoid-variables45" xml:space="preserve">a d q c m x b g p o k t f z h</variables> </figure> </div> </div> <div xml:id="echoid-div320" type="section" level="0" n="0"> <head xml:id="echoid-head329" xml:space="preserve" style="it">26. Si linea reflexionis æquans ſua parte inſcripta ſemidiametrum circuli (qui est communis <lb/>ſectio ſuperficierum reflexionis & ſpeculi ſphærici conuexi) terminetur in peripheria non appa <lb/>rente: perpẽdicularis incidẽtiæ, ſecãs peripheriã inter lineã reflexionis, & rectã à uiſu ſpeculũ <lb/>tangentẽ: habebit quaſdam imagines intra, quaſdam extra ſpeculũ: unam in ſuperficie. 31 p 6.</head> <p> <s xml:id="echoid-s8248" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s8249" xml:space="preserve"> dico, quòd in arcu o g, quęcunque <lb/> <anchor type="figure" xlink:label="fig-0145-02a" xlink:href="fig-0145-02"/> ſumatur diameter, continebit loca imagi-<lb/>num:</s> <s xml:id="echoid-s8250" xml:space="preserve"> & intra ſpeculum quaſdã:</s> <s xml:id="echoid-s8251" xml:space="preserve"> & unã in ſu <lb/>perficie:</s> <s xml:id="echoid-s8252" xml:space="preserve"> & alias extra ſpeculũ.</s> <s xml:id="echoid-s8253" xml:space="preserve"> Sumatur ergo pun-<lb/>ctum l:</s> <s xml:id="echoid-s8254" xml:space="preserve"> & protrahatur diameter b l, quouſq;</s> <s xml:id="echoid-s8255" xml:space="preserve"> ſecet <lb/>a t in puncto e:</s> <s xml:id="echoid-s8256" xml:space="preserve"> & producatur linea a l, ſecans ſphæ <lb/>ram in puncto r.</s> <s xml:id="echoid-s8257" xml:space="preserve"> Palàm, quòd r l minor eſt t b:</s> <s xml:id="echoid-s8258" xml:space="preserve"> quia <lb/>[per 15 p 3] eſt minor m o:</s> <s xml:id="echoid-s8259" xml:space="preserve"> quæ eſt ęqualis ſemidia <lb/>metro [ex theſi.</s> <s xml:id="echoid-s8260" xml:space="preserve">] Si ergo ab a ducatur linea ad dia <lb/>metrum b l:</s> <s xml:id="echoid-s8261" xml:space="preserve"> cuius pars interiacens inter circulũ & <lb/>diametrum, ſit æqualis parti diametri à puncto, in <lb/>quod cadit, uſq;</s> <s xml:id="echoid-s8262" xml:space="preserve"> ad centrũ:</s> <s xml:id="echoid-s8263" xml:space="preserve"> cadet inter l & b.</s> <s xml:id="echoid-s8264" xml:space="preserve"> Si e-<lb/>nim inter l & e ceciderit:</s> <s xml:id="echoid-s8265" xml:space="preserve"> erit r l maior l b:</s> <s xml:id="echoid-s8266" xml:space="preserve"> oĩs enim <lb/>linea interiacens inter centrũ, & illam partẽ lineæ <lb/>reflexionis, illi parti diametri ęqualem:</s> <s xml:id="echoid-s8267" xml:space="preserve"> erit maior <lb/>parte diametri, qua terminatur, ſecundum proba-<lb/>tionem aſsignatam in explanatione metæ imagi-<lb/>num [23 & proximo numeris.</s> <s xml:id="echoid-s8268" xml:space="preserve">] Sit ergo punctum, <lb/>in quod linea æqualis cadit:</s> <s xml:id="echoid-s8269" xml:space="preserve"> i.</s> <s xml:id="echoid-s8270" xml:space="preserve"> Dico, quòd in quo-<lb/>libet puncto lineę e i eſt locus imaginis:</s> <s xml:id="echoid-s8271" xml:space="preserve"> & erit ea-<lb/>dem demonſtratio, quę fuit in t o [præcedente nu <lb/>mero.</s> <s xml:id="echoid-s8272" xml:space="preserve">] Igitur quędã imagines in diametro e b ſor <lb/>tiuntur loca intra ſpeculũ:</s> <s xml:id="echoid-s8273" xml:space="preserve"> quędam extra ſpeculũ:</s> <s xml:id="echoid-s8274" xml:space="preserve"> <lb/>una ſola in ſuperficie:</s> <s xml:id="echoid-s8275" xml:space="preserve"> ſcilicet in puncto l.</s> <s xml:id="echoid-s8276" xml:space="preserve"> Et ita po <lb/>teris demonſtrare in qualibet diametro per puncta arcus o g tranſeunte.</s> <s xml:id="echoid-s8277" xml:space="preserve"/> </p> <div xml:id="echoid-div320" type="float" level="0" n="0"> <figure xlink:label="fig-0145-02" xlink:href="fig-0145-02a"> <variables xml:id="echoid-variables46" xml:space="preserve">a d k u m r h b g i l f e o z t y</variables> </figure> </div> </div> <div xml:id="echoid-div322" type="section" level="0" n="0"> <head xml:id="echoid-head330" xml:space="preserve" style="it">27. Si linea reflexionis, æquans ſua parte in ſcripta ſemidiametrum circuli (qui eſt commu-<lb/>nis ſectio ſuperficierum reflexionis & ſpeculi ſphærici conuexi) terminetur in peripheria nõ ap-<lb/>parente: perpendicularis incidentiæ ſecans peripheriam inter terminos lineæ reflexionis & <lb/>quadr antis peripheriæ, à puncto tact{us}, rectæ à uiſu ſpeculum tangentis, inchoati, habebit i-<lb/>magines extra ſpeculum. 32 p 6.</head> <p> <s xml:id="echoid-s8278" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s8279" xml:space="preserve"> ſumpta quacunq;</s> <s xml:id="echoid-s8280" xml:space="preserve"> diametro in arcu o h:</s> <s xml:id="echoid-s8281" xml:space="preserve"> locus imaginis in eo erit extra ſpeculũ.</s> <s xml:id="echoid-s8282" xml:space="preserve"> Suma <lb/> <pb o="140" file="0146" n="146" rhead="ALHAZEN"/> tur diameter b q:</s> <s xml:id="echoid-s8283" xml:space="preserve"> & concurrat cum cõtingente in puncto p [concurrit enim per lemma Procli ad 29 <lb/>p 1:</s> <s xml:id="echoid-s8284" xml:space="preserve">] & ducatur linea a u q ſecãs ſphęram in puncto u.</s> <s xml:id="echoid-s8285" xml:space="preserve"> Iam dictum eſt, quòd m o eſt æqualis o b [per <lb/>theſin communẽ 20.</s> <s xml:id="echoid-s8286" xml:space="preserve"> 21.</s> <s xml:id="echoid-s8287" xml:space="preserve"> 22.</s> <s xml:id="echoid-s8288" xml:space="preserve"> 23.</s> <s xml:id="echoid-s8289" xml:space="preserve"> 24.</s> <s xml:id="echoid-s8290" xml:space="preserve"> 25.</s> <s xml:id="echoid-s8291" xml:space="preserve"> 26.</s> <s xml:id="echoid-s8292" xml:space="preserve"> 27 n.</s> <s xml:id="echoid-s8293" xml:space="preserve">] Sed [per 15 p 3] u q eſt <lb/> <anchor type="figure" xlink:label="fig-0146-01a" xlink:href="fig-0146-01"/> maior m o:</s> <s xml:id="echoid-s8294" xml:space="preserve"> quare u q eſt maior o b, id eſt b q.</s> <s xml:id="echoid-s8295" xml:space="preserve"> Et linea ducta à circum-<lb/>ferentia ad diametrum p b, ęqualis parti p b, interiacenti inter ipſam <lb/>& centrum:</s> <s xml:id="echoid-s8296" xml:space="preserve"> non cadet inter q & b.</s> <s xml:id="echoid-s8297" xml:space="preserve"> Si enim ceciderit:</s> <s xml:id="echoid-s8298" xml:space="preserve"> ſecundũ ſupra-<lb/>dictam probationem [23 & præcedente numeris] erit u q minor q b.</s> <s xml:id="echoid-s8299" xml:space="preserve"> <lb/>Reſtat ergo, ut linea ęqualis cadat inter p & q.</s> <s xml:id="echoid-s8300" xml:space="preserve"> Et quòd non cadatin <lb/>punctũ p:</s> <s xml:id="echoid-s8301" xml:space="preserve"> palàm per hoc:</s> <s xml:id="echoid-s8302" xml:space="preserve"> quia angulus p g b eſt rectus [per 18 p 3.</s> <s xml:id="echoid-s8303" xml:space="preserve">] I-<lb/>gitur [per 19 p 1] p b maius eſt p g.</s> <s xml:id="echoid-s8304" xml:space="preserve"> Cadet ergo citra punctum p:</s> <s xml:id="echoid-s8305" xml:space="preserve"> Sit <lb/>punctum, in quod cadit:</s> <s xml:id="echoid-s8306" xml:space="preserve"> s.</s> <s xml:id="echoid-s8307" xml:space="preserve"> Erit ergo s meta locorum imaginum [per <lb/>23 n:</s> <s xml:id="echoid-s8308" xml:space="preserve">] & quodlibet punctũ inter p & s erit locus imaginum.</s> <s xml:id="echoid-s8309" xml:space="preserve"> Et eadẽ <lb/>eſt probatio, quæ ſuprà [25.</s> <s xml:id="echoid-s8310" xml:space="preserve"> 26 n.</s> <s xml:id="echoid-s8311" xml:space="preserve">] Palàm ex his, quòd imagines dia-<lb/>metrorum arcus h o, omnes ſunt extra ſuperficiem ſpeculi:</s> <s xml:id="echoid-s8312" xml:space="preserve"> imaginũ <lb/>diametri f y, una in ſuperficie ſpeculi:</s> <s xml:id="echoid-s8313" xml:space="preserve"> quę eſt in l:</s> <s xml:id="echoid-s8314" xml:space="preserve"> aliæ intra, ſcilicet in <lb/>i l:</s> <s xml:id="echoid-s8315" xml:space="preserve"> aliæ omnes extra, ſcilicet in l e.</s> <s xml:id="echoid-s8316" xml:space="preserve"> Omniũ aũt imaginum diametri ar-<lb/>cus o g, quædam intra ſpeculum:</s> <s xml:id="echoid-s8317" xml:space="preserve"> quędã extra:</s> <s xml:id="echoid-s8318" xml:space="preserve"> quędam in ſuperficie.</s> <s xml:id="echoid-s8319" xml:space="preserve"/> </p> <div xml:id="echoid-div322" type="float" level="0" n="0"> <figure xlink:label="fig-0146-01" xlink:href="fig-0146-01a"> <variables xml:id="echoid-variables47" xml:space="preserve">a d u m b g o e q s z h p</variables> </figure> </div> </div> <div xml:id="echoid-div324" type="section" level="0" n="0"> <head xml:id="echoid-head331" xml:space="preserve" style="it">28. Perpendicularis incidentiæ ſecans occult ãperipheriam cir <lb/>culι (quieſt communis ſectio ſuperficierum reflexionis & ſpeculi <lb/>ſphærici conuexi) inter terminos rectæ per centra uiſ{us} ac ſpeculi <lb/>ductæ, & quadrantis peripheriæ, à puncto tact{us} rectæ à uiſu ſpe-<lb/>culum tangentis, inchoati: imaginem nullam habet. 33 p 6.</head> <p> <s xml:id="echoid-s8320" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s8321" xml:space="preserve"> in arcu h z non poteſt ſumi diameter, in qua eſt locus imaginis.</s> <s xml:id="echoid-s8322" xml:space="preserve"> Quoniam nulla dia-<lb/>meter ibi ſumpta concurrit cũ contingente a p.</s> <s xml:id="echoid-s8323" xml:space="preserve"> [Quia enim g h eſt quadrans totius periphe <lb/>riæ ex theſi:</s> <s xml:id="echoid-s8324" xml:space="preserve"> rectus eſt angulus h b g per 33 p 6:</s> <s xml:id="echoid-s8325" xml:space="preserve"> & ſimiliter b g <lb/> <anchor type="figure" xlink:label="fig-0146-02a" xlink:href="fig-0146-02"/> p per 18 p 3.</s> <s xml:id="echoid-s8326" xml:space="preserve"> Quare perpẽdicularis incidentię, cadens in peripheriã <lb/>h z, facit cum b g angulũ obtuſum:</s> <s xml:id="echoid-s8327" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s8328" xml:space="preserve"> cũ tangente a g p non cõ <lb/>curret ad partes h & p:</s> <s xml:id="echoid-s8329" xml:space="preserve"> ſecus duæ rectæ ſpatium cõprehenderẽt cõ-<lb/>tra 12 ax.</s> <s xml:id="echoid-s8330" xml:space="preserve">] Et à quocunq;</s> <s xml:id="echoid-s8331" xml:space="preserve"> puncto illius talis diametri ducatur linea <lb/>ad ſphærã:</s> <s xml:id="echoid-s8332" xml:space="preserve"> cadet quidem in portionẽ g z c, & nulla in portionẽ g d <lb/>c, niſi ſecando ſphęram.</s> <s xml:id="echoid-s8333" xml:space="preserve"> Quare nulla forma alicuius puncti talis dia <lb/>metri ueniet ad portionem uiſui apparentem.</s> <s xml:id="echoid-s8334" xml:space="preserve"> Quod aũt dictum eſt <lb/>in arcu g h z:</s> <s xml:id="echoid-s8335" xml:space="preserve"> poteſt eodem modo demonſtrari in parte arcus c z eã <lb/>reſpiciente.</s> <s xml:id="echoid-s8336" xml:space="preserve"> Et ſumpto arcu citra z, æquali h z:</s> <s xml:id="echoid-s8337" xml:space="preserve"> in nulla diametro il-<lb/>lius arcus erit imaginis locus.</s> <s xml:id="echoid-s8338" xml:space="preserve"> Idẽ eſt demonſtrandi modus in quo-<lb/>cunq;</s> <s xml:id="echoid-s8339" xml:space="preserve"> circulo.</s> <s xml:id="echoid-s8340" xml:space="preserve"> Quare ſi linea h b moueatur, eodem manente angu-<lb/>lo h b z:</s> <s xml:id="echoid-s8341" xml:space="preserve"> ſignabit motu ſuo portionem ſphæræ, in cuius diametris <lb/>nullus ſit imaginis locus.</s> <s xml:id="echoid-s8342" xml:space="preserve"> Si uerò h b immota, moueatur o h:</s> <s xml:id="echoid-s8343" xml:space="preserve"> deſcri-<lb/>betur portio, cuius oẽs imagines extra ſpeculum ſunt.</s> <s xml:id="echoid-s8344" xml:space="preserve"> Moto aũt ar <lb/>cu o g:</s> <s xml:id="echoid-s8345" xml:space="preserve"> fiet portio, cuius quędam imagines ſuntin ſuperficie:</s> <s xml:id="echoid-s8346" xml:space="preserve"> quędã <lb/>extra ſpeculum:</s> <s xml:id="echoid-s8347" xml:space="preserve"> quędam intra.</s> <s xml:id="echoid-s8348" xml:space="preserve"> Verũ uiſus nõ comprehendit, quæ <lb/>imagines ſint in ſuperficie ſphęræ, aut quę extra:</s> <s xml:id="echoid-s8349" xml:space="preserve"> nec certificatur in <lb/>comprehenſione earum:</s> <s xml:id="echoid-s8350" xml:space="preserve"> niſi quòd ſint ultra portionem apparentẽ.</s> <s xml:id="echoid-s8351" xml:space="preserve"> <lb/>Iam ergo determinata ſunt in his ſpeculis imaginum loca.</s> <s xml:id="echoid-s8352" xml:space="preserve"/> </p> <div xml:id="echoid-div324" type="float" level="0" n="0"> <figure xlink:label="fig-0146-02" xlink:href="fig-0146-02a"> <variables xml:id="echoid-variables48" xml:space="preserve">a d u m c g b o t q p n z h</variables> </figure> </div> </div> <div xml:id="echoid-div326" type="section" level="0" n="0"> <head xml:id="echoid-head332" xml:space="preserve" style="it">29. Ab uno ſpeculi ſphærici conuexi puncto, unum uiſibilis punctum adunũ uiſum reflecti-<lb/>tur. Ita uni{us} punctiuna uidetur imago. 16 p 6.</head> <p> <s xml:id="echoid-s8353" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s8354" xml:space="preserve"> Puncti uiſi forma nõ poteſt in hoc ſpeculo ad unũ uiſum reflecti, niſi ab uno ſolo pũ <lb/>cto ſpeculi.</s> <s xml:id="echoid-s8355" xml:space="preserve"> Sit enim punctũ uiſum b:</s> <s xml:id="echoid-s8356" xml:space="preserve"> a centrũ uiſus:</s> <s xml:id="echoid-s8357" xml:space="preserve"> & nõ ſit a in perpẽdiculari ducta ad cẽtrũ <lb/>ſphęrę.</s> <s xml:id="echoid-s8358" xml:space="preserve"> Dico, quòd b reflectitur ad a ab uno ſolo ſpeculi puncto:</s> <s xml:id="echoid-s8359" xml:space="preserve"> & unã ſolã oſtendit uiſui ima <lb/>ginẽ in hoc ſpeculo.</s> <s xml:id="echoid-s8360" xml:space="preserve"> Palàm [per 25 n 4] quòd ab aliquo puncto poteſt reflecti forma eius:</s> <s xml:id="echoid-s8361" xml:space="preserve"> ſit illud g:</s> <s xml:id="echoid-s8362" xml:space="preserve"> <lb/>& ducantur b g, a g:</s> <s xml:id="echoid-s8363" xml:space="preserve"> & ſit n centrum ſphęrę:</s> <s xml:id="echoid-s8364" xml:space="preserve"> & ducatur diameter b n, ſecans ſuperficiem ſphæræ in <lb/>puncto l:</s> <s xml:id="echoid-s8365" xml:space="preserve"> & termini portionis uiſui oppoſitæ ſint d, e:</s> <s xml:id="echoid-s8366" xml:space="preserve"> & ſecet linea a g perpẽdicularem in puncto q:</s> <s xml:id="echoid-s8367" xml:space="preserve"> <lb/>quod eſt locus imaginum [per 3 uel 16 n.</s> <s xml:id="echoid-s8368" xml:space="preserve">] Palàm, quòd a, n, b ſint in eadẽ ſuperficie orthogonali ſuք <lb/>ſphæram [per 13.</s> <s xml:id="echoid-s8369" xml:space="preserve"> 23 n 4.</s> <s xml:id="echoid-s8370" xml:space="preserve">] Et cum omnes ſuperficies orthogonales ſuper ſphærã, in quibus fuerint b, <lb/>n, ſecent ſe ſuper b n:</s> <s xml:id="echoid-s8371" xml:space="preserve"> & nõ poſsit ſuperficies, in qua b n linea, extendi ad punctũ a, niſi una tantũ:</s> <s xml:id="echoid-s8372" xml:space="preserve"> [ꝗa <lb/>punctum a indiuiduũ eſt.</s> <s xml:id="echoid-s8373" xml:space="preserve">] Palàm, quòd a, & b, & n ſunt in una ſuperficie tantùm, orthogonali ſuper <lb/>ſphęrã, non in plurib.</s> <s xml:id="echoid-s8374" xml:space="preserve"> & cũ neceſſe ſit, [per 13.</s> <s xml:id="echoid-s8375" xml:space="preserve"> 23 n 4] ut omne punctũ uiſum, & a ſint in eadẽ ſuperfi-<lb/>cie orthogonali ſuper punctũ reflexionis:</s> <s xml:id="echoid-s8376" xml:space="preserve"> palàm, quòd non fiet reflexio puncti b ad uiſum, niſi in cir <lb/>culo ſphęrę, qui eſt in ſuperficie a n b.</s> <s xml:id="echoid-s8377" xml:space="preserve"> Sit ergo circulus d g e.</s> <s xml:id="echoid-s8378" xml:space="preserve"> Dico igitur, quòd à nullo puncto huius <lb/>circuli pręterꝗ̃ à g, fiet reflexio.</s> <s xml:id="echoid-s8379" xml:space="preserve"> Si enim dicatur, quòd à pũcto l:</s> <s xml:id="echoid-s8380" xml:space="preserve"> cum b n ſit ſuք ſuքficiẽ ſpeculi per-<lb/>pendicularis:</s> <s xml:id="echoid-s8381" xml:space="preserve"> [ut oſtẽſum eſt 25 n 4] & a l nõ ſit perpẽdicularis:</s> <s xml:id="echoid-s8382" xml:space="preserve"> [ꝗa nõ tranſit per centrü:</s> <s xml:id="echoid-s8383" xml:space="preserve">] & forma <lb/>per perpẽdicularẽ ueniens, neceſſariò ք perpendicularẽ reflectatur:</s> <s xml:id="echoid-s8384" xml:space="preserve"> [ք 11 n 4:</s> <s xml:id="echoid-s8385" xml:space="preserve">] palã, quòd non refle <lb/>ctetur b ad a à puncto l, Planum etiam eſt, quòd non reflectetur ab alio puncto arcus l e:</s> <s xml:id="echoid-s8386" xml:space="preserve"> quìa ad <lb/> <pb o="141" file="0147" n="147" rhead="OPTICAE LIBER V."/> quodcunq;</s> <s xml:id="echoid-s8387" xml:space="preserve"> punctum illius arcus ducatur linea à puncto b:</s> <s xml:id="echoid-s8388" xml:space="preserve"> tenebit cũ contingente illius puncti an-<lb/>gulum obtuſum ex parte e:</s> <s xml:id="echoid-s8389" xml:space="preserve"> [Nam ſemidiameter circuli ad rectam lineam per punctum illud, ſpecu <lb/>lum tangẽtem educta, declinat à puncto b uerſus <lb/> <anchor type="figure" xlink:label="fig-0147-01a" xlink:href="fig-0147-01"/> e, & facit cum tangente angulum rectum per 18 p <lb/>3.</s> <s xml:id="echoid-s8390" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s8391" xml:space="preserve"> angulus tangentis & lineæ rectę à puncto <lb/>b ad pũctum tactus ductę, maior eſt recto:</s> <s xml:id="echoid-s8392" xml:space="preserve"> ide o q́;</s> <s xml:id="echoid-s8393" xml:space="preserve"> <lb/>per 11 d 1 obtuſus:</s> <s xml:id="echoid-s8394" xml:space="preserve">] & linea ducta à puncto a ad il.</s> <s xml:id="echoid-s8395" xml:space="preserve"> <lb/>lud punctum, tenebit cũ contingente illa angulũ <lb/>acutum ex parte l:</s> <s xml:id="echoid-s8396" xml:space="preserve"> [quia angulus ſemidiametri & <lb/>tangentis rectus eſt per 18 p 3:</s> <s xml:id="echoid-s8397" xml:space="preserve"> & recta à puncto a <lb/>ad illud punctũ ducta ſecat circulũ d e g.</s> <s xml:id="echoid-s8398" xml:space="preserve">] Quare <lb/>ſi ab illo puncto fieret reflexio:</s> <s xml:id="echoid-s8399" xml:space="preserve"> eſſet angulus acu-<lb/>tus æqualis obtuſo [per 12 n 4.</s> <s xml:id="echoid-s8400" xml:space="preserve">] Iterũ à nullo pun <lb/>cto arcus g l poteſt fieri reflexio.</s> <s xml:id="echoid-s8401" xml:space="preserve"> Sumatur enim <lb/>punctum quodcunq;</s> <s xml:id="echoid-s8402" xml:space="preserve">: & ſit z:</s> <s xml:id="echoid-s8403" xml:space="preserve"> & ducatur linea a z o, <lb/>ſecans perpendicularẽ in puncto o:</s> <s xml:id="echoid-s8404" xml:space="preserve"> & ducatur li-<lb/>nea contingens circulum in puncto z:</s> <s xml:id="echoid-s8405" xml:space="preserve"> [per 17 p 3] <lb/>quæ cadit neceſſariò inter b g, & b l:</s> <s xml:id="echoid-s8406" xml:space="preserve"> [quia punctũ <lb/>z eſt inter puncta g & l] & ſit m z:</s> <s xml:id="echoid-s8407" xml:space="preserve"> & f g circulũ cõ-<lb/>tingat in puncto g.</s> <s xml:id="echoid-s8408" xml:space="preserve"> Palã ex ſuperiorib.</s> <s xml:id="echoid-s8409" xml:space="preserve"> [18 n] quòd <lb/>proportio b n ad n q, ſicut b f ad f q.</s> <s xml:id="echoid-s8410" xml:space="preserve"> Eodem modo <lb/>proportio b n ad n o, ſicut proportio b m ad m o:</s> <s xml:id="echoid-s8411" xml:space="preserve"> <lb/>Sed [per 9 ax.</s> <s xml:id="echoid-s8412" xml:space="preserve"> 8 p 5] maior eſt proportio b n ad n <lb/>q, quã b n ad n o.</s> <s xml:id="echoid-s8413" xml:space="preserve"> Igitur [per 11 p 5] maior eſt pro-<lb/>portio b fad f q, quàm b m ad m o.</s> <s xml:id="echoid-s8414" xml:space="preserve"> Quod planè im <lb/>poſsibile:</s> <s xml:id="echoid-s8415" xml:space="preserve"> cum [per 9 ax] b f ſit minor b m, & f q <lb/>maior m o.</s> <s xml:id="echoid-s8416" xml:space="preserve"> [ideoq́;</s> <s xml:id="echoid-s8417" xml:space="preserve"> ratio b f ad f q minor eſt ratio-<lb/>ne b m ad m o, ut patet ex 8 p 5.</s> <s xml:id="echoid-s8418" xml:space="preserve">] Reſtat ergo, ut à puncto z non fiat reflexio.</s> <s xml:id="echoid-s8419" xml:space="preserve"> Verùm quòd ab aliquo <lb/>puncto arcus g d non fiat reflexio:</s> <s xml:id="echoid-s8420" xml:space="preserve"> ſic conſtabit.</s> <s xml:id="echoid-s8421" xml:space="preserve"> Sumatur quodcunq;</s> <s xml:id="echoid-s8422" xml:space="preserve"> punctum:</s> <s xml:id="echoid-s8423" xml:space="preserve"> & ſit t:</s> <s xml:id="echoid-s8424" xml:space="preserve"> & ducatur li-<lb/>nea b t:</s> <s xml:id="echoid-s8425" xml:space="preserve"> & linea a t h, ſecans b n in pũcto h & [per 17 p 3] ducatur cõtingens circulũ in pũcto t:</s> <s xml:id="echoid-s8426" xml:space="preserve"> quę ſit <lb/>p t.</s> <s xml:id="echoid-s8427" xml:space="preserve"> Erit ergo ex ſuperiorib.</s> <s xml:id="echoid-s8428" xml:space="preserve"> [18 n] ꝓportio b n ad n h, ſicut b p ad p h:</s> <s xml:id="echoid-s8429" xml:space="preserve"> & b n ad n q, ſicut b fad f q:</s> <s xml:id="echoid-s8430" xml:space="preserve"> Sed <lb/>[per 9 ax.</s> <s xml:id="echoid-s8431" xml:space="preserve"> 8 p 5] b n ad n h maior eſt, quá b n ad n q:</s> <s xml:id="echoid-s8432" xml:space="preserve"> ergo [per 11 p 5] maior eſt proportio b p ad p h, ꝗ̃ <lb/>b fad f q:</s> <s xml:id="echoid-s8433" xml:space="preserve"> quod planè falſum:</s> <s xml:id="echoid-s8434" xml:space="preserve"> cum [per 9 ax] b f ſit maior b p, & p h maiorf q.</s> <s xml:id="echoid-s8435" xml:space="preserve"> [ideoq́;</s> <s xml:id="echoid-s8436" xml:space="preserve"> ratio b p ad <lb/>p h minor eſt ratione b f ad f q, ut conſtat ex 8 p 5.</s> <s xml:id="echoid-s8437" xml:space="preserve">] Reſtat ergo, ut à nullo puncto arcus g d fiat refle-<lb/>xio puncti b.</s> <s xml:id="echoid-s8438" xml:space="preserve"> Quare quodlibet punctum ab uno ſolo puncto ſpeculi reflectitur ad uiſum.</s> <s xml:id="echoid-s8439" xml:space="preserve"> Ergo una <lb/>ſola erit linea reflexionis cuiuslibet puncti uiſi.</s> <s xml:id="echoid-s8440" xml:space="preserve"> Quare unica unius puncti imago.</s> <s xml:id="echoid-s8441" xml:space="preserve"> Si aũt punctum b <lb/>fuerit in perpendiculari uiſuali:</s> <s xml:id="echoid-s8442" xml:space="preserve"> palàm [per 11 n 4] quòd reflectetur ab uno ſolo puncto, per quod <lb/>perpendicularis, tantũ:</s> <s xml:id="echoid-s8443" xml:space="preserve"> & unica erit eius imago:</s> <s xml:id="echoid-s8444" xml:space="preserve"> & erit propter continuitatem aliorum punctorum, <lb/>in loco imaginis proprio.</s> <s xml:id="echoid-s8445" xml:space="preserve"/> </p> <div xml:id="echoid-div326" type="float" level="0" n="0"> <figure xlink:label="fig-0147-01" xlink:href="fig-0147-01a"> <variables xml:id="echoid-variables49" xml:space="preserve">b k a p f m e l z g t r o q h n d</variables> </figure> </div> </div> <div xml:id="echoid-div328" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables50" xml:space="preserve">b k u a p e g t q n d</variables> </figure> <head xml:id="echoid-head333" xml:space="preserve" style="it">30. Siduo perpendicularis incidentiæ pun-<lb/> cta, à ſpeculo ſphærico conuexo ad unum uiſum reflectantur: loc{us} tum imaginis tum reflexio- nis, puncti centro ſpeculi propinquioris erit re- motior: imaginis ab eodem centro: reflexionis à uiſu. 17 p 6.</head> <p> <s xml:id="echoid-s8446" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s8447" xml:space="preserve"> ſi in aliqua diametro ſumãtur duo <lb/>puncta ex parte centri eadem:</s> <s xml:id="echoid-s8448" xml:space="preserve"> locus ima-<lb/>ginis centro propinquioris, erit remotior <lb/>à centro ſphęrę, loco imaginis puncti remotioris <lb/>à cẽtro ſphęrę.</s> <s xml:id="echoid-s8449" xml:space="preserve"> Verbi gratia dico, quòd locus ima-<lb/>ginis puncti p, remotior eſt à centro, loco imagi-<lb/>nis puncti b:</s> <s xml:id="echoid-s8450" xml:space="preserve"> & punctum reflexionis puncti p re-<lb/>motius ab a puncto uiſus, puncto reflexionis pun <lb/>cti b, quod eſt punctum g.</s> <s xml:id="echoid-s8451" xml:space="preserve"> Dico, quòd punctum p <lb/>non reflectitur, niſi ab aliquo puncto arcus g l.</s> <s xml:id="echoid-s8452" xml:space="preserve"> <lb/>Palàm enim, quòd non reflectitur ab aliquo pun-<lb/>cto arcus le, niſi à puncto l:</s> <s xml:id="echoid-s8453" xml:space="preserve"> [ſicq́;</s> <s xml:id="echoid-s8454" xml:space="preserve"> per 11 n 4 refle-<lb/>ctetur per perpendicularem b l n, nõ ad uiſum, in <lb/>a poſitum] nec à puncto g:</s> <s xml:id="echoid-s8455" xml:space="preserve"> cũ b reflectatur ab eo, <lb/>[ad uiſum ſcilicet a ex theſi:</s> <s xml:id="echoid-s8456" xml:space="preserve"> ideoq́ue nullum ali-<lb/>ud punctum, ut p, ab eodem puncto g reflectetur <lb/>ad eundẽ uiſum a ք pręcedẽtẽ numerũ.</s> <s xml:id="echoid-s8457" xml:space="preserve">] Et ſi dica <lb/>tur, qđ ab aliquo pũcto arcus g d:</s> <s xml:id="echoid-s8458" xml:space="preserve"> ſit illud pũctũ t:</s> <s xml:id="echoid-s8459" xml:space="preserve"> <lb/> <pb o="142" file="0148" n="148" rhead="ALHAZEN"/> & ſit t p linea, per quam forma mouetur ad ſpeculũ:</s> <s xml:id="echoid-s8460" xml:space="preserve"> & ducatur perpendicularis n t u:</s> <s xml:id="echoid-s8461" xml:space="preserve"> quę neceſſariò <lb/>diuidet angulum p t a per æqualia:</s> <s xml:id="echoid-s8462" xml:space="preserve"> [ut oſtenſum eſt 13 n 4] & ducatur perpendicularis n g k:</s> <s xml:id="echoid-s8463" xml:space="preserve"> erit [ք <lb/>21 p 1] angulus n t a maior n g a:</s> <s xml:id="echoid-s8464" xml:space="preserve"> reſtat ergo [per 13 p 1] angulus u t a minor angulo k g a.</s> <s xml:id="echoid-s8465" xml:space="preserve"> Quare angu-<lb/>lus p t u<gap/>minor angulo b g k:</s> <s xml:id="echoid-s8466" xml:space="preserve"> [angulus enim k g a æquatur angulo b g k, per 12 n 4.</s> <s xml:id="echoid-s8467" xml:space="preserve">] Sed [per 32 p <lb/>1] angulus p t u ualet angulum t n p, & t p n:</s> <s xml:id="echoid-s8468" xml:space="preserve"> quia exterior:</s> <s xml:id="echoid-s8469" xml:space="preserve"> & angulus b g k ualet angulũ g n b, & an-<lb/>gulum g b n Erunt ergo duo anguli t n p, t p n minores duobus angulis g b n, g n b:</s> <s xml:id="echoid-s8470" xml:space="preserve"> quod [per 9 ax] <lb/>eſt impoſsibile:</s> <s xml:id="echoid-s8471" xml:space="preserve"> cum angulus p n t contineat angulum g n b tanquam partem:</s> <s xml:id="echoid-s8472" xml:space="preserve"> & [per 16 p 1] angulus <lb/>t p n ſit maior g b n.</s> <s xml:id="echoid-s8473" xml:space="preserve"> Reſtat ergo, ut punctum p non reflectatur, niſi à punctis inter g & l intermedijs.</s> <s xml:id="echoid-s8474" xml:space="preserve"> <lb/>Et omnes lineæ à puncto a per hæc puncta ductæ ad diametrum b n, cadunt in puncta ſphęræ à cen <lb/>tro uiſus magis elongata, puncto g.</s> <s xml:id="echoid-s8475" xml:space="preserve"> Et ita patet propoſitum.</s> <s xml:id="echoid-s8476" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div329" type="section" level="0" n="0"> <head xml:id="echoid-head334" xml:space="preserve" style="it">31. Viſa & uiſibilia à centro ſpeculi ſphærici conuexi æquabiliter diſtantib{us}: punctum refle-<lb/>xionis inuenire. 20 p 6.</head> <p> <s xml:id="echoid-s8477" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s8478" xml:space="preserve"> dato ſpeculo, & dato puncto uiſo:</s> <s xml:id="echoid-s8479" xml:space="preserve"> eſt inuenire punctum reflexionis.</s> <s xml:id="echoid-s8480" xml:space="preserve"> Sit enim b pun-<lb/>ctum uiſum:</s> <s xml:id="echoid-s8481" xml:space="preserve"> a centrum uiſus:</s> <s xml:id="echoid-s8482" xml:space="preserve"> & ducantur ab eis duæ lineæ ad <lb/> <anchor type="figure" xlink:label="fig-0148-01a" xlink:href="fig-0148-01"/> centrum ſpeculi.</s> <s xml:id="echoid-s8483" xml:space="preserve"> Si fuerint duæ illæ lineæ æquales:</s> <s xml:id="echoid-s8484" xml:space="preserve"> erit facile <lb/>inuenire:</s> <s xml:id="echoid-s8485" xml:space="preserve"> quoniam ſumetur circulus ſphæræ in ſuperficie duarum il-<lb/>larum linearum.</s> <s xml:id="echoid-s8486" xml:space="preserve"> Et ſcimus [per 29 n] quòd ab unico ſolo puncto il <lb/>lius circuli fit unius puncti reflexio.</s> <s xml:id="echoid-s8487" xml:space="preserve"> Diuidatur ergo [per 9 p 1] angu-<lb/>lus, quem continent in centro duæ illæ lineę, per æqualia:</s> <s xml:id="echoid-s8488" xml:space="preserve"> & ducatur <lb/>linea diuidens angulum, extra ſphæram:</s> <s xml:id="echoid-s8489" xml:space="preserve"> erit quidem [per 18 p 3] per-<lb/>pendicularis ſuper lineam contingentem hunc circulum in puncto, <lb/>per quod tranſit.</s> <s xml:id="echoid-s8490" xml:space="preserve"> Et ſi ducantur ad illud punctum duę lineę:</s> <s xml:id="echoid-s8491" xml:space="preserve"> una à cen <lb/>tro uiſus:</s> <s xml:id="echoid-s8492" xml:space="preserve"> alia à punctu uiſo:</s> <s xml:id="echoid-s8493" xml:space="preserve"> efficient cum perpendiculari illa & dua-<lb/>bus primis lineis, duo triangula:</s> <s xml:id="echoid-s8494" xml:space="preserve"> quorum duo latera duobus laterib.</s> <s xml:id="echoid-s8495" xml:space="preserve"> <lb/>æqualia, & angulus angulo [ideoq́;</s> <s xml:id="echoid-s8496" xml:space="preserve"> per 4.</s> <s xml:id="echoid-s8497" xml:space="preserve"> 13 p 1.</s> <s xml:id="echoid-s8498" xml:space="preserve"> 3 ax angulus a e d æ-<lb/>quatur angulo b e d.</s> <s xml:id="echoid-s8499" xml:space="preserve">] Et ita punctum circuli, per quod perpendicula <lb/>ris illa tranſit:</s> <s xml:id="echoid-s8500" xml:space="preserve"> eſt punctum reflexionis:</s> <s xml:id="echoid-s8501" xml:space="preserve"> [quia c e d bifariam ſecat an-<lb/>gulum ab incidentię & reflexionis lineis comprehenſum, ut patuit 13 <lb/>n 4.</s> <s xml:id="echoid-s8502" xml:space="preserve">] Si nero linea à puncto uiſo ad centrum ſphærę ducta, fuerit inę <lb/>qualis lineę à centro uiſus ad idem centrum ductę:</s> <s xml:id="echoid-s8503" xml:space="preserve"> oportet nos quę-<lb/>dam antecedentia præponere:</s> <s xml:id="echoid-s8504" xml:space="preserve"> quorum unum eſt.</s> <s xml:id="echoid-s8505" xml:space="preserve"/> </p> <div xml:id="echoid-div329" type="float" level="0" n="0"> <figure xlink:label="fig-0148-01" xlink:href="fig-0148-01a"> <variables xml:id="echoid-variables51" xml:space="preserve">b d a f e g c</variables> </figure> </div> </div> <div xml:id="echoid-div331" type="section" level="0" n="0"> <head xml:id="echoid-head335" xml:space="preserve" style="it">32. À<unsure/> puncto dimidiatæ peripheriæ medio, ducere lineam re-<lb/>ctam, ut ſegmentum ei{us} conterminum continuatæ diametro, æquetur datæ lineæ rectæ. 128 p 1.</head> <p> <s xml:id="echoid-s8506" xml:space="preserve">SVmpta circuli diametro, & ſumpto in circumferentia puncto:</s> <s xml:id="echoid-s8507" xml:space="preserve"> eſt ducere ab eo ad diametrũ ex-<lb/>tra productam, lineam, quę à puncto, in quo ſecat circulum, uſq;</s> <s xml:id="echoid-s8508" xml:space="preserve"> ad cõcurſum cum diametro, ſit <lb/>æqualis lineæ datæ Verbi gratia, ſit q e data linea:</s> <s xml:id="echoid-s8509" xml:space="preserve"> g b diameter circuli a b g:</s> <s xml:id="echoid-s8510" xml:space="preserve"> a punctũ datum.</s> <s xml:id="echoid-s8511" xml:space="preserve"> Di <lb/>co, quòd à puncto a ducam lineam, quæ à pũcto, in quo ſecuerit circulum, uſq;</s> <s xml:id="echoid-s8512" xml:space="preserve"> ad diametrum g b, ſit <lb/>æqualis lineæ q e:</s> <s xml:id="echoid-s8513" xml:space="preserve"> quod ſic conſtabit.</s> <s xml:id="echoid-s8514" xml:space="preserve"> Ducantur duę lineæ a b, a g:</s> <s xml:id="echoid-s8515" xml:space="preserve"> quę aut erunt æ quales:</s> <s xml:id="echoid-s8516" xml:space="preserve"> aut inęqua <lb/>les.</s> <s xml:id="echoid-s8517" xml:space="preserve"> Sint ęquales:</s> <s xml:id="echoid-s8518" xml:space="preserve"> & adiungatur lineæ q e linea talis, ut illud, quod fit ex ductu totius cum adiuncta <lb/>in ad ũctam, ſit <lb/> <anchor type="figure" xlink:label="fig-0148-02a" xlink:href="fig-0148-02"/> <anchor type="figure" xlink:label="fig-0148-03a" xlink:href="fig-0148-03"/> <anchor type="figure" xlink:label="fig-0148-04a" xlink:href="fig-0148-04"/> ęquale quadra <lb/>to a g.</s> <s xml:id="echoid-s8519" xml:space="preserve"> [id uerò <lb/>expeditè fiet:</s> <s xml:id="echoid-s8520" xml:space="preserve"> ſi <lb/>linea q e fiat dia <lb/>meter circuli, <lb/>cuius periphe-<lb/>riam tangens re <lb/>cta linea æqua-<lb/>lis a g, cõcurrat <lb/>cum continua-<lb/>ta diametro q e:</s> <s xml:id="echoid-s8521" xml:space="preserve"> <lb/>ſic enim oblongum cõprehenſum ſub continuata diametro & exte-<lb/>riore eius ſegmento ęquabitur quadrato lineę a g per 36 p 3.</s> <s xml:id="echoid-s8522" xml:space="preserve">] Et ſit li <lb/>nea adiũcta e z.</s> <s xml:id="echoid-s8523" xml:space="preserve"> Cũ igitur illud, qđ fit ex ductu q z in e z ſit ęquale ei, <lb/>quod fit ex ductu a g in ſe:</s> <s xml:id="echoid-s8524" xml:space="preserve"> erit q z maior a g, & e z minor eadem.</s> <s xml:id="echoid-s8525" xml:space="preserve"> Si e-<lb/>nim e z fuerit æqualis, aut maior a g:</s> <s xml:id="echoid-s8526" xml:space="preserve"> eſt impoſsibile, ut ductus q z in <lb/>e z ſit æqualis quadrato a g [ſic enim oblongum comprehẽſum ſub <lb/>q z & e z ſemper maius eſſet quadrato a g:</s> <s xml:id="echoid-s8527" xml:space="preserve"> quia linea q z eſſet maior <lb/>e z, ut totum ſua parte.</s> <s xml:id="echoid-s8528" xml:space="preserve">] Si autem minor:</s> <s xml:id="echoid-s8529" xml:space="preserve"> palàm, quòd q z eſt maior a <lb/>g.</s> <s xml:id="echoid-s8530" xml:space="preserve"> Producatur ergo ad ęqualitatem:</s> <s xml:id="echoid-s8531" xml:space="preserve"> & ſit a g t:</s> <s xml:id="echoid-s8532" xml:space="preserve"> & poſito pede circini ſuper a, fiat circulus ſecundum <lb/>quantitatem a g t:</s> <s xml:id="echoid-s8533" xml:space="preserve"> qui quidem circulus ſecabit diametrum b g:</s> <s xml:id="echoid-s8534" xml:space="preserve"> [infinitè uerſus t continuatam] & <lb/> <pb o="143" file="0149" n="149" rhead="OPTICAE LIBER V."/> ſecet in puncto d:</s> <s xml:id="echoid-s8535" xml:space="preserve"> & ducatur linea a d:</s> <s xml:id="echoid-s8536" xml:space="preserve"> quæ ſecabit neceſſariò circũlum.</s> <s xml:id="echoid-s8537" xml:space="preserve"> Si enim contingeretin <lb/>puncto a:</s> <s xml:id="echoid-s8538" xml:space="preserve"> eſſet æquidiſtans b g, & nunquam concurreret cum ea.</s> <s xml:id="echoid-s8539" xml:space="preserve"> [Nam ex theſi g a, b a æ-<lb/>quantur.</s> <s xml:id="echoid-s8540" xml:space="preserve"> Itaque ſemidiameter à centro ad a ducta, efficiet per 8 p.</s> <s xml:id="echoid-s8541" xml:space="preserve"> 10 d 1 angulos cum b g rectos.</s> <s xml:id="echoid-s8542" xml:space="preserve"> <lb/>Similiter angulus lineæ d a tãgentis & ſemidiametri rectus eſt per 18 p 3:</s> <s xml:id="echoid-s8543" xml:space="preserve"> ergo per 28 p 1 b d, a d eſ-<lb/>ſent parallelæ:</s> <s xml:id="echoid-s8544" xml:space="preserve"> quæ tamen concurrunt in puncto d, è fabricatione.</s> <s xml:id="echoid-s8545" xml:space="preserve">] Secet ergo in puncto h:</s> <s xml:id="echoid-s8546" xml:space="preserve"> & du-<lb/>catur linea g h.</s> <s xml:id="echoid-s8547" xml:space="preserve"> Palàm [per 22 p 3] cum a b g h ſit quadrangulum intra circulum:</s> <s xml:id="echoid-s8548" xml:space="preserve"> a b g, a h g an-<lb/>gulos oppoſitos ualere duos rectos:</s> <s xml:id="echoid-s8549" xml:space="preserve"> ſed [per 5 p 1] a g b eſt æqualis angulo a b g, cum reſpiciant <lb/>æqualia latera ex hypotheſi.</s> <s xml:id="echoid-s8550" xml:space="preserve"> Erit igitur angulus a h g æqualis angulo d g a:</s> <s xml:id="echoid-s8551" xml:space="preserve"> [per 13 p 1] & angu-<lb/>lus h a g communis triangulo totali a d g, & partiali a h g:</s> <s xml:id="echoid-s8552" xml:space="preserve"> reſtat ergo [per 32 p 1] ut angulus <lb/>h d g ſit æqualis angulo h g a:</s> <s xml:id="echoid-s8553" xml:space="preserve"> & triangulum ſimile triangulo [per 4 p.</s> <s xml:id="echoid-s8554" xml:space="preserve"> 1 d 6.</s> <s xml:id="echoid-s8555" xml:space="preserve">] Quare proportio <lb/>d a ad a g, ſicut a g ad a h:</s> <s xml:id="echoid-s8556" xml:space="preserve"> ergo [per 17 p 6] quod fit ex ductu d a in a h, eſt æquale quadrato a g:</s> <s xml:id="echoid-s8557" xml:space="preserve"> <lb/>Sed [per 15 d 1] d a eſt æqualis t a:</s> <s xml:id="echoid-s8558" xml:space="preserve"> igitur [per 1 ax] eſt æqualis q z:</s> <s xml:id="echoid-s8559" xml:space="preserve"> & erit a h æqualis e z:</s> <s xml:id="echoid-s8560" xml:space="preserve"> [quia <lb/>è prima fabricatione oblongum comprehenſum ſub q z & e z æquatur quadrato a g:</s> <s xml:id="echoid-s8561" xml:space="preserve"> cui æquale o-<lb/>ſtenſum eſt oblongum comprehenſum ſub d a & a h:</s> <s xml:id="echoid-s8562" xml:space="preserve"> & d a æquaturipſi q z] & [per 3 ax] d h æqua-<lb/>lis q e:</s> <s xml:id="echoid-s8563" xml:space="preserve"> quæ eſt data linea.</s> <s xml:id="echoid-s8564" xml:space="preserve"> Et ita eſt propoſitum.</s> <s xml:id="echoid-s8565" xml:space="preserve"/> </p> <div xml:id="echoid-div331" type="float" level="0" n="0"> <figure xlink:label="fig-0148-02" xlink:href="fig-0148-02a"> <variables xml:id="echoid-variables52" xml:space="preserve">q a e g</variables> </figure> <figure xlink:label="fig-0148-03" xlink:href="fig-0148-03a"> <variables xml:id="echoid-variables53" xml:space="preserve">a z g e b q</variables> </figure> <figure xlink:label="fig-0148-04" xlink:href="fig-0148-04a"> <variables xml:id="echoid-variables54" xml:space="preserve">d <gap/> q g h <gap/> a z b</variables> </figure> </div> </div> <div xml:id="echoid-div333" type="section" level="0" n="0"> <head xml:id="echoid-head336" xml:space="preserve" style="it">33. À<unsure/> puncto dimidiatæ peripheriæ non medio, ducere lineam rectam: ut ſegmentum ei{us} <lb/>conterminum continuatæ diametro, æquetur datæ lineæ rectæ. 130 p 1.</head> <p> <s xml:id="echoid-s8566" xml:space="preserve">SI uerò a b & a g non ſint æquales:</s> <s xml:id="echoid-s8567" xml:space="preserve"> protrahatur [per 31 p 1] à puncto g linea æquidiſtans a b:</s> <s xml:id="echoid-s8568" xml:space="preserve"> <lb/>quæ ſit g n:</s> <s xml:id="echoid-s8569" xml:space="preserve"> & ſumatur linea, quæcunque ſit:</s> <s xml:id="echoid-s8570" xml:space="preserve"> z t:</s> <s xml:id="echoid-s8571" xml:space="preserve"> & [per 23 p 1] ſuper punctum z fiat angulus ę-<lb/>qualis angulo a g d per lineam z f:</s> <s xml:id="echoid-s8572" xml:space="preserve"> & [per 31 p 1] ducatur à puncto t linea æquidiſtans z f:</s> <s xml:id="echoid-s8573" xml:space="preserve"> & ſit <lb/>t m:</s> <s xml:id="echoid-s8574" xml:space="preserve"> & [per 23 p 1] ex angulo t z f ſecetur angulus æqualis angulo d g n per lineam z m.</s> <s xml:id="echoid-s8575" xml:space="preserve"> Hæc <lb/> <anchor type="figure" xlink:label="fig-0149-01a" xlink:href="fig-0149-01"/> <anchor type="figure" xlink:label="fig-0149-02a" xlink:href="fig-0149-02"/> igitur linea neceſſariò cõcurrit cum <lb/>t m:</s> <s xml:id="echoid-s8576" xml:space="preserve"> [per lemma Procli ad 29 p 1] <lb/>cum ſit inter æquidiſtantes.</s> <s xml:id="echoid-s8577" xml:space="preserve"> Sit pun-<lb/>ctum concurſus m:</s> <s xml:id="echoid-s8578" xml:space="preserve"> reſtat ergo [per <lb/>3 ax] angulus m z f æqualis angu-<lb/>lo a g n.</s> <s xml:id="echoid-s8579" xml:space="preserve"> Et à puncto t ducatur li-<lb/>nea æquidiſtans lineæ z m:</s> <s xml:id="echoid-s8580" xml:space="preserve"> [per 31 <lb/>p 1] quæ ſit t o:</s> <s xml:id="echoid-s8581" xml:space="preserve"> quæ quidem neceſſa-<lb/>riò concurret cum f z:</s> <s xml:id="echoid-s8582" xml:space="preserve"> [per lemma <lb/>Procli ad 29 p 1] & ſit concurſus in <lb/>puncto k:</s> <s xml:id="echoid-s8583" xml:space="preserve"> & ſumatur [per 12 p 6] li-<lb/>nea, cuius proportio ad lineam z t, ſi-<lb/>cut b g ad q e lineam datam:</s> <s xml:id="echoid-s8584" xml:space="preserve"> & ſit <lb/>i.</s> <s xml:id="echoid-s8585" xml:space="preserve"> Deinde fiat ſuper punctum m ſe-<lb/>ctio pyramidalis, quemadmodũ do-<lb/>cet Apollonius in libro ſecundo de <lb/>pyramidalibus, propoſitiõe quarta:</s> <s xml:id="echoid-s8586" xml:space="preserve"> <lb/>& ſit u c m:</s> <s xml:id="echoid-s8587" xml:space="preserve"> quæ quidem ſectio non <lb/>ſecat lineas k o, k f:</s> <s xml:id="echoid-s8588" xml:space="preserve"> & in hac ſectione <lb/>ducatur linea æqualis lineę i:</s> <s xml:id="echoid-s8589" xml:space="preserve"> ſcilicet <lb/>m c:</s> <s xml:id="echoid-s8590" xml:space="preserve"> & producatur uſque ad lineas k t, k f:</s> <s xml:id="echoid-s8591" xml:space="preserve"> & ſint puncta ſectio num o, l.</s> <s xml:id="echoid-s8592" xml:space="preserve"> Igitur, ſicut ibidem [8 th 2 <lb/>coni conicorum] probatur:</s> <s xml:id="echoid-s8593" xml:space="preserve"> erit o m æqualis c l:</s> <s xml:id="echoid-s8594" xml:space="preserve"> & à puncto t ducatur linea æquidiſtans c m:</s> <s xml:id="echoid-s8595" xml:space="preserve"> [per <lb/>31 p 1,] quæ ſit t f:</s> <s xml:id="echoid-s8596" xml:space="preserve"> & [per 23 p 1] ſuper punctum a fiat angulus æqualis angulo z f t per lineam a n <lb/>d.</s> <s xml:id="echoid-s8597" xml:space="preserve"> Palàm, quòd hæc linea concurret cum g d:</s> <s xml:id="echoid-s8598" xml:space="preserve"> cum angulus a g n ſit ęqualis f z m angulo:</s> <s xml:id="echoid-s8599" xml:space="preserve"> [per con-<lb/>cluſionem] & angulus g a n angulo z f t [per fabricationem:</s> <s xml:id="echoid-s8600" xml:space="preserve"> & totus angulus f z t æquatus ſit toti <lb/>angulo d g a:</s> <s xml:id="echoid-s8601" xml:space="preserve"> & per 32 p 1 anguli ad z & f ſint minores duobus rectis.</s> <s xml:id="echoid-s8602" xml:space="preserve"> Ergo anguli ad g & a ipſis æ-<lb/>quales, minores erunt duobus rectis.</s> <s xml:id="echoid-s8603" xml:space="preserve"> Itaque per 11 ax.</s> <s xml:id="echoid-s8604" xml:space="preserve"> g d, a d concurrent.</s> <s xml:id="echoid-s8605" xml:space="preserve">] Igitur a d linea aut tan-<lb/>get circulum:</s> <s xml:id="echoid-s8606" xml:space="preserve"> aut ſecabit ipſum.</s> <s xml:id="echoid-s8607" xml:space="preserve"> Quoniam ſi non tetigerit, & arcus a b fuerit maior arcu a g:</s> <s xml:id="echoid-s8608" xml:space="preserve"> ſeca-<lb/>bit arcum a b:</s> <s xml:id="echoid-s8609" xml:space="preserve"> & ſi a b fuerit minor:</s> <s xml:id="echoid-s8610" xml:space="preserve"> ſecabit arcum a g.</s> <s xml:id="echoid-s8611" xml:space="preserve"> Tangat igitur in puncto a.</s> <s xml:id="echoid-s8612" xml:space="preserve"> Cum igitur [per <lb/>fabricationem] angulus g a n ſit æqualis angulo z f t, & angulus a g n angulo f z y:</s> <s xml:id="echoid-s8613" xml:space="preserve"> erit [per <lb/>32 p 1] tertius tertio æqualis:</s> <s xml:id="echoid-s8614" xml:space="preserve"> & erit triangulum a g n ſimile triangulo z f y.</s> <s xml:id="echoid-s8615" xml:space="preserve"> Similiter cum [per <lb/>fabricationem] a g d ſit æqualis angulo f z t:</s> <s xml:id="echoid-s8616" xml:space="preserve"> erit [per 32 p 1.</s> <s xml:id="echoid-s8617" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s8618" xml:space="preserve"> 1 d 6] triangulum a g d ſimile <lb/>triangulo f z t.</s> <s xml:id="echoid-s8619" xml:space="preserve"> Igitur quæ eſt proportio a n ad a g, ea eſt proportio f y ad f z:</s> <s xml:id="echoid-s8620" xml:space="preserve"> & quæ eſt propor-<lb/>tio a g ad g d, ea eſt f z ad z t.</s> <s xml:id="echoid-s8621" xml:space="preserve"> Quare [per 22 p 5] quæ eſt proportio a n ad g d, ea eſt f y ad <lb/>z t.</s> <s xml:id="echoid-s8622" xml:space="preserve"> Verùm cum [per fabricationem] t m ſit æquidiſtans f l, & f t ſit æquidiſtans l m:</s> <s xml:id="echoid-s8623" xml:space="preserve"> eſt <gap/> per <lb/>34 p 1] f t æqualis l m.</s> <s xml:id="echoid-s8624" xml:space="preserve"> Quare [per 2 ax] erit æqualis c o:</s> <s xml:id="echoid-s8625" xml:space="preserve"> cum [per 8 th 2 conicorum Apol-<lb/>lonij] m o ſit æqualis l c:</s> <s xml:id="echoid-s8626" xml:space="preserve"> ſed [per 34 p 1] m o eſt æqualis y t:</s> <s xml:id="echoid-s8627" xml:space="preserve"> cum [per fabricationem] ſit ipſt <lb/>æquidiſtans, & y m æquidiſtans t o.</s> <s xml:id="echoid-s8628" xml:space="preserve"> Reſtat ergo [per 3 ax] f y æqualis c m:</s> <s xml:id="echoid-s8629" xml:space="preserve"> ſed [per fabrica-<lb/>tionem] c m eſt æqualis i.</s> <s xml:id="echoid-s8630" xml:space="preserve"> Quare [per 1 ax] f y eſt æqualis i:</s> <s xml:id="echoid-s8631" xml:space="preserve"> ſed [per fabricationem] propor-<lb/>tio i [id eſt, per 7 p 5 f y] ad z t, ſicut b g ad e q.</s> <s xml:id="echoid-s8632" xml:space="preserve"> Igitur [per 11 p 5] proportio a n ad g d, ſi-<lb/>cut b g ad e q.</s> <s xml:id="echoid-s8633" xml:space="preserve"> Verùm angulus g a n eſt æqualis angulo g b a:</s> <s xml:id="echoid-s8634" xml:space="preserve"> ſicut probat Euclides in ter-<lb/> <pb o="144" file="0150" n="150" rhead="ALHAZEN"/> tio [32 propoſitione] ſed [per 29 p 1] angulus n g d eſt æqualis angulo a b g:</s> <s xml:id="echoid-s8635" xml:space="preserve"> cum [per fabri-<lb/>cationem] n g ſit æquidiſtans a b.</s> <s xml:id="echoid-s8636" xml:space="preserve"> Igitur [per 1 ax] angulus n g d æqualis eſt angulo n a g, & an-<lb/>gulus n d g communis.</s> <s xml:id="echoid-s8637" xml:space="preserve"> Quare [per 32 p 1] tertius tertio eſt æqualis.</s> <s xml:id="echoid-s8638" xml:space="preserve"> Quare [per 4 p.</s> <s xml:id="echoid-s8639" xml:space="preserve"> 1 d 6] triangu-<lb/>lum n d g ſimile triangulo a d g.</s> <s xml:id="echoid-s8640" xml:space="preserve"> Igitur proportio a d ad d g, ſicut g d ad d n.</s> <s xml:id="echoid-s8641" xml:space="preserve"> Quare [per 17 p 6] quod <lb/>fit ex ductu a d in d n eſt æquale quadrato d g.</s> <s xml:id="echoid-s8642" xml:space="preserve"> Verùm quadratum a d eſt æquale ei, quod fit ex du-<lb/>ctu b d in d g:</s> <s xml:id="echoid-s8643" xml:space="preserve"> ſicut probat Euclides 36 propoſitione [libri tertij,] & quadratum a d eſt æquale ei, <lb/>quod fit ex ductu a d in d n, & ei quod fit ex ductu a d in n a [per 2 p 2:</s> <s xml:id="echoid-s8644" xml:space="preserve">] & illud, quod fit ex ductu b d <lb/>in d g, eſt æquale quadrato d g, & ei quod fit ex ductu b g in g d:</s> <s xml:id="echoid-s8645" xml:space="preserve"> ſicut probat Euclides [3 p 2.</s> <s xml:id="echoid-s8646" xml:space="preserve">] Abla-<lb/>tis ergo æqualibus [quadrato nempe d g & rectangulo a d n] reſtat [per 3 ax,] ut, quòd fit ex du-<lb/>ctu a d in a n, ſit æquale ei, quod fit ex b g in g d.</s> <s xml:id="echoid-s8647" xml:space="preserve"> Igitur [per 16 p 6] proportio primæ lineæ ad ſecun <lb/>dam, eſt ſicut tertiæ ad quartam [nempe ut a d ad g d, ſic b g ad a n:</s> <s xml:id="echoid-s8648" xml:space="preserve"> & alternè [per 16 p 5] ut a d ad <lb/>b g, ſic g d ad a n.</s> <s xml:id="echoid-s8649" xml:space="preserve">] Quare [per conſectarium 4 p 5] proportio a n ad g d, ſicut b g ad a d.</s> <s xml:id="echoid-s8650" xml:space="preserve"> Sediam <lb/>dictum eſt, quòd proportio a n ad g d eſt, ſicut b g ad e q.</s> <s xml:id="echoid-s8651" xml:space="preserve"> Igitur [per 9 p 5] e q eſt æqualis a d.</s> <s xml:id="echoid-s8652" xml:space="preserve"> Quod <lb/>eſt propoſitum.</s> <s xml:id="echoid-s8653" xml:space="preserve"> Quòd ſi a d non tetigerit circulum, ſed ſecuerit, & <lb/> <anchor type="figure" xlink:label="fig-0150-01a" xlink:href="fig-0150-01"/> fuerit a g maior a b:</s> <s xml:id="echoid-s8654" xml:space="preserve"> ſecabit quidem arcũ a g.</s> <s xml:id="echoid-s8655" xml:space="preserve"> Secet ergo in puncto h:</s> <s xml:id="echoid-s8656" xml:space="preserve"> <lb/>& ducatur linea h g.</s> <s xml:id="echoid-s8657" xml:space="preserve"> Palàm [per 22 p 3] quòd duo anguli a h g, a b g <lb/>ualent duos rectos:</s> <s xml:id="echoid-s8658" xml:space="preserve"> ſed angulus n g d æqualis eſt a b g [per 29 p 1:</s> <s xml:id="echoid-s8659" xml:space="preserve"> <lb/>quia n g parallela ducta eſt ipſi a b.</s> <s xml:id="echoid-s8660" xml:space="preserve">] Igitur angulus a h g & angulus n <lb/>g d ſunt ęquales duobus rectis.</s> <s xml:id="echoid-s8661" xml:space="preserve"> Quare [per 13 p 1.</s> <s xml:id="echoid-s8662" xml:space="preserve"> 3 ax] angulus n g d <lb/>eſt ęqualis angulo n h g:</s> <s xml:id="echoid-s8663" xml:space="preserve"> & angulus n d g communis.</s> <s xml:id="echoid-s8664" xml:space="preserve"> Quare [per 32 p <lb/>1] tertius angulus tertio angulo eſt æqualis:</s> <s xml:id="echoid-s8665" xml:space="preserve"> & triangulum h g d ſimi <lb/>le triangulo n d g [per 4 p.</s> <s xml:id="echoid-s8666" xml:space="preserve"> 1 d 6.</s> <s xml:id="echoid-s8667" xml:space="preserve">] Igitur proportio h d ad d g eſt, ſicut <lb/>proportio d g ad d n.</s> <s xml:id="echoid-s8668" xml:space="preserve"> Quare [per 17 p 6] illud, quod fit ex ductu h d <lb/>in d n, eſt ęquale quadrato d g:</s> <s xml:id="echoid-s8669" xml:space="preserve"> ſed quod fit ex ductu a d in h d, eſt æ-<lb/>quale ei, quod fit ex ductu b d in d g, ſicut probat Euclides [cõſecta-<lb/>rio 36 p 3] & [per 1 p 2] illud, quod fit ex ductu a d in d h, eſt ęquale ei, <lb/>quod fit ex ductu d h in d n, & d h in a n:</s> <s xml:id="echoid-s8670" xml:space="preserve"> & [per 3 p 2] qđ fit ex ductu <lb/>b d in d g, eſt æquale ei, quod fit ex ductu b g in g d & quadrato d g.</s> <s xml:id="echoid-s8671" xml:space="preserve"> <lb/>Ablatis igitur æqualibus, ſcilicet quadrato d g, & eo, quod fit ex du-<lb/>ctu d h in d n:</s> <s xml:id="echoid-s8672" xml:space="preserve"> reſtat [per 3 ax] ut illud, quod fit ex ductu d h in a n, ſit <lb/>ęquale ei, quod fit ex ductu b g in d g.</s> <s xml:id="echoid-s8673" xml:space="preserve"> Quare proportio ſecundę lineę <lb/>ad quartam, id eſt a n ad g d, ſicut tertiæ ad primã, id eſt b g ad d h [eſt <lb/>enim per 16 p 6 ut d h ad d g, ſic b g ad a n:</s> <s xml:id="echoid-s8674" xml:space="preserve"> & per 16 p 5, ut d h ad b g, <lb/>ſic d g ad a n, & per conſectarium 4 p 5, ut a n ad d g, ſic b g ad d h.</s> <s xml:id="echoid-s8675" xml:space="preserve">] Sed iam probatum eſt, quòd pro-<lb/>portio a n ad d g, ſicut b g ad e q.</s> <s xml:id="echoid-s8676" xml:space="preserve"> Igitur [per 9 p 5] e q eſt ęqualis d h.</s> <s xml:id="echoid-s8677" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0150-02a" xlink:href="fig-0150-02"/> Et ita eſt propoſitum.</s> <s xml:id="echoid-s8678" xml:space="preserve"> Si uerò a g ſit minor a b:</s> <s xml:id="echoid-s8679" xml:space="preserve"> & ſecet a d arcum a b:</s> <s xml:id="echoid-s8680" xml:space="preserve"> <lb/>ſit ſectionis punctum h:</s> <s xml:id="echoid-s8681" xml:space="preserve"> & ducatur linea h g.</s> <s xml:id="echoid-s8682" xml:space="preserve"> Palàm [per fabricatio <lb/>nem primam & 29 p 1] quòd angulus n g d eſt æqualis angulo a b g:</s> <s xml:id="echoid-s8683" xml:space="preserve"> <lb/>ſed [per 27 p 3] anguli a b g, a h g ſunt æquales:</s> <s xml:id="echoid-s8684" xml:space="preserve"> quia cadunt in eundẽ <lb/>arcum.</s> <s xml:id="echoid-s8685" xml:space="preserve"> Igitur [per 1 ax] angulus n g d eſt æqualis angulo a h g, & an-<lb/>gulus n d g communis.</s> <s xml:id="echoid-s8686" xml:space="preserve"> Quare [per 32 p 1] tertius tertio æqualis:</s> <s xml:id="echoid-s8687" xml:space="preserve"> & <lb/>triangula ſimilia [per 4 p.</s> <s xml:id="echoid-s8688" xml:space="preserve"> 1 d 6.</s> <s xml:id="echoid-s8689" xml:space="preserve">] Igitur proportio h d ad d g, ſicut d g <lb/>ad d n.</s> <s xml:id="echoid-s8690" xml:space="preserve"> Quare [per 17 p 6] quod fit ex ductu h d in d n, eſt æquale qua <lb/>drato d g:</s> <s xml:id="echoid-s8691" xml:space="preserve"> ſed quod fit ex ductu h d in d a, eſt æquale ei, quod fit ex du <lb/>ctu b d in d g [per conſectarium Campani ad 36 p 3] & [per 1 p 2] qđ <lb/>fit ex ductu h d in d a, eſt æquale ei, quod fit ex ductu d n in h d & a n <lb/>in h d:</s> <s xml:id="echoid-s8692" xml:space="preserve"> & [per 3 p 2] ductus b d in d g ualet quadratum d g, & ductum <lb/>b g in d g.</s> <s xml:id="echoid-s8693" xml:space="preserve"> Igitur remotis æqualibus:</s> <s xml:id="echoid-s8694" xml:space="preserve"> [rectangulo nimirum h d n, & <lb/>quadrato d g] erit [per 3 ax] ductus h d in a n, ſicut b g in d g.</s> <s xml:id="echoid-s8695" xml:space="preserve"> Igitur <lb/>[per 16 p 6.</s> <s xml:id="echoid-s8696" xml:space="preserve"> 16 p.</s> <s xml:id="echoid-s8697" xml:space="preserve"> 13 d 5] proportio a n ad d g, ſicut b g ad h d.</s> <s xml:id="echoid-s8698" xml:space="preserve"> Sed iam <lb/>dictum eſt, quòd proportio a n ad d g eſt, ſicut b g ad e q.</s> <s xml:id="echoid-s8699" xml:space="preserve"> Igitur [per <lb/>9 p 5] e q eſt æqualis h d.</s> <s xml:id="echoid-s8700" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s8701" xml:space="preserve"> Quare à puncto a da <lb/>to, duximus lineam, ſecantem circulum, & à puncto ſectionis ad dia <lb/>metrum eſt æqualis lineæ datæ.</s> <s xml:id="echoid-s8702" xml:space="preserve"/> </p> <div xml:id="echoid-div333" type="float" level="0" n="0"> <figure xlink:label="fig-0149-01" xlink:href="fig-0149-01a"> <variables xml:id="echoid-variables55" xml:space="preserve">k t o z m u y f c l z</variables> </figure> <figure xlink:label="fig-0149-02" xlink:href="fig-0149-02a"> <variables xml:id="echoid-variables56" xml:space="preserve">q d <gap/> g e a b</variables> </figure> <figure xlink:label="fig-0150-01" xlink:href="fig-0150-01a"> <variables xml:id="echoid-variables57" xml:space="preserve">q d n e g h a b</variables> </figure> <figure xlink:label="fig-0150-02" xlink:href="fig-0150-02a"> <variables xml:id="echoid-variables58" xml:space="preserve">d q n g a e h b</variables> </figure> </div> </div> <div xml:id="echoid-div335" type="section" level="0" n="0"> <head xml:id="echoid-head337" xml:space="preserve" style="it">34. À<unsure/> puncto peripheriæ circuli extra datam diametrum dato, ducere lineam rectam, it æ <lb/>ſectam data diametro, ut ſegmentum inter diametrum & punctum peripheriæ dato puncto op <lb/>poſitum, æquetur datæ rectæ, minori circuli diametro. 133 p 1.</head> <p> <s xml:id="echoid-s8703" xml:space="preserve">AMplius à puncto dato in circulo, extra diametrum eius, eſt ducere lineam per diametrum ad <lb/>circulum, ut pars eius à diametro ad circulum, ſit æqualis lineę datæ.</s> <s xml:id="echoid-s8704" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s8705" xml:space="preserve"> a b g ſit da <lb/>tus circulus:</s> <s xml:id="echoid-s8706" xml:space="preserve"> b g diameter:</s> <s xml:id="echoid-s8707" xml:space="preserve"> a punctum datum:</s> <s xml:id="echoid-s8708" xml:space="preserve"> h z linea data.</s> <s xml:id="echoid-s8709" xml:space="preserve"> Dico, quòd à puncto a eſt duce <lb/>re lineam, tranſeuntem per diametrum b g, cuius pars à diametro ad circulum ſit æqualis lineæ h z.</s> <s xml:id="echoid-s8710" xml:space="preserve"> <lb/>Ducantur lineæ a b, a g:</s> <s xml:id="echoid-s8711" xml:space="preserve"> & [per 23 p 1] ſuper punctum h fiat angulus ęqualis angulo a g b per lineam <lb/>m h:</s> <s xml:id="echoid-s8712" xml:space="preserve"> & ſuper idem punctum fiat angulus ęqualis angulo a b g per lineam h l:</s> <s xml:id="echoid-s8713" xml:space="preserve"> & [per 31 p 1] à puncto z <lb/> <pb o="145" file="0151" n="151" rhead="OPTICAE LIBER V."/> ducatur æquidiſtans lineæ h m:</s> <s xml:id="echoid-s8714" xml:space="preserve"> quæ ſit z n:</s> <s xml:id="echoid-s8715" xml:space="preserve"> quæ quidem ſecabit h l [per lemma Procli ad 29 p 1] & <lb/>à puncto z ducatur æquidiſtans h l:</s> <s xml:id="echoid-s8716" xml:space="preserve"> quæ ſit z t:</s> <s xml:id="echoid-s8717" xml:space="preserve"> & ſecet h m in puncto t:</s> <s xml:id="echoid-s8718" xml:space="preserve"> & à puncto t ducatur ſectio <lb/>pyramidis t p, quam aſsignauit Apollonius in libro pyramidum [4 th 2:</s> <s xml:id="echoid-s8719" xml:space="preserve">] quæ quidem ſectio non <lb/>continget aliquam linearum z n, h l, inter quas iacet.</s> <s xml:id="echoid-s8720" xml:space="preserve"> Similiter fiat ſectio pyramidis ei oppoſita <lb/>inter eaſdem lineas:</s> <s xml:id="echoid-s8721" xml:space="preserve"> quæ ſit c u.</s> <s xml:id="echoid-s8722" xml:space="preserve"> Cum igitur li-<lb/> <anchor type="figure" xlink:label="fig-0151-01a" xlink:href="fig-0151-01"/> <anchor type="figure" xlink:label="fig-0151-02a" xlink:href="fig-0151-02"/> nea minima ex lineis à puncto t ad ſectionem <lb/>c u ductis, fuerit æqualis diametro b g:</s> <s xml:id="echoid-s8723" xml:space="preserve"> circu-<lb/>lus factus ſecundum hanc minimam lineam, <lb/>poſito pede circini ſuper punctum t:</s> <s xml:id="echoid-s8724" xml:space="preserve"> contin-<lb/>get ſectionem c u.</s> <s xml:id="echoid-s8725" xml:space="preserve"> Si uerò minima ex lineis à <lb/>puncto t ad ſectionem c u ductis, fuerit minor <lb/>diametro b g:</s> <s xml:id="echoid-s8726" xml:space="preserve"> circulus factus modo prædicto <lb/>ſecundum quãtitatem b g, ſecabit ſectionem <lb/>in duobus punctis.</s> <s xml:id="echoid-s8727" xml:space="preserve"> Sit ergo t c minima, & æ-<lb/>qualis diametro b g:</s> <s xml:id="echoid-s8728" xml:space="preserve"> quæ quidem ſecabit z n <lb/>& h l:</s> <s xml:id="echoid-s8729" xml:space="preserve"> cum ducatur ad ſectionem, quæ inter <lb/>eas interiacet:</s> <s xml:id="echoid-s8730" xml:space="preserve"> & [per 31 p 1] ducatur à puncto <lb/>z æ quidiſtans huic:</s> <s xml:id="echoid-s8731" xml:space="preserve"> quæ quidem ſecabit h m, <lb/>h l [per lemma Procli ad 29 p 1] ſicut ſua ęqui <lb/>diſtans t c.</s> <s xml:id="echoid-s8732" xml:space="preserve"> Secet ergo in punctis m l:</s> <s xml:id="echoid-s8733" xml:space="preserve"> & ſit m z <lb/>l:</s> <s xml:id="echoid-s8734" xml:space="preserve"> & punctum ſectionis, in quo t c ſecat z n, ſit <lb/>q:</s> <s xml:id="echoid-s8735" xml:space="preserve"> & [per 23 p 1] ſuper diametrum g b fiat an-<lb/>gulus ęqualis angulo h l m:</s> <s xml:id="echoid-s8736" xml:space="preserve"> qui ſit d g b, & du-<lb/>cantur lineę duę a d, d b.</s> <s xml:id="echoid-s8737" xml:space="preserve"> Palàm ergo, cũ [per <lb/>31 p 3] angulus g a b ſit rectus:</s> <s xml:id="echoid-s8738" xml:space="preserve"> alij duo anguli <lb/>trianguli a g b ualent rectũ [per 32 p 1.</s> <s xml:id="echoid-s8739" xml:space="preserve">] Quare <lb/>angulus l h m eſt rectus:</s> <s xml:id="echoid-s8740" xml:space="preserve"> [conſtat enim è duo-<lb/>bus angulis per fabricationẽ ęqualib.</s> <s xml:id="echoid-s8741" xml:space="preserve"> angulis <lb/>a g b, a b g rectũ ęquantibus] & eſt æqualis an <lb/>gulo g d b:</s> <s xml:id="echoid-s8742" xml:space="preserve"> & [per fabricationem] angulus h l <lb/>m eſt æqualis angulo d g b:</s> <s xml:id="echoid-s8743" xml:space="preserve"> Igitur [per 32 p 1] <lb/>tertius tertio:</s> <s xml:id="echoid-s8744" xml:space="preserve"> & triangulum ſimile triangulo <lb/>[per 4 p.</s> <s xml:id="echoid-s8745" xml:space="preserve"> 1 d 6.</s> <s xml:id="echoid-s8746" xml:space="preserve">] Quare ꝓportio b g ad b d eſt, <lb/>ſicut l m ad m h.</s> <s xml:id="echoid-s8747" xml:space="preserve"> Sed quoniam [per 27 p 3] angulus a d b æqualis eſt angulo b g a:</s> <s xml:id="echoid-s8748" xml:space="preserve"> quia cadunt in <lb/>eundem arcum:</s> <s xml:id="echoid-s8749" xml:space="preserve"> & angulus b g a æqualis angulo m h z:</s> <s xml:id="echoid-s8750" xml:space="preserve"> [per fabricationem] eſt ergo [per 1 ax.</s> <s xml:id="echoid-s8751" xml:space="preserve">] an-<lb/>gulus a d b æqualis angulo m h z.</s> <s xml:id="echoid-s8752" xml:space="preserve"> Et iam habemus, quòd angulus d b g eſt æqualis angulo h m z.</s> <s xml:id="echoid-s8753" xml:space="preserve"> I-<lb/>gitur [per 32 p 1] tertius tertio:</s> <s xml:id="echoid-s8754" xml:space="preserve"> & triangulum d e b ſimile triangulo m h z.</s> <s xml:id="echoid-s8755" xml:space="preserve"> [per 4 p.</s> <s xml:id="echoid-s8756" xml:space="preserve"> 1 d 6] Sit autem <lb/>e punctum, in quo linea a d ſecat diametrum b g.</s> <s xml:id="echoid-s8757" xml:space="preserve"> Igitur proportio b d ad d e, ſicut m h ad h z.</s> <s xml:id="echoid-s8758" xml:space="preserve"> Verùm <lb/>Apollonius [16 th 2] probat:</s> <s xml:id="echoid-s8759" xml:space="preserve"> quòd cum fuerint duæ ſectiones oppoſitæ, & producatur linea à ſe-<lb/>ctione ad aliam:</s> <s xml:id="echoid-s8760" xml:space="preserve"> pars eius, quæ interiacet inter unam ſectionẽ, & unam ex lineis, eſt ęqualis alij par-<lb/>ti, quæ interiacet inter aliam ſectionem, & aliam lineam.</s> <s xml:id="echoid-s8761" xml:space="preserve"> Quare q t æqualis eſt c f.</s> <s xml:id="echoid-s8762" xml:space="preserve"> Sed [per 34 p 1] t <lb/>q eſt æqualis m z:</s> <s xml:id="echoid-s8763" xml:space="preserve"> cum ſit illi æquidiſtans, inter duas æquidiſtantes.</s> <s xml:id="echoid-s8764" xml:space="preserve"> Igitur [per 1 ax.</s> <s xml:id="echoid-s8765" xml:space="preserve">] m z æqualis f <lb/>c:</s> <s xml:id="echoid-s8766" xml:space="preserve"> & [per 34 p 1] z l æqualis t f.</s> <s xml:id="echoid-s8767" xml:space="preserve"> Igitur [per 2 ax.</s> <s xml:id="echoid-s8768" xml:space="preserve">] m l æqualis t c.</s> <s xml:id="echoid-s8769" xml:space="preserve"> Quare proportio t c ad h z, ſicut m <lb/>l ad h z.</s> <s xml:id="echoid-s8770" xml:space="preserve"> [per 7 p 5] Quare proportio g b ad e d, ſicut t c ad h z.</s> <s xml:id="echoid-s8771" xml:space="preserve"> [demõſtratũ enim eſt, ut g b ad b d, ſic <lb/>l m ad m h:</s> <s xml:id="echoid-s8772" xml:space="preserve"> item ut b d ad d e, ſic m h ad h z:</s> <s xml:id="echoid-s8773" xml:space="preserve"> ergo per 22 p 5, ut g b ad d e, ſic l m ad h z:</s> <s xml:id="echoid-s8774" xml:space="preserve"> ſed ut l m ad h <lb/>z, ſic t c a d h z:</s> <s xml:id="echoid-s8775" xml:space="preserve"> quare per 11 p 5 ut g b ad d e, ſic t c ad h z.</s> <s xml:id="echoid-s8776" xml:space="preserve">] Et cum t c ſit æqualis b g [ex theſi] erit [per <lb/>14 p 5] e d æqualis h z.</s> <s xml:id="echoid-s8777" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s8778" xml:space="preserve"> Si autem lineat c ad ſectionem c u ducta, & minima:</s> <s xml:id="echoid-s8779" xml:space="preserve"> <lb/>fuerit minor diametro b g:</s> <s xml:id="echoid-s8780" xml:space="preserve"> producatur ultra ſectionem, donec ſit æqualis, & ſecundum quantitatẽ <lb/>eius fiat circulus:</s> <s xml:id="echoid-s8781" xml:space="preserve"> qui quidem circulus ſecabit ſectionem in duobus punctis:</s> <s xml:id="echoid-s8782" xml:space="preserve"> à quibus lineæ ductæ <lb/>ad t, erunt æquales b g:</s> <s xml:id="echoid-s8783" xml:space="preserve"> [per 15 d 1] & à puncto z ducatur ęquidiſtãs utriq.</s> <s xml:id="echoid-s8784" xml:space="preserve"> Et tunc erit ducere à pun <lb/>cto a modo prædicto duas lineas, æquales lineæ datæ:</s> <s xml:id="echoid-s8785" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s8786" xml:space="preserve"> idem penitus probandi modus.</s> <s xml:id="echoid-s8787" xml:space="preserve"/> </p> <div xml:id="echoid-div335" type="float" level="0" n="0"> <figure xlink:label="fig-0151-01" xlink:href="fig-0151-01a"> <variables xml:id="echoid-variables59" xml:space="preserve">a g e b d</variables> </figure> <figure xlink:label="fig-0151-02" xlink:href="fig-0151-02a"> <variables xml:id="echoid-variables60" xml:space="preserve">h n t f x q c u p m z ſ</variables> </figure> </div> </div> <div xml:id="echoid-div337" type="section" level="0" n="0"> <head xml:id="echoid-head338" xml:space="preserve" style="it">35. À<unsure/> puncto dato in altero laterum trianguli rectanguli angulum rectum continẽtium, <lb/>ducere per lat{us} angulo recto oppoſitum, rectam, cui{us} ſegmentum conterminum reliquo late-<lb/>ri infinito, habeat ad ſegmentum lateris angulo recto oppoſiti, conterminum primo lateri, ratio <lb/>nem in duab{us} rectis datam. 134 p 1.</head> <p> <s xml:id="echoid-s8788" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s8789" xml:space="preserve"> dato triangulo orthogonio, & dato puncto in uno laterum angulum rectum conti-<lb/>nentium, eſt ducere à puncto illo lineam, ad aliud laterum continentium rectum, ſecantem <lb/>tertium oppoſitum recto, ita ut pars huius lineę interiacens interpunctum ſectionis & latus, <lb/>in quo non eſt punctum datum, ſe habeat ad partem lateris oppoſiti recto, quæ eſt de ſectione ad la <lb/>tus, in quo eſt punctum datum, ſicut data linea ad datam lineam.</s> <s xml:id="echoid-s8790" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s8791" xml:space="preserve"> eſt triangulum datũ <lb/>a b g, cuius angulus a b g rectus:</s> <s xml:id="echoid-s8792" xml:space="preserve"> & in latere g b eſt punctum datum d, extra triangulum, aut intra.</s> <s xml:id="echoid-s8793" xml:space="preserve"> Di <lb/>co, quòd à puncto d eſt ducere lineam, ſecantem latus a g, & concurrentem cum latere a b:</s> <s xml:id="echoid-s8794" xml:space="preserve"> ita ut <lb/> <pb o="146" file="0152" n="152" rhead="ALHAZEN"/> pars eius interiacens inter latera a b, a g, ſit eius proportionis ad partem lateris a g, quæ eſt ab illa li-<lb/>nea uſq;</s> <s xml:id="echoid-s8795" xml:space="preserve"> ad punctum g, ſicut ſe habet e ad z, quæ ſunt datę lineæ.</s> <s xml:id="echoid-s8796" xml:space="preserve"> Sit punctum d in ipſo triangulo <lb/>a b g:</s> <s xml:id="echoid-s8797" xml:space="preserve"> & [per 31 p 1] ducatur ab eo linea æquidiſtans a b:</s> <s xml:id="echoid-s8798" xml:space="preserve"> quæ ſit d m:</s> <s xml:id="echoid-s8799" xml:space="preserve"> & [per 5 p 4] ſuper tria puncta <lb/>g, m, d fiat circulus:</s> <s xml:id="echoid-s8800" xml:space="preserve"> & protrahatur linea a d.</s> <s xml:id="echoid-s8801" xml:space="preserve"> Quoniam [per 29 p 1] <lb/> <anchor type="figure" xlink:label="fig-0152-01a" xlink:href="fig-0152-01"/> planum eſt, quòd angulus g m d eſt æqualis angulo g a b:</s> <s xml:id="echoid-s8802" xml:space="preserve"> erit [per <lb/>9 ax.</s> <s xml:id="echoid-s8803" xml:space="preserve">] maior g a d.</s> <s xml:id="echoid-s8804" xml:space="preserve"> Secetur ex eo æqualis per lineam m n:</s> <s xml:id="echoid-s8805" xml:space="preserve"> & ſit d n <lb/>m:</s> <s xml:id="echoid-s8806" xml:space="preserve"> & ſit [per 12 p 6] h linea, ad quam ſe habeat a d, ſicut ſe habet e <lb/>ad z:</s> <s xml:id="echoid-s8807" xml:space="preserve"> & à puncto n, quod eſt punctum circuli, ducatur linea ad dia-<lb/>metrum g m æqualis lineæ h, ſecundum ſuprà dicta [32 uel 33 n] & <lb/>ſit n l:</s> <s xml:id="echoid-s8808" xml:space="preserve"> [ita ut ſegmentum l c inter continuatam diametrum & peri <lb/>pheriam æquetur lineæ h] & punctum, in quo ſecat circulum, ſit c:</s> <s xml:id="echoid-s8809" xml:space="preserve"> <lb/>& ducatur linea g c:</s> <s xml:id="echoid-s8810" xml:space="preserve"> & à puncto d ducatur linea ad punctum c:</s> <s xml:id="echoid-s8811" xml:space="preserve"> quę, <lb/>cum cadat inter duas æquidiſtantes, tenens angulum acutum cum <lb/>altera, ſi producatur, neceſſariò concurret cum alia [per lẽma Pro-<lb/>cli ad 29 p 1.</s> <s xml:id="echoid-s8812" xml:space="preserve">] Cõcurrat igitur:</s> <s xml:id="echoid-s8813" xml:space="preserve"> & ſit punctum concurſus q.</s> <s xml:id="echoid-s8814" xml:space="preserve"> Planum <lb/>[per 27 p 3] quòd angulus g m d eſt æqualis angulo g c d:</s> <s xml:id="echoid-s8815" xml:space="preserve"> quia ca-<lb/>dunt in eundem arcum:</s> <s xml:id="echoid-s8816" xml:space="preserve"> & [per 29 p 1] angulus g m d eſt æqualis <lb/>angulo g a b:</s> <s xml:id="echoid-s8817" xml:space="preserve"> reſtat igitur [per 1 ax.</s> <s xml:id="echoid-s8818" xml:space="preserve"> 13 p 1.</s> <s xml:id="echoid-s8819" xml:space="preserve"> 3 ax.</s> <s xml:id="echoid-s8820" xml:space="preserve">] ut angulus g c q ſit <lb/>æqualis angulo g a q.</s> <s xml:id="echoid-s8821" xml:space="preserve"> Sit t punctum, in quo d q ſecat a g:</s> <s xml:id="echoid-s8822" xml:space="preserve"> & [per 15 <lb/>p 1] angulus g t c eſt æqualis angulo a t q:</s> <s xml:id="echoid-s8823" xml:space="preserve"> igitur [per 32 p 1] tertius <lb/>tertio.</s> <s xml:id="echoid-s8824" xml:space="preserve"> Quare triangulum a t q ſimile triangulo t c g [per 4 p.</s> <s xml:id="echoid-s8825" xml:space="preserve"> 1 d 6.</s> <s xml:id="echoid-s8826" xml:space="preserve">] <lb/>Igitur proportio q t ad t g, ſicut a t ad t c.</s> <s xml:id="echoid-s8827" xml:space="preserve"> Verùm [per fabricationẽ] <lb/>angulus n m d eſt æqualis angulo t a d:</s> <s xml:id="echoid-s8828" xml:space="preserve"> & [per 27 p 3] angulo n c d <lb/>[id eſt per 15 p 1 angulo l c t.</s> <s xml:id="echoid-s8829" xml:space="preserve">] Quare [per 1 ax.</s> <s xml:id="echoid-s8830" xml:space="preserve">] l c t æqualis t a d:</s> <s xml:id="echoid-s8831" xml:space="preserve"> & angulus c t l communis duobus <lb/>triangulis:</s> <s xml:id="echoid-s8832" xml:space="preserve"> quare [per 32 p 1] tertius tertio:</s> <s xml:id="echoid-s8833" xml:space="preserve"> & triangulum ſimile triangulo, ſcilicet t l c triangulot <lb/>a d [per 4 p.</s> <s xml:id="echoid-s8834" xml:space="preserve"> 1 d 6.</s> <s xml:id="echoid-s8835" xml:space="preserve">] Igitur proportio t a ad t c, ſicut proportio a d ad l c.</s> <s xml:id="echoid-s8836" xml:space="preserve"> Quare [per 11 p 5] proportio <lb/>a d ad l c, ſicut q t ad t g [patuit enim, ut q t ad t g, ſic a t ad t c.</s> <s xml:id="echoid-s8837" xml:space="preserve">] Sed [per fabricationem] l c eſt æqua <lb/>lis lineæ h:</s> <s xml:id="echoid-s8838" xml:space="preserve"> & [per fabricationem] proportio a d ad h, ſicut e ad z.</s> <s xml:id="echoid-s8839" xml:space="preserve"> Ergo [per 7.</s> <s xml:id="echoid-s8840" xml:space="preserve"> 11 p 5] proportio q t <lb/>ad t g, ſicut e ad z.</s> <s xml:id="echoid-s8841" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s8842" xml:space="preserve"> Si uerò d ſumatur in illo latere extra triangulum:</s> <s xml:id="echoid-s8843" xml:space="preserve"> produ-<lb/>catur [per 31 p 1] à puncto d, æquidiſtans a b:</s> <s xml:id="echoid-s8844" xml:space="preserve"> & ſit d m:</s> <s xml:id="echoid-s8845" xml:space="preserve"> & ducatur a g, donec concurrat cum d m in <lb/>puncto m, [concurrat autem per lemma Procli ad 29 p 1.</s> <s xml:id="echoid-s8846" xml:space="preserve">] Et fiat circulus tranſiens per tria pun-<lb/>cta g, d, m:</s> <s xml:id="echoid-s8847" xml:space="preserve"> & ducatur linea a d:</s> <s xml:id="echoid-s8848" xml:space="preserve"> erit <lb/> <anchor type="figure" xlink:label="fig-0152-02a" xlink:href="fig-0152-02"/> <anchor type="figure" xlink:label="fig-0152-03a" xlink:href="fig-0152-03"/> quidem [per 16 p 1] angulus g a d ma <lb/>ior angulo g m d:</s> <s xml:id="echoid-s8849" xml:space="preserve"> fiat [per 23 p 1] ei æ-<lb/>qualis:</s> <s xml:id="echoid-s8850" xml:space="preserve"> & ſit n m d:</s> <s xml:id="echoid-s8851" xml:space="preserve"> & à pũcto n, quod <lb/>eſt punctum circuli, ducatur linea æ-<lb/>qualis h lineæ [id uerò fiet per 33 uel <lb/>34 n:</s> <s xml:id="echoid-s8852" xml:space="preserve"> ita ut non tota linea à puncto n <lb/>ducta, ſed pars eius contermina dia-<lb/>metro extrà continuatæ, æquetur ipſi <lb/>h] ad quam lineã h ſe habeat a d, ſicut <lb/>e ad z, & ſit n c l [tota nimirum linea, <lb/>cuius pars c l æquetur lineæ h] ſuper <lb/>diametrum m g:</s> <s xml:id="echoid-s8853" xml:space="preserve"> & concurſus ſit l.</s> <s xml:id="echoid-s8854" xml:space="preserve"> Cũ <lb/>igitur [per 22 p 3] angulus n m d & an <lb/>gulus n c d ualeant duos rectos:</s> <s xml:id="echoid-s8855" xml:space="preserve"> & <lb/>[per fabricationem] angulus n m d <lb/>ſit ęqualis angulo t a d:</s> <s xml:id="echoid-s8856" xml:space="preserve"> erũt duo trian <lb/>gula t c l, t a d ſimilia.</s> <s xml:id="echoid-s8857" xml:space="preserve"> [Quia enim an-<lb/>guli n c d & l c d æquantur duobus re <lb/>ctis per 13 p 1, quibus item ex conclu-<lb/>ſo æquantur n c d & t a d:</s> <s xml:id="echoid-s8858" xml:space="preserve"> communi igitur n c d ſubducto, æquabitur reliquus l c d reliquo t a d:</s> <s xml:id="echoid-s8859" xml:space="preserve"> & <lb/>anguli ad uerticem t æquantur per 15 p 1, & per 32 p 1, tertius tertio.</s> <s xml:id="echoid-s8860" xml:space="preserve"> Quare triangula t c l, t a d ſunt ſi-<lb/>milia per 4 p.</s> <s xml:id="echoid-s8861" xml:space="preserve"> 1 d 6.</s> <s xml:id="echoid-s8862" xml:space="preserve">] Et cum [per 27 p 3] duo anguli g c d, g m d ſint æquales:</s> <s xml:id="echoid-s8863" xml:space="preserve"> erunt duo triangula gt <lb/>c, t a q ſimilia:</s> <s xml:id="echoid-s8864" xml:space="preserve"> [Nam cum angulus g m d æquetur angulo t a q per 29 p 1 (parallelæ enim ſunt d m <lb/>b a per fabricationem) æquabitur per 1 ax.</s> <s xml:id="echoid-s8865" xml:space="preserve"> angulus t a q angulo t c g, & ad uerticem t æquantur per <lb/>15 p 1:</s> <s xml:id="echoid-s8866" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s8867" xml:space="preserve"> per 32 p 1 triangula g t c, t a q ſunt æquiangula, & per 4 p.</s> <s xml:id="echoid-s8868" xml:space="preserve"> 1 d 6 ſimilia] & erit proportio <lb/>a d ad c l (quæ eſt æqualis h) ſicut q t ad t g:</s> <s xml:id="echoid-s8869" xml:space="preserve"> & ita eſt e ad z, ſicut q t ad t g.</s> <s xml:id="echoid-s8870" xml:space="preserve"> [Quia enim triangula t a d, <lb/>t c l ſunt æquiangula:</s> <s xml:id="echoid-s8871" xml:space="preserve"> erit per 4 p 6, ut a d ad c l, ſic a t ad t c.</s> <s xml:id="echoid-s8872" xml:space="preserve"> Rurſus quia triangula a t q, t c g funt æ-<lb/>quiangula:</s> <s xml:id="echoid-s8873" xml:space="preserve"> erit per 4 p 6, a t ad t c, ſicut q t ad t g.</s> <s xml:id="echoid-s8874" xml:space="preserve"> Quare per 11 p 5 ut a d ad cl (id eſt e ad z) ſic q t <lb/>ad t g.</s> <s xml:id="echoid-s8875" xml:space="preserve">] Quod eſt propoſitum.</s> <s xml:id="echoid-s8876" xml:space="preserve"/> </p> <div xml:id="echoid-div337" type="float" level="0" n="0"> <figure xlink:label="fig-0152-01" xlink:href="fig-0152-01a"> <variables xml:id="echoid-variables61" xml:space="preserve">q ſ a e z h a t d m c b d g n</variables> </figure> <figure xlink:label="fig-0152-02" xlink:href="fig-0152-02a"> <variables xml:id="echoid-variables62" xml:space="preserve">ſ a e z h d g c t b q a d n m</variables> </figure> <figure xlink:label="fig-0152-03" xlink:href="fig-0152-03a"> <variables xml:id="echoid-variables63" xml:space="preserve">d b q a ſ e z h <gap/> g c a m n d</variables> </figure> </div> </div> <div xml:id="echoid-div339" type="section" level="0" n="0"> <head xml:id="echoid-head339" xml:space="preserve" style="it">36. Duob{us} punctis extra circuli peripheriam, uel uno extra, reliquo intra datis: inuenire in <lb/>peripheria punctum, in quo recta linea ipſam tangẽs, bif ariam ſecet angulum comprehenſum <lb/> <pb o="147" file="0153" n="153" rhead="OPTICAE LIBER V."/> duab{us} rectis, à dictis punctis ad punctum tact{us} ductis. 135 p 1.</head> <p> <s xml:id="echoid-s8877" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s8878" xml:space="preserve"> duobus punctis datis, ſcilicet e, d, & dato circulo:</s> <s xml:id="echoid-s8879" xml:space="preserve"> eſt inuenire punctum in eo, ut an-<lb/>gulum contentum à lineis, à punctis prædictis ad illud punctum ductis, diuidat per æqualia, <lb/>linea circulum contingens in illo puncto.</s> <s xml:id="echoid-s8880" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s8881" xml:space="preserve"> ducatur à puncto e ad centrum circu <lb/>li dati, linea e g:</s> <s xml:id="echoid-s8882" xml:space="preserve"> & producatur uſq;</s> <s xml:id="echoid-s8883" xml:space="preserve"> ad circumferentiam:</s> <s xml:id="echoid-s8884" xml:space="preserve"> & ſit e s:</s> <s xml:id="echoid-s8885" xml:space="preserve"> deinde ducatur linea g d.</s> <s xml:id="echoid-s8886" xml:space="preserve"> Et ſit [per <lb/>10 p 6] m i linea diuiſa in puncto c, ut ſit proportio i c ad e m, ſicut e g ad g d:</s> <s xml:id="echoid-s8887" xml:space="preserve"> & [per 10 p 1] diuidatur <lb/>m i per æqualia in puncto n:</s> <s xml:id="echoid-s8888" xml:space="preserve"> & [per 11 p 1] ducatur perpendicularis n o:</s> <s xml:id="echoid-s8889" xml:space="preserve"> & ſuper punctum m fiat an-<lb/>gulus ęqualis medietati anguli d g s [per 9 & 23 p 1] per lineã m o.</s> <s xml:id="echoid-s8890" xml:space="preserve"> Palàm, quòd erit minor recto.</s> <s xml:id="echoid-s8891" xml:space="preserve"> [Nã <lb/>anguli ad g deinceps æquantur duobus rectis per 13 p 1:</s> <s xml:id="echoid-s8892" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s8893" xml:space="preserve"> d g s ijſdem eſt minor:</s> <s xml:id="echoid-s8894" xml:space="preserve"> quare d g s dimi <lb/>diatus minor eſt recto] & o n m rectus:</s> <s xml:id="echoid-s8895" xml:space="preserve"> igitur [per 11 ax.</s> <s xml:id="echoid-s8896" xml:space="preserve">] cõcurret cum n o:</s> <s xml:id="echoid-s8897" xml:space="preserve"> concurrat aũt in puncto <lb/>o:</s> <s xml:id="echoid-s8898" xml:space="preserve"> & ducatur à puncto c [per præcedentem numerum] linea ad triangulũ:</s> <s xml:id="echoid-s8899" xml:space="preserve"> quę ſit c k f:</s> <s xml:id="echoid-s8900" xml:space="preserve"> ita ut propor <lb/>tio k f ad m f ſit, ſicut proportio e g ad g s:</s> <s xml:id="echoid-s8901" xml:space="preserve"> & [per 23 p 1] ſuper punctum g [terminum lineæ e g] fiat <lb/>angulus æqualis angulo m f k, per lineam uſq;</s> <s xml:id="echoid-s8902" xml:space="preserve"> ad circulum ductam:</s> <s xml:id="echoid-s8903" xml:space="preserve"> quæ ſit a g:</s> <s xml:id="echoid-s8904" xml:space="preserve"> & ſit angulus a g e:</s> <s xml:id="echoid-s8905" xml:space="preserve"> & <lb/>ducantur lineæ a g, a d.</s> <s xml:id="echoid-s8906" xml:space="preserve"> Dico, quòd a eſt punctum, quod quęrimus.</s> <s xml:id="echoid-s8907" xml:space="preserve"> Ducatur linea e a.</s> <s xml:id="echoid-s8908" xml:space="preserve"> Cum igitur an <lb/>gulus m f k [per fabricationem] ſit æqualis angulo a g e:</s> <s xml:id="echoid-s8909" xml:space="preserve"> & [per fabricationem] proportio k f ad m <lb/>f, ſicut g e ad g a:</s> <s xml:id="echoid-s8910" xml:space="preserve"> cum [per 15 d 1] g a ſit æqualis g s:</s> <s xml:id="echoid-s8911" xml:space="preserve"> erit triangulum a g e ſimile triangulo m f k [per <lb/>6.</s> <s xml:id="echoid-s8912" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s8913" xml:space="preserve"> 1 d 6.</s> <s xml:id="echoid-s8914" xml:space="preserve">] Igitur angulus f m k eſt ęqualis angulo e a g, & angulus a e g æqualis angulo m k f.</s> <s xml:id="echoid-s8915" xml:space="preserve"> Iam <lb/>[per 23 p 1] à puncto a ducatur linea, tenens cum linea a e angulum æqualem angulo n m k:</s> <s xml:id="echoid-s8916" xml:space="preserve"> & ſit li-<lb/>nea a z:</s> <s xml:id="echoid-s8917" xml:space="preserve"> quę neceſſariò concurret cum linea e g.</s> <s xml:id="echoid-s8918" xml:space="preserve"> Quoniam, quæ eſt proportio k fad m f, ea eſt e g ad <lb/>g a, & angulus g a z ęqualis <lb/> <anchor type="figure" xlink:label="fig-0153-01a" xlink:href="fig-0153-01"/> <anchor type="figure" xlink:label="fig-0153-02a" xlink:href="fig-0153-02"/> eſt angulo f m c.</s> <s xml:id="echoid-s8919" xml:space="preserve"> [ęqualis e-<lb/>nim concluſus eſt angulus <lb/>f m k angulo e a g] Igitur ſi-<lb/>cut linea m o concurrit cũ <lb/>f k in puncto f:</s> <s xml:id="echoid-s8920" xml:space="preserve"> ſic concur-<lb/>ret a z cum e g.</s> <s xml:id="echoid-s8921" xml:space="preserve"> Sit concur-<lb/>ſus in puncto z:</s> <s xml:id="echoid-s8922" xml:space="preserve"> & produ-<lb/>catur a z uſq;</s> <s xml:id="echoid-s8923" xml:space="preserve"> ad punctũ q:</s> <s xml:id="echoid-s8924" xml:space="preserve"> <lb/>ita ut linea a z ſe habeat ad <lb/>z q, ſicut m c ad c i:</s> <s xml:id="echoid-s8925" xml:space="preserve"> [per 12 p <lb/>6] & ducatur linea e q.</s> <s xml:id="echoid-s8926" xml:space="preserve"> De-<lb/>inde [per 31 p 1] à puncto a <lb/>ducatur æquidiſtans e q:</s> <s xml:id="echoid-s8927" xml:space="preserve"> <lb/>quę ſit a t:</s> <s xml:id="echoid-s8928" xml:space="preserve"> erit quidem [per <lb/>29 p 1] angulus a q e æqua-<lb/>lis angulo q a t.</s> <s xml:id="echoid-s8929" xml:space="preserve"> Et quoniam duo anguli z e a, e a t ſunt minores duobus rectis [quia per 29 p 1 angu-<lb/>li q e a, e a t æquantur duobus rectis] concurret a t neceſſariò cum e z [per 11 ax.</s> <s xml:id="echoid-s8930" xml:space="preserve">] Sit concurſus pun <lb/>ctum t.</s> <s xml:id="echoid-s8931" xml:space="preserve"> Palàm [ex prius demonſtratis] quòd angulus a e g eſt æqualis angulo m k f.</s> <s xml:id="echoid-s8932" xml:space="preserve"> Ducta autem à <lb/>puncto e linea perpẽdiculari ſuper a z:</s> <s xml:id="echoid-s8933" xml:space="preserve"> quæ ſit e l:</s> <s xml:id="echoid-s8934" xml:space="preserve"> erit [per 32 p 1] angulus a e l æqualis angulo m k n:</s> <s xml:id="echoid-s8935" xml:space="preserve"> <lb/>cum [per fabricationem] angulus e a l ſit æqualis angulo k m n, & angulus a l e ęqualis m n k:</s> <s xml:id="echoid-s8936" xml:space="preserve"> quia <lb/>uterq;</s> <s xml:id="echoid-s8937" xml:space="preserve"> rectus:</s> <s xml:id="echoid-s8938" xml:space="preserve"> reſtat ergo [per 13 p 1.</s> <s xml:id="echoid-s8939" xml:space="preserve"> 3 ax.</s> <s xml:id="echoid-s8940" xml:space="preserve">] l e z æqualis angulo n k c:</s> <s xml:id="echoid-s8941" xml:space="preserve"> & angulus e l z rectus, ęqualis an <lb/>gulo k n c:</s> <s xml:id="echoid-s8942" xml:space="preserve"> reſtat [per 32 p 1] ut angulus e z l ſit ęqualis k c n:</s> <s xml:id="echoid-s8943" xml:space="preserve"> igitur [per 13 p 1.</s> <s xml:id="echoid-s8944" xml:space="preserve"> 3 ax.</s> <s xml:id="echoid-s8945" xml:space="preserve">] e z q æqualis angu <lb/>lo k c i.</s> <s xml:id="echoid-s8946" xml:space="preserve"> Palàm ergo [per 4 p.</s> <s xml:id="echoid-s8947" xml:space="preserve"> 1 d 6] quòd triangulum e a g ſimile eſt triangulo f m k:</s> <s xml:id="echoid-s8948" xml:space="preserve"> & triangulũ e a l <lb/>ſimile triangulo k m n:</s> <s xml:id="echoid-s8949" xml:space="preserve"> & triangulũ e l z ſimile k n c:</s> <s xml:id="echoid-s8950" xml:space="preserve"> & triangulũ e a z triangulo k m c [Nam ք fabri-<lb/>cationem angulus e a l æquatur angulo k m n, & angulus e z l ęqualis oſtenſus eſt angulo k c n:</s> <s xml:id="echoid-s8951" xml:space="preserve"> ergo <lb/>per 32 p 1 reliquus reliquo:</s> <s xml:id="echoid-s8952" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s8953" xml:space="preserve"> per 4 p.</s> <s xml:id="echoid-s8954" xml:space="preserve"> 1 d 6 triangula e a z, k m c ſunt ſimilia] Ergo proportio a z <lb/>ad z e, ſicut m c ad c k, & [per fabricationem] proportio a z ad z q, ſicut m c ad c i:</s> <s xml:id="echoid-s8955" xml:space="preserve"> Igitur [per 22 p 5] <lb/>proportio q z ad e z, ſicut i c ad c k.</s> <s xml:id="echoid-s8956" xml:space="preserve"> Quare [per 6.</s> <s xml:id="echoid-s8957" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s8958" xml:space="preserve"> 1 d 6] triangulum q z e ſimile triangulo i c k:</s> <s xml:id="echoid-s8959" xml:space="preserve"> & <lb/>triangulũ q l e ſimile triangulo i n k [quia iam patuit triangulum e l z ſimile eſſe triangulo k n c:</s> <s xml:id="echoid-s8960" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s8961" xml:space="preserve"> <lb/>cum partes partibus ſimiles ſint:</s> <s xml:id="echoid-s8962" xml:space="preserve"> totum triãgulum q l e toti i k n ſimile erit.</s> <s xml:id="echoid-s8963" xml:space="preserve"> Quare per 1 d 6, ut q l ad <lb/>l e, ſic i n ad n k:</s> <s xml:id="echoid-s8964" xml:space="preserve"> & ſimiliter ob triangulorum a e l, k m n ſimilitudinem eſt, ut e l ad a l, ſic k n ad m n] <lb/>erit ergo [per 22 p & conſectarium 4 p 5] proportio m n ad n i, ſicut a l ad l q:</s> <s xml:id="echoid-s8965" xml:space="preserve"> & ita a l æqualis l q [ꝗ̃a <lb/>m n æquata eſt ipſi n i] & [per 4 p 1] e q erit ęqualis e a:</s> <s xml:id="echoid-s8966" xml:space="preserve"> & angulus e q z æqualis angulo l a e:</s> <s xml:id="echoid-s8967" xml:space="preserve"> & [per <lb/>fabricationem & 29 p 1] angulus e q z ęqualis angulo z a t:</s> <s xml:id="echoid-s8968" xml:space="preserve"> igitur [per 15.</s> <s xml:id="echoid-s8969" xml:space="preserve"> 32 p 1] tertius tertio ęqualis.</s> <s xml:id="echoid-s8970" xml:space="preserve"> <lb/>Quare [per 4 p 6] proportio q z ad z a, ſicut e z ad z t, & ſicut e q ad at:</s> <s xml:id="echoid-s8971" xml:space="preserve"> & [per 7 p 5] ſicut a e ad a t.</s> <s xml:id="echoid-s8972" xml:space="preserve"> <lb/>Sed q z ad z a, ſicut e g ad d g [fuit enim per fabricationem e g ad g d, ſicut i c ad c m:</s> <s xml:id="echoid-s8973" xml:space="preserve"> item ut c m ad i <lb/>c, ſic a z a d z q, & per cõſectarium 4 p 5 ut i c ad c m, ſic z q ad a z:</s> <s xml:id="echoid-s8974" xml:space="preserve"> ergo per 11 p 5, ut e g ad g d, ſic z q ad <lb/>a z.</s> <s xml:id="echoid-s8975" xml:space="preserve">] Igitur [per 11 p 5] a e ad a t, ſicut e g ad g d.</s> <s xml:id="echoid-s8976" xml:space="preserve"> Fiat autem [per 23 p 1] ſuper punctum a angulus æ-<lb/>qualis angulo g a e:</s> <s xml:id="echoid-s8977" xml:space="preserve"> qui ſit u a g.</s> <s xml:id="echoid-s8978" xml:space="preserve"> Palàm, quòd angulus g a l eſt medietas anguli u a t:</s> <s xml:id="echoid-s8979" xml:space="preserve"> [Quia enim ex <lb/>concluſo anguli z a t, z a e æquantur eidem z q e:</s> <s xml:id="echoid-s8980" xml:space="preserve"> ipſi inter ſe æquantur.</s> <s xml:id="echoid-s8981" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s8982" xml:space="preserve"> ſi æqualib.</s> <s xml:id="echoid-s8983" xml:space="preserve"> æqualia ad-<lb/>dantur, æquabitur angulus g a l duobus angulis u a g, z a t.</s> <s xml:id="echoid-s8984" xml:space="preserve"> Quare totus u a t duplus erit anguli g a l] <lb/>Sed eſt medietas d g u:</s> <s xml:id="echoid-s8985" xml:space="preserve"> [quia angulus g a l ęqualis concluſus eſt angulo f m c:</s> <s xml:id="echoid-s8986" xml:space="preserve"> qui per fabricationem <lb/>eſt dimidius anguli d g u.</s> <s xml:id="echoid-s8987" xml:space="preserve">] Quare angulus u a t eſt ęqualis angulo d g u:</s> <s xml:id="echoid-s8988" xml:space="preserve"> [per 6 ax.</s> <s xml:id="echoid-s8989" xml:space="preserve">] ſed anguli u a t, <lb/> <pb o="148" file="0154" n="154" rhead="ALHAZEN"/> & t u a ſunt minores duobus rectis [per 17 p 1] cum a t & t u concurrant.</s> <s xml:id="echoid-s8990" xml:space="preserve"> Quare duo anguli t u a & d <lb/>g u ſunt minores duobus rectis:</s> <s xml:id="echoid-s8991" xml:space="preserve"> igitur [per 11 ax.</s> <s xml:id="echoid-s8992" xml:space="preserve">] a u concurret cum d g.</s> <s xml:id="echoid-s8993" xml:space="preserve"> Dico, quòd concurret in <lb/>puncto d:</s> <s xml:id="echoid-s8994" xml:space="preserve"> quoniam efficiet cum lineis u g, g d triangulum ſimile triãgulo a u t:</s> <s xml:id="echoid-s8995" xml:space="preserve"> habebunt enim angu <lb/>lum a u g communem:</s> <s xml:id="echoid-s8996" xml:space="preserve"> & angulus t a u eſt æqualis angulo d g u [per concluſionẽ.</s> <s xml:id="echoid-s8997" xml:space="preserve">] Igitur [per 4 p 6] <lb/>proportio a u ad a t, ſicut u g ad lineam, quã ſecat a u ex d g:</s> <s xml:id="echoid-s8998" xml:space="preserve"> & [per 3 p 6] proportio e a ad a u, ſicute <lb/>gad g u:</s> <s xml:id="echoid-s8999" xml:space="preserve"> cũ ſit angulus u a g ęqualis angulo g a e [per fabricationem.</s> <s xml:id="echoid-s9000" xml:space="preserve">] Cum ergo eadem ſit propor-<lb/>tio e a ad a t, ficut e g ad g d [ex concluſo] & proportio e a ad a t, ſit compoſita ex proportione e a ad <lb/>a u & a u ad a t [ratio enim extremorum cõponitur ex omnib.</s> <s xml:id="echoid-s9001" xml:space="preserve"> rationibus intermedijs, ut demonſtra <lb/>uit Theon ad 5 d 6] erit proportio e g ad g d cõpoſita ex ijſdem.</s> <s xml:id="echoid-s9002" xml:space="preserve"> Quare erit cõpacta ex proportione <lb/>e g ad g u & g u ad lineã, quã ſecat a u ex d g.</s> <s xml:id="echoid-s9003" xml:space="preserve"> Sed [ratio e g ad g d] eſt cõpacta ex proportionib.</s> <s xml:id="echoid-s9004" xml:space="preserve"> e g ad <lb/>g u & g u ad g d.</s> <s xml:id="echoid-s9005" xml:space="preserve"> Igitur linea, quã ſecat a u ex g d, eſt linea g d:</s> <s xml:id="echoid-s9006" xml:space="preserve"> igitur a u ſecat d g in puncto d.</s> <s xml:id="echoid-s9007" xml:space="preserve"> Produca <lb/>tur ergo [per 17 p 3] à puncto a cõtingens:</s> <s xml:id="echoid-s9008" xml:space="preserve"> quę ſit h a:</s> <s xml:id="echoid-s9009" xml:space="preserve"> erit ergo [per 18 p 3] g a h rectus:</s> <s xml:id="echoid-s9010" xml:space="preserve"> ſed g a l eſt me <lb/>dietas anguli d g u:</s> <s xml:id="echoid-s9011" xml:space="preserve"> igitur angulus l a h eſt medietas anguli d g e:</s> <s xml:id="echoid-s9012" xml:space="preserve"> cũ illi duo [d g u, d g e] ualeãt duos <lb/>rectos [per 13 p 1.</s> <s xml:id="echoid-s9013" xml:space="preserve">] Sed cũ angulus t a u ſit æqualis angulo d g u:</s> <s xml:id="echoid-s9014" xml:space="preserve"> erit angulus t a d ęqualis d g e [per 13 <lb/>p 1.</s> <s xml:id="echoid-s9015" xml:space="preserve"> 3 ax.</s> <s xml:id="echoid-s9016" xml:space="preserve">] Igitur angulus l a h eſt medietas anguli t a d:</s> <s xml:id="echoid-s9017" xml:space="preserve"> & angulus e a l medietas anguli e a t [quia, ut <lb/>patuit, e a l æquatur ipſi l a t:</s> <s xml:id="echoid-s9018" xml:space="preserve">] igitur angulus e a h medietas anguli e a d.</s> <s xml:id="echoid-s9019" xml:space="preserve"> Quare a h diuidit angulum <lb/>e a d per ęqualia.</s> <s xml:id="echoid-s9020" xml:space="preserve"> Quod eſt propoſitũ.</s> <s xml:id="echoid-s9021" xml:space="preserve"> Si uerò a u (cum ſit angulus ſuper punctum a ęqualis angu <lb/>lo g a e) non cadit ſuper lineam e s extra circulum, uel intra:</s> <s xml:id="echoid-s9022" xml:space="preserve"> ſit ergo æquidiſtans.</s> <s xml:id="echoid-s9023" xml:space="preserve"> Igitur [ք 29 p 1] an <lb/>gulus u a g ęqualis eſt angulo a g e:</s> <s xml:id="echoid-s9024" xml:space="preserve"> ſed idem eſt æqualis angulo g a e [ex theſi.</s> <s xml:id="echoid-s9025" xml:space="preserve">] Quare [per 1 ax.</s> <s xml:id="echoid-s9026" xml:space="preserve">] an <lb/>gulus g a e eſt æqua-<lb/> <anchor type="figure" xlink:label="fig-0154-01a" xlink:href="fig-0154-01"/> lis angulo a g e:</s> <s xml:id="echoid-s9027" xml:space="preserve"> igi-<lb/>tur [per 6 p 1] e g eſt <lb/>æqualis a e.</s> <s xml:id="echoid-s9028" xml:space="preserve"> Simili <lb/>ter angulus t a d erit <lb/>ęqualis angulo a t g <lb/>[per 29 p 1.</s> <s xml:id="echoid-s9029" xml:space="preserve">] Sed iam <lb/>dictum eſt [in primo <lb/>caſu huius numeri] <lb/>quòd angulus t a d <lb/>eſt ęqualis angulo d <lb/>g t.</s> <s xml:id="echoid-s9030" xml:space="preserve"> Igitur angulus a t g eſt ęqualis angulo d g t:</s> <s xml:id="echoid-s9031" xml:space="preserve"> & ſimiliter [per 29 p 1] duo anguli a d g, d g t ſunt ę-<lb/>quales:</s> <s xml:id="echoid-s9032" xml:space="preserve"> igitur duo anguli a d g, a t g ſunt ęquales.</s> <s xml:id="echoid-s9033" xml:space="preserve"> Sequetur ergo ex his, quòd linea, quam ſecat a u ex <lb/>d g, ſit ęqualis lineæ a t [nam cũ anguli a t g, d g t:</s> <s xml:id="echoid-s9034" xml:space="preserve"> itẽ a d g, t a d ęquentur:</s> <s xml:id="echoid-s9035" xml:space="preserve"> ęquabitur per 6 p 1 t m ipſi <lb/>m g:</s> <s xml:id="echoid-s9036" xml:space="preserve"> item m d ipſi m a.</s> <s xml:id="echoid-s9037" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s9038" xml:space="preserve"> ſi ęqualibus ęqualia addantur:</s> <s xml:id="echoid-s9039" xml:space="preserve"> ęquabitur d g ipſi a t.</s> <s xml:id="echoid-s9040" xml:space="preserve">] Et iam dictum eſt, <lb/>quòd e g ęqualis ſit a e.</s> <s xml:id="echoid-s9041" xml:space="preserve"> Igitur [per 7 p 5] proportio e g ad lineam, quã ſecat a u e x d g, eſt ſicut a e ad <lb/>a t.</s> <s xml:id="echoid-s9042" xml:space="preserve"> Sed iam dictum eſt ut a e ad a t, ſic e g ad g d:</s> <s xml:id="echoid-s9043" xml:space="preserve"> igitur linea, quã ſecat a u ex d g, eſt d g.</s> <s xml:id="echoid-s9044" xml:space="preserve"> Et cum t a d <lb/>ſit æqualis d g t:</s> <s xml:id="echoid-s9045" xml:space="preserve"> erit l a h medietas anguli t a d, ſicut dictum eſt ſuprà, & e a l medietas e a t.</s> <s xml:id="echoid-s9046" xml:space="preserve"> Erit ergo <lb/>e a h medietas anguli e a d.</s> <s xml:id="echoid-s9047" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s9048" xml:space="preserve"/> </p> <div xml:id="echoid-div339" type="float" level="0" n="0"> <figure xlink:label="fig-0153-01" xlink:href="fig-0153-01a"> <variables xml:id="echoid-variables64" xml:space="preserve">d a h ſ s u g e z t q</variables> </figure> <figure xlink:label="fig-0153-02" xlink:href="fig-0153-02a"> <variables xml:id="echoid-variables65" xml:space="preserve">o k f i l<unsure/> n m</variables> </figure> <figure xlink:label="fig-0154-01" xlink:href="fig-0154-01a"> <variables xml:id="echoid-variables66" xml:space="preserve">d a u m l t z c g s h q</variables> </figure> </div> </div> <div xml:id="echoid-div341" type="section" level="0" n="0"> <head xml:id="echoid-head340" xml:space="preserve" style="it">37. À<unsure/> dato extra circulum puncto, ducere ad datam diametrũ, lineã rectã: cui{us} pars inter <lb/>peripheriam & datam diametrum æquetur parti diametri centro circuli conterminæ. 136 p 1.</head> <p> <s xml:id="echoid-s9049" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s9050" xml:space="preserve"> dato circulo, cuius centrum g:</s> <s xml:id="echoid-s9051" xml:space="preserve"> & data in eo diametro b g:</s> <s xml:id="echoid-s9052" xml:space="preserve"> & dato e puncto extra cir-<lb/>culum:</s> <s xml:id="echoid-s9053" xml:space="preserve"> eſt ducere à puncto e ad diametrum b g, lineã ſecãtem circulum, ita ut pars eius à cir-<lb/>culo uſq;</s> <s xml:id="echoid-s9054" xml:space="preserve"> ad diametrũ ſit ęqualis parti diametri, interiacenti inter ipſam & centrum.</s> <s xml:id="echoid-s9055" xml:space="preserve"> Verbi <lb/>gratia:</s> <s xml:id="echoid-s9056" xml:space="preserve"> ducatur à puncto e perpendicularis ſuper diametrum:</s> <s xml:id="echoid-s9057" xml:space="preserve"> & ſit e c:</s> <s xml:id="echoid-s9058" xml:space="preserve"> & ducatur linea e g:</s> <s xml:id="echoid-s9059" xml:space="preserve"> & ſuma-<lb/>tur linea q t æqualis e c:</s> <s xml:id="echoid-s9060" xml:space="preserve"> & [per 33 p 3] fiat ſuper q t portio circuli, ut quilibet angulus cadens in hanc <lb/>portionem, ſit ęqualis angul<gap/> e g b:</s> <s xml:id="echoid-s9061" xml:space="preserve"> & compleatur circulus [per 25 p 3] & à medio puncto q t duca-<lb/>tur ex utraq;</s> <s xml:id="echoid-s9062" xml:space="preserve"> parte perpẽdicularis uſq;</s> <s xml:id="echoid-s9063" xml:space="preserve"> ad circulũ:</s> <s xml:id="echoid-s9064" xml:space="preserve"> erit quidẽ [per coſectarium 1 p 3] dιameter huius <lb/>circuli:</s> <s xml:id="echoid-s9065" xml:space="preserve"> & à puncto q ducatur linea ad hanc diametrũ, ſecans eam in puncto f, & producatur uſq;</s> <s xml:id="echoid-s9066" xml:space="preserve"> ad <lb/>punctum p circuli, ita ut f p ſit æqualis medietati g b [per 34 n] & ducatur linea p t, & linea t f, Et du <lb/>catur à puncto p linea <lb/> <anchor type="figure" xlink:label="fig-0154-02a" xlink:href="fig-0154-02"/> <anchor type="figure" xlink:label="fig-0154-03a" xlink:href="fig-0154-03"/> ęquidiſtans diametro:</s> <s xml:id="echoid-s9067" xml:space="preserve"> <lb/>quæ ſit p u:</s> <s xml:id="echoid-s9068" xml:space="preserve"> cõcurratq́;</s> <s xml:id="echoid-s9069" xml:space="preserve"> <lb/>cũ t f in puncto u:</s> <s xml:id="echoid-s9070" xml:space="preserve"> [con <lb/>curret autem per lem-<lb/>ma Procli ad 29 p 1] & <lb/>à puncto u ducatur æ-<lb/>quidiſtãs t q:</s> <s xml:id="echoid-s9071" xml:space="preserve"> quæ ſit u <lb/>o:</s> <s xml:id="echoid-s9072" xml:space="preserve"> & à pũcto t ducatur <lb/>perpendicularis ſuper <lb/>p q:</s> <s xml:id="echoid-s9073" xml:space="preserve"> quæ ſit t n:</s> <s xml:id="echoid-s9074" xml:space="preserve"> & à pun <lb/>cto t ducatur æquidiſtans p q:</s> <s xml:id="echoid-s9075" xml:space="preserve"> quæ ſit t s:</s> <s xml:id="echoid-s9076" xml:space="preserve"> & à puncto u perpendicularis ſuper p q:</s> <s xml:id="echoid-s9077" xml:space="preserve"> quæ ſit u h.</s> <s xml:id="echoid-s9078" xml:space="preserve"> Dein <lb/>de [per 23 p 1] ex angulo b g e ſecetur angulus æqualis angulo q p u:</s> <s xml:id="echoid-s9079" xml:space="preserve"> [id aũt fieri poteſt, cum totus <lb/>angulus q p t ęquetur ք theſin angulo b g e:</s> <s xml:id="echoid-s9080" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s9081" xml:space="preserve"> pars illius ab hoc toto detrahi poteſt] ꝗ ſit b g d:</s> <s xml:id="echoid-s9082" xml:space="preserve"> <lb/> <pb o="149" file="0155" n="155" rhead="OPTICAE LIBER V."/> & ducatur linea e d z.</s> <s xml:id="echoid-s9083" xml:space="preserve"> Dico, quòd d z eſt ęqualis z g:</s> <s xml:id="echoid-s9084" xml:space="preserve"> & ducatur à puncto d perpendicularis ſuper b <lb/>g:</s> <s xml:id="echoid-s9085" xml:space="preserve"> quæ ſit d i.</s> <s xml:id="echoid-s9086" xml:space="preserve"> & ducatur [per 17 p 3] à puncto d contingens:</s> <s xml:id="echoid-s9087" xml:space="preserve"> quę ſit d k.</s> <s xml:id="echoid-s9088" xml:space="preserve"> Palàm, cũ diameter fl ſit per-<lb/>pendicularis ſuper q t [per fabricationem] & ſuper o u [per 29 p 1] & p u ſit ęquidiſtans ei:</s> <s xml:id="echoid-s9089" xml:space="preserve"> erit [per <lb/>29 p 1] angulus o u p rectus:</s> <s xml:id="echoid-s9090" xml:space="preserve"> & cum o u diuidatur à diametro per æqualia & orthogonaliter:</s> <s xml:id="echoid-s9091" xml:space="preserve"> [Nam <lb/>per fabricationẽ, 29.</s> <s xml:id="echoid-s9092" xml:space="preserve"> 32 p 1 triangula f q l, f o m:</s> <s xml:id="echoid-s9093" xml:space="preserve"> itẽ f l t, f m u ſunt ęquiangula.</s> <s xml:id="echoid-s9094" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s9095" xml:space="preserve"> per 4 p 6 ut q l ad <lb/>l f, ſic o m ad m f:</s> <s xml:id="echoid-s9096" xml:space="preserve"> & ut fl ad l t, ſic f m ad m u:</s> <s xml:id="echoid-s9097" xml:space="preserve"> ergo per 22 p 5 ut q l ad l t, ſic o m ad m u:</s> <s xml:id="echoid-s9098" xml:space="preserve"> atqui per fabri-<lb/>cationem q l ęquatur ipſi l t:</s> <s xml:id="echoid-s9099" xml:space="preserve"> ergo o m ęquatur ipſi m u] erit [per 4 p 1] f o ęqualis fu, & angulus f o u <lb/>ęqualis angulo f u o.</s> <s xml:id="echoid-s9100" xml:space="preserve"> Sed cũ duo anguli p o u, o p u ualeant rectum [per 32 p 1:</s> <s xml:id="echoid-s9101" xml:space="preserve"> quia angulus ad u re-<lb/>ctus oſtenſus eſt] erit angulus f u p ęqualis angulo f p u.</s> <s xml:id="echoid-s9102" xml:space="preserve"> [angulus enim f o u ęquatur angulo f u o ex <lb/>concluſo, & anguli p o u & o p u ęquantur uni recto.</s> <s xml:id="echoid-s9103" xml:space="preserve"> Quare anguli f u o f p u æquantur uni recto:</s> <s xml:id="echoid-s9104" xml:space="preserve"> & <lb/>angulus o u p rectus eſt:</s> <s xml:id="echoid-s9105" xml:space="preserve"> ſubducto igitur cõmuni angulo f u o:</s> <s xml:id="echoid-s9106" xml:space="preserve"> reliquus f u p æquabitur reliquo f p u <lb/>per 3 ax.</s> <s xml:id="echoid-s9107" xml:space="preserve">] Quare [per 6 p 1] f p ęqualis eſt f u:</s> <s xml:id="echoid-s9108" xml:space="preserve"> & ita [per 1 ax.</s> <s xml:id="echoid-s9109" xml:space="preserve">] æqualis f o:</s> <s xml:id="echoid-s9110" xml:space="preserve"> & ita p o ęqualis b g:</s> <s xml:id="echoid-s9111" xml:space="preserve"> [quia <lb/>f p æquatur per fabricationem dimidię g b, & ex concluſo ipſi f o:</s> <s xml:id="echoid-s9112" xml:space="preserve"> tota igitur p o ęquatur toti b g] & <lb/>æqualis g d:</s> <s xml:id="echoid-s9113" xml:space="preserve"> [per 15 d 1] & ita [per 7 p 5] e c ad g d, ſicut t q ad p o.</s> <s xml:id="echoid-s9114" xml:space="preserve"> Sed cũ angulus k d g ſit rectus [per <lb/>18 p 3] æqualis angulo g i d:</s> <s xml:id="echoid-s9115" xml:space="preserve"> & angulus i g d cõmunis:</s> <s xml:id="echoid-s9116" xml:space="preserve"> erit triangulum i g d ſimile triangulo k d g:</s> <s xml:id="echoid-s9117" xml:space="preserve"> [ք <lb/>32 p 1.</s> <s xml:id="echoid-s9118" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s9119" xml:space="preserve"> 1 d 6] & proportio g d ad d i, ſicut g k a d k d.</s> <s xml:id="echoid-s9120" xml:space="preserve"> Sed [per fabricationẽ] angulus k g d eſt ęqua-<lb/>lis angulo o p u, & angulus k d g rectus, æqualis o u p:</s> <s xml:id="echoid-s9121" xml:space="preserve"> & ita triangulum k g d ſimile triangulo o u p:</s> <s xml:id="echoid-s9122" xml:space="preserve"> <lb/>& [per 1 d 6] proportio g k ad k d, ſicut o p ad o u.</s> <s xml:id="echoid-s9123" xml:space="preserve"> Igitur [per 11 p 5] d g ad d i, ſicut o p ad o u.</s> <s xml:id="echoid-s9124" xml:space="preserve"> Igitur <lb/>proportio e c ad d i, ſicut t q ad o u [demõſtratum enim eſt, ut e c ad g d, ſic t q ad p o:</s> <s xml:id="echoid-s9125" xml:space="preserve"> & ut g d ad d i, <lb/>ſic p o ad o u:</s> <s xml:id="echoid-s9126" xml:space="preserve"> ergo per 22 p 5, ut e c ad d i, ſic t q ad o u.</s> <s xml:id="echoid-s9127" xml:space="preserve">] Sed proportio q t ad o u, ſicut t f, ad f u:</s> <s xml:id="echoid-s9128" xml:space="preserve"> cũ triã <lb/>gulum t f q ſit ſimile triangulo o f u [per 29 p 1.</s> <s xml:id="echoid-s9129" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s9130" xml:space="preserve"> 1 d 6.</s> <s xml:id="echoid-s9131" xml:space="preserve">] Verũ [per fabricationem & 29 p 1] angulus <lb/>u t s æqualis angulo h f u:</s> <s xml:id="echoid-s9132" xml:space="preserve"> quia coalternus ei:</s> <s xml:id="echoid-s9133" xml:space="preserve"> & angulus u s t rectus, æqualis angulo f h u:</s> <s xml:id="echoid-s9134" xml:space="preserve"> erit trian-<lb/>gulum u s t ſimile triangulo h u f:</s> <s xml:id="echoid-s9135" xml:space="preserve"> & ita proportio t u ad u f, ſicut s u ad u h:</s> <s xml:id="echoid-s9136" xml:space="preserve"> [quare per 18 p 5 t fad u f, <lb/>ſicut s h ad u h.</s> <s xml:id="echoid-s9137" xml:space="preserve">] Sed [per 34 p 1] t n æqualis s h:</s> <s xml:id="echoid-s9138" xml:space="preserve"> cum ſit ei ęquidiſtans [per 28 p 1:</s> <s xml:id="echoid-s9139" xml:space="preserve"> quia anguli ad h & <lb/>n interiores ſunt recti per fabricationẽ] & ſint inter duas æquidiſtãtes.</s> <s xml:id="echoid-s9140" xml:space="preserve"> Igitur [per 7 p 5] proportio <lb/>t f ad u f, ſicut t n ad u h.</s> <s xml:id="echoid-s9141" xml:space="preserve"> Quare proportio q t ad o u, ſicut t n ad u h:</s> <s xml:id="echoid-s9142" xml:space="preserve"> & e c ad d i, ſicut t n ad u h [Nã o-<lb/>ſtenſum eſt, ut e c ad d i, ſic t q ad o u:</s> <s xml:id="echoid-s9143" xml:space="preserve"> itẽ ut t q ad o u, ſic t f ad u f:</s> <s xml:id="echoid-s9144" xml:space="preserve"> & ut t f ad u f, ſic s h, id eſt, t n ad u h:</s> <s xml:id="echoid-s9145" xml:space="preserve"> <lb/>ergo per 11 p 5 ut e c ad d i, ſic t n ad u h.</s> <s xml:id="echoid-s9146" xml:space="preserve">] Sed cum [per fabricationem] angulus g i d ſit rectus, æqua-<lb/>lis angulo p h u, & angulus i g d ęqualis angulo h p u:</s> <s xml:id="echoid-s9147" xml:space="preserve"> eſt triangulũ i g d ſimile h p u triangulo:</s> <s xml:id="echoid-s9148" xml:space="preserve"> & [per <lb/>1 d 6] proportio i d ad g d, ſicut h u ad u p:</s> <s xml:id="echoid-s9149" xml:space="preserve"> quare proportio e c ad g d, ſicut t n ad u p [oſtenſum enim <lb/>eſt proximè ut e c ad d i, ſic t n ad u h:</s> <s xml:id="echoid-s9150" xml:space="preserve"> & ut d i ad g d, ſic u h ad u p:</s> <s xml:id="echoid-s9151" xml:space="preserve"> ergo ex æquo ut e c ad d g, ſic t n ad <lb/>u p.</s> <s xml:id="echoid-s9152" xml:space="preserve">] Sed cum [per fabricationem] c g e ſit ęqualis angulo n p t, & angulus g c e rectus, ęqualis p n t:</s> <s xml:id="echoid-s9153" xml:space="preserve"> <lb/>erιt [ք 32 p 1.</s> <s xml:id="echoid-s9154" xml:space="preserve"> 4 p 6] g e ad e c, ſicut p t ad t n.</s> <s xml:id="echoid-s9155" xml:space="preserve"> Igitur g e ad g d, ſicut p t ad u p:</s> <s xml:id="echoid-s9156" xml:space="preserve"> [patuit enim, ut g e ad e c, <lb/>ſic p t ad t n:</s> <s xml:id="echoid-s9157" xml:space="preserve"> & ut e c ad d g, ſic t n ad u p:</s> <s xml:id="echoid-s9158" xml:space="preserve"> ergo ք 22 p 5, ut g e ad d g, ſic p t ad u p.</s> <s xml:id="echoid-s9159" xml:space="preserve">] Sed [ք fabricationẽ, <lb/>3 ax.</s> <s xml:id="echoid-s9160" xml:space="preserve">] angulus d g e eſt ęqualis angulo u p t.</s> <s xml:id="echoid-s9161" xml:space="preserve"> Igitur triangulũ d g e ſimile triangulo u p t:</s> <s xml:id="echoid-s9162" xml:space="preserve"> [ք 6.</s> <s xml:id="echoid-s9163" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s9164" xml:space="preserve"> 1 d <lb/>6] ergo angulus g d e ęqualis angulo p u t:</s> <s xml:id="echoid-s9165" xml:space="preserve"> reſtat ergo [per 13 p 1.</s> <s xml:id="echoid-s9166" xml:space="preserve"> 3 ax.</s> <s xml:id="echoid-s9167" xml:space="preserve">] angulus g d z ęqualis angulo <lb/>f u p:</s> <s xml:id="echoid-s9168" xml:space="preserve"> [& per fabricationẽ angulus f p u æquatur angulo z g d] quare tertius tertio [per 32 p 1:</s> <s xml:id="echoid-s9169" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s9170" xml:space="preserve"> <lb/>triangula z g d, f p u erunt æquiangula] & [per 4 p 6] proportio d z ad z g, ſicut u f ad f p:</s> <s xml:id="echoid-s9171" xml:space="preserve"> ſed u f ęqua <lb/>lis eſt f p.</s> <s xml:id="echoid-s9172" xml:space="preserve"> Ergo d z ęqualis z g.</s> <s xml:id="echoid-s9173" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s9174" xml:space="preserve"/> </p> <div xml:id="echoid-div341" type="float" level="0" n="0"> <figure xlink:label="fig-0154-02" xlink:href="fig-0154-02a"> <variables xml:id="echoid-variables67" xml:space="preserve">k b d z e i c g x</variables> </figure> <figure xlink:label="fig-0154-03" xlink:href="fig-0154-03a"> <variables xml:id="echoid-variables68" xml:space="preserve">p <gap/> n f o m u q ſ <gap/></variables> </figure> </div> </div> <div xml:id="echoid-div343" type="section" level="0" n="0"> <head xml:id="echoid-head341" xml:space="preserve" style="it">38. À<unsure/> puncto dato in altero laterũ trianguli rectanguli, angulũ rectũ continentiũ, ducere <lb/>ad lat{us} angulo recto oppoſitũ, rectã cõcurrẽtẽ cũ reliquo latere infinito: ita, ut tota ad ſegmẽtũ <lb/>lateris angulo recto oppoſiti, cõterminũ primo lateri, habeat rationẽ in duab. rectis datã. 137 p 1.</head> <p> <s xml:id="echoid-s9175" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s9176" xml:space="preserve"> dato triangulo orthogonio a b g:</s> <s xml:id="echoid-s9177" xml:space="preserve"> cuius angulus a b g rectus:</s> <s xml:id="echoid-s9178" xml:space="preserve"> & dato in b g, uel a b pũ <lb/>cto d:</s> <s xml:id="echoid-s9179" xml:space="preserve"> eſt ducere lineã à puncto d ad latus a g, concurrentẽ in puncto, quod ſit q:</s> <s xml:id="echoid-s9180" xml:space="preserve"> & ex alia par <lb/>te cõcurrentẽ cũ alio latere:</s> <s xml:id="echoid-s9181" xml:space="preserve"> ut ipſa totalis ſe habeat ad g q, ſicut eſt e ad z.</s> <s xml:id="echoid-s9182" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s9183" xml:space="preserve"> duca-<lb/>tur à puncto d ęquidiſtãs a b:</s> <s xml:id="echoid-s9184" xml:space="preserve"> quæ ſit d m:</s> <s xml:id="echoid-s9185" xml:space="preserve"> & [ք <lb/> <anchor type="figure" xlink:label="fig-0155-01a" xlink:href="fig-0155-01"/> 5 p 4] fiat circulus, trãſiens per tria puncta d, <lb/>m, g:</s> <s xml:id="echoid-s9186" xml:space="preserve"> erit m g diameter [per conſectariũ 5 p 4] <lb/>& ducatur linea a d:</s> <s xml:id="echoid-s9187" xml:space="preserve"> & ſit [per 12 p 6] h linea, ad <lb/>quã ſe habet a d, ſicut e ad z.</s> <s xml:id="echoid-s9188" xml:space="preserve"> Et cũ [per 29 p 1] <lb/>angulus d m g ſit ęqualis b a g:</s> <s xml:id="echoid-s9189" xml:space="preserve"> ſecetur ex eo ę-<lb/>qualis angulo d a g:</s> <s xml:id="echoid-s9190" xml:space="preserve"> & ſit c m d:</s> <s xml:id="echoid-s9191" xml:space="preserve"> & ducatur m c, <lb/>quouſq;</s> <s xml:id="echoid-s9192" xml:space="preserve"> contingat circulũ in puncto c:</s> <s xml:id="echoid-s9193" xml:space="preserve"> à quo <lb/>ducatur [per 34 n] linea ad diametrũ m g uſq;</s> <s xml:id="echoid-s9194" xml:space="preserve"> <lb/>ad circulum:</s> <s xml:id="echoid-s9195" xml:space="preserve"> ita quòd l n ſit æqualis lineæ h:</s> <s xml:id="echoid-s9196" xml:space="preserve"> & <lb/>ducatur linea n g, & linea d n cõcurrens cũ a g <lb/>in puncto q, & cũ a b in puncto t.</s> <s xml:id="echoid-s9197" xml:space="preserve"> Cũ igitur [ք <lb/>27 p 3] angulus d m c ſit æqualis angulo d n c:</s> <s xml:id="echoid-s9198" xml:space="preserve"> quia ſuper eundẽ arcũ:</s> <s xml:id="echoid-s9199" xml:space="preserve"> erit [per 1 ax.</s> <s xml:id="echoid-s9200" xml:space="preserve">] angulus q n l æ-<lb/>qualis angulo d a q, & [per 15 p 1] n q l æqualis angulo d q a.</s> <s xml:id="echoid-s9201" xml:space="preserve"> Quare [per 32 p 1.</s> <s xml:id="echoid-s9202" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s9203" xml:space="preserve"> 1 d 6] triangulum <lb/>n q l ſimile triangulo d q a:</s> <s xml:id="echoid-s9204" xml:space="preserve"> ergo a q ad q n, ſicut a d ad n l.</s> <s xml:id="echoid-s9205" xml:space="preserve"> Verũ cũ angulus d m g ſit ęqualis angulo d <lb/>n g [per 27 p 3] erit [per 1 ax.</s> <s xml:id="echoid-s9206" xml:space="preserve">] q n g ęqualis t a q.</s> <s xml:id="echoid-s9207" xml:space="preserve"> Sit t punctũ, in quo d n concurrit cũ a b:</s> <s xml:id="echoid-s9208" xml:space="preserve"> & [ք 15 p 1] <lb/>angulus t q a ęqualis angulo n q g:</s> <s xml:id="echoid-s9209" xml:space="preserve"> erit triangulũ t q a ſimile triãgulo n q g:</s> <s xml:id="echoid-s9210" xml:space="preserve"> & [per 1 d 6.</s> <s xml:id="echoid-s9211" xml:space="preserve"> 16 p 5] erit ꝓ-<lb/>portio a q ad q n, ſicut t q ad q g.</s> <s xml:id="echoid-s9212" xml:space="preserve"> Igitur [per 11 p 5] proportio t q ad q g, ſicut a d ad n l:</s> <s xml:id="echoid-s9213" xml:space="preserve"> ſed [per fabri-<lb/> <pb o="150" file="0156" n="156" rhead="ALHAZEN"/> cationẽ] n l ęqualis h:</s> <s xml:id="echoid-s9214" xml:space="preserve"> & a d ad h, ſicut e ad z.</s> <s xml:id="echoid-s9215" xml:space="preserve"> Igitur [ք 11 p 5] t q ad q g, ſicut e ad z.</s> <s xml:id="echoid-s9216" xml:space="preserve"> Qđ eſt ꝓpoſitum.</s> <s xml:id="echoid-s9217" xml:space="preserve"> <lb/>Põt aũt cõtĩgere:</s> <s xml:id="echoid-s9218" xml:space="preserve"> quòd à pũcto c erit ducere lineas duas, ſimiles c l n:</s> <s xml:id="echoid-s9219" xml:space="preserve"> & tũc erit ducere duas lineas à <lb/>puncto d, ſimiles t q, ut utriuſq;</s> <s xml:id="echoid-s9220" xml:space="preserve"> ad partẽ, ꝗ̃ ſecat ex a g, ſit ꝓportio, ſicut e ad z:</s> <s xml:id="echoid-s9221" xml:space="preserve"> & erit eadẽ probatio.</s> <s xml:id="echoid-s9222" xml:space="preserve"/> </p> <div xml:id="echoid-div343" type="float" level="0" n="0"> <figure xlink:label="fig-0155-01" xlink:href="fig-0155-01a"> <variables xml:id="echoid-variables69" xml:space="preserve">a a n m e z h q ſ b d g d t c</variables> </figure> </div> </div> <div xml:id="echoid-div345" type="section" level="0" n="0"> <head xml:id="echoid-head342" xml:space="preserve" style="it">39. Viſu & uiſibili à centro ſpeculi ſphærici conuexi inæquabiliter diſtantib{us}, punctum re-<lb/>flexionis inuenire. 22 p 6.</head> <p> <s xml:id="echoid-s9223" xml:space="preserve">PRędictis habitis, dato ſpeculo ſphærico:</s> <s xml:id="echoid-s9224" xml:space="preserve"> erit inuenire punctũ reflexionis.</s> <s xml:id="echoid-s9225" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s9226" xml:space="preserve"> ſit a cẽ-<lb/>trũ uiſus:</s> <s xml:id="echoid-s9227" xml:space="preserve"> b punctũ uiſum:</s> <s xml:id="echoid-s9228" xml:space="preserve"> g centrũ ſphærę:</s> <s xml:id="echoid-s9229" xml:space="preserve"> & ducantur lineę a g, b g:</s> <s xml:id="echoid-s9230" xml:space="preserve"> & ſumatur ſuperficies, in <lb/>qua ſunt hę duę lineæ [ſunt enim in eadẽ ſuքficie ք 23 n 4] & ſumatur circulus, cõmunis huic <lb/>ſuperficiei & ſpeculo.</s> <s xml:id="echoid-s9231" xml:space="preserve"> Inuenietur ergo punctũ reflexionis in hoc circulo.</s> <s xml:id="echoid-s9232" xml:space="preserve"> Et ſumatur linea alia m k:</s> <s xml:id="echoid-s9233" xml:space="preserve"> <lb/>& [ք 10 p 6] diuidatur in pũcto f, ut m f ſe habeat ad f k, ſicut b g ad g a:</s> <s xml:id="echoid-s9234" xml:space="preserve"> & [per 10 p 1] diuidatur m k ք <lb/>æqualia in puncto o:</s> <s xml:id="echoid-s9235" xml:space="preserve"> & [per 11 p 1] ducatur à puncto o perpẽdicularis:</s> <s xml:id="echoid-s9236" xml:space="preserve"> quę ſit c o:</s> <s xml:id="echoid-s9237" xml:space="preserve"> & ducatur à pũcto <lb/>k linea ad c o, tenens cũ ea angulũ æqualẽ medietati anguli b g a:</s> <s xml:id="echoid-s9238" xml:space="preserve"> [hoc aũt fiet:</s> <s xml:id="echoid-s9239" xml:space="preserve"> ſi linea e g bifariã ſe-<lb/>cans angulũ b g a, & o c infinitę intelligãtur, educta à puncto b perpendiculari ſuper e g, fiat angu-<lb/>lus o k c ęqualis angulo e b g:</s> <s xml:id="echoid-s9240" xml:space="preserve"> tũc enim (quia anguli ad o & e ſunt recti) ęquabitur angulus o c k an-<lb/>gulo e g b per 32 p 1] quę ſit k c:</s> <s xml:id="echoid-s9241" xml:space="preserve"> & à pũcto f ducatur linea ad c k:</s> <s xml:id="echoid-s9242" xml:space="preserve"> quę ſit f p:</s> <s xml:id="echoid-s9243" xml:space="preserve"> & cõcurrat cũ c o in pũcto <lb/>s, ita ut proportio s p ad p k ſit, ſicut b g ad ſemidiametrũ g d [per pręcedentẽ numerũ.</s> <s xml:id="echoid-s9244" xml:space="preserve">] Et [ք 23 p 1] <lb/>ex angulo b g a ſecetur ęqualis angulo f p k:</s> <s xml:id="echoid-s9245" xml:space="preserve"> [Id aũt fieri poſſe hinc cõſtat.</s> <s xml:id="echoid-s9246" xml:space="preserve"> Quia enim angulus s c p, <lb/>maior angulo c s p per 18 p 1 (cũ latus p s maius ſit latere c p:</s> <s xml:id="echoid-s9247" xml:space="preserve"> ſecus propoſitũ problema per lineã m k <lb/>expediri nõ poſſet) ęquetur per fabricationẽ dimidiato angulo b g a:</s> <s xml:id="echoid-s9248" xml:space="preserve"> ergo c s p eodẽ dimidiato mi-<lb/>nor eſt.</s> <s xml:id="echoid-s9249" xml:space="preserve"> Quare duo anguli s c p, c s p minores ſunt angulo b g a:</s> <s xml:id="echoid-s9250" xml:space="preserve"> at per 32 p 1 duob.</s> <s xml:id="echoid-s9251" xml:space="preserve"> angulis s c p, c s p <lb/>ęquatur angulus s p k:</s> <s xml:id="echoid-s9252" xml:space="preserve"> idcirco s p k minor eſt angulo b g a.</s> <s xml:id="echoid-s9253" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s9254" xml:space="preserve"> ab hoc ęqualis illi detrahi poteſt] ſci <lb/>licet d g b:</s> <s xml:id="echoid-s9255" xml:space="preserve"> & ducãtur lineę s k, b d:</s> <s xml:id="echoid-s9256" xml:space="preserve"> erit igitur [ք fabricationẽ] ꝓportio b g ad g d, ſicut s p ad p k:</s> <s xml:id="echoid-s9257" xml:space="preserve"> & i-<lb/>ta [per 6.</s> <s xml:id="echoid-s9258" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s9259" xml:space="preserve"> 1 d 6] triangulũ <lb/> <anchor type="figure" xlink:label="fig-0156-01a" xlink:href="fig-0156-01"/> <anchor type="figure" xlink:label="fig-0156-02a" xlink:href="fig-0156-02"/> s p k ſimile triãgulo b g d:</s> <s xml:id="echoid-s9260" xml:space="preserve"> & <lb/>erit angulus s k p ęqualis an <lb/>gulo b d g.</s> <s xml:id="echoid-s9261" xml:space="preserve"> Sed forſan ſecun-<lb/>dũ prędicta [34.</s> <s xml:id="echoid-s9262" xml:space="preserve"> 38 n] poteri-<lb/>mus à puncto f ducere aliã li <lb/>neã ad c k, ſimilẽ s p:</s> <s xml:id="echoid-s9263" xml:space="preserve"> ut ſit ꝓ-<lb/>portio eius ad partẽ, ꝗ̃ ſeca-<lb/>bit ex c k, ſicut s p ad p k:</s> <s xml:id="echoid-s9264" xml:space="preserve"> & <lb/>tũc à pũcto k ad o s ducetur <lb/>alia linea ꝗ̃ s k, aliũ cũ c k an-<lb/>gulũ tenẽs maiorẽ uel mino <lb/>rem angulo c k s Si maior ex <lb/>his angulis non fuerit maior <lb/>recto:</s> <s xml:id="echoid-s9265" xml:space="preserve"> nõ licebit inuenire pũ <lb/>ctũ reflexionis [ut mox oſtẽ <lb/>detur.</s> <s xml:id="echoid-s9266" xml:space="preserve">] Sit ergo angulus c k s maior recto:</s> <s xml:id="echoid-s9267" xml:space="preserve"> erit angulus b d g [ꝗ illi ęqualis eſt oſtẽſus] maior recto.</s> <s xml:id="echoid-s9268" xml:space="preserve"> <lb/>& inuenitur punctũ ſic.</s> <s xml:id="echoid-s9269" xml:space="preserve"> Ducatur [ք 17 p 3] cõtingens n d y.</s> <s xml:id="echoid-s9270" xml:space="preserve"> Et quia angulus p k o eſt minor recto [ք <lb/>32 p 1:</s> <s xml:id="echoid-s9271" xml:space="preserve"> ꝗ a c o k rectus eſt ք fabricationẽ] ſecetur [per 23 p 1] ex angulo b d g [ꝗ recto maior eſt ex con-<lb/>cluſione] æqualis ei:</s> <s xml:id="echoid-s9272" xml:space="preserve"> quι ſit q d g:</s> <s xml:id="echoid-s9273" xml:space="preserve"> eſt igitur triangulũ f p k ſimile triãgulo q g d [ęquatus.</s> <s xml:id="echoid-s9274" xml:space="preserve"> n.</s> <s xml:id="echoid-s9275" xml:space="preserve"> eſt angu-<lb/>lus s p k angulo b g d, & q d g angulo p k f:</s> <s xml:id="echoid-s9276" xml:space="preserve"> reliquus igitur ad f ęquatur reliquo ad q ք 32 p 1, & ք 4 p.</s> <s xml:id="echoid-s9277" xml:space="preserve"> 1 <lb/>d 6 triãgula f p k, q d g ſunt ſimilia] & erit angulus d q b ęqualis angulo k f s [ք 13 p 1.</s> <s xml:id="echoid-s9278" xml:space="preserve"> 3 ax.</s> <s xml:id="echoid-s9279" xml:space="preserve">] & trιãgulũ <lb/>d q b ſimile triangulo k f s.</s> <s xml:id="echoid-s9280" xml:space="preserve"> [totus.</s> <s xml:id="echoid-s9281" xml:space="preserve"> n.</s> <s xml:id="echoid-s9282" xml:space="preserve"> angulus p k s ęquatur toti b d g, ut patuit:</s> <s xml:id="echoid-s9283" xml:space="preserve"> & p k f ęquatur ipſi q d <lb/>g ք proximã fabricationẽ:</s> <s xml:id="echoid-s9284" xml:space="preserve"> ergo per 3 ax.</s> <s xml:id="echoid-s9285" xml:space="preserve"> reliquus f k s ęquatur reliquo q d b:</s> <s xml:id="echoid-s9286" xml:space="preserve"> & ք 32 p 1 tertius tertio.</s> <s xml:id="echoid-s9287" xml:space="preserve"> <lb/>Itaq;</s> <s xml:id="echoid-s9288" xml:space="preserve"> per 4 p.</s> <s xml:id="echoid-s9289" xml:space="preserve"> 1 d 6 d q b, k f s triangula ſunt ſimilia.</s> <s xml:id="echoid-s9290" xml:space="preserve">] Producatur d q, & [per 12 p 1] ducatur à puncto b <lb/>perpẽdicularis ſuք ipſam:</s> <s xml:id="echoid-s9291" xml:space="preserve"> quę ſit b z:</s> <s xml:id="echoid-s9292" xml:space="preserve"> erit [per 13 p 1] angulus b q z ęqualis angulo s f o & angulus b z <lb/>q rectus, ęqualis angulo s o f:</s> <s xml:id="echoid-s9293" xml:space="preserve"> & ita triãgulũ b q z ſimile triangulo s f o.</s> <s xml:id="echoid-s9294" xml:space="preserve"> Ducatur d z uſq;</s> <s xml:id="echoid-s9295" xml:space="preserve"> ad punctũ i:</s> <s xml:id="echoid-s9296" xml:space="preserve"> <lb/>& ſit z i ęqualis z d [per 3 p 1.</s> <s xml:id="echoid-s9297" xml:space="preserve">] Palã [è triangulorũ z q b, s o f:</s> <s xml:id="echoid-s9298" xml:space="preserve"> itẽ q d b, k f s ſimilitudine] quòd z q ad <lb/>q b, & q b ad q d, ſicut o f ad f s, & f s ad f k [ideoq́;</s> <s xml:id="echoid-s9299" xml:space="preserve"> per 22 p 5, ut z q ad q d, ſic o f ad f k] & ex hoc [per <lb/>18 p 5] z d ad q d, ſicut o k ad f k:</s> <s xml:id="echoid-s9300" xml:space="preserve"> & ita [ſumendo antecedentiũ dupla per 15 p 5] i d ad q d, ſicut m k <lb/>ad f k:</s> <s xml:id="echoid-s9301" xml:space="preserve"> & ita [per 17 p 5] i q ad q d, ſicut m f ad f k:</s> <s xml:id="echoid-s9302" xml:space="preserve"> & [per 11 p 5] i q ad q d, ſicut b g ad g a [eſt enim per <lb/>fabricationẽ m f ad f k, ſicut b g ad g a.</s> <s xml:id="echoid-s9303" xml:space="preserve">] Ducatur aũt linea b i:</s> <s xml:id="echoid-s9304" xml:space="preserve"> & ei æquidiſtãs d l:</s> <s xml:id="echoid-s9305" xml:space="preserve"> erit triangulũ l d q <lb/>ſimile triãgulo b q i:</s> <s xml:id="echoid-s9306" xml:space="preserve"> [Nã per 29 p 1 angulus q d l ęquatur angulo b i q, & per 15 p 1 d q lipſi b q i:</s> <s xml:id="echoid-s9307" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s9308" xml:space="preserve"> <lb/>per 32 p 1 reliquus reliquo:</s> <s xml:id="echoid-s9309" xml:space="preserve"> & per 4 p.</s> <s xml:id="echoid-s9310" xml:space="preserve"> 1 d 6 triãgula d q l, b q i erunt ſimilia] & ꝓportio i q ad q d, ſicut <lb/>i b ad d l.</s> <s xml:id="echoid-s9311" xml:space="preserve"> Et cum i z ſit ęqualis z d, & b z perpendicularis:</s> <s xml:id="echoid-s9312" xml:space="preserve"> erit [per 4 p 1] b d æqualis b i.</s> <s xml:id="echoid-s9313" xml:space="preserve"> Quare e-<lb/>rit [per 7.</s> <s xml:id="echoid-s9314" xml:space="preserve"> 11 p 5] b d ad d l, ſicut b g ad g a.</s> <s xml:id="echoid-s9315" xml:space="preserve"> Ducatur à puncto d linea:</s> <s xml:id="echoid-s9316" xml:space="preserve"> quę ſit d h, æqualem tenens angu <lb/>lum cũ linea l d, angulo b g a:</s> <s xml:id="echoid-s9317" xml:space="preserve"> & cũ h l & d l concurrant:</s> <s xml:id="echoid-s9318" xml:space="preserve"> erunt [per 17 p 1] l h d, l d h minores duobus <lb/>rectis:</s> <s xml:id="echoid-s9319" xml:space="preserve"> & ita duo anguli a g h, d h g, eis ęquales, ſunt minores duobus rectis:</s> <s xml:id="echoid-s9320" xml:space="preserve"> quare [ք 11 ax.</s> <s xml:id="echoid-s9321" xml:space="preserve">] h d cõcur <lb/>ret cũ g a.</s> <s xml:id="echoid-s9322" xml:space="preserve"> Dico quòd cõcurret in pũcto a.</s> <s xml:id="echoid-s9323" xml:space="preserve"> Palã [per 18 p 3] quòd angulus g d n rectus, eſt ęqualis duo <lb/>bus angulis o c k & o k c:</s> <s xml:id="echoid-s9324" xml:space="preserve"> [quia ęqualis eſt angulo m o c recto, ęquali eiſdẽ angulis per 32 p 1] & an-<lb/>gulus o k c ęqualis angulo g d q:</s> <s xml:id="echoid-s9325" xml:space="preserve"> per fabricationem] reſtat [per 3 ax.</s> <s xml:id="echoid-s9326" xml:space="preserve">] angulus q d n ęqualis angu-<lb/> <pb o="151" file="0157" n="157" rhead="OPTICAE LIBER V."/> lo o c k:</s> <s xml:id="echoid-s9327" xml:space="preserve"> & ita q d n medietas anguli b g a, & ita medietas anguli h d l [æquati angulo b g a.</s> <s xml:id="echoid-s9328" xml:space="preserve">] Sed [ք <lb/>3 p 6] angulus q d b eſt medietas anguli b d l:</s> <s xml:id="echoid-s9329" xml:space="preserve"> quoniã ꝓportio b q ad q l, ſicut b d ad d l:</s> <s xml:id="echoid-s9330" xml:space="preserve"> cũ triangulũ <lb/>d l q ſit ſimile triangulo b q i [ex cõcluſo] & b d æqualis b i, [ut patuit.</s> <s xml:id="echoid-s9331" xml:space="preserve">] Reſtat ergo, ut angulus n d <lb/>b ſit medietas anguli h d b:</s> <s xml:id="echoid-s9332" xml:space="preserve"> & ita b d n æqualis n d h.</s> <s xml:id="echoid-s9333" xml:space="preserve"> Producatur g d ultra d a d punctũ f.</s> <s xml:id="echoid-s9334" xml:space="preserve"> Quia igitur <lb/>[per 18 p 3] anguli f d n, g d n ſunt recti:</s> <s xml:id="echoid-s9335" xml:space="preserve"> ergo [ք 3 ax.</s> <s xml:id="echoid-s9336" xml:space="preserve">] reſtat b d f æqualis angulo h d g:</s> <s xml:id="echoid-s9337" xml:space="preserve"> Sed angulus <lb/>h d g æqualis angulo f d a contrà poſito [per 15 p 1.</s> <s xml:id="echoid-s9338" xml:space="preserve">] Quare b d f æqualis f d a.</s> <s xml:id="echoid-s9339" xml:space="preserve"> Et ita d eſt punctũ re-<lb/>flexionis [per 12 n 4.</s> <s xml:id="echoid-s9340" xml:space="preserve">] Ita dico:</s> <s xml:id="echoid-s9341" xml:space="preserve"> ſi a d cõcurrat cũ a g in pũcto a:</s> <s xml:id="echoid-s9342" xml:space="preserve"> quod quidẽ ſic patebit.</s> <s xml:id="echoid-s9343" xml:space="preserve"> Ducatur [per <lb/>31 p 1] linea h t æquidiſtãs b d.</s> <s xml:id="echoid-s9344" xml:space="preserve"> Palàm [è proximè demõſtratis] quòd angulus b d f æqualis eſt an-<lb/>gulo h d g:</s> <s xml:id="echoid-s9345" xml:space="preserve"> ſed [per 29 p 1] b d f eſt æqualis angulo h t d [ergo per 1 ax.</s> <s xml:id="echoid-s9346" xml:space="preserve"> h d g, h t d æquãtur.</s> <s xml:id="echoid-s9347" xml:space="preserve">] Quare <lb/>[per 6 p 1] h t erit æqualis h d.</s> <s xml:id="echoid-s9348" xml:space="preserve"> Sed proportio b d ad h t, ſicut b g ad g h.</s> <s xml:id="echoid-s9349" xml:space="preserve"> [ſunt enim triangula b d g, h <lb/>t g æquiangula:</s> <s xml:id="echoid-s9350" xml:space="preserve"> quãdoquidẽ angulus ad g cõmunis eſt, & g h t æquatur g b d per 29 p 1:</s> <s xml:id="echoid-s9351" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s9352" xml:space="preserve"> per <lb/>32 p 1 tertius tertio.</s> <s xml:id="echoid-s9353" xml:space="preserve"> Quare per 4 p 6 habẽt latera æqualib.</s> <s xml:id="echoid-s9354" xml:space="preserve"> angulis oppoſita homologa.</s> <s xml:id="echoid-s9355" xml:space="preserve">] Igitur [per <lb/>7 p 5] proportio b d ad d h, ſicut b g ad g h.</s> <s xml:id="echoid-s9356" xml:space="preserve"> Sed h d producta cõcurret cũ g a [ut mõſtratũ eſt] & fiet <lb/>triangulũ ſimile triangulo h d l:</s> <s xml:id="echoid-s9357" xml:space="preserve"> cũ habeant angulũ l h d cõmunẽ, & angulus h d l ſit æqualis angulo <lb/>h g a [per fabricationẽ:</s> <s xml:id="echoid-s9358" xml:space="preserve"> & per 32 p 1 reliquus reliquo.</s> <s xml:id="echoid-s9359" xml:space="preserve">] Igitur [per 4 p 6] proportio h d ad d l, ſicut h <lb/>g ad lineã, ꝗ̃ ſecat h d ex g a:</s> <s xml:id="echoid-s9360" xml:space="preserve"> & ꝓportio b d ad d l cõſtat ex ꝓportiõe b d ad d h, & d h ad d l [ratio.</s> <s xml:id="echoid-s9361" xml:space="preserve"> n.</s> <s xml:id="echoid-s9362" xml:space="preserve"> <lb/>extremorũ cõponitur ex omnib.</s> <s xml:id="echoid-s9363" xml:space="preserve"> ratiõib.</s> <s xml:id="echoid-s9364" xml:space="preserve"> intermedijs, ut oſtẽdit Theon ad 5 d 6.</s> <s xml:id="echoid-s9365" xml:space="preserve">] Igitur cõſtat ex b g <lb/>ad g h, & g h ad lineã, ꝗ̃ ſecath d ex g a:</s> <s xml:id="echoid-s9366" xml:space="preserve"> ſed b d ad d l, ſicut b g ad g a [ut patuit.</s> <s xml:id="echoid-s9367" xml:space="preserve">] Igitur ꝓportio b g ad <lb/>g a cõſtat ex ꝓportionib.</s> <s xml:id="echoid-s9368" xml:space="preserve"> b g ad g h & g h ad lineã, ꝗ̃ ſecat h d ex g a:</s> <s xml:id="echoid-s9369" xml:space="preserve"> ſed cõſtat ex ꝓportionib.</s> <s xml:id="echoid-s9370" xml:space="preserve"> b g ad <lb/>g h, & g h ad g a.</s> <s xml:id="echoid-s9371" xml:space="preserve"> Igitur g a eſt linea ꝗ̃ ſecat h d ex g a:</s> <s xml:id="echoid-s9372" xml:space="preserve"> & ita cõcurret cũ ea in pũcto a.</s> <s xml:id="echoid-s9373" xml:space="preserve"> Qđ eſt ꝓpoſitũ.</s> <s xml:id="echoid-s9374" xml:space="preserve"/> </p> <div xml:id="echoid-div345" type="float" level="0" n="0"> <figure xlink:label="fig-0156-01" xlink:href="fig-0156-01a"> <variables xml:id="echoid-variables70" xml:space="preserve">c p r m o f k y s</variables> </figure> <figure xlink:label="fig-0156-02" xlink:href="fig-0156-02a"> <variables xml:id="echoid-variables71" xml:space="preserve">b f e m h u d a i z q c t y g ſ</variables> </figure> </div> </div> <div xml:id="echoid-div347" type="section" level="0" n="0"> <head xml:id="echoid-head343" xml:space="preserve" style="it">40. Si radi{us} à uiſibili ſpeculo ſphærico cõuexo obliquè incidens, cum ſemidiametro eiuſdem an-<lb/>gulũ nõ maiorẽ recto coprehendat: non reflectetur ad uiſum ab illo incidẽtiæ puncto. 21. 22 p 6.</head> <p> <s xml:id="echoid-s9375" xml:space="preserve">SI uerò angulus c k s nõ fuerit maior recto.</s> <s xml:id="echoid-s9376" xml:space="preserve"> Dico, qđ nõ fiet reflexio ab aliquo pũcto ſpeculi ad <lb/>uiſum.</s> <s xml:id="echoid-s9377" xml:space="preserve"> Si enim dicatur, quòd poteſt:</s> <s xml:id="echoid-s9378" xml:space="preserve"> Sit d punctũ reflexionis:</s> <s xml:id="echoid-s9379" xml:space="preserve"> & producatur linea a d uſq;</s> <s xml:id="echoid-s9380" xml:space="preserve"> ad h <lb/>punctũ in diametro b g.</s> <s xml:id="echoid-s9381" xml:space="preserve"> Et [per 23 p 1] fiat angulus l d h æqualis angulo a g b:</s> <s xml:id="echoid-s9382" xml:space="preserve"> & producatur <lb/>cõtingens n d y:</s> <s xml:id="echoid-s9383" xml:space="preserve"> & fiat angu <lb/> <anchor type="figure" xlink:label="fig-0157-01a" xlink:href="fig-0157-01"/> <anchor type="figure" xlink:label="fig-0157-02a" xlink:href="fig-0157-02"/> lus q d n æqualis medietati <lb/>anguli a g b.</s> <s xml:id="echoid-s9384" xml:space="preserve"> Palàm, quòd <lb/>triangulũ h d l ſimile eſt tri-<lb/>angulo h g a [quia enim an-<lb/>gulus h d l æquatus eſt an-<lb/>gulo h g a:</s> <s xml:id="echoid-s9385" xml:space="preserve"> & d h g eſt com-<lb/>munis:</s> <s xml:id="echoid-s9386" xml:space="preserve"> æquabitur per 32 p 1 <lb/>tertius tertio:</s> <s xml:id="echoid-s9387" xml:space="preserve"> & per 4 p.</s> <s xml:id="echoid-s9388" xml:space="preserve"> 1 d <lb/>6 triangula erunt ſimilia.</s> <s xml:id="echoid-s9389" xml:space="preserve">] <lb/>Quare proportio d h ad d l, <lb/>ſicut h g ad g a:</s> <s xml:id="echoid-s9390" xml:space="preserve"> ſed b d ad d <lb/>h, ſicut b g ad g h:</s> <s xml:id="echoid-s9391" xml:space="preserve"> qđ pate-<lb/>bit per æquidiſtantẽ h t ipſi <lb/>b d.</s> <s xml:id="echoid-s9392" xml:space="preserve"> [ſic enim triangula b g <lb/>d, h g t fient æquiangula.</s> <s xml:id="echoid-s9393" xml:space="preserve"> Et <lb/>h d ęquatur ipſi h t.</s> <s xml:id="echoid-s9394" xml:space="preserve"> Nam quia d per theſin eſt punctum reflexionis, & e g perpendicularis plano ſpe <lb/>culũ in reflexionis puncto tãgenti per 25 n 4:</s> <s xml:id="echoid-s9395" xml:space="preserve"> ęquabitur angulus b d e angulo a d e per 12 n 4:</s> <s xml:id="echoid-s9396" xml:space="preserve"> & per <lb/>29 p 1 b d e, id eſt a d e, id eſt ք 15 p 1 h d t ęquatur ipſi h t d:</s> <s xml:id="echoid-s9397" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s9398" xml:space="preserve"> per 6 p 1 latus h d æquatur lateri h t.</s> <s xml:id="echoid-s9399" xml:space="preserve">] <lb/>Igitur b d ad d l, ſicut b g ad g a [quia enim ex cõcluſo eſt, ut b d ad d h, ſic b g ad g h:</s> <s xml:id="echoid-s9400" xml:space="preserve"> itẽ ut d h ad d l, <lb/>ſic g h ad g a:</s> <s xml:id="echoid-s9401" xml:space="preserve"> erit per 22 p 5, ut b d ad d l, ſic b g ad g a.</s> <s xml:id="echoid-s9402" xml:space="preserve">] Sed cũ angulus b d e ſit æqualis angulo h d g:</s> <s xml:id="echoid-s9403" xml:space="preserve"> <lb/>[ex cõcluſo] erit angulus b d n medietas anguli b d h [nã angulι n d e, n d g recti per 18 p 3, æquãtur <lb/>per 10 ax:</s> <s xml:id="echoid-s9404" xml:space="preserve"> & b d e ipſi h d g:</s> <s xml:id="echoid-s9405" xml:space="preserve"> ergo per 3 ax.</s> <s xml:id="echoid-s9406" xml:space="preserve"> reliquus b d n reliquo h d n æquatur.</s> <s xml:id="echoid-s9407" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s9408" xml:space="preserve"> b d n dimidius <lb/>eſt ipſius b d h.</s> <s xml:id="echoid-s9409" xml:space="preserve">] Sed n d q eſt medietas anguli h d l [eſt enim per fabricationẽ dimidius anguli a g b, <lb/>cui æquatus eſt h d l.</s> <s xml:id="echoid-s9410" xml:space="preserve">] Igitur b d q medietas anguli b d l.</s> <s xml:id="echoid-s9411" xml:space="preserve"> Quare [per 3 p 6] proportio b q ad q l, ſicut <lb/>b d ad d l.</s> <s xml:id="echoid-s9412" xml:space="preserve"> Ducatur [ք 31 p 1] à pũcto b ęquidiſtãs d l:</s> <s xml:id="echoid-s9413" xml:space="preserve"> & ſit b i:</s> <s xml:id="echoid-s9414" xml:space="preserve"> & cõcurrat d q cũ e a in pũcto i:</s> <s xml:id="echoid-s9415" xml:space="preserve"> [cõcur-<lb/>ret aũt per lẽma Procli ad 29 p 1] & [ք 10 p 1] diuidatur d i in æqualia in pũcto z:</s> <s xml:id="echoid-s9416" xml:space="preserve"> & ducatur b z:</s> <s xml:id="echoid-s9417" xml:space="preserve"> e-<lb/>rit [ք 29.</s> <s xml:id="echoid-s9418" xml:space="preserve"> 15.</s> <s xml:id="echoid-s9419" xml:space="preserve"> 32 p 1.</s> <s xml:id="echoid-s9420" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s9421" xml:space="preserve"> 1 d 6] triangulũ b q i ſimile triangulo q d l.</s> <s xml:id="echoid-s9422" xml:space="preserve"> Igitur ut b q ad q l, ſic b i ad d l [at <lb/>oſtẽſum eſt, ut b q ad q l, ſic b d ad d l:</s> <s xml:id="echoid-s9423" xml:space="preserve"> ergo per 11 p 5 ut b i ad d l, ſic b d ad d l] & ita [per 9 p 5] b i ę-<lb/>qualis b d:</s> <s xml:id="echoid-s9424" xml:space="preserve"> & i q ad q d, ſicut m f ad f k:</s> <s xml:id="echoid-s9425" xml:space="preserve"> [eſt enim ob triangulorum b q i, q d l ſimilitudinem, ut i q ad <lb/>q d, ſic b q ad q l:</s> <s xml:id="echoid-s9426" xml:space="preserve"> & ut b q ad q l, ſic b i, id eſt, b d ad d l:</s> <s xml:id="echoid-s9427" xml:space="preserve"> & ut b d ad d l, ſic b g ad g a ex cõcluſo:</s> <s xml:id="echoid-s9428" xml:space="preserve"> & ut b g <lb/>ad g a, ſic m f ad f k per fabricationẽ:</s> <s xml:id="echoid-s9429" xml:space="preserve"> ergo per 11 p 5, ut i q ad q d, ſic m f ad f k] & ita [per 18 p 5] i d ad <lb/>q d, ſicut m k ad f k:</s> <s xml:id="echoid-s9430" xml:space="preserve"> & ita [ſumendo antecedentiũ dimidia per 15 p 5] d z ad q d, ſicut o k ad f k:</s> <s xml:id="echoid-s9431" xml:space="preserve"> & ita <lb/>[per 17 p 5] z q ad q d, ſicut o f ad fk.</s> <s xml:id="echoid-s9432" xml:space="preserve"> Palàm, quòd b z eſt perpendicularis:</s> <s xml:id="echoid-s9433" xml:space="preserve"> [quia enim b i æquatur <lb/>b d ex concluſo, & i z ipſi z d per fabricationem, & b z communis eſt:</s> <s xml:id="echoid-s9434" xml:space="preserve"> erũt triangula i b z, d b z æqui-<lb/>angula ք 8 p 1:</s> <s xml:id="echoid-s9435" xml:space="preserve"> & angulus b z i æquabitur angulo b z d:</s> <s xml:id="echoid-s9436" xml:space="preserve"> ſuntq́;</s> <s xml:id="echoid-s9437" xml:space="preserve"> deinceps:</s> <s xml:id="echoid-s9438" xml:space="preserve"> Quare per 10 d 1 b z perpẽdi <lb/>cularis eſt i d] ꝓducatur, donec cõcurrat cũ d g in pũcto x:</s> <s xml:id="echoid-s9439" xml:space="preserve"> qđ quidẽ poſsibile eſt [per 11 ax.</s> <s xml:id="echoid-s9440" xml:space="preserve">] cũ an-<lb/>gulus d z x ſit rectus, z d x minor recto.</s> <s xml:id="echoid-s9441" xml:space="preserve"> Et palã, qđ ꝓportio b g ad g d, ſicut s p ad p k:</s> <s xml:id="echoid-s9442" xml:space="preserve"> [ք fabricatio-<lb/>nẽ.</s> <s xml:id="echoid-s9443" xml:space="preserve">] Cũ ergo angulus c k s dicatur nõ eſſe maior recto:</s> <s xml:id="echoid-s9444" xml:space="preserve"> dico, qđ ſuք pũctũ k fiet maior recto, ք lineã <lb/> <pb o="152" file="0158" n="158" rhead="ALHAZEN"/> cõcurrentẽ cũ co in pũcto, à quo ducetur linea ad ck, trãſiens ք pũctũ f, retinẽs proportionẽ ad par <lb/>tẽ p k, ſicut b g ad g d.</s> <s xml:id="echoid-s9445" xml:space="preserve"> Quòd aũt hoc poſsibile, planum eſt:</s> <s xml:id="echoid-s9446" xml:space="preserve"> cũ angulus q d n ſit æ qualis angulo k c o:</s> <s xml:id="echoid-s9447" xml:space="preserve"> <lb/>[eſt enim uterq;</s> <s xml:id="echoid-s9448" xml:space="preserve"> dimidius duorũ æ qualiũ b g a, h d l, ut mõſtratũ eſt] erit angulus q d g æ qualis an-<lb/>gulo cko.</s> <s xml:id="echoid-s9449" xml:space="preserve"> [quia enim trianguli c o k angulus ad o rectus eſt:</s> <s xml:id="echoid-s9450" xml:space="preserve"> reliqui o c k, o k c æ quãtur uni recto ք <lb/>32 p 1:</s> <s xml:id="echoid-s9451" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s9452" xml:space="preserve"> ք 10 ax.</s> <s xml:id="echoid-s9453" xml:space="preserve"> angulo n d g recto per 18 p 3:</s> <s xml:id="echoid-s9454" xml:space="preserve"> & o c k æ quatur q d n:</s> <s xml:id="echoid-s9455" xml:space="preserve"> ergo per 3 ax.</s> <s xml:id="echoid-s9456" xml:space="preserve"> reliquus o k c <lb/>ęquatur reliquo q d g.</s> <s xml:id="echoid-s9457" xml:space="preserve">] Fiat ergo ſuք pũctũ k angulus ęqualis b d q:</s> <s xml:id="echoid-s9458" xml:space="preserve"> & ponatur, quòd linea hũc angu <lb/>lũ tenẽs, cõcurrat cũ c o in pũcto s:</s> <s xml:id="echoid-s9459" xml:space="preserve"> & ducatur s fp.</s> <s xml:id="echoid-s9460" xml:space="preserve"> Planũ eſt, cũ angulus b z d rectus, ęqualis angu-<lb/>lo s o k:</s> <s xml:id="echoid-s9461" xml:space="preserve"> qđ erit triangulũ b z d ſimile s o k [per proximã fabricationẽ.</s> <s xml:id="echoid-s9462" xml:space="preserve"> 32 p 1.</s> <s xml:id="echoid-s9463" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s9464" xml:space="preserve"> 1 d 6] & b z ad b d, <lb/>ſicut o s ad s k, & b z ad z d, ſicut s o ad o k:</s> <s xml:id="echoid-s9465" xml:space="preserve"> ſed q z ad q d, ſicut o f ad fk.</s> <s xml:id="echoid-s9466" xml:space="preserve"> [ex cõcluſo:</s> <s xml:id="echoid-s9467" xml:space="preserve"> & ք cõſectariũ 4 <lb/>p 5, ut q d ad q z, ſic fk ad o f:</s> <s xml:id="echoid-s9468" xml:space="preserve"> & per 18 p 5, ut z d ad q z, ſic o k ad o f:</s> <s xml:id="echoid-s9469" xml:space="preserve"> eſt autẽ ut b z ad z d, ſic s o ad o <lb/>k:</s> <s xml:id="echoid-s9470" xml:space="preserve"> & ut z d ad q z, ſic o k ad o f:</s> <s xml:id="echoid-s9471" xml:space="preserve"> ergo per 22 p 5, ut b z ad z q, ſic s o ad o f.</s> <s xml:id="echoid-s9472" xml:space="preserve"> Triangula igitur b z q, s o f <lb/>ſunt æ quiangula per 6.</s> <s xml:id="echoid-s9473" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s9474" xml:space="preserve"> 1 d 6 ſimilia.</s> <s xml:id="echoid-s9475" xml:space="preserve">] erit ergo angulus z b q æ qualis angulo o s f:</s> <s xml:id="echoid-s9476" xml:space="preserve"> & angulus q b <lb/>d æ qualis angulo f s k.</s> <s xml:id="echoid-s9477" xml:space="preserve"> [per 3 ax.</s> <s xml:id="echoid-s9478" xml:space="preserve"> totus enim angulus o s k toti z b d æquatur, ob triangulorũ o s k, z <lb/>b d ſimilitudinẽ iã demõſtratã.</s> <s xml:id="echoid-s9479" xml:space="preserve">] Quare triangulũ b g d ſimile triangulo s p k.</s> <s xml:id="echoid-s9480" xml:space="preserve"> [Nã angulus q d g æ-<lb/>qualis oſtẽſus eſt angulo p k f:</s> <s xml:id="echoid-s9481" xml:space="preserve"> & angulus f k s ęquatus eſt angulo b d q:</s> <s xml:id="echoid-s9482" xml:space="preserve"> totus igitur p k s toti b d g ę-<lb/>quatur.</s> <s xml:id="echoid-s9483" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s9484" xml:space="preserve"> per 32 p 1 reliquus reliquo.</s> <s xml:id="echoid-s9485" xml:space="preserve"> Suntigitur per 4 p.</s> <s xml:id="echoid-s9486" xml:space="preserve"> 1 d 6 triangula b g d, s p k ſimilia.</s> <s xml:id="echoid-s9487" xml:space="preserve">] Igitur <lb/>ꝓ portio s p ad p k, ſicut b g ad g d.</s> <s xml:id="echoid-s9488" xml:space="preserve"> [Quare ſi ad lineã b g, eiusq́;</s> <s xml:id="echoid-s9489" xml:space="preserve"> terminũ g, per ſemidiametrũ ſpecu <lb/>li ſphęrici g u angulus æ quetur angulo s p k ſecũdo:</s> <s xml:id="echoid-s9490" xml:space="preserve"> erit u punctũ reflexionis.</s> <s xml:id="echoid-s9491" xml:space="preserve"> Quia igitur an gulus <lb/>ad p primus, maior eſt angulo ad p ſecũdo per 16 p 1:</s> <s xml:id="echoid-s9492" xml:space="preserve"> perſpicuũ eſt è primò demõſtratis, uiſibile h à <lb/>duobus pũctis ſpeculi d & u ad eundẽ uiſum reflecti, cõtra 29 n.</s> <s xml:id="echoid-s9493" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s9494" xml:space="preserve"> angulus c k s, cuius beneficio <lb/>reflexionis punctũ inueniendũ eſt, neceſſariò eſt obtuſus.</s> <s xml:id="echoid-s9495" xml:space="preserve">] Quod eſt propoſitũ.</s> <s xml:id="echoid-s9496" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s9497" xml:space="preserve"> impoſsi-<lb/>bile eſt, ut duorũ angulorũ ſuper m o cõſtitutorum, ſit uterq;</s> <s xml:id="echoid-s9498" xml:space="preserve"> maior recto.</s> <s xml:id="echoid-s9499" xml:space="preserve"> Si enim uterq;</s> <s xml:id="echoid-s9500" xml:space="preserve"> talium fue <lb/>rit maior recto, cum ſuperidẽ centrum fiat angulus æ qualis angulo s k m:</s> <s xml:id="echoid-s9501" xml:space="preserve"> fiet ſuperidẽ centrum a-<lb/>lius angulus diuerſus ab iſto, quem efficit ſuper k m alia linea ſimilis s k:</s> <s xml:id="echoid-s9502" xml:space="preserve"> & ita à puncto d, & ab alio <lb/>puncto illius circuli fiet reflexio:</s> <s xml:id="echoid-s9503" xml:space="preserve"> quod eſt impoſsibile:</s> <s xml:id="echoid-s9504" xml:space="preserve"> cum iam probatum ſit [29 n] quòd unum <lb/>uni uiſui ſit reflexionis punctum:</s> <s xml:id="echoid-s9505" xml:space="preserve"> & iam oſtenſum ſit, quomodo inueniri poſsit.</s> <s xml:id="echoid-s9506" xml:space="preserve"/> </p> <div xml:id="echoid-div347" type="float" level="0" n="0"> <figure xlink:label="fig-0157-01" xlink:href="fig-0157-01a"> <variables xml:id="echoid-variables72" xml:space="preserve">c p p m o f k s s</variables> </figure> <figure xlink:label="fig-0157-02" xlink:href="fig-0157-02a"> <variables xml:id="echoid-variables73" xml:space="preserve">b e n h d a i z q u t y g ſ x</variables> </figure> </div> </div> <div xml:id="echoid-div349" type="section" level="0" n="0"> <head xml:id="echoid-head344" xml:space="preserve" style="it">41. Viſibile à duob. ſpeculi ſphærici cõuexi pũctis ad utrũ uisũ reflexũ, unã habet imaginẽ. 34 p 6.</head> <p> <s xml:id="echoid-s9507" xml:space="preserve">DVobus aũt uiſibus, licet duo ſint reflexiõis pũcta:</s> <s xml:id="echoid-s9508" xml:space="preserve"> tamẽ unica erit imago ſenſuali ſyllo giſmo, <lb/>& unus imaginis locus.</s> <s xml:id="echoid-s9509" xml:space="preserve"> Et hoc probabimus, quãdo duæ lineæ à cẽtris.</s> <s xml:id="echoid-s9510" xml:space="preserve"> oculorũ ad cẽtrũ cir-<lb/>culi ductę, ſunt æ quales.</s> <s xml:id="echoid-s9511" xml:space="preserve"> Si ergo ſitus pũcti uiſi, reſpectu utriuſq;</s> <s xml:id="echoid-s9512" xml:space="preserve"> uiſus, ſit idẽ, ut lineæ à pun-<lb/>cto uiſo ad cẽtra oculorũ, ſint æ quales:</s> <s xml:id="echoid-s9513" xml:space="preserve"> facilis erit probatio.</s> <s xml:id="echoid-s9514" xml:space="preserve"> Quoniã <lb/> <anchor type="figure" xlink:label="fig-0158-01a" xlink:href="fig-0158-01"/> diametriuiſuales ſecãt ex circulo arcus reflexionis, & tenẽt angulos <lb/>æquales cũ linea, à puncto uiſo ad cẽtrũ ſphæræ ducta, & arcus inter <lb/>hác lineã & diametros uiſuales interiacẽtes, ſunt æquales.</s> <s xml:id="echoid-s9515" xml:space="preserve"> [Cũ enim <lb/>ex theſi uterq;</s> <s xml:id="echoid-s9516" xml:space="preserve"> uiſus æ quabiliter diſtet tũ à uiſibili tũ à ſpeculi cẽtro:</s> <s xml:id="echoid-s9517" xml:space="preserve"> <lb/>ducta igitur perpẽdiculari incidẽtiæ:</s> <s xml:id="echoid-s9518" xml:space="preserve"> fient duo triangula æ quilatera, <lb/>ideoq́;</s> <s xml:id="echoid-s9519" xml:space="preserve"> per 8 p 1 æ quiangula.</s> <s xml:id="echoid-s9520" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s9521" xml:space="preserve"> æ qualib.</s> <s xml:id="echoid-s9522" xml:space="preserve"> in cẽtro angulis æ quales <lb/>arcus ſubtẽdẽtur per 26 p 3.</s> <s xml:id="echoid-s9523" xml:space="preserve">] Et ſi ſumãtur pũcta reflexionis:</s> <s xml:id="echoid-s9524" xml:space="preserve"> ſecũdũ <lb/>ſuprà dictã probationẽ, arcus circuli interiacẽtes inter hæc pũcta, & <lb/>punctũ circuli, qđ eſt in perpẽdiculari, à puncto uiſo ducta:</s> <s xml:id="echoid-s9525" xml:space="preserve"> erunt æ-<lb/>quales:</s> <s xml:id="echoid-s9526" xml:space="preserve"> [Nã propter utriuſq;</s> <s xml:id="echoid-s9527" xml:space="preserve"> uiſus æquabilẽ tũ à uiſibili tũ à ſpeculi <lb/>cẽtro diſtãtiã:</s> <s xml:id="echoid-s9528" xml:space="preserve"> perpẽdiculares per reflexionũ pũcta ductę, cõprehen-<lb/>dunt cũ perpẽdiculari incidẽtię æ quales angulos in cẽtro, quib.</s> <s xml:id="echoid-s9529" xml:space="preserve"> per <lb/>26 p 3 æquales arcus ſubtẽduntur] qđ facile patebit, iterata ſuperio-<lb/>re probatione:</s> <s xml:id="echoid-s9530" xml:space="preserve"> & hoc:</s> <s xml:id="echoid-s9531" xml:space="preserve"> ſiue pũcta reflexion is ſint in eadẽ ſuperficie re <lb/>flexionis, ſiue in diuerſis:</s> <s xml:id="echoid-s9532" xml:space="preserve"> erũttamẽ arcus illi æquales:</s> <s xml:id="echoid-s9533" xml:space="preserve"> & lineæ ductę <lb/>à cẽtris oculorũ ad pũcta reflexionũ æquales:</s> <s xml:id="echoid-s9534" xml:space="preserve"> & lineę à pũcto uiſo ad <lb/>eadẽ pũcta, æ quales.</s> <s xml:id="echoid-s9535" xml:space="preserve"> [Quia enim anguliab opticis diametris ex theſi <lb/>æqualibus, & ſpeculi ſemidiametris cõprehenſi, æquales demõſtrati <lb/>ſunt:</s> <s xml:id="echoid-s9536" xml:space="preserve"> æquabũtur igitur ք 4 p 1 tũ reflexionis tũ incidẽtiæ lineæ inter <lb/>ſe.</s> <s xml:id="echoid-s9537" xml:space="preserve">] Et lineæ à cẽtris oculorũ ad reflexionũ pũcta procedẽtes, neceſſariò ſe ſecabũt [per 11 ax:</s> <s xml:id="echoid-s9538" xml:space="preserve"> angu-<lb/>li enim ք reflexionũ lineas in utroq;</s> <s xml:id="echoid-s9539" xml:space="preserve"> uiſu facti, ſunt minores duob.</s> <s xml:id="echoid-s9540" xml:space="preserve"> rectis.</s> <s xml:id="echoid-s9541" xml:space="preserve">] Et euidẽs eſt ꝓbatio, qđ <lb/>ſuper idẽ punctũ perpẽdicularis à pũcto uiſo ductę, erit ſectio ambarũ linearũ reflexionis.</s> <s xml:id="echoid-s9542" xml:space="preserve"> [Nã an-<lb/>gulorum reflexionis oſtenſam æquabilitatem conſequitur æquabilitas angulorum incidentiæ per <lb/>12 n 4:</s> <s xml:id="echoid-s9543" xml:space="preserve"> & anguli comprehenſi à lineis incidẽtię & perpendiculari æquales probati ſunt.</s> <s xml:id="echoid-s9544" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s9545" xml:space="preserve"> per 32 <lb/>p 1 triangula comprehenſa à lineis incidentię, cõtinuatione linearum reflexionis, & communi per-<lb/>pendiculari incidentię, ſunt ęquiangula.</s> <s xml:id="echoid-s9546" xml:space="preserve"> Quare per 4 p 6, ut ſunt lineę incidentię, ſic ſunt cõtinua-<lb/>tiones linearum reflexionis:</s> <s xml:id="echoid-s9547" xml:space="preserve"> at illę ęquantur:</s> <s xml:id="echoid-s9548" xml:space="preserve"> igitur & hæ.</s> <s xml:id="echoid-s9549" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s9550" xml:space="preserve"> in un o perpendicularis puncto con <lb/>currunt.</s> <s xml:id="echoid-s9551" xml:space="preserve">] Et in hoc puncto utriq;</s> <s xml:id="echoid-s9552" xml:space="preserve"> uiſui apparebit imago:</s> <s xml:id="echoid-s9553" xml:space="preserve"> & una ſola.</s> <s xml:id="echoid-s9554" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s9555" xml:space="preserve"/> </p> <div xml:id="echoid-div349" type="float" level="0" n="0"> <figure xlink:label="fig-0158-01" xlink:href="fig-0158-01a"> <variables xml:id="echoid-variables74" xml:space="preserve">a b c p g l m g h o j k d e f</variables> </figure> </div> </div> <div xml:id="echoid-div351" type="section" level="0" n="0"> <head xml:id="echoid-head345" xml:space="preserve" style="it">42. In ſpeculo ſphærico conuexo puncta imaginis, punctis uiſibilis ſitu & ordine, in utro uiſu <lb/>reſpondent. 35 p 6.</head> <p> <s xml:id="echoid-s9556" xml:space="preserve">ESt aũt ordinatio imaginũ, ſicut ordinatio pũctorũ uiſorũ.</s> <s xml:id="echoid-s9557" xml:space="preserve"> Sienim in re uiſa ſumatur linea, à <lb/>cuius capitib.</s> <s xml:id="echoid-s9558" xml:space="preserve"> ducãtur duæ lineæ ad cẽtrũ ſphærę:</s> <s xml:id="echoid-s9559" xml:space="preserve"> fiet triangulũ, in quo cõtine buntur imagi-<lb/>nes omniũ punctorũ illius lineæ.</s> <s xml:id="echoid-s9560" xml:space="preserve"> Et ſi ſit in illa linea punctũ nõ eiuſdẽ ſitus, reſpectu amborũ <lb/>uiſuũ:</s> <s xml:id="echoid-s9561" xml:space="preserve"> Imago puncti remotioris ab eo, erit in diametro remotiore ab eius diametro:</s> <s xml:id="echoid-s9562" xml:space="preserve"> & propinquio-<lb/> <pb o="153" file="0159" n="159" rhead="OPTICAE LIBER V."/> re.</s> <s xml:id="echoid-s9563" xml:space="preserve"> Etita obſeruatur ſitus partiũ in imaginibus, ſicut fuit in punctis uiſis.</s> <s xml:id="echoid-s9564" xml:space="preserve"> Sumpta aũt linea, in qua <lb/>eſt punctũ eiuſdẽ ſitus:</s> <s xml:id="echoid-s9565" xml:space="preserve"> quodlibet punctũ illius lineę eiuſdẽ ſitus erit, reſpectu duorũ oculorũ ſecũ-<lb/>dũ modũ prędictũ:</s> <s xml:id="echoid-s9566" xml:space="preserve"> & unicã habebit imaginẽ, propter æqualitatẽ angulorũ illius lineę cũ lineis ui-<lb/>ſualibus.</s> <s xml:id="echoid-s9567" xml:space="preserve"> Si aũt ſumatur linea, quæ angulũ, quẽ cõtinent duæ lineæ à cẽtris oculorũ ad punctum ui <lb/>ſum, diuidat per æqualia:</s> <s xml:id="echoid-s9568" xml:space="preserve"> ſitus cuiuslibet puncti lineæ quãtumlibet productæ, eritidẽ utriq;</s> <s xml:id="echoid-s9569" xml:space="preserve"> uifui<gap/> <lb/>ſicut ſuit uni.</s> <s xml:id="echoid-s9570" xml:space="preserve"> Et idẽ eſt probationis modus.</s> <s xml:id="echoid-s9571" xml:space="preserve"> Præter has duas lineas nõ eſt ſumere aliã, eundem ob-<lb/>ſeruantem ſitum.</s> <s xml:id="echoid-s9572" xml:space="preserve"> Vnde, cum punctum uiſum comprehendatur in perpendiculari [per 3 n] cadet <lb/>imago eius in diuerſis punctis illius perpendicularis, ſed imperceptibiliter à ſe remotis:</s> <s xml:id="echoid-s9573" xml:space="preserve"> & imago <lb/>cuiuslibet puncti à quotcunq;</s> <s xml:id="echoid-s9574" xml:space="preserve"> uideatur oculis, ſemper obſeruat identitatem partis.</s> <s xml:id="echoid-s9575" xml:space="preserve"> Vnde apparet <lb/>unitas imaginis, ſicut dictum eſt in uiſu directo [27 n 1] quòd formæ, licet in diuerſa cadant loca:</s> <s xml:id="echoid-s9576" xml:space="preserve"> <lb/>propter tamen diſtantiã earum inſenſibilem nõ diuerſiſicant apparentiam, niſi diuerſificent partẽ.</s> <s xml:id="echoid-s9577" xml:space="preserve"> <lb/>Similiter hic, quando remotio puncti ab uno uiſu fuerit modicò maior, quàm ab alio:</s> <s xml:id="echoid-s9578" xml:space="preserve"> eruntlocai-<lb/>maginum imperceptibiliter remota.</s> <s xml:id="echoid-s9579" xml:space="preserve"> Vnde apparent ſimul, & ex eis una imago compacta:</s> <s xml:id="echoid-s9580" xml:space="preserve"> quando-<lb/>quidem imaginum loca aliquando non totaliter diſtant, ſed partialiter.</s> <s xml:id="echoid-s9581" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div352" type="section" level="0" n="0"> <head xml:id="echoid-head346" xml:space="preserve" style="it">43. Si cõmunis ſectio ſuperficierũ reſlexiõis & ſpeculi cylindracei cõuexi fuerit latus cylindri, <lb/>uel circul{us}: loca, tum reflexionum tum imaginum eodem modo ſehabebunt, ut in ſpeculis pla-<lb/>no & ſphærico conuexo. 42. 43 p 7.</head> <p> <s xml:id="echoid-s9582" xml:space="preserve">IN ſpeculis columnaribus exterioribus aliquãdo linea cõmunis ſuperficiei reflexiõis & ſuperfi-<lb/>ciei ſpeculi, eſt linea recta:</s> <s xml:id="echoid-s9583" xml:space="preserve"> aliquãdo circulus:</s> <s xml:id="echoid-s9584" xml:space="preserve"> aliquãdo ſectio columnaris.</s> <s xml:id="echoid-s9585" xml:space="preserve"> Cũ fuerit linea cõmu-<lb/>nis, linea recta:</s> <s xml:id="echoid-s9586" xml:space="preserve"> erit locus imaginis in perpendiculari à puncto uiſo ducta ſuper ſuperficiem ſpe-<lb/>culi, tantum diſtans à linea communi, quantum punctum uiſum ab eadem.</s> <s xml:id="echoid-s9587" xml:space="preserve"> Et eadem eſt probatio, <lb/>quæ dicta eſt in ſpeculo plano [11 n.</s> <s xml:id="echoid-s9588" xml:space="preserve">] Cum autem communis linea fuerit circulus:</s> <s xml:id="echoid-s9589" xml:space="preserve"> erit aliquando <lb/>imaginis locus intra circulum:</s> <s xml:id="echoid-s9590" xml:space="preserve"> aliquando extra:</s> <s xml:id="echoid-s9591" xml:space="preserve"> aliquando in ipſa circumferentia.</s> <s xml:id="echoid-s9592" xml:space="preserve"> Eius rei eadem <lb/>penitus aſsignatio, quæ in ſpeculo exteriore ſphærico [22 n.</s> <s xml:id="echoid-s9593" xml:space="preserve">]</s> </p> </div> <div xml:id="echoid-div353" type="section" level="0" n="0"> <head xml:id="echoid-head347" xml:space="preserve" style="it">44. Siperpendicularis incidentiæ ſecetur à lineis: reflexionis, intra ellipſin (quæ est communis <lb/>ſectio ſuperficierum reflexionis & ſpeculi cylindracei conuexi) & tangente in reflexionis pun-<lb/>cto: erit ut tota perpendicularis adinferum ſegmentum, ſic ſuperum adintermedium. Et infe-<lb/>rum mai{us} erit ſegmento lineæ reflexionis. 47.48 p 7.</head> <p> <s xml:id="echoid-s9594" xml:space="preserve">SIuerò linea cõmunis fuerit ſectio colũnaris:</s> <s xml:id="echoid-s9595" xml:space="preserve"> dico, quòd imaginũ quędã ſunt intra ſpeculũ:</s> <s xml:id="echoid-s9596" xml:space="preserve"> quę <lb/>dã in ſuքficie ſpeculi:</s> <s xml:id="echoid-s9597" xml:space="preserve"> quędã extra ſpeculũ:</s> <s xml:id="echoid-s9598" xml:space="preserve"> quę in ſingulari explanabũtur.</s> <s xml:id="echoid-s9599" xml:space="preserve"> Sit a b c ſectio colũ-<lb/>naris:</s> <s xml:id="echoid-s9600" xml:space="preserve"> b ſit pũctũ reflexionis:</s> <s xml:id="echoid-s9601" xml:space="preserve"> e pũctũ uiſum:</s> <s xml:id="echoid-s9602" xml:space="preserve"> d cẽtrũ uiſus:</s> <s xml:id="echoid-s9603" xml:space="preserve"> & [ք 12 p 11] ducatur à puncto b per-<lb/>pendicularis ſuper ſuperficiẽ cõtingentẽ ſpeculũ:</s> <s xml:id="echoid-s9604" xml:space="preserve"> quæ ſit g b q:</s> <s xml:id="echoid-s9605" xml:space="preserve"> & [ք 11 p 11] ducatur à puncto e per-<lb/>pendicularis ſuper ſuperficiẽ, cõtingentẽ ſpeculũ:</s> <s xml:id="echoid-s9606" xml:space="preserve"> quę ſit e k q:</s> <s xml:id="echoid-s9607" xml:space="preserve"> & linea cõtingẽs ſpeculũ in pũcto b:</s> <s xml:id="echoid-s9608" xml:space="preserve"> <lb/>ſit t u:</s> <s xml:id="echoid-s9609" xml:space="preserve"> linea cõtingẽs ſpeculũ in pũcto k:</s> <s xml:id="echoid-s9610" xml:space="preserve"> ſit k m.</s> <s xml:id="echoid-s9611" xml:space="preserve"> Dico, quòd duę perpẽdiculares g b q, e k q cõcurrẽt.</s> <s xml:id="echoid-s9612" xml:space="preserve"> <lb/>Ducãtur lineę e b, d b:</s> <s xml:id="echoid-s9613" xml:space="preserve"> & ducatur linea k b.</s> <s xml:id="echoid-s9614" xml:space="preserve"> Palàm, qđ k m cadet in figurã e k b, & linea b t in figurã <lb/>eandẽ [quia recta linea ſecãs angulũ trianguli, ſecat baſim angulo ſubtẽſam:</s> <s xml:id="echoid-s9615" xml:space="preserve"> ſecus nõ ſecaret angu <lb/>lũ.</s> <s xml:id="echoid-s9616" xml:space="preserve">] Igitur b t ſecabite k:</s> <s xml:id="echoid-s9617" xml:space="preserve"> ſecetin pũcto t.</s> <s xml:id="echoid-s9618" xml:space="preserve"> Palàm, quòd angulus g b k eſt maior recto, & angulus e k b <lb/>ſimiliter maior recto [quia g b q, e k q ſunt քpendiculares ipſis t u, k m.</s> <s xml:id="echoid-s9619" xml:space="preserve">] Quare [per 13 p 1.</s> <s xml:id="echoid-s9620" xml:space="preserve"> 11 ax.</s> <s xml:id="echoid-s9621" xml:space="preserve">] g b, <lb/>e k cõcurrẽt.</s> <s xml:id="echoid-s9622" xml:space="preserve"> Sit cõcurſus punctũ q.</s> <s xml:id="echoid-s9623" xml:space="preserve"> Similiter d b k maior recto:</s> <s xml:id="echoid-s9624" xml:space="preserve"> igitur d b, e k cõcurrẽt.</s> <s xml:id="echoid-s9625" xml:space="preserve"> Sit cõcurſus <lb/>punctũ h.</s> <s xml:id="echoid-s9626" xml:space="preserve"> Igitur h eſt locus imaginis [ք 4 n.</s> <s xml:id="echoid-s9627" xml:space="preserve">] Dico <lb/> <anchor type="figure" xlink:label="fig-0159-01a" xlink:href="fig-0159-01"/> etiã, quòd proportio e q ad q h, ſicut e t ad th:</s> <s xml:id="echoid-s9628" xml:space="preserve"> & etiã <lb/>quòd q h eſt maior h b.</s> <s xml:id="echoid-s9629" xml:space="preserve"> Ducatur [ք 31 p 1] h f æquidi <lb/>ſtãs e b.</s> <s xml:id="echoid-s9630" xml:space="preserve"> Palàm, quòd angulus e b t eſt ę qualis angulo <lb/>d b u [ք 12 n 4:</s> <s xml:id="echoid-s9631" xml:space="preserve">] eſt igitur [ք 15 p 1.</s> <s xml:id="echoid-s9632" xml:space="preserve"> 1 ax.</s> <s xml:id="echoid-s9633" xml:space="preserve">] æqualis an-<lb/>gulo t b h:</s> <s xml:id="echoid-s9634" xml:space="preserve"> reſtat e b g æqualis angulo h b q:</s> <s xml:id="echoid-s9635" xml:space="preserve"> cũ g b t, t <lb/>b q ſint recti.</s> <s xml:id="echoid-s9636" xml:space="preserve"> Cũ igitur t b diuidat angulũ e b h ք æ-<lb/>qualia:</s> <s xml:id="echoid-s9637" xml:space="preserve"> erit [ք 3 p 6] et ad t h, ſicut e b ad b h:</s> <s xml:id="echoid-s9638" xml:space="preserve"> Sed an-<lb/>gulus e b g eſt æqualis angulo h ſb [ք 29 p 1:</s> <s xml:id="echoid-s9639" xml:space="preserve">] quare <lb/>h f, h b ſunt æqualia.</s> <s xml:id="echoid-s9640" xml:space="preserve"> [angulus enim e b g ęqualis con <lb/>cluſus eſt angulo h b f:</s> <s xml:id="echoid-s9641" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s9642" xml:space="preserve"> anguli h f b, h b f æquan-<lb/>tur:</s> <s xml:id="echoid-s9643" xml:space="preserve"> quare ք 6 p 1 latera h f, h b ęquantur:</s> <s xml:id="echoid-s9644" xml:space="preserve"> ergo ք 7 p 5, <lb/>ut e t ad th, ſic e b ad h f] Sed e b ad h f, ſicut e q ad q h <lb/>[ք 4 p 6:</s> <s xml:id="echoid-s9645" xml:space="preserve"> ꝗa enim h f parallela ducta eſt ipſi e b:</s> <s xml:id="echoid-s9646" xml:space="preserve"> ſunt <lb/>trιangula e b q, h f q æquiãgula ք 29.</s> <s xml:id="echoid-s9647" xml:space="preserve"> 32 p 1.</s> <s xml:id="echoid-s9648" xml:space="preserve">] Erit ergo <lb/>[per 11 p 5] et ad th, ſicute q ad q h.</s> <s xml:id="echoid-s9649" xml:space="preserve"> Qđ eſt propoſi-<lb/>tũ.</s> <s xml:id="echoid-s9650" xml:space="preserve"> Et ex hoc:</s> <s xml:id="echoid-s9651" xml:space="preserve"> cũ ſit ꝓportio e q ad q h, ſicut e b ad h <lb/>f [& h f æquetur ipſi h b:</s> <s xml:id="echoid-s9652" xml:space="preserve"> erit ք 7 p 5, e q ad q h, ſicute <lb/>b ad b h] & e q ſit maior e b [ք 19 p 1:</s> <s xml:id="echoid-s9653" xml:space="preserve"> ꝗa angulus e b q recto maior eſt] erit [ք 14 p 5] q h maior h b <lb/>Quod eſt ꝓpoſitũ.</s> <s xml:id="echoid-s9654" xml:space="preserve"> Palàm exhoc, quòd ſi ſuper ſectionẽ a b c ducatur քpendicul aris ſuք ſuperficiẽ <lb/>cõtingentẽ ſectionẽ:</s> <s xml:id="echoid-s9655" xml:space="preserve"> cõcurret cũ g b.</s> <s xml:id="echoid-s9656" xml:space="preserve"> Et hęc quidẽ patẽt, cũpunctũ uiſum nõ fuerit in քpẽdiculari <lb/>uiſuali.</s> <s xml:id="echoid-s9657" xml:space="preserve"> Palàm enim ex ſuperioribus [19 n] quòd unius ſolius pũcti forma ք perpẽdicularẽ accedit <lb/>ad ſpeculũ, & ſecũdũ eundẽ reflectitur.</s> <s xml:id="echoid-s9658" xml:space="preserve"> Et eſt pũctũ քpendicularis, exiſtẽs in ſuքficie uiſus:</s> <s xml:id="echoid-s9659" xml:space="preserve"> punctũ <lb/>enim ultra uiſum ſumptũ nõ poteſt reflecti ſuք hãc քpendicularẽ:</s> <s xml:id="echoid-s9660" xml:space="preserve"> ꝗa nõ põt accedere ad ſpeculũ ſu <lb/> <pb o="154" file="0160" n="160" rhead="ALHAZEN"/> per perpendicularẽ, propter prædictã ibidẽ rationẽ.</s> <s xml:id="echoid-s9661" xml:space="preserve"> Et ſimiliter non poterit reflecti ab alio puncto <lb/>ſpeculi, quã à puncto perpendicularis huius:</s> <s xml:id="echoid-s9662" xml:space="preserve"> quia accideret duas perpẽdiculares cõcurrere, & effi-<lb/>ficere triangulum, cuius duo anguli recti, ſicut ſuprà patuit.</s> <s xml:id="echoid-s9663" xml:space="preserve"/> </p> <div xml:id="echoid-div353" type="float" level="0" n="0"> <figure xlink:label="fig-0159-01" xlink:href="fig-0159-01a"> <variables xml:id="echoid-variables75" xml:space="preserve">e g d t m b u k h f q a c</variables> </figure> </div> </div> <div xml:id="echoid-div355" type="section" level="0" n="0"> <head xml:id="echoid-head348" xml:space="preserve" style="it">45. Si cõmunis ſectio ſuperficierũ, reflexionis & ſpeculi cylindracei cõuexi fuerit ellipſis: imago <lb/>uiſibilis obliquè reflexi, aliâs in ſuքficie ſpeculi: aliâs intra: aliâs extra ſpeculũ uidebitur. 49 p 7.</head> <p> <s xml:id="echoid-s9664" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s9665" xml:space="preserve"> ſumatur ſectio columnaris:</s> <s xml:id="echoid-s9666" xml:space="preserve"> & ſumatur in ea punctũ a:</s> <s xml:id="echoid-s9667" xml:space="preserve"> & ducatur contingens ſectio-<lb/>nẽ:</s> <s xml:id="echoid-s9668" xml:space="preserve"> quæ ſit e a t:</s> <s xml:id="echoid-s9669" xml:space="preserve"> & ſumatur perpendicularis ſuper a tintra ſpeculũ:</s> <s xml:id="echoid-s9670" xml:space="preserve"> quę ſit d a.</s> <s xml:id="echoid-s9671" xml:space="preserve"> Palàm, quòd d <lb/>a diuidit ſectionẽ in duas partes, in quarũ utraq;</s> <s xml:id="echoid-s9672" xml:space="preserve"> eſt punctũ unicũ, cuius puncti linea cõtin-<lb/>gens, erit æ quidiſtans a d.</s> <s xml:id="echoid-s9673" xml:space="preserve"> Sit ergo aliud punctũ g, cuius cõtingens cõcurrat cũ linea a d in puncto <lb/>h:</s> <s xml:id="echoid-s9674" xml:space="preserve"> & ducatur perpendicularis ſuper hãc cõtingentẽ:</s> <s xml:id="echoid-s9675" xml:space="preserve"> quæ ſit q g:</s> <s xml:id="echoid-s9676" xml:space="preserve"> & hæc quidẽ neceſſariò cõcurret cũ <lb/>h d, ſicut oſtenſum eſt in præcedente figura [eſt enim angulus q g h per fabricationẽ rectus:</s> <s xml:id="echoid-s9677" xml:space="preserve"> ergo q <lb/>g a maior eſt recto:</s> <s xml:id="echoid-s9678" xml:space="preserve"> & ob id h a g recto maior eſt.</s> <s xml:id="echoid-s9679" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s9680" xml:space="preserve"> per 13 p 1 d g a, d a g ſunt minores duobus re-<lb/>ctis.</s> <s xml:id="echoid-s9681" xml:space="preserve"> Quare per 11 ax.</s> <s xml:id="echoid-s9682" xml:space="preserve"> q g, h a cotinuatæ cõcurrent.</s> <s xml:id="echoid-s9683" xml:space="preserve">] Sit concurſus in puncto d:</s> <s xml:id="echoid-s9684" xml:space="preserve"> & ducatur linea g a <lb/>uſq;</s> <s xml:id="echoid-s9685" xml:space="preserve"> ad p:</s> <s xml:id="echoid-s9686" xml:space="preserve"> & ducatur linea q a.</s> <s xml:id="echoid-s9687" xml:space="preserve"> Igitur angulus q a h aut eſt æqualis angulo h a p:</s> <s xml:id="echoid-s9688" xml:space="preserve"> aut maior:</s> <s xml:id="echoid-s9689" xml:space="preserve"> aut mi-<lb/>nor.</s> <s xml:id="echoid-s9690" xml:space="preserve"> Sit ęqualis.</s> <s xml:id="echoid-s9691" xml:space="preserve"> Procedet igitur forma puncti q ad <lb/> <anchor type="figure" xlink:label="fig-0160-01a" xlink:href="fig-0160-01"/> a, & reflectetur ad p [per 12 n 4] quod uiſus ſit:</s> <s xml:id="echoid-s9692" xml:space="preserve"> & <lb/>locus imaginis erit punctũ ſectionis columnaris, <lb/>ſcilicet g [per 4 n.</s> <s xml:id="echoid-s9693" xml:space="preserve">] Si uerò ſupra punctum q ſuma <lb/>tur aliquod punctũ, ut punctũ f:</s> <s xml:id="echoid-s9694" xml:space="preserve"> erit quidẽ angulus <lb/>f a h minor angulo h a p [quia angulus h a p æqua-<lb/>tur h a q, qui per 9 ax.</s> <s xml:id="echoid-s9695" xml:space="preserve"> maior eſtangulo h a f.</s> <s xml:id="echoid-s9696" xml:space="preserve">] Fiat ei <lb/>æqualis n a h:</s> <s xml:id="echoid-s9697" xml:space="preserve"> cõcurret quidẽ n a cũg q [per 11 ax.</s> <s xml:id="echoid-s9698" xml:space="preserve"> <lb/>ut antea] intra columnã.</s> <s xml:id="echoid-s9699" xml:space="preserve"> [quia punctũ n ſublimi-<lb/>us eſt puncto p.</s> <s xml:id="echoid-s9700" xml:space="preserve">] Sit in puncto k.</s> <s xml:id="echoid-s9701" xml:space="preserve"> Palã ergo, quòd <lb/>imago puncti f erit in puncto k [per 4 n] & imagi-<lb/>nes omniũ punctorũ lineæ q fultra punctũ q, intra <lb/>columnã.</s> <s xml:id="echoid-s9702" xml:space="preserve"> Si uerò inter q & t ſumatur punctum ali-<lb/>quod:</s> <s xml:id="echoid-s9703" xml:space="preserve"> ut punctũ r:</s> <s xml:id="echoid-s9704" xml:space="preserve"> erit angulus r a h maior angulo <lb/>h a p [quia h a p æquatur h a q, quo angulus h a r <lb/>maior eſt per 9 ax.</s> <s xml:id="echoid-s9705" xml:space="preserve">] Fiat ei ęqualis h a m.</s> <s xml:id="echoid-s9706" xml:space="preserve"> Palàm, qđ <lb/>m a cadet ſupra lineã g q, & extra ſectionem [cũ e-<lb/>nim linea p a (quæ cũ h a cõtinet angulũ æqualem <lb/>h a q) cõcurrat cũ ſectione in puncto g:</s> <s xml:id="echoid-s9707" xml:space="preserve"> & punctũ m ſit inferius puncto p:</s> <s xml:id="echoid-s9708" xml:space="preserve"> linea igitur m a cõtinua-<lb/>ta cõcurret cũ g q extra ſectionẽ.</s> <s xml:id="echoid-s9709" xml:space="preserve">] Sit in pũcto o.</s> <s xml:id="echoid-s9710" xml:space="preserve"> Erit igitur imago r in pũcto o [per 4 n.</s> <s xml:id="echoid-s9711" xml:space="preserve">] Et omniũ <lb/>punctorũ inter t, q interiacentiũ imagines, erũt extra ſectionẽ inter o & g.</s> <s xml:id="echoid-s9712" xml:space="preserve"> Siuerò angulus q a h fue <lb/>rit minor angulo h a p:</s> <s xml:id="echoid-s9713" xml:space="preserve"> ſecetur ex eo æqualis:</s> <s xml:id="echoid-s9714" xml:space="preserve"> & ſit h a n.</s> <s xml:id="echoid-s9715" xml:space="preserve"> Palàm, quòd imago q erit in puncto k:</s> <s xml:id="echoid-s9716" xml:space="preserve"> & o-<lb/>mniũ punctorũ ſuperiorũ imagines erũt intra ſectionẽ.</s> <s xml:id="echoid-s9717" xml:space="preserve"> Si uerò inferius ſumatur r punctũ, ut angu-<lb/>lus r a h ſit ęqualis angulo h a p:</s> <s xml:id="echoid-s9718" xml:space="preserve"> erit imago r in ſectione:</s> <s xml:id="echoid-s9719" xml:space="preserve"> & oẽs inter r & q intra:</s> <s xml:id="echoid-s9720" xml:space="preserve"> oẽs inter r & t extra.</s> <s xml:id="echoid-s9721" xml:space="preserve"> <lb/>Si uerò angulus q a h fuerit maior angulo h a p:</s> <s xml:id="echoid-s9722" xml:space="preserve"> fiat ei æqualis h a m.</s> <s xml:id="echoid-s9723" xml:space="preserve"> Palàm, quòd m a ſecabit ſectio <lb/>nẽ:</s> <s xml:id="echoid-s9724" xml:space="preserve"> [quia e a t tangit] & ſecet in puncto b:</s> <s xml:id="echoid-s9725" xml:space="preserve"> & ducatur cõtingẽs ſuper punctũ b:</s> <s xml:id="echoid-s9726" xml:space="preserve"> quę cõcurret cũ d h, <lb/>utin puncto l [ducta enim recta d b:</s> <s xml:id="echoid-s9727" xml:space="preserve"> erit angulus d b l rectus, & b d lacutus:</s> <s xml:id="echoid-s9728" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s9729" xml:space="preserve"> tãgens ſectionẽ in <lb/>pũcto b cõcurret cũ d h per 11 ax.</s> <s xml:id="echoid-s9730" xml:space="preserve">] eritq́;</s> <s xml:id="echoid-s9731" xml:space="preserve"> [per 17 p 1] angulus d l b acutus, & angulus h l b obtuſus:</s> <s xml:id="echoid-s9732" xml:space="preserve"> <lb/>[per 13 p 1] & l b cõcurrẽs cũ h g faciet cũ ea acutũ [per 32 p 1:</s> <s xml:id="echoid-s9733" xml:space="preserve"> quia angulus h l b eſt obtuſus.</s> <s xml:id="echoid-s9734" xml:space="preserve">] Duca <lb/>tur perpẽdicularis à pũcto b ſuper l b:</s> <s xml:id="echoid-s9735" xml:space="preserve"> quę ſit s b:</s> <s xml:id="echoid-s9736" xml:space="preserve"> ſecabit quidẽ h g, utin pũcto x:</s> <s xml:id="echoid-s9737" xml:space="preserve"> & faciet angulũ a-<lb/>cutũ cũ ea [per 15 p 1] quoniã angulus cõtrapoſitus ſimiliter erit acutus [ք 32 p 1:</s> <s xml:id="echoid-s9738" xml:space="preserve"> quia angulus ad b <lb/>rectus eſt] & h g ſecat q a:</s> <s xml:id="echoid-s9739" xml:space="preserve"> ſit punctũ ſectionis u:</s> <s xml:id="echoid-s9740" xml:space="preserve"> & facit acutũ angulũ cũ ea ſuper punctum u [cum <lb/>enim h g cõcurrat cum q a:</s> <s xml:id="echoid-s9741" xml:space="preserve"> & q a cum fd, & angulus h g q ſit rectus:</s> <s xml:id="echoid-s9742" xml:space="preserve"> erit per 32 p 1 angulus q u g acu-<lb/>tus.</s> <s xml:id="echoid-s9743" xml:space="preserve">] Quare s b & q u concurrunt [quia enim angulis s x h, qu g acutis cõcluſis æquãtur anguli ad <lb/>uerticẽ per 15 p 1.</s> <s xml:id="echoid-s9744" xml:space="preserve"> Ergo per 11 ax.</s> <s xml:id="echoid-s9745" xml:space="preserve"> q u & s b cõcurrũt.</s> <s xml:id="echoid-s9746" xml:space="preserve">] Sit cõcurſus in z.</s> <s xml:id="echoid-s9747" xml:space="preserve"> Palàm ergo, quòd forma pun <lb/>cti z mouebitur ad ſpeculũ per z a, & reflectetur per a m:</s> <s xml:id="echoid-s9748" xml:space="preserve"> & locus imaginis, b:</s> <s xml:id="echoid-s9749" xml:space="preserve"> & imagines punctorũ <lb/>lineæ z s ultra z, erunt intra ſectionẽ:</s> <s xml:id="echoid-s9750" xml:space="preserve"> & punctorũ citra z, extra ſectionem.</s> <s xml:id="echoid-s9751" xml:space="preserve"> Quod fuit propoſitum.</s> <s xml:id="echoid-s9752" xml:space="preserve"/> </p> <div xml:id="echoid-div355" type="float" level="0" n="0"> <figure xlink:label="fig-0160-01" xlink:href="fig-0160-01a"> <variables xml:id="echoid-variables76" xml:space="preserve">s f h q n x r p l z u t m a b o e g k d</variables> </figure> </div> </div> <div xml:id="echoid-div357" type="section" level="0" n="0"> <head xml:id="echoid-head349" xml:space="preserve" style="it">46. Si cõmunis ſectio ſuperficierũ, reflexiõis & ſpeculi cylindracei conuexi, fuerit lat{us} cylindri, <lb/>uel circul{us} baſib. parallel9<unsure/>: ab uno pũcto unũ uiſibilis pũctũ ad unũ uisũ reflectetur. 26. 27 p 7.</head> <p> <s xml:id="echoid-s9753" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s9754" xml:space="preserve"> ab uno ſolo pũcto ſpeculi colũnaris fit reflexio ad cẽtrũ uiſus:</s> <s xml:id="echoid-s9755" xml:space="preserve"> utpote pũctũ b refle-<lb/>ctatur ad a à pũcto g.</s> <s xml:id="echoid-s9756" xml:space="preserve"> Dico, quòd nõ reflectetur ad ipſum ab alio puncto ſpeculi, quã à pũcto <lb/>g.</s> <s xml:id="echoid-s9757" xml:space="preserve"> Quoniã, ſi in ſuperficie reflexionis, quæ eſt a b g, ſit totus axis ſpeculi:</s> <s xml:id="echoid-s9758" xml:space="preserve"> erit linea cõmunis <lb/>ſuperficiei ſpeculi & ſuperficiei reflexionis linea lõgitudinis ſpeculi [per 29 n 4.</s> <s xml:id="echoid-s9759" xml:space="preserve">] Et cũ in ſuperfi-<lb/>cie reflexionis ſit cẽtrũ uiſus, pũctũ uiſum, punctũ reflexiõis, & punctũ axis, in qđ cadit perpẽdicu <lb/>laris:</s> <s xml:id="echoid-s9760" xml:space="preserve"> [per 23.</s> <s xml:id="echoid-s9761" xml:space="preserve"> 34 n 4] una ſola ſuperficies ſumi poteſt, in qua ſit linea illa longitudinis, axis, & pun-<lb/>cta a, b, g.</s> <s xml:id="echoid-s9762" xml:space="preserve"> Quare non poteſt ſieri reflexio ad a, niſi ab aliquo puncto lineę longitudinis:</s> <s xml:id="echoid-s9763" xml:space="preserve"> ſed iam pro-<lb/>batũ eſt [51 n 4 generatim de quolibet ſpeculo, & 14 n ſpeciatim de ſpeculo plano] quòd nõ poteſt <lb/>fieri reflexio ad a ab alio puncto, quã à puncto g.</s> <s xml:id="echoid-s9764" xml:space="preserve"> Quare in hoc ſitu ab uno ſolo pũcto ſpeculi fit ad <lb/>a reflexio.</s> <s xml:id="echoid-s9765" xml:space="preserve"> Si uerò ſuperficies a b g ſit æquidiſtans baſi colũnæ:</s> <s xml:id="echoid-s9766" xml:space="preserve"> erit linea cõmunis, circulus æquidi-<lb/> <pb o="155" file="0161" n="161" rhead="OPTICAE LIBER V."/> ſtans baſi [per 5 th Sereni de ſectione cylindri.</s> <s xml:id="echoid-s9767" xml:space="preserve">] Et iam patuit [29 n] quòd ab alio pũcto illius cir-<lb/>culi non poteſt fieri ad a reflexio.</s> <s xml:id="echoid-s9768" xml:space="preserve"> Et ſi ab alio <lb/> <anchor type="figure" xlink:label="fig-0161-01a" xlink:href="fig-0161-01"/> puncto ſpeculi fiat reflexio perpẽdicularis du <lb/>cta à puncto illo, cadet orthogonaliter ſuper <lb/>axẽ.</s> <s xml:id="echoid-s9769" xml:space="preserve"> [Nã cũ per 34 n 4 perpẽdicularis illa in-<lb/>tus cõtinuata fiat diameter circuli baſibus pa <lb/>ralleli:</s> <s xml:id="echoid-s9770" xml:space="preserve"> erit per 21 d 11.</s> <s xml:id="echoid-s9771" xml:space="preserve"> 29 p 1 ad axem perpendi <lb/>cularis] & ſecabit lineã a b in puncto aliquo.</s> <s xml:id="echoid-s9772" xml:space="preserve"> <lb/>À<unsure/> pũcto illo ducatur linea ad axem in ſuper-<lb/>ficie, æquidiſtante baſi colũnæ:</s> <s xml:id="echoid-s9773" xml:space="preserve"> erit quidẽ or-<lb/>thogonalis ſuper axem [per 21 d 11.</s> <s xml:id="echoid-s9774" xml:space="preserve"> 29 p 1.</s> <s xml:id="echoid-s9775" xml:space="preserve">] Et <lb/>ita duæ perpẽdiculares efficient cũ axe trian-<lb/>gulum, cuius duo anguli ſunt recti:</s> <s xml:id="echoid-s9776" xml:space="preserve"> quod eſt <lb/>impoſsibile [& contra 32 p 1.</s> <s xml:id="echoid-s9777" xml:space="preserve">] Palàm ergo, quòd in hoc ſitu non reflectetur b ad a, niſi à puncto g.</s> <s xml:id="echoid-s9778" xml:space="preserve"/> </p> <div xml:id="echoid-div357" type="float" level="0" n="0"> <figure xlink:label="fig-0161-01" xlink:href="fig-0161-01a"> <variables xml:id="echoid-variables77" xml:space="preserve">a q k b f l n g c e l d h</variables> </figure> </div> </div> <div xml:id="echoid-div359" type="section" level="0" n="0"> <head xml:id="echoid-head350" xml:space="preserve" style="it">47. Si communis ſectio ſuperficierum, reflexionis & ſpeculi cylindracei conuexi fuerit elli-<lb/>pſis: ab uno puncto unum uiſibilis punctum ad unum uiſum reflectetur. 28 p 7.</head> <p> <s xml:id="echoid-s9779" xml:space="preserve">SIuerò ſuperficies a b g ſecet ſpeculũ ſectione columnari:</s> <s xml:id="echoid-s9780" xml:space="preserve"> dico, quòd à ſolo pũcto g fit reflexio.</s> <s xml:id="echoid-s9781" xml:space="preserve"> <lb/>Ducatur à puncto a ſuperficies æquidiſtans baſi columnæ:</s> <s xml:id="echoid-s9782" xml:space="preserve"> [ductis nimirũ duabus perpen di-<lb/>cularibus ſuper axem ſe interſecãtibus:</s> <s xml:id="echoid-s9783" xml:space="preserve"> una quidẽ à puncto a per 12 p 1:</s> <s xml:id="echoid-s9784" xml:space="preserve"> altera uerò ab axis pun <lb/>cto, in quod illa cadit per 11 p 1.</s> <s xml:id="echoid-s9785" xml:space="preserve"> Sic enim axis, qui per 21 d 11 eſt perpendicularis baſi:</s> <s xml:id="echoid-s9786" xml:space="preserve"> erit per 4 p 11 <lb/>perpendicularis plano ductarũ perpen diculariũ.</s> <s xml:id="echoid-s9787" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s9788" xml:space="preserve"> per 14 p 11 baſis & hoc planũ erũt parallela] <lb/>quæ ſit e z i:</s> <s xml:id="echoid-s9789" xml:space="preserve"> & à puncto g ſimiliter ſuperficies æquidiſtans baſi ſpeculi:</s> <s xml:id="echoid-s9790" xml:space="preserve"> in qua ducatur ab axe linea <lb/>ad pũctũ g:</s> <s xml:id="echoid-s9791" xml:space="preserve"> quæ ſit t g:</s> <s xml:id="echoid-s9792" xml:space="preserve"> erit quidẽ perpẽdicularis ſuper ſuperficiẽ, cõtingẽtẽ ſpeculũ in pũcto g [per <lb/>34 n 4:</s> <s xml:id="echoid-s9793" xml:space="preserve"> quia eſt diameter circuli baſibus cylindri paralleli] & cõcurrat cũ a b in puncto k [cõcurret <lb/>aũt:</s> <s xml:id="echoid-s9794" xml:space="preserve"> quia diuidit angulũ a g b] & ducatur à puncto g linea lõgitudinis ſpeculi:</s> <s xml:id="echoid-s9795" xml:space="preserve"> [educto nẽpe plano <lb/>per axem & per rectã, cũ ipſo à puncto g utlibet cõcurrentẽ:</s> <s xml:id="echoid-s9796" xml:space="preserve"> erit enim huius plani & cylindraceæ ſu <lb/>perficiei cõmunis ſectio latus cylindri per 21 d 11] quæ ſit g z:</s> <s xml:id="echoid-s9797" xml:space="preserve"> & ſit axis t q:</s> <s xml:id="echoid-s9798" xml:space="preserve"> & à puncto b perpẽdicu <lb/>laris ducatur ad ſuperficiẽ e z i:</s> <s xml:id="echoid-s9799" xml:space="preserve"> quę ſit b h:</s> <s xml:id="echoid-s9800" xml:space="preserve"> & ducãtur lineę a z, h z:</s> <s xml:id="echoid-s9801" xml:space="preserve"> & ducatur à pũcto z in ſuperficie <lb/>illa ad axem linea, quæ ſit z q:</s> <s xml:id="echoid-s9802" xml:space="preserve"> erit quidẽ perpẽdicularis ſuper axem [per 3 d 11] cũ axis ſit perpẽdi-<lb/>cularis ſuper hãc ſuperficiẽ [per 21 d 11] & erit perpẽdicularis ſuper ſuperficiẽ, cõtingentẽ ſpeculũ <lb/>in puncto z [ut paulò antè oſtẽſum eſt] & cõcurrat cũ linea a k in pũcto l.</s> <s xml:id="echoid-s9803" xml:space="preserve"> [cõcurret uerò, quia ſe-<lb/>cat angulũ a z h.</s> <s xml:id="echoid-s9804" xml:space="preserve">] Dico, quòd forma puncti h reflectetur ad a, à puncto z.</s> <s xml:id="echoid-s9805" xml:space="preserve"> Ducatur à pũcto a æ quidi-<lb/>ſtãs lineę k g:</s> <s xml:id="echoid-s9806" xml:space="preserve"> quę ſit a m:</s> <s xml:id="echoid-s9807" xml:space="preserve"> quę quidẽ cõcurret cũ b g.</s> <s xml:id="echoid-s9808" xml:space="preserve"> [per lẽma Procli ad 29 p 1.</s> <s xml:id="echoid-s9809" xml:space="preserve">] Sit cõcurſus in pun <lb/>cto m.</s> <s xml:id="echoid-s9810" xml:space="preserve"> Palàm [per 6 p 11] quòd g z eſt æquidiſtãs lineæ b h:</s> <s xml:id="echoid-s9811" xml:space="preserve"> cũ utraq;</s> <s xml:id="echoid-s9812" xml:space="preserve"> ſit orthogonalis ſuper ſuperfi <lb/>ciẽ æquidiſtantẽ baſibus colũnæ.</s> <s xml:id="echoid-s9813" xml:space="preserve"> Quare [per 7 p 11] linea b g m eſt in ſuperficie harũ linearũ.</s> <s xml:id="echoid-s9814" xml:space="preserve"> Igitur <lb/>tria pũcta m, z, h ſunt in hac <lb/> <anchor type="figure" xlink:label="fig-0161-02a" xlink:href="fig-0161-02"/> ſuքficie.</s> <s xml:id="echoid-s9815" xml:space="preserve"> Sed iterũ a m eſt æ-<lb/>quidiſtans k g [per fabrica-<lb/>tionẽ] & l z æquidiſtãs k g:</s> <s xml:id="echoid-s9816" xml:space="preserve"> <lb/>quoniã g z æquidiſtãs t q & <lb/>inter ſuperficies æquidiſtan <lb/>tes.</s> <s xml:id="echoid-s9817" xml:space="preserve"> [nã per 21 d 11 latus z g & <lb/>axis q t paralleli & æquales, <lb/>circulis oppoſitis & paral-<lb/>lelis terminantur, in quibus <lb/>ſemidiametritg, q z ſunt pa <lb/>rallelę per 33 p 1:</s> <s xml:id="echoid-s9818" xml:space="preserve"> & t g conti-<lb/>nuata eſt in k.</s> <s xml:id="echoid-s9819" xml:space="preserve">] Igitur l z æ-<lb/>quidiſtãs a m [ք 30 p 1:</s> <s xml:id="echoid-s9820" xml:space="preserve"> ſunt <lb/>enim m a, z l eidẽ t g k paral-<lb/>lelæ.</s> <s xml:id="echoid-s9821" xml:space="preserve">] Quare ſunt in eadem <lb/>ſuperficie [per 35 d 1] & in ea eſt linea a h [per 7 p 11:</s> <s xml:id="echoid-s9822" xml:space="preserve"> quia cõnectit m a, z l parallelas.</s> <s xml:id="echoid-s9823" xml:space="preserve">] Igitur in hac <lb/>ſuperficie ſunt tria puncta, m, z, h:</s> <s xml:id="echoid-s9824" xml:space="preserve"> & iã patuit, quòd ſint in ſuperficie b m h:</s> <s xml:id="echoid-s9825" xml:space="preserve"> igitur ſunt in linea cõmu <lb/>ni his duabus ſuperficiebus.</s> <s xml:id="echoid-s9826" xml:space="preserve"> Igitur [per 3 p 11] h z m eſt linea recta.</s> <s xml:id="echoid-s9827" xml:space="preserve"> Palàm igitur, cum g ſit punctum <lb/>reflexionis:</s> <s xml:id="echoid-s9828" xml:space="preserve"> erit [per 12 n 4] angulus a g k æqualis angulo k g b:</s> <s xml:id="echoid-s9829" xml:space="preserve"> & ita [per 29 p 1.</s> <s xml:id="echoid-s9830" xml:space="preserve">1 ax.</s> <s xml:id="echoid-s9831" xml:space="preserve">] ęqualis an-<lb/>gulo a m g:</s> <s xml:id="echoid-s9832" xml:space="preserve"> ſed [per 29 p 1] eſt æqualis m a g:</s> <s xml:id="echoid-s9833" xml:space="preserve"> quia coalternus.</s> <s xml:id="echoid-s9834" xml:space="preserve"> Igitur [per 6 p 1] a g, m g ſunt æ qua <lb/>les.</s> <s xml:id="echoid-s9835" xml:space="preserve"> Sed quoniam g z eſt orthogonalis ſuper quãlibet lineã ſuperficiei z a h:</s> <s xml:id="echoid-s9836" xml:space="preserve"> [per 3 d 11] erit quadra <lb/>tũ m g æquale quadratis m z, g z [per 47 p 1] erit igitur a z æqualis m z [Nam propter eandẽ cauſ-<lb/>ſam quadratum a g æquatur quadratis a z, g z:</s> <s xml:id="echoid-s9837" xml:space="preserve"> at quadrata a g, m g æquãtur:</s> <s xml:id="echoid-s9838" xml:space="preserve"> quia ipſorum latera a g, <lb/>m g æquãtur:</s> <s xml:id="echoid-s9839" xml:space="preserve"> communi igitur quadrato g z ablato, reliquum quadratũ a z ęquabitur quadrato m z:</s> <s xml:id="echoid-s9840" xml:space="preserve"> <lb/>quare ipſorũ latera m z, a z ęquabuntur.</s> <s xml:id="echoid-s9841" xml:space="preserve">] Quare [per 5 p 1] angulus a m z eſt æqualis angulo m a z:</s> <s xml:id="echoid-s9842" xml:space="preserve"> <lb/>ſed [per 29 p 1] angulus a m z eſt æqualis angulo l z h:</s> <s xml:id="echoid-s9843" xml:space="preserve"> & angulus z a m eſt æqualis l z a:</s> <s xml:id="echoid-s9844" xml:space="preserve"> quia coal-<lb/>ternus.</s> <s xml:id="echoid-s9845" xml:space="preserve"> Igitur angulus a z l eſt æqualis angulo l z h.</s> <s xml:id="echoid-s9846" xml:space="preserve"> Quare forma puncti h accedẽs ad punctũ z, re-<lb/> <pb o="156" file="0162" n="162" rhead="ALHAZEN"/> flectetur ad punctum a.</s> <s xml:id="echoid-s9847" xml:space="preserve"> [per 12 n 4.</s> <s xml:id="echoid-s9848" xml:space="preserve">] Si ergo dicatur, quòd ab alio puncto, quàm à puncto g, poteſt <lb/>forma b reflecti ad a:</s> <s xml:id="echoid-s9849" xml:space="preserve"> illud aliud punctũ aut erit in linea lõgitudinis, quæ eſt g z:</s> <s xml:id="echoid-s9850" xml:space="preserve"> aut in alia.</s> <s xml:id="echoid-s9851" xml:space="preserve"> Si eſt in li <lb/>nea g z:</s> <s xml:id="echoid-s9852" xml:space="preserve"> ducatur ab eo perpẽdicularis:</s> <s xml:id="echoid-s9853" xml:space="preserve"> quę neceſſariò ſecabit lineã a k [quia ſecar angulũ lineis inci <lb/>dẽtię & reflexionis cõprehenſum, ut patet per 13 n 4] & [per 28 p 1] erit æquidiſtãs lineę a m:</s> <s xml:id="echoid-s9854" xml:space="preserve"> & li-<lb/>nea ducta à puncto b ad illud punctũ neceſſariò cõcurret cũ a m:</s> <s xml:id="echoid-s9855" xml:space="preserve"> [per lemma Procli ad 29 p 1] & e-<lb/>rit punctũ illud, & punctũ m in eadẽ ſuperficie:</s> <s xml:id="echoid-s9856" xml:space="preserve"> & linea illa aut cadet ſuper pũctũ m:</s> <s xml:id="echoid-s9857" xml:space="preserve"> autſuper aliud.</s> <s xml:id="echoid-s9858" xml:space="preserve"> <lb/>Si ſuper punctũ m:</s> <s xml:id="echoid-s9859" xml:space="preserve"> erit ducere à puncto b ad punctũ m duas lineas rectas:</s> <s xml:id="echoid-s9860" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s9861" xml:space="preserve"> [ſic <lb/>enim duę rectę lineę ſpatiũ cõprehenderẽt cõtra 12 ax.</s> <s xml:id="echoid-s9862" xml:space="preserve">] Si aũtad aliud punctũ lineę a m:</s> <s xml:id="echoid-s9863" xml:space="preserve"> ducatur à <lb/>puncto illo linea ad punctũ z:</s> <s xml:id="echoid-s9864" xml:space="preserve"> & probabitur, quòd hęc linea cũ h z facit lineã rectã, ſicut probatũ eſt <lb/>de linea z m:</s> <s xml:id="echoid-s9865" xml:space="preserve"> & ita à puncto h erit ducere duas lineas rectas, per punctũ z trãſeuntes in diuerſa pun-<lb/>cta lineę a m cadẽtes:</s> <s xml:id="echoid-s9866" xml:space="preserve"> quod eſt impoſsibile [& cõtra 1 p 11:</s> <s xml:id="echoid-s9867" xml:space="preserve"> hocq́;</s> <s xml:id="echoid-s9868" xml:space="preserve"> modo duarü rectarũ linearũ eſſet <lb/>cõmune ſegmentum contra lineę rectę definitionẽ.</s> <s xml:id="echoid-s9869" xml:space="preserve">] Palàm ergo, quòd à nullo puncto lineę g z, niſi <lb/>à g, poteſt b reflecti ad a.</s> <s xml:id="echoid-s9870" xml:space="preserve"> Si dicatur, quòd à puncto extra hãc lineam ſumpto:</s> <s xml:id="echoid-s9871" xml:space="preserve"> ducatur ſuper punctũ <lb/>illud linea longitudinis ſpeculi:</s> <s xml:id="echoid-s9872" xml:space="preserve"> [per 7 th.</s> <s xml:id="echoid-s9873" xml:space="preserve"> Sereni de ſectione cylindri] & à puncto circuli e z i, in <lb/>quod cadit hęc linea, probabitur h reflecti ad a ſecundũ ſuprà dictã probationẽ:</s> <s xml:id="echoid-s9874" xml:space="preserve"> ſed iã probatũ eſt.</s> <s xml:id="echoid-s9875" xml:space="preserve"> <lb/>quòd h à puncto z reflectitur ad a.</s> <s xml:id="echoid-s9876" xml:space="preserve"> Etita impoſsibile:</s> <s xml:id="echoid-s9877" xml:space="preserve"> [quia ita à duobus ſpeculi punctis forma e-<lb/>iuſdem uiſibilis ad eundem uiſum reflecteretur, contra 51 n 4, & 29 n.</s> <s xml:id="echoid-s9878" xml:space="preserve">] Reſtat ergo ut à ſolo puncto <lb/>ſpeculi reflectatur b ad a.</s> <s xml:id="echoid-s9879" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s9880" xml:space="preserve"/> </p> <div xml:id="echoid-div359" type="float" level="0" n="0"> <figure xlink:label="fig-0161-02" xlink:href="fig-0161-02a"> <variables xml:id="echoid-variables78" xml:space="preserve">a ſ f K b h d z g e s n q o t m i p</variables> </figure> </div> </div> <div xml:id="echoid-div361" type="section" level="0" n="0"> <head xml:id="echoid-head351" xml:space="preserve" style="it">48. Si communis ſectio ſuperſicierum, reflexionis & ſpeculi cylindracei conuexi fuerit elli-<lb/>pſis: uiſu & uiſibili datis, punctum reflexionis inucnire. 29 p 7.</head> <p> <s xml:id="echoid-s9881" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s9882" xml:space="preserve"> dato pũcto b, quod reflectatur ad a:</s> <s xml:id="echoid-s9883" xml:space="preserve"> erit in uenire punctũ reflexionis:</s> <s xml:id="echoid-s9884" xml:space="preserve"> & hoc patebit <lb/>per reuolutionẽ prędictę probationis.</s> <s xml:id="echoid-s9885" xml:space="preserve"> Ducatur à puncto a ſuperficies æquidiſtãs baſi colu-<lb/>mnę:</s> <s xml:id="echoid-s9886" xml:space="preserve"> quę quidẽ ſecabit columnã ſuper circulũ:</s> <s xml:id="echoid-s9887" xml:space="preserve"> [per 5 th.</s> <s xml:id="echoid-s9888" xml:space="preserve"> Sereni de ſectione cylindri] qui ſit <lb/>e z i:</s> <s xml:id="echoid-s9889" xml:space="preserve"> & ducatur à puncto b perpẽdicularis ſuperhãc ſuperficiẽ:</s> <s xml:id="echoid-s9890" xml:space="preserve"> quę ſit b h:</s> <s xml:id="echoid-s9891" xml:space="preserve"> & inueniatur in hac ſu-<lb/>perficie punctũ, à quo fit reflexio h ad a:</s> <s xml:id="echoid-s9892" xml:space="preserve"> [ut traditũ eſt 31 uel 39 n] quod ſit z:</s> <s xml:id="echoid-s9893" xml:space="preserve"> & à puncto z ducatur <lb/>linea longitudinis:</s> <s xml:id="echoid-s9894" xml:space="preserve"> [per 7 th.</s> <s xml:id="echoid-s9895" xml:space="preserve"> Sereni de ſectione cylindri] quę ſit z g:</s> <s xml:id="echoid-s9896" xml:space="preserve"> & à pũcto z perpẽdicularis z <lb/>l:</s> <s xml:id="echoid-s9897" xml:space="preserve"> & huic æquidiſtãs à pũcto a:</s> <s xml:id="echoid-s9898" xml:space="preserve"> quę ſit a m:</s> <s xml:id="echoid-s9899" xml:space="preserve"> & etiã linea h z producatur, quouſq;</s> <s xml:id="echoid-s9900" xml:space="preserve"> cõcurrat cũea:</s> <s xml:id="echoid-s9901" xml:space="preserve"> [con <lb/>curret uerò per lemma Procli ad 29 p 1] & ſit cõcurſus in pũcto m:</s> <s xml:id="echoid-s9902" xml:space="preserve"> & à pũcto m ducatur linea ad b:</s> <s xml:id="echoid-s9903" xml:space="preserve"> <lb/>quę neceſſariò ſecabit lineã z g:</s> <s xml:id="echoid-s9904" xml:space="preserve"> cũ ſit in eadẽ ſuperficie cũ ea:</s> <s xml:id="echoid-s9905" xml:space="preserve"> quoniã cũ b h ſit æquidiſtãs g z:</s> <s xml:id="echoid-s9906" xml:space="preserve"> [per <lb/>6 p 11:</s> <s xml:id="echoid-s9907" xml:space="preserve"> eſt enim utraq;</s> <s xml:id="echoid-s9908" xml:space="preserve"> ipſarũ perpẽdicularis circulo e zi] erit h z m in ſuperficie illarũ:</s> <s xml:id="echoid-s9909" xml:space="preserve"> [per 7 p 11:</s> <s xml:id="echoid-s9910" xml:space="preserve"> <lb/>quia cõnectit parallelas] & ita b m in eadẽ:</s> <s xml:id="echoid-s9911" xml:space="preserve"> quę, ſi ſecuerit z g in puncto g:</s> <s xml:id="echoid-s9912" xml:space="preserve"> erit g punctum reflexio-<lb/>nis:</s> <s xml:id="echoid-s9913" xml:space="preserve"> quod quidem, ſi reuoluas probationem prædictam, uidere poteris.</s> <s xml:id="echoid-s9914" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div362" type="section" level="0" n="0"> <head xml:id="echoid-head352" xml:space="preserve" style="it">49. Si communis ſectio ſuperficierum, reflexionis & ſpeculi conici conuexi fuerit lat{us} coni: <lb/>locatum reflexionum tum imaginum eodem modo ſe habebunt, ut in ſpeculo plano. 42 p 7.</head> <p> <s xml:id="echoid-s9915" xml:space="preserve">IN ſpeculis exteriorib.</s> <s xml:id="echoid-s9916" xml:space="preserve"> pyramidalibus, ſi linea cõmunis ſupficiei reflexiõis & ſpeculi, fuerit linea <lb/>lõgitudinis ſpeculi:</s> <s xml:id="echoid-s9917" xml:space="preserve"> erit locus imaginis, ſicut aſsignatus eſt in ſpeculis planis.</s> <s xml:id="echoid-s9918" xml:space="preserve"> Et eadẽ eſt ꝓbatio.</s> <s xml:id="echoid-s9919" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div363" type="section" level="0" n="0"> <head xml:id="echoid-head353" xml:space="preserve" style="it">50. Cõmunis ſectio ſuperficierũ, reflexiõis et ſpeculi conici cõuexi nõ eſt circul{us}. 12 p 7. Idẽ 41 n 4.</head> <p> <s xml:id="echoid-s9920" xml:space="preserve">QVòd aũt nõ poſsit eſſe linea cõmunis, circulus:</s> <s xml:id="echoid-s9921" xml:space="preserve"> palàm per hoc:</s> <s xml:id="echoid-s9922" xml:space="preserve"> q đ ſuperficies reflexionis or <lb/>thogonalis eſt ſuper ſuperficiẽ, cõtingentẽ ſpeculũ in pũcto reflexionis [per 13 n 4] & cir-<lb/>culus neceſſariò eſt æquidiſtans baſi.</s> <s xml:id="echoid-s9923" xml:space="preserve"> [per cõuerſionẽ 4 th 1 conicorũ Apollonij] Superfi-<lb/>cies ergo hęc æquidiſtãs baſi, nõ erit orthogonalis ſuper ſuperficiẽ, cõtingentẽ ſpeculũ.</s> <s xml:id="echoid-s9924" xml:space="preserve"> [Nam pla-<lb/>nũ tangẽs conũ, tangit in latere per 35 n 4, ad baſim & circulũ ipſi parallelũ obliquo:</s> <s xml:id="echoid-s9925" xml:space="preserve"> quia eſt latus <lb/>trianguli acutanguli facti à plano conũ per uerticẽ ſecante, per 3 th 1 conicorũ Apollonij.</s> <s xml:id="echoid-s9926" xml:space="preserve"> Quare cir <lb/>culus erit extra reflexionis ſuperficiem:</s> <s xml:id="echoid-s9927" xml:space="preserve"> neq;</s> <s xml:id="echoid-s9928" xml:space="preserve"> idcirco uiſibile ab ipſo ad uiſum reflectetur.</s> <s xml:id="echoid-s9929" xml:space="preserve">]</s> </p> </div> <div xml:id="echoid-div364" type="section" level="0" n="0"> <head xml:id="echoid-head354" xml:space="preserve" style="it">51. Si cõmunis ſectio ſuperficierũ reflexiõis & ſpeculi conici cõuexi fuerit ellipſis: imago uiſibilis <lb/>obliquè reflexi, aliâs in ſuperficie ſpeculi: aliâs intra: aliâs extra ſpeculũ uidebitur. 49 p 7.</head> <p> <s xml:id="echoid-s9930" xml:space="preserve">SI uerò cõmunis linea fuerit ſectio pyramidalis:</s> <s xml:id="echoid-s9931" xml:space="preserve"> imagines quędam erunt in ſuperficie ſpeculi:</s> <s xml:id="echoid-s9932" xml:space="preserve"> <lb/>quędã intra ſpeculũ:</s> <s xml:id="echoid-s9933" xml:space="preserve"> quędã extra.</s> <s xml:id="echoid-s9934" xml:space="preserve"> Etidẽ eſt aſsignationis modus, qui fuit in ſpeculo columna-<lb/>ri exteriore:</s> <s xml:id="echoid-s9935" xml:space="preserve"> [44 n] & eadẽ ꝓbatio.</s> <s xml:id="echoid-s9936" xml:space="preserve"> Et (ſicut eſt in colũnari exteriore) [44 n] penperpẽdi <lb/>cularẽ uiſualẽ nõ reflectetur forma ad oculũ, niſi pũcti ſuperficiei oculi tãtũ:</s> <s xml:id="echoid-s9937" xml:space="preserve"> & hoc ab uno ſolo ſpe-<lb/>culi pũcto:</s> <s xml:id="echoid-s9938" xml:space="preserve"> & locus imaginis eius erit cõtinuus locis aliarũ imaginũ, ſicut patuit ſuperius [44 n.</s> <s xml:id="echoid-s9939" xml:space="preserve">]</s> </p> </div> <div xml:id="echoid-div365" type="section" level="0" n="0"> <head xml:id="echoid-head355" xml:space="preserve" style="it">52. Si à puncto in communi ſectione ſuperficierum, reflexionis & ſpeculi conici conuexi dato, re <lb/>flexio fiat: poſſunt uiſ{us} & uiſibile ſic collocari, ut ab eodem puncto, tanquam puncto circuli ba-<lb/>ſi paralleli ad uiſum reflexio fiat. 32 p 7.</head> <p> <s xml:id="echoid-s9940" xml:space="preserve">REſtat in his ſpeculis declarare:</s> <s xml:id="echoid-s9941" xml:space="preserve"> quòd ab uno ſolo puncto eius fiat reflexio:</s> <s xml:id="echoid-s9942" xml:space="preserve"> quod ſic patebit.</s> <s xml:id="echoid-s9943" xml:space="preserve"> <lb/>Sit uiſus a:</s> <s xml:id="echoid-s9944" xml:space="preserve"> b punctũ uiſum:</s> <s xml:id="echoid-s9945" xml:space="preserve"> g punctũ reflexionis:</s> <s xml:id="echoid-s9946" xml:space="preserve"> & ducatur ſuper punctũ g ſuperficies æqui <lb/>diſtãs baſi:</s> <s xml:id="echoid-s9947" xml:space="preserve"> [ductis nimirũ duabus perpẽdicularibus ſuper axem ſe interſecãtibus:</s> <s xml:id="echoid-s9948" xml:space="preserve"> una qui-<lb/>dẽ à reflexionis puncto per 12 p 1:</s> <s xml:id="echoid-s9949" xml:space="preserve"> altera uerò ab axis puncto, in quod illa cadit, per 11 p 1.</s> <s xml:id="echoid-s9950" xml:space="preserve"> Sic enim <lb/>axis, qui per 18 d 11 perpendicularis eſt baſi:</s> <s xml:id="echoid-s9951" xml:space="preserve"> erit per 4 p 11 perpendicularis plano ductarũ perpendi <lb/>culariũ.</s> <s xml:id="echoid-s9952" xml:space="preserve"> Quare per 14 p 11 baſis & hoc planũ erunt parallela] quę quidẽ ſecabit pyramidẽ ſuper cir-<lb/> <pb o="157" file="0163" n="163" rhead="OPTICAE LIBER V."/> culum [per 4 th.</s> <s xml:id="echoid-s9953" xml:space="preserve"> 1 conicorũ Apollonij] ꝗ ſit p g:</s> <s xml:id="echoid-s9954" xml:space="preserve"> & ducãtur lineę a g, b g, a b:</s> <s xml:id="echoid-s9955" xml:space="preserve"> & à pũcto g ducatur ad <lb/>cẽtrũ circuli linea:</s> <s xml:id="echoid-s9956" xml:space="preserve"> q̃ ſit g t:</s> <s xml:id="echoid-s9957" xml:space="preserve"> & uertex pyramidis ſit e:</s> <s xml:id="echoid-s9958" xml:space="preserve"> à quo ducatur axis:</s> <s xml:id="echoid-s9959" xml:space="preserve"> ꝗ erit e t.</s> <s xml:id="echoid-s9960" xml:space="preserve"> [per 3 d 1 coni.</s> <s xml:id="echoid-s9961" xml:space="preserve"> A-<lb/>pol.</s> <s xml:id="echoid-s9962" xml:space="preserve">] Et ducatur [per 12 p 11] perpẽdicularis ſuper ſuperficiẽ, cõtingentẽ fpeculũ in pũcto g:</s> <s xml:id="echoid-s9963" xml:space="preserve"> q̃ ſit h <lb/>g:</s> <s xml:id="echoid-s9964" xml:space="preserve"> q̃ cũ diuidat angulũ a g b per æqualia, [per 13 n 4] cadet ſuper a b:</s> <s xml:id="echoid-s9965" xml:space="preserve"> pũctũ caſus ſit z.</s> <s xml:id="echoid-s9966" xml:space="preserve"> Et à uertice py <lb/>ramidis ducatur linea lõgitudinis ſpeculi ad punctũ g:</s> <s xml:id="echoid-s9967" xml:space="preserve"> [educto nẽpe plano per axem, & perrectã à <lb/>puncto g, cũ ipſo utlibet cõcurrentẽ:</s> <s xml:id="echoid-s9968" xml:space="preserve"> cõmunis enim fectio huius plani & conicæ ſuperficiei erit la-<lb/>tus coni, ք 18 d 11, uel 3 th.</s> <s xml:id="echoid-s9969" xml:space="preserve"> 1 coni.</s> <s xml:id="echoid-s9970" xml:space="preserve"> Apol.</s> <s xml:id="echoid-s9971" xml:space="preserve">] quę ſit e g:</s> <s xml:id="echoid-s9972" xml:space="preserve"> cui lineæ ducatur æquidiſtãs à pũcto a:</s> <s xml:id="echoid-s9973" xml:space="preserve"> [per 31 p 1] <lb/>quę neceſſariò ſecabit ſuperficiẽ circuli g p:</s> <s xml:id="echoid-s9974" xml:space="preserve"> [ſi enim circulũ cũ diametro infinitè extẽſum cogites:</s> <s xml:id="echoid-s9975" xml:space="preserve"> <lb/>diameter ſecãs e g conilatus, ſecabit etiã rectã lateri parallelã, per lẽma Procli ad 29 p 1.</s> <s xml:id="echoid-s9976" xml:space="preserve"> Quare eadẽ <lb/>parallela circulũ ipſum quoq;</s> <s xml:id="echoid-s9977" xml:space="preserve"> ſecabit] ſecet in pũcto n:</s> <s xml:id="echoid-s9978" xml:space="preserve"> & ſit n a.</s> <s xml:id="echoid-s9979" xml:space="preserve"> Similiter à pũcto b ducatur æqui-<lb/>diſtãs eidẽ e g, ſcilicet b m:</s> <s xml:id="echoid-s9980" xml:space="preserve"> quę ſecet ſuperficiẽ p g in pũcto m.</s> <s xml:id="echoid-s9981" xml:space="preserve"> Et à pũcto n ducatur ę ꝗ diſtãs ipſi g t:</s> <s xml:id="echoid-s9982" xml:space="preserve"> <lb/>quę ſit n f:</s> <s xml:id="echoid-s9983" xml:space="preserve"> & ducãtur lineæ n g, m g, n m.</s> <s xml:id="echoid-s9984" xml:space="preserve"> Palàm, quòd t g ſecabit m n:</s> <s xml:id="echoid-s9985" xml:space="preserve"> [per lẽma Procli ad 29 p 1] ſe-<lb/>cetin pũcto q.</s> <s xml:id="echoid-s9986" xml:space="preserve"> Palàm etiã, quòd m g ſecabit n f:</s> <s xml:id="echoid-s9987" xml:space="preserve"> cũ ſecet ei æquidiſtãtẽ:</s> <s xml:id="echoid-s9988" xml:space="preserve"> ſit pũctũ ſectionis f.</s> <s xml:id="echoid-s9989" xml:space="preserve"> Et à pun <lb/>cto a ducatur æquidiſtãs h z:</s> <s xml:id="echoid-s9990" xml:space="preserve"> quę ſit a l.</s> <s xml:id="echoid-s9991" xml:space="preserve"> Palàm [per lẽma Procli ad 29 p 1] quòd b g cõcurret cũ a l:</s> <s xml:id="echoid-s9992" xml:space="preserve"> <lb/>ſit cõcurſus l.</s> <s xml:id="echoid-s9993" xml:space="preserve"> Deinde ducatur linea cõmunis ſuperficiei, <lb/> <anchor type="figure" xlink:label="fig-0163-01a" xlink:href="fig-0163-01"/> cõtingẽti ſpeculũ in puncto g, & ſuperficiei circuli p g:</s> <s xml:id="echoid-s9994" xml:space="preserve"> q̃ <lb/>ſit g o.</s> <s xml:id="echoid-s9995" xml:space="preserve"> Palàm [per 18 p 3] quòd erit orthogonalis ſuper <lb/>g t:</s> <s xml:id="echoid-s9996" xml:space="preserve"> & ſimiliter [ք 29 p 1] ſuper n f.</s> <s xml:id="echoid-s9997" xml:space="preserve"> Sumatur etiã linea cõ-<lb/>munis ſuperficiei, cõtingẽti ſpeculũ, & ſuperficiei reflexi <lb/>onis:</s> <s xml:id="echoid-s9998" xml:space="preserve"> quę ſit g d:</s> <s xml:id="echoid-s9999" xml:space="preserve"> q̃ quidẽ cũ ſecet g h, ſecabit a l.</s> <s xml:id="echoid-s10000" xml:space="preserve"> [per lẽma <lb/>Procli ad 29 p 1.</s> <s xml:id="echoid-s10001" xml:space="preserve">] Sit punctũ ſectionis d:</s> <s xml:id="echoid-s10002" xml:space="preserve"> & erit orthogo-<lb/>nalis ſuper a l.</s> <s xml:id="echoid-s10003" xml:space="preserve"> [Quia enim h g perpẽdicularis eſt plano, <lb/>tãgẽti ſpeculũ in pũcto reflexionis g, ք fabricationẽ:</s> <s xml:id="echoid-s10004" xml:space="preserve"> erit <lb/>ք 3 d 11 perpẽdicularis rectæ lineæ g d ipſam in puncto g <lb/>tãgẽti.</s> <s xml:id="echoid-s10005" xml:space="preserve"> Et quoniã a l, h z ſunt parallelę, ք fabricationẽ:</s> <s xml:id="echoid-s10006" xml:space="preserve"> erit <lb/>g d perpẽdicularis ipſi a l per 29 p 1.</s> <s xml:id="echoid-s10007" xml:space="preserve">] Palàm ex prędictis, <lb/>quoniã n f eſt æquidiſtãs g t, & a l ęquidiſtãs g h:</s> <s xml:id="echoid-s10008" xml:space="preserve"> igitur [ք <lb/>15 p 11] ſuperficies, in qua ſunt n f, al, eſt ęquidiſtãs ſuper-<lb/>ficiei g t h:</s> <s xml:id="echoid-s10009" xml:space="preserve"> ſed linea e g æquidiſtat b m [ք fabricationẽ] <lb/>quare ſunt in eadẽ ſuperficie [ք 35 d 1] q̃ ſuperficies ſecat <lb/>preędictas æquidiſtãtes:</s> <s xml:id="echoid-s10010" xml:space="preserve"> unã ſuper lineã e g:</s> <s xml:id="echoid-s10011" xml:space="preserve"> aliã ſuper li-<lb/>neã fl.</s> <s xml:id="echoid-s10012" xml:space="preserve"> Quare [ք 16 p 11] fl eſt æquidiſtãs e g:</s> <s xml:id="echoid-s10013" xml:space="preserve"> ſed a n æqui <lb/>diſtat eidẽ.</s> <s xml:id="echoid-s10014" xml:space="preserve"> Igitur [ք 30 p 1] fl eſt æquidiſtãs an.</s> <s xml:id="echoid-s10015" xml:space="preserve"> Verũ ſu <lb/>perficies cõtingẽs ſpeculũ in pũcto g, ſecat ſuperficies e-<lb/>aſdẽ æquidiſtãtes:</s> <s xml:id="echoid-s10016" xml:space="preserve"> unã in linea e g:</s> <s xml:id="echoid-s10017" xml:space="preserve"> aliã in linea o d.</s> <s xml:id="echoid-s10018" xml:space="preserve"> Igitur <lb/>[ք 16 p 11] o d eſt æquidiſtãs e g.</s> <s xml:id="echoid-s10019" xml:space="preserve"> Igitur [ք 30 p 1] eſt æ ꝗ-<lb/>diſtãs a n & l f.</s> <s xml:id="echoid-s10020" xml:space="preserve"> Et à pũcto f ducatur linea æ quidiſtãs l a, <lb/>ſecãs d o in k, & a n in i:</s> <s xml:id="echoid-s10021" xml:space="preserve"> ergo f k æqualis l d, & k i æqualis <lb/>d a.</s> <s xml:id="echoid-s10022" xml:space="preserve"> [ք 34 p 1.</s> <s xml:id="echoid-s10023" xml:space="preserve">] Quare erit ꝓ portio a d ad d l, ſicut n o ad <lb/>o f.</s> <s xml:id="echoid-s10024" xml:space="preserve"> [nã ք 7 p 5 eſt, ut a d ad d l, ſic i k ad k f:</s> <s xml:id="echoid-s10025" xml:space="preserve"> ſed ք 2 p 6, ut <lb/>i k ad k f, ſic n o ad o f:</s> <s xml:id="echoid-s10026" xml:space="preserve"> ergo ք 11 p 5, ut a d ad d l, ſic n o ad o <lb/>f.</s> <s xml:id="echoid-s10027" xml:space="preserve">] Palã etiã, quòd angulus b g z æqualis eſt angulo z g a:</s> <s xml:id="echoid-s10028" xml:space="preserve"> [recta enim linea g z bifariã ſecat angulũ a <lb/>g b, ut patuit] & etiã angulo g l a:</s> <s xml:id="echoid-s10029" xml:space="preserve"> [interiori & oppoſito per 29 p 1] & etiã angulo g a l:</s> <s xml:id="echoid-s10030" xml:space="preserve"> [alterno ք 29 <lb/>p 1.</s> <s xml:id="echoid-s10031" xml:space="preserve">] Quare [per 1 ax.</s> <s xml:id="echoid-s10032" xml:space="preserve">] g a l, g l a ſunt æquales:</s> <s xml:id="echoid-s10033" xml:space="preserve"> & [ք 6 p 1] g a, g l æquales:</s> <s xml:id="echoid-s10034" xml:space="preserve"> & g d քpẽdicularis ſuper <lb/>al:</s> <s xml:id="echoid-s10035" xml:space="preserve"> [per cõcluſionẽ] erit [per 26 p 1] a d æqualis d l.</s> <s xml:id="echoid-s10036" xml:space="preserve"> Erit igitur n o ęqualis o f:</s> <s xml:id="echoid-s10037" xml:space="preserve"> [demõſtratũ enim eſt, <lb/>ut a d ad d l, ſic n o ad o f:</s> <s xml:id="echoid-s10038" xml:space="preserve"> & alternè, ut a d ad n o, ſic d l ad o f:</s> <s xml:id="echoid-s10039" xml:space="preserve"> ſed a d æquatur ipſi d l:</s> <s xml:id="echoid-s10040" xml:space="preserve"> ergo ք 14 p 5 n o <lb/>æquabitur ipſi o f] & g o perpẽdicularis ſuper n f:</s> <s xml:id="echoid-s10041" xml:space="preserve"> [parallelæ enim ſunt n f, g t ք fabricationẽ, & g o <lb/>perpẽdicularis eſt ipſi gt per 18 p 3:</s> <s xml:id="echoid-s10042" xml:space="preserve"> ergo per 29 p 1 g o eſt perpendicularis ipſi n f:</s> <s xml:id="echoid-s10043" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s10044" xml:space="preserve"> angulus ad <lb/>o uterq;</s> <s xml:id="echoid-s10045" xml:space="preserve"> rectus eſt] erit [per 4 p 1] angulus o f g ęqualis angulo o n g.</s> <s xml:id="echoid-s10046" xml:space="preserve"> Erit igitur angulus n g q ęqua <lb/>lis angulo m g q.</s> <s xml:id="echoid-s10047" xml:space="preserve"> [Nã cũ t q, f n ductę ſint parallelę:</s> <s xml:id="echoid-s10048" xml:space="preserve"> æquabitur ք 29 p 1 angulus m g q angulo n f g:</s> <s xml:id="echoid-s10049" xml:space="preserve"> ք <lb/>æqualis cõcluſus eſt ipſi f n g:</s> <s xml:id="echoid-s10050" xml:space="preserve"> æquali angulo n g q alterno per 29 p 1.</s> <s xml:id="echoid-s10051" xml:space="preserve"> Quare anguli m g q, n g q inter <lb/>ſe ęquãtur.</s> <s xml:id="echoid-s10052" xml:space="preserve">] Igitur [per 12 n 4] à puncto circuli p g, quod eſt g, poteſt punctum m reflecti ad n, nõ <lb/>impediente pyramide.</s> <s xml:id="echoid-s10053" xml:space="preserve"> [Hęc concluſio uidetur repugnare 41 n 4 & 50 n, quibus demonſtratum eſt <lb/>communem ſectionem ſuperficierum reflexionis & ſpeculi conici cõuexi non eſfe circulum.</s> <s xml:id="echoid-s10054" xml:space="preserve"> Qua-<lb/>re punctum g circuli p g, à quo hic reflexio fieri concluditur, intelligendum eſt punctum circuli, <lb/>qui eſt communis ſectio ſphæræuel cylindri, quos mens intra conum fingit ac concipit.</s> <s xml:id="echoid-s10055" xml:space="preserve">]</s> </p> <div xml:id="echoid-div365" type="float" level="0" n="0"> <figure xlink:label="fig-0163-01" xlink:href="fig-0163-01a"> <variables xml:id="echoid-variables79" xml:space="preserve">f d a e p t m f k h i g z o q n b</variables> </figure> </div> </div> <div xml:id="echoid-div367" type="section" level="0" n="0"> <head xml:id="echoid-head356" xml:space="preserve" style="it">53. Si communis ſectio ſuperficierum, reflexionis, & ſpeculi conici cõuexifuerit latus conicũ: <lb/>ab uno puncto unum uiſibilis punctum ad unum uiſum reflectetur. 33 p 7.</head> <p> <s xml:id="echoid-s10056" xml:space="preserve">DIco igitur, quòd punctũ b à ſolo g reflectitur ad a.</s> <s xml:id="echoid-s10057" xml:space="preserve"> Si enim dicatur, quòd ab alio pũcto poteſt <lb/>reflecti:</s> <s xml:id="echoid-s10058" xml:space="preserve"> illud aut erit in linea lõgitudinis:</s> <s xml:id="echoid-s10059" xml:space="preserve"> quę eſt e g:</s> <s xml:id="echoid-s10060" xml:space="preserve"> aut nõ.</s> <s xml:id="echoid-s10061" xml:space="preserve"> Sit in ea:</s> <s xml:id="echoid-s10062" xml:space="preserve"> & ſit x:</s> <s xml:id="echoid-s10063" xml:space="preserve"> & ab eo ducatur <lb/>perpẽdicularis ſuper ſuperficiẽ, cõtingẽtẽ ſpeculũ in pũcto illo:</s> <s xml:id="echoid-s10064" xml:space="preserve"> [per 12 p 11] q̃ quidẽ perpẽdi-<lb/>cularis, erit [ք 6 p 11] ęquidiſtãs z g:</s> <s xml:id="echoid-s10065" xml:space="preserve"> & ita [per 30 p 1] æquidiſtãs a l.</s> <s xml:id="echoid-s10066" xml:space="preserve"> Igitur a l eſt in ſuperficie reflexio-<lb/>nis huius perpẽdicularis:</s> <s xml:id="echoid-s10067" xml:space="preserve"> [per 35 d 1] & eſt ſimiliter in ſuperficie reflexionis perpẽdicularis z g:</s> <s xml:id="echoid-s10068" xml:space="preserve"> [ք <lb/>35 d 1:</s> <s xml:id="echoid-s10069" xml:space="preserve"> parallela enim ducta eſt a l ipſi z g] igitur illæ duæ ſuperficies reflexiõis ſecãt ſe ſuper lineam <lb/> <pb o="158" file="0164" n="164" rhead="ALHAZEN"/> al:</s> <s xml:id="echoid-s10070" xml:space="preserve"> ſed ſecãt ſe ſuper pũctũ b:</s> <s xml:id="echoid-s10071" xml:space="preserve"> [quia uiſibile eſt in qualibet reflexionis ſuperficie ք 23 n 4] qđ eſt im-<lb/>poſsibile.</s> <s xml:id="echoid-s10072" xml:space="preserve"> Quoniã b nõ eſt in linea a l:</s> <s xml:id="echoid-s10073" xml:space="preserve"> qđ patet ք hoc:</s> <s xml:id="echoid-s10074" xml:space="preserve"> quoniã fl æquidiſtat b m.</s> <s xml:id="echoid-s10075" xml:space="preserve"> [ut patu it proximo <lb/>numero per fabricationẽ & 30 p 1.</s> <s xml:id="echoid-s10076" xml:space="preserve">] Reſtat ergo, ut à nullo pũcto lineæ e g, pręterquã à g, poſsit re-<lb/>flecti b a d a.</s> <s xml:id="echoid-s10077" xml:space="preserve"> Si aũt ab aliquo pũcto extra lineã e g:</s> <s xml:id="echoid-s10078" xml:space="preserve"> ſit illud u:</s> <s xml:id="echoid-s10079" xml:space="preserve"> & ducatur linea lõgitudinis e u o:</s> <s xml:id="echoid-s10080" xml:space="preserve"> & ſu-<lb/>matur ſuperficies æquidiſtãs baſi, trãſiẽs ք pũctũ u.</s> <s xml:id="echoid-s10081" xml:space="preserve"> [ut dictũ eſt proximo numero.</s> <s xml:id="echoid-s10082" xml:space="preserve">] Palã, quòd a n <lb/>ſecabit hãc ſuperficiẽ:</s> <s xml:id="echoid-s10083" xml:space="preserve"> [quia e g parallela ipſi a n, eandẽ ſecat] ſit punctũ ſectionis y.</s> <s xml:id="echoid-s10084" xml:space="preserve"> Similiter b m ſe-<lb/>cabit eandẽ:</s> <s xml:id="echoid-s10085" xml:space="preserve"> ſit punctũ ſectionis k:</s> <s xml:id="echoid-s10086" xml:space="preserve"> & ducãtur lineæ k u, y <lb/> <anchor type="figure" xlink:label="fig-0164-01a" xlink:href="fig-0164-01"/> u, y k.</s> <s xml:id="echoid-s10087" xml:space="preserve"> Et cũ ſuperficies illa ſecet pyramidẽ ſuper circulũ, <lb/>trãſeuntẽ per u [per 4 th 1 coni.</s> <s xml:id="echoid-s10088" xml:space="preserve"> Apol.</s> <s xml:id="echoid-s10089" xml:space="preserve">] ducatur à pũcto u <lb/>linea ad cẽtrũ huius circuli, quę extra circulũ ꝓducta, ſit <lb/>r u:</s> <s xml:id="echoid-s10090" xml:space="preserve"> & ducãtur lineę e k, e y:</s> <s xml:id="echoid-s10091" xml:space="preserve"> quę quidẽ ſecabũt ſuperficiẽ <lb/>circuli p g:</s> <s xml:id="echoid-s10092" xml:space="preserve"> [quia ſecãt circulũ ipſi parallelũ, per u trãſeun <lb/>tẽ] & ſint pũcta ſectionũ s, i:</s> <s xml:id="echoid-s10093" xml:space="preserve"> & ducãtur lineæ i c, s c.</s> <s xml:id="echoid-s10094" xml:space="preserve"> Sicut <lb/>igitur probatũ eſt [proximo numero] de pũcto m:</s> <s xml:id="echoid-s10095" xml:space="preserve"> quòd, <lb/>nõ impediente pyramide, poteſt reflecti ad n à pũcto g:</s> <s xml:id="echoid-s10096" xml:space="preserve"> <lb/>ita ꝓbabitur de pũcto k:</s> <s xml:id="echoid-s10097" xml:space="preserve"> qđ poteſt reflecti à puncto u ad <lb/>punctũ y:</s> <s xml:id="echoid-s10098" xml:space="preserve"> & eadẽ eſt ꝓbatio:</s> <s xml:id="echoid-s10099" xml:space="preserve"> & ita angulus r u y erit ęqua <lb/>lis angulo r u k [per 12 n 4.</s> <s xml:id="echoid-s10100" xml:space="preserve">] Palàm, quoniã b k eſt æquidi <lb/>ſtãs e g:</s> <s xml:id="echoid-s10101" xml:space="preserve"> [Nã b m parallela ipſi e g ք fabricationẽ, cõtinua-<lb/>ta eſt in pũctũ k] & linea, cõmunis ſuքficiei b g e k, & ſuք-<lb/>ficiei circùli p g, eſt linea m g.</s> <s xml:id="echoid-s10102" xml:space="preserve"> Igitur linea e k cũ ſit in hac <lb/>ſuperficie, & ſecet ſuperficiẽ circuli p g:</s> <s xml:id="echoid-s10103" xml:space="preserve"> [in pũcto s, ut pa <lb/>tuit] cadet ſuper lineã cõmunẽ, quę eſt m g.</s> <s xml:id="echoid-s10104" xml:space="preserve"> Erit igitur s <lb/>m g linea recta.</s> <s xml:id="echoid-s10105" xml:space="preserve"> Eodẽ modo cũ ſuperficies n y e g ſecet ſu <lb/>perficiẽ circuli p g, ſuper lineã n g:</s> <s xml:id="echoid-s10106" xml:space="preserve"> linea e y cõcurret cũ li <lb/>nea n g.</s> <s xml:id="echoid-s10107" xml:space="preserve"> [in pũcto i, ut patuit.</s> <s xml:id="echoid-s10108" xml:space="preserve">] Igitur i n g linea eſt recta.</s> <s xml:id="echoid-s10109" xml:space="preserve"> <lb/>Palã etiã, quòd ſuքficies i e c ſecat ſuperficiẽ circuli p g, <lb/>ſuper lineã i c, & ſecat ſuperficiẽ huic æquidiſtãtem, quæ <lb/>trãſit ք u, ſuper lineã y u.</s> <s xml:id="echoid-s10110" xml:space="preserve"> Ergo [per 16 p 11] y u æquidiſtat <lb/>i c.</s> <s xml:id="echoid-s10111" xml:space="preserve"> Similiter ſuperficies s e c ſecat ſuperficies illas æqui-<lb/>diſtãtes, ſuper duas lineas s c, k u.</s> <s xml:id="echoid-s10112" xml:space="preserve"> Ergo [per 16 p 11] s c ę-<lb/>quidiſtat k u.</s> <s xml:id="echoid-s10113" xml:space="preserve"> Similiter ſi ſumatur ſuperficies, ſecãs ſpecu <lb/>lũ ſuper lineã lõgitudinis e c, in qua ſuքficie ſuntru, c M:</s> <s xml:id="echoid-s10114" xml:space="preserve"> <lb/>ſecabit illas ſuքficies æquidiſtãtes [nẽpe circulos ք u & c eductos] ſuք duas lineas M c, r u.</s> <s xml:id="echoid-s10115" xml:space="preserve"> Igitur [ք <lb/>16 p 11] hę duę lineæ ſunt æquidiſtãtes.</s> <s xml:id="echoid-s10116" xml:space="preserve"> Igitur angulus s c M æqualis eſt angulo k u r, & angulus M c <lb/>i æqualis angulo r u y.</s> <s xml:id="echoid-s10117" xml:space="preserve"> [ք 10 p 11.</s> <s xml:id="echoid-s10118" xml:space="preserve">] Sed iã patuit, qđ angulus k u r æqualis eſt r u y.</s> <s xml:id="echoid-s10119" xml:space="preserve"> Igitur [ք 1 ax.</s> <s xml:id="echoid-s10120" xml:space="preserve">] an-<lb/>gulus s c M æqualis eſt angulo M c i.</s> <s xml:id="echoid-s10121" xml:space="preserve"> Quare pũctũ s poteſt reflecti ad i à puncto c, nõ impediente py <lb/>ramide:</s> <s xml:id="echoid-s10122" xml:space="preserve"> ſed iã probatũ eſt [proximo numero] qđ punctũ m reflecti põt ad i à pũcto g.</s> <s xml:id="echoid-s10123" xml:space="preserve"> [cadunt.</s> <s xml:id="echoid-s10124" xml:space="preserve"> n.</s> <s xml:id="echoid-s10125" xml:space="preserve"> <lb/>pũcta i, n, g in eandẽ rectã lineã, ut mõſtratũ eſt.</s> <s xml:id="echoid-s10126" xml:space="preserve">] Igitur punctũ s reflectitur ad i à duob.</s> <s xml:id="echoid-s10127" xml:space="preserve"> punctis cir-<lb/>culi p g.</s> <s xml:id="echoid-s10128" xml:space="preserve"> [nimirũ g & c] qđ eſt impoſsibile [& cõtra 51 n 4.</s> <s xml:id="echoid-s10129" xml:space="preserve"> 29.</s> <s xml:id="echoid-s10130" xml:space="preserve"> 46 n.</s> <s xml:id="echoid-s10131" xml:space="preserve">] Reſtat ergo, ut primũ ſit impoſ <lb/>ſibile, ſcilicet, ut punctũ b reflectatur ad a ab aliquo puncto alio ſpeculi, ꝗ̃ à g.</s> <s xml:id="echoid-s10132" xml:space="preserve"> Quod eſt propoſitũ.</s> <s xml:id="echoid-s10133" xml:space="preserve"/> </p> <div xml:id="echoid-div367" type="float" level="0" n="0"> <figure xlink:label="fig-0164-01" xlink:href="fig-0164-01a"> <variables xml:id="echoid-variables80" xml:space="preserve">l d a e f <gap/> x u y t k p r c z o h g M n q m i b s</variables> </figure> </div> </div> <div xml:id="echoid-div369" type="section" level="0" n="0"> <head xml:id="echoid-head357" xml:space="preserve" style="it">54. Viſu & uiſibili inter baſim ſpeculi conici conuexi, & planum per uerticem ductum, ba-<lb/>ſí parallelum poſitis: punctum reflexionis inuenire. 35 p 7.</head> <p> <s xml:id="echoid-s10134" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s10135" xml:space="preserve"> dato ſpeculo pyramidali:</s> <s xml:id="echoid-s10136" xml:space="preserve"> eſt inuenire punctũ reflexionis.</s> <s xml:id="echoid-s10137" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s10138" xml:space="preserve"> ſit g uertex <lb/>pyramidalis ſpeculi:</s> <s xml:id="echoid-s10139" xml:space="preserve"> & ſuper ipſum fiat ſuperficies æquidiſtãs baſi pyramidis:</s> <s xml:id="echoid-s10140" xml:space="preserve"> [ut oſtenfum <lb/>eſt 52 n] quę ſit m n g:</s> <s xml:id="echoid-s10141" xml:space="preserve"> a ſit pũctũ uiſum:</s> <s xml:id="echoid-s10142" xml:space="preserve"> b cẽtrũ uiſus.</s> <s xml:id="echoid-s10143" xml:space="preserve"> A & b aut erũt citra illã ſuperficiẽ:</s> <s xml:id="echoid-s10144" xml:space="preserve"> aut <lb/>ultra:</s> <s xml:id="echoid-s10145" xml:space="preserve"> aut in ipſa ſuperficie:</s> <s xml:id="echoid-s10146" xml:space="preserve"> aut unũ citra, aliud ultra:</s> <s xml:id="echoid-s10147" xml:space="preserve"> aut unũ in ſuperficie, aliud citra uel ultra.</s> <s xml:id="echoid-s10148" xml:space="preserve"> Sint <lb/>citra ſuperficiẽ:</s> <s xml:id="echoid-s10149" xml:space="preserve"> & à puncto a ducatur ſuperficies, ſecãs pyramidẽ ęquidiſtãter baſi:</s> <s xml:id="echoid-s10150" xml:space="preserve"> & ducatur à pun <lb/>cto g linea ad punctũ b:</s> <s xml:id="echoid-s10151" xml:space="preserve"> quę ꝓducta cadet in ſuperficiẽ ab a ductã, cũ ſit inter ſuperficies æquidiſtã-<lb/>tes:</s> <s xml:id="echoid-s10152" xml:space="preserve"> [quarũ una ք uerticẽ, altera ք uiſibile a ducitur] punctũ, in qđ cadit hęc linea, ſit h.</s> <s xml:id="echoid-s10153" xml:space="preserve"> Probatur <lb/>aũt modo ſuprà dicto [52 n] qđ a reflectitur ad h ab aliquo pũcto circuli, quẽ efficit ſuperficies, ſe-<lb/>cãs pyramidẽ, ducta à pũctis a, h:</s> <s xml:id="echoid-s10154" xml:space="preserve"> & inueniatur in circulo illo punctũ reflexionis:</s> <s xml:id="echoid-s10155" xml:space="preserve"> [ք 31 uel 39 n] & <lb/>ſit e:</s> <s xml:id="echoid-s10156" xml:space="preserve"> & ducatur linea a b:</s> <s xml:id="echoid-s10157" xml:space="preserve"> & linea lõgitudinis pyramidis g e:</s> <s xml:id="echoid-s10158" xml:space="preserve"> & axis pyramidis g t:</s> <s xml:id="echoid-s10159" xml:space="preserve"> & ducatur à pun-<lb/>cto e linea ad centrũ circuli:</s> <s xml:id="echoid-s10160" xml:space="preserve"> quę quidẽ cadet ſuper axem:</s> <s xml:id="echoid-s10161" xml:space="preserve"> [ք 4 th 1 coni.</s> <s xml:id="echoid-s10162" xml:space="preserve"> Apol.</s> <s xml:id="echoid-s10163" xml:space="preserve"> quia cẽtrũ circuli eſt <lb/>in axe] & ſit e t:</s> <s xml:id="echoid-s10164" xml:space="preserve"> & erit [ք 18 p 3] orthogonalis ſuper lineã, cõtingentẽ circulũ illũ in pũcto e:</s> <s xml:id="echoid-s10165" xml:space="preserve"> & du-<lb/>ctis lineis a e, h e:</s> <s xml:id="echoid-s10166" xml:space="preserve"> ſecabit angulũ earũ ք æqualia:</s> <s xml:id="echoid-s10167" xml:space="preserve"> [ut oſtẽſum eſt 13 n 4] & diuidet lineã a h:</s> <s xml:id="echoid-s10168" xml:space="preserve"> [քa ſecat <lb/>angulũ ipſi ſubtẽſum:</s> <s xml:id="echoid-s10169" xml:space="preserve"> ſunt enim e h, e a, h a in eadẽ reflexionis ſuperficie ք 23 n 4] ſit pũctũ diuiſio-<lb/>nis r.</s> <s xml:id="echoid-s10170" xml:space="preserve"> Palàm, quoniã g e, e t efficiũt ſuperficiẽ, ſecantẽ lineã a b:</s> <s xml:id="echoid-s10171" xml:space="preserve"> ſit pũctũ ſectionis f:</s> <s xml:id="echoid-s10172" xml:space="preserve"> & à pũcto f duca-<lb/>tur perpẽdicularis ſuper lineã g e [ք 12 p 1] & ſit f q:</s> <s xml:id="echoid-s10173" xml:space="preserve"> quę quidẽ erit orthogonalis ſuper ſuperficiẽ, <lb/>cõtingẽtẽ pyramidẽ ſuper lineã g e.</s> <s xml:id="echoid-s10174" xml:space="preserve"> [quia enim f q perpẽdicularis eſt duab.</s> <s xml:id="echoid-s10175" xml:space="preserve"> rectis interſectis, in cõ-<lb/>muni ipſarũ ſectione (qđ eſt punctũ q) lateri nẽpe conico e g ք fabricationẽ proximã, & rectę pe-<lb/>ripheriã circuli ք punctũ q deſcripti, in extrema diametro tãgenti ք 18 p 3:</s> <s xml:id="echoid-s10176" xml:space="preserve"> erit perpẽdicularis plano <lb/>ք ipſas ducto ք 4 p 11, id eſt plano in latere conũ tãgente per 35 n 4.</s> <s xml:id="echoid-s10177" xml:space="preserve">] Deinde à pũcto a ducatur ęqui <lb/>diſtãs lineæ f q:</s> <s xml:id="echoid-s10178" xml:space="preserve"> & ſit a l:</s> <s xml:id="echoid-s10179" xml:space="preserve"> f q aũt cõcurrat cũ axe in pũcto k:</s> <s xml:id="echoid-s10180" xml:space="preserve"> [qđ enim cõcurrat, patet ք 11 ax.</s> <s xml:id="echoid-s10181" xml:space="preserve"> quia per <lb/> <pb o="159" file="0165" n="165" rhead="OPTICAE LIBER V."/> 18 d 11 & 32 p 1 angulus ab axe & latere e g cõprehẽſus, eſt acutus] & à pũcto a ducatur ęquidiſtãs li-<lb/>n e æ r t:</s> <s xml:id="echoid-s10182" xml:space="preserve"> quę ſit a s:</s> <s xml:id="echoid-s10183" xml:space="preserve"> & ducatur à pũcto e linea, cõmunis ſuperficiei reflexionis a e h & ſuperficiei, con <lb/>tingẽti pyra midẽ in linea g e:</s> <s xml:id="echoid-s10184" xml:space="preserve"> quæ ſit e o.</s> <s xml:id="echoid-s10185" xml:space="preserve"> Cadet quidẽ orthogonaliter ſuper a s:</s> <s xml:id="echoid-s10186" xml:space="preserve"> cũ ſit orthogonalis <lb/>ſuper e t:</s> <s xml:id="echoid-s10187" xml:space="preserve"> [quia enim e o tãgit peripheriã circuli in e:</s> <s xml:id="echoid-s10188" xml:space="preserve"> erit <lb/> <anchor type="figure" xlink:label="fig-0165-01a" xlink:href="fig-0165-01"/> ք 18 p 3 perpẽdicularis ipſi r e t, cui a s parallela eſt ք fabri <lb/>cationẽ.</s> <s xml:id="echoid-s10189" xml:space="preserve"> Quare ք 29 p 1 e o perpendicularis eſt ipſi a s] & <lb/>ducatur linea b q:</s> <s xml:id="echoid-s10190" xml:space="preserve"> quæ ꝓ ducta, neceſſariò cõcurret cũ li-<lb/>nea a l:</s> <s xml:id="echoid-s10191" xml:space="preserve"> [ք lẽma Procli ad 29 p 1] ſit pũctũ cõcurſus l:</s> <s xml:id="echoid-s10192" xml:space="preserve"> & <lb/>ducatur à pũcto q linea, cõmunis ſuperficiei cõtingenti, <lb/>[ſpeculũ in latere conico e g] & ſuperficiei a b l:</s> <s xml:id="echoid-s10193" xml:space="preserve"> quæ ſit <lb/>o p:</s> <s xml:id="echoid-s10194" xml:space="preserve"> & ducãtur l s, p o.</s> <s xml:id="echoid-s10195" xml:space="preserve"> Palàm, quoniam ſuperficies a l s eſt <lb/>ęquidiſtãs ſuperficiei g e k:</s> <s xml:id="echoid-s10196" xml:space="preserve"> [Nã quia e t ſemidiameter cir <lb/>culi, eſt perpẽdicularis axi ք 18.</s> <s xml:id="echoid-s10197" xml:space="preserve"> 3 d 11:</s> <s xml:id="echoid-s10198" xml:space="preserve"> & angulus g q k re-<lb/>ctus ք fabricationẽ:</s> <s xml:id="echoid-s10199" xml:space="preserve"> ergo ք 32 p 1 angulus g k q eſt acutus, <lb/>& reliquus t k q obtuſus.</s> <s xml:id="echoid-s10200" xml:space="preserve"> Quare e t, f k ultra axẽ cõtinua-<lb/>tæ efficient angulos duob.</s> <s xml:id="echoid-s10201" xml:space="preserve"> rectis minores ք 13 p 1, & ք 11 <lb/>ax.</s> <s xml:id="echoid-s10202" xml:space="preserve"> cõcurrent:</s> <s xml:id="echoid-s10203" xml:space="preserve"> His uerò parallelæ a l, a s cõcurrũt in pun-<lb/>cto a:</s> <s xml:id="echoid-s10204" xml:space="preserve"> ſuntq́;</s> <s xml:id="echoid-s10205" xml:space="preserve"> binę in diuerſis planis.</s> <s xml:id="echoid-s10206" xml:space="preserve"> Ergo ք 15 p 11 ipſarũ <lb/>plana ſunt parallela] & lineæ q e, p o ſunt in ſuքficie con <lb/>tingẽte:</s> <s xml:id="echoid-s10207" xml:space="preserve"> quę ſuքficies ſecat illas ſuperficies ęquidiſtãtes, <lb/>ſuper duas lineas q e, p o:</s> <s xml:id="echoid-s10208" xml:space="preserve"> Igitur [ք 16 p 11] q e æquidiſtat <lb/>p o.</s> <s xml:id="echoid-s10209" xml:space="preserve"> Ducatur aũt linea h e, donec cõcurrat cũ h s in pũcto <lb/>s [cõcurret aũt ք lẽma Procli ad 29 p 1.</s> <s xml:id="echoid-s10210" xml:space="preserve">] Palã [ք 1 p 11] q đ <lb/>linea e s eſt in ſuperficie h e g:</s> <s xml:id="echoid-s10211" xml:space="preserve"> & in eadẽ eſt linea b l:</s> <s xml:id="echoid-s10212" xml:space="preserve"> [ք 2 <lb/>p 11] & hęc ſuperficies ſecat prædictas ſuperficies æquidi <lb/>ſtãtes, in duabus lineis e q, l s.</s> <s xml:id="echoid-s10213" xml:space="preserve"> Igitur [ք 16 p 11] e q eſt æ-<lb/>diſtãs l s:</s> <s xml:id="echoid-s10214" xml:space="preserve"> erit igitur [<gap/>ք 30 p 1] p o ęquidiſtãs l s.</s> <s xml:id="echoid-s10215" xml:space="preserve"> Quare [ք <lb/>2 p 6] a o ad o s, ſicut a p ad p l:</s> <s xml:id="echoid-s10216" xml:space="preserve"> ſed palã [per 12 n 4] quod <lb/>angulus h e r æ qualis eſt angulo r e a:</s> <s xml:id="echoid-s10217" xml:space="preserve"> erit angulus e s a ę-<lb/>qualis angulo e a s:</s> <s xml:id="echoid-s10218" xml:space="preserve"> [Nã cũ r t ſit parallela ipſi a s per fabri <lb/>cationẽ:</s> <s xml:id="echoid-s10219" xml:space="preserve"> æquabitur tũ angulus h e r exterior, angulo e s a interiori & oppoſito, tũ e a s alterno r e a <lb/>per 29 p 1.</s> <s xml:id="echoid-s10220" xml:space="preserve"> Quare ք 1 ax.</s> <s xml:id="echoid-s10221" xml:space="preserve"> angulus e s a æquabitur angulo e a s] & e o eſt perpẽdicularis ſuper a s:</s> <s xml:id="echoid-s10222" xml:space="preserve"> [ut <lb/>oſtẽſum eſt] erit ergo [per 26 p 1] a o æqualis o s:</s> <s xml:id="echoid-s10223" xml:space="preserve"> erit ergo a p æqualis p l [demõſtratũ enim eſt, ut a o <lb/>ad o s, ſic a p ad p l] & q p perpẽdicularis eſt ſuper a l.</s> <s xml:id="echoid-s10224" xml:space="preserve"> cũ ſit perpẽdicularis ſuper f k.</s> <s xml:id="echoid-s10225" xml:space="preserve"> [Quia enim f k <lb/>քpẽdicularis eſt plano tãgẽti, ut patuit, in quo eſt q p:</s> <s xml:id="echoid-s10226" xml:space="preserve"> cũ ſit illius, & plani a b l cõmunis ſectio:</s> <s xml:id="echoid-s10227" xml:space="preserve"> ergo <lb/>ք 3 d 11 f k eſt քpẽdicularis ipſi p q, & ք 29 p 1 ipſi a l parallelę.</s> <s xml:id="echoid-s10228" xml:space="preserve">] Igitur [ք 4 p 1] q l ęqualis a q:</s> <s xml:id="echoid-s10229" xml:space="preserve"> & angul9<unsure/> <lb/>q l a æqualis angulo l a q.</s> <s xml:id="echoid-s10230" xml:space="preserve"> Erit ergo angulus b q f æqualis angulo a q f:</s> <s xml:id="echoid-s10231" xml:space="preserve"> [Quia enim q f parallela eſt <lb/>ipſi a l:</s> <s xml:id="echoid-s10232" xml:space="preserve"> ęquabitur exterior angulus b q finteriori & oppoſito q l a:</s> <s xml:id="echoid-s10233" xml:space="preserve"> & q a l alterno a q f ք 29 p 1.</s> <s xml:id="echoid-s10234" xml:space="preserve"> Qua-<lb/>re b q f æquabitur a q f.</s> <s xml:id="echoid-s10235" xml:space="preserve">] Igitur a reflectetur ad b à puncto q [per 12 n 4.</s> <s xml:id="echoid-s10236" xml:space="preserve">] Quod eſt propoſitum.</s> <s xml:id="echoid-s10237" xml:space="preserve"/> </p> <div xml:id="echoid-div369" type="float" level="0" n="0"> <figure xlink:label="fig-0165-01" xlink:href="fig-0165-01a"> <variables xml:id="echoid-variables81" xml:space="preserve">g m n b f q k l <gap/> <gap/> e p o h r a</variables> </figure> </div> </div> <div xml:id="echoid-div371" type="section" level="0" n="0"> <head xml:id="echoid-head358" xml:space="preserve" style="it">55. Viſu & uiſibili in plano per uerticem ſpeculi conici <lb/>conuexi ducto, baſí par allelo, poſitis: punctũ reflexio-<lb/>nis inuenire. 36 p 7.</head> <figure> <variables xml:id="echoid-variables82" xml:space="preserve">g m q n t e b r a</variables> </figure> <p> <s xml:id="echoid-s10238" xml:space="preserve">SI uerò cẽtrũ uiſus & punctũ uiſum fuerint in ſuperfi <lb/>cie m g n:</s> <s xml:id="echoid-s10239" xml:space="preserve"> ſit unũ in puncto m, aliud in pũcto n:</s> <s xml:id="echoid-s10240" xml:space="preserve"> & du <lb/>cãtur lineæ m g, n g, m n:</s> <s xml:id="echoid-s10241" xml:space="preserve"> & diuidatur angulus m g n <lb/>per æqualia, per lineã q g [per 9 p 1.</s> <s xml:id="echoid-s10242" xml:space="preserve">] Palã [per 12 n 4] qđ <lb/>n à puncto g reflectitur ad m.</s> <s xml:id="echoid-s10243" xml:space="preserve"> Palã etiã, quòd linea q g & <lb/>axis pyramidis ſunt in ſuperficie, ſecãte pyramidẽ ſuper <lb/>lineã longitudinis:</s> <s xml:id="echoid-s10244" xml:space="preserve"> [ſunt enim axis & latus in uno plano, <lb/>ut è 18 d 11 intelligitur, & in eodẽ plano eſt recta linea q g <lb/>per 2 p 11] à pũcto q ducatur orthogonalis ſuper hãc lineã <lb/>lõgitudinis g e:</s> <s xml:id="echoid-s10245" xml:space="preserve"> quę ſit q e:</s> <s xml:id="echoid-s10246" xml:space="preserve"> & ſuper pũctũ e fiat ſuperficies <lb/>æquidiſtãs baſi:</s> <s xml:id="echoid-s10247" xml:space="preserve"> [ut dictũ eſt 52 n] quæ ſecabit pyramidẽ <lb/>ſuper circulũ [per 4 th 1 coni.</s> <s xml:id="echoid-s10248" xml:space="preserve"> Apol.</s> <s xml:id="echoid-s10249" xml:space="preserve">] linea cõmunis ſuք-<lb/>ficiei q e g, & huic circulo ſit e t.</s> <s xml:id="echoid-s10250" xml:space="preserve"> Palã, quoniã cadet ſuper <lb/>axem & ſuper cẽtrũ circuli.</s> <s xml:id="echoid-s10251" xml:space="preserve"> [Quia enim conus ſectus eſt <lb/>duplici plano:</s> <s xml:id="echoid-s10252" xml:space="preserve"> uno per axem, altero ad baſim parallelo:</s> <s xml:id="echoid-s10253" xml:space="preserve"> & <lb/>illius quidẽ & coni cõmunis ſectio eſt triágulũ, per 3 th 1 <lb/>coni.</s> <s xml:id="echoid-s10254" xml:space="preserve"> Apol.</s> <s xml:id="echoid-s10255" xml:space="preserve"> huius uerò circulus per 4 th eiuſdẽ:</s> <s xml:id="echoid-s10256" xml:space="preserve"> ergo per <lb/>cõſectariũ 4 th comunis ſectio circuli & trianguli eſt dia <lb/>meter circuli, cuius cẽtrum eſt in axe.</s> <s xml:id="echoid-s10257" xml:space="preserve">] Deinde à pũcto m <lb/>ducatur ęquidiſtãs lineę e g:</s> <s xml:id="echoid-s10258" xml:space="preserve"> quę quidẽ in ſuperficie illius <lb/>circuli cadat in pũctũ b:</s> <s xml:id="echoid-s10259" xml:space="preserve"> [cadet aũt, ꝗa eſt interplana pa-<lb/>rallela.</s> <s xml:id="echoid-s10260" xml:space="preserve">] Similiter à pũcto n ducatur æquidiſtans g e:</s> <s xml:id="echoid-s10261" xml:space="preserve"> quæ <lb/>cadat in pũctũ a:</s> <s xml:id="echoid-s10262" xml:space="preserve"> & ducatur a b:</s> <s xml:id="echoid-s10263" xml:space="preserve"> & e t ſecet eá in punctor.</s> <s xml:id="echoid-s10264" xml:space="preserve"> <lb/> <pb o="160" file="0166" n="166" rhead="ALHAZEN"/> [ſecabitaũt:</s> <s xml:id="echoid-s10265" xml:space="preserve"> quia cũ ſint in uno plano ք 23 n 4:</s> <s xml:id="echoid-s10266" xml:space="preserve"> & et ſecet angulũ a e b:</s> <s xml:id="echoid-s10267" xml:space="preserve"> cõtinuata ſecabit etiã baſim <lb/>angulo ſubtẽſam.</s> <s xml:id="echoid-s10268" xml:space="preserve">] Palàm, quoniã m b æquidiſtat g e:</s> <s xml:id="echoid-s10269" xml:space="preserve"> eſt in eadẽ ſuperficie cũ ipſa:</s> <s xml:id="echoid-s10270" xml:space="preserve"> [ք 35 d 1] quę ſu <lb/>perficies ſecat ſuperficiẽ m g n & ſuperficiẽ b e a ęquidiſtãtes, ſuper duas lineas m g, b e:</s> <s xml:id="echoid-s10271" xml:space="preserve"> ergo [ք 16 p <lb/>11] m g æquidiſtãs eſt b e.</s> <s xml:id="echoid-s10272" xml:space="preserve"> Similiter n a, ge ſunt in ſuperficie ſecante illas ſuperficies æquidiſtãte s, ſu <lb/>per n g, a e:</s> <s xml:id="echoid-s10273" xml:space="preserve"> igitur [per 16 p 11] n g æquidiſtat a e.</s> <s xml:id="echoid-s10274" xml:space="preserve"> Similiter ſuperficies q g e ſecat eaſdẽ ſuperficies, ſu <lb/>per duas lineas r e, q g:</s> <s xml:id="echoid-s10275" xml:space="preserve"> igitur [ք 16 p 11] r e, q g æquidiſtãt.</s> <s xml:id="echoid-s10276" xml:space="preserve"> Igitur q g & m g æquidiſtãt b e, r e.</s> <s xml:id="echoid-s10277" xml:space="preserve"> Quare <lb/>[ք 10 p 11] angulus m g q æqualis angulo b e r:</s> <s xml:id="echoid-s10278" xml:space="preserve"> & angulus q g n æqualis angulo r e a:</s> <s xml:id="echoid-s10279" xml:space="preserve"> & angulus b e r <lb/>æqualis angulo r e a.</s> <s xml:id="echoid-s10280" xml:space="preserve"> [ꝗa angulus m g q æquatus eſt angulo n g q.</s> <s xml:id="echoid-s10281" xml:space="preserve">] Et ita pũctũ a põt reflecti ad pun <lb/>ctũ b à pũcto e [ք 12 n 4.</s> <s xml:id="echoid-s10282" xml:space="preserve">] Si ergo à pũcto a ducatur ęꝗdiſtãs q e, & alia ęꝗ diſtãs r e:</s> <s xml:id="echoid-s10283" xml:space="preserve"> & ducatur b e, do <lb/>nec cõcurrat cũ linea æꝗdiſtãte ipſi q e:</s> <s xml:id="echoid-s10284" xml:space="preserve"> & ducãtur lineę cõmunes, ut prius, & m e, n e:</s> <s xml:id="echoid-s10285" xml:space="preserve"> & iteretur ꝓ-<lb/>batio p̃dicta:</s> <s xml:id="echoid-s10286" xml:space="preserve"> patebit, qđ n põtreflecti ad m à pũcto e.</s> <s xml:id="echoid-s10287" xml:space="preserve"> Erit igitur e pũctũ reflexiõis.</s> <s xml:id="echoid-s10288" xml:space="preserve"> Qđ eſt ꝓpoſitũ.</s> <s xml:id="echoid-s10289" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div372" type="section" level="0" n="0"> <head xml:id="echoid-head359" xml:space="preserve" style="it">56. Viſu & uiſibili ultra planum per uerticem ſpeculi conici conuexi ductum, baſí paralle <lb/>lum, poſitis: punctum reflexionis inuenire. 37 p 7.</head> <p> <s xml:id="echoid-s10290" xml:space="preserve">SI uerò ambo fuerint ultra m g n:</s> <s xml:id="echoid-s10291" xml:space="preserve"> fiat pyramis huic oppoſita:</s> <s xml:id="echoid-s10292" xml:space="preserve"> & eſt, ut ꝓtrahãtur lineæ lõgitudi <lb/>nis pyramidis iã factæ [ut è 1 d 1 coni.</s> <s xml:id="echoid-s10293" xml:space="preserve"> Apol.</s> <s xml:id="echoid-s10294" xml:space="preserve"> intelligitur] & à pũcto a ducatur ſuperficies, ſecans <lb/>hãc ultimã pyramidẽ ſuք circulũ y z:</s> <s xml:id="echoid-s10295" xml:space="preserve"> [ut oſtẽſum eſt 52 n:</s> <s xml:id="echoid-s10296" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s10297" xml:space="preserve"> hic circulus parallelus utriuſq;</s> <s xml:id="echoid-s10298" xml:space="preserve"> <lb/>coni baſib.</s> <s xml:id="echoid-s10299" xml:space="preserve"> ք cõuerſionẽ 4 th 1 coni.</s> <s xml:id="echoid-s10300" xml:space="preserve"> Apol.</s> <s xml:id="echoid-s10301" xml:space="preserve">] B aũt erit in <lb/> <anchor type="figure" xlink:label="fig-0166-01a" xlink:href="fig-0166-01"/> hac ſuքficie:</s> <s xml:id="echoid-s10302" xml:space="preserve"> aut nõ.</s> <s xml:id="echoid-s10303" xml:space="preserve"> Si fuerit:</s> <s xml:id="echoid-s10304" xml:space="preserve"> fiat operatio à pũcto b.</s> <s xml:id="echoid-s10305" xml:space="preserve"> [ut <lb/>54 n.</s> <s xml:id="echoid-s10306" xml:space="preserve">] Si nõ:</s> <s xml:id="echoid-s10307" xml:space="preserve"> ducatur linea g b, uſq;</s> <s xml:id="echoid-s10308" xml:space="preserve"> dũ cõcurrat cũ hac ſu <lb/>perficie:</s> <s xml:id="echoid-s10309" xml:space="preserve"> [cõcurret aũt:</s> <s xml:id="echoid-s10310" xml:space="preserve"> quia eſt inter plana parallela] & <lb/>ſit cõcurſus in pũcto d.</s> <s xml:id="echoid-s10311" xml:space="preserve"> Palàm, qđ a reflectitur ad d ab ali <lb/>quo pũcto circuli y z interiore [per 40 n 4.</s> <s xml:id="echoid-s10312" xml:space="preserve">] Inueniatur <lb/>pũctũ illud:</s> <s xml:id="echoid-s10313" xml:space="preserve"> ſicut deinceps ꝓbabimus & docebimus, nõ <lb/>ex anterioribus:</s> <s xml:id="echoid-s10314" xml:space="preserve"> & ſit z:</s> <s xml:id="echoid-s10315" xml:space="preserve"> & ducãtur lineæ a z, d z, a d:</s> <s xml:id="echoid-s10316" xml:space="preserve"> & li-<lb/>nea p z diuidat angulũ illũ ք æq̀ualia:</s> <s xml:id="echoid-s10317" xml:space="preserve"> [ք 9 p 1] & à pun-<lb/>cto g ducatur g z linea lõgitudinis:</s> <s xml:id="echoid-s10318" xml:space="preserve"> [ut oſtẽſum eſt 52 n] <lb/>& ducatur a b:</s> <s xml:id="echoid-s10319" xml:space="preserve"> & producatur linea z g ad aliã pyramidẽ:</s> <s xml:id="echoid-s10320" xml:space="preserve"> <lb/>quę quidẽ perueniet ad ſuperficiẽ eius:</s> <s xml:id="echoid-s10321" xml:space="preserve"> & erit linea lõgi-<lb/>tudinis:</s> <s xml:id="echoid-s10322" xml:space="preserve"> [ut patet è 1 d 1 coni.</s> <s xml:id="echoid-s10323" xml:space="preserve"> Apollo.</s> <s xml:id="echoid-s10324" xml:space="preserve">] & ſit z g e.</s> <s xml:id="echoid-s10325" xml:space="preserve"> Palàm, <lb/>quòd ſuperficies p z e ſecabit lineã a b:</s> <s xml:id="echoid-s10326" xml:space="preserve"> ſecet in puncto q:</s> <s xml:id="echoid-s10327" xml:space="preserve"> <lb/>& ducatur à pũcto q perpẽdicularis ſuper lineã g e:</s> <s xml:id="echoid-s10328" xml:space="preserve"> [ք 12 <lb/>p 1] & cadat in pũctũ e:</s> <s xml:id="echoid-s10329" xml:space="preserve"> & erit perpẽdicularis ſuper ſuper <lb/>ficiẽ, cõtingentẽ pyramidẽ ſuper lineã g e:</s> <s xml:id="echoid-s10330" xml:space="preserve"> [ք 3 d 11] & ſu-<lb/>perpũctũ e fiat ſuperficies, æquidiſtãs baſi:</s> <s xml:id="echoid-s10331" xml:space="preserve"> quę ſit f e h:</s> <s xml:id="echoid-s10332" xml:space="preserve"> & <lb/>ducatur à pũcto d linea æquidiſtãs z e:</s> <s xml:id="echoid-s10333" xml:space="preserve"> quę ſit d h, cõcur-<lb/>rẽs cũ ſuperficie illa in pũcto h:</s> <s xml:id="echoid-s10334" xml:space="preserve"> [cõcurret aũt:</s> <s xml:id="echoid-s10335" xml:space="preserve"> quia cõcur <lb/>rit cũ plano ipſi parallelo] & eidẽ lineæ ſit æ quidiſtãs a f.</s> <s xml:id="echoid-s10336" xml:space="preserve"> <lb/>Palàm, quoniam d h eſt æquidiſtãs z e:</s> <s xml:id="echoid-s10337" xml:space="preserve"> quòd ſunt in eadẽ <lb/>ſuperficie:</s> <s xml:id="echoid-s10338" xml:space="preserve"> [ք 35 d 1] quę ſuperficies ſecat ſuperficies æ-<lb/>quidiſtantes, ſuper duas lineas d z, h e:</s> <s xml:id="echoid-s10339" xml:space="preserve"> igitur [ք 16 p 11] h <lb/>e, d z ſunt ęquidiſtátes.</s> <s xml:id="echoid-s10340" xml:space="preserve"> Similiter a z, fe ſunt ęquidiſtãtes.</s> <s xml:id="echoid-s10341" xml:space="preserve"> <lb/>Similiter, quoniã p z trãſit per cêtrũ circuli y z:</s> <s xml:id="echoid-s10342" xml:space="preserve"> [ducta e-<lb/>nim recta linea circulũ in pũcto z tãgẽte ք 17 p 3:</s> <s xml:id="echoid-s10343" xml:space="preserve"> quoniã angulus a z d bifariã ſectus eſt à linea p z:</s> <s xml:id="echoid-s10344" xml:space="preserve"> & <lb/>anguli incidẽtiæ & reflexiõis æquãtur ք 10 n 4:</s> <s xml:id="echoid-s10345" xml:space="preserve"> anguli igitur deinceps lineę p z & tãgẽtis ęquãtur ք <lb/>2 ax.</s> <s xml:id="echoid-s10346" xml:space="preserve"> & ita ք 10 d 1 uterq;</s> <s xml:id="echoid-s10347" xml:space="preserve"> rectus eſt.</s> <s xml:id="echoid-s10348" xml:space="preserve"> Quare ք 19 p 3 p z eſt diameter circuli y z] ſimiliter r e t ք cẽtrũ al <lb/>terius circuli, ſuper quẽ ſuperficies a e h ſecat pyramidẽ.</s> <s xml:id="echoid-s10349" xml:space="preserve"> Igitur ſuperficies p z e r ſecat duas ſuperfi-<lb/>cies æquidiſtãtes, ſuper duas lineas p z, r e:</s> <s xml:id="echoid-s10350" xml:space="preserve"> igitur [ք 16 p 11] p z æquidiſtat r e.</s> <s xml:id="echoid-s10351" xml:space="preserve"> Quare [ք 10 p 11] an-<lb/>gulus a z p æ qualis angulo f e r:</s> <s xml:id="echoid-s10352" xml:space="preserve"> & angulus d z p angulo h e r:</s> <s xml:id="echoid-s10353" xml:space="preserve"> & ita erit angulus f e r æqualis angulo <lb/>r e h.</s> <s xml:id="echoid-s10354" xml:space="preserve"> [ꝗa a z p æquatus eſt d z p.</s> <s xml:id="echoid-s10355" xml:space="preserve">] Quare f reflectetur ad h à pũcto e.</s> <s xml:id="echoid-s10356" xml:space="preserve"> Igitur ſi à pũcto f ꝓtraxerimus <lb/>ęquidiſtãtẽ q e, & aliã æquidiſtãtẽ r e:</s> <s xml:id="echoid-s10357" xml:space="preserve"> & lineas cõmunes, ſicut ſuprà:</s> <s xml:id="echoid-s10358" xml:space="preserve"> & iterauerimus modũ ꝓbandi <lb/>prædictũ:</s> <s xml:id="echoid-s10359" xml:space="preserve"> patebit, quòd punctum a reflectetur ad b à puncto e.</s> <s xml:id="echoid-s10360" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s10361" xml:space="preserve"/> </p> <div xml:id="echoid-div372" type="float" level="0" n="0"> <figure xlink:label="fig-0166-01" xlink:href="fig-0166-01a"> <variables xml:id="echoid-variables83" xml:space="preserve">z y a p d q b m n g t e f r h</variables> </figure> </div> </div> <div xml:id="echoid-div374" type="section" level="0" n="0"> <head xml:id="echoid-head360" xml:space="preserve" style="it">57. Viſu in plano per uerticem ſpeculi conici conuexi ducto, baſí parallelo, uiſibili citraidẽ <lb/>poſitis: punctum reflexionis inuenire. 38 p 7.</head> <p> <s xml:id="echoid-s10362" xml:space="preserve">SI uerò centrũ uiſus fuerit in ſuperficie æquidiſtante, quæ eſt ſupra uerticẽ, ſcilicet g:</s> <s xml:id="echoid-s10363" xml:space="preserve"> & punctũ <lb/>uiſum citra hãc ſuperficiẽ:</s> <s xml:id="echoid-s10364" xml:space="preserve"> erit inuenire punctũ reflexionis hoc modo.</s> <s xml:id="echoid-s10365" xml:space="preserve"> Sit enim cẽtrũ uiſus m:</s> <s xml:id="echoid-s10366" xml:space="preserve"> <lb/>pũctũ uiſum a:</s> <s xml:id="echoid-s10367" xml:space="preserve"> & ſit m n g ſuperficies æquidiſtãs baſi pyramidis:</s> <s xml:id="echoid-s10368" xml:space="preserve"> & à pũcto a ducatur ſuperfici-<lb/>es æquidiſtãs baſi pyramidis:</s> <s xml:id="echoid-s10369" xml:space="preserve"> [ut mõſtratũ eſt 52 n] quæ ſecabit pyramidẽ ſuper circulũ [per 4 th.</s> <s xml:id="echoid-s10370" xml:space="preserve"> 1 <lb/>coni.</s> <s xml:id="echoid-s10371" xml:space="preserve"> Apol.</s> <s xml:id="echoid-s10372" xml:space="preserve">] ꝗ ſit d e k:</s> <s xml:id="echoid-s10373" xml:space="preserve"> cuius cẽtrũ t:</s> <s xml:id="echoid-s10374" xml:space="preserve"> & à pũcto m ducatur perpẽdicularis ſuper hãc ſuperficiẽ:</s> <s xml:id="echoid-s10375" xml:space="preserve"> [ք 12 <lb/>p 1] quę ſit m h:</s> <s xml:id="echoid-s10376" xml:space="preserve"> & ducatur axis g t:</s> <s xml:id="echoid-s10377" xml:space="preserve"> & linea h t:</s> <s xml:id="echoid-s10378" xml:space="preserve"> & ducatur ab m ad a linea recta m a:</s> <s xml:id="echoid-s10379" xml:space="preserve"> & à puncto a du <lb/>catur ad lineã h t, intra circulũ, linea a e q, & e q ſit æqualis q t ſecũdũ ſupradicta:</s> <s xml:id="echoid-s10380" xml:space="preserve"> [37 n] & ducatur <lb/>linea t e i:</s> <s xml:id="echoid-s10381" xml:space="preserve"> & à pũcto h ducatur æquidiſtãs te, & æqualis:</s> <s xml:id="echoid-s10382" xml:space="preserve"> [ք 31.</s> <s xml:id="echoid-s10383" xml:space="preserve"> 3 p 1] q̃ ſit h b:</s> <s xml:id="echoid-s10384" xml:space="preserve"> & ducãtur lineæ m b, <lb/>b e, g e.</s> <s xml:id="echoid-s10385" xml:space="preserve"> Palã, qđ ſuperficies g t e ſecabit lineã a m:</s> <s xml:id="echoid-s10386" xml:space="preserve"> ſit punctũ ſectionis f:</s> <s xml:id="echoid-s10387" xml:space="preserve"> & ducatur à pũcto fperpẽdi <lb/>cularis ſuper lineã e g:</s> <s xml:id="echoid-s10388" xml:space="preserve"> [ք 12 p 1] & ꝓducatur ad axem:</s> <s xml:id="echoid-s10389" xml:space="preserve"> [cũ quo cõcurret, ut oſtẽſum eſt 54 n] cadẽs <lb/>in pũctũ o:</s> <s xml:id="echoid-s10390" xml:space="preserve"> quę ſit f o p:</s> <s xml:id="echoid-s10391" xml:space="preserve"> & ducãtur lineæ m o, a o.</s> <s xml:id="echoid-s10392" xml:space="preserve"> Dico, qđ o eſt punctũ reflexionis.</s> <s xml:id="echoid-s10393" xml:space="preserve"> Palã, quoniã h b <lb/>æquidiſtãs & æqualis t e:</s> <s xml:id="echoid-s10394" xml:space="preserve"> [per fabricationẽ] igitur h t æquidiſtãs & æqualis e b.</s> <s xml:id="echoid-s10395" xml:space="preserve"> [per 33 p 1.</s> <s xml:id="echoid-s10396" xml:space="preserve">] Sed m h <lb/> <pb o="161" file="0167" n="167" rhead="OPTICAE LIBER V."/> æqualis & æquidiſtãs g t:</s> <s xml:id="echoid-s10397" xml:space="preserve"> cũ utraq;</s> <s xml:id="echoid-s10398" xml:space="preserve"> ſit perpẽdicularis:</s> <s xml:id="echoid-s10399" xml:space="preserve"> [duob.</s> <s xml:id="echoid-s10400" xml:space="preserve"> planis ք m n g & ք a ductis.</s> <s xml:id="echoid-s10401" xml:space="preserve"> Nã utraq;</s> <s xml:id="echoid-s10402" xml:space="preserve"> <lb/>քpẽdicularis eſt plano ք a ducto:</s> <s xml:id="echoid-s10403" xml:space="preserve"> m h quidẽ ք fabricationẽ:</s> <s xml:id="echoid-s10404" xml:space="preserve"> g t uerò <lb/> <anchor type="figure" xlink:label="fig-0167-01a" xlink:href="fig-0167-01"/> ք 18 d 11:</s> <s xml:id="echoid-s10405" xml:space="preserve"> ꝗa eſt axis.</s> <s xml:id="echoid-s10406" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s10407" xml:space="preserve"> ք cõuerſam 14 p 11.</s> <s xml:id="echoid-s10408" xml:space="preserve"> utraq;</s> <s xml:id="echoid-s10409" xml:space="preserve"> perpẽdicularis eſt <lb/>plano m n g parallelo ք fabricationẽ plano ք a ducto:</s> <s xml:id="echoid-s10410" xml:space="preserve"> quare ք 28.</s> <s xml:id="echoid-s10411" xml:space="preserve"> 33 <lb/>p 1 m h, g t ſunt parallelæ & æquales] igitur h t eſt æquidiſtans & æ-<lb/>qualis m g:</s> <s xml:id="echoid-s10412" xml:space="preserve"> [ք 33 p 1] igitur m g æ quidiſtãs & æqualis b e [per 30 p 1.</s> <s xml:id="echoid-s10413" xml:space="preserve"> <lb/>1 ax.</s> <s xml:id="echoid-s10414" xml:space="preserve">] Quare m b æ quidiſtãs & æqualis g e [ք 33 p 1.</s> <s xml:id="echoid-s10415" xml:space="preserve">] Palã etiã, qđ an-<lb/>gulus q t e æqualis eſt angulo q e t:</s> <s xml:id="echoid-s10416" xml:space="preserve"> [ք 5 p 1:</s> <s xml:id="echoid-s10417" xml:space="preserve"> ꝗ a e q, q t æ quatę ſunt] & <lb/>ita [ք 15 p 1] æqualis angulo a e i:</s> <s xml:id="echoid-s10418" xml:space="preserve"> ſed q t e eſt æqualis angulo i e b.</s> <s xml:id="echoid-s10419" xml:space="preserve"> [ք <lb/>29 p 1:</s> <s xml:id="echoid-s10420" xml:space="preserve"> ꝗ a e b, h t ſunt parallelæ.</s> <s xml:id="echoid-s10421" xml:space="preserve">] Igitur [ք 1 ax.</s> <s xml:id="echoid-s10422" xml:space="preserve">] i e b æqualis eſt i e a:</s> <s xml:id="echoid-s10423" xml:space="preserve"> <lb/>Quare a reflectitur ad b à pũcto e [ք 12 n 4.</s> <s xml:id="echoid-s10424" xml:space="preserve">] Et cũ linea b m æquidi-<lb/>ſtãs ſit lineæ g e:</s> <s xml:id="echoid-s10425" xml:space="preserve"> ſi à pũcto a ducatur æquidiſtãs f o p, & æquidiſtans <lb/>it:</s> <s xml:id="echoid-s10426" xml:space="preserve"> & iteretur figura ſupradicta, & ꝓbatio:</s> <s xml:id="echoid-s10427" xml:space="preserve"> [54 n] palàm, quòd a re-<lb/>flectetur ad m à puncto o.</s> <s xml:id="echoid-s10428" xml:space="preserve"> Etita eſt propoſitum.</s> <s xml:id="echoid-s10429" xml:space="preserve"/> </p> <div xml:id="echoid-div374" type="float" level="0" n="0"> <figure xlink:label="fig-0167-01" xlink:href="fig-0167-01a"> <variables xml:id="echoid-variables84" xml:space="preserve">m n g p o f i b a h e q d t k</variables> </figure> </div> </div> <div xml:id="echoid-div376" type="section" level="0" n="0"> <head xml:id="echoid-head361" xml:space="preserve" style="it">58. Viſu in plano per uerticẽ ſpeculi conici conuexiducto, haſí <lb/>parallelo, uiſibili ultra idẽ poſitis: pũctũ reflexiõis inuenire. 39 p 7.</head> <p> <s xml:id="echoid-s10430" xml:space="preserve">SI uerò m ſit in ſuperficie, & a ultra ſuperficiem:</s> <s xml:id="echoid-s10431" xml:space="preserve"> fiet pyramis alia <lb/>huic oppoſita:</s> <s xml:id="echoid-s10432" xml:space="preserve"> & fiat ſuք a ſuքficies æquidiſtãs baſi huius pyra-<lb/>midis:</s> <s xml:id="echoid-s10433" xml:space="preserve"> & inueniatur in circulo buius ſuքficiei pũctũ reflexiõis <lb/>ex punctis interiorib.</s> <s xml:id="echoid-s10434" xml:space="preserve"> & ducatur à pũcto illo linea ad g:</s> <s xml:id="echoid-s10435" xml:space="preserve"> & ꝓducatur:</s> <s xml:id="echoid-s10436" xml:space="preserve"> <lb/>& inuenιetur pũctũ ſecũdũ ſuքiora:</s> <s xml:id="echoid-s10437" xml:space="preserve"> [56 n] & idẽ eſt ꝓbãdi modus.</s> <s xml:id="echoid-s10438" xml:space="preserve"/> </p> <figure> <variables xml:id="echoid-variables85" xml:space="preserve">y z m q p a n g t e f r h</variables> </figure> </div> <div xml:id="echoid-div377" type="section" level="0" n="0"> <head xml:id="echoid-head362" xml:space="preserve" style="it">59. Viſu citra planum per uerticem ſpeculi conici cõuexi ductum, <lb/>baſí parallelum: uiſibili ultra idem poſitis, uel contrà: punctum <lb/>reflexionis inuenire. 40 p 7.</head> <p> <s xml:id="echoid-s10439" xml:space="preserve">SIaũt pũcta, ſcilicet cẽtrũ uiſus & pũctũ uiſum ita diſponãtur, ut <lb/>unũ ſit citra ſuքficiẽ uerticis, aliud ultra:</s> <s xml:id="echoid-s10440" xml:space="preserve"> ſit unũ b:</s> <s xml:id="echoid-s10441" xml:space="preserve"> aliud a:</s> <s xml:id="echoid-s10442" xml:space="preserve"> ſuքfi <lb/>cies uerticis m g n:</s> <s xml:id="echoid-s10443" xml:space="preserve"> & ducatur à pũcto a ſuperficies ęꝗdiſtãs ba-<lb/>ſi:</s> <s xml:id="echoid-s10444" xml:space="preserve"> [ut mõſtratũ eſt 52 n] ſecabit pyramidẽ ſuք circulũ:</s> <s xml:id="echoid-s10445" xml:space="preserve"> [ք 4 th.</s> <s xml:id="echoid-s10446" xml:space="preserve"> 1 coni.</s> <s xml:id="echoid-s10447" xml:space="preserve"> <lb/>Apol.</s> <s xml:id="echoid-s10448" xml:space="preserve">] ꝗ ſit d e:</s> <s xml:id="echoid-s10449" xml:space="preserve"> centrũ eius ſit t:</s> <s xml:id="echoid-s10450" xml:space="preserve"> & ducatur axis g t:</s> <s xml:id="echoid-s10451" xml:space="preserve"> & ducatur linea b <lb/>g:</s> <s xml:id="echoid-s10452" xml:space="preserve"> cõcurret quidẽ cũ ſuperficie a e d:</s> <s xml:id="echoid-s10453" xml:space="preserve"> [ꝗa cõcurrit cũ plano ipſi paral-<lb/>lelo] ſit cõcurſus k, & in circulo d e inueniatur punctũ, qđ ſit e:</s> <s xml:id="echoid-s10454" xml:space="preserve"> ita, ut <lb/>cõtingẽs ducta à pũcto illo, quę ſit s e, diuidat ք æqualia angulũ, quẽ <lb/>cõtinẽt lineæ k e, a e:</s> <s xml:id="echoid-s10455" xml:space="preserve"> [ք 36 n] & ducatur linea lõgitudinis g e:</s> <s xml:id="echoid-s10456" xml:space="preserve"> & à pũ-<lb/>cto b ducatur linea æquidiſtãs g e:</s> <s xml:id="echoid-s10457" xml:space="preserve"> quę neceſſariò cõcurret cũ linea k <lb/>e:</s> <s xml:id="echoid-s10458" xml:space="preserve"> [ք lẽma Procli ad 29 p 1] ſit cõcurſus h.</s> <s xml:id="echoid-s10459" xml:space="preserve"> Palàm [ք 1 p 11] qđ h eſt in <lb/>ſuperficie g e k:</s> <s xml:id="echoid-s10460" xml:space="preserve"> & b h in eadẽ ſuperficie [ք 35 d 1] ꝗa æ ꝗ diſtãs eſt g e:</s> <s xml:id="echoid-s10461" xml:space="preserve"> <lb/>& ducatur linea t e i.</s> <s xml:id="echoid-s10462" xml:space="preserve"> Palã, qđ ſuperficies g t e ſecat lineã b a:</s> <s xml:id="echoid-s10463" xml:space="preserve"> ſecetin <lb/>pũcto u:</s> <s xml:id="echoid-s10464" xml:space="preserve">à quo ducatur քpẽdicularis ſuք ſuperficiẽ cõtingẽtẽ:</s> <s xml:id="echoid-s10465" xml:space="preserve"> [ſpecu <lb/>lũ in latere conico ge] q̃ ſit u o p:</s> <s xml:id="echoid-s10466" xml:space="preserve"> & ducãtur lineę a o, b o.</s> <s xml:id="echoid-s10467" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0167-03a" xlink:href="fig-0167-03"/> Palã [è fabricatione] qđ angulus a e s ęqualis eſt angu-<lb/>lo s e k:</s> <s xml:id="echoid-s10468" xml:space="preserve"> & cũ [per 18 p 3] angulus i e s ſit rectus, & s e t re-<lb/>ctus:</s> <s xml:id="echoid-s10469" xml:space="preserve"> [ideoq́;</s> <s xml:id="echoid-s10470" xml:space="preserve"> ք 10 ax.</s> <s xml:id="echoid-s10471" xml:space="preserve"> æquales:</s> <s xml:id="echoid-s10472" xml:space="preserve"> & per 3 ax.</s> <s xml:id="echoid-s10473" xml:space="preserve"> reliquus a e t æ-<lb/>quatur reliquo k e i] erit i e a æqualis angulo t e k:</s> <s xml:id="echoid-s10474" xml:space="preserve"> [per 2 <lb/>ax.</s> <s xml:id="echoid-s10475" xml:space="preserve">] & ita angulus a e i æqualis angulo i e h.</s> <s xml:id="echoid-s10476" xml:space="preserve"> [Nã angulus <lb/>t e k æqualis angulo a e i, æquatur angulo i e h per 15 p 1:</s> <s xml:id="echoid-s10477" xml:space="preserve"> <lb/>ideoq́;</s> <s xml:id="echoid-s10478" xml:space="preserve"> ք 1 ax.</s> <s xml:id="echoid-s10479" xml:space="preserve"> angulus a e i æquatur angulo i e h] Quare a <lb/>reflectetur ad h à pũcto e.</s> <s xml:id="echoid-s10480" xml:space="preserve"> Si ergo à pũcto a ducatur ęqui <lb/>diſtãs u o, & æquidiſtãs i t:</s> <s xml:id="echoid-s10481" xml:space="preserve"> & iteretur ꝓbatio:</s> <s xml:id="echoid-s10482" xml:space="preserve"> [54 n] pa-<lb/>tebit, qđ reflectetur a à pũcto o ad b.</s> <s xml:id="echoid-s10483" xml:space="preserve"> Et ita patet ꝓpoſi-<lb/>tũ.</s> <s xml:id="echoid-s10484" xml:space="preserve"> Palã ergo, qũo ſit inuenire pũctũ refſexionis.</s> <s xml:id="echoid-s10485" xml:space="preserve"> Et hęc, <lb/>quędicta ſunt, de unico uiſu intelligẽda ſunt:</s> <s xml:id="echoid-s10486" xml:space="preserve"> in duplici <lb/>aũt uiſuidẽ accidit:</s> <s xml:id="echoid-s10487" xml:space="preserve"> quoniã eadẽforma, & idẽlocus for-<lb/>mæ cõprehenditur ab utroq;</s> <s xml:id="echoid-s10488" xml:space="preserve"> uiſu.</s> <s xml:id="echoid-s10489" xml:space="preserve"> Et (ſicut dictũ eſt [41 <lb/>n] in ſpeculo ſphærico exteriore) formę à duob.</s> <s xml:id="echoid-s10490" xml:space="preserve"> oculis <lb/>cõprehẽſę, in his ſpeculis propter cõtiguitatẽ uldẽtur u-<lb/>na:</s> <s xml:id="echoid-s10491" xml:space="preserve"> & aliquãdo ſimul ſunt in loco:</s> <s xml:id="echoid-s10492" xml:space="preserve"> & aliquãdo cõmiſcen-<lb/>tur earum loca in parte:</s> <s xml:id="echoid-s10493" xml:space="preserve"> aliquando ſeparãtur, ſed modi-<lb/>cùm.</s> <s xml:id="echoid-s10494" xml:space="preserve"> Forma autem, quæ per perpendicularẽ in his ſpe-<lb/>culis deſcendit, ſecundum eandem regreditur, ſicut ſu-<lb/>prà patuit:</s> <s xml:id="echoid-s10495" xml:space="preserve"> [11 n 4] & forma illa ab uno oculo ſuper per-<lb/>pendicularem, percipitur ab alio oculo ſecundum line-<lb/>am reflexionis, ſed loca formarum continua ſunt.</s> <s xml:id="echoid-s10496" xml:space="preserve"> Vnde <lb/>eadem apparet uiſui forma.</s> <s xml:id="echoid-s10497" xml:space="preserve"/> </p> <div xml:id="echoid-div377" type="float" level="0" n="0"> <figure xlink:label="fig-0167-03" xlink:href="fig-0167-03a"> <variables xml:id="echoid-variables86" xml:space="preserve">a s t d k i e h o p u m g n b</variables> </figure> </div> <pb o="162" file="0168" n="168" rhead="ALHAZEN"/> </div> <div xml:id="echoid-div379" type="section" level="0" n="0"> <head xml:id="echoid-head363" xml:space="preserve" style="it">60. In ſpeculo ſphærico cauo, imago uidetur aliâs in reflexionis puncto: aliâs in uiſu: aliâs ul-<lb/>tra: aliâs citra ſpeculum: aliâs inter uiſum & ſpeculum. 11 p 8.</head> <p> <s xml:id="echoid-s10498" xml:space="preserve">IN ſpeculis ſphæricis concauis aliquãdo perpendicularis à pũcto uiſo ducta ſecat lineã reflexio-<lb/>nis:</s> <s xml:id="echoid-s10499" xml:space="preserve"> aliquãdo eſt æquidiſtãs ei.</s> <s xml:id="echoid-s10500" xml:space="preserve"> Quãdo ſecat:</s> <s xml:id="echoid-s10501" xml:space="preserve"> erit locus formæ aliquãdo in ſpeculo:</s> <s xml:id="echoid-s10502" xml:space="preserve"> aliquãdo ul-<lb/>tra ſpeculũ:</s> <s xml:id="echoid-s10503" xml:space="preserve"> aliquãdo citra.</s> <s xml:id="echoid-s10504" xml:space="preserve"> Et cũ fuerit locus formæ citra ſpeculũ:</s> <s xml:id="echoid-s10505" xml:space="preserve"> aliquando erit inter uiſum & <lb/>ſpeculũ:</s> <s xml:id="echoid-s10506" xml:space="preserve"> aliquãdo in cẽtro uiſus:</s> <s xml:id="echoid-s10507" xml:space="preserve"> aliquãdo citra cẽtrũ uiſus.</s> <s xml:id="echoid-s10508" xml:space="preserve"> Et nos h æc demõſtrabimus.</s> <s xml:id="echoid-s10509" xml:space="preserve"> Sit a cẽtrũ <lb/>uiſus:</s> <s xml:id="echoid-s10510" xml:space="preserve"> d cẽtrũ ſpeculi:</s> <s xml:id="echoid-s10511" xml:space="preserve"> & fiat ſuperficies ſuper hæc puncta:</s> <s xml:id="echoid-s10512" xml:space="preserve"> quæ ſecabit ſpeculũ ſuք circulũ [per 1 th.</s> <s xml:id="echoid-s10513" xml:space="preserve"> 1 <lb/>ſphær.</s> <s xml:id="echoid-s10514" xml:space="preserve"> Theodo.</s> <s xml:id="echoid-s10515" xml:space="preserve">] ꝗ circulus ſit h b f g:</s> <s xml:id="echoid-s10516" xml:space="preserve"> erit qúidẽ ſuքficies hęc, ſuքficies reflexionis:</s> <s xml:id="echoid-s10517" xml:space="preserve"> quoniã eſt ortho <lb/>gonalis ſuք quãlibet ſuperficiẽ, cõtingentẽ circulũ:</s> <s xml:id="echoid-s10518" xml:space="preserve"> [ք 13 n 4] & ducatur linea a d:</s> <s xml:id="echoid-s10519" xml:space="preserve"> & à pũcto a duca <lb/>tur linea ad circulũ maior ꝗ̃ a d:</s> <s xml:id="echoid-s10520" xml:space="preserve"> quę ſit a e:</s> <s xml:id="echoid-s10521" xml:space="preserve"> & à pũcto d ducatur ad circulũ æquidiſtãs a e:</s> <s xml:id="echoid-s10522" xml:space="preserve"> q̃ ſit d h:</s> <s xml:id="echoid-s10523" xml:space="preserve"> & <lb/>ꝓducatur a d uſq;</s> <s xml:id="echoid-s10524" xml:space="preserve"> in pũcta b, i:</s> <s xml:id="echoid-s10525" xml:space="preserve"> [qñ nimirũ uiſus fuerit intra circulũ h b f g:</s> <s xml:id="echoid-s10526" xml:space="preserve"> qđ ſi fuerit in peripheria <lb/>uel extra:</s> <s xml:id="echoid-s10527" xml:space="preserve"> linea a d ab una tãtùm parte in peripheriã cõtinuabitur:</s> <s xml:id="echoid-s10528" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s10529" xml:space="preserve"> eadẽ demõſtrãdi ratio:</s> <s xml:id="echoid-s10530" xml:space="preserve"> ꝗa <lb/>a e ſemք maior eſſe debet a d] & ducatur linea d e.</s> <s xml:id="echoid-s10531" xml:space="preserve"> Palã, qđ angulus a e d eſt minor recto:</s> <s xml:id="echoid-s10532" xml:space="preserve"> [cõnexis <lb/>enim rectis e i, e b:</s> <s xml:id="echoid-s10533" xml:space="preserve"> erit angulus i e b rectus ք 31 p 3, ideoq́;</s> <s xml:id="echoid-s10534" xml:space="preserve"> a e d acutus] quoniã e d ſemidiameter:</s> <s xml:id="echoid-s10535" xml:space="preserve"> & <lb/>quęlibet linea in circulo cũ diametro facit angulũ acutũ [ut patet ք 31 p 3.</s> <s xml:id="echoid-s10536" xml:space="preserve"> 32 p 1 uel 9 ax.</s> <s xml:id="echoid-s10537" xml:space="preserve">] Et ſuք pun <lb/>ctũ e fiat angulus æqualis angulo a e d [ք 23 p 1] ꝗ ſit d e t.</s> <s xml:id="echoid-s10538" xml:space="preserve"> Palã, qđ e t cadet intra circulũ:</s> <s xml:id="echoid-s10539" xml:space="preserve"> [ſi.</s> <s xml:id="echoid-s10540" xml:space="preserve"> n.</s> <s xml:id="echoid-s10541" xml:space="preserve"> cade <lb/>ret extra:</s> <s xml:id="echoid-s10542" xml:space="preserve"> uel tãgeret peripheriã, efficeretq́;</s> <s xml:id="echoid-s10543" xml:space="preserve"> cũ ſemidiametro de angulũ rectũ ք 18 p 3:</s> <s xml:id="echoid-s10544" xml:space="preserve"> uel ſecaret, & <lb/>efficeret obtuſum:</s> <s xml:id="echoid-s10545" xml:space="preserve"> quorũ uterq;</s> <s xml:id="echoid-s10546" xml:space="preserve"> cũ acuto a e d maior ſit ք 11 12 d 1:</s> <s xml:id="echoid-s10547" xml:space="preserve"> mãdato ſatis factũ nõ eſſet] & ſe-<lb/>cabit lineã d h:</s> <s xml:id="echoid-s10548" xml:space="preserve"> [ք lẽma Procli ad 29 p 1:</s> <s xml:id="echoid-s10549" xml:space="preserve"> ꝗa ſecat a e ipſi parallelã] ſit pũctũ ſectiõis t.</s> <s xml:id="echoid-s10550" xml:space="preserve"> Palã etiã, qđ an <lb/>gulus a d e maior eſt angulo d e t:</s> <s xml:id="echoid-s10551" xml:space="preserve"> [ꝗa cũ a e maior ſit a d, maior erit ք <lb/> <anchor type="figure" xlink:label="fig-0168-01a" xlink:href="fig-0168-01"/> 18 p 1 angulus a d e angulo a e d, cui ęquatus eſt angulus d e t] & ita et <lb/>ſecabit a b:</s> <s xml:id="echoid-s10552" xml:space="preserve"> [ꝗa.</s> <s xml:id="echoid-s10553" xml:space="preserve"> n.</s> <s xml:id="echoid-s10554" xml:space="preserve"> anguli a d e, e d b æquãtur duob.</s> <s xml:id="echoid-s10555" xml:space="preserve"> rectis per 13 p 1:</s> <s xml:id="echoid-s10556" xml:space="preserve"> & <lb/>angulus a d e maior eſt angulo d et:</s> <s xml:id="echoid-s10557" xml:space="preserve"> anguli igitur e d b, d e t minores <lb/>ſunt duob.</s> <s xml:id="echoid-s10558" xml:space="preserve"> rectis:</s> <s xml:id="echoid-s10559" xml:space="preserve"> quare e t, d b cõcurrẽt ք 11 ax.</s> <s xml:id="echoid-s10560" xml:space="preserve">] ſecet in pũcto z.</s> <s xml:id="echoid-s10561" xml:space="preserve"> De-<lb/>inde à pũcto a ducatur ad arcũ e h linea:</s> <s xml:id="echoid-s10562" xml:space="preserve"> q̃ ſit a n:</s> <s xml:id="echoid-s10563" xml:space="preserve"> & ducatur linea d n:</s> <s xml:id="echoid-s10564" xml:space="preserve"> <lb/>& ſuper pũctũ n fiat angulus æqualis angulo d n a, ք lineã m n:</s> <s xml:id="echoid-s10565" xml:space="preserve"> q̃ ne-<lb/>ceſſariò cadet intra circulũ:</s> <s xml:id="echoid-s10566" xml:space="preserve"> [ob cauſſam ꝓximè expoſitã] & ſecabit <lb/>d h:</s> <s xml:id="echoid-s10567" xml:space="preserve"> [Nã cũ angulus a n d à ſemidiametro d n & recta linea a n cõpre <lb/>henſus, ſit acutus, ut patuit:</s> <s xml:id="echoid-s10568" xml:space="preserve"> erit angulus d n m ipſi æquatus, acutus:</s> <s xml:id="echoid-s10569" xml:space="preserve"> <lb/>ſed & n d m eſt acutus, ꝗa pars eſt acuti e d t:</s> <s xml:id="echoid-s10570" xml:space="preserve"> anguli igitur d n m, n d <lb/>m ſunt minores duob.</s> <s xml:id="echoid-s10571" xml:space="preserve"> rectis.</s> <s xml:id="echoid-s10572" xml:space="preserve"> Quare n m, d h cõcurrent ք 11 ax.</s> <s xml:id="echoid-s10573" xml:space="preserve">] ſecet <lb/>in pũcto m.</s> <s xml:id="echoid-s10574" xml:space="preserve"> Palã etiã, qđ a n cõcurret cũ d h extra circulũ:</s> <s xml:id="echoid-s10575" xml:space="preserve"> [per lẽma <lb/>Procli ad 29 p 1] ſit cõcurſus in l.</s> <s xml:id="echoid-s10576" xml:space="preserve"> Ducatur etiã à pũcto a linea ad arcũ <lb/>e i f:</s> <s xml:id="echoid-s10577" xml:space="preserve"> q̃ ſit a g:</s> <s xml:id="echoid-s10578" xml:space="preserve"> & ducatur d g:</s> <s xml:id="echoid-s10579" xml:space="preserve"> & fiat angulus d g q æqualis angulo a g d.</s> <s xml:id="echoid-s10580" xml:space="preserve"> <lb/>Palã, qđ q g ſecabit d h:</s> <s xml:id="echoid-s10581" xml:space="preserve"> [ut patuit] ſit pũctũ ſectionis q.</s> <s xml:id="echoid-s10582" xml:space="preserve"> Palã etiã, qđ <lb/>a g cõcurret cũ d h ex parte f:</s> <s xml:id="echoid-s10583" xml:space="preserve"> [Nã qđ cõcurrat, cõſtatè lẽmate Pro-<lb/>cli ad 29 p 1:</s> <s xml:id="echoid-s10584" xml:space="preserve"> qđ uerò uerſus f, è paulò antè demõſtratis քſpicuũ eſt] <lb/>ſit cõcurſus o.</s> <s xml:id="echoid-s10585" xml:space="preserve"> Qđ aũt g q cadat inter d & h, palã:</s> <s xml:id="echoid-s10586" xml:space="preserve"> cũ arcus, quẽ ſecat g <lb/>o ex circulo, ſit maior arcu g h:</s> <s xml:id="echoid-s10587" xml:space="preserve"> ſi.</s> <s xml:id="echoid-s10588" xml:space="preserve"> n.</s> <s xml:id="echoid-s10589" xml:space="preserve"> ducatur linea g h:</s> <s xml:id="echoid-s10590" xml:space="preserve"> angulus h g d <lb/>maiorẽ reſpiciet arcũ angulo a g d.</s> <s xml:id="echoid-s10591" xml:space="preserve"> [ideoq́;</s> <s xml:id="echoid-s10592" xml:space="preserve"> ք 33 p 6 angulus h g d e-<lb/>rit maior angulo a g d:</s> <s xml:id="echoid-s10593" xml:space="preserve"> at angulo a g d, æquatus eſt angulus q g d:</s> <s xml:id="echoid-s10594" xml:space="preserve"> an-<lb/>gulus igitur h g d maior eſt angulo q g d:</s> <s xml:id="echoid-s10595" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s10596" xml:space="preserve"> linea q g ſecans angu-<lb/>lũ q g d, ſecabit baſim h d angulo q g d ſubtẽſam.</s> <s xml:id="echoid-s10597" xml:space="preserve">] Iterũ à pũcto a du <lb/>catur ad arcũ fb, linea a k, ſecãs d h in pũcto s:</s> <s xml:id="echoid-s10598" xml:space="preserve"> [qđ.</s> <s xml:id="echoid-s10599" xml:space="preserve"> n.</s> <s xml:id="echoid-s10600" xml:space="preserve"> ſecet, patet è lẽ <lb/>mate Procli ad 29 p 1] ut ſit k s maior s d:</s> <s xml:id="echoid-s10601" xml:space="preserve"> [ſecta nẽpe d f bifariã ք 10 ք <lb/>1, & ab a ducta linea a k ք ſectionis pũctũ, uel ք quodcũq;</s> <s xml:id="echoid-s10602" xml:space="preserve"> aliud uer-<lb/>ſus d:</s> <s xml:id="echoid-s10603" xml:space="preserve"> utroq;</s> <s xml:id="echoid-s10604" xml:space="preserve">. n.</s> <s xml:id="echoid-s10605" xml:space="preserve"> modo erit s k maior fs ք 7 p 3:</s> <s xml:id="echoid-s10606" xml:space="preserve"> & ob id maior s d] & ducatur k d.</s> <s xml:id="echoid-s10607" xml:space="preserve"> Palã, qđ angulus d k <lb/>a eſt acutus.</s> <s xml:id="echoid-s10608" xml:space="preserve"> [ut in principio huius numeri oſtẽſum eſt] Fiat [ք 23 p 1] ei æqualis:</s> <s xml:id="echoid-s10609" xml:space="preserve"> ꝗ ſit d k u.</s> <s xml:id="echoid-s10610" xml:space="preserve"> Palã, qđ, <lb/>cũ angulus k d s ſit maior angulo d k s:</s> <s xml:id="echoid-s10611" xml:space="preserve"> [ք 18 p 1:</s> <s xml:id="echoid-s10612" xml:space="preserve"> ꝗ a s k maior eſt s d ք fabricationẽ]k u cõcurret cũ d <lb/>h:</s> <s xml:id="echoid-s10613" xml:space="preserve"> [Nã cũ anguli h d k, k d s æquẽtur duob rectis ք 13 p 1:</s> <s xml:id="echoid-s10614" xml:space="preserve"> & k d s ſit maior d k s è cõcluſione:</s> <s xml:id="echoid-s10615" xml:space="preserve"> erit k d s <lb/>etiã maior d k u æquali d k s:</s> <s xml:id="echoid-s10616" xml:space="preserve"> anguli igitur h d k, d k u ſũt minores duob.</s> <s xml:id="echoid-s10617" xml:space="preserve"> rectis.</s> <s xml:id="echoid-s10618" xml:space="preserve"> Quare h d, ku cõcur <lb/>rẽt ք 11 ax.</s> <s xml:id="echoid-s10619" xml:space="preserve">] ſit cõcurſus in pũcto u.</s> <s xml:id="echoid-s10620" xml:space="preserve"> Palã ſecũdũ ſupra dicta [& 12 n 4] qđ pũctũ t mouetur ad e, & re <lb/>flectitur ad a:</s> <s xml:id="echoid-s10621" xml:space="preserve"> & քpẽdicularis à pũcto t ducta, eſt t d:</s> <s xml:id="echoid-s10622" xml:space="preserve"> q̃ քpẽdicularis eſt ſuք ſuքficiẽ, cõtingẽtẽ ſpecu <lb/>lũ:</s> <s xml:id="echoid-s10623" xml:space="preserve"> [ք 25 n 4] & eſt æquidiſtãs lineæ reflexiõis, q̃ eſt a e:</s> <s xml:id="echoid-s10624" xml:space="preserve"> [ք fabricationẽ] unde nõ cõcurret cũ e a:</s> <s xml:id="echoid-s10625" xml:space="preserve"> [ք <lb/>35 d 1.</s> <s xml:id="echoid-s10626" xml:space="preserve"> Imago igitur pũcti tuidebitur in reflexiõis pũcto e] Pũctũ aũt z mouetur ad e, & reflectitur ad <lb/>a:</s> <s xml:id="echoid-s10627" xml:space="preserve"> & քpẽdicularis ducta à pũcto z, eſt a z:</s> <s xml:id="echoid-s10628" xml:space="preserve"> q̃ cõcurrit cũ a e in pũcto a.</s> <s xml:id="echoid-s10629" xml:space="preserve"> Vnde locus formę pũcti z erit a.</s> <s xml:id="echoid-s10630" xml:space="preserve"> <lb/>[ք 3 n.</s> <s xml:id="echoid-s10631" xml:space="preserve">] Pũctũ uerò m mouetur ad n, & reflectitur ad a:</s> <s xml:id="echoid-s10632" xml:space="preserve"> & քpẽdicularis ducta à pũcto m, quę eſt m d.</s> <s xml:id="echoid-s10633" xml:space="preserve"> <lb/>cocurrit cũ a n in pũcto l, qđ eſt ultra ſpeculũ:</s> <s xml:id="echoid-s10634" xml:space="preserve"> & locus formę pũcti m erit l.</s> <s xml:id="echoid-s10635" xml:space="preserve"> Forma uerò pũcti q m o <lb/>uetur ad g, & reflectitur ad a:</s> <s xml:id="echoid-s10636" xml:space="preserve"> & locus eius erit o:</s> <s xml:id="echoid-s10637" xml:space="preserve"> qui eſt ultra uiſum.</s> <s xml:id="echoid-s10638" xml:space="preserve"> Et forma puncti u mouetur ad <lb/>k, & reflectitur ad a:</s> <s xml:id="echoid-s10639" xml:space="preserve"> & perpendicularis ab eo, eſt k d:</s> <s xml:id="echoid-s10640" xml:space="preserve"> & locus imaginis s.</s> <s xml:id="echoid-s10641" xml:space="preserve"> [inter uiſum & ſpeculũ.</s> <s xml:id="echoid-s10642" xml:space="preserve">] <lb/>Palàm ergo ex prædictis, quòd imaginum quædam inter uiſum & ſpeculum:</s> <s xml:id="echoid-s10643" xml:space="preserve"> quædam in ipſo uiſus <lb/>quædam citra uiſum:</s> <s xml:id="echoid-s10644" xml:space="preserve"> quædam ultra uiſum apparent.</s> <s xml:id="echoid-s10645" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s10646" xml:space="preserve"/> </p> <div xml:id="echoid-div379" type="float" level="0" n="0"> <figure xlink:label="fig-0168-01" xlink:href="fig-0168-01a"> <variables xml:id="echoid-variables87" xml:space="preserve">l g e n h m t q u i a <gap/> s z b k y f p o</variables> </figure> </div> </div> <div xml:id="echoid-div381" type="section" level="0" n="0"> <head xml:id="echoid-head364" xml:space="preserve" style="it">61. In ſpeculo ſphærico cauo imago prouario eius ſitu at loco uariè uidetur. 12 p 8.</head> <p> <s xml:id="echoid-s10647" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s10648" xml:space="preserve"> palàm, quoniã uiſus քfectius acquirit formas ſibi oppoſitas.</s> <s xml:id="echoid-s10649" xml:space="preserve"> [per 21.</s> <s xml:id="echoid-s10650" xml:space="preserve"> 38 n 1.</s> <s xml:id="echoid-s10651" xml:space="preserve"> 17 n 3.</s> <s xml:id="echoid-s10652" xml:space="preserve">] Vn <lb/>de cũlocus imaginis fuerit ultra ſpeculũ [ut in puncto l] autinter uiſum & ſpeculum:</s> <s xml:id="echoid-s10653" xml:space="preserve"> [ut in <lb/> <pb o="163" file="0169" n="169" rhead="OPTICAE LIBER V."/> punctos] cõprehẽditur ueritas illius imaginis.</s> <s xml:id="echoid-s10654" xml:space="preserve"> Cũ aũt քpẽdicularis à pũcto uiſo ducta, fuerit ęqui <lb/>diſtãs lineæ reflexiõis:</s> <s xml:id="echoid-s10655" xml:space="preserve"> apparebit imago in pũcto reflexiõis.</s> <s xml:id="echoid-s10656" xml:space="preserve"> [utin e.</s> <s xml:id="echoid-s10657" xml:space="preserve">] Quoniã cũ pũctũ illud ſit ſen <lb/>ſuale [ut patet è 16 n 4] ſumpto pũcto eius intellectuali medio:</s> <s xml:id="echoid-s10658" xml:space="preserve"> imago cuiuſcũq;</s> <s xml:id="echoid-s10659" xml:space="preserve"> partis illius puncti <lb/>ſenſualis, ultra mediũ ſumptæ, erit ultra ſpeculũ:</s> <s xml:id="echoid-s10660" xml:space="preserve"> & imago partis citra mediũ erit inter uiſum & ſpe <lb/>culũ.</s> <s xml:id="echoid-s10661" xml:space="preserve"> Et cũ totalis forma ex ulteriorib.</s> <s xml:id="echoid-s10662" xml:space="preserve"> & citeriorib.</s> <s xml:id="echoid-s10663" xml:space="preserve"> partibus uideatur una & continua:</s> <s xml:id="echoid-s10664" xml:space="preserve"> neceſſariò <lb/>forma illius puncti ſenſualis uidebitur in ipſo ſpeculo, in loco reflexionis.</s> <s xml:id="echoid-s10665" xml:space="preserve"> Verũ in imaginib.</s> <s xml:id="echoid-s10666" xml:space="preserve"> quarũ <lb/>locus fuerit in cẽtro uiſus, non cõprehẽditur ueritas earũ:</s> <s xml:id="echoid-s10667" xml:space="preserve"> unde ſæpius error accidit in his ſpeculis.</s> <s xml:id="echoid-s10668" xml:space="preserve"> <lb/>Vt aũt hoc pateat:</s> <s xml:id="echoid-s10669" xml:space="preserve"> erigatur ſuք ſuքficiẽ ſpeculi lignũ perpẽdiculariter, minus medietate ſemidiame <lb/>tri ſpeculi:</s> <s xml:id="echoid-s10670" xml:space="preserve"> & circa caput huius ligni, ſit cẽtrũ uiſus:</s> <s xml:id="echoid-s10671" xml:space="preserve"> & dirigatur uiſus ad pũctũ ſpeculi, cuius lõgitu, <lb/>do à ligno ſit maior, ꝗ̃ lõgitudo cẽtri uiſus à diametro, ք lignũ trãſeunte:</s> <s xml:id="echoid-s10672" xml:space="preserve"> uidebitur ꝗ dẽ imago illius <lb/>ligni ultra uiſum, nec erit certa cõprehẽſio eius:</s> <s xml:id="echoid-s10673" xml:space="preserve"> imò apparebit arcuata:</s> <s xml:id="echoid-s10674" xml:space="preserve"> cũ nõ ſit.</s> <s xml:id="echoid-s10675" xml:space="preserve"> In his ergo ſpecu-<lb/>lis nõ cõprehẽditur ueritas imaginis, niſi cuius locus fuerit ultra ſpeculũ:</s> <s xml:id="echoid-s10676" xml:space="preserve"> aut inter uiſum & ſpecu-<lb/>lũ.</s> <s xml:id="echoid-s10677" xml:space="preserve"> Cũ aũt cẽtrũ uiſus fuerit in քpẽdiculari ք lignũ trãſeũte:</s> <s xml:id="echoid-s10678" xml:space="preserve"> nõ plenè cõprehẽdit formã illius ligni.</s> <s xml:id="echoid-s10679" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div382" type="section" level="0" n="0"> <head xml:id="echoid-head365" xml:space="preserve" style="it">62. Vιſus in centro ſpeculi ſphærici caui poſitus: <lb/>ſeipſum tantùm uidet. 4 p 8. Idem 44 n 4.</head> <figure> <variables xml:id="echoid-variables88" xml:space="preserve">b c a e d</variables> </figure> <p> <s xml:id="echoid-s10680" xml:space="preserve">SIuerò uiſus fuerit ín diametro ſphęrę, & in cẽ <lb/>tro eius (cũ quęlibet linea ab eo ad ſpeculum <lb/>ducta ſit perpẽdicularis ſuper ſpeculũ) [quia <lb/>perpẽdicularis eſt plano ſpeculum tangẽti ք 4 th 1.</s> <s xml:id="echoid-s10681" xml:space="preserve"> <lb/>ſphęr.</s> <s xml:id="echoid-s10682" xml:space="preserve"> uel 25 n 4:</s> <s xml:id="echoid-s10683" xml:space="preserve"> eaq́;</s> <s xml:id="echoid-s10684" xml:space="preserve"> de cauſſa in ſe ipſam reflecti-<lb/>tur per 11 n 4] nõ cõprehẽdetur forma alicuius pũ <lb/>cti, niſi puncti portionis oculi, interiacentis latera <lb/>pyramidis uiſualis, quę à cẽtro ſpeculi intelligitur <lb/>ꝓtẽdi.</s> <s xml:id="echoid-s10685" xml:space="preserve"> Quoniã forma cuiuslibet alterius pũcti ca <lb/>det in ſpeculũ ſuք lineã declinatã, & neceſſariò re-<lb/>flectetur ſuք declinatã.</s> <s xml:id="echoid-s10686" xml:space="preserve"> Quare linea reflexionis nõ <lb/>trãſibit per centrũ:</s> <s xml:id="echoid-s10687" xml:space="preserve"> & ita nõ cõtinget centrũ uiſus.</s> <s xml:id="echoid-s10688" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div383" type="section" level="0" n="0"> <head xml:id="echoid-head366" xml:space="preserve" style="it">63. Semidiameter ſpeculi ſphærici caui, in qua <lb/>eſt uiſ{us} extra cẽtrũ: nullum ſui punctũ obliquè <lb/>ſpeculo incidẽs ad uiſum reflectit: reliqua uerò ſemidiameter prædictæ cõtinua, reflectit. 5 p 8.</head> <p> <s xml:id="echoid-s10689" xml:space="preserve">SIuerò fuerit uiſus in diametro:</s> <s xml:id="echoid-s10690" xml:space="preserve"> non comprehendet formam alterius puncti ſemidiametri, in <lb/>qua eſt.</s> <s xml:id="echoid-s10691" xml:space="preserve"> Quoniã angulus, quem efficient duæ lineæ à puncto ſumpto in ſemidiametro, & à cen-<lb/>tro uiſus ιn idẽ ſpeculi punctũ, non diuidetur per perpendicularem ab illo puncto ſpeculi du-<lb/>ctam:</s> <s xml:id="echoid-s10692" xml:space="preserve"> cum illa perpendicularis tendat ad centrum ſpeculi:</s> <s xml:id="echoid-s10693" xml:space="preserve"> [per 4 th.</s> <s xml:id="echoid-s10694" xml:space="preserve"> 1 ſphær.</s> <s xml:id="echoid-s10695" xml:space="preserve">] Sed formam alicuius <lb/>puncti alterius ſemidiametri percipere poterit.</s> <s xml:id="echoid-s10696" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div384" type="section" level="0" n="0"> <head xml:id="echoid-head367" xml:space="preserve" style="it">64. In ſpeculo ſphærico cauo perpendiculari incidentiæ, & linea reflexionis concurrentib{us}: <lb/>eſt: ut perpendicularis incidentiæ ad rectam inter centrum ſpeculi & locum imaginis: ſic re-<lb/>cta inter uiſibile & finem contingentiæ, adrectam inter finem contingentiæ & locum ima-<lb/>ginis. 13 p 8.</head> <p> <s xml:id="echoid-s10697" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s10698" xml:space="preserve"> uiſo pũcto in huiuſmodi ſpeculo, <lb/> <anchor type="figure" xlink:label="fig-0169-02a" xlink:href="fig-0169-02"/> cũ non fuerit perpendicularis ęquidiſtans <lb/>lineę reflexionis:</s> <s xml:id="echoid-s10699" xml:space="preserve"> linea à centro ſpeculi ad <lb/>punctũ uifum ducta, ſe habebit ad lineã ab eodem <lb/>centro ad locũ imaginis ductam, ſicut linea à pun <lb/>ctò uiſo ad punctum, (quod diximus) contingen-<lb/>tię [17 n] ſe habet ad lineam à puncto contingen <lb/>tiæ, ad locum imaginis ductam.</s> <s xml:id="echoid-s10700" xml:space="preserve"> Verbigratia:</s> <s xml:id="echoid-s10701" xml:space="preserve"> ſit e <lb/>centrum ſpeculi:</s> <s xml:id="echoid-s10702" xml:space="preserve"> b punctum uiſum:</s> <s xml:id="echoid-s10703" xml:space="preserve"> a centrum ui-<lb/>ſus:</s> <s xml:id="echoid-s10704" xml:space="preserve"> g punctum reflexionis:</s> <s xml:id="echoid-s10705" xml:space="preserve"> linea contingentiæ z <lb/>g.</s> <s xml:id="echoid-s10706" xml:space="preserve"> z g autẽ aut concurret cum e b:</s> <s xml:id="echoid-s10707" xml:space="preserve"> aut erit æquidi-<lb/>ſtãs ei.</s> <s xml:id="echoid-s10708" xml:space="preserve"> Cõcurrat in puncto t.</s> <s xml:id="echoid-s10709" xml:space="preserve"> Linea uerô e b cõcur <lb/>rit cũ a g [ex theſi,] ſed non in puncto g:</s> <s xml:id="echoid-s10710" xml:space="preserve"> cũ b e, a g <lb/>ſint duę lineę.</s> <s xml:id="echoid-s10711" xml:space="preserve"> Igitur aut cõcurrit ultra g:</s> <s xml:id="echoid-s10712" xml:space="preserve"> aut inter <lb/>g & a:</s> <s xml:id="echoid-s10713" xml:space="preserve"> aut in a:</s> <s xml:id="echoid-s10714" xml:space="preserve"> aut ultra a.</s> <s xml:id="echoid-s10715" xml:space="preserve"> Sit ultra g, & in pũcto h.</s> <s xml:id="echoid-s10716" xml:space="preserve"> <lb/>Dico ergo, quòd eſt proportio e b ad e h, ſicut b t <lb/>ad t h.</s> <s xml:id="echoid-s10717" xml:space="preserve"> Produeatur perpẽdicularis e g:</s> <s xml:id="echoid-s10718" xml:space="preserve"> & à puncto <lb/>h ducatur ęquidiſtans lineæ b g:</s> <s xml:id="echoid-s10719" xml:space="preserve"> [per 31 p 1] quę cõ <lb/>curret cũ e g:</s> <s xml:id="echoid-s10720" xml:space="preserve"> [per lẽma Procli ad 29 p 1] ſit cõcur-<lb/>ſus l:</s> <s xml:id="echoid-s10721" xml:space="preserve"> & à puncto b ducatur ęquidiſtãs g h:</s> <s xml:id="echoid-s10722" xml:space="preserve"> [quę ne <lb/>ceſſariò cõcurret cũ z t:</s> <s xml:id="echoid-s10723" xml:space="preserve"> [per dictũ lẽma] ſit cõcur-<lb/>ſus q.</s> <s xml:id="echoid-s10724" xml:space="preserve"> Palã [per 12 n 4] quòd angulus b g e eſt ęqua <lb/>lis a g e:</s> <s xml:id="echoid-s10725" xml:space="preserve"> ſed angulus b g e eſt æqualis angulo g l h:</s> <s xml:id="echoid-s10726" xml:space="preserve"> <lb/>[exterior interiori & oppoſito ք 29 p 1] & [ք 15 p <lb/>1] angulus a g e ęqualis angulo l g h:</s> <s xml:id="echoid-s10727" xml:space="preserve"> ergo angulus g l h ęqualis eſt angulo l g h.</s> <s xml:id="echoid-s10728" xml:space="preserve"> Igitur [ք 6 p 1] lh ęqua <lb/> <pb o="164" file="0170" n="170" rhead="ALHAZEN"/> lis eſt g h.</s> <s xml:id="echoid-s10729" xml:space="preserve"> Similiter angulus b g q ęqualis eſt angulo a g z.</s> <s xml:id="echoid-s10730" xml:space="preserve"> [Nã cũ angulus e g q æquetur angulo e g z:</s> <s xml:id="echoid-s10731" xml:space="preserve"> <lb/>ꝗa ք 18 p 3 uterq;</s> <s xml:id="echoid-s10732" xml:space="preserve"> rectus eſt, & e g b ipſi e g a, ut patuit:</s> <s xml:id="echoid-s10733" xml:space="preserve"> reliquus igitur b g q ęquatur reliquo a g z ք 3 <lb/>ax.</s> <s xml:id="echoid-s10734" xml:space="preserve">] & angulus a g z ęqualis eſt angulo g q b [exterior interiori oppoſito ք 29 p 1:</s> <s xml:id="echoid-s10735" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s10736" xml:space="preserve"> b g q ęquatur <lb/>g q b:</s> <s xml:id="echoid-s10737" xml:space="preserve"> & ita [ք 6 p 1] b q ęqualis eſt b g.</s> <s xml:id="echoid-s10738" xml:space="preserve"> Quare [ք 7 p 5] ꝓportio b g ad h l, ſicut b q ad h g.</s> <s xml:id="echoid-s10739" xml:space="preserve"> Sed quoniã <lb/>angulus g h t eſt ęqualis angulo t b q:</s> <s xml:id="echoid-s10740" xml:space="preserve"> [per 29 p 1] erit triangulũ t b q ſimile triãgulo g h t.</s> <s xml:id="echoid-s10741" xml:space="preserve"> [Nã anguli <lb/>ad t æquãtur ք 15 p 1, & ք 32 p 1 tertius tertio.</s> <s xml:id="echoid-s10742" xml:space="preserve"> Quare ք 4 p.</s> <s xml:id="echoid-s10743" xml:space="preserve"> 1 d 6 triãgula t b q.</s> <s xml:id="echoid-s10744" xml:space="preserve"> g h t ſunt ſimilia.</s> <s xml:id="echoid-s10745" xml:space="preserve">] Igitur <lb/>ꝓportio q b ad h g, ſicut b t ad t h:</s> <s xml:id="echoid-s10746" xml:space="preserve"> & ita [per 7 p 5] b g ad h l, ſicut b t ad t h.</s> <s xml:id="echoid-s10747" xml:space="preserve"> Sed cũ triangulũ b g e ſit <lb/>ſimile triangulo h e l:</s> <s xml:id="echoid-s10748" xml:space="preserve"> [angulus enim ad e cõmunis eſt, & exteriores ad g & b ęquãtur interiorib, op <lb/>poſitis ad l & h per 29 p 1.</s> <s xml:id="echoid-s10749" xml:space="preserve"> Quare ք 4 p.</s> <s xml:id="echoid-s10750" xml:space="preserve"> 1 d 6 triangula b g e, h e l ſunt ſimilia] erit ꝓportio b g ad h l, <lb/> <anchor type="figure" xlink:label="fig-0170-01a" xlink:href="fig-0170-01"/> <anchor type="figure" xlink:label="fig-0170-02a" xlink:href="fig-0170-02"/> ſicut e b ad e h:</s> <s xml:id="echoid-s10751" xml:space="preserve"> & ita [ք 11 p 5] e b ad e h, ſicut b <lb/> <anchor type="figure" xlink:label="fig-0170-03a" xlink:href="fig-0170-03"/> t ad t h.</s> <s xml:id="echoid-s10752" xml:space="preserve"> Qđ eſt ꝓpoſitũ.</s> <s xml:id="echoid-s10753" xml:space="preserve"> Eadẽ erit ꝓbatio, ſi lo <lb/>cus imaginis fuerit inter a & g:</s> <s xml:id="echoid-s10754" xml:space="preserve"> aut in a:</s> <s xml:id="echoid-s10755" xml:space="preserve"> aut ul <lb/>tra.</s> <s xml:id="echoid-s10756" xml:space="preserve"> Si uerò linea cõtingẽtiæ z g ſit æquidiſtãs <lb/>perpẽdiculari, q̃ eſt b e h:</s> <s xml:id="echoid-s10757" xml:space="preserve"> ducatur perpẽdicu-<lb/>laris g e:</s> <s xml:id="echoid-s10758" xml:space="preserve"> [à pũcto g ſuper z g] quę cũ ſit քpen-<lb/>dicularis ſuper g z:</s> <s xml:id="echoid-s10759" xml:space="preserve"> erit perpẽdicularis ſuք b h <lb/> <anchor type="figure" xlink:label="fig-0170-04a" xlink:href="fig-0170-04"/> [per 29 p 1] & erit angulus b e g ęqualis angu <lb/>lo h e g:</s> <s xml:id="echoid-s10760" xml:space="preserve"> & [per 12 n 4] angulus b g e æqualis <lb/>eſt angulo e g h:</s> <s xml:id="echoid-s10761" xml:space="preserve"> reſtat triãgulũ b g e ſimile triã <lb/>gulo e g h.</s> <s xml:id="echoid-s10762" xml:space="preserve"> [ęquabitur.</s> <s xml:id="echoid-s10763" xml:space="preserve"> n.</s> <s xml:id="echoid-s10764" xml:space="preserve"> ք 32 p 1 reliquus angu <lb/>lus ad b, reliquo ad h:</s> <s xml:id="echoid-s10765" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s10766" xml:space="preserve"> ք 4 p.</s> <s xml:id="echoid-s10767" xml:space="preserve"> 1 d 6 triãgu-<lb/>la b g e, h g e erunt ſimilia.</s> <s xml:id="echoid-s10768" xml:space="preserve">] lgitur proportio b <lb/>e ad h e, ſicut b g ad g h.</s> <s xml:id="echoid-s10769" xml:space="preserve"> Qđ eſt propoſitum.</s> <s xml:id="echoid-s10770" xml:space="preserve"> <lb/>Quare in hoc caſu non põt ſumi aliud punctũ <lb/>cõtingẽtię, ꝗ̃punctũ g, eo modo, quo punctũ <lb/>contingentię ſuprà [17 n] appellauimus.</s> <s xml:id="echoid-s10771" xml:space="preserve"/> </p> <div xml:id="echoid-div384" type="float" level="0" n="0"> <figure xlink:label="fig-0169-02" xlink:href="fig-0169-02a"> <variables xml:id="echoid-variables89" xml:space="preserve">l b z c g q a b e</variables> </figure> <figure xlink:label="fig-0170-01" xlink:href="fig-0170-01a"> <variables xml:id="echoid-variables90" xml:space="preserve">b l a e h q g f z</variables> </figure> <figure xlink:label="fig-0170-02" xlink:href="fig-0170-02a"> <variables xml:id="echoid-variables91" xml:space="preserve">l t b e a q g z</variables> </figure> <figure xlink:label="fig-0170-03" xlink:href="fig-0170-03a"> <variables xml:id="echoid-variables92" xml:space="preserve">t f g q a c b</variables> </figure> <figure xlink:label="fig-0170-04" xlink:href="fig-0170-04a"> <variables xml:id="echoid-variables93" xml:space="preserve">z g q h c b</variables> </figure> </div> </div> <div xml:id="echoid-div386" type="section" level="0" n="0"> <head xml:id="echoid-head368" xml:space="preserve" style="it">65. Viſu & uiſibili in diametro ſpeculi ſphærici caui æquabiliter à cẽtro diſtantib{us}: poteſt fie-<lb/>rireflexio à tota peripheria circuli, quẽ ſemidiameter perpẽdicularis ad dictã diametrum, cõ-<lb/>uerſa deſcribit. 14 p 8.</head> <p> <s xml:id="echoid-s10772" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s10773" xml:space="preserve"> ſit circulus a b g d:</s> <s xml:id="echoid-s10774" xml:space="preserve"> & h centrum ui-<lb/> <anchor type="figure" xlink:label="fig-0170-05a" xlink:href="fig-0170-05"/> ſus intra ſpeculum:</s> <s xml:id="echoid-s10775" xml:space="preserve"> e centrum ſpeculi:</s> <s xml:id="echoid-s10776" xml:space="preserve"> z pun <lb/>ctum uiſum:</s> <s xml:id="echoid-s10777" xml:space="preserve"> & ducatur diameter b e d.</s> <s xml:id="echoid-s10778" xml:space="preserve"> Si fue <lb/>rit z in ſemidiametro b e:</s> <s xml:id="echoid-s10779" xml:space="preserve"> poterit eſſe reflexio ab ali <lb/>quo puncto ſemicirculi b a d, & ab aliquo pũcto ſe-<lb/>micirculi ei oppoſiti.</s> <s xml:id="echoid-s10780" xml:space="preserve"> Quoniam quocunq;</s> <s xml:id="echoid-s10781" xml:space="preserve"> puncto <lb/>ſemidiametri b e ſumpto:</s> <s xml:id="echoid-s10782" xml:space="preserve"> ſi ab eo ducatur linea ad <lb/>aliquod punctum ſemicirculi, & à puncto h ad idẽ <lb/>punctum ducatur alia linea:</s> <s xml:id="echoid-s10783" xml:space="preserve"> illæ duę lineæ efficient <lb/>angulum, quem diuidet per æqualia ſemidiameter <lb/>ducta à puncto e ad illud punctum [quia enim ſe-<lb/>midiameter illa extheſi eſt perpendicularis diame-<lb/>tro, in qua uiſus & uiſibile ęquabiliter à centro ſpe-<lb/>culi diſtantia collocantur:</s> <s xml:id="echoid-s10784" xml:space="preserve"> ita que ſi à uiſu & uiſibi-<lb/>li duę rectæ lineæ cum dicta ſemidiametro in peri-<lb/>pheria cõcurrant:</s> <s xml:id="echoid-s10785" xml:space="preserve"> erunt anguli ad cõcurſus punctũ <lb/>ęquales per 4 p 1.</s> <s xml:id="echoid-s10786" xml:space="preserve"> Quare per 12 n 4 ipſum eſt reflexionis punctũ.</s> <s xml:id="echoid-s10787" xml:space="preserve">] Similiter in ſemicirculo oppoſito.</s> <s xml:id="echoid-s10788" xml:space="preserve"/> </p> <div xml:id="echoid-div386" type="float" level="0" n="0"> <figure xlink:label="fig-0170-05" xlink:href="fig-0170-05a"> <variables xml:id="echoid-variables94" xml:space="preserve">b z a c g h d</variables> </figure> </div> <pb o="165" file="0171" n="171" rhead="OPTICAE LIBER V."/> </div> <div xml:id="echoid-div388" type="section" level="0" n="0"> <head xml:id="echoid-head369" xml:space="preserve" style="it">66. Viſ{us} & uiſibile in diuerſis dimetris circuli (qui eſt commu nis ſectio ſuperficierum refle-<lb/>xionis & ſpeculi ſphærici caui) inter ſe reflectuntur, tum à perip heria inter ſemidiametros, in <lb/>quibus ſunt: tum ab alia huic oppoſita: à reliquis uerò duab{us} minimè. 20 p 8.</head> <p> <s xml:id="echoid-s10789" xml:space="preserve">SI uerò b punctũ uiſum fuerit extra diametrum d a g, ducatur diameter tranſiens per b:</s> <s xml:id="echoid-s10790" xml:space="preserve"> quæ fit <lb/>t q.</s> <s xml:id="echoid-s10791" xml:space="preserve"> Dico, quòd b poteſt reflecti ad uiſum a per arcum interiacentẽ diametros, in quibus ſunt a <lb/>& b, & ſimiliter per eius oppoſitum, id eſt, per arcum t d, & per arcum g q:</s> <s xml:id="echoid-s10792" xml:space="preserve"> & non poterit refle-<lb/>cti ab aliquo puncto arcus g t uelarcus q d.</s> <s xml:id="echoid-s10793" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s10794" xml:space="preserve"> ſumatur punctum in arcu g t, propet, quod <lb/>ſit k:</s> <s xml:id="echoid-s10795" xml:space="preserve"> & du cantur lineæ a k, k b:</s> <s xml:id="echoid-s10796" xml:space="preserve"> donec cadat k b ſuper diametrum d g in puncto o.</s> <s xml:id="echoid-s10797" xml:space="preserve"> Cum igitur o & a <lb/>ſint ex eadẽ parte centri circuli, quod eſt e:</s> <s xml:id="echoid-s10798" xml:space="preserve"> perpẽdicularis ducta à puncto k ad e, nõ diuidet angulũ <lb/>o k a.</s> <s xml:id="echoid-s10799" xml:space="preserve"> Et ita b non reflectetur ad a à puncto k.</s> <s xml:id="echoid-s10800" xml:space="preserve"> Simili-<lb/> <anchor type="figure" xlink:label="fig-0171-01a" xlink:href="fig-0171-01"/> ter ſumpto alio pũcto, quod ſit f:</s> <s xml:id="echoid-s10801" xml:space="preserve"> patebit, quòd per-<lb/>pendicularis e f non diuidet angulum a fb.</s> <s xml:id="echoid-s10802" xml:space="preserve"> Et ita nõ <lb/>reflectetur b ad a à puncto f.</s> <s xml:id="echoid-s10803" xml:space="preserve"> Quòd autem à puncto <lb/>arcus t d, uel arcus g q poſsit fieri reflexio:</s> <s xml:id="echoid-s10804" xml:space="preserve"> palàm per <lb/>hoc.</s> <s xml:id="echoid-s10805" xml:space="preserve"> Sit m punctum arcus t d:</s> <s xml:id="echoid-s10806" xml:space="preserve"> & ducantur lineę a m, <lb/>m b:</s> <s xml:id="echoid-s10807" xml:space="preserve"> fiet quadrangulum a m b e.</s> <s xml:id="echoid-s10808" xml:space="preserve"> Igitur perpendicu-<lb/>laris e m diuidet angulum a m b.</s> <s xml:id="echoid-s10809" xml:space="preserve"> Simili modo ſit h <lb/>punctum arcus g q:</s> <s xml:id="echoid-s10810" xml:space="preserve"> Linea a h ſecabit diametrum t q <lb/>in puncto c:</s> <s xml:id="echoid-s10811" xml:space="preserve"> & linea h b eundem in puncto b.</s> <s xml:id="echoid-s10812" xml:space="preserve"> Et ſunt <lb/>hæc etiam duo puncta ex diuerſis partibus centri.</s> <s xml:id="echoid-s10813" xml:space="preserve"> <lb/>Quare linea e h diuidet illum angulum.</s> <s xml:id="echoid-s10814" xml:space="preserve"> Pari modo, <lb/>ſi fuerit b in ſuperficie ſpeculi:</s> <s xml:id="echoid-s10815" xml:space="preserve"> aut extra ſpeculum, <lb/>dum a ſit intra ſpeculum:</s> <s xml:id="echoid-s10816" xml:space="preserve"> idem erit probãdi modus, <lb/>qui prius.</s> <s xml:id="echoid-s10817" xml:space="preserve"> Similiter ſi a fuerit in ſuperficie ſpeculi, b <lb/>interius, aut exterius.</s> <s xml:id="echoid-s10818" xml:space="preserve"> Si uerò a fuerit extra ſpecu-<lb/>lum, b intra:</s> <s xml:id="echoid-s10819" xml:space="preserve"> patebit, quod diximus.</s> <s xml:id="echoid-s10820" xml:space="preserve"> Ducantur enim lineæ à puncto a contingentes circulum d t g <lb/>[per 17 p 1] quæ ſint a h, a z:</s> <s xml:id="echoid-s10821" xml:space="preserve"> & ducantur duę diametri <lb/> <anchor type="figure" xlink:label="fig-0171-02a" xlink:href="fig-0171-02"/> a e g, t e q:</s> <s xml:id="echoid-s10822" xml:space="preserve"> & b in diametro t e q:</s> <s xml:id="echoid-s10823" xml:space="preserve"> reflectetur b ad a ab <lb/>aliquo puncto arcus t d:</s> <s xml:id="echoid-s10824" xml:space="preserve"> [ut conftat è iam demonſtra <lb/>tis.</s> <s xml:id="echoid-s10825" xml:space="preserve">] Sed palàm, quòd non ab aliquo puncto arcus z <lb/>d.</s> <s xml:id="echoid-s10826" xml:space="preserve"> [ductis enim duabus rectis e z, b z:</s> <s xml:id="echoid-s10827" xml:space="preserve"> erit angulus e z <lb/>a rectus per 18 p 3, & e z b acutus, ut oſtenſum eſt 60 <lb/>n.</s> <s xml:id="echoid-s10828" xml:space="preserve"> Quare ob angulorum inæquabilitatem, à puncto <lb/>z, ad uiſum a nulla fiet reflexio:</s> <s xml:id="echoid-s10829" xml:space="preserve"> multò igitur minus à <lb/>punctis inter z & d intermedijs:</s> <s xml:id="echoid-s10830" xml:space="preserve"> quia angulorum ad <lb/>lineã z a factorũ, unius quidẽ acuti, alterius uerò ob-<lb/>tuſi per 16 p 1, multò maior futura eſt inęquabilitas.</s> <s xml:id="echoid-s10831" xml:space="preserve">] <lb/>Igitur ab aliquo pũcto arcus t z:</s> <s xml:id="echoid-s10832" xml:space="preserve"> & ſimiliter ab aliquo <lb/>pũcto arcus oppoſiti ipſi t d, ſcilicet arcus g q refle-<lb/>xio fiet.</s> <s xml:id="echoid-s10833" xml:space="preserve"> Sed ab arcu t g, uel d q nõ fiet reflexio ſecun-<lb/>dũ ſuprà dictũ modũ.</s> <s xml:id="echoid-s10834" xml:space="preserve"> Si uerò b fuerit extra hanc dia <lb/>metrũ, & ſuper aliã, quæ ſimiliter ſit t e q:</s> <s xml:id="echoid-s10835" xml:space="preserve"> fiet reflexio <lb/>ab arcu t d:</s> <s xml:id="echoid-s10836" xml:space="preserve"> & à ſola parte eiust z, & ab arcu oppoſito, <lb/>qui eſt g q:</s> <s xml:id="echoid-s10837" xml:space="preserve"> ſed ab arcu t g, uel d q non fiet reflexio.</s> <s xml:id="echoid-s10838" xml:space="preserve"/> </p> <div xml:id="echoid-div388" type="float" level="0" n="0"> <figure xlink:label="fig-0171-01" xlink:href="fig-0171-01a"> <variables xml:id="echoid-variables95" xml:space="preserve">t k m b f d a o e g c h q</variables> </figure> <figure xlink:label="fig-0171-02" xlink:href="fig-0171-02a"> <variables xml:id="echoid-variables96" xml:space="preserve">a z m d h f b t b e q q g</variables> </figure> </div> </div> <div xml:id="echoid-div390" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables97" xml:space="preserve">l p m t n b d a c g x s u q</variables> </figure> <head xml:id="echoid-head370" xml:space="preserve" style="it">67. Si uiſu & uiſibili in diuerſis diametris circuli (qui eſt communis ſectio ſuperficierum, re-<lb/>flexionis & ſpeculi ſphærici caui) ſitis: linea à uiſu parallela dia-<lb/> metro uiſibilis, ſecet dicti circuli peripheriam. Imago reflexa à peripheria inter parallelam & uiſibilis diametrum, uidebitur extra ſpeculum: à peripheria inter par allelam & diametrum ui- ſ{us}, ultra uiſum: à peripheria uerò oppoſita, inter uiſum & ſpe- culum. 21 p 8.</head> <p> <s xml:id="echoid-s10839" xml:space="preserve">VErũ ſi â puncto a ducatur æquidiſtans t e:</s> <s xml:id="echoid-s10840" xml:space="preserve"> quæ ſit a p:</s> <s xml:id="echoid-s10841" xml:space="preserve"> loca ima <lb/>ginũ reflexarũ à punctis arcus t p, erunt extra ſpeculũ:</s> <s xml:id="echoid-s10842" xml:space="preserve"> loca au <lb/>tẽ imaginũ arcus p d, ultra centrũ uiſus, quod eſt a:</s> <s xml:id="echoid-s10843" xml:space="preserve"> loca au-<lb/>tem imaginum arcus q g ſunt inter centrum uiſus & ſpeculum.</s> <s xml:id="echoid-s10844" xml:space="preserve"> Et <lb/>quod ſuprà [60.</s> <s xml:id="echoid-s10845" xml:space="preserve"> 61 n] dictum eſt de locis imaginum:</s> <s xml:id="echoid-s10846" xml:space="preserve"> idem intelligẽ-<lb/>dum, ducta a m æquidiſtante lineæ t q.</s> <s xml:id="echoid-s10847" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div391" type="section" level="0" n="0"> <head xml:id="echoid-head371" xml:space="preserve" style="it">68. In quolibet puncto diametri circuli (qui eſt com-<lb/>munis ſectio ſuperficierum, reflexionis & ſpeculi ſphæri-<lb/>ci caui) quantumlibet continuatæ, poteſt imago uideri. <lb/>22 p 8.</head> <p> <s xml:id="echoid-s10848" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s10849" xml:space="preserve"> ſumpta diametro circuli in ſphærico ſpeculo cõcau:</s> <s xml:id="echoid-s10850" xml:space="preserve"> quodlibet punctũ illius diametri, <lb/> <pb o="166" file="0172" n="172" rhead="ALHAZEN"/> quantum cunq;</s> <s xml:id="echoid-s10851" xml:space="preserve"> productæ, poteſt eſſe locus imaginum.</s> <s xml:id="echoid-s10852" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s10853" xml:space="preserve"> ſit a g diameter circuli a m g:</s> <s xml:id="echoid-s10854" xml:space="preserve"> cu-<lb/>ius d centrum.</s> <s xml:id="echoid-s10855" xml:space="preserve"> Sumatur in hac diametro punctum z:</s> <s xml:id="echoid-s10856" xml:space="preserve"> e cẽtrum uiſus.</s> <s xml:id="echoid-s10857" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0172-01a" xlink:href="fig-0172-01"/> Dico, quòd z poteſt eſſe locus imaginis.</s> <s xml:id="echoid-s10858" xml:space="preserve"> Ducatur linea e t z per t pun <lb/>ctum circuli:</s> <s xml:id="echoid-s10859" xml:space="preserve"> & ducatur linea d t:</s> <s xml:id="echoid-s10860" xml:space="preserve"> erit angulus e t d acutus:</s> <s xml:id="echoid-s10861" xml:space="preserve"> [ut demõ-<lb/>ſtratum eſt 60 n.</s> <s xml:id="echoid-s10862" xml:space="preserve">] Fiat aũt ei ęqualis [ք 23 p 1] ꝗ ſit d t l.</s> <s xml:id="echoid-s10863" xml:space="preserve"> Palã [per 12 n <lb/>4] quòd l reflectetur ad e à puncto t:</s> <s xml:id="echoid-s10864" xml:space="preserve"> & eius imago erit z [ք 6 n.</s> <s xml:id="echoid-s10865" xml:space="preserve">] Simi <lb/>liter ſumpto l puncto:</s> <s xml:id="echoid-s10866" xml:space="preserve"> patebit quod eſt locus imaginis.</s> <s xml:id="echoid-s10867" xml:space="preserve"> Ducatur.</s> <s xml:id="echoid-s10868" xml:space="preserve"> n.</s> <s xml:id="echoid-s10869" xml:space="preserve"> li-<lb/>neale uſq;</s> <s xml:id="echoid-s10870" xml:space="preserve"> in b punctũ circuli:</s> <s xml:id="echoid-s10871" xml:space="preserve"> & ducatur linea b d:</s> <s xml:id="echoid-s10872" xml:space="preserve"> erit [ut prius] an-<lb/>gulus e b d acutus.</s> <s xml:id="echoid-s10873" xml:space="preserve"> Fiat ei ęqualis:</s> <s xml:id="echoid-s10874" xml:space="preserve"> qui ſit d b p:</s> <s xml:id="echoid-s10875" xml:space="preserve"> reflectetur quidẽ pun-<lb/>ctum p ad e à puncto b:</s> <s xml:id="echoid-s10876" xml:space="preserve"> [per 12 n 4] & locus imaginis eius erit l:</s> <s xml:id="echoid-s10877" xml:space="preserve"> [per <lb/>6 n.</s> <s xml:id="echoid-s10878" xml:space="preserve">] Et ita ſumpto quocunq;</s> <s xml:id="echoid-s10879" xml:space="preserve"> alio puncto:</s> <s xml:id="echoid-s10880" xml:space="preserve"> erit eadem probatio.</s> <s xml:id="echoid-s10881" xml:space="preserve"/> </p> <div xml:id="echoid-div391" type="float" level="0" n="0"> <figure xlink:label="fig-0172-01" xlink:href="fig-0172-01a"> <variables xml:id="echoid-variables98" xml:space="preserve">z t a l m e d b p g</variables> </figure> </div> </div> <div xml:id="echoid-div393" type="section" level="0" n="0"> <head xml:id="echoid-head372" xml:space="preserve" style="it">69. Si uiſu et uiſibili in eadẽ diametro circuli (ꝗ eſt cõmunis ſectio <lb/>ſuperficierũ, reflexionis & ſpeculi ſphærici caui) ſitis: imago uidea <lb/>tur in ipſo uiſu: ab uno ſemicirculi, uel à quolibet alteri{us} definiti <lb/>circuli puncto poteſt ad uiſum reflexio fieri. 23 p 8.</head> <p> <s xml:id="echoid-s10882" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s10883" xml:space="preserve"> punctorum, quæ cõprehẽduntur in his ſpeculis:</s> <s xml:id="echoid-s10884" xml:space="preserve"> quo-<lb/>rundã imagines quaturor loca ſortiuntur:</s> <s xml:id="echoid-s10885" xml:space="preserve"> quorundã tria:</s> <s xml:id="echoid-s10886" xml:space="preserve"> quo-<lb/>rundã duo:</s> <s xml:id="echoid-s10887" xml:space="preserve"> quorundã unum.</s> <s xml:id="echoid-s10888" xml:space="preserve"> Punctũ, cuius imago in quatuor <lb/>ceciderit loca:</s> <s xml:id="echoid-s10889" xml:space="preserve"> â quatuor pũctis determinatis reflectitur, nõ ab alijs, <lb/>uel plurib.</s> <s xml:id="echoid-s10890" xml:space="preserve"> Punctum, cuius imago tria ſibi uſurpat loca:</s> <s xml:id="echoid-s10891" xml:space="preserve"> à tribus pun-<lb/>ctis ſpeculi reflectitur, nõ à plurib.</s> <s xml:id="echoid-s10892" xml:space="preserve"> cuius duo:</s> <s xml:id="echoid-s10893" xml:space="preserve"> à duobus.</s> <s xml:id="echoid-s10894" xml:space="preserve"> Puncti aũt, cuius imago in unicum caditlo-<lb/>cũ:</s> <s xml:id="echoid-s10895" xml:space="preserve"> poterit eſſe:</s> <s xml:id="echoid-s10896" xml:space="preserve"> quòd ab uno tãtùm puncto ſit reflexio:</s> <s xml:id="echoid-s10897" xml:space="preserve"> & poterit eſſe:</s> <s xml:id="echoid-s10898" xml:space="preserve"> quòd à quolibet circuli deter-<lb/>minati puncto, non ab alio.</s> <s xml:id="echoid-s10899" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s10900" xml:space="preserve"> ſit e cẽtrum uiſus:</s> <s xml:id="echoid-s10901" xml:space="preserve"> h ſit punctũ uiſum in eadẽ diametro:</s> <s xml:id="echoid-s10902" xml:space="preserve"> d ſit <lb/>centrum circuli.</s> <s xml:id="echoid-s10903" xml:space="preserve"> Ducatur diameter z e h a:</s> <s xml:id="echoid-s10904" xml:space="preserve"> aut e d eſt æqualis d h:</s> <s xml:id="echoid-s10905" xml:space="preserve"> aut nõ.</s> <s xml:id="echoid-s10906" xml:space="preserve"> Sit æqualis:</s> <s xml:id="echoid-s10907" xml:space="preserve"> & ſuper e h du <lb/>catur à puncto d perpendiculariter diameter g d b:</s> <s xml:id="echoid-s10908" xml:space="preserve"> & ducãtur lineę h g, g e, h b, b e.</s> <s xml:id="echoid-s10909" xml:space="preserve"> Palã [per 4 p 1] <lb/>quòd triangulũ h g d æquale triãgulo e d g, & ęqua-<lb/> <anchor type="figure" xlink:label="fig-0172-02a" xlink:href="fig-0172-02"/> le triãgulo h b d, & triãgulo e b d.</s> <s xml:id="echoid-s10910" xml:space="preserve"> Palàm, quòd, cum <lb/>angulus h g e diuiſus ſit per ęqualia, h à puncto g re <lb/>flectetur ad e:</s> <s xml:id="echoid-s10911" xml:space="preserve"> [per 12 n 4] & locus imaginis eius eſt <lb/>e [per 6 n.</s> <s xml:id="echoid-s10912" xml:space="preserve">] Similiter h à puncto b reflectetur ad e:</s> <s xml:id="echoid-s10913" xml:space="preserve"> <lb/>& locus imaginis eius e.</s> <s xml:id="echoid-s10914" xml:space="preserve"> Si igitur diametro z e h a <lb/>immota, moueatur ſemicirculus a g z per ſphæram <lb/>ſpeculi aut ſolũ triangulũ h g e:</s> <s xml:id="echoid-s10915" xml:space="preserve"> deſcribet quidẽ pun <lb/>ctum g motu ſuo circulũ:</s> <s xml:id="echoid-s10916" xml:space="preserve"> & à quolibet puncto circu <lb/>li reflectetur h ad e:</s> <s xml:id="echoid-s10917" xml:space="preserve"> & locus imaginis eius ſemper <lb/>erit punctũ e.</s> <s xml:id="echoid-s10918" xml:space="preserve"> Et ita patet propoſitum.</s> <s xml:id="echoid-s10919" xml:space="preserve"> Quòd aũt ab <lb/>alio puncto, ꝗ̃ aliquo illius circuli, nõ poſsit fieri re-<lb/>flexio puncti h ad e:</s> <s xml:id="echoid-s10920" xml:space="preserve"> palã per hoc.</s> <s xml:id="echoid-s10921" xml:space="preserve"> Sumatur punctũ <lb/>c, & ducatur e c, c h:</s> <s xml:id="echoid-s10922" xml:space="preserve"> erit quidẽ [per 7 p 3] e c maior <lb/>linea e g, & linea h c minor linea h g.</s> <s xml:id="echoid-s10923" xml:space="preserve"> Quare non erit <lb/>proportio e c ad c h, ſicut e d ad d h.</s> <s xml:id="echoid-s10924" xml:space="preserve"> [Quia.</s> <s xml:id="echoid-s10925" xml:space="preserve"> n.</s> <s xml:id="echoid-s10926" xml:space="preserve"> ex the <lb/>ſi punctum h reflectitur ad e à puncto g:</s> <s xml:id="echoid-s10927" xml:space="preserve"> erit per 12 <lb/>n 4.</s> <s xml:id="echoid-s10928" xml:space="preserve"> 3 p 6 e g ad g h, ſicut e d ad d h.</s> <s xml:id="echoid-s10929" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s10930" xml:space="preserve"> cum e c, h c <lb/>ſint inęquales ipſis e g, g h:</s> <s xml:id="echoid-s10931" xml:space="preserve"> nõ erunt proportionales ipſis e d, d h.</s> <s xml:id="echoid-s10932" xml:space="preserve">] Igitur [per 3 p 6] linea d c nõ diui-<lb/>det angulum e c h per æqualia.</s> <s xml:id="echoid-s10933" xml:space="preserve"> Quare h à puncto c non põtreflecti ad e.</s> <s xml:id="echoid-s10934" xml:space="preserve"> [Idẽ breuius cõcludetur ք <lb/>4 ꝓ.</s> <s xml:id="echoid-s10935" xml:space="preserve"> geometrię Iordani.</s> <s xml:id="echoid-s10936" xml:space="preserve"> Quia.</s> <s xml:id="echoid-s10937" xml:space="preserve"> n.</s> <s xml:id="echoid-s10938" xml:space="preserve"> triãguli e c h latera e c, h c ſuntinęqualia:</s> <s xml:id="echoid-s10939" xml:space="preserve"> & recta c d ab ipſorũ angu <lb/>lo eſt in mediũ baſis e h ex theſi:</s> <s xml:id="echoid-s10940" xml:space="preserve"> erit ք allegatã 4 ꝓpoſitionẽ angulus e c d minorangulo h c d.</s> <s xml:id="echoid-s10941" xml:space="preserve"> Qua <lb/>re h à puncto c ad uiſum e nõ reflectetur.</s> <s xml:id="echoid-s10942" xml:space="preserve"> Quòd aũt e c, h c latera ſint inæqualia, patet:</s> <s xml:id="echoid-s10943" xml:space="preserve"> quia per 7 p 3 e <lb/>c maior eſt e g, id eſt, h g (ęquales.</s> <s xml:id="echoid-s10944" xml:space="preserve"> n.</s> <s xml:id="echoid-s10945" xml:space="preserve"> ſunt è cõcluſo) & h g maior h c:</s> <s xml:id="echoid-s10946" xml:space="preserve"> erit e c multò maior h c.</s> <s xml:id="echoid-s10947" xml:space="preserve">] Eadem <lb/>erit probatio, ſi ſumatur c inter g & z.</s> <s xml:id="echoid-s10948" xml:space="preserve"> Si uerò e d fuerit maior d h:</s> <s xml:id="echoid-s10949" xml:space="preserve"> mutetur figura:</s> <s xml:id="echoid-s10950" xml:space="preserve"> & addatur lineæ <lb/>d h, linea h q, ut productum ex e q in q h, ſit æquale quadrato d q.</s> <s xml:id="echoid-s10951" xml:space="preserve"> Erit igitur proportio e q ad d q, ſi-<lb/>cut d q ad h q, ſicut probat Euclides [17 p 6.</s> <s xml:id="echoid-s10952" xml:space="preserve">] Fiat circulus ad quãtitatẽ ſemidiametri q d:</s> <s xml:id="echoid-s10953" xml:space="preserve"> cuius q cẽ <lb/>trum:</s> <s xml:id="echoid-s10954" xml:space="preserve"> g, b loca ſectionis duorũ circulorum:</s> <s xml:id="echoid-s10955" xml:space="preserve"> & ducãtur lineę e g, e b, q g, q b, d g, d b, h g, h b.</s> <s xml:id="echoid-s10956" xml:space="preserve"> Palã ergo, <lb/>quòd erit proportio e q ad q g, ſicut q g ad q h:</s> <s xml:id="echoid-s10957" xml:space="preserve"> [ęquales enim ſunt q g, d q ք 15 d 1:</s> <s xml:id="echoid-s10958" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s10959" xml:space="preserve"> e q ad d q & q <lb/>g eãdẽ habet rationẽ ք 7 p 5:</s> <s xml:id="echoid-s10960" xml:space="preserve"> & iã patuit, ut e q ad d q, ſic d q ad h q:</s> <s xml:id="echoid-s10961" xml:space="preserve"> ergo per 7 p 5, ute q ad g q, ſic g q <lb/>ad h q] & angulus g q h cõmunis utriq;</s> <s xml:id="echoid-s10962" xml:space="preserve"> triãgulo e q g, h q g.</s> <s xml:id="echoid-s10963" xml:space="preserve"> Igitur illa duo triãgula ſunt ſimilia.</s> <s xml:id="echoid-s10964" xml:space="preserve"> [ք 6.</s> <s xml:id="echoid-s10965" xml:space="preserve"> <lb/>4 p.</s> <s xml:id="echoid-s10966" xml:space="preserve"> 1 d 6.</s> <s xml:id="echoid-s10967" xml:space="preserve">] Erit igitur proportio e q ad q g, ſicute e g ad g h.</s> <s xml:id="echoid-s10968" xml:space="preserve"> Erit igitur e d ad d h, ſicut e g ad g h.</s> <s xml:id="echoid-s10969" xml:space="preserve"> [oſten <lb/>ſũ.</s> <s xml:id="echoid-s10970" xml:space="preserve"> n.</s> <s xml:id="echoid-s10971" xml:space="preserve"> eſt, ut tota e q ad totã d q, ſic ablata d q ad ablatã h q:</s> <s xml:id="echoid-s10972" xml:space="preserve"> ergo ք 19 p 5, uttota e q ad totã d q, id eſt q <lb/>g.</s> <s xml:id="echoid-s10973" xml:space="preserve"> ſic reliqua e d ad reliquã d h:</s> <s xml:id="echoid-s10974" xml:space="preserve"> ſed ut e q ad q g, ſic e g ad g h:</s> <s xml:id="echoid-s10975" xml:space="preserve"> ergo ք 11 p 5 ut e d ad d h, ſice g ad g h.</s> <s xml:id="echoid-s10976" xml:space="preserve">] <lb/>Quare [ք 3 p 6] linea d g diuidet angulũ e g h ք ęqualia.</s> <s xml:id="echoid-s10977" xml:space="preserve"> Vnde punctũ h à pũcto g reflectetur ad e:</s> <s xml:id="echoid-s10978" xml:space="preserve"> [ք <lb/>12 n 4] & locus imaginis eius pũctũ e [ք 6 n.</s> <s xml:id="echoid-s10979" xml:space="preserve">] Similiter h à pũcto b reflectetur ad e:</s> <s xml:id="echoid-s10980" xml:space="preserve"> & locus imaginis <lb/>eſt pũctũ e.</s> <s xml:id="echoid-s10981" xml:space="preserve"> Si ergo moueatur triãgulũ e g h, pũctis e, h immotis:</s> <s xml:id="echoid-s10982" xml:space="preserve"> pũctũ g deſcribet in ſphęra circulũ, <lb/>à cuius quolibet pũcto reflectetur h ad e:</s> <s xml:id="echoid-s10983" xml:space="preserve"> & ſemք erit locus imaginis e.</s> <s xml:id="echoid-s10984" xml:space="preserve"> Et qđ ab alio pũcto, ꝗ̃ aliquo <lb/>illius circuli, nõ poſsit h reflecti ad e:</s> <s xml:id="echoid-s10985" xml:space="preserve"> palã, ut prius.</s> <s xml:id="echoid-s10986" xml:space="preserve"> Si.</s> <s xml:id="echoid-s10987" xml:space="preserve"> n.</s> <s xml:id="echoid-s10988" xml:space="preserve"> ſumatur c inter g & a:</s> <s xml:id="echoid-s10989" xml:space="preserve"> erit e c maior e g, & h c <lb/> <pb o="167" file="0173" n="173" rhead="OPTICAE LIBER V."/> minor h g:</s> <s xml:id="echoid-s10990" xml:space="preserve"> [per 7 p 3] non ergo erit ꝓportio e c ad h c, ſicut e d ad d h:</s> <s xml:id="echoid-s10991" xml:space="preserve"> & ita [ք 3 p 6] d c nó diuidet an <lb/>gulũ e c h ք ęqualia.</s> <s xml:id="echoid-s10992" xml:space="preserve"> Similiter, ſic ſumatur inter g & z, poterit improbari.</s> <s xml:id="echoid-s10993" xml:space="preserve"> Et ita patet ꝓpoſitũ.</s> <s xml:id="echoid-s10994" xml:space="preserve"> Notã-<lb/>dum tñ, quòd e eſt punctum intellectuale:</s> <s xml:id="echoid-s10995" xml:space="preserve"> & circulus ille (cuius e eſt polus) eſt circulus intellectua-<lb/>lis:</s> <s xml:id="echoid-s10996" xml:space="preserve"> & h punctũ intellectuale.</s> <s xml:id="echoid-s10997" xml:space="preserve"> Vn-<lb/> <anchor type="figure" xlink:label="fig-0173-01a" xlink:href="fig-0173-01"/> de, quod dictũ eſt, ſecundũ geo-<lb/>metricã demõſtrationẽ eſt intel-<lb/>ligendum, nõ ſecundum uiſus ꝓ <lb/>bationẽ:</s> <s xml:id="echoid-s10998" xml:space="preserve"> cũ intellectualia uiſum <lb/>lateant.</s> <s xml:id="echoid-s10999" xml:space="preserve"> Sed quoniã forma h con <lb/>tinua uidetur formis aliorũ pun <lb/>ctorũ:</s> <s xml:id="echoid-s11000" xml:space="preserve"> uidebitur quidẽ à uiſu for <lb/>ma, cuius punctũ medium h:</s> <s xml:id="echoid-s11001" xml:space="preserve"> & <lb/>locus puncti medij illius formæ <lb/>erit e:</s> <s xml:id="echoid-s11002" xml:space="preserve"> & reflectetur h forma à lo-<lb/>co ſpeculi circulari, cuiusmediũ <lb/>erit circulus p̃dictus, & e polus <lb/>eius.</s> <s xml:id="echoid-s11003" xml:space="preserve"> Cum aũt e d fuerit maior d <lb/>h:</s> <s xml:id="echoid-s11004" xml:space="preserve"> in tãtum poterit eſſe maior, ut <lb/>nõ reflectatur h ad e à puncto g.</s> <s xml:id="echoid-s11005" xml:space="preserve"> <lb/>Sciendum, quòd, niſi fuerit pro-<lb/>portio e a ad a h maior, quã e d <lb/>ad d h:</s> <s xml:id="echoid-s11006" xml:space="preserve"> nõ poterit h reflecti ad e.</s> <s xml:id="echoid-s11007" xml:space="preserve"> <lb/>Si enim poteſt reflecti:</s> <s xml:id="echoid-s11008" xml:space="preserve"> reflectatur à puncto:</s> <s xml:id="echoid-s11009" xml:space="preserve"> quod ſit g:</s> <s xml:id="echoid-s11010" xml:space="preserve"> erit quidem g d h minor recto, cũ reſpiciat ſe-<lb/>ctionem minorẽ quarta.</s> <s xml:id="echoid-s11011" xml:space="preserve"> [quadrans enim peripherię ab angulo recto in cẽtro ſubtẽditur per 33 p 6.</s> <s xml:id="echoid-s11012" xml:space="preserve"> <lb/>Vel angulus g d h minor eſt recto, quia ſemidiametro q d & recta g d cõprehenditur, ut demõſtratũ <lb/>eſt 60 n.</s> <s xml:id="echoid-s11013" xml:space="preserve">] Ducatur à puncto g cõtingens [per 17 p 3] quę neceſſariò cõcurret cũ e a:</s> <s xml:id="echoid-s11014" xml:space="preserve"> [per 11 ax:</s> <s xml:id="echoid-s11015" xml:space="preserve"> quia <lb/>anguli interiores ad g & d ſunt minores duobus rectis:</s> <s xml:id="echoid-s11016" xml:space="preserve"> cum angulus ad g ſit rectus per 18 p 3, ad d ue <lb/>rò acutus] ſit cõcurſus f.</s> <s xml:id="echoid-s11017" xml:space="preserve"> Erit quidẽ proportio e f ad f h, ſicut e d ad d h:</s> <s xml:id="echoid-s11018" xml:space="preserve"> [eſt enim per 64 n d h ad d e, <lb/>ſicut h fad e f:</s> <s xml:id="echoid-s11019" xml:space="preserve"> & per cõſectariũ 4 p 5, ut e f ad f h, ſic e d ad d h] ſed maior eſt proportio e a ad a h, quã <lb/>e f ad fh.</s> <s xml:id="echoid-s11020" xml:space="preserve"> [Quia enim a h minor eſt h f:</s> <s xml:id="echoid-s11021" xml:space="preserve"> erit ratio e h ad a h maior, quã ad h f per 8 p 5:</s> <s xml:id="echoid-s11022" xml:space="preserve"> & per 18 p 5, e a <lb/>ad a h maior, quã e f ad h f.</s> <s xml:id="echoid-s11023" xml:space="preserve">] Igitur maior eſt e a ad a h, ꝗ̃ e d ad d h:</s> <s xml:id="echoid-s11024" xml:space="preserve"> & ita neceſſariò:</s> <s xml:id="echoid-s11025" xml:space="preserve"> ſi h reflectitur ad <lb/>e:</s> <s xml:id="echoid-s11026" xml:space="preserve"> erit proportio e a ad a h maior, quàm e d ad d h.</s> <s xml:id="echoid-s11027" xml:space="preserve"> Patent ergo, quæ dicta ſunt:</s> <s xml:id="echoid-s11028" xml:space="preserve"> cum centrum uiſus & <lb/>punctum uiſum fuerint in eadem diametro.</s> <s xml:id="echoid-s11029" xml:space="preserve"/> </p> <div xml:id="echoid-div393" type="float" level="0" n="0"> <figure xlink:label="fig-0172-02" xlink:href="fig-0172-02a"> <variables xml:id="echoid-variables99" xml:space="preserve">g c z e d h a b</variables> </figure> <figure xlink:label="fig-0173-01" xlink:href="fig-0173-01a"> <variables xml:id="echoid-variables100" xml:space="preserve">g c f q a h d e z b</variables> </figure> </div> </div> <div xml:id="echoid-div395" type="section" level="0" n="0"> <head xml:id="echoid-head373" xml:space="preserve" style="it">70. Viſu & uiſibili extra circulum (qui eſt cõmunis ſectio ſuperficierũ, reflexionis & ſpeculi <lb/>ſphærici caui) ſitis in diuerſis diametris: ab uno puncto fit reflexio, et una uidetur imago. 24 p 8.</head> <p> <s xml:id="echoid-s11030" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s11031" xml:space="preserve"> cum punctũ uiſum & centrũ uiſus non fuerint in eadẽ diametro, & fuerint extra ſpe <lb/>culum:</s> <s xml:id="echoid-s11032" xml:space="preserve"> non reflectetur punctũ uiſum ad centrũ uiſus, niſi ab uno tantùm ſpeculi puncto.</s> <s xml:id="echoid-s11033" xml:space="preserve"> Ver <lb/>bi gratia:</s> <s xml:id="echoid-s11034" xml:space="preserve"> ſit t punctũ uiſum:</s> <s xml:id="echoid-s11035" xml:space="preserve"> h centrũ uiſus:</s> <s xml:id="echoid-s11036" xml:space="preserve"> d centrũ ſphęræ:</s> <s xml:id="echoid-s11037" xml:space="preserve"> & ducãtur lineę h d, d t, h t.</s> <s xml:id="echoid-s11038" xml:space="preserve"> Suքfi <lb/>cies quidẽ h d t ſecat ſphærã ſuper circulum:</s> <s xml:id="echoid-s11039" xml:space="preserve"> [per 1 th.</s> <s xml:id="echoid-s11040" xml:space="preserve"> 1 ſphęr.</s> <s xml:id="echoid-s11041" xml:space="preserve">] qui ſit e b q g.</s> <s xml:id="echoid-s11042" xml:space="preserve"> Palàm, quòd t non refle <lb/>ctetur ad h, niſi ab aliquo puncto huius circuli:</s> <s xml:id="echoid-s11043" xml:space="preserve"> [quia ipſe eſt reflexionis ſuperficies.</s> <s xml:id="echoid-s11044" xml:space="preserve">] Producãtur er <lb/>go h d, t d uſq;</s> <s xml:id="echoid-s11045" xml:space="preserve"> ad circumferentiã circuli.</s> <s xml:id="echoid-s11046" xml:space="preserve"> Palã, quòd nõ reflectetur ab <lb/> <anchor type="figure" xlink:label="fig-0173-02a" xlink:href="fig-0173-02"/> arcu q g, uel b a ſecundum modũ prędictum [66 n.</s> <s xml:id="echoid-s11047" xml:space="preserve">] Reflectetur ergo <lb/>aut ab arcu g b:</s> <s xml:id="echoid-s11048" xml:space="preserve"> aut a q.</s> <s xml:id="echoid-s11049" xml:space="preserve"> Diuidatur [per 9 p 1] angulus t d h per ęqua-<lb/>lia, per lineã l e d z:</s> <s xml:id="echoid-s11050" xml:space="preserve"> & à puncto e ducatur contingens:</s> <s xml:id="echoid-s11051" xml:space="preserve"> [per 17 p 3] quę <lb/>ſit k e f.</s> <s xml:id="echoid-s11052" xml:space="preserve"> Si puncta t, h fuerint ſuper illam contingentẽ:</s> <s xml:id="echoid-s11053" xml:space="preserve"> nõ reflectetur <lb/>t ad h ab aliquo pũcto arcus b g.</s> <s xml:id="echoid-s11054" xml:space="preserve"> Cũ enim à puncto t ducetur linea ad <lb/>aliquod interius punctũ huius arcus:</s> <s xml:id="echoid-s11055" xml:space="preserve"> linea à puncto h ad idẽ punctũ <lb/>ducta, cadet ſuper ipſum exterius, interius nõ.</s> <s xml:id="echoid-s11056" xml:space="preserve"> Et ideo non erit refle-<lb/>xio [à caua ſpeculi ſuperficie.</s> <s xml:id="echoid-s11057" xml:space="preserve">] Et quòd ab uno puncto tãtùm arcus <lb/>a q fiat reflexio:</s> <s xml:id="echoid-s11058" xml:space="preserve"> palã erit ex hoc.</s> <s xml:id="echoid-s11059" xml:space="preserve"> Ducãtur enim lineæ t z, h z.</s> <s xml:id="echoid-s11060" xml:space="preserve"> Cũ angu <lb/>lus t d h diuiſus ſit per ęqualia:</s> <s xml:id="echoid-s11061" xml:space="preserve"> erit t d z ęqualis angulo h d z.</s> <s xml:id="echoid-s11062" xml:space="preserve"> [per 13 p <lb/>1.</s> <s xml:id="echoid-s11063" xml:space="preserve">] Lineæ igitur t d, h d aut ſunt æquales:</s> <s xml:id="echoid-s11064" xml:space="preserve"> aut nõ ſunt æquales.</s> <s xml:id="echoid-s11065" xml:space="preserve"> Si ſunt <lb/>æquales, & d z cõmunis:</s> <s xml:id="echoid-s11066" xml:space="preserve"> erit [per 4 p 1] triãgulũ t z d æquale triangu <lb/>lo h z d:</s> <s xml:id="echoid-s11067" xml:space="preserve"> & angulus t z h diuiſus per ęqualia, per lineã d z.</s> <s xml:id="echoid-s11068" xml:space="preserve"> Et ita t refle-<lb/>cterur ad h à puncto z.</s> <s xml:id="echoid-s11069" xml:space="preserve"> [per 12 n.</s> <s xml:id="echoid-s11070" xml:space="preserve">] Quòd aũt ab alio puncto nõ poſsit:</s> <s xml:id="echoid-s11071" xml:space="preserve"> <lb/>ſic cõſtabit.</s> <s xml:id="echoid-s11072" xml:space="preserve"> Sumatur punctũ o:</s> <s xml:id="echoid-s11073" xml:space="preserve"> & ducãtur lineæ t o, h o:</s> <s xml:id="echoid-s11074" xml:space="preserve"> & linea o d m <lb/>per cẽtrum d diuidat angulum illum per ęqualia.</s> <s xml:id="echoid-s11075" xml:space="preserve"> Planũ [per 8 p 3] qđ <lb/>t z minor eſt t o, & h o minor h z:</s> <s xml:id="echoid-s11076" xml:space="preserve"> & proportio t z ad h z, ſicut t l ad l h:</s> <s xml:id="echoid-s11077" xml:space="preserve"> <lb/>[per 3 p 6:</s> <s xml:id="echoid-s11078" xml:space="preserve"> eſt enim angulus t z h bifariã ſectus à recta linea z l] & erit <lb/>[per eandẽ] proportio t o ad h o, ſicut t m ad m h:</s> <s xml:id="echoid-s11079" xml:space="preserve"> ſed minor eſt ꝓpor <lb/>tio h o ad t o, quã h z ad t z.</s> <s xml:id="echoid-s11080" xml:space="preserve"> [quia enim è quatuor lineis h o, t o, h z, t z prima minor eſt quã tertia, ſe-<lb/>cunda maior ꝗ̃ quarta:</s> <s xml:id="echoid-s11081" xml:space="preserve"> erit ratio primæ ad ſecundã minor, ꝗ̃ tertię ad quartã, ut patet ex 8 p 5] Ergo <lb/>[per 11 p 5] minor eſt proportio h m ad m t, ꝗ̃ h l ad l t:</s> <s xml:id="echoid-s11082" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s11083" xml:space="preserve"> [Nã cum è quatuor lineis <lb/>h m, m t, h l, l t prima h m maior ſit, ꝗ̃ tertia h l:</s> <s xml:id="echoid-s11084" xml:space="preserve"> ſecũda uerò m t minor, ꝗ̃ quarta l t:</s> <s xml:id="echoid-s11085" xml:space="preserve"> erit ratio h m ad m t <lb/> <pb o="168" file="0174" n="174" rhead="ALHAZEN"/> maior, ꝗ̃ h l ad l t, ut cõſtat ex 8 p 5.</s> <s xml:id="echoid-s11086" xml:space="preserve">] Palã igitur, quòd ſit & h ęqualiter diſtẽt à cẽtro, & fuerint ſuper <lb/>contingentẽ:</s> <s xml:id="echoid-s11087" xml:space="preserve"> non reflectetur t ad h, niſi ab uno ſpeculi puncto tãtùm:</s> <s xml:id="echoid-s11088" xml:space="preserve"> & unicus erit eius imaginis lo <lb/>cus.</s> <s xml:id="echoid-s11089" xml:space="preserve"> Si uerò t d, h d ſunt inæquales:</s> <s xml:id="echoid-s11090" xml:space="preserve"> ſecentur ad æqualitatẽ [per 3 p 1] & fiat demonſtratio, ut antea.</s> <s xml:id="echoid-s11091" xml:space="preserve"/> </p> <div xml:id="echoid-div395" type="float" level="0" n="0"> <figure xlink:label="fig-0173-02" xlink:href="fig-0173-02a"> <variables xml:id="echoid-variables101" xml:space="preserve">h l m t k g e b f d p q o z a</variables> </figure> </div> </div> <div xml:id="echoid-div397" type="section" level="0" n="0"> <head xml:id="echoid-head374" xml:space="preserve" style="it">71. Si angulum comprebẽſum à duab{us} diametris, in centro circuli (qui eſt communis ſectio <lb/>ſuperficierum, reflexionis & ſpeculi ſphærici caui) tertia bifariã ſecet: & ab eius termino in pe-<lb/>ripheria dicto angulo ſubtenſa, ſint perpendiculares ſuper dictas diametros: puncta diametro-<lb/>rum, tum in quæ perpendiculares cadunt: tũ citr a hæc, à ſpeculi centro æquabiliter diſtãtia, à <lb/>ſecantis diametri terminis tantùm inter ſe mutuò reflectẽtur: duaś babebũt imagines. 25 p 8.</head> <p> <s xml:id="echoid-s11092" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s11093" xml:space="preserve"> b d q, a d g ſint duæ diametri ſphæræ:</s> <s xml:id="echoid-s11094" xml:space="preserve"> & diameter e d z diuidat angulũ b d g per ęqua-<lb/>lia:</s> <s xml:id="echoid-s11095" xml:space="preserve"> & à puncto e ducãtur duæ perpendiculares, ſuper duas diametros b d, d g:</s> <s xml:id="echoid-s11096" xml:space="preserve"> [per 12 p 1] quę <lb/>ſint e t, e h.</s> <s xml:id="echoid-s11097" xml:space="preserve"> Palàm [per 26 p 1] quòd triangulũ e t d æquale eſt triangulo e h d, & angulus t e d <lb/>angulo h e d, latusq́;</s> <s xml:id="echoid-s11098" xml:space="preserve"> t d lateri h d, & latus e t lateri e h:</s> <s xml:id="echoid-s11099" xml:space="preserve"> cũ e d ſit cõmunis utriq;</s> <s xml:id="echoid-s11100" xml:space="preserve">. tigitur reflectetur ad <lb/>h à puncto e.</s> <s xml:id="echoid-s11101" xml:space="preserve"> [per 12 n 4.</s> <s xml:id="echoid-s11102" xml:space="preserve">] Eodẽ modo à puncto z [Quia enim angulus b d g bifariã ſectus eſt per li-<lb/>neã e d:</s> <s xml:id="echoid-s11103" xml:space="preserve"> erit angulus t d z æqualis angulo h d z per 13 p 1, & t d æquatur ex cõcluſo ipſi h d, latusq́;</s> <s xml:id="echoid-s11104" xml:space="preserve"> d z <lb/>cõmune:</s> <s xml:id="echoid-s11105" xml:space="preserve"> angulus igitur t z d æquatur angulo h z d per 4 p 1.</s> <s xml:id="echoid-s11106" xml:space="preserve"> Quare per 12 n 4 t & h reflectẽtur inter <lb/>ſe à puncto z.</s> <s xml:id="echoid-s11107" xml:space="preserve">] Et palã [per 66 n] quòd t non reflectetur ad h, ab aliquo puncto arcus a b, uel arcus g <lb/>q:</s> <s xml:id="echoid-s11108" xml:space="preserve"> nec reflectetur ab alio puncto arcus a q, ꝗ̃ à puncto z ſecundũ ſupradictam probationẽ:</s> <s xml:id="echoid-s11109" xml:space="preserve"> [numero <lb/>præcedẽte.</s> <s xml:id="echoid-s11110" xml:space="preserve">] Verũ quòd ab alio puncto arcus b g, ꝗ̃ à puncto e, nõ poſsit reflecti:</s> <s xml:id="echoid-s11111" xml:space="preserve"> patebit ſic.</s> <s xml:id="echoid-s11112" xml:space="preserve"> Detur o <lb/>punctũ:</s> <s xml:id="echoid-s11113" xml:space="preserve"> & ducantur lineę o d, h o, t o:</s> <s xml:id="echoid-s11114" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s11115" xml:space="preserve"> circulus ad quantitatẽ lineæ d e, trãſiens per tria puncta, <lb/>t, d, h:</s> <s xml:id="echoid-s11116" xml:space="preserve"> [tranſibit aũt per conuerſionẽ 31 p 3 demonſtratã à Theone in cõm entarijs in 3 librũ magnę cõ <lb/>ſtructionis Ptolemęi, & à Cãpano ad 31 p 13:</s> <s xml:id="echoid-s11117" xml:space="preserve">] cuius quidẽ circuli linea d e erit diameter:</s> <s xml:id="echoid-s11118" xml:space="preserve"> cũ angulus e <lb/>t d, quẽ reſpicit, ſit rectus Igitur circulus ille tranſibit per punctũ e.</s> <s xml:id="echoid-s11119" xml:space="preserve"> Cum igitur e ſit cõmunis utriq;</s> <s xml:id="echoid-s11120" xml:space="preserve"> <lb/>circulo, & ſit ſuper eandẽ diametrum:</s> <s xml:id="echoid-s11121" xml:space="preserve"> continget circulus minor maiorem in puncto e:</s> <s xml:id="echoid-s11122" xml:space="preserve"> ſicut probat <lb/>Euclidis [13 p 3.</s> <s xml:id="echoid-s11123" xml:space="preserve">] Igitur circulus iſte ſecabit lineam d o, [ſecus tangeret maiorem circulũ in puncto <lb/>o:</s> <s xml:id="echoid-s11124" xml:space="preserve"> ſicq́;</s> <s xml:id="echoid-s11125" xml:space="preserve"> in duobus punctis e & o tangeret contra 13 p 3] ſecet in puncto l:</s> <s xml:id="echoid-s11126" xml:space="preserve"> & ducantur lineæ t l, h l.</s> <s xml:id="echoid-s11127" xml:space="preserve"> Iam <lb/>patet [è ſuperioribus] quod t d eſt ęqualis h d [ergo per 28 p 3 peripheria t d æquatur peripheriæ h <lb/>d.</s> <s xml:id="echoid-s11128" xml:space="preserve">] Igitur angulus t l d æqualis angulo d l h [per 27 p <lb/> <anchor type="figure" xlink:label="fig-0174-01a" xlink:href="fig-0174-01"/> 3] quia ſuper æquales arcus.</s> <s xml:id="echoid-s11129" xml:space="preserve"> Reſtat [per 13 p 1] t l o æ-<lb/>qualis angulo h l o:</s> <s xml:id="echoid-s11130" xml:space="preserve"> & angulus l o t ęqualis angulo l o <lb/>h ex hypotheſi:</s> <s xml:id="echoid-s11131" xml:space="preserve"> [quia ſunt anguli incidentiæ & refle-<lb/>xionis] & l o commune latus:</s> <s xml:id="echoid-s11132" xml:space="preserve"> erit [per 26 p 1] triangu <lb/>lum t l o æquale triangulo h l o:</s> <s xml:id="echoid-s11133" xml:space="preserve"> & erit t o ęqualis h o:</s> <s xml:id="echoid-s11134" xml:space="preserve"> <lb/>quod eſt impoſsibile:</s> <s xml:id="echoid-s11135" xml:space="preserve"> quoniam [per 7 p 3] h o maior <lb/>h e, & t o minor t e:</s> <s xml:id="echoid-s11136" xml:space="preserve"> & t e, ſicut prius probatum eſt, æ-<lb/>qualis eſt h e:</s> <s xml:id="echoid-s11137" xml:space="preserve"> [linea igitur h o maior eſt linea t o.</s> <s xml:id="echoid-s11138" xml:space="preserve">] Re <lb/>ſtat ergo, ut t nõ reflectatur ad h, ab alio puncto, quã <lb/>ab e uel à z.</s> <s xml:id="echoid-s11139" xml:space="preserve"> Item à puncto e ducatur linea ſuper dia-<lb/>metrum t d:</s> <s xml:id="echoid-s11140" xml:space="preserve"> quæ ſit e m:</s> <s xml:id="echoid-s11141" xml:space="preserve"> & ſecetur à linea h d pars, æ-<lb/>qualis m d:</s> <s xml:id="echoid-s11142" xml:space="preserve"> quæ fit n d:</s> <s xml:id="echoid-s11143" xml:space="preserve"> & ducantur e m, e n.</s> <s xml:id="echoid-s11144" xml:space="preserve"> Palàm <lb/>[per 16 p 1] quòd e m d maior eſt recto:</s> <s xml:id="echoid-s11145" xml:space="preserve"> [quia angu-<lb/>lus e t d rectus eſt per fabricationem] ſecetur ex eo <lb/>æqualis recto per lineam p m [per 23 p 1] quæ cõcur-<lb/>ret cum d e:</s> <s xml:id="echoid-s11146" xml:space="preserve"> [per lemma Procli ad 29 p 1] ſit concur-<lb/>ſus punctum p:</s> <s xml:id="echoid-s11147" xml:space="preserve"> & ducatur n p:</s> <s xml:id="echoid-s11148" xml:space="preserve"> & fiat circulus ad quantitatem p d, tranſiens per tria puncta m, d, n.</s> <s xml:id="echoid-s11149" xml:space="preserve"> <lb/>Cum p m d ſit rectus [ex fabricatione] erit p d diameter [per conſectarium 5 p 4] & tranſibit circu <lb/>lus per p, [ut oſtenſum eſt.</s> <s xml:id="echoid-s11150" xml:space="preserve">] Palàm ergo, quòd m reflectetur ad n à puncto e:</s> <s xml:id="echoid-s11151" xml:space="preserve"> [cum en:</s> <s xml:id="echoid-s11152" xml:space="preserve"> m per 4 p 1 tri <lb/>angulum d m p ſit æquilaterum & æquiangulum triangulo d n p:</s> <s xml:id="echoid-s11153" xml:space="preserve"> æquabitur m p ipſi n p, & angulus <lb/>d p m angulo d p n:</s> <s xml:id="echoid-s11154" xml:space="preserve"> ergo per 13 p 1.</s> <s xml:id="echoid-s11155" xml:space="preserve"> 3 ax.</s> <s xml:id="echoid-s11156" xml:space="preserve"> angulus m p e æquatur angulo n p e, latusq́ue p e commune <lb/>eſt:</s> <s xml:id="echoid-s11157" xml:space="preserve"> angulus igitur m e p æquatur angulo n e p per 4 p 1.</s> <s xml:id="echoid-s11158" xml:space="preserve"> Quare per 12 n 4 m & n à puncto e inter ſe <lb/>mutuò reflectuntur] & ſimiliter à puncto z:</s> <s xml:id="echoid-s11159" xml:space="preserve"> & non ab aliquo puncto arcus a b, uel g q:</s> <s xml:id="echoid-s11160" xml:space="preserve"> [per 66 n.</s> <s xml:id="echoid-s11161" xml:space="preserve">] <lb/>Et palàm, quòd non ab alio puncto arcus a q, quã à puncto z:</s> <s xml:id="echoid-s11162" xml:space="preserve"> & quòd non ab alio puncto arcus b g, <lb/>quàm à puncto e ſecundum modum prædictum.</s> <s xml:id="echoid-s11163" xml:space="preserve"> Sumpto enim puncto, & ductis lineis à punctis t, <lb/>d, h:</s> <s xml:id="echoid-s11164" xml:space="preserve"> & ſumpto puncto, in quo circulus ultimus ſecabit diametrum:</s> <s xml:id="echoid-s11165" xml:space="preserve"> & à punctis ſectionis ductis li-<lb/>neis ad puncta t, h:</s> <s xml:id="echoid-s11166" xml:space="preserve"> eadem erit improbatio, quæ prius.</s> <s xml:id="echoid-s11167" xml:space="preserve"> Palàm ergo ex prædictis:</s> <s xml:id="echoid-s11168" xml:space="preserve"> quòd ſi angulum <lb/>contentum duabus diametris, per æqualia diuidat tertia diameter:</s> <s xml:id="echoid-s11169" xml:space="preserve"> & à termino illius diametri du-<lb/>cantur perpendiculares ad illas diametros:</s> <s xml:id="echoid-s11170" xml:space="preserve"> puncta diametrorum, in quæ cadunt, ad ſe inuicem re-<lb/>flectuntur à duobus punctis ſpeculi tantùm.</s> <s xml:id="echoid-s11171" xml:space="preserve"> P unctorum aũt diametrorum citra hos terminos per-<lb/>pen dicularium ſumptorum, id eſt uerſus centrum:</s> <s xml:id="echoid-s11172" xml:space="preserve"> reflectitur quodlibet à duobus punctis tantùm:</s> <s xml:id="echoid-s11173" xml:space="preserve"> <lb/>& unũ reflectitur ad illud, quod æqualiter diſtat à cẽtro:</s> <s xml:id="echoid-s11174" xml:space="preserve"> & omniũ talium duplex eſt imaginis locus.</s> <s xml:id="echoid-s11175" xml:space="preserve"/> </p> <div xml:id="echoid-div397" type="float" level="0" n="0"> <figure xlink:label="fig-0174-01" xlink:href="fig-0174-01a"> <variables xml:id="echoid-variables102" xml:space="preserve">e p o l g h n d m t b q a z</variables> </figure> </div> </div> <div xml:id="echoid-div399" type="section" level="0" n="0"> <head xml:id="echoid-head375" xml:space="preserve" style="it">72. Si angulũ cõprehenſum à duabus diametris in cẽtro circuli (qui eſt cõmunis ſectio ſuper-<lb/>ficierũ, reflexionis & ſpeculi ſphæricicaui) tertia bifariã ſecet: et ab eius termino in peripheria <lb/>dicto angulo ſubtẽſa, ſint քpẽdiculares ſuք dict{as} diametros: pũcta diametrorũ inter քipheriã et <lb/> <pb o="169" file="0175" n="175" rhead="OPTICAE LIBER V."/> perpendicularium terminos à centro ſpeculi æquabiliter diſtantia, à quatuor peripheriæ pũctis <lb/>inter ſe mutuo reflectentur, & quatuor habebunt imagines. 26 p 8.</head> <p> <s xml:id="echoid-s11176" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s11177" xml:space="preserve"> ſumptis duabus diametris b q, a g:</s> <s xml:id="echoid-s11178" xml:space="preserve"> & e z diuidente angulum earum per æqualia:</s> <s xml:id="echoid-s11179" xml:space="preserve"> ſu-<lb/>matur in b d punctum t ſupra punctum, in quod cadit perpendicularis, ducta à puncto e:</s> <s xml:id="echoid-s11180" xml:space="preserve"> & <lb/>in d g ſumatur d h æqualis d t:</s> <s xml:id="echoid-s11181" xml:space="preserve"> [per 3 p 1] & ducanturte, h e.</s> <s xml:id="echoid-s11182" xml:space="preserve"> Reflectetur quidem t ad h à pun-<lb/>cto e, & ſimiliter à puncto z, non ab alio puncto arcus a q:</s> <s xml:id="echoid-s11183" xml:space="preserve"> nec ab aliquo puncto arcus a b uel g q [per <lb/>66 n.</s> <s xml:id="echoid-s11184" xml:space="preserve">] Deinde à puncto t ducatur perpẽdicularis ſuper t d:</s> <s xml:id="echoid-s11185" xml:space="preserve"> [per 11 p 1] quæ quidem concurret cũ d e <lb/>extra circulum ſphærę, cũ angulus b d e ſit acutus [ut oſtenſum eſt 36 n:</s> <s xml:id="echoid-s11186" xml:space="preserve"> quare d e & perpẽdicularis <lb/>ſuper t d per 11 ax:</s> <s xml:id="echoid-s11187" xml:space="preserve"> concurrent:</s> <s xml:id="echoid-s11188" xml:space="preserve"> & quidem extra circulum b z g.</s> <s xml:id="echoid-s11189" xml:space="preserve"> Quia cum hæc perpẽdicularis, & ea, <lb/>quæ à puncto e ſuper eandem ſemidiametrum d b ducitur, ſint parallelę per 28 p 1:</s> <s xml:id="echoid-s11190" xml:space="preserve"> nunquã cõcurrẽt <lb/>per 35 d 1.</s> <s xml:id="echoid-s11191" xml:space="preserve"> Quare perpendicularis à puncto t continuata, cadet extra circulũ ultra punctũ e.</s> <s xml:id="echoid-s11192" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s11193" xml:space="preserve"> cõ-<lb/>curret cum ſemidiametro e d extra circulum b z g.</s> <s xml:id="echoid-s11194" xml:space="preserve">] Cõcurrat ergo in puncto o:</s> <s xml:id="echoid-s11195" xml:space="preserve"> & ducãtur lineę to, <lb/>h o.</s> <s xml:id="echoid-s11196" xml:space="preserve"> Et fiat circulus tranſiens per tria puncta t, d, h:</s> <s xml:id="echoid-s11197" xml:space="preserve"> qui neceſſariò tranſibit per punctum o [ut pręce-<lb/>dente numero demonſtratum eſt] & erit d o diameter eius:</s> <s xml:id="echoid-s11198" xml:space="preserve"> [per conſectarium 5 p 4] & ducatur li-<lb/>nea cõtingens circulũ b z g, in puncto e [per 17 p 3] quę ſit k e.</s> <s xml:id="echoid-s11199" xml:space="preserve"> Palàm, quòd ultimus circulus ſecabit <lb/>primum, ſcilicet b z g in duobus punctis:</s> <s xml:id="echoid-s11200" xml:space="preserve"> [per 10 p 3] ſint illa puncta l, m:</s> <s xml:id="echoid-s11201" xml:space="preserve"> & ducantur lineæ t l, h l, <lb/>l d, t m, d m, h m.</s> <s xml:id="echoid-s11202" xml:space="preserve"> Cũ ergo arcus t d ſit æqualis arcui h <lb/> <anchor type="figure" xlink:label="fig-0175-01a" xlink:href="fig-0175-01"/> d:</s> <s xml:id="echoid-s11203" xml:space="preserve"> [per 28 p 3:</s> <s xml:id="echoid-s11204" xml:space="preserve"> quia rectę d t, d h ſunt ęquales per fa-<lb/>bricationem] erit [per 27 p 3] angulus t l d æqualis <lb/>angulo d l h.</s> <s xml:id="echoid-s11205" xml:space="preserve"> Et ita t reflectetur ad h à puncto l [per <lb/>12 n 4.</s> <s xml:id="echoid-s11206" xml:space="preserve">] Similiter angulus t m d æqualis angulo d m <lb/>h [per 27 p 3.</s> <s xml:id="echoid-s11207" xml:space="preserve">] Et ita t reflectetur ad h à puncto m.</s> <s xml:id="echoid-s11208" xml:space="preserve"> Pa <lb/>làm igitur, quòd t reflectitur à quatuor pũctis a d h:</s> <s xml:id="echoid-s11209" xml:space="preserve"> <lb/>ſcilicet e, z, l, m:</s> <s xml:id="echoid-s11210" xml:space="preserve"> & quadruplex erit locus imaginis e-<lb/>ius.</s> <s xml:id="echoid-s11211" xml:space="preserve"> Et non poteſt t reflecti ad h ab alio puncto, quã <lb/>ab aliquo iſtorum.</s> <s xml:id="echoid-s11212" xml:space="preserve"> Detur enim f punctum:</s> <s xml:id="echoid-s11213" xml:space="preserve"> & ducan <lb/>tur lineæ t f, h f, d f:</s> <s xml:id="echoid-s11214" xml:space="preserve"> & producatur d f, quouſque cõ-<lb/>currat cum contingente k e:</s> <s xml:id="echoid-s11215" xml:space="preserve"> [concurret autem per <lb/>11 ax:</s> <s xml:id="echoid-s11216" xml:space="preserve"> quia angulus k e d rectus eſt per 18 p 3, & f d e <lb/>acutus, quia pars acuti b d e] & ſit concurſus k:</s> <s xml:id="echoid-s11217" xml:space="preserve"> & du <lb/>cantur lineæ t k, h k.</s> <s xml:id="echoid-s11218" xml:space="preserve"> Igitur angulus t f d æqualis an-<lb/>gulo d f h ex hypotheſi:</s> <s xml:id="echoid-s11219" xml:space="preserve"> [& 12 n 4] reſtat [per 13 p 1] <lb/>angulus t f k æqualis angulo k fh.</s> <s xml:id="echoid-s11220" xml:space="preserve"> Sed angulus t k f <lb/>eſt æqualis angulo f k h [per 27 p.</s> <s xml:id="echoid-s11221" xml:space="preserve"> 3] quia ſuper ęqua <lb/>les arcus:</s> <s xml:id="echoid-s11222" xml:space="preserve"> & f k communis:</s> <s xml:id="echoid-s11223" xml:space="preserve"> erit [per 26 p 1] triangulum æquale triangulo:</s> <s xml:id="echoid-s11224" xml:space="preserve"> & ita t k æqualis k h:</s> <s xml:id="echoid-s11225" xml:space="preserve"> quod <lb/>eſt impoſsibile:</s> <s xml:id="echoid-s11226" xml:space="preserve"> quoniam h k maior h o, & t k minor to [per 7 p 3] & t o ęqualis h o.</s> <s xml:id="echoid-s11227" xml:space="preserve"> [Nam quia recta <lb/>d t æquatur ipſi d h per fabricationem, & angulus t d o ipſi h d o per theſim, & latus o d commune:</s> <s xml:id="echoid-s11228" xml:space="preserve"> <lb/>ergo per 4 p 1 latus t o æquatur lateri h o:</s> <s xml:id="echoid-s11229" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s11230" xml:space="preserve"> t k minor eſt h k.</s> <s xml:id="echoid-s11231" xml:space="preserve">] Palàm igitur, quòd non eſt refle-<lb/>xio ab aliquo puncto, quam à punctis quatuor.</s> <s xml:id="echoid-s11232" xml:space="preserve"> Igitur ſi in diuerſis diametris ſumantur duo puncta, <lb/>ſcilicet t, h, ęqualiter à centro diſtantia:</s> <s xml:id="echoid-s11233" xml:space="preserve"> ſi fuerint ſuper punctis diametrorum, in quę cadunt perpen <lb/>diculares, ductę à termino diametri diuidentis per æqualia angulum duarum diametrorũ:</s> <s xml:id="echoid-s11234" xml:space="preserve"> aut fue-<lb/>rint inter centrum & puncta illa, id eſt citra perpendiculares, dum æqualiter diſtent à centro:</s> <s xml:id="echoid-s11235" xml:space="preserve"> refle-<lb/>ctetur quidem t ad h à duobus punctis tantùm.</s> <s xml:id="echoid-s11236" xml:space="preserve"> Si uerò fuerint t & h à locis perpendicularium uſq;</s> <s xml:id="echoid-s11237" xml:space="preserve"> <lb/>ad circulum:</s> <s xml:id="echoid-s11238" xml:space="preserve"> reflectetur quidem t ad h à quatuor punctis.</s> <s xml:id="echoid-s11239" xml:space="preserve"> Si uerò fuerint in circulo, uel extra:</s> <s xml:id="echoid-s11240" xml:space="preserve"> tamẽ <lb/>citra contingentem k e:</s> <s xml:id="echoid-s11241" xml:space="preserve"> reflectetur quidem t ad h à duobus punctis tantùm.</s> <s xml:id="echoid-s11242" xml:space="preserve"> Si uerò ſupra contingẽ <lb/>tem fuerint:</s> <s xml:id="echoid-s11243" xml:space="preserve"> reflectetur quidem t ad h ab uno puncto tantùm.</s> <s xml:id="echoid-s11244" xml:space="preserve"> Et hæc quidem accidunt, dum t ęqua-<lb/>liter diſtat à centro cum puncto h.</s> <s xml:id="echoid-s11245" xml:space="preserve"/> </p> <div xml:id="echoid-div399" type="float" level="0" n="0"> <figure xlink:label="fig-0175-01" xlink:href="fig-0175-01a"> <variables xml:id="echoid-variables103" xml:space="preserve">o e k m f l g h d t b q a z</variables> </figure> </div> </div> <div xml:id="echoid-div401" type="section" level="0" n="0"> <head xml:id="echoid-head376" xml:space="preserve" style="it">73. Viſu & uiſibili in diuerſis diametris circuli (qui eſt communis ſectio ſuperficierum refle-<lb/>xionis & ſpeculi ſphæricicaui) à centro inæquabiliter diſtantibus: ab uno puncto peripheriæ in-<lb/>ter ſemidiametros, extra quas ſunt uiſus & uiſibile, reflexio fieripoteſt. 27 p 8. 120 p 1.</head> <p> <s xml:id="echoid-s11246" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s11247" xml:space="preserve"> t, h ſi fuerint in diuerſis diametris:</s> <s xml:id="echoid-s11248" xml:space="preserve"> & longitudo eorum à centro fuerit inęqualis:</s> <s xml:id="echoid-s11249" xml:space="preserve"> re-<lb/>flexio fiet ab uno puncto.</s> <s xml:id="echoid-s11250" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s11251" xml:space="preserve"> ducantur diametri a d g, b d q:</s> <s xml:id="echoid-s11252" xml:space="preserve"> & e z diuidat angulum <lb/>eorum per æqualia:</s> <s xml:id="echoid-s11253" xml:space="preserve"> & t propinquius ſit centro d, quàm h.</s> <s xml:id="echoid-s11254" xml:space="preserve"> Et ſumatur linea l y:</s> <s xml:id="echoid-s11255" xml:space="preserve"> & [per 10 p 6] <lb/>diuidatur in puncto m, ut ſit proportio y m ad m l, ſicut h d ad d t:</s> <s xml:id="echoid-s11256" xml:space="preserve"> & diuidatur l y in æqualia in pun-<lb/>cto n [per 10 p 1] & à puncto n ducatur perpendicularis n k:</s> <s xml:id="echoid-s11257" xml:space="preserve"> [per 11 p 1] & ſuper punctum l fiat angu <lb/>lus ęqualis medietati a d t per lineã f l:</s> <s xml:id="echoid-s11258" xml:space="preserve"> erit quidẽ angulus f l y acutus:</s> <s xml:id="echoid-s11259" xml:space="preserve"> [quia æquatus eſt dimidiato <lb/>angulo a d t acuto, ut oſtenſum eſt 36 n.</s> <s xml:id="echoid-s11260" xml:space="preserve">] Quare [per 11 ax] fl cõcurret cum n k:</s> <s xml:id="echoid-s11261" xml:space="preserve"> [quia l n k rectus eſt <lb/>per fabricationem] concurrant in puncto f:</s> <s xml:id="echoid-s11262" xml:space="preserve"> & [per 35 n] à puncto m ducatur linea ad latus fl, cõcur <lb/>rens cum latere n k in puncto, quod ſit k:</s> <s xml:id="echoid-s11263" xml:space="preserve"> & ſecet linea illa latus fl in puncto c, ut ſit proportio k c ad <lb/>c l, ſicut h d ad d z.</s> <s xml:id="echoid-s11264" xml:space="preserve"> Deinde ſuper pũctum d fiat angulus æqualis angulo l c m:</s> <s xml:id="echoid-s11265" xml:space="preserve"> [per 23 p 1] qui ſit i d a:</s> <s xml:id="echoid-s11266" xml:space="preserve"> <lb/>& ſit i punctum circuli ſupra z, aut infra:</s> <s xml:id="echoid-s11267" xml:space="preserve"> & ſuper i punctũ fiat angulus ęqualis c l m:</s> <s xml:id="echoid-s11268" xml:space="preserve"> qui ſit o i d:</s> <s xml:id="echoid-s11269" xml:space="preserve"> & ſu <lb/>per hanc lineam o i [continuatã] ducatur perpendicularis à puncto h [per 12 p 1] quæ ſit h r:</s> <s xml:id="echoid-s11270" xml:space="preserve"> & pro-<lb/> <pb o="170" file="0176" n="176" rhead="ALHAZEN"/> ducatur r x æqualis lineę ri:</s> <s xml:id="echoid-s11271" xml:space="preserve"> & ducantur lineę h x, h i.</s> <s xml:id="echoid-s11272" xml:space="preserve"> Palàm ſecundum prædicta, quòd à puncto m <lb/>non poteſt linea duci ad latus fl, diuidens ipſum eo modo, quo diuidit lineam m c k, præter hanc ſo <lb/>lam lineam m c k.</s> <s xml:id="echoid-s11273" xml:space="preserve"> Si enim poſsit:</s> <s xml:id="echoid-s11274" xml:space="preserve"> ſit m p o.</s> <s xml:id="echoid-s11275" xml:space="preserve"> Palàm, quòd p o minor erit c k:</s> <s xml:id="echoid-s11276" xml:space="preserve"> quod quidẽ patebit ducta <lb/>linea p q æquidiſtante c k:</s> <s xml:id="echoid-s11277" xml:space="preserve"> quę erit minor c k:</s> <s xml:id="echoid-s11278" xml:space="preserve"> [cũ enim triangula k c f, q p f ſint æquiangula per 29.</s> <s xml:id="echoid-s11279" xml:space="preserve"> <lb/>32 p 1:</s> <s xml:id="echoid-s11280" xml:space="preserve"> erit per 4 p 6 ut k f ad q f, ſic k c ad q p:</s> <s xml:id="echoid-s11281" xml:space="preserve"> ſed k f maior eſt q f per 9 ax.</s> <s xml:id="echoid-s11282" xml:space="preserve"> ergo k c maior eſt q p] & <lb/>maior p o:</s> <s xml:id="echoid-s11283" xml:space="preserve"> [quia maior eſt q p, quæ per 19 p 1 maior eſt o p, cũ angulus p o q ſit obtuſus per 32.</s> <s xml:id="echoid-s11284" xml:space="preserve"> 13 p 1] <lb/>& p l maior c l [per 9 ax:</s> <s xml:id="echoid-s11285" xml:space="preserve">] Igitur non erit proportio p o ad p l, ſicut k c ad c l.</s> <s xml:id="echoid-s11286" xml:space="preserve"> [Si enim ſit ut k c ad cl, <lb/>ſic o p ad p l:</s> <s xml:id="echoid-s11287" xml:space="preserve"> erit per 14 p 5 c l maior l p, contra 9 ax:</s> <s xml:id="echoid-s11288" xml:space="preserve"> quia k c maior eſt o p.</s> <s xml:id="echoid-s11289" xml:space="preserve">] Quare non erit propor-<lb/>tio p o ad p l, ſicut h d ad d t [per 11 p 5.</s> <s xml:id="echoid-s11290" xml:space="preserve">] Reſtat ergo ut à puncto m non ducatur alia, quàm m c k, ſimi <lb/>lis ei.</s> <s xml:id="echoid-s11291" xml:space="preserve"> Verùm cũ o d i ſit ęqualis angulo l c m, & angulus o i d ęqualis angulo c l m:</s> <s xml:id="echoid-s11292" xml:space="preserve"> [per fabricationẽ] <lb/>erit triãgulum c l m ſimile triangulo i o d [per 32 p 1.</s> <s xml:id="echoid-s11293" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s11294" xml:space="preserve"> 1 d 6.</s> <s xml:id="echoid-s11295" xml:space="preserve">] Igitur angulus i o d erit æqualis angu <lb/>lo l m c:</s> <s xml:id="echoid-s11296" xml:space="preserve"> reſtat [per 13 p 1] angulus r o h æqualis angulo k m n:</s> <s xml:id="echoid-s11297" xml:space="preserve"> & angulus h r o rectus ęqualis erit an-<lb/>gulo k n m:</s> <s xml:id="echoid-s11298" xml:space="preserve"> reſtat [per 32 p 1] angulus n k m æqualis angulo r h o.</s> <s xml:id="echoid-s11299" xml:space="preserve"> Ducta autem linea d i, donec con-<lb/>currat cum h r in puncto s:</s> <s xml:id="echoid-s11300" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0176-01a" xlink:href="fig-0176-01"/> <anchor type="figure" xlink:label="fig-0176-02a" xlink:href="fig-0176-02"/> [concurret aũt per 11 ax:</s> <s xml:id="echoid-s11301" xml:space="preserve"> ꝗa <lb/>angulus ad r rectus eſt, ad i <lb/>uerò acutus] erit angulus s <lb/>d h æqualis angulo k c f [ք <lb/>15 p 1.</s> <s xml:id="echoid-s11302" xml:space="preserve"> 1 ax:</s> <s xml:id="echoid-s11303" xml:space="preserve">] & erit triangulũ <lb/>s d h ſimile triãgulo c k f [ք <lb/>32 p 1.</s> <s xml:id="echoid-s11304" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s11305" xml:space="preserve"> 1 d 6.</s> <s xml:id="echoid-s11306" xml:space="preserve">] Igitur pro-<lb/>portio s d ad d h, ſicut f c ad <lb/>c k:</s> <s xml:id="echoid-s11307" xml:space="preserve"> ſed [per 7 p 5] h d ad d i <lb/>[æqualẽ ipſi d z per 15 d 1] <lb/>ſicut k c ad c l [ք fabricatio <lb/>nẽ.</s> <s xml:id="echoid-s11308" xml:space="preserve">] Igitur [per 22 p 5] s d <lb/>ad d i, ſicut f c ad c l:</s> <s xml:id="echoid-s11309" xml:space="preserve"> igitur <lb/>[per 18 p 5] s i ad d i, ſicut fl <lb/>ad c l:</s> <s xml:id="echoid-s11310" xml:space="preserve"> ſed d i ad i o, ſicut c l <lb/>ad l m:</s> <s xml:id="echoid-s11311" xml:space="preserve"> cũ triangulũ d i o ſit <lb/>ſimile triangulo c l m.</s> <s xml:id="echoid-s11312" xml:space="preserve"> Igitur [per 22 p 5] s i ad i o, ſicut fl ad l m [& ք conſectariũ 4 p 5, ut i o ad s i, ſic <lb/>l m ad fl.</s> <s xml:id="echoid-s11313" xml:space="preserve">] Sed proportio s i ad i r, ſicut fl ad l n:</s> <s xml:id="echoid-s11314" xml:space="preserve"> quoniam triangulũ s i r ſimile eſt triangulo fl n [an-<lb/>gulus enim r i s æquatur angulo fl n per fabricationem, & s r i rectus fn l recto:</s> <s xml:id="echoid-s11315" xml:space="preserve"> ergo per 32 p 1.</s> <s xml:id="echoid-s11316" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s11317" xml:space="preserve"> 1 <lb/>d 6 triangula s i r, fl n ſunt ſimilia.</s> <s xml:id="echoid-s11318" xml:space="preserve">] Igitur [per 22 p 5] proportio i o ad i r, ſicut l m ad l n.</s> <s xml:id="echoid-s11319" xml:space="preserve"> [& percon <lb/>ſectarium 4 p 5 ut i r ad i o, ſic l n ad l m.</s> <s xml:id="echoid-s11320" xml:space="preserve">] Igitur proportio y m ad l m, ſicut x o ad i o.</s> <s xml:id="echoid-s11321" xml:space="preserve"> [Quia enim xi <lb/>dupla eſt ipſius i r, & y l dupla ipſius l n:</s> <s xml:id="echoid-s11322" xml:space="preserve"> erit igitur per 15 p 5 ut x i ad i o, ſic y l ad l m, & ք 17 p 5 ut x o <lb/>ad i o, ſic y m ad l m.</s> <s xml:id="echoid-s11323" xml:space="preserve">] Ducta autem à puncto i ęquidiſtante linea u i, lineę h x, & producta linea d a, <lb/>donec concurrat cum u i [concurret autem per lemma Procli ad 29 p 1] concurrat in puncto u:</s> <s xml:id="echoid-s11324" xml:space="preserve"> erit <lb/>triangulum o u i triangulo h o x ſimile [per 15.</s> <s xml:id="echoid-s11325" xml:space="preserve"> 29.</s> <s xml:id="echoid-s11326" xml:space="preserve"> 32 p 1.</s> <s xml:id="echoid-s11327" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s11328" xml:space="preserve"> 1 d 6.</s> <s xml:id="echoid-s11329" xml:space="preserve">] Igitur erit proportio h o ad o u, ſi-<lb/>cut y m ad m l:</s> <s xml:id="echoid-s11330" xml:space="preserve"> [Quia ob triãgulorum o u i, h o x oſtenſam ſimilitu dinẽ eſt, ut h o ad o u, ſic x o ad i o, <lb/>& ut x o ad i o, ſic y m ad m l ex cõcluſo:</s> <s xml:id="echoid-s11331" xml:space="preserve"> ergo per 11 p 5, ut h o ad o u, ſic y m ad m l] & ita h o ad o u, <lb/>ſicut h d ad d t.</s> <s xml:id="echoid-s11332" xml:space="preserve"> [Fuit enim per fabricationẽ h d ad d t, ſicut y m ad m l.</s> <s xml:id="echoid-s11333" xml:space="preserve">] Sed quoniã [per 4 p 1] triãgu <lb/>lũ h r i æquale eſt triangulo h r x:</s> <s xml:id="echoid-s11334" xml:space="preserve"> cũ h r ſit perpendicularis [per fabricationẽ:</s> <s xml:id="echoid-s11335" xml:space="preserve"> & x r æquetur ipſi r i, <lb/>latusq́;</s> <s xml:id="echoid-s11336" xml:space="preserve"> r h cõmune ſit.</s> <s xml:id="echoid-s11337" xml:space="preserve">] Igitur angulus h x r æqualis eſt angulo r i h:</s> <s xml:id="echoid-s11338" xml:space="preserve"> & ita r i h æqualis eſt angulo u i <lb/>o [quia u i o æquatur ipſι h x o propter ſimilitu dinem triangulorum u i o, h o x.</s> <s xml:id="echoid-s11339" xml:space="preserve">] Quare [per 3 p 6] <lb/>proportio h o ad o u, ſicut h i ad i u:</s> <s xml:id="echoid-s11340" xml:space="preserve"> & ita [per 11 p 5] h i ad i u, ſicut h d ad d t.</s> <s xml:id="echoid-s11341" xml:space="preserve"> Verùm angulus u i d ma <lb/>ior eſt angulo d i h:</s> <s xml:id="echoid-s11342" xml:space="preserve"> [quia æqualis concluſus eſt angulo o i h] ſecetur ab eo æqualis:</s> <s xml:id="echoid-s11343" xml:space="preserve"> & ſit p i d:</s> <s xml:id="echoid-s11344" xml:space="preserve"> & du-<lb/>catur linea p t:</s> <s xml:id="echoid-s11345" xml:space="preserve"> & p ſit punctum diametri d a.</s> <s xml:id="echoid-s11346" xml:space="preserve"> Palàm, quòd proportio h i ad u i cõſtat ex proportione <lb/>h i ad i p, & p i ad u i:</s> <s xml:id="echoid-s11347" xml:space="preserve"> [quia ratio extremorum componitur ex omnibus rationibus intermedijs, ut <lb/>Theon demonſtrauit ad 5 d 6.</s> <s xml:id="echoid-s11348" xml:space="preserve">] & [per 3 p 6] proportio h i ad i p, ſicut d h ad d p:</s> <s xml:id="echoid-s11349" xml:space="preserve"> quoniam d i diuidit <lb/>angulum p i h per ęqualia.</s> <s xml:id="echoid-s11350" xml:space="preserve"> Igitur proportio h i ad u i (quæ eſt h d ad d t) conſtat ex proportione h d <lb/>ad d p, & d p ad d t.</s> <s xml:id="echoid-s11351" xml:space="preserve"> Igitur proportio d p ad d t, ſicut p i ad u i.</s> <s xml:id="echoid-s11352" xml:space="preserve"> Verùm angulus o i h eſt medietas angu <lb/>li u i h:</s> <s xml:id="echoid-s11353" xml:space="preserve"> [ex concluſo] ſed angulus d i h medietas eſt anguli p i h:</s> <s xml:id="echoid-s11354" xml:space="preserve"> reſtat angulus d i o medietas anguli <lb/>p i u.</s> <s xml:id="echoid-s11355" xml:space="preserve"> Sed angulus d i o eſt medietas anguli t d p:</s> <s xml:id="echoid-s11356" xml:space="preserve"> quia eſt æqualis angulo fl m [qui ęquatus eſt dimi-<lb/>diato angulo a d t ſeu p d t.</s> <s xml:id="echoid-s11357" xml:space="preserve">] Igitur angulus p i u eſt ęqualis angulo t d p:</s> <s xml:id="echoid-s11358" xml:space="preserve"> & proportio d p ad d t, ſicut <lb/>p i ad u i.</s> <s xml:id="echoid-s11359" xml:space="preserve"> Igitur triangulũ u i p ſimile triangulo t p d:</s> <s xml:id="echoid-s11360" xml:space="preserve"> [per 6.</s> <s xml:id="echoid-s11361" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s11362" xml:space="preserve"> 1 d 6] & angulus u p i æqualis t p d:</s> <s xml:id="echoid-s11363" xml:space="preserve"> <lb/>erit igitur [per 14 p 1] t p i linea recta:</s> <s xml:id="echoid-s11364" xml:space="preserve"> quia angulus d p t cum angulo t p o ualet duos rectos:</s> <s xml:id="echoid-s11365" xml:space="preserve"> & ita an <lb/>gulus o p i cum angulo o p t ualet duos rectos.</s> <s xml:id="echoid-s11366" xml:space="preserve"> [Idem uerò patet per conuerſionem 15 p 1 à Proclo <lb/>demonſtratã.</s> <s xml:id="echoid-s11367" xml:space="preserve">] Et ita [per 12 n 4] treflectetur ad h à puncto i.</s> <s xml:id="echoid-s11368" xml:space="preserve"> [quia linea t p i eſt linea incidentię, & <lb/>anguli t i d, h i d ſunt æquales per fabricationem.</s> <s xml:id="echoid-s11369" xml:space="preserve">] Et eadem erit probatio, ſiue ſit t extra circulum, <lb/>ſiue intra.</s> <s xml:id="echoid-s11370" xml:space="preserve"> Et ſimiliter ſumpto puncto h extra uel intra:</s> <s xml:id="echoid-s11371" xml:space="preserve"> dum inęqualiter diſtent à centro.</s> <s xml:id="echoid-s11372" xml:space="preserve"/> </p> <div xml:id="echoid-div401" type="float" level="0" n="0"> <figure xlink:label="fig-0176-01" xlink:href="fig-0176-01a"> <variables xml:id="echoid-variables104" xml:space="preserve">b u a x r o i c p e d z s h g q</variables> </figure> <figure xlink:label="fig-0176-02" xlink:href="fig-0176-02a"> <variables xml:id="echoid-variables105" xml:space="preserve">l m c k p q o f n y</variables> </figure> </div> </div> <div xml:id="echoid-div403" type="section" level="0" n="0"> <head xml:id="echoid-head377" xml:space="preserve" style="it">74. Si angulum comprehenſum à duabus diametris in centro circuli (qui eſt cõmunis ſectio <lb/>ſuperficierum reflexionis & ſpeculi ſphærici caui) tertia bifariam ſecet: puncta in dictis dia-<lb/> <pb o="171" file="0177" n="177" rhead="OPTICAE LIBER V."/> metris à centro inæquabiliter diſtantia, reflectuntur à quolibet puncto peripheriæ inter ſemidia <lb/>metros, extra quas ſunt, comprehenſæ: excepto eo, in quo ſecans diameter terminatur. 28 p 8.</head> <p> <s xml:id="echoid-s11373" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s11374" xml:space="preserve"> ductis diametris b q, a g:</s> <s xml:id="echoid-s11375" xml:space="preserve"> & diametro e z diuidente angulum b d g per æqualia.</s> <s xml:id="echoid-s11376" xml:space="preserve"> Dico, <lb/>quòd quodcũq;</s> <s xml:id="echoid-s11377" xml:space="preserve"> punctũ ſumatur in arcu a q, pręter punctũ z:</s> <s xml:id="echoid-s11378" xml:space="preserve"> [à pũcto enim z reflectũtur tan <lb/>tùm pũcta diametrorũ à centro æquabiliter diſtania, ut conſtat è ſuperioribus numeris] ab <lb/>illo poterunt reflecti infinita paria punctorũ, inæqualiter à centro diſtantiũ.</s> <s xml:id="echoid-s11379" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s11380" xml:space="preserve"> ſumatur h <lb/>punctum:</s> <s xml:id="echoid-s11381" xml:space="preserve"> & ſumatur in ſemidiametro g d punctũ l:</s> <s xml:id="echoid-s11382" xml:space="preserve"> & à ſemidiametro b d ſecetur m d, æqualis l d:</s> <s xml:id="echoid-s11383" xml:space="preserve"> <lb/>[per 3 p 1] & ducantur lineę l m, l h, m h, d h.</s> <s xml:id="echoid-s11384" xml:space="preserve"> Punctũ, in quo e z diuidit l m, ſit f:</s> <s xml:id="echoid-s11385" xml:space="preserve"> erit [per 4 p 1] f l æqua-<lb/>lis f m:</s> <s xml:id="echoid-s11386" xml:space="preserve"> & ducatur h d, quouſq;</s> <s xml:id="echoid-s11387" xml:space="preserve"> cadat ſuper l m in pun-<lb/> <anchor type="figure" xlink:label="fig-0177-01a" xlink:href="fig-0177-01"/> cto n:</s> <s xml:id="echoid-s11388" xml:space="preserve"> erit igitur l n minor m n.</s> <s xml:id="echoid-s11389" xml:space="preserve"> Verùm cum angulus <lb/>m d f ſit æqualis f d l [ex theſi] & angulo q d z:</s> <s xml:id="echoid-s11390" xml:space="preserve"> [per <lb/>15 p 1] & angulus m d a æqualis angulo l d q:</s> <s xml:id="echoid-s11391" xml:space="preserve"> [per ean <lb/>dem] & angulus a d h æqualis angulo n d l:</s> <s xml:id="echoid-s11392" xml:space="preserve"> erit angu-<lb/>lus l d h maior angulo m d h:</s> <s xml:id="echoid-s11393" xml:space="preserve"> [Quia enim m d n, id eſt <lb/>per 15 p 1 h d q maior eſt n d l, id eſt a d h, & m d a æ-<lb/>quatur ipſi l d q:</s> <s xml:id="echoid-s11394" xml:space="preserve"> angulus igitur l d h maior eſt angulo <lb/>m d h] igitur [per 24 p 1] l h erit maior m h:</s> <s xml:id="echoid-s11395" xml:space="preserve"> cum m d, <lb/>d h æqualia ſint l d, d h.</s> <s xml:id="echoid-s11396" xml:space="preserve"> Erit ergo angulus d h l minor <lb/>angulo d h m.</s> <s xml:id="echoid-s11397" xml:space="preserve"> Si enim eſſet æqualis:</s> <s xml:id="echoid-s11398" xml:space="preserve"> eſſet proportio l <lb/>h ad h m, ſicut l n ad n m:</s> <s xml:id="echoid-s11399" xml:space="preserve"> [per 3 p 6] qđ eſt impoſsibi <lb/>le.</s> <s xml:id="echoid-s11400" xml:space="preserve"> [Sic enim maior l h ad minorẽ h m eandẽ haberet <lb/>rationẽ, quam minor l n ad maiorẽ n m.</s> <s xml:id="echoid-s11401" xml:space="preserve">] Si autẽ fue-<lb/>rit maior:</s> <s xml:id="echoid-s11402" xml:space="preserve"> ſecetur ex e o æqualis:</s> <s xml:id="echoid-s11403" xml:space="preserve"> & improbabitur eo-<lb/>dẽ modo.</s> <s xml:id="echoid-s11404" xml:space="preserve"> Igitur eſt minor.</s> <s xml:id="echoid-s11405" xml:space="preserve"> Secetur igitur ab angulo <lb/>m h d ęqualis illi [d h l:</s> <s xml:id="echoid-s11406" xml:space="preserve">] qui ſit t h d.</s> <s xml:id="echoid-s11407" xml:space="preserve"> Igitur t reflectetur <lb/>ad l à puncto h [per 12 n 4.</s> <s xml:id="echoid-s11408" xml:space="preserve">] Et linea t d eſt minor l d.</s> <s xml:id="echoid-s11409" xml:space="preserve"> <lb/>[quia minor eſt d m, pertheſin æquali ipſi d l.</s> <s xml:id="echoid-s11410" xml:space="preserve">] Similiter ſi ſumãtur in ſemidiametris b d, g d alia pun <lb/>cta, quàm l, m, æqualiter à puncto d diſtantia:</s> <s xml:id="echoid-s11411" xml:space="preserve"> probabitur ſimiliter, quòd à puncto h fit reflexio<gap/> pun-<lb/>ctorum ad inuicem, in æqualiter diſtantium à centro:</s> <s xml:id="echoid-s11412" xml:space="preserve"> & ita de infinitis punctis in his diametris ſum-<lb/>ptis ſimilis erit probatio:</s> <s xml:id="echoid-s11413" xml:space="preserve"> & à quocunq;</s> <s xml:id="echoid-s11414" xml:space="preserve"> puncto arcus a q ſumpto, præter quàm à puncto z.</s> <s xml:id="echoid-s11415" xml:space="preserve"/> </p> <div xml:id="echoid-div403" type="float" level="0" n="0"> <figure xlink:label="fig-0177-01" xlink:href="fig-0177-01a"> <variables xml:id="echoid-variables106" xml:space="preserve">b a m h e f t d z n p l g q</variables> </figure> </div> </div> <div xml:id="echoid-div405" type="section" level="0" n="0"> <head xml:id="echoid-head378" xml:space="preserve" style="it">75. Si uiſus & uiſibile in diuerſis diametris circuli (qui eſt cõmunis ſectio ſuperficierũ refle-<lb/>xiõis, et ſpeculι ſphærici caui) à cẽtro inæquabilιter diſtãtia, à pũcto aliquo peripheriæinter ſemi <lb/>diametros, extr a quas ſunt, inter ſe mutuò reflectãtur: ab uno tatùm puncto reflectẽtur. 29 p 8.</head> <p> <s xml:id="echoid-s11416" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s11417" xml:space="preserve"> ſumptis punctis t, l in diametris:</s> <s xml:id="echoid-s11418" xml:space="preserve"> quorũ inæqualis ſit lõgitudo à centro:</s> <s xml:id="echoid-s11419" xml:space="preserve"> reflectantur <lb/>ipla ad inuicẽ à puncto h:</s> <s xml:id="echoid-s11420" xml:space="preserve"> nõ poterit reflecti t ad l ab alio puncto arcus a q, quàm à puncto h.</s> <s xml:id="echoid-s11421" xml:space="preserve"> <lb/>Si enim ab alio:</s> <s xml:id="echoid-s11422" xml:space="preserve"> ſit illud k:</s> <s xml:id="echoid-s11423" xml:space="preserve"> & ducantur, t k, l k, <lb/> <anchor type="figure" xlink:label="fig-0177-02a" xlink:href="fig-0177-02"/> d k, l t, t h, l h, n d h:</s> <s xml:id="echoid-s11424" xml:space="preserve"> & producatur d k;</s> <s xml:id="echoid-s11425" xml:space="preserve"> quouſq;</s> <s xml:id="echoid-s11426" xml:space="preserve"> con-<lb/>currat cum l t in puncto p [concurret autem, quia <lb/>ſecat angulũ t k l à baſi t l ſubtenſum.</s> <s xml:id="echoid-s11427" xml:space="preserve">] Palàm, quòd <lb/>proportio l h ad h t, ſicut l n ad n t.</s> <s xml:id="echoid-s11428" xml:space="preserve"> [per 3 p 6:</s> <s xml:id="echoid-s11429" xml:space="preserve"> quia e-<lb/>nim t ex theſi eſt reflexionis punctũ:</s> <s xml:id="echoid-s11430" xml:space="preserve"> æquabitur per <lb/>12 n 4 angulus l h n angulo t h n.</s> <s xml:id="echoid-s11431" xml:space="preserve">] Et ſimiliter cũ an-<lb/>gulus t p k ſit æqualis l k p, exhypotheſi:</s> <s xml:id="echoid-s11432" xml:space="preserve"> erit [per 3 p <lb/>6] proportio l k ad k t, ſicut l p ad p t:</s> <s xml:id="echoid-s11433" xml:space="preserve"> Sed [per 7 <lb/>p 3] l h maior l k, & t h minor t k:</s> <s xml:id="echoid-s11434" xml:space="preserve"> igitur maior eſt pro <lb/>portio l h ad t h, quàm l k ad t k.</s> <s xml:id="echoid-s11435" xml:space="preserve"> [ut patet per 8 p 5.</s> <s xml:id="echoid-s11436" xml:space="preserve">] <lb/>Quare maior erit proportio l n ad n t, quàm l p ad p <lb/>t:</s> <s xml:id="echoid-s11437" xml:space="preserve"> quod planè impoſsibile.</s> <s xml:id="echoid-s11438" xml:space="preserve"> [Quia enim l n minor eſt <lb/>l p per 9 ax:</s> <s xml:id="echoid-s11439" xml:space="preserve"> & n t maior p t:</s> <s xml:id="echoid-s11440" xml:space="preserve"> erit ratio l n ad n t mi-<lb/>nor, quàm ratio l p ad p t, ut conſtat ex 8 p 5.</s> <s xml:id="echoid-s11441" xml:space="preserve">] Re-<lb/>ſtat, ut ab alio puncto arcus a q, quàm à puncto h <lb/>non poſsit t reflecti ad l.</s> <s xml:id="echoid-s11442" xml:space="preserve"> Palàm igitur, quæ acci-<lb/>dunt in arcu a q.</s> <s xml:id="echoid-s11443" xml:space="preserve"/> </p> <div xml:id="echoid-div405" type="float" level="0" n="0"> <figure xlink:label="fig-0177-02" xlink:href="fig-0177-02a"> <variables xml:id="echoid-variables107" xml:space="preserve">b a t h e p d z n l k g q</variables> </figure> </div> </div> <div xml:id="echoid-div407" type="section" level="0" n="0"> <head xml:id="echoid-head379" xml:space="preserve" style="it">76. Viſu in diametro circuli (qui eſt communis ſectio ſuperficierũ, reflexionis & ſpeculi ſphæ-<lb/>rici caui) intra peripheriam poſito: uiſibile cum uiſu à centro utlibet diſtans: à quolιbet ſemicir-<lb/>culi puncto ad ipſum reflecti poteſt. 30 p 8.</head> <p> <s xml:id="echoid-s11444" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s11445" xml:space="preserve"> ſit a centrũ uiſus:</s> <s xml:id="echoid-s11446" xml:space="preserve"> b centrum ſpeculi:</s> <s xml:id="echoid-s11447" xml:space="preserve"> & ducatur diameter d a b g:</s> <s xml:id="echoid-s11448" xml:space="preserve"> & ſumatur ſuperfi.</s> <s xml:id="echoid-s11449" xml:space="preserve"> <lb/>cies, in qua ſit a b quocunq;</s> <s xml:id="echoid-s11450" xml:space="preserve"> modo:</s> <s xml:id="echoid-s11451" xml:space="preserve"> quæ ſecabit ſphærã ſuper circulum:</s> <s xml:id="echoid-s11452" xml:space="preserve"> [per 1 th.</s> <s xml:id="echoid-s11453" xml:space="preserve"> 1 ſphær.</s> <s xml:id="echoid-s11454" xml:space="preserve">] qui <lb/>ſit d l g.</s> <s xml:id="echoid-s11455" xml:space="preserve"> Dico, quòd à quolibet puncto ſemicirculi d l g reflectuntur puncta ad a, inæqualis <lb/>longitudinis à centro, cum eo.</s> <s xml:id="echoid-s11456" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s11457" xml:space="preserve"> ſumatur punctũ e:</s> <s xml:id="echoid-s11458" xml:space="preserve"> & ducantur lineę e a, e b.</s> <s xml:id="echoid-s11459" xml:space="preserve"> Palàm, quòd <lb/>angulus a e b erit acutus:</s> <s xml:id="echoid-s11460" xml:space="preserve"> quia cadet in minorem arcum ſemicirculo.</s> <s xml:id="echoid-s11461" xml:space="preserve"> [Nam angulus inſiſtens in pe-<lb/>ripheriam ſemicirculi rectus eſt per 31 p 3.</s> <s xml:id="echoid-s11462" xml:space="preserve"> Vel etiã angulus a e b acutus eſt per ea, quæ 60 n demon-<lb/> <pb o="172" file="0178" n="178" rhead="ALHAZEN"/> ſtrata ſunt.</s> <s xml:id="echoid-s11463" xml:space="preserve">] Fiat ei æqualis [per 23 p 1] & ſit p e b:</s> <s xml:id="echoid-s11464" xml:space="preserve"> & producatur linea b e quantumlibet.</s> <s xml:id="echoid-s11465" xml:space="preserve"> Palàm, quòd <lb/>quo dlibet punctum illius lineę reflectetur ad a à pun <lb/> <anchor type="figure" xlink:label="fig-0178-01a" xlink:href="fig-0178-01"/> cto e:</s> <s xml:id="echoid-s11466" xml:space="preserve"> [per 12 n 4.</s> <s xml:id="echoid-s11467" xml:space="preserve">] Ducta autem à puncto b ad li-<lb/>neam p e perpendiculari:</s> <s xml:id="echoid-s11468" xml:space="preserve"> [per 12 p 1] aut erit perpen-<lb/>dicularis illa æqualis b a:</s> <s xml:id="echoid-s11469" xml:space="preserve"> aut maior:</s> <s xml:id="echoid-s11470" xml:space="preserve"> aut minor.</s> <s xml:id="echoid-s11471" xml:space="preserve"> Si fue <lb/>rit æqualis:</s> <s xml:id="echoid-s11472" xml:space="preserve"> lineæ omnes ductę à puncto b ad lineam <lb/>p e, præter illam perpendicularem, erunt maiores li-<lb/>nea b a:</s> <s xml:id="echoid-s11473" xml:space="preserve"> [quia per 19 p 1 maiores ſunt perpendiculari, <lb/>æquali b a] & ita quodlibet punctum lineæ p e, uno <lb/>excepto [puncto nimirum perpendicularis] in æqua <lb/>liter diſta bit à centro, cum puncto a.</s> <s xml:id="echoid-s11474" xml:space="preserve"> Si uerò perpen-<lb/>dicularis fuerit maior:</s> <s xml:id="echoid-s11475" xml:space="preserve"> omnia puncta lineę illius plus <lb/>diſtabunt à centro, quàm a punctum.</s> <s xml:id="echoid-s11476" xml:space="preserve"> Si autẽ perpen <lb/>dicularis fuerit minor:</s> <s xml:id="echoid-s11477" xml:space="preserve"> erit poſsibile ducere à puncto <lb/>b duas lineas ex diuerſis partibus perpendicularis, <lb/>æquales lineę b a:</s> <s xml:id="echoid-s11478" xml:space="preserve"> & omnes alię lineę [ductæ à pun-<lb/>cto b ad lineam e p] aut minores erunt, aut maiores <lb/>[b a.</s> <s xml:id="echoid-s11479" xml:space="preserve">] Palàm igitur, [per 74 n] quòd à puncto e refle-<lb/>ctuntur puncta ad a:</s> <s xml:id="echoid-s11480" xml:space="preserve"> quorũ longitudo à centro in æ-<lb/>qualis eſt longitudini a ab eodem.</s> <s xml:id="echoid-s11481" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s11482" xml:space="preserve"/> </p> <div xml:id="echoid-div407" type="float" level="0" n="0"> <figure xlink:label="fig-0178-01" xlink:href="fig-0178-01a"> <variables xml:id="echoid-variables108" xml:space="preserve">l e p d a b g</variables> </figure> </div> </div> <div xml:id="echoid-div409" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables109" xml:space="preserve">h d t b q g</variables> </figure> <head xml:id="echoid-head380" xml:space="preserve" style="it">77. Si à uiſu duæ rectæ lineæ tangant circulum (qui eſt communis ſectio ſuperficierum, refle <lb/>xionis & ſpeculi ſphærici caui) tertia per centrũ ſecet: uiſibile cũ <lb/> uiſu à centro ſpeculi inæquabiliter diſtãs, poteſt reflecti à quolibet pũcto peripheriæ inter tactus punct a ultra centrũ interiectæ: ex- ceptis tactus punctis & ſecantis diametri termino. 31 p 8.</head> <p> <s xml:id="echoid-s11483" xml:space="preserve">COnſtat ex his:</s> <s xml:id="echoid-s11484" xml:space="preserve"> quòd ſi ſumatur uiſus extra circulũ:</s> <s xml:id="echoid-s11485" xml:space="preserve"> & ſit h:</s> <s xml:id="echoid-s11486" xml:space="preserve"> & <lb/>ducatur diameter h b d g:</s> <s xml:id="echoid-s11487" xml:space="preserve"> & duę cõtingentes h t, h q:</s> <s xml:id="echoid-s11488" xml:space="preserve"> [per 17 <lb/>p 3] à quolibet pũcto arcus t g q, pręterquã à punctis t, g, q po <lb/>teſt fieri reflexio ad h punctorum, inæqualiter diſtantium à centro <lb/>cum puncto h.</s> <s xml:id="echoid-s11489" xml:space="preserve"> [nam à peripheria t d q & punctis t, g, q nullam ad ui <lb/>ſum h reflexionem fieri conſtat tum per 70 n:</s> <s xml:id="echoid-s11490" xml:space="preserve"> tum quia angulus ta-<lb/>ctus indiuiduus eſt:</s> <s xml:id="echoid-s11491" xml:space="preserve"> tum ex ijs, quę 45 n 4 demonſtrata ſunt.</s> <s xml:id="echoid-s11492" xml:space="preserve">] Et ea-<lb/>dem erit probatio [quæ fuit 70 n.</s> <s xml:id="echoid-s11493" xml:space="preserve">]</s> </p> </div> <div xml:id="echoid-div410" type="section" level="0" n="0"> <head xml:id="echoid-head381" xml:space="preserve" style="it">78. Si uiſus & uiſibile intra circulum (qui eſt communis ſectio <lb/>ſuperficierũ, reflexionis & ſpeculi ſphærici caui) à centro inæqua-<lb/>biliter diſtantia, inter ſe reflect antur: angulus exterior à diame <lb/>tris uiſus & uiſibilis factus, aliâs maior: aliâs minor eſt angulo <lb/>incidentiæ & reflexionis ſimul utro. 32 p 8.</head> <p> <s xml:id="echoid-s11494" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s11495" xml:space="preserve"> ex his conſtabit, quòd, facta reflexione ad a à puncto e, uel alio puncto inæqualiter <lb/>diſtante à centro, cũ puncto a:</s> <s xml:id="echoid-s11496" xml:space="preserve"> diameter, in qua fuerit punctũ reflexũ, cum diametro a b g fa-<lb/>cit duos angulos, unũ reſpicientẽ angulum reflexionis, alium ei collateralem:</s> <s xml:id="echoid-s11497" xml:space="preserve"> qui quidẽ col-<lb/>lateralis aliquã do erit maior angulo, cõſtãte ex angu <lb/> <anchor type="figure" xlink:label="fig-0178-03a" xlink:href="fig-0178-03"/> lo incidentię & reflexionis:</s> <s xml:id="echoid-s11498" xml:space="preserve"> aliquando minor.</s> <s xml:id="echoid-s11499" xml:space="preserve"> Verbi <lb/>gratia:</s> <s xml:id="echoid-s11500" xml:space="preserve"> ducatur perpendicularis f b ſuper e o [per 12 <lb/>p 1] b a aut erit perpendicularis ſuper e a, aut non.</s> <s xml:id="echoid-s11501" xml:space="preserve"> Sit <lb/>perpen dicularis:</s> <s xml:id="echoid-s11502" xml:space="preserve"> erũt ergo duo anguli f b a, f e a ęqua <lb/>les duobus rectis [per theſin & 32 p 1.</s> <s xml:id="echoid-s11503" xml:space="preserve">] Ducta aũt li-<lb/>nea b o:</s> <s xml:id="echoid-s11504" xml:space="preserve"> erũt duo anguli o b a, o e a minores duobus <lb/>rectis.</s> <s xml:id="echoid-s11505" xml:space="preserve"> Igitur [per 13 p 1] erit angulus o b g maior angu <lb/>lo o e a, qui eſt angulus conſtãs ex angulo incidentię <lb/>& reflexiõis.</s> <s xml:id="echoid-s11506" xml:space="preserve"> Et cũ triangulũ e b f ſit æquale triangu <lb/>lo e b a:</s> <s xml:id="echoid-s11507" xml:space="preserve"> [quia enim anguli ad a & frecti ſunt per the-<lb/>ſin & fabricationẽ:</s> <s xml:id="echoid-s11508" xml:space="preserve"> & per 12 n 4 anguli fe b, a e b æ-<lb/>quãtur, & cõmune latus eſt e b:</s> <s xml:id="echoid-s11509" xml:space="preserve"> erũt triangula e b f, e <lb/>b a æquilatera & ęqualia per 26 p 1] & erit b f æqualis <lb/>b a:</s> <s xml:id="echoid-s11510" xml:space="preserve"> & ita o b maior b a [quia maior eſt f b per 19 p 1, cũ <lb/>ſubtẽdat angulũ rectũ in triangulo o f b.</s> <s xml:id="echoid-s11511" xml:space="preserve">] Ducta aũt <lb/>linea b n:</s> <s xml:id="echoid-s11512" xml:space="preserve"> erunt duo anguli n b a, n e a maiores duob.</s> <s xml:id="echoid-s11513" xml:space="preserve"> <lb/>rectis:</s> <s xml:id="echoid-s11514" xml:space="preserve"> [quia fb a, f e a æquãtur duobus rectis, ut patuit] erit ergo angulus n b g minor angulo n e a:</s> <s xml:id="echoid-s11515" xml:space="preserve"> <lb/>[Nam cũ anguli n b a, n b g æquẽtur duobus rectis per 13 p 1, & n b a, n e a maiores duobus rectis per <lb/>concluſionẽ:</s> <s xml:id="echoid-s11516" xml:space="preserve"> erit angulus n b g minor angulo n e a] & n b maior b a [quia maior b f ք 19 p 1.</s> <s xml:id="echoid-s11517" xml:space="preserve">] Et ita n <lb/> <pb o="173" file="0179" n="179" rhead="OPTICAE LIBER V."/> & o reflectũtur ad a à puncto e, & inæqualiter diſtant à centro cũ puncto a:</s> <s xml:id="echoid-s11518" xml:space="preserve"> & diameter o b cũ diame <lb/>tro a b g ex parte g facit angulũ maiorẽ angulo reflexionis & incidentiæ:</s> <s xml:id="echoid-s11519" xml:space="preserve"> & diameter n b minorẽ.</s> <s xml:id="echoid-s11520" xml:space="preserve"> Et <lb/>ita patet ꝓpoſitũ.</s> <s xml:id="echoid-s11521" xml:space="preserve"> Si uerò b a nõ fuerit perpẽdicularis ſuք e a:</s> <s xml:id="echoid-s11522" xml:space="preserve"> ducatur [per 12 p 1] perpẽdicularis:</s> <s xml:id="echoid-s11523" xml:space="preserve"> quę <lb/>ſit b k:</s> <s xml:id="echoid-s11524" xml:space="preserve"> quę quidẽ ſiue cadat ſupra a b, aut ſub:</s> <s xml:id="echoid-s11525" xml:space="preserve"> eadẽ erit ꝓbatio.</s> <s xml:id="echoid-s11526" xml:space="preserve"> Et b f ſit perpendicularis ſuper e o:</s> <s xml:id="echoid-s11527" xml:space="preserve"> & <lb/>ducatur f t æqualis a k:</s> <s xml:id="echoid-s11528" xml:space="preserve"> & ducatur b t.</s> <s xml:id="echoid-s11529" xml:space="preserve"> Palàm, quòd in triangulo k e b angulus e k b rectus, ęqualis eſt <lb/>angulo e f b, & [per 12 n 4] angulus k e b ęqualis angulo reflexiõis f e b:</s> <s xml:id="echoid-s11530" xml:space="preserve"> reſtat [per 32 p 1] tertius tertio <lb/>ęqualis:</s> <s xml:id="echoid-s11531" xml:space="preserve"> & cũ latus e b ſit cõmune utriq;</s> <s xml:id="echoid-s11532" xml:space="preserve"> triãgulo:</s> <s xml:id="echoid-s11533" xml:space="preserve"> erũt [per 26 p 1] triãgula æqualia:</s> <s xml:id="echoid-s11534" xml:space="preserve"> & erit f b æqualis <lb/>k b:</s> <s xml:id="echoid-s11535" xml:space="preserve"> ſed [ք fabricationẽ] a k eſt æqualis ft:</s> <s xml:id="echoid-s11536" xml:space="preserve"> erit ergo [per 4 p 1] a b æqualis b t, & angulus a b k æqualis <lb/>angulo f b t:</s> <s xml:id="echoid-s11537" xml:space="preserve"> addito igitur cõmuni angulo f b a:</s> <s xml:id="echoid-s11538" xml:space="preserve"> erit k b f æqualis t b a:</s> <s xml:id="echoid-s11539" xml:space="preserve"> Sed k b f & fe a ualent duos re-<lb/> <anchor type="figure" xlink:label="fig-0179-01a" xlink:href="fig-0179-01"/> <anchor type="figure" xlink:label="fig-0179-02a" xlink:href="fig-0179-02"/> ctos:</s> <s xml:id="echoid-s11540" xml:space="preserve"> [per 32 p 1:</s> <s xml:id="echoid-s11541" xml:space="preserve"> quia in quadrilatero e b anguli ad f & k recti ſunt.</s> <s xml:id="echoid-s11542" xml:space="preserve">] Quare t b a, t e a ualent duos re-<lb/>ctos:</s> <s xml:id="echoid-s11543" xml:space="preserve"> & ita t b g æqualis eſt angulo t e a:</s> <s xml:id="echoid-s11544" xml:space="preserve"> [quia t b g & t b a æquantur duobus rectis per 13 p 1] qui eſt <lb/>angulus conſtans ex angulo incidentiæ & reflexionis.</s> <s xml:id="echoid-s11545" xml:space="preserve"> Si igitur à puncto b ad lineam e t, ducatur li-<lb/>nea ultra t:</s> <s xml:id="echoid-s11546" xml:space="preserve"> faciet cum b g ex parte g, angulum minorẽ angulo conſtante ex angulo incidentiæ & re-<lb/>flexionis:</s> <s xml:id="echoid-s11547" xml:space="preserve"> & erit linea illa maior a b:</s> <s xml:id="echoid-s11548" xml:space="preserve"> quoniã t b [qua illa per 19 p 1 maior eſt] æqualis eſt a b.</s> <s xml:id="echoid-s11549" xml:space="preserve"> Et quæli <lb/>bet linea à puncto b ad e t ducta citra t:</s> <s xml:id="echoid-s11550" xml:space="preserve"> faciet angulũ t b g ex parte g, maiorẽ angulo cõſtante ex an-<lb/>gulo incidẽtiæ & reflexionis:</s> <s xml:id="echoid-s11551" xml:space="preserve"> & erit minor a b [quia minor æquali b t per 19 p 1.</s> <s xml:id="echoid-s11552" xml:space="preserve">] Et ita eſt propoſitũ.</s> <s xml:id="echoid-s11553" xml:space="preserve"/> </p> <div xml:id="echoid-div410" type="float" level="0" n="0"> <figure xlink:label="fig-0178-03" xlink:href="fig-0178-03a"> <variables xml:id="echoid-variables110" xml:space="preserve">e o f n p d a b g</variables> </figure> <figure xlink:label="fig-0179-01" xlink:href="fig-0179-01a"> <variables xml:id="echoid-variables111" xml:space="preserve">e o f t p d a b g k</variables> </figure> <figure xlink:label="fig-0179-02" xlink:href="fig-0179-02a"> <variables xml:id="echoid-variables112" xml:space="preserve">e o f t p k d a b g</variables> </figure> </div> </div> <div xml:id="echoid-div412" type="section" level="0" n="0"> <head xml:id="echoid-head382" xml:space="preserve" style="it">79. Si uiſus & uiſibile in diuerſis diametris circuli (qui eſt communis ſectio ſuperficierum, <lb/>reflexionis & ſpeculi ſphærici caui) à centro inæquabiliter diſtantia, inter ſe reflectantur: angu-<lb/>lus exterior à diametris uiſus & uiſibilis factus, eſt inæqualis angulo incidentiæ & reflexionis <lb/>ſimul utri. 33 p 8.</head> <p> <s xml:id="echoid-s11554" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s11555" xml:space="preserve"> ſit b centrum uiſus:</s> <s xml:id="echoid-s11556" xml:space="preserve"> g centrum ſphæræ:</s> <s xml:id="echoid-s11557" xml:space="preserve"> ducatur diameter z b g d:</s> <s xml:id="echoid-s11558" xml:space="preserve"> & ſumatur ſuperfi-<lb/>cies, in qua ſit diameter ſecans ſphęram ſuper circulũ [per 1 th 1 ſphæ.</s> <s xml:id="echoid-s11559" xml:space="preserve">] qui ſit e z h.</s> <s xml:id="echoid-s11560" xml:space="preserve"> Dico, quòd <lb/>ſi punctum a reflectitur ad b ab aliquo puncto circuli:</s> <s xml:id="echoid-s11561" xml:space="preserve"> & inæqualis eſt diſtantia puncti a à cen-<lb/> <anchor type="figure" xlink:label="fig-0179-03a" xlink:href="fig-0179-03"/> <anchor type="figure" xlink:label="fig-0179-04a" xlink:href="fig-0179-04"/> tro, & puncti b ab eodem:</s> <s xml:id="echoid-s11562" xml:space="preserve"> diameter a g cum diametro g d, ex parte d faciet angulũ, quem impoſsibi-<lb/>le eſt eſſe æqualẽ angulo conſtanti ex angulo incidentiæ & reflexionis.</s> <s xml:id="echoid-s11563" xml:space="preserve"> Sit enim æqualis:</s> <s xml:id="echoid-s11564" xml:space="preserve"> & t ſit pun <lb/>ctum reflexionis:</s> <s xml:id="echoid-s11565" xml:space="preserve"> & ſit a g inæqualis b g:</s> <s xml:id="echoid-s11566" xml:space="preserve"> & ducantur lineæ t a, t g, t b:</s> <s xml:id="echoid-s11567" xml:space="preserve"> & fiat circulus tranſiẽs per tria <lb/>puncta a, g, b:</s> <s xml:id="echoid-s11568" xml:space="preserve"> [per 5 p 4] qui neceſſariò tranſibit per punctũ t.</s> <s xml:id="echoid-s11569" xml:space="preserve"> Si enim cadit extra:</s> <s xml:id="echoid-s11570" xml:space="preserve"> ductis lineis à pun <lb/> <pb o="174" file="0180" n="180" rhead="ALHAZEN"/> ctis a, b ad idẽ pũctũ illius circuli extrà:</s> <s xml:id="echoid-s11571" xml:space="preserve"> fiet [per 21 p 1] angulus minor angulo a t b:</s> <s xml:id="echoid-s11572" xml:space="preserve"> & probabitur eſſe <lb/>æqualis.</s> <s xml:id="echoid-s11573" xml:space="preserve"> Quoniã [per 22 p 3] cũ angulo a g b ualebit duos rectos, & anguli a g b & a g d ualent duos <lb/>rectos:</s> <s xml:id="echoid-s11574" xml:space="preserve"> [per 13 p 1] & angulus a t b eſt æqualis angulo a g d ex hypotheſi:</s> <s xml:id="echoid-s11575" xml:space="preserve"> ergo angulus a t b cum angu <lb/>lo a g b ualet duos rectos.</s> <s xml:id="echoid-s11576" xml:space="preserve"> Et ita impoſsibile [cõtra 21 p 1.</s> <s xml:id="echoid-s11577" xml:space="preserve">] Similiter ſi circulus citra t ceciderit, eadẽ <lb/>erit improbatio.</s> <s xml:id="echoid-s11578" xml:space="preserve"> Reſtat ergo, ut tranſeat per punctum t.</s> <s xml:id="echoid-s11579" xml:space="preserve"> Cum igitur [per 12 n 4] angulus a t g ſit æqua <lb/>lis angulo b t g:</s> <s xml:id="echoid-s11580" xml:space="preserve"> erit [per 26 p 3] arcus a g æqualis arcui b g:</s> <s xml:id="echoid-s11581" xml:space="preserve"> & ita [per 29 p 3] a g erit æqualis b g:</s> <s xml:id="echoid-s11582" xml:space="preserve"> & po-<lb/>ſitum eſt eſſe eas inæquales.</s> <s xml:id="echoid-s11583" xml:space="preserve"> Et ita eſt propoſitum.</s> <s xml:id="echoid-s11584" xml:space="preserve"/> </p> <div xml:id="echoid-div412" type="float" level="0" n="0"> <figure xlink:label="fig-0179-03" xlink:href="fig-0179-03a"> <variables xml:id="echoid-variables113" xml:space="preserve">t z e b a g h d</variables> </figure> <figure xlink:label="fig-0179-04" xlink:href="fig-0179-04a"> <variables xml:id="echoid-variables114" xml:space="preserve">t z e b a g h d</variables> </figure> </div> </div> <div xml:id="echoid-div414" type="section" level="0" n="0"> <head xml:id="echoid-head383" xml:space="preserve" style="it">80. Si uiſus & uiſibile in diuerſis diametris circuli (qui eſt cõmunis ſectio ſuperficierũ, refle-<lb/>xionis & ſpeculi ſphærici caui) à centro inæquabiliter diſtantia inter ſe reflectãtur à duobus pun <lb/>ctis peripheriæ, cõprehenſæ inter ſemidiametros, in quibus ipſa ſunt: nõ erit uter angulus cõpo <lb/>ſit us ex angulo incidẽtiæ & reflexionis, minor angulo exteriore à dictis diametris facto. 34 p 8.</head> <p> <s xml:id="echoid-s11585" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s11586" xml:space="preserve"> ſumptis in duabus diametris e g h, z g d, duobus pũctis a, b, ut b g ſit maior a g.</s> <s xml:id="echoid-s11587" xml:space="preserve"> Dico, <lb/>quòd ſi punctũ a reflectatur ad b à duobus punctis arcus e z:</s> <s xml:id="echoid-s11588" xml:space="preserve"> nõ erit uterq;</s> <s xml:id="echoid-s11589" xml:space="preserve"> angulus conſtans <lb/>ex angulo incidentię & reflexiõis, minor angulo a g d.</s> <s xml:id="echoid-s11590" xml:space="preserve"> Sumãtur enim duo puncta t, q ιn arcu <lb/>e z, à quib.</s> <s xml:id="echoid-s11591" xml:space="preserve"> a reflectatur ad b:</s> <s xml:id="echoid-s11592" xml:space="preserve"> & ducãtur lineę b t, g t, a t, b q, g q, a q:</s> <s xml:id="echoid-s11593" xml:space="preserve"> & ſi angulus a t b minor eſt angu <lb/>lo a g d:</s> <s xml:id="echoid-s11594" xml:space="preserve"> dico, quòd angulus a q b nõ erit minor a g d.</s> <s xml:id="echoid-s11595" xml:space="preserve"> Sit enim minor:</s> <s xml:id="echoid-s11596" xml:space="preserve"> & ducatur linea g n, diuidẽs an <lb/>gulũ diametrorũ per ęqualia:</s> <s xml:id="echoid-s11597" xml:space="preserve"> [per 9 p 1] & ducatur linea a b, quã diuidat g n per punctũ f.</s> <s xml:id="echoid-s11598" xml:space="preserve"> Palàm [per <lb/>3 p 6] quòd proportio b g ad g a, ſicut b f ad f a:</s> <s xml:id="echoid-s11599" xml:space="preserve"> ſed cũ b g maior ſit g a:</s> <s xml:id="echoid-s11600" xml:space="preserve"> [ex theſi] erit b f maior f a.</s> <s xml:id="echoid-s11601" xml:space="preserve"> Diui <lb/>datur a b per mediũ in puncto k:</s> <s xml:id="echoid-s11602" xml:space="preserve"> [per 10 p 1] & fiat [per 5 p 4] circulus tranſiens per tria puncta a, b, t:</s> <s xml:id="echoid-s11603" xml:space="preserve"> <lb/>qui quidẽ circulus nõ tranſibit per g:</s> <s xml:id="echoid-s11604" xml:space="preserve"> quoniã anguli a g b, b t a eſſent æquales duobus rectis [per 22 p <lb/>3] & palàm, quòd ſunt minores:</s> <s xml:id="echoid-s11605" xml:space="preserve"> cũ [per theſin] angu <lb/> <anchor type="figure" xlink:label="fig-0180-01a" xlink:href="fig-0180-01"/> lus b t a ſit minor angulo a g d [qui cũ angulo a g b ę-<lb/>quatur duobus rectis per 13 p 1.</s> <s xml:id="echoid-s11606" xml:space="preserve">] Igitur trãſibit ſupra <lb/>g.</s> <s xml:id="echoid-s11607" xml:space="preserve"> Similiter nõ trãſibit per q:</s> <s xml:id="echoid-s11608" xml:space="preserve"> quoniã ſumpto puncto <lb/>circuli, in quo linea g q ſecat ipſũ, ſcilicet m:</s> <s xml:id="echoid-s11609" xml:space="preserve"> eſſet ar-<lb/>cus a m æqualis arcui b m [per 26 p 3] cũ reſpiciãt æ-<lb/>quales angulos ſuper q:</s> <s xml:id="echoid-s11610" xml:space="preserve"> [per theſin & 12 n 4:</s> <s xml:id="echoid-s11611" xml:space="preserve"> quia q <lb/>eſt reflexiõis punctũ] quod manet impoſsibile.</s> <s xml:id="echoid-s11612" xml:space="preserve"> Quo <lb/>niam ſumpto puncto o, in quo linea g t ſecat hũc cir.</s> <s xml:id="echoid-s11613" xml:space="preserve"> <lb/>culũ:</s> <s xml:id="echoid-s11614" xml:space="preserve"> erit arcus a o ęqualis arcui o b:</s> <s xml:id="echoid-s11615" xml:space="preserve"> [per 26 p 3] quia <lb/>reſpiciũt ęquales angulos ſuք t [per theſin & 12 n 4:</s> <s xml:id="echoid-s11616" xml:space="preserve"> <lb/>& ſic peripheria b o maior eſſet peripheria b m, pars <lb/>ſuo toto cõtra 9 ax:</s> <s xml:id="echoid-s11617" xml:space="preserve">] Reſtat, ut hic circulus tranſeat <lb/>ſupra q:</s> <s xml:id="echoid-s11618" xml:space="preserve"> ſi enim infra:</s> <s xml:id="echoid-s11619" xml:space="preserve"> eadẽ erit improbatio.</s> <s xml:id="echoid-s11620" xml:space="preserve"> Ducatur <lb/>aũt linea à puncto o ad punctũ k:</s> <s xml:id="echoid-s11621" xml:space="preserve"> quæ quidẽ cum di-<lb/>uidat chordã a b per ęqualia:</s> <s xml:id="echoid-s11622" xml:space="preserve"> [per fabricationẽ] & ſi-<lb/>militer arcũ a b:</s> <s xml:id="echoid-s11623" xml:space="preserve"> [quia peripheria a o æqualis oſtenſa <lb/>eſt ipſi b o] erit perpendicularis ſuper a b.</s> <s xml:id="echoid-s11624" xml:space="preserve"> [rectæ e-<lb/>nim lineæ ſubtendentes peripherias a o, b o, æquales ſunt per 29 p 3, & b k æquatur ipſi k a, & cõmu <lb/>ne latus eſt k o.</s> <s xml:id="echoid-s11625" xml:space="preserve"> Quare per 8 p.</s> <s xml:id="echoid-s11626" xml:space="preserve"> 10 d 1, o k perpendicularis eſt ipſi a b.</s> <s xml:id="echoid-s11627" xml:space="preserve">] Verùm angulus b a g maior an-<lb/>gulo a b g:</s> <s xml:id="echoid-s11628" xml:space="preserve"> [per 18 p 1] cũ b g ſit maior g a:</s> <s xml:id="echoid-s11629" xml:space="preserve"> [ex theli] & angulus b f g ualet duos angulos fa g, f g a [per <lb/>32 p 1] & angulus a f g ualet duos angulos f b g, f g b:</s> <s xml:id="echoid-s11630" xml:space="preserve"> ſed a g f ęqualis eſt f g b:</s> <s xml:id="echoid-s11631" xml:space="preserve"> [per fabricationẽ] & f a g <lb/>maior f b g.</s> <s xml:id="echoid-s11632" xml:space="preserve"> Igitur angulus b f g maior eſt angulo a f g:</s> <s xml:id="echoid-s11633" xml:space="preserve"> igitur b f g maior eſt recto:</s> <s xml:id="echoid-s11634" xml:space="preserve"> [per 13 p 1] quare n f <lb/>b minor eſt recto.</s> <s xml:id="echoid-s11635" xml:space="preserve"> [per 13 p 1.</s> <s xml:id="echoid-s11636" xml:space="preserve">] Sed o k ſuper f b facit angulũ rectũ:</s> <s xml:id="echoid-s11637" xml:space="preserve"> ergo producta cõcurret cũ g n [per <lb/>11 ax:</s> <s xml:id="echoid-s11638" xml:space="preserve">] ſupra b f, & inferius nunꝗ̃.</s> <s xml:id="echoid-s11639" xml:space="preserve"> [ſecus per 3 p 6 b k <lb/> <anchor type="figure" xlink:label="fig-0180-02a" xlink:href="fig-0180-02"/> fieret maior k a, cui eſt æquata.</s> <s xml:id="echoid-s11640" xml:space="preserve">] Facto autem circulo <lb/>trãſeunte per tria pũcta a, q, b:</s> <s xml:id="echoid-s11641" xml:space="preserve"> trãſibit ſupra g.</s> <s xml:id="echoid-s11642" xml:space="preserve"> [Quia <lb/>ſi trãſiret per punctũ g:</s> <s xml:id="echoid-s11643" xml:space="preserve"> eſſent anguli a q b, a g b æqua <lb/>les duob.</s> <s xml:id="echoid-s11644" xml:space="preserve"> rectis per 22 p 3:</s> <s xml:id="echoid-s11645" xml:space="preserve"> & anguli a g b, a g d æquan <lb/>tur duob.</s> <s xml:id="echoid-s11646" xml:space="preserve"> rectis per 13 p 1.</s> <s xml:id="echoid-s11647" xml:space="preserve"> Quare per 3 ax.</s> <s xml:id="echoid-s11648" xml:space="preserve"> a q b æqua-<lb/>retur a g d:</s> <s xml:id="echoid-s11649" xml:space="preserve"> cõtra præcedentẽ numerũ] & g q diuidet <lb/>arcũ eius a b per æqualia [quia enim q ex theſi eſt re-<lb/>flexiõis punctũ:</s> <s xml:id="echoid-s11650" xml:space="preserve"> æquãtur anguli g q a, g q b per 12 n 4 <lb/>& per 26 p 3 peripheria a b bifariã ſecabitur à recta g <lb/>q] ſed k o diuidit chordã a b per æqualia [per fabrica <lb/>tionẽ.</s> <s xml:id="echoid-s11651" xml:space="preserve">] Ergo k o cõcurret cũ g n infra b f, & ſupra pũ-<lb/>ctũ g.</s> <s xml:id="echoid-s11652" xml:space="preserve"> Igitur k o cõcurrens cũ b a, prius cõcurret cum <lb/>g n infra b f:</s> <s xml:id="echoid-s11653" xml:space="preserve"> & iam improbatũ eſt.</s> <s xml:id="echoid-s11654" xml:space="preserve"> Reſtat ergo, ut an-<lb/>gulus a q b nõ ſit minor angulo a g d:</s> <s xml:id="echoid-s11655" xml:space="preserve"> aut quòd a nõ <lb/>reflectetur ad b à pũcto q [cõtra theſin.</s> <s xml:id="echoid-s11656" xml:space="preserve">] Similis erit <lb/>improbatio, ſumpto quolibet puncto arcus e n.</s> <s xml:id="echoid-s11657" xml:space="preserve"> Sũ-<lb/>pto aũt puncto in arcu n z:</s> <s xml:id="echoid-s11658" xml:space="preserve"> qui ſit p:</s> <s xml:id="echoid-s11659" xml:space="preserve"> fiat reflexio puncti a ad b à puncto p, ut angulus cõſtans ex angu <lb/>lo incidẽtię & reflexiõis ſuprap, ſit minor angulo a g d, ſicut angulus cõſtãs ex angulo incidẽtię & re <lb/>flexionis ſuprat, minor eſt eodẽ.</s> <s xml:id="echoid-s11660" xml:space="preserve"> Improbabitur aũt hoc modo, Ducãtur a p, b p, g p:</s> <s xml:id="echoid-s11661" xml:space="preserve"> oportet ergo ne-<lb/> <pb o="175" file="0181" n="181" rhead="OPTICAE LIBER V."/> ceſſariò, ut g p diuidat k o propter arcum a b, qué diuidit ex circulo a b t linea g t per æqualia:</s> <s xml:id="echoid-s11662" xml:space="preserve"> [peri-<lb/>pheria enim b o æquatur peripheriæ c a ex concluſo:</s> <s xml:id="echoid-s11663" xml:space="preserve">] & ſimiliter linea k o.</s> <s xml:id="echoid-s11664" xml:space="preserve"> Sit ergo punctum con-<lb/>curſus lineæ g p cum k o, punctum l:</s> <s xml:id="echoid-s11665" xml:space="preserve"> & ducatur linea t p.</s> <s xml:id="echoid-s11666" xml:space="preserve"> Cum igitur duæ lineæ g p, g t ſint æquales:</s> <s xml:id="echoid-s11667" xml:space="preserve"> <lb/>[per15 d 1] erũt [per 5 p 1] duo anguli g p t, g t p æquales:</s> <s xml:id="echoid-s11668" xml:space="preserve"> & [per 32 p 1] uterq;</s> <s xml:id="echoid-s11669" xml:space="preserve"> acutus.</s> <s xml:id="echoid-s11670" xml:space="preserve"> Ductaigitur <lb/>perpendiculari ſuper g t à punctot:</s> <s xml:id="echoid-s11671" xml:space="preserve"> [per 11 p 1] cõtingetcirculum ſpeculi [per conſectarium 16 p 3] <lb/>& producta, cadet ſuper terminum diametri minoris circuli:</s> <s xml:id="echoid-s11672" xml:space="preserve"> cum angulus, quem efficit cum g t, re-<lb/>ipiciat arcum ſemicirculi minoris circuli:</s> <s xml:id="echoid-s11673" xml:space="preserve"> [per 31 p 3] & cũ to cadatſuprako, & k o producta tran-<lb/>ſeat per cẽtrum minoris circuli:</s> <s xml:id="echoid-s11674" xml:space="preserve"> [per conſectarium 1 p 3, quia recta linea o k bifariam, & ad angulos <lb/>rectos ſecat rectam a b] neceſſario illa perpendicularis cadet ſuper terminum k o producta:</s> <s xml:id="echoid-s11675" xml:space="preserve"> [per 31 <lb/>p 3] & p t eſt inſerior illa perpẽdiculari, habito reſpectu ad n.</s> <s xml:id="echoid-s11676" xml:space="preserve"> Igitur quæcung;</s> <s xml:id="echoid-s11677" xml:space="preserve"> linea ducatur à pun-<lb/>cto g ad lineam t p, ſecans diametrum illius circuli, quæ eſt o k:</s> <s xml:id="echoid-s11678" xml:space="preserve"> cadet in punctum aliquod lineæ t p, <lb/>citra illam perpendicularem.</s> <s xml:id="echoid-s11679" xml:space="preserve"> Cum igitur g p cadat in p, & ſecet o k:</s> <s xml:id="echoid-s11680" xml:space="preserve"> erit p citra perpendicularem, & <lb/>inſra arcum illius perpendicularis.</s> <s xml:id="echoid-s11681" xml:space="preserve"> Facto igitur circulo tranſeunte per tria puncta a, b, p:</s> <s xml:id="echoid-s11682" xml:space="preserve"> tranſibit <lb/>quidem per l, & ſecabit circulum a b t in duobus punctis a, b:</s> <s xml:id="echoid-s11683" xml:space="preserve"> & cum exeat à puncto b, & iterum re:</s> <s xml:id="echoid-s11684" xml:space="preserve"> <lb/>deat in punctum p, inferius punctot, cum p ſit citra illum circulum:</s> <s xml:id="echoid-s11685" xml:space="preserve"> neceſſariò ſecabit illum in ter-<lb/>tio puncto:</s> <s xml:id="echoid-s11686" xml:space="preserve"> quod eſt impoſsibile [& contra 10 p 3.</s> <s xml:id="echoid-s11687" xml:space="preserve">] Reſtat ergo, ut punctum a non reflectatur ad b à <lb/>duobus punctis arcus, interiacentis eorum diametros, id eſt arcus e z, ut uterq;</s> <s xml:id="echoid-s11688" xml:space="preserve"> angulus conſtans <lb/>ex angulo incidentiæ & reflexionis ſit minor angulo a g d.</s> <s xml:id="echoid-s11689" xml:space="preserve"/> </p> <div xml:id="echoid-div414" type="float" level="0" n="0"> <figure xlink:label="fig-0180-01" xlink:href="fig-0180-01a"> <variables xml:id="echoid-variables115" xml:space="preserve">z t n q p i b k f e l a n m g h d</variables> </figure> <figure xlink:label="fig-0180-02" xlink:href="fig-0180-02a"> <variables xml:id="echoid-variables116" xml:space="preserve">z t n q b k f a e o g h d</variables> </figure> </div> </div> <div xml:id="echoid-div416" type="section" level="0" n="0"> <head xml:id="echoid-head384" xml:space="preserve" style="it">81. Duo punctain diuerſis diametris circuli (qui eſt cõmunis ſectio ſuperficierum, reflexio-<lb/>nis, & ſpeculi ſphærici caui) à centro inæquabiliter diſtantia: à duobus punctis peripheriæ com-<lb/>prehenſæ inter ſemidiametros, in quibus ipſa ſunt, inter ſe mutuò reflecti poſſunt. 35 p 8.</head> <p> <s xml:id="echoid-s11690" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s11691" xml:space="preserve"> dico quòd poſſunt reflecti duo puncta ad ſe, inæqualis longitudinis à centro, à duo-<lb/>bus punctis arcus ipſa reſpicientis, id eſt diametros, in quibus ſunt puncta illa, interiacẽtis.</s> <s xml:id="echoid-s11692" xml:space="preserve"> <lb/>Verbi gratia:</s> <s xml:id="echoid-s11693" xml:space="preserve"> ſumptis duabus ſemidiametris in circulo ſpheræ, ſcilicet b d, g d:</s> <s xml:id="echoid-s11694" xml:space="preserve"> diuidatur an-<lb/>gulus earũ p æqualia, perſemidiametrũ e d:</s> <s xml:id="echoid-s11695" xml:space="preserve"> [per 9 p 1] & in b d ſumatur punctũ m, ſupra punctũ, <lb/>in quod cadet perpendicularis ducta à puncto e ſuper b d:</s> <s xml:id="echoid-s11696" xml:space="preserve"> & ſumatur [per 3 p 1] n d æqualis m d:</s> <s xml:id="echoid-s11697" xml:space="preserve"> & <lb/>[per 5 p 4] fiat circulus tranſiens per tria puncta d, m, n:</s> <s xml:id="echoid-s11698" xml:space="preserve"> neceſſariò circulus ille tranſibit extra e.</s> <s xml:id="echoid-s11699" xml:space="preserve"> Si <lb/>enim per e:</s> <s xml:id="echoid-s11700" xml:space="preserve"> fieret quadrangulũ à quatuor punctis d, n, e, m:</s> <s xml:id="echoid-s11701" xml:space="preserve"> & duo anguli illius qua dranguli ſibi op-<lb/>poſiti ſunt æquales duobus rectis:</s> <s xml:id="echoid-s11702" xml:space="preserve"> [per 22 p 3] quod quidẽ non eſſet:</s> <s xml:id="echoid-s11703" xml:space="preserve"> cum linea e m ſit ſupra perpen <lb/>dicularem:</s> <s xml:id="echoid-s11704" xml:space="preserve"> & ideo angulus e m d acutus:</s> <s xml:id="echoid-s11705" xml:space="preserve"> [per 16 p.</s> <s xml:id="echoid-s11706" xml:space="preserve"> 12 d 1] & ſimiliter ei oppoſitus ſuper n, acutus:</s> <s xml:id="echoid-s11707" xml:space="preserve"> <lb/>quia e n ſupra perpendicularem eſt.</s> <s xml:id="echoid-s11708" xml:space="preserve"> [Quare in quadrilatero circulo inſcripto oppoſiti anguli eſſent <lb/>minores duobus rectis contra 22 p 3.</s> <s xml:id="echoid-s11709" xml:space="preserve">] Similis erit improbatio:</s> <s xml:id="echoid-s11710" xml:space="preserve"> ſi tranſeat circulus citra e.</s> <s xml:id="echoid-s11711" xml:space="preserve"> Tranſibit <lb/>ergo extra, & [per 10 p 3] ſecabit circulũ ſphæræ in duobus punctis, ſicut t, l:</s> <s xml:id="echoid-s11712" xml:space="preserve"> & ducantur lineæ m t, <lb/>d t, n t, m i, d l, n l:</s> <s xml:id="echoid-s11713" xml:space="preserve"> & ducatur linea m n ſecans t d in puncto f, lineam e d in puncto p.</s> <s xml:id="echoid-s11714" xml:space="preserve"> Palàm, cum m d <lb/>ſit æqualis n d [per fabricationem] & p d cõmunis, & angulus n d p æqualis angulo m d p:</s> <s xml:id="echoid-s11715" xml:space="preserve"> [per fa-<lb/>bricationem] erit [per 4 p 1] triangulum æquale triangulo:</s> <s xml:id="echoid-s11716" xml:space="preserve"> & erit angulus f p d rectus:</s> <s xml:id="echoid-s11717" xml:space="preserve"> [per 10 d 1] <lb/>igitur angulus p f d acutus [per 32 p 1] Ducatur [per 11 p 1] à pũcto f perpendicularis ſupert d:</s> <s xml:id="echoid-s11718" xml:space="preserve"> quæ <lb/>ſit k f.</s> <s xml:id="echoid-s11719" xml:space="preserve"> Palàm, quòd aliquod punctũ lineę n l, erit infe-<lb/> <anchor type="figure" xlink:label="fig-0181-01a" xlink:href="fig-0181-01"/> rius pũcto k, ſumpta inferioritate reſpectu n:</s> <s xml:id="echoid-s11720" xml:space="preserve"> ſitillud <lb/>punctũ z:</s> <s xml:id="echoid-s11721" xml:space="preserve"> & ducatur t z linea uſq;</s> <s xml:id="echoid-s11722" xml:space="preserve">; ad circulũ, cadẽs in <lb/>punctũ circuli:</s> <s xml:id="echoid-s11723" xml:space="preserve"> quod ſit o.</s> <s xml:id="echoid-s11724" xml:space="preserve"> Arcus n o aut minor eſt ar-<lb/>cu tl:</s> <s xml:id="echoid-s11725" xml:space="preserve"> aut nõ Sinõ fuerit minor:</s> <s xml:id="echoid-s11726" xml:space="preserve"> ſumatur ex eo arcus <lb/>minor;</s> <s xml:id="echoid-s11727" xml:space="preserve"> & ad terminũ illius arcus ducatur linea à pun <lb/>cto t:</s> <s xml:id="echoid-s11728" xml:space="preserve"> & erit idẽ, ac ſi arcus n o eſſet minor arcutl.</s> <s xml:id="echoid-s11729" xml:space="preserve"> Sit <lb/>igitur n o minortl.</s> <s xml:id="echoid-s11730" xml:space="preserve"> Palàm [per 33 p 6] angulus t n l <lb/>erit maior angulo o t n, quia reſpicit maiorẽ arcum.</s> <s xml:id="echoid-s11731" xml:space="preserve"> <lb/>Secetur ex eo æqualis:</s> <s xml:id="echoid-s11732" xml:space="preserve"> & ſit i n z:</s> <s xml:id="echoid-s11733" xml:space="preserve"> & ſuper punctum t <lb/>lineæ t m, fiat angulus, æqualis angulo o t n [ք 23 p 1] <lb/>qui ſit q t m.</s> <s xml:id="echoid-s11734" xml:space="preserve"> Cum igiturangulus t m l ſit maior angu-<lb/>lo m t q:</s> <s xml:id="echoid-s11735" xml:space="preserve"> [ք 33 p 6:</s> <s xml:id="echoid-s11736" xml:space="preserve"> quia peripheria t l ſubtenſa angulo <lb/>t m l, maior eſt extheſi, peripheria n o, ſubtẽſa angu-<lb/>lo n t o, cui æquatus eſt angulus m t q] cõcurret linea <lb/>t q cũ linea l m:</s> <s xml:id="echoid-s11737" xml:space="preserve"> cõcurrat in puncto q.</s> <s xml:id="echoid-s11738" xml:space="preserve"> Cum igitur an-<lb/>gulus l m t ſit æqualis duob.</s> <s xml:id="echoid-s11739" xml:space="preserve"> angulis m q t, m t q [per <lb/>32 p 1] & angulus l n t ſit ęqualis l m t [ք 27 p 3] ꝗa ſunt ſuք eũdẽ arcũ:</s> <s xml:id="echoid-s11740" xml:space="preserve"> [l t] & ang<gap/><gap/>us in z ſit ęqualis <lb/>in t q:</s> <s xml:id="echoid-s11741" xml:space="preserve"> [ք ſabricationẽ] erit angulus int æqualis angulo m q t:</s> <s xml:id="echoid-s11742" xml:space="preserve"> & ita triangulũ m q t ſimile triangulo <lb/>int [eſt enim angulus m t q æquatus angulo o t n:</s> <s xml:id="echoid-s11743" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s11744" xml:space="preserve"> ք 32 p 1 triãgula m t q, i t n ſunt æquiangula:</s> <s xml:id="echoid-s11745" xml:space="preserve"> <lb/>& ք 4 p.</s> <s xml:id="echoid-s11746" xml:space="preserve"> 1 d 6 ſimilia.</s> <s xml:id="echoid-s11747" xml:space="preserve">] Et ſimiliter triangulũ i n z eſt ſimile triãgulo t n z:</s> <s xml:id="echoid-s11748" xml:space="preserve"> [cõmunis enim eſt angulus <lb/>n z t:</s> <s xml:id="echoid-s11749" xml:space="preserve"> & z n i æquatus eſt ipſi o t n:</s> <s xml:id="echoid-s11750" xml:space="preserve"> ergo ք 32 p 1.</s> <s xml:id="echoid-s11751" xml:space="preserve">4 p.</s> <s xml:id="echoid-s11752" xml:space="preserve">1 d 6 triãgula ſunt ſimilia] & ita ꝓportio n t ad t q, <lb/>ſicut n i ad m q:</s> <s xml:id="echoid-s11753" xml:space="preserve"> & ſimiliter ꝓportio t n ad t z, ſicut in ad n z.</s> <s xml:id="echoid-s11754" xml:space="preserve"> Sed t z maior t q:</s> <s xml:id="echoid-s11755" xml:space="preserve"> qđ ſic patet.</s> <s xml:id="echoid-s11756" xml:space="preserve"> Sit r pun-<lb/>ctũ, in quo t z ſecat k f.</s> <s xml:id="echoid-s11757" xml:space="preserve"> Angulus t freſtrectus:</s> <s xml:id="echoid-s11758" xml:space="preserve"> [nã k f քpẽdicularis ducta eſt ſuք t d] quare [ք 32 p 1] <lb/>angul<emph style="sub">9</emph> ſtracutus.</s> <s xml:id="echoid-s11759" xml:space="preserve"> Igitur angul<emph style="sub">9</emph> qtfei ęqualis.</s> <s xml:id="echoid-s11760" xml:space="preserve"> [Quia enim ex theſi recta d m æquatur ipſi d n:</s> <s xml:id="echoid-s11761" xml:space="preserve"> æqua <lb/>bitur peripheria d m peripheriæ d n ք 28 p 3:</s> <s xml:id="echoid-s11762" xml:space="preserve"> & angulus d t m angulo d t n:</s> <s xml:id="echoid-s11763" xml:space="preserve"> & m t q æquatus eſt o t n.</s> <s xml:id="echoid-s11764" xml:space="preserve"> <lb/> <pb o="176" file="0182" n="182" rhead="ALHAZEN"/> Totusigitur ſtr æquatur toti ft q] eſt acutus:</s> <s xml:id="echoid-s11765" xml:space="preserve"> & k f perpendicularis ſupert d.</s> <s xml:id="echoid-s11766" xml:space="preserve"> Quare [per 11 ax.</s> <s xml:id="echoid-s11767" xml:space="preserve">] k f <lb/>producta concurret cum t q:</s> <s xml:id="echoid-s11768" xml:space="preserve"> ſit concurſus s:</s> <s xml:id="echoid-s11769" xml:space="preserve"> & linea t s ducta à puncto t ad punctum concurſus, cu-<lb/>ius lineæ pars eſt t q:</s> <s xml:id="echoid-s11770" xml:space="preserve"> erit æqualis lineæ t r:</s> <s xml:id="echoid-s11771" xml:space="preserve"> [quia anguli ad ſſunt recti, & ft r, ft s æquantur, latusq́;</s> <s xml:id="echoid-s11772" xml:space="preserve"> t <lb/>fcommune:</s> <s xml:id="echoid-s11773" xml:space="preserve"> æquabitur r t ipſit s per 26 p 1] & ita t q minortz [quia minor eſt ipſatr, quæ pars eſt <lb/>ipſius t z.</s> <s xml:id="echoid-s11774" xml:space="preserve">] Quare [per 8 p 5] maior eſt proportio n t ad t q, quàm n t ad tz.</s> <s xml:id="echoid-s11775" xml:space="preserve"> Igitur maior eſt propor-<lb/>tio in ad m q, quàm in ad n z.</s> <s xml:id="echoid-s11776" xml:space="preserve"> Quare [per 10 p 5] m q minor eſt n z.</s> <s xml:id="echoid-s11777" xml:space="preserve"> Secetur igitur exn z æqualis ei <lb/>[per 3 p 1] quæ ſit n x.</s> <s xml:id="echoid-s11778" xml:space="preserve"> Quoniam [per 22 p 3] angulus l n d cum angulo l m d ualet duos rectos:</s> <s xml:id="echoid-s11779" xml:space="preserve"> erit <lb/>[per 13 p 1.</s> <s xml:id="echoid-s11780" xml:space="preserve"> 3 ax] angulus l n d æqualis q m d:</s> <s xml:id="echoid-s11781" xml:space="preserve"> & x n, n d, æqualia q m, m d.</s> <s xml:id="echoid-s11782" xml:space="preserve"> Igitur [per 4 p 1] q d æqua-<lb/>lis x d.</s> <s xml:id="echoid-s11783" xml:space="preserve"> Sed z d maior x d:</s> <s xml:id="echoid-s11784" xml:space="preserve"> quoniam angulus l n d cum angulo l m d ualet duos rectos:</s> <s xml:id="echoid-s11785" xml:space="preserve"> [per 22 p 3] ſed <lb/>angulus l m d acutus:</s> <s xml:id="echoid-s11786" xml:space="preserve"> cum angulus e m d ſit acutus [per 16 p.</s> <s xml:id="echoid-s11787" xml:space="preserve"> 12 d 1.</s> <s xml:id="echoid-s11788" xml:space="preserve">] Igitur angulus l n d maior eſt <lb/>recto:</s> <s xml:id="echoid-s11789" xml:space="preserve"> igitur z d maior x d [quia enim angulus l n d eſt obtuſus:</s> <s xml:id="echoid-s11790" xml:space="preserve"> erit per 32 p 1 n x d acutus, & per 13.</s> <s xml:id="echoid-s11791" xml:space="preserve"> <lb/>32 p 1 z x d obtuſus, x z d acutus:</s> <s xml:id="echoid-s11792" xml:space="preserve"> quare per 19 p 1 z d maior eſt x d.</s> <s xml:id="echoid-s11793" xml:space="preserve">] Quare z d<unsure/> maior q d.</s> <s xml:id="echoid-s11794" xml:space="preserve"> Igitur q re-<lb/>flectitur ad z à duobus punctis t, l:</s> <s xml:id="echoid-s11795" xml:space="preserve"> & q & z ſuntinęqualis longitudinis à centro, & in diuerſis dia-<lb/>metris.</s> <s xml:id="echoid-s11796" xml:space="preserve"> Et quòd non ſint in eadem diametro, palàm:</s> <s xml:id="echoid-s11797" xml:space="preserve"> quoniam angulus x d n æqualis eſt angulo q d <lb/>m:</s> <s xml:id="echoid-s11798" xml:space="preserve"> addito ergo communiangulo x d m, erit angulus n d m æqualis angulo x d q:</s> <s xml:id="echoid-s11799" xml:space="preserve"> & minor duobus <lb/>rectis.</s> <s xml:id="echoid-s11800" xml:space="preserve"> Quare magis angulus z d q [pars anguli x d q] minor duobus rectis.</s> <s xml:id="echoid-s11801" xml:space="preserve"> Quare q & z non ſunt in <lb/>eadem diametro, ſed in diuerſis.</s> <s xml:id="echoid-s11802" xml:space="preserve"/> </p> <div xml:id="echoid-div416" type="float" level="0" n="0"> <figure xlink:label="fig-0181-01" xlink:href="fig-0181-01a"> <variables xml:id="echoid-variables117" xml:space="preserve">k e <gap/> t o z r l g b x n p f m q d s n a</variables> </figure> </div> </div> <div xml:id="echoid-div418" type="section" level="0" n="0"> <head xml:id="echoid-head385" xml:space="preserve" style="it">82. Siduo punctain diuerſis diametris circuli (qui eſt communis ſectio ſuperficierum, refle-<lb/>xionis & ſpeculi ſphærici caui) à centro inæquabiliter diſtantia, à duobus punctis peripheriæ <lb/>comprebenſæ inter ſemidiametros, in quibus ipſa ſunt, inter ſe mutuò reflect antur: à nullo alio <lb/>eiuſdem peripheriæ puncto reflecti poſſunt. 36 p 8.</head> <p> <s xml:id="echoid-s11803" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s11804" xml:space="preserve"> ſumptis duobus punctis, quæ ſint o, k, & inæqualiter diſtantibus à cẽtro:</s> <s xml:id="echoid-s11805" xml:space="preserve"> reflectetur <lb/>quidẽ unum ad aliud à duob pũctis arcus, reſpiciẽtis ſemidiametros, in quib.</s> <s xml:id="echoid-s11806" xml:space="preserve"> ſunt:</s> <s xml:id="echoid-s11807" xml:space="preserve"> ſed nõ ab <lb/>alio pũcto illius arcus, quàm ab illis duob.</s> <s xml:id="echoid-s11808" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s11809" xml:space="preserve"> d ſit centrũ:</s> <s xml:id="echoid-s11810" xml:space="preserve"> k remotius à d quàm o à <lb/>d:</s> <s xml:id="echoid-s11811" xml:space="preserve"> g d, b d ſemidiam etri:</s> <s xml:id="echoid-s11812" xml:space="preserve"> t punctũ unũ reflexionis.</s> <s xml:id="echoid-s11813" xml:space="preserve"> Palàm ex ſuperioribus, quod uterq;</s> <s xml:id="echoid-s11814" xml:space="preserve"> angulus con-<lb/>ſtans ex angulo incidẽtiæ & reflexionis, nõ erit min or angulo o d a:</s> <s xml:id="echoid-s11815" xml:space="preserve"> [ք 80 n] nec ęqualis [per 79 n] <lb/>alter ergo erit maior.</s> <s xml:id="echoid-s11816" xml:space="preserve"> Sit angulus cõſtans ex angulo incidẽtiæ & reflexionis, qui eſt ſuք t, maior an-<lb/>gulo o d a:</s> <s xml:id="echoid-s11817" xml:space="preserve"> & ducãtur lineæ o t, d t, k t:</s> <s xml:id="echoid-s11818" xml:space="preserve"> & ex angulo illo ſecetur angulus æqualis angulo o d a:</s> <s xml:id="echoid-s11819" xml:space="preserve"> [ք 23 <lb/>p 1] quiſit o t ſ:</s> <s xml:id="echoid-s11820" xml:space="preserve"> & diuidatur angulus f t k per æqualia per lineã t e [per 9 p 1] & à puncto k ducatur <lb/>æquidiſtãs t f:</s> <s xml:id="echoid-s11821" xml:space="preserve"> [per 31 p 1] quæ quidẽ cõcurret cũ te:</s> <s xml:id="echoid-s11822" xml:space="preserve"> [per lemma Procli ad 29 p 1] cõcurrat in pũcto <lb/>z:</s> <s xml:id="echoid-s11823" xml:space="preserve"> & ducatur linea o k:</s> <s xml:id="echoid-s11824" xml:space="preserve"> & diuidatur angulus o d k peræqualia, per lineã d u, ſecantẽlineã o k in pun-<lb/>cto p:</s> <s xml:id="echoid-s11825" xml:space="preserve"> & eſt k d maior o d [extheſi.</s> <s xml:id="echoid-s11826" xml:space="preserve">] Cũ igitur [per 3 p 6] ſit ꝓportio k d ad d o, ſicut k p ad p o:</s> <s xml:id="echoid-s11827" xml:space="preserve"> erit <lb/>k p maior p o.</s> <s xml:id="echoid-s11828" xml:space="preserve"> Itẽlinead t ſecet lineã o k in puncton.</s> <s xml:id="echoid-s11829" xml:space="preserve"> Dico, quod p cadit inter n & k, nõ inter n & o, <lb/>quod ſic patebit.</s> <s xml:id="echoid-s11830" xml:space="preserve"> Angulus k p dualet duos angulos p d o, p o d:</s> <s xml:id="echoid-s11831" xml:space="preserve"> & angulus o p d ualet duos angulos <lb/>p k d & p d k [per 32 p 1.</s> <s xml:id="echoid-s11832" xml:space="preserve">] Sed angulus p d o æqualis eſt angulo p d k:</s> <s xml:id="echoid-s11833" xml:space="preserve"> [per fabricationem] & [per <lb/>the ſim & 18 p 1] angulus k o d maior angulo o k d:</s> <s xml:id="echoid-s11834" xml:space="preserve"> igitur angulus k p d maior angulo o p d:</s> <s xml:id="echoid-s11835" xml:space="preserve"> igitur <lb/>[ք 13 p 1] angulus k p d maior recto:</s> <s xml:id="echoid-s11836" xml:space="preserve"> & angulus k n d <lb/> <anchor type="figure" xlink:label="fig-0182-01a" xlink:href="fig-0182-01"/> acutus:</s> <s xml:id="echoid-s11837" xml:space="preserve"> quod ſic cõſtabit:</s> <s xml:id="echoid-s11838" xml:space="preserve"> ſi fiat circulus per tria pũcta <lb/>o, t, k:</s> <s xml:id="echoid-s11839" xml:space="preserve"> [per 5 p 4] tranſibit infra d.</s> <s xml:id="echoid-s11840" xml:space="preserve"> Quoniã ſi tranſeat <lb/>per d:</s> <s xml:id="echoid-s11841" xml:space="preserve"> cũ angulus o t k ſit maior angulo o d a:</s> <s xml:id="echoid-s11842" xml:space="preserve"> [ք the-<lb/>ſin] erũtduo anguli o t k, o d k maiores duobus rectis <lb/>[cõtra 22 p 3.</s> <s xml:id="echoid-s11843" xml:space="preserve">] Si tranſeat ſupra d:</s> <s xml:id="echoid-s11844" xml:space="preserve"> eadẽ eſt demõſtra-<lb/>tio.</s> <s xml:id="echoid-s11845" xml:space="preserve"> Et linea n d diuidet arcũ illius circuli, qui eſt o k, ք <lb/>æqualia infra d.</s> <s xml:id="echoid-s11846" xml:space="preserve"> [Quia cum t ſit reflexionis punctũ ex <lb/>theſi:</s> <s xml:id="echoid-s11847" xml:space="preserve"> æquabuntur anguli k t d, d t o per 12 n 4, & peri-<lb/>pheriæ illis ſubtenſæ per 26 p 3.</s> <s xml:id="echoid-s11848" xml:space="preserve">] Si autẽ à pũcto diui-<lb/>ſionis ducatur linea ad mediũ punctũ lineæ o k:</s> <s xml:id="echoid-s11849" xml:space="preserve"> quæ <lb/>eſt chorda illius arcus:</s> <s xml:id="echoid-s11850" xml:space="preserve"> erit linea illa perpendicularis <lb/>ſuper o k:</s> <s xml:id="echoid-s11851" xml:space="preserve"> [rectę enim lineæ à puncto medio periphe-<lb/>riæ k o, ductæ ad puncta k & o, æquantur per 29 p3:</s> <s xml:id="echoid-s11852" xml:space="preserve"> & <lb/>recta, quę ab eodem puncto connectit medium rectæ <lb/>k o, æquatur ſibijpſi.</s> <s xml:id="echoid-s11853" xml:space="preserve"> Quare per 8 p.</s> <s xml:id="echoid-s11854" xml:space="preserve"> 10 d 1 ipſa perpen <lb/>dicularis eſt ad k o] & cadet inter p & k:</s> <s xml:id="echoid-s11855" xml:space="preserve"> cũ p k ſit ma-<lb/>ior p o:</s> <s xml:id="echoid-s11856" xml:space="preserve"> [excõcluſo] & angulus ſuper n à parte illius perpẽdicularis & ex parte p erit acutus:</s> <s xml:id="echoid-s11857" xml:space="preserve"> [per <lb/>32 p 1] & angulus ſuper p ex parte o eſt acutus [per 13 p 1:</s> <s xml:id="echoid-s11858" xml:space="preserve"> oſtenſum enim eſt angulum k p d eſſe ob-<lb/>tuſum.</s> <s xml:id="echoid-s11859" xml:space="preserve">] Si ergo p cadit inter n & o:</s> <s xml:id="echoid-s11860" xml:space="preserve"> impoſsibile erit perpẽdicularem illam cadere inter n & p:</s> <s xml:id="echoid-s11861" xml:space="preserve"> quia <lb/>ſecatet d p, & fieret triangulũ, cuius unus angulus rectus, alius obtuſus [contra 32 p 1.</s> <s xml:id="echoid-s11862" xml:space="preserve">] Cadet ergo <lb/>intern & k:</s> <s xml:id="echoid-s11863" xml:space="preserve"> & erit angulus n ex parte perpendicularis acutus:</s> <s xml:id="echoid-s11864" xml:space="preserve"> igitur ex parte o obtuſus [per 13 p 1:</s> <s xml:id="echoid-s11865" xml:space="preserve">] <lb/>ergo p non cadit inter n & o:</s> <s xml:id="echoid-s11866" xml:space="preserve"> quia ita erit triangulum, cuius duo anguli obtuſi [eſt enim angulus k <lb/>p d obtuſus concluſus.</s> <s xml:id="echoid-s11867" xml:space="preserve">] Palàm, quòd angulus k t d eſt medietas anguli k t o:</s> <s xml:id="echoid-s11868" xml:space="preserve"> [per theſin & 12 n 4:</s> <s xml:id="echoid-s11869" xml:space="preserve"> <lb/>quia t eſt punctũ reflexionis:</s> <s xml:id="echoid-s11870" xml:space="preserve"> & d t perpendicularis eſt plano ſpeculum in puncto t tangenti per 25 <lb/>n 4] ſed k t e eſtmedietas anguli k t f [per fabricationem.</s> <s xml:id="echoid-s11871" xml:space="preserve">] Reſtat e t d medietas angulift o:</s> <s xml:id="echoid-s11872" xml:space="preserve"> ſed fto <lb/> <pb o="177" file="0183" n="183" rhead="OPTICAE LIBER V."/> æqualis eſt angulo o d a [per ſabricationem:</s> <s xml:id="echoid-s11873" xml:space="preserve">] igitur e t d medietas anguli o d a:</s> <s xml:id="echoid-s11874" xml:space="preserve"> ſed angulus o d a cũ <lb/>angulo o d f ualet duos rectos [per 13 p 1] & [per 32 p 1] tres anguli trianguli e t d duos rectos:</s> <s xml:id="echoid-s11875" xml:space="preserve"> ab-<lb/>lato e d t cõmuni:</s> <s xml:id="echoid-s11876" xml:space="preserve"> reſtat angulus t e d æqualis medietati anguli o d a, & angulo o d n [nam poſt ſub-<lb/>ductionem communis anguli t d e, relinquũtur anguli d t e, t e d æquales angulis o d t, o d a:</s> <s xml:id="echoid-s11877" xml:space="preserve"> ſed d t e <lb/>æquatur dimidiato angulo o d a, ut patuit:</s> <s xml:id="echoid-s11878" xml:space="preserve"> reliquus igitur t e d ęquatur dimidiato angulo o d a & an <lb/>gulo o d n ſimul utriq;</s> <s xml:id="echoid-s11879" xml:space="preserve">.] Sed angulus o d p cũ medietate anguli o d a eſt rectus:</s> <s xml:id="echoid-s11880" xml:space="preserve"> [ꝗ a enim anguli o d <lb/>k, o d a æquãtur duobus rectis per 13 p 1:</s> <s xml:id="echoid-s11881" xml:space="preserve"> & angulus o d u eſt dimidius anguli o d k per fabricationẽ:</s> <s xml:id="echoid-s11882" xml:space="preserve"> <lb/>duo igitur dimidiati anguli duorũ rectorũ æquãtur uni recto] igitur angulus t e d eſt acutus:</s> <s xml:id="echoid-s11883" xml:space="preserve"> [quia <lb/>enim angulus o d p cum dimidiato angulo o d a æquatur unirecto ex concluſo:</s> <s xml:id="echoid-s11884" xml:space="preserve"> & maior eſt angulo <lb/>o d n:</s> <s xml:id="echoid-s11885" xml:space="preserve"> quia, ut patuit, n cadit inter p & o:</s> <s xml:id="echoid-s11886" xml:space="preserve"> ergo angulus t e d æqualis angulo o d n, & dimidiato o d a, <lb/>erit minor recto:</s> <s xml:id="echoid-s11887" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s11888" xml:space="preserve"> acutus] quare ei contrapoſitus eſt acutus [per 15 p 1.</s> <s xml:id="echoid-s11889" xml:space="preserve">] Igitur ſi à puncto k <lb/>ducatur perpendicularis ad t z:</s> <s xml:id="echoid-s11890" xml:space="preserve"> [per 12 p 1] cadet inter e & z.</s> <s xml:id="echoid-s11891" xml:space="preserve"> Si enim ſupra e ceciderit, cum angulus <lb/>t e k ſit obtuſus:</s> <s xml:id="echoid-s11892" xml:space="preserve"> [per 13 p 1:</s> <s xml:id="echoid-s11893" xml:space="preserve"> acutus enim concluſus eſt t e d] accidet triangulũ habere duos angulos <lb/>rectum & obtuſum [contra 32 p 1.</s> <s xml:id="echoid-s11894" xml:space="preserve">] Sit ergo perpendicularis k q.</s> <s xml:id="echoid-s11895" xml:space="preserve"> Dico, quo d k t ſe habet ad t f, ſicut <lb/>k d ad d o, t o enim aut eſt æquidiſtans k d:</s> <s xml:id="echoid-s11896" xml:space="preserve"> aut concurrit cum ea.</s> <s xml:id="echoid-s11897" xml:space="preserve"> Sit æquidiſtans:</s> <s xml:id="echoid-s11898" xml:space="preserve"> erit ergo [per 29 <lb/> <anchor type="figure" xlink:label="fig-0183-01a" xlink:href="fig-0183-01"/> <anchor type="figure" xlink:label="fig-0183-02a" xlink:href="fig-0183-02"/> p 1] angulus o d a æqualis angulo t o d:</s> <s xml:id="echoid-s11899" xml:space="preserve"> & ita t o d æqualis angulo o t f [æquatus enim eſt o t f ipſi <lb/>o d a.</s> <s xml:id="echoid-s11900" xml:space="preserve">] Et o d, t ſaut ſunt æquidiſtãtes:</s> <s xml:id="echoid-s11901" xml:space="preserve"> aut cõcurrunt.</s> <s xml:id="echoid-s11902" xml:space="preserve"> Si æquidiſtantes, cũ cadant inter æquidiſtan-<lb/>tes [k d, t o] erũt [per 34 p 1] æquales.</s> <s xml:id="echoid-s11903" xml:space="preserve"> Si uerò cõcurrunt:</s> <s xml:id="echoid-s11904" xml:space="preserve"> faciẽt triangulũ, cuius latera æqualia [per <lb/>6 p 1] quia reſpiciunt æquales angulos:</s> <s xml:id="echoid-s11905" xml:space="preserve"> [f t o, & d o t] & f d ſecat illa latera æquidiſtanter baſi.</s> <s xml:id="echoid-s11906" xml:space="preserve"> Erit <lb/>ergo [per 2 p 6.</s> <s xml:id="echoid-s11907" xml:space="preserve">18 p 5] proportio unius laterum ad d o, ſicut alterius ad f t:</s> <s xml:id="echoid-s11908" xml:space="preserve"> & ita t f æqualis d o [per <lb/>9 p 5.</s> <s xml:id="echoid-s11909" xml:space="preserve">] Ethoc dico, ſilineæ illæ concurrant ſub k d.</s> <s xml:id="echoid-s11910" xml:space="preserve"> Et ſi cõcurrant ſub t o:</s> <s xml:id="echoid-s11911" xml:space="preserve"> eadem erit probatio:</s> <s xml:id="echoid-s11912" xml:space="preserve"> quia <lb/>ſiet triangulum, cuius unũ latus eſt t o, & alia duo latera æqualia:</s> <s xml:id="echoid-s11913" xml:space="preserve"> [per 6 p 1] & erit [per 2 p 6.</s> <s xml:id="echoid-s11914" xml:space="preserve">18 p 5] <lb/>proportio unius laterum ad d o, ſicut alterius a d t f:</s> <s xml:id="echoid-s11915" xml:space="preserve"> & ita [per 9 p 5] t ſ æqualis d o.</s> <s xml:id="echoid-s11916" xml:space="preserve"> Item angulus t d <lb/>k eſt æqualis angulo d t o [per 29 p 1] quia d tinter æquidiſtantes:</s> <s xml:id="echoid-s11917" xml:space="preserve"> [ex theſi:</s> <s xml:id="echoid-s11918" xml:space="preserve"> nempe k d, t o] igitur <lb/>eſt æqualis angulo d t k:</s> <s xml:id="echoid-s11919" xml:space="preserve"> [qui ex theſi & 12 n 4 æquatur angulo d t o] quare [per 6 p 1] d k æqualis <lb/>eſt t k.</s> <s xml:id="echoid-s11920" xml:space="preserve"> Igitur [per 7 p 5] proportio tkad t f, ſicut k d ad d o.</s> <s xml:id="echoid-s11921" xml:space="preserve"> Siuero to concurrit cum k d:</s> <s xml:id="echoid-s11922" xml:space="preserve"> concurrat <lb/>ex parte a in puncto l.</s> <s xml:id="echoid-s11923" xml:space="preserve"> Scimus [è demõſtratis à Theo <lb/> <anchor type="figure" xlink:label="fig-0183-03a" xlink:href="fig-0183-03"/> ne ad 5 d 6] quòd proportio k t ad t ſ compacta eſt ex <lb/>proportione k t ad tl, & tl ad t f:</s> <s xml:id="echoid-s11924" xml:space="preserve"> ſed [per 3 p 6] k t ad <lb/>tleſt, ſicut k d ad d l:</s> <s xml:id="echoid-s11925" xml:space="preserve"> quoniam d t diuidit angulum k <lb/>to per æqualia:</s> <s xml:id="echoid-s11926" xml:space="preserve"> & proportio tladtf, ſicut d l ad d o:</s> <s xml:id="echoid-s11927" xml:space="preserve"> <lb/>quoniã angulus o d leſt æqualis angulo l t f [perſa-<lb/>bricationem] & angulus ſuperl communis:</s> <s xml:id="echoid-s11928" xml:space="preserve"> [trian-<lb/>gulis l t f, o d l] erit partiale triangulum ſimile totali <lb/>[per 32 p 1.</s> <s xml:id="echoid-s11929" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s11930" xml:space="preserve"> 1 d 6.</s> <s xml:id="echoid-s11931" xml:space="preserve">] Igitur proportio k t ad t f cõſtat <lb/>ex proportione k d ad d l, & proportione dl ad d o:</s> <s xml:id="echoid-s11932" xml:space="preserve"> <lb/>ſed proportio k d ad d o conſtat exijſdem [aſſumpta <lb/>dlmedia interk d & d o.</s> <s xml:id="echoid-s11933" xml:space="preserve">] Quare proportio k t ad t f, <lb/>ſicut k d ad d o.</s> <s xml:id="echoid-s11934" xml:space="preserve"> Si uerò to concurrat cum k d ex par-<lb/>te g:</s> <s xml:id="echoid-s11935" xml:space="preserve"> ſit concurſus s.</s> <s xml:id="echoid-s11936" xml:space="preserve"> Et à puncto d ducatur æquidi-<lb/>ſtans lineæ k t:</s> <s xml:id="echoid-s11937" xml:space="preserve"> [per 31 p 1] quæ ſit d r, cõcurrens cum <lb/>to in puncto r:</s> <s xml:id="echoid-s11938" xml:space="preserve"> igitur [per 29 p 1] angulus k t d eſt <lb/>æqualis angulo t d r:</s> <s xml:id="echoid-s11939" xml:space="preserve"> ſed idẽ eſt æqualis angulo d t o <lb/>[pertheſin & 12 n 4.</s> <s xml:id="echoid-s11940" xml:space="preserve">] Quare [ք 6 p 1] d r eſt æqualis <lb/>tr.</s> <s xml:id="echoid-s11941" xml:space="preserve"> Sed quia triangulũ s t k ſimile eſt triangulo s r d:</s> <s xml:id="echoid-s11942" xml:space="preserve"> [per 29.</s> <s xml:id="echoid-s11943" xml:space="preserve"> 32 p 1.</s> <s xml:id="echoid-s11944" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s11945" xml:space="preserve"> 1 d 6] erit proportio d rad sr, <lb/>ſicut k t ad t s:</s> <s xml:id="echoid-s11946" xml:space="preserve"> & ita r t ad r s, ſicut kt ad ts:</s> <s xml:id="echoid-s11947" xml:space="preserve"> [ք 7 p 5:</s> <s xml:id="echoid-s11948" xml:space="preserve"> æqualis enim cõcluſa eſt tripſi d r] ſed r t ad r s, <lb/> <pb o="178" file="0184" n="184" rhead="ALHAZEN"/> ſicut d k ad d s [eſt enim per 2 p 6, ut s t ad tr, ſic s k ad kd:</s> <s xml:id="echoid-s11949" xml:space="preserve"> & per 18 p 5, ut s r ad rt, ſic s d ad d k, & <lb/>per conſectarium 4 p 5, utrtadrs, ſic d k ad d s.</s> <s xml:id="echoid-s11950" xml:space="preserve">] Igitur [per 11 p 5] ktadts, ſicut d k ad d s.</s> <s xml:id="echoid-s11951" xml:space="preserve"> Sed que <lb/>niam angulus ft o æqualis eſt angulo o d a:</s> <s xml:id="echoid-s11952" xml:space="preserve"> [perfabrica-<lb/> <anchor type="figure" xlink:label="fig-0184-01a" xlink:href="fig-0184-01"/> tionem] erit [per 13 p 1] angulus o d s ęqualis angulo fts <lb/>[& angulus ad s ęquatur ſibijpſi:</s> <s xml:id="echoid-s11953" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s11954" xml:space="preserve"> per 32 p 1 trian gula <lb/>s t f, d s o ſunt æquiangula.</s> <s xml:id="echoid-s11955" xml:space="preserve">] Igitur [ք 4 p 6] stad t f, ſicut <lb/>d s ad d o:</s> <s xml:id="echoid-s11956" xml:space="preserve"> & eſt k t ad t s, ſicut d k ad d s:</s> <s xml:id="echoid-s11957" xml:space="preserve"> & ts ad t f, ſicut <lb/>d s ad d o:</s> <s xml:id="echoid-s11958" xml:space="preserve"> quare [ք 22 p 5] ktadtſ, ſicut k d ad d o.</s> <s xml:id="echoid-s11959" xml:space="preserve"> Quod <lb/>eſt propoſitũ.</s> <s xml:id="echoid-s11960" xml:space="preserve"> Sed quoniã k z æquidiſtat t f:</s> <s xml:id="echoid-s11961" xml:space="preserve"> [per fabrica-<lb/>tionem] erit [per 29 p 1] angulus k z e ęqualis angulo e t <lb/>f:</s> <s xml:id="echoid-s11962" xml:space="preserve"> & ita triangulũ k z e ſimile triangulo e t ſ.</s> <s xml:id="echoid-s11963" xml:space="preserve"> [Nam anguli <lb/>ad e æquãtur per 15 p 1;</s> <s xml:id="echoid-s11964" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s11965" xml:space="preserve"> per 32 p 1.</s> <s xml:id="echoid-s11966" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s11967" xml:space="preserve"> 1 d 6 triangula <lb/>k z e, e t f ſunt ſimilia.</s> <s xml:id="echoid-s11968" xml:space="preserve">] Quare ꝓportio k e ad e ſ, ſicut k z <lb/>ad t f:</s> <s xml:id="echoid-s11969" xml:space="preserve"> ſed [per 3 p 6] k e ad e f, ſicut k t ad t f, propter angu-<lb/>lum ſuper t diuiſum ք æqualia.</s> <s xml:id="echoid-s11970" xml:space="preserve"> Igitur [ք 9 p 5] k z æqua-<lb/>lis eſt k t.</s> <s xml:id="echoid-s11971" xml:space="preserve"> Verùm quoniã k q eſt perpẽdicularis ſuper e z:</s> <s xml:id="echoid-s11972" xml:space="preserve"> <lb/>[per ſabricationẽ] erũt omnes eius anguli recti:</s> <s xml:id="echoid-s11973" xml:space="preserve"> ſed an-<lb/>gulus e t d eſt acutus:</s> <s xml:id="echoid-s11974" xml:space="preserve"> quoniã eſt medietas anguli [fto, ut <lb/>patuit.</s> <s xml:id="echoid-s11975" xml:space="preserve">] Igitur k q cõcurret cũ t d [ք 11 ax.</s> <s xml:id="echoid-s11976" xml:space="preserve">] Sit cõcurſus <lb/>h:</s> <s xml:id="echoid-s11977" xml:space="preserve"> & ducatur linea e h:</s> <s xml:id="echoid-s11978" xml:space="preserve"> & [per 31 p 1] à pũcto e ducature æ-<lb/>quidiſtans h k, ꝓducta uſq;</s> <s xml:id="echoid-s11979" xml:space="preserve"> ad d h:</s> <s xml:id="echoid-s11980" xml:space="preserve"> quæ ſit e x:</s> <s xml:id="echoid-s11981" xml:space="preserve"> & mutetur <lb/>figura propter intricationẽ linearũ:</s> <s xml:id="echoid-s11982" xml:space="preserve"> & [per 5 p 4] fiat cir-<lb/>culus, trãſiens per tria puncta x, t, e:</s> <s xml:id="echoid-s11983" xml:space="preserve"> & ꝓducatur k d uſq;</s> <s xml:id="echoid-s11984" xml:space="preserve"> <lb/>in circulũ, cadens in punctũ m:</s> <s xml:id="echoid-s11985" xml:space="preserve"> & educatur m t:</s> <s xml:id="echoid-s11986" xml:space="preserve"> erit [per <lb/>27 p 3] angulus t m e æqualis angulo t x e:</s> <s xml:id="echoid-s11987" xml:space="preserve"> quia cadunt in <lb/>eundẽ arcũ:</s> <s xml:id="echoid-s11988" xml:space="preserve"> [e f t] & [ք 29 p 1] angulus t x e æqualis an <lb/>gulo t h k:</s> <s xml:id="echoid-s11989" xml:space="preserve"> erit t m e æqualis angulo t h k.</s> <s xml:id="echoid-s11990" xml:space="preserve"> Secetur ab an-<lb/>gulo t m e, æqualis angulo d h e:</s> <s xml:id="echoid-s11991" xml:space="preserve"> [id uerò fieri poteſt:</s> <s xml:id="echoid-s11992" xml:space="preserve"> ꝗa <lb/>angulus t h k maior eſt angulo d h e per 9 ax:</s> <s xml:id="echoid-s11993" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s11994" xml:space="preserve"> t m e <lb/>eodẽ maior eſt] ꝗ ſit ſ m d:</s> <s xml:id="echoid-s11995" xml:space="preserve"> & punctũ, in quo ſ m ſecat t x, ſit i.</s> <s xml:id="echoid-s11996" xml:space="preserve"> Palàm quòd triangulũ i m d ſimile eſt <lb/>triangulo e d h [quia enim angulus f m d æquatus eſt angulo d h e, & anguli ad d æquãtur per 15 p 1<gap/> <lb/>ergo ք 32 p 1 triangula ſunt æquiangula, & ք 4 p.</s> <s xml:id="echoid-s11997" xml:space="preserve"> 1 d <lb/> <anchor type="figure" xlink:label="fig-0184-02a" xlink:href="fig-0184-02"/> 6 ſimilia.</s> <s xml:id="echoid-s11998" xml:space="preserve">] Quare proportio h d ad d m, ſicut e h ad i <lb/>m.</s> <s xml:id="echoid-s11999" xml:space="preserve"> Et ſimiliter triangulũ t m d ſimile triangulo k h d:</s> <s xml:id="echoid-s12000" xml:space="preserve"> <lb/>[Nã angulus t m d æqualis cõcluſus eſt angulo th k:</s> <s xml:id="echoid-s12001" xml:space="preserve"> <lb/>& anguli ad d ęquãtur ք 1 5 p 1.</s> <s xml:id="echoid-s12002" xml:space="preserve"> Quare ut prius trian-<lb/>gula ſunt ſimilia] & ꝓportio k d ad d t, ſicut h d ad d <lb/>m:</s> <s xml:id="echoid-s12003" xml:space="preserve"> & ita [ք 11 p 5] k d ad d t, ſicut e h ad im.</s> <s xml:id="echoid-s12004" xml:space="preserve"> Sed pro-<lb/>portio k d ad d t nota:</s> <s xml:id="echoid-s12005" xml:space="preserve"> quoniã ſemք una & eadẽ per-<lb/>manet, quodcũq;</s> <s xml:id="echoid-s12006" xml:space="preserve"> punctũ reflexionis ſit t in arcu b g:</s> <s xml:id="echoid-s12007" xml:space="preserve"> <lb/>quia ſemper linea t d eſt una:</s> <s xml:id="echoid-s12008" xml:space="preserve"> [quia eſt ſemidiameter <lb/>circuli, qui eſt cõmunis ſectio ſuperſicierũ reflexio-<lb/>nis & ſpeculi] & k d ſimiliter [quia eſt diſtãtia pũcti <lb/>reflexi à cẽtro ſpeculi.</s> <s xml:id="echoid-s12009" xml:space="preserve">] Linea etiam e h unà in qua-<lb/>cunq;</s> <s xml:id="echoid-s12010" xml:space="preserve"> reflexione permanet, & nõ mutatur eius quã-<lb/>titas [quia angulus o d a idẽ ſemper քmanet:</s> <s xml:id="echoid-s12011" xml:space="preserve"> eiusq́;</s> <s xml:id="echoid-s12012" xml:space="preserve"> <lb/>dimidius eſt angulus e t d:</s> <s xml:id="echoid-s12013" xml:space="preserve"> ꝗ a, ut patuit, dimidius eſt <lb/>angulifto, æquati angulo o d a.</s> <s xml:id="echoid-s12014" xml:space="preserve">] Quare linea im <lb/>ſemper erit una:</s> <s xml:id="echoid-s12015" xml:space="preserve"> quare punctũ ſnotũ & determina-<lb/>tum [quia per lineam i m longitudine ſemper eandẽ, continuatam in peripheriam oſtenditur.</s> <s xml:id="echoid-s12016" xml:space="preserve">] Si <lb/>ergo à tribus punctis arcus b g fieri poſſet reflexio:</s> <s xml:id="echoid-s12017" xml:space="preserve"> eſſet ducere à puncto fad circulum x t e tres li-<lb/>neas æquales:</s> <s xml:id="echoid-s12018" xml:space="preserve"> quarum cuiuslibet pars interiacens diametrum tx & circumferentiam circuli eſſet <lb/>æqualis lineæ i m:</s> <s xml:id="echoid-s12019" xml:space="preserve"> quia ſemper erit proportio k d ad d t, ſicut e h ad quamlibet illarum.</s> <s xml:id="echoid-s12020" xml:space="preserve"> Et patet ex <lb/>ſuperioribus [34 n] quòd non, niſi duæ æquales poſſunt.</s> <s xml:id="echoid-s12021" xml:space="preserve"> Quare à duobus tantùm punctis fiet re-<lb/>flexio.</s> <s xml:id="echoid-s12022" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s12023" xml:space="preserve"/> </p> <div xml:id="echoid-div418" type="float" level="0" n="0"> <figure xlink:label="fig-0182-01" xlink:href="fig-0182-01a"> <variables xml:id="echoid-variables118" xml:space="preserve">b <gap/> o <gap/> p n g k e f d a q l m</variables> </figure> <figure xlink:label="fig-0183-01" xlink:href="fig-0183-01a"> <variables xml:id="echoid-variables119" xml:space="preserve">b t o u p n g k e f d a q z m</variables> </figure> <figure xlink:label="fig-0183-02" xlink:href="fig-0183-02a"> <variables xml:id="echoid-variables120" xml:space="preserve">b u t o p n g k e f d a q z m</variables> </figure> <figure xlink:label="fig-0183-03" xlink:href="fig-0183-03a"> <variables xml:id="echoid-variables121" xml:space="preserve">u t b p n o g k e f d l<unsure/> a q m z</variables> </figure> <figure xlink:label="fig-0184-01" xlink:href="fig-0184-01a"> <variables xml:id="echoid-variables122" xml:space="preserve" style="it">s g z k t e f d o b r a</variables> </figure> <figure xlink:label="fig-0184-02" xlink:href="fig-0184-02a"> <variables xml:id="echoid-variables123" xml:space="preserve">t f i k e d m q z x h</variables> </figure> </div> </div> <div xml:id="echoid-div420" type="section" level="0" n="0"> <head xml:id="echoid-head386" xml:space="preserve" style="it">83. Datis duobus punctis in diuerſis diametris circuli (quieſt cõmunis ſectio ſuperficierum, <lb/>reflexionis & ſpeculi ſphærici caui) à centro inæquabiliter diſtantibus: inuenire in peripberia <lb/>comprebenſa inter ſemidiametros, in quibus ipſa ſunt, duo reflexionis puncta. 37 p 8.</head> <p> <s xml:id="echoid-s12024" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s12025" xml:space="preserve"> datis duobus punctis k, o in diuerſis diametris, inæ qualiter diſtantibus à centro:</s> <s xml:id="echoid-s12026" xml:space="preserve"> eſt <lb/>inuenire punctum reflexionis.</s> <s xml:id="echoid-s12027" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s12028" xml:space="preserve"> ſumatur linea z t:</s> <s xml:id="echoid-s12029" xml:space="preserve"> & [per 10 p 6] diuidatur in <lb/>puncto e, ut ſit proportio z e ad et, ſicut k d ad d o [in primo diagrammate præcedentis <lb/>numeri.</s> <s xml:id="echoid-s12030" xml:space="preserve">] Quoniam k d maior d o [extheſi præcedẽtis numeri] erit z e maior e t:</s> <s xml:id="echoid-s12031" xml:space="preserve"> diuidatur z t per <lb/>æqualia in puncto q:</s> <s xml:id="echoid-s12032" xml:space="preserve"> [per 10 p 1] & à puncto q ducatur perpendicularis ſuper z t:</s> <s xml:id="echoid-s12033" xml:space="preserve"> [per 11 p 1] & fiat <lb/> <pb o="179" file="0185" n="185" rhead="OPTICAE LIBER V."/> ãngulus e t d æqualis medietati anguli o d a:</s> <s xml:id="echoid-s12034" xml:space="preserve"> erit quidem acutus:</s> <s xml:id="echoid-s12035" xml:space="preserve"> [quia æquatur angulo rectilineo <lb/>dimidiato, ut oſtenſum eſt 36 n] igitur t d concur-<lb/> <anchor type="figure" xlink:label="fig-0185-01a" xlink:href="fig-0185-01"/> <anchor type="figure" xlink:label="fig-0185-02a" xlink:href="fig-0185-02"/> ret cum perpendiculari [per 11 ax.</s> <s xml:id="echoid-s12036" xml:space="preserve">] Sit concurſus <lb/>in puncto h:</s> <s xml:id="echoid-s12037" xml:space="preserve"> & [per 38 n] ducatur linea d e k, ut ſit <lb/>proportio k d ad d t, ſicut k d ad ſemidiametrum <lb/>ſphæræ [erit igitur ſemidiameter ſphæræ æqualis <lb/>d t per 9 p 5.</s> <s xml:id="echoid-s12038" xml:space="preserve">] Etangulo, quem habemus k d t fiat <lb/>[per 23 p 1] in ſpeculo angulus æqualis, ſcilicet k d <lb/>t.</s> <s xml:id="echoid-s12039" xml:space="preserve"> Dico, quò d t eſt punctum reflexionis, Et ſi prædi-<lb/>ctam probationẽ replicaueris, manifeſtè uidebis.</s> <s xml:id="echoid-s12040" xml:space="preserve"/> </p> <div xml:id="echoid-div420" type="float" level="0" n="0"> <figure xlink:label="fig-0185-01" xlink:href="fig-0185-01a"> <variables xml:id="echoid-variables124" xml:space="preserve"><gap/> k e d q h z</variables> </figure> <figure xlink:label="fig-0185-02" xlink:href="fig-0185-02a"> <variables xml:id="echoid-variables125" xml:space="preserve">l b k d o</variables> </figure> </div> </div> <div xml:id="echoid-div422" type="section" level="0" n="0"> <head xml:id="echoid-head387" xml:space="preserve" style="it">84. Siduo puncta extra circulum (quieſt com-<lb/>munis ſectio ſuperficierum reflexionis & ſpecu-<lb/>li ſphæricicaui) uel alterum intra, reliquum ex-<lb/>tra, in diuerſis diametris, à centro inæquabiliter <lb/>diſtantia, reflectantur à peripheria comprehen-<lb/>ſa inter ſemidiametros, extra quas ipſa ſunt: ab <lb/>uno puncto tantùm reflectentur. 38 p 8.</head> <p> <s xml:id="echoid-s12041" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s12042" xml:space="preserve"> ſumptis duobus punctis in diuerſis diametris, quæ puncta inæqualis ſint longitu.</s> <s xml:id="echoid-s12043" xml:space="preserve"> <lb/>dinis à centro:</s> <s xml:id="echoid-s12044" xml:space="preserve"> ſi fuerint extra circulum, & reflectantur ab a-<lb/> <anchor type="figure" xlink:label="fig-0185-03a" xlink:href="fig-0185-03"/> liquo puncto arcus oppoſiti diametris:</s> <s xml:id="echoid-s12045" xml:space="preserve"> non reflectentur ab <lb/>alio eiuſdem arcus.</s> <s xml:id="echoid-s12046" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s12047" xml:space="preserve"> ſint a, b puncta in diuerſis diame-<lb/>tris, extra circulum:</s> <s xml:id="echoid-s12048" xml:space="preserve"> g centrum:</s> <s xml:id="echoid-s12049" xml:space="preserve"> t punctum reflexionis:</s> <s xml:id="echoid-s12050" xml:space="preserve"> & ducantur <lb/>b t, a t, t g:</s> <s xml:id="echoid-s12051" xml:space="preserve"> b t ſecabit arcum circuli:</s> <s xml:id="echoid-s12052" xml:space="preserve"> ſit punctum ſectionis q:</s> <s xml:id="echoid-s12053" xml:space="preserve"> a t ſeca-<lb/>bit ſimiliter arcum circuli:</s> <s xml:id="echoid-s12054" xml:space="preserve"> ſit punctum ſectionis m.</s> <s xml:id="echoid-s12055" xml:space="preserve"> Quoniam angu-<lb/>lus b t g ęqualis eſt angulo a t g:</s> <s xml:id="echoid-s12056" xml:space="preserve"> [per 12 n 4:</s> <s xml:id="echoid-s12057" xml:space="preserve"> quia t eſt reflexionis pun <lb/>ctum ex theſi] cadent in arcus circuli æquales:</s> <s xml:id="echoid-s12058" xml:space="preserve"> [per 26 p 3] quod p <gap/> <lb/>tebit producta ſemidiametro t g in p.</s> <s xml:id="echoid-s12059" xml:space="preserve"> Erit ergo arcus q p æqualis ar-<lb/>cui m p.</s> <s xml:id="echoid-s12060" xml:space="preserve"> Si igitur b reflectitur ab alio puncto:</s> <s xml:id="echoid-s12061" xml:space="preserve"> ſit illud h:</s> <s xml:id="echoid-s12062" xml:space="preserve"> & ducantur <lb/>lineæ b h, a h, g h.</s> <s xml:id="echoid-s12063" xml:space="preserve"> Secet b h circulum in puncto l:</s> <s xml:id="echoid-s12064" xml:space="preserve"> a h in puncto n:</s> <s xml:id="echoid-s12065" xml:space="preserve"> & <lb/>producatur h g in k.</s> <s xml:id="echoid-s12066" xml:space="preserve"> Secundum igitur prædictam probationem e-<lb/>rit l k æqualis n k:</s> <s xml:id="echoid-s12067" xml:space="preserve"> ſed iam habemus, quòd q p æqualis p m:</s> <s xml:id="echoid-s12068" xml:space="preserve"> quod eſt <lb/>impoſsibile [& contra 9 ax.</s> <s xml:id="echoid-s12069" xml:space="preserve">] Reſtat ut b non reflectatur ad a, à pun <lb/>cto h, uel ab alio puncto arcus oppoſiti diametris, præterquam à t.</s> <s xml:id="echoid-s12070" xml:space="preserve"> <lb/>Similiter ſi fuerit alterum punctorum in circulo, alterum extra:</s> <s xml:id="echoid-s12071" xml:space="preserve"> ab <lb/>uno tantùm puncto arcus poterit reflecti ad aliud.</s> <s xml:id="echoid-s12072" xml:space="preserve"/> </p> <div xml:id="echoid-div422" type="float" level="0" n="0"> <figure xlink:label="fig-0185-03" xlink:href="fig-0185-03a"> <variables xml:id="echoid-variables126" xml:space="preserve">a b n m <gap/> k l q g d h <gap/> e</variables> </figure> </div> </div> <div xml:id="echoid-div424" type="section" level="0" n="0"> <head xml:id="echoid-head388" xml:space="preserve" style="it">85. Sirecta linea connectens duo puncta in diuerſis diametris <lb/>circuli (qui eſt communis ſectio ſuperficierum reflexionis & ſpecu <lb/>li ſphæricicaui) à centro inæquabiliter diſtantia, tangat peripheriam dicti circuli, uelſit extra <lb/>ipſam: ab uno tantùm puncto reflexio fiet. 39 p 8.</head> <p> <s xml:id="echoid-s12073" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s12074" xml:space="preserve"> ſilinea ducta ab uno duorũ puncto-<lb/> <anchor type="figure" xlink:label="fig-0185-04a" xlink:href="fig-0185-04"/> rum, cõtingat circulũ, aut tota ſit extra:</s> <s xml:id="echoid-s12075" xml:space="preserve"> ſum-<lb/>pto quocũq;</s> <s xml:id="echoid-s12076" xml:space="preserve"> puncto in atcu oppoſito diame-<lb/>tris:</s> <s xml:id="echoid-s12077" xml:space="preserve"> [in quibus ſunt data puncta] altera linearum à <lb/>punctorum duorum altero, ad illud punctum ducta <lb/>rum, tota erit extra circulum:</s> <s xml:id="echoid-s12078" xml:space="preserve"> & ſic neutrum puncto <lb/>rum ad aliud reflectetur ab aliquo puncto illius ar-<lb/>cus:</s> <s xml:id="echoid-s12079" xml:space="preserve"> [m l] & ab uno ſolo puncto ſpeculi [in periphe-<lb/>ria d n ſumpto per 73 & præcedentem numeros.</s> <s xml:id="echoid-s12080" xml:space="preserve">]</s> </p> <div xml:id="echoid-div424" type="float" level="0" n="0"> <figure xlink:label="fig-0185-04" xlink:href="fig-0185-04a"> <variables xml:id="echoid-variables127" xml:space="preserve">b a b a m <gap/> f g d n</variables> </figure> </div> </div> <div xml:id="echoid-div426" type="section" level="0" n="0"> <head xml:id="echoid-head389" xml:space="preserve" style="it">86. Sirecta linea connectens duo puncta in di-<lb/>uerſis diametris circuli (qui eſt communis ſectio <lb/>ſuperficierũ reflexiõis & ſpeculi ſphærici caui) à cẽ-<lb/>troinæquabiliter diſtantia, continuata eundem <lb/>ſecet: poſſunt dicta puncta ab uno, duobus, tri-<lb/>bus, aut quatuor punctis ſpeculi inter ſe reflecti. <lb/>40 p 8.</head> <p> <s xml:id="echoid-s12081" xml:space="preserve">SI uerò linea ducta ab uno pũcto ad aliud, ſecet circulũ:</s> <s xml:id="echoid-s12082" xml:space="preserve"> fiat circulus ք centrũ ſpeculi & illa duo <lb/>pũcta [ք 5 p 4.</s> <s xml:id="echoid-s12083" xml:space="preserve">] circulus ille aut tot<emph style="sub">9</emph> erit intra circulũ:</s> <s xml:id="echoid-s12084" xml:space="preserve"> aut cõtingetipſum intrinſecus:</s> <s xml:id="echoid-s12085" xml:space="preserve"> aut ſeca <lb/>bit.</s> <s xml:id="echoid-s12086" xml:space="preserve"> Sit tot<emph style="sub">9</emph> intra:</s> <s xml:id="echoid-s12087" xml:space="preserve"> & ducãtur duę<unsure/> lineę<unsure/> à duob.</s> <s xml:id="echoid-s12088" xml:space="preserve"> pũctis ad aliquod pũctũ arcus oppofiti:</s> <s xml:id="echoid-s12089" xml:space="preserve"> angul<emph style="sub">9</emph>, <lb/> <pb o="180" file="0186" n="186" rhead="ALHAZEN"/> quem facient [qui ſit at b] erit minorangulo [b g d] quem una diameter facit cum alia, exparte c<gap/>-<lb/>tri:</s> <s xml:id="echoid-s12090" xml:space="preserve"> [Nam ſi à puncto f, in quo g t ſecat peripheriam <lb/> <anchor type="figure" xlink:label="fig-0186-01a" xlink:href="fig-0186-01"/> circuli a b g, ducantur rectæ f a, f b:</s> <s xml:id="echoid-s12091" xml:space="preserve"> ęquabuntur angu <lb/>li ad f & g duobus rectis per 22 p 3:</s> <s xml:id="echoid-s12092" xml:space="preserve"> quibus etiam æ-<lb/>quantur anguli ad g deinceps per 13 p 1:</s> <s xml:id="echoid-s12093" xml:space="preserve"> quare per 3 <lb/>ax.</s> <s xml:id="echoid-s12094" xml:space="preserve"> b g d ęquatur a f b:</s> <s xml:id="echoid-s12095" xml:space="preserve"> qui per 21 p 1 maior eſt angulo <lb/>a t b.</s> <s xml:id="echoid-s12096" xml:space="preserve"> Angulus igitur a t b minor eſt angulo b g d.</s> <s xml:id="echoid-s12097" xml:space="preserve">] Et <lb/>quilibet angulus ſic factus ſuper arcum oppoſitum <lb/>[l m] minor erit illo angulo.</s> <s xml:id="echoid-s12098" xml:space="preserve"> Quoniam angulus fa-<lb/>ctus in interiore circulo, per lineas à punctis ad ar-<lb/>cum eius interiacentem ductas, erιt æqualis illi an-<lb/>gulo:</s> <s xml:id="echoid-s12099" xml:space="preserve"> quoniam cum angulo diametrorum ſuper cen <lb/>trũ ualet duos angulos rectos [per 22 p 3.</s> <s xml:id="echoid-s12100" xml:space="preserve">] Sed [per <lb/>21 p 1] angulus arcus minoris circuli [angulus nem-<lb/>pe in ipſius peripheria] maior eſt angulo arcus ſpe-<lb/>culi [eo nempe, qui fit in peripheria circuli:</s> <s xml:id="echoid-s12101" xml:space="preserve"> qui eſt <lb/>communis ſectio ſuperficierum, reflexionis & ſpe-<lb/>culi.</s> <s xml:id="echoid-s12102" xml:space="preserve">] Igitur in arcu ſpeculi nõ fiet reflexio, niſiab u-<lb/>no puncto:</s> <s xml:id="echoid-s12103" xml:space="preserve"> cum iam dictum ſit [80 n] quòd non eſt <lb/>poſsibile reflexionem à duobus punctis fieri, ut ſit uterque angulus, conſtans ex angulo incidentiæ <lb/>& reflexionis, minor angulo diametrorum ex alia parte centri.</s> <s xml:id="echoid-s12104" xml:space="preserve"> Siuerò circulus ille contingat intrin <lb/>ſecus circulum ſpeculi:</s> <s xml:id="echoid-s12105" xml:space="preserve"> angulus factus à lineis, ab il-<lb/> <anchor type="figure" xlink:label="fig-0186-02a" xlink:href="fig-0186-02"/> lis punctis ad punctum contactus ductis, erit æqua-<lb/>lis angulo diametrorum ex alia parte centri [angu-<lb/>li enim ad h & g æquantur duobus rectis per 22 p 3:</s> <s xml:id="echoid-s12106" xml:space="preserve"> <lb/>quibus etiam æquantur anguli ad g deinceps per 13 <lb/>p 1:</s> <s xml:id="echoid-s12107" xml:space="preserve"> angulus igitur a h d æquatur angulo b g d per 13 <lb/>ax.</s> <s xml:id="echoid-s12108" xml:space="preserve">] Quare ab illo puncto contactus non fiet refle-<lb/>xio [per 79 n.</s> <s xml:id="echoid-s12109" xml:space="preserve">].</s> <s xml:id="echoid-s12110" xml:space="preserve"> Et angulus factus ſuper quodcunq;</s> <s xml:id="echoid-s12111" xml:space="preserve"> <lb/>punctum aliud maioris circuli, erit minorillo [ut ſi <lb/>angulus fiat ſuper pũctum t:</s> <s xml:id="echoid-s12112" xml:space="preserve"> erit a f b maior a t b per <lb/>21 p 1:</s> <s xml:id="echoid-s12113" xml:space="preserve"> ſed a f b æquatur a h b per 21 p 3:</s> <s xml:id="echoid-s12114" xml:space="preserve"> quare a h b ma <lb/>ior eſt a t b:</s> <s xml:id="echoid-s12115" xml:space="preserve"> eodemq́;</s> <s xml:id="echoid-s12116" xml:space="preserve"> modo de quocunq;</s> <s xml:id="echoid-s12117" xml:space="preserve"> angulo de-<lb/>monſtrabitur.</s> <s xml:id="echoid-s12118" xml:space="preserve">] Quare à duobus punctis arcus non <lb/>fiet reflexio ſecundum prædicta [80 n.</s> <s xml:id="echoid-s12119" xml:space="preserve">] Si uerò cir-<lb/>culus interior ſecet circulum ſpeculi:</s> <s xml:id="echoid-s12120" xml:space="preserve"> duo puncta <lb/>[peripheria enim peripheriam in duobus punctis <lb/>tantùm ſecat per 10 p 3] aut erunt extra circulũ:</s> <s xml:id="echoid-s12121" xml:space="preserve"> aut <lb/>intra:</s> <s xml:id="echoid-s12122" xml:space="preserve"> aut unum intra, aliud extra:</s> <s xml:id="echoid-s12123" xml:space="preserve"> aut unum in cir-<lb/>cumferentia, aliud extra, uel intra.</s> <s xml:id="echoid-s12124" xml:space="preserve"> Si fuerint extra:</s> <s xml:id="echoid-s12125" xml:space="preserve"> circulus ſecans, non ſecabit arcum circuli ſpecu-<lb/>li, interiacentem diametros.</s> <s xml:id="echoid-s12126" xml:space="preserve"> Et iam probatum eſt in præcedente figura [pręcedentis numeri] quòd <lb/>hęc puncta ab uno ſolo puncto arcus <lb/> <anchor type="figure" xlink:label="fig-0186-03a" xlink:href="fig-0186-03"/> <anchor type="figure" xlink:label="fig-0186-04a" xlink:href="fig-0186-04"/> interiacentis diametros poterunt re-<lb/>flecti [quod eſt punctum peripheriæ <lb/>inter ſemidiametros, extra quas ſunt <lb/>reflexa puncta, ut patuit pręcedẽte nu <lb/>mero.</s> <s xml:id="echoid-s12127" xml:space="preserve">] Si uerò unum fuerit in circũ-<lb/>ferentia, aliud extra:</s> <s xml:id="echoid-s12128" xml:space="preserve"> circulus ſecans <lb/>ſecabit arcum circuli ſpeculi, diame-<lb/>tros interiacentem in unico puncto.</s> <s xml:id="echoid-s12129" xml:space="preserve"> <lb/>Et quilibet angulus factus ſuper arcũ <lb/>illum:</s> <s xml:id="echoid-s12130" xml:space="preserve"> erit maior angulo diametrorũ <lb/>ex alia parte centri:</s> <s xml:id="echoid-s12131" xml:space="preserve"> & ſic [per 80 n] ab <lb/>uno puncto, uel à duobus poteſt fieri <lb/>reflexio.</s> <s xml:id="echoid-s12132" xml:space="preserve"> Si uerò duo pũcta fuerint in-<lb/>tra:</s> <s xml:id="echoid-s12133" xml:space="preserve"> ſecabit circulus interior arcũ in-<lb/>teriacentem in duobus punctis:</s> <s xml:id="echoid-s12134" xml:space="preserve"> & re-<lb/>ſtabunt ex eo duo arcus ex diuerſis <lb/>partibus.</s> <s xml:id="echoid-s12135" xml:space="preserve"> Et omnes anguli facti ſuper <lb/>arcum, interiacentem duo puncta ſectionis, erunt maiores angulo diametrorum ex alia parte cen-<lb/>tri:</s> <s xml:id="echoid-s12136" xml:space="preserve"> [ut patet in angulo a e b per 22 p 3.</s> <s xml:id="echoid-s12137" xml:space="preserve"> 13.</s> <s xml:id="echoid-s12138" xml:space="preserve"> 21 p 1.</s> <s xml:id="echoid-s12139" xml:space="preserve">] Etab hoc arcu poſſet fieri reflexio forſitan ab u-<lb/>no puncto tantùm:</s> <s xml:id="echoid-s12140" xml:space="preserve"> forſitan à duobus [per 80 n.</s> <s xml:id="echoid-s12141" xml:space="preserve">] Et ſi à duobus arcubus fiat reflexio, quireſtant <lb/>ex arcu totali, & ex diuerſis partibus:</s> <s xml:id="echoid-s12142" xml:space="preserve"> omnes anguli erunt minores angulo diametrorum:</s> <s xml:id="echoid-s12143" xml:space="preserve"> [per <lb/>22 p 3.</s> <s xml:id="echoid-s12144" xml:space="preserve"> 13.</s> <s xml:id="echoid-s12145" xml:space="preserve"> 21 p 1] & tantùm ab uno eorum puncto fiet reflexio [per 80 n.</s> <s xml:id="echoid-s12146" xml:space="preserve">] Et in hoc ſitu poterit <lb/> <pb o="181" file="0187" n="187" rhead="OPTICAE LIBER V."/> fieri reflexio à duobus punctis arcus interiacentis diametros, aut à tribus.</s> <s xml:id="echoid-s12147" xml:space="preserve"> Palàm etiam [per 73.</s> <s xml:id="echoid-s12148" xml:space="preserve"> 75 <lb/>n] quòd ab uno tantùm puncto arcus oppoſiti [n d] fiet reflexio.</s> <s xml:id="echoid-s12149" xml:space="preserve"> Et <lb/> <anchor type="figure" xlink:label="fig-0187-01a" xlink:href="fig-0187-01"/> ita in hoc ſitu, aliquando à tribus, aliquando à quatuor punctis fiet <lb/>reflexio.</s> <s xml:id="echoid-s12150" xml:space="preserve"> Si uerò unum punctorum fuerit intra circulũ, aliud in cir-<lb/>cumferentia, uel extra:</s> <s xml:id="echoid-s12151" xml:space="preserve"> ſecabit circulus arcum interiacentem in uni-<lb/>co puncto:</s> <s xml:id="echoid-s12152" xml:space="preserve"> & reſtabit unus arcus tantùm.</s> <s xml:id="echoid-s12153" xml:space="preserve"> Et omnes anguli facti in <lb/>parte illius arcus, incluſa à ſecante circulo:</s> <s xml:id="echoid-s12154" xml:space="preserve"> erũt maiores angulo dia-<lb/>metrorũ [per 22 p 3.</s> <s xml:id="echoid-s12155" xml:space="preserve"> 13.</s> <s xml:id="echoid-s12156" xml:space="preserve"> 21 p 1.</s> <s xml:id="echoid-s12157" xml:space="preserve">] Et poterit fieri reflexio à duobus pun-<lb/>ctis illius partis, uel ab uno [per 80 n.</s> <s xml:id="echoid-s12158" xml:space="preserve">] Omnes uerò anguli alterius <lb/>partis interiacentis [quæ eſt tl] erunt minores angulo diametro-<lb/>rum [ut oſtenſum eſt.</s> <s xml:id="echoid-s12159" xml:space="preserve">] Et ab uno tantùm pũcto illius partis fiet re-<lb/>flexio [per 80 n.</s> <s xml:id="echoid-s12160" xml:space="preserve">] Etita, cũ ab uno puncto arcus oppoſiti [n d] ſem-<lb/>per fiat reflexio in hoc ſitu:</s> <s xml:id="echoid-s12161" xml:space="preserve"> [per 73.</s> <s xml:id="echoid-s12162" xml:space="preserve">75 n] aliquãdo à tribus, aliquan-<lb/>do à quatuor:</s> <s xml:id="echoid-s12163" xml:space="preserve"> & nõ à pluribus poterit eſſe reflexio.</s> <s xml:id="echoid-s12164" xml:space="preserve"> Palàm ergo, quòd <lb/>puncta inæqualis longitudinis à centro, aliquando ab uno puncto <lb/>tantùm:</s> <s xml:id="echoid-s12165" xml:space="preserve"> aliquando à duobus:</s> <s xml:id="echoid-s12166" xml:space="preserve"> aliquando à tribus:</s> <s xml:id="echoid-s12167" xml:space="preserve"> aliquando à qua-<lb/>tuor:</s> <s xml:id="echoid-s12168" xml:space="preserve"> nunquam à pluribus reflectuntur.</s> <s xml:id="echoid-s12169" xml:space="preserve"/> </p> <div xml:id="echoid-div426" type="float" level="0" n="0"> <figure xlink:label="fig-0186-01" xlink:href="fig-0186-01a"> <variables xml:id="echoid-variables128" xml:space="preserve">m t h <gap/> f b p a g d n</variables> </figure> <figure xlink:label="fig-0186-02" xlink:href="fig-0186-02a"> <variables xml:id="echoid-variables129" xml:space="preserve">m t h <gap/> b <gap/> a g d n</variables> </figure> <figure xlink:label="fig-0186-03" xlink:href="fig-0186-03a"> <variables xml:id="echoid-variables130" xml:space="preserve">a b l<unsure/> m l<unsure/> t a b m g n d n d</variables> </figure> <figure xlink:label="fig-0186-04" xlink:href="fig-0186-04a"> <variables xml:id="echoid-variables131" xml:space="preserve">f e t h k o b m <gap/> a g n d</variables> </figure> <figure xlink:label="fig-0187-01" xlink:href="fig-0187-01a"> <variables xml:id="echoid-variables132" xml:space="preserve">f e t b m f a g d n</variables> </figure> </div> </div> <div xml:id="echoid-div428" type="section" level="0" n="0"> <head xml:id="echoid-head390" xml:space="preserve" style="it">87. Sirecta linea connectens duo puncta in diuerſis diametris circuli (quieſt communis ſe-<lb/>ctio ſuperficierum, reflexionis & ſpeculi ſphærici caui) à centro æquabiliter diſtantia, cõtinua-<lb/>ta eundẽ ſecet: poſſunt dicta puncta ab uno, duobus uel quatuor punctis ſpeculi inter ſe reflecti: <lb/>nunquam uerò à tribus tantùm. 41 p 8.</head> <p> <s xml:id="echoid-s12170" xml:space="preserve">CVm autem puncta eiuſdẽ longitudinis fuerint:</s> <s xml:id="echoid-s12171" xml:space="preserve"> poterit fieri reflexio aut ab uno tantûm pun <lb/>cto;</s> <s xml:id="echoid-s12172" xml:space="preserve"> [ut oſtenſum eſt 70 n] aut à duobus:</s> <s xml:id="echoid-s12173" xml:space="preserve"> [ut 71 n] aut à quatuor:</s> <s xml:id="echoid-s12174" xml:space="preserve"> [ut 72 n] nunquam uerò <lb/>à tribus.</s> <s xml:id="echoid-s12175" xml:space="preserve"> [Quia ſi reflexio fiat à tribus punctis, fiet etiam à quatuor.</s> <s xml:id="echoid-s12176" xml:space="preserve"> Nam cum è tribus iſtis <lb/>reflexionum punctis duo in eandẽ peripheriam ca-<lb/> <anchor type="figure" xlink:label="fig-0187-02a" xlink:href="fig-0187-02"/> dant, ut in præcedentibus numeris patuit:</s> <s xml:id="echoid-s12177" xml:space="preserve"> periphe-<lb/>ria igitur inter duo illa reflexionũ puncta interiecta, <lb/>per 30 p 3 bifariam ſecta, ductisq́;</s> <s xml:id="echoid-s12178" xml:space="preserve"> rectis à centro, & <lb/>datis punctis, in diuerſis diametris ęquabiliter à cẽ-<lb/>tro diſtantibus, ad ſectionis punctum:</s> <s xml:id="echoid-s12179" xml:space="preserve"> erunt anguli <lb/>ad ipſum facti æquales per 27 p 3.</s> <s xml:id="echoid-s12180" xml:space="preserve"> 4 p 1.</s> <s xml:id="echoid-s12181" xml:space="preserve"> Quare ipſum <lb/>eſt reflexionis punctum per 12 n 4:</s> <s xml:id="echoid-s12182" xml:space="preserve"> atqui in periphe-<lb/>ria priori l m oppoſita, ſcilicet n d, eſt etiam unũ re-<lb/>flexionis punctũ per 73.</s> <s xml:id="echoid-s12183" xml:space="preserve"> 75 n.</s> <s xml:id="echoid-s12184" xml:space="preserve"> A<gap/> quatuor igitur pun-<lb/>ctis, nõ à tribus tantùm fit reflexio.</s> <s xml:id="echoid-s12185" xml:space="preserve">] Vbi ab uno pun <lb/>cto fit reflexio, una apparet imago:</s> <s xml:id="echoid-s12186" xml:space="preserve"> ibi â duobus, <lb/>duæ:</s> <s xml:id="echoid-s12187" xml:space="preserve"> ubi à tribus, tres:</s> <s xml:id="echoid-s12188" xml:space="preserve"> ibi à quatuor, quatuor.</s> <s xml:id="echoid-s12189" xml:space="preserve"> Siue-<lb/>rò punctum uiſum & cẽtrum uiſus fuerintin eadem <lb/>diametro:</s> <s xml:id="echoid-s12190" xml:space="preserve"> fiet reflexio à circulo toto:</s> <s xml:id="echoid-s12191" xml:space="preserve"> & locus imagi-<lb/>nis erit cẽtrum uiſus [ut oſtenſum eſt 65 n.</s> <s xml:id="echoid-s12192" xml:space="preserve">] Verùm <lb/>ſi centrum uiſus fuerit in centro ſpeculi:</s> <s xml:id="echoid-s12193" xml:space="preserve"> nihil uidet <lb/>[præter ſe ipſum, ut patuit 44 n 4.</s> <s xml:id="echoid-s12194" xml:space="preserve"> 62 n.</s> <s xml:id="echoid-s12195" xml:space="preserve">] Si uerò pũ-<lb/>ctum uiſum fuerit in centro ſpeculi:</s> <s xml:id="echoid-s12196" xml:space="preserve"> non uidebitur:</s> <s xml:id="echoid-s12197" xml:space="preserve"> <lb/>quoniam forma eius accedet ad ſpeculum ſuper perpendicularem, nec reflecti poterit, niſi ſuper <lb/>perpendicularem [per 11 n 4.</s> <s xml:id="echoid-s12198" xml:space="preserve">] Cum autem centrum uiſus & punctum uiſum fuerint in diuerſis li-<lb/>neis extra centrum:</s> <s xml:id="echoid-s12199" xml:space="preserve"> lineæ illę ad centrum productæ, ſecabunt in diuerſis partibus ex circulo ſphæ-<lb/>ræ duos arcus:</s> <s xml:id="echoid-s12200" xml:space="preserve"> ab uno puncto unius tantùm fiet reflexio:</s> <s xml:id="echoid-s12201" xml:space="preserve"> ab alio forſitan à quatuor.</s> <s xml:id="echoid-s12202" xml:space="preserve"> Quòd ſi cẽtrum <lb/>ſphæræ fuerit ex una parte:</s> <s xml:id="echoid-s12203" xml:space="preserve"> centrum uiſus & punctum uiſum ex una:</s> <s xml:id="echoid-s12204" xml:space="preserve"> arcus, quem ſecant diametri, <lb/>propter oppoſitionem capitis abſcondetur.</s> <s xml:id="echoid-s12205" xml:space="preserve"> Vnde tunc à tribus tantùm punctis fiet reflexio.</s> <s xml:id="echoid-s12206" xml:space="preserve"> Et ſi <lb/>dirigatur in hoc ſitu uiſus ad arcum unius reflexionis tantùm:</s> <s xml:id="echoid-s12207" xml:space="preserve"> abſcondetur alius trium reflexionũ, <lb/>& unica apparebit imago.</s> <s xml:id="echoid-s12208" xml:space="preserve"> Item:</s> <s xml:id="echoid-s12209" xml:space="preserve"> Si integrum fuerit ſpeculum:</s> <s xml:id="echoid-s12210" xml:space="preserve"> nõ erit ibi perceptio:</s> <s xml:id="echoid-s12211" xml:space="preserve"> oportet igitur, <lb/>ut in eo ſit abſciſsio.</s> <s xml:id="echoid-s12212" xml:space="preserve"> Et accidet nonnunquam arcum interiacẽtem diametros abſciſſum eſſe:</s> <s xml:id="echoid-s12213" xml:space="preserve"> & tunc <lb/>nihil in eo uideri.</s> <s xml:id="echoid-s12214" xml:space="preserve"> Quare rarò eueniet quatuor imagines in hoc ſpeculo comprehendi.</s> <s xml:id="echoid-s12215" xml:space="preserve"> Vnde ſi quis <lb/>hanc pluralitatem imaginum uoluerit uidere:</s> <s xml:id="echoid-s12216" xml:space="preserve"> diſponat uiſum intra ſpeculum circa ipſum, ut modi <lb/>cam partem eius abſcondat mole capitis, & totam ſpeculi ſuperficiem uiſu diſcurrat.</s> <s xml:id="echoid-s12217" xml:space="preserve"/> </p> <div xml:id="echoid-div428" type="float" level="0" n="0"> <figure xlink:label="fig-0187-02" xlink:href="fig-0187-02a"> <variables xml:id="echoid-variables133" xml:space="preserve">l m a b g n d</variables> </figure> </div> </div> <div xml:id="echoid-div430" type="section" level="0" n="0"> <head xml:id="echoid-head391" xml:space="preserve" style="it">88. In ſpeculo ſphærico cauo imago eiuſdem uiſibilis utrog uiſu aliâs una, aliâs gemina uide-<lb/>tur. 59 p 8.</head> <p> <s xml:id="echoid-s12218" xml:space="preserve">CVm autem aliquid in hoc ſpeculo percipietur duplici uiſu:</s> <s xml:id="echoid-s12219" xml:space="preserve"> ſi linea reflexionis fuerit æqui-<lb/>diſtans perpendiculari:</s> <s xml:id="echoid-s12220" xml:space="preserve"> [incidentiæ] erit locus imaginis punctum reflexionis [per 60 n.</s> <s xml:id="echoid-s12221" xml:space="preserve">] <lb/>Et cum diſtant à ſe puncta reflexionis reſpectu duorum uiſuum:</s> <s xml:id="echoid-s12222" xml:space="preserve"> apparebũt duobus uiſibus <lb/>duæ imagines eiuſdem puncti:</s> <s xml:id="echoid-s12223" xml:space="preserve"> & locus cuiuſq;</s> <s xml:id="echoid-s12224" xml:space="preserve"> imaginis eſt in puncto ſuæ reflexionis.</s> <s xml:id="echoid-s12225" xml:space="preserve"> Si uerò linea <lb/>reflexionis non ſit æquidiſtans perpendiculari:</s> <s xml:id="echoid-s12226" xml:space="preserve"> [incidentiæ] & punctum uiſum tantùm diſtet ab <lb/> <pb o="182" file="0188" n="188" rhead="ALHAZEN"/> uno uiſu, quantum ab alio, uel modica ſit differentia:</s> <s xml:id="echoid-s12227" xml:space="preserve"> erit locus imaginis reſpectu utríuſque uiſus <lb/>idem, aut diuerſus, ſed modicùm diſtans.</s> <s xml:id="echoid-s12228" xml:space="preserve"> Vnde aut una apparebit imago, aut ferè una:</s> <s xml:id="echoid-s12229" xml:space="preserve"> ſicut proba-<lb/>tum eſt in ſpeculis ſphæricis exterioribus.</s> <s xml:id="echoid-s12230" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div431" type="section" level="0" n="0"> <head xml:id="echoid-head392" xml:space="preserve" style="it">89. Communis ſectio ſuperficierum, reflexionis & ſpeculi cylindracei caui aliâs eſt latus cy-<lb/>lindri: aliâs circulus: aliâs ellipſis. 1 p 9.</head> <p> <s xml:id="echoid-s12231" xml:space="preserve">IN ſpeculis columnaribus concauis aliquando linea communis eſt linea recta:</s> <s xml:id="echoid-s12232" xml:space="preserve"> cum ſuperficies <lb/>reflexionis tranſit per axem:</s> <s xml:id="echoid-s12233" xml:space="preserve"> [per 21 d 11] aliquando linea communis erit circulus, cum ſuperfi-<lb/>cies illa eſt æquidiſtans baſibus:</s> <s xml:id="echoid-s12234" xml:space="preserve"> [per 5 th.</s> <s xml:id="echoid-s12235" xml:space="preserve"> Sereni de ſectione cylindri] aliquando linea commu-<lb/>nis eſt ſectio columnaris.</s> <s xml:id="echoid-s12236" xml:space="preserve"> Quando fuerit linea recta:</s> <s xml:id="echoid-s12237" xml:space="preserve"> erit locus imaginis & modus reflexionis, ſicut <lb/>in ſpeculis planis.</s> <s xml:id="echoid-s12238" xml:space="preserve"> Quando fuerit circulus:</s> <s xml:id="echoid-s12239" xml:space="preserve"> erit idem modus, qui in ſphæricis concauis.</s> <s xml:id="echoid-s12240" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div432" type="section" level="0" n="0"> <head xml:id="echoid-head393" xml:space="preserve" style="it">90. Sicommunis ſectio ſuperficierum, reflexionis & ſpeculi cylindracei caui fuerit ellipſis: <lb/>image uidebitur, aliâs ultra ſpeculum: aliâs in ſuperficie: aliâs citra uiſum: aliâs in uiſu: aliâs <lb/>inter uiſum & ſpeculum. 10 p 9.</head> <p> <s xml:id="echoid-s12241" xml:space="preserve">CVm uerò linea communis fuerit columnaris ſectio:</s> <s xml:id="echoid-s12242" xml:space="preserve"> aut erit locus imaginis ultra ſpeculum:</s> <s xml:id="echoid-s12243" xml:space="preserve"> <lb/>aut citra uiſum:</s> <s xml:id="echoid-s12244" xml:space="preserve"> aut in centro uiſus:</s> <s xml:id="echoid-s12245" xml:space="preserve"> aut inter ſpeculum & uiſum:</s> <s xml:id="echoid-s12246" xml:space="preserve"> aut in ipſo ſpeculo:</s> <s xml:id="echoid-s12247" xml:space="preserve"> quod ſic <lb/>patebit.</s> <s xml:id="echoid-s12248" xml:space="preserve"> Sit a b g ſectio:</s> <s xml:id="echoid-s12249" xml:space="preserve"> ducatur perpendicularis in hac ſectione:</s> <s xml:id="echoid-s12250" xml:space="preserve"> [ſuper planum tangens ſpe <lb/>culum in reflexionis puncto] quæ ſit d g:</s> <s xml:id="echoid-s12251" xml:space="preserve"> quam ſecundum prædicta patet eſſe diametrum circuli.</s> <s xml:id="echoid-s12252" xml:space="preserve"> <lb/>[Quia enim planum tangens cylindrum, tangit in latere per 26 n 4:</s> <s xml:id="echoid-s12253" xml:space="preserve"> ergo per 3 d 11 linea recta, per-<lb/>pendicularis plano tangenti, erit perpendicularis lateri, quod eſt parallelum axi per 21 d 11.</s> <s xml:id="echoid-s12254" xml:space="preserve"> Quar<gap/> <lb/>per 29 p 1 perpendicularis plano tangenti, perpendicula-<lb/> <anchor type="figure" xlink:label="fig-0188-01a" xlink:href="fig-0188-01"/> ris eſt axi.</s> <s xml:id="echoid-s12255" xml:space="preserve"> Planum uerò baſi parallelum & per dictam per-<lb/>pendicularem ductum eſt circulus, cẽtrum habens in axe <lb/>per 5th Sereni de ſectione cylindri.</s> <s xml:id="echoid-s12256" xml:space="preserve"> Recta igitur linea per <lb/>pendicularis plano, cylindrum in reflexionis puncto tan-<lb/>genti, eſt diameter circuli per reflexionis punctum ducti] <lb/>& unicam poſſe eſſe:</s> <s xml:id="echoid-s12257" xml:space="preserve"> cum ab alio puncto ſectionis nõ poſ-<lb/>ſit duci perpendicularis ſuper ſuperficiem contingentem.</s> <s xml:id="echoid-s12258" xml:space="preserve"> <lb/>[Nam cum communis ſectio circuli & ellipſis per reflexio <lb/>nis punctum ſe ſecantium, ſit perpẽdicularis, tum ad pla-<lb/>num in eodem reflexionis puncto cylindrum tangẽs, tum <lb/>ad axem, ut iam patuit:</s> <s xml:id="echoid-s12259" xml:space="preserve"> rectæ igitur lineæ ab alijs ſectionis <lb/>punctis ad axem ductę, ad ipſum obliquæ erunt:</s> <s xml:id="echoid-s12260" xml:space="preserve"> ſecus per <lb/>4 p 11 axis eſſet perpendicularis plano ellipſis:</s> <s xml:id="echoid-s12261" xml:space="preserve"> contra 9 th <lb/>Sereni de ſectione cylindri.</s> <s xml:id="echoid-s12262" xml:space="preserve">] Sumatur aliud punctũ, & ſit <lb/>b:</s> <s xml:id="echoid-s12263" xml:space="preserve"> & ducatur ab eo in ſectione linea perpendicularis ſuper <lb/>lineam, contingẽtem ſectionem in puncto b:</s> <s xml:id="echoid-s12264" xml:space="preserve"> quæ quidem <lb/>linea ſecundũ prędicta neceſſariò concurret cum perpen-<lb/>diculari g d.</s> <s xml:id="echoid-s12265" xml:space="preserve"> Concurrat in puncto d:</s> <s xml:id="echoid-s12266" xml:space="preserve"> & ſumptum ſit b circa <lb/>punctum g, ut angulus b d g ſit acutus.</s> <s xml:id="echoid-s12267" xml:space="preserve"> Deinde [per 31 p 1] <lb/>à puncto g ducatur in ſectione linea æquidiſtans b d:</s> <s xml:id="echoid-s12268" xml:space="preserve"> quæ <lb/>ſit g h:</s> <s xml:id="echoid-s12269" xml:space="preserve"> quę quidẽ cadet intra columnarem ſectionem:</s> <s xml:id="echoid-s12270" xml:space="preserve"> quia <lb/>angulus h g d erit acutus, cum ſit æqualis g d b:</s> <s xml:id="echoid-s12271" xml:space="preserve"> [per 29 p 1] <lb/>& à puncto ginter d & h ducatur linea:</s> <s xml:id="echoid-s12272" xml:space="preserve"> quæ neceſſariò cõ-<lb/>curret cum b d:</s> <s xml:id="echoid-s12273" xml:space="preserve"> [per lemma Procli ad 29 p 1] concurrat in <lb/>puncto n:</s> <s xml:id="echoid-s12274" xml:space="preserve"> & inter n & g ſumatur punctũ quodcunq;</s> <s xml:id="echoid-s12275" xml:space="preserve">: quod <lb/>ſit o:</s> <s xml:id="echoid-s12276" xml:space="preserve"> ultra punctum n ſumatur punctum t.</s> <s xml:id="echoid-s12277" xml:space="preserve"> Item à puncto g <lb/>ducatur ſupra g h, alia linea g z, tamen intra ſectionem:</s> <s xml:id="echoid-s12278" xml:space="preserve"> quæ neceſſariò concurret cũ b d ex alia par-<lb/>re:</s> <s xml:id="echoid-s12279" xml:space="preserve"> [per lemma Procli ad 29 p 1] ſit concurſus e.</s> <s xml:id="echoid-s12280" xml:space="preserve"> Ducatur g q linea, ut angulus q g d ſit æ qualis z g d <lb/>[per 23 p 1] & fiat angulus l g d æqualis angulo h g d:</s> <s xml:id="echoid-s12281" xml:space="preserve"> & angulus m g d æqualis angulo n g d.</s> <s xml:id="echoid-s12282" xml:space="preserve"> Palàm, <lb/>[per 12 n 4] quòd ſi fuerit uiſus in puncto z:</s> <s xml:id="echoid-s12283" xml:space="preserve"> reflectetur punctũ q ad ipſum, à puncto g:</s> <s xml:id="echoid-s12284" xml:space="preserve"> & punctum <lb/>imaginis eſt e:</s> <s xml:id="echoid-s12285" xml:space="preserve"> [per 6 n] & ſi uiſus fuerit in puncto h:</s> <s xml:id="echoid-s12286" xml:space="preserve"> reflectetur ad ipſum l à puncto g:</s> <s xml:id="echoid-s12287" xml:space="preserve"> & erit locus <lb/>imaginis g:</s> <s xml:id="echoid-s12288" xml:space="preserve"> ſi uerò fuerit uiſus in puncto o:</s> <s xml:id="echoid-s12289" xml:space="preserve"> reflectetur ad ipſum, punctum m:</s> <s xml:id="echoid-s12290" xml:space="preserve"> & locus imaginis erit <lb/>n:</s> <s xml:id="echoid-s12291" xml:space="preserve"> ſi autem fuerit in n:</s> <s xml:id="echoid-s12292" xml:space="preserve"> erit locus imaginis puncti m in centro uiſus, id eſt in n:</s> <s xml:id="echoid-s12293" xml:space="preserve"> ſi autem fuerit in t:</s> <s xml:id="echoid-s12294" xml:space="preserve"> erit <lb/>locus imaginis tunc inter uiſum & ſpeculum:</s> <s xml:id="echoid-s12295" xml:space="preserve"> quia in n.</s> <s xml:id="echoid-s12296" xml:space="preserve"> Et ita patet propoſitum.</s> <s xml:id="echoid-s12297" xml:space="preserve"/> </p> <div xml:id="echoid-div432" type="float" level="0" n="0"> <figure xlink:label="fig-0188-01" xlink:href="fig-0188-01a"> <variables xml:id="echoid-variables134" xml:space="preserve">e b <gap/> g q <gap/> m d a o <gap/> z <gap/> h k <gap/></variables> </figure> </div> </div> <div xml:id="echoid-div434" type="section" level="0" n="0"> <head xml:id="echoid-head394" xml:space="preserve" style="it">91. Si uiſus & uiſibile fuerint in eadẽ recta linea, perpendiculari plano ſpeculum cylindra-<lb/>ceum cauum tangenti: aliâs ab uno: aliâs à duobus ſpeculi punctis reflexio fiet: & imago uide-<lb/>bitur in centro uiſus. 11 p 9.</head> <p> <s xml:id="echoid-s12298" xml:space="preserve">HAec quidem iam dicta intelligenda ſunt, cum punctum uiſum nõ fuerit ſuper perpendicu-<lb/>larem cum ipſo uiſu.</s> <s xml:id="echoid-s12299" xml:space="preserve"> Tunc enim cum infinitæ ſuperficies poſsintintelligi, quarum quælibet <lb/>orthogonalis ſit ſuper ſuperficiem, contingentem ſpeculum [per 18 p 11:</s> <s xml:id="echoid-s12300" xml:space="preserve"> quia ſuperficies illæ <lb/>ducuntur per rectam plano ſpeculum tangenti perpendicularem] & omnes ſecent ſe ſuper illam <lb/> <pb o="183" file="0189" n="189" rhead="OPTICAE LIBER V."/> perpendicularem:</s> <s xml:id="echoid-s12301" xml:space="preserve"> quædam illarum ſuperficierum efficiet lineam cõmunem, lineam rectam:</s> <s xml:id="echoid-s12302" xml:space="preserve"> & non <lb/>fiet reflexio, niſi ſuper illam perpendicularem:</s> <s xml:id="echoid-s12303" xml:space="preserve"> [per 11 n 4] & locus imaginis erit centrum uiſus:</s> <s xml:id="echoid-s12304" xml:space="preserve"> & <lb/>non uidebitur punctum, niſi quod fuerit in ſuperficie uiſus [per 13 n.</s> <s xml:id="echoid-s12305" xml:space="preserve">] Quædam autẽillarũ ſuper-<lb/>ficierum efficiet lineam communem, circulum:</s> <s xml:id="echoid-s12306" xml:space="preserve"> & tunc puncta, inter quæ & uiſum fuerit centrum <lb/>circuli:</s> <s xml:id="echoid-s12307" xml:space="preserve"> poterunt reflecti ad uiſum, ſingula à duobus punctis circuli:</s> <s xml:id="echoid-s12308" xml:space="preserve"> cum à ſingulis ducantur lineæ <lb/>facientes angulũ cum ſuperficie contingente, quem <lb/> <anchor type="figure" xlink:label="fig-0189-01a" xlink:href="fig-0189-01"/> per æqualia diuidit perpendicularis ducta ad cen-<lb/>trum.</s> <s xml:id="echoid-s12309" xml:space="preserve"> [Nam cum a b ſit diameter circuli, & f g axis <lb/>cylindri:</s> <s xml:id="echoid-s12310" xml:space="preserve"> erit per 3 d 11 e f perpẽdicularis f g:</s> <s xml:id="echoid-s12311" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s12312" xml:space="preserve"> an-<lb/>guli ad f erunt recti:</s> <s xml:id="echoid-s12313" xml:space="preserve"> at ex theſic e æquatur ipſi e d:</s> <s xml:id="echoid-s12314" xml:space="preserve"> & <lb/>communis eſt e f:</s> <s xml:id="echoid-s12315" xml:space="preserve"> ergo per 4 p 1 triangula d f e, c f e <lb/>ſunt æquiangula.</s> <s xml:id="echoid-s12316" xml:space="preserve"> Quare perpẽdicularis fe bifariam <lb/>ſecat angulum c e d:</s> <s xml:id="echoid-s12317" xml:space="preserve"> eodemq́;</s> <s xml:id="echoid-s12318" xml:space="preserve"> modo oſtẽdetur per-<lb/>pendicularem g f bifariam ſecare angulũ d g c.</s> <s xml:id="echoid-s12319" xml:space="preserve">] Et <lb/>hæc quidem dico de punctis, quę ſunt in illa perpen <lb/>diculari:</s> <s xml:id="echoid-s12320" xml:space="preserve"> & loca imaginũ erunt in centro circuli:</s> <s xml:id="echoid-s12321" xml:space="preserve"> alia <lb/>puncta illius perpendicularis nõ reflectentur ad ui-<lb/>ſum, præter punctum, quod eſt in ſuperficie uiſus:</s> <s xml:id="echoid-s12322" xml:space="preserve"> & <lb/>illud per illam perpendicularem [per 11 n 4.</s> <s xml:id="echoid-s12323" xml:space="preserve">] Cum <lb/>autem fuerit linea cõmunis, ſectio columnaris:</s> <s xml:id="echoid-s12324" xml:space="preserve"> non <lb/>poterunt puncta perpendicularis reflecti ab aliqui-<lb/>bus alijs punctis ſectionis:</s> <s xml:id="echoid-s12325" xml:space="preserve"> cum forma accedens ſu-<lb/>per perpendicularem, reflectatur ſuper perpendicularem:</s> <s xml:id="echoid-s12326" xml:space="preserve"> & in ſectione una ſit perpendicularis [ut <lb/>proximo numero oſtenſum eſt.</s> <s xml:id="echoid-s12327" xml:space="preserve">] Quare per hanc ſolam perpendicularem fiet reflexio:</s> <s xml:id="echoid-s12328" xml:space="preserve"> & ſolũ pun-<lb/>ctu m ſuperficiei uiſus uidebitur:</s> <s xml:id="echoid-s12329" xml:space="preserve"> & locus imaginis erit centrum uiſus.</s> <s xml:id="echoid-s12330" xml:space="preserve"/> </p> <div xml:id="echoid-div434" type="float" level="0" n="0"> <figure xlink:label="fig-0189-01" xlink:href="fig-0189-01a"> <variables xml:id="echoid-variables135" xml:space="preserve">a s c p c f d <gap/> d e b</variables> </figure> </div> </div> <div xml:id="echoid-div436" type="section" level="0" n="0"> <head xml:id="echoid-head395" xml:space="preserve" style="it">92. Siuiſus fuerit in centro circuli ſpeculi cylindracei caui: reflectetur ab eiuſdẽ circuli peri-<lb/>pheria, ſimili peripheriæ circuli per centrũ uiſus ducti: & imago uidebitur in cẽtro uiſus. 12 p 9.</head> <p> <s xml:id="echoid-s12331" xml:space="preserve">SI uerò fuerit uiſus in cẽtro circuli:</s> <s xml:id="echoid-s12332" xml:space="preserve"> reflectetur portio uiſus, quam ſecant perpendiculares, du-<lb/>ctæ à centro uiſus ad circulum, [per cẽtrum uiſus ductum] à portione ſimili circulo, [ſpeculi] <lb/>quam ſecant ſimiliter eædem perpendiculares.</s> <s xml:id="echoid-s12333" xml:space="preserve"> Quia cum quælibet linea ducta à centro uiſus <lb/>ad circulum, ſit perpendicularis:</s> <s xml:id="echoid-s12334" xml:space="preserve"> [ſuper ſuperficies uiſus & ſpeculi per 25 n 4:</s> <s xml:id="echoid-s12335" xml:space="preserve"> quia tranſit per cen-<lb/>tra uiſus & ſpeculi] fiet reflexio ſuper perpendicularem:</s> <s xml:id="echoid-s12336" xml:space="preserve"> [per 11 n 4] & locus imaginis erit cẽtrum <lb/>uiſus:</s> <s xml:id="echoid-s12337" xml:space="preserve"> quod eſt centrum circuli.</s> <s xml:id="echoid-s12338" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div437" type="section" level="0" n="0"> <head xml:id="echoid-head396" xml:space="preserve" style="it">93. Si communis ſectio ſuperficierum, reflexionis & ſpeculi cylindracei cauifuerit ellipſis: à <lb/>pluribus punct is idem uiſibile ad eundem uiſum reflecti <lb/>poteſt. 9 p 9.</head> <figure> <variables xml:id="echoid-variables136" xml:space="preserve">e b <gap/> g q l m d o a z n <gap/> h k</variables> </figure> <p> <s xml:id="echoid-s12339" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s12340" xml:space="preserve"> ſuper punctum a fiat angulus acutus quo <gap/> <lb/>quo modo:</s> <s xml:id="echoid-s12341" xml:space="preserve"> qui ſit f a g.</s> <s xml:id="echoid-s12342" xml:space="preserve"> Palàm, quòd cõcurret f a cũ <lb/>g z:</s> <s xml:id="echoid-s12343" xml:space="preserve"> [quia g z cadens intra ellipſin ex theſi 90 n ef-<lb/>ficit angulum z g d acutum:</s> <s xml:id="echoid-s12344" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s12345" xml:space="preserve"> cũ anguli z g d, f a g duo-<lb/>bus rectis ſint minores:</s> <s xml:id="echoid-s12346" xml:space="preserve"> rectæ a f, g z concurrent ad partes <lb/>z, per 11 a x] ſit concurſus in puncto z:</s> <s xml:id="echoid-s12347" xml:space="preserve"> & [per 23 p 1] fiat an-<lb/>gulus c a g æqualis angulo f a g:</s> <s xml:id="echoid-s12348" xml:space="preserve"> concurret equidẽ a c cum <lb/>g q:</s> <s xml:id="echoid-s12349" xml:space="preserve"> [per 11 ax.</s> <s xml:id="echoid-s12350" xml:space="preserve"> Nam quia angulus q g d æquatus eſt angulo <lb/>z g a acuto, ut patuit 90 n:</s> <s xml:id="echoid-s12351" xml:space="preserve"> & modo angulus c a g æquatur <lb/>z a g:</s> <s xml:id="echoid-s12352" xml:space="preserve"> anguli q g d, c a g ſunt minores duobus rectis] ſit cõ-<lb/>curſus in puncto c.</s> <s xml:id="echoid-s12353" xml:space="preserve"> Palàm [per 12 n 4] quòd c reflectetur <lb/>ad z à puncto g:</s> <s xml:id="echoid-s12354" xml:space="preserve"> & ita reflectetur à puncto a ad z, & non ab <lb/>alio puncto ſectionis.</s> <s xml:id="echoid-s12355" xml:space="preserve"> Quia non poterit reflecti, niſi à ter-<lb/>mino perpendicularis:</s> <s xml:id="echoid-s12356" xml:space="preserve"> & una eſt in ſectione illa perpendi-<lb/>cularis [ut oſtenſum eſt 90 n] ſcilicet g a.</s> <s xml:id="echoid-s12357" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div438" type="section" level="0" n="0"> <head xml:id="echoid-head397" xml:space="preserve" style="it">94. Si duo puncta ſumantur in axeſpeculi cylindra-<lb/>ceicaui: poſſunt à tota circuli peripheria inter ſe mutuò <lb/>reflecti: & imago uidebitur in peripheria circuliextra <lb/>ſpeculi ſuperficiem deſcripti. 13 p 9.</head> <p> <s xml:id="echoid-s12358" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s12359" xml:space="preserve"> ſumptis duobus punctis in axe columnæ:</s> <s xml:id="echoid-s12360" xml:space="preserve"> <lb/>poterit unum reflecti ad aliud ab uno circulo colu-<lb/>mnæ toto:</s> <s xml:id="echoid-s12361" xml:space="preserve"> & locus imaginis erit circulus quidam <lb/>extra columnam.</s> <s xml:id="echoid-s12362" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s12363" xml:space="preserve"> ſit e z axis:</s> <s xml:id="echoid-s12364" xml:space="preserve"> t, h puncta ſum-<lb/>pta in axe:</s> <s xml:id="echoid-s12365" xml:space="preserve"> a g, b d baſes.</s> <s xml:id="echoid-s12366" xml:space="preserve"> Diuidatur t h per æqualia in puncto q [per 10 p 1] & fiat circulus, cuius q <lb/>centrum:</s> <s xml:id="echoid-s12367" xml:space="preserve"> eius diameter l m:</s> <s xml:id="echoid-s12368" xml:space="preserve"> qui erit æquidiſtans baſibus:</s> <s xml:id="echoid-s12369" xml:space="preserve"> [per 5 th.</s> <s xml:id="echoid-s12370" xml:space="preserve"> Sereni de ſectione cylindri] la-<lb/>tera columnæ b l a, d m g.</s> <s xml:id="echoid-s12371" xml:space="preserve"> Fiat etiam circulus k p, cuius h centrum, p k diameter:</s> <s xml:id="echoid-s12372" xml:space="preserve"> & ducantur lineæ <lb/> <pb o="184" file="0190" n="190" rhead="ALHAZEN"/> t l, t m, h l, h m.</s> <s xml:id="echoid-s12373" xml:space="preserve"> Palàm, quòd quatuor angulorum ſuper q quilibet eſt rectus [per 3 d 11:</s> <s xml:id="echoid-s12374" xml:space="preserve"> quia axis pe<gap/><gap/> <lb/>pendicularis eſt circulo l m per 21 d 11] & t q æqualis q h <lb/> <anchor type="figure" xlink:label="fig-0190-01a" xlink:href="fig-0190-01"/> [per fabricationem] & q l æqualis q m:</s> <s xml:id="echoid-s12375" xml:space="preserve"> [per 15 d 1] erũt <lb/>illa triangula ſimilia [per 4 p 1.</s> <s xml:id="echoid-s12376" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s12377" xml:space="preserve"> 1 d 6] & anguli t l q, q l <lb/>h æquales:</s> <s xml:id="echoid-s12378" xml:space="preserve"> ſimiliter anguli t m q, q m h æquales.</s> <s xml:id="echoid-s12379" xml:space="preserve"> Siergo <lb/>fuerit t centrum uiſus:</s> <s xml:id="echoid-s12380" xml:space="preserve"> reflectetur quidem h ad punctum <lb/>t à punctol:</s> <s xml:id="echoid-s12381" xml:space="preserve"> & ſimiliter à puncto m [per 12 n 4.</s> <s xml:id="echoid-s12382" xml:space="preserve">] Si ergo <lb/>moueatur triangulũ t l h, immoto axe t h:</s> <s xml:id="echoid-s12383" xml:space="preserve"> deſcribet pun-<lb/>ctum l circulũ:</s> <s xml:id="echoid-s12384" xml:space="preserve"> & ſemper duo anguli t l q, q l h manebunt <lb/>æquales:</s> <s xml:id="echoid-s12385" xml:space="preserve"> & ſemper in hoc motu reflectetur h ad t.</s> <s xml:id="echoid-s12386" xml:space="preserve"> Pro-<lb/>ducatur autem linea p h k, donec cõcurrat cum linea t l:</s> <s xml:id="echoid-s12387" xml:space="preserve"> <lb/>[concurret autẽ per lemma Procli ad 29 p 1:</s> <s xml:id="echoid-s12388" xml:space="preserve"> quia m l, c k <lb/>ſunt parallelę per 29 p 1] & ſit cõcurſus f.</s> <s xml:id="echoid-s12389" xml:space="preserve"> Palàm [per 7 n] <lb/>quòd ferit locus imaginis.</s> <s xml:id="echoid-s12390" xml:space="preserve"> Et motu trianguli t l h, mo-<lb/>uebitur triangulum t f h:</s> <s xml:id="echoid-s12391" xml:space="preserve"> & hoc motu punctum f deſcri-<lb/>bet circulum extra columnam:</s> <s xml:id="echoid-s12392" xml:space="preserve"> & totus ille circulus erit <lb/>locus imaginis.</s> <s xml:id="echoid-s12393" xml:space="preserve"> Et hoc eſt propoſitum Idẽ erit probandi <lb/>modus, ſumptis quibuslibet duobus punctis in axe.</s> <s xml:id="echoid-s12394" xml:space="preserve"/> </p> <div xml:id="echoid-div438" type="float" level="0" n="0"> <figure xlink:label="fig-0190-01" xlink:href="fig-0190-01a"> <variables xml:id="echoid-variables137" xml:space="preserve">d z b t m l q r p h k f g e a</variables> </figure> </div> </div> <div xml:id="echoid-div440" type="section" level="0" n="0"> <head xml:id="echoid-head398" xml:space="preserve" style="it">95. Si communis ſectio ſuperficierum, reflexionis <lb/>& ſpeculi cylindracei caui fuerit circulus, uelellipſis: <lb/>reflexio fiet aliâs ab uno: aliâs à duobus: aliâs àtri-<lb/>bus: aliâs à quatuor ſpeculipũctis: totideḿ uidebun-<lb/>tur imagines. 14. 15 p 9.</head> <p> <s xml:id="echoid-s12395" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s12396" xml:space="preserve"> punctorum extra perpẽdicularem uiſus <lb/>ſumptorum quædam unicam habent imaginem:</s> <s xml:id="echoid-s12397" xml:space="preserve"> <lb/>quædam duas:</s> <s xml:id="echoid-s12398" xml:space="preserve"> quædam tres:</s> <s xml:id="echoid-s12399" xml:space="preserve"> quædam quatuor:</s> <s xml:id="echoid-s12400" xml:space="preserve"> <lb/>& non plures.</s> <s xml:id="echoid-s12401" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s12402" xml:space="preserve"> ſit a punctum uiſum extra <lb/>perpendicularem uiſus:</s> <s xml:id="echoid-s12403" xml:space="preserve"> & fiat ſuperficies tranſiens per a æquidiſtans baſibus ſpeculi:</s> <s xml:id="echoid-s12404" xml:space="preserve"> [ut oftẽſum <lb/>eſt 47 n] faciet quidem [per 5 th.</s> <s xml:id="echoid-s12405" xml:space="preserve"> Sereni de ſectione cylindri] circulum in columna.</s> <s xml:id="echoid-s12406" xml:space="preserve"> Sit centrum il-<lb/>lius circuli h:</s> <s xml:id="echoid-s12407" xml:space="preserve"> & ſumatur in ſuperficie circuli aliud <lb/> <anchor type="figure" xlink:label="fig-0190-02a" xlink:href="fig-0190-02"/> punctum, quod ſit b:</s> <s xml:id="echoid-s12408" xml:space="preserve"> & ducãtur dιametri a h, b h.</s> <s xml:id="echoid-s12409" xml:space="preserve"> <lb/>Palàm ex eis, quæ dicta ſunt in ſpeculis ſphæricis <lb/>concauis [86 n] quòd ab uno puncto arcus, quẽ <lb/>intercipiunt hæ duæ diametri, poteſt a reflecti ad <lb/>b:</s> <s xml:id="echoid-s12410" xml:space="preserve"> forſitan à duobus punctis, aut tribus, ſed non à <lb/>pluribus:</s> <s xml:id="echoid-s12411" xml:space="preserve"> ab arcu autem oppoſito, nõ niſi ab uno <lb/>puncto.</s> <s xml:id="echoid-s12412" xml:space="preserve"> Sit ergo, quòd a reflectatur ad b à tribus <lb/>punctιs interciſi arcus:</s> <s xml:id="echoid-s12413" xml:space="preserve"> & ſint puncta illa g, d, e:</s> <s xml:id="echoid-s12414" xml:space="preserve"> & <lb/>ducãtur lineæ a g, h g, b g, h d, b d, a d, a e, h e, b e:</s> <s xml:id="echoid-s12415" xml:space="preserve"> & <lb/>à puncto a ducantur in eadẽ ſuperficie tres lineæ <lb/>æquidiſtantes tribus diametris h g, h d, h e:</s> <s xml:id="echoid-s12416" xml:space="preserve"> quæ <lb/>ſint a k, a f, a n.</s> <s xml:id="echoid-s12417" xml:space="preserve"> Cum igitur a k ſit æquidiſtans h g:</s> <s xml:id="echoid-s12418" xml:space="preserve"> <lb/>cõcurret b g cum a k:</s> <s xml:id="echoid-s12419" xml:space="preserve"> [per lemma Procli ad 29 p 1] <lb/>concurrat in puncto k.</s> <s xml:id="echoid-s12420" xml:space="preserve"> Similiter b d concurret cũ <lb/>a f:</s> <s xml:id="echoid-s12421" xml:space="preserve"> ſit cõcurſus in puncto f.</s> <s xml:id="echoid-s12422" xml:space="preserve"> Similiter b e cum a n:</s> <s xml:id="echoid-s12423" xml:space="preserve"> <lb/>ſit concurſus in puncto n.</s> <s xml:id="echoid-s12424" xml:space="preserve"> Deinde à puncto h eri-<lb/>gatur axis:</s> <s xml:id="echoid-s12425" xml:space="preserve"> qui ſit h x:</s> <s xml:id="echoid-s12426" xml:space="preserve"> & à puncto b perpendicula-<lb/>ris ſuper ſuperficiem circuli:</s> <s xml:id="echoid-s12427" xml:space="preserve"> [per 12 p 11] quæ erit <lb/>æquidiſtans axi [per 6 p 11] quæ ſit b t:</s> <s xml:id="echoid-s12428" xml:space="preserve"> & ſumatur <lb/>in ea punctum quodcunq;</s> <s xml:id="echoid-s12429" xml:space="preserve">: quod ſit t:</s> <s xml:id="echoid-s12430" xml:space="preserve"> & ducantur <lb/>tres lineæ t k, t f, t n:</s> <s xml:id="echoid-s12431" xml:space="preserve"> & [per 12 p 11] à tribus punctis <lb/>g, d, e erigantur tres perpẽdiculares ſuper ſuper-<lb/>ficiem circuli:</s> <s xml:id="echoid-s12432" xml:space="preserve"> g m, d l, e q:</s> <s xml:id="echoid-s12433" xml:space="preserve"> erunt quidẽ [per 6 p 11] <lb/>æquidiſtãtes t b, e q:</s> <s xml:id="echoid-s12434" xml:space="preserve"> igitur erũt in ſuperficie trian <lb/>guli t b n:</s> <s xml:id="echoid-s12435" xml:space="preserve"> [per 35 d 1.</s> <s xml:id="echoid-s12436" xml:space="preserve"> 1 p 11] igitur e q ſecabit t n:</s> <s xml:id="echoid-s12437" xml:space="preserve"> <lb/>[per lemma Procli ad 29 p 1] ſecet in puncto q:</s> <s xml:id="echoid-s12438" xml:space="preserve"> d l <lb/>ſecett fin puncto l:</s> <s xml:id="echoid-s12439" xml:space="preserve"> g m ſecet t k in puncto m.</s> <s xml:id="echoid-s12440" xml:space="preserve"> Et <lb/>erunt hæ tres perpẽdiculares, lineæ longitudinis <lb/>columnæ [ut patet è 21 d 11.</s> <s xml:id="echoid-s12441" xml:space="preserve">] À<unsure/> puncto q ducatur <lb/>æquidiſtans lineæ n a:</s> <s xml:id="echoid-s12442" xml:space="preserve"> [per 31 p 1] quæ quidẽ con-<lb/>curret cum axe x h:</s> <s xml:id="echoid-s12443" xml:space="preserve"> [per<gap/>lemma Procli<gap/>ad 29 p 1] <lb/>quoniam erit æquidiſtans e h:</s> <s xml:id="echoid-s12444" xml:space="preserve"> [per 30 p 1] ſit con <lb/>curſus in puncto u:</s> <s xml:id="echoid-s12445" xml:space="preserve"> & ducatur linea t a:</s> <s xml:id="echoid-s12446" xml:space="preserve"> quam ſeca <lb/>bit q u:</s> <s xml:id="echoid-s12447" xml:space="preserve"> quoniam q u ducitur à latere trianguli [tbn] & linea e q equidiſtãte baſi [t b.</s> <s xml:id="echoid-s12448" xml:space="preserve">] Sit punctum <lb/> <pb o="185" file="0191" n="191" rhead="OPTICAE LIBER V."/> ſectionisi:</s> <s xml:id="echoid-s12449" xml:space="preserve"> & ducatur linea q a.</s> <s xml:id="echoid-s12450" xml:space="preserve"> Palàm, quòd angulus b e h æqualis eſt angulo e n a [per 29 p 1:</s> <s xml:id="echoid-s12451" xml:space="preserve"> quia <lb/>a n, h e ſunt parallelæ per fabricationem] & angulus h e a æqualis angulo e a n:</s> <s xml:id="echoid-s12452" xml:space="preserve"> & [per 12 n 4] angu <lb/>lus b e h æqualis angulo h e a:</s> <s xml:id="echoid-s12453" xml:space="preserve"> erit angulus e a n æqualis angulo e n a:</s> <s xml:id="echoid-s12454" xml:space="preserve"> quare [per 6 p 1] e n æqualis <lb/>e a, & e q perpendicularis:</s> <s xml:id="echoid-s12455" xml:space="preserve"> [duabus rectis e a, e n per 3 d 11:</s> <s xml:id="echoid-s12456" xml:space="preserve"> quia perpendicularis eſt per fabricationẽ <lb/>triangulo a e n] erit [per 4 p 1] triangulũ q e a æquale triangulo q e n:</s> <s xml:id="echoid-s12457" xml:space="preserve"> & erit q n æqualis q a:</s> <s xml:id="echoid-s12458" xml:space="preserve"> & erit <lb/>[per 5 p 1] q n a æqualis angulo q a n:</s> <s xml:id="echoid-s12459" xml:space="preserve"> ſed angulus t q i æqualis angulo q n a, & angulus i q a æqua-<lb/>lis angulo q a n:</s> <s xml:id="echoid-s12460" xml:space="preserve"> [per 29 p 1:</s> <s xml:id="echoid-s12461" xml:space="preserve"> quia q i, a n ſunt parallelæ per fabricationem] erit angulus i q t æqualis <lb/>angulo i q a.</s> <s xml:id="echoid-s12462" xml:space="preserve"> Quare a reflectetur adt à puncto columnæ, quod eſt q [per 12 n 4.</s> <s xml:id="echoid-s12463" xml:space="preserve">] Eodem modo pro-<lb/>babitur, quòd reflectetur a ad t â punctis l, m:</s> <s xml:id="echoid-s12464" xml:space="preserve"> Et ita à tribus punctis columnæ ex eadem parte:</s> <s xml:id="echoid-s12465" xml:space="preserve"> nec <lb/>poteſt à pluribus.</s> <s xml:id="echoid-s12466" xml:space="preserve"> Detur enim aliud:</s> <s xml:id="echoid-s12467" xml:space="preserve"> ducto latere [cylindri, ut oſtenſum eſt 47 n] ab illo puncto:</s> <s xml:id="echoid-s12468" xml:space="preserve"> <lb/>cadet in circulum, quem habemus:</s> <s xml:id="echoid-s12469" xml:space="preserve"> & probabitur, quòd à pũcto caſus, qui eſt in circulo, poterit re-<lb/>flecti a ad t, repetita ꝓbatione:</s> <s xml:id="echoid-s12470" xml:space="preserve"> quod eſt impoſsibile [ut oſtẽſum eſt 86 n.</s> <s xml:id="echoid-s12471" xml:space="preserve">] Ex arcu uerò circuli op-<lb/>poſito [arcui g d e] poterit reflecti a ad b ab uno puncto:</s> <s xml:id="echoid-s12472" xml:space="preserve"> [per 73 n] ſit illud z:</s> <s xml:id="echoid-s12473" xml:space="preserve"> & ducatur diameter <lb/>h z:</s> <s xml:id="echoid-s12474" xml:space="preserve"> & [per 31 p 1] ei æquidiſtans a s:</s> <s xml:id="echoid-s12475" xml:space="preserve"> & ducatur b z:</s> <s xml:id="echoid-s12476" xml:space="preserve"> quæ concurrat cum a s in puncto s:</s> <s xml:id="echoid-s12477" xml:space="preserve"> [cõcurret au-<lb/>tem per lemma Procli ad 29 p 1] & erigatur perpendicularis:</s> <s xml:id="echoid-s12478" xml:space="preserve"> [ſuper circulũ, cuius centrũ eſt h] quę <lb/>ſit o z:</s> <s xml:id="echoid-s12479" xml:space="preserve"> quæ erit latus [per 21 d 11] & [per 6 p 11] æquidiſtans t b:</s> <s xml:id="echoid-s12480" xml:space="preserve"> & ducatur t s:</s> <s xml:id="echoid-s12481" xml:space="preserve"> quę ſecabitur à linea o <lb/>z:</s> <s xml:id="echoid-s12482" xml:space="preserve"> [per lemma Procli ad 29 p 1.</s> <s xml:id="echoid-s12483" xml:space="preserve">] Sit ſectio in pũcto o.</s> <s xml:id="echoid-s12484" xml:space="preserve"> Probabitur modo prędicto, quòd a reflectetur <lb/>ad t à puncto o.</s> <s xml:id="echoid-s12485" xml:space="preserve"> Et ſi ſumatur exilla parte punctum aliud columnæ, à quo poſsit reflecti:</s> <s xml:id="echoid-s12486" xml:space="preserve"> per repli-<lb/>cationem probationis probabitur, quòd ab alio puncto circuli, quàm z, poteſt reflecti ex parte illa:</s> <s xml:id="echoid-s12487" xml:space="preserve"> <lb/>quod eſt impoſsibile [ut demonſtratũ eſt 75 n.</s> <s xml:id="echoid-s12488" xml:space="preserve">] Si ergo a ab uno puncto circuli reflectitur ad b ex <lb/>aliqua parte:</s> <s xml:id="echoid-s12489" xml:space="preserve"> reflectetur ab uno columnæ ex eadem ad t:</s> <s xml:id="echoid-s12490" xml:space="preserve"> ſi à duobus, à duobus:</s> <s xml:id="echoid-s12491" xml:space="preserve"> ſi à tribus, à tribus:</s> <s xml:id="echoid-s12492" xml:space="preserve"> <lb/>nec poteſt amplius ab illa parte:</s> <s xml:id="echoid-s12493" xml:space="preserve"> ab oppoſita uerò parte non niſi ab uno puncto circuli tantùm, & <lb/>ab uno columnæ tantùm.</s> <s xml:id="echoid-s12494" xml:space="preserve"> Item t b æquidiſtat u h:</s> <s xml:id="echoid-s12495" xml:space="preserve"> [ut ab initio demõſtratum eſt:</s> <s xml:id="echoid-s12496" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s12497" xml:space="preserve"> per 35 d 1 ſunt <lb/>in eadem ſuperficie, quæ eſt t b u h] nec poteſt ſumi ſuperficies æqualis, in qua ſit t cum u h, præter <lb/>ſuperficiem t b u h.</s> <s xml:id="echoid-s12498" xml:space="preserve"> Similiter non poteſt ſuperficies ſumi, in qua ſit a cum u h, præter ſuperficiem a u <lb/>h, quæ eſt perpendicularis [circulo, cuius cẽtrum h, per 18 p 11.</s> <s xml:id="echoid-s12499" xml:space="preserve">] Igitur t non eſt in eadem ſuperficie <lb/>perpendiculari cum a, necin eodem circulo, nec eſt in axe, quia eſt in linea ei æ quidiſtante.</s> <s xml:id="echoid-s12500" xml:space="preserve"> Super-<lb/>ficies igitur, in qua a reflectitur ad t, eft ſectio colũnaris [per 9 th.</s> <s xml:id="echoid-s12501" xml:space="preserve"> Sereni de ſectione cylindri.</s> <s xml:id="echoid-s12502" xml:space="preserve">] Ve-<lb/>rùm producta ſit t a ultra t, & a ex utraq;</s> <s xml:id="echoid-s12503" xml:space="preserve"> parte:</s> <s xml:id="echoid-s12504" xml:space="preserve"> & ſit r p.</s> <s xml:id="echoid-s12505" xml:space="preserve"> Cum quatuor ſint ſuperficies reflexionis:</s> <s xml:id="echoid-s12506" xml:space="preserve"> <lb/>quia à quatuor punctis [q, l, m;</s> <s xml:id="echoid-s12507" xml:space="preserve"> o] ſit reflexio, & in qualibet harum ſint duo puncta t, a:</s> <s xml:id="echoid-s12508" xml:space="preserve"> erit r p com-<lb/>munis quatuor ſuperficiebus reflexionis:</s> <s xml:id="echoid-s12509" xml:space="preserve"> [per 1 p 11:</s> <s xml:id="echoid-s12510" xml:space="preserve"> quia uiſus & uiſibile, quæ ſunt in linea r p, ſunt <lb/>in qualibet reflexionis ſuperficie per 23 n 4] & quælibet harũ ſuperficierum ſecat ſuperficiem, con-<lb/>tingentem ſpeculum in puncto ſ<gap/>æ reflexionis, ſuper ſuam lineam communem, nõ ſuper eandem <lb/>[quia cum puncta reflexionis ſint diuerſa, etiam communes ſectiones illarum ſuperficierum (quæ <lb/>ſuntrectæ lineæ per 3 d 11) diuerſæ erunt.</s> <s xml:id="echoid-s12511" xml:space="preserve">] Linea ergo r p perpendicularis eſt ſuper unam linearum <lb/>quatuor cõmunium, non ſuper duas:</s> <s xml:id="echoid-s12512" xml:space="preserve"> eſſet enim perpendicularis ſuper ſuperficiem contingentem:</s> <s xml:id="echoid-s12513" xml:space="preserve"> <lb/>[per 3 d 11] & ita perueniret ad axem.</s> <s xml:id="echoid-s12514" xml:space="preserve"> [Quia enim per 21 d 11 latus cylindraceum ęquidiſtat axi:</s> <s xml:id="echoid-s12515" xml:space="preserve"> & r p <lb/>perpendicularis plano tangenti ex cõcluſo, ſimul perpendicularis eſt lateri per 3 d 11:</s> <s xml:id="echoid-s12516" xml:space="preserve"> ergo per lem-<lb/>ma Procli ad 29 p 1 r p (quæ paulo antè oſtẽſa eſt extra axẽ eſſe) cõtinuata ſecabit axẽ;</s> <s xml:id="echoid-s12517" xml:space="preserve"> quod eſt ab:</s> <s xml:id="echoid-s12518" xml:space="preserve"> <lb/>ſurdum.</s> <s xml:id="echoid-s12519" xml:space="preserve">] Sunt ergo diuerſæ perpendiculares à puncto t ad has quatuor lineas communes:</s> <s xml:id="echoid-s12520" xml:space="preserve"> nec eſt <lb/>niſi una perpendicularis tantùm, quę tranſit per a.</s> <s xml:id="echoid-s12521" xml:space="preserve"> Et perpendicularis aut eſt æquidiſtans lineæ re-<lb/>flexionis:</s> <s xml:id="echoid-s12522" xml:space="preserve"> aut concurrit cum ea ultra ſpeculum, uel intra.</s> <s xml:id="echoid-s12523" xml:space="preserve"> Si fuerit æquidiſtans:</s> <s xml:id="echoid-s12524" xml:space="preserve"> erit locus imaginis <lb/>punctum reflexionis, ut probatum eſt<gap/>[91.</s> <s xml:id="echoid-s12525" xml:space="preserve"> n] Et cum quatuor ſint reflexionis puncta:</s> <s xml:id="echoid-s12526" xml:space="preserve"> erunt qua-<lb/>tuor imagines.</s> <s xml:id="echoid-s12527" xml:space="preserve"> Si concurrit, cum quatuor ſunt perpendiculares:</s> <s xml:id="echoid-s12528" xml:space="preserve"> erunt concurſus quatuor, & qua-<lb/>tuor im agines.</s> <s xml:id="echoid-s12529" xml:space="preserve"/> </p> <div xml:id="echoid-div440" type="float" level="0" n="0"> <figure xlink:label="fig-0190-02" xlink:href="fig-0190-02a"> <variables xml:id="echoid-variables138" xml:space="preserve">s z o r x a h k g m u b d e t l f q p n</variables> </figure> </div> </div> <div xml:id="echoid-div442" type="section" level="0" n="0"> <head xml:id="echoid-head399" xml:space="preserve" style="it">96. Viſu & uiſibili datis, in ſpeculo cylindraceo cauo punctum reflexionis inuenire. 16 p 9.</head> <p> <s xml:id="echoid-s12530" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s12531" xml:space="preserve"> datis puncto uiſo, & puncto uiſus:</s> <s xml:id="echoid-s12532" xml:space="preserve"> erit inuenire punctum reflexionis.</s> <s xml:id="echoid-s12533" xml:space="preserve"> Verbi gratia;</s> <s xml:id="echoid-s12534" xml:space="preserve"> <lb/>ſit a punctum uiſum:</s> <s xml:id="echoid-s12535" xml:space="preserve"> b centrum uiſus.</s> <s xml:id="echoid-s12536" xml:space="preserve"> Fiat ſuperficies ſecans columnam æquidiſtanter baſi <lb/>[ut oſtenſum eſt 47 n] trànſiens per a:</s> <s xml:id="echoid-s12537" xml:space="preserve"> & [per 5 th.</s> <s xml:id="echoid-s12538" xml:space="preserve"> Sereni de ſectione cylindri] faciet circu-<lb/>lum.</s> <s xml:id="echoid-s12539" xml:space="preserve"> b aut eſt in ſuperficie huius circuli:</s> <s xml:id="echoid-s12540" xml:space="preserve"> aut nõ.</s> <s xml:id="echoid-s12541" xml:space="preserve"> Si fuerit:</s> <s xml:id="echoid-s12542" xml:space="preserve"> inueniemus punctũ reflexionis in illo cir-<lb/>culo, ſicut dictum eſt in ſphærico concauo [73 n.</s> <s xml:id="echoid-s12543" xml:space="preserve">] Si nõ fuerit:</s> <s xml:id="echoid-s12544" xml:space="preserve"> ducatur [per 11 p 11] à puncto b perpẽ-<lb/>dicularis ſuper ſuperficiem huius circuli:</s> <s xml:id="echoid-s12545" xml:space="preserve"> & replicetur ſuprà dicta probatio:</s> <s xml:id="echoid-s12546" xml:space="preserve"> & inuenietur pũctum <lb/>reflexionis.</s> <s xml:id="echoid-s12547" xml:space="preserve"> Duplici autem uiſu adhibito, una imago in ueritate, efficientur duæ, ſed contiguæ uel <lb/>admixtæ:</s> <s xml:id="echoid-s12548" xml:space="preserve"> unde uidebitur una.</s> <s xml:id="echoid-s12549" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div443" type="section" level="0" n="0"> <head xml:id="echoid-head400" xml:space="preserve" style="it">97. Cõmunis ſectio ſuperficierũ, reflexionis & ſpeculiconici caui eſt latus coni, aut ellipſis. 2 p 9.</head> <p> <s xml:id="echoid-s12550" xml:space="preserve">IN ſpeculis pyramidalibus concauis linea, communis ſuperficiei reflexionis & ſuperficiei ſpecu <lb/>li, aut erit linea lõgitudinis ſpeculi:</s> <s xml:id="echoid-s12551" xml:space="preserve"> aut erit ſectio pyramidalis.</s> <s xml:id="echoid-s12552" xml:space="preserve"> Si fuerit linea longitudinis:</s> <s xml:id="echoid-s12553" xml:space="preserve"> erũt <lb/>loca imaginum in ipſo ſpeculo.</s> <s xml:id="echoid-s12554" xml:space="preserve"> Si fuerit ſectio pyramidalis:</s> <s xml:id="echoid-s12555" xml:space="preserve"> erunt loca imaginum aliquando ci-<lb/>tra uiſum:</s> <s xml:id="echoid-s12556" xml:space="preserve"> aliquando in uiſu:</s> <s xml:id="echoid-s12557" xml:space="preserve"> aliquando inter uiſum & ſpeculum:</s> <s xml:id="echoid-s12558" xml:space="preserve"> & aliquando ultra ſpeculum, ſi-<lb/>cut oſtenſum eſt in ſpeculo columnari concauo.</s> <s xml:id="echoid-s12559" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div444" type="section" level="0" n="0"> <head xml:id="echoid-head401" xml:space="preserve" style="it">98. Siuiſus ſit in communi ſectione axis & rectæ lineæ perpendicularis plano, ſpeculum co-<lb/>nicum cauum tangẽti: reflectetur à tota peripheria circuli (cuius centrum eſt dict a communis <lb/>ſectio) per lineas perpendiculares: & imago uidebitur in centro uiſus. 17 p 9.</head> <pb o="186" file="0192" n="192" rhead="ALHAZEN"/> <p> <s xml:id="echoid-s12560" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s12561" xml:space="preserve"> ſi in perpendiculari ducta à centro uiſus ad ſuperficiem contingentem pyramide<gap/><gap/>, <lb/>ſumatur punctum corporeum inter uiſum & ſpeculum:</s> <s xml:id="echoid-s12562" xml:space="preserve"> non refle-<lb/> <anchor type="figure" xlink:label="fig-0192-01a" xlink:href="fig-0192-01"/> ctetur forma eius ad uiſum per perpẽdicularem:</s> <s xml:id="echoid-s12563" xml:space="preserve"> quoniam punctũ <lb/>illud occultabit terminũ perpendicularis illius, & ob hoc non reflectetur <lb/>ab eo.</s> <s xml:id="echoid-s12564" xml:space="preserve"> Si autem nullum fuerit punctum in perpendiculari illa:</s> <s xml:id="echoid-s12565" xml:space="preserve"> reflectetur <lb/>quidem ad uiſum per hanc perpendicularem punctum uiſus, quod ſecat <lb/>perpendicularis ex eo:</s> <s xml:id="echoid-s12566" xml:space="preserve"> & illud ſolum.</s> <s xml:id="echoid-s12567" xml:space="preserve"> Verùm uiſu exiſtẽte in hac perpen-<lb/>diculari & in axe:</s> <s xml:id="echoid-s12568" xml:space="preserve"> efficietur circulus, ad cuius quodlibet punctum linea <lb/>ducta à uiſu, erit perpendicularis ſuper ſuperficiem contingentem.</s> <s xml:id="echoid-s12569" xml:space="preserve"> Vnde <lb/>â quolibet puncto illius circuli fieri poterit reflexio ad uiſum, ſecundum <lb/>perpendiculares.</s> <s xml:id="echoid-s12570" xml:space="preserve"> Et fiet reflexio partis uiſus, quam ſecant perpendicula-<lb/>res duæ, maiorem angulum in eo continentes.</s> <s xml:id="echoid-s12571" xml:space="preserve"> Si uerò inter uiſum & ſpe-<lb/>culum fuerit axis:</s> <s xml:id="echoid-s12572" xml:space="preserve"> non fiet ad ipſum reflexio per perpendicularem, niſi <lb/>puncti eius, quod ſecant perpendiculares.</s> <s xml:id="echoid-s12573" xml:space="preserve"/> </p> <div xml:id="echoid-div444" type="float" level="0" n="0"> <figure xlink:label="fig-0192-01" xlink:href="fig-0192-01a"> <variables xml:id="echoid-variables139" xml:space="preserve">a b h</variables> </figure> </div> </div> <div xml:id="echoid-div446" type="section" level="0" n="0"> <head xml:id="echoid-head402" xml:space="preserve" style="it">99. Siuiſus & uiſibile fuerint in axe ſpeculi conici caui: poſſunt à <lb/>tota alicuius circuli peripheria inter ſe reflecti: & ιmago uidetur in <lb/>peripheria circuli, extra ſpeculi ſuperficiem deſcripti. 18 p 9.</head> <p> <s xml:id="echoid-s12574" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s12575" xml:space="preserve"> exiſtente uiſu & puncto uiſo in axe:</s> <s xml:id="echoid-s12576" xml:space="preserve"> poterit reflecti unum <lb/>ad aliud.</s> <s xml:id="echoid-s12577" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s12578" xml:space="preserve"> ſit h centrum uiſus:</s> <s xml:id="echoid-s12579" xml:space="preserve"> t punctum uiſum.</s> <s xml:id="echoid-s12580" xml:space="preserve"> Fiat <lb/>ſuperficies ſecans pyramidem, trãſiens ſuper axis longitudinem:</s> <s xml:id="echoid-s12581" xml:space="preserve"> <lb/>quę ſit a b g h:</s> <s xml:id="echoid-s12582" xml:space="preserve"> a h axis:</s> <s xml:id="echoid-s12583" xml:space="preserve"> a b, a g latera pyramidis:</s> <s xml:id="echoid-s12584" xml:space="preserve"> à puncto t du-<lb/> <anchor type="figure" xlink:label="fig-0192-02a" xlink:href="fig-0192-02"/> catur perpendicularis ſuper lineam a b [per 12 p 1] quæ ſit t q:</s> <s xml:id="echoid-s12585" xml:space="preserve"> <lb/>& producatur quouſq;</s> <s xml:id="echoid-s12586" xml:space="preserve"> q l ſit æqualis q t:</s> <s xml:id="echoid-s12587" xml:space="preserve"> & à puncto h duca-<lb/>tur linea ad punctum l:</s> <s xml:id="echoid-s12588" xml:space="preserve"> quæ ſecabit lineam longitudinis, quæ <lb/>eſt a b:</s> <s xml:id="echoid-s12589" xml:space="preserve"> ſecet in puncto b:</s> <s xml:id="echoid-s12590" xml:space="preserve"> & à puncto b ducatur æquidiſtans <lb/>lineæ t q [per 31 p 1] quæ neceſſariò perueniet ad axem:</s> <s xml:id="echoid-s12591" xml:space="preserve"> [ut <lb/>oſtenſum eſt 54 n] perueniat in pũcto d:</s> <s xml:id="echoid-s12592" xml:space="preserve"> & ducatur linea t b.</s> <s xml:id="echoid-s12593" xml:space="preserve"> <lb/>Palàm, cum t q ſit perpendicularis ſuper a b, & t q æqualis q l:</s> <s xml:id="echoid-s12594" xml:space="preserve"> <lb/>erit [per 4 p 1] b t q triangulum æquale triangulo b q l:</s> <s xml:id="echoid-s12595" xml:space="preserve"> & erit <lb/>angulus q l b æqualis angulo q t b:</s> <s xml:id="echoid-s12596" xml:space="preserve"> ſed [per 29 p 1] angulus q t <lb/>b æqualis eſt angulo t b d:</s> <s xml:id="echoid-s12597" xml:space="preserve"> & angulus d b h æqualis eſt angu-<lb/>lo q l b:</s> <s xml:id="echoid-s12598" xml:space="preserve"> igitur angulus t b d æqualis eſt angulo d b h.</s> <s xml:id="echoid-s12599" xml:space="preserve"> Et ita <lb/>[per 12 n 4] t reflectitur ad h à puncto b:</s> <s xml:id="echoid-s12600" xml:space="preserve"> & locus imaginis eſt <lb/>l [per 7 n.</s> <s xml:id="echoid-s12601" xml:space="preserve">] Igitur moto triangulo t l h:</s> <s xml:id="echoid-s12602" xml:space="preserve"> deſcribet punctum b <lb/>circulum in pyramide:</s> <s xml:id="echoid-s12603" xml:space="preserve"> & à quolibet puncto illius circuli re-<lb/>flectetur t ad h:</s> <s xml:id="echoid-s12604" xml:space="preserve"> l uerò extra lpeculum deſcribet circulum, qui <lb/>totus erit locus imaginis puncti t.</s> <s xml:id="echoid-s12605" xml:space="preserve"/> </p> <div xml:id="echoid-div446" type="float" level="0" n="0"> <figure xlink:label="fig-0192-02" xlink:href="fig-0192-02a"> <variables xml:id="echoid-variables140" xml:space="preserve">a l c q g d b h</variables> </figure> </div> </div> <div xml:id="echoid-div448" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables141" xml:space="preserve">a g e u <gap/> m q d o n z h p l</variables> </figure> <head xml:id="echoid-head403" xml:space="preserve" style="it">100. Si cõmunis ſectio ſuperficierum, reflexionis & ſpe-<lb/> culi conici caui fuerit ellipſis: uiſus & uiſibile extra axẽ in ba- ſi, aut plano ipſi parallelo, reflectentur inter ſe: aliâs ab uno: aliâs à duobus: aliâs à tribus: aliâs à quatuor ſpeculipunctis: tot́ erunt imagines, quot reflexionum puncta. 19 p 9.</head> <p> <s xml:id="echoid-s12606" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s12607" xml:space="preserve"> ſumptis duobus punctis & extra perpendicula-<lb/>rem uiſus, & extra axem in hoc ſpeculo:</s> <s xml:id="echoid-s12608" xml:space="preserve"> ſcilicetz, e.</s> <s xml:id="echoid-s12609" xml:space="preserve"> Fiat <lb/>ſuperficies æquidiſtans baſi ſuperz:</s> <s xml:id="echoid-s12610" xml:space="preserve"> [ut oſtẽſum eſt 52 n] <lb/>faciet circulum in ſpeculo [per 4 th.</s> <s xml:id="echoid-s12611" xml:space="preserve"> 1 coni.</s> <s xml:id="echoid-s12612" xml:space="preserve"> Apoll.</s> <s xml:id="echoid-s12613" xml:space="preserve">] e aut erit in <lb/>hoc circulo, aut in alia ſuperficie ipſi æquidiſtante.</s> <s xml:id="echoid-s12614" xml:space="preserve"> Sit in ſuperfi-<lb/>cieillius circuli:</s> <s xml:id="echoid-s12615" xml:space="preserve"> & ducatur linea e z.</s> <s xml:id="echoid-s12616" xml:space="preserve"> Palàm [per demonſtrata in <lb/>ſpeculis ſphæricis cauis 86 n] quòd z reflectetur ad e à circulo <lb/>illo ex una parte, aut ab uno pũcto:</s> <s xml:id="echoid-s12617" xml:space="preserve"> aut à duobus:</s> <s xml:id="echoid-s12618" xml:space="preserve"> aut à tribus:</s> <s xml:id="echoid-s12619" xml:space="preserve"> ex <lb/>alia uerò ab uno.</s> <s xml:id="echoid-s12620" xml:space="preserve"> Sumatur igitur punctum circuli, à quo reflecti-<lb/>tur ad ipſum:</s> <s xml:id="echoid-s12621" xml:space="preserve"> & ſit h:</s> <s xml:id="echoid-s12622" xml:space="preserve"> centrum circuli t:</s> <s xml:id="echoid-s12623" xml:space="preserve"> & ducantur lineæ z h, e h:</s> <s xml:id="echoid-s12624" xml:space="preserve"> <lb/>& diameter t h diuidet quidem angulum illum per æqualia:</s> <s xml:id="echoid-s12625" xml:space="preserve"> [per <lb/>13 n 4] & ſecabit lineam e z:</s> <s xml:id="echoid-s12626" xml:space="preserve"> [quia ſecat angulum ipſi e z ſubten-<lb/>ſum] ſecet in puncto q:</s> <s xml:id="echoid-s12627" xml:space="preserve"> & ſit a uertex pyramidis:</s> <s xml:id="echoid-s12628" xml:space="preserve"> a h linea longi-<lb/>tudinis.</s> <s xml:id="echoid-s12629" xml:space="preserve"> À<unsure/> puncto q ducatur linea perpẽdicularis ſuper lineam <lb/>a h:</s> <s xml:id="echoid-s12630" xml:space="preserve"> [per 12 p 1] quæ ſit q m:</s> <s xml:id="echoid-s12631" xml:space="preserve"> quæ quidem perueniet ad axem:</s> <s xml:id="echoid-s12632" xml:space="preserve"> [ut <lb/>oſtẽſum eſt 54 n] qui eſt a d:</s> <s xml:id="echoid-s12633" xml:space="preserve"> & cadat in ipſum in puncto d:</s> <s xml:id="echoid-s12634" xml:space="preserve"> & du-<lb/>cantur lineæ z m, e m:</s> <s xml:id="echoid-s12635" xml:space="preserve"> à puncto z ducatur in ſuperficie circuli li-<lb/>nea æquidiſtans lineæ q h:</s> <s xml:id="echoid-s12636" xml:space="preserve"> [per 31 p 1] quæ ſit z l:</s> <s xml:id="echoid-s12637" xml:space="preserve"> concurret qui-<lb/>dem [per lemma Procli ad 29 p 1] e h cũilla:</s> <s xml:id="echoid-s12638" xml:space="preserve"> ſit cõcurſus in pun <lb/>ctol:</s> <s xml:id="echoid-s12639" xml:space="preserve"> & à puncto h ducatur perpendicularis ſuper l z:</s> <s xml:id="echoid-s12640" xml:space="preserve"> quæ ſit h p.</s> <s xml:id="echoid-s12641" xml:space="preserve"> <lb/>Deinde in ſuperficie e m z ducatur linea æquidiſtans lineæ q m:</s> <s xml:id="echoid-s12642" xml:space="preserve"> <lb/>quæ ſit z o:</s> <s xml:id="echoid-s12643" xml:space="preserve"> & cõcurrat e m cum ea in puncto o:</s> <s xml:id="echoid-s12644" xml:space="preserve"> [cõcurret aũt per lemma Procli ad 29 p 1] & ducatu<gap/> <lb/> <pb o="187" file="0193" n="193" rhead="OPTICAE LIBER V."/> linea l o:</s> <s xml:id="echoid-s12645" xml:space="preserve"> & à puncto p ducatur æquidiſtans l o:</s> <s xml:id="echoid-s12646" xml:space="preserve"> quæ ſit p n:</s> <s xml:id="echoid-s12647" xml:space="preserve"> & ducatur linea m n.</s> <s xml:id="echoid-s12648" xml:space="preserve"> Palàm [per theſin & <lb/>12 n 4] quòd angulus e h q æqualis eſt angulo q h z:</s> <s xml:id="echoid-s12649" xml:space="preserve"> & [per 29 p 1] angulo h l z:</s> <s xml:id="echoid-s12650" xml:space="preserve"> & angulus q h z æqua-<lb/>lis eſt angulo coalterno h z l [ideoq́;</s> <s xml:id="echoid-s12651" xml:space="preserve"> angulus h l z æquatur angulo h z l.</s> <s xml:id="echoid-s12652" xml:space="preserve">] Eritigitur [per 6 p 1] h l æ-<lb/>qualis h z:</s> <s xml:id="echoid-s12653" xml:space="preserve"> & h p perpendicularis eſt ſuper l z:</s> <s xml:id="echoid-s12654" xml:space="preserve"> [per fabricationé] erit triangulũ l p h æquale triangulo <lb/>p h z, & erit l p æqualis p z:</s> <s xml:id="echoid-s12655" xml:space="preserve"> [per 26 p 1:</s> <s xml:id="echoid-s12656" xml:space="preserve"> quia anguli ad l & z æquantur, & ad precti ſunt per fabricatio <lb/>nem, & h z æquatur h l] & p n æquidiſtans eſt o l:</s> <s xml:id="echoid-s12657" xml:space="preserve"> erit [per 2 p 6] proportio l p ad p z, ſicut o n ad n z.</s> <s xml:id="echoid-s12658" xml:space="preserve"> <lb/>Quare o n æqualis n z.</s> <s xml:id="echoid-s12659" xml:space="preserve"> Item cum o z ſit æquidiſtans q m [per fabricationẽ] & h q æquidiſtans l z:</s> <s xml:id="echoid-s12660" xml:space="preserve"> erit <lb/>[per 15 p 11] ſuperficies z o l æquidiſtans ſuperficiei q m h:</s> <s xml:id="echoid-s12661" xml:space="preserve"> & ſuperficies e o l ſecatillas duas, ſuper li-<lb/>neas cõmunes, [per 3 p 11] quę quidẽ[per 16 p 11] erunt æquidiſtãtes, ſcilicet m h, l o:</s> <s xml:id="echoid-s12662" xml:space="preserve"> quare [per 30 p 1] <lb/>h m, p n ſunt æquidiſtantes.</s> <s xml:id="echoid-s12663" xml:space="preserve"> Et quoniã h p cadit inter l z, h q æquidiſtantes:</s> <s xml:id="echoid-s12664" xml:space="preserve"> & eſt perpendicularis ſu <lb/>per l z:</s> <s xml:id="echoid-s12665" xml:space="preserve"> [angulus igitur p h trectus eſt:</s> <s xml:id="echoid-s12666" xml:space="preserve"> quia per 29 p 1 æquatur alterno h p l] quare [per conſectariũ 16 <lb/>p 3] p h continget circulũ:</s> <s xml:id="echoid-s12667" xml:space="preserve"> quare ſuperficies a h p eſt ſuperficies contingens pyramidẽ.</s> <s xml:id="echoid-s12668" xml:space="preserve"> In hac ſuperſi <lb/>cie eſt p n & m n:</s> <s xml:id="echoid-s12669" xml:space="preserve"> [Nam cũ h m ſit in plano a h p conũ tangẽte, & illi parallela ſit n p, ut patuit:</s> <s xml:id="echoid-s12670" xml:space="preserve"> erit igi <lb/>tur n p in eodẽ plano per 35 d 1:</s> <s xml:id="echoid-s12671" xml:space="preserve"> m n uerò, quia utranq;</s> <s xml:id="echoid-s12672" xml:space="preserve"> h m & n p connectit, in eodẽ eſt cũ ipſis plano <lb/>per 7 p 11] & ſuper hanc ſuperficiẽ eſt perpendicularis linea d m [per demõſtrata 54 n.</s> <s xml:id="echoid-s12673" xml:space="preserve">] Igitur [per 3 d <lb/>11] perpendicularis eſt ſuper lineam m n:</s> <s xml:id="echoid-s12674" xml:space="preserve"> quare [per 29 p 1] m n eſt perpendicularis ſuper o z, & o n æ-<lb/>qualis n z:</s> <s xml:id="echoid-s12675" xml:space="preserve"> [ex cõcluſo] erit [per 4 p 1] m o æqualis m z:</s> <s xml:id="echoid-s12676" xml:space="preserve"> & [per 7 p 5] e m ad m o, ſicut e m ad m z:</s> <s xml:id="echoid-s12677" xml:space="preserve"> ſed [ք <lb/>2 p 6] e m ad m o, ſicut e h ad h l:</s> <s xml:id="echoid-s12678" xml:space="preserve"> [nã h m ex cõcluſo parallela eſt ipſi o l] & [per 7 p 5] e h ad h l, ſicut e <lb/>h ad h z:</s> <s xml:id="echoid-s12679" xml:space="preserve"> [æquales enim demõſtratę ſunt h l, h z] & [per 3 p 6] e h ad h z, ſicute q ad q z [angulus enim <lb/>e h z bifariã ſectus eſt à linea h q.</s> <s xml:id="echoid-s12680" xml:space="preserve">] Igitur [per 11 p 5] e m ad m z, ſicut e q ad q z.</s> <s xml:id="echoid-s12681" xml:space="preserve"> Quare [per 3 p 6] angu-<lb/>lus e m q æqualis angulo q m z.</s> <s xml:id="echoid-s12682" xml:space="preserve"> Quare [per 12 n 4] z reflectitur ad e à puncto m.</s> <s xml:id="echoid-s12683" xml:space="preserve"> Siigitur z reflectitur <lb/>ad e à puncto circuli h:</s> <s xml:id="echoid-s12684" xml:space="preserve">reflectetur ad ipſum à puncto pyramidis m:</s> <s xml:id="echoid-s12685" xml:space="preserve"> & ſi à duobus circuli, à duobus <lb/>pyramidis:</s> <s xml:id="echoid-s12686" xml:space="preserve"> ſi à tribus, à tribus:</s> <s xml:id="echoid-s12687" xml:space="preserve"> ſi à pluribus, à pluribus.</s> <s xml:id="echoid-s12688" xml:space="preserve"> Eodem modo ex alia parte circuli fiet proba-<lb/>tio:</s> <s xml:id="echoid-s12689" xml:space="preserve"> quòd ab uno puncto pyramidis, ſicut ab uno circuli, reflexio fiat.</s> <s xml:id="echoid-s12690" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div449" type="section" level="0" n="0"> <head xml:id="echoid-head404" xml:space="preserve" style="it">101. Sicõmunis ſectio ſuperficierum, reflexionis & ſpeculi conici cauifuerit ellipſis: uiſus & ui <lb/>ſibile intra ſpeculum, extra tum axem tum baſim uel planum ipſi parallelum: reflectentur inter <lb/>ſe: aliâs ab uno: aliâs à duobus: aliâs à tribus: aliâs à quatuor ſpeculi punctιs: tot́ erunt imagi-<lb/>nes, quot reflexionum puncta. 20 p 9.</head> <p> <s xml:id="echoid-s12691" xml:space="preserve">SIuerò e nõ fuerit in circulo ęquidiſtãte baſi, trãſeũte ſuper z:</s> <s xml:id="echoid-s12692" xml:space="preserve"> erit quidẽ ſuprà uel infrà.</s> <s xml:id="echoid-s12693" xml:space="preserve"> Sit ſuprà:</s> <s xml:id="echoid-s12694" xml:space="preserve"> <lb/>quia utrobiq;</s> <s xml:id="echoid-s12695" xml:space="preserve"> eadẽ eſt probatio.</s> <s xml:id="echoid-s12696" xml:space="preserve"> Ducatur linea à uertice a per punctũ e, donec ſecet ſuperficiẽ <lb/>illius circuli:</s> <s xml:id="echoid-s12697" xml:space="preserve"> & ſit punctũ ſectionis h:</s> <s xml:id="echoid-s12698" xml:space="preserve"> q centrũ circuli.</s> <s xml:id="echoid-s12699" xml:space="preserve"> Palàm [per demonſtrata in ſpeculis ſphæ <lb/>ricis cauis 66 n] quòd h poteſt reflecti ad z ab aliquo pũcto circuli:</s> <s xml:id="echoid-s12700" xml:space="preserve"> ſit illud t:</s> <s xml:id="echoid-s12701" xml:space="preserve"> & ducatur diameter q t:</s> <s xml:id="echoid-s12702" xml:space="preserve"> <lb/>& linea h z ſecabit hãc diametrũ in puncto:</s> <s xml:id="echoid-s12703" xml:space="preserve"> quod ſit n:</s> <s xml:id="echoid-s12704" xml:space="preserve"> [Nam quia per theſin t eſt reflexiõis punctũ:</s> <s xml:id="echoid-s12705" xml:space="preserve"> <lb/>ergo per 12 n 4 ſemidiameter q t bifariã ſecat angulũ h t z:</s> <s xml:id="echoid-s12706" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s12707" xml:space="preserve"> & baſim h z angulo ſubtẽſam] & du <lb/>catur e z:</s> <s xml:id="echoid-s12708" xml:space="preserve"> & linea longitudinis a t.</s> <s xml:id="echoid-s12709" xml:space="preserve"> Palàm, cũ punctũ z ſit ex una parte diametri q t, & ex alia e:</s> <s xml:id="echoid-s12710" xml:space="preserve"> linea e <lb/>z ſecabit ſuperficiẽ a q t:</s> <s xml:id="echoid-s12711" xml:space="preserve"> ſecet in puncto o:</s> <s xml:id="echoid-s12712" xml:space="preserve"> & à puncto o ducatur perpendicularis ſuper lineã a t:</s> <s xml:id="echoid-s12713" xml:space="preserve"> [per <lb/>12 p 1] quę ſit o p:</s> <s xml:id="echoid-s12714" xml:space="preserve"> quę neceſſariò cadet ſuper axem:</s> <s xml:id="echoid-s12715" xml:space="preserve"> [ut oſtenſum eſt 54 n] cadat in puncto d:</s> <s xml:id="echoid-s12716" xml:space="preserve"> & ducan <lb/>turlineę e p, z p.</s> <s xml:id="echoid-s12717" xml:space="preserve"> Dico, quòd z reflectetur ad e à puncto p.</s> <s xml:id="echoid-s12718" xml:space="preserve"> Ducatur à pũcto z linea æquidiſtãs q t[per <lb/>31 p 1] quæ ſit z f:</s> <s xml:id="echoid-s12719" xml:space="preserve"> & producatur linea h t, donec cõcurrat cũ illa:</s> <s xml:id="echoid-s12720" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0193-01a" xlink:href="fig-0193-01"/> [cõcurret aũt per lem ma Procli ad 29 p 1] ſit cõcurſus in puncto <lb/>f.</s> <s xml:id="echoid-s12721" xml:space="preserve"> Similiter à puncto z ducatur æquidiſtãs lineę o p:</s> <s xml:id="echoid-s12722" xml:space="preserve"> quę ſit z k:</s> <s xml:id="echoid-s12723" xml:space="preserve"> & <lb/>producatur linea e p, donec cõcurrat cũ illa:</s> <s xml:id="echoid-s12724" xml:space="preserve"> ſit cõcurſus in pun <lb/>cto k:</s> <s xml:id="echoid-s12725" xml:space="preserve"> & ducantur lineæ k f, k h.</s> <s xml:id="echoid-s12726" xml:space="preserve"> Palàm [per 15 p 11] cũ linea z f ſit <lb/>æquidiſtás q t, & z k ęquidiſtans o p:</s> <s xml:id="echoid-s12727" xml:space="preserve"> [& z f, z k cõcurrant in pun <lb/>cto z:</s> <s xml:id="echoid-s12728" xml:space="preserve"> & p o, t q cõtinuatæ concurrãt per 11 ax:</s> <s xml:id="echoid-s12729" xml:space="preserve"> quia angulus o p t <lb/>rectus eſt è fabricatiõe, & q t p acutus ք 18 d 11] quòd erit ſuperfi <lb/>cies z k f ęquidiſtãs o p t:</s> <s xml:id="echoid-s12730" xml:space="preserve"> quę eſt ſuperficies a q t:</s> <s xml:id="echoid-s12731" xml:space="preserve"> [quia enim p o <lb/>cadit in axem, ut patuit:</s> <s xml:id="echoid-s12732" xml:space="preserve"> eſt igitur in a q t plano per 1 p 11:</s> <s xml:id="echoid-s12733" xml:space="preserve"> in quo <lb/>etiã eſt linea q t:</s> <s xml:id="echoid-s12734" xml:space="preserve"> planũ igitur o p q t eſt pars plani a q t] & ſuperfi <lb/>cies h k f ſecat has duas ſuperficies, ſuper lineas p t, k f.</s> <s xml:id="echoid-s12735" xml:space="preserve"> Igitur[ք <lb/>16 p 11] p t, k f ſunt æquidiſtantes.</s> <s xml:id="echoid-s12736" xml:space="preserve"> Ducatur à puncto t perpendi-<lb/>cularis ſuper lineã z f [per 12 p 1] quę ſit t s.</s> <s xml:id="echoid-s12737" xml:space="preserve"> Palàm, cũ cadatinter <lb/>duas æquidiſtantes [q t, z f:</s> <s xml:id="echoid-s12738" xml:space="preserve">] erit angulus q t s rectus [per 29 p 1] <lb/>& ita [per cõſectariũ 16 p 3] cõtinget circulũ:</s> <s xml:id="echoid-s12739" xml:space="preserve"> [cuius cẽtrũ eſt q.</s> <s xml:id="echoid-s12740" xml:space="preserve">] <lb/>Igitur ſuperficies a t s contingit pyramidẽ ſuper lineã a t:</s> <s xml:id="echoid-s12741" xml:space="preserve"> [per 35 <lb/>n 4] & linea o p eſt perpendicularis ſuper hãc ſuperficiẽ [ut de-<lb/>monſtratũ eſt 54 n.</s> <s xml:id="echoid-s12742" xml:space="preserve">] Superficies igitur a t q erit orthogonalis ſu <lb/>per ſuperficiem a t s:</s> <s xml:id="echoid-s12743" xml:space="preserve"> [per 18 p 11] & ſuperficies a t s ſecat duas ſu <lb/>perficies a t q, z k f<gap/>quæ ſunt æquidiſtantes:</s> <s xml:id="echoid-s12744" xml:space="preserve"> igitur[per 16 p 11] li-<lb/>neę cõmunes ſectionũ ſunt æquidiſtantes.</s> <s xml:id="echoid-s12745" xml:space="preserve"> Vnaharũ linearũ eſt <lb/>p t:</s> <s xml:id="echoid-s12746" xml:space="preserve"> alia ſit s i.</s> <s xml:id="echoid-s12747" xml:space="preserve"> Sed iam patuit, quod p t æquidiſtans eſt k f:</s> <s xml:id="echoid-s12748" xml:space="preserve"> igitur <lb/>[per 30 p 1] s i eſt æquidiſtans k f.</s> <s xml:id="echoid-s12749" xml:space="preserve"> Sed planũ eſt, quòd angulus n <lb/>t z ęqualis eſt angulo t z f, & angulus h t n ęqualis angulo t f z:</s> <s xml:id="echoid-s12750" xml:space="preserve"> [ք <lb/>29 p 1:</s> <s xml:id="echoid-s12751" xml:space="preserve"> quia q t & z fſunt parallelę per fabricationẽ] & t s perpen <lb/>cicularis [ſuper z f perfabricationem] erit f s æqualis s z.</s> <s xml:id="echoid-s12752" xml:space="preserve"> [Quia <lb/> <pb o="188" file="0194" n="194" rhead="ALHAZEN"/> enim anguli t z f, t f z æquantur angulis z t n & h t n per 12 n 4 æqualibus, cum t ex theſi ſit reflexio <lb/>nis punctum:</s> <s xml:id="echoid-s12753" xml:space="preserve"> ipſi igitur inter ſe æquantur:</s> <s xml:id="echoid-s12754" xml:space="preserve"> & anguli ad s recti ſunt:</s> <s xml:id="echoid-s12755" xml:space="preserve"> & t s commune latus eſt.</s> <s xml:id="echoid-s12756" xml:space="preserve"> Quare <lb/>per 26 p 1 f s æquatur s z.</s> <s xml:id="echoid-s12757" xml:space="preserve">] Sed [per 2 p 6] proportio f s ad s z, ſicut k i ad i z:</s> <s xml:id="echoid-s12758" xml:space="preserve"> erit ergo k i ęqualisi z.</s> <s xml:id="echoid-s12759" xml:space="preserve"> Du <lb/>cta autem linea p i:</s> <s xml:id="echoid-s12760" xml:space="preserve"> cum ſuperficies a t f ſit orthogonalis ſuper ſuperficiem z k f:</s> <s xml:id="echoid-s12761" xml:space="preserve"> erit [per 4 d 11] p i or-<lb/>thogonalis ſuper z k:</s> <s xml:id="echoid-s12762" xml:space="preserve"> & erit [per 4 p 1] angulus p k z æqualis angulo k z p:</s> <s xml:id="echoid-s12763" xml:space="preserve"> ſed [ք 29 p 1] angulus e p o <lb/>æqualis angulo p k z, & angulus o p z æqualis angulo p z k.</s> <s xml:id="echoid-s12764" xml:space="preserve"> Quare angulus e p o æqualis eſt angulo <lb/>o p z.</s> <s xml:id="echoid-s12765" xml:space="preserve"> Etita z reflectitur ad e à puncto p [per 12 n 4.</s> <s xml:id="echoid-s12766" xml:space="preserve">] Quod eſt propoſitum.</s> <s xml:id="echoid-s12767" xml:space="preserve"> Si autem ſumatur aliud <lb/>punctũ ſn circulo, à quo z reflectatur ad h:</s> <s xml:id="echoid-s12768" xml:space="preserve"> probabitur, quòd ab alio puncto pyramidis, quàm p, refle <lb/>ctetur z ad e.</s> <s xml:id="echoid-s12769" xml:space="preserve"> Et ſi reflectatur z ad h à tribus punctis circuli:</s> <s xml:id="echoid-s12770" xml:space="preserve"> reflectetur z a d e à tribus pũctis pyrami <lb/>dis:</s> <s xml:id="echoid-s12771" xml:space="preserve"> ſi à quatuor, à quatuor.</s> <s xml:id="echoid-s12772" xml:space="preserve"> Si uerò dicatur, quòd à pluribus pũctis pyramidis, ꝗ̃ quatuor, poſsit pun <lb/>ctũ z reflecti ad e:</s> <s xml:id="echoid-s12773" xml:space="preserve"> per cõuerſionẽ prædictæ probatiõis poterit oſtendi, quòd punctũ z reflectitur ad <lb/>h à pluribus punctis circuli quàm quatuor [contra 86 n.</s> <s xml:id="echoid-s12774" xml:space="preserve">] Et ubi accidet punctũ z reflecti ad h ab ali <lb/>quot punctis circuli, uel ab uno tantùm:</s> <s xml:id="echoid-s12775" xml:space="preserve"> accidet punctũ z reflecti ad e à totidem punctis pyramidis, <lb/>aut ab uno tantùm, aut è contrario.</s> <s xml:id="echoid-s12776" xml:space="preserve"> Quòd ſi dicatur contrarium:</s> <s xml:id="echoid-s12777" xml:space="preserve"> poterit improbari prædicto modo.</s> <s xml:id="echoid-s12778" xml:space="preserve"> <lb/>Palàm ergo, quòd punctorũ quædam unicã habent imaginẽ:</s> <s xml:id="echoid-s12779" xml:space="preserve"> quędam duas:</s> <s xml:id="echoid-s12780" xml:space="preserve"> quædã tres:</s> <s xml:id="echoid-s12781" xml:space="preserve"> quædã qua-<lb/>tuor:</s> <s xml:id="echoid-s12782" xml:space="preserve"> ſed nõ poſsibile plures.</s> <s xml:id="echoid-s12783" xml:space="preserve"> Verùm duplici uiſu adhibito, ſpeculo:</s> <s xml:id="echoid-s12784" xml:space="preserve"> eiuſdem imaginis diuerſa erunt <lb/>loca:</s> <s xml:id="echoid-s12785" xml:space="preserve"> quæ diuerſitas propter ſuam imperceptibilitatem non inducit errorem.</s> <s xml:id="echoid-s12786" xml:space="preserve"/> </p> <div xml:id="echoid-div449" type="float" level="0" n="0"> <figure xlink:label="fig-0193-01" xlink:href="fig-0193-01a"> <variables xml:id="echoid-variables142" xml:space="preserve">a e u g d o p h q n k z i s t f</variables> </figure> </div> </div> <div xml:id="echoid-div451" type="section" level="0" n="0"> <head xml:id="echoid-head405" xml:space="preserve" style="it">102. Viſu & uiſibili datis, in ſpeculo conico cauo punctum reflexionis inuenire. 21 p 9.</head> <p> <s xml:id="echoid-s12787" xml:space="preserve">PVnctum autem reflexionis, à quo z reflectitur ad e, facile eſt inuenire:</s> <s xml:id="echoid-s12788" xml:space="preserve"> inuento puncto circuli, <lb/>à quo punctum z reflectitur ad h.</s> <s xml:id="echoid-s12789" xml:space="preserve"> Et erit inuentio modo prædicto.</s> <s xml:id="echoid-s12790" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div452" type="section" level="0" n="0"> <head xml:id="echoid-head406" xml:space="preserve">ALHAZEN FILII</head> <head xml:id="echoid-head407" xml:space="preserve">ALHAYZEN OPTICAE</head> <head xml:id="echoid-head408" xml:space="preserve">LIBER SEXTVS.</head> <p style="it"> <s xml:id="echoid-s12791" xml:space="preserve">Llber ſextus in nouẽ partes diuiditur.</s> <s xml:id="echoid-s12792" xml:space="preserve"> Pars prima eſt titulus libri.</s> <s xml:id="echoid-s12793" xml:space="preserve"> Secunda, quòd er-<lb/>ror accidat uiſui propter reflexionem.</s> <s xml:id="echoid-s12794" xml:space="preserve"> Tertia de errore eueniente in ſpeculis planis.</s> <s xml:id="echoid-s12795" xml:space="preserve"> <lb/>Quarta de errore, qui oritur in ſpeculis ſphæricis exteriorib.</s> <s xml:id="echoid-s12796" xml:space="preserve"> Quinta de errore in ſpeculis <lb/>columnaribus exterioribus.</s> <s xml:id="echoid-s12797" xml:space="preserve"> Sexta de errore in pyramidalibus exterioribus.</s> <s xml:id="echoid-s12798" xml:space="preserve"> Septima de <lb/>errore in ſphæricis concauis.</s> <s xml:id="echoid-s12799" xml:space="preserve"> Octaua de errore in columnaribus concauis.</s> <s xml:id="echoid-s12800" xml:space="preserve"> Nona de errore <lb/>in pyramidalibus concauis.</s> <s xml:id="echoid-s12801" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div453" type="section" level="0" n="0"> <head xml:id="echoid-head409" xml:space="preserve">PROOEMIVM LIBRI. CAP. I.</head> <p> <s xml:id="echoid-s12802" xml:space="preserve">PAtuit ex ſuperioribus libris modus acquiſitionis formarum in ſpeculis per uiſum, ſitus <lb/>linearum reflexionis & acceſſus, ſitus imaginum:</s> <s xml:id="echoid-s12803" xml:space="preserve"> & loca ipſarũ.</s> <s xml:id="echoid-s12804" xml:space="preserve"> Verùm per reflexionẽ <lb/>non ſemper comprehenditur formæ ueritas.</s> <s xml:id="echoid-s12805" xml:space="preserve"> In ſpeculis enim concauis apparet imago <lb/>faciei diſtorta, & occultatur uiſui diſpoſitio eius uera.</s> <s xml:id="echoid-s12806" xml:space="preserve"> Vnde planum eſt, errorẽ incide.</s> <s xml:id="echoid-s12807" xml:space="preserve"> <lb/>re in comprehenſione formarum propter reflexionem.</s> <s xml:id="echoid-s12808" xml:space="preserve"> Huius erroris modum, & modi <lb/>cauſſam, propoſitum eſt in libro præſente explanare, & ſecundum diuerſitates ſpeculorum diſcurre <lb/>re uarietates errorum.</s> <s xml:id="echoid-s12809" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div454" type="section" level="0" n="0"> <head xml:id="echoid-head410" xml:space="preserve">QVO'D ERROR ACCIDAT VISVI PROPTER RE-<lb/>flexionem. Cap. II.</head> <head xml:id="echoid-head411" xml:space="preserve" style="it">1. Viſus reflexus ſimiliter allucinatur, ut directus: ſed uebementius & frequentius. 7 p 5.</head> <p> <s xml:id="echoid-s12810" xml:space="preserve">COmprehenſionem formarũ in uiſu directo liber ſecundus docuit:</s> <s xml:id="echoid-s12811" xml:space="preserve"> & ſingula, quę propter e-<lb/>greſſum à tẽperantia in uiſu illo errorẽ inducũt, liber tertius diligẽter expoſuit.</s> <s xml:id="echoid-s12812" xml:space="preserve"> Fit aũt com-<lb/>prehenſio formarũ per reflexionẽ, ſicut & directè:</s> <s xml:id="echoid-s12813" xml:space="preserve"> & quorũ fit acquiſitio in directione, fit eriã <lb/>in reflexione, utpote lucis, coloris, figurę, magnitudinis, diſtantiæ, & ſimiliũ.</s> <s xml:id="echoid-s12814" xml:space="preserve"> Et quemadmodũ in di <lb/>rectione rerum præfixarũ & cognitarũ ad alias fit collatio, & inde oritur cõiecturatio, & ſumitur iu-<lb/>diciũ in anima:</s> <s xml:id="echoid-s12815" xml:space="preserve"> ſimiliter accidit in reflexione.</s> <s xml:id="echoid-s12816" xml:space="preserve"> Vnde quæcunq;</s> <s xml:id="echoid-s12817" xml:space="preserve"> temperamentũ egreſſa, in uiſu dire-<lb/>cto errorẽ efficiunt, in reflexione ſimiliter inducunt.</s> <s xml:id="echoid-s12818" xml:space="preserve"> Et ſecundũ ſingula maior accidit error in refle <lb/>xione, propter lucẽ debilem, quã debilitat ipſa reflexio.</s> <s xml:id="echoid-s12819" xml:space="preserve"> Vt aũt generaliter loquamur, nõ poteſt in re <lb/>flexione coprehendi ueritas formę, ſicut in directione, propter triplex impedιmentũ reflexioni ſpe-<lb/>ciale.</s> <s xml:id="echoid-s12820" xml:space="preserve"> Primũ eſt, quòd in reflexione apparet rei forma præ oculis uiſui oppoſita, cum non ſit reuera.</s> <s xml:id="echoid-s12821" xml:space="preserve"> <lb/>Secundũ, quòd lux & color corporis uiſi miſcentur cum colore ſpeculi, quã mixturã uiſus percipit, <lb/>nõ uerũ rei uiſę colorẽ uel lucem.</s> <s xml:id="echoid-s12822" xml:space="preserve"> Tertiũ, quòd ipſa reflexio, ut in ſuperioribus [4 n 4] eſt aſsignatũ, <lb/>lucẽ & colorẽ debilitat.</s> <s xml:id="echoid-s12823" xml:space="preserve"> Quare in reflexione latebit uiſum ueritas lucis & coloris plus, ꝗ̃ in directio-<lb/>ne.</s> <s xml:id="echoid-s12824" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s12825" xml:space="preserve"> ſuperiorà docuerunt, quòd quantitas temperamenti eorũ, quę in uiſu directo errorem <lb/>inducunt, fortitudinẽ lucis & coloris reſpicit:</s> <s xml:id="echoid-s12826" xml:space="preserve"> fortiore enim luce uel colore erit maior, debiliore mi-<lb/>hor.</s> <s xml:id="echoid-s12827" xml:space="preserve"> Cum autem per reflexionem debilitentur lux & color:</s> <s xml:id="echoid-s12828" xml:space="preserve"> erit latitudo temperamenti ſingulorum <lb/>errorẽ inducentium minor in reflexione, quàm in directione:</s> <s xml:id="echoid-s12829" xml:space="preserve"> & temperantiæ diminuta latitudo plu <lb/><gap/>alitatem erroris inducit.</s> <s xml:id="echoid-s12830" xml:space="preserve"> Præterea quædam minutiæ corporum comprehendi poterunt per dire-<lb/> <pb o="189" file="0195" n="195" rhead="OPTICAE LIBER VI."/> ctionem, quę nullatenus ſunt comprehenſibiles per reflexionem.</s> <s xml:id="echoid-s12831" xml:space="preserve"> Palàm ergo, quòd directionem ſu <lb/>perat reflexio in maioritate errorum & numero.</s> <s xml:id="echoid-s12832" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div455" type="section" level="0" n="0"> <head xml:id="echoid-head412" xml:space="preserve">DE ERRORE, QVI ACCIDIT IN SPECVLIS</head> <head xml:id="echoid-head413" xml:space="preserve">planis. Cap. III.</head> <head xml:id="echoid-head414" xml:space="preserve" style="it">2. In ſpeculo plano imago æquatur uiſibili. 52 p 5.</head> <p> <s xml:id="echoid-s12833" xml:space="preserve">IN ſingulis ſpeculis erronea formarum accidit comprehenſio, ſed iuxta uarietatem ſpeculorũ fi <lb/>uarietas errorum.</s> <s xml:id="echoid-s12834" xml:space="preserve"> In ſpeculis planis minor accidit error, quàm in alijs.</s> <s xml:id="echoid-s12835" xml:space="preserve"> In his etenim comprehen <lb/>ditur ueritas figuræ, & quantitatis, ſicut & in directione, quod per probationẽ patebit.</s> <s xml:id="echoid-s12836" xml:space="preserve"> Propona <lb/>tur ſpeculũ planum:</s> <s xml:id="echoid-s12837" xml:space="preserve"> & ſit a b linea in ſuperficie illius ſpe-<lb/>culi, cõmunis ſuperficiei ſpeculi & ſuperficiei orthogona <lb/> <anchor type="figure" xlink:label="fig-0195-01a" xlink:href="fig-0195-01"/> li ſuper ſuperficiem ſpeculi, [id eſt ſuperficiei reflexionis, <lb/>quę per 13 n 4 perpendicularis eſt plano ſpeculo, uel pla-<lb/>no ſpeculũ obliquũ in reflexionis puncto tangenti.</s> <s xml:id="echoid-s12838" xml:space="preserve">] Sint <lb/>l, f duo puncta in ſuperficie illa orthogonali:</s> <s xml:id="echoid-s12839" xml:space="preserve"> e centrum ui <lb/>ſus:</s> <s xml:id="echoid-s12840" xml:space="preserve"> & à puncto l ducatur perpendicularis ſuper ſuperfi-<lb/>ciem ſpeculi[per 11 p 11] quæ ſit l h:</s> <s xml:id="echoid-s12841" xml:space="preserve"> & producatur, ut h g ſit <lb/>æqualis l h.</s> <s xml:id="echoid-s12842" xml:space="preserve"> Similiter producatur perpendicularis f z, ut d <lb/>f ſit æqualis z f.</s> <s xml:id="echoid-s12843" xml:space="preserve"> Palàm ex ſuperioribus [2.</s> <s xml:id="echoid-s12844" xml:space="preserve">3 n 4] quòd l re <lb/>flectitur ad e ab aliquo puncto ſpeculi:</s> <s xml:id="echoid-s12845" xml:space="preserve"> & locus imaginis <lb/>eſt g [per 2 n 5] tantùm diſtans à ſuperficie ſpeculi, quan-<lb/>tùm l [per 11 n 5.</s> <s xml:id="echoid-s12846" xml:space="preserve">] Similiter f reflectitur ad e:</s> <s xml:id="echoid-s12847" xml:space="preserve"> & locus ima-<lb/>ginis eſt d [per 2 n 5.</s> <s xml:id="echoid-s12848" xml:space="preserve">] Ducta autẽ linea f l:</s> <s xml:id="echoid-s12849" xml:space="preserve"> & ſimiliter g d:</s> <s xml:id="echoid-s12850" xml:space="preserve"> <lb/>quodcunq;</s> <s xml:id="echoid-s12851" xml:space="preserve"> punctum lineæ f l reflectetur ad e:</s> <s xml:id="echoid-s12852" xml:space="preserve"> locus ima-<lb/>ginis eius eſt tantùm diſtans à ſuperficie ſpeculi, quantũ <lb/>ipſum punctũ.</s> <s xml:id="echoid-s12853" xml:space="preserve"> Et ita quo dlibet punctũ linéæ f l tantùm <lb/>uidetur diſtare, quantùm diſtat! Vnde ſi linea f l fuerit re <lb/>cta:</s> <s xml:id="echoid-s12854" xml:space="preserve"> erit linea d g recta:</s> <s xml:id="echoid-s12855" xml:space="preserve"> ſi fuerit arcus:</s> <s xml:id="echoid-s12856" xml:space="preserve"> erit d g arcus & eiuſdem curuitatis.</s> <s xml:id="echoid-s12857" xml:space="preserve"> Quare linea l fapparebit e-<lb/>iuſdem quantitatis, eiuſdem figuræ, cuius fuerit.</s> <s xml:id="echoid-s12858" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s12859" xml:space="preserve"/> </p> <div xml:id="echoid-div455" type="float" level="0" n="0"> <figure xlink:label="fig-0195-01" xlink:href="fig-0195-01a"> <variables xml:id="echoid-variables143" xml:space="preserve">f f e a z b h d g</variables> </figure> </div> </div> <div xml:id="echoid-div457" type="section" level="0" n="0"> <head xml:id="echoid-head415" xml:space="preserve" style="it">3. Viſus in reflexione præcipuè allucinatur propter lucis immoderationẽ: ſitus diuerſitatem: <lb/>uiſus & uiſibilis à ſpeculo diſtantiam. 7 p 5.</head> <p> <s xml:id="echoid-s12860" xml:space="preserve">VErùm ſi in punctis lineæ f l fuerit uarietas colorum minutim uariata:</s> <s xml:id="echoid-s12861" xml:space="preserve"> forſitan nõ diſcernetur <lb/>uariatio, ſed una prætendetur uiſui coloris conſuſio.</s> <s xml:id="echoid-s12862" xml:space="preserve"> Vnde error erit in luce & colore.</s> <s xml:id="echoid-s12863" xml:space="preserve"> Et hoc <lb/>in numero, propter reflexionẽ.</s> <s xml:id="echoid-s12864" xml:space="preserve"> Illa etenim colorum & lucium uarietas forſitan comprehen-<lb/>di poſſet directè, ſed egreſſus eſt color à tẽperantia reſpectu reflexionis, nõ reſpectu directionis.</s> <s xml:id="echoid-s12865" xml:space="preserve"> Si-<lb/>militer particulæ minutæ occultantur, aut confunduntur in reflexione, quæ diſcerni poſſent in dire <lb/>ctione.</s> <s xml:id="echoid-s12866" xml:space="preserve"> Et propter debilitatẽ lucis uel coloris ex reflexione, accidit error in longitudine, qui quidẽ <lb/>nõ accideret directè.</s> <s xml:id="echoid-s12867" xml:space="preserve"> In ſitu manifeſtè accidit error ex reflexione ſola.</s> <s xml:id="echoid-s12868" xml:space="preserve"> In imagine enim, ſiniſtra com <lb/>prehendimus ea, quæ in corpore uiſo (ſi eſſet in loco imaginis) dextra uideremus.</s> <s xml:id="echoid-s12869" xml:space="preserve"> Cũ enim aliquid <lb/>alij opponitur, contrarius eis ſitus eſt ad inuicem:</s> <s xml:id="echoid-s12870" xml:space="preserve"> quod enim uni fuerit dextrum, alij erit ſiniſtrum.</s> <s xml:id="echoid-s12871" xml:space="preserve"> <lb/>Igitur quod rei uiſæ dextrum, eſt imagini ſiniſtrum:</s> <s xml:id="echoid-s12872" xml:space="preserve"> & ſiniſtrum in imagine, dextrum eſt uidenti.</s> <s xml:id="echoid-s12873" xml:space="preserve"> Et <lb/>generabter in modo lucis, uel coloris, uel ſitus ſemper error accidit ex ſola reflexione.</s> <s xml:id="echoid-s12874" xml:space="preserve"> Et in alijs, <lb/>quæ errorem inducunt directè:</s> <s xml:id="echoid-s12875" xml:space="preserve"> inducunt ſimiliter in reflexione:</s> <s xml:id="echoid-s12876" xml:space="preserve"> & facilius:</s> <s xml:id="echoid-s12877" xml:space="preserve"> quoniam temperamen-<lb/>tum ſingulorum minus eſt in uiſu reflexo, quàm in directo.</s> <s xml:id="echoid-s12878" xml:space="preserve"> Horum omniũ unum proponatur exem <lb/>plum, & idem in cæteris intelligatur.</s> <s xml:id="echoid-s12879" xml:space="preserve"> In uiſu directo cum fuerit corpus uiſum, remotum ab axibus <lb/>uiſualibus, accidit ipſum uider<gap/> duo:</s> <s xml:id="echoid-s12880" xml:space="preserve"> [ut demonſtratum eſt 11 n 3] idem euenit in ſpeculis, re uiſa ab <lb/>axibus elongata.</s> <s xml:id="echoid-s12881" xml:space="preserve"> In ſpeculis ab aliqua lõgitudine uidebitur corpus minus, quàm ſit, quod forſan di-<lb/>rectè à tanta longitudine uideretur etiã minus, quàm eſſet in ueritate, ſed non adeò minus.</s> <s xml:id="echoid-s12882" xml:space="preserve"> Et hoe <lb/>minoritatis additamentum in ſpeculis, prouenit propter minus in longitudine temperamentũ.</s> <s xml:id="echoid-s12883" xml:space="preserve"> In <lb/>figura nonnunquam accidit error in ſpeculis propter cauſſas, propter quas in uiſu directo:</s> <s xml:id="echoid-s12884" xml:space="preserve"> ſed ma-<lb/>ior & frequentior propter ſitum.</s> <s xml:id="echoid-s12885" xml:space="preserve"> Si aliquid ab aliqua longitudine opponatur ſpeculo, & eius capita <lb/>non percipiantur à uiſu, ut funis, del aliquid tale:</s> <s xml:id="echoid-s12886" xml:space="preserve"> uidebitur forſitan continuũ ſpeculo.</s> <s xml:id="echoid-s12887" xml:space="preserve"> Idem accidit <lb/>in uiſu directo, ſi opponatur funis aliquιs foramini, & non uideantur capita funis:</s> <s xml:id="echoid-s12888" xml:space="preserve"> non apparebit di <lb/>ſtantia inter funem & foramen, licet magna ſit.</s> <s xml:id="echoid-s12889" xml:space="preserve"> Et eſt propter ſitum.</s> <s xml:id="echoid-s12890" xml:space="preserve"> Si autem alterum capitum ui-<lb/>deatur, alterum uerò non:</s> <s xml:id="echoid-s12891" xml:space="preserve"> uidebitur fortaſsis illud caput continuum.</s> <s xml:id="echoid-s12892" xml:space="preserve"> Et in ſingulis ubi directè er-<lb/>ror accidit:</s> <s xml:id="echoid-s12893" xml:space="preserve"> ſimiliter in reflexione.</s> <s xml:id="echoid-s12894" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div458" type="section" level="0" n="0"> <head xml:id="echoid-head416" xml:space="preserve">DE ERRORE, QVI ACCIDIT IN SPECVLIS SPHAE-<lb/>ricis conuexis. Cap. IIII.</head> <head xml:id="echoid-head417" xml:space="preserve" style="it">4. In ſpeculo ſphærico cõuexo idẽ eſt ſitus, eadeḿ diſpoſitio partiũ imaginis & uiſibilis. 35 p 6.</head> <p> <s xml:id="echoid-s12895" xml:space="preserve">VNiuerſitas errorũ in ſpeculis planis accidentiũ, euenit ſimiliter in ſphæricis exterioribus.</s> <s xml:id="echoid-s12896" xml:space="preserve"> Et <lb/>præter hoc res uiſa uidetur minor ꝗ̃ ſit.</s> <s xml:id="echoid-s12897" xml:space="preserve"> Et generaliter in his ſpeculis nihil exre uiſa compre-<lb/> <pb o="190" file="0196" n="196" rhead="ALHAZEN"/> henditur in ueritate, præter ordinationẽ partiũ, quæ talis apparet in ſpeculo, qualis eſt in imagine.</s> <s xml:id="echoid-s12898" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div459" type="section" level="0" n="0"> <head xml:id="echoid-head418" xml:space="preserve" style="it">5. In ſpeculo ſphærico conuexo, imago uiſibilis, cuius uera magnitudo uiſione directa percipi <lb/>poteſt, minor eſt uiſibili. 39 p 6.</head> <p> <s xml:id="echoid-s12899" xml:space="preserve">QVòd autem res ſemper uideatur minor, quàm ſit:</s> <s xml:id="echoid-s12900" xml:space="preserve"> probatur.</s> <s xml:id="echoid-s12901" xml:space="preserve"> Sit a b linea uiſa:</s> <s xml:id="echoid-s12902" xml:space="preserve"> z x ſpeculum:</s> <s xml:id="echoid-s12903" xml:space="preserve"> d <lb/>centrum:</s> <s xml:id="echoid-s12904" xml:space="preserve"> e punctum uiſus.</s> <s xml:id="echoid-s12905" xml:space="preserve"> a reflectatur ad e à puncto h:</s> <s xml:id="echoid-s12906" xml:space="preserve"> b à puncto n.</s> <s xml:id="echoid-s12907" xml:space="preserve"> a b producta aut tranſi <lb/>bit per centrum ſpeculi, aut non.</s> <s xml:id="echoid-s12908" xml:space="preserve"> Tranſeat:</s> <s xml:id="echoid-s12909" xml:space="preserve"> & ducatur à puncto n linea contingens circulum <lb/>[per 17 p 3] quæ ſit n l:</s> <s xml:id="echoid-s12910" xml:space="preserve"> à puncto h contingens circulum, h m:</s> <s xml:id="echoid-s12911" xml:space="preserve"> & ducantur lineę acceſſus & reflexionis <lb/>b n, e n, a h, e h:</s> <s xml:id="echoid-s12912" xml:space="preserve"> & producãtur lineæ e h, e n, donec cadant in perpendicalarẽ, quæ eſt a d:</s> <s xml:id="echoid-s12913" xml:space="preserve"> & puncta ca <lb/>ſus ſint, t, q.</s> <s xml:id="echoid-s12914" xml:space="preserve"> Palàm [ք 3 n 5] quòd t eſt locus imaginis a:</s> <s xml:id="echoid-s12915" xml:space="preserve"> q eſt locus imaginis b.</s> <s xml:id="echoid-s12916" xml:space="preserve"> Dico, quòd a b maior <lb/>eſt q t.</s> <s xml:id="echoid-s12917" xml:space="preserve"> Palàm ex ſuperioribus [18 n 5] quòd proportio a d ad d t, ſicut a m ad m t.</s> <s xml:id="echoid-s12918" xml:space="preserve"> Similiter ꝓportio b <lb/>d ad d q, ſicut proportio b l ad l q:</s> <s xml:id="echoid-s12919" xml:space="preserve"> ſed [per 9 ax:</s> <s xml:id="echoid-s12920" xml:space="preserve">] a d maior d b, & d t mi <lb/>nor d q:</s> <s xml:id="echoid-s12921" xml:space="preserve"> ergo maior eſt proportio a m ad m t:</s> <s xml:id="echoid-s12922" xml:space="preserve"> quàm b l ad l q.</s> <s xml:id="echoid-s12923" xml:space="preserve"> [Quia e-<lb/> <anchor type="figure" xlink:label="fig-0196-01a" xlink:href="fig-0196-01"/> nim è quatuor lineis a d prima maior eſt b d tertia, & d t ſecunda mi-<lb/>nor d q quarta:</s> <s xml:id="echoid-s12924" xml:space="preserve"> erit ratio a d ad d t maior quàm b d ad d q, ut patet ex <lb/>8 p 5:</s> <s xml:id="echoid-s12925" xml:space="preserve"> & per 11 p 5 ratio a m ad m t maior quàm b l ad l q.</s> <s xml:id="echoid-s12926" xml:space="preserve">] Secetur [per <lb/>12 p 6] a m in pũcto f, ut proportio fm ad m t ſit, ſicut b l ad l q:</s> <s xml:id="echoid-s12927" xml:space="preserve"> erit er <lb/>go minor proportio b m ad m t, quàm b l ad l q.</s> <s xml:id="echoid-s12928" xml:space="preserve"> [Nam cum m t ſit ma-<lb/>ior l q:</s> <s xml:id="echoid-s12929" xml:space="preserve"> erit ք 14 p 5 f m maior b l:</s> <s xml:id="echoid-s12930" xml:space="preserve"> quare per 8 p 5 ratio f m ad m t maior <lb/>eſt, quàm b l ad eandem m t:</s> <s xml:id="echoid-s12931" xml:space="preserve"> ratio igitur b l ad m t minor eſt, quàm b l <lb/>ad l q:</s> <s xml:id="echoid-s12932" xml:space="preserve"> ergo ratio b m ad m t multo minor erit, ꝗ̃ b l ad l q.</s> <s xml:id="echoid-s12933" xml:space="preserve">] Secetur <lb/>[per 12 p 6] m t in puncto k, ut proportio b m ad m k ſit, ſicut b l ad l q.</s> <s xml:id="echoid-s12934" xml:space="preserve"> <lb/>k cadet neceſſariò inter m & q:</s> <s xml:id="echoid-s12935" xml:space="preserve"> quia l q minor m q, & b l maior b m.</s> <s xml:id="echoid-s12936" xml:space="preserve"> <lb/>Cũ igitur f m ad m t, ſicut b l ad l q, & ſicut b m ad m k:</s> <s xml:id="echoid-s12937" xml:space="preserve"> erit [per 19 p 5] <lb/>proportio f b ad k t, ſicut b l ad l q:</s> <s xml:id="echoid-s12938" xml:space="preserve"> ſed b l, maior l q:</s> <s xml:id="echoid-s12939" xml:space="preserve"> [concluſum enim <lb/>eſt ut b d ad d q, ſic b l ad l q:</s> <s xml:id="echoid-s12940" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s12941" xml:space="preserve"> cum b d ſit maior d q, erit b l maior <lb/>l q] ergo f b maior k t.</s> <s xml:id="echoid-s12942" xml:space="preserve"> Quare a b maior q t.</s> <s xml:id="echoid-s12943" xml:space="preserve"> [quia a b maior eſt f b, quæ <lb/>maior oſtẽſa eſt k t, & k t maior eſt q t.</s> <s xml:id="echoid-s12944" xml:space="preserve"> Quare a b multò maior eſt q t.</s> <s xml:id="echoid-s12945" xml:space="preserve">] <lb/>Quod eſt propoſitum.</s> <s xml:id="echoid-s12946" xml:space="preserve"> Si uerò linea a b producta non perueniat ad <lb/>centrum:</s> <s xml:id="echoid-s12947" xml:space="preserve"> ducatur à puncto a linea ad cẽtrum:</s> <s xml:id="echoid-s12948" xml:space="preserve"> quæ ſit a d:</s> <s xml:id="echoid-s12949" xml:space="preserve"> & ſit d cen-<lb/>trum:</s> <s xml:id="echoid-s12950" xml:space="preserve"> & à puncto b ducatur linea b d:</s> <s xml:id="echoid-s12951" xml:space="preserve"> & locus imaginis a ſit punctum <lb/>g:</s> <s xml:id="echoid-s12952" xml:space="preserve"> locus imaginis b ſit p:</s> <s xml:id="echoid-s12953" xml:space="preserve"> & ducatur linea g p:</s> <s xml:id="echoid-s12954" xml:space="preserve"> quæ quidem eſt imago lineæ a b.</s> <s xml:id="echoid-s12955" xml:space="preserve"> Dico quòd a b maior <lb/>eſt g p:</s> <s xml:id="echoid-s12956" xml:space="preserve"> quoniam g p aut eſt æquidiſtans a b, aut nõ.</s> <s xml:id="echoid-s12957" xml:space="preserve"> Si fuerit æquidiſtans, planum:</s> <s xml:id="echoid-s12958" xml:space="preserve"> quòd eſt minor.</s> <s xml:id="echoid-s12959" xml:space="preserve"> <lb/>[Nam per 29.</s> <s xml:id="echoid-s12960" xml:space="preserve">32 p 1 triangula a d b, & g d p ſunt æquiangula:</s> <s xml:id="echoid-s12961" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s12962" xml:space="preserve"> per 4 p 6, ut a d ad d g, ſic a b ad <lb/> <anchor type="figure" xlink:label="fig-0196-02a" xlink:href="fig-0196-02"/> <anchor type="figure" xlink:label="fig-0196-03a" xlink:href="fig-0196-03"/> g p:</s> <s xml:id="echoid-s12963" xml:space="preserve"> ſed per 9 ax.</s> <s xml:id="echoid-s12964" xml:space="preserve"> a d maior eſt d g:</s> <s xml:id="echoid-s12965" xml:space="preserve"> ergo a b maior eſt g p.</s> <s xml:id="echoid-s12966" xml:space="preserve">] Si non fuerit æquidiſtans, producatur, <lb/>quouſque concurrat cum ea:</s> <s xml:id="echoid-s12967" xml:space="preserve"> ſit concurſus z:</s> <s xml:id="echoid-s12968" xml:space="preserve"> & [per 31 p 1] à puncto p producatur æquidiſtans <lb/>a b:</s> <s xml:id="echoid-s12969" xml:space="preserve"> quæ ſit p h.</s> <s xml:id="echoid-s12970" xml:space="preserve"> Angulus p g h aut eſt acutus:</s> <s xml:id="echoid-s12971" xml:space="preserve"> aut rectus:</s> <s xml:id="echoid-s12972" xml:space="preserve"> aut maior.</s> <s xml:id="echoid-s12973" xml:space="preserve"> Sit rectus uel maior:</s> <s xml:id="echoid-s12974" xml:space="preserve"> erit <lb/>[per 19 p 1] latus p h maius p g:</s> <s xml:id="echoid-s12975" xml:space="preserve"> ſed [per 29.</s> <s xml:id="echoid-s12976" xml:space="preserve"> 32 p 1.</s> <s xml:id="echoid-s12977" xml:space="preserve"> 4 p 6] p h minus a b:</s> <s xml:id="echoid-s12978" xml:space="preserve"> [ideoq́;</s> <s xml:id="echoid-s12979" xml:space="preserve"> recta p g multò mi-<lb/>nor eſt a b.</s> <s xml:id="echoid-s12980" xml:space="preserve">] Et ita eſt propoſitũ.</s> <s xml:id="echoid-s12981" xml:space="preserve"> Si fuerit acutus:</s> <s xml:id="echoid-s12982" xml:space="preserve"> poteſt accidere ut forma ſit maior ipſa re, cuius eſt <lb/>forma:</s> <s xml:id="echoid-s12983" xml:space="preserve"> [quando nimirum angulus p g h minor eſt angulo p h g] quam licet, excedat:</s> <s xml:id="echoid-s12984" xml:space="preserve"> rarò accidet.</s> <s xml:id="echoid-s12985" xml:space="preserve"> <lb/>Et ſi acciderit, forſitan comprehendetur forma à longitudine tali, quòd minor uidebitur quàm ſit:</s> <s xml:id="echoid-s12986" xml:space="preserve"> <lb/>quoniam ipſum corpus ab hac longitudine forſitan uidebitur minus.</s> <s xml:id="echoid-s12987" xml:space="preserve"/> </p> <div xml:id="echoid-div459" type="float" level="0" n="0"> <figure xlink:label="fig-0196-01" xlink:href="fig-0196-01a"> <variables xml:id="echoid-variables144" xml:space="preserve">a f b m <gap/> k <gap/> q n e t h d <gap/> z</variables> </figure> <figure xlink:label="fig-0196-02" xlink:href="fig-0196-02a"> <variables xml:id="echoid-variables145" xml:space="preserve">b a e p g d</variables> </figure> <figure xlink:label="fig-0196-03" xlink:href="fig-0196-03a"> <variables xml:id="echoid-variables146" xml:space="preserve">a b h z e p g d</variables> </figure> </div> </div> <div xml:id="echoid-div461" type="section" level="0" n="0"> <head xml:id="echoid-head419" xml:space="preserve" style="it">6. In ſpeculo ſphærico conuexo, imagouiſibilis, cuius uera magnitudo uiſione directa propter <lb/>immoder at am diſtantiam percipi non poteſt: aliâs eſt æquabilis uiſibili: aliâs maior. 38 p 6.</head> <pb o="191" file="0197" n="197" rhead="OPTICAE LIBER VI."/> <p> <s xml:id="echoid-s12988" xml:space="preserve">QVòd aũt forma in his ſpeculis aliquando uideatur maior re uiſa:</s> <s xml:id="echoid-s12989" xml:space="preserve"> ſcilicet cum cõprehenditur <lb/>à tali longitudine, à qua eius certa quantitas nõ poſsit diſcerni:</s> <s xml:id="echoid-s12990" xml:space="preserve"> declarabitur.</s> <s xml:id="echoid-s12991" xml:space="preserve"> Sit a centrum <lb/>ſpeculi:</s> <s xml:id="echoid-s12992" xml:space="preserve"> & ſuperficies ſumatur reflexionis:</s> <s xml:id="echoid-s12993" xml:space="preserve"> quæ ſecabit ſpeculum ſuper circulum:</s> <s xml:id="echoid-s12994" xml:space="preserve"> [per 1 th.</s> <s xml:id="echoid-s12995" xml:space="preserve"> 1 <lb/>ſphær.</s> <s xml:id="echoid-s12996" xml:space="preserve">] ſit circulus ille e d b:</s> <s xml:id="echoid-s12997" xml:space="preserve"> e d diameter illius circuli:</s> <s xml:id="echoid-s12998" xml:space="preserve"> & producatur diameter e d uſq;</s> <s xml:id="echoid-s12999" xml:space="preserve"> ad z, ut multi <lb/>plicatio e z in z d ſit æqualis quadrato a d:</s> <s xml:id="echoid-s13000" xml:space="preserve"> quod planũ eſt, cum ſit poſsibile diametro e d talem addi <lb/>lineam, ut ductus totalis in partem additam, ſit æqualis quadrato a d:</s> <s xml:id="echoid-s13001" xml:space="preserve"> [id uerò quomodo expeditè <lb/>fiat, oſtenſum eſt 32 n 5] & diuidatur linea z d in partes æquales, in puncto h [per 10 p 1.</s> <s xml:id="echoid-s13002" xml:space="preserve">] Erit igi-<lb/>tur a h medietas e z.</s> <s xml:id="echoid-s13003" xml:space="preserve"> [Nam ſi a d, a e per 15 d 1 æquales, addantur æqualibus h d, h z:</s> <s xml:id="echoid-s13004" xml:space="preserve"> æquabitur a h <lb/>ipſis z h & a e.</s> <s xml:id="echoid-s13005" xml:space="preserve"> Tota igitur e z dupla eſt ipſius a h.</s> <s xml:id="echoid-s13006" xml:space="preserve">] Ductus ergo a h in h d erit æqualis quartæ parti <lb/>quadrati a d.</s> <s xml:id="echoid-s13007" xml:space="preserve"> [Quia enim oblongum comprehenſum ſub e z & z d æquatur quadrato a d per fabri-<lb/>cationem:</s> <s xml:id="echoid-s13008" xml:space="preserve"> ergo quod comprehend:</s> <s xml:id="echoid-s13009" xml:space="preserve"> tur ſub a h dimidiata baſi & z d altitudine eadem, æquatur di-<lb/>midiato quadrato a d per 1 p 6:</s> <s xml:id="echoid-s13010" xml:space="preserve"> rurſusq́;</s> <s xml:id="echoid-s13011" xml:space="preserve"> oblongũ comprehenſum ſub a h baſi eadem & h d altitudi-<lb/>ne dimidiata, æquatur dimidiato oblongo ſub a h & z d.</s> <s xml:id="echoid-s13012" xml:space="preserve"> Quare æquatur quadranti quadrati a d.</s> <s xml:id="echoid-s13013" xml:space="preserve">] Et <lb/>quoniam ductus a h in h d maior eſt quadrato h d:</s> <s xml:id="echoid-s13014" xml:space="preserve"> [quia per 3 p 2 æquatur quadrato h d, & oblongo <lb/>comprehenſo ſub a d, & d h] ſit ductus a h in h t, æqualis quadrato h d [fiet autem æqualis, ſi ipſis <lb/>a h & h d tertiam proportionalem per 11 p 6 inueneris:</s> <s xml:id="echoid-s13015" xml:space="preserve"> tum enim per 17 p 6 oblongum extremarum <lb/>æquabitur quadrato mediæ h d.</s> <s xml:id="echoid-s13016" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s13017" xml:space="preserve"> ſi de h d detraxeris æqualem inuentæ proportionali, manda-<lb/>tum executus fueris.</s> <s xml:id="echoid-s13018" xml:space="preserve">] Fiat circulus ſecundum quantitatem a h:</s> <s xml:id="echoid-s13019" xml:space="preserve"> & à puncto h producatur chorda, <lb/>æqualis medietati lineæ h d:</s> <s xml:id="echoid-s13020" xml:space="preserve"> [per 1 p 4] quæ ſit h q:</s> <s xml:id="echoid-s13021" xml:space="preserve"> & producantur lineæ q a, q t:</s> <s xml:id="echoid-s13022" xml:space="preserve"> & [per 23 p 1] ſuper <lb/>punctũ q fiat angulus, æqualis angulo q a h:</s> <s xml:id="echoid-s13023" xml:space="preserve"> qui ſit h q n.</s> <s xml:id="echoid-s13024" xml:space="preserve"> Cum ergo in his duobus triangulis hi duo <lb/>anguli ſint æquales, & unus cõmunis, ſcilicet q h a:</s> <s xml:id="echoid-s13025" xml:space="preserve"> erit [per 32 p 1] tertius tertio æqualis, ſcilicet a q h <lb/>angulo h n q:</s> <s xml:id="echoid-s13026" xml:space="preserve"> & erũt triangula ſimilia:</s> <s xml:id="echoid-s13027" xml:space="preserve"> [per 4 p.</s> <s xml:id="echoid-s13028" xml:space="preserve"> 1 d 6] & erit proportio a h ad h q, ſi cut h q ad h n.</s> <s xml:id="echoid-s13029" xml:space="preserve"> Igi-<lb/>tur [per 17 p 6] qđ fit ex ductu a h in h n, æquale eſt quadrato h q:</s> <s xml:id="echoid-s13030" xml:space="preserve"> ſed, [per conſectariũ 4 p 2] quadra <lb/>tum h q eſt quarta pars quadrati h d:</s> <s xml:id="echoid-s13031" xml:space="preserve"> cũ h q ſit medietas h d [per fabricationẽ.</s> <s xml:id="echoid-s13032" xml:space="preserve">] Igitur multiplicatio <lb/>a h in h n, ęqualis eſt quartæ parti multiplicationis a h in h t.</s> <s xml:id="echoid-s13033" xml:space="preserve"> Quare h n eſt quarta pars h t [per 1 p 6.</s> <s xml:id="echoid-s13034" xml:space="preserve">] <lb/>Igitur n cadit inter h & t:</s> <s xml:id="echoid-s13035" xml:space="preserve"> reſtat, ut ductus h t in t n ſint tres quartæ quadrati h t.</s> <s xml:id="echoid-s13036" xml:space="preserve"> [Quia enim h n eſt <lb/>quadrans ipſius h t:</s> <s xml:id="echoid-s13037" xml:space="preserve"> reliqua igitur n t eſt dodrans, ſeu tres quartæ h t.</s> <s xml:id="echoid-s13038" xml:space="preserve"> Et quoniam rectangula com-<lb/>prehenſa ſub tota h t & ſegmentis n t & n h æquantur quadrato h t per 2 p 2:</s> <s xml:id="echoid-s13039" xml:space="preserve"> rectangulum igitur <lb/>comprehenſum ſub tota h t & ſegmento t n (quod eſt dodrans totius h t) æquatur dodranti quadra <lb/>ti h t.</s> <s xml:id="echoid-s13040" xml:space="preserve">] Verùm angulus q h a acutus eſt:</s> <s xml:id="echoid-s13041" xml:space="preserve"> [ut oſtenſum eſt 60 n 5] & [per 5 p 1] æqualis angulo h q a:</s> <s xml:id="echoid-s13042" xml:space="preserve"> <lb/>quia reſpiciunt æqualia latera in triangulo maiori.</s> <s xml:id="echoid-s13043" xml:space="preserve"> Igitur angulus q h n æqualis angulo h n q:</s> <s xml:id="echoid-s13044" xml:space="preserve"> [æqua <lb/>lis enim concluſus eſt angulus a q h angulo h n q] & ita [per 6 p 1] h q æqualis q n, & angulus h n q <lb/>acutus:</s> <s xml:id="echoid-s13045" xml:space="preserve"> quare [per 13 p 1] angulus q n t obtuſus.</s> <s xml:id="echoid-s13046" xml:space="preserve"> Quadratum igitur t q ſuperat quadratum q n & qua-<lb/>dratum t n, ductu lineæ t n in h n.</s> <s xml:id="echoid-s13047" xml:space="preserve"> Quoniam, ut dicit Euclides [12 p 2] quadratum lateris oppoſiti ob-<lb/>ruſo ſuperat quadrata duorũ laterũ, quantũ eſt, quod fit ex ductu unius lateris bis in partẽ ei adiun-<lb/>ctam, procedentẽ uſq;</s> <s xml:id="echoid-s13048" xml:space="preserve"> ad locũ caſus perpendicularis à capite alterius lateris ductæ.</s> <s xml:id="echoid-s13049" xml:space="preserve"> Nam ſi à pũctò q <lb/>ducatur perpendicularis ſuper lineam h t:</s> <s xml:id="echoid-s13050" xml:space="preserve"> cadet in punctũ me-<lb/> <anchor type="figure" xlink:label="fig-0197-01a" xlink:href="fig-0197-01"/> dium lineæ h n:</s> <s xml:id="echoid-s13051" xml:space="preserve"> [non enim cadit extra puncta h & n:</s> <s xml:id="echoid-s13052" xml:space="preserve"> ſecus per <lb/>16 p 1 angulus acutus maior eſſet recto contra 12 d 1:</s> <s xml:id="echoid-s13053" xml:space="preserve"> caditigitur <lb/>in medium rectæ h n per 26 p 1] & [per 1 p 2] ductus t n in medie-<lb/>tatem h n bis, æquipollet ductui t n in h n.</s> <s xml:id="echoid-s13054" xml:space="preserve"> Igitur quadratum t q <lb/>ſuperat quadrata q n, t n, ductu tn in n h.</s> <s xml:id="echoid-s13055" xml:space="preserve"> Sed [per 3 p 2] ductus <lb/>t n in h n, cum quadrato tn, æqualis eſt ductui h t in tn.</s> <s xml:id="echoid-s13056" xml:space="preserve"> Igitur <lb/>[ſubducto quadrato tn] ductus h t in t n eſt exceſſus quadrati <lb/>t q ſupra quadratum h q.</s> <s xml:id="echoid-s13057" xml:space="preserve"> [nam quadratum q h æquatur quadra-<lb/>to q n:</s> <s xml:id="echoid-s13058" xml:space="preserve"> quia rectæ q h, q n æquales oſtenſæ ſunt.</s> <s xml:id="echoid-s13059" xml:space="preserve">] Amplius:</s> <s xml:id="echoid-s13060" xml:space="preserve"> ſit <lb/>proportio a i ad a h, ſicut q t ad q h:</s> <s xml:id="echoid-s13061" xml:space="preserve"> [per 12 p 6] erit [per 22 p 6] <lb/>quadratum [ai] ad quadratum [a h] ficut quadratum [qt] ad <lb/>quadratum [qh] & erit [per 17 p 5] proportio exceſſus quadra-<lb/>ti a i ſupra quadratum a h, ad quadratum a h, ſicut ductus h t in <lb/>t n, ad quadratum q h.</s> <s xml:id="echoid-s13062" xml:space="preserve"> [Nam a i maior eſt a h:</s> <s xml:id="echoid-s13063" xml:space="preserve"> quia q t maior eſt <lb/>q h:</s> <s xml:id="echoid-s13064" xml:space="preserve"> cum quadratum q t ſit maius quadrato q h:</s> <s xml:id="echoid-s13065" xml:space="preserve"> & oblongũ com <lb/>prehenſum ſub h t, t n eſt exuperantia quadrati q t ſupra quadra <lb/>tum q h.</s> <s xml:id="echoid-s13066" xml:space="preserve">] Et quoniã quadratum q h quater ſumptũ, efficit quadratum h d:</s> <s xml:id="echoid-s13067" xml:space="preserve"> [per conſectarium 4 p 2:</s> <s xml:id="echoid-s13068" xml:space="preserve"> <lb/>quia q h dimidia eſt ipſius h d per fabricationẽ] & ductus h t in t n quater ſumptus, efficit triplũ qua-<lb/>drati h t:</s> <s xml:id="echoid-s13069" xml:space="preserve"> [oſtenſum eſt enim rectangulũ cõprehenſum ſub h t & t n, eſſe dodrantẽ quadrati h t:</s> <s xml:id="echoid-s13070" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s13071" xml:space="preserve"> <lb/>quater ſumptũ, erit triplũ quadrati h t] erit [per 15 p 5] ductus h t in t n ad quadratũ h q, ſicut triplum <lb/>quadrati h t ad quadratum h d.</s> <s xml:id="echoid-s13072" xml:space="preserve"> Sit autem h o tripla ad h t:</s> <s xml:id="echoid-s13073" xml:space="preserve"> erit ductus h o in h t triplus ad quadratum <lb/>h t [per 1 p 6.</s> <s xml:id="echoid-s13074" xml:space="preserve">] Sed quoniã proportio a h ad h d eſt, ſicut h d ad h t:</s> <s xml:id="echoid-s13075" xml:space="preserve"> [Nam per theſin rectangulum com <lb/>prehenſum ſub a h & h t ęquatur quadrato h d:</s> <s xml:id="echoid-s13076" xml:space="preserve"> ergo per 17 p 6, ut a h ad h d, ſic h d ad h t] erit [per cõ-<lb/>ſectaria 20 p 6.</s> <s xml:id="echoid-s13077" xml:space="preserve"> 4 p 5] h t ad a h, ſicut quadratũ h t ad quadratum h d.</s> <s xml:id="echoid-s13078" xml:space="preserve"> Verùm proportio o h ad a h, ſi-<lb/>cut ductus o h in h t ad ductum a h in h t [per 1 p 6] & [per 11 p 5] proportio o h ad h a, ſicut propor-<lb/>tio tripli quadrati h t ad quadratũ h d.</s> <s xml:id="echoid-s13079" xml:space="preserve"> Sed hæc erat ꝓportio exceſſus quadrati a i ſupra quadratũ a <lb/>h ad quadratũ a h.</s> <s xml:id="echoid-s13080" xml:space="preserve"> Igitur o h ad a h, ſicut exceſſus quadrati a i ſupra quadratũ a h ad quadratum a h.</s> <s xml:id="echoid-s13081" xml:space="preserve"> <lb/> <pb o="192" file="0198" n="198" rhead="ALHAZEN"/> Igitur coniumctim [per 18 p 5] proportio o a ad a h, ſicut quadrati a i ad quadratũ a h:</s> <s xml:id="echoid-s13082" xml:space="preserve"> exceſſus enin<gap/> <lb/>quadrati a i ſupra quadratũ a h, cum quadrato a h efficit quadratum a i:</s> <s xml:id="echoid-s13083" xml:space="preserve"> igitur [per conuerſionẽ cõſe-<lb/>ctarij ad 20 p 6] i a erit media in proportione inter o a & a h.</s> <s xml:id="echoid-s13084" xml:space="preserve"> Igitur proportio o a ad i a, ſicut i a ad h a:</s> <s xml:id="echoid-s13085" xml:space="preserve"> <lb/>& [per 19 p 5] eadem erit proportio reſidui ad reſiduum:</s> <s xml:id="echoid-s13086" xml:space="preserve"> id eſt o i ad i h.</s> <s xml:id="echoid-s13087" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s13088" xml:space="preserve"> ductus a d in h d <lb/>minor eſt quarta parte quadrati a d:</s> <s xml:id="echoid-s13089" xml:space="preserve"> [demonſtratum enim eſt rectangulum comprehenſum ſub a h <lb/>& h d, æquari quadranti quadrati a d:</s> <s xml:id="echoid-s13090" xml:space="preserve"> & a d minor eſt quàm a h per 9 ax:</s> <s xml:id="echoid-s13091" xml:space="preserve">] igitur h d eſt minor quarta <lb/>parte lineæ a d.</s> <s xml:id="echoid-s13092" xml:space="preserve"> [nam ſi æqualis eſſet:</s> <s xml:id="echoid-s13093" xml:space="preserve"> rectangulũ comprehenſum ſub a d & h d, æquaretur quadranti <lb/>quadrati a d per 1 p 6.</s> <s xml:id="echoid-s13094" xml:space="preserve">] Igitur h d eſt minor quinta parte a h.</s> <s xml:id="echoid-s13095" xml:space="preserve"> Cũ ergo a h ſit maior quàm quintupla ad <lb/>h d, & ductus eius in h t efficiat quadratũ h d:</s> <s xml:id="echoid-s13096" xml:space="preserve"> [per theſin] erit h t minor quinta parte h d:</s> <s xml:id="echoid-s13097" xml:space="preserve"> [nam per <lb/>theſin & 17 p 6 eſt, ut a h ad h d, ſic h d ad h t:</s> <s xml:id="echoid-s13098" xml:space="preserve"> ſed per proximã concluſionẽ a h maior eſt, quàm quintu <lb/>pla ipſius h d:</s> <s xml:id="echoid-s13099" xml:space="preserve"> ergo h d maior eſt quàm quintupla ipſius h t:</s> <s xml:id="echoid-s13100" xml:space="preserve"> ide<gap/>q́;</s> <s xml:id="echoid-s13101" xml:space="preserve"> h t minor quinta parte ipſius h d] <lb/>& ita h t minor uiceſima quinta parte h a.</s> <s xml:id="echoid-s13102" xml:space="preserve"> [Quia enim ratio h a ad h d, & h d ad h t maior eſt ꝗ̃ quintu <lb/>pla, ut patuit:</s> <s xml:id="echoid-s13103" xml:space="preserve"> erit per 10 d 5 ratio a h ad h t maior, ꝗ̃ uicecupla quintupla:</s> <s xml:id="echoid-s13104" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s13105" xml:space="preserve"> h t minor uiceſima <lb/>quinta parte ipſius a h.</s> <s xml:id="echoid-s13106" xml:space="preserve">] Sed proportio o i ad i h, ſicut i a ad a h, ut dictũ eſt.</s> <s xml:id="echoid-s13107" xml:space="preserve"> Igitur cõiunctim [per 18 p <lb/>5] o h ad i h, ſicut i a cũ a h ad a h.</s> <s xml:id="echoid-s13108" xml:space="preserve"> Igitur [per 15 p 5] tertia primę ad ſecũdã, ſicut tertia tertię ad quartã:</s> <s xml:id="echoid-s13109" xml:space="preserve"> <lb/>ſed h t eſt tertia pars lineæ o h [nam per theſin h o tripla eſt ipſius h t.</s> <s xml:id="echoid-s13110" xml:space="preserve">] Igitur t h ad i h eſt, ſicut tertia <lb/>pars lineę i a, cũ tertia parte a h, ad lineã a h.</s> <s xml:id="echoid-s13111" xml:space="preserve"> Igitur t h ad i a, ſicut duę tertię lineę a h, cum tertia lineę i <lb/>h, ad lineã a h.</s> <s xml:id="echoid-s13112" xml:space="preserve"> Sed quoniã linea o i eſt maior i h:</s> <s xml:id="echoid-s13113" xml:space="preserve"> [oſtenſum enim eſt, ut o a ad ia, ſic o i ad i h:</s> <s xml:id="echoid-s13114" xml:space="preserve"> at per 9 <lb/>ax:</s> <s xml:id="echoid-s13115" xml:space="preserve"> o a maior eſt i a:</s> <s xml:id="echoid-s13116" xml:space="preserve"> ergo o i maior eſt i h] erit i h minor medietate o h:</s> <s xml:id="echoid-s13117" xml:space="preserve"> & erit tertia i h minor ſexta par <lb/>te o h:</s> <s xml:id="echoid-s13118" xml:space="preserve"> & ita tertia i h erit minor medietate t h.</s> <s xml:id="echoid-s13119" xml:space="preserve"> Igitur duæ tertiæ a h, cum minore parte, quàm ſit me-<lb/>dietas h t, ſe habebunt ad a h, ſicut t h ad i h.</s> <s xml:id="echoid-s13120" xml:space="preserve"> Igitur [per conſectariũ 4 p 5] i h ad h t, ſicut a h ad duas <lb/>ſui tertias cum minore, quàm ſit medietas h t:</s> <s xml:id="echoid-s13121" xml:space="preserve"> ſed h t minor uiceſima quinta a h:</s> <s xml:id="echoid-s13122" xml:space="preserve"> & eius medietas mi <lb/>nor quàm medietas uiceſimæ quintæ partis.</s> <s xml:id="echoid-s13123" xml:space="preserve"> Sed linea a h in uigintiquinq;</s> <s xml:id="echoid-s13124" xml:space="preserve"> partes diuiſa:</s> <s xml:id="echoid-s13125" xml:space="preserve"> duæ tertiæ <lb/>cum medietate uiceſimæ quintæ partis non efficiunt octodecim eius partes.</s> <s xml:id="echoid-s13126" xml:space="preserve"> [Nam ex arithmeticæ <lb/>regulis intelliges {2/3} de 25 eſſe 16 integra, & ſupereſſe {2/3}, quæ additæ cum eo, quod minus eſt {1/2} uel etiã <lb/>cum {1/2}, efficiunt 1{1/6}.</s> <s xml:id="echoid-s13127" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s13128" xml:space="preserve"> {2/3} cum {1/2} de 25, ſunt 17{1/6}.</s> <s xml:id="echoid-s13129" xml:space="preserve">] Igitur proportio i h ad h t maior eſt, quàm ſit pro-<lb/>portio 25 ad 18.</s> <s xml:id="echoid-s13130" xml:space="preserve"> Item cum h t ſit minor uiceſima quinta parte a h:</s> <s xml:id="echoid-s13131" xml:space="preserve"> erit at maior uigintiquatuor parti-<lb/>bus, quarum a h eſt uigintiquinq;</s> <s xml:id="echoid-s13132" xml:space="preserve">. Sed linea i h minor eſt medietate o h:</s> <s xml:id="echoid-s13133" xml:space="preserve"> & ita minor medietate h t:</s> <s xml:id="echoid-s13134" xml:space="preserve"> <lb/>[quia h t & eius ſemiſsis efficiunt ſemiſſem ipſius o h:</s> <s xml:id="echoid-s13135" xml:space="preserve"> quo i h minor concluſa eſt] & ita minor una <lb/>& dimidia uigintiquinq;</s> <s xml:id="echoid-s13136" xml:space="preserve"> partium a h:</s> <s xml:id="echoid-s13137" xml:space="preserve"> & i a ita minor 26{1/2}, ſumptis partibus ſecundum diuiſionem a <lb/>h.</s> <s xml:id="echoid-s13138" xml:space="preserve"> Ergo proportio i a ad a t, ſicut minoris lineæ 26{1/2} ad maiorem 24.</s> <s xml:id="echoid-s13139" xml:space="preserve"> Igitur proportio i a ad a t mi-<lb/>nor eſt, quàm 26{1/2} ad 24.</s> <s xml:id="echoid-s13140" xml:space="preserve"> Sed proportio i h ad h t maior eſt, quàm 25 ad 18:</s> <s xml:id="echoid-s13141" xml:space="preserve"> igitur proportio i h ad h t <lb/>maior eſt, quàm i a ad a t [ratio enim 25 ad 18 maior eſt, ꝗ̃ 26{1/2} ad 24, ut patet ex arithmethica.</s> <s xml:id="echoid-s13142" xml:space="preserve">] Sit <lb/>proportio i m ad m t, ſicut i a ad a t:</s> <s xml:id="echoid-s13143" xml:space="preserve"> [id autem efficies:</s> <s xml:id="echoid-s13144" xml:space="preserve"> ſirectæ ex i a & a t compoſitæ ſegmenta ſu-<lb/>mas i a, a t:</s> <s xml:id="echoid-s13145" xml:space="preserve"> i t uerò inſectam ſimiliter ſeces per 10 p 6] cadet quidem m inter i & h.</s> <s xml:id="echoid-s13146" xml:space="preserve"> [Quia enim ra-<lb/>tio lineæ i h ad h t maior eſt, quàm i m ad m t:</s> <s xml:id="echoid-s13147" xml:space="preserve"> erit i m minor i h:</s> <s xml:id="echoid-s13148" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s13149" xml:space="preserve"> punctum m cadit inter i & h:</s> <s xml:id="echoid-s13150" xml:space="preserve"> e-<lb/>ritq́;</s> <s xml:id="echoid-s13151" xml:space="preserve"> per 9 ax:</s> <s xml:id="echoid-s13152" xml:space="preserve"> m t maior m h.</s> <s xml:id="echoid-s13153" xml:space="preserve">] Item maior erit proportio i m ad m h, quàm i a ad a t:</s> <s xml:id="echoid-s13154" xml:space="preserve"> [Quia enim li-<lb/>nea m t maior eſt m h è proxima concluſione:</s> <s xml:id="echoid-s13155" xml:space="preserve"> erit per 8 p 5 ratio i m ad m h maior, quàm ad m t:</s> <s xml:id="echoid-s13156" xml:space="preserve"> at <lb/>ratio i m ad m t, eſt ratio i a ad a t per fabricationẽ.</s> <s xml:id="echoid-s13157" xml:space="preserve"> Qua-<lb/> <anchor type="figure" xlink:label="fig-0198-01a" xlink:href="fig-0198-01"/> re per 11 p 5 ratio i m ad m h maior eſt, quàm i a ad a t] <lb/>& ita maior, quàm i a ad a h.</s> <s xml:id="echoid-s13158" xml:space="preserve"> [Quoniam enim ratio i m <lb/>ad m h maior eſt, quàm i a ad a t è ſuperiore conclu-<lb/>ſione:</s> <s xml:id="echoid-s13159" xml:space="preserve"> ratio uerò i a ad a t maior eſt, quàm ad a h per <lb/>8 p 5:</s> <s xml:id="echoid-s13160" xml:space="preserve"> cum a t ſit maior ipſa a h per 9 ax.</s> <s xml:id="echoid-s13161" xml:space="preserve"> Ratio igitur i m <lb/>ad m h multò maior eſt, quàm ratio i a ad a h.</s> <s xml:id="echoid-s13162" xml:space="preserve">] Sit igi-<lb/>tur proportio il ad lh, ſicut i a ad a h:</s> <s xml:id="echoid-s13163" xml:space="preserve"> [per 10 p 6] ca-<lb/>det quidem linter m & i.</s> <s xml:id="echoid-s13164" xml:space="preserve"> Amplius à punctis l, m ducan-<lb/>tur contingentes l b, m g [per 17 p 3] & ducantur lineæ <lb/>i b, h b, i g, t g, a b, a g:</s> <s xml:id="echoid-s13165" xml:space="preserve"> quæ duæ ultimæ producantur uſ-<lb/>que ad exteriorem circulum:</s> <s xml:id="echoid-s13166" xml:space="preserve"> & habebitur ex quarto li-<lb/>bro, quòd angulus i b z ſit æqualis angulo h b a [conti-<lb/>nuata enim h b in x:</s> <s xml:id="echoid-s13167" xml:space="preserve"> æquabuntur anguli i b z & x b z <lb/>per 12 n 4:</s> <s xml:id="echoid-s13168" xml:space="preserve"> item x b z & h b a per 15 p 1:</s> <s xml:id="echoid-s13169" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s13170" xml:space="preserve"> per 1 ax.</s> <s xml:id="echoid-s13171" xml:space="preserve"> an-<lb/>guli i b z, h b a æquantur.</s> <s xml:id="echoid-s13172" xml:space="preserve">] Cum igitur ſit proportio il <lb/>ad l h, ſicut i a ad a h [per ſuperiorem fabricatio-<lb/>nem] erit [per 18 n 5] h locus imaginis i, dum reflecti-<lb/>tur à puncto b.</s> <s xml:id="echoid-s13173" xml:space="preserve"> Et ſi dicatur cõtrarium, & ſumatur alius <lb/>locus imaginis i:</s> <s xml:id="echoid-s13174" xml:space="preserve"> probabis per impoſsibile, ſumpta im-<lb/>poſsibilitate à proportione, quam non eſt uerum eſſe i <lb/>a ad lineam à puncto imaginis ductam ad punctum a, <lb/>ſicut i l ad lineam à puncto l ad locum imaginis.</s> <s xml:id="echoid-s13175" xml:space="preserve"> Cum <lb/>igitur h ſit locus imaginis:</s> <s xml:id="echoid-s13176" xml:space="preserve"> & l b contingat circulum in <lb/>b:</s> <s xml:id="echoid-s13177" xml:space="preserve"> producta a b faciet angulum l b z æqualem ſuo collaterali [a b l:</s> <s xml:id="echoid-s13178" xml:space="preserve"> quia uterq;</s> <s xml:id="echoid-s13179" xml:space="preserve"> per 18 p 3 rectus eſt.</s> <s xml:id="echoid-s13180" xml:space="preserve">] <lb/>Et quoniã l b perpendicularis ſuper a b z [per 18 p 3] reſtabit angulus i b l æ qualis angulo l b h.</s> <s xml:id="echoid-s13181" xml:space="preserve"> [Nam <lb/> <pb o="193" file="0199" n="199" rhead="OPTICAE LIBER VI."/> recti l b z, a b l æquantur per 10 ax:</s> <s xml:id="echoid-s13182" xml:space="preserve"> & i b z æqualis cõcluſus eſt ipſi h b a:</s> <s xml:id="echoid-s13183" xml:space="preserve"> reliquus igitur i b l æquatur <lb/>reliquo l b h.</s> <s xml:id="echoid-s13184" xml:space="preserve">] Eodẽ modo erit angulus i g z æqualis angulo t g a.</s> <s xml:id="echoid-s13185" xml:space="preserve"> [Quia enim m g tangit, & per fabri-<lb/>cationẽ eſt, ut i m ad m t, ſic i a ad a t:</s> <s xml:id="echoid-s13186" xml:space="preserve"> erit per 18 n 5 t locus imaginis pũcti i, reflexi à puncto ſpeculi g.</s> <s xml:id="echoid-s13187" xml:space="preserve"> <lb/>Quare cõtinuata t g in x:</s> <s xml:id="echoid-s13188" xml:space="preserve"> æquabũtur per 12 n 4 anguli i g z, x g z:</s> <s xml:id="echoid-s13189" xml:space="preserve"> & per 15 p 1 x g z, t g a:</s> <s xml:id="echoid-s13190" xml:space="preserve"> quare i g z, t g a <lb/>æquãtur.</s> <s xml:id="echoid-s13191" xml:space="preserve">] Et cũ m g ſit perpendicularis ſuper a g z:</s> <s xml:id="echoid-s13192" xml:space="preserve"> [per 18 p 3] erit angulus i g m æqualis angulo m g <lb/>t [quia enim anguli m g z, m g a per 18 p 3 recti ęquãtur per 10 ax:</s> <s xml:id="echoid-s13193" xml:space="preserve"> & i g z t g a æquales cõcluſi ſunt:</s> <s xml:id="echoid-s13194" xml:space="preserve"> reli <lb/>qui igitur i g m, t g m ęquabũtur.</s> <s xml:id="echoid-s13195" xml:space="preserve">] Amplius:</s> <s xml:id="echoid-s13196" xml:space="preserve"> ducatur à pũcto h ad lineã a b linea ęquidiſtãs i b [ք 31 p 1] <lb/>quę ſit h p:</s> <s xml:id="echoid-s13197" xml:space="preserve"> & à pũcto t æquidiſtãs i g ad lineã a g:</s> <s xml:id="echoid-s13198" xml:space="preserve"> quę ſit t r:</s> <s xml:id="echoid-s13199" xml:space="preserve"> erit [ք 29 p 1] angulus i b z æqualis angulo <lb/>h p b:</s> <s xml:id="echoid-s13200" xml:space="preserve"> Sed angulus i b z ęqualis angulo h b a, ut dictũ eſt:</s> <s xml:id="echoid-s13201" xml:space="preserve"> & ita duo anguli h b a, h p b ſũt ęquales.</s> <s xml:id="echoid-s13202" xml:space="preserve"> Qua <lb/>re [ք 6 p 1] duo latera h b, h p ſunt ęqualia:</s> <s xml:id="echoid-s13203" xml:space="preserve"> ſimiliter t r ęqualis t g.</s> <s xml:id="echoid-s13204" xml:space="preserve"> Verũ angulus h p b eſt acutus:</s> <s xml:id="echoid-s13205" xml:space="preserve"> cũ ſit <lb/>æqualis angulo i b z:</s> <s xml:id="echoid-s13206" xml:space="preserve"> [qui minor eſt recto l b z] erit igitur angulus h p a obtuſus:</s> <s xml:id="echoid-s13207" xml:space="preserve"> [ք 13 p 1] & erit [ք 19 <lb/>p 1] a h maior h p.</s> <s xml:id="echoid-s13208" xml:space="preserve"> Similiter erit t a maior t g.</s> <s xml:id="echoid-s13209" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s13210" xml:space="preserve"> quoniã h p æquidiſtat i b:</s> <s xml:id="echoid-s13211" xml:space="preserve"> erit [per 29 p 1.</s> <s xml:id="echoid-s13212" xml:space="preserve"> 4 p 6] <lb/>i a ad a h, ſicut a b ad a p:</s> <s xml:id="echoid-s13213" xml:space="preserve"> erit ſimiliter proportio i a ad a t, ſicut a g ad a r:</s> <s xml:id="echoid-s13214" xml:space="preserve"> & erit [per conſectariũ 4 p 5] <lb/>proportio a h ad i a, ſicut a p ad a b:</s> <s xml:id="echoid-s13215" xml:space="preserve"> ſed i a ad a t, ſicut a b ad a r (cum a b ſit æqualis a g) [per 15 d 1.</s> <s xml:id="echoid-s13216" xml:space="preserve">] <lb/>Igitur [per 22 p 5] erit proportio a h ad a t, ſicut a p ad a r.</s> <s xml:id="echoid-s13217" xml:space="preserve"> Verùm cum angulus h p a ſit obtuſus [ut <lb/>patuit] quadratum h a excedet quadratum h p & quadratum a p, multiplicatione a p in lineam du-<lb/>ctam à puncto p uſq;</s> <s xml:id="echoid-s13218" xml:space="preserve"> ad locum perpendicularis, ductæ à puncto h, bis [per 12 p 2.</s> <s xml:id="echoid-s13219" xml:space="preserve">] Sed perpendicu-<lb/>laris ducta à puncto h, cadet in medium lineæ p b:</s> <s xml:id="echoid-s13220" xml:space="preserve"> [non enim cadit extra puncta p, b:</s> <s xml:id="echoid-s13221" xml:space="preserve"> ſecus angulus <lb/>acutus eſſet maior recto per 16 p 1:</s> <s xml:id="echoid-s13222" xml:space="preserve"> cadit igitur inter puncta p, b, & in medium lineæ p b per 26 p 1] <lb/>cum h b, h p ſint æquales:</s> <s xml:id="echoid-s13223" xml:space="preserve"> & ita [per 1 p 2] quadratum h a excedet quadratum h p, & quadratum a p, <lb/>in multiplicatione a p in p b:</s> <s xml:id="echoid-s13224" xml:space="preserve"> & ita quadratum a h excedit quadratum h p in multiplicatione a b in <lb/>a p:</s> <s xml:id="echoid-s13225" xml:space="preserve"> quoniam [per 3 p 2] ductus a p in p b cum quadrato a p, ualet ductum a b in a p.</s> <s xml:id="echoid-s13226" xml:space="preserve"> Similiter qua-<lb/>dratum a t excedit quadratum tr, in ductu a g in a r, ſiue a b in a r:</s> <s xml:id="echoid-s13227" xml:space="preserve"> quod idem eſt.</s> <s xml:id="echoid-s13228" xml:space="preserve"> [æquales enim <lb/>ſunt a g, a b per 15 d 1.</s> <s xml:id="echoid-s13229" xml:space="preserve">] Ducatur igitur linea a b in duas lineas a p & a r, & prouenient duo exceſſus.</s> <s xml:id="echoid-s13230" xml:space="preserve"> <lb/>Igitur proportio exceſſus ad exceſſum, ſicut a p ad a r.</s> <s xml:id="echoid-s13231" xml:space="preserve"> [nam eadem altitudo a b multiplicans baſes <lb/>a p & a r, facit duo rectangula æquantia duos exceſſus, proportionalia baſibus per 1 p 6.</s> <s xml:id="echoid-s13232" xml:space="preserve">] Erit ergo <lb/>proportio exceſſus quadrati a h ſupra quadratum h p;</s> <s xml:id="echoid-s13233" xml:space="preserve"> ad exceſſum quadrati a t ſupra quadratum t r, <lb/>ſicut a h ad a t [patuit enim a p & a r proportionales eſſe ipſis a h & a t.</s> <s xml:id="echoid-s13234" xml:space="preserve">] Et cum h p ſit æqualis h b, <lb/>& t r, t g:</s> <s xml:id="echoid-s13235" xml:space="preserve"> erit [per 7 p 5] proportio exceſſus quadrati a h ſupra quadratum h b, ad exceſſum quadra-<lb/>ti a t ſupra quadratum t g, ſicut a h ad a t.</s> <s xml:id="echoid-s13236" xml:space="preserve"> Sed multiplicatio e h in h d eſt æqualis quadrato lineæ, à <lb/>puncto h ad circulum d b e contingenter ductæ:</s> <s xml:id="echoid-s13237" xml:space="preserve"> [per 36 p 3] & erit [tangens] minor h b.</s> <s xml:id="echoid-s13238" xml:space="preserve"> [Quia enim <lb/>h b continuata ſecat peripheriam d b e:</s> <s xml:id="echoid-s13239" xml:space="preserve"> æquabitur oblongum comprehenſum ſub tota ſecante & <lb/>exteriore ſegmento, quadrato rectæ ab eodem puncto h peripheriam tangentis per 36 p 3.</s> <s xml:id="echoid-s13240" xml:space="preserve"> Itaque <lb/>per 17 p 6 ut exterius ſegmentum ad tangentem, ſic tangens ad totam ſecantem:</s> <s xml:id="echoid-s13241" xml:space="preserve"> at per 8 p 3 exte-<lb/>rius ſegmentum minus eſt tangente:</s> <s xml:id="echoid-s13242" xml:space="preserve"> quare tangens minor eſt ſecante] & ita multiplicatio e h in h d <lb/>minor eſt quadrato h b.</s> <s xml:id="echoid-s13243" xml:space="preserve"> Et fiat ductus a h in h u æqualis quadrato h b [ut oſtenſum eſt 32 n 5.</s> <s xml:id="echoid-s13244" xml:space="preserve">] Er-<lb/>go h u minor eſt h a.</s> <s xml:id="echoid-s13245" xml:space="preserve"> [Quia enim oblongum comprehenſum ſub h a & & h u æquatum eſt quadrato <lb/>h b:</s> <s xml:id="echoid-s13246" xml:space="preserve"> erit per 17 p 6, ut h a ad h b, ſic h b ad h u:</s> <s xml:id="echoid-s13247" xml:space="preserve"> at h a maior eſt h b, ut patuit:</s> <s xml:id="echoid-s13248" xml:space="preserve"> ergo h b maior eſt h u:</s> <s xml:id="echoid-s13249" xml:space="preserve"> qua-<lb/>re h a multò maior eſt h u] & quadratum a h eſt ęquale multiplicationi a h in a u & h u:</s> <s xml:id="echoid-s13250" xml:space="preserve"> [per 2 p 2.</s> <s xml:id="echoid-s13251" xml:space="preserve">] Igi <lb/>tur multiplicatio a h in a u erit exceſſus quadrati h a, ſupra quadratum h b.</s> <s xml:id="echoid-s13252" xml:space="preserve"> Igitur proportio a h ad a <lb/>t, ſicut proportio multiplicationis a h in a u, ad exceſſum quadrati a t, ſupra quadratum t g.</s> <s xml:id="echoid-s13253" xml:space="preserve"> Et ſi duæ <lb/>lineæ a h, a t ducantur in a u:</s> <s xml:id="echoid-s13254" xml:space="preserve"> erit proportio a h ad a t, ſicut proportio multiplicationis a h in a u, ad <lb/>multiplicationem a t in a u [per 1 p 6:</s> <s xml:id="echoid-s13255" xml:space="preserve"> quia eadem altitudo a u multiplicat baſes a h & h t.</s> <s xml:id="echoid-s13256" xml:space="preserve">] Igitur mul <lb/>tiplicatio a t in a u, eſt exceſſus quadrati a t ſupra quadratum t g:</s> <s xml:id="echoid-s13257" xml:space="preserve"> erit ergo multiplicatio h a in h u, æ-<lb/>qualis quadrato h b:</s> <s xml:id="echoid-s13258" xml:space="preserve"> & multiplicatio a t in t u æqualis quadrato t g.</s> <s xml:id="echoid-s13259" xml:space="preserve"> [Quia enim per 2 p 2 quadra-<lb/>tum a t æquatur oblongis comprehenſis ſub a t & t u, item ſub a t & a u:</s> <s xml:id="echoid-s13260" xml:space="preserve"> & oblongũ comprehenſum <lb/>ſub a t & a u, æquatur exuperantiæ quadrati a t ſupra quadratum t g per proximam concluſionẽ:</s> <s xml:id="echoid-s13261" xml:space="preserve"> re-<lb/>liquum igitur oblongum comprehenſum ſub a t & t u æquatur quadrato t g.</s> <s xml:id="echoid-s13262" xml:space="preserve">] Amplius:</s> <s xml:id="echoid-s13263" xml:space="preserve"> arcus b g di <lb/>uidatur per æqualia in puncto o [per 30 p 3] & ducatur a o:</s> <s xml:id="echoid-s13264" xml:space="preserve"> & [per 12 p 1] ducantur tres perpendicula <lb/>res ſuper lineam h a:</s> <s xml:id="echoid-s13265" xml:space="preserve"> ſcilicet b f, o y, g k:</s> <s xml:id="echoid-s13266" xml:space="preserve"> & [per 31 p 1] à puncto g ducatur æquidiſtans h a:</s> <s xml:id="echoid-s13267" xml:space="preserve"> quę ſit g s:</s> <s xml:id="echoid-s13268" xml:space="preserve"> & <lb/>[per 11 p 1] à puncto b ducatur perpendicularis ſuper a g:</s> <s xml:id="echoid-s13269" xml:space="preserve"> quæ ſit b c:</s> <s xml:id="echoid-s13270" xml:space="preserve"> hæc quidem b c, ſi produceretur <lb/>uſq;</s> <s xml:id="echoid-s13271" xml:space="preserve"> ad circulum [id eſt peripheriam circuli d b e] diuideret linea a g ipſam per æqualia [per 3 p 3] & <lb/>arcum, cuius eſſet chorda:</s> <s xml:id="echoid-s13272" xml:space="preserve"> & ita ſecaretur alius arcus, ęqualis arcui b g:</s> <s xml:id="echoid-s13273" xml:space="preserve"> quoniam illum arcum reſpi-<lb/>ceret angulus c b g:</s> <s xml:id="echoid-s13274" xml:space="preserve"> & ita angulus c b g eſt medietas anguli ſuper centrũ reſpicientis eundẽ arcũ, fe-<lb/>cundũ Euclidẽ [20 p 3.</s> <s xml:id="echoid-s13275" xml:space="preserve">] Igitur angulus c b g eſt medietas anguli g a b, [æquatur enim angulo ſubten <lb/>denti peripheriã æqualem ipſi b g per 27 p 3] quẽ diuidit linea a o per ęqualia.</s> <s xml:id="echoid-s13276" xml:space="preserve"> Igitur angulus c b g eſt <lb/>æqualis angulo o a g:</s> <s xml:id="echoid-s13277" xml:space="preserve"> Duo autem anguli b s g, b c g recti ſunt.</s> <s xml:id="echoid-s13278" xml:space="preserve"> Si igitur intelligatur circulus ſuper b g <lb/>tranſiens per s, tranſibit per c[per conuerſionẽ 31 p 3 demonſtratam à Theone in cõmentarijs in 3 li-<lb/>brum magnę cõſtructionis Ptolemei] & fiet arcus s c, ſuper quẽ cadent duo anguli c b s, c g s:</s> <s xml:id="echoid-s13279" xml:space="preserve"> igitur <lb/>[per 27 p 3] hi duo anguli ſunt æquales.</s> <s xml:id="echoid-s13280" xml:space="preserve"> Sed angulus g a y æqualis eſt angulo c g s [per 29 p 1] propter <lb/>æquidiſtantiá linearũ:</s> <s xml:id="echoid-s13281" xml:space="preserve"> [g s & y a] & ita angulus g a y æqualis angulo c b s.</s> <s xml:id="echoid-s13282" xml:space="preserve"> Et, ut dictũ eſt, angulus g b <lb/>c ęqualis angulo o a g:</s> <s xml:id="echoid-s13283" xml:space="preserve"> erit angulus o a y æqualis angulo g b s:</s> <s xml:id="echoid-s13284" xml:space="preserve"> & erit triangulũ o a y ſimile triangulo <lb/>g b s.</s> <s xml:id="echoid-s13285" xml:space="preserve"> Igitur proportio g b ad b s, ſicut o a ad a y, & proportio g b ad g s, ſicut o a ad o y.</s> <s xml:id="echoid-s13286" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s13287" xml:space="preserve"> cum <lb/>angulus a h b ſit acutus [ut oſtenſum eſt 60 n 5] quadratũ a b minus eſt quadratis a h, h b, quantũ eſt <lb/> <pb o="194" file="0200" n="200" rhead="ALHAZEN"/> illud, quod fit ex ductu a h in h f bis, ſecundũ quod dicit Euclides [13 p 2.</s> <s xml:id="echoid-s13288" xml:space="preserve">] Igitur quadratũ a h cũ qua <lb/>drato h b, ſuperat quadratum a d quæ eſt æqualis a b) in ductu a h in h f bis:</s> <s xml:id="echoid-s13289" xml:space="preserve"> & ita [per 1 p 2] in ductu <lb/>a h in h d bis, & a h in d f bis:</s> <s xml:id="echoid-s13290" xml:space="preserve"> Sed [per 7 p 2] multiplicatio a h in h d bis, cum quadrato a d, eſt æqua-<lb/>lis quadrato a h cum quadrato h d:</s> <s xml:id="echoid-s13291" xml:space="preserve"> & ita ablato cõmuni quadrato a d, cũ ductu a h in h d bis:</s> <s xml:id="echoid-s13292" xml:space="preserve"> reſtabit <lb/>quadratũ h d cũ ductu a h in f d bis, æquale quadrato h b.</s> <s xml:id="echoid-s13293" xml:space="preserve"> Sed [per fabricationẽ] multiplicatio a h in <lb/>h t æqualis eſt quadrato h d:</s> <s xml:id="echoid-s13294" xml:space="preserve"> & multiplicatio a h in h u, æqualis quadrato h b:</s> <s xml:id="echoid-s13295" xml:space="preserve"> erit ergo multiplicatio <lb/>a h in h u, æqualis multiplicationi a h in h t, & multiplicationi a h in d f bis, ſubtractoq́;</s> <s xml:id="echoid-s13296" xml:space="preserve"> ductu a h in <gap/> <lb/>t(quẽ communẽ ponimus utriq;</s> <s xml:id="echoid-s13297" xml:space="preserve"> multiplicationi.</s> <s xml:id="echoid-s13298" xml:space="preserve">) [Quia enim oblonga cõprehenſa ſub a h t & ſub <lb/>a h & t u, æquãtur oblongo cõprehenſo ſub a h u perip 2:</s> <s xml:id="echoid-s13299" xml:space="preserve"> ergo æquãtur oblõgis cõprehenſis ſub a h <lb/>t & ſub a h & d f bis:</s> <s xml:id="echoid-s13300" xml:space="preserve"> cõmune igitur eſt oblongũ cõprehenſum ſub a h t] reſtabit multiplicatio a h in <lb/>t u ęqualis multiplicationi a h in d f bis.</s> <s xml:id="echoid-s13301" xml:space="preserve"> Igitur t u eſt dupla d f:</s> <s xml:id="echoid-s13302" xml:space="preserve"> [Quia enim oblongũ comprehenſum <lb/>ſub altitudine a h & baſi t u, æquatur duplici oblongo, comprehenſo ſub eadem altitudine & baſi d <lb/>f:</s> <s xml:id="echoid-s13303" xml:space="preserve">erit per 1 p 6 baſis t u dupla baſis d f.</s> <s xml:id="echoid-s13304" xml:space="preserve">] Amplius:</s> <s xml:id="echoid-s13305" xml:space="preserve"> cũ angulus a t g ſit acutus [ut oſtenſũ eſt 60 n 5] erit <lb/>ſecundũ prædictũ modũ, quadratũ a t cum quadrato t g, æquale quadrato a d, cũ ductu a t in t k bis:</s> <s xml:id="echoid-s13306" xml:space="preserve"> <lb/>& ita [per 1 p 2] cũ ductu a t in t d bis, & in d k bis.</s> <s xml:id="echoid-s13307" xml:space="preserve"> Et probabitur modo prædicto, quòd quadratũ t g <lb/>æquale eſt quadrato t d, cũ ductu a t in d k bis:</s> <s xml:id="echoid-s13308" xml:space="preserve"> ſed ductus a t in t u, æqualis eſt quadrato t g [excõclu <lb/>ſo] & ita æqualis quadrato t d, cũ ductu a t in d k bis.</s> <s xml:id="echoid-s13309" xml:space="preserve"> Sit aũt ductus a t in t æ æqualis quadrato t d [ut <lb/>oſtenſũ eſt in principio huius numeri] reſtat ergo, ut ductus a t in æ u, ſit ęqualis ductui a t in d k bis, <lb/>per ablationẽ cõmunis, qui eſt ductus a t in t æ [nam oblonga cõprehenſa ſub a t æ, itẽ ſub a t & æ u, <lb/>æquãtur oblongo cõprehenſo ſub a t u per 1 p 2:</s> <s xml:id="echoid-s13310" xml:space="preserve"> ergo æquãtur oblongis cõprehenſis ſub a t æ ſemel, <lb/>& ſub a t & d k bis.</s> <s xml:id="echoid-s13311" xml:space="preserve"> Cõmune igitur eſt a t æ, quo ſublato:</s> <s xml:id="echoid-s13312" xml:space="preserve"> reliquũ oblongũ coprehenſum ſub at & æ u <lb/>æquatur oblongo ſub a t & d k bis cõprehenſo.</s> <s xml:id="echoid-s13313" xml:space="preserve">] Igitur æ u eſt dupla k d [per 1 p 6] ſed iam dictũ eſt, <lb/>quòd t u eſt dupla d f:</s> <s xml:id="echoid-s13314" xml:space="preserve"> reſtat ergo t æ dupla k f.</s> <s xml:id="echoid-s13315" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s13316" xml:space="preserve"> proportio a h ad h t eſt, ſicut a h ad h d dupli <lb/>cata [per 10 d 5] h d enim media eſt in proportione interillas:</s> <s xml:id="echoid-s13317" xml:space="preserve"> cũ eius quadratũ ſit æquale ductui a h <lb/>in h t [per fabricationẽ.</s> <s xml:id="echoid-s13318" xml:space="preserve">] Et ſimiliter proportio a t ad t æ, ſicut a t ad t d duplicata [eſt enim ex ſabri-<lb/>catione & 17 p 6 a t ad t d, ſicut t d ad t æ.</s> <s xml:id="echoid-s13319" xml:space="preserve">] Sed maior eſt proportio a t ad t d, quàm a h ad h d.</s> <s xml:id="echoid-s13320" xml:space="preserve"> [Quia <lb/>enim h t minor eſt quinta parte h d, ut patuit:</s> <s xml:id="echoid-s13321" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s13322" xml:space="preserve"> ſi a t, uerbi gratia, ipſam t d quater contineat:</s> <s xml:id="echoid-s13323" xml:space="preserve"> a h <lb/>eandem t d quater continebit, & h d ſemel.</s> <s xml:id="echoid-s13324" xml:space="preserve"> Quare a h nõ continebit h d quater.</s> <s xml:id="echoid-s13325" xml:space="preserve"> Ratio igitur a t ad t d <lb/>maior eſt, quàm a h ad h d.</s> <s xml:id="echoid-s13326" xml:space="preserve">] Et cum a h ſit maior a t:</s> <s xml:id="echoid-s13327" xml:space="preserve"> [per 9 ax:</s> <s xml:id="echoid-s13328" xml:space="preserve">] erit h t maior t æ [quia enim a h maior <lb/>eſt a t:</s> <s xml:id="echoid-s13329" xml:space="preserve"> erit ք 8 p 5 ratio a h ad t æ maior, quàm a t ad t æ:</s> <s xml:id="echoid-s13330" xml:space="preserve"> ſed ratio a t ad t æ maior eſt, quàm a h ad h t.</s> <s xml:id="echoid-s13331" xml:space="preserve"> <lb/>Ergo per 11 p 5 ratio a h ad t æ maior eſt, quàm a h ad h t.</s> <s xml:id="echoid-s13332" xml:space="preserve"> Quare ք 10 p 5 h t maior eſt t æ.</s> <s xml:id="echoid-s13333" xml:space="preserve">] Sed t æ du-<lb/>pla ad k f:</s> <s xml:id="echoid-s13334" xml:space="preserve"> ergo h t maior eſt, quàm dupla ad k f.</s> <s xml:id="echoid-s13335" xml:space="preserve"> Item.</s> <s xml:id="echoid-s13336" xml:space="preserve"> Vt dictũ eſt, proportio b g ad g s, ſicut o a ad o y, <lb/>erit [per 16 p 5] b g ad o a, ſicut g s ad o y:</s> <s xml:id="echoid-s13337" xml:space="preserve"> ſed o a ęqualis b a [per 15 d 1] & g s ęqualis f k [per 34 p 1] pro-<lb/>pter ęquidiſtantiã:</s> <s xml:id="echoid-s13338" xml:space="preserve"> erit [per 7 p 5] proportio b g ad b a, ſicut f k ad o y.</s> <s xml:id="echoid-s13339" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s13340" xml:space="preserve"> quia i h minor eſt me-<lb/>dietate o h [ut patuit] & o h tripla th:</s> <s xml:id="echoid-s13341" xml:space="preserve"> eriti h minor h t, & medietate ipſius:</s> <s xml:id="echoid-s13342" xml:space="preserve"> ſed h t minor quinta parte <lb/>h d.</s> <s xml:id="echoid-s13343" xml:space="preserve"> Igitur i h minor eſt t d:</s> <s xml:id="echoid-s13344" xml:space="preserve"> quare i h multò minor n d:</s> <s xml:id="echoid-s13345" xml:space="preserve"> quare m i multò minor n d [quia m i minor eſt <lb/>i h, quæ minor eſt n d.</s> <s xml:id="echoid-s13346" xml:space="preserve">] Et palàm per hoc, quòd i cadit inter h & z.</s> <s xml:id="echoid-s13347" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s13348" xml:space="preserve"> quod fit ex ductu e z in z <lb/>d, eſt æquale quadrato a d:</s> <s xml:id="echoid-s13349" xml:space="preserve"> [per theſin] igitur quod fit ex ductu e m in m d, eſt minus quadrato a d.</s> <s xml:id="echoid-s13350" xml:space="preserve"> <lb/>Sed quoniam m g circulum d b e cõtingit, quod fit ex ductu e m in m d, eſt æquale quadrato m g, ſe-<lb/>cundũ quod dicit Euclides [36 p 3.</s> <s xml:id="echoid-s13351" xml:space="preserve">] Igitur m g eſt minor a d:</s> <s xml:id="echoid-s13352" xml:space="preserve"> igitur minor eſt a g.</s> <s xml:id="echoid-s13353" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s13354" xml:space="preserve"> triangula <lb/>a g m, m g k habent unum angulũ communem [a d m] & utrunq;</s> <s xml:id="echoid-s13355" xml:space="preserve"> eorum habet unũ angulum rectũ <lb/>[ad g & k.</s> <s xml:id="echoid-s13356" xml:space="preserve">] Igitur [per 32 p 1.</s> <s xml:id="echoid-s13357" xml:space="preserve"> 4 p.</s> <s xml:id="echoid-s13358" xml:space="preserve"> 1 d 6] ſunt ſimilia.</s> <s xml:id="echoid-s13359" xml:space="preserve"> Quare proportio m k ad k g, ſicut m g ad g a:</s> <s xml:id="echoid-s13360" xml:space="preserve"> & ita <lb/>m k minor eſt k g [eſt enim m g minor g a ex concluſo.</s> <s xml:id="echoid-s13361" xml:space="preserve">] Et cum [per 15 p 3] o y ſit maior g k:</s> <s xml:id="echoid-s13362" xml:space="preserve"> erit h d <lb/>minor o y [quia h d minor eſt m k, & m k minor k g, & k g minor o y.</s> <s xml:id="echoid-s13363" xml:space="preserve">] Amplius:</s> <s xml:id="echoid-s13364" xml:space="preserve"> quia a h ad h d, ſicut h <lb/>d ad h t:</s> <s xml:id="echoid-s13365" xml:space="preserve"> [per theſin & 17 p 6] erit ſic [per 15 p 5] medietas h d ad medietatem h t:</s> <s xml:id="echoid-s13366" xml:space="preserve"> & ita a h ad h d, ſicut <lb/>q h ad medietatem h t:</s> <s xml:id="echoid-s13367" xml:space="preserve"> cum q h ſit medietas h d:</s> <s xml:id="echoid-s13368" xml:space="preserve"> [per fabricationem] & ita a h ad q h, ſicut h d ad <lb/>medietatem h t:</s> <s xml:id="echoid-s13369" xml:space="preserve"> & ita [per conſectarium 4 p 5] q h ad a h, ſicut medietas h t ad h d.</s> <s xml:id="echoid-s13370" xml:space="preserve"> Sed medietas h t <lb/>maior eſt f k [demonſtratũ enim eſt ipſam h t maiorẽ eſſe, quàm duplam ipſius k f] & h d minor o y.</s> <s xml:id="echoid-s13371" xml:space="preserve"> <lb/>Erit igitur proportio medietatis h t ad h d maior, quàm f k ad o y [ut conſtat ex 8 p 5.</s> <s xml:id="echoid-s13372" xml:space="preserve">] Quare [per 11 p <lb/>5] erit proportio q h ad a h maior, quàm f k ad o y.</s> <s xml:id="echoid-s13373" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s13374" xml:space="preserve"> linea a q ſecat circulum e b d:</s> <s xml:id="echoid-s13375" xml:space="preserve"> ſit punctũ <lb/>ſectiõis œ:</s> <s xml:id="echoid-s13376" xml:space="preserve"> & ducatur linea d œ:</s> <s xml:id="echoid-s13377" xml:space="preserve"> quę erit æquidiſtãs q h:</s> <s xml:id="echoid-s13378" xml:space="preserve"> [Quia enim tota a h æquatur toti a q, & pars <lb/>a d parti a œ per 15 d 1:</s> <s xml:id="echoid-s13379" xml:space="preserve"> reliqua igitur d h ęquatur reliquę œ q:</s> <s xml:id="echoid-s13380" xml:space="preserve"> quare per 7 p 5, ut a d ad d h, ſic a œ ad œ <lb/>q.</s> <s xml:id="echoid-s13381" xml:space="preserve"> Ita q;</s> <s xml:id="echoid-s13382" xml:space="preserve"> per 2 p 6 œ d parallela eſt ipſi q h] eritq́;</s> <s xml:id="echoid-s13383" xml:space="preserve"> per 29 p 1.</s> <s xml:id="echoid-s13384" xml:space="preserve"> 4 p 6 proportio q h ad h a, ſicut œ ad d a:</s> <s xml:id="echoid-s13385" xml:space="preserve"> & <lb/>ita proportio œ ad d a maior, quàm f k ad o y.</s> <s xml:id="echoid-s13386" xml:space="preserve"> Sed fk ad o y, ſicut g b ad b a [ex concluſo.</s> <s xml:id="echoid-s13387" xml:space="preserve">] Erit igitur <lb/>maior proportio œ d ad d a, quàm b g ad b a [id eſt ad d a:</s> <s xml:id="echoid-s13388" xml:space="preserve"> æquales enim ſunt d a & b a per 15 d 1] & ita <lb/>œ d maior b g:</s> <s xml:id="echoid-s13389" xml:space="preserve"> [per 10 p 5] & arcus œ d maior arcu g b [per 28 p 3.</s> <s xml:id="echoid-s13390" xml:space="preserve">] Amplius:</s> <s xml:id="echoid-s13391" xml:space="preserve"> producatur a q uſq;</s> <s xml:id="echoid-s13392" xml:space="preserve"> ad <lb/>punctũ s, ut ſit a s æqualis a i:</s> <s xml:id="echoid-s13393" xml:space="preserve"> [per 3 p 1] & ducatur linea s i:</s> <s xml:id="echoid-s13394" xml:space="preserve"> quę erit æquidiſtãs q h:</s> <s xml:id="echoid-s13395" xml:space="preserve"> [eodẽ argumẽto, <lb/>quo œ d parallela cõcluſa eſt ipſi q h] & erit [per 29 p 1.</s> <s xml:id="echoid-s13396" xml:space="preserve"> 4 p 6] si ad q h, ſicut i a ad h a.</s> <s xml:id="echoid-s13397" xml:space="preserve"> Sed ſuprà poſi-<lb/>tũ eſt, quòd i a ad a h, ſicut t q ad q h:</s> <s xml:id="echoid-s13398" xml:space="preserve"> erit igitur [per 9 p 5] si æqualis t q.</s> <s xml:id="echoid-s13399" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s13400" xml:space="preserve"> mutetur figura ad <lb/>euitandam linearũ intricationẽ multiplicẽ, & propter defectũ literarũ ad diſtinctionẽ linearũ.</s> <s xml:id="echoid-s13401" xml:space="preserve"> Cum <lb/>ergo i a ſit æqualis lineę, quã diximus a s:</s> <s xml:id="echoid-s13402" xml:space="preserve"> fiat circulus ſecundũ quantitatẽ ipſarũ, & loco s ponatur li <lb/>tera n:</s> <s xml:id="echoid-s13403" xml:space="preserve"> & producantur a g i ab uſq;</s> <s xml:id="echoid-s13404" xml:space="preserve"> ad circulũ hunc:</s> <s xml:id="echoid-s13405" xml:space="preserve"> & ſint a b c, a g r:</s> <s xml:id="echoid-s13406" xml:space="preserve"> & loco literæ œ ponamus f.</s> <s xml:id="echoid-s13407" xml:space="preserve"> Di-<lb/>ctum eſt, quòd arcus d f maior eſt arcu b g:</s> <s xml:id="echoid-s13408" xml:space="preserve"> ſit arcus b m æqualis arcui d f:</s> <s xml:id="echoid-s13409" xml:space="preserve"> [fiet uerò æqualis, ſiad re-<lb/>ctam a b eiusq́;</s> <s xml:id="echoid-s13410" xml:space="preserve"> punctũ a cõſtituatur per 23 p 1 angulus b a m æqualis angulo d a f:</s> <s xml:id="echoid-s13411" xml:space="preserve"> ſic enim per 33 p 6 <lb/> <pb o="195" file="0201" n="201" rhead="OPTICAE LIBER VI."/> peripheriæ b m & d f æquabũtur] & ducatur linea a m u:</s> <s xml:id="echoid-s13412" xml:space="preserve"> & lineę i b, i g, i m, n m:</s> <s xml:id="echoid-s13413" xml:space="preserve"> & linea q m:</s> <s xml:id="echoid-s13414" xml:space="preserve"> quę pro-<lb/>ducatur uſq;</s> <s xml:id="echoid-s13415" xml:space="preserve"> ad exteriorẽ circulũ:</s> <s xml:id="echoid-s13416" xml:space="preserve"> & cadat in punctũ z:</s> <s xml:id="echoid-s13417" xml:space="preserve"> & ducantur lineę z a, z g.</s> <s xml:id="echoid-s13418" xml:space="preserve"> Cum aũt arcus b m <lb/>ſit æqualis arcui d f:</s> <s xml:id="echoid-s13419" xml:space="preserve"> addito cõmuni:</s> <s xml:id="echoid-s13420" xml:space="preserve"> [m d] erit arcus m f æqualis arcui d b:</s> <s xml:id="echoid-s13421" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s13422" xml:space="preserve"> [per 27 p 3] angulus <lb/>n a m æqualis angulo i a b, & latera lateribus æqualia [per 15 d 1] erit [per 4 p 1] m n æqualis i b:</s> <s xml:id="echoid-s13423" xml:space="preserve"> & an-<lb/>gulus n m a æqualis angulo i b a:</s> <s xml:id="echoid-s13424" xml:space="preserve"> & [per 13 <lb/> <anchor type="figure" xlink:label="fig-0201-01a" xlink:href="fig-0201-01"/> p 1] angulus n m u angulo i b c.</s> <s xml:id="echoid-s13425" xml:space="preserve"> Et cũ poſi <lb/>ta ſit ſuprà [in ſecunda figura] a q æqualis <lb/>a h:</s> <s xml:id="echoid-s13426" xml:space="preserve"> erunt a q, a m latera æqualia a h, a b:</s> <s xml:id="echoid-s13427" xml:space="preserve"> & <lb/>angulus [q a m] angulo [h a b per proxi-<lb/>mam cõcluſionẽ] erit [per 4 p 1] q m ęqua <lb/>lis h b:</s> <s xml:id="echoid-s13428" xml:space="preserve"> & erit angulus q m a æqualis h b a, <lb/>& q m n æqualis angulo h b i:</s> <s xml:id="echoid-s13429" xml:space="preserve"> [per 8 p 1] <lb/>quoniã duo eius latera duobus illius æ-<lb/>qualia:</s> <s xml:id="echoid-s13430" xml:space="preserve"> [nam m n æqualis cõcluſa eſt ipſi <lb/>i b, & q m ipſi h b] & baſis, quæ eſt q n, eſt <lb/>æqualis baſi h i:</s> <s xml:id="echoid-s13431" xml:space="preserve"> [nam a n, a i æquãtur per <lb/>15 d 1:</s> <s xml:id="echoid-s13432" xml:space="preserve"> itẽ a q, a h per theſin:</s> <s xml:id="echoid-s13433" xml:space="preserve"> reliqua igitur <lb/>q n æquatur reliquæ h i] & angulus n m u <lb/>æqualis angulo i b c, & i b c æqualis angu <lb/>lo h b a:</s> <s xml:id="echoid-s13434" xml:space="preserve"> [ut oſtenſum eſt in ſecũda figura:</s> <s xml:id="echoid-s13435" xml:space="preserve"> <lb/>ubi angulus i b z eſt hic i b c] & angulus h <lb/>b a ęqualis angulo q m a:</s> <s xml:id="echoid-s13436" xml:space="preserve"> ergo n m u ęqua <lb/>lis q m a.</s> <s xml:id="echoid-s13437" xml:space="preserve"> Et quoniam, ut poſuimus, q m z <lb/>eſt linea recta:</s> <s xml:id="echoid-s13438" xml:space="preserve"> erit angulus q m a æqualis angulo u m z [ք 15 p 1:</s> <s xml:id="echoid-s13439" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s13440" xml:space="preserve"> anguli n m u, z m u æquãtur.</s> <s xml:id="echoid-s13441" xml:space="preserve"> <lb/>Quare punctũn reflectitur ad z à puncto m:</s> <s xml:id="echoid-s13442" xml:space="preserve"> [per 12 n 4] & locus imaginis ipſius q [per 3 n 5.</s> <s xml:id="echoid-s13443" xml:space="preserve">] Hocta-<lb/>men deeſt probationi, ut pateat m z totã eſſe extra circulũ:</s> <s xml:id="echoid-s13444" xml:space="preserve"> quod ſic patebit.</s> <s xml:id="echoid-s13445" xml:space="preserve"> Palàm, quod contingẽs <lb/>ducta à pũcto b cadat inter i & h:</s> <s xml:id="echoid-s13446" xml:space="preserve"> [demõſtratũ enim eſt in prima figura punctũ l alterũ terminũ rectę <lb/>tangentis peripheriã d b e in puncto b, cadere inter puncta i & h] & tanta eſt remotio puncti b à pun <lb/>cto h, quãta eſt puncti m à pũcto q[æquales enim cõcluſę ſunt h b, q m] & i h æqualis n q.</s> <s xml:id="echoid-s13447" xml:space="preserve"> igitur con-<lb/>tingẽs, ducta à puncto m cadet inter n & q.</s> <s xml:id="echoid-s13448" xml:space="preserve"> Igitur q m ſecat circulũ [quia tangẽte inferior eſt.</s> <s xml:id="echoid-s13449" xml:space="preserve">] Qua-<lb/>re tota m z eſt extra circulũ.</s> <s xml:id="echoid-s13450" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s13451" xml:space="preserve"> quoniã angulus n m u æ qualis eſt angulo u m z:</s> <s xml:id="echoid-s13452" xml:space="preserve"> erit arcus n u <lb/>æqualis arcui u z.</s> <s xml:id="echoid-s13453" xml:space="preserve"> [Quia enim m n ęquatur ipſi m z:</s> <s xml:id="echoid-s13454" xml:space="preserve"> cõnexę igitur n u & u z æquãtur per 4 p 1.</s> <s xml:id="echoid-s13455" xml:space="preserve"> Quare <lb/>per 28 p 3 peripherię n u, u z æquãtur] & erit angulus n a u æqualis angulo u a z [per 27 p 3.</s> <s xml:id="echoid-s13456" xml:space="preserve">] Sed iam <lb/>patuit, quòd angulus n a u æqualis eſt angulo i a c:</s> <s xml:id="echoid-s13457" xml:space="preserve"> igitur angulus i a c erit æqualis angulo u a z.</s> <s xml:id="echoid-s13458" xml:space="preserve"> An-<lb/>gulus uerò b a g aut erit æqualis angulo g a m:</s> <s xml:id="echoid-s13459" xml:space="preserve"> aut minor:</s> <s xml:id="echoid-s13460" xml:space="preserve"> aut maior.</s> <s xml:id="echoid-s13461" xml:space="preserve"> Sit æqualis.</s> <s xml:id="echoid-s13462" xml:space="preserve"> Si igitur ab angulo <lb/>i a c ſubtrahatur angulus b a g, & ab angulo z a u angulus m a g:</s> <s xml:id="echoid-s13463" xml:space="preserve"> remanebit angulus i a g æqualis an-<lb/>gulo z a g:</s> <s xml:id="echoid-s13464" xml:space="preserve"> & erit per 4 p 1 i g æqualis z g, & triangulũ triangulo:</s> <s xml:id="echoid-s13465" xml:space="preserve"> & erit angulus i g a æqualis angulo <lb/>z a g:</s> <s xml:id="echoid-s13466" xml:space="preserve"> reſtabit igitur [per 13 p 1] angulus i g r æqualis angulo z g r.</s> <s xml:id="echoid-s13467" xml:space="preserve"> Fiat igitur angulo i g r æqualis angu <lb/>lus t g a:</s> <s xml:id="echoid-s13468" xml:space="preserve"> per 23 p 1] erit angulus t g a æqualis angulo z g r.</s> <s xml:id="echoid-s13469" xml:space="preserve"> Si igitur t g producatur:</s> <s xml:id="echoid-s13470" xml:space="preserve"> ueniet ad z [ք con <lb/>uerſionẽ 15 p 1 à Proclo ibidẽ demonſtratã.</s> <s xml:id="echoid-s13471" xml:space="preserve">] Quare t g z linea recta [per 14 p 1.</s> <s xml:id="echoid-s13472" xml:space="preserve">] Igitur i à puncto g re <lb/>flectitur ad z:</s> <s xml:id="echoid-s13473" xml:space="preserve"> & locus imaginis eius eſt punctũ t.</s> <s xml:id="echoid-s13474" xml:space="preserve"> Si ergo z ſit uiſus:</s> <s xml:id="echoid-s13475" xml:space="preserve"> reflectẽtur ad ipſum duo pũctai, <lb/>n à duobus punctis m, g:</s> <s xml:id="echoid-s13476" xml:space="preserve"> & loca imaginũ puncta t, q.</s> <s xml:id="echoid-s13477" xml:space="preserve"> Igitur linea t q erit imago lineæ i n.</s> <s xml:id="echoid-s13478" xml:space="preserve"> Probatũ aũt <lb/>eſt ſuprà, quòd t q æqualis eſt i n.</s> <s xml:id="echoid-s13479" xml:space="preserve"> Et ita poteſt accidere in his ſpeculis imaginẽ eſſe æqualẽ rei uiſę.</s> <s xml:id="echoid-s13480" xml:space="preserve"> Si <lb/>uerò angulus b a g fuerit maior angulo <lb/> <anchor type="figure" xlink:label="fig-0201-02a" xlink:href="fig-0201-02"/> g a m:</s> <s xml:id="echoid-s13481" xml:space="preserve"> erit angulus z a g maior àngulo i <lb/>a g [mutua angulorum ſubductione, ut <lb/>prius facta.</s> <s xml:id="echoid-s13482" xml:space="preserve">] Sit angulus k a g æqualis <lb/>angulo i a g.</s> <s xml:id="echoid-s13483" xml:space="preserve"> Quonia pũctũ k demiſsius <lb/>puncto z, & punctũ m demiſsius pũcto <lb/>g:</s> <s xml:id="echoid-s13484" xml:space="preserve">linea k g ſecabit lineã z m:</s> <s xml:id="echoid-s13485" xml:space="preserve"> ſecet in pũ-<lb/>ctol.</s> <s xml:id="echoid-s13486" xml:space="preserve"> Igitur exiſtẽte uiſu in puncto l, re-<lb/>flectetur n ad ipſum à pũcto m:</s> <s xml:id="echoid-s13487" xml:space="preserve"> & locus <lb/>imaginis q.</s> <s xml:id="echoid-s13488" xml:space="preserve"> Similiter i reflectetur ad i-<lb/>pſum:</s> <s xml:id="echoid-s13489" xml:space="preserve"> & locus imaginis eſt t ſecundum <lb/>priorẽ probationẽ.</s> <s xml:id="echoid-s13490" xml:space="preserve"> Et ita t q imago eſt i <lb/>n.</s> <s xml:id="echoid-s13491" xml:space="preserve"> Quod eſt propoſitũ.</s> <s xml:id="echoid-s13492" xml:space="preserve"> Si uerò angulus <lb/>b a g f uerit minor angulo g a m:</s> <s xml:id="echoid-s13493" xml:space="preserve"> erit an <lb/>gulus z a g minor angulo i a g.</s> <s xml:id="echoid-s13494" xml:space="preserve"> Sit angu <lb/>lus o a g æqualis angulo i a g:</s> <s xml:id="echoid-s13495" xml:space="preserve"> & duca-<lb/>tur linea o g.</s> <s xml:id="echoid-s13496" xml:space="preserve"> Palàm, quòd i reflectitur <lb/>ad o à puncto g.</s> <s xml:id="echoid-s13497" xml:space="preserve"> Linea o g aut ſecabit li <lb/>neam z m q extra circulũ ſpeculi:</s> <s xml:id="echoid-s13498" xml:space="preserve"> aut nõ.</s> <s xml:id="echoid-s13499" xml:space="preserve"> Si ſecet extra, & uiſus fuerit inpuncto ſectionis:</s> <s xml:id="echoid-s13500" xml:space="preserve"> reflectẽtur <lb/>ad ipſum duo puncta n, i:</s> <s xml:id="echoid-s13501" xml:space="preserve"> & loca imaginũ erunt t q.</s> <s xml:id="echoid-s13502" xml:space="preserve"> Et ita redit propoſitũ [quod erat imaginẽ æquari <lb/>uiſibili.</s> <s xml:id="echoid-s13503" xml:space="preserve">] Si forſan linea o g ſecet lineã z m q intra circulũ:</s> <s xml:id="echoid-s13504" xml:space="preserve"> nõ poterit applicari prędicta probatio.</s> <s xml:id="echoid-s13505" xml:space="preserve"> Sed <lb/> <pb o="196" file="0202" n="202" rhead="ALHAZEN"/> dico, quòd extra hanc totalẽ ſuperficiem licebit inuenire punctũ, ad quod reflectãtur duo pũcta i, n <lb/>à duobus ſpeculi punctis:</s> <s xml:id="echoid-s13506" xml:space="preserve"> & imago erit t q.</s> <s xml:id="echoid-s13507" xml:space="preserve"> Verbi gratia.</s> <s xml:id="echoid-s13508" xml:space="preserve"> Palàm, quòd angulus n a z duplus eſt ad an-<lb/>gulũ c a b:</s> <s xml:id="echoid-s13509" xml:space="preserve"> [oſtenſum enim eſt peripherias n u & u z ęquari:</s> <s xml:id="echoid-s13510" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s13511" xml:space="preserve"> n z dupla eſt ipſius u z, & per 33 p 6 <lb/>angulus n a z duplus ad angulũ n a u, ideoq́;</s> <s xml:id="echoid-s13512" xml:space="preserve"> duplus ad æqualẽ i a b] & angulus i a o duplus ad angu <lb/>lum i a g, ſecundũ prædicta:</s> <s xml:id="echoid-s13513" xml:space="preserve"> [æquatus enim eſt o a g ipſi i a g:</s> <s xml:id="echoid-s13514" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s13515" xml:space="preserve"> totus i a o duplus eſt ad i a g] & an <lb/>gulus n a z nõ excedit angulũ i a o in angulo maiore angulo n a i.</s> <s xml:id="echoid-s13516" xml:space="preserve"> [Quia enim anguli n a z & i a o du-<lb/>pli ſunt angulorũ i a b & i a g:</s> <s xml:id="echoid-s13517" xml:space="preserve"> & i a b exuperat angulũ i a g, angulo g a b (qui per theſin minor eſt an-<lb/>gulo g a m) ergo angulus g a b minor eſt dimidiato angulo b a m (qui per 33 p 6 ęquatur angulo n a i, <lb/>ob peripherias f d & m b æquales) angulus igitur g a b minor eſt dimidiato angulo n a i.</s> <s xml:id="echoid-s13518" xml:space="preserve"> Quare angu <lb/>lus n a z exuperans angulũ i a o, duplo angulo g a b, nõ exuperat maiore angulo ꝗ̃ ſit n a i] & duo an-<lb/>guli i a o, i a n maiores tertio, qui eſt n a z:</s> <s xml:id="echoid-s13519" xml:space="preserve"> & duo z a n, n a i maiores tertio i a o:</s> <s xml:id="echoid-s13520" xml:space="preserve"> & duo n a z, i a o maio <lb/>res tertio n a i.</s> <s xml:id="echoid-s13521" xml:space="preserve"> Habemus ergo tres angulos [n a i, n a z, i a o] quorũ quilibet duo maiores ſunt tertio, <lb/>& oẽs ſimul quatuor rectis minores:</s> <s xml:id="echoid-s13522" xml:space="preserve"> [quia non totũ circa centrum a locum replent.</s> <s xml:id="echoid-s13523" xml:space="preserve">] Igitur [per 23 p <lb/>11] exillis licet facere angulũ corporalẽ.</s> <s xml:id="echoid-s13524" xml:space="preserve"> Fiat angulus ille ſuper a:</s> <s xml:id="echoid-s13525" xml:space="preserve"> & ſit linea s a erecta ſuper a:</s> <s xml:id="echoid-s13526" xml:space="preserve"> & angu <lb/>lus i a s ſit ęqualis angulo i a o:</s> <s xml:id="echoid-s13527" xml:space="preserve"> & angulus n a s ęqualis angulo n a z:</s> <s xml:id="echoid-s13528" xml:space="preserve"> angulus n a i manebit immotus:</s> <s xml:id="echoid-s13529" xml:space="preserve"> <lb/>& fiat linea a s æqualis lineæ a n uel a i:</s> <s xml:id="echoid-s13530" xml:space="preserve"> quæ oẽs ſunt æquales:</s> <s xml:id="echoid-s13531" xml:space="preserve"> & ꝓducãtur lineę t s, q s.</s> <s xml:id="echoid-s13532" xml:space="preserve"> Palàm, quo-<lb/>niã angulus t a s eſt æqualis angulo t a o [eſt enim t a pars lineæ i a [& duo latera [t a, & a o] lateribus <lb/>duob.</s> <s xml:id="echoid-s13533" xml:space="preserve"> [t a & a s] erit [per 4 p 1] baſis t s ęqualis baſi t o, & triangulũ triãgulo:</s> <s xml:id="echoid-s13534" xml:space="preserve"> & ita angulus g t a æqua <lb/>lis angulo s t a [ꝗ a g t pars eſt lineæ o t.</s> <s xml:id="echoid-s13535" xml:space="preserve">] Si-<lb/> <anchor type="figure" xlink:label="fig-0202-01a" xlink:href="fig-0202-01"/> militer angulus q a s ęqualis angulo q a z, & <lb/>latera [q a, a s] lateribus:</s> <s xml:id="echoid-s13536" xml:space="preserve"> [q a, a z] & [per 4 p <lb/>1] triangulũ æquale triangulo:</s> <s xml:id="echoid-s13537" xml:space="preserve"> & angulus m <lb/>q a æqualis angulo s q a [eſt enim m q pars <lb/>lineę z q.</s> <s xml:id="echoid-s13538" xml:space="preserve">] Diuidatur angulus t a s per æqua <lb/>lia per lineam a y [per 9 p 1.</s> <s xml:id="echoid-s13539" xml:space="preserve">] Sit y punctũ, in <lb/>quo linea illa ſecabit lineã t s.</s> <s xml:id="echoid-s13540" xml:space="preserve"> Palàm, cũ an-<lb/>gulus i a g ſit medietas angulii a o:</s> <s xml:id="echoid-s13541" xml:space="preserve"> erit an-<lb/>gulus t a g æqualis angulo t a y, & angulus <lb/>g t a æqualis y t a:</s> <s xml:id="echoid-s13542" xml:space="preserve"> & unũ latus cõmune, ſcili <lb/>cet t a:</s> <s xml:id="echoid-s13543" xml:space="preserve"> erit [per 26 p 1] t g æqualis t y, & trian <lb/>gulũ [y t a] triangulo:</s> <s xml:id="echoid-s13544" xml:space="preserve"> [g t a] & erit a y æqua <lb/>lis a g:</s> <s xml:id="echoid-s13545" xml:space="preserve"> & ita y in ſuperficie ſpeculi:</s> <s xml:id="echoid-s13546" xml:space="preserve"> [cũ enim <lb/>puncta g & y à centro a æquabiliter diſtent <lb/>per concluſionẽ proximam:</s> <s xml:id="echoid-s13547" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s13548" xml:space="preserve"> g ex theſi <lb/>in ſpeculi ſuperficie:</s> <s xml:id="echoid-s13549" xml:space="preserve"> erit y in eadẽ.</s> <s xml:id="echoid-s13550" xml:space="preserve">] Erit etiã <lb/>angulus i a g æqualis angulo i a y, & latera <lb/>[i a, a g] lateribus [i a, a y] & [per 4 p 1] trian-<lb/>gulum i a g triangulo [i a y] æquale:</s> <s xml:id="echoid-s13551" xml:space="preserve"> & erit angulus a g i ęqualis angulo a y i:</s> <s xml:id="echoid-s13552" xml:space="preserve"> & linea i y ꝓducta, æqua <lb/>lis i g.</s> <s xml:id="echoid-s13553" xml:space="preserve"> Et producatur a y extra ſphęrã uſq;</s> <s xml:id="echoid-s13554" xml:space="preserve"> ad punctũ p:</s> <s xml:id="echoid-s13555" xml:space="preserve"> reſtabit angulus i g r æqualis angulo i y p [ք 13 <lb/>p 1.</s> <s xml:id="echoid-s13556" xml:space="preserve">] Verũ cum t s ſit æqualis t o, & t y æqualis t g:</s> <s xml:id="echoid-s13557" xml:space="preserve"> [per cõcluſionẽ] reſtat g o æqualis y s.</s> <s xml:id="echoid-s13558" xml:space="preserve"> Igitur a y, y s <lb/>æqualia, a g, g o:</s> <s xml:id="echoid-s13559" xml:space="preserve"> & baſis a s æqualis baſi a o:</s> <s xml:id="echoid-s13560" xml:space="preserve"> erit [per 8 p 1] triangulũ [a y s] ęquale triangulo:</s> <s xml:id="echoid-s13561" xml:space="preserve"> [a g o] & <lb/>erit angulus a y s æqualis angulo a g o:</s> <s xml:id="echoid-s13562" xml:space="preserve"> reſtat [per 13 p 1] angulus s y p æqualis angulo o g r.</s> <s xml:id="echoid-s13563" xml:space="preserve"> Igitur duo <lb/>anguli i g r, o g r æquales ſunt duobus angulis i y p, s y p.</s> <s xml:id="echoid-s13564" xml:space="preserve"> Verùm linea a s ſecabit ſphęrã:</s> <s xml:id="echoid-s13565" xml:space="preserve"> ſit punctum <lb/>ſectionis e.</s> <s xml:id="echoid-s13566" xml:space="preserve"> Igitur tria pũcta e, y, d ſunt in ſuperficie ſphæræ.</s> <s xml:id="echoid-s13567" xml:space="preserve"> Quare linea e y d eſt pars circuli ſphærę:</s> <s xml:id="echoid-s13568" xml:space="preserve"> <lb/>& eſt linea comunis ſuperficiei ſphærę & ſuperficiei reflexionis t s p.</s> <s xml:id="echoid-s13569" xml:space="preserve"> Quare punctũ i reflectitur ad <lb/>punctũ s à puncto y:</s> <s xml:id="echoid-s13570" xml:space="preserve"> & locus imaginis eſt t.</s> <s xml:id="echoid-s13571" xml:space="preserve"> Similiter diuiſo angulo n a s per æqualia per ax:</s> <s xml:id="echoid-s13572" xml:space="preserve"> probabi <lb/>tur modo prædicto, quòd q x æqualis eſt q m, & a x æqualis a m, & x s æqualis m z:</s> <s xml:id="echoid-s13573" xml:space="preserve"> & duo anguli n x <lb/>æ & s x æ æquales duobus angulis n m u, z m u.</s> <s xml:id="echoid-s13574" xml:space="preserve"> Et ita n reflectetur ad s à puncto x:</s> <s xml:id="echoid-s13575" xml:space="preserve"> & locus imaginis <lb/>q:</s> <s xml:id="echoid-s13576" xml:space="preserve"> & ita t q imago i n:</s> <s xml:id="echoid-s13577" xml:space="preserve"> [& ſic imago, ut prius, erit æqualis uiſibili:</s> <s xml:id="echoid-s13578" xml:space="preserve"> cum t q æqualis concluſa ſit ipſi i n.</s> <s xml:id="echoid-s13579" xml:space="preserve">] <lb/>Quod eſt propoſitũ.</s> <s xml:id="echoid-s13580" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s13581" xml:space="preserve"> ſi à puncto i ducatur perpendicularis ſuper n a:</s> <s xml:id="echoid-s13582" xml:space="preserve"> cadet inter n & q, non <lb/>extra n:</s> <s xml:id="echoid-s13583" xml:space="preserve"> cũ angulus i n a ſit acutus:</s> <s xml:id="echoid-s13584" xml:space="preserve"> quoniá æqualis angulo n i a [ducta enim recta in, ęquabuntur an-<lb/>guli ad baſim i n per 5 p 1] & ſi caderet քpendicularis illa extra n:</s> <s xml:id="echoid-s13585" xml:space="preserve"> eſſet acutus maior recto [per 16 p 1.</s> <s xml:id="echoid-s13586" xml:space="preserve">] <lb/>Faciet ergo perpendicularis illa angulũ rectũ ſuper n q, quẽ angulũ reſpicit linea i n.</s> <s xml:id="echoid-s13587" xml:space="preserve"> Quare [ք 19 p 1] <lb/>linea in maior eſt illa perpendiculari.</s> <s xml:id="echoid-s13588" xml:space="preserve"> Quare perpendicularis illa minor t q [ęquali ipſi in per cõclu <lb/>ſionẽ.</s> <s xml:id="echoid-s13589" xml:space="preserve">] Punctũ igitur lineę n q, in quod cadit perpẽdicularis, reflectitur ad punctũ s:</s> <s xml:id="echoid-s13590" xml:space="preserve"> imago uerò eius <lb/>cadet in lineã n a [per 3 n 5] ſupra punctũ q.</s> <s xml:id="echoid-s13591" xml:space="preserve"> Quia quantò remotiora ſunt puncta, quę reflectũtur, tan <lb/>tò loca imaginũ magis accedunt ad centrũ circuli [per 30 n 5.</s> <s xml:id="echoid-s13592" xml:space="preserve">] Et quæcunq;</s> <s xml:id="echoid-s13593" xml:space="preserve"> linea ducetur à puncto t <lb/>[quod eſt imago puncti i, reflexi à puncto ſpeculi y] ad aliquod punctũ n q ſupra q:</s> <s xml:id="echoid-s13594" xml:space="preserve"> erit maior t q [per <lb/>19 p 1.</s> <s xml:id="echoid-s13595" xml:space="preserve">] Igitur imago perpendicularis erit maior ipſa perpendiculari.</s> <s xml:id="echoid-s13596" xml:space="preserve"> [Quia enim t q æquatur ipſi i n, <lb/>quę maior cõcluſa eſt perpendiculari:</s> <s xml:id="echoid-s13597" xml:space="preserve"> ergo t q imago perpendicularis eadẽ maior eſt.</s> <s xml:id="echoid-s13598" xml:space="preserve">] Eodẽ modo <lb/>quęcunq;</s> <s xml:id="echoid-s13599" xml:space="preserve"> linea ducetur à puncto i ad n q, inter hanc perpendicularẽ & in:</s> <s xml:id="echoid-s13600" xml:space="preserve"> erit imago ipſius maior i-<lb/>pſa.</s> <s xml:id="echoid-s13601" xml:space="preserve"> Verùm determinentur hęc certius.</s> <s xml:id="echoid-s13602" xml:space="preserve"> Punctũ n quia reflectitur ad z à puncto m:</s> <s xml:id="echoid-s13603" xml:space="preserve"> & locus imaginis <lb/>eſt q:</s> <s xml:id="echoid-s13604" xml:space="preserve"> linea z m q ſecat circulũ in puncto, quod eſt 3:</s> <s xml:id="echoid-s13605" xml:space="preserve"> cõtingens ergo ducta à pũcto z ad circulũ:</s> <s xml:id="echoid-s13606" xml:space="preserve"> cadet <lb/>ſuper punctũ aliquod arcus m 3 [ſi enim z m tangeret:</s> <s xml:id="echoid-s13607" xml:space="preserve"> angulus z m a eſſet rectus per 18 p 3:</s> <s xml:id="echoid-s13608" xml:space="preserve"> quare per <lb/> <pb o="197" file="0203" n="203" rhead="OPTICAE LIBER VI."/> 40 n 5 nulla fieret à pũcto m reflexio:</s> <s xml:id="echoid-s13609" xml:space="preserve"> multò igitur minus tangẽs à pũcto z, tanget citra punctũ m] ſi <lb/>uerò caderet in punctũ 3, ſecaret peripheriã, nõ tangeret:</s> <s xml:id="echoid-s13610" xml:space="preserve"> cadit igitur in peripheriá m 3.</s> <s xml:id="echoid-s13611" xml:space="preserve"> & contingẽs <lb/>illa cadet ſupra q:</s> <s xml:id="echoid-s13612" xml:space="preserve"> quoniá punctũ, in quod cadit, erit finis contingentiæ, & finis imaginũ:</s> <s xml:id="echoid-s13613" xml:space="preserve"> [per 17 n 5] <lb/>& puncta ſub puncto illo, quod eſt finis cõtingentię, nõ poterũt reflecti:</s> <s xml:id="echoid-s13614" xml:space="preserve"> ſuperiora uerò poterũt.</s> <s xml:id="echoid-s13615" xml:space="preserve"> Igi-<lb/>tur perpendicularis ducta à puncto i ſuper n q, ſi ceciderit ſupra punctũ, quod eſt finis cõtingentiæ:</s> <s xml:id="echoid-s13616" xml:space="preserve"> <lb/>reflectetur punctũ, in quod cadit:</s> <s xml:id="echoid-s13617" xml:space="preserve"> & erit imago perpendicularis maior perpendiculari.</s> <s xml:id="echoid-s13618" xml:space="preserve"> Siuerò per-<lb/>pendicularis cadat in punctũ contingentiæ, aut infra:</s> <s xml:id="echoid-s13619" xml:space="preserve"> non reflectetur punctũ, in quod cadit.</s> <s xml:id="echoid-s13620" xml:space="preserve"> Quare <lb/>nulla erit imago perpendicularis.</s> <s xml:id="echoid-s13621" xml:space="preserve"> Veruntamen quoniá finis cõtingentiæ eſt infra n:</s> <s xml:id="echoid-s13622" xml:space="preserve"> erunt inter finẽ <lb/>cõtingentiæ & n infinita pũcta:</s> <s xml:id="echoid-s13623" xml:space="preserve"> quorũ quodlibet reflectetur:</s> <s xml:id="echoid-s13624" xml:space="preserve"> & erit imago cuiuslibet ſuper n q:</s> <s xml:id="echoid-s13625" xml:space="preserve"> & cu <lb/>iuslibet lineę ductę à puncto i ad quodlibet illorũ punctorũ, erit imago maior linea, cuius fuerit ima <lb/>go.</s> <s xml:id="echoid-s13626" xml:space="preserve"> Igitur accidit in his ſpeculis imaginem aliquando æqualem rei uiſæ:</s> <s xml:id="echoid-s13627" xml:space="preserve"> aliquando maiorem eſſe.</s> <s xml:id="echoid-s13628" xml:space="preserve"> <lb/>Quod erat explanandum.</s> <s xml:id="echoid-s13629" xml:space="preserve"> Huius autem rei explanationem nec ſcriptam legimus, nec aliquem, qui <lb/>dixiſſet, aut intellexiſſet, audiuimus.</s> <s xml:id="echoid-s13630" xml:space="preserve"/> </p> <div xml:id="echoid-div461" type="float" level="0" n="0"> <figure xlink:label="fig-0197-01" xlink:href="fig-0197-01a"> <variables xml:id="echoid-variables147" xml:space="preserve">o z <gap/> l h m n q t d a b e</variables> </figure> <figure xlink:label="fig-0198-01" xlink:href="fig-0198-01a"> <variables xml:id="echoid-variables148" xml:space="preserve"><gap/> z i l <gap/> m h n t d z a <gap/> k g y c f b z r s u p a <gap/> e x</variables> </figure> <figure xlink:label="fig-0201-01" xlink:href="fig-0201-01a"> <variables xml:id="echoid-variables149" xml:space="preserve">i u r c z h <gap/> t m g b n q f a</variables> </figure> <figure xlink:label="fig-0201-02" xlink:href="fig-0201-02a"> <variables xml:id="echoid-variables150" xml:space="preserve">i u r k c z l b d t m g n q f a</variables> </figure> <figure xlink:label="fig-0202-01" xlink:href="fig-0202-01a"> <variables xml:id="echoid-variables151" xml:space="preserve">l u r c z o d t <gap/> m g b n k q f a s p x e <gap/> s</variables> </figure> </div> </div> <div xml:id="echoid-div463" type="section" level="0" n="0"> <head xml:id="echoid-head420" xml:space="preserve" style="it">7. Si duo uiſibilis pũcta à centro ſpeculi ſphærici cõuexi æquabiliter, à uiſu uerò inæquabiliter <lb/>diſtẽt: imago & finis cõtingẽtiæ pũcti lõginquioris à uiſu, erũt lõginquiores à cẽtro ſpeculi. 4 p 6.</head> <p> <s xml:id="echoid-s13631" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s13632" xml:space="preserve"> in his ſpeculis lineæ rectæ uidentur curuæ, & in pluribus curuitate quidẽ ſpeculũ nõ <lb/>reſpiciente, ſed ei aduerſa.</s> <s xml:id="echoid-s13633" xml:space="preserve"> Similiter curuæ apparebũt in his ſpeculis curuæ:</s> <s xml:id="echoid-s13634" xml:space="preserve"> & ſi curuitas ſpe-<lb/>culum reſpexerit, cõtrario ſitu apparebit.</s> <s xml:id="echoid-s13635" xml:space="preserve"> Et hoc quidẽ intelligendũ nõ in omnibus, ſed in pluribus.</s> <s xml:id="echoid-s13636" xml:space="preserve"> <lb/>Ad cuius rei explanationẽ neceſſe eſt quędam antecedentia præmittere:</s> <s xml:id="echoid-s13637" xml:space="preserve"> quorũ unum eſt.</s> <s xml:id="echoid-s13638" xml:space="preserve"> Si fuerint <lb/>duo puncta eiuſdẽ longitudinis à centro ſpeculi, & inæqualis lõgitudinis à centro uiſus:</s> <s xml:id="echoid-s13639" xml:space="preserve"> imago pun <lb/>cti remotioris à centro uiſus erit remotior à centro ſpeculi, ꝗ̃ propinquioris:</s> <s xml:id="echoid-s13640" xml:space="preserve"> & finis cõtingentię re-<lb/>motioris erit remotior à centro ſpeculi, ꝗ̃ finis cõtingentię propinquioris:</s> <s xml:id="echoid-s13641" xml:space="preserve"> ſiue puncta illa ſint in ea-<lb/>dem ſuperficie cum centro uiſus, ſiue in diuerſis.</s> <s xml:id="echoid-s13642" xml:space="preserve"> Sintt, d duo puncta æqualiter à g cẽtro ſpeculi re-<lb/>mota:</s> <s xml:id="echoid-s13643" xml:space="preserve"> e centrũ uiſus:</s> <s xml:id="echoid-s13644" xml:space="preserve"> & d propinquius uiſui ꝗ̃ t.</s> <s xml:id="echoid-s13645" xml:space="preserve"> Superficies cõmunis ſectionis d t g ſecabit ſpeculũ <lb/>ſuper circulũ [per 1 th.</s> <s xml:id="echoid-s13646" xml:space="preserve"> 1 ſphæ.</s> <s xml:id="echoid-s13647" xml:space="preserve">] qui ſit a b:</s> <s xml:id="echoid-s13648" xml:space="preserve"> & ſit angulus e g d æqualis angulo t g z:</s> <s xml:id="echoid-s13649" xml:space="preserve"> angulus e g t æqua-<lb/>lis angulo t g h:</s> <s xml:id="echoid-s13650" xml:space="preserve"> & ſumatur in circulo punctũ, à quo t reflectatur ad z:</s> <s xml:id="echoid-s13651" xml:space="preserve"> [per 31.</s> <s xml:id="echoid-s13652" xml:space="preserve"> uel 39 n 5] quod ſit q.</s> <s xml:id="echoid-s13653" xml:space="preserve"> Di <lb/>co, quòd t non reflectitur ad h ab aliquo puncto b q.</s> <s xml:id="echoid-s13654" xml:space="preserve"> Palàm, quòd non à puncto b [quia cũ ea ſit per-<lb/>pendicularis ſpeculo, reflectetur in ſeipſam, nõ ad h per 11 n 4.</s> <s xml:id="echoid-s13655" xml:space="preserve">] Si aũt ſumatur punctũ quodcunq;</s> <s xml:id="echoid-s13656" xml:space="preserve"> in <lb/>b q:</s> <s xml:id="echoid-s13657" xml:space="preserve"> linea ducta à puncto h ad illud punctũ, ſecabit lineá q z.</s> <s xml:id="echoid-s13658" xml:space="preserve"> Igitur ad illud punctũ ſectionis reflecti-<lb/>tur t ab aliquo puncto, ſumpto in b q:</s> <s xml:id="echoid-s13659" xml:space="preserve"> & ad idẽ ſectionis punctũ reflectitur à puncto q.</s> <s xml:id="echoid-s13660" xml:space="preserve"> Igitur t refle-<lb/>ctitur ad idem punctum à duobus punctis illius circuli:</s> <s xml:id="echoid-s13661" xml:space="preserve"> quod impoſsibile in his ſpeculis, ut in libro <lb/>quinto [29 n] patuit.</s> <s xml:id="echoid-s13662" xml:space="preserve"> Reſtat ergo, ut t reflectatur ad h ab aliquo puncto q a:</s> <s xml:id="echoid-s13663" xml:space="preserve"> ſit illud m:</s> <s xml:id="echoid-s13664" xml:space="preserve"> & [per 17 p 3] <gap/> <lb/>puncto m ducatur contingens circulum uſq;</s> <s xml:id="echoid-s13665" xml:space="preserve"> ad li <lb/> <anchor type="figure" xlink:label="fig-0203-01a" xlink:href="fig-0203-01"/> neam g t:</s> <s xml:id="echoid-s13666" xml:space="preserve"> quæ ſit m n.</s> <s xml:id="echoid-s13667" xml:space="preserve"> Erit n finis contingentiæ t, <lb/>reſpectu h:</s> <s xml:id="echoid-s13668" xml:space="preserve"> [per 17 n 5] & à puncto q ducatur cõtin <lb/>gens:</s> <s xml:id="echoid-s13669" xml:space="preserve"> quę ſit q o:</s> <s xml:id="echoid-s13670" xml:space="preserve"> quę quidẽ neceſſariò cadet ſub m <lb/>n:</s> <s xml:id="echoid-s13671" xml:space="preserve"> [quòd enim non cadar in punctũ n, inde perſpi <lb/>cuum eſt:</s> <s xml:id="echoid-s13672" xml:space="preserve"> quia ductis ſemidiametrιs g q, g m:</s> <s xml:id="echoid-s13673" xml:space="preserve"> angu <lb/>li n q g, n m g per 18 p 3 recti, eſſent inęquales per 21 <lb/>p 1 contra 10 ax:</s> <s xml:id="echoid-s13674" xml:space="preserve"> Si uerò cadat ultra n:</s> <s xml:id="echoid-s13675" xml:space="preserve"> erit per 21 p 1 <lb/>angulus rectus obtuſo maior cótra 11 p 1] & produ <lb/>catur z q uſq;</s> <s xml:id="echoid-s13676" xml:space="preserve"> dum cadat ſuper g t in puncto p.</s> <s xml:id="echoid-s13677" xml:space="preserve"> [ca <lb/>det aũt per 3 uel 16 n 5.</s> <s xml:id="echoid-s13678" xml:space="preserve">] Erit p locus imaginis z.</s> <s xml:id="echoid-s13679" xml:space="preserve"> E-<lb/>rit ergo [per 18 n 5] proportio g t ad p g, ſicut t o ad <lb/>o p:</s> <s xml:id="echoid-s13680" xml:space="preserve"> igitur maior erit proportio g t ad t n, quàm g t <lb/>ad t o [per 8 p 5:</s> <s xml:id="echoid-s13681" xml:space="preserve"> quia t o màior eſt t n.</s> <s xml:id="echoid-s13682" xml:space="preserve">] Ergo multò <lb/>maior g t ad t n, quàm g p ad p n.</s> <s xml:id="echoid-s13683" xml:space="preserve"> [Quia enim ra-<lb/>tio g t ad t n maior eſt, ꝗ̃ ad t o ex concluſo:</s> <s xml:id="echoid-s13684" xml:space="preserve"> eſtq́;</s> <s xml:id="echoid-s13685" xml:space="preserve"> <lb/>gt ad to, ſicut p g ad p o per 16 p 5.</s> <s xml:id="echoid-s13686" xml:space="preserve"> Ratio igitur t <lb/>g ad tn maior eſt, quàm p g ad p o:</s> <s xml:id="echoid-s13687" xml:space="preserve"> ſed ratio p g ad <lb/>p o maior eſt, quàm ad p n per 8 p 5.</s> <s xml:id="echoid-s13688" xml:space="preserve"> Ratio igitur <lb/>g t ad t n multò maior eſt, quàm p g ad p n.</s> <s xml:id="echoid-s13689" xml:space="preserve">] Sit er-<lb/>go [per 10 p 6] g t ad t n, ſicut g l ad i n.</s> <s xml:id="echoid-s13690" xml:space="preserve"> Erit g l ma-<lb/>ior g p.</s> <s xml:id="echoid-s13691" xml:space="preserve"> Et eritl locus imaginis h [per 18 n 5:</s> <s xml:id="echoid-s13692" xml:space="preserve"> eſt e-<lb/>nim per 16 p 5 g t ad g l, ſicut t n ad n l.</s> <s xml:id="echoid-s13693" xml:space="preserve">] Sint ergo h <lb/>g, e g, z g lineę æ quales:</s> <s xml:id="echoid-s13694" xml:space="preserve"> g f æqualis g p:</s> <s xml:id="echoid-s13695" xml:space="preserve"> g s æqualis g o.</s> <s xml:id="echoid-s13696" xml:space="preserve"> Cũ igitur angulus e g d ſit ęqualis angulo t g z <lb/>[per fabricationẽ] & remotio d à puncto è, ſicut z à puncto t:</s> <s xml:id="echoid-s13697" xml:space="preserve"> [Quia enim rectæ e g, d g, z g, t g:</s> <s xml:id="echoid-s13698" xml:space="preserve"> itẽ an-<lb/>guli e g d, z g t æquantur per fabricationẽ:</s> <s xml:id="echoid-s13699" xml:space="preserve"> baſes e d, z t ęquabuntur per 4 p 1:</s> <s xml:id="echoid-s13700" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s13701" xml:space="preserve"> puncta d, z ęqua <lb/>biliter diſtabũt à punctis e & t] erit imago d reſpectu e tantùm eleuata in linea g d, quantùm imago t <lb/>reſpectu z in linea g t:</s> <s xml:id="echoid-s13702" xml:space="preserve"> erit igitur imago d in puncto f:</s> <s xml:id="echoid-s13703" xml:space="preserve"> & ſimiliter finis cõtingentię d, reſpectu e erit al <lb/>titudinis eiuſdẽ, cuius eſt finis cõtingẽtię pũcti t, reſpectu z.</s> <s xml:id="echoid-s13704" xml:space="preserve"> Quare erit finis cõtingẽtiæ d in pũcto s.</s> <s xml:id="echoid-s13705" xml:space="preserve"> <lb/>Verùm quoniá angulus e g t æqualis eſt angulo t g h, & h g æqualis e g:</s> <s xml:id="echoid-s13706" xml:space="preserve"> [per fabricationẽ] erit l ima-<lb/>got, reſpectue, ſicut eſt reſpectu puncti h:</s> <s xml:id="echoid-s13707" xml:space="preserve"> & n finis cõtingentiæ reſpectu e, ſicut eſt reſpectu pũcti h.</s> <s xml:id="echoid-s13708" xml:space="preserve"> <lb/> <pb o="198" file="0204" n="204" rhead="ALHAZEN"/> Quare imago puncti remotioris ab e remotior eſt à centro, imagine propinquioris:</s> <s xml:id="echoid-s13709" xml:space="preserve"> & finis contin-<lb/>gentiæ remotioris remotior à centro, fine propinquioris.</s> <s xml:id="echoid-s13710" xml:space="preserve"> Quod erat propoſitum.</s> <s xml:id="echoid-s13711" xml:space="preserve"/> </p> <div xml:id="echoid-div463" type="float" level="0" n="0"> <figure xlink:label="fig-0203-01" xlink:href="fig-0203-01a"> <variables xml:id="echoid-variables152" xml:space="preserve">d t e <gap/> h s n q b l q m f p a g</variables> </figure> </div> </div> <div xml:id="echoid-div465" type="section" level="0" n="0"> <head xml:id="echoid-head421" xml:space="preserve" style="it">8. Si data recta in duob{us} punctis ſecta, ſit ad alterũ extremorũ ſegmentorũ, ut reliquũ ex-<lb/>tremum ad intermediũ: & ab altero ipſi{us} termino, ſectionuḿ punctis tres rectæ in eodẽ pun <lb/>cto cõcurrant: recta à reliquo termino ſecãs cõcurrentes, ſecabitur proportionaliter datæ. 123 p 1.</head> <p> <s xml:id="echoid-s13712" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s13713" xml:space="preserve"> propoſita linea a b, & diuiſa in punctis g, d, ut ſit proportio a b ad b d, ſicut a g ad g d:</s> <s xml:id="echoid-s13714" xml:space="preserve"> <lb/>ſi à punctis ſectionũ ducantur tres lineæ concurrentes in punctum unum, ſcilicet g e, d e, b e:</s> <s xml:id="echoid-s13715" xml:space="preserve"> <lb/>& à puncto a ducatur linea ſecans illas tres lineas:</s> <s xml:id="echoid-s13716" xml:space="preserve"> Dico, quòd linea illa diuiſa erit ſecundum <lb/>prædictam proportionẽ.</s> <s xml:id="echoid-s13717" xml:space="preserve"> Ducatur linea a c ſecans tria latera g e, d e, b e in tribus punctis z, h, c.</s> <s xml:id="echoid-s13718" xml:space="preserve"> Dico <lb/>quòd proportio a c ad c h, ſicut a z ad z h.</s> <s xml:id="echoid-s13719" xml:space="preserve"> Ducatur [per 31 p 1] à puncto h æquidiſtans a b:</s> <s xml:id="echoid-s13720" xml:space="preserve"> quæ ſit h q.</s> <s xml:id="echoid-s13721" xml:space="preserve"> <lb/>Palàm [è demonſtratis à Theone ad 5 d 6] quòd proportio a b ad b d, conſtat ex proportionibus a b <lb/>ad h q, & h q ad b d.</s> <s xml:id="echoid-s13722" xml:space="preserve"> Sed quoniã q h æquidiſtat a b:</s> <s xml:id="echoid-s13723" xml:space="preserve"> erit triangulũ c q h ſimile triangulo c a b:</s> <s xml:id="echoid-s13724" xml:space="preserve">] per 29 p <lb/>1.</s> <s xml:id="echoid-s13725" xml:space="preserve">4 p.</s> <s xml:id="echoid-s13726" xml:space="preserve"> 1 d 6] & erit proportio a b ad q h, ſicut a c ad c h.</s> <s xml:id="echoid-s13727" xml:space="preserve"> Similiter triangulũ q e h ſimile triangulo b e d:</s> <s xml:id="echoid-s13728" xml:space="preserve"> <lb/>igitur erit porportio q h ad b d, ſicut h e ad e d.</s> <s xml:id="echoid-s13729" xml:space="preserve"> Ergo <lb/> <anchor type="figure" xlink:label="fig-0204-01a" xlink:href="fig-0204-01"/> proportio a b ad b d, cõſtat ex proportionibus a c ad <lb/>c h & h e ad e d.</s> <s xml:id="echoid-s13730" xml:space="preserve"> Producatur q h, uſq;</s> <s xml:id="echoid-s13731" xml:space="preserve"> dum cadat ſuper <lb/>e g in puncto m.</s> <s xml:id="echoid-s13732" xml:space="preserve"> [cadet aũt per lemma Procli ad 29 p <lb/>1] Proportio igitur a g ad g d, conſtat ex proportioni-<lb/>bus a g ad h m, & h m ad g d.</s> <s xml:id="echoid-s13733" xml:space="preserve"> Sed cum [per 29 p 1] <lb/>angulus e m h ſit æqualis angulo z g d:</s> <s xml:id="echoid-s13734" xml:space="preserve"> erit [per 13 p 1] <lb/>angulus h m zæqualis angulo z g a:</s> <s xml:id="echoid-s13735" xml:space="preserve"> & erit triangulũ <lb/>a z g ſimile triangulo h m z [quia enim anguli aduer-<lb/>ticem z æquantur per 15 p 1:</s> <s xml:id="echoid-s13736" xml:space="preserve"> æquabitur per 32 p 1 ter-<lb/>tius m h z tertio g a z.</s> <s xml:id="echoid-s13737" xml:space="preserve"> Quare per 4 p.</s> <s xml:id="echoid-s13738" xml:space="preserve">1 d 6 triangula h <lb/>m z, a g z ſunt ſimilia.</s> <s xml:id="echoid-s13739" xml:space="preserve">] Et erit proportio a z ad z h ſi-<lb/>cut a g ad h m.</s> <s xml:id="echoid-s13740" xml:space="preserve"> Sed [per 29 p 1.</s> <s xml:id="echoid-s13741" xml:space="preserve">4 p.</s> <s xml:id="echoid-s13742" xml:space="preserve">1 d 6] triangulum h <lb/>e m ſimile eſt triangulo g e d:</s> <s xml:id="echoid-s13743" xml:space="preserve"> erit igitur proportio h <lb/>m ad g d, ſicut h e ad e d.</s> <s xml:id="echoid-s13744" xml:space="preserve"> Igitur proportio a g ad g d, <lb/>conſtat ex proportione a z ad z h, & h e ad e d:</s> <s xml:id="echoid-s13745" xml:space="preserve"> & eadẽ <lb/>eſt a g ad g d, quæ a b ad b d [pertheſin.</s> <s xml:id="echoid-s13746" xml:space="preserve">] Igitur illa ea-<lb/>dem cõſtat ex proportionibus a z ad z h & h e ad e d.</s> <s xml:id="echoid-s13747" xml:space="preserve"> <lb/>Igitur [ſubducta utrinq;</s> <s xml:id="echoid-s13748" xml:space="preserve"> ratione h c ad e d] eadẽ erit <lb/>proportio a c ad c h, quę eſt a z ad z h.</s> <s xml:id="echoid-s13749" xml:space="preserve"> Et ita eſt propoſitũ.</s> <s xml:id="echoid-s13750" xml:space="preserve"> Eadem erit probatio, quęcunq;</s> <s xml:id="echoid-s13751" xml:space="preserve"> linea duca <lb/>rur à puncto a, ſecans lineas illas tres concurrentes.</s> <s xml:id="echoid-s13752" xml:space="preserve"> Et ſi ducantur aliæ tres lineę à tribus punctis g, <lb/>d, b, ad aliud punctũ quàm e cõcurrentes, & à puncto a ducatur linea quæcũq;</s> <s xml:id="echoid-s13753" xml:space="preserve">, ſecans eas:</s> <s xml:id="echoid-s13754" xml:space="preserve"> diuidetur <lb/>ſecundum prædictã proportionẽ.</s> <s xml:id="echoid-s13755" xml:space="preserve"> Et ita quocunq;</s> <s xml:id="echoid-s13756" xml:space="preserve"> modo concurrãt tres lineę.</s> <s xml:id="echoid-s13757" xml:space="preserve"> Et ſi tres lineę e g, e d, <lb/>e b producantur ultra tria puncta b, d, g ex alia parte:</s> <s xml:id="echoid-s13758" xml:space="preserve"> & à puncto a ducantur lineæ, ſecantes eas ex il <lb/>la alia parte:</s> <s xml:id="echoid-s13759" xml:space="preserve"> nunquam illæ lineæ diuidentur ſecundum prædictam proportionem.</s> <s xml:id="echoid-s13760" xml:space="preserve"/> </p> <div xml:id="echoid-div465" type="float" level="0" n="0"> <figure xlink:label="fig-0204-01" xlink:href="fig-0204-01a"> <variables xml:id="echoid-variables153" xml:space="preserve">e c <gap/> h m z b d <gap/> a</variables> </figure> </div> </div> <div xml:id="echoid-div467" type="section" level="0" n="0"> <head xml:id="echoid-head422" xml:space="preserve" style="it">9. Si duæ rectæ facientes angulum, ſimiliteŕ in duob{us} punctis ita ſectæ (ut tota ſit ad alterũ <lb/>extremorũ ſegmentorũ, ſicut reliquum extremum ad intermedium) baſi infinita cõnect antur: <lb/>rectæ per pũcta ſectionũ utriuſ, cũ baſi & inter ſe cõcurrẽtes, in eodẽ puncto cõcurrẽt. 124 p 1.</head> <p> <s xml:id="echoid-s13761" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s13762" xml:space="preserve"> data linea a b prædicto modo diuiſa:</s> <s xml:id="echoid-s13763" xml:space="preserve"> ſi à puncto a ducatur alia linea, uelut a c, quæ di <lb/>uidatur iuxta eandem proportionẽ:</s> <s xml:id="echoid-s13764" xml:space="preserve"> & à punctis diuiſionũ a b ducantur lineę ad puncta diui <lb/>ſionũ a c, quę quidẽ nõ ſint æquidiſtãtes:</s> <s xml:id="echoid-s13765" xml:space="preserve"> Dico quòd <lb/> <anchor type="figure" xlink:label="fig-0204-02a" xlink:href="fig-0204-02"/> illæ tres concurrent in uno & eodẽ puncto.</s> <s xml:id="echoid-s13766" xml:space="preserve"> Sit pro-<lb/>portio ac ad c h, ſicut a z ad z h.</s> <s xml:id="echoid-s13767" xml:space="preserve"> Et quia b c, d h non <lb/>ſunt æquidiſtantes [ex theſi] igitur concurrent in ali <lb/>quo puncto:</s> <s xml:id="echoid-s13768" xml:space="preserve"> quod ſit e.</s> <s xml:id="echoid-s13769" xml:space="preserve"> Linea g z aut concurret ad <lb/>idem punctũ:</s> <s xml:id="echoid-s13770" xml:space="preserve"> aut non.</s> <s xml:id="echoid-s13771" xml:space="preserve"> Si ad idem:</s> <s xml:id="echoid-s13772" xml:space="preserve"> habemus propoſi <lb/>tum.</s> <s xml:id="echoid-s13773" xml:space="preserve"> Si nõ, ducatur linea e g:</s> <s xml:id="echoid-s13774" xml:space="preserve"> ſecabit quidem lineã a <lb/>c in alio puncto quàm z:</s> <s xml:id="echoid-s13775" xml:space="preserve"> ſit illud punctũ l.</s> <s xml:id="echoid-s13776" xml:space="preserve"> Erit ergo <lb/>proportio a c ad c h, ſicut a l ad l h iuxta priorẽ pro-<lb/>bationem [præcedentis numeri] ſed poſitum eſt a <lb/>c ad ch, ſicut a z ad z h.</s> <s xml:id="echoid-s13777" xml:space="preserve"> Et ita impoſsibile [nempe to-<lb/>tum æquari ſuæ parti.</s> <s xml:id="echoid-s13778" xml:space="preserve"> Quia enim per præcedentem <lb/>numerum eſt, ut a l ad l h, ſic a c ad c h, & ex theſi, ut <lb/>a c ad ch, ſic a z ad z h:</s> <s xml:id="echoid-s13779" xml:space="preserve"> erit per 11 p 5, ut a l ad l h, ſic a z <lb/>ad z h & per 18 p 5, ut a h ad h l, ſic a h ad h z.</s> <s xml:id="echoid-s13780" xml:space="preserve"> Quare <lb/>cum a h ad duas rectas h l, h z eandem habeat ratio-<lb/>nem, æquabuntur ipſæ inter ſe per 9 p 5:</s> <s xml:id="echoid-s13781" xml:space="preserve"> & ſic tota h <lb/>l erit æqualis parti h z.</s> <s xml:id="echoid-s13782" xml:space="preserve">] Similiter, ſi ponatur, quòd li <lb/>nea g z concurrat cum d h ad punctum e:</s> <s xml:id="echoid-s13783" xml:space="preserve"> probabitur hoc modo, quòd linea b c concurrat ad idem <lb/> <pb o="199" file="0205" n="205" rhead="OPTICAE LIBER VI."/> Similiter ſi ponatur, quòd g z, b c concurrant ad punctum e:</s> <s xml:id="echoid-s13784" xml:space="preserve"> probabitur, quòd d h concurret <lb/>ad idem.</s> <s xml:id="echoid-s13785" xml:space="preserve"/> </p> <div xml:id="echoid-div467" type="float" level="0" n="0"> <figure xlink:label="fig-0204-02" xlink:href="fig-0204-02a"> <variables xml:id="echoid-variables154" xml:space="preserve">e n c <gap/> z <gap/> b d g a</variables> </figure> </div> </div> <div xml:id="echoid-div469" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables155" xml:space="preserve">c h z b d g a</variables> </figure> <head xml:id="echoid-head423" xml:space="preserve" style="it">10. Si data recta in duob{us} punctis ſecta, ſit ad alterum extremorum ſegmẽtorum, ſicut re-<lb/>liquum extremum ad intermedium: & ab altero <lb/> ipſi{us} termino, ſectionuḿ punctis tres rectæ li- neæ ſint parallelæ: recta à reliquo termino ſecan s parallel{as}, ſecabitur proportionaliter datæ. 122 p 1.</head> <p> <s xml:id="echoid-s13786" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s13787" xml:space="preserve"> diuiſa ab ſecundum hanc proportio-<lb/>nem:</s> <s xml:id="echoid-s13788" xml:space="preserve"> ſi fuerint lineæ g z, d h, b c æquidiſtãtes:</s> <s xml:id="echoid-s13789" xml:space="preserve"> <lb/>& ducatur ac diuidens illas:</s> <s xml:id="echoid-s13790" xml:space="preserve"> erit ac diuiſa ſe-<lb/>cundum hanc proportionem.</s> <s xml:id="echoid-s13791" xml:space="preserve"> Cum d h ſit æquidi-<lb/>ſtans g z:</s> <s xml:id="echoid-s13792" xml:space="preserve"> erit [per 2 p 6] proportio a z ad z h, ſicut a g <lb/>ad g d:</s> <s xml:id="echoid-s13793" xml:space="preserve"> & cum b c ſit æquidiſtãs d h:</s> <s xml:id="echoid-s13794" xml:space="preserve"> erit [per 2 p 6.</s> <s xml:id="echoid-s13795" xml:space="preserve"> 18 <lb/>p 5] a b ad b d, ſicut a c ad c h:</s> <s xml:id="echoid-s13796" xml:space="preserve"> ſed [ex theſi] a b ad b d, <lb/>ſicut a g ad g d:</s> <s xml:id="echoid-s13797" xml:space="preserve"> erit [per 11 p 5] a c ad ch, ſicut a z ad <lb/>z h.</s> <s xml:id="echoid-s13798" xml:space="preserve"> Et ita patet propoſitum.</s> <s xml:id="echoid-s13799" xml:space="preserve"> His præmiſsis, acceda-<lb/>mus ad propoſitum.</s> <s xml:id="echoid-s13800" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div470" type="section" level="0" n="0"> <head xml:id="echoid-head424" xml:space="preserve" style="it">11. Sirecta linea à uiſu ſit perpendicularis ſu-<lb/>perficiei incidentiæ: imago perιpheriæ concentricæ <lb/>peripheriæ circuli (qui eſt communis ſectio ſuperficierum reflexionis & ſpeculi ſphærici cõuexi) <lb/>uidebitur curua, & par allela ipſi peripheriæ concentricæ. 46 p 6.</head> <p> <s xml:id="echoid-s13801" xml:space="preserve">PRimum de arcu declaremus, qnomodo in his ſpeculis imago <lb/> <anchor type="figure" xlink:label="fig-0205-02a" xlink:href="fig-0205-02"/> eius ſit curua, curuitate quidem ſpeculum non reſpiciente, ſed <lb/>centrũ.</s> <s xml:id="echoid-s13802" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s13803" xml:space="preserve"> ſit ab arcus oppoſitus ſpeculo:</s> <s xml:id="echoid-s13804" xml:space="preserve"> & ſit g cen-<lb/>trum illius arcus, & ſimiliter centrum ſpeculi:</s> <s xml:id="echoid-s13805" xml:space="preserve"> d cẽtrum uiſus:</s> <s xml:id="echoid-s13806" xml:space="preserve"> & du-<lb/>cantur lineæ d g, a g, b g:</s> <s xml:id="echoid-s13807" xml:space="preserve"> & ſumatur e in arcu a b quocunq;</s> <s xml:id="echoid-s13808" xml:space="preserve"> modo:</s> <s xml:id="echoid-s13809" xml:space="preserve"> & <lb/>ducatur linea e g.</s> <s xml:id="echoid-s13810" xml:space="preserve"> Linea uerò d g non ſit in ſuperficie a b g.</s> <s xml:id="echoid-s13811" xml:space="preserve"> Linea igi-<lb/>tur d g aut erit orthogonalis ſuper ſuperficiẽ a b g:</s> <s xml:id="echoid-s13812" xml:space="preserve"> aut declinata.</s> <s xml:id="echoid-s13813" xml:space="preserve"> Sit <lb/>orthogonalis:</s> <s xml:id="echoid-s13814" xml:space="preserve"> erunt anguli d g a, d g e, d g b æquales [quia per 3 d 11 <lb/>recti ſunt] & [per 15 d 1] latera lateribus.</s> <s xml:id="echoid-s13815" xml:space="preserve"> Quare [per 4 p 1] baſes baſi-<lb/>bus.</s> <s xml:id="echoid-s13816" xml:space="preserve"> Igitur omnia puncta arcus a b eiuſdem longitudinis erũt à cen-<lb/>tro uiſus.</s> <s xml:id="echoid-s13817" xml:space="preserve"> Quare imagines omniũ punctorũ, eiuſdẽ longitudinis ſunt <lb/>â cẽtro:</s> <s xml:id="echoid-s13818" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s13819" xml:space="preserve"> q, m, l imagines ipſorũ a, e, b.</s> <s xml:id="echoid-s13820" xml:space="preserve"> Erit igitur g q ęqualis g m, <lb/>g l.</s> <s xml:id="echoid-s13821" xml:space="preserve"> Quare q m l erit arcus:</s> <s xml:id="echoid-s13822" xml:space="preserve"> [per 9 p 3] & cõuexitas ipſius reſpectu cen <lb/>tri, non reſpectu ſpeculi, ſiue loci reflexionis.</s> <s xml:id="echoid-s13823" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s13824" xml:space="preserve"/> </p> <div xml:id="echoid-div470" type="float" level="0" n="0"> <figure xlink:label="fig-0205-02" xlink:href="fig-0205-02a"> <variables xml:id="echoid-variables156" xml:space="preserve">b e a d h <gap/> z <gap/> m <gap/> g</variables> </figure> </div> </div> <div xml:id="echoid-div472" type="section" level="0" n="0"> <head xml:id="echoid-head425" xml:space="preserve" style="it">12. Si recta linea à uiſu ſit obliqua ſuperficiei incidentiæ: ima-<lb/>go peripheriæ concentricæ peripheriæ circuli (qui eſt communis ſe-<lb/>ctio ſuperficierum, reflexionis & ſpeculi ſphærici conucxi) uidebi-<lb/>tur curua, non parallela peripheriæ concentricæ. 47 p 6.</head> <p> <s xml:id="echoid-s13825" xml:space="preserve">SI uerò linea d g non fuerit perpẽdicularis ſuper ſuperficiem a b g:</s> <s xml:id="echoid-s13826" xml:space="preserve"> ducta perpendiculari à pun-<lb/>cto d ſuper hanc ſuperficiem:</s> <s xml:id="echoid-s13827" xml:space="preserve"> [per 11 p 11] cum [per 5 n 5] illa perpendicularis ſit minor omni-<lb/>bus lineis ductis à puncto d ad hanc ſuperficiem:</s> <s xml:id="echoid-s13828" xml:space="preserve"> erit angulus, quem continet hæc perpendi-<lb/>cularis uerſus g, minor quolibet angulo uerſus punctũ g intellecto, quem continet alia linea à pun-<lb/>cto d ad hanc ſuperficiem ducta [per 16 p 1.</s> <s xml:id="echoid-s13829" xml:space="preserve">] Et linea ducta à puncto d ad hanc ſuperficiem, quan-<lb/>tò remotior erit à perpendiculari, tantò maior erit, & continebit maiorem angulum uerſus g [per <lb/>21 p 1.</s> <s xml:id="echoid-s13830" xml:space="preserve">] Si ergo hæc perpendicularis non cadat in arcum a e b, ſed ex parte una:</s> <s xml:id="echoid-s13831" xml:space="preserve"> erunt omnes lineæ <lb/>ductæ à puncto d ad hunc arcum, declinatæ ad partem unam:</s> <s xml:id="echoid-s13832" xml:space="preserve"> & remotiores maiores, & maiorem <lb/>angulum continentes uerſus g.</s> <s xml:id="echoid-s13833" xml:space="preserve"> Sit ergo:</s> <s xml:id="echoid-s13834" xml:space="preserve"> & ſumantur tria puncta in arcu, ſcilicet e, c, b:</s> <s xml:id="echoid-s13835" xml:space="preserve"> finis contin-<lb/>gentiæ puncti b ſit l:</s> <s xml:id="echoid-s13836" xml:space="preserve"> finis contingẽtiæ puncti c, ſit m.</s> <s xml:id="echoid-s13837" xml:space="preserve"> Quoniam igitur c propinquius d, quam b:</s> <s xml:id="echoid-s13838" xml:space="preserve"> erit <lb/>ιn propinquius g quàm l:</s> <s xml:id="echoid-s13839" xml:space="preserve"> [per 7 n] & ita c m maior b l [quia gc, g b ęquantur per 15 d 1] q ſit imago <lb/>c:</s> <s xml:id="echoid-s13840" xml:space="preserve"> timago b:</s> <s xml:id="echoid-s13841" xml:space="preserve"> & ducatur t q:</s> <s xml:id="echoid-s13842" xml:space="preserve"> & ducantur lineæ c b, m l:</s> <s xml:id="echoid-s13843" xml:space="preserve"> quæ quidem productæ concurrent.</s> <s xml:id="echoid-s13844" xml:space="preserve"> Si enim à <lb/>puncto m duceretur æquidiſtans c b, ſecaret ex g b lineam æqualem c m [eſſet enim per 2 p 6 18 p 5, <lb/>ut g c ad c m, ſic g b ad rectam, quam ſecat parallela à pũcto m ducta ex g b:</s> <s xml:id="echoid-s13845" xml:space="preserve"> itaq, cum g c, g b æquen-<lb/>tur per 15 d 1:</s> <s xml:id="echoid-s13846" xml:space="preserve"> æquaretur c m, ſectæ per parallelam ex g b:</s> <s xml:id="echoid-s13847" xml:space="preserve"> ſed c m, ut patuit, maior eſt b l:</s> <s xml:id="echoid-s13848" xml:space="preserve"> quare c b, <lb/>m l productæ concurrent.</s> <s xml:id="echoid-s13849" xml:space="preserve">] Concurrant in puncto o.</s> <s xml:id="echoid-s13850" xml:space="preserve"> Et quoniam proportio g c ad c m, ſicut g q ad <lb/>q m [eſt enim per 18 n 5, ut c g ad g q, ſic c m ad m q:</s> <s xml:id="echoid-s13851" xml:space="preserve"> ergo per 16 p 5, ut c g ad c m, ſic g q ad q m.</s> <s xml:id="echoid-s13852" xml:space="preserve">] Si-<lb/>militer g b ad b l, ſicut g t ad t l:</s> <s xml:id="echoid-s13853" xml:space="preserve"> ergo linea q t concurret cum lineis c b, m l [per 9 n.</s> <s xml:id="echoid-s13854" xml:space="preserve">] Sit con-<lb/>curſus in puncto o.</s> <s xml:id="echoid-s13855" xml:space="preserve"> Finis contingentiæ puncti e ſit n.</s> <s xml:id="echoid-s13856" xml:space="preserve"> Quoniam punctum n demiſsius eſt puncto <lb/>m:</s> <s xml:id="echoid-s13857" xml:space="preserve"> [per 7 n] erit e n maior c m:</s> <s xml:id="echoid-s13858" xml:space="preserve"> ductis ergo lineis e c, m n, concurrent [ut antea.</s> <s xml:id="echoid-s13859" xml:space="preserve">] Sit concurſus in <lb/> <pb o="200" file="0206" n="206" rhead="ALHAZEN"/> puncto p:</s> <s xml:id="echoid-s13860" xml:space="preserve"> & ducatur linea q p:</s> <s xml:id="echoid-s13861" xml:space="preserve"> & procedat, donec cadat ſuper e g in pũcto f:</s> <s xml:id="echoid-s13862" xml:space="preserve"> & ducatur linea t q uſq;</s> <s xml:id="echoid-s13863" xml:space="preserve"> <lb/>ad e g:</s> <s xml:id="echoid-s13864" xml:space="preserve"> & cadat in pũctum k.</s> <s xml:id="echoid-s13865" xml:space="preserve"> Pa-<lb/> <anchor type="figure" xlink:label="fig-0206-01a" xlink:href="fig-0206-01"/> làm, quòd k erit ſupra f [ꝗa pun-<lb/>ctum n humilius eſt puncto m.</s> <s xml:id="echoid-s13866" xml:space="preserve">] <lb/>Verùm cũ proportio g c ad c m, <lb/>ſicut g q ad q m [ut patuit] & à <lb/>punctis diuiſionũ ducantur tres <lb/>lineæ concurrẽtes, in aliam par-<lb/>tem productæ ſecabunt lineam <lb/>e g ſecundum prædictã propor-<lb/>tionẽ [per 8 n.</s> <s xml:id="echoid-s13867" xml:space="preserve">] Quare propor-<lb/>tio g e ad e n, ſicut g f ad f n:</s> <s xml:id="echoid-s13868" xml:space="preserve"> ſed n <lb/>eſt finis cõtingentiæ.</s> <s xml:id="echoid-s13869" xml:space="preserve"> Quare flo-<lb/>cus eſt imaginis [per 18 n 5.</s> <s xml:id="echoid-s13870" xml:space="preserve">] Igi-<lb/>tur linea f q t erit imago arcus e <lb/>c b:</s> <s xml:id="echoid-s13871" xml:space="preserve"> & erit linea curua, non recta:</s> <s xml:id="echoid-s13872" xml:space="preserve"> <lb/>quoniam t q k eſt recta:</s> <s xml:id="echoid-s13873" xml:space="preserve"> & curui-<lb/>tas lineæ non eſt ex parte ſpecu-<lb/>li.</s> <s xml:id="echoid-s13874" xml:space="preserve"> Similiter ſi perpendicularis à <lb/>puncto d cadat ex alia parte arcus:</s> <s xml:id="echoid-s13875" xml:space="preserve"> ſimilis erit probatio.</s> <s xml:id="echoid-s13876" xml:space="preserve"> Si uerò cadat perpendicularis in medium <lb/>arcus a b:</s> <s xml:id="echoid-s13877" xml:space="preserve"> lineæ à puncto d ex diuerſis partibus ad arcum ductæ, æqualiter diſtantes à perpendicu-<lb/>lari:</s> <s xml:id="echoid-s13878" xml:space="preserve"> erunt æquales, & æquales angulos continebunt uerſus g:</s> <s xml:id="echoid-s13879" xml:space="preserve"> & imagines à g æqualiter diſtabunt:</s> <s xml:id="echoid-s13880" xml:space="preserve"> <lb/>& fines contingentiæ ſimiliter.</s> <s xml:id="echoid-s13881" xml:space="preserve"> Et licebit probare prædicto modo de utraq;</s> <s xml:id="echoid-s13882" xml:space="preserve"> parte arcus per ſe, ſe-<lb/>cundum quod diuiditur à perpendiculari:</s> <s xml:id="echoid-s13883" xml:space="preserve"> quòd eius imago ſit linea curua modo prædicto.</s> <s xml:id="echoid-s13884" xml:space="preserve"> Quod <lb/>eſt propoſitum.</s> <s xml:id="echoid-s13885" xml:space="preserve"/> </p> <div xml:id="echoid-div472" type="float" level="0" n="0"> <figure xlink:label="fig-0206-01" xlink:href="fig-0206-01a"> <variables xml:id="echoid-variables157" xml:space="preserve">p o b c e l m t n a q k f d g</variables> </figure> </div> </div> <div xml:id="echoid-div474" type="section" level="0" n="0"> <head xml:id="echoid-head426" xml:space="preserve" style="it">13. Si uiſ{us} ſit extra ſuperficiem incidentiæ: imago peripheriæ eccentricæ peripheriæ circuli <lb/>(qui eſt communis ſectio ſuperficierum, reflex ionis & ſpeculi ſphærici conuexi) uidebitur magis <lb/>curua, quàm imago peripheriæ concentricæ. 48 p 6.</head> <p> <s xml:id="echoid-s13886" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s13887" xml:space="preserve"> ſumatur circulus, cuius centrum non ſit centrum ſpeculi, ueruntamen ſit in eadem <lb/>ſuperficie cum centro ſpeculi.</s> <s xml:id="echoid-s13888" xml:space="preserve"> Dico, quòd ſi in hoc circulo <lb/> <anchor type="figure" xlink:label="fig-0206-02a" xlink:href="fig-0206-02"/> exteriore ſumatur arcus ex parte cẽtri ſpeculi, propinquior <lb/>ei ſecundum medium eius punctum, erit imago eius curua.</s> <s xml:id="echoid-s13889" xml:space="preserve"> Dato <lb/>enim hoc arcu:</s> <s xml:id="echoid-s13890" xml:space="preserve"> ducatur linea à centro ſpeculi ad centrum exterio-<lb/>ris circuli:</s> <s xml:id="echoid-s13891" xml:space="preserve"> & producatur hæc linea uſq;</s> <s xml:id="echoid-s13892" xml:space="preserve"> ad arcum datum:</s> <s xml:id="echoid-s13893" xml:space="preserve"> linea du-<lb/>cta à centro ſpeculi ad hunc arcum, quæ eſt pars diametri maioris <lb/>circuli, erit breuior omnibus lineis ductis ab eodem centro ſpecu-<lb/>li ad illum arcum [per 7 p 3.</s> <s xml:id="echoid-s13894" xml:space="preserve">] Et à centro ſpeculi poſſunt duci ad ar-<lb/>cum datũ duæ lineæ æquales à diuerſis partibus huius breuis [per <lb/>7 p 3] quæ quidem maiores erũt illa breui.</s> <s xml:id="echoid-s13895" xml:space="preserve"> Et ſi ſecundum alteram <lb/>illarum fiat circulus, cuius centrum ſit ſpeculi centrum:</s> <s xml:id="echoid-s13896" xml:space="preserve"> tranſibit <lb/>per capita harum duarum linearum arcus excedens arcum datum.</s> <s xml:id="echoid-s13897" xml:space="preserve"> <lb/>Et palàm, quòd imago huius arcus excedentis, erit linea curua ſe-<lb/>cundum prædicta [11.</s> <s xml:id="echoid-s13898" xml:space="preserve">12 n:</s> <s xml:id="echoid-s13899" xml:space="preserve">] Et imagines punctorum huic arcui & <lb/>arcui dato communium eædem:</s> <s xml:id="echoid-s13900" xml:space="preserve"> & medium punctum arcus exce-<lb/>dentis eſt remotius à centro ſpeculi, quam punctũ arcus dati, quod <lb/>ipſum reſpicit.</s> <s xml:id="echoid-s13901" xml:space="preserve"> Quare eius imago propinquior eſt centro, quã ima-<lb/>go puncti arcus dati illum reſpicientis.</s> <s xml:id="echoid-s13902" xml:space="preserve"> Et ita cuiuslibet puncti ar-<lb/>cus exterioris imago propinquior eſt cẽtro, imagine puncti arcus <lb/>dati, quod ipſum reſpicit.</s> <s xml:id="echoid-s13903" xml:space="preserve"> Quare imago arcus dati curuior, quã imago arcus exterioris.</s> <s xml:id="echoid-s13904" xml:space="preserve"> Quare ima-<lb/>go arcus dati curua eſt.</s> <s xml:id="echoid-s13905" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s13906" xml:space="preserve"/> </p> <div xml:id="echoid-div474" type="float" level="0" n="0"> <figure xlink:label="fig-0206-02" xlink:href="fig-0206-02a"> <variables xml:id="echoid-variables158" xml:space="preserve">b d a e h t z g f</variables> </figure> </div> </div> <div xml:id="echoid-div476" type="section" level="0" n="0"> <head xml:id="echoid-head427" xml:space="preserve" style="it">14. Si uiſ{us} ſit extra ſuperficiem incidentiæ: imago lineæ rectæ, parallelæ rectæ tangẽti peri-<lb/>pheriam circuli (qui eſt communis ſectio ſuperficierum, reflexionis & ſpeculi ſphærici conuexi) <lb/>uidebitur curua. 49 p 6.</head> <p> <s xml:id="echoid-s13907" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s13908" xml:space="preserve"> quòd lineę rectæ imago in his ſpeculis ſit curua, probatur ſic.</s> <s xml:id="echoid-s13909" xml:space="preserve"> Sit a b linea uiſa:</s> <s xml:id="echoid-s13910" xml:space="preserve"> g cen <lb/>trum ſpeculi:</s> <s xml:id="echoid-s13911" xml:space="preserve"> ducantur lineæ a g, b g.</s> <s xml:id="echoid-s13912" xml:space="preserve"> Hæ aut ſunt æquales:</s> <s xml:id="echoid-s13913" xml:space="preserve"> aut non.</s> <s xml:id="echoid-s13914" xml:space="preserve"> Si æquales:</s> <s xml:id="echoid-s13915" xml:space="preserve"> fiat circulus, <lb/>cuius g centrum, ſecundum quantitatem illarum:</s> <s xml:id="echoid-s13916" xml:space="preserve"> qui ſit a e b:</s> <s xml:id="echoid-s13917" xml:space="preserve"> cadet quidem linea a b intra <lb/>circulum.</s> <s xml:id="echoid-s13918" xml:space="preserve"> Palàm ex prædictis [11.</s> <s xml:id="echoid-s13919" xml:space="preserve">12 n] quòd imago arcus a e b erit curua.</s> <s xml:id="echoid-s13920" xml:space="preserve"> Sit igitur imago eius z t h:</s> <s xml:id="echoid-s13921" xml:space="preserve"> <lb/>imago a ſit z:</s> <s xml:id="echoid-s13922" xml:space="preserve"> imago b ſit h:</s> <s xml:id="echoid-s13923" xml:space="preserve"> imago e ſit t:</s> <s xml:id="echoid-s13924" xml:space="preserve"> & ducatur g e ſecans a b in puncto f.</s> <s xml:id="echoid-s13925" xml:space="preserve"> Palàm, quòd e eſt in ea-<lb/>dem linea cum f, remotior à centro g.</s> <s xml:id="echoid-s13926" xml:space="preserve"> Erit ergo eius imago propinquior centro, quàm fimago [per <lb/>30 n 5.</s> <s xml:id="echoid-s13927" xml:space="preserve">] Sit ergo m.</s> <s xml:id="echoid-s13928" xml:space="preserve"> Palàm ergo, quòd linea z m h eſt imago lineæ a b:</s> <s xml:id="echoid-s13929" xml:space="preserve"> [imagines enim punctorum a <lb/>& b communium eædem permanent] & eſt linea curua.</s> <s xml:id="echoid-s13930" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s13931" xml:space="preserve"/> </p> <pb o="201" file="0207" n="207" rhead="OPTICAE LIBER VI."/> </div> <div xml:id="echoid-div477" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables159" xml:space="preserve">e b f a d m h t z g</variables> </figure> <figure> <variables xml:id="echoid-variables160" xml:space="preserve">q e a b d m h z <gap/></variables> </figure> <head xml:id="echoid-head428" xml:space="preserve" style="it">15. Si uiſ{us} ſit extra ſuperficiem incidẽtiæ: imago lineæ rectæ infinitæ, nec parallelæ, nec tan-<lb/>gentis, nec ſecantis peripheriam cir <lb/> culi (qui eſt communis ſectio ſuper- ficierum, reflexionis & ſpeculi ſphæ- rici cõuexi) uidebitur curua. 50 p 6.</head> <p> <s xml:id="echoid-s13932" xml:space="preserve">SI uerò lineæ a g, b g fuerint inæ-<lb/>quales:</s> <s xml:id="echoid-s13933" xml:space="preserve"> linea a b protracta aut ſe-<lb/>cabit ſpeculũ:</s> <s xml:id="echoid-s13934" xml:space="preserve"> aut non.</s> <s xml:id="echoid-s13935" xml:space="preserve"> Sit quòd <lb/>nõ ſecet:</s> <s xml:id="echoid-s13936" xml:space="preserve"> & ſit a g maior g b:</s> <s xml:id="echoid-s13937" xml:space="preserve"> & fiat cir-<lb/>culus ſuper g ad quantitatem a g:</s> <s xml:id="echoid-s13938" xml:space="preserve"> qui <lb/>ſit a e q:</s> <s xml:id="echoid-s13939" xml:space="preserve"> & producatur a b, quouſq;</s> <s xml:id="echoid-s13940" xml:space="preserve"> ca <lb/>datin circulũ ex parte b:</s> <s xml:id="echoid-s13941" xml:space="preserve"> cadat in pun <lb/>ctum e.</s> <s xml:id="echoid-s13942" xml:space="preserve"> Patet ex ſuperioribus [11 uel <lb/>12 n] quòd imago arcus a e eſt curua.</s> <s xml:id="echoid-s13943" xml:space="preserve"> <lb/>Punctum imaginis a ſit z:</s> <s xml:id="echoid-s13944" xml:space="preserve"> punctũ ima <lb/>ginis e ſit m:</s> <s xml:id="echoid-s13945" xml:space="preserve"> erit z m imago arcus a e.</s> <s xml:id="echoid-s13946" xml:space="preserve"> <lb/>Et quoniam imago pũcti b remotior <lb/>eſt à cẽtro, quàm imago puncti e:</s> <s xml:id="echoid-s13947" xml:space="preserve"> erit <lb/>imago lineæ a b curua:</s> <s xml:id="echoid-s13948" xml:space="preserve"> quod etiã per <lb/>puncta media arcus a e & lineæ a b fa <lb/>ciliter poterit probari.</s> <s xml:id="echoid-s13949" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s13950" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div478" type="section" level="0" n="0"> <head xml:id="echoid-head429" xml:space="preserve" style="it">16. Si uiſ{us} ſit extra ſuperficiem incidentiæ: imago lineæ rectæ infinitæ, tangentis periphe-<lb/>riam circuli (qui eſt communis ſectio ſuperficierum, reflexionis & ſpeculi ſphærici conuexi) <lb/>uidebitur curua. 51 p 6.</head> <p> <s xml:id="echoid-s13951" xml:space="preserve">SI uerò linea a b tangit ſpeculum:</s> <s xml:id="echoid-s13952" xml:space="preserve"> aut ſecabit:</s> <s xml:id="echoid-s13953" xml:space="preserve"> aut continget.</s> <s xml:id="echoid-s13954" xml:space="preserve"> Tangat primò:</s> <s xml:id="echoid-s13955" xml:space="preserve"> & g ſit cẽtrum ſpe-<lb/>culi:</s> <s xml:id="echoid-s13956" xml:space="preserve"> & ducantur lineæ a g, b g.</s> <s xml:id="echoid-s13957" xml:space="preserve"> Superficies a b g ſecabit ſpeculum ſuper circulum communem <lb/>[per 1 th.</s> <s xml:id="echoid-s13958" xml:space="preserve"> 1 ſphær.</s> <s xml:id="echoid-s13959" xml:space="preserve">] qui ſit e h z.</s> <s xml:id="echoid-s13960" xml:space="preserve"> Palàm, quòd linea a b continget ſpeculum in hoc circulo [ſunt <lb/>enim peripheria e h z, & recta a b in eodem incidentiæ plano:</s> <s xml:id="echoid-s13961" xml:space="preserve"> & a b continuata tangit ſpeculum ex <lb/>theſi.</s> <s xml:id="echoid-s13962" xml:space="preserve"> Quare tangit in peripheria e h z.</s> <s xml:id="echoid-s13963" xml:space="preserve">] Contingat in puncto e.</s> <s xml:id="echoid-s13964" xml:space="preserve"> Protrahatur ergo a b uſq;</s> <s xml:id="echoid-s13965" xml:space="preserve"> ad e:</s> <s xml:id="echoid-s13966" xml:space="preserve"> d ſit <lb/>centrum uiſus.</s> <s xml:id="echoid-s13967" xml:space="preserve"> Superficies, in qua ſunt lineæ d g, a g ſecabit ſpeculum ſuper circulum, communem <lb/>ſuperficiei reflexionis & ſpeculi [per 1 th.</s> <s xml:id="echoid-s13968" xml:space="preserve"> 1 ſphær.</s> <s xml:id="echoid-s13969" xml:space="preserve">] Sit arcus illius circuli z p:</s> <s xml:id="echoid-s13970" xml:space="preserve"> ſimiliter linea com-<lb/>munis ſuperficiei reflexionis & ſpeculi, in qua ſunt d g, b g:</s> <s xml:id="echoid-s13971" xml:space="preserve"> & arcus illius circuli ſit h p.</s> <s xml:id="echoid-s13972" xml:space="preserve"> Palam [è 29 <lb/>n 5] quòd b reflectitur ad d ab aliquo puncto arcus h p.</s> <s xml:id="echoid-s13973" xml:space="preserve"> Si à puncto illo ducatur contingens:</s> <s xml:id="echoid-s13974" xml:space="preserve"> ſecabit <lb/>lineam b g, & punctum ſectionis erit finis contingentiæ [per 17 n 5.</s> <s xml:id="echoid-s13975" xml:space="preserve">] Sit punctum illud m.</s> <s xml:id="echoid-s13976" xml:space="preserve"> Palàm <lb/>etiam, quòd ſi à puncto m ducatur contingens arcum circuli e h:</s> <s xml:id="echoid-s13977" xml:space="preserve"> cadet contingens illa citra e:</s> <s xml:id="echoid-s13978" xml:space="preserve"> quo-<lb/>niam a b contingit in puncto e, & punctũ b eſt altius puncto m.</s> <s xml:id="echoid-s13979" xml:space="preserve"> Cadat igitur in punctum f:</s> <s xml:id="echoid-s13980" xml:space="preserve"> quæ con <lb/>tingens producta ſecabit lineam a e:</s> <s xml:id="echoid-s13981" xml:space="preserve"> ſecet in puncto t:</s> <s xml:id="echoid-s13982" xml:space="preserve"> ex alia parte ſecabit lineam a g:</s> <s xml:id="echoid-s13983" xml:space="preserve"> [per 11 ax] ſe-<lb/>cet in puncto c.</s> <s xml:id="echoid-s13984" xml:space="preserve"> Fiat [per 23 p 1] angulus b g s æqualis angulo b g d:</s> <s xml:id="echoid-s13985" xml:space="preserve"> & producatur g s uſq;</s> <s xml:id="echoid-s13986" xml:space="preserve"> ad pun-<lb/>ctum l ad æqualitatem lineæ d g:</s> <s xml:id="echoid-s13987" xml:space="preserve"> erit ergo [per 26 p 3] arcus h s æqualis arcui h p.</s> <s xml:id="echoid-s13988" xml:space="preserve"> Et ſicut reflecti-<lb/>tur b ad d, ab aliquo puncto arcus h p:</s> <s xml:id="echoid-s13989" xml:space="preserve"> ſic reflectetur <lb/> <anchor type="figure" xlink:label="fig-0207-03a" xlink:href="fig-0207-03"/> ad l, ab aliquo puncto arcus h s.</s> <s xml:id="echoid-s13990" xml:space="preserve"> Et erit reflexio à pun-<lb/>cto f, ſicut in arcu h p eſt reflexio à puncto, à quo du-<lb/>citur contingẽs ad punctũm.</s> <s xml:id="echoid-s13991" xml:space="preserve"> Et illa duo puncta ſunt <lb/>eiuſdem longitudinis à pũcto m.</s> <s xml:id="echoid-s13992" xml:space="preserve"> [Si enim duæ rectæ <lb/>ab eodem puncto peripheriam tangentes, duabus ſe-<lb/>midiametris connectantur:</s> <s xml:id="echoid-s13993" xml:space="preserve"> recta à centro ad idẽ pun <lb/>ctum ducta bifariam ſecabit angulum in centro per 2 <lb/>conſectarium 36 p 3.</s> <s xml:id="echoid-s13994" xml:space="preserve">15 d.</s> <s xml:id="echoid-s13995" xml:space="preserve"> 8 p 1:</s> <s xml:id="echoid-s13996" xml:space="preserve"> & peripheriæ angulis in <lb/>centro æqualibus ſubtenſæ, & rectę eaſdem periphe-<lb/>rias ſubtendentes æquabuntur per 26.</s> <s xml:id="echoid-s13997" xml:space="preserve"> 29 p 3:</s> <s xml:id="echoid-s13998" xml:space="preserve"> quare <lb/>prædicta duo reflexionũ puncta à puncto m æquabi-<lb/>liter diſtabunt.</s> <s xml:id="echoid-s13999" xml:space="preserve">] Ducantur ergo lineę b f, l f.</s> <s xml:id="echoid-s14000" xml:space="preserve"> Item a re-<lb/>flectatur ad d ab aliquo puncto arcus z p [per 29 n 5.</s> <s xml:id="echoid-s14001" xml:space="preserve">] <lb/>Verùm in triangulo h z p duo arcus h z, h p maiores <lb/>ſunt tertio z p:</s> <s xml:id="echoid-s14002" xml:space="preserve"> [ք 5 th.</s> <s xml:id="echoid-s14003" xml:space="preserve"> 1 ſphæricorũ Menelai] ſed h p eſt æqualis h s [ex cõcluſo.</s> <s xml:id="echoid-s14004" xml:space="preserve">] Igitur z p eſt mi-<lb/>nor z s.</s> <s xml:id="echoid-s14005" xml:space="preserve"> Reſcindatur z s ad æqualitatẽ in pũcto y [id uerò promptè præſtiterís, ſi latus anguli ad ter-<lb/>minũ g rectæ g z, æquati angulo z g p, in peripheriã cõtinuaueris:</s> <s xml:id="echoid-s14006" xml:space="preserve"> ſic enim peripheria angulo æqua-<lb/>to ſubtenſa æquabitur peripheriæ z p per 33 p 6] & ducatur linea g y, quę ꝓducta ad æqualitatẽ g d, <lb/>neceſſariò ſecabit lineã f l [quia ſecat angulũ z g l.</s> <s xml:id="echoid-s14007" xml:space="preserve">] Secet in pũcto x:</s> <s xml:id="echoid-s14008" xml:space="preserve"> & ſit g x k æqualis g d.</s> <s xml:id="echoid-s14009" xml:space="preserve"> Palàm, <lb/>quòd ſicut a reflectitur ad d, ab aliquo puncto arcus z p:</s> <s xml:id="echoid-s14010" xml:space="preserve"> ſimiliter reflectitur ad k, ab aliquo puncto <lb/>arcus z y.</s> <s xml:id="echoid-s14011" xml:space="preserve"> Dico, quòd non reflectetur ad ipſum, niſi à pũcto, quod eſt citra f, ex parte z.</s> <s xml:id="echoid-s14012" xml:space="preserve"> Si enim dica-<lb/>tur, quòd poſsit à puncto f, uel alio puncto arcus f y:</s> <s xml:id="echoid-s14013" xml:space="preserve"> linea ducta à pũcto a ad punctũ reflexionis, ſe-<lb/>cabit lineam b f:</s> <s xml:id="echoid-s14014" xml:space="preserve"> & ad idem punctum ſectionis reflectetur punctum k, & ad idẽ punctum reflectetur <lb/> <pb o="202" file="0208" n="208" rhead="ALHAZEN"/> punctum b.</s> <s xml:id="echoid-s14015" xml:space="preserve"> Et ita duo puncta in his ſpeculis reflectentur ad idem punctum ex eadem parte:</s> <s xml:id="echoid-s14016" xml:space="preserve"> quod <lb/>eſt impoſsibile [& contra 29 n 5.</s> <s xml:id="echoid-s14017" xml:space="preserve">] Reſtat, ut punctum a reflectatur ad k, ab aliquo puncto arcus z f.</s> <s xml:id="echoid-s14018" xml:space="preserve"> <lb/>Si ab illo puncto ducatur contingens:</s> <s xml:id="echoid-s14019" xml:space="preserve"> ſecabit lineam a z, & cadet inter z & c:</s> <s xml:id="echoid-s14020" xml:space="preserve"> quoniam punctum f <lb/>demiſsius eſt quolibet puncto arcus z f:</s> <s xml:id="echoid-s14021" xml:space="preserve"> & ita contingens à puncto f altior alijs, à punctis arcus z f <lb/>ductis.</s> <s xml:id="echoid-s14022" xml:space="preserve"> Cadat ergo contingens illa in punctum n:</s> <s xml:id="echoid-s14023" xml:space="preserve"> & ducatur linea m n:</s> <s xml:id="echoid-s14024" xml:space="preserve"> quę quidem linea cum tran-<lb/>ſeat per acumen trianguli b m t, & producta diuidat angulum, neceſſariò ſecabit b t.</s> <s xml:id="echoid-s14025" xml:space="preserve"> Secet in pun-<lb/>cto q:</s> <s xml:id="echoid-s14026" xml:space="preserve"> & ducatur linea g q.</s> <s xml:id="echoid-s14027" xml:space="preserve"> Sit autem i imago puncti a:</s> <s xml:id="echoid-s14028" xml:space="preserve"> o ſit imago puncti b:</s> <s xml:id="echoid-s14029" xml:space="preserve"> r ſit imago puncti q.</s> <s xml:id="echoid-s14030" xml:space="preserve"> Pa-<lb/>làm, cum b ſit propinquius puncto g, quàm a:</s> <s xml:id="echoid-s14031" xml:space="preserve"> erit o remotior à puncto g, quàm c [per 7 n.</s> <s xml:id="echoid-s14032" xml:space="preserve">] Ducatur <lb/>ergo linea i o.</s> <s xml:id="echoid-s14033" xml:space="preserve"> Palàm etiam [per 18 n 5.</s> <s xml:id="echoid-s14034" xml:space="preserve">16 p 5] quòd proportio a g ad a n, ſicut g i ad i n:</s> <s xml:id="echoid-s14035" xml:space="preserve"> & proportio <lb/>b<gap/>g ad b m, ſicut g o ad o m.</s> <s xml:id="echoid-s14036" xml:space="preserve"> Cum ergo lineæ a g, b g diuidantur ſecundũ hanc proportionem, utraq;</s> <s xml:id="echoid-s14037" xml:space="preserve"> <lb/>in duobus punctis, & à punctis diuiſionum ducantur lineæ, quarum duæ, ſcilicet a b, n m concur-<lb/>rant ad idem punctum, ſcilicet q:</s> <s xml:id="echoid-s14038" xml:space="preserve"> tertia neceſſariò concurret ad idem punctum [per 9 n.</s> <s xml:id="echoid-s14039" xml:space="preserve">] Igitur i o <lb/>ꝓducta cadet ſuper q.</s> <s xml:id="echoid-s14040" xml:space="preserve"> Quare i o q eſt recta linea.</s> <s xml:id="echoid-s14041" xml:space="preserve"> Igitur i o r nõ erit recta:</s> <s xml:id="echoid-s14042" xml:space="preserve"> ſed i o r eſt imago lineę a q.</s> <s xml:id="echoid-s14043" xml:space="preserve"> <lb/>Quare imago lineę a q erit curua.</s> <s xml:id="echoid-s14044" xml:space="preserve"> Poſito autẽ puncto b loco pũcti q, & aliquo pũcto lineæ a b poſito <lb/>loco pũcti b:</s> <s xml:id="echoid-s14045" xml:space="preserve"> erit eodẽ penitus modo probare, quòd imago lineæ a b eſt curua.</s> <s xml:id="echoid-s14046" xml:space="preserve"> Et hoc eſt propoſitũ.</s> <s xml:id="echoid-s14047" xml:space="preserve"/> </p> <div xml:id="echoid-div478" type="float" level="0" n="0"> <figure xlink:label="fig-0207-03" xlink:href="fig-0207-03a"> <variables xml:id="echoid-variables161" xml:space="preserve">l k x s y e t q b a f u r m h o m <gap/> <gap/> z g p d</variables> </figure> </div> </div> <div xml:id="echoid-div480" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables162" xml:space="preserve">ſ k x b a s <gap/> t c q f m <gap/> o h z i g p d</variables> </figure> <head xml:id="echoid-head430" xml:space="preserve" style="it">17. Si uiſ{us} ſit extra ſuperficem incidentiæ: imago lineæ rectæ infinitæ, ſecantis inæquabili-<lb/>ter peripheriam circuli (qui eſt communis ſectio <lb/> ſuperficierum, reflexionis & ſpeculi ſphærici con- uexi) uidebitur curua. 52 p 6.</head> <p> <s xml:id="echoid-s14048" xml:space="preserve">SI uerò a b ſecet circulum:</s> <s xml:id="echoid-s14049" xml:space="preserve"> ſecet in puncto e:</s> <s xml:id="echoid-s14050" xml:space="preserve"> m <lb/>finis contingentiæ lineæ contingentis circulũ <lb/>e h z, à puncto f productæ ad lineã b g:</s> <s xml:id="echoid-s14051" xml:space="preserve"> b igitur <lb/>reflectitur ad d ab aliquo pũcto arcus h p.</s> <s xml:id="echoid-s14052" xml:space="preserve"> Arcus ab <lb/>illo puncto reflexionis uſq;</s> <s xml:id="echoid-s14053" xml:space="preserve"> ad h, aut eſt æqualis ar-<lb/>cui h e:</s> <s xml:id="echoid-s14054" xml:space="preserve"> aut maior:</s> <s xml:id="echoid-s14055" xml:space="preserve"> aut minor.</s> <s xml:id="echoid-s14056" xml:space="preserve"> Si æqualis:</s> <s xml:id="echoid-s14057" xml:space="preserve"> palàm <lb/>quòd arcus ille eſt æqualis arcui h f [ut patuit præ-<lb/>cedente numero.</s> <s xml:id="echoid-s14058" xml:space="preserve">] Sit q punctum circuli, in quod <lb/>cadit contingẽs ducta à puncto m exparte e.</s> <s xml:id="echoid-s14059" xml:space="preserve"> Igitur <lb/>a e tranſit per punctũ q:</s> <s xml:id="echoid-s14060" xml:space="preserve"> & ita m q ſecat a e per pun-<lb/>ctum e [quia in hoc caſu q & e coniungũtur, unũq́;</s> <s xml:id="echoid-s14061" xml:space="preserve"> <lb/>punctum fiũt.</s> <s xml:id="echoid-s14062" xml:space="preserve">] Si uerò arcus ille minor eſt arcu h e:</s> <s xml:id="echoid-s14063" xml:space="preserve"> <lb/>ſecabit quidem m q lineam a e ultra punctum q:</s> <s xml:id="echoid-s14064" xml:space="preserve"> ſe-<lb/>cet in t, ut efficiatur triangulum e q t.</s> <s xml:id="echoid-s14065" xml:space="preserve"> Si uerò arcus ille fuerit maior arcu h e:</s> <s xml:id="echoid-s14066" xml:space="preserve"> ſecabit quidem linea in <lb/>q lineam a e citra punctum q.</s> <s xml:id="echoid-s14067" xml:space="preserve"> Siue hoc, ſiue illud fuerit:</s> <s xml:id="echoid-s14068" xml:space="preserve"> iteretur probatio, & eodem penitus modo <lb/>probabitur, quòd imago lineæ a b eſt curua.</s> <s xml:id="echoid-s14069" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s14070" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div481" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables163" xml:space="preserve">d a b e h z g</variables> </figure> <head xml:id="echoid-head431" xml:space="preserve" style="it">18. Si uiſ{us} ſit in ſuperficie incidentiæ, extra rectam lineam infinitam per centrum circuli <lb/>(qui eſt communis ſectio ſuperficierum, reflexionis & ſpeculi ſphæ-<lb/> riciconuexi) trãſeuntis: imago illi{us} lineæ uidebitur recta. 53 p 6.</head> <p> <s xml:id="echoid-s14071" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s14072" xml:space="preserve"> ſi in ſuperficie, in qua ſunt linea uiſa, & cẽtrum ſphæ-<lb/>ræ, fuerit uiſus:</s> <s xml:id="echoid-s14073" xml:space="preserve"> (ſuperiora enim dicta ſunt, non exiſtente uiſu <lb/>in illa ſuperficie) linea uiſa recta, aut concurret cum circulo <lb/>communi illi ſuperficiei & ſpeculo:</s> <s xml:id="echoid-s14074" xml:space="preserve"> aut non concurret.</s> <s xml:id="echoid-s14075" xml:space="preserve"> Si concurrat:</s> <s xml:id="echoid-s14076" xml:space="preserve"> <lb/>angulus illarum linearum [quem nimirum efficiunt diameter opti-<lb/>ca g d & data recta a b continuata per centrum g] cadet ſuper centrũ <lb/>ſpeculi:</s> <s xml:id="echoid-s14077" xml:space="preserve"> quæ quidem linea uidebitur recta.</s> <s xml:id="echoid-s14078" xml:space="preserve"> Imago enim cuiuslibet <lb/>puncti illius lineæ apparet in ipſa linea [per 6 n 5.</s> <s xml:id="echoid-s14079" xml:space="preserve">] Et ita imago il-<lb/>lius lineæ eſt recta.</s> <s xml:id="echoid-s14080" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div482" type="section" level="0" n="0"> <head xml:id="echoid-head432" xml:space="preserve" style="it">19. Si uiſ{us} ſit in ſuperficie incidẽtiæ: imago lineæ rectæ, infini-<lb/>tæ peripheriam circuli (qui eſt communis ſectio ſuperficierum, re-<lb/>flexionis & ſpeculi ſphærici conuexi) tangentis, & ad partem ui-<lb/>ſui oppoſitam obliquatæ, uidebitur punctum. 54 p 6.</head> <p> <s xml:id="echoid-s14081" xml:space="preserve">SI uerò linea ꝓpoſita declinata fuerit:</s> <s xml:id="echoid-s14082" xml:space="preserve"> aut erit declinatio ex par-<lb/>te uiſus:</s> <s xml:id="echoid-s14083" xml:space="preserve"> aut ex alia parte.</s> <s xml:id="echoid-s14084" xml:space="preserve"> Si ex alia parte:</s> <s xml:id="echoid-s14085" xml:space="preserve"> ſumatur punctum cir-<lb/>culi, à quo reflectatur aliquid uiſum:</s> <s xml:id="echoid-s14086" xml:space="preserve"> [per 39 n 5] & ſumatur li-<lb/>nea reflexionis aliqua.</s> <s xml:id="echoid-s14087" xml:space="preserve"> Aliqua linearum declinatarum cadet forſitan <lb/>ſuper hanc lineam reflexionis:</s> <s xml:id="echoid-s14088" xml:space="preserve"> quòd ſi fuerit:</s> <s xml:id="echoid-s14089" xml:space="preserve"> non uidebitur quidem hæc linea declinata, niſi ſecun-<lb/>dum unum punctum [ducta enim a g ſecante peripheriam circuli in puncto z:</s> <s xml:id="echoid-s14090" xml:space="preserve"> peripheria inter <lb/>punctum, à quo b reflectitur, & punctum z, continebit puncta reflexionis totius lineæ a b, ut pa-<lb/>tuit 16 n.</s> <s xml:id="echoid-s14091" xml:space="preserve">] Protracta igitur à centro uiſus ad centrum ſpeculi linea:</s> <s xml:id="echoid-s14092" xml:space="preserve"> ſumatur in arcu circuli citra <lb/>hanc lineam punctum, à quo reflectatur ad uiſum aliquod punctum lineæ declinatæ:</s> <s xml:id="echoid-s14093" xml:space="preserve"> ſed illud <lb/>punctum reflectitur à puncto prius aſsignato, quod eſt terminus lineæ reflexionis, cum li-<lb/>nea declinata ſit ſupra lineam reflexionis.</s> <s xml:id="echoid-s14094" xml:space="preserve"> Et ita illud punctum lineæ declinatæ reflectitur ad <lb/> <pb o="203" file="0209" n="209" rhead="OPTICAE LIBER VI."/> uiſum à duobus punctis arcus:</s> <s xml:id="echoid-s14095" xml:space="preserve"> quod eſt impoſsibile [& contra 29 n 5.</s> <s xml:id="echoid-s14096" xml:space="preserve">] Licet autẽ reflectatur pun-<lb/>ctum illud à puncto primùm ſumpto:</s> <s xml:id="echoid-s14097" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0209-01a" xlink:href="fig-0209-01"/> <anchor type="figure" xlink:label="fig-0209-02a" xlink:href="fig-0209-02"/> non tamen ũidetur, cum ſit in linea re <lb/>flexionis, quæ occultatur per præce-<lb/>dentia puncta.</s> <s xml:id="echoid-s14098" xml:space="preserve"> Et ita linea adiacens li-<lb/>neæ reflexionis non uidetur.</s> <s xml:id="echoid-s14099" xml:space="preserve"/> </p> <div xml:id="echoid-div482" type="float" level="0" n="0"> <figure xlink:label="fig-0209-01" xlink:href="fig-0209-01a"> <variables xml:id="echoid-variables164" xml:space="preserve">d a b e h z g</variables> </figure> <figure xlink:label="fig-0209-02" xlink:href="fig-0209-02a"> <variables xml:id="echoid-variables165" xml:space="preserve">a d b b g</variables> </figure> </div> </div> <div xml:id="echoid-div484" type="section" level="0" n="0"> <head xml:id="echoid-head433" xml:space="preserve" style="it">20. Si uiſ{us} ſit in ſuperficie inci-<lb/>dentiæ: imago lineæ rectæ infinitæ, <lb/>peripheriam circuli (qui eſt commu-<lb/>nis ſectio ſuperficierũ reflexionis & <lb/>ſpeculi ſphærici conuexi) ſiue tangen <lb/>tis, ſiue non, & ad uiſ{us} partemobli-<lb/>quatæ, nulla uidebitur. 55 p 6.</head> <p> <s xml:id="echoid-s14100" xml:space="preserve">SI uerò ſumatur linea declinata, <lb/>cuius declinatio ſit ex parte ui-<lb/>ſus, iacẽs ſub linea reflexionis, & <lb/>ſecans ipſam in puncto circuli Dico, <lb/>quòd nullũ punctum illius lineæ uide <lb/>bitur.</s> <s xml:id="echoid-s14101" xml:space="preserve"> Sumpto enim pũcto:</s> <s xml:id="echoid-s14102" xml:space="preserve"> ſi dicatur, <lb/>quòd punctum illud poſsit reflecti ab <lb/>aliquo puncto arcus, interiacentis lineam reflexionis, & lineam à centro uiſus ad centrum ſpeculi <lb/>ductam:</s> <s xml:id="echoid-s14103" xml:space="preserve"> & ducatur linea ab illo puncto ad punctũ arcus ſumptum:</s> <s xml:id="echoid-s14104" xml:space="preserve"> hæc ſecabit lineam reflexionis:</s> <s xml:id="echoid-s14105" xml:space="preserve"> <lb/>& punctum ſectionis reflectetur ad uiſum, à duobus punctis arcus ſpeculi:</s> <s xml:id="echoid-s14106" xml:space="preserve"> quod eſt impoſsibile [& <lb/>contra 29 n 5.</s> <s xml:id="echoid-s14107" xml:space="preserve">] Si uerò dicatur, quòd punctum ſumptum in linea, reflectatur à puncto arcus circuli, <lb/>qui eſt ſub ipſa linea:</s> <s xml:id="echoid-s14108" xml:space="preserve"> erit impoſsibile:</s> <s xml:id="echoid-s14109" xml:space="preserve"> quia ille totus arcus occultatur à linea.</s> <s xml:id="echoid-s14110" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div485" type="section" level="0" n="0"> <head xml:id="echoid-head434" xml:space="preserve" style="it">21. Si uiſ{us} ſit in ſuperficie incidentiæ: ιmago lιneæ rectæ infinitæ; peripheriam circuli (qui <lb/>eſt communis ſectio ſuperficierum, reflexionis & ſpeculi ſphærici conuexi) nec tangentis nec per <lb/>centrum ſecantis, & ad partem uιſuι oppoſitam obliquatæ, uidebitur curua. 56 p 6.</head> <p> <s xml:id="echoid-s14111" xml:space="preserve">SI uerò linea ſumpta nõ attingat circulum:</s> <s xml:id="echoid-s14112" xml:space="preserve"> poterit quidem uideri:</s> <s xml:id="echoid-s14113" xml:space="preserve"> ſed modicum eſt.</s> <s xml:id="echoid-s14114" xml:space="preserve"> Si uerò ſu-<lb/>matur linea declinata prædicta inter lineam reflexionis, & lineam per punctũ reflexionis pri-<lb/>mò ſumptum tranſeuntem ad centrum:</s> <s xml:id="echoid-s14115" xml:space="preserve"> poterit quidem uideri hæclinea:</s> <s xml:id="echoid-s14116" xml:space="preserve"> & imminuetur cur-<lb/>uitas imaginis huius lineæ, ſecundum quod magis acceſſerit ad lineã tranſeuntem ad centrum, per <lb/>punctum reflexionis.</s> <s xml:id="echoid-s14117" xml:space="preserve"> Si uerò ſumantur lineæ inter lineam ad centrũ <lb/> <anchor type="figure" xlink:label="fig-0209-03a" xlink:href="fig-0209-03"/> tranſeuntem per punctum reflexionis:</s> <s xml:id="echoid-s14118" xml:space="preserve"> uidebuntur quidem, ſiue de-<lb/>clinatio earum ſit ex parte uiſus, ſiue non:</s> <s xml:id="echoid-s14119" xml:space="preserve"> & modus uiſus earũ, ſimi-<lb/>lis modo uiſus linearum inter lineam reflexionis & lineam ad cen-<lb/>trum tranſeuntem.</s> <s xml:id="echoid-s14120" xml:space="preserve"> Et hæc quidem intelligenda ſunt de lineis con-<lb/>currentibus in arcu circuli, qui apparet uiſui, id eſt, in arcu, qui inter-<lb/>iacet duas contingentes, ductas à centro uiſus ad circulum.</s> <s xml:id="echoid-s14121" xml:space="preserve"> Linearũ <lb/>autem concurrentium cum circulo in parte circuli occulta uiſui:</s> <s xml:id="echoid-s14122" xml:space="preserve"> ali-<lb/>qua erit æquidiſtans lineæ reflexionis:</s> <s xml:id="echoid-s14123" xml:space="preserve"> & illa quidem non uidebitur.</s> <s xml:id="echoid-s14124" xml:space="preserve"> <lb/>Similiter conterminalis æquidiſtanti, quæ eſt ſub æquidiſtante, oc-<lb/>cultabitur:</s> <s xml:id="echoid-s14125" xml:space="preserve"> ſed conterminalis æquidiſtanti, ſupra ipſam exiſtens, po-<lb/>terit uideri.</s> <s xml:id="echoid-s14126" xml:space="preserve"> Si uerò ſumatur linea inter æquidiſtantes, nõ contermi-<lb/>nalis alicui earum:</s> <s xml:id="echoid-s14127" xml:space="preserve"> ſi fuerit eius declinatio ex parte uiſus, uidebitur:</s> <s xml:id="echoid-s14128" xml:space="preserve"> <lb/>ſi ex alia parte, aliquando uidebitur, aliquãdo non Quoniam ſi à ter-<lb/>mino eius ducatur æquidiſtans lineæ reflexionis:</s> <s xml:id="echoid-s14129" xml:space="preserve"> ſi fuerit linea ſub <lb/>æquidiſtante:</s> <s xml:id="echoid-s14130" xml:space="preserve"> non uidebitur:</s> <s xml:id="echoid-s14131" xml:space="preserve"> ſi ſupra eã, uideri poterit.</s> <s xml:id="echoid-s14132" xml:space="preserve"> Si uerò lineæ <lb/>non concurrant cum circulo, aut ſecabunt lineam ductam à centro <lb/>uiſus ad cẽtrum ſpeculi:</s> <s xml:id="echoid-s14133" xml:space="preserve"> aut æquidiſtabunt ei.</s> <s xml:id="echoid-s14134" xml:space="preserve"> Si ſecet aliqua earum:</s> <s xml:id="echoid-s14135" xml:space="preserve"> <lb/>linea illa aut ſecabit illam ex parte uiſus, id eſt, inter uiſum & ſpecu-<lb/>lum:</s> <s xml:id="echoid-s14136" xml:space="preserve"> aut ultra ſpeculũ.</s> <s xml:id="echoid-s14137" xml:space="preserve"> Si ultra:</s> <s xml:id="echoid-s14138" xml:space="preserve"> occultabitur linea illa, ſed forſan ap-<lb/>parebunt eius capita.</s> <s xml:id="echoid-s14139" xml:space="preserve"> Si uerò ſecet lineam uiſualem ex parte uiſus, apparebit quidem ſimiliter.</s> <s xml:id="echoid-s14140" xml:space="preserve"> Si <lb/>fuerit æquidιſtans lineæ uiſuali:</s> <s xml:id="echoid-s14141" xml:space="preserve"> poterit uideri.</s> <s xml:id="echoid-s14142" xml:space="preserve"> Omnium autem harum linearum imagines curuæ.</s> <s xml:id="echoid-s14143" xml:space="preserve"> <lb/>Vιſu autem exiſtente in eadem ſuperficie cum centro ſpeculi & lineis uiſis, diminuta eſt apparẽtia:</s> <s xml:id="echoid-s14144" xml:space="preserve"> <lb/>& quæ ſit, quæ manifeſtius apparet, eſt illa, quæ declinata eſt maxima declinatione, & illa uiſum re-<lb/>ſpiciente.</s> <s xml:id="echoid-s14145" xml:space="preserve"> Pari modo arcuum in his ſpeculis apparentium, & in eadem ſuperficie cum cẽtro ſpecu-<lb/>li, & uiſu exiſtẽtium, imagines quidẽ curuæ ſunt curuitate ſpeculũ reſpiciente.</s> <s xml:id="echoid-s14146" xml:space="preserve"> Hæc aũt intelligẽda <lb/>ſunt duplici uiſu exiſtẽte in eadẽ ſuperficie cũ cẽtro ſpeculi, & re uiſa.</s> <s xml:id="echoid-s14147" xml:space="preserve"> Si enim alter uiſus modicùm <lb/>declinetur, quò ad ipſum, alio modo res uiſa comprehendetur.</s> <s xml:id="echoid-s14148" xml:space="preserve"> Et uiſu exiſtente extra ſuperficiem <lb/>rei uiſæ & centrum ſpeculi, certior erit ipſius rei comprehenſio, quam exiſtente in ea.</s> <s xml:id="echoid-s14149" xml:space="preserve"/> </p> <div xml:id="echoid-div485" type="float" level="0" n="0"> <figure xlink:label="fig-0209-03" xlink:href="fig-0209-03a"> <variables xml:id="echoid-variables166" xml:space="preserve">a d f b ſ m e <gap/> c z g</variables> </figure> </div> </div> <div xml:id="echoid-div487" type="section" level="0" n="0"> <head xml:id="echoid-head435" xml:space="preserve" style="it">22. Si uiſ{us} ſit in ſuperficie incidentiæ: imago lineæ rectæ infinitæ, quæ uel non concurrens <lb/> <pb o="204" file="0210" n="210" rhead="ALHAZEN"/> cum ſuperficie ſpeculi ſphærici cõuexi, parallela eſt rectæ connectenti centra ſpeculi & uiſ{us}, uel <lb/>quæ cum eadem connectente extra ſpeculum, uerſ{us} uiſum concurrit: uidebitur curua. 57 p 6.</head> <p> <s xml:id="echoid-s14150" xml:space="preserve">QVòd autẽ imago rei uiſæ ſit curua, uiſu exi-<lb/> <anchor type="figure" xlink:label="fig-0210-01a" xlink:href="fig-0210-01"/> ſtente in ſuperficie cẽtri ſpeculi & rei uiſæ, <lb/>probabitur.</s> <s xml:id="echoid-s14151" xml:space="preserve"> Sit d centrũ uiſus:</s> <s xml:id="echoid-s14152" xml:space="preserve"> g centrũ ſpe-<lb/>culi:</s> <s xml:id="echoid-s14153" xml:space="preserve"> h e ſit linea uiſa:</s> <s xml:id="echoid-s14154" xml:space="preserve"> quæ quidẽ h e non cõcurrat cũ <lb/>circulo ſpeculi, ſed ſit æquidiſtãs lineę d g:</s> <s xml:id="echoid-s14155" xml:space="preserve"> uel ſecet <lb/>eã ex parte d.</s> <s xml:id="echoid-s14156" xml:space="preserve"> Superficies incidentiæ ſit, in qua ſint <lb/>lineæ d g, h e.</s> <s xml:id="echoid-s14157" xml:space="preserve"> Circulus cõmunis huic ſuperficiei & <lb/>ſpeculo ſit a b.</s> <s xml:id="echoid-s14158" xml:space="preserve"> Producatur linea h g, & punctum in <lb/>ipſa z ſit imago h:</s> <s xml:id="echoid-s14159" xml:space="preserve"> punctũ circuli à quo reflectitur h <lb/>ad d, ſit b.</s> <s xml:id="echoid-s14160" xml:space="preserve"> Et [per 17 p 3] à pũcto b ducatur linea cõ-<lb/>tingẽs, quæ ſecet lineã h g ſuper punctũ t:</s> <s xml:id="echoid-s14161" xml:space="preserve"> erit t finis <lb/>contingẽtiæ [ք 17 n 5.</s> <s xml:id="echoid-s14162" xml:space="preserve">] Ducatur linea b g:</s> <s xml:id="echoid-s14163" xml:space="preserve"> quę pro-<lb/>ducta neceſſario concurret cũ h e.</s> <s xml:id="echoid-s14164" xml:space="preserve"> Si enim h e fuerit <lb/>æquidiſtãs d g:</s> <s xml:id="echoid-s14165" xml:space="preserve"> cõcurret quidẽ:</s> <s xml:id="echoid-s14166" xml:space="preserve"> [ք lemma Procli ad <lb/>29 p 1] ſi uerò d g cõcurrat cũ h e:</s> <s xml:id="echoid-s14167" xml:space="preserve"> multò fortius g b <lb/>cũ eadẽ cõcurret.</s> <s xml:id="echoid-s14168" xml:space="preserve"> Cõcurſus ille aut erit in linea h e:</s> <s xml:id="echoid-s14169" xml:space="preserve"> <lb/>aut ultra hãc lineã.</s> <s xml:id="echoid-s14170" xml:space="preserve"> Sit ultra:</s> <s xml:id="echoid-s14171" xml:space="preserve"> cõcurrat in puncto m:</s> <s xml:id="echoid-s14172" xml:space="preserve"> <lb/>imago pũcti m ſit q:</s> <s xml:id="echoid-s14173" xml:space="preserve"> finis contingẽtiæ ſit s:</s> <s xml:id="echoid-s14174" xml:space="preserve"> & duca-<lb/>tur linea z q, & ſimiliter linea t s:</s> <s xml:id="echoid-s14175" xml:space="preserve"> & d g ſecet circulũ in a:</s> <s xml:id="echoid-s14176" xml:space="preserve"> & [per 17 p 3] ducatur à puncto a cõtingen<gap/> <lb/>a u.</s> <s xml:id="echoid-s14177" xml:space="preserve"> Palàm [è 24 n 4] quòd a b eſt minor quarta circuli:</s> <s xml:id="echoid-s14178" xml:space="preserve"> cum d uideat ex circulo minus medietate.</s> <s xml:id="echoid-s14179" xml:space="preserve"> <lb/>Quare angulus a g b eſt acutus:</s> <s xml:id="echoid-s14180" xml:space="preserve"> [ք 33 p 6] & [per 18 <lb/> <anchor type="figure" xlink:label="fig-0210-02a" xlink:href="fig-0210-02"/> p 3] angulus u a g eſt rectus.</s> <s xml:id="echoid-s14181" xml:space="preserve"> Igitur a u cõcurret cum <lb/>b g [per 11 ax.</s> <s xml:id="echoid-s14182" xml:space="preserve">] cõcurrat in puncto u.</s> <s xml:id="echoid-s14183" xml:space="preserve"> Dico, quòd pun <lb/>ctum u cadet ſupra punctũ s.</s> <s xml:id="echoid-s14184" xml:space="preserve"> Cũ enim m reflectatur <lb/>à puncto aliquo arcus a b [per 29 n 5] & a ſit demiſ-<lb/>ſius illo puncto:</s> <s xml:id="echoid-s14185" xml:space="preserve"> erit finis contingentiæ a, altior fine <lb/>contingentiæ illius puncti:</s> <s xml:id="echoid-s14186" xml:space="preserve"> & ita s demiſsius pũcto <lb/>u.</s> <s xml:id="echoid-s14187" xml:space="preserve"> Procedat ergo t s, donec concurrat cum linea a u:</s> <s xml:id="echoid-s14188" xml:space="preserve"> <lb/>[cõcurret aũt per 11 ax] & ſit cõcurſus in pũcto k:</s> <s xml:id="echoid-s14189" xml:space="preserve"> & <lb/>ducatur linea g k:</s> <s xml:id="echoid-s14190" xml:space="preserve"> quę producta concurrat cũ h m in <lb/>pũcto c:</s> <s xml:id="echoid-s14191" xml:space="preserve"> [cõcurret autẽ per lemma Procli ad 29 p 1] <lb/>Punctũ c reflectitur ad d ab aliquo puncto arcus a b <lb/>[per 29 n 5.</s> <s xml:id="echoid-s14192" xml:space="preserve">] Sit illud punctũ f:</s> <s xml:id="echoid-s14193" xml:space="preserve"> à quo ducatur linea <lb/>contingẽs uſq;</s> <s xml:id="echoid-s14194" xml:space="preserve"> ad g c, quę quidẽ erit demiſsior linea <lb/>a k:</s> <s xml:id="echoid-s14195" xml:space="preserve"> & erit punctũ o demiſsius pũcto k.</s> <s xml:id="echoid-s14196" xml:space="preserve"> Sit o finis cõ-<lb/>tingẽtiæ.</s> <s xml:id="echoid-s14197" xml:space="preserve"> Ducatur linea d f, quouſq;</s> <s xml:id="echoid-s14198" xml:space="preserve"> cadat ſuper g c:</s> <s xml:id="echoid-s14199" xml:space="preserve"> <lb/>cadat in punctũ r:</s> <s xml:id="echoid-s14200" xml:space="preserve"> & producatur z q uſq;</s> <s xml:id="echoid-s14201" xml:space="preserve"> ad lineã g c:</s> <s xml:id="echoid-s14202" xml:space="preserve"> <lb/>& cadat in punctum l.</s> <s xml:id="echoid-s14203" xml:space="preserve"> Dico quòd l eſt ſupra r.</s> <s xml:id="echoid-s14204" xml:space="preserve"> Lineæ <lb/>enim h c, t k, z l autſunt æquidiſtantes:</s> <s xml:id="echoid-s14205" xml:space="preserve"> aut cõcurrunt.</s> <s xml:id="echoid-s14206" xml:space="preserve"> Sint æquidiſtantes.</s> <s xml:id="echoid-s14207" xml:space="preserve"> Cũ ergo hæ æquidiſtãtes <lb/>ſecent lineam g c ſuper tria pun-<lb/> <anchor type="figure" xlink:label="fig-0210-03a" xlink:href="fig-0210-03"/> cta c, k, l, & ſecent utram q;</s> <s xml:id="echoid-s14208" xml:space="preserve"> linea-<lb/>rum m g, h g:</s> <s xml:id="echoid-s14209" xml:space="preserve"> & [per 18 n 5.</s> <s xml:id="echoid-s14210" xml:space="preserve"> 16 p <lb/>5] ꝓportio h g ad h t, ſicut g z ad <lb/>z t:</s> <s xml:id="echoid-s14211" xml:space="preserve"> ſimiliter m g ad m s, ſicut g q <lb/>ad q s:</s> <s xml:id="echoid-s14212" xml:space="preserve"> erit [ք 10 n] ꝓportio eadẽ <lb/>g c ad c k, ſicut l g ad l k.</s> <s xml:id="echoid-s14213" xml:space="preserve"> Sed pa-<lb/>làm [per 3 n 5] quòd r eſt imago <lb/>c:</s> <s xml:id="echoid-s14214" xml:space="preserve"> linea enim d f, linea reflexio-<lb/>nis, concurrit cum c g in puncto <lb/>r:</s> <s xml:id="echoid-s14215" xml:space="preserve"> & o finis contingentiæ.</s> <s xml:id="echoid-s14216" xml:space="preserve"> Quare <lb/>[per 18 n 5.</s> <s xml:id="echoid-s14217" xml:space="preserve"> 16 p 5] proportio g c <lb/>ad c o, ſicut g r ad r o:</s> <s xml:id="echoid-s14218" xml:space="preserve"> ſed [per 8 p <lb/>5] maior eſt proportio g c ad c k, <lb/>quàm g c ad c o:</s> <s xml:id="echoid-s14219" xml:space="preserve"> & ita maior g l <lb/>ad l k, quã g r ad r o:</s> <s xml:id="echoid-s14220" xml:space="preserve"> ergo maior <lb/>eſt proportio o r ad r g, quàm l k <lb/>ad l g:</s> <s xml:id="echoid-s14221" xml:space="preserve"> [quia per 26 p Cãpani in <lb/>quintũ librum elementorum, ratio l k ad g l minor eſt, quã ratio o r ad r g] & ita [per 18 p 5] maior <lb/>eſt proportio o g ad r g, quàm k g ad l g.</s> <s xml:id="echoid-s14222" xml:space="preserve"> Sed [per 9 ax.</s> <s xml:id="echoid-s14223" xml:space="preserve"> k g maior eſt o g.</s> <s xml:id="echoid-s14224" xml:space="preserve">] Quare [per 14 p 5] l g ma-<lb/>ior r g.</s> <s xml:id="echoid-s14225" xml:space="preserve"> Igitur r demiſsius eſt puncto l.</s> <s xml:id="echoid-s14226" xml:space="preserve"> Sed z q l eſt linea recta:</s> <s xml:id="echoid-s14227" xml:space="preserve"> igitur z q r eſt linea curua.</s> <s xml:id="echoid-s14228" xml:space="preserve"> Et ita imago <lb/>lineæ h c eſt curua.</s> <s xml:id="echoid-s14229" xml:space="preserve"> Poſito ergo aliquo puncto lineæ h e loco puncti m, & puncto e loco puncti c:</s> <s xml:id="echoid-s14230" xml:space="preserve"> e-<lb/>rit probare, quòd imago h e eſt curua.</s> <s xml:id="echoid-s14231" xml:space="preserve"> Si uerò lineæ h c, t s, z q concurrant:</s> <s xml:id="echoid-s14232" xml:space="preserve"> aut erit concurſus ex par <lb/> <pb o="205" file="0211" n="211" rhead="OPTICAE LIBER VI."/> te d:</s> <s xml:id="echoid-s14233" xml:space="preserve"> aut ex parte h g.</s> <s xml:id="echoid-s14234" xml:space="preserve"> Sit ex parte d:</s> <s xml:id="echoid-s14235" xml:space="preserve"> & ſit concurſus in puncto c:</s> <s xml:id="echoid-s14236" xml:space="preserve"> erit z q t linea recta:</s> <s xml:id="echoid-s14237" xml:space="preserve"> quare z q r erit <lb/>curua.</s> <s xml:id="echoid-s14238" xml:space="preserve"> Et ita imago lineæ h e curua.</s> <s xml:id="echoid-s14239" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s14240" xml:space="preserve"/> </p> <div xml:id="echoid-div487" type="float" level="0" n="0"> <figure xlink:label="fig-0210-01" xlink:href="fig-0210-01a"> <variables xml:id="echoid-variables167" xml:space="preserve">h e m c u t s k o b z ſ q r f g a d</variables> </figure> <figure xlink:label="fig-0210-02" xlink:href="fig-0210-02a"> <variables xml:id="echoid-variables168" xml:space="preserve">h e m c u s t b o q z r f g a d</variables> </figure> <figure xlink:label="fig-0210-03" xlink:href="fig-0210-03a"> <variables xml:id="echoid-variables169" xml:space="preserve">i h e m c t z u s b o k q <gap/> r f g a d</variables> </figure> </div> </div> <div xml:id="echoid-div489" type="section" level="0" n="0"> <head xml:id="echoid-head436" xml:space="preserve" style="it">23. Imago peripheriæ cum uiſu in eodem planoſitæ, intra ſpeculum ſphæricum conuexum ſen <lb/>ſiliter uiſa, curua uidetur. 58. 62 p 6.</head> <p> <s xml:id="echoid-s14241" xml:space="preserve">SI uerò proponatur arcus extra ſpeculum:</s> <s xml:id="echoid-s14242" xml:space="preserve"> erit probare de eo, quòd imago ſit curua, ſicut proba <lb/>tum eſt, uiſu non exiſtente in eadẽ ſuperficie cũ arcu & centro ſpeculi.</s> <s xml:id="echoid-s14243" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s14244" xml:space="preserve"> <lb/>Igitur in his ſpeculis lineæ rectæ apparent curuæ, & ſimiliter curuæ apparẽt ſimiliter curuę.</s> <s xml:id="echoid-s14245" xml:space="preserve"> Si <lb/>autem proponatur uiſui in his ſpeculis corpus curuũ, ſed longũ, modicam habens latitudinem:</s> <s xml:id="echoid-s14246" xml:space="preserve"> ap-<lb/>parebit quidem corporis illius curuitas manifeſtè, cũ ipſa diſcerni poſsit per ea, quæ ſupra corpus <lb/>ſunt, aut infra.</s> <s xml:id="echoid-s14247" xml:space="preserve"> Non enim planè diſcernitur curuitas, niſi magna, ubi occultæ fuerint extremitates lõ <lb/>gitudinis & latitudinis.</s> <s xml:id="echoid-s14248" xml:space="preserve"> Vnde propoſito uiſui corpore conuexitatis modicæ & quantitatis magnæ, <lb/>nõ planè diſcernitur eius conuexitas, licet imago ipſius ſit cõuexa, cũ non appareant termini cor-<lb/>poris in longitudine uel latitudine.</s> <s xml:id="echoid-s14249" xml:space="preserve"> Amplius:</s> <s xml:id="echoid-s14250" xml:space="preserve"> errores in ſpeculis planis accidentes, omnes accidunt <lb/>& in his:</s> <s xml:id="echoid-s14251" xml:space="preserve"> & præterillos, accidit imagines linearum rectarum eſſe curuas:</s> <s xml:id="echoid-s14252" xml:space="preserve"> quod à ſpeculis planis eſt <lb/>remotum.</s> <s xml:id="echoid-s14253" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div490" type="section" level="0" n="0"> <head xml:id="echoid-head437" xml:space="preserve">DE ERRORIBVS, QVI ACCIDVNT IN SPECVLIS CO-<lb/>lumnaribus conuexis. Cap. V.</head> <head xml:id="echoid-head438" xml:space="preserve" style="it">24. Si à duob{us} ellipſis cylindraceæ punctis ſint duæ perpendiculares: prima axi, continens <lb/>cum recta à ſecundo puncto, ad idem axis punctum ducta acutum angulum: ſecunda rectæ el-<lb/>lipſin in ſecundo puncto tangenti: ultra axem & dictum acutum angulum concurrent. 114 <lb/>p 1. 44 p 7.</head> <p> <s xml:id="echoid-s14254" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s14255" xml:space="preserve"> in ſpeculis columnaribus exterioribus errores accidunt ijdem, qui in ſpeculis ſphę-<lb/>ricis exterioribus.</s> <s xml:id="echoid-s14256" xml:space="preserve"> Lineæ enim rectæ uidentur curuæ, & diminuta apparet rei quantitas:</s> <s xml:id="echoid-s14257" xml:space="preserve"> ſed <lb/>longè fortius in his, quàm in eis.</s> <s xml:id="echoid-s14258" xml:space="preserve"> Quoniam in ſphęricis res magna apparebit quidem minor, <lb/>ſed non multò minor:</s> <s xml:id="echoid-s14259" xml:space="preserve"> ſed in his res etiam maxima uidebitur minima.</s> <s xml:id="echoid-s14260" xml:space="preserve"> Similiter linea recta appare-<lb/>bit curua in ſpeculis ſphæricis, ſed modicæ curuitatis:</s> <s xml:id="echoid-s14261" xml:space="preserve"> in columnaribus-maximæ curuitatis.</s> <s xml:id="echoid-s14262" xml:space="preserve"> Vnde <lb/>multiplicantur errores columnaris ſpeculi ſuper errores ſphærici.</s> <s xml:id="echoid-s14263" xml:space="preserve"> Verùm in columnaribus aliquan <lb/>do fit reflexio à linea recta, ſcilicet à longitudine ſpeculi:</s> <s xml:id="echoid-s14264" xml:space="preserve"> aliquando à circulo:</s> <s xml:id="echoid-s14265" xml:space="preserve"> aliquando à ſectione.</s> <s xml:id="echoid-s14266" xml:space="preserve"> <lb/>Quando linea uiſa fuerit æquidiſtans longitudini ſpeculi, fiet reflexio à linea longitudinis:</s> <s xml:id="echoid-s14267" xml:space="preserve"> & linea <lb/>uiſa apparebit recta, modicæ curuitatis.</s> <s xml:id="echoid-s14268" xml:space="preserve"> Et hæc quidem probabuntur:</s> <s xml:id="echoid-s14269" xml:space="preserve"> ad quorum probationẽ ne-<lb/>ceſſe quiddam præmitti:</s> <s xml:id="echoid-s14270" xml:space="preserve"> quod huiuſmodi eſt.</s> <s xml:id="echoid-s14271" xml:space="preserve"> Sumpta columnari ſectione, & ſumpto in ea puncto, <lb/>quod non ſit punctum reflexionis:</s> <s xml:id="echoid-s14272" xml:space="preserve"> ſi ab illo puncto ducatur linea ad perpendicularẽ, quæ eſt à pun-<lb/>cto reflexionis ad axem, & linea illa faciat angulum acutum cum perpendiculari:</s> <s xml:id="echoid-s14273" xml:space="preserve"> ſi ducatur à pun-<lb/>cto ſumpto linea, quæ ſit orthogonalis ſuper contingentem illud punctum:</s> <s xml:id="echoid-s14274" xml:space="preserve"> hæc linea concurret cũ <lb/>perpendiculari ſub axe, & ſub concurſu prioris lineæ cum perpendiculari.</s> <s xml:id="echoid-s14275" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s14276" xml:space="preserve"> ſit a e b ſe-<lb/>ctio:</s> <s xml:id="echoid-s14277" xml:space="preserve"> e punctum datum:</s> <s xml:id="echoid-s14278" xml:space="preserve"> n punctum uiſum:</s> <s xml:id="echoid-s14279" xml:space="preserve"> b punctum reflexionis:</s> <s xml:id="echoid-s14280" xml:space="preserve"> b d perpẽdicularis:</s> <s xml:id="echoid-s14281" xml:space="preserve"> e d b angulus <lb/>acutus:</s> <s xml:id="echoid-s14282" xml:space="preserve"> q e l contingens.</s> <s xml:id="echoid-s14283" xml:space="preserve"> Super b fiat circulus æquidiſtans baſi columnæ [ut oſtenſum eſt 47 n 5] ſci <lb/>licet b t o:</s> <s xml:id="echoid-s14284" xml:space="preserve"> & ducatur à puncto e linea longitudinis columnæ [ut eodem numero demonſtratũ eſt] <lb/>ſcilicet e t:</s> <s xml:id="echoid-s14285" xml:space="preserve"> ducatur axis d h:</s> <s xml:id="echoid-s14286" xml:space="preserve"> & [per 11 p 1] ducatur linea d g perpendicularis ſuper lineam b d, in ſu-<lb/>perficie circuli.</s> <s xml:id="echoid-s14287" xml:space="preserve"> Palàm, quod ſuperficies h d g eſt orthogonalis ſuper ſuperficiem circuli [per 18 p 11:</s> <s xml:id="echoid-s14288" xml:space="preserve"> <lb/>quia ducitur per axem perpendicularem circulo per 21 d 11.</s> <s xml:id="echoid-s14289" xml:space="preserve">] Superficies uerò contingens columnã <lb/>in puncto b, erit æquidiſtans huic ſuperficiei:</s> <s xml:id="echoid-s14290" xml:space="preserve"> quoniam linea longitudinis ducta à puncto b eſt ęqui <lb/>diſtans axi [per 21 d 11.</s> <s xml:id="echoid-s14291" xml:space="preserve">] Et contingens circulum ſuper b eſt æquidiſtans d g [per 28 p 1:</s> <s xml:id="echoid-s14292" xml:space="preserve"> recti enim <lb/>ſunt anguli g d b per fabricationem, & comprehenſus ſub tangente in puncto b & ſemidiametro cir <lb/>culi d b per 18 p 3.</s> <s xml:id="echoid-s14293" xml:space="preserve">] Igitur ſuperficies, in qua ſunt lineæ l e, e t non eſt æquidiſtans ſuperficiei h d g <lb/>[quia non eſt parallela ſuperficiei tangenti ellipſin in puncto b:</s> <s xml:id="echoid-s14294" xml:space="preserve"> cum angulus e d b ſit acutus ex the-<lb/>ſi.</s> <s xml:id="echoid-s14295" xml:space="preserve">] Concurretigitur cum ea.</s> <s xml:id="echoid-s14296" xml:space="preserve"> Concurrat in linea l g:</s> <s xml:id="echoid-s14297" xml:space="preserve"> & ducatur linea t g:</s> <s xml:id="echoid-s14298" xml:space="preserve"> quæ quidem erit contin-<lb/>gens:</s> <s xml:id="echoid-s14299" xml:space="preserve"> cum ſuperficies l e t ſit contingens.</s> <s xml:id="echoid-s14300" xml:space="preserve"> Du-<lb/> <anchor type="figure" xlink:label="fig-0211-01a" xlink:href="fig-0211-01"/> cta autem linea t d:</s> <s xml:id="echoid-s14301" xml:space="preserve"> erit angulus g t d rectus:</s> <s xml:id="echoid-s14302" xml:space="preserve"> <lb/>[per 18 p 3] quoniam t d diameter, [& t g tan-<lb/>git peripheriam in ipſius termino t.</s> <s xml:id="echoid-s14303" xml:space="preserve">] Fiat au-<lb/>tem ſuper e circulus æquidiſtans baſi colu-<lb/>mnæ [ut demonſtratum eſt 47 n 5] ſcilicet e <lb/>s p:</s> <s xml:id="echoid-s14304" xml:space="preserve"> punctum axis in hoc circulo ſit k:</s> <s xml:id="echoid-s14305" xml:space="preserve"> & du-<lb/>catur linea k e.</s> <s xml:id="echoid-s14306" xml:space="preserve"> Ducatur etiam linea d l:</s> <s xml:id="echoid-s14307" xml:space="preserve"> quæ <lb/>quidem ſecabit ſuperficiem circuli e s p:</s> <s xml:id="echoid-s14308" xml:space="preserve"> ſe-<lb/>cet in puncto f:</s> <s xml:id="echoid-s14309" xml:space="preserve"> ubicunque ſit punctum extra <lb/>circumferentiam uel intra:</s> <s xml:id="echoid-s14310" xml:space="preserve"> & ducantur lineæ <lb/>k f, e f:</s> <s xml:id="echoid-s14311" xml:space="preserve"> & [per 11 p 11] à puncto f ducaturper-<lb/>pendicularis ſuper ſuperficiem circuli b t o:</s> <s xml:id="echoid-s14312" xml:space="preserve"> quæſit f m:</s> <s xml:id="echoid-s14313" xml:space="preserve"> & ducatur linea t m.</s> <s xml:id="echoid-s14314" xml:space="preserve"> Palàm, quòd k d æqui-<lb/> <pb o="206" file="0212" n="212" rhead="ALHAZEN"/> diſtans eſt & æqualis f m:</s> <s xml:id="echoid-s14315" xml:space="preserve"> [Nam cum axis k d & recta f m ſint perpendiculares circulo b t o:</s> <s xml:id="echoid-s14316" xml:space="preserve"> ille per <lb/>21 d 11, hæc per fabricationem:</s> <s xml:id="echoid-s14317" xml:space="preserve"> erunt ipſæ inter ſe parallelæ per 6 p 11:</s> <s xml:id="echoid-s14318" xml:space="preserve"> & æquales per 34 p 1:</s> <s xml:id="echoid-s14319" xml:space="preserve"> quia circu <lb/>li b t o, e s p ſunt paralleli] & ita [per 33 p 1] k f æquidiſtans & æqualis d m.</s> <s xml:id="echoid-s14320" xml:space="preserve"> Similiter f m æquidiſtans <lb/>& æqualis e t:</s> <s xml:id="echoid-s14321" xml:space="preserve"> [per 30 p 1:</s> <s xml:id="echoid-s14322" xml:space="preserve"> quia e t latus cylindraceum parallelum eſt axi k d per 21 d 11] & k e æqualis <lb/>& æquidiſtans d t:</s> <s xml:id="echoid-s14323" xml:space="preserve"> & ita e f erit æquidiſtans & æqualis t m [per 33 p 1.</s> <s xml:id="echoid-s14324" xml:space="preserve">] Verùm ſuperficies k d l eſt or-<lb/>thogonalis ſuper ſuperficiem ſectionis b e t:</s> <s xml:id="echoid-s14325" xml:space="preserve"> [quia per axem ducitur, & angulus g d b in e llipſis pla-<lb/>no rectus eſt ex theſi] & eſt orthogonalis ſuper ſuperficiem circuli e s p [per 18 p 11:</s> <s xml:id="echoid-s14326" xml:space="preserve"> quia tranſit per <lb/>axem, perpendicularem circulo per 21 d 11.</s> <s xml:id="echoid-s14327" xml:space="preserve">] Ergo eſt perpendicularis ſuper lineam, communem ſe-<lb/>ctioni & circulo [per 19 p 11] quæ eſt e f.</s> <s xml:id="echoid-s14328" xml:space="preserve"> Igitur [per 3 d 11] angulus e f k rectus.</s> <s xml:id="echoid-s14329" xml:space="preserve"> Similiter angulus t m <lb/>d rectus [per 10 p 11:</s> <s xml:id="echoid-s14330" xml:space="preserve"> ſunt enim e f, f k parallelæ ipſis t m, m d, ut patuit, & in circulis parallelis.</s> <s xml:id="echoid-s14331" xml:space="preserve">] Cũ <lb/>igitur angulus d m t ſit rectus:</s> <s xml:id="echoid-s14332" xml:space="preserve"> & g t d rectus:</s> <s xml:id="echoid-s14333" xml:space="preserve"> [per 18 p 3] multiplicatio d m in m g erit, ſicut t m in <lb/>ſe.</s> <s xml:id="echoid-s14334" xml:space="preserve"> [Nam quia ab angulo g t d recto ducta eſt t m, perpẽdicularis baſi g d:</s> <s xml:id="echoid-s14335" xml:space="preserve"> erit per 8 p 6, ut d m ad m t, <lb/>ſic m t ad m g.</s> <s xml:id="echoid-s14336" xml:space="preserve"> Ita que per 17 p 6 rectangulum comprehenſum ſub extremis d m, g m æquatur qua-<lb/>drato mediæ t m.</s> <s xml:id="echoid-s14337" xml:space="preserve">] Sed quoniam f m æquidiſtat g l:</s> <s xml:id="echoid-s14338" xml:space="preserve"> [Nam cum g l ſit communis ſectio duorum pla-<lb/>norum, quorum alterum l e t g ſpeculum tangit, reliquum h d g l per axem ſecat:</s> <s xml:id="echoid-s14339" xml:space="preserve"> utrũque uerò per-<lb/>pendiculare eſt circulo b t o per 21 d.</s> <s xml:id="echoid-s14340" xml:space="preserve"> 18 p 11:</s> <s xml:id="echoid-s14341" xml:space="preserve"> erit ipſa g l eidem circulo perpendicularis per 19 p 11.</s> <s xml:id="echoid-s14342" xml:space="preserve"> <lb/>Quare per 6 p 11 erit parallela axi:</s> <s xml:id="echoid-s14343" xml:space="preserve"> ideoque per 30 p 1 ipſi f m] erit [per 2 p 6] proportio d f ad f l, ſi-<lb/>cut d m ad m g.</s> <s xml:id="echoid-s14344" xml:space="preserve"> Sed d f maior d m [per 19 p 1:</s> <s xml:id="echoid-s14345" xml:space="preserve"> quia angulus ad m rectus eſt perfabricationem.</s> <s xml:id="echoid-s14346" xml:space="preserve">] Igi-<lb/>tur fl maior m g [per 14 p 5.</s> <s xml:id="echoid-s14347" xml:space="preserve">] Igitur maior eſt multiplicatio d f in f l, quàm d m in m g:</s> <s xml:id="echoid-s14348" xml:space="preserve"> ergo maior ꝗ̃ <lb/>t m in ſe.</s> <s xml:id="echoid-s14349" xml:space="preserve"> Quare cum t m ſit æqualis e f [ex concluſo] erit multiplicatio d f in f l maior ductu lineæ e <lb/>fin ſe.</s> <s xml:id="echoid-s14350" xml:space="preserve"> Quare angulus l e d maior recto.</s> <s xml:id="echoid-s14351" xml:space="preserve"> Si enim rectus eſſet, cum linea e f ſit perpendicularis ſuper <lb/>l d [rectus enim demonſtratus eſt angulus e f k] eſſet ductus d fin fl æqualis quadrato e f [per 8.</s> <s xml:id="echoid-s14352" xml:space="preserve">17 <lb/>p 6.</s> <s xml:id="echoid-s14353" xml:space="preserve">] Reſtat ergo [per 13 p 1] ut angulus d e q ſit acutus.</s> <s xml:id="echoid-s14354" xml:space="preserve"> Igitur orthogonalis ducta à puncto e, ortho <lb/>gonalis, inquam, ſuper contingentem q l, cadet ſub linea e d, & concurret cum perpendiculari b d <lb/>ſub puncto d.</s> <s xml:id="echoid-s14355" xml:space="preserve"> [Quòd enim perpendicularis illa & b d concurrant, patet per 11 ax:</s> <s xml:id="echoid-s14356" xml:space="preserve"> quia anguli, e d b <lb/>& comprehenſus ab e d & dicta perpendiculari, ſunt acuti:</s> <s xml:id="echoid-s14357" xml:space="preserve"> ille per theſin, hic, quia pars eſt recti, cõ-<lb/>prehenſi à tangente e q & dicta perpendiculari.</s> <s xml:id="echoid-s14358" xml:space="preserve">] Quod eſt propoſitum.</s> <s xml:id="echoid-s14359" xml:space="preserve"> His præmiſsis accedẽdum <lb/>eſt ad propoſitum.</s> <s xml:id="echoid-s14360" xml:space="preserve"/> </p> <div xml:id="echoid-div490" type="float" level="0" n="0"> <figure xlink:label="fig-0211-01" xlink:href="fig-0211-01a"> <variables xml:id="echoid-variables170" xml:space="preserve">n q e ſ g t f m o K d h c a s u p z b</variables> </figure> </div> </div> <div xml:id="echoid-div492" type="section" level="0" n="0"> <head xml:id="echoid-head439" xml:space="preserve" style="it">25. Si uiſ{us}, & linea recta, axi ſpeculi cylindracei conuexi parallela, fuerint in eodem plana: <lb/>à toto cylindri latere ad uiſum reflecti poteſt: & imago uidetur linea recta, æqualis par alle-<lb/>læ. 50 p 7.</head> <p> <s xml:id="echoid-s14361" xml:space="preserve">PRoponatur columna:</s> <s xml:id="echoid-s14362" xml:space="preserve"> [ut in ſequente numero] linea æquidiſtans axi ſit t h.</s> <s xml:id="echoid-s14363" xml:space="preserve"> erit quidem æqui-<lb/>diſtans lineæ longitudinis columnæ [per 21 d 11.</s> <s xml:id="echoid-s14364" xml:space="preserve"> 30 p 1.</s> <s xml:id="echoid-s14365" xml:space="preserve">] Si ergo uiſus fuerit in eadem ſuperfi-<lb/>cie cum axe & linea t h:</s> <s xml:id="echoid-s14366" xml:space="preserve"> poterit quidem reflecti linea, & erit reflexio à linea longitudinis colu-<lb/>mnæ, quæ eſt linea communis ſuperficiei, in qua ſunt uiſus & axis, & ſuperficiei columnę, ſicut oſtẽ <lb/>ſum eſt in libro quinto [43.</s> <s xml:id="echoid-s14367" xml:space="preserve"> 89 n.</s> <s xml:id="echoid-s14368" xml:space="preserve">] Sicigitur uidebitur linea t h linea recta.</s> <s xml:id="echoid-s14369" xml:space="preserve"> Quoniam quælibet per-<lb/>pendicularis ducta à puncto lineæ t h, erit in eadem ſuperficie cum uiſu & axe.</s> <s xml:id="echoid-s14370" xml:space="preserve"> Et probabitur imagi <lb/>nem lineæ t h eſſe rectam, ſicut probatum eſt in ſpeculis planis de rectis lineis [2 n.</s> <s xml:id="echoid-s14371" xml:space="preserve">]</s> </p> </div> <div xml:id="echoid-div493" type="section" level="0" n="0"> <head xml:id="echoid-head440" xml:space="preserve" style="it">26. Si uiſ{us} ſit extra planum lineæ rectæ, axi ſpeculi cylindracei conuexi parallelæ: à latere cy <lb/>lindri fit reflexio. 30 p 7.</head> <p> <s xml:id="echoid-s14372" xml:space="preserve">SI autem uiſus ſit extra ſuperficiem lineæ t h, & axis:</s> <s xml:id="echoid-s14373" xml:space="preserve"> & t h æquidiſtet axi:</s> <s xml:id="echoid-s14374" xml:space="preserve"> qui axis ſit z k:</s> <s xml:id="echoid-s14375" xml:space="preserve"> fiat ſu-<lb/>perficies per uiſum tranſiens, ſecans ſuperficiem columnæ æquidiſtanter baſi:</s> <s xml:id="echoid-s14376" xml:space="preserve"> [ut oſtenſum eſt <lb/>47 n 5] ſecabit quidem ſecundum circulum [per 5 th.</s> <s xml:id="echoid-s14377" xml:space="preserve"> Sereni de ſectione cylindri.</s> <s xml:id="echoid-s14378" xml:space="preserve">] Sit circu-<lb/>lus ille b f.</s> <s xml:id="echoid-s14379" xml:space="preserve"> Aliquod igitur punctum lineæ h t reflectitur ad uiſum, ab aliquo puncto huius circuli:</s> <s xml:id="echoid-s14380" xml:space="preserve"> ſit <lb/>punctum b:</s> <s xml:id="echoid-s14381" xml:space="preserve"> & uiſus ſit e:</s> <s xml:id="echoid-s14382" xml:space="preserve"> punctum illud lineæ t h, ſit q:</s> <s xml:id="echoid-s14383" xml:space="preserve"> & ducantur lineæ e b, q b, q e.</s> <s xml:id="echoid-s14384" xml:space="preserve"> Et ducatur à pũ <lb/>cto b linea longitudinis [ut monſtratum eſt 47 n 5] quæ ſit a b g:</s> <s xml:id="echoid-s14385" xml:space="preserve"> & ducatur à puncto b perpendicu-<lb/>laris, cadens ſuper axem in puncto l [cadet uerò per lẽma Procli ad 29 p 1:</s> <s xml:id="echoid-s14386" xml:space="preserve"> quia latus cylindraceũ & <lb/>axis ſunt paralleli per 21 d 11] quæ ſit m l:</s> <s xml:id="echoid-s14387" xml:space="preserve"> & ducatur à puncto e linea æquidiſtans l m:</s> <s xml:id="echoid-s14388" xml:space="preserve"> quæ ſit e o:</s> <s xml:id="echoid-s14389" xml:space="preserve"> & <lb/>ducatur q b, quouſque concurrat [concurret autem per allegatum Procli lemma] ſit concurſus in <lb/>puncto o.</s> <s xml:id="echoid-s14390" xml:space="preserve"> Palàm, quòd angulus q b m eſt æqualis angulo e b m:</s> <s xml:id="echoid-s14391" xml:space="preserve"> [anguli enim m b g, m b a recti per <lb/>fabricationem & 29 p 1, æquantur per 10 ax.</s> <s xml:id="echoid-s14392" xml:space="preserve"> itemq́ue q b g, e b a per 12 n 4:</s> <s xml:id="echoid-s14393" xml:space="preserve"> quare reliqui q b m, e b m <lb/>æquantur.</s> <s xml:id="echoid-s14394" xml:space="preserve">] Sed [per 29 p 1] angulus q b m æqualis eſt angulo b o e:</s> <s xml:id="echoid-s14395" xml:space="preserve"> quia l m æquidiſtans o e.</s> <s xml:id="echoid-s14396" xml:space="preserve"> Simi <lb/>liter [per eandem 29] angulus m b e æqualis angulo b e o:</s> <s xml:id="echoid-s14397" xml:space="preserve"> quia coalternus.</s> <s xml:id="echoid-s14398" xml:space="preserve"> Igitur angulus b o e æ-<lb/>qualis eſt angulo b e o.</s> <s xml:id="echoid-s14399" xml:space="preserve"> Quare [per 6 p 1] latera b o, b e æqualia.</s> <s xml:id="echoid-s14400" xml:space="preserve"> Sumaturautem aliud punctum in <lb/>linea t h:</s> <s xml:id="echoid-s14401" xml:space="preserve"> quod ſit t:</s> <s xml:id="echoid-s14402" xml:space="preserve"> & ducatur linea t o.</s> <s xml:id="echoid-s14403" xml:space="preserve"> Palàm, quòd linea t h æquidiſtat lineæ longitudinis, quæ eſt <lb/>a g [per 30 p 1:</s> <s xml:id="echoid-s14404" xml:space="preserve"> quia t h ex theſi parallela eſt axi, cui latus cylindraceum parallelum eſt per 21 d 11.</s> <s xml:id="echoid-s14405" xml:space="preserve">] <lb/>Ergo ſunt in eadem ſuperficie:</s> <s xml:id="echoid-s14406" xml:space="preserve"> [per 35 d 1] & in illa ſuperficie eſt linea q b o [per 7 p 11:</s> <s xml:id="echoid-s14407" xml:space="preserve"> quia conne-<lb/>ctit t h & a g.</s> <s xml:id="echoid-s14408" xml:space="preserve">] Quare in eadem erit linea t q [per 1 p 11.</s> <s xml:id="echoid-s14409" xml:space="preserve">] Secabit igitur lineam a g.</s> <s xml:id="echoid-s14410" xml:space="preserve"> Secet in puncto <lb/>g.</s> <s xml:id="echoid-s14411" xml:space="preserve"> Ducatur linea e g.</s> <s xml:id="echoid-s14412" xml:space="preserve"> Palàm etiam [per 8 p 11] quòd linea a g eſt perpendicularis ſuper ſuperficiem <lb/>circuli b f, ſicut axis, cui æquidiſtat, [per 21 d 11.</s> <s xml:id="echoid-s14413" xml:space="preserve">] Et ſuperficies illius circuli, eſt pars ſuperficiei, e <lb/>o b f, ſecans ſcilicet columnam æquidiſtanter baſi.</s> <s xml:id="echoid-s14414" xml:space="preserve"> Igitur [per 3 d 11] angulus g b o eſt rectus, & an-<lb/> <pb o="207" file="0213" n="213" rhead="OPTICAE LIBER VI."/> gulus g b e eſt rectus.</s> <s xml:id="echoid-s14415" xml:space="preserve"> Ergo [per 47 p 1] quadratum lineæ g o ualet quadratum lineæ b g & qua-<lb/>dratum lineæ b o.</s> <s xml:id="echoid-s14416" xml:space="preserve"> Similiter quadratum g e ua <lb/> <anchor type="figure" xlink:label="fig-0213-01a" xlink:href="fig-0213-01"/> let quadrata g b & b e.</s> <s xml:id="echoid-s14417" xml:space="preserve"> Et quoniam b e & b o <lb/>ſunt æquales:</s> <s xml:id="echoid-s14418" xml:space="preserve"> [per concluſionem] & g b com <lb/>munis:</s> <s xml:id="echoid-s14419" xml:space="preserve"> erit g o ęqualis g e [quia ipſarum qua-<lb/>drata æqualia.</s> <s xml:id="echoid-s14420" xml:space="preserve">] Igitur [per 5 p 1] angulus g o <lb/>e ęqualis angulo g e o.</s> <s xml:id="echoid-s14421" xml:space="preserve"> Ducta autem perpen-<lb/>diculari ſuper axem z g n:</s> <s xml:id="echoid-s14422" xml:space="preserve"> æquidiſtãs erit e o:</s> <s xml:id="echoid-s14423" xml:space="preserve"> <lb/>[per 30 p 1] cum ſit æquidiſtans m b l.</s> <s xml:id="echoid-s14424" xml:space="preserve"> Igitur <lb/>[per 29 p 1] angulus t g n æqualis angulo g o <lb/>e:</s> <s xml:id="echoid-s14425" xml:space="preserve"> & angulus n g e æqualis angulo g e o:</s> <s xml:id="echoid-s14426" xml:space="preserve"> quare <lb/>angulus t g n æqualis n g e.</s> <s xml:id="echoid-s14427" xml:space="preserve"> Cum autem t g o, <lb/>n g z ſint in eadem ſuperficie, in qua g.</s> <s xml:id="echoid-s14428" xml:space="preserve"> Ergo <lb/>puncta o, g, terunt in eadẽ ſuperficie:</s> <s xml:id="echoid-s14429" xml:space="preserve"> & ita in <lb/>eadẽ ſuperficie ſunt lineę e g, o g t g [ք 1 p 11.</s> <s xml:id="echoid-s14430" xml:space="preserve">] <lb/>Igitur t reflectitur ad e à pũcto g.</s> <s xml:id="echoid-s14431" xml:space="preserve"> Sumpto aũt <lb/>in linea th puncto h eiuſdem longitudinis à puncto q, cuius eſt punctũ t, & linea ducta h o:</s> <s xml:id="echoid-s14432" xml:space="preserve"> tranſibit <lb/>quidẽ per punctũ lineæ a g:</s> <s xml:id="echoid-s14433" xml:space="preserve"> tranſeat per punctũ a:</s> <s xml:id="echoid-s14434" xml:space="preserve"> ductaq́;</s> <s xml:id="echoid-s14435" xml:space="preserve"> à puncto a ſuper axẽ perpendiculari d a, <lb/>& linea e a:</s> <s xml:id="echoid-s14436" xml:space="preserve"> erit, ſicut prius, probare:</s> <s xml:id="echoid-s14437" xml:space="preserve"> quòd duo anguli a b o, a b e recti:</s> <s xml:id="echoid-s14438" xml:space="preserve"> & duo latera a o, a e æqualia:</s> <s xml:id="echoid-s14439" xml:space="preserve"> <lb/>& duo anguli h a r, e a r æquales:</s> <s xml:id="echoid-s14440" xml:space="preserve"> & ita h reflectetur ad e à puncto a.</s> <s xml:id="echoid-s14441" xml:space="preserve"> Similiter ſumpto quocunq, pun <lb/>cto lineę t h:</s> <s xml:id="echoid-s14442" xml:space="preserve"> erit probare, quòd reflectatur ab aliquo puncto lineę a g.</s> <s xml:id="echoid-s14443" xml:space="preserve"> Quare linea th reflectetur à <lb/>linea longitudinis, quæ eſt a g.</s> <s xml:id="echoid-s14444" xml:space="preserve"/> </p> <div xml:id="echoid-div493" type="float" level="0" n="0"> <figure xlink:label="fig-0213-01" xlink:href="fig-0213-01a"> <variables xml:id="echoid-variables171" xml:space="preserve">t n q z g m b ſ f h r a d e k o</variables> </figure> </div> </div> <div xml:id="echoid-div495" type="section" level="0" n="0"> <head xml:id="echoid-head441" xml:space="preserve" style="it">27. Si uiſ{us} ſit extra planum lineæ rectæ, axi ſpeculi cylindracei conuexi parallelæ: imago ui-<lb/>debitur parum curua, & minor ipſaparallela. 51 p 7.</head> <p> <s xml:id="echoid-s14445" xml:space="preserve">REſtat probare imaginem lineę t h eſſe curuã.</s> <s xml:id="echoid-s14446" xml:space="preserve"> Palàm ex prædictis, quòd q reflectitur ad e à pun <lb/>cto b, quod eſt punctum circuli.</s> <s xml:id="echoid-s14447" xml:space="preserve"> Sed cum ſic reflectatur à circulo:</s> <s xml:id="echoid-s14448" xml:space="preserve"> ſi ducatur linea à puncto q, <lb/>ad centrum illius circuli:</s> <s xml:id="echoid-s14449" xml:space="preserve"> concurret cum perpendiculari ducta à puncto b:</s> <s xml:id="echoid-s14450" xml:space="preserve"> [quia perpendicu <lb/>laris illa tranſit per eiuſdem circuli centrum, ut oſtenſum eſt 16 n 5] & erit cõcurſus in puncto axis.</s> <s xml:id="echoid-s14451" xml:space="preserve"> <lb/>Ducatur ergo q l, concurrens cum m l in puncto axis:</s> <s xml:id="echoid-s14452" xml:space="preserve"> quod eſt l:</s> <s xml:id="echoid-s14453" xml:space="preserve"> & eſt centrum circuli f b:</s> <s xml:id="echoid-s14454" xml:space="preserve"> & produ-<lb/>catur e b, quouſq;</s> <s xml:id="echoid-s14455" xml:space="preserve"> concurrat cum q l.</s> <s xml:id="echoid-s14456" xml:space="preserve"> Sit concurſus in puncto c.</s> <s xml:id="echoid-s14457" xml:space="preserve"> Erit c imago q:</s> <s xml:id="echoid-s14458" xml:space="preserve"> & eſt c in ſuperficie, <lb/>in qua ſunt lineæ q h, & axis, & linea longitudinis a g [per 1 p 11.</s> <s xml:id="echoid-s14459" xml:space="preserve">] Palàm etiam [è 31 n 4] quod t refle <lb/>ctitur ad e, à puncto ſectionis columnaris, ſcilicet à puncto g.</s> <s xml:id="echoid-s14460" xml:space="preserve"> Eſt autem à puncto t unam ducere per <lb/>pendicularem, ſuper lineam contingentem in aliquo puncto ſectionem:</s> <s xml:id="echoid-s14461" xml:space="preserve"> quæ quidem concurret cũ <lb/>perpendiculari ducta à puncto g:</s> <s xml:id="echoid-s14462" xml:space="preserve"> quæ eſt n g z, ſub axe, id eſt, ſub puncto z:</s> <s xml:id="echoid-s14463" xml:space="preserve"> quod eſt concurſus per-<lb/>pendicularis n z & axis [per 24 n.</s> <s xml:id="echoid-s14464" xml:space="preserve">] Quoniam ducta linea t z:</s> <s xml:id="echoid-s14465" xml:space="preserve"> erit angulus t z n acutus:</s> <s xml:id="echoid-s14466" xml:space="preserve"> [quia conti-<lb/>nuato axe k z ultra z in y:</s> <s xml:id="echoid-s14467" xml:space="preserve"> erit angulus n z y rectus per fabricationẽ & 29 p 1.</s> <s xml:id="echoid-s14468" xml:space="preserve">] Producatur n z ultra z <lb/>in x.</s> <s xml:id="echoid-s14469" xml:space="preserve"> Ducatur ergo t x, concurrens cum n z in puncto x:</s> <s xml:id="echoid-s14470" xml:space="preserve"> & producatur e g, donec concurrat cum <lb/>t x in puncto i.</s> <s xml:id="echoid-s14471" xml:space="preserve"> Erit i imago puncti t [per 4 n 5.</s> <s xml:id="echoid-s14472" xml:space="preserve">] Similiter ducta à puncto h linea, quæ ſit orthogona <lb/>lis ſuper lineam, contingentem ſpeculum in puncto aliquo ſectionis, à quo h reflectitur ad e:</s> <s xml:id="echoid-s14473" xml:space="preserve"> cõcur-<lb/>ret cum perpendiculari d a r, ſub puncto d, quod eſt punctum axis [per 24 n.</s> <s xml:id="echoid-s14474" xml:space="preserve">] Concurrat in puncto <lb/>p:</s> <s xml:id="echoid-s14475" xml:space="preserve"> & producatur e a, donec concurrat cũ h p in puncto s.</s> <s xml:id="echoid-s14476" xml:space="preserve"> Erit imago puncti h punctum s [per 4 n 5.</s> <s xml:id="echoid-s14477" xml:space="preserve">] <lb/> <anchor type="figure" xlink:label="fig-0213-02a" xlink:href="fig-0213-02"/> Ducatur autem linea s t.</s> <s xml:id="echoid-s14478" xml:space="preserve"> Palàm, cum linea t i concurrat cum perpendiculari n z, quæ eſt æquidiſtãs <lb/>lineę e o:</s> <s xml:id="echoid-s14479" xml:space="preserve"> concurret cum linea e o [per lemma Procli ad 29 p 1.</s> <s xml:id="echoid-s14480" xml:space="preserve">] Sit concurſus in u.</s> <s xml:id="echoid-s14481" xml:space="preserve"> Similiter linea h <lb/>s, quoniam concurrit cum perpendiculari d a r, quæ eſt æquidiſtans e o:</s> <s xml:id="echoid-s14482" xml:space="preserve"> cõcurret cum e o.</s> <s xml:id="echoid-s14483" xml:space="preserve"> Sed quo-<lb/>niam ſitus t, reſpectu puncti e, idem eſt cum ſitu h & eadem longitudo:</s> <s xml:id="echoid-s14484" xml:space="preserve"> [quia th parallela eſt axi ex <lb/>theſi.</s> <s xml:id="echoid-s14485" xml:space="preserve">] Similiter ſitus puncti t & puncti h ad punctum q idem [ut præcedente numero patuit] & pũ <lb/>ctorum i, s, reſpectu o, etiam eſt idem:</s> <s xml:id="echoid-s14486" xml:space="preserve"> erit idem ſitus linearum t i, h s, reſpectu lineæ e o.</s> <s xml:id="echoid-s14487" xml:space="preserve"> Igitur li-<lb/> <pb o="208" file="0214" n="214" rhead="ALHAZEN"/> neæ t i, h s cõcurrent ſuper idem punctũ lineæ e o.</s> <s xml:id="echoid-s14488" xml:space="preserve"> Concurrant in puncto u.</s> <s xml:id="echoid-s14489" xml:space="preserve"> Erit ergo t u h triangu-<lb/>lum, & in ſuperficie huius trianguli erit linea i s.</s> <s xml:id="echoid-s14490" xml:space="preserve"> Axis autem non eſt in eadem ſuperficie:</s> <s xml:id="echoid-s14491" xml:space="preserve"> uerùm t h <lb/>eſt in eadem ſuperficie cum axe.</s> <s xml:id="echoid-s14492" xml:space="preserve"> [ex theſi.</s> <s xml:id="echoid-s14493" xml:space="preserve">] Igitur ſuperficies illa ſecat ſuperficiem trianguli, ſuper <lb/>lineam communem:</s> <s xml:id="echoid-s14494" xml:space="preserve"> quæ eſt t h, non ſuper aliam.</s> <s xml:id="echoid-s14495" xml:space="preserve"> Cum ergo punctum c ſit in ſuperficie lineæ t h & <lb/>axis, & non ſit in linea t h:</s> <s xml:id="echoid-s14496" xml:space="preserve"> non eſt in ſuperficie trianguli t u h:</s> <s xml:id="echoid-s14497" xml:space="preserve"> & duo puncta i, s ſunt in ſuperficie il-<lb/>lius trianguli.</s> <s xml:id="echoid-s14498" xml:space="preserve"> Quare linea i c s eſt linea curua:</s> <s xml:id="echoid-s14499" xml:space="preserve"> & imago lineæ t h erit curua.</s> <s xml:id="echoid-s14500" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s14501" xml:space="preserve"> <lb/>Sed eius curuitas eſt modica:</s> <s xml:id="echoid-s14502" xml:space="preserve"> quia perpendicularis ducta à puncto c ad punctum ſectionis lineæ i s <lb/>& ſuperficiei circuli, eſt ualde parua.</s> <s xml:id="echoid-s14503" xml:space="preserve"> Et quantò maior fuerit linea uiſa, æquidiſtans lineæ longitudi <lb/>nis ſpeculi:</s> <s xml:id="echoid-s14504" xml:space="preserve"> tantò imago eius erit minus curua:</s> <s xml:id="echoid-s14505" xml:space="preserve"> & quantò minor, tantò magis.</s> <s xml:id="echoid-s14506" xml:space="preserve"/> </p> <div xml:id="echoid-div495" type="float" level="0" n="0"> <figure xlink:label="fig-0213-02" xlink:href="fig-0213-02a"> <variables xml:id="echoid-variables172" xml:space="preserve">t i y n q g z x m b c ſ f h s r a d p e k o u</variables> </figure> </div> </div> <div xml:id="echoid-div497" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables173" xml:space="preserve">f d b g t e h e</variables> </figure> <head xml:id="echoid-head442" xml:space="preserve" style="it">28. Si uiſ{us} ſit in communi ſectione planorum, lineæ rectæ & axis ſpeculi cylindracei conuexi, <lb/>inter ſeperpendicularium: fiet reflexio à peripheria circuli, qui eſt <lb/> communis ſectio plani lineæ & ſuperficiei ſpeculi: & imago uidebi- tur curua. 52 p 7.</head> <p> <s xml:id="echoid-s14507" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s14508" xml:space="preserve"> ſi linea t h ſecet ſuperficiem, in qua ſunt centrum ui-<lb/>ſus & axis, & ſit orthogonalis ſuper eam.</s> <s xml:id="echoid-s14509" xml:space="preserve"> Viſus aut erit in illa <lb/>ſuperficie lineæ t h, ſecante orthogonaliter ſuperficiem axis <lb/>& uiſus:</s> <s xml:id="echoid-s14510" xml:space="preserve"> aut extra.</s> <s xml:id="echoid-s14511" xml:space="preserve"> Si fuerit in ſuperficie illa:</s> <s xml:id="echoid-s14512" xml:space="preserve"> aut ſupra lineam t h:</s> <s xml:id="echoid-s14513" xml:space="preserve"> aut <lb/>infra.</s> <s xml:id="echoid-s14514" xml:space="preserve"> Si ſupra, cum illa linea ſit corporalis, occultabit uiſui ſpeculũ:</s> <s xml:id="echoid-s14515" xml:space="preserve"> <lb/>& ita non reflectetur, ſed forſan capita eius apparebunt & reflecten-<lb/>tur à circulo columnæ, qui communis eſt ſuperficiei lineæ t h, ſecanti <lb/>columnam, & columnæ.</s> <s xml:id="echoid-s14516" xml:space="preserve"> Et erit horum capitum imago, ſicut in ſphæ <lb/>ricis exterioribus [21 n.</s> <s xml:id="echoid-s14517" xml:space="preserve">] Similiter ſi uiſus fuerit ſub linea t h:</s> <s xml:id="echoid-s14518" xml:space="preserve"> occul-<lb/>tabitur pars eius propter caput, in quo eſt uiſus.</s> <s xml:id="echoid-s14519" xml:space="preserve"> Pars aũt lineæ uiſæ <lb/>reflectitur à circulo, eodẽ penitus modo, quo in exteriorib.</s> <s xml:id="echoid-s14520" xml:space="preserve"> ſphęricis.</s> <s xml:id="echoid-s14521" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div498" type="section" level="0" n="0"> <head xml:id="echoid-head443" xml:space="preserve" style="it">29. Si uiſ{us} æquabiliter diſtans à terminis lineæ rectæ, ſit extra <lb/>eiuſdem planum, perpendiculare plano axis ſpeculi cylindracei cõ-<lb/>uexi: imago maximè curua uidebitur. 53 p 7.</head> <p> <s xml:id="echoid-s14522" xml:space="preserve">SI uerò uiſus fuerit extra ſuperficiem lineę t h, orthogonaliter ſe-<lb/>cantem ſuperficiem uiſus & axis:</s> <s xml:id="echoid-s14523" xml:space="preserve"> ſit e uiſus:</s> <s xml:id="echoid-s14524" xml:space="preserve"> & b g x columna:</s> <s xml:id="echoid-s14525" xml:space="preserve"> reflectetur h ad e ab aliquo pun-<lb/>cto columnæ:</s> <s xml:id="echoid-s14526" xml:space="preserve"> ſit à puncto b:</s> <s xml:id="echoid-s14527" xml:space="preserve"> & ſit t eiuſdem longitudinis à puncto e, cuius eſt h.</s> <s xml:id="echoid-s14528" xml:space="preserve"> Dico, quòd t <lb/>reflectetur ad e ab aliquo puncto columnæ.</s> <s xml:id="echoid-s14529" xml:space="preserve"> Et cum puncta h, t ſint eiuſdem ſitus & eiuſdem lon-<lb/>gitudinis à puncto e:</s> <s xml:id="echoid-s14530" xml:space="preserve"> erunt ſimiliter puncta reflexionum, ſcilicet b, g eiuſdem longitudinis & eiuſ-<lb/>dem ſitus à puncto e.</s> <s xml:id="echoid-s14531" xml:space="preserve"> Igitur duo puncta b, g erunt in circulo.</s> <s xml:id="echoid-s14532" xml:space="preserve"> Sit circulus b z g:</s> <s xml:id="echoid-s14533" xml:space="preserve"> eius centrum d:</s> <s xml:id="echoid-s14534" xml:space="preserve"> & <lb/>ducantur lineæ h b, b e, t g, g e:</s> <s xml:id="echoid-s14535" xml:space="preserve"> & à centro ducantur perpendiculares, ſuper contingentes circulum <lb/>in punctis b, g, ſcilicet d b o, d g s:</s> <s xml:id="echoid-s14536" xml:space="preserve"> & ducatur linea e d.</s> <s xml:id="echoid-s14537" xml:space="preserve"> Cum puncta h, e ſint eiuſdem ſitus & longitu <lb/>dinis, reſpectu e, & reſpectu d:</s> <s xml:id="echoid-s14538" xml:space="preserve"> & ſimiliter puncta b, g, eiuſdem ſitus, reſpectu e & reſpectu d:</s> <s xml:id="echoid-s14539" xml:space="preserve"> habe-<lb/>bunt lineæ h b, t g eundem ſitum, reſpectu lineæ e d.</s> <s xml:id="echoid-s14540" xml:space="preserve"> Et ita concurrent in idem punctum illius li-<lb/>neę.</s> <s xml:id="echoid-s14541" xml:space="preserve"> Sit concurſus in puncto l.</s> <s xml:id="echoid-s14542" xml:space="preserve"> Fiat linea longitudinis columnæ, [ut oſtenſum eſt 47 p 5] in qua <lb/>punctum z:</s> <s xml:id="echoid-s14543" xml:space="preserve"> & ſit hæc linea in ſuperficie uiſus & axis:</s> <s xml:id="echoid-s14544" xml:space="preserve"> quæ ſit a z:</s> <s xml:id="echoid-s14545" xml:space="preserve"> & ducantur lineæ l z n, d z c:</s> <s xml:id="echoid-s14546" xml:space="preserve"> q ſit <lb/>punctum lineæ t h, punctum ſcilicet, quod eſt in ſuperſicie uiſus & axis:</s> <s xml:id="echoid-s14547" xml:space="preserve"> & à puncto q ducatur linea <lb/>æquidiſtans lineę d z c [per 31 p 1] cadet quidem hęc linea ſuper axem:</s> <s xml:id="echoid-s14548" xml:space="preserve"> [per lemm a Procli ad 29 <lb/>p 1] & l z n cadet in hanc lineam ſupra pũctum q.</s> <s xml:id="echoid-s14549" xml:space="preserve"> Cadat in punctum n.</s> <s xml:id="echoid-s14550" xml:space="preserve"> Palàm ex prædictis [12 n 4] <lb/>quòd angulus h b o ęqualis eſt o b e:</s> <s xml:id="echoid-s14551" xml:space="preserve"> ſed [per 15 p 1] angulus h b o æqualis eſt angulo l b d, per con-<lb/>trapoſitionem:</s> <s xml:id="echoid-s14552" xml:space="preserve"> & [per 32 p 1] angulus o b e æqualis eſt duobus angulis b e d, b d e:</s> <s xml:id="echoid-s14553" xml:space="preserve"> quia extrinſe-<lb/>cus.</s> <s xml:id="echoid-s14554" xml:space="preserve"> Ergo angulus l b d ęqualis eſt duobus angulis b e d, b d e.</s> <s xml:id="echoid-s14555" xml:space="preserve"> Fiat ergo angulus m b d æqualis <lb/>angulo b d e [per 23 p 1] remanet angulus m b l ęqualis angulo b e l.</s> <s xml:id="echoid-s14556" xml:space="preserve"> Quare ductus e m in m l æ-<lb/>qualis quadrato b m [triangula enim m e b, m b l ſunt ęquiangula:</s> <s xml:id="echoid-s14557" xml:space="preserve"> quia angulus m b l ęqualis con-<lb/>cluſus eſt angulo m e b, & communis utriuſque trianguli eſt b m e:</s> <s xml:id="echoid-s14558" xml:space="preserve"> reliquus igitur m l b ęquatur <lb/>reliquo l b e per 32 p 1.</s> <s xml:id="echoid-s14559" xml:space="preserve"> Quare per 4 p 6 erit, ut e m ad m b, ſic m b ad m l.</s> <s xml:id="echoid-s14560" xml:space="preserve"> Ergo per 17 p 6 rectangu-<lb/>lum comprehenſum ſub extremis e m & m l, ęquatur quadrato medię m b.</s> <s xml:id="echoid-s14561" xml:space="preserve">] Ducatur linea m z.</s> <s xml:id="echoid-s14562" xml:space="preserve"> <lb/>Quoniam igitur angulus b d m maior eſt angulo z d m:</s> <s xml:id="echoid-s14563" xml:space="preserve"> [Nam propter ſimilem ſitum punctorum <lb/>reflexionis b & g, ęquatur angulus s d e angulo o d e:</s> <s xml:id="echoid-s14564" xml:space="preserve"> ſed angulus s d e maior eſt angulo z d m per <lb/>9 ax.</s> <s xml:id="echoid-s14565" xml:space="preserve"> Quare angulus o d e, id eſt, b d m maior eſt angulo z d m] & duo latera z d, d m ęqualia duo-<lb/>bus lateribus b d, d m:</s> <s xml:id="echoid-s14566" xml:space="preserve"> [ęquantur enim z d, b d per 15 d 1:</s> <s xml:id="echoid-s14567" xml:space="preserve"> & d m eſt communis] erit [per 24 p 1] <lb/>m b maior m z:</s> <s xml:id="echoid-s14568" xml:space="preserve"> quare ductus e m in m l maior eſt quadrato z m.</s> <s xml:id="echoid-s14569" xml:space="preserve"> Sit ductus e m in m i æqualis <lb/>quadrato m z:</s> <s xml:id="echoid-s14570" xml:space="preserve"> [per 11 p 6, ut demonſtratum eſt 6 n] & ducantur lineę i b, i z.</s> <s xml:id="echoid-s14571" xml:space="preserve"> Erit ergo angulus <lb/>m z i ęqualis angulo z e i, [eſt enim per proximam fabricationem & 17 p 6, ut e m ad m z, ſic m z ad <lb/>m i.</s> <s xml:id="echoid-s14572" xml:space="preserve"> Sunt igitur duo triangula e m z, i m z lateribus circa communem angulum i m z propor-<lb/>tionalia:</s> <s xml:id="echoid-s14573" xml:space="preserve"> itaque per 6 p 6 ſunt ęquiangula, & angulus m z i ęquatur angulo z e i.</s> <s xml:id="echoid-s14574" xml:space="preserve">] Quare m z l ma-<lb/>ior angulo z e d.</s> <s xml:id="echoid-s14575" xml:space="preserve"> Sed quoniam angulus m b d poſitus eſt ęqualis augulo b d m:</s> <s xml:id="echoid-s14576" xml:space="preserve"> erit [per 6 p 1] <lb/>linea m b ęqualis lineę m d:</s> <s xml:id="echoid-s14577" xml:space="preserve"> ſed m b maior m z, [ut patuit.</s> <s xml:id="echoid-s14578" xml:space="preserve">] Quare m d maior m z.</s> <s xml:id="echoid-s14579" xml:space="preserve"> Igitur <lb/> <pb o="209" file="0215" n="215" rhead="OPTICAE LIBER VI."/> [per 18 p 1] angulus m z d maior angulo m d z.</s> <s xml:id="echoid-s14580" xml:space="preserve"> Igitur d z l maior duobus angulis z d e, z e d.</s> <s xml:id="echoid-s14581" xml:space="preserve"> <lb/>[conſtat enim è duobus angulis m z l & m z d, <lb/> <anchor type="figure" xlink:label="fig-0215-01a" xlink:href="fig-0215-01"/> quorum ille angulo z e d, hic angulo z d e maior <lb/>eſt concluſus.</s> <s xml:id="echoid-s14582" xml:space="preserve">] Sed angulus d z l ęqualis eſt an-<lb/>gulo n z c [per 15 p 1] & angulus e z c ęqualis <lb/>duobus angulis z d e, z e d [per 32 p 1.</s> <s xml:id="echoid-s14583" xml:space="preserve">] Quare an <lb/>gulus n z c maior eſt angulo e z c:</s> <s xml:id="echoid-s14584" xml:space="preserve"> ſecetur ad ę-<lb/>qualitatem per lineam f z:</s> <s xml:id="echoid-s14585" xml:space="preserve"> quę quidem concur-<lb/>ret cum linea n q:</s> <s xml:id="echoid-s14586" xml:space="preserve"> [per lemma Procli ad 29 p 1:</s> <s xml:id="echoid-s14587" xml:space="preserve"> <lb/>quia n q, c d ſunt parallelę per fabricationem.</s> <s xml:id="echoid-s14588" xml:space="preserve">] <lb/>Concurrat ſuper punctum f.</s> <s xml:id="echoid-s14589" xml:space="preserve"> Cum ergo angulus <lb/>f z c ſit ęqualis angulo c z e:</s> <s xml:id="echoid-s14590" xml:space="preserve"> reflectetur f ad e à <lb/>puncto z.</s> <s xml:id="echoid-s14591" xml:space="preserve"> [per 12 n 4] q uerò reflectetur ad e à <lb/>puncto lineę longitudinis, quę tranſit per z à pũ <lb/>cto, quod eſt ultra z.</s> <s xml:id="echoid-s14592" xml:space="preserve"> Si enim à puncto citra z, id <lb/>eſt propinquiore e:</s> <s xml:id="echoid-s14593" xml:space="preserve"> linea ducta à puncto q ad <lb/>punctum illud reflexionis, ſecabit lineam f z:</s> <s xml:id="echoid-s14594" xml:space="preserve"> & <lb/>ita punctum ſectionis reflectetur ad e à duobus <lb/>punctis:</s> <s xml:id="echoid-s14595" xml:space="preserve"> quod eſt impoſsibile [& contra 46 n 5.</s> <s xml:id="echoid-s14596" xml:space="preserve">] <lb/>Sumatur ergo ultra punctum z pũctum k, à quo <lb/>reflectatur q ad e:</s> <s xml:id="echoid-s14597" xml:space="preserve"> & ducatur linea e k, donec cõ <lb/>currat cum linea n q, in puncto p [concurret au-<lb/>tem per lemma Procli ad 29 p 1.</s> <s xml:id="echoid-s14598" xml:space="preserve">] Erit p imago <lb/>q [per 4 n 5.</s> <s xml:id="echoid-s14599" xml:space="preserve">] Sed h reflectitur ad e à puncto <lb/>ſectionis columnę [ſunt enim h & e in diuerſis <lb/>planis.</s> <s xml:id="echoid-s14600" xml:space="preserve">] Si ergo à puncto h ducatur perpen-<lb/>dicularis ſuper lineam, cõtingentem ſectionem <lb/>in aliquo puncto:</s> <s xml:id="echoid-s14601" xml:space="preserve"> perpendicularis illa concur-<lb/>ret cum perpendiculari c z d ſub axe [per 24 n.</s> <s xml:id="echoid-s14602" xml:space="preserve">] <lb/>Concurrat in puncto u.</s> <s xml:id="echoid-s14603" xml:space="preserve"> Similiter à puncto l eſt <lb/>ducere unam perpendicularem ſuper ſectio-<lb/>nem, à cuius puncto reflectatur t ad e.</s> <s xml:id="echoid-s14604" xml:space="preserve"> Et quo-<lb/>niam [ex theſi] puncta h, t ſunt eiuſdem ſitus, <lb/>reſpectu lineæ e d, & puncta ſectionis ſimiliter, <lb/>per quæ tranſeunt perpendiculares ab ipſis du-<lb/>ctæ.</s> <s xml:id="echoid-s14605" xml:space="preserve"> Igitur illæ duæ perpendiculares concurrent in idem punctum lineę e d.</s> <s xml:id="echoid-s14606" xml:space="preserve"> Concurrant ergo in <lb/>puncto u.</s> <s xml:id="echoid-s14607" xml:space="preserve"> Et quia linea e b concurrit cum h u:</s> <s xml:id="echoid-s14608" xml:space="preserve"> ſit concurſus in puncto r.</s> <s xml:id="echoid-s14609" xml:space="preserve"> Similiter e g concurrat <lb/>cum t u in puncto y:</s> <s xml:id="echoid-s14610" xml:space="preserve"> & ducatur linea r y.</s> <s xml:id="echoid-s14611" xml:space="preserve"> Palàm [per 4 n 5] quòd r eſt imago h:</s> <s xml:id="echoid-s14612" xml:space="preserve"> & y eſt imago t:</s> <s xml:id="echoid-s14613" xml:space="preserve"> & <lb/>habemus triangulum e r y:</s> <s xml:id="echoid-s14614" xml:space="preserve"> extra ſuperficiem huius trianguli eſt punctum z:</s> <s xml:id="echoid-s14615" xml:space="preserve"> & in ſuperficie huius <lb/>trianguli altior eſt linea e p:</s> <s xml:id="echoid-s14616" xml:space="preserve"> & ita p eſt extra.</s> <s xml:id="echoid-s14617" xml:space="preserve"> Quare linea r p y erit curua:</s> <s xml:id="echoid-s14618" xml:space="preserve"> & illa eſt imago lineæ t h.</s> <s xml:id="echoid-s14619" xml:space="preserve"> <lb/>Et eſt quidem hęc imago curuitatis non modicæ.</s> <s xml:id="echoid-s14620" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s14621" xml:space="preserve"> Palàm ergo, quòd in his <lb/>ſpeculis, ſi linea recta uiſa ęquidiſtans fuerit lineę longitudinis columnæ:</s> <s xml:id="echoid-s14622" xml:space="preserve"> erit imago eius recta, aut <lb/>accedens ad rectitudinem.</s> <s xml:id="echoid-s14623" xml:space="preserve"> Siuerò linea recta uiſa ęquidiſtans fuerit columnæ:</s> <s xml:id="echoid-s14624" xml:space="preserve"> erit imago eius cur-<lb/>ua, curuitate non modica.</s> <s xml:id="echoid-s14625" xml:space="preserve"> Lineę autem inter has duas ſitę, quę magis accedunt ad ſitum lineę ęqui-<lb/>diſtantis, reſpectu columnę, habebunt imagines ſuas rectitudini magis uicinas:</s> <s xml:id="echoid-s14626" xml:space="preserve"> & imagines earũ, <lb/>quæ propinquiores ſunt ſitui ęquidiſtantium latitudini, erunt magis curuę:</s> <s xml:id="echoid-s14627" xml:space="preserve"> & minuetur, uel augmẽ <lb/>tabitur curuitas imaginum ſecundum acceſſum uel elongationem linearum ad alterum horum ſi-<lb/>tuum.</s> <s xml:id="echoid-s14628" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s14629" xml:space="preserve"/> </p> <div xml:id="echoid-div498" type="float" level="0" n="0"> <figure xlink:label="fig-0215-01" xlink:href="fig-0215-01a"> <variables xml:id="echoid-variables174" xml:space="preserve">e c s ſ o f i g m b k z d t q p h y n r u a x</variables> </figure> </div> </div> <div xml:id="echoid-div500" type="section" level="0" n="0"> <head xml:id="echoid-head444" xml:space="preserve">DE ERRORIBVS, QVI ACCIDVNT IN SPECVLIS <lb/>pyramidalibus conuexis. Cap. VI.</head> <head xml:id="echoid-head445" xml:space="preserve" style="it">30. Si duæ rectæ à duob{us} punctis ellipſis conicæ, inæquabiliter à uertice diſtantib{us}, ſint per-<lb/>pendiculares duab{us} rectis, ellipſin in dictis punctis tangentib{us}: ultra axem concurrent. Opor <lb/>tet autem ut perpendicularis à puncto propinquiore, & recta à longinquiore ad axem ductæ, <lb/>acutum angulum comprehendant. 113 p 1. 45 p 7.</head> <p> <s xml:id="echoid-s14630" xml:space="preserve">AMplius:</s> <s xml:id="echoid-s14631" xml:space="preserve"> in ſpeculis pyramidalibus exterioribus ij dem errores accidunt, qui in ſphæricis ex-<lb/>terioribus eueniunt.</s> <s xml:id="echoid-s14632" xml:space="preserve"> Lineę enim uiſę ęquidiſtantes, reſpectu pyramidis, aut rectę uidentur, <lb/>aut fortè ęquidiſtantes latitudini curuę:</s> <s xml:id="echoid-s14633" xml:space="preserve"> & intermedię augmentant uel diminuunt curuita-<lb/>tem ſecundum propinquitatem earum uel remotionem.</s> <s xml:id="echoid-s14634" xml:space="preserve"> Et hoc probabitur.</s> <s xml:id="echoid-s14635" xml:space="preserve"> Quiddam tamen prę-<lb/>mittendum proponamùs:</s> <s xml:id="echoid-s14636" xml:space="preserve"> & eſt.</s> <s xml:id="echoid-s14637" xml:space="preserve"> Si ſumatur in ſuperficie pyramidis, punctum reflexionis:</s> <s xml:id="echoid-s14638" xml:space="preserve"> & fiat ſe-<lb/>ctio tranſiens per punctum illud:</s> <s xml:id="echoid-s14639" xml:space="preserve"> & in ſectione ſumatur punctum remotius à uertice pyramidis, <lb/>puncto reflexionis:</s> <s xml:id="echoid-s14640" xml:space="preserve"> & à puncto ſumpto ducatur perpendicularis ſuper contingentem ſectionem:</s> <s xml:id="echoid-s14641" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0215-02a" xlink:href="fig-0215-02"/> <pb o="210" file="0216" n="216" rhead="ALHAZEN"/> hæc perpendicularis concurret cum perpendiculari ſuper contin gentem ſectionem ducta à pun-<lb/>cto reflexionis, ſub axe.</s> <s xml:id="echoid-s14642" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s14643" xml:space="preserve"> ſit a b g z pyramis erecta ſuper baſim ſuam:</s> <s xml:id="echoid-s14644" xml:space="preserve"> a uertex pyrami-<lb/>dis:</s> <s xml:id="echoid-s14645" xml:space="preserve"> b f z ſectio:</s> <s xml:id="echoid-s14646" xml:space="preserve"> e punctum reflexionis:</s> <s xml:id="echoid-s14647" xml:space="preserve"> z punctũ ſectionis remotius à puncto a quàm e.</s> <s xml:id="echoid-s14648" xml:space="preserve"> Super pun-<lb/>ctum z fiat ſuperficies ſecans pyramidem æquidiſtanter baſi [ut oſtenſum eſt 52 n 5.</s> <s xml:id="echoid-s14649" xml:space="preserve">] Secabit qui-<lb/>dem ſuper circulum communem [per 4 th.</s> <s xml:id="echoid-s14650" xml:space="preserve"> 1 coni.</s> <s xml:id="echoid-s14651" xml:space="preserve"> Apol.</s> <s xml:id="echoid-s14652" xml:space="preserve">] Sit circulus ille g b r z:</s> <s xml:id="echoid-s14653" xml:space="preserve"> & ducantur lineæ <lb/>a z, a e:</s> <s xml:id="echoid-s14654" xml:space="preserve"> & producatur a e, donec ſit æqualis a z:</s> <s xml:id="echoid-s14655" xml:space="preserve"> ueniet quidem ad circulum [per 18 d 11:</s> <s xml:id="echoid-s14656" xml:space="preserve"> quia eſt la-<lb/>tus conicum.</s> <s xml:id="echoid-s14657" xml:space="preserve">] Cadat ergo in punctum eius o:</s> <s xml:id="echoid-s14658" xml:space="preserve"> & c ſit centrum circuli:</s> <s xml:id="echoid-s14659" xml:space="preserve"> & ducatur axis a c:</s> <s xml:id="echoid-s14660" xml:space="preserve"> & à pun-<lb/>cto e ducatur perpendicularis ſuper ſuperficiem contingentem pyramidem [per 12 p 11.</s> <s xml:id="echoid-s14661" xml:space="preserve">] Concur-<lb/>ret quidem [per 11 ax.</s> <s xml:id="echoid-s14662" xml:space="preserve">] cum axe citra cẽtrum circuli, quod eſt c:</s> <s xml:id="echoid-s14663" xml:space="preserve"> ſit in puncto d:</s> <s xml:id="echoid-s14664" xml:space="preserve"> & ducatur linea d z, <lb/>continens angulum acutum cum perpendiculari e d:</s> <s xml:id="echoid-s14665" xml:space="preserve"> & à puncto o ducatur perpendicularis ſuper <lb/>lineam a o, concurrens cum axe in puncto k:</s> <s xml:id="echoid-s14666" xml:space="preserve"> & ducatur linea k z:</s> <s xml:id="echoid-s14667" xml:space="preserve"> & ſuper punctum z ducatur con-<lb/>tingens ſectionem, quæ ſit t q:</s> <s xml:id="echoid-s14668" xml:space="preserve"> & alia contingens circulum b g z:</s> <s xml:id="echoid-s14669" xml:space="preserve"> [per 17 p 3] quæ ſit z y:</s> <s xml:id="echoid-s14670" xml:space="preserve"> & ducatur <lb/>linea b c z:</s> <s xml:id="echoid-s14671" xml:space="preserve"> & à puncto c ducatur perpendicularis ſuper lineam b c z:</s> <s xml:id="echoid-s14672" xml:space="preserve"> [per 11 p 1] quæ ſit c r.</s> <s xml:id="echoid-s14673" xml:space="preserve"> Erit qui-<lb/>dem perpendicularis ſuper axem:</s> <s xml:id="echoid-s14674" xml:space="preserve"> [per 3 d 11] cum axis ſit perpendicularis ſuper ſuperficiem circu-<lb/>li:</s> <s xml:id="echoid-s14675" xml:space="preserve"> [per 18 d 11.</s> <s xml:id="echoid-s14676" xml:space="preserve">] Quare [per 4 p 11] c r eſt perpendicularis ſuper ſuperficiem a c z:</s> <s xml:id="echoid-s14677" xml:space="preserve"> & erit æquidi-<lb/>diſtans z y cõtingenti [per 28 p 1:</s> <s xml:id="echoid-s14678" xml:space="preserve"> quia anguli interiores ad c & z ſunt recti:</s> <s xml:id="echoid-s14679" xml:space="preserve"> ille per fabricationem, <lb/>hic per 18 p 3.</s> <s xml:id="echoid-s14680" xml:space="preserve">] Quare z y eſt perpendicularis ſuper <lb/> <anchor type="figure" xlink:label="fig-0216-01a" xlink:href="fig-0216-01"/> ſuperficiem a c z [per 8 p 11.</s> <s xml:id="echoid-s14681" xml:space="preserve">] Quare t q non eſt per-<lb/>pendicularis ſuper eandem ſuperficiem.</s> <s xml:id="echoid-s14682" xml:space="preserve"> Verùm <lb/>quoniam k eſt p olus circuli b r z:</s> <s xml:id="echoid-s14683" xml:space="preserve"> [quia eſt in axe co <lb/>nico per fabricationem] palàm, cum lineæ k o, k z <lb/>ſint æquales [per 5 defin.</s> <s xml:id="echoid-s14684" xml:space="preserve"> 1 ſphæricorum Theodo-<lb/>ſij,] & axis a k communis, & a o æqualis a z [per 18 <lb/>d 11:</s> <s xml:id="echoid-s14685" xml:space="preserve"> quia utraque eſt latus conicum] quòd erit an-<lb/>gulus a o k æqualis angulo a z k [per 8 p 1] & ita an-<lb/>gulus a z k rectus:</s> <s xml:id="echoid-s14686" xml:space="preserve"> [quia a o k illi ęqualis, rectus eſt:</s> <s xml:id="echoid-s14687" xml:space="preserve"> <lb/>cum k o ſit perpendicularis a o per fabricationem.</s> <s xml:id="echoid-s14688" xml:space="preserve">] <lb/>Cum ergo linea k z ſit perpẽdicularis ſuper a z, quæ <lb/>eſt linea longitudinis:</s> <s xml:id="echoid-s14689" xml:space="preserve"> erit perpendicularis ſuper ſu <lb/>perficiem, contingentem pyramidem, ſuper hanc li <lb/>neam longitudinis [ut demonſtratum eſt 54 n 5.</s> <s xml:id="echoid-s14690" xml:space="preserve">] <lb/>Sed t q eſt in ſuperficie contingente:</s> <s xml:id="echoid-s14691" xml:space="preserve"> quia eſt cõmu-<lb/>nis ſectio ſuperficiei contingenti & ſectioni.</s> <s xml:id="echoid-s14692" xml:space="preserve"> Igitur <lb/>k z eſt perpendicularis ſuper t q [per 3 d 11.</s> <s xml:id="echoid-s14693" xml:space="preserve">] Duca-<lb/>tur autem h z in ſuperficie ſectionis perpendicula-<lb/>ris ſuper lineam t q [per 11 p 1.</s> <s xml:id="echoid-s14694" xml:space="preserve">] Cum autẽ linea k z <lb/>ſit extra ſuperficiem ſectionis:</s> <s xml:id="echoid-s14695" xml:space="preserve"> ſecabit lineã h z, nec <lb/>erit una linea [per 1 p 11.</s> <s xml:id="echoid-s14696" xml:space="preserve">] Quare illa ſuperficies k z h <lb/>ſecat ſuperficiem ſectionis, ſuper lineam h z com-<lb/>munem:</s> <s xml:id="echoid-s14697" xml:space="preserve"> & ſecat lineam t q ſuper punctum z:</s> <s xml:id="echoid-s14698" xml:space="preserve"> & <lb/>ſuperficies h z k ſecat ſuperficiem d z k, ſuper lineam communem k z:</s> <s xml:id="echoid-s14699" xml:space="preserve"> uerùm d z eſt in ſuperficie ſe-<lb/>ctionis, & ſecatur à linea k z in puncto z:</s> <s xml:id="echoid-s14700" xml:space="preserve"> & punctum t eſt ſupra ſuperficiem k z h, punctum q infra:</s> <s xml:id="echoid-s14701" xml:space="preserve"> & <lb/>ita ſuperficies k z h ſecabit ſuperficiem d z q ſuper lineam communem:</s> <s xml:id="echoid-s14702" xml:space="preserve"> & illa linea communis eſt <lb/>perpendicularis ſuper lineam t q:</s> <s xml:id="echoid-s14703" xml:space="preserve"> quia linea illa eſt in ſuperficie h z k, ſuper quam eſt perpendicula-<lb/>ris t q [ut oſtenſum eſt.</s> <s xml:id="echoid-s14704" xml:space="preserve">] Et quoniam ſuperficies h z k ſecat ſuperficiem d z q:</s> <s xml:id="echoid-s14705" xml:space="preserve"> & declinatio ſuperfi-<lb/>ciei h z k à ſuperficie ſectionis fit ex parte z c:</s> <s xml:id="echoid-s14706" xml:space="preserve"> erit linea communis ſectioni illarum ſuperficierũ in-<lb/>ter lineas q z, d z.</s> <s xml:id="echoid-s14707" xml:space="preserve"> Et ita concurret cum perpendiculari ſub axe.</s> <s xml:id="echoid-s14708" xml:space="preserve"> Et quòd neceſſariò concurrat, pro-<lb/>batum eſt in libro quinto [quia anguli e d z, d z p ſunt acuti:</s> <s xml:id="echoid-s14709" xml:space="preserve"> ille per theſin, hic, quia pars ẽſt recti <lb/>t z p.</s> <s xml:id="echoid-s14710" xml:space="preserve">] Et ita eſt propoſitum.</s> <s xml:id="echoid-s14711" xml:space="preserve"/> </p> <div xml:id="echoid-div500" type="float" level="0" n="0"> <figure xlink:label="fig-0215-02" xlink:href="fig-0215-02a"> <description xml:id="echoid-description3" xml:space="preserve">CIN EMATH EQUE FRANCAISE</description> <description xml:id="echoid-description4" xml:space="preserve">BIBLIOTHEQUE MUSEE</description> </figure> <figure xlink:label="fig-0216-01" xlink:href="fig-0216-01a"> <variables xml:id="echoid-variables175" xml:space="preserve">a e t o f z h g d j c p k b q r</variables> </figure> </div> </div> <div xml:id="echoid-div502" type="section" level="0" n="0"> <head xml:id="echoid-head446" xml:space="preserve" style="it">31. Linea recta tota ab uno ſpeculi conici conuexi latere ad uiſum reflecti po-<lb/>teſt. 41 p 7.</head> <p> <s xml:id="echoid-s14712" xml:space="preserve">SIt ergo pyramis:</s> <s xml:id="echoid-s14713" xml:space="preserve"> cuius uertex a<gap/>axis a h:</s> <s xml:id="echoid-s14714" xml:space="preserve"> linea longitudinis a z.</s> <s xml:id="echoid-s14715" xml:space="preserve"> Et à puncto z ducatur perpendi-<lb/>cularis ſuper ſuperficiem, contingentem pyramidem in linea a z [per 12 p 11] quæ neceſſariò <lb/>concurret cum axe [per 11 ax.</s> <s xml:id="echoid-s14716" xml:space="preserve"> quia angulus h a z eſt acutus per 17 p 1:</s> <s xml:id="echoid-s14717" xml:space="preserve"> cum a d z ſit rectus per <lb/>18 d 11.</s> <s xml:id="echoid-s14718" xml:space="preserve">] Sit linea t z h.</s> <s xml:id="echoid-s14719" xml:space="preserve"> Ducatur à puncto a linea extra pyramidem, ultra ſuperficiem contingentem <lb/>pyramidem in linea a z, faciens angulum acutum cum axe & cum linea longitudinis a z:</s> <s xml:id="echoid-s14720" xml:space="preserve"> quæ ſit a n.</s> <s xml:id="echoid-s14721" xml:space="preserve"> <lb/>Et in ſuperficie a h n à puncto h ducatur linea, cum axe faciens angulum æqualem angulo a h z:</s> <s xml:id="echoid-s14722" xml:space="preserve"> quæ <lb/>linea neceſſariò concurret cum linea a n:</s> <s xml:id="echoid-s14723" xml:space="preserve"> [per 11 ax.</s> <s xml:id="echoid-s14724" xml:space="preserve"> quia anguli n a h & a h z ex theſi acuti, ſunt mi-<lb/>nores duobus rectis] quæ ſit h o.</s> <s xml:id="echoid-s14725" xml:space="preserve"> Et facto ſuper punctum z circulo æquidiſtante baſi:</s> <s xml:id="echoid-s14726" xml:space="preserve"> [ut oſtenſum <lb/>eſt 52 n 5] tranſibit h o per circulum, ſicut h z tranſit per ipſum.</s> <s xml:id="echoid-s14727" xml:space="preserve"> Ducatur linea o z:</s> <s xml:id="echoid-s14728" xml:space="preserve"> & producatur <lb/>ad punctum f.</s> <s xml:id="echoid-s14729" xml:space="preserve"> Quoniam linea o z ſecat ſuperficiem, contingentem pyramidem in linea a z:</s> <s xml:id="echoid-s14730" xml:space="preserve"> cum li-<lb/>nea h z ſit perpendicularis ſuper illam ſuperficiem:</s> <s xml:id="echoid-s14731" xml:space="preserve"> [per fabricationem] erit angulus o z h maior <lb/>recto:</s> <s xml:id="echoid-s14732" xml:space="preserve"> quia a z h rectus eſt [per fabricationẽ.</s> <s xml:id="echoid-s14733" xml:space="preserve">] Igitur [per 13 p 1] angulus f z h acutus.</s> <s xml:id="echoid-s14734" xml:space="preserve"> À<unsure/> puncto z du <lb/> <pb o="211" file="0217" n="217" rhead="OPTICAE LIBER VI."/> catur contingens circulum [per 17 p 3] quæ ſit m z:</s> <s xml:id="echoid-s14735" xml:space="preserve"> & à puncto f ducatur perpẽdicularis ſupera z, <lb/>[per 12 p 1] cadens in punctũ eius e:</s> <s xml:id="echoid-s14736" xml:space="preserve"> quæ producta cõcurrat cũ a o [per 11 ax.</s> <s xml:id="echoid-s14737" xml:space="preserve">] quoniã angulus o a z <lb/>eſt acutus [ex theſi.</s> <s xml:id="echoid-s14738" xml:space="preserve">] Concurrat <lb/> <anchor type="figure" xlink:label="fig-0217-01a" xlink:href="fig-0217-01"/> igitur in puncto n.</s> <s xml:id="echoid-s14739" xml:space="preserve"> Et [per 31 p 1] à <lb/>pũcto e ducatur æquidiſtãs lineę <lb/>t h:</s> <s xml:id="echoid-s14740" xml:space="preserve"> & ſit q e:</s> <s xml:id="echoid-s14741" xml:space="preserve"> & à puncto e ducatur <lb/>ęquidiſtãs m z:</s> <s xml:id="echoid-s14742" xml:space="preserve"> quę ſit e l.</s> <s xml:id="echoid-s14743" xml:space="preserve"> Palã [ք <lb/>lemma ad 37 th.</s> <s xml:id="echoid-s14744" xml:space="preserve"> opticorum Eucli <lb/>dis:</s> <s xml:id="echoid-s14745" xml:space="preserve"> uel per 42 th 6 libri σ{υν}αγωγῶ<gap/> <lb/>μαθκματικῶμ Pappi] quòd m z eſt <lb/>perpendicularis ſuper a e:</s> <s xml:id="echoid-s14746" xml:space="preserve"> quo-<lb/>niam a h eſt perpendicularis ſu-<lb/>per circulum, per z tranſeuntem, <lb/>[per 18 d 11] & m z ſuper diame-<lb/>trum illius circuli [per 18 p 3] <lb/>quia contingit.</s> <s xml:id="echoid-s14747" xml:space="preserve"> Igitur l e eſt per-<lb/>pendicularis ſuper a e [per 29 <lb/>p 1] & producatur q e ultra e:</s> <s xml:id="echoid-s14748" xml:space="preserve"> hęc <lb/>concurret quidem cum axe:</s> <s xml:id="echoid-s14749" xml:space="preserve"> [per <lb/>lemma Procli ad 29 p 1] concur-<lb/>rat in d.</s> <s xml:id="echoid-s14750" xml:space="preserve"> Fiat autẽ ſuperficies l e, <lb/>d q ſecans pyramidem:</s> <s xml:id="echoid-s14751" xml:space="preserve"> erit quidem ſectio pyramidalis:</s> <s xml:id="echoid-s14752" xml:space="preserve"> [per 5 th.</s> <s xml:id="echoid-s14753" xml:space="preserve"> 1 con.</s> <s xml:id="echoid-s14754" xml:space="preserve"> Apoll.</s> <s xml:id="echoid-s14755" xml:space="preserve"> quia l d q planum ob-<lb/>liquum eſt ad axem.</s> <s xml:id="echoid-s14756" xml:space="preserve">] Cum ergo a e ſit perpendicularis ſuper f n, & ſuper q d, & ſuper l e:</s> <s xml:id="echoid-s14757" xml:space="preserve"> erit f n in <lb/>ſuperficie illa ſecante pyramidem [per 5 p 11.</s> <s xml:id="echoid-s14758" xml:space="preserve">] Fiat ergo in illa ſuperficie p f æquidiſtans q e:</s> <s xml:id="echoid-s14759" xml:space="preserve"> erit æ-<lb/>quidiſtans t z [per 9 p 11] uerùm cum angulus f z h ſit acutus:</s> <s xml:id="echoid-s14760" xml:space="preserve"> [per concluſionẽ] erit angulus t z <gap/> <lb/>obtuſus [per 13 p 1.</s> <s xml:id="echoid-s14761" xml:space="preserve">] Ducatur à puncto z linea, faciens cum t z angulum, æqualem angulo o z t:</s> <s xml:id="echoid-s14762" xml:space="preserve"> quę <lb/>quidem linea neceſſariò ſecabit f p [per lemma Procli ad 29 p 1:</s> <s xml:id="echoid-s14763" xml:space="preserve"> quia z t, f p ſunt parallelæ.</s> <s xml:id="echoid-s14764" xml:space="preserve">] Secet <lb/>in puncto p:</s> <s xml:id="echoid-s14765" xml:space="preserve"> & ducatur linea p e.</s> <s xml:id="echoid-s14766" xml:space="preserve"> Cum ergo p z, o z ſint in eadem ſuperficie, & angulus o z t ęqua-<lb/>lis angulo t z p [per fabricationem] reflectetur o ad p à puncto ſpeculi z [per 12 n 4.</s> <s xml:id="echoid-s14767" xml:space="preserve">] Et quia an-<lb/>gulus o z t æqualis eſt angulo z f p:</s> <s xml:id="echoid-s14768" xml:space="preserve"> [per 29 p 1, & t z p æqualis z p f per eandem:</s> <s xml:id="echoid-s14769" xml:space="preserve"> quare z f p, <lb/>& z p f æquantur] erunt latera z p, z f æqualia [per 6 p 1.</s> <s xml:id="echoid-s14770" xml:space="preserve">] Et quia angulus f e z rectus [quia a <gap/> <lb/>perpendicularis eſt ipſ<gap/> f e n] quadratum f z ualet quadrata e z, e f:</s> <s xml:id="echoid-s14771" xml:space="preserve"> & quadratum p z ualet <lb/>quadrata e z, e p [per 47 p 1.</s> <s xml:id="echoid-s14772" xml:space="preserve">] Igitur p e, f e æqualia:</s> <s xml:id="echoid-s14773" xml:space="preserve"> [Quia enim z p, z f æquales iam concluſæ <lb/>ſunt:</s> <s xml:id="echoid-s14774" xml:space="preserve"> erunt ipſarum quadrata æqualia:</s> <s xml:id="echoid-s14775" xml:space="preserve"> ſubducto igitur communi quadrato z e:</s> <s xml:id="echoid-s14776" xml:space="preserve"> relinquentur qua-<lb/>drata e p, e f æqualia:</s> <s xml:id="echoid-s14777" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s14778" xml:space="preserve"> ipſorum latera e p, e f] & ita [per 5 p 1] e p f, e f p anguli erunt <lb/>æquales.</s> <s xml:id="echoid-s14779" xml:space="preserve"> Quare anguli n e q, q e p æquales.</s> <s xml:id="echoid-s14780" xml:space="preserve"> [nam per 29 p 1 anguli n e q, e f p:</s> <s xml:id="echoid-s14781" xml:space="preserve"> item p e q, e p f <lb/>æquantur:</s> <s xml:id="echoid-s14782" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s14783" xml:space="preserve"> per 1 ax.</s> <s xml:id="echoid-s14784" xml:space="preserve"> n e q, p e q æquantur.</s> <s xml:id="echoid-s14785" xml:space="preserve">] Et cum in eadem ſuperficie ſint, quæ eſt p en:</s> <s xml:id="echoid-s14786" xml:space="preserve"> refle-<lb/>ctetur n ad p à puncto e [per 12 n 4.</s> <s xml:id="echoid-s14787" xml:space="preserve">] Similiter ſi ducatur quæcunq;</s> <s xml:id="echoid-s14788" xml:space="preserve"> linea à puncto f a d aliquod pũ-<lb/>ctum z e, & producatur uſque ad o n:</s> <s xml:id="echoid-s14789" xml:space="preserve"> probabitur de puncto lineæ o n, in quod cadit, quòd refle-<lb/>ctetur ad p à puncto lineæ z e, quod ſecat illa linea.</s> <s xml:id="echoid-s14790" xml:space="preserve"> Simili modo & omnium huiuſmodi linea-<lb/>rum probatio ſumet initium à perpendiculari, quæ eſt f e, & à parte lineæ e z:</s> <s xml:id="echoid-s14791" xml:space="preserve"> quæ erit com-<lb/>munis omnibus illis triangulis.</s> <s xml:id="echoid-s14792" xml:space="preserve"> Et ita quo dlibet punctum lineę o n reflectetur ad p ab aliquo pun-<lb/>cto lineæ e z.</s> <s xml:id="echoid-s14793" xml:space="preserve"/> </p> <div xml:id="echoid-div502" type="float" level="0" n="0"> <figure xlink:label="fig-0217-01" xlink:href="fig-0217-01a"> <variables xml:id="echoid-variables176" xml:space="preserve">a o u m h z t s n d ſ e q f p</variables> </figure> </div> </div> <div xml:id="echoid-div504" type="section" level="0" n="0"> <head xml:id="echoid-head447" xml:space="preserve" style="it">32. Si linea recta obliquè inciderit uertici ſpeculi conici conuexi: reflectetur à latere coni-<lb/>co ad uiſum inter dictam lineam & ſpeculi ſuperficiem ſitum: eiuś imago parum curua ui-<lb/>debitur. 55 p 7.</head> <p> <s xml:id="echoid-s14794" xml:space="preserve">HOc declarato dicamus.</s> <s xml:id="echoid-s14795" xml:space="preserve"> Cum uiſus comprehenderit lineas rectas, tranſeuntes per uerti-<lb/>cem ſpeculi pyramidalis conuexi recti, obliquas ſuper axem ſpeculi:</s> <s xml:id="echoid-s14796" xml:space="preserve"> tunc formæ earum e-<lb/>runt parùm conuexæ.</s> <s xml:id="echoid-s14797" xml:space="preserve"> Sit ergo ſpeculum pyramidale erectum a b c:</s> <s xml:id="echoid-s14798" xml:space="preserve"> cuius uertex ſit a:</s> <s xml:id="echoid-s14799" xml:space="preserve"> <lb/>& cuius axis ſit a d:</s> <s xml:id="echoid-s14800" xml:space="preserve"> & extrahamus in ſuperficie eius lineam a z [ut oſtenſum eſt 52 n 5] quocun-<lb/>que modo ſit:</s> <s xml:id="echoid-s14801" xml:space="preserve"> in qua ſignetur punctum z, quocunque modo ſit.</s> <s xml:id="echoid-s14802" xml:space="preserve"> Et tranfeat per z ſuperficies æ-<lb/>quidiſtans baſi pyramidis:</s> <s xml:id="echoid-s14803" xml:space="preserve"> & faciat circulum z u [faciet autem per 4 th 1 con.</s> <s xml:id="echoid-s14804" xml:space="preserve"> Apol.</s> <s xml:id="echoid-s14805" xml:space="preserve">] Et extraha-<lb/>mus ex z perpendicularem z h ſuper a z [per 11 p 1.</s> <s xml:id="echoid-s14806" xml:space="preserve">] Hæc ergo linea concurret cum axe pyrami-<lb/>dis [per 11 ax.</s> <s xml:id="echoid-s14807" xml:space="preserve"> ut patuit præcedente numero.</s> <s xml:id="echoid-s14808" xml:space="preserve">] Concurrat ergo in h.</s> <s xml:id="echoid-s14809" xml:space="preserve"> Et extrahamus ex z line-<lb/>am contingentem circulum:</s> <s xml:id="echoid-s14810" xml:space="preserve"> [per 17 p 3] & ſit z m:</s> <s xml:id="echoid-s14811" xml:space="preserve"> & extrahamus ex a lineam continentem cum <lb/>utraque linea a z, h a angulum acutum:</s> <s xml:id="echoid-s14812" xml:space="preserve"> & ſit extra ſuperficiem, contingentem pyramidem, tran-<lb/>ſeuntem per lineam a z.</s> <s xml:id="echoid-s14813" xml:space="preserve"> Et hoc eſt poſsibile:</s> <s xml:id="echoid-s14814" xml:space="preserve"> [quia angulus h a z eſt acutus per 18 d 11.</s> <s xml:id="echoid-s14815" xml:space="preserve"> 32 p 1.</s> <s xml:id="echoid-s14816" xml:space="preserve">] Sit <lb/>ergo a n:</s> <s xml:id="echoid-s14817" xml:space="preserve"> & extrahamus ex puncto h lineam in ſuperficie, in qua ſunt a n, a h, continentem cum <lb/>a h angulum æqualem angulo a h z.</s> <s xml:id="echoid-s14818" xml:space="preserve"> Hæc ergo linea concurret cum <gap/> o:</s> <s xml:id="echoid-s14819" xml:space="preserve"> [per 11 ax.</s> <s xml:id="echoid-s14820" xml:space="preserve">] nam <lb/>duo anguli ad a, h ſunt acuti.</s> <s xml:id="echoid-s14821" xml:space="preserve"> Concurrant ergo in o.</s> <s xml:id="echoid-s14822" xml:space="preserve"> Linea ergo h o concurret cum cir-<lb/>cumferentiã circuli z u.</s> <s xml:id="echoid-s14823" xml:space="preserve"> Nam angulus a h o eſt æqualis angulo a h z.</s> <s xml:id="echoid-s14824" xml:space="preserve"> Concurrat ergo in <lb/>u:</s> <s xml:id="echoid-s14825" xml:space="preserve"> & extrahamus a u rectè:</s> <s xml:id="echoid-s14826" xml:space="preserve"> & extrahamus perpendicularem h z ad t:</s> <s xml:id="echoid-s14827" xml:space="preserve"> & continuemus o z, <lb/>& extrahamus rectè ad f:</s> <s xml:id="echoid-s14828" xml:space="preserve"> & extrahatur a z ad e.</s> <s xml:id="echoid-s14829" xml:space="preserve"> Angulus igitur f z h erit acutus:</s> <s xml:id="echoid-s14830" xml:space="preserve"> quia <lb/> <pb o="212" file="0218" n="218" rhead="ALHAZEN"/> linea o z ſecat ſuperficiem, contingentem pyramidem, trãſeuntem per a z:</s> <s xml:id="echoid-s14831" xml:space="preserve"> linea ergo a z eſt ſub dif-<lb/>ferentia communi inter ſuperficiem o z h & ſuperficiem contingentem.</s> <s xml:id="echoid-s14832" xml:space="preserve"> Et hæc differentia conti-<lb/>net cum linea h z angulum rectum, [per fabricationem.</s> <s xml:id="echoid-s14833" xml:space="preserve">] Angulus ergo e z h obtuſus:</s> <s xml:id="echoid-s14834" xml:space="preserve"> ergo angu-<lb/>lus f z h acutus [per 13 p 1.</s> <s xml:id="echoid-s14835" xml:space="preserve">] Ponatur ergo in z f punctum f:</s> <s xml:id="echoid-s14836" xml:space="preserve"> à quo extrahatur perpendicularis f e ſu-<lb/>per a e:</s> <s xml:id="echoid-s14837" xml:space="preserve"> & extrahatur rectè.</s> <s xml:id="echoid-s14838" xml:space="preserve"> Concurret ergo cum linea a o:</s> <s xml:id="echoid-s14839" xml:space="preserve"> [per 11 ax.</s> <s xml:id="echoid-s14840" xml:space="preserve">] nam angulus o a e eſt acutus <lb/>[per theſin, & ad e rectus eſt.</s> <s xml:id="echoid-s14841" xml:space="preserve">] Concurrat ergo in n.</s> <s xml:id="echoid-s14842" xml:space="preserve"> Et extrahatur ex e linea e d æquidiſtans z h li-<lb/>neæ [per 17 p 3.</s> <s xml:id="echoid-s14843" xml:space="preserve">] Erit ergo [per 8 p 11] e d perpendicularis ſuper ſuperficiem, contingentem pyra-<lb/>midem, tranſeuntem per a e:</s> <s xml:id="echoid-s14844" xml:space="preserve"> & extrahatur ex e linea æquidiſtans lineæ z m:</s> <s xml:id="echoid-s14845" xml:space="preserve"> & ſit e l.</s> <s xml:id="echoid-s14846" xml:space="preserve"> Et extrahatur <lb/>ſuperficies, in qua ſunt lineæ l e, e d.</s> <s xml:id="echoid-s14847" xml:space="preserve"> Secabit ergo ſuperficiem pyramidis, & faciet ſectionem [per <lb/>5 th.</s> <s xml:id="echoid-s14848" xml:space="preserve"> 1.</s> <s xml:id="echoid-s14849" xml:space="preserve"> con.</s> <s xml:id="echoid-s14850" xml:space="preserve"> Apoll.</s> <s xml:id="echoid-s14851" xml:space="preserve">] Nam hæc ſuperficies eſt obliqua ſuper axem a d.</s> <s xml:id="echoid-s14852" xml:space="preserve"> Sit ergo ſectio d e c:</s> <s xml:id="echoid-s14853" xml:space="preserve"> & m z eſt <lb/>perpẽdicularis ſuper ſuperficiem <lb/> <anchor type="figure" xlink:label="fig-0218-01a" xlink:href="fig-0218-01"/> a z h:</s> <s xml:id="echoid-s14854" xml:space="preserve"> & hoc declaratũ eſt in præ-<lb/>dictis.</s> <s xml:id="echoid-s14855" xml:space="preserve"> [præcedente numero, per <lb/>lemma ad 37 theor.</s> <s xml:id="echoid-s14856" xml:space="preserve"> opticor.</s> <s xml:id="echoid-s14857" xml:space="preserve"> Eucli <lb/>dis.</s> <s xml:id="echoid-s14858" xml:space="preserve">] Ergo linea l e eſt perpendi-<lb/>cularis ſuper ſuperficiẽ a e d [per <lb/>8 p 11.</s> <s xml:id="echoid-s14859" xml:space="preserve">] Ergo angulus a e l eſt re-<lb/>ctus.</s> <s xml:id="echoid-s14860" xml:space="preserve"> Et ſimiliter angulus a e d re-<lb/>ctus eſt [per 29 p 1] & a e n ſimili-<lb/>ter rectus.</s> <s xml:id="echoid-s14861" xml:space="preserve"> Ergo [per 5 p 11] lineæ <lb/>l e, n e, d e ſunt in eadem ſuperfi-<lb/>cie.</s> <s xml:id="echoid-s14862" xml:space="preserve"> Ergo linea fen eſt in ſuperfi-<lb/>cie ſectionis.</s> <s xml:id="echoid-s14863" xml:space="preserve"> Et extrahatur ex f li-<lb/>nea æquidiſtans lineæ d e:</s> <s xml:id="echoid-s14864" xml:space="preserve"> [per 31 <lb/>p 1] & ſit f r.</s> <s xml:id="echoid-s14865" xml:space="preserve"> Hęc ergo linea æqui-<lb/>diſtat lineæ h z [per 30 p 1.</s> <s xml:id="echoid-s14866" xml:space="preserve">] Et <lb/>extrahatur ex z in ſuperficie o z h, <lb/>linea continens cum z t angulum, <lb/>æqualem angulo o z t.</s> <s xml:id="echoid-s14867" xml:space="preserve"> [per 23 p 1.</s> <s xml:id="echoid-s14868" xml:space="preserve">] <lb/>Hæc ergo linea concurret cum f r [per lemma Procli ad 29 p 1] quia ſecat z h æquidiſtantem f r:</s> <s xml:id="echoid-s14869" xml:space="preserve"> & <lb/>eſt in ſuperficie eius:</s> <s xml:id="echoid-s14870" xml:space="preserve"> quia z f eſt in ſuperficie eius [per 35 d 1.</s> <s xml:id="echoid-s14871" xml:space="preserve">] Concurrat ergo in r.</s> <s xml:id="echoid-s14872" xml:space="preserve"> Ergo duo an-<lb/>guli, qui ſunt apud r, f, ſunt æquales:</s> <s xml:id="echoid-s14873" xml:space="preserve"> ſunt enim æquales duobus angulis, qui ſunt apud z [nam per <lb/>29 p 1 o z t, z f r:</s> <s xml:id="echoid-s14874" xml:space="preserve"> item t z r, z r f æquantur.</s> <s xml:id="echoid-s14875" xml:space="preserve">] Duæ ergo lineæ r z, f z ſunt æquales [per 6 p 1.</s> <s xml:id="echoid-s14876" xml:space="preserve">] Et de-<lb/>claratum eſt, quòd linea f e n eſt in ſuperficie ſectionis:</s> <s xml:id="echoid-s14877" xml:space="preserve"> & linea f r eſt æquidiſtans e d:</s> <s xml:id="echoid-s14878" xml:space="preserve"> eſt ergo <lb/>in ſuperficie ſectionis [per 35 d 1.</s> <s xml:id="echoid-s14879" xml:space="preserve">] Et continuemus r e:</s> <s xml:id="echoid-s14880" xml:space="preserve"> erit ergo [per 7 p 11] in ſuperficie ſectio-<lb/>nis:</s> <s xml:id="echoid-s14881" xml:space="preserve"> & extrahatur d e ad k.</s> <s xml:id="echoid-s14882" xml:space="preserve"> Et declaratum eſt, quòd e a eſt perpendicularis ſuper ſuperficiem ſe-<lb/>ctionis:</s> <s xml:id="echoid-s14883" xml:space="preserve"> uterque ergo angulorum a e r, a e f rectus eſt:</s> <s xml:id="echoid-s14884" xml:space="preserve"> [per 3 d 11] & duæ lineæ f z, r z ſunt ęqua-<lb/>les [per concluſionem.</s> <s xml:id="echoid-s14885" xml:space="preserve">] Ergo duæ lineæ r e, f e ſunt ęquales.</s> <s xml:id="echoid-s14886" xml:space="preserve"> [Quia enim anguli a e r, a e f ſunt <lb/>recti:</s> <s xml:id="echoid-s14887" xml:space="preserve"> quadrata z e, e f æquantur quadrato z f per 47 p 1:</s> <s xml:id="echoid-s14888" xml:space="preserve"> item q́ue quadrata z e, e r quadrato z r:</s> <s xml:id="echoid-s14889" xml:space="preserve"> <lb/>at quadrata laterum z f, z r æqualium æquantur:</s> <s xml:id="echoid-s14890" xml:space="preserve"> quare ablato communi quadrato z e:</s> <s xml:id="echoid-s14891" xml:space="preserve"> quadrata <lb/>e f, e r, ideo q́ue latera e f, e r æquabuntur.</s> <s xml:id="echoid-s14892" xml:space="preserve">] Ergo [per 5 p 1] duo anguli e r f, e f r ſunt æqua-<lb/>les.</s> <s xml:id="echoid-s14893" xml:space="preserve"> Ergo forma n reflectetur ad r ex e:</s> <s xml:id="echoid-s14894" xml:space="preserve"> [per 12 n 4:</s> <s xml:id="echoid-s14895" xml:space="preserve"> quia anguli n e k, r e k æquantur, cum per <lb/>29 p 1 æquentur æqualibus ad f & r] & forma o reflectetur ad r ex z.</s> <s xml:id="echoid-s14896" xml:space="preserve"> Et omnis linea extracta ex <lb/>f ad aliquod punctum lineæ o n, ſecabit a e.</s> <s xml:id="echoid-s14897" xml:space="preserve"> Et patet, quòd linea illa erit æqualis lineæ extractæ <lb/>ex r ad idem punctum.</s> <s xml:id="echoid-s14898" xml:space="preserve"> Nam a e eſt perpendicularis ſuper ſuperficiem, in qua ſunt lineæ r e, f e:</s> <s xml:id="echoid-s14899" xml:space="preserve"> <lb/>nam hæc ſuperficies eſt ſuperficies ſectionis:</s> <s xml:id="echoid-s14900" xml:space="preserve"> & duæ lineæ r e, f e ſunt æquales.</s> <s xml:id="echoid-s14901" xml:space="preserve"> Ergo omnes duæ <lb/>lineæ extractæ ex r, f ad unum aliquod punctum lineæ a e, ſunt æquales.</s> <s xml:id="echoid-s14902" xml:space="preserve"> Patet ergo, quòd forma <lb/>puncti, quod eſt in o n, reflectetur ad r exillo puncto, quod ſecatur in z e.</s> <s xml:id="echoid-s14903" xml:space="preserve"> Et ſimiliter de omni <lb/>puncto poſito in a n ultra n, ſi copulatum fuerit cum f per lineam rectam, illa linea ſecabit a e ul-<lb/>tra e.</s> <s xml:id="echoid-s14904" xml:space="preserve"> Patet ergo ex hoc, quòd forma lineæ a n, & quicquid continuatur cum ipſa, reflectetur ad <lb/>r à ſuperficie pyramidis a b g ex linea recta.</s> <s xml:id="echoid-s14905" xml:space="preserve"> Et ſimiliter omnis linea extracta ex a, obliqua ſuper <lb/>axem.</s> <s xml:id="echoid-s14906" xml:space="preserve"> Et continuemus n d:</s> <s xml:id="echoid-s14907" xml:space="preserve"> ſecabit ergo circumferentiam ſectionis:</s> <s xml:id="echoid-s14908" xml:space="preserve"> nam duo puncta d, n ſunt in <lb/>ſuperficie ſectionis, & n eſt extra circumferentiam ſectionis:</s> <s xml:id="echoid-s14909" xml:space="preserve"> & d eſt intra ſectionem.</s> <s xml:id="echoid-s14910" xml:space="preserve"> Secet ergo <lb/>circũferẽtiã ſectionis in c.</s> <s xml:id="echoid-s14911" xml:space="preserve"> Et quia triangulũ a o h eſt in eadẽ ſuperficie [per 2 p 11] erit [per 1 p 11] n d <lb/>in ſuperficie trianguli a o h:</s> <s xml:id="echoid-s14912" xml:space="preserve"> c ergo eſt in ſuperficie trianguli a o h:</s> <s xml:id="echoid-s14913" xml:space="preserve"> & duo puncta a, u ſunt in ſu-<lb/>perficie trianguli huius a o h:</s> <s xml:id="echoid-s14914" xml:space="preserve"> ſed puncta a, u, c ſunt in ſuperficie pyramidis.</s> <s xml:id="echoid-s14915" xml:space="preserve"> Ergo puncta a, u, c ſunt <lb/>in differentia communi ſuperficiei pyramidis, & ſuperficiei a u d:</s> <s xml:id="echoid-s14916" xml:space="preserve"> ſed hæc differentia eſt linea re-<lb/>cta [per 18 d 11.</s> <s xml:id="echoid-s14917" xml:space="preserve">] Ergo puncta a, u, c ſunt in linea recta.</s> <s xml:id="echoid-s14918" xml:space="preserve"> Extrahatur ergo a u rectè ad c:</s> <s xml:id="echoid-s14919" xml:space="preserve"> & extra-<lb/>hatur r z rectè:</s> <s xml:id="echoid-s14920" xml:space="preserve"> ſecabit ergo o h [quia ſecat angulum z h o baſi h o ſubtenſum, & utraque z r & h o <lb/>ſunt in uno plano.</s> <s xml:id="echoid-s14921" xml:space="preserve">] Secet ergo in puncto p.</s> <s xml:id="echoid-s14922" xml:space="preserve"> Eſt ergo p in ſuperficie trianguli a o h.</s> <s xml:id="echoid-s14923" xml:space="preserve"> Continuetur <lb/>ergo a p, & tranſeat rectè.</s> <s xml:id="echoid-s14924" xml:space="preserve"> Secabit ergo n d in g [quia ſecat angulum d a n.</s> <s xml:id="echoid-s14925" xml:space="preserve">] Et quia f non eſt in ſu-<lb/>perficie pyramidẽ contingente, trãſeunte per lineã a z:</s> <s xml:id="echoid-s14926" xml:space="preserve"> [ex concluſo] erit angulus fe d acutus.</s> <s xml:id="echoid-s14927" xml:space="preserve"> [Nã <lb/>quia per concluſionem punctũ f eſt in plano ſectionis ſeu ellipſis, obliquo ad a d e planũ axis, per 5 <lb/>th.</s> <s xml:id="echoid-s14928" xml:space="preserve"> 1 con.</s> <s xml:id="echoid-s14929" xml:space="preserve"> Apol.</s> <s xml:id="echoid-s14930" xml:space="preserve"> & angulus a e frectus eſt cõcluſus:</s> <s xml:id="echoid-s14931" xml:space="preserve"> erit angulus f e d acutus:</s> <s xml:id="echoid-s14932" xml:space="preserve"> & angulus d e n eſt obtu <lb/> <pb o="213" file="0219" n="219" rhead="OPTICAE LIBER VI."/> ſus [per 13 p 1.</s> <s xml:id="echoid-s14933" xml:space="preserve">] Igitur angulus e n d eſt acutus [per 32 p 1.</s> <s xml:id="echoid-s14934" xml:space="preserve">] Et ſit linea c x cõtingens ſectionẽ in pun <lb/>cto c.</s> <s xml:id="echoid-s14935" xml:space="preserve"> Patet ergo, ut in prædicta figura [30 n] quòd angulus d c x eſt obtuſus:</s> <s xml:id="echoid-s14936" xml:space="preserve"> & qđ perpẽdicularis <lb/>extracta ex c ſuper c x, ſecabit angulũ d c x:</s> <s xml:id="echoid-s14937" xml:space="preserve"> & cõcurret cũ e d ſub d.</s> <s xml:id="echoid-s14938" xml:space="preserve"> Ergo hæc perpendicularis ſecet <lb/>e d in s.</s> <s xml:id="echoid-s14939" xml:space="preserve"> Perpẽdicularis ergo extracta ex n ſuք lineã cõtingentẽ ſectionẽ, ſecabit ſectionẽ ultra s:</s> <s xml:id="echoid-s14940" xml:space="preserve"> ſed <lb/>remotius à d quã s:</s> <s xml:id="echoid-s14941" xml:space="preserve"> nã iſtę perpendiculares cõcurrent ultra circũferentiã ſectionis.</s> <s xml:id="echoid-s14942" xml:space="preserve"> Perpẽdicularis <lb/>ergo extracta ex puncto n ſuper lineã contingentẽ ſectionẽ, non ſecabit angulũ d c x:</s> <s xml:id="echoid-s14943" xml:space="preserve"> erit ergo <gap/>e-<lb/>motior ab n e, quàm ſit n d.</s> <s xml:id="echoid-s14944" xml:space="preserve"> Ergo hæc perpendicularis ſecat a d ſupra d.</s> <s xml:id="echoid-s14945" xml:space="preserve"> Sit ergo perpẽdicularis ex-<lb/>tracta ex n ſuper lineam cõtingentẽ ſectionẽ, linea n q.</s> <s xml:id="echoid-s14946" xml:space="preserve"> Et r e ſecat e n, & ſecat circumferẽtiã ſectio-<lb/>nis:</s> <s xml:id="echoid-s14947" xml:space="preserve"> & eſt in ſuperficie eius:</s> <s xml:id="echoid-s14948" xml:space="preserve"> & n q eſt in ſuperficie ſectionis.</s> <s xml:id="echoid-s14949" xml:space="preserve"> Si ergo r e extrahatur rectè, ſecabit n q <lb/>[quia cõtinuata ſecat angulũ n e q.</s> <s xml:id="echoid-s14950" xml:space="preserve">] Secet ergo in y:</s> <s xml:id="echoid-s14951" xml:space="preserve"> & ſuperficies a n d ſecabit ſuperficiẽ ſectionis.</s> <s xml:id="echoid-s14952" xml:space="preserve"> <lb/>Itẽ quia punctũ e eſt extra ſuperficiẽ a n d:</s> <s xml:id="echoid-s14953" xml:space="preserve"> (nã ſuperficies a n d nõ eſt ſuperficies ſectionis [in qua <lb/>eſt punctũ e] quia punctũ a eſt extra ſuperficiẽ ſectionis:</s> <s xml:id="echoid-s14954" xml:space="preserve"> & quia a e eſt perpẽdicularis ſuper ſuperfi <lb/>ciẽ ſectionis, & e eſt in circumferentia illius) ergo n c d eſt differentia cõmunis ſuperficiei a n d & <lb/>ſuperficiei ſectionis:</s> <s xml:id="echoid-s14955" xml:space="preserve"> & n q concurrit cũ ſectione ultra c [ut patuit.</s> <s xml:id="echoid-s14956" xml:space="preserve">] Ergo n q eſt ultra ſuperficiem <lb/>a n d:</s> <s xml:id="echoid-s14957" xml:space="preserve"> y ergo eſt ultra lineam a p g [quæ nõ eſt in ſuperficie a n d.</s> <s xml:id="echoid-s14958" xml:space="preserve">] Si ergo uiſus fuerit in r, & forma <lb/>alicuius uiſibilis reflectatur à linea longitudinis:</s> <s xml:id="echoid-s14959" xml:space="preserve"> tunc p erit imago o:</s> <s xml:id="echoid-s14960" xml:space="preserve"> [per 4 n 5] & y erit imago n:</s> <s xml:id="echoid-s14961" xml:space="preserve"> <lb/>& a uidebitur in ſuo loco:</s> <s xml:id="echoid-s14962" xml:space="preserve"> quia eſt in uertice pyramidis.</s> <s xml:id="echoid-s14963" xml:space="preserve"> Et erit imago lineæ a o n linea tranſiens per <lb/>pũcta a, p, y:</s> <s xml:id="echoid-s14964" xml:space="preserve"> ſed hæc linea eſt cõuexa:</s> <s xml:id="echoid-s14965" xml:space="preserve"> quia eſt ultra lineã a p g.</s> <s xml:id="echoid-s14966" xml:space="preserve"> Sit ergo linea a p y.</s> <s xml:id="echoid-s14967" xml:space="preserve"> Et patuitiã, quòd <lb/>formæ omniũ punctorũ, quæ ſunt in a n, reflectantur ad r ex a e.</s> <s xml:id="echoid-s14968" xml:space="preserve"> Lineæ ergo radiales, per quas refle <lb/>ctuntur illæ formæ, ſunt in ſuperficie trianguli r a e.</s> <s xml:id="echoid-s14969" xml:space="preserve"> Omnes ergo imagines lineæ a n ſunt in hac ſu-<lb/>perficie.</s> <s xml:id="echoid-s14970" xml:space="preserve"> Ergo linea a p y conuexa eſt in hac ſuperficie:</s> <s xml:id="echoid-s14971" xml:space="preserve"> & p eſt propinquius r quàm y.</s> <s xml:id="echoid-s14972" xml:space="preserve"> Et erit <lb/>conuexitas imaginis huius ex parte uiſus:</s> <s xml:id="echoid-s14973" xml:space="preserve"> & erit conuexitas parua:</s> <s xml:id="echoid-s14974" xml:space="preserve"> & diameter huius imaginis e-<lb/>rit minor ipſa linea, modica quantitate.</s> <s xml:id="echoid-s14975" xml:space="preserve"> Imagines ergo linearum rectarum, quæ extrahuntur ex <lb/>uertice pyramidis obliquè ſuper axem:</s> <s xml:id="echoid-s14976" xml:space="preserve"> comprehenduntur à uiſu in tali ſpeculo conuexæ.</s> <s xml:id="echoid-s14977" xml:space="preserve"> Et for-<lb/>mę harum linearum reflectuntur à lineis rectis extenſis in longitudine pyramidis.</s> <s xml:id="echoid-s14978" xml:space="preserve"> Et hoc eſt, quod <lb/>uoluimus declarare.</s> <s xml:id="echoid-s14979" xml:space="preserve"/> </p> <div xml:id="echoid-div504" type="float" level="0" n="0"> <figure xlink:label="fig-0218-01" xlink:href="fig-0218-01a"> <variables xml:id="echoid-variables177" xml:space="preserve">a o u p m h z t x b n y c q s l d g e K f r</variables> </figure> </div> </div> <div xml:id="echoid-div506" type="section" level="0" n="0"> <head xml:id="echoid-head448" xml:space="preserve" style="it">33. Si recta linea ſit parallela latitudini ſpeculi conici conuexi: & uiſ{us} ſit extra planum di-<lb/>ctæ lineæ baſi parallelum: reflectetur ab ellipſi: & imago uidebitur maximè curua. 56 p 7.</head> <p> <s xml:id="echoid-s14980" xml:space="preserve">FOrmæ uerò linearũ æquidiſtantiũ latitudini ſpeculi pyramidalis cõuexi, reflectuntur à lineis <lb/>conuexis in ſuperficie ſpeculi:</s> <s xml:id="echoid-s14981" xml:space="preserve"> & conuexitas harum linearum patet, ut in ſpeculo columnari <lb/>conuexo [29 n.</s> <s xml:id="echoid-s14982" xml:space="preserve">] Et per illam eandem uiam etiam ſimiliter patebit, quòd imagines harum li-<lb/>nearum erunt nimium cõuexæ & manifeſtæ ſenſui.</s> <s xml:id="echoid-s14983" xml:space="preserve"> Et erit centrum uiſus extra ſuperficies, in qui-<lb/>bus eſt cõuexitas formarum harum linearum.</s> <s xml:id="echoid-s14984" xml:space="preserve"> Et erunt diametri imaginum harum linearum mul-<lb/>tò minores ipſis lineis.</s> <s xml:id="echoid-s14985" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div507" type="section" level="0" n="0"> <head xml:id="echoid-head449" xml:space="preserve" style="it">34. Si recta linea nec uertici ſpeculi conici conuexi obliquè incidat, nec latitudini ei{us} ſit paral <lb/>lela: imaginem uariæ obliquitatis prouario ſit u uiſui offeret. 57 p 7.</head> <p> <s xml:id="echoid-s14986" xml:space="preserve">DE lineis uerò obliquis exiſtentibus inter hos duos modos, quę appropinquant in ſuo motu <lb/>lineis extenſis in longitudine pyramidis, habent formas parũ conuexas:</s> <s xml:id="echoid-s14987" xml:space="preserve"> quę uerò appropin <lb/>quant lineis æquidiſtantibus latitudini pyramidis, habent formas manifeſtè conuexas.</s> <s xml:id="echoid-s14988" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div508" type="section" level="0" n="0"> <head xml:id="echoid-head450" xml:space="preserve" style="it">35. In ſpeculo conico conuexo imago conica uidetur. 58 p 7. 40 p 6.</head> <p> <s xml:id="echoid-s14989" xml:space="preserve">SEd tamen lineæ tortuoſæ, quæ appropinquant uertici pyramidis, habent formas minores, & <lb/>ſtrictiores & conuexiores.</s> <s xml:id="echoid-s14990" xml:space="preserve"> Quæ uerò appropinquant baſi pyramidis, habent formas amplio-<lb/>res, propter illud, quod declaratum fuit in ſpeculus ſphæricis conuexis:</s> <s xml:id="echoid-s14991" xml:space="preserve"> ſcilicet quòd quantò <lb/>minus fuerit ſpeculum, tantò minores erunt circuli, qui cadunt in ſuperficiem eius:</s> <s xml:id="echoid-s14992" xml:space="preserve"> & ſic ima-<lb/>gines erunt propinquiores centro:</s> <s xml:id="echoid-s14993" xml:space="preserve"> idcirco erunt minores.</s> <s xml:id="echoid-s14994" xml:space="preserve"> Et ſimiliter ſectiones, quæ cadunt <lb/>in ſpeculũ pyramidale, quæ ſunt ex parte uerticis pyramidis, ſunt ſtrictiores & minores:</s> <s xml:id="echoid-s14995" xml:space="preserve"> & ſic ima-<lb/>go erit propinquior puncto, in quo cõcurrunt perpendiculares, exeuntes à linea uiſibili perpendi-<lb/>culariter ſuper lineas contingentes ſectiones, quæ ſunt differentiæ communes:</s> <s xml:id="echoid-s14996" xml:space="preserve"> & ideo iſtę ima-<lb/>gines erunt minores.</s> <s xml:id="echoid-s14997" xml:space="preserve"> Sectiones uerò, quæ ſunt ex parte baſis pyramidis, è contrario.</s> <s xml:id="echoid-s14998" xml:space="preserve"> Vnde ac-<lb/>cidit, ut forma comprehenſa in ſpeculo pyramidali conuexo ſit pyramidata:</s> <s xml:id="echoid-s14999" xml:space="preserve"> quod ſcilicet fuerit ex <lb/>parte uerticis ſpeculi, erit ſtrictius, & quod ex parte baſis, erit amplius:</s> <s xml:id="echoid-s15000" xml:space="preserve"> & conuexitas latitudinis <lb/>formæ erit manifeſta.</s> <s xml:id="echoid-s15001" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div509" type="section" level="0" n="0"> <head xml:id="echoid-head451" xml:space="preserve" style="it">36. Imago uiſibilis propinqui ſpeculo conico conuexo, maior: longinqui, minor uidetur. 59 p 7.</head> <p> <s xml:id="echoid-s15002" xml:space="preserve">ET accidit etiam in his ſpeculis, quòd quantò magis res uiſa appropinquauerit ſpeculo, tantò <lb/>uidebitur maior:</s> <s xml:id="echoid-s15003" xml:space="preserve"> & quantò magis erit remota, tantò uidebitur minor.</s> <s xml:id="echoid-s15004" xml:space="preserve"> Fallaciæ ergo, quæ ac-<lb/>cidunt in his ſpeculis, ſunt ſimiles in omnibus diſpoſitionibus, illis, quæ accidunt in ſpecu-<lb/>lis columnaribus conuexis, præterquam in pyramidatione formæ.</s> <s xml:id="echoid-s15005" xml:space="preserve"/> </p> <pb o="214" file="0220" n="220" rhead="ALHAZEN"/> </div> <div xml:id="echoid-div510" type="section" level="0" n="0"> <head xml:id="echoid-head452" xml:space="preserve" style="it">37. Imago figuratur quodammodo à ſuo ſpeculo. 38 p 5.</head> <p> <s xml:id="echoid-s15006" xml:space="preserve">ET omnino forma rei uiſæ, quæ comprehenditur per reflexionẽ, ſemper aſsimilabitur formæ <lb/>ſuperficiei ſpeculi, à qua reflectitur forma.</s> <s xml:id="echoid-s15007" xml:space="preserve"> Et huius cauſſa eſt, quòd ſemper locus imaginis <lb/>eſt ex forma ſuperficiei ſpeculi & ex loco concurſus perpendicularium.</s> <s xml:id="echoid-s15008" xml:space="preserve"> Ideo ſemper ſuperfi-<lb/>cies ſpeculi habet aliquam dignitatem in forma rei uiſæ, quæ comprehenditur in ſpeculo.</s> <s xml:id="echoid-s15009" xml:space="preserve"> Fallaciæ <lb/>uerò compoſitæ in hoc ſpeculo, ſimiles ſunt fallacijs in prædictis ſpeculis.</s> <s xml:id="echoid-s15010" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div511" type="section" level="0" n="0"> <head xml:id="echoid-head453" xml:space="preserve">DE ERRORIBVS, QVI ACCIDVNT IN SPECVLIS <lb/>ſphæricis concauis. Cap. VII.</head> <head xml:id="echoid-head454" xml:space="preserve" style="it">38. In ſpeculo cauo allucinationes frequentiores & maiores accidunt, quàm in plano & con-<lb/>uexo. Vitell. in proœmio 8 libri.</head> <p> <s xml:id="echoid-s15011" xml:space="preserve">IN his uerò plures errores accidunt, quã in omnibus ſpeculis cõuexis & ſuperficialibus.</s> <s xml:id="echoid-s15012" xml:space="preserve"> Accidũt <lb/>enim in ijs, quę in illis accidunt, ſcilicet debilitas lucis & coloris:</s> <s xml:id="echoid-s15013" xml:space="preserve"> & diuerſitas ſitus & remotiõis.</s> <s xml:id="echoid-s15014" xml:space="preserve"> <lb/>Nã cauſſa huius eſt tãtũ reflexio, nõ forma ſpeculi.</s> <s xml:id="echoid-s15015" xml:space="preserve"> Accidit etiã in his ſpeculis ex diuerſitate quã-<lb/>titatis, plus erroris, quã in ſpeculis cõuexis.</s> <s xml:id="echoid-s15016" xml:space="preserve"> Nã in cõuexis in maiore parte res cõprehẽditur minor:</s> <s xml:id="echoid-s15017" xml:space="preserve"> <lb/>in cõcauis uerò quãdoq;</s> <s xml:id="echoid-s15018" xml:space="preserve"> cõprehẽditur maior:</s> <s xml:id="echoid-s15019" xml:space="preserve"> quãdoq;</s> <s xml:id="echoid-s15020" xml:space="preserve"> minor:</s> <s xml:id="echoid-s15021" xml:space="preserve"> quãdoq;</s> <s xml:id="echoid-s15022" xml:space="preserve"> ſecũdũ qđ eſt:</s> <s xml:id="echoid-s15023" xml:space="preserve"> & hoc ſecun <lb/>dũ diuerſitatẽ poſitionũ ex ſpeculo & ex uiſu, ꝓut nos declarabimus in hoc capitulo.</s> <s xml:id="echoid-s15024" xml:space="preserve"> Accidit etiã <lb/>in his ſpeculis, qđ unũ uiſibile uidetur duo, & tria, & quatuor:</s> <s xml:id="echoid-s15025" xml:space="preserve"> & nõ eſt ita in ſpeculis ſuperficialib.</s> <s xml:id="echoid-s15026" xml:space="preserve"> <lb/>& cõuexis.</s> <s xml:id="echoid-s15027" xml:space="preserve"> Vnũ enim uiſibile nõ cõprehẽditur in illis, niſi unũ:</s> <s xml:id="echoid-s15028" xml:space="preserve"> in cõcauis uerò nõ.</s> <s xml:id="echoid-s15029" xml:space="preserve"> Itẽ ordinatio par <lb/>tiũ rei uiſæ cõprehẽditur in ſpeculis cõuexis & ſuperficialibus, ſecũdũ qđ eſt:</s> <s xml:id="echoid-s15030" xml:space="preserve"> in ſpeculis uerò cõca <lb/>uis in pluribus ſitib.</s> <s xml:id="echoid-s15031" xml:space="preserve"> alio modo.</s> <s xml:id="echoid-s15032" xml:space="preserve"> Et hęc duo:</s> <s xml:id="echoid-s15033" xml:space="preserve"> ſcilicet cõprehenſio unius ut unũ:</s> <s xml:id="echoid-s15034" xml:space="preserve"> & cõprehẽſio ordina <lb/>tionis partiũ, ſecũdũ qđ eſt, nõ habet aliquã deceptionẽ in ſpeculis ſphæricis cõuexis.</s> <s xml:id="echoid-s15035" xml:space="preserve"> Et cũ in his <lb/>ſpeculis ſphęricis cõcauis accidit deceptio:</s> <s xml:id="echoid-s15036" xml:space="preserve"> patet, qđ nihil cõprehẽditur in huiuſmodi ſpeculis, niſi <lb/>cũ fallacia, aut ſemper, aut aliqua hora ſecũdũ diuerſitatẽ poſitiõis.</s> <s xml:id="echoid-s15037" xml:space="preserve"> Debilitas uerò lucis & coloris, <lb/>& diuerſitas poſitionis, & diſtãtia ac cidũt in his ſpeculis, ſicut in alijs ſemper, & in omni poſitione.</s> <s xml:id="echoid-s15038" xml:space="preserve"> <lb/>Quãtitas uerò, & forma, & numerus habẽt deceptionẽ in his ſpeculis in aliquib.</s> <s xml:id="echoid-s15039" xml:space="preserve"> ſitibus, ꝓut decla-<lb/>rabimus.</s> <s xml:id="echoid-s15040" xml:space="preserve"> De numero uerò declaratũ eſt in capitulo de imagine [66.</s> <s xml:id="echoid-s15041" xml:space="preserve">67.</s> <s xml:id="echoid-s15042" xml:space="preserve">69.</s> <s xml:id="echoid-s15043" xml:space="preserve">70.</s> <s xml:id="echoid-s15044" xml:space="preserve">71.</s> <s xml:id="echoid-s15045" xml:space="preserve">72 n 5] quòd unũ <lb/>uiſum in ſpeculis cõcauis habet unã imaginẽ, & duas, & tres, & quatuor:</s> <s xml:id="echoid-s15046" xml:space="preserve"> & quòd forma rei uiſæ ſem <lb/>per cõprehenditur in loco imaginis.</s> <s xml:id="echoid-s15047" xml:space="preserve"> Verũ unũ uiſum cõprehenſum in ſpeculis ſphæricis concauis <lb/>etiã fortè cõprehenditur unũ, & fortè duo, & fortè tria, & fortè quatuor:</s> <s xml:id="echoid-s15048" xml:space="preserve"> quod nõ actidit in ſpecu-<lb/>lis ſphæricis cõuexis & ſuperficialibus.</s> <s xml:id="echoid-s15049" xml:space="preserve"> De ordinatione uerò partiũ rei uiſæ dictũ eſt in capitulo de <lb/>imagine [65 n 5] quòd forma unius puncti reflectitur ex circũferentia unius circuli:</s> <s xml:id="echoid-s15050" xml:space="preserve"> & quòd uiſi-<lb/>bilia, quorũ imagines retrò poſt uiſum, & antè, & in cẽtro uiſus, apparẽt dubia nõ certificata:</s> <s xml:id="echoid-s15051" xml:space="preserve"> & qđ <lb/>eſt huiuſmodi, nõ habet ordinationẽ partiũ, ſicut ipſa res uiſa habet.</s> <s xml:id="echoid-s15052" xml:space="preserve"> Et hoc etiã eſt in his ſpeculis <lb/>aliter, quã ſit in ſpeculis cõuexis & ſuperficialibus.</s> <s xml:id="echoid-s15053" xml:space="preserve"> Cauſſæ aũt huius rei declaratæ ſunt in capitulo <lb/>de imagine.</s> <s xml:id="echoid-s15054" xml:space="preserve"> Reſtat ergo declarare, quòd illud, quod cõprehenditur in his ſpeculis, fortè cõprehen-<lb/>ditur maius:</s> <s xml:id="echoid-s15055" xml:space="preserve"> & fortè minus:</s> <s xml:id="echoid-s15056" xml:space="preserve"> & fortè ęquale:</s> <s xml:id="echoid-s15057" xml:space="preserve"> & quòd in quibuſdã poſitionibus cõprehendetur con-<lb/>uerſum, & in quibuſdã erectũ:</s> <s xml:id="echoid-s15058" xml:space="preserve"> & quòd erectũ in huiuſmodi ſpeculis cõprehendetur concauum, & <lb/>conuexum, & rectum:</s> <s xml:id="echoid-s15059" xml:space="preserve"> & quòd conuexũ & concauũ cõprehẽduntur etiam aliter quàm ſint.</s> <s xml:id="echoid-s15060" xml:space="preserve"> Et hæc <lb/>etiã ſunt ex diuerſitate ordinationis partiũ rei uiſæ.</s> <s xml:id="echoid-s15061" xml:space="preserve"> Et nos declarabimus hæc hoc modo.</s> <s xml:id="echoid-s15062" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div512" type="section" level="0" n="0"> <head xml:id="echoid-head455" xml:space="preserve" style="it">39. Si uiſ{us} & uiſibile fuerint intra ſpeculũ ſphæricum cauũ, in recta linea extremis ſuis à <lb/>centro æquabiliter diſtante: imago uidebitur ultra ſpeculũ, maior uiſibili. 46 p 8.</head> <p> <s xml:id="echoid-s15063" xml:space="preserve">SIt ſpeculũ ſphæricum concauũ, cuius centrũ a:</s> <s xml:id="echoid-s15064" xml:space="preserve"> & ſecetur ſuperficie plana, tranſeunte per cen-<lb/>trũ:</s> <s xml:id="echoid-s15065" xml:space="preserve"> & faciat circulũ b g [faciet aũt per 1 th 1 ſphær.</s> <s xml:id="echoid-s15066" xml:space="preserve">] Extrahatur ab ipſius cẽtro linea quocũq;</s> <s xml:id="echoid-s15067" xml:space="preserve"> <lb/>modo ſit:</s> <s xml:id="echoid-s15068" xml:space="preserve"> & diuidatur in duo æqualia:</s> <s xml:id="echoid-s15069" xml:space="preserve"> [per 10 p 1] & ponatur a centrũ, & in diſtantia a o facia-<lb/>mus circulũ:</s> <s xml:id="echoid-s15070" xml:space="preserve"> & ſit e z:</s> <s xml:id="echoid-s15071" xml:space="preserve"> & ponatur in linea o u punctũ t caſualiter, quocũq;</s> <s xml:id="echoid-s15072" xml:space="preserve"> modo ſit:</s> <s xml:id="echoid-s15073" xml:space="preserve"> & ext extrahan-<lb/>tur lineæ t n, t m, rectę ſuper lineã a u:</s> <s xml:id="echoid-s15074" xml:space="preserve"> [per 11 p 1] & extrahantur ext lineæ t e, t z tangentes circulũ <lb/>e z:</s> <s xml:id="echoid-s15075" xml:space="preserve"> [per 17 p 1] & continuemus a e, a z, & tranſeant ad b, g:</s> <s xml:id="echoid-s15076" xml:space="preserve"> & continuemus t b, b g:</s> <s xml:id="echoid-s15077" xml:space="preserve"> & [per 31 p 1] <lb/>protrahamus b m æquidiſtantẽ ad a u, & g n etiam æquidiſtantẽ a u:</s> <s xml:id="echoid-s15078" xml:space="preserve"> & cõtinuemus a n, a m, & extra <lb/>hantur rectè.</s> <s xml:id="echoid-s15079" xml:space="preserve"> Quia ergo a o eſt, ſicut o u:</s> <s xml:id="echoid-s15080" xml:space="preserve"> erit a e, ſicut e b, & a z, ſicut z g.</s> <s xml:id="echoid-s15081" xml:space="preserve"> [diametri enim circuli b g <lb/>bifariam ſectæ ſunt in punctis e, o, z, per peripheriam e o z.</s> <s xml:id="echoid-s15082" xml:space="preserve">] Et quia t e tangit circulũ e z:</s> <s xml:id="echoid-s15083" xml:space="preserve"> erit [per <lb/>18 p 3] t e perpendicularis ſuper a b:</s> <s xml:id="echoid-s15084" xml:space="preserve"> & ſimiliter t z perpendicularis ſuper a g.</s> <s xml:id="echoid-s15085" xml:space="preserve"> Linea ergo b t eſt, ſicut <lb/>t a, & t g, ſicut t a:</s> <s xml:id="echoid-s15086" xml:space="preserve"> & angulus t b a, ſicut angulus t a b, & angulus t g a, ſicut angulus t a g.</s> <s xml:id="echoid-s15087" xml:space="preserve"> [per 4 p 1:</s> <s xml:id="echoid-s15088" xml:space="preserve"> ꝗa <lb/>duo latera a e, e b ęquãtur ex cõcluſo, & cõmune eſt e t, anguliq́;</s> <s xml:id="echoid-s15089" xml:space="preserve"> ad e deinceps recti ſũt ք 18 p 3:</s> <s xml:id="echoid-s15090" xml:space="preserve"> itẽq́;</s> <s xml:id="echoid-s15091" xml:space="preserve"> <lb/>duo latera a z, z g, & commune t z, anguliq́;</s> <s xml:id="echoid-s15092" xml:space="preserve"> ad z recti.</s> <s xml:id="echoid-s15093" xml:space="preserve">] Et quia b m eſt æquidiſtans a u:</s> <s xml:id="echoid-s15094" xml:space="preserve"> [è fabrica-<lb/>tione] erit [per 29 p 1] m b a, ſicut angulus b a t.</s> <s xml:id="echoid-s15095" xml:space="preserve"> Ergo angulus m b a eſt, ſicut angulus a b t:</s> <s xml:id="echoid-s15096" xml:space="preserve"> & ſimi-<lb/>liter angulus t g a, ſicut angulus a g n.</s> <s xml:id="echoid-s15097" xml:space="preserve"> Cum ergo uiſus fuerit in t:</s> <s xml:id="echoid-s15098" xml:space="preserve"> & m b fuerit aliquod uiſibile:</s> <s xml:id="echoid-s15099" xml:space="preserve"> tunc <lb/>forma m exten detur per lineam m b, & reflectetur ad uiſum per lineam b t:</s> <s xml:id="echoid-s15100" xml:space="preserve"> & forma n extendetur <lb/>per lineam n g, & reflectetur per g t.</s> <s xml:id="echoid-s15101" xml:space="preserve"> Viſus ergo t comprehendet puncta m, n ex punctis b, g, & lineã <lb/>m n ex arcu b g [per 66 n 5.</s> <s xml:id="echoid-s15102" xml:space="preserve">] Et quia m t eſt perpendicularis ſuper a t:</s> <s xml:id="echoid-s15103" xml:space="preserve"> [per fabricationẽ] erit angu-<lb/>lus m t b acutus:</s> <s xml:id="echoid-s15104" xml:space="preserve"> [per 32 p 1] & quia angulus b m t eſt, ſicut angulus m t u.</s> <s xml:id="echoid-s15105" xml:space="preserve"> [per 29 p 1:</s> <s xml:id="echoid-s15106" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s15107" xml:space="preserve"> angu-<lb/>lus b m t rectus eſt, cũ m t u ſit rectus per fabricationẽ.</s> <s xml:id="echoid-s15108" xml:space="preserve">] Ergo [per 19 p 1] t b eſt maior b m, & linea t <lb/>b eſt æqualis lineæ a t:</s> <s xml:id="echoid-s15109" xml:space="preserve"> [per concluſionẽ.</s> <s xml:id="echoid-s15110" xml:space="preserve">] ergo linea a t eſt maior linea b m, & ſunt æquidiſtãtes.</s> <s xml:id="echoid-s15111" xml:space="preserve"> Er-<lb/> <pb o="215" file="0221" n="221" rhead="OPTICAE LIBER VI."/> go t b concurret cum a m.</s> <s xml:id="echoid-s15112" xml:space="preserve"> [ſi enim ex trapezio a m b t fiat parallelogrammũ (æquato nẽpe latere <lb/>b m ipſi t a, cumq́ue eodem connexo) patebit per lemma Procli ad 29 p 1, a m concurrere cum t b:</s> <s xml:id="echoid-s15113" xml:space="preserve"> <lb/>quia concurrit cum ipſius parallela.</s> <s xml:id="echoid-s15114" xml:space="preserve">] Concurrant ergo in f:</s> <s xml:id="echoid-s15115" xml:space="preserve"> fergo eſt imago m.</s> <s xml:id="echoid-s15116" xml:space="preserve"> [per 6 n 5.</s> <s xml:id="echoid-s15117" xml:space="preserve">] Et ſic <lb/>declarabitur, quòd t g concurret cum a n.</s> <s xml:id="echoid-s15118" xml:space="preserve"> Concurrat in q:</s> <s xml:id="echoid-s15119" xml:space="preserve"> q <lb/> <anchor type="figure" xlink:label="fig-0221-01a" xlink:href="fig-0221-01"/> ergo erit imago n.</s> <s xml:id="echoid-s15120" xml:space="preserve"> Et continuemus f q:</s> <s xml:id="echoid-s15121" xml:space="preserve"> quæ eſt diameter i-<lb/>maginis m b.</s> <s xml:id="echoid-s15122" xml:space="preserve"> Et quia t e, t z ſunt æquales:</s> <s xml:id="echoid-s15123" xml:space="preserve"> [per conſectariũ <lb/>Campani ad 36 p 3] erunt anguli t a e, t a z æquales [per 8 <lb/>p 1:</s> <s xml:id="echoid-s15124" xml:space="preserve"> quia a e, a z æquantur per 15 d 1, & a t eſt cõmune latus] <lb/>& erunt lineæ t b, t g æquales [per 4 p 1:</s> <s xml:id="echoid-s15125" xml:space="preserve"> quia a b, a g æquan <lb/>tur per 15 d 1] & lineæ b m, g n æquales.</s> <s xml:id="echoid-s15126" xml:space="preserve"> [Quia enim b a, g a <lb/>æquantur per 15 d 1, & a t eſt cõmunis, angulusq́;</s> <s xml:id="echoid-s15127" xml:space="preserve"> b a t æqua <lb/>lis concluſus eſt angulo g a t:</s> <s xml:id="echoid-s15128" xml:space="preserve"> æquabitur per 4 p 1 angulus <lb/>b t a angulo g t a, ideoq́;</s> <s xml:id="echoid-s15129" xml:space="preserve"> per 13 p 1 angulus u t b angulo u t g.</s> <s xml:id="echoid-s15130" xml:space="preserve"> <lb/>Quare cum anguli a d t deinceps recti ſint per fabricationẽ:</s> <s xml:id="echoid-s15131" xml:space="preserve"> <lb/>æquabitur per 3 ax.</s> <s xml:id="echoid-s15132" xml:space="preserve"> angulus b t m angulo g t n, & anguli ad <lb/>m & n recti per 29 p 1, æquantur per 10 ax.</s> <s xml:id="echoid-s15133" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s15134" xml:space="preserve"> per 26 p 1 b <lb/>m æquatur g n:</s> <s xml:id="echoid-s15135" xml:space="preserve"> & m tipſi n t] & lineæ a m, a n æquales [per <lb/>4 p 1:</s> <s xml:id="echoid-s15136" xml:space="preserve"> quia latera m t, n t ęqualia concluſa ſunt, & commune <lb/>eſt a t, anguliq́;</s> <s xml:id="echoid-s15137" xml:space="preserve"> a d t deinceps recti] & proportio a f ad f m, <lb/>ſicut proportio a t ad m b [per 4 p 6:</s> <s xml:id="echoid-s15138" xml:space="preserve"> quia triangula a t f, m b f ſunt æquiangula per 29.</s> <s xml:id="echoid-s15139" xml:space="preserve"> 32 p 1.</s> <s xml:id="echoid-s15140" xml:space="preserve">] Et <lb/>proportio a q ad q n eſt, ſicut proportio a t ad n g.</s> <s xml:id="echoid-s15141" xml:space="preserve"> Ergo proportio a fad f m eſt, ſicut proportio a q <lb/>ad q n [per 7 p 5:</s> <s xml:id="echoid-s15142" xml:space="preserve"> quia ratio a t ad b m & ad g n eadem eſt, cum b m æqualis oſtenſa ſit ipſi g n] & a <lb/>m eſt ſicut a n [per concluſionem.</s> <s xml:id="echoid-s15143" xml:space="preserve">] Ergo a f eſt ſicut a q.</s> <s xml:id="echoid-s15144" xml:space="preserve"> [Quia enim per concluſionem eſt, ut a f ad <lb/>f m, ſic a q ad q n:</s> <s xml:id="echoid-s15145" xml:space="preserve"> erit per 16 p 5, ut f a ad a q, ſic f m ad q n:</s> <s xml:id="echoid-s15146" xml:space="preserve"> ergo per 19 p 5 ut a m ad a n, ſic a f ad a q:</s> <s xml:id="echoid-s15147" xml:space="preserve"> <lb/>ſed a m æqualis oſtenſa eſt ipſi a n.</s> <s xml:id="echoid-s15148" xml:space="preserve"> Quare a f æqualis eſt a q.</s> <s xml:id="echoid-s15149" xml:space="preserve">] Ergo f q æquidiſtat n m [per proxi-<lb/>mam concluſionem & 2 p 6.</s> <s xml:id="echoid-s15150" xml:space="preserve">] Ergo f q eſt maior m n [per 4 p 6:</s> <s xml:id="echoid-s15151" xml:space="preserve"> quia a f ad a m, ſicut f q ad m n:</s> <s xml:id="echoid-s15152" xml:space="preserve"> ſed a f <lb/>maior eſt a m ք 9 ax:</s> <s xml:id="echoid-s15153" xml:space="preserve"> ergo f q maior eſt m n:</s> <s xml:id="echoid-s15154" xml:space="preserve"> ſed f q eſt diameter imaginis n m.</s> <s xml:id="echoid-s15155" xml:space="preserve"> Ergo ſi uiſus fuerit in <lb/>t, & linea m n fuerit in aliquo uiſibili:</s> <s xml:id="echoid-s15156" xml:space="preserve"> tunc uiſus comprehendet formam maiorem, quàm ſit.</s> <s xml:id="echoid-s15157" xml:space="preserve">]</s> </p> <div xml:id="echoid-div512" type="float" level="0" n="0"> <figure xlink:label="fig-0221-01" xlink:href="fig-0221-01a"> <variables xml:id="echoid-variables178" xml:space="preserve">f u q b <gap/> <gap/> m t n e o z a</variables> </figure> </div> </div> <div xml:id="echoid-div514" type="section" level="0" n="0"> <head xml:id="echoid-head456" xml:space="preserve" style="it">40. Si uiſ{us} fuerit ſublimior uiſibili intra ſpeculum ſphæricum cauum extremis ſuis à cen-<lb/>tro æquabiliter diſtante: imago uidebitur ultra ſpeculum, maior uiſibili. 47 p 8.</head> <p> <s xml:id="echoid-s15158" xml:space="preserve">ITem:</s> <s xml:id="echoid-s15159" xml:space="preserve"> iteremus circulum b g:</s> <s xml:id="echoid-s15160" xml:space="preserve"> & lineam a u:</s> <s xml:id="echoid-s15161" xml:space="preserve"> & lineas a b, a g, t b, t g:</s> <s xml:id="echoid-s15162" xml:space="preserve"> & ſuper punctum t ſit perpen-<lb/>dicularis ſuper ſuperficiem circuli b g [per 12 p 11] & ſit t k:</s> <s xml:id="echoid-s15163" xml:space="preserve"> continuemus k a, k b, k g.</s> <s xml:id="echoid-s15164" xml:space="preserve"> Superfici-<lb/>es ergo k b a, k g a ſecant ſphæram ſuper centrum ſuum perpendiculariter, & ſuperficies tangen <lb/>tes ipſam [per 18 p 11.</s> <s xml:id="echoid-s15165" xml:space="preserve">] Ex ipſis ergo reflectitur forma:</s> <s xml:id="echoid-s15166" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0221-02a" xlink:href="fig-0221-02"/> & duæ differentiæ cõmunes inter has duas ſuperficies <lb/>& ſphærã, ſunt circuli magni [per 1 th 1 ſphęr.</s> <s xml:id="echoid-s15167" xml:space="preserve">] à quorũ <lb/>circũferentia reflectũtur formæ.</s> <s xml:id="echoid-s15168" xml:space="preserve"> Et extrah amus b m in <lb/>ſuperficie b k a æquidiſtantẽ a k:</s> <s xml:id="echoid-s15169" xml:space="preserve"> & ſit minor, quã a k:</s> <s xml:id="echoid-s15170" xml:space="preserve"> & <lb/>cõtinuemus a m, & extrahatur rectè:</s> <s xml:id="echoid-s15171" xml:space="preserve"> & extrahatur k b, <lb/>donec cõcnrrat cum a m in f [cõcurret aũt, ut proximo <lb/>numero oſtẽſum eſt:</s> <s xml:id="echoid-s15172" xml:space="preserve"> quia b m minor eſt a k per ſabrica-<lb/>tionẽ.</s> <s xml:id="echoid-s15173" xml:space="preserve">] Et extrahatur n g in ſuperficie k g a:</s> <s xml:id="echoid-s15174" xml:space="preserve"> & ſit æqui-<lb/>diſtãs a k:</s> <s xml:id="echoid-s15175" xml:space="preserve"> & ponatur æqualis b m:</s> <s xml:id="echoid-s15176" xml:space="preserve"> & cõtinuemus a n, & <lb/>extrahatur rectè, donec cõcurrat in q:</s> <s xml:id="echoid-s15177" xml:space="preserve"> & cõtinuemus m <lb/>n, f q.</s> <s xml:id="echoid-s15178" xml:space="preserve"> Quia ergo b t eſt ſicut t a [ut ſuperiore numero <lb/>demonſtratũ eſt] erit b k, ſicut k a [per 4 p 1:</s> <s xml:id="echoid-s15179" xml:space="preserve"> nã t k com <lb/>mune latus eſt utriuſq;</s> <s xml:id="echoid-s15180" xml:space="preserve"> trianguli b t k, a t k, & anguli ad <lb/>t recti per 3 d 11] & g k, ſicut k a:</s> <s xml:id="echoid-s15181" xml:space="preserve"> ergo b k eſt, ſicut g k:</s> <s xml:id="echoid-s15182" xml:space="preserve"> & <lb/>[per 5 p 1] angulus k a b eſt, ſicut angulus k b a:</s> <s xml:id="echoid-s15183" xml:space="preserve"> & ſimi-<lb/>liter angulus k g a eſt, ſicut angulus k a g.</s> <s xml:id="echoid-s15184" xml:space="preserve"> Ergo angulus <lb/>a b m eſt, ſicut angulus a b k [quia per 29 p 1 angulus a <lb/>b m æquatur angulo k a b, cui æqualis cõcluſus eſt a b k] & angulus a g n eſt, ſicut angulus a g k.</s> <s xml:id="echoid-s15185" xml:space="preserve"> [Nã <lb/>per 29 p 1 angulus a g n æquatur angulo k a g, cui æqualis oſtẽſus eſt angulus a g k.</s> <s xml:id="echoid-s15186" xml:space="preserve">] Ergo erit angu <lb/>lus a b m, ſicut angulus a g n.</s> <s xml:id="echoid-s15187" xml:space="preserve"> [Quia enim g k æqualis concluſa eſt ipſi b k:</s> <s xml:id="echoid-s15188" xml:space="preserve"> & a g, a b æquantur <lb/>per 15 d 1:</s> <s xml:id="echoid-s15189" xml:space="preserve"> & cõmmunis eſt a k:</s> <s xml:id="echoid-s15190" xml:space="preserve"> æquabũtur anguli a b k, a g k per 8 p 1:</s> <s xml:id="echoid-s15191" xml:space="preserve"> & his ęquãtur per proximã cõ <lb/>cluſionẽ a b m, a g n.</s> <s xml:id="echoid-s15192" xml:space="preserve"> Quare a b m, a g n æquãtur] & linea b m, ſicut linea g n:</s> <s xml:id="echoid-s15193" xml:space="preserve"> [ex fabricatione] tũc li <lb/>nea a m erit, ſicut linea a n:</s> <s xml:id="echoid-s15194" xml:space="preserve"> [ք 4 p 1:</s> <s xml:id="echoid-s15195" xml:space="preserve"> quia a b, b m ęquãtur ipſis a g, g n, & angulus a b m angulo a g n] <lb/>tũc duę lineæ f q, m n erũt æquidiſtãtes:</s> <s xml:id="echoid-s15196" xml:space="preserve"> [per 2 p 6, ut proximo numero demõſtratũ eſt] tũc f q erit <lb/>maior linea m n.</s> <s xml:id="echoid-s15197" xml:space="preserve"> Tunc quando uiſus fuerit ſuper punctum k, & fuerit linea m n in aliquo uiſibili in-<lb/>feriore:</s> <s xml:id="echoid-s15198" xml:space="preserve"> tunc forma m extendetur ſuper lineam m b, & reflectetur per lineam b k in ſuperficie circu <lb/>li, tranſeuntis per puncta b, a, k:</s> <s xml:id="echoid-s15199" xml:space="preserve"> & forma puncti n extendetur ſuper lineam n g, & reſlectetur ſuper <lb/>lineam g k in ſuperficie circuli, tranſeuntis per tria puncta g, a, k.</s> <s xml:id="echoid-s15200" xml:space="preserve"> Et erit imago puncti f punctum m:</s> <s xml:id="echoid-s15201" xml:space="preserve"> <lb/>[per 6 n 5] & punctum q erit imago puncti n:</s> <s xml:id="echoid-s15202" xml:space="preserve"> & erit linea f q diameter imaginis n m.</s> <s xml:id="echoid-s15203" xml:space="preserve"> Etiam decla-<lb/> <pb o="216" file="0222" n="222" rhead="ALHAZEN"/> rauimus [ſuperiore numero] quòd linea f q eſt maior linea m n.</s> <s xml:id="echoid-s15204" xml:space="preserve"> T unc quando uiſus fuerit ſuper <lb/>punctum k, & fuerit linea m n in aliquo uiſibili:</s> <s xml:id="echoid-s15205" xml:space="preserve"> tunc uiſus apprehẽdet formam maiorem re uiſa.</s> <s xml:id="echoid-s15206" xml:space="preserve"> Et <lb/>ſic, ſi reuoluerimus totam figuram in circuitu lineæ a u, ipſa immobili:</s> <s xml:id="echoid-s15207" xml:space="preserve"> tunc punctum k faciet circu <lb/>lum perpendicularem ſuper lineam a u.</s> <s xml:id="echoid-s15208" xml:space="preserve"> Et ſic omne punctum illius circuli habebit ſitum, reſpectu <lb/>lineæ comparis m n, ſicut eſt ſitus k reſpectu m n.</s> <s xml:id="echoid-s15209" xml:space="preserve"> Si ergo uiſus fuerit in aliquo puncto circumferen <lb/>tiæ huius circuli, & linea compar lineæ m n, fuerit in ſuperficie alicuius rei uiſæ:</s> <s xml:id="echoid-s15210" xml:space="preserve"> tunc uiſus compre <lb/>hendet formam illius lineæ maiorem.</s> <s xml:id="echoid-s15211" xml:space="preserve"> Et ſimiliter ſi extrahamus t k rectè, & poſuerimus in ipſa ali-<lb/>quod punctum præter k, & extraxerimus lineas ſemper ab illo puncto, quod eſt quaſi punctum k:</s> <s xml:id="echoid-s15212" xml:space="preserve"> <lb/>erit modus eius ſicut modus puncti k.</s> <s xml:id="echoid-s15213" xml:space="preserve"> Ex his ergo duabus figuris patet, quòd in ſphæricis ſpeculis <lb/>con cauis & multa & ex multis ſitibus comprehenduntur maiora.</s> <s xml:id="echoid-s15214" xml:space="preserve"/> </p> <div xml:id="echoid-div514" type="float" level="0" n="0"> <figure xlink:label="fig-0221-02" xlink:href="fig-0221-02a"> <variables xml:id="echoid-variables179" xml:space="preserve">f q b u g m c n K p a</variables> </figure> </div> </div> <div xml:id="echoid-div516" type="section" level="0" n="0"> <head xml:id="echoid-head457" xml:space="preserve" style="it">41. In ſpeculo ſphærico cauo imago interdum æquatur uiſibili: & quæ inter uiſum & ſpecu-<lb/>lum, euerſa, quæ pone uiſum, erecta eſt. 48 p 8.</head> <p> <s xml:id="echoid-s15215" xml:space="preserve">ITem:</s> <s xml:id="echoid-s15216" xml:space="preserve"> ſit ſpeculum ſphæricum a b circa centrum e:</s> <s xml:id="echoid-s15217" xml:space="preserve"> & extrahamus ſuperficiem tranſeuntem per <lb/>e:</s> <s xml:id="echoid-s15218" xml:space="preserve"> & faciat circulũ a b:</s> <s xml:id="echoid-s15219" xml:space="preserve"> & extrahamus ex e lineã e z, quocunq;</s> <s xml:id="echoid-s15220" xml:space="preserve"> modo fuerit, uſq;</s> <s xml:id="echoid-s15221" xml:space="preserve"> ad g:</s> <s xml:id="echoid-s15222" xml:space="preserve"> & ex g extra <lb/>hamus g d perpendicularem ſuper ſuperficiem circuli a b:</s> <s xml:id="echoid-s15223" xml:space="preserve"> [per 12 p 11] & in ipſa ſignemus pun-<lb/>ctum d, quocunq;</s> <s xml:id="echoid-s15224" xml:space="preserve"> modo fuerit:</s> <s xml:id="echoid-s15225" xml:space="preserve"> & continuemus d e:</s> <s xml:id="echoid-s15226" xml:space="preserve"> & extrahamus ipſam uſq;</s> <s xml:id="echoid-s15227" xml:space="preserve"> ad o:</s> <s xml:id="echoid-s15228" xml:space="preserve"> & extrahamus <lb/>e b ita, ut contineat cum e d angulum obtuſum:</s> <s xml:id="echoid-s15229" xml:space="preserve"> & extrahamus e a ita, ut contineat cum e d angulũ, <lb/>æqualem angulo d e b:</s> <s xml:id="echoid-s15230" xml:space="preserve"> & continuemus d a, d b.</s> <s xml:id="echoid-s15231" xml:space="preserve"> Sic ergo ſuperficies duorum triangulorũ d a e, d b e <lb/>ſecant ſe ſuper lineam d e:</s> <s xml:id="echoid-s15232" xml:space="preserve"> & duo anguli acuti d b e, d a e erunt æquales.</s> <s xml:id="echoid-s15233" xml:space="preserve"> [per 4 p 1:</s> <s xml:id="echoid-s15234" xml:space="preserve"> nam ſemidiame-<lb/>tri e a, e b æquantur per 15 d 1, & d e communis eſt:</s> <s xml:id="echoid-s15235" xml:space="preserve"> anguliq́;</s> <s xml:id="echoid-s15236" xml:space="preserve"> d e a, d e b æquantur per fabricationẽ.</s> <s xml:id="echoid-s15237" xml:space="preserve">] <lb/>Extrahamus ergo ex b lineam in ſuperficie trianguli d e b, continentem cum e b angulum, æqualẽ <lb/>angulo d b e.</s> <s xml:id="echoid-s15238" xml:space="preserve"> Hæc ergo linea cõcurret cum linea d e:</s> <s xml:id="echoid-s15239" xml:space="preserve"> quia angulus b e d eſt obtuſus, & angulus, qui <lb/>eſt apud b, eſt acutus.</s> <s xml:id="echoid-s15240" xml:space="preserve"> [quia enim angulus d e b eſt obtuſus per fabricationem, reliquus b e o eſt a-<lb/>cutus per 13 p 1, & e b o acutus, quia ęquatus eſt d b e acuto.</s> <s xml:id="echoid-s15241" xml:space="preserve"> Quare d e, b o cõcurrent per 11 ax.</s> <s xml:id="echoid-s15242" xml:space="preserve">] Con <lb/>currant in o:</s> <s xml:id="echoid-s15243" xml:space="preserve"> & extrahamus etiam ex a lineam in ſuperficie trianguli d a e, cõtinentẽ cũ a e angulũ, <lb/>æqualem angulo d a e.</s> <s xml:id="echoid-s15244" xml:space="preserve"> Cõcurret ergo cũ d e in o:</s> <s xml:id="echoid-s15245" xml:space="preserve"> quia duo an-<lb/> <anchor type="figure" xlink:label="fig-0222-01a" xlink:href="fig-0222-01"/> guli a e o, b e o ſunt æquales [per fabricationem & 13 p 1] & an-<lb/>guli, qui ſunt apud a, b, ſunt æquales [itaq;</s> <s xml:id="echoid-s15246" xml:space="preserve"> per 26 p 1 b o, a o æ-<lb/>quantur:</s> <s xml:id="echoid-s15247" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s15248" xml:space="preserve"> concurruntin eodem puncto cõtinuatæ lineæ <lb/>d e.</s> <s xml:id="echoid-s15249" xml:space="preserve">] Et extrahamus e t ita, ut cõtineat cum e b angulum rectũ:</s> <s xml:id="echoid-s15250" xml:space="preserve"> <lb/>& extrahamus t e ex parte e, & b o ex parte o:</s> <s xml:id="echoid-s15251" xml:space="preserve"> & concurrant in <lb/>h, [concurrent autẽ per 11 ax:</s> <s xml:id="echoid-s15252" xml:space="preserve"> quia angulus h e b rectus eſt per <lb/>fabricationem, & e b o acutus per concluſionem] & erit e t æ-<lb/>qualis e h [per 26 p 1:</s> <s xml:id="echoid-s15253" xml:space="preserve"> anguli enim ad e deinceps recti æquan-<lb/>tur:</s> <s xml:id="echoid-s15254" xml:space="preserve"> item q́;</s> <s xml:id="echoid-s15255" xml:space="preserve"> ad b per fabricationem:</s> <s xml:id="echoid-s15256" xml:space="preserve"> & b e commune latus eſt u-<lb/>triuſq;</s> <s xml:id="echoid-s15257" xml:space="preserve"> triangulι b e t, b e h] & b t æqualis b h.</s> <s xml:id="echoid-s15258" xml:space="preserve"> Et ſimiliter extra <lb/>hamus e k ita, ut contineat cum e a angulum rectum:</s> <s xml:id="echoid-s15259" xml:space="preserve"> & extra-<lb/>hamus illã ex parte e:</s> <s xml:id="echoid-s15260" xml:space="preserve"> & extrahamus a o, & concurrant in l [con <lb/>current autem per 11 ax.</s> <s xml:id="echoid-s15261" xml:space="preserve"> ut proximè oſtenſum eſt.</s> <s xml:id="echoid-s15262" xml:space="preserve">] Sic ergo k e <lb/>erit æqualis e l, & k a æqualis a l, & t e æqualis e h [per 26 p 1, ut <lb/>patuit.</s> <s xml:id="echoid-s15263" xml:space="preserve">] Et continuemus t k, l h.</s> <s xml:id="echoid-s15264" xml:space="preserve"> Erũt ergo æquales [duo enim <lb/>latera e l, e h æqualia concluſa ſunt duobus lateribus e k, e t, & <lb/>angulus l e h æquatur angulo k e t per 15 p 1.</s> <s xml:id="echoid-s15265" xml:space="preserve"> Quare per 4 p 1 l h, <lb/>k t æquantur.</s> <s xml:id="echoid-s15266" xml:space="preserve">] Si ergo uiſus fuerit in d, & l h fuerit in aliquo ui <lb/>ſibili:</s> <s xml:id="echoid-s15267" xml:space="preserve"> tunc d comprehendet l h in ſpeculo a b:</s> <s xml:id="echoid-s15268" xml:space="preserve"> & erit t imago h:</s> <s xml:id="echoid-s15269" xml:space="preserve"> <lb/>& k imago l [per 6 n 5.</s> <s xml:id="echoid-s15270" xml:space="preserve">] Sic igitur t k erit diameter imaginis l h:</s> <s xml:id="echoid-s15271" xml:space="preserve"> <lb/>& eſt ei ęqualis.</s> <s xml:id="echoid-s15272" xml:space="preserve"> Si ergo reuoluerimus totam figuram, l h immo <lb/>bili:</s> <s xml:id="echoid-s15273" xml:space="preserve"> tunc d faciet circulum.</s> <s xml:id="echoid-s15274" xml:space="preserve"> Et ſi uiſus fuerit in aliquo puncto il <lb/>lius circumferentiæ, poterit comprehendere aliquod uiſibile, <lb/>comparlineę l h:</s> <s xml:id="echoid-s15275" xml:space="preserve"> & erit imago eius æqualis ei.</s> <s xml:id="echoid-s15276" xml:space="preserve"> Et ſimiliter ſi ui-<lb/>ſus fuerit in o, & res uiſa fuerit t k:</s> <s xml:id="echoid-s15277" xml:space="preserve"> erit imago æqualis rei uiſæ.</s> <s xml:id="echoid-s15278" xml:space="preserve"> <lb/>Sed tamen cum res uiſa fuerit l h, & uiſus fuerit d, fueritq́;</s> <s xml:id="echoid-s15279" xml:space="preserve"> imago t k:</s> <s xml:id="echoid-s15280" xml:space="preserve"> erit imago conuerſa:</s> <s xml:id="echoid-s15281" xml:space="preserve"> ſi h fuerit <lb/>in dextra, erit t in ſiniſtra:</s> <s xml:id="echoid-s15282" xml:space="preserve"> & ſi h fuerit in ſiniſtra, erit t in dextra:</s> <s xml:id="echoid-s15283" xml:space="preserve"> & ſi h fuerit ſupra lineam, erit t infra <lb/>lineam:</s> <s xml:id="echoid-s15284" xml:space="preserve"> & ſimiliter l.</s> <s xml:id="echoid-s15285" xml:space="preserve"> Et ſi res uiſa fuerit t k, & uiſus fuerit o, & imago fuerit l h:</s> <s xml:id="echoid-s15286" xml:space="preserve"> forma eſt recta.</s> <s xml:id="echoid-s15287" xml:space="preserve"> Nam <lb/>imago l h erit retro uiſum, & comprehendetur ante rem uiſam, ſicut declarauimus in capitulo im a <lb/>ginis quinti tractatus [60 n.</s> <s xml:id="echoid-s15288" xml:space="preserve">] Et uiſus comprehendet h, quod eſt imago t in linea h o, & l, quod eſt <lb/>imago k, in l o.</s> <s xml:id="echoid-s15289" xml:space="preserve"> Patet ergo, quòd in ſpeculis concauis cõprehendatur res uiſa quãdoq;</s> <s xml:id="echoid-s15290" xml:space="preserve"> æqualis ſibi.</s> <s xml:id="echoid-s15291" xml:space="preserve"/> </p> <div xml:id="echoid-div516" type="float" level="0" n="0"> <figure xlink:label="fig-0222-01" xlink:href="fig-0222-01a"> <variables xml:id="echoid-variables180" xml:space="preserve">d g t K z b e a o ſ h</variables> </figure> </div> </div> <div xml:id="echoid-div518" type="section" level="0" n="0"> <head xml:id="echoid-head458" xml:space="preserve" style="it">42. In ſpeculo ſphærico cauo imago inter uiſum & ſpeculum aliquando minor eſt uiſibili & <lb/>euerſa: pone uiſum aliquando maior eſt, & erecta. 49 p 8.</head> <p> <s xml:id="echoid-s15292" xml:space="preserve">ITem:</s> <s xml:id="echoid-s15293" xml:space="preserve"> extrahamus b h rectè:</s> <s xml:id="echoid-s15294" xml:space="preserve"> & in ipſa ſignemus r, & cõtinemus r e.</s> <s xml:id="echoid-s15295" xml:space="preserve"> Sic ergo angulus r e b erit ob-<lb/>tuſus:</s> <s xml:id="echoid-s15296" xml:space="preserve"> [quia h e b rectus eſt per fabricationẽ] & extrahamus r e ad n.</s> <s xml:id="echoid-s15297" xml:space="preserve"> Sic ergo t b erit maior b n:</s> <s xml:id="echoid-s15298" xml:space="preserve"> <lb/>[Quia enim angulus b e r obtuſus eſt:</s> <s xml:id="echoid-s15299" xml:space="preserve"> ergo r e continuata ultra e faciet cum e b angulum acutũ <lb/> <pb o="217" file="0223" n="223" rhead="OPTICAE LIBER VI."/> per 13 p 1, minorẽ recto b e t, & terminabitur in linea b d inter pũcta b & t.</s> <s xml:id="echoid-s15300" xml:space="preserve"> Quare b t erit maior b n] er <lb/>go linea r b eſt maior b n.</s> <s xml:id="echoid-s15301" xml:space="preserve"> [ſuperiore enim numero t b æqualis cõcluſa eſt ipſi b h, & r b maior eſt b h <lb/>ք 9 ax:</s> <s xml:id="echoid-s15302" xml:space="preserve"> ergo r b maior eſt t b.</s> <s xml:id="echoid-s15303" xml:space="preserve"> Quare eadẽ multò maior eſt b n] & [per 3 p 6] proportio r b ad b n eſt, ſi-<lb/>cut proportio r e ad e n.</s> <s xml:id="echoid-s15304" xml:space="preserve"> [angulus enim n b r bifariã ſecatur ք <lb/> <anchor type="figure" xlink:label="fig-0223-01a" xlink:href="fig-0223-01"/> lineã b e, ut patuit ꝓximo numero.</s> <s xml:id="echoid-s15305" xml:space="preserve">] Quare linea r e eſt maior <lb/>quàm linea e n.</s> <s xml:id="echoid-s15306" xml:space="preserve"> Et extrahamus a l rectè in m:</s> <s xml:id="echoid-s15307" xml:space="preserve"> & ſit a m æqua-<lb/>lis b r:</s> <s xml:id="echoid-s15308" xml:space="preserve"> & continuemus m e, & tranſeat uſq;</s> <s xml:id="echoid-s15309" xml:space="preserve"> ad u.</s> <s xml:id="echoid-s15310" xml:space="preserve"> Erit ergo m e <lb/>maior quàm e u [Quia enim latera e a, m a æquantur duobus <lb/>lateribus e b, r b per 15 d 1, & proximam fabricationem, & <lb/>angulus e a m æqualis concluſus eſt ſuperiore numero angu <lb/>lo e b r:</s> <s xml:id="echoid-s15311" xml:space="preserve"> erit per 4 p 1 baſis m e æqualis baſi r e, & angulus m <lb/>e a ęqualis angulo r e b, per concluſionem obtuſo:</s> <s xml:id="echoid-s15312" xml:space="preserve"> ergo m e a <lb/>eſt obtuſus, & a e u acutus per 13 p 1.</s> <s xml:id="echoid-s15313" xml:space="preserve"> Quare cũ angulus a e u <lb/>ſit minor angulo m e a, & u a e ęqualis e a m per cõcluſionẽ:</s> <s xml:id="echoid-s15314" xml:space="preserve"> <lb/>reliquus a u e maior erit reliquo a m e per 32 p 1:</s> <s xml:id="echoid-s15315" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s15316" xml:space="preserve"> per 19 <lb/>p 1 in triangulo a u m latus m a maius latere a u:</s> <s xml:id="echoid-s15317" xml:space="preserve"> ſed ut m a ad <lb/>a u, ſic m e ad e u per 3 p 6:</s> <s xml:id="echoid-s15318" xml:space="preserve"> quia angulus m a u bifariam ſectus <lb/>eſt per rectam a e, ut patuit proximo numero.</s> <s xml:id="echoid-s15319" xml:space="preserve"> Quare m e ma <lb/>ior eſt e u.</s> <s xml:id="echoid-s15320" xml:space="preserve">] Et continuemus m r, n u:</s> <s xml:id="echoid-s15321" xml:space="preserve"> erit ergo m r maior quã <lb/>n u [Nam quia anguli e a u, e b n æquales concluſi ſunt, & an-<lb/>gulus a e u æquatur angulo b e n per 13 p 1:</s> <s xml:id="echoid-s15322" xml:space="preserve"> quia anguli m e a, <lb/>r e b æquales demõſtrati ſunt, & a e ipſi e b:</s> <s xml:id="echoid-s15323" xml:space="preserve"> ęquabitur e u ipſi <lb/>e n per 26 p 1:</s> <s xml:id="echoid-s15324" xml:space="preserve"> & m e æquatur ipſi r e per concluſionem, & an-<lb/>gulus u e n angulo m e r per 13 p 1:</s> <s xml:id="echoid-s15325" xml:space="preserve"> erit per 7 p 5 m e ad r e, ſi-<lb/>cut u e ad n e.</s> <s xml:id="echoid-s15326" xml:space="preserve"> Quare cum triangula m e r, u e n ſint per 6 p 6 <lb/>æquiangula:</s> <s xml:id="echoid-s15327" xml:space="preserve"> erit per 4 p 6, ut m e ad e u, ſic m r ad u n.</s> <s xml:id="echoid-s15328" xml:space="preserve"> Itaque <lb/>cum m e maior ſit per concluſionem ipſa e u, erit m r maior <lb/>u n.</s> <s xml:id="echoid-s15329" xml:space="preserve">] Si ergo m r fuerit in aliquo uiſibili, & uiſus fuerit in d:</s> <s xml:id="echoid-s15330" xml:space="preserve"> <lb/>erit n u diameter imaginis m r:</s> <s xml:id="echoid-s15331" xml:space="preserve"> & n u eſt minor quàm m r.</s> <s xml:id="echoid-s15332" xml:space="preserve"> Et <lb/>ſi uiſus fuerit in o, & u n fuerit in aliquo uiſibili:</s> <s xml:id="echoid-s15333" xml:space="preserve"> erit m r ima-<lb/>go n u:</s> <s xml:id="echoid-s15334" xml:space="preserve"> & eſt maior quàm n u.</s> <s xml:id="echoid-s15335" xml:space="preserve"> Sed cũ m r fuerit uiſibile, & n u fuerit imago, & d uiſus:</s> <s xml:id="echoid-s15336" xml:space="preserve"> erit imago cõ-<lb/>uerſa.</s> <s xml:id="echoid-s15337" xml:space="preserve"> Et ſi res uiſa fuerit n u, & uiſus o:</s> <s xml:id="echoid-s15338" xml:space="preserve"> imago m r erit recta.</s> <s xml:id="echoid-s15339" xml:space="preserve"> Nam imago ſi fuerit ultra uiſum, uide-<lb/>bitur ante.</s> <s xml:id="echoid-s15340" xml:space="preserve"> Et omne punctum imaginis uidebitur in linea, in qua eſt de lineis radialibus.</s> <s xml:id="echoid-s15341" xml:space="preserve"/> </p> <div xml:id="echoid-div518" type="float" level="0" n="0"> <figure xlink:label="fig-0223-01" xlink:href="fig-0223-01a"> <variables xml:id="echoid-variables181" xml:space="preserve">d g t k n z u e b a o ſ<unsure/> h m r</variables> </figure> </div> </div> <div xml:id="echoid-div520" type="section" level="0" n="0"> <head xml:id="echoid-head459" xml:space="preserve" style="it">43. In ſpeculo ſphærico cauo imago inter uiſum & ſpeculum aliquando maior eſt uiſibili, & <lb/>euerſa: pone uiſum aliquando minor eſt, & erecta. 50 p 8.</head> <p> <s xml:id="echoid-s15342" xml:space="preserve">IT ẽ:</s> <s xml:id="echoid-s15343" xml:space="preserve"> ſignemus in linea o h punctum q:</s> <s xml:id="echoid-s15344" xml:space="preserve"> & cõtinuemus q e:</s> <s xml:id="echoid-s15345" xml:space="preserve"> & trãſeat ad p:</s> <s xml:id="echoid-s15346" xml:space="preserve"> & ſit o f æqualis o q:</s> <s xml:id="echoid-s15347" xml:space="preserve"> [per <lb/>3 p 1] & continuemus e f, & tranſeat ad i.</s> <s xml:id="echoid-s15348" xml:space="preserve"> Erunt ergo duę li-<lb/> <anchor type="figure" xlink:label="fig-0223-02a" xlink:href="fig-0223-02"/> neæ p e, e i maiores duabus lineis e f, e q:</s> <s xml:id="echoid-s15349" xml:space="preserve"> [Quia enim angu <lb/>lus a e l rectus eſt, ut patuit 4 n:</s> <s xml:id="echoid-s15350" xml:space="preserve"> erit a e f acutus.</s> <s xml:id="echoid-s15351" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s15352" xml:space="preserve"> f e con-<lb/>tinuata ultra e, fac<unsure/>iet cũ a e angulũ obtuſum per 13 p 1, & cadet <lb/>ultra e k.</s> <s xml:id="echoid-s15353" xml:space="preserve"> Erit igitur a i maior a k:</s> <s xml:id="echoid-s15354" xml:space="preserve"> ſed a k æqualis concluſa eſt ci <lb/>tato numero ipſi a l:</s> <s xml:id="echoid-s15355" xml:space="preserve"> ergo a i maior eſt a l, ideoq́;</s> <s xml:id="echoid-s15356" xml:space="preserve"> multò maior <lb/>ipſa a f.</s> <s xml:id="echoid-s15357" xml:space="preserve"> Et quia angulus i a f bifariã ſectus eſt per rectã a e:</s> <s xml:id="echoid-s15358" xml:space="preserve"> erit <lb/>per 3 p 6 uti a ad a f, ſic i e ad e f:</s> <s xml:id="echoid-s15359" xml:space="preserve"> ſed cum i a maior ſit a f:</s> <s xml:id="echoid-s15360" xml:space="preserve"> erit i e <lb/>maior e f.</s> <s xml:id="echoid-s15361" xml:space="preserve"> Eodẽ argumento p e maior demonſtrabituripſa e q] <lb/>& erit linea p i maior quàm linea f q [cum enim duobus ſupe-<lb/>rioribus numeris æqualitas tum rectarum e h, e l, tum angulo-<lb/>rum e h q, e l f demonſtrata ſit:</s> <s xml:id="echoid-s15362" xml:space="preserve"> & l f æquetur h q:</s> <s xml:id="echoid-s15363" xml:space="preserve"> quia tota a l æ-<lb/>qualis eſt toti b h è concluſo duorũ numerorũ præcedẽtium, <lb/>& pars o f parti o h per theſin:</s> <s xml:id="echoid-s15364" xml:space="preserve"> æquabitur reliqua l f reliquę h q <lb/>per 19 p 5:</s> <s xml:id="echoid-s15365" xml:space="preserve"> & erit per 4 p 1 e f æqualis e q, & angulus l e fangulo <lb/>h e q.</s> <s xml:id="echoid-s15366" xml:space="preserve"> Et quia anguli recti a e l, b e h:</s> <s xml:id="echoid-s15367" xml:space="preserve"> itẽ a e o, b e o ęquantur:</s> <s xml:id="echoid-s15368" xml:space="preserve"> re-<lb/>liquus l e o æquabitur reliquo h e o, & l e f æqualis oſtenſus eſt <lb/>ipſi h e q:</s> <s xml:id="echoid-s15369" xml:space="preserve"> ergo f e o æquatur q e o, & ք 15 p 1, 1 ax.</s> <s xml:id="echoid-s15370" xml:space="preserve"> d e i ipſi d e p, <lb/>& d e a æquatus eſt d e b, 41 n:</s> <s xml:id="echoid-s15371" xml:space="preserve"> reliquus igitur i e a æquatur reli <lb/>quo p e b, & i a e æqualis concluſus eſt ipſi p b e, & a e æqualis <lb/>ipſi b e per 15 d 1.</s> <s xml:id="echoid-s15372" xml:space="preserve"> Quare per 26 p 1 i e æquaturipſi p e, & angu-<lb/>lus i e p angulo f e q per 15 p 1.</s> <s xml:id="echoid-s15373" xml:space="preserve"> Ergo ք 7 p 5.</s> <s xml:id="echoid-s15374" xml:space="preserve">6 p 6 triangula i e p, <lb/>f e q ſunt ęquiangula, & per 4 p 6, ut i e ad e f, ſic p i ad f q:</s> <s xml:id="echoid-s15375" xml:space="preserve"> ſed i e <lb/>maior eſt e f è cõcluſo:</s> <s xml:id="echoid-s15376" xml:space="preserve"> ergo p i maior eſt f q.</s> <s xml:id="echoid-s15377" xml:space="preserve">] Si ergo uiſus fue-<lb/>rit in o, & p i in aliquo uiſibili:</s> <s xml:id="echoid-s15378" xml:space="preserve"> erit f q imago p i:</s> <s xml:id="echoid-s15379" xml:space="preserve"> & f q eſt minor <lb/>quã p i:</s> <s xml:id="echoid-s15380" xml:space="preserve"> & f q uidebitur ſuper duas lineas a o, b o.</s> <s xml:id="echoid-s15381" xml:space="preserve"> Erit ergo for-<lb/>ma retro uiſum, & minor ꝗ̃ res uiſa:</s> <s xml:id="echoid-s15382" xml:space="preserve"> & erit recta.</s> <s xml:id="echoid-s15383" xml:space="preserve"> Et ſi uiſus fue <lb/>rit in d, & f q fuerit in aliquo uiſibili:</s> <s xml:id="echoid-s15384" xml:space="preserve"> erit p i imago f q:</s> <s xml:id="echoid-s15385" xml:space="preserve"> & eſt maior ꝗ̃ f q:</s> <s xml:id="echoid-s15386" xml:space="preserve"> & erit forma ante uiſum con <lb/>uerſa.</s> <s xml:id="echoid-s15387" xml:space="preserve"> Patet ergo, quòd in ſpeculis cõcauis cõprehẽditur forma rei uiſæ minor, & maior, & æqualis.</s> <s xml:id="echoid-s15388" xml:space="preserve"/> </p> <div xml:id="echoid-div520" type="float" level="0" n="0"> <figure xlink:label="fig-0223-02" xlink:href="fig-0223-02a"> <variables xml:id="echoid-variables182" xml:space="preserve">d g p i t k b e a o l f q h</variables> </figure> </div> <pb o="218" file="0224" n="224" rhead="ALHAZEN"/> </div> <div xml:id="echoid-div522" type="section" level="0" n="0"> <head xml:id="echoid-head460" xml:space="preserve" style="it">44. Si uiſ{us} ſit citra centrum ſpeculi ſphærici caui, uiſibile ultra: imago tum uiſibilis, tum ui-<lb/>dentis, euerſa & minor uidebitur. 51 p 8.</head> <p> <s xml:id="echoid-s15389" xml:space="preserve">ITem:</s> <s xml:id="echoid-s15390" xml:space="preserve"> ſit ſpeculum concauum a b:</s> <s xml:id="echoid-s15391" xml:space="preserve"> & centrũ g:</s> <s xml:id="echoid-s15392" xml:space="preserve"> & habeat ſuperficiem planam, tranſeuntem per cẽ <lb/>trum, & faciat circulum a b:</s> <s xml:id="echoid-s15393" xml:space="preserve"> & extrahamus lineam g d, quocunque modo ſit:</s> <s xml:id="echoid-s15394" xml:space="preserve"> & tranſeat ex parte <lb/>gad e:</s> <s xml:id="echoid-s15395" xml:space="preserve"> & ſit uiſus in e:</s> <s xml:id="echoid-s15396" xml:space="preserve"> & ſit t in ſuperficie uiſus:</s> <s xml:id="echoid-s15397" xml:space="preserve"> & extrahamus t h perpendiculariter ſuper lineã <lb/>e d:</s> <s xml:id="echoid-s15398" xml:space="preserve"> [per 11 p 1] & ſit z t ęqualis t h:</s> <s xml:id="echoid-s15399" xml:space="preserve"> & comprehendat e punctum h ex a:</s> <s xml:id="echoid-s15400" xml:space="preserve"> & g h producta in p, compre-<lb/>hendat arcum a p maiorem quarta circuli.</s> <s xml:id="echoid-s15401" xml:space="preserve"> Sic ergo erunt duo puncta a, h, à duobus lateribus puncti <lb/>g.</s> <s xml:id="echoid-s15402" xml:space="preserve"> Nam ſi in eodem eſſent:</s> <s xml:id="echoid-s15403" xml:space="preserve"> tunc linea, quæ exiret à ſpeculo ad a, non diuideret angulum, quem conti <lb/>nent duæ lineę radiales, per ęqualia [ſicq́;</s> <s xml:id="echoid-s15404" xml:space="preserve">, ut oſtenſum eſt 66 n 5, reflexio nulla fieret.</s> <s xml:id="echoid-s15405" xml:space="preserve">] Et extraha-<lb/>mus lineas e a, a h, g a, g h:</s> <s xml:id="echoid-s15406" xml:space="preserve"> & tranſeat g h rectè ad k:</s> <s xml:id="echoid-s15407" xml:space="preserve"> duo ergo anguli apud a erunt ęquales:</s> <s xml:id="echoid-s15408" xml:space="preserve"> [per the-<lb/>ſin & 12 n 4] & erit k imago h [per 6 n 5.</s> <s xml:id="echoid-s15409" xml:space="preserve">] Et ſit arcus b d ęqualis arcui d a:</s> <s xml:id="echoid-s15410" xml:space="preserve"> [fiet autem ęqualis per 33 <lb/>p 6, ſi per 23 p 1 ęquaueris angulum d g b angulo d g a] & continuemus lineas e b, b z, b g:</s> <s xml:id="echoid-s15411" xml:space="preserve"> & extra-<lb/>hamus z g ad l:</s> <s xml:id="echoid-s15412" xml:space="preserve"> & ſecet z b diametrum d g in f.</s> <s xml:id="echoid-s15413" xml:space="preserve"> Erunt ergo duo anguli apud b ęquales:</s> <s xml:id="echoid-s15414" xml:space="preserve"> [Quia enim <lb/>a g, b g ęquantur per 15 d 1, & communis eſt g f, angulusq́ue a g f ęquatus eſt angulo b g f:</s> <s xml:id="echoid-s15415" xml:space="preserve"> ęquabi-<lb/>tur baſis a f, baſi b f, & angulus f a g angulo f b g per 4 p 1.</s> <s xml:id="echoid-s15416" xml:space="preserve"> Eadem de cauſſa e a g, e b g æquãtur, quia <lb/>angulus b g e ęquatur angulo a g e per 13 p 1.</s> <s xml:id="echoid-s15417" xml:space="preserve"> Quare cum anguli ad a ęquentur, anguli ad b ęquabun-<lb/>tur.</s> <s xml:id="echoid-s15418" xml:space="preserve">] & comprehendetur z à uiſu ex b:</s> <s xml:id="echoid-s15419" xml:space="preserve"> [per 12 n 4] & erit punctum l imago z:</s> <s xml:id="echoid-s15420" xml:space="preserve"> [per 6 n 5] & cõtinue <lb/>mus k l:</s> <s xml:id="echoid-s15421" xml:space="preserve"> erit ergo k l diameter imaginis z h.</s> <s xml:id="echoid-s15422" xml:space="preserve"> Et quia t h eſt perpendicularis ſuper d e, & z t eſt ę-<lb/>qualis t h:</s> <s xml:id="echoid-s15423" xml:space="preserve"> erunt duę lineę e a, a h ęquales duabus li-<lb/> <anchor type="figure" xlink:label="fig-0224-01a" xlink:href="fig-0224-01"/> neis e b, b z:</s> <s xml:id="echoid-s15424" xml:space="preserve"> [Quia enim t h, z t ęquantur per fabrica <lb/>tionem, & t f communis, anguliq́ue ad t recti ſunt:</s> <s xml:id="echoid-s15425" xml:space="preserve"> ę-<lb/>quabitur baſis h f baſi z f per 4 p 1:</s> <s xml:id="echoid-s15426" xml:space="preserve"> & a fiam antè ę-<lb/>qualis concluſa eſt ipſi b f:</s> <s xml:id="echoid-s15427" xml:space="preserve"> itaque tota a h ęquatur to <lb/>ti b z, & a e, b e ęquantur è cõcluſo] & duo anguli a-<lb/>pud a ſunt ęquales duobus angulis apud b:</s> <s xml:id="echoid-s15428" xml:space="preserve"> erit h e <lb/>ęqualis z e:</s> <s xml:id="echoid-s15429" xml:space="preserve"> [per 4 p 1] & linea g h eſt ęqualis lineę <lb/>z h [per 4 p 1:</s> <s xml:id="echoid-s15430" xml:space="preserve"> quia z t, t h ęquantur per fabricatio-<lb/>nem, & communis eſt t g, anguliq́ue ad t recti ſunt.</s> <s xml:id="echoid-s15431" xml:space="preserve">] <lb/>Ergo duæ lineę a g, g h ſunt ęquales duabus lineis <lb/>b g, g z, & baſis a h eſt ęqualis baſi b z:</s> <s xml:id="echoid-s15432" xml:space="preserve"> ergo [per 8 p <lb/>1] angulus a h k eſt ęqualis angulo b z l, & angulus h <lb/>a k eſt ęqualis z b l:</s> <s xml:id="echoid-s15433" xml:space="preserve"> ergo h k eſt ęqualis z l [per 26 p 1:</s> <s xml:id="echoid-s15434" xml:space="preserve"> <lb/>quia z b ęqualis concluſa eſt ipſi h a] & linea h g eſt <lb/>ęqualis z g:</s> <s xml:id="echoid-s15435" xml:space="preserve"> [è concluſo] ergo g k eſt æqualis g l:</s> <s xml:id="echoid-s15436" xml:space="preserve"> [per <lb/>19 p 5] ergo k l eſt ęquidiſtans z h, [per 27 p 1:</s> <s xml:id="echoid-s15437" xml:space="preserve"> nam <lb/>cum anguli ad uerticem g ęquentur per 15 p 1:</s> <s xml:id="echoid-s15438" xml:space="preserve"> ſitq́ue <lb/>per 7 p 5 l g ad g k, ſicut g z ad g h:</s> <s xml:id="echoid-s15439" xml:space="preserve"> ęquabitur per 6 p 6 angulus z l k angulo l z h.</s> <s xml:id="echoid-s15440" xml:space="preserve">] Item angulus h g a <lb/>eſt obtuſus [ex theſi & 33 p 6] & duo anguli apud a ſunt æquales:</s> <s xml:id="echoid-s15441" xml:space="preserve"> ergo linea g h eſt maior linea g k:</s> <s xml:id="echoid-s15442" xml:space="preserve"> <lb/>[Nam quia angulus a g h obtuſus:</s> <s xml:id="echoid-s15443" xml:space="preserve"> erit per 13 p 1 angulus a g k acutus, & h a g, g a k ſunt ęquales ex <lb/>theſi<unsure/>:</s> <s xml:id="echoid-s15444" xml:space="preserve"> quia punctum a eſt punctum reflexionis:</s> <s xml:id="echoid-s15445" xml:space="preserve"> quare per 32 p 1 angulus a k g maior eſt angulo a h k:</s> <s xml:id="echoid-s15446" xml:space="preserve"> <lb/>& per 19 p 1 in triangulo a h k latus a h maius eſt a k:</s> <s xml:id="echoid-s15447" xml:space="preserve"> ſed ut a h ad a k, ſic h g ad g k per 3 p 6:</s> <s xml:id="echoid-s15448" xml:space="preserve"> quia angu <lb/>li ad a æquales.</s> <s xml:id="echoid-s15449" xml:space="preserve"> Itaque cum a h maior ſit a k:</s> <s xml:id="echoid-s15450" xml:space="preserve"> erit h g maior g k] & ſimiliter z g eſt maior, quàm g l.</s> <s xml:id="echoid-s15451" xml:space="preserve"> Li-<lb/>nea ergo k l eſt minor, quàm z h [cum enim triangula k g l, h g z ſint ęquiangula per 15 p 1.</s> <s xml:id="echoid-s15452" xml:space="preserve"> 6 p 6:</s> <s xml:id="echoid-s15453" xml:space="preserve"> erit ք <lb/>4 p 6, ut g k ad g h, ſic k l ad z h:</s> <s xml:id="echoid-s15454" xml:space="preserve"> & cum g k ſit minor g h, erit k l minor z h.</s> <s xml:id="echoid-s15455" xml:space="preserve">] Sed k l eſt diameterima-<lb/>ginis z h:</s> <s xml:id="echoid-s15456" xml:space="preserve"> ergo z h uidetur minor, quàm ſit ſecundum ueritatem:</s> <s xml:id="echoid-s15457" xml:space="preserve"> & linea z h eſt ſuperficies faciei a-<lb/><gap/>picientis.</s> <s xml:id="echoid-s15458" xml:space="preserve"> Si ergo reuoluerimus circulum a d b, e d immobili:</s> <s xml:id="echoid-s15459" xml:space="preserve"> fiet ex duobus punctis a, b circulus <lb/>in ſuperficie ſpeculi:</s> <s xml:id="echoid-s15460" xml:space="preserve"> & erit ſitus uiſus e, reſpectu cuiuslibet comparis lineæ z h ex illo circulo, quem <lb/>ſignant puncta z, h, & ex omni arcu compari arcui a b ex portione ſpeculi, quam diuidit circulus, <lb/>quem ſignant duo puncta a, b, ſicut eſt ſitus, quem uiſus e habet ex linea z h, & ex arcu a b.</s> <s xml:id="echoid-s15461" xml:space="preserve"> Et ſimili-<lb/>ter declarabitur, ſi poſuerimus lineã z h maiorem, aut minorẽ.</s> <s xml:id="echoid-s15462" xml:space="preserve"> Patet ergo ex his omnib.</s> <s xml:id="echoid-s15463" xml:space="preserve"> quòd diame <lb/>ter ſuperficiei faciei aſpicientis cõprehenditur in ſpeculo cõcauo minor, ꝗ̃ ſit.</s> <s xml:id="echoid-s15464" xml:space="preserve"> Sciendum ergo, quòd <lb/>ſi fuerit uiſus in e:</s> <s xml:id="echoid-s15465" xml:space="preserve"> tunc aſpiciens comprehẽdet formam ſuam minorem, ꝗ̃ ſit.</s> <s xml:id="echoid-s15466" xml:space="preserve"> Et quia k eſt imago h, <lb/>& l eſt imago z:</s> <s xml:id="echoid-s15467" xml:space="preserve"> erit imago cõuerſa.</s> <s xml:id="echoid-s15468" xml:space="preserve"> Et ſic uifus e cõprehendet ſuam formam ſecundum quod eſt de-<lb/>xtrũ in ſiniſtro, & ſurſum deorſum, & è contrario.</s> <s xml:id="echoid-s15469" xml:space="preserve"> Similiter ſi uiſus fuerit in quolibet puncto, inter <lb/>quod & ſuperficiẽ ſpeculi fuerit centrũ ſpeculi:</s> <s xml:id="echoid-s15470" xml:space="preserve"> cõprehendet formã ſuã conuerſam.</s> <s xml:id="echoid-s15471" xml:space="preserve"> Et hoc eſt quod <lb/>uoluimus.</s> <s xml:id="echoid-s15472" xml:space="preserve"> Patet ergo ex his quatuor figuris, quòd in ſpeculo concauo imago quandoq;</s> <s xml:id="echoid-s15473" xml:space="preserve"> comprehẽ-<lb/>ditur maior:</s> <s xml:id="echoid-s15474" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s15475" xml:space="preserve"> minor:</s> <s xml:id="echoid-s15476" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s15477" xml:space="preserve"> ęqualis:</s> <s xml:id="echoid-s15478" xml:space="preserve"> & nunc recta, nunc conuerſa.</s> <s xml:id="echoid-s15479" xml:space="preserve"> Et in capitulo de ima <lb/>gine [72 n 5] diximus, quòd in ſpeculo cõcauo imago quandoq;</s> <s xml:id="echoid-s15480" xml:space="preserve"> erit una:</s> <s xml:id="echoid-s15481" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s15482" xml:space="preserve"> duę:</s> <s xml:id="echoid-s15483" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s15484" xml:space="preserve"> <lb/>tres:</s> <s xml:id="echoid-s15485" xml:space="preserve"> & quandoq;</s> <s xml:id="echoid-s15486" xml:space="preserve"> quatuor:</s> <s xml:id="echoid-s15487" xml:space="preserve"> & hoc idem accidit in his prędictis.</s> <s xml:id="echoid-s15488" xml:space="preserve"> Illud ergo, quod habet imaginem ſe <lb/>maiorem, fortè habebit alias minores & ęquales:</s> <s xml:id="echoid-s15489" xml:space="preserve"> & quod imaginem habet minorem, fortè habebit <lb/>alias maiores & minores.</s> <s xml:id="echoid-s15490" xml:space="preserve"> Et quod rectum uidebitur, fortè uidebitur ſub alia imagine conuerſum, & <lb/>è contrario.</s> <s xml:id="echoid-s15491" xml:space="preserve"> Reſtat ergo declarare formas eorum, quæ comprehenduntur in his ſpeculis.</s> <s xml:id="echoid-s15492" xml:space="preserve"/> </p> <div xml:id="echoid-div522" type="float" level="0" n="0"> <figure xlink:label="fig-0224-01" xlink:href="fig-0224-01a"> <variables xml:id="echoid-variables183" xml:space="preserve">p d h t z f b g a ſ<unsure/> e k q</variables> </figure> </div> </div> <div xml:id="echoid-div524" type="section" level="0" n="0"> <head xml:id="echoid-head461" xml:space="preserve" style="it">45. In ſpeculo ſphærico cauo imago lineæ rectæ aliquando uidetur recta. Et ſiduo lineæ rectæ <lb/>termini reflectantur à duob{us} punctis peripheriæ circuli (qui eſt communis ſectio ſuperficie-<lb/> <pb o="219" file="0225" n="225" rhead="OPTICAE LIBER VI."/> rum, reflexionis & ſpeculi ſphærici caui) puncta dictæ rectæ intermedia à punctis dictæ peri-<lb/>pheriæ intermedijs reflectentur. 54. 42 p 8.</head> <p> <s xml:id="echoid-s15493" xml:space="preserve">SIt ergo ſpeculum ſphæricum concauum a b:</s> <s xml:id="echoid-s15494" xml:space="preserve"> & extrahamus in ipſo ſpeculo ſuperficiem planã, <lb/>tranſeuntem per centrũ:</s> <s xml:id="echoid-s15495" xml:space="preserve"> & faciat circulũ a b circa centrũ e [faciet autem per 1 th.</s> <s xml:id="echoid-s15496" xml:space="preserve"> 1 ſphær.</s> <s xml:id="echoid-s15497" xml:space="preserve">] & <lb/>extrahamus in hoc circulo duas diametros ſe ſecãtes a e o, b e d:</s> <s xml:id="echoid-s15498" xml:space="preserve"> & ſpeculum nõ excedat arcũ <lb/>b a d o:</s> <s xml:id="echoid-s15499" xml:space="preserve"> & ponamus in b e punctum z, quocunq;</s> <s xml:id="echoid-s15500" xml:space="preserve"> modo ſit:</s> <s xml:id="echoid-s15501" xml:space="preserve"> & ponamus in linea a e punctum k:</s> <s xml:id="echoid-s15502" xml:space="preserve"> & ſit <lb/>a k maior quàm k e:</s> <s xml:id="echoid-s15503" xml:space="preserve"> & continuemus z k:</s> <s xml:id="echoid-s15504" xml:space="preserve"> & tranſeat ad f:</s> <s xml:id="echoid-s15505" xml:space="preserve"> & continuemus e f:</s> <s xml:id="echoid-s15506" xml:space="preserve"> & ſit angulus g f e æqua <lb/>lis angulo z f e.</s> <s xml:id="echoid-s15507" xml:space="preserve"> [per 23 p 1.</s> <s xml:id="echoid-s15508" xml:space="preserve">] Quia igitur [per 7 p 3] f k eſt maior k a, & k a eſt maior quàm k e:</s> <s xml:id="echoid-s15509" xml:space="preserve"> [ex <lb/>theſi] erit f k maior quàm k e:</s> <s xml:id="echoid-s15510" xml:space="preserve"> angulus ergo f e k maior eſt angulo e f k:</s> <s xml:id="echoid-s15511" xml:space="preserve"> [per 18 p 1] ergo eſt maior <lb/>angulo e f g.</s> <s xml:id="echoid-s15512" xml:space="preserve"> Linea ergo f g concurret cũ linea k e.</s> <s xml:id="echoid-s15513" xml:space="preserve"> [ſi enim non concurrat:</s> <s xml:id="echoid-s15514" xml:space="preserve"> erit ad ipſam parallela:</s> <s xml:id="echoid-s15515" xml:space="preserve"> <lb/>itaq;</s> <s xml:id="echoid-s15516" xml:space="preserve"> per 29 p 1 angulus e f g æquabitur angulo f e k, quo minor eſt concluſus.</s> <s xml:id="echoid-s15517" xml:space="preserve">] Concurrant ergo in <lb/>g.</s> <s xml:id="echoid-s15518" xml:space="preserve"> Duæ ergo lineæ z f, f g reflectuntur propter angu <lb/> <anchor type="figure" xlink:label="fig-0225-01a" xlink:href="fig-0225-01"/> los æquales z f e, g f e:</s> <s xml:id="echoid-s15519" xml:space="preserve"> [per 12 n 4] k ergo eſt imago <lb/>g, ſi uiſus fuerit in z [per 6 n 5.</s> <s xml:id="echoid-s15520" xml:space="preserve">] Et extrahamus li-<lb/>neam z l h quocunq;</s> <s xml:id="echoid-s15521" xml:space="preserve"> modo ſit:</s> <s xml:id="echoid-s15522" xml:space="preserve"> & cõtinuemus e h, <lb/>h g, z g:</s> <s xml:id="echoid-s15523" xml:space="preserve"> & extrahamus f e uſq;</s> <s xml:id="echoid-s15524" xml:space="preserve"> ad m.</s> <s xml:id="echoid-s15525" xml:space="preserve"> Proportio er-<lb/>go z m ad m g eſt, ſicut ꝓportio z f ad f g [per 3 p 6:</s> <s xml:id="echoid-s15526" xml:space="preserve"> <lb/>quia angulus g f z bifariam ſectus eſt per rectã e f] <lb/>& [per 7 p 3] z h eſt maior quàm z f, & g h eſt mi-<lb/>nor quàm g f.</s> <s xml:id="echoid-s15527" xml:space="preserve"> Ergo proportio z h ad g h eſt maior, <lb/>quàm proportio z f ad f g:</s> <s xml:id="echoid-s15528" xml:space="preserve"> [ut conſtat ex 8 p 5] eſt <lb/>ergo maior quàm proportio z m ad m g.</s> <s xml:id="echoid-s15529" xml:space="preserve"> Ergo [per <lb/>3 p 6] linea, quę diuidit angulũ z h g in duo æqua-<lb/>lia, ſecat lineã m g:</s> <s xml:id="echoid-s15530" xml:space="preserve"> ſecat ergo lineã e g.</s> <s xml:id="echoid-s15531" xml:space="preserve"> Secet ergo <lb/>lineam e g in r:</s> <s xml:id="echoid-s15532" xml:space="preserve"> ergo angulus g h e maior eſt angulo <lb/>z h e:</s> <s xml:id="echoid-s15533" xml:space="preserve"> & h z ſecet a e in l.</s> <s xml:id="echoid-s15534" xml:space="preserve"> Ergo duæ lineæ z h, h r re-<lb/>flectũtur propter angulos æquales:</s> <s xml:id="echoid-s15535" xml:space="preserve"> [r h e, z h e per <lb/>12 n 4] & erit l imago r.</s> <s xml:id="echoid-s15536" xml:space="preserve"> Dico ergo, quòd forma cu-<lb/>iuslibet puncti lineæ g r reflectitur ad uiſum z ex <lb/>puncto aliquo arcus f h, & non ex alio.</s> <s xml:id="echoid-s15537" xml:space="preserve"> Huius rei demonſtratio eſt, quoniam in capitulo de imagi-<lb/>ne, quinto tractatu in duabus figuris [66 n] dictum eſt, quòd duo arcus a b, d o non poſſunt eſſe ta <lb/>les, quòd ex illis reflectatur aliquid de linea e o ad z:</s> <s xml:id="echoid-s15538" xml:space="preserve"> & arcus e o non eſt de ſpeculo:</s> <s xml:id="echoid-s15539" xml:space="preserve"> [nam ex theſi <lb/>ab arcu ſpeculi b a d o fit reflexio, cũ ille tantùm ſub uiſum in diametro d b poſitum cadat] nõ ergo <lb/>remanet niſi arcus a d.</s> <s xml:id="echoid-s15540" xml:space="preserve"> Sed in triceſima quinta figura [66 n 5] dictum eſt, quòd forma cuiuslibet <lb/>pũcti diametri e o reflectitur ab aliquo puncto arcus a d.</s> <s xml:id="echoid-s15541" xml:space="preserve"> Et in triceſima ſexta, capitulo de imagine <lb/>[73 n 5] patuit, quòd nunquã reflectitur forma puncti lineæ g r ad z ex arcu a d, niſi ex ſolo puncto.</s> <s xml:id="echoid-s15542" xml:space="preserve"> <lb/>Forma ergo cuiuslibet puncti lineæ g r reflectitur ad z ex uno ſolo puncto arcus a d.</s> <s xml:id="echoid-s15543" xml:space="preserve"> Et ponamus <lb/>in linea g r punctum c.</s> <s xml:id="echoid-s15544" xml:space="preserve"> Dico ergo, quòd illud punctũ <lb/> <anchor type="figure" xlink:label="fig-0225-02a" xlink:href="fig-0225-02"/> non erit, niſi in arcu fh.</s> <s xml:id="echoid-s15545" xml:space="preserve"> Sin autem reflectatur forma c <lb/>ad z ex u, quod eſt in arcu a f:</s> <s xml:id="echoid-s15546" xml:space="preserve"> & continuemus lineas <lb/>z u, e u, g u, c u.</s> <s xml:id="echoid-s15547" xml:space="preserve"> Linea ergo g u erit maior g f [per 7 p 3] <lb/>& z u eſt minor quàm z f.</s> <s xml:id="echoid-s15548" xml:space="preserve"> Ergo [ut cõſtat ex 8 p 5] ꝓ-<lb/>portio g u ad z u eſt maior proportione g f ad f z:</s> <s xml:id="echoid-s15549" xml:space="preserve"> ergo <lb/>maior proportione g m ad m z [quia enim angulus <lb/>g f z bifariam ſectus eſt per rectam f m:</s> <s xml:id="echoid-s15550" xml:space="preserve"> erit per 3 p 6 g f <lb/>ad f z, ſicut g m ad m z.</s> <s xml:id="echoid-s15551" xml:space="preserve">] Linea ergo, q̃ diuidit angulũ <lb/>g u z per æqualia, ſecat lineam z m:</s> <s xml:id="echoid-s15552" xml:space="preserve"> ſecat ergo z e:</s> <s xml:id="echoid-s15553" xml:space="preserve"> angu <lb/>lus ergo g u e eſt minor angulo e u z:</s> <s xml:id="echoid-s15554" xml:space="preserve"> ergo angulus c u <lb/>e multò minor eſt angulo e u z.</s> <s xml:id="echoid-s15555" xml:space="preserve"> [Itaq;</s> <s xml:id="echoid-s15556" xml:space="preserve"> cum anguli inci-<lb/>dentiæ & reflexionis ſint inæquales:</s> <s xml:id="echoid-s15557" xml:space="preserve"> nulla à puncto u <lb/>ad uiſum z fiet reflexio, ut patet per 12 n 4.</s> <s xml:id="echoid-s15558" xml:space="preserve">] Et ſimili-<lb/>ter de quolibet puncto arcus a u.</s> <s xml:id="echoid-s15559" xml:space="preserve"> Forma ergo c non re-<lb/>flectitur ad z, niſi ex arcu h f.</s> <s xml:id="echoid-s15560" xml:space="preserve"> Et dico, quòd non po-<lb/>teſt reflecti ex arcu h d.</s> <s xml:id="echoid-s15561" xml:space="preserve"> Quod ſi fuerit poſsibile:</s> <s xml:id="echoid-s15562" xml:space="preserve"> refle-<lb/>ctatur ex q, quod eſt in arcu h d:</s> <s xml:id="echoid-s15563" xml:space="preserve"> & continuemus lineas z q, c q, r q, e q, z r:</s> <s xml:id="echoid-s15564" xml:space="preserve"> & extrahamus e h ad n.</s> <s xml:id="echoid-s15565" xml:space="preserve"> Li-<lb/>nea ergo z q eſt maior quã z h [per 7 p 3], & linea q r eſt minor quàm h r:</s> <s xml:id="echoid-s15566" xml:space="preserve"> ergo proportio z q ad q r <lb/>eſt maior proportione z h ad h r:</s> <s xml:id="echoid-s15567" xml:space="preserve"> [ut patet per 8 p 5] quæ eſt, ſicut proportio z n ad n r [per 3 p 6:</s> <s xml:id="echoid-s15568" xml:space="preserve"> <lb/>quia angulus r h z bifariam ſectus eſt per rectam h n.</s> <s xml:id="echoid-s15569" xml:space="preserve">] Linea ergo, quæ diuidit angulum z q r in duo <lb/>æqualia, ſecat lineam n r:</s> <s xml:id="echoid-s15570" xml:space="preserve"> ſecat ergo lineam e r:</s> <s xml:id="echoid-s15571" xml:space="preserve"> angulus ergo r q e eſt maior angulo e q z:</s> <s xml:id="echoid-s15572" xml:space="preserve"> angulus er <lb/>go c q e eſt multò maior angulo e q z.</s> <s xml:id="echoid-s15573" xml:space="preserve"> Hoc idem ſequitur in omni puncto arcus h d.</s> <s xml:id="echoid-s15574" xml:space="preserve"> Forma ergo c <lb/>non reflectitur ad z ex arcu h d:</s> <s xml:id="echoid-s15575" xml:space="preserve"> neque ex arcu a f.</s> <s xml:id="echoid-s15576" xml:space="preserve"> Sed iam patuit, quòd omnino debet reflecti ex ar-<lb/>cu a d.</s> <s xml:id="echoid-s15577" xml:space="preserve"> Forma ergo c non reflectitur ad z, niſi ex aliquo puncto arcus f h [nam quòd à punctis h & <lb/>freflexio nulla fiat, patet per 74.</s> <s xml:id="echoid-s15578" xml:space="preserve"> 75 n 5.</s> <s xml:id="echoid-s15579" xml:space="preserve">] Reflectatur ergo ex t:</s> <s xml:id="echoid-s15580" xml:space="preserve"> & continuemus lineas c t, & z t.</s> <s xml:id="echoid-s15581" xml:space="preserve"> Quia <lb/> <pb o="220" file="0226" n="226" rhead="ALHAZEN"/> ergo t eſt inter duo pũcta f, h:</s> <s xml:id="echoid-s15582" xml:space="preserve"> erit linea z t inter duas lineas z f, z h.</s> <s xml:id="echoid-s15583" xml:space="preserve"> linea ergo z t ſecat lineam k l:</s> <s xml:id="echoid-s15584" xml:space="preserve"> ſe-<lb/>cet ergo lineam ipſam in i:</s> <s xml:id="echoid-s15585" xml:space="preserve"> i igitur eſt imago t [per 6 n 5] & t nullam habet imaginem niſi i.</s> <s xml:id="echoid-s15586" xml:space="preserve"> [quia ab <lb/>uno tantùm puncto peripheriæ f h fit reflexio per 73 n 5.</s> <s xml:id="echoid-s15587" xml:space="preserve">] Et ſic declarabitur, quòd imago cuiusli-<lb/>bet puncti lineę g r eſt punctum lineæ k l:</s> <s xml:id="echoid-s15588" xml:space="preserve"> k l ergo eſt imago g r:</s> <s xml:id="echoid-s15589" xml:space="preserve"> & k l eſt linea recta:</s> <s xml:id="echoid-s15590" xml:space="preserve"> quia eſt pars ſe-<lb/>midiametri circuli, a e:</s> <s xml:id="echoid-s15591" xml:space="preserve"> & g r eſt linea recta, quia eſt pars ſemidiametri circuli, o e.</s> <s xml:id="echoid-s15592" xml:space="preserve"> Ergo comprehen <lb/>dit formam g r rectè in ſpeculo ſphærico a b.</s> <s xml:id="echoid-s15593" xml:space="preserve"> Et hoc eſt quod uoluimus.</s> <s xml:id="echoid-s15594" xml:space="preserve"/> </p> <div xml:id="echoid-div524" type="float" level="0" n="0"> <figure xlink:label="fig-0225-01" xlink:href="fig-0225-01a"> <variables xml:id="echoid-variables184" xml:space="preserve">t f h a ſ i k d r e z b c m o g</variables> </figure> <figure xlink:label="fig-0225-02" xlink:href="fig-0225-02a"> <variables xml:id="echoid-variables185" xml:space="preserve">q h f d u o g c r e a n m z b</variables> </figure> </div> </div> <div xml:id="echoid-div526" type="section" level="0" n="0"> <head xml:id="echoid-head462" xml:space="preserve" style="it">46. In ſpeculo ſphærico cauo imagines linearum: conuexæ, cauæ, aliquando uidentur cõuexæ, <lb/>cauæ: eadeḿ obliquitate uiſum, qua ipſæ lineæ ſpeculum, reſpiciunt. 55 p 8.</head> <p> <s xml:id="echoid-s15595" xml:space="preserve">ET iteremus figurã, & conſtituamus ſuper lineam g r à duobus lateribus duos arcus, quomo-<lb/>docunq;</s> <s xml:id="echoid-s15596" xml:space="preserve"> ſint, ſcilicet g n r, g q r:</s> <s xml:id="echoid-s15597" xml:space="preserve"> & ſit arcus g n r non ſecans lineam g h:</s> <s xml:id="echoid-s15598" xml:space="preserve"> & ponamus in linea g r <lb/>punctum m, quomodocunq;</s> <s xml:id="echoid-s15599" xml:space="preserve"> ſit.</s> <s xml:id="echoid-s15600" xml:space="preserve"> Forma ergo m reflectitur ad z ex pũcto aliquo arcus f h [per <lb/>proximum numerum.</s> <s xml:id="echoid-s15601" xml:space="preserve">] reflectatur ergo ex t:</s> <s xml:id="echoid-s15602" xml:space="preserve"> & con <lb/> <anchor type="figure" xlink:label="fig-0226-01a" xlink:href="fig-0226-01"/> tinuemus lineas z t, & m t.</s> <s xml:id="echoid-s15603" xml:space="preserve"> Duo ergo anguli z t e, <lb/>e t m ſunt æquales [per theſin & 12 n 4.</s> <s xml:id="echoid-s15604" xml:space="preserve">] Linea ergo <lb/>m t ſecabit arcũ g n r:</s> <s xml:id="echoid-s15605" xml:space="preserve"> ſecet ergo in n:</s> <s xml:id="echoid-s15606" xml:space="preserve"> & extrahamus <lb/>lineã t m in parte m:</s> <s xml:id="echoid-s15607" xml:space="preserve"> ſecabit ergo g q r:</s> <s xml:id="echoid-s15608" xml:space="preserve"> ſecet ergo in <lb/>pũcto q:</s> <s xml:id="echoid-s15609" xml:space="preserve"> & cõtinuemus n e:</s> <s xml:id="echoid-s15610" xml:space="preserve"> & extrahatur rectè:</s> <s xml:id="echoid-s15611" xml:space="preserve"> ſeca <lb/>bit ergo z t ſub linea k l:</s> <s xml:id="echoid-s15612" xml:space="preserve"> ſecet ergo illã in i:</s> <s xml:id="echoid-s15613" xml:space="preserve"> & cõtinue <lb/>mus q e:</s> <s xml:id="echoid-s15614" xml:space="preserve"> & extrahamus ipſam rectè:</s> <s xml:id="echoid-s15615" xml:space="preserve"> ſecabit ergo z t <lb/>ſupra k l:</s> <s xml:id="echoid-s15616" xml:space="preserve"> ſecet ergo ipſam in p.</s> <s xml:id="echoid-s15617" xml:space="preserve"> Quia ergo duo angu <lb/>li ad t ſunt æquales:</s> <s xml:id="echoid-s15618" xml:space="preserve"> [per theſin & 12 n 4] erit i ima <lb/>go n:</s> <s xml:id="echoid-s15619" xml:space="preserve"> [per 6 n 5] & duo puncta k, l imagines duo-<lb/>rum punctorum g, r.</s> <s xml:id="echoid-s15620" xml:space="preserve"> Imago ergo arcus g n r, eſt linea <lb/>tranſiens per puncta k, i, l, ut linea k i l.</s> <s xml:id="echoid-s15621" xml:space="preserve"> Sed linea k i l <lb/>eſt conuexa ex parte uiſus z:</s> <s xml:id="echoid-s15622" xml:space="preserve"> & arcus g n r eſt con-<lb/>uexus ex parte ſpeculi.</s> <s xml:id="echoid-s15623" xml:space="preserve"> Ergo uiſus z comprehendet <lb/>formam lineæ g n r conuexæ, lineam conuexam.</s> <s xml:id="echoid-s15624" xml:space="preserve"> Et <lb/>quia duo anguli apud t ſunt æquales [nimirũ p t e, <lb/>q t e per theſin & 12 n 4] erit p etiam imago q [per <lb/>6 n 5] & erit linea l p k ex parte uiſus cõcaua:</s> <s xml:id="echoid-s15625" xml:space="preserve"> & eſt imago arcus g q r, cõcaui ex parte ſuperficiei ſp<gap/> <lb/>culi.</s> <s xml:id="echoid-s15626" xml:space="preserve"> Ergo uiſus z comprehendet formam arcus g q r concaui, lineam concauam.</s> <s xml:id="echoid-s15627" xml:space="preserve"> In ſpeculis ergo <lb/>concauis ex quibuſdam ſitibus comprehenditur linea conuexa, conuexa:</s> <s xml:id="echoid-s15628" xml:space="preserve"> & concaua, concaua.</s> <s xml:id="echoid-s15629" xml:space="preserve"/> </p> <div xml:id="echoid-div526" type="float" level="0" n="0"> <figure xlink:label="fig-0226-01" xlink:href="fig-0226-01a"> <variables xml:id="echoid-variables186" xml:space="preserve">t f h a p k l i d e z b n r m o g q</variables> </figure> </div> </div> <div xml:id="echoid-div528" type="section" level="0" n="0"> <head xml:id="echoid-head463" xml:space="preserve" style="it">47. In ſpeculo ſphærico cauo lineæ: recta, & curua conuexa parte ſpeculum reſpiciens, habent <lb/>aliquando imagines curuas: recta quatuor: curua unam: omneś caua parte uiſum reſpi-<lb/>ciunt. 56 p 8.</head> <p> <s xml:id="echoid-s15630" xml:space="preserve">ITem:</s> <s xml:id="echoid-s15631" xml:space="preserve"> ſit ſpeculum concauum:</s> <s xml:id="echoid-s15632" xml:space="preserve"> in quo ſit circulus a b d maximus:</s> <s xml:id="echoid-s15633" xml:space="preserve"> & centrum g:</s> <s xml:id="echoid-s15634" xml:space="preserve"> & extrahamus <lb/>lineam b g, quomodocunq;</s> <s xml:id="echoid-s15635" xml:space="preserve"> ſit:</s> <s xml:id="echoid-s15636" xml:space="preserve"> & diuidamus ex ipſa lineam g t maiorem medietate:</s> <s xml:id="echoid-s15637" xml:space="preserve"> & extraha-<lb/>mus ext lineam e t z perpendicularẽ ſuper b g:</s> <s xml:id="echoid-s15638" xml:space="preserve"> & ſit utraq;</s> <s xml:id="echoid-s15639" xml:space="preserve"> e t, t z æqualis t g [per 3 p 1.</s> <s xml:id="echoid-s15640" xml:space="preserve">] Et cõti-<lb/>nuemus e g, g z:</s> <s xml:id="echoid-s15641" xml:space="preserve"> & deſcribamus circa triangulũ e g z circulũ:</s> <s xml:id="echoid-s15642" xml:space="preserve"> [per 5 p 4] ſecabit ergo circulũ a b d in <lb/>duobus punctis:</s> <s xml:id="echoid-s15643" xml:space="preserve"> [per 10 p 3] nam punctũ t eſt centrũ huius circuli [per 9 p 3:</s> <s xml:id="echoid-s15644" xml:space="preserve"> æquatæ enim ſunt <lb/>rectæ e t, t z, t g] & t g eſt maior t b.</s> <s xml:id="echoid-s15645" xml:space="preserve"> Secet ergo circulus iſte circulum a b d in punctis a, d:</s> <s xml:id="echoid-s15646" xml:space="preserve"> & conti-<lb/>nuemus lineas g a, g d, e a, e b, e d, z a, z b, z d.</s> <s xml:id="echoid-s15647" xml:space="preserve"> Quia ergo duæ lineæ e t, t z ſunt æquales:</s> <s xml:id="echoid-s15648" xml:space="preserve"> erunt duæ <lb/>lineæ e g, g z æquales:</s> <s xml:id="echoid-s15649" xml:space="preserve"> [per 4 p 1:</s> <s xml:id="echoid-s15650" xml:space="preserve"> quia t g communis eſt, & anguli ad t per fabricationem recti ſunt] <lb/>& ſimiliter e b, b z æquales.</s> <s xml:id="echoid-s15651" xml:space="preserve"> Et quia duo arcus e g, g z ſunt æquales:</s> <s xml:id="echoid-s15652" xml:space="preserve"> [per 28 p 3:</s> <s xml:id="echoid-s15653" xml:space="preserve"> quia ſubtenduntur <lb/>a b æqualibus rectis e g, g z] duæ lineæ e a, a z reflectentur inter ſe propter angulos æquales [nam <lb/>anguli e a g, z a g per 27 p 3 æquantur] & duæ lineæ e b, b z reflectentur inter ſe propter angulos [<gap/> <lb/>b g, z b g] æquales [per 27 p 3.</s> <s xml:id="echoid-s15654" xml:space="preserve">] Et quia g t eſt maior quàm t b:</s> <s xml:id="echoid-s15655" xml:space="preserve"> [ex theſi] erit g e maior quàm e b.</s> <s xml:id="echoid-s15656" xml:space="preserve"> <lb/>[Quia enim anguli ad t ſunt recti per fabricationem, æquabitur per 47 p 1 quadratum e g quadra-<lb/>tis g t, e t:</s> <s xml:id="echoid-s15657" xml:space="preserve"> item quadratum e b quadratis b t, e t:</s> <s xml:id="echoid-s15658" xml:space="preserve"> itaque cum quadratum g t ſit maius quadrato t b:</s> <s xml:id="echoid-s15659" xml:space="preserve"> <lb/>quia g t maior eſt t b ex theſi:</s> <s xml:id="echoid-s15660" xml:space="preserve"> ſubducto communi e t:</s> <s xml:id="echoid-s15661" xml:space="preserve"> erit per 5 ax.</s> <s xml:id="echoid-s15662" xml:space="preserve"> quadratum e g maius quadra-<lb/>to e b:</s> <s xml:id="echoid-s15663" xml:space="preserve"> ideoq́ue latus e g maius latere e b.</s> <s xml:id="echoid-s15664" xml:space="preserve">] Angulus ergo e b g eſt maior angulo e g b [per 18 p 1] & <lb/>angulus e g b eſt ſemirectus.</s> <s xml:id="echoid-s15665" xml:space="preserve"> [Quia enim angulus ad t rectus eſt per fabricationem, & t e g, t g e æ-<lb/>quales per 5 p 1:</s> <s xml:id="echoid-s15666" xml:space="preserve"> quia e t, g t æquales poſitæ ſunt:</s> <s xml:id="echoid-s15667" xml:space="preserve"> erit eorum quilibet dimidius unius recti per 32 p 1.</s> <s xml:id="echoid-s15668" xml:space="preserve">] <lb/>Igitur duo anguli e g b, e b g ſimul ſunt maiores recto:</s> <s xml:id="echoid-s15669" xml:space="preserve"> ergo angulus b e g eſt recto minor:</s> <s xml:id="echoid-s15670" xml:space="preserve"> [ք 32 p 1] <lb/>& angulus e g z eſt rectus [ք 31 p 3.</s> <s xml:id="echoid-s15671" xml:space="preserve">] Ergo duæ lineæ e b, g z cõcurrẽt extra circulũ in parte b z [ք 11 <lb/>ax.</s> <s xml:id="echoid-s15672" xml:space="preserve">] Cõcurrant ergo in l.</s> <s xml:id="echoid-s15673" xml:space="preserve"> Et quia e d eſt intra triangulũ l e g:</s> <s xml:id="echoid-s15674" xml:space="preserve"> cõcurret cũlinea g m:</s> <s xml:id="echoid-s15675" xml:space="preserve"> cõcurrat ergo in <lb/>m.</s> <s xml:id="echoid-s15676" xml:space="preserve"> Et quia g b trãſit per centrũ z e g circuli:</s> <s xml:id="echoid-s15677" xml:space="preserve"> erit portio a g minor ſemicirculo:</s> <s xml:id="echoid-s15678" xml:space="preserve"> ergo [ք 31 p 3] angulus <lb/>a e g eſt obtuſus, & angulus e g z eſt rectus.</s> <s xml:id="echoid-s15679" xml:space="preserve"> Ergo illæ duæ lineæ a e, z g cõcurrẽt in parte e g [erunt <lb/>enim anguli ad e & g dictis angulis deinceps, minores duobus rectis per 13 p 1.</s> <s xml:id="echoid-s15680" xml:space="preserve"> Quare cõcurrent ex <lb/>parte e g per 11 ax.</s> <s xml:id="echoid-s15681" xml:space="preserve">] Concurrant ergo in f.</s> <s xml:id="echoid-s15682" xml:space="preserve"> Si ergo uiſus fuerit in e, & z in aliquo uiſibili:</s> <s xml:id="echoid-s15683" xml:space="preserve"> tunc puncta <lb/> <pb o="221" file="0227" n="227" rhead="OPTICAE LIBER VI."/> m, l, f eruntimagines punctiz.</s> <s xml:id="echoid-s15684" xml:space="preserve"> Sic ergo z comprehendetur in tribus locis [quia à tribus punctis a, <lb/>b, d reflectitur ad uiſum e.</s> <s xml:id="echoid-s15685" xml:space="preserve">] Item extrahamus ex e lineam ad arcum d z, quomodocunque ſit:</s> <s xml:id="echoid-s15686" xml:space="preserve"> & <lb/>ſit e k:</s> <s xml:id="echoid-s15687" xml:space="preserve"> & continuemus g k:</s> <s xml:id="echoid-s15688" xml:space="preserve"> & ſecet arcum d z in k:</s> <s xml:id="echoid-s15689" xml:space="preserve"> & continuemus lineam k z, Quia ergo arcus <lb/>e g, g z ſunt æquales:</s> <s xml:id="echoid-s15690" xml:space="preserve"> [ex concluſo] erunt [per <lb/> <anchor type="figure" xlink:label="fig-0227-01a" xlink:href="fig-0227-01"/> 27 p 3] duo anguli e k g, g k z æquales.</s> <s xml:id="echoid-s15691" xml:space="preserve"> Et conti-<lb/>nuemus g k in r:</s> <s xml:id="echoid-s15692" xml:space="preserve"> & extrahamus e r, z r.</s> <s xml:id="echoid-s15693" xml:space="preserve"> Ergo an-<lb/>gulus e r g eſt maior angulo g r z.</s> <s xml:id="echoid-s15694" xml:space="preserve"> [Quia enim <lb/>anguli e k g, z k g æquales ſunt concluſi:</s> <s xml:id="echoid-s15695" xml:space="preserve"> æqua-<lb/>buntur anguli e k r, z k r per 13 p 1.</s> <s xml:id="echoid-s15696" xml:space="preserve"> Poſitis igitur <lb/>angulis ad r æqualibus:</s> <s xml:id="echoid-s15697" xml:space="preserve"> erunt triangula e k r, z k r <lb/>æquiangula per 32 p 1:</s> <s xml:id="echoid-s15698" xml:space="preserve"> & per 4 p 6 r k ad duasre-<lb/>ctas k e, k z eandem habebit rationem.</s> <s xml:id="echoid-s15699" xml:space="preserve"> Quare <lb/>ipſæ erunt æquales per 9 p 5:</s> <s xml:id="echoid-s15700" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s15701" xml:space="preserve"> & periphe-<lb/>riæ e a d k & k z ipſis ſubtenſæ per 28 p 3:</s> <s xml:id="echoid-s15702" xml:space="preserve"> quod <lb/>fieri non poteſt.</s> <s xml:id="echoid-s15703" xml:space="preserve"> Nam quia rectæ a g, d g æquan-<lb/>tur per 15 d 1:</s> <s xml:id="echoid-s15704" xml:space="preserve"> æquabuntur peripheriæ a g, d g <lb/>ipſis ſubtenſæ per 28 p 3:</s> <s xml:id="echoid-s15705" xml:space="preserve"> & e g æqualis conclu-<lb/>ſa eſtipſi z g, reliqua igitur a e æquatur reliquæ <lb/>d z:</s> <s xml:id="echoid-s15706" xml:space="preserve"> ergo e a maior eſt k z per 9 ax:</s> <s xml:id="echoid-s15707" xml:space="preserve"> ergo e a d k <lb/>multò maior eſt k z.</s> <s xml:id="echoid-s15708" xml:space="preserve"> Quare angulus e r g non <lb/>eſt æqualis angulo g r z:</s> <s xml:id="echoid-s15709" xml:space="preserve"> nec eſt eo minor:</s> <s xml:id="echoid-s15710" xml:space="preserve"> quod <lb/>eodem argumento oſtendetur.</s> <s xml:id="echoid-s15711" xml:space="preserve"> Angulus igitur <lb/>e r g maior eſt angulo g r z] Sit ergo angulus <lb/>g r n æqualis angulo e r g [per 23 p 1.</s> <s xml:id="echoid-s15712" xml:space="preserve">] Duæ er-<lb/>go lineæ e r, r n reflectentur inter ſe, propter an-<lb/>gulos æquales [per 12 n 4] & extrahamus e r ad <lb/>q:</s> <s xml:id="echoid-s15713" xml:space="preserve"> erιt ergo q imago n reſpectu e.</s> <s xml:id="echoid-s15714" xml:space="preserve"> Et imaginemur <lb/>ſuperficiem exeuntem à linea m g f, perpendicu-<lb/>lariter ſuper circulum a b d:</s> <s xml:id="echoid-s15715" xml:space="preserve"> & extrahamus ex z <lb/>lineam in hac ſuperficie, perpendicularem ſuper <lb/>g z, & tranſeat in utranque partem.</s> <s xml:id="echoid-s15716" xml:space="preserve"> Sit ergo c z p:</s> <s xml:id="echoid-s15717" xml:space="preserve"> <lb/>& ponamus g centrum:</s> <s xml:id="echoid-s15718" xml:space="preserve"> & in longitudine g n fa-<lb/>ciamus arcum circuli c n p:</s> <s xml:id="echoid-s15719" xml:space="preserve"> ſecabit ergo lineam <lb/>c z p in duobus punctis:</s> <s xml:id="echoid-s15720" xml:space="preserve"> & ſint c, p:</s> <s xml:id="echoid-s15721" xml:space="preserve"> & continue-<lb/>mus lineas g c, g p.</s> <s xml:id="echoid-s15722" xml:space="preserve"> Erunt ergo in ſuperficie per-<lb/>pendiculari ſuper ſuperficiem a b d:</s> <s xml:id="echoid-s15723" xml:space="preserve"> & extraha-<lb/>mus g c, g p rectè:</s> <s xml:id="echoid-s15724" xml:space="preserve"> & ſuper g, & in longitudine <lb/>g q faciamus arcum circuli:</s> <s xml:id="echoid-s15725" xml:space="preserve"> ſecabit ergo duas li-<lb/>neas g c, g p:</s> <s xml:id="echoid-s15726" xml:space="preserve"> ſecet in s, o.</s> <s xml:id="echoid-s15727" xml:space="preserve"> Quia ergo ſuperficies <lb/>a b d circuli eſt perpendicularis ſuper ſuperficiem duarum linearum g c, g p:</s> <s xml:id="echoid-s15728" xml:space="preserve"> erunt duo anguli <lb/>e g s, e g o recti [per 4 d 11] & e g perpendicularis ſuper ſuperficiem g c p:</s> <s xml:id="echoid-s15729" xml:space="preserve"> erit ergo [per 18 p 11] <lb/>utraque ſuperficies e g s, e g o perpendicularis ſuper ſuperficiem s g o:</s> <s xml:id="echoid-s15730" xml:space="preserve"> & utraque iſtarum dua-<lb/>rum ſuperficierum facit in ſpeculo circulum magnum, [per 1 th.</s> <s xml:id="echoid-s15731" xml:space="preserve"> 1 ſphær.</s> <s xml:id="echoid-s15732" xml:space="preserve">] comparem circulo a b d.</s> <s xml:id="echoid-s15733" xml:space="preserve"> <lb/>Punctum ergo circuli compar puncto r, eſt, quod facit ſuperficies e g s.</s> <s xml:id="echoid-s15734" xml:space="preserve"> Ergo concurrunt ex ipſo <lb/>ſecundum angulos æquales duæ lineæ inter duo puncta e, c:</s> <s xml:id="echoid-s15735" xml:space="preserve"> & ſimiliter inter duo puncta e, p:</s> <s xml:id="echoid-s15736" xml:space="preserve"> & li-<lb/>neæ g c, g p ſunt æquales [per 15 d 1] & lineæ g s, g q, g o ſunt æquales:</s> <s xml:id="echoid-s15737" xml:space="preserve"> & q eſt imago n:</s> <s xml:id="echoid-s15738" xml:space="preserve"> & s ima-<lb/>go c:</s> <s xml:id="echoid-s15739" xml:space="preserve"> & o imago p.</s> <s xml:id="echoid-s15740" xml:space="preserve"> Imago ergo arcus c n p conuexi ex parte ſpeculi, eſt arcus s q o concauus ex <lb/>parte uiſus:</s> <s xml:id="echoid-s15741" xml:space="preserve"> & leſt imago z:</s> <s xml:id="echoid-s15742" xml:space="preserve"> & duo puncta s, o ſunt imagines c, p.</s> <s xml:id="echoid-s15743" xml:space="preserve"> Imago ergo lineæ c z p eſt linea <lb/>tranſiens per puncta s, l, o:</s> <s xml:id="echoid-s15744" xml:space="preserve"> & talis eſt concaua ex parte uiſus.</s> <s xml:id="echoid-s15745" xml:space="preserve"> Et ſignemus lineam tranſeuntem <lb/>per puncta s, l, o:</s> <s xml:id="echoid-s15746" xml:space="preserve"> & extrahamus lineam e g a d h.</s> <s xml:id="echoid-s15747" xml:space="preserve"> Si ergo ſpeculum non peruenit ad duo puncta b, <lb/>h, ſed alter ſuorum terminorum fuerit inter duo puncta b, h, & reliquus fuerit infra h, & uiſus fue-<lb/>rit in e:</s> <s xml:id="echoid-s15748" xml:space="preserve"> & duæ lineæ p z c, p n c fuerint in aliquo uiſibili:</s> <s xml:id="echoid-s15749" xml:space="preserve"> tunc forma lineæ p z c rectæ, erit conca-<lb/>ua, ſcilicet s l o:</s> <s xml:id="echoid-s15750" xml:space="preserve"> & forma arcus p n c conuexi erit etiam linea concaua, ſcilicet s q o.</s> <s xml:id="echoid-s15751" xml:space="preserve"> Et p z c re-<lb/>cta habebit unam imaginem:</s> <s xml:id="echoid-s15752" xml:space="preserve"> & arcus p n c habebit unam imaginem.</s> <s xml:id="echoid-s15753" xml:space="preserve"> Item extrahamus b g ad i:</s> <s xml:id="echoid-s15754" xml:space="preserve"> <lb/>& continuemus lineas e i, i z:</s> <s xml:id="echoid-s15755" xml:space="preserve"> iſtæ ergo duæ lineæ reflectuntur ſecundum angulos æquales:</s> <s xml:id="echoid-s15756" xml:space="preserve"> [Quia <lb/>enim e b, z b æquales ſunt concluſæ, & communis eſt b i:</s> <s xml:id="echoid-s15757" xml:space="preserve"> anguliq́ue e b i, z b i æquales per 27 <lb/>p 3, ut patuit:</s> <s xml:id="echoid-s15758" xml:space="preserve"> æquabuntur per 4 p 1 anguli e i b, z i b] & e i ſecabit f g:</s> <s xml:id="echoid-s15759" xml:space="preserve"> ſecet ergo in u:</s> <s xml:id="echoid-s15760" xml:space="preserve"> u ergo <lb/>erit imago z [per 6 n 5.</s> <s xml:id="echoid-s15761" xml:space="preserve">] Puncta ergo m, l, u, f ſunt imagines z.</s> <s xml:id="echoid-s15762" xml:space="preserve"> Et ſi ſpeculum exceſſerit duo pun-<lb/>cta a, d, & uiſus fuerit in e, & dorſum aſpicientis fuerit ex parte arcus a i, & comprehenderit to-<lb/>tum arcum i d a:</s> <s xml:id="echoid-s15763" xml:space="preserve"> tunc z uidebitur in quatuor locis, ſcilicet l, m, u, f:</s> <s xml:id="echoid-s15764" xml:space="preserve"> & uidebit duo puncta p, cin <lb/>duobus punctis s, o:</s> <s xml:id="echoid-s15765" xml:space="preserve"> & ſic linea recta p z c habebit quatuor imagines concauas:</s> <s xml:id="echoid-s15766" xml:space="preserve"> una tranſibit per <lb/>puncta s, m, o, ſcilicet linea s m o:</s> <s xml:id="echoid-s15767" xml:space="preserve"> ſecunda tranſibit per puncta s, l, o, ſcilicet linea s l o:</s> <s xml:id="echoid-s15768" xml:space="preserve"> tertia tran-<lb/>ſibit per puncta s, u, o, ſcilicet linea s u o:</s> <s xml:id="echoid-s15769" xml:space="preserve"> quarta tranſibit per puncta s, f, o, linea ſcilicet s f o.</s> <s xml:id="echoid-s15770" xml:space="preserve"> Pa-<lb/> <pb o="222" file="0228" n="228" rhead="ALHAZEN"/> tet ergo ex hac figura, quòd linea recta in ſpeculis concauis comprehendatur concaua:</s> <s xml:id="echoid-s15771" xml:space="preserve"> & con-<lb/>uexa comprehendatur concaua:</s> <s xml:id="echoid-s15772" xml:space="preserve"> & quòd recta habet plures formas concauas.</s> <s xml:id="echoid-s15773" xml:space="preserve"/> </p> <div xml:id="echoid-div528" type="float" level="0" n="0"> <figure xlink:label="fig-0227-01" xlink:href="fig-0227-01a"> <variables xml:id="echoid-variables187" xml:space="preserve">ſ m s q c d r b n <gap/> p t a h e g u i f</variables> </figure> </div> </div> <div xml:id="echoid-div530" type="section" level="0" n="0"> <head xml:id="echoid-head464" xml:space="preserve" style="it">48. Si duo uiſibilis puncta à duob{us} ſpeculi ſphærici caui punctis adunum uiſum reflexa, <lb/>in eadem ſpeculi diametro imagines ſu{as} habeant: recta inter centrum ſpeculi & imaginem <lb/>longinquiorem, ad rectam inter idem centrum & punctum uiſibilis à ſpeculi centro lon-<lb/>ginqui{us}, maiorem rationem habet: quàm recta inter ſpeculi centrum & imaginem pro-<lb/>pinquiorem, ad rectam inter idem centrum & punctum uiſibilis centro ſpeculi propin-<lb/>quius. 43 p 8.</head> <p> <s xml:id="echoid-s15774" xml:space="preserve">ITem:</s> <s xml:id="echoid-s15775" xml:space="preserve"> ſit ſpeculum concauum, per cuius centrum tranſeat plana ſuperficies:</s> <s xml:id="echoid-s15776" xml:space="preserve"> & faciat circu-<lb/>lum a b g [faciet autem per 1 th.</s> <s xml:id="echoid-s15777" xml:space="preserve"> 1 ſphær.</s> <s xml:id="echoid-s15778" xml:space="preserve">] & ſit centrum d:</s> <s xml:id="echoid-s15779" xml:space="preserve"> & extrahamus ex d lineam, quo-<lb/>cunque modo ſit:</s> <s xml:id="echoid-s15780" xml:space="preserve"> & ſit d g:</s> <s xml:id="echoid-s15781" xml:space="preserve"> & tranſeat extra circulum:</s> <s xml:id="echoid-s15782" xml:space="preserve"> & extrahamus ex d in ſuperficie huius <lb/>circuli lineam perpendicularem ſuper lineam d g [per 11 p 1] & ſit d a:</s> <s xml:id="echoid-s15783" xml:space="preserve"> & abſcindamus de angu-<lb/>lo a d g recto particulam paruam, quomodocunque ſit:</s> <s xml:id="echoid-s15784" xml:space="preserve"> & ſit angulus g d e, ita ut inter angu-<lb/>lum rectum & angulum a d e ſit multiplum anguli e d g:</s> <s xml:id="echoid-s15785" xml:space="preserve"> [id quod fieri poteſt continua anguli <lb/>recti biſſectione, donec angulus a d e ſit multiplex ad angulum e d g] & diuidamus angulum <lb/>a d e in duo æqualia, per lineam d b [per 9 p 1] & abſcindamus de angulo b d a æqualem an-<lb/>gulo e d g, per lineam z d:</s> <s xml:id="echoid-s15786" xml:space="preserve"> & extrahamus ex d lineam continentem cum b d angulum rectum:</s> <s xml:id="echoid-s15787" xml:space="preserve"> <lb/>& ſit d x:</s> <s xml:id="echoid-s15788" xml:space="preserve"> & extrahamus a d in parte d:</s> <s xml:id="echoid-s15789" xml:space="preserve"> & ſit d k:</s> <s xml:id="echoid-s15790" xml:space="preserve"> & extrahamus ex z lineam continentem cum z d <lb/>angulum, æqualem angulo k d x:</s> <s xml:id="echoid-s15791" xml:space="preserve"> & ſit z h.</s> <s xml:id="echoid-s15792" xml:space="preserve"> Hæc ergo linea concurret cum d a:</s> <s xml:id="echoid-s15793" xml:space="preserve"> [per 11 ax.</s> <s xml:id="echoid-s15794" xml:space="preserve">] Nam <lb/>duo anguli k d x, a d z ſunt minores duobus rectis [ideoq́ue a d z, h z d ijſdem ſunt minores:</s> <s xml:id="echoid-s15795" xml:space="preserve"> quia <lb/>h z d æquatus eſt angulo k d x.</s> <s xml:id="echoid-s15796" xml:space="preserve">] Concurrant ergo in h.</s> <s xml:id="echoid-s15797" xml:space="preserve"> Angulus ergo z h d eſt æqualis angulo <lb/>z d x.</s> <s xml:id="echoid-s15798" xml:space="preserve"> [Quia enim tres anguli z d h, z d x, k d x æquantur duobus rectis per 13 p 1:</s> <s xml:id="echoid-s15799" xml:space="preserve"> quibus item <lb/>æquantur tres anguli trianguli z d h per 32 p 1:</s> <s xml:id="echoid-s15800" xml:space="preserve"> tres igitur illi tribus his æquantur.</s> <s xml:id="echoid-s15801" xml:space="preserve"> Itaque cum <lb/>z d h communis æquetur ſibi ipſi, & d z h æquatus ſit ipſi k d x:</s> <s xml:id="echoid-s15802" xml:space="preserve"> reliquus z h d æquabitur reli-<lb/>quo z d x.</s> <s xml:id="echoid-s15803" xml:space="preserve">] Et extrahamus ex z lineam conti-<lb/> <anchor type="figure" xlink:label="fig-0228-01a" xlink:href="fig-0228-01"/> nentem cum z h angulum, æqualem angulo b d <lb/>k obtuſo:</s> <s xml:id="echoid-s15804" xml:space="preserve"> & ſit z l.</s> <s xml:id="echoid-s15805" xml:space="preserve"> Duo ergo anguli l z d, b d z <lb/>ſunt minores duobus rectis.</s> <s xml:id="echoid-s15806" xml:space="preserve"> [Quia enim angu-<lb/>li b d k, b d a æquantur duobus rectis per 13 p 1:</s> <s xml:id="echoid-s15807" xml:space="preserve"> <lb/>erunt anguli, b d k, id eſt, per fabricationem, <lb/>l z h, & b d z minores duobus rectis:</s> <s xml:id="echoid-s15808" xml:space="preserve"> ideoq́ue <lb/>l z d, b d z ijſdem multò minores erunt.</s> <s xml:id="echoid-s15809" xml:space="preserve">] Li-<lb/>nea ergo z l concurret cum d b [per 11 ax.</s> <s xml:id="echoid-s15810" xml:space="preserve">] <lb/>Concurrant ergo in l:</s> <s xml:id="echoid-s15811" xml:space="preserve"> & continuemus l h:</s> <s xml:id="echoid-s15812" xml:space="preserve"> & [per <lb/>5 p 4] circa triangulum h l d faciamus circu-<lb/>lum d h l:</s> <s xml:id="echoid-s15813" xml:space="preserve"> tranſibit ergo per z [per conuerſio-<lb/>nem 22 p 3] quia duo anguli l z h, l d h ſunt æ-<lb/>quales duobus rectis [quia æquantur duobus <lb/>angulis b d k, l d h æqualibus duobus rectis <lb/>per 13 p 1.</s> <s xml:id="echoid-s15814" xml:space="preserve">] Anguli ergo l h z, l d z ſunt æquales <lb/>[per 27 p 3] quia baſis eorum eſt idem arcus:</s> <s xml:id="echoid-s15815" xml:space="preserve"> <lb/>[l z] ſed angulus z h d eſt æqualis angulo z d <lb/>x:</s> <s xml:id="echoid-s15816" xml:space="preserve"> [per concluſionem] remanet ergo angulus <lb/>l h d æqualis angulo l d x:</s> <s xml:id="echoid-s15817" xml:space="preserve"> & angulus l d x eſt <lb/>rectus:</s> <s xml:id="echoid-s15818" xml:space="preserve"> [per fabricationem] ergo angulus l h d <lb/>eſt rectus.</s> <s xml:id="echoid-s15819" xml:space="preserve"> Et abſcindamus exlinea d e lineam <lb/>d m, æqualem d h [per 3 p 1] & continuemus l m.</s> <s xml:id="echoid-s15820" xml:space="preserve"> <lb/>Angulus ergo l m d eſt rectus.</s> <s xml:id="echoid-s15821" xml:space="preserve"> [quia per 4 p 1 <lb/>æquatur angulo l h d recto concluſo:</s> <s xml:id="echoid-s15822" xml:space="preserve"> duo enim <lb/>latera h d, l d æquantur duobus lateribus m d, <lb/>l d, & angulus h d l angulo m d l per fabricatio-<lb/>nem.</s> <s xml:id="echoid-s15823" xml:space="preserve">] Circulus ergo l h d tranſit per m [per <lb/>conuerſionem 31 p 3 demonſtratam à Theone in <lb/>commentarijs in 3 librum magnæ conſtructio-<lb/>nis Ptolemæi] & ſecat arcum b e in compari pun <lb/>cto z.</s> <s xml:id="echoid-s15824" xml:space="preserve"> Secet ergo in f:</s> <s xml:id="echoid-s15825" xml:space="preserve"> & continuemus d f.</s> <s xml:id="echoid-s15826" xml:space="preserve"> An-<lb/>gulus ergo l d f erit æqualis angulo l d z:</s> <s xml:id="echoid-s15827" xml:space="preserve"> [per 27 <lb/>p 3:</s> <s xml:id="echoid-s15828" xml:space="preserve"> quia arcus l m eſt æqualis arcui l h.</s> <s xml:id="echoid-s15829" xml:space="preserve"> [Quia <lb/>enim triangulo l m d circulus circumſcriptus <lb/>eſt, & angulus ad m rectus ex concluſo:</s> <s xml:id="echoid-s15830" xml:space="preserve"> erit l d diameter circuli per conſectarium 5 p 4, ſeu <lb/> <pb o="223" file="0229" n="229" rhead="OPTICAE LIBER VI."/> 31 p 3.</s> <s xml:id="echoid-s15831" xml:space="preserve"> Quare ſemiperipheria l f d æquatur ſemiperipheriæ l z d:</s> <s xml:id="echoid-s15832" xml:space="preserve"> & peripheria d m æquatur periphe-<lb/>riæ d h per 28 p 3, quia d m, d h æquatæ ſunt:</s> <s xml:id="echoid-s15833" xml:space="preserve"> reliqua igitur l f m æquatur reliquę l z h] & arcus <lb/>m f eſt æqualis arcui z h.</s> <s xml:id="echoid-s15834" xml:space="preserve"> [Nam propter æqualitatem ſemidiametrorum d f, & d z, ęquantur periphe <lb/>riæ d m f, d h z per 28 p 3:</s> <s xml:id="echoid-s15835" xml:space="preserve"> & per eandem peripheriæ d m & d h ęquales concluſæ ſunt:</s> <s xml:id="echoid-s15836" xml:space="preserve"> reliqua igitur <lb/>m f æquatur reliquæ z h.</s> <s xml:id="echoid-s15837" xml:space="preserve">] Ergo arcus l f eſt ęqualis arcui l z [per 3 ax:</s> <s xml:id="echoid-s15838" xml:space="preserve"> quare per 27 p 3 anguli l d f, l d <lb/>z ęquabuntur.</s> <s xml:id="echoid-s15839" xml:space="preserve">] Et continuemus lineas h b, h f, f m, b m, f z, f b.</s> <s xml:id="echoid-s15840" xml:space="preserve"> Angulus ergo b h d eſt acutus [quia <lb/>l h d rectus eſt concluſus] & angulus g d h rectus [per fabricationem.</s> <s xml:id="echoid-s15841" xml:space="preserve">] Ergo linea h b concurret cũ <lb/>linea d g extra circulum [per 11 ax.</s> <s xml:id="echoid-s15842" xml:space="preserve">] Concurrant ergo in q:</s> <s xml:id="echoid-s15843" xml:space="preserve"> h f ergo concurret etiam cum d g extra <lb/>circulum [eadem de cauſſa.</s> <s xml:id="echoid-s15844" xml:space="preserve">] Concurrant ergo in n.</s> <s xml:id="echoid-s15845" xml:space="preserve"> Et extrahamus f b, quouſque ſecet arcum l z:</s> <s xml:id="echoid-s15846" xml:space="preserve"> <lb/>ſecet ergo in r:</s> <s xml:id="echoid-s15847" xml:space="preserve"> & continuemus r m:</s> <s xml:id="echoid-s15848" xml:space="preserve"> angulus ergo f r m, qui eſt in circumferentia, reſpicit arcum f m:</s> <s xml:id="echoid-s15849" xml:space="preserve"> <lb/>& [per 16 p 1] angulus f b m eſt maior angulo f r m:</s> <s xml:id="echoid-s15850" xml:space="preserve"> & angulus f b m eſt in circumferentia a b g.</s> <s xml:id="echoid-s15851" xml:space="preserve"> Ergo <lb/>ſi b m linea extrahatur ex parte m:</s> <s xml:id="echoid-s15852" xml:space="preserve"> abſcindet de circulo a b g arcum maiorem ſimili arcui f m circuli <lb/>l h d [per 33 p 6] & arcus f m eſt ſimilis duplo arcus f e:</s> <s xml:id="echoid-s15853" xml:space="preserve"> [angulus enim duplus anguli f d e in periphe <lb/>ria circuli a b g conſtituti, inſiſtit in peripheriam duplam peripheriæ f e per 33 p 6] & arcus f e eſt æ-<lb/>qualis arcui a z:</s> <s xml:id="echoid-s15854" xml:space="preserve"> [quia enim anguli a d b, e d b ęquantur propter angulum a d e per rectam b d bifa-<lb/>riam ſectum:</s> <s xml:id="echoid-s15855" xml:space="preserve"> & z d b, f d b per concluſionem:</s> <s xml:id="echoid-s15856" xml:space="preserve"> ęquabitur reliquus a d z reliquo f d e:</s> <s xml:id="echoid-s15857" xml:space="preserve"> ideoq́ue peri-<lb/>pherię a z peripherię f e per 26 p 3] & arcus a z eſt ęqualis arcui e g [per 26 p 3:</s> <s xml:id="echoid-s15858" xml:space="preserve"> quia angulus a d z <lb/>ęquatus eſt angulo e d g.</s> <s xml:id="echoid-s15859" xml:space="preserve">] Ergo arcus f e eſt ęqualis arcui e g:</s> <s xml:id="echoid-s15860" xml:space="preserve"> ergo arcus g f eſt duplus arcus g e:</s> <s xml:id="echoid-s15861" xml:space="preserve"> er <lb/>go arcus g f eſt ſimilis arcui f m.</s> <s xml:id="echoid-s15862" xml:space="preserve"> Si ergo b m extrahatur rectè in partem m:</s> <s xml:id="echoid-s15863" xml:space="preserve"> abſcindet de circulo a b g <lb/>arcum ultra punctum g, maiorem arcu f g.</s> <s xml:id="echoid-s15864" xml:space="preserve"> Linea ergo b m ſecabit lineam d g inter duo puncta g, d.</s> <s xml:id="echoid-s15865" xml:space="preserve"> <lb/>Secet ergo in o:</s> <s xml:id="echoid-s15866" xml:space="preserve"> & extrahamus lineam f m:</s> <s xml:id="echoid-s15867" xml:space="preserve"> & ſecet d o in u:</s> <s xml:id="echoid-s15868" xml:space="preserve"> [ſecabit autem:</s> <s xml:id="echoid-s15869" xml:space="preserve"> quia ſecat angulum <lb/>d m o à baſi d o ſubtenſum] & extrahamus b m in parte b:</s> <s xml:id="echoid-s15870" xml:space="preserve"> & ſecet arcum l r in c:</s> <s xml:id="echoid-s15871" xml:space="preserve"> & continuemus <lb/>c d.</s> <s xml:id="echoid-s15872" xml:space="preserve"> Quia ergo angulus b f z eſt in circumferentia a b g:</s> <s xml:id="echoid-s15873" xml:space="preserve"> erit [per 20 p 3] angulus b f z dimidius angu <lb/>li b d z:</s> <s xml:id="echoid-s15874" xml:space="preserve"> ſed angulus b d z eſt multiplus anguli z d a:</s> <s xml:id="echoid-s15875" xml:space="preserve"> [è fabricatione] ergo angulus b f z eſt multi-<lb/>plus anguli z d h:</s> <s xml:id="echoid-s15876" xml:space="preserve"> ergo [per 33 p 6] arcus r z eſt multiplus arcus z h:</s> <s xml:id="echoid-s15877" xml:space="preserve"> & arcus c z eſt maior arcu r z <lb/>[per 9 axiom.</s> <s xml:id="echoid-s15878" xml:space="preserve">] ergo arcus c z eſt multiplus arcus z h.</s> <s xml:id="echoid-s15879" xml:space="preserve"> Et continuemus c h:</s> <s xml:id="echoid-s15880" xml:space="preserve"> angulus ergo c h d cum <lb/>angulo c m d, eſt æqualis duobus rectis:</s> <s xml:id="echoid-s15881" xml:space="preserve"> [per 22 p 3] ergo angulus c h d eſt æqualis angulo b m e.</s> <s xml:id="echoid-s15882" xml:space="preserve"> <lb/>[Nam per 13 p 1 anguli c m d, c m e ęquantur duobus rectis, quibus etiam ęquantur per proximam <lb/>concluſionem c h d, c m d:</s> <s xml:id="echoid-s15883" xml:space="preserve"> communi igitur c m d ſubducto, reliquus c h d æquabitur reliquo c m e <lb/>ſeu b m e.</s> <s xml:id="echoid-s15884" xml:space="preserve">] Sed angulus z h d addit ſuper angulum c h d, angulum c h z, qui eſt æqualis angulo c <lb/>d z:</s> <s xml:id="echoid-s15885" xml:space="preserve"> [per 27 p 3:</s> <s xml:id="echoid-s15886" xml:space="preserve"> quia uterque inſiſtit in eandem peripheriam c z] & angulus c d z eſt multiplus an-<lb/>guli z d a, [per 33 p 6:</s> <s xml:id="echoid-s15887" xml:space="preserve"> quia peripheria c z multiplex oſtenſa eſt peripheriæ z h.</s> <s xml:id="echoid-s15888" xml:space="preserve">] Ergo angulus c h z <lb/>eſt multiplus anguli e d g:</s> <s xml:id="echoid-s15889" xml:space="preserve"> [quia multiplex eſt ad angulum z d h, æqualem ipſi e d g.</s> <s xml:id="echoid-s15890" xml:space="preserve">] Ergo angu-<lb/>lus z h d excedit angulum c h d multiplo anguli e d g.</s> <s xml:id="echoid-s15891" xml:space="preserve"> Angulus ergo z h d eſt æqualis angulo f m d:</s> <s xml:id="echoid-s15892" xml:space="preserve"> <lb/>quia arcus f m d eſt ęqualis arcui z h d [per concluſionem.</s> <s xml:id="echoid-s15893" xml:space="preserve"> Itaque per 2 ax.</s> <s xml:id="echoid-s15894" xml:space="preserve"> peripheria z f d, in quam <lb/>inſiſtit angulus z h d, ęquabitur peripheriæ f z d, in quam inſ<gap/>ſtit angulus f m d:</s> <s xml:id="echoid-s15895" xml:space="preserve"> & idcirco z h d æ-<lb/>quabitur f m d per 27 p 3] & angulus c h d, ut declarauimus, eſt ęqualis angulo b m e.</s> <s xml:id="echoid-s15896" xml:space="preserve"> Ergo angu-<lb/>lus f m d excedit angulum b m e multiplo anguli e d g:</s> <s xml:id="echoid-s15897" xml:space="preserve"> ergo angulus f m d excedit angulum o m d <lb/>multiplo anguli e d g:</s> <s xml:id="echoid-s15898" xml:space="preserve"> [quia angulus o m d ęquatur angulo b m e per 15 p 1] & angulus m o g exce-<lb/>dit angulum o m d angulo e d g [nam angulus m o g æquatur angulis o m d & e d g per 32 p 1.</s> <s xml:id="echoid-s15899" xml:space="preserve">] Er-<lb/>go angulus f m d excedit angulum m o g, multiplo anguli e d g:</s> <s xml:id="echoid-s15900" xml:space="preserve"> & angulus f m d excedit angulum <lb/>m u d, angulo e d g ſolo:</s> <s xml:id="echoid-s15901" xml:space="preserve"> [quia per 32 p 1 æquatur angulis m d u ſeu e d g & m u d] ergo angulus m u d <lb/>eſt maior angulo m o g:</s> <s xml:id="echoid-s15902" xml:space="preserve"> ergo angulus m o u eſt maior angulo m u o:</s> <s xml:id="echoid-s15903" xml:space="preserve"> [Nam quia anguli ad u dein-<lb/>ceps ęquantur angulis ad o deinceps per 13 p 1:</s> <s xml:id="echoid-s15904" xml:space="preserve"> & m u d maior concluſus m o g:</s> <s xml:id="echoid-s15905" xml:space="preserve"> reliquus igitur m o u <lb/>maior eſt reliquo m u o] ergo [per 19 p 1] linea m u eſt maior linea m o.</s> <s xml:id="echoid-s15906" xml:space="preserve"> Et quia arcus z h d eſt ęqua <lb/>lis arcui f m d:</s> <s xml:id="echoid-s15907" xml:space="preserve"> erunt duo anguli h f d, m f d æquales [per 27 p 3:</s> <s xml:id="echoid-s15908" xml:space="preserve"> quia peripheriæ h d, m d æquales <lb/>ſunt concluſæ.</s> <s xml:id="echoid-s15909" xml:space="preserve">] Duæ ergo lineæ h f, f u reflectentur æqualiter:</s> <s xml:id="echoid-s15910" xml:space="preserve"> & ſimiliter h b, b o reflectentur ęqua <lb/>liter [propter concluſam æqualitatem angulorum h b d, o b d] q ergo eſt imago o:</s> <s xml:id="echoid-s15911" xml:space="preserve"> & n imago u [per <lb/>6 n 5.</s> <s xml:id="echoid-s15912" xml:space="preserve">] Et extrahamus ex m lineam æquidiſtantem lineæ h q [per 31 p 1] & ſit m s:</s> <s xml:id="echoid-s15913" xml:space="preserve"> & extrahamus ex <lb/>m etiam lineam æquidiſtantem lineæ h n:</s> <s xml:id="echoid-s15914" xml:space="preserve"> & ſit m p.</s> <s xml:id="echoid-s15915" xml:space="preserve"> Quia ergo [per 16 p 1] angulus h n d eſt maior <lb/>angulo h q d:</s> <s xml:id="echoid-s15916" xml:space="preserve"> erit angulus m p o maior angulo m s o.</s> <s xml:id="echoid-s15917" xml:space="preserve"> [nam per 29 p 1 angulus m s o æ quatur angu-<lb/>lo ad q, & angulus m p o æquatur angulo ad n] p ergo erit inter duo puncta s, u.</s> <s xml:id="echoid-s15918" xml:space="preserve"> Et quia angulus <lb/>h d n eſt rectus [ex theſi:</s> <s xml:id="echoid-s15919" xml:space="preserve">] erit angulus h n d acutus [per 32 p 1] ergo angulus m p d eſt acutus:</s> <s xml:id="echoid-s15920" xml:space="preserve"> ergo <lb/>[per 13 p 1] angulus m p s eſt obtuſus:</s> <s xml:id="echoid-s15921" xml:space="preserve"> ergo [per 19 p 1] linea m s eſt maior, quàm m p:</s> <s xml:id="echoid-s15922" xml:space="preserve"> ſed m u eſt ma-<lb/>ior, quàm m o, ut diximus:</s> <s xml:id="echoid-s15923" xml:space="preserve"> ergo proportio s m ad m o eſt maior, quàm proportio p m ad m u:</s> <s xml:id="echoid-s15924" xml:space="preserve"> [ut <lb/>patet per 8 p 5] & [per 29 p 1.</s> <s xml:id="echoid-s15925" xml:space="preserve">4 p 6] proportio s m ad m o eſt, ſicut proportio q b ad b o:</s> <s xml:id="echoid-s15926" xml:space="preserve"> quia m s eſt <lb/>æquidiſtans b q:</s> <s xml:id="echoid-s15927" xml:space="preserve"> & ſimiliter proportio p m ad m u eſt, ſicut proportio n f a d f u:</s> <s xml:id="echoid-s15928" xml:space="preserve"> ergo [per 11 p 5] pro-<lb/>portio q b ad b o eſt maior, quàm proportio n f ad f u:</s> <s xml:id="echoid-s15929" xml:space="preserve"> & proportio q b ad b o eſt, ſicut proportio q d <lb/>ad d o:</s> <s xml:id="echoid-s15930" xml:space="preserve"> & proportio n f ad f u eſt, ſicut proportio n d ad d u, ut declarauimus in capitulo de imagine <lb/>[64 n 5.</s> <s xml:id="echoid-s15931" xml:space="preserve">] Ergo proportio q d ad d o eſt maior, quàm proportio n d ad d u.</s> <s xml:id="echoid-s15932" xml:space="preserve"> [Eſt autem q, imago pun-<lb/>cti o, à centro ſpeculi d longinquior:</s> <s xml:id="echoid-s15933" xml:space="preserve"> & o punctum uiſibilis ab eodem centro eſt lon-<lb/>ginquius.</s> <s xml:id="echoid-s15934" xml:space="preserve"> n uerò, imago puncti u centro ſpeculi d eſt propinquior:</s> <s xml:id="echoid-s15935" xml:space="preserve"> & u <lb/>alterum uiſibilis punctum eodem centro d eſt propinquius.</s> <s xml:id="echoid-s15936" xml:space="preserve">] <lb/>Quare patet propoſitum.</s> <s xml:id="echoid-s15937" xml:space="preserve"/> </p> <div xml:id="echoid-div530" type="float" level="0" n="0"> <figure xlink:label="fig-0228-01" xlink:href="fig-0228-01a"> <variables xml:id="echoid-variables188" xml:space="preserve">q s n p e f o <gap/> x u m l b <gap/> <gap/> z k d h a</variables> </figure> </div> <pb o="224" file="0230" n="230" rhead="ALHAZEN"/> </div> <div xml:id="echoid-div532" type="section" level="0" n="0"> <head xml:id="echoid-head465" xml:space="preserve" style="it">49. In ſpeculo ſphærico cauo imago lineæ rectæ aliquando uidetur conuexa. 57 p 8.</head> <p> <s xml:id="echoid-s15938" xml:space="preserve">HIs præoſtenſis, iteremus circulum, & perficiamus demonſtrationem, ne multiplicentur & li-<lb/>neæ, & dubitentur literæ.</s> <s xml:id="echoid-s15939" xml:space="preserve"> Sit ergo circulus in ſecunda figura a b g:</s> <s xml:id="echoid-s15940" xml:space="preserve"> & centrum d:</s> <s xml:id="echoid-s15941" xml:space="preserve"> & extraha-<lb/>mus lineam d q:</s> <s xml:id="echoid-s15942" xml:space="preserve"> & ſit d b æqualis d b in prima figura:</s> <s xml:id="echoid-s15943" xml:space="preserve"> & d o æqualis d o in prima figura:</s> <s xml:id="echoid-s15944" xml:space="preserve"> & d q <lb/>ſit compar ſibi in prima figura:</s> <s xml:id="echoid-s15945" xml:space="preserve"> & ſimiliter d u:</s> <s xml:id="echoid-s15946" xml:space="preserve"> & extrahamus ſuper d q perpendicularem ſuper ſu-<lb/>perficiem circuli [per 12 p 11] & ſit d h æqualis ſibi in prima figura.</s> <s xml:id="echoid-s15947" xml:space="preserve"> Angulus ergo h d q erit rectus:</s> <s xml:id="echoid-s15948" xml:space="preserve"> <lb/>[per 3 d 11] & circulus, quem facit h d q in ſpeculo, erit ex circulis, ex quibus forma punctorum o, u <lb/>reflectitur:</s> <s xml:id="echoid-s15949" xml:space="preserve"> & erit arcus, quem menſurant lineæ h d, d q, æqualis arcui a g in primo circulo:</s> <s xml:id="echoid-s15950" xml:space="preserve"> [per 33 p <lb/>6:</s> <s xml:id="echoid-s15951" xml:space="preserve"> quia uterque ſubtendit angulum rectum] & ex duobus punctis iſtius arcus, comparibus duobus <lb/>punctis b, f, reflectentur duo puncta lineæ u p ad duo puncta n, q æqualiter.</s> <s xml:id="echoid-s15952" xml:space="preserve"> Erit ergo q imago o, & <lb/>n imago u.</s> <s xml:id="echoid-s15953" xml:space="preserve"> Et extrahamus ex u perpendicularem lineam in ſuperficie circuli a b g, ſuper lineam d u <lb/>[per 11 p 1] & ſit z u e:</s> <s xml:id="echoid-s15954" xml:space="preserve"> & ſit d centrum:</s> <s xml:id="echoid-s15955" xml:space="preserve"> & in longitudine d o faciamus arcum circuli:</s> <s xml:id="echoid-s15956" xml:space="preserve"> ſecabit ergo li-<lb/>neam z u e in duobus punctis:</s> <s xml:id="echoid-s15957" xml:space="preserve"> [quia punctum o <lb/> <anchor type="figure" xlink:label="fig-0230-01a" xlink:href="fig-0230-01"/> altius eſt puncto u, ex prima theſi] ſecet ergo in <lb/>z, e:</s> <s xml:id="echoid-s15958" xml:space="preserve"> & ſit arcus z o e:</s> <s xml:id="echoid-s15959" xml:space="preserve"> & continuemus d z, d e:</s> <s xml:id="echoid-s15960" xml:space="preserve"> & <lb/>extrahamus extra circulum:</s> <s xml:id="echoid-s15961" xml:space="preserve"> & à d & in longitu-<lb/>dine d q faciamus arcum t q:</s> <s xml:id="echoid-s15962" xml:space="preserve"> ſecabit ergo duas <lb/>lineas d z, d e in t, k:</s> <s xml:id="echoid-s15963" xml:space="preserve"> & continuemus t k:</s> <s xml:id="echoid-s15964" xml:space="preserve"> ſecabit <lb/>ergo lineam d q in l.</s> <s xml:id="echoid-s15965" xml:space="preserve"> Quia ergo h d eſt perpendi-<lb/>cularis ſuper ſuperficiem circuli:</s> <s xml:id="echoid-s15966" xml:space="preserve"> uterque angu-<lb/>lus h d t, h d k erit rectus:</s> <s xml:id="echoid-s15967" xml:space="preserve"> [per 3 d 11] & utraque <lb/>ſuperficies h d t, h d k faciet in ſuperficie ſpecu-<lb/>li circulum [per 1 th.</s> <s xml:id="echoid-s15968" xml:space="preserve"> 1 ſphær.</s> <s xml:id="echoid-s15969" xml:space="preserve">] & arcus, qui eſt in-<lb/>ter duas lineas h d, d t erit æqualis arcui, qui eſt <lb/>inter duas lineas h d, d q:</s> <s xml:id="echoid-s15970" xml:space="preserve"> & ſimiliter arcus, qui <lb/>eſt inter duas lineas h d, d k & utraque linea d z, <lb/>d e eſt æqualis lineę d o [per 15 d 1.</s> <s xml:id="echoid-s15971" xml:space="preserve">] Ergo hi duo <lb/>arcus ſunt huiuſmodi, quòd ex illis reflectentur <lb/>ſecundum angulos æquales duo puncta z, e:</s> <s xml:id="echoid-s15972" xml:space="preserve"> [ut <lb/>demonſtratum eſt 66 n 5] & duæ lineæ d t, d k <lb/>ſunt æquales lineæ d q [per 15 d 1.</s> <s xml:id="echoid-s15973" xml:space="preserve">] Ergo pun-<lb/>ctum t eſt imago z, & k eſt imago e.</s> <s xml:id="echoid-s15974" xml:space="preserve"> Et quia li-<lb/>neæ d t, d q, d k ſunt æquales:</s> <s xml:id="echoid-s15975" xml:space="preserve"> & lineæ d z, d o, <lb/>d e ſunt æquales:</s> <s xml:id="echoid-s15976" xml:space="preserve"> erit [per 7 p 5] proportio d t ad <lb/>d z, ſicut proportio q d ad d o, & ſicut proportio <lb/>k d ad d e.</s> <s xml:id="echoid-s15977" xml:space="preserve"> Sed proportio q d ad d o, ut in prima <lb/>figura [præcedentis numeri] præoſtendimus, <lb/>eſt maior proportione n d ad d u.</s> <s xml:id="echoid-s15978" xml:space="preserve"> Ergo propor-<lb/>tio d t ad d z eſt maior proportione n d ad d u:</s> <s xml:id="echoid-s15979" xml:space="preserve"> & <lb/>ſimiliter k d ad d e.</s> <s xml:id="echoid-s15980" xml:space="preserve"> Et quia duæ lineę z d, d e ſunt <lb/>æquales, & duę lineæ d t, d k ſunt æquales:</s> <s xml:id="echoid-s15981" xml:space="preserve"> erit li <lb/>nea t k æquidiſtans z e [per 2 p 6:</s> <s xml:id="echoid-s15982" xml:space="preserve"> eſt enim per 7 <lb/>p 5 d t ad d z, ſicut d k ad d e:</s> <s xml:id="echoid-s15983" xml:space="preserve"> & per 17 p 5, ut t z ad <lb/>z d, ſic k e ad e d.</s> <s xml:id="echoid-s15984" xml:space="preserve">] Ergo [per 2 p 6.</s> <s xml:id="echoid-s15985" xml:space="preserve"> 18 p 5] utraq;</s> <s xml:id="echoid-s15986" xml:space="preserve"> <lb/>proportio d t ad d z, & k d ad d e erit, ſicut pro-<lb/>portio l d ad d u.</s> <s xml:id="echoid-s15987" xml:space="preserve"> Ergo proportio l d ad d u eſt maior proportione n d ad d u:</s> <s xml:id="echoid-s15988" xml:space="preserve"> ergo linea l d eſt maior <lb/>linea n d [per 10 p 5.</s> <s xml:id="echoid-s15989" xml:space="preserve">] Ergo n eſt inter l, u.</s> <s xml:id="echoid-s15990" xml:space="preserve"> Sed n eſt imago u:</s> <s xml:id="echoid-s15991" xml:space="preserve"> & duo puncta t, k ſunt imagines z, e.</s> <s xml:id="echoid-s15992" xml:space="preserve"> Er <lb/>go imago lineæ z u e rectæ, eſt linea tranſiens per puncta t n k:</s> <s xml:id="echoid-s15993" xml:space="preserve"> & linea, quæ tranſit per hæc puncta, <lb/>eſt conuexa.</s> <s xml:id="echoid-s15994" xml:space="preserve"> Ex quibus patet, quòd linea in ſpeculis concauis quandoque uidetur conuexa in <lb/>quibuſdam ſitibus.</s> <s xml:id="echoid-s15995" xml:space="preserve"/> </p> <div xml:id="echoid-div532" type="float" level="0" n="0"> <figure xlink:label="fig-0230-01" xlink:href="fig-0230-01a"> <variables xml:id="echoid-variables189" xml:space="preserve">k q t ſ n ſ g b o e u z d h a</variables> </figure> </div> </div> <div xml:id="echoid-div534" type="section" level="0" n="0"> <head xml:id="echoid-head466" xml:space="preserve" style="it">50. In ſpeculo ſphærico cauo imagines linearum: cauæ, conuexæ, aliquando uiden-<lb/>tur cauæ. 58 p 8.</head> <p> <s xml:id="echoid-s15996" xml:space="preserve">ITem:</s> <s xml:id="echoid-s15997" xml:space="preserve"> ponamus in linea z u punctum m, quocun que modo ſit:</s> <s xml:id="echoid-s15998" xml:space="preserve"> & circa centrum m, & in longitu-<lb/>dine m u faciamus arcum r u f.</s> <s xml:id="echoid-s15999" xml:space="preserve"> Iſte ergo arcus ſecabit arcum u o e in duobus punctis:</s> <s xml:id="echoid-s16000" xml:space="preserve"> [per 10 p <lb/>3] ſecet in r, f:</s> <s xml:id="echoid-s16001" xml:space="preserve"> & continuemus lineas d r, d f:</s> <s xml:id="echoid-s16002" xml:space="preserve"> & tranſeant rectè, quouſque concurrant in arcu <lb/>t q k, in p, i.</s> <s xml:id="echoid-s16003" xml:space="preserve"> Superficies ergo duarum linearum h d, d p faciet in ſpeculo circulum, à cuius circum-<lb/>ferentia reflectentur lineę ad r:</s> <s xml:id="echoid-s16004" xml:space="preserve"> & ſimiliter ſuperficies duarum linearum h d, d i faciet in ſpeculo cir-<lb/>culum, à cuius circumferentia reflectentur lineæ ad f.</s> <s xml:id="echoid-s16005" xml:space="preserve"> p ergo eſt imago r, & i eſt imago f:</s> <s xml:id="echoid-s16006" xml:space="preserve"> & n eſt ima <lb/>go u.</s> <s xml:id="echoid-s16007" xml:space="preserve"> Imago ergo arcus r u f, eſt linea tranſiens per i, p, n.</s> <s xml:id="echoid-s16008" xml:space="preserve"> Sed hęc linea erit concaua ex parte uiſus, <lb/>& arcus r u f eſt concauus ex parte ſuperficiei ſpeculi.</s> <s xml:id="echoid-s16009" xml:space="preserve"> Cum ergo uiſus fuerit in h, & unaquęque li-<lb/>nearum z u e, z o e, r u f fuerit in aliquo uiſibili:</s> <s xml:id="echoid-s16010" xml:space="preserve"> tunc linea z u e recta comprehendetur conuexa:</s> <s xml:id="echoid-s16011" xml:space="preserve"> & <lb/> <pb o="225" file="0231" n="231" rhead="OPTICAE LIBER VI."/> linea z o e conuexa, comprehendetur concaua:</s> <s xml:id="echoid-s16012" xml:space="preserve"> & r u f concaua:</s> <s xml:id="echoid-s16013" xml:space="preserve"> conuexa.</s> <s xml:id="echoid-s16014" xml:space="preserve"> Si ergo unaquęque linea <lb/>rum z u e, z o e, r u f habuerit unam imaginem:</s> <s xml:id="echoid-s16015" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0231-01a" xlink:href="fig-0231-01"/> tunc forma illarum linearum erit eodem modo, <lb/>quo declarauimus:</s> <s xml:id="echoid-s16016" xml:space="preserve"> & ſi habuerit alias imagines:</s> <s xml:id="echoid-s16017" xml:space="preserve"> <lb/>fortè erunt ſimiles alijs imaginibus, & fortè di-<lb/>uerſæ.</s> <s xml:id="echoid-s16018" xml:space="preserve"> Patet ergo ex iſtis figuris, quòd lineę re-<lb/>ctæ in ſpeculis concauis quandoque compre-<lb/>henduntur rectæ:</s> <s xml:id="echoid-s16019" xml:space="preserve"> quandoque conuexæ:</s> <s xml:id="echoid-s16020" xml:space="preserve"> quan-<lb/>doque concauæ:</s> <s xml:id="echoid-s16021" xml:space="preserve"> & lineæ conuexæ quandoque <lb/>comprehenduntur conuexæ:</s> <s xml:id="echoid-s16022" xml:space="preserve"> quandoque con-<lb/>cauę:</s> <s xml:id="echoid-s16023" xml:space="preserve"> & concauę quandoque comprehendun-<lb/>tur conuexę:</s> <s xml:id="echoid-s16024" xml:space="preserve"> quandoque concauę.</s> <s xml:id="echoid-s16025" xml:space="preserve"> Formę ergo <lb/>ſuperficierum uiſibilium comprehenduntur a-<lb/>liter, quàm ſunt, in huiuſmodi ſpeculis.</s> <s xml:id="echoid-s16026" xml:space="preserve"> Nam li-<lb/>neę rectę non ſunt, niſi in ſuperficiebus rectis:</s> <s xml:id="echoid-s16027" xml:space="preserve"> & <lb/>cum linea recta, quę exiſtit in ſuperficie plana, <lb/>comprehenditur conuexa aut concaua:</s> <s xml:id="echoid-s16028" xml:space="preserve"> tunc ſu-<lb/>perficies, in qua ipſa linea eſt, comprehendetur <lb/>conuexa aut concaua.</s> <s xml:id="echoid-s16029" xml:space="preserve"> Cum ergo uiſus compre-<lb/>hendat lineas conuexas & concauas, & rectas <lb/>aliter, quàm ſint:</s> <s xml:id="echoid-s16030" xml:space="preserve"> comprehendet ſuperficies, in <lb/>quibus ſunt, aliter, quàm ſint.</s> <s xml:id="echoid-s16031" xml:space="preserve"> Patet ergo ex prę <lb/>dictis, quòd in omnibus, quæ in ſpeculis con-<lb/>cauis comprehenduntur, accidit fallacia:</s> <s xml:id="echoid-s16032" xml:space="preserve"> ſed in <lb/>quibuſdam accidit ſemper, & in omni poſitio-<lb/>ne, in quibuſdam accidit in aliqua poſitione.</s> <s xml:id="echoid-s16033" xml:space="preserve"> Fal <lb/>lacię autem compoſitæ accidunt in his ſpeculis <lb/>eo modo, quo incompoſitæ.</s> <s xml:id="echoid-s16034" xml:space="preserve"> Et hoc uoluimus <lb/>declarare.</s> <s xml:id="echoid-s16035" xml:space="preserve"/> </p> <div xml:id="echoid-div534" type="float" level="0" n="0"> <figure xlink:label="fig-0231-01" xlink:href="fig-0231-01a"> <variables xml:id="echoid-variables190" xml:space="preserve">k q p <gap/> t ſ n g b o r f e u m z d h a</variables> </figure> </div> </div> <div xml:id="echoid-div536" type="section" level="0" n="0"> <head xml:id="echoid-head467" xml:space="preserve">DE ERRORIBVS, QVI ACCI-<lb/>dunt in ſpeculis columnaribus <lb/>concauis. Cap. VIII.</head> <p> <s xml:id="echoid-s16036" xml:space="preserve">IN his autem accidunt ſimiles eis, qui accidũt in ſphęricis concauis.</s> <s xml:id="echoid-s16037" xml:space="preserve"> Accidunt enim fallaciæ, quę <lb/>proueniunt ex reflexione, ſcilicet debilitas lucis & coloris:</s> <s xml:id="echoid-s16038" xml:space="preserve"> & diuerſitas ſitus, & remotionis, quę <lb/>accidunt omnibus ſpeculis.</s> <s xml:id="echoid-s16039" xml:space="preserve"> Accidit autem eis ex diuerſitate quantitatis ſimile illi, quod accidit <lb/>in ſpeculis ſphęricis concauis.</s> <s xml:id="echoid-s16040" xml:space="preserve"> Et uidetur etiam unum uiſibile, unum:</s> <s xml:id="echoid-s16041" xml:space="preserve"> & duo:</s> <s xml:id="echoid-s16042" xml:space="preserve"> & tria:</s> <s xml:id="echoid-s16043" xml:space="preserve"> & quatuor:</s> <s xml:id="echoid-s16044" xml:space="preserve"> & <lb/>rectum & conuexum ſecundum diuerſos ſitus:</s> <s xml:id="echoid-s16045" xml:space="preserve"> & planum uidetur concauum & conuexum.</s> <s xml:id="echoid-s16046" xml:space="preserve"> Oſten <lb/>demus ergo qualiter in his ſpeculis diuerſatur quantitas & numerus rei uiſæ:</s> <s xml:id="echoid-s16047" xml:space="preserve"> & qualiter apparet re <lb/>ctum & conuerſum eo modo, quo in ſpeculis ſphęricis concauis declarauimus.</s> <s xml:id="echoid-s16048" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div537" type="section" level="0" n="0"> <head xml:id="echoid-head468" xml:space="preserve" style="it">51. Siuiſ{us} ſit extra planũ lineærectæ, parallelæ axi ſpeculi cylindraceicaui: imago aliàs ui-<lb/>debitur recta & maior ipſa linea: aliâs caua: aliâs cõuexa: aliâs ſimplex: aliâs multiplex. 25 p 9.</head> <p> <s xml:id="echoid-s16049" xml:space="preserve">ITeremus ergo primam figuram ex duabus figuris pręmiſsis in fallacijs ſpeculorum columnariũ <lb/>conuexorum, & ijſdem literis.</s> <s xml:id="echoid-s16050" xml:space="preserve"> In illa autem figura [quę eſt 26 n] patuit, quòd lineę e g, g t, e b, q b, <lb/>e a, a h reflectuntur ſecundum angulos æquales:</s> <s xml:id="echoid-s16051" xml:space="preserve"> & quòd lineę e k, h a, q b, t g coniunguntur in o:</s> <s xml:id="echoid-s16052" xml:space="preserve"> <lb/>& quòd linea a b g eſt linea recta extenſa in longitudine ſpeculi:</s> <s xml:id="echoid-s16053" xml:space="preserve"> & quòd lineę g z, b l, a d ſunt perpẽ-<lb/>diculares ſuper ſuperficiẽ, contingentẽ ſuperficiem, quæ trãſit per lineã a b g:</s> <s xml:id="echoid-s16054" xml:space="preserve"> & quòd linea a b g eſt <lb/>perpẽdicularis ſuք ſuperficiẽ, in qua eſt triãgulũ e b o:</s> <s xml:id="echoid-s16055" xml:space="preserve"> & quòd linea t q eſt æqualis q h, & a b ęqualis <lb/>b g:</s> <s xml:id="echoid-s16056" xml:space="preserve"> & quòd s c, i ſunt imagines h, q, t:</s> <s xml:id="echoid-s16057" xml:space="preserve"> & quòd c eſt propinquius puncto e, quàm linea s i:</s> <s xml:id="echoid-s16058" xml:space="preserve"> & quòd li-<lb/>nea s i eſt in ſuperficie trianguli u h t:</s> <s xml:id="echoid-s16059" xml:space="preserve"> & quòd duæ lineæ u h, u t ſunt æquales:</s> <s xml:id="echoid-s16060" xml:space="preserve"> & quòd u s & u i ſunt <lb/>æquales:</s> <s xml:id="echoid-s16061" xml:space="preserve"> & quòd duæ lineæ e s, e i ſunt æquales.</s> <s xml:id="echoid-s16062" xml:space="preserve"> Et continuemus c u:</s> <s xml:id="echoid-s16063" xml:space="preserve"> & ſecet s i in æ:</s> <s xml:id="echoid-s16064" xml:space="preserve"> diuidet ergo i-<lb/>pſam in duo æqualia:</s> <s xml:id="echoid-s16065" xml:space="preserve"> nam h t eſt diuiſa in duo æqualia in q:</s> <s xml:id="echoid-s16066" xml:space="preserve"> [& linea i s parallela eſt ipſi t h:</s> <s xml:id="echoid-s16067" xml:space="preserve"> quia cũ <lb/>tota t u æqualis concluſa ſit toti h u, & pars i u parti s u:</s> <s xml:id="echoid-s16068" xml:space="preserve"> erit reliqua t i æqualis reliquæ h s:</s> <s xml:id="echoid-s16069" xml:space="preserve"> eſt igitur <lb/>per 7 p 5, ut u i ad i t, ſic u s ad s h:</s> <s xml:id="echoid-s16070" xml:space="preserve"> ergo per 2 p 6 h t & s i ſunt parallelæ.</s> <s xml:id="echoid-s16071" xml:space="preserve"> Itaque triangula t u q, i u æ:</s> <s xml:id="echoid-s16072" xml:space="preserve"> i-<lb/>tem q u h, æ u s ſunt æquiãgula per 29 p 1:</s> <s xml:id="echoid-s16073" xml:space="preserve"> & per 4 p 6, ut t q ad q u, ſic i æ ad æ u:</s> <s xml:id="echoid-s16074" xml:space="preserve"> & ut q u ad q h, ſic ę u <lb/>ad ę s:</s> <s xml:id="echoid-s16075" xml:space="preserve"> ergo per 22 p 5, ut t q ad q h, ſic i æ ad æ s.</s> <s xml:id="echoid-s16076" xml:space="preserve"> Quare cũ 26.</s> <s xml:id="echoid-s16077" xml:space="preserve"> 27 n, t q ęquata ſit ipſi q h:</s> <s xml:id="echoid-s16078" xml:space="preserve"> ęquabitur i æ <lb/>ipſi ę s] & erit c u in ſuperficie trianguli q u e, quæ eſt ſuperficies circuli b f, ęquidiſtantis baſi ſpecu-<lb/>li:</s> <s xml:id="echoid-s16079" xml:space="preserve"> ergo c erit in ſuperficie trianguli c u e:</s> <s xml:id="echoid-s16080" xml:space="preserve"> & eſt in ſuperficie trianguli c e i:</s> <s xml:id="echoid-s16081" xml:space="preserve"> ergo c eſt in linea, quæ eſt <lb/>differentia cõmunis his duabus ſuperficieb.</s> <s xml:id="echoid-s16082" xml:space="preserve"> ſed hęc differẽtia eſt linea e b:</s> <s xml:id="echoid-s16083" xml:space="preserve"> [ք 3 p 11] ergo c eſt in recti <lb/>tudine e b:</s> <s xml:id="echoid-s16084" xml:space="preserve"> & duę lineę h u, t u ſunt ſub duob.</s> <s xml:id="echoid-s16085" xml:space="preserve"> pũctis d, z:</s> <s xml:id="echoid-s16086" xml:space="preserve"> nã duę lineę h u, t u ſunt perpẽdiculares exe <lb/>untes ex h, t ſuper duas lineas, cõtingẽtes duas portiones, in quarũ circuferẽtia ſunt puncta a, g.</s> <s xml:id="echoid-s16087" xml:space="preserve"> Su-<lb/>perficies ergo triãguli u h t eſt ſub axe d l z.</s> <s xml:id="echoid-s16088" xml:space="preserve"> Sed nullũ pũctũ huius axis, quãuis exeat in infinitũ, erit <lb/>in ſuperficie trianguli u h t.</s> <s xml:id="echoid-s16089" xml:space="preserve"> Nam ſi eſſet:</s> <s xml:id="echoid-s16090" xml:space="preserve"> tunc ſi continuaretur cũ aliquo puncto lineæ h t linea re-<lb/> <pb o="226" file="0232" n="232" rhead="ALHAZEN"/> cta:</s> <s xml:id="echoid-s16091" xml:space="preserve"> tuncilla ſuperficies, in qua eſſet illa linea recta & linea h t eſſet ſuperficies trianguli u h t:</s> <s xml:id="echoid-s16092" xml:space="preserve"> <lb/>& illa ſuperficies eſſet illa, in qua ſunt duæ lineæ æquidiſtantes h t, d z:</s> <s xml:id="echoid-s16093" xml:space="preserve"> & ſic ſuperficies, in qua ſunt <lb/>duę lineæ h t, d z, eſſet ſuperficies trianguli h u t:</s> <s xml:id="echoid-s16094" xml:space="preserve"> & ſic axis eſſet in ſuperficie trianguli h u t:</s> <s xml:id="echoid-s16095" xml:space="preserve"> ſed axis <lb/>eſt æquidiſtans lineæ h t poſitione.</s> <s xml:id="echoid-s16096" xml:space="preserve"> Et axis ſecat duas lineas h u, t u:</s> <s xml:id="echoid-s16097" xml:space="preserve"> & linea t h eſt in ſuperficie trian <lb/>guli u e h, quæ eſt ſuperficies reflexionis:</s> <s xml:id="echoid-s16098" xml:space="preserve"> & linea communis huic ſuperficiei & ſuperficiei columnę, <lb/>eſt aliqua ſectio columnaris.</s> <s xml:id="echoid-s16099" xml:space="preserve"> Superficies ergo e u h ſecat axem columnæ in uno puncto, ſcilicet in d, <lb/>ut præoſten-<lb/> <anchor type="figure" xlink:label="fig-0232-01a" xlink:href="fig-0232-01"/> dimus [27 n.</s> <s xml:id="echoid-s16100" xml:space="preserve">] <lb/>Et ſi axis ſe-<lb/>cet lineá h u:</s> <s xml:id="echoid-s16101" xml:space="preserve"> <lb/>punctum ſe-<lb/>ctionis cum <lb/>linea h u erit <lb/>in ſuperficie <lb/>trianguli u e <lb/>h:</s> <s xml:id="echoid-s16102" xml:space="preserve"> ſed in hac <lb/>ſuperficie nõ <lb/>eſt punctum, <lb/>per quod a-<lb/>xis tranſeat, <lb/>præter d:</s> <s xml:id="echoid-s16103" xml:space="preserve"> er-<lb/>go linea h u <lb/>iecat axem in d:</s> <s xml:id="echoid-s16104" xml:space="preserve"> & iam oſtendimus [24 n] quòd h u ſecat eum in puncto ſub d:</s> <s xml:id="echoid-s16105" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s16106" xml:space="preserve"> <lb/>Ergo axis d z eſt extra ſuperficiem u h t, & propinquior puncto e, quàm ſuperficies h u t.</s> <s xml:id="echoid-s16107" xml:space="preserve"> Superfi-<lb/>cies ergo, in qua ſunt lineæ h t, d z, eſt propinquior puncto e, quàm ſuperficies u h t:</s> <s xml:id="echoid-s16108" xml:space="preserve"> & c eſt in ſuper-<lb/>ficie, in qua ſunt h t, d z:</s> <s xml:id="echoid-s16109" xml:space="preserve"> quia eſt in linea q l:</s> <s xml:id="echoid-s16110" xml:space="preserve"> & q l eſt in ſuperficie, in qua ſunt h t, d z:</s> <s xml:id="echoid-s16111" xml:space="preserve"> [per 7 p 11] er-<lb/>go c eſt propinquius e, quàm s i:</s> <s xml:id="echoid-s16112" xml:space="preserve"> ſed c eſt in rectitudine e b [ut patuit.</s> <s xml:id="echoid-s16113" xml:space="preserve">] Si ergo e b exiue-<lb/>rit in parte b:</s> <s xml:id="echoid-s16114" xml:space="preserve"> perueniet ad c:</s> <s xml:id="echoid-s16115" xml:space="preserve"> perueniet ergo ad c.</s> <s xml:id="echoid-s16116" xml:space="preserve"> His præoſtenſis, dico quòd linea s i, quæ eſt æ-<lb/>quidiſtans axi ſpeculi, cum fuerit in aliquo uiſibili, & uiſus fuerit in o ex parte concauitatis co-<lb/>lumnæ, & ſuperficies ſpeculata fuerit ſuperficies concaua:</s> <s xml:id="echoid-s16117" xml:space="preserve"> tunc s i comprehendetur ex o m ſpeculo <lb/>concauo a b g à linea a b g:</s> <s xml:id="echoid-s16118" xml:space="preserve"> & diuerſabuntur imagines eius ſecundum diuerſitatem diſtãtiæ ab axe, <lb/>cuius demonſtratio eſt.</s> <s xml:id="echoid-s16119" xml:space="preserve"> Quia angulus e b m eſt acutus [quia m b a eſt rectus ex theſi 26 n] ergo [per <lb/>15 p 1] l b c eſt acutus:</s> <s xml:id="echoid-s16120" xml:space="preserve"> & linea e b c eſt in ſuperficie circuli b f:</s> <s xml:id="echoid-s16121" xml:space="preserve"> & l b eſt diameter huius circuli [per 34 <lb/>n 4.</s> <s xml:id="echoid-s16122" xml:space="preserve">] Ergo e b c ſecat circulum:</s> <s xml:id="echoid-s16123" xml:space="preserve"> ergo c b eſtintra concauitatem ſpeculi:</s> <s xml:id="echoid-s16124" xml:space="preserve"> & ſimiliter o b erit intra cõ-<lb/>cauitatem ſpeculi:</s> <s xml:id="echoid-s16125" xml:space="preserve"> quia angulus o b l eſt acutus, & duo anguli o b l, c b l ſunt æquales duobus angu <lb/>lis e b m, q b m:</s> <s xml:id="echoid-s16126" xml:space="preserve"> [quia per 15 p 1 æquantur angulis e b m, q b m, ęqualibus concluſis 27 n] & l b eſt <lb/>perpendicularis ſuper ſuperficiem, contingentem columnam, quæ tranſit per b.</s> <s xml:id="echoid-s16127" xml:space="preserve"> Forma ergo c <lb/>extenditur per c b, & peruenit ad b, & reflectitur per b o, & comprehenditur à uiſu in o [per 7 n <lb/>5.</s> <s xml:id="echoid-s16128" xml:space="preserve">] Item in quinto capitulo [27 n] cum fuimus locuti de ſpeculis columnaribus conuexis, decla-<lb/>rauimus, quod ſuperficies contingens columnam m g, erit ſub e:</s> <s xml:id="echoid-s16129" xml:space="preserve"> ergo e g ſecat ſuperficiem contin-<lb/>gentem:</s> <s xml:id="echoid-s16130" xml:space="preserve"> ſecat ergo lineam contingentem circum ferentiam ſectionis in g:</s> <s xml:id="echoid-s16131" xml:space="preserve"> ſecat ergo ſectionem, & <lb/>cadit intra ipſam:</s> <s xml:id="echoid-s16132" xml:space="preserve"> cadet ergo intra concauitatẽ ſpeculi:</s> <s xml:id="echoid-s16133" xml:space="preserve"> ergo duæ lineæ o g, g i ſunt intra concauita-<lb/>tem ſpeculi:</s> <s xml:id="echoid-s16134" xml:space="preserve"> & z g eſt perpendicularis ſuper ſuperficiem, contingentcm columnam in g [quia ex <lb/>theſi perpendicularis eſt a g lateri cylindraceo:</s> <s xml:id="echoid-s16135" xml:space="preserve"> & duo anguli o g z, i g z ſunt æquales:</s> <s xml:id="echoid-s16136" xml:space="preserve"> quia per 15 p 1 <lb/>æquantur angulis e g n, t g n, æqualibus per 4 p 1.</s> <s xml:id="echoid-s16137" xml:space="preserve">] Ergo forma i extenditur per i g, & peruenit ad <lb/>g, & reflectitur per g o, & comprehenditur in o per lineam g o.</s> <s xml:id="echoid-s16138" xml:space="preserve"> Et ſimiliter s extenditur per s a, & <lb/>peruenit ad a, & reflectitur per a o, & comprehenditur in o.</s> <s xml:id="echoid-s16139" xml:space="preserve"> Et iam declarauimus, cum tractaui-<lb/>mus de fallacijs ſpeculorum columnarium conuexorum [27 n] quòd duæ lineæ h u, t u ſunt per-<lb/>pendiculares ſuper ſuperficies, contingentes ſectiones, tranſeuntes per duo puncta a, g.</s> <s xml:id="echoid-s16140" xml:space="preserve"> Imago er-<lb/>go s eſt in linea h u, & a o linea radialis, quæ extenditur ex uiſu ad punctum reflexionis:</s> <s xml:id="echoid-s16141" xml:space="preserve"> ergo ima-<lb/>go s eſt in a o:</s> <s xml:id="echoid-s16142" xml:space="preserve"> h ergo eſt imago s:</s> <s xml:id="echoid-s16143" xml:space="preserve"> [per 7 n 5] & ſic patet, quòd t eſt imago i.</s> <s xml:id="echoid-s16144" xml:space="preserve"> Et continuemus c l.</s> <s xml:id="echoid-s16145" xml:space="preserve"> <lb/>Quiaergo c reflectitur ad o ex circumferentiæ puncto b:</s> <s xml:id="echoid-s16146" xml:space="preserve"> erit imago c in line a cl:</s> <s xml:id="echoid-s16147" xml:space="preserve"> & o b eſt linea ra-<lb/>dialis, quæ extenditur inter uiſum & punctum reflexionis.</s> <s xml:id="echoid-s16148" xml:space="preserve"> Ergo imago c eſt in puncto communi <lb/>c l & o b [per 7 n 5] nempe in puncto q.</s> <s xml:id="echoid-s16149" xml:space="preserve"> Sed in capitulo de imagine, cum tractauimus de imagini-<lb/>bus ſpeculorum ſphæricorum concauorum [60 n 5] patuit, quòd imago puncti, cuius forma refle-<lb/>ctitur à concauitate circuli, fortè concurret cum radiali linea, quæ eſt inter uiſum & punctum refle-<lb/>xionis, ultra ſpeculum:</s> <s xml:id="echoid-s16150" xml:space="preserve"> & fortè inter uiſum & ſpeculum:</s> <s xml:id="echoid-s16151" xml:space="preserve"> & fortè in centro uiſus:</s> <s xml:id="echoid-s16152" xml:space="preserve"> & fortè ultra cen-<lb/>trum uiſus:</s> <s xml:id="echoid-s16153" xml:space="preserve"> & fortè c l æquidiſtans erit o b.</s> <s xml:id="echoid-s16154" xml:space="preserve"> Et in illo capitulo [86 n 5] patuit, quòd fortè imago <lb/>erit unum punctum:</s> <s xml:id="echoid-s16155" xml:space="preserve"> aut duo:</s> <s xml:id="echoid-s16156" xml:space="preserve"> aut tria:</s> <s xml:id="echoid-s16157" xml:space="preserve"> aut quatuor.</s> <s xml:id="echoid-s16158" xml:space="preserve"> Imago ergo fortè erit in b q:</s> <s xml:id="echoid-s16159" xml:space="preserve"> fortè ultra o q:</s> <s xml:id="echoid-s16160" xml:space="preserve"> & <lb/>fortè in b o:</s> <s xml:id="echoid-s16161" xml:space="preserve"> & fortè in o:</s> <s xml:id="echoid-s16162" xml:space="preserve"> & fortè ultra:</s> <s xml:id="echoid-s16163" xml:space="preserve"> & fortè imago t q erit unum punctum:</s> <s xml:id="echoid-s16164" xml:space="preserve"> aut duo:</s> <s xml:id="echoid-s16165" xml:space="preserve"> aut tria:</s> <s xml:id="echoid-s16166" xml:space="preserve"> aut <lb/>quatuor.</s> <s xml:id="echoid-s16167" xml:space="preserve"> Si ergo imago c fuerit q:</s> <s xml:id="echoid-s16168" xml:space="preserve"> tũc h q t erit diameter imaginis s i.</s> <s xml:id="echoid-s16169" xml:space="preserve"> Si ergo omnes imagines s i fue <lb/>rint in linea h q t:</s> <s xml:id="echoid-s16170" xml:space="preserve"> tunc forma eius erit linea recta:</s> <s xml:id="echoid-s16171" xml:space="preserve"> nã mediũ eius eſt in rectitudine duarũ extremita-<lb/>tũ h t.</s> <s xml:id="echoid-s16172" xml:space="preserve"> Si aũt imago c fueritultra q:</s> <s xml:id="echoid-s16173" xml:space="preserve"> tunc imago s i erit ferè cõcaua ex parte uiſus.</s> <s xml:id="echoid-s16174" xml:space="preserve"> Et ſi imago c fuerint <lb/>plura puncta:</s> <s xml:id="echoid-s16175" xml:space="preserve"> tunc imago cerunt plures lineæ, quarum omnium extremitates cõiungentur in duo-<lb/> <pb o="227" file="0233" n="233" rhead="OPTICAE LIBER VI."/> bus punctis h, t:</s> <s xml:id="echoid-s16176" xml:space="preserve"> & media earum erunt diſtincta & ſeparata:</s> <s xml:id="echoid-s16177" xml:space="preserve"> & h t eſt diameter imaginis s i, quocun-<lb/>que modo fuerit imago:</s> <s xml:id="echoid-s16178" xml:space="preserve"> & diameter eſt cõmunis omnibus imaginibus eius, ſi plures habuerit ima-<lb/>gines:</s> <s xml:id="echoid-s16179" xml:space="preserve"> & linea h t eſt maior, quàm si, modica quantitate.</s> <s xml:id="echoid-s16180" xml:space="preserve"> Patet ergo, quòd cum lineæ rectæ, æquidi-<lb/>ſtantes axi columnaris ſpeculi concaui fuerint in aliquo uiſibili:</s> <s xml:id="echoid-s16181" xml:space="preserve"> imago earum fortè erit recta aut <lb/>concaua, & fortè una, aut plures.</s> <s xml:id="echoid-s16182" xml:space="preserve"/> </p> <div xml:id="echoid-div537" type="float" level="0" n="0"> <figure xlink:label="fig-0232-01" xlink:href="fig-0232-01a"> <variables xml:id="echoid-variables191" xml:space="preserve">t i n g y z x q m b c œ <gap/> f h z r a d p e K o <gap/></variables> </figure> </div> </div> <div xml:id="echoid-div539" type="section" level="0" n="0"> <head xml:id="echoid-head469" xml:space="preserve" style="it">52. Si uiſ{us} à terminis lineæ rectæ æquabiliter diſtans, ſit extra ipſi{us} planum, perpendicula <lb/>re plano axis ſpeculi cylindr acei caui: imago uidebitur maximè caua. 27 p 9.</head> <p> <s xml:id="echoid-s16183" xml:space="preserve">ITem:</s> <s xml:id="echoid-s16184" xml:space="preserve"> iteremus ſecundam figuram de fallacijs ſpeculorum columnarium conuexorum [29 n.</s> <s xml:id="echoid-s16185" xml:space="preserve">] <lb/>In hac autem figura dictum eſt:</s> <s xml:id="echoid-s16186" xml:space="preserve"> quòd duæ lineæ e b, h b reflectuntur ſecũdum angulos æquales:</s> <s xml:id="echoid-s16187" xml:space="preserve"> <lb/>& quòd duæ lineæ e g, t g reflectuntur ſecundum angulos æquales:</s> <s xml:id="echoid-s16188" xml:space="preserve"> & quòd h b, t g perueniunt <lb/>a d l:</s> <s xml:id="echoid-s16189" xml:space="preserve"> & h b continet cum b o angulum acutum.</s> <s xml:id="echoid-s16190" xml:space="preserve"> Ergo h b ſecat ſuperficiem, contingentem columnam <lb/>in b:</s> <s xml:id="echoid-s16191" xml:space="preserve"> b l ergo eſt ſub concauitate columnæ:</s> <s xml:id="echoid-s16192" xml:space="preserve"> & ſimiliter g l:</s> <s xml:id="echoid-s16193" xml:space="preserve"> & ſimiliter duæ lineæ b r, g y:</s> <s xml:id="echoid-s16194" xml:space="preserve"> & duo angu-<lb/>li l b d, d b r ſunt æquales [quia per 15 p 1 æquantur angulis e b o, h b o æqualibus] & ſimiliter l g d, g <lb/>d y ſunt æquales.</s> <s xml:id="echoid-s16195" xml:space="preserve"> Si ergo r y fuerit in aliquo uiſibili, & uiſus fuerit in l, & ſuperficies concaua colu-<lb/>mnæ fuerit terſa:</s> <s xml:id="echoid-s16196" xml:space="preserve"> tunc forma r extenditur per r b, & peruenit ad b, & reflectitur ք b l:</s> <s xml:id="echoid-s16197" xml:space="preserve"> & perueniet ad <lb/>l, & comprehendetur in l.</s> <s xml:id="echoid-s16198" xml:space="preserve"> Et linea h u eſt perpendicularis ſuper lineam, contingentem ſectionem, <lb/>ex cuius circumferentia reflectentur duæ lineæ b r, b l:</s> <s xml:id="echoid-s16199" xml:space="preserve"> h ergo eſt imago r [per 7 n 5.</s> <s xml:id="echoid-s16200" xml:space="preserve">] Similiter decla <lb/>rabitur, quòd forma y extenditur per y g, & reflectitur ք g l:</s> <s xml:id="echoid-s16201" xml:space="preserve"> & imago eius eſt t.</s> <s xml:id="echoid-s16202" xml:space="preserve"> Et continuemus q u:</s> <s xml:id="echoid-s16203" xml:space="preserve"> <lb/>ſecabit ergo r y in m:</s> <s xml:id="echoid-s16204" xml:space="preserve"> m ergo eſt in ſuperficie tranſeunte per axem & per l:</s> <s xml:id="echoid-s16205" xml:space="preserve"> nam l & q ſunt in hac ſu-<lb/>perficie, [ut demonſtratum eſt 29 n.</s> <s xml:id="echoid-s16206" xml:space="preserve">] Ergo q u eſt in hac ſuperficie [nam 29 n oſten ſum eſt, quòd <lb/>planum ductum per uiſum & axem ſpeculi, in quo eſt linea e l d, ſecat lineam h t in puncto q:</s> <s xml:id="echoid-s16207" xml:space="preserve"> eſtq́ue <lb/>punctum u in linea e l d:</s> <s xml:id="echoid-s16208" xml:space="preserve"> linea igitur q u eſt in plano per uiſum & axem ſpeculi ducto per 1 p 11:</s> <s xml:id="echoid-s16209" xml:space="preserve"> ideõ-<lb/>que & punctum m.</s> <s xml:id="echoid-s16210" xml:space="preserve">] Et quia duo puncta m, l ſunt in ſuperficie tranſeunte per axem columnæ:</s> <s xml:id="echoid-s16211" xml:space="preserve"> ideo <lb/>forma m reflectetur ad l in hac ſuperficie.</s> <s xml:id="echoid-s16212" xml:space="preserve"> Et quia a z eſt differentia communis inter columnę ſuper-<lb/>ficiem, & ſuperficiem, tranſeuntem per ſuum axem, & per l:</s> <s xml:id="echoid-s16213" xml:space="preserve"> forma ergo m reflectetur à linea a z.</s> <s xml:id="echoid-s16214" xml:space="preserve"> Et <lb/>continuemus e m, quæ eſt in hac ſuperficie:</s> <s xml:id="echoid-s16215" xml:space="preserve"> & e l <lb/> <anchor type="figure" xlink:label="fig-0233-01a" xlink:href="fig-0233-01"/> eſt in hac ſuperficie:</s> <s xml:id="echoid-s16216" xml:space="preserve"> & punctum e eſt elongatum à <lb/>ſuperficie contingente ſuperficiem columnæ in li-<lb/>nea a z [ut patuit 29 n.</s> <s xml:id="echoid-s16217" xml:space="preserve">] Ergo ſi a z extrahatur re-<lb/>ctè in parte z:</s> <s xml:id="echoid-s16218" xml:space="preserve"> concurret cum duabus lineis e m, <lb/>e l.</s> <s xml:id="echoid-s16219" xml:space="preserve"> Concurrat ergo cum e m in i, & cum e l in n:</s> <s xml:id="echoid-s16220" xml:space="preserve"> <lb/>ergo n eſt inter duo puncta e, l:</s> <s xml:id="echoid-s16221" xml:space="preserve"> quia l eſt intra con <lb/>cauitatem columnæ, & n eſt in ſuperficie colu-<lb/>mnæ:</s> <s xml:id="echoid-s16222" xml:space="preserve"> & e eſt elõgatum à columna:</s> <s xml:id="echoid-s16223" xml:space="preserve"> & in dem on-<lb/>ſtratione huius figuræ [29 n] patuit, quòd circu-<lb/>lus b g eſt medius inter lineam h t, & ſuperficiem <lb/>exeuntem ex e, æ quidiſtantem baſibus columnæ:</s> <s xml:id="echoid-s16224" xml:space="preserve"> <lb/>& perpendicularis, quæ exit ex e ſuper a z, eſt in ſu <lb/>perficie exeunte ex e, æ quidiſtante columnæ.</s> <s xml:id="echoid-s16225" xml:space="preserve"> Er-<lb/>go perpendicularis, quæ exit ex e ſuper lineam a <lb/>z n, cadit extra triangulum e i n, & in parte n:</s> <s xml:id="echoid-s16226" xml:space="preserve"> angu <lb/>lus ergo e i n eſt acutus:</s> <s xml:id="echoid-s16227" xml:space="preserve"> [per 32 p 1] ergo [per 15 p <lb/>1] angulus m i a eſt acutus:</s> <s xml:id="echoid-s16228" xml:space="preserve"> ergo m i n obtuſus [per <lb/>13 p 1.</s> <s xml:id="echoid-s16229" xml:space="preserve">] Extrahamus ergo ex m perpendicularem <lb/>ſuper a i [per 12 p 1] & ſit m k:</s> <s xml:id="echoid-s16230" xml:space="preserve"> k ergo erit ultra i, <lb/>reſpectu 11.</s> <s xml:id="echoid-s16231" xml:space="preserve"> [ſi enim caderet inter i & n:</s> <s xml:id="echoid-s16232" xml:space="preserve"> eſſent triã-<lb/>guli tres anguli maiores duobus rectis contra 32 <lb/>p 1:</s> <s xml:id="echoid-s16233" xml:space="preserve"> quia angulus m i n obtuſus eſt concluſus.</s> <s xml:id="echoid-s16234" xml:space="preserve">] Et <lb/>extrahamus m k ex parte k, in s:</s> <s xml:id="echoid-s16235" xml:space="preserve"> & diuidamus k s <lb/>ad æqualitatem k m:</s> <s xml:id="echoid-s16236" xml:space="preserve"> ergo s erit extra ſuperficiem <lb/>ſpeculi, & ultra concauitatem eius, & l erit ſub concauitate eius.</s> <s xml:id="echoid-s16237" xml:space="preserve"> Et continuemus l s:</s> <s xml:id="echoid-s16238" xml:space="preserve"> ſecabit ergo <lb/>n k in f:</s> <s xml:id="echoid-s16239" xml:space="preserve"> & ex f extrahamus f x ad æquidiſtantiam m k.</s> <s xml:id="echoid-s16240" xml:space="preserve"> Cum ergo [per 29 p 1] f x ſit perpendicularis <lb/>ſuper a n, & in ſuperficie tranſeunte per axem & per l:</s> <s xml:id="echoid-s16241" xml:space="preserve"> ergo eſt diameter circuli exeuntis ex f & æ-<lb/>quidiſtantis baſi columnæ [per 34 n 4.</s> <s xml:id="echoid-s16242" xml:space="preserve">] Linea ergo f x eſt perpendicularis ſuper ſuperficiem, con-<lb/>tingentem columnam, tranſeuntem per a z [ſicut oſtenſum eſt 54 n 5.</s> <s xml:id="echoid-s16243" xml:space="preserve">] Et continuemus m f:</s> <s xml:id="echoid-s16244" xml:space="preserve"> erit <lb/>ergo æqualis f s:</s> <s xml:id="echoid-s16245" xml:space="preserve"> [per 4 p 1:</s> <s xml:id="echoid-s16246" xml:space="preserve"> quia k s, k m æquantur per fabricationem, & communis eſt k f, anguli-<lb/>que ad k recti] & duo anguli qui ſunt, apud m, s erunt æquales:</s> <s xml:id="echoid-s16247" xml:space="preserve"> [per 5 p 1.</s> <s xml:id="echoid-s16248" xml:space="preserve">] Et quia x f eſt æquidi-<lb/>ſtans m g:</s> <s xml:id="echoid-s16249" xml:space="preserve"> erunt [per 29 p 1] duo anguli apud f æquales duobus angulis, qui ſunt apud s, m [ideó-<lb/>que anguli x f m & x f l æquabuntur.</s> <s xml:id="echoid-s16250" xml:space="preserve">] Duæ ergo lineę m f, f l reflectuntur ſecundum angu-<lb/>los æquales:</s> <s xml:id="echoid-s16251" xml:space="preserve"> & x f eſt perpendicularis ſuper ſuperficiem, contingentem ſpeculum in f.</s> <s xml:id="echoid-s16252" xml:space="preserve"> For-<lb/>ma ergo m extenditur per m f, & reflectitur per f l:</s> <s xml:id="echoid-s16253" xml:space="preserve"> & imago eius erit s [per 7 n 5.</s> <s xml:id="echoid-s16254" xml:space="preserve">] Et quia <lb/>duæ lineæ r y, h t ſunt æquidiſtantes, & perpendiculares ſuper ſuperficiem tranſeuntem per <lb/>axem, & per l:</s> <s xml:id="echoid-s16255" xml:space="preserve"> quia h t fuit poſita talis:</s> <s xml:id="echoid-s16256" xml:space="preserve"> [29 n] ideo duæ ſuperficies exeuntes à duabus li-<lb/> <pb o="228" file="0234" n="234" rhead="ALHAZEN"/> neis h t, r y, erunt æquidiſtantes & perpendiculares [per 18 p 11.</s> <s xml:id="echoid-s16257" xml:space="preserve">] Et quia r y eſt perpendicularis ſu-<lb/>per ſuperficiem tranſeuntem per axem & per l:</s> <s xml:id="echoid-s16258" xml:space="preserve"> ideo [per 18 p 11] ſuperficies duarum linearum rm, <lb/>m s erit perpendicularis ſuper ſuperficiem, tranſeuntem per axem & per l:</s> <s xml:id="echoid-s16259" xml:space="preserve"> & erit m s differentia cõ-<lb/>munis his duabus ſuperficiebus.</s> <s xml:id="echoid-s16260" xml:space="preserve"> Et quia a k eſt in ſuperficie tranſeunte per axem:</s> <s xml:id="echoid-s16261" xml:space="preserve"> [per 21 d 11:</s> <s xml:id="echoid-s16262" xml:space="preserve"> quia <lb/>pars eſt lateris cylindracei] & eſt perpendicularis ſuper m s [per fabricationem] quę eſt differentia <lb/>communis inter ſuperficiem, trãſeuntem per axem, & inter ſuperficiem duarum linearum r m, m s:</s> <s xml:id="echoid-s16263" xml:space="preserve"> <lb/>erit a k n perpendicularis ſuper ſuperficiem duarum linearum r m, m s:</s> <s xml:id="echoid-s16264" xml:space="preserve"> & linea a n eſt æquidiſtans <lb/>axi columnæ [per 21 d 11:</s> <s xml:id="echoid-s16265" xml:space="preserve">] ergo [per 8 p 11] axis columnæ eſt perpendicularis ſuper ſuperficiem, in <lb/>qua ſunt r m, m s.</s> <s xml:id="echoid-s16266" xml:space="preserve"> Superficies ergo iſta eſt perpendicularis ſuper axem columnæ:</s> <s xml:id="echoid-s16267" xml:space="preserve"> s ergo eſt in ſuper-<lb/>ficie exeunte ex linea r y, perpendiculari ſuper axem columnæ:</s> <s xml:id="echoid-s16268" xml:space="preserve"> ſed linea h t eſt in ſuperficie perpen-<lb/>diculari ſuper axem, æquidiſtante ſuperficiei ex linea r y:</s> <s xml:id="echoid-s16269" xml:space="preserve"> s ergo eſt extra h t, & propinquius l, quàm <lb/>ſint h & t:</s> <s xml:id="echoid-s16270" xml:space="preserve"> & duo puncta h, t ſunt imagines r, y:</s> <s xml:id="echoid-s16271" xml:space="preserve"> & punctum s eſt imago m:</s> <s xml:id="echoid-s16272" xml:space="preserve"> imago ergo lineæ r m y eſt <lb/>linea tranſiens per h, s, t:</s> <s xml:id="echoid-s16273" xml:space="preserve"> ſed talis eſt linea arcualis:</s> <s xml:id="echoid-s16274" xml:space="preserve"> quia s eſt extra h t.</s> <s xml:id="echoid-s16275" xml:space="preserve"> Et tranſeat per puncta h, s, t li-<lb/>nea h s t arcualis.</s> <s xml:id="echoid-s16276" xml:space="preserve"> Et quia h t ſecundum poſitionem [29 n] fuit elongata à conuexo columnæ:</s> <s xml:id="echoid-s16277" xml:space="preserve"> erit <lb/>h t ultra ſuperficiem ſpeculi, reſpectu l:</s> <s xml:id="echoid-s16278" xml:space="preserve"> & iam declarauimus, quòd s eſt ultra concauitatem ſpeculi, <lb/>reſpectul.</s> <s xml:id="echoid-s16279" xml:space="preserve"> Ergo tota linea h s t erit ultra concauitatem ſuperficiei ſpeculi:</s> <s xml:id="echoid-s16280" xml:space="preserve"> & e l eſt ſub concauitate <lb/>ſpeculi:</s> <s xml:id="echoid-s16281" xml:space="preserve"> ergo l eſt extra ſuperficiẽ, in qua eſt linea h s t:</s> <s xml:id="echoid-s16282" xml:space="preserve"> arcualitas igitur lineæ h s t apparebit uiſuil <lb/>manifeſtè.</s> <s xml:id="echoid-s16283" xml:space="preserve"> Et quia f eſt in ſuperficie columnæ, & t h ultra columnam, eſt in ſuperficie trianguli l h t:</s> <s xml:id="echoid-s16284" xml:space="preserve"> <lb/>erit linea l f s altior quàm ſuperficies trianguli l h t.</s> <s xml:id="echoid-s16285" xml:space="preserve"> Linea ergo l s erit altior duabus lineis l h, h t, re-<lb/>ſpectu uiſus.</s> <s xml:id="echoid-s16286" xml:space="preserve"> Ergo s eſt altius, quàm duo puncta h, t.</s> <s xml:id="echoid-s16287" xml:space="preserve"> Linea ergo h s t apparebit uiſuil concaua.</s> <s xml:id="echoid-s16288" xml:space="preserve"/> </p> <div xml:id="echoid-div539" type="float" level="0" n="0"> <figure xlink:label="fig-0233-01" xlink:href="fig-0233-01a"> <variables xml:id="echoid-variables192" xml:space="preserve">u r h d x b y m ſ o n f g i k q z t c c s a</variables> </figure> </div> </div> <div xml:id="echoid-div541" type="section" level="0" n="0"> <head xml:id="echoid-head470" xml:space="preserve" style="it">53. Si uiſ{us} ſit in plano lineæ rectæ, obliquo adplanum axis ſpeculi cylindracei caui: imago <lb/>uidebitur caua & euerſa. 28 p 9.</head> <p> <s xml:id="echoid-s16289" xml:space="preserve">ITem:</s> <s xml:id="echoid-s16290" xml:space="preserve"> ſecemus columnam per ſuperficiem decliuem ſuper axem eius:</s> <s xml:id="echoid-s16291" xml:space="preserve"> faciet ergo ſectionem co-<lb/>lumnarem [per 9 th.</s> <s xml:id="echoid-s16292" xml:space="preserve"> cylindricorum Sereni.</s> <s xml:id="echoid-s16293" xml:space="preserve">] Sit ergo a b g.</s> <s xml:id="echoid-s16294" xml:space="preserve"> Sed in prima figura de columnis con <lb/>cauis [91 n 5] declaratum eſt, quòd in ſuperficie cuiuslibet ſectionis columnæ exit à puncto re-<lb/>flexionis perpendicularis ſuper ſuperficiem contingentem, ex cuius extremitatibus reflectuntur <lb/>formæ.</s> <s xml:id="echoid-s16295" xml:space="preserve"> Sit ergo perpendicularis g a:</s> <s xml:id="echoid-s16296" xml:space="preserve"> & ſit b e k perpendicularis ſuper lineam, contingentem circũ-<lb/>ferentiam ſectionis in b:</s> <s xml:id="echoid-s16297" xml:space="preserve"> & ſit b prope g.</s> <s xml:id="echoid-s16298" xml:space="preserve"> b k ergo ſecabit perpendicularem g a ſub axe, & continebit <lb/>cum ipſa angulum acutum.</s> <s xml:id="echoid-s16299" xml:space="preserve"> [per 24 n:</s> <s xml:id="echoid-s16300" xml:space="preserve"> punctum enim b tam propin quum ipſi g ſumitur, ut recta à <lb/>puncto b, & perpendicularis à reflexionis puncto in axe angulum acutum cõprehendant.</s> <s xml:id="echoid-s16301" xml:space="preserve">] Secet <lb/>ergo in e.</s> <s xml:id="echoid-s16302" xml:space="preserve"> Angulus ergo b e g erit acutus [per 32 p 1.</s> <s xml:id="echoid-s16303" xml:space="preserve">] Et extrahamus ex g lineam ad æquidiſtan-<lb/>tiam lineæ b k:</s> <s xml:id="echoid-s16304" xml:space="preserve"> & ſit g d.</s> <s xml:id="echoid-s16305" xml:space="preserve"> Angulus ergo d g e erit acutus:</s> <s xml:id="echoid-s16306" xml:space="preserve"> [quia <lb/> <anchor type="figure" xlink:label="fig-0234-01a" xlink:href="fig-0234-01"/> per proximam fabricationem & 29 p 1 æquatur angulo b e g <lb/>acuto] ergo g d e erit intra concauitatem columnæ.</s> <s xml:id="echoid-s16307" xml:space="preserve"> Et pona <lb/>mus angulum e g l æqualem angulo e g d:</s> <s xml:id="echoid-s16308" xml:space="preserve"> [per 23 p 1] g l ergo <lb/>concurret cum b e in l:</s> <s xml:id="echoid-s16309" xml:space="preserve"> [per 11 ax.</s> <s xml:id="echoid-s16310" xml:space="preserve"> quia anguli ad g & e acuti, <lb/>minores ſunt duobus rectis] & ſignemus punctum m in li-<lb/>nea l e:</s> <s xml:id="echoid-s16311" xml:space="preserve"> erit ergo m a g acutus:</s> <s xml:id="echoid-s16312" xml:space="preserve"> [quia per 16 p 1 minor eſt an-<lb/>gulo g e m acuto] ergo a m eſt intra ſectionem.</s> <s xml:id="echoid-s16313" xml:space="preserve"> Et ponamus <lb/>angulum g a d æqualem angulo g a m:</s> <s xml:id="echoid-s16314" xml:space="preserve"> ergo a d concurret cũ <lb/>g d:</s> <s xml:id="echoid-s16315" xml:space="preserve"> [per 11 ax.</s> <s xml:id="echoid-s16316" xml:space="preserve">] nam duo anguli, qui ſunt apud a, g, ſunt acuti, <lb/>Concurrant ergo in d.</s> <s xml:id="echoid-s16317" xml:space="preserve"> a d igitur ſecabit b k [per lemma Pro-<lb/>cli ad 29 p 1.</s> <s xml:id="echoid-s16318" xml:space="preserve">] Secet ergo in t.</s> <s xml:id="echoid-s16319" xml:space="preserve"> Cum ergo l k fuerit in aliquo ui <lb/>ſibili, & uiſus fuerit in d:</s> <s xml:id="echoid-s16320" xml:space="preserve"> tunc forma l uidebitur in g:</s> <s xml:id="echoid-s16321" xml:space="preserve"> [ut oſtẽ <lb/>ſum eſt 90 n 5] quia forma l reflectetur ad d ex g, & quia d g <lb/>eſt æquidiſtans perpendiculari b l k:</s> <s xml:id="echoid-s16322" xml:space="preserve"> Et forma m uidebitur in t:</s> <s xml:id="echoid-s16323" xml:space="preserve"> quia forma m reflectitur ad d ex a:</s> <s xml:id="echoid-s16324" xml:space="preserve"> & <lb/>t imago eſt m.</s> <s xml:id="echoid-s16325" xml:space="preserve"> Et tranſeat per d ſuperficies æquidiſtans baſi columnæ:</s> <s xml:id="echoid-s16326" xml:space="preserve"> [ut oſtenſum eſt 47 n 5] ſeca <lb/>bit ergo ſectionem a b g, & faciet in ſuperficie columnæ circulum p o r [per 5 theor:</s> <s xml:id="echoid-s16327" xml:space="preserve"> cylindricorum <lb/>Sereni.</s> <s xml:id="echoid-s16328" xml:space="preserve">] Superficies ergo huius circuli ſecabit b k:</s> <s xml:id="echoid-s16329" xml:space="preserve"> ſecat enim g d, quæ eſt ei æquidiſtans.</s> <s xml:id="echoid-s16330" xml:space="preserve"> Ergo ſe-<lb/>cet b k in k:</s> <s xml:id="echoid-s16331" xml:space="preserve"> & ſit centrum circuli p o r, punctũ h:</s> <s xml:id="echoid-s16332" xml:space="preserve"> & continuemus d h, & tranſeat ad r:</s> <s xml:id="echoid-s16333" xml:space="preserve"> & cõtinuemus <lb/>k h, & tranſeat ad p.</s> <s xml:id="echoid-s16334" xml:space="preserve"> Forma ergo k reflectitur ad d ex circumferentia arcus r p, ut patuit de imagini-<lb/>bus ſpeculorum [73 n 5.</s> <s xml:id="echoid-s16335" xml:space="preserve">] Reflectatur ergo ex o:</s> <s xml:id="echoid-s16336" xml:space="preserve"> & cõtinuemus k o, d o, h o.</s> <s xml:id="echoid-s16337" xml:space="preserve"> Anguli ergo, qui ſunt a-<lb/>pud o, ſunt æquales:</s> <s xml:id="echoid-s16338" xml:space="preserve"> [per 12 n 4] & d o ſecabit h p in n.</s> <s xml:id="echoid-s16339" xml:space="preserve"> n ergo eſt imago k.</s> <s xml:id="echoid-s16340" xml:space="preserve"> Et cõtinuemus k d:</s> <s xml:id="echoid-s16341" xml:space="preserve"> k d er <lb/>go erit differentia communis inter circulum r p & ſectionem a b g.</s> <s xml:id="echoid-s16342" xml:space="preserve"> Nam duo puncta k, d ſunt in u-<lb/>traque ſuperficie, & nihil de ſuperficie ſectionis a b g eſt in ſuperficie circuli r p, niſi linea k d:</s> <s xml:id="echoid-s16343" xml:space="preserve"> g ergo <lb/>eſt extra circulum:</s> <s xml:id="echoid-s16344" xml:space="preserve"> & ſimiliter b:</s> <s xml:id="echoid-s16345" xml:space="preserve"> & ſunt in ſuperficie ſectionis:</s> <s xml:id="echoid-s16346" xml:space="preserve"> & n eſt in ſuperficie circuli r p:</s> <s xml:id="echoid-s16347" xml:space="preserve"> & for-<lb/>ma l m k tranſit per puncta g, t, n:</s> <s xml:id="echoid-s16348" xml:space="preserve"> & linea, quæ tranſit per hæc puncta, eſt arcualis:</s> <s xml:id="echoid-s16349" xml:space="preserve"> ſed ſuperficies <lb/>ſectionis eſt decliuis ſuper ſuperficiem columnæ:</s> <s xml:id="echoid-s16350" xml:space="preserve"> [per 9 th.</s> <s xml:id="echoid-s16351" xml:space="preserve"> cylindricorum Sereni] axis ergo ſectio <lb/>nis non tranſit per totum axem columnæ, neque eſt æquidiſtans baſi columnæ.</s> <s xml:id="echoid-s16352" xml:space="preserve"> Patet ergo ex hac <lb/>figura & duabus præmiſsis, quòd lineæ rectæ æquidiſtantes axi columnæ, & æquidiſtantes baſi e-<lb/>ius:</s> <s xml:id="echoid-s16353" xml:space="preserve"> & etiam illæ lineæ, quæ obliquantur ſuper ſuperficiem eius:</s> <s xml:id="echoid-s16354" xml:space="preserve"> fortè uidebuntur arcuales, fortè re <lb/>ctæ, fortè conuerſæ.</s> <s xml:id="echoid-s16355" xml:space="preserve"> Et quia t eſt imago m, & n imago k:</s> <s xml:id="echoid-s16356" xml:space="preserve"> erit forma m k conuerſa.</s> <s xml:id="echoid-s16357" xml:space="preserve"> Et ſi linea etiam <lb/>fuerit in ſuperficie circuli, æquidiſtante baſibus columnæ, cuius ſuperficies tranſit per centrũ uiſus, <lb/> <pb o="229" file="0235" n="235" rhead="OPTICAE LIBER VI."/> ut dictum eſt de imaginibus ſpeculorum concauorum in ſeptimo capitulo huius tractatus:</s> <s xml:id="echoid-s16358" xml:space="preserve"> forma <lb/>fortè erit æqualis recta:</s> <s xml:id="echoid-s16359" xml:space="preserve"> fortè conuerſa.</s> <s xml:id="echoid-s16360" xml:space="preserve"> Patet ergo, quòd forma eorum, quæ comprehenduntur in <lb/>ſpeculis columnaribus concauis, fortè erit recta, fortè conuerſa.</s> <s xml:id="echoid-s16361" xml:space="preserve"/> </p> <div xml:id="echoid-div541" type="float" level="0" n="0"> <figure xlink:label="fig-0234-01" xlink:href="fig-0234-01a"> <variables xml:id="echoid-variables193" xml:space="preserve">p b <gap/> o n m d r h c t a K</variables> </figure> </div> </div> <div xml:id="echoid-div543" type="section" level="0" n="0"> <head xml:id="echoid-head471" xml:space="preserve" style="it">54. Siuiſ{us} ſit in plano lineæ rectæ, perpendiculari plano axis ſpeculi cylindracei caui: imago <lb/>uidebitur recta & euerſa: aliâ s maior: aliâs minor: aliâs æqualis ipſi lineæ: aliâs ſimplex: aliâs <lb/>multiplex. 29 p 9.</head> <p> <s xml:id="echoid-s16362" xml:space="preserve">ITem:</s> <s xml:id="echoid-s16363" xml:space="preserve"> iteremus formã tertiæ figurę de fallacijs ſpeculorũ cõcauorũ ijſdẽ literis exiſtentibus:</s> <s xml:id="echoid-s16364" xml:space="preserve"> [41.</s> <s xml:id="echoid-s16365" xml:space="preserve"> <lb/>42.</s> <s xml:id="echoid-s16366" xml:space="preserve"> 43 n] & ſit circulus b z a in ſuperficie ſpeculi columnaris cõcaui:</s> <s xml:id="echoid-s16367" xml:space="preserve"> & ſit uiſus in d.</s> <s xml:id="echoid-s16368" xml:space="preserve"> Erit ergo ex-<lb/>tra ſuperficiẽ circuli:</s> <s xml:id="echoid-s16369" xml:space="preserve"> & erunt duæ lineæ e a, e b perpendiculares ſuper ſuperficies, cõtingentes ſu <lb/>perficiẽ colũnę:</s> <s xml:id="echoid-s16370" xml:space="preserve"> & erit ſuperficies trianguli d g e perpẽdicularis ſuք ſuperficiẽ circuli [ք 18 p 11] ꝗa g d <lb/>eſt perpẽdicularis ſuper ſuperficiẽ circuli [ut oſtẽſum eſt 41 n.</s> <s xml:id="echoid-s16371" xml:space="preserve">] Superficies ergo trianguli d g e trãſit <lb/>per totũ axẽ:</s> <s xml:id="echoid-s16372" xml:space="preserve"> & in neutra ſuperficie d b o, d a o eſt aliquid de axe colũnæ, niſi e, qđ eſt centrũ circuli.</s> <s xml:id="echoid-s16373" xml:space="preserve"> <lb/>Et utraq;</s> <s xml:id="echoid-s16374" xml:space="preserve"> ſuperficies d b o, d a o facit in ſuperficie columnę ſectio <lb/> <anchor type="figure" xlink:label="fig-0235-01a" xlink:href="fig-0235-01"/> nem:</s> <s xml:id="echoid-s16375" xml:space="preserve"> [per 9 th.</s> <s xml:id="echoid-s16376" xml:space="preserve"> cylindricorum Sereni] & formæ reflectuntur ex <lb/>his ſectionibus à duobus punctis a, b [ut patuit 41 n.</s> <s xml:id="echoid-s16377" xml:space="preserve">] Forma ergo <lb/>r reflectitur ad d ex b:</s> <s xml:id="echoid-s16378" xml:space="preserve"> & forma m reflectitur ex a:</s> <s xml:id="echoid-s16379" xml:space="preserve"> & n u erit diame <lb/>ter imaginis m r:</s> <s xml:id="echoid-s16380" xml:space="preserve"> [ſunt enim puncta n & u imagines punctorum <lb/>r & m per 7 n 5] & eſt minor quàm m r:</s> <s xml:id="echoid-s16381" xml:space="preserve"> [ut demonſtratum eſt 42 <lb/>n.</s> <s xml:id="echoid-s16382" xml:space="preserve">] Et ſimiliter duo puncta h, l reflectentur ad d ex duobus pun-<lb/>ctis à, b:</s> <s xml:id="echoid-s16383" xml:space="preserve"> & erit t k diameter imaginis l h:</s> <s xml:id="echoid-s16384" xml:space="preserve"> & erit e i æqualis [ut pa-<lb/>tuit 41 n] & erit p i diameter imaginis f q:</s> <s xml:id="echoid-s16385" xml:space="preserve"> & eſt maior illa.</s> <s xml:id="echoid-s16386" xml:space="preserve"> Et o-<lb/>mnes iſtæ imagines erunt conuerſæ [ut oſtenſum eſt 43 n.</s> <s xml:id="echoid-s16387" xml:space="preserve">] Et ſi <lb/>uiſus fuerit in o, & lineæ p i, t k, n u fuerint uiſibiles:</s> <s xml:id="echoid-s16388" xml:space="preserve"> erunt è con-<lb/>trario:</s> <s xml:id="echoid-s16389" xml:space="preserve"> tunc enim diameter imaginis p i erit minor ipſa:</s> <s xml:id="echoid-s16390" xml:space="preserve"> & diame-<lb/>ter imaginis n u erit maior ipſa:</s> <s xml:id="echoid-s16391" xml:space="preserve"> & diameter t k erit ei æqualis.</s> <s xml:id="echoid-s16392" xml:space="preserve"> Et <lb/>oẽs imagines erũt rectæ.</s> <s xml:id="echoid-s16393" xml:space="preserve"> Et omnia iſta oſtẽſa ſunt in prædicto ca <lb/>pitulo.</s> <s xml:id="echoid-s16394" xml:space="preserve"> Item cum utraq;</s> <s xml:id="echoid-s16395" xml:space="preserve"> extremitas alicuius harũ habuerit unam <lb/>imaginẽ, & aliquod punctũ in medio habuerit plures imagines:</s> <s xml:id="echoid-s16396" xml:space="preserve"> <lb/>tũc illa linea habebit totimagines, quot punctũ mediũ habet.</s> <s xml:id="echoid-s16397" xml:space="preserve"> Et <lb/>ſi utraq;</s> <s xml:id="echoid-s16398" xml:space="preserve"> extremitas, uel altera habuerit plures imagines, & pun-<lb/>ctum mediũ habuerit unã:</s> <s xml:id="echoid-s16399" xml:space="preserve"> tunc linea tot habebit imagines, quot <lb/>habet punctũ extremũ.</s> <s xml:id="echoid-s16400" xml:space="preserve"> Et ſi utraq;</s> <s xml:id="echoid-s16401" xml:space="preserve"> extremitas uel altera habue-<lb/>rit multas imagines, & pũctũ mediũ habuerit multas imagines:</s> <s xml:id="echoid-s16402" xml:space="preserve"> <lb/>tunc linea tot habebit imagines ſecundum maiorem numerum.</s> <s xml:id="echoid-s16403" xml:space="preserve"> <lb/>Et hoc patebit, ut de imaginibus patuit ſpeculorum ſphærico-<lb/>rum concauorum.</s> <s xml:id="echoid-s16404" xml:space="preserve"> In ſpeculis ergo columnaribus concauis <lb/>accidit fallacia in omnibus, quæ in eis comprehenduntur, ſicut <lb/>accidit in ſpeculis ſphæricis concauis:</s> <s xml:id="echoid-s16405" xml:space="preserve"> ſcilicet de formis ſpecie-<lb/>rum uiſibilium, & de quantitatibus:</s> <s xml:id="echoid-s16406" xml:space="preserve"> & de numero ſuarum ima-<lb/>ginum:</s> <s xml:id="echoid-s16407" xml:space="preserve"> & de rectitudine, & de cõuerſione, cum fallacijs, quę appropriantur reflexioni.</s> <s xml:id="echoid-s16408" xml:space="preserve"> Et fallaciæ e-<lb/>runt inhis, ut in ſpeculis prædictis.</s> <s xml:id="echoid-s16409" xml:space="preserve"> Ethoc eſt, quod uoluimus declarare in hoc capitulo.</s> <s xml:id="echoid-s16410" xml:space="preserve"/> </p> <div xml:id="echoid-div543" type="float" level="0" n="0"> <figure xlink:label="fig-0235-01" xlink:href="fig-0235-01a"> <variables xml:id="echoid-variables194" xml:space="preserve">d g p i t k n u b e a o f q l h m r</variables> </figure> </div> </div> <div xml:id="echoid-div545" type="section" level="0" n="0"> <head xml:id="echoid-head472" xml:space="preserve">DE ERRORIBVS, QVI ACCIDVNT IN SPECVLIS <lb/>pyramidalibus concauis. Cap. IX.</head> <p> <s xml:id="echoid-s16411" xml:space="preserve">IN his autem accidunt illæ fallaciæ, quæ accidunt in ſpeculis columnaribus concauis.</s> <s xml:id="echoid-s16412" xml:space="preserve"> Debilitas <lb/>uerò coloris & lucis:</s> <s xml:id="echoid-s16413" xml:space="preserve"> & diuerſitas poſitionis, & remotionis accidunt in his, ſicut in omnibus ſpe <lb/>culis:</s> <s xml:id="echoid-s16414" xml:space="preserve"> nam cauſſa huius eſt reflexio.</s> <s xml:id="echoid-s16415" xml:space="preserve"> Accidit etiam in his ſpeculis multitudo imaginum, ſicut in <lb/>ſpeculis columnaribus & ſphæricis concauis dictũ in capitulo [ſecũdo libri quinti] de imaginibus.</s> <s xml:id="echoid-s16416" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div546" type="section" level="0" n="0"> <head xml:id="echoid-head473" xml:space="preserve" style="it">55. Si lineæ: recta uel curua obliquè incidant uertici ſpeculi conici caui: reflectentur à latere <lb/>conico ad uiſum inter ipſas & ſpeculi ſuperficiem poſitum: & imago rectæ uidebitur parum cur-<lb/>ua: curuæ, recta. 31 p 9.</head> <p> <s xml:id="echoid-s16417" xml:space="preserve">ACcidit etiam in eis, quod in columnaribus concauis, ſcilicet ut rectum uideatur conuexum <lb/>& concauum.</s> <s xml:id="echoid-s16418" xml:space="preserve"> Huius autem demonſtratio eſt:</s> <s xml:id="echoid-s16419" xml:space="preserve"> quod rectæ lineæ, quæ extenduntur in longi-<lb/>tudine ſpeculi, quæ tranſeunt per uerticem pyramidis, & quæ ſunt prope illas, uidentur con <lb/>uexæ, & fortè rectæ.</s> <s xml:id="echoid-s16420" xml:space="preserve"> Et demonſtratio ſuper hoc eſt, ut demonſtratio in ſpeculis columnaribus con-<lb/>cauis.</s> <s xml:id="echoid-s16421" xml:space="preserve"> Nam ſi itera uerimus ſecundam figuram de fallacijs ſpeculorum pyramidalium conuexorum <lb/>[quæ eſt 32 n] inueniemus diametrum imaginis lineæ rectæ poſitæ in illo ſpeculo, quæ eſt illic linea <lb/>a n, intra concauitatẽ ſpeculi pyramidalis:</s> <s xml:id="echoid-s16422" xml:space="preserve"> & inueniemus punctũ, quod eſt ſub ſuperficie contingen <lb/>te pyramidẽ, tranſeuntẽ per lineã longitudinis, à qua reflectitur forma lineę rectę ad uiſum:</s> <s xml:id="echoid-s16423" xml:space="preserve"> quod il-<lb/>lic punctum f.</s> <s xml:id="echoid-s16424" xml:space="preserve"> Si igitur fuerit punctum illud centrum uiſus:</s> <s xml:id="echoid-s16425" xml:space="preserve"> erunt omnia puncta, quę ſunt in diame-<lb/> <pb o="230" file="0236" n="236" rhead="ALHAZEN"/> tro imaginis reflexa ad pũctũ f:</s> <s xml:id="echoid-s16426" xml:space="preserve"> & <lb/> <anchor type="figure" xlink:label="fig-0236-01a" xlink:href="fig-0236-01"/> imagines duarum extremitatum <lb/>a p y erunt extremitates lineæ <lb/>rectæ a n:</s> <s xml:id="echoid-s16427" xml:space="preserve"> & loca imaginis puncti <lb/>p, quod eſt in medio a y, diuerſa-<lb/>buntur.</s> <s xml:id="echoid-s16428" xml:space="preserve"> Et hoc declarabitur eadẽ <lb/>uia, qua proceſsimus in demon-<lb/>ſtratione primę figurę ſpeculorũ <lb/>columnarium concauorũ.</s> <s xml:id="echoid-s16429" xml:space="preserve"> Patet <lb/>ergo ex hoc, quòd ſi a p y fuerit in <lb/>aliquo uiſibili, & uiſus fuerit f:</s> <s xml:id="echoid-s16430" xml:space="preserve"> <lb/>tunc imago fortè uidebitur con-<lb/>uexa, & fortè cõcaua.</s> <s xml:id="echoid-s16431" xml:space="preserve"> Et patet e-<lb/>tiam in figura ſecũda de fallacijs <lb/>ſpeculorum columnarium conca <lb/>uorum [52 n] quòd lineæ poſitæ <lb/>in latitudine ſpeculi apparebunt <lb/>concauæ concauitate mirabili:</s> <s xml:id="echoid-s16432" xml:space="preserve"> & <lb/>quòd imagines linearũ, quæ ſunt <lb/>in ſuperficiebus tranſeuntibus per axem & per centrum uiſus, erunt rectæ.</s> <s xml:id="echoid-s16433" xml:space="preserve"/> </p> <div xml:id="echoid-div546" type="float" level="0" n="0"> <figure xlink:label="fig-0236-01" xlink:href="fig-0236-01a"> <variables xml:id="echoid-variables195" xml:space="preserve">a h p u <gap/> m z t x b n <gap/> c q s d g ſ <gap/> K f r</variables> </figure> </div> </div> <div xml:id="echoid-div548" type="section" level="0" n="0"> <head xml:id="echoid-head474" xml:space="preserve" style="it">56. Si uiſ{us} ſit in communi ſectione planorum: lineæ rectæ & axis ſpeculi conici caui, inter ſe <lb/>perpendicularium: imago uidebitur recta & euerſa: aliâs maior: aliâs æqualis: aliâs minor ιpſa <lb/>line a: aliâs ſimplex: aliâs multiplex. 34 p 9.</head> <p> <s xml:id="echoid-s16434" xml:space="preserve">ITem:</s> <s xml:id="echoid-s16435" xml:space="preserve"> iteremus tertiam figuram de fallacijs ſpeculorum ſphæricorum concauorum ijſdem lite-<lb/>ris [quæ fuit 41 n.</s> <s xml:id="echoid-s16436" xml:space="preserve">] Si ergo aliquod punctum fuerit in axe pyramidis:</s> <s xml:id="echoid-s16437" xml:space="preserve"> & duæ lineæ e a, e b fuerint <lb/>perpendiculares ſuper ſuperficies, contingentes pyramidem:</s> <s xml:id="echoid-s16438" xml:space="preserve"> & hoc eſt poſsibile:</s> <s xml:id="echoid-s16439" xml:space="preserve"> quia ſunt æ-<lb/>quales:</s> <s xml:id="echoid-s16440" xml:space="preserve"> poſſunt enim cum axe continere duos angulos acutos <lb/> <anchor type="figure" xlink:label="fig-0236-02a" xlink:href="fig-0236-02"/> æquales.</s> <s xml:id="echoid-s16441" xml:space="preserve"> Cum ergo hæ duę lineæ fuerint perpendiculares, & fue <lb/>rit uiſus d:</s> <s xml:id="echoid-s16442" xml:space="preserve"> tune ſuperficies, in qua ſunt g e, e d, tranſibit per totũ <lb/>axem, & per centrũ uiſus:</s> <s xml:id="echoid-s16443" xml:space="preserve"> & utraq;</s> <s xml:id="echoid-s16444" xml:space="preserve"> ſuperficies d a o, d b o erit de-<lb/>cliuis ſuper axem pyramidis:</s> <s xml:id="echoid-s16445" xml:space="preserve"> & erunt differentię earũ duæ ſectio <lb/>nes pyramidis [per 5 th.</s> <s xml:id="echoid-s16446" xml:space="preserve"> 1 conicorum A pollonij] & erunt formæ <lb/>punctorum h, r, q reflexę ad d ex b:</s> <s xml:id="echoid-s16447" xml:space="preserve"> & formæ punctorum l, m, f re-<lb/>flectẽtur ad d ex a.</s> <s xml:id="echoid-s16448" xml:space="preserve"> Cum ergo lineę m l f, r h q fuerint in aliqua ſu-<lb/>perficie uiſibili, & uiſus fuerit in d:</s> <s xml:id="echoid-s16449" xml:space="preserve"> tunc n u erit imago m r:</s> <s xml:id="echoid-s16450" xml:space="preserve"> & t k <lb/>erit imago l h:</s> <s xml:id="echoid-s16451" xml:space="preserve"> & p i erit imago f q [ut oſtenſum eſt 54 n.</s> <s xml:id="echoid-s16452" xml:space="preserve">] Sic ergo <lb/>imago m r erit minor ſe ipſa:</s> <s xml:id="echoid-s16453" xml:space="preserve"> & imago f q maior ſeipſa:</s> <s xml:id="echoid-s16454" xml:space="preserve"> & imago <lb/>l h æqualis ſibi ipſi.</s> <s xml:id="echoid-s16455" xml:space="preserve"> Et omnes imagines erunt conuerſæ.</s> <s xml:id="echoid-s16456" xml:space="preserve"> Et ſi ui-<lb/>ſus fuerit in o, & n u, t k, p i fuerint in ſuperficiebus uiſibilium:</s> <s xml:id="echoid-s16457" xml:space="preserve"> <lb/>tunc imagines earum erunt m r, l h, f q.</s> <s xml:id="echoid-s16458" xml:space="preserve"> Sic ergo erit imago f q ſei-<lb/>pſa minor:</s> <s xml:id="echoid-s16459" xml:space="preserve"> & imago n u maior:</s> <s xml:id="echoid-s16460" xml:space="preserve"> & imago t k æqualis.</s> <s xml:id="echoid-s16461" xml:space="preserve"> Et iſtæ imagi <lb/>nes erunt rectæ.</s> <s xml:id="echoid-s16462" xml:space="preserve"> Nam iſtæ imagines erunt ultra centrum uiſus, & <lb/>comprehen duntur ante uiſum ſuper lineas radiales.</s> <s xml:id="echoid-s16463" xml:space="preserve"> Puncta ergo <lb/>m, l, f comprehenduntur in linea a o:</s> <s xml:id="echoid-s16464" xml:space="preserve"> & puncta r, h, q comprehen-<lb/>duntur in o b:</s> <s xml:id="echoid-s16465" xml:space="preserve"> & ſic forma reflectetur recta.</s> <s xml:id="echoid-s16466" xml:space="preserve"> Patet ergo ex his, quę <lb/>diximus in hoc capitulo:</s> <s xml:id="echoid-s16467" xml:space="preserve"> quòd lineæ rectę quandoq;</s> <s xml:id="echoid-s16468" xml:space="preserve"> uidentur in <lb/>his ſpeculis conuexæ:</s> <s xml:id="echoid-s16469" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s16470" xml:space="preserve"> concauæ:</s> <s xml:id="echoid-s16471" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s16472" xml:space="preserve"> rectæ:</s> <s xml:id="echoid-s16473" xml:space="preserve"> & <lb/>quandoq;</s> <s xml:id="echoid-s16474" xml:space="preserve"> maiores:</s> <s xml:id="echoid-s16475" xml:space="preserve"> & minores:</s> <s xml:id="echoid-s16476" xml:space="preserve"> & æquales:</s> <s xml:id="echoid-s16477" xml:space="preserve"> & quãdoq;</s> <s xml:id="echoid-s16478" xml:space="preserve"> rectę, con <lb/>uerſæ.</s> <s xml:id="echoid-s16479" xml:space="preserve"> Et in capitulo [ſecundo libri quinti] de imagine declara-<lb/>uimus, quòd omne punctum uiſibile in huiuſmodi ſpeculis <lb/>quandoque habet unam imaginem:</s> <s xml:id="echoid-s16480" xml:space="preserve"> quandoque duas:</s> <s xml:id="echoid-s16481" xml:space="preserve"> & tres:</s> <s xml:id="echoid-s16482" xml:space="preserve"> & <lb/>quatuor.</s> <s xml:id="echoid-s16483" xml:space="preserve"> In omnibus ergo, quæ comprehenduntur in his ſpecu-<lb/>lis, accidit fallacia, ut in columnaribus concauis:</s> <s xml:id="echoid-s16484" xml:space="preserve"> acciduntq́;</s> <s xml:id="echoid-s16485" xml:space="preserve"> e-<lb/>tiam in eis fallaciæ compoſitæ, ſicut in cæteris ſpeculis:</s> <s xml:id="echoid-s16486" xml:space="preserve"> & exempla, & declaratio eorum ſunt, ſicut <lb/>in ſpeculis planis.</s> <s xml:id="echoid-s16487" xml:space="preserve"> Et hoc intendimus declarare in hoc capitulo:</s> <s xml:id="echoid-s16488" xml:space="preserve"> nunc autem <lb/>finiamus ſextum tractatum.</s> <s xml:id="echoid-s16489" xml:space="preserve"/> </p> <div xml:id="echoid-div548" type="float" level="0" n="0"> <figure xlink:label="fig-0236-02" xlink:href="fig-0236-02a"> <variables xml:id="echoid-variables196" xml:space="preserve">d g p i t k n z u b e a ſ o q l h m r</variables> </figure> </div> <pb o="231" file="0237" n="237"/> </div> <div xml:id="echoid-div550" type="section" level="0" n="0"> <head xml:id="echoid-head475" xml:space="preserve">ALHAZEN FILII</head> <head xml:id="echoid-head476" xml:space="preserve">ALHAYZEN OPTICAE</head> <head xml:id="echoid-head477" xml:space="preserve">LIBER SEPTIMVS.</head> <p style="it"> <s xml:id="echoid-s16490" xml:space="preserve">SEptimi tractat{us} ſunt ſeptem partes.</s> <s xml:id="echoid-s16491" xml:space="preserve"> Prima pars eſt proœmium.</s> <s xml:id="echoid-s16492" xml:space="preserve"> Secunda, quòd lux <lb/>tranſeat diaphana corpora ſecundum uerticationes linearum rectarum, & refrin <lb/>gatur, cum occurrit corpori, cui{us} diaphanit{as} fuerit diuerſa à diaphanitate corporis, in <lb/>quo existit.</s> <s xml:id="echoid-s16493" xml:space="preserve"> Tertia de qualitate refractionis luminum in diaphanis corporib{us}.</s> <s xml:id="echoid-s16494" xml:space="preserve"> Quarta, <lb/>quòd quicquid comprehenditur à uiſu ultra diaphana corpora, quorũ diaphanit{as} dif-<lb/>fert à diaphanitate corporis, in quo uiſ{us} exiſtit, cum fuerit decliue à perpendicularib{us}, <lb/>exeuntib{us} ſuper ſuperficiem eorum, comprehenditur ſecundum refractionem.</s> <s xml:id="echoid-s16495" xml:space="preserve"> Quinta <lb/>de phantaſmatib{us}.</s> <s xml:id="echoid-s16496" xml:space="preserve"> Sexta, quomodo uiſ{us} comprehendat uiſibilia ſecundum refractio-<lb/>nem.</s> <s xml:id="echoid-s16497" xml:space="preserve"> Septima de fallacijs uiſ{us}, quæ accidunt ex refractione.</s> <s xml:id="echoid-s16498" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div551" type="section" level="0" n="0"> <head xml:id="echoid-head478" xml:space="preserve">PROOEMIVM LIBRI. CAP. I.</head> <head xml:id="echoid-head479" xml:space="preserve" style="it">1. Viſio fit trifariam: rectè, reflexè & refractè. In præfat. 1. 10 libr. Idem 1 n 4.</head> <p> <s xml:id="echoid-s16499" xml:space="preserve">PRædictum eſt in proœmio quarti tractatus huius libri [1 n] quòd uiſus tribus modis <lb/>comprehendat uiſibilia, uidelicet ſecundum rectitudinem:</s> <s xml:id="echoid-s16500" xml:space="preserve"> ſecundum reflexionem <lb/>à terſis corporibus:</s> <s xml:id="echoid-s16501" xml:space="preserve"> & ſecundũ refractionẽ ultra diaphana corpora, quæ differunt in <lb/>diaphanitate à diaphanitate aeris:</s> <s xml:id="echoid-s16502" xml:space="preserve"> & quòd uiſus nihil cõprehẽdit ex uiſibilibus, niſi <lb/>aliquo iſtorũ triũ modorũ:</s> <s xml:id="echoid-s16503" xml:space="preserve"> & quòd quolibet iſtorũ modorum cõprehendit uiſus uiſi <lb/>bilia & omnes res, quæ ſunt in uiſibilibus, & omnibus modis uiſionis, quorũ diſtin-<lb/>ctio declarata eſt in ultima differentia ſecundi tractatus.</s> <s xml:id="echoid-s16504" xml:space="preserve"> In præcedentibus autem tractatibus decla <lb/>ratum eſt, qualiter uiſus comprehendat uiſibilia ſecundum rectitudinem, & ſecundum reflexionẽ:</s> <s xml:id="echoid-s16505" xml:space="preserve"> <lb/>& oſtendimus diuerſitatem comprehenſionis uiſus ad uiſibilia ſecundum utrumq;</s> <s xml:id="echoid-s16506" xml:space="preserve"> iſtorum modo-<lb/>rum.</s> <s xml:id="echoid-s16507" xml:space="preserve"> Remanet ergo declararare, quomodo uiſus comprehendat uiſibilia ſecundum refractionem <lb/>ultra corpora diaphana.</s> <s xml:id="echoid-s16508" xml:space="preserve"> Nos autem in iſto tractatu ſolummodo de refractione tractabimus:</s> <s xml:id="echoid-s16509" xml:space="preserve"> & ma-<lb/>nifeſtabimus formam refractionis:</s> <s xml:id="echoid-s16510" xml:space="preserve"> & diſtinguemus eius modos:</s> <s xml:id="echoid-s16511" xml:space="preserve"> & diuidemus proprietates eius:</s> <s xml:id="echoid-s16512" xml:space="preserve"> & <lb/>declarabimus, quomodo accidat uiſui in huiuſmodi uiſione.</s> <s xml:id="echoid-s16513" xml:space="preserve"> Et primò proponemus fundamenta, <lb/>quæ certificant, quicquid dependet ab hac re.</s> <s xml:id="echoid-s16514" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div552" type="section" level="0" n="0"> <head xml:id="echoid-head480" xml:space="preserve">QVÒ<unsure/>D LVX PERTRANSEAT PER DIAPHANA CORPORA SECVN <lb/>dum uerticationes linearũ rectarum, & refringatur, cum occurrit cor-<lb/>pori, cuius diaphanitas fuerit diuerſa à diaphanitate <lb/>corporis, in quo exiſtit. Cap. II.</head> <head xml:id="echoid-head481" xml:space="preserve" style="it">2. Constructio organi refractionis. 1 p 2.</head> <p> <s xml:id="echoid-s16515" xml:space="preserve">QVòd lumen quidem tranſeat in aerem, & extendatur ſecundum lineas rectas, declaratũ eſt <lb/>in tractatu primo huius operis:</s> <s xml:id="echoid-s16516" xml:space="preserve"> [14.</s> <s xml:id="echoid-s16517" xml:space="preserve"> 17.</s> <s xml:id="echoid-s16518" xml:space="preserve"> 28 n] aer autem eſt unum de corporibus diaphanis:</s> <s xml:id="echoid-s16519" xml:space="preserve"> <lb/>per aquam autem, uitrũ, & diaphanos lapides lumen tranſit, & extenditur ſecundum lineas <lb/>rectas:</s> <s xml:id="echoid-s16520" xml:space="preserve"> hoc autem comprehenditur per experientiam.</s> <s xml:id="echoid-s16521" xml:space="preserve"> Si quis ergo experiri uoluerit:</s> <s xml:id="echoid-s16522" xml:space="preserve"> accipiat lami-<lb/>nam ex ære rotundam, cuius diameter nõ ſit minor uno cubito:</s> <s xml:id="echoid-s16523" xml:space="preserve"> & ſit ſpiſsitudo eius aliquantulum <lb/>fortis:</s> <s xml:id="echoid-s16524" xml:space="preserve"> & habeat oras rotundas, perpendiculares ſuper ſuperficiem eius:</s> <s xml:id="echoid-s16525" xml:space="preserve"> & ſit altitudo orarum eius <lb/>non minor latitudine duorum digitorũ.</s> <s xml:id="echoid-s16526" xml:space="preserve"> In medio autem dorſi laminæ ſit aliquod corpus paruum, <lb/>columnare, rotundũ, cuius longitudo non minor latitudine trium digitorum:</s> <s xml:id="echoid-s16527" xml:space="preserve"> & ſit perpendiculare <lb/>ſuper ſuperficiem laminæ.</s> <s xml:id="echoid-s16528" xml:space="preserve"> Et ponamus hoc inſtrumentũ in tornatorio, in quo tornant tornarij in-<lb/>ſtrumenta cupri, & ponamus alterum dentem tornatorij in medio laminæ, & reliquũ in medio ex-<lb/>tremitatis corporis, quod eſt in dorſo laminæ:</s> <s xml:id="echoid-s16529" xml:space="preserve"> & radamus reuoluendo hoc inſtrumentum abraſio-<lb/>ne uera, quouſq;</s> <s xml:id="echoid-s16530" xml:space="preserve"> uerificetur rotunditas orarum ſuarum intus & extrà, & adæquetur ſuperficies inte <lb/>rior & exterior, & fiant duæ ſuperficies æquidiſtantes:</s> <s xml:id="echoid-s16531" xml:space="preserve"> & abrademus etiam corpus, quod eſt in dor-<lb/>ſo, donec fiat rotundum.</s> <s xml:id="echoid-s16532" xml:space="preserve"> Cum ergo hoc inſtrumentum fuerit perfectum per abraſionem:</s> <s xml:id="echoid-s16533" xml:space="preserve"> ſignemus <lb/>in ſuperficie eius interiore duas diametros ſecantes ſe perpendiculariter:</s> <s xml:id="echoid-s16534" xml:space="preserve"> & ſunt tranſeũtes per cen <lb/>trum eius:</s> <s xml:id="echoid-s16535" xml:space="preserve"> deinde ſignemus punctum in baſi oræ inſtrumenti, cuius diſtantia ab extremitate alte-<lb/>rius duarum diametrorum ſecantium ſe, eſt latitudo unius digiti:</s> <s xml:id="echoid-s16536" xml:space="preserve"> deinde extrahamus exiſto pun-<lb/>cto tertiam diametrum tranſeuntem per centrum laminę:</s> <s xml:id="echoid-s16537" xml:space="preserve"> quę quidem diameter extendatur in tota <lb/>ſuperficie eius:</s> <s xml:id="echoid-s16538" xml:space="preserve"> deinde extrahamus à duobus extremis huius diametri duas lineas in ſuperficie oræ <lb/> <pb o="232" file="0238" n="238" rhead="ALHAZEN"/> inſtrumenti, perpendiculares ſuper ſuperficiem laminæ.</s> <s xml:id="echoid-s16539" xml:space="preserve"> Deinde diuidemus ex altera iſtarum dua-<lb/>rum linearum tres lineas paruas, æquales, quarum prima ſequitur ſuperficiem laminæ:</s> <s xml:id="echoid-s16540" xml:space="preserve"> & longitu-<lb/>dò cuiuslibet harum ſit in quantitate medietatis grani hordeacei:</s> <s xml:id="echoid-s16541" xml:space="preserve"> fient ergo ſuper lineam perpendi <lb/>cularẽ tria puncta, quę ſunt fines illarũ linearũ.</s> <s xml:id="echoid-s16542" xml:space="preserve"> Deinde reducamus hoc inſtrum entũ ad tornatoriũ, <lb/>& ſignemus in ipſo tres circulos æquidiſtãtes, tranſ-<lb/> <anchor type="figure" xlink:label="fig-0238-01a" xlink:href="fig-0238-01"/> euntes per tria puncta, quæ ſunt ſuper lineam per-<lb/>pendicularem ſuper extremitatem diametri:</s> <s xml:id="echoid-s16543" xml:space="preserve"> ſecabi-<lb/>tur ergo alia perpendicularis, quæ eſt perpendicula-<lb/>ris ſuper aliam extremitatẽ huius diametri, per iſtos <lb/>tres circulos, & fient in ipſa tria puncta, & fient in u-<lb/>noquoq;</s> <s xml:id="echoid-s16544" xml:space="preserve"> trium circulorũ duo puncta oppoſita, quæ <lb/>ſunt extrema alicuius diametri ex ipſorũ diametris.</s> <s xml:id="echoid-s16545" xml:space="preserve"> <lb/>Deinde diuidamus medium circulum ex iſtis tribus <lb/>circulis in 360 partes, & ſi poſsibile fuerit, in minuta:</s> <s xml:id="echoid-s16546" xml:space="preserve"> <lb/>deinde perforemus in ora inſtrumenti foramen ro-<lb/>tundum, cuius centrum ſit medium punctum trium <lb/>punctorum, quæ ſunt ſuper alteram duarum linea-<lb/>rum, perpendicularium ſuper extremitatem diame-<lb/>tri laminæ:</s> <s xml:id="echoid-s16547" xml:space="preserve"> & ſit medietas diametri eius in quantita-<lb/>te diſtantiæ, quę eſt inter circulos:</s> <s xml:id="echoid-s16548" xml:space="preserve"> perueniet ergo cir <lb/>cumferentia foraminis inter duos circulos æquidi-<lb/>ſtantes, qui ſunt in extremitatibus.</s> <s xml:id="echoid-s16549" xml:space="preserve"> Poſtea accipia <lb/>mus laminam ſubtilem quadratam, aliquantulæ ſpiſsitudinis:</s> <s xml:id="echoid-s16550" xml:space="preserve"> cuius longitudo ſit in quantitate alti-<lb/>tudinis oræ inſtrumenti:</s> <s xml:id="echoid-s16551" xml:space="preserve"> & cuius latitudo ſit prope hoc:</s> <s xml:id="echoid-s16552" xml:space="preserve"> & adæquetur ſuperficies eius, quantùm po <lb/>teſt:</s> <s xml:id="echoid-s16553" xml:space="preserve"> & adæquetur ſpiſsitudo eius etiam, quæ ſequitur alteram extremitatem eius, quouſq;</s> <s xml:id="echoid-s16554" xml:space="preserve"> differen-<lb/>tia communis inter ſuperficiem faciei eius, & inter ſuperficiem ſpiſsitudinis eius, ſit linea recta:</s> <s xml:id="echoid-s16555" xml:space="preserve"> quã <lb/>lineam diuidemus in duo æqualia:</s> <s xml:id="echoid-s16556" xml:space="preserve"> à cuius medio extrahamus lineam rectam in ſuperficie faciei e-<lb/>ius perpendicularem ſuper lineam rectam, quę eſt communis differentia.</s> <s xml:id="echoid-s16557" xml:space="preserve"> Deinde diuidamus ex hac <lb/>linea perpendiculari ex parte extremitatis, quæ eſt ſuper communem dif-<lb/> <anchor type="figure" xlink:label="fig-0238-02a" xlink:href="fig-0238-02"/> ferentiam, tres lineas, æquales inter ſe, & æquales unicuiq;</s> <s xml:id="echoid-s16558" xml:space="preserve"> paruarum li-<lb/>nearum, quæ diſtinctæ ſunt ſuper perpendicularem lineam in ora laminæ:</s> <s xml:id="echoid-s16559" xml:space="preserve"> <lb/>fient igitur ſuper lineam perpendicularem in facie laminæ paruæ tria pun-<lb/>cta.</s> <s xml:id="echoid-s16560" xml:space="preserve"> Deinde perforabimus hanc paruam laminam foramine rotundo, cu-<lb/>ius centrum ſit medium punctum punctorum, quę diſtinguunt lineas, quę <lb/>ſunt in ea:</s> <s xml:id="echoid-s16561" xml:space="preserve"> & ſit medietas diametri eius æqualis alicui uni linearum parua-<lb/>rum:</s> <s xml:id="echoid-s16562" xml:space="preserve"> erit ergo hoc foramen æquale foramini, quod eſt in ora inſtrumenti.</s> <s xml:id="echoid-s16563" xml:space="preserve"> <lb/>Deinde ſignabimus ſuper diametrũ laminæ, ſuper cuius extremitates ſunt <lb/>duæ lineæ perpendiculares:</s> <s xml:id="echoid-s16564" xml:space="preserve"> punctum in medio lineæ, quæ eſt inter centrũ <lb/>laminæ & extremitatem diametri, quæ eſt in parte foraminis:</s> <s xml:id="echoid-s16565" xml:space="preserve"> & faciamus tranſire ſuper hoc pun-<lb/>ctum lineam perpendicularem ſuper diametrum:</s> <s xml:id="echoid-s16566" xml:space="preserve"> deinde ponamus baſim laminę paruæ ſuper han c <lb/>lineam, quouſq;</s> <s xml:id="echoid-s16567" xml:space="preserve"> differentia communis, quę eſt in parua lamina, ſuperponatur huic lineæ perpendi <lb/>culari ſuper diametrum:</s> <s xml:id="echoid-s16568" xml:space="preserve"> & erit punctum, quod diuidit differentiam communem, quę eſt in parua la <lb/>mina, in duo æqualia, poſitum ſuper punctum ſignatum in diametro laminæ.</s> <s xml:id="echoid-s16569" xml:space="preserve"> Hoc autem facto, ap-<lb/>plicetur parua lamina cum maiore, completa applicatione & conſolidatione:</s> <s xml:id="echoid-s16570" xml:space="preserve"> tunc ergo foramen, <lb/>quod eſt in parua lamina, erit oppoſitum foramini, quod eſt in ora inſtrumenti.</s> <s xml:id="echoid-s16571" xml:space="preserve"> Et erit linea intelle-<lb/>cta, quæ copulat centra duorum foraminum, in ſuperficie circuli medij trium circulorum, qui ſunt <lb/>in interiore ora inſtrumenti:</s> <s xml:id="echoid-s16572" xml:space="preserve"> & erit æquidiſtans diametro laminæ:</s> <s xml:id="echoid-s16573" xml:space="preserve"> & erit lamina parua, quæ appli-<lb/>cabitur puncto, quaſi ora aſtrolabij.</s> <s xml:id="echoid-s16574" xml:space="preserve"> Hoc autem completo, ſecetur de ora inſtrumenti quarta, quæ <lb/>ſequitur quartam, in qua eſt foramen ex quatuor quartis diſtinctis per duas primas diametros, per-<lb/>pendiculariter ſe ſecantes, & adæquetur locus ſectionis, donec fiat unus cum ſuperficie laminæ.</s> <s xml:id="echoid-s16575" xml:space="preserve"> <lb/>Deinde accipiamus regulam æris, cuius longitudo non ſit minor, ſed maior uno cubito, & quadra-<lb/>tæ figurę, quam circundent quatuor ſuperficies æquales in latitudine duorum digitorum:</s> <s xml:id="echoid-s16576" xml:space="preserve"> & adæ-<lb/>quentur ſuperficies eius, in quantum poteſt, donec fiant æquales & habentes angulos rectos.</s> <s xml:id="echoid-s16577" xml:space="preserve"> Dein <lb/>de perforetur in medio alicuius ſuperficiei e-<lb/> <anchor type="figure" xlink:label="fig-0238-03a" xlink:href="fig-0238-03"/> ius foramen rotundum, cuius amplitudo ſit <lb/>tãta, ut poſsit recipere corpus, quod eſt in dor-<lb/>ſo inſtrumenti, utreuoluatur in ipſo non leui <lb/>reuolutione, ſed difficili:</s> <s xml:id="echoid-s16578" xml:space="preserve"> & ſit foramen perpen <lb/>diculare ſuper ſuperficiem regulæ, & tranſiens <lb/>ad aliam partem regulæ:</s> <s xml:id="echoid-s16579" xml:space="preserve"> deinde ponamus inſtrumentum ſuper regulam, & mittamus corpus, <lb/>quod eſt in inſtrumenti dorſo, in foramen, quod eſt in medio regulæ, donec ſuperponatur ſuperfi-<lb/>cies inſtrumenti ſuperficiei regulæ.</s> <s xml:id="echoid-s16580" xml:space="preserve"> Hoc autem facto, ſecetur illud, quod ſuperfluit ex extremitati-<lb/>bus regulæ ſuper diametrum laminæ:</s> <s xml:id="echoid-s16581" xml:space="preserve"> nam regula longior eſt, quàm diameter laminæ, quia ſic po-<lb/>ſuimus eam.</s> <s xml:id="echoid-s16582" xml:space="preserve"> Cum ergo ſecuerimus duas ſuperfluitates ex duabus extremitatibus regulæ, reduce-<lb/> <pb o="233" file="0239" n="239" rhead="OPTICAE LIBER VII."/> mus has duas ſuperfluitates, & ponemus illas ſuper duas extremitates regulę, ita ut ponamus duas <lb/>extremitates ſuperfluitatum ſuper duas extremitates illius, quod remanſit de regula, & applicabi-<lb/>mus ſuperficiem extremitatum cum ſuperficie dorſi inſtrumenti:</s> <s xml:id="echoid-s16583" xml:space="preserve"> & erit illud, quod ponetur ex u-<lb/>traq;</s> <s xml:id="echoid-s16584" xml:space="preserve"> duarum ſuperfluitatum ſuper reſiduum regulæ æquale latitudini unius digiti.</s> <s xml:id="echoid-s16585" xml:space="preserve"> Hac autem po-<lb/>ſitione conſiderata eminebunt duæ ſuperfluitates ſuper duas extremitates regulę.</s> <s xml:id="echoid-s16586" xml:space="preserve"> Et ſi perforatum <lb/>fuerit illud, quod ſuperfluit de corpore in dorſo inſtrumenti, & immiſſus fuerit in foramen eius ſti-<lb/>lus ferreus, qui ipſum prohibeat exire, erit melius.</s> <s xml:id="echoid-s16587" xml:space="preserve"> Hoc autem perfecto, perfectum erit inſtrumen-<lb/>tum.</s> <s xml:id="echoid-s16588" xml:space="preserve"> Deinde accipiat experimentator regulam cupream paruæ latitudinis, cuius latitudo ſit dupla <lb/>diametri foraminis, quod eſt in ora inſtrumenti:</s> <s xml:id="echoid-s16589" xml:space="preserve"> & cuius ſpiſsitudo ſit æqualis diametro foraminis, <lb/>& cuius longitudo non ſit minor medietate cubiti:</s> <s xml:id="echoid-s16590" xml:space="preserve"> & uerificabitur regula iſta, donec fiat ualde re-<lb/>cta & uera:</s> <s xml:id="echoid-s16591" xml:space="preserve"> & fiant ſuperficies elus æquales & æquidiſtantes.</s> <s xml:id="echoid-s16592" xml:space="preserve"> Deinde obliquè ſecabimus altera par <lb/>te làtitudinem eius, quouſq;</s> <s xml:id="echoid-s16593" xml:space="preserve"> finis longitudinis eius contineat cum fine latitudinis eius angulum a-<lb/>cutum, ut poſsit ſic facilius declinare & mouere eam quocunq;</s> <s xml:id="echoid-s16594" xml:space="preserve"> quis uoluerit:</s> <s xml:id="echoid-s16595" xml:space="preserve"> & ponet latitudinem <lb/>eius ex alia extremitate perpendicularem ſuper finem longitudinis eius.</s> <s xml:id="echoid-s16596" xml:space="preserve"> Deinde diuidemus hanc <lb/>latitudinem in duo æqualia, & extrahemus ex loco diuiſionis lineam in ſuperficie faciei regulæ, <lb/>quæ extendatur in longitudine eius, & erit perpendicularis ſuper latitudinem eius.</s> <s xml:id="echoid-s16597" xml:space="preserve"> Cum ergo hæc <lb/>regula fuerit ſuperpoſita ſuperficiei laminæ, erit ſuperficies eius ſuperior in ſuperficie circuli me-<lb/>dij trium circulorum figuratorum in interiore ora inſtrumenti.</s> <s xml:id="echoid-s16598" xml:space="preserve"> Nam ſpiſsitudo huius regulæ eſt æ-<lb/>qualis diametro foraminis, & diameter foraminis eſt æqualis perpendiculari exeunti è centro fo-<lb/>raminis, quod eſt in ora inſtrumenti ad ſuperficiem laminæ:</s> <s xml:id="echoid-s16599" xml:space="preserve"> quia diameter foraminis eſt æqua-<lb/>lis duabus lineis trium linearum paruarum, quæ diſtinctæ ſunt de linea perpendiculari in interio-<lb/>re ora inſtrumenti.</s> <s xml:id="echoid-s16600" xml:space="preserve"> Cum ergo hæc regula fuerit erecta ſuper oram ipſius, & fuerit ſuperficies latitu-<lb/>dinis eius ſuper ſuperficiem laminę:</s> <s xml:id="echoid-s16601" xml:space="preserve"> tunc linea deſcripta in medio eius, erit in ſuperficie medij cir-<lb/>culi prædicti:</s> <s xml:id="echoid-s16602" xml:space="preserve"> quia perpendicularis, quę egreditur à quolibet puncto huius lineę ad finem longitu-<lb/>dinis regulę, eſt æqualis perpendiculari, quę egreditur à centro foraminis ad ſuperficiem laminę:</s> <s xml:id="echoid-s16603" xml:space="preserve"> <lb/>nam utraq;</s> <s xml:id="echoid-s16604" xml:space="preserve"> iſtarum perpendicularium eſt æqualis diametro foraminis.</s> <s xml:id="echoid-s16605" xml:space="preserve"/> </p> <div xml:id="echoid-div552" type="float" level="0" n="0"> <figure xlink:label="fig-0238-01" xlink:href="fig-0238-01a"> <variables xml:id="echoid-variables197" xml:space="preserve">h n m ſ a <gap/> s x t r c e d z b g o p q k</variables> </figure> <figure xlink:label="fig-0238-02" xlink:href="fig-0238-02a"> <variables xml:id="echoid-variables198" xml:space="preserve">u g z y x r s t</variables> </figure> <figure xlink:label="fig-0238-03" xlink:href="fig-0238-03a"> <image file="0238-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0238-03"/> </figure> </div> </div> <div xml:id="echoid-div554" type="section" level="0" n="0"> <head xml:id="echoid-head482" xml:space="preserve" style="it">3. Radius medio denſiori perpendicularis, irrefract{us} penetrat. 42. p 2. Idem 17 n 1.</head> <p> <s xml:id="echoid-s16606" xml:space="preserve">CVm ergo experimentator uoluerit experiri tranſitum luminis in aqua per hoc inſtrumen-<lb/>tum:</s> <s xml:id="echoid-s16607" xml:space="preserve"> accipiet uas rectarum orarum, ut cadum cupreum, aut ollam figulinam, aut conſimile:</s> <s xml:id="echoid-s16608" xml:space="preserve"> <lb/>& ſit altitudo orarum eius non minor medietate cubiti:</s> <s xml:id="echoid-s16609" xml:space="preserve"> & ſit diameter circumferentię eius <lb/>non minor diametro inſtrumenti:</s> <s xml:id="echoid-s16610" xml:space="preserve"> & adæquentur orę eius, donec ſuperficies, quę tranſit per oras <lb/>eius, ſit ſuperficies æqualis:</s> <s xml:id="echoid-s16611" xml:space="preserve"> & ponamus in fundo eius corpus diuerſarum partium aut diuerſorum <lb/>colorum, ut annulum, aut argentum depictum, aut depingatur in fundo eius pictura manifeſta:</s> <s xml:id="echoid-s16612" xml:space="preserve"> de-<lb/>inde infundatur aqua clara in uas, donec impleatur:</s> <s xml:id="echoid-s16613" xml:space="preserve"> & expectetur donec motus eius quieſcat.</s> <s xml:id="echoid-s16614" xml:space="preserve"> Cum <lb/>ergo motus eius quieuerit, erigatur aſpiciens, aut ſedeat erectus, & aſpiciat ad uas, & apponat ui-<lb/>ſum ſuum corpori, quod eſt in fundo aquę, aut picturę, quę eſt in fundo aquę, donec linea inter ui-<lb/>ſum & medium illius corporis aut picturę illius, ſit perpendicularis ſuper ſuperficiem aquę quò ad <lb/>ſenſum, & aſpiciat corpus, quod eſt in fundo, aut picturam:</s> <s xml:id="echoid-s16615" xml:space="preserve"> tunc inueniet illam eo modo, quo eſt, & <lb/>inueniet ordinationem ſuarum partium inter ſe eo modo, quo ordinarentur, ſi aſpiceret illud, cum <lb/>uas eſſet uacuum.</s> <s xml:id="echoid-s16616" xml:space="preserve"> Hoc autem declarato, certificatur, quòd illud, quod comprehenditur in fundo a-<lb/>quæ, cum aſpexerit illud eadem poſitione, qua aſpexit corpus, quòd eſt in fundo aquæ, aut pictu-<lb/>ram:</s> <s xml:id="echoid-s16617" xml:space="preserve"> comprehenditur ſecundum ordinationem ſuarum partium.</s> <s xml:id="echoid-s16618" xml:space="preserve"> Hoc autem certificato, ſi quis uo-<lb/>luerit experiri tranſitum lucis:</s> <s xml:id="echoid-s16619" xml:space="preserve"> eligat locum, ſuper quem oritur lux ſolis, in quo ponat uas, & obſer-<lb/>uet, ut ſuperficies circumferentię uaſis ſit æquidiſtans horizonti:</s> <s xml:id="echoid-s16620" xml:space="preserve"> hoc autem poteſt obſeruari hoc <lb/>modo:</s> <s xml:id="echoid-s16621" xml:space="preserve"> ut ſit circumferentia ſuperficiei aquę æquidiſtans circumferentię uaſis:</s> <s xml:id="echoid-s16622" xml:space="preserve"> & ſi intus in uaſè <lb/>prope circumferentiam eius fuerit ſignatus circulus, æquidiſtans circumferentię uaſis, erit melius <lb/>ad hoc, ut circumferentia ſuperficiei aquę comparetur ad circumferentiam circuli.</s> <s xml:id="echoid-s16623" xml:space="preserve"> Deinde expe-<lb/>rimentator debet imponere inſtrumentum rotundum intra hoc uas, ita ut duę regulę paruę po-<lb/>ſitæ ſuper duo extrema regulæ maioris, ſuperponantur oræ uaſis ex utraque parte:</s> <s xml:id="echoid-s16624" xml:space="preserve"> tunc medietas <lb/>inſtrumenti, & regula extenſa in longitudine inſtrumenti erunt intra uas:</s> <s xml:id="echoid-s16625" xml:space="preserve"> deinde addatur aqua, aut <lb/>diminuatur de ea, donec fiat ſuperficies aquę una cum centro inſtrumenti:</s> <s xml:id="echoid-s16626" xml:space="preserve"> & ſit aqua clara:</s> <s xml:id="echoid-s16627" xml:space="preserve"> deinde <lb/>reuoluatur inſtrumentum in circuitu uaſis, donec obumbretur illud, quod eſt intra aquam ex oris <lb/>eius:</s> <s xml:id="echoid-s16628" xml:space="preserve"> tunc teneatur regula altera manu, & reuoluatur reliqua manu inſtrumentum ſuper ſe in cir-<lb/>cuitu centri eius, donec foramen, quod eſt in ora inſtrumenti, ſit oppoſitum corpori ſolis, & tran-<lb/>ſeat lumen ſolis per foramen oræ inſtrumenti, & perueniat ad alterum foramen tabulę paruæ, & <lb/>tranſeat per illud.</s> <s xml:id="echoid-s16629" xml:space="preserve"> Cum ergo pertranſierit forma lucis per duo foramina, perueniet ad fundum a-<lb/>quæ:</s> <s xml:id="echoid-s16630" xml:space="preserve"> tunc experimentator obſeruabit, ut ſitus lucis in regula de ſecundo foramine, ſit ſitus æqua-<lb/>lis:</s> <s xml:id="echoid-s16631" xml:space="preserve"> hoc autem ſitu præſeruato, & luce perueniente ad ſuperficiem aquæ, auferat experimentator <lb/>manus ſuas ab inſtrumento, & ſtet uel ſedeat erectus, & inſpiciat ad fundum aquæ, ex quarta, cu-<lb/>ius oræ ſunt abſciſſę, & ſeruet poſitionem, quam ſeruauerat, cum aſpexerat corpus, quod erat in <lb/>fundo aquæ, ut ſit certus, quòd illud, quod uidet, eſt, ſecundum quod eſt:</s> <s xml:id="echoid-s16632" xml:space="preserve"> tunc ergo cum intue-<lb/>bitur illud, quod eſt intra aquam de ora inſtrumenti:</s> <s xml:id="echoid-s16633" xml:space="preserve"> inueniet lumen pertranſiens ex duobus fo-<lb/> <pb o="234" file="0240" n="240" rhead="ALHAZEN"/> raminibus ſuper ſuperficiem oræ inſtrumenti, quæ eſt intra aquam:</s> <s xml:id="echoid-s16634" xml:space="preserve"> & inueniet lumen inter duos <lb/>circulos æquidiſtãtes extremos de tribus circulis ſignatis in interiore parte inſtrumẽti oræ:</s> <s xml:id="echoid-s16635" xml:space="preserve"> aut ad-<lb/>detur ſuper diſtantiam, quæ eſt inter circulos, modicùm:</s> <s xml:id="echoid-s16636" xml:space="preserve"> & erit additio eius ex duobus lateribus cir <lb/>culorum æqualis.</s> <s xml:id="echoid-s16637" xml:space="preserve"> Sequitur ergo ex poſitione, quòd punctum, quod eſt in medio luminis appa-<lb/>rentis intra aquam, quod eſt ſuper interiorem partem oræ inſtrumenti, ſit per medium circulum <lb/>trium circulorum æquidiſtantium, qui ſunt in interiore parte oræ inſtrumenti.</s> <s xml:id="echoid-s16638" xml:space="preserve"> Et hoc lumen, quod <lb/>eſt intra aquam, erit manifeſtum, quòd ora ſuperior inſtrumenti, quæ circumdat ſuperius fora-<lb/>men, obumbrat interiorem partem oræ inſtrumenti, quæ circundat lumen, quod eſt in interiore <lb/>parte oræ inſtrumenti.</s> <s xml:id="echoid-s16639" xml:space="preserve"> Et ſic in illo loco non erit ex interiore parte oræ inſtrumenti aliquid de lu-<lb/>mine ſolis, niſi lumen, quod exit ex duobus foraminibus.</s> <s xml:id="echoid-s16640" xml:space="preserve"> Deinde experimentator accipiat li-<lb/>gnum minutum, ſiue acum, & applicet eam in exteriore parte ſuperioris foraminis, quod eſt in o-<lb/>ra inſtrumenti, & obſeruet, ut acus tranſeat per medium foraminis:</s> <s xml:id="echoid-s16641" xml:space="preserve"> Deinde aſpiciat ſupra uas, & <lb/>ſeruet poſitionem, quam menſurauit prius:</s> <s xml:id="echoid-s16642" xml:space="preserve"> tunc uidebit umbram acus in medio lucis:</s> <s xml:id="echoid-s16643" xml:space="preserve"> deinde in-<lb/>curuet acum attrahendo ipſam, donec extremitas eius ſit in medio foraminis, & intueatur lumen, <lb/>quod eſt intra aquam, & quod eſt in ſuperficie aquæ:</s> <s xml:id="echoid-s16644" xml:space="preserve"> tunc inueniet umbram extremitatis acus in <lb/>medio lucis, quæ eſt intra aquam, & in medio lucis, quæ eſt in ſuperficie aquæ.</s> <s xml:id="echoid-s16645" xml:space="preserve"> Deinde mutet po-<lb/>ſitionem acus, & ponat extremitatem eius etiam apud medium foraminis, & intueatur umbram:</s> <s xml:id="echoid-s16646" xml:space="preserve"> <lb/>tunc inueniet umbram extremitatis acus apud medium lucis:</s> <s xml:id="echoid-s16647" xml:space="preserve"> deinde leuet acum:</s> <s xml:id="echoid-s16648" xml:space="preserve"> & inueniet lu-<lb/>cem redeuntem ad ſuum ſtatum intra aquam, & in ſuperficie aquæ.</s> <s xml:id="echoid-s16649" xml:space="preserve"> Deinde applicet acum in late-<lb/>re foraminis, & ponat eam chordam in foramine, non diametrum, & intueatur lumen, quod eſt in-<lb/>tra aquam, & in ſuperficie aquæ:</s> <s xml:id="echoid-s16650" xml:space="preserve"> tunc inueniet in utroque illorum umbram, quæ eſt chorda:</s> <s xml:id="echoid-s16651" xml:space="preserve"> deìn-<lb/>de leuet acum:</s> <s xml:id="echoid-s16652" xml:space="preserve"> tunc inueniet lumen rediens ad locum ſuum:</s> <s xml:id="echoid-s16653" xml:space="preserve"> & ſi mutauerit ſitum acus in lateribus <lb/>foraminis:</s> <s xml:id="echoid-s16654" xml:space="preserve"> inueniet umbram ſemper in latere luminis.</s> <s xml:id="echoid-s16655" xml:space="preserve"> Declarabitur ergo ex hac experientia, quòd <lb/>ad punctum, quod eſt in medio lucis, quæ eſt intra aquam, in circumferentia medij circuli, non exi <lb/>uit lux, niſi ex puncto, quod eſt medium lucis, quæ eſt in ſuperficie aquæ:</s> <s xml:id="echoid-s16656" xml:space="preserve"> & quòd ad punctum, <lb/>quod eſt medium lucis, quæ eſt in ſuperficie aquæ, non exiuit lux, niſi ex puncto, quod eſt centrum <lb/>foraminis ſuperioris, & tranſiuit per centrum foraminis inferioris, quod eſt in oris alijs.</s> <s xml:id="echoid-s16657" xml:space="preserve"> Nam ſi <lb/>non tranſiſſet per centrum foraminis inferioris, non manifeſtaretur medium lucis, quæ eſt in ſuper <lb/>ficie aquæ, cum acus eſſet in medio foraminis inferioris, ſed non manifeſtaretur de luce, quæ eſt <lb/>in ſuperficie aquæ, niſi locus alius à centro eius.</s> <s xml:id="echoid-s16658" xml:space="preserve"> Lux ergo, quæ peruenit ad punctum, quod eſt cen-<lb/>trum lucis, quæ eſt in ſuperficie aquæ, & lux, quæ extenditur in aere, non extenditur niſi ſecun-<lb/>dum lineas rectas.</s> <s xml:id="echoid-s16659" xml:space="preserve"> Luxergo, quæ tranſit per centra duorum foraminum, extenditur ſecundum <lb/>rectitudinem lineę tranſeuntis per centra duorum foraminum:</s> <s xml:id="echoid-s16660" xml:space="preserve"> hæc autem lux eſt illa, quę peruenit <lb/>ad medium lucis, quæ eſt in ſuperficie aquæ.</s> <s xml:id="echoid-s16661" xml:space="preserve"> Punctum ergo, quod eſt in medio lucis, quæ eſt in ſu-<lb/>perficie aquæ, eſt in linea recta tranſeunte per centra duorum foraminum.</s> <s xml:id="echoid-s16662" xml:space="preserve"> Et hæc linea eſt in ſu-<lb/>perficie medij circuli de tribus circulis ſignatis in interiore parte oræ inſtrumenti:</s> <s xml:id="echoid-s16663" xml:space="preserve"> & eſt illius dia-<lb/>meter, quia hæc linea eſt æquidiſtans diametro circuli, qui eſt in ſuperficie laminæ.</s> <s xml:id="echoid-s16664" xml:space="preserve"> Cum ergo pun-<lb/>ctum, quod eſt in medio lucis, quæ eſt in ſuperficie aquæ, fuerit ſuper hanc lineam:</s> <s xml:id="echoid-s16665" xml:space="preserve"> tunc illud <lb/>punctum eſt in ſuperficie circuli medij prædicti:</s> <s xml:id="echoid-s16666" xml:space="preserve"> punctum autem, quod eſt in medio lucis, quæ eſt <lb/>intra aquam, eſt in circumferentia medij circuli:</s> <s xml:id="echoid-s16667" xml:space="preserve"> ergo hæc duo puncta ſunt in ſuperficie medij circu <lb/>li.</s> <s xml:id="echoid-s16668" xml:space="preserve"> Si ergo lux, quæ eſt in ſuperficie aquæ, latuerit, & non fuerit bene manifeſta:</s> <s xml:id="echoid-s16669" xml:space="preserve"> tunc experimenta-<lb/>tor mittet illam minorem regulam in aquam, & applicet oram eius in ſuperficie laminæ, & ponat <lb/>ſuperficiem, in qua ſignata eſt linea, ſequentem ſuperficiem aquæ, & moueat eam, donec ſuperfi-<lb/>cies eius fiat cum ſuperficie aquæ.</s> <s xml:id="echoid-s16670" xml:space="preserve"> Cum ergo ſuperficies regulæ fuerit cum ſuperficie aquæ, & fue-<lb/>rit regula erecta ſuper oram eius:</s> <s xml:id="echoid-s16671" xml:space="preserve"> tunc linea, quæ eſt in ſuperficie ipſius, erit in ſuperficie circuli <lb/>medij, qui tranſit per centra duorum foraminum:</s> <s xml:id="echoid-s16672" xml:space="preserve"> hac autem poſitione præſeruata:</s> <s xml:id="echoid-s16673" xml:space="preserve"> apparebit lux, <lb/>quæ eſt in ſuperficie aquæ, ſuper ſuperficiem regulæ, & inueniet medium lucis ſuper lineam, <lb/>quæ eſt in medio regulæ.</s> <s xml:id="echoid-s16674" xml:space="preserve"> Et ſi aeus ſit poſita ſuper medium ſuperioris foraminis:</s> <s xml:id="echoid-s16675" xml:space="preserve"> tunc linea, quę eſt <lb/>in medio regulæ, obumbrabitur:</s> <s xml:id="echoid-s16676" xml:space="preserve"> & ſi extremitas acus fuerit poſita ſuper centrum foraminis, ap-<lb/>parebit umbra extremitatis acus in medio lucis, quæ eſt ſuper regulam:</s> <s xml:id="echoid-s16677" xml:space="preserve"> & ſi acus fuerit ablata, <lb/>redibit lux, ſicut erat.</s> <s xml:id="echoid-s16678" xml:space="preserve"> Cum hac ergo regula apparebit lux, quæ eſt in ſuperficie aquæ, apparitio-<lb/>ne manifeſta, & manifeſtabitur, quòd eſt ſupra lineam tranſeuntem per centra duorum forami-<lb/>num:</s> <s xml:id="echoid-s16679" xml:space="preserve"> & iam poſueramus ſuperficiem aquæ apud centrum laminæ.</s> <s xml:id="echoid-s16680" xml:space="preserve"> Cum ergo ſuperficies regulæ <lb/>fuerit cum ſuperficie aquæ:</s> <s xml:id="echoid-s16681" xml:space="preserve"> tranſibit ſuperficies regulæ per centrum laminæ:</s> <s xml:id="echoid-s16682" xml:space="preserve"> & tunc erit remo-<lb/>tio centri lucis à centro laminæ æqualis medietati latitudinis regulæ, quæ eſt æqualis perpendicu-<lb/>lari cadenti à centro foraminis ſuper ſuperficiem laminæ:</s> <s xml:id="echoid-s16683" xml:space="preserve"> & ſic erit centrum lucis, quę eſt in ſuper-<lb/>ficie regulę, centrum circuli medij.</s> <s xml:id="echoid-s16684" xml:space="preserve"> Deinde oportet experimentatorem auferre regulam ſubtilem, <lb/>& mittere eam iterum in aquam, & applicare ſuperficiem latitudinis eius cum ſuperficie lami-<lb/>næ, & ponere angulum eius acutum apud centrum lucis, quæ eſt intra aquam, ſcilicet angulum, <lb/>qui eſt in ſuperficie eius ſuperiore:</s> <s xml:id="echoid-s16685" xml:space="preserve"> deinde moueat regulam, donec acuitas eius inferior tranſeat <lb/>per centrum laminæ, & ſic acuitas eius ſuperior tranſibit per centrum circuli medij.</s> <s xml:id="echoid-s16686" xml:space="preserve"> Punctum <lb/>ergo ex linea ſuperiore regulę, quod eſt in ſuperficie aquę, eſt centrum circuli medij:</s> <s xml:id="echoid-s16687" xml:space="preserve"> eſt ergo cen-<lb/>trum lucis, quæ eſt in ſuperficie aquæ:</s> <s xml:id="echoid-s16688" xml:space="preserve"> & erit longitudo eius diametri ex diametris medij circuli.</s> <s xml:id="echoid-s16689" xml:space="preserve"> <lb/>Hac autem ratione præſeruata, accipiat experimentator acum longam:</s> <s xml:id="echoid-s16690" xml:space="preserve"> & mittat eam in aquam:</s> <s xml:id="echoid-s16691" xml:space="preserve"> <lb/> <pb o="235" file="0241" n="241" rhead="OPTICAE LIBER VII."/> & ponat caput ſuum in punctum ultimitatis regulæ:</s> <s xml:id="echoid-s16692" xml:space="preserve"> & intueatur lucem, quæ eſt intra aquam:</s> <s xml:id="echoid-s16693" xml:space="preserve"> <lb/>tunc inueniet umbram acus ſecantem lucem:</s> <s xml:id="echoid-s16694" xml:space="preserve"> & inueniet umbram capitis acus apud cornu regu-<lb/>læ, quod eſt apud medium lucis.</s> <s xml:id="echoid-s16695" xml:space="preserve"> Deinde mutet poſitionem acus, & caput eius ſit in loco eius ex <lb/>fine regulæ:</s> <s xml:id="echoid-s16696" xml:space="preserve"> tunc mutabitur ſitus umbræ ex luce, quæ eſt intra aquam:</s> <s xml:id="echoid-s16697" xml:space="preserve"> & erit umbra capitis acus <lb/>inſeparabilis à medio lucis:</s> <s xml:id="echoid-s16698" xml:space="preserve"> deinde auferat acum & redibit lux ad locum ſuum.</s> <s xml:id="echoid-s16699" xml:space="preserve"> Deinde mittat a-<lb/>cum in aquam iterum, & ponat caput eius in alio puncto finis regulę, & intueatur umbram, donec <lb/>inueniat ſecãtem lucem, quæ eſt intra aquam:</s> <s xml:id="echoid-s16700" xml:space="preserve"> & inueniet umbram capitis acus in medio lucis.</s> <s xml:id="echoid-s16701" xml:space="preserve"> De-<lb/>inde mutet poſitionem acus ſuper multitudinem punctorum ex acuitate regulæ:</s> <s xml:id="echoid-s16702" xml:space="preserve"> & inueniet um-<lb/>bram capitis eius ſemper in medio lucis.</s> <s xml:id="echoid-s16703" xml:space="preserve"> Declarabitur ergo ex hac experientia declaratione mani-<lb/>feſta, quòd lux, quæ eſt in puncto mediante lucem, quæ eſt intra aquam, quę eſt ſuper circumferen-<lb/>tiam medij circuli:</s> <s xml:id="echoid-s16704" xml:space="preserve"> peruenit ad illud punctum à puncto, quod eſt mediũ lucis, quæ eſt in ſuperficie <lb/>aquæ.</s> <s xml:id="echoid-s16705" xml:space="preserve"> Et declarabitur cum hoc, quòd hæc lux extenditur ſuper lineam rectam, quę eſt finis regulæ.</s> <s xml:id="echoid-s16706" xml:space="preserve"> <lb/>Nam experientia eius per extremitatem acus ex diuerſis locis in fine regulę oſtendit illã tranſeun-<lb/>tem per omne punctum finis regulæ.</s> <s xml:id="echoid-s16707" xml:space="preserve"> Hac ergo uia experimentabitur tranſitus lucis per corpus <lb/>aquæ:</s> <s xml:id="echoid-s16708" xml:space="preserve"> ex quo declarabitur, quòd extenſio lucis per corpus aquæ eſt ſecundum uerticationes re-<lb/>ctarum linearum.</s> <s xml:id="echoid-s16709" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div555" type="section" level="0" n="0"> <head xml:id="echoid-head483" xml:space="preserve" style="it">4. Radi{us} medio denſiori obliqu{us}, refringitur ad perpendicularem à refractionis puncto <lb/>excitatam. 43 p 2. Idem 17 n 1.</head> <p> <s xml:id="echoid-s16710" xml:space="preserve">DEinde oportebit experimentatorem ponere ſuper centrum lucis ſignum fixum cũ ſculptio-<lb/>ne:</s> <s xml:id="echoid-s16711" xml:space="preserve"> deinde quan do experimentator intuebitur punctum, quod eſt in medio lucis, quę eſt in-<lb/>tra aquam:</s> <s xml:id="echoid-s16712" xml:space="preserve"> inueniet ipſum nõ æquidiſtans duabus extremitatibus diametri laminæ, ſed ex-<lb/>tra duas lineas perpendiculares, quę ſunt ſuper extremitatẽ diametri laminæ, quę eſt intra aquam:</s> <s xml:id="echoid-s16713" xml:space="preserve"> <lb/>& inueniet declinationem eius ab iſta linea ad partẽ, in qua eſt ſol:</s> <s xml:id="echoid-s16714" xml:space="preserve"> & inueniet inter punctum, quod <lb/>eſt centrum mediæ lucis, & punctum;</s> <s xml:id="echoid-s16715" xml:space="preserve"> quod eſt communis differentia lineæ perpendiculari ſuper <lb/>extremitatem diametri laminæ, & puncto medio, quod eſt extremitas diametri medij circuli, tran-<lb/>ſeuntis per centrum foraminis:</s> <s xml:id="echoid-s16716" xml:space="preserve"> inueniet dico, diſtantiam ſenſibilem.</s> <s xml:id="echoid-s16717" xml:space="preserve"> Hoc declarato, oportet mitte-<lb/>re regulam ſubtilem in a quam, & applicare eam cum ſuperficie laminæ, & ponere terminum regu-<lb/>læ ſuper centrum laminę, & mouere regulam, quouſq;</s> <s xml:id="echoid-s16718" xml:space="preserve"> acuitas eius ſit perpendicularis ſuper ſuper-<lb/>ficiem aquæ, quò ad ſenſum:</s> <s xml:id="echoid-s16719" xml:space="preserve"> tune igitur inueniet centrũ lucis, quę eſt intra aquam, inter acuitatem <lb/>regulæ & lineam perpendicularem ſuper diametrum laminæ.</s> <s xml:id="echoid-s16720" xml:space="preserve"> Declarabitur ergo ex hoc, quòd hæc <lb/>refractio eſt ad partem perpendicularis, exẽuntis à loco refractionis perpendicularis ſuper ſuper-<lb/>ficiem aquæ.</s> <s xml:id="echoid-s16721" xml:space="preserve"> Cum ergo certus fuerit experimentator de hoc:</s> <s xml:id="echoid-s16722" xml:space="preserve"> oportebit eum ſignare apud extremi-<lb/>tatem regulæ, quę eſt ſuper circumferentiam medij circuli, quę eſt extremitas perpendicularis, ex-<lb/>euntis à centro medij circuli perpendicularis ſuper ſuperficiem aquæ, ſignum fixum, ut primum, <lb/>quod ſignatum eſt apud centrum lucis.</s> <s xml:id="echoid-s16723" xml:space="preserve"> Et iam declaratum eſt, quòd lux, quę peruenit ad punctum, <lb/>quod eſt centrum lucis, quæ eſt intra aquam, eſt lux extenſa ſecundum rectitudinem lineæ conti-<lb/>nuantis duo centra foraminum:</s> <s xml:id="echoid-s16724" xml:space="preserve"> & hæc linea peruenit ad cẽtrum medij circuli æquidiſtantis ſuper-<lb/>ficiei laminæ:</s> <s xml:id="echoid-s16725" xml:space="preserve"> & eſt illius diameter.</s> <s xml:id="echoid-s16726" xml:space="preserve"> Si hęc linea fuerit extenſa in imaginatione ſecundum rectitudi-<lb/>nem intra aquam, donec perueniat ad oram laminæ:</s> <s xml:id="echoid-s16727" xml:space="preserve"> tunc igitur erit æquidiſtans diametro laminæ, <lb/>& perueniet ad lineam perpendicularem in interiore parte oræ laminæ.</s> <s xml:id="echoid-s16728" xml:space="preserve"> Et cum centrum lucis, quę <lb/>eſt intra aquam, non eſt ſuper perpendicularem lineam oræ laminæ:</s> <s xml:id="echoid-s16729" xml:space="preserve"> tunc lux, quę extenditur à me-<lb/>dio lucis, quę eſt in ſuperficie aquæ, ad medium lucis, quæ eſt intra aquam, non extenditur ſecun-<lb/>dum rectitudinem lineæ tranſeuntis per centra duorum foraminum, ſed refringitur.</s> <s xml:id="echoid-s16730" xml:space="preserve"> Declaratum <lb/>eſt autem, quòd hæc lux extẽditur rectè à medio lucis, quę eſt in ſuperficie aquæ, ad medium lucis, <lb/>quæ eſt intra aquam.</s> <s xml:id="echoid-s16731" xml:space="preserve"> Ergo refractio huius lucis eſt apud ſuperficiem aquæ.</s> <s xml:id="echoid-s16732" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div556" type="section" level="0" n="0"> <head xml:id="echoid-head484" xml:space="preserve" style="it">5. Radij incidentiæ & refractionis ſunt in uno plano. 46 p 2.</head> <p> <s xml:id="echoid-s16733" xml:space="preserve">ET iam declaratum eſt, quòd hæc lux tranſit per centra duorum foraminum, & per medium <lb/>lucis, quæ eſt in ſuperficie aquæ, quod eſt centrum circuli medij, æquidiſtantis ſuperficiei la-<lb/>minæ, & per medium lucis, quæ eſt intrà aquam, quod eſt in circumferentia medij circuli.</s> <s xml:id="echoid-s16734" xml:space="preserve"> Ex <lb/>quo patet, quòd lumen perueniens ad centrum lucis, quæ eſt intra aquam, dum extenditur in aere, <lb/>& poſtquam refringitur intra aquã, eſt in eadem ſuperficie æquali, ſcilicet in ſuperficie circuli me-<lb/>dij trium circulorũ, qui ſunt in interiore parte oræ inſtrumenti.</s> <s xml:id="echoid-s16735" xml:space="preserve"> Et refractio hæc inuenitur, quando <lb/>linea tranſiens per centra foraminum fuerit decliuis ſuper ſuperficiem aquæ, non perpendicularis.</s> <s xml:id="echoid-s16736" xml:space="preserve"> <lb/>Et nunquam erit hæc linea perpẽdicularis ſuper ſuperficiem aquæ in hora tranſitus lucis ſolis, niſi <lb/>quando fuerit ſol in uertice capitis:</s> <s xml:id="echoid-s16737" xml:space="preserve"> & hoc erit in aliquibus locis, & non in omnibus:</s> <s xml:id="echoid-s16738" xml:space="preserve"> & in quibuſ-<lb/>dam temporibus, non in omnibus:</s> <s xml:id="echoid-s16739" xml:space="preserve"> neq;</s> <s xml:id="echoid-s16740" xml:space="preserve"> tranſit ſol per uerticem capitis habitantium in pluribus lo-<lb/>cis habitationis:</s> <s xml:id="echoid-s16741" xml:space="preserve"> & in quibus tranſit:</s> <s xml:id="echoid-s16742" xml:space="preserve"> in iſtis locis diſtinguetur hęc experimentatio in omni tempo-<lb/>re:</s> <s xml:id="echoid-s16743" xml:space="preserve"> illi autem ſuper quorum zenit h tranſit ſol, ſi uoluerint hoc experiri, cauebũt tempus, in quo ſol <lb/>tranſit per capita eorum.</s> <s xml:id="echoid-s16744" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div557" type="section" level="0" n="0"> <head xml:id="echoid-head485" xml:space="preserve" style="it">6. Radi{us} medio rariori perpendιcularis, irrefract{us} penetrat. 44 p 2.</head> <p> <s xml:id="echoid-s16745" xml:space="preserve">ITẽ accipiat exքimẽtator fruſta uitri clari, quorũ figuræ ſint cubicæ:</s> <s xml:id="echoid-s16746" xml:space="preserve"> & ſit lõgitudo uniuſcuiuſq;</s> <s xml:id="echoid-s16747" xml:space="preserve"> <lb/> <pb o="236" file="0242" n="242" rhead="ALHAZEN"/> eorum dupla diametri foraminis, quod eſt in ora inſtrumenti:</s> <s xml:id="echoid-s16748" xml:space="preserve"> & adæquẽtur ſuperficies eorum ue-<lb/>hementer per cõfricationem, quouſq;</s> <s xml:id="echoid-s16749" xml:space="preserve"> ſint æquales & æquidiſtantes, & latera ſint recta:</s> <s xml:id="echoid-s16750" xml:space="preserve"> deinde po-<lb/>liantur.</s> <s xml:id="echoid-s16751" xml:space="preserve"> Hoc autem completo, ſignetur in medio laminæ linea recta tranſiens per cẽtrum eius:</s> <s xml:id="echoid-s16752" xml:space="preserve"> & ſit <lb/>perpendicularis ſuper diametrum eius, ſuper cuius extrema ſunt lineæ duæ perpendiculares in in-<lb/>teriore p̀arte oræ inſtrumenti, & tranſeat in utramq;</s> <s xml:id="echoid-s16753" xml:space="preserve"> partẽ:</s> <s xml:id="echoid-s16754" xml:space="preserve"> & ſignetur hæc linea ferro, ut deſcendat <lb/>in corpus laminæ, & remaneat ibi.</s> <s xml:id="echoid-s16755" xml:space="preserve"> Deinde ponàt unum uitrorum cubicorum ſuper ſuperficiem la-<lb/>minæ, & applicet unum latus ſuorum laterum cum hac perpendiculari, & ponat mediũ lateris ui-<lb/>tri uerè ſuper centrum laminæ, & ponat corpus uitri ex parte foraminum.</s> <s xml:id="echoid-s16756" xml:space="preserve"> Tranſibit ergo diameter <lb/>laminæ, ſuper cuius extrema ſunt duæ lineæ perpendiculares, per mediũ ſuperficiei uitri ſuperpo-<lb/>ſitæ laminæ.</s> <s xml:id="echoid-s16757" xml:space="preserve"> Hac poſitione præſeruata, applicetur uitrum applicatione fixa per glutinũ tali modo, <lb/>ut poſsit euelli:</s> <s xml:id="echoid-s16758" xml:space="preserve"> deinde accipiatur alterũ uitrum, & ponatur ultra primum, ſcilicet ex parte forami-<lb/>num, & applicetur aliqua ſuperficierũ eius ſuperficiei primi uitri:</s> <s xml:id="echoid-s16759" xml:space="preserve"> hoc præſeruato, applicetur ſecun <lb/>dum uitrum laminæ applicatione fixa:</s> <s xml:id="echoid-s16760" xml:space="preserve"> deinde accipiatur tertium uitrum, & applicetur ſecundo ui-<lb/>tro, & adæquetur ſuperficies eius cum duabus ſuperficiebus laterum ſecũdi uitri, & applicetur la-<lb/>minæ:</s> <s xml:id="echoid-s16761" xml:space="preserve"> & ſic fiat de pluribus uitris, quouſq;</s> <s xml:id="echoid-s16762" xml:space="preserve"> perueniãt uitra ad oram perpendicularium ſuper ſuper-<lb/>ficiem inſtrumẽti, aut prope.</s> <s xml:id="echoid-s16763" xml:space="preserve"> Cum ergo uitra fuerint applicata ſuperficiei laminæ ſecundũ poſitio-<lb/>nem prædictam:</s> <s xml:id="echoid-s16764" xml:space="preserve"> tranſibit diameter laminæ, ſuper cuius extremitates ſunt duæ lineæ perpẽdicula-<lb/>res in extremitate inſtrumẽtí, per mediam ſuperficiem uitrorum ſuperpoſitorum laminæ:</s> <s xml:id="echoid-s16765" xml:space="preserve"> altitudo <lb/>autem iſtorum uitrorũ in latitudine eſt dupla diametri foraminis:</s> <s xml:id="echoid-s16766" xml:space="preserve"> ſed diameter foraminis eſt æqua <lb/>lis perpendiculari exeunti à centro foraminis ſuper ſuperficiẽ laminæ & ſuper diametrũ eius:</s> <s xml:id="echoid-s16767" xml:space="preserve"> ergo <lb/>unaquæq;</s> <s xml:id="echoid-s16768" xml:space="preserve"> perpendiculariũ exeuntium à centris ſuperficierum uitrorũ, ſcιlicet ſuperficierum per-<lb/>pendicularium ſuper ſuperficiẽ laminæ, ſecãtium diametrum oppoſitam duobus foramínibus, eſt <lb/>æqualís perpẽdiculari exeunti à centro foraminis ſuper ſuperficiẽ laminæ, & ſuper diametrũ lami-<lb/>næ:</s> <s xml:id="echoid-s16769" xml:space="preserve"> & cadẽt perpẽdiculares exeuntes à cẽtris ſuperficierũ uitrorũ ad ſuperficiũ laminæ, ſuper dia-<lb/>metrum laminæ, ſuper cuius extremitates eſt perpẽdicularis, egrediens à centro foraminis.</s> <s xml:id="echoid-s16770" xml:space="preserve"> Linea <lb/>ergo tranſiens per centra dubrũ foraminum, ſi extendatur in imaginatione ſecundũ rectitudinem, <lb/>tranſibit per cẽtra ſuperficierum uitrorum, ſcilicet ſuperficierum perpendiculariũ ſuper ſuperficiẽ <lb/>laminæ oppoſitæ duobus foraminibus.</s> <s xml:id="echoid-s16771" xml:space="preserve"> Deinde experimẽtator accipiat regulã ſubtilẽ prædictã:</s> <s xml:id="echoid-s16772" xml:space="preserve"> & <lb/>erigat eam ſuper oram ipſius in ſuperficie laminę:</s> <s xml:id="echoid-s16773" xml:space="preserve"> & ponat faciẽ eius, in qua ſignata eſt linea ex par-<lb/>te primi uitri, quod eſt ſuper centrum laminæ, & ponat regulam prope uitrum, & ponat finem lon-<lb/>gitudinis regulæ ſecantem diametrum laminæ perpendiculariter.</s> <s xml:id="echoid-s16774" xml:space="preserve"> Hoc autem præſeruato, applicet <lb/>regulam laminæ applicatione fixa, íta ut poſsit euelli:</s> <s xml:id="echoid-s16775" xml:space="preserve"> hac autẽ poſitione præſeruata in regula:</s> <s xml:id="echoid-s16776" xml:space="preserve"> tunc <lb/>linea, quæ eſt in ſuperficie regulæ, erit in ſuperficie medij circuli ex tribus circulis, ſignatis in inte-<lb/>riore parte oræ inſtrnmenti:</s> <s xml:id="echoid-s16777" xml:space="preserve"> & tranſibit linea recta per cẽtra duorum foraminum, & per media ſu-<lb/>perficierum uitrorum, ſecans lineam, quę eſt in regula.</s> <s xml:id="echoid-s16778" xml:space="preserve"> Hoc toto completo, ponatur inſtrumentum <lb/>in uas prædictum:</s> <s xml:id="echoid-s16779" xml:space="preserve"> ſit autem uas uacuum aqua:</s> <s xml:id="echoid-s16780" xml:space="preserve"> & ponatur uas in ſole, & moueatur inſtrumentum, <lb/>quouſq;</s> <s xml:id="echoid-s16781" xml:space="preserve"> lux ſolis tranſeat per duo foramina:</s> <s xml:id="echoid-s16782" xml:space="preserve"> & ſit lux apud ſecundum foramen æqualis:</s> <s xml:id="echoid-s16783" xml:space="preserve"> tunc igitur <lb/>intueatur experimẽtator ſuperficiem regulæ oppoſitam uitro:</s> <s xml:id="echoid-s16784" xml:space="preserve"> & inueniet lucem exeuntem à duo-<lb/>bus foraminibus ſuper ſuperficiem regulæ:</s> <s xml:id="echoid-s16785" xml:space="preserve"> & inueniet illud, quod circundat lucem ex ſuperficie <lb/>regulæ, obumbratum umbra oræ inſtrumenti:</s> <s xml:id="echoid-s16786" xml:space="preserve"> & inueniet centrum uiſus ſuper lineam, quæ eſt in <lb/>ſuperficiè regulæ.</s> <s xml:id="echoid-s16787" xml:space="preserve"> Hoc autem declarato, accipiat feſtucam ſubtilem, uel acum, & ponat illam ſuper <lb/>ſuperius foramen, & ponat extremitatem perpendiculariter ſuper centrum foraminis, & intueatur <lb/>lucem, quæ eſt ſuper regulam:</s> <s xml:id="echoid-s16788" xml:space="preserve"> tunc inueniet umbram extremitatis feſtucæ ſuper centrum lucis, & <lb/>inueniet illam ſuper lineam, quę eſt in ſuperficie regulæ.</s> <s xml:id="echoid-s16789" xml:space="preserve"> Tunc ergo accipiat experimentator pen-<lb/>nam intinctam incauſto, & ſignet ſuper extremitatẽ umbrę, quę eſt in medio lucis, quę eſt ſuper re-<lb/>gulam, punctum:</s> <s xml:id="echoid-s16790" xml:space="preserve"> ergo erit iſtud punctum ſuper lineam, quę eſt in ſuperficie regulæ:</s> <s xml:id="echoid-s16791" xml:space="preserve"> deinde auferat <lb/>acum à ſuperiore foramine:</s> <s xml:id="echoid-s16792" xml:space="preserve"> & ponat ipſam ſuper inferius foramen, ſcilicet quod eſt in ora:</s> <s xml:id="echoid-s16793" xml:space="preserve"> & ponat <lb/>extremitatem acus ſuper centrum foraminis:</s> <s xml:id="echoid-s16794" xml:space="preserve"> & intueatur lucem, quę eſt ſuper regulam:</s> <s xml:id="echoid-s16795" xml:space="preserve"> tunc inue-<lb/>niet umbram extremitatis acus ſuper punctum, quod eſt in ſuperficie regulę:</s> <s xml:id="echoid-s16796" xml:space="preserve"> deinde auferat acum, <lb/>& redibit umbra ad ſuum locum.</s> <s xml:id="echoid-s16797" xml:space="preserve"> Declarabitur ergo ex hac experimentatione, quòd lux, quę eſt ſu-<lb/>per punctum, quod eſt in ſuperficie regulæ, eſt lux, quę tranſit per cẽtra duorũ foraminum.</s> <s xml:id="echoid-s16798" xml:space="preserve"> Deinde <lb/>accipiat experimentator calamũ tinctum incauſto, & ſignet punctũ in uero medio ſuperficiei uitri <lb/>ex parte regulæ:</s> <s xml:id="echoid-s16799" xml:space="preserve"> ſi uerò nõ comprehẽdat mediũ uitri, quò ad ſenſum:</s> <s xml:id="echoid-s16800" xml:space="preserve"> ſignet in ipſo duas diametros <lb/>ſecãtes ſe, & locus ſectionis eſt medium ſuperficiei uitri.</s> <s xml:id="echoid-s16801" xml:space="preserve"> Hoc autem facto, intueatur lucem, quę eſt <lb/>ſuper regulam:</s> <s xml:id="echoid-s16802" xml:space="preserve"> & inueniet umbram puncti, quod eſt in medio uitri ſuper punctum, quod eſt in ſu-<lb/>perficie regulæ.</s> <s xml:id="echoid-s16803" xml:space="preserve"> Declarabitur ergo exhoc, quòd lux, quæ tranſit per duo centra duorũ foraminum, <lb/>tranſibit per punctum, quod eſt in medio uitri.</s> <s xml:id="echoid-s16804" xml:space="preserve"> Hoc autem declarato oportet experimẽtatorem ui-<lb/>trum primum euellere, & ſignare in ſuperficie ſecundi uitri punctum medium, ut prius, & compo-<lb/>nere inſtrumentum ſecundò, & moueat ípſum, quouſq;</s> <s xml:id="echoid-s16805" xml:space="preserve"> luxtranſeat per duo foramina:</s> <s xml:id="echoid-s16806" xml:space="preserve"> deinde in-<lb/>tueatur:</s> <s xml:id="echoid-s16807" xml:space="preserve"> & inueniet lucem peruenientẽ ad centrum lucis, quę eſt in ſuperficie regulę:</s> <s xml:id="echoid-s16808" xml:space="preserve"> & eſt lux, quę <lb/>trãſit per cẽtra duorũ foraminũ.</s> <s xml:id="echoid-s16809" xml:space="preserve"> Declarabitur igitur ex hoc, quòd lux, quę trãſit per cẽtra duorũ fo-<lb/>raminũ, trãſit etiã ք punctũ, quod eſt in medio ſuperficiei ſecũdi uitri:</s> <s xml:id="echoid-s16810" xml:space="preserve"> & ſitus eius eſt ſitus lucis trã-<lb/>ſeuntis ք cẽtra duorũ foraminũ de ſuքficiebus uitrorũ in prima experimẽtatione:</s> <s xml:id="echoid-s16811" xml:space="preserve"> & cũhoc quãdo <lb/>lux trãſit per punctũ, quod eſt in medio uitri ſecũdi:</s> <s xml:id="echoid-s16812" xml:space="preserve"> tũc lux, quæ trãſit per cẽtra duorũ foraminũ in <lb/> <pb o="237" file="0243" n="243" rhead="OPTICAE LIBER VII."/> prima experimentatione, tranſit etiam per punctum, quod eſt in medio uitri ſecũdi.</s> <s xml:id="echoid-s16813" xml:space="preserve"> Deinde opor-<lb/>tet experimentatorem euellere ſecundũ uitrum, & experiri tertium, & ſic de cęteris uſq;</s> <s xml:id="echoid-s16814" xml:space="preserve"> ad ultimũ.</s> <s xml:id="echoid-s16815" xml:space="preserve"> <lb/>Patebit ergo experimẽtatione hac, quòd lux quæ tranſit per centra duorum foraminũ, perueniens <lb/>ad ſuperficiẽ regulæ, tranſit per centra ſuperficierũ uitrorum omniũ poſitorum ſuper ſuperficiẽ la-<lb/>minæ.</s> <s xml:id="echoid-s16816" xml:space="preserve"> Manifeſtũ eſt ergo, quòd ſit in rectitudine lineæ tranſeuntis per centra duorũ foraminum:</s> <s xml:id="echoid-s16817" xml:space="preserve"> & <lb/>lux, quæ tranſit per centra duorũ foraminum in experimentatione omniũ uitrorum, extenditur in <lb/>rectitudine lineæ continuantis centra duorũ foraminum.</s> <s xml:id="echoid-s16818" xml:space="preserve"> Manifeſtũ eſt ergo, quòd lux, quæ tranſit <lb/>per lineã rectam, tranſeuntẽ per cẽtra duorũ foraminũ, tranſit etiã per centra ſuperficierũ uitrorũ.</s> <s xml:id="echoid-s16819" xml:space="preserve"> <lb/>Ex quo patet, quòd lux tranſit in corpus uitri, in quo extenditur, poſtquã tranſit, ſecundũ lineas re-<lb/>ctas:</s> <s xml:id="echoid-s16820" xml:space="preserve"> & quòd lux, quæ tranſit per centra duorũ foraminum, extenditur etiã in corpus uitri ſecũdum <lb/>rectitudinem lineæ, per quam extendebatur in aere, antequam pertranſiret uitrum:</s> <s xml:id="echoid-s16821" xml:space="preserve"> & illa linea, per <lb/>quam extenditur lux in aere, eſt perpẽdicularis ſuper ſuperficiẽ uitri oppoſitã foramini [per 8 p 11.</s> <s xml:id="echoid-s16822" xml:space="preserve">] <lb/>Nam linea, quæ tranſit per centra duorũ foraminum, eſt æquidiſtans diametro laminæ, quę eſt per-<lb/>pendicularis ſuper primam ſuperficiem ſuperficierum uitrorum:</s> <s xml:id="echoid-s16823" xml:space="preserve"> quia eſt perpẽdicularis ſuper dif-<lb/>ferentiam communem inter ſuperficiem uitri, & ſuperficiem laminæ.</s> <s xml:id="echoid-s16824" xml:space="preserve"> Item accipiat experimẽtator <lb/>medietatem ſphęræ uitreæ mundæ claræ, ut cryſtallinæ, cuius ſemidiameter ſit minor diſtantia in-<lb/>ter tabulam & centrum laminæ, & inueniat centrum baſis eius, ſuper quod ſignet lineam ſubtilem <lb/>cum incauſto:</s> <s xml:id="echoid-s16825" xml:space="preserve"> poſtea ſeparet ex hac linea ex parte centri baſis, quod eſt centrũ ſphæræ, lineã æqua-<lb/>lem diametro foraminis, quod eſt in ora inſtrumẽti:</s> <s xml:id="echoid-s16826" xml:space="preserve"> erit ergo hæc linea æqualis lineæ, quæ eſt inter <lb/>centrũ foraminis, quod eſt in ora inſtrumẽti, quæ eſt perpẽdicularis ſuper ſu-<lb/> <anchor type="figure" xlink:label="fig-0243-01a" xlink:href="fig-0243-01"/> perficiem laminæ.</s> <s xml:id="echoid-s16827" xml:space="preserve"> Deinde ſtatuamus ſuper extremitatẽ lineæ ſeparatæ à dia-<lb/>metro lineã perpendicularem, & extrahamus illam in utramq;</s> <s xml:id="echoid-s16828" xml:space="preserve"> partẽ:</s> <s xml:id="echoid-s16829" xml:space="preserve"> deinde <lb/>ſecemus uitrum ſuper hác lineam in confrictorio uel in tornatorio, donec lo <lb/>cus ſectionis fiat ſuperficies æqualis, & perpendicularis ſuper ſuperficiẽ baſis <lb/>ſemicirculi, & mẽſuremus angulũ, qui eſt inter duas ſuperficies, per angulũ rectum factũ ex cupro, <lb/>donec uerificetur ſuperficies iſta:</s> <s xml:id="echoid-s16830" xml:space="preserve"> & tunc differentia communis huic ſuperficiei & ſuperficiei baſis <lb/>ſphęræ erit linea recta:</s> <s xml:id="echoid-s16831" xml:space="preserve"> & linea copulans centrũ ſphęræ cum hac linea, erit perpẽdicularis ſuper ſu-<lb/>perficiem factã:</s> <s xml:id="echoid-s16832" xml:space="preserve"> poſtea ſumatur in medio huius lineæ, quę eſt cõmunis differentia, particula parua, <lb/>quæ eſt ſignũ medij eius.</s> <s xml:id="echoid-s16833" xml:space="preserve"> Hoc completo, poliatur uitrũ uehemẽtiſsimè, & ponatur ſuper ſuperficiẽ <lb/>laminæ, & gibboſitas eius ſit ex parte foraminũ, & ſit pars facta in uitro ſuper ſuperficiẽ laminæ, & <lb/>ſuperponatur linea recta, quæ eſt cõmunis differentia duabus ſuperficiebus æqualibus, quę ſunt in <lb/>uitro, ſuper lineã ſcilicet ſignatã in lamina, ſecantẽ diametrũ perpendiculariter, & ponatur medium <lb/>lineæ ſuper centrũ laminæ.</s> <s xml:id="echoid-s16834" xml:space="preserve"> Hac ergo poſitione præſeruata, applicetur uitrum laminæ applicatione <lb/>fixa:</s> <s xml:id="echoid-s16835" xml:space="preserve"> deinde ponamus regulã ſubtilem ſuper ſuperficiẽ inſtrumẽti, ſicut ponebamus in experimẽta-<lb/>tione uitrorũ cubicorũ, & ponamus ſuperficiẽ regulæ, in qua eſt linea recta latitudinis, ſit ex parte <lb/>uitri, & prope illud:</s> <s xml:id="echoid-s16836" xml:space="preserve"> deinde ponatur inſtrumentũ in prædictũ uas:</s> <s xml:id="echoid-s16837" xml:space="preserve"> & ponatur uas in ſole, uacuũ ſine <lb/>aqua:</s> <s xml:id="echoid-s16838" xml:space="preserve"> & moueatur inſtrumentũ, donec lux ſolis trãſeat per duo foramina:</s> <s xml:id="echoid-s16839" xml:space="preserve"> & ſit ſitus lucis de ſecun-<lb/>do foramine ſitus mediocris, & intueatur experimẽtator regulã:</s> <s xml:id="echoid-s16840" xml:space="preserve"> & inueniet lucẽ tranſeuntẽ ք duo <lb/>foramina, ſuք ſuperficiẽ regulæ:</s> <s xml:id="echoid-s16841" xml:space="preserve"> deinde applicet ſtilũ ſuperiori foramini, & ponat extremitatẽ ſtili <lb/>ſuք centrũ foraminis, & intueatur lucẽ, quę eſt in regula:</s> <s xml:id="echoid-s16842" xml:space="preserve"> tũc inueniet umbrã extremitatis ſtili apud <lb/>centrũ lucis:</s> <s xml:id="echoid-s16843" xml:space="preserve"> dein de auferat ſtilũ, & redibit lux ad ſuũ locum.</s> <s xml:id="echoid-s16844" xml:space="preserve"> Poſtea applicet ſtilũ ad ſecundũ fora-<lb/>men, & ponat extremitatẽ eius apud centrũ ſecundũ, & intueatur lucẽ, quę eſt in regula:</s> <s xml:id="echoid-s16845" xml:space="preserve"> tũc inue-<lb/>niet umbrá extremitatis ſtili apud centrũ lucis.</s> <s xml:id="echoid-s16846" xml:space="preserve"> Poſtea ponat extremitatẽ ſtili apud centrũ baſis ui-<lb/>tri (quod eſt centrũ ſphęræ) & intueatur lucẽ, quę eſt ſuք regulã:</s> <s xml:id="echoid-s16847" xml:space="preserve"> inueniet umbrã extremitatis ſtili <lb/>ſuper centrũ lucis.</s> <s xml:id="echoid-s16848" xml:space="preserve"> Deinde ponat ſtilũ in medio lucis, quæ eſt ſuք conuexũ uitri oppoſiti foramini <lb/>ſecũdo, quod eſt propè illud, & intueatur lucẽ, quę eſt ſuper regulã:</s> <s xml:id="echoid-s16849" xml:space="preserve"> & inueniet umbrã extremitatis <lb/>ſtili apud centrũ lucis.</s> <s xml:id="echoid-s16850" xml:space="preserve"> Ex quo patet, quòd lux, quę tranſit per centra duorũ foraminũ, trãſit etiã per <lb/>centrũ baſis uitri, & per mediũ ſuperficiei lucis, quę eſt in cõuexo uitri.</s> <s xml:id="echoid-s16851" xml:space="preserve"> Manifeſtũ eſt igitur qđ lux, <lb/>quę trãſit in corpus uitri, extẽditur ſecundũ rectitudinẽ lineę trãſeuntis per cẽtra duorũ foraminũ:</s> <s xml:id="echoid-s16852" xml:space="preserve"> <lb/>hęc aũt linea eſt diameter ſphęræ uitreæ.</s> <s xml:id="echoid-s16853" xml:space="preserve"> Nã perpẽdicularis exiens à cẽtro baſis uitri ad laminã, eſt <lb/>æqualis diametro foraminis:</s> <s xml:id="echoid-s16854" xml:space="preserve"> diameter autẽ foraminis eſt æqualis perpẽdiculari exeunti à cẽtro fo-<lb/>raminis ad ſuperficiẽ laminę:</s> <s xml:id="echoid-s16855" xml:space="preserve"> ergo perpẽdicularis à cẽtro foraminis baſis uitri ſuք ſuperficiẽ lami-<lb/>næ, eſt æqualis perpẽdiculari exeũti à cẽtro foraminis ad ſuperficiẽ laminę:</s> <s xml:id="echoid-s16856" xml:space="preserve"> & hæ duę perpẽdicula-<lb/>res cadũt ſuper diametrũ laminę.</s> <s xml:id="echoid-s16857" xml:space="preserve"> Linea ergo, quę trãſit per cẽtra duorũ foraminũ, ſi fuerit extẽſa in <lb/>rectitudine, perueniet ad centrũ ſphęræ uitreæ:</s> <s xml:id="echoid-s16858" xml:space="preserve"> erit ergo diameter huius ſphęræ:</s> <s xml:id="echoid-s16859" xml:space="preserve"> eſt ergo perpẽdi-<lb/>cularis ſuք ſuperficiẽ huius ſphęræ [ut demonſtratũ eſt 25 n 4.</s> <s xml:id="echoid-s16860" xml:space="preserve">] Experimẽtatione aũt uitrorũ cu-<lb/>bicorũ patuit, quòd lux, quę extẽditur in corpus uitri, eſt in rectitudine lineę, ք quã extẽdebatur in <lb/>aere:</s> <s xml:id="echoid-s16861" xml:space="preserve"> & linea, ք quã extẽdebatur in aere, erat illic perpẽdicularis ſuք ſuperficiẽ uitri.</s> <s xml:id="echoid-s16862" xml:space="preserve"> Et oportet ex-<lb/>perimentatorẽ auferre regulã ſubtilẽ, applicatã ad ſuperficiẽ laminę:</s> <s xml:id="echoid-s16863" xml:space="preserve"> & cõponat inſtrumentũ ſecũ-<lb/>dò, & moueat ipſum, quouſq;</s> <s xml:id="echoid-s16864" xml:space="preserve"> lux trãſeat ք duo foramina, & intueatur orã inſtrumẽti, quæ eſt intra <lb/>uas:</s> <s xml:id="echoid-s16865" xml:space="preserve"> & inueniet lucẽ ſuper orã inſtrumẽti, & inueniet centrũ lucis in pũcto, quod eſt differẽtia com <lb/>munis inter circumferentiã circuli medij & lineã perpẽdicularem in ora inſtrumẽti, quod eſt extre-<lb/>mitas diametri circuli medij, trãſeuntis per cẽtra duorũ foraminũ:</s> <s xml:id="echoid-s16866" xml:space="preserve"> & lux, quæ extẽditur ք hãc lineã, <lb/>erit differentia cõmunis perueniens ad centrum ſphęræ uitreę.</s> <s xml:id="echoid-s16867" xml:space="preserve"> Centrum ergo lucis, quę eſt in ora <lb/> <pb o="238" file="0244" n="244" rhead="ALHAZEN"/> inſtrumenti, & centrum ſphæræ uitreæ, & centrum duorum foraminum ſunt in eadem linea recta.</s> <s xml:id="echoid-s16868" xml:space="preserve"> <lb/>Ex quo patet, quòd lux, quæ tranſit in corpus uitri, perueniens ad cẽtrum ſphæræ eius, cum extra-<lb/>hitur in aerem, extenditur in rectitudine lineæ, per quam extendebatur in corpore uitri.</s> <s xml:id="echoid-s16869" xml:space="preserve"> Hæc au-<lb/>tem linea eſt perpendicularis ſuper ſuperficiem baſis uitri, quæ eſt æquidiſtans diametro laminæ, <lb/>quæ eſt perpendicularis ſuper ſuperficiem baſis uitri:</s> <s xml:id="echoid-s16870" xml:space="preserve"> quia eſt perpendicularis ſuper lineã rectam, <lb/>quæ eſt differentia communis duabus ſuperficiebus uitri æqualibus, quarum altera eſt ſuperpoſi-<lb/>ta ſuperficiei laminæ, & reliqua erecta ſuper ſuperficiem laminæ.</s> <s xml:id="echoid-s16871" xml:space="preserve"> Linea igitur tranſiens per centra <lb/>duorum foraminum & per centrum ſphæræ uitreæ eſt perpen dicularis ſuper ſuperficiem uitri:</s> <s xml:id="echoid-s16872" xml:space="preserve"> eſt <lb/>ergo perpendicularis ſuper ſuperficiem aeris, qui tangit hanc ſuperficiem.</s> <s xml:id="echoid-s16873" xml:space="preserve"> Et ſi experimentator in-<lb/>fuderit aquam in uas, remanente uitro in ſua poſitione, & poſuerit aquam ſupra cẽtrum uitri, & in-<lb/>ſpexerit lucem, quæ eſt in ora in ſtrumenti:</s> <s xml:id="echoid-s16874" xml:space="preserve"> inueniet centrum lucis ſuper extremitatẽ diametri me-<lb/>dij circuli.</s> <s xml:id="echoid-s16875" xml:space="preserve"> Et ſi euulſerit uitrum, & poſuerit illud in lamina è contrario huic ordinationi, ſcilicet, ut <lb/>ſuperficies æqualis ſit ex parte foraminum, & conuexitas uitri ſit ex parte interiore uaſis:</s> <s xml:id="echoid-s16876" xml:space="preserve"> & ſuper-<lb/>poſuerit lineam rectam, quæ eſt in uitro, quæ eſt differentia communis duabus ſuis ſuperficiebus <lb/>æqualibus, ſuper lineam rectam, quæ eſt in lamina, ſecatem perpendiculariter diametrum laminæ, <lb/>& poſuerit medium huius lineæ, ſcilicet, quæ eſt in uitro, ſuper centrũ laminæ, & inſpexerit lucem, <lb/>ſicut fecit in prima poſitione:</s> <s xml:id="echoid-s16877" xml:space="preserve"> inueniet lucem cadentem ſuper oram inſtrumenti, & inueniet cen-<lb/>trum lucis ſuper punctum, quod eſt differentia cõmunis medij circuli, & lineæ ſtanti in ora inſtru-<lb/>menti.</s> <s xml:id="echoid-s16878" xml:space="preserve"> Ex quibus declarabitur, quòd lux ſolis, quæ tranſit per centra duorum foraminum, tranſit <lb/>etiam in corpus uirri ſecundum rectitudinem lineæ, per quam extendebatur in aere:</s> <s xml:id="echoid-s16879" xml:space="preserve"> & poſtquam <lb/>egreditur corpus uitri, extenditur etiam in aere ſecundum rectitudinem lineæ, per quam extende-<lb/>batur in uitro:</s> <s xml:id="echoid-s16880" xml:space="preserve"> lineaq́;</s> <s xml:id="echoid-s16881" xml:space="preserve">, quæ tranſit per centra duorum foraminum, eſt in hac poſitione etiã perpen-<lb/>dicularis ſuper ſuperficiem uitri, oppoſitam foramini, ſeilicet ſuperficiẽ, quæ eſt baſis hemilphærij.</s> <s xml:id="echoid-s16882" xml:space="preserve"> <lb/>Et hæc linea eſt etiam perpendicularis ſuper ſuperficiem cõuexam:</s> <s xml:id="echoid-s16883" xml:space="preserve"> nam in hac poſitione etiam eſt <lb/>diameter ſphæræ:</s> <s xml:id="echoid-s16884" xml:space="preserve"> eſt ergo perpendicularis ſuper ſuperficiem aeris contingentis ſuperficiem ſphæ-<lb/>ræ.</s> <s xml:id="echoid-s16885" xml:space="preserve"> Et ſi experimentator infuderit aquam in uas, & reliquerit uitrum in ſua poſitione, & poſuerit <lb/>aquam infra centrum uitri, & aſpexerit lucem, quę eſt in ora inſtrumenti:</s> <s xml:id="echoid-s16886" xml:space="preserve"> inueniet centrum lucis in <lb/>extremitate diametri medij circuli.</s> <s xml:id="echoid-s16887" xml:space="preserve"> Ex his ergo experimentationibus, quæ fiunt per cubicum & <lb/>ſphęricum uitrum, patet, quòd ſi lux occurrerit corpori diaphano diuerſæ diaphanitatis à corpore, <lb/>in quo eſt, & linea, per quam extenditur, fuerit perpendicularis ſuper ſuperficiem ſecũdi corporis:</s> <s xml:id="echoid-s16888" xml:space="preserve"> <lb/>tunc lax extenditur in ſecundo corpore in rectitudine lineæ, per quam extendebatur in corpore <lb/>primo:</s> <s xml:id="echoid-s16889" xml:space="preserve"> nec differt, ſi ſecundum corpus fuerit groſsius primo aut ſubtilius.</s> <s xml:id="echoid-s16890" xml:space="preserve"/> </p> <div xml:id="echoid-div557" type="float" level="0" n="0"> <figure xlink:label="fig-0243-01" xlink:href="fig-0243-01a"> <image file="0243-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0243-01"/> </figure> </div> </div> <div xml:id="echoid-div559" type="section" level="0" n="0"> <head xml:id="echoid-head486" xml:space="preserve" style="it">7. Radi<emph style="sub">9</emph> medio rariori obliqu{us}, refringitur à քpẽdiculari à refractiõis pũcto excitata. 45 p 2.</head> <p> <s xml:id="echoid-s16891" xml:space="preserve">ITem oportet experimentatorẽ euellere uitrũ, & referre illud ad laminã, & ponere mediũ lineæ <lb/>rectæ, quæ eſt in eo, ſuper centrũ laminæ, & ponere ſuperficiẽ æqualem ex parte duorũ forami-<lb/>num, & lineã, quæ eſt in uitro, quæ eſt differẽtia cõmunis duabus ſuis ſuperficiebus, obliquã ſu-<lb/>per diametrũ laminæ qualibet obliquatione, & ponere obliquationẽ diametri laminæ ſuper hãc li-<lb/>neam ad illam partẽ, ad quã declinabat apud experimentationẽ aquæ.</s> <s xml:id="echoid-s16892" xml:space="preserve"> Neceſſe eſt igitur, ut perpen-<lb/>dicularis, quæ egreditur à centro uitri, quæ eſt ſuper ſuperficiẽ uitri perpendicularis, quę extẽditur <lb/>in corpore uitri, obliqua ſit a linea tranſeunte per cẽtra duorum foraminũ ad partẽ, in qua ſunt duo <lb/>foramina.</s> <s xml:id="echoid-s16893" xml:space="preserve"> Et applicet experimentator uitrũ ſecundũ hunc ſitum applicatione fixa, & ponat inſtru-<lb/>mentũ in uas, & uas in ſole, & moueat inſtrumentũ, donec lux trãſeat per duo foramina, & intuea-<lb/>tur lucẽ, quæ eſt intra uas:</s> <s xml:id="echoid-s16894" xml:space="preserve"> tunc inueniet illã in interiore ora inſtrumẽti, & inueniet centrũ lucis in <lb/>circumferentia medij circuli:</s> <s xml:id="echoid-s16895" xml:space="preserve"> ſed extra punctũ, quod eſt differentia cõmunis circumferẽtiæ circuli <lb/>medij, & lineæ ſtanti in ora inſtrumenti:</s> <s xml:id="echoid-s16896" xml:space="preserve"> & declinatio eius erit ad partem, in qua eſt ſol:</s> <s xml:id="echoid-s16897" xml:space="preserve"> erit ergo ad <lb/>partem perpẽdicularis, exeuntis à loco refractionis.</s> <s xml:id="echoid-s16898" xml:space="preserve"> Et hæc lux extenditur in aere in rectitudine li-<lb/>neæ, tranſeuntis per centra duorũ foraminũ:</s> <s xml:id="echoid-s16899" xml:space="preserve"> & hęc linea in hoc ſitu perueniet ad centrũ ſphęræ ui-<lb/>treæ, & erit obliqua ſuper ſuperficiẽ æqualem.</s> <s xml:id="echoid-s16900" xml:space="preserve"> Huius autẽ lucis terminatio extẽſionis in uitro eſt à <lb/>cẽtro uitri:</s> <s xml:id="echoid-s16901" xml:space="preserve"> extẽditur igitur in corpore uitri ſecundũ lineam rectã, exeuntem à centro ſphæræ:</s> <s xml:id="echoid-s16902" xml:space="preserve"> ergo <lb/>illius eſt diameter:</s> <s xml:id="echoid-s16903" xml:space="preserve"> hęc igitur lux extẽditur in corpore uitri ſecundũ uerticationẽ diametri alicuius <lb/>eius.</s> <s xml:id="echoid-s16904" xml:space="preserve"> Cũ ergo peruenerit ad ſphęricam ſuperficiẽ, erit perpendicularis ſuper illã:</s> <s xml:id="echoid-s16905" xml:space="preserve"> & cum extrahetur <lb/>in aerem, erit perpendicularis ſuper aerem contingentẽ ſuperficiem ſphęricam.</s> <s xml:id="echoid-s16906" xml:space="preserve"> Non ergo refringi-<lb/>tur in aere, neq;</s> <s xml:id="echoid-s16907" xml:space="preserve"> extẽditur rectè:</s> <s xml:id="echoid-s16908" xml:space="preserve"> ergo refringitur, ſed nõ in corpore uitri, neq;</s> <s xml:id="echoid-s16909" xml:space="preserve"> in cõuexo eius, neq;</s> <s xml:id="echoid-s16910" xml:space="preserve"> in <lb/>primo aere, neq;</s> <s xml:id="echoid-s16911" xml:space="preserve"> in ſecũdo:</s> <s xml:id="echoid-s16912" xml:space="preserve"> ergo refringitur apud centrum uitri:</s> <s xml:id="echoid-s16913" xml:space="preserve"> & hęc lux eſt obliqua ſuper ſuper-<lb/>ficiem ęqualem, in qua eſt centrum uitri.</s> <s xml:id="echoid-s16914" xml:space="preserve"> Ex quibus patet, quòd, cum lux extenditur in aere & tran-<lb/>ſit in uitrum, & fuerit obliqua ſuper ſuperficiem uitri:</s> <s xml:id="echoid-s16915" xml:space="preserve"> refringetur, & non tranſibit rectè:</s> <s xml:id="echoid-s16916" xml:space="preserve"> & refra-<lb/>ctio eius erit ad partem, in qua eſt perpendicularis, exiens à loco refractionis:</s> <s xml:id="echoid-s16917" xml:space="preserve"> & corpus uitri groſ-<lb/>ſius eſt corpore aeris.</s> <s xml:id="echoid-s16918" xml:space="preserve"> Manifeſtum eſt igitur ex hac experimentatione, & prima de refractione lu-<lb/>cis ab aere ad aquam (luce exiſtente obliqua ſuper ſuperficiem aquę) quòd, cum lux fuerit extenſa <lb/>in corpore ſubtiliore, & occurrerit illi groſsius corpus:</s> <s xml:id="echoid-s16919" xml:space="preserve"> refringetur ab ipſo:</s> <s xml:id="echoid-s16920" xml:space="preserve"> & erit refractio eius ad <lb/>partem, in qua eſt linea exiens à loco refractionis, quę eſt perpendicularis ſuper ſuperficiẽ corporis <lb/>groſsioris.</s> <s xml:id="echoid-s16921" xml:space="preserve"> Item oportet experimentatorem euellere uitrum, & ponere ipſum è contrario:</s> <s xml:id="echoid-s16922" xml:space="preserve"> ſcilicet <lb/>ut ſuperficies conuexa ſit ex parte foraminum, & ponat medium differentię communis, quę eſt in <lb/>uitro ſuper centrum laminę, & ponat d<gap/>ſſerẽtiam communem obliquã ſuper diametrum laminę, & <lb/> <pb o="239" file="0245" n="245" rhead="OPTICAE LIBER VII."/> applicet uitrum applicatione fixa, & extrahat à centro laminæ lineã in ſuperficie perpendicula rem <lb/>ſuper differẽtiam communẽ, quæ eſt in uitro:</s> <s xml:id="echoid-s16923" xml:space="preserve"> erit hæc linea perpendicularis ſuper ſup erficiẽ uitri.</s> <s xml:id="echoid-s16924" xml:space="preserve"> <lb/>Nam ſuperficies uitri æqualis, eſt perpendicularis ſuper ſuperficiẽ laminæ.</s> <s xml:id="echoid-s16925" xml:space="preserve"> Deinde experimẽtator <lb/>ponat inſtrumentũ in uaſe exiſtente ſine aqua, & moueat inſtrumentũ, quouſq;</s> <s xml:id="echoid-s16926" xml:space="preserve"> lux trãſeat per duo <lb/>foramina, & intueatur lucem, quæ eſt intra uas:</s> <s xml:id="echoid-s16927" xml:space="preserve"> tune inueniet illam in interiore ora inſtrumenti, & <lb/>inueniet centrum lucis in circumferentia medij circuli, & extra punctum, quod eſt differentia com <lb/>munis circumferentiæ medij circuli, & lineæ perpendiculari, in ora inſtrumenti:</s> <s xml:id="echoid-s16928" xml:space="preserve"> quod punctum eſt <lb/>extremitas diametri medij circuli:</s> <s xml:id="echoid-s16929" xml:space="preserve"> & inueniet declinationem eius ad cõtrariam partẽ illi, in qua oſt <lb/>perpendicularis.</s> <s xml:id="echoid-s16930" xml:space="preserve"> Hæc autẽ lux extenditur in uitro ſecundũ rectitudinem lineæ tranſeuntis per cen <lb/>tra duorum foraminũ:</s> <s xml:id="echoid-s16931" xml:space="preserve"> quia hæc linea eſt diameter uitri in hac etiã poſitione, quia tranſit per centrũ <lb/>uitri.</s> <s xml:id="echoid-s16932" xml:space="preserve"> In hac ergo poſitione refractio lucis etiam eſt apud centrum uitri:</s> <s xml:id="echoid-s16933" xml:space="preserve"> & hęc lux eſt obliqua ſuper <lb/>ſuperficiem uitri æqualem, & ſuperficiem aeris contingentem uitrum.</s> <s xml:id="echoid-s16934" xml:space="preserve"> Ex quibus patet, quòd, cum <lb/>lux extẽditur in uitro, & egreditur ad aerem, & fuerit obliqua ſuper ſuperficiem aeris:</s> <s xml:id="echoid-s16935" xml:space="preserve"> refringetur:</s> <s xml:id="echoid-s16936" xml:space="preserve"> <lb/>& refractio eius erit in ſuperficie circuli medij, & ad partem contrariam illi, in qua eſt linea exiens à <lb/>loco refractionis, quæ eſt perpendicularis ſuper ſuperficiem aeris.</s> <s xml:id="echoid-s16937" xml:space="preserve"> Et ſi experimentator infuderit <lb/>aquam in uas (exiſtente uitro in ſua poſitione) & poſuerit aquam ſuper centrum uitri, & aſpexe-<lb/>rit lucem, quæ eſt intra uas:</s> <s xml:id="echoid-s16938" xml:space="preserve"> inueniet lucem in interiore parte oræ inſtrumenti, & inueniet centrum <lb/>lucis in circumferentia medij circuli, & inueniet illud extra extremitatem diametri med ij circuli, <lb/>obliquum ad partem contrariam illi, ſuper quam eadit perpendicularis:</s> <s xml:id="echoid-s16939" xml:space="preserve"> & inueniet diſtãtiam cen-<lb/>tri lucis ab extremitate diametri medij circuli minorem diſtantia centri lucis ab hoc puncto, in ex-<lb/>perientia egreſſus lucis à cẽtro ad aerem:</s> <s xml:id="echoid-s16940" xml:space="preserve"> quia aer eſt ſubtilior aqua, aqua autem eſt ſubtilior uitro.</s> <s xml:id="echoid-s16941" xml:space="preserve"> <lb/>Ex hac autem experimentatione, & prædicta, patet, quòd quando lux extenditur in corpore groſ-<lb/>ſiore, & occurrerit corpori ſubtiliori, & fuerit obliqua ſuper ſuperficiem corporis ſubtilioris:</s> <s xml:id="echoid-s16942" xml:space="preserve"> refrin <lb/>getur, & non tranſibit rectè:</s> <s xml:id="echoid-s16943" xml:space="preserve"> & refractio eius erit ad partem contrariã illi, in qua eſt perpendicularis <lb/>exiens à loco refractionis, quæ eſt perpendicularis ſuper ſuperficiem corporis ſubtilioris:</s> <s xml:id="echoid-s16944" xml:space="preserve"> & tantò <lb/>magis declinabit à perpendiculari, quantò corpus erit ſubtilius.</s> <s xml:id="echoid-s16945" xml:space="preserve"> Item oportet experimentatorem <lb/>euellere uitrum, & ponere etiam ipſum in ſuperficie laminæ, & ſuperponat lineam rectam, quæ eſt <lb/>in eo, ſuper lineam rectam, quæ eſt in lamina, & ponat ſuperficiem eius conuexam ex parte duo-<lb/>rum foraminum, & lineam rectam, quę eſt in uitro, extra centrum laminæ, & coniungat uitrum be-<lb/>ne, & ponat regulam ſubtilem ſuper ſuperficiem laminæ, & erigat eam ſuper oram eius, & ponat <lb/>ſuperficiem eius, in qua ſignatur linea, ex parte uitri, & terminus eius ſecet diametrum laminæ per-<lb/>pendiculariter, & applicetur hoc modo.</s> <s xml:id="echoid-s16946" xml:space="preserve"> Sic ergo linea, quæ tranſit per centra duorum foraminum, <lb/>non tranſit per centrum ſp<gap/>æræ, ſed per aliud punctum ſuperficiei uitri æqualis:</s> <s xml:id="echoid-s16947" xml:space="preserve"> & erit obliqua ſu-<lb/>per ſphæricam ſuperficiem.</s> <s xml:id="echoid-s16948" xml:space="preserve"> Deinde oportet experimentatorem ponere inſtrumentum in uaſe, & <lb/>uas in ſole:</s> <s xml:id="echoid-s16949" xml:space="preserve"> & moueat inſtrumentum, quouſque lux tranſeat per duo foramina, & intueatur ſuper-<lb/>ficiem regulæ:</s> <s xml:id="echoid-s16950" xml:space="preserve"> tunc inueniet lucem ſuper ſuperficiem regulæ, & centrum eius ſuper lineam, quæ <lb/>eſt in ſuperficie regulæ, & centrum lucis extra rectitudinem lineæ, quæ tranſit per centra duorum <lb/>foraminum:</s> <s xml:id="echoid-s16951" xml:space="preserve"> & inueniet declinationem eius ad partem, in qua eſt centrum uitri:</s> <s xml:id="echoid-s16952" xml:space="preserve"> & inueniet lineam, <lb/>quæ tranſit per centra duorum foraminum, perpendicularẽ ſuper ſuperficiem uitri æqualem [per <lb/>8 p 11] eſt enim æquidiſtans diametro, & diameter laminæ eſt perpendicularis ſuper ſuperficiem <lb/>uitri æqualem.</s> <s xml:id="echoid-s16953" xml:space="preserve"> Et ſi lux tranſiſſet per centra duorum foraminum, & extenderetur ſecundum recti-<lb/>tudinem ad ſuperficiem æqualem:</s> <s xml:id="echoid-s16954" xml:space="preserve"> tunc extenderetur in rectitudine in aere:</s> <s xml:id="echoid-s16955" xml:space="preserve"> ſed cum centrum lu-<lb/>cis, quę eſt in regula, non ſit in rectitudine huius lineæ:</s> <s xml:id="echoid-s16956" xml:space="preserve"> ergo lux nõ extenditur in rectitudine ipſius <lb/>ad ſuperficiem æqualem:</s> <s xml:id="echoid-s16957" xml:space="preserve"> & lux in corpore uitri extenditur rectè:</s> <s xml:id="echoid-s16958" xml:space="preserve"> ergo lux, quæ extenditur in cor-<lb/>pore uitri, non eſt in rectitudine lineæ, quæ tranſit per cẽtra duorum foraminum:</s> <s xml:id="echoid-s16959" xml:space="preserve"> ergo eſt refracta:</s> <s xml:id="echoid-s16960" xml:space="preserve"> <lb/>ſed non in aere, neque in corpore uitritergo refringitur apud ſphæricam ſuperficiem uitri.</s> <s xml:id="echoid-s16961" xml:space="preserve"> Et linea, <lb/>quæ tranſit per centra duorum foraminum, nõ tranſit per centrum uitri:</s> <s xml:id="echoid-s16962" xml:space="preserve"> & hæc lux, cum egreditur <lb/>à ſuperficie uitri æquali, refringitur.</s> <s xml:id="echoid-s16963" xml:space="preserve"> Sed cum regula ſubtilis fuerit ualde propinqua ſuperficiei ui-<lb/>tri:</s> <s xml:id="echoid-s16964" xml:space="preserve"> tunc declinatio centri lucis, quæ eſt in regula, à rectitudine lineæ, quę extenditur in corpore ui-<lb/>tri, non latebit in tantùm, ut poſsit occultare refractionem lucis in corpore uitri aut partem eius.</s> <s xml:id="echoid-s16965" xml:space="preserve"> <lb/>Et hæc refractio erit ad partem, in qua eſt centrum uitri:</s> <s xml:id="echoid-s16966" xml:space="preserve"> ergo eſt ad perpendicularem exeuntem à <lb/>loco refractionis, perpendicularem ſuper ſuperficiem uitri ſphæricam:</s> <s xml:id="echoid-s16967" xml:space="preserve"> quia linea exiens à centro <lb/>uitri ad punctũ refractionis, eſt perpendicularis exiens à loco refractionis ſuper ſuperficiem ſphæ-<lb/>ricam.</s> <s xml:id="echoid-s16968" xml:space="preserve"> Deinde oportet experimentatorem euellere uitrum, & ponere è contrario huic poſitioni:</s> <s xml:id="echoid-s16969" xml:space="preserve"> <lb/>ſcilicet ut ponat ſuperficiem uitri æqualem ex parte duorum foraminũ, & ponat differentiam com <lb/>munem duabus ſuperficiebus æqualibus uitri, ſuper lineam ſecantem diametrum laminę perpen-<lb/>diculariter, & ponat medium differentiæ cõmunis extra centrũ laminæ.</s> <s xml:id="echoid-s16970" xml:space="preserve"> Vitro autẽ coniuncto hoc-<lb/>modo:</s> <s xml:id="echoid-s16971" xml:space="preserve"> linea, quæ tranſit per centra duorũ foraminum, non tranſit per centrũ uitri, ſed perueniet ad <lb/>punctum de ſuperficie eius æquali, in qua eſt centrũ eius, extra punctũ centri:</s> <s xml:id="echoid-s16972" xml:space="preserve"> & erit perpendicula-<lb/>ris ſuper ſuperficiem æqualẽ, ſicut ſupradictũ eſt.</s> <s xml:id="echoid-s16973" xml:space="preserve"> Et cũ linea, quæ tranſit per centra duorũ forami-<lb/>num, extẽſa fuerit rectè in imaginatione:</s> <s xml:id="echoid-s16974" xml:space="preserve"> perueniet ad punctũ, quod eſt extremitas diametri circuli <lb/>medij.</s> <s xml:id="echoid-s16975" xml:space="preserve"> Et cũ experimentator poſuerit uitrũ hoc modo, ponet inſtrumẽtum in uaſe, & uas in ſole, & <lb/>moueat inſtrumentũ, donec lux tranſeat per duo foramina, & intueatur oram inſtrumẽti:</s> <s xml:id="echoid-s16976" xml:space="preserve"> & inue-<lb/>niet lucem in interiore parte oræ inſtrumenti, & inueniet centrum lucis in circumferentia circuli <lb/> <pb o="240" file="0246" n="246" rhead="ALHAZEN"/> medij, & extra punctum, quod eſt extremitas diametri circuli medij:</s> <s xml:id="echoid-s16977" xml:space="preserve"> & declinans ad partem, in qua <lb/>eſt centrum ſphæræ uitreæ.</s> <s xml:id="echoid-s16978" xml:space="preserve"> Et linea, quæ egreditur à centro huius ſphæræ in imaginatione ad lo-<lb/>cum refractionis, eſt perpendicularis ſuper ſuperficiem huius ſphæræ:</s> <s xml:id="echoid-s16979" xml:space="preserve"> eſt ergo perpendicularis ſu-<lb/>per ſuperficiem aeris, qui contingit ſuperficiem ſphæræ.</s> <s xml:id="echoid-s16980" xml:space="preserve"> Hęc ergo refractio eſt ad partẽ contrariam <lb/>illi, in qua eſt perpendicularis, exiens à loco refractionis ſuper ſuperficiem aeris contingẽtis ſuper-<lb/>ficiem ſphæræ.</s> <s xml:id="echoid-s16981" xml:space="preserve"> Lux autem, quæ tranſit per centra duorum foraminum, tranſit in corpus uitri rectè:</s> <s xml:id="echoid-s16982" xml:space="preserve"> <lb/>quia eſt perpendicularis ſuper ſuperficiem uitri æqualem, oppoſitam duobus foraminibus:</s> <s xml:id="echoid-s16983" xml:space="preserve"> & per-<lb/>ueniet ad conuexitatem ſphæræ uitreæ:</s> <s xml:id="echoid-s16984" xml:space="preserve"> & cum peruenerit ad illam ſuperficiem, non erit perpen-<lb/>dicularis ſuper illam:</s> <s xml:id="echoid-s16985" xml:space="preserve"> cũ hon ſit diameter in ſphærà.</s> <s xml:id="echoid-s16986" xml:space="preserve"> Et omnis perpendicularis ſuper ſphæræ ſuper-<lb/>ficiem, eſt diameter illius, aut ſecundum rectitudinem diametri illius [ut cõſtat è 4 th 1 ſphæ.</s> <s xml:id="echoid-s16987" xml:space="preserve">] Sed <lb/>lux, quæ extenditur in corpore uitri hoc modo, non eſt perpendicularis ſuper ſuperficiẽ aeris con-<lb/>tingentis conuexum uitri:</s> <s xml:id="echoid-s16988" xml:space="preserve"> & hæc lux inuenitur refracta:</s> <s xml:id="echoid-s16989" xml:space="preserve"> ergo refringitur apud conuexum ſphæræ.</s> <s xml:id="echoid-s16990" xml:space="preserve"> <lb/>Et ſi experimentator infuderit aquam intra uas, (uitro remanente in ſuo ſitu) & poſuerit aquam <lb/>infra centrum laminæ, & aſpexerit lucem, quæ eſt in ora inſtrumenti:</s> <s xml:id="echoid-s16991" xml:space="preserve"> inueniet lucem refractam ad <lb/>partem, in qua eſt centrum uitri:</s> <s xml:id="echoid-s16992" xml:space="preserve"> ergo ad partem contrariam illi, in qua eſt perpendicularis, exiens <lb/>à loco refractionis, quæ extenditur à corpore uitri in corpore aeris perpendicularis ſuper concaui-<lb/>tatem aeris, contingentis conuexum uitri.</s> <s xml:id="echoid-s16993" xml:space="preserve"> Ex omnibus ergo his experimentationibus patet, quòd <lb/>lux ſolis tranſit in omne corpus diaphanum ſecũdum uerticationes linearum rectarum:</s> <s xml:id="echoid-s16994" xml:space="preserve"> & cum oc-<lb/>currit corpori diaphano diuerſæ diaphanitatis à diaphanitate corporis, in quo eſt, lineæq́ue, per <lb/>quas extenditur in primo corpore, fuerint declinãtes ſuper ſuperficiem ſecundi corporis:</s> <s xml:id="echoid-s16995" xml:space="preserve"> tunc lux <lb/>refringitur in corpore ſecundo in uerticatione linearũ rectarum aliarum à primis, per quas exten-<lb/>debatur in primo corpore.</s> <s xml:id="echoid-s16996" xml:space="preserve"> Et ſi lineæ rectæ, per quas extendebatur in primo corpore, fuerint per-<lb/>pendiculares ſuper ſuperficiem ſecundi corporis:</s> <s xml:id="echoid-s16997" xml:space="preserve"> tunc lux extenditur in rectitudine eius, & nõ re-<lb/>fringitur.</s> <s xml:id="echoid-s16998" xml:space="preserve"> Et cum lux obliqua fuerit, & exierit à corpore ſubtiliore ad groſsius, refringetur ad par-<lb/>tem perpendicularis, exeuntis à loco refractionis perpendicularis ſuper ſuperficiem ſecundi cor-<lb/>poris.</s> <s xml:id="echoid-s16999" xml:space="preserve"> Cum uerò lux obliqua, fuerit extenſa à groſsiore ad ſubtilius:</s> <s xml:id="echoid-s17000" xml:space="preserve"> refringetur ad partem contra-<lb/>riam perpendicularis exeuntis à loco refractionis ſuper ſuperficiem ſecundi corporis.</s> <s xml:id="echoid-s17001" xml:space="preserve"> Cum ergo <lb/>lux tranſeat per omnia diaphana ſecundum lineas rectas:</s> <s xml:id="echoid-s17002" xml:space="preserve"> ergo omnes luces extendentur in omni-<lb/>bus corporibus diaphanis:</s> <s xml:id="echoid-s17003" xml:space="preserve"> quia declaratum eſt in primo tractatu huius libri [14.</s> <s xml:id="echoid-s17004" xml:space="preserve"> 17.</s> <s xml:id="echoid-s17005" xml:space="preserve"> 28 n] quòd pro <lb/>prium lucis eſt extendi ſemper ſecundum lineas rectas, ſiue lux fuerit eſſentialis, ſiue accidentalis, <lb/>ſiue fortis, ſiue debilis.</s> <s xml:id="echoid-s17006" xml:space="preserve"> Præterea poteſt experimentator experiri luces accidentales in illo prædicto <lb/>inſtrumento, & illis uijs prædictis:</s> <s xml:id="echoid-s17007" xml:space="preserve"> ſi in aliqua domo, in quam intret lux diei per aliquod foramen <lb/>alicuius quantitatis, clauſerit ianuam, & poſuerit inſtrumẽtum in oppoſitione foraminis, & inſpe-<lb/>xerit lucem, quæ eſt intra aquam, & ultra uitrum in ora inſtrumenti, & proceſſerit per uias præo-<lb/>ſtenſas in experimentatione lucis ſolis.</s> <s xml:id="echoid-s17008" xml:space="preserve"> Cum ergo experimentator expertus fuerit lucem acciden-<lb/>talem his prædictis uijs:</s> <s xml:id="echoid-s17009" xml:space="preserve"> inueniet lucem accidentalem tranſeuntem per corpus aquæ & per corpus <lb/>uitri, & inueniet extenſionem eius in uitro ſecũdum uerticationes linearũ rectarum:</s> <s xml:id="echoid-s17010" xml:space="preserve"> & refractam, <lb/>ſi fuerit obliqua ſuper ſuperficiem ſecundi corporis:</s> <s xml:id="echoid-s17011" xml:space="preserve"> & rectam, ſi fuerit perpẽdicularis ſuper ſuper-<lb/>ficiem corporis ſecundi.</s> <s xml:id="echoid-s17012" xml:space="preserve"> In primo autem tractatu declaratum eſt, quòd lux omnis ſiue eſſentialis, <lb/>ſiue accidentalis, ſiue fortis, ſiue debilis, ſemper extenditur à quolibet puncto cuiuslibet corporis <lb/>ſecundum lineam rectam.</s> <s xml:id="echoid-s17013" xml:space="preserve"> Ex iſtis ergo omnibus, quæ declarauimus experientia & ratione:</s> <s xml:id="echoid-s17014" xml:space="preserve"> patet, <lb/>quòd omnis lux in corpore lucido eſſentialiter aut accidentaliter, fortiter aut debiliter extenditur <lb/>à quolibet puncto illius per corpus diaphanum, contingẽs illud corpus, per omnẽ lineam rectam, <lb/>per quam poterit extendi, ſiue illud corpus contingens ſit aer, aut aqua, aut lapis diaphanus.</s> <s xml:id="echoid-s17015" xml:space="preserve"> Et ſi <lb/>luces extenſæ per corpus contingens lucem, quæ eſt principium eius, occurrerint corpori diuerſæ <lb/>diaphanitatis à diaphanitate corporis, in quo exiſtit, & fuerint in lineis perpendicularibus ſuper <lb/>ſuperficiem ſecundi corporis:</s> <s xml:id="echoid-s17016" xml:space="preserve"> extendentur rectè in ſecundo corpore:</s> <s xml:id="echoid-s17017" xml:space="preserve"> & ſi fuerint in obliquis lineis <lb/>ſuper ſuperficiẽ ſecundi corporis, refringentur in ſecundo corpore:</s> <s xml:id="echoid-s17018" xml:space="preserve"> tum in ſecundo corpore exten-<lb/>dentur in uerticatione linearum rectarum aliarum à primis.</s> <s xml:id="echoid-s17019" xml:space="preserve"> Et ſi lux fuerit refracta:</s> <s xml:id="echoid-s17020" xml:space="preserve"> tunc linea, per <lb/>quam extendebatur lux in primo corpore, & linea per quam refringebatur in ſecundo:</s> <s xml:id="echoid-s17021" xml:space="preserve"> erunt in ea-<lb/>dem æquali ſuperficie [ut oſtenſum eſt 5 n] & refractio eius, cum egreſſa fuerit à corpore ſubtilio-<lb/>re ad groſsius:</s> <s xml:id="echoid-s17022" xml:space="preserve"> erit ad partem perpendicularis, exeuntis à loco refractionis ſuper ſuperficiem groſ-<lb/>ſioris corporis:</s> <s xml:id="echoid-s17023" xml:space="preserve"> & cũ egreſſa fuerit à groſsiore corpore ad ſubtilius:</s> <s xml:id="echoid-s17024" xml:space="preserve"> tũc refractio eius erit ad partem <lb/>cõtrariã illi, in quá eſt perpẽdicularis exiẽs à loco refractionis ſuper ſuperficiẽ ſubtilioris corporis.</s> <s xml:id="echoid-s17025" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div560" type="section" level="0" n="0"> <head xml:id="echoid-head487" xml:space="preserve" style="it">8. Radi{us} medio perpẽdicularis, irrefract{us} penetrat, obliqu{us} refringitur: in denſiore qui-<lb/>dem ad perpendicularem: in rariore uerò à perpẽdiculari è refractionis puncto excitata. 47 p 2.</head> <p> <s xml:id="echoid-s17026" xml:space="preserve">QVare autẽ refringatur lux, quando occurrit corpori diaphano diuerſæ diaphanitatis, cauſſa <lb/>hæc eſt:</s> <s xml:id="echoid-s17027" xml:space="preserve"> quia tranſitus lucis per corpora diaphana fit per motum uelociſsimum, ut declara-<lb/>uimus in tractatu ſecundo.</s> <s xml:id="echoid-s17028" xml:space="preserve"> Luces ergo, quę extenduntur per corpora diaphana, extendun-<lb/>tur motu ueloci, qui non patet ſenſui propter ſuam uelocitatem.</s> <s xml:id="echoid-s17029" xml:space="preserve"> Præterea motus earum in ſubtili-<lb/>bus corporibus, ſcilicet in illis;</s> <s xml:id="echoid-s17030" xml:space="preserve"> quæ ualde ſunt diaphana, uelocior eſt motu earum in ijs, quæ ſunt <lb/>groſsiora illis, ſcilicet quæ minus ſunt diaphana.</s> <s xml:id="echoid-s17031" xml:space="preserve"> Omne enim corpus diaphanum, cum lux trãſit in <lb/>ipſum, reſiſtit luci aliquantulum, ſecũdum quod habet de groſsitie.</s> <s xml:id="echoid-s17032" xml:space="preserve"> Nam in omni corpore naturali <lb/> <pb o="241" file="0247" n="247" rhead="OPTICAE LIBER VII."/> neceſſe eſt, ut ſit aliqua groſsities:</s> <s xml:id="echoid-s17033" xml:space="preserve"> nam corpus paruæ diaphanitatis nõ habet finem in imaginatio-<lb/>ne, quæ eſt imaginatio lucidæ diaphanitatis:</s> <s xml:id="echoid-s17034" xml:space="preserve"> & omnia corpora naturalia perueniũt ad finem, quem <lb/>non poſſunt tranſire.</s> <s xml:id="echoid-s17035" xml:space="preserve"> Corporà ergo naturalia diaphana non poſſunt euadere aliquam groſsitiem.</s> <s xml:id="echoid-s17036" xml:space="preserve"> <lb/>Luces ergo cum tranſeunt per corpora diaphana, tranſeunt ſecundũ diaphanitatem, quæ eſt in eis, <lb/>& ſic impediunt lucem ſecundum groſsitiem, quæ eſt in eis.</s> <s xml:id="echoid-s17037" xml:space="preserve"> Cum ergo lux tranſiuerit per corpus <lb/>diaphanum, & occurrit alij corpori groſsiori primo:</s> <s xml:id="echoid-s17038" xml:space="preserve"> tunc corpus groſsius reſiſtit luci uehemẽtιus, <lb/>quàm primum reſiſtebat:</s> <s xml:id="echoid-s17039" xml:space="preserve"> & omne motum cum mouetur ad aliquam partem eſſentialiter aut acci-<lb/>dentaliter, ſi occurrerit reſiſtenti, neceſſe eſt, ut motus eius tranſmutetur:</s> <s xml:id="echoid-s17040" xml:space="preserve"> & ſi reſiſtentia fuerit for-<lb/>tis:</s> <s xml:id="echoid-s17041" xml:space="preserve"> tunc motus ille refringetur ad contrariam partem:</s> <s xml:id="echoid-s17042" xml:space="preserve"> ſi uerò debilis, nõ refringetur ad contrariam <lb/>partem, nec poterit per illã procedere, per quam incœperat:</s> <s xml:id="echoid-s17043" xml:space="preserve"> ſed motus eius mutabitur.</s> <s xml:id="echoid-s17044" xml:space="preserve"> Omnium <lb/>autem motorum naturaliter, quæ rectè mouentur per aliquod corpus paſsibile:</s> <s xml:id="echoid-s17045" xml:space="preserve"> trãſitus ſuper per-<lb/>pendicularem, quæ eſt in ſuperficie corporis, in quo eſt trãſitus, erιt facilior.</s> <s xml:id="echoid-s17046" xml:space="preserve"> Et hoc uidetur in cor-<lb/>poribus naturalibus.</s> <s xml:id="echoid-s17047" xml:space="preserve"> Si enim aliquis acceperit tabulã ſubtilem, & paxιllauerit illam ſuper aliquod <lb/>foramen amplum, & ſteterit in oppoſitione tabulæ, & acceperit pilam ferream, & eiecerit eã ſuper <lb/>tabulam fortiter, & obſeruauerit, ut motus pilæ ſit ſuper perpendicularem ſuper ſuperficiem tabu-<lb/>læ:</s> <s xml:id="echoid-s17048" xml:space="preserve"> tunc tabula cedet pilæ aut frangetur, ſi tabula ſubtilis fuerit, & uis, qua ſphæra mouetur, fuerit <lb/>fortis.</s> <s xml:id="echoid-s17049" xml:space="preserve"> Et ſi ſteterit in parte obliqua ab oppoſitione tabulę, & in illa eadẽ diſtantia, in qua prius erat, <lb/>& eiecerit pilam ſuper tabulam illam eandem, in quam prius eiecerat:</s> <s xml:id="echoid-s17050" xml:space="preserve"> tunc ſphæra labetur de tabu-<lb/>la, ſi tabula non fuerit ualde ſubtilis, nec mouebitur ad illam partem, ad quam primò mouebatur, <lb/>ſed declinabit ad aliquam partem aliam.</s> <s xml:id="echoid-s17051" xml:space="preserve"> Et ſimiliter, ſi acceperit enſem, & poſuerit corã ſe lignum, <lb/>& percuſſeri t cum enſe, ita ut enſis ſit perpendicularis ſuper ſuperficiem ligni:</s> <s xml:id="echoid-s17052" xml:space="preserve"> tunc lignum ſecabi-<lb/>tur magis:</s> <s xml:id="echoid-s17053" xml:space="preserve"> & ſi fuerit obliquus, & percuſſerit obliquè lignum:</s> <s xml:id="echoid-s17054" xml:space="preserve"> tunc lignum non ſecabitur omnino, <lb/>ſed fortè ſecabitur in parte, aut fortè enſis errabit deuiando:</s> <s xml:id="echoid-s17055" xml:space="preserve"> & quanto magis fuerit enſis obliquus, <lb/>tantò minus aget in lignum:</s> <s xml:id="echoid-s17056" xml:space="preserve"> & alia multa ſunt ſimilia:</s> <s xml:id="echoid-s17057" xml:space="preserve"> ex quibus patet, quòd motus ſuper perpen-<lb/>dicularem eſt fortior & facilior:</s> <s xml:id="echoid-s17058" xml:space="preserve"> & quòd de obliquis motibus ille, qui uιcinior eſt perpendiculari, <lb/>eſt facilior remotiore.</s> <s xml:id="echoid-s17059" xml:space="preserve"> Lux ergo, ſi occurrit corpori diaphano groſsiori illo corpore, in quo exiſtit:</s> <s xml:id="echoid-s17060" xml:space="preserve"> <lb/>tunc impedietur ab eo, ita quòd non tranſibit in partem, in quam mouebatur, ſed quia non fortiter <lb/>reſiſtit, non redibit in partem, ad quam mouebatur.</s> <s xml:id="echoid-s17061" xml:space="preserve"> Si ergo motus lucis tranſiuerit ſuper perpendi-<lb/>cularem, tranſibit rectè propter fortitudinem motus ſuper perpendicularem:</s> <s xml:id="echoid-s17062" xml:space="preserve"> & ſi motus eius fue-<lb/>rit ſuper lineam obliquam:</s> <s xml:id="echoid-s17063" xml:space="preserve"> tunc nõ poterit tranſire propter debilitatem motus:</s> <s xml:id="echoid-s17064" xml:space="preserve"> accidit ergo, ut de-<lb/>clinetur ad partem motus, in quam facilius mouebitur, quàm in partem, in quam mouebatur:</s> <s xml:id="echoid-s17065" xml:space="preserve"> ſed <lb/>facilior motuum eſt ſuper perpendicularem:</s> <s xml:id="echoid-s17066" xml:space="preserve"> & quod uicinius eſt perpendiculari, eſt facilius remo-<lb/>tiore.</s> <s xml:id="echoid-s17067" xml:space="preserve"> Et motus in corpore, in quod tranſit, ſi fuerit obliquus ſuper ſuperficiem illius corporis, com <lb/>ponitur ex motu in par<gap/>e perpendicuiaris trãſeuntis in corpus, in quo eſt motus, & ex motu in par-<lb/>te lineæ, quæ eſt perpendicularis ſuper perpendicularem, quæ tranſit in ipſum.</s> <s xml:id="echoid-s17068" xml:space="preserve"> Cum ergo lux fue-<lb/>rit mota in corpore diaphano groſſo ſuper lineã obliquam:</s> <s xml:id="echoid-s17069" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0247-01a" xlink:href="fig-0247-01"/> tunc trãſitus eius in illo corpore diaphano erit per motum <lb/>compoſitum ex duobus prædictis motibus.</s> <s xml:id="echoid-s17070" xml:space="preserve"> Et quia groſsi-<lb/>ties corporis reſiſtit ei ad uerticationem, quam intẽdebat, <lb/>& reſiſtentia eius non eſt ualde fortis:</s> <s xml:id="echoid-s17071" xml:space="preserve"> ex quo ſequeretur, <lb/>quòd declinaret ad partẽ, ad quam facilius tranſiret:</s> <s xml:id="echoid-s17072" xml:space="preserve"> & mo-<lb/>tus ſuper perpendicularem eſt facilimus motuum:</s> <s xml:id="echoid-s17073" xml:space="preserve"> neceſſe <lb/>eſt ergo, ut lux quæ extẽditur ſuper lineam obliquam, mo-<lb/>ueatur ſuper perpendicularem, exeuntem à puncto, in quo <lb/>lux occurrit ſuperficiei corporis diaphani groſsi.</s> <s xml:id="echoid-s17074" xml:space="preserve"> Et quia <lb/>motus eius eſt compoſitus ex duobus motibus, quorũ al-<lb/>ter eſt ſuper lineam perpendicularem ſuper ſuperficiẽ cor-<lb/>poris groſsi, & reliquus ſuper lineam perpendicularem ſu-<lb/>per perpendicularem hanc:</s> <s xml:id="echoid-s17075" xml:space="preserve"> & motus compoſitus, qui eſt in <lb/>ipſo, nõ omnino dimittitur, ſed ſolummodo impeditur:</s> <s xml:id="echoid-s17076" xml:space="preserve"> ne-<lb/>ceſſe eſt, ut lux declinet ad partẽ faciliorem parte, ad quam <lb/>prius mouebatur, remanente in ipſo motu compoſito:</s> <s xml:id="echoid-s17077" xml:space="preserve"> ſed <lb/>pars facilior parte, ad quam mouebatur remanente motu <lb/>in ipſo, eſt illa pars, quę eſt uicinor perpẽdiculari.</s> <s xml:id="echoid-s17078" xml:space="preserve"> Vnde lux, <lb/>quę extẽditur in corpore diaphano, ſi occurrit corpori dia-<lb/>phano groſsiori corpore, in quo exiſtit:</s> <s xml:id="echoid-s17079" xml:space="preserve"> refringetur per li-<lb/>neam propinquiorem perpendiculari, exeunti à puncto, in <lb/>quo occurrit corpori groſsiori, quæ extenditur in corpore <lb/>groſsiore per aliam lineam quàm ſit linea, per quam moue-<lb/>batur.</s> <s xml:id="echoid-s17080" xml:space="preserve"> Hęc ergo cauſſa eſt refractionis ſplendoris in corpo-<lb/>ribus diaphanis, quæ ſunt groſsiora corporibus diaphanis, <lb/>in quibus exiſtunt:</s> <s xml:id="echoid-s17081" xml:space="preserve"> & ideo refractio propriè eſt inuẽta in lucibus obliquis.</s> <s xml:id="echoid-s17082" xml:space="preserve"> Cum ergo lux extendi-<lb/>tur in corpore diaphano, & occurrerit corpori diaphano diuerſæ diaphanitatis a corpore, in quo <lb/>exiſtit, & groſsiori, & fuerit obliqua ſuper ſuperficiem corporis diaphani cui occurrit:</s> <s xml:id="echoid-s17083" xml:space="preserve"> refringetur <lb/> <pb o="242" file="0248" n="248" rhead="ALHAZEN"/> ad partẽ perpendicularis ſuper ſuperficiem corporis diaphani extẽſæ in corpore groſsiore.</s> <s xml:id="echoid-s17084" xml:space="preserve"> Cauſſa <lb/>autem, quæ facit refractionem lucis à corpore groſsiore ad corpus ſubtilius ad partem contrariam <lb/>parti perpẽdicularis, eſt:</s> <s xml:id="echoid-s17085" xml:space="preserve"> quia cum lux mota fuerit in corpore diaphano, repellet eam aliqua repul-<lb/>ſione, & corpus groſsius repellet eam maiore repulſione, ſicut lapis, cũ mouetur in aere, mouetur <lb/>facilius & uelocius, quàm ſi moueretur in aqua:</s> <s xml:id="echoid-s17086" xml:space="preserve"> eò quòd aqua repellit ipſum maiore repulſione, <lb/>quàm aer.</s> <s xml:id="echoid-s17087" xml:space="preserve"> Cum ergo lux exierit à corpore groſsiore in ſubtilius:</s> <s xml:id="echoid-s17088" xml:space="preserve"> tunc motus eius erit uelocior.</s> <s xml:id="echoid-s17089" xml:space="preserve"> Et <lb/>cum lux fuerit obliqua ſuper duas ſuperficies corporis diaphani, quod eſt differentia cõmunis am-<lb/>bobus corporibus:</s> <s xml:id="echoid-s17090" xml:space="preserve"> tunc motus eius erit ſuper lineam exiſtẽtem inter perpendicularem, exeuntem <lb/>à principio motus eius, & inter perpendicularem ſuper lineam perpendicularem, exeuntem etiam <lb/>à principio motus.</s> <s xml:id="echoid-s17091" xml:space="preserve"> Reſiſtentia ergo corporis groſsioris erit à parte, ad quam exit ſecunda perpen-<lb/>dicularis.</s> <s xml:id="echoid-s17092" xml:space="preserve"> Cum ergo lux exiuerit à corpore groſsiore, & peruenerit ad corpus ſubtilius:</s> <s xml:id="echoid-s17093" xml:space="preserve"> tunc reſi-<lb/>ſtentia corporis ſubtilioris facta luci, quæ eſt in parte, ad quam ſecunda exit perpendicularis, erit <lb/>minor prima reſiſtentia:</s> <s xml:id="echoid-s17094" xml:space="preserve"> & fit motus lucis ad partem, à qua reſiſtebatur, maior.</s> <s xml:id="echoid-s17095" xml:space="preserve"> Et ſic eſt de luce in <lb/>corpore ſubtiliore ad partem contrariam parti perpendicularis.</s> <s xml:id="echoid-s17096" xml:space="preserve"/> </p> <div xml:id="echoid-div560" type="float" level="0" n="0"> <figure xlink:label="fig-0247-01" xlink:href="fig-0247-01a"> <variables xml:id="echoid-variables199" xml:space="preserve">a h e d c b k q l g f</variables> </figure> </div> </div> <div xml:id="echoid-div562" type="section" level="0" n="0"> <head xml:id="echoid-head488" xml:space="preserve">DE QVALITATE REFRACTIONIS LVCIS IN <lb/>corporibus diaphanis. Cap. III.</head> <head xml:id="echoid-head489" xml:space="preserve" style="it">9. Superficies refractionis eſt perpendicularis ſuperficiei refractiui. 2 p 10.</head> <p> <s xml:id="echoid-s17097" xml:space="preserve">IN prædicto capitulo [5 n] declaratũ eſt, quòd omnis lux, quæ reſringitur à corpore diaphano <lb/>ad aliud corpus diaphanum, ſemper erit in una ſuperficie æquali.</s> <s xml:id="echoid-s17098" xml:space="preserve"> Linea ergo recta, per quã ex-<lb/>tenditur lux in aere, & linea recta, per quam refringitur in aqua, ſemper erũt in eadem ſuperficie <lb/>æquali.</s> <s xml:id="echoid-s17099" xml:space="preserve"> Hæc autem ſuperficies apud ιnſpectionem inſtrumenti prædicti, eſt medius circulus ille ex <lb/>tribus ſignatis in interiore parte oræ inſtrumẽti & ille circulus eſt æquidiſtans ſuperficiei interio-<lb/>ris laminæ:</s> <s xml:id="echoid-s17100" xml:space="preserve"> ſed ſuperficies interioris laminæ eſt æquidiſtans ſuperficiei dorſi, cui ſuperponitur ſu-<lb/>perficies regulæ quadratæ:</s> <s xml:id="echoid-s17101" xml:space="preserve"> ergo ſuperficies circuli medij eſt æquidiſtãs ſuperficiei regulæ quadra-<lb/>tæ:</s> <s xml:id="echoid-s17102" xml:space="preserve"> & ſuperficies regulæ quadratæ, quæ eſt ſuperpoſita dorſo laminæ, eſt perpendicularis ſuper al-<lb/>teram ſuperficiem, ſecantem ſuperficiem ſuperpoſitam:</s> <s xml:id="echoid-s17103" xml:space="preserve"> & hæc ſuperficies regulæ ſuperponitur ſu-<lb/>perficiei duarum differẽtiarum ſibi applicatarum in duabus extremitatibus regulæ:</s> <s xml:id="echoid-s17104" xml:space="preserve"> ſed ſuperficies <lb/>duarum differentiarum ſuperponitur oræ inſtrumenti.</s> <s xml:id="echoid-s17105" xml:space="preserve"> Ergo ſuperficies medij circuli eſt perpen-<lb/>dicularis ſuper ſuperficiem tranſeuntem ſuper oram inſtrumenti.</s> <s xml:id="echoid-s17106" xml:space="preserve"> Et hæc ſuperficies tranſiens per <lb/>oram inſtrumenti, eſt æquidiſtans horizonti apud experimẽtationem.</s> <s xml:id="echoid-s17107" xml:space="preserve"> Superficies ergo medij cir-<lb/>culi eſt perpendicularis ſuper ſuperficiem horizontis.</s> <s xml:id="echoid-s17108" xml:space="preserve"> Cum ergo declaratũ ſit [4.</s> <s xml:id="echoid-s17109" xml:space="preserve"> 7.</s> <s xml:id="echoid-s17110" xml:space="preserve"> 8 n] quòd lux, <lb/>quæ eſt in aere, & refringitur in aqua, eſt apud experim entationem in circumferẽtia medij circuli:</s> <s xml:id="echoid-s17111" xml:space="preserve"> <lb/>manifeſtum, quòd lux, quę extenditur in aere, & refringitur in aqua, eſt ſemper in eadem ſuperficie <lb/>æquali ſuper ſuperficiem horizontis.</s> <s xml:id="echoid-s17112" xml:space="preserve"> Et etiam imaginemur lineam à <lb/> <anchor type="figure" xlink:label="fig-0248-01a" xlink:href="fig-0248-01"/> centro medij circuli ad centrum mundi:</s> <s xml:id="echoid-s17113" xml:space="preserve"> ſic ergo linea hæc erit per-<lb/>pendicularis ſuper ſuperficiem aquæ [ut oſtenſum eſt 25 n 4] quia <lb/>eſt diameter mundi:</s> <s xml:id="echoid-s17114" xml:space="preserve"> ſed hęc linea eſt in ſuperficie medij circuli:</s> <s xml:id="echoid-s17115" xml:space="preserve"> ergo <lb/>eſt in ſuperficie refractionis.</s> <s xml:id="echoid-s17116" xml:space="preserve"> Ergo ſuperficies refractionis eſt perpen <lb/>dicularis ſuper ſuperficiem aquæ.</s> <s xml:id="echoid-s17117" xml:space="preserve"> Et iam declaratũ eſt, quòd cũ lux <lb/>refringitur ex aere ad aquam:</s> <s xml:id="echoid-s17118" xml:space="preserve"> erit inter primam lineã, per quã exten-<lb/>ditur in aere, quæ eſt inter diametrũ medij circuli, & inter perpendi-<lb/>cularem, exeuntẽ à cẽtro medij circuli ſuper ſuperficiẽ aquæ.</s> <s xml:id="echoid-s17119" xml:space="preserve"> Et iam <lb/>declaratum eſt etiã, quòd lux, quæ eſt in puncto, quod eſt centrũ lu-<lb/>cis, quæ eſt intra aquã, non peruenit ad ipſum, niſi ex luce, quæ extẽ <lb/>ditur à cẽtro medij circuli.</s> <s xml:id="echoid-s17120" xml:space="preserve"> Lux ergo, quę refringitur ex aere ad aquã, <lb/>refringitur in ſuperficie perpendiculari ſuper ſuperficiẽ aquæ.</s> <s xml:id="echoid-s17121" xml:space="preserve"> Et re-<lb/>fractio eius erit ad partẽ perpẽdicularis exeuntis à loco refractionis <lb/>ſuper ſuperficiẽ aquæ & nõ perueniet ad perpendicularẽ.</s> <s xml:id="echoid-s17122" xml:space="preserve"> Refractio <lb/>autẽ lucis ab aere ad uitrũ hoc modo fit.</s> <s xml:id="echoid-s17123" xml:space="preserve"> Declaratũ eſt enim in expe-<lb/>rimentatione uitri, quòd cũ linea, quæ tranſit per centra duorũ fora-<lb/>minũ, fuerit obliqua ſuper ſuperficiẽ uitri æqualẽ, & tranſiuerit per <lb/>centrũ uitri, & ſuperficies uitri æqualis fuerit ex parte foraminum:</s> <s xml:id="echoid-s17124" xml:space="preserve"> <lb/>tunc refringetur apud centrũ uitri:</s> <s xml:id="echoid-s17125" xml:space="preserve"> & refractio eius erit in ſuperficie <lb/>circuli medij ad partẽ, in qua eſt perpendicularis, exiens à cẽtro uitri <lb/>ſuper ſuperficiẽ uitri æqualẽ.</s> <s xml:id="echoid-s17126" xml:space="preserve"> Et declaratũ eſt etiã, quòd cũ linea, quę <lb/>tranſit per cẽtra duorũ foraminũ, fuerit obliqua ſuper ſuperficiẽ ui-<lb/>tri ſphæricã:</s> <s xml:id="echoid-s17127" xml:space="preserve"> & ſuperficies ſphærica fuerit ex parte foraminũ:</s> <s xml:id="echoid-s17128" xml:space="preserve"> tũc lux <lb/>refringetur in corpore uitri, & apud ſuperficiẽ uitri ſphæricã:</s> <s xml:id="echoid-s17129" xml:space="preserve"> & erit <lb/>refractio eius in ſuperficie medij circuli, & ad partẽ perpendicularis, <lb/>exeuntis à loco refractionis ſuper ſuperficiẽ uitri ſphæricam.</s> <s xml:id="echoid-s17130" xml:space="preserve"> Et ſu-<lb/>perficies uitri æqualis, in qua eſt centrũ uitrei circuli, eſt perpendi-<lb/>cularis ſuper ſuperficiem laminæ.</s> <s xml:id="echoid-s17131" xml:space="preserve"> Eſt ergo perpendicularis ſuper ſu-<lb/>perficiem medij circuli.</s> <s xml:id="echoid-s17132" xml:space="preserve"> Superficies ergo medij circuli eſt perpendicularis ſuper ſuperficiem uitri <lb/> <pb o="243" file="0249" n="249" rhead="OPTICAE LIBER VII."/> æqualem.</s> <s xml:id="echoid-s17133" xml:space="preserve"> Et ſuperficies circuli medij tranſit etiã per centrũ ſphęræ uitreæ in omnibus experimen-<lb/>tationibus uitri.</s> <s xml:id="echoid-s17134" xml:space="preserve"> Ergo eſt perpẽdicularis ſuper ſuperficiẽ uitri ſphæricã.</s> <s xml:id="echoid-s17135" xml:space="preserve"> Lux ergo, quę extẽditur in <lb/>aere, & refringitur in corpore uitri apud extenſionẽ eius in aere, poſtquã iterũ refringitur in uitro, <lb/>ſemper eſt in ſuperficie perpẽdiculari ſuper ſuperficiẽ uitri.</s> <s xml:id="echoid-s17136" xml:space="preserve"> Et ſemper refractio eius erit ad partem <lb/>perpẽdicularis, exeũtis à loco refractionis ſuper ſuperficiẽ uitri, ſiue ſuperficies uitri fuerit æqualis, <lb/>ſiue ſphęrica.</s> <s xml:id="echoid-s17137" xml:space="preserve"> Itẽ declaratũ eſt etiã, quòd linea, quę trãſit per duo cẽtra foraminũ, cũ fuerit perpẽdi-<lb/>cularis ſuք ſuperficiẽ uitri, & extẽſa fuerit in corpus uitri ſecundũ rectitudinẽ, & ſuperficies ſphæ-<lb/>rica fuerit ex parte foraminũ, & fuerit hęc linea, ſcilicet quę trãſit per centra duorũ foraminũ, decli-<lb/>nãs ſuper ſuperficiẽ uitri æqualẽ, & trãſiuerit per centrũ uitri, & refracta fuerit in corpore aeris cõ <lb/>tingẽtis ſuperficiẽ uitri æqualẽ, & apud centrũ uitri:</s> <s xml:id="echoid-s17138" xml:space="preserve"> tũc refractio eius erit in ſuperficie circuli me-<lb/>dij, & ad contrariã partẽ illi, in qua eſt perpẽdicularis, exiẽs à cẽtro uitri ſuper ſuperficiẽ uitri æqua-<lb/>lem.</s> <s xml:id="echoid-s17139" xml:space="preserve"> Et declaratũ eſt etiã, quòd linea, quę trãſit per cẽtra duorũ foraminũ, cũ fuerit perpendicularis <lb/>ſuper ſuperficiẽ uitri æqualẽ, & ſi fuerit extenſa in corpore uitri ſecundũ rectitudinẽ, & ſuperficies <lb/>æqualis fuerit ex parte foraminũ, & hęc linea, ſcilicet quę trãſit per cẽtra duorũ foraminũ, fuerit ob-<lb/>liqua ſuper ſuperficiẽ uitri ſphęricã, & nõ trãſiens per centrũ eius, & fuerit refracta apud ſuperficiẽ <lb/>uitri ſphæricã in corpore aeris contingẽtis ſuperficiẽ ſphęricam:</s> <s xml:id="echoid-s17140" xml:space="preserve"> tũc refractio eius erit in ſuperficie <lb/>medij circuli, & ad partẽ contrariã illi, in qua eſt perpẽdicularis, exiens à loco refractionis ſuper ſu-<lb/>perficiem ſecũdi corporis.</s> <s xml:id="echoid-s17141" xml:space="preserve"> Et in his duobus ſitibus ſuperficies etiã medij circuli eſt perpẽdicularis <lb/>ſuper ſuperficiẽ uitri æqualẽ & ſphæricã.</s> <s xml:id="echoid-s17142" xml:space="preserve"> Lux ergo, quę extẽditur in corpore uitri, & refringitur in <lb/>aere, dũ extẽditur in uitro, & refringitur in aere, ſemper eſt in ſuperficie perpẽdiculari ſuper ſuper-<lb/>ficiem aeris:</s> <s xml:id="echoid-s17143" xml:space="preserve"> & ſemper refractio erit ad partẽ contrariam illi, in qua eſt perpẽdicularis exiens à loco <lb/>refractionis ſuք ſuperficiẽ aeris.</s> <s xml:id="echoid-s17144" xml:space="preserve"> Ex omnibus ergo iſtis prædeclaratis patet, quòd omnis lux refra-<lb/>cta à corpore diaphano ad aliud corpus, ſemper refringitur in ſuperficie perpẽdiculari ſuper ſuper-<lb/>ficiem ſecũdi corporis.</s> <s xml:id="echoid-s17145" xml:space="preserve"> Et ſi ſecundũ corpus fuerit groſsius primo:</s> <s xml:id="echoid-s17146" xml:space="preserve"> tũc refractio eius erit ad partem <lb/>perpẽdicularis, exeuntis à loco refractionis ſuper ſuperficiẽ ſecũdi corporis, & nõ peruenit ad per-<lb/>pendicularem.</s> <s xml:id="echoid-s17147" xml:space="preserve"> Et ſi ſecundũ corpus fuerit ſubtilius primo:</s> <s xml:id="echoid-s17148" xml:space="preserve"> refractio erit ad partẽ contrariam illi, in <lb/>qua eſt perpẽdicularis, exiens à loco refractionis ſuper ſuperficiẽ ſecundi corporis, ſecundũ diuer-<lb/>ſitatem figurarũ ſuperficierum corporũ diaphanorũ.</s> <s xml:id="echoid-s17149" xml:space="preserve"> Et ex his etiã patet, quòd cum lux refringitur <lb/>à corpore diaphano ad ſecundũ corpus diaphanũ, & de ſecũdo ad tertiũ:</s> <s xml:id="echoid-s17150" xml:space="preserve"> refringetur etiã in ſuper-<lb/>ficie tertij, ſi diaphanitas tertij differt à diaphanitate ſecũdi:</s> <s xml:id="echoid-s17151" xml:space="preserve"> ſi uerò tertiũ fuerit groſsius ſecũdo:</s> <s xml:id="echoid-s17152" xml:space="preserve"> tũc <lb/>refractio lucis erit ad partẽ perpẽdicularis exeuntis à loco refractionis ſuper ſuperficiẽ tertij:</s> <s xml:id="echoid-s17153" xml:space="preserve"> ſi aũt <lb/>tertiũ fuerit ſubtilius ſecũdo:</s> <s xml:id="echoid-s17154" xml:space="preserve"> tũc refractio lucis erit ad partẽ cõtrariã illi, in qua eſt perpẽdicularis.</s> <s xml:id="echoid-s17155" xml:space="preserve"> <lb/>Similiter ſi lux refracta fuerit ad quartũ corpus, & ad quintum, aut ad plurá.</s> <s xml:id="echoid-s17156" xml:space="preserve"> Hoc aũt declarauimus <lb/>quidẽ in hoc capitulo, qualiter omnes luces refringãtur in corporibus diaphanis diuerſæ diaphani <lb/>tatis.</s> <s xml:id="echoid-s17157" xml:space="preserve"> Quare aũt fiat refractio in ſuperficie perpẽdiculari ſuper ſuperficiẽ corporis diaphani, hęc eſt:</s> <s xml:id="echoid-s17158" xml:space="preserve"> <lb/>quia linea, per quã extẽditur lux in primo diaphano corpore, refringitur ad partẽ perpẽdicularis in <lb/>hac ſuperficie, ſcilicet, in qua eſt perpẽdicularis & prima linea:</s> <s xml:id="echoid-s17159" xml:space="preserve"> pars enim perpẽdicularis eſt in hac <lb/>ſuperficie:</s> <s xml:id="echoid-s17160" xml:space="preserve"> ideo refractio fit in ſuperficie perpendiculari ſuper ſuperficiem corporis diaphani.</s> <s xml:id="echoid-s17161" xml:space="preserve"/> </p> <div xml:id="echoid-div562" type="float" level="0" n="0"> <figure xlink:label="fig-0248-01" xlink:href="fig-0248-01a"> <variables xml:id="echoid-variables200" xml:space="preserve">a d c g b e f</variables> </figure> </div> </div> <div xml:id="echoid-div564" type="section" level="0" n="0"> <head xml:id="echoid-head490" xml:space="preserve" style="it">10. Magnitudines angulorũ refractiõis ab aere ad aquãorgano refractiõis explorare. 5 p 10.</head> <p> <s xml:id="echoid-s17162" xml:space="preserve">QVantitates autẽ angulorũ refractionis differũt ſecundũ quantitates angulorũ, quos conti-<lb/>nent prima linea, per quã extenditur lux in primo corpore, & perpẽdicularis exiens à loco <lb/>refractionis ſuper ſuperficiẽ ſecũdi corporis, ſecundũ diaphanitatem ſecũdi corporis.</s> <s xml:id="echoid-s17163" xml:space="preserve"> Nam <lb/>quanto magis creſcit angulus, quẽ cõtinent prima linea & perpẽdicularis, tantò creſcit angulus re-<lb/>fractionis:</s> <s xml:id="echoid-s17164" xml:space="preserve"> & quantò magis decreſcit ille angulus, quẽ continẽt perpẽdicularis & prima linea, tantò <lb/>decreſcit angulus refractionis.</s> <s xml:id="echoid-s17165" xml:space="preserve"> Sed anguli refractionũ nõ obſeruãt eandẽ proportionẽ ad angulos, <lb/>quos cõtinet prima linea cũ perpẽdiculari, ſed differũt hæ ꝓportiones in eodẽ corpore diaphano.</s> <s xml:id="echoid-s17166" xml:space="preserve"> <lb/>Cũ ergo prima linea, per quã lux extẽditur in primo corpore, cõtinuerit cũ perpẽdiculari duos an-<lb/>gulos inæquales, in duobus diuerſis tẽporibus, aut in duobus locis diuerſis:</s> <s xml:id="echoid-s17167" xml:space="preserve"> tũc ꝓportio anguli re-<lb/>fractionis, quæ eſt ab angulo minore ad angulũ minorẽ, minor erit ꝓportione anguli refractionis <lb/>anguli maioris ad angulũ maiorẽ.</s> <s xml:id="echoid-s17168" xml:space="preserve"> Cũ ergo experimẽtator uoluerit experiri illos angulos, diuidat à <lb/>circulo medio, qui eſt in circũferentia inſtrumẽti, ex parte cẽtri foraminis, quod eſt in circũferentia <lb/>inſtrumẽti, arcum decẽ partium ex illis partibus, quibus medius circulus diuiditur 360:</s> <s xml:id="echoid-s17169" xml:space="preserve"> deinde ex-<lb/>trahamus à loco differẽtiæ lineã rectã, perpendicularẽ ſuper ſuperficiẽ laminæ, & copulemus extre <lb/>mitatem eius, quæ eſt in lamina, cũ centro laminæ per lineã rectã, & protrahamus ipſam in aliã par-<lb/>tem:</s> <s xml:id="echoid-s17170" xml:space="preserve"> deinde diuidamus in circumferẽtia medij circuli etiã arcum ſequentẽ primum, cuius quãtitas <lb/>ſit 90 partiũ:</s> <s xml:id="echoid-s17171" xml:space="preserve"> & ſignemus in extremitate huius arcus ſignũ.</s> <s xml:id="echoid-s17172" xml:space="preserve"> Linea ergo, quæ exit à centro medij cir-<lb/>culi ad hoc ſignũ, erit perpẽdicularis ſuper lineã exeuntem à centro medij circuli ad primum ſignũ, <lb/>quod eſt in circũferentia medij circuli [per 33 p 6:</s> <s xml:id="echoid-s17173" xml:space="preserve"> quia hæ duæ lineæ quadrantẽ totius peripheriæ <lb/>comprehẽdunt] & erit arcus reſiduus, qui eſt inter ſignũ & extremitatẽ diametri medij circuli, quę <lb/>tranſit per centra duorũ foraminũ, 80 partiũ.</s> <s xml:id="echoid-s17174" xml:space="preserve"> Signemus in extremitate huius diametri etiã ſignum:</s> <s xml:id="echoid-s17175" xml:space="preserve"> <lb/>deinde ponamus inſtrumentũ in uaſe, & obſeruemus ut circumferentia uaſis ſit æquidiſtans hori-<lb/>zonti, & incipiamus experiri ab hora ortus ſolis, & infundamus in uas aquam claram, quouſq;</s> <s xml:id="echoid-s17176" xml:space="preserve"> per-<lb/>ueniat ad centrum laminæ, & moueamus inſtrumẽtum, donec prima linea ſignata in ſuperficie la-<lb/>minæ, contingat ſuperficiem aquæ:</s> <s xml:id="echoid-s17177" xml:space="preserve"> in hoc ergo ſitu linea, quę tranſit per centrũ circuli medij, æqui-<lb/> <pb o="244" file="0250" n="250" rhead="ALHAZEN"/> diſtans eſt primæ lineæ ſignatæ in ſuperficie laminæ, cuius extremitas peruenit ad primũ ſignum, <lb/>ſignatum in circumferentia medij circuli, & tanget <lb/> <anchor type="figure" xlink:label="fig-0250-01a" xlink:href="fig-0250-01"/> etiam ſuperficiem aquæ:</s> <s xml:id="echoid-s17178" xml:space="preserve"> locus enim harum duarum <lb/>linearũ non differt in reſpectu ſuperficiei aquæ, quò <lb/>ad ſenſum.</s> <s xml:id="echoid-s17179" xml:space="preserve"> Et hæc linea continet cum linea exeunte <lb/>à centro medij circuli ad ſignum, quod eſt in circum <lb/>ferentia medij circuli, perpendiculari ſuper ſuperfi-<lb/>ciem aquæ, angulum rectum:</s> <s xml:id="echoid-s17180" xml:space="preserve"> & diameter medιj cir-<lb/>culi, quæ tranſit per cẽtra duorum foraminum, con-<lb/>tinet cum hac perpendiculari exeunte à centro me-<lb/>dij circuli ſuper ſuperficiem aquæ, angulum, cuius <lb/>quantitas erit 80 partium.</s> <s xml:id="echoid-s17181" xml:space="preserve"> Hunc enim angulũ chor-<lb/>dat arcus medij circuli, qui eſt inter ſecundũ & ter-<lb/>tium ſignum:</s> <s xml:id="echoid-s17182" xml:space="preserve"> arcus autem, qui eſt inter centrum fo-<lb/>raminis & primum ſignum, qui eſt 10 partium, chor-<lb/>dat angulum declinationis.</s> <s xml:id="echoid-s17183" xml:space="preserve"> Deinde oportet experi-<lb/>mentatorem conſiderare ſolẽ, & mutare inſtrumen-<lb/>tum, doneclux tranſeat per duo foramina:</s> <s xml:id="echoid-s17184" xml:space="preserve"> & tunc <lb/>aſpiciat lucem, quæ eſt in ora inſtrumenti, quæ eſt intra aquam, & ſignet ſuper centrũ lucis ſignum:</s> <s xml:id="echoid-s17185" xml:space="preserve"> <lb/>hoc ergo ſignum erit in circumferentia medij circuli:</s> <s xml:id="echoid-s17186" xml:space="preserve"> deinde auferat inſtrumentum, & aſpicιat ter-<lb/>tium ſignum, quod eſt inter extremitatem medij circuli, & inter ſecundum ſignum, quod eſt extre-<lb/>mitas perpendicularis, exeuntis à cẽtro medιj circuli ſuper ſuperficiem aquæ.</s> <s xml:id="echoid-s17187" xml:space="preserve"> Ex hac ergo experi-<lb/>mentatione patebit, quòd angulus refractionis eſt ille, quem chordat arcus, qui eſt inter centrũ lu-<lb/>cis & tertium ſignum, quod eſt extremitas lineæ tranſeuntis per centra duorum foraminum, per <lb/>quam extendebatur lux:</s> <s xml:id="echoid-s17188" xml:space="preserve"> & ex numero partium huius arcus patebit quantitas anguli refractionis, <lb/>& quantitas proportionis anguli refractionis ad 80 partes, quæ eſt angulus, quẽ continet linea, per <lb/>quam extendebatur lux, cum perpendiculari exeunte à puncto refractionis ſuper ſuperficiẽ aquæ.</s> <s xml:id="echoid-s17189" xml:space="preserve"> <lb/>Deinde oportet experimentatorem delere ſignum & lineam ſignatam in lamina, & diſtinguere in-<lb/>ter circumferentiam medij circuli ex parte centri foraminis, quod eſt in ora inſtrumenti arcum, cu-<lb/>ius quãtitas ſit 20 partium:</s> <s xml:id="echoid-s17190" xml:space="preserve"> & ſignet in extremitate eius ſignum, & extrahat ab hoc ſigno perpendi-<lb/>cularem ſuper ſuperficiem laminæ, & extrahat ab eιus extremitate lineam ad centrum laminæ:</s> <s xml:id="echoid-s17191" xml:space="preserve"> & <lb/>protrahamus illam in utramq;</s> <s xml:id="echoid-s17192" xml:space="preserve"> partem:</s> <s xml:id="echoid-s17193" xml:space="preserve"> & diuidamus arcum ſequente<gap/> illum (cuius quantitas 20) <lb/>in partes 90:</s> <s xml:id="echoid-s17194" xml:space="preserve"> & ſignemus in ipſo ſignum:</s> <s xml:id="echoid-s17195" xml:space="preserve"> & erit arcus, qui eſt inter ſignum ſecundum & extremita-<lb/>tem lineæ tranſeuntis per centra duorum foraminum, 70 partium:</s> <s xml:id="echoid-s17196" xml:space="preserve"> & ſignemus in extremitate hu-<lb/>ius lineæ ſignum.</s> <s xml:id="echoid-s17197" xml:space="preserve"> Dein de ponamus inſtrumentũ in uas, & reuoluamus illud, quouſq;</s> <s xml:id="echoid-s17198" xml:space="preserve"> linea ſignata <lb/>in lamina tangat ſuperficiem aquæ.</s> <s xml:id="echoid-s17199" xml:space="preserve"> Linea ergo, quæ exit à centro circuli medιj ad ſecundũ ſignum, <lb/>erit perpendicularis ſuper ſuperficiem aquæ, ut prædictum eſt:</s> <s xml:id="echoid-s17200" xml:space="preserve"> & linea, quæ tranſit per centra duo-<lb/>rum foraminum, continet cum hac perpendiculari angulum 70 partium.</s> <s xml:id="echoid-s17201" xml:space="preserve"> Deinde experimentator <lb/>conſideret ſolem, & moueat inſtrumentum, quouſq;</s> <s xml:id="echoid-s17202" xml:space="preserve"> lux tranſeat per duo foramina, & ſignet ſuper <lb/>centrum lucis ſignum, & auferat inſtrumentum, & aſpiciat ſigna, quæ ſunt in circumferentia medij <lb/>circuli:</s> <s xml:id="echoid-s17203" xml:space="preserve"> ex qua experimentatione habebit quãtiratem anguli refractionis, & proportionem eius ad <lb/>angulum, quem continet linea, per quam extenditur lux, cum perpendiculari exeunte à loco refra-<lb/>ctionis, qui eſt in hoc ſtatu 70 partiũ.</s> <s xml:id="echoid-s17204" xml:space="preserve"> Deinde experimentator auferat inſtrumentũ, & deleat ſigna, <lb/>& lineam, quæ eſt in lamina, & diuidat arcum ex parte foraminis, cuius quantitas ſit 30 partium, & <lb/>procedat, ut in primis ablationibus:</s> <s xml:id="echoid-s17205" xml:space="preserve"> & ſic habebit quãtitatem anguli refractionis & proportionem <lb/>eius ad angulum, quem continet linea, per quam extendebatur lux cum perpendiculari exeunte à <lb/>loco refractionis, qui eſt in hoc ſitu 60 partium.</s> <s xml:id="echoid-s17206" xml:space="preserve"> Deinde diuidamus arcum, cuius quantitas ſit 40 <lb/>partium:</s> <s xml:id="echoid-s17207" xml:space="preserve"> deinde arcum, cuius quantitas ſit 50 partium:</s> <s xml:id="echoid-s17208" xml:space="preserve"> deinde 60:</s> <s xml:id="echoid-s17209" xml:space="preserve"> deinde 70:</s> <s xml:id="echoid-s17210" xml:space="preserve"> deinde 80:</s> <s xml:id="echoid-s17211" xml:space="preserve"> & conſide <lb/>ret unumquemq;</s> <s xml:id="echoid-s17212" xml:space="preserve"> iſtorum arcuum:</s> <s xml:id="echoid-s17213" xml:space="preserve"> & ſic habebit quãtitates angulorum refractionis, & angulorum <lb/>declinationis, quos chordant primi arcus diſtincti ex parte centri foraminis:</s> <s xml:id="echoid-s17214" xml:space="preserve"> & habebit proportio-<lb/>nes angulorum refractionis ad angulos, quos continent primæ lineæ, per quas extendebatur lux, <lb/>cum perpendiculari, quæ eſt in ſuperficie aquæ, qui creſcunt per decẽ.</s> <s xml:id="echoid-s17215" xml:space="preserve"> Et ſi experimentator uolue-<lb/>rit, ut anguli creſcant per quinq;</s> <s xml:id="echoid-s17216" xml:space="preserve">, bene poterit facere:</s> <s xml:id="echoid-s17217" xml:space="preserve"> & ſi uoluerit per minus, quàm per quinque, <lb/>bene poterit facere prædicto ordine.</s> <s xml:id="echoid-s17218" xml:space="preserve"/> </p> <div xml:id="echoid-div564" type="float" level="0" n="0"> <figure xlink:label="fig-0250-01" xlink:href="fig-0250-01a"> <variables xml:id="echoid-variables201" xml:space="preserve">k n m x b l p <gap/> f s u z y t</variables> </figure> </div> </div> <div xml:id="echoid-div566" type="section" level="0" n="0"> <head xml:id="echoid-head491" xml:space="preserve" style="it">11. Magnitudines angulorum refractionis ab aere uel aqua ad uitra planum uel conuexum, <lb/>& contrà, organo refractionis inuenire. 6 p 10.</head> <p> <s xml:id="echoid-s17219" xml:space="preserve">ET cum experimentator uoluerit experiri per uitrum:</s> <s xml:id="echoid-s17220" xml:space="preserve"> diuidat arcus, & ſignet prædicta ſigna, <lb/>& ſuperponat uitrum prædictum ſuperficiei laminæ, & ſuperponat differentiam eius cõmu-<lb/>nem lineæ ſignatæ in lamina, & ponat ſuperficiem uitri æqualem ex parte foraminum, & ap-<lb/>plicet uitrum bene, & ponat inſtrumentum in uaſe, & moueat ipſum, quouſq, lux tranſeat per duo <lb/>foramina, & ſignet ſuper centrum lucis ſignum, & auferat inſtrumentum, & intueatur arcus, & de-<lb/>inde deleat ſigna, & diuidat alios arcus, & ſignet alia ſigna, & inſpiciat arcus ꝓut aſpexit per aquã:</s> <s xml:id="echoid-s17221" xml:space="preserve"> <lb/>& ſic habebit quantitates angulorum refractionis in tranſitu lucis de aere ad uitrum.</s> <s xml:id="echoid-s17222" xml:space="preserve"> Et ſi uoluerit <lb/> <pb o="245" file="0251" n="251" rhead="OPTICAE LIBER VII."/> experiri refractiones lucis de uitro ad aerem, & ad aquam:</s> <s xml:id="echoid-s17223" xml:space="preserve"> applicet uitrum è contrario primi ſitus:</s> <s xml:id="echoid-s17224" xml:space="preserve"> <lb/>ſcilicet, ut ponat conuexum eius ex parte duorum foraminum, & ponat medium communis diffe-<lb/>rentiæ, quæ eſt in uitro, ſuper centrum laminæ.</s> <s xml:id="echoid-s17225" xml:space="preserve"> Tunc ergo lux, quæ tranſit per centra duorum fora-<lb/>minum, peruenit rectè ad cẽtrum uitri, & refringitur apud illud de uitro ad aerem.</s> <s xml:id="echoid-s17226" xml:space="preserve"> Deinde diuidat <lb/>arcus ſucceſsiuè, & mutet poſitionem uitri:</s> <s xml:id="echoid-s17227" xml:space="preserve"> & ſic habebit angulos refractionũ particulares, & pro-<lb/>portiones eorum ad angulos, quos continet prima linea, per quam extenditur lux, cum linea per-<lb/>pendiculari ſuper ſuperficiem contingentẽ ſuperficiem uitri.</s> <s xml:id="echoid-s17228" xml:space="preserve"> Et cum experimẽtator expertus fue-<lb/>rit hos duos prædictos ſitus:</s> <s xml:id="echoid-s17229" xml:space="preserve"> uidebit quòd quãtitates angulorum refractionis de aere ad uitrum, & <lb/>de uitro ad aerem ſemper erunt æquales:</s> <s xml:id="echoid-s17230" xml:space="preserve"> cum angulus, quem continet linea, per quam extenditur <lb/>lux ad locum refractionis cum linea perpendiculari, cum refringitur de aere ad uitrum, æqualis ſit <lb/>angulo, quem cõtinet linea, per quam extenditur lux à loco refractionis cum perpendiculari, cum <lb/>reflectitur à uitro ad aerem.</s> <s xml:id="echoid-s17231" xml:space="preserve"> Et ſi quis uoluerit experiri quantitates angulorũ refractionis, qui ſunt <lb/>apud conuexum uitri:</s> <s xml:id="echoid-s17232" xml:space="preserve"> diuidat de circumferentia medij circuli ex parte cẽtri foraminis, quod eſt in <lb/>ora inſtrumenti, arcum, cuius quantitas ſit 10 partium, & extrahat ab extremitate eius perpendicu-<lb/>larem ſuper ſuperficiem laminæ in ſuperficie oræ inſtrumenti, ſicut prius fecerat:</s> <s xml:id="echoid-s17233" xml:space="preserve"> deinde diuidat ex <lb/>hac linea incipiens à centro laminæ lineam æqualem ſemidiametro uitri, & ab extremitate huius <lb/>lineæ extrahat perpẽdicularem ſuper diametrum laminæ, ſuper cuius extremitates ſunt duæ lineæ <lb/>perpendiculares in ora inſtrumenti:</s> <s xml:id="echoid-s17234" xml:space="preserve"> & protrahat hãc perpendicularem in utramq;</s> <s xml:id="echoid-s17235" xml:space="preserve"> partem:</s> <s xml:id="echoid-s17236" xml:space="preserve"> deinde <lb/>ſuperponat uitrum ſuper ſuperficiẽ laminæ, & ſuper-<lb/> <anchor type="figure" xlink:label="fig-0251-01a" xlink:href="fig-0251-01"/> ponat differentiam eius cõmunem prędictæ perpen-<lb/>diculari, & ponat medium differẽtiæ cõmunis ſuper <lb/>punctum, à quo extracta fuerit perpendicularis:</s> <s xml:id="echoid-s17237" xml:space="preserve"> & ſic <lb/>erit centrum uitri in ſuperficie medij circuli, & linea, <lb/>quæ tranſit per centra duorũ foraminum, erit perpen <lb/>dicularis ſuper ſuperficiẽ uitri æqualẽ:</s> <s xml:id="echoid-s17238" xml:space="preserve"> [per 8 p 11] eſt <lb/>enim æquidiſtans diametro laminæ, quæ eſt perpen-<lb/>dicularis ſuper illam ſuperficiem & differẽtiam com-<lb/>munem, quę eſt in uitro:</s> <s xml:id="echoid-s17239" xml:space="preserve"> & centrum circuli medij erit <lb/>in conuexo uitri.</s> <s xml:id="echoid-s17240" xml:space="preserve"> Nam linea, quæ exit à centro circuli <lb/>medij ad cẽtrum laminæ, eſt æqualis lineæ exeunti à <lb/>centro uitri ad mediũ differentiæ cõmunis:</s> <s xml:id="echoid-s17241" xml:space="preserve"> & utraq;</s> <s xml:id="echoid-s17242" xml:space="preserve"> <lb/>iſtarum linearũ eſt perpẽdicularis ſuper ſuperficiem <lb/>laminæ:</s> <s xml:id="echoid-s17243" xml:space="preserve"> ergo duæ lineæ ſunt æquales & æquidiſtãtes:</s> <s xml:id="echoid-s17244" xml:space="preserve"> <lb/>[per 33 p 1] & linea, quę copulat centrũ ultri cũ centro <lb/>circuli medij, eſt æqualis lineæ, quę copulat centrũ la-<lb/>minæ, & mediũ differentiæ cõmunis, quæ eſt in uitro:</s> <s xml:id="echoid-s17245" xml:space="preserve"> <lb/>hæc autem linea æqualis poſita fuit ſemidiametro uitri.</s> <s xml:id="echoid-s17246" xml:space="preserve"> Centrum ergo medij circuli eſt in conuexo <lb/>uitri.</s> <s xml:id="echoid-s17247" xml:space="preserve"> Linea ergo, quę tranſit per centra duorum foraminum, quæ tranſit per medij circuli centrum, <lb/>tenet cum linea, exeunte à centro uitri, angulum æqualem angulo, qui eſt apud centrũ laminæ.</s> <s xml:id="echoid-s17248" xml:space="preserve"> Ex-<lb/>tendantur ergo duæ lineæ in imaginatione rectè in utramq;</s> <s xml:id="echoid-s17249" xml:space="preserve"> partẽ, ſcilicet diameter prædicta uitri, <lb/>& linea, quæ tranſit per centra duorum foraminum:</s> <s xml:id="echoid-s17250" xml:space="preserve"> peruenient ergo ad circumferẽtiam medij cir-<lb/>culi:</s> <s xml:id="echoid-s17251" xml:space="preserve"> ſunt enim ambæ in ſuperficie medij circuli.</s> <s xml:id="echoid-s17252" xml:space="preserve"> Ergo duę lineæ diuident à circũferentia medij cir-<lb/>culi ex utraq;</s> <s xml:id="echoid-s17253" xml:space="preserve"> parte arcum, cuius quantitas eſt 10 partium:</s> <s xml:id="echoid-s17254" xml:space="preserve"> & extremitates lineæ, quę tranſit per cen <lb/>tra duorũ foraminum, ſunt notæ:</s> <s xml:id="echoid-s17255" xml:space="preserve"> altera enim earũ eſt centrum foraminis, & altera punctũ oppoſi-<lb/>tum centro foraminis:</s> <s xml:id="echoid-s17256" xml:space="preserve"> & altera duarũ extremitatum lineæ, quę tranſit per centrũ uitri, eſt extremi-<lb/>tas arcus, quẽ ſeparauerat à circũferentia medij circuli, qui diſtat à cẽtro foraminis 10 partibus:</s> <s xml:id="echoid-s17257" xml:space="preserve"> re-<lb/>liqua ergo extremitas lineæ, quę tranſit per centrũ uitri, diſtat à linea, quę tranſit per centra duorũ <lb/>foraminũ, decẽ partibus in parte oppoſita primo ſigno.</s> <s xml:id="echoid-s17258" xml:space="preserve"> Signemus ergo extremitatẽ huius diame-<lb/>tri, & extremitatẽ lineæ, quę tranſit per centra duorũ foraminum, quoniã locus iſte eſt notus:</s> <s xml:id="echoid-s17259" xml:space="preserve"> quia <lb/>eſt ſuper lineã perpendicularem in ora inſtrumenti:</s> <s xml:id="echoid-s17260" xml:space="preserve"> & intueatur experimentator ſignũ:</s> <s xml:id="echoid-s17261" xml:space="preserve"> & inueniet <lb/>illud remotius ab extremitate lineæ, quæ tranſit per centra duorũ foraminum.</s> <s xml:id="echoid-s17262" xml:space="preserve"> Hæc ergo refractio <lb/>eſt ad partem contrariam perpendiculari à loco refractionis:</s> <s xml:id="echoid-s17263" xml:space="preserve"> quia perpẽdicularis exiens à loco re-<lb/>fractionis, eſt linea, quæ tranſit per centrũ uitri:</s> <s xml:id="echoid-s17264" xml:space="preserve"> & arcus circumferentiæ medij circuli, qui eſt inter <lb/>cẽtrum lucis & extremitatem lineæ, quę tranſit per centra duorũ foraminum, eſt quantitas anguli <lb/>refractionis:</s> <s xml:id="echoid-s17265" xml:space="preserve"> angulus enim refractionis eſt apud centrum medij circuli.</s> <s xml:id="echoid-s17266" xml:space="preserve"> Lux enim extenditur ſuper <lb/>lineam tranſeuntem per centra duorum foraminum rectè, donec perueniat ad conuexum uitri & <lb/>ſphęricum.</s> <s xml:id="echoid-s17267" xml:space="preserve"> Angulus ergo refractionis erit apud centrũ circuli medij, qui eſt ſuper conuexum uitri:</s> <s xml:id="echoid-s17268" xml:space="preserve"> <lb/>& arcus, qui eſt inter cẽtrum lucis & extremitatem lineæ, quę tranſit per centra duorũ foraminum, <lb/>eſt ille, qui chordat angulũ refractionis, qui eſt 10 partium.</s> <s xml:id="echoid-s17269" xml:space="preserve"> Deinde oportet experimentatorẽ euel-<lb/>lere uitrum, & diuidere à centro foraminis arcum, qui ſit 20 partium, & procedat ut prius:</s> <s xml:id="echoid-s17270" xml:space="preserve"> & ſic ha-<lb/>bebit quantitatẽ anguli refractionis differentem à quantitate anguli, qui eſt 20 partium.</s> <s xml:id="echoid-s17271" xml:space="preserve"> Et ſic diui-<lb/>dat alios arcus ſucceſsiuè:</s> <s xml:id="echoid-s17272" xml:space="preserve"> & experiatur refractiones eorum ſicut in primis:</s> <s xml:id="echoid-s17273" xml:space="preserve"> & habebit quantitates <lb/>angulorum refractionis, qui ſunt apud conuexum uitri.</s> <s xml:id="echoid-s17274" xml:space="preserve"> Et eædem ſunt quantitates angulorum re.</s> <s xml:id="echoid-s17275" xml:space="preserve"> <lb/>fractionis lucis de aere ad uitrum:</s> <s xml:id="echoid-s17276" xml:space="preserve"> hoc enim declaratum eſt in prædictis experimentationibus:</s> <s xml:id="echoid-s17277" xml:space="preserve"> ſed <lb/> <pb o="246" file="0252" n="252" rhead="ALHAZEN"/> refractio de aere ad uitrũ eſt ad partẽ քpẽdicularis:</s> <s xml:id="echoid-s17278" xml:space="preserve"> refractio uerò de uitro ad aerẽ eſt ad partẽ cõtra <lb/>riam perpẽdiculari.</s> <s xml:id="echoid-s17279" xml:space="preserve"> Et ſi quis uoluerit experiri uitrum & aquã, & à cõuexo uitri & à ſuperficie eius <lb/>æquali, habebit quãtitates angulorũ refractionis de uitro ad aquã:</s> <s xml:id="echoid-s17280" xml:space="preserve"> aqua enim ponitur in loco aeris.</s> <s xml:id="echoid-s17281" xml:space="preserve"/> </p> <div xml:id="echoid-div566" type="float" level="0" n="0"> <figure xlink:label="fig-0251-01" xlink:href="fig-0251-01a"> <variables xml:id="echoid-variables202" xml:space="preserve">k n b l <gap/> o q f g u z</variables> </figure> </div> </div> <div xml:id="echoid-div568" type="section" level="0" n="0"> <head xml:id="echoid-head492" xml:space="preserve" style="it">12. Magnitudines angulorum refractionis ab aere uel aqua ad uitrum cauum, & contrà, <lb/>organo refractionis inueſtigare. 7. 8 p 10.</head> <p> <s xml:id="echoid-s17282" xml:space="preserve">ET ſi quis uoluerit experiri quãtitates angulorũ refractionis apud concauũ uitri:</s> <s xml:id="echoid-s17283" xml:space="preserve"> accipiat ui-<lb/>trum concauum concauitate columnari in quantitate ſemicolumnę:</s> <s xml:id="echoid-s17284" xml:space="preserve"> & ſit figura uniuerſi ui-<lb/>tri æquidiſtãtium ſuperficierũ:</s> <s xml:id="echoid-s17285" xml:space="preserve"> & longitudo eius ſit maior diametro uitri ſphærici uno grano <lb/>hordei:</s> <s xml:id="echoid-s17286" xml:space="preserve"> & latitudo eius ſit ſimiliter:</s> <s xml:id="echoid-s17287" xml:space="preserve"> & ſpiſsitudo eius ſit dupla diametri foraminis, quod eſt in ora <lb/>inſtrumẽti:</s> <s xml:id="echoid-s17288" xml:space="preserve"> & concauitas ſit in uno ſuorũ laterum:</s> <s xml:id="echoid-s17289" xml:space="preserve"> columnaris ſcilicet in ſuperficie una quadrata:</s> <s xml:id="echoid-s17290" xml:space="preserve"> & <lb/>longitudo columnæ ſit in lõgitudine uitri:</s> <s xml:id="echoid-s17291" xml:space="preserve"> & ſemidiameter baſis columnæ ſit in quãtitate ſemidia-<lb/>metri uitri ſphęrici:</s> <s xml:id="echoid-s17292" xml:space="preserve"> & ſint fines uitri lineæ rectę ueriſsimæ.</s> <s xml:id="echoid-s17293" xml:space="preserve"> Hoc autẽ inſtrumentum ſic bene poteſt <lb/>fieri ſuper formam:</s> <s xml:id="echoid-s17294" xml:space="preserve"> ita ut forma fiat eadẽ doctrina prædicta, & diſſoluatur uitrũ, & infundatur ſuper <lb/>formam prædictam.</s> <s xml:id="echoid-s17295" xml:space="preserve"> Si ergo experimentator uoluerit experiri refractionem hoc inſtrumẽto:</s> <s xml:id="echoid-s17296" xml:space="preserve"> diui-<lb/>dat de circumferentia medij circuli arcum, cuius quãtitas ſit illa, quam uult experiri, & extrahat ab <lb/>extremitate arcus perpendicularẽ ſuper ſuperficiẽ laminę, ut prędictũ eſt, & copulet extrem t<gap/>tem <lb/>perpendicularis cũ centro laminæ linea recta, quam protrahat in alteram partẽ, & diuidat ex hac li-<lb/>nea in altera parte, ſcilicet in qua ſunt duo foramina, lineam æqualẽ ſemidiametro baſis columnæ, <lb/>& extrahat ab extremitate eius perpendicularẽ ſuper diametrũ laminę, & protrahat illã in utramq;</s> <s xml:id="echoid-s17297" xml:space="preserve"> <lb/>partem.</s> <s xml:id="echoid-s17298" xml:space="preserve"> Deinde ſuperponat uitrum laminę, & ponat dorſum cõcauitatis ex parte duorũ foraminũ, <lb/>& ſuperponat duas ſuperfluitates, quę ſuperfluunt ſuper diametrũ columnæ, huic perpendiculari, <lb/>obſeruetq́;</s> <s xml:id="echoid-s17299" xml:space="preserve">, ut ſint diſtantiæ duarũ extremitatum diametri baſis cõcauitatis à puncto, à quo exiuit <lb/>perpendicularis, diſtantiæ æquales.</s> <s xml:id="echoid-s17300" xml:space="preserve"> Erit ergo centrũ baſis cõcauitatis columnaris ſuper punctum, <lb/>à quo exiuit perpendicularis, ſuperq́;</s> <s xml:id="echoid-s17301" xml:space="preserve"> punctum, cuius diſtantia à centro laminæ, eſt in quantitate <lb/>ſemidiametri baſis cõcauitatis.</s> <s xml:id="echoid-s17302" xml:space="preserve"> Hoc ſitu obſeruato, applicet uitrum fixa applicatione:</s> <s xml:id="echoid-s17303" xml:space="preserve"> & erit ſuper-<lb/>ficies medij circuli ſecãs foramen columnæ & æqui <lb/> <anchor type="figure" xlink:label="fig-0252-01a" xlink:href="fig-0252-01"/> diſtans baſi eius:</s> <s xml:id="echoid-s17304" xml:space="preserve"> nã baſis eius in hac dιſpoſitione eſt <lb/>in ſuperficie laminæ.</s> <s xml:id="echoid-s17305" xml:space="preserve"> Superficies ergo circuli medij <lb/>facit in ſuperficie columnari concaua ſemicirculum <lb/>[per 5 th.</s> <s xml:id="echoid-s17306" xml:space="preserve"> cylindricorum Sereni] & eſt diameter hu-<lb/>ius ſemicirculi æquidiſtans diametro baſis concaui-<lb/>tatis.</s> <s xml:id="echoid-s17307" xml:space="preserve"> Erit ergo linea, quæ egrediturà cẽtro huius ſe-<lb/>micirculi ad centrum baſis concauitatis, quę eſt per-<lb/>pendicularis ſuper ſuperficiem laminæ, æqualis per-<lb/>pendiculari exeunti à cẽtro circuli medij perpendi-<lb/>culari ſuper ſuperficiem laminę:</s> <s xml:id="echoid-s17308" xml:space="preserve"> & perpendicularis, <lb/>quę exit à centro circuli medij ad cẽtrum laminę, eſt <lb/>æqualis ſemidiametro baſis colũnæ.</s> <s xml:id="echoid-s17309" xml:space="preserve"> Ergo linea, quæ <lb/>exit à centro circuli medij ad cẽtrum ſemicirculi, qui <lb/>fit in ſuperficie columnæ, eſt æqualis ſemidiametro <lb/>huius ſemicirculi [per 33 p 1.</s> <s xml:id="echoid-s17310" xml:space="preserve">] Centrum ergo circuli <lb/>medij eſt in circumferentia ſemicirculi facti:</s> <s xml:id="echoid-s17311" xml:space="preserve"> eſt ergo <lb/>in concauo columnæ.</s> <s xml:id="echoid-s17312" xml:space="preserve"> Et quia terminus uitri ſuper-<lb/>ponitur lineę perpẽdiculari ſuper punctũ laminę:</s> <s xml:id="echoid-s17313" xml:space="preserve"> erit diameter laminę perpẽdicularis ſuper ſuper-<lb/>ficiem uitri æqualem.</s> <s xml:id="echoid-s17314" xml:space="preserve"> Nã ſuperficies uitri æquales, ſunt perpẽdiculares inter ſe.</s> <s xml:id="echoid-s17315" xml:space="preserve"> Erit ergo linea, quę <lb/>tranſit per centra duorũ foraminum perpẽdicularis ſuper ſuperficiẽ uitri æqualem, quę eſt in parte <lb/>conuexa uitri [per 8 p 11] quia eſt æquidiſtans diametro laminę:</s> <s xml:id="echoid-s17316" xml:space="preserve"> & hęc ſuperficies uitri æqualis, eſt <lb/>ex parte foraminum.</s> <s xml:id="echoid-s17317" xml:space="preserve"> In hoc ergo ſitu lux, quę extenditur ſuper lineã, quę tranſit per cẽtra duorum <lb/>foraminũ, extenditur in corpore uitri rectè, donec perueniat ad concauum uitri:</s> <s xml:id="echoid-s17318" xml:space="preserve"> & tũc refringitur <lb/>apud concauum uitri:</s> <s xml:id="echoid-s17319" xml:space="preserve"> cum non tranſeat per centrum circuli, qui eſt in concauo uitri:</s> <s xml:id="echoid-s17320" xml:space="preserve"> neq;</s> <s xml:id="echoid-s17321" xml:space="preserve"> eſt per-<lb/>pendicularis ſuper concauum uitri:</s> <s xml:id="echoid-s17322" xml:space="preserve"> ergo refringitur in concauo uitri:</s> <s xml:id="echoid-s17323" xml:space="preserve"> ergo differẽtia cõmunis huic <lb/>lineæ & concauo uitri eſt centrum circuli medij.</s> <s xml:id="echoid-s17324" xml:space="preserve"> Ergo lux, quę extenditur ſuper lineam, quę tranſit <lb/>per centra duorum foraminũ, refringitur apud centrum medij circuli:</s> <s xml:id="echoid-s17325" xml:space="preserve"> ergo arcus, qui eſt inter cen-<lb/>trum lucis & extremitatem lineæ, quę tranſit per centra duorum foraminum, chordat angulum re-<lb/>fractionis.</s> <s xml:id="echoid-s17326" xml:space="preserve"> Hac igitur uia poſſet quis experiri quantitates angulorum refractionis, qui fiunt in con-<lb/>cauo uitri, addendo in arcubus parum.</s> <s xml:id="echoid-s17327" xml:space="preserve"> Et hæc refractio eſt à uitro concauo ad aerem:</s> <s xml:id="echoid-s17328" xml:space="preserve"> & eruntan-<lb/>guli acquiſiti hac refractione ijdem illis, qui fiunt ex aere ad uitrum in concauo uitri.</s> <s xml:id="echoid-s17329" xml:space="preserve"> Declaratum <lb/>eſt autem paulò antè, quòd angulus refractionis à uitro ad aerem, & ab aere ad uitrum, eſt idem <lb/>cum angulo, quem continet prima linea, per quam extenditur lux, & perpendicularis exiens à lo-<lb/>co refractionis.</s> <s xml:id="echoid-s17330" xml:space="preserve"> Hac ergo uia poſſet quis habere quantitates angulorum refractionis de aere ad <lb/>aquam, & de aere ad uitrum, & de uitro ad aerem, & de uitro ad aquam à ſuperficie æquali, & con-<lb/>caua & conuexa.</s> <s xml:id="echoid-s17331" xml:space="preserve"> His ergo angulis experimentatis & proportionibus eorum notis, experimẽtator <lb/>inueniet duos angulos, quorum utrumq;</s> <s xml:id="echoid-s17332" xml:space="preserve"> continet prima linea, per quã extenditur lux, & perpẽdi-<lb/> <pb o="247" file="0253" n="253" rhead="OPTICAE LIBER VII."/> cularis, exiens à loco refractionis ſuper ſuperficiem corporis diaphani:</s> <s xml:id="echoid-s17333" xml:space="preserve"> inueniet dico in eiſdem cor-<lb/>poribus diaphanis:</s> <s xml:id="echoid-s17334" xml:space="preserve"> & erunt duo anguli diuerſi.</s> <s xml:id="echoid-s17335" xml:space="preserve"> Nam angulus refractionis ab angulo maiore ex il-<lb/>lis, erit maior duobus angulis refractionis ab angulo minore:</s> <s xml:id="echoid-s17336" xml:space="preserve"> & exceſſus anguli refractionis ſuper <lb/>angulum refractionis, erit minor exceſſu anguli maioris, quem continet prima linea cum perpendi <lb/>culari ſuper angulum minorem, quem continet prima linea cum perpendiculari.</s> <s xml:id="echoid-s17337" xml:space="preserve"> Et proportio an-<lb/>guli refractionis ab angulo maiore ad angulum maiorem, erit maior proportione anguli refractio-<lb/>nis ab angulo minore ad angulum minorem:</s> <s xml:id="echoid-s17338" xml:space="preserve"> Et illud, quod reſtat poſt angulum refractionis de an-<lb/>gulo maiore, eſt maius illo, quod remanet poſt angulum refractionis de angulo minore.</s> <s xml:id="echoid-s17339" xml:space="preserve"> Et remo-<lb/>tio anguli refractionis, cum lux exiuerit de corpore ſubtiliore ad groſsius, ſemper erit minor an-<lb/>gulo, quem continet linea, per quam extenditur lux ad locum refractionis cum perpendiculari ex-<lb/>eunte à loco refractionis.</s> <s xml:id="echoid-s17340" xml:space="preserve"> Et ſi lux exiuerit à corpore groſsiore ad ſubtilius:</s> <s xml:id="echoid-s17341" xml:space="preserve"> tunc angulus refractio-<lb/>nis erit medietas duorum angulorum coniunctorum.</s> <s xml:id="echoid-s17342" xml:space="preserve"> Et ſi comparaueris angulos refractionis, qui <lb/>ſunt inter aliquod iſtorum corporum diaphanorum, & aliud corpus groſsius illis, ad angulos refra-<lb/>ctionis, qui ſunt inter illud idem corpus diaphanũ ſubtilius & aliud corpus groſsius primo groſſo:</s> <s xml:id="echoid-s17343" xml:space="preserve"> <lb/>inuenies proportiones maiores angulorum refractionis ad angulos, quos continet prima linea & <lb/>perpendicularis, qui ſunt inter corpus ſubtilius & groſsius, quod magis groſſum eſt, proportioni-<lb/>bus angulorum refractionis, quos continet prima linea & perpendicularis, qui ſunt inter idem cor-<lb/>pus ſubtilius & corpus groſsius, quod minus eſt groſſum.</s> <s xml:id="echoid-s17344" xml:space="preserve"> Quoniam ſi fuerint duo anguli æquales, <lb/>quorum utrumlibet continet prima linea, per quam extenditur lux, & perpendicularis, quæ exit à <lb/>loco refractionis:</s> <s xml:id="echoid-s17345" xml:space="preserve"> quorum alter eſt inter corpus ſubtilius & corpus groſsius illo, & alter inter illud <lb/>idem corpus ſubtilius & corpus groſsius primo groſſo:</s> <s xml:id="echoid-s17346" xml:space="preserve"> tunc angulus refractionis, qui eſt in corpore <lb/>groſsiore, erit maior angulo refractionis, qui eſt in corpore groſsiore, quod eſt minus groſſum.</s> <s xml:id="echoid-s17347" xml:space="preserve"> Et ſi-<lb/>militer ſi refractio fuerit à corpore groſsiore ad ſubtilius, quod eſt magis ſubtile:</s> <s xml:id="echoid-s17348" xml:space="preserve"> maior erit angulo <lb/>refractionis, qui eſt ab illo eodẽ corpore groſsiore ad corpus ſubtilius, quod eſt minus ſubtile.</s> <s xml:id="echoid-s17349" xml:space="preserve"> Hæc <lb/>ergo ſunt omnia, quæ pertinent ad qualitatem refractionis lucis à corporibus diaphanis.</s> <s xml:id="echoid-s17350" xml:space="preserve"/> </p> <div xml:id="echoid-div568" type="float" level="0" n="0"> <figure xlink:label="fig-0252-01" xlink:href="fig-0252-01a"> <variables xml:id="echoid-variables203" xml:space="preserve">k n m b l d p o q f g u <gap/></variables> </figure> </div> </div> <div xml:id="echoid-div570" type="section" level="0" n="0"> <head xml:id="echoid-head493" xml:space="preserve">QVÒ<unsure/>D QVICQVID COMPREHENDITVR VLTRA CORPORA</head> <head xml:id="echoid-head494" xml:space="preserve">diaphana, quæ differuntin diaphanitate à corpore, in quo eſt uiſus, cum <lb/>fuerit obliquum à lineis perpendicularibus ſuper ſuperficiem <lb/>eorum, comprehenditur ſecundum refractio-<lb/>nem. Cap. IIII.</head> <head xml:id="echoid-head495" xml:space="preserve" style="it">13. Viſibile medio diuerſo perpendiculare, rectè: obliquum refractè uidetur. 3 p 10.</head> <p> <s xml:id="echoid-s17351" xml:space="preserve">IN prædicto autem capitulo patuit, quòd lux tranſit de uitro ad aerem, & de aere ad uitrum, & <lb/>de aere ad aquam.</s> <s xml:id="echoid-s17352" xml:space="preserve"> Et cum tranſit de uitro ad aerem & ad aquam:</s> <s xml:id="echoid-s17353" xml:space="preserve"> conſtat quòd tranſibit de aqua <lb/>ad aerem:</s> <s xml:id="echoid-s17354" xml:space="preserve"> aqua enim eſt ſubtilior uitro, cum fuerit clara:</s> <s xml:id="echoid-s17355" xml:space="preserve"> & cum tranſit de aere ad uitrum, tranſi-<lb/>bit de aqua ad uitrum, cum aqua ſit groſsior aere.</s> <s xml:id="echoid-s17356" xml:space="preserve"> Præterea patuιt, quòd luces omnes accidentales <lb/>& eſſentiales, fortes & debiles tranſeunt per hæc corpora diaphana.</s> <s xml:id="echoid-s17357" xml:space="preserve"> His ergo modis omne corpus <lb/>lucidum quacunq;</s> <s xml:id="echoid-s17358" xml:space="preserve"> luce, mittit lucem ſuam in omne corpus diaphanũ:</s> <s xml:id="echoid-s17359" xml:space="preserve"> & ſi occurrerit aliud corpus <lb/>diaphanũ:</s> <s xml:id="echoid-s17360" xml:space="preserve"> trãſibit in alio corpore aut refractè aut rectè.</s> <s xml:id="echoid-s17361" xml:space="preserve"> Et in primo libro [14.</s> <s xml:id="echoid-s17362" xml:space="preserve"> 18.</s> <s xml:id="echoid-s17363" xml:space="preserve"> 19 n] declaratũ eſt, <lb/>quòd à quolibet puncto cuiuslibet corporis lucidi oritur lux per quamcunq;</s> <s xml:id="echoid-s17364" xml:space="preserve"> lineam rectã, quæ po <lb/>reſt extendi ex illo puncto.</s> <s xml:id="echoid-s17365" xml:space="preserve"> Ex quibus patet, quòd à quolibet puncto cuiuslibet corporis diaphani <lb/>contingentis aliquod corpus lucidum quacunq;</s> <s xml:id="echoid-s17366" xml:space="preserve"> luce, oritur lux per omnem lineam rectam, quæ <lb/>poterit extendi ex illo puncto, & tranſit in corpor<gap/> diaphano tangente illud punctũ.</s> <s xml:id="echoid-s17367" xml:space="preserve"> Et ſi occurrerit <lb/>aliud corpus diaphanum diuerſæ diaphanitatis à diaphanitate corporis tangentis illud, tranſibit e-<lb/>tiam in ipſum aut refractè aut rectè, ſiue primum corpus ſit ſubtilius ſecundo, ſiue ſecundũ ſit ſub-<lb/>tilius primo.</s> <s xml:id="echoid-s17368" xml:space="preserve"> Et etiã primo libro [14.</s> <s xml:id="echoid-s17369" xml:space="preserve"> 18.</s> <s xml:id="echoid-s17370" xml:space="preserve"> 19.</s> <s xml:id="echoid-s17371" xml:space="preserve"> 28 n] declaratũ eſt, quòd ab omni corpore colorato lucido <lb/>color oritur cumluce, qui eſt mixtus cum luce:</s> <s xml:id="echoid-s17372" xml:space="preserve"> & quòd uiſus cum comprehendit lucem, compre-<lb/>hendit formam coloris mixtam ſibi.</s> <s xml:id="echoid-s17373" xml:space="preserve"> Ex quibus patet, quòd corpora colorata, quæ ſunt in aqua & ul <lb/>tra corpora diaphana, quæ differunt in diaphanitate à diaphanιtate aeris, cum in eis fuerit lux eſſen <lb/>tialis, aut accidentalis fortis aut debilis:</s> <s xml:id="echoid-s17374" xml:space="preserve"> tunc lux, quę eſt in eis, oritur à quolibet puncto cum forma <lb/>coloris, qui eſt in illo puncto, & tranſit lux mixta cum colore in corpore aquæ, & in omni corpore <lb/>diaphano contingente ipſum:</s> <s xml:id="echoid-s17375" xml:space="preserve"> & extenditur lux in corpore aquæ & in omni corpore diaphano cum <lb/>forma coloris per lineas rectas, donec perueniat ad ſuperficiem aquæ, aut illius corporis diapha-<lb/>ni.</s> <s xml:id="echoid-s17376" xml:space="preserve"> Et cum fuerit aer aut aliud corpus diaphanum tangens aquam:</s> <s xml:id="echoid-s17377" xml:space="preserve"> tunc in illud corpus diapha-<lb/>num tranſibit lux cum forma mixta ſibi in aere aut in alio corpore diaphano per lineas rectas.</s> <s xml:id="echoid-s17378" xml:space="preserve"> Et <lb/>hæ lineæ ſecundæ in maiore parte ſecabunt primas lineas, per quas extendebatur:</s> <s xml:id="echoid-s17379" xml:space="preserve"> & quædam <lb/>earum erunt in rectitudine primarum linearum.</s> <s xml:id="echoid-s17380" xml:space="preserve"> Et omnia corpora, quæ ſunt in aqua & ultra dia-<lb/>phana corpora, quæ differunt à diaphanitate aeris, cum fuerint in loco lucido, ſcilicet cum <lb/>lux orta fuerit ſuper aquam, in qua ſunt:</s> <s xml:id="echoid-s17381" xml:space="preserve"> tunc lux perueniet ad ipſa.</s> <s xml:id="echoid-s17382" xml:space="preserve"> Manifeſtum eſt enim, quòd <lb/>omnis lux tranſit in omne corpus in aqua exiſtens aut in alio corpore diaphano, cum ſuper a-<lb/>quam illam aut corpus illud diaphanum ceciderit:</s> <s xml:id="echoid-s17383" xml:space="preserve"> & à quolibet puncto ipſius corporis orietur <lb/>formalucis, quæ eſtin ipſo cum forma coloris, & extendetur in uniuerſo illius aquæ aut illius cor-<lb/>poris diaphani per omnem lineam rectam, quæ poterit extendi ab ipſo puncto, donec perue-<lb/> <pb o="248" file="0254" n="254" rhead="ALHAZEN"/> niat lux cum forma coloris, qui eſt in illo puncto ad ſuperficiem aquæ aut ad ſuperficiem illius cor-<lb/>poris diaphani.</s> <s xml:id="echoid-s17384" xml:space="preserve"> Sed non poteſt extrahi ab eodem puncto alicuius ſuperficiei ad eandem ſuperficiẽ <lb/>linea perpendicularis niſi una [per 13 p 11.</s> <s xml:id="echoid-s17385" xml:space="preserve">] Ergo à quolibet puncto cuiuslibet corporis colorati exi-<lb/>ſtentis in corpore diaphano oritur forma lucis cum forma coloris in uniuerſo corporis diaphani, <lb/>in quo exiſtit, ſecundum lineas rectas:</s> <s xml:id="echoid-s17386" xml:space="preserve"> & peruenit forma ad uniuerſum oppoſitum de ſuperficie <lb/>corporis diaphani:</s> <s xml:id="echoid-s17387" xml:space="preserve"> & una illarum linearum erit perpendicularis ſuper ſuperficiem corporis dia-<lb/>phani uel ſuperficiem continuam cum ſuperficie corporis diaphani, reliquæ autem lineæ erunt ob-<lb/>liquæ ſuper ſuperficiem corporis diaphani.</s> <s xml:id="echoid-s17388" xml:space="preserve"> Sed in præcedente capitulo [3.</s> <s xml:id="echoid-s17389" xml:space="preserve"> 6.</s> <s xml:id="echoid-s17390" xml:space="preserve"> 8.</s> <s xml:id="echoid-s17391" xml:space="preserve"> 4.</s> <s xml:id="echoid-s17392" xml:space="preserve"> 7 n] declaratum <lb/>eſt, quòd lux, cum extẽditur in corpore diaphano, & occurrerit alij corpori diaphano diuerſo à dia <lb/>phanitate primi corporis, & linea, per quam extenſa eſt lux in primo corpore, fuerit perpendicula-<lb/>ris ſuper ſuperficiem ſecundi corporis:</s> <s xml:id="echoid-s17393" xml:space="preserve"> tunc lux extendetur in rectitudine eius in ſecundo corpore:</s> <s xml:id="echoid-s17394" xml:space="preserve"> <lb/>& ſi linea, per quam extenditur lux, fuerit obliqua ſuper ſuperficiem ſecundi corporis:</s> <s xml:id="echoid-s17395" xml:space="preserve"> tunc lux re-<lb/>fringetur.</s> <s xml:id="echoid-s17396" xml:space="preserve"> Et cuiuslibet puncti cuiuslibet corporis colorati, & lucidi exiſtentis in corpore dia-<lb/>phano forma lucis & coloris extenditur in uniuerſo corpore diaphano, & peruenit ad oppoſitam <lb/>ſuperficiem corporis diaphani.</s> <s xml:id="echoid-s17397" xml:space="preserve"> Et ſi fuerit aliud corpus oppoſitum contingens diaphanum, & fue-<lb/>rit alterius diaphanitatis:</s> <s xml:id="echoid-s17398" xml:space="preserve"> tunc forma, quæ peruenit ad ſuperficiem illius corporis diaphani, tranſit <lb/>in corpus ipſum contingens:</s> <s xml:id="echoid-s17399" xml:space="preserve"> & omnes erunt refractæ, præterquam forma, quæ eſt in perpendicu-<lb/>lari:</s> <s xml:id="echoid-s17400" xml:space="preserve"> extenditur enim ſecundum rectitudinem in corpore contingente.</s> <s xml:id="echoid-s17401" xml:space="preserve"> Et ſi fortè perpendicula-<lb/>ris ceciderit ſuper punctum ſuperficiei continuæ cum ſuperficie corporis, quod non eſt in ipſo cor-<lb/>pore diaphano:</s> <s xml:id="echoid-s17402" xml:space="preserve"> tunc illa forma delebitur, & tunc omnes formæ, quæ tranſeunt in corpus contin-<lb/>gens, erunt refractæ.</s> <s xml:id="echoid-s17403" xml:space="preserve"> Ergo formæ omnium uiſibilium, quæ ſunt in aqua, & in cœlo, & in omni-<lb/>bus corporibus diaphanis contingentibus aerem, quæ differunt à diaphanitate aeris, extendun-<lb/>tur in uniuerſo aere oppoſito ſecundum lineas rectas:</s> <s xml:id="echoid-s17404" xml:space="preserve"> & illæ lineæ, quæ fuerint ex iſtis lineis decli-<lb/>natæ, per quas extenduntur formæ ſuper ſuperficiem aeris contingentis ſuperficiem corporis dia-<lb/>phani, habebunt formas refractas:</s> <s xml:id="echoid-s17405" xml:space="preserve"> & quæ fuerint ex illis perpendiculares ſuper ſuperficiem ae-<lb/>ris, contingentis ſuperficiem corporis diaphani, habebunt formas extenſas ſecundum rectitudi-<lb/>nem ipſarum.</s> <s xml:id="echoid-s17406" xml:space="preserve"> Et cum iam declaratum ſit, quòd à quolibet puncto cuiuslibet corporis colorati & <lb/>lucidi extenditur forma lucis, & coloris in uniuerſo corpore diaphano, & peruenit ad ſuperficiem <lb/>eius, & refringitur à ſuperficie eius:</s> <s xml:id="echoid-s17407" xml:space="preserve"> ergo forma, quæ extenditur ab uno puncto ad ſuperficiem cor-<lb/>poris diaphani, erit continua & coniuncta.</s> <s xml:id="echoid-s17408" xml:space="preserve"> Et cum forma fuerit continua, & ſuperficies corpo-<lb/>ris diaphani fuerit continua coniuncta, & forma fuerit refracta in alio corpore diaphano:</s> <s xml:id="echoid-s17409" xml:space="preserve"> tunc re-<lb/>fringetur continua.</s> <s xml:id="echoid-s17410" xml:space="preserve"> Et cum forma refracta fuerit continua:</s> <s xml:id="echoid-s17411" xml:space="preserve"> & occurrerit corpus denſum:</s> <s xml:id="echoid-s17412" xml:space="preserve"> tunc for-<lb/>ma perueniet ad illud corpus diaphanum:</s> <s xml:id="echoid-s17413" xml:space="preserve"> & ſic locus corporis diaphani, per quem extenditur for-<lb/>ma puncti, quod eſt in primo corpore, quæ refringitur à ſuperficie primi corporis, ad illum locum, <lb/>cum fuerit lucidus coloratus, mittet formam lucis & coloris à quolibet puncto ipſius per omnem <lb/>lineam rectam, quæ poterit extendi ex illo puncto.</s> <s xml:id="echoid-s17414" xml:space="preserve"> Accidit ergo ex hoc, quòd ſint lineæ refra-<lb/>ctæ ad illum locum exlineis, per quas extenditur forma illius loci:</s> <s xml:id="echoid-s17415" xml:space="preserve"> & iam extendebatur forma cu-<lb/>iuslibet puncti illius loci per unam illarum linearum refractarum.</s> <s xml:id="echoid-s17416" xml:space="preserve"> Forma ergo illius loci ex cor-<lb/>pore denſo colorato lucido erit in loco ex ſuperficie corporis diaphani, apud quem refringitur for-<lb/>ma unius puncti extenſi ad illum locum ſuperficiei corporis diaphani, quæ refringitur ad eundem <lb/>locum corporis denſi.</s> <s xml:id="echoid-s17417" xml:space="preserve"> Ex quo ſequitur, quòd forma loci corporis denſi, quæ extenditur ad illum <lb/>locum corporis diaphani, refringitur ad eaſdem lineas extenſas ab uno puncto ad illum locum cor <lb/>poris diaphani.</s> <s xml:id="echoid-s17418" xml:space="preserve"> Et cum formaloci corporis diaphani fuerit refracta ſuper illas eaſdem lineas:</s> <s xml:id="echoid-s17419" xml:space="preserve"> tunc <lb/>perueniet ad illud idem punctum.</s> <s xml:id="echoid-s17420" xml:space="preserve"> Ex quo declaratur, quòd ſi imaginatus fuerit aliquis pyramidem <lb/>extenſam à quolibet puncto aeris ſecundum lineas rectas, & pyramis fuerit coniuncta continua, <lb/>& peruenerit illa pyramis ad ſuperficiem corporis diaphani diuerſæ diaphanitatis ab aere, & ima-<lb/>ginatus fuerit omnem lineam rectam, quæ poſsit extendi ex illa pyramide, refringi apud ſuperfi-<lb/>ciem corporis diaphani in loco, quem exigit eius declinatio:</s> <s xml:id="echoid-s17421" xml:space="preserve"> & ſi aliqua fuerit perpendicularis, ex-<lb/>tendetur rectè:</s> <s xml:id="echoid-s17422" xml:space="preserve"> tunc efficitur & hoc corpus continuum refractum in corpore diaphano, quod dif-<lb/>fert à diaphanitate aeris.</s> <s xml:id="echoid-s17423" xml:space="preserve"> Et cum hoc corpus refractum peruenerit ad corpus denſum:</s> <s xml:id="echoid-s17424" xml:space="preserve"> tunc illud <lb/>corpus denſum, ſi fuerit coloratum & lucidum, mittet formam lucis & coloris, quæ ſunt in ipſo, in <lb/>hoc corpore refracto imaginato per quamlibet lineam rectam, quæ poterit extendi in hoc corpo-<lb/>re refracto à linea extenſa in corpore pyramidis à puncto, quod eſt in aere.</s> <s xml:id="echoid-s17425" xml:space="preserve"> Nam omne corpus co-<lb/>loratum lucidum propriè mittit formam ſuam à quolibet puncto ipſius per omnem lineam rectam, <lb/>quæ poterit extendi ab illo puncto.</s> <s xml:id="echoid-s17426" xml:space="preserve"> Erit ergo forma puncti illius loci corporis denſi extenſa per <lb/>quamlibet linearum refractarum ad illum locum corporis denſi.</s> <s xml:id="echoid-s17427" xml:space="preserve"> Perueniet ergo illius forma à cor-<lb/>pore denſo, colorato, lucido ad locum ſuperficiei corporis diaphani, in quem refringuntur illæ <lb/>lineæ.</s> <s xml:id="echoid-s17428" xml:space="preserve"> Et cum peruenerit forma ad illum locum ſuperficiei corporis diaphani, neceſſariò refringe-<lb/>tur per eaſdem lineas extenſas ad illum locum ab uno puncto, quod eſt in aere:</s> <s xml:id="echoid-s17429" xml:space="preserve"> forma autem, quæ <lb/>eſt forma loci colorati corporis denſi, quod eſt in corpore diaphano, quod differt à diaphanitate <lb/>aeris (& eſt ſuper lineam, quæ eſt de numero illarum linearum, per quas extenditur forma ad cen-<lb/>trum uiſus) forma, dico, quæ extenditur per illam lineam:</s> <s xml:id="echoid-s17430" xml:space="preserve"> peruenit ad centrum uiſus rectè.</s> <s xml:id="echoid-s17431" xml:space="preserve"> Formæ <lb/>autem, quæ extenduntur per omnes alias lineas, quæ conſtituunt pyramidem extenſam à centro <lb/>uiſus, erunt refractæ, non directæ.</s> <s xml:id="echoid-s17432" xml:space="preserve"> Et in primo tractatu [14.</s> <s xml:id="echoid-s17433" xml:space="preserve"> 17.</s> <s xml:id="echoid-s17434" xml:space="preserve"> 28 n] declaratum eſt, quòd aer re-<lb/> <pb o="249" file="0255" n="255" rhead="OPTICAE LIBER VII."/> cipit formam uiſibilium, & reddit eam omni corpori oppoſito:</s> <s xml:id="echoid-s17435" xml:space="preserve"> & quòd aer deferens formam cum <lb/>tetigerit uiſum:</s> <s xml:id="echoid-s17436" xml:space="preserve"> tranſibit forma, quæ eſt in ipſo, in corpus uiſus:</s> <s xml:id="echoid-s17437" xml:space="preserve"> & ſic uiſus comprehendit uiſibilia, <lb/>quæ aer reddit uiſui.</s> <s xml:id="echoid-s17438" xml:space="preserve"> Ex omnibus ergo iſtis patet, quod forma omnis corporis colorati, lucidi, exi-<lb/>ſtentis in corpore diaphano diuerſæ diaphanitatis à diaphanitate aeris, extenditur in corpore dia-<lb/>phano, in quo exiſtit, & refringitur in aere, & extenditur in aere ſecundum lineas rectas:</s> <s xml:id="echoid-s17439" xml:space="preserve"> & quòd <lb/>quædam lιnearum rectarum, per quas forma refringitur in aere, coniunguntur apud idem pun-<lb/>ctum aeris.</s> <s xml:id="echoid-s17440" xml:space="preserve"> Et cum centrum uiſus fuerit apud illud punctum:</s> <s xml:id="echoid-s17441" xml:space="preserve"> tunc uiſus comprehendit illud ui-<lb/>ſum ſecundum refractionem:</s> <s xml:id="echoid-s17442" xml:space="preserve"> & ſi aliquid ipſius comprehenditur rectè:</s> <s xml:id="echoid-s17443" xml:space="preserve"> non erit niſi unum pun-<lb/>ctum tantùm.</s> <s xml:id="echoid-s17444" xml:space="preserve"> Hoc ergo modo comprehendit uiſus res, quæ ſunt in aqua, & in cœlo, & omnia uiſi-<lb/>bilia, quæ ſunt ultra corpora diaphana, quę differunt à diaphanitate aeris.</s> <s xml:id="echoid-s17445" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div571" type="section" level="0" n="0"> <head xml:id="echoid-head496" xml:space="preserve" style="it">14. Imago refracti uiſibilis à medio quidem denſiore, inclinat ad perpendicularem à refra-<lb/>ctionis puncto excitatam: à rariore uerò ab eadem declinat. 4 p 10.</head> <p> <s xml:id="echoid-s17446" xml:space="preserve">QVòd autem hoc uerum ſit, ſic poterit experimentari.</s> <s xml:id="echoid-s17447" xml:space="preserve"> Accipiat ergo experimentator prędi-<lb/>ctum inſtrumentum, & ponat in uaſe, & ponat uas in loco lucido quacunque luce, ita ut <lb/>lux perueniat ad interius uaſis, & infundat in uas aquam, quouſque perueniat ad centrum <lb/>laminæ:</s> <s xml:id="echoid-s17448" xml:space="preserve"> deinde diminuat foramina cum cera, ita ut non remaneat de foraminibus, niſi modicũ in <lb/>medio eorum, & mittat in duobus foraminibus unum calamum, ita ut ſpatium, quod eſt inter duo <lb/>foramina, ſit determinatum:</s> <s xml:id="echoid-s17449" xml:space="preserve"> deinde moueat inſtrumentum, donec diameter laminæ, ſuper cu-<lb/>ius extremitates ſunt duæ lineæ perpendiculares in ora inſtrumenti, ſit perpendicularis ſuper ſu-<lb/>perficiem aquæ.</s> <s xml:id="echoid-s17450" xml:space="preserve"> Deinde accipiat ſtilum ſubtilem album, & mittat eum in uas, & eius extremita-<lb/>tem ponat in puncto medij circuli, quod eſt differentia communis circumferentiæ medij circuli <lb/>& lineæ perpendiculari in ora inſtrumenti, quod eſt extremitas diametri circuli, quę tranſit per cen <lb/>tra duorum foraminum.</s> <s xml:id="echoid-s17451" xml:space="preserve"> Deinde ponat experimentator alterum uiſum ſuper ſuperius foramen, & <lb/>claudat reliquum, & intueatur oram inſtrumenti, quæ eſt intra aquam:</s> <s xml:id="echoid-s17452" xml:space="preserve"> tunc enim uidebit extremi-<lb/>tatem ſtili.</s> <s xml:id="echoid-s17453" xml:space="preserve"> Declarabitur ergo ex hac experimentatione, quòd comprehenſio eius ad extremita-<lb/>tem ſtili eſt ſecundum rectitudinem perpendicularis, egredientis ab extremitate ſtili ſuper ſuper-<lb/>ficiem aquæ.</s> <s xml:id="echoid-s17454" xml:space="preserve"> Nam linea, quæ tranſit per centra duorum foraminum, in qua eſt centrum uiſus, & ex-<lb/>tremitas ſtili, ex cuius uerticatione comprehendit uiſus extremitatem ſtili, ſunt perpendiculares <lb/>ſuper ſuperficiem aquæ.</s> <s xml:id="echoid-s17455" xml:space="preserve"> In primo autem libro [18.</s> <s xml:id="echoid-s17456" xml:space="preserve"> 19 n] patuit, quòd uiſus nihil comprehendit, ni-<lb/>ſi ſecundum rectitudinem linearum, quæ extenduntur per centrum uiſus.</s> <s xml:id="echoid-s17457" xml:space="preserve"> Viſus ergo comprehen-<lb/>dit extremitatem ſtili à uerticatione lineæ, quæ tranſit per centra duorum foraminum.</s> <s xml:id="echoid-s17458" xml:space="preserve"> Et hæc li-<lb/>nea extenditur ad extremitatem ſtili rectè:</s> <s xml:id="echoid-s17459" xml:space="preserve"> & eſt perpendicularis ſuper ſuperficiem aquæ.</s> <s xml:id="echoid-s17460" xml:space="preserve"> Deinde <lb/>oportet experimentatorem declinare inſtrumentum, donec linea, quæ tranſit per centra duorum <lb/>foraminum, ſit obliqua ſuper ſuperficiem aquæ, & mittat ſtilum in aquam, & ponat extremitatem <lb/>eius ſuper primum punctum, ſcilicet ſuper extremitatem diametri circuli medij, quę tranſit per cen <lb/>tra duorum foraminum, & ponat uiſum ſuum ſuper ſuperius foramen, & intueatur oram inſtrumen <lb/>ti, quæ eſt intra aquam:</s> <s xml:id="echoid-s17461" xml:space="preserve"> tunc enim non uidebit extremitatem ſtili:</s> <s xml:id="echoid-s17462" xml:space="preserve"> deinde moueat ſtilum ad partem <lb/>contrariam illi, in qua eſt uiſus:</s> <s xml:id="echoid-s17463" xml:space="preserve"> & moueat extremitatem ſtili per circumferentiam circuli medij ſua-<lb/>uiter, & molliter, & intueatur oram inſtrumẽti:</s> <s xml:id="echoid-s17464" xml:space="preserve"> tunc enim uidebit extremitatem ſtili:</s> <s xml:id="echoid-s17465" xml:space="preserve"> tunc figat ex-<lb/>tremitatem ſtili in ſuo loco.</s> <s xml:id="echoid-s17466" xml:space="preserve"> Deinde pręcipiat alij, ut mittat in uas lignum aliquod uel acum perpen <lb/>dicularem, neq;</s> <s xml:id="echoid-s17467" xml:space="preserve"> groſſam, neq;</s> <s xml:id="echoid-s17468" xml:space="preserve"> gracilem, & ponat illam <lb/> <anchor type="figure" xlink:label="fig-0255-01a" xlink:href="fig-0255-01"/> apud ſuperficiem aquæ in oppoſitione ſecundi fora-<lb/>minis, ut ſit apud centrum circuli medij, & intueatur <lb/>experimentator interius uaſis:</s> <s xml:id="echoid-s17469" xml:space="preserve"> tunc non uidebit extre <lb/>mitatem ſtili:</s> <s xml:id="echoid-s17470" xml:space="preserve"> deinde præcipiat auferre lignum:</s> <s xml:id="echoid-s17471" xml:space="preserve"> & tunc <lb/>uidebit extremitatem ſtili:</s> <s xml:id="echoid-s17472" xml:space="preserve"> deinde figat extremitatem <lb/>ſtili in ſuo loco, & leuet uiſum ſuum à foramine, & au-<lb/>ferat inſtrumentũ ſuum à uaſe, exiſtente extremitate <lb/>ſtili in ſuo loco, & intueatur locum, in quo eſt extremi <lb/>tas ſtili:</s> <s xml:id="echoid-s17473" xml:space="preserve"> tunc enim uidebit inter ipſum & diametrum <lb/>circuli medij diſtantiam ſenſibilem.</s> <s xml:id="echoid-s17474" xml:space="preserve"> Et ſi miſerit regu-<lb/>lam ſubtilem in aquam in hora experimentationis, & <lb/>acumen eius fecerit tranſire per centrum laminę, & ſi-<lb/>gnauerit locum circuli medij, qui eſt apud extremita-<lb/>tem regulæ, ſigno, & abſtulerit inſtrumentum, & aſpe-<lb/>xerit locum extremitatis ſtili:</s> <s xml:id="echoid-s17475" xml:space="preserve"> uidebit locum extremi-<lb/>tatis ſtili etiam medium inter locum extremitatis regulæ & diametrum circuli medij.</s> <s xml:id="echoid-s17476" xml:space="preserve"> Deinde opor <lb/>tet eum auferre inſtrumentum, & infundere aquam in uas, & applicare uitrum laminæ, & ponere <lb/>ſuperficiem uitri ęqualem ex parte foraminum, & ponere differentiam communem, quę eſt in ipſo, <lb/>ſuper lineam ſecantem diametrum laminæ perpendiculariter.</s> <s xml:id="echoid-s17477" xml:space="preserve"> Sic ergo linea, quę tranſit per centra <lb/>duorum foraminũ, erit perpendicularis ſuper ſuperficiẽ uitri æqualem & ſuper ſuperficiẽ eius con-<lb/>uexã.</s> <s xml:id="echoid-s17478" xml:space="preserve"> Deinde ponat experimentator inſtrumentũ in aquã, & mittat ſtilũ in uas, & ponat extremita <lb/> <pb o="250" file="0256" n="256" rhead="ALHAZEN"/> tem ſtili ſuper extremitatem diametri circuli medij, & ponat uiſum ſuum ſuper ſuperius foramen, <lb/>& intueatur oram inſtrumenti:</s> <s xml:id="echoid-s17479" xml:space="preserve"> tunc uidebit extremitatem ſtili.</s> <s xml:id="echoid-s17480" xml:space="preserve"> Et ſi mouerit extremitatem ſtili, <lb/>& extraxerit illam à puncto, quod eſt extremitas diametri medij circuli, non uidebit extremitatem <lb/>ſtili.</s> <s xml:id="echoid-s17481" xml:space="preserve"> Ex quo patet, quòd extremitatem ſtili comprehendit rectè.</s> <s xml:id="echoid-s17482" xml:space="preserve"> Nam duo centra foraminum, & <lb/>extremitas diametri circuli medij ſunt in eadem linea recta:</s> <s xml:id="echoid-s17483" xml:space="preserve"> & experimentator non comprehen-<lb/>dit extremitatem ſtili in hoc ſitu, cum extremitas ſtili non fuerit ſuper extremitatem diametri.</s> <s xml:id="echoid-s17484" xml:space="preserve"> Et <lb/>ſi euulſerit uitrum, & poſuerit ipſum è contrario, ſcilicet ut ponat conuexum uitri ex parte duo-<lb/>rum foraminum, & differentiam eius communem ſuper primum locum, & expertus fuerit extre-<lb/>mitatem ſtili:</s> <s xml:id="echoid-s17485" xml:space="preserve"> etiam uidebit illam, cum fuerit in extremitate diametri circuli medij:</s> <s xml:id="echoid-s17486" xml:space="preserve"> ideo in hoc ſi-<lb/>tu etiam linea, quæ tranſit per centra duorum foraminum, ex cuius uerticatione comprehendit <lb/>uiſus extremitatem ſtili:</s> <s xml:id="echoid-s17487" xml:space="preserve"> erit perpendicularis ſuper ſuperficiem uitri æqualem, & ſuperficiem eius <lb/>conuexam.</s> <s xml:id="echoid-s17488" xml:space="preserve"> Deinde oportet experimentatorem euellere uitrum, & extrahere à centro laminæ li-<lb/>neam rectam in ſuperficie laminæ, quæ contineat cum diametro laminæ, ſuper cuius extremitates <lb/>ſunt duæ lineæ perpendiculares in ora inſtrumenti, angulum obtuſum:</s> <s xml:id="echoid-s17489" xml:space="preserve"> & extrahat illam, donec <lb/>perueniat ad oram inſtrumenti:</s> <s xml:id="echoid-s17490" xml:space="preserve"> deinde extrahat à centro laminæ lineam in ſuperficie laminæ, <lb/>quæ contineat cum prima linea angulum rectum:</s> <s xml:id="echoid-s17491" xml:space="preserve"> & protrahat illam in utramque partem:</s> <s xml:id="echoid-s17492" xml:space="preserve"> tunc <lb/>hæc linea continebit cum diametro laminæ angulum acutum:</s> <s xml:id="echoid-s17493" xml:space="preserve"> & diameter laminæ erit obliqua ſu-<lb/>per hanc lineam.</s> <s xml:id="echoid-s17494" xml:space="preserve"> Deinde ſuperponat uitrum laminæ, & ponat differentiam eius communem ſu-<lb/>per lineam, quam ultimò ſignauit in ſuperficie laminæ, & ponat ſuperficiem uitri æqualem ex par-<lb/>te duorum foraminum, & ponat medium differentiæ communis ſuper centrum laminæ.</s> <s xml:id="echoid-s17495" xml:space="preserve"> Sic ergo <lb/>erit centrum uitri ſuper centrum circuli medij, ut prius declaratum eſt:</s> <s xml:id="echoid-s17496" xml:space="preserve"> & linea, quæ tranſit per cen <lb/>tra duorum foraminum, tranſibit per centrum uitri.</s> <s xml:id="echoid-s17497" xml:space="preserve"> Et hæc linea erit obliqua ſuper ſuperficiem ui-<lb/>tri æqualem:</s> <s xml:id="echoid-s17498" xml:space="preserve"> nam diameter laminæ illi æquidiſtans, eſt obliqua ſuper differentiam communem, <lb/>quæ eſt in uitro.</s> <s xml:id="echoid-s17499" xml:space="preserve"> Et hæc linea erit perpendicularis ſuper ſuperficiem uitri conuexam, [ut oſten-<lb/>ſum eſt 25 n 4] quia tranſit per centrum eius.</s> <s xml:id="echoid-s17500" xml:space="preserve"> Deinde extrahat experimentator ab extremita-<lb/>te lineæ, quam primò ſignauit in lamina, lineam perpendicularem in ora inſtrumenti:</s> <s xml:id="echoid-s17501" xml:space="preserve"> & ducat il-<lb/>lam ad circumferentiam circuli medij:</s> <s xml:id="echoid-s17502" xml:space="preserve"> & ſint hæ lineæ nigræ.</s> <s xml:id="echoid-s17503" xml:space="preserve"> Erit ergo linea cum ab illo puncto <lb/>extracta fuerit ad centrum circuli medij, quod eſt centrum uitri, perpendicularis ſuper ſuperficiem <lb/>uitri æqualem, & ſuper ſuperficiem uitri ſphæricam.</s> <s xml:id="echoid-s17504" xml:space="preserve"> Super ſuperficiem autem uitri æqualem eſt <lb/>perpendicularis, [per 8 p 11] quia eſt æquidiſtans primæ lineæ ſignatæ in lamina ſuper differen-<lb/>tiam communem, quæ eſt in uitro:</s> <s xml:id="echoid-s17505" xml:space="preserve"> ſuper ſphæricam uerò [per 25 n 4] quia tranſit per centrum e-<lb/>ius.</s> <s xml:id="echoid-s17506" xml:space="preserve"> Punctum ergo, ad quod peruenit linea extracta in ora inſtrumenti, quod eſt ſuper circumferen-<lb/>tiam circuli medij, eſt caſus, in quem cadit perpendicularis, exiens à centro uitri ſuper ſuperficiem <lb/>uitri planam.</s> <s xml:id="echoid-s17507" xml:space="preserve"> Deinde oportet experimentatorem ponere inſtrumentum in uas, & ponere extremi-<lb/>tatem ſtili in puncto, quod eſt extremitas diametri circuli medij, & ponat experimentator ſuum ui <lb/>ſum ſuper ſuperius foramen, & intueatur oram inſtrumenti:</s> <s xml:id="echoid-s17508" xml:space="preserve"> tunc non uidebit extremitatem ſtili:</s> <s xml:id="echoid-s17509" xml:space="preserve"> <lb/>deinde moueat ſtilum ad partem contrariam illi, in qua eſt caſus perpendicularis:</s> <s xml:id="echoid-s17510" xml:space="preserve"> & tunc etiam nõ <lb/>uidebit extremitatem ſtili:</s> <s xml:id="echoid-s17511" xml:space="preserve"> deinde moueat ſtilum ad partem illam, in qua eſt caſus perpendicula-<lb/>ris, & per circumferentiam circuli medij:</s> <s xml:id="echoid-s17512" xml:space="preserve"> tunc enim, ſi motus fuerit ſuauis, uidebit extremitatem <lb/>ſtili in ſuo loco, in quo apparuit.</s> <s xml:id="echoid-s17513" xml:space="preserve"> Deinde præcipiat alicui cooperire centrum uitri tenui & ſubtili li-<lb/>gno:</s> <s xml:id="echoid-s17514" xml:space="preserve"> & tunc non uidebit extremitatem ſtili:</s> <s xml:id="echoid-s17515" xml:space="preserve"> & ſi abſtulerit coopertorium, uidebit ipſum.</s> <s xml:id="echoid-s17516" xml:space="preserve"> Ex hac <lb/>ergo experimentatione patet, quòd cum uiſus comprehendit extremitatem ſtili, eſt ſecundum re-<lb/>fractionem:</s> <s xml:id="echoid-s17517" xml:space="preserve"> & quòd refractio eſt à centro uitri:</s> <s xml:id="echoid-s17518" xml:space="preserve"> & quòd forma refracta eſt in ſuperficie circuli me-<lb/>dij, quæ eſt perpendicularis ſuper ſuperficiem uitri æqualem, apud quam fit refractio ad perpendi-<lb/>cularem, ut prius declaratum eſt [5 n.</s> <s xml:id="echoid-s17519" xml:space="preserve">] Et ſi experimentator aſpexerit locum extremitatis ſtili:</s> <s xml:id="echoid-s17520" xml:space="preserve"> in-<lb/>ueniet ipſum inter caſum perpendicularis & extremitatem diametri circuli medij, quæ tranſit per <lb/>centra duorum foraminum.</s> <s xml:id="echoid-s17521" xml:space="preserve"> Linea ergo, quæ exit ab extremitate ſtili ad centrum uitri, cum exten-<lb/>ſa fuerit rectè in aere:</s> <s xml:id="echoid-s17522" xml:space="preserve"> perpendicularis exiens à centro uitri ſuper ſuperficiem uitri æqualem, erit <lb/>media inter perpendicularem & lineam, quæ tranſit per centra duorum foraminum.</s> <s xml:id="echoid-s17523" xml:space="preserve"> Et forma ex-<lb/>tremitatis ſtili, quæ extenſa eſt ab extremitate ſtili ad centrum uitri, extenſa eſt ſuper hanc lineam, <lb/>& extenſa eſt in rectitudine eius ad centrum uitri.</s> <s xml:id="echoid-s17524" xml:space="preserve"> Hæc enim linea eſt perpendicularis ſuper ſuper-<lb/>ficiem uitri ſphæricam, quæ eſt ex parte extremitatis.</s> <s xml:id="echoid-s17525" xml:space="preserve"> Deinde cum hæc forma fuerit refracta ſuper <lb/>lineam, quæ tranſit per centra duorum foraminum:</s> <s xml:id="echoid-s17526" xml:space="preserve"> lineæ radiales, quæ exeunt in hoc ſitu à uiſu, <lb/>non perueniunt ad uitrum, præter lineam, quæ tranſit per centra duorum foraminum:</s> <s xml:id="echoid-s17527" xml:space="preserve"> calamus e-<lb/>nim, qui extenditur inter duo foramina, ſecat omnem in eam à uiſu exeuntem ad uitrum, præter-<lb/>quam lineam, quę tranſit per centra duorum foraminum.</s> <s xml:id="echoid-s17528" xml:space="preserve"> Viſus autem non comprehendit for-<lb/>mas, niſi ex uerticationibus harum linearum tantùm:</s> <s xml:id="echoid-s17529" xml:space="preserve"> ergo formæ non extenduntur niſi rectè:</s> <s xml:id="echoid-s17530" xml:space="preserve"> er-<lb/>go uiſus non comprehendit hanc formam, niſi ex uerticatione huius lineę perpendicularis.</s> <s xml:id="echoid-s17531" xml:space="preserve"> Er-<lb/>go quę extenditur rectè in aere, eſt perpendicularis ſuper ſuperficiem aeris contingentis ſuperfi-<lb/>ciem uitri ęqualem.</s> <s xml:id="echoid-s17532" xml:space="preserve"> Ergo hęc refractio erit ad partem contrariam parti perpendicularis, exeuntis <lb/>à loco refractionis ſuper ſuperficiem aeris.</s> <s xml:id="echoid-s17533" xml:space="preserve"> Nam linea, quę tranſit per centra duorum foraminum, <lb/>magis diſtat à perpendiculari, quę extenditur in aere, quàm linea, quę exit ab extremitate ſti-<lb/>li ad centrum uitri, quę extenditur in aere.</s> <s xml:id="echoid-s17534" xml:space="preserve"> Et hęc forma exit à uitro, & refringitur in aere:</s> <s xml:id="echoid-s17535" xml:space="preserve"> & <lb/>aer eſt ſubtilior uitro.</s> <s xml:id="echoid-s17536" xml:space="preserve"> Ethoc modo fiet refractio formę de aqua ad aerem.</s> <s xml:id="echoid-s17537" xml:space="preserve"> Viſus enim compre-<lb/> <pb o="251" file="0257" n="257" rhead="OPTICAE LIBER VII."/> hendit extremitatem ſtili in aqua ab iſto loco, ſcilicet quia comprehendit extremitatem ſtili, quan-<lb/>do fuit inter caſum perpendicularis & extremitatem diametri circuli medij, quæ tranſit per cen-<lb/>tra duorum foraminum.</s> <s xml:id="echoid-s17538" xml:space="preserve"> Et illa forma etiam exiuit ab aqua, & refracta eſt in aere:</s> <s xml:id="echoid-s17539" xml:space="preserve"> & aer eſt ſubti-<lb/>lior aqua.</s> <s xml:id="echoid-s17540" xml:space="preserve"> Deinde oportet experimentatorem euellere uitrum, & ponere ipſum ſupra laminam <lb/>extra huiuſmodi ſitum, ſcilicet, ut ponat conuexum eius ex parte duorum foraminum, & ponat <lb/>differentiam eius communem ſuper lineam æqualem in ſuperficie laminæ, in qua poſuerat illam <lb/>in prædicto ſitu, & ponat medium differentiæ communis ſuper centrum laminæ:</s> <s xml:id="echoid-s17541" xml:space="preserve"> & ſic linea, quæ <lb/>tranſit per centra duorum foraminum, erit obliqua ſuper ſuperficiem uitri æqualem, & perpendi-<lb/>cularis ſuper ſuperficiem eius conuexam:</s> <s xml:id="echoid-s17542" xml:space="preserve"> & applicet uitrum in hoc ſitu, & ponat inſtrumentum <lb/>in uas, & ponat extremitatem ſtili ſuper extremitatem diametri circuli medij, ut prius fecerat, & <lb/>ponat uiſum ſuum ſuper ſuperius foramen, & intueatur oram inſtrumenti:</s> <s xml:id="echoid-s17543" xml:space="preserve"> non enim uidebit tunc <lb/>extremitatem ſtili:</s> <s xml:id="echoid-s17544" xml:space="preserve"> deinde moueat ſtilum ad partem caſus perpendicularis:</s> <s xml:id="echoid-s17545" xml:space="preserve"> & tunc non uidebit ex-<lb/>tremitatem ſtili:</s> <s xml:id="echoid-s17546" xml:space="preserve"> deinde moueat eundem ad partem contrariam illi, in qua eſt caſus perpendicula-<lb/>ris per circumferentiam medij circuli, & ſuauiter:</s> <s xml:id="echoid-s17547" xml:space="preserve"> tunc enim uidebit extremitatem ſtili.</s> <s xml:id="echoid-s17548" xml:space="preserve"> Sic ergo li-<lb/>nea recta, quæ exit ab extremitate ſtili ad centrum uitri, cum fuerit extenſa rectè in corpore uitri, <lb/>& extenſa fuerit cum ipſa perpendicularis exiens à centro uitri:</s> <s xml:id="echoid-s17549" xml:space="preserve"> erit linea, quæ tranſit per centra <lb/>duorum foraminum, media inter duas lineas.</s> <s xml:id="echoid-s17550" xml:space="preserve"> Et forma extremitatis ſtili, quæ extenditur ſuper <lb/>hanclineam, cum fuerit extenſa ad centrum uitri:</s> <s xml:id="echoid-s17551" xml:space="preserve"> refringetur ſuper lineam, quæ tranſit per centra <lb/>duorum foraminum.</s> <s xml:id="echoid-s17552" xml:space="preserve"> Erit ergo refractio iſta ad partem perpendicularis exeuntis à loco refractio-<lb/>nis ſuper ſuperficiem uitri.</s> <s xml:id="echoid-s17553" xml:space="preserve"> Et hæc forma exit ab aere, & refringitur in uitro:</s> <s xml:id="echoid-s17554" xml:space="preserve"> & uitrum eſt groſsius <lb/>aere.</s> <s xml:id="echoid-s17555" xml:space="preserve"> Ex omnibus ergo iſtis experimentationibus patet, quòd uiſus comprehendit uiſibilia, quæ <lb/>ſunt in aqua, & ultra corpora diaphana, quæ differunt à diaphanitate aeris, ſecundum refractio-<lb/>nem, præterquam illa, quæ ſunt ſuper lineas perpendiculares ſuper ſuperficiem corporis diapha-<lb/>ni, in quo exiſtit:</s> <s xml:id="echoid-s17556" xml:space="preserve"> & quòd refractio formarum ipſorum eſt in ſuperficiebus perpendicularibus ſu-<lb/>per ſuperficies corporum diaphanorum.</s> <s xml:id="echoid-s17557" xml:space="preserve"> Omne enim quod experimentatum eſt per prædictum in-<lb/>ſtrumentum, inuenitur refringi in ſuperficie medij circuli, de quo patuit, [5 n] quòd eſt perpendi-<lb/>cularis ſuper ſuperficies corporum diaphanorum, & ſuper ſuperficies corporum contingentium <lb/>ſuperficies eorum.</s> <s xml:id="echoid-s17558" xml:space="preserve"> Ex hac ergo experimentatione declarabitur etiam, quòd formæ, quæ compre-<lb/>henduntur à uiſu ſecundum refractionem, quæ exeunt à groſsiore corpore diaphano ad ſubtilius, <lb/>refringuntur ad partem contrariam illi, in qua eſt perpendicularis exiens à loco refractionis ſuper <lb/>ſuperficiem corporis diaphani:</s> <s xml:id="echoid-s17559" xml:space="preserve"> & quæ exeunt à ſubtiliore ad groſsius, refringuntur ad partem, in <lb/>qua eſt perpendicularis prædicta.</s> <s xml:id="echoid-s17560" xml:space="preserve"/> </p> <div xml:id="echoid-div571" type="float" level="0" n="0"> <figure xlink:label="fig-0255-01" xlink:href="fig-0255-01a"> <variables xml:id="echoid-variables204" xml:space="preserve">k b d o <gap/> f u g z r e a</variables> </figure> </div> </div> <div xml:id="echoid-div573" type="section" level="0" n="0"> <head xml:id="echoid-head497" xml:space="preserve" style="it">15. Stella uidetur refractè. 49 p 10.</head> <p> <s xml:id="echoid-s17561" xml:space="preserve">STellæ autem comprehenduntur etiam ſecundum refractionem:</s> <s xml:id="echoid-s17562" xml:space="preserve"> nam corpus cœli eſt ſubtilius <lb/>corpore aeris, id eſt maioris diaphanitatis.</s> <s xml:id="echoid-s17563" xml:space="preserve"> Hoc autem poteſt experimentari experimentatio-<lb/>ne, quæ oſtendet, quòd ſtellæ comprehendantur ſecundum refractionem:</s> <s xml:id="echoid-s17564" xml:space="preserve"> ex quo patebit e-<lb/>tiam, quòd corpus cœli eſt magis diaphanum corpore aeris.</s> <s xml:id="echoid-s17565" xml:space="preserve"> Et cum quis hoc uoluerit experiri, ac-<lb/>cipiat inſtrumentum de armillis, & ponat illud in loco eminente, in quo poterit apparere hori-<lb/>zon orientalis, & ponat inſtrumentum armillarum ſuo modo proprio:</s> <s xml:id="echoid-s17566" xml:space="preserve"> ſcilicet ut ponat armillam, <lb/>quæ eſt in loco circuli meridionalis, in ſuperficie circuli meridiei, & polus eius ſit exaltatus à terra <lb/>ſecundum altitudinem poli mundi ſupra horizontem loci, in quo ponitur inſtrumentum:</s> <s xml:id="echoid-s17567" xml:space="preserve"> & in no-<lb/>cte obſeruet aliquam ſtellarum fixarum magnarum, quæ tranſit per uerticem capitis illius loci, aut <lb/>prope, & obſeruet illam ab ortu ſuo in oriente:</s> <s xml:id="echoid-s17568" xml:space="preserve"> ſtella autem orta, reuoluat armillam, quæ reuo lui-<lb/>tur in circuitu poli æquinoctialis, donec fiat æquidiſtans ſtellæ, & certificetur locus ſtellæ exar-<lb/>milla:</s> <s xml:id="echoid-s17569" xml:space="preserve"> & ſic habebit longitudinem ſtellæ à polo mundi.</s> <s xml:id="echoid-s17570" xml:space="preserve"> Deinde obſeruet ſtellam, quouſque per-<lb/>uenerit ad circulum meridiei, & reuoluat armillam, quam prius mouerat, donec fiat æquidiſtans <lb/>ſtellæ:</s> <s xml:id="echoid-s17571" xml:space="preserve"> & ſic habebit longitudinem ſtellæ à polo mundi, cum ſtella fuerit in uertice capitis.</s> <s xml:id="echoid-s17572" xml:space="preserve"> Hoc au-<lb/>tem facto, inueniet remotionem ſtellæ à polo mundi in aſcenſione, minorem remotione eius à po-<lb/>lo mundi in hora exiſtentiæ eius in uertice capitis.</s> <s xml:id="echoid-s17573" xml:space="preserve"> Ex quo patet, quòd uiſus comprehendit ſtellas <lb/>refractè, non rectè:</s> <s xml:id="echoid-s17574" xml:space="preserve"> Stella enim fixa ſemper mouetur per eundẽ circulũ de circulis æquidiſtantibus <lb/>æquatori, & nunquam exit ab ipſo, ita ut appareat, niſi in longiſsimo tempore.</s> <s xml:id="echoid-s17575" xml:space="preserve"> Et ſi ſtella compre-<lb/>henderetur rectè:</s> <s xml:id="echoid-s17576" xml:space="preserve"> tunc lineæ radiales extenderentur à uiſu rectè ad ſtellas, & extenderentur for-<lb/>mæ ſtellarum per lineas radiales rectè, quouſque peruenirent ad uiſum.</s> <s xml:id="echoid-s17577" xml:space="preserve"> Et ſi forma extendere-<lb/>tur à ſtella recte ad uiſum:</s> <s xml:id="echoid-s17578" xml:space="preserve"> tunc uiſus comprehenderet eam in ſuo loco:</s> <s xml:id="echoid-s17579" xml:space="preserve"> & ſic inueniret diſtantiam <lb/>ſtellæ fixæ à polo mundi in eadem nocte eandem:</s> <s xml:id="echoid-s17580" xml:space="preserve"> Sed diſtantia ſtellæ mutatur eadem nocte à po-<lb/>lo mundi:</s> <s xml:id="echoid-s17581" xml:space="preserve"> ergo uiſus non rectè comprehendit ſtellam.</s> <s xml:id="echoid-s17582" xml:space="preserve"> In cœlo autem non eſt corpus denſum ter-<lb/>ſum, nec in aere, à quo poſsint formæ reflecti.</s> <s xml:id="echoid-s17583" xml:space="preserve"> Et cum uiſus non comprehendat ſtellam rectè, <lb/>nec ſecundum reflexionem:</s> <s xml:id="echoid-s17584" xml:space="preserve"> ergo ſecundum refractionem, cùm his ſolis tribus modis compre-<lb/>hendantur res à uiſu [per 1 n 4.</s> <s xml:id="echoid-s17585" xml:space="preserve"> 1 n.</s> <s xml:id="echoid-s17586" xml:space="preserve">] Ex diuerſitate ergo diſtantiæ eiuſdem ſtellæ in eadem no-<lb/>cte à polo mundi, patet procul dubio, quòd uiſus comprehendat ſtellas refractè:</s> <s xml:id="echoid-s17587" xml:space="preserve"> Ergo corpus, <lb/>in quo ſunt ſtellæ fixæ, differt in diaphanitate ab aere.</s> <s xml:id="echoid-s17588" xml:space="preserve"> Præterea poteſt experimentari diapha-<lb/>nitas corporis cœli per experimentationem lunæ.</s> <s xml:id="echoid-s17589" xml:space="preserve"> Nam cum æquaueris locum lunæ in aliqua ho-<lb/>ra prope ortum eius, & pòſt in nocte nota, & in loco noto uerificaueris locum eius à polo mundi, <lb/> <pb o="252" file="0258" n="258" rhead="ALHAZEN"/> deinde poſueris inſtrumentum horarum in illa nocte ante ortum lunę, & ſciueris altitudinem lunę, <lb/>& obſeruaueris lunam uſq;</s> <s xml:id="echoid-s17590" xml:space="preserve"> ad ortum eius, & perueniat tempus in inſtrumento ad minutum idem <lb/>eiuſdem horæ, quod habet luna, & obſeruaueris altitudinem lunæ, quam habet in illa hora à uerti-<lb/>ce capitis, & obſeruaueris, ut inſtrumentum eleuationis ſit diuiſum per minuta, & per minora mi-<lb/>nutis, ſi poſsibile eſt:</s> <s xml:id="echoid-s17591" xml:space="preserve"> tunc inuenies diſtantiam lunæ à uertice capitis in illa hora per inſtrumen-<lb/>tum, minorem ſpatio remotionis à uertice capitis in illa hora per computationem.</s> <s xml:id="echoid-s17592" xml:space="preserve"> Ergo lux lunæ <lb/>non extenditur per duo foramina inſtrumenti, per quæ ſumpta eſt eleuatio rectè:</s> <s xml:id="echoid-s17593" xml:space="preserve"> tunc enim diſtan <lb/>tia eius à uertice capitis eſſet eadem cum illa, quę eſt inuenta per computationem:</s> <s xml:id="echoid-s17594" xml:space="preserve"> Sed diſtantia in-<lb/>uenta per computationem, differt à diſtantia per inſtrumentum.</s> <s xml:id="echoid-s17595" xml:space="preserve"> Ergo lux lunæ non extenditur à <lb/>cœlo ad aerem per lineas rectas:</s> <s xml:id="echoid-s17596" xml:space="preserve"> ergo ſecundum refractionem.</s> <s xml:id="echoid-s17597" xml:space="preserve"> Ex his ergo experimentationibus <lb/>patet, quòd uiſus comprehendit omnes ſtellas, quæ ſunt in cœlo refractè.</s> <s xml:id="echoid-s17598" xml:space="preserve"> Ergo uniuerſum cœlum <lb/>differt à diaphanitate aeris.</s> <s xml:id="echoid-s17599" xml:space="preserve"> Reſtat ergo declarare, quòd corpus cœli differt in ſubtilitate ab aere:</s> <s xml:id="echoid-s17600" xml:space="preserve"> & <lb/>hoc declarabitur per experimentationem prædictam.</s> <s xml:id="echoid-s17601" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div574" type="section" level="0" n="0"> <head xml:id="echoid-head498" xml:space="preserve" style="it">16. Cœlum rari{us} eſt aere & igne. 50 p 10.</head> <p> <s xml:id="echoid-s17602" xml:space="preserve">SIt ergo circulus meridiei in loco experimentationis circulus a b g:</s> <s xml:id="echoid-s17603" xml:space="preserve"> & zenith capitis b:</s> <s xml:id="echoid-s17604" xml:space="preserve"> & polus <lb/>mundi d:</s> <s xml:id="echoid-s17605" xml:space="preserve"> & centrum mundi e:</s> <s xml:id="echoid-s17606" xml:space="preserve"> & continuemus b cum e:</s> <s xml:id="echoid-s17607" xml:space="preserve"> & ſit locus uiſus z:</s> <s xml:id="echoid-s17608" xml:space="preserve"> & circulus æquidi-<lb/>ſtans æquinoctiali (cuius diſtantia à poli mundi eſt illa, in qua inuenitur ſtella in hora certifica <lb/>tionis diſtantię primæ) circulus h t:</s> <s xml:id="echoid-s17609" xml:space="preserve"> & ſit locus ſtellæ in illa hora h:</s> <s xml:id="echoid-s17610" xml:space="preserve"> & ſit circulus æquidiſtans æqui <lb/>noctiali (cuius diſtantia à polo eſt illa, in qua inuenitur ſtella in ſecunda hora) circulus k b:</s> <s xml:id="echoid-s17611" xml:space="preserve"> iſte ergo <lb/>circulus erit ille, in quo requieſcet ſtella ſecundum uerticationem.</s> <s xml:id="echoid-s17612" xml:space="preserve"> Nam cum ſtella fuerit in uertice <lb/>capitis, aut ualde prope:</s> <s xml:id="echoid-s17613" xml:space="preserve"> tunc uiſus comprehendet illam rectè:</s> <s xml:id="echoid-s17614" xml:space="preserve"> [per 13 n] quia linea recta, quæ tranſit <lb/>per uiſum & per uerticem capitis, eſt perpendicularis ſuper concauum ſphæræ cœli & perpendicu <lb/>laris ſuper conuexum aeris:</s> <s xml:id="echoid-s17615" xml:space="preserve"> & cum ſit perpendicularis ſuper utrumq;</s> <s xml:id="echoid-s17616" xml:space="preserve"> corpus:</s> <s xml:id="echoid-s17617" xml:space="preserve"> ergo uiſus compre-<lb/>hendit ſtellam, quæ eſt ſuper lineam hanc rectè, ſiue hæc duo corpora cœli & aeris fuerint diuerſæ <lb/>diaphanitatis, ſiue conſimilis.</s> <s xml:id="echoid-s17618" xml:space="preserve"> Cum ergo ſtella fuerit in uertice capitis, aut prope:</s> <s xml:id="echoid-s17619" xml:space="preserve"> uiſus comprehen-<lb/>dit illam in ſuo uero circulo æquidiſtante æquinoctiali, ſuper quẽ mouebatur ab initio noctis, quo-<lb/>uſq;</s> <s xml:id="echoid-s17620" xml:space="preserve"> peruenit ad circulum meridiei.</s> <s xml:id="echoid-s17621" xml:space="preserve"> Circulus ergo k b g eſt ille, in quo erat ſtella in experimentatio-<lb/>ne prima:</s> <s xml:id="echoid-s17622" xml:space="preserve"> & ſit circulus uerticationis, qui tranſit per ſtellam in hora experimentationis primæ cir-<lb/>culus b h k:</s> <s xml:id="echoid-s17623" xml:space="preserve"> & ſecet ille circulus circulum k b g in puncto k, & circulum h t in puncto h.</s> <s xml:id="echoid-s17624" xml:space="preserve"> Et quia di-<lb/>ſtantia ſtellę à polo mundi fuit in prima experimen <lb/> <anchor type="figure" xlink:label="fig-0258-01a" xlink:href="fig-0258-01"/> tatione minor, quàm in ſecũda:</s> <s xml:id="echoid-s17625" xml:space="preserve"> erit circulus h t pro <lb/>pinquior polo, circulo k b g:</s> <s xml:id="echoid-s17626" xml:space="preserve"> ergo punctũ h eſt pro-<lb/>pinquius zenith capitis, quàm punctum k:</s> <s xml:id="echoid-s17627" xml:space="preserve"> & conti <lb/>nuemus duas lineas h z, k z.</s> <s xml:id="echoid-s17628" xml:space="preserve"> Quia ergo ſtella com-<lb/>prehenditur à uiſu in hora experimentationis pri-<lb/>mę in puncto h:</s> <s xml:id="echoid-s17629" xml:space="preserve"> & tunc erat in ſuperficie circuli b h <lb/>k uerticalis:</s> <s xml:id="echoid-s17630" xml:space="preserve"> & ſtella erat in illa hora in circumferen <lb/>tia k b g:</s> <s xml:id="echoid-s17631" xml:space="preserve"> ergo ſtella erat in illa hora in puncto k:</s> <s xml:id="echoid-s17632" xml:space="preserve"> & <lb/>comprehenditur à uiſu in puncto h, & per rectitudi <lb/>nem lineæ z h:</s> <s xml:id="echoid-s17633" xml:space="preserve"> uiſus enim nihil comprehendit, niſi <lb/>per uerticationes linearum radialium, per quas for <lb/>mæ perueniunt ad uiſum.</s> <s xml:id="echoid-s17634" xml:space="preserve"> Viſus ergo cõprehendit <lb/>ſtellam in puncto h:</s> <s xml:id="echoid-s17635" xml:space="preserve"> quia forma peruenit ad illũ in <lb/>rectitudine lineæ h z.</s> <s xml:id="echoid-s17636" xml:space="preserve"> Et cum uiſus cõprehendat illam in rectitudine h z:</s> <s xml:id="echoid-s17637" xml:space="preserve"> & linea recta, quæ eſt inter <lb/>ſtellam & uiſum, ſit linea k z:</s> <s xml:id="echoid-s17638" xml:space="preserve"> manifeſtum eſt ergo, quòd uiſus non comprehendit ſtellam, quæ eſt in <lb/>puncto k rectè:</s> <s xml:id="echoid-s17639" xml:space="preserve"> ergo refractè.</s> <s xml:id="echoid-s17640" xml:space="preserve"> Sit ergo locus refractionis m:</s> <s xml:id="echoid-s17641" xml:space="preserve"> & continuemus k m:</s> <s xml:id="echoid-s17642" xml:space="preserve"> & protrahamus ab <lb/>m rectã uſq;</s> <s xml:id="echoid-s17643" xml:space="preserve"> ad z.</s> <s xml:id="echoid-s17644" xml:space="preserve"> Forma ergo ſtellæ, quæ peruenit ad z, ex qua uiſus comprehendit ſtellam:</s> <s xml:id="echoid-s17645" xml:space="preserve"> extendi <lb/>tur à ſtella perlineam k m, & refringitur per lineam m z:</s> <s xml:id="echoid-s17646" xml:space="preserve"> & non refringuntur formæ, niſi cum occur-<lb/>rit corpus diuerſæ diaphanitatis à diaphanitate corporis, in quo exiſtit.</s> <s xml:id="echoid-s17647" xml:space="preserve"> Ergo corpus, in quo eſt ſtel <lb/>la, ſcilicet cœlum, eſt diaphanũ differens in diaphanitate ab aere.</s> <s xml:id="echoid-s17648" xml:space="preserve"> Et quia locus refractionis eſt apud <lb/>ſuperficiem, quæ tranſit in duo corpora, quæ differunt in diaphanitate:</s> <s xml:id="echoid-s17649" xml:space="preserve"> punctum ergo m eſt pun-<lb/>ctum in concauitate cœli.</s> <s xml:id="echoid-s17650" xml:space="preserve"> Et continuemus lineam inter e, m:</s> <s xml:id="echoid-s17651" xml:space="preserve"> & ſit diameter ſphærę cœli:</s> <s xml:id="echoid-s17652" xml:space="preserve"> erit ergo li-<lb/>nea e m perpendicularis ſuper ſuperficiem cœli concauam contingentem aerem, & ſuper ſuperfi-<lb/>ciem aeris conuexam:</s> <s xml:id="echoid-s17653" xml:space="preserve"> [ut demonſtratum eſt 25 n 4.</s> <s xml:id="echoid-s17654" xml:space="preserve">] Et cum forma ſtellæ, quæ eſt in puncto k, exten <lb/>datur per lineam m k, & refringatur in aere per lineam m z:</s> <s xml:id="echoid-s17655" xml:space="preserve"> patet, quòd hæc refractio eſt ad lineam, <lb/>in qua eſt perpendicularis e m, quæ tranſit per punctum refractionis, quæ eſt perpendicularis ſu-<lb/>per ſuperficiem aeris.</s> <s xml:id="echoid-s17656" xml:space="preserve"> Et cum refractio in aere ſit ad partem perpendicularis exeuntis à loco refra-<lb/>ctionis:</s> <s xml:id="echoid-s17657" xml:space="preserve"> ergo corpus aeris eſt groſsius corpore cœli.</s> <s xml:id="echoid-s17658" xml:space="preserve"> Patet ergo, quòd hoc, qùod inuenimus per ex-<lb/>perimentationẽ ſtellarũ, ſignificat demõſtratiuè, quòd uiſus nõ comprehendit ſtellas, niſi refractè:</s> <s xml:id="echoid-s17659" xml:space="preserve"> <lb/>& quòd corpus aeris eſt groſsius corpore cœli:</s> <s xml:id="echoid-s17660" xml:space="preserve"> & quòd corpus cœli eſt ſubtilius corpore aeris.</s> <s xml:id="echoid-s17661" xml:space="preserve"> Ex <lb/>his ergo omnibus patet, quòd omnia, quę cõprehendũtur à uiſu ultra corpora diaphana, quorũ dia <lb/>phanitas differt à diaphanitate aeris (ſi uiſus fuerit obliquus à perpendicularibus egredientibus ex <lb/>ipſis ſuper ſuperficiem diaphanorum corporum, in quibus conſiſtunt) comprehenduntur refractè.</s> <s xml:id="echoid-s17662" xml:space="preserve"/> </p> <div xml:id="echoid-div574" type="float" level="0" n="0"> <figure xlink:label="fig-0258-01" xlink:href="fig-0258-01a"> <variables xml:id="echoid-variables205" xml:space="preserve">k h b m z d e a t i g</variables> </figure> </div> <pb o="253" file="0259" n="259" rhead="OPTICAE LIBER VII."/> </div> <div xml:id="echoid-div576" type="section" level="0" n="0"> <head xml:id="echoid-head499" xml:space="preserve">DE IMAGINIBVS. CAP. V.</head> <head xml:id="echoid-head500" xml:space="preserve" style="it">17. Imago (quæ eſt forma refracti uiſibilis à medio diuerſo) extra uiſibilis locum uidetur. <lb/>in defin. 11 p 10.</head> <p> <s xml:id="echoid-s17663" xml:space="preserve">IMago eſt forma rei uiſibilis, quam uiſus comprehendit ultra diaphanum corpus, quod differt in <lb/>ſua diaphanitate à diaphanitate aeris, cum uiſus fuerit obliquus à perpendicular b.</s> <s xml:id="echoid-s17664" xml:space="preserve"> exeuntib.</s> <s xml:id="echoid-s17665" xml:space="preserve"> ab <lb/>illo uiſibili ad ſuperficiem illius corporis diaphani.</s> <s xml:id="echoid-s17666" xml:space="preserve"> Nam forma, quam cõprehendit uiſus in cor-<lb/>pore diaphano de re uiſa, quæ eſt ultra ipſum corpus, non eſt ipſa res uiſa:</s> <s xml:id="echoid-s17667" xml:space="preserve"> quoniam uiſus tunc non <lb/>comprehendit rem uiſam in ſuo loco, neque in ſua forma, ſed in alio loco & in alio modo, ſcilicet re <lb/>fracte:</s> <s xml:id="echoid-s17668" xml:space="preserve"> & cum hoc comprehendit illam rem in ſua oppoſitione:</s> <s xml:id="echoid-s17669" xml:space="preserve"> hęc autem forma dicitur imago.</s> <s xml:id="echoid-s17670" xml:space="preserve"> Hoc <lb/>autem comprehenditur ratione & experientia.</s> <s xml:id="echoid-s17671" xml:space="preserve"> Ratione, quoniam ex prędicto capitulo patet, quòd <lb/>uiſum, quod eſt in diaphano corpore diuerſę diaphanitatis ab aere, comprehenditur à uiſu refractè, <lb/>cum uiſus fuerit decliuis à perpendicularib.</s> <s xml:id="echoid-s17672" xml:space="preserve"> exeuntibus à re uiſa ſuper ſuperficiem corporis diapha <lb/>ni.</s> <s xml:id="echoid-s17673" xml:space="preserve"> Et cum uiſus comprehendit huiuſmodi uiſum refractè, nec eſt in oppoſitione eius, non cõprehẽ <lb/>dit ipſum rectè, nec ſentit ſe comprehẽdere ipſum refractè:</s> <s xml:id="echoid-s17674" xml:space="preserve"> patet, quòd comprehendit ipſum extra <lb/>ſuum locum.</s> <s xml:id="echoid-s17675" xml:space="preserve"> Per experientiam uero ſic poteſt cognoſci.</s> <s xml:id="echoid-s17676" xml:space="preserve"> Nam ſi aliquis acceperit uas habens oras <lb/>erectas perpendiculares, in cuius medio poſuerit aliquod uiſum manifeſtum, ut obolum aut dena-<lb/>rium, & ſteterit à longè, quouſq;</s> <s xml:id="echoid-s17677" xml:space="preserve"> uiderit rem uiſam in profundo uaſis:</s> <s xml:id="echoid-s17678" xml:space="preserve"> deinde elongauerit ſe à re ui-<lb/>ſa, quouſque non uideat rem paulatim:</s> <s xml:id="echoid-s17679" xml:space="preserve"> tunc in initio occultationis ſtet in ſuo loco, & præcipiat alte <lb/>ri infundere aquam in uas ipſo exiſtente in ſuo loco, nec moueat uiſum, nec mutet ſitum:</s> <s xml:id="echoid-s17680" xml:space="preserve"> tunc enim <lb/>cum aſpexerit aquam, quæ eſt in uaſe:</s> <s xml:id="echoid-s17681" xml:space="preserve"> uidebit rem uiſam, poſtquam non uiderat eam, & uidebit eã <lb/>in eius oppoſitione.</s> <s xml:id="echoid-s17682" xml:space="preserve"> Ex quo patet, quòd forma, quam uidet in aqua, nõ eſt in loco uiſi.</s> <s xml:id="echoid-s17683" xml:space="preserve"> Nam ſi forma <lb/>eſſet in loco uiſi:</s> <s xml:id="echoid-s17684" xml:space="preserve"> tunc uiſus comprehẽderet rem uiſam non exiſtente aqua in uaſe:</s> <s xml:id="echoid-s17685" xml:space="preserve"> uiſus enim in ſe-<lb/>cundo ſtatu comprehendit rem uiſam in ſua oppoſitione, ipſa non exiſtẽte in ſua oppoſitione.</s> <s xml:id="echoid-s17686" xml:space="preserve"> Hoc <lb/>ergo modo declarabitur utroque modo, ratione uidelicet & experientia, quòd imago rei uiſæ, quã <lb/>uiſus comprehendit refractè, non eſt in loco rei uiſæ.</s> <s xml:id="echoid-s17687" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div577" type="section" level="0" n="0"> <head xml:id="echoid-head501" xml:space="preserve" style="it">18. Imago uideturin concurſu linearum refractionis, & perpendicularis incidentiæ. 15 p 10.</head> <p> <s xml:id="echoid-s17688" xml:space="preserve">DEinde dico, quòd imago cuiuslibet puncti, quod uiſus comprehendit refractè, eſt in puncto, <lb/>quod eſt differentia communis lineæ, per quam forma peruenit ad uiſum, & perpendicula-<lb/>ri, exeunti ab illo puncto uiſo ſuper ſuperficiem diaphani corporis.</s> <s xml:id="echoid-s17689" xml:space="preserve"> Hoc autem declarabitur <lb/>per experientiam hoc modo.</s> <s xml:id="echoid-s17690" xml:space="preserve"> Accipiat aliquis circulum ligneum, cuius diam eter non ſit minor uno <lb/>cubito, altitudo duorum ueltrium digitorum, & adęquet ſuperficies eius quantumcunque poterit:</s> <s xml:id="echoid-s17691" xml:space="preserve"> <lb/>& inueniat centrum eius, & extrahat in ipſo diametros ſeſe interſecantes quomodocunque uolue-<lb/>rit, & ſignentur ferro, ut appareant, & impleat lineas illas corpore albo, ut ceruſa mixta lacte:</s> <s xml:id="echoid-s17692" xml:space="preserve"> & pun <lb/>ctum centri ſit nigrum.</s> <s xml:id="echoid-s17693" xml:space="preserve"> Hoc autem perfecto, accipiat uas amplum, ut peluim habens oras eleuatas, <lb/>& ponat uas in loco luminoſo, & infundat in uas aquam claram, & ſit altitudo aquæ minor diame-<lb/>tro circuli, & maior ſemidiam etro eius, & menſuretur hoc ipſo circulo, quouſque aqua tranſeat cen <lb/>trum circuli aliquot digitis, duabus ſcilicet diametris aut pluribus ſignatis in ipſo uaſe, ſcilicet, ut <lb/>ſit aqua cooperiens aliquam partem utriuſque diametri, & remaneat altera pars extra aquã, & ex-<lb/>pectet, donec aqua quieſcat in uaſe, & tunc mittat circulum ligneum in uas, & erigat circulum ſu-<lb/>per oram ipſius, & ponat ſuperficiem ipſius, in qua ſunt lineæ ſignatæ, ex parte uiſus:</s> <s xml:id="echoid-s17694" xml:space="preserve"> deinde moue-<lb/>at circulum, donec aliqua ſuarum diametrorum ſit perpendicularis ſuper ſuperficiem aquæ:</s> <s xml:id="echoid-s17695" xml:space="preserve"> dein-<lb/>de dimittat uiſum ſuum, & erigat uas, quouſque uiſus ſimul appropinquet æquidiſtantiæ ſuperfi-<lb/>ciei aquæ, & extra oram uaſis, & ſupra ſuperficiem aquæ in tantùm, ut poſsit uidere centrum circu-<lb/>li:</s> <s xml:id="echoid-s17696" xml:space="preserve"> experientia enim ſecundum hunc modum erit manifeſtior.</s> <s xml:id="echoid-s17697" xml:space="preserve"> Hoc ergo facto, intueatur centrũ cir-<lb/>culi & diametrum circuli perpendicularem ſuper ſuperficiem aquæ:</s> <s xml:id="echoid-s17698" xml:space="preserve"> tunc enim inueniet centrum <lb/>circuli in rectitudine diame<gap/>ri perpendicularis.</s> <s xml:id="echoid-s17699" xml:space="preserve"> Deιnde intueatur diametrum circuli decliuem, cu-<lb/>ius pars eminet ſupra aquam:</s> <s xml:id="echoid-s17700" xml:space="preserve"> tunc enim inueniet ipſam incuruatam:</s> <s xml:id="echoid-s17701" xml:space="preserve"> cuius incuruatio erit apud ſu-<lb/>perficiem aquæ:</s> <s xml:id="echoid-s17702" xml:space="preserve"> & illa pars, quæ eſt intra aquam, continet cum illa, quæ eſt extra aquam, angulum <lb/>obtuſum:</s> <s xml:id="echoid-s17703" xml:space="preserve"> & inueniet angulum ex parte diametri perpendicularis:</s> <s xml:id="echoid-s17704" xml:space="preserve"> & inueniet illud, quod eſt intra <lb/>aquam, rectum & continuum.</s> <s xml:id="echoid-s17705" xml:space="preserve"> Ex quo patet, quòd forma puncti, quod eſt centrum circuli, ſcilicet <lb/>forma, quam uiſus comprehendit, non eſt apud centrum circuli.</s> <s xml:id="echoid-s17706" xml:space="preserve"> Nam ſi eſſet apud centrum circu-<lb/>li:</s> <s xml:id="echoid-s17707" xml:space="preserve"> tunc eſſet in rectitudine diametri decliu<gap/>s:</s> <s xml:id="echoid-s17708" xml:space="preserve"> nam in rei ueritate talem habet ſitum.</s> <s xml:id="echoid-s17709" xml:space="preserve"> Cum ergo uiſus <lb/>comprehendit hoc punctum extra rectitudinem diametri decliuis, & anguli, quem continent par-<lb/>tes diametri decliuis, ſequuntur diametrum perpendicularem:</s> <s xml:id="echoid-s17710" xml:space="preserve"> tunc punctum, quod eſt forma cen-<lb/>tri, eſt eleuatum à centro.</s> <s xml:id="echoid-s17711" xml:space="preserve"> Et quia uiſus comprehendit hoc punctum in rectitudine diametri, per-<lb/>pendicularis ſuper ſuperficiem aquæ:</s> <s xml:id="echoid-s17712" xml:space="preserve"> erit hoc punctum, quod eſt forma puncti, quod eſt in centro, <lb/>eleuatum à centro:</s> <s xml:id="echoid-s17713" xml:space="preserve"> & cum hoc, eſt in rectitudine perpendicularis, exeuntis à centro ſuper ſuperfi-<lb/>ciem aquæ.</s> <s xml:id="echoid-s17714" xml:space="preserve"> Et declarabitur ex incuruatione diametri decliuis apud ſuperficiem aquæ, & rectitu-<lb/>dine eius, quod eſt ntra aquam ex diametro, & continuatione eius:</s> <s xml:id="echoid-s17715" xml:space="preserve"> quòd omne punctum partis, <lb/>quæ eſt intra aquã ex diametro decliui, eſt eleuatum à ſuo loco.</s> <s xml:id="echoid-s17716" xml:space="preserve"> Deinde oportet experimẽtatorem <lb/>reuoluere cιrculum ligneum, quouſque diameter decliuis fiat perpendicularis ſuper ſuperficiem <lb/> <pb o="254" file="0260" n="260" rhead="ALHAZEN"/> aquæ, & diameter, quæ erat perpendicularis, fiat decliuis:</s> <s xml:id="echoid-s17717" xml:space="preserve"> deinde dimittat uiſum ſuum, & intueatur <lb/>centrum:</s> <s xml:id="echoid-s17718" xml:space="preserve"> & tunc inueniet formam centri in rectitudine diametri, quæ nunc eſt perpendicularis ſu-<lb/>per ſuperficiem aquę, extra cuius rectitudinem erat forma centri, quando erat decliuis:</s> <s xml:id="echoid-s17719" xml:space="preserve"> & inueniet <lb/>formam extra rectitudinem diametri, quæ eſt nunc decliuis, quæ prius erat perpendicularis ſuper <lb/>ſuperficiem aquæ:</s> <s xml:id="echoid-s17720" xml:space="preserve"> & inueniet diametrum decliuem incuruatam apud ſuperficiem aquæ:</s> <s xml:id="echoid-s17721" xml:space="preserve"> & angulus <lb/>incuruationis erit ex parte diametri decliuis.</s> <s xml:id="echoid-s17722" xml:space="preserve"> Et ſi fuerint in circulo plures diametri, & reuoluerit <lb/>experimentator circulum, quouſque unaquęque earum fuerit perpendicularis ſuper ſuperficiem <lb/>aquæ ſucceſsiuè, & fuerit diameter, quæ ſequitur illam diametrum, decliuis, & aliqua pars eius fue-<lb/>rit extra aquam:</s> <s xml:id="echoid-s17723" xml:space="preserve"> tunc inueniet formam puncti, quod eſt centrum circuli, ſemper in rectitudine dia-<lb/>metri perpendicularis, & eleuatam à rectitudine diametri decliuis, & ſemper inueniet illud, quod <lb/>eſt intra aquam, rectum.</s> <s xml:id="echoid-s17724" xml:space="preserve"> Ex omnibus ergo iſtis patet, quòd forma cuiuslibet puncti comprehenſi à <lb/>uiſu in corpore diaphano groſsiore corpore aeris:</s> <s xml:id="echoid-s17725" xml:space="preserve"> comprehenditur extra ſuum locum & eleuatum <lb/>à ſuo loco, & in rectitudine perpendicularis exeuntis ab illo puncto ſuper ſuperficiem corporis dia <lb/>phani:</s> <s xml:id="echoid-s17726" xml:space="preserve"> cum linea, quæ continuat centrum uiſus cum illo puncto, non fuerit perpendicularis ſuper <lb/>ſuperficiem corporis diaphani:</s> <s xml:id="echoid-s17727" xml:space="preserve"> omne autem punctum comprehenditur à uiſu in eius oppoſitio-<lb/>ne, & in rectitudine lineæ rectæ, per quam extenditur forma ad uiſum, [per 19.</s> <s xml:id="echoid-s17728" xml:space="preserve"> 21.</s> <s xml:id="echoid-s17729" xml:space="preserve"> 38 n 1.</s> <s xml:id="echoid-s17730" xml:space="preserve"> 13 n.</s> <s xml:id="echoid-s17731" xml:space="preserve">] Pun-<lb/>cta ergo, quę comprehendit uiſus refractè, comprehenduntur in eius oppoſitione, & in rectitudine <lb/>lineæ rectę, per quam forma peruenit ad uiſum.</s> <s xml:id="echoid-s17732" xml:space="preserve"> Hoc autem declarabitur per experimentationem <lb/>comprehenſionis rerum uiſibilium ſecundum refractionem per illud inſtrumentum prædictum.</s> <s xml:id="echoid-s17733" xml:space="preserve"> <lb/>Nam ſi experimentator clauſerit ſecundum foramen, quod eſt in inſtrumento:</s> <s xml:id="echoid-s17734" xml:space="preserve"> tunc non compre-<lb/>hendet rem uiſam, quam comprehendebat ſecundum refractionem:</s> <s xml:id="echoid-s17735" xml:space="preserve"> & cum clauſerit ſecundum fo-<lb/>ramen, nihil aliud facit, niſi ſecare lineam rectam imaginabilem, quæ exit à centro uiſus ad locum <lb/>refractionis.</s> <s xml:id="echoid-s17736" xml:space="preserve"> Ex quo patet, quòd forma, quæ extenditur à uiſu in corpore diaphano, in quo res ui-<lb/>ſa eſt, & refringitur in corpore diaphano, in quo eſt uiſus:</s> <s xml:id="echoid-s17737" xml:space="preserve"> extenditur per lineam rectam, quæ exit à <lb/>centro uiſus ad locum refractionis:</s> <s xml:id="echoid-s17738" xml:space="preserve"> & quod omne punctum, quod comprehenditur à uiſu in corpo <lb/>re diaphano magis groſſo, quàm ſit corpus aeris (ſi centrum uiſus fuerit extra perpendicularem, <lb/>exeuntem ab illo puncto ſuper corpus diaphanum) comprehenditur in puncto, quod eſt differen-<lb/>tia communis lineę, ſuper quam peruenit forma ad uiſum, & perpendiculari, exeunti à puncto ui-<lb/>ſo ſuper ſuperficiem corporis diaphani, quod eſt ex parte uiſus.</s> <s xml:id="echoid-s17739" xml:space="preserve"> Si autem experimentator uolue-<lb/>rit experiri imaginem rei uiſæ, cuius forma refrin-<lb/> <anchor type="figure" xlink:label="fig-0260-01a" xlink:href="fig-0260-01"/> gitur à corpore ſubtiliore ad corpus groſsius:</s> <s xml:id="echoid-s17740" xml:space="preserve"> acci-<lb/>piat fruſtum uitri, cuius ſuperficies ſint æquatæ & <lb/>æquidiſtantes, habens in longitudine octo digi-<lb/>tos, & in altitudine quatuor, & in ſpiſsitudine qua-<lb/>tuor:</s> <s xml:id="echoid-s17741" xml:space="preserve"> & accipiat circulum ligneum prædictum, & <lb/>ſignet in dorſo eius chordam in longitudine decẽ <lb/>digitorum, & diuidat illam in duo æqualia, & con-<lb/>tinuet locum diuiſionis cum cẽtro cιrculi linea re <lb/>cta, quæ tranſeat in utram que partem:</s> <s xml:id="echoid-s17742" xml:space="preserve"> hæc ergo li <lb/>nea erit perpẽdicularis ſuper lineam primam [per <lb/>3 p 3.</s> <s xml:id="echoid-s17743" xml:space="preserve">] Deinde continuet alteram extremitatem <lb/>chordæ cum centro circuli linea recta, quæ etiam <lb/>tranſeat in utramque partem.</s> <s xml:id="echoid-s17744" xml:space="preserve"> Et hæ duæ diame-<lb/>tri ſint ſignatę ferro, quarum alteram impleat cor-<lb/>pore albo, & aliam alterius modi colore.</s> <s xml:id="echoid-s17745" xml:space="preserve"> Deinde <lb/>ponat uitrum longum ſuper dorſum inſtrumenti <lb/>circuli lignei, & ſuperponat alteram extremitatem <lb/>longitudinis e<gap/>us medietati chordæ, & diſtinguat de uitro tres digitos, ex quibus duo erunt ex par <lb/>te diametri decliuis extra circulum, & remanebit de longitudine uitri unus digitus:</s> <s xml:id="echoid-s17746" xml:space="preserve"> qui erit ultra <lb/>diametrum perpendicularem ſuper chordam:</s> <s xml:id="echoid-s17747" xml:space="preserve"> & ſit corpus uitri ex parte centri:</s> <s xml:id="echoid-s17748" xml:space="preserve"> & applicet uitrum <lb/>ſecundum hunc ſitum circulo ligneo applicatione fixa.</s> <s xml:id="echoid-s17749" xml:space="preserve"> Sic ergo diameter perpendicularis ſuper <lb/>chordam, erit perpendicularis ſuper extremitates uitri ęquidiſtantes, & altera diameter erit decli-<lb/>uis ſuper has duas ſuperficies.</s> <s xml:id="echoid-s17750" xml:space="preserve"> Deinde oportet, ut experimentator ponat oram circuli, in qua eſt <lb/>extremitas uitri eminens ex parte ſui uiſus, & ponat alterum uiſum in differentia communi circũ-<lb/>ferentiæ & extremitati uitri, quæ eſt extremitas diametri decliuis, & appropinquet uiſum ſuum ui-<lb/>tro, quantum poterit, ita, ut non poſsit per illum uidere ex ſuperficie aliquid, pręter extremitatem <lb/>diametri decliuis:</s> <s xml:id="echoid-s17751" xml:space="preserve"> reliquus autem uiſus ſit in parte, in qua eſt uitrum & circulus:</s> <s xml:id="echoid-s17752" xml:space="preserve"> deinde cooperiat <lb/>illud, quod opponitur alteri uiſui ex ſuperficie uitri cum bombace:</s> <s xml:id="echoid-s17753" xml:space="preserve"> quam applicet ſuper aliquam <lb/>partem uitri, ita ut comprehendat diametrum decliuem, quæ eſt ultima linea per unum uiſum, qui <lb/>contingit uitrum:</s> <s xml:id="echoid-s17754" xml:space="preserve"> & non uideat ultra hanc lineam, & uideat lineam albam perpendicularem utro-<lb/>que uiſu.</s> <s xml:id="echoid-s17755" xml:space="preserve"> Ipſo autem exiſtente in hoc ſitu, intueatur centrum circuli, & inueniet illud in rectitudi-<lb/>nelineæ albæ, quę eſt perpendicularis ſuper ſuperficiem uitri:</s> <s xml:id="echoid-s17756" xml:space="preserve"> & intueatur diametrum decliuem, <lb/>apud cuius extremitatem tenet uiſum ſuum:</s> <s xml:id="echoid-s17757" xml:space="preserve"> & tunc uidebit eam incuruatam apud ſuperficiem ui-<lb/>tri, quæ eſt ex parte centri, & inueniet angulum incuruationis ex parte circumferentiæ:</s> <s xml:id="echoid-s17758" xml:space="preserve"> uiſus au-<lb/> <pb o="255" file="0261" n="261" rhead="OPTICAE LIBER VII."/> tem comprehendet partem huius diametri decliuis, quæ eſt ſub uitro in rectitudine.</s> <s xml:id="echoid-s17759" xml:space="preserve"> Et quia uiſus <lb/>tangit ſuperficiem uitri, & diametri perpendicularis una pars eſt ſub uitro, alia extra uitrum ex par <lb/>te centri, altera extra uitrum ex parte extremitatis diametri:</s> <s xml:id="echoid-s17760" xml:space="preserve"> pars igitur, quæ ſub uitro eſt, compre-<lb/>henditur à uiſu extra uitrum ſecundum refraction em:</s> <s xml:id="echoid-s17761" xml:space="preserve"> & pars, quæ eſt parte extremitatis diametri, <lb/>comprehenditur à uiſu extra uitrum:</s> <s xml:id="echoid-s17762" xml:space="preserve"> qui uiſus eſt extra uitrum rectè & ſine refractione:</s> <s xml:id="echoid-s17763" xml:space="preserve"> pars au-<lb/>tem quæ eſt ex parte centri, comprehenditur ab utroque uiſu ſecundum refractionem.</s> <s xml:id="echoid-s17764" xml:space="preserve"> Nam lineę, <lb/>quæ exeunt à centro uiſus contingentis uitrum, & extenduntur in corpore uitri, quando perue-<lb/>niunt ad ſuperficiem uitri, quæ eſt ex parte extremitatis centri, omnes erunt decliues ſuper ſuper-<lb/>ficiem uitri.</s> <s xml:id="echoid-s17765" xml:space="preserve"> Pars ergo, quæ eſt ex parte centri ex diametro perpendicularis, comprehenditur à uiſu <lb/>contingente uitrum ſecundum refractionem.</s> <s xml:id="echoid-s17766" xml:space="preserve"> Lineæ uerò, quæ exeunt à reliquo uiſu ad ſuperio-<lb/>rem ſuperficiem uitri, erunt decliues ſuper ſuperficiem uitri ſuperiorem:</s> <s xml:id="echoid-s17767" xml:space="preserve"> & cum extenduntur ſu-<lb/>per ſuperficiem aliam uitri, quæ eſt ex parte centri, erunt etiam decliues:</s> <s xml:id="echoid-s17768" xml:space="preserve"> reliquus ergo uiſus com-<lb/>prehendit partem diametri perpendicularis, quæ eſt ex parte centri, duabus refractionibus:</s> <s xml:id="echoid-s17769" xml:space="preserve"> par-<lb/>tem autem, quæ eſt ſub uitro, una ſola refractione:</s> <s xml:id="echoid-s17770" xml:space="preserve"> & cum hoc toto, uiſus comprehendit hanc dia-<lb/>metrum rectam.</s> <s xml:id="echoid-s17771" xml:space="preserve"> Et ſi experimentator cooperuerit alterum uiſum, & aſpexerit per uiſum, qui ex <lb/>parte uitri:</s> <s xml:id="echoid-s17772" xml:space="preserve"> comprehendet perpendicularem rectam.</s> <s xml:id="echoid-s17773" xml:space="preserve"> Et ſi eleuauerit uiſum ſuum à uitro, & intu-<lb/>ens ſuerit diametrum perpendicularem ultra uitrum:</s> <s xml:id="echoid-s17774" xml:space="preserve"> comprehendet ipſam rectam, cum hoc, quòd <lb/>comprehendit ipſam ſecundum refractionem.</s> <s xml:id="echoid-s17775" xml:space="preserve"> Cauſſa autem huius eſt, quòd omne punctum dia-<lb/>metri perpendicularis, quando comprehenditur à uiſu ſecundum refractionem, comprehenditur <lb/>non in ſuo loco, ſed tamen comprehenditur in loco, qui eſt in rectitudine perpendicularis, quæ <lb/>exit ab illo ſuper ſuperficiem uitri:</s> <s xml:id="echoid-s17776" xml:space="preserve"> & iſta diameter eſt perpendicularis, quæ exit à quolibet puncto <lb/>eius ad ſuperficiem uitri:</s> <s xml:id="echoid-s17777" xml:space="preserve"> & nullum punctum comprehenditur refractè, niſi ſuper ipſam.</s> <s xml:id="echoid-s17778" xml:space="preserve"> Cum er-<lb/>go uiſus comprehendit hanc diametrum rectam, & comprehendit formam centri in rectitudine hu <lb/>ius diametri:</s> <s xml:id="echoid-s17779" xml:space="preserve"> forma centri, quam uiſus comprehendit ultra uitrum, quando uiſus tangit uitrum, eſt <lb/>in rectitudine perpendicularis exeuntis à centro ſuper ſuperficiẽ uitri.</s> <s xml:id="echoid-s17780" xml:space="preserve"> Et cum cõprehenderit dia-<lb/>metrũ decliuem incuruatam:</s> <s xml:id="echoid-s17781" xml:space="preserve"> cõprehendet partem eius, quæ exit à centro, quæ eſt ex parte centri, <lb/>non in ſuo loco:</s> <s xml:id="echoid-s17782" xml:space="preserve"> & punctum centri non comprehenditur à uiſu, niſi præter fuuum locum.</s> <s xml:id="echoid-s17783" xml:space="preserve"> Et cum <lb/>angulus incuruationis fuerit ex parte circumferentiæ:</s> <s xml:id="echoid-s17784" xml:space="preserve"> tunc punctum, quod eſt forma centri, eſt <lb/>ſub centro.</s> <s xml:id="echoid-s17785" xml:space="preserve"> Ex quo patet, quòd imago cuiuslibet puncti comprehenſi à uiſu ultra corpus diapha-<lb/>num, ſubtilius corpore diaphano, quod eſt in parte uiſus, eſt in rectitudine lineæ, quæ exit ab illo <lb/>puncto, perpendicularis ſuper ſuperficiem corporis diaphani, quod eſt in parte uiſus:</s> <s xml:id="echoid-s17786" xml:space="preserve"> & eſt remo-<lb/>tior à ſuperficie corporis diaphani, quod eſt in parte uiſus, quàm ipſum punctum.</s> <s xml:id="echoid-s17787" xml:space="preserve"> Et omne pun-<lb/>ctum comprehenſum à uiſu, eſt in rectitudine lineæ, per quam forma peruenit ad uiſum.</s> <s xml:id="echoid-s17788" xml:space="preserve"> Et imago <lb/>cuiuslibet puncti comprehenſi à uiſu ultra corpus diaphanum, ſubtilius corpore diaphano, quod <lb/>eſt ex parte uiſus, eſt in differentia communi lineæ, per quam forma peruenit ad uiſum, & perpen-<lb/>diculari, quæ exit à puncto uiſo ſuper ſuperficiem corporis diaphani, quod eſt ex parte uiſus.</s> <s xml:id="echoid-s17789" xml:space="preserve"> Ex <lb/>omnibus ergo iſtis declaratis in hoc capitulo patet, quòd imago cuiuslibet puncti uiſi, comprehen-<lb/>ſi à uiſu ultra corpus diaphanum diuerſæ diaphanitatis à diaphanitate corporis, quod eſt in parte <lb/>uiſus (cum uiſus fuerit decliuis à perpendicularib exeuntibus ab illa reſuper ſuperficiem corporis <lb/>diaphani, quod eſt in parte uiſus) eſt in differentia communilineæ, per quam forma illius pun-<lb/>cti peruenit ad uiſum, & perpendiculari, quæ exit ab illo puncto ſuper ſuperficiem corporis dia-<lb/>phani, quod eſt in parte uiſus:</s> <s xml:id="echoid-s17790" xml:space="preserve"> ſiue corpus diaphanum, quod eſt in parte uiſus, ſit ſubtilius corpo-<lb/>re diaphano, quod eſt in parte rei uiſæ:</s> <s xml:id="echoid-s17791" xml:space="preserve"> ſiue groſsius.</s> <s xml:id="echoid-s17792" xml:space="preserve"> Quare autem uiſus comprehendat rem ui-<lb/>ſam in loco imaginis, & quare imago ſit in loco ſectionis inter lineam, per quam forma peruenit <lb/>ad uiſum, & inter perpendicularem, quæ exit à puncto uiſo ad ſuperficiem corporis diaphani, po-<lb/>ſtea dicetur.</s> <s xml:id="echoid-s17793" xml:space="preserve"/> </p> <div xml:id="echoid-div577" type="float" level="0" n="0"> <figure xlink:label="fig-0260-01" xlink:href="fig-0260-01a"> <variables xml:id="echoid-variables206" xml:space="preserve">h m k o n q e f p g i</variables> </figure> </div> </div> <div xml:id="echoid-div579" type="section" level="0" n="0"> <head xml:id="echoid-head502" xml:space="preserve" style="it">19. Imago uidetur tum in linea refractionis, tum in perpendiculari incidentiæ. 12. <lb/>13. 18 p 10.</head> <p> <s xml:id="echoid-s17794" xml:space="preserve">QVòd autem uiſus comprehendat formam puncti uiſi, quam comprehẽdit refractè, etiam in <lb/>rectitudine lineæ, per quam forma peruenit ad uiſum, manifeſtum eſt:</s> <s xml:id="echoid-s17795" xml:space="preserve"> & cauſſa eius decla-<lb/>rata eſt in prædictis tractatibus:</s> <s xml:id="echoid-s17796" xml:space="preserve"> & eſt:</s> <s xml:id="echoid-s17797" xml:space="preserve"> quoniam uiſus nihil comprehendit, niſi in rectitudi-<lb/>ne linearum radialium:</s> <s xml:id="echoid-s17798" xml:space="preserve"> non enim patitur, niſi in uerticationibus iſtarum linearum.</s> <s xml:id="echoid-s17799" xml:space="preserve"> Quare autem <lb/>comprehendat formam per perpendiculares, exeuntes à re uiſa ſuper ſuperficiem corporis diapha-<lb/>ni:</s> <s xml:id="echoid-s17800" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s17801" xml:space="preserve"> quia, ut in ſecundo libro declarauimus:</s> <s xml:id="echoid-s17802" xml:space="preserve"> quando lux extenditur in corpore diaphano, exten-<lb/>ditur per motum uelociſsimum:</s> <s xml:id="echoid-s17803" xml:space="preserve"> & in quarto capitulo huius tractatus [8 n] declarauimus, quòd <lb/>motus lucis in corpore diaphano ſuper lineam decliuem ſuper ſuperficiem illius corporis, eſt com-<lb/>poſitus ex motu ſuper perpendicularem, exeuntem à puncto, in quo extenditur lux, ſuper ſuperfi-<lb/>ciem illius corporis diaphani, & ex motu ſuper lineam, quæ eſt perpendicularis ſuper hanc perpen <lb/>dicularem.</s> <s xml:id="echoid-s17804" xml:space="preserve"> Forma autem, quæ extenditur à puncto uiſo refractè ad locum refractionis (quæ eſt for <lb/>ma lucis exiſtens in puncto uiſo mixta cum forma coloris) ſemper extenditur ſuper lineam decli-<lb/>uem ſuper ſuperficiem corporis diaphani.</s> <s xml:id="echoid-s17805" xml:space="preserve"> Hæc igitur forma extenditur ad locum refractionis mo-<lb/>tu compoſito ex motu ſuper perpendicularem, quæ exit à puncto uiſo ſuper ſuperficiem corporis <lb/>diaphani, & ex motu ſuper lineam, quæ eſt perpendicularis ſuper hanc perpendicularem.</s> <s xml:id="echoid-s17806" xml:space="preserve"> Eſt ergo <lb/> <pb o="256" file="0262" n="262" rhead="ALHAZEN"/> motus formæ, quæ mouetur, aut ſuper perpendicularem, quæ eſt ſuper ſuperficiem corporis dia-<lb/>phani, & deinde translata eſt ab hac perpendiculari alio motu:</s> <s xml:id="echoid-s17807" xml:space="preserve"> aut ſuper perpendicularem, quæ exi <lb/>ſtit ſuper primam perpendicularem, & translata eſt poſt motum ipſius ſuper primam perpendicu-<lb/>larem motu compoſito ex prædictis duobus motibus.</s> <s xml:id="echoid-s17808" xml:space="preserve"> Hoc autem punctum comprehenditur à ui-<lb/>ſu in rectitudine lineæ, per quam forma peruenit ad uiſum.</s> <s xml:id="echoid-s17809" xml:space="preserve"> Forma ergo exiſtens in loco refractio-<lb/>nis peruenit ad ipſum per motũ formæ, quæ mouetur ſuper lineã perpendicularẽ ſuper ſuperficiẽ <lb/>corporis diaphani:</s> <s xml:id="echoid-s17810" xml:space="preserve"> deinde translata eſt ab hac perpendiculari per motum in rectitudine lineæ, per <lb/>quam forma peruenit ad uiſum.</s> <s xml:id="echoid-s17811" xml:space="preserve"> Forma autem, quæ eſt ſuper perpendicularem exiſtentem ſuper <lb/>ſuperficiem corporis diaphani:</s> <s xml:id="echoid-s17812" xml:space="preserve"> & deinde mouetur in rectitudine lineæ, per quam forma extendi-<lb/>tur ad uiſum:</s> <s xml:id="echoid-s17813" xml:space="preserve"> eſt forma, quæ extenditur à puncto uiſo ſuper ſuperficiem corporis diaphani, donec <lb/>perueniat ad punctum ſectionis inter hanc perpendicularem, & lineam, per quam forma extendi-<lb/>turad uiſum.</s> <s xml:id="echoid-s17814" xml:space="preserve"> Forma igitur puncti, quam uiſus comprehendit refractè ultra corpus diaphanum, eſt <lb/>per motum formæ, quæ peruenit ad uiſum à loco imaginis.</s> <s xml:id="echoid-s17815" xml:space="preserve"> Viſus autem comprehendit hanc for-<lb/>mam ex loco imaginis:</s> <s xml:id="echoid-s17816" xml:space="preserve"> quia eſt per motum formæ, quam uiſus comprehendit rectè, & ſine refractio <lb/>ne:</s> <s xml:id="echoid-s17817" xml:space="preserve"> & eſt locus, qui diſtat tantùm à uiſu, quantùm punctum imaginis:</s> <s xml:id="echoid-s17818" xml:space="preserve"> cuius ſitus, in reſpectu uiſus, <lb/>eſt ſitus formæ, quę eſt in loco imaginis:</s> <s xml:id="echoid-s17819" xml:space="preserve"> unde uiſus comprehendit illud punctum ſecundum refra-<lb/>ctionem in loco imaginis.</s> <s xml:id="echoid-s17820" xml:space="preserve"> Hęc autem eſt cauſſa, propter quam uiſus comprehendit rem uiſam ultra <lb/>corpus diaphanum in loco imaginis, & propter quam imago cuiuslibet puncti rei uiſæ comprehen <lb/>ſæ ſecundum refractionem, eſt in loco, in quo linea, per quam forma peruenit ad uiſum, ſecat per-<lb/>pendicularem, exeuntem à puncto illo ſuper ſuperficiem corporis diaphani.</s> <s xml:id="echoid-s17821" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div580" type="section" level="0" n="0"> <head xml:id="echoid-head503" xml:space="preserve" style="it">20. Viſibile refractum à medio (quod ſectum plano, facit communem ſectionem lineam re-<lb/>ctam aut peripheriam) unam habet imaginem. 29. 30 p 10.</head> <p> <s xml:id="echoid-s17822" xml:space="preserve">HOc autem declarato:</s> <s xml:id="echoid-s17823" xml:space="preserve"> dicamus quòd omne uiſum comprehenſum à uiſu ultra aliquod cor-<lb/>pus diaphanũ, quod differt in diaphanitate à corpore, quod eſt in parte uiſus (ſi corpus fue-<lb/>rit ex corporibus communibus) non habet, niſi unam imaginem.</s> <s xml:id="echoid-s17824" xml:space="preserve"> Corpora autem diaphana <lb/>aſſueta ſunt cœlum, & aer, & aqua, & uitrum, & lapides diaphani:</s> <s xml:id="echoid-s17825" xml:space="preserve"> & ſuperficies cœli, quæ eſt ex par-<lb/>te uiſus, eſt ſphærica & concaua.</s> <s xml:id="echoid-s17826" xml:space="preserve"> Vnde omnis ſuperficies plana, quę ſecat eam, facit in ea lineam cir <lb/>cularem, cuius concauitas eſt ex parte uiſus.</s> <s xml:id="echoid-s17827" xml:space="preserve"> Superficies autem aeris, quæ tangit illam, eſt ſphærica <lb/>conuexa.</s> <s xml:id="echoid-s17828" xml:space="preserve"> Vnde ſi ſecetur à ſuperficie æqualι:</s> <s xml:id="echoid-s17829" xml:space="preserve"> fiet in ipſa linea circularis, [per 1 th 1 ſphær.</s> <s xml:id="echoid-s17830" xml:space="preserve">] cuius con <lb/>uexum eſt ex parte cœli.</s> <s xml:id="echoid-s17831" xml:space="preserve"> Superficies uerò aquæ, quæ eſt ex parte uiſus, eſt ſphærica conuexa:</s> <s xml:id="echoid-s17832" xml:space="preserve"> & ſi <lb/>ſecetur à ſuperficie æquali, fiet in ipſa linea circularis:</s> <s xml:id="echoid-s17833" xml:space="preserve"> cuius conuexum eſt ex parte uiſus.</s> <s xml:id="echoid-s17834" xml:space="preserve"> Vitro-<lb/>rum autem & lapidum diaphanorum figuræ aſſuetæ ſunt rotundæ, aut planæ.</s> <s xml:id="echoid-s17835" xml:space="preserve"> Vnde ſi ſecentur à <lb/>planis ſuperficiebus, fient in illis aut circuli, aut lineæ rectę.</s> <s xml:id="echoid-s17836" xml:space="preserve"> Et uniuerſaliter dicimus, quòd omne <lb/>punctum comprehenſum à uiſu ultra quodcunque corpus diaphanum, (cuius ſuperficies, quæ op <lb/>ponitur uiſui, eſt unica ſuperficies, & ſi ſecetur à ſuքficie ęquali, fiat in ſuperficie eius linea recta, aut <lb/>circularis) non habet, niſi unã imaginem:</s> <s xml:id="echoid-s17837" xml:space="preserve"> nec comprehenditur à uiſu, niſi unum punctum tantùm.</s> <s xml:id="echoid-s17838" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div581" type="section" level="0" n="0"> <head xml:id="echoid-head504" xml:space="preserve" style="it">21. Si communis ſectio ſuperficierum, refractionis & refractiui fuerit linea recta: uiſibile in <lb/>perpendiculari ſuper refractiuum à uiſu duct a: rectè, & unum uidebitur. 19 p 10.</head> <p> <s xml:id="echoid-s17839" xml:space="preserve">SIt ergo uiſus a:</s> <s xml:id="echoid-s17840" xml:space="preserve"> & punctum uiſibile b:</s> <s xml:id="echoid-s17841" xml:space="preserve"> & corpus diaphanum ul-<lb/> <anchor type="figure" xlink:label="fig-0262-01a" xlink:href="fig-0262-01"/> tra, quod eſt b ſit illud, in cuius ſuperficie eſt g:</s> <s xml:id="echoid-s17842" xml:space="preserve"> & ſit diaphani <lb/>tas huius corporis groſsior diaphanitate corporis, quod eſt ex <lb/>parte uiſus:</s> <s xml:id="echoid-s17843" xml:space="preserve"> & ſit ſuperficies eius, quæ eſt ex parte uiſus, æqualis:</s> <s xml:id="echoid-s17844" xml:space="preserve"> & <lb/>[per 11 p 11] extrahamus ſuper ipſam à puncto a perpendicularem <lb/>a g c.</s> <s xml:id="echoid-s17845" xml:space="preserve"> Punctum ergo b aut erit ſuper lineam a g c:</s> <s xml:id="echoid-s17846" xml:space="preserve"> aut extra ipſam.</s> <s xml:id="echoid-s17847" xml:space="preserve"> Si <lb/>ergo punctum b fuerit in linea g c:</s> <s xml:id="echoid-s17848" xml:space="preserve"> tunc uiſus a comprehendet b re-<lb/>ctè & ſine refractione [per 13 n.</s> <s xml:id="echoid-s17849" xml:space="preserve">] Nam forma b, quando extenditur <lb/>per b g, exit ad corpus, quod eſt in parte a in rectιtudine b g:</s> <s xml:id="echoid-s17850" xml:space="preserve"> nam b g <lb/>eſt perpendicularis ſuper ſuperficiem corporis diaphani, quod eſt <lb/>exparte uiſus [per theſin.</s> <s xml:id="echoid-s17851" xml:space="preserve">] Viſus ergo a comprehendit b inſuo lo <lb/>co, & in rectitudine a g b.</s> <s xml:id="echoid-s17852" xml:space="preserve"> Dicimus ergo, quòd punctum b extra hãc <lb/>lineam nunquam refringetur ad a.</s> <s xml:id="echoid-s17853" xml:space="preserve"> Quòd ſi ſit poſsibile:</s> <s xml:id="echoid-s17854" xml:space="preserve"> refringatur <lb/>forma b a d a ex puncto p:</s> <s xml:id="echoid-s17855" xml:space="preserve"> & extrahamus ſuperficiem, in qua eſt per-<lb/>pendicularis a g b & punctum p:</s> <s xml:id="echoid-s17856" xml:space="preserve"> faciet ergo [per 3 p 11] in ſuperficie <lb/>corporis diaphani lineam rectam:</s> <s xml:id="echoid-s17857" xml:space="preserve"> ſit ergo g p d:</s> <s xml:id="echoid-s17858" xml:space="preserve"> & [per 11 p 1] extra-<lb/>hamus à puncto p perpendicularem ſuper lineam d p g:</s> <s xml:id="echoid-s17859" xml:space="preserve"> & ſit k p l:</s> <s xml:id="echoid-s17860" xml:space="preserve"> e <lb/>rit ergo k p l perpendicularis ſuper ſuperficiem corporis diaphani:</s> <s xml:id="echoid-s17861" xml:space="preserve"> <lb/>[per conuerſionem 4 d 11.</s> <s xml:id="echoid-s17862" xml:space="preserve"> Nam a g p refractionis planum eſt ad per-<lb/>pendiculum plano refractiui per 9 n:</s> <s xml:id="echoid-s17863" xml:space="preserve">] & continuemus b p, & extra-<lb/>hamus ad h:</s> <s xml:id="echoid-s17864" xml:space="preserve"> erit ergo angulus k p h ille, quem continet linea, per quam extenditur forma, & perpen <lb/>dicularis, exiens à loco refractionis ſuper ſuperficiem corporis diaphani.</s> <s xml:id="echoid-s17865" xml:space="preserve"> Quia ergo corpus, quod <lb/>eſt ex parte a, eſt ſubtilius illo, quod eſt ex parte b:</s> <s xml:id="echoid-s17866" xml:space="preserve"> cum b peruenerit ad p, refringetur ad partem con <lb/>trariam illi, in qua eſt perpendicularis p k, [per 14 n:</s> <s xml:id="echoid-s17867" xml:space="preserve">] nõ ergo perueniet forma refracta ad lineã a b:</s> <s xml:id="echoid-s17868" xml:space="preserve"> <lb/>ſed [ex hypotheſi] eſt refracta ad punctum a:</s> <s xml:id="echoid-s17869" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s17870" xml:space="preserve"> Non ergo refringetur forma b ad <lb/> <pb o="257" file="0263" n="263" rhead="OPTICAE LIBER VII."/> a ex p, neque ex alio puncto:</s> <s xml:id="echoid-s17871" xml:space="preserve"> a ergo non comprehendit b, niſi in rectitudine lineæ a g b:</s> <s xml:id="echoid-s17872" xml:space="preserve"> non ergo cõ <lb/>prehendit ipſum, niſi puncto uno tantùm.</s> <s xml:id="echoid-s17873" xml:space="preserve"/> </p> <div xml:id="echoid-div581" type="float" level="0" n="0"> <figure xlink:label="fig-0262-01" xlink:href="fig-0262-01a"> <variables xml:id="echoid-variables207" xml:space="preserve">a k h g p d b c l</variables> </figure> </div> </div> <div xml:id="echoid-div583" type="section" level="0" n="0"> <head xml:id="echoid-head505" xml:space="preserve" style="it">22. Si communis ſectio ſuperficierum, refractionis & refractiui denſioris fuerit linea rect a: <lb/>uiſibile extra perpendicularem à uiſu ſuper refractiuum ductam, ab uno puncto refringetur, & <lb/>unam habebit imaginem. 20 p 10.</head> <p> <s xml:id="echoid-s17874" xml:space="preserve">SIuerò b fuerit extra a g c:</s> <s xml:id="echoid-s17875" xml:space="preserve"> extrahamus ſuperficiem, in qua eſt a g c linea, & punctum b:</s> <s xml:id="echoid-s17876" xml:space="preserve"> ergo [per <lb/>18 p 11] erit perpendicularis ſuper ſuperficiem corporis diaphani:</s> <s xml:id="echoid-s17877" xml:space="preserve"> & fiat in ſuperficie huius cor <lb/>poris linea g d ſectio communis:</s> <s xml:id="echoid-s17878" xml:space="preserve"> ergo [per 3 p 11] g d eſt recta:</s> <s xml:id="echoid-s17879" xml:space="preserve"> non ergo refringetur forma b ad <lb/>a, niſi in ſuperficie, in qua eſt g d [per 5.</s> <s xml:id="echoid-s17880" xml:space="preserve">9 n:</s> <s xml:id="echoid-s17881" xml:space="preserve">] non enim tranſit per duo puncta a, b ſuperficies perpẽ-<lb/>dicularis ſuper ſuperficiem corporis diaphani, niſi ſuperficies tranſiens per perpendicularem a c:</s> <s xml:id="echoid-s17882" xml:space="preserve"> & <lb/>per punctum b & per pendicularem a c non tranſit ſuperficies æqualis, niſi una ſola tantùm.</s> <s xml:id="echoid-s17883" xml:space="preserve"> Forma <lb/>ergo b non refringitur ad a, niſi ex linea g d.</s> <s xml:id="echoid-s17884" xml:space="preserve"> Refringatur ergo forma b ad a à puncto e:</s> <s xml:id="echoid-s17885" xml:space="preserve"> & continue-<lb/>mus duas lineas b e, e a:</s> <s xml:id="echoid-s17886" xml:space="preserve"> & [per 11 p 1] extrahamus ex e perpendicularem ſuper lineam g e d:</s> <s xml:id="echoid-s17887" xml:space="preserve"> ſit ergo <lb/>h e z:</s> <s xml:id="echoid-s17888" xml:space="preserve"> erit ergo h e z perpendicularis ſuper duas ſuperficies duorum corporum diaphanorum:</s> <s xml:id="echoid-s17889" xml:space="preserve"> [per <lb/>9 n & conuerſionem 4 d 11] & extrahamus b e rectè ad p:</s> <s xml:id="echoid-s17890" xml:space="preserve"> erit ergo e p inter duas lineas e h, e a:</s> <s xml:id="echoid-s17891" xml:space="preserve"> nam <lb/>corpus diaphanum, quod eſt ex parte a, eſt ſubtilius illo, quod eſt ex parte b, [ex theſi.</s> <s xml:id="echoid-s17892" xml:space="preserve">] Forma ergo <lb/>b, quæ extenditur per lineam b e, cum peruenerit ad e, refringetur ad partem contrariam parti per-<lb/>pendicularis z e h [per 14 n] ideo <lb/> <anchor type="figure" xlink:label="fig-0263-01a" xlink:href="fig-0263-01"/> erit linea e p inter duas lineas e h <lb/>e a:</s> <s xml:id="echoid-s17893" xml:space="preserve"> & [per 12 p 1] extrahamus ex <lb/>b perpendicularem ſuper lineam <lb/>g d:</s> <s xml:id="echoid-s17894" xml:space="preserve"> ſcilicet b k:</s> <s xml:id="echoid-s17895" xml:space="preserve"> erit ergo b k per-<lb/>pendicularis ſuper ſuperficiẽ dia <lb/>phani corporis, quod eſt ex par-<lb/>re b:</s> <s xml:id="echoid-s17896" xml:space="preserve"> [per 9 n & conuerſionem 4 <lb/>d 11:</s> <s xml:id="echoid-s17897" xml:space="preserve">] & extrahamus a e rectè, ut <lb/>ſecet angulũ b e k:</s> <s xml:id="echoid-s17898" xml:space="preserve"> & ſecet lineã <lb/>b k in m:</s> <s xml:id="echoid-s17899" xml:space="preserve"> m ergo erit imago pun-<lb/>cti b [per 18 n]:</s> <s xml:id="echoid-s17900" xml:space="preserve"> & angulus p e a e-<lb/>rit angulus refractionis.</s> <s xml:id="echoid-s17901" xml:space="preserve"> Dico er <lb/>go, quòd b nõ habebit aliã imagi <lb/>nem, pręter m.</s> <s xml:id="echoid-s17902" xml:space="preserve"> Quoniã enim de-<lb/>mõſtratum eſt [19 n] quòd b nõ <lb/>comprehẽditur à uiſu, niſi ſuper <lb/>perpendicularem b k:</s> <s xml:id="echoid-s17903" xml:space="preserve"> Si ergo b <lb/>aliam habuerit imaginem:</s> <s xml:id="echoid-s17904" xml:space="preserve"> erit in linea b k, & inter duo pũcta b, k:</s> <s xml:id="echoid-s17905" xml:space="preserve"> corpus enim, quod eſt ex parte b, <lb/>eſt groſsius illo, quod eſt ex parte a.</s> <s xml:id="echoid-s17906" xml:space="preserve"> Sit ergo illa alia imago, ſi poſsibile eſt, punctum n:</s> <s xml:id="echoid-s17907" xml:space="preserve"> erit ergo aut <lb/>inter duo pũcta m, k:</s> <s xml:id="echoid-s17908" xml:space="preserve"> aut inter duo puncta m, b:</s> <s xml:id="echoid-s17909" xml:space="preserve"> ſit inter m, k:</s> <s xml:id="echoid-s17910" xml:space="preserve"> & cõtinuemus a n:</s> <s xml:id="echoid-s17911" xml:space="preserve"> ſecabit ergo lineam <lb/>g d in puncto o:</s> <s xml:id="echoid-s17912" xml:space="preserve"> & continuemus b o:</s> <s xml:id="echoid-s17913" xml:space="preserve"> & trãſeat uſq;</s> <s xml:id="echoid-s17914" xml:space="preserve"> ad l:</s> <s xml:id="echoid-s17915" xml:space="preserve"> erit ergo o pũctũ refractionis:</s> <s xml:id="echoid-s17916" xml:space="preserve"> quia linea b o l <lb/>eſt illa, per quã extenditur forma, quę eſt apud b:</s> <s xml:id="echoid-s17917" xml:space="preserve"> & erit angulus l o a angulus refractiõis:</s> <s xml:id="echoid-s17918" xml:space="preserve"> & [ք 11 p 1] <lb/>extrahamus ex o perpẽdicularem ſuք lineã g d:</s> <s xml:id="echoid-s17919" xml:space="preserve"> & ſit f o q:</s> <s xml:id="echoid-s17920" xml:space="preserve"> erit ergo linea f o q perpendicularis ſuper <lb/>ſuperficiẽ corporis diaphani [ք 9 n & conuerſionẽ 4 d 11] & erit angulus l o f ſicù<unsure/>t angulus, quẽ con <lb/>tinet perpendicularis, & linea, ք quã extenditur forma ad locũ refractionis [ք 15 p 1.</s> <s xml:id="echoid-s17921" xml:space="preserve">] Si igitur n fue <lb/>rit inter duo puncta m, k:</s> <s xml:id="echoid-s17922" xml:space="preserve"> tũc o erit inter duo pũcta e, k:</s> <s xml:id="echoid-s17923" xml:space="preserve"> angulus ergo e b k eſt maior angulo o b k [ք <lb/>9 ax.</s> <s xml:id="echoid-s17924" xml:space="preserve">] angulus ergo p e h eſt maior angulo l o f:</s> <s xml:id="echoid-s17925" xml:space="preserve"> [Quia.</s> <s xml:id="echoid-s17926" xml:space="preserve"> n.</s> <s xml:id="echoid-s17927" xml:space="preserve"> h e z, k b & f o q ſunt քpẽdiculares ipſi g d ք <lb/>fabricationẽ:</s> <s xml:id="echoid-s17928" xml:space="preserve"> erũt per 28 p 1 paral-<lb/> <anchor type="figure" xlink:label="fig-0263-02a" xlink:href="fig-0263-02"/> lelę:</s> <s xml:id="echoid-s17929" xml:space="preserve"> & ք 29 p 1 angulus p e h ęqua <lb/>bitur angulo e b k:</s> <s xml:id="echoid-s17930" xml:space="preserve"> eadẽq́;</s> <s xml:id="echoid-s17931" xml:space="preserve"> de cau-<lb/>ſa l o f ęquabitur o b k.</s> <s xml:id="echoid-s17932" xml:space="preserve"> Quare ſum <lb/>ptis ꝓ e b k, o b k:</s> <s xml:id="echoid-s17933" xml:space="preserve"> ęqualib.</s> <s xml:id="echoid-s17934" xml:space="preserve"> p e h, l o <lb/>f:</s> <s xml:id="echoid-s17935" xml:space="preserve"> erit angulus p e h maior angulo <lb/>l o f] & angulus p e a eſt angulus <lb/>refractionis ex angulo p e h:</s> <s xml:id="echoid-s17936" xml:space="preserve"> & an <lb/>gulus l o a eſt angulus refractiõis <lb/>ex angulo l o f:</s> <s xml:id="echoid-s17937" xml:space="preserve"> angulus ergo p e a <lb/>eſt maior angulo l o a, ut declara-<lb/>tũ eſt in tertio capite huius tracta <lb/>tus [12 n:</s> <s xml:id="echoid-s17938" xml:space="preserve">] angulus ergo a e h eſt <lb/>maior angulo a o f:</s> <s xml:id="echoid-s17939" xml:space="preserve"> qđ eſt impoſ-<lb/>ſibile.</s> <s xml:id="echoid-s17940" xml:space="preserve"> [Quia.</s> <s xml:id="echoid-s17941" xml:space="preserve"> n.</s> <s xml:id="echoid-s17942" xml:space="preserve"> anguli h e g, f o g <lb/>ęquantur ք 10 ax:</s> <s xml:id="echoid-s17943" xml:space="preserve"> & per 16 p 1 an-<lb/>gulus a e g maior eſt angulo a o g:</s> <s xml:id="echoid-s17944" xml:space="preserve"> <lb/>reliquus igitur a e h minor eſt re-<lb/>liquo a o f.</s> <s xml:id="echoid-s17945" xml:space="preserve">] Si aũt n fuerit inter duo puncta m, b:</s> <s xml:id="echoid-s17946" xml:space="preserve"> tunc punctũ e erit inter duo puncta o, k:</s> <s xml:id="echoid-s17947" xml:space="preserve"> & erit an-<lb/> <pb o="258" file="0264" n="264" rhead="ALHAZEN"/> gulus e b k minor angulo o b k [per 9 axio.</s> <s xml:id="echoid-s17948" xml:space="preserve">] erit ergo angulus p e h minor angulo l o f:</s> <s xml:id="echoid-s17949" xml:space="preserve"> erit ergo angu <lb/>lus p e a, qui eſt angulus refractionis, minor angulo l o a, qui eſt angulus refractionis:</s> <s xml:id="echoid-s17950" xml:space="preserve"> angulus ergo <lb/>a e h eſt maior angulo a o f:</s> <s xml:id="echoid-s17951" xml:space="preserve"> quod eſt impoſsibile, [ut ꝓximè oſtenſum eſt.</s> <s xml:id="echoid-s17952" xml:space="preserve">] Ergo impoſsibile eſt, ut <lb/>punctum n ſit imago puncti b:</s> <s xml:id="echoid-s17953" xml:space="preserve"> neque aliud punctum eſt præter m.</s> <s xml:id="echoid-s17954" xml:space="preserve"> Ergo punctum b, reſpectu uiſus <lb/>a, nullam habet imaginem, præterquam punctum m:</s> <s xml:id="echoid-s17955" xml:space="preserve"> & hoc declarare uoluimus.</s> <s xml:id="echoid-s17956" xml:space="preserve"/> </p> <div xml:id="echoid-div583" type="float" level="0" n="0"> <figure xlink:label="fig-0263-01" xlink:href="fig-0263-01a"> <variables xml:id="echoid-variables208" xml:space="preserve">a p h f l g e o k a n m e z q b</variables> </figure> <figure xlink:label="fig-0263-02" xlink:href="fig-0263-02a"> <variables xml:id="echoid-variables209" xml:space="preserve">a <gap/> f h p g o e k d m n c q z b</variables> </figure> </div> </div> <div xml:id="echoid-div585" type="section" level="0" n="0"> <head xml:id="echoid-head506" xml:space="preserve" style="it">23. Si cõmunis ſectio ſuperficierũ refractionis & refractiui rarioris fuerit linea recta: uiſibi-<lb/>le extra perpendicularem, à uiſu ſuper refractiuum ductã: ab uno puncto refringetur: & unam <lb/>habebit imaginem. 21 p 10.</head> <p> <s xml:id="echoid-s17957" xml:space="preserve">ET iterũ:</s> <s xml:id="echoid-s17958" xml:space="preserve"> ſit corpus groſsius ex parte uiſus, & ſubtilius ex parte rei uiſę:</s> <s xml:id="echoid-s17959" xml:space="preserve"> & ſit differẽtia cõmunis <lb/>inter hãc ſuperficiẽ & ſuperficiẽ corporis diaphani linea g d:</s> <s xml:id="echoid-s17960" xml:space="preserve"> & [ք 12 p 1] extrahamus ex b li-<lb/>neã perpẽdicularẽ ſuper lineã g d:</s> <s xml:id="echoid-s17961" xml:space="preserve"> & ſit b k:</s> <s xml:id="echoid-s17962" xml:space="preserve"> erit ergo b k քpendicularis ſuք ſuperficiẽ corporis <lb/>diaphani:</s> <s xml:id="echoid-s17963" xml:space="preserve"> [per 9 n & cõuerſionẽ 4 d 11] & refringatur forma b ad a ex e:</s> <s xml:id="echoid-s17964" xml:space="preserve"> & cõtinuemus lineas b e, e a:</s> <s xml:id="echoid-s17965" xml:space="preserve"> <lb/>& extrahamus perpendicularẽ h e:</s> <s xml:id="echoid-s17966" xml:space="preserve"> & extrahamus b e rectè ad p:</s> <s xml:id="echoid-s17967" xml:space="preserve"> erit ergo a e linea media inter duas <lb/>lineas e p, e h.</s> <s xml:id="echoid-s17968" xml:space="preserve"> Nam prima linea, per <lb/> <anchor type="figure" xlink:label="fig-0264-01a" xlink:href="fig-0264-01"/> quã extẽditur forma ad locũ refra-<lb/>ctionis, eſt linea b e p:</s> <s xml:id="echoid-s17969" xml:space="preserve"> refractio.</s> <s xml:id="echoid-s17970" xml:space="preserve"> n.</s> <s xml:id="echoid-s17971" xml:space="preserve"> <lb/>eſt ad partẽ perpẽdicularis e h:</s> <s xml:id="echoid-s17972" xml:space="preserve"> [ք <lb/>14 n] nã corpus, quod eſt ex parte <lb/>a, eſt groſsius illo, qđ eſt ex parte <lb/>b [ex theſi.</s> <s xml:id="echoid-s17973" xml:space="preserve">] Linea ergo a e eſt me-<lb/>dia inter duas lineas e p, e h:</s> <s xml:id="echoid-s17974" xml:space="preserve"> & ex-<lb/>trahamus a e, directè ad partem e, <lb/>quouſq;</s> <s xml:id="echoid-s17975" xml:space="preserve"> occurrat lineę b k:</s> <s xml:id="echoid-s17976" xml:space="preserve"> ſecat.</s> <s xml:id="echoid-s17977" xml:space="preserve"> n.</s> <s xml:id="echoid-s17978" xml:space="preserve"> <lb/>h e z.</s> <s xml:id="echoid-s17979" xml:space="preserve"> [Itaq;</s> <s xml:id="echoid-s17980" xml:space="preserve"> ſecabit k b ipſi h e z per <lb/>6 p 11 parallelã, per lẽma Procli ad <lb/>29 p 1] occurrat ergo illi in puncto <lb/>m:</s> <s xml:id="echoid-s17981" xml:space="preserve"> m ergo erit imago pũcti b:</s> <s xml:id="echoid-s17982" xml:space="preserve"> [per <lb/>18 n] nã corpus, qđ eſt ex parte b, <lb/>eſt ſubtilius illo, quod eſt ex parte <lb/>a.</s> <s xml:id="echoid-s17983" xml:space="preserve"> Dico igitur, qđ b nõ habet ima-<lb/>ginẽ, niſi m.</s> <s xml:id="echoid-s17984" xml:space="preserve"> Habeat enim n:</s> <s xml:id="echoid-s17985" xml:space="preserve"> ſi poſ-<lb/>ſibile eſt:</s> <s xml:id="echoid-s17986" xml:space="preserve"> n ergo erit in perpendiculari b k, [per 19 n] & infra punctũ b:</s> <s xml:id="echoid-s17987" xml:space="preserve"> quia corpus, quod eſt in par-<lb/>te b, eſt ſubtilius illo, quod eſt ex parte a.</s> <s xml:id="echoid-s17988" xml:space="preserve"> Eſt ergo aut inter duo puncta m, b:</s> <s xml:id="echoid-s17989" xml:space="preserve"> aut infra m:</s> <s xml:id="echoid-s17990" xml:space="preserve"> & cõtinue-<lb/>mus a n:</s> <s xml:id="echoid-s17991" xml:space="preserve"> ſecabit ergo lineã d g in o:</s> <s xml:id="echoid-s17992" xml:space="preserve"> o ergo eſt punctũ refractionis.</s> <s xml:id="echoid-s17993" xml:space="preserve"> Et cõtinuemus b o:</s> <s xml:id="echoid-s17994" xml:space="preserve"> & trãſeat uſq;</s> <s xml:id="echoid-s17995" xml:space="preserve"> <lb/>a d l:</s> <s xml:id="echoid-s17996" xml:space="preserve"> & [ք 11 p 1] extrahamus ex o perpẽdicularẽ f o q.</s> <s xml:id="echoid-s17997" xml:space="preserve"> Linea ergo b o eſt linea, ք quã extẽditur forma <lb/>ad locũ refractionis:</s> <s xml:id="echoid-s17998" xml:space="preserve"> ergo linea o a erit inter duas lineas o l, o f:</s> <s xml:id="echoid-s17999" xml:space="preserve"> refractio enim eſt ad partẽ perpẽdicu <lb/>laris [ք theſin & 14 n.</s> <s xml:id="echoid-s18000" xml:space="preserve">] Si ergo fuerit n inter duo pũcta m, b:</s> <s xml:id="echoid-s18001" xml:space="preserve"> tũc punctũ o erit inter duo puncta e, k:</s> <s xml:id="echoid-s18002" xml:space="preserve"> er <lb/>go erit angulus o b k minor angulo e b k:</s> <s xml:id="echoid-s18003" xml:space="preserve"> [ք 9 ax.</s> <s xml:id="echoid-s18004" xml:space="preserve">] ergo angulus l o f eſt minor angulo p e h, [ut de-<lb/>monſtratũ eſt ſuperiore numero] ergo [ք 12 n] angulus l o a qui eſt angulus refractionis) eſt minor <lb/>angulo p e a, qui eſt angulus refractionis:</s> <s xml:id="echoid-s18005" xml:space="preserve"> & angulus a o f, qui remanet poſt angulum refractionis, eſt <lb/>minor angulo a e h, qui remanet poſt angulũ refractionis [per 12 n] ſed [per 29 p 1] angulus a o f eſt <lb/>æqualis angulo a n k, & angulus a e h eſt æqualis angulo a m k:</s> <s xml:id="echoid-s18006" xml:space="preserve"> ergo angulus a n k eſt minor angulo <lb/>a m k:</s> <s xml:id="echoid-s18007" xml:space="preserve"> quod eſt impoſsibile [& cõtra 16 p 1.</s> <s xml:id="echoid-s18008" xml:space="preserve">] Si aũt n fuerit infra m:</s> <s xml:id="echoid-s18009" xml:space="preserve"> tũc erit e inter duo puncta o, k:</s> <s xml:id="echoid-s18010" xml:space="preserve"> & <lb/>erit angulus o b k maior angulo e b <lb/> <anchor type="figure" xlink:label="fig-0264-02a" xlink:href="fig-0264-02"/> k:</s> <s xml:id="echoid-s18011" xml:space="preserve"> angulus ergo l o f erit maior an-<lb/>gulo p e h:</s> <s xml:id="echoid-s18012" xml:space="preserve"> [ut patuit proximo nu-<lb/>mero] ergo angulus l o a eſt maior <lb/>angulo p e a:</s> <s xml:id="echoid-s18013" xml:space="preserve"> & angulus a o f eſt ma <lb/>ior angulo a e h:</s> <s xml:id="echoid-s18014" xml:space="preserve"> [ք 12 n] ergo angu <lb/>lus a n k eſt maior angulo a m k:</s> <s xml:id="echoid-s18015" xml:space="preserve"> qđ <lb/>eſt impoſsibile:</s> <s xml:id="echoid-s18016" xml:space="preserve"> [& cõtra 16 p 1] n er <lb/>go non eſt imago b:</s> <s xml:id="echoid-s18017" xml:space="preserve"> nec aliud pun-<lb/>ctũ, præterquã m:</s> <s xml:id="echoid-s18018" xml:space="preserve"> b ergo non habet <lb/>imaginem, niſi m.</s> <s xml:id="echoid-s18019" xml:space="preserve"> Et hoc eſt, quod <lb/>uoluimus declarare.</s> <s xml:id="echoid-s18020" xml:space="preserve"/> </p> <div xml:id="echoid-div585" type="float" level="0" n="0"> <figure xlink:label="fig-0264-01" xlink:href="fig-0264-01a"> <variables xml:id="echoid-variables210" xml:space="preserve">a f h p l g o e k d b m c q z n</variables> </figure> <figure xlink:label="fig-0264-02" xlink:href="fig-0264-02a"> <variables xml:id="echoid-variables211" xml:space="preserve">a <gap/> f l p g e o k d b n m c z <gap/></variables> </figure> </div> </div> <div xml:id="echoid-div587" type="section" level="0" n="0"> <head xml:id="echoid-head507" xml:space="preserve" style="it">24. Si duæ rectæ lineæ circulo <lb/>inſcriptæ interſecentur: angul{us} <lb/>ſectionis quilibet æquatur angulo <lb/>in peripheria, inſiſtẽti in periphe-<lb/>riam æqualẽ duab{us} peripherijs <lb/>eidem angulo, & ad uerticem oppoſito ſubtenſis 54 p 1.</head> <p> <s xml:id="echoid-s18021" xml:space="preserve">AD duas aũt lineas circulares conuexã & cõcauã pręmittemus hęc.</s> <s xml:id="echoid-s18022" xml:space="preserve"> Cũ duę chordę ſeſe ſecuerint <lb/> <pb o="259" file="0265" n="265" rhead="OPTICAE LIBER VII."/> in circulo:</s> <s xml:id="echoid-s18023" xml:space="preserve"> angulus ſectionis erit æqualis angulo, qui eſt apud circumferentiam, quam chordant <lb/>duo arcus, quos diſtinguunt illæ duæ chordæ.</s> <s xml:id="echoid-s18024" xml:space="preserve"> Et ſi duæ hneæ ſecuerint circulum, & ſecuerint ſe <lb/>extra circulum:</s> <s xml:id="echoid-s18025" xml:space="preserve"> angulus ſectionis erit æqualis angulo, <lb/> <anchor type="figure" xlink:label="fig-0265-01a" xlink:href="fig-0265-01"/> qui eſt apud circũferentiam, quã chordat exceſſus ma <lb/>ioris illorum duorũ arcuũ, quos diſtinguunt illæ duæ <lb/>lineæ, ſupra reliquũ.</s> <s xml:id="echoid-s18026" xml:space="preserve"> Verbi gratia:</s> <s xml:id="echoid-s18027" xml:space="preserve"> in circulo a b c d ſe-<lb/>cent ſe duæ chordę a c, b d in e.</s> <s xml:id="echoid-s18028" xml:space="preserve"> Dico igitur, quòd angu <lb/>lus a e b eſt æqualis angulo, qui eſt apud circumferen-<lb/>rẽtiam, quam reſpiciunt duo arcus a b, c d:</s> <s xml:id="echoid-s18029" xml:space="preserve"> & quòd an-<lb/>gulus b e c eſt æqualis angulo in circumferẽtia, quam <lb/>reſpiciũt duo arcus d g a, b z c.</s> <s xml:id="echoid-s18030" xml:space="preserve"> Extrahamus enim ex b <lb/>lineam b z æquidiſtantẽ lineæ a c [ք 31 p 1] arcus ergo <lb/>c z eſt æqualis arcui a b [Ducta enim recta a z:</s> <s xml:id="echoid-s18031" xml:space="preserve"> æquabi <lb/>tur angulus c a z angulo a z b ք 29 p 1:</s> <s xml:id="echoid-s18032" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s18033" xml:space="preserve"> periphe-<lb/>ria c z peripherię a b ք 26 p 3:</s> <s xml:id="echoid-s18034" xml:space="preserve">] & arcusc d eſt cõmunis:</s> <s xml:id="echoid-s18035" xml:space="preserve"> <lb/>ergo arcus d z eſt æqualis duobus arcubus, a b, c d:</s> <s xml:id="echoid-s18036" xml:space="preserve"> ſed <lb/>arcus d z reſpicit angulũ d b z [ք 8 d 3] ergo d z reſpicit <lb/>arcus æquales duob.</s> <s xml:id="echoid-s18037" xml:space="preserve"> arcubus a b, c d:</s> <s xml:id="echoid-s18038" xml:space="preserve"> & [ք 29 p 1] an-<lb/>gulus d b z eſt æqualis angulo a e b:</s> <s xml:id="echoid-s18039" xml:space="preserve"> ergo angulus a e b <lb/>eſt æqualis angulo, qui eſt in circum ferẽtia, quã reſpi-<lb/>ciunt duo arcus a b, c d.</s> <s xml:id="echoid-s18040" xml:space="preserve"> Et hoc eſt quod uoluimus.</s> <s xml:id="echoid-s18041" xml:space="preserve"> Itẽ continuemus d z:</s> <s xml:id="echoid-s18042" xml:space="preserve"> & producamus z b in h:</s> <s xml:id="echoid-s18043" xml:space="preserve"> e-<lb/>rit ergo [per 32 p 1] angulus h b d æqualis duob.</s> <s xml:id="echoid-s18044" xml:space="preserve"> angulis b d z, b z d, & [per 8 d 3] duo anguli b z d, <lb/>b d z reſpiciuntur à duobus arcubus b g d, b f z:</s> <s xml:id="echoid-s18045" xml:space="preserve"> angu-<lb/> <anchor type="figure" xlink:label="fig-0265-02a" xlink:href="fig-0265-02"/> lus ergo h b d eſt æqualis angulo, quem reſpicit arcus <lb/>d b z:</s> <s xml:id="echoid-s18046" xml:space="preserve"> & arcus a b eſt æqualis arcui z c, [ex cõcluſo:</s> <s xml:id="echoid-s18047" xml:space="preserve">] re <lb/>go arcus d b z eſt æqualis duobus arcubus d g a, b z c:</s> <s xml:id="echoid-s18048" xml:space="preserve"> <lb/>ergo angulus h b e eſt æqualis angulo, quẽ reſpiciunt <lb/>duo arcus d g a, b z c:</s> <s xml:id="echoid-s18049" xml:space="preserve"> & [per 29 p 1] angulus h b e eſt ę-<lb/>qualis angulo b e c.</s> <s xml:id="echoid-s18050" xml:space="preserve"> Ergo angulus b e c eſt æqualis an-<lb/>gulo, qui eſt in circumferẽtia, quã reſpiciũt duo arcus <lb/>d g a, b z c.</s> <s xml:id="echoid-s18051" xml:space="preserve"> Et hoc eſt, quod uoluimus declarare.</s> <s xml:id="echoid-s18052" xml:space="preserve"> Et ſi li <lb/>nea h b z contingat circulum:</s> <s xml:id="echoid-s18053" xml:space="preserve"> tunc [per 32 p 3] angu-<lb/>lus e b z erit æqualis angulo cadẽti in portionẽ b a d:</s> <s xml:id="echoid-s18054" xml:space="preserve"> & <lb/>ſic arcus b c d reſpicit angulum apud circumferẽtiam, <lb/>æqualem angulo e b z:</s> <s xml:id="echoid-s18055" xml:space="preserve"> & [per 29 p 1] angulus e b z eſt <lb/>æqualis angulo b e a:</s> <s xml:id="echoid-s18056" xml:space="preserve"> ergo angulus b e a eſt æqualis an <lb/>gulo, qui eſt apud circumferentiam, quẽ reſpicit arcus <lb/>b c d:</s> <s xml:id="echoid-s18057" xml:space="preserve"> & arcus b c eſt æqualis arcui b a:</s> <s xml:id="echoid-s18058" xml:space="preserve"> quia diameter, <lb/>quæ exit ex b, eſt perpendicularis ſuper lineã a c:</s> <s xml:id="echoid-s18059" xml:space="preserve"> [Nã <lb/>diameter per punctũ b educta, eſt perpẽdicularis tan-<lb/>gẽti per 18 p 3:</s> <s xml:id="echoid-s18060" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s18061" xml:space="preserve"> per 29 p 1 eſt perpẽdicularis ipſi a c ad tangentẽ parallelę] quare [per 3 p 3] diui <lb/>ditipſam in duo æqualia:</s> <s xml:id="echoid-s18062" xml:space="preserve"> ergo arcus a b æqualis erit arcui b c:</s> <s xml:id="echoid-s18063" xml:space="preserve"> [ductis enim rectis a b, b c:</s> <s xml:id="echoid-s18064" xml:space="preserve"> erũt ipſæ <lb/>ք 4 p 1 æquales:</s> <s xml:id="echoid-s18065" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s18066" xml:space="preserve"> peripheriæ a b, b c ipſis ſubtẽſæ, per 28 p 3:</s> <s xml:id="echoid-s18067" xml:space="preserve">] arcus ergo b c d eſt ęqualis duo-<lb/>bus arcubus a b, c d:</s> <s xml:id="echoid-s18068" xml:space="preserve"> ergo angulus b e a eſt æqualis angulo, ꝗ eſt apud circũferẽtiam, quẽ reſpiciunt <lb/>duo arcus a b, c d.</s> <s xml:id="echoid-s18069" xml:space="preserve"> Et ſimiliter declarabitur, quòd angulus b e c eſt æqualis angulo, qui eſt apud cir-<lb/>cumferentiam, quem reſpiciunt duo arcus b c, a d.</s> <s xml:id="echoid-s18070" xml:space="preserve"> Et hoc eſt quod uoluimus.</s> <s xml:id="echoid-s18071" xml:space="preserve"/> </p> <div xml:id="echoid-div587" type="float" level="0" n="0"> <figure xlink:label="fig-0265-01" xlink:href="fig-0265-01a"> <variables xml:id="echoid-variables212" xml:space="preserve">h a b g e f d e z</variables> </figure> <figure xlink:label="fig-0265-02" xlink:href="fig-0265-02a"> <variables xml:id="echoid-variables213" xml:space="preserve">h a b e d c z</variables> </figure> </div> </div> <div xml:id="echoid-div589" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables214" xml:space="preserve">e a b d f c</variables> </figure> <head xml:id="echoid-head508" xml:space="preserve" style="it">25. Si duæ rectæ lineæ circulo inſcriptæ, extrà cõtinuatæ cõcurrant: angulus concurſ{us} æqua-<lb/>tur angulo in peripheria, inſiſtenti in peripheriã, qua maior peri-<lb/> pheriarum inter inſcript{as} cõprehenſarũ exuperat minorẽ. 55 p 1.</head> <p> <s xml:id="echoid-s18072" xml:space="preserve">ITem:</s> <s xml:id="echoid-s18073" xml:space="preserve"> ſit e extra circulũ a b c d:</s> <s xml:id="echoid-s18074" xml:space="preserve"> & extrahamus ex e duas lineas ſe-<lb/>cantes circulũ a b c d:</s> <s xml:id="echoid-s18075" xml:space="preserve"> & ſint e a d, e b c.</s> <s xml:id="echoid-s18076" xml:space="preserve"> Dico ergo, quòd angulus <lb/>c e d eſt æqualis angulo, ꝗ eſt apud circũferẽtiã circuli, quẽ reſpi-<lb/>cit arcus exceſſus d c ſupera arcũ a b.</s> <s xml:id="echoid-s18077" xml:space="preserve"> Extrahamus enim lineã æquidi-<lb/>ſtantẽ lineæ b c [per 31 p 1] erit ergo [ut paulò antè oſtẽſum eſt] ar-<lb/>cus f c æqualis arcui a b:</s> <s xml:id="echoid-s18078" xml:space="preserve"> erit ergo arcus d f exceſſus arcus d c ſupra <lb/>arcũ a b:</s> <s xml:id="echoid-s18079" xml:space="preserve"> ſed [per 8 d 3] arcus d freſpicit angulũ d a f:</s> <s xml:id="echoid-s18080" xml:space="preserve"> & [per 29 p 1] <lb/>angulus d a f eſt æqualis angulo c e d:</s> <s xml:id="echoid-s18081" xml:space="preserve"> ergo c e d eſt æqualis angulo, <lb/>qui eſt apud circumferentiam d f.</s> <s xml:id="echoid-s18082" xml:space="preserve"> Et hoc eſt quod uoluimus.</s> <s xml:id="echoid-s18083" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div590" type="section" level="0" n="0"> <head xml:id="echoid-head509" xml:space="preserve" style="it">26. Sι cõmunis ſectio ſuperficierũ refractionis & refractiui con <lb/>uexi fuerit peripheria: uiſibile in perpendiculari à uiſu ſuper re-<lb/>fractiuum duct a: rectè, & unum uidebitur. 22 p 10.</head> <p> <s xml:id="echoid-s18084" xml:space="preserve">HIs ergo declaratis, ſit uiſus punctũ a:</s> <s xml:id="echoid-s18085" xml:space="preserve"> & ſit pũctum b in aliquo <lb/>uiſo:</s> <s xml:id="echoid-s18086" xml:space="preserve"> & ſit ultra corpus diaphanũ groſsius corpore, qđ eſt in <lb/> <pb o="260" file="0266" n="266" rhead="ALHAZEN"/> parte uiſus:</s> <s xml:id="echoid-s18087" xml:space="preserve"> & ſit ſuperficies corporis diaphani, quod eſt ex parte b, ſuperficies circularis cõuexa ex <lb/>parte uiſus.</s> <s xml:id="echoid-s18088" xml:space="preserve"> Ergo ք duo pũcta, a, b tranſit ſuperficies perpendicularis ſuper ſuperficiẽ corporis dia-<lb/>phani, [per 9 n:</s> <s xml:id="echoid-s18089" xml:space="preserve"> quia ſuperficies per a & b educta, eſt ſuperficies refractionis:</s> <s xml:id="echoid-s18090" xml:space="preserve">] & non tranſit per illa, <lb/>ſuperficies perpendicularis ſuper ſuperficiẽ corporis, in qua refringitur forma b ad a, niſi una tan-<lb/>tùm.</s> <s xml:id="echoid-s18091" xml:space="preserve"> Hãc ergo ſuperficiẽ corporis diaphani ſignet circulus c e d:</s> <s xml:id="echoid-s18092" xml:space="preserve"> cuius centrum ſit z:</s> <s xml:id="echoid-s18093" xml:space="preserve"> & continue-<lb/>mus a c z d:</s> <s xml:id="echoid-s18094" xml:space="preserve"> linea ergo c z d erit perpẽdicularis ſuper ſuperficiẽ corporis diaphani [per 4 th 1 ſphę-<lb/>ricorum:</s> <s xml:id="echoid-s18095" xml:space="preserve"> quia perpendicularis eſt plano tangenti.</s> <s xml:id="echoid-s18096" xml:space="preserve">] Punctum autẽ b aut erit extra lineam c d:</s> <s xml:id="echoid-s18097" xml:space="preserve"> aut in <lb/>ipſa.</s> <s xml:id="echoid-s18098" xml:space="preserve"> Si igitur b fuerit in linea c d:</s> <s xml:id="echoid-s18099" xml:space="preserve"> tunc uiſus a comprehendet b rectè, & ſine refractione [per 13 n.</s> <s xml:id="echoid-s18100" xml:space="preserve">] <lb/>Nam forma, quæ extenditur per lineam c d, extenditur rectè in corpore diaphano, quod eſt ex par-<lb/>te uiſus:</s> <s xml:id="echoid-s18101" xml:space="preserve"> quia linea c d eſt perpendicularis ſuper ſuperficiem corporis diaphani, quod eſt ex parte <lb/>uiſus.</s> <s xml:id="echoid-s18102" xml:space="preserve"> Viſus ergo a comprehendit b in ſuo loco, & rectè.</s> <s xml:id="echoid-s18103" xml:space="preserve"> Dico ergo, quòd forma punctib, quod eſt <lb/>in c d linea, nunquam refringetur ad a.</s> <s xml:id="echoid-s18104" xml:space="preserve"> Quoniam punctum b aut <lb/> <anchor type="figure" xlink:label="fig-0266-01a" xlink:href="fig-0266-01"/> erit in centro:</s> <s xml:id="echoid-s18105" xml:space="preserve"> aut extra centrum.</s> <s xml:id="echoid-s18106" xml:space="preserve"> Si ergo fuerit in centro:</s> <s xml:id="echoid-s18107" xml:space="preserve"> tunc o-<lb/>mnis linea, per quam extenditur forma b ad circumferẽtiam c e d, <lb/>extenditur in rectitudine eius in corpore diaphano, quod eſt ex <lb/>parte uiſus.</s> <s xml:id="echoid-s18108" xml:space="preserve"> Nam omnis linea exiens à centro circuli c e d eſt per-<lb/>pendicularis ſuper ſuperficiem corporis diaphani, [ut oſtenſum <lb/>eſt 25 n 4:</s> <s xml:id="echoid-s18109" xml:space="preserve">] & non exit à centro circuli c e d ad uiſum a linea recta, <lb/>niſi linea z a.</s> <s xml:id="echoid-s18110" xml:space="preserve"> Ergo forma puncti b, quod eſt in centro, non refringi-<lb/>tur ad a ex circumferentia c e d.</s> <s xml:id="echoid-s18111" xml:space="preserve"> Ergo forma b nunquam refringe-<lb/>tur ad a, ſi b fuerit in centro.</s> <s xml:id="echoid-s18112" xml:space="preserve"> Si uerò fuerit extra centrum:</s> <s xml:id="echoid-s18113" xml:space="preserve"> aut erit <lb/>in linea z c, aut in z d:</s> <s xml:id="echoid-s18114" xml:space="preserve"> ſit ergo primò in linea z c.</s> <s xml:id="echoid-s18115" xml:space="preserve"> Dico, quòd forma b <lb/>non refringatur ad a.</s> <s xml:id="echoid-s18116" xml:space="preserve"> Quod ſi fuerit poſsibile:</s> <s xml:id="echoid-s18117" xml:space="preserve"> refringatur ex ipſo <lb/>e:</s> <s xml:id="echoid-s18118" xml:space="preserve"> & continuemus b e:</s> <s xml:id="echoid-s18119" xml:space="preserve"> & extrahamus illud ad h:</s> <s xml:id="echoid-s18120" xml:space="preserve"> & continuemus <lb/>z e:</s> <s xml:id="echoid-s18121" xml:space="preserve"> & extrahamus ipſam ad p:</s> <s xml:id="echoid-s18122" xml:space="preserve"> erit ergo linea z e p perpendicu-<lb/>laris ſuper ſuperficiem corporis diaphani [per 25 n 4,] quod eſt <lb/>ex parte uiſus.</s> <s xml:id="echoid-s18123" xml:space="preserve"> Forma ergo b, quan do extenditur ad lineam b e, & <lb/>refringitur in puncto e:</s> <s xml:id="echoid-s18124" xml:space="preserve"> tranſit à perpendiculari p e ad partem h <lb/>contrariam illi, in qua eſt perpendicularis [per 14 n:</s> <s xml:id="echoid-s18125" xml:space="preserve">] forma ergo <lb/>b non perueniet ad a ſecũdum refractionem, ſi b fuerit in linea z c.</s> <s xml:id="echoid-s18126" xml:space="preserve"> <lb/>Item ſit b in linea d z.</s> <s xml:id="echoid-s18127" xml:space="preserve"> Dico ergo, quòd forma b non refringetur ad <lb/>a.</s> <s xml:id="echoid-s18128" xml:space="preserve"> Quod ſi eſt poſsibile:</s> <s xml:id="echoid-s18129" xml:space="preserve"> refringatur ex e:</s> <s xml:id="echoid-s18130" xml:space="preserve"> & continuemus b e:</s> <s xml:id="echoid-s18131" xml:space="preserve"> & extrahamus b e lineam ad r:</s> <s xml:id="echoid-s18132" xml:space="preserve"> & co<gap/> <lb/>tinuemus z e, & extrahamus lineam uſque ad p:</s> <s xml:id="echoid-s18133" xml:space="preserve"> & refringatur forma b ad a per lineam e a:</s> <s xml:id="echoid-s18134" xml:space="preserve"> Sic ergo <lb/>angulus r e a erit angulus refractionis:</s> <s xml:id="echoid-s18135" xml:space="preserve"> angulus autem r e p erit angulus, quem continet linea, per <lb/>quam extenditur forma, & perpendicularis exiens à loco refractionis:</s> <s xml:id="echoid-s18136" xml:space="preserve"> angulus ergo r e a eſt minor <lb/>angulo r e p [per 12 n] & linea b z aut eſt minor linea z e, aut æqualis ei:</s> <s xml:id="echoid-s18137" xml:space="preserve"> nam b aut eſt inter duo <lb/>puncta d, z:</s> <s xml:id="echoid-s18138" xml:space="preserve"> aut in puncto d:</s> <s xml:id="echoid-s18139" xml:space="preserve"> ergo angulus e b z aut eſt maior angulo b e z [per 18 p 1] aut æqualis <lb/>ei:</s> <s xml:id="echoid-s18140" xml:space="preserve"> [per 5 p 1] ſed [per 16 p 1] angulus a e r eſt maior angulo e b z:</s> <s xml:id="echoid-s18141" xml:space="preserve"> ergo angulus a e r eſt maior angu <lb/>lo r e p.</s> <s xml:id="echoid-s18142" xml:space="preserve"> [Nam quia a e r maior eſt e b z, qui maior eſt, uel æqualis ipſi b e z:</s> <s xml:id="echoid-s18143" xml:space="preserve"> erit etiam maior ipſo <lb/>b e z:</s> <s xml:id="echoid-s18144" xml:space="preserve"> at ipſi b e z æquatur r e p per 15 p 1:</s> <s xml:id="echoid-s18145" xml:space="preserve"> quare a e r maior eſt r e p] quo prius erat minor:</s> <s xml:id="echoid-s18146" xml:space="preserve"> quod eſt <lb/>impoſsibile.</s> <s xml:id="echoid-s18147" xml:space="preserve"> Ergo forma b non refringetur ad a ex e:</s> <s xml:id="echoid-s18148" xml:space="preserve"> nec ex alio puncto circumferentiæ c e d:</s> <s xml:id="echoid-s18149" xml:space="preserve"> ne-<lb/>que ex alia circumferentia circulorum, qui fiunt in ſuperficie corporis diaphani, in quo eſt b.</s> <s xml:id="echoid-s18150" xml:space="preserve"> Igitur <lb/>b exiſtente in linea c d:</s> <s xml:id="echoid-s18151" xml:space="preserve"> non comprehendetur ipſum à uiſu per refractionem.</s> <s xml:id="echoid-s18152" xml:space="preserve"> Quare non compre-<lb/>henditur, niſi unum ſolum punctum.</s> <s xml:id="echoid-s18153" xml:space="preserve"/> </p> <div xml:id="echoid-div590" type="float" level="0" n="0"> <figure xlink:label="fig-0266-01" xlink:href="fig-0266-01a"> <variables xml:id="echoid-variables215" xml:space="preserve">a r c p e h b z b d</variables> </figure> </div> </div> <div xml:id="echoid-div592" type="section" level="0" n="0"> <head xml:id="echoid-head510" xml:space="preserve" style="it">27. Si communis ſectio ſuperficierum, refractionis & refractiui conuexi denſioris fuerit <lb/>peripheria: uiſibιle extra perpendicularem à uiſu ſuper refractiuum ductam, ab uno puncto re <lb/>fringetur, unaḿ habebit imaginem, uariè, pro uaria uiſ{us} uel uiſibilis poſitione, ſitam. <lb/>23 p 10.</head> <p> <s xml:id="echoid-s18154" xml:space="preserve">ITem:</s> <s xml:id="echoid-s18155" xml:space="preserve"> ſit b extra lineam c d:</s> <s xml:id="echoid-s18156" xml:space="preserve"> & extrahamus ſuperficiem, in qua eſt perpẽdicularis, & punctum b.</s> <s xml:id="echoid-s18157" xml:space="preserve"> <lb/>Hæc ergo ſuperficies erit perpendicularis ſuper ſuperficiem corporis diaphani:</s> <s xml:id="echoid-s18158" xml:space="preserve"> [per 9 n:</s> <s xml:id="echoid-s18159" xml:space="preserve"> quia <lb/>planum ductum per perpendicularẽ a c d & uiſibile b, eſt planum refractionis] & punctum b <lb/>non refringetur ad a, niſi in hac ſuperficie:</s> <s xml:id="echoid-s18160" xml:space="preserve"> non enim tranſit per duo puncta a, b ſuperficies perpen-<lb/>dicularis ſuper ſuperficiem corporis diaphani, niſi illa, quæ tranſit per lineam a d:</s> <s xml:id="echoid-s18161" xml:space="preserve"> & non exit ex <lb/>linea a d ſuperficies, quæ tranſit per b, niſi una tantùm.</s> <s xml:id="echoid-s18162" xml:space="preserve"> Hæc ergo ſuperficies ſignet in ſuperficie <lb/>corporis diaphani circulum c e d:</s> <s xml:id="echoid-s18163" xml:space="preserve"> forma ergo b non refringetur ad a, niſi ex circumferentia c e d:</s> <s xml:id="echoid-s18164" xml:space="preserve"> <lb/>refringatur ergo ex e.</s> <s xml:id="echoid-s18165" xml:space="preserve"> Dico ergo, quòd nõ refringetur ex alio puncto quàm e.</s> <s xml:id="echoid-s18166" xml:space="preserve"> Refringatur enim (ſi <lb/>poſsibile eſt) ex alio puncto:</s> <s xml:id="echoid-s18167" xml:space="preserve"> quod, ut dictũ eſt, erit in circũferentia c e d:</s> <s xml:id="echoid-s18168" xml:space="preserve"> Sit ergo m:</s> <s xml:id="echoid-s18169" xml:space="preserve"> & cõtinuemus <lb/>lineas b e, e a, b m, m a, z e, z m:</s> <s xml:id="echoid-s18170" xml:space="preserve"> & ſecẽt ſe lineæ b m, z e in pũcto g:</s> <s xml:id="echoid-s18171" xml:space="preserve"> & extrahamus b e uſq;</s> <s xml:id="echoid-s18172" xml:space="preserve"> ad h:</s> <s xml:id="echoid-s18173" xml:space="preserve"> & b m <lb/>ad n:</s> <s xml:id="echoid-s18174" xml:space="preserve"> & e z ad p:</s> <s xml:id="echoid-s18175" xml:space="preserve"> & z m a d l.</s> <s xml:id="echoid-s18176" xml:space="preserve"> Erit ergo angulus h e p ille, quem continet linea, per quam extenditur <lb/>forma, & perpendicularis exiens à loco refractionis:</s> <s xml:id="echoid-s18177" xml:space="preserve"> & angulus h e a erit angulus refractionis:</s> <s xml:id="echoid-s18178" xml:space="preserve"> & <lb/>n m l angulus ille, quem continet linea, per quam extenditur ſorma, & perpendicularis exiens à <lb/>loco refractionis:</s> <s xml:id="echoid-s18179" xml:space="preserve"> & angulus n m a erit angulus refractionis.</s> <s xml:id="echoid-s18180" xml:space="preserve"> Angulus igitur h e p aut erit æqua-<lb/>lis angulo n m l:</s> <s xml:id="echoid-s18181" xml:space="preserve"> aut erit minor:</s> <s xml:id="echoid-s18182" xml:space="preserve"> aut maior.</s> <s xml:id="echoid-s18183" xml:space="preserve"> Si æqualis:</s> <s xml:id="echoid-s18184" xml:space="preserve"> angulus h e a, qui eſt angulus refractionis:</s> <s xml:id="echoid-s18185" xml:space="preserve"> <lb/> <pb o="261" file="0267" n="267" rhead="OPTICAE LIBER VII."/> erit æqualis angulo n m a, qui eſt angulus refractionis [per 12 n:</s> <s xml:id="echoid-s18186" xml:space="preserve">] angulus ergo a m b erit æqua-<lb/>lis angulo a e b [per 13 p 1.</s> <s xml:id="echoid-s18187" xml:space="preserve"> 3 ax.</s> <s xml:id="echoid-s18188" xml:space="preserve">] quod eſt impoſsibile.</s> <s xml:id="echoid-s18189" xml:space="preserve"> [Ducta enim recta linea a b:</s> <s xml:id="echoid-s18190" xml:space="preserve"> erit angulus <lb/>a m b maior angulo a e b per 21 p 1.</s> <s xml:id="echoid-s18191" xml:space="preserve">] Si minor:</s> <s xml:id="echoid-s18192" xml:space="preserve"> erit [per 12 n] angulus h e a minor angulo n m a:</s> <s xml:id="echoid-s18193" xml:space="preserve"> <lb/>angulus ergo a m b erit minor angulo a e b [per 13 p 1] <lb/> <anchor type="figure" xlink:label="fig-0267-01a" xlink:href="fig-0267-01"/> quod eſt impoſsibile [& contra 21 p 1.</s> <s xml:id="echoid-s18194" xml:space="preserve">] Si maior:</s> <s xml:id="echoid-s18195" xml:space="preserve"> extra-<lb/>hamus lineam e b in partem b ad f:</s> <s xml:id="echoid-s18196" xml:space="preserve"> & extrahamus m b <lb/>uſque ad o:</s> <s xml:id="echoid-s18197" xml:space="preserve"> angulus ergo e m b erit æqualis angulo, qui <lb/>eſt apud circumferentiam, quem reſpiciunt duo arcus <lb/>e m, f o [per 24 n.</s> <s xml:id="echoid-s18198" xml:space="preserve">] Et cum [ex hypotheſi] angulus <lb/>h e p ſit maior angulo n m l:</s> <s xml:id="echoid-s18199" xml:space="preserve"> erit [per 15 p 1] angulus z <lb/>e b maior angulo n m l:</s> <s xml:id="echoid-s18200" xml:space="preserve"> & cum angulus z e b ſit maior <lb/>angulo n m l:</s> <s xml:id="echoid-s18201" xml:space="preserve"> angulus m z p erit maior angulo m b e.</s> <s xml:id="echoid-s18202" xml:space="preserve"> <lb/>[Nam quia in triangulis e b g, m z g, angulus b e g ma-<lb/>ior eſt angulo z m g per theſin & 15 p 1:</s> <s xml:id="echoid-s18203" xml:space="preserve"> & anguli ad g æ-<lb/>quantur per eandem:</s> <s xml:id="echoid-s18204" xml:space="preserve"> erit reliquus m z p maior reliquo <lb/>m b e per 32 p 1:</s> <s xml:id="echoid-s18205" xml:space="preserve">] & exceſſus anguli m z e ſupra angu-<lb/>lum m b e, erit æqualis exceſſui anguli z e b ſupra an-<lb/>gulum z m b:</s> <s xml:id="echoid-s18206" xml:space="preserve"> nam duo anguli apud gſunt æquales [per <lb/>15 p 1.</s> <s xml:id="echoid-s18207" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s18208" xml:space="preserve"> cum per 32 p 1 anguli trianguli z m g æquen-<lb/>tur angulis trianguli b e g:</s> <s xml:id="echoid-s18209" xml:space="preserve"> erunt exuperantiæ angulo-<lb/>rum m z e, z e b ſupra angulos m b e, z m b æquales.</s> <s xml:id="echoid-s18210" xml:space="preserve">] <lb/>Arcus uero, qui reſpicit angulum m z e, cũ fuerit apud <lb/>circumferentiam, erit duplus ad arcum m e.</s> <s xml:id="echoid-s18211" xml:space="preserve"> [Quia enim <lb/>angulus m z e duplus eſt anguli in peripheria, in ean-<lb/>dem peripheriam m e inſiſtentis per 20 p 3:</s> <s xml:id="echoid-s18212" xml:space="preserve"> ergo angulus <lb/>m z e in peripheria conſtitutus, inſiſtet in duplam peri-<lb/>pheriam m e per 33 p 6.</s> <s xml:id="echoid-s18213" xml:space="preserve">] Si ergo angulus m z e fuerit maior angulo m b e:</s> <s xml:id="echoid-s18214" xml:space="preserve"> tunc arcus m e dupli-<lb/>catus erit maior duobus arcubus m e, f o:</s> <s xml:id="echoid-s18215" xml:space="preserve"> & erit exceſſus arcus m e duplicati ſupra duos arcus <lb/>m e, f o, æqualis exceſſui arcus m e ſupra arcum f o [ſubducta enim communi peripheria m e, ſu <lb/>pereſt eadem exuperantia.</s> <s xml:id="echoid-s18216" xml:space="preserve">] Exceſſus ergo anguli m z e ſupra angulum m b e eſt iſte, quem re-<lb/>ſpicit apud circumferentiam exceſſus arcus m e ſupra arcum f o:</s> <s xml:id="echoid-s18217" xml:space="preserve"> ſed exceſſus arcus m e ſupra ar-<lb/>cum f o eſt minor duobus arcubus m e, f o [per 9 ax.</s> <s xml:id="echoid-s18218" xml:space="preserve">] Ergo exceſſus anguli m z e ſupra angu-<lb/>lum m b e, eſt minor angulo m b e [per 33 p 6.</s> <s xml:id="echoid-s18219" xml:space="preserve">] Exceſſus igitur anguli z e b ſupra angulum z m <lb/>b eſt minor angulo m b e:</s> <s xml:id="echoid-s18220" xml:space="preserve"> ergo [per 15 p 1] exceſſus anguli h e p ſupra angulum n m l eſt minor <lb/>angulo m b e.</s> <s xml:id="echoid-s18221" xml:space="preserve"> Ergo [per 12 n] exceſſus anguli h e a, qui eſt angulus refractionis, ſupra angulum <lb/>n m a, qui eſt angulus refractionis, eſt multò minor angulo m b e.</s> <s xml:id="echoid-s18222" xml:space="preserve"> Sed exceſſus anguli h e a ſu-<lb/>pra angulum n m a, eſt exceſſus anguli a m b ſupra angulum a e b [per 13 p 1.</s> <s xml:id="echoid-s18223" xml:space="preserve">] Ergo exceſſus an-<lb/>guli a m b ſupra angulum a e b eſt minor angulo m b e.</s> <s xml:id="echoid-s18224" xml:space="preserve"> Sed exceſſus anguli a m b ſupra angu-<lb/>lum a e b, ſunt duo anguli m a e, m b e.</s> <s xml:id="echoid-s18225" xml:space="preserve"> [Nam connexa recta a b & continuata e m ultra m in x:</s> <s xml:id="echoid-s18226" xml:space="preserve"> <lb/>æquabitur per 32 p 1 angulus a m x duobus interioribus ad a & e:</s> <s xml:id="echoid-s18227" xml:space="preserve"> itemq́ue b m x duobus interiori-<lb/>bus ad b & e.</s> <s xml:id="echoid-s18228" xml:space="preserve"> Totus igitur a m b exuperat totum a e b duobus angulis m a e, m b e.</s> <s xml:id="echoid-s18229" xml:space="preserve">] Ergo duo an-<lb/>guli m a e, m b e ſunt minores angulo m b e:</s> <s xml:id="echoid-s18230" xml:space="preserve"> quod eſt impoſsibile [& cõtra 9 ax.</s> <s xml:id="echoid-s18231" xml:space="preserve">] Forma ergo b non <lb/>refringetur ad a ex alio puncto, præterquam ex e.</s> <s xml:id="echoid-s18232" xml:space="preserve"> Et hoc eſt quod uoluimus.</s> <s xml:id="echoid-s18233" xml:space="preserve"> Cum ergo b non re-<lb/>fringatur ad a, niſi ex uno puncto:</s> <s xml:id="echoid-s18234" xml:space="preserve"> nec habebit, niſi unam imaginem.</s> <s xml:id="echoid-s18235" xml:space="preserve"> Sed locus imaginis diuerſatur <lb/>ſecundum diuerſitatem loci, in quo eſt b.</s> <s xml:id="echoid-s18236" xml:space="preserve"> Continuemus enim b z:</s> <s xml:id="echoid-s18237" xml:space="preserve"> linea ergo b z aut concurret cum <lb/>linea e a:</s> <s xml:id="echoid-s18238" xml:space="preserve"> aut erit ei æquidiſtans:</s> <s xml:id="echoid-s18239" xml:space="preserve"> & concurſus aut erit in parte e b, ut in k:</s> <s xml:id="echoid-s18240" xml:space="preserve"> aut in parte a, ut in r.</s> <s xml:id="echoid-s18241" xml:space="preserve"> Et <lb/>cum b z fuerit æquidiſtans lineæ e a:</s> <s xml:id="echoid-s18242" xml:space="preserve"> erit ut linea b z ſit media inter duas lineas k b z, b z r.</s> <s xml:id="echoid-s18243" xml:space="preserve"> Si uerò <lb/>concurſus harum duarum linearum fuerit in k:</s> <s xml:id="echoid-s18244" xml:space="preserve"> erit imago ante uiſum, & erit forma manifeſta & <lb/>comprehenſa à uiſu in k [per 18 n.</s> <s xml:id="echoid-s18245" xml:space="preserve">] Si uerò concurſus fuerit in r:</s> <s xml:id="echoid-s18246" xml:space="preserve"> erit imago punctum r:</s> <s xml:id="echoid-s18247" xml:space="preserve"> & tunc for <lb/>ma comprehendetur à uiſu in eius oppoſitione:</s> <s xml:id="echoid-s18248" xml:space="preserve"> ſed non tam manifeſtè, quia comprehenditur à ui-<lb/>ſu extra ſuum locum.</s> <s xml:id="echoid-s18249" xml:space="preserve"> Hoc autem declaratum eſt in loco, in quo locuti ſumus de reflexiõe [61 n 5.</s> <s xml:id="echoid-s18250" xml:space="preserve">] <lb/>Si linea b z fuerit æquidiſtans lineæ e a:</s> <s xml:id="echoid-s18251" xml:space="preserve"> tunc imago erit indeterminata, & forma comprehendetur <lb/>in loco refractionis.</s> <s xml:id="echoid-s18252" xml:space="preserve"> Huius autem cauſſa ſimilis eſt illi, quam diximus in loco reflexionis [61 n 5] <lb/>cum fuerit reflexio per lineam æquidiſtantem perpendiculari.</s> <s xml:id="echoid-s18253" xml:space="preserve"> Ex prædictis ergo patet, quòd res, <lb/>quæ comprehenditur à uiſu ultra corpus diaphanum groſsius corpore, quod eſt ex parte uiſus:</s> <s xml:id="echoid-s18254" xml:space="preserve"> nõ <lb/>habet, niſi unam imaginem, neq;</s> <s xml:id="echoid-s18255" xml:space="preserve"> comprehenditur, niſi unum tantùm.</s> <s xml:id="echoid-s18256" xml:space="preserve"> Hæc uerò refractio eſt à con-<lb/>cauitate corporis diaphani ex parte uιſus contingentis conuexum corporis diaphani, quod eſt ex <lb/>parte rei uiſæ.</s> <s xml:id="echoid-s18257" xml:space="preserve"> Et hoc eſt quod uoluimus.</s> <s xml:id="echoid-s18258" xml:space="preserve"/> </p> <div xml:id="echoid-div592" type="float" level="0" n="0"> <figure xlink:label="fig-0267-01" xlink:href="fig-0267-01a"> <variables xml:id="echoid-variables216" xml:space="preserve">a n r l c x m h e p z g b b f d o k</variables> </figure> </div> </div> <div xml:id="echoid-div594" type="section" level="0" n="0"> <head xml:id="echoid-head511" xml:space="preserve" style="it">28. Si communis ſectio ſuperficierum refractionis & refractiui conuexi rarioris fuerit peri <lb/>pherιa: uiſibile extra perpendicularem à uiſu ſuper refractiuum ductam: ab uno puncto refrin <lb/>getur, unaḿ habebit imaginem, uariè pro uaria uiſ{us} ueluiſibilis poſitione ſit am. 24 p 10.</head> <p> <s xml:id="echoid-s18259" xml:space="preserve">ET ſi corpus diaphanum fuerit groſsius ex parte uiſus, & ſubtilius ex parte rei uiſæ:</s> <s xml:id="echoid-s18260" xml:space="preserve"> tunc <lb/> <pb o="262" file="0268" n="268" rhead="ALHAZEN"/> uiſus non uidebit niſi unam ſolam imaginem.</s> <s xml:id="echoid-s18261" xml:space="preserve"> Nam tunc uiſus erit ut b:</s> <s xml:id="echoid-s18262" xml:space="preserve"> & res uiſa ut a.</s> <s xml:id="echoid-s18263" xml:space="preserve"> Et cum for-<lb/>ma a refringetur ad b:</s> <s xml:id="echoid-s18264" xml:space="preserve"> refractio erit in ſuperficie perpendiculari ſuper ſuperficiem corporis diapha <lb/>ni [per 9 n] & erit differentia communis inter illam ſuperficiem & ſuperficiem corporis diaphani <lb/>circulus [per 1 th 1 ſphæricorum,] ut circulus c e d:</s> <s xml:id="echoid-s18265" xml:space="preserve"> & erit punctum refractionis, ut e:</s> <s xml:id="echoid-s18266" xml:space="preserve"> & erit linea re <lb/>fracta, ut a e k.</s> <s xml:id="echoid-s18267" xml:space="preserve"> Sequitur ergo, ut forma, quę extendetur per lineam a e, & refringetur per b e:</s> <s xml:id="echoid-s18268" xml:space="preserve"> exten-<lb/>datur ex b per lineam b e, & refringatur per lineam a e.</s> <s xml:id="echoid-s18269" xml:space="preserve"> Si ergo forma a refringitur ad b ex alio pun-<lb/>cto quàm ex e:</s> <s xml:id="echoid-s18270" xml:space="preserve"> ſequetur quòd forma b refringetur ad a ex illo puncto.</s> <s xml:id="echoid-s18271" xml:space="preserve"> [Quia lineæ incidentiæ & <lb/>refractionis eædem permanent, nominibus tantùm mutatis.</s> <s xml:id="echoid-s18272" xml:space="preserve">] Sed iam declaratum eſt [ſuperiore <lb/>numero] quòd cum forma extenſa fuerit per lineam b e, & refracta per lineam a e:</s> <s xml:id="echoid-s18273" xml:space="preserve"> nunquam refrin <lb/>getur, niſi ex puncto uno, nec habebit niſi unam imaginem.</s> <s xml:id="echoid-s18274" xml:space="preserve"> Et ſi a fuerit in perpendiculari exeunte <lb/>ex b ad centrum ſphæræ:</s> <s xml:id="echoid-s18275" xml:space="preserve"> tunc b comprehendet a in rectitudine perpendicularis [per 13 n] & patet, <lb/>quòd forma a non refringetur ad b.</s> <s xml:id="echoid-s18276" xml:space="preserve"> Ex quo patuit, quòd forma b, cum fuerit in perpendiculari, nõ <lb/>refringetur ad a.</s> <s xml:id="echoid-s18277" xml:space="preserve"> Cum ergo groſsius corpus fuerit ex parte uiſus, & ſubtilius ex parte rei uiſæ:</s> <s xml:id="echoid-s18278" xml:space="preserve"> tunc <lb/>res uiſa non habebit, niſi unam imaginem & unam formam tantùm.</s> <s xml:id="echoid-s18279" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div595" type="section" level="0" n="0"> <head xml:id="echoid-head512" xml:space="preserve" style="it">29. Si uiſ{us} ſit extra circulum (qui eſt communis ſectio ſuperficierum, refractionis & re-<lb/>fractiui ſphærici conuexi denſioris) linea recta in definito ſitu poteſt à ſegmento peripheriæ nõ <lb/>magnæ refringi: & aliquod ei{us} punctum rectè: è reliquis plura refractè uideri: & locus to-<lb/>ti{us} imaginis est in ipſo uiſu. 25 p 10.</head> <p> <s xml:id="echoid-s18280" xml:space="preserve">ITem:</s> <s xml:id="echoid-s18281" xml:space="preserve"> iteremus figuram ponentes in circumferentia g e d, punctum ex parte g:</s> <s xml:id="echoid-s18282" xml:space="preserve"> & ſit e:</s> <s xml:id="echoid-s18283" xml:space="preserve"> ex quo <lb/>extrahamus lineam æquidiſtantem lineæ a b [per 31 p 1:</s> <s xml:id="echoid-s18284" xml:space="preserve">] & ſit linea e t:</s> <s xml:id="echoid-s18285" xml:space="preserve"> & continuemus z e, & <lb/>extrahamus illam uſque ad h:</s> <s xml:id="echoid-s18286" xml:space="preserve"> & ſit proportio anguli z e k ad angulum k e t duplicatum maxima <lb/>proportio, quam angulus, quem continet linea, per quam extenditur forma cum perpendiculari, <lb/>poſsit habere ad angulum refractionis, quem exigit ille angulus, quò ad ſenſum.</s> <s xml:id="echoid-s18287" xml:space="preserve"> [Id autem per 10.</s> <s xml:id="echoid-s18288" xml:space="preserve"> <lb/>11.</s> <s xml:id="echoid-s18289" xml:space="preserve"> 12 n præſtari poteſt, quibus anguli refractionum à medio craſsiore ad ſubtilius & contrà, inuen-<lb/>ti ſunt.</s> <s xml:id="echoid-s18290" xml:space="preserve">] Anguli enim refractionis, qui fuerint inter duo corpora diuerſa in diaphanitate, à luce <lb/>tranſeunte per illa diuerſantur:</s> <s xml:id="echoid-s18291" xml:space="preserve"> quorum diuerſitas, quò ad ſenſum, habet finem:</s> <s xml:id="echoid-s18292" xml:space="preserve"> quem ſi exceſſerit:</s> <s xml:id="echoid-s18293" xml:space="preserve"> <lb/>ſenſus non comprehendet quantitatem refractionis:</s> <s xml:id="echoid-s18294" xml:space="preserve"> comprehendet enim centrum lucis in rectitu <lb/>dine lineæ, per quam lux extenditur, cum uidelicet experimentatus fuerit hoc per inſtrumentum.</s> <s xml:id="echoid-s18295" xml:space="preserve"> <lb/>Et ponamus angulum d z t æqualem angulo k e t [per 23 p 1] erit ergo angulus z k e duplus ad an-<lb/>gulum k e t.</s> <s xml:id="echoid-s18296" xml:space="preserve"> [Quia enim e t, z b ſunt parallelæ per fabricationem:</s> <s xml:id="echoid-s18297" xml:space="preserve"> æquatur angulus k b z angulo <lb/>k e t per 29 p 1:</s> <s xml:id="echoid-s18298" xml:space="preserve"> cui iam æquatus eſt k z b:</s> <s xml:id="echoid-s18299" xml:space="preserve"> anguli igitur k b z, k z b ſunt æquales:</s> <s xml:id="echoid-s18300" xml:space="preserve"> quibus cum æque-<lb/>tur z k e per 32 p 1:</s> <s xml:id="echoid-s18301" xml:space="preserve"> erit duplus ad utrumlibet:</s> <s xml:id="echoid-s18302" xml:space="preserve"> itaque duplus ad ęqua-<lb/> <anchor type="figure" xlink:label="fig-0268-01a" xlink:href="fig-0268-01"/> lẽ k e t] & ſic proportio anguli z e k ad angulũ z k e erit maxima pro-<lb/>portio inter angulum, quem continet prima linea & perpendicula-<lb/>ris, exiens à puncto refractionis, & inter angulum refractionis.</s> <s xml:id="echoid-s18303" xml:space="preserve"> Sed <lb/>linea e k concurret cum linea a d:</s> <s xml:id="echoid-s18304" xml:space="preserve"> [per lemma Procli ad 29 p 1] con-<lb/>currant ergo in b:</s> <s xml:id="echoid-s18305" xml:space="preserve"> & extrahamus ex e lineam æquidiſtantem t z:</s> <s xml:id="echoid-s18306" xml:space="preserve"> con <lb/>curret ergo [ut antè] cum z g extra circulum ex parte g:</s> <s xml:id="echoid-s18307" xml:space="preserve"> ſit concur-<lb/>ſus in a:</s> <s xml:id="echoid-s18308" xml:space="preserve"> & extrahamus b e uſq;</s> <s xml:id="echoid-s18309" xml:space="preserve"> ad l:</s> <s xml:id="echoid-s18310" xml:space="preserve"> erit ergo [per 29 p 1] angulus l <lb/>e a æqualis angulo z k e:</s> <s xml:id="echoid-s18311" xml:space="preserve"> & [per 15 p 1] angulus l e h æqualis angulo <lb/>z e k.</s> <s xml:id="echoid-s18312" xml:space="preserve"> Erit ergo angulus l e a angulus refractionis, quẽ exigit angulus <lb/>l e h [angulus enim z e k, qui ք 15 p 1 æquatur angulo l e h, talis eſt ex <lb/>theſi.</s> <s xml:id="echoid-s18313" xml:space="preserve">] Si ergo b fuerit in aliquo uiſo:</s> <s xml:id="echoid-s18314" xml:space="preserve"> & corpus diaphanũ, cuius con <lb/>uexum eſt ex parte a, fuerit continuatum ex e uſque ad b, & nõ fue-<lb/>rit diſtinctum apud circum ferentiam g e d ex parte b:</s> <s xml:id="echoid-s18315" xml:space="preserve"> tunc forma b <lb/>extendetur per lineam b e, & refringetur per lineam e a, & compre-<lb/>hendetur à uiſu a peruerticationem a e.</s> <s xml:id="echoid-s18316" xml:space="preserve"> Et quia angulus a e h poteſt <lb/>diuidi pluribus proportionibus earum, quæ fuerint inter angulos re <lb/>fractionis, & angulos, quos continent perpendiculares cum primis <lb/>lincis, quæ fuerint inter duo corpora diaphana:</s> <s xml:id="echoid-s18317" xml:space="preserve"> ſic ergo in linea d b <lb/>erunt plura puncta, quorum formæ extenduntur ad arcum e g, & re-<lb/>fringuntur ad a:</s> <s xml:id="echoid-s18318" xml:space="preserve"> & forma totius lineæ, in qua ſunt illa puncta, refrin-<lb/>getur ad a ex arcu g e.</s> <s xml:id="echoid-s18319" xml:space="preserve"> Cum ergo uiſus fuerit in corpore diaphano, <lb/>& res uiſa fuerit in alio diaphano groſsiore, & fuerit ſuperficies dia-<lb/>phani groſsioris, quæ eſt ex parte uiſus, ſphærica conuexa, & uiſus <lb/>fuerit extra circulum, cuius conuexum eſt ex parte uiſus, & fuerit il-<lb/>le circulus remotior à uiſu, quàm punctum remotius ex duobus pun <lb/>ctis, ſectionis factæ inter perpendicularem & circumferentiam, & <lb/>corpus diaphanum groſſum, quod eſt ex parte uiſus, fuerit conti-<lb/>nuum uſque ad locum, in quo eſt res uiſa, & non fuerit deciſum a-<lb/>pud circulum, qui eſt ex parte rei uiſæ:</s> <s xml:id="echoid-s18320" xml:space="preserve"> tunc uiſus poterit comprehendere illam rem uiſam & re-<lb/> <pb o="263" file="0269" n="269" rhead="OPTICAE LIBER VII."/> fractè & rectè:</s> <s xml:id="echoid-s18321" xml:space="preserve"> & huius rei uiſæ imago erit centrum uiſus [per 13 n.</s> <s xml:id="echoid-s18322" xml:space="preserve">] Item ſi fixerimus lineam <lb/>a g b, & reuoluerimus figuram a e b in circuitu a b, & pars ſuperficiei corporis diaphani, quod <lb/>eſt ex parte rei uiſæ, fuerit ſphærica:</s> <s xml:id="echoid-s18323" xml:space="preserve"> tuncpunctum e ſignabit circumferentiam in ſuperficie cir-<lb/>culari conuexa, quæ eſt ex parte uiſus, ex qua circumferentia refringetur b ad a:</s> <s xml:id="echoid-s18324" xml:space="preserve"> ſed imago in to-<lb/>ta circumferentia refractionis erit una, ſcilicet centrum uiſus.</s> <s xml:id="echoid-s18325" xml:space="preserve"> Imago ergo rei uiſæ etiam erit u-<lb/>na.</s> <s xml:id="echoid-s18326" xml:space="preserve"> Sed ex hac poſitione accidit, ut uiſus comprehendat formam rei uiſæ apud locum refra-<lb/>ctionis ea de cauſſa, quam diximus in reflexione ex ſpeculis, [61 n 5] cum fuerit reflexio à <lb/>circumferentia in aliqua ſphæra, & fuerit imago centrum uiſus.</s> <s xml:id="echoid-s18327" xml:space="preserve"> Ergo huius rei uiſæ forma à <lb/>uiſu circularis comprehenditur apud circulum refractionis:</s> <s xml:id="echoid-s18328" xml:space="preserve"> & punctum eius ſuperius circa d ui-<lb/>detur in rectitudine perpendicularis, tranſeuntis per uiſum & rem uiſam ſimul.</s> <s xml:id="echoid-s18329" xml:space="preserve"> Et hoc eſt quod <lb/>uoluimus.</s> <s xml:id="echoid-s18330" xml:space="preserve"/> </p> <div xml:id="echoid-div595" type="float" level="0" n="0"> <figure xlink:label="fig-0268-01" xlink:href="fig-0268-01a"> <variables xml:id="echoid-variables217" xml:space="preserve">a l g h e z d k b t</variables> </figure> </div> </div> <div xml:id="echoid-div597" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables218" xml:space="preserve">e a g e z b</variables> </figure> <head xml:id="echoid-head513" xml:space="preserve" style="it">30. Si communis ſectio ſuperficierum, refractionis & refractiui caui, denſioris fuerit peri-<lb/>pheria: uiſibile in perpendiculari à uiſu ſuper refractiuum ducta, re <lb/> ctè: & unum uidebitur. 26 p 10.</head> <p> <s xml:id="echoid-s18331" xml:space="preserve">ITem:</s> <s xml:id="echoid-s18332" xml:space="preserve"> ſit a uiſus:</s> <s xml:id="echoid-s18333" xml:space="preserve"> & ſit b in aliquo uiſo, & ultra corpus diaphanum <lb/>groſsius illo, in quo eſt uiſus:</s> <s xml:id="echoid-s18334" xml:space="preserve"> & ſit ſuperficies corporis, quod eſt ex <lb/>parte uiſus, circularis concaua:</s> <s xml:id="echoid-s18335" xml:space="preserve"> cuius concauitas ſit ex parte uiſus.</s> <s xml:id="echoid-s18336" xml:space="preserve"> <lb/>Dico ergo, quòd b unam ſolam habebit imaginem, & unam tãtùm for-<lb/>mam apud a.</s> <s xml:id="echoid-s18337" xml:space="preserve"> Et ſit centrum concauitatis g:</s> <s xml:id="echoid-s18338" xml:space="preserve"> & continuemus a g:</s> <s xml:id="echoid-s18339" xml:space="preserve"> & ex-<lb/>trahamus ipſam rectè uſque ad z.</s> <s xml:id="echoid-s18340" xml:space="preserve"> Erit ergo a z perpendicularis ſuper ſu <lb/>perficiem concauam:</s> <s xml:id="echoid-s18341" xml:space="preserve"> [ut oſtenſum eſt 25 n 4:</s> <s xml:id="echoid-s18342" xml:space="preserve">] & b aut erit in a z, aut <lb/>extra.</s> <s xml:id="echoid-s18343" xml:space="preserve"> Sit ergo primò in linea a z.</s> <s xml:id="echoid-s18344" xml:space="preserve"> A ergo comprehendet b in rectitudi-<lb/>ne a b, cum a b ſit perpendicularis ſuper ſuperficiem concauam, & nun <lb/>quam refractè [per 13 n.</s> <s xml:id="echoid-s18345" xml:space="preserve">] Quòd ſi eſt poſsibile, refringatur forma b ad <lb/>a ex e, & continuemus b e, g e, & extrahamus b e uſque ad t:</s> <s xml:id="echoid-s18346" xml:space="preserve"> angulus er-<lb/>go t e g eſt ille, quem continet linea, per quam extenditur forma, & per-<lb/>pendicularis exiens à loco refractionis.</s> <s xml:id="echoid-s18347" xml:space="preserve"> Et quia corpus, quod eſt ex par <lb/>te a, ſubtilius eſt illo, quod eſt ex parte b:</s> <s xml:id="echoid-s18348" xml:space="preserve"> erit [per 14 n] refractio ad par <lb/>tem contrariam illi, in qua eſt e g.</s> <s xml:id="echoid-s18349" xml:space="preserve"> Linea ergo e t, quan do refringitur, re-<lb/>mouetur à linea e g:</s> <s xml:id="echoid-s18350" xml:space="preserve"> & non concurret cum linea b a aliquo modo.</s> <s xml:id="echoid-s18351" xml:space="preserve"> For-<lb/>ma ergo b non refringetur ad a:</s> <s xml:id="echoid-s18352" xml:space="preserve"> non ergo comprehẽdetur refractè, ſed <lb/>rectè:</s> <s xml:id="echoid-s18353" xml:space="preserve"> ergo non habebit apud uiſum, niſi unam formam tantùm.</s> <s xml:id="echoid-s18354" xml:space="preserve"> Et hoc <lb/>eſt quod uoluimus.</s> <s xml:id="echoid-s18355" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div598" type="section" level="0" n="0"> <head xml:id="echoid-head514" xml:space="preserve" style="it">31. Si communis ſectio ſuperficierum, refractionis & refractiui <lb/>caui, denſioris fuerit peripheria: uiſibile extra perpendicularem à ui <lb/>ſu ſuper refractiuum ductam, ab uno puncto refringetur, unaḿ <lb/>habebit imaginem, uariè pro uaria uiſ{us} uel uiſibilis poſitione ſi-<lb/>tam. 27 p 10.</head> <p> <s xml:id="echoid-s18356" xml:space="preserve">ITem:</s> <s xml:id="echoid-s18357" xml:space="preserve"> iteremus figurã, & ſit b extra lineam a z, & extrahamus ſuperficiem, in qua eſt a z b:</s> <s xml:id="echoid-s18358" xml:space="preserve"> Hæc <lb/>ergo ſuperficies erit perpendicularis ſuper ſuperficiem concauam [per 9 n] & non refringetur <lb/>forma b ad a, niſi in hac ſuperficie.</s> <s xml:id="echoid-s18359" xml:space="preserve"> Non enim erigitur perpendicularis ſuper ſuperficiem con-<lb/>cauam alia ſuperficies æqualis, quæ tranſit per a, niſi illa, quæ tranſit per a z:</s> <s xml:id="echoid-s18360" xml:space="preserve"> ſed per a z & per b non <lb/>tranſit, niſi una ſola tantùm.</s> <s xml:id="echoid-s18361" xml:space="preserve"> Forma ergo b non refringetur ad a, niſi in ſuperficie tranſeunte per li-<lb/>neam a z, & per b.</s> <s xml:id="echoid-s18362" xml:space="preserve"> Et ſit differentia communis inter hanc ſuperficiem & ſuperficiem concauam ar-<lb/>cus h d e, & refringatur forma b ad a ex h.</s> <s xml:id="echoid-s18363" xml:space="preserve"> Dico ergo, quòd non refringetur ex alio puncto.</s> <s xml:id="echoid-s18364" xml:space="preserve"> Quòd ſi <lb/>poſsibile fuerit, refringatur ex m, & continuemus lineas a h, b h, g h, a m, b m, g m, & extrahamus h b <lb/>rectè uſque ad c, & b m rectè uſque ad n, & g h rectè uſq;</s> <s xml:id="echoid-s18365" xml:space="preserve"> ad l, & g m rectè uſque ad p, & perficiamus <lb/>circumferentiam h e d, & ſecet lineam a g in k.</s> <s xml:id="echoid-s18366" xml:space="preserve"> A ergo aut erit in linea k d:</s> <s xml:id="echoid-s18367" xml:space="preserve"> aut extrà in parte k, [quia <lb/>ea pars obiecta eſt cauæ refractiui ſuperficiei, à qua refractio fit ad uiſum a.</s> <s xml:id="echoid-s18368" xml:space="preserve">] ſi ergo a fuerit in k d, <lb/>aut erit in g, aut in altera duarum linearumg d, g k.</s> <s xml:id="echoid-s18369" xml:space="preserve"> Si ergo fuerit a in g:</s> <s xml:id="echoid-s18370" xml:space="preserve"> tunc forma b non refrin-<lb/>getur a d a [per præcedentem numerum:</s> <s xml:id="echoid-s18371" xml:space="preserve">] lineæ enim, quæ continuant corpus circulare cum g, <lb/>ſunt perpendiculares ſuper ſuperficiem corporis, [per 25 n 4,] quod eſt ex parte a:</s> <s xml:id="echoid-s18372" xml:space="preserve"> Refractio au-<lb/>tem non fit per ipſam perpendicularem, ſed extra ipſam.</s> <s xml:id="echoid-s18373" xml:space="preserve"> Forma ergo b non refringetur ad a, ſi a <lb/>fuerit in g.</s> <s xml:id="echoid-s18374" xml:space="preserve"> Et ſi a fuerit in g d:</s> <s xml:id="echoid-s18375" xml:space="preserve"> tunc linea h c erit inter duas lineas h a, h g:</s> <s xml:id="echoid-s18376" xml:space="preserve"> & ideo linea n m erit <lb/>inter duas lineas m a, m g.</s> <s xml:id="echoid-s18377" xml:space="preserve"> Nam refractio eſt ad partem contrariam partι perpendicularis, [per <lb/>14 n] nam corpus diaphanum, quod eſt ex parte uiſus, eſt ſubtilius illo, quod eſt ex parte rei ui-<lb/>ſæ.</s> <s xml:id="echoid-s18378" xml:space="preserve"> Et ſi linea h c fuerit inter duas lineas h a, h g, & a fuerit in linea g d:</s> <s xml:id="echoid-s18379" xml:space="preserve"> tunc angulus b h a e-<lb/>rit ex parte d:</s> <s xml:id="echoid-s18380" xml:space="preserve"> & ſimiliter angulus b m a erit ex parte d:</s> <s xml:id="echoid-s18381" xml:space="preserve"> & erit b ultra lineam g h l, uidelicet ex par-<lb/>te k, à linea h g l.</s> <s xml:id="echoid-s18382" xml:space="preserve"> Et erit angulus c h g ille, quem continet linea, per quam extenditur forma cum <lb/> <pb o="264" file="0270" n="270" rhead="ALHAZEN"/> perpendiculari exeunte à loco refractionis:</s> <s xml:id="echoid-s18383" xml:space="preserve"> & ſimiliter angulus n m g:</s> <s xml:id="echoid-s18384" xml:space="preserve"> & erit angulus c h a angulus <lb/>refractionis:</s> <s xml:id="echoid-s18385" xml:space="preserve"> & ſimiliter angulus n m a.</s> <s xml:id="echoid-s18386" xml:space="preserve"> Angulus autem n m g aut erit æqualis angulo c h g, aut ma-<lb/>ior, aut minor.</s> <s xml:id="echoid-s18387" xml:space="preserve"> Si æqualis:</s> <s xml:id="echoid-s18388" xml:space="preserve"> erit [per 12 n] n m a æqualis <lb/> <anchor type="figure" xlink:label="fig-0270-01a" xlink:href="fig-0270-01"/> angulo a h c:</s> <s xml:id="echoid-s18389" xml:space="preserve"> ergo [per 13 p 1.</s> <s xml:id="echoid-s18390" xml:space="preserve"> 3 ax.</s> <s xml:id="echoid-s18391" xml:space="preserve">] angulus b h a erit æ-<lb/>qualis angulo b m a:</s> <s xml:id="echoid-s18392" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s18393" xml:space="preserve"> [Ducta enim <lb/>recta b a:</s> <s xml:id="echoid-s18394" xml:space="preserve"> erit angulus b m a maior angulo b h a per 21 <lb/>p 1.</s> <s xml:id="echoid-s18395" xml:space="preserve">] Si maior:</s> <s xml:id="echoid-s18396" xml:space="preserve"> tunc [per 12 n] angulus n m a erit maior <lb/>angulo a h c:</s> <s xml:id="echoid-s18397" xml:space="preserve"> & ſic [per 13 p 1.</s> <s xml:id="echoid-s18398" xml:space="preserve"> 3 ax.</s> <s xml:id="echoid-s18399" xml:space="preserve">] angulus b m a erit mi-<lb/>nor angulo b h a:</s> <s xml:id="echoid-s18400" xml:space="preserve"> quod eſt impoſsibile [& contra 21 p 1.</s> <s xml:id="echoid-s18401" xml:space="preserve">] <lb/>Si minor:</s> <s xml:id="echoid-s18402" xml:space="preserve"> tunc [per 12 n] angulus n m a erit minor angu <lb/>lo a h c:</s> <s xml:id="echoid-s18403" xml:space="preserve"> & ſic totus angulus a m g erit minor toto angulo <lb/>a h g:</s> <s xml:id="echoid-s18404" xml:space="preserve"> & erit [per 12 n] diminutio anguli n m a, ab angu-<lb/>lo a h c minor, quàm diminutio anguli a m g, ab angulo <lb/>a h g:</s> <s xml:id="echoid-s18405" xml:space="preserve"> Sed diminutio anguli a m g ab angulo a h g, eſt æ-<lb/>qualis diminutioni anguli h g m ab angulo h a m:</s> <s xml:id="echoid-s18406" xml:space="preserve"> duo <lb/>enim anguli, qui ſunt in ſectione linearum a h, m g ſunt <lb/>æquales [per 15 p 1:</s> <s xml:id="echoid-s18407" xml:space="preserve"> & per 32 p 1 reliquus ſimul uterque <lb/>trianguli h g fæquatur reliquo ſimul utrique trianguli <lb/>m a f.</s> <s xml:id="echoid-s18408" xml:space="preserve"> Itaque quantò minor eſt angulus a m g angulo a h <lb/>g:</s> <s xml:id="echoid-s18409" xml:space="preserve"> tãtò minor erit angulus h g m angulo h a m per 32 p 1.</s> <s xml:id="echoid-s18410" xml:space="preserve">] <lb/>Ergo diminutio anguli n m a ab angulo a h c minor eſt, <lb/>quàm diminutio anguli h g m ab angulo h a m.</s> <s xml:id="echoid-s18411" xml:space="preserve"> Et extra-<lb/>hamus duas a h, m a ad duo puncta e, o:</s> <s xml:id="echoid-s18412" xml:space="preserve"> erit ergo [per 24 <lb/>n] angulus h a m ille, quem reſpiciunt in circumferen-<lb/>tia duo arcus h m, e o:</s> <s xml:id="echoid-s18413" xml:space="preserve"> & angulũ h g m reſpicit in circũ-<lb/>ferentia arcus h m duplicatus [angulus enim h g m du-<lb/>plus eſt anguli in peripheria conſtituti, & in eandẽ peri-<lb/>pheriã h m inſiſtentis per 20 p 3.</s> <s xml:id="echoid-s18414" xml:space="preserve"> Si igitur angulus, æqua-<lb/>lis angulo h g m in peripheria conſtituatur:</s> <s xml:id="echoid-s18415" xml:space="preserve"> inſiſtet in pe <lb/>ripheriam duplã peripheriæ h m per 33 p 6.</s> <s xml:id="echoid-s18416" xml:space="preserve">] Et cum angulus h g m ſit minor angulo h a m:</s> <s xml:id="echoid-s18417" xml:space="preserve"> [angu-<lb/>lus enim a h g maior eſt concluſus angulo a m g:</s> <s xml:id="echoid-s18418" xml:space="preserve"> & ad uerticem f ęquantur per 15 p 1:</s> <s xml:id="echoid-s18419" xml:space="preserve"> reliquus igitur <lb/>h g m minor eſt reliquo h a m per 32 p 1] erit arcus h m duplicatus minor duobus arcubus h m, e o <lb/>[per 33 p 6:</s> <s xml:id="echoid-s18420" xml:space="preserve">] & erit dimin utio arcus h m duplicati à duobus arcubus h m, e o, ſicut diminutio ar-<lb/>cus h m ab arcu e o [quia h m communis eſt.</s> <s xml:id="echoid-s18421" xml:space="preserve">] Ergo diminutio anguli n m a ab angulo a h c erit mi <lb/>nor angulo, quem reſpicit apud circumferentiam dimi-<lb/> <anchor type="figure" xlink:label="fig-0270-02a" xlink:href="fig-0270-02"/> nutio arcus h m ab arcu e o.</s> <s xml:id="echoid-s18422" xml:space="preserve"> Sed angulus, quẽ reſpicit a-<lb/>pud circumferẽtiam diminutio arcus h m ab arcu e o, eſt <lb/>minor angulo h a m.</s> <s xml:id="echoid-s18423" xml:space="preserve"> Eſt ergo diminutio anguli n m a ab <lb/>angulo a h c minor angulo h a m.</s> <s xml:id="echoid-s18424" xml:space="preserve"> Exceſſus ergo anguli <lb/>b m a ſupra angulũ b h a eſt minor, quàm angulus h a m.</s> <s xml:id="echoid-s18425" xml:space="preserve"> <lb/>[Nam per 13 p 1 exuperantia anguli b m a ſupra angulum <lb/>b h a eſt exuperantia anguli a h c ſupra angulum n m a, <lb/>quæ minor eſt concluſa angulo h a m.</s> <s xml:id="echoid-s18426" xml:space="preserve">] Sed exceſſus an-<lb/>guli b m a ſupra angulum b h a ſunt duo anguli h a m, h b <lb/>m, [ut oſtenſum eſt 27 n.</s> <s xml:id="echoid-s18427" xml:space="preserve">] Ergo iſtí duo anguli ſimul <lb/>ſunt minores angulo h a m:</s> <s xml:id="echoid-s18428" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s18429" xml:space="preserve"> Et <lb/>ſi a fuerit in linea g k:</s> <s xml:id="echoid-s18430" xml:space="preserve"> tunc linea h c erit inter duas lineas <lb/>h g, h a:</s> <s xml:id="echoid-s18431" xml:space="preserve"> & ſimiliter linea m n erit inter duas lineas m g, <lb/>m a:</s> <s xml:id="echoid-s18432" xml:space="preserve"> Erit ergo angulus b h a ex parte k:</s> <s xml:id="echoid-s18433" xml:space="preserve"> & ſimiliter angu-<lb/>lus b m a erit ex parte k:</s> <s xml:id="echoid-s18434" xml:space="preserve"> & erit b infra lineam g m p, ſci-<lb/>licet ex parte d, à linea g m p:</s> <s xml:id="echoid-s18435" xml:space="preserve"> & uterque angulus c h g.</s> <s xml:id="echoid-s18436" xml:space="preserve"> n <lb/>m g eſt ille, quem continet linea, per quam extẽditur for-<lb/>ma, & perpendicularis exiens à loco refractionis:</s> <s xml:id="echoid-s18437" xml:space="preserve"> & uter-<lb/>que angulus c h a, n m a erit angulus refractionis.</s> <s xml:id="echoid-s18438" xml:space="preserve"> Si ergo <lb/>c h g fuerit æqualis n m g:</s> <s xml:id="echoid-s18439" xml:space="preserve"> tunc [per 12 n] angulus c h a e-<lb/>rit æqualis angulo n m a:</s> <s xml:id="echoid-s18440" xml:space="preserve"> & ſic [per 13 p 1] angulus b h a <lb/>erit æqualis angulo b m a:</s> <s xml:id="echoid-s18441" xml:space="preserve"> quod eſt impoſsibile [& con-<lb/>tra 21 p 1, connexa recta b a.</s> <s xml:id="echoid-s18442" xml:space="preserve">] Et ſi fuerit maior:</s> <s xml:id="echoid-s18443" xml:space="preserve"> tunc [per <lb/>12 n] angulus c h a erit maior angulo n m a:</s> <s xml:id="echoid-s18444" xml:space="preserve"> & ſic [per 13 <lb/>p 1] angulus b h a erit minor angulo b m a:</s> <s xml:id="echoid-s18445" xml:space="preserve"> quod eſt im-<lb/>poſsibile.</s> <s xml:id="echoid-s18446" xml:space="preserve"> Et ſi fuerit minor:</s> <s xml:id="echoid-s18447" xml:space="preserve"> tunc [per 12 n] angulus c h a <lb/>erit minor angulo n m a:</s> <s xml:id="echoid-s18448" xml:space="preserve"> & ſic totus angulus g h a erit minor toto angulo g m a:</s> <s xml:id="echoid-s18449" xml:space="preserve"> Ergo [ut ſuprà o-<lb/>ſtenſum eſt] erit angulus h g m minor angulo h a m.</s> <s xml:id="echoid-s18450" xml:space="preserve"> Et erit diminutio anguli h g m ab angulo h a m <lb/>minor, quàm angulus g m a, ut prius declarauimus.</s> <s xml:id="echoid-s18451" xml:space="preserve"> Et diminutio anguli c h a ab angulo n m a eſt <lb/> <pb o="265" file="0271" n="271" rhead="OPTICAE LIBER VII."/> minor, quàm diminutio anguli g h a ab angulo g m a:</s> <s xml:id="echoid-s18452" xml:space="preserve"> eſt ergo minor, quàm diminutio anguli h g m <lb/>ab angulo h a m:</s> <s xml:id="echoid-s18453" xml:space="preserve"> ergo diminutio anguli c h a ab angulo <lb/> <anchor type="figure" xlink:label="fig-0271-01a" xlink:href="fig-0271-01"/> n m a eſt minor, quàm angulus g m a:</s> <s xml:id="echoid-s18454" xml:space="preserve"> Sed diminutio an-<lb/>guli c h a ab angulo n m a, eſt exceſſus anguli b h a ſu-<lb/>per angulum b m a [per 13 p 1,] qui ſunt duo anguli h a m, <lb/>h b m [ut patuit 27 n.</s> <s xml:id="echoid-s18455" xml:space="preserve">] Ergo iſti duo anguli ſimul ſunt mi <lb/>nores angulo h a m:</s> <s xml:id="echoid-s18456" xml:space="preserve"> qđ eſt impoſsibile.</s> <s xml:id="echoid-s18457" xml:space="preserve"> Si uerò a fuerit <lb/>extra lineã k d ad partem k:</s> <s xml:id="echoid-s18458" xml:space="preserve"> & corpus, in quo eſt a, fuerit <lb/>cõtinuũ uſq;</s> <s xml:id="echoid-s18459" xml:space="preserve"> ad a:</s> <s xml:id="echoid-s18460" xml:space="preserve"> cõtinuabimus duas lineas a h, a m:</s> <s xml:id="echoid-s18461" xml:space="preserve"> & <lb/>ſecabũt circumferentiã in q & in r.</s> <s xml:id="echoid-s18462" xml:space="preserve"> Et ſi angulus c h g fue <lb/>rit æqualis angulo n m g:</s> <s xml:id="echoid-s18463" xml:space="preserve"> tunc [per 12 n] angulus b h a <lb/>erit æqualis angulo b m a:</s> <s xml:id="echoid-s18464" xml:space="preserve"> quod eſt impoſsibile [ut ſu-<lb/>prà.</s> <s xml:id="echoid-s18465" xml:space="preserve">] Et ſi fuerit maior:</s> <s xml:id="echoid-s18466" xml:space="preserve"> tunc angulus c h a erit maior an-<lb/>gulo n m a:</s> <s xml:id="echoid-s18467" xml:space="preserve"> & ſic [per 13 p 1] angulus b h a erit minor <lb/>angulo b m a:</s> <s xml:id="echoid-s18468" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s18469" xml:space="preserve"> Si uerò fuerit mi-<lb/>nor:</s> <s xml:id="echoid-s18470" xml:space="preserve"> tunc angulus c h a erit minor angulo n m a:</s> <s xml:id="echoid-s18471" xml:space="preserve"> & to-<lb/>tus angulus g h a erit minor toto angulo g m a.</s> <s xml:id="echoid-s18472" xml:space="preserve"> Ergo an-<lb/>gulus h g m erit minor angulo h a m [ut ſuprà:</s> <s xml:id="echoid-s18473" xml:space="preserve">] ſed an-<lb/>gulus h g m eſt ille, quem reſpicit apud circumferẽtiam <lb/>arcus h m duplicatus:</s> <s xml:id="echoid-s18474" xml:space="preserve"> & angulus h a m eſt ille, quem re-<lb/>ſpicit in circumferentia exceſſus arcus h m ſupra arcum <lb/>r q [per 25 n.</s> <s xml:id="echoid-s18475" xml:space="preserve">] Ergo arcus h m duplicatus eſt minor <lb/>exceſſu arcus h m, ſupra arcum r q:</s> <s xml:id="echoid-s18476" xml:space="preserve"> quod eſt impoſsibi-<lb/>le [& contra 9 ax.</s> <s xml:id="echoid-s18477" xml:space="preserve">] Ergo ſi punctum b fuerit extra lineã <lb/>a k g:</s> <s xml:id="echoid-s18478" xml:space="preserve"> tunc forma eius non refringetur ad a, niſi ex uno <lb/>puncto tantùm.</s> <s xml:id="echoid-s18479" xml:space="preserve"> Quapropter non habebit, niſi unami-<lb/>maginem:</s> <s xml:id="echoid-s18480" xml:space="preserve"> quæ imago aut erit ante uiſum, aut retro, <lb/>aut in loco refractionis, ut in præcedentibus declaraui-<lb/>mus.</s> <s xml:id="echoid-s18481" xml:space="preserve"> Et hoc eſt quod uoluimus declarare.</s> <s xml:id="echoid-s18482" xml:space="preserve"/> </p> <div xml:id="echoid-div598" type="float" level="0" n="0"> <figure xlink:label="fig-0270-01" xlink:href="fig-0270-01a"> <variables xml:id="echoid-variables219" xml:space="preserve">k o g e c n a d z f h m l p b</variables> </figure> <figure xlink:label="fig-0270-02" xlink:href="fig-0270-02a"> <variables xml:id="echoid-variables220" xml:space="preserve">e o k a c n g d z h m l p b</variables> </figure> <figure xlink:label="fig-0271-01" xlink:href="fig-0271-01a"> <variables xml:id="echoid-variables221" xml:space="preserve">a k r q c n g h l m d p z b</variables> </figure> </div> </div> <div xml:id="echoid-div600" type="section" level="0" n="0"> <head xml:id="echoid-head515" xml:space="preserve" style="it">32. Si communis ſectio ſuperficierum, refractionis & refractiui cauirarioris, fuerit peri-<lb/>pheria: uiſibile extra perpendicularem à uiſu ſuper refractiuum ductam, ab uno puncto refrin <lb/>getur, unaḿ habebit imaginem, uariè pro uaria uiſ{us} uel uiſibilis poſitione ſitam. 28 p 10.</head> <p> <s xml:id="echoid-s18483" xml:space="preserve">SI uerò corpus diaphanum groſsius fuerit ex parte uiſus, & ſubtilius ex parte rei uiſæ, ijſdem <lb/>manentibus figuris:</s> <s xml:id="echoid-s18484" xml:space="preserve"> tunc etiam res uiſa non habebit niſi unam imaginem ſolam:</s> <s xml:id="echoid-s18485" xml:space="preserve"> & hoc decla-<lb/>rabitur, ut in conuerſa ſeptimæ figuræ [quæ fuit 27.</s> <s xml:id="echoid-s18486" xml:space="preserve"> 28 n.</s> <s xml:id="echoid-s18487" xml:space="preserve">] Et omnia, quæ declarauimus in re-<lb/>fractionibus à conuexo & concauo circuli:</s> <s xml:id="echoid-s18488" xml:space="preserve"> ſequũtur in ſuperficiebus ſphæricis & columnaribus:</s> <s xml:id="echoid-s18489" xml:space="preserve"> <lb/>præter refractionem circularem, à circumferentia circuli, quæ non fit, niſi in ſuperficiebus ſphæri-<lb/>cis tantùm.</s> <s xml:id="echoid-s18490" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div601" type="section" level="0" n="0"> <head xml:id="echoid-head516" xml:space="preserve" style="it">33. Viſibile refractum à refractiuo uariæ uel figuræ uel perſpicuitatis, uel ſimul utriuſ: <lb/>uari{as} & monſtrific{as} uarijs in locis imagines habet. 29. 30 p 10.</head> <p> <s xml:id="echoid-s18491" xml:space="preserve">HÆc autem, quæ diximus, ſunt imagines uiſibilium, quæ comprehenduntur à uiſu ultra <lb/>corpora diaphana ſimplicia, quæ ſunt unius ſubſtantiæ, & quorum figura, quæ eſt ex par-<lb/>te uiſus, eſt una figura.</s> <s xml:id="echoid-s18492" xml:space="preserve"> Si uerò corpus diaphanum fuerit diuerſum, aut non conſimilis dia-<lb/>phanitatis:</s> <s xml:id="echoid-s18493" xml:space="preserve"> tunc imagines rei uiſæ diuerſantur.</s> <s xml:id="echoid-s18494" xml:space="preserve"> Et ſi ſuperficies corporis diaphani, quæ eſt ex par-<lb/>te uiſus, fuerit diuerſa:</s> <s xml:id="echoid-s18495" xml:space="preserve"> tunc loca etiam imaginum rei uiſæ diuerſantur, cum formę refractionum ex <lb/>ſuperficie corporis diaphani diuerſentur etiam.</s> <s xml:id="echoid-s18496" xml:space="preserve"> Et ſi aliquis reſpexerit ad paruam ſphæram, aut ali <lb/>quod corpus rotundum paruum, aut columnare uitri aut cryſtalli, ultra quod corpus fuerit ali-<lb/>quod uiſibile:</s> <s xml:id="echoid-s18497" xml:space="preserve"> inueniet imaginem illius alio modo, quàm ſit res uiſa in ſe:</s> <s xml:id="echoid-s18498" xml:space="preserve"> & fortè inueniet rei ui-<lb/>ſæ imaginem aliam:</s> <s xml:id="echoid-s18499" xml:space="preserve"> & ſic dubitabitur ſuper ea.</s> <s xml:id="echoid-s18500" xml:space="preserve"> Sed huiuſmodi refractio non eſt una, ſed duæ:</s> <s xml:id="echoid-s18501" xml:space="preserve"> for-<lb/>ma enim rei uiſæ extenditur à re uiſa ad ſphæram, aut ad aliud corpus rotundum columnare, & <lb/>refringitur à conuexo ſphæræ aut columnæ ad interius corporis, & extenditur intra corpus, quo-<lb/>uſque perueniat ad ſuperficiem eius:</s> <s xml:id="echoid-s18502" xml:space="preserve"> & deinde refringitur à ſphæra aut columna apud concaui-<lb/>tatem aeris contingentis ſphæram aut columnam.</s> <s xml:id="echoid-s18503" xml:space="preserve"> Et ſic comprehenſio huiuſmodi rerum erit dua-<lb/>bus diuerſis refractionibus.</s> <s xml:id="echoid-s18504" xml:space="preserve"> Quapropter imago eius erit diuerſa ab imagine eius, quod compre-<lb/>henditur una refractione.</s> <s xml:id="echoid-s18505" xml:space="preserve"> Nos autem loquemur de hoc parum, quando tracta-<lb/>bimus de deceptionibus uiſus, quæ fiunt per <lb/>refractionem.</s> <s xml:id="echoid-s18506" xml:space="preserve"/> </p> <pb o="266" file="0272" n="272" rhead="ALHAZEN"/> </div> <div xml:id="echoid-div602" type="section" level="0" n="0"> <head xml:id="echoid-head517" xml:space="preserve">QVOMODO VISVS COMPREHENDAT VISIBILIA SE-<lb/>cundum refractionem. Cap. VI.</head> <head xml:id="echoid-head518" xml:space="preserve" style="it">34. Si uiſ{us} & uiſibile in diuerſis medijs ſua loca inter ſe permutent: nomina linearum <lb/>in cidentiæ & refractionis mutantur. 9 p 10.</head> <p> <s xml:id="echoid-s18507" xml:space="preserve">IN præcedentibus iam declarauimus, quòd, cum forma refringitur ab aliquo corpore diapha-<lb/>no, ad aliud corpus diuerſæ diaphanitatis:</s> <s xml:id="echoid-s18508" xml:space="preserve"> extenditur per lineam rectam, donec perueniat ad <lb/>ſuperficiem corporis diaphani, in quo eſt:</s> <s xml:id="echoid-s18509" xml:space="preserve"> deinde refringitur in illo alio corpore diaphano per <lb/>lineam aliam rectam, quæ continet cum prima linea angulum.</s> <s xml:id="echoid-s18510" xml:space="preserve"> Et cum forma extenditur per hanc <lb/>aliam lineam, ſuper quam refringitur forma in ſecundo corpore, alia quæcunque forma ſit in ſe-<lb/>cundo corpore uſque ad punctum ſectionis, inter duas lineas rectas, refringetur per primam li-<lb/>neam rectam.</s> <s xml:id="echoid-s18511" xml:space="preserve"> Et eſt manifeſtum per experientiam, quòd ſi aliquis inſpexerit aliquod corpus dia-<lb/>phanum, quod differt in ſua diaphanitate à diaphanitate aeris:</s> <s xml:id="echoid-s18512" xml:space="preserve"> comprehendet omnia, quæ ſunt ul <lb/>trà de illis, quæ opponuntur uiſui.</s> <s xml:id="echoid-s18513" xml:space="preserve"> Et ſi cooperuerit alterum uiſum, & aſpexerit reliquo:</s> <s xml:id="echoid-s18514" xml:space="preserve"> compre-<lb/>hendet etiam, quæcunque ſunt ultrà, ſiue illud corpus ſit aer, ſiue aqua, ſiue uitrum.</s> <s xml:id="echoid-s18515" xml:space="preserve"> Et ſimiliter ſi <lb/>homo poſuerit uiſum in aliquo corpore groſsiore aere, ut uitro aut cryſtallo:</s> <s xml:id="echoid-s18516" xml:space="preserve"> uidebit omnia, quæ <lb/>ſunt ultrà de illis, quæ ſunt in aere.</s> <s xml:id="echoid-s18517" xml:space="preserve"> Et ſi aſpiciẽs mouerit uiſum ſuum dextrorſum aut ſiniſtrorſum, <lb/>& in omnem partem, & non remouerit ipſum multum à ſuo primo loco:</s> <s xml:id="echoid-s18518" xml:space="preserve"> tunc comprehẽdet etiam <lb/>omnia, quæ prius comprehẽdebat, ſiue motus uiſus fuerit in aere, ſiue in uitro.</s> <s xml:id="echoid-s18519" xml:space="preserve"> Sed iam declaraui-<lb/>mus experientia & demonſtratione, quòd uiſus nihil comprehẽdit de illis, quæ ſunt ultra corpora <lb/>diaphana, quæ differunt in diaphanitate ab aere, niſi ſecundum refractionem, præterquam unum <lb/>punctum, quod eſt in perpendiculari exeunte à centro uiſus ſuper ſuperficiem corporis diaphani.</s> <s xml:id="echoid-s18520" xml:space="preserve"> <lb/>Ergo omne punctum comprehenſum à uiſu ultra corpus diaphanum, præter illud punctum præ-<lb/>dictum, comprehenditur ex forma, quæ extenditur ex illo puncto ad ſuperficiem corporis diapha-<lb/>ni, ultra quod eſt, & refringitur à ſuperficie illius corporis ad uiſum.</s> <s xml:id="echoid-s18521" xml:space="preserve"> Et cum unus uiſus compre-<lb/>hendat omnia, quæ ſunt ultra corpus diaphanum:</s> <s xml:id="echoid-s18522" xml:space="preserve"> forma omnis puncti exiſtentis ultra corpus il-<lb/>lud diaphanum, extenditur per lineam rectam ad ſuperficiem illius corporis diaphani, & refringi-<lb/>tur ad illum uiſum unum, præterquam illud punctum prædictum.</s> <s xml:id="echoid-s18523" xml:space="preserve"> Et cum formę omnium puncto-<lb/>rum, quæ ſunt in omnibus uiſibilibus exiſtentibus ultra corpus diaphanum, refringantur in eo-<lb/>dem tempore ad centrum uiſus unius:</s> <s xml:id="echoid-s18524" xml:space="preserve"> forma puncti, quod exiſtit apud centrum uiſus illius, cum <lb/>fuerit in aliquo uiſibili, refringetur ad omnia puncta, quæ ſunt in omnibus uiſibilibus exiſtentibus <lb/>ultra corpus diaphanum, oppoſitum uiſui in eodem tempore & eodem modo.</s> <s xml:id="echoid-s18525" xml:space="preserve"> Et ſimiliter eſt <lb/>de omni puncto propinquo puncto, quod eſt apud centrum uiſus.</s> <s xml:id="echoid-s18526" xml:space="preserve"> Nam ſi uiſus motus fuerit ad <lb/>omnem partem, & non fuerit remotus à ſuo ſitu:</s> <s xml:id="echoid-s18527" xml:space="preserve"> comprehendet uiſibilia.</s> <s xml:id="echoid-s18528" xml:space="preserve"> Ergo forma cuiusli-<lb/>bet puncti cuiuslibet uiſi, cum fuerit ultra aliquod corpus diaphanum, extendetur ad ſuperficiem <lb/>corporis diaphani, ultra quod eſt, & refringetur ad uniuerſum eius, quod opponitur ei ex corpo-<lb/>re aeris.</s> <s xml:id="echoid-s18529" xml:space="preserve"> Et non eſt aliquod tempus magis appropriatum huic, quàm aliud:</s> <s xml:id="echoid-s18530" xml:space="preserve"> ſed hoc eſt proprium <lb/>naturæ lucis & coloris, quæſunt in uiſibilibus:</s> <s xml:id="echoid-s18531" xml:space="preserve"> ſcilicet, ut ſemper extendãtur à quolibet puncto cu <lb/>iuslibet corporis lucidi, per lineam rectam, quæ extenditur ab illo puncto, & refringantur in omni <lb/>corpore diaphano diuerſo, præterquam punctum, quod eſt in perpendiculari.</s> <s xml:id="echoid-s18532" xml:space="preserve"> Et omnis forma <lb/>cuiuslibet puncti uiſibilis exiſtentis in aliquo corpore diuerſo ab aere:</s> <s xml:id="echoid-s18533" xml:space="preserve"> extendetur in illo corpore, <lb/>in quo exiſtit, & refringetur in uniuerſo corpore aeris ſibi oppoſito, & illa forma exit ad quodlibet <lb/>punctum aeris.</s> <s xml:id="echoid-s18534" xml:space="preserve"> Quapropter forma totius rei uiſæ coniungitur apud quodlibet punctum aeris:</s> <s xml:id="echoid-s18535" xml:space="preserve"> & <lb/>forma totius cuiuslibet uiſi exiſtentis in aliquo corpore diuerſo ab aere, exiſtit apud unumquod-<lb/>que punctum aeris oppoſiti illi rei uiſæ:</s> <s xml:id="echoid-s18536" xml:space="preserve"> & forma illa extenditur à quolibet puncto rei uiſæ in cor-<lb/>pore, in quo eſt, & refringitur apud ſuperficiem illius corporis, & peruenit ad illud punctum ae-<lb/>ris.</s> <s xml:id="echoid-s18537" xml:space="preserve"> Et ideo ſi uiſus aſpexerit aliquod corpus diaphanum diuerſum ab aere, ultra quod fuerit ali-<lb/>qua res uiſibilis:</s> <s xml:id="echoid-s18538" xml:space="preserve"> uiſus comprehendit illam rem.</s> <s xml:id="echoid-s18539" xml:space="preserve"> Nam forma illius exiſtit apud punctum, apud <lb/>quod exiſtit centrum uiſus.</s> <s xml:id="echoid-s18540" xml:space="preserve"> Propter hoc, quòd & ſi uiſus comprehenderit aliquam rem uiſibilem <lb/>ultra aliquod corpus diaphanum diuerſum ab aere:</s> <s xml:id="echoid-s18541" xml:space="preserve"> deinde motus fuerit à loco ſuo dextrorſum, <lb/>aut ſiniſtrorſum:</s> <s xml:id="echoid-s18542" xml:space="preserve"> dum in ſuo motu fuerit oppoſitus corpori diaphano, & rei uiſæ, quæ eſt ultrà:</s> <s xml:id="echoid-s18543" xml:space="preserve"> <lb/>ſemper comprehendet illam rẽ.</s> <s xml:id="echoid-s18544" xml:space="preserve"> Vnde etiam plures aſpicientes comprehendũt unam rem in cœlo, <lb/>& in aqua, & in uno & eodem tempore.</s> <s xml:id="echoid-s18545" xml:space="preserve"> Et hoc etiam eſt in eodem corpore diaphano:</s> <s xml:id="echoid-s18546" xml:space="preserve"> ſcilicet, <lb/>quòd forma uiſi congregatur apud quodlibet punctum corporis, in quo eſt:</s> <s xml:id="echoid-s18547" xml:space="preserve"> nam forma puncti cu-<lb/>iuslibet eius extenditur per lineam rectam:</s> <s xml:id="echoid-s18548" xml:space="preserve"> & inter quodlibet punctum corporis, in quo eſt ui-<lb/>ſus, & quo dlibet punctum rei uiſæ, eſt linea recta.</s> <s xml:id="echoid-s18549" xml:space="preserve"> Forma ergo cuiuslibet puncti rei uiſæ extendi-<lb/>tur ad quodlibet punctum corporis diaphani, in quo eſt res uiſa:</s> <s xml:id="echoid-s18550" xml:space="preserve"> & forma cuiuslibet rei lucidæ <lb/>congregatur apud quodlibet punctum cuiuslibet corporis, in quo exiſtit, & congregatur apud <lb/>quodlibet punctum corporis cuiuslibet diaphani diuerſi à corpore, in quo exiſtit, quando inter <lb/>rem uiſam, & illud corpus diaphanum diuerſum non interfuerit aliquod impedimentum.</s> <s xml:id="echoid-s18551" xml:space="preserve"> Et for-<lb/>ma rei uiſæ, quæ eſt apud quodlibet punctum corporis diaphani, in quo extenditur, extenditur ad <lb/>illud punctum rectè:</s> <s xml:id="echoid-s18552" xml:space="preserve"> & forma illius apud quodlibet punctum corporis diaphani diuerſi, extendi-<lb/>tur ad illud punctum refractè:</s> <s xml:id="echoid-s18553" xml:space="preserve"> quia inter quodlibet punctum aeris & quamlibet rem uiſibilem exi-<lb/> <pb o="267" file="0273" n="273" rhead="OPTICAE LIBER VII."/> ſtentem in aliquo corpore diaphano diuerſo ab aere:</s> <s xml:id="echoid-s18554" xml:space="preserve"> fit pyramis refracta, cuius caput eſt punctum <lb/>in aere, & baſis eſt illa res uiſa:</s> <s xml:id="echoid-s18555" xml:space="preserve"> & erit refractio eius apud ſuperficiem corporis ab aere diuerſi.</s> <s xml:id="echoid-s18556" xml:space="preserve"> O-<lb/>mnis ergo res uiſa in corpore diaphano diuerſo ab aere, quando comprehenditur à uiſu:</s> <s xml:id="echoid-s18557" xml:space="preserve"> compre-<lb/>henditur à forma extenſa in pyramide refracta, adunata apud punctum exiſtens in cẽtro uiſus.</s> <s xml:id="echoid-s18558" xml:space="preserve"> Hoc <lb/>ergo modo comprchendit uiſus ea, quæ refractè comprehendit.</s> <s xml:id="echoid-s18559" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div603" type="section" level="0" n="0"> <head xml:id="echoid-head519" xml:space="preserve" style="it">35. Imago uiſibilis refracti aßimilatur figuræ refractiui. 46 p 10.</head> <p> <s xml:id="echoid-s18560" xml:space="preserve">IN capitulo autem imaginis declarauimus, quòd omne uiſum comprehenditur à uiſu ultra ima-<lb/>ginem:</s> <s xml:id="echoid-s18561" xml:space="preserve"> & locus imaginis eſt punctum, in quo ſecant ſe linea radialis, per quam extenditur for-<lb/>ma ad uiſum, & perpẽdicularis exiens à puncto uiſo.</s> <s xml:id="echoid-s18562" xml:space="preserve"> Si ergo imaginati fuerimus, quòd ab uno-<lb/>quoq;</s> <s xml:id="echoid-s18563" xml:space="preserve"> puncto rei uiſæ exit perpendicularis ad ſuperficiem corporis diaphani, in quo eſt res uiſa:</s> <s xml:id="echoid-s18564" xml:space="preserve"> <lb/>tunc habebimus quoddam corpus, exiens à uiſu ad ſuperficiem corporis diaphani:</s> <s xml:id="echoid-s18565" xml:space="preserve"> unde ſequitur <lb/>quòd iſtud corpus ſecet pyramidem refractam, & illa ſuperficies, in qua ſecãt ſe, eſt imago illius rei <lb/>uiſæ.</s> <s xml:id="echoid-s18566" xml:space="preserve"> Si ergo ſuperficies corporis diaphani, in quo eſt res uiſa, fuerit æ qualis:</s> <s xml:id="echoid-s18567" xml:space="preserve"> tunc corpus imagina-<lb/>tum continens omnes perpendiculares, erit æqualis ſuperficiei.</s> <s xml:id="echoid-s18568" xml:space="preserve"> Quare imago addit parum ſuper <lb/>rem uiſam.</s> <s xml:id="echoid-s18569" xml:space="preserve"> Et ſi corpus fuerit ſphæricum, & conuexum eius ex parte uiſus, & centrum eius fuerit <lb/>ſuper illam rem uiſam:</s> <s xml:id="echoid-s18570" xml:space="preserve"> tunc corpus imaginatum erit pyramidale, cuius caput eſt centrum ſphæræ:</s> <s xml:id="echoid-s18571" xml:space="preserve"> <lb/>& quantò magis exten ditur à ſuperficie corporis ſphærici, tantò magis amplificabitur:</s> <s xml:id="echoid-s18572" xml:space="preserve"> & ſi ſectio <lb/>fuerit inter rem uiſam & ſuperficiem ſphæricam:</s> <s xml:id="echoid-s18573" xml:space="preserve"> tunc imago erit amplior illa re uiſa:</s> <s xml:id="echoid-s18574" xml:space="preserve"> Si autẽ ſectio <lb/>fuerit ultra rem uiſam:</s> <s xml:id="echoid-s18575" xml:space="preserve"> tunc imago erit ſtrictior re uiſa.</s> <s xml:id="echoid-s18576" xml:space="preserve"> Si uerò res uiſa fuerit ultra ſuperficiem ſphę <lb/>ricam:</s> <s xml:id="echoid-s18577" xml:space="preserve"> tunc corpus imaginatum, erunt duæ pyramides oppoſitæ, quarum caput centrum ſphæræ.</s> <s xml:id="echoid-s18578" xml:space="preserve"> <lb/>Quare cum loca ſectionis inter corpus imaginatum & pyramidem poſsint eſſe diuerſa:</s> <s xml:id="echoid-s18579" xml:space="preserve"> fortè locus <lb/>ſectionis, in quo eſt imago, erit maior uiſo, fortè minor, fortè æqualis.</s> <s xml:id="echoid-s18580" xml:space="preserve"> Si uerò corpus diaphanum <lb/>fuerit ſphæricum, & concauitas eius fuerit ex parte uiſus:</s> <s xml:id="echoid-s18581" xml:space="preserve"> tunc corpus imaginatum erit pyramis, <lb/>cuius caput eſt centrum ſphæræ.</s> <s xml:id="echoid-s18582" xml:space="preserve"> Quantò ergo magis extenditur hoc corpus in partem ſuperficiei <lb/>ſpheræ, tantò magis adunatur & conſtringitur, & quantò magis extenditur in aliam partem, tantò <lb/>magis amplificatur:</s> <s xml:id="echoid-s18583" xml:space="preserve"> ſuperficies enim continua parua, erit media inter centrum eius, & ſphæram.</s> <s xml:id="echoid-s18584" xml:space="preserve"> Si <lb/>nerò locus ſectionis huius corporis cum pyramide refracta fuerit propinquior centro concauita-<lb/>tis ſphæræ, quàm res uiſa:</s> <s xml:id="echoid-s18585" xml:space="preserve"> erit imago minor ipſa re uiſa.</s> <s xml:id="echoid-s18586" xml:space="preserve"> Si aũt fuerit remotior à centro cõcauitatis, <lb/>quàm res uiſa:</s> <s xml:id="echoid-s18587" xml:space="preserve"> erit imago maior, quàm res uiſa.</s> <s xml:id="echoid-s18588" xml:space="preserve"> Et cum una res uiſa comprehenditur à pluribus uiſi <lb/>bus in uno momento:</s> <s xml:id="echoid-s18589" xml:space="preserve"> omnes imagines, quas illi uiſus comprehendunt, erunt in illo tempore in u-<lb/>no imaginato, quod eſt perpendiculare ſuper ſuperficiem corporis diaphani.</s> <s xml:id="echoid-s18590" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div604" type="section" level="0" n="0"> <head xml:id="echoid-head520" xml:space="preserve" style="it">36. Vtro uiſu una refracti uiſibilis imago uidetur. 47 p 10.</head> <p> <s xml:id="echoid-s18591" xml:space="preserve">ET una res uiſibilis comprehenditur ab uno homine in uno tempore, ultra corpus diaphanũ <lb/>diuerſum à diaphanitate corporis, in quo eſt uiſus, utro q;</s> <s xml:id="echoid-s18592" xml:space="preserve"> uiſu:</s> <s xml:id="echoid-s18593" xml:space="preserve"> & tamen comprehendit rem <lb/>illam unam.</s> <s xml:id="echoid-s18594" xml:space="preserve"> Si enim homo comprehenderit aliquid de eis, quæ ſunt in cœlo, aut in a qua, aut <lb/>ultra uitrum, & cooperuerit alterum uiſum:</s> <s xml:id="echoid-s18595" xml:space="preserve"> nihilo minus cõprehendet illud reliquo.</s> <s xml:id="echoid-s18596" xml:space="preserve"> Ex quo patet, <lb/>quòd una res uiſa exiſtens ultra corpus diaphanum, diuerſum ab aere, comprehendetur utroq;</s> <s xml:id="echoid-s18597" xml:space="preserve"> ui-<lb/>ſu, & altero uiſu.</s> <s xml:id="echoid-s18598" xml:space="preserve"> Cauſſa autem huius eſt, ut in tertio libro [9.</s> <s xml:id="echoid-s18599" xml:space="preserve"> 14 n] diximus:</s> <s xml:id="echoid-s18600" xml:space="preserve"> quoniã in omni pun-<lb/>cto cuiuslibet uiſi comprehenſibilis rectè & utroq;</s> <s xml:id="echoid-s18601" xml:space="preserve"> uiſu, in quo cõiuncti fuerint duo radij utriuſq;</s> <s xml:id="echoid-s18602" xml:space="preserve"> <lb/>uiſus conſimilis poſitionis, quantùm ad duos axes uiſuum:</s> <s xml:id="echoid-s18603" xml:space="preserve"> comprehendetur unum:</s> <s xml:id="echoid-s18604" xml:space="preserve"> & ſi in ipſo ag-<lb/>gregati fuerintra dij diuerſæ poſitionis, quantùm ad duos axes uiſuum:</s> <s xml:id="echoid-s18605" xml:space="preserve"> comprehendentur duo:</s> <s xml:id="echoid-s18606" xml:space="preserve"> & <lb/>in maiore parte, eorum quæ comprehenduntur, poſitio eſt conſimilis.</s> <s xml:id="echoid-s18607" xml:space="preserve"> Hæc autem, quæ ſunt diuer-<lb/>ſæ poſitionis, reſpectu utriuſque uiſus, ſunt ualderara, ut in tertio diximus tractatu.</s> <s xml:id="echoid-s18608" xml:space="preserve"> Et illud, quod <lb/>comprehenditur refractè, comprehenditur in loco imaginis:</s> <s xml:id="echoid-s18609" xml:space="preserve"> forma autem, quæ eſt in loco imagi-<lb/>nis, comprehẽditur à uiſu rectè, poſitio autem huius formæ, quæ eſt imago reſpectu uiſus:</s> <s xml:id="echoid-s18610" xml:space="preserve"> eſt, ſicut <lb/>poſitio alterius rei uiſæ earum, quæ uidentur rectè.</s> <s xml:id="echoid-s18611" xml:space="preserve"> Vnde poſitio harum imaginum, reſpectu uiſus, <lb/>eſt in maiore parte conſimilis:</s> <s xml:id="echoid-s18612" xml:space="preserve"> & in omni puncto imaginis congregantur duo radij duorum uiſuũ <lb/>conſimilis poſitionis.</s> <s xml:id="echoid-s18613" xml:space="preserve"> Quare una res uiſa uidetur una utroq;</s> <s xml:id="echoid-s18614" xml:space="preserve"> uiſu.</s> <s xml:id="echoid-s18615" xml:space="preserve"> Et ut hoc euidentius declaretur:</s> <s xml:id="echoid-s18616" xml:space="preserve"> <lb/>dicamus, quodiam diximus:</s> <s xml:id="echoid-s18617" xml:space="preserve"> quòd omne punctum eius, quod comprehenditur refractè:</s> <s xml:id="echoid-s18618" xml:space="preserve"> compre-<lb/>henditur in loco imaginis, qui eſt inter punctum ſectionis ex perpendiculari, exeunte ab illo pun-<lb/>cto ſuper ſuperficiem corporis diaphani, in quo eſt res uiſa, & inter lineam radialem, per quã exten <lb/>ditur forma ad uiſum.</s> <s xml:id="echoid-s18619" xml:space="preserve"> Cum ergo aſpiciens comprehenderit punctum alicuius rei utroq;</s> <s xml:id="echoid-s18620" xml:space="preserve"> uiſu:</s> <s xml:id="echoid-s18621" xml:space="preserve"> ima-<lb/>go illius puncti reſpectu utriuſq;</s> <s xml:id="echoid-s18622" xml:space="preserve"> uiſus eſt in perpendiculari, exeunte exillo pũcto, quæ eſt eadem <lb/>linea.</s> <s xml:id="echoid-s18623" xml:space="preserve"> Et cum forma illius puncti peruenerit ad duo puncta ſuperficierũ uiſuũ, quorum ſitus reſpe-<lb/>ctu axis uiſus eſt conſimilis:</s> <s xml:id="echoid-s18624" xml:space="preserve"> tunc duæ lineæ, per quas formę extendũtur ad utrũq;</s> <s xml:id="echoid-s18625" xml:space="preserve"> uiſum:</s> <s xml:id="echoid-s18626" xml:space="preserve"> perueni-<lb/>unt ad duo centra duorum uiſuũ.</s> <s xml:id="echoid-s18627" xml:space="preserve"> Sunt ergo axes, aut habentes ex axibus poſitionem conſimilem:</s> <s xml:id="echoid-s18628" xml:space="preserve"> <lb/>& duo axes uiſuũ ſemper ſunt in eadem ſuperficie:</s> <s xml:id="echoid-s18629" xml:space="preserve"> & omnes lineæ exeuntes à cẽtro duorum uiſuũ <lb/>habentes poſitionem conſimilem ab axe communi, erunt in eadem ſuperficie:</s> <s xml:id="echoid-s18630" xml:space="preserve"> axis enim commu-<lb/>nis ſemper eſt in eadem ſuperficie.</s> <s xml:id="echoid-s18631" xml:space="preserve"> Nam ſi aliquid comprehenditur utroq;</s> <s xml:id="echoid-s18632" xml:space="preserve"> uiſu in eodem tempore <lb/>uera comprehenſione:</s> <s xml:id="echoid-s18633" xml:space="preserve"> tũc axes concurrunt in uno puncto illius rei [per 10.</s> <s xml:id="echoid-s18634" xml:space="preserve"> 15 n 3.</s> <s xml:id="echoid-s18635" xml:space="preserve">] Quare ſunt in <lb/>eadem ſuperficie.</s> <s xml:id="echoid-s18636" xml:space="preserve"> Item poſitio uiſuum naturalis eſt conſimilis, & non exit à naturali poſitione, niſi <lb/>per accidens, aut per uiolentiam:</s> <s xml:id="echoid-s18637" xml:space="preserve"> quare axes eorum ſunt in eadem ſuperficie.</s> <s xml:id="echoid-s18638" xml:space="preserve"> Principium enim <lb/> <pb o="268" file="0274" n="274" rhead="ALHAZEN"/> axium eſt unum punctum, quod eſt in medio concauitatis communis nerui, à quo exit communis <lb/>axis.</s> <s xml:id="echoid-s18639" xml:space="preserve"> Exiſtentibus ergo duobus uiſibus in ſua naturali poſitione, ſemper axes erũt in eadem ſuper-<lb/>ficie, ſiue ſint moti, ſiue quieſcentes.</s> <s xml:id="echoid-s18640" xml:space="preserve"> Si autem poſitio alterius uiſuũ mutata fuerit, reſpectu reliqui <lb/>propter aliquod impedimentum:</s> <s xml:id="echoid-s18641" xml:space="preserve"> tunc res uiſa uidebitur duplex, ut in primo libro declarauimus.</s> <s xml:id="echoid-s18642" xml:space="preserve"> <lb/>Duo ergo axes in maiore parte ſunt in eadem ſuperficie.</s> <s xml:id="echoid-s18643" xml:space="preserve"> Quare omnes duo radij habentes poſitio-<lb/>nem ſimilem ex duo bus axibus, erunt in eadem ſuperficie.</s> <s xml:id="echoid-s18644" xml:space="preserve"> Duæ ergo lineę, per quas extenduntur <lb/>formę unius puncti ad duo loca cõſimilis poſitionis, ſuntin eadem ſuperficie.</s> <s xml:id="echoid-s18645" xml:space="preserve"> Sed imagines illius, <lb/>reſpectu duorum uiſuũ, ſunt in illis duabus lineis.</s> <s xml:id="echoid-s18646" xml:space="preserve"> Ergo ſunt in eadem ſuperficie.</s> <s xml:id="echoid-s18647" xml:space="preserve"> Sed imagines illi-<lb/>us puncti ſunt in perpendiculari exeunte ex illo puncto.</s> <s xml:id="echoid-s18648" xml:space="preserve"> Ergo ſunt in loco ſectionis inter ſuperfi-<lb/>ciem, in qua ſunt lineę radiales, quę eſt una ſuperficies, & inter perpendicularem, quę eſt una linea.</s> <s xml:id="echoid-s18649" xml:space="preserve"> <lb/>Sectio autem unius ſuperficiei cum una linea eſt unum punctum.</s> <s xml:id="echoid-s18650" xml:space="preserve"> Ergo imagines unius puncti, re-<lb/>ſpectu duorum uiſuum, quando perueniunt ad duo loca conſimilis poſitionis, ſunt punctum unũ.</s> <s xml:id="echoid-s18651" xml:space="preserve"> <lb/>Ex quo patet, quòd imago totius rei uiſæ, reſpectu duorum uiſuũ, erit una:</s> <s xml:id="echoid-s18652" xml:space="preserve"> ſi poſitio imaginis fuerit <lb/>conſimilis.</s> <s xml:id="echoid-s18653" xml:space="preserve"> Quare res comprehenditur una utroq;</s> <s xml:id="echoid-s18654" xml:space="preserve"> uiſu.</s> <s xml:id="echoid-s18655" xml:space="preserve"> Si uerò poſitio fuerit parum diuerſa:</s> <s xml:id="echoid-s18656" xml:space="preserve"> uide-<lb/>bitur res una:</s> <s xml:id="echoid-s18657" xml:space="preserve"> ſed non uerè, ſed cauilloſè.</s> <s xml:id="echoid-s18658" xml:space="preserve"> Si autem diuerſitas poſitionis fuerit multa:</s> <s xml:id="echoid-s18659" xml:space="preserve"> tunc forma rei <lb/>uidebuntur duæ:</s> <s xml:id="echoid-s18660" xml:space="preserve"> ſed hoc fit rariſsimè.</s> <s xml:id="echoid-s18661" xml:space="preserve"> Hæc eſt ergo qualitas comprehenſionis uiſus de uiſibilibus <lb/>ſecundum refractionem.</s> <s xml:id="echoid-s18662" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div605" type="section" level="0" n="0"> <head xml:id="echoid-head521" xml:space="preserve" style="it">37. Viſio diſtincta fit rectis lineis à uiſibili ad uiſum perpendicularib{us}. Et uiſio omnis fit re-<lb/>fractè. 17. 18 p 3.</head> <p> <s xml:id="echoid-s18663" xml:space="preserve">HOc autem declarato:</s> <s xml:id="echoid-s18664" xml:space="preserve"> dicamus uniuerſaliter, quòd omnia, quæ comprehendũtur à uiſu, com <lb/>prehenduntur refractè, ſiue uiſus & uiſum fuerint in eodem diaphano, ſiue in diuerſis, ſiue <lb/>uiſum ſit in oppoſitione uiſus, ſiue comprehendatur ab ipſo reflexè.</s> <s xml:id="echoid-s18665" xml:space="preserve"> Nihil enim comprehen <lb/>ditur ſine refractione facta apud ſuperficiem uiſus.</s> <s xml:id="echoid-s18666" xml:space="preserve"> Nam tunicæ uiſus, quæ ſunt cornea, albuginea, <lb/>glacialis, ſunt etiam diaphanæ & ſpiſsiores aere.</s> <s xml:id="echoid-s18667" xml:space="preserve"> Et iam declaratum eſt, quòd formæ eorũ, quæ ſunt <lb/>in aere & in alijs corporibus diaphanis, extenduntur in illis corporibus:</s> <s xml:id="echoid-s18668" xml:space="preserve"> & ſi occurrerint corpori-<lb/>bus diuerſæ diaphanitatis ab eo, in quo ſunt:</s> <s xml:id="echoid-s18669" xml:space="preserve"> refringuntur in illo corpore diaphano:</s> <s xml:id="echoid-s18670" xml:space="preserve"> forma ergo e-<lb/>ius, quæ eſt in aere, ſemper extenditur in aere.</s> <s xml:id="echoid-s18671" xml:space="preserve"> Cum ergo aer tangit ſuperficiem alicuius uiſus:</s> <s xml:id="echoid-s18672" xml:space="preserve"> tunc <lb/>illa forma, quæ eſt in aere, refringitur in ſuperficie uiſus:</s> <s xml:id="echoid-s18673" xml:space="preserve"> & tunc refringitur omni modo in corpore <lb/>corneæ & albugineæ.</s> <s xml:id="echoid-s18674" xml:space="preserve"> Refractio enim propriè eſt de numero formarum:</s> <s xml:id="echoid-s18675" xml:space="preserve"> recipere autem formas & <lb/>refractiones eſt proprium corporibus diaphanis.</s> <s xml:id="echoid-s18676" xml:space="preserve"> Form æ ergo eorum, quæ opponuntur uiſui, ſem-<lb/>per refring untur in tunicis uiſus.</s> <s xml:id="echoid-s18677" xml:space="preserve"> Et iam patuit, quòd cum formę extenduntur ſuper lineas perpen <lb/>diculares ſuper ſecundum corpus:</s> <s xml:id="echoid-s18678" xml:space="preserve"> pertranſeunt rectè in ſecundo corpore.</s> <s xml:id="echoid-s18679" xml:space="preserve"> Formę ergo eorum, quę <lb/>opponuntur ſuperficiei uiſus, refringuntur omnes in tunicis uiſus:</s> <s xml:id="echoid-s18680" xml:space="preserve"> & quæ fuerint ex eis in extremi <lb/>tatibus linearum radialium, perpendicularium ſuper ſuperficiem uiſus, pertranſeunt rectè, cum re <lb/>fractione formarum earum in tunicis uiſus.</s> <s xml:id="echoid-s18681" xml:space="preserve"> Parti enim ſuperficiei uiſus, quæ opponitur foramini <lb/>uueæ, multa opponuntur uiſibilia, quorum alia ſunt apud extremitates linearum radialium, & a-<lb/>lia extra.</s> <s xml:id="echoid-s18682" xml:space="preserve"> Omnes enim lineæ radiales, quę ſunt perpendi culares ſuper ſuperficies tunicarum uiſus, <lb/>continentur in pyramide, cuius caput eſt centrum uiſus, & cuius baſis eſt circumferentia uueæ fo-<lb/>raminis.</s> <s xml:id="echoid-s18683" xml:space="preserve"> Et quantò magis extenditur hæc pyramis, & remouetur à uiſu, tantò magis amplificatur:</s> <s xml:id="echoid-s18684" xml:space="preserve"> <lb/>& omnes formæ eorum, quæ ſunt intra pyramidem, extenduntur in rectitudine linearum radialiũ, <lb/>& pertranſeunt in tunicis uiſus rectè.</s> <s xml:id="echoid-s18685" xml:space="preserve"> Et hæc pyramis dicitur pyramis radialis.</s> <s xml:id="echoid-s18686" xml:space="preserve"> Lineæ autem, quæ <lb/>extenduntur in hac pyramide, quarum extremitates ſunt apud centrum uiſus, dicuntur lineæ ra-<lb/>diales.</s> <s xml:id="echoid-s18687" xml:space="preserve"> Formæ uerò eorum, quæ ſunt extra hanc pyramidem, nunquam extenduntur per aliquam <lb/>linearum radialium:</s> <s xml:id="echoid-s18688" xml:space="preserve"> tamen extenduntur per lineas rectas, quæ ſuntinter ipſam ſuperficiem uiſus, <lb/>quæ opponuntur foramini uueæ:</s> <s xml:id="echoid-s18689" xml:space="preserve"> & formæ, quæ extenduntur per has lineas, refringuntur à diapha <lb/>nitate tunicarum uiſus.</s> <s xml:id="echoid-s18690" xml:space="preserve"> Et forma cuiuslibet puncti eorum, quæ ſunt intra pyramidẽ radialẽ, exten-<lb/>ditur ad ſuperficiem uiſus, quæ opponitur foramini uueæ in pyramide, cuius caput eſt illud pun-<lb/>ctum, & cuius baſis eſt ſuperficies, quæ opponitur foramini uueæ:</s> <s xml:id="echoid-s18691" xml:space="preserve"> & una linea earum, quæ imagi-<lb/>natur in hac pyramide, eſt linea radialis:</s> <s xml:id="echoid-s18692" xml:space="preserve"> c æteræ autem omnes, quę non ſunt in hac pyramide, non <lb/>ſunt radiales:</s> <s xml:id="echoid-s18693" xml:space="preserve"> & nulla earum eſt perpendicularis ſuper ſuperficies tunicarũ uiſus.</s> <s xml:id="echoid-s18694" xml:space="preserve"> Et forma cuiusli-<lb/>bet puncti eorum, quæ ſunt intra pyramidem radialem, extenditur ſuper lineam omnem, quę po-<lb/>teſt cadere in illam pyramidem, cuius caput eſt illud punctum, & cuius baſis eſt ſuperficies rei uiſę, <lb/>quę opponitur foramini uueę:</s> <s xml:id="echoid-s18695" xml:space="preserve"> & per unam iſtarum linearum tranſit forma, quę extenditur per illã <lb/>in tunicis uiſus ſecundum rectitudinem:</s> <s xml:id="echoid-s18696" xml:space="preserve"> & omnes formę alię extenſę in reſiduo pyramidis, refrin-<lb/>guntur in tunicis uiſus, & non pertranſeunt rectè.</s> <s xml:id="echoid-s18697" xml:space="preserve"> Omnia ergo, quę opponuntur parti ſuperficiei <lb/>uiſus, quę opponitur foramini uueę, ex illis quę ſunt in aere, aut in cœlo, aut in aqua, aut in conſimi <lb/>libus, & ex illis, quę reflectuntur à terſis corporibus, quæ perueniunt ad hanc partem ſuperficiei ui <lb/>ſus, refringuntur in tunicis uiſus.</s> <s xml:id="echoid-s18698" xml:space="preserve"> Et formę eorum, quę ſunt intra pyramidem, pertranſeunt rectè in <lb/>tunicis uiſus, cum refractione form arum earũ, quę extenduntur ſuper pyramidem, quę remanent <lb/>in uniuerſo huius partis ſuperficiei uiſus.</s> <s xml:id="echoid-s18699" xml:space="preserve"> Reſtat ergo declarare, quòd formę, quę refringuntur in <lb/>tunicis uiſus, comprehenduntur à uiſu, & ſentiuntur à uirtute ſenſibili.</s> <s xml:id="echoid-s18700" xml:space="preserve"> In primo autem tractatu <lb/>[15.</s> <s xml:id="echoid-s18701" xml:space="preserve"> 18.</s> <s xml:id="echoid-s18702" xml:space="preserve"> 19 25 n] declarauimus, quòd ſi membrum ſenſibile ſentiret ex quolibet puncto ſuę ſuper-<lb/>ficiei omnem formam ad ſe uenientem:</s> <s xml:id="echoid-s18703" xml:space="preserve"> tunc ſentiret rerum formas mixtas.</s> <s xml:id="echoid-s18704" xml:space="preserve"> Vnde membrũ ſenſibi-<lb/> <pb o="269" file="0275" n="275" rhead="OPTICAE LIBER VII."/> le non ſentit formas, niſi ex rectitudine linearum perpendicularium ſuper ſuperficiem ipſius tan-<lb/>tùm.</s> <s xml:id="echoid-s18705" xml:space="preserve"> Quare tranſeunt formæ uiſibilium, nec ad miſcentur apud ipſum.</s> <s xml:id="echoid-s18706" xml:space="preserve"> In hoc uerò tractatu mõſtra-<lb/>uimus, quòd form æ refractę nunquam comprehenduntur, niſi in perpendicularibus exeuntibus à <lb/>uiſibilibus ſuper ſuperficies corporum diaphanorũ.</s> <s xml:id="echoid-s18707" xml:space="preserve"> Er go formæ refractæ in tunicis uiſus nõ com-<lb/>prehen duntur à uiſu, niſi in perpendicularibus exeuntibus à uiſibilibus ſuper ſuperficies tunicarũ <lb/>uiſus:</s> <s xml:id="echoid-s18708" xml:space="preserve"> & hæ perpendiculares lineæ ſunt exeuntes à centro uiſus.</s> <s xml:id="echoid-s18709" xml:space="preserve"> Formæ ergo omnes refractę in tu-<lb/>nicis uiſus comprehendũtur à uiſu in rectitudine linearum exeuntium à centro uiſus.</s> <s xml:id="echoid-s18710" xml:space="preserve"> Formæ ergo <lb/>omnium uiſibilium, quæ opponuntur parti ſuperficiei uiſus, quę opponitur foramini uueæ, & exi-<lb/>ſtũt in hac parte ſuperficiei uiſus:</s> <s xml:id="echoid-s18711" xml:space="preserve"> refringũtur in diaphanitate tunicarũ uiſus, & perueniũt ad mem <lb/>brũ ſenſibile, quòd eſt humor glacialis, & cõprehenduntur à uirtute ſenſibili per lineas rectas, quę <lb/>cõtinuãt centrũ uiſus cũ ipſis uiſibilibus, ſcilicet quòd forma cuiuslibet pũcti cuiuslibet uiſi, oppo <lb/>ſiti ſuperficiei uiſus, quæ opponitur foramini uueæ, exiſtit in uniuerſo ſuperficiei huius partis, & <lb/>refringitur à tota ſuperficie, & peruenit ad humorem glacialem:</s> <s xml:id="echoid-s18712" xml:space="preserve"> & tũc ille humor ſentit formam ad <lb/>ſe uenientem:</s> <s xml:id="echoid-s18713" xml:space="preserve"> & uirtus ſenſibilis comprehendit omnia, quæ perueniunt ad glacialem ex forma ui-<lb/>ſus pũcti ſuper unam lineam continuantem centrũ uiſus cũ illo puncto.</s> <s xml:id="echoid-s18714" xml:space="preserve"> Hoc ergo modo compre-<lb/>hẽdit uiſus omnia uiſibilia.</s> <s xml:id="echoid-s18715" xml:space="preserve"> In hoc autem capitulo diximus, quòd eorũ, quæ opponũtur ſuperficiei <lb/>uiſus, alia ſunt intra pyramidem, & alia extra:</s> <s xml:id="echoid-s18716" xml:space="preserve"> & cũ dico ſuperficiem uiſus:</s> <s xml:id="echoid-s18717" xml:space="preserve"> intelligere oportet nunc <lb/>& ammodo partem oppoſitam ſuperficiei uueæ.</s> <s xml:id="echoid-s18718" xml:space="preserve"> Viſibilia ergo, quæ ſunt intra pyramidem radialẽ, <lb/>comprehendũtur à uiſu ex rectitudine linearũ radialiũ rectè, ex formis eorũ, quæ extendũtur ad ui <lb/>ſum in rectitudine harũ linearũ.</s> <s xml:id="echoid-s18719" xml:space="preserve"> Et hæ lineæ ſunt perpendiculares, quę exeunt à pũctis uiſibilibus, <lb/>quæ ſunt intra pyramidem ſuper ſuperficies tunicarũ uiſus:</s> <s xml:id="echoid-s18720" xml:space="preserve"> illa autem, quæ ſunt extra pyramidem <lb/>radialẽ, cõprehendũtur à uiſu ex formis refractis, & in rectitudine linearũ exeuntiũ à centro uiſus, <lb/>exiſtentiũ extra pyramidẽ radialẽ.</s> <s xml:id="echoid-s18721" xml:space="preserve"> Et hæ lineæ, quæ ſunt extra pyramidẽ radialẽ, poſſunt etiam di <lb/>ci lineæ radiales tranſſumptiuè:</s> <s xml:id="echoid-s18722" xml:space="preserve"> aſsimilantur enim lineis radialibus in eo, quòd exeunt à cẽtro ui-<lb/>ſus.</s> <s xml:id="echoid-s18723" xml:space="preserve"> Reſtat ergo declarare per experientiam, quòd uiſus comprehẽdit ea, quę ſunt extra pyramidẽ <lb/>radialem.</s> <s xml:id="echoid-s18724" xml:space="preserve"> Dicimus ergo, quòd manifeſtũ eſt, quòd lachrymalia, & ea, quæ continẽt circulum, ſunt <lb/>extra pyramidem, cuius caput centrũ uiſus eſt, & cuius baſis eſt circũferẽtia foraminis uueæ, quod <lb/>eſt paruũ foramẽ in medio nigredinis oculi.</s> <s xml:id="echoid-s18725" xml:space="preserve"> Et ſi aliquis ſump ſerit acũ ſubtilem gracilem, & poſue-<lb/>rit extremitatẽ eius in poſtremo oculi, & inter palpebras, & quieuerit uiſus:</s> <s xml:id="echoid-s18726" xml:space="preserve"> tũc uidebit extremita-<lb/>tem eius:</s> <s xml:id="echoid-s18727" xml:space="preserve"> & ſimiliter ſi poſuerit extremitatẽ acus in lachrymali, & ſi miſerit illã in oculo, & applica-<lb/>uerit extremitatem in latere nigredinis oculi aut prope, uidebit extremitatem acus.</s> <s xml:id="echoid-s18728" xml:space="preserve"> Item omnia, <lb/>quæ æquidiſtant ſuperficiei rei uifæ, ex locis continentibus uifum, ſunt extra pyramidem radialẽ.</s> <s xml:id="echoid-s18729" xml:space="preserve"> <lb/>Et cum dico loca continẽtia uiſum:</s> <s xml:id="echoid-s18730" xml:space="preserve"> intelligo illa, à quibus lineæ exeuntes ad mediũ ſuperficiei ui-<lb/>ſus, ſecant axem pyramidis radialis.</s> <s xml:id="echoid-s18731" xml:space="preserve"> Et ſi homo erexeritindicem ſuum exparte ſuæ faciei & pro-<lb/>pe palpebram:</s> <s xml:id="echoid-s18732" xml:space="preserve"> uidebit indicem.</s> <s xml:id="echoid-s18733" xml:space="preserve"> Et ſimiliter ſi applicauerit indicem cum inferiore palpebra, ita.</s> <s xml:id="echoid-s18734" xml:space="preserve"> <lb/>ut ſuperior ſuperficies eius indicis ſit æquidiſtans ſuperficiei uiſus, quantùm ad ſenſum:</s> <s xml:id="echoid-s18735" xml:space="preserve"> uidebit <lb/>ſuperficiem indicis.</s> <s xml:id="echoid-s18736" xml:space="preserve"> Sed omnia iſta loca ſunt extra pyramidem radialem:</s> <s xml:id="echoid-s18737" xml:space="preserve"> & hoc patebit.</s> <s xml:id="echoid-s18738" xml:space="preserve"> Nam py-<lb/>ramis radialis, quam continet foramẽ uueæ, eſt ualde ſubtilis, & extenditur rectè, & pyramidalitas <lb/>eius non eſt ampla:</s> <s xml:id="echoid-s18739" xml:space="preserve"> unde nihil ex ipſa peruenit ad loca, quæ circundant oculum, & appropinquant <lb/>corpori oculi, et æquidiſtant ſuperficiei oculi:</s> <s xml:id="echoid-s18740" xml:space="preserve"> & inter omnia loca continentia oculum, & æquidi-<lb/>ſtantia ſuperficiei uiſus, & inter ſuperficiem uiſus, ſunt lineæ rectæ, propter refractionẽ earũ à cor-<lb/>poribus denſis, cum aer, qui eſt inter ipſa & ſuperficiem uiſus, fuerit continuus:</s> <s xml:id="echoid-s18741" xml:space="preserve"> tunc forma horum <lb/>uiſibilium peruenit ad ſuperficiem uiſus ſuper has lineas, quæ ſunt extra pyramidem.</s> <s xml:id="echoid-s18742" xml:space="preserve"> Et cum hæc <lb/>forma perueniat ad uiſum non per lineas radiales, & tamen comprehendatur à uiſu:</s> <s xml:id="echoid-s18743" xml:space="preserve"> patet, quòd ui-<lb/>ſus comprehendat illam refractè.</s> <s xml:id="echoid-s18744" xml:space="preserve"> Ex hac ergo experientia patet, quòd uiſus comprehendit multa <lb/>eorum, quę ſunt extra pyramidem radialem, refractè.</s> <s xml:id="echoid-s18745" xml:space="preserve"> Inductione autem poſſumus oſtendere, quòd <lb/>uiſus comprehendit illa, quæ ſunt intra pyramidem radialem, refractè, cum hoc, quod comprehen-<lb/>dit illa rectè, hoc modo.</s> <s xml:id="echoid-s18746" xml:space="preserve"> Accipias acum ſubtilem, & ſedeas in loco oppoſito albo parieti, & coope-<lb/>rias alterum oculorum, & ponas acũ in oppoſitione alterius oculi, & facias acum appropinquare, <lb/>ita ut applicetur palpebræ, & ponas acum in oppoſitione medij uiſus, & aſpicias parietem oppoſi-<lb/>tum:</s> <s xml:id="echoid-s18747" xml:space="preserve"> tunc enim uidebis acum, quaſi corpus diaphanum, in quo eſt aliquantula denſitas:</s> <s xml:id="echoid-s18748" xml:space="preserve"> & uidebis <lb/>quicquid eſt ultra acum ex pariete, & apud acum quaſi corpus latum, cuius latitudo eſt multiplex <lb/>ad latitudinem acus.</s> <s xml:id="echoid-s18749" xml:space="preserve"> Cauſſa autem huius in ſecundo tractatu declarata eſt:</s> <s xml:id="echoid-s18750" xml:space="preserve"> ſcilicet quòd ſi res uiſi-<lb/>bilis fuerit multùm propinqua uiſui:</s> <s xml:id="echoid-s18751" xml:space="preserve"> uidebitur maior, quàm ſit:</s> <s xml:id="echoid-s18752" xml:space="preserve"> & quantò magis fuerit propinqua, <lb/>tantò magis uidebitur maior.</s> <s xml:id="echoid-s18753" xml:space="preserve"> Diaphanitas autem eius eſt, quia uiſus comprehendit quicquid eſt <lb/>ultrà:</s> <s xml:id="echoid-s18754" xml:space="preserve"> acus autem eſt corpus denſum cooperiens, quod eſt ultrà:</s> <s xml:id="echoid-s18755" xml:space="preserve"> & quia acus eſt ualde propinqua <lb/>uiſui:</s> <s xml:id="echoid-s18756" xml:space="preserve"> ideo cooperuit de pariete multiplex ad ſuam latitudinẽ.</s> <s xml:id="echoid-s18757" xml:space="preserve"> Baſis enim pyramidis (cuius caput <lb/>eſt centrum uiſus, & baſis eſt altitudo acus) erit multiplex ad latitudinem acus:</s> <s xml:id="echoid-s18758" xml:space="preserve"> & cum hoc, uiſus <lb/>comprehendit quicquid eſt ultra acum, nec cooperitur à uiſu aliquid de pariete, ſed comprehen-<lb/>dit quod eſt ultrà, quaſi ultra corpus diaphanum.</s> <s xml:id="echoid-s18759" xml:space="preserve"> Et cum acus fuerit oppoſita medio uiſui:</s> <s xml:id="echoid-s18760" xml:space="preserve"> tunc nõ <lb/>cooperiet totam ſuperficiem uiſus, propter ſubtilitatem eius, ſed aliquam partem, quanta eſt lati-<lb/>tu do eius:</s> <s xml:id="echoid-s18761" xml:space="preserve"> & remanet ex ſuperficie uiſus aliquid à lateribus acus:</s> <s xml:id="echoid-s18762" xml:space="preserve"> & exit forma cius ad illud, quod <lb/>eſt à lateribus acus de ſuperficie uiſus.</s> <s xml:id="echoid-s18763" xml:space="preserve"> Forma autem exiens ad acum, nũquam perueniet ad uiſum, <lb/>nec comprehendetur ab ipſo:</s> <s xml:id="echoid-s18764" xml:space="preserve"> forma autem, quæ peruenit ad latera ſuperficiei uiſus, refringitur ad <lb/> <pb o="270" file="0276" n="276" rhead="ALHAZEN"/> uiſum, cum non rectè perueniat ad centrum uiſus.</s> <s xml:id="echoid-s18765" xml:space="preserve"> Si ergo uiſus non comprehenderet illud, quod <lb/>opponitur ex pariete acui, niſi rectè:</s> <s xml:id="echoid-s18766" xml:space="preserve"> tunc illud, quod opponitur acui ex pariete, eſſet coopertum à <lb/>uiſu.</s> <s xml:id="echoid-s18767" xml:space="preserve"> Cũ ergo comprehendatur, & non rectè:</s> <s xml:id="echoid-s18768" xml:space="preserve"> patet ipſum comprehendi refractè performam, quæ <lb/>refringitur à lateribus acus ex ſuperficie uiſus.</s> <s xml:id="echoid-s18769" xml:space="preserve"> Et hoc iam manifeſtatur etiam, quòd ſi experimẽta-<lb/>tor poſuerit loco acus aliquod corpus latũ, cuius latitudo ſit maior latitudine foraminis uueæ:</s> <s xml:id="echoid-s18770" xml:space="preserve"> tũc <lb/>enim nihil uidebit omnino de pariete, nec uidebit illud corpus diaphanũ, ſed dẽſum.</s> <s xml:id="echoid-s18771" xml:space="preserve"> Ex hoc ergo, <lb/>quòd paries comprehenditur ultra acum ex gracilitate eius, & non comprehenditur ultra corpus <lb/>latũ:</s> <s xml:id="echoid-s18772" xml:space="preserve"> ſcimus quòd illa comprehẽſio eſt ex forma, quæ peruenit ad acũ ex ſuperficie uilu<gap/>, & refrin-<lb/>gitur in tunicis uiſus.</s> <s xml:id="echoid-s18773" xml:space="preserve"> Et quia quic quid à uiſu comprehenditur refractè, comprehenditur in rectitu <lb/>dine perpendicularium:</s> <s xml:id="echoid-s18774" xml:space="preserve"> ideo illud, quod comprehẽdit, comprehẽdit refractè ex forma eius, quod <lb/>opponitur acui per rectitudinem linearũ, exeuntiũ à cẽtro uiſus, cũ eo, quod opponitur acui ex pa <lb/>riete:</s> <s xml:id="echoid-s18775" xml:space="preserve"> & hæ lineæ ſecantur acu, & uiſus comprehendit illud, quod eſt ultra acũ etiam in rectitudine <lb/>harũ linearũ, & comprehendit acũ etiam in rectitudine illarũ.</s> <s xml:id="echoid-s18776" xml:space="preserve"> Quare totam formam quaſi compre-<lb/>hendet ultra corpus diaphanũ, in quo eſt aliquãtula denſitas.</s> <s xml:id="echoid-s18777" xml:space="preserve"> Et ſi experimẽtator ſcripſerit in bom-<lb/>bace ſubtiliter, & applicauerit ipſum parieti, & remotus fuerit à pariete, in quantũ poſsit legere ſcri <lb/>pturam, & poſueritacũ in oppoſitione medij uiſus, ut primò fecit, & aſpexerit bombacem:</s> <s xml:id="echoid-s18778" xml:space="preserve"> tũc po-<lb/>terit legere ſcripturam, ſed tamen uidebit eam quaſi ultra uitrũ aut ultra corpus diaphanũ, in quo <lb/>eſt aliqua dẽſitas.</s> <s xml:id="echoid-s18779" xml:space="preserve"> Si ergo uiſus non comprehẽdit illud, quod opponitur acui de bom bace ſecũdum <lb/>refractionem:</s> <s xml:id="echoid-s18780" xml:space="preserve"> tũc aliquid lateret de ſcriptura:</s> <s xml:id="echoid-s18781" xml:space="preserve"> acus enim debet cooperire de ſcriptura multò magis <lb/>ſe in quantitate latitudinis diaphanitatis, quàm tũc comprehendit, propter remotionem bomba-<lb/>cis à uiſu.</s> <s xml:id="echoid-s18782" xml:space="preserve"> Sed quia uiſui non patet aliquid de ſcriptura:</s> <s xml:id="echoid-s18783" xml:space="preserve"> patet ipſum comprehẽdere illud, quod op-<lb/>ponitur acui:</s> <s xml:id="echoid-s18784" xml:space="preserve"> ſed hoc non poteſt fieri rectè:</s> <s xml:id="echoid-s18785" xml:space="preserve"> reſtat ergo, ut fiat refractè.</s> <s xml:id="echoid-s18786" xml:space="preserve"> Et ſi experimẽtator abſtule-<lb/>rit acum, non deſtruetur refractio, quę prius erat:</s> <s xml:id="echoid-s18787" xml:space="preserve"> non enim propter acum erat refractio, fed creſcit <lb/>refractio, eò quòd refringitur ex loco acus.</s> <s xml:id="echoid-s18788" xml:space="preserve"> Et cũ experimẽtator abſtulerit acum:</s> <s xml:id="echoid-s18789" xml:space="preserve"> comprehẽdet il-<lb/>lud, quod opponitur uiſui, manifeſtius.</s> <s xml:id="echoid-s18790" xml:space="preserve"> Nam comprehẽdet illud rectè, quod cooperiebatur acu:</s> <s xml:id="echoid-s18791" xml:space="preserve"> cũ <lb/>hoc, quod comprehẽdit illud refractè, ſicut comprehẽdebat cum cooperiebatur:</s> <s xml:id="echoid-s18792" xml:space="preserve"> & propter hanc <lb/>additionem comprehendit illud manifeſtius, quàm antequam auferretacum.</s> <s xml:id="echoid-s18793" xml:space="preserve"> Ex qua experientia <lb/>patet, quòd illud quod opponitur uiſui de illis, quæ ſunt intra pyramidem radialem, comprehẽdi-<lb/>tur refractè & rectè.</s> <s xml:id="echoid-s18794" xml:space="preserve"> Ex his ergo omnibus declaratur, quòd omnia, quæ comprehenduntur à uiſu, <lb/>quorum formæ perueniunt ad uiſum rectè, aut reflexè, aut refractè:</s> <s xml:id="echoid-s18795" xml:space="preserve"> comprehenduntur ſecundum <lb/>refractionẽ factam apud ſuperficiem uiſus:</s> <s xml:id="echoid-s18796" xml:space="preserve"> & quòd illorũ quę comprehendũtur ſecũdum refractio <lb/>nem factam à ſuperficie uiſus:</s> <s xml:id="echoid-s18797" xml:space="preserve"> quædam comprehẽduntur refractè & rectè ſimul:</s> <s xml:id="echoid-s18798" xml:space="preserve"> & ideo illud, qđ <lb/>opponitur medio uiſus, eſt manifeſtius illo, quod eſt in circuitu medij.</s> <s xml:id="echoid-s18799" xml:space="preserve"> Et cum uiſus comprehẽde-<lb/>rit aliquid latum, comprehẽdet illud, quod eft in medio, manifeſtius illo, quod eſt in lateribus.</s> <s xml:id="echoid-s18800" xml:space="preserve"> Hoc <lb/>aũt declaratũ eſt in ſecũ do tractatu, in quo declarauimus, quo modo hoc poſſet experimẽtari:</s> <s xml:id="echoid-s18801" xml:space="preserve"> & di <lb/>ximus, quòd cauſſa huius eſt propter lineas radiales:</s> <s xml:id="echoid-s18802" xml:space="preserve"> & hoc eſt in illis, quę ſunt intra pyramidem ra <lb/>dialẽ:</s> <s xml:id="echoid-s18803" xml:space="preserve"> In alijs aũt, quæ ſunt extrà, eſt cauſſa refractio.</s> <s xml:id="echoid-s18804" xml:space="preserve"> Cauſſa aũt uniuerſalis in hoc, quòd illud, quod <lb/>opponitur medio uiſus, eſt manifeſtius, quã illud, quod eſt in circuitu:</s> <s xml:id="echoid-s18805" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s18806" xml:space="preserve"> quoniã illud quòd oppo <lb/>nitur medio uiſus, comprehẽditur rectè & refractè ſimul.</s> <s xml:id="echoid-s18807" xml:space="preserve"> Hoc aũt, quòd quicquid comprehẽditur <lb/>à uiſu, comprehendatur refractè, à nullo antiquorum dictum eſt.</s> <s xml:id="echoid-s18808" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div606" type="section" level="0" n="0"> <head xml:id="echoid-head522" xml:space="preserve">DE FALLACIIS VISVS, QVAE ACCIDVNT <lb/>ex refractione. Cap. VII.</head> <head xml:id="echoid-head523" xml:space="preserve" style="it">38. Refractio debilit at lucem & colorem uiſibilis: ita totam imaginem confuſam uiſui of-<lb/>fert. 10 p 10.</head> <p> <s xml:id="echoid-s18809" xml:space="preserve">FAllaciæ, quæ accidunt ſecũdum refractionem:</s> <s xml:id="echoid-s18810" xml:space="preserve"> ſimiles ſunt ijs, quæ accidunt per reflexionem.</s> <s xml:id="echoid-s18811" xml:space="preserve"> <lb/>Quod enim comprehẽditur refractè, comprehẽditur non in ſuo loco, cũ comprehendatur in <lb/>loco imaginis:</s> <s xml:id="echoid-s18812" xml:space="preserve"> quapropter poſitio formæ comprehenſæ erit alia à poſitione rei uiſæ.</s> <s xml:id="echoid-s18813" xml:space="preserve"> Item re-<lb/>fractio debilitat formam refractam, ſcilicet, formam lucis & coloris, quæ ſunt in re uiſa.</s> <s xml:id="echoid-s18814" xml:space="preserve"> Et hoc po-<lb/>teſt intelligi:</s> <s xml:id="echoid-s18815" xml:space="preserve"> quoniam ſi aſpexeris aliquid exiſtens in aqua, & tu ſis obliquus à perpendicularibus, <lb/>exeuntibus à re uiſa ſuper ſuperficiem aquę multa obliquatione, & intuearis illud uerè, deinde mo <lb/>uearis, & moueas uiſum, donec ponas ipſum in aliqua perpẽdiculari, exeunte à re uiſa ſuper ſuper-<lb/>ficiem aquæ, & aſpexeris:</s> <s xml:id="echoid-s18816" xml:space="preserve"> tunc uidebis illud manifeſtius, quàm cum eras obliquus:</s> <s xml:id="echoid-s18817" xml:space="preserve"> & nulla eſt dif-<lb/>ferentia inter duos ſitus, niſi quia in primo, forma, quæ exit ad uiſum, eſt refracta & multùm obli-<lb/>qua:</s> <s xml:id="echoid-s18818" xml:space="preserve"> in ſecundo autem forma exit rectè, aut quædam pars ipſius exit rectè, & quædam modicũ obli <lb/>què aut ferè rectè.</s> <s xml:id="echoid-s18819" xml:space="preserve"> Ex hac ergo experimentatione declaratur, quòd refractio debilitat formas refra <lb/>ctas.</s> <s xml:id="echoid-s18820" xml:space="preserve"> Item ea, quę ſunt in aqua, & ultra uitrum & conſimilia, quando refringuntur ad uiſum, deferũt <lb/>ſecum colorem corporis, in quo exiſtunt.</s> <s xml:id="echoid-s18821" xml:space="preserve"> In illis ergo, quæ comprehendũtur refractè ultra corpora <lb/>diaphana, accidunt propter refractionem fallaciæ, quæ non accidunt in eis, quæ uidentur rectè, ſci <lb/>licet diuerſitas poſitionis & diſtantiæ, & debilitas lucis & coloris.</s> <s xml:id="echoid-s18822" xml:space="preserve"> Præterea accidunt eis iſta, quæ <lb/>accidunt illis, quæ rectè uidentur.</s> <s xml:id="echoid-s18823" xml:space="preserve"> Formæ enim eorum, quæ comprehenduntur refractè, compre-<lb/>henduntur in oppoſitione uiſus & in rectitudine linearũ radialiũ.</s> <s xml:id="echoid-s18824" xml:space="preserve"> Quicquid ergo accidit eis, quæ <lb/>uidentur in rectitudine linearũ radialiũ, accidit iſtis.</s> <s xml:id="echoid-s18825" xml:space="preserve"> Et in tertio libro declarauimus oẽs illas falla-<lb/> <pb o="271" file="0277" n="277" rhead="OPTICAE LIBER VII."/> cias & cauſſas earum:</s> <s xml:id="echoid-s18826" xml:space="preserve"> & quę ſunt etiam cauſſæ iſtarum:</s> <s xml:id="echoid-s18827" xml:space="preserve"> ſed in his accidit magis, & citius propter de-<lb/>bilitatem harum formarum.</s> <s xml:id="echoid-s18828" xml:space="preserve"> Particulares autem deceptiones, quæ accidunt propter figuras ſuper-<lb/>ficierum corporum diaphanorum, ſunt multimodæ, ſed accidunt rarò uiſui.</s> <s xml:id="echoid-s18829" xml:space="preserve"> Ea enim, quę compre-<lb/>henduntur ultra corpora diaphana, diuerſa ab aere, ſunt ſtellæ, & ea, quæ ſunt in aqua:</s> <s xml:id="echoid-s18830" xml:space="preserve"> illa autem, <lb/>quæ ſunt ultra ultrum, & lapides diaphanos diuerſarum figurarum rarò comprehenduntur à uiſu:</s> <s xml:id="echoid-s18831" xml:space="preserve"> <lb/>& non eſt ita de iſtis corporibus diaphanis, ut de ſpeculis:</s> <s xml:id="echoid-s18832" xml:space="preserve"> ſpecula enim ſæpius aſpiciuntur ab homi <lb/>nibus, ut uideant in eis ſuas formas, & habentur in domibus.</s> <s xml:id="echoid-s18833" xml:space="preserve"> Et ſimiliter quando homo inſpexerit <lb/>in quodlibet corpus terſum:</s> <s xml:id="echoid-s18834" xml:space="preserve"> etiam uidebit formam eorum, quæ ſunt in oppoſitione.</s> <s xml:id="echoid-s18835" xml:space="preserve"> Et ſimiliter ſi <lb/>aſpexerit a quam:</s> <s xml:id="echoid-s18836" xml:space="preserve"> uidebit formam ſuam in ea, & uidebit, quæ ſunt in oppoſitione.</s> <s xml:id="echoid-s18837" xml:space="preserve"> Et non eſt ita il-<lb/>lud, quod uidebit ultra uitrum, & lapides diaphanos:</s> <s xml:id="echoid-s18838" xml:space="preserve"> quia homines rarò aſpiciunt ad illud, quod eſt <lb/>ultra uitrum, & lapides diaphanos.</s> <s xml:id="echoid-s18839" xml:space="preserve"> Et quia ita eſt, dicamus de deceptionibus refractionis particu-<lb/>laribus, quæ ſemper accidunt & ſine difficultate, ſcilicet quæ accidunt in eis, quę uidentur in cœlo, <lb/>& in aqua:</s> <s xml:id="echoid-s18840" xml:space="preserve"> & dicemus parum de his, quæ uidentur ultra uitrũ, & lapides.</s> <s xml:id="echoid-s18841" xml:space="preserve"> Dicamus ergo, quòd ſemք <lb/>uiſus fallitur in eis, quæ comprehen duntur ultra corpus diaphanum, diuerſum ab aere, præſertim <lb/>in poſitione & remotione, in coloribus & lucibus eorum, & in magnitudine eorum & figuris quo-<lb/>rundam.</s> <s xml:id="echoid-s18842" xml:space="preserve"> Ea enim, quæ uidentur in aqua, & ultra uitrum, & lapides diaphanos, uidẽtur maiora:</s> <s xml:id="echoid-s18843" xml:space="preserve"> ſtel-<lb/>læ autem, & diſtantiæ inter ſtellas, quandoq;</s> <s xml:id="echoid-s18844" xml:space="preserve"> uidentur maiores, quandoque minores.</s> <s xml:id="echoid-s18845" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div607" type="section" level="0" n="0"> <head xml:id="echoid-head524" xml:space="preserve" style="it">39. Si communis ſectio ſuperficierum, refractionis & refractiui fuerit linea recta, & uiſ{us} <lb/>ſit in perpendiculari duct a à medio uiſibilis par alleli communi ſectioni: imago maior uidebitur <lb/>uiſibili. 31 p 10.</head> <p> <s xml:id="echoid-s18846" xml:space="preserve">SIt ergo uiſus a:</s> <s xml:id="echoid-s18847" xml:space="preserve"> & ſit b c ultra corpus diaphanum, groſsius aere:</s> <s xml:id="echoid-s18848" xml:space="preserve"> Dico, quòd b cuidetur maior, <lb/>quàm ſit.</s> <s xml:id="echoid-s18849" xml:space="preserve"> Sit ergo primò ſuperficies corporis diaphani plana.</s> <s xml:id="echoid-s18850" xml:space="preserve"> A aut eſt in perpendiculari, exe-<lb/>unte à medio b c ſuper ſuperficiem corporis:</s> <s xml:id="echoid-s18851" xml:space="preserve"> aut extra.</s> <s xml:id="echoid-s18852" xml:space="preserve"> Sit ergo in primis, in ipſa:</s> <s xml:id="echoid-s18853" xml:space="preserve"> & [per 12 p 1] <lb/>ſit illa perpendicularis a m z:</s> <s xml:id="echoid-s18854" xml:space="preserve"> & extrahamus ſuperficiem, in qua ſunt lineæ a z, b c:</s> <s xml:id="echoid-s18855" xml:space="preserve"> & faciet in ſuperfi <lb/>cie corporis diaphani lineam d m e [per 3 p 11:</s> <s xml:id="echoid-s18856" xml:space="preserve">] & [per 9 n] ſuperficies, in qua ſunt duæ lineę a z, b c, <lb/>crit perpendicularis ſuper ſuperficiem corporis diaphani.</s> <s xml:id="echoid-s18857" xml:space="preserve"> Et non tranſit per a & per aliquod pun-<lb/>ctum lineæ b c ſuperficies, quæ ſit perpendicularis ſuper ſuperficiem corporis diaphani, niſi illa, in <lb/>qua ſunt lineæ a z, b c.</s> <s xml:id="echoid-s18858" xml:space="preserve"> Non enim tranſit per a ſuperficies perpendicularis ſuper ſuperficiem corpo-<lb/>ris diaphani, niſi illa, quæ tranſit per a z:</s> <s xml:id="echoid-s18859" xml:space="preserve"> quæ linea eſt perpendicularis ſuper ſuperficiem corporis:</s> <s xml:id="echoid-s18860" xml:space="preserve"> <lb/>[per 9 n & conuerſionem 4 d 11] nec exit ex a perpendicularis ſuper ſuperficiem corporis diapha-<lb/>ni, niſi linea a z.</s> <s xml:id="echoid-s18861" xml:space="preserve"> Non ergo per a tranſit ſuperficies, quæ ſit perpendicularis ſuper ſuperficiem corpo <lb/>ris diaphani, niſi illa, quæ tranſit per lineam a z:</s> <s xml:id="echoid-s18862" xml:space="preserve"> & non tranſit per aliquod punctum lineæ b c & per <lb/>lineam a z, niſi illa ſuperficies, in qua ſunt duæ lineæ a z, b c.</s> <s xml:id="echoid-s18863" xml:space="preserve"> Non ergo tranſit per a & per aliquod <lb/>punctum lineæ b c ſuperficies perpen dicularis ſuper ſu-<lb/> <anchor type="figure" xlink:label="fig-0277-01a" xlink:href="fig-0277-01"/> perficiẽ corporis diaphani, niſi illa, in qua ſunt lineæ a z, <lb/>b c.</s> <s xml:id="echoid-s18864" xml:space="preserve"> Non ergo refringetur forma alicuius puncti eorum, <lb/>quæ ſunt in b c, niſi ex linea d e.</s> <s xml:id="echoid-s18865" xml:space="preserve"> Et [per 11 p 1] extraha-<lb/>mus ex b & c duas perpendiculares:</s> <s xml:id="echoid-s18866" xml:space="preserve"> cadent ergo in lineã <lb/>d e in duobus punctis d e, [per lemma Procli ad 29 p 1:</s> <s xml:id="echoid-s18867" xml:space="preserve"> <lb/>quia b c, d e ſunt parallelæ ex theſi] ſcilicet b d, c e.</s> <s xml:id="echoid-s18868" xml:space="preserve"> Et ſit <lb/>b c in primis æ quidiſtans lineę d e:</s> <s xml:id="echoid-s18869" xml:space="preserve"> & refringatur forma <lb/>b ad a ex p:</s> <s xml:id="echoid-s18870" xml:space="preserve"> & forma c ad a ex h:</s> <s xml:id="echoid-s18871" xml:space="preserve"> & cõtinuemus lineas b p, <lb/>p a, c h, h a:</s> <s xml:id="echoid-s18872" xml:space="preserve"> item a b, a c:</s> <s xml:id="echoid-s18873" xml:space="preserve"> & extrahamus a p ad l, & a h ad k.</s> <s xml:id="echoid-s18874" xml:space="preserve"> <lb/>[Nam quòd a p, a h concurrant cum b d, c e patet per lem <lb/>ma Procli ad 29 p 1.</s> <s xml:id="echoid-s18875" xml:space="preserve">] Quia ergo z poſitum fuit in medio <lb/>lineæ b c, poſitio b ex a erit ęqualis poſitioni c exa:</s> <s xml:id="echoid-s18876" xml:space="preserve"> & ſic <lb/>diſtantia p ex a erit ſicut diſtantia h ex a.</s> <s xml:id="echoid-s18877" xml:space="preserve"> [Quia enim a z <lb/>bifariam ſecans b c, eſt ad eandem perpendicularis per <lb/>theſin, ipſaq́;</s> <s xml:id="echoid-s18878" xml:space="preserve"> a z communis, æquatur ſibijpſi:</s> <s xml:id="echoid-s18879" xml:space="preserve"> erit per 4 p 1 <lb/>a b æqualis a c.</s> <s xml:id="echoid-s18880" xml:space="preserve"> Itaque cum b c, d e ſint parallelæ ex theſi, <lb/>& puncta b & c à uiſu a æ quabiliter diſtent:</s> <s xml:id="echoid-s18881" xml:space="preserve"> ab eodem æ-<lb/>quabiliter diſtabuntrefractionum puncta p & h, propter <lb/>æquabilem in eodem & æquabili medio punctorum o-<lb/>mnium diffuſionem.</s> <s xml:id="echoid-s18882" xml:space="preserve"> Quare a p æquatur ipſi h a:</s> <s xml:id="echoid-s18883" xml:space="preserve">] & ſic <lb/>[per 5.</s> <s xml:id="echoid-s18884" xml:space="preserve"> 15 p 1] angulus d p l erit æqualis angulo e h k, ſed <lb/>[per 29 p 1] duo anguli d, e ſunt recti:</s> <s xml:id="echoid-s18885" xml:space="preserve"> & linea d p eſt æqua <lb/>lis lineæ e h:</s> <s xml:id="echoid-s18886" xml:space="preserve"> quia p m eſt ęqualis m h.</s> <s xml:id="echoid-s18887" xml:space="preserve"> [Nam quia per the <lb/>ſin, fabricationem & 34 p 1 tota m d ęquatur toti m e:</s> <s xml:id="echoid-s18888" xml:space="preserve"> & an <lb/>guli ad m deinceps recti per 29 p 1, & ad p & h ęquales per <lb/>concluſionem, latusq́;</s> <s xml:id="echoid-s18889" xml:space="preserve"> a m commune:</s> <s xml:id="echoid-s18890" xml:space="preserve"> æquabitur per 26 <lb/>p 1 m p ipſim h.</s> <s xml:id="echoid-s18891" xml:space="preserve"> Quare reliqua d p æ quabitur reliquæ e h per 19 p 5] ergo [per 26 p 1] d l eſt æqualis <lb/>e k:</s> <s xml:id="echoid-s18892" xml:space="preserve"> & continuemus l k:</s> <s xml:id="echoid-s18893" xml:space="preserve"> erit ergo [per 33 p 1] æqualis lineæ b c:</s> <s xml:id="echoid-s18894" xml:space="preserve"> angulus ergo c a b erit minor angu-<lb/>lo k a l.</s> <s xml:id="echoid-s18895" xml:space="preserve"> [Nam recta l k ſecãs latera a b, a c, facit duos angulos exteriores, maiores interioribus oppo <lb/> <pb o="272" file="0278" n="278" rhead="ALHAZEN"/> ſitis ad l & k per 16 p 1:</s> <s xml:id="echoid-s18896" xml:space="preserve"> ſed angulis exterioribus à rectis a b, a c & ſecante k l factis æquantur interio-<lb/>res ad b & c trianguli a b c per 29 p 1.</s> <s xml:id="echoid-s18897" xml:space="preserve"> Anguli igitur ad b & c ſunt maiores angulis a d l & k.</s> <s xml:id="echoid-s18898" xml:space="preserve"> Quare per <lb/>32 p 1 reliquus a b c minor eſt reliquo l a k:</s> <s xml:id="echoid-s18899" xml:space="preserve">] & linea l k eſt diameter imaginis b c.</s> <s xml:id="echoid-s18900" xml:space="preserve"> Nam omne punctũ <lb/>lineæ b c refringitur ab aliquo puncto p h.</s> <s xml:id="echoid-s18901" xml:space="preserve"> Nam ſi forma b refringitur ex p:</s> <s xml:id="echoid-s18902" xml:space="preserve"> punctum, quod eſt inter <lb/>b & z, refringitur ab aliquo puncto inter p & m:</s> <s xml:id="echoid-s18903" xml:space="preserve"> & ponamus ſuper lineam b z punctũ n.</s> <s xml:id="echoid-s18904" xml:space="preserve"> Si ergo for-<lb/>ma n refringeretur ab aliquo puncto extra lineam m p exparte d:</s> <s xml:id="echoid-s18905" xml:space="preserve"> tunc linea, per quam extenditur <lb/>forman, ſecaret lineam b p:</s> <s xml:id="echoid-s18906" xml:space="preserve"> & ſic forma puncti ſectionis refringeretur ad a ex duobus punctis [p & <lb/>g,] quod eſt impoſsibile, ut diximus in capitulo quinto huius libri de imagine:</s> <s xml:id="echoid-s18907" xml:space="preserve"> [19 n] n ergo non re <lb/>fringitur ad a, niſi ex aliquo puncto inter p m.</s> <s xml:id="echoid-s18908" xml:space="preserve"> Et ſimiliter omne punctum in z c, non refringetur ad <lb/>a, niſi ex linea m h.</s> <s xml:id="echoid-s18909" xml:space="preserve"> Linea ergo l k eſt diameter imaginis lineę b c:</s> <s xml:id="echoid-s18910" xml:space="preserve"> [per 18 n] forma ergo b c uidebitur <lb/>in l k.</s> <s xml:id="echoid-s18911" xml:space="preserve"> Item iam declarauimus [numero præcedente] quòd forma refracta eſt debilior recta:</s> <s xml:id="echoid-s18912" xml:space="preserve"> ergo for <lb/>ma b c, quę comprehenditur refractè, eſt debilior forma eius, quę comprehenditur rectè:</s> <s xml:id="echoid-s18913" xml:space="preserve"> & propter <lb/>debilitatem formæ rei, uiſus aſsimilat eam formæ rei, quæ uidetur à maiore remotione:</s> <s xml:id="echoid-s18914" xml:space="preserve"> maior enim <lb/>diſtantia debilitat formam.</s> <s xml:id="echoid-s18915" xml:space="preserve"> Et iam declarauimus in ſecundo libro [38 n] quòd uiſus comprehendit <lb/>imaginem rei uiſæ ſecundum quantitatem anguli, reſpectu remotionis & poſitionis rei uiſæ apud <lb/>uiſum:</s> <s xml:id="echoid-s18916" xml:space="preserve"> & angulus k a l eſt maior angulo c a b [ex concluſo,] & poſitio l k eſt ſicut poſitio c b, & b c ui <lb/>detur in l k, & l k comprehenditur in maiore quaſi diſtantia, diſtantia b c, propter debilitatem for-<lb/>mæ.</s> <s xml:id="echoid-s18917" xml:space="preserve"> Viſus ergo comprehendit b c refractè ex comparatione anguli maioris angulo c a b ad diſtan-<lb/>tiam maiorem diſtantia b c, & ad poſitionem æqualem poſitioni b c.</s> <s xml:id="echoid-s18918" xml:space="preserve"> Quapropter b c comprehendi-<lb/>tur refractè maior:</s> <s xml:id="echoid-s18919" xml:space="preserve"> & hoc duabus de cauſsis, ſcilicet magnitudine anguli, & debilitate formæ.</s> <s xml:id="echoid-s18920" xml:space="preserve"> Cauſ-<lb/>ſa autem magnitudinis anguli, eſt propinquitas anguli ad uiſum:</s> <s xml:id="echoid-s18921" xml:space="preserve"> & cauſſa propin quitatis anguli eſt <lb/>refractio.</s> <s xml:id="echoid-s18922" xml:space="preserve"> Cauſſa ergo, qua b c comprehenditur maior, eſt refractio.</s> <s xml:id="echoid-s18923" xml:space="preserve"/> </p> <div xml:id="echoid-div607" type="float" level="0" n="0"> <figure xlink:label="fig-0277-01" xlink:href="fig-0277-01a"> <variables xml:id="echoid-variables222" xml:space="preserve">a<unsure/>d m <gap/> g p h l k q bn<unsure/> z c</variables> </figure> </div> </div> <div xml:id="echoid-div609" type="section" level="0" n="0"> <head xml:id="echoid-head525" xml:space="preserve" style="it">40. Si communis ſectio ſuperficierum, refractionis & refractiui fuerit linea recta, & uiſ{us} <lb/>ſit in perpendiculari duct a à medio uiſibilis obliqui ad communem ſectionem: imago maior ui-<lb/>debitur uiſibili. 32 p 10.</head> <p> <s xml:id="echoid-s18924" xml:space="preserve">ITem:</s> <s xml:id="echoid-s18925" xml:space="preserve"> iteremus figurã:</s> <s xml:id="echoid-s18926" xml:space="preserve"> & ſit b c nõ æquidiſtans lineę d e:</s> <s xml:id="echoid-s18927" xml:space="preserve"> & extrahamus à remotiore extremitatũ <lb/>b c lineam æquidiſtantẽ lineæ d e:</s> <s xml:id="echoid-s18928" xml:space="preserve"> [per 31 p 1] & ſit c q:</s> <s xml:id="echoid-s18929" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0278-01a" xlink:href="fig-0278-01"/> & extrahamus a z ad o:</s> <s xml:id="echoid-s18930" xml:space="preserve"> erit ergo o in medio c q.</s> <s xml:id="echoid-s18931" xml:space="preserve"> [Quia <lb/>enim per fabricationem a z parallela d b, continuata eſt <lb/>in o, & d b in q:</s> <s xml:id="echoid-s18932" xml:space="preserve"> erit ք 2 p 6, ut b z ad z c, ſic q o ad o c:</s> <s xml:id="echoid-s18933" xml:space="preserve"> ſed b <lb/>z ęquatur z c ex theſi:</s> <s xml:id="echoid-s18934" xml:space="preserve"> ergo q o ęquabitur o c:</s> <s xml:id="echoid-s18935" xml:space="preserve"> o igitur erit <lb/>medium punctũ lineę q c:</s> <s xml:id="echoid-s18936" xml:space="preserve">] quare z eſt in medio b c:</s> <s xml:id="echoid-s18937" xml:space="preserve"> quia <lb/>b q eſt æ quidiſtans z o:</s> <s xml:id="echoid-s18938" xml:space="preserve"> & [per 2 p 6] proportio q o ad o c, <lb/>ſicut b z ad z c.</s> <s xml:id="echoid-s18939" xml:space="preserve"> Et refringatur forma q ad a exp:</s> <s xml:id="echoid-s18940" xml:space="preserve"> & forma <lb/>c ad a ex h:</s> <s xml:id="echoid-s18941" xml:space="preserve"> & continuemus a p, & pertranſeat uſque ad l:</s> <s xml:id="echoid-s18942" xml:space="preserve"> <lb/>& continuemus a h, & pertranſeat uſque ad k:</s> <s xml:id="echoid-s18943" xml:space="preserve"> & conti-<lb/>nuemus l k:</s> <s xml:id="echoid-s18944" xml:space="preserve"> erit ergo l k diameter imaginis q c:</s> <s xml:id="echoid-s18945" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s18946" xml:space="preserve"> an-<lb/>gulus k a l maior angulo c a q:</s> <s xml:id="echoid-s18947" xml:space="preserve"> [ut oſtenſum eſt pręceden <lb/>te numero] a ergo comprehendet imaginem q c maiorẽ <lb/>q c, ut prius diximus.</s> <s xml:id="echoid-s18948" xml:space="preserve"> Linea autem q p ſecab it lineam b c <lb/>in r:</s> <s xml:id="echoid-s18949" xml:space="preserve"> r ergo refringetur ad a ex p:</s> <s xml:id="echoid-s18950" xml:space="preserve"> ergo b refringetur ad a <lb/>ex puncto inter duo puncta p, d.</s> <s xml:id="echoid-s18951" xml:space="preserve"> Nam ſi refrin geretur ex <lb/>puncto inter p, m:</s> <s xml:id="echoid-s18952" xml:space="preserve"> accideret prædictum impoſsibile [nu <lb/>mero pręcedente:</s> <s xml:id="echoid-s18953" xml:space="preserve"> quod erat, idem punctũ uiſibilis à duo <lb/>bus refractiui punctis refringi non poſſe.</s> <s xml:id="echoid-s18954" xml:space="preserve">] Refringatur <lb/>ergo b ad a ex f, & continuemus a f, & pertranſeat ad i, & <lb/>cõtinuemus i k:</s> <s xml:id="echoid-s18955" xml:space="preserve"> ergo i k erit diameter imaginis b c:</s> <s xml:id="echoid-s18956" xml:space="preserve"> & po-<lb/>ſitio i k in reſpectu a, eſt ſimilis poſitioni b c, quia i k aut <lb/>erit ęquidiſtans ad b c, aut non erit inter illam & æ quidi-<lb/>ſtantem diuerſitas, quæ mutet poſitionem:</s> <s xml:id="echoid-s18957" xml:space="preserve"> non eſt enim <lb/>inter diſtantiam i k & diſtantiã b c à uiſu grandis diuerſi-<lb/>tas:</s> <s xml:id="echoid-s18958" xml:space="preserve"> quare declinatio i k à linea æquidiſtante b c, quę exit <lb/>ex k, erit ualde parua.</s> <s xml:id="echoid-s18959" xml:space="preserve"> Ergo angulus i a k eſt maior angu-<lb/>lo b a c:</s> <s xml:id="echoid-s18960" xml:space="preserve"> & poſitio i k eſt ſimilis poſitioni b c:</s> <s xml:id="echoid-s18961" xml:space="preserve"> & i k comprehenditur quaſi remotior, propter debilita-<lb/>tem formæ eius.</s> <s xml:id="echoid-s18962" xml:space="preserve"> Linea ergo k i uidetur maior, quã b c, utin præcedente figura declarauimus:</s> <s xml:id="echoid-s18963" xml:space="preserve"> Sed <lb/>i k eſt imago b c:</s> <s xml:id="echoid-s18964" xml:space="preserve"> ergo b c uidebitur maior, quàm ſit:</s> <s xml:id="echoid-s18965" xml:space="preserve"> & hoc eſt quod uoluimus.</s> <s xml:id="echoid-s18966" xml:space="preserve"/> </p> <div xml:id="echoid-div609" type="float" level="0" n="0"> <figure xlink:label="fig-0278-01" xlink:href="fig-0278-01a"> <variables xml:id="echoid-variables223" xml:space="preserve">a d e i f p m h l k b z q o c</variables> </figure> </div> </div> <div xml:id="echoid-div611" type="section" level="0" n="0"> <head xml:id="echoid-head526" xml:space="preserve" style="it">41. Si communis ſectio ſuperficierum, refractionis & refractiui fuerit linea recta: & uiſ{us} <lb/>ſit extra planum perpendicularium à terminis uiſibilis, par alleli communiſectioni ſuper refra-<lb/>ctiuum duct arum: imago uidebitur maior uiſibili. 33 p 10.</head> <p> <s xml:id="echoid-s18967" xml:space="preserve">ITem:</s> <s xml:id="echoid-s18968" xml:space="preserve"> ſit uiſus a:</s> <s xml:id="echoid-s18969" xml:space="preserve"> & res uiſa b c:</s> <s xml:id="echoid-s18970" xml:space="preserve"> extrahamus perpendiculares b d, c e:</s> <s xml:id="echoid-s18971" xml:space="preserve"> & continuemus d e:</s> <s xml:id="echoid-s18972" xml:space="preserve"> & ſit <lb/>b c æquidiſtans d e:</s> <s xml:id="echoid-s18973" xml:space="preserve"> & ſit a extra, ſuperficiem b d c e, cum co quod continuatur cum ipſa:</s> <s xml:id="echoid-s18974" xml:space="preserve"> & <lb/>[per 10 p 1] diuidamus b c in duo æqualia in z:</s> <s xml:id="echoid-s18975" xml:space="preserve"> & extrahamus perpendicularem a h ſuper ſuperfi <lb/> <pb o="273" file="0279" n="279" rhead="OPTICAE LIBER VII."/> ciem b c d e:</s> <s xml:id="echoid-s18976" xml:space="preserve"> & cõtinuemus a z:</s> <s xml:id="echoid-s18977" xml:space="preserve"> & ſit a z poſita perpẽdiculariter ſuper b z c.</s> <s xml:id="echoid-s18978" xml:space="preserve"> Poſitio ergo b reſpectu a <lb/>eſt ſimilis poſitioni c, reſpectu a:</s> <s xml:id="echoid-s18979" xml:space="preserve"> & diſtantia b ex a eſt æqualis diſtantiæ c ex a, [ut 39 n oſtẽſum eſt.</s> <s xml:id="echoid-s18980" xml:space="preserve">] <lb/>Et refringatur b ad a ex p:</s> <s xml:id="echoid-s18981" xml:space="preserve"> & c ad a ex k.</s> <s xml:id="echoid-s18982" xml:space="preserve"> Poſitio ergo p, reſpectu a, eſt ſimilis poſitioni k, reſpectu a:</s> <s xml:id="echoid-s18983" xml:space="preserve"> & <lb/>diſtantia p ex a, ſicut diſtantia k ex a:</s> <s xml:id="echoid-s18984" xml:space="preserve"> & continuemus lineas b p, p a, c k, k a.</s> <s xml:id="echoid-s18985" xml:space="preserve"> Eſt ergo [per 9 n] ſu-<lb/>perficies, in qua ſunt duę lineæ, a p, b p perpendicularis ſuper ſuperficiem corporis diaphani:</s> <s xml:id="echoid-s18986" xml:space="preserve"> quia <lb/>eſt ſuperficies refractionis:</s> <s xml:id="echoid-s18987" xml:space="preserve"> perpendicularis ergo b d erit <lb/> <anchor type="figure" xlink:label="fig-0279-01a" xlink:href="fig-0279-01"/> in hac ſuperficie:</s> <s xml:id="echoid-s18988" xml:space="preserve"> & perpendicularis, quæ exit ex p, erit <lb/>in illa ſuperficie:</s> <s xml:id="echoid-s18989" xml:space="preserve"> linea ergo a p ſecabit b d [per lemma <lb/>Procli ad 29 p 1:</s> <s xml:id="echoid-s18990" xml:space="preserve">] extrahatur ergo a p, & ſecet b d in l:</s> <s xml:id="echoid-s18991" xml:space="preserve"> & <lb/>extrahatur a k, & ſecet c e in o:</s> <s xml:id="echoid-s18992" xml:space="preserve"> erit ergo a l, ſicut a o:</s> <s xml:id="echoid-s18993" xml:space="preserve"> [ꝓ-<lb/>pter ſimilem poſitionem punctorum l & o ad punctum <lb/>a:</s> <s xml:id="echoid-s18994" xml:space="preserve">] & erit b l, ſicut c o:</s> <s xml:id="echoid-s18995" xml:space="preserve"> & continuemus l o, quæ eſt dia-<lb/>meter imaginis b c:</s> <s xml:id="echoid-s18996" xml:space="preserve"> & [per 33 p 1] erit l o æqualis b c:</s> <s xml:id="echoid-s18997" xml:space="preserve"> & <lb/>continuemus a b, a c.</s> <s xml:id="echoid-s18998" xml:space="preserve"> Vtraque ergo ſuperficies a l b, a o c <lb/>eſt perpendicularis ſuper ſuperficiem corporis diaphani <lb/>[per 9 n:</s> <s xml:id="echoid-s18999" xml:space="preserve">] & tres ſuperficies perpendiculares ſuper ſu <lb/>perficiem corporis diaphani, quæ tranſeunt per puncta <lb/>b, z, c, [nempe a l b:</s> <s xml:id="echoid-s19000" xml:space="preserve"> a m z:</s> <s xml:id="echoid-s19001" xml:space="preserve"> a o c] ſecant ſe in perpendicu-<lb/>lari exeunte ex a ſuper ſuperficiẽ corporis diaphani [per <lb/>19 p 11:</s> <s xml:id="echoid-s19002" xml:space="preserve">] & erit angulus b p l angulus refractionis:</s> <s xml:id="echoid-s19003" xml:space="preserve"> & linea <lb/>b l d perpendicularis eſt ſuper ſuperficiem corporis:</s> <s xml:id="echoid-s19004" xml:space="preserve"> ergo <lb/>[per 13 p 11] linea a l eſt obliqua ſuper ipſam.</s> <s xml:id="echoid-s19005" xml:space="preserve"> Linea ergo <lb/>a p continet cum perpendiculari exeunte ex p ſuper ſu-<lb/>perficiem corporis angulum acutum ex parte l:</s> <s xml:id="echoid-s19006" xml:space="preserve"> & extra-<lb/>hamus perpendicularem:</s> <s xml:id="echoid-s19007" xml:space="preserve"> & ſit p g:</s> <s xml:id="echoid-s19008" xml:space="preserve"> ergo [per 6 p 11] erit ę-<lb/>quidiſtans l d:</s> <s xml:id="echoid-s19009" xml:space="preserve"> angulus ergo p l d eſt acutus [per 29 p 1:</s> <s xml:id="echoid-s19010" xml:space="preserve">] <lb/>ergo [per 13 p 1] angulus a l b eſt obtuſus.</s> <s xml:id="echoid-s19011" xml:space="preserve"> Linea ergo a l <lb/>eſt minor, quàm linea a b [per 19 p 1.</s> <s xml:id="echoid-s19012" xml:space="preserve">] Et ſimiliter declara <lb/>tur, quòd a o erit minor a c:</s> <s xml:id="echoid-s19013" xml:space="preserve"> ſed lineæ a l, a o ſunt æquales, <lb/>& a b, a c ſunt æquales, & linea l o eſt ęqualis lineæ c b:</s> <s xml:id="echoid-s19014" xml:space="preserve"> er <lb/>go angulus o a l eſt maior angulo c a b:</s> <s xml:id="echoid-s19015" xml:space="preserve"> [ut patuit 39 n] & <lb/>poſitio l o eſt conſimilis poſitioni b c:</s> <s xml:id="echoid-s19016" xml:space="preserve"> quia linea, quę exit <lb/>exa ad medium l o, eſt perpendicularis ſuper lineam l o, quia [per 29 p 1] l o eſt æquidiſtans b c, & <lb/>b c eſt perpendicularis ſuper ſuperficiem, in qua ſunt a z, d b:</s> <s xml:id="echoid-s19017" xml:space="preserve">ergo [per 8 p 11] l o eſt perpendicularis <lb/>ſuper eandem ſuperficiem.</s> <s xml:id="echoid-s19018" xml:space="preserve"> Linea ergo l o eſt perpendicularis ſuper ſuperficiem, quæ continuat a <lb/>cum medio l o.</s> <s xml:id="echoid-s19019" xml:space="preserve"> Poſitio ergo l o reſpectu a eſt, ſicut poſitio b c reſpectu a:</s> <s xml:id="echoid-s19020" xml:space="preserve"> Sed l o comprehenditur re-<lb/>motior, propter debilitatem formæ:</s> <s xml:id="echoid-s19021" xml:space="preserve"> ergo l o uidebitur maior quàm b c:</s> <s xml:id="echoid-s19022" xml:space="preserve"> ſed l o eſt imago b c.</s> <s xml:id="echoid-s19023" xml:space="preserve"> Ergo <lb/>b c uidebitur maior, quàm ſit.</s> <s xml:id="echoid-s19024" xml:space="preserve"/> </p> <div xml:id="echoid-div611" type="float" level="0" n="0"> <figure xlink:label="fig-0279-01" xlink:href="fig-0279-01a"> <variables xml:id="echoid-variables224" xml:space="preserve">a p k d m e l o g h b z c</variables> </figure> </div> </div> <div xml:id="echoid-div613" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables225" xml:space="preserve">a q p k d m e g l o b z f c</variables> </figure> <head xml:id="echoid-head527" xml:space="preserve" style="it">42. Si communis ſectio ſuperficierum, refractionis et <lb/> refractiui fuerit linea recta: & uiſ{us} ſit extr a planum perpendicularium à terminis uiſibilis obliqui ad com- munem ſectionem, ſuper refractiuum ductarum: ima- go maior uidebitur uiſibili. 34 p 10.</head> <p> <s xml:id="echoid-s19025" xml:space="preserve">ITem iteremus figuram:</s> <s xml:id="echoid-s19026" xml:space="preserve"> & ſit b c non æquidiſtans d e:</s> <s xml:id="echoid-s19027" xml:space="preserve"> <lb/>& extrahamus c f æquidiſtantem lineæ d e:</s> <s xml:id="echoid-s19028" xml:space="preserve"> & conti-<lb/>nuemus a f:</s> <s xml:id="echoid-s19029" xml:space="preserve"> & ſit p punctum, ex quo refringatur f ad a:</s> <s xml:id="echoid-s19030" xml:space="preserve"> <lb/>b autem refringatur ad a ex q:</s> <s xml:id="echoid-s19031" xml:space="preserve"> & continuemus a q:</s> <s xml:id="echoid-s19032" xml:space="preserve"> & ꝓ-<lb/>trahamus illam ad g.</s> <s xml:id="echoid-s19033" xml:space="preserve"> Sic ergo erit g altius quàm l:</s> <s xml:id="echoid-s19034" xml:space="preserve"> nam b <lb/>eſt ultra lineam a f:</s> <s xml:id="echoid-s19035" xml:space="preserve">unde linea a g eſt ultra lineam a l:</s> <s xml:id="echoid-s19036" xml:space="preserve"> ergo <lb/>g eſt altius, quàm l:</s> <s xml:id="echoid-s19037" xml:space="preserve"> & continuemus g o:</s> <s xml:id="echoid-s19038" xml:space="preserve"> erit ergo g o dia-<lb/>meter imaginis b g:</s> <s xml:id="echoid-s19039" xml:space="preserve"> & erit [per 19 p 1] g o maior l o [angu <lb/>lus enim g l o eſt rectus per fabricationem & 29 p 1:</s> <s xml:id="echoid-s19040" xml:space="preserve">] & a g <lb/>minor à l [per 19 p 1:</s> <s xml:id="echoid-s19041" xml:space="preserve"> quia angulus a g l eſt obtuſus, ut oſtẽ-<lb/>ſum eſt 40 n] & duæ lineæ a g, a o ſunt in duabus ſuperfi-<lb/>ciebus ſecantibus ſe, ſcilicet a g b, a o c:</s> <s xml:id="echoid-s19042" xml:space="preserve"> & differentia com <lb/>munis inter duas has ſuperficies tranſit per a:</s> <s xml:id="echoid-s19043" xml:space="preserve"> & duæ li-<lb/>neæ, quæ exeunt ex a perpendiculariter ſuper illam ſu-<lb/>perficiem corporis diaphani, ſunt extra hãc communem <lb/>differentiam in his duabus ſuperficiebus, & ſunt altiores <lb/>duabus lineis a g, a o:</s> <s xml:id="echoid-s19044" xml:space="preserve"> ergo angulus g a o eſt maior angulo <lb/>b a c:</s> <s xml:id="echoid-s19045" xml:space="preserve"> [ut oſtenſum eſt 39 n] & remotiones g o, b c ex a <lb/>non differũt multũ:</s> <s xml:id="echoid-s19046" xml:space="preserve"> quia linea g o aut erit æquidiſtãs b c, <lb/> <pb o="274" file="0280" n="280" rhead="ALHAZEN"/> aut non erit ibi differentia ſenſibilis in poſitione.</s> <s xml:id="echoid-s19047" xml:space="preserve"> Poſitio ergo g o, reſpectu a eſt, ſicut poſitio b c, re-<lb/>ſpectu a:</s> <s xml:id="echoid-s19048" xml:space="preserve"> & inter diſtantias g o, b c reſpectu a, non eſt diuerſitas ſenſibilis.</s> <s xml:id="echoid-s19049" xml:space="preserve"> Quapropter g o uidebi-<lb/>tur maior quàm b c:</s> <s xml:id="echoid-s19050" xml:space="preserve"> ſed g o eſt imago b c.</s> <s xml:id="echoid-s19051" xml:space="preserve"> Ergo b c uidetur maior quàm ſit.</s> <s xml:id="echoid-s19052" xml:space="preserve"> Et hoc eſt quod uo-<lb/>luimus.</s> <s xml:id="echoid-s19053" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div614" type="section" level="0" n="0"> <head xml:id="echoid-head528" xml:space="preserve" style="it">43. Si tota imago refracti uiſibilis à refractiuo plano, uideatur maior uiſibili: uidebitur & <lb/>pars imaginis maior parte uiſibilis proportionali. 35 p 10.</head> <p> <s xml:id="echoid-s19054" xml:space="preserve">ITem:</s> <s xml:id="echoid-s19055" xml:space="preserve"> iteremus figuram primam huius capituli:</s> <s xml:id="echoid-s19056" xml:space="preserve"> [39 n] & ſit perpẽdicularis, ſecans lineam l k, a m <lb/>o z:</s> <s xml:id="echoid-s19057" xml:space="preserve"> erit ergo l o medietas l k:</s> <s xml:id="echoid-s19058" xml:space="preserve"> & punctum z uidebitur in o:</s> <s xml:id="echoid-s19059" xml:space="preserve"> quia uidetur in perpendiculari z m:</s> <s xml:id="echoid-s19060" xml:space="preserve"> er <lb/>go b c uidebitur in linea l k:</s> <s xml:id="echoid-s19061" xml:space="preserve"> & b z eſt medietas b c:</s> <s xml:id="echoid-s19062" xml:space="preserve"> & l o eſt medietas l k:</s> <s xml:id="echoid-s19063" xml:space="preserve"> & l k uidetur maior quã <lb/>b c.</s> <s xml:id="echoid-s19064" xml:space="preserve"> ergo l o uidebitur maior quàm b z.</s> <s xml:id="echoid-s19065" xml:space="preserve"> Cauſſa autem magnitudinis b c eſt refractio:</s> <s xml:id="echoid-s19066" xml:space="preserve"> ergo cauſſa ma-<lb/>gnitudinis b z eſt refractio.</s> <s xml:id="echoid-s19067" xml:space="preserve"> a autem eſt in perpendiculari a z, quæ exit ab extremitate b z ſuper ſu-<lb/>perficiem corporis diaphani.</s> <s xml:id="echoid-s19068" xml:space="preserve"> Et hoc idem ſequitur in tribus figuris ſequentibus primam, ſcilicet in <lb/>ſecunda, in tertia, & quarta huius capituli:</s> <s xml:id="echoid-s19069" xml:space="preserve"> ſcilicet quòd <lb/> <anchor type="figure" xlink:label="fig-0280-01a" xlink:href="fig-0280-01"/> uiſus comprehendit medietates uiſibilium maiores, <lb/>quàm ſint:</s> <s xml:id="echoid-s19070" xml:space="preserve"> & uiſus eſt in perpendiculari exeunte ab ex-<lb/>tremitate medietatis ſuper ſuperficiem corporis diapha <lb/>ni, aut ſuper ſuperficiem tranſeuntem per extremitatem <lb/>medietatis perpendicularis ſuper ſuperficiem corporis.</s> <s xml:id="echoid-s19071" xml:space="preserve"> <lb/>Nam punctum, quod eſt medium imaginis, eſt in perpen <lb/>diculari exeunte à medio rei uiſæ, ſiue res uiſa ſit ęquidi-<lb/>ſtans ſuperficiei corporis diaphani, ſiue non.</s> <s xml:id="echoid-s19072" xml:space="preserve"> Item b n ſit <lb/>quædam pars lineę b z:</s> <s xml:id="echoid-s19073" xml:space="preserve"> & extrahamus perpendicularem <lb/>n g:</s> <s xml:id="echoid-s19074" xml:space="preserve"> imago ergo n erit in linea n g:</s> <s xml:id="echoid-s19075" xml:space="preserve"> [per 19 n] ſit ergo gi-<lb/>mago n:</s> <s xml:id="echoid-s19076" xml:space="preserve"> g ergo aut erit in linea l g, aut prope illam.</s> <s xml:id="echoid-s19077" xml:space="preserve"> Qua-<lb/>propter l g aut erit æqualis lineæ b n, aut ferè.</s> <s xml:id="echoid-s19078" xml:space="preserve"> Sed in pri-<lb/>ma figura huius capituli [39 n] declarauimus, quòd b c <lb/>comprehenditur maior, quàm ſit.</s> <s xml:id="echoid-s19079" xml:space="preserve"> Et cauſſa huius eſt re-<lb/>fractio:</s> <s xml:id="echoid-s19080" xml:space="preserve"> & refractiones formarum, quæ remotiores ſunt <lb/>â perpendiculari, cadente à centro uiſus ſuper ſuperfi-<lb/>ciem corporis diaphani, ſunt maiores refractionibus for <lb/>marum, quæ ſunt propinquiores perpendiculari:</s> <s xml:id="echoid-s19081" xml:space="preserve"> refra-<lb/>ctio ergo formæ b n ad a eſt maior quàm refractio formę <lb/>partis z n ad a.</s> <s xml:id="echoid-s19082" xml:space="preserve"> Cauſſa ergo, quæ facit imaginem b z ui-<lb/>deri maiorem, facit, ut b n habeat maiorem proportio-<lb/>nem ad ipſam, quàm illa, quam habet b z ad b n:</s> <s xml:id="echoid-s19083" xml:space="preserve"> ergo l g <lb/>(quæ eſt imago b n) comprehenditur maior, quàm b n.</s> <s xml:id="echoid-s19084" xml:space="preserve"> <lb/>Item ſi a non comprehenderit imaginem b n maiorem, <lb/>quàm ipſam b n:</s> <s xml:id="echoid-s19085" xml:space="preserve"> non comprehendet imagines cætera-<lb/>rum partium lineæ b n, quæ ſunt propinquiores a d z, ma <lb/>iores ipſis partibus.</s> <s xml:id="echoid-s19086" xml:space="preserve"> Nam formæ cæterarum partium ſunt minoris refractionis, quàm forma b z:</s> <s xml:id="echoid-s19087" xml:space="preserve"> <lb/>ſed refractio eſt cauſſa magnitudinis imaginis:</s> <s xml:id="echoid-s19088" xml:space="preserve"> ergo a non comprehenderet l o maiorem, quàm b z:</s> <s xml:id="echoid-s19089" xml:space="preserve"> <lb/>a ergo comprehendet maiorem b n, quàm ſit.</s> <s xml:id="echoid-s19090" xml:space="preserve"> Et idem accidit, ſi a extra perpendicularem eſt exe-<lb/>untem ex b z ſuper ſuperficiem corporis diaphani, & linea, quæ exit ex a ad mediũ b z, non eſt per-<lb/>pendicularis ſuper b z.</s> <s xml:id="echoid-s19091" xml:space="preserve"> Et hoc idem ſequitur in tribus figuris, in ſecunda ſcilicet, tertia & quarta <lb/>huius capituli:</s> <s xml:id="echoid-s19092" xml:space="preserve"> [40.</s> <s xml:id="echoid-s19093" xml:space="preserve"> 41.</s> <s xml:id="echoid-s19094" xml:space="preserve"> 42 numeris.</s> <s xml:id="echoid-s19095" xml:space="preserve">] Omne ergo, quod comprehenditur à uiſu ultra corpus <lb/>diaphanum groſsius aere, cuius ſuperficies fuerit plana, comprehenditur maius, quàm ſit, ſiue ſit <lb/>uiſus in aliqua perpendiculari exeunte exillo uiſu ſuper ſuperficiem corporis, ſiue ſit extra:</s> <s xml:id="echoid-s19096" xml:space="preserve"> & in-<lb/>differenter, ſiue diameter rei uiſæ fuerit æquidiſtans ſuperficiei corporis, ſiue non æquidiſtans.</s> <s xml:id="echoid-s19097" xml:space="preserve"/> </p> <div xml:id="echoid-div614" type="float" level="0" n="0"> <figure xlink:label="fig-0280-01" xlink:href="fig-0280-01a"> <variables xml:id="echoid-variables226" xml:space="preserve">a d p m h e ſ g o k b n z c</variables> </figure> </div> </div> <div xml:id="echoid-div616" type="section" level="0" n="0"> <head xml:id="echoid-head529" xml:space="preserve" style="it">44. Si uiſ{us} ſit in continuat a diametro circuli (qui eſt communis ſectio ſuperficierum, re-<lb/>fractionis & refractiui conuexi denſioris) uiſibile uerò inter ipſi{us} centrum & uiſum, ab eodem <lb/>centro æquabiliter diſtet: imago uidebitur maior uiſibili. 36 p 10.</head> <p> <s xml:id="echoid-s19098" xml:space="preserve">ITem:</s> <s xml:id="echoid-s19099" xml:space="preserve"> ſit ſuperficies corporis ſphærica, cuius conuexum ſit ex parte uiſus, & groſsius aere:</s> <s xml:id="echoid-s19100" xml:space="preserve"> & ſit <lb/>uiſus a:</s> <s xml:id="echoid-s19101" xml:space="preserve"> & res uiſa b c:</s> <s xml:id="echoid-s19102" xml:space="preserve"> & ſit centrum ſphæræ ultra b c, in reſpectu uiſus:</s> <s xml:id="echoid-s19103" xml:space="preserve"> & ſit centrum d:</s> <s xml:id="echoid-s19104" xml:space="preserve"> z me-<lb/>dium b c:</s> <s xml:id="echoid-s19105" xml:space="preserve"> & continuemus d b, d z, d c:</s> <s xml:id="echoid-s19106" xml:space="preserve"> & extrahamus has lineas, quouſq;</s> <s xml:id="echoid-s19107" xml:space="preserve"> concurrant cũ ſuperfi-<lb/>cie ſphæræ a d e, m, n:</s> <s xml:id="echoid-s19108" xml:space="preserve"> & extrahamus z m in parte m:</s> <s xml:id="echoid-s19109" xml:space="preserve"> & primò ſit uiſus in linea z m:</s> <s xml:id="echoid-s19110" xml:space="preserve"> erit ergo a m z <lb/>linea recta:</s> <s xml:id="echoid-s19111" xml:space="preserve"> & primò ſit b d æqualis c d:</s> <s xml:id="echoid-s19112" xml:space="preserve"> Sic ergo [per 8 p 1.</s> <s xml:id="echoid-s19113" xml:space="preserve"> 10 d 1] erit a z perpẽdicularis ſuper b c.</s> <s xml:id="echoid-s19114" xml:space="preserve"> Po <lb/>ſitio ergo b, reſpectu a, erit ſimilis poſitioni c reſpectu a.</s> <s xml:id="echoid-s19115" xml:space="preserve"> Et extrahamus ſuperficiem, in qua ſunt de, <lb/>d n, d m:</s> <s xml:id="echoid-s19116" xml:space="preserve"> faciet ergo [per 1 th.</s> <s xml:id="echoid-s19117" xml:space="preserve"> 1 ſphęricorum] in ſuperficie ſphęrica arcũ circuli magni:</s> <s xml:id="echoid-s19118" xml:space="preserve"> ſit ergo arcus <lb/>e m n:</s> <s xml:id="echoid-s19119" xml:space="preserve"> & hæc ſuperficies eſt perpẽdicularis ſuք ſuperficiem ſphæricã [per 9 n:</s> <s xml:id="echoid-s19120" xml:space="preserve"> quia eſt ſuperficies re <lb/> <pb o="275" file="0281" n="281" rhead="OPTICAE LIBER VII."/> fractiõis]nec fit refractio extra hãc ſuperficiẽ:</s> <s xml:id="echoid-s19121" xml:space="preserve"> nã a z eſt քpẽdicularis ſuք ſuքficiẽ ſphæricã corporis <lb/>Nõ ergo refringetur forma alicuius partis b c ad a, niſi ex <lb/> <anchor type="figure" xlink:label="fig-0281-01a" xlink:href="fig-0281-01"/> circũferẽtia e m n Refringatur ergo b ad ad a ex h:</s> <s xml:id="echoid-s19122" xml:space="preserve"> & c ad a <lb/>ex g.</s> <s xml:id="echoid-s19123" xml:space="preserve"> Poſitio ergo h reſpectu a, & diſtantia eius eſt ęqualis <lb/>poſitiõi & diſtãtię g.</s> <s xml:id="echoid-s19124" xml:space="preserve"> Et cõtinuemus b h, h a, c g, g a:</s> <s xml:id="echoid-s19125" xml:space="preserve"> & extra <lb/>hamus a h ad k, & a g ad l:</s> <s xml:id="echoid-s19126" xml:space="preserve"> & cõtinuemus k l:</s> <s xml:id="echoid-s19127" xml:space="preserve"> erit ergo a k ę-<lb/>qualis a l.</s> <s xml:id="echoid-s19128" xml:space="preserve"> [Quia enim anguli ad z recti ſunt è cõcluſione, <lb/>& b z æqualis c z, & z d communis:</s> <s xml:id="echoid-s19129" xml:space="preserve"> erunt anguli b d z, <lb/>c d z æquales per 4 p 1.</s> <s xml:id="echoid-s19130" xml:space="preserve"> Et cum puncta h & g à puncto a ę-<lb/>quabiliter diſtent, propter æquabilem punctorum b & c, <lb/>à puncto a diſtantiam:</s> <s xml:id="echoid-s19131" xml:space="preserve"> æquabiliter etiam à puncto m di-<lb/>ſtabũt, quia m eſt in peripheria e m n, in recta linea a m z:</s> <s xml:id="echoid-s19132" xml:space="preserve"> <lb/>itaq;</s> <s xml:id="echoid-s19133" xml:space="preserve"> peripheria h m æquabitur peripherię g m:</s> <s xml:id="echoid-s19134" xml:space="preserve"> & conne-<lb/>xis rectis d h, d g:</s> <s xml:id="echoid-s19135" xml:space="preserve"> æquabitur angulus h d m angulo g d m <lb/>per 27 p 3:</s> <s xml:id="echoid-s19136" xml:space="preserve"> & per 15 d.</s> <s xml:id="echoid-s19137" xml:space="preserve"> 4 p 1 angulus d a h angulo d a g.</s> <s xml:id="echoid-s19138" xml:space="preserve"> Qua <lb/>re cum triangula d a k, d a l habeãt duos angulos duobus <lb/>angulis æquales ad cõmune latus d a:</s> <s xml:id="echoid-s19139" xml:space="preserve"> erunt ipſa æquilate <lb/>ra per 26 p 1:</s> <s xml:id="echoid-s19140" xml:space="preserve"> itaque latus a k ęquabitur lateri a l, & d k ipſi <lb/>d l:</s> <s xml:id="echoid-s19141" xml:space="preserve">] & erit l k imago b c:</s> <s xml:id="echoid-s19142" xml:space="preserve"> & erit ęquidiſtans b c:</s> <s xml:id="echoid-s19143" xml:space="preserve"> [Nã quia <lb/>d k æqualis concluſa eſt ipſi d l, & d b æqualis d c ex theſi:</s> <s xml:id="echoid-s19144" xml:space="preserve"> <lb/>erit b k ęqualis c l:</s> <s xml:id="echoid-s19145" xml:space="preserve"> & per 7 p 5 ut d b ad b k, ſic d c ad c l:</s> <s xml:id="echoid-s19146" xml:space="preserve"> I-<lb/>taq;</s> <s xml:id="echoid-s19147" xml:space="preserve"> per 2 p 6 l k parallela eſt c b:</s> <s xml:id="echoid-s19148" xml:space="preserve">] erit ergo maior quã b c:</s> <s xml:id="echoid-s19149" xml:space="preserve"> <lb/>[Nam pr opter triangulorum l d k, c d b ſimilitudinem è <lb/>29.</s> <s xml:id="echoid-s19150" xml:space="preserve"> 32 p 1 manifeſtã:</s> <s xml:id="echoid-s19151" xml:space="preserve"> eſt, ut l d ad c d, ſic l k ad c b:</s> <s xml:id="echoid-s19152" xml:space="preserve"> ſed per 9 <lb/>ax.</s> <s xml:id="echoid-s19153" xml:space="preserve"> l d maior eſt c d:</s> <s xml:id="echoid-s19154" xml:space="preserve"> ergo l k maior eſt c b:</s> <s xml:id="echoid-s19155" xml:space="preserve">] & cõtinuemus <lb/>a b, a c:</s> <s xml:id="echoid-s19156" xml:space="preserve"> erit ergo [ut patuit 39 n] angulus k a l maior angulo <lb/>b a c:</s> <s xml:id="echoid-s19157" xml:space="preserve"> & erit poſitio k l ſimilis poſitioni b c:</s> <s xml:id="echoid-s19158" xml:space="preserve"> & inter l k & c <lb/>b non eſt differentia in diſtantia, ut in præcedentib.</s> <s xml:id="echoid-s19159" xml:space="preserve"> diximus:</s> <s xml:id="echoid-s19160" xml:space="preserve"> ergo k l uidebitur maior quàm b c:</s> <s xml:id="echoid-s19161" xml:space="preserve"> ſed <lb/>k l eſt imago b c:</s> <s xml:id="echoid-s19162" xml:space="preserve"> ergo b c uidebitur maior, quàm ſit:</s> <s xml:id="echoid-s19163" xml:space="preserve"> quia imago eius eſt maior ſe:</s> <s xml:id="echoid-s19164" xml:space="preserve"> & hoc eſt, quia for, <lb/>ma eius eſt debilior, quã ueraforma.</s> <s xml:id="echoid-s19165" xml:space="preserve"> Et hoc eſt quod uoluimus.</s> <s xml:id="echoid-s19166" xml:space="preserve"/> </p> <div xml:id="echoid-div616" type="float" level="0" n="0"> <figure xlink:label="fig-0281-01" xlink:href="fig-0281-01a"> <variables xml:id="echoid-variables227" xml:space="preserve">a h m g e n k z b c ſ d</variables> </figure> </div> </div> <div xml:id="echoid-div618" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables228" xml:space="preserve">a h g<unsure/> m x e n k z l b c d</variables> </figure> <figure> <variables xml:id="echoid-variables229" xml:space="preserve">a h g<unsure/> f m r e n k b p q d c ſ</variables> </figure> <head xml:id="echoid-head530" xml:space="preserve" style="it">45. Si uiſ{us} ſit in continuata diametro circuli (quieſt cõmunis ſectio ſuperficierum refractio-<lb/>nis et refractiui cõuexi dẽſioris) uiſibile uerò inter ipſi{us} centrũ & uiſum ab eodẽ cẽtro inæqua-<lb/>biliter diſtet: <lb/> imago uιdebi tur maior uiſi bili. 37 p 10.</head> <p> <s xml:id="echoid-s19167" xml:space="preserve">SIuerò b d, b <lb/>c fuerĩt inę <lb/>quales:</s> <s xml:id="echoid-s19168" xml:space="preserve"> tũc a k <lb/>a l erũt inęqua <lb/>les:</s> <s xml:id="echoid-s19169" xml:space="preserve"> & ſic b c, k l <lb/>erunt obliquę <lb/>ſuք lineã a d:</s> <s xml:id="echoid-s19170" xml:space="preserve"> <lb/>erit ergo k l, ut <lb/>in ſecũda figu <lb/>ra huius capi-<lb/>tis [40 n] dixi <lb/>mus, maior ꝗ̃ <lb/>b c in uiſu.</s> <s xml:id="echoid-s19171" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div619" type="section" level="0" n="0"> <head xml:id="echoid-head531" xml:space="preserve" style="it">46. Si cõmu <lb/>nis ſectio ſuք-<lb/>ficierũ refra-<lb/>ctionis & re-<lb/>fractiui cõue-<lb/>xi dẽſioris fue <lb/>rit քipheria: et uiſ{us} ſit extra planum perpendicularium duct arũ à terminis uiſibilis inter cen <lb/>trũ refractiui & uiſum, ab eodem centro ſiue æquabiliter ſiue in æquabiliter diſtantis: imago ui-<lb/>debitur maior uiſibili. 38. 39 p 10.</head> <p> <s xml:id="echoid-s19172" xml:space="preserve">ITem:</s> <s xml:id="echoid-s19173" xml:space="preserve"> ſi a fuerit extra ſup erficiem b z c:</s> <s xml:id="echoid-s19174" xml:space="preserve"> & b d, c d fuerint æquales autinæquales, declarabitur, ut <lb/> <pb o="276" file="0282" n="282" rhead="ALHAZEN"/> in tertia figura & quarta huius capituli [41.</s> <s xml:id="echoid-s19175" xml:space="preserve"> 42 n] quòd k l uidebitur maior quàm b c.</s> <s xml:id="echoid-s19176" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div620" type="section" level="0" n="0"> <head xml:id="echoid-head532" xml:space="preserve" style="it">47. Si tota imago refracti uiſibilis à refractiuo conuexo, uideatur maior uiſibili: uidebitur <lb/>& pars imaginis maior parte uiſibilis proportionali. 41 p 10.</head> <p> <s xml:id="echoid-s19177" xml:space="preserve">SEd ſecet linea d <lb/> <anchor type="figure" xlink:label="fig-0282-01a" xlink:href="fig-0282-01"/> <anchor type="figure" xlink:label="fig-0282-02a" xlink:href="fig-0282-02"/> m lineã k lin o:</s> <s xml:id="echoid-s19178" xml:space="preserve"> <lb/>erit ergo k o ima-<lb/>go b z:</s> <s xml:id="echoid-s19179" xml:space="preserve"> & erit angu <lb/>lus k a o maior an-<lb/>gulo b a z [ք 9 ax.</s> <s xml:id="echoid-s19180" xml:space="preserve">] <lb/>& poſitio k o eſt ſi-<lb/>milis poſitiõi b z:</s> <s xml:id="echoid-s19181" xml:space="preserve"> <lb/>& diſtantię k o, b z <lb/>re ſpectu a nõ dιffe <lb/>runt multũ.</s> <s xml:id="echoid-s19182" xml:space="preserve"> Qua-<lb/>propter k o uidebi <lb/>tur maior ꝗ̃ b z:</s> <s xml:id="echoid-s19183" xml:space="preserve"> Et <lb/>a eſt in perpẽdicu-<lb/>lari z m exeũte ab <lb/>extremitate b z ſu <lb/>per ſuperficiẽ cor-<lb/>poris:</s> <s xml:id="echoid-s19184" xml:space="preserve"> ſit ergo b f <lb/>pars b z:</s> <s xml:id="echoid-s19185" xml:space="preserve"> & ſit k r i-<lb/>mago b f:</s> <s xml:id="echoid-s19186" xml:space="preserve"> Ergo, ut <lb/>in quinta figura hu <lb/>ius capituli [43 n] <lb/>diximus:</s> <s xml:id="echoid-s19187" xml:space="preserve"> patet qđ <lb/>k r uidebitur ma-<lb/>ior quã b f.</s> <s xml:id="echoid-s19188" xml:space="preserve"> Si aũt <lb/>a eſt extra ſuperfi-<lb/>ciẽ, in qua ſunt oẽs perpendiculares exeuntes ex b c ſuper ſuperficiem corporis diaphani (nã linea, <lb/>quæ exit ex a ad medium b c perpendiculariter, non eſt idcirco perpendicularis ſuper ſuperficiẽ li-<lb/>neæ b c) idem patebit.</s> <s xml:id="echoid-s19189" xml:space="preserve"> Nam quia b c, k l ſunt erectæ ſuper lineam a z d, aut ſuper ſuperficiẽ, quæ trã-<lb/>ſit per lineam m d:</s> <s xml:id="echoid-s19190" xml:space="preserve"> & k o eſt imago b z:</s> <s xml:id="echoid-s19191" xml:space="preserve"> & l o eſt imago c, & angulus, quem reſpicit k o apud centrum <lb/>uiſus, eſt maior angulo, quẽ reſpicit b z apud centrum uiſus:</s> <s xml:id="echoid-s19192" xml:space="preserve"> & ſimiliter angulus, quẽ reſpicit o l, eſt <lb/>maior angulo, quẽ reſpicit z c:</s> <s xml:id="echoid-s19193" xml:space="preserve"> ergo k o uidebitur maior quã c z:</s> <s xml:id="echoid-s19194" xml:space="preserve"> & ſimiliter k r uidebitur maior quã <lb/>b f.</s> <s xml:id="echoid-s19195" xml:space="preserve"> Et omnia hæc declarantur in quinta figura huius capituli [43 n.</s> <s xml:id="echoid-s19196" xml:space="preserve">] Sed in hac poſitione eſt quæ-<lb/>dam additio, ſcilicet quòd k l, quæ eſt imago b c, eſt maior in ueritate quàm b c, & k r eſt maior b f.</s> <s xml:id="echoid-s19197" xml:space="preserve"> <lb/>In prima aũt poſitione, ſcilicet in plana ſuperficie [refractiui:</s> <s xml:id="echoid-s19198" xml:space="preserve"> qualis fuit 39.</s> <s xml:id="echoid-s19199" xml:space="preserve"> 40.</s> <s xml:id="echoid-s19200" xml:space="preserve"> 41.</s> <s xml:id="echoid-s19201" xml:space="preserve"> 42.</s> <s xml:id="echoid-s19202" xml:space="preserve"> 43 n] duæ ima <lb/>gines ſunt æquales duobus uiſibilib, apparent aũt uiſui eſſe maiores.</s> <s xml:id="echoid-s19203" xml:space="preserve"> Imago uerò kl, & imago ko, in <lb/>ſuքficie ſphęrica, à qua fit refractio, ſunt maiores in uiſu ipſis rebus:</s> <s xml:id="echoid-s19204" xml:space="preserve"> & ſic ſunt in ueritate.</s> <s xml:id="echoid-s19205" xml:space="preserve"> Et patet, <lb/>quòd angulus, quem reſpicit k l apud centrum uiſus, eſt maior angulo, quem reſpicit b c apud cen-<lb/>trum uiſus:</s> <s xml:id="echoid-s19206" xml:space="preserve"> & angulus, quem reſpicit k o apud centrũ uiſus, eſt maior angulo, quem reſpicit b z, cũ <lb/>uiſus fuerit extra ſuperficiem, in qua ſunt d e, d z, ut in quarta figura huius capituli [42 n] diximus.</s> <s xml:id="echoid-s19207" xml:space="preserve"> <lb/>Ergo ſi uiſus comprehenderit aliquid ultra corpus groſsius aere, cuius ſuperficies fuerit ſphęrica, <lb/>& cuius conuexum fuerit ex parte uiſus, & cuius centrum fuerit ultra rem uiſam, quãtùm ad uiſum:</s> <s xml:id="echoid-s19208" xml:space="preserve"> <lb/>comprehendet illud maius, quàm ſit ſecundum ueritatem, & etiam ſecundum apparẽtiam in uiſu:</s> <s xml:id="echoid-s19209" xml:space="preserve"> <lb/>ſiue fuerit uiſus in perpendiculari, exeunte à re uiſa ſuper ſuperficiem ſphæricam, ſiue extra, ſiue li-<lb/>nea, quæ exit à centro uiſus ad mediũ rei uiſæ, fuerit perpendicularis ſuper rem uiſam, ſiue obliqua.</s> <s xml:id="echoid-s19210" xml:space="preserve"> <lb/>Et hoc eſt quod uoluimus declarare.</s> <s xml:id="echoid-s19211" xml:space="preserve"/> </p> <div xml:id="echoid-div620" type="float" level="0" n="0"> <figure xlink:label="fig-0282-01" xlink:href="fig-0282-01a"> <variables xml:id="echoid-variables230" xml:space="preserve">a f h m g e n <lb/>k b p q d c l</variables> </figure> <figure xlink:label="fig-0282-02" xlink:href="fig-0282-02a"> <variables xml:id="echoid-variables231" xml:space="preserve">a h m g e r o n k b s z c l d</variables> </figure> </div> </div> <div xml:id="echoid-div622" type="section" level="0" n="0"> <head xml:id="echoid-head533" xml:space="preserve" style="it">48. Imago refracti uiſibilis ab aqua ad aerem, uidetur maior uiſibili. 42 p 10.</head> <p> <s xml:id="echoid-s19212" xml:space="preserve">ET hoc accidit in eis, quæ uidentur in aqua:</s> <s xml:id="echoid-s19213" xml:space="preserve"> nam conuexum ſuperficiei a quæ ſphæricum eſt ex <lb/>parte uiſus, & centrum ſuperficiei aquæ eſt ultra illa, quæ comprehenduntur in a qua, & aqua <lb/>eſt groſsior aere:</s> <s xml:id="echoid-s19214" xml:space="preserve"> Sed illud, quod uidetur in aqua, ſi aqua fuerit clara & pauca, fortè non com-<lb/>prehenditur à uiſu eſſe maius in aqua, quàm ſi eſſet in aere.</s> <s xml:id="echoid-s19215" xml:space="preserve"> Non enim differt quantitas eius tunc, <lb/>quantùm ad ſenſum, ſcilicet quantitas eius in aqua & aere:</s> <s xml:id="echoid-s19216" xml:space="preserve"> tunc enim illa additio in aqua crit par-<lb/>ua, & ideo ſenſus non diſtinguet tunc illam additionem:</s> <s xml:id="echoid-s19217" xml:space="preserve"> tamen experientia poteſt comprehendi <lb/>hoc modo.</s> <s xml:id="echoid-s19218" xml:space="preserve"> Accipe corpus columnare, cuius longitudo non ſit minor uno cubito:</s> <s xml:id="echoid-s19219" xml:space="preserve"> & ſit aliquantæ <lb/>groſsiciei:</s> <s xml:id="echoid-s19220" xml:space="preserve"> album:</s> <s xml:id="echoid-s19221" xml:space="preserve"> nam albedo in aqua manifeſtius diſtinguitur:</s> <s xml:id="echoid-s19222" xml:space="preserve"> & ſit ſuperficies baſis eius plana, <lb/>ita ut per ſe ſtet æqualiter ſuper ſuperficiem terræ.</s> <s xml:id="echoid-s19223" xml:space="preserve"> Hoc obſeruato, accipe uas amplum, & ſit ſu-<lb/>perficies eius plana, & infunde in uas aquam claram in altitudine minore longitudine corpo-<lb/>ris columnaris:</s> <s xml:id="echoid-s19224" xml:space="preserve"> deinde mitte illud corpus columnare in aquam, & pone ipſum ſuper ſuam ba-<lb/>ſim in medio uaſis:</s> <s xml:id="echoid-s19225" xml:space="preserve"> erit ergo aliqua pars huius corporis extra aquam:</s> <s xml:id="echoid-s19226" xml:space="preserve"> nam altitudo aquæ eſt mi-<lb/> <pb o="277" file="0283" n="283" rhead="OPTICAE LIBER VII."/> nor longitudine huius corporis.</s> <s xml:id="echoid-s19227" xml:space="preserve"> Tunc enim, cum quieuerit aqua, uidebis partẽ corporis, quę eſt in-<lb/>tra aquã groſsiorem illa, quę eſt extra aquam.</s> <s xml:id="echoid-s19228" xml:space="preserve"> Patet ergo ex hac experientia, quòd omne uiſum com <lb/>prehenſum in aqua, cõprehenditur maius, quàm ſit in ueritate.</s> <s xml:id="echoid-s19229" xml:space="preserve"> Item ſit corpus ſphæricũ, cuius con-<lb/>uexum ſit ex parte uiſus, & res uiſa ſit ultra centrum ſuperficiei ſphæricę, & ſit illud corpus groſsius <lb/>aere:</s> <s xml:id="echoid-s19230" xml:space="preserve"> Sed in aſſuetis uiſibilibus non eſt tale ali quid, quod uideatur ultra corpus diaphanũ ſphæricũ <lb/>groſsius aere, ultra centrũ ſphæræ, & res uiſa cum hoc ſit intra corpus ſphæricũ:</s> <s xml:id="echoid-s19231" xml:space="preserve"> hoc enim nõ fit, niſi <lb/>corpus ſphæricũ fuerit uitreum aut lapideũ, & fuerit totum corpus ſphæricum ſolidum, & res uiſa <lb/>fuerit intra ipſum, aut ut corpus ſphæricum ſit portio ſphæræ maior ſemiſphæra, & res uiſa ſit appli <lb/>cata cum baſi eius:</s> <s xml:id="echoid-s19232" xml:space="preserve"> ſed hi duo ſitus rarò accidunt:</s> <s xml:id="echoid-s19233" xml:space="preserve"> huiuſmodi ergo res nõ ſunt de aſſuetis uiſibilibus:</s> <s xml:id="echoid-s19234" xml:space="preserve"> <lb/>non ergo debemus occupari circa ea, quæ accidunt huiuſmodi uiſibilibus.</s> <s xml:id="echoid-s19235" xml:space="preserve"> Sed ſunt quædam aſſue-<lb/>ta, quæ uidentur ultra corpus diaphanum ſphæricũ groſsius aere, cuius conuexum erit ex parte ui-<lb/>ſus, cum res uiſa fuerit ultra ſphæram cryſtallinam, aut uitream in aere, non intra ſphæram.</s> <s xml:id="echoid-s19236" xml:space="preserve"> Poſitio-<lb/>nes aũt huiuſmodi uiſibiliũ ſunt multimodæ:</s> <s xml:id="echoid-s19237" xml:space="preserve"> Sed hæc rarò cõprehenduntur:</s> <s xml:id="echoid-s19238" xml:space="preserve"> & ſi cõprehendantur, <lb/>rarò uidentur.</s> <s xml:id="echoid-s19239" xml:space="preserve"> Non eſt ergo cõueniens diſtinguere oẽs illas poſitiones.</s> <s xml:id="echoid-s19240" xml:space="preserve"> Simus ergo cõtenti una ſola <lb/>poſitione, ſcilicet ut uiſus & res uiſa ſint in eadẽ perpendiculari ſuper ſuperficiẽ corporis ſphærici.</s> <s xml:id="echoid-s19241" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div623" type="section" level="0" n="0"> <head xml:id="echoid-head534" xml:space="preserve" style="it">49. Siuiſ{us}, centrum refractiui conuexi denſioris & uiſibile ultra refractiuum poſitum, fue-<lb/>rint in e adem recta linea: imago uidebitur corona ſeu armilla: & maior uiſibili. 43 p 10.</head> <p> <s xml:id="echoid-s19242" xml:space="preserve">SIt ergo uiſus a:</s> <s xml:id="echoid-s19243" xml:space="preserve"> & corpus ſphæricum b g z d:</s> <s xml:id="echoid-s19244" xml:space="preserve"> & centrũ eius ſit e:</s> <s xml:id="echoid-s19245" xml:space="preserve"> & continuemus a e, & extraha-<lb/>mus eam rectè:</s> <s xml:id="echoid-s19246" xml:space="preserve"> & ſecet ſuperficiem ſphæræ in duobus punctis b, d:</s> <s xml:id="echoid-s19247" xml:space="preserve"> & extrahamus ipſam in par <lb/>te d uſq;</s> <s xml:id="echoid-s19248" xml:space="preserve"> ad h:</s> <s xml:id="echoid-s19249" xml:space="preserve"> & extrahamus exlinea h b a ſuperficiem æqualem ſecantem ſphæram:</s> <s xml:id="echoid-s19250" xml:space="preserve"> faciet er-<lb/>go [per 1 th 1 ſphæricorum] in ſuperficie ſphærę circulum b g z d.</s> <s xml:id="echoid-s19251" xml:space="preserve"> Octaua autem figura in capitulo de <lb/>imagine [29 n] diximus, quòd in linea b d ſunt plura puncta, quorum formę refringuntur ad a ex cir <lb/>cumferentia b g z d:</s> <s xml:id="echoid-s19252" xml:space="preserve"> & quòd forma totius illius lineæ refringitur ad a, ſi b g z d fuerit continuum & <lb/>non fractum in parte b.</s> <s xml:id="echoid-s19253" xml:space="preserve"> Refringatur ergo h l ad a ex circumferentia b g z d, & refringatur h ad a ex g:</s> <s xml:id="echoid-s19254" xml:space="preserve"> <lb/>& l ad a exp:</s> <s xml:id="echoid-s19255" xml:space="preserve"> forma ergo h l refringetur ad a ex arcu g p:</s> <s xml:id="echoid-s19256" xml:space="preserve"> & continuemus lineas g m h, g a, l z p, p a:</s> <s xml:id="echoid-s19257" xml:space="preserve"> h <lb/>ergo extenditur per g h, & refringitur per g a:</s> <s xml:id="echoid-s19258" xml:space="preserve"> & l extenditur per l p, & refringitur per p a:</s> <s xml:id="echoid-s19259" xml:space="preserve"> & cõtinue <lb/>mus lineas e g, e m, e z:</s> <s xml:id="echoid-s19260" xml:space="preserve"> & extrahamus e m ad c:</s> <s xml:id="echoid-s19261" xml:space="preserve"> & e z ad f forma ergo, quæ extenditur per a g, refrin-<lb/>gitur per g h, & peruenit ad h:</s> <s xml:id="echoid-s19262" xml:space="preserve"> & forma, quæ extenditur per a p, refrin-<lb/> <anchor type="figure" xlink:label="fig-0283-01a" xlink:href="fig-0283-01"/> gitur per p l, & peruenit ad l:</s> <s xml:id="echoid-s19263" xml:space="preserve"> hoc eſt, ſi corpus diaphanum fuerit conti <lb/>nuum uſq;</s> <s xml:id="echoid-s19264" xml:space="preserve"> ad g.</s> <s xml:id="echoid-s19265" xml:space="preserve"> Si uerò corpus ſphæricum fuerit ſignatũ apud ſuper <lb/>ficiem ſphæricam:</s> <s xml:id="echoid-s19266" xml:space="preserve"> tunc formà, quæ extenditur per a g:</s> <s xml:id="echoid-s19267" xml:space="preserve"> refringitur per <lb/>g m in partem perpendicularis, quę eſt e h:</s> <s xml:id="echoid-s19268" xml:space="preserve"> & cum forma perueniet ad <lb/>m:</s> <s xml:id="echoid-s19269" xml:space="preserve"> refringetur ſecundò in partem contrariam perpendicularis, quæ <lb/>eſt e m c:</s> <s xml:id="echoid-s19270" xml:space="preserve"> refringatur ergo ad k.</s> <s xml:id="echoid-s19271" xml:space="preserve"> Et ideo etiam forma, quæ extenditur <lb/>per a p, refringitur ք p z:</s> <s xml:id="echoid-s19272" xml:space="preserve"> & cũ fuerit refracta ad z, refringetur ſecundò <lb/>ad contrariã partẽ perpendicularis, quæ eſt e z f:</s> <s xml:id="echoid-s19273" xml:space="preserve"> ſit ergo refractio for-<lb/>mę, quę peruenit ad z, per lineã z o.</s> <s xml:id="echoid-s19274" xml:space="preserve"> Forma ergo k extenditur per k m, <lb/>& refringitur ք m g:</s> <s xml:id="echoid-s19275" xml:space="preserve"> deinde refringitur ſecũdò per ga.</s> <s xml:id="echoid-s19276" xml:space="preserve"> Et ſimiliter for <lb/>ma o extenditur per o z, & refringitur per z p:</s> <s xml:id="echoid-s19277" xml:space="preserve"> deinde ſecũdò refringi-<lb/>tur ք p a.</s> <s xml:id="echoid-s19278" xml:space="preserve"> Forma ergo totius k o refringitur ad a ex arcu g p.</s> <s xml:id="echoid-s19279" xml:space="preserve"> Et ſi linea <lb/>a k o fuerit fixa, & imaginati fuerimus figurã a g p k circũuolui circa à <lb/>k o:</s> <s xml:id="echoid-s19280" xml:space="preserve">tunc arcus g p faciet figurã circularẽ, ut armillam, à cuius uniuer-<lb/>ſo refringetur forma k o ad a:</s> <s xml:id="echoid-s19281" xml:space="preserve"> & erit imago k o apud centrum uiſus, <lb/>quod eſt a.</s> <s xml:id="echoid-s19282" xml:space="preserve"> Forma ergo k o uidebitur in tota ſuperficie circulari, quæ <lb/>eſt locus refractionis, quæ eſt in rectitudine linearum radialium, quæ <lb/>eſt figura armillæ.</s> <s xml:id="echoid-s19283" xml:space="preserve"> Forma ergo k o uidebitur maior ſeipſa:</s> <s xml:id="echoid-s19284" xml:space="preserve"> & erit figu-<lb/>ra formæ diuerſæ à figura k o.</s> <s xml:id="echoid-s19285" xml:space="preserve"> Hoc autẽ poteſt experimentari ſic.</s> <s xml:id="echoid-s19286" xml:space="preserve"> Ac-<lb/>cipe ſphæram cryſtallinam aut uitream rotundiſsimam, & accipe cor <lb/>pus paruum:</s> <s xml:id="echoid-s19287" xml:space="preserve"> ut granum ciceris uel ceram paruam:</s> <s xml:id="echoid-s19288" xml:space="preserve"> nam experientia <lb/>per corpus paruum erit manifeſtior, & tingas ipſam colore nigro, & <lb/>ſit figura ceræ ſphærica:</s> <s xml:id="echoid-s19289" xml:space="preserve"> deinde ponas ipſam in capite a cus, & ponas <lb/>ſphæram cryſtallinam in oppoſitione alterius oculorum, & claude al <lb/>terum oculum, & eleua acum ultra ſphærã, & aſpice ad medium ſphæ <lb/>ræ, & pone ceram in oppoſitione medij formæ, ita ut ſit oppoſità me-<lb/>dio ſphæræ in una linea recta, quò ad ſenſum, & reſpice ad ſuperficiẽ <lb/>ſphæræ:</s> <s xml:id="echoid-s19290" xml:space="preserve"> tunc enim uidebis in illa ſuperficie ſphæræ nigredinẽ rotun-<lb/>dam in figura armillæ.</s> <s xml:id="echoid-s19291" xml:space="preserve"> Si uerò non uideas eam:</s> <s xml:id="echoid-s19292" xml:space="preserve"> moue ceram antè & <lb/>pòſt, donec uideas nigredinem rotundam, tunc aufer ceram, & abſcindetur nigredo:</s> <s xml:id="echoid-s19293" xml:space="preserve"> deinde redeat <lb/>cera ad ſuum locum, & iterum uidebis illam nigredinem rotundam.</s> <s xml:id="echoid-s19294" xml:space="preserve"> Ex hac ergo experientia pate-<lb/>bit, quòd ſi res uiſa fuerit ultra corpus diaphanum ſphæricum groſsius aere, & uiſus, & res uiſa, & <lb/>centrum corporis ſphærici fuerint in eadem linea recta:</s> <s xml:id="echoid-s19295" xml:space="preserve"> tunc uiſus comprehendit illam rem uiſam <lb/>in figura armillæ.</s> <s xml:id="echoid-s19296" xml:space="preserve"/> </p> <div xml:id="echoid-div623" type="float" level="0" n="0"> <figure xlink:label="fig-0283-01" xlink:href="fig-0283-01a"> <variables xml:id="echoid-variables232" xml:space="preserve">a b g p e d z m h o h <gap/> l c</variables> </figure> </div> </div> <div xml:id="echoid-div625" type="section" level="0" n="0"> <head xml:id="echoid-head535" xml:space="preserve" style="it">50. Siuiſ{us}, centrum circuli in refractiuo cylindraceo conuexo denſiore, & uiſibιle ultra re-<lb/> <pb o="278" file="0284" n="284" rhead="ALHAZEN"/> fractiuum poſitum fuerint in eadem recta linea: imago uidebitur duplicata. 44 p 10.</head> <p> <s xml:id="echoid-s19297" xml:space="preserve">SIuerò b g z d fuerit in corpore columnari, & corpus fuerit groſsius aere:</s> <s xml:id="echoid-s19298" xml:space="preserve"> tunc form a k o uidebi <lb/>tur apud arcum g p & apud arcum ſibi æqualem, & ſibi reſpondentem exarcu b d:</s> <s xml:id="echoid-s19299" xml:space="preserve"> Sed hæc for <lb/>ma non erit circularis:</s> <s xml:id="echoid-s19300" xml:space="preserve"> quia figura a h p g cum fuerit circumuoluta circa a k:</s> <s xml:id="echoid-s19301" xml:space="preserve"> nõ tranſibit per li-<lb/>neam illam arcus g p per totam ſuperficiem columnarem:</s> <s xml:id="echoid-s19302" xml:space="preserve"> Sed refringetur fortè forma ex aliquibus <lb/>portionibus columnaribus, & erit continua in una parte & ſimiliter in alia.</s> <s xml:id="echoid-s19303" xml:space="preserve"> Nam ſuperficies ex l k, <lb/>quæ etiam tranſit per axem columnæ, facit in ſuperficie columnę, quę eſt ex parte a, lineam rectam, <lb/>quæ tranſit per b, & extenditur in longitudine columnæ:</s> <s xml:id="echoid-s19304" xml:space="preserve"> & non refringitur forma k o ex illa linea <lb/>recta:</s> <s xml:id="echoid-s19305" xml:space="preserve"> nam k b erit perpendicularis ſuper illam lineam rectam.</s> <s xml:id="echoid-s19306" xml:space="preserve"> Non ergo erit forma rotunda, ſi fue-<lb/>rit corpus columnare:</s> <s xml:id="echoid-s19307" xml:space="preserve"> ſed erunt duæ formæ, quarum altera refringitur ſuper alteram.</s> <s xml:id="echoid-s19308" xml:space="preserve"> Videbitur <lb/>ergo k o eſſe duo, quorum utrumq;</s> <s xml:id="echoid-s19309" xml:space="preserve"> erit maius k o:</s> <s xml:id="echoid-s19310" xml:space="preserve"> & forma utriuſque erit diuerſa à forma k o:</s> <s xml:id="echoid-s19311" xml:space="preserve"> & ta-<lb/>men illæ duæ formæ erunt apud idem punctum, ſcilicet centrum uiſus.</s> <s xml:id="echoid-s19312" xml:space="preserve"> In uiſibilibus autem aſſue-<lb/>tis nihil eſt, quod comprehendatur à uiſu ultra diaphanum corpus, ſphæricum, groſsius aere, cu-<lb/>ius concauum ſit ex parte uiſus.</s> <s xml:id="echoid-s19313" xml:space="preserve"> Nam ſi fuerit ex uitro aut aliquo lapide:</s> <s xml:id="echoid-s19314" xml:space="preserve"> oportet, ut ſit portio ſphæ-<lb/>ræ concaua, & ut res uiſa ſit intra illam ſphæram, aut ut ſuperficies eius, quæ eſt ultra concauitatem, <lb/>ſit plana, & res uiſa adhæreat illi.</s> <s xml:id="echoid-s19315" xml:space="preserve"> Et illi duo ſitus non inueniuntur, aut rarò:</s> <s xml:id="echoid-s19316" xml:space="preserve"> non ergo ſolicitemur <lb/>circa huiuſmodi.</s> <s xml:id="echoid-s19317" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div626" type="section" level="0" n="0"> <head xml:id="echoid-head536" xml:space="preserve" style="it">51. Stella in horizonte ut plurimum uidetur maior, quàm in medio cæli. 54 p 10.</head> <p> <s xml:id="echoid-s19318" xml:space="preserve">ITem:</s> <s xml:id="echoid-s19319" xml:space="preserve"> non inuenitur aliquod corpus ſubtilius aere, cuius ſuperficies, quæ eſt ex parte uiſus, ſit <lb/>plana aut conuexa.</s> <s xml:id="echoid-s19320" xml:space="preserve"> Et nõ inuenitur aliquid ſubtilius aere, ultra quod comprehendatur aliquid, <lb/>niſi corpus cœli & ignis.</s> <s xml:id="echoid-s19321" xml:space="preserve"> Et non diuiditur à corpore aeris ſuperficies, quæ diſtinguit unam par-<lb/>tem ab alia, ſed quanto magis appropinquat aer cœlo, tantò magis purificatur, donec fiat ignis.</s> <s xml:id="echoid-s19322" xml:space="preserve"> <lb/>Subtilitas ergo eius fit ordinatè ſecundum ſucceſsionem, non in differentia terminata.</s> <s xml:id="echoid-s19323" xml:space="preserve"> Formę ergo <lb/>eorum, quæ ſunt in cœlo, quando extenduntur ad uiſum, non refringuntur apud concauitatẽ ſphæ-<lb/>ræignis, cum non ſit ibi ſuperficies concaua determinata.</s> <s xml:id="echoid-s19324" xml:space="preserve"> Nullum ergo inuenitur corpus ſubtilius <lb/>aere, in quo extendantur formæ uiſibilium, & refringantur apud ſuperficiem eius, niſi corpus cœle <lb/>ſte:</s> <s xml:id="echoid-s19325" xml:space="preserve"> & corpus cœleſte eſt ſphęricum concauum ex parte uiſus.</s> <s xml:id="echoid-s19326" xml:space="preserve"> Ergo omnes ſtellæ, quæ ſunt in cœlo, <lb/>extendũtur in corpore cœli, & refringuntur apud cõcauitatẽ cœli, & extenduntur ιn corpore ignis, <lb/>& in corpore aeris rectè, donec perueniant ad uiſum.</s> <s xml:id="echoid-s19327" xml:space="preserve"> Et centrum concauitatis cœli eſt centrum ter-<lb/>ræ.</s> <s xml:id="echoid-s19328" xml:space="preserve"> Dico ergo quòd ſtellæ in maiore parte comprehenduntur in ſuis locis:</s> <s xml:id="echoid-s19329" xml:space="preserve"> & quòd ſemper compre-<lb/>henduntur non in ſuis magnitudinibus:</s> <s xml:id="echoid-s19330" xml:space="preserve"> & cum hoc diuerſatur magnitudo uniuſcuiuſq;</s> <s xml:id="echoid-s19331" xml:space="preserve"> earum, ſe-<lb/>cundum locorum diuerſitatem.</s> <s xml:id="echoid-s19332" xml:space="preserve"> Diuerſitas autem locorum eſt propter radiorum refractorum po-<lb/>ſitionem, ut prius diximus.</s> <s xml:id="echoid-s19333" xml:space="preserve"> Diuerſitas autem quantitatum eſt propter remotionem:</s> <s xml:id="echoid-s19334" xml:space="preserve"> nam propter <lb/>remotionem comprehenduntur minores, quàm ſint in ueritate, ut diximus in tertio tractatu, ſcili-<lb/>cet quòd illa, quæ in maxima remotione ſunt, comprehenduntur minora.</s> <s xml:id="echoid-s19335" xml:space="preserve"> Diuerſitas autem quanti <lb/>tatum ſecundum diuerſitatem locorum, accidit propter refractionem, cuius cauſſam hic declaraui-<lb/>mus:</s> <s xml:id="echoid-s19336" xml:space="preserve"> & in quarto capitulo [15 n] declarauimus, quòd formæ ſtellarum, quæ comprehenduntur à ui-<lb/>ſu, ſunt refractæ.</s> <s xml:id="echoid-s19337" xml:space="preserve"> Dico ergo, quòd omnis ſtella comprehenditur ex omnibus locis cœli, per quos <lb/>mouetur, minore quantitate, quàm ſit in ueritate, ſecundum quod exigit remotio eius, (ſcilicet mi-<lb/>nor, ſi uiſa fuerit rectè) cum nõ fuerit inter illam & uiſum aliqua nubes, aut uapor groſſus.</s> <s xml:id="echoid-s19338" xml:space="preserve"> Et omnis <lb/>ſtella in uertice capitis aſpicientis exiſtens uidetur minor, quàm in alio loco cœli:</s> <s xml:id="echoid-s19339" xml:space="preserve"> & quantò magis <lb/>remouetur à uertice capitis, tantò magis apparet maior:</s> <s xml:id="echoid-s19340" xml:space="preserve"> ita ut in horizonte appareat maior, quàm <lb/>in alio loco.</s> <s xml:id="echoid-s19341" xml:space="preserve"> Et hoc eſt commune omnibus ſtellis remotis & propinquis.</s> <s xml:id="echoid-s19342" xml:space="preserve"> Item ſi in aere fuerit uapor <lb/>groſſus, ultra quem ſuerit aliqua ſtella:</s> <s xml:id="echoid-s19343" xml:space="preserve"> tunc comprehendetur maior, quàm ſi eſſet ſine illo uapore:</s> <s xml:id="echoid-s19344" xml:space="preserve"> <lb/>& multoties accidit, ut uapor groſſus ſit in horizonte.</s> <s xml:id="echoid-s19345" xml:space="preserve"> Vnde ſtellæ in maiore parte uidentur in hori-<lb/>zonte maiores, quàm in medio cœli.</s> <s xml:id="echoid-s19346" xml:space="preserve"> Et hoc apparet in diſtantijs, quæ ſunt inter illas, magis, quàm <lb/>In magnitudinibus ipſarum ſtellarum:</s> <s xml:id="echoid-s19347" xml:space="preserve"> nam quantitas ſtellæ, quò ad uiſum, eſt parua, ſed exceſſus in <lb/>diuerſitate diſtantiæ inter ſtellas, cum fuerint in horizonte, eſt grandis & manifeſtus ſenſui, & maxi <lb/>mè in diſtantijs ſpatioſis, & maximè, ſi in horizonte fuerit uapor groſſus.</s> <s xml:id="echoid-s19348" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div627" type="section" level="0" n="0"> <head xml:id="echoid-head537" xml:space="preserve" style="it">52. Diameter ſtellæ uertici propinquæ, & duarum inter ſe diſtantia, refractè uiſa, minor: <lb/>rectè, maior uidetur. 51 p 10.</head> <p> <s xml:id="echoid-s19349" xml:space="preserve">SIt ergo circulus meridiei in aliquo horizonte, b k:</s> <s xml:id="echoid-s19350" xml:space="preserve"> & differentia communis inter hunc circu-<lb/>lum & concauitatem cœli, circulus m e z:</s> <s xml:id="echoid-s19351" xml:space="preserve"> & ſit centrum mundi g:</s> <s xml:id="echoid-s19352" xml:space="preserve"> & centrum uiſus t:</s> <s xml:id="echoid-s19353" xml:space="preserve"> & extra-<lb/>hamus g t in partem t:</s> <s xml:id="echoid-s19354" xml:space="preserve"> & occurrat circulo meridiei in b:</s> <s xml:id="echoid-s19355" xml:space="preserve"> & ſecet circulum, qui eſt in concauitate <lb/>orbis, in e:</s> <s xml:id="echoid-s19356" xml:space="preserve"> erit ergo b uertex capitis, quò ad uiſum t.</s> <s xml:id="echoid-s19357" xml:space="preserve"> Sit k l diameter alicuius ſtellæ, aut diſtantia <lb/>inter aliquas duas ſtellas:</s> <s xml:id="echoid-s19358" xml:space="preserve"> & linea t b tranſeat per medium k l, & ſecet illam in c:</s> <s xml:id="echoid-s19359" xml:space="preserve"> ergo erit arcus k b <lb/>æqualis arcui b l:</s> <s xml:id="echoid-s19360" xml:space="preserve"> [Nam quia t b bifariam ſecans k l ex theſi ſecat ad angulos rectos per 3 p 3:</s> <s xml:id="echoid-s19361" xml:space="preserve"> con-<lb/>nexæ igitur rectæ k b, b l æquabuntur per 4 p 1.</s> <s xml:id="echoid-s19362" xml:space="preserve"> Quare per 28 p 3 peripheria k b æquatur periphe-<lb/>riæ b l] & continuemus duas lineas t k, t l:</s> <s xml:id="echoid-s19363" xml:space="preserve"> erit ergo angulus k t l ille, à quo t comprehendit k l, ſi ré-<lb/>ctè comprehenderet:</s> <s xml:id="echoid-s19364" xml:space="preserve"> & refringatur k ad t ex m, & l ad t ex z:</s> <s xml:id="echoid-s19365" xml:space="preserve"> & continuemus g m, g z:</s> <s xml:id="echoid-s19366" xml:space="preserve"> & pertranſeant <lb/>ad f, o:</s> <s xml:id="echoid-s19367" xml:space="preserve"> & cõtinuemus lineas k m, m t, l z, z t.</s> <s xml:id="echoid-s19368" xml:space="preserve"> Forma, aũt quę extenditur ex k per m k, refringitur ք m t:</s> <s xml:id="echoid-s19369" xml:space="preserve"> <lb/> <pb o="279" file="0285" n="285" rhead="OPTICAE LIBER VII."/> & g m eſt perpendicularis, exiens ex m (quod eſt punctum refractionis) ſuper ſuperficiem corpo-<lb/>ris, quod eſt in parte t [ut oſtenſum eſt 25 n 4.</s> <s xml:id="echoid-s19370" xml:space="preserve">] Et quia corpus z m eſt ſubtilius corpore g t [per 16 <lb/>n] erit refractio m t ad partem perpendicularis m g:</s> <s xml:id="echoid-s19371" xml:space="preserve"> [per 14 n] m ergo erit inter duas lineas t b, t k.</s> <s xml:id="echoid-s19372" xml:space="preserve"> <lb/>Nam ſim eſſet ultra t k:</s> <s xml:id="echoid-s19373" xml:space="preserve"> tunc perpendicularis, quæ exit ex g, eſſet ultra t:</s> <s xml:id="echoid-s19374" xml:space="preserve"> & forma k cum extendere-<lb/>tur ad illud punctum:</s> <s xml:id="echoid-s19375" xml:space="preserve"> refringeretur ad partem perpendicularis g m, & non perueniret ad perpendi <lb/>cularem g e:</s> <s xml:id="echoid-s19376" xml:space="preserve"> & ſic non perueniret ad t.</s> <s xml:id="echoid-s19377" xml:space="preserve"> M ergo eſt inter duas lineas t b, t k.</s> <s xml:id="echoid-s19378" xml:space="preserve"> Et ſimiliter declarabitur <lb/>quòd z eſt inter duas lineas t b, t l.</s> <s xml:id="echoid-s19379" xml:space="preserve"> Et extrahamus t m ad <lb/> <anchor type="figure" xlink:label="fig-0285-01a" xlink:href="fig-0285-01"/> q, & t z ad r:</s> <s xml:id="echoid-s19380" xml:space="preserve"> erit ergo arcus q k æqualis arcui l r:</s> <s xml:id="echoid-s19381" xml:space="preserve"> [Quia <lb/>enim puncta k & l æquabiliter à uiſu diſtant per theſin:</s> <s xml:id="echoid-s19382" xml:space="preserve"> <lb/>puncta refractionis m & z in refractiuo m e z æquabili-<lb/>ter à puncto e diſtabunt:</s> <s xml:id="echoid-s19383" xml:space="preserve"> ideoq́;</s> <s xml:id="echoid-s19384" xml:space="preserve"> peripheria m e æquabi-<lb/>tur peripheriæ z e:</s> <s xml:id="echoid-s19385" xml:space="preserve"> & per 33 p 6 angulus b t q angulo b t <lb/>r, & peripheria b q peripheriæ b r (eſt enim uiſus t, ut in <lb/>aſtrologia demonſtratur, tanquam centrum mundi) at <lb/>tota peripheria b k æqualis concluſa eſt peripheriæ b l:</s> <s xml:id="echoid-s19386" xml:space="preserve"> <lb/>reliqua igitur q k æquatur reliquæ r l] & angulus q t r <lb/>eſt ille, per quem t comprehendit k l refractè:</s> <s xml:id="echoid-s19387" xml:space="preserve"> & angu-<lb/>lus k t l eſt ille, per quem t comprehenderet k l, ſi rectè <lb/>cõprehenderet.</s> <s xml:id="echoid-s19388" xml:space="preserve"> Sed remotio k l à uiſu eſt maxima:</s> <s xml:id="echoid-s19389" xml:space="preserve"> qua-<lb/>propter quantitas eius non certificatur.</s> <s xml:id="echoid-s19390" xml:space="preserve"> Quare t exiſti-<lb/>mat remotionem k l, ſicut in ſecundo libro diximus [24.</s> <s xml:id="echoid-s19391" xml:space="preserve"> <lb/>25 n.</s> <s xml:id="echoid-s19392" xml:space="preserve">] Sed æſtimatio eius quando comprehendit refra-<lb/>ctè, nõ differt ab æſtimatione eius quando comprehen-<lb/>dit rectè, niſi quòd putat ſe rectè comprehendere cum <lb/>refractè comprehendat.</s> <s xml:id="echoid-s19393" xml:space="preserve"> t ergo comprehendit k l refractè ex angulo minore illo, ex quo comprehen <lb/>dit illam rectè, & ſecundum comparationem ad illam eandem remotionem, ad quam compararet <lb/>illam, ſi rectè comprehenderet.</s> <s xml:id="echoid-s19394" xml:space="preserve"> Sed uiſus comprehendit magnitudinem ex quantitate anguli reſpe <lb/>ctu remotionis [per 38 n 2.</s> <s xml:id="echoid-s19395" xml:space="preserve">] tergo comprehendit quantitatem k l refractè minorem, quàm ſi com-<lb/>prehenderet illam rectè.</s> <s xml:id="echoid-s19396" xml:space="preserve"> Et ſi circumuoluamus figuram k t l circa t b immobilem, faciet circulum:</s> <s xml:id="echoid-s19397" xml:space="preserve"> <lb/>& erũt anguli, qui ſunt apud t, quos continent duæ lineæ k t, t l, & ſui compares, æquales:</s> <s xml:id="echoid-s19398" xml:space="preserve"> t ergo com <lb/>prehendit k l refractè in omni ſitu, in reſpectu circuli meridiei, cum fuerit in uertice capitis, minorẽ, <lb/>quàm ſi cõprehenderet eam rectè.</s> <s xml:id="echoid-s19399" xml:space="preserve"> Et ſi t b ſecuerit k l in duo æqualia:</s> <s xml:id="echoid-s19400" xml:space="preserve"> tunc duo puncta q, r erunt in-<lb/>ter duo puncta k, l:</s> <s xml:id="echoid-s19401" xml:space="preserve"> & erit angulus q t r minor angulo k t l:</s> <s xml:id="echoid-s19402" xml:space="preserve"> & erit omnis angulus eius exiens à pun-<lb/>cto t, ſecans ſtellam:</s> <s xml:id="echoid-s19403" xml:space="preserve"> & linea, quæ exit ex t in ſuperficie illius circuli, ſecabit circulum, & comprehen <lb/>detur minor, quàm ſit:</s> <s xml:id="echoid-s19404" xml:space="preserve"> & ſic tota ſtella uidebitur minor, quàm ſit.</s> <s xml:id="echoid-s19405" xml:space="preserve"> Stella ergo in uertice capitis com-<lb/>prehenditur minor, quàm ſi comprehenderetur rectè.</s> <s xml:id="echoid-s19406" xml:space="preserve"> Et ſimiliter diſtantia inter duas ſtellas, cum <lb/>uertex fuerit inter duas extremitates diſtantiæ, comprehendetur in omnibus poſitionibus minor, <lb/>quàm ſi rectè comprehenderetur.</s> <s xml:id="echoid-s19407" xml:space="preserve"> Et hoc eſt, quod uoluimus.</s> <s xml:id="echoid-s19408" xml:space="preserve"/> </p> <div xml:id="echoid-div627" type="float" level="0" n="0"> <figure xlink:label="fig-0285-01" xlink:href="fig-0285-01a"> <variables xml:id="echoid-variables233" xml:space="preserve">k q f b o r c l m e z f g</variables> </figure> </div> </div> <div xml:id="echoid-div629" type="section" level="0" n="0"> <head xml:id="echoid-head538" xml:space="preserve" style="it">53. Diameter ſtellæ, uel duarum ſtellarum diſtantia in horizonte, aut inter horizontem & <lb/>meridianum, ad horizontem parallela, refractè uiſa, minor: rectè, maior uidetur. 52 p 10.</head> <p> <s xml:id="echoid-s19409" xml:space="preserve">ITem:</s> <s xml:id="echoid-s19410" xml:space="preserve"> ſit ſtella ſiue diſtantia in horizonte, aut inter horizonta & uerticem capitis, æquidiſtans ho <lb/>rizonti:</s> <s xml:id="echoid-s19411" xml:space="preserve"> & ſit uiſus a:</s> <s xml:id="echoid-s19412" xml:space="preserve"> & uertex capitis b:</s> <s xml:id="echoid-s19413" xml:space="preserve"> & continuemus a b:</s> <s xml:id="echoid-s19414" xml:space="preserve"> & ſit diameter ſtellæ aut diſtantia d <lb/>e æquidiſtans horizonti:</s> <s xml:id="echoid-s19415" xml:space="preserve"> & ſit circulus uerticalis, qui tranſit per alteram extremitatem diametri <lb/>uel diſtantię, circulus b d:</s> <s xml:id="echoid-s19416" xml:space="preserve"> & ille, qui tranſit per aliam <lb/> <anchor type="figure" xlink:label="fig-0285-02a" xlink:href="fig-0285-02"/> extremitatem, circulus b e:</s> <s xml:id="echoid-s19417" xml:space="preserve"> & ſint duæ differentiæ <lb/>communes inter duos circulos & inter concauita-<lb/>tem orbis duo circuli h g, g z.</s> <s xml:id="echoid-s19418" xml:space="preserve"> Forma ergo d refringa-<lb/>tur ad a ex h:</s> <s xml:id="echoid-s19419" xml:space="preserve"> & e ad a ex z:</s> <s xml:id="echoid-s19420" xml:space="preserve"> & continuemus lineas a h, <lb/>h d, a z, z e, a d, a e:</s> <s xml:id="echoid-s19421" xml:space="preserve"> & ſit centrum mundi m:</s> <s xml:id="echoid-s19422" xml:space="preserve"> & conti-<lb/>nuemus m h, m z, & pertranſeant ad f, n:</s> <s xml:id="echoid-s19423" xml:space="preserve"> erit ergo m <lb/>h perpendicularis, exiens ex h ad ſuperficiem corpo <lb/>ris diaphani:</s> <s xml:id="echoid-s19424" xml:space="preserve"> [ut demonſtratum eſt 25 n 4] & erit h a <lb/>refracta ad partem h m:</s> <s xml:id="echoid-s19425" xml:space="preserve"> erit ergo refracta ad partem <lb/>contrariam illi, in qua eſt [f h:</s> <s xml:id="echoid-s19426" xml:space="preserve"> per 14 n] h ergo eſt al-<lb/>tius, quàm a d.</s> <s xml:id="echoid-s19427" xml:space="preserve"> Et ſimiliter declarabitur, quòd z eſt al <lb/>tius quã a e:</s> <s xml:id="echoid-s19428" xml:space="preserve"> ergo duo puncta f, n ſunt inter duo pun-<lb/>cta d, e & zenith capitis:</s> <s xml:id="echoid-s19429" xml:space="preserve"> & angulus refractionis, qui <lb/>eſt apud h, eſt æqualis angulo refractionis qui eſt a-<lb/>pud z:</s> <s xml:id="echoid-s19430" xml:space="preserve"> poſitio enim duorum punctorum d, e reſpectu <lb/>a eſt conſimilis.</s> <s xml:id="echoid-s19431" xml:space="preserve"> Tantùm ergo diſtat f à d, quantùm n <lb/>ab e:</s> <s xml:id="echoid-s19432" xml:space="preserve"> & extrahamus a h ad t, & a z ad k.</s> <s xml:id="echoid-s19433" xml:space="preserve"> Diſtabit ergo <lb/>t à d tantùm, quantùm k ab e:</s> <s xml:id="echoid-s19434" xml:space="preserve"> & continuemus t k:</s> <s xml:id="echoid-s19435" xml:space="preserve"> erit ergo æquidiſtans d e:</s> <s xml:id="echoid-s19436" xml:space="preserve"> eſt ergo minor:</s> <s xml:id="echoid-s19437" xml:space="preserve"> [quorũ <lb/>utrumq;</s> <s xml:id="echoid-s19438" xml:space="preserve"> demonſtratum eſt à Campano 14 p 12] & lineę a t, a k, a f, a e ſunt æquales:</s> <s xml:id="echoid-s19439" xml:space="preserve"> quia a eſt quaſi <lb/> <pb o="280" file="0286" n="286" rhead="ALHAZEN"/> centrum mundi & duorum circulorum b d, b e.</s> <s xml:id="echoid-s19440" xml:space="preserve"> Duæ ergo lineæ a t, a k ſunt æquales duabus lineis <gap/> <lb/>d, a e, & baſis t k eſt minor quàm baſis d e:</s> <s xml:id="echoid-s19441" xml:space="preserve"> ergo [per 25 p 1] angulus t a k eſt minor angulo d a e:</s> <s xml:id="echoid-s19442" xml:space="preserve"> & an-<lb/>gulus t a k eſt ille, quo d e cõprehenditur refractè:</s> <s xml:id="echoid-s19443" xml:space="preserve"> & angulus d a e eſt ille, quo d e cõprehenditur re-<lb/>ctè.</s> <s xml:id="echoid-s19444" xml:space="preserve"> Si ergo ſtella fuerit in horizonte, aut inter horizonta & circulũ meridiei:</s> <s xml:id="echoid-s19445" xml:space="preserve"> & fuerit diameter eius <lb/>æquidiſtans horizonti:</s> <s xml:id="echoid-s19446" xml:space="preserve"> uidebitur minor, quàm ſi uideretur rectè.</s> <s xml:id="echoid-s19447" xml:space="preserve"> Et hocidem eſt de diſtantia inter <lb/>duas ſtellas, ſi diſtantia fuerit æquidiſtans horizonti.</s> <s xml:id="echoid-s19448" xml:space="preserve"/> </p> <div xml:id="echoid-div629" type="float" level="0" n="0"> <figure xlink:label="fig-0285-02" xlink:href="fig-0285-02a"> <variables xml:id="echoid-variables234" xml:space="preserve">b g f t n d h k z a m e</variables> </figure> </div> </div> <div xml:id="echoid-div631" type="section" level="0" n="0"> <head xml:id="echoid-head539" xml:space="preserve" style="it">54. Diameter ſtellæ, uel duarum ſtellarum dιſtantia in circulo altitudinis refractè uiſa, mi-<lb/>nor: rectè, maior uidetur. 53 p 10.</head> <p> <s xml:id="echoid-s19449" xml:space="preserve">ITem:</s> <s xml:id="echoid-s19450" xml:space="preserve"> iteremus figuram:</s> <s xml:id="echoid-s19451" xml:space="preserve"> & ſit diameter aut diſtantia erecta ſcilicet in eodem circulo uerticali:</s> <s xml:id="echoid-s19452" xml:space="preserve"> & <lb/>ſit illa diameter aut diſtantia linea d e in circulo uerticali b d e:</s> <s xml:id="echoid-s19453" xml:space="preserve"> & ſit differentia communis inter <lb/>hunc circulum & inter concauitatem orbis, circulus g h z:</s> <s xml:id="echoid-s19454" xml:space="preserve"> & continuemus a d, a e:</s> <s xml:id="echoid-s19455" xml:space="preserve"> & refringatur <lb/>d ad a ex h, & e ad a ex z.</s> <s xml:id="echoid-s19456" xml:space="preserve"> Patet ergo, ut in præcedente figura, quòd h eſt altius quàm a d, & quòd z eſt <lb/>al<gap/>ius quàm a e:</s> <s xml:id="echoid-s19457" xml:space="preserve"> & continuemus lineas a h, h d, a z, z <lb/> <anchor type="figure" xlink:label="fig-0286-01a" xlink:href="fig-0286-01"/> e, m h, m z:</s> <s xml:id="echoid-s19458" xml:space="preserve"> & extrahamus m h a d t, & m z a d k.</s> <s xml:id="echoid-s19459" xml:space="preserve"> Erit <lb/>ergo angulus a z m ualde paruus.</s> <s xml:id="echoid-s19460" xml:space="preserve"> [Nam ſemidiame <lb/>ter terræ ad ſemidiametrũ cœli, rationem ſenſilem <lb/>nullam habet, ut docetur in aſtrologia] & angulus <lb/>refractionis erit pars illius Erit ergo [per 12 n] angu <lb/>lus e z k acutus:</s> <s xml:id="echoid-s19461" xml:space="preserve"> & ſimiliter d h t acutus:</s> <s xml:id="echoid-s19462" xml:space="preserve"> & [ք 13 p 1] <lb/>uterq;</s> <s xml:id="echoid-s19463" xml:space="preserve"> angulus a h d, a z e obtuſus.</s> <s xml:id="echoid-s19464" xml:space="preserve"> z autem aut erit <lb/>in horizonte, aut altius:</s> <s xml:id="echoid-s19465" xml:space="preserve"> ſi in horizonte:</s> <s xml:id="echoid-s19466" xml:space="preserve"> erit ergo in <lb/>extremitate perpendicularis exeuntis ex a ſuper a <lb/>b, aut altius illa:</s> <s xml:id="echoid-s19467" xml:space="preserve"> & h eſt altius quàm z:</s> <s xml:id="echoid-s19468" xml:space="preserve"> ergo angulus <lb/>a h m erit minor angulo a z m.</s> <s xml:id="echoid-s19469" xml:space="preserve"> [Nam conſtitutis ad <lb/>puncta m & a angulis a m p, g a q ęqualibus angulis <lb/>z m a, h a g per 23 p 1, connexιsq́;</s> <s xml:id="echoid-s19470" xml:space="preserve"> rectis a p, h p:</s> <s xml:id="echoid-s19471" xml:space="preserve"> erunt <lb/>anguli m p h, m h p æquales per 15 d.</s> <s xml:id="echoid-s19472" xml:space="preserve"> 5 p 1:</s> <s xml:id="echoid-s19473" xml:space="preserve"> & a p ma-<lb/>ior a h:</s> <s xml:id="echoid-s19474" xml:space="preserve"> քa per 7 p 3 maior eſt a q:</s> <s xml:id="echoid-s19475" xml:space="preserve"> & per 18 p 1 angulus <lb/>a h p maior angulo a p h.</s> <s xml:id="echoid-s19476" xml:space="preserve"> Quare angulus a h m, mi-<lb/>nor erit angulo ap m, cui ęqualis eſt angulus a z m ք <lb/>15 d.</s> <s xml:id="echoid-s19477" xml:space="preserve"> 4 p 1.</s> <s xml:id="echoid-s19478" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s19479" xml:space="preserve"> angulus a h m minor erit angulo a z m] ergo [per 12 n] angulus d h t eſt minor angulo <lb/>e z k:</s> <s xml:id="echoid-s19480" xml:space="preserve"> ergo angulus a h deſt maior angulo a z e [per 12 n.</s> <s xml:id="echoid-s19481" xml:space="preserve"> 13 p 1:</s> <s xml:id="echoid-s19482" xml:space="preserve">] & duę lineę m t, m k ſunt ſemidiametri <lb/>circuli b d e:</s> <s xml:id="echoid-s19483" xml:space="preserve"> & duę lineæ m h, m z ſunt ſemidiametri circuli g h z:</s> <s xml:id="echoid-s19484" xml:space="preserve"> ergo [per 15 d 1] m t eſt æqualis m k, <lb/>& m h eſt æqualis m z:</s> <s xml:id="echoid-s19485" xml:space="preserve"> ergo [per 3 ax] h t eſt æqualis z k, & angulus d h t eſt minor angulo e z k:</s> <s xml:id="echoid-s19486" xml:space="preserve"> ergo <lb/>linea d h eſt minor quàm e z.</s> <s xml:id="echoid-s19487" xml:space="preserve"> [Nam linea æqualis ipſi d h (quę cũ k z continet angulũ æqualẽ angulo <lb/>d h t) minor eſt linea e z per 7 p 3] & duæ lineæ a d, a e ſunt æquales, ſimiliter duæ a h, a z ſunt æqua-<lb/>les:</s> <s xml:id="echoid-s19488" xml:space="preserve"> quia a eſt quaſi centrum circuli b d e, & circuli g h z.</s> <s xml:id="echoid-s19489" xml:space="preserve"> Ergo circulus, qui continet triangulum a <lb/>h d, maior eſt circulo, qui cõtinet trian gulũ a e z, quia angulus a h d eſt maior angulo a z e, & linea h <lb/>d eſt minor, ut declaratum eſt, quàm z e.</s> <s xml:id="echoid-s19490" xml:space="preserve"> Ergo h d diſtinguit de circulo continente triangulum a h <lb/>d, arcum minorem arcu, ſimili arcui, quem diuidit z e ex circulo continente a e z:</s> <s xml:id="echoid-s19491" xml:space="preserve"> angulus ergo h a d <lb/>minor eſt angulo z a e:</s> <s xml:id="echoid-s19492" xml:space="preserve"> ſit ergo z a d communis:</s> <s xml:id="echoid-s19493" xml:space="preserve"> ergo angulus h a z eſt minor angulo d a e:</s> <s xml:id="echoid-s19494" xml:space="preserve"> & angu-<lb/>lus h a z eſt ille, ſub quo a comprehendit refractè d e:</s> <s xml:id="echoid-s19495" xml:space="preserve"> & angulus d a e eſt ille, ſub quo comprehendit <lb/>d e rectè:</s> <s xml:id="echoid-s19496" xml:space="preserve"> ſi comprehenderet:</s> <s xml:id="echoid-s19497" xml:space="preserve"> a ergo comprehendit d e refractè minorem, quàm rectè.</s> <s xml:id="echoid-s19498" xml:space="preserve"> Et hęc demon <lb/>ſtratio ſequitur, ſi circulus b d e fuerit circulus meridiei.</s> <s xml:id="echoid-s19499" xml:space="preserve"> Diameter ergo ſtellæ cum fuerit directa & <lb/>recta, & diſtantia inter duas ſtellas recta:</s> <s xml:id="echoid-s19500" xml:space="preserve"> comprehenditur refractè minor quàm rectè.</s> <s xml:id="echoid-s19501" xml:space="preserve"> Et hoc eſt <lb/>quod uoluimus.</s> <s xml:id="echoid-s19502" xml:space="preserve"/> </p> <div xml:id="echoid-div631" type="float" level="0" n="0"> <figure xlink:label="fig-0286-01" xlink:href="fig-0286-01a"> <variables xml:id="echoid-variables235" xml:space="preserve">b <gap/> d g q h n k z o a p e m</variables> </figure> </div> </div> <div xml:id="echoid-div633" type="section" level="0" n="0"> <head xml:id="echoid-head540" xml:space="preserve" style="it">55. Stella uidetur circularis: maior in horizonte, quàm in medio cæli: ſimiliteŕ duarum ſio <lb/>ſitarum inter ſe diſtantia. 54 p 10. Idem 51 n.</head> <p> <s xml:id="echoid-s19503" xml:space="preserve">ET omnis ſtella in cœlo comprehenditur rotunda:</s> <s xml:id="echoid-s19504" xml:space="preserve"> quia diametri eius comprehenduntur æ-<lb/>quales.</s> <s xml:id="echoid-s19505" xml:space="preserve"> Et cum ſit manifeſtum, quòd utraq;</s> <s xml:id="echoid-s19506" xml:space="preserve"> diameter eius recta & tranſuerſa ſecundum lati-<lb/>tudinem comprehenditur minor, quàm ſi comprehenderetur rectè:</s> <s xml:id="echoid-s19507" xml:space="preserve"> ergo utraq;</s> <s xml:id="echoid-s19508" xml:space="preserve"> diameter e-<lb/>ius decliuis comprehenditur æqualiter minor, quàm ſi comprehenderetur rectè.</s> <s xml:id="echoid-s19509" xml:space="preserve"> Et ſimiliter diſtan <lb/>tiæ inter ſtellas comprehenduntur in omnibus locis & in omnιbus ſitibus minores, quàm ſi com-<lb/>prehenderentur rectè.</s> <s xml:id="echoid-s19510" xml:space="preserve"> Item diximus [51 n] quòd omnis ſtella in uertice capitis comprehenditur <lb/>minor, quàm in omnibus alijs partibus cœli:</s> <s xml:id="echoid-s19511" xml:space="preserve"> & quantò fuerit remotior à uertice capitis, tantò com-<lb/>prehendetur maior:</s> <s xml:id="echoid-s19512" xml:space="preserve"> & quàm maxima comprehenditur, quando comprehenditur in horizonte.</s> <s xml:id="echoid-s19513" xml:space="preserve"> Re-<lb/>ſtat ergo declarare cauſſam, quare hoc ſit.</s> <s xml:id="echoid-s19514" xml:space="preserve"> Dico, quòd in ſecundo tractatu huius libri declarauimus, <lb/>cum tractauimus de magnitudine [38 n:</s> <s xml:id="echoid-s19515" xml:space="preserve">] quòd ſi uiſus comprehenderit magnitudines uiſibilium:</s> <s xml:id="echoid-s19516" xml:space="preserve"> <lb/>comprehendit illas ex quantitatibus angulorum, quos reſpiciunt uiſibilia apud centrum uiſus, & <lb/>ex quantitatibus remot onum, & ex comparatione angulorum ad remotiones.</s> <s xml:id="echoid-s19517" xml:space="preserve"> Et declarauimus, <lb/>quòd uiſus nun quam comprehendit uiſibilium quantitates, niſi remotiones eorum ſint in rectitu-<lb/>dine corporum propinquorum continuorum:</s> <s xml:id="echoid-s19518" xml:space="preserve"> &, quòd ſi uiſus non certificarit remotiones uiſibi-<lb/>lium, non certificabit quantitates uiſibilium.</s> <s xml:id="echoid-s19519" xml:space="preserve"> Et declarauimus illic etiam, quòd uiſus, ſi non certifi-<lb/> <pb o="281" file="0287" n="287" rhead="OPTICAE LIBER VII."/> cauerit diſtantiam uiſi, poteſt perpendere diſtantiam eius, & aſsimilare eam diſtantijs uiſibilium aſ-<lb/>fuetorũ, quibus tale uiſibile comprehenditur, in tali forma & in tali figura:</s> <s xml:id="echoid-s19520" xml:space="preserve"> dein de cõprehendit ma-<lb/>gnitudinem illius ex quantate anguli, quem reſpicit illud uiſibile apud centrũ uiſus, reſpectu remo-<lb/>tionis, quam perpendit:</s> <s xml:id="echoid-s19521" xml:space="preserve"> & remotiones ſtellarum nõ ſunt in rectitudine corporum propinquorum.</s> <s xml:id="echoid-s19522" xml:space="preserve"> <lb/>Quare uiſus nõ comprehendit quantitates earum, neq;</s> <s xml:id="echoid-s19523" xml:space="preserve"> certificat diſtantias earum.</s> <s xml:id="echoid-s19524" xml:space="preserve"> Viſus ergo per-<lb/>pendit diſtantias ſtellarum, & aſsimilat illas diſtantijs eorum, quæ ſunt terreſtria, quæ comprehen-<lb/>duntur ex diſtantia maxima, & perpendit quantitates eorum.</s> <s xml:id="echoid-s19525" xml:space="preserve"> Corpus autem cœli non uidetur ſen-<lb/>ſui, quòd ſit ſphæricum, & concauum eius ſit ex parte uiſus, neq;</s> <s xml:id="echoid-s19526" xml:space="preserve"> uiſus ſentit corporeitatẽ cœli, neq;</s> <s xml:id="echoid-s19527" xml:space="preserve"> <lb/>uiſus ſentit de cœlo, niſi colorem glaucum ſolummodo:</s> <s xml:id="echoid-s19528" xml:space="preserve"> corporeitas uerò & extenſio ſecundũ tres <lb/>dimenſiones, & rotunditas & concauitas nullo modo poſſunt cõprehendi.</s> <s xml:id="echoid-s19529" xml:space="preserve"> Et ſi uiſus non certifica-<lb/>uerit aliquid:</s> <s xml:id="echoid-s19530" xml:space="preserve"> tunc aſsimilabit ipſum alicui de rebus aſſuetis:</s> <s xml:id="echoid-s19531" xml:space="preserve"> unde comprehendit ſolem & lunã pla <lb/>nos, & corpora conuexa & concaua à maxima diſtantia, plana:</s> <s xml:id="echoid-s19532" xml:space="preserve"> & arcus quorum conuexum aut con <lb/>cauum eſt ex parte uiſus, comprehendet lineas rectas.</s> <s xml:id="echoid-s19533" xml:space="preserve"> Nam ſi non comprehenderit propinquitatẽ <lb/>medij, & remotionẽ extremitatum in conuexis, & remotionẽ medij & propinquitatem extremita-<lb/>tum in concauls:</s> <s xml:id="echoid-s19534" xml:space="preserve"> tunc aſsimilabit ſuperficies conuexas, & concauas ſuperficiebus planis, & aſsimi-<lb/>labit arcus lineis rectis:</s> <s xml:id="echoid-s19535" xml:space="preserve"> aſſueta enim uiſibilia in maiore parte ſunt plana & recta.</s> <s xml:id="echoid-s19536" xml:space="preserve"> Nec uiſus, cum for-<lb/>ma ſtellę peruenit ad ipſum, ſentit quòd illa forma ſit refracta, aut quòd refringatur ex ſuperficie cõ-<lb/>caua, & quòd corpus, in quo ſtella eſt, ſit ſubtilius corpore, in quo eſt uiſus:</s> <s xml:id="echoid-s19537" xml:space="preserve"> ſed forma ſtellæ compre <lb/>henditur, ſicut formæ aliarũ rerum, quæ comprehen duntur in aere rectè.</s> <s xml:id="echoid-s19538" xml:space="preserve"> Et formæ uiſibiliũ non re-<lb/>fringuntur, quando occurrunt corpori diuerſo ab aere, propter uiſum:</s> <s xml:id="echoid-s19539" xml:space="preserve"> necuiſus ſentit refractionẽ <lb/>eorũ, nec ſuperficiem, à qua refringuntur formæ in corporibus diuerſis in diaphanitate, niſi proprie <lb/>tate naturali formę lucis & coloris, quę extenduntur in corporibus diaphanis.</s> <s xml:id="echoid-s19540" xml:space="preserve"> Formæ ergo ſtellarũ <lb/>refractarũ perueniunt ad uiſum, ſicut perueniũt formę eorũ, quę ſunt in aere, ad uiſum, & non com-<lb/>prehenduntur, ſicut comprehenduntur in aere.</s> <s xml:id="echoid-s19541" xml:space="preserve"> Viſus aũt comprehendit colorẽ cœli, nec tamen cer <lb/>tificat formã eius nudo ſenſu.</s> <s xml:id="echoid-s19542" xml:space="preserve"> Et cum uiſus comprehenderit colorẽ aliquẽ in longitudine & latitudi <lb/>ne:</s> <s xml:id="echoid-s19543" xml:space="preserve"> ſuper hoc, quod cõprehendit figuram & formã:</s> <s xml:id="echoid-s19544" xml:space="preserve"> comprehendet ipſum planũ:</s> <s xml:id="echoid-s19545" xml:space="preserve"> aſsimilabit enim i-<lb/>pſum aliquibus ſuperficiebus aſſuetis, ut parieti & alijs.</s> <s xml:id="echoid-s19546" xml:space="preserve"> Et hoc modo cõprehendit ſuperficies con-<lb/>uexas & cõcauas in remotione maxima.</s> <s xml:id="echoid-s19547" xml:space="preserve"> Viſus ergo comprehendit planiciem terrę planã omnino, <lb/>nec ſentit conuexitatẽ eius, niſi fuerint ibi montes & ualles.</s> <s xml:id="echoid-s19548" xml:space="preserve"> Viſus ergo cõprehendit ſuperficiẽ cœli <lb/>planã, & comprehendit ſtellas, ſicut comprehendit uiſibilia aſſueta ſeparata, quę ſunt in locis ſpatio <lb/>ſis.</s> <s xml:id="echoid-s19549" xml:space="preserve"> Et cum uiſus comprehenderit aliqua uiſibilia aſſueta in loco aliquo ſpatioſo, & comprehenderit <lb/>illa angulis æqualibus, & cõprehenderit quantitates diſtantiarũ uiſibiliũ:</s> <s xml:id="echoid-s19550" xml:space="preserve"> tunc illud, quod eſt remo <lb/>tius, comprehen detur maius.</s> <s xml:id="echoid-s19551" xml:space="preserve"> Nam quantitates remotionis magnitudinis cõprehenduntur ex com <lb/>paratione anguli, quẽ reſpicit illa remotio apud centrũ uiſus, ad diſtantiam remotã:</s> <s xml:id="echoid-s19552" xml:space="preserve"> & comprehen-<lb/>dit uiſus quantitatẽ magnitudinis propin quæ ex cõparatione anguli;</s> <s xml:id="echoid-s19553" xml:space="preserve"> quẽ reſpicit illud propin quũ, <lb/>qui eſt æqualis angulo, quem reſpicit diſtantia ad diſtantiã propinquã.</s> <s xml:id="echoid-s19554" xml:space="preserve"> Et hoc patet, & eſſe, teſtatur <lb/>ei:</s> <s xml:id="echoid-s19555" xml:space="preserve"> ſcilicet:</s> <s xml:id="echoid-s19556" xml:space="preserve"> quòd duorũ uiſibilium, quæ à uiſu comprehenduntur duobus angulis æqualibus, quorũ <lb/>diſtantię ſunt diuerſæ;</s> <s xml:id="echoid-s19557" xml:space="preserve"> ſenſibiliter:</s> <s xml:id="echoid-s19558" xml:space="preserve"> remotius uidebitur maius.</s> <s xml:id="echoid-s19559" xml:space="preserve"> Nam ſi homo oppoſuerit ſe ſpatioſo <lb/>parieti, deinde eleuauerit manum, donec apponat illam uiſui, & cooperuerit alterum uiſum;</s> <s xml:id="echoid-s19560" xml:space="preserve"> & aſpe <lb/>xerit reliquo, & poſuerit manũ mediam inter uiſum ſuum & illum parietẽ;</s> <s xml:id="echoid-s19561" xml:space="preserve"> tunc manus eius coope-<lb/>riet portionem & latitudinẽ illius parietis, & comprehendet manum ſuam & parietem ſimul.</s> <s xml:id="echoid-s19562" xml:space="preserve"> Com <lb/>prehendet ergo manum ſuam angulo acuto:</s> <s xml:id="echoid-s19563" xml:space="preserve"> & in hoc ſtatu comprehendet latitudinẽ parietis maio <lb/>rem, quã latitudinem manus multiplicem:</s> <s xml:id="echoid-s19564" xml:space="preserve"> deinde ſi mouerit manũ ita, ut detegatur illud, quod ma-<lb/>nus cooperuerat de pariete, & aſpexerit ad manũ:</s> <s xml:id="echoid-s19565" xml:space="preserve"> uidebit illud, quod detectũ eſt de pariete, maius, <lb/>quàm ſit ſua manus, multipliciter:</s> <s xml:id="echoid-s19566" xml:space="preserve"> & ipſe comprehendet manum ſuam & parietem duobus angulis <lb/>æqualibus.</s> <s xml:id="echoid-s19567" xml:space="preserve"> Ex quo patet, quòd uiſus comprehendit magnitudinẽ ex comparatione anguli ad remo <lb/>tionem.</s> <s xml:id="echoid-s19568" xml:space="preserve"> Viſus ergo comprehendit ſuperficiem cœli planam, nec ſentit concauitatẽ eius, & compre-<lb/>hendit ſtellas ſeparatas ιn ipſo.</s> <s xml:id="echoid-s19569" xml:space="preserve"> Comprehendit ergo ſtellas æquales, ſeparatas inæquales:</s> <s xml:id="echoid-s19570" xml:space="preserve"> nam com <lb/>parat angulum, quẽ reſpicit ſtella extrema, propinqua horizonti apud centrum uiſus, ad diſtantiam <lb/>remotã, & comparat angulum, quem reſpicit ſtella in medio cœli, & propinqua medio, remotionl <lb/>propinquæ.</s> <s xml:id="echoid-s19571" xml:space="preserve"> Et ſimiliter comprehendit ſtellam, quæ eſt in horizonte aut prope, maiorem ea, quæ eſt <lb/>in medio cœli aut prope.</s> <s xml:id="echoid-s19572" xml:space="preserve"> Comprehendit ergo eandem ſtellam & diſtantiã in diuerſis locis cœli, di-<lb/>uerſæ quantitatis.</s> <s xml:id="echoid-s19573" xml:space="preserve"> Sic ergo comprehendit eandem ſtellam & diſtantiã in horizante aut prope.</s> <s xml:id="echoid-s19574" xml:space="preserve"> Nam <lb/>cõparat angulum, quẽ reſpicit illa ſtella apud centrum uiſus, ſtella exiſtente in horizonte, diſtantiæ <lb/>remotæ:</s> <s xml:id="echoid-s19575" xml:space="preserve"> & comparat angulum, quẽ reſpicit illa ſtella apud centrum uiſus, exiſtente ſtella in medio <lb/>cœli, diſtantiæ propinquę.</s> <s xml:id="echoid-s19576" xml:space="preserve"> Sed inter angulum, quẽ reſpicit ſtella apud centrũ uiſus, ſtella exiſtente <lb/>in medio cœli, & inter angulum, quem reſpicit ſtella apud centrum uiſus, ſtella exiſtente in horizon <lb/>te, non eſt maxima diuerſitas, ſed duo anguli ſunt propinqui, quamuis diuerſit & ſimiliter diſtantiæ <lb/>inter ſtellas.</s> <s xml:id="echoid-s19577" xml:space="preserve"> Et cum ſenſus comparauerit duos angulos propinquos in magnitudine ad duas diuer <lb/>ſas diſtantias in magnitudine:</s> <s xml:id="echoid-s19578" xml:space="preserve"> tunc remotior comprehenditur maior.</s> <s xml:id="echoid-s19579" xml:space="preserve"> Et quod certificat hanc cauſ-<lb/>ſam:</s> <s xml:id="echoid-s19580" xml:space="preserve"> eſt:</s> <s xml:id="echoid-s19581" xml:space="preserve"> quòd anguli, quos eadem ſtella reſpicit apud centrũ uiſus ex omnibus partibus cœli (cum <lb/>lineæ, quę continẽt ipſos, fuerint refractæ) ſunt quaſi anguli, per quos cõprehenderetur rectè:</s> <s xml:id="echoid-s19582" xml:space="preserve"> quo-<lb/>niam locus uiſus eſt centrum cœli, & refractiones formarum ſtellarum nõ diminuuntur ex illis an-<lb/>gulis diminutione maxima.</s> <s xml:id="echoid-s19583" xml:space="preserve"> Et cum iſtæ diminutiones non ſint maximę:</s> <s xml:id="echoid-s19584" xml:space="preserve"> tunc diuerſitas inter an-<lb/>gulos refractos, quibus ſtella comprehenditur, & inter remotionẽ inter ſtellas à locis diuerſis cœli, <lb/> <pb o="282" file="0288" n="288" rhead="ALHAZEN"/> hon erit maxima diuerſitas.</s> <s xml:id="echoid-s19585" xml:space="preserve"> Et cum diuerſitas iſtorũ angulorum non eſt maxima:</s> <s xml:id="echoid-s19586" xml:space="preserve"> tunc magnitudo <lb/>ſtellæ non comprehendetur diuerſa maxima diuerſitate:</s> <s xml:id="echoid-s19587" xml:space="preserve"> & quod demonſtrat diminutiones angulo <lb/>rum refractionis ad angulos, quos continent lineæ rectæ, non eſt maximæ magnitudinis.</s> <s xml:id="echoid-s19588" xml:space="preserve"> Et quòd <lb/>ſunt ualde paruę:</s> <s xml:id="echoid-s19589" xml:space="preserve"> eſt<gap/>quòd dictũ eſt in prędicta experientia in capitulo refractionis [15 n] in quo de-<lb/>clarauimus, quòd uiſus cõprehendit ſtellã refractè, & uidet ſtellã fixam ex polo mundi, & remotio <lb/>eius eſt ab ipſo in una reuolutione:</s> <s xml:id="echoid-s19590" xml:space="preserve"> nam hæc diuerſitas inuenitur parua:</s> <s xml:id="echoid-s19591" xml:space="preserve"> ex quo patet, quòd anguli <lb/>refractionis ſunt parui.</s> <s xml:id="echoid-s19592" xml:space="preserve"> Vnde per illã diuerſitatẽ, quæ eſt inter ipſos, non diuerſantur anguli, quibus <lb/>ſtella cõprehenditur in locis diuerſis cœli, maxima diuerſitate.</s> <s xml:id="echoid-s19593" xml:space="preserve"> Sed magnitudo ſtellæ & diſtantiæ <lb/>ſtellarũ differunt multùm, cum ſunt in horizonte & in medio cœli.</s> <s xml:id="echoid-s19594" xml:space="preserve"> Ergo cauſſa diuerſitatis ſtellæ & <lb/>diſtantiæ in magnitudine, in locis diuerſis cœli, non eſt diuerſitas angulorũ refractionis.</s> <s xml:id="echoid-s19595" xml:space="preserve"> Et iam de-<lb/>clarauimus, quòd uiſus comprehendit magnitudinẽ comparando angulos remotionis ad remotio <lb/>nes.</s> <s xml:id="echoid-s19596" xml:space="preserve"> Ergo ſi diuerſitas inter angulos fuerit modica, & inter diſtantias & remotiones multa:</s> <s xml:id="echoid-s19597" xml:space="preserve"> tunc res <lb/>uidebitur ex maiore diſtantia maior.</s> <s xml:id="echoid-s19598" xml:space="preserve"> Cauſſa ergo, propter quam uidentur diſtantiæ ſtellarũ in hori-<lb/>zonte maiores quàm in medio cœli aut prope:</s> <s xml:id="echoid-s19599" xml:space="preserve"> eſt illud:</s> <s xml:id="echoid-s19600" xml:space="preserve"> quòd ſenſus ęſtimat illas diſtare magis in ho <lb/>rizonte, quàm in medio cœli.</s> <s xml:id="echoid-s19601" xml:space="preserve"> Et hoc, quòd uiſus cõprehendit ſtellas in diuerſis locis cœli diuerſas <lb/>in magnitudine:</s> <s xml:id="echoid-s19602" xml:space="preserve"> eſt error perpetuus:</s> <s xml:id="echoid-s19603" xml:space="preserve"> quia cauſſa eſt perpetua:</s> <s xml:id="echoid-s19604" xml:space="preserve"> & eſt:</s> <s xml:id="echoid-s19605" xml:space="preserve"> quoniã uiſus comprehendit ſu-<lb/>perficiem cœli planã, nec ſentit concauitatẽ eius & æqualitatẽ diſtantiæ à uiſu.</s> <s xml:id="echoid-s19606" xml:space="preserve"> Et conſtat in anima, <lb/>quòd in ſuperficie plana, quæ extenditur ad omnẽ partem, differũt diſtantię eius in uiſu:</s> <s xml:id="echoid-s19607" xml:space="preserve"> & id, quod <lb/>eſt propin quius, eſt illud, quod eſt proximũ capiti.</s> <s xml:id="echoid-s19608" xml:space="preserve"> Comprehendit ergo illud, quod eſt in horizonte <lb/>remotius, quàm illud, quod eſt in medio cœli:</s> <s xml:id="echoid-s19609" xml:space="preserve"> & quòd anguli, quos reſpicit eadẽ ſtella apud centrũ <lb/>uiſus ex omnibus partibus cœli, non maximè diuerſantur:</s> <s xml:id="echoid-s19610" xml:space="preserve"> & quòd uiſus cõprehendit magnitudinẽ <lb/>rei ex cõparatione anguli, quẽ res reſpicit ad remotionẽ illius rei à uiſu.</s> <s xml:id="echoid-s19611" xml:space="preserve"> Comprehendit ergo quanti <lb/>tatem ſtellæ, & quantitatẽ diſtantiæ, quæ eſt inter ſtellas, cum fuerint in horizonte aut prope, cõpa-<lb/>ratione anguli ad diſtantiã remotã:</s> <s xml:id="echoid-s19612" xml:space="preserve"> & cum fuerint in medio cœli, aut prope, ex cõparatione anguli <lb/>æqualis primo aut ferè, ad diſtantiã propinquã:</s> <s xml:id="echoid-s19613" xml:space="preserve"> & inter ipſam & inter diſtantiã horizontis uidetur <lb/>maxima diuerſitas.</s> <s xml:id="echoid-s19614" xml:space="preserve"> Hæc eſt igitur cauſſa, propter quã errat uiſus in diuerſitate magnitudinis ſtella-<lb/>rum & diſtantiarũ:</s> <s xml:id="echoid-s19615" xml:space="preserve"> & hæc cauſſa fixa eſt & perpetua & immutabilis.</s> <s xml:id="echoid-s19616" xml:space="preserve"> Et uiſus coprehendit ſtellas par <lb/>uas propter remotionẽ earum:</s> <s xml:id="echoid-s19617" xml:space="preserve"> reſpiciunt enim apud centrũ uiſus angulos paruos.</s> <s xml:id="echoid-s19618" xml:space="preserve"> Sed & ſenſus nõ <lb/>certificat quantitatẽ remotionis ſtellæ, ſed æſtimat & comparat remotiones ſtellarũ cum remotio-<lb/>nibus uiſibiliũ aſſuetorũ, quę ſunt in terra:</s> <s xml:id="echoid-s19619" xml:space="preserve"> ita quòd opinatur, quòd remotio ſtellæ eſt, ſicut remotio <lb/>alicuius maximè remoti in terra.</s> <s xml:id="echoid-s19620" xml:space="preserve"> Comparat ergo angulũ, quẽ facit ſtella apud uiſum, qui eſt paruus <lb/>ad remotionẽ, ſicut remotio eſt eorũ, quæ ſunt in terra.</s> <s xml:id="echoid-s19621" xml:space="preserve"> Et ſic cõprehendit ſtellam, propter hanc cõ-<lb/>parationem, paruã.</s> <s xml:id="echoid-s19622" xml:space="preserve"> Et ſi uiſus eſſet certus de quantitate remotionis ſtellæ:</s> <s xml:id="echoid-s19623" xml:space="preserve"> tunc cõprehenderet eam <lb/>magnã.</s> <s xml:id="echoid-s19624" xml:space="preserve"> Et ſimiliter eſt de omnibus, quę ſunt ſuper terrã, maximè remotis, ſi cõprehendantur, parua <lb/>ſunt:</s> <s xml:id="echoid-s19625" xml:space="preserve"> quia nõ certificatur remotio eorũ.</s> <s xml:id="echoid-s19626" xml:space="preserve"> Et iam declarauimus hoc perfectè in tertio tractatu huius li <lb/>bri [23 n.</s> <s xml:id="echoid-s19627" xml:space="preserve">] Et ſicut uiſus errat in quantitate remotionis ſtellæ:</s> <s xml:id="echoid-s19628" xml:space="preserve"> quia nõ eſt certus de ipſa, & aſsimilat <lb/>ipſam remotionibus, quę ſunt ſuper terrã:</s> <s xml:id="echoid-s19629" xml:space="preserve"> ſic errat in hoc, quòd diſtantiæ earũ in locis diuerſis cœli <lb/>ſint diuerſę, cum ſint æquales:</s> <s xml:id="echoid-s19630" xml:space="preserve"> quia aſsimilat eas etiã diſtantijs diuerſis, quæ ſunt ſuper terrã, de qui-<lb/>bus non eſt dubiũ eas eſſe diuerſas.</s> <s xml:id="echoid-s19631" xml:space="preserve"> Et ſicut error in remotione & magnitudine ſtellę eſt perpetuus:</s> <s xml:id="echoid-s19632" xml:space="preserve"> <lb/>ſic error in diuerſitate diſtantiarũ ſtellarum in locis diuerſis cœli & in diuerſitate magnitudinis, eſt <lb/>perpetuus.</s> <s xml:id="echoid-s19633" xml:space="preserve"> Nam formæ earũ diſtantiarum non diuerſantur apud uiſum in diuerſis temporibus, ſed <lb/>femper ſunt eodem modo:</s> <s xml:id="echoid-s19634" xml:space="preserve"> & uiſus aſsimilat eas diſtantijs aſſuetarũ rerum, quę maximè diſtant à ui <lb/>ſu ſuper ſuperficiẽ terræ.</s> <s xml:id="echoid-s19635" xml:space="preserve"> Accedit etiã eis, quæ ſunt in cœlo alia cauſſa, ad hoc, quòd uideantur maio <lb/>ra in horizonte, in maiore parte:</s> <s xml:id="echoid-s19636" xml:space="preserve"> ſcilicet uapores groſsi, qui ſunt oppoſiti inter uiſum & ſtellam.</s> <s xml:id="echoid-s19637" xml:space="preserve"> Et <lb/>cum uapor fuerit in horizonte aut prope, & nõ fuerit cõtinuus uſq;</s> <s xml:id="echoid-s19638" xml:space="preserve"> ad mediũ cœli:</s> <s xml:id="echoid-s19639" xml:space="preserve">erit portio ſphæ-<lb/>ræ, cuius centrũ erit centrum mundi, qui cõtinet terrã:</s> <s xml:id="echoid-s19640" xml:space="preserve"> & ſic abſcindetur ex parte medij cœli, & erit <lb/>ſuperficies eius, quæ eſt ex parte uiſus, plana.</s> <s xml:id="echoid-s19641" xml:space="preserve"> Quare formę aut diſtantię, quæ ſunt ultra illũ uaporẽ, <lb/>uidebuntur maiores, quàm ſine illo uapore.</s> <s xml:id="echoid-s19642" xml:space="preserve"> In illo enim loco concauitatis cœli, ex quo loco refringi <lb/>tur forma ſtellæ ad uiſum, forma ſtellę exiſtit, & ex ipſo extenditur rectè ad uiſum, ſi in horizonte nõ <lb/>fuerit uapor groſſus.</s> <s xml:id="echoid-s19643" xml:space="preserve"> Si uerò fuerit uapor groſſus:</s> <s xml:id="echoid-s19644" xml:space="preserve"> tunc hęc forma extendetur ad ſuperficiẽ uaporis, <lb/>quę eſt ex parte cœli, & exiſtet in illa ſuperficie:</s> <s xml:id="echoid-s19645" xml:space="preserve"> & ſic uiſus cõprehendet illã, ſicut comprehendit ea, <lb/>quę ſunt in uapore:</s> <s xml:id="echoid-s19646" xml:space="preserve"> ſcilicet, quòd illa forma extenditur in uapore groſſo rectè:</s> <s xml:id="echoid-s19647" xml:space="preserve"> deinde refringitur a-<lb/>pud ſuperficiem uaporis ad contrariã partem perpendicularis, exeuntis ſuper ſuperficiem uaporis, <lb/>quæ eſt plana.</s> <s xml:id="echoid-s19648" xml:space="preserve"> Nam aer, qui eſt ex parte uiſus, eſt ſubtilior illo uapore:</s> <s xml:id="echoid-s19649" xml:space="preserve"> ex quo ſequitur, quòd forma <lb/>uidetur maior, quàm ſi uideretur rectè, ut in prima figura huius capituli [39 n] diximus:</s> <s xml:id="echoid-s19650" xml:space="preserve"> & eſt, cum <lb/>corpus ſubtilius fuerit ex parte uiſus, & groſsius ex parte rei uiſæ, erit ſuperficies corporis groſsio-<lb/>ris plana.</s> <s xml:id="echoid-s19651" xml:space="preserve"> Forma ergo, quę peruenit ad ſuperficiem uaporis, quę eſt ex parte cœli, eſt res uiſa;</s> <s xml:id="echoid-s19652" xml:space="preserve"> & cor-<lb/>pus, in quo extenditur forma, eſt uapor groſſus, & aer, in quo eſt uiſus, eſt ſubtilior illo.</s> <s xml:id="echoid-s19653" xml:space="preserve"> Cauſſa uerò <lb/>principalis, quare ſtellę & diſtantię ſtellarum uideantur in horizonte maiores, quàm in medio cœli, <lb/>eſt illa prædicta:</s> <s xml:id="echoid-s19654" xml:space="preserve"> & eſt fixa & perpetua.</s> <s xml:id="echoid-s19655" xml:space="preserve"> Si uerò acciderit, ut ſit uapor groſſus, creſcit magnitudo ea-<lb/>rum:</s> <s xml:id="echoid-s19656" xml:space="preserve"> ſed hæc cauſa eſt in quibuſdam locis ſemper, & in quibuſdam quandoq;</s> <s xml:id="echoid-s19657" xml:space="preserve">. Omnia ergo, quę dixi <lb/>mus in hoc capitulo de illis, quę accidunt uiſui propter refractionẽ:</s> <s xml:id="echoid-s19658" xml:space="preserve"> ſunt deceptiones illæ, quę ſem-<lb/>per accidũt aut in maiore parte:</s> <s xml:id="echoid-s19659" xml:space="preserve"> & ſufficiunt in hoc, quo indigemus de deceptionibus, quarũ cauſſe <lb/>eſt refractio.</s> <s xml:id="echoid-s19660" xml:space="preserve"> Nunc autem terminemus hunc tractatum, qui eſt finis libri.</s> <s xml:id="echoid-s19661" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div634" type="section" level="0" n="0"> <head xml:id="echoid-head541" xml:space="preserve">ALHAZEN FILII ALHAYZEN OPTICAE FINIS.</head> <pb o="283" file="0289" n="289"/> </div> <div xml:id="echoid-div635" type="section" level="0" n="0"> <head xml:id="echoid-head542" xml:space="preserve">ALHAZEN FILII</head> <head xml:id="echoid-head543" xml:space="preserve">ALHAYZEN DE CREPVSCVLIS</head> <head xml:id="echoid-head544" xml:space="preserve">ET NVBIVM ASCENSIONIBVS LIBER VNVS.</head> <head xml:id="echoid-head545" xml:space="preserve">Gerardo Cremonenſi interprete.</head> <head xml:id="echoid-head546" xml:space="preserve" style="it">NVMERI.</head> <head xml:id="echoid-head547" xml:space="preserve" style="it">1. Crepuſculum matutinum incipit, ac ueſpertinum deſinit, ſole ante ortum & poſt occaſum <lb/>ſuum 19 partib{us}, in peripheria circuli per uerticem regionis ſoliś locum tranſeuntis, ſub <lb/>horizontem demerſo.</head> <p> <s xml:id="echoid-s19662" xml:space="preserve">OStendere uolo in hoc tractatu quid ſit crepuſculum, & quæ cauſſa neceſſariò faciens <lb/>eius apparitionem:</s> <s xml:id="echoid-s19663" xml:space="preserve"> inde uerò progrediar ad cognoſcendum ultimum, quod eleuatur <lb/>à ſuperficie terræ, de uaporibus ſubtilibus aſcendentibus ex ea.</s> <s xml:id="echoid-s19664" xml:space="preserve"> Dico ergo, quòd cre-<lb/>puſculum matutinum & crepuſculum ueſpertinum ſunt ſimilis figuræ:</s> <s xml:id="echoid-s19665" xml:space="preserve"> unum namq;</s> <s xml:id="echoid-s19666" xml:space="preserve"> <lb/>eorum ex acceſsione luminis ſolis, & alterũ ex ipſius receſsione contingit.</s> <s xml:id="echoid-s19667" xml:space="preserve"> Vtrorumq;</s> <s xml:id="echoid-s19668" xml:space="preserve"> <lb/>uerò colorẽs diuerſi ſunt, propter diuerſitatem horizontum;</s> <s xml:id="echoid-s19669" xml:space="preserve"> in quibus ſol eſt apparens.</s> <s xml:id="echoid-s19670" xml:space="preserve"> Quoniam <lb/>ſol quando eſt in horizonte orientali, non multum eleuatus, eſt illic color eius alius à colore ipſius <lb/>in uiſibus, quando eſt ſecundum æqualitatem illius altitudinis in horizonte occidentall.</s> <s xml:id="echoid-s19671" xml:space="preserve"> Et ſimili-<lb/>ter radij eius, qui uidẽtur in crepuſculo, & quod uidetur in æthere de luminibus eius.</s> <s xml:id="echoid-s19672" xml:space="preserve"> Et ipſe æther <lb/>coloratus eſt, ſequens illud, ſecundum quod eſt ſol in utriſque partibus eius.</s> <s xml:id="echoid-s19673" xml:space="preserve"> Nam qui ex illo eſt in <lb/>oriente color, eſt albedo & claritas:</s> <s xml:id="echoid-s19674" xml:space="preserve"> & qui eſt in occidẽte, ad rubedinem aliquantulùm uergit.</s> <s xml:id="echoid-s19675" xml:space="preserve"> Quæ <lb/>res uerò ſit illud illuminans, & qualiter ſit apparens illic, & quæ cauſſa neceſſariò faciat ipſum, ad il-<lb/>lud præmittemus propoſitiones, exponentes illud, cuius uolumus declarationem.</s> <s xml:id="echoid-s19676" xml:space="preserve"> Ex illo quidem <lb/>eſt, quòd ſphæra orbis [è terra & aqua conſtantis] tota ſemper eſt ſplendida & luminoſa ex lumi-<lb/>nari maiori (quod eſt ſol) niſi quantum obtegunt tenebræ contingẽtes ex terra, in figura pyrami-<lb/>dis, quæ eſt nox.</s> <s xml:id="echoid-s19677" xml:space="preserve"> Et ego non ſignifico in hoc libro per illud, quod accidit de huiuſmodi receptionè <lb/>luminis ex ſphæris ſtellarum, niſi quòd cum ſphæra, propter claritatem aeris & ſubtilitatem æthe-<lb/>ris, & tenuitatem eius non ſuſpenditur aliquid de lumine ſolis, ſicut uidemus ipſum ſuſpendi cum <lb/>corporibus altis (quę ſunt ſtellæ) quia illuminantur & deferunt nobis illud, quod recipiunt exlu-<lb/>mine, & conſequuntur ipſum uiſus noſtri in eis:</s> <s xml:id="echoid-s19678" xml:space="preserve"> & quamuis diſſentiant in ſtellis, in lumine tamen <lb/>non diſſentiunt.</s> <s xml:id="echoid-s19679" xml:space="preserve"> Viſus autem noſtri non conſequũtur, quod in eis eſt de luminibus:</s> <s xml:id="echoid-s19680" xml:space="preserve"> niſi quòd ipſæ <lb/>procul dubio ſunt ſpiſsioris & uehementioris corporeitatis;</s> <s xml:id="echoid-s19681" xml:space="preserve"> quàm æther, in quo ſunt.</s> <s xml:id="echoid-s19682" xml:space="preserve"> Et hoc patet <lb/>per ſignificationes, quòd quædam earum tegunt nobis quaſdam, quia eclipſant eas:</s> <s xml:id="echoid-s19683" xml:space="preserve"> aer uerò non <lb/>tegit nobis aliquid ex eis, quæ ſunt poſt ipſum.</s> <s xml:id="echoid-s19684" xml:space="preserve"> Et propterea uidemus, quòd tota nox eſt ſecundum <lb/>habitudinem unam, in qua non illuminatur nobis ex æthere aliquid:</s> <s xml:id="echoid-s19685" xml:space="preserve"> quamuis ſciamus ſecundum <lb/>ſcientiam noſtram, quòd quàmplurimum eius ætheris eſt luminoſum, non tectum à ſole.</s> <s xml:id="echoid-s19686" xml:space="preserve"> Et uide-<lb/>mus quòd illud, quod ex eo ſoli apparet, & nihil aliud tegit, eſt in uiſione, ſicut illud, quod terra te-<lb/>git, quod pyramis tenebrarum continet.</s> <s xml:id="echoid-s19687" xml:space="preserve"> Et non facit neceſſariò æqualitatem utriuſq;</s> <s xml:id="echoid-s19688" xml:space="preserve"> apud uiſus <lb/>noſtros, niſi illud, quod diximus de ſubtilitate aeris, & quòd non perducit illuminationem eius, & <lb/>perducit nobis tenebroſitatem ipſius.</s> <s xml:id="echoid-s19689" xml:space="preserve"> Tunc autem non ceſſat habitudo umbræ apparere nobis ſe-<lb/>cundum ſimilitudinem ipſius, quouſq;</s> <s xml:id="echoid-s19690" xml:space="preserve"> incipiat ab oriente ſplendor diluculi & lumen ſparſum, cu-<lb/>ius principium eſt in primis cum ſuperficie horizontis:</s> <s xml:id="echoid-s19691" xml:space="preserve"> & illius principij non eſt nobis cauffa, niſi <lb/>ſol:</s> <s xml:id="echoid-s19692" xml:space="preserve"> cum ſit cauſſa illuminationũ.</s> <s xml:id="echoid-s19693" xml:space="preserve"> Et non eſt nobis principium illud ſol ipſe, nec radius eius tantùm, <lb/>quoniam iam præmiſimus, quòd radij eius pertranſeunt uſq;</s> <s xml:id="echoid-s19694" xml:space="preserve"> ad ætherem totum, quem uidemus, <lb/>aut ad plurimum eius:</s> <s xml:id="echoid-s19695" xml:space="preserve"> & nõ eſt diuerſa eius habitudo in illa hora ab alia habitudine ante illud.</s> <s xml:id="echoid-s19696" xml:space="preserve"> Ve-<lb/>runtamen radij eius ſuſpenduntur tunc cum aliquo corpore ſpiſsiore aere:</s> <s xml:id="echoid-s19697" xml:space="preserve"> ducit ergo nobis cum <lb/>ſua ſpiſsitudine radium, quem induit.</s> <s xml:id="echoid-s19698" xml:space="preserve"> Et dico, quòd illud, quo ſuſpenſus eſt radius in illa hora, non <lb/>eſt terra, neq;</s> <s xml:id="echoid-s19699" xml:space="preserve"> extremitates plagarum eius diſtinctæ à nobis:</s> <s xml:id="echoid-s19700" xml:space="preserve"> quoniam cum uidens eſt ſuper æqua-<lb/>litatem terræ, non peruenit eius uiſus, niſi quaſi ad 23 milliaria [Italica] ab omni parte.</s> <s xml:id="echoid-s19701" xml:space="preserve"> Et ſi acci-<lb/>dit ei, ut ſit ſuper altiorem montium, qui eſſe poteſt (& ille non pertranſit octo milliaria, ſecundum <lb/>quod dixerunt ſapientes, intendentes hoc) uiſus non pertranſit tunc, niſi 250 milliaria ferè.</s> <s xml:id="echoid-s19702" xml:space="preserve"> Et hoc <lb/>manifeſtum eſt ex eo, quòd noctẽ facit forma terræ:</s> <s xml:id="echoid-s19703" xml:space="preserve"> ſed altitudo loci uiſus à ſuperficie eius, hoc eſt <lb/>ſpatium, quod diximus, abſcondit orbẽ in quarta horæ.</s> <s xml:id="echoid-s19704" xml:space="preserve"> Oportet ergo, ut oriatur ſol paululùm poſt <lb/>crepuſculum matutinum per quartam horæ ad minus:</s> <s xml:id="echoid-s19705" xml:space="preserve"> illud ergo, quod eſt inter apparitionem cre-<lb/>puſculi & apparitionẽ ſolis, eſt plus hora multò.</s> <s xml:id="echoid-s19706" xml:space="preserve"> Hoc autẽ, quod diximus, nõ eſt, niſi propinquitas, <lb/>propter eũ, qui non eſt exercitatus in geometricis.</s> <s xml:id="echoid-s19707" xml:space="preserve"> In ueritate uerò uiſus nõ peruenit ad punctum <lb/>terrę, quod iã illuminatũ eſt à ſole, niſi cũ ipſe peruenerit & cõprehẽderit cornu ipſius ſolis:</s> <s xml:id="echoid-s19708" xml:space="preserve"> quoniã <lb/>duæ lineæ contingẽtes unũ punctũ circuli à duabus partib.</s> <s xml:id="echoid-s19709" xml:space="preserve"> diuerſis cõiunctæ, ſunt linea una ſecun-<lb/>dũ rectitudinẽ [ք 14 p 1:</s> <s xml:id="echoid-s19710" xml:space="preserve"> quia ſemidiameter circuli ad tactus punctũ ducta, efficiet cũ utraq;</s> <s xml:id="echoid-s19711" xml:space="preserve"> angu-<lb/>los rectos ք 18 p 3.</s> <s xml:id="echoid-s19712" xml:space="preserve">] Quãdo ergo illuminatũ apparet nobis, tũ non eſt illud terra ipſa, ꝓpter id, quod <lb/> <pb o="284" file="0290" n="290" rhead="ALHAZEN"/> diximus:</s> <s xml:id="echoid-s19713" xml:space="preserve"> nec eſt aer implens totam ſphęram:</s> <s xml:id="echoid-s19714" xml:space="preserve"> quoniam, ut præmiſimus, ſuper totum aerem aut plu-<lb/>rimum eius, ſemper cadit radius ſolis nocte & die:</s> <s xml:id="echoid-s19715" xml:space="preserve"> & nõ apparet illud in ipſo, propter ipſius ſubtili-<lb/>tatem.</s> <s xml:id="echoid-s19716" xml:space="preserve"> Et ſuper terram non eſt corpus ſpiſsius aere, niſi uapores aſcendẽtes, quibus non deeſt ſem-<lb/>per, quin illuminentur à ſole.</s> <s xml:id="echoid-s19717" xml:space="preserve"> Tunc uerò, quando pyramis umbræ ab eo remouetur, quod de uapo-<lb/>rum ſphæra terram continente uiſus noſtri conſequuntur, & recipit eos corpus ſolis, & cadunt ſu-<lb/>per eos radij eius, ſuſpenditur cum eo radius, & defert ipſum nobis, & conſequuntur ipſum uiſus <lb/>noſtri, & uidetur à nobis eius lumen, ſicut uidemus ipſum apparere in nubibus ex coloratione hu-<lb/>miditatum aſcen dentiũ, & ſicut colores, qui in roribus uidentur, in forma portionis circuli, & alio-<lb/>rum modorum.</s> <s xml:id="echoid-s19718" xml:space="preserve"> Quãdo ergo uolumus ſcire, quanta ſit ultima eleuatio illorũ uaporum à ſuperficie <lb/>terræ:</s> <s xml:id="echoid-s19719" xml:space="preserve"> tunc ad eam cognitionem præmittũtur quatuor res, quarum nulla excuſatur, & præter ipſas <lb/>nulla alia re indigemus, ita ut nõ poſsit fieri per minus, nec ſit neceſſarium plus.</s> <s xml:id="echoid-s19720" xml:space="preserve"> Illa autem quatuor <lb/>ſunt:</s> <s xml:id="echoid-s19721" xml:space="preserve"> corpus terræ:</s> <s xml:id="echoid-s19722" xml:space="preserve"> corpus ſolis:</s> <s xml:id="echoid-s19723" xml:space="preserve"> longitudo centri ſolis à centro terræ in omni ſitu:</s> <s xml:id="echoid-s19724" xml:space="preserve"> & quanta ſit de-<lb/>preſsio ſolis ab horizonte, donec appareat crepuſculum matutinum.</s> <s xml:id="echoid-s19725" xml:space="preserve"> Corpus autem terræ eſt ſicut <lb/>inſtrumentum omnium aliorum:</s> <s xml:id="echoid-s19726" xml:space="preserve"> & quantitas circuli magni continentis eam, ſecũdum quod dixe-<lb/>runt ſapientes, & ſignificauerunt illud per propoſitiones certas, eſt 24000 milliaria.</s> <s xml:id="echoid-s19727" xml:space="preserve"> Et dixerunt, <lb/>quòd per quãtitatem, qua medietas diametri terræ eſt pars una, eſt medietas diametri ſolis quinq;</s> <s xml:id="echoid-s19728" xml:space="preserve"> <lb/>partes, & medietas partis:</s> <s xml:id="echoid-s19729" xml:space="preserve"> & per eam eſt longitudo centri ſolis à cẽtro terræ in longitudine media, <lb/>(non in omni ſitu) mille & centum & circiter decem partes:</s> <s xml:id="echoid-s19730" xml:space="preserve"> & quòd depreſsio ſolis ab horizonte, <lb/>cum oritur crepuſculum, eſt 18 gradus:</s> <s xml:id="echoid-s19731" xml:space="preserve"> & iã inuenitur ſuper 19:</s> <s xml:id="echoid-s19732" xml:space="preserve"> & ſuper hoc fabricabo ſupputatio-<lb/>nem noſtram:</s> <s xml:id="echoid-s19733" xml:space="preserve"> quoniã cum narrator rei eſt cũ additione in ea, dignior eſt, ut recipiatur ſermo eius, <lb/>cum non contradicit ei alius:</s> <s xml:id="echoid-s19734" xml:space="preserve"> quandoquidem narrator cũ additione ſcit, quod non ſcit alius, & con <lb/>ſequitur, quod non conſequitur alius.</s> <s xml:id="echoid-s19735" xml:space="preserve"> Nã qui narrat de aliquo, quod uiderit illud, antequam uiderit <lb/>ipſum alius, dignior eſt, ut conſequatur, quod intendit, quando nõ exiſtimatur de eo ſuſpicio.</s> <s xml:id="echoid-s19736" xml:space="preserve"> Præ-<lb/>mittam igitur ad illud, quod inter manus meas eſt, propoſitiones quaſdam multi iuuaminis.</s> <s xml:id="echoid-s19737" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div636" type="section" level="0" n="0"> <head xml:id="echoid-head548" xml:space="preserve" style="it">2. Si ſphæricũ luminoſum illuminet opacum æquale: hemiſphæriũ illuminabit. Vitell. 26 p 2.</head> <p> <s xml:id="echoid-s19738" xml:space="preserve">DIco ergo, quòd omnium duarum ſphærarum æqualium, inter <lb/> <anchor type="figure" xlink:label="fig-0290-01a" xlink:href="fig-0290-01"/> quas non eſt aliud corpus, quod unam earum alteri abſcondat:</s> <s xml:id="echoid-s19739" xml:space="preserve"> <lb/>illud, quod ex unaquaq;</s> <s xml:id="echoid-s19740" xml:space="preserve"> earum uerſa facie reſpicit alteram, eſt <lb/>medietas eius æqualiter.</s> <s xml:id="echoid-s19741" xml:space="preserve"> Et ſignifico per uerſam faciẽ unius reſpectu <lb/>alterius:</s> <s xml:id="echoid-s19742" xml:space="preserve"> quòd ſi una earum eſt luminoſa, & altera recipiẽs lumen, illu-<lb/>minatur, & relucet medietas recipientis lumen.</s> <s xml:id="echoid-s19743" xml:space="preserve"> Cuius exemplum eſt, <lb/>ut ſint duæ ſphæræ a & b æquales:</s> <s xml:id="echoid-s19744" xml:space="preserve"> & pono, ut aliqua ſuperficies plana <lb/>tranſeat per centrũ utriuſq;</s> <s xml:id="echoid-s19745" xml:space="preserve">: ſecabit ergo duas ſphæras ſuper duos cir-<lb/>culos æquales, & in ſuperficie una [per 1th.</s> <s xml:id="echoid-s19746" xml:space="preserve"> 1 ſphær.</s> <s xml:id="echoid-s19747" xml:space="preserve"> Theodoſij.</s> <s xml:id="echoid-s19748" xml:space="preserve">] Sint <lb/>ergo illi duo circuli a g h, b d c:</s> <s xml:id="echoid-s19749" xml:space="preserve"> & cõtinuabo a cum b:</s> <s xml:id="echoid-s19750" xml:space="preserve"> & protrahã duas <lb/>lineas a g, b d perpendiculares ſuper lineam a b:</s> <s xml:id="echoid-s19751" xml:space="preserve"> [per 11 p 1] ergo ipſæ <lb/>ſunt æquidiſtantes [per 28 p 1] & continuabo g cum d.</s> <s xml:id="echoid-s19752" xml:space="preserve"> Et quoniã duæ <lb/>lineæ a g, b d ſunt ęquales [per 15 d 1:</s> <s xml:id="echoid-s19753" xml:space="preserve"> quia ſunt ſemidiametri ęqualium <lb/>circulorum] & æquidiſtantes [è cõcluſo] duæ lineæ a b, g d ſimiliter <lb/>erunt æquales & æquidiſtantes:</s> <s xml:id="echoid-s19754" xml:space="preserve"> [per 33 p 1] ergo duo] anguli ad g & d <lb/>ſunt recti:</s> <s xml:id="echoid-s19755" xml:space="preserve"> [per ſecundam partem 34 p 1] ergo linea g d eſt contingens <lb/>duos circulos [per conſectarium 16 p 3.</s> <s xml:id="echoid-s19756" xml:space="preserve">] Et quando nos protrahemus <lb/>g a & b d ſecundum rectitudinem, ad duas circumferentias duorũ cir-<lb/>culorum, uſq;</s> <s xml:id="echoid-s19757" xml:space="preserve"> ad duo puncta e & z, deinde cõtinuabimus e cum z:</s> <s xml:id="echoid-s19758" xml:space="preserve"> erit <lb/>recta linea e z contingens duos circulos [ijſdem de cauſsis, quibus d g <lb/>tangere oſtenſa eſt:</s> <s xml:id="echoid-s19759" xml:space="preserve">] & erit una quæq;</s> <s xml:id="echoid-s19760" xml:space="preserve"> duarum portionum g h e, d c z, <lb/>quarum una eſt uerſa facie ad alterã, medietas circuli [per 17 d 1] quo-<lb/>niam unam quamq;</s> <s xml:id="echoid-s19761" xml:space="preserve"> earum fecat diameter circuli.</s> <s xml:id="echoid-s19762" xml:space="preserve"> Et ſimiliter cõtingit <lb/>in omnibus ſuperficiebus planis, quæ tranſeunt per duo centra duarũ <lb/>ſphærarum.</s> <s xml:id="echoid-s19763" xml:space="preserve"> Iam igitur declaratum eſt, quòd lineæ egredientes ex una duarum ſphærarum ad alte-<lb/>ram, contingunt utraſq;</s> <s xml:id="echoid-s19764" xml:space="preserve"> ſimul, & comprehendunt ex unaquaque earum medietatem.</s> <s xml:id="echoid-s19765" xml:space="preserve"> Et illud eſt, <lb/>quod declarare uoluimus.</s> <s xml:id="echoid-s19766" xml:space="preserve"/> </p> <div xml:id="echoid-div636" type="float" level="0" n="0"> <figure xlink:label="fig-0290-01" xlink:href="fig-0290-01a"> <variables xml:id="echoid-variables236" xml:space="preserve">g a e h c d b z</variables> </figure> </div> </div> <div xml:id="echoid-div638" type="section" level="0" n="0"> <head xml:id="echoid-head549" xml:space="preserve" style="it">3. Si ſphæricum luminoſum illuminet opacum min{us}: pl{us} hemiſphærio illuminabit. Vi-<lb/>tell. 27 p 2.</head> <p> <s xml:id="echoid-s19767" xml:space="preserve">QVòd ſi una duarum ſphærarum eſt maior altera:</s> <s xml:id="echoid-s19768" xml:space="preserve"> tũc illud, quod ex minore uerſa facie reſpi-<lb/>cit maiorem, eſt plus medietate minoris:</s> <s xml:id="echoid-s19769" xml:space="preserve"> & quod ex maiore uerſa facie reſpicit minorem, <lb/>eſt minus medietate maioris.</s> <s xml:id="echoid-s19770" xml:space="preserve"> Cuius exemplum eſt, ut ſint duæ ſphæræ a & b:</s> <s xml:id="echoid-s19771" xml:space="preserve"> & ſphæra a ſit <lb/>maior.</s> <s xml:id="echoid-s19772" xml:space="preserve"> Protrahã ergo ſuperficiẽ planã, tranſeuntẽ per cẽtra utriuſq;</s> <s xml:id="echoid-s19773" xml:space="preserve">: ſecabit ergo utrãq;</s> <s xml:id="echoid-s19774" xml:space="preserve"> earũ in duo <lb/>media ſuք duos circulos a g d, b e z [per 1 the.</s> <s xml:id="echoid-s19775" xml:space="preserve"> 1 ſphęr.</s> <s xml:id="echoid-s19776" xml:space="preserve">] & cõtinuabo a cũ b, & protrahã ipſam ſecũdũ <lb/>rectitudinẽ in partẽ h:</s> <s xml:id="echoid-s19777" xml:space="preserve"> & ponã proportionẽ medietatis diametri circuli a g d ad medietatẽ diametri <lb/>circuli b e z, ſicut ꝓportio a h ad b h.</s> <s xml:id="echoid-s19778" xml:space="preserve"> Eius uerò acceptio eſt prõpta ex tractatu ſexto & ꝗnto Eucli-<lb/>dis [ſi enim trib.</s> <s xml:id="echoid-s19779" xml:space="preserve"> rectis datis, differẽtia nẽpe ſemidiametrorũ circulorũ a & b:</s> <s xml:id="echoid-s19780" xml:space="preserve"> ſemidiametro b c mi-<lb/>noris circuli, & ipſa a b, inueniatur ք 12 p 6 quarta ꝓportionalis b h:</s> <s xml:id="echoid-s19781" xml:space="preserve"> erit ք 18 p 5 ut a d ſemidiameter <lb/> <pb o="285" file="0291" n="291" rhead="DE CREPVSCVLIS LIBER."/> maioris circuli ad b c ſemidiametrum minoris b c:</s> <s xml:id="echoid-s19782" xml:space="preserve"> ſic a h ad b h.</s> <s xml:id="echoid-s19783" xml:space="preserve">] Et protraham à puncto h lineam <lb/>contingẽtem circulũ a g d [per 17 p 3] quæ ſit h c d.</s> <s xml:id="echoid-s19784" xml:space="preserve"> Dico ergo, quòd ipſa contingit etiã circulũ b e z:</s> <s xml:id="echoid-s19785" xml:space="preserve"> <lb/>quod patet:</s> <s xml:id="echoid-s19786" xml:space="preserve"> quia cõtinuabo a cum d per lineam a d:</s> <s xml:id="echoid-s19787" xml:space="preserve"> ergo eſt perpendi-<lb/> <anchor type="figure" xlink:label="fig-0291-01a" xlink:href="fig-0291-01"/> cularis ſuper lineam h d [per 18 p 3] & protraham à puncto b perpen-<lb/>dicularem ſuper lineam h c d [per 11 p 1] quæ ſit b c.</s> <s xml:id="echoid-s19788" xml:space="preserve"> Et quoniam duæ <lb/>lineæ b c, a d ſunt perpendiculares ſuper lineam h d [è fabricatione & <lb/>concluſo] ſunt æ quidiſtantes [per 28 p 1.</s> <s xml:id="echoid-s19789" xml:space="preserve">] Et quia linea b c eſt æqui-<lb/>diſtans ipſi a d, quæ eſt baſis trianguli:</s> <s xml:id="echoid-s19790" xml:space="preserve"> erit ergo proportio a d ad b c, <lb/>ſicut ꝓportio a h ad h b [per 4 p 6:</s> <s xml:id="echoid-s19791" xml:space="preserve"> quia triangula a h d, b h c ſunt æqui-<lb/>angula per 29.</s> <s xml:id="echoid-s19792" xml:space="preserve"> 32 p 1] & iam poſuimus proportionem a h ad h b, ſicut <lb/>proportionem medietatis diametri circuli a g d, ad medietatẽ diame-<lb/>tri b e z:</s> <s xml:id="echoid-s19793" xml:space="preserve"> ergo linea b c eſt medietas diametri circuli b e z:</s> <s xml:id="echoid-s19794" xml:space="preserve"> ergo punctũ <lb/>c eſt ſuper circumferẽtiam circuli b e z [per 17 d 1] & duos angulos ad <lb/>d & c poſuimus rectos:</s> <s xml:id="echoid-s19795" xml:space="preserve"> ergo linea h c d contingit minorem circulum <lb/>[per conſectarium 16 p 3] nos uerò iam protraximus eam contingen-<lb/>tem maiorẽ:</s> <s xml:id="echoid-s19796" xml:space="preserve"> ergo ipſa eſt contingens utroſq;</s> <s xml:id="echoid-s19797" xml:space="preserve"> ſimul.</s> <s xml:id="echoid-s19798" xml:space="preserve"> Et protraham ſimi <lb/>liter ex puncto h lineam, contingentem duos circulos ſimiliter in par-<lb/>te z, quæ ſit linea h z k.</s> <s xml:id="echoid-s19799" xml:space="preserve"> Eſt ergo, quòd ex circulo a maiore uerſa facie <lb/>reſpicit circulum b minorem, portio d g k:</s> <s xml:id="echoid-s19800" xml:space="preserve"> & eſt minor medietate cir-<lb/>culi:</s> <s xml:id="echoid-s19801" xml:space="preserve"> quoniam angulus h a d eſt minor recto [per 32 p 1] quoniam ipſe <lb/>eſt in trian gulo uno, & eſt triangulum d a h cum angulo a d h recto.</s> <s xml:id="echoid-s19802" xml:space="preserve"> <lb/>Ergo eſt portio d g minor quarta circuli [per 33 p 6] & ſimiliter por-<lb/>tio g k, æqualis e [quòd autem g k ſit æqualis d g, patet, ducta ſemidia-<lb/>metro a k.</s> <s xml:id="echoid-s19803" xml:space="preserve"> Quia enim rectæ d h, k h tangentes æquantur per conſecta-<lb/>rium 36 p 3 & ſemidiametri a d, a k per 15 d 1, eſtq́;</s> <s xml:id="echoid-s19804" xml:space="preserve"> communis a h:</s> <s xml:id="echoid-s19805" xml:space="preserve"> ęqua-<lb/>bitur angulus h a d angulo h a k per 8 p 1:</s> <s xml:id="echoid-s19806" xml:space="preserve"> quare per 26 p 3 peripheria d <lb/>g æquabitur peripheriæ g k.</s> <s xml:id="echoid-s19807" xml:space="preserve">] Ergo portio d g k eſt minor medietate <lb/>circuli.</s> <s xml:id="echoid-s19808" xml:space="preserve"> Et quoniam linea b c eſt æquidiſtans lineæ a d [è concluſo] eſt angulus c b h æqualis an-<lb/>gulo d a h [per 29 p 1] ergo erit portio c l ſimilis portioni d g, & tota portio c l z ſimilis portioni d g <lb/>k [per 33 p 6.</s> <s xml:id="echoid-s19809" xml:space="preserve">] Ergo unaquęq;</s> <s xml:id="echoid-s19810" xml:space="preserve"> earũ eſt minor medietate circuli:</s> <s xml:id="echoid-s19811" xml:space="preserve"> remanet ergo portio c e z maior me <lb/>dietate circuli:</s> <s xml:id="echoid-s19812" xml:space="preserve"> & illud eſt, quod ex circulo minore uerſa facie reſpicit circulum maiorem.</s> <s xml:id="echoid-s19813" xml:space="preserve"> Ergo duę <lb/>portiones c e z, & d g k ſunt ex duobus circulis, qui uerſa facie ſe reſpiciũt.</s> <s xml:id="echoid-s19814" xml:space="preserve"> Et ſignifico quidem per <lb/>hoc, quòd aliquid portionis unius nõ cooperitur ex circulo altero:</s> <s xml:id="echoid-s19815" xml:space="preserve"> & portio c e z eſt maior medie-<lb/>tate circuli, & portio d g k minor.</s> <s xml:id="echoid-s19816" xml:space="preserve"> Etillud eſt, quod uoluimus declarare.</s> <s xml:id="echoid-s19817" xml:space="preserve"/> </p> <div xml:id="echoid-div638" type="float" level="0" n="0"> <figure xlink:label="fig-0291-01" xlink:href="fig-0291-01a"> <variables xml:id="echoid-variables237" xml:space="preserve">d a k g e c b z h</variables> </figure> </div> </div> <div xml:id="echoid-div640" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables238" xml:space="preserve">e d a n b g m q t k z h l</variables> </figure> <head xml:id="echoid-head550" xml:space="preserve" style="it">4. Si peripheri{as} duorum circulorum æqualium duæ rectæ lιneæ tangant: punct a ſemiperi-<lb/>pheriarum cõuexis partib{us} ſe reſpicientium ſingula ſingulis appa-<lb/> rent, reliquarum uerò ſemiperipheriarum conuexis partib{us} ſenon reſpicientium latent.</head> <p> <s xml:id="echoid-s19818" xml:space="preserve">ET dico, quòd quando ſunt duo circuli æquales, & protrahuntur <lb/>duæ lineæ, quarum unaquæq;</s> <s xml:id="echoid-s19819" xml:space="preserve"> contingit duos circulos ſimul, ſe-<lb/>cundum formam, quam præmiſimus:</s> <s xml:id="echoid-s19820" xml:space="preserve"> tunc in unaquaq;</s> <s xml:id="echoid-s19821" xml:space="preserve"> duarum <lb/>portionum, quarum una uerſa facie reſpicit alteram, non eſt locus, qui <lb/>abſcõdat aliquid ex circulo uno circulo alteri:</s> <s xml:id="echoid-s19822" xml:space="preserve"> & quòd in reliquis dua-<lb/>bus portionibus duorum circulorum, quę non facie ad faciem ſe reſpi-<lb/>ciunt, non eſt locus, qui appareat circulo alteri.</s> <s xml:id="echoid-s19823" xml:space="preserve"> Cuius exemplum eſt, <lb/>quòd ſint duo circuli a b g d e, & z h t k l:</s> <s xml:id="echoid-s19824" xml:space="preserve"> & protrahantur duę lineæ b h, <lb/>& d k contingentes duos circulos ſimul:</s> <s xml:id="echoid-s19825" xml:space="preserve"> ergo duæ portiones b g d, & <lb/>h t k ſunt, quæ ſe facie ad faciem reſpiciunt:</s> <s xml:id="echoid-s19826" xml:space="preserve"> earum portiones b e d, & h <lb/>l k ſunt, quæ ſe non facie ad faciem reſpiciunt.</s> <s xml:id="echoid-s19827" xml:space="preserve"> Dico ergo, quòd non eſt <lb/>in portione b g d punctum, quod aliquid ex circulo z h abſcondat cir-<lb/>culo a b:</s> <s xml:id="echoid-s19828" xml:space="preserve"> & quòd non eſt in portione b e d punctum, quod appareat pe-<lb/>nitus circulo z h:</s> <s xml:id="echoid-s19829" xml:space="preserve"> & quòd tota ipſa portio eſt abſcondita circulo z h:</s> <s xml:id="echoid-s19830" xml:space="preserve"> & <lb/>quòd neq;</s> <s xml:id="echoid-s19831" xml:space="preserve"> eſt in portione h l k punctũ, quod appareat circulo a b.</s> <s xml:id="echoid-s19832" xml:space="preserve"> Cu-<lb/>ius demonſtratio eſt:</s> <s xml:id="echoid-s19833" xml:space="preserve"> quòd ego continuabo a cum z, per lineam a g z, <lb/>& ſignabo ſuper arcum b g d punctum, qualiter uelim, quod ſit punctũ <lb/>m.</s> <s xml:id="echoid-s19834" xml:space="preserve"> Si ergo fuerit punctum m à puncto g ad partem b:</s> <s xml:id="echoid-s19835" xml:space="preserve"> tunc protraham <lb/>ex puncto m lineã æquidiſtantem lineæ b h [per 31 p 1] & ſi fuerit pun-<lb/>ctum m à puncto g ad partem d:</s> <s xml:id="echoid-s19836" xml:space="preserve"> tunc protraham ex puncto m lineam <lb/>æquidiſtãtem lineæ d k:</s> <s xml:id="echoid-s19837" xml:space="preserve"> ſit ergo m t.</s> <s xml:id="echoid-s19838" xml:space="preserve"> Dico igitur quòd linea m t tota eſt <lb/>extra circulũ b m g d e, de qua nõ cadit aliquid in eo.</s> <s xml:id="echoid-s19839" xml:space="preserve"> Cuius demõſtratio eſt:</s> <s xml:id="echoid-s19840" xml:space="preserve"> quòd ego cõtinuabo a <lb/>cũ b, & protrahã lineã m t ſecundũ rectitudinẽ, donec cõcurrat cũ linea b a ſuper punctũ n [cõcur-<lb/>ret aũt per lẽma Procli ad 29 p1:</s> <s xml:id="echoid-s19841" xml:space="preserve"> ꝗa m t parallela ducta eſt ipſi b h, quę cõcurrit cũ a b in b] ergo duo <lb/>rũ angulorũ ad n unuſquiſq;</s> <s xml:id="echoid-s19842" xml:space="preserve"> eſt rectus [ꝗa enim angulus n b h rectus eſt ք 18 p 3, & ipſi b h parallela <lb/>ducta eſt t m n:</s> <s xml:id="echoid-s19843" xml:space="preserve"> ęquabitur per 29 p 1 angulus t n b angulo n b h, ideoq́;</s> <s xml:id="echoid-s19844" xml:space="preserve"> rectus, & per 13 p 1 an t rectus] <lb/> <pb o="286" file="0292" n="292" rhead="ALHAZEN"/> & cõtinuabo m cũ a.</s> <s xml:id="echoid-s19845" xml:space="preserve"> Angulus igitur trianguli a n m eſt rectus:</s> <s xml:id="echoid-s19846" xml:space="preserve"> & iá protractú eſt latus n m ſecundú <lb/>rectitudiné uſq;</s> <s xml:id="echoid-s19847" xml:space="preserve"> ad t, & prouenit angulus a m t extra triangulũ, qui eſt maior recto [per 16 p 1] ſcili-<lb/>cet angulo n.</s> <s xml:id="echoid-s19848" xml:space="preserve"> Et quãdo protrahitur ab extremitate diametri circuli linea, quæ cũ ipla cõtineat plus <lb/>angulo recto:</s> <s xml:id="echoid-s19849" xml:space="preserve"> tũc illa linea nõ ſecat circulũ, nec cadit de ea intra ipſum aliquid:</s> <s xml:id="echoid-s19850" xml:space="preserve"> ergo de linea m t nó <lb/>cadit in circulo a m aliquid.</s> <s xml:id="echoid-s19851" xml:space="preserve"> Ergo punctũ m facie ad facié reſpicit circulũ z, & nõ abſcondit aliquid <lb/>ei:</s> <s xml:id="echoid-s19852" xml:space="preserve"> quoniã quando nõ abſcondit ei aliquid ex corpore iſtiuſmetſphæræ a m:</s> <s xml:id="echoid-s19853" xml:space="preserve"> tunc nulla alia res tegit <lb/>illud:</s> <s xml:id="echoid-s19854" xml:space="preserve"> quoniá nos poſuimus, ut inter duas ſphæras nõ ſit corpus aliud ab eis, quod tegat unam earũ <lb/>alteri.</s> <s xml:id="echoid-s19855" xml:space="preserve"> Et ſimiliter oſtẽdetur hoc in omni pũcto ſuք arcũ h t k.</s> <s xml:id="echoid-s19856" xml:space="preserve"> Et dico iterũ, quòd nõ eſt in arcu b e d <lb/>punctú, quod appareat circulo z:</s> <s xml:id="echoid-s19857" xml:space="preserve"> nec eſt poſsibile, ut continuetur cũ aliquo de circulo z p ք lineá, niſt <lb/>& illa linea ſecet circulũ a b, & cadat intra ipſum.</s> <s xml:id="echoid-s19858" xml:space="preserve"> Quod ſi poſsibile eſt:</s> <s xml:id="echoid-s19859" xml:space="preserve"> ꝓtrahamus à pũcto e lineam <lb/>peruenienté ad aliꝗ d de circũferentia circuli h t k l:</s> <s xml:id="echoid-s19860" xml:space="preserve"> & nó ſecet aliꝗd de circulo a e d:</s> <s xml:id="echoid-s19861" xml:space="preserve"> & ſi fuerit poſ-<lb/>ſibile, ſit linea e q l:</s> <s xml:id="echoid-s19862" xml:space="preserve"> & ꝓtrahá lineá d k in utraſq;</s> <s xml:id="echoid-s19863" xml:space="preserve"> partes duarũ extremitatũ eius:</s> <s xml:id="echoid-s19864" xml:space="preserve"> neceſſe eſt ergo, ut <lb/>occurrat lineæ e q l in duob.</s> <s xml:id="echoid-s19865" xml:space="preserve"> locis:</s> <s xml:id="echoid-s19866" xml:space="preserve"> quoniá linea d k, quá iá poſuimus contingenté duos circulos, nõ <lb/>eſt poſsibile, ut ſecet unũ duorũ circulorum, nec cadat inter utroſq;</s> <s xml:id="echoid-s19867" xml:space="preserve"> [per 16 p 3:</s> <s xml:id="echoid-s19868" xml:space="preserve">] & quoniã nó cadit <lb/>inter ipſos, tunc ſecabit lineam e l in duobus locis:</s> <s xml:id="echoid-s19869" xml:space="preserve"> ergo iam ſunt duæ lineæ rectæ continentes ſu-<lb/>perficiem:</s> <s xml:id="echoid-s19870" xml:space="preserve">illud autem eſt contrarium & impoſsibile [per 12 axioma.</s> <s xml:id="echoid-s19871" xml:space="preserve">]</s> </p> </div> <div xml:id="echoid-div641" type="section" level="0" n="0"> <head xml:id="echoid-head551" xml:space="preserve" style="it">5. Deperipheria maximi in terra circuli ſol illuminat partes 180, ſcrupula prima 27, ſcru-<lb/>pula ſecunda 52. Vitell. 59 p 10.</head> <p> <s xml:id="echoid-s19872" xml:space="preserve">QVod aũt oportet nos facere ſecundũ illud, quod pręmiſimus, ut inueniamus, quãta ſit quã-<lb/>titas arcus terræ illuminati à ſole:</s> <s xml:id="echoid-s19873" xml:space="preserve"> quã iã poſuimus maiorẽ eſſe medietate terræ:</s> <s xml:id="echoid-s19874" xml:space="preserve"> ponã ergo <lb/>duos circulos ſolis & terræ, ſuper quos ſecat utroſq;</s> <s xml:id="echoid-s19875" xml:space="preserve"> una ſuperficies plana, quales ſunt a b c <lb/>d e, f h g.</s> <s xml:id="echoid-s19876" xml:space="preserve"> Circulus ergo a ſit terræ, & circulus ſolis f:</s> <s xml:id="echoid-s19877" xml:space="preserve"> & protrahã duas lineas contingẽtes unũquenq;</s> <s xml:id="echoid-s19878" xml:space="preserve"> <lb/>eorũ, ſicut diximus, quæ ſint duæ lineæ b h & e g.</s> <s xml:id="echoid-s19879" xml:space="preserve"> Igitur portio b c d e exterra, eſt illuminata à ſole, <lb/>ſicut iam oſtendimus [3 n] & illud eſt plus me-<lb/> <anchor type="figure" xlink:label="fig-0292-01a" xlink:href="fig-0292-01"/> dietate circuli.</s> <s xml:id="echoid-s19880" xml:space="preserve"> Quando ergo uolumus ſcire <lb/>quantitatẽ eius, tũc nos cõtinuabimus a cum b <lb/>& cũ f, & f cũ h:</s> <s xml:id="echoid-s19881" xml:space="preserve"> ergo b a & h fſunt æquidiſtãtes <lb/>[per 28 p 1] quoniã utræq;</s> <s xml:id="echoid-s19882" xml:space="preserve"> ſunt perpẽdiculares <lb/>ſuper lineã b h, contingentẽ duos circulos [per <lb/>18 p 3.</s> <s xml:id="echoid-s19883" xml:space="preserve">] Et ſecabo ex linea h f, quod ſit æquale li-<lb/>neæ b a [id uerò fieri poteſt, quia f h ex theſi ma <lb/>ior eſt a b] & ſit linea h k:</s> <s xml:id="echoid-s19884" xml:space="preserve"> & continuabo a cũ k:</s> <s xml:id="echoid-s19885" xml:space="preserve"> <lb/>ergo a k eſt perpẽdicularis ſuper h f [per 29 p 1] <lb/>quoniã eſt æquidiſtãs ipſi b h:</s> <s xml:id="echoid-s19886" xml:space="preserve"> cũ cõtinuet totũ, <lb/>quod eſt inter extremitates duarũ linearũ b a, <lb/>& h k æqualiũ & æquidiſtantiũ:</s> <s xml:id="echoid-s19887" xml:space="preserve"> ergo angulus k <lb/>eſt rectus.</s> <s xml:id="echoid-s19888" xml:space="preserve"> Et ꝓpterea quòd linea h f eſt quinq;</s> <s xml:id="echoid-s19889" xml:space="preserve"> <lb/>partes & medιetas partis, ք quãtitatẽ, qua linea <lb/>b a eſt pars una [ut dictũ eſt 1 n] remanet linea <lb/>k f quatuor partium & medietatis unius partis <lb/>ex illa quãtitate:</s> <s xml:id="echoid-s19890" xml:space="preserve"> & per eandẽ inuenitur linea a <lb/>f 1110, in medijs lõgitudinibus [ſole cõſtituto.</s> <s xml:id="echoid-s19891" xml:space="preserve">] <lb/>Ergo per quantitatẽ, qua linea a f ſubtẽſa angu-<lb/>lo recto, eſt 60 grad.</s> <s xml:id="echoid-s19892" xml:space="preserve"> eſt linea k f 14 minuta & <lb/>tres quintæ unius minuti:</s> <s xml:id="echoid-s19893" xml:space="preserve"> ergo angulus k a f eſt <lb/>14 min.</s> <s xml:id="echoid-s19894" xml:space="preserve"> excepta tertia parte ꝗntæ partis unius <lb/>minuti, [id eſt 13 minu.</s> <s xml:id="echoid-s19895" xml:space="preserve"> & 56 ſec.</s> <s xml:id="echoid-s19896" xml:space="preserve"> Nam ſecũdum <lb/>pręcepta arithmetices quin cunx ſeu ꝗnta pars <lb/>unius minuti ſunt 12 ſecunda, quorũ tertia pars <lb/>per diuiſionẽ inuẽta, ſunt 4 ſecun.</s> <s xml:id="echoid-s19897" xml:space="preserve"> quibus ſub-<lb/>ductis à 14 minutis, rectã 13 minuta & 56 ſecun-<lb/>da] per quãtitatem, qua angulus rectus eſt 90 <lb/>grad.</s> <s xml:id="echoid-s19898" xml:space="preserve"> & illud eſt quãtitas arcus c d:</s> <s xml:id="echoid-s19899" xml:space="preserve"> ſed arcus b c <lb/>eſt 90 grad.</s> <s xml:id="echoid-s19900" xml:space="preserve"> quoniã angulus b a c eſt rectus.</s> <s xml:id="echoid-s19901" xml:space="preserve"> Er-<lb/>go arcus b d eſt 90 grad.</s> <s xml:id="echoid-s19902" xml:space="preserve"> 14 min.</s> <s xml:id="echoid-s19903" xml:space="preserve"> excepta tertia <lb/>parte quintæ partis unius minuti:</s> <s xml:id="echoid-s19904" xml:space="preserve"> & arcus d e <lb/>eſt ęqualis arcui b d.</s> <s xml:id="echoid-s19905" xml:space="preserve"> [Ducta enim à pũcto a pa-<lb/>rallela ip̀ſi e g:</s> <s xml:id="echoid-s19906" xml:space="preserve"> erit angulus à ſemidiametro e a & parallela cõprehenſus, rectus per 29 p 1, & æqualis <lb/>angulo b a c per 10 ax.</s> <s xml:id="echoid-s19907" xml:space="preserve"> Et quia ducta parallela ſecat de ſemidiametro f g uerſus f æqualẽ ipſi f k ք 15 <lb/>d.</s> <s xml:id="echoid-s19908" xml:space="preserve"> 34 p 1.</s> <s xml:id="echoid-s19909" xml:space="preserve"> 1 ax:</s> <s xml:id="echoid-s19910" xml:space="preserve"> & angulus à parallela & ſemidiametro f g cõprehẽſus, rectus eſt per 29 uel 34 p 1:</s> <s xml:id="echoid-s19911" xml:space="preserve"> ęqua-<lb/>buntur quadrata parallelæ & ſectæ de ſemidiametro f g uerſus f, quadrato f a per 47 p 1, cui per ean-<lb/>dem æquantur quadrata ipſarũ a k & k f:</s> <s xml:id="echoid-s19912" xml:space="preserve"> ſubductis igitur quadratis æqualibus ipſarũ f k & ſectæ d e <lb/>ſemidiametro f g uerſus f, relinquẽtur quadrata ipſarũ a k & ductæ parallelæ æqualia, ideoq́;</s> <s xml:id="echoid-s19913" xml:space="preserve"> recta <lb/>a k æqualis erit ductæ parallelæ:</s> <s xml:id="echoid-s19914" xml:space="preserve"> & per 8 p 1 angulus d a c æquabitur angulo ab f a & parallela ad cẽ-<lb/>trum a cõprehenſo.</s> <s xml:id="echoid-s19915" xml:space="preserve"> ſed angulo c ab æqualis cõcluſus eſt angulus à ſemidiametro e a & parallela cõ-<lb/> <pb o="287" file="0293" n="293" rhead="DE CREPVSCVLIS LIBER."/> prehenſus.</s> <s xml:id="echoid-s19916" xml:space="preserve"> Quare ſi æqualibus angulis æquales addãtur, æquabitur per 2 axio:</s> <s xml:id="echoid-s19917" xml:space="preserve"> totus angulus b a d <lb/>toti angulo e a d:</s> <s xml:id="echoid-s19918" xml:space="preserve"> & per 26 p 3 peripheria b d peripheriæ e d.</s> <s xml:id="echoid-s19919" xml:space="preserve">] Ergo totus arcus b c d e illuminatus à <lb/>ſole, eſt 180 partes & 27 minuta & quatuorquintæ & tertia quintæ unius minuti cũ propinquitate <lb/>[id eſt 52 ſecunda:</s> <s xml:id="echoid-s19920" xml:space="preserve"> nã ex arithmeticæ regulis {4/5} unius minuti ſunt 48 ſerupula ſecũda, & quinta pars <lb/>unius minuti ſunt 12 ſcrupula ſecunda, quorũ tertia pars, 4 ſcilicet ſcrupula ſecunda addιta cum 48 <lb/>ſcrupulis ſecundis, efficiunt 52 ſcrupulà ſecunda.</s> <s xml:id="echoid-s19921" xml:space="preserve">] Et illud eſt, quod uoluimus declarare.</s> <s xml:id="echoid-s19922" xml:space="preserve"/> </p> <div xml:id="echoid-div641" type="float" level="0" n="0"> <figure xlink:label="fig-0292-01" xlink:href="fig-0292-01a"> <variables xml:id="echoid-variables239" xml:space="preserve">f g k h d c e a b</variables> </figure> </div> </div> <div xml:id="echoid-div643" type="section" level="0" n="0"> <head xml:id="echoid-head552" xml:space="preserve" style="it">6. Poſit a peripheria maximi in terra circuli 2 4000 milliarium Italicorum: erit ſumma ua-<lb/>porum in nubem coactorum à terra altιtudo 5 2000 paſſuum. Vitell. 60 p 10.</head> <p> <s xml:id="echoid-s19923" xml:space="preserve">INcipiamus ergo nũc ex eo, quod intẽdimus de cauſſa apparition is crepuſculi, & formæ appari-<lb/>tionis eius nobis, & figurationis ipſius in horizonte oriẽtali.</s> <s xml:id="echoid-s19924" xml:space="preserve"> Ponam ergo circulũ ſignatum ſu-<lb/>per ſphærã terræ, & ſuper quã abſcindit terrã ſuperficies plana, trãſiens per zenιth capitũ & per <lb/><gap/> centrũ terræ & ſolis circulũ a b, & locũ uiſus a:</s> <s xml:id="echoid-s19925" xml:space="preserve"> & faciã trãſire ſuper punctũ a lineam contingentẽ <lb/>circulũ [per 17 p 3] & prolongabo duas extremitates eius in duas partes, ſuper quas ſint d, e.</s> <s xml:id="echoid-s19926" xml:space="preserve"> Mani-<lb/>feſtum eſt igitur, quòd ſuper totũ;</s> <s xml:id="echoid-s19927" xml:space="preserve"> quod cadit ſub linea d a e ad partẽ b, nõ cadit uiſus, quoniã terra <lb/>abſcondit illud nobis:</s> <s xml:id="echoid-s19928" xml:space="preserve"> quia extẽſio uiſus nõ eſt, niſi ſuper lineã rectam [per primã hypotheſin opti-<lb/>corum Euclidis.</s> <s xml:id="echoid-s19929" xml:space="preserve">] Et Euclιdes quidẽiam declarauit [16 p 3] quòd nõ egreditur à puncto cõtactus <lb/>linea inter lineã cõtingent ẽ& circulũ.</s> <s xml:id="echoid-s19930" xml:space="preserve"> Viſus ergo nõ cadit ſub linea d a e, ſed cadit ſuper illud, quod <lb/>eleuatur ab ea.</s> <s xml:id="echoid-s19931" xml:space="preserve"> Et ponã formã pyramidis tenebrarũ euenientiũ ex umbra terræ, parum ante crepu-<lb/>ſculum, quãdo eſt depreſsio ſolis plus 19 gradibus per minutũ unũ, uerbi gratia;</s> <s xml:id="echoid-s19932" xml:space="preserve"> aut circiter:</s> <s xml:id="echoid-s19933" xml:space="preserve"> ſuper <lb/>quam ſint g, e, f, c:</s> <s xml:id="echoid-s19934" xml:space="preserve"> totũ enim, quod cadit in hac pyramιde deſignata (cuius caput eſt f, & baſis ipſius <lb/>terra) eſt rectum ſoli, nõ apparẽs ei, neq;</s> <s xml:id="echoid-s19935" xml:space="preserve"> illuminatũ ab eo, & eſt in ueritate tenebroſum:</s> <s xml:id="echoid-s19936" xml:space="preserve"> & quod ca-<lb/>dit exterius ab ea, eſt apparẽs ſoli, & ſuper ipſum cadũt radij eius & lumẽ eius.</s> <s xml:id="echoid-s19937" xml:space="preserve"> Veruntamẽ quod ex <lb/>corporib.</s> <s xml:id="echoid-s19938" xml:space="preserve"> eſt ſubtile ualde, nõ perducit ad uiſus noſtros illud, quod <lb/> <anchor type="figure" xlink:label="fig-0293-01a" xlink:href="fig-0293-01"/> ex radιo induit, ꝓpterea quòd æquãtur in uiſibus noſtris illud, qđ <lb/>ex aere ſubtile eſt intra pyramidẽ, & qđ eſt extra ipſum:</s> <s xml:id="echoid-s19939" xml:space="preserve"> & uidetur <lb/>æther totus in forma luminis & tenebrarum.</s> <s xml:id="echoid-s19940" xml:space="preserve"> Et nos quidẽ ſcimus, <lb/>quòd illud, quod cõtinet nos ex aere, & quod eſt propinquũ nobis, <lb/>eſt tenebroſum, nõ apparẽs ſoli:</s> <s xml:id="echoid-s19941" xml:space="preserve"> & quod procedιt in inceſſu in altũ, <lb/>aut dextrorſum, aut ſiniſtrorſum, & anterius & poſterius, eſt lumi-<lb/>noſum, apparẽs ſoli:</s> <s xml:id="echoid-s19942" xml:space="preserve"> & ſunt ambo cũ illo apud nos æqualiter in tota <lb/>cõprehenſione uiſus:</s> <s xml:id="echoid-s19943" xml:space="preserve"> & nõ apparet aliquid uiſibus noſtris ante ortũ <lb/>ſolis, & poſt occaſum ſolis, niſi ſit eleuatũ à ſuperficie horizontis, & <lb/>niſi ſit extra pyramidẽ umbræ, & niſi ſit ſpiſsius aere ſubtili.</s> <s xml:id="echoid-s19944" xml:space="preserve"> Manife-<lb/>ſtum eſt igitur, quòd nõ apparet uiſibus noſtris aliquid in habitudi-<lb/>ne ſplẽdoris & illuminationis, niſi per aggregationẽ triũ conditio-<lb/>num in eo:</s> <s xml:id="echoid-s19945" xml:space="preserve"> quarũ una eſt, ut nõ ſit ſub lιnea d a e:</s> <s xml:id="echoid-s19946" xml:space="preserve"> quoniã ſi eſt ſub ea, <lb/>prohibet ſphęra terræ inter ipſum & uiſum:</s> <s xml:id="echoid-s19947" xml:space="preserve"> quia nõ comprẽhendιt <lb/>ipſum uiſus luminoſum neq;</s> <s xml:id="echoid-s19948" xml:space="preserve"> tenebroſum.</s> <s xml:id="echoid-s19949" xml:space="preserve"> Et alia eſt, ut nõ ſit in py-<lb/>ramide umbræ:</s> <s xml:id="echoid-s19950" xml:space="preserve"> nã ſi eſt in ea, eſt tenebroſum, propterea quòd priua <lb/>tũ eſt facie ſolis & illuminatione ſua ab eo.</s> <s xml:id="echoid-s19951" xml:space="preserve"> Et alia eſt ut ſit ſpiſsius <lb/>aere ſubtili implẽte ſphæram:</s> <s xml:id="echoid-s19952" xml:space="preserve"> quoniã iam ſciuimus, quòd aer altior <lb/>extra pyramidẽ, cadit ſuper lineã d a e:</s> <s xml:id="echoid-s19953" xml:space="preserve"> & cũ illo non apparet nobis <lb/>in eo aliquid luminis, propter tenuitatem & ſubtilitatẽ ſuam, & pro <lb/>pterea quod uidemus in hoc loco, & eſt parum ante crepuſculũ, il-<lb/>lud, quod comprehẽdimus de ſphæra, tectum, nõ illuminatũ, & non <lb/>diuerſificatur pars eius à parte.</s> <s xml:id="echoid-s19954" xml:space="preserve"> Et ſcimus, quòd nõ eſt in eo punctũ <lb/>neq;</s> <s xml:id="echoid-s19955" xml:space="preserve"> locus unus, in quo aggregentur iſtæ cõditiones tres.</s> <s xml:id="echoid-s19956" xml:space="preserve"> Sed pun <lb/>ctum e eſt:</s> <s xml:id="echoid-s19957" xml:space="preserve"> ubi occurrit ultιmo ſtatui pyramidis linea d a e:</s> <s xml:id="echoid-s19958" xml:space="preserve"> & iã po-<lb/>ſuimus in eo duas conditiones:</s> <s xml:id="echoid-s19959" xml:space="preserve"> quoniã nõ eſt ſub linea d a e, nec in-<lb/>tra pyramidẽ:</s> <s xml:id="echoid-s19960" xml:space="preserve"> ergo cadit ſuper ipſum radius ſolis.</s> <s xml:id="echoid-s19961" xml:space="preserve"> Nõ ergo facit ne-<lb/>ceſſariam tenebroſitatẽ eius in oculis noſtris tũc, niſi priuatio eius à conditione tertia, quę eſt ſpiſ-<lb/>ſitudo.</s> <s xml:id="echoid-s19962" xml:space="preserve"> Iam ergo certificatur, quòd aer, ubi eſt punctũ e, in hoc loco eſt ſubtilis, & non perueniũt ad <lb/>ipſum uapores ſpiſsi, aſeendentes de terra, qui ſunt ſpiſsiores aere.</s> <s xml:id="echoid-s19963" xml:space="preserve"> Deinde poſtquã eleuatur ſol pa-<lb/>rum, & fit depreſsio eius ab horizonte 19 graduũ tantùm, & fit forma pyramidis & figura eius, ſicut <lb/>illa, ſuper quã ſunt i, m, h, k, & apparet in horizõte res luminoſa, & nõ fuerat antè illic res lum inoſa:</s> <s xml:id="echoid-s19964" xml:space="preserve"> <lb/>ſeimus quòd ille eſt primus locorũ & hoſpitiorũ, in quo aggregãtur cõditiones tres prędictæ:</s> <s xml:id="echoid-s19965" xml:space="preserve"> quo-<lb/>niã ante illud parũ per illud, cuι nõ eſt quantitas, nõ fuit illic aliquid de lumine:</s> <s xml:id="echoid-s19966" xml:space="preserve"> & primus locorũ, in <lb/>quo aggregatur, ut non ſit ſub linea d a e, nec intret pyramidein tenebrarum, eſt punctum m.</s> <s xml:id="echoid-s19967" xml:space="preserve"> Ergo <lb/>punctũ m eſt primus locorũ, in quo inuẽta eſt cõditio rertia, & eſt illic ſpiſsitudo aeris.</s> <s xml:id="echoid-s19968" xml:space="preserve"> Ergo pũctũ <lb/>in eſt ultimus ſtatus uaporũ, & ſumma aſcẽſio eorũ:</s> <s xml:id="echoid-s19969" xml:space="preserve"> & nõ abbreuiãtur ab eo, neq;</s> <s xml:id="echoid-s19970" xml:space="preserve"> pertrãſeũt ipſum.</s> <s xml:id="echoid-s19971" xml:space="preserve"> <lb/>Quoniã ſi abbreuiarẽtur ab eo, eſſet punctũ m in aere ſubt li, & nõ appareret nobis in eo aliquid de <lb/>lumine, ſicut nõ apparet in eo, qui eſt poſt ipſum, ad partem e:</s> <s xml:id="echoid-s19972" xml:space="preserve"> & ſi pertrãſirent ipſum, illuminaretur <lb/>nobis punctũ e ante hoc:</s> <s xml:id="echoid-s19973" xml:space="preserve"> quoniã nõ ponimus in eo, quod eſt inter m & e, in his duobus locis rẽ ſen-<lb/> <pb o="288" file="0294" n="294" rhead="ALHAZEN OPTIC. LIB. VII."/> ſibilẽ.</s> <s xml:id="echoid-s19974" xml:space="preserve"> Ergo punctũ m eſt ultimus ſtatus, ad quẽ perueniũt uapores aſcendentes in altũ, & occurſus <lb/>lineæ d a e cõtingentis ſphærã terræ cũ linea h i.</s> <s xml:id="echoid-s19975" xml:space="preserve"> Quando ergo uolumus ſcire longitudinẽ eius à fa-<lb/>cie terrę, tũc nos deſcribemus altitudinis circulũ, tranſeuntẽ per centrũ ſolis, quãdo eius depreſsio <lb/>ab horizõte eſt 19 graduũ:</s> <s xml:id="echoid-s19976" xml:space="preserve"> & illud eſt a pud ortũ crepuſculi, ſuper quẽ ſint a, b, c, d:</s> <s xml:id="echoid-s19977" xml:space="preserve"> ſecabit ergo<gap/>ſphę-<lb/>ram terræ ſuper circulũ e f g h [per 1 the.</s> <s xml:id="echoid-s19978" xml:space="preserve"> 1 ſphær.</s> <s xml:id="echoid-s19979" xml:space="preserve"> Theodoſij] & linea a e k pertrãſeat per zenith ca-<lb/>pitum & per centrũ terræ, perpẽdicularis ſuper lineam b k d [per 11 p 1] ergo linea b k d ſecat terrã <lb/>in duo media, [per 17 d 1] apparẽs & occultũ.</s> <s xml:id="echoid-s19980" xml:space="preserve"> Apparẽs ergo eſt illud, quod eſt ſupra ipſam, ad partẽ <lb/>a, & occultum, quod eſt ad partẽ g:</s> <s xml:id="echoid-s19981" xml:space="preserve"> & nõ dicimus hoc, niſi dilatãdo & appropinquãdo.</s> <s xml:id="echoid-s19982" xml:space="preserve"> Veritas uerò <lb/>eſt, quòd apparẽs nõ eſt, niſi illud, quod eſt ſuper lineã p e q o protractã, contingentem ſphærã ſuper <lb/>punctũ uiſus:</s> <s xml:id="echoid-s19983" xml:space="preserve"> ueruntamen nõ eſt apud hũc or-<lb/> <anchor type="figure" xlink:label="fig-0294-01a" xlink:href="fig-0294-01"/> bẽ terræ magna quãtitas.</s> <s xml:id="echoid-s19984" xml:space="preserve"> Et ponã arcum b c 19 <lb/>graduũ, qui ſunt depreſsio ſolis apud ortũ cre-<lb/>puſculi.</s> <s xml:id="echoid-s19985" xml:space="preserve"> Super punctũ ergo c eſt centrum ſolis:</s> <s xml:id="echoid-s19986" xml:space="preserve"> <lb/>faciã igitur illic ſuper ipſum punctũ, circulũ, cũ <lb/>lõgitudine quintupli & medietatis eius, quod <lb/>eſt æquale lineę e k:</s> <s xml:id="echoid-s19987" xml:space="preserve"> qui ſit circulus l m:</s> <s xml:id="echoid-s19988" xml:space="preserve"> & ſuper <lb/>ipſum ſcilicet punctũ c ſecat ſolẽ orbis a b c d:</s> <s xml:id="echoid-s19989" xml:space="preserve"> <lb/>& continuabo lineã k g:</s> <s xml:id="echoid-s19990" xml:space="preserve"> deinde protrahã duas <lb/>lineas contingẽtes duos circulos ſolis & terræ <lb/>[per 17 p 3] continẽtes illuminatũ terræ à ſole, <lb/>quæ ſint m h n, l f n, cõtingẽtes terrã ſuper duo <lb/>puncta h & f:</s> <s xml:id="echoid-s19991" xml:space="preserve"> & ſunt termini pyramidis umbrę.</s> <s xml:id="echoid-s19992" xml:space="preserve"> <lb/>Ergo linea m h n occurrit lineæ p o ſuper pun-<lb/>ctum q [per lẽma Procli ad 29 p 1:</s> <s xml:id="echoid-s19993" xml:space="preserve"> quia cõcur-<lb/>rit cũ b k d parallela ipſi p o per 28 p 1] ergo pũ-<lb/>ctum q, ſecundũ quod oſtẽdimus in figura, quę <lb/>eſt ante hãc, eſt locus luminoſus apud ortũ cre <lb/>puſculi:</s> <s xml:id="echoid-s19994" xml:space="preserve"> & eſt ultimus ſtatus aſcenſionis uapo-<lb/>rum.</s> <s xml:id="echoid-s19995" xml:space="preserve"> Cum ergo uolumus cognoſcere longitu-<lb/>dinem eius à ſuperficie terræ:</s> <s xml:id="echoid-s19996" xml:space="preserve"> tũc continuabi-<lb/>mus k cũ q per lineã k r q:</s> <s xml:id="echoid-s19997" xml:space="preserve"> & continuabo k cum <lb/>h.</s> <s xml:id="echoid-s19998" xml:space="preserve"> Ergo portio h g f eſt illuminata:</s> <s xml:id="echoid-s19999" xml:space="preserve"> quia facie ad <lb/>faciẽ reſpicit ſolem.</s> <s xml:id="echoid-s20000" xml:space="preserve"> Iam ergo oſtẽdimus [præ-<lb/>cedente numero] quòd ea eſt 180 grad.</s> <s xml:id="echoid-s20001" xml:space="preserve"> & 27 <lb/>min.</s> <s xml:id="echoid-s20002" xml:space="preserve"> & 52 ſecũd.</s> <s xml:id="echoid-s20003" xml:space="preserve"> & arcus g h eſt medietas eius:</s> <s xml:id="echoid-s20004" xml:space="preserve"> <lb/>[Quia enim l n, m n tangunt peripheriã circuli <lb/>e f g h in punctis f & h per fabricationem, erunt <lb/>anguli ad f & h recti per 18 p 3.</s> <s xml:id="echoid-s20005" xml:space="preserve"> Si igitur ſemidia-<lb/>metros k l, k m circuli a b c d ductas cogites:</s> <s xml:id="echoid-s20006" xml:space="preserve"> <lb/>æquabuntur quadrata linearũ f l, f k quadrato <lb/>ſemidiametri k l per 47 p 1, per quam etiã qua-<lb/>drata linearum h m, h k æquabuntur quadrato <lb/>ſemidiametri k m:</s> <s xml:id="echoid-s20007" xml:space="preserve"> ſubductis igitur quadratis <lb/>ipſarũ f k, h k per 5 d 1 æqualibus, à quadratis k l, k m ſimiliter per 15 d 1 æqualibus:</s> <s xml:id="echoid-s20008" xml:space="preserve"> relinquẽtur qua-<lb/>drata ipſarũ f l, h m æqualia, & iccirco rectę f l, h m æquales.</s> <s xml:id="echoid-s20009" xml:space="preserve"> Quare cũ triangula f k l, h k m ſint æqui-<lb/>latera, erunt æquiangula, & angulus f k l æqualis angulo h k m per 8 p 1.</s> <s xml:id="echoid-s20010" xml:space="preserve"> Rurſus ſi ſemidiametros l c, <lb/>m c circuli l m ductas animo concipias:</s> <s xml:id="echoid-s20011" xml:space="preserve"> erũt triangula l k c, m k c æquilatera & æquiangula, & angu-<lb/>lus l k c æqualis angulo m k c.</s> <s xml:id="echoid-s20012" xml:space="preserve"> Quamobrem ſi angulis f k l, h k m è concluſo æqualibus addas angu-<lb/>los l k c, m k c æquales:</s> <s xml:id="echoid-s20013" xml:space="preserve"> totus angulus f k g æquabitur toti angulo h k g per 2 axio:</s> <s xml:id="echoid-s20014" xml:space="preserve"> & peripheria f g <lb/>peripheriæ h g per 26 p 3] & eſt grad.</s> <s xml:id="echoid-s20015" xml:space="preserve"> 90 & 13 min.</s> <s xml:id="echoid-s20016" xml:space="preserve"> & 56 ſecun.</s> <s xml:id="echoid-s20017" xml:space="preserve"> & illud eſt quãtitas anguli h k g:</s> <s xml:id="echoid-s20018" xml:space="preserve"> & iã <lb/>fuit angulus b k c 19 grad quoniã eſt depreſsio ſolis:</s> <s xml:id="echoid-s20019" xml:space="preserve"> ergo remanet angulus h k b 71 grad.</s> <s xml:id="echoid-s20020" xml:space="preserve"> 13 min.</s> <s xml:id="echoid-s20021" xml:space="preserve"> 56 <lb/>ſecun.</s> <s xml:id="echoid-s20022" xml:space="preserve"> ſed angulus e k b eſt 90:</s> <s xml:id="echoid-s20023" xml:space="preserve"> quia rectus exiſtit.</s> <s xml:id="echoid-s20024" xml:space="preserve"> Ergo remanet angulus e k h 18 grad.</s> <s xml:id="echoid-s20025" xml:space="preserve"> 46 min.</s> <s xml:id="echoid-s20026" xml:space="preserve"> 4 ſe-<lb/>cun.</s> <s xml:id="echoid-s20027" xml:space="preserve"> Et quia linea k q diuidit eũ in duo media, & illud eſt manifeſtũ:</s> <s xml:id="echoid-s20028" xml:space="preserve"> [Quia enim e k, h k:</s> <s xml:id="echoid-s20029" xml:space="preserve"> item e q, h q <lb/>æquãtur:</s> <s xml:id="echoid-s20030" xml:space="preserve"> illæ per 15 d 1, quia circuli e f g h ſunt ſemidiametri:</s> <s xml:id="echoid-s20031" xml:space="preserve"> hæ per ſecundũ conſectariũ 36 p 3, quia <lb/>ab eodẽ puncto q peripheriã e f g h tangunt:</s> <s xml:id="echoid-s20032" xml:space="preserve"> & cõmunis eſt k q:</s> <s xml:id="echoid-s20033" xml:space="preserve"> æquabitur angulus e k q angulo h k <lb/>q per 8 p 1.</s> <s xml:id="echoid-s20034" xml:space="preserve"> Quare angulus e k h bifariã ſectus eſt per rectã q k] angulus igitur q k e eſt 9 grad.</s> <s xml:id="echoid-s20035" xml:space="preserve"> 23.</s> <s xml:id="echoid-s20036" xml:space="preserve"> mi.</s> <s xml:id="echoid-s20037" xml:space="preserve"> <lb/>2 ſecund.</s> <s xml:id="echoid-s20038" xml:space="preserve"> ergo angulus k q e eſt cõplementũ recti [per 32 p 1:</s> <s xml:id="echoid-s20039" xml:space="preserve"> quia angulus ad e rectus eſt per 18 p 3] <lb/>& illud eſt 80 grad.</s> <s xml:id="echoid-s20040" xml:space="preserve"> 36 min.</s> <s xml:id="echoid-s20041" xml:space="preserve"> 58 ſecun.</s> <s xml:id="echoid-s20042" xml:space="preserve"> Chorda ergo eius, quę eſt linea e k, eſt 59 grad.</s> <s xml:id="echoid-s20043" xml:space="preserve"> 11 min.</s> <s xml:id="echoid-s20044" xml:space="preserve"> 48 ſecun.</s> <s xml:id="echoid-s20045" xml:space="preserve"> <lb/>per quantitatẽ, qua eſt linea k q 60 grad.</s> <s xml:id="echoid-s20046" xml:space="preserve"> [ut monſtrat tabula rectarũ ſubtenſarũ in circulo.</s> <s xml:id="echoid-s20047" xml:space="preserve">] Verun <lb/>tamen per quantitatẽ, qua eſt linea k e 60 grad.</s> <s xml:id="echoid-s20048" xml:space="preserve"> erit q r k 60 grad.</s> <s xml:id="echoid-s20049" xml:space="preserve"> & 48 min.</s> <s xml:id="echoid-s20050" xml:space="preserve"> & quinq;</s> <s xml:id="echoid-s20051" xml:space="preserve"> ſextorũ unius <lb/>minuti:</s> <s xml:id="echoid-s20052" xml:space="preserve"> ſed linea k r ex illis eſt 60 grad.</s> <s xml:id="echoid-s20053" xml:space="preserve"> ergo remanet r q 48 min.</s> <s xml:id="echoid-s20054" xml:space="preserve"> & 50 ſecun.</s> <s xml:id="echoid-s20055" xml:space="preserve"> & eſt illud ex miliari-<lb/>bus (quibus circumferentia terræ continet 24000) milliaria, 51 & 47 minut.</s> <s xml:id="echoid-s20056" xml:space="preserve"> & 34 ſecun.</s> <s xml:id="echoid-s20057" xml:space="preserve"> & 6 <lb/>partes ex 11 partib.</s> <s xml:id="echoid-s20058" xml:space="preserve"> ſecundis.</s> <s xml:id="echoid-s20059" xml:space="preserve"> Et illud eſt ultimũ, ad quod eleuantur & perueniũt <lb/>uapores aſcendentes ex terra.</s> <s xml:id="echoid-s20060" xml:space="preserve"> Et illud eſt, quod uoluimus.</s> <s xml:id="echoid-s20061" xml:space="preserve"/> </p> <div xml:id="echoid-div643" type="float" level="0" n="0"> <figure xlink:label="fig-0293-01" xlink:href="fig-0293-01a"> <variables xml:id="echoid-variables240" xml:space="preserve">h <gap/> d a m e c k z g b</variables> </figure> <figure xlink:label="fig-0294-01" xlink:href="fig-0294-01a"> <variables xml:id="echoid-variables241" xml:space="preserve">n a d p e q o r f k h g b l c m</variables> </figure> </div> </div> <div xml:id="echoid-div645" type="section" level="0" n="0"> <head xml:id="echoid-head553" xml:space="preserve">FINIS.</head> <pb file="0295" n="295"/> </div> <div xml:id="echoid-div646" type="section" level="0" n="0"> <head xml:id="echoid-head554" xml:space="preserve">VITELLONIS THV-<lb/>RINGOPOLONI OPTI-<lb/>CAE LIBRI DECEM.</head> <head xml:id="echoid-head555" xml:space="preserve">Inſtaurati, figuris nouis illuſtrati atque aucti: infinitis q́; erroribus, <lb/>quibus antea ſcatebant, expurgati.</head> <head xml:id="echoid-head556" xml:space="preserve">À<unsure/> <lb/><emph style="sc">Federico</emph> <emph style="sc">Risnero.</emph></head> <figure> <image file="0295-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0295-01"/> </figure> </div> <div xml:id="echoid-div647" type="section" level="0" n="0"> <head xml:id="echoid-head557" xml:space="preserve">BASILEAE.</head> <pb file="0296" n="296"/> <pb file="0297" n="297"/> </div> <div xml:id="echoid-div648" type="section" level="0" n="0"> <head xml:id="echoid-head558" xml:space="preserve">FEDERICI RISNE-</head> <head xml:id="echoid-head559" xml:space="preserve">RI IN VITELLONIS</head> <head xml:id="echoid-head560" xml:space="preserve">OPTICAM PRAEFATIO</head> <head xml:id="echoid-head561" xml:space="preserve">AD</head> <head xml:id="echoid-head562" xml:space="preserve" style="it">ILLVSTRISSIMAM REGINAM CA-</head> <head xml:id="echoid-head563" xml:space="preserve" style="it">tharinam Mediceam, matrem regis Gallia <lb/>Caroli noni.</head> <p> <s xml:id="echoid-s20062" xml:space="preserve">ALHAZENVS opticas ſuas opes, regina illuſtriſsi-<lb/>ma, noſtris laboribus uigilijsq́;</s> <s xml:id="echoid-s20063" xml:space="preserve"> explicatas tibi <lb/>nũcupauit:</s> <s xml:id="echoid-s20064" xml:space="preserve"> Vitello Alhazenũ ducem, quamuis <lb/>antea ſibi proignoto tacitoq́;</s> <s xml:id="echoid-s20065" xml:space="preserve"> præteritum, atta-<lb/>men ueluti conſcientia præeuntis in eo uirtutis <lb/>permotus, conſequitur, ſeq́;</s> <s xml:id="echoid-s20066" xml:space="preserve"> Alhazeni diſcipulum eſſe con-<lb/>fitetur.</s> <s xml:id="echoid-s20067" xml:space="preserve"> Etenim cum opticorum longè maximam nobiliſsi-<lb/>mamq́;</s> <s xml:id="echoid-s20068" xml:space="preserve"> partem, quam ex Alhazeno deſumpſiſſet, tibi deuo-<lb/>tam dicatamq́;</s> <s xml:id="echoid-s20069" xml:space="preserve"> cerneret, qua tandem coloris ſpecie purpu-<lb/>ram eandem, aut quo aêris ſitu permutatã alijs pro ſua uen-<lb/>ditaret?</s> <s xml:id="echoid-s20070" xml:space="preserve"> Certè ingenua animiliberaliq́;</s> <s xml:id="echoid-s20071" xml:space="preserve"> inductione tam prę-<lb/>ſtantem patronam potius eandem adoptabit, ſeq́;</s> <s xml:id="echoid-s20072" xml:space="preserve"> in regiæ <lb/>maieſtatis tuæ fidem clientelamq́;</s> <s xml:id="echoid-s20073" xml:space="preserve"> conferet.</s> <s xml:id="echoid-s20074" xml:space="preserve"> Ergo iam libe-<lb/>rius exponamus quis ſit Vitello, & quid in tanto inſupero-<lb/>pere contenderit.</s> <s xml:id="echoid-s20075" xml:space="preserve"> E<gap/> Sarmatarum gente (qui Poloni hodie <lb/>nominantur) ille fuit.</s> <s xml:id="echoid-s20076" xml:space="preserve"> Ait enim libro 10 theoremate 74, in <lb/>noſtra terra ſcilicet Poloniæ habitabili, &c.</s> <s xml:id="echoid-s20077" xml:space="preserve"> Ideoq́;</s> <s xml:id="echoid-s20078" xml:space="preserve"> intitulo <lb/>optici operis cognominatur filius Polonorum & Thurin-<lb/>gorum, patre uidelicet Polono & matre Thuringa, aut con-<lb/>trà procreatus:</s> <s xml:id="echoid-s20079" xml:space="preserve"> qualia cognomenta habentarabicæ inſcri-<lb/>ptiones Alhazenus filius Alhayzeni & ſimiles.</s> <s xml:id="echoid-s20080" xml:space="preserve"> Regiomon-<lb/>tanus autem in pręfatione Alphragani uidetur eum Germa-<lb/>num efficere, inquit enim, Vitello autem noſter Thuringus, <lb/>&c.</s> <s xml:id="echoid-s20081" xml:space="preserve"> inq́;</s> <s xml:id="echoid-s20082" xml:space="preserve"> eandem opinionem Gualtherus Regiomontani di <lb/>ſcipulus diſcedit, cum in ſuis obſeruationibus aſtronomi-<lb/>cis ait, & Vitello noſter, &c.</s> <s xml:id="echoid-s20083" xml:space="preserve"> uterque tamen commune artis <lb/>ſtudium, non patriæ commune ſolum hic ſpectaſſe potuit.</s> <s xml:id="echoid-s20084" xml:space="preserve"> <lb/>Sed de tempore, quo Vitello floruerit, res magis controuer-<lb/>ſa eſt.</s> <s xml:id="echoid-s20085" xml:space="preserve"> Tanſtetterus in epiſtola opticis Vitellonis antea editis <lb/>præpoſita opinatur Vitellonem annis abhincſexcentis ui-<lb/>xiſſe, ſed opinione deceptus eſt.</s> <s xml:id="echoid-s20086" xml:space="preserve"> Nam frater Guilielmus de <lb/> <pb file="0298" n="298" rhead="FEDERICI RISNERI"/> Morbeta (cui Vitello opticam ſuam nuncupauit) uixit anno <lb/>Chriſti 1269, ut ille ipſe de Morbeta teſtificatur in ſua geo-<lb/>mantia (quam manuſcriptam legimus) eodem etiam anno <lb/>ſectionibus octo collecta, magiſtroq́;</s> <s xml:id="echoid-s20087" xml:space="preserve"> Arnolpho nepotiſuo <lb/>dedicata:</s> <s xml:id="echoid-s20088" xml:space="preserve"> & in hanc quo que temporis ætatem doctiſſimi <lb/>uiri & excellentiſſimi mathematici Eraſmus Reinholdus & <lb/>Gaſparus Peucerus Vitellonem retulerunt.</s> <s xml:id="echoid-s20089" xml:space="preserve"> Quapropterlo-<lb/>cupletioribus teſtimonijs cõſtat Vitellonem incidiſſe in an-<lb/>num Chriſti circiter 1270, annis nempe anteactis propemo-<lb/>dum trecentis.</s> <s xml:id="echoid-s20090" xml:space="preserve"> Verùm id de tempore.</s> <s xml:id="echoid-s20091" xml:space="preserve"> Locus autem, ubi ſtu-<lb/>dia hæc excoluerit, minimè uidetur Sarmatia fuiſſe.</s> <s xml:id="echoid-s20092" xml:space="preserve"> Quædã <lb/>ſuntin opticis notæ Vitellonem in Italiam ueniſſe, Italiæq́;</s> <s xml:id="echoid-s20093" xml:space="preserve"> <lb/>bibliothecis adiutum fuiſſe.</s> <s xml:id="echoid-s20094" xml:space="preserve"> Etenim Vitello ipſe de ſe teſtis <lb/>eſt libro 10 theoremate 42 ſe primùm omniũ in Italia ad Cu-<lb/>balum (quilocus eſt inter Paduam & Vincentiam) contem-<lb/>platione aquæ tenuiſſimæ ac limpidiſſimæ ad opticas artes <lb/>incenſum atq;</s> <s xml:id="echoid-s20095" xml:space="preserve"> inflammatum eſſe:</s> <s xml:id="echoid-s20096" xml:space="preserve"> harum enim form arum <lb/>intuitu (ait) & mirabili tranſmutatione primum nos am or <lb/>huius ſtudij allexit:</s> <s xml:id="echoid-s20097" xml:space="preserve"> & libro 10 theoremate 67, ubi ſcribit ex <lb/>iride, quam in aqua è ſcopulo Viterbio proximo uehemen-<lb/>tius præcipitata ſæpenumero uidiſſet, pleraſque iridis affe-<lb/>ctiones & proprietates ſibi animaduerſas & obſeruatas eſſe:</s> <s xml:id="echoid-s20098" xml:space="preserve"> <lb/>illud (inquit) nobis principium cogitationis fuit, ut præſen <lb/>ti negotio ſtudium applicaremus.</s> <s xml:id="echoid-s20099" xml:space="preserve"> At quòd Vitello in Italia, <lb/>quòd Romæ tum cæteris liberalibus honeſtisq́;</s> <s xml:id="echoid-s20100" xml:space="preserve"> ſtudijs, tum <lb/>uerò opticis operam nauarit, maius fortaſſe argumentũ ui-<lb/>deatur, quòd Guilielmo de Morbeta (quitum romani pon-<lb/>tificis pœnitentiarum, utappellant, Romæ agebat) ſuaſore <lb/>& hortatore, utipſe in proœmio teſtatur, optica primũ con-<lb/>ſcribenda ſuſceperit, eidemq́;</s> <s xml:id="echoid-s20101" xml:space="preserve"> abſoluta poſtea nuncuparit.</s> <s xml:id="echoid-s20102" xml:space="preserve"> <lb/>Verumenimuerò fuerit Vitello Sarmata:</s> <s xml:id="echoid-s20103" xml:space="preserve"> uixerit tempore nõ <lb/>admodum literarũ, præſertim tam reconditarum ſtudijs de-<lb/>dito:</s> <s xml:id="echoid-s20104" xml:space="preserve"> bibliothecas Italię puluere obſitas, & in ijs ſepultos o-<lb/>pticos offenderit:</s> <s xml:id="echoid-s20105" xml:space="preserve"> attamen quid & quãtum uiribus ingenij <lb/>perfecerit, pręclara eius monimenta ſempiterno teſtimonio <lb/>erunt:</s> <s xml:id="echoid-s20106" xml:space="preserve"> non ſolùm in phyſiologicis, quæ citat libro 5 theore-<lb/>mate 18, & libro 10 theoremate 80:</s> <s xml:id="echoid-s20107" xml:space="preserve"> in libris de ordine entiũ:</s> <s xml:id="echoid-s20108" xml:space="preserve"> <lb/>de elementatis cõcluſionibus, quinominantur in præfatio-<lb/>ne & libro 1 theoremate 28:</s> <s xml:id="echoid-s20109" xml:space="preserve"> in libris de ſcientia motuum cœ <lb/> <pb file="0299" n="299" rhead="PRAEFATIO."/> leſtiũ, quos allegatlib 10 theor.</s> <s xml:id="echoid-s20110" xml:space="preserve"> 53:</s> <s xml:id="echoid-s20111" xml:space="preserve"> ſed multò maximè in de-<lb/>cẽlibris opticis:</s> <s xml:id="echoid-s20112" xml:space="preserve"> quos, ut ex Alhazeno inprimis, deinde è grę <lb/>corũ authorũ fontibus hauſerit, certè mirandis acceſsioni-<lb/>bus amplificauit.</s> <s xml:id="echoid-s20113" xml:space="preserve"> Alhazeni, Euclidis, Ptolemęi axiomata, hy <lb/>potheſes, theoremata omnia collegit:</s> <s xml:id="echoid-s20114" xml:space="preserve"> id laboris infiniti fu-<lb/>it.</s> <s xml:id="echoid-s20115" xml:space="preserve"> Sed ex Apollonio, Theodoſio, Menelao, Theone, Pappo, <lb/>Proclo & alijs firmamenta permultarũ demonſtrationũ ſin-<lb/>gulari iudicio repetiuit:</s> <s xml:id="echoid-s20116" xml:space="preserve"> ſingulari ordine, maximè naturali, <lb/>perſua genera, ſpeciesq́;</s> <s xml:id="echoid-s20117" xml:space="preserve"> opticã, catoptricã, meſopticã diſpo <lb/>ſuit, artẽq́;</s> <s xml:id="echoid-s20118" xml:space="preserve"> totã mirabiliter abſoluit.</s> <s xml:id="echoid-s20119" xml:space="preserve"> Quid plura?</s> <s xml:id="echoid-s20120" xml:space="preserve"> Si artis opi <lb/>fex atq;</s> <s xml:id="echoid-s20121" xml:space="preserve"> author habendus ſit, qui arti formã, animãq́;</s> <s xml:id="echoid-s20122" xml:space="preserve"> dedit:</s> <s xml:id="echoid-s20123" xml:space="preserve"> <lb/>Vitello iure optimo opticæ artis author habeatur.</s> <s xml:id="echoid-s20124" xml:space="preserve"> Atq;</s> <s xml:id="echoid-s20125" xml:space="preserve"> hæc <lb/>quidẽ de Vitellone, eiusq́;</s> <s xml:id="echoid-s20126" xml:space="preserve"> optico opere ita dicta ſint:</s> <s xml:id="echoid-s20127" xml:space="preserve"> quid <lb/>uerò ipſe operæ, induſtriæ, ac diligẽtiæ in eo renouãdo atq;</s> <s xml:id="echoid-s20128" xml:space="preserve"> <lb/>inſtaurãdo poſuerim, quantũq́;</s> <s xml:id="echoid-s20129" xml:space="preserve"> in eo reſtituendo cõforman <lb/>doq́;</s> <s xml:id="echoid-s20130" xml:space="preserve"> elaborarim, uix quenquã cogitaturũ arbitror, niſi qui <lb/>uetus exẽplar cum noſtro cõtulerit.</s> <s xml:id="echoid-s20131" xml:space="preserve"> Dicã parcè de me & bre <lb/>uiter.</s> <s xml:id="echoid-s20132" xml:space="preserve"> In Vitellone adhuc edito publicatoq́;</s> <s xml:id="echoid-s20133" xml:space="preserve"> nullũ omnino <lb/>theorema fuit, in cuius demõſtratione literæ nõ fuerint mul <lb/>tifariã permutatæ, nõ alię pro alijs repoſitę:</s> <s xml:id="echoid-s20134" xml:space="preserve"> in pleriſq;</s> <s xml:id="echoid-s20135" xml:space="preserve"> etiam <lb/>demonſtrationibus ε'κθέσ{ει}ς atq;</s> <s xml:id="echoid-s20136" xml:space="preserve"> expoſitiones nullæ fuerunt:</s> <s xml:id="echoid-s20137" xml:space="preserve"> <lb/>uerba itẽ multa, im ò uerò integræ etiã clauſulæ, eæq́;</s> <s xml:id="echoid-s20138" xml:space="preserve"> cõplu <lb/>res defuerũt:</s> <s xml:id="echoid-s20139" xml:space="preserve"> quæ omnia uetuſtis exẽplaribus manuſcriptis, <lb/>quæ P.</s> <s xml:id="echoid-s20140" xml:space="preserve"> Ramus undiq;</s> <s xml:id="echoid-s20141" xml:space="preserve"> cõ quiſierat, adiutus reſtitui:</s> <s xml:id="echoid-s20142" xml:space="preserve"> & in uni-<lb/>uerſo opere errata in rebus ſentẽtijsq́;</s> <s xml:id="echoid-s20143" xml:space="preserve"> 3645 (tot enim anim-<lb/>aduertere potui) cęteraq́;</s> <s xml:id="echoid-s20144" xml:space="preserve"> leuiora, quæ innumera fuerũt, cor <lb/>rexi & emendaui.</s> <s xml:id="echoid-s20145" xml:space="preserve"> Sed præter literas in demonſtrationibus <lb/>tranſpoſitas;</s> <s xml:id="echoid-s20146" xml:space="preserve"> præteruoces plurimas, præter etiã totas ſenten <lb/>tias omiſſas, figuræ, quæ perſe ſine literis, ſine uocibus, ſine <lb/>ſcriptis ſententijs rem poterant intelligẽti demõſtrare, ple-<lb/>ræq;</s> <s xml:id="echoid-s20147" xml:space="preserve"> erãt malè figuratæ, nec demonſtrationibus cõgruẽtes, <lb/>& quod etiam fœdius eſt, nõ ſuis, ſed alienis theorematis ſæ <lb/>pius accõmodatæ:</s> <s xml:id="echoid-s20148" xml:space="preserve"> in quibuſdam theorematis nullæ omni-<lb/>no fuerunt.</s> <s xml:id="echoid-s20149" xml:space="preserve"> Figuras igitur uniuerſas de integro conforma-<lb/>ui, ſtudioſeq́;</s> <s xml:id="echoid-s20150" xml:space="preserve"> egi, ne qua iſtarum offenſionum remora poſ-<lb/>ſet in reliquum optices ſtudioſos remorari:</s> <s xml:id="echoid-s20151" xml:space="preserve"> poſtremò locos <lb/>ueterum geometrarum & opticorum, unde pleraque de-<lb/>ſumpta eſſent, indicaui:</s> <s xml:id="echoid-s20152" xml:space="preserve"> denique neruis omnibus conten-<lb/>di, ut opticarum rerum fructus, quanticunque ſint, qui <lb/>ſanè maximi ſunt (ut Alhazeni præfatio iam prius attigit) <lb/> <pb file="0300" n="300" rhead="PRAEFATIO."/> omnes certè non ſolum pleniores atque uberiores, ſed gra-<lb/>tiores & faciliores eſſent.</s> <s xml:id="echoid-s20153" xml:space="preserve"> Quamobrem, illuſtriſſima regina, <lb/>ſi laboribus noſtris uota reſponderint, ſpero opticæ artis <lb/>ſtudioſos te ſummis ac ſempiternis laudibus in mathema-<lb/>tico puluere ad cœlum elaturos eſſe:</s> <s xml:id="echoid-s20154" xml:space="preserve"> quòd tuis felicibus au-<lb/>ſpicijs duos opticos excellentiſsimos Alhazenum <lb/>& Vitellonem uelut ab inferis excitatos, & <lb/>publicis priuatisq́;</s> <s xml:id="echoid-s20155" xml:space="preserve"> ſcholis commu-<lb/>nicatos habeant.</s> <s xml:id="echoid-s20156" xml:space="preserve"/> </p> <pb file="0301" n="301" rhead="ERRATA."/> </div> <div xml:id="echoid-div649" type="section" level="0" n="0"> <head xml:id="echoid-head564" xml:space="preserve" style="it">Primus numerus paginam, ſecundus lineam indicat.</head> <p> <s xml:id="echoid-s20157" xml:space="preserve">Pagina 4.</s> <s xml:id="echoid-s20158" xml:space="preserve"> linea 36 lineæ.</s> <s xml:id="echoid-s20159" xml:space="preserve"> 10.</s> <s xml:id="echoid-s20160" xml:space="preserve">44 poſt conuexã tolle comma.</s> <s xml:id="echoid-s20161" xml:space="preserve"> Ibidem, ultima, quolibet.</s> <s xml:id="echoid-s20162" xml:space="preserve"> Pag:</s> <s xml:id="echoid-s20163" xml:space="preserve"> <lb/>13.</s> <s xml:id="echoid-s20164" xml:space="preserve">30 poſt angulo tolle b, & repone poſt angulus.</s> <s xml:id="echoid-s20165" xml:space="preserve"> Ibidem 45 poſt 17 p 6 adde, Et.</s> <s xml:id="echoid-s20166" xml:space="preserve"> 32.</s> <s xml:id="echoid-s20167" xml:space="preserve">39 cen-<lb/>trum.</s> <s xml:id="echoid-s20168" xml:space="preserve"> 35.</s> <s xml:id="echoid-s20169" xml:space="preserve">51 ſuarum.</s> <s xml:id="echoid-s20170" xml:space="preserve"> Ibidem 55 rotundam.</s> <s xml:id="echoid-s20171" xml:space="preserve"> 37.</s> <s xml:id="echoid-s20172" xml:space="preserve">34 poſt conica pone comma, & adde ubi:</s> <s xml:id="echoid-s20173" xml:space="preserve"> & poſt <lb/>ſupremo dele comma.</s> <s xml:id="echoid-s20174" xml:space="preserve"> 48.</s> <s xml:id="echoid-s20175" xml:space="preserve">8 proportionem.</s> <s xml:id="echoid-s20176" xml:space="preserve"> 54.</s> <s xml:id="echoid-s20177" xml:space="preserve">56 Sint enim ut.</s> <s xml:id="echoid-s20178" xml:space="preserve"> 64.</s> <s xml:id="echoid-s20179" xml:space="preserve">54 eidem.</s> <s xml:id="echoid-s20180" xml:space="preserve"> 65.</s> <s xml:id="echoid-s20181" xml:space="preserve"> ult.</s> <s xml:id="echoid-s20182" xml:space="preserve"> terreæ.</s> <s xml:id="echoid-s20183" xml:space="preserve"> <lb/>67.</s> <s xml:id="echoid-s20184" xml:space="preserve">4 poſt baſis tolle comma, & repone poſt ſuperficie.</s> <s xml:id="echoid-s20185" xml:space="preserve"> 70.</s> <s xml:id="echoid-s20186" xml:space="preserve">25 puncti.</s> <s xml:id="echoid-s20187" xml:space="preserve"> Ibidem 54 abſcindatur.</s> <s xml:id="echoid-s20188" xml:space="preserve"> <lb/>Pag.</s> <s xml:id="echoid-s20189" xml:space="preserve">74 in figura 37 theorematis ad concurſum lineæ a z cum peripheria h l k pone literam f.</s> <s xml:id="echoid-s20190" xml:space="preserve"> <lb/>Pag.</s> <s xml:id="echoid-s20191" xml:space="preserve">80.</s> <s xml:id="echoid-s20192" xml:space="preserve">40 ductæ.</s> <s xml:id="echoid-s20193" xml:space="preserve"> Ibid.</s> <s xml:id="echoid-s20194" xml:space="preserve"> 48 ſuæ.</s> <s xml:id="echoid-s20195" xml:space="preserve"> 82.</s> <s xml:id="echoid-s20196" xml:space="preserve">9 poſt participans adde de.</s> <s xml:id="echoid-s20197" xml:space="preserve"> 89.</s> <s xml:id="echoid-s20198" xml:space="preserve">8 conſtantis.</s> <s xml:id="echoid-s20199" xml:space="preserve"> 90.</s> <s xml:id="echoid-s20200" xml:space="preserve">12 poſt <lb/>tranſiens adde per, & poſt huius dele comma.</s> <s xml:id="echoid-s20201" xml:space="preserve"> 91.</s> <s xml:id="echoid-s20202" xml:space="preserve">2 poſsint.</s> <s xml:id="echoid-s20203" xml:space="preserve"> 92.</s> <s xml:id="echoid-s20204" xml:space="preserve">ulti.</s> <s xml:id="echoid-s20205" xml:space="preserve"> pro formæ, forma.</s> <s xml:id="echoid-s20206" xml:space="preserve"> Pag.</s> <s xml:id="echoid-s20207" xml:space="preserve"> <lb/>95.</s> <s xml:id="echoid-s20208" xml:space="preserve">1 pro perunitatem paruitatem.</s> <s xml:id="echoid-s20209" xml:space="preserve"> 111.</s> <s xml:id="echoid-s20210" xml:space="preserve">43 uifibilium.</s> <s xml:id="echoid-s20211" xml:space="preserve"> 112.</s> <s xml:id="echoid-s20212" xml:space="preserve">7 illarum.</s> <s xml:id="echoid-s20213" xml:space="preserve"> 122.</s> <s xml:id="echoid-s20214" xml:space="preserve">10 quantitates.</s> <s xml:id="echoid-s20215" xml:space="preserve"> 132.</s> <s xml:id="echoid-s20216" xml:space="preserve">46 <lb/>diuerſitatis.</s> <s xml:id="echoid-s20217" xml:space="preserve"> Pag.</s> <s xml:id="echoid-s20218" xml:space="preserve">141 in ultima figura ducantur lineæ g b & d b.</s> <s xml:id="echoid-s20219" xml:space="preserve"> 145.</s> <s xml:id="echoid-s20220" xml:space="preserve">19 quæ.</s> <s xml:id="echoid-s20221" xml:space="preserve"> Ibidem 48 poſt <lb/>poſsibile adde ut.</s> <s xml:id="echoid-s20222" xml:space="preserve"> 160.</s> <s xml:id="echoid-s20223" xml:space="preserve">21 huius.</s> <s xml:id="echoid-s20224" xml:space="preserve"> 181.</s> <s xml:id="echoid-s20225" xml:space="preserve">15 poſt uiſu adde ex.</s> <s xml:id="echoid-s20226" xml:space="preserve"> 195.</s> <s xml:id="echoid-s20227" xml:space="preserve">25 linea.</s> <s xml:id="echoid-s20228" xml:space="preserve"> 200.</s> <s xml:id="echoid-s20229" xml:space="preserve">14 patet.</s> <s xml:id="echoid-s20230" xml:space="preserve"> 204.</s> <s xml:id="echoid-s20231" xml:space="preserve">43 <lb/>lineæ.</s> <s xml:id="echoid-s20232" xml:space="preserve"> 222.</s> <s xml:id="echoid-s20233" xml:space="preserve">56 poſt punctum adde literam e.</s> <s xml:id="echoid-s20234" xml:space="preserve"> 238.</s> <s xml:id="echoid-s20235" xml:space="preserve">6 poſt a b k pone colon.</s> <s xml:id="echoid-s20236" xml:space="preserve"> 244.</s> <s xml:id="echoid-s20237" xml:space="preserve">27 alterum fi-<lb/>gnum parentheſeos poſt circulum dele, & pone poſt circuli.</s> <s xml:id="echoid-s20238" xml:space="preserve"> Ibid.</s> <s xml:id="echoid-s20239" xml:space="preserve"> poſt cathetus tolle comma.</s> <s xml:id="echoid-s20240" xml:space="preserve"> <lb/>Pag.</s> <s xml:id="echoid-s20241" xml:space="preserve">264 in figura continuetur recta z q in l.</s> <s xml:id="echoid-s20242" xml:space="preserve"> Pag.</s> <s xml:id="echoid-s20243" xml:space="preserve"> 266 figuræ 60 & 61 theorematum permu-<lb/>tatæ ſunt.</s> <s xml:id="echoid-s20244" xml:space="preserve"> 267.</s> <s xml:id="echoid-s20245" xml:space="preserve">56 quomodocunque.</s> <s xml:id="echoid-s20246" xml:space="preserve"> 272.</s> <s xml:id="echoid-s20247" xml:space="preserve">15 æquidiſtante.</s> <s xml:id="echoid-s20248" xml:space="preserve"> 282.</s> <s xml:id="echoid-s20249" xml:space="preserve">49 pro 21,1.</s> <s xml:id="echoid-s20250" xml:space="preserve"> 290.</s> <s xml:id="echoid-s20251" xml:space="preserve">ult.</s> <s xml:id="echoid-s20252" xml:space="preserve">pro linea <lb/>repone ſuperficie.</s> <s xml:id="echoid-s20253" xml:space="preserve"> 295.</s> <s xml:id="echoid-s20254" xml:space="preserve">24 in qua.</s> <s xml:id="echoid-s20255" xml:space="preserve"> 306.</s> <s xml:id="echoid-s20256" xml:space="preserve">48 ergo perpendicularis.</s> <s xml:id="echoid-s20257" xml:space="preserve"> 312.</s> <s xml:id="echoid-s20258" xml:space="preserve">40 poſt &, adde ducatur.</s> <s xml:id="echoid-s20259" xml:space="preserve"> <lb/>320.</s> <s xml:id="echoid-s20260" xml:space="preserve">56 linea z h.</s> <s xml:id="echoid-s20261" xml:space="preserve"> 327.</s> <s xml:id="echoid-s20262" xml:space="preserve"> ulti.</s> <s xml:id="echoid-s20263" xml:space="preserve">pro 2,3.</s> <s xml:id="echoid-s20264" xml:space="preserve"> 343.</s> <s xml:id="echoid-s20265" xml:space="preserve">28 angulo b g d.</s> <s xml:id="echoid-s20266" xml:space="preserve"> 346.</s> <s xml:id="echoid-s20267" xml:space="preserve">58 quodam arcu, fimili arcui.</s> <s xml:id="echoid-s20268" xml:space="preserve"> <lb/>Pag.</s> <s xml:id="echoid-s20269" xml:space="preserve"> 352 in prima figura litera è regione m obſcurior, eſt r.</s> <s xml:id="echoid-s20270" xml:space="preserve"> Ibid.</s> <s xml:id="echoid-s20271" xml:space="preserve"> ult.</s> <s xml:id="echoid-s20272" xml:space="preserve">linea uerò.</s> <s xml:id="echoid-s20273" xml:space="preserve"> 357.</s> <s xml:id="echoid-s20274" xml:space="preserve">20 pro 11, 1.</s> <s xml:id="echoid-s20275" xml:space="preserve"> <lb/>Ibid.</s> <s xml:id="echoid-s20276" xml:space="preserve"> in figura litera proximè infra r in linea g r obſcurior, eſt k.</s> <s xml:id="echoid-s20277" xml:space="preserve"> 358.</s> <s xml:id="echoid-s20278" xml:space="preserve">43 linea.</s> <s xml:id="echoid-s20279" xml:space="preserve"> 326.</s> <s xml:id="echoid-s20280" xml:space="preserve">26 primum.</s> <s xml:id="echoid-s20281" xml:space="preserve"> <lb/>fignũ parentheſeos ante quia inuerſum corrige.</s> <s xml:id="echoid-s20282" xml:space="preserve"> Ibid.</s> <s xml:id="echoid-s20283" xml:space="preserve"> ult.</s> <s xml:id="echoid-s20284" xml:space="preserve"> lineæ x g ad lineam g s.</s> <s xml:id="echoid-s20285" xml:space="preserve"> 370.</s> <s xml:id="echoid-s20286" xml:space="preserve">19 quod <lb/>eſt c.</s> <s xml:id="echoid-s20287" xml:space="preserve"> 374.</s> <s xml:id="echoid-s20288" xml:space="preserve">15 ipſa.</s> <s xml:id="echoid-s20289" xml:space="preserve"> 379.</s> <s xml:id="echoid-s20290" xml:space="preserve">43 ſitq́ue.</s> <s xml:id="echoid-s20291" xml:space="preserve"> 380.</s> <s xml:id="echoid-s20292" xml:space="preserve">46 p z k ſint.</s> <s xml:id="echoid-s20293" xml:space="preserve"> 395.</s> <s xml:id="echoid-s20294" xml:space="preserve">24 tolle literam m.</s> <s xml:id="echoid-s20295" xml:space="preserve"> 410.</s> <s xml:id="echoid-s20296" xml:space="preserve">54 ſuper.</s> <s xml:id="echoid-s20297" xml:space="preserve"> <lb/>415.</s> <s xml:id="echoid-s20298" xml:space="preserve">48 poſt forma adde extenditur.</s> <s xml:id="echoid-s20299" xml:space="preserve"> 438.</s> <s xml:id="echoid-s20300" xml:space="preserve">36 lineæ a k & a l.</s> <s xml:id="echoid-s20301" xml:space="preserve"> 472.</s> <s xml:id="echoid-s20302" xml:space="preserve">42 diſpofitam.</s> <s xml:id="echoid-s20303" xml:space="preserve"/> </p> <pb file="0302" n="302"/> <pb o="1" file="0303" n="303"/> </div> <div xml:id="echoid-div650" type="section" level="0" n="0"> <head xml:id="echoid-head565" xml:space="preserve">VERITATIS AMA-<lb/>TORI FRATRI GVILIELMO DE</head> <head xml:id="echoid-head566" xml:space="preserve">MORBETA, VITELLO FILIVS THVRINGORVM ET</head> <head xml:id="echoid-head567" xml:space="preserve" style="it">Polonorum, æternæ lucis irrefracto mentis radio fælicem intuitum, <lb/>& intellectum perſpicuum ſubſcriptorum.</head> <p> <s xml:id="echoid-s20304" xml:space="preserve">VNIVERSALIVM entium ſtudioſus amor te uinctum de-<lb/>tinens, me tibi, utidem appetentem, ſic coniunxit, ut uoluntas <lb/>tua mihi ſit imperium:</s> <s xml:id="echoid-s20305" xml:space="preserve"> me uoluntas quoq;</s> <s xml:id="echoid-s20306" xml:space="preserve"> tua arceat ab affe-<lb/>ctibus tibi diſplicentium paſsionum.</s> <s xml:id="echoid-s20307" xml:space="preserve"> Quia ergo tibi, ut totius <lb/>entis ſedulo ſcrutatori (dũ ens intelligibile à primis ſuis pro-<lb/>diens principijs, entibus indiuiduis ſenſibilib.</s> <s xml:id="echoid-s20308" xml:space="preserve"> per modum cau <lb/>ſæ, actu mẽtis coniungeres, & ſingulorum cauſſas ſingulas in-<lb/>dagares) occurrit diuinarum uirtutum influentiam inferiori-<lb/>bus rebus corporalib.</s> <s xml:id="echoid-s20309" xml:space="preserve"> per uirtutes corporales ſuperiores modo mirabilifieri.</s> <s xml:id="echoid-s20310" xml:space="preserve"> Nec <lb/>enim res corporeæ inferiores in ordine partium uniuerſi, diuinæ uirtutis incorpo-<lb/>raliter ſunt participes, ſed per ſuperiora ſui ordinis, contractam uirtutem partici-<lb/>pant, ut poſſunt:</s> <s xml:id="echoid-s20311" xml:space="preserve"> ſicut & in alio ſubſtantiarum intellectiuarũ ordine inferiores ſub-<lb/>ſtantias perſuperiorum ſui ordinis illuſtrationem à fonte diuinæ bonitatis deriua-<lb/>tam, prout uniuſcuiuſque natura fert, per modum intelligibilium influentiarũ fie-<lb/>ri, mentis acumine perſpexiſti:</s> <s xml:id="echoid-s20312" xml:space="preserve"> Sic, ut omnis rerum entitas à diuina profluat entita <lb/>te, & omnis intelligibilitas ab intelligentia diuina, omnisq́;</s> <s xml:id="echoid-s20313" xml:space="preserve"> uitalitas à diuina uita:</s> <s xml:id="echoid-s20314" xml:space="preserve"> <lb/>quarum influentiarum diuinum lumen per modum intelligibilem eſt principium, <lb/>medium & finis:</s> <s xml:id="echoid-s20315" xml:space="preserve"> ut à quo, & per quod, & ad quod omnia diſponuntur.</s> <s xml:id="echoid-s20316" xml:space="preserve"> Corpora-<lb/>lium uerò influentiarum lumen ſenſibile, eſt medium, ſuperioribus corporib.</s> <s xml:id="echoid-s20317" xml:space="preserve"> per-<lb/>petuis ſecundum ſubſtantiam ſolum in potentia ad ubi exiſtentibus, infima corpo <lb/>ra (quæ ſecundum formas, & ubi uariantur) mirificè aſſimilans & connectens.</s> <s xml:id="echoid-s20318" xml:space="preserve"> Eſt <lb/>enim lumen ſupremarum formarum corporalium diffuſio per naturam corpora-<lb/>lis formæ materijs inferiorum corporum ſe applicans, & ſecum delatas formas di <lb/>uinorum & in diuiſibilium artificum per modum diuiſibilem caducis corporibus <lb/>imprimens, ſuiq́;</s> <s xml:id="echoid-s20319" xml:space="preserve"> cum illis incorporatione nouas ſemper formas ſpecificas aut in-<lb/>diuiduas producens, in quibus reſultat per actum luminis diuinum artificium tam <lb/>motorum orbium quàm mouentium uirtutum.</s> <s xml:id="echoid-s20320" xml:space="preserve"> Quia itaque lumen corporalis <lb/>formæ actum habet:</s> <s xml:id="echoid-s20321" xml:space="preserve"> corporalibus dimenſionib.</s> <s xml:id="echoid-s20322" xml:space="preserve"> corporum (quibus influit) ſe coę <lb/>quat, & extenſione capacium corporum ſe extendit:</s> <s xml:id="echoid-s20323" xml:space="preserve"> attamen quia fontem (à quo <lb/>profluit) habet ſemper ſecundum ſuæ uirtutis exordium:</s> <s xml:id="echoid-s20324" xml:space="preserve"> proſpicere dimenſionem <lb/>diſtantiæ (quæ eſt linea recta) per accidens aſſumit, ſicq́;</s> <s xml:id="echoid-s20325" xml:space="preserve"> ſibi nomen radij coaptat.</s> <s xml:id="echoid-s20326" xml:space="preserve"> <lb/>Et quoniam linea recta naturalis ſemper eſt in aliqua ſuperficie naturali:</s> <s xml:id="echoid-s20327" xml:space="preserve"> ſuperficie <lb/>rum uerò paſſio (quæ per terminantes lineas eis accidit) eſt angulus:</s> <s xml:id="echoid-s20328" xml:space="preserve"> ideo radio lu <lb/>minoſo conſideratio adiacet angularis:</s> <s xml:id="echoid-s20329" xml:space="preserve"> & rectis angulis radiorum perpendiculari <lb/>tas cſt cauſſa.</s> <s xml:id="echoid-s20330" xml:space="preserve"> Obliquatio uerò irradiantis corporis ſuper irradiatum corpus, acu-<lb/>tos cauſſat angulos & obtuſos:</s> <s xml:id="echoid-s20331" xml:space="preserve"> & ſecundum huiuſino di luminarium influentiæ ua <lb/>riantur.</s> <s xml:id="echoid-s20332" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s20333" xml:space="preserve"> tui ſolertis diligentia ingenij, ſecundum hæc, cæleſtium influẽ-<lb/>tiarum diuinam uirtutem reſpectu rerum capacium mutari proſpiceret, & non ſo-<lb/>lum ſecundum uirtutes agentes, ſed ſecundum diuerſitatem modi actionis, res a-<lb/>ctas diuerſari uideret:</s> <s xml:id="echoid-s20334" xml:space="preserve"> placuit tibi in illius rei occulta indagine uerſari, eiusq́;</s> <s xml:id="echoid-s20335" xml:space="preserve"> diligẽ <lb/>ti inquiſitioni ſtudioſam animam applicare.</s> <s xml:id="echoid-s20336" xml:space="preserve"> Libros itaq;</s> <s xml:id="echoid-s20337" xml:space="preserve"> ueterum tibi ſuper hoc <lb/>negotio pcrquirenti, occurrit tædium uerboſitatis arabicæ, implicationis græcæ, <lb/>paucitas quoq;</s> <s xml:id="echoid-s20338" xml:space="preserve"> exarationis latinæ, præſertim quia tibi commiſſum officiũ pœnitẽ-<lb/>tiariæ romanæ eccleſiæ, cuius curæ partem geris, credens plus intellectu practico <lb/>quàm ſpeculatiuo, pœnitẽtibus ſuccurrere, te cohibuit à multitudine uidendorũ:</s> <s xml:id="echoid-s20339" xml:space="preserve"> <lb/> <pb o="2" file="0304" n="304"/> maluiſti enim languentium animarum diuino antidoto languoribus ſuccurrere, <lb/>quàm ipſorum hominum ignorantias releuare:</s> <s xml:id="echoid-s20340" xml:space="preserve"> meq́;</s> <s xml:id="echoid-s20341" xml:space="preserve"> putans uacare otio, ſub amo <lb/>ris nexu, quo tibi coniungor, uoluiſti conſtringere, ut hoclaboris tibi placiti onus <lb/>ſubirem, hisq́;</s> <s xml:id="echoid-s20342" xml:space="preserve"> materijs mihi nondum cognitis, animum applicarem.</s> <s xml:id="echoid-s20343" xml:space="preserve"> Atego, qui <lb/>cunctis iuſſionibus tuis obtemperare deſidero, uelle tuum ſuſcipiens pro manda-<lb/>to, maioris negotij, quòd de ordine entium olim conſcribendum ſuſceperam ca-<lb/>pitulum, in tempus ſemoui, præſentisq́;</s> <s xml:id="echoid-s20344" xml:space="preserve"> operis diſpendium pro meæ poſſibilitatis <lb/>uiribus (quibus hic impar, fateor) adij conſcrib endum.</s> <s xml:id="echoid-s20345" xml:space="preserve"> Attendens quoq;</s> <s xml:id="echoid-s20346" xml:space="preserve">, quia ea-<lb/>dem uis formæ immittitur in contrarium & in ſenſum, & quòd lumen ſit primum <lb/>omnium formarum ſenſibilium, quodq́;</s> <s xml:id="echoid-s20347" xml:space="preserve"> rerum ſenſibilium omniũ cauſſas efficien <lb/>tes intendamus perquirere, quarum plurimas differẽtias uiſus nobis oſtendit:</s> <s xml:id="echoid-s20348" xml:space="preserve"> prę-<lb/>miſſorum permodum entium uiſibilium perſcrutatio placuit, ſicut & eadem uiris, <lb/>qui ante nos plurimi tractauerunt huius ſcientiæ negotiũ, PERSPECTIVORVM <lb/>nomine nuncupantibus, quorum & ego nominationem (ut placitam) approbo:</s> <s xml:id="echoid-s20349" xml:space="preserve"> li <lb/>cet plus ad naturalium formarum actionis modum occultiſſimum pertractãdum, <lb/>ut opus præſens tuis affectibus reſpondeat, ſcribentis intentio ſe declinet.</s> <s xml:id="echoid-s20350" xml:space="preserve"> Quòd <lb/>enim in ſenſu uiſus plus perceptibiliter agitur, hoc in ipſius ſenſus abſentia in reb.</s> <s xml:id="echoid-s20351" xml:space="preserve"> <lb/>naturalibus nullatenus euitatur.</s> <s xml:id="echoid-s20352" xml:space="preserve"> Senſus enim præſentia nihil addit actionibus na-<lb/>turalium formarum.</s> <s xml:id="echoid-s20353" xml:space="preserve"> Omnem itaq;</s> <s xml:id="echoid-s20354" xml:space="preserve"> modum uiſionis mathematica uel naturali de-<lb/>monſtratione tranſcurrendo, ea quæ de naturalibus formarum actionibus permo <lb/>dum paſsionum uiſibilium iuxta triplicem uidendi modum pro meę poſſibilitatis <lb/>modulo tractabo.</s> <s xml:id="echoid-s20355" xml:space="preserve"> In omnibus enim illis uidẽdi modis, formæ naturales ad uiſum <lb/>ſe diffundunt, radijq́;</s> <s xml:id="echoid-s20356" xml:space="preserve"> uiſuales non exeunt ad capeſſendas formas rerum.</s> <s xml:id="echoid-s20357" xml:space="preserve"> Vnde ſi <lb/>præſentiæ formarum diffuſarum per corpora naturalia ipſarum ſuſceptibilia, uiſus <lb/>non affuerit, non propter hoc naturalis actio non erit, ſed formæ in ſubiecta corpo <lb/>ra ſibi diſſimilia, impriment quãtum poſſunt.</s> <s xml:id="echoid-s20358" xml:space="preserve"> Tuitaq;</s> <s xml:id="echoid-s20359" xml:space="preserve"> uir deſideriorum omnis ſci-<lb/>entialis boni, ſuſcipe quod fieri mandaſti, in quo ſi quid incultum inueneris, perſpi <lb/>caciori ingenio modereris.</s> <s xml:id="echoid-s20360" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div651" type="section" level="0" n="0"> <head xml:id="echoid-head568" xml:space="preserve" style="it">[Sequentia deſunt in uetuſto exemplari.]</head> <head xml:id="echoid-head569" xml:space="preserve">TOTIVS OPERIS IN DECEM</head> <head xml:id="echoid-head570" xml:space="preserve">Libros diuiſio, & quid in ſingulis tractetur.</head> <p style="it"> <s xml:id="echoid-s20361" xml:space="preserve">PRAESENS ita negotium decem libris partialibus duximus diſtinguendũ.</s> <s xml:id="echoid-s20362" xml:space="preserve"> <lb/>Volentes enim omne ens uiſibile, utſuæ uiſibilitati paßio accidit, mathematica <lb/>demonſtratione concludere, & hac uia eatenus (ut nobis eſt poßibile) certius am-<lb/>bulare:</s> <s xml:id="echoid-s20363" xml:space="preserve"> librum hunc per ſe ſtantem effecimus, exceptis his, quæ ex elementis Eu-<lb/>clidis, & paucis, quæ ex conicis elementis Apollonij Pergæi dependent, quæ ſunt <lb/>ſolum duo, quibus in hac ſcientia ſumus uſi, ut in proceſſu poſtmodum patebit.</s> <s xml:id="echoid-s20364" xml:space="preserve"> In primo itaque <lb/>huius ſcientiæ libro axiomata præmittimus, quæ præter elementa Euclidis huic ſcientiæ ſunt ne <lb/>ceſſaria:</s> <s xml:id="echoid-s20365" xml:space="preserve"> & in hoc ea duo, quæ demonſtrata ſunt ab Apollonio, declaramus.</s> <s xml:id="echoid-s20366" xml:space="preserve"> Plurima tamen & <lb/>horum, quæ in hoc libro præmittimus, continentur in eo libro, quem de elementatis concluſioni <lb/>bus nominamus, in quo uniuerſaliter omnia conſcripſimus, quæ nobis uiſa ſunt, & quæ ad nos <lb/>peruenerunt à uiris poſterioribus Euclide, pro particularium neceßitate ſcientiarum uniuerſa-<lb/>liter concluſa.</s> <s xml:id="echoid-s20367" xml:space="preserve"> In ſecundo libro, de modo proiectionis radiorum per medium unius diaphani, <lb/>uelplurium, ſuper figuras corporum diuerſas:</s> <s xml:id="echoid-s20368" xml:space="preserve"> necnon de proiectione umbrarum & figuratione <lb/>lucis cadentis per fenestras tractauimus, ut de his, quæ præambula ſunt actioni ſenſibili forma <lb/>rum natur alium, & quæ fiunt non exiſtente ſenſu.</s> <s xml:id="echoid-s20369" xml:space="preserve"> In tertio uerò libro de organo uiſus, dé eſ-<lb/>ſentiali modo uidendi ſuo modo tract auimus, ut patitur ſcientia opticorum.</s> <s xml:id="echoid-s20370" xml:space="preserve"> In quarto quoque <lb/>libro percurrimus deceptiones, quæ accidunt uiſui ſecundum directum modum uidendi per u-<lb/>num medium, ſiue ſint paßiones mathematicæ, ſiue etiam naturales.</s> <s xml:id="echoid-s20371" xml:space="preserve"> In quinto autẽ libro nos <lb/>ad alium modum uidendi, qui fit per reflexiones à politis corporibus, quæ ſpecula dicimus, tràſ-<lb/>ferentes, tract auimus de paßionibus communibus omni ſpeculo, ſiue ſit planum, ſiue ſphæricũ;</s> <s xml:id="echoid-s20372" xml:space="preserve"> <lb/>columnare ſiue pyramidale, concauum uel conuexum.</s> <s xml:id="echoid-s20373" xml:space="preserve"> Hæc enim ſunt omnia ſpecula, à quibus <lb/> <pb o="3" file="0305" n="305"/> regularis poteſt fieri reflexio, ut nos declarabimus ſuo loco:</s> <s xml:id="echoid-s20374" xml:space="preserve"> nectamen intelligimus per hæc ſpe-<lb/>cula ſolùm corpor a polita artificio, ſed potius per naturam.</s> <s xml:id="echoid-s20375" xml:space="preserve"> Quia dum demonſtrationem his ſpe <lb/>culis applicamus, natur alia corpor a eiuſdem figuræ intelligimus.</s> <s xml:id="echoid-s20376" xml:space="preserve"> Quòd enim in artificialib.</s> <s xml:id="echoid-s20377" xml:space="preserve">cor <lb/>poribus exemplariter accidit, in corporibus natur alibus certius accidere neceſſe est.</s> <s xml:id="echoid-s20378" xml:space="preserve"> Et dum ſic <lb/>per figur as ſpeculorum diſcurrimus, cæleſtes & omnes natur ales influentias à ſubiectis corpo-<lb/>ribus ſub quodam reflexionis modo ad alia corpora declaramus.</s> <s xml:id="echoid-s20379" xml:space="preserve"> In his enim diuerſit atibus la-<lb/>tens eſt naturæ operatio:</s> <s xml:id="echoid-s20380" xml:space="preserve"> & ab eiſdem agentibus, ſecundum huius diuerſit atis modum, fit diuer <lb/>ſitas formarum, & accidit uiſibus, ſi ad locũ reflexionis deueniant, ut ad ipſos fiat reflexio:</s> <s xml:id="echoid-s20381" xml:space="preserve"> quo-<lb/>niam uiſibus, ut quodam poſteriori formis natur alibus & corporibus exiſtentibus, ipſorum præ <lb/>ſentia rebus natur alibus nibiladdit.</s> <s xml:id="echoid-s20382" xml:space="preserve"> Horum it a ſpeculorum communes paßiones, & omnes <lb/>proprietates ſpeculorum planorum in quinto libro propoſuimus.</s> <s xml:id="echoid-s20383" xml:space="preserve"> In ſexto uerò libro demonſtra-<lb/>uimus paßiones, quæ accidunt uiſibus & rebus ex reflexione facta à ſpeculis ſphæricis conuexis.</s> <s xml:id="echoid-s20384" xml:space="preserve"> <lb/>In ſeptimo uerò poſuimus paßiones accidentium à ſpeculis columnaribus uel pyramidalibus <lb/>conuexis:</s> <s xml:id="echoid-s20385" xml:space="preserve"> & hæc duo ſpecula ſimul coniunximus propter conformitatem plurium paßionum.</s> <s xml:id="echoid-s20386" xml:space="preserve"> <lb/>In octauo, de reflexionibus, quæ fiunt à ſpeculis ſphæricis concauis, prolixius tract auimus.</s> <s xml:id="echoid-s20387" xml:space="preserve"> In <lb/>nono quo, de his, quæ fiunt à ſpeculis columnaribus uel pyramidalibus concauis:</s> <s xml:id="echoid-s20388" xml:space="preserve"> & in eodem de <lb/>ſpeculis quibuſdam irregularibus, à quorum tot ali ſuperficie fit reflexio lucis & uirtutis ad pun <lb/>ctum unum (quæ ſpecula comburentia dicimus) adiunximus tract atum.</s> <s xml:id="echoid-s20389" xml:space="preserve"> In decimo uerò libro <lb/>huius ſcientiæ, egimus de tertio modo uidendi, quie eſt per medium alterius diaphani, ut cum per <lb/>aerem fit uiſio, ſub aqua, uel ſub uitro.</s> <s xml:id="echoid-s20390" xml:space="preserve"> Et de deceptionibus, quæ ex hoc accidunt uiſui:</s> <s xml:id="echoid-s20391" xml:space="preserve"> nam & ſi <lb/>uiſus non fuerit, eædem paßiones uirtuti accidunt agenti.</s> <s xml:id="echoid-s20392" xml:space="preserve"> Et in hoc quo decimo tractatu adie-<lb/>cimus paßionem ſoli uiſui accidentem ex diuerſitate mediorum, ut eſt impreßio arcus dæmonis, <lb/>qui dicitur iris:</s> <s xml:id="echoid-s20393" xml:space="preserve"> quoniam & illius generatio ex hac præſenti ſcientia ortum habet:</s> <s xml:id="echoid-s20394" xml:space="preserve"> ſić quaſio-<lb/>mnium uiſibilium gener alibus paßionibus pertractatis, operi finem damus.</s> <s xml:id="echoid-s20395" xml:space="preserve"> Patet it a expræ-<lb/>mißis, quòd triplex eſt modus uidendi:</s> <s xml:id="echoid-s20396" xml:space="preserve"> quidam per unum medium tantùm, qui eſt uiſio directa:</s> <s xml:id="echoid-s20397" xml:space="preserve"> <lb/>quidam uerò per reflexionem formarum uiſibilium à corporibus politis:</s> <s xml:id="echoid-s20398" xml:space="preserve"> quidam uerò per refra-<lb/>ctionem formarum uiſibilium propter diuerſitatẽ mediorum.</s> <s xml:id="echoid-s20399" xml:space="preserve"> Hiquo tres modiuidendi, ſignũ <lb/>ſunt triplicis actionis formarum & omnium uirtutum cœleſtium & naturalium.</s> <s xml:id="echoid-s20400" xml:space="preserve"> Quædame-<lb/>nim agunt directè in obiectum ſuſceptibile, & hæc actio eſt fortior, quoniam eſt directè intenta <lb/>per naturam, & fit ſecũdum lineas rectas.</s> <s xml:id="echoid-s20401" xml:space="preserve"> Accidit aũt illi uirtuti corporalis debilitas, propter <lb/>remotionem maiorem agentis ab ipſo acto:</s> <s xml:id="echoid-s20402" xml:space="preserve"> ſolenim non adeò calefacit remotiora, ſicut propin-<lb/>quiora calefactibilia, quæ ſunt eiuſdem diſpoſitionis.</s> <s xml:id="echoid-s20403" xml:space="preserve"> Alia uerò natur alis actio fit per reflexio-<lb/>nem à corporibus alijs, ut radij ſolis à corpore lunæ reflectuntur:</s> <s xml:id="echoid-s20404" xml:space="preserve"> quamuis enim propter rarit atẽ <lb/>lunaris corporis quiddam ſolaris tranſeat uirtutis:</s> <s xml:id="echoid-s20405" xml:space="preserve"> plurimi tamen radiorum reflectuntur infe-<lb/>rius, ut à ſpeculo ſphærico conuexo.</s> <s xml:id="echoid-s20406" xml:space="preserve"> Eſt ergo illi actioni conueniens omne, quod diximus in paßio <lb/>nibus ſpeculorum, aßimilante ſe figura corporis (à quo fit reflexio) figuræ ſpeculari.</s> <s xml:id="echoid-s20407" xml:space="preserve"> Tertia uerò <lb/>maneries natur alium actionum, eſt per plur a media diuerſorum diaphanorum, quæ ſimiliter <lb/>in ſuo modo agendi diuerſitatem accipit, quam uiſibus accidere dicemus.</s> <s xml:id="echoid-s20408" xml:space="preserve"> In his it a naturalib.</s> <s xml:id="echoid-s20409" xml:space="preserve"> <lb/>actionibus uiſus ſignum eſt, non cauſa, niſi fortè deceptio ſit per ſe proueniens in uiſu:</s> <s xml:id="echoid-s20410" xml:space="preserve"> quoniam <lb/>non exiſtente perceptione uiſiua, ijdem modi ſunt omnium natur alium actionum.</s> <s xml:id="echoid-s20411" xml:space="preserve"> His itaque <lb/>præmißis, aggrediamur intentum.</s> <s xml:id="echoid-s20412" xml:space="preserve"> Hoctamen legentem latere nolumus, quia dum ex libro ele-<lb/>mentorum Euclidis arguimus, ſola nominatione numeri libri & theorematis conten-<lb/>ti ſumus:</s> <s xml:id="echoid-s20413" xml:space="preserve"> dum uerò aliquid ex hoc noſlro libro adducimus, & nume-<lb/>rum & theorema huius libri nominamus.</s> <s xml:id="echoid-s20414" xml:space="preserve"/> </p> <pb o="4" file="0306" n="306"/> </div> <div xml:id="echoid-div652" type="section" level="0" n="0"> <head xml:id="echoid-head571" xml:space="preserve">VITELLONIS FI-<lb/>LII THVRINGORVM ET PO-<lb/>LONORVM OPTICAE LIBER PRIMVS.</head> <head xml:id="echoid-head572" xml:space="preserve" style="it">DEFINITIONES.</head> <p> <s xml:id="echoid-s20415" xml:space="preserve">OVAE uerò per modum principiorum huic primo libro præ-<lb/>mittimus, ſuntiſta.</s> <s xml:id="echoid-s20416" xml:space="preserve"> 1.</s> <s xml:id="echoid-s20417" xml:space="preserve"> Cathetum dicimus lineam perpendi-<lb/>cularẽ ſuper ſuperficiem aliquam, erctam.</s> <s xml:id="echoid-s20418" xml:space="preserve"> 2.</s> <s xml:id="echoid-s20419" xml:space="preserve"> Polum dicimus <lb/>omnem punctum lineæ ſuper ſuperficiem circuli à centro or-<lb/>thogonaliter erectæ.</s> <s xml:id="echoid-s20420" xml:space="preserve"> 3.</s> <s xml:id="echoid-s20421" xml:space="preserve"> Conuexam lineam uel ſuperficiem di-<lb/>cimus, quæ extrinſecus aliquam regularem curuitatem habet.</s> <s xml:id="echoid-s20422" xml:space="preserve"> <lb/>4.</s> <s xml:id="echoid-s20423" xml:space="preserve"> Lineam cõcauam uel ſuperficiem dicimus, quæ intrinſecus <lb/>aliquam regularem curuitatem habet.</s> <s xml:id="echoid-s20424" xml:space="preserve"> 5.</s> <s xml:id="echoid-s20425" xml:space="preserve"> Lineam ſuper ſuperficiem conuexam uel <lb/>concauam perpendicularem dicimus, quæ ſuper planam ſuperficiẽ in puncto ſuæ <lb/>incidentiæ ſuperficiem conuexam uel concauam contingentem eſt erecta.</s> <s xml:id="echoid-s20426" xml:space="preserve"> 6.</s> <s xml:id="echoid-s20427" xml:space="preserve"> Cir-<lb/>culi ſeinuicem ſecantes dicuntur, quorum diametris eſt aliqua linea communis, u-<lb/>no reliquum non continente.</s> <s xml:id="echoid-s20428" xml:space="preserve"> 7.</s> <s xml:id="echoid-s20429" xml:space="preserve"> Circulus magnus ſphęræ dicitur, qui tranſiens cen <lb/>trum ſphæræ, diuiditipſam in duo æqualia.</s> <s xml:id="echoid-s20430" xml:space="preserve"> 8.</s> <s xml:id="echoid-s20431" xml:space="preserve"> Minor uerò circulus ſphæræ dicitur, <lb/>qui neque tranſit centrum ſphæræ, neque diuiditipſam in duo æqualia.</s> <s xml:id="echoid-s20432" xml:space="preserve"> 9.</s> <s xml:id="echoid-s20433" xml:space="preserve"> Sphæras <lb/>æquales dicimus, quarum diametri ſunt æquales.</s> <s xml:id="echoid-s20434" xml:space="preserve"> 10.</s> <s xml:id="echoid-s20435" xml:space="preserve"> Sphæras uel circulos ſeinui-<lb/>cem continentes, ęquidiſtantes dicimus, inter quas à centro maioris ductæ lineæ <lb/>à conuexo minoris ad concauum maioris ſunt æquales.</s> <s xml:id="echoid-s20436" xml:space="preserve"> 11.</s> <s xml:id="echoid-s20437" xml:space="preserve"> Sphæras ſe inuicem cõ <lb/>tingentes dicimus, quæ ſe tangentes extrinſecus uelintrinſecus nõ ſecant.</s> <s xml:id="echoid-s20438" xml:space="preserve"> 12.</s> <s xml:id="echoid-s20439" xml:space="preserve"> Sphę <lb/>ras ſeinuicem interſecantes dicimus, cùm ſphęris ſe non continentibus, diameter <lb/>unius per alteram reſecatur.</s> <s xml:id="echoid-s20440" xml:space="preserve"> 13.</s> <s xml:id="echoid-s20441" xml:space="preserve"> Sphęras intrinſecus ſe interſecantes dicimus, qua-<lb/>rum maior pars unius in altera continetur.</s> <s xml:id="echoid-s20442" xml:space="preserve"> 14.</s> <s xml:id="echoid-s20443" xml:space="preserve"> Superficiem planam ſphæram con-<lb/>tingere dicimus, quæ cum ſphæram tangat, ad omnem partem educta, non ſecat.</s> <s xml:id="echoid-s20444" xml:space="preserve"> <lb/>15.</s> <s xml:id="echoid-s20445" xml:space="preserve"> Denominatio proportionis primi ad ſecundum, dicitur quantitas, quę ducta in <lb/>minorem producit maiorem:</s> <s xml:id="echoid-s20446" xml:space="preserve"> uel quæ maiorem diuidit ſecundum minorem.</s> <s xml:id="echoid-s20447" xml:space="preserve"> <lb/>16.</s> <s xml:id="echoid-s20448" xml:space="preserve"> Proportio dicitur componi ex duabus proportionibus, quando denominatio <lb/>illius proportionis producitur ex ductu denominationum illarum proportio-<lb/>num, unius in alteram.</s> <s xml:id="echoid-s20449" xml:space="preserve"/> </p> <figure> <variables xml:id="echoid-variables242" xml:space="preserve">b a c d</variables> </figure> </div> <div xml:id="echoid-div653" type="section" level="0" n="0"> <head xml:id="echoid-head573" xml:space="preserve" style="it">PETITIONES.</head> <p> <s xml:id="echoid-s20450" xml:space="preserve">Petimus autem hæc.</s> <s xml:id="echoid-s20451" xml:space="preserve"> 1.</s> <s xml:id="echoid-s20452" xml:space="preserve"> Aequales angulos ſuperidem <lb/>punctum conſtitutos, æqualem continere diſtantiam æ-<lb/>qualium linearum:</s> <s xml:id="echoid-s20453" xml:space="preserve"> ut ſi anguli a b c, & c b d ſint æquales, <lb/>& linea a b & b d ſint æquales:</s> <s xml:id="echoid-s20454" xml:space="preserve"> tantum diſtabit linea a b à li-<lb/>nea b c, quãtum linea b d diſtat ab eadem linea b c.</s> <s xml:id="echoid-s20455" xml:space="preserve"> 2.</s> <s xml:id="echoid-s20456" xml:space="preserve"> Item <lb/>inter quælibet duo puncta lineam, & inter quaslibet duas <lb/>lineas ſuperficiem poſſe extendi.</s> <s xml:id="echoid-s20457" xml:space="preserve"> 3.</s> <s xml:id="echoid-s20458" xml:space="preserve"> Item, cum duæ planæ <lb/>ſuperficies ſe contingunt, unã ex eis fieri ſuperficiem.</s> <s xml:id="echoid-s20459" xml:space="preserve"> 4.</s> <s xml:id="echoid-s20460" xml:space="preserve"> I-<lb/>tem duas planas ſuperficies corpus non includere.</s> <s xml:id="echoid-s20461" xml:space="preserve"> 5.</s> <s xml:id="echoid-s20462" xml:space="preserve"> Item <lb/>omnes eaſdem proportiones ex ſimilibus proportioni-<lb/>bus componi, & in ſimiles proportiones diuidi, & eaſdem habe-<lb/>re denominationes.</s> <s xml:id="echoid-s20463" xml:space="preserve"/> </p> <pb o="5" file="0307" n="307"/> </div> <div xml:id="echoid-div654" type="section" level="0" n="0"> <head xml:id="echoid-head574" xml:space="preserve">THEOREMATA<gap/></head> <head xml:id="echoid-head575" xml:space="preserve" style="it">1. Omnes lineæ æquidiſt antes in eadem ſuperficie plana neceſſariò conſiſtunt. <lb/>E' 35 definit. 1 element.</head> <p> <s xml:id="echoid-s20464" xml:space="preserve">Sint duæ lineæ æquidiſtantes, quæ a b & c d utcunque diſpoſitæ:</s> <s xml:id="echoid-s20465" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0307-01a" xlink:href="fig-0307-01"/> dico quòd ipſæ ſunt in eadem ſuperficie plana:</s> <s xml:id="echoid-s20466" xml:space="preserve"> copulentur enim per <lb/>lineam b d.</s> <s xml:id="echoid-s20467" xml:space="preserve"> Quoniam ergo lineæ a b & b d angulariter coniungun-<lb/>tur:</s> <s xml:id="echoid-s20468" xml:space="preserve"> palàm quoniam ipſæ ſunt in eadem ſuperficie per 2 p 11.</s> <s xml:id="echoid-s20469" xml:space="preserve"> Simi-<lb/>liter, quia lineę e d & b d angulariter coniunguntur, eruntipſæ in ea-<lb/>dem ſuperficie:</s> <s xml:id="echoid-s20470" xml:space="preserve"> Sed linea b d eſt in una tantum ſuperficie plana, quo-<lb/>niam ipſius partem eſſe in ſublimi, partem in plano, eſt impoſsibile ք <lb/>1 p 11.</s> <s xml:id="echoid-s20471" xml:space="preserve"> Palàm ergo, quoniam lineę a b & c d neceſſariò conſiſtunt in ea-<lb/>dem plana ſuperficie contenta inter eas & inter lineas, extremitates <lb/>illarum linearum copulantes:</s> <s xml:id="echoid-s20472" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s20473" xml:space="preserve"/> </p> <div xml:id="echoid-div654" type="float" level="0" n="0"> <figure xlink:label="fig-0307-01" xlink:href="fig-0307-01a"> <variables xml:id="echoid-variables243" xml:space="preserve">a c b d</variables> </figure> </div> </div> <div xml:id="echoid-div656" type="section" level="0" n="0"> <head xml:id="echoid-head576" xml:space="preserve" style="it">2. Lineam à puncto unius linearum æquidiſtantium in eadem <lb/>ſuperficie protr actam, cum alter a indefinitæ quantitatis concurre <lb/>re eſt neceſſe. Lemma Procli ad 29 p relement.</head> <p> <s xml:id="echoid-s20474" xml:space="preserve">Sint duæ lineæ æquidiſtantes, quæ a b & c d:</s> <s xml:id="echoid-s20475" xml:space="preserve"> quarum unam, ſcilicet a b, ſecet linea b e in puncte <lb/>b.</s> <s xml:id="echoid-s20476" xml:space="preserve"> Dico, quòd linea b e ſecabit etiam lineam c d.</s> <s xml:id="echoid-s20477" xml:space="preserve"> Quia enim linea c d <lb/> <anchor type="figure" xlink:label="fig-0307-02a" xlink:href="fig-0307-02"/> indefinitæ quantitatis eſſe ſupponitur, protrahatur uerſus ipſam li-<lb/>nea b e:</s> <s xml:id="echoid-s20478" xml:space="preserve"> quę ſi concurrit cum c d, habetur propoſitum.</s> <s xml:id="echoid-s20479" xml:space="preserve"> Sinon concur-<lb/>rat:</s> <s xml:id="echoid-s20480" xml:space="preserve"> palàm per definitionem æquidiſtantium linearum, quoniam linea <lb/>b e eſt æquidiſtans lineæ c d:</s> <s xml:id="echoid-s20481" xml:space="preserve"> & quia lineæ a b & b e ambę ſunt æquidi-<lb/>ſtãtes lineę c d:</s> <s xml:id="echoid-s20482" xml:space="preserve"> erit per 30 p 1 linea e b ęquidiſtans lineę a b:</s> <s xml:id="echoid-s20483" xml:space="preserve"> ſed palã ex <lb/>hypotheſi, quoniam concurrunt, ut in puncto b:</s> <s xml:id="echoid-s20484" xml:space="preserve"> non ergo ęquidiſtat li <lb/>nea b e lineę c d:</s> <s xml:id="echoid-s20485" xml:space="preserve"> ergo neceſſariò cõcurrit linea b e cum linea c d:</s> <s xml:id="echoid-s20486" xml:space="preserve"> quod <lb/>eſt propoſitum.</s> <s xml:id="echoid-s20487" xml:space="preserve"/> </p> <div xml:id="echoid-div656" type="float" level="0" n="0"> <figure xlink:label="fig-0307-02" xlink:href="fig-0307-02a"> <variables xml:id="echoid-variables244" xml:space="preserve">c a b d e</variables> </figure> </div> </div> <div xml:id="echoid-div658" type="section" level="0" n="0"> <head xml:id="echoid-head577" xml:space="preserve" style="it">3. Datis tribus lineis, cuilibet tertiæ ſecundum proportionẽ alia-<lb/>rum duarum proportionalem inuenire. E' 12 p 6 element.</head> <p> <s xml:id="echoid-s20488" xml:space="preserve">Sint datæ tres lineæ, quę ſint a b, c d, e f, quarum uni ut a b, ſecun-<lb/>dum proportionem aliarum duarum, quę ſunt c d & e f, quarta propor <lb/>tionalis debeat inueniri.</s> <s xml:id="echoid-s20489" xml:space="preserve"> Duæ itaque lineæ æquales duabus lineis, <lb/>quæ ſunt c d & e f, ab una linea continua abſcin dantur, quę ſit a e f per <lb/>3 p 1, & illi lineę a e fangulariter tertia data ſcilicet a b coniungatur in puncto a:</s> <s xml:id="echoid-s20490" xml:space="preserve"> & à puncto commu <lb/>ni diſtinguẽte duas lineas reſectas, (quod ſit punctum e) ducatur li-<lb/> <anchor type="figure" xlink:label="fig-0307-03a" xlink:href="fig-0307-03"/> nea e b a d extremitatem tertię datarum, quę eſt a b:</s> <s xml:id="echoid-s20491" xml:space="preserve"> & à puncto f du-<lb/>catur linea ęquidiſtans lineę e b per 31 p 1, quę ſit f g.</s> <s xml:id="echoid-s20492" xml:space="preserve"> Deinde protraha <lb/>tur linea a b in cõtinuum & directum, quouſque ſecet lineã f g:</s> <s xml:id="echoid-s20493" xml:space="preserve"> ſeca-<lb/> <anchor type="figure" xlink:label="fig-0307-04a" xlink:href="fig-0307-04"/> bit aũt per pręmiſſam:</s> <s xml:id="echoid-s20494" xml:space="preserve"> ſit itaq;</s> <s xml:id="echoid-s20495" xml:space="preserve"> punctus cõcurſus g.</s> <s xml:id="echoid-s20496" xml:space="preserve"> Dico, quod per 2 <lb/>p 6 eadem eſt proportio lineę a b ad lineam b g, quę eſt lineę a e datę <lb/>ad lineam e f datam.</s> <s xml:id="echoid-s20497" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s20498" xml:space="preserve"> de qualibet aliarum reſpectu re <lb/>liquarum duarum demonſtrari poteſt:</s> <s xml:id="echoid-s20499" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s20500" xml:space="preserve"/> </p> <div xml:id="echoid-div658" type="float" level="0" n="0"> <figure xlink:label="fig-0307-03" xlink:href="fig-0307-03a"> <variables xml:id="echoid-variables245" xml:space="preserve">a b c d e f</variables> </figure> <figure xlink:label="fig-0307-04" xlink:href="fig-0307-04a"> <variables xml:id="echoid-variables246" xml:space="preserve">a e b f g</variables> </figure> </div> </div> <div xml:id="echoid-div660" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables247" xml:space="preserve">a b c d g c d g f</variables> </figure> <head xml:id="echoid-head578" xml:space="preserve" style="it">4. Cum duabus lineis inæqualibus notæ proportionis, æqualiũ <lb/>linearum facta fuerit ad <lb/> ditio: maioris adminorẽ minuitur proportio. Ex 8 p 5 element.</head> <p> <s xml:id="echoid-s20501" xml:space="preserve">Sint duæ lineæ a b & c d <lb/>inæquales, notæ propor-<lb/>tionis:</s> <s xml:id="echoid-s20502" xml:space="preserve"> ſitq́ue linea a b ma-<lb/>ior quàm linea c d:</s> <s xml:id="echoid-s20503" xml:space="preserve"> adda-<lb/>tur quoq;</s> <s xml:id="echoid-s20504" xml:space="preserve"> linea b e ipſi a b, <lb/>& linea d f ipſi c d:</s> <s xml:id="echoid-s20505" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s20506" xml:space="preserve"> lineę b e & d f ęquales.</s> <s xml:id="echoid-s20507" xml:space="preserve"> Dico, quòd minor eſt proportio lineę a e ad lineam <lb/>c f, quàm lineę a b ad lineam c d.</s> <s xml:id="echoid-s20508" xml:space="preserve"> Quoniam enim datę ſunt tres lineę, quę ſunt a b & c d & b e:</s> <s xml:id="echoid-s20509" xml:space="preserve"> inue-<lb/>niatur per pręcedẽtem linea proportionalis lineę b e, ſecundum proportionem linearum a b & c d, <lb/>quę ſit d g.</s> <s xml:id="echoid-s20510" xml:space="preserve"> Quia ergo linea a b eſt maior quàm linea c d, patet, quia linea b e eſt maior quã linea d g:</s> <s xml:id="echoid-s20511" xml:space="preserve"> <lb/>ergo & linea d f eſt maior quã linea d g.</s> <s xml:id="echoid-s20512" xml:space="preserve"> Abſcindatur ergo per 3 p 1 è linea d f ęqualis ipſi d g.</s> <s xml:id="echoid-s20513" xml:space="preserve"> Quia er-<lb/>go eſt proportio lineę a b ad lineam c d, ſicut lineę b e ad lineam d g:</s> <s xml:id="echoid-s20514" xml:space="preserve"> erit per 15 p 5 proportio totius <lb/>lineę a e ad totalem lineã c g, ſicut lineę a b ad lineam c d:</s> <s xml:id="echoid-s20515" xml:space="preserve"> ſed per 8 p 5 minor eſt proportio lineæ a e <lb/> <pb o="6" file="0308" n="308" rhead="VITELLONIS OPTICAE"/> ad lineam c f maiorem, quàm ad lineam c g minorem:</s> <s xml:id="echoid-s20516" xml:space="preserve"> eſt ergo maior proportio lineæ a b ad linea m <lb/>c d, quàm lineę a e ad lineam c f:</s> <s xml:id="echoid-s20517" xml:space="preserve"> & hoc eſt propoſitum.</s> <s xml:id="echoid-s20518" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div661" type="section" level="0" n="0"> <head xml:id="echoid-head579" xml:space="preserve" style="it">5. Cum fuerit proportio primi ad ſecundum, tanquam tertij ad quartũ: erit è contrario pro-<lb/>portio ſecundi ad primum, ſicut quarti ad tertium. E' 13 def. & conſectario 4 p 5 element.</head> <p> <s xml:id="echoid-s20519" xml:space="preserve">Sit enim a primum, & b ſecundum, & ctertium, & d quartum:</s> <s xml:id="echoid-s20520" xml:space="preserve"> & ſit <lb/>proportio a ad b, ſicut c ad d.</s> <s xml:id="echoid-s20521" xml:space="preserve"> Dico, quòd erit è contrario proportio b ad <lb/> <anchor type="figure" xlink:label="fig-0308-01a" xlink:href="fig-0308-01"/> a, ſicut d ad c.</s> <s xml:id="echoid-s20522" xml:space="preserve"> Quoniam enim eſt proportio a ad b, ſicut c ad d:</s> <s xml:id="echoid-s20523" xml:space="preserve"> erit per 16 <lb/>p 5 permutatim proportio a ad c, ſicut b ad d:</s> <s xml:id="echoid-s20524" xml:space="preserve"> eſt ergo proportio b ad d, <lb/>ſicut a ad c:</s> <s xml:id="echoid-s20525" xml:space="preserve"> ergo iterum per 16 p 5 erit permutatim proportio b ad a, ſi-<lb/>cut d ad c, ſecundi uidelicet ad primum, ſicut quarti ad tertium:</s> <s xml:id="echoid-s20526" xml:space="preserve"> quod eſt <lb/>propoſitum.</s> <s xml:id="echoid-s20527" xml:space="preserve"/> </p> <div xml:id="echoid-div661" type="float" level="0" n="0"> <figure xlink:label="fig-0308-01" xlink:href="fig-0308-01a"> <variables xml:id="echoid-variables248" xml:space="preserve">a b c d</variables> </figure> </div> </div> <div xml:id="echoid-div663" type="section" level="0" n="0"> <head xml:id="echoid-head580" xml:space="preserve" style="it">6. Cum fuerit quatuor quantitatum proportio primæ ad ſecundã <lb/>maior, quãtertiæ ad quartam: erit è contr ario minor proportio ſecun-<lb/>dæ ad primam, quàm quartæ ad tertiam. 26 p 5 element. in Campano.</head> <p> <s xml:id="echoid-s20528" xml:space="preserve">Eſto proportio lineæ a ad lineam b maior, quàm lineæ c ad lineam d.</s> <s xml:id="echoid-s20529" xml:space="preserve"> Dico, quôd erit è contrario <lb/>minor proportio lineæ b ad lineam a, quàm lineæ d ad li-<lb/> <anchor type="figure" xlink:label="fig-0308-02a" xlink:href="fig-0308-02"/> neam c.</s> <s xml:id="echoid-s20530" xml:space="preserve"> Sit enim per 3 huius, ut, quæ eſt proportio lineæ c <lb/>ad lineam d, eadem ſit lineæ e ad lineam b.</s> <s xml:id="echoid-s20531" xml:space="preserve"> Quia ergo ma-<lb/>ior eſt proportio lineæ a ad lineam b, quàm lineæ c ad li-<lb/>neam d ex hypotheſi:</s> <s xml:id="echoid-s20532" xml:space="preserve"> patet, quòd minor eſt proportio li-<lb/>neæ e ad lineam b, quam lineę a ad lineã b:</s> <s xml:id="echoid-s20533" xml:space="preserve"> ergo per 10 p 5 <lb/>linea a eſt maior quã linea e.</s> <s xml:id="echoid-s20534" xml:space="preserve"> Et quia eſt proportio lineæ e <lb/>ad lineam b, ſicut lineę c ad lineam d, erit per præmiſſam <lb/>eadem proportio lineę b ad lineã e, quę lineæ d ad lineam <lb/>c.</s> <s xml:id="echoid-s20535" xml:space="preserve"> Eſt autem per 8 p 5 minor proportio lineæ b ad lineam a, <lb/>quàm ad lineam e:</s> <s xml:id="echoid-s20536" xml:space="preserve"> eſt ergo minor proportio lineæ b ad lineam a, quàm lineę d ad lineam c:</s> <s xml:id="echoid-s20537" xml:space="preserve"> quod <lb/>eſt propoſitum.</s> <s xml:id="echoid-s20538" xml:space="preserve"/> </p> <div xml:id="echoid-div663" type="float" level="0" n="0"> <figure xlink:label="fig-0308-02" xlink:href="fig-0308-02a"> <variables xml:id="echoid-variables249" xml:space="preserve">a b e c d</variables> </figure> </div> </div> <div xml:id="echoid-div665" type="section" level="0" n="0"> <head xml:id="echoid-head581" xml:space="preserve" style="it">7. Si quatuor quantitatum proportion alium prima fuerit maior quãſecunda, & tertia ma-<lb/>ior quã quarta: erit euerſim eadem proportio primæ ad augmentum ſui ſuper ſecundam, quæ ter <lb/>tiæ ad augmentum ſui ſuper quartam. E' 16 definit. & conſectario 19 p 5.</head> <p> <s xml:id="echoid-s20539" xml:space="preserve">Sint quatuor lineę proportionales a c prima:</s> <s xml:id="echoid-s20540" xml:space="preserve"> b c ſecunda:</s> <s xml:id="echoid-s20541" xml:space="preserve"> d ftertia:</s> <s xml:id="echoid-s20542" xml:space="preserve"> & e f quarta.</s> <s xml:id="echoid-s20543" xml:space="preserve"> Sitq́ue linea a b <lb/>maior quàm linea b c, & linea d f maior, quàm linea e f:</s> <s xml:id="echoid-s20544" xml:space="preserve"> ex-<lb/> <anchor type="figure" xlink:label="fig-0308-03a" xlink:href="fig-0308-03"/> cedat quoque linea a c lineam b c, in linea a b, & linea d f <lb/>lineam e f, in linea d e.</s> <s xml:id="echoid-s20545" xml:space="preserve"> Dico, quòd eadem erit proportio <lb/>lineę a c ad lineam a b, quę lineę d f ad lineam d e.</s> <s xml:id="echoid-s20546" xml:space="preserve"> Quo-<lb/>niam enim eſt proportio lineę a c ad lineam b c, ſicut lineę <lb/>d f ad lineam e f:</s> <s xml:id="echoid-s20547" xml:space="preserve"> eſt ergo per 16 p 5 permutatim proportio <lb/>lineę a c ad lineam d f, ſicut lineę b c ad lineam e f:</s> <s xml:id="echoid-s20548" xml:space="preserve"> ergo per <lb/>19 p 5 erit proportio lineę a b ad lineam d e, ſicut lineę a c <lb/>ad lineam d f:</s> <s xml:id="echoid-s20549" xml:space="preserve"> ergo per 16 p 5 erit proportio lineę a b ad lineam a c.</s> <s xml:id="echoid-s20550" xml:space="preserve"> ſicut lineę d e ad lineam d f.</s> <s xml:id="echoid-s20551" xml:space="preserve"> Ergo <lb/>per 5 huius erit proportio lineę a c ad lineam a b, ſicut lineę d fad lineam d e:</s> <s xml:id="echoid-s20552" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s20553" xml:space="preserve"/> </p> <div xml:id="echoid-div665" type="float" level="0" n="0"> <figure xlink:label="fig-0308-03" xlink:href="fig-0308-03a"> <variables xml:id="echoid-variables250" xml:space="preserve">a b c d e f</variables> </figure> </div> </div> <div xml:id="echoid-div667" type="section" level="0" n="0"> <head xml:id="echoid-head582" xml:space="preserve" style="it">8. Si quatuor quantit atum prima fuerit maior ſecunda, & tertia maior quarta: erit maior <lb/>proportio primæ ad quartam, quàm ſecundæ ad tertiam. Conſectarium ex 8 p 5 element.</head> <p> <s xml:id="echoid-s20554" xml:space="preserve">Sint quatuor lineę a, b, c, d:</s> <s xml:id="echoid-s20555" xml:space="preserve"> & ſit a prima maior quàm b ſecũda, & ſit c tertia maior, quàm d quar-<lb/>ta.</s> <s xml:id="echoid-s20556" xml:space="preserve"> Dico, quòd maior eſt proportio lineæ a, ad lineam d, quàm <lb/> <anchor type="figure" xlink:label="fig-0308-04a" xlink:href="fig-0308-04"/> lineę b ad lineam c.</s> <s xml:id="echoid-s20557" xml:space="preserve"> Quia enim linea c eſt maior quàm linea d <lb/>ex hypotheſi:</s> <s xml:id="echoid-s20558" xml:space="preserve"> patet per 8 p 5:</s> <s xml:id="echoid-s20559" xml:space="preserve"> quoniam maior eſt proportio li-<lb/>neę a ad lineam d, quàm ad lineam c:</s> <s xml:id="echoid-s20560" xml:space="preserve"> minor uero eſt proportio <lb/>lineę b ad lineã c, quàm lineę a, ad lineam c per eandem 8 p 5:</s> <s xml:id="echoid-s20561" xml:space="preserve"> <lb/>quoniam ut pręmiſſum eſt linea a eſt maior quàm linea b.</s> <s xml:id="echoid-s20562" xml:space="preserve"> Et <lb/>quoniam quicquid eſt maius maiore, eſt maius minore:</s> <s xml:id="echoid-s20563" xml:space="preserve"> patet, <lb/>quòd maior eſt proportio lineę a ad lineam d, quàm lineę b ad <lb/>lineam c:</s> <s xml:id="echoid-s20564" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s20565" xml:space="preserve"/> </p> <div xml:id="echoid-div667" type="float" level="0" n="0"> <figure xlink:label="fig-0308-04" xlink:href="fig-0308-04a"> <variables xml:id="echoid-variables251" xml:space="preserve">a b c d</variables> </figure> </div> </div> <div xml:id="echoid-div669" type="section" level="0" n="0"> <head xml:id="echoid-head583" xml:space="preserve" style="it">9. Cum quatuor quantitatum prima fuerit maior quàm <lb/>tertia, & ſecunda minor quàm quarta: maior erit proportio primæ ad ſecundam, quàm tertiæ <lb/>ad quartam. Conſectarium ex 8 p 5 element.</head> <p> <s xml:id="echoid-s20566" xml:space="preserve">Sint quatuor lineę a prima:</s> <s xml:id="echoid-s20567" xml:space="preserve"> b ſecũda:</s> <s xml:id="echoid-s20568" xml:space="preserve"> c tertia:</s> <s xml:id="echoid-s20569" xml:space="preserve"> d quarta:</s> <s xml:id="echoid-s20570" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s20571" xml:space="preserve"> a maior quàm c, & ſit b minor quã d.</s> <s xml:id="echoid-s20572" xml:space="preserve"> <lb/> <pb o="7" file="0309" n="309" rhead="LIBER I."/> Dico, quòd maior eſt proportio a ad b, quàm c ad d.</s> <s xml:id="echoid-s20573" xml:space="preserve"> Quoniã enim linea a eſt maior quàm linea c, pa <lb/>tet per 8 p 5, quoniã maior eſt ꝓportio lineę a ad lineã b quàm lineę c ad lineam b:</s> <s xml:id="echoid-s20574" xml:space="preserve"> ſed quia exhypo <lb/>theſi linea b eſt minor quàm linea d:</s> <s xml:id="echoid-s20575" xml:space="preserve"> patet per 8 p 5, quo-<lb/> <anchor type="figure" xlink:label="fig-0309-01a" xlink:href="fig-0309-01"/> niam maior eſt proportio lineæ c ad lineam b, quàm ad li-<lb/>neam d.</s> <s xml:id="echoid-s20576" xml:space="preserve"> Eſt ergo maior proportio lineæ a primæ ad lineã <lb/>b ſecũdã, ꝗ̃ lineæ c tertię ad d quartã:</s> <s xml:id="echoid-s20577" xml:space="preserve"> & hoc eſt propoſitũ.</s> <s xml:id="echoid-s20578" xml:space="preserve"/> </p> <div xml:id="echoid-div669" type="float" level="0" n="0"> <figure xlink:label="fig-0309-01" xlink:href="fig-0309-01a"> <variables xml:id="echoid-variables252" xml:space="preserve">a b c d</variables> </figure> </div> </div> <div xml:id="echoid-div671" type="section" level="0" n="0"> <head xml:id="echoid-head584" xml:space="preserve" style="it">10. Siquatuor quantitatum fuerit maior propor-<lb/>tio primæ ad ſecundam, quàm tertiæ ad quartam: erit <lb/>permutatim maior proportio primæ ad tertiam, quàm <lb/>ſecundæ ad quartam. E' 12 definit. 16 p 5. 27 p 5 elem. in <lb/>Campano.</head> <p> <s xml:id="echoid-s20579" xml:space="preserve">Sint quatuor lineæ a, b, c, d:</s> <s xml:id="echoid-s20580" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s20581" xml:space="preserve"> proportio a ad b maior, quàm c ad d.</s> <s xml:id="echoid-s20582" xml:space="preserve"> Dico, quòd erit permuta-<lb/>tim maior proportio lineæ a ad lineam c, quàm lineę b ad <lb/> <anchor type="figure" xlink:label="fig-0309-02a" xlink:href="fig-0309-02"/> lineam d.</s> <s xml:id="echoid-s20583" xml:space="preserve"> Sit enim per 3 huius proportio lineæ e ad lineã <lb/>b, ſicut lineæ c ad lineam d:</s> <s xml:id="echoid-s20584" xml:space="preserve"> erit ergo ex hypotheſi & ex 10 <lb/>p 5 linea e minor quã linea a:</s> <s xml:id="echoid-s20585" xml:space="preserve"> ergo per 8 p 5 maior eſt pro-<lb/>portio lineæ a ad lineam c, quàm lineæ e ad lineam c.</s> <s xml:id="echoid-s20586" xml:space="preserve"> Eſt <lb/>autem ex præmiſsis & per 16 p 5 proportio lineę e ad li-<lb/>neam c, ſicut lineę b ad lineam d.</s> <s xml:id="echoid-s20587" xml:space="preserve"> Palàm ergo, quoniã ma-<lb/>ior eſt proportio lineę a ad lineã c, quàm lineę b ad lineã <lb/>d:</s> <s xml:id="echoid-s20588" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s20589" xml:space="preserve"/> </p> <div xml:id="echoid-div671" type="float" level="0" n="0"> <figure xlink:label="fig-0309-02" xlink:href="fig-0309-02a"> <variables xml:id="echoid-variables253" xml:space="preserve">a b e c d</variables> </figure> </div> </div> <div xml:id="echoid-div673" type="section" level="0" n="0"> <head xml:id="echoid-head585" xml:space="preserve" style="it">11. Cum quatuor quantitatum maior fuerit propor <lb/>tio primæ ad ſecundam quàm tertiæ ad quartam: erit <lb/>coniunctim maior proportio primæ & ſecundæ ad ſecũdam, quàm tertiæ & quartæ ad quartã. <lb/>E' 14 definit. 18 p 5 element. 28 p 5 ele. in Campano.</head> <p> <s xml:id="echoid-s20590" xml:space="preserve">Eſto quatuor linearum a, b, c, d maior proportio a ad b, quàm c ad d.</s> <s xml:id="echoid-s20591" xml:space="preserve"> Dico, quòd totius lineę a b <lb/>ad lineã b maior erit proportio, quàm totius lineę c d ad <lb/> <anchor type="figure" xlink:label="fig-0309-03a" xlink:href="fig-0309-03"/> lineam d.</s> <s xml:id="echoid-s20592" xml:space="preserve"> Sit enim per 3 huius proportio lineę e ad lineam <lb/>b, quæ lineę c ad lineam d:</s> <s xml:id="echoid-s20593" xml:space="preserve"> eſt ergo ex hypotheſi maior ꝓ-<lb/>portio lineę a ad lineam b, quàm lineæ e ad lineam b:</s> <s xml:id="echoid-s20594" xml:space="preserve"> ergo <lb/>per 10 p 5 linea a eſt maior quàm lineae.</s> <s xml:id="echoid-s20595" xml:space="preserve"> Tota ergo linea <lb/>a b eſt maior quàm tota linea e b:</s> <s xml:id="echoid-s20596" xml:space="preserve"> ergo per 8 p 5 maior eſt <lb/>proportio totius lineæ a b ad lineã b, quàm totius lineę <lb/>e b ad lineã b:</s> <s xml:id="echoid-s20597" xml:space="preserve"> per 18 uerò 5 eſt proportio lineę e b ad line-<lb/>am b, quę lineę c d ad lineam d:</s> <s xml:id="echoid-s20598" xml:space="preserve"> eſt enim ex pręmiſsis pro-<lb/>portio lineę e ad lineam b, ſicut lineę c ad lineam d.</s> <s xml:id="echoid-s20599" xml:space="preserve"> Eſt <lb/>ergo maior proportio lineę a b ad lineã b, quàm lineę c d <lb/>ad lineam d:</s> <s xml:id="echoid-s20600" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s20601" xml:space="preserve"/> </p> <div xml:id="echoid-div673" type="float" level="0" n="0"> <figure xlink:label="fig-0309-03" xlink:href="fig-0309-03a"> <variables xml:id="echoid-variables254" xml:space="preserve">a b e c d</variables> </figure> </div> </div> <div xml:id="echoid-div675" type="section" level="0" n="0"> <head xml:id="echoid-head586" xml:space="preserve" style="it">12. Si quatuor quantitatum proportio primæ & ſecundæ ad ſecundam ſit maior, quàm ter-<lb/>tiæ & quartæ ad quartam: erit diſiunctim maior proportio primæ ad ſecundam, quàm tertiæ <lb/>ad quartam. E' 15 definit. 17 p 5 element. 29 p 5 elem. in Campano.</head> <p> <s xml:id="echoid-s20602" xml:space="preserve">Sit proportio totius lineę a b ad eius partem lineam b maior, quàm totius lineæ c d ad eius par-<lb/>tem d.</s> <s xml:id="echoid-s20603" xml:space="preserve"> Dico, quòd erit diſiunctim proportio lineę a ad line-<lb/> <anchor type="figure" xlink:label="fig-0309-04a" xlink:href="fig-0309-04"/> a m b maior, quàm lineę c ad lineam d.</s> <s xml:id="echoid-s20604" xml:space="preserve"> Sit en im per 3 huius <lb/>proportio lineę e b ad lineam b, ſicut lineę c d ad lineam d:</s> <s xml:id="echoid-s20605" xml:space="preserve"> <lb/>erit ergo ex hypotheſi maior proportio lineę a b ad lineam <lb/>b, quàm lineę e b ad eandem lineam b:</s> <s xml:id="echoid-s20606" xml:space="preserve"> ergo per 10 p 5 erit <lb/>linea a b maior quàm linea e b:</s> <s xml:id="echoid-s20607" xml:space="preserve"> a blata ergo utrobique linea <lb/>b communi, relinquitur linea a maior quàm linea e.</s> <s xml:id="echoid-s20608" xml:space="preserve"> Eſt er-<lb/>go per 8 p 5 maior proportio lineę a ad lineam b, quàm li-<lb/>neę e ad eandem lineam b:</s> <s xml:id="echoid-s20609" xml:space="preserve"> ſed per pręmiſſa eſt proportio li <lb/>neę e b ad lineam b, ſicut lineę c d ad lineam d:</s> <s xml:id="echoid-s20610" xml:space="preserve"> ergo per 17 <lb/>p 5 eſt proportio lineę e ad lineã b, ſicut lineę c ad lineam d.</s> <s xml:id="echoid-s20611" xml:space="preserve"> Erit ergo maior proportio lineę a ad li-<lb/>neam b.</s> <s xml:id="echoid-s20612" xml:space="preserve"> quàm lineę c ad lineam d:</s> <s xml:id="echoid-s20613" xml:space="preserve"> & hoc eſt propoſitum.</s> <s xml:id="echoid-s20614" xml:space="preserve"/> </p> <div xml:id="echoid-div675" type="float" level="0" n="0"> <figure xlink:label="fig-0309-04" xlink:href="fig-0309-04a"> <variables xml:id="echoid-variables255" xml:space="preserve">a b c e d</variables> </figure> </div> </div> <div xml:id="echoid-div677" type="section" level="0" n="0"> <head xml:id="echoid-head587" xml:space="preserve" style="it">13. Quarumlibet trium quantitatum quocun ordine diſpoſitarum, quarum mediæ ad <lb/>utram extremarum nota ſit proportio: erit proportio primæ adtertiam compoſit a ex propor-<lb/>tione primæ ad ſecũdam, & ſecundæ ad tertiam. Ex quo patet, quòd proportio extremorum ad <lb/>inuicem componitur ſemper ex proportione mediorum ad inuicem & adipſa extrema. E' ſcho-<lb/> <pb o="8" file="0310" n="310" rhead="VITELLONIS OPTICAE"/> lio Theonis ad 5 definit. 6 element. & commentarijs in 1 librum magnæ cõſtructionis Ptolemæi. <lb/>Item è commentarijs Eutocij in 8 theor. 2 de ſphæra & cylindro Archimedis.</head> <p> <s xml:id="echoid-s20615" xml:space="preserve">Sint extra gradus tres lineæ, quæ a, b, g, quarum prima (quæ eſt a) ſit maior quàm media (quæ <lb/>eſt b) & b ſit maior quàm tertia, quæ eſt g:</s> <s xml:id="echoid-s20616" xml:space="preserve"> ſit q́;</s> <s xml:id="echoid-s20617" xml:space="preserve"> ipſius b ad ambas extremas proportio nota.</s> <s xml:id="echoid-s20618" xml:space="preserve"> Dico, <lb/>quòd proportio lineæ a ad lineam g tertiam componitur ex proportione lineæ a ad lineam b, & ex <lb/>proportione lineæ b ad lineam g.</s> <s xml:id="echoid-s20619" xml:space="preserve"> Quoniam enim proportio lineæ a ad lineam b eſt nota:</s> <s xml:id="echoid-s20620" xml:space="preserve"> ſit quanti-<lb/>tas d denominatio illius proportionis:</s> <s xml:id="echoid-s20621" xml:space="preserve"> & ſimiliter quia proportio lineæ b ad lineam g eſt nota:</s> <s xml:id="echoid-s20622" xml:space="preserve"> ſit <lb/>denominatio illius proportionis quantitas e:</s> <s xml:id="echoid-s20623" xml:space="preserve"> & ſit quantitas z denominatio proportionis lineæ a <lb/>ad lineam g.</s> <s xml:id="echoid-s20624" xml:space="preserve"> Dico, quòd ex ductu e in d fit z.</s> <s xml:id="echoid-s20625" xml:space="preserve"> Quoniam enim per 15 definitionem huius ex ductu z <lb/>denominationis proportionis lineæ a ad lineam g in ipſam lineam g minorem, quàm ſit a, fit linea <lb/>a:</s> <s xml:id="echoid-s20626" xml:space="preserve"> & ſimiliter ex ductu d in lineam b fit linea a:</s> <s xml:id="echoid-s20627" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0310-01a" xlink:href="fig-0310-01"/> ponatur itaq;</s> <s xml:id="echoid-s20628" xml:space="preserve"> z primum & d ſecundum, linea b <lb/>tertiũ & linea g quartũ.</s> <s xml:id="echoid-s20629" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s20630" xml:space="preserve"> illud, quod <lb/>fit ex ductu primi in quartum, eſt ęquale ei, qđ <lb/>fit ex ductu ſecũdi in tertium:</s> <s xml:id="echoid-s20631" xml:space="preserve"> patet per 16 p 6 <lb/>quoniam eſt proportio primi ad ſecundum, ſi-<lb/>cut tertij ad quartum:</s> <s xml:id="echoid-s20632" xml:space="preserve"> eſt ergo proportio z ad <lb/>d, ſicut lineæ b ad lineam g:</s> <s xml:id="echoid-s20633" xml:space="preserve"> ergo denominatio <lb/>proportionis z ad d ex 5 ſuppoſitione eſt eadẽ <lb/>cum denominatione proportionis lineæ b ad <lb/>lineam g:</s> <s xml:id="echoid-s20634" xml:space="preserve"> ſed denominatio proportionis lineæ b ad lineam g eſt quantitas e:</s> <s xml:id="echoid-s20635" xml:space="preserve"> ergo denominatio ꝓ-<lb/>portionis z ad d eſt idẽ e:</s> <s xml:id="echoid-s20636" xml:space="preserve"> ergo ex ductu e in d fit z.</s> <s xml:id="echoid-s20637" xml:space="preserve"> Quia ergo denominatio proportionis lineę a ad <lb/>lineam g, quæ eſt z, producitur ex ductu denominationis proportionis lineæ a ad lineam b in de-<lb/>nominationem proportionis lineæ b ad lineam g:</s> <s xml:id="echoid-s20638" xml:space="preserve"> patet per 16 definitionem huius, quoniam pro-<lb/>portio lineæ a primæ ad lineam g tertiam componitur ex proportione lineæ a primæ ad lineam b <lb/>ſecundam, & ex proportione lineæ b ſecundæ ad lineam g tertiam:</s> <s xml:id="echoid-s20639" xml:space="preserve"> quod eſt propoſitum primum.</s> <s xml:id="echoid-s20640" xml:space="preserve"> <lb/>Eodem quoq;</s> <s xml:id="echoid-s20641" xml:space="preserve"> modo poteſt faciliter demonſtrari de quotcunq;</s> <s xml:id="echoid-s20642" xml:space="preserve"> medijs inter quęlibet duo extrema <lb/>collocatis:</s> <s xml:id="echoid-s20643" xml:space="preserve"> ſemper enim proportio extremorum ad inuicem componitur ex omnibus proportioni <lb/>bus mediorum ad inuicem, & ad ipſa extrema.</s> <s xml:id="echoid-s20644" xml:space="preserve"> Similiter demonſtrandum uia diuiſionis, ſi mediam <lb/>contingat eſſe maiorem qualibet extremarum:</s> <s xml:id="echoid-s20645" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s20646" xml:space="preserve"/> </p> <div xml:id="echoid-div677" type="float" level="0" n="0"> <figure xlink:label="fig-0310-01" xlink:href="fig-0310-01a"> <variables xml:id="echoid-variables256" xml:space="preserve">a b g d e z</variables> </figure> </div> </div> <div xml:id="echoid-div679" type="section" level="0" n="0"> <head xml:id="echoid-head588" xml:space="preserve" style="it">14. Si linea recta ſuper duas rect{as} ceciderit, fecerit́ angulos coalternos inæquales, aut <lb/>duos intrinſecos minores duobus rectis, uel extrinſecum inæqualem intrinſeco: illas duas lineas <lb/>ad minorum angulorum partem concurrere eſt neceſſe, ad aliam uerò partem impoßibile: & ſi <lb/>lineæ concurrunt, neceſſe est dictos angulos aliquo propoſitorum modorum ſe habere. E' 27.28 <lb/>p 1 element. Lemma Procli ad 16 p 1 elem.</head> <p> <s xml:id="echoid-s20647" xml:space="preserve">Sint duæ lineæ a b & c d, quas ſecet linea e fſecundum quod proponitur.</s> <s xml:id="echoid-s20648" xml:space="preserve"> Dico, quoniam lineæ <lb/>a b & c d concurrent.</s> <s xml:id="echoid-s20649" xml:space="preserve"> Si enim nõ concurrant, patet quòd ſunt æ quidiſtantes:</s> <s xml:id="echoid-s20650" xml:space="preserve"> ergo per 29 p 1 ſequi-<lb/>tur contrarium hypothe.</s> <s xml:id="echoid-s20651" xml:space="preserve"> quòd eſt inconueniens:</s> <s xml:id="echoid-s20652" xml:space="preserve"> concur <lb/> <anchor type="figure" xlink:label="fig-0310-02a" xlink:href="fig-0310-02"/> runt ergo.</s> <s xml:id="echoid-s20653" xml:space="preserve"> Ad partem uerò minorum angulorum cõcur-<lb/>rere eſt neceſſarium:</s> <s xml:id="echoid-s20654" xml:space="preserve"> quoniam ſi ad partem maiorum an-<lb/>gulorum concurrant, ſequetur angulum extrinſecum tri <lb/>goni contenti fieri minorẽ angulo intrinſeco:</s> <s xml:id="echoid-s20655" xml:space="preserve"> & eſt con-<lb/>tra 16 & 32 p 1.</s> <s xml:id="echoid-s20656" xml:space="preserve"> Et quia per præmiſſas probationes ad par-<lb/>tes minorum angulorum concurrunt:</s> <s xml:id="echoid-s20657" xml:space="preserve"> ſi ex conceſſo ad <lb/>partes maiorum angulorum concurrerent, ſequeretur <lb/>duas rectas lineas ſuperficiem includere:</s> <s xml:id="echoid-s20658" xml:space="preserve"> quod eſt impoſ <lb/>ſibile.</s> <s xml:id="echoid-s20659" xml:space="preserve"> Eſt ergo impoſsibile, ut ad partes maiorum angu-<lb/>lorum concurrant:</s> <s xml:id="echoid-s20660" xml:space="preserve"> quod eſt propoſitum primum.</s> <s xml:id="echoid-s20661" xml:space="preserve"> Sed & <lb/>ſi detur, quòd illæ lineæ concurrant, neceſſe eſt angulos aliquo propofitorum modorum ſe habere <lb/>per 32 p 1:</s> <s xml:id="echoid-s20662" xml:space="preserve"> patet ergo totum, quod proponebatur, ſeruata ſemper hypotheſi.</s> <s xml:id="echoid-s20663" xml:space="preserve"/> </p> <div xml:id="echoid-div679" type="float" level="0" n="0"> <figure xlink:label="fig-0310-02" xlink:href="fig-0310-02a"> <variables xml:id="echoid-variables257" xml:space="preserve">e a b c d f</variables> </figure> </div> </div> <div xml:id="echoid-div681" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables258" xml:space="preserve">a d e c b</variables> </figure> <head xml:id="echoid-head589" xml:space="preserve" style="it">15. Cumlineis, ſe inter duas line{as} æquidiſtantes, à <lb/>quarum terminis producuntur, ſecantibus, ex utra <lb/> parte ſectionis partes eiuſdẽ lineæ inter ſe fuerint æqua les: neceſſe eſt lineas, inter quas fit ſectio, æquales eſſe.</head> <p> <s xml:id="echoid-s20664" xml:space="preserve">Verbi gratia:</s> <s xml:id="echoid-s20665" xml:space="preserve"> ſit, ut duæ lineæ a b & c d inter duas line-<lb/>as æquidiſtantes, à quarũ terminis producũtur, quę ſint a <lb/>d & c b, ſecent ſe in puncto e, ita, quòd linea a e ſit æqualis <lb/>lineæ e b, & linea c e ſit æqualis ipſi e d.</s> <s xml:id="echoid-s20666" xml:space="preserve"> Dico, quòd linea <lb/>a d eſt æqualis lineæ c b.</s> <s xml:id="echoid-s20667" xml:space="preserve"> Quoniam enim per 15 p 1 angu-<lb/>lus a e d eſt æqualis angulo c e b, erit ex hypotheſi & per <lb/>4 p 1 linea a d æqualis lineæ c b:</s> <s xml:id="echoid-s20668" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s20669" xml:space="preserve"/> </p> <pb o="9" file="0311" n="311" rhead="LIBER I."/> </div> <div xml:id="echoid-div682" type="section" level="0" n="0"> <head xml:id="echoid-head590" xml:space="preserve" style="it">16. Si per terminos duarum linearum æquidiſtantium & inæqualium, rectæproducantur, <lb/>illas ad partem minoris lineæ concurrere est neceſſe.</head> <p> <s xml:id="echoid-s20670" xml:space="preserve">Sint duæ lineæ a b & c d æquidiſtantes & inæquales:</s> <s xml:id="echoid-s20671" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s20672" xml:space="preserve"> linea c d minor quàm linea a b:</s> <s xml:id="echoid-s20673" xml:space="preserve"> <lb/>producãturq́;</s> <s xml:id="echoid-s20674" xml:space="preserve"> per terminos ipſarum, lineę a c <lb/> <anchor type="figure" xlink:label="fig-0311-01a" xlink:href="fig-0311-01"/> & b d.</s> <s xml:id="echoid-s20675" xml:space="preserve"> Dico, quòd illæ lineæ a c & b d concur <lb/>rent ultra lineam c d.</s> <s xml:id="echoid-s20676" xml:space="preserve"> Producatur enim linea <lb/>c d ultra punctum d ad punctum e, fiatq́;</s> <s xml:id="echoid-s20677" xml:space="preserve"> per <lb/>3 p 1 linea c e æqualis lineę a b, & ducatur li-<lb/>nea b e.</s> <s xml:id="echoid-s20678" xml:space="preserve"> Hic itaque linea b e per 33 p 1 eſt æqui <lb/>diſtans lineę a c:</s> <s xml:id="echoid-s20679" xml:space="preserve"> ergo per 2 huius cum linea <lb/>b d concurrat cum linea b e in puncto b:</s> <s xml:id="echoid-s20680" xml:space="preserve"> pa-<lb/>tet, quòd ipſa concurret cum linea a c, quę æ-<lb/>quidiſtat lineę b e:</s> <s xml:id="echoid-s20681" xml:space="preserve"> ſed & ad partem lineę c d, <lb/>quę eſt minor quàm linea a b, concurrere eſt <lb/>neceſſe per 14 huius, uel per 2 p 6:</s> <s xml:id="echoid-s20682" xml:space="preserve"> patet ergo <lb/>propoſitum:</s> <s xml:id="echoid-s20683" xml:space="preserve"> punctus enim concurſus eius, (qui ſit f) erit ultra lineam c d.</s> <s xml:id="echoid-s20684" xml:space="preserve"/> </p> <div xml:id="echoid-div682" type="float" level="0" n="0"> <figure xlink:label="fig-0311-01" xlink:href="fig-0311-01a"> <variables xml:id="echoid-variables259" xml:space="preserve">a c f d b e</variables> </figure> </div> </div> <div xml:id="echoid-div684" type="section" level="0" n="0"> <head xml:id="echoid-head591" xml:space="preserve" style="it">17. Lineæ rectæ continentes angulos æquales cum linea recta, cui ad unum punctum inci-<lb/>dunt, ſimuliunctæ, ſunt breuiores omnibus lineis ab eiſdem terminis ſuper eandem lineam <lb/>adunum punctum alium productis, continentibus cum eadem linea angulos inæquales, ſi-<lb/>muliunctis.</head> <p> <s xml:id="echoid-s20685" xml:space="preserve">Sit linea recta, quę a b c f:</s> <s xml:id="echoid-s20686" xml:space="preserve"> & ſint duo puncta g, & d, â quibus duę lineę g b & d b productę ſuper <lb/>lineam a b c f, contineant angulos æquales, <lb/> <anchor type="figure" xlink:label="fig-0311-02a" xlink:href="fig-0311-02"/> ita, ut angulus a b g ſit æqualis angulo c b d.</s> <s xml:id="echoid-s20687" xml:space="preserve"> <lb/>Dico, quòd ſi à pũctis d & g ad aliquod aliud <lb/>punctum lineæ a b c f (quod ſitc) lineę du-<lb/>ctę contineant inęquales angulos, ita, ut an-<lb/>gulus g c a ſit minor angulo f c d:</s> <s xml:id="echoid-s20688" xml:space="preserve"> quòd lineę <lb/>g b & b d ſimul iunctę ſunt minores duabus <lb/>lineis g c & d c ſimul iunctis.</s> <s xml:id="echoid-s20689" xml:space="preserve"> Ducatur enim <lb/>à puncto g ſuper lineam a f perpendicularis <lb/>per 12 p 1, quę ſit g h:</s> <s xml:id="echoid-s20690" xml:space="preserve"> & producatur linea g h <lb/>ultra punctum h:</s> <s xml:id="echoid-s20691" xml:space="preserve"> & producatur d b, donec <lb/>concurrat cum linea g h producta:</s> <s xml:id="echoid-s20692" xml:space="preserve"> concur-<lb/>rent autem per 14 huius:</s> <s xml:id="echoid-s20693" xml:space="preserve"> ſit ergo punctus concurſus k:</s> <s xml:id="echoid-s20694" xml:space="preserve"> & coniungatur linea k c.</s> <s xml:id="echoid-s20695" xml:space="preserve"> Et quoniam angu-<lb/>lus d b c eſt æqualis angulo g b h exhypotheſi, & angulo h b k, ex 15 p 1:</s> <s xml:id="echoid-s20696" xml:space="preserve"> palàm, quòd angulus <lb/>h b k eſt ęqualis g b h:</s> <s xml:id="echoid-s20697" xml:space="preserve"> ſed anguli g h b & k h b ſunt ęquales:</s> <s xml:id="echoid-s20698" xml:space="preserve"> quia recti:</s> <s xml:id="echoid-s20699" xml:space="preserve"> ergo per 32 p 1 trigoni <lb/>g h b & k h b ſunt ęquianguli.</s> <s xml:id="echoid-s20700" xml:space="preserve"> Ergo per 4 p 6, cum linea h b ſit communis & ęqualis ſibijpſi, erit <lb/>linea g b ęqualis lineę k b, & linea g h ęqualis lineę h k.</s> <s xml:id="echoid-s20701" xml:space="preserve"> Et eadem ratione per 4 p 1 erit linea g c <lb/>ęqualis lineę k c.</s> <s xml:id="echoid-s20702" xml:space="preserve"> Quia uerò per 20 p 1 linea k d in trigono k d c minor eſt ambabus lineis d c & <lb/>k c ſimuliunctis, & linea g b ęqualis eſt lineę b k, & linea g c ęqualis eſt lineę k c:</s> <s xml:id="echoid-s20703" xml:space="preserve"> palàm, quia <lb/>ambę lineę g b & d b ſimul iunctę, minores ſunt ambabus lineis d c & g c ſimul iunctis.</s> <s xml:id="echoid-s20704" xml:space="preserve"> Simi-<lb/>liter quoque de quibuſcunque lineis à punctis g & d ad lineam a fproductis eſt demonſtrandum:</s> <s xml:id="echoid-s20705" xml:space="preserve"> <lb/>patet ergo propoſitum.</s> <s xml:id="echoid-s20706" xml:space="preserve"/> </p> <div xml:id="echoid-div684" type="float" level="0" n="0"> <figure xlink:label="fig-0311-02" xlink:href="fig-0311-02a"> <variables xml:id="echoid-variables260" xml:space="preserve">g d a h b c f k</variables> </figure> </div> </div> <div xml:id="echoid-div686" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables261" xml:space="preserve">g d e a z b f c</variables> </figure> <head xml:id="echoid-head592" xml:space="preserve" style="it">18. Lineæ rectæ continentes angulos æ-<lb/> quales cumlinea conuexa, cui ad unum pun- ctum incidunt, ſimuliunctæ, ſunt breuiores omnibus lineis ab eiſdem terminis ſuper ean- dem lineam adunum punctum alium produ- ctis, continentibus cum eadem linea angulos inæquales, ſimuliunctis.</head> <p> <s xml:id="echoid-s20707" xml:space="preserve">Sit linea curua a b c, ſuper cuius conuexum <lb/>â punctis g & d incidant lineę d a & g a, conti-<lb/>nentes angulos ęquales, ita, ut angulus c a g ſit <lb/>ęqualis angulo b a d.</s> <s xml:id="echoid-s20708" xml:space="preserve"> Dico, quòd ſi ducantur <lb/>alię lineę à punctis g & d ſuper lineam a b c, <lb/>ut g b & d b, continentes angulos inęquales <lb/>cum linea a b c:</s> <s xml:id="echoid-s20709" xml:space="preserve"> quòd ambę lineę g a & d a ſi-<lb/>mul iunctę, erunt breuiores duabus lineis g b & <lb/>d b ſimul iũctis, Ducatur enim linea e f, cõtingẽs <lb/> <anchor type="figure" xlink:label="fig-0311-04a" xlink:href="fig-0311-04"/> <pb o="10" file="0312" n="312" rhead="VITELLONIS OPTICAE"/> arcum a b c in puncto a per 17 p 3:</s> <s xml:id="echoid-s20710" xml:space="preserve"> anguli ergo contingentiæ, qui ſunt e a c & f a b ſunt æquales <lb/>per 16 p 3:</s> <s xml:id="echoid-s20711" xml:space="preserve"> ſed anguli g a c & d a b ſunt æquales ex hypotheſi:</s> <s xml:id="echoid-s20712" xml:space="preserve"> erunt ergo anguli g a e & d a f æqua-<lb/>les.</s> <s xml:id="echoid-s20713" xml:space="preserve"> Et ad punctum, ubi linea g b ſecat lineam e f(quod ſit z) ducatur linea d z:</s> <s xml:id="echoid-s20714" xml:space="preserve"> ergo per præceden <lb/>tem ambæ lineæ g a & d a ſunt breuiores ambabus lineis g z & d z:</s> <s xml:id="echoid-s20715" xml:space="preserve"> cum angulus g z a ſit minor an-<lb/>gulo g a e, & angulus d z f ſit maior angulo d a f per 16 p 1.</s> <s xml:id="echoid-s20716" xml:space="preserve"> Sed linea g b eſt maior quàm linea <lb/>g z, ut totum parte, & linea d b eſt maior quàm linea d z per 19 p 1, quoniam angulus d z b eſt <lb/>maior angulus ſui trigoni.</s> <s xml:id="echoid-s20717" xml:space="preserve"> Patet ergo propoſitum in arcu circuli conuexo:</s> <s xml:id="echoid-s20718" xml:space="preserve"> & eodem modo demon <lb/>ſtrandum in quacunque alia columnali uel pyramidali ſectione ſecũdum ipſius conuexum:</s> <s xml:id="echoid-s20719" xml:space="preserve"> patet <lb/>ergo propoſitum.</s> <s xml:id="echoid-s20720" xml:space="preserve"/> </p> <div xml:id="echoid-div686" type="float" level="0" n="0"> <figure xlink:label="fig-0311-04" xlink:href="fig-0311-04a"> <image file="0311-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0311-04"/> </figure> </div> </div> <div xml:id="echoid-div688" type="section" level="0" n="0"> <head xml:id="echoid-head593" xml:space="preserve" style="it">19. Vna linea recta in duabus ſuperficiebus planis exiſtente, neceſſe est, ut illæ duæ ſuperſi-<lb/>cies ſecundum illam lineam ſe ſecent. E' 3 p 11 element.</head> <p> <s xml:id="echoid-s20721" xml:space="preserve">Sint duæ ſuperficies planæ a b c d & c d e f:</s> <s xml:id="echoid-s20722" xml:space="preserve"> in quarum utraque ſit linea c d.</s> <s xml:id="echoid-s20723" xml:space="preserve"> Dico, quòd illæ <lb/>duæ ſuperficies ſecant ſe ſuper lineam e d.</s> <s xml:id="echoid-s20724" xml:space="preserve"> Si enim illæ duæ ſuperfici-<lb/> <anchor type="figure" xlink:label="fig-0312-01a" xlink:href="fig-0312-01"/> es ad lineam c d, ut ad communem terminum per modum unius ſu-<lb/>perficiei continuè copulentur:</s> <s xml:id="echoid-s20725" xml:space="preserve"> tunc patet, quòd ipſæ ſunt partes uni-<lb/>us ſuperficiei, & non duæ ſuperficies:</s> <s xml:id="echoid-s20726" xml:space="preserve"> quod eſt contra hypotheſim.</s> <s xml:id="echoid-s20727" xml:space="preserve"> <lb/>Quòd ſi ipſæ ſuperficies datam lineam c d pertranſeant, nec ad ipſam, <lb/>ut ad communem terminum copulentur:</s> <s xml:id="echoid-s20728" xml:space="preserve"> palàm per 3 p 11, cum ipſæ <lb/>ad inuicem ſe ſecent, quòd ipſis aliqua linea eſt communis.</s> <s xml:id="echoid-s20729" xml:space="preserve"> Aut ergo <lb/>ſecant ſe ſuper lineam c d:</s> <s xml:id="echoid-s20730" xml:space="preserve"> & habetur propoſitum:</s> <s xml:id="echoid-s20731" xml:space="preserve"> aut ſuper aliam <lb/>quamcunque datam:</s> <s xml:id="echoid-s20732" xml:space="preserve"> & tunc, cum illa ſit ambabus propoſitis ſuper-<lb/>ficiebus communis per prænominatam 3 p 11, & eiſdem ſit linea c d <lb/>communis ex hypotheſi:</s> <s xml:id="echoid-s20733" xml:space="preserve"> ſequetur, ut duæ planæ ſuperficies illas du-<lb/>as lineas interiacentes corpus includãt:</s> <s xml:id="echoid-s20734" xml:space="preserve"> quod eſt impoſsibile, & con-<lb/>tra 4 ſuppoſitionem huius:</s> <s xml:id="echoid-s20735" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s20736" xml:space="preserve"/> </p> <div xml:id="echoid-div688" type="float" level="0" n="0"> <figure xlink:label="fig-0312-01" xlink:href="fig-0312-01a"> <variables xml:id="echoid-variables262" xml:space="preserve">a b c d e f</variables> </figure> </div> </div> <div xml:id="echoid-div690" type="section" level="0" n="0"> <head xml:id="echoid-head594" xml:space="preserve" style="it">20. Ab uno puncto in aere dato, ſuper unamquam ſubſtratã <lb/>planam uel conuexam ſuperficiem, una tantũ perpendicularis du-<lb/>ci potest. E' 11 & 13 p 11 elem.</head> <p> <s xml:id="echoid-s20737" xml:space="preserve">Sit data ſuperficies plana a b c d, & datus in aere punctus e.</s> <s xml:id="echoid-s20738" xml:space="preserve"> Dico, quòd à puncto e ad ſubſtra-<lb/>tam ſuperficiem, unam tantùm perpendicularem duci eſt poſsibi-<lb/> <anchor type="figure" xlink:label="fig-0312-02a" xlink:href="fig-0312-02"/> le.</s> <s xml:id="echoid-s20739" xml:space="preserve"> Sienim poſsibile, ſit ut ſuper ſuperficiem planam datam, quæ a <lb/>b c d, ducantur à puncto e duæ perpendiculares, quæ ſint e f & e g.</s> <s xml:id="echoid-s20740" xml:space="preserve"> <lb/>Quia itaq;</s> <s xml:id="echoid-s20741" xml:space="preserve"> lineę e f & e g angulariter cõiunguntur in puncto e, pa <lb/>tet per 2 p 11, quoniam illæ duæ lineæ ſunt in eadem ſuperficie:</s> <s xml:id="echoid-s20742" xml:space="preserve"> & <lb/>quoniam lineæ illæ ſunt perpendiculares ſuper ſuperficiem a b c d, <lb/>erit ſuperficies, in qua ſunt lineæ illæ, erecta ſuper ſuperficiem a b <lb/>c d.</s> <s xml:id="echoid-s20743" xml:space="preserve"> Huius itaq;</s> <s xml:id="echoid-s20744" xml:space="preserve"> ſuperficiei & ſuperficiei a b c d communis ſectio <lb/>eſt linea f g per præmiſſam:</s> <s xml:id="echoid-s20745" xml:space="preserve"> in trigono itaque e f g ſunt duo angu <lb/>li recti, ſcilicet e f g & e g f per definitionem lineæ erectæ ſuper ſu <lb/>perficiem 3 definit.</s> <s xml:id="echoid-s20746" xml:space="preserve"> 11:</s> <s xml:id="echoid-s20747" xml:space="preserve"> hoc autem eſt impoſsibile, & contra 32 p 1.</s> <s xml:id="echoid-s20748" xml:space="preserve"> <lb/>Hoc autem etiam patet in ſuperficiebus conuexis:</s> <s xml:id="echoid-s20749" xml:space="preserve"> quia enim, per <lb/>5 definitionem huius omnis linea perpendicularis ſuper quam cun <lb/>que ſuperficiem conuexam, eſt perpendicularis ſuper planam ſu-<lb/>perficiem ipſam conuexam ſuperficiem in puncto incidentię lineę <lb/>illius contingentem:</s> <s xml:id="echoid-s20750" xml:space="preserve"> patet, quia in omni ſuperficie conuexaidem <lb/>accidit impoſsibile.</s> <s xml:id="echoid-s20751" xml:space="preserve"> Si enim ſit ſuperficies ſphærica cõuexa, in qua <lb/>ſit arcus f g:</s> <s xml:id="echoid-s20752" xml:space="preserve"> ſit ut ipſam contingat in puncto fſuperficies plana, in <lb/>qua ducatur linea h f k, & in puncto g ſuperficies plana, in qua ſit li-<lb/>nea l g m.</s> <s xml:id="echoid-s20753" xml:space="preserve"> Palàm ergo ex pręmiſsis, quia anguli e f k & e g l ſunt re-<lb/>cti.</s> <s xml:id="echoid-s20754" xml:space="preserve"> Producta quoq;</s> <s xml:id="echoid-s20755" xml:space="preserve"> chorda f g:</s> <s xml:id="echoid-s20756" xml:space="preserve"> palàm quia anguli e f g & e g f ſunt maiores duobus rectis, quod eſt <lb/>impoſsibile.</s> <s xml:id="echoid-s20757" xml:space="preserve"> Non eſt ergo poſsibile ab uno puncto dato plus una perpendiculari duci ad ſuperficiẽ <lb/>planam uel conuexam.</s> <s xml:id="echoid-s20758" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s20759" xml:space="preserve"> quoniam in quibuſcunque alijs conuexis ſuperfi-<lb/>ciebus eſt eodem modo demonſtrandum.</s> <s xml:id="echoid-s20760" xml:space="preserve"/> </p> <div xml:id="echoid-div690" type="float" level="0" n="0"> <figure xlink:label="fig-0312-02" xlink:href="fig-0312-02a"> <variables xml:id="echoid-variables263" xml:space="preserve">e a b k l f g h m c d</variables> </figure> </div> </div> <div xml:id="echoid-div692" type="section" level="0" n="0"> <head xml:id="echoid-head595" xml:space="preserve" style="it">21. Omnium linearum ab eodem puncto adeandem ſuperficiem planamuel conuexam pro-<lb/>ductarum, minima eſt perpendicularis. Albazen 5 n 5.</head> <p> <s xml:id="echoid-s20761" xml:space="preserve">Eſto ſuperficies plana b c d i:</s> <s xml:id="echoid-s20762" xml:space="preserve"> & punctum extrà ſignatum a, à quo ducantur plurimæ lineæ ad ſu-<lb/>perficiem datam, ut contingit, ſcilicet a e, a f, a g, a h, ſola tamen a e ſit perpendicularis.</s> <s xml:id="echoid-s20763" xml:space="preserve"> Dico, quòd li <lb/>nea a e eſt omnium aliarum breuiſsima.</s> <s xml:id="echoid-s20764" xml:space="preserve"> Ducantur enim lineæ e f, e g, e h, & componantur tri-<lb/>gona orthogonia.</s> <s xml:id="echoid-s20765" xml:space="preserve"> Palàm itaque (cum per 32 p 1 angulus rectus ſit maior in qualibet trigono <lb/> <anchor type="figure" xlink:label="fig-0312-03a" xlink:href="fig-0312-03"/> <pb o="11" file="0313" n="313" rhead="LIBER I."/> orthogonio) quoniam linea a e per 19 p 1 breuior eſt qualibet linearum a f, a g, a h, & etiam <lb/>aliarum quarumcunq;</s> <s xml:id="echoid-s20766" xml:space="preserve"> ſic productarum:</s> <s xml:id="echoid-s20767" xml:space="preserve"> patet ergo propoſitum in planis.</s> <s xml:id="echoid-s20768" xml:space="preserve"> Sed & in conuexis patet <lb/>idem:</s> <s xml:id="echoid-s20769" xml:space="preserve"> quoniam ſi perpendicularis ſuper conuexam <lb/> <anchor type="figure" xlink:label="fig-0313-01a" xlink:href="fig-0313-01"/> ſuperficiem ſit a e, & ſit b c d i ſuperficies plana con <lb/>tingens ſuperficiem conuexam ſecundum punctũ <lb/>e, ducanturq́;</s> <s xml:id="echoid-s20770" xml:space="preserve"> lineæ a f, a g, a h ſuper ſuperficiem pla <lb/>nam:</s> <s xml:id="echoid-s20771" xml:space="preserve"> erunt omnes illę maiores perpendiculari:</s> <s xml:id="echoid-s20772" xml:space="preserve"> er-<lb/>go eædem productæ ad ſuperficiem conuexã ſunt <lb/>multo maiores:</s> <s xml:id="echoid-s20773" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s20774" xml:space="preserve"/> </p> <div xml:id="echoid-div692" type="float" level="0" n="0"> <figure xlink:label="fig-0312-03" xlink:href="fig-0312-03a"> <image file="0312-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0312-03"/> </figure> <figure xlink:label="fig-0313-01" xlink:href="fig-0313-01a"> <variables xml:id="echoid-variables264" xml:space="preserve">a b c e f g h d i</variables> </figure> </div> </div> <div xml:id="echoid-div694" type="section" level="0" n="0"> <head xml:id="echoid-head596" xml:space="preserve" style="it">22. Ducta linea à ſupremo termino lineæ ſu-<lb/>per ſuperficiem erectæ, ad lineam perpendicularẽ <lb/>cuicun lineæ à puncto incidẽtiæ lineæ erectæ in <lb/>ſubiecta ſuperficie protractæ: neceſſe eſt protractã <lb/>lineam ſuperiacenti perpendicularem eſſe. Lem-<lb/>ma ad 37 theorema opticorum Euclidis: item 42 <lb/>theor. 6 libri μαθκματικυεμ συναγωγυεμ Pappi.</head> <p> <s xml:id="echoid-s20775" xml:space="preserve">Sit punctũ in aere datum, quod ſit a, à quo ad ſu-<lb/>perficiem planã ſubiectam, quæ ſit b c d, erigatur li-<lb/>nea per 12 p 11, quæ ſit a b, incidens datæ ſuperficiei in puncto b:</s> <s xml:id="echoid-s20776" xml:space="preserve"> & in ſuperficie b c d ducatur linea <lb/>d c, ut placuerit, & à puncto b ducatur perpendicularis ſuper lineam <lb/> <anchor type="figure" xlink:label="fig-0313-02a" xlink:href="fig-0313-02"/> d c, quæ ſit b d:</s> <s xml:id="echoid-s20777" xml:space="preserve"> & copuletur linea a d.</s> <s xml:id="echoid-s20778" xml:space="preserve"> Dico, quòd a d eſt perpendi-<lb/>cularis ſuper lineã d c.</s> <s xml:id="echoid-s20779" xml:space="preserve"> Sumatur enim in linea d c quodcunq;</s> <s xml:id="echoid-s20780" xml:space="preserve"> punctũ, <lb/>ut c, & ducantur lineæ a c, b c.</s> <s xml:id="echoid-s20781" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s20782" xml:space="preserve"> linea a b eſt erecta ſuper ſu-<lb/>perficiẽ b c d, patet ք definitionẽ lineę erectę 3 defin.</s> <s xml:id="echoid-s20783" xml:space="preserve"> 11, quoniã angu-<lb/>lus a b c eſt rectus:</s> <s xml:id="echoid-s20784" xml:space="preserve"> ergo ք 47 p 1, quadratũ lineę a c eſt æquale duob.</s> <s xml:id="echoid-s20785" xml:space="preserve"> <lb/>quadratis linearũ a b & b c:</s> <s xml:id="echoid-s20786" xml:space="preserve"> ſed & quadratũ lineę b c eſt æquale duob.</s> <s xml:id="echoid-s20787" xml:space="preserve"> <lb/>quadratis c d & b d per 47 p 1, quia linea b d eſt perpẽdicularis ſuper <lb/>lineam c d ex hypotheſi.</s> <s xml:id="echoid-s20788" xml:space="preserve"> Quadratum itaq;</s> <s xml:id="echoid-s20789" xml:space="preserve"> lineæ a c eſt æquale tribus <lb/>quadratis trium linearum, quæ ſunt a b & b d & c d:</s> <s xml:id="echoid-s20790" xml:space="preserve"> ſed quadratum li-<lb/>neæ a d eſt æquale duobus quadratis duarum linearum a b & b d:</s> <s xml:id="echoid-s20791" xml:space="preserve"> <lb/>quadratum ergo lineæ a c eſt æquale duobus quadratis duarum li-<lb/>nearum a d & d c.</s> <s xml:id="echoid-s20792" xml:space="preserve"> Ergo per 48 p 1 angulus a d c eſt rectus.</s> <s xml:id="echoid-s20793" xml:space="preserve"> Patet er-<lb/>go, quòd linea a d eſt perpendicularis ſuper lineam d c:</s> <s xml:id="echoid-s20794" xml:space="preserve"> quod eſt <lb/>propoſitum.</s> <s xml:id="echoid-s20795" xml:space="preserve"/> </p> <div xml:id="echoid-div694" type="float" level="0" n="0"> <figure xlink:label="fig-0313-02" xlink:href="fig-0313-02a"> <variables xml:id="echoid-variables265" xml:space="preserve">a c b d</variables> </figure> </div> </div> <div xml:id="echoid-div696" type="section" level="0" n="0"> <head xml:id="echoid-head597" xml:space="preserve" style="it">23. Duabus planis ſuperficiebus æquidiſtantibus, una linea rect a incidente, quæ ad alteram <lb/>earũ erit perpendicularis, erit quo ad reliquã perpendicularis. Conuerſa 14 p 11 elem.</head> <p> <s xml:id="echoid-s20796" xml:space="preserve">Sit, ut duabus ſuperficiebus planis & æquidiſtantibus incidatun a linea, quæ a b, uni ipſarum <lb/>in puncto a, & reliquæ in puncto b.</s> <s xml:id="echoid-s20797" xml:space="preserve"> Dico, quòd ſi linea a b fuerit <lb/> <anchor type="figure" xlink:label="fig-0313-03a" xlink:href="fig-0313-03"/> perpendicularis ſuper unam iſtarum ſuperficierum, quòd erit per-<lb/>pendicularis & ſuper reliquam.</s> <s xml:id="echoid-s20798" xml:space="preserve"> Nam à puncto a ducatur in altera ſu-<lb/>perficierum illarum linea recta, quæ a c, & in reliqua à puncto b du-<lb/>catur linea b d.</s> <s xml:id="echoid-s20799" xml:space="preserve"> Palàm itaque, quoniam lineæ a c & b d æquidiſtant:</s> <s xml:id="echoid-s20800" xml:space="preserve"> <lb/>in infinitum enim protractæ non concurrent, quia & ſuperficies in <lb/>quibus ſunt, non concurrunt.</s> <s xml:id="echoid-s20801" xml:space="preserve"> Si itaque alter angulorum, qui b a c <lb/>uel a b d fueritrectus:</s> <s xml:id="echoid-s20802" xml:space="preserve"> palàm ſemper per 29 p 1, quoniam & reli-<lb/>quus ipſorum erit rectus.</s> <s xml:id="echoid-s20803" xml:space="preserve"> Et quoniam eodem modo poteſt hoc de-<lb/>clarari de omnibus lineis in ſuperficiebus hinc inde ductis à punctis <lb/>a & b:</s> <s xml:id="echoid-s20804" xml:space="preserve"> patet, quòd linea a b cum ſingulis ſibi conterminalibus lineis <lb/>in utraque ſuperficierum illarum productis angulos rectos facit.</s> <s xml:id="echoid-s20805" xml:space="preserve"> Si <lb/>eſt ergo linea a b perpendicularis ſuper alteram ſuperficierum, pa-<lb/>làm, quia erit perpendicularis ſuper reliquam ipſarum:</s> <s xml:id="echoid-s20806" xml:space="preserve"> & hoc eſt <lb/>propoſitum.</s> <s xml:id="echoid-s20807" xml:space="preserve"/> </p> <div xml:id="echoid-div696" type="float" level="0" n="0"> <figure xlink:label="fig-0313-03" xlink:href="fig-0313-03a"> <variables xml:id="echoid-variables266" xml:space="preserve">c d a b</variables> </figure> </div> </div> <div xml:id="echoid-div698" type="section" level="0" n="0"> <head xml:id="echoid-head598" xml:space="preserve" style="it">24. Si duæ ſuperficies uni ſuperficiei æquidiſtantes fuerint, eædem inter ſe erunt æquidiſtan <lb/>tes: ſuperficies quoque concurrens cum una æquidiſtantium ſuperficierum & cum reliqua con-<lb/>curret. E' 30 p 1 & 9 p 11 elementorum.</head> <p> <s xml:id="echoid-s20808" xml:space="preserve">Sint duæ ſuperficies a b c & g h k æquidiſtantes uni ſuperficiei, quæ d e f.</s> <s xml:id="echoid-s20809" xml:space="preserve"> Dico, quòd <lb/>illæ duæ ſuperficies a b c & g h k neceſſariò adinuicem æquidiſtabunt.</s> <s xml:id="echoid-s20810" xml:space="preserve"> Educatur enim à pun-<lb/>cto l ſuperficiei a b c linea perpendicularis ſuper illam ſuperficiem per 12 p undecimi, quæ <lb/> <pb o="12" file="0314" n="314" rhead="VITELLONIS OPTIC AE"/> ſit l m.</s> <s xml:id="echoid-s20811" xml:space="preserve"> Palàm itaque per præmiſſſam, quoniã illa linea l m erit perpendicularis ſuper ſuperficiẽ d e f <lb/>æquieiſtantẽ ſuperficiei a b c.</s> <s xml:id="echoid-s20812" xml:space="preserve"> Producta ergo linea l m ultra alterutrũ <lb/> <anchor type="figure" xlink:label="fig-0314-01a" xlink:href="fig-0314-01"/> ſuorũ terminorũ, erit ipſa ք eandẽ pręmiſſam քpendicularis ſuper ſu <lb/>perficiẽ g h k, æquidiſtãtẽ ſuքficiei a b c.</s> <s xml:id="echoid-s20813" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s20814" xml:space="preserve"> una linea l m ſuք <lb/>duas ſuperficies a b c & g h k orthogonaliter inſiſtit, patet per 14 p 11, <lb/>quòd illę duę ſuperficies, etiam ſi in infinitũ protrahantur, nunquã <lb/>concurrent.</s> <s xml:id="echoid-s20815" xml:space="preserve"> Sunt ergo ęquidiſtantes:</s> <s xml:id="echoid-s20816" xml:space="preserve"> patet ergo propoſitum primũ:</s> <s xml:id="echoid-s20817" xml:space="preserve"> <lb/>& per hoc & per 2 huius patet etiam ſecundum propoſitum.</s> <s xml:id="echoid-s20818" xml:space="preserve"/> </p> <div xml:id="echoid-div698" type="float" level="0" n="0"> <figure xlink:label="fig-0314-01" xlink:href="fig-0314-01a"> <variables xml:id="echoid-variables267" xml:space="preserve">b c l a e f d h k m g</variables> </figure> </div> </div> <div xml:id="echoid-div700" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables268" xml:space="preserve">k a e i l g b c ſ h d</variables> </figure> <head xml:id="echoid-head599" xml:space="preserve" style="it">25. Omnes lineæ perpendiculares inter lineas uel ſuperficies æ-<lb/>quidiſtãtes du <lb/> ctæ, ſunt æqui diſtantes & æ- quales: & ſi li- neærectæ line- is uel ſuperficie bus æquidiſt an tibus ad angu- los æquales in- cidant, ſunt æ- quales.</head> <p> <s xml:id="echoid-s20819" xml:space="preserve">Sint duę lineę a b & c d æquidiſtãtes, inter quas ducãtur lineę perpẽdiculares, quę ſint e f & g h.</s> <s xml:id="echoid-s20820" xml:space="preserve"> <lb/>Dico, quòd lineæ e f & g h ſunt ęquidiſtantes & æquales.</s> <s xml:id="echoid-s20821" xml:space="preserve"> Quòd enim ſunt ęquidiſtãtes, hoc patet ք <lb/>28 p 1:</s> <s xml:id="echoid-s20822" xml:space="preserve"> quòd etiã ſunt ęquales, patet per 34 p 1.</s> <s xml:id="echoid-s20823" xml:space="preserve"> Et eodẽ modo demonſtrãdũ eſt, ſi lineę a b & c d ſint <lb/>in ſuperficiebus ęquidiſtantibus ſignatę.</s> <s xml:id="echoid-s20824" xml:space="preserve"> Quòd ſi lineę e f & g h non perpendiculariter, ſed ad angu <lb/>los ęquales incidãt, ductis lineis uel ſuperficiebus, ita, ut angulus g h c ſit ęqualis angulo e f d, erũt <lb/>etiam lineę g h & e f ęquales:</s> <s xml:id="echoid-s20825" xml:space="preserve"> concurrent enim per 14 huius:</s> <s xml:id="echoid-s20826" xml:space="preserve"> ſit ergo punctus concurſus k.</s> <s xml:id="echoid-s20827" xml:space="preserve"> Quia ita-<lb/>que angulus k f h eſt ęqualis angulo k h f, ex hypotheſi:</s> <s xml:id="echoid-s20828" xml:space="preserve"> erit per 6 p 1 trigoni k f h latus k f ęquale la-<lb/>teri k h.</s> <s xml:id="echoid-s20829" xml:space="preserve"> Sed per 29 & 26 p 1 erit trigoni k i llatus k i ęquale lateri k l:</s> <s xml:id="echoid-s20830" xml:space="preserve"> relinquitur ergo linea i f ęqualis <lb/>lineæ l h:</s> <s xml:id="echoid-s20831" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s20832" xml:space="preserve"> In ſuperficiebus quoq;</s> <s xml:id="echoid-s20833" xml:space="preserve"> æquidiſtantibus ſignatis lineis a b & c d ea-<lb/>dem eſt demonſtratio:</s> <s xml:id="echoid-s20834" xml:space="preserve"> patet ergo illud, quod proponebatur.</s> <s xml:id="echoid-s20835" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div701" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables269" xml:space="preserve">d e b f h g l a k c</variables> </figure> <head xml:id="echoid-head600" xml:space="preserve" style="it">26. Cuilibet angulo dato baſim, æqualem datæ lineæ, ſub-<lb/> tendere.</head> <p> <s xml:id="echoid-s20836" xml:space="preserve">Eſto angulus datus a b c, & linea data d e:</s> <s xml:id="echoid-s20837" xml:space="preserve"> ſeparetur itaque à li-<lb/>nea b c, ex parte puncti b linea b f, non maior medietate lineæ d e <lb/>per 3 p 1, & in puncto f poſito pede circini immobili, deſcribatur cir-<lb/>culus ſecundum quantitatem ſemidiametri d e:</s> <s xml:id="echoid-s20838" xml:space="preserve"> hic itaq;</s> <s xml:id="echoid-s20839" xml:space="preserve"> ſecabit ne-<lb/>ceſſariò latus b a per 20 p 1, cum latus b f non ſit maius medietate li-<lb/>neæ d e.</s> <s xml:id="echoid-s20840" xml:space="preserve"> Sit ergo, ut ſecet ipſum in puncto g, & ducatur linea g f:</s> <s xml:id="echoid-s20841" xml:space="preserve"> hęc <lb/>itaque neceſſariò erit æqualis lineæ d e per circuli definitionem 15 <lb/>defin:</s> <s xml:id="echoid-s20842" xml:space="preserve"> 1 elemen:</s> <s xml:id="echoid-s20843" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s20844" xml:space="preserve"> Poteſt & idem aliter demon <lb/>ſtrari.</s> <s xml:id="echoid-s20845" xml:space="preserve"> A' puncto enim b ducatur linea b h angulariter, ut conting it, <lb/>ſuper lineam a b, quæ per 3 p 1 fiat æqualis datæ lineę d e:</s> <s xml:id="echoid-s20846" xml:space="preserve"> & à puncto <lb/>h ducatur æquidiſtans lineę a b per 31 p 1, quæ per 2 huius neceſſariò <lb/>concurret cum linea b c.</s> <s xml:id="echoid-s20847" xml:space="preserve"> Sit punctus concurſus k, & à puncto k du-<lb/>catur linea æquidiſtans lineæ b h, quæ ſit k l:</s> <s xml:id="echoid-s20848" xml:space="preserve"> erit quo que ſuperficies <lb/>b h k l æquidiſtantium laterum:</s> <s xml:id="echoid-s20849" xml:space="preserve"> ergo per 34 p 1, linea l k eſt æqualis <lb/>lineæ b h:</s> <s xml:id="echoid-s20850" xml:space="preserve"> ergo & lineæ datæ, quæ eſt d e:</s> <s xml:id="echoid-s20851" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s20852" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div702" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables270" xml:space="preserve">b a g c e d f</variables> </figure> <head xml:id="echoid-head601" xml:space="preserve" style="it">27. Datis duobus angulis inæqualibus, ex maiore <lb/> ipſorum æquum minorireſecare. E' 23 p 1 element.</head> <p> <s xml:id="echoid-s20853" xml:space="preserve">Sint duo anguli dati a b c, d e f:</s> <s xml:id="echoid-s20854" xml:space="preserve"> ſit a b c maior & d e f mi <lb/>nor.</s> <s xml:id="echoid-s20855" xml:space="preserve"> Propoſitum eſt, ut ex angulo a b c reſecetur angulus <lb/>æqualis angulo d e f:</s> <s xml:id="echoid-s20856" xml:space="preserve"> hoc autem fiet per 23 p 1, ſi ſuper b ter <lb/>minum lineæ a b intra angulum a b c fiat angulus æqualis <lb/>angulo d e f, qui ſit a b g:</s> <s xml:id="echoid-s20857" xml:space="preserve"> & hoc eſt propoſitum.</s> <s xml:id="echoid-s20858" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div703" type="section" level="0" n="0"> <head xml:id="echoid-head602" xml:space="preserve" style="it">28. Datum angulum rectum in tres partes æqua-<lb/>les diuidere.</head> <p> <s xml:id="echoid-s20859" xml:space="preserve">Nõ indiguimus quò ad præſens propoſitum diuiſione <lb/>aliorum angulorum in partes tres æquales, ſed ſolum recto:</s> <s xml:id="echoid-s20860" xml:space="preserve"> & ob hoc non proponimus hic, niſi de <lb/> <pb o="13" file="0315" n="315" rhead="LIBER I."/> recto:</s> <s xml:id="echoid-s20861" xml:space="preserve"> in uniuerſaliori ſcientia, ut in ea, quę de elementatis concluſionibus, uniuerſaliorem dignã <lb/>propoſitione exiſtimantes.</s> <s xml:id="echoid-s20862" xml:space="preserve"> Sit ita que angu-<lb/> <anchor type="figure" xlink:label="fig-0315-01a" xlink:href="fig-0315-01"/> lus rectus a b c, quem in partes tres ęquales uo <lb/>lumus diuidere:</s> <s xml:id="echoid-s20863" xml:space="preserve"> aſſumatur ergo linea quęcun-<lb/>que, & ſit d e:</s> <s xml:id="echoid-s20864" xml:space="preserve"> ſuper quam conſtituatur trigo nũ <lb/>ęquilaterum per 1 p 1:</s> <s xml:id="echoid-s20865" xml:space="preserve"> quòd ſit d f e, cuius angu-<lb/>lus d f e diuidatur per ęqualia per 9 p 1 ducta li <lb/>nea f g:</s> <s xml:id="echoid-s20866" xml:space="preserve"> erit ergo angulus d f g tertia pars unius <lb/>recti, cum ipſe ſit ſexta pars duorum rectorum <lb/>per 32 p 1:</s> <s xml:id="echoid-s20867" xml:space="preserve"> ergo per pręcedentem ab angulo re-<lb/>cto a b c reſecetur angulus a b h ęqualis angu-<lb/>lo d f g, & diuidatur angulus h b c per ęqualia per 9 p 1:</s> <s xml:id="echoid-s20868" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s20869" xml:space="preserve"/> </p> <div xml:id="echoid-div703" type="float" level="0" n="0"> <figure xlink:label="fig-0315-01" xlink:href="fig-0315-01a"> <variables xml:id="echoid-variables271" xml:space="preserve">b a h c ſ d g e</variables> </figure> </div> </div> <div xml:id="echoid-div705" type="section" level="0" n="0"> <head xml:id="echoid-head603" xml:space="preserve" style="it">29. Linea diuidens angulum alicuius trigoni, producta, baſim ſubtenſam illi angulo neceſſa <lb/>riò ſecabit: & ſi linea ſecans baſim, ad punctum concurſ{us} laterum trigoni producatur: illa an-<lb/>gulum baſi oppoſitum ſecabit.</head> <p> <s xml:id="echoid-s20870" xml:space="preserve">Sit, ut linea b d ſecet angulum a b c trigoni a b c.</s> <s xml:id="echoid-s20871" xml:space="preserve"> Dico, quòd eadem linea b d producta, neceſſa-<lb/>riò ſecabit baſim a c illi angulo ſubtenſam.</s> <s xml:id="echoid-s20872" xml:space="preserve"> Si enim non ſecabit baſim a c, concurret tamen cũ pro-<lb/>ducta a c per 14 huius:</s> <s xml:id="echoid-s20873" xml:space="preserve"> ideo quia anguli b a c & a b f ſunt <lb/> <anchor type="figure" xlink:label="fig-0315-02a" xlink:href="fig-0315-02"/> minores duobus rectis ex hypotheſi & per 32 p 1:</s> <s xml:id="echoid-s20874" xml:space="preserve"> ſit ergo <lb/>concurſus in puncto fultra punctum c.</s> <s xml:id="echoid-s20875" xml:space="preserve"> Eſt ergo trigono-<lb/>rum a b c & a b f angulus b a c cõmunis, & angulus b c a <lb/>maior angulo b f c per 16 p 1:</s> <s xml:id="echoid-s20876" xml:space="preserve"> erit ergo per 32 p 1 angulus a <lb/>b f maior angulo a b c:</s> <s xml:id="echoid-s20877" xml:space="preserve"> non ergo ſecat linea b f angulum <lb/>a b c:</s> <s xml:id="echoid-s20878" xml:space="preserve"> cadet itaq;</s> <s xml:id="echoid-s20879" xml:space="preserve"> neceſſariò inter puncta a & c:</s> <s xml:id="echoid-s20880" xml:space="preserve"> & ita ſeca-<lb/>bit baſim a c:</s> <s xml:id="echoid-s20881" xml:space="preserve"> quia ſi etiam caderet in punctũ a, uel in pun-<lb/>ctum c, non adhuc diuideret angulum a b c:</s> <s xml:id="echoid-s20882" xml:space="preserve"> patet ergo ꝓ-<lb/>poſitum primum.</s> <s xml:id="echoid-s20883" xml:space="preserve"> Patet etiã & reliquum propoſitorum:</s> <s xml:id="echoid-s20884" xml:space="preserve"> <lb/>quoniam ſi linea b d ſecet baſim trigoni a b c, & applice-<lb/>tur puncto b, quod eſt punctus concurſus laterum a b & c b:</s> <s xml:id="echoid-s20885" xml:space="preserve"> patet, quòd linea b d ſecabit angulum <lb/>a b c:</s> <s xml:id="echoid-s20886" xml:space="preserve"> ſit enim per 16 p 1 angulus a d b maior angulo b a c b:</s> <s xml:id="echoid-s20887" xml:space="preserve"> ſed angulus a c eſt cõmunis ambobus tri <lb/>gonis a b c & a b d:</s> <s xml:id="echoid-s20888" xml:space="preserve"> ergo per 32 p 1 angulus a b d eſt minor angulo a b c.</s> <s xml:id="echoid-s20889" xml:space="preserve"> Eſt ergo ſectus angulus a b c <lb/>per lineam b d:</s> <s xml:id="echoid-s20890" xml:space="preserve"> quod eſt ſecundum propoſitorum.</s> <s xml:id="echoid-s20891" xml:space="preserve"/> </p> <div xml:id="echoid-div705" type="float" level="0" n="0"> <figure xlink:label="fig-0315-02" xlink:href="fig-0315-02a"> <variables xml:id="echoid-variables272" xml:space="preserve">b a d c f</variables> </figure> </div> </div> <div xml:id="echoid-div707" type="section" level="0" n="0"> <head xml:id="echoid-head604" xml:space="preserve" style="it">30. Ab angulo dati trigoni linea perpendiculariter ad baſim producta, ſirectangulum ſub <lb/>partibus baſis contentum, maius fuerit quadrato perpendicularis: neceſſe est angulum (à quo <lb/>fit ductio) obtuſum eſſe: ſi minus, acutum: ſi æquale, rectum.</head> <p> <s xml:id="echoid-s20892" xml:space="preserve">Sit datus trigonus a b c, à cuius angulo b a c ducatur linea perpendicularis ſuper baſim b c:</s> <s xml:id="echoid-s20893" xml:space="preserve"> ſe-<lb/>cetq́;</s> <s xml:id="echoid-s20894" xml:space="preserve"> ipſam in puncto d:</s> <s xml:id="echoid-s20895" xml:space="preserve"> & ſit a d:</s> <s xml:id="echoid-s20896" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s20897" xml:space="preserve"> illud, quod fit ex ductu b d in d c maius quadrato lineæ a d.</s> <s xml:id="echoid-s20898" xml:space="preserve"> <lb/>Dico, quòd angulus b a c eſt obtuſus.</s> <s xml:id="echoid-s20899" xml:space="preserve"> Patet e-<lb/> <anchor type="figure" xlink:label="fig-0315-03a" xlink:href="fig-0315-03"/> <anchor type="figure" xlink:label="fig-0315-04a" xlink:href="fig-0315-04"/> <anchor type="figure" xlink:label="fig-0315-05a" xlink:href="fig-0315-05"/> <anchor type="figure" xlink:label="fig-0315-06a" xlink:href="fig-0315-06"/> nim per 17 p 6, quia non eſt proportio lineæ <lb/>b d ad lineam a d, quæ lineæ a d ad lineam d c.</s> <s xml:id="echoid-s20900" xml:space="preserve"> <lb/>ſit ergo per 12 p 6, ut quæ eſt proportio lineæ <lb/>b d ad lineam a d, eadem ſit lineæ a d ad lineã <lb/>g e:</s> <s xml:id="echoid-s20901" xml:space="preserve"> erit ergo illud, quod fit ex ductu lineæ b d <lb/>in lineam g e æquale quadrato lineæ a d per <lb/>17 p 6:</s> <s xml:id="echoid-s20902" xml:space="preserve"> quia illud, quod fit ex ductu lineę b d in <lb/>lineam d c, eſt maius quadrato lineę a d:</s> <s xml:id="echoid-s20903" xml:space="preserve"> patet, <lb/>quòd linea g e eſt minor quàm linea d c per 1 <lb/>p 6.</s> <s xml:id="echoid-s20904" xml:space="preserve"> Abſcindatur ergo à linea d c æqualis lineę <lb/>g e per 3 p 1, & ſit d f, ducaturq́;</s> <s xml:id="echoid-s20905" xml:space="preserve"> linea a f.</s> <s xml:id="echoid-s20906" xml:space="preserve"> Quia <lb/>itaq;</s> <s xml:id="echoid-s20907" xml:space="preserve"> illud, quod fit ex ductu lineæ b d in lineam d f, eſt æquale quadrato lineæ a d:</s> <s xml:id="echoid-s20908" xml:space="preserve"> patet per 17 p 6, <lb/>quoniam eſt proportio lineæ b d ad lineã a d, ſicut lineæ a d ad lineã d f:</s> <s xml:id="echoid-s20909" xml:space="preserve"> erit ergo per conuerſam 8 <lb/>p 6 angulus b a f rectus.</s> <s xml:id="echoid-s20910" xml:space="preserve"> Ergo angulus b a c eſt eſt maior recto.</s> <s xml:id="echoid-s20911" xml:space="preserve"> Similiterq́;</s> <s xml:id="echoid-s20912" xml:space="preserve"> demonſtrandum, quòd ſi <lb/>illud, quòd fit ex ductu b d in d c ſit minus quadrato a d, quoniam angulus b a c eſt acutus:</s> <s xml:id="echoid-s20913" xml:space="preserve"> nam per <lb/>eadem fit demonftratio.</s> <s xml:id="echoid-s20914" xml:space="preserve"> Pater etiam per eandem conuerſam 8 p 6, quoniam ſi illud, quod fit ex du <lb/>ctu lineæ b d in lineam d c, ſit æquale quadrato lineæ a d, quoniam angulus b a c eſt rectus:</s> <s xml:id="echoid-s20915" xml:space="preserve"> patet <lb/>ergo propoſitum.</s> <s xml:id="echoid-s20916" xml:space="preserve"/> </p> <div xml:id="echoid-div707" type="float" level="0" n="0"> <figure xlink:label="fig-0315-03" xlink:href="fig-0315-03a"> <variables xml:id="echoid-variables273" xml:space="preserve">a b d c</variables> </figure> <figure xlink:label="fig-0315-04" xlink:href="fig-0315-04a"> <variables xml:id="echoid-variables274" xml:space="preserve">g e</variables> </figure> <figure xlink:label="fig-0315-05" xlink:href="fig-0315-05a"> <variables xml:id="echoid-variables275" xml:space="preserve">a b d c</variables> </figure> <figure xlink:label="fig-0315-06" xlink:href="fig-0315-06a"> <variables xml:id="echoid-variables276" xml:space="preserve">a b d f c</variables> </figure> </div> </div> <div xml:id="echoid-div709" type="section" level="0" n="0"> <head xml:id="echoid-head605" xml:space="preserve" style="it">31. Abangulo iſoſcelis ducta perpendicularis ſuper baſim in duos partiales ſimiles trigo-<lb/>nos diuidit iſoſcelem. Ex quo patet, quòd linea perpendicularis ad medium punctum baſis ne-<lb/>ceſſariò pertingit.</head> <p> <s xml:id="echoid-s20917" xml:space="preserve">Sit iſoſceles a b c, cuius latera a b & a c ſint æqualia:</s> <s xml:id="echoid-s20918" xml:space="preserve"> & ab angulo b a c ducatur ſuper ba-<lb/> <pb o="14" file="0316" n="316" rhead="VITELLONIS OPTICAE"/> ſim b c perpendicularis a d.</s> <s xml:id="echoid-s20919" xml:space="preserve"> Dico, quòd propoſitus iſoſceles diuiſus eſt in duos trigonos par-<lb/>tiales ſimiles.</s> <s xml:id="echoid-s20920" xml:space="preserve"> Quoniam enim per 5 p 1 angulus a b d eſt æqualis angulo a c d, ſed & per definitio-<lb/>nem perpendicularis 10 defin.</s> <s xml:id="echoid-s20921" xml:space="preserve"> 1.</s> <s xml:id="echoid-s20922" xml:space="preserve"> elem.</s> <s xml:id="echoid-s20923" xml:space="preserve"> anguli a d b & a d c ſunt æqua-<lb/> <anchor type="figure" xlink:label="fig-0316-01a" xlink:href="fig-0316-01"/> les, quia recti:</s> <s xml:id="echoid-s20924" xml:space="preserve"> patet per 32 p 1, quòd anguli b a d & c a d ſunt æquales.</s> <s xml:id="echoid-s20925" xml:space="preserve"> <lb/>Ergo trigoni a b d & a c d ſunt æquianguli:</s> <s xml:id="echoid-s20926" xml:space="preserve"> ergo per 4 p 6 latera illo-<lb/>rum trigonorũ æquos angulos reſpicientia, ſunt proportionalia:</s> <s xml:id="echoid-s20927" xml:space="preserve"> ſunt <lb/>ergo illa trigona partialia, quæ a b d & a c d ſimilia per definitionem <lb/>ſimilium trigonorum:</s> <s xml:id="echoid-s20928" xml:space="preserve"> patet ergo propoſitum primum.</s> <s xml:id="echoid-s20929" xml:space="preserve"> Et quoniam <lb/>illa trigona a b d & a c d ſunt ſimilia, & eorum latera a b & a c ſunt æ-<lb/>qualia, & latus a d cõmune:</s> <s xml:id="echoid-s20930" xml:space="preserve"> patet, quòd etiam latera c d & b d ſunt æ-<lb/>qualia.</s> <s xml:id="echoid-s20931" xml:space="preserve"> Linea ergo քpendicularis, quę a d, neceſſariò pertingit ad me-<lb/>dium punctum lineæ b c:</s> <s xml:id="echoid-s20932" xml:space="preserve"> quod eſt propoſitum ſecundum.</s> <s xml:id="echoid-s20933" xml:space="preserve"/> </p> <div xml:id="echoid-div709" type="float" level="0" n="0"> <figure xlink:label="fig-0316-01" xlink:href="fig-0316-01a"> <variables xml:id="echoid-variables277" xml:space="preserve">a b d c</variables> </figure> </div> </div> <div xml:id="echoid-div711" type="section" level="0" n="0"> <head xml:id="echoid-head606" xml:space="preserve" style="it">32. Linea ducta à quocun puncto unius lateris trigoni produ-<lb/>cti, ultr a trigonum ſecans latus ab illo puncto remotius, & propin-<lb/>quius illi neceſſariò ſecabit.</head> <p> <s xml:id="echoid-s20934" xml:space="preserve">Sit trigonum a b c, cuius latus a b producatur ultra punctum b ad <lb/>punctum d:</s> <s xml:id="echoid-s20935" xml:space="preserve"> & à puncto d ducatur linea d e ſecans latus trigoni a c in puncto e.</s> <s xml:id="echoid-s20936" xml:space="preserve"> Dico, quòd d e ne-<lb/>ceſſariò ſecabit latus b c.</s> <s xml:id="echoid-s20937" xml:space="preserve"> Si enim non ſecabit latus b c, ſed ſolum latus <lb/> <anchor type="figure" xlink:label="fig-0316-02a" xlink:href="fig-0316-02"/> a c, ducatur linea d c, & producatur in continuum & directum:</s> <s xml:id="echoid-s20938" xml:space="preserve"> ſecabit <lb/>itaq;</s> <s xml:id="echoid-s20939" xml:space="preserve"> linea d c in aliquo puncto lineam d e:</s> <s xml:id="echoid-s20940" xml:space="preserve"> quoniam cum linea d c exeat <lb/>â puncto d, à quo exit etiam linea d e, & terminetur ad pũctum c inter-<lb/>iacens punctum e, neceſſariò illam ſecabit:</s> <s xml:id="echoid-s20941" xml:space="preserve"> ſit punctus ſectionis f.</s> <s xml:id="echoid-s20942" xml:space="preserve"> Pa-<lb/>làm itaq;</s> <s xml:id="echoid-s20943" xml:space="preserve">, quoniam duæ rectæ lineæ, quæ ſunt d f & d e f includunt ſu-<lb/>perficiem:</s> <s xml:id="echoid-s20944" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s20945" xml:space="preserve"> Idem quoque accidit, ſi linea d e duca-<lb/>tur extra lineam b c ultra punctum a:</s> <s xml:id="echoid-s20946" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s20947" xml:space="preserve"/> </p> <div xml:id="echoid-div711" type="float" level="0" n="0"> <figure xlink:label="fig-0316-02" xlink:href="fig-0316-02a"> <variables xml:id="echoid-variables278" xml:space="preserve">d b a e c f f</variables> </figure> </div> </div> <div xml:id="echoid-div713" type="section" level="0" n="0"> <head xml:id="echoid-head607" xml:space="preserve" style="it">33. Si à punctis terminalibus unius lateris trianguli duæ rectæ <lb/>exeuntes, intr a trigonum ad punctum unum conueniant: erit angu <lb/>lus inferior æqualis ſuperiori, & duobus angulis inter lineas duct as <lb/>ad alia duo later a trigoni contentis.</head> <p> <s xml:id="echoid-s20948" xml:space="preserve">Sit trigonum a b c, à cuius unius laterum a b punctis terminalibus, <lb/>quæ ſunt a & b, ducantur lineæ taliter, ut intra trigonum a b c concur-<lb/>rant in puncto d.</s> <s xml:id="echoid-s20949" xml:space="preserve"> Dico, quòd angulus a d b eſt æqualis angulo a c b, & <lb/>inſuper duobus angulis c a d & c b d.</s> <s xml:id="echoid-s20950" xml:space="preserve"> Quòd enim angulus a d b ſit maior angulo a c b, hoc patet per <lb/>21 p 1.</s> <s xml:id="echoid-s20951" xml:space="preserve"> Producatur itaq;</s> <s xml:id="echoid-s20952" xml:space="preserve"> linea c d ultra punctum d uſq;</s> <s xml:id="echoid-s20953" xml:space="preserve"> ad punctum e.</s> <s xml:id="echoid-s20954" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0316-03a" xlink:href="fig-0316-03"/> Eſt itaq;</s> <s xml:id="echoid-s20955" xml:space="preserve"> per 32 p 1 angulus e d a æqualis duobus angulis d c a & d a c:</s> <s xml:id="echoid-s20956" xml:space="preserve"> <lb/>& ſimiliter angulus e d b æqualis eſt duobus angulis d b c & d c b.</s> <s xml:id="echoid-s20957" xml:space="preserve"> To-<lb/>tus ergo angulus a d b ęqualis eſt angulo a c b, & angulis d a c & d b c:</s> <s xml:id="echoid-s20958" xml:space="preserve"> <lb/>quod eſt propoſitum.</s> <s xml:id="echoid-s20959" xml:space="preserve"/> </p> <div xml:id="echoid-div713" type="float" level="0" n="0"> <figure xlink:label="fig-0316-03" xlink:href="fig-0316-03a"> <variables xml:id="echoid-variables279" xml:space="preserve">c d e a b</variables> </figure> </div> </div> <div xml:id="echoid-div715" type="section" level="0" n="0"> <head xml:id="echoid-head608" xml:space="preserve" style="it">34. Linea æqualis & æquidiſtans baſi alicuius trigoni, uicini-<lb/>or angulo ſupremo, maiori angulo neceſſariò ſubtenditur.</head> <p> <s xml:id="echoid-s20960" xml:space="preserve">Eſto trigonum a b c, cuius baſi a c:</s> <s xml:id="echoid-s20961" xml:space="preserve"> uicinior angulo a b c duca-<lb/>tur linea æqualis & æquidiſtans, quæ ſit d e.</s> <s xml:id="echoid-s20962" xml:space="preserve"> Dico, quòd ſi à puncto <lb/>b ducantur lineæ b d & b e, quòd angulus d b e eſt maior angulo a b <lb/>c.</s> <s xml:id="echoid-s20963" xml:space="preserve"> Quia enim linea d e eſt æqualis lineæ a c, palàm, quòd ipſa ſic pro-<lb/>ducta ſecat lineas a b & b c, argumento 16 huius:</s> <s xml:id="echoid-s20964" xml:space="preserve"> quòd etiã patet ex a-<lb/>lijs.</s> <s xml:id="echoid-s20965" xml:space="preserve"> Nam omnis linea cadens intra trigonum ſecans latera eius & æ-<lb/>quidiſtans, eſt minor baſi per 29 p 1 & 4 p 6.</s> <s xml:id="echoid-s20966" xml:space="preserve"> Secet ergo linea d e latus <lb/>b a in puncto f, & latus b c in puncto g.</s> <s xml:id="echoid-s20967" xml:space="preserve"> Quia ita que per 16 p 1 angulus b g f eſt maior angulo b e <lb/>g:</s> <s xml:id="echoid-s20968" xml:space="preserve"> erit per 29 p 1 angulus b c a maior angulo b e d:</s> <s xml:id="echoid-s20969" xml:space="preserve"> & ea-<lb/> <anchor type="figure" xlink:label="fig-0316-04a" xlink:href="fig-0316-04"/> dem ratione angulus b a c eſt maior angulo b d e:</s> <s xml:id="echoid-s20970" xml:space="preserve"> ne-<lb/>ceſſariò ergo per 32 p 1 erit angulus d b e cum angulis mi-<lb/>noribus ualens duos rectos, maior angulo a b c, ualente <lb/>cum duobus angulis maioribus duos rectos:</s> <s xml:id="echoid-s20971" xml:space="preserve"> patet ergo <lb/>propoſitum.</s> <s xml:id="echoid-s20972" xml:space="preserve"/> </p> <div xml:id="echoid-div715" type="float" level="0" n="0"> <figure xlink:label="fig-0316-04" xlink:href="fig-0316-04a"> <variables xml:id="echoid-variables280" xml:space="preserve">b d f g e a c</variables> </figure> </div> </div> <div xml:id="echoid-div717" type="section" level="0" n="0"> <head xml:id="echoid-head609" xml:space="preserve" style="it">35. In trigono orthogonio ab uno reliquorum an-<lb/>gulorum producta linea ad baſim: erit remotioris an-<lb/>guli ad propinquiorem recto minor proportio, quàm <lb/> <pb o="15" file="0317" n="317" rhead="LIBER I."/> partis b aſis remotioris ad propinquiorem. 5 p geometriæ Iordani.</head> <p> <s xml:id="echoid-s20973" xml:space="preserve">Sit trigonum orthogonium a b c, cuius angulus b a c ſit rectus:</s> <s xml:id="echoid-s20974" xml:space="preserve"> & à puncto b ducatur ad latus <lb/>a c (quod eſt baſis anguli a b c) linea recta, quæ ſit b d.</s> <s xml:id="echoid-s20975" xml:space="preserve"> Dico, quòd minor eſt proportio anguli <lb/>c b d remotioris ab angulo recto, ad angulum d b a propinquiorem ipſi recto, quàm partis baſis <lb/>remotioris ab angulo recto (quæ eſt c d) ad latus d a propinquius ipſi angulo recto.</s> <s xml:id="echoid-s20976" xml:space="preserve"> Quoniam <lb/>enim angulus b a c eſt rectus, patet, quia angulus b d a eſt acutus per <lb/> <anchor type="figure" xlink:label="fig-0317-01a" xlink:href="fig-0317-01"/> 32 p 1:</s> <s xml:id="echoid-s20977" xml:space="preserve"> ergo per 13 p 1, angulus b d c eſt obtuſus:</s> <s xml:id="echoid-s20978" xml:space="preserve"> ergo per 19 p 1 latus <lb/>b d eſt maius latere a b, & minus latere b c.</s> <s xml:id="echoid-s20979" xml:space="preserve"> A' centro itaque b ſe-<lb/>cundum quantitatem ſemidiametri b d deſcribatur arcus circuli ſe-<lb/>cans lineam b c in puncto e:</s> <s xml:id="echoid-s20980" xml:space="preserve"> & ad ipſum producatur linea b a, in pun <lb/>ctum f:</s> <s xml:id="echoid-s20981" xml:space="preserve"> factiq́ue erunt duo ſectores b d e minor trigono b d c, & <lb/>b d f maior trigono b d a.</s> <s xml:id="echoid-s20982" xml:space="preserve"> Et quoniam eſt proportio ſectoris ad ſe-<lb/>ctorem, ſicut arcus f d ad arcum d e, ut patet per modum demon-<lb/>ſtrationis 1 p 6:</s> <s xml:id="echoid-s20983" xml:space="preserve"> quoniam omnes ſectores eiuſdem circuli, ſunt eiuſdẽ <lb/>altitudinis, & æquemultiplicia arcuum faciunt æquemultiplicia <lb/>ipſorum ſectorum:</s> <s xml:id="echoid-s20984" xml:space="preserve"> proportio uerò arcus d fad arcum d e eſt ſicut <lb/>anguli d b f ad angulum d b e per 33 p 6.</s> <s xml:id="echoid-s20985" xml:space="preserve"> Cum itaque trigonum c <lb/>d b ſit maius quàm ſector e d b, & ſector f d b ſit maior trigonoa <lb/>d b:</s> <s xml:id="echoid-s20986" xml:space="preserve"> erit per 9 huius trigoni c d b primi ad trigonum d b a ſecũdum <lb/>maior proportio, quàm ſectoris e b d tertij ad ſectorem d b f quar-<lb/>tum.</s> <s xml:id="echoid-s20987" xml:space="preserve"> Eſt autem per 1 p 6 trigoni c b d ad trigonum d b a, ſicut baſis <lb/>c d ad baſim d a:</s> <s xml:id="echoid-s20988" xml:space="preserve"> ſectoris uerò e d f ad ſectorem d b f, ut patet expræmiſsis, eſt proportio ſicut <lb/>anguli e b d ad angulũ d b f.</s> <s xml:id="echoid-s20989" xml:space="preserve"> Patet ergo, quòd maior eſt proportio lineæ c d ad lineam d a, quàm an-<lb/>guli c b d ad angulum d b a.</s> <s xml:id="echoid-s20990" xml:space="preserve"> Ergo minor eſt proportio anguli c b d ad angulum d b a, quàm lateris <lb/>c d ad latus d a:</s> <s xml:id="echoid-s20991" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s20992" xml:space="preserve"/> </p> <div xml:id="echoid-div717" type="float" level="0" n="0"> <figure xlink:label="fig-0317-01" xlink:href="fig-0317-01a"> <variables xml:id="echoid-variables281" xml:space="preserve">c d f e a b</variables> </figure> </div> </div> <div xml:id="echoid-div719" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables282" xml:space="preserve">e d b a c</variables> </figure> <head xml:id="echoid-head610" xml:space="preserve" style="it">36. Cuiuslibet trigoni duo latera producta, aliud trigonum <lb/> priori ſimile principiant, lateribus poſitione & ſitu tranſmutatis.</head> <p> <s xml:id="echoid-s20993" xml:space="preserve">Sit trigonum a b c, cuius latus a b ſit dextrum, & latus b c ſiniſtrũ, <lb/>quæ producantur ultra punctum b:</s> <s xml:id="echoid-s20994" xml:space="preserve"> & proportionaliter prioribus la-<lb/>teribus abſcindantur per 12 p 6, linea ſcilicet a b in puncto d, & linea <lb/>c b in puncto e:</s> <s xml:id="echoid-s20995" xml:space="preserve"> & coniungatur linea d e.</s> <s xml:id="echoid-s20996" xml:space="preserve"> Erit ita que trigonum d b e <lb/>ſimile trigono a b c:</s> <s xml:id="echoid-s20997" xml:space="preserve"> ſed & latus d b erit ſiniſtrum, & latus e b dextrũ.</s> <s xml:id="echoid-s20998" xml:space="preserve"> <lb/>Sunt ita que latera iſtorum trigonorum poſitione, & ſitu tranſmuta-<lb/>ta:</s> <s xml:id="echoid-s20999" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s21000" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div720" type="section" level="0" n="0"> <head xml:id="echoid-head611" xml:space="preserve" style="it">37. Omnium duorum trigonorum rectangulorum, quorum <lb/>unius unum laterum rectos angulos continentium fuerit maius <lb/>altero alterius, reliquum uerò minus reliquo: erit angulus acu-<lb/>tus unius maius latus reſpiciens, maior angulo alterius ſuum rela-<lb/>tiuum latus reſpiciente.</head> <p> <s xml:id="echoid-s21001" xml:space="preserve">Verbi gratia:</s> <s xml:id="echoid-s21002" xml:space="preserve"> ſint duo trianguli rectanguli a b c & a c d:</s> <s xml:id="echoid-s21003" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0317-03a" xlink:href="fig-0317-03"/> ſintq́;</s> <s xml:id="echoid-s21004" xml:space="preserve"> anguli a b c & a d c recti:</s> <s xml:id="echoid-s21005" xml:space="preserve"> & ſit latus b c trianguli a b c <lb/>maius latere c d trianguli a c d, & reliquum laterum rectos <lb/>angulos continentium a b unius ſit minus reliquo latere al-<lb/>terius, quod eſt a d, ut patet in propoſita figuratione, ſi linea <lb/>a b intelligatur erecta ſuper lineam b c & ſuperficiem eius, <lb/>& linea b d intelligatur perpendicularis ſuper lineam d c in <lb/>eadem ſuperficie iacentem:</s> <s xml:id="echoid-s21006" xml:space="preserve"> tunc enim erit linea a d perpen-<lb/>dicularis ſuper lineam d c per 22 huius:</s> <s xml:id="echoid-s21007" xml:space="preserve"> quod etiam patet, ſi <lb/>in ſuperficie iacente ducatur linea b e æquidiſtanter lineæ <lb/>d c per 31 p 1.</s> <s xml:id="echoid-s21008" xml:space="preserve"> Et quoniam linea a b eſt perpendicularis ſuper <lb/>ſuperficiem iacentem, in qua ſunt lineæ b d, d c, b e, palàm <lb/>per definitionem lineæ erectæ, quoniam angulus a b e eſt <lb/>rectus:</s> <s xml:id="echoid-s21009" xml:space="preserve"> ſed & angulus e b d eſt rectus per 29 p 1, cum angu-<lb/>lus b d c ſit rectus per 22 huius, & lineæ b e & d c æquidiſtẽt:</s> <s xml:id="echoid-s21010" xml:space="preserve"> <lb/>ergo per 4 p 11 linea b e eſt erecta ſuper ſuperficiem trigoni <lb/>a b d:</s> <s xml:id="echoid-s21011" xml:space="preserve"> ergo per 8 p 11 linea d c eſt perpen dicularis ſuper ean-<lb/>dem ſuperficiem trigoni a b d:</s> <s xml:id="echoid-s21012" xml:space="preserve"> angulus ergo a d c eſt rectus:</s> <s xml:id="echoid-s21013" xml:space="preserve"> <lb/>ſed & latus a d maius eſt latere a b per 19 p 1:</s> <s xml:id="echoid-s21014" xml:space="preserve"> quoniam angulus a b d eſt rectus.</s> <s xml:id="echoid-s21015" xml:space="preserve"> Dico ergo, quòd <lb/>angulus a c d eſt maior angulo a c b.</s> <s xml:id="echoid-s21016" xml:space="preserve"> quoniam enim latus a d eſt maius latere b a per 19 p 1, cum an-<lb/>gulus a b d ſit rectus:</s> <s xml:id="echoid-s21017" xml:space="preserve"> patet, quòd præſens figuratio eſt cõformis hypotheſi.</s> <s xml:id="echoid-s21018" xml:space="preserve"> Reſecetur ergo per 3 p 1 <lb/> <pb o="16" file="0318" n="318" rhead="VITELLONIS OPTICAE"/> à latere d a æquale lateri b a, quod ſit linea d f.</s> <s xml:id="echoid-s21019" xml:space="preserve"> Et quia linea d c eſt minor latere b c per 19 p 1:</s> <s xml:id="echoid-s21020" xml:space="preserve"> quo-<lb/>niã angulus b d c eſt rectus:</s> <s xml:id="echoid-s21021" xml:space="preserve"> protrahatur linea d c, & reſecetur in pũcto g taliter, ut ſit linea d g ęqua <lb/>lis lineæ b c.</s> <s xml:id="echoid-s21022" xml:space="preserve"> Quia ergo trigoni f d g duo latera f d & d g ſunt æqualia duobus lateribus a b & b c tri-<lb/>goni a b c, & angulus f d g æqualis eſt angulo a b c:</s> <s xml:id="echoid-s21023" xml:space="preserve"> quia uterq;</s> <s xml:id="echoid-s21024" xml:space="preserve"> rectus:</s> <s xml:id="echoid-s21025" xml:space="preserve"> erit per 4 p 1 baſis f g æqualis <lb/>baſi a c, & reliqui anguli reliquis angulis:</s> <s xml:id="echoid-s21026" xml:space="preserve"> angulus ergo f g d æqualis erit angulo a c b.</s> <s xml:id="echoid-s21027" xml:space="preserve"> Quia uerò <lb/>puncta a & fſunt in linea a d, & puncta c & g ſunt in linea d g:</s> <s xml:id="echoid-s21028" xml:space="preserve"> palàm, quia lineæ a c & f g ſunt in una <lb/>ſuperficie, quæ eſt a d g per 2 p 11:</s> <s xml:id="echoid-s21029" xml:space="preserve"> ergo interſecant ſe lineæ g f & c a:</s> <s xml:id="echoid-s21030" xml:space="preserve"> ſit earũ interſectio in puncto h.</s> <s xml:id="echoid-s21031" xml:space="preserve"> <lb/>Quia uerò in trigono c h g latus g c protrahitur, palàm ex 16 p 1, quoniã angulus h c d maior eſt an-<lb/>gulo h g c:</s> <s xml:id="echoid-s21032" xml:space="preserve"> ergo & eius æquali, ſcilicet angulo a c b:</s> <s xml:id="echoid-s21033" xml:space="preserve"> angulus ergo a c d maior eſt angulo a c b:</s> <s xml:id="echoid-s21034" xml:space="preserve"> quod <lb/>eſt propoſitũ.</s> <s xml:id="echoid-s21035" xml:space="preserve"> Similiterq́;</s> <s xml:id="echoid-s21036" xml:space="preserve"> demonſtrandũ in alijs:</s> <s xml:id="echoid-s21037" xml:space="preserve"> ſi enim trigona propoſita fuerint in diuerſis locis <lb/>conſtituta, palàm, quia ipſis æqualia & æquiangula trigona ſic poſſunt ordinari, ut in figura diſpo-<lb/>nuntur, & demonſtratio facta de ijs ſe extendit ad alia.</s> <s xml:id="echoid-s21038" xml:space="preserve"> Patet ergo uniuerſaliter propoſitum.</s> <s xml:id="echoid-s21039" xml:space="preserve"> Et ex <lb/>hoc patet, quòd angulus b a c eſt maior angulo d a c per 32 p 1.</s> <s xml:id="echoid-s21040" xml:space="preserve"/> </p> <div xml:id="echoid-div720" type="float" level="0" n="0"> <figure xlink:label="fig-0317-03" xlink:href="fig-0317-03a"> <variables xml:id="echoid-variables283" xml:space="preserve">a f h b e d c g</variables> </figure> </div> </div> <div xml:id="echoid-div722" type="section" level="0" n="0"> <head xml:id="echoid-head612" xml:space="preserve" style="it">38. Omnium duorum trigonorum rectangulorũ, quorũ latus ſubtenſum recto angulo unius <lb/>ad minus latus eiuſdem proportionem habuerit maiorem, quàm latus ſubtenſum recto angulo <lb/>alterius ad minus latus eiuſdem: erit angulus linearum maioris proportionis maior angulo li-<lb/>nearum minoris proportionis: & econuerſo.</head> <p> <s xml:id="echoid-s21041" xml:space="preserve">Sint duo trigona rectangula a b c & d e f, quorũ anguli a b c & d e f ſint recti:</s> <s xml:id="echoid-s21042" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s21043" xml:space="preserve"> latus b c minus <lb/>latere a b, & latus e f minus latere d e:</s> <s xml:id="echoid-s21044" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s21045" xml:space="preserve"> maior proportio lineæ a c ad lineam f e.</s> <s xml:id="echoid-s21046" xml:space="preserve"> Dico, quòd an-<lb/>gulus a c b maior eſt angulo d f e.</s> <s xml:id="echoid-s21047" xml:space="preserve"> Quia enim maior eſt proportio lineæ a c ad lineã c b, quàm lineæ <lb/>d f ad lineam f e:</s> <s xml:id="echoid-s21048" xml:space="preserve"> ſed per 47 p 1 quadratũ lineæ <lb/> <anchor type="figure" xlink:label="fig-0318-01a" xlink:href="fig-0318-01"/> <anchor type="figure" xlink:label="fig-0318-02a" xlink:href="fig-0318-02"/> <anchor type="figure" xlink:label="fig-0318-03a" xlink:href="fig-0318-03"/> a c ualet quadrata duarum linearũ a b & c b:</s> <s xml:id="echoid-s21049" xml:space="preserve"> & <lb/>quadratũ lineæ d fualet quadrata duarũ linea <lb/>rum, quæ ſunt d e & f e:</s> <s xml:id="echoid-s21050" xml:space="preserve"> & quia per 20 p 6 pro-<lb/>portio quadratorũ eſt proportio duplicata la-<lb/>terũ:</s> <s xml:id="echoid-s21051" xml:space="preserve"> patet, quòd maior eſt proportio quadra.</s> <s xml:id="echoid-s21052" xml:space="preserve"> <lb/>tia c ad quadratum c b, quàm quadrati d f ad <lb/>quadratũ f e:</s> <s xml:id="echoid-s21053" xml:space="preserve"> eſt ergo per 11 huius maior pro-<lb/>portio amborũ quadratorũ linearũ a b & b c <lb/>ad quadra tũ b c, quàm am borũ quadratorũ li-<lb/>nearũ d e & f e a d quadratũ f e:</s> <s xml:id="echoid-s21054" xml:space="preserve"> ergo per 12 hu-<lb/>ius maior eſt proportio quadrati a b ad qua-<lb/>dratum b c, quàm quadrati d e ad quadratũ e f:</s> <s xml:id="echoid-s21055" xml:space="preserve"> eſt ergo per 22 p 6 maior proportio lineę a b ad line-<lb/>am b c, quàm lineæ d e ad lineã e f.</s> <s xml:id="echoid-s21056" xml:space="preserve"> Eſto, ut, quæ eſt proportio lineæ d e ad lineã e f, eadẽ ſit alicuius <lb/>lineæ, ut g h ad lineam c b per 3 huius:</s> <s xml:id="echoid-s21057" xml:space="preserve"> erit ergo linea g h minor quàm linea a b per 10 p 5.</s> <s xml:id="echoid-s21058" xml:space="preserve"> Reſecetur <lb/>ergo per 3 p 1 ex linea a b æqualis lineæ g h:</s> <s xml:id="echoid-s21059" xml:space="preserve"> & ſit b k, & continuetur linea c k:</s> <s xml:id="echoid-s21060" xml:space="preserve"> erunt ergo per 6 p 6 <lb/>trigona d e f & k b c æquiangula:</s> <s xml:id="echoid-s21061" xml:space="preserve"> angulus itaq;</s> <s xml:id="echoid-s21062" xml:space="preserve"> b c k eſt æqualis angulo e f d:</s> <s xml:id="echoid-s21063" xml:space="preserve"> ſed angulus b c a eſt <lb/>maior angulo b c k, totũ parte.</s> <s xml:id="echoid-s21064" xml:space="preserve"> Angulus itaq;</s> <s xml:id="echoid-s21065" xml:space="preserve"> a c b maior eſt angulo d f e:</s> <s xml:id="echoid-s21066" xml:space="preserve"> & hoc eſt ꝓpoſitũ:</s> <s xml:id="echoid-s21067" xml:space="preserve"> ex quo <lb/>etiã patet, quòd eius cõuerſa eſt uera:</s> <s xml:id="echoid-s21068" xml:space="preserve"> quoniã in talibus trigonis lineæ maiores angulos continen-<lb/>tes, maiorem habent ad ſeinuicem proportionem.</s> <s xml:id="echoid-s21069" xml:space="preserve"/> </p> <div xml:id="echoid-div722" type="float" level="0" n="0"> <figure xlink:label="fig-0318-01" xlink:href="fig-0318-01a"> <variables xml:id="echoid-variables284" xml:space="preserve">a k b c</variables> </figure> <figure xlink:label="fig-0318-02" xlink:href="fig-0318-02a"> <variables xml:id="echoid-variables285" xml:space="preserve">d e f</variables> </figure> <figure xlink:label="fig-0318-03" xlink:href="fig-0318-03a"> <variables xml:id="echoid-variables286" xml:space="preserve">h g</variables> </figure> </div> </div> <div xml:id="echoid-div724" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables287" xml:space="preserve">a c e f b d</variables> </figure> <head xml:id="echoid-head613" xml:space="preserve" style="it">39. A puncto in aere dato ad ſubſtratam planãſuperficiẽ una linea perpendiculariter, alia <lb/>obliquè incidente, & linea recta inter pũcta incidentiæ in ipſa ſu <lb/> perficie protracta: erit angulus à non perpendiculari cũ iacẽte li- nea contentus, minimus omnium angulorum ſub illa obliqua & quacun linea in ſubſtrata ſuperſicie protracta contentorum: & omnis angulus illi propinquior, eſt minor remotiore: & duo ex utra parte æqualiter approximantes, ſunt æquales. Lemma ad 37 the. opticorum Euclidis. 43 theor 6 libri συναγωγυζμ μαθκμα- τικυζμ Pappi.</head> <p> <s xml:id="echoid-s21070" xml:space="preserve">Sit punctus in aere datus a, cui ſit ſub ſtrata ſuperficies plana, quę <lb/>b c d, fuper quã ab illo puncto ducatur obliquè linea a b, ducaturq́;</s> <s xml:id="echoid-s21071" xml:space="preserve"> <lb/>perpendiculariter linea a c, & copuletur linea b c.</s> <s xml:id="echoid-s21072" xml:space="preserve"> Dico, quòd angu-<lb/>lus a b c eſt minimus omnium angulorũ contentorũ ſub linea obli-<lb/>qua a b, & ſub unaquaq;</s> <s xml:id="echoid-s21073" xml:space="preserve"> linearũ à puncto b ductarũ in ſuperficie b <lb/>c d:</s> <s xml:id="echoid-s21074" xml:space="preserve"> & quòd ſemper propinquior ipſi eſt minor quàm remotior:</s> <s xml:id="echoid-s21075" xml:space="preserve"> & <lb/>quòd duo anguli æquales ſolũ ex utraq;</s> <s xml:id="echoid-s21076" xml:space="preserve"> parte ipſius cõſiſtunt.</s> <s xml:id="echoid-s21077" xml:space="preserve"> Duca <lb/>tur enim in data plana ſuperficie, utcunq;</s> <s xml:id="echoid-s21078" xml:space="preserve"> contingit, linea b d, & à <lb/>puncto c ducatur in eadem ſuperficie linea perpendicularis ſuper lineam b d per 11 p 1, quæ ſit c d, <lb/>& copuletur à puncto a linea a d:</s> <s xml:id="echoid-s21079" xml:space="preserve"> eſt ita q;</s> <s xml:id="echoid-s21080" xml:space="preserve"> per 22 huius linea a d perpẽdicularis ſuper lineam b d.</s> <s xml:id="echoid-s21081" xml:space="preserve"> Et <lb/>quoniam angulus a c d eſt rectus, palàm per 19 p 1, quoniam obliqua linea a d maior eſt catheto a c:</s> <s xml:id="echoid-s21082" xml:space="preserve"> <lb/>linea itaq;</s> <s xml:id="echoid-s21083" xml:space="preserve"> b a ad lineam a c maiorẽ habet proportionẽ quàm ad lineã a d per 8 p 5:</s> <s xml:id="echoid-s21084" xml:space="preserve"> & anguli b c a & <lb/> <pb o="17" file="0319" n="319" rhead="LIBER I."/> b d a ſunt recti:</s> <s xml:id="echoid-s21085" xml:space="preserve"> erit itaq;</s> <s xml:id="echoid-s21086" xml:space="preserve"> ք præ cedẽtẽ proximã angulus b a c maior angulo b a d:</s> <s xml:id="echoid-s21087" xml:space="preserve"> erit ergo per 32 p 1 <lb/>angulus a b c minor angulo a b d.</s> <s xml:id="echoid-s21088" xml:space="preserve"> Similiterq́;</s> <s xml:id="echoid-s21089" xml:space="preserve"> patet, quoniã angulus a b c minimus eſt omniũ angu-<lb/>lorũ cõtẽtorũ ſub linea obliquè incidẽte à pũcto a lineę b c, & ſub ipſa linea b c.</s> <s xml:id="echoid-s21090" xml:space="preserve"> Propinquior quoq;</s> <s xml:id="echoid-s21091" xml:space="preserve"> <lb/>illi eſt minor remotiore.</s> <s xml:id="echoid-s21092" xml:space="preserve"> Ducatur enim à pũcto b in ſubſtrata ſuperficie linea, ut cõtingit, quę ſit b e, <lb/>& à pũcto c ducatur in eadẽ ſuperficie linea քpẽdicularis ſuper lineã b e, q̃ ſit linea c e, & ꝓducatur <lb/>linea a e, quę ք 22 huius erit perpẽdicularis ſuper lineã b e.</s> <s xml:id="echoid-s21093" xml:space="preserve"> Et quoniã angulus b d c eſt rectus, & an-<lb/>gulus c e b rectus, & angulus b c d maior eſt angulo b c e per cõuerſam pręmiſſæ, quoniã linea e c ad <lb/>lineã c b maiorẽ habet ꝓportionẽ ꝗ̃ linea d c ad lineã c b.</s> <s xml:id="echoid-s21094" xml:space="preserve"> Linea itaq;</s> <s xml:id="echoid-s21095" xml:space="preserve"> e c eſt multõ maior ꝗ̃ linea c d:</s> <s xml:id="echoid-s21096" xml:space="preserve"> <lb/>ſed cathetus a c քpendiculariter incidit lineis c e & c d ք definitionẽ lineę erectæ:</s> <s xml:id="echoid-s21097" xml:space="preserve"> maior ergo eſt li-<lb/>nea a e ꝗ̃ linea a d ք 47 p 1:</s> <s xml:id="echoid-s21098" xml:space="preserve"> linea enim c e eſt maior ꝗ̃ linea c d.</s> <s xml:id="echoid-s21099" xml:space="preserve"> Linea itaq;</s> <s xml:id="echoid-s21100" xml:space="preserve"> b a ad lineã a d maiorẽ ha-<lb/>bet proportionẽ ꝗ̃ ad lineã e a ք 8 p 5:</s> <s xml:id="echoid-s21101" xml:space="preserve"> & anguli a d b & a e b ſunt recti:</s> <s xml:id="echoid-s21102" xml:space="preserve"> angulus itaq;</s> <s xml:id="echoid-s21103" xml:space="preserve"> b a d eſt maior <lb/>angulo b a e per præcedentẽ:</s> <s xml:id="echoid-s21104" xml:space="preserve"> ergo per 32 p 1 angulus a b d minor eſt angulo a b e.</s> <s xml:id="echoid-s21105" xml:space="preserve"> Similiter quoque <lb/>demonſtrandũ, quòd ſemper angulus propinquior, minor eſt remotiore:</s> <s xml:id="echoid-s21106" xml:space="preserve"> ſolũ uerò duo ex utraque <lb/>parte æquales cõſiſtunt:</s> <s xml:id="echoid-s21107" xml:space="preserve"> ſuper punctũ enim b terminũ lineæ c b in ſubiecta ſuperficie conſtituatur <lb/>angulus æqualis angulo d b c per 23 p 1, qui ſit c b f:</s> <s xml:id="echoid-s21108" xml:space="preserve"> & à puncto c ducatur linea c f perpendiculariter <lb/>ſuper lineã b f per 12 p 1, & ducatur linea a f.</s> <s xml:id="echoid-s21109" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s21110" xml:space="preserve"> angulus c b d eſt æqualis angulo c b f ex hypo <lb/>theſi, & angulus c d b eſt rectus æqualis angulo c f b recto, & linea c b eſt cõmunis ambobus trigo-<lb/>nis b c d & b c f:</s> <s xml:id="echoid-s21111" xml:space="preserve"> palàm per 26 p 1, quoniam latus b d eſt æquale lateri b f, & latus d c eſt æquale lateri <lb/>c f:</s> <s xml:id="echoid-s21112" xml:space="preserve"> ſed quia linea a c eſt cathetus ſuper ſuperficiẽ b c d, eſt per pendicularis ſuper ambas lineas d c & <lb/>f c.</s> <s xml:id="echoid-s21113" xml:space="preserve"> Eſt itaq;</s> <s xml:id="echoid-s21114" xml:space="preserve"> linea a d æqualis lineæ a f.</s> <s xml:id="echoid-s21115" xml:space="preserve"> Quoniã itaq;</s> <s xml:id="echoid-s21116" xml:space="preserve"> æqualis eſt linea d b lineæ b f, & linea b a eſt cõ-<lb/>munis ambobus trigonis d b a & b a f, & linea d a æqualis lineæ a f, erit angulus a b d æqualis angu-<lb/>lo a b f per 8 p 1.</s> <s xml:id="echoid-s21117" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s21118" xml:space="preserve"> demonſtrandũ, quoniã angulo a b d non erit aliquis alius æqualis.</s> <s xml:id="echoid-s21119" xml:space="preserve"> <lb/>Eſt ergo angulus a b c minimus, &c.</s> <s xml:id="echoid-s21120" xml:space="preserve"> ut proponitur:</s> <s xml:id="echoid-s21121" xml:space="preserve"> patet itaq;</s> <s xml:id="echoid-s21122" xml:space="preserve"> intentum.</s> <s xml:id="echoid-s21123" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div725" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables288" xml:space="preserve">b f c h e k a g d</variables> </figure> <head xml:id="echoid-head614" xml:space="preserve" style="it">40. Omnium ſuperficierum æquidiſtantiũ laterũ <lb/>diagonij per æqualia ſe ſecãt: ex quo patet, quòd pun <lb/> ctum interſectionis diagoniorum eſt medium pun- ctum eiuſdem ſuperficiei.</head> <p> <s xml:id="echoid-s21124" xml:space="preserve">Sit ſuperficies æquidiſtantiũ laterũ, ſiue ſit quadra <lb/>ta, ſiue altera parte longior, quæ a b c d, in qua ducan-<lb/>tur diagonij, quæſint a c & b d, ſecantes ſe in puncto e.</s> <s xml:id="echoid-s21125" xml:space="preserve"> <lb/>Dico, quòd diagonij ſecant ſe adinuicem per ęqualia:</s> <s xml:id="echoid-s21126" xml:space="preserve"> <lb/>& quòd punctũ e eſt mediũ punctũ ſuperficiei a b c d.</s> <s xml:id="echoid-s21127" xml:space="preserve"> <lb/>Palàm enim, quia trigona b e c & a e d per 15 & 29 p 1 <lb/>ſunt æquiangula:</s> <s xml:id="echoid-s21128" xml:space="preserve"> & erit angulus e b c æqualis angulo <lb/>e d a, ꝗa ſunt coalterni.</s> <s xml:id="echoid-s21129" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s21130" xml:space="preserve"> angulus e c b, <lb/>eſt æ qualis angulo e a d:</s> <s xml:id="echoid-s21131" xml:space="preserve"> ergo per 4 p 6 erit proportio <lb/>lineæ b e ad lineam e d, ſicut lineæ c e ad lineam e a:</s> <s xml:id="echoid-s21132" xml:space="preserve"> & <lb/>ſicut lineæ b c ad lineã a d:</s> <s xml:id="echoid-s21133" xml:space="preserve"> ſed linea b c eſt æqualis li-<lb/>neæ a d per 34 p 1.</s> <s xml:id="echoid-s21134" xml:space="preserve"> Linea ergo b e eſt æqualis lineę e d, <lb/>& linea c e æqualis lineę e a.</s> <s xml:id="echoid-s21135" xml:space="preserve"> Illę ergo diagonij diuidũt <lb/>ſe adinuicẽ per æqualia.</s> <s xml:id="echoid-s21136" xml:space="preserve"> Et ք hoc manifeſtũ eſt corollariũ:</s> <s xml:id="echoid-s21137" xml:space="preserve"> punctũ enim e æqualiter diſtat ab omni-<lb/>bus extremis:</s> <s xml:id="echoid-s21138" xml:space="preserve"> in quo tñ ſi aliquod dubiũ fuerit, ducãtur à pũcto e lineę æquidiſtantes lateribus ſu-<lb/>perficiei propoſitę per 31 p 1, quę ſint f g & h k:</s> <s xml:id="echoid-s21139" xml:space="preserve"> ſequeturq́;</s> <s xml:id="echoid-s21140" xml:space="preserve"> propter æqualitatem partiũ ipſarũ diago-<lb/>niorũ modo prædicto argumẽtãdo, lineã f e æqualẽ fieri lineę e g, & lineã h e æqualẽ fieri lineæ e k.</s> <s xml:id="echoid-s21141" xml:space="preserve"> <lb/>Patet itaq;</s> <s xml:id="echoid-s21142" xml:space="preserve">, quoniã ſecundum omnem modum, punctum e æqualiter diſtat à punctis extrem arum <lb/>linearum:</s> <s xml:id="echoid-s21143" xml:space="preserve"> directè igitur oppoſitum eſt:</s> <s xml:id="echoid-s21144" xml:space="preserve"> ergo medium inter illa:</s> <s xml:id="echoid-s21145" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s21146" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div726" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables289" xml:space="preserve">a b n l e p m d c</variables> </figure> <head xml:id="echoid-head615" xml:space="preserve" style="it">41. Datæ ſuperficiei æquidiſtantium laterũ ſimilem ſuperficiẽ, <lb/> cuius latera æquidiſtent datæ ſuperficiei laterib{us}, inſcribere.</head> <p> <s xml:id="echoid-s21147" xml:space="preserve">Data ſuperficies ęquidiſtãtiũ laterũ, cui altera inſcribi modo prę-<lb/>dicto debeat, ſit a b c d, in qua ducãtur diagonij a c & b d, ſecãtes ſe in <lb/>puncto e:</s> <s xml:id="echoid-s21148" xml:space="preserve"> palamq́;</s> <s xml:id="echoid-s21149" xml:space="preserve"> per proximã pręcedentẽ, quoniã illæ diagonij per <lb/>æqualia ſe ſecantin puncto e:</s> <s xml:id="echoid-s21150" xml:space="preserve"> ſed & ipſæ adinuicẽ ſunt æquales:</s> <s xml:id="echoid-s21151" xml:space="preserve"> & ſi <lb/>quidẽ data ſuperficies fuerit rectangula:</s> <s xml:id="echoid-s21152" xml:space="preserve"> tunc patet per 34 & 47 p 1, <lb/>quoniã ipſarũ diagonij ſunt æquales, & ipſarũ medietates æquales.</s> <s xml:id="echoid-s21153" xml:space="preserve"> <lb/>A' puncto itaq;</s> <s xml:id="echoid-s21154" xml:space="preserve"> e, à medietatibus diagoniorũ partes æquales abſcin <lb/>dantur ք 3 p 1.</s> <s xml:id="echoid-s21155" xml:space="preserve"> Et ſi data ſuperficies nõ fuerit rectangula:</s> <s xml:id="echoid-s21156" xml:space="preserve"> tũc erũt dia <lb/>gonij forſitan inęquales:</s> <s xml:id="echoid-s21157" xml:space="preserve"> ab illis ergo partes proportionales refecen <lb/>tur, ſecundũ 3 huius:</s> <s xml:id="echoid-s21158" xml:space="preserve"> utcunq;</s> <s xml:id="echoid-s21159" xml:space="preserve"> autẽ hoc contingat, abſcindantur illæ <lb/>partes ex parte puncti e, quæ ſint e l, e m, e n, e p, & ducantur lineæ <lb/>l m, l n, n p, m p.</s> <s xml:id="echoid-s21160" xml:space="preserve"> Dico itaq;</s> <s xml:id="echoid-s21161" xml:space="preserve">, quòd ſuperficies l m p n eſt datæ ſuperfi-<lb/>ciei ſimilis, & quòd latera ipſius æquidiſtant lateribus datę ſuperfi-<lb/>ciei.</s> <s xml:id="echoid-s21162" xml:space="preserve"> Quoniã enim in trigono b e c reſecta ſunt latera b e & c e in pun <lb/>ctis l & m, & eſt proportio b l ad l e, ſicut c m ad m e:</s> <s xml:id="echoid-s21163" xml:space="preserve"> patet ergo per 2 <lb/>p 6, quoniam linea l m æquidiſtat lineæ b c.</s> <s xml:id="echoid-s21164" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s21165" xml:space="preserve"> linea l n <lb/> <pb o="18" file="0320" n="320" rhead="VITELLONIS OPTICAE"/> æ quidiſtat lateri a b, & linea n p lateri a d, & linea p m lateri c d.</s> <s xml:id="echoid-s21166" xml:space="preserve"> Ergo ք 29 p 1 anguli ſuperficiei l m <lb/>p n ſunt æquales angulis datæ ſuperficiei a b c d, & latera eorum ſunt proportionalia per 4 p 6.</s> <s xml:id="echoid-s21167" xml:space="preserve"> Pa-<lb/>tet ergo, quòd illæ ſuperficies ſunt ſimiles:</s> <s xml:id="echoid-s21168" xml:space="preserve"> & hoc proponebatur faciendũ:</s> <s xml:id="echoid-s21169" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s21170" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div727" type="section" level="0" n="0"> <head xml:id="echoid-head616" xml:space="preserve" style="it">42. Omnis angulus à diametro & quacun linea ſuper circumferentiam circuli contẽtus, <lb/>neceſſariò est acutus. Alhazen 60 n 5.</head> <p> <s xml:id="echoid-s21171" xml:space="preserve">Sit circulus a b c, cuius diameter a b, & ducatur linea a c, utcunq;</s> <s xml:id="echoid-s21172" xml:space="preserve"> contingit.</s> <s xml:id="echoid-s21173" xml:space="preserve"> Dico, quòd angulus <lb/>b a c neceſſariò eſt acutus.</s> <s xml:id="echoid-s21174" xml:space="preserve"> Producatur enim linea b c <lb/> <anchor type="figure" xlink:label="fig-0320-01a" xlink:href="fig-0320-01"/> ad peripheriam in pũctum c.</s> <s xml:id="echoid-s21175" xml:space="preserve"> Et quoniã angulus a c b <lb/>eſt rectus per 31 p 3, patet per 32 p 1, quia angulus b a c <lb/>eſt acutus:</s> <s xml:id="echoid-s21176" xml:space="preserve"> & ſimiliter angulus a b c.</s> <s xml:id="echoid-s21177" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s21178" xml:space="preserve"> propo <lb/>ſitum:</s> <s xml:id="echoid-s21179" xml:space="preserve"> & de hoc theoremate nõ ſeruimus intellectui, <lb/>ſed breuitati, quia hanc demonſtrationem toties, ut <lb/>occurrit, repetere, tædium fuit.</s> <s xml:id="echoid-s21180" xml:space="preserve"/> </p> <div xml:id="echoid-div727" type="float" level="0" n="0"> <figure xlink:label="fig-0320-01" xlink:href="fig-0320-01a"> <variables xml:id="echoid-variables290" xml:space="preserve">c a b</variables> </figure> </div> </div> <div xml:id="echoid-div729" type="section" level="0" n="0"> <head xml:id="echoid-head617" xml:space="preserve" style="it">43. Omnes angulos æqualium ucl ſimilium por-<lb/>tionum eiuſdem circuli ſub arcu & recta contentos <lb/>æquales: angulos uerò cuiuſcun minoris portionis <lb/>minores, & maioris maiores eſſe neceſſe eſt. Ex quo <lb/>patet, omnes angulos ſemicir culorum æquales eſſe.</head> <p> <s xml:id="echoid-s21181" xml:space="preserve">Sit circulus, cuius centrum a, & diameter g f:</s> <s xml:id="echoid-s21182" xml:space="preserve"> & in <lb/>c o ſignentur arcus æquales, qui ſint b c & d e, produ-<lb/>ctis chordis b c & d e.</s> <s xml:id="echoid-s21183" xml:space="preserve"> Dico, quòd anguli g b c, & f d e, <lb/>ſub arcubus & chordis contenti ſunt æquales.</s> <s xml:id="echoid-s21184" xml:space="preserve"> Duca-<lb/>tur enim à puncto b linea contingens circulum per 17 p 3, quæ ſit b l, & à puncto d linea d m:</s> <s xml:id="echoid-s21185" xml:space="preserve"> & du-<lb/>cantur à centro lineę a b, a c, a d, a e, eruntq́;</s> <s xml:id="echoid-s21186" xml:space="preserve"> per 5 p 1 anguli a b c & a c b æquales:</s> <s xml:id="echoid-s21187" xml:space="preserve"> & anguli a d e & <lb/>a e d æquales:</s> <s xml:id="echoid-s21188" xml:space="preserve"> ſed trigona a b c & a d e ſunt æquiangula per 4 p 1:</s> <s xml:id="echoid-s21189" xml:space="preserve"> angulus enim b a c eſt æqualis an-<lb/>gulo d a e, per 27 p 3:</s> <s xml:id="echoid-s21190" xml:space="preserve"> angulus quoq;</s> <s xml:id="echoid-s21191" xml:space="preserve"> a b l eſt æqualis angulo <lb/> <anchor type="figure" xlink:label="fig-0320-02a" xlink:href="fig-0320-02"/> a d m, quoniam uterq;</s> <s xml:id="echoid-s21192" xml:space="preserve"> eorũ eſt rectus per 18 p 3:</s> <s xml:id="echoid-s21193" xml:space="preserve"> ſed angulus <lb/>contingentiæ l b g eſt æqualis angulo contingentiæ m d f:</s> <s xml:id="echoid-s21194" xml:space="preserve"> <lb/>quoniam uterq;</s> <s xml:id="echoid-s21195" xml:space="preserve"> ipſorum eſt minimus acutorum per 18 p 3.</s> <s xml:id="echoid-s21196" xml:space="preserve"> <lb/>Relin quitur ergo angulus g b c a b arcu b g, & recta b c con <lb/>tentus, æqualis angulo f d e, ab arcu f d, & recta d e conten-<lb/>to:</s> <s xml:id="echoid-s21197" xml:space="preserve"> ſed & angulus g c b eſt ęqualis angulo g b c eadem ratio-<lb/>ne:</s> <s xml:id="echoid-s21198" xml:space="preserve"> ſimiliter quoq;</s> <s xml:id="echoid-s21199" xml:space="preserve"> angulus f e d eſt æqualis angulo f d e.</s> <s xml:id="echoid-s21200" xml:space="preserve"> O-<lb/>mnes itaq;</s> <s xml:id="echoid-s21201" xml:space="preserve"> hi anguli ſunt æquales.</s> <s xml:id="echoid-s21202" xml:space="preserve"> Sit quoq;</s> <s xml:id="echoid-s21203" xml:space="preserve"> arcus minor ar <lb/>cu b c, quireſecetur ab arcu b c, qui ſit arcus n o, & ducãtur <lb/>lineæ a n, a o:</s> <s xml:id="echoid-s21204" xml:space="preserve"> ducatur quoq;</s> <s xml:id="echoid-s21205" xml:space="preserve"> chorda n o:</s> <s xml:id="echoid-s21206" xml:space="preserve"> & ducantur contin <lb/>gẽtes n q & o r.</s> <s xml:id="echoid-s21207" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s21208" xml:space="preserve"> trigoni a n o anguli ad baſim ſunt <lb/>æquales per 5 p 1, & angulus o a n minor angulo c a b, per <lb/>33 p 6:</s> <s xml:id="echoid-s21209" xml:space="preserve"> erit per 32 p 1 quilibet angulorum a n o & a o n maior <lb/>quolibet angulorum a b c & a c b.</s> <s xml:id="echoid-s21210" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s21211" xml:space="preserve"> angulus o n a m a <lb/>ior angulo c b a:</s> <s xml:id="echoid-s21212" xml:space="preserve"> ſed angulus contingentię q n g eſt ęqualis <lb/>angulo cõtingentię l b g:</s> <s xml:id="echoid-s21213" xml:space="preserve"> relinquitur ergo angulus g n o mi-<lb/>nor angulo g b c, cum anguli l b a & q n a ſint æquales:</s> <s xml:id="echoid-s21214" xml:space="preserve"> quia <lb/>uterq;</s> <s xml:id="echoid-s21215" xml:space="preserve"> rectus per 18 p 3.</s> <s xml:id="echoid-s21216" xml:space="preserve"> Sit iam arcus maior arcu b c, qui ſit s c, & ducatur chorda s c:</s> <s xml:id="echoid-s21217" xml:space="preserve"> & quia angulus <lb/>c a s eſt maior angulo c a b per 33 p 6:</s> <s xml:id="echoid-s21218" xml:space="preserve"> patet tũc, quòd angulus a s c eſt minor angulo a b c:</s> <s xml:id="echoid-s21219" xml:space="preserve"> & ita con-<lb/>cludetur, ut prius, quoniã angulus g s c contentus ſub arcu g s, & chorda s c eſt maior angulo g b c:</s> <s xml:id="echoid-s21220" xml:space="preserve"> <lb/>ergo & angulo g n o.</s> <s xml:id="echoid-s21221" xml:space="preserve"> Patet & hocidem de ſimilibus arcubus quibuſcunq;</s> <s xml:id="echoid-s21222" xml:space="preserve"> eorundem circulorum, <lb/>quoniam per definitionem ſimilium arcuũ ipſi angulos ſuſcipiunt æquales per 10 defin.</s> <s xml:id="echoid-s21223" xml:space="preserve"> 3.</s> <s xml:id="echoid-s21224" xml:space="preserve"> Ex quo <lb/>patet corollarium, quoniam omnes anguli ſemicirculorum ſunt æquales:</s> <s xml:id="echoid-s21225" xml:space="preserve"> omnes enim ſemicirculi <lb/>ſunt ſimiles:</s> <s xml:id="echoid-s21226" xml:space="preserve"> & eiuſdem circuli ſimiles & ęquales:</s> <s xml:id="echoid-s21227" xml:space="preserve"> hoc itaq;</s> <s xml:id="echoid-s21228" xml:space="preserve"> proponebatur.</s> <s xml:id="echoid-s21229" xml:space="preserve"/> </p> <div xml:id="echoid-div729" type="float" level="0" n="0"> <figure xlink:label="fig-0320-02" xlink:href="fig-0320-02a"> <variables xml:id="echoid-variables291" xml:space="preserve">ſ q r n g o b c s c a d e f m</variables> </figure> </div> </div> <div xml:id="echoid-div731" type="section" level="0" n="0"> <head xml:id="echoid-head618" xml:space="preserve" style="it">44. Si idem angulus ſuper centrum unius æqualium circulorum, & ſuper peripheriam alte-<lb/>rius conſiſtat, arcus reſpondens angulo ſuper peripheriã conſtituto, reliquo arcui duplus erit. In <lb/>circulis uerò inæqualibus illorũ arcuum proportio ad ſuas totales peripherias duplicatur.</head> <p> <s xml:id="echoid-s21230" xml:space="preserve">Sint duo circuli æquales, unus a b c d, cuius centrum g:</s> <s xml:id="echoid-s21231" xml:space="preserve"> & alius e f g, cuius centrum b, punctum <lb/>peripheriæ circuli a b c d:</s> <s xml:id="echoid-s21232" xml:space="preserve"> & producantur lineę a b & c b, ſecantes circulum e g f in punctis e & f.</s> <s xml:id="echoid-s21233" xml:space="preserve"> Pa-<lb/>làm itaq;</s> <s xml:id="echoid-s21234" xml:space="preserve"> quoniam angulus a b c erit ſuper peripheriam circuli a b c & ſuper centrum circuli e g f.</s> <s xml:id="echoid-s21235" xml:space="preserve"> <lb/>Dico, quòd arcus a d c capiens angulũ a b c ſuper circũferentiam ſui circuli, eſt duplus arcui e g f, ca <lb/>pienti eundẽ angulũ ſuper eius centrũ b.</s> <s xml:id="echoid-s21236" xml:space="preserve"> Sit enim, ut linea b a ſecet circulũ e g f in puncto e, & linea <lb/>b cin puncto f:</s> <s xml:id="echoid-s21237" xml:space="preserve"> ducatur quoq;</s> <s xml:id="echoid-s21238" xml:space="preserve"> linea e f, & ducta linea g h ſuper centrũ g, fiat per 23 p 1 angulus æqua <lb/>lis angulo a b c, qui ſit h g l, ductis lineis g h & g l ad circumferentiam circuli a b c d:</s> <s xml:id="echoid-s21239" xml:space="preserve"> & ducantur li-<lb/>neę b h, b l, h l.</s> <s xml:id="echoid-s21240" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s21241" xml:space="preserve"> per 20 p 3, q́uoniam angulus h g l eſt duplus angulo h b l:</s> <s xml:id="echoid-s21242" xml:space="preserve"> ergo etiam an-<lb/>gulus a b c eſt duplus eidem:</s> <s xml:id="echoid-s21243" xml:space="preserve"> ergo per 33 p 6 arcus a d c eſt duplus arcui h d l:</s> <s xml:id="echoid-s21244" xml:space="preserve"> ſed arcus h d l <lb/> <pb o="19" file="0321" n="321" rhead="LIBER PRIMVS."/> eſt æqualis arcui e g f per 26 p 3:</s> <s xml:id="echoid-s21245" xml:space="preserve"> erit ergo arcus a d c duplus arcui e g f:</s> <s xml:id="echoid-s21246" xml:space="preserve"> quod eſt propoſitum primũ.</s> <s xml:id="echoid-s21247" xml:space="preserve"> <lb/>Quòd ſi circulus a b c d ſit minor circulo e g f, & angulus m g n ſit æ-<lb/> <anchor type="figure" xlink:label="fig-0321-01a" xlink:href="fig-0321-01"/> qualis angulo a g c, facto angulo p b q ſuper centrum b, per 23 p 1 æ-<lb/>quali angulo a g c, & ductis lineis g p, g q, b p, b q:</s> <s xml:id="echoid-s21248" xml:space="preserve"> erit angulus p b q <lb/>duplus angulo p g q, per 20 p 3.</s> <s xml:id="echoid-s21249" xml:space="preserve"> Ergo angulus a g c eſt duplus angulo <lb/>p g q.</s> <s xml:id="echoid-s21250" xml:space="preserve"> Proportio itaq;</s> <s xml:id="echoid-s21251" xml:space="preserve"> arcus m f n ad ſui totam circumferentiã dupli-<lb/>catur reſpectu arcus a c ad totam ſui peripheriam.</s> <s xml:id="echoid-s21252" xml:space="preserve"> Quoniã enim an-<lb/>gulus m g n eſt duplus angulo p g q, erit per 33 p 6 arcus m f n duplus <lb/>arcui p f q:</s> <s xml:id="echoid-s21253" xml:space="preserve"> ſed arcus p f q eiuſdem eſt proportionis ad ſui peripheriã, <lb/>cuius eſt arcus a d c ad ſuam:</s> <s xml:id="echoid-s21254" xml:space="preserve"> arcus enim a d c ſi fuerit quinq;</s> <s xml:id="echoid-s21255" xml:space="preserve"> partiũ <lb/>reſpectu ſuæ circum ferentiæ:</s> <s xml:id="echoid-s21256" xml:space="preserve"> erit arcus m f n decem partium reſpe-<lb/>ctu ſuæ peripheriæ:</s> <s xml:id="echoid-s21257" xml:space="preserve"> & hoc eſt propoſitum.</s> <s xml:id="echoid-s21258" xml:space="preserve"/> </p> <div xml:id="echoid-div731" type="float" level="0" n="0"> <figure xlink:label="fig-0321-01" xlink:href="fig-0321-01a"> <variables xml:id="echoid-variables292" xml:space="preserve">h d l a c e g f p q b n d n a c g b</variables> </figure> </div> </div> <div xml:id="echoid-div733" type="section" level="0" n="0"> <head xml:id="echoid-head619" xml:space="preserve" style="it">45. À<unsure/> terminis lineæ intra circulum collocatæ partib. æqualib. <lb/>reſectis, & à punctis ſectionum perpendicularibus ſuper illam li-<lb/>neam ad circumferentiam productis: neceſſe eſt ductas perpen-<lb/>diculares æquales eſſe. Et ſi ductæ perpẽdiculares ſunt æquales: ne-<lb/>ceſſariũ eſt à terminis illius lineæ partes reſectas æquales eſſe.</head> <p> <s xml:id="echoid-s21259" xml:space="preserve">Sit circulus a k d, cuius cẽtrum r:</s> <s xml:id="echoid-s21260" xml:space="preserve"> in quo circulo collocata ſit linea <lb/>a d:</s> <s xml:id="echoid-s21261" xml:space="preserve"> à cuius terminis a & d reſecentur lineæ a b & d g æquales:</s> <s xml:id="echoid-s21262" xml:space="preserve"> & à <lb/>prædictis b & g erigantur duæ lineæ perpẽdiculares ſuper lineã d a, <lb/>quę productę ad circũferentiã, ſint g k & b c.</s> <s xml:id="echoid-s21263" xml:space="preserve"> Dico, quòd linea g k eſt ęqualis lineę b c.</s> <s xml:id="echoid-s21264" xml:space="preserve"> Ducatur enim <lb/>â centror linea æquidiſtans lineæ a d per 31 p 1, quæ ſit l m diameter:</s> <s xml:id="echoid-s21265" xml:space="preserve"> & diuidatur linea d a in duo æ-<lb/>qualia in puncto e per 10 p 1, & à puncto e, ducatur per-<lb/> <anchor type="figure" xlink:label="fig-0321-02a" xlink:href="fig-0321-02"/> pendicularis ſuper l m per 12 p 1:</s> <s xml:id="echoid-s21266" xml:space="preserve"> hęc ergo per 1 p 3 tran-<lb/>ſibit cẽtrum circuli, quod eſt punctũ r:</s> <s xml:id="echoid-s21267" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s21268" xml:space="preserve"> linea e r.</s> <s xml:id="echoid-s21269" xml:space="preserve"> <lb/>Educatur aũt linea k g ultra punctum g ad diametrum <lb/>l m in punctũ n, & linea c b in punctũ f, & copulẽtur li-<lb/>neę k r & c r.</s> <s xml:id="echoid-s21270" xml:space="preserve"> Quia ita q;</s> <s xml:id="echoid-s21271" xml:space="preserve"> linea d e eſt ęqualis lineæ a e, & <lb/>lineę d g & b a ex hypotheſi ſunt ęquales:</s> <s xml:id="echoid-s21272" xml:space="preserve"> remanet ergo <lb/>linea g e æqualis lineę e b:</s> <s xml:id="echoid-s21273" xml:space="preserve"> ſed per 34 p 1, linea g e eſt æ-<lb/>qualis lineæ n r, & linea e b ęqualis lineę r f:</s> <s xml:id="echoid-s21274" xml:space="preserve"> ſunt ergo <lb/>lineæ n r & r f æquales:</s> <s xml:id="echoid-s21275" xml:space="preserve"> ſed per 47 p 1, quadratum lineę <lb/>r k ualet duo quadrata linearum k n & r n:</s> <s xml:id="echoid-s21276" xml:space="preserve"> quia ex præ-<lb/>miſsis angulus k n r eſt rectus:</s> <s xml:id="echoid-s21277" xml:space="preserve"> & ſimiliter quadratum <lb/>lineę c r ualet duo quadrata linearũ c f & r f:</s> <s xml:id="echoid-s21278" xml:space="preserve"> eſt aũt qua <lb/>dratum lineę k r ęquale quadrato lineæ c r, quoniã li-<lb/>nea k r eſt ęqualis lineæ c r per definitionem circuli:</s> <s xml:id="echoid-s21279" xml:space="preserve"> & <lb/>quadratũ lineæ n r eſt ęquale quadrato lineæ f r.</s> <s xml:id="echoid-s21280" xml:space="preserve"> Relin <lb/>quitur ergo quadratũ lineæ k n ęquale quadrato lineæ c f.</s> <s xml:id="echoid-s21281" xml:space="preserve"> E ſt ergo linea k n æqualis lineę c f:</s> <s xml:id="echoid-s21282" xml:space="preserve"> ſed per <lb/>25 huius linea g n eſt æqualis b f.</s> <s xml:id="echoid-s21283" xml:space="preserve"> Relinquitur ergo linea k g ęqualis lineę c b:</s> <s xml:id="echoid-s21284" xml:space="preserve"> quod eſt primũ propo-<lb/>ſitũ.</s> <s xml:id="echoid-s21285" xml:space="preserve"> Conuerſa etiã patet, manente totali diſpoſitione, ut prius.</s> <s xml:id="echoid-s21286" xml:space="preserve"> Quia enim linea g n eſt æqualis lineæ <lb/>b f per 34 p 1, & linea k g æqualis lineæ c b ex hypotheſi:</s> <s xml:id="echoid-s21287" xml:space="preserve"> erit tota linea k n ęqualis toti lineę c f.</s> <s xml:id="echoid-s21288" xml:space="preserve"> Ergo <lb/>per 47 p 1 erit linea n r ęqualis lineę r f.</s> <s xml:id="echoid-s21289" xml:space="preserve"> Ergo & linea g e ipſi lineę e b ęqualis erit, & linea d g ipſi li-<lb/>neę b a:</s> <s xml:id="echoid-s21290" xml:space="preserve"> quod eſt propoſitum ſecundum.</s> <s xml:id="echoid-s21291" xml:space="preserve"> Patet ergo, quod proponebatur.</s> <s xml:id="echoid-s21292" xml:space="preserve"/> </p> <div xml:id="echoid-div733" type="float" level="0" n="0"> <figure xlink:label="fig-0321-02" xlink:href="fig-0321-02a"> <variables xml:id="echoid-variables293" xml:space="preserve">k c d g e b a l n r f m</variables> </figure> </div> </div> <div xml:id="echoid-div735" type="section" level="0" n="0"> <head xml:id="echoid-head620" xml:space="preserve" style="it">46. In duobus circulis inæqualibus duobus ſimilib. arcubus ſumptis, productiś, præter illos, <lb/>ad arcus alios ſimiles, ſemidiametris: ſi à punctis extra circulos proportionaliter ſemidiametris <lb/>diſtantibus ab utriſ extremitatibus amborum arcuum, per terminos ſimilium arcuum, li-<lb/>neæ ad diametros ducantur: pars diametri interiacens lineas arcus circuli maioris eſt maior <lb/>parte interiacente lineas arcus circuli minoris.</head> <p> <s xml:id="echoid-s21293" xml:space="preserve">Sint duo circuli inæquales, quorum maior ſit a b c, & eius centrum d, & ſemidiameter d a:</s> <s xml:id="echoid-s21294" xml:space="preserve"> minor <lb/>uerò ſit e f g.</s> <s xml:id="echoid-s21295" xml:space="preserve"> cuius centrum h, & ſemidiameter h e:</s> <s xml:id="echoid-s21296" xml:space="preserve"> ſignenturq́;</s> <s xml:id="echoid-s21297" xml:space="preserve"> in ipſis arcus ſimiles, in maiori circu <lb/>lo arcus b c, & in minori arcus f g:</s> <s xml:id="echoid-s21298" xml:space="preserve"> ſitq́ue arcus a b ſimilis arcui e f:</s> <s xml:id="echoid-s21299" xml:space="preserve"> ſit q́;</s> <s xml:id="echoid-s21300" xml:space="preserve"> punctũ k extra circulũ maio-<lb/>rem, & punctum l extra circulum minorem, taliter data, utilla puncta ſecundum proportionem ſe-<lb/>midiametri d a, ad ſemidiametrum h e diſtent ab utriſque terminis dictorum arcuum:</s> <s xml:id="echoid-s21301" xml:space="preserve"> erit ergo pro-<lb/>portio lineę k b ad lineam l f, & lineæ k c ad lineam l g, ſicut ſemidiametri a d ad h e:</s> <s xml:id="echoid-s21302" xml:space="preserve"> & producãtur li <lb/>neę ad ſemidiametros, k b in punctum m, & k c in punctum n.</s> <s xml:id="echoid-s21303" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s21304" xml:space="preserve"> producatur linea l f <lb/>in punctum o, & l g in punctum p.</s> <s xml:id="echoid-s21305" xml:space="preserve"> Dico, quòd linea m n, pars ſemidiametri a d, eſt maior quã linea <lb/>o p, pars ſemidiametri e h.</s> <s xml:id="echoid-s21306" xml:space="preserve"> Ducantur enim chordę b c & f g:</s> <s xml:id="echoid-s21307" xml:space="preserve"> & copulentur à centris lineæ d b, d c, <lb/>h f, h g:</s> <s xml:id="echoid-s21308" xml:space="preserve"> palamq́;</s> <s xml:id="echoid-s21309" xml:space="preserve"> propter inæqualitatem circulorum, quoniam linea d b eſt maior quã linea h f:</s> <s xml:id="echoid-s21310" xml:space="preserve"> ſed <lb/>propter ſimilitudinem arcuum angulus b d c eſt ęqualis angulo f h g:</s> <s xml:id="echoid-s21311" xml:space="preserve"> ergo per 5 p 1 trigona b c d & <lb/>f g h ſunt ęquiangula.</s> <s xml:id="echoid-s21312" xml:space="preserve"> Ergo per 4 p 6 latera ſunt proportionalia:</s> <s xml:id="echoid-s21313" xml:space="preserve"> eſt ergo proportio lineæ b c ad li-<lb/>neam f g, ſicut lineę b d ad lineam f h:</s> <s xml:id="echoid-s21314" xml:space="preserve"> ergo ex hypotheſi & per 11 p 5, ſicut k b ad l f, & ſicut k c ad l g:</s> <s xml:id="echoid-s21315" xml:space="preserve"> <lb/> <pb o="20" file="0322" n="322" rhead="VITELLONIS OPTICAE"/> ergo per 5 p 6 angulus b k c eſt ęqualis angulo f l g:</s> <s xml:id="echoid-s21316" xml:space="preserve"> & angulus k b c æqualis angulo l f g:</s> <s xml:id="echoid-s21317" xml:space="preserve"> ſed exprę-<lb/> <anchor type="figure" xlink:label="fig-0322-01a" xlink:href="fig-0322-01"/> <anchor type="figure" xlink:label="fig-0322-02a" xlink:href="fig-0322-02"/> miſsis anguli d b c & h f g ſunt ęqua <lb/>les:</s> <s xml:id="echoid-s21318" xml:space="preserve"> eſt ergo angulus d b k æqualis <lb/>angulo h f l.</s> <s xml:id="echoid-s21319" xml:space="preserve"> Ducãtur ergo lineæ d k <lb/>& h l.</s> <s xml:id="echoid-s21320" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s21321" xml:space="preserve"> in trigonis b d k & <lb/>f h l anguli ęquales (qui d b k & h f l) <lb/>ſunt laterib.</s> <s xml:id="echoid-s21322" xml:space="preserve"> ꝓportiõalib.</s> <s xml:id="echoid-s21323" xml:space="preserve"> cõtẽti, pa-<lb/>tet ք 6 p 6, quoniã illa trigona ſunt <lb/>æquiangula:</s> <s xml:id="echoid-s21324" xml:space="preserve"> ergo angulus b k d eſt <lb/>ęqualis angulo fl h, & angulus b d k <lb/>ęqualis angulo f h l:</s> <s xml:id="echoid-s21325" xml:space="preserve"> ſed angulus a d <lb/>b eſt æqualis angulo e h f ex hypo-<lb/>theſi, propter ſimilitudinem arcuũ <lb/>a b & e f.</s> <s xml:id="echoid-s21326" xml:space="preserve"> Totus ergo angulus m d k <lb/>eſt æqualis toti angulo o h l:</s> <s xml:id="echoid-s21327" xml:space="preserve"> ergo ք <lb/>32 p 1 trigona d k m & h l o ſunt ęqui <lb/>angula, & angulus k m d eſt ęqualis <lb/>angulo l o h:</s> <s xml:id="echoid-s21328" xml:space="preserve"> ergo per 4 p 6 erit pro <lb/>portio lineę m k ad lineã o l, ſicut lineę k d ad lineã l h:</s> <s xml:id="echoid-s21329" xml:space="preserve"> ergo ք 11 p 5 ſicut lineę a d ad lineam e h.</s> <s xml:id="echoid-s21330" xml:space="preserve"> Quia <lb/>itaq;</s> <s xml:id="echoid-s21331" xml:space="preserve"> ex pręmiſsis angulus m k n eſt ęqualis angulo o l p, & angulus k m n ęqualis angulo l o p:</s> <s xml:id="echoid-s21332" xml:space="preserve"> patet <lb/>per 32 p 1, quoniã trigona k m n & l o p ſunt ęquiangula:</s> <s xml:id="echoid-s21333" xml:space="preserve"> ergo per 4 p 6 eſt proportio lineę m n ad li-<lb/>neam o p, ſicut lineæ m k ad lineã o l:</s> <s xml:id="echoid-s21334" xml:space="preserve"> ergo per 11 p 5, ſicut lineæ a d ad lineã e h.</s> <s xml:id="echoid-s21335" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s21336" xml:space="preserve"> a d ſemidia <lb/>meter maior eſt ſemidiametro e h:</s> <s xml:id="echoid-s21337" xml:space="preserve"> erit linea m n maior quã linea o p:</s> <s xml:id="echoid-s21338" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s21339" xml:space="preserve"/> </p> <div xml:id="echoid-div735" type="float" level="0" n="0"> <figure xlink:label="fig-0322-01" xlink:href="fig-0322-01a"> <variables xml:id="echoid-variables294" xml:space="preserve"><gap/> b c a m n d</variables> </figure> <figure xlink:label="fig-0322-02" xlink:href="fig-0322-02a"> <variables xml:id="echoid-variables295" xml:space="preserve">l f g e o p h</variables> </figure> </div> </div> <div xml:id="echoid-div737" type="section" level="0" n="0"> <head xml:id="echoid-head621" xml:space="preserve" style="it">47. À<unsure/> quocun puncto diametri circuli producta linea adperipheriam, ſi maior, quã illa, <lb/>fuerit una pars diametri: erit pars illa, maior reli-<lb/>qua ſui parte: & ſiminor, minor.</head> <figure> <variables xml:id="echoid-variables296" xml:space="preserve">c a d b</variables> </figure> <p> <s xml:id="echoid-s21340" xml:space="preserve">Eſto circulus a b c, cuius diameter a b:</s> <s xml:id="echoid-s21341" xml:space="preserve"> in qua ſuma-<lb/>tur punctũ d, utcunq;</s> <s xml:id="echoid-s21342" xml:space="preserve"> cõtingit:</s> <s xml:id="echoid-s21343" xml:space="preserve"> & ducatur linea d c ad <lb/>circũferentiam, ita quòd pars diametri, quę eſt a d, ſit <lb/>maior ꝗ̃ linea d c.</s> <s xml:id="echoid-s21344" xml:space="preserve"> Dico, quòd linea a d eſt maior quã li <lb/>nea d b, quę eſt reliqua pars ipſius diametri:</s> <s xml:id="echoid-s21345" xml:space="preserve"> quod pa-<lb/>tet, ſi copulẽtur lineę a c & b c.</s> <s xml:id="echoid-s21346" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s21347" xml:space="preserve"> linea a d ma <lb/>ior eſt quã linea d c ex hypotheſi:</s> <s xml:id="echoid-s21348" xml:space="preserve"> ergo ք 18 p 1 angulus <lb/>a c d maior eſt angulo c a d, & angulus a c b eſt rectus <lb/>per 31 p 3:</s> <s xml:id="echoid-s21349" xml:space="preserve"> palã ergo per 32 p 1, quoniã angulus c b d ma <lb/>ior eſt angulo d c b.</s> <s xml:id="echoid-s21350" xml:space="preserve"> Quia enim angulus c b d cũ angu-<lb/>lo c a b ualet rectũ, & angulus d c b cũ angulo a c d, qui <lb/>eſt maior angulo c a d, ualet rectũ:</s> <s xml:id="echoid-s21351" xml:space="preserve"> patet, quòd angu-<lb/>lus c b d eſt maior angulo d c b:</s> <s xml:id="echoid-s21352" xml:space="preserve"> ergo per 19 p 1 erit la-<lb/>tus d c maius latere d b:</s> <s xml:id="echoid-s21353" xml:space="preserve"> ſed latus a d eſt maius latere d c.</s> <s xml:id="echoid-s21354" xml:space="preserve"> Ergo multo maius erit latus a d quã latus <lb/>d b.</s> <s xml:id="echoid-s21355" xml:space="preserve"> Et hoc eſt unum propoſitorum.</s> <s xml:id="echoid-s21356" xml:space="preserve"> Eodem quoq;</s> <s xml:id="echoid-s21357" xml:space="preserve"> modo demonſtrandum, ſi pars diametri, quæ eſt <lb/>a d, ſit minor quã linea d c:</s> <s xml:id="echoid-s21358" xml:space="preserve"> quoniã erit linea a d minor quã linea d b:</s> <s xml:id="echoid-s21359" xml:space="preserve"> & hoc proponebatur.</s> <s xml:id="echoid-s21360" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div738" type="section" level="0" n="0"> <head xml:id="echoid-head622" xml:space="preserve" style="it">48. Si à quocun puncto diametri circuli duæ lineæ (quarum ſemper una ſit maior reliqua) <lb/>ad circuli peripheriã ducantur: erit pars diametri, <lb/>cuimaior linea propinquior ducitur, maior reliqua <lb/>ſui parte.</head> <figure> <variables xml:id="echoid-variables297" xml:space="preserve">c g f e a h d b</variables> </figure> <p> <s xml:id="echoid-s21361" xml:space="preserve">Sit circulus a b e c, cuius diameter ſit a b:</s> <s xml:id="echoid-s21362" xml:space="preserve"> in qua ſu-<lb/>matur punctus d, ut libuerit:</s> <s xml:id="echoid-s21363" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s21364" xml:space="preserve"> à puncto d li-<lb/>neę, d c maior & d e minor:</s> <s xml:id="echoid-s21365" xml:space="preserve"> ſit aũt c ſuperior uerſus a, <lb/>& e inferior uerſus b.</s> <s xml:id="echoid-s21366" xml:space="preserve"> Dico, quòd pars diametri, quę eſt <lb/>a d, maior eſt quã d b.</s> <s xml:id="echoid-s21367" xml:space="preserve"> Ducatur enim linea c e, & ſuper <lb/>lineam c e ducatur à puncto d per 12 p 1 linea perpẽdi-<lb/>cularis, quę ſit d f.</s> <s xml:id="echoid-s21368" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s21369" xml:space="preserve"> quadratũ lineę d c per 47 <lb/>p 1 ualet ambo quadrata linearũ d f & f c, & quadratũ <lb/>d e ualet ambo quadrata duarũ linearũ d f & f e, qua-<lb/>dratũ uerò lineę d c maius eſt quadrato lineę d e:</s> <s xml:id="echoid-s21370" xml:space="preserve"> i deo, <lb/>quia linea d c eſt maior ꝗ̃ linea d e:</s> <s xml:id="echoid-s21371" xml:space="preserve"> ablato itaq;</s> <s xml:id="echoid-s21372" xml:space="preserve"> quadra <lb/>to lineæ d f:</s> <s xml:id="echoid-s21373" xml:space="preserve"> relinquitur quadratũ lineæ c f, maius qua-<lb/>drato lineæ f e.</s> <s xml:id="echoid-s21374" xml:space="preserve"> Diuidatur itaq;</s> <s xml:id="echoid-s21375" xml:space="preserve"> linea c e in partes æqua <lb/>les in puncto g per 10 p 1, & ab illo puncto g ducatur <lb/>linea g h ad diametrum æquidiſtanter lineæ d f per 31 p 1:</s> <s xml:id="echoid-s21376" xml:space="preserve"> erititaque per 29 p 1 linea h g perpendicu-<lb/>laris ſuper lineam c e:</s> <s xml:id="echoid-s21377" xml:space="preserve"> ſecat autem h g ipſam c e in duo ęqualia:</s> <s xml:id="echoid-s21378" xml:space="preserve"> tranſit ergo linea h g ք centrũ circuli <lb/> <pb o="21" file="0323" n="323" rhead="LIBER PRIMVS."/> per 1 p 3.</s> <s xml:id="echoid-s21379" xml:space="preserve"> Et quoniam punctum h cadit in diametrum a b:</s> <s xml:id="echoid-s21380" xml:space="preserve"> palàm, quia ipſum punctum h eſt centrum <lb/>circuli.</s> <s xml:id="echoid-s21381" xml:space="preserve"> Eſt ergo linea a d, pars diametri a b, maior quàm linea d b:</s> <s xml:id="echoid-s21382" xml:space="preserve"> & hoc eſt propoſitum.</s> <s xml:id="echoid-s21383" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div739" type="section" level="0" n="0"> <head xml:id="echoid-head623" xml:space="preserve" style="it">49. Si ab angulis duorum trigonorum ad medietates ſuarũ baſiũ æqualiũ una perpendicula <lb/>riter, alia obliquè æquales lineæ duc antur, ſit́ quælibet duct arum maior medietate ſuæ baſis: <lb/>erit angulus trigoni, à quo ducitur perpendicularis, maior angulo alterius trigoni, à quo linea <lb/>ducitur obliqua.</head> <p> <s xml:id="echoid-s21384" xml:space="preserve">Sint duo trigona a b c & d e f, quorum baſes b c, & e f, ſint æquales:</s> <s xml:id="echoid-s21385" xml:space="preserve"> quæ ſecentur per 10 p 1 in par-<lb/>tes æquales, b c in puncto g, & e f in puncto h:</s> <s xml:id="echoid-s21386" xml:space="preserve"> & ducantur ab angulis ad baſes lineæ a g & d h, quæ <lb/>ſint ęquales:</s> <s xml:id="echoid-s21387" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s21388" xml:space="preserve"> linea a g ք-<lb/> <anchor type="figure" xlink:label="fig-0323-01a" xlink:href="fig-0323-01"/> <anchor type="figure" xlink:label="fig-0323-02a" xlink:href="fig-0323-02"/> pẽdicularis ſuper lineã b c, li-<lb/>nea uerò d h nõ ſit perpẽdicu <lb/>laris ſuք lineã e f.</s> <s xml:id="echoid-s21389" xml:space="preserve"> Sitq́;</s> <s xml:id="echoid-s21390" xml:space="preserve"> linea <lb/>perpendicularis a g maior li-<lb/>nea b g parte baſis:</s> <s xml:id="echoid-s21391" xml:space="preserve"> item obli-<lb/>qua d h maior linea e h parte <lb/>baſis.</s> <s xml:id="echoid-s21392" xml:space="preserve"> Dico, quod angulus b a <lb/>c eſt maior angulo e d f.</s> <s xml:id="echoid-s21393" xml:space="preserve"> Cir-<lb/>cũſcribatur enim trigono a b <lb/>c circulus per 5 p 4, & produ-<lb/>catur linea a g ad circũferen-<lb/>tiã in punctũ k:</s> <s xml:id="echoid-s21394" xml:space="preserve"> hoc aũt poſsi-<lb/>bile.</s> <s xml:id="echoid-s21395" xml:space="preserve"> Quoniã uerò ſuppoſitũ <lb/>eſt lineã a g eſſe maiorẽ linea <lb/>g b, erit per 47 huius linea a g <lb/>maior ꝗ̃ linea g k:</s> <s xml:id="echoid-s21396" xml:space="preserve"> ergo per 1 p 3 centrũ circuli eſt in linea a g inter pũcta a & g:</s> <s xml:id="echoid-s21397" xml:space="preserve"> & erit a k diameter, & <lb/>per 7 p 3 linea g a eſt lõgiſsima omnium linearũ à puncto g ad circũferentiã productarũ:</s> <s xml:id="echoid-s21398" xml:space="preserve"> & linea g k <lb/>erit omniũ linearũ illarum minima:</s> <s xml:id="echoid-s21399" xml:space="preserve"> & quęlibet propinquior lineę g a eſt maior remotiore.</s> <s xml:id="echoid-s21400" xml:space="preserve"> Fiat itaq;</s> <s xml:id="echoid-s21401" xml:space="preserve"> <lb/>per 23 p 1 ſuper punctũ g termini lineę, c g angulus ęqualis angulo f h d minori angulo d h e, qui ſit l <lb/>g c, producta linea g l uſq;</s> <s xml:id="echoid-s21402" xml:space="preserve"> ad peripheriã circuli.</s> <s xml:id="echoid-s21403" xml:space="preserve"> Palã itaq;</s> <s xml:id="echoid-s21404" xml:space="preserve"> ex 7 p 3, quoniã linea g a eſt maior ꝗ̃ linea <lb/>g l:</s> <s xml:id="echoid-s21405" xml:space="preserve"> ergo & linea d h, quę ex hypotheſi eſt ęqualis lineę a g, eſt maior ꝗ̃ linea g l.</s> <s xml:id="echoid-s21406" xml:space="preserve"> Producatur itaq;</s> <s xml:id="echoid-s21407" xml:space="preserve"> li-<lb/>nea g l, quouſq;</s> <s xml:id="echoid-s21408" xml:space="preserve"> ſit ęqualis lineę d h per 3 p 1, & ſit linea g m ęqualis lineę d h:</s> <s xml:id="echoid-s21409" xml:space="preserve"> & ducantur lineę m b <lb/>& m c:</s> <s xml:id="echoid-s21410" xml:space="preserve"> angulus itaq;</s> <s xml:id="echoid-s21411" xml:space="preserve"> b m c eſt ęqualis angulo e d f ex hypotheſi per 4.</s> <s xml:id="echoid-s21412" xml:space="preserve"> 13 p 1.</s> <s xml:id="echoid-s21413" xml:space="preserve"> Sed angulus b a c eſt ma <lb/>ior angulo b m c.</s> <s xml:id="echoid-s21414" xml:space="preserve"> Producantur enim lineę b l & c l:</s> <s xml:id="echoid-s21415" xml:space="preserve"> palã, quia angulus b l c eſt maior angulo b m c per <lb/>21 p 1:</s> <s xml:id="echoid-s21416" xml:space="preserve"> ſed angulus b a c eſt æqualis angulo b l c per 21 p 3.</s> <s xml:id="echoid-s21417" xml:space="preserve"> Erit ergo angulus b a c maior angulo b m c:</s> <s xml:id="echoid-s21418" xml:space="preserve"> <lb/>ergo & angulo e d f:</s> <s xml:id="echoid-s21419" xml:space="preserve"> & hoc proponebatur.</s> <s xml:id="echoid-s21420" xml:space="preserve"/> </p> <div xml:id="echoid-div739" type="float" level="0" n="0"> <figure xlink:label="fig-0323-01" xlink:href="fig-0323-01a"> <variables xml:id="echoid-variables298" xml:space="preserve">a b ſ m g c k</variables> </figure> <figure xlink:label="fig-0323-02" xlink:href="fig-0323-02a"> <variables xml:id="echoid-variables299" xml:space="preserve">d e h f</variables> </figure> </div> </div> <div xml:id="echoid-div741" type="section" level="0" n="0"> <head xml:id="echoid-head624" xml:space="preserve" style="it">50. Si ab angulis duorum trigonorum ad medietates ſuarum baſium æqualium una perpẽdi-<lb/>culariter, alia obliquè, æquales lineæ ducantur, ſit́ quælibet ductarum minor medietate baſis <lb/>ſuæ: erit angulus trigoni, à quo ducitur perpendicularis, minor angulo alterius trigoni, à quo <lb/>linea ducitur obliqua.</head> <p> <s xml:id="echoid-s21421" xml:space="preserve">Remaneat diſpoſitio pręcedentis, niſi quòd perpendicularis a g ſit minor medietate baſis b g.</s> <s xml:id="echoid-s21422" xml:space="preserve"> Di <lb/>co, qđ angulus b a c eſt mi-<lb/> <anchor type="figure" xlink:label="fig-0323-03a" xlink:href="fig-0323-03"/> <anchor type="figure" xlink:label="fig-0323-04a" xlink:href="fig-0323-04"/> nor angulo e d f.</s> <s xml:id="echoid-s21423" xml:space="preserve"> Sit enim, <lb/>ut prius, angulus c g l ęqua-<lb/>lis angulo d h f.</s> <s xml:id="echoid-s21424" xml:space="preserve"> Et quoniã li <lb/>nea a g eſt minor quã linea <lb/>b g, & linea a k eſt diame-<lb/>ter:</s> <s xml:id="echoid-s21425" xml:space="preserve"> palã per 47 huius, quo-<lb/>niam cẽtrũ circuli eſt inter <lb/>puncta g & k:</s> <s xml:id="echoid-s21426" xml:space="preserve"> ergo per 7 p 3 <lb/>linea g a eſt minima omniũ <lb/>linearũ à puncto gad peri-<lb/>pheriã circuli productarũ:</s> <s xml:id="echoid-s21427" xml:space="preserve"> <lb/>eſt ergo linea g l maior ꝗ̃ li-<lb/>nea g a:</s> <s xml:id="echoid-s21428" xml:space="preserve"> ergo & maior quã li <lb/>nea d h.</s> <s xml:id="echoid-s21429" xml:space="preserve"> Fiat itaq;</s> <s xml:id="echoid-s21430" xml:space="preserve"> per 3 p 1 li <lb/>nea g n ęqualis lineæ d h:</s> <s xml:id="echoid-s21431" xml:space="preserve"> & <lb/>copulẽtur lineę bn & c n:</s> <s xml:id="echoid-s21432" xml:space="preserve"> erit itaq;</s> <s xml:id="echoid-s21433" xml:space="preserve">, ut in pręmiſſa, angulus e d f æqualis angulo b n c:</s> <s xml:id="echoid-s21434" xml:space="preserve"> ſed angulus b <lb/>n e maior eſt angulo b l c per 21 p 1, & angulus b l c æqualis angulo b a c per 21 p 3.</s> <s xml:id="echoid-s21435" xml:space="preserve"> Erit ergo angulus <lb/>b a c minor angulo b n c:</s> <s xml:id="echoid-s21436" xml:space="preserve"> ergo & eιus æquali, angulo e d f:</s> <s xml:id="echoid-s21437" xml:space="preserve"> & hoc eſt propoſitum.</s> <s xml:id="echoid-s21438" xml:space="preserve"/> </p> <div xml:id="echoid-div741" type="float" level="0" n="0"> <figure xlink:label="fig-0323-03" xlink:href="fig-0323-03a"> <variables xml:id="echoid-variables300" xml:space="preserve">a l n b g c k</variables> </figure> <figure xlink:label="fig-0323-04" xlink:href="fig-0323-04a"> <variables xml:id="echoid-variables301" xml:space="preserve">d c h f</variables> </figure> </div> </div> <div xml:id="echoid-div743" type="section" level="0" n="0"> <head xml:id="echoid-head625" xml:space="preserve" style="it">51. Si ab angulis duorum trigonorum ad medietates ſuarum baſium æqualium duæ lineæ æ-<lb/>quales, obliquè incidant ad angulos inæquales, & ſi quælibet linearum incidentium maior fue-<lb/>rit medietate ſuæ baſis: erit angulus ſuperior illius trigoni, cuius incidens linea maiorem angu-<lb/> <pb o="22" file="0324" n="324" rhead="VITELLONIS OPTICAE"/> lum cum baſi continet, maior angulo ſuperiori alterius: & ſi minor, minor.</head> <p> <s xml:id="echoid-s21439" xml:space="preserve">Sint itẽ duo trianguli a b c & d e f, habentes baſes b c & e f æquales:</s> <s xml:id="echoid-s21440" xml:space="preserve"> diuidaturq́;</s> <s xml:id="echoid-s21441" xml:space="preserve"> baſis b c ք ęqua-<lb/>lia in puncto g, & baſis e f in <lb/> <anchor type="figure" xlink:label="fig-0324-01a" xlink:href="fig-0324-01"/> <anchor type="figure" xlink:label="fig-0324-02a" xlink:href="fig-0324-02"/> pũcto h:</s> <s xml:id="echoid-s21442" xml:space="preserve"> & ducãtur lineę a g, <lb/>d h, quę ſint ęquales, & utraq;</s> <s xml:id="echoid-s21443" xml:space="preserve"> <lb/>ipſarum incidat obliquè ſuæ <lb/>baſi:</s> <s xml:id="echoid-s21444" xml:space="preserve"> ſit aũt angulus a g c ma-<lb/>ior angulo d h f.</s> <s xml:id="echoid-s21445" xml:space="preserve"> Dico, quòd ſi <lb/>maior ſit linea a g, ꝗ̃ linea g c:</s> <s xml:id="echoid-s21446" xml:space="preserve"> <lb/>erit angulus b a c maior an <lb/>gulo e d f:</s> <s xml:id="echoid-s21447" xml:space="preserve"> & ſi linea a g ſit mi-<lb/>nor, ꝗ̃ linea g c, erit angulus <lb/>b a c minor angulo e d f.</s> <s xml:id="echoid-s21448" xml:space="preserve"> Cir-<lb/>cum ſcribatur enim per 5 p 4 <lb/>trigono a b c circulus:</s> <s xml:id="echoid-s21449" xml:space="preserve"> & duca <lb/>tur à puncto g perpendicula-<lb/>ris ſuper lineã b c per 11 p 1:</s> <s xml:id="echoid-s21450" xml:space="preserve"> <lb/>quæ producta ad circũferen-<lb/>tiam, ſit g k.</s> <s xml:id="echoid-s21451" xml:space="preserve"> Erit itaq;</s> <s xml:id="echoid-s21452" xml:space="preserve"> g k per 1 p 3 pars diametri circuli propoſiti, quę cõpleta, ſit k l.</s> <s xml:id="echoid-s21453" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s21454" xml:space="preserve">, ut prius, <lb/>linea a g maior ꝗ̃ linea g c:</s> <s xml:id="echoid-s21455" xml:space="preserve"> eſt aũt linea k g maior, ꝗ̃ linea g l per 48 huius.</s> <s xml:id="echoid-s21456" xml:space="preserve"> In linea ergo g k eſt centrũ <lb/>circuli:</s> <s xml:id="echoid-s21457" xml:space="preserve"> eſt ergo linea k g maior ꝗ̃ linea a g per 7 p 3:</s> <s xml:id="echoid-s21458" xml:space="preserve"> ergo & maior ꝗ̃ linea d h, quę eſt ęqualis ipſi a g ex <lb/>hypotheſi.</s> <s xml:id="echoid-s21459" xml:space="preserve"> Fiat itaq;</s> <s xml:id="echoid-s21460" xml:space="preserve"> per 23 p 1 ſuper punctũ g terminũ lineę g c, angulus ęqualis angulo d h f, qui ſit <lb/>m g c:</s> <s xml:id="echoid-s21461" xml:space="preserve"> cadatq́;</s> <s xml:id="echoid-s21462" xml:space="preserve"> pũctũ m in peripheriã circuli.</s> <s xml:id="echoid-s21463" xml:space="preserve"> E ſt itaq;</s> <s xml:id="echoid-s21464" xml:space="preserve"> ք 7 p 3 linea a g maior ꝗ̃ linea m g:</s> <s xml:id="echoid-s21465" xml:space="preserve"> ergo & linea <lb/>d h eſt maior ꝗ̃ linea m g.</s> <s xml:id="echoid-s21466" xml:space="preserve"> Producatur itaq;</s> <s xml:id="echoid-s21467" xml:space="preserve">, donec linea g m ſit ęqualis lineę d h:</s> <s xml:id="echoid-s21468" xml:space="preserve"> & ducãtur lineę n c <lb/>& n b.</s> <s xml:id="echoid-s21469" xml:space="preserve"> Erit itaq;</s> <s xml:id="echoid-s21470" xml:space="preserve"> angulus b n c ęqualis angulo e d f:</s> <s xml:id="echoid-s21471" xml:space="preserve"> ſed angulus b m c eſt maior angulo b n c:</s> <s xml:id="echoid-s21472" xml:space="preserve"> eſt ergo <lb/>angulus b a c maior angulo e d f per modum pręoſtẽſum.</s> <s xml:id="echoid-s21473" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s21474" xml:space="preserve"> demonſtrandũ, ſi linea a g <lb/>ſit minor ꝗ̃ linea g c, quòd minor eſt angulus b a c angulo e d f:</s> <s xml:id="echoid-s21475" xml:space="preserve"> quod proponebatur demonſtrandũ.</s> <s xml:id="echoid-s21476" xml:space="preserve"/> </p> <div xml:id="echoid-div743" type="float" level="0" n="0"> <figure xlink:label="fig-0324-01" xlink:href="fig-0324-01a"> <variables xml:id="echoid-variables302" xml:space="preserve">k a n m b g c l</variables> </figure> <figure xlink:label="fig-0324-02" xlink:href="fig-0324-02a"> <variables xml:id="echoid-variables303" xml:space="preserve">d e h f</variables> </figure> </div> </div> <div xml:id="echoid-div745" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables304" xml:space="preserve">l n m d f e a g c h o k d f e b</variables> </figure> <head xml:id="echoid-head626" xml:space="preserve" style="it">52. Siduas lineas rectas ſecantes circulũ, æqua <lb/> les arcus interiaceant, illæ neceſſariò ſunt æquidi- ſtantes: ideḿ accidit, ſi una earum fuerit ſecans & alia contingens.</head> <p> <s xml:id="echoid-s21477" xml:space="preserve">Sit circulus a b c, cuius centrum ſit punctum o:</s> <s xml:id="echoid-s21478" xml:space="preserve"> ſe-<lb/>centq́;</s> <s xml:id="echoid-s21479" xml:space="preserve"> duæ lineę a c & d e illum circulum taliter, ut ar <lb/>cus d a ſit ęqualis arcui e c.</s> <s xml:id="echoid-s21480" xml:space="preserve"> Dico, quòd lineæ a c & d e <lb/>ſunt ęquidiſtantes.</s> <s xml:id="echoid-s21481" xml:space="preserve"> Autitaq;</s> <s xml:id="echoid-s21482" xml:space="preserve"> o centrũ circuli eſt in al-<lb/>tera illarum linearum, aut in neurra:</s> <s xml:id="echoid-s21483" xml:space="preserve"> & tuncuel inter <lb/>utraſq;</s> <s xml:id="echoid-s21484" xml:space="preserve">, uel extra utraſq;</s> <s xml:id="echoid-s21485" xml:space="preserve">. Si ſit in altera ipſarum:</s> <s xml:id="echoid-s21486" xml:space="preserve"> eſto <lb/>quòd ſit in linea a c, & à centro o ducatur linea perpẽ <lb/>dicularis ſuper a c per 11 p 1, & producatur ad circũfe <lb/>rentiã, ſitq́;</s> <s xml:id="echoid-s21487" xml:space="preserve"> o b ſecans lineã d e in puncto f:</s> <s xml:id="echoid-s21488" xml:space="preserve"> & ducan-<lb/>tur lineę o d, o e, quę cum ſint ęquales, erunt per 5 p 1, <lb/>anguli o d f & o e f æquales:</s> <s xml:id="echoid-s21489" xml:space="preserve"> ſed angulus f o a eſt ęqua <lb/>lis angulo f o c, ꝗ a ſunt recti:</s> <s xml:id="echoid-s21490" xml:space="preserve"> angulus uerò d o a ęqua <lb/>lis eſt angulo e o c per 27 p 3, cum ex hypotheſi arcus d a ſit æqualis arcui e c:</s> <s xml:id="echoid-s21491" xml:space="preserve"> erit ergo angulus d o f <lb/>æqualis angulo e o f:</s> <s xml:id="echoid-s21492" xml:space="preserve"> ergo per 32 p 1 erit angulus d f o ęqualis angulo e f o:</s> <s xml:id="echoid-s21493" xml:space="preserve"> eſt ergo linea of perpendi <lb/>cularis ſuper lineã d e.</s> <s xml:id="echoid-s21494" xml:space="preserve"> Erunt ergo per 28 p 1 lineę d e, <lb/> <anchor type="figure" xlink:label="fig-0324-04a" xlink:href="fig-0324-04"/> & a c ęquidiſtãtes.</s> <s xml:id="echoid-s21495" xml:space="preserve"> Si uerò centrũ o fuerit inter ipſas <lb/>lineas a c & d e:</s> <s xml:id="echoid-s21496" xml:space="preserve"> ductis lineis à centro perpẽdicularib.</s> <s xml:id="echoid-s21497" xml:space="preserve"> <lb/>ſuper utranq;</s> <s xml:id="echoid-s21498" xml:space="preserve"> illarũ, quę ſint o f, & o g, & ductis lineis <lb/>ad terminos linearum a c & d e, à cẽtro o, quę ſint o a, <lb/>o c, o d, o e, & diametro h k:</s> <s xml:id="echoid-s21499" xml:space="preserve"> fient ex utraq;</s> <s xml:id="echoid-s21500" xml:space="preserve"> parte cen-<lb/>tri o quatuor anguli ęquales duobus rectis ideo quia <lb/>anguli circa centrum ualent quatuor rectos, quo, ex <lb/>ęquo diuidit quælibet diameter:</s> <s xml:id="echoid-s21501" xml:space="preserve"> ſed angulus e o c eſt <lb/>ęqualis angulo d o a per 27 p 3:</s> <s xml:id="echoid-s21502" xml:space="preserve"> remanet ergo angulus <lb/>d o e ęqualis angulo a o c:</s> <s xml:id="echoid-s21503" xml:space="preserve"> per definitionẽ ergo circu-<lb/>li & per 6 p 6 trianguli d o e & a o c ſunt inuicẽ ęquiã <lb/>guli:</s> <s xml:id="echoid-s21504" xml:space="preserve"> ergo erit angulus g c o æqualis angulo o d f:</s> <s xml:id="echoid-s21505" xml:space="preserve"> ſed <lb/>angulus o g c eſt ęqualis angulo o f d:</s> <s xml:id="echoid-s21506" xml:space="preserve"> quia uterq;</s> <s xml:id="echoid-s21507" xml:space="preserve"> re-<lb/>ctus ex pręmiſsis:</s> <s xml:id="echoid-s21508" xml:space="preserve"> ergo per 32 p 1 trigona g o c, d o f <lb/>ſunt æquiangula:</s> <s xml:id="echoid-s21509" xml:space="preserve"> ergo per 14 p 1 lineę d o & o c con-<lb/>iunctæ ſunt linea una:</s> <s xml:id="echoid-s21510" xml:space="preserve"> quia anguli c o h & d o h ex præmiſsis ſunt ęquales duobus rectis.</s> <s xml:id="echoid-s21511" xml:space="preserve"> Ergo <lb/>per 27 p 1 patet propoſitum.</s> <s xml:id="echoid-s21512" xml:space="preserve"> Quòd ſi centrum o fuerit extra utraſque:</s> <s xml:id="echoid-s21513" xml:space="preserve"> ducatur perpendicu-<lb/>laris à centro o ſuperipſarum alteram:</s> <s xml:id="echoid-s21514" xml:space="preserve"> & ſit linea o g perpendicularis ſuper lineam a c, quæ diuidet <lb/> <pb o="23" file="0325" n="325" rhead="LIBER PRIMVS."/> ipſam a c in duo æqualia per 3 p 3, producaturq́;</s> <s xml:id="echoid-s21515" xml:space="preserve"> linea o g, ut ſecet lineam d e in puncto f:</s> <s xml:id="echoid-s21516" xml:space="preserve"> & ductis li-<lb/>neis o a, o c, o d, o e:</s> <s xml:id="echoid-s21517" xml:space="preserve"> palàm per 4 p 1, cum in trigonis a g o & c g o duo latera a g & g c ſint æqualia, & <lb/>latus g o commune, & anguli ad g recti ex hypotheſi:</s> <s xml:id="echoid-s21518" xml:space="preserve"> quòd angulus a o g eſt æqualis angulo c o g:</s> <s xml:id="echoid-s21519" xml:space="preserve"> <lb/>ſed angulus a o d æqualis eſt angulo c o e per 27 p 3:</s> <s xml:id="echoid-s21520" xml:space="preserve"> relin quitur ergo angulus d o f æqualis angulo <lb/>f o e:</s> <s xml:id="echoid-s21521" xml:space="preserve"> ſed latus d o æquale lateri e o, & latus o f commune:</s> <s xml:id="echoid-s21522" xml:space="preserve"> erit ergo per 4 p 1 angulus o f d æqualis an <lb/>gulo o fe:</s> <s xml:id="echoid-s21523" xml:space="preserve"> uterq;</s> <s xml:id="echoid-s21524" xml:space="preserve"> ergo eſt rectus.</s> <s xml:id="echoid-s21525" xml:space="preserve"> Eſt ergo angulus o f d æqualis angulo o g a:</s> <s xml:id="echoid-s21526" xml:space="preserve"> ergo per 28 p 1 lineę d e <lb/>& a c ſunt æquidiſtantes:</s> <s xml:id="echoid-s21527" xml:space="preserve"> quod eſt propoſitum primum.</s> <s xml:id="echoid-s21528" xml:space="preserve"> Quòd ſi una illarum duarum linearum ſe-<lb/>cet circulum, & alia ipſum contingat:</s> <s xml:id="echoid-s21529" xml:space="preserve"> ſi ſecans tranſit centrũ, & ſit diameter, quæ h k, & linea l m con <lb/>tingat in puncto n:</s> <s xml:id="echoid-s21530" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s21531" xml:space="preserve"> arcus n h æqualis arcui n k:</s> <s xml:id="echoid-s21532" xml:space="preserve"> palàm, quòd illorum arcuum quilibet eſt quar-<lb/>ta circuli:</s> <s xml:id="echoid-s21533" xml:space="preserve"> ducatur ita que linea n o:</s> <s xml:id="echoid-s21534" xml:space="preserve"> ergo per 18 p 3 angulus l n o eſt rectus:</s> <s xml:id="echoid-s21535" xml:space="preserve"> ſed & angulus n o h eſt re-<lb/>ctus:</s> <s xml:id="echoid-s21536" xml:space="preserve"> ergo per 28 p 1 lineæ l m & h k ęquidiſtant:</s> <s xml:id="echoid-s21537" xml:space="preserve"> quod eſt ſecundũ propoſitum.</s> <s xml:id="echoid-s21538" xml:space="preserve"> Quòd ſi linea l m cir-<lb/>culum contingente in puncto n, linea d e ſecet circulum nõ per centrũ:</s> <s xml:id="echoid-s21539" xml:space="preserve"> ducantur lineę o d l & o e m, <lb/>& à centro o ad punctum contactus, quod eſt n, ducatur linea o n ſecãs lineam d e in puncto f.</s> <s xml:id="echoid-s21540" xml:space="preserve"> Quia <lb/>ita que arcus n d eſt æqualis arcui n e:</s> <s xml:id="echoid-s21541" xml:space="preserve"> erit per 27 p 3 angulus l o n ęqualis angulo m o n:</s> <s xml:id="echoid-s21542" xml:space="preserve"> ſed per 18 p 3 <lb/>angulus o n l eſt æqualis angulo o n m:</s> <s xml:id="echoid-s21543" xml:space="preserve"> quia ambo ſunt recti.</s> <s xml:id="echoid-s21544" xml:space="preserve"> Item per 4 p 1 angulus o f d eſt æqualis <lb/>angulo o f e:</s> <s xml:id="echoid-s21545" xml:space="preserve"> ſunt ergo recti.</s> <s xml:id="echoid-s21546" xml:space="preserve"> Ergo per 28 p 1 patet propoſitum tertium.</s> <s xml:id="echoid-s21547" xml:space="preserve"/> </p> <div xml:id="echoid-div745" type="float" level="0" n="0"> <figure xlink:label="fig-0324-04" xlink:href="fig-0324-04a"> <variables xml:id="echoid-variables305" xml:space="preserve">a o c d f e b</variables> </figure> </div> </div> <div xml:id="echoid-div747" type="section" level="0" n="0"> <head xml:id="echoid-head627" xml:space="preserve" style="it">53. Lineas æquidiſt antes trans circuli ſuperficiem product{as}, ſiue ambæ ſecent, ſiue ambæ cõ-<lb/>tingant, ſiue una ſecet & alia contingat, arcus interiacent æquales.</head> <p> <s xml:id="echoid-s21548" xml:space="preserve">Sit circulus a c b d, cuius centrum e:</s> <s xml:id="echoid-s21549" xml:space="preserve"> contingantq́;</s> <s xml:id="echoid-s21550" xml:space="preserve"> ipſum duæ lineæ ęquidiſtãtes f g in puncto d, <lb/>& h q in puncto c:</s> <s xml:id="echoid-s21551" xml:space="preserve"> & à puncto contingentiæ, quod eſt d, <lb/> <anchor type="figure" xlink:label="fig-0325-01a" xlink:href="fig-0325-01"/> ducatur linea d e ad centrum e.</s> <s xml:id="echoid-s21552" xml:space="preserve"> Eſt ergo per 18 p 3 linea <lb/>d e perpendicularis ſuper lineam in illo puncto contin-<lb/>gentem, quæ f g.</s> <s xml:id="echoid-s21553" xml:space="preserve"> Ducatur quoque linea c e à puncto cõ <lb/>tingentiæ ad centrum e:</s> <s xml:id="echoid-s21554" xml:space="preserve"> erit ergo linea c e perpendicu-<lb/>laris ſuper lineam h q contingentem in puncto c.</s> <s xml:id="echoid-s21555" xml:space="preserve"> Duca <lb/>tur quoq;</s> <s xml:id="echoid-s21556" xml:space="preserve"> à centro e linea ęquidiſtans lineę f g per 31 p 1, <lb/>quæ ſit n m:</s> <s xml:id="echoid-s21557" xml:space="preserve"> hæc quoq;</s> <s xml:id="echoid-s21558" xml:space="preserve"> etiam æquidiſtabit lineæ h q per <lb/>30 p 1:</s> <s xml:id="echoid-s21559" xml:space="preserve"> ergo per 29 p 1 angulus m e d eſt æqualis angulo <lb/>m e c:</s> <s xml:id="echoid-s21560" xml:space="preserve"> ergo per 14 p 1 lineæ d e & e c cõiunctæ, ſunt linea <lb/>una:</s> <s xml:id="echoid-s21561" xml:space="preserve"> eſt ergo linea d c diameter circuli, cum trãſeat per <lb/>centrum e:</s> <s xml:id="echoid-s21562" xml:space="preserve"> arcus itaque d a c eſt ſemicirculus æqualis <lb/>ſemicirculo d b c.</s> <s xml:id="echoid-s21563" xml:space="preserve"> Sed & ſi linea a b ſecet circulum æqui <lb/>diſtans lineæ h q contingenti in puncto e, erit iterum ar <lb/>cus a c æqualis arcui c b.</s> <s xml:id="echoid-s21564" xml:space="preserve"> Quia enim ſemidiameter e c <lb/>ſecat lineam contingẽtem, quæ h q:</s> <s xml:id="echoid-s21565" xml:space="preserve"> palàm per 2 huius, <lb/>quoniam ſecabit & eius æquidiſtantem, quæ eſt linea <lb/>a b:</s> <s xml:id="echoid-s21566" xml:space="preserve"> ſit, ut ſecet ipſam in puncto o.</s> <s xml:id="echoid-s21567" xml:space="preserve"> Et quia angulus h c e <lb/>eſtrectus per 18 p 3, palàm per 29 p 1, quoniam angulus <lb/>b o e eſt rectus:</s> <s xml:id="echoid-s21568" xml:space="preserve"> ergo per 3 p 3 linea a b diuiditur per æqualia in puncto o.</s> <s xml:id="echoid-s21569" xml:space="preserve"> Ducantur itaq;</s> <s xml:id="echoid-s21570" xml:space="preserve"> lineę a c & <lb/>c b:</s> <s xml:id="echoid-s21571" xml:space="preserve"> palamq́;</s> <s xml:id="echoid-s21572" xml:space="preserve"> per 4 p 1, quoniã illę erunt æquales:</s> <s xml:id="echoid-s21573" xml:space="preserve"> ergo per 28 p 3 arcus a c eſt æqualis arcui b c.</s> <s xml:id="echoid-s21574" xml:space="preserve"> Quòd <lb/>ſi linea æquidiſtans lineę b a ſecet circulũ:</s> <s xml:id="echoid-s21575" xml:space="preserve"> quæ ſit k l:</s> <s xml:id="echoid-s21576" xml:space="preserve"> palàm, quoniam ſemidiameter e c producta ſe-<lb/>cabit lineam k l per ęqualia per 29 p 1.</s> <s xml:id="echoid-s21577" xml:space="preserve"> 3 p 3:</s> <s xml:id="echoid-s21578" xml:space="preserve"> ſecet ergo ipſam per æqualia & orthogonaliter in puncto <lb/>p:</s> <s xml:id="echoid-s21579" xml:space="preserve"> & ducãtur lineæ p a, p b, k a, l b:</s> <s xml:id="echoid-s21580" xml:space="preserve"> erit ergo in trigonis p a c, p b c ք præmiſſa, & 4 p 1 latus p a ęquale <lb/>lateri p b:</s> <s xml:id="echoid-s21581" xml:space="preserve"> & angulus p b c æqualis angulo a p c:</s> <s xml:id="echoid-s21582" xml:space="preserve"> relin quitur ergo angulus k p a æqualis angulo b p l:</s> <s xml:id="echoid-s21583" xml:space="preserve"> <lb/>ſed linea k p eſt æqualis lineæ p l:</s> <s xml:id="echoid-s21584" xml:space="preserve"> erit ergo per 4 p 1 linea k a æqualis lineæ l b.</s> <s xml:id="echoid-s21585" xml:space="preserve"> Ergo per 28 p 3 erit ar-<lb/>cus k a æqualis arcui l b:</s> <s xml:id="echoid-s21586" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s21587" xml:space="preserve"/> </p> <div xml:id="echoid-div747" type="float" level="0" n="0"> <figure xlink:label="fig-0325-01" xlink:href="fig-0325-01a"> <variables xml:id="echoid-variables306" xml:space="preserve">f m a h k d p e o c l g n b q</variables> </figure> </div> <figure> <variables xml:id="echoid-variables307" xml:space="preserve">a h b g e f d c <gap/></variables> </figure> </div> <div xml:id="echoid-div749" type="section" level="0" n="0"> <head xml:id="echoid-head628" xml:space="preserve" style="it">54. Duabus chordis in aliquo circulo ſe ſecanti-<lb/>bus: erit quilibet angulus ſectionis æqualis angulo <lb/>apud circumferentiam, cadenti in arcum æqua-<lb/>lem duobus arcubus ſcilicet eidem angulo & ſuo cõ <lb/>trapoſito ſubtenſis. Albazen 24 n 7.</head> <p> <s xml:id="echoid-s21588" xml:space="preserve">Sit circulus a b c d, in quo ſecẽt ſe duę chordę a c & <lb/>b d:</s> <s xml:id="echoid-s21589" xml:space="preserve"> & ſit pũctũ ſectionis e.</s> <s xml:id="echoid-s21590" xml:space="preserve"> Dico, quòd angulus a e b <lb/>eſt æqualis angulo, qui eſt in circumferentia, quam <lb/>ſubtẽdunt duo arcus a b & c d:</s> <s xml:id="echoid-s21591" xml:space="preserve"> & quòd angulus b e c <lb/>eſt ęqualis angulo in circumferẽtia, quã ſubtendunt <lb/>duo arcus d g a & b z c.</s> <s xml:id="echoid-s21592" xml:space="preserve"> Ducatur enim à puncto b li-<lb/>nea b z ęquidiſtanter lineę a c per 31 p 1.</s> <s xml:id="echoid-s21593" xml:space="preserve"> Si ergo linea <lb/>b z ſecat circulum, palã, quia arcus c z eſt ęqualis ar-<lb/>cui a b per præcedentem:</s> <s xml:id="echoid-s21594" xml:space="preserve"> arcus itaq;</s> <s xml:id="echoid-s21595" xml:space="preserve"> z d æqualis eſt <lb/>ambobus arcubus a b & d c:</s> <s xml:id="echoid-s21596" xml:space="preserve"> quoniam arcus d c utro-<lb/>biq;</s> <s xml:id="echoid-s21597" xml:space="preserve"> eſt cõmunis:</s> <s xml:id="echoid-s21598" xml:space="preserve"> fed arcus d z reſpicit angulũ d b z, <lb/> <pb o="24" file="0326" n="326" rhead="VITELLONIS OPTICAE"/> qui eſt æqualis angulo a e b per 29 p 1:</s> <s xml:id="echoid-s21599" xml:space="preserve"> angulus itaque a e b eſt æqualis angulo in circumferentia, ca <lb/>denti in arcum æqualem duobus arcubus b a, & c d.</s> <s xml:id="echoid-s21600" xml:space="preserve"> Item d ucatur linea d z, & producatur linea z b <lb/>extra circulum in punctum h:</s> <s xml:id="echoid-s21601" xml:space="preserve"> erit ergo angulus h b d ext rinſecus æqualis duobus angulis intrinſe-<lb/>cis b d z, & b z d per 32 p 1:</s> <s xml:id="echoid-s21602" xml:space="preserve"> ſed duo anguli b z d & b d z re ſpiciuntur à duobus arcubus b g d, & b f z:</s> <s xml:id="echoid-s21603" xml:space="preserve"> <lb/>angulus ergo h b d eſt æqualis angulo, quem reſpiciunt duo arcus b g d & b f z:</s> <s xml:id="echoid-s21604" xml:space="preserve"> hic autem eſt arcus <lb/>d a z:</s> <s xml:id="echoid-s21605" xml:space="preserve"> ſed arcus a b eſt æqualis arcui z c:</s> <s xml:id="echoid-s21606" xml:space="preserve"> arcus itaque d a z eſt æqualis duobus arcubus d g a & b z c.</s> <s xml:id="echoid-s21607" xml:space="preserve"> <lb/>Cum itaque per 29 p 1 angulus h b e ſit æqualis angulo <lb/> <anchor type="figure" xlink:label="fig-0326-01a" xlink:href="fig-0326-01"/> b e c:</s> <s xml:id="echoid-s21608" xml:space="preserve"> patet, quia angulus b e c eſt æqualis angulo, quem <lb/>in circũferentia reſpiciunt duo arcus d g a & b z c.</s> <s xml:id="echoid-s21609" xml:space="preserve"> Quòd <lb/>ſi linea h b z contingit circulum, & non ſecat:</s> <s xml:id="echoid-s21610" xml:space="preserve"> tunc patet <lb/>per 32 p 3, quia angulus e b z eſt æqualis angulo cadenti <lb/>in portionem circuli, quę eſt b a d, & angulus e b h eſt ę-<lb/>qualis angulo cadenti in portionem circuli b c d:</s> <s xml:id="echoid-s21611" xml:space="preserve"> ſed an <lb/>gulus e b z eſt æqualis angulo b e a per 29 p 1.</s> <s xml:id="echoid-s21612" xml:space="preserve"> Angulus <lb/>itaque b e a eſt æqualis angulo, qui apud circumferen-<lb/>tiam cadit in arcum b c d:</s> <s xml:id="echoid-s21613" xml:space="preserve"> ſed arcus b c eſt æqualis arcui <lb/>b a per proximam pręcedentem:</s> <s xml:id="echoid-s21614" xml:space="preserve"> arcus ergo b c d eſt æ-<lb/>qualis duobus arcubus b a & c d.</s> <s xml:id="echoid-s21615" xml:space="preserve"> Angulus itaq;</s> <s xml:id="echoid-s21616" xml:space="preserve"> b e a eſt <lb/>æqualis angulo, qui apud circumferẽtiam reſpicit duos <lb/>arcus a b & c d:</s> <s xml:id="echoid-s21617" xml:space="preserve"> quoniam angulus cadens in arcum b c d <lb/>eſt conſiſtens in portione circuli, quæ eſt b g d.</s> <s xml:id="echoid-s21618" xml:space="preserve"> Simi-<lb/>liter quoque poteſt declarari, quòd angulus b e c eſt <lb/>æqualis angulo apud circumferentiam, quem reſpiciũt <lb/>duo arcus b c & a d:</s> <s xml:id="echoid-s21619" xml:space="preserve"> quoniam angulus b e c eſt æqualis <lb/>angulo h b d, cuius ęqualitas per 32 p 3 cadit in portionem circuli b c d, hoc eſt in arcum b a d:</s> <s xml:id="echoid-s21620" xml:space="preserve"> eſt au-<lb/>tem ex præmiſsis arcus a b æqualis arcui b c:</s> <s xml:id="echoid-s21621" xml:space="preserve"> patet itaque propoſitum.</s> <s xml:id="echoid-s21622" xml:space="preserve"/> </p> <div xml:id="echoid-div749" type="float" level="0" n="0"> <figure xlink:label="fig-0326-01" xlink:href="fig-0326-01a"> <variables xml:id="echoid-variables308" xml:space="preserve">h a b e d z c</variables> </figure> </div> </div> <div xml:id="echoid-div751" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables309" xml:space="preserve">e a b d f c</variables> </figure> <head xml:id="echoid-head629" xml:space="preserve" style="it">55. Angulus à duabus lineis ab uno puncto extra circulum dato, circulum ſecantibus con-<lb/>tentus, æqualis eſt angulo ſuper circumferẽtiam cadenti in arcũ, <lb/> quo maior arcuum inter illas duas lineas comprehenſus, excedit minorem. Alhazen 25 n 7.</head> <p> <s xml:id="echoid-s21623" xml:space="preserve">Eſto circulus a b c d, extra quem ſit datum punctum e:</s> <s xml:id="echoid-s21624" xml:space="preserve"> & ducan-<lb/>tur à puncto e duę lineę ſecantes circulum, quæ ſint e a d & e b c.</s> <s xml:id="echoid-s21625" xml:space="preserve"> Di-<lb/>co itaq;</s> <s xml:id="echoid-s21626" xml:space="preserve">, quòd angulus d e c eſt æqualis angulo, qui eſt apud circum-<lb/>ferentiam circuli, quem reſpicit arcus, in quo arcus d c excedit arcũ <lb/>a b.</s> <s xml:id="echoid-s21627" xml:space="preserve"> À<unsure/> pũcto enim a ducatur per circulum linea a f ęquidiſtans lineę <lb/>b c per 31 p 1:</s> <s xml:id="echoid-s21628" xml:space="preserve"> erit ergo per 53 huius arcus f c ęqualis arcui a b.</s> <s xml:id="echoid-s21629" xml:space="preserve"> Eſt itaq;</s> <s xml:id="echoid-s21630" xml:space="preserve"> <lb/>arcus d f exceſſus arcus d c ſuper arcum a b:</s> <s xml:id="echoid-s21631" xml:space="preserve"> ſed angulus d a f apud <lb/>circumferentiã exiſtens cadit in arcum d f:</s> <s xml:id="echoid-s21632" xml:space="preserve"> & angulus d a f eſt æqua-<lb/>lis angulo d e c per 29 p 1.</s> <s xml:id="echoid-s21633" xml:space="preserve"> Ergo angulus d e c eſt æqualis angulo ca-<lb/>denti ſuper circumferentiam in arcum d f:</s> <s xml:id="echoid-s21634" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s21635" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div752" type="section" level="0" n="0"> <head xml:id="echoid-head630" xml:space="preserve" style="it">56. In dato ſemicirculo ad unum punctũ circumferentiæ, dua-<lb/>bus lineis: una à termino diametri, & alia à centro ductis: ab eiſ-<lb/>dem punctis ad aliud punctum quodcun ſemicirculi dati lineas <lb/>duas prioribus duabus proportionales duci eſt impoßibile: in diuerſis uerò ſemicirculis hoc eſt <lb/>poßibile.</head> <p> <s xml:id="echoid-s21636" xml:space="preserve">Eſto datus ſemicirculus a d b:</s> <s xml:id="echoid-s21637" xml:space="preserve"> cuius diameter a b:</s> <s xml:id="echoid-s21638" xml:space="preserve"> centrum uerò c:</s> <s xml:id="echoid-s21639" xml:space="preserve"> & ſit aliquod punctum circũ-<lb/>ferentiæ d:</s> <s xml:id="echoid-s21640" xml:space="preserve"> & ducatur à puncto a termino dia-<lb/> <anchor type="figure" xlink:label="fig-0326-03a" xlink:href="fig-0326-03"/> metri ad punctum d linea a d:</s> <s xml:id="echoid-s21641" xml:space="preserve"> & à cẽtro c linea <lb/>c d.</s> <s xml:id="echoid-s21642" xml:space="preserve"> Dico, quòd ſi à punctis a & c duæ lineæ ad <lb/>aliud punctum ſemicirculi ducantur:</s> <s xml:id="echoid-s21643" xml:space="preserve"> quòd illę <lb/>duę ductę lineę duabus lineis a d & c d propor <lb/>tionales non erunt.</s> <s xml:id="echoid-s21644" xml:space="preserve"> Sit enim, ſi poſsibile eſt, <lb/>ut à punctis a & c ducantur ad punctum g duę <lb/>lineæ a g & c g, & quę eſt proportio lineę a d ad <lb/>lineam c d, eadem ſit lineæ a g ad lineam c g, e-<lb/>rit permutatim per 16 p 5 proportio lineæ a d <lb/>ad lineam a g, ſicut lineę c d ad lineam c g:</s> <s xml:id="echoid-s21645" xml:space="preserve"> ſe d li <lb/>nea c d eſt æqualis lineę c g:</s> <s xml:id="echoid-s21646" xml:space="preserve"> quoniã ambę ſunt <lb/>ex cẽtro ſemicirculi:</s> <s xml:id="echoid-s21647" xml:space="preserve"> ergo linea a d ęqualis erit lineę a g:</s> <s xml:id="echoid-s21648" xml:space="preserve"> hoc aũt eſt impoſsibile ex 7 p 3 & 19 p 1:</s> <s xml:id="echoid-s21649" xml:space="preserve"> ma-<lb/>iori enim angulo ſubtẽditur linea a d ꝗ̃ linea a g:</s> <s xml:id="echoid-s21650" xml:space="preserve"> & eſt uicinior diametro.</s> <s xml:id="echoid-s21651" xml:space="preserve"> Patet ergo ꝓpoſitũ primũ:</s> <s xml:id="echoid-s21652" xml:space="preserve"> <lb/>quia à quocũq;</s> <s xml:id="echoid-s21653" xml:space="preserve"> pũcto alio dato idẽ accidit impoſsibile, & eodẽ modo deducẽdũ.</s> <s xml:id="echoid-s21654" xml:space="preserve"> In diuerſis uerò ſe-<lb/> <pb o="25" file="0327" n="327" rhead="LIBER PRIMVS."/> micirculis hoc eſt poſsibile.</s> <s xml:id="echoid-s21655" xml:space="preserve"> Si enim ſemicirculi æquales fuerint:</s> <s xml:id="echoid-s21656" xml:space="preserve"> tunc in centro alterius ſemicirculi <lb/>ſuper ſemidiametrum conſtituto æquali angulo a c d, per 23 p 1, compleatur propoſitum ex 4 p 1, & <lb/>4 p 6.</s> <s xml:id="echoid-s21657" xml:space="preserve"> Quòd ſi alter ſemicirculus minor fuerit dato ſemicirculo:</s> <s xml:id="echoid-s21658" xml:space="preserve"> inſcribatur æqualis illi ſemicirculo <lb/>ad idem centrum:</s> <s xml:id="echoid-s21659" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s21660" xml:space="preserve"> æquidiſtans primo, & in punctum, ubi linea c d ipſum ſecabit, (quod ſit f) <lb/>ducatur linea à termino ſuæ ſemidiametri, quæ ſit e f:</s> <s xml:id="echoid-s21661" xml:space="preserve"> & patet propoſitum per definitionem circuli <lb/>& 29 p 1, & 4 p 6.</s> <s xml:id="echoid-s21662" xml:space="preserve"> Quòd ſi dato ſemicirculo alter fuerit maior, circumſcribatur æquidiſtãter eidem, <lb/>& producta linea à centro primi ſemicirculi ad datum punctum d, quouſq;</s> <s xml:id="echoid-s21663" xml:space="preserve"> tangat peripheriam al-<lb/>terius ſemicirculi, & coniungatur à puncto contactus alia linea ad terminum diametri:</s> <s xml:id="echoid-s21664" xml:space="preserve"> & deinde <lb/>compleatur, ut prius, demonſtratio:</s> <s xml:id="echoid-s21665" xml:space="preserve"> & patet propoſitum.</s> <s xml:id="echoid-s21666" xml:space="preserve"/> </p> <div xml:id="echoid-div752" type="float" level="0" n="0"> <figure xlink:label="fig-0326-03" xlink:href="fig-0326-03a"> <variables xml:id="echoid-variables310" xml:space="preserve">g d f a e c h b</variables> </figure> </div> </div> <div xml:id="echoid-div754" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables311" xml:space="preserve">d a g b f e c</variables> </figure> <head xml:id="echoid-head631" xml:space="preserve" style="it">57. À<unsure/> puncto uno ad datum ſemicir culum unam tantum lineam contingentẽ poßibile eſt <lb/>duci. Ex quo patet, quòd omnis linea ab eodẽ puncto ſub contingẽte ducta, <lb/> ſecat ſemicirculũ in uno pũcto ſupr a punctũ cõtingẽtiæ, & in alio ſub ipſo.</head> <p> <s xml:id="echoid-s21667" xml:space="preserve">Eſto datus ſemicirculus a b c, cuius cẽtrum e:</s> <s xml:id="echoid-s21668" xml:space="preserve"> & ſit extrà datus punctus d:</s> <s xml:id="echoid-s21669" xml:space="preserve"> à <lb/>quo ad ſemicirculũ ducatur linea contingẽs, quæ ſit d b.</s> <s xml:id="echoid-s21670" xml:space="preserve"> Dico, quòd à puncto <lb/>d ad ſemicirculum a b c, aliam contingẽtem;</s> <s xml:id="echoid-s21671" xml:space="preserve"> quàm lineam d b duci eſt impoſ-<lb/>ſibile.</s> <s xml:id="echoid-s21672" xml:space="preserve"> Si enim hoc ſit poſsibile, ducatur:</s> <s xml:id="echoid-s21673" xml:space="preserve"> hæc ergo contingens aut cadet ultra <lb/>punctum b, aut citra:</s> <s xml:id="echoid-s21674" xml:space="preserve"> ſit primò, ut cadat ultra punctum b, uerſus c in punctũ f, <lb/>& ſit d f:</s> <s xml:id="echoid-s21675" xml:space="preserve"> ducantur itaq;</s> <s xml:id="echoid-s21676" xml:space="preserve"> à centro e ad puncta contingentiæ, lineæ e f, e b, & pro <lb/>ducatur diameter c e a uſq;</s> <s xml:id="echoid-s21677" xml:space="preserve"> ad punctum d.</s> <s xml:id="echoid-s21678" xml:space="preserve"> Palàm ergo per 18 p 3, quoniam an-<lb/>gulus e b d eſt rectus:</s> <s xml:id="echoid-s21679" xml:space="preserve"> ſimiliter angulus e f d eſt rectus.</s> <s xml:id="echoid-s21680" xml:space="preserve"> Sunt itaq;</s> <s xml:id="echoid-s21681" xml:space="preserve"> æquales, & <lb/>cadunt in trigonum e f d:</s> <s xml:id="echoid-s21682" xml:space="preserve"> quod eſt contra 21 p 1.</s> <s xml:id="echoid-s21683" xml:space="preserve"> Idẽ quoq;</s> <s xml:id="echoid-s21684" xml:space="preserve"> accidit impoſsibile, <lb/>ſi linea contingẽs ducta à puncto d ad ſemicirculum a b c, cadat inter puncta <lb/>b & a:</s> <s xml:id="echoid-s21685" xml:space="preserve"> ut linea d g.</s> <s xml:id="echoid-s21686" xml:space="preserve"> Palàm ergo corollarium:</s> <s xml:id="echoid-s21687" xml:space="preserve"> quoniam enim linea d g non con-<lb/>tingit ſemicirculum:</s> <s xml:id="echoid-s21688" xml:space="preserve"> ergo ipſa producta ſecat ipſum:</s> <s xml:id="echoid-s21689" xml:space="preserve"> & hoc eſt propoſitum.</s> <s xml:id="echoid-s21690" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div755" type="section" level="0" n="0"> <head xml:id="echoid-head632" xml:space="preserve" style="it">58. Quælibet duæ lineæ ab uno puncto productæ circulum contingẽtes, <lb/>ſunt æquales: & arcus interiacens puncta contingentiæ eſt minor ſemicir-<lb/>culo. Linea quo diuidens angulum illarum per æqualia: & arcum inter-<lb/>iacentem diuidit per æqualia: & linea per æqualia diuidens arcum, hæc <lb/>producta per æqualia diuidit & angulum à lineis contingentibus conten-<lb/>tum. Conſectarium ſecundum Campani ad 36 p 3.</head> <p> <s xml:id="echoid-s21691" xml:space="preserve">Sit circulus a b c, cuius centrum f:</s> <s xml:id="echoid-s21692" xml:space="preserve"> & ſit, ut à puncto e ducantur duę lineæ circulum contingen-<lb/>tes per 17 p 3, quæ ſint e a & e c.</s> <s xml:id="echoid-s21693" xml:space="preserve"> Dico, quòd lineæ e a, e c ſunt æqua-<lb/> <anchor type="figure" xlink:label="fig-0327-02a" xlink:href="fig-0327-02"/> les:</s> <s xml:id="echoid-s21694" xml:space="preserve"> & quòd arcus a b c interiacens puncta contingentiæ eſt mi-<lb/>nor ſemicirculo:</s> <s xml:id="echoid-s21695" xml:space="preserve"> & ſi producatur à puncto e linea e b, diuidẽs angu-<lb/>lum a e c per æqualia:</s> <s xml:id="echoid-s21696" xml:space="preserve"> dico;</s> <s xml:id="echoid-s21697" xml:space="preserve"> quòd linea e b in puncto b diuidet arcum <lb/>a c per æqualia:</s> <s xml:id="echoid-s21698" xml:space="preserve"> & ſi linea d e diuidat arcum a c per æqualia, diuidet <lb/>etiam angulum a e c per æqualia.</s> <s xml:id="echoid-s21699" xml:space="preserve"> Ducatur enim primò linea e f diui-<lb/>dens a e c, quæ producta ſecabit circulũ:</s> <s xml:id="echoid-s21700" xml:space="preserve"> ſecet ergo ipſum in punctis <lb/>b & d.</s> <s xml:id="echoid-s21701" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s21702" xml:space="preserve"> per 36 p 3, quoniam illud, quod fit ex ductu lineæ <lb/>d e in lineam e b, æquale eſt quadrato lineæ a e:</s> <s xml:id="echoid-s21703" xml:space="preserve"> & eadẽ ratione qua-<lb/>drato lineæ e c.</s> <s xml:id="echoid-s21704" xml:space="preserve"> Ergo quadratum lineæ a c eſt ęquale quadrato lineæ <lb/>e c.</s> <s xml:id="echoid-s21705" xml:space="preserve"> Ergo & linea a e eſt æqualis lineæ c e:</s> <s xml:id="echoid-s21706" xml:space="preserve"> & hoc eſt primum propoſi-<lb/>torũ.</s> <s xml:id="echoid-s21707" xml:space="preserve"> Sed quia ductis lineis f a & f c, erunt anguli f c e & f a e recti, per <lb/>18 p 3:</s> <s xml:id="echoid-s21708" xml:space="preserve"> ſunt ergo æquales:</s> <s xml:id="echoid-s21709" xml:space="preserve"> ergo per 4 p 1 linea f e diuidit angulum a e c <lb/>per æqualia.</s> <s xml:id="echoid-s21710" xml:space="preserve"> Et quia lineæ c e & a e concurrunt in puncto e:</s> <s xml:id="echoid-s21711" xml:space="preserve"> palàm <lb/>per 32 p 1, quoniã anguli e f c & e f a ſunt minores duobus rectis.</s> <s xml:id="echoid-s21712" xml:space="preserve"> Ar-<lb/>cus ergo a b c eſt minor ſemicirculo per 33 p 6:</s> <s xml:id="echoid-s21713" xml:space="preserve"> quod eſt ſecundum.</s> <s xml:id="echoid-s21714" xml:space="preserve"> <lb/>Ducatur quoq;</s> <s xml:id="echoid-s21715" xml:space="preserve"> linea a c ſecans lineam e d in puncto g:</s> <s xml:id="echoid-s21716" xml:space="preserve"> & ducantur <lb/>lineæ a b & a c.</s> <s xml:id="echoid-s21717" xml:space="preserve"> Quia ergo linea e g ſecat angulum a e c per æqualia:</s> <s xml:id="echoid-s21718" xml:space="preserve"> <lb/>patet per 4 p 1, cum linea a e ſit æqualis lineæ e c, & latus e g ſit com-<lb/>mune, quoniã linea a g eſt æqualis lineæ c g, & angulus e g a eſt æqua <lb/>lis angulo e g c.</s> <s xml:id="echoid-s21719" xml:space="preserve"> Sed & trigonis a b g & c b g latus b g eſt comune:</s> <s xml:id="echoid-s21720" xml:space="preserve"> ergo per 4 p 1 erit linea a b æqua-<lb/>lis lineæ b c:</s> <s xml:id="echoid-s21721" xml:space="preserve"> ergo per 28 p 3 arcus a b eſt æqualis arcui b c.</s> <s xml:id="echoid-s21722" xml:space="preserve"> Eodem quoq;</s> <s xml:id="echoid-s21723" xml:space="preserve"> modo patet, quòd ſi linea <lb/>g e ſecat arcum a c per æqualia in puncto b, quòd ipſa etiam diuidet per æqualia angulũ a e c.</s> <s xml:id="echoid-s21724" xml:space="preserve"> Quia <lb/>enim trigona a e b & c e b ſunt æquilatera, ut patet:</s> <s xml:id="echoid-s21725" xml:space="preserve"> palam ergo per 8 p 1, quoniam angulus a e b eſt <lb/>æqualis angulo c e b:</s> <s xml:id="echoid-s21726" xml:space="preserve"> & hoc eſt totum quod proponebatur.</s> <s xml:id="echoid-s21727" xml:space="preserve"/> </p> <div xml:id="echoid-div755" type="float" level="0" n="0"> <figure xlink:label="fig-0327-02" xlink:href="fig-0327-02a"> <variables xml:id="echoid-variables312" xml:space="preserve">e b a g c f d</variables> </figure> </div> </div> <div xml:id="echoid-div757" type="section" level="0" n="0"> <head xml:id="echoid-head633" xml:space="preserve" style="it">59. Arcubus æqualibus, minoribus quolibet, quarta circuli ex utra parte diametri cir-<lb/>culi reſectis: à terminis illorũ arcuum ductas contingentes in uno puncto eductæ diametri con-<lb/>currere eſt neceſſe: & ab uno puncto diametri ductas contingẽtes in terminis æqualiũ arcuum <lb/>contingere eſt neceſſe. Ex quo patet, quoniam omnem angulum & arcum à lineis contingenti-<lb/>bus contentum diuidit diameter educta per æqualia.</head> <pb o="26" file="0328" n="328" rhead="VITELLONIS OPTICAE"/> <p> <s xml:id="echoid-s21728" xml:space="preserve">Eſto circulus a b c, cuius centrum ſit d, & eius diameter c e, quæ producatur indefinitè ad pun-<lb/>ctum f:</s> <s xml:id="echoid-s21729" xml:space="preserve"> & ab unaquaq;</s> <s xml:id="echoid-s21730" xml:space="preserve"> parte puncti e ſint a e & b e arcus æquales:</s> <s xml:id="echoid-s21731" xml:space="preserve"> & à punctis a & b ducantur lineæ <lb/>circulũ contingentes per 17 p 3.</s> <s xml:id="echoid-s21732" xml:space="preserve"> Dico, quòd illæ duę lineæ concurrẽt <lb/> <anchor type="figure" xlink:label="fig-0328-01a" xlink:href="fig-0328-01"/> in uno puncto eductæ diametri e f.</s> <s xml:id="echoid-s21733" xml:space="preserve"> Quod ſi dicatur ipſas nõ concur-<lb/>rere in puncto uno diametri, concurrent tamen ambæ contingentes <lb/>cũ diametro d f:</s> <s xml:id="echoid-s21734" xml:space="preserve"> productis enim lineis d a, d b:</s> <s xml:id="echoid-s21735" xml:space="preserve"> erũtanguli ad puncta <lb/>a & b recti:</s> <s xml:id="echoid-s21736" xml:space="preserve"> ſed anguli e d a & e d b ſunt acuti per 33 p 6:</s> <s xml:id="echoid-s21737" xml:space="preserve"> arcus enim a <lb/>e, b e ſunt minores, quilibet, quarta circuli:</s> <s xml:id="echoid-s21738" xml:space="preserve"> ergo per 14 huius linearũ <lb/>contingentium utraq;</s> <s xml:id="echoid-s21739" xml:space="preserve"> concurret cum linea d f.</s> <s xml:id="echoid-s21740" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s21741" xml:space="preserve"> non fit hoc in <lb/>eodem puncto:</s> <s xml:id="echoid-s21742" xml:space="preserve"> ſit, ut linea contingẽs ducta à puncto a, concurrat cũ <lb/>linea d f in puncto g:</s> <s xml:id="echoid-s21743" xml:space="preserve"> & contingens ducta à puncto b concurrat cum <lb/>d fin puncto h, quod ſit ultra punctum g:</s> <s xml:id="echoid-s21744" xml:space="preserve"> & ducatur linea a h:</s> <s xml:id="echoid-s21745" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s21746" xml:space="preserve"> <lb/>per 27 p 3, & exhypotheſi angulus h d a æqualis angulo h d b:</s> <s xml:id="echoid-s21747" xml:space="preserve"> ergo <lb/>per 4 p 1 erit angulus h a d æqualis angulo h b a:</s> <s xml:id="echoid-s21748" xml:space="preserve"> & per 18 p 3 uterq;</s> <s xml:id="echoid-s21749" xml:space="preserve"> <lb/>ipſorũ eſt rectus.</s> <s xml:id="echoid-s21750" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s21751" xml:space="preserve"> angulus d a g eſt rectus per 18 p 3:</s> <s xml:id="echoid-s21752" xml:space="preserve"> patet, <lb/>quòd ipſe eſt ęqualis angulo d a h recto, & angulus a d g eſt commu-<lb/>nis:</s> <s xml:id="echoid-s21753" xml:space="preserve"> erit ergo per 32 p 1 angulus a g d ęqualis angulo a h d, extrinſecus <lb/>ſcilicet intrinſeco in trigono a h g:</s> <s xml:id="echoid-s21754" xml:space="preserve"> quod eſt contra 16 p 1 & impoſsi-<lb/>bile.</s> <s xml:id="echoid-s21755" xml:space="preserve"> Patet ergo primum.</s> <s xml:id="echoid-s21756" xml:space="preserve"> Sed & ſi à puncto diametri h ducantur duæ <lb/>lineæ circulum contingentes in punctis a & b:</s> <s xml:id="echoid-s21757" xml:space="preserve"> erunt arcus a e & b e <lb/>æquales:</s> <s xml:id="echoid-s21758" xml:space="preserve"> trigona enim a h d & h b d ſunt æquilatera per præcedentẽ:</s> <s xml:id="echoid-s21759" xml:space="preserve"> <lb/>ergo ſunt æquiangula per 8 p 1:</s> <s xml:id="echoid-s21760" xml:space="preserve"> eſt ergo angulus a d h æqualis angu-<lb/>lo b d h.</s> <s xml:id="echoid-s21761" xml:space="preserve"> Ergo per 26 p 3 arcus a e eſt æqual s arcui b e:</s> <s xml:id="echoid-s21762" xml:space="preserve"> quod eſt propoſitum:</s> <s xml:id="echoid-s21763" xml:space="preserve"> & patet corollarium.</s> <s xml:id="echoid-s21764" xml:space="preserve"/> </p> <div xml:id="echoid-div757" type="float" level="0" n="0"> <figure xlink:label="fig-0328-01" xlink:href="fig-0328-01a"> <variables xml:id="echoid-variables313" xml:space="preserve">f h g a e b d c</variables> </figure> </div> </div> <div xml:id="echoid-div759" type="section" level="0" n="0"> <head xml:id="echoid-head634" xml:space="preserve" style="it">60. Si intra duas lineas circulum contιngẽtes ab uno puncto ductas, aliæ duæ lineæ eundem <lb/>circulam contingentes ducantur: cadent puncta contingẽtiæ interiorum intra puncta contin-<lb/>gentiæ exteriorum: & ſiarcus hinc inde interiacentes puncta contingentiæ, fuerint æquales, <lb/>erit utrarum concurſus ſemper in eadẽ diametro circuli educta: interiores quo ad utram <lb/>partem productæ cum exterioribus neceſſariò concurrent.</head> <p> <s xml:id="echoid-s21765" xml:space="preserve">Eſto circulus a b c d e, cuius cẽtrũ k:</s> <s xml:id="echoid-s21766" xml:space="preserve"> & eius diameter e h educatur:</s> <s xml:id="echoid-s21767" xml:space="preserve"> & ſit, ut ab aliquo puncto ſuo, <lb/>quod ſit f, lineæ f a & f d contingentes circulũ ducantur:</s> <s xml:id="echoid-s21768" xml:space="preserve"> & inter lineas f a & f d ducantur ab aliquo <lb/>puncto ſuperficiei a f d, quod ſit g, lineæ g b & g c circulũ contingen-<lb/> <anchor type="figure" xlink:label="fig-0328-02a" xlink:href="fig-0328-02"/> tes in punctis b & c Dico, quòd puncta b & c cadent intra pũcta a & <lb/>d.</s> <s xml:id="echoid-s21769" xml:space="preserve"> Si enim nõ caduntintra puncta a & d:</s> <s xml:id="echoid-s21770" xml:space="preserve"> aut cadũt in illa puncta aut <lb/>extra:</s> <s xml:id="echoid-s21771" xml:space="preserve"> ſi in illa, ducãtur lineæ k a & k d à cẽtro k ad puncta contingen <lb/>tiæ a & d:</s> <s xml:id="echoid-s21772" xml:space="preserve"> erit itaq;</s> <s xml:id="echoid-s21773" xml:space="preserve"> per 18 p 3 angulus k a frectus:</s> <s xml:id="echoid-s21774" xml:space="preserve"> & ſimiliter angulus <lb/>k a g rectus:</s> <s xml:id="echoid-s21775" xml:space="preserve"> & ſic rectus maior recto.</s> <s xml:id="echoid-s21776" xml:space="preserve"> Itẽ inter contingentẽ f a & cir-<lb/>culum, alia linea capitur, ut g a:</s> <s xml:id="echoid-s21777" xml:space="preserve"> hoc autẽ eſt cõtra 16 p 3.</s> <s xml:id="echoid-s21778" xml:space="preserve"> Palàm ergo, <lb/>quoniã impoſsibile.</s> <s xml:id="echoid-s21779" xml:space="preserve"> Si uerò detur, quòd puncta b & c cadant extra <lb/>pũcta a & d:</s> <s xml:id="echoid-s21780" xml:space="preserve"> ſit punctũ b ultra a punctũ, ſecabitq́;</s> <s xml:id="echoid-s21781" xml:space="preserve"> linea g b producta <lb/>lineam f a per 14 huius.</s> <s xml:id="echoid-s21782" xml:space="preserve"> Et quoniã eſt contingẽs ſolum in puncto b, <lb/>erit punctus ſectionis extra circulũ:</s> <s xml:id="echoid-s21783" xml:space="preserve"> ſit ille punctus m.</s> <s xml:id="echoid-s21784" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s21785" xml:space="preserve">, <lb/>quoniã lineæ m a & m b ab uno pũcto m productæ ſemicirculũ con-<lb/>tingunt:</s> <s xml:id="echoid-s21786" xml:space="preserve"> quod eſt contra 57 huius.</s> <s xml:id="echoid-s21787" xml:space="preserve"> Non ergo cadit punctum b ultra <lb/>punctũ a, ſed intra.</s> <s xml:id="echoid-s21788" xml:space="preserve"> Similiterq́;</s> <s xml:id="echoid-s21789" xml:space="preserve"> demonſtrabitur, quia punctũ c cadit <lb/>intra punctum d.</s> <s xml:id="echoid-s21790" xml:space="preserve"> Cadũt ergo puncta contingẽtiæ interiorum intra <lb/>puncta contingẽtiæ exteriorũ.</s> <s xml:id="echoid-s21791" xml:space="preserve"> Sed & arcubus a b & c d exiſtẽtibus <lb/>ęqualibus, punctũ g neceſſariò cadit in diametro e h f.</s> <s xml:id="echoid-s21792" xml:space="preserve"> Si enim extra <lb/>illã, ducatur linea k g ſecãs circũferentiã in pũcto p.</s> <s xml:id="echoid-s21793" xml:space="preserve"> Quia ergo arcus <lb/>b p eſt æqualis arcui p c per præcedentẽ:</s> <s xml:id="echoid-s21794" xml:space="preserve"> arcus quoq;</s> <s xml:id="echoid-s21795" xml:space="preserve"> a b eſt æqualis <lb/>arcui c d ex hypotheſi:</s> <s xml:id="echoid-s21796" xml:space="preserve"> remanet ergo arcus c h æqualis arcui h b:</s> <s xml:id="echoid-s21797" xml:space="preserve"> ſed <lb/>arcus h b eſt maior arcu p b:</s> <s xml:id="echoid-s21798" xml:space="preserve"> ergo arcus c h eſt maior arcu c p, pars ſuo toto:</s> <s xml:id="echoid-s21799" xml:space="preserve"> qđ eſt impoſsibile.</s> <s xml:id="echoid-s21800" xml:space="preserve"> Nõ <lb/>ergo cadit pũctũ g extra diametrũ e h f.</s> <s xml:id="echoid-s21801" xml:space="preserve"> Palàm etiã eſt ք 14 huius, quoniã linea g b ꝓducta ultra pũ-<lb/>ctũ b, neceſſariò cõcurret cũ linea f a, & linea c g ꝓducta ultra pũctũ c, cõcurret neceſſariò cũ linea <lb/>f d:</s> <s xml:id="echoid-s21802" xml:space="preserve"> linea enim k c rectũ angulũ cõtinẽs cũ linea a g, cõtinet acutũ cũ linea f d:</s> <s xml:id="echoid-s21803" xml:space="preserve"> patet ergo ꝓpoſitũ.</s> <s xml:id="echoid-s21804" xml:space="preserve"/> </p> <div xml:id="echoid-div759" type="float" level="0" n="0"> <figure xlink:label="fig-0328-02" xlink:href="fig-0328-02a"> <variables xml:id="echoid-variables314" xml:space="preserve">f g g m b p h c a k d b e</variables> </figure> </div> </div> <div xml:id="echoid-div761" type="section" level="0" n="0"> <head xml:id="echoid-head635" xml:space="preserve" style="it">61. Si ad mediũ punctũ arcus interiacẽtis punct a contingẽtiæ duarũ linearũ, abuno puncto <lb/>ad circulũ productarũ, linea cõtingens circulũ ad alias contingẽtes producatur: illa in puncto <lb/>ſuæ contingentiæ per æqualia diuiditur: & ab alys lineis cõtingentib. partes abſcindit æquales.</head> <p> <s xml:id="echoid-s21805" xml:space="preserve">Sit circulus a b c, quẽ contingãt duæ lineæ d a & d c, à puncto d productæ:</s> <s xml:id="echoid-s21806" xml:space="preserve"> producatur ergo dia-<lb/>meter g b d:</s> <s xml:id="echoid-s21807" xml:space="preserve"> & palàm ք 59 huius, quoniã ipſa diuidit angulũ a d c, & arcũ a c per æqualia in pũcto b.</s> <s xml:id="echoid-s21808" xml:space="preserve"> <lb/>À<unsure/> puncto itaq;</s> <s xml:id="echoid-s21809" xml:space="preserve"> b producatur linea contingens circulũ per 17 p 3:</s> <s xml:id="echoid-s21810" xml:space="preserve"> h æ c itaq;</s> <s xml:id="echoid-s21811" xml:space="preserve"> quoniã eſt orthogonalis <lb/>ſuper diametrum g b, ut patet per 18 p 3:</s> <s xml:id="echoid-s21812" xml:space="preserve"> palàm per 14 huius, quia ipſa producta ſecabit lineas d a & <lb/>d c:</s> <s xml:id="echoid-s21813" xml:space="preserve"> ſit ergo ut ſecet lineam d a in puncto e, & lineam d c in puncto f.</s> <s xml:id="echoid-s21814" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s21815" xml:space="preserve"> e d b & f d b anguli <lb/><gap/>unt æquales per 59 huius, & anguli d b e & d b f ſunt recti:</s> <s xml:id="echoid-s21816" xml:space="preserve"> palàm, quia trigona e b d & f d b ſunt <lb/> <pb o="27" file="0329" n="329" rhead="LIBER PRIMVS."/> æquiangula per 32 p 1:</s> <s xml:id="echoid-s21817" xml:space="preserve"> ergo per 4 p 6 latera ſunt proportionalia:</s> <s xml:id="echoid-s21818" xml:space="preserve"> ſed latus d b eſt æquale ſibi:</s> <s xml:id="echoid-s21819" xml:space="preserve"> erit er-<lb/>go linea e b æqualis lineæ b f, & linea d e ęqualis <lb/> <anchor type="figure" xlink:label="fig-0329-01a" xlink:href="fig-0329-01"/> lineæ d f.</s> <s xml:id="echoid-s21820" xml:space="preserve"> Quod etiam ſic patere poteſt.</s> <s xml:id="echoid-s21821" xml:space="preserve"> Quia e-<lb/>nim à puncto e ducuntur duæ lineæ contingen-<lb/>tes circulum, ſcilicet e a & e b, patet per 58 huius, <lb/>quòd ipſæ ſunt æquales.</s> <s xml:id="echoid-s21822" xml:space="preserve"> Omnes ergo lineæ a e, <lb/>e b, b f, f c ſunt æquales.</s> <s xml:id="echoid-s21823" xml:space="preserve"> Ergo lineæ e d & f d <lb/>ſunt æquales:</s> <s xml:id="echoid-s21824" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s21825" xml:space="preserve"/> </p> <div xml:id="echoid-div761" type="float" level="0" n="0"> <figure xlink:label="fig-0329-01" xlink:href="fig-0329-01a"> <variables xml:id="echoid-variables315" xml:space="preserve">a e g b d c f</variables> </figure> </div> </div> <div xml:id="echoid-div763" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables316" xml:space="preserve">g f h k b l a c e m d n</variables> </figure> <head xml:id="echoid-head636" xml:space="preserve" style="it">62. A duobus puuctis æqualiter diſtanti-<lb/>bus ab uno termino eductæ diametri, & à li-<lb/>nea circulum in termino propiore diametri cõ <lb/>tingente, duabus lineis ad alium terminũ dia-<lb/>metri productis: arcus interiacẽtes illarum line arum alter am & diametrum, ſunt æquales: il-<lb/>lis uerò ad alium punctum circumferentiæ produ-<lb/> ctis, arcus interiacent inæquales.</head> <p> <s xml:id="echoid-s21826" xml:space="preserve">Sit circulus a b c d, cuius centrum e:</s> <s xml:id="echoid-s21827" xml:space="preserve"> diameterq́;</s> <s xml:id="echoid-s21828" xml:space="preserve"> e-<lb/>ius d b educatur ad punctũ f:</s> <s xml:id="echoid-s21829" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s21830" xml:space="preserve"> duo puncta g & h <lb/>ęqualiter diſtãtia à pũcto f eductę diametri:</s> <s xml:id="echoid-s21831" xml:space="preserve"> ducãtúr-<lb/>que duę lineę g d & h d adaliũ terminũ diametri ſecã-<lb/>tes circulũ:</s> <s xml:id="echoid-s21832" xml:space="preserve"> linea g d in pũcto a, & linea h d in pũcto c:</s> <s xml:id="echoid-s21833" xml:space="preserve"> <lb/>& à puncto b ducatur linea cõtingens circulũ, quę ſit <lb/>k b l, à qua ęqualiter diſtẽt pũcta g & h.</s> <s xml:id="echoid-s21834" xml:space="preserve"> Dico, quòd ar-<lb/>cus a b & b c ſunt æquales.</s> <s xml:id="echoid-s21835" xml:space="preserve"> Ducatur enim linea g f h:</s> <s xml:id="echoid-s21836" xml:space="preserve"> <lb/>erit ergo ex hypotheſi linea g f æqualis lineę h f:</s> <s xml:id="echoid-s21837" xml:space="preserve"> ideo, <lb/>quia puncta g & h ęqualiter diſtãt à puncto f:</s> <s xml:id="echoid-s21838" xml:space="preserve"> & ducã-<lb/>turlineę h l & g k perpẽdiculariter ſuper lineã k b l cõ <lb/>tingẽtẽ, ք 12 p 1:</s> <s xml:id="echoid-s21839" xml:space="preserve"> erũt ergo ex hypotheſi & illę ęquales:</s> <s xml:id="echoid-s21840" xml:space="preserve"> <lb/>ergo ք 33 p 1 linea g h ęꝗdiſtat lineę k l.</s> <s xml:id="echoid-s21841" xml:space="preserve"> Ergo ք 18 p 3 & <lb/>29 p 1 anguli d f h & d f g ſunt recti:</s> <s xml:id="echoid-s21842" xml:space="preserve"> ergo ք 4 p 1 anguli <lb/>g d f & h d f ſunt ęquales.</s> <s xml:id="echoid-s21843" xml:space="preserve"> Ergo ք 26 p 3 arcus a b eſt ę-<lb/>qualis arcui b c.</s> <s xml:id="echoid-s21844" xml:space="preserve"> Patet quoq;</s> <s xml:id="echoid-s21845" xml:space="preserve"> manifeſtè, quòd ſi à pũctis g & h lineę ad aliud pũctũ circũferentię quã <lb/>ad pũctũ d ꝓducãtur, ut ad pũcta m ueln:</s> <s xml:id="echoid-s21846" xml:space="preserve"> quòd illę lineę arcus reſecabũt inęquales:</s> <s xml:id="echoid-s21847" xml:space="preserve"> quęlibet enim <lb/>illarũ, quę ſecat diametrũ, abſcindit minorẽ arcum, & alia maiorẽ:</s> <s xml:id="echoid-s21848" xml:space="preserve"> & hoc eſt, quod proponebatur.</s> <s xml:id="echoid-s21849" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div764" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables317" xml:space="preserve">g b c a f d e</variables> </figure> <head xml:id="echoid-head637" xml:space="preserve" style="it">63. Diameter circuli diuidens hexagonum, eidẽ cir-<lb/> culo inſcriptum, ab oppoſitis angulis per æqualia, duob. lateribus medijs hexagoni erit æquidiſtans.</head> <p> <s xml:id="echoid-s21850" xml:space="preserve">Sit circulus, cuius centrũ ſit punctũ a:</s> <s xml:id="echoid-s21851" xml:space="preserve"> inſcriptus hexa-<lb/>gonus, qui b c d e f g:</s> <s xml:id="echoid-s21852" xml:space="preserve"> & ab oppoſitis angulis illius hexago <lb/>ni ducatur diameter b a e.</s> <s xml:id="echoid-s21853" xml:space="preserve"> Dico, quòd illa diameter æqui-<lb/>diſtat duobus medijs lateribus hexagoni, quæ ſunt c d & <lb/>g f.</s> <s xml:id="echoid-s21854" xml:space="preserve"> Ducantur enim lineæ a c & a d.</s> <s xml:id="echoid-s21855" xml:space="preserve"> Quia itaque lineę b c <lb/>& c d, (quę ſunt latera hexagoni) ſunt inter ſe ęqualia, & <lb/>utrunq;</s> <s xml:id="echoid-s21856" xml:space="preserve"> ipſorũ eſt ęquale ſemidiametro circuli per 15 p 4:</s> <s xml:id="echoid-s21857" xml:space="preserve"> <lb/>patetergo, quòd trigona a b c & a c d ſunt ęquilatera:</s> <s xml:id="echoid-s21858" xml:space="preserve"> er-<lb/>go per 8 p 1 ipſa ſunt ęquiangula:</s> <s xml:id="echoid-s21859" xml:space="preserve"> erit ergo angulus c a b ę-<lb/>qualis angulo a c d.</s> <s xml:id="echoid-s21860" xml:space="preserve"> Ergo per 27 p 1 lineæ a b & c d ęquidi <lb/>ſtant.</s> <s xml:id="echoid-s21861" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s21862" xml:space="preserve"> poteſt demonſtrari de lineis a b & <lb/>f g.</s> <s xml:id="echoid-s21863" xml:space="preserve"> Patet ergo, quoniam diameter b e ęquidiſtat medijs la <lb/>teribus hexagoni:</s> <s xml:id="echoid-s21864" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s21865" xml:space="preserve"/> </p> <figure> <variables xml:id="echoid-variables318" xml:space="preserve">g f c b d a</variables> </figure> </div> <div xml:id="echoid-div765" type="section" level="0" n="0"> <head xml:id="echoid-head638" xml:space="preserve" style="it">64. Duobus circulis inæqualibus ſe ſecantibus, it a ut minor pertrã-<lb/>ſeat centrum maioris: arcum minor is interiacentem peripheriã ma-<lb/>ioris in centro maioris per æqualia diuidi eſt neceſſe.</head> <p> <s xml:id="echoid-s21866" xml:space="preserve">Sint duo circuli c f d maior, cuius centrũ ſit a:</s> <s xml:id="echoid-s21867" xml:space="preserve"> & c g d minor, cuius cen <lb/>trum ſit b:</s> <s xml:id="echoid-s21868" xml:space="preserve"> ſecentq́;</s> <s xml:id="echoid-s21869" xml:space="preserve"> ſe hi circuli in punctis c & d:</s> <s xml:id="echoid-s21870" xml:space="preserve"> tranſeatq́;</s> <s xml:id="echoid-s21871" xml:space="preserve"> minor (qui c <lb/>g d) per centrũ maioris, quod eſt a:</s> <s xml:id="echoid-s21872" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s21873" xml:space="preserve"> arcus c a d minoris circuli con <lb/>tentus intra peripheriam maioris.</s> <s xml:id="echoid-s21874" xml:space="preserve"> Dico, quòd arcus c a d diuiditur per <lb/>æqualia in puncto a.</s> <s xml:id="echoid-s21875" xml:space="preserve"> Ducatur enim linea copulans centra, quę ſit a b:</s> <s xml:id="echoid-s21876" xml:space="preserve"> & <lb/>hec producta compleat diametrũ minoris circuli, quæ ſit a b g:</s> <s xml:id="echoid-s21877" xml:space="preserve"> & ad pũ-<lb/>cta ſectionum c & d, ducantur lineæ a d, a c, b d, b c.</s> <s xml:id="echoid-s21878" xml:space="preserve"> Quia itaque in trigo-<lb/>nis a b c & a b d, duo latera a b & b c unius ſunt æqualia duobus laterib.</s> <s xml:id="echoid-s21879" xml:space="preserve"> <lb/>a b & b d alterius:</s> <s xml:id="echoid-s21880" xml:space="preserve"> quoniam omnes ſunt rectę ex puncto b centro circuli <lb/> <pb o="28" file="0330" n="330" rhead="VITELLONIS OPTICAE"/> minoris ductæ ad peripheriam, & baſis a c eſt æqualis baſi a d:</s> <s xml:id="echoid-s21881" xml:space="preserve"> quoniam ſunt ex centro circuli maio <lb/>ris.</s> <s xml:id="echoid-s21882" xml:space="preserve"> Ergo per 8 p 1 anguli æquis lateribus contenti ſunt ęquales:</s> <s xml:id="echoid-s21883" xml:space="preserve"> angulus ergo c a b eſt æqualis angu <lb/>lo d a b:</s> <s xml:id="echoid-s21884" xml:space="preserve"> ergo per 26 p 3 arcus c g eſt ęqualis arcui d g:</s> <s xml:id="echoid-s21885" xml:space="preserve"> reliqui ergo arcus ſemicirculorum, qui ſunt a c <lb/>& a d, ſunt ęquales.</s> <s xml:id="echoid-s21886" xml:space="preserve"> Arcus ergo c a d diuiditur per æqualia in puncto a:</s> <s xml:id="echoid-s21887" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s21888" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div766" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables319" xml:space="preserve">e a d b c</variables> </figure> <head xml:id="echoid-head639" xml:space="preserve" style="it">65. Omnes lineæ rectæ ductæ à polo ad peripheriam ſui circuli <lb/> ſunt æquales. 5 def. 1 ſphæ. Theodo.</head> <p> <s xml:id="echoid-s21889" xml:space="preserve">Eſto circulus a b c, cuius centrum d:</s> <s xml:id="echoid-s21890" xml:space="preserve"> & erigatur perpendiculariter <lb/>ſuper circulum à centro linea d e, ita, ut per definitionem polus cir-<lb/>culi ſit punctũ e:</s> <s xml:id="echoid-s21891" xml:space="preserve"> & ducantur lineæ e a, e b, e c.</s> <s xml:id="echoid-s21892" xml:space="preserve"> Dico, quòd ipſæ oẽs <lb/>ſunt æquales.</s> <s xml:id="echoid-s21893" xml:space="preserve"> Ducantur enim lineę a d, b d, c d.</s> <s xml:id="echoid-s21894" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s21895" xml:space="preserve"> quadratũ <lb/>lineę a e eſt ęquale quadrato lineę e d & lineę d a:</s> <s xml:id="echoid-s21896" xml:space="preserve"> quadratum quoq;</s> <s xml:id="echoid-s21897" xml:space="preserve"> <lb/>lineæ b e æquale eſt quadrato lineæ e d & lineæ d b per 47 p 1:</s> <s xml:id="echoid-s21898" xml:space="preserve"> qua-<lb/>dratum uerò lineæ e d eſt æquale ſibijpſi, & quadratũ lineę d a ęqua-<lb/>le quadrato lineæ d b per circuli definitionem:</s> <s xml:id="echoid-s21899" xml:space="preserve"> palàm quia quadra-<lb/>tum lineæ a e eſt æquale quadrato lineę b e, & ſimiliter quadrato li-<lb/>neæ c e.</s> <s xml:id="echoid-s21900" xml:space="preserve"> Palàm ergo, quoniam lineę a e, b e, c e, & quæcunq;</s> <s xml:id="echoid-s21901" xml:space="preserve"> ſimiliter <lb/>ductæ, ſunt æquales:</s> <s xml:id="echoid-s21902" xml:space="preserve"> & hoc eſt propoſitum.</s> <s xml:id="echoid-s21903" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div767" type="section" level="0" n="0"> <head xml:id="echoid-head640" xml:space="preserve" style="it">66. Omnis linea centrum ſphæræ cum centro circuli non magni <lb/>illius ſphæræ continuans eſt perpẽdicularis ſuper ſuperficiem illius <lb/>circuli. 7 & 23 th. 1 ſphæ. Theodo.</head> <p> <s xml:id="echoid-s21904" xml:space="preserve">Sit centrum ſphærę punctum z, ſitq́;</s> <s xml:id="echoid-s21905" xml:space="preserve"> punctum e centrum circuli non magni illius ſphæræ, qui ſit <lb/>a b g d, & ducatur linea z e.</s> <s xml:id="echoid-s21906" xml:space="preserve"> Dico, quòd linea z e eſt perpendicularis ſuper ſuperficiem circuli a b g d.</s> <s xml:id="echoid-s21907" xml:space="preserve"> <lb/>Ducantur enim lineę a e, b e, quę productæ cõpleant duas <lb/> <anchor type="figure" xlink:label="fig-0330-02a" xlink:href="fig-0330-02"/> diametros circuli, quæ ſint a g, & b d:</s> <s xml:id="echoid-s21908" xml:space="preserve"> & ducantur lineę z a <lb/>& z b & z d & z g, quę omnes erunt æquales per definitio-<lb/>nem ſphæræ:</s> <s xml:id="echoid-s21909" xml:space="preserve"> ſed & lineæ e a, e b, e d, e g ſunt æquales per <lb/>definitionem circuli:</s> <s xml:id="echoid-s21910" xml:space="preserve"> linea itaq;</s> <s xml:id="echoid-s21911" xml:space="preserve"> z e exiſtente communi, pa <lb/>tet quòd trigona z a e, z b e, z d e, z g e omnia ſunt ęquilate-<lb/>ra:</s> <s xml:id="echoid-s21912" xml:space="preserve"> ergo per 8 p 1 ipſorum anguli ęqualibus laterib.</s> <s xml:id="echoid-s21913" xml:space="preserve"> conten-<lb/>ti ſunt ęquales.</s> <s xml:id="echoid-s21914" xml:space="preserve"> Oęs ergo anguli z e a, z e g, z e b, z e d ſunt <lb/>ęquales:</s> <s xml:id="echoid-s21915" xml:space="preserve"> ſunt ergo recti.</s> <s xml:id="echoid-s21916" xml:space="preserve"> Eodemq́;</s> <s xml:id="echoid-s21917" xml:space="preserve"> modo poteſt demõſtra.</s> <s xml:id="echoid-s21918" xml:space="preserve"> <lb/>ri de omnibus angulis contentis ſub linea z e & omni ſemi <lb/>diametro circuli a b g d.</s> <s xml:id="echoid-s21919" xml:space="preserve"> Linea ergo z e eſt perpendicularis <lb/>ſuper ſuperficiem circuli a b g d:</s> <s xml:id="echoid-s21920" xml:space="preserve"> & hoc eſt propoſitum.</s> <s xml:id="echoid-s21921" xml:space="preserve"/> </p> <div xml:id="echoid-div767" type="float" level="0" n="0"> <figure xlink:label="fig-0330-02" xlink:href="fig-0330-02a"> <variables xml:id="echoid-variables320" xml:space="preserve">b z g a e d</variables> </figure> </div> </div> <div xml:id="echoid-div769" type="section" level="0" n="0"> <head xml:id="echoid-head641" xml:space="preserve" style="it">67. À<unsure/> centro ſphæræ ductã perpendicularẽ ſuք ſuper-<lb/>ficiẽ circuli non magni ipſius ſphæræ, eiuſdẽ circuli cẽtro <lb/>incidere eſt neceſſe. Cõſectariũ ſecundũ 1 th. 1 ſphæ. Theo.</head> <p> <s xml:id="echoid-s21922" xml:space="preserve">Sit, ut in præmiſſa, centrum ſphęræ punctum z:</s> <s xml:id="echoid-s21923" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s21924" xml:space="preserve"> punctum e centrum circuli non magni illius <lb/>ſphęrę, qui ſit a b g d:</s> <s xml:id="echoid-s21925" xml:space="preserve"> & ducatur à puncto z centro ſphærę linea perpendiculariter ſuper ſuperficiẽ <lb/>circuli a b g d, quæ ſit z e.</s> <s xml:id="echoid-s21926" xml:space="preserve"> Dico, quòd punctum e eſt centrum circuli a b g d.</s> <s xml:id="echoid-s21927" xml:space="preserve"> Ducantur enim lineæ <lb/>z a, z b, z g, quæ erũt ęquales per definitionẽ ſphęrę.</s> <s xml:id="echoid-s21928" xml:space="preserve"> Quoniã ergo anguli a e z, b e z, d e z, g e z ſunt re <lb/>cti:</s> <s xml:id="echoid-s21929" xml:space="preserve"> patet per 47 p 1 quoniam quadratũ lineę z a ualet quadrata linearum a e & z e, & quadratum li-<lb/>neę z b ualet ambo quadrata linearum b e & z e:</s> <s xml:id="echoid-s21930" xml:space="preserve"> & ſimiliter quadratũ lineę z g ualet ambo quadra-<lb/>ta linearum g e & z e:</s> <s xml:id="echoid-s21931" xml:space="preserve"> lineę uerò z a, z b, z g ſunt ęquales, & quadrata ipſarum ęqualia:</s> <s xml:id="echoid-s21932" xml:space="preserve"> ablato itaque <lb/>quadrato lineę z e omnib.</s> <s xml:id="echoid-s21933" xml:space="preserve"> cõmuni, relinquitur ut quadrata linearum <lb/> <anchor type="figure" xlink:label="fig-0330-03a" xlink:href="fig-0330-03"/> a e, b e, g e ſint ęqualia:</s> <s xml:id="echoid-s21934" xml:space="preserve"> ergo & ipſę lineę a e, b e, g e ſunt ęquales.</s> <s xml:id="echoid-s21935" xml:space="preserve"> Ergo <lb/>per 9 p 3 punctum e eſt centrum circuli a b g d:</s> <s xml:id="echoid-s21936" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s21937" xml:space="preserve"/> </p> <div xml:id="echoid-div769" type="float" level="0" n="0"> <figure xlink:label="fig-0330-03" xlink:href="fig-0330-03a"> <variables xml:id="echoid-variables321" xml:space="preserve">b f c a d g e</variables> </figure> </div> </div> <div xml:id="echoid-div771" type="section" level="0" n="0"> <head xml:id="echoid-head642" xml:space="preserve" style="it">68. Aequidiſtantium in ſphæra circulorum centra in eadẽ dia <lb/>metro ſphæræ conſiſtere eſt neceſſe. Ex quo patet, quòd omnes circu-<lb/>li in ſphæra æquidiſtantes eoſdem habent polos: & ſi eoſdem habent <lb/>polos, ſunt æquidiſtantes. 1 & 2 th. 2 ſphæ. Theodo.</head> <p> <s xml:id="echoid-s21938" xml:space="preserve">Sit ſphęra, cuius centrũ ſit punctũ a, & in ipſa ſint duo circuli ęquidi <lb/>ſtãtes:</s> <s xml:id="echoid-s21939" xml:space="preserve"> b c, cuius cẽtrũ ſit f:</s> <s xml:id="echoid-s21940" xml:space="preserve"> & d e, cuius cẽtrũ g:</s> <s xml:id="echoid-s21941" xml:space="preserve"> & ducatur linea a f, quę <lb/>ꝓducta erit diameter ſphęrę, cũ ipſa trãſeat centrũ ſphęrę a:</s> <s xml:id="echoid-s21942" xml:space="preserve"> ergo ք 66 <lb/>huius lineá a f eſt erecta ſup ſupficiẽ circuli b c:</s> <s xml:id="echoid-s21943" xml:space="preserve"> ergo ք 23 huius erit ea <lb/>dẽ diameter erecta ſuք ſuքficiẽ circuli d e:</s> <s xml:id="echoid-s21944" xml:space="preserve"> ergo ք pmiſſam ipſa trãſit ք <lb/>centrũ circuli d e.</s> <s xml:id="echoid-s21945" xml:space="preserve"> Sunt ergo centra illorũ circulorũ in eadẽ diametro <lb/>ſphęrę:</s> <s xml:id="echoid-s21946" xml:space="preserve"> qđ eſt ꝓpoſitũ.</s> <s xml:id="echoid-s21947" xml:space="preserve"> Et exhoc patet, qđ illi circuli eoſdẽ habẽt po-<lb/> <pb o="29" file="0331" n="331" rhead="LIBER PRIMVS."/> los per definitionẽ poli.</s> <s xml:id="echoid-s21948" xml:space="preserve"> Et ſi aliqui circuli eoſdẽ habent polos, patet per 14 p 11, quòd ipſi ſunt æqui-<lb/>diſtantes:</s> <s xml:id="echoid-s21949" xml:space="preserve"> & hoc proponebatur.</s> <s xml:id="echoid-s21950" xml:space="preserve"> Quòd ſi etiã reliquus circulorũ æquidiſtantium eſſet circulus ma-<lb/>gnus, eadem eſſet demonſtratio.</s> <s xml:id="echoid-s21951" xml:space="preserve"> Duo uerò circuli magni eiuſdem ſphęræ ſibi inuicem æquidiſtare <lb/>non poſſunt:</s> <s xml:id="echoid-s21952" xml:space="preserve"> quoniam amborum illorum eſt idem centrum, quod eſt centrum ſphæræ.</s> <s xml:id="echoid-s21953" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div772" type="section" level="0" n="0"> <head xml:id="echoid-head643" xml:space="preserve" style="it">69. Si plana ſuperficies ſecet ſphærã, cõmunis ſectio erit circulus. Ex quo patet, quoniã à quo-<lb/>libet puncto in diametro uel ſuperficie ſphærica dato, eſt poſsibile totali ſuperficiei ſphæricæ circu-<lb/>lumcircumduci, alij etiam circulo illius æquidiſtantem. 1 th. 1 ſphær. Theodoſy.</head> <p> <s xml:id="echoid-s21954" xml:space="preserve">Sit ſphęra, cuius centrũ a, ſeceturq́;</s> <s xml:id="echoid-s21955" xml:space="preserve"> per planam ſuperficiẽ.</s> <s xml:id="echoid-s21956" xml:space="preserve"> Dico, quòd cõmunis ſectio ſuperficiei <lb/>ſphęricæ & planæ eſt circulus.</s> <s xml:id="echoid-s21957" xml:space="preserve"> Si enim fiat ſectio ք centrũ <lb/> <anchor type="figure" xlink:label="fig-0331-01a" xlink:href="fig-0331-01"/> a:</s> <s xml:id="echoid-s21958" xml:space="preserve"> tũc patet, quòd oẽs lineæ ductæ à cẽtro a ad ſphæræ ſu-<lb/>perficiẽ, quę ſunt in illa plana ſuքficie ſecãte, & terminan-<lb/>tur ad cõmunem terminũ illorũ, ſunt æquales per defini-<lb/>tionẽ ſphęræ:</s> <s xml:id="echoid-s21959" xml:space="preserve"> ergo per definitionẽ circuli, illa cõmunis ſe-<lb/>ctio eſt circulus.</s> <s xml:id="echoid-s21960" xml:space="preserve"> Si aũt ſuperficies plana ſecet ſphærã datã <lb/>nõ per centrũ a:</s> <s xml:id="echoid-s21961" xml:space="preserve"> ducatur per 11 p 11 à centro a perpẽdicula-<lb/>ris ſuper ſuperficiẽ ſecantẽ, quę ſit a b, & cõtinuẽtur lineæ <lb/>a c, a d, a e, a f, & quot quis uoluerit ad illã ſectionem com-<lb/>munem à cẽtro ipſius ſphęræ:</s> <s xml:id="echoid-s21962" xml:space="preserve"> ducãtur quoq;</s> <s xml:id="echoid-s21963" xml:space="preserve"> lineę c b, d b, <lb/>e b, f b, in ipſa ſuperficie ſecãte, ad puncta, quibus incidũt <lb/>lineę ex centro ſphęræ ductæ.</s> <s xml:id="echoid-s21964" xml:space="preserve"> Palàm ergo per 47 p 1, quo-<lb/>niã quadratũ lineæ a c eſt ęquale duobus quadratis linea-<lb/>rum a b & b c:</s> <s xml:id="echoid-s21965" xml:space="preserve"> & ſimiliter quadratum lineę a d eſt æquale <lb/>duob.</s> <s xml:id="echoid-s21966" xml:space="preserve"> quadratis linearũ a b & b d:</s> <s xml:id="echoid-s21967" xml:space="preserve"> ſed quadratũ lineæ a c <lb/>eſt æquale quadrato lineæ a d:</s> <s xml:id="echoid-s21968" xml:space="preserve"> quoniã linea a c eſt æqualis <lb/>lineæ a d per definitionẽ ſphęræ, & quadratũ lineæ a b eſt ęquale ſibijpſi:</s> <s xml:id="echoid-s21969" xml:space="preserve"> relinquitur ergo quadratũ <lb/>lineæ c b æquale quadrato lineæ d b:</s> <s xml:id="echoid-s21970" xml:space="preserve"> eſt ergo linea c b æqualis lineæ d b:</s> <s xml:id="echoid-s21971" xml:space="preserve"> & ſimiliter erit linea d b <lb/>æqualis lineis e b & f b:</s> <s xml:id="echoid-s21972" xml:space="preserve"> eadẽ enim eſt demonſtratio, quotcunq;</s> <s xml:id="echoid-s21973" xml:space="preserve"> alijs lineis à cẽtro ſphærę a ad illam <lb/>communẽ ſectionem productis.</s> <s xml:id="echoid-s21974" xml:space="preserve"> Omnes itaq;</s> <s xml:id="echoid-s21975" xml:space="preserve"> lineæ à puncto b ad illã communem ſectionẽ ductæ, <lb/>ſunt æquales:</s> <s xml:id="echoid-s21976" xml:space="preserve"> ergo per 9 p 3 & per definitionẽ circuli, ut prius, punctũ b eſt centrũ circuli.</s> <s xml:id="echoid-s21977" xml:space="preserve"> Cõmunis <lb/>ergo ſectio iſtarũ ſuperficierũ eſt circulus:</s> <s xml:id="echoid-s21978" xml:space="preserve"> & hoc eſt propoſitũ.</s> <s xml:id="echoid-s21979" xml:space="preserve"> Patet etiã ex hoc corollariũ:</s> <s xml:id="echoid-s21980" xml:space="preserve"> quoniã <lb/>à pũcto dato per 12 p 1 producta perpẽdiculari ſuper diametrũ ſphęræ, imaginetur ſuperficies plana <lb/>ſecãs ſphærã ſecundũ illã perpendicularẽ:</s> <s xml:id="echoid-s21981" xml:space="preserve"> & patet propoſitũ per præmiſſa.</s> <s xml:id="echoid-s21982" xml:space="preserve"> Quòd ſi alicui circulo in <lb/>ſphęra ſignato æquidiſtãs duci debeat:</s> <s xml:id="echoid-s21983" xml:space="preserve"> à dato pũcto ducatur perpẽdicularis ſuper ſphęræ diametrũ <lb/>tranſeuntẽ circuli centrũ, cui æquidiſtãs debet duci circulus, & ꝓducatur in continuũ uſq;</s> <s xml:id="echoid-s21984" xml:space="preserve"> ad aliã <lb/>ſphęræ ſuperficiẽ, & ducatur alia linea à pũcto diametri utcũq;</s> <s xml:id="echoid-s21985" xml:space="preserve"> ſuք productã, & orthogonaliter ſu-<lb/>per diametrũ ſphęræ, imagineturq́;</s> <s xml:id="echoid-s21986" xml:space="preserve"> ſuperficies plana trãſiens terminos iſtarũ linearũ in ipſa ſuper-<lb/>ficie ſphęræ faciẽs ſectionẽ:</s> <s xml:id="echoid-s21987" xml:space="preserve"> quę per præmiſſa neceſſariò erit circulus:</s> <s xml:id="echoid-s21988" xml:space="preserve"> quia ք 4 p 11 diameter ſphęrę, <lb/>ſuper quã ducitur linea à pũcto dato, erit perpẽdicularis ſuper ſuperficiẽ in punctis illis, ut præmit-<lb/>titur, ſphæram ſecantem:</s> <s xml:id="echoid-s21989" xml:space="preserve"> unde à centro ſphæræ ductis lineis, ut prius, patet quod proponebatur.</s> <s xml:id="echoid-s21990" xml:space="preserve"/> </p> <div xml:id="echoid-div772" type="float" level="0" n="0"> <figure xlink:label="fig-0331-01" xlink:href="fig-0331-01a"> <variables xml:id="echoid-variables322" xml:space="preserve">d f b c e d</variables> </figure> </div> </div> <div xml:id="echoid-div774" type="section" level="0" n="0"> <head xml:id="echoid-head644" xml:space="preserve" style="it">70. À<unsure/> dato puncto ad datam ſphæram lineam contingentem ducere.</head> <p> <s xml:id="echoid-s21991" xml:space="preserve">Sit enim datũ punctũ a, & centrũ datę ſphę-<lb/> <anchor type="figure" xlink:label="fig-0331-02a" xlink:href="fig-0331-02"/> ræ ſit punctũ b:</s> <s xml:id="echoid-s21992" xml:space="preserve"> & ducatur linea a b:</s> <s xml:id="echoid-s21993" xml:space="preserve"> & à cẽtro <lb/>ſphæræ, quod eſt b, ducatur linea b c, ut cõtin-<lb/>git, & copuletur linea a c:</s> <s xml:id="echoid-s21994" xml:space="preserve"> palamq́;</s> <s xml:id="echoid-s21995" xml:space="preserve"> ք 2 p 11, quo <lb/>niam trigonũ a b c eſt in una ſuperficie plana:</s> <s xml:id="echoid-s21996" xml:space="preserve"> <lb/>hęcitaq;</s> <s xml:id="echoid-s21997" xml:space="preserve"> per præcedẽtem ſecabit ſphęrã ſecũ-<lb/>dũ circulũ, cui per 17 p 3 à pũcto a ducatur cõ-<lb/>tingẽs in pũcto d, quæ ſit a d:</s> <s xml:id="echoid-s21998" xml:space="preserve"> & patet ꝓpoſitũ.</s> <s xml:id="echoid-s21999" xml:space="preserve"/> </p> <div xml:id="echoid-div774" type="float" level="0" n="0"> <figure xlink:label="fig-0331-02" xlink:href="fig-0331-02a"> <variables xml:id="echoid-variables323" xml:space="preserve">c a d b</variables> </figure> </div> </div> <div xml:id="echoid-div776" type="section" level="0" n="0"> <head xml:id="echoid-head645" xml:space="preserve" style="it">71. Omnis ſuperficies plana contingens <lb/>ſphæram, ſecundũ unicum punctum eſt con-<lb/>tingens. 3 th. 1 ſphær. Theodoſij.</head> <p> <s xml:id="echoid-s22000" xml:space="preserve">Ducatur in plana ſuperficie contingente ſphæram, linea recta trans locum cõtactus, & in ſuper-<lb/>ficie ſphęræ circulus magnus.</s> <s xml:id="echoid-s22001" xml:space="preserve"> Si ergo ſuperficies plana contingit ſphæram ſecundum aliud quàm <lb/>ſecundum punctum, & linea recta continget circulum ſecundum idem:</s> <s xml:id="echoid-s22002" xml:space="preserve"> non ergo ſecundum pun-<lb/>ctum continget linea recta circulum:</s> <s xml:id="echoid-s22003" xml:space="preserve"> quod eſt contra 16 p 3:</s> <s xml:id="echoid-s22004" xml:space="preserve"> palàm ergo propoſitum.</s> <s xml:id="echoid-s22005" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div777" type="section" level="0" n="0"> <head xml:id="echoid-head646" xml:space="preserve" style="it">72. À<unsure/> dato pũcto ſuքficiei ſphæricæ ſuքficiẽ planã cõtingentẽ ducere. Ex quo patet, ꝗ omnis <lb/>linea centrũ ſphæræ trãſiens, eſt perpẽdicularis ſuք eius ſuperficiẽ: & ſieſt perpendicularis ſuper <lb/>ſphæricam ſuperficiem, neceſſariò tranſit centrũ ſphæræ. È<unsure/> 4 th. 1 ſphær. Theodoſy. Alh. 25 n 4.</head> <p> <s xml:id="echoid-s22006" xml:space="preserve">Eſto ſphęra, cuius centrũ ſit a, & circulus eius magnus b d c:</s> <s xml:id="echoid-s22007" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s22008" xml:space="preserve"> linea a b à cẽtro ad circũ-<lb/>ferentiã:</s> <s xml:id="echoid-s22009" xml:space="preserve"> & à pũcto b ducatur linea cõtingẽs circulũ, quę ſit f b e ք 17 p 3:</s> <s xml:id="echoid-s22010" xml:space="preserve"> erũt ergo anguli a b e & a b f <lb/>recti.</s> <s xml:id="echoid-s22011" xml:space="preserve"> Imaginatis quoq;</s> <s xml:id="echoid-s22012" xml:space="preserve"> ք 69 huius circulis quotcũq;</s> <s xml:id="echoid-s22013" xml:space="preserve"> in ſuքficie ſphęrę ſecantib.</s> <s xml:id="echoid-s22014" xml:space="preserve"> ſe in pũcto b, & du-<lb/>ctis lineis, cõtingentib.</s> <s xml:id="echoid-s22015" xml:space="preserve"> illos circulos in pũcto b:</s> <s xml:id="echoid-s22016" xml:space="preserve"> palàm ք 18 p 3, quoniã linea b a cũ omnib.</s> <s xml:id="echoid-s22017" xml:space="preserve"> illis lineis <lb/>cõtinetangulos rectos.</s> <s xml:id="echoid-s22018" xml:space="preserve"> Ergo oẽs illę lineæ ſunt in una ſuքficie plana ք 2 p 11.</s> <s xml:id="echoid-s22019" xml:space="preserve"> Illa itaq;</s> <s xml:id="echoid-s22020" xml:space="preserve"> ſuքficies con-<lb/> <pb o="30" file="0332" n="332" rhead="VITELLONIS OPTICAE"/> tingit ſphęrã ք definitionẽ ſuքficiei planę ſphęrã cõtingẽtis.</s> <s xml:id="echoid-s22021" xml:space="preserve"> Ex hoc itaq;</s> <s xml:id="echoid-s22022" xml:space="preserve"> patet, quoniã omnis linea <lb/>à cẽtro ſphęræ ducta, ſit erecta ſuք planã ſuքficiẽ, ſphęrã ipſam in pũ-<lb/> <anchor type="figure" xlink:label="fig-0332-01a" xlink:href="fig-0332-01"/> cto ſuæ incidẽtię cõtingentẽ, & anguli incidẽtiæ ſint æquales:</s> <s xml:id="echoid-s22023" xml:space="preserve"> quoniã <lb/>ipſa eſt perpẽdicularis ſuք ſphęrę ſuperficiẽ, ք definitionẽ perpẽdicu-<lb/>laris:</s> <s xml:id="echoid-s22024" xml:space="preserve"> anguli enim ſemicirculorũ oẽs ſunt æquales ք 43 huius.</s> <s xml:id="echoid-s22025" xml:space="preserve"> Et quo-<lb/>niã linea ab ꝓducta ad punctũ g, eſt adhuc erecta ſuք ſuքficiẽ planã, <lb/>ſphęrã cõtingentẽ in pũcto b:</s> <s xml:id="echoid-s22026" xml:space="preserve"> palã, ք a linea g b, & quęcũq;</s> <s xml:id="echoid-s22027" xml:space="preserve"> alia քpẽdi-<lb/>cularis erigi poteſt ſuք ſuքficiẽ planã in pũcto b, cõtingẽtẽ ſphęrã, trã-<lb/>ſit cẽtrũ ſphęræ a:</s> <s xml:id="echoid-s22028" xml:space="preserve"> ꝗ a ſi à pũcto b poſsit alia linea erigi ſuք ſuքficiẽ cõ-<lb/>tingẽtẽ, nõ trãſiẽs cetrũ ſphærę a:</s> <s xml:id="echoid-s22029" xml:space="preserve"> ſit illa h b d, & ſit angul<emph style="sub">9</emph> h b e rectus:</s> <s xml:id="echoid-s22030" xml:space="preserve"> <lb/>ſed angul<emph style="sub">9</emph> g b e eſt rectus ք 13 p 1, cũ angul<emph style="sub">9</emph> a b e ſit rect<emph style="sub">9</emph> ex hypotheſi:</s> <s xml:id="echoid-s22031" xml:space="preserve"> <lb/>erit itaq;</s> <s xml:id="echoid-s22032" xml:space="preserve"> rectus maior recto:</s> <s xml:id="echoid-s22033" xml:space="preserve"> qđ eſt impoſsibile:</s> <s xml:id="echoid-s22034" xml:space="preserve"> patet ergo ꝓpoſitũ.</s> <s xml:id="echoid-s22035" xml:space="preserve"/> </p> <div xml:id="echoid-div777" type="float" level="0" n="0"> <figure xlink:label="fig-0332-01" xlink:href="fig-0332-01a"> <variables xml:id="echoid-variables324" xml:space="preserve">g h e b f d a</variables> </figure> </div> </div> <div xml:id="echoid-div779" type="section" level="0" n="0"> <head xml:id="echoid-head647" xml:space="preserve" style="it">73. Omnium ſphærarum, quarum conuexæ ſuperficies æquidi-<lb/>ſtant, uel ſecundũ ſe tot{as} ſe contingunt, neceſſariò eſt idẽ centrum.</head> <p> <s xml:id="echoid-s22036" xml:space="preserve">Sint duę ſphęræ, quarũ cõuexæ ſuքficies æquidiſtẽt, ſectæ ք æqua-<lb/>lia ք unã planã ſuքficiẽ:</s> <s xml:id="echoid-s22037" xml:space="preserve"> cõmunis ergo ſectio ſuperficierũ illarũ ſphæ <lb/>ricarũ & huius planæ erũt circuli:</s> <s xml:id="echoid-s22038" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s22039" xml:space="preserve"> magnus circulus maioris ſphęræ a b, & centrũ eius e:</s> <s xml:id="echoid-s22040" xml:space="preserve"> mino-<lb/>ris uerò ſphęrę circulus magnus ſit c d.</s> <s xml:id="echoid-s22041" xml:space="preserve"> Dico, quòd idẽ <lb/> <anchor type="figure" xlink:label="fig-0332-02a" xlink:href="fig-0332-02"/> punctũ e etiã erit cẽtrũ circuli c d.</s> <s xml:id="echoid-s22042" xml:space="preserve"> Ducatur enim linea <lb/>a e b taliter, ut ſi e nõ ſit cẽtrũ amborũ circulorũ, linea <lb/>tñ a e b trãſeat ք ambo cẽtra (qđ poteſt fieri cõtinua-<lb/>tis cẽtris ք lineã rectã) & ꝓducta illa ad քipheriã ma-<lb/>ioris ſphęrę:</s> <s xml:id="echoid-s22043" xml:space="preserve"> hęc itaq;</s> <s xml:id="echoid-s22044" xml:space="preserve"> erit diameter circuli a b.</s> <s xml:id="echoid-s22045" xml:space="preserve"> Et quo-<lb/>niã circuli a b & c d ſunt in eadẽ ſuքficie:</s> <s xml:id="echoid-s22046" xml:space="preserve"> ſit ut diame-<lb/>ter a b ſecet քipheriã circuli c d in pũctis c & d:</s> <s xml:id="echoid-s22047" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s22048" xml:space="preserve"> <lb/>recta c d diameter circuli c d.</s> <s xml:id="echoid-s22049" xml:space="preserve"> Quia ergo ꝓpter æqui-<lb/>diſtantiã circulorũ linea a c eſt æqualis lineæ b d, & li-<lb/>nea a e eſt æqualis lineæ e b:</s> <s xml:id="echoid-s22050" xml:space="preserve"> remanet linea c e æqualis <lb/>lineę e d.</s> <s xml:id="echoid-s22051" xml:space="preserve"> Et ꝗ a diameter c d diuiditur ք ęqualia in pũ-<lb/>cto e:</s> <s xml:id="echoid-s22052" xml:space="preserve"> patet, quòd pũctus e eſt cẽtrũ circuli c d.</s> <s xml:id="echoid-s22053" xml:space="preserve"> Si enim <lb/>nõ ſit pũctus e centrũ circuli c d:</s> <s xml:id="echoid-s22054" xml:space="preserve"> ſit cẽtrũ eius pũctus <lb/>h:</s> <s xml:id="echoid-s22055" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s22056" xml:space="preserve"> ք definitionẽ circuli linea h d æqualis lineæ a <lb/>c:</s> <s xml:id="echoid-s22057" xml:space="preserve"> erit ergo linea h a æqualis lineæ h b:</s> <s xml:id="echoid-s22058" xml:space="preserve"> ſed linea h a eſt <lb/>maior quàm linea a e:</s> <s xml:id="echoid-s22059" xml:space="preserve"> ergo h b eſt maior quã linea e b, <lb/>pars ſuo toto:</s> <s xml:id="echoid-s22060" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s22061" xml:space="preserve"> Eſt ergo pũctus e <lb/>neceſſariò cẽtrum circuli c d.</s> <s xml:id="echoid-s22062" xml:space="preserve"> Et quia circulus c d eſt magnus circulus ſuę ſphęræ, patet quòd æqui-<lb/>diſtantium ſphęrarum eſt idem cẽtrum:</s> <s xml:id="echoid-s22063" xml:space="preserve"> quod eſt propoſitum primum.</s> <s xml:id="echoid-s22064" xml:space="preserve"> Et eodem modo de ſphæris <lb/>ſecundum totas ſuas ſuperficies contingentibus, eſt demonſtrandum.</s> <s xml:id="echoid-s22065" xml:space="preserve"> Lineæ enim ductæ à centro <lb/>ad concauum maioris & ad cõuexum minoris, ſunt ęquales:</s> <s xml:id="echoid-s22066" xml:space="preserve"> patet ergo illud quod proponebatur.</s> <s xml:id="echoid-s22067" xml:space="preserve"/> </p> <div xml:id="echoid-div779" type="float" level="0" n="0"> <figure xlink:label="fig-0332-02" xlink:href="fig-0332-02a"> <variables xml:id="echoid-variables325" xml:space="preserve">a c e h d b</variables> </figure> </div> </div> <div xml:id="echoid-div781" type="section" level="0" n="0"> <head xml:id="echoid-head648" xml:space="preserve" style="it">74. Si duæ ſphæræ fuerint æquidiſtãtes, uel ſecundũ totas ſuքficies ſe cõtingẽtes: quæcũ lineæ <lb/>ſuք unius earũ ſuperficiẽ perpẽdicularis fuerit, ſuք alterius quo ſuperficiẽ perpẽdicularis erit.</head> <p> <s xml:id="echoid-s22068" xml:space="preserve">Iſtud faciliter patet.</s> <s xml:id="echoid-s22069" xml:space="preserve"> Quoniã enim ex præmiſſa tales ſphęræ idẽ centrum habere neceſſariò com-<lb/>probantur:</s> <s xml:id="echoid-s22070" xml:space="preserve"> ergo per 72 huius linea perpendicularis ſuper alteram iſtarum ſphęrarum, centrũ ipſius <lb/>tranſit:</s> <s xml:id="echoid-s22071" xml:space="preserve"> ſed centrum ipſius eſt cẽtrum alterius.</s> <s xml:id="echoid-s22072" xml:space="preserve"> Ergo per eandem 72 huius ſuper alterius etiã ſphæ-<lb/>ræ ſuperficiem illa linea perpendicularis erit:</s> <s xml:id="echoid-s22073" xml:space="preserve"> & hoc eſt propoſitum.</s> <s xml:id="echoid-s22074" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div782" type="section" level="0" n="0"> <head xml:id="echoid-head649" xml:space="preserve" style="it">75. Si duæ ſphæræ cẽtra diuerſa habuerint: impoßibile eſt, ut lineæ քpẽdiculares ſuք unius ſu-<lb/>perficiẽ, ſint perpẽdiculares ſuper alterius ſuperficiẽ, niſi unatantũ, quæ trãſit cẽtra ambarum.</head> <p> <s xml:id="echoid-s22075" xml:space="preserve">Quocũq;</s> <s xml:id="echoid-s22076" xml:space="preserve"> modo ſe habẽtibus adinuicẽ ſphęris, ſiue extrinſecus ſiue intrinſecus ſe cõtingẽtibus, <lb/>uel etiam ſe nõ contingẽtibus, uel etiã ſe adinuicẽ ſecãtibus, ſemper patet ex 72 huius, quoniã linea <lb/>tranſiens per cẽtra ipſarũ, eſt perpẽdicularis ſuper ſuperficiẽ utriuſq;</s> <s xml:id="echoid-s22077" xml:space="preserve">;:</s> <s xml:id="echoid-s22078" xml:space="preserve"> aliã quoq;</s> <s xml:id="echoid-s22079" xml:space="preserve"> lineã ſuper utriuſq;</s> <s xml:id="echoid-s22080" xml:space="preserve"> <lb/>ſuperficiẽ perpendicularẽ eſſe, eſt impoſsibile.</s> <s xml:id="echoid-s22081" xml:space="preserve"> Si enim ſit poſsibile:</s> <s xml:id="echoid-s22082" xml:space="preserve"> ducatur aliqua alia perpẽdicu-<lb/>lariter ſuper utriuſq;</s> <s xml:id="echoid-s22083" xml:space="preserve"> ſphęræ ſuperficiẽ:</s> <s xml:id="echoid-s22084" xml:space="preserve"> palamq́;</s> <s xml:id="echoid-s22085" xml:space="preserve"> erit ex eadẽ 72 huius ipſam per utriuſq;</s> <s xml:id="echoid-s22086" xml:space="preserve"> centrũ trã-<lb/>ſire:</s> <s xml:id="echoid-s22087" xml:space="preserve"> quod eſt oppoſitũ hypotheſi.</s> <s xml:id="echoid-s22088" xml:space="preserve"> Patet ergo, quoniã nullã aliam lineã, præter eã, quę tranſit centra <lb/>ambarũ, perpẽdiculariter duci ſuք utriuſq;</s> <s xml:id="echoid-s22089" xml:space="preserve"> ſphęrarũ ſuperficies eſt poſsibile.</s> <s xml:id="echoid-s22090" xml:space="preserve"> Et hoc eſt propoſitũ.</s> <s xml:id="echoid-s22091" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div783" type="section" level="0" n="0"> <head xml:id="echoid-head650" xml:space="preserve" style="it">76. Si ſphæra ſphærã intrinſec<emph style="sub">9</emph> aut extrinſec<emph style="sub">9</emph> cõtingat: in uno tãtũ pũcto cõtingere eſt neceſſe.</head> <p> <s xml:id="echoid-s22092" xml:space="preserve">Si enim ſphęræ contingẽtes ſe intrinſecus, nõ in puncto ſe contingant:</s> <s xml:id="echoid-s22093" xml:space="preserve"> neceſſe eſt circulos ſuos <lb/>maiores a dinuicem applicatos non ſe in puncto contingere:</s> <s xml:id="echoid-s22094" xml:space="preserve"> quod eſt contra 13 p 3, & impoſsibile.</s> <s xml:id="echoid-s22095" xml:space="preserve"> <lb/>Quòd ſi ſphęræ extrinſecus ſe contingentes, non ſe contingant in puncto:</s> <s xml:id="echoid-s22096" xml:space="preserve"> etiam hoc eſt contra na-<lb/>turam circulorum extrinſecus ſe contingentium, & contra eandẽ 13 p 3.</s> <s xml:id="echoid-s22097" xml:space="preserve"> Poteſt & hoc aliter demon-<lb/>ſtrari.</s> <s xml:id="echoid-s22098" xml:space="preserve"> Si enim inter illas ſphęras, quę ſe extrinſecus contingunt, imaginata fuerit ſuperficies plana:</s> <s xml:id="echoid-s22099" xml:space="preserve"> <lb/>palàm ex 71 huius, quoniam utraq;</s> <s xml:id="echoid-s22100" xml:space="preserve"> illarum ſphęrarum illã ſuperficiem planam contingit in puncto.</s> <s xml:id="echoid-s22101" xml:space="preserve"> <lb/>Ergo & ſeinuicem in puncto contingẽt:</s> <s xml:id="echoid-s22102" xml:space="preserve"> & propinquior eſt utriq;</s> <s xml:id="echoid-s22103" xml:space="preserve"> ſphærarum ipſa plana ſuperficies <lb/>interpoſita, quàm ſphæræ inter ſe.</s> <s xml:id="echoid-s22104" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s22105" xml:space="preserve"/> </p> <pb o="31" file="0333" n="333" rhead="LIBER PRIMVS."/> </div> <div xml:id="echoid-div784" type="section" level="0" n="0"> <head xml:id="echoid-head651" xml:space="preserve" style="it">77. Sphærarum ſe contingentium, centra diuerſa eſſe eſt neceſſe.</head> <p> <s xml:id="echoid-s22106" xml:space="preserve">Signentur enim in utralibet ſphærarum à puncto contactus duo circuli maiores per 69 huius, <lb/>ſecantes eorum ſuperficiebus planis ſphæras per ſua centra, & per puncta contactuum.</s> <s xml:id="echoid-s22107" xml:space="preserve"> Et quia cen <lb/>tra horum circulorum ſunt centra ſphærarum ſuarum per definitionem circulorum magnorũ:</s> <s xml:id="echoid-s22108" xml:space="preserve"> hos <lb/>autem circulos centra diuerſa habere eſt concluſio 6 p 3.</s> <s xml:id="echoid-s22109" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s22110" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div785" type="section" level="0" n="0"> <head xml:id="echoid-head652" xml:space="preserve" style="it">78. Centrorum, ſphærarum ſe extrinſecus contingentium, diſtantiam ſecundum lineam com <lb/>poſitam ex ambarum ſphærarum ſemidiametris. intrinſecus uerò ſe contingentium, ſecundum <lb/>exceſſum ſemidiametri maioris ad ſemidiametrum minoris eſſe, palàm est.</head> <p> <s xml:id="echoid-s22111" xml:space="preserve">Hoc patet per 76 huius.</s> <s xml:id="echoid-s22112" xml:space="preserve"> Quoniam enim contactus ſphærarum fit ſecundum unum tantùm pun-<lb/>ctum:</s> <s xml:id="echoid-s22113" xml:space="preserve"> punctus uerò eſt, cui pars nõ eſt:</s> <s xml:id="echoid-s22114" xml:space="preserve"> tunc euidẽs eſt, quòd punctus ille cõmunis in utraq;</s> <s xml:id="echoid-s22115" xml:space="preserve"> interſe <lb/>ctione nihil adimit de diametrorum quantitate:</s> <s xml:id="echoid-s22116" xml:space="preserve"> indiuiſibile enim (cum non ſit pars quanti) nec ad-<lb/>dit nec minuit aliquid de quanto.</s> <s xml:id="echoid-s22117" xml:space="preserve"> Et ſic patet propoſitum.</s> <s xml:id="echoid-s22118" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div786" type="section" level="0" n="0"> <head xml:id="echoid-head653" xml:space="preserve" style="it">79. Si concauũ alicuius ſphæræ, ſuperficiem aliquam ſecundum eam totam contingat: neceſſe <lb/>eſt ſuperficiem contactam partem ſphæræ minoris eſſe.</head> <p> <s xml:id="echoid-s22119" xml:space="preserve">Sit, ut aliqua ſphæra ſecundũ ſuum concauũ contingat aliquã ſuperficiem ſecundũ oẽs illius par <lb/>tes, ſicut uas ſphæricũ ſuperficiem aquę contentę.</s> <s xml:id="echoid-s22120" xml:space="preserve"> Dico, quòd uerũ eſt quod proponitur.</s> <s xml:id="echoid-s22121" xml:space="preserve"> Ducantur <lb/>enim lineę plurimę à centro ſphærę ad locum contactus ſui cum illa ſuperficie.</s> <s xml:id="echoid-s22122" xml:space="preserve"> Et quia omnes lineę <lb/>productæ ad cõcauũ ſphærę ſunt æquales inter ſe ex definitione ſphæræ, & ſunt æquales productis <lb/>lineis ad conuexũ ſuperficiei cõtactę:</s> <s xml:id="echoid-s22123" xml:space="preserve"> patet ex dicta definitiõe, quoniã illa ſuքficies eſt pars ſphærę:</s> <s xml:id="echoid-s22124" xml:space="preserve"> <lb/>& quælibet intellecta exten di ſecundũ cõcauũ ambientis ſphærę, ſphærã minorẽ cõplebit.</s> <s xml:id="echoid-s22125" xml:space="preserve"> Eſt ergo <lb/>pars minoris ſphærę.</s> <s xml:id="echoid-s22126" xml:space="preserve"> Linea quoq;</s> <s xml:id="echoid-s22127" xml:space="preserve"> in illa ſuperficie ſignata, eſt pars circuli ex 9 p 3, idem habens cen <lb/>trum cum circulo, cui applicatur.</s> <s xml:id="echoid-s22128" xml:space="preserve"> Et ſic illa ſuperficies eſt pars minoris ſphærę.</s> <s xml:id="echoid-s22129" xml:space="preserve"> Quod eſt propoſitũ.</s> <s xml:id="echoid-s22130" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div787" type="section" level="0" n="0"> <head xml:id="echoid-head654" xml:space="preserve" style="it">80. Si ſphæra ſphæram interſecet, communis ſectio ſuperficierum ſphæricarum ſe interſecan-<lb/>tium erit peripheria circuli.</head> <p> <s xml:id="echoid-s22131" xml:space="preserve">Quod hic proponitur, patet.</s> <s xml:id="echoid-s22132" xml:space="preserve"> Imaginetur enim ſuperficies ſecans ambas ſphæras ſecundum lineã <lb/>cõmunẽ ſectionis ſphærarũ, qualiſcũq;</s> <s xml:id="echoid-s22133" xml:space="preserve"> fuerit.</s> <s xml:id="echoid-s22134" xml:space="preserve"> Hæc ergo ſuperficies propter ſimilitudinẽ corporũ ſe <lb/>interſecantiũ plana erit:</s> <s xml:id="echoid-s22135" xml:space="preserve"> cõmunis ergo ſectio illius ſuperficiei & utriuſq;</s> <s xml:id="echoid-s22136" xml:space="preserve"> ſphærarũ erit circulus per <lb/>69 huius.</s> <s xml:id="echoid-s22137" xml:space="preserve"> Palàm ergo, quòd cõmunis linea interſectionis ſuperficierũ ſphærarum illarum erit peri-<lb/>pheria circuli, in qua incluſa ſuperficies, erit circulus communis illi ſectioni:</s> <s xml:id="echoid-s22138" xml:space="preserve"> quoniam aliàs corpus, <lb/>quo utræq;</s> <s xml:id="echoid-s22139" xml:space="preserve"> ſphærę communicant, eſt corpus cõmune ſphærarum interſectioni:</s> <s xml:id="echoid-s22140" xml:space="preserve"> & eſt corpus irregu <lb/>lare, duabus ſcilicet ſuperficiebus ſphæricis contentum & diuerſis, ſecundum diſpoſitionẽ ſe inter-<lb/>ſecantium ſphærarum.</s> <s xml:id="echoid-s22141" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s22142" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div788" type="section" level="0" n="0"> <head xml:id="echoid-head655" xml:space="preserve" style="it">81. Sphærarum ſe interſecantium, maiores circulos ſe inuicem ſecare palàm est. Ex quo patet <lb/>interſecantium ſe ſphærarum centra diuerſa eſſe.</head> <p> <s xml:id="echoid-s22143" xml:space="preserve">Primum patet ex definitione ſphærarum ſe interſecantium.</s> <s xml:id="echoid-s22144" xml:space="preserve"> Quoniam enim interſecantibus ſe <lb/>ſphæris, diameter unius per alteram abſcinditur, & maiorum circulorũ diametri ſunt etiam diame-<lb/>tri ſuarum ſphærarum (diuidunt enim circuli magni ſuas ſphæras per æqualia) tunc patet, quòd cir-<lb/>culis unius ſphæræ & alterius ſe interſecantium aliqua linea eſt cõmunis.</s> <s xml:id="echoid-s22145" xml:space="preserve"> Cum ergo unus circulus <lb/>aliũ non cõtineat, quia nec una ſphæra ſphæram aliam continet:</s> <s xml:id="echoid-s22146" xml:space="preserve"> palàm, quia tales circuli ſe inuicem <lb/>ſecant ex definitione taliũ circulorũ.</s> <s xml:id="echoid-s22147" xml:space="preserve"> Quia uerò ex 5 p 3 circulorũ ſe inuicem ſecantiũ centra eſſe di <lb/>uerſa neceſſe eſt, & idem eſt centrũ ſphærę, quod eſt centrũ circuli magni in illa ſphæra:</s> <s xml:id="echoid-s22148" xml:space="preserve"> patet corol-<lb/>arium, ſcilicet, quia interſecantium ſe ſphærarum centra ſunt diuerſa.</s> <s xml:id="echoid-s22149" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s22150" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div789" type="section" level="0" n="0"> <head xml:id="echoid-head656" xml:space="preserve" style="it">82. Si ſphæra ſphæram interſecet: linea, quæ centra illarum ſphærarum tranſit, centrũ circuli <lb/>peripheriæ cõmunis ſectionis tranſire, & ſuper ipſius ſuperficiem perpendicularẽ eſſe, neceſſe eſt.</head> <p> <s xml:id="echoid-s22151" xml:space="preserve">Circulus cõmunis ſectiõis ſphærarũ aut eſt circulus maior alterius ſpherarũ ſe interſecantiũ, aut <lb/>minor:</s> <s xml:id="echoid-s22152" xml:space="preserve"> ſi maior:</s> <s xml:id="echoid-s22153" xml:space="preserve"> hoc erit ſolũ, cũ maior ſphæra minorẽ interſecat.</s> <s xml:id="echoid-s22154" xml:space="preserve"> Si enim æquales ſphærę ſecundũ <lb/>circulũ maiorẽ ſe interſecarẽt, nõ eſſet ſphærarũ interſectio, ſed unius ſphærę ex duobus hemiſphæ <lb/>rijs æqualibus cõpoſitio.</s> <s xml:id="echoid-s22155" xml:space="preserve"> Si ergo circulus cõis ſectionis ſphęrarũ ſit circulus maior, nõ erit ille circu <lb/>lus maior, niſi in ſphæris inæqualibus ſe interſecãtibus, circulus ſphærę minoris:</s> <s xml:id="echoid-s22156" xml:space="preserve"> quoniã ipſum eſſe <lb/>circulũ maiorẽ ſphærę maioris eſt impoſsibile:</s> <s xml:id="echoid-s22157" xml:space="preserve"> quoniã maior circulus ſphærę maioris nõ poteſt ca-<lb/>dere in ſuperficiẽ ſphęrę minoris.</s> <s xml:id="echoid-s22158" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s22159" xml:space="preserve"> circulus talis a b c:</s> <s xml:id="echoid-s22160" xml:space="preserve"> & ſit centrũ maioris ſphærę d:</s> <s xml:id="echoid-s22161" xml:space="preserve"> ſphærę <lb/>uerò minoris e:</s> <s xml:id="echoid-s22162" xml:space="preserve"> erit quoq;</s> <s xml:id="echoid-s22163" xml:space="preserve"> e centrũ circuli a b c ex hypotheſi.</s> <s xml:id="echoid-s22164" xml:space="preserve"> Ducatur ergo linea d e:</s> <s xml:id="echoid-s22165" xml:space="preserve"> & patebit pro-<lb/>poſitum primum.</s> <s xml:id="echoid-s22166" xml:space="preserve"> Item ducantur lineę d a, b d, d c, & lineę a e, b e, c e:</s> <s xml:id="echoid-s22167" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s22168" xml:space="preserve"> triangulorum d a e & <lb/>d b e latera æqualia:</s> <s xml:id="echoid-s22169" xml:space="preserve"> ideo, quoniam linea d e latus eſt commune, & latus d a æquale eſt lateri d b ex <lb/>definitione ſphærę:</s> <s xml:id="echoid-s22170" xml:space="preserve"> latus quoque a e ęquale eſt lateri b e ex definitione circuli:</s> <s xml:id="echoid-s22171" xml:space="preserve"> ergo per 8 p 1 anguli <lb/>ęquis lateribus contenti, erunt ęquales.</s> <s xml:id="echoid-s22172" xml:space="preserve"> Angulus ergo d e b ęqualis erit angulo d e a:</s> <s xml:id="echoid-s22173" xml:space="preserve"> ſimiliter an <lb/>gulus d e c erit ęqualis angulo d e b:</s> <s xml:id="echoid-s22174" xml:space="preserve"> & uniuerſaliter à quocunq;</s> <s xml:id="echoid-s22175" xml:space="preserve"> puncto circuli a b c ducantur lineę <lb/>ad e centrum ſphærę, anguli ſuper centrum e ſemper erunt æquales.</s> <s xml:id="echoid-s22176" xml:space="preserve"> Et quia ſuper eandem diame-<lb/>trum oppoſitis punctis ſignatis linea d e æquales angulos conſtituit:</s> <s xml:id="echoid-s22177" xml:space="preserve"> patet per definitionem per-<lb/>pendicularis, quoniam ipſa linea d e ſuper omnes diametros perpendicularis erit.</s> <s xml:id="echoid-s22178" xml:space="preserve"> Ergo per 4 p 11 <lb/>linea d e ſuper ſuperficiem circuli a b c erecta eſt, & ſupeream perpendicularis.</s> <s xml:id="echoid-s22179" xml:space="preserve"> Si uerò circu-<lb/> <pb o="32" file="0334" n="334" rhead="VITELLONIS OPTICAE"/> lus a b c non ſit circulus maior alicuius ſphærarũ ſe interſecantiũ, ſed minor:</s> <s xml:id="echoid-s22180" xml:space="preserve"> intelligatur in ipſo pro-<lb/>tracta diameter, quæ ſit l f per pũcta l & f, & utraq;</s> <s xml:id="echoid-s22181" xml:space="preserve"> ſphæra <lb/> <anchor type="figure" xlink:label="fig-0334-01a" xlink:href="fig-0334-01"/> rum imaginetur ſecta per ſuperficiem planam trans cen-<lb/>trũ, & ք puncta f & l, quę ſunt in ſuperficie utriuſq;</s> <s xml:id="echoid-s22182" xml:space="preserve"> ſphæ <lb/>rę.</s> <s xml:id="echoid-s22183" xml:space="preserve"> Erit ergo per præmiſſa quilibet illorum circulorum <lb/>circulus maior in utraq;</s> <s xml:id="echoid-s22184" xml:space="preserve"> ſphærarum ſe interſecantiũ, ſe-<lb/>cabitq́;</s> <s xml:id="echoid-s22185" xml:space="preserve"> circulum a b c uterq;</s> <s xml:id="echoid-s22186" xml:space="preserve"> illorum circulorũ maiorum <lb/>per æqualia:</s> <s xml:id="echoid-s22187" xml:space="preserve"> quoniam arcus f l eſt medietas circumferen <lb/>tię circuli a b c:</s> <s xml:id="echoid-s22188" xml:space="preserve"> tranſeunt ergo ambo illi circuli maiores <lb/>per centrũ illius circuli a b c, quod eſt e.</s> <s xml:id="echoid-s22189" xml:space="preserve"> Imaginẽtur item <lb/>duo circuli alιj maiores in eiſdem ſphæris, quorum quili-<lb/>bet ſecet portionẽ circuli maioris ſuę ſphærę erectã ſuper <lb/>circulum a b c per æqualia:</s> <s xml:id="echoid-s22190" xml:space="preserve"> quod fieri poterit ex 30 p 3, di-<lb/>uiſo arcu f l utriuſq;</s> <s xml:id="echoid-s22191" xml:space="preserve"> circuli ſphærarum ſe interſecantium <lb/>per ęqualia, & à puncto ſectionis utriuſq;</s> <s xml:id="echoid-s22192" xml:space="preserve"> circuli imagina <lb/>ta ſuperficie plana tranſeunte centrum ſphærę utriuſq;</s> <s xml:id="echoid-s22193" xml:space="preserve">. <lb/>Fiat itaq;</s> <s xml:id="echoid-s22194" xml:space="preserve"> ſectio arcus ſphęrę maioris in puncto g:</s> <s xml:id="echoid-s22195" xml:space="preserve"> & ſe-<lb/>ctio arcus ſphæræ minoris in puncto h:</s> <s xml:id="echoid-s22196" xml:space="preserve"> & ſiue hi cir-<lb/>culi maiores cum illis circulis, quos ſecãt, angulos æqua-<lb/>les ſphærales uel inæquales contineant, patet, cum à po-<lb/>lo circuli a b c per centra ſphærarum ambarum tranſeant, quoniam ambo ſecabunt circulum a b c <lb/>per æqualia.</s> <s xml:id="echoid-s22197" xml:space="preserve"> Tranſibunt ergo per centrum ipſius, quod eſt e.</s> <s xml:id="echoid-s22198" xml:space="preserve"> Linea ergo d g, quę per definitionem <lb/>maiorum circulorum, & 3 p 11 eſt communis ſectio duorum circulorum maiorũ in ſphęra maiori ſe <lb/>ſecantium, tranſit per centrum e:</s> <s xml:id="echoid-s22199" xml:space="preserve"> quoniã cum centrum e ſit in ſuperficie utriuſq;</s> <s xml:id="echoid-s22200" xml:space="preserve"> illorũ circulorum, <lb/>neceſſe eſt, ut ſit in linea cõmuni utriſq;</s> <s xml:id="echoid-s22201" xml:space="preserve">. Similiter etiã linea e h (quę eſt cõmunis ſectio circulorum <lb/>maiorũ in ſphæra minori ſe interſecantiũ) tranſit per centrũ e.</s> <s xml:id="echoid-s22202" xml:space="preserve"> Sed quia lineę e h, & lineę d g per defi <lb/>nitionem circulorũ ſe ſecantiũ, eſt aliqua linea recta cõmunis, ut e g, erit illa per 1 p 11 in eadẽ ſuperfi <lb/>cie cum illis:</s> <s xml:id="echoid-s22203" xml:space="preserve"> ergo erunt linea una.</s> <s xml:id="echoid-s22204" xml:space="preserve"> Tota ergo linea d e g h eſt linea una tranſiens per ambo centra <lb/>ſphærarum ſe interſecantiũ, & per centrum circuli, qui eſt cõmunis ſectio, cuius centrum eſt in peri <lb/>pheria cõmunis ſectionis ſuperficierum ſphęricarum ſe interſecantium.</s> <s xml:id="echoid-s22205" xml:space="preserve"> Patet ergo propoſitum pri <lb/>mum.</s> <s xml:id="echoid-s22206" xml:space="preserve"> Secundũ uerò patet ex pręmiſsis.</s> <s xml:id="echoid-s22207" xml:space="preserve"> Circuli enim maiores per ęqualia diuidentes circulum mi-<lb/>norem orthogonaliter eum ſecant, & eorum communis ſectio, ut linea d h per 19 p 11 ſuper eundem <lb/>circulum perpendicularis erit.</s> <s xml:id="echoid-s22208" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s22209" xml:space="preserve"> Poteſt & idem per 66 & 67 huius facilius de-<lb/>monſtrari diligentiam adhibenti.</s> <s xml:id="echoid-s22210" xml:space="preserve"/> </p> <div xml:id="echoid-div789" type="float" level="0" n="0"> <figure xlink:label="fig-0334-01" xlink:href="fig-0334-01a"> <variables xml:id="echoid-variables326" xml:space="preserve">l h g b e c k a d f</variables> </figure> </div> </div> <div xml:id="echoid-div791" type="section" level="0" n="0"> <head xml:id="echoid-head657" xml:space="preserve" style="it">83. Si ſphæra ſphærã interſecet: lineã tranſeuntẽ centrũ circuli peripheriæ cõmunis ſectionis <lb/>perpendiculariter ſuper ipſius ſuperficiẽ inſiſtentẽ, ambarũ ſphærarũ centra tranſire neceſſe eſt.</head> <p> <s xml:id="echoid-s22211" xml:space="preserve">Hęc eſt cõuerſa pręcedẽtis, nec oportet in ipſius demonſtratiõe aliter immorari.</s> <s xml:id="echoid-s22212" xml:space="preserve"> Si enim ſit poſsi-<lb/>bile, ducatur linea per e centrũ circuli cõmunis ſectiõis ſphęrarũ, (qui eſt a b c) perpendiculariter ſu <lb/>per ipſius ſuperficiẽ ad aliũ aliquẽ punctũ, pręter centum ambarũ, uel alterius ſphęrarũ:</s> <s xml:id="echoid-s22213" xml:space="preserve"> & ſit linea <lb/>e k:</s> <s xml:id="echoid-s22214" xml:space="preserve"> & ducatur item per centra ambarũ ſphęrarũ alia linea, quę ſit d h.</s> <s xml:id="echoid-s22215" xml:space="preserve"> Patet aũt per pręcedentẽ, quo-<lb/>niam hęc erit tranſiens per centrũ e, & erit perpendicularis ſuper ſuperficiẽ circuli a b c.</s> <s xml:id="echoid-s22216" xml:space="preserve"> Ab eodem <lb/>ergo pũcto ſuperficiei circuli a b c, utpote centro e, duę exeũt perpendiculares ſuper eandẽ circuli <lb/>ſuperficiem a b c, quę ſunt e d & e k:</s> <s xml:id="echoid-s22217" xml:space="preserve"> quod eſt contra 13 p 11, & impoſsibile.</s> <s xml:id="echoid-s22218" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s22219" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div792" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables327" xml:space="preserve">f c c l a</variables> </figure> <head xml:id="echoid-head658" xml:space="preserve" style="it">84. Si ſphæra ſphærã intrinſecus interſecet: neceſſe eſt centra illarũ ſphærarũ, reſpectu ſitus ſui <lb/>contactus ſecundum quantitatẽ peripheriæ circuli, qui eſt cõmu-<lb/> nis ſectio ſuarum ſuperficierũ plus diſtare: centrum ſphæræ con- tinentis plus profundari.</head> <p> <s xml:id="echoid-s22220" xml:space="preserve">Sphærę datę interſecare ſe debẽtes, ſi ęquales fuerint, & taliter ad <lb/>inuicẽ collocentur, ut nõ ſe interſecẽt:</s> <s xml:id="echoid-s22221" xml:space="preserve"> tunc ipſarũ idẽ erit centrũ:</s> <s xml:id="echoid-s22222" xml:space="preserve"> fa-<lb/>cta uerò interſectiõe ipſarũ, cẽtra diuerſantur per 81 huius:</s> <s xml:id="echoid-s22223" xml:space="preserve"> & ſecun-<lb/>dũ quod circuli քipheria, quę eſt cõmunis ſectio illarũ ſuperficierũ <lb/>ſphęricarũ, fit maior uel minor:</s> <s xml:id="echoid-s22224" xml:space="preserve"> ſecũdũ hoc plus uel minus diſtabũt <lb/>centra.</s> <s xml:id="echoid-s22225" xml:space="preserve"> Quòd ſi ſphęrę fuerint inęquales, quarũ una alterã intrinſe-<lb/>cus cõtingere poterit:</s> <s xml:id="echoid-s22226" xml:space="preserve"> tunc in ſitu ſuę cõtingentię centrorũ ſuorũ di <lb/>ſtantia ք 78 huius eſt exceſſus ſemidiametri ſphęrę maioris ad ſemi <lb/>diametrum minoris.</s> <s xml:id="echoid-s22227" xml:space="preserve"> Demus ergo, quòd centrum maioris ſit a, cen-<lb/>trũ minoris b, punctus cõtactus ſit c.</s> <s xml:id="echoid-s22228" xml:space="preserve"> Et quia cõtactus fit in puncto <lb/>per 76 huius, interſectio uerò fit ſecundũ circulũ per 80 huius:</s> <s xml:id="echoid-s22229" xml:space="preserve"> palã, <lb/>quia facta interſectione ſphærarum, abſcindet ſphęra a diametrũ b c <lb/>in puncto alio quàm in termino ſuo, qui eſt punctus c.</s> <s xml:id="echoid-s22230" xml:space="preserve"> Sit ergo pun-<lb/>ctus, in quo ipſum abſcindit, punctus e:</s> <s xml:id="echoid-s22231" xml:space="preserve"> ponaturq́;</s> <s xml:id="echoid-s22232" xml:space="preserve">, ut linea f e ſit æ-<lb/>qualis diametro ſphęrę b.</s> <s xml:id="echoid-s22233" xml:space="preserve"> Quoniam itaq;</s> <s xml:id="echoid-s22234" xml:space="preserve"> linea a c excedit lineam b <lb/>c in linea a b:</s> <s xml:id="echoid-s22235" xml:space="preserve"> linea uerò f e eſt ęqualis ſemidiametro b c:</s> <s xml:id="echoid-s22236" xml:space="preserve"> quoniam <lb/>ſunt ſemidiametri eiuſdem ſphęrę.</s> <s xml:id="echoid-s22237" xml:space="preserve"> Linea ergo a c excedit lineam <lb/> <pb o="33" file="0335" n="335" rhead="LIBER PRIMVS."/> f e in linea a b:</s> <s xml:id="echoid-s22238" xml:space="preserve"> ſed linea f e eſt maior quàm linea e c:</s> <s xml:id="echoid-s22239" xml:space="preserve"> ergo a e, in qua linea a c excedit lineam e c, eſt <lb/>maior quàm linea a b.</s> <s xml:id="echoid-s22240" xml:space="preserve"> Plus ergo diſtant centra ſphærarum in interſectione, quàm in ſitu contactus:</s> <s xml:id="echoid-s22241" xml:space="preserve"> <lb/>& ſecundum quòd peripheria circuli, quæ eſt communis ſectio ſuarum ſuperficierum, minoratu<gap/>, <lb/>ſecundum hoc diſtantia centrorum augetur:</s> <s xml:id="echoid-s22242" xml:space="preserve"> & ſecundum quòd illa peripheria augetur, ſecundum <lb/>hoc diſtantia centrorum minuitur:</s> <s xml:id="echoid-s22243" xml:space="preserve"> & reſpectu partis uniuerſi, ad quam fit interſectio, plus profun-<lb/>datur centrum ſphæræ continentis, reſpectu contactus, in tanto, quantò linea a e fit maior quàm li <lb/>nea a b.</s> <s xml:id="echoid-s22244" xml:space="preserve"> Et hoc eſt, quod proponebatur.</s> <s xml:id="echoid-s22245" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div793" type="section" level="0" n="0"> <head xml:id="echoid-head659" xml:space="preserve" style="it">85. Si duæ ſphæræ intra tertiam ſecundum circulũ æqualem circulo maiori ſphæræ, intra quã <lb/>fit interſectio, ſe interſecent: utra illarum ſphærarum ſphæram, intra quam fit interſectio, in-<lb/>terſecabit: et omniũ illarũ ſuperficierũ ſphæricarũ cõmunis ſectio erit peripheria circuli unius.</head> <p> <s xml:id="echoid-s22246" xml:space="preserve">Verbi gratia:</s> <s xml:id="echoid-s22247" xml:space="preserve"> ſit, ut ſphæra, cuius centrum a, interſecet ſphæram, cuius centrum ſit b, intra ſphæ-<lb/>ram, cuius centrum ſit c, ſecundum circulũ æqualẽ circulo maiori ſphę-<lb/> <anchor type="figure" xlink:label="fig-0335-01a" xlink:href="fig-0335-01"/> rę c.</s> <s xml:id="echoid-s22248" xml:space="preserve"> Dico, quòd ſphæra a & ſphæra b interſecabũt ſphæram c:</s> <s xml:id="echoid-s22249" xml:space="preserve"> & omniũ <lb/>ſuperficierum ſphæricarum illarum ſphærarum erit communis ſectio <lb/>peripheria circuli illius, ſecundum quẽ ſphærarum a & b fiebat interſe-<lb/>ctio, hoc eſt cuiuſdam circuli magni ſphæræ c.</s> <s xml:id="echoid-s22250" xml:space="preserve"> Quoniam enim circulus <lb/>maior diuidit ſphæram per æqualia, quia tranſit per centrũ eius ex defi-<lb/>nitione:</s> <s xml:id="echoid-s22251" xml:space="preserve"> tũc patet, quòd æqualis eidẽ, (undecunq;</s> <s xml:id="echoid-s22252" xml:space="preserve"> contingat eũ in ſphæ <lb/>ra produci) diuidet eã per æqualia:</s> <s xml:id="echoid-s22253" xml:space="preserve"> & ſic interſecabit ſecun dũ illum cir <lb/>culum utraq;</s> <s xml:id="echoid-s22254" xml:space="preserve"> ſphærarum, ſcilicet a & b ſphæram c.</s> <s xml:id="echoid-s22255" xml:space="preserve"> Sphæra autem a in-<lb/>terſecante ſphæram b, communis ſectio eſt peripheria circuli per 80 hu <lb/>ius:</s> <s xml:id="echoid-s22256" xml:space="preserve"> diuidit autem iſte circulus ſphæram c per æqualia:</s> <s xml:id="echoid-s22257" xml:space="preserve"> ergo interſecat.</s> <s xml:id="echoid-s22258" xml:space="preserve"> <lb/>Eſt ergo eius peripheria in ſuperficie ſphærę c:</s> <s xml:id="echoid-s22259" xml:space="preserve"> ſed & eadem peripheria <lb/>eſt in ſuperficiebus ſphærarum a & b.</s> <s xml:id="echoid-s22260" xml:space="preserve"> In omniũ ergo ſphærarum illarũ <lb/>triũ ſuperficieb.</s> <s xml:id="echoid-s22261" xml:space="preserve"> eſt illa circuli peripheria.</s> <s xml:id="echoid-s22262" xml:space="preserve"> Eſt ergo ipſa cõmunis ſectio <lb/>omnium ſuperficierum dictarum ſphærarum.</s> <s xml:id="echoid-s22263" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s22264" xml:space="preserve"/> </p> <div xml:id="echoid-div793" type="float" level="0" n="0"> <figure xlink:label="fig-0335-01" xlink:href="fig-0335-01a"> <variables xml:id="echoid-variables328" xml:space="preserve">b c a</variables> </figure> </div> </div> <div xml:id="echoid-div795" type="section" level="0" n="0"> <head xml:id="echoid-head660" xml:space="preserve" style="it">86. Lineam à centro ſphæræ per centrum circuli ſphæram ſecantis, orthogonaliter ductam<gap/> <lb/>medio abſciſſæ portionis eſt neceſſarium applicari.</head> <p> <s xml:id="echoid-s22265" xml:space="preserve">Sit ſphæra, cuius centrum a, & ſit circulus b c d, cuius centrum ſit <lb/> <anchor type="figure" xlink:label="fig-0335-02a" xlink:href="fig-0335-02"/> e, abſcindens portionem ſphærę:</s> <s xml:id="echoid-s22266" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s22267" xml:space="preserve"> linea a e, & producatur <lb/>uſq;</s> <s xml:id="echoid-s22268" xml:space="preserve"> ad ſuperficiem ſphæricam, cui incidat in puncto f.</s> <s xml:id="echoid-s22269" xml:space="preserve"> Dico, quòd li <lb/>nea a e neceſſariò applicatur puncto, qui eſt medium abſciſſę portio <lb/>nis ſphærę in conuexo uel concauo ipſius:</s> <s xml:id="echoid-s22270" xml:space="preserve"> & quòd hoc eſt punctum <lb/>f.</s> <s xml:id="echoid-s22271" xml:space="preserve"> Ducantur enim lineæ a b, a c & a d, & copulentur lineę e b, e c, e d:</s> <s xml:id="echoid-s22272" xml:space="preserve"> <lb/>erunt itaq;</s> <s xml:id="echoid-s22273" xml:space="preserve"> trigona a e b, a e c & a e d omnia ſecundum latera ęquales <lb/>angulos reſpiciẽtia adinuicem proportionalia:</s> <s xml:id="echoid-s22274" xml:space="preserve"> quoniam illa ipſorũ <lb/>latera ſunt adinuicẽ æqualia, ut patet per ſphęrę & circuli definitio-<lb/>nes, & quia latus a e eſt omnibus commune:</s> <s xml:id="echoid-s22275" xml:space="preserve"> anguli itaq;</s> <s xml:id="echoid-s22276" xml:space="preserve"> b a e, c a e, d <lb/>a e omnes ſunt æquales per 5 p 6:</s> <s xml:id="echoid-s22277" xml:space="preserve"> ergo per 26 p 3 arcus b f, c f, d f ſunt <lb/>æquales.</s> <s xml:id="echoid-s22278" xml:space="preserve"> Et quoniam productis quibuslibet lineis à centro ſphæræ <lb/>a ad peripheriam circuli b c d, idem ſemper accidit:</s> <s xml:id="echoid-s22279" xml:space="preserve"> palàm, quia pun <lb/>ctus f eſt in medio portiõis abſciſſę de ſphęra.</s> <s xml:id="echoid-s22280" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s22281" xml:space="preserve"/> </p> <div xml:id="echoid-div795" type="float" level="0" n="0"> <figure xlink:label="fig-0335-02" xlink:href="fig-0335-02a"> <variables xml:id="echoid-variables329" xml:space="preserve">c f b e d a</variables> </figure> </div> </div> <div xml:id="echoid-div797" type="section" level="0" n="0"> <head xml:id="echoid-head661" xml:space="preserve" style="it">87. Proportionem partis ſuperficiei ſphæricæ ad totalem ſuperficiem ſuæ ſphæræ, ſicut anguli <lb/>ſolidi in ipſam à centro ſphæræ cadentis, ad octo rectos ſolidos neceſſe eſt eſſe. È<unsure/> Nicolao Caba-<lb/>ſilla in 3 librum magnæ conſtructionis Ptolemæi.</head> <figure> <variables xml:id="echoid-variables330" xml:space="preserve">a b d c</variables> </figure> <p> <s xml:id="echoid-s22282" xml:space="preserve">Verbi gratia:</s> <s xml:id="echoid-s22283" xml:space="preserve"> ſit a b c pars ſuperficiei ſphæricę ali-<lb/>cuius ſphærę, cuius centum ſit d:</s> <s xml:id="echoid-s22284" xml:space="preserve"> & ducantur lineæ <lb/>a d, b d, c d:</s> <s xml:id="echoid-s22285" xml:space="preserve"> & in ipſa ſuperficie ducantur lineæ a b, b <lb/>c, a c:</s> <s xml:id="echoid-s22286" xml:space="preserve"> fietq́;</s> <s xml:id="echoid-s22287" xml:space="preserve"> pyramis, cuius uertex eſt punctum d, & <lb/>baſis a b c.</s> <s xml:id="echoid-s22288" xml:space="preserve"> Palàm quoq;</s> <s xml:id="echoid-s22289" xml:space="preserve">, quoniã angulus circa pun-<lb/>ctum d eſt ſolidus, tribus angulis ſuperficialibus cõ-<lb/>tentus.</s> <s xml:id="echoid-s22290" xml:space="preserve"> Dico, quòd quę eſt proportio illius anguli <lb/>ad 8 rectos angulos ſolidos, qui replent locum ſoli-<lb/>dum circa centrum d, eadem erit proportio ſuperfi-<lb/>ciei ſphæricæ, quæ eſt a b c, ad totam ſphæricam ſu-<lb/>perficiem ſuę ſphæræ.</s> <s xml:id="echoid-s22291" xml:space="preserve"> Imaginentur enim plurimi <lb/>circuli magni, tranſeuntes per omnia puncta illius <lb/>ſuperficiei, non ſecantes ſe ſuper illam.</s> <s xml:id="echoid-s22292" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s22293" xml:space="preserve">, <lb/>quoniã aliqui arcus illorum circulorũ determinãtur <lb/>per lineas terminales illius ſuperficiei:</s> <s xml:id="echoid-s22294" xml:space="preserve"> omniũ aũt il-<lb/>lorũ arcuũ partialiũ ad totos ſuos circulos eſt ꝓpor <lb/> <pb o="34" file="0336" n="336" rhead="VITELLONIS OPTICAE"/> tio, ſicut angulorum contentorum ſub lineis à centro d ad ipſorum terminos productis ad 4 rectos <lb/>ſuperficiales per 33 p 6.</s> <s xml:id="echoid-s22295" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s22296" xml:space="preserve"> Et etiam poteſt patere ex hoc, quoniam ſicut ille an-<lb/>gulus correſpõdet illi parti ſuperficiei ſphæricæ:</s> <s xml:id="echoid-s22297" xml:space="preserve"> ſic reſiduum 8 ſolidorum angulorũ rectorũ totali <lb/>reſiduo ſuperficiei illius ſphæræ reſpondet:</s> <s xml:id="echoid-s22298" xml:space="preserve"> ergo per 16 p 5 erit permutatim anguli ad angulum, ſi-<lb/>cut ſuperficiei ad ſuperficiem, & per 18 p 5 coniunctim, & per 5 huius è contrario patet propoſitũ.</s> <s xml:id="echoid-s22299" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div798" type="section" level="0" n="0"> <head xml:id="echoid-head662" xml:space="preserve" style="it">88. Si inter duas quartas circulorũ æqualium in ſphæræ ſuperficie ſe ſecantium, ad extremi-<lb/>tates arcuum æqualium lineæ rectæ ducantur: illæ erũt æquidiſtantes: & remotior à puncto ſe-<lb/>ctionis erit longior. È<unsure/> 14 p 12 ele. in Campano.</head> <p> <s xml:id="echoid-s22300" xml:space="preserve">Sint arcus magnorum circulorũ in ſuperficie alicuius ſphæræ ſe ſecantiũ, qui a b c & a d e, ſecan-<lb/>tes ſe in puncto a:</s> <s xml:id="echoid-s22301" xml:space="preserve"> in quibus ſignentur arcus æquales, ita, ut arcus a b ſit æqualis arcui a d, & arcus b <lb/>c ſit æqualis arcui d e, & cõtinuentur lineæ rectę, quę <lb/> <anchor type="figure" xlink:label="fig-0336-01a" xlink:href="fig-0336-01"/> ſint b d & c e.</s> <s xml:id="echoid-s22302" xml:space="preserve"> Dico, quòd lineæ c e & b d ſunt æquidi-<lb/>ſtantes:</s> <s xml:id="echoid-s22303" xml:space="preserve"> & quòd linea c e eſt maior ꝗ̃ linea b d.</s> <s xml:id="echoid-s22304" xml:space="preserve"> Quia <lb/>itaq;</s> <s xml:id="echoid-s22305" xml:space="preserve"> arcus a b eſt æqualis arcui a d:</s> <s xml:id="echoid-s22306" xml:space="preserve"> palàm ք 29 p 3 & <lb/>per 65 huius, quoniã punctus a eſt polus circuli trãſ-<lb/>euntis per pũcta d & b:</s> <s xml:id="echoid-s22307" xml:space="preserve"> ideo quòd rectę lineę, quę a d <lb/>& a b, ſunt æquales:</s> <s xml:id="echoid-s22308" xml:space="preserve"> & ſimiliter eſt de circulo trãſeũte <lb/>per pũcta c & e.</s> <s xml:id="echoid-s22309" xml:space="preserve"> Circũducatur ergo ſuperficiei ſphęrę <lb/>per puncta d, b circulus erectus ſuper diametrũ ſphæ <lb/>rę per 69 huius, & ſimiliter per puncta e & c.</s> <s xml:id="echoid-s22310" xml:space="preserve"> Erũt er-<lb/>go illi circuli æquidiſtãtes per 14 p 11.</s> <s xml:id="echoid-s22311" xml:space="preserve"> Erunt ergo li-<lb/>neæ c e & b d æquidiſtantes per 16 p 11, imaginata ſu-<lb/>perficie plana, in qua ſunt puncta b, c, d, e, circulos ſe-<lb/>cundum illas lineas ſecãte.</s> <s xml:id="echoid-s22312" xml:space="preserve"> Sed & linea c e eſt maior <lb/>quàm linea d b.</s> <s xml:id="echoid-s22313" xml:space="preserve"> Si enim ſit æqualis, cũ ſit æquidiſtãs:</s> <s xml:id="echoid-s22314" xml:space="preserve"> <lb/>palàm, quia circuli a b c & a e d æquidiſtantes erunt:</s> <s xml:id="echoid-s22315" xml:space="preserve"> <lb/>quod eſt cõtra hypotheſim:</s> <s xml:id="echoid-s22316" xml:space="preserve"> ſupponũtur enim ſe ſeca <lb/>re in puncto a:</s> <s xml:id="echoid-s22317" xml:space="preserve"> aut ſequetur circulum tranſeuntẽ per <lb/>puncta b & d æqualem fieri circulo tranſeunti per puncta c & d, quorum circulorum polus eſt pun <lb/>ctum a:</s> <s xml:id="echoid-s22318" xml:space="preserve"> quod iterum eſt impoſsibile.</s> <s xml:id="echoid-s22319" xml:space="preserve"> Et ſi linea c e ſit minor quàm linea b d, concurrent circuli a b c <lb/>& a d e ultra lineam c e potius quàm ultra lineam b d.</s> <s xml:id="echoid-s22320" xml:space="preserve"> Eſt ergo linea b d remotior à puncto ſectiõis.</s> <s xml:id="echoid-s22321" xml:space="preserve"> <lb/>Quod eſt propoſitum hypotheſis.</s> <s xml:id="echoid-s22322" xml:space="preserve"/> </p> <div xml:id="echoid-div798" type="float" level="0" n="0"> <figure xlink:label="fig-0336-01" xlink:href="fig-0336-01a"> <variables xml:id="echoid-variables331" xml:space="preserve">a b d c e</variables> </figure> </div> </div> <div xml:id="echoid-div800" type="section" level="0" n="0"> <head xml:id="echoid-head663" xml:space="preserve" style="it">89. Omnes lineæ longitudinis unius pyramidis rotundæ, ſunt æquales: & cum ſemidiametris <lb/>baſis æquales, ſed acutos angulos continentes. Ex quo patet omnem pũctum uerticis pyramidis <lb/>eſſe polum circuli ſuæ b a ſis: omneḿ lineam longitudinis eſſe in eadẽ ſuperficie cum axe: ipſum <lb/>quo axem centrum circuli baſis orthogonaliter attingere. È<unsure/> 18 defin. 11 element.</head> <p> <s xml:id="echoid-s22323" xml:space="preserve">Quoniã enim per principium 11 Euclidis pyramis rotunda fit per trãſitum trianguli rectanguli, <lb/>alterutro ſuorum laterum rectum angulum continentiũ fixo, donec <lb/> <anchor type="figure" xlink:label="fig-0336-02a" xlink:href="fig-0336-02"/> ad locum ſuum, unde in cœpit, redeat, triangulo ipſo circumducto:</s> <s xml:id="echoid-s22324" xml:space="preserve"> <lb/>qui triangulus, ſi fuerit duorum laterum æqualium:</s> <s xml:id="echoid-s22325" xml:space="preserve"> & unum laterũ <lb/>æqualium rectum angulum continentium fuerit fixum, cauſſabitur <lb/>pyramis rectãgula:</s> <s xml:id="echoid-s22326" xml:space="preserve"> ideo, quòd angulus duplicati ſui trianguli ad uer <lb/>ticem pyramidis eſt rectus per 5 & 32 p 1.</s> <s xml:id="echoid-s22327" xml:space="preserve"> Et ſi fixũ latus fuerit minus <lb/>latere moto, erit pyramis amblygonia:</s> <s xml:id="echoid-s22328" xml:space="preserve"> quoniã per 19 p 1 angulus ad <lb/>uerticem fit obtuſus.</s> <s xml:id="echoid-s22329" xml:space="preserve"> Et ſi latus fixum fuerit maius latere moto, erit <lb/>pyramis oxygonia:</s> <s xml:id="echoid-s22330" xml:space="preserve"> quia per eandem 19 p 1 angulus eius ad uerticem <lb/>remanet acutus, adiuuãte ſemper 32 p 1.</s> <s xml:id="echoid-s22331" xml:space="preserve"> Sic ergo diuerſantur formæ <lb/>pyramidum ſecundum diuerſitatem proportionis lateris fixi ad alte <lb/>rum latus motum rectum angulum cõtinens cum fixo.</s> <s xml:id="echoid-s22332" xml:space="preserve"> Et quia latus <lb/>ſubtenſum angulo recto, cauſſat omnes lineas longitudinis in quali <lb/>bet pyramide:</s> <s xml:id="echoid-s22333" xml:space="preserve"> palàm, quòd omnes lineæ longitudinis totius rotun-<lb/>dæ pyramidis uni lineæ ſunt æquales ei, ſcilicet q̃ in trigono rectan-<lb/>gulo opponitur angulo recto.</s> <s xml:id="echoid-s22334" xml:space="preserve"> Ergo & oẽs inter ſe ſunt æquales.</s> <s xml:id="echoid-s22335" xml:space="preserve"> Si <lb/>ergo trigonũ orthogoniũ cauſſans pyramidẽ, ſit a b c, cuius angulus <lb/>a b c ſit rectus:</s> <s xml:id="echoid-s22336" xml:space="preserve"> erit per 32 p 1 angulus a c b acutus:</s> <s xml:id="echoid-s22337" xml:space="preserve"> & eſt a c b angulus, <lb/>cui omnes anguli cõtenti à lineis lõgitudinis & ſemidiametris baſis, <lb/>ſunt æquales:</s> <s xml:id="echoid-s22338" xml:space="preserve"> & hoc proponebatur.</s> <s xml:id="echoid-s22339" xml:space="preserve"> Patet etiã ex ijs, quoniã punctus <lb/>uerticis pyramidis cuiuslibet eſt polus circuli ſuę baſis per 65 huius.</s> <s xml:id="echoid-s22340" xml:space="preserve"> Et quoniã linea a c eſt in eadẽ <lb/>ſuperficie trigona cum linea a b, patet, quoniam omnes lineæ longitudinis ſuntin eadem ſuperficie <lb/>cum axe a b.</s> <s xml:id="echoid-s22341" xml:space="preserve"> Et quoniam linea b c motu ſuo deſcribit circulum baſis, patet, quòd axis a b centrum <lb/>circuli baſis orthogonaliter attingit per 8 p 1:</s> <s xml:id="echoid-s22342" xml:space="preserve"> quia ex circuli definitione & prima parte præſen-<lb/> <pb o="35" file="0337" n="337" rhead="LIBER PRIMVS."/> tis, axe exiſtente cõmuni, omnes anguli ad centrum b cõſtituti ſunt æquales.</s> <s xml:id="echoid-s22343" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s22344" xml:space="preserve"/> </p> <div xml:id="echoid-div800" type="float" level="0" n="0"> <figure xlink:label="fig-0336-02" xlink:href="fig-0336-02a"> <variables xml:id="echoid-variables332" xml:space="preserve">a d b c</variables> </figure> </div> </div> <div xml:id="echoid-div802" type="section" level="0" n="0"> <head xml:id="echoid-head664" xml:space="preserve" style="it">90. Omnis ſuperficiei planæ ſecantis pyramidem rotundam uel lateratam ſecundum a-<lb/>xis longitudinem & ſuperficiei conicæ communis ſectio eſt trigonum duab{us} lineis longitudi-<lb/>nis pyramid{ιs} & diametro baſis contentũ. Ex quo patet, quoniam illa ſuperficies dιuidit pyra-<lb/>midem per æqualia: & quòd ſuperficies, quæpyramidem ſecundum lineam longitudinis per æ-<lb/>qualia ſecuerit, ſecundum axem neceſſariò ſecabit. È<unsure/> 18 defin. 11 element. item 3. theor. 1 Co-<lb/>nicorum Apollonij.</head> <p> <s xml:id="echoid-s22345" xml:space="preserve">Eſto pyramis rotunda a b c, cuius uertex a:</s> <s xml:id="echoid-s22346" xml:space="preserve"> & diameter baſis b c:</s> <s xml:id="echoid-s22347" xml:space="preserve"> & ſit centrum baſis d.</s> <s xml:id="echoid-s22348" xml:space="preserve"> Et palàm <lb/>per pręmiſſam, quoniã linea a d eſt axis illius pyramidis.</s> <s xml:id="echoid-s22349" xml:space="preserve"> Superficies <lb/> <anchor type="figure" xlink:label="fig-0337-01a" xlink:href="fig-0337-01"/> itaq;</s> <s xml:id="echoid-s22350" xml:space="preserve"> plana ſecans pyramidem rotundam ſecundum axis longitudi-<lb/>nem, pertranſit puncta a & d:</s> <s xml:id="echoid-s22351" xml:space="preserve"> erit itaq;</s> <s xml:id="echoid-s22352" xml:space="preserve"> illa ſuperficies plana orthogo-<lb/>naliter erecta ſuper baſim pyramidis per 18 p 11.</s> <s xml:id="echoid-s22353" xml:space="preserve"> Communis itaq;</s> <s xml:id="echoid-s22354" xml:space="preserve"> ſe-<lb/>ctio baſis pyramidis & illιus ſuperficiei planę eſt linea recta per 3 p 11, <lb/>quæ eſt diameter baſis:</s> <s xml:id="echoid-s22355" xml:space="preserve"> & ſit hæc b c.</s> <s xml:id="echoid-s22356" xml:space="preserve"> Trigonũ itaq;</s> <s xml:id="echoid-s22357" xml:space="preserve"> a b c eſt in ſuper-<lb/>ficie ſecante:</s> <s xml:id="echoid-s22358" xml:space="preserve"> ſed & idem trigonum eſt in ſuperficie conica pyr mi-<lb/>dis.</s> <s xml:id="echoid-s22359" xml:space="preserve"> Et quoniam trigonum orthogonium b a d eſt illud, ex cuius per-<lb/>tranſitu deſcribιtur pyramis a b c, & trigonum a b c eſt duplum illi <lb/>per 1 p 6, patet illud, quod primò proponitur de pyramide rotunda.</s> <s xml:id="echoid-s22360" xml:space="preserve"> <lb/>Patet etiam, quòd illa ſuperficies taliter pyramidem ſecans, diuidit <lb/>ipſam per æqualia:</s> <s xml:id="echoid-s22361" xml:space="preserve"> quoniam tranſiens uerticem & concluſa diame-<lb/>tro, per æqualia diuidit & baſim.</s> <s xml:id="echoid-s22362" xml:space="preserve"> In laterata uerò pyramide, aut ſu-<lb/>perficies plana ſecans tranſit latus aut angulum:</s> <s xml:id="echoid-s22363" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s22364" xml:space="preserve"> productis li-<lb/>neis ad terminum axis pyramidis, illa communis ſectio ſemper trigo <lb/>nus maior uel minor.</s> <s xml:id="echoid-s22365" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s22366" xml:space="preserve"> quoniam & conuerſa <lb/>per ſe & ex præmιſsis patet.</s> <s xml:id="echoid-s22367" xml:space="preserve"/> </p> <div xml:id="echoid-div802" type="float" level="0" n="0"> <figure xlink:label="fig-0337-01" xlink:href="fig-0337-01a"> <variables xml:id="echoid-variables333" xml:space="preserve">a b d c</variables> </figure> </div> </div> <div xml:id="echoid-div804" type="section" level="0" n="0"> <head xml:id="echoid-head665" xml:space="preserve" style="it">91. Omnis pyramidis rotundæ uel lateratæ lineæ lõgitudinis ſu <lb/>per axem in uertice tantùm ſe interſecant: productæ quo aliam <lb/>ſimilem pyramidem principiant, cui{us} lineæ longitudinis ſecun-<lb/>dum poſitionem & ſitum priori pyramidi modo contrario ſe habent. È<unsure/> 18 defin. 11 elemen. item <lb/>1 defin. 1 Conicorum Apollonij.</head> <p> <s xml:id="echoid-s22368" xml:space="preserve">Quòd omnes lineę longitudinis pyramidis cuiuſcunq;</s> <s xml:id="echoid-s22369" xml:space="preserve"> prod ctę ſe ſuper axem in uertice ſecent, <lb/>euidens eſt:</s> <s xml:id="echoid-s22370" xml:space="preserve"> quonιam concurrunt omnes in illo puncto uerticis.</s> <s xml:id="echoid-s22371" xml:space="preserve"> Et quonιam omnes ſunt æquales <lb/>per 89 huius:</s> <s xml:id="echoid-s22372" xml:space="preserve"> patet, quia citra uerticem nulla ipſarum aliam interſe-<lb/> <anchor type="figure" xlink:label="fig-0337-02a" xlink:href="fig-0337-02"/> cat.</s> <s xml:id="echoid-s22373" xml:space="preserve"> Quòd etiam product æ aliam pyramιdem priori ſimilem princi-<lb/>pient, patet.</s> <s xml:id="echoid-s22374" xml:space="preserve"> Secet enιm ſuperficies plana pyramidem ſecundũ axis <lb/>longitudinem:</s> <s xml:id="echoid-s22375" xml:space="preserve"> erit ergo per præcedentem communis ſectio iſtius <lb/>ſuperficiei & ſuperficiei conicę pyramidis, trigonum æquum duplo <lb/>trigoni rectanguli pyramidem cauſſantis:</s> <s xml:id="echoid-s22376" xml:space="preserve"> ſed palàm per 36 huius, <lb/>quòd latera cuiuslibet trigoni producta principiant alium trigonũ <lb/>priori ſimile, cuius latera poſitionem & ſitum prioris trigoni lateri-<lb/>bus contrarium habent.</s> <s xml:id="echoid-s22377" xml:space="preserve"> Et quoniam tot poſſunt imaginari planæ ſu <lb/>perficies trans axem pyramidem ſecantes, quot ſunt lineæ longitu-<lb/>dinis imaginabiles in medietate pyramidis, pater, quoniam omnes <lb/>lineæ longitudinis productæ, principiant aliam pyramidem priori <lb/>ſimilem, lineis longitudinis à dextro prioris prodeuntibus in ſini-<lb/>ſtrum poſterioris, & à ſiniſtro prioris in dextrũ poſterioris, & è con-<lb/>uerſo.</s> <s xml:id="echoid-s22378" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s22379" xml:space="preserve"/> </p> <div xml:id="echoid-div804" type="float" level="0" n="0"> <figure xlink:label="fig-0337-02" xlink:href="fig-0337-02a"> <variables xml:id="echoid-variables334" xml:space="preserve">d e a b c</variables> </figure> </div> </div> <div xml:id="echoid-div806" type="section" level="0" n="0"> <head xml:id="echoid-head666" xml:space="preserve" style="it">92. Omnes lineæ longitudinis uni{us} columnæ rotundæ ſunt æ-<lb/>quales, rectos angulos cum ſemidiametris ſuarum baſium conti-<lb/>nentes, & in eadem ſuperficie cum axe exiſtentes. Ex quo patet, <lb/>quoniam axis cui{us}lιbet columnæ rotundæ centris ſuaru baſium <lb/>orthogonaliter inſiſtit. È<unsure/> 21 defin. 11 element.</head> <p> <s xml:id="echoid-s22380" xml:space="preserve">Hoc non indiget demonſtratione alia, niſi ſimili illi, quæ fit in 89 huius.</s> <s xml:id="echoid-s22381" xml:space="preserve"> Sicut enim trigonum <lb/>orthogonium altero laterum rectum angulum continentium fixo, per reuolutionem ſuam cauſ-<lb/>ſat pyramidem rotundum:</s> <s xml:id="echoid-s22382" xml:space="preserve"> ſic quadrilaterum rectangulum altero ſuorum laterum fixo manente, <lb/>alijs tribus, quouſque ad locum ſuum redeant, circumductis, cauſſat motu ſuo figuram colu-<lb/>mnarem rotundam.</s> <s xml:id="echoid-s22383" xml:space="preserve"> fiet ergo probatio omnium eorum, quæ proponunttur hîc, ut in illa:</s> <s xml:id="echoid-s22384" xml:space="preserve"> quia pa-<lb/>tet totum euidenter.</s> <s xml:id="echoid-s22385" xml:space="preserve"/> </p> <pb o="36" file="0338" n="338" rhead="VITELLONIS OPTICAE"/> </div> <div xml:id="echoid-div807" type="section" level="0" n="0"> <head xml:id="echoid-head667" xml:space="preserve" style="it">93. Omnis ſuperficiei planæ ſecantis columnam rotundam ſecundum axis longitudinem & <lb/>ſuperficiei columnæ communis ſectio eſt rectangulum ſub duab{us} lineis longitudinis columnæ, <lb/>& duab{us} diametris baſium contentum. Ex quo patet, quoniam illa ſuperficies per æqualia diui <lb/>dit columnam. È<unsure/> 21 defin. 11. element.</head> <p> <s xml:id="echoid-s22386" xml:space="preserve">Columna rotunda ſit, cuius axis e f:</s> <s xml:id="echoid-s22387" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s22388" xml:space="preserve"> ipſam per e f ſuperfi-<lb/> <anchor type="figure" xlink:label="fig-0338-01a" xlink:href="fig-0338-01"/> cies plana, ſitq́;</s> <s xml:id="echoid-s22389" xml:space="preserve"> communis ſectio ſecundum puncta a, b, c, d.</s> <s xml:id="echoid-s22390" xml:space="preserve"> Dico, <lb/>quòd ſectio a b c d eſt quadrangula rectangula ſub lineis longitudi-<lb/>nis columnæ, & duabus diametris baſium contenta.</s> <s xml:id="echoid-s22391" xml:space="preserve"> Ducatur enim <lb/>linea e a in baſi columnæ & in ſuperficie ſecante:</s> <s xml:id="echoid-s22392" xml:space="preserve"> hæc eſt ergo ſemi-<lb/>diameter circuli baſis columnæ.</s> <s xml:id="echoid-s22393" xml:space="preserve"> Producatur itaq;</s> <s xml:id="echoid-s22394" xml:space="preserve"> taliter, ut linea e g <lb/>compleat diametrum baſis columnæ, cadetq́;</s> <s xml:id="echoid-s22395" xml:space="preserve"> linea e g in ſuperficie <lb/>plana columnam ſecante.</s> <s xml:id="echoid-s22396" xml:space="preserve"> Si enim linea e g nõ eſt ducta in ſuperficie <lb/>plana columnam ſecante:</s> <s xml:id="echoid-s22397" xml:space="preserve"> ducatur linea b e in illa ſuperficie ſecante.</s> <s xml:id="echoid-s22398" xml:space="preserve"> <lb/>Lineæ ergo b e & e a ſunt linea una:</s> <s xml:id="echoid-s22399" xml:space="preserve"> quoniam ſunt in una ſuperficie <lb/>productæ ambæ orthogonaliter ſuper axem e f cõtinuè:</s> <s xml:id="echoid-s22400" xml:space="preserve"> ſimiliterq́;</s> <s xml:id="echoid-s22401" xml:space="preserve"> <lb/>quia linea e g complet diametrum a e, non in ſuperficie ſecante, ſed <lb/>alia:</s> <s xml:id="echoid-s22402" xml:space="preserve"> erit ergo lineæ a g pars in plano, pars in ſublimi:</s> <s xml:id="echoid-s22403" xml:space="preserve"> quod eſt con-<lb/>tra 1 p 11.</s> <s xml:id="echoid-s22404" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s22405" xml:space="preserve">, quoniam linea a b eſt diameter baſis, & quòd <lb/>punctus g cadit ſuper punctum b.</s> <s xml:id="echoid-s22406" xml:space="preserve"> Similiterq́;</s> <s xml:id="echoid-s22407" xml:space="preserve"> declarandum de linea <lb/>c d, quoniam eſt diameter alterius baſis.</s> <s xml:id="echoid-s22408" xml:space="preserve"> Lineæ quoq;</s> <s xml:id="echoid-s22409" xml:space="preserve"> a c & b d ſunt <lb/>lineæ longitudinis columnæ.</s> <s xml:id="echoid-s22410" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s22411" xml:space="preserve"> Ex hoc itaq;</s> <s xml:id="echoid-s22412" xml:space="preserve"> pa <lb/>tet, quoniã cum illa ſectio diuidat per æqualia baſes columnæ, quòd <lb/>etiam diuidit per æqualia columnam.</s> <s xml:id="echoid-s22413" xml:space="preserve"/> </p> <div xml:id="echoid-div807" type="float" level="0" n="0"> <figure xlink:label="fig-0338-01" xlink:href="fig-0338-01a"> <variables xml:id="echoid-variables335" xml:space="preserve">g a m e n b h i c p f o d k l</variables> </figure> </div> </div> <div xml:id="echoid-div809" type="section" level="0" n="0"> <head xml:id="echoid-head668" xml:space="preserve" style="it">94. Superficiei ſecantis columnam rotundam æquidistanter ſuperficiei per axem ſecanti & <lb/>ſuperficiei columnaris, cõmunis ſectio eſt rectangulum ſub duab{us} lineis longitudinis columnæ, <lb/>& duab{us} lineis minorib{us} diametris baſium contentum. È<unsure/> 21 defin. 11 elem.</head> <p> <s xml:id="echoid-s22414" xml:space="preserve">Sit, ut in præcedenti propoſitione, columna ſecta per planam ſuperficiem ſecundum ſectionem, <lb/>rectangula a b c d:</s> <s xml:id="echoid-s22415" xml:space="preserve"> cuius axis ſit e f:</s> <s xml:id="echoid-s22416" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s22417" xml:space="preserve"> nunc ſuperficies plana columnã ſecans, æquidiſtans ſuper-<lb/>ficiei a b c d, cuius communis ſectio cum ſuperficie columnæ ſit h i k l:</s> <s xml:id="echoid-s22418" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s22419" xml:space="preserve"> à punctis h & i li <lb/>neæ perpendiculares ſuper diametrum a b per 12 p 1, quæ ſint h m, i n.</s> <s xml:id="echoid-s22420" xml:space="preserve"> Erit itaq;</s> <s xml:id="echoid-s22421" xml:space="preserve"> linea m n æqualis li-<lb/>neæ h i, ut patet per 34 p 1:</s> <s xml:id="echoid-s22422" xml:space="preserve"> lineæ enim a b & h i ſunt æquidiſtantes ex hypotheſi, & lineæ h m & i n <lb/>ſunt æquidiſtantes per 28 p 1.</s> <s xml:id="echoid-s22423" xml:space="preserve"> Eſt ergo linea h i minor diametro a b.</s> <s xml:id="echoid-s22424" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s22425" xml:space="preserve"> l k minor eſt dia <lb/>metro c d, ductis perpendicularibus lineis, quæ l o & k p:</s> <s xml:id="echoid-s22426" xml:space="preserve"> ſed lineæ h k & i l ſunt lineæ longitudinis <lb/>columnæ.</s> <s xml:id="echoid-s22427" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s22428" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div810" type="section" level="0" n="0"> <head xml:id="echoid-head669" xml:space="preserve" style="it">95. Omnis ſuperficies plana contingens pyramidem, uel columnam rotundam: ſecundum li-<lb/>neam longitudinis eſt contingens.</head> <p> <s xml:id="echoid-s22429" xml:space="preserve">Non enim ſecundum punctũ contingit ſuperficies plana propoſita corpora ſicut ſphæram:</s> <s xml:id="echoid-s22430" xml:space="preserve"> quo-<lb/>niam in ipſis eſt longitudo, quæ non eſt in ſphæra:</s> <s xml:id="echoid-s22431" xml:space="preserve"> ſed nec contingit ipſa ſecundũ ſuperficiem:</s> <s xml:id="echoid-s22432" xml:space="preserve"> quo-<lb/>niam cum in quolibet iſtorum corporũ ſint infiniti circuli ſuis baſibus æquidiſtantes & ipſæ baſes:</s> <s xml:id="echoid-s22433" xml:space="preserve"> <lb/>accideret illos ſecundum lineas in ſuperficie plana contingente ductas ad ipſorum contactum, non <lb/>contingi ſecundum punctũ, ſed ſecari:</s> <s xml:id="echoid-s22434" xml:space="preserve"> quod eſt contra 16 p 3, & impoſsibile.</s> <s xml:id="echoid-s22435" xml:space="preserve"> Non ergo continget ſu-<lb/>perficies plana propoſita corpora ſecundũ ſuperficiem.</s> <s xml:id="echoid-s22436" xml:space="preserve"> Reſtat ergo, <lb/> <anchor type="figure" xlink:label="fig-0338-02a" xlink:href="fig-0338-02"/> ut ſecundũ lineam contingat.</s> <s xml:id="echoid-s22437" xml:space="preserve"> Et quia contingit in pyramide uerti-<lb/>cem & baſim & in columna ambas baſes:</s> <s xml:id="echoid-s22438" xml:space="preserve"> patet, quòd utrunq;</s> <s xml:id="echoid-s22439" xml:space="preserve"> illo-<lb/>rum ſecundum lineas ſuarum longitudinum eſt contingens.</s> <s xml:id="echoid-s22440" xml:space="preserve"> Patet <lb/>ergo propoſitum.</s> <s xml:id="echoid-s22441" xml:space="preserve"/> </p> <div xml:id="echoid-div810" type="float" level="0" n="0"> <figure xlink:label="fig-0338-02" xlink:href="fig-0338-02a"> <variables xml:id="echoid-variables336" xml:space="preserve">a e d c g b</variables> </figure> </div> </div> <div xml:id="echoid-div812" type="section" level="0" n="0"> <head xml:id="echoid-head670" xml:space="preserve" style="it">96. Omnis linea perpendicularis ſuper curuam ſuperficiem py <lb/>rami dis, uel columnæ rotundæ: neceſſariò trãſit per ipſarũ axem.</head> <p> <s xml:id="echoid-s22442" xml:space="preserve">Pyramis rotunda uel columna ſit, cuius linea longitudinis ſit a b:</s> <s xml:id="echoid-s22443" xml:space="preserve"> <lb/>& eius axis a g:</s> <s xml:id="echoid-s22444" xml:space="preserve"> & ſit linea d e perpendicularis ſuper curuam illius ſu <lb/>perficiẽ.</s> <s xml:id="echoid-s22445" xml:space="preserve"> Dico, quòd linea e d tranſit per axem a g.</s> <s xml:id="echoid-s22446" xml:space="preserve"> Ducatur enim ſe-<lb/>midiameter baſis, quæ ſit b g.</s> <s xml:id="echoid-s22447" xml:space="preserve"> Quia ergo linea e d eſt perpendicula-<lb/>ris ſuper curuam ſuperficiem propoſitam:</s> <s xml:id="echoid-s22448" xml:space="preserve"> palàm per definitionem, <lb/>quoniã linea e d eſt perpendiculariter erecta ſuper ſuperficiem con-<lb/>tingentem pyramidem ſecundum aliquam lineam ſuę longitudinis:</s> <s xml:id="echoid-s22449" xml:space="preserve"> <lb/>ſit hoc ſecundum lineam a b.</s> <s xml:id="echoid-s22450" xml:space="preserve"> Cadit ergo linea e d ſuper lineam a b.</s> <s xml:id="echoid-s22451" xml:space="preserve"> <lb/>Palàm ergo per 2 p 11, quoniam lineę d e & a b ſunt in eadem ſuperfi-<lb/>cie.</s> <s xml:id="echoid-s22452" xml:space="preserve"> Et quia linea d e eſt perpendicularis ſuper curuam ſuperficiem <lb/>pyramidis:</s> <s xml:id="echoid-s22453" xml:space="preserve"> patet, quòd illa ſuperficies erit erecta ſuper ſuperficiem <lb/>conicam pyramidis, & in ipſa eſt linea a b.</s> <s xml:id="echoid-s22454" xml:space="preserve"> Producta ergo transpyra <lb/>midem, ſecabit ipſam ſecundũ lineam longitudinis a b per æqualia diuidens pyramidem, & tranſi-<lb/> <pb o="37" file="0339" n="339" rhead="LIBER PRIMVS."/> bit per axem a g per 90 huius.</s> <s xml:id="echoid-s22455" xml:space="preserve"> Trigonum ergo a b g cum linea d e eſt in eadem ſuperficie.</s> <s xml:id="echoid-s22456" xml:space="preserve"> Quia ergo <lb/>linea e d cum uno latere trigoni b a g, quod eſt a b, continet angulũ rectum, qui eſt d e a:</s> <s xml:id="echoid-s22457" xml:space="preserve"> angulus ue-<lb/>rò e a g eſt acutus:</s> <s xml:id="echoid-s22458" xml:space="preserve"> palàm, quia linea d e concurret cum linea a g per 14 huius.</s> <s xml:id="echoid-s22459" xml:space="preserve"> Tranſit ergo per axem <lb/>pyramidis uel columnæ rotundę.</s> <s xml:id="echoid-s22460" xml:space="preserve"> Quod eſt propoſitum:</s> <s xml:id="echoid-s22461" xml:space="preserve"> quoniã in columna rotunda eodem modo <lb/>demonſtandũ.</s> <s xml:id="echoid-s22462" xml:space="preserve"> In illa enim, quia linea longitudinis a b æquidiſtat axi, & lineę d e & a b & axis ſunt <lb/>in eadem ſuperficie:</s> <s xml:id="echoid-s22463" xml:space="preserve"> patet per 2 huius, quia linea d e concurrẽs cum una linearum æquidiſtantium, <lb/>ideo cum a b & cum axe neceſſariò concurret.</s> <s xml:id="echoid-s22464" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s22465" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div813" type="section" level="0" n="0"> <head xml:id="echoid-head671" xml:space="preserve" style="it">97. Omnis ſuperficies plana ſuperficiei contingenti pyramidem uel columnam in loco con-<lb/>tact{us} orthogonaliter inſiſtens, neceſſariò ſecat pyramidem uel columnam per ipſi{us} axem.</head> <p> <s xml:id="echoid-s22466" xml:space="preserve">Sit pyramis uel columna rotunda, quam contingat ſuperficies plana.</s> <s xml:id="echoid-s22467" xml:space="preserve"> Palàm ergo per 95 huius, <lb/>quoniã continget illam ſecundũ lineã longitudinis.</s> <s xml:id="echoid-s22468" xml:space="preserve"> Superficies itaq;</s> <s xml:id="echoid-s22469" xml:space="preserve"> huic ſuperficiei orthogonali-<lb/>ter in loco contactus inſiſtẽs, eſt perpendicularis ſuper ſuperficiẽ curuam pyramidis uel columnę:</s> <s xml:id="echoid-s22470" xml:space="preserve"> <lb/>& ipſarũ cõmunis ſectio eſt linea longitudinis, ſuper quã in ſuperficie erecta ducantur perpendicu-<lb/>lares.</s> <s xml:id="echoid-s22471" xml:space="preserve"> Eæ itaq;</s> <s xml:id="echoid-s22472" xml:space="preserve"> lineæ per præmiſſam tranſibunt axem pyramidis uel columnæ rotundæ.</s> <s xml:id="echoid-s22473" xml:space="preserve"> Er go & ſu-<lb/>perficies illa axem tranſiens, ſecabit pyramidẽ uel columnã ſecundum axem.</s> <s xml:id="echoid-s22474" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s22475" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div814" type="section" level="0" n="0"> <head xml:id="echoid-head672" xml:space="preserve" style="it">98. Omnis ſuperficiei planæ ſecantis pyramidem rotundam non per uerticẽ, & ſuperficiei co-<lb/>nicæ pyramidis communem ſectionem figuram triangularem eſſe impoßibile.</head> <p> <s xml:id="echoid-s22476" xml:space="preserve">Eſto pyramis, cuius uertex a, diameter baſis b c, centrũ baſis d, & axis a d, quã ſecundum axis lon <lb/>gitudinem ſecet ſuperficies plana ſecundum trigonũ a b c per 90 huius:</s> <s xml:id="echoid-s22477" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s22478" xml:space="preserve"> ipſam alia ſuperfi-<lb/>cies erecta ſuper trigonũ a <lb/> <anchor type="figure" xlink:label="fig-0339-01a" xlink:href="fig-0339-01"/> b c, nõ per uerticem, ſecun-<lb/>dum ſectionẽ, quæ ſit e f g, <lb/>cuius ſupremus pũctus ſit <lb/>f, & ſit linea e g æquidiſtãs <lb/>alicui diametro baſis pyra-<lb/>midis, cuius medius pun-<lb/>ctus ſit h:</s> <s xml:id="echoid-s22479" xml:space="preserve"> & ducatur linea f <lb/>h à ſupremo puncto ſectio-<lb/>nis ad mediũ ſuæ baſis.</s> <s xml:id="echoid-s22480" xml:space="preserve"> Et <lb/>quia linea e g eſt linea re-<lb/>cta, quę eſt æquidiſtãs dia-<lb/>metro baſis pyramidis, & <lb/>punctũ f ſignatum eſt in ſu-<lb/>perficie conica in ſupremo, <lb/>ſuperficies e f g ſecat coni-<lb/>cam ſuperficiẽ.</s> <s xml:id="echoid-s22481" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s22482" xml:space="preserve"> ſe-<lb/>ctio e f g ſit trigonũ ſcilicet <lb/>rectilineum:</s> <s xml:id="echoid-s22483" xml:space="preserve"> patet, quoniã duæ lineæ longitudinis pyramidis, quæ ſunt e f & g f, concurrunt in pun-<lb/>cto f, præter uerticem pyramidis, quod eſt impoſsibile & cõtra 91 huius.</s> <s xml:id="echoid-s22484" xml:space="preserve"> Trigonũ quoq;</s> <s xml:id="echoid-s22485" xml:space="preserve"> curuilineũ <lb/>fieri eſt impoſsibile:</s> <s xml:id="echoid-s22486" xml:space="preserve"> quoniã ſuperficies ſecans ſupponitur eſſe plana, & ſuperficies illius trigoni eſt <lb/>curua, ut patet ex definitione.</s> <s xml:id="echoid-s22487" xml:space="preserve"> Erit ergo linea e f g linea una.</s> <s xml:id="echoid-s22488" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s22489" xml:space="preserve"> illa ſectio ſit linea una:</s> <s xml:id="echoid-s22490" xml:space="preserve"> dica-<lb/>tur ſectio conica uel pyramidalis.</s> <s xml:id="echoid-s22491" xml:space="preserve"> Si itaq́ axis pyramidis, qui eſt a d, ſit æqualis ſemidiametro baſis, <lb/>quæ eſt d b:</s> <s xml:id="echoid-s22492" xml:space="preserve"> palàm, quia pyramis a b c eſt orthogonia, quoniam angulus b a c trigoni a b c eſt rectus.</s> <s xml:id="echoid-s22493" xml:space="preserve"> <lb/>Si ergo linea f h, quæ eſt communis ſectio ſuperficiei e f g, & trigoni a b c æquidiſtet lineæ a c, quæ <lb/>eſt latus trigoni, & linea longitudinis pyramidis:</s> <s xml:id="echoid-s22494" xml:space="preserve"> palàm per 29 p 1, cum angulus b a c ſit rectus, quòd <lb/>etiam angulus b f h erit rectus:</s> <s xml:id="echoid-s22495" xml:space="preserve"> & ſimiliter angulus h f a:</s> <s xml:id="echoid-s22496" xml:space="preserve"> tunc itaq;</s> <s xml:id="echoid-s22497" xml:space="preserve"> ſectio e f g dicetur ſectio rectangu <lb/>la, uel parabola:</s> <s xml:id="echoid-s22498" xml:space="preserve"> & eſt illa, quam Arabes dicunt mukefi.</s> <s xml:id="echoid-s22499" xml:space="preserve"> Si uerò lineæ h f & a c non æquidiſtent, ſed <lb/>concurrant:</s> <s xml:id="echoid-s22500" xml:space="preserve"> ſi concurſus fiat ad partem puncti a, qui eſt uertex pyramidis:</s> <s xml:id="echoid-s22501" xml:space="preserve"> tunc patet per 14 huius, <lb/>quòd angulus h f a erit obtuſus:</s> <s xml:id="echoid-s22502" xml:space="preserve"> & tunc ſectio e f g dicetur amblygonia uel hyperbole uel mukefi <lb/>addita.</s> <s xml:id="echoid-s22503" xml:space="preserve"> Si uerò lineæ h f & a c concurrant uerſus punctum c, qui non eſt uertex pyramidis:</s> <s xml:id="echoid-s22504" xml:space="preserve"> tunc per <lb/>14 huius erit angulus h f a acutus:</s> <s xml:id="echoid-s22505" xml:space="preserve"> & tunc ſectio e f g dicetur oxygonia, uel ellipſis uel mukefi dimi-<lb/>nuta.</s> <s xml:id="echoid-s22506" xml:space="preserve"> Et ſecundum hunc modum iſtæ ſectiones & earum paſsiones ampliſsimè uariantur.</s> <s xml:id="echoid-s22507" xml:space="preserve"/> </p> <div xml:id="echoid-div814" type="float" level="0" n="0"> <figure xlink:label="fig-0339-01" xlink:href="fig-0339-01a"> <variables xml:id="echoid-variables337" xml:space="preserve">d f f f g g b h h d c h e e c</variables> </figure> </div> </div> <div xml:id="echoid-div816" type="section" level="0" n="0"> <head xml:id="echoid-head673" xml:space="preserve" style="it">99. Omnis ſuperficiei planæ ſecantis pyramidem uel columnam lateratã trans axem æquidi-<lb/>stanter baſi & ſuperficiei pyramidalis uel columnaris cõmunis ſectio eſt ſimilis peripheriæ baſis: <lb/>& ſi illa ſectio peripheriæ baſis eſt ſimilis, ſuperficies ſecans æquidistat baſi pyramidis uel colũnæ.</head> <p> <s xml:id="echoid-s22508" xml:space="preserve">Si enim illa ſectio baſi æquidiſtat, omnes trigoni laterales totius pyramidis & partiales trigoni <lb/>ſunt æquianguli per 29 p 1.</s> <s xml:id="echoid-s22509" xml:space="preserve"> Patet ergo per 4 p 6, quòd tota peripheria ſectionis eſt ſimilis baſi pyra-<lb/>midis, quoniam omnia latera trigonorum totalium & partialium erunt proportionalia.</s> <s xml:id="echoid-s22510" xml:space="preserve"> Et ſi illa <lb/>ſectio eſt baſi ſimilis, eſt etiam baſi æquidiſtans.</s> <s xml:id="echoid-s22511" xml:space="preserve"> Quoniam ſi nõ eſt æquidiſtans, erit alia ſecundum <lb/>idem punctum ſecans axem, æquidiſtans baſi, ſimilis peripheriæ baſis per præmiſſa, Sequitur itaq;</s> <s xml:id="echoid-s22512" xml:space="preserve"> <lb/>ut una ſimilis, alia quoq;</s> <s xml:id="echoid-s22513" xml:space="preserve"> non ſimilis, ſecundum idem punctum ſecent axem pyramidis.</s> <s xml:id="echoid-s22514" xml:space="preserve"> Alia uerò <lb/> <pb o="38" file="0340" n="340" rhead="VITELLONIS OPTICAE"/> æquidiſtans baſi fieri poterit per 31 p 1, ducta ab uno puncto primæ ſectiõis linea æquidiſtante alicui <lb/>linearum baſis pyramidis, & à terminis illius alijs lineis æquidiſtantibus reliquis lineis baſis produ <lb/>ctis.</s> <s xml:id="echoid-s22515" xml:space="preserve">) Ex hoc autem accidit impoſsibile, quoniã ſequitur ex hypotheſi angulum extrinſecum pro-<lb/>pter trigonorum ſimilitudinem æqualem fieri intrinſeco:</s> <s xml:id="echoid-s22516" xml:space="preserve"> cum ab uno puncto exeant duæ lineæ æ-<lb/>quales angulos cõtinentes angulis illis, qui fiunt per lineã aliquã longitudinis & per lineam aliquã <lb/>peripherię baſis.</s> <s xml:id="echoid-s22517" xml:space="preserve"> Patet ergo propoſitum in pyramidibus.</s> <s xml:id="echoid-s22518" xml:space="preserve"> Et eodem modo demonſtrandũ eſt in co-<lb/>lumnis lateratis, & facilius propter æqualitatem linearum per 34 primi.</s> <s xml:id="echoid-s22519" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div817" type="section" level="0" n="0"> <head xml:id="echoid-head674" xml:space="preserve" style="it">100. Omnis ſuperficiei planæ ſecantis pyramidem uel columnam rotundam trans axem æ-<lb/>quidiſtanter baſi, & curuæ ſuperficiei pyramidis uel columnæ communis ſectio eſt circulus: & ſi <lb/>illa ſectio eſt circulus, ſuperficies ſecans eſt æquidiſtans baſi. Ex quo patet, quòd omnis plana ſu-<lb/>perficies æquidiſtanter baſi ſecans pyramidem uel columnam, nouam pyramidem conſtituit uel <lb/>columnam. 4 theor. 1 Conicorum Apollonij, & 5 the. Cylindricorum Sereni.</head> <p> <s xml:id="echoid-s22520" xml:space="preserve">Sit pyramis rotũda a b c, cuius uertex a:</s> <s xml:id="echoid-s22521" xml:space="preserve"> diameter baſis b c, & centrũ baſis d:</s> <s xml:id="echoid-s22522" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s22523" xml:space="preserve"> ipſam ſuperfi <lb/>cies plana æquidiſtanter baſi:</s> <s xml:id="echoid-s22524" xml:space="preserve"> & ſit cõmunis ſectio ſuperficiei illius & ſuperficiei conicæ pyramidis <lb/>linea e f g.</s> <s xml:id="echoid-s22525" xml:space="preserve"> Dico, quòd linea e f g eſt peripheria circuli.</s> <s xml:id="echoid-s22526" xml:space="preserve"> Secet enim alia ſuperficies plana pyramidem <lb/>per uerticem & per axem, qui eſt a d.</s> <s xml:id="echoid-s22527" xml:space="preserve"> Cõmunis itaq;</s> <s xml:id="echoid-s22528" xml:space="preserve"> illius ſuperficiei & pyramidis ſectio eſt trigonũ <lb/>(quod ſit a b c) per 90 huius:</s> <s xml:id="echoid-s22529" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s22530" xml:space="preserve"> ſuperficies e f g axem a d in puncto h:</s> <s xml:id="echoid-s22531" xml:space="preserve"> & trigonum a b c ſecet ſu-<lb/>perficiem e f g in linea e h f.</s> <s xml:id="echoid-s22532" xml:space="preserve"> Erit ergo linea e h æquidiſtans lineæ b d <lb/> <anchor type="figure" xlink:label="fig-0340-01a" xlink:href="fig-0340-01"/> per 16 p 11:</s> <s xml:id="echoid-s22533" xml:space="preserve"> eſt ergo per 29 p 1 & 4 p 6 proportio lineæ b a ad e a, ſi-<lb/>cut lineæ c a ad lineam a f:</s> <s xml:id="echoid-s22534" xml:space="preserve"> ergo per 7 huius erit euerſim proportio <lb/>lineę b a ad lineam b e, ſicut lineę c a ad lineam c f:</s> <s xml:id="echoid-s22535" xml:space="preserve"> ergo per 16 p 5 erit <lb/>permutatim proportio lineę b a ad lineam c a, ſicut lineę b e ad lineã <lb/>c f.</s> <s xml:id="echoid-s22536" xml:space="preserve"> Sed linea b a eſt ęqualis ipſi c a per 89 huius:</s> <s xml:id="echoid-s22537" xml:space="preserve"> ergo erit linea b e æ-<lb/>qualis lineę c f.</s> <s xml:id="echoid-s22538" xml:space="preserve"> Ducantur itaq;</s> <s xml:id="echoid-s22539" xml:space="preserve"> lineæ d e, d f.</s> <s xml:id="echoid-s22540" xml:space="preserve"> Et quoniã per 89 huius, <lb/>anguli, quos continent lineę longitudinis pyramidum cum ſemidia <lb/>metris baſium, ſunt æquales:</s> <s xml:id="echoid-s22541" xml:space="preserve"> palàm per 4 p 1, quia linea d e eſt æqua-<lb/>lis lineæ d f:</s> <s xml:id="echoid-s22542" xml:space="preserve"> & angulus e d b eſt æqualis angulo f d c.</s> <s xml:id="echoid-s22543" xml:space="preserve"> Quia uerò an-<lb/>gulus h d b æqualis angulo h d c:</s> <s xml:id="echoid-s22544" xml:space="preserve"> quoniã ambo ſunt recti:</s> <s xml:id="echoid-s22545" xml:space="preserve"> & angulus <lb/>e d b æqualis angulo f d c:</s> <s xml:id="echoid-s22546" xml:space="preserve"> remanet angulus e d h æqualis angulo f d <lb/>h:</s> <s xml:id="echoid-s22547" xml:space="preserve"> quoniã ſunt reſiduę partes rectorũ ſupér angulos æquales.</s> <s xml:id="echoid-s22548" xml:space="preserve"> Palàm <lb/>ergo per 4 p 1 quoniã linea e h eſt ęquà<unsure/>lis lineę h f.</s> <s xml:id="echoid-s22549" xml:space="preserve"> Similiterq́;</s> <s xml:id="echoid-s22550" xml:space="preserve"> ductis <lb/>lineis h g & d g, & cõpleta, prout in præmiſsis, figuratione, declara-<lb/>bitur, quoniã linea f h eſt æqualis lineæ g h:</s> <s xml:id="echoid-s22551" xml:space="preserve"> ſunt enim trigona æqui-<lb/>angula, ut patet intendenti.</s> <s xml:id="echoid-s22552" xml:space="preserve"> Ergo per 9 p 3 punctum h eſt centrũ cir-<lb/>culi.</s> <s xml:id="echoid-s22553" xml:space="preserve"> Eſt ergo e f g linea circũferentia circuli.</s> <s xml:id="echoid-s22554" xml:space="preserve"> Quod eſt propoſitũ.</s> <s xml:id="echoid-s22555" xml:space="preserve"> Et <lb/>ſi ſectio e f g eſt circulus, palàm quoniã ſuperficies plana ſecundum <lb/>illum circulum ſecans pyramidem, eſt æquidiſtans baſi:</s> <s xml:id="echoid-s22556" xml:space="preserve"> erit enim e a <lb/>f pyramis, cuius axis a h, & centrum baſis h:</s> <s xml:id="echoid-s22557" xml:space="preserve"> erit itaq;</s> <s xml:id="echoid-s22558" xml:space="preserve"> linea longitudinis, quę eſt e a, æqualis lineę f a <lb/>per 89 huius:</s> <s xml:id="echoid-s22559" xml:space="preserve"> ſed & linea b a æqualis eſt ipſi c a:</s> <s xml:id="echoid-s22560" xml:space="preserve"> remanet ergo linea b e æqualis ipſi e f.</s> <s xml:id="echoid-s22561" xml:space="preserve"> Erit quoq;</s> <s xml:id="echoid-s22562" xml:space="preserve"> li-<lb/>nea e d æqualis lineę f d per 4 p 1.</s> <s xml:id="echoid-s22563" xml:space="preserve"> Et quia trigona e h d & f h d ſunt æqualia inter ſe latera habentia:</s> <s xml:id="echoid-s22564" xml:space="preserve"> <lb/>ergo per 8 p 1 angulus e h d eſt æqualis angulo f h d.</s> <s xml:id="echoid-s22565" xml:space="preserve"> Ergo per definitionem lineæ ſuper ſuperficiem <lb/>erectę patet, quod linea d h erecta eſt ſuper ſuperficiem e f g:</s> <s xml:id="echoid-s22566" xml:space="preserve"> ſed eadem linea h d eſt erecta ſuper ba-<lb/>ſim pyramidis, cuius diameter eſt b c.</s> <s xml:id="echoid-s22567" xml:space="preserve"> Ergo per 14 p 11 ſuperficies e f g eſt æquidiſtans baſi datę pyra <lb/>midis.</s> <s xml:id="echoid-s22568" xml:space="preserve"> Quod eſt propoſitum:</s> <s xml:id="echoid-s22569" xml:space="preserve"> quoniam ſimpliciter ſecundum præmiſſum in pyramidibus modum, <lb/>in columnis quoq;</s> <s xml:id="echoid-s22570" xml:space="preserve"> rotundis poteſt demonſtrari, & propter æquidiſtantiam linearum longitudinis <lb/>columnę facilitas accedit demõſtrationi.</s> <s xml:id="echoid-s22571" xml:space="preserve"> Fiunt enim lineę d f, d g, d e æquales:</s> <s xml:id="echoid-s22572" xml:space="preserve"> ergo & lineę h e, h g, <lb/>h f:</s> <s xml:id="echoid-s22573" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s22574" xml:space="preserve"> ſectio e g f circulus per 9 p 3.</s> <s xml:id="echoid-s22575" xml:space="preserve"> Et conuerſa ſimpliciter patet per 14 p 11, ut prius.</s> <s xml:id="echoid-s22576" xml:space="preserve"> Et hoc pro-<lb/>ponebatur.</s> <s xml:id="echoid-s22577" xml:space="preserve"> Per hæc itaq;</s> <s xml:id="echoid-s22578" xml:space="preserve"> patet manifeſtè, quoniam omnis plana ſuperficies ſecans quamcunq;</s> <s xml:id="echoid-s22579" xml:space="preserve"> py-<lb/>ramidem ęquidiſtanter ſuę baſi, nouã conſtituit pyramidem, cuius in pyramide rotunda, baſis eſt <lb/>circulus, & in laterata pyramide ſuperficies ſimilis baſi illius ſectę pyramidis, ut patet per 99 huius.</s> <s xml:id="echoid-s22580" xml:space="preserve"> <lb/>Semper tamen uertex illius pyramidis abſciſſę eſt idem cum uertice prioris, & axis abſciſſę, pars a-<lb/>xis ipſius prioris datę:</s> <s xml:id="echoid-s22581" xml:space="preserve"> baſis quoq;</s> <s xml:id="echoid-s22582" xml:space="preserve"> æquidiſtat baſi.</s> <s xml:id="echoid-s22583" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s22584" xml:space="preserve"> fit in columnis rotundis uel late <lb/>ratis:</s> <s xml:id="echoid-s22585" xml:space="preserve"> ſuperficies enim ęquidiſtanter baſibus ſecans quamcunq;</s> <s xml:id="echoid-s22586" xml:space="preserve"> columnam, nouam efficit columnã <lb/>rotundam uel lateratam:</s> <s xml:id="echoid-s22587" xml:space="preserve"> imò duas, ſcilicet abſciſſam & ipſam reſiduam:</s> <s xml:id="echoid-s22588" xml:space="preserve"> quod non accidit in pyrami <lb/>dibus.</s> <s xml:id="echoid-s22589" xml:space="preserve"> Patet ergo totum, quod proponebatur.</s> <s xml:id="echoid-s22590" xml:space="preserve"/> </p> <div xml:id="echoid-div817" type="float" level="0" n="0"> <figure xlink:label="fig-0340-01" xlink:href="fig-0340-01a"> <variables xml:id="echoid-variables338" xml:space="preserve">a e h f g b d c</variables> </figure> </div> </div> <div xml:id="echoid-div819" type="section" level="0" n="0"> <head xml:id="echoid-head675" xml:space="preserve" style="it">101. In qualibet columna uel pyramide à dato in eius ſuperficie puncto, lineam longitudinis <lb/>ducere. 7 theo. Cylindricorum Sereni.</head> <p> <s xml:id="echoid-s22591" xml:space="preserve">Imaginetur enim ſuperficies plana ſecãs pyramidem uel columnã trans illius punctum & trans <lb/>axem:</s> <s xml:id="echoid-s22592" xml:space="preserve"> quod fiet, ſi à puncto dato ducatur linea recta ſuper axẽ:</s> <s xml:id="echoid-s22593" xml:space="preserve"> illa ergo linea & axis ſunt in una ſu-<lb/>perficie per 2 p 11:</s> <s xml:id="echoid-s22594" xml:space="preserve"> quę ſuperficies ſecabit pyramidem ſecundum lineam longitudinis per illud pun-<lb/>ctum tranſeuntem per 90 huius:</s> <s xml:id="echoid-s22595" xml:space="preserve"> columnam quoq;</s> <s xml:id="echoid-s22596" xml:space="preserve"> per 92 huius.</s> <s xml:id="echoid-s22597" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s22598" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div820" type="section" level="0" n="0"> <head xml:id="echoid-head676" xml:space="preserve" style="it">102. À<unsure/> dato puncto, ſiue in axe, ſiue in ſuperficie curua datæ pyramidis rotundæ uel colũnæ, <lb/>circulum circumducere.</head> <pb o="39" file="0341" n="341" rhead="LIBER PRIMVS."/> <p> <s xml:id="echoid-s22599" xml:space="preserve">Eſto pyramis, cuius uertex punctũ a, axis uerò a d:</s> <s xml:id="echoid-s22600" xml:space="preserve"> in quo ſit datus punctus e, à quo debemus cir <lb/>culum totali ſuperficiei conicæ circunducere.</s> <s xml:id="echoid-s22601" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s22602" xml:space="preserve">, ut ſuperficies plana ſecet pyramidẽ ſecundũ <lb/>axem a d trans punctũ e:</s> <s xml:id="echoid-s22603" xml:space="preserve"> cõmunis itaq;</s> <s xml:id="echoid-s22604" xml:space="preserve"> ſectio illius ſuperficiei planæ & ſuperficiei conicæ erit trigo <lb/>num per 90 huius:</s> <s xml:id="echoid-s22605" xml:space="preserve"> cuius baſis ſit b c, quę erit diameter baſis pyrami-<lb/> <anchor type="figure" xlink:label="fig-0341-01a" xlink:href="fig-0341-01"/> dis.</s> <s xml:id="echoid-s22606" xml:space="preserve"> In hac itaq;</s> <s xml:id="echoid-s22607" xml:space="preserve"> ſuperficie per 11 p 1 ducatur à puncto e linea perpendi <lb/>culariter ſuper axem a d, quæ producta ad conicã ſuperficiem ſit e f:</s> <s xml:id="echoid-s22608" xml:space="preserve"> <lb/>& item ab eodẽ puncto e ducatur linea e g perpendiculariter ſuper <lb/>axẽ a d:</s> <s xml:id="echoid-s22609" xml:space="preserve"> cadatq́;</s> <s xml:id="echoid-s22610" xml:space="preserve"> punctũ g in conica pyramidis ſuperficie:</s> <s xml:id="echoid-s22611" xml:space="preserve"> & ſimiliter <lb/>ducatur linea e h perpendiculariter ſuper axem a d:</s> <s xml:id="echoid-s22612" xml:space="preserve"> cadatq́;</s> <s xml:id="echoid-s22613" xml:space="preserve"> punctus <lb/>h in conica ſuperficie.</s> <s xml:id="echoid-s22614" xml:space="preserve"> Quia ergo linea a e ſuper cõmunem terminum <lb/>linearũ e f, e g, e h orthogonaliter inſiſtit, palàm per 5 p 11, quoniã illæ <lb/>lineę ſunt in una ſuperficie:</s> <s xml:id="echoid-s22615" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s22616" xml:space="preserve"> per 4 p 11 linea a e perpẽdiculariter <lb/>erecta ſuper illã ſuperficiẽ f g h.</s> <s xml:id="echoid-s22617" xml:space="preserve"> Et quoniã linea a d erecta eſt perpen-<lb/>diculariter ſuper baſim pyramidis per 89 huius, & per definitionẽ p y <lb/>ramidis:</s> <s xml:id="echoid-s22618" xml:space="preserve"> patet per 14 p 11, quoniã ſuperficies f g h æquidiſtat baſi pyra <lb/>midis.</s> <s xml:id="echoid-s22619" xml:space="preserve"> Eſt ergo per 100 huius f g h circulus.</s> <s xml:id="echoid-s22620" xml:space="preserve"> Quòd ſi pũctus datus ſit <lb/>in ſuperficie conica, ſit ille punctus f:</s> <s xml:id="echoid-s22621" xml:space="preserve"> & ducatur à puncto f perpendi-<lb/>cularis ſuper axem a d, quę ſit f e, per 12 p 1:</s> <s xml:id="echoid-s22622" xml:space="preserve"> educanturq́;</s> <s xml:id="echoid-s22623" xml:space="preserve"> à puncto e li-<lb/>neæ e g & e h perpendiculares ſuper axem a d per 11 p 1:</s> <s xml:id="echoid-s22624" xml:space="preserve"> & deinde, ut <lb/>prius, compleatur demonſtratio.</s> <s xml:id="echoid-s22625" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s22626" xml:space="preserve"> propoſitum:</s> <s xml:id="echoid-s22627" xml:space="preserve"> quoniã ſim <lb/>pliciter eodem modo negotiandum eſt in columnis.</s> <s xml:id="echoid-s22628" xml:space="preserve"/> </p> <div xml:id="echoid-div820" type="float" level="0" n="0"> <figure xlink:label="fig-0341-01" xlink:href="fig-0341-01a"> <variables xml:id="echoid-variables339" xml:space="preserve">a f e g h b d c</variables> </figure> </div> </div> <div xml:id="echoid-div822" type="section" level="0" n="0"> <head xml:id="echoid-head677" xml:space="preserve" style="it">103. Omnis ſuperficiei ſecantis pyramidem uel columnã rotun-<lb/>dam trans axem non æquidiſtanter baſibus, & ſuperficiei curuæ <lb/>communem ſectionem circulum eſſe eſt impoßibile. 5 theo. 1 Conicorum Apollonij. item 9 theor. <lb/>Cylindricorum Sereni.</head> <p> <s xml:id="echoid-s22629" xml:space="preserve">Sit pyramis, cuius uertex a, diameter baſis b c:</s> <s xml:id="echoid-s22630" xml:space="preserve"> & centrum baſis d, & axis a d:</s> <s xml:id="echoid-s22631" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s22632" xml:space="preserve"> ipſam ſuper-<lb/>ficies plana trans axem a d in puncto e, nõ æquidiſtanter baſi:</s> <s xml:id="echoid-s22633" xml:space="preserve"> & ſit cõmunis ſectio huius ſuperficiei <lb/>planæ & ſuperficiei conicæ linea f g h k.</s> <s xml:id="echoid-s22634" xml:space="preserve"> Dico quòd hæc ſectio non eſt poſsibile, ut ſit circulus.</s> <s xml:id="echoid-s22635" xml:space="preserve"> Eſto <lb/>enim, ut circa punctum e in pyramidis conica ſuperficie ducatur circulus per præmiſſam:</s> <s xml:id="echoid-s22636" xml:space="preserve"> hic itaq;</s> <s xml:id="echoid-s22637" xml:space="preserve"> <lb/>æquidiſtabit baſi per 100 huius:</s> <s xml:id="echoid-s22638" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s22639" xml:space="preserve"> f g l m:</s> <s xml:id="echoid-s22640" xml:space="preserve"> & ſignentur lineę longi-<lb/> <anchor type="figure" xlink:label="fig-0341-02a" xlink:href="fig-0341-02"/> tudinis pyramidis a f, a g, a l, a m.</s> <s xml:id="echoid-s22641" xml:space="preserve"> Eæ itaq;</s> <s xml:id="echoid-s22642" xml:space="preserve"> omnes erunt æquales per <lb/>89 huius, ideo quòd ſuperficies æquidiſtans baſi pyramidis nouã py <lb/>ramidem abſcindit per 100 huius.</s> <s xml:id="echoid-s22643" xml:space="preserve"> Et quoniã ſectio f g h k nõ æquidi-<lb/>ſtat baſi pyramidis, patet quòd non æqualiter diſtat à uertice pyrami <lb/>dis, qui eſt punctus a:</s> <s xml:id="echoid-s22644" xml:space="preserve"> ſit itaq;</s> <s xml:id="echoid-s22645" xml:space="preserve"> punctus h remotior à uertice a, & cadat <lb/>in linea a l producta, & punctus k ſit propinquior uertici a, & cadat <lb/>in linea a m.</s> <s xml:id="echoid-s22646" xml:space="preserve"> Erit itaq;</s> <s xml:id="echoid-s22647" xml:space="preserve"> linea a h maior quàm linea a l, & linea a k mi-<lb/>nor eſt quàm linea a m:</s> <s xml:id="echoid-s22648" xml:space="preserve"> & continuentur lineę h e, k e, f e, g e, & lineæ <lb/>e l, e m.</s> <s xml:id="echoid-s22649" xml:space="preserve"> Et quoniã angulus a l e eſt acutus per 89 huius, erit angulus <lb/>h l e obtuſus per 13 p 1.</s> <s xml:id="echoid-s22650" xml:space="preserve"> Ergo per 19 p 1 latus h e trigoni h e l eſt maius <lb/>latere e l:</s> <s xml:id="echoid-s22651" xml:space="preserve"> ſed latus e l eſt æquale lateri e f per definitionẽ circuli.</s> <s xml:id="echoid-s22652" xml:space="preserve"> Li-<lb/>nea uerò e f uenit à puncto axis ad punctũ ſectionis:</s> <s xml:id="echoid-s22653" xml:space="preserve"> quia eſt cõmu-<lb/>nis ſectio circuli & ſuperficiei obliquè pyramidem ſecantis:</s> <s xml:id="echoid-s22654" xml:space="preserve"> inæ qua-<lb/>les itaq;</s> <s xml:id="echoid-s22655" xml:space="preserve"> lineę ab hoc puncto e producuntur ad peripheriã ſectionis.</s> <s xml:id="echoid-s22656" xml:space="preserve"> <lb/>Non eſt ergo ſectio illa circulus per circuli definitionẽ.</s> <s xml:id="echoid-s22657" xml:space="preserve"> Dicemus er-<lb/>go illam ſectionẽ in pyramidibus pyramidalem, & in columnis colu <lb/>mnalem.</s> <s xml:id="echoid-s22658" xml:space="preserve"> Eſt tamẽ illa ſectio in pyramidibus in 98 huius prius dicta <lb/>ſectio oxygonia uel ellipſis.</s> <s xml:id="echoid-s22659" xml:space="preserve"> Et quoniam talis ſectio eſt figuræ oblon <lb/>gæ, patet quòd ipſa habet diametros plurimas omnes inæquales, & <lb/>per idem punctum axis ſecti corporis tranſeuntes, ipſam quoq;</s> <s xml:id="echoid-s22660" xml:space="preserve"> ſectionem per æqualia diuidentes:</s> <s xml:id="echoid-s22661" xml:space="preserve"> <lb/>quarum maxima eſt, quæ tranſit longitudinem ſectionis, minima uerò eſt, quæ pertranſit latitudi-<lb/>nem:</s> <s xml:id="echoid-s22662" xml:space="preserve"> & eſt ſuper maximam diametrum orthogonaliter erecta.</s> <s xml:id="echoid-s22663" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s22664" xml:space="preserve"> propoſitum.</s> <s xml:id="echoid-s22665" xml:space="preserve"/> </p> <div xml:id="echoid-div822" type="float" level="0" n="0"> <figure xlink:label="fig-0341-02" xlink:href="fig-0341-02a"> <variables xml:id="echoid-variables340" xml:space="preserve">a k f l e m h g b d c</variables> </figure> </div> </div> <div xml:id="echoid-div824" type="section" level="0" n="0"> <head xml:id="echoid-head678" xml:space="preserve" style="it">104. Omnium duarum planarum ſuperficierũ ſecantium pyramidem uel columnam rotun-<lb/>dam trans idem punctum axis, ſi una æquidiſtanter baſi, & alia nõ æquidiſtanter ſecuerit: com <lb/>munis ſectio eſt linea recta tranſiens pyramidem uel columnam, orhogonalis ſuper axem. Ex <lb/>quo patet, quòd ſiue circulι peripheria, ſiue ſectio alia quæcun non in eadem ſuperficie, quam-<lb/>cun ſecuerit ſectionem, in duobus tantùm punctis ipſam interſecabit.</head> <p> <s xml:id="echoid-s22666" xml:space="preserve">Sit, ut pyramis, cuius uertex a:</s> <s xml:id="echoid-s22667" xml:space="preserve"> & axis a d ſecetur ſecundum punctum axis e, per<gap/>duas planas ſu-<lb/>perficies, quarum una ſecet æquidiſtanter baſi, ut f g h, alia uerò non æquidiſtanter, ut f g k l.</s> <s xml:id="echoid-s22668" xml:space="preserve"> Di-<lb/>co, quòd communis ſectio iſtarum ſuperficierum eſt linea tranſiens pyramidem, orthogonalis ſu-<lb/>per axem, ut eſt linea f e g.</s> <s xml:id="echoid-s22669" xml:space="preserve"> Quòd enim illæ ſuperficies ſe interſecent, patet per hoc, quòd aliquæ li-<lb/> <pb o="40" file="0342" n="342" rhead="VITELLONIS OPTICAE"/> neæ in ipſis productę, ad unum communem terminum copulantur, <lb/> <anchor type="figure" xlink:label="fig-0342-01a" xlink:href="fig-0342-01"/> & in illo ſe interſecant, ut in puncto e.</s> <s xml:id="echoid-s22670" xml:space="preserve"> Quòd enim illarum ſuperficie <lb/>rum communis ſectio ſit linea recta, patet per 3 p 11:</s> <s xml:id="echoid-s22671" xml:space="preserve"> quòd autem illa <lb/>linea (quæ eſt illarum linearum communis ſectio) ſit orthogonalis <lb/>ſuper axem pyramidis, qui eſt a d:</s> <s xml:id="echoid-s22672" xml:space="preserve"> patet:</s> <s xml:id="echoid-s22673" xml:space="preserve"> quoniam per 14 p 11 axis a d <lb/>eſt քpendicularis ſuper baſim pyramidis & ſuք ſuperficiẽ f g h:</s> <s xml:id="echoid-s22674" xml:space="preserve"> quo-<lb/>niam illæ ſuperficies ſunt ex hypotheſi æquidiſtantes.</s> <s xml:id="echoid-s22675" xml:space="preserve"> Ergo per defi <lb/>nitionem lineæ ſuper ſuperficiem erectæ, omnis linea ducta à pun-<lb/>cto axis e in ſuperficie f g h eſt perpendicularis ſuper axem a d.</s> <s xml:id="echoid-s22676" xml:space="preserve"> Li-<lb/>nea uerò, quę eſt communis ſectio iſtarum ſuperficierũ ſecantium, <lb/>neceſſariò cadit in ſuperficie f g h:</s> <s xml:id="echoid-s22677" xml:space="preserve"> alioquin nõ eſſet cõmunis ſectio.</s> <s xml:id="echoid-s22678" xml:space="preserve"> <lb/>Palàm ergo propoſitum primum:</s> <s xml:id="echoid-s22679" xml:space="preserve"> quoniam communis ſectio ſuper-<lb/>ficierum taliter, ut proponitur, pyramidem ſecantium, eſt orthogo-<lb/>nalis ſuper axem pyramidis.</s> <s xml:id="echoid-s22680" xml:space="preserve"> Et eodem modo demonſtrando, idem <lb/>patet in columnis rotundis.</s> <s xml:id="echoid-s22681" xml:space="preserve"> Ex quo patet & corollarium:</s> <s xml:id="echoid-s22682" xml:space="preserve"> quoniam <lb/>communis ſectio talium ſuperficierum eſt linea recta.</s> <s xml:id="echoid-s22683" xml:space="preserve"> In duobus au-<lb/>tem tantùm punctis, qui ſunt termini illius lineæ, fiet interſectio il-<lb/>larum ſectionum, quamuis in pluribus punctis hoc ſit fieri poſsibi-<lb/>le, cum ſe interſecant in eadẽ plana ſuperficie.</s> <s xml:id="echoid-s22684" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s22685" xml:space="preserve"/> </p> <div xml:id="echoid-div824" type="float" level="0" n="0"> <figure xlink:label="fig-0342-01" xlink:href="fig-0342-01a"> <variables xml:id="echoid-variables341" xml:space="preserve">a l f e h k g b d c</variables> </figure> </div> </div> <div xml:id="echoid-div826" type="section" level="0" n="0"> <head xml:id="echoid-head679" xml:space="preserve" style="it">105. Ex aliquo puncto baſis peripheriæ columnæ rotundæ ſemicirculo in ſuperficie cõuexa uel <lb/>cõcaua columnari circumducto: neceſſe eſt lineam ſemicirculum il-<lb/>lum per æqualia diuidentem ad ſuperficiem baſis erect am eſſe.</head> <figure> <variables xml:id="echoid-variables342" xml:space="preserve">d a b c</variables> </figure> <p> <s xml:id="echoid-s22686" xml:space="preserve">Sit, ut ex aliquo puncto peripheriæ baſis colũnæ rotundę, qđ ſit a, <lb/>circumducatur ſemicirculus in ſuperficie columnæ concaua uel con <lb/>uexa, qui ſit b c d, & eius centrum erit punctum a:</s> <s xml:id="echoid-s22687" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s22688" xml:space="preserve"> ita, ut linea a d <lb/>diuidat illum ſemicirculum per æqualia in puncto d.</s> <s xml:id="echoid-s22689" xml:space="preserve"> Dico, quòd linea <lb/>a d eſt erecta ſuper ſuperficiem baſis columnę.</s> <s xml:id="echoid-s22690" xml:space="preserve"> Quoniam enim arcus <lb/>d b eſt æqualis arcui d c:</s> <s xml:id="echoid-s22691" xml:space="preserve"> patet, quòd angulus d a b eſt æqualis angulo <lb/>d a c per 27 p 3.</s> <s xml:id="echoid-s22692" xml:space="preserve"> Eſt igitur linea a d pars unius linearũ longitudinis co-<lb/>lũnę.</s> <s xml:id="echoid-s22693" xml:space="preserve"> Eſt ergo erecta ſuper baſim per 92 huius.</s> <s xml:id="echoid-s22694" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s22695" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div827" type="section" level="0" n="0"> <head xml:id="echoid-head680" xml:space="preserve" style="it">106. Datæ pyramidirotundæ pyramidem eiuſdem uel diuerſæ al <lb/>titudinis inſcribere. Ex quo patet inſcriptæ angulum ad baſim, an-<lb/>gulo circumſcribentis maiorẽ eſse: & ſi inſcripta pyramis ad aliam <lb/>baſim priori baſi æquidiſtantem producatur, anguli productæ ad <lb/>baſim, angulis datæ pyramidis maiores erunt: & quantum-<lb/>cun anguli ad baſim augment antur, tantum anguli ad uerti <lb/>cem minuuntur.</head> <figure> <variables xml:id="echoid-variables343" xml:space="preserve">a x e i b g d h c k f o l n m p</variables> </figure> <p> <s xml:id="echoid-s22696" xml:space="preserve">Eſto exempli gratia, ut pyramis, cui alia eiuſdem altitudinis de-<lb/>bet inſcribi, ſit orthogonia, & ſit a b, a c, a e, a f lineis ſuæ longit udi <lb/>nis ſignata:</s> <s xml:id="echoid-s22697" xml:space="preserve"> & axis eius ſit a d:</s> <s xml:id="echoid-s22698" xml:space="preserve"> abſcindatur itaq;</s> <s xml:id="echoid-s22699" xml:space="preserve"> ſemidiameter ba-<lb/>ſis, quæ eſt d c, ut libuerit, & ſit abſciſſa in puncto h:</s> <s xml:id="echoid-s22700" xml:space="preserve"> producaturq́;</s> <s xml:id="echoid-s22701" xml:space="preserve"> <lb/>linea a h, & habetur triãgulus a d h, cuius latera a h, d h latere a d fi-<lb/>xo manente, reuoluantur ad locũ, unde moueri incœperũt, ꝓue-<lb/>nietq́;</s> <s xml:id="echoid-s22702" xml:space="preserve"> pyramisa g h i k, cuius axis a d.</s> <s xml:id="echoid-s22703" xml:space="preserve"> Et ſic poteſt fieri inſcriptio <lb/>ad quodcũq;</s> <s xml:id="echoid-s22704" xml:space="preserve"> punctũ lineæ d c.</s> <s xml:id="echoid-s22705" xml:space="preserve"> Et hoc eſt, qđ ꝓponebatur primũ.</s> <s xml:id="echoid-s22706" xml:space="preserve"> <lb/>Quod ſi diuerſę altitudinis pyramidẽ ad baſim cõmunẽ inſcribere <lb/>placuerit ſimilem priori datæ:</s> <s xml:id="echoid-s22707" xml:space="preserve"> ſignato puncto, ubi uolueris, in li-<lb/>nea axis a d, uel extra:</s> <s xml:id="echoid-s22708" xml:space="preserve"> tum intra corpus pyramidis, quod ſit x, pro-<lb/>ducantur lineæ à puncto x ad totam peripheriam, ut x b, x c, x e, x <lb/>f.</s> <s xml:id="echoid-s22709" xml:space="preserve"> Et patet propoſitum.</s> <s xml:id="echoid-s22710" xml:space="preserve"> Similiter erit faciendum, ſi quis inſcribere <lb/>uoluerit pyramidem ad baſim minorem baſi pyramidis datæ.</s> <s xml:id="echoid-s22711" xml:space="preserve"> Pa-<lb/>tet autem ex præmiſsis, cum omnes anguli cuiuſcunq;</s> <s xml:id="echoid-s22712" xml:space="preserve"> pyramidis <lb/>ad baſim ſint æquales per 89 huius, quoniã ex motu anguli unius <lb/>trianguli, omnes illi anguli cauſſantur:</s> <s xml:id="echoid-s22713" xml:space="preserve"> palàm, quòd quicquid in <lb/>triangulo cauſſante maiorem pyramidem reſpectu trianguli cauſ-<lb/>ſantis minorem pyramidem proueniet, in omnibus ſimilibus & <lb/>æqualibus triangulis maioris pyramidis ad ſimiles triangulos mi <lb/>noris prouenire neceſſe eſt.</s> <s xml:id="echoid-s22714" xml:space="preserve"> Cum ergo in triangulo d h a angulus <lb/>a h d ſit per 16 p 1 maior angulo a c d trianguli d c a:</s> <s xml:id="echoid-s22715" xml:space="preserve"> quoniã eſt ex-<lb/>trinſecus:</s> <s xml:id="echoid-s22716" xml:space="preserve"> patet, quòd omnes anguli pyramidis a g h i k ad baſim <lb/> <pb o="41" file="0343" n="343" rhead="LIBER PRIMVS."/> ſunt maiores omnibus angulis pyramidis a b c e f ad baſim exiſtentibus.</s> <s xml:id="echoid-s22717" xml:space="preserve"> Et eodẽ modo poteſt de-<lb/>monſtrari in pyramide inſcripta pyramidi a g h i k.</s> <s xml:id="echoid-s22718" xml:space="preserve"> Et hoc eſt ſecundum propoſitũ.</s> <s xml:id="echoid-s22719" xml:space="preserve"> Quòd ſi linea lon <lb/>gitudinis, quæ eſt a h, protrahatur ad punctum m, & axis a d ad punctum n, fiatq́;</s> <s xml:id="echoid-s22720" xml:space="preserve"> angulus a n m re-<lb/>ctus, & ſecundum eum compleatur pyramis a l m o p ſuper axem a n:</s> <s xml:id="echoid-s22721" xml:space="preserve"> patet tertium propoſitũ, quòd <lb/>anguli productæ pyramidis, qui fiunt ad baſim, erunt maiores angulis ad baſim primæ datæ pyrami <lb/>dis:</s> <s xml:id="echoid-s22722" xml:space="preserve"> quoniam ex 29 p 1 angulus n m a ęqualis eſt angulo d h a, & angulus d h a maior eſt angulo d c a:</s> <s xml:id="echoid-s22723" xml:space="preserve"> <lb/>ergo angulus n m a maior eſt angulo d c a.</s> <s xml:id="echoid-s22724" xml:space="preserve"> Omnes ergo anguli ad baſim pyramidis a l m o p angulis <lb/>ad baſim pyramidis a b c e f ſunt maiores, quilibet ſcilicet ſuo correſpondenti.</s> <s xml:id="echoid-s22725" xml:space="preserve"> Eodem autẽ modo <lb/>demonſtrari poterit, & ſi pyramis inſcripta pyramidi a g h i k, producatur ad baſim dictæ pyramidis <lb/>priori baſi æquidiſtantem:</s> <s xml:id="echoid-s22726" xml:space="preserve"> eſt enim idem modus.</s> <s xml:id="echoid-s22727" xml:space="preserve"> Patetq́;</s> <s xml:id="echoid-s22728" xml:space="preserve"> ex prædictis ultimum propoſitũ, ſcilicet, <lb/>quia quantùm anguli ad baſim ampliantur, tantùm anguli ad uerticem eiuſdem pyramidis minuun <lb/>tur:</s> <s xml:id="echoid-s22729" xml:space="preserve"> quilibet enim anguli cuiuslibet trianguli cum ſint ęquales duobus rectis per 32 p 1:</s> <s xml:id="echoid-s22730" xml:space="preserve"> angulo ergo <lb/>recto in omnibus permanente, reliqui duo ualent unum rectum:</s> <s xml:id="echoid-s22731" xml:space="preserve"> quod ergo in uno illorum additur, <lb/>neceſſe eſt, ut in reliquo minuatur.</s> <s xml:id="echoid-s22732" xml:space="preserve"> Et hoc eſt totum quod proponebatur.</s> <s xml:id="echoid-s22733" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div828" type="section" level="0" n="0"> <head xml:id="echoid-head681" xml:space="preserve" style="it">107. Si pyramis rotunda pyramidi rotundæ inſcribatur ſic, ut ambarum eadem baſi exiſtente <lb/>diuerſi ſint axes: centrũ axis, & uertices ambarũ pyramidum in eadẽ linea cõſiſtere eſt neceſſe.</head> <p> <s xml:id="echoid-s22734" xml:space="preserve">Eſto pyramis data, quæ ſit a b c e f:</s> <s xml:id="echoid-s22735" xml:space="preserve"> cuius baſis ſit circulus b c e f:</s> <s xml:id="echoid-s22736" xml:space="preserve"> & eius centum d:</s> <s xml:id="echoid-s22737" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s22738" xml:space="preserve"> axis pyra-<lb/>ramidis a d:</s> <s xml:id="echoid-s22739" xml:space="preserve"> & ſit exempli gratia orthogonia:</s> <s xml:id="echoid-s22740" xml:space="preserve"> inſcribaturq́;</s> <s xml:id="echoid-s22741" xml:space="preserve"> ei per præcedentem ad eandem baſim py <lb/>ramis breuioris axis taliter, quòd intra illam cõtineatur.</s> <s xml:id="echoid-s22742" xml:space="preserve"> Dico, quòd <lb/> <anchor type="figure" xlink:label="fig-0343-01a" xlink:href="fig-0343-01"/> centrum circuli baſis ambarum pyramidum, quod eſt d, & uertex <lb/>datæ pyramidis, qui eſt a, & uertex inſcriptæ pyramidis, qui ſit g, o-<lb/>mnes erunt in eadem linea a d.</s> <s xml:id="echoid-s22743" xml:space="preserve"> Et hoc quidem patet de punctis a & <lb/>d.</s> <s xml:id="echoid-s22744" xml:space="preserve"> Quòd autem punctum g in eadem ſit linea, probatur.</s> <s xml:id="echoid-s22745" xml:space="preserve"> Si enim non <lb/>eſt in eadem:</s> <s xml:id="echoid-s22746" xml:space="preserve"> ergo ad aliquam partem extra illam lineam declinat:</s> <s xml:id="echoid-s22747" xml:space="preserve"> ſit <lb/>ergo nunc eius declinatio ad partem dextram uerſus lineam a c in <lb/>ſuperficie trianguli a d c.</s> <s xml:id="echoid-s22748" xml:space="preserve"> Producatur linea g d.</s> <s xml:id="echoid-s22749" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s22750" xml:space="preserve"> per 89 hu-<lb/>ius omnes lineæ longitudinis eiuſdem pyramidis ſunt æquales:</s> <s xml:id="echoid-s22751" xml:space="preserve"> <lb/>patet, quòd latera g b & g c ſunt æqualia:</s> <s xml:id="echoid-s22752" xml:space="preserve"> ſed & b d eſt æqualis ipſi c <lb/>d, & axis g d cõmunis:</s> <s xml:id="echoid-s22753" xml:space="preserve"> ergo per 8 p 1 angulus g d c eſt æqualis angulo <lb/>g d b:</s> <s xml:id="echoid-s22754" xml:space="preserve"> uterq;</s> <s xml:id="echoid-s22755" xml:space="preserve"> ergo eſt rectus.</s> <s xml:id="echoid-s22756" xml:space="preserve"> Sicut autem angulus a d c eſt rectus, ſic <lb/>& angulus g d c erit rectus.</s> <s xml:id="echoid-s22757" xml:space="preserve"> Ergo rectus eſt pars recti:</s> <s xml:id="echoid-s22758" xml:space="preserve"> hoc autem eſt <lb/>impoſsibile.</s> <s xml:id="echoid-s22759" xml:space="preserve"> Patet ergo, cum ubicunq;</s> <s xml:id="echoid-s22760" xml:space="preserve"> extra lineam a d ſignato pun-<lb/>cto g, ſemper idem accidat impoſsibile, quoniam punctus g neceſſa-<lb/>riò erit in linea a d.</s> <s xml:id="echoid-s22761" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s22762" xml:space="preserve"> Quòd ſi à puncto g ad ba-<lb/>ſim pyramidis productus axis dicatur nõ cadere in punctum d, cen-<lb/>trum circuli baſis:</s> <s xml:id="echoid-s22763" xml:space="preserve"> ſequetur aliud impoſsibile contra hypotheſim, ſci <lb/>licet quòd ad eandem baſim illa pyramis non ſit inſcripta:</s> <s xml:id="echoid-s22764" xml:space="preserve"> quod eſt <lb/>cõtra præmiſſa:</s> <s xml:id="echoid-s22765" xml:space="preserve"> uel ſequetur, quòd lineæ ductæ à centro ad circum-<lb/>ferentiam non ſint æquales:</s> <s xml:id="echoid-s22766" xml:space="preserve"> quod totum eſt impoſsibile.</s> <s xml:id="echoid-s22767" xml:space="preserve"> Patet ergo illud quod proponebatur.</s> <s xml:id="echoid-s22768" xml:space="preserve"/> </p> <div xml:id="echoid-div828" type="float" level="0" n="0"> <figure xlink:label="fig-0343-01" xlink:href="fig-0343-01a"> <variables xml:id="echoid-variables344" xml:space="preserve">a g g e b d c f</variables> </figure> </div> </div> <div xml:id="echoid-div830" type="section" level="0" n="0"> <head xml:id="echoid-head682" xml:space="preserve" style="it">108. Duarum pyr amidum rotundarũ uel later at arum æqualium baſium & inæqualium alti <lb/>tudinum, uerticem altioris acutioris anguli eſſe neceſſe eſt.</head> <p> <s xml:id="echoid-s22769" xml:space="preserve">Duarum pyramidum rotundarum uel lateratarum ſit a b c altior, cuius axis a d, & uertex a:</s> <s xml:id="echoid-s22770" xml:space="preserve"> & py <lb/>ramis e f g, cuius uertex f, & axis f h <lb/> <anchor type="figure" xlink:label="fig-0343-02a" xlink:href="fig-0343-02"/> <anchor type="figure" xlink:label="fig-0343-03a" xlink:href="fig-0343-03"/> ſit baſsior:</s> <s xml:id="echoid-s22771" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s22772" xml:space="preserve"> ipſarum baſes b c <lb/>& e g æquales:</s> <s xml:id="echoid-s22773" xml:space="preserve"> & axis f h breuior a-<lb/>xe a d.</s> <s xml:id="echoid-s22774" xml:space="preserve"> Dico, quòd angulus b a c eſt <lb/>minor angulo e f g.</s> <s xml:id="echoid-s22775" xml:space="preserve"> Reſecetur enim <lb/>ab axe a d æqualis axi f h:</s> <s xml:id="echoid-s22776" xml:space="preserve"> qui ſit d k:</s> <s xml:id="echoid-s22777" xml:space="preserve"> <lb/>& ducãtur lineæ b k & c k:</s> <s xml:id="echoid-s22778" xml:space="preserve"> erit itaq;</s> <s xml:id="echoid-s22779" xml:space="preserve"> <lb/>pyramis b c k æqualis e f g:</s> <s xml:id="echoid-s22780" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s22781" xml:space="preserve"> <lb/>ſuperficies plana ambas pyramides <lb/>a b c & b k c per axem:</s> <s xml:id="echoid-s22782" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s22783" xml:space="preserve"> per <lb/>90 huius communes ipſarũ ſectio-<lb/>nes trigoni.</s> <s xml:id="echoid-s22784" xml:space="preserve"> Sit ergo ut ſecetur pyra <lb/>mis a b c ſecundum trigonum b a c, <lb/>& pyramis b k c ſecundum trigonũ <lb/>b k c:</s> <s xml:id="echoid-s22785" xml:space="preserve"> erit ergo angulus b k c maior <lb/>angulo b a c, per 33 huius:</s> <s xml:id="echoid-s22786" xml:space="preserve"> ductisq́;</s> <s xml:id="echoid-s22787" xml:space="preserve"> <lb/>alijs ſuperficiebus ſecantibus:</s> <s xml:id="echoid-s22788" xml:space="preserve"> erũt <lb/>ſemper trigona iſtis æqualia & æ-<lb/>quiangula.</s> <s xml:id="echoid-s22789" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s22790" xml:space="preserve"/> </p> <div xml:id="echoid-div830" type="float" level="0" n="0"> <figure xlink:label="fig-0343-02" xlink:href="fig-0343-02a"> <variables xml:id="echoid-variables345" xml:space="preserve">a k b d c</variables> </figure> <figure xlink:label="fig-0343-03" xlink:href="fig-0343-03a"> <variables xml:id="echoid-variables346" xml:space="preserve">f e h g</variables> </figure> </div> </div> <div xml:id="echoid-div832" type="section" level="0" n="0"> <head xml:id="echoid-head683" xml:space="preserve" style="it">109. Si à uerticibus duarũ pyramidum rotundarũ uel later atarũ inæqualium altitudinũ & <lb/>æqualium baſium, duæ pyramides æqualis inter ſe altitudinis abſcindantur: neceſſe eſt baſim py <lb/> <pb o="42" file="0344" n="344" rhead="VITELLONIS OPTICAE"/> ramidis abſciſſæ ab altiori, baſi alterius abſciſſæ minorem eſſe.</head> <p> <s xml:id="echoid-s22791" xml:space="preserve">Duarum pyramidũ rotundarum ambarũ, uel lateratarũ ambarum, ęqualiũ baſium, ſit altior a b c, <lb/>cuius axis ſit a d, & uertex a:</s> <s xml:id="echoid-s22792" xml:space="preserve"> & baſsior pyramis ſit e f g, cuius axis ſit f h, & uertex f:</s> <s xml:id="echoid-s22793" xml:space="preserve"> a b ſcindaturq́;</s> <s xml:id="echoid-s22794" xml:space="preserve"> a b <lb/>axe a d linea a k æqualis lineę f l abſciſſæ ab axe f h.</s> <s xml:id="echoid-s22795" xml:space="preserve"> Secetur itaq;</s> <s xml:id="echoid-s22796" xml:space="preserve"> pyramis altior per ſuperficiẽ planã <lb/>per axem:</s> <s xml:id="echoid-s22797" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s22798" xml:space="preserve"> per 90 huius ſectio <lb/> <anchor type="figure" xlink:label="fig-0344-01a" xlink:href="fig-0344-01"/> <anchor type="figure" xlink:label="fig-0344-02a" xlink:href="fig-0344-02"/> cõmunis trigonus, qui ſit a b c.</s> <s xml:id="echoid-s22799" xml:space="preserve"> Et ſi-<lb/>militer ſecetur altera pyramis per <lb/>axem:</s> <s xml:id="echoid-s22800" xml:space="preserve"> & ſit ſectio trigonus e f g:</s> <s xml:id="echoid-s22801" xml:space="preserve"> & à <lb/>puncto k ducatur linea k m æquidi-<lb/>ſtanter baſi b d.</s> <s xml:id="echoid-s22802" xml:space="preserve"> Et ſimiliter à puncto <lb/>l ducatur linea l o æquidiſtanter baſi <lb/>e h per 31 p 1:</s> <s xml:id="echoid-s22803" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s22804" xml:space="preserve"> per 29 p 1 & 4 p 6 <lb/>proportio lineæ b d ad lineam k m, <lb/>ſicut lineæ d a ad lineã a k:</s> <s xml:id="echoid-s22805" xml:space="preserve"> & propor <lb/>tio lineæ e h ad lineam o l, ſicut lineę <lb/>h f ad lineam f l:</s> <s xml:id="echoid-s22806" xml:space="preserve"> eſt aũt linea a k ęqua <lb/>lis lineę f l, & linea d a maior quàm li <lb/>nea f h ex hypotheſi.</s> <s xml:id="echoid-s22807" xml:space="preserve"> Ergo per 8 p 5 <lb/>maior eſt ꝓportio lineę d a ad lineã <lb/>a k, ꝗ̃ ſit linea h f ad lineã f l:</s> <s xml:id="echoid-s22808" xml:space="preserve"> eſt ergo <lb/>maior proportio lineę b d ad lineam <lb/>m k, ꝗ̃ lineæ e h ad lineã o l:</s> <s xml:id="echoid-s22809" xml:space="preserve"> ſed linea <lb/>b d eſt æqualis ipſi e h ex hypotheſi.</s> <s xml:id="echoid-s22810" xml:space="preserve"> Ergo per 10 p 5 linea o l eſt maior ꝗ̃ linea k m.</s> <s xml:id="echoid-s22811" xml:space="preserve"> Et ſimiliter pro-<lb/>ducta m k ad latus trigoni a c, & linea o l ad latus trigoni f g, ſequetur lineã l p eſſe maiorẽ, ꝗ̃ ſit linea <lb/>k n:</s> <s xml:id="echoid-s22812" xml:space="preserve"> & tota linea o p erit maior, quàm linea m n.</s> <s xml:id="echoid-s22813" xml:space="preserve"> Circũducãtur itaq;</s> <s xml:id="echoid-s22814" xml:space="preserve"> per 102 huius pyramidibus datis <lb/>duo circuli, quorũ unius diameter ſit m n, & alterius o p:</s> <s xml:id="echoid-s22815" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s22816" xml:space="preserve"> circulus o p maior circulo m n.</s> <s xml:id="echoid-s22817" xml:space="preserve"> Et ꝗa <lb/>circuli illi æquidiſtant baſibus pyramidium, patet per 100 huius, quoniã à uerticibus abſcindunt py <lb/>ramides, quarũ axes ſunt a k & f l, quę ex pręmiſsis ſunt æquales.</s> <s xml:id="echoid-s22818" xml:space="preserve"> I demq́;</s> <s xml:id="echoid-s22819" xml:space="preserve"> penitus accidit in lateratis <lb/>pyramidibus aſſumptis trigonis, & ductis lineis æquidiſtantibus baſibus trigoni, hoc eſt lateribus <lb/>baſis datę pyramidis & lineis ad axes æquidiſtãtibus, ꝗbuſdã lineis ꝓductis à terminis laterũ baſiũ <lb/>ipſarũ pyramidum ad punctum terminantẽ axem ſuper baſim.</s> <s xml:id="echoid-s22820" xml:space="preserve"> Patet ergo propoſitũ per 99 huius.</s> <s xml:id="echoid-s22821" xml:space="preserve"/> </p> <div xml:id="echoid-div832" type="float" level="0" n="0"> <figure xlink:label="fig-0344-01" xlink:href="fig-0344-01a"> <variables xml:id="echoid-variables347" xml:space="preserve">a m k n b d c</variables> </figure> <figure xlink:label="fig-0344-02" xlink:href="fig-0344-02a"> <variables xml:id="echoid-variables348" xml:space="preserve">f o l p p h g</variables> </figure> </div> </div> <div xml:id="echoid-div834" type="section" level="0" n="0"> <head xml:id="echoid-head684" xml:space="preserve" style="it">110. Si pyramis rotunda ſphæram interſecet, nec eius conica ſuperficies à ſuperficie ſphæræ <lb/>interſecetur: communis ſectio ſuperficierum ſphæræ & pyramidis erit circumferentia circu-<lb/>li baſis pyramidis.</head> <p> <s xml:id="echoid-s22822" xml:space="preserve">Quoniam enim per 69 huius ſuperficies plana ſecundum circulum ſecat ſphærã, baſisq́;</s> <s xml:id="echoid-s22823" xml:space="preserve"> pyrami-<lb/>dis ſuperficies plana eſt, quia circulus:</s> <s xml:id="echoid-s22824" xml:space="preserve"> palàm, quòd illa baſis ſphæram ſecundum circulum interſe-<lb/>cabit:</s> <s xml:id="echoid-s22825" xml:space="preserve"> interſecat autem pyramis ſphæræ ſuperficiem ſecundum totam ſuam baſim:</s> <s xml:id="echoid-s22826" xml:space="preserve"> quia ſuperficies <lb/>eius cõuexa conica à ſuperficie ſphæræ non interſecatur, ut patet per hypotheſim.</s> <s xml:id="echoid-s22827" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s22828" xml:space="preserve">, quòd <lb/>communis ſectio ſuperficierum dictarum erit circumferentia circuli baſis pyramidis, ſuperficiesq́;</s> <s xml:id="echoid-s22829" xml:space="preserve"> <lb/>illa circumferentia contenta (quæ eſt circulus, qui eſt baſis pyramidis) erit ſuperficies communis:</s> <s xml:id="echoid-s22830" xml:space="preserve"> <lb/>quamuis aliàs corpuſculum (quod eſt pars ſphæræ) reſectum à ſphæra per illam ſuperficiem, ſit cor-<lb/>pus utriq;</s> <s xml:id="echoid-s22831" xml:space="preserve"> dictorum corporum commune.</s> <s xml:id="echoid-s22832" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div835" type="section" level="0" n="0"> <head xml:id="echoid-head685" xml:space="preserve" style="it">111. Si pyramis ſphæram interſecet ſic, ut circulus baſis pyramidis in ſphæræ ſuperficie circu-<lb/>lo maiori ſphæræ æquidiſtet: diametrum ſphæræ ſuper illum circulum maiorem erectã, centrum <lb/>circuli baſis pyramidis orthogonaliter tranſire neceſſe eſt. Ex quo manifeſtum eſt, diametrum <lb/>ſphæræ & axem pyramidis coniuncta eſſe lineam unam.</head> <p> <s xml:id="echoid-s22833" xml:space="preserve">Quia enim per præcedentem circulus (qui eſt baſis pyramidis) communis eſt ſphæræ, ſicut pyra-<lb/>midi:</s> <s xml:id="echoid-s22834" xml:space="preserve"> tunc per 68 huius patet propoſitum.</s> <s xml:id="echoid-s22835" xml:space="preserve"> Quia enim circulus (qui eſt baſis pyramidis) æquidiſtat <lb/>circulo magno ſphæræ, & ij circuli æquidiſtãtes ſunt ambo in ſuperficie ſphærę:</s> <s xml:id="echoid-s22836" xml:space="preserve"> erit diameter ſphæ <lb/>ræ centrũ circuli baſis pyramidis orthogonaliter tranſiens:</s> <s xml:id="echoid-s22837" xml:space="preserve"> tranſit enim orthogonaliter centra am-<lb/>borum illorum circulorum.</s> <s xml:id="echoid-s22838" xml:space="preserve"> Et quoniam à termino alicuius lineæ ductæ à centro communis circuli <lb/>ad circumferentiam, exeunt duæ lineæ orthogonaliter ſuper ipſam inſiſtentes, ſcilicet axis pyrami-<lb/>dis, ut patet per 89 huius, & diameter ſphæræ, ut præmiſſum eſt:</s> <s xml:id="echoid-s22839" xml:space="preserve"> patet ex 14 p 1, quoniam illę duæ li-<lb/>neæ coniunctæ, ſunt linea una.</s> <s xml:id="echoid-s22840" xml:space="preserve"> Diametrum ergo ſphærę & axem pyramidis coniuncta eſſe lineam <lb/>unam neceſſe eſt.</s> <s xml:id="echoid-s22841" xml:space="preserve"> Et hoc eſt quod proponebatur.</s> <s xml:id="echoid-s22842" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div836" type="section" level="0" n="0"> <head xml:id="echoid-head686" xml:space="preserve" style="it">112. Omnium linearum perpendicularium ſuper peripheriam oxygoniæ ſectionis product a <lb/>rum trans eius ſuperficiem, unιca eſt perpendicularis ſuper ſecti corporis axem: & ipſa eſt mini-<lb/>ma diametrorum ſectionis.</head> <p> <s xml:id="echoid-s22843" xml:space="preserve">Sicut enim patet per 104 huius, communis ſectio ſuperficiei ipſius ſectionis oxygoniæ & circuli <lb/>ſecundum idem punctum axem ſecantium, eſt linea orthogonalis ſuper axem ſecti corporis:</s> <s xml:id="echoid-s22844" xml:space="preserve"> in alijs <lb/> <pb o="43" file="0345" n="345" rhead="LIBER PRIMVS."/> autem omnibus punctis ſectionis perpendiculares ſuper ſectionẽ productæ obliquè incidunt axi:</s> <s xml:id="echoid-s22845" xml:space="preserve"> <lb/>quoniam ſi aliqua ipſarum ipſi axi perpẽdiculariter inciderit:</s> <s xml:id="echoid-s22846" xml:space="preserve"> tunc per 4 p 11 axis ſuper ſuperficiem <lb/>ſectionis perpendicularis erit:</s> <s xml:id="echoid-s22847" xml:space="preserve"> quod eſt contra naturam ſectionis.</s> <s xml:id="echoid-s22848" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s22849" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div837" type="section" level="0" n="0"> <head xml:id="echoid-head687" xml:space="preserve" style="it">113. In ſectione pyramidali tranſeunte punctum datum ſuperficiei pyramidis rotundæ, à <lb/>puncto dato perpendicularem in ſuperficie ſectionis ductam ſuper ſuperficiem pyramidis, cum <lb/>perpendiculari ducta à puncto eiuſdem ſectionis remotiore à uertice pyramidis ſuper lineam in <lb/>illo puncto ſectionem contingentem, ſub axe pyramidis concurrere eſt neceſſe: dum tamen linea <lb/>ducta à puncto inferiori cum perpendiculari ducta à puncto ſuperiori ſuper axem pyramidis, <lb/>angulum contineat acutum. Alhazen 30 n 6.</head> <p> <s xml:id="echoid-s22850" xml:space="preserve">Eſto pyramis, cuius uertex ſit a, & eius axis ſit a c k:</s> <s xml:id="echoid-s22851" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s22852" xml:space="preserve"> in ſuperficie conica huius pyramidis <lb/>ſignatus punctus e, quem pertranſeat ſectio pyramidalis, quæ ſit e f z, in qua etiam ſit punctus z re-<lb/>motior à puncto a uertice pyramidis, quã ſit punctus e:</s> <s xml:id="echoid-s22853" xml:space="preserve"> contineatq́;</s> <s xml:id="echoid-s22854" xml:space="preserve"> linea ducta à puncto z ad axem <lb/>cum perpẽdiculari ducta à puncto e angulum acutum.</s> <s xml:id="echoid-s22855" xml:space="preserve"> Dico, quòd ſi ducatur à puncto z linea per-<lb/>pendicularis ſuper lineam in illo puncto z ipſam ſectionem oxygoniam contingentem:</s> <s xml:id="echoid-s22856" xml:space="preserve"> & alia per-<lb/>pendicularis ſuper ſuperficiem contingentem pyramidem in puncto e ducatur à puncto e, quòd <lb/>illæ duæ perpendiculares concurrent ſub axe a c k.</s> <s xml:id="echoid-s22857" xml:space="preserve"> Sit enim, ut ſuperficies plana ſecet pyramidem <lb/>ſuper punctum z æquidiſtanter baſi:</s> <s xml:id="echoid-s22858" xml:space="preserve"> & hæc quidem per 100 huius ſecabit eam ſecũdum circulum:</s> <s xml:id="echoid-s22859" xml:space="preserve"> <lb/>ſit ille circulus g b r z, cuius cẽtrum ſit c:</s> <s xml:id="echoid-s22860" xml:space="preserve"> communisq́;</s> <s xml:id="echoid-s22861" xml:space="preserve"> ſectio huius circuli & ſectionis oxygoniæ ſit <lb/>diameter ut chorda circuli, qui eſt g b r z per 104 huius:</s> <s xml:id="echoid-s22862" xml:space="preserve"> & à pũcto uerticis pyramidis per 101 huius <lb/> <anchor type="figure" xlink:label="fig-0345-01a" xlink:href="fig-0345-01"/> ducantur per ſignata in ſuperficie pyramidis puncta <lb/>e & z lineæ longitudinis pyramidis, quæ ſint lineæ a <lb/>z & a e:</s> <s xml:id="echoid-s22863" xml:space="preserve"> & producatur linea a e, donecipſa ſit æqualis <lb/>lineæ a z.</s> <s xml:id="echoid-s22864" xml:space="preserve"> Veniet quidem ad circulum, eò quòd eſt li-<lb/>nea longitudinis, & quia punctus e propinquior eſt <lb/>uertici pyramidis, quàm ſit punctus z.</s> <s xml:id="echoid-s22865" xml:space="preserve"> Cadat ergo li-<lb/>nea a e producta in punctum circuli o:</s> <s xml:id="echoid-s22866" xml:space="preserve"> & à pũcto da-<lb/>to (qui eſt e) ducatur linea perpẽdicularis ſuper ſu-<lb/>perficiem contingentem pyramidem:</s> <s xml:id="echoid-s22867" xml:space="preserve"> hæc quidẽ per <lb/>96 huius concurret cum axe pyramidis, qui eſt a c k.</s> <s xml:id="echoid-s22868" xml:space="preserve"> <lb/>Concurrat ergo in puncto d:</s> <s xml:id="echoid-s22869" xml:space="preserve"> & ſit illa perpendicula.</s> <s xml:id="echoid-s22870" xml:space="preserve"> <lb/>ris e d:</s> <s xml:id="echoid-s22871" xml:space="preserve"> copuletur quoq;</s> <s xml:id="echoid-s22872" xml:space="preserve"> linea z d, cõtinens angulum <lb/>acutum cum perpẽdiculari e d, qui ſit angulus z d e.</s> <s xml:id="echoid-s22873" xml:space="preserve"> <lb/>Et quoniam linea d z eſt in ſuperficie ſectionis per 1 <lb/>p 11, ſicut & puncta d & z:</s> <s xml:id="echoid-s22874" xml:space="preserve"> tunc à puncto o lineæ lon-<lb/>gitudinis a e o ducatur perpẽdicularis ſuper lineam <lb/>a o per 11 p 1, & ducatur à cẽtro circuli g b r z, qđ eſt c, <lb/>ſemidiameter c o.</s> <s xml:id="echoid-s22875" xml:space="preserve"> Quia ergo per 89 huius angulus <lb/>c o a eſt acutus, patet, quòd perpendicularis ſuper <lb/>lineam a o ducta à puncto o, cadet ſub cẽtro circuli, <lb/>quod eſt c, in aliud punctum axis.</s> <s xml:id="echoid-s22876" xml:space="preserve"> Sit ergo ut cõcur-<lb/>rat cum axe in puncto k:</s> <s xml:id="echoid-s22877" xml:space="preserve"> & erit o k ęquidiſtans lineæ <lb/>e d per 6 p 11:</s> <s xml:id="echoid-s22878" xml:space="preserve"> & ducatur linea k z:</s> <s xml:id="echoid-s22879" xml:space="preserve"> & ducatur linea cõ-<lb/>tingens ſectionem in puncto z, quę ſit t q:</s> <s xml:id="echoid-s22880" xml:space="preserve"> & ducatur <lb/>alia contingens circulum b g z in puncto z per 17 p 3, quæ ſit z y:</s> <s xml:id="echoid-s22881" xml:space="preserve"> & ducatur diameter circuli, quę ſit <lb/>b c z:</s> <s xml:id="echoid-s22882" xml:space="preserve"> & à centro c ducatur ſemidiameter perpendicularis ſuper diametrum b c z, quæ ſit c r.</s> <s xml:id="echoid-s22883" xml:space="preserve"> Et quia <lb/>axis a c k orthogonaliter erigitur ſuper centrum circuli b g z per 89 huius, erit linea c r perpendi-<lb/>cularis ſuper axem a c k, quoniam eſt ſemidiameter circuli.</s> <s xml:id="echoid-s22884" xml:space="preserve"> Ergo per 4 p 11 linea c r eſt perpendicu-<lb/>laris ſuper ſuperficiem a c z ſecantem pyramidem per axem:</s> <s xml:id="echoid-s22885" xml:space="preserve"> ſed & linea c r eſt æquidiſtans lineæ <lb/>contingenti circulum in puncto z, quæ eſt y z, per 28 p 1.</s> <s xml:id="echoid-s22886" xml:space="preserve"> Ergo per 8 p 11 linea z y eſt perpendicularis <lb/>ſuper ſuperficiem a c z.</s> <s xml:id="echoid-s22887" xml:space="preserve"> Linea ergo t q contingens ſectionem oxygoniam e f z in puncto z, continet <lb/>angulum acutum cum linea y z.</s> <s xml:id="echoid-s22888" xml:space="preserve"> Et quia linea t q continet angulum acutũ cum y z:</s> <s xml:id="echoid-s22889" xml:space="preserve"> patet, quòd linea <lb/>t q non eſt perpẽdicularis ſuper illam ſuperficiẽ a c z.</s> <s xml:id="echoid-s22890" xml:space="preserve"> Verùm, quia punctus k (qui eſt punctus axis) <lb/>ut patet per 89 huius & per definitionẽ poli factã in principio, eſt polus ad circulũ b r z:</s> <s xml:id="echoid-s22891" xml:space="preserve"> palàm per <lb/>65 huius, quia lineæ k o & k z ſunt æquales, & axis a k cõmunis:</s> <s xml:id="echoid-s22892" xml:space="preserve"> ſed & linea a o eſt æqualis lineæ a z <lb/>per 89 huius, cũ ſint lineæ longitudinis, ut patet per præmiſſa.</s> <s xml:id="echoid-s22893" xml:space="preserve"> Ergo per 8 p 1 trianguli a o k & a z k <lb/>ſunt ęquianguli:</s> <s xml:id="echoid-s22894" xml:space="preserve"> erit ergo angulus a o k ęqualis angulo a z k.</s> <s xml:id="echoid-s22895" xml:space="preserve"> Et quoniã angul<emph style="sub">9</emph> a o k eſt rectus, ideo <lb/>quòd linea o k ducta eſt perpẽdiculariter ſuper lineã a o, ut patet ք præmiſſa:</s> <s xml:id="echoid-s22896" xml:space="preserve"> erit ergo etiã angulus <lb/>a z k rectus.</s> <s xml:id="echoid-s22897" xml:space="preserve"> Cum ergo linea k z ſit perpẽ licularis ſuք lineã a z, quæ eſt linea lõgitudinis pyramidis:</s> <s xml:id="echoid-s22898" xml:space="preserve"> <lb/>palàm, quia linea k z erit perpẽdicularis ſuper ſuperficiẽ contingentem pyramidẽ ſecundum lineã a <lb/>z lineã longitudinis:</s> <s xml:id="echoid-s22899" xml:space="preserve"> ſed linea t q eſt in ſuperficie illa contingẽte, quia eſt cõmunis ſectio ſuperficiei <lb/>contingẽtis & ſuperficiei ſectionis e f z, quoniã eſt in ſuperficie contingente pyramidẽ, ducta con-<lb/>tingens ſectionem.</s> <s xml:id="echoid-s22900" xml:space="preserve"> Eſtigitur linea k z perpendicularis ſuper lineam t q per definitionem lineæ ſu-<lb/> <pb o="44" file="0346" n="346" rhead="VITELLONIS OPTICAE"/> per ſuperficiem erectæ.</s> <s xml:id="echoid-s22901" xml:space="preserve"> Ducatur quoq;</s> <s xml:id="echoid-s22902" xml:space="preserve"> à puncto z in ipſa ſuperficie ſectionis per 11 p 1 perpendicu-<lb/>laris ſuper lineam t q, quæ ſit linea z h.</s> <s xml:id="echoid-s22903" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s22904" xml:space="preserve"> linea k z ſit extra ſuperficiem ſectionis cõcurrens <lb/>cum linea h z in puncto z:</s> <s xml:id="echoid-s22905" xml:space="preserve"> palàm, quòd ipſa ſecabit lineam h z, nec erit una linea cum illa per 1 p 11.</s> <s xml:id="echoid-s22906" xml:space="preserve"> <lb/>Sunt itaq;</s> <s xml:id="echoid-s22907" xml:space="preserve"> lineæ k z & h z in una ſuperficie per 2 p 11.</s> <s xml:id="echoid-s22908" xml:space="preserve"> Superficies ergo k z h ſecat ſuperficiem ſectio-<lb/>nis ſuper lineam eis ambabus communem, quæ eſt h z, per 19 huius:</s> <s xml:id="echoid-s22909" xml:space="preserve"> & ſecat lineam t q in puncto <lb/>z:</s> <s xml:id="echoid-s22910" xml:space="preserve"> & ſuperficies h z k ſecat ſuperficiem d z h ſuper lineam communem ambabus illis ſuperficiebus, <lb/>quæ eſt linea h z p.</s> <s xml:id="echoid-s22911" xml:space="preserve"> Verùm linea d z eſt in ſuperficie ſectionis, ut ſuprà patuit, & ſecatur à linea k z in <lb/>puncto z:</s> <s xml:id="echoid-s22912" xml:space="preserve"> & punctus t eſt ſupra ſuperficiẽ k z h, & punctus q infra illam:</s> <s xml:id="echoid-s22913" xml:space="preserve"> & ita ſuperficies k z h ſecat <lb/>ſuperficiem d z q ſuper lineam communem, quæ eſt perpendicularis ſuper lineam t q:</s> <s xml:id="echoid-s22914" xml:space="preserve"> & eſt linea z <lb/>h:</s> <s xml:id="echoid-s22915" xml:space="preserve"> quia linea illa eſt in ſuperficie h z k, & ſuper eam eſt perpendicularis linea t q, ut patet ex præmiſ-<lb/>ſis.</s> <s xml:id="echoid-s22916" xml:space="preserve"> Et quoniam ſuperficies h z k ſecat ſuperficiem d z q, & declinatio ſuperficiei h z k à ſuperficie ſe-<lb/>ctionis, cuius pars eſt ſuperficies d z q, fit ex parte ſemidiametri z c:</s> <s xml:id="echoid-s22917" xml:space="preserve"> erit linea, quæ eſt cõmunis ſe-<lb/>ctio illarum ſuperficierum (& eſt linea h z p) cadens inter lineas q z & d z.</s> <s xml:id="echoid-s22918" xml:space="preserve"> Et ita linea z h, quæ eſt à <lb/>puncto z ducta perpendiculariter ſuper lineam ſectionem oxygoniam e f z in illo puncto contin-<lb/>gentem, concurret cum perpendiculari e d ſub axe a c k.</s> <s xml:id="echoid-s22919" xml:space="preserve"> Quoniam perpendicularis e d ſecat axem <lb/>pyramidis, quæ eſt a c k in puncto d.</s> <s xml:id="echoid-s22920" xml:space="preserve"> Quòd autem concurrant, patet per 14 huius.</s> <s xml:id="echoid-s22921" xml:space="preserve"> Producatur enim <lb/>linea h z ultra punctum z intra ſectionem in punctum p.</s> <s xml:id="echoid-s22922" xml:space="preserve"> Quia ergo angulus z d e eſt acutus, & an-<lb/>gulus d z p acutus:</s> <s xml:id="echoid-s22923" xml:space="preserve"> palàm, quoniam concurrent lineæ z h & e d ſub puucto d:</s> <s xml:id="echoid-s22924" xml:space="preserve"> & ſit concurſus pun-<lb/>ctum p.</s> <s xml:id="echoid-s22925" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s22926" xml:space="preserve"/> </p> <div xml:id="echoid-div837" type="float" level="0" n="0"> <figure xlink:label="fig-0345-01" xlink:href="fig-0345-01a"> <variables xml:id="echoid-variables349" xml:space="preserve">a e t g o f z h d c p y k b r q</variables> </figure> </div> </div> <div xml:id="echoid-div839" type="section" level="0" n="0"> <head xml:id="echoid-head688" xml:space="preserve" style="it">114. Ab altero duorum punctorum in ſectione columnari ſignatorum ducta perpẽdiculari <lb/>ſuper axem columnæ in ipſa ſuperficie ſectionis, & à reliquo puncto ducta linea acutum angu-<lb/>lum cum illa perpendiculari ſuper axem columnæ continente: ſi ab eodem puncto reliquo duca-<lb/>tur perpendicularis ſuper ipſam ſectionem: hæc concurret cum priori perpendiculari ſub axe, <lb/>& ſub puncto concurſus prioris lineæ cum perpendiculari. Alhazen 24 n 6.</head> <p> <s xml:id="echoid-s22927" xml:space="preserve">Sit ſectio columnaris, quæ a b c e:</s> <s xml:id="echoid-s22928" xml:space="preserve"> in qua ſignati ſint duo puncti, qui ſint b & e:</s> <s xml:id="echoid-s22929" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s22930" xml:space="preserve"> columnæ, in <lb/>cuius ſuperficie cadit illa ſectio, axis linea h d k:</s> <s xml:id="echoid-s22931" xml:space="preserve"> & ab altero ſignatorum punctorum, ut à puncto b, <lb/>ducatur in ipſa ſuperficie ſectionis linea b d, perpendiculariter ſuper axem incidens puncto d:</s> <s xml:id="echoid-s22932" xml:space="preserve"> & <lb/>ducatur item in ſuperficie ſectionis à reliquo datorum punctorum, quod eſt e, linea e d acutum an-<lb/>gulum continens cũ perpendiculari d b, qui ſit e d b:</s> <s xml:id="echoid-s22933" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s22934" xml:space="preserve"> linea cõtingens ſectionẽ in puncto e, quæ <lb/>ſit exempli cauſſa, linea l e q.</s> <s xml:id="echoid-s22935" xml:space="preserve"> Dico, quòd perpendicularis à puncto e ducta ſuper lineam l e q, con-<lb/>curret cum perpendiculari b d ſub axe h k, & ſub puncto d, qui eſt punctus cõcurſus lineæ e d cum <lb/>perpendiculari b d.</s> <s xml:id="echoid-s22936" xml:space="preserve"> Fiat enim per 102 huius ſuper punctũ ſectionis, quod eſt b, circulus ęquidiſtans <lb/>baſibus columnæ, qui ſit b t o, cuius centrũ ſit d:</s> <s xml:id="echoid-s22937" xml:space="preserve"> & ducatur à puncto e linea longitudinis columnæ <lb/>per 101 huius, quæ ſit e t:</s> <s xml:id="echoid-s22938" xml:space="preserve"> & à puncto d per 11 p 1 ducatur linea d g perpendicularis ſuper lineam b d <lb/>in ipſa circuli ſuperficie.</s> <s xml:id="echoid-s22939" xml:space="preserve"> Palàm ergo, quòd ſuperficies h d g cum per axem tranſeat (qui erectus eſt <lb/>ſuper circuli ſuperficiem) perpendicularis eſt ſuper eandem circuli ſuperficiem per 18 p 11.</s> <s xml:id="echoid-s22940" xml:space="preserve"> Super-<lb/> <anchor type="figure" xlink:label="fig-0346-01a" xlink:href="fig-0346-01"/> ficies uerò contingens columnam in puncto <lb/>b, erit æquidiſtãs ſuperficiei b d g.</s> <s xml:id="echoid-s22941" xml:space="preserve"> Ideo enim, <lb/>quia linea lõgitudinis columnæ ducta à pun-<lb/>cto b eſt æquidiſtãs axi h k per 92 huius, & 28 <lb/>p 1, & linea circulum b t o contingens ſuper <lb/>punctũ b, eſt æquidiſtans lineæ d g per 28 p 1:</s> <s xml:id="echoid-s22942" xml:space="preserve"> <lb/>angulus enim g d b eſt rectus ex pręmiſsis, & <lb/>angulus contentus ſub linea d b, & ſub linea <lb/>contingente in puncto b eſt rectus per 18 p 3.</s> <s xml:id="echoid-s22943" xml:space="preserve"> <lb/>Ergo illæ ſuperficies æquidiſtant per 15 p 11.</s> <s xml:id="echoid-s22944" xml:space="preserve"> <lb/>Igitur ſuperficies, in qua ſunt lineæ l e & et <lb/>non eſt æquidiſtans ſuperficiei b d g per 24 <lb/>huius:</s> <s xml:id="echoid-s22945" xml:space="preserve"> quoniam ſuperficies contingẽs ſectionem oxygoniam in puncto b, non eſt æquidiſtans ſu-<lb/>perficiei contingenti eandem ſectionem in puncto e, in qua ſunt lineæ, l e q contingens ſectionem, <lb/>& linea longitudinis, quæ eſt e t:</s> <s xml:id="echoid-s22946" xml:space="preserve"> angulus enim e d b eſt acutus ex hypotheſi.</s> <s xml:id="echoid-s22947" xml:space="preserve"> Superficies ergo b d g <lb/>non æquidiſtat ſuperficiei l e t.</s> <s xml:id="echoid-s22948" xml:space="preserve"> Ergo concurret cum illa.</s> <s xml:id="echoid-s22949" xml:space="preserve"> Concurrat ergo in linea l g per 3 p 11:</s> <s xml:id="echoid-s22950" xml:space="preserve"> & du-<lb/>catur linea g t:</s> <s xml:id="echoid-s22951" xml:space="preserve"> quæ neceſſariò erit contingens circulum b t o, cuius ſuperficies, in qua ipſa ducitur, <lb/>columnam fit contingens.</s> <s xml:id="echoid-s22952" xml:space="preserve"> Ducta autem linea t d, erit angulus g t d rectus per 18 p 3:</s> <s xml:id="echoid-s22953" xml:space="preserve"> quoniam linea <lb/>t d eſt ſemidiameter circuli, & linea g t contingit circulum in puncto t.</s> <s xml:id="echoid-s22954" xml:space="preserve"> Fiat quoq;</s> <s xml:id="echoid-s22955" xml:space="preserve">, ut prius, ſuper e <lb/>punctum ſectionis circulus æquidiſtans baſibus columnæ, qui ſit e s z p, & cẽtrum huius circuli ſit <lb/>punctus axis, qui k:</s> <s xml:id="echoid-s22956" xml:space="preserve"> & ducatur linea k e:</s> <s xml:id="echoid-s22957" xml:space="preserve"> & ducatur etiam linea d l:</s> <s xml:id="echoid-s22958" xml:space="preserve"> quæ quidem ſecabit ſuperficiem <lb/>e s p:</s> <s xml:id="echoid-s22959" xml:space="preserve"> ſecet ergo illã in puncto f.</s> <s xml:id="echoid-s22960" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s22961" xml:space="preserve"> punctũ d eſt in ſuperficie ſectionis, ut patet ex præmiſsis <lb/>& exhypotheſi, & punctũ l, quod eſt punctũ lineæ contingẽtis ſectionem, eſt in eadẽ ſuperficie ſe-<lb/>ctionis:</s> <s xml:id="echoid-s22962" xml:space="preserve"> ergo per 1 p 11 tota linea d l eſt in ſuperficie ſectionis.</s> <s xml:id="echoid-s22963" xml:space="preserve"> Punctũ ergo f eſt in ſuperficie ſectionis <lb/>& circuli e s z p:</s> <s xml:id="echoid-s22964" xml:space="preserve"> ſed & punctum e eſt in eiſdem ambabus ſuperficiebus:</s> <s xml:id="echoid-s22965" xml:space="preserve"> ergo per 1 p 11 linea e f pro-<lb/>ducta erit in ambabus illis ſuperficiebus.</s> <s xml:id="echoid-s22966" xml:space="preserve"> Ergo per 19 huius ſecundum lineam e f ſecantſe ſuper-<lb/>ficies ſectionis & circuli e s z p.</s> <s xml:id="echoid-s22967" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s22968" xml:space="preserve"> linea k f:</s> <s xml:id="echoid-s22969" xml:space="preserve"> & à puncto f ducatur linea perpendicularis <lb/> <pb o="45" file="0347" n="347" rhead="LIBER PRIMVS."/> ſuper ſuperficiem circuli b t o per 11 p 11, quæ ſit f m:</s> <s xml:id="echoid-s22970" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s22971" xml:space="preserve"> punctus m in linea d g, ut patet ex præ-<lb/>miſsis:</s> <s xml:id="echoid-s22972" xml:space="preserve"> & ducatur linea t m.</s> <s xml:id="echoid-s22973" xml:space="preserve"> Palàm ergo, quoniam linea k d æqualis & æquidiſtans eſt lineæ f m per <lb/>25 huius.</s> <s xml:id="echoid-s22974" xml:space="preserve"> Sunt enim lineæ k d & f m ambæ perpendiculares ſuper ſuperficiem circuli b t o & ſuper <lb/>ſuperficiem circuli e s z p:</s> <s xml:id="echoid-s22975" xml:space="preserve"> quoniam illi circuli æquidiſtant per 24 huius:</s> <s xml:id="echoid-s22976" xml:space="preserve"> utraq;</s> <s xml:id="echoid-s22977" xml:space="preserve"> enim ipſarum æqui-<lb/>diſtat ambabus baſibus columnæ per 100 huius.</s> <s xml:id="echoid-s22978" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s22979" xml:space="preserve"> linea f m eſt æqualis & æquidiſtans li-<lb/>neæ d k, quæ eſt pars axis:</s> <s xml:id="echoid-s22980" xml:space="preserve"> ergo per 33 p 1 linea k f æqualis & æquidiſtans eſt lineæ d m.</s> <s xml:id="echoid-s22981" xml:space="preserve"> Et ſimiliter <lb/>erit linea f m æqualis & æquidiſtans lineæ longitudinis, quæ eſt e t per 30 p 1:</s> <s xml:id="echoid-s22982" xml:space="preserve"> quoniam linea e t eſt <lb/>æqualis & æquidiſtans axi k d per 92 huius, cũ ſit linea longitudinis:</s> <s xml:id="echoid-s22983" xml:space="preserve"> & erit, ut prius, linea k e æqua-<lb/>lis & æquidiſtans lineæ d t, & linea e f æqualis & æquidiſtans lineæ t m per eandem 33 p 1.</s> <s xml:id="echoid-s22984" xml:space="preserve"> Verùm <lb/>etiam ſuperficies k d l (quia tranſit axem columnę, & angulus g d b eſt rectus) eſt orthogonalis ſu-<lb/>per ſuperficiem ſectionis oxygoniæ a e c b, per definitionem ſuperficiei erectę ſuper ſuperficiem:</s> <s xml:id="echoid-s22985" xml:space="preserve"> & <lb/>eadem ſuperficies k d l eſt orthogonalis ſuper ſuperficiem circuli e s p.</s> <s xml:id="echoid-s22986" xml:space="preserve"> Quoniam enim illa ſuperfi-<lb/>cies k d l trãſiens per axem, per 18 p 11 erecta eſt ſuper baſes columnæ:</s> <s xml:id="echoid-s22987" xml:space="preserve"> ergo & ſuper ſuperficiem cir-<lb/>culi e s p æqui diſtantem baſibus columnæ, erecta eſt eadem ſuperficies k d l.</s> <s xml:id="echoid-s22988" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s22989" xml:space="preserve"> dicta ſuper-<lb/>ficies k d l eſt erecta ſuper ſuperficiem ſectionis oxygoniæ & circuli e s p:</s> <s xml:id="echoid-s22990" xml:space="preserve"> ergo per 19 p 11 eſt ipſa or-<lb/>thogonalis ſuper lineam communem dictæ ſectioni & circulo, quæ eſt linea e f.</s> <s xml:id="echoid-s22991" xml:space="preserve"> Et quia linea e f eſt <lb/>erecta ſuper ſuperficiem k d l, in qua ducta eſt linea k f:</s> <s xml:id="echoid-s22992" xml:space="preserve"> igitur per definitionem lineæ ſuper ſuperfi-<lb/>ciem erectæ, angulus e f k eſt rectus:</s> <s xml:id="echoid-s22993" xml:space="preserve"> ergo angulus t m d eſt rectus per 10 p 11:</s> <s xml:id="echoid-s22994" xml:space="preserve"> latera enim illos angu-<lb/>los continẽtia in æquidiſtantibus circulorum ſuperficiebus protracta, æqualia ſunt & æquidiſtan-<lb/>tia, ut patet ex præmiſsis.</s> <s xml:id="echoid-s22995" xml:space="preserve"> Cum ergo angulus d m t ſit rectus, & angulus g d t ſit rectus per 18 p 3:</s> <s xml:id="echoid-s22996" xml:space="preserve"> in <lb/>trigono ergo orthogonio d t g ducta eſt ab angulo ad baſim perpendicularis, quæ t m:</s> <s xml:id="echoid-s22997" xml:space="preserve"> ergo per 8 & <lb/>17 p 6 illud, quod fit ex ductu lineæ d m in lineam g m eſt æquale quadrato lineæ m t.</s> <s xml:id="echoid-s22998" xml:space="preserve"> Et quoniam <lb/>linea g t contingit circulum b t o, cum ſit in ſuperficie contingente ducta ad punctum contingẽtiæ, <lb/>quod eſt t:</s> <s xml:id="echoid-s22999" xml:space="preserve"> palàm, quoniam linea l g eſt æquidiſtans axi k d.</s> <s xml:id="echoid-s23000" xml:space="preserve"> Quoniam enim ſuperficies ſecũdum li-<lb/>neam longitudinis columnam contingẽs, quæ eſt l e t g, & ſuperficies ſecans columnã trans axem, <lb/>quæ eſt h d g l, ſunt erectæ ſuper baſium columnæ ſuperficies per 92 huius, & per 18 p 11.</s> <s xml:id="echoid-s23001" xml:space="preserve"> Ergo per 19 <lb/>p 11 earum communis ſectio, quæ eſt in propoſito, linea l g, ſuper eaſdem ſuperficies baſium perpen <lb/>dicularis erit.</s> <s xml:id="echoid-s23002" xml:space="preserve"> Aequidiſtabit ergo axi h k per 6 p 11:</s> <s xml:id="echoid-s23003" xml:space="preserve"> ergo etiam æquidiſtat lineæ f m per 30 p 1.</s> <s xml:id="echoid-s23004" xml:space="preserve"> Quia <lb/>ergo in trigono l d g linea f m æquidiſtat baſi l g, patet per 2 p 6, quòd linea f m ſecat illa latera pro-<lb/>portionaliter:</s> <s xml:id="echoid-s23005" xml:space="preserve"> eſt ergo proportio lineæ d f a d lineam f l, ſicut lineæ d m ad lineam m g:</s> <s xml:id="echoid-s23006" xml:space="preserve"> ergo permu-<lb/>tatim per 16 p 5 erit proportio lineæ d f ad lineã d m, ſicut lineæ f l ad lineam m g:</s> <s xml:id="echoid-s23007" xml:space="preserve"> ſed linea d f maior <lb/>eſt quàm linea d m per 19 p 1, quoniam in trigono f d m angulus f d m eſt rectus per 8 p 11:</s> <s xml:id="echoid-s23008" xml:space="preserve"> ergo & li-<lb/>nea f l eſt maior quàm linea m g.</s> <s xml:id="echoid-s23009" xml:space="preserve"> Ergo illud, quod fit ex ductu lineæ f d in lineam f l, maius eſt illo, <lb/>quod fit ex ductu lineæ d m in lineã m g.</s> <s xml:id="echoid-s23010" xml:space="preserve"> Ergo & quadrato lineæ t m:</s> <s xml:id="echoid-s23011" xml:space="preserve"> ſed linea t m eſt æqualis lineæ <lb/>e f, ut patet ex præmiſsis.</s> <s xml:id="echoid-s23012" xml:space="preserve"> Ergo illud, quod fit ex ductu lineæ d f in lineam f l maius eſt quadrato li-<lb/>neæ e f.</s> <s xml:id="echoid-s23013" xml:space="preserve"> Eſt ergo in trigono d e l angulus l e d maior recto per 30 huius:</s> <s xml:id="echoid-s23014" xml:space="preserve"> quia ſi eſſet rectus, cum linea <lb/>e f ſit perpendicularis ſuper lineã d l:</s> <s xml:id="echoid-s23015" xml:space="preserve"> eſſet per 8 & 17 p 6 illud, quod fit ex ductu lineæ d f in lineam f <lb/>l æquale quadrato lineæ e f.</s> <s xml:id="echoid-s23016" xml:space="preserve"> Reſtat ergo ut linea perpendicularis ſuper lineam contingẽtem ſectio-<lb/>nem a e c b (quæ eſt q l, ducta à puncto e) cadat ſub linea e d, nõ perueniens in punctum d.</s> <s xml:id="echoid-s23017" xml:space="preserve"> Sit ergo <lb/>illa perpendicularis linea e u.</s> <s xml:id="echoid-s23018" xml:space="preserve"> Et quia angulus e d b eſt acutus, & angulus d e u eſt acutus:</s> <s xml:id="echoid-s23019" xml:space="preserve"> quoniam <lb/>angulus u e q eſt rectus.</s> <s xml:id="echoid-s23020" xml:space="preserve"> Ergo per 14 huius lineæ e u & d b productæ concurrent in puncto aliquo <lb/>ſub axe h k, & ſub concurſu lineæ e d cum linea d b:</s> <s xml:id="echoid-s23021" xml:space="preserve"> quod eſt euidens.</s> <s xml:id="echoid-s23022" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s23023" xml:space="preserve"> per-<lb/>pendicularis enim ſuper lineam ſectionem contingentem, eſt perpendicularis ſuper ipſam ſectio-<lb/>nem columnarem per 5 definitionẽ factam in principio huius libri.</s> <s xml:id="echoid-s23024" xml:space="preserve"/> </p> <div xml:id="echoid-div839" type="float" level="0" n="0"> <figure xlink:label="fig-0346-01" xlink:href="fig-0346-01a"> <variables xml:id="echoid-variables350" xml:space="preserve">n q e t o l g f m d K d h c a s u p z b</variables> </figure> </div> <figure> <variables xml:id="echoid-variables351" xml:space="preserve">e b h a f c l m k d g</variables> </figure> </div> <div xml:id="echoid-div841" type="section" level="0" n="0"> <head xml:id="echoid-head689" xml:space="preserve" style="it">115. Omnis recta perpẽdicularis ſuper oxygoniam ſectionem, <lb/>productataliter diuidet ſectionem, ut in unaqua illarum par-<lb/>tium unic{us} tantùm ſit punct{us}, à quo ducta contingens æquidi-<lb/>ſtet ipſi perpendiculari.</head> <p> <s xml:id="echoid-s23025" xml:space="preserve">Eſto ſectio oxygonia, quę a b c d:</s> <s xml:id="echoid-s23026" xml:space="preserve"> quã perpẽdicularis e b d ſecet in <lb/>duas partes, quæ ſint b c d & b a d.</s> <s xml:id="echoid-s23027" xml:space="preserve"> Dico quòd in unaquaq;</s> <s xml:id="echoid-s23028" xml:space="preserve"> illarum <lb/>partium eſt unicus tantùm punctus, à quo ducta contingens æqui-<lb/>diſtat perpendiculari e b d.</s> <s xml:id="echoid-s23029" xml:space="preserve"> Quoniam enim perpẽdicularis e b d di-<lb/>uidit ſectionem, diuidatur eius pars b d cadens intra ſectionem per <lb/>æqualia per 10 p 1 in puncto f:</s> <s xml:id="echoid-s23030" xml:space="preserve"> & ab illo pũcto f erigatur per 11 p 1 per <lb/>pendicularis ſuper lineam b d:</s> <s xml:id="echoid-s23031" xml:space="preserve"> quę producta ad peripheriam ſectio-<lb/>nis in punctum c, ſit f c:</s> <s xml:id="echoid-s23032" xml:space="preserve"> & à puncto c ducatur perpendicularis ſuper <lb/>lineam f c, quæ ſit g c h:</s> <s xml:id="echoid-s23033" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s23034" xml:space="preserve"> linea g c h contingens ſectionem:</s> <s xml:id="echoid-s23035" xml:space="preserve"> quo-<lb/>niam ad utranq;</s> <s xml:id="echoid-s23036" xml:space="preserve"> partẽ producta non ſecabit illam.</s> <s xml:id="echoid-s23037" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s23038" xml:space="preserve">, quo-<lb/>niam linea g c n æquidiſtat perpendiculari ſuper ſectionem, quæ eſt <lb/>e b d per 28 p 1.</s> <s xml:id="echoid-s23039" xml:space="preserve"> Quòd ſi ab alio aliquo puncto partis ſectionis, quæ <lb/>b c d, ut à puncto k, producatur linea contingens ſectionem, quæ <lb/>ſit k l:</s> <s xml:id="echoid-s23040" xml:space="preserve"> patet, quoniam illa concurret cum linea g c h per 14 huíus:</s> <s xml:id="echoid-s23041" xml:space="preserve"> <lb/>quia ducta linea recta c k à puncto contactus c ad illum alium punctum k:</s> <s xml:id="echoid-s23042" xml:space="preserve"> fient anguli c k l & k c g <lb/>minores duobus rectis, ideo quòd angulus f c g eſt rectus, & linea k l cũ aliqua linea ſecante lineam <lb/> <pb o="46" file="0348" n="348" rhead="VITELLONIS OPTICAE"/> b d, continet angulum rectum, ut fortè cum linea k m.</s> <s xml:id="echoid-s23043" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s23044" xml:space="preserve"> anguli c k l & k c g ſunt minores <lb/>duobus rectis:</s> <s xml:id="echoid-s23045" xml:space="preserve"> concurret linea k l cum perpendiculari h c g per 14 huius.</s> <s xml:id="echoid-s23046" xml:space="preserve"> Ergo per 2 huius illa linea <lb/>contingens, quę k l, concurret cum perpendiculari e b d.</s> <s xml:id="echoid-s23047" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s23048" xml:space="preserve"> in parte ſectionis, quæ eſt <lb/>b a d, facta deductione, patet propoſitum.</s> <s xml:id="echoid-s23049" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div842" type="section" level="0" n="0"> <head xml:id="echoid-head690" xml:space="preserve" style="it">116. Omnes oxygoniæ pyramidales ſectiones ampliantur exparte baſis pyramidis: quod nõ <lb/>accidit in columnis.</head> <p> <s xml:id="echoid-s23050" xml:space="preserve">Hoc quod proponitur, accidit propter corporis pyramidalis acuitatẽ, & propter columnarum <lb/>æqualitatem.</s> <s xml:id="echoid-s23051" xml:space="preserve"> Si enim ſecundum punctum axis pyramidis, cui incidit linea perpendicularis ſuper <lb/>ſectionem pyramidalem, circumducatur pyramidi circulus per 102 huius, & imaginetur columna, <lb/>cuius baſis ſit ille circulus:</s> <s xml:id="echoid-s23052" xml:space="preserve"> patet, quòd inferior pars pyramidis excedit illam columnam, & colu-<lb/>mna excedit ſuperiorem partem pyramidis:</s> <s xml:id="echoid-s23053" xml:space="preserve"> & ſic inferior pars ſectionis pyramidalis continebit <lb/>inferiorem partem ſectionis columnaris, & ſuperior pars ſectionis columnaris cõtinebit ſuperio-<lb/>rem ſectionis partem pyramidalis.</s> <s xml:id="echoid-s23054" xml:space="preserve"> Partes autem ſectionis columnaris ſunt æquales propter æqua-<lb/>litatem corporis & angulorum ſuper axem per 92 huius.</s> <s xml:id="echoid-s23055" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s23056" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div843" type="section" level="0" n="0"> <head xml:id="echoid-head691" xml:space="preserve" style="it">117. Omnis ſuperficiei planæ ſuper axem fixum reuolutæ, donec ad locũ, unde exiuit, redeat, <lb/>linea mota deſcribit ſuperficiem corporis ſibi ſimilem, cui{us} ſuperficiei corporis & ſuperficiei <lb/>planæ ipſum corp{us} per axem ſecantis, communis ſectio eſt linea ſimilis motæ lineæ illam ſuper-<lb/>ficiem cauſſanti.</head> <p> <s xml:id="echoid-s23057" xml:space="preserve">Quod hic proponitur, patet ſatis euidenter in lineis rectis motis:</s> <s xml:id="echoid-s23058" xml:space="preserve"> quælibet enim illarum linea-<lb/>rum circa axem aliquem mota deſcribit ſuperficiem, cuius omnes lineæ ſunt ſimiles ipſi lineæ mo-<lb/>tæ cauſſanti motu ſuo illam ſuperficiem.</s> <s xml:id="echoid-s23059" xml:space="preserve"> Hoc enim patet in ſuperficie rectangula, quæ uno latere <lb/>fixo ſuo & alijs tribus motis deſcribit columnam rotundã, cuius ſuperficiei & ſuperficiei planę co-<lb/>lumnam per axem ſecantis, communis ſectio eſt linea ſimilis lineæ priori motæ.</s> <s xml:id="echoid-s23060" xml:space="preserve"> Et hoc idem patet <lb/>in triangulo moto, qui motu ſuorum duorum laterum, fixo tertio, efficit pyramidem rotundam:</s> <s xml:id="echoid-s23061" xml:space="preserve"> &, <lb/>ut patet per 90 huius, omnis ſuperficiei planæ ſecantis ipſam pyramidem per axem & ſuperficiei <lb/>conicæ pyramidis, communis ſectio eſt triangulus continens lineas ſimiles prioribus lineis motis <lb/>& axi.</s> <s xml:id="echoid-s23062" xml:space="preserve"> Hoc idem etiã in ſemicirculo moto, cuius diametro fixa deſcribitur ſphæra, & omnis ſuper-<lb/>ficiei planæ ſecantis ſphæram per axem, qui eſt diameter, & ſuperficiei ſphæricæ communis ſectio <lb/>eſt circulus, ut patent hæc omnia ex principijs lib.</s> <s xml:id="echoid-s23063" xml:space="preserve"> 11.</s> <s xml:id="echoid-s23064" xml:space="preserve"> Quòd ſi linea mota circa axem fixum (qui ſit <lb/> <anchor type="figure" xlink:label="fig-0348-01a" xlink:href="fig-0348-01"/> <anchor type="figure" xlink:label="fig-0348-02a" xlink:href="fig-0348-02"/> fg) fuerit compoſi-<lb/>ta ex lineis rectis, <lb/>ut ex a b & b c & c d <lb/>& d e, continẽtibus <lb/>angulos a b c, b c d, <lb/>c d e:</s> <s xml:id="echoid-s23065" xml:space="preserve"> uel ſi linea mo <lb/>ta fuerit compoſita <lb/>ex lineis rectis & <lb/>curuis actu, ut ſi a b <lb/>& c d ſint rectę, qua <lb/>rũ media b c utrãq;</s> <s xml:id="echoid-s23066" xml:space="preserve"> <lb/>rectarum illarũ co-<lb/>pulans, ſit curua, fiatq́;</s> <s xml:id="echoid-s23067" xml:space="preserve"> motus circa axẽ fixum, qui e f, fiet adhuc ſuperficies corporis deſcripti ſimi-<lb/>les habẽs lineas ipſis lineis cauſſantibus illam rotundam ſuperficiem motu ſuo.</s> <s xml:id="echoid-s23068" xml:space="preserve"> Quòd ſi linea mo-<lb/>ta fuerit compoſita eſſentialiter ex natura linearum rectarũ & curuarum, ut ſunt multæ lineæ, quæ <lb/> <anchor type="figure" xlink:label="fig-0348-03a" xlink:href="fig-0348-03"/> fiunt per motũ, uerbi gratia, aliqua ſectio co-<lb/>nica, ut ſi ſectionis parabolæ medietas, quæ <lb/>mouetur, ſit a b g, cuius axis a d, & ſit linea g <lb/>d perpẽdicularis ſuper ipſum axem a d, figa-<lb/>turq́;</s> <s xml:id="echoid-s23069" xml:space="preserve"> axis a d, & reuoluatur ſectio a b g, do-<lb/>nec redeat ad locum, à quo exiuit:</s> <s xml:id="echoid-s23070" xml:space="preserve"> tũc fiet ex <lb/>motu illius lineæ ſuperficies cõcaua uel con-<lb/>uexa, cuius baſis erit circulus proueniens ex <lb/>motu lineæ rectæ, quæ eſt d g:</s> <s xml:id="echoid-s23071" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s23072" xml:space="preserve"> ille circu-<lb/>lus g e z, & eius centrum eſt punctum d:</s> <s xml:id="echoid-s23073" xml:space="preserve"> quo-<lb/>niam punctum g motu ſuo illius circuli peri-<lb/>pheriam deſcribit, eritq́;</s> <s xml:id="echoid-s23074" xml:space="preserve"> uertex illius cauſſati <lb/>corporis pũctum a.</s> <s xml:id="echoid-s23075" xml:space="preserve"> Egrediatur quoq;</s> <s xml:id="echoid-s23076" xml:space="preserve"> ex axe <lb/>illius corporis, qui eſt a d, ſuperficies plana, <lb/>utcũq;</s> <s xml:id="echoid-s23077" xml:space="preserve"> id ſit poſsibile accidere, & ſecet illius corporis ſuperficiẽ.</s> <s xml:id="echoid-s23078" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s23079" xml:space="preserve"> per 3 p 11, quoniã illius <lb/>ſuperficiei & ſuperficiei corporis cõmunis eſt linea, quę ſit a h e.</s> <s xml:id="echoid-s23080" xml:space="preserve"> Dico, quòd linea a h e eſt ſectio pa-<lb/>rabola ęqualis & ſimilis ſectioni a b g.</s> <s xml:id="echoid-s23081" xml:space="preserve"> Ducatur enim linea d e, & imaginetur moueri ſectio a b g cir-<lb/>ca axẽ a d.</s> <s xml:id="echoid-s23082" xml:space="preserve"> Cum ergo punctũ g քuenit ad punctũ e, cooperit tota ſuքficies a b g d totã ſuperficiẽ a h <lb/>e d, & fiẽt ſuքficies una.</s> <s xml:id="echoid-s23083" xml:space="preserve"> Et quoniã ſectio a b g d facit euenire ſuքficiẽ concauã uel cõuexam:</s> <s xml:id="echoid-s23084" xml:space="preserve"> palàm, <lb/> <pb o="47" file="0349" n="349" rhead="LIBER PRIMVS."/> quoniam linea a b g d ſemper, ubicunq;</s> <s xml:id="echoid-s23085" xml:space="preserve"> reuoluatur ſectio, eſt cõmunis differẽtia inter ſuperficiem <lb/>ſibi continuam & inter ſuperficiem planam ſecantem.</s> <s xml:id="echoid-s23086" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s23087" xml:space="preserve"> ſuperponitur ſectio a b g d ſectio-<lb/>ni a h e d, erit communis ſectio inter ſuperficiem ſecantem & ſuperficiem corporis linea a b g d:</s> <s xml:id="echoid-s23088" xml:space="preserve"> ſed <lb/>& eadem cõmunis ſectio eſt linea a h e d.</s> <s xml:id="echoid-s23089" xml:space="preserve"> Linea ergo a b g d & linea a h e d ſibi adinuicem ſuperpo-<lb/>ſitæ ſunt linea una.</s> <s xml:id="echoid-s23090" xml:space="preserve"> Linea ergo a h e eſt peripheria ſectionis parabolæ æqualis & ſimilis lineæ a b g.</s> <s xml:id="echoid-s23091" xml:space="preserve"> <lb/>Superficies ergo a h e d eſt ſectio parabola.</s> <s xml:id="echoid-s23092" xml:space="preserve"> Et idẽ patet in omnibus lineis illius corporis, quę ſunt <lb/>communes ſectiones ſuperficiei planæ ſecãtis corpus per axem a d, & omnis ſuperficiei illius cor-<lb/>poris.</s> <s xml:id="echoid-s23093" xml:space="preserve"> Patet ergo propoſitum in illis ſectionibus conicis quibuſcunq;</s> <s xml:id="echoid-s23094" xml:space="preserve"> Patet etiam eodẽ modo pro-<lb/>poſitum de quacunq;</s> <s xml:id="echoid-s23095" xml:space="preserve"> linea regulari uel irregulari.</s> <s xml:id="echoid-s23096" xml:space="preserve"> Et hoc eſt propoſitum principale.</s> <s xml:id="echoid-s23097" xml:space="preserve"/> </p> <div xml:id="echoid-div843" type="float" level="0" n="0"> <figure xlink:label="fig-0348-01" xlink:href="fig-0348-01a"> <variables xml:id="echoid-variables352" xml:space="preserve">b d a c e f g</variables> </figure> <figure xlink:label="fig-0348-02" xlink:href="fig-0348-02a"> <variables xml:id="echoid-variables353" xml:space="preserve">a b c d e f</variables> </figure> <figure xlink:label="fig-0348-03" xlink:href="fig-0348-03a"> <variables xml:id="echoid-variables354" xml:space="preserve">a h b z d <gap/> g</variables> </figure> </div> </div> <div xml:id="echoid-div845" type="section" level="0" n="0"> <head xml:id="echoid-head692" xml:space="preserve" style="it">118. Omnis ſuperficies conuexa uel concaua regularis, aut eſt pars ſuperficiei ſphæræ: aut co-<lb/>lumnæ: aut pyramidis rotundæ.</head> <p> <s xml:id="echoid-s23098" xml:space="preserve">Omnis enim linea regularis, quę uniformis eſt in qualibet ſui parte, aut eſt circulus:</s> <s xml:id="echoid-s23099" xml:space="preserve"> aut linea re-<lb/>cta.</s> <s xml:id="echoid-s23100" xml:space="preserve"> Circulus uerò motu ſuo facit ſphæram:</s> <s xml:id="echoid-s23101" xml:space="preserve"> quoniam ſphæra eſt tranſitus circumferentiæ dimidij <lb/>circuli, ut patet ex principio 11.</s> <s xml:id="echoid-s23102" xml:space="preserve"> Linea uerò recta una motu ſuo non poteſt cauſſare niſi pyramidem, <lb/>cum eſt latus trigoni, uel columnam, cum eſt latus quadranguli:</s> <s xml:id="echoid-s23103" xml:space="preserve"> quoniam in omnibus alijs figuris <lb/>motis, uno latere remanente fixo, eſt angulus cauſſans diuerſitatem formæ in ſuperficie figuræ pro <lb/>ductæ.</s> <s xml:id="echoid-s23104" xml:space="preserve"> Non ergo efficit conuexam ſuperficiem uel concauam regularem.</s> <s xml:id="echoid-s23105" xml:space="preserve"> Patet ergo, quòd omnis <lb/>ſuperficies conuexa uel concaua regularis eſt talis, ut proponitur.</s> <s xml:id="echoid-s23106" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div846" type="section" level="0" n="0"> <head xml:id="echoid-head693" xml:space="preserve" style="it">119. Lineã datam ſecundũ quamlibet proportionẽ duarum datarũ diuidere. 10 p 6 element.</head> <figure> <variables xml:id="echoid-variables355" xml:space="preserve">c d e f a g k h b</variables> </figure> <p> <s xml:id="echoid-s23107" xml:space="preserve">Sit linea a b data, quæ debeat diuidi ſecundũ proportionem dua-<lb/>rum datarum linearum c d & e f.</s> <s xml:id="echoid-s23108" xml:space="preserve"> A puncto itaq;</s> <s xml:id="echoid-s23109" xml:space="preserve"> a datæ lineæ a b du-<lb/>catur linea indefinitè angulariter coniuncta cum linea a b:</s> <s xml:id="echoid-s23110" xml:space="preserve"> & à pun-<lb/>cto a incipiendo abſcindatur æqualis lineæ c d per 3 p 1, quę ſit a g, & <lb/>à puncto g incipiendo abſcindatur linea g h æqualis lineæ e f:</s> <s xml:id="echoid-s23111" xml:space="preserve"> & du-<lb/>catur linea b h:</s> <s xml:id="echoid-s23112" xml:space="preserve"> & à puncto g ducatur linea æquidiſtanter lineæ b h <lb/>per 31 p 1:</s> <s xml:id="echoid-s23113" xml:space="preserve"> hęc itaq;</s> <s xml:id="echoid-s23114" xml:space="preserve"> producta ſecabit lineam a b per 2 huius:</s> <s xml:id="echoid-s23115" xml:space="preserve"> ſecet ergo <lb/>in puncto k.</s> <s xml:id="echoid-s23116" xml:space="preserve"> Linea itaq;</s> <s xml:id="echoid-s23117" xml:space="preserve"> a b indiuiſa propoſita erit diuiſa ſecundum <lb/>modũ diuiſionis lineæ a h diuiſæ:</s> <s xml:id="echoid-s23118" xml:space="preserve"> erit enim per 2 p 6 proportio lineæ <lb/>a k ad lineam k b, ſicut lineæ a g ad lineam g h.</s> <s xml:id="echoid-s23119" xml:space="preserve"> Ergo ſicut lineæ c d ad <lb/>lineam e f per 7 p 5.</s> <s xml:id="echoid-s23120" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s23121" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div847" type="section" level="0" n="0"> <head xml:id="echoid-head694" xml:space="preserve" style="it">120. Ducta à puncto dato linea, aliam lineam ſecũdum datam <lb/>proportionem partium illarum linearum ſecãte: ab eodem puncto <lb/>inter eaſdem rectas, quæ pri{us} diuiſam ab eiſdem terminis ſerua-<lb/>ta denominatione proportionis, ſecundum eandem proportionem <lb/>ſecet, aliam lineam duci eſt impoßibile.</head> <p> <s xml:id="echoid-s23122" xml:space="preserve">Verbi gratia:</s> <s xml:id="echoid-s23123" xml:space="preserve"> ſit, ut linea a b ducta à dato puncto a, ſecet lineam d e <lb/>in puncto c ſecundum aliquam datam proportionem.</s> <s xml:id="echoid-s23124" xml:space="preserve"> Dico, quòd à <lb/>puncto a non poteſt duci alia linea ad lineam d c, quę ipſam ſecet ſe-<lb/>cundum eandem datam proportionem, ita, ut denominatio proportionis, ſeruetur ab eiſdem ter-<lb/>minis lineæ d e.</s> <s xml:id="echoid-s23125" xml:space="preserve"> Si enim à puncto a lineam aliam duci taliter ſit poſsibile, fiat ſuper punctum d ter-<lb/>minum lineæ e d per 23 p 1 angulus maior recto uerſus punctum b terminum lineæ a b:</s> <s xml:id="echoid-s23126" xml:space="preserve"> & produca-<lb/>tur linea b d, fiatq́;</s> <s xml:id="echoid-s23127" xml:space="preserve"> angulus c d b obtuſus:</s> <s xml:id="echoid-s23128" xml:space="preserve"> & producatur linea d b in continuum uerſus punctum at <lb/> <anchor type="figure" xlink:label="fig-0349-02a" xlink:href="fig-0349-02"/> & à puncto a ducatur linea perpendicularis ſuper li-<lb/>neam d b, quæ ſit a f:</s> <s xml:id="echoid-s23129" xml:space="preserve"> & ducatur linea a g ſecans lineã <lb/>e d in puncto h ſecundũ proportionem prius datam, <lb/>quę eſt lineæ d c ad lineã c e:</s> <s xml:id="echoid-s23130" xml:space="preserve"> & ducatur linea h i æqui-<lb/>diſtans lineæ c b per 31 p 1.</s> <s xml:id="echoid-s23131" xml:space="preserve"> Erit itaque linea h i maior <lb/>quã linea h g per 19 p 1.</s> <s xml:id="echoid-s23132" xml:space="preserve"> Angulus enim i g h eſt maior <lb/>recto b f a per 16 p 1:</s> <s xml:id="echoid-s23133" xml:space="preserve"> angulus uerò b f a rectus eſt ma-<lb/>ior angulo f b a per 32 p 1:</s> <s xml:id="echoid-s23134" xml:space="preserve"> ſed angulus g i h eſt per 29 <lb/>p 1 æqualis angulo f b a:</s> <s xml:id="echoid-s23135" xml:space="preserve"> angulus ergo i g h eſt maior <lb/>angulo g i h.</s> <s xml:id="echoid-s23136" xml:space="preserve"> Ergo per 19 p 1 linea i h eſt maior quàm <lb/>linea h g.</s> <s xml:id="echoid-s23137" xml:space="preserve"> Et ducatur à puncto h linea h k æquidiſtans <lb/>lineæ d b:</s> <s xml:id="echoid-s23138" xml:space="preserve"> erit ergo per 34 p 1 linea b k æqualis lineæ i <lb/>h:</s> <s xml:id="echoid-s23139" xml:space="preserve"> ſed linea b c eſt maior quàm linea k b:</s> <s xml:id="echoid-s23140" xml:space="preserve"> ergo linea c b <lb/>eſt maior quàm linea h i:</s> <s xml:id="echoid-s23141" xml:space="preserve"> ergo c b eſt maior quã linea <lb/>h g:</s> <s xml:id="echoid-s23142" xml:space="preserve"> ſed & linea h e maior eſt quã linea c e, quoniã totũ <lb/>maius eſt ſua parte:</s> <s xml:id="echoid-s23143" xml:space="preserve"> erit ergo per 9 huius maior pro-<lb/>portio lineæ b c ad lineam c e, quàm lineę g h ad lineã <lb/>h e.</s> <s xml:id="echoid-s23144" xml:space="preserve"> Non eſt ergo eadẽ proportio:</s> <s xml:id="echoid-s23145" xml:space="preserve"> quod eſt contra hypotheſim:</s> <s xml:id="echoid-s23146" xml:space="preserve"> aut ſequetur lineam e c eſſe maiorem <lb/>quàm ſit linea e h per 14 p 5:</s> <s xml:id="echoid-s23147" xml:space="preserve"> quod totũ eſt impoſsibile.</s> <s xml:id="echoid-s23148" xml:space="preserve"> Faciliter uerò idẽ patet in linea d e, cũ linea <lb/>d h ſit minor quã linea d c, & h e ſit maior quã c e:</s> <s xml:id="echoid-s23149" xml:space="preserve"> per 9 ergo huius cõcludatur, ut prius.</s> <s xml:id="echoid-s23150" xml:space="preserve"> Nõ eſt ergo <lb/>poſsibile à puncto a duci aliã lineã ſecantẽ lineã d e ſecundũ datã proportionẽ.</s> <s xml:id="echoid-s23151" xml:space="preserve"> Quod eſt ppoſitu.</s> <s xml:id="echoid-s23152" xml:space="preserve"/> </p> <div xml:id="echoid-div847" type="float" level="0" n="0"> <figure xlink:label="fig-0349-02" xlink:href="fig-0349-02a"> <variables xml:id="echoid-variables356" xml:space="preserve">e a c k h b i g d f</variables> </figure> </div> <pb o="48" file="0350" n="350" rhead="VITELLONIS OPTICAE"/> </div> <div xml:id="echoid-div849" type="section" level="0" n="0"> <head xml:id="echoid-head695" xml:space="preserve" style="it">121. Lineam datam in duob{us} punctis taliter ſecare, ut ſui toti{us} proportio ad unã ſuarum <lb/>extremarum partium ſit ſimilis proportioni alteri{us} extremæ partis ad eam partẽ, quæ utraſ <lb/>interiacet ſectiones. E 10 p 6 element.</head> <p> <s xml:id="echoid-s23153" xml:space="preserve">Eſto data linea a b, quam ſecundum modum propoſitum debemus diuidere.</s> <s xml:id="echoid-s23154" xml:space="preserve"> Diuidatur itaq;</s> <s xml:id="echoid-s23155" xml:space="preserve"> ſe-<lb/>cundum proportionem, quam libuerit:</s> <s xml:id="echoid-s23156" xml:space="preserve"> & ſit diuiſa in puncto c:</s> <s xml:id="echoid-s23157" xml:space="preserve"> & ſit pars eius a c maior quàm pars <lb/> <anchor type="figure" xlink:label="fig-0350-01a" xlink:href="fig-0350-01"/> cius c b.</s> <s xml:id="echoid-s23158" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s23159" xml:space="preserve"> propoſitæ ſunt nobis tres <lb/>lineæ a b, a c, c b:</s> <s xml:id="echoid-s23160" xml:space="preserve"> diuidatur ergo per 119 huius <lb/>linea a c ſecundum portionem lineæ a b ad li-<lb/>neam c b:</s> <s xml:id="echoid-s23161" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s23162" xml:space="preserve"> diuiſio in puncto d ita, ut ſit proportio lineæ a d ad lineam d c, ſicut lineæ totius a b <lb/>ad lineã c b.</s> <s xml:id="echoid-s23163" xml:space="preserve"> Palàm ergo, quòd linea a b eſt modo propoſito diuiſa:</s> <s xml:id="echoid-s23164" xml:space="preserve"> eſt enim proportio totius lineæ <lb/>a b ad unam extremarum ſuarum partium, quæ eſt c b, ſicut reliquæ ſuæ partis extremę, quæ eſt a d, <lb/>ad partem, quę utraſq;</s> <s xml:id="echoid-s23165" xml:space="preserve"> interiacet ſectiones, quę eſt d c.</s> <s xml:id="echoid-s23166" xml:space="preserve"> Patet ergo factum eſſe, quod proponebatur.</s> <s xml:id="echoid-s23167" xml:space="preserve"/> </p> <div xml:id="echoid-div849" type="float" level="0" n="0"> <figure xlink:label="fig-0350-01" xlink:href="fig-0350-01a"> <variables xml:id="echoid-variables357" xml:space="preserve">a d c b</variables> </figure> </div> </div> <div xml:id="echoid-div851" type="section" level="0" n="0"> <head xml:id="echoid-head696" xml:space="preserve" style="it">122. Diuiſa linea recta taliter, ut ſuitoti{us} proportio ad unam ſuarum extremarũ partium <lb/>ſit ſimilis proportioni partis alteri{us} extremæ ad eam ſui partem, quæutraſ interiacet ſectio-<lb/>nes: ſi fuerint lineæ ductæ abuno termino datæ lineæ, & à punctis ſectionum æquidiſt antes in-<lb/>ter ſe: à terminó reliquo datæ lineæ producatur linea ſecans illas tres æquidiſtantes: erit linea <lb/>producta ſecundum eandem proportionem diuiſa. Alhazen 10 n 6.</head> <figure> <variables xml:id="echoid-variables358" xml:space="preserve">c h z b d g d</variables> </figure> <p> <s xml:id="echoid-s23168" xml:space="preserve">Sit linea a b diuiſa in punctis g & d taliter, ut lineę <lb/>a b ad lineam d b ſit proportio, ſicut lineæ a g ad li-<lb/>neam g d:</s> <s xml:id="echoid-s23169" xml:space="preserve"> & ab uno termino datę lineæ, qui eſt b, & <lb/>à punctis ſectionũ g & d per 31 primi ducantur lineæ <lb/>ad inuicem æquidiſtantes, quæ ſint b c, d h, g z:</s> <s xml:id="echoid-s23170" xml:space="preserve"> & ab <lb/>altero termino datæ lineæ, quę eſt a, producatur li-<lb/>nea ſecans illas æquidiſtantes in punctis z, h, c, quæ <lb/>ſit a z h c.</s> <s xml:id="echoid-s23171" xml:space="preserve"> Dico, quòd linea a c ſecundũ hanc propor-<lb/>tionem erit diuiſa.</s> <s xml:id="echoid-s23172" xml:space="preserve"> Cũ enim linea d h ſit æquidiſtans <lb/>lineæ g z ex hypotheſi, erit ex 2 p 6 proportio lineæ <lb/>a z ad lineã z h, ſicut lineæ a g ad lineam g d.</s> <s xml:id="echoid-s23173" xml:space="preserve"> Et cum <lb/>linea b c ſit æquidiſtans lineæ d h, erit per eandem 2 <lb/>p 6 & 18 p 5 proportio lineæ a b ad lineam b d, ſicut <lb/>lineæ a c ad lineam c h:</s> <s xml:id="echoid-s23174" xml:space="preserve"> ſed ex hypotheſi fuit propor-<lb/>tio lineæ a b ad lineam b d, ſicut lineæ a g ad lineam <lb/>d g.</s> <s xml:id="echoid-s23175" xml:space="preserve"> Erit ergo per 11 p 5 proportio lineæ a c ad lineam <lb/>c h, ſicut lineæ a z ad lineam z h.</s> <s xml:id="echoid-s23176" xml:space="preserve"> Linea ergo a c, quæ <lb/>producitur à puncto a termino lineæ datæ, ſecat du-<lb/>ctas lineas æquidiſtantes b c, d h, g z, & ſecatur per illas ſecundum proportionem partium diuiſio-<lb/>nis lineæ datæ a b.</s> <s xml:id="echoid-s23177" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s23178" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div852" type="section" level="0" n="0"> <head xml:id="echoid-head697" xml:space="preserve" style="it">123. Linea in duob{us} punctis taliter diuiſa, ut ſui toti{us} proportio adunam ſuarum extre-<lb/>marum partium ſimilis ſit proportioni alteri{us} extremæ partis ad eam ſui partem, quæ utraſ <lb/>interiacet ſectiones: ſi ab uno termino illi{us} lineæ, & à punctis ſectionis ducantur tres lineæ con <lb/>currentes in punctum unum, & ab alio termino producatur linea ſecans illas tres ductas: erit <lb/>linea producta ſecundum prædictum modum pro <lb/>portionaliter diuiſa. Alhazen 8 n 6.</head> <figure> <variables xml:id="echoid-variables359" xml:space="preserve">e c q h m z b d g a</variables> </figure> <p> <s xml:id="echoid-s23179" xml:space="preserve">Eſto linea propoſita a b taliter diuiſa in punctis g <lb/>& d, ut ſit proportio totius lineæ a b ad lineam b d, <lb/>ſicut lineæ a g ad lineam g d:</s> <s xml:id="echoid-s23180" xml:space="preserve"> & à puncto b, & à pun-<lb/>ctis ſectionũ g & d ducantur tres lineæ concurren-<lb/>tes in unum punctũ e, quę ſint g e, d e, b e:</s> <s xml:id="echoid-s23181" xml:space="preserve"> & à pũcto <lb/>a ducatur linea, quæ ſit a c, ſecãs illas tres lineas, ſci-<lb/>licet g e in puncto z, & d e in puncto h, & b e in pun-<lb/>cto c.</s> <s xml:id="echoid-s23182" xml:space="preserve"> Dico, quòd erit proportio lineæ a c ad lineam <lb/>c h, ſicut lineæ a z ad lineã z h.</s> <s xml:id="echoid-s23183" xml:space="preserve"> Ducatur enim à pun-<lb/>cto h linea æquidiſtans lineæ a b per 31 p 1, quę ſit q <lb/>h.</s> <s xml:id="echoid-s23184" xml:space="preserve"> Palàm ergo per 13 huius, quoniã <gap/> proportio lineæ <lb/>a b ad lineam b d, conſtat ex proportionibus lineæ <lb/>a b ad lineam h q, & lineæ h q ad lineã b d.</s> <s xml:id="echoid-s23185" xml:space="preserve"> Sed quo-<lb/>niam linea q h ęquidiſtat lineę a b, erit per 29 p 1 an <lb/>gulus c q h ęqualis angulo c b a:</s> <s xml:id="echoid-s23186" xml:space="preserve"> ſed angulus c b a eſt <lb/>communis ambobus trigonis a b c & q h c:</s> <s xml:id="echoid-s23187" xml:space="preserve"> ergo per <lb/>32 p 1 illa trigona ſunt ęquiangula.</s> <s xml:id="echoid-s23188" xml:space="preserve"> Ergo per 4 p 6 erit proportio lineę a b ad lineam q h, ſicut lineę a <lb/>cad lineam c h.</s> <s xml:id="echoid-s23189" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s23190" xml:space="preserve"> trigona q e h & b e d ſunt ſimilia.</s> <s xml:id="echoid-s23191" xml:space="preserve"> Eſt ergo proportio lineę q h ad li-<lb/>neam b d, ſicut lineę h e ad lineam d e.</s> <s xml:id="echoid-s23192" xml:space="preserve"> Proportio ergo lineę a b ad lineam b d per 13 huius componi-<lb/> <pb o="49" file="0351" n="351" rhead="LIBER PRIMVS."/> tur ex proportione lineæ a c ad lineam e h, & lineę h e ad lineam e d.</s> <s xml:id="echoid-s23193" xml:space="preserve"> Producatur itaque in directum <lb/>linea q h ad lineam ge, quã ſecet in puncto m.</s> <s xml:id="echoid-s23194" xml:space="preserve"> Proportio itaq;</s> <s xml:id="echoid-s23195" xml:space="preserve"> lineę a g ad lineam g d per 13 huius cõ <lb/>ſtat ex proportione lineæ a g ad lineã h m, & lineæ h m ad lineam g d.</s> <s xml:id="echoid-s23196" xml:space="preserve"> Sed cũ angulus e m h ſit ęqua-<lb/>lis angulo z g d per 29 p 1, erit per 13 & 29 p 1 angulus h m z æqualis angulo z g a:</s> <s xml:id="echoid-s23197" xml:space="preserve"> ergo per 15 & 32 p 1 <lb/>triangulus a g z erit æquiangulus triangulo h z m.</s> <s xml:id="echoid-s23198" xml:space="preserve"> Ergo per 4 p 6 erit proportio lineæ a z ad lineam <lb/>h z, ſicut lineæ a g ad lineam h m:</s> <s xml:id="echoid-s23199" xml:space="preserve"> ſed triangulus h e m, ut ſuprà patuit, ſimilis erit triangulo g e d:</s> <s xml:id="echoid-s23200" xml:space="preserve"> erit <lb/>ergo proportio lineæ h m ad lineam d g, ſicut lineę h e ad lineam d e.</s> <s xml:id="echoid-s23201" xml:space="preserve"> Ergo proportio lineæ a g ad li-<lb/>neam d g conſtat ex proportione lineę a z ad lineam z h, & lineæ h e ad lineam e d:</s> <s xml:id="echoid-s23202" xml:space="preserve"> ſed ex hypotheſi <lb/>eadem eſt proportio lineæ a b ad lineam b d, quæ lineæ a g ad lineam g d.</s> <s xml:id="echoid-s23203" xml:space="preserve"> Proportio igitur lineæ a b <lb/>ad lineam b d conſtat ex proportione lineæ a z ad lineam z h, & lineæ h e ad lineam e d:</s> <s xml:id="echoid-s23204" xml:space="preserve"> conſtabat au <lb/>tem ex proportione lineæ a c ad lineam c h, & lineæ h e ad lineam e d.</s> <s xml:id="echoid-s23205" xml:space="preserve"> Ablata ergo utrinque propor <lb/>tione lineæ h e ad lineam d e:</s> <s xml:id="echoid-s23206" xml:space="preserve"> reſtat, ut ſit eadem proportio lineæ a c ad lineam c h, quæ lineæ a z ad <lb/>lineam z h.</s> <s xml:id="echoid-s23207" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s23208" xml:space="preserve"> Non tamẽ oportet, quòd lineę a b & a c ſint eiuſdem ſpeciei pro-<lb/>portionis reſpectu ſuarum partium:</s> <s xml:id="echoid-s23209" xml:space="preserve"> quoniam cum ex præmiſsis lineæ a b ad lineam q h ſit propor-<lb/>tio, quæ lineæ a c ad lineam c h, & linea q h ſit minor quã linea b d per 4 p 6:</s> <s xml:id="echoid-s23210" xml:space="preserve"> palàm per 8 p 5, quoniã <lb/>minor eſt proportio lineæ a b ad lineam b d quàm ſit lineę a c ad lineam c h.</s> <s xml:id="echoid-s23211" xml:space="preserve"> Sunt ergo proportiona <lb/>les ſecundum generalem ſimilitudinem proportionis.</s> <s xml:id="echoid-s23212" xml:space="preserve"> Eadem quoque demonſtratio eſt, quęcunq;</s> <s xml:id="echoid-s23213" xml:space="preserve"> <lb/>lineæ ducantur à puncto a, ſecantes illas tres lineas à tribus punctis a, d, g ad quodcunque punctũ <lb/>productas, ut ſupra e, uel ſub e, uel etiam ad aliam partem lineę a b:</s> <s xml:id="echoid-s23214" xml:space="preserve"> ſemper enim linea ducta à pun-<lb/>cto a ſecans illas tres lineas, ſecabitur modo dicto.</s> <s xml:id="echoid-s23215" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s23216" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div853" type="section" level="0" n="0"> <head xml:id="echoid-head698" xml:space="preserve" style="it">124. Duab{us} lineis angulariter coniunctis, diuiſiś ſic ambab{us}, ut cui{us}libet ipſarum pro-<lb/>portio adunam ſuarum extremarum partium ſit, ſicut alteri{us} extremæ partis ad illa ſui par-<lb/>tem, quæutraſ interiacet ſectiones: ſi producta baſi à punctis diuiſionis uni{us} ducantur lineæ <lb/>ad puncta diuiſionis alteri{us}, non æquidiſtantes adinuicẽ, ne baſi: neceſſe eſt productas lineas <lb/>ambas concurrere cum baſi, producta in puncto uno. Alhazen 9 n 6.</head> <p> <s xml:id="echoid-s23217" xml:space="preserve">Sit data linea a b taliter, ut proponitur, diuiſa in punctis d & g ſcilicet, ut ſit proportio totius li-<lb/>neæ a b ad lineam b d, ſicut lineę a g ad lineam g d, a diunctaq́;</s> <s xml:id="echoid-s23218" xml:space="preserve"> ſibi angulariter linea a c eodem modo <lb/> <anchor type="figure" xlink:label="fig-0351-01a" xlink:href="fig-0351-01"/> diuiſa in punctis h, z ita, ut ſit proportio lineæ a c ad <lb/>c h, ſicut lineæ a z, ad z h:</s> <s xml:id="echoid-s23219" xml:space="preserve"> ſi producatur baſis b c, ut <lb/>fiat triangulus b c a, & protrahatur b c in directum, <lb/>& ducantur lineæ à punctis ſectionum unius ad pũ <lb/>ctum ſectionis alterius, ut d h, g z, protrahanturq́ue <lb/>omnes illæ lineę in continuum & directum.</s> <s xml:id="echoid-s23220" xml:space="preserve"> Dico, <lb/>quòd omnes concurrent in puncto uno.</s> <s xml:id="echoid-s23221" xml:space="preserve"> Cum enim <lb/>lineæ b c & d h non ſint æquidiſtantes ex hypothe-<lb/>ſi, patet quòd neceſſariò concurrent:</s> <s xml:id="echoid-s23222" xml:space="preserve"> concurrant er <lb/>go in puncto, quod ſit e:</s> <s xml:id="echoid-s23223" xml:space="preserve"> linea quoque g z neceſſariò <lb/>concurret cum illis:</s> <s xml:id="echoid-s23224" xml:space="preserve"> cum non æquidiſtet alicui illa-<lb/>rum.</s> <s xml:id="echoid-s23225" xml:space="preserve"> Aut ergo ad idem punctum e.</s> <s xml:id="echoid-s23226" xml:space="preserve"> Et ſic habemus <lb/>propoſitum.</s> <s xml:id="echoid-s23227" xml:space="preserve"> Aut ad alium punctum cum aliqua il-<lb/>larum concurret:</s> <s xml:id="echoid-s23228" xml:space="preserve"> ſit illud punctum n, in quo concur <lb/>rit cum linea d e.</s> <s xml:id="echoid-s23229" xml:space="preserve"> Ducatur itaque linea e g:</s> <s xml:id="echoid-s23230" xml:space="preserve"> ſecabit <lb/>ergo linea e g lineam a c in alio pũcto, ꝗ̃ in puncto <lb/>z:</s> <s xml:id="echoid-s23231" xml:space="preserve"> quoniã in pũcto z ſecat ipſam linea n g:</s> <s xml:id="echoid-s23232" xml:space="preserve"> ſit illud pũ <lb/>ctũ l.</s> <s xml:id="echoid-s23233" xml:space="preserve"> Erit ergo per pręmiſſa proportio lineæ a c ad li <lb/>neã c h, ſicut lineæ a l ad lineã l h:</s> <s xml:id="echoid-s23234" xml:space="preserve"> fuit aũt ex hypotheſi proportio lineæ a c ad lineam c h, ſicut lineę a <lb/>z ad lineã z h:</s> <s xml:id="echoid-s23235" xml:space="preserve"> ergo per 11 p 5 erit proportio lineæ a l ad lineam l h, ſicut lineæ a z ad lineã z h:</s> <s xml:id="echoid-s23236" xml:space="preserve"> ergo ք <lb/>18 p 5 erit proportio lineæ a h ad lineã h z, ſicut lineæ a h ad lineã h l:</s> <s xml:id="echoid-s23237" xml:space="preserve"> erit ergo per 9 p 5 linea h z æqua <lb/>lis lineæ h l, maior minori:</s> <s xml:id="echoid-s23238" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s23239" xml:space="preserve"> Idẽ etiã patet per 120 huius, quoniã à puncto g ꝓdu <lb/>ctæ ſunt duæ lineę ſecantes lineã a h.</s> <s xml:id="echoid-s23240" xml:space="preserve"> Palàm ergo, quòd linea g z nõ concurret cũ lineis b c, d h in a-<lb/>lio puncto quã in puncto e.</s> <s xml:id="echoid-s23241" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s23242" xml:space="preserve"> Similiter ſi ponatur quòd linea g z concurrat cũ <lb/>linea d h in puncto e:</s> <s xml:id="echoid-s23243" xml:space="preserve"> erit prædicto modo demonſtrandum, quòd linea b c concurret cum ambabus <lb/>illis in puncto e.</s> <s xml:id="echoid-s23244" xml:space="preserve"> Et ſi lineæ b c & g z concurrant in puncto e, cõcurret linea d h cum eiſdem in eo-<lb/>dem puncto e.</s> <s xml:id="echoid-s23245" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s23246" xml:space="preserve"/> </p> <div xml:id="echoid-div853" type="float" level="0" n="0"> <figure xlink:label="fig-0351-01" xlink:href="fig-0351-01a"> <variables xml:id="echoid-variables360" xml:space="preserve">e n c h z l b d g a</variables> </figure> </div> </div> <div xml:id="echoid-div855" type="section" level="0" n="0"> <head xml:id="echoid-head699" xml:space="preserve" style="it">125. Linea taliter diuiſa, ut ſui toti{us} ad alteram ſuarum extremarum partiũ ſit proportio, <lb/>ſicut alteri{us} ſuæ partis extremæ ad eam ſui partem, quæutraſ interiacet ſectiones: ſi à puncto <lb/>concurſ{us} linearum à termino, & à duob{us} punctis ſectionis product arum in puncto concurſ{us} <lb/>æquales angulos cõtinentium, linea ad alium ei{us} terminũ ducatur: neceſſe eſt ipſam ſuper me-<lb/>diam productarum perpendicularem eſſe.</head> <p> <s xml:id="echoid-s23247" xml:space="preserve">Sit linea b k in punctis c & d taliter diuiſa, ut proponitur:</s> <s xml:id="echoid-s23248" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s23249" xml:space="preserve"> proportio lineæ b k ad lineam k d, <lb/>ſicut lineę b c ad lineam c d:</s> <s xml:id="echoid-s23250" xml:space="preserve"> producanturq́ue à punctis b, c, d lineę nõ æquidiſtantes:</s> <s xml:id="echoid-s23251" xml:space="preserve"> quę per proxi-<lb/> <pb o="50" file="0352" n="352" rhead="VITELLONIS OPTICAE"/> mam concurrent in puncto uno:</s> <s xml:id="echoid-s23252" xml:space="preserve"> ſit punctus concurſus z:</s> <s xml:id="echoid-s23253" xml:space="preserve"> & lineæ productę ſint b z, c z, d z:</s> <s xml:id="echoid-s23254" xml:space="preserve"> ſitq́ue an <lb/>gulus b z c ęqualis angulo c z d:</s> <s xml:id="echoid-s23255" xml:space="preserve"> & ducatur linea z k.</s> <s xml:id="echoid-s23256" xml:space="preserve"> Dico, quòd angulus c z k eſt rectus.</s> <s xml:id="echoid-s23257" xml:space="preserve"> A puncto <lb/> <anchor type="figure" xlink:label="fig-0352-01a" xlink:href="fig-0352-01"/> enim c ducatur per 31 p 1 linea ęquidiſtãs lineę <lb/>z k, quæ ſit c h:</s> <s xml:id="echoid-s23258" xml:space="preserve"> quæ producta ſecabit lineam <lb/>z b per 2 huius:</s> <s xml:id="echoid-s23259" xml:space="preserve"> ſecet ergo ipſam in puncto g:</s> <s xml:id="echoid-s23260" xml:space="preserve"> <lb/>& producatur linea z d, donec concurrat cum <lb/>linea g c h (concurret autem per 2 huius) & ſit <lb/>cõcurſus punctus h.</s> <s xml:id="echoid-s23261" xml:space="preserve"> Quia igitur ex hypotheſi <lb/>eſt proportio lineę b k ad lineam k d, ſicut li-<lb/>neę b c ad lineam c d, erit per 16 p 5 permuta-<lb/>tim proportio lineę b k ad lineam b c, ſicut li-<lb/>neę k d ad lineam d c:</s> <s xml:id="echoid-s23262" xml:space="preserve"> ſed per 29 p 1 trigona b z <lb/>k & b g c ſunt ęquiangula:</s> <s xml:id="echoid-s23263" xml:space="preserve"> ergo per 4 p 6 eſt proportio lineę b k ad lineam b c, quę eſt lineę z k ad li-<lb/>neam g c:</s> <s xml:id="echoid-s23264" xml:space="preserve"> ergo per 11 p 5 erit proportio lineæ z k ad lineam g c, ſicut lineæ k d ad lineam d c:</s> <s xml:id="echoid-s23265" xml:space="preserve"> ſed quæ <lb/>eſt proportio lineæ k d ad lineam d c, eadem eſt lineæ k z ad lineam c h per 15 & 29 p 1 & per 4 p 6:</s> <s xml:id="echoid-s23266" xml:space="preserve"> ꝗa <lb/>trigona k d z & c d h ſunt æquiangula.</s> <s xml:id="echoid-s23267" xml:space="preserve"> Habet itaque linea z k ad ambas lineas g c & h c eandem pro-<lb/>portionem:</s> <s xml:id="echoid-s23268" xml:space="preserve"> ergo per 9 p 5 linea g c eſt æqualis lineæ c h:</s> <s xml:id="echoid-s23269" xml:space="preserve"> ſed per 3 p 6 eſt proportio lineæ g c ad lineã <lb/>c h, ſicut lineæ g z ad lineam z h, cum linea z c diuidat angulum g z h per æqualia.</s> <s xml:id="echoid-s23270" xml:space="preserve"> Eſt ergo linea g z <lb/>æqualis lineę z h.</s> <s xml:id="echoid-s23271" xml:space="preserve"> Et quoniam linea g c eſt æqualis lineæ c h, & linea g z ęqualis lineæ z h, & latus c z <lb/>eſt commune ambobus trigonis g z c & h z c:</s> <s xml:id="echoid-s23272" xml:space="preserve"> erit per 8 p 1 angulus z c h æqualis angulo z c g:</s> <s xml:id="echoid-s23273" xml:space="preserve"> uter-<lb/>que ergo ipſorũ eſt rectus.</s> <s xml:id="echoid-s23274" xml:space="preserve"> Ergo per 29 p 1 angulus k z c eſt rectus:</s> <s xml:id="echoid-s23275" xml:space="preserve"> lineæ enim z k & c h ſunt æquidi-<lb/>ſtantes.</s> <s xml:id="echoid-s23276" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s23277" xml:space="preserve"/> </p> <div xml:id="echoid-div855" type="float" level="0" n="0"> <figure xlink:label="fig-0352-01" xlink:href="fig-0352-01a"> <variables xml:id="echoid-variables361" xml:space="preserve">z b g c d k h</variables> </figure> </div> </div> <div xml:id="echoid-div857" type="section" level="0" n="0"> <head xml:id="echoid-head700" xml:space="preserve" style="it">126. Diuiſa linea per inæqualia: poßibile est minoriſüæ parti lineam adiungi, ita, ut illud, <lb/>quod fit ex ductu toti{us} lineæ diuiſæ cum adiecta in ipſam adiectam, æquale ſit quadrato ei{us}, <lb/>quæ constat ex minore & adiecta.</head> <p> <s xml:id="echoid-s23278" xml:space="preserve">Sit data linea a b diuiſa per inęqualia in puncto c:</s> <s xml:id="echoid-s23279" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s23280" xml:space="preserve"> linea a c maior quã linea b c.</s> <s xml:id="echoid-s23281" xml:space="preserve"> Dico, quòd eſt <lb/>poſsibile inuenire quandam lineam, quæ adiecta ipſi lineæ b c, id efficiat, ut hoc, quod fit ex ductu <lb/>lineę compoſitæ ex linea a b, & ex adiecta in ipſam adiectam ſit æquale quadrato lineæ, quæ conſtat <lb/>ex b c parte minore, & ex adiecta.</s> <s xml:id="echoid-s23282" xml:space="preserve"> Aſſumatur enim quædam alia linea æqualis, uel minor linea a b, <lb/>quæ ſit d e, & quæ eſt proportio lineæ a c ad lineam b c, eadem ſit proportio lineæ d e ad quandam <lb/> <anchor type="figure" xlink:label="fig-0352-02a" xlink:href="fig-0352-02"/> <anchor type="figure" xlink:label="fig-0352-03a" xlink:href="fig-0352-03"/> aliam lineã per <lb/>3 huius:</s> <s xml:id="echoid-s23283" xml:space="preserve"> quæ ſit <lb/>e f:</s> <s xml:id="echoid-s23284" xml:space="preserve"> aſſumatúr-<lb/>que linea d f ę-<lb/>qualis lineæ a <lb/>b.</s> <s xml:id="echoid-s23285" xml:space="preserve"> Et quoniam <lb/>exlineis d e, e <lb/>f, d f quęcun q;</s> <s xml:id="echoid-s23286" xml:space="preserve"> <lb/>duę ſimul iun-<lb/>ctæ maiores ſunt tertia, ut patet ex præmiſsis, poſsibile eſt conſtitui triangulum per 22 p 1.</s> <s xml:id="echoid-s23287" xml:space="preserve"> Conſti-<lb/>tuatur ergo, & ſit d e f.</s> <s xml:id="echoid-s23288" xml:space="preserve"> Super terminum itaque lineæ a b, qui eſt a, conſtituatur angulus æqualis an-<lb/>gulo e d f per 23 p 1, qui ſit g a b:</s> <s xml:id="echoid-s23289" xml:space="preserve"> & reſecetur linea a g ad ęqualitatem lineæ d e, & ducatur linea g b.</s> <s xml:id="echoid-s23290" xml:space="preserve"> Er <lb/>go per 4 p 1, cum linea d f ſit æqualis lineæ a b, & linea a g æqualis lineę d e, & angulus g a b ſit ęqua-<lb/>lis angulo e d f:</s> <s xml:id="echoid-s23291" xml:space="preserve"> erit linea g b æqualis lineæ e f, & reliqui anguli trigoni a g b æquales erunt reliquis <lb/>angulis trigoni d e f.</s> <s xml:id="echoid-s23292" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s23293" xml:space="preserve"> linea g c.</s> <s xml:id="echoid-s23294" xml:space="preserve"> Et quoniam proportio lineæ d e ad lineã e f, ſicut lineæ <lb/>a c ad lineam c b:</s> <s xml:id="echoid-s23295" xml:space="preserve"> erit proportio lineæ a g ad lineam g b, ſicut lineæ a c ad lineam c b per 7 p 5:</s> <s xml:id="echoid-s23296" xml:space="preserve"> ergo ք <lb/>3 p 6 angulus a g b diuiſus eſt per æqualia:</s> <s xml:id="echoid-s23297" xml:space="preserve"> palã autem, quòd angulus g c b eſt acutus:</s> <s xml:id="echoid-s23298" xml:space="preserve"> ſienim ſit re-<lb/>ctus, tunc trianguli a g c & g c b æquianguli per 32 p 1, quoniam ad punctum g duo ipſorum anguli <lb/>ſunt æquales:</s> <s xml:id="echoid-s23299" xml:space="preserve"> ergo latera eorum ſunt proportionalia per 4 p 6:</s> <s xml:id="echoid-s23300" xml:space="preserve"> erit ergo proportio lateris a c ad c b, <lb/>ſicut lateris g c ad ſeipſum:</s> <s xml:id="echoid-s23301" xml:space="preserve"> æqualis eſt ergo linea a c lineę c b:</s> <s xml:id="echoid-s23302" xml:space="preserve"> quod eſt contra hypotheſin & impoſ-<lb/>ſibile.</s> <s xml:id="echoid-s23303" xml:space="preserve"> Si uerò angulus g c b detur eſſe obtuſus, maior angulo g c a, palã per 32 p 1, quoniam angulus <lb/>g b c eſt minor angulo g a c.</s> <s xml:id="echoid-s23304" xml:space="preserve"> Ergo per 19 p 1 in trigono a g b latus g b maius eſt latere a g.</s> <s xml:id="echoid-s23305" xml:space="preserve"> Et quia eſt <lb/>proportio lineę b g ad lineam g a, ſicut lineę b c ad lineam c a:</s> <s xml:id="echoid-s23306" xml:space="preserve"> erit per 5 huius, per proportionem ſci-<lb/>licet, è contrario latus b c maius quàm latus a c:</s> <s xml:id="echoid-s23307" xml:space="preserve"> quod eſt contra hypotheſim.</s> <s xml:id="echoid-s23308" xml:space="preserve"> Palàm ergo, quoniam <lb/>angulus g c b eſt acutus.</s> <s xml:id="echoid-s23309" xml:space="preserve"> Ducatur itaque per 31 p 1, à pũcto c linea c h ęquidiſtans lineę g a, ſecans li-<lb/>neam g b in puncto h:</s> <s xml:id="echoid-s23310" xml:space="preserve"> erit ergo per 29 p 1 angulus g c h æqualis angulo c g a:</s> <s xml:id="echoid-s23311" xml:space="preserve"> ergo & angulo c g h:</s> <s xml:id="echoid-s23312" xml:space="preserve"> e-<lb/>rit quoque angulus h c b ęqualis angulo g a c.</s> <s xml:id="echoid-s23313" xml:space="preserve"> Super punctum itaque g terminum lineę b g fiat per <lb/>23 p 1 angulus ęqualis angulo g a c:</s> <s xml:id="echoid-s23314" xml:space="preserve"> ergo & angulo h c b, qui ſit b g i.</s> <s xml:id="echoid-s23315" xml:space="preserve"> Et quia angulus g c b eſt æqua-<lb/>lis duobus angulis c g a & c a g, ut patet ex pręmiſsis, & per 32 p 1:</s> <s xml:id="echoid-s23316" xml:space="preserve"> erit angulùs i g c ęqualis angulo <lb/>g c b.</s> <s xml:id="echoid-s23317" xml:space="preserve"> Et quoniam angulus g c b eſt acutus:</s> <s xml:id="echoid-s23318" xml:space="preserve"> palã ergo per 14 huius, quoniam lineę g i & c b concurrẽt:</s> <s xml:id="echoid-s23319" xml:space="preserve"> <lb/>ſit punctus concurſus i.</s> <s xml:id="echoid-s23320" xml:space="preserve"> Ergo per 6 p 1 erit latus g i ęquale lateri c i.</s> <s xml:id="echoid-s23321" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s23322" xml:space="preserve"> angulus b g i eſt ęqua-<lb/>lis angulo g a i, & angulus g i a communis ambobus trigonis a g i & b g i:</s> <s xml:id="echoid-s23323" xml:space="preserve"> erit per 32 p 1 angulus a g i <lb/>æqualis angulo g b i.</s> <s xml:id="echoid-s23324" xml:space="preserve"> Ergo per 4 p 6 erit proportio lineę a i ad lineam i g, ſicut lineæ i g ad lineam <lb/> <pb o="51" file="0353" n="353" rhead="LIBER PRIMVS."/> b i:</s> <s xml:id="echoid-s23325" xml:space="preserve"> ſed linea i c eſt ęqualis lineæ g i:</s> <s xml:id="echoid-s23326" xml:space="preserve"> ergo per 7 p 5 eſt proportio lineę a i ad lineam c i, ſicut lineę c i ad <lb/>lineam b i.</s> <s xml:id="echoid-s23327" xml:space="preserve"> Ergo per 17 p 6 illud, quod fit ex ductu lineæ a i in lineam b i eſt ęquale quadrato lineę c i:</s> <s xml:id="echoid-s23328" xml:space="preserve"> <lb/>eſt autem linea b i lineę b c adiecta.</s> <s xml:id="echoid-s23329" xml:space="preserve"> Palàm ergo propoſitum.</s> <s xml:id="echoid-s23330" xml:space="preserve"/> </p> <div xml:id="echoid-div857" type="float" level="0" n="0"> <figure xlink:label="fig-0352-02" xlink:href="fig-0352-02a"> <variables xml:id="echoid-variables362" xml:space="preserve">e d f</variables> </figure> <figure xlink:label="fig-0352-03" xlink:href="fig-0352-03a"> <variables xml:id="echoid-variables363" xml:space="preserve">g h a c b i</variables> </figure> </div> </div> <div xml:id="echoid-div859" type="section" level="0" n="0"> <head xml:id="echoid-head701" xml:space="preserve" style="it">127. Propoſitis duab{us} lineis: poßibile eſt uni ipſarum lineam aliam adiungere, ita, ut illud, <lb/>quodfit ex ductu toti{us} lineæ cum adiunctain adiunctam, æquale ſit quadrato reliquæ datarũ. <lb/>E 36 p 3 element.</head> <p> <s xml:id="echoid-s23331" xml:space="preserve">Verbi gratia, proponantur duæ lineæ q e & a g.</s> <s xml:id="echoid-s23332" xml:space="preserve"> Dico, quòd poſsibile eſt uni ipſarum, ut lineę q e, <lb/> <anchor type="figure" xlink:label="fig-0353-01a" xlink:href="fig-0353-01"/> <anchor type="figure" xlink:label="fig-0353-02a" xlink:href="fig-0353-02"/> adiungere quãdam a-<lb/>liam lineam cuiuſcũq;</s> <s xml:id="echoid-s23333" xml:space="preserve"> <lb/>ſit quãtitatis, ita quòd <lb/>id, quod fit ex ductu li-<lb/>neę q e, cũ adiuncta in <lb/>ipſam adiunctã, ęqua-<lb/>le ſit quadrato lineæ <lb/>a g.</s> <s xml:id="echoid-s23334" xml:space="preserve"> Quadretur ergo li-<lb/>nea a g per 46 p 1, & ſit <lb/>eius quadratum a h:</s> <s xml:id="echoid-s23335" xml:space="preserve"> & <lb/>linea a g producta reſe <lb/>cetur in pũcto f ita, ut <lb/>linea g f ſit æqualis lineæ a g:</s> <s xml:id="echoid-s23336" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s23337" xml:space="preserve"> linea h f.</s> <s xml:id="echoid-s23338" xml:space="preserve"> Palã, quoniam triangulus a h f æqualis eſt quadra-<lb/>to a h:</s> <s xml:id="echoid-s23339" xml:space="preserve"> eſt enim parallelogrammum a h duplum trigoni a h g per 41 p 1, & trigonum a h f eſt duplum <lb/>eiuſdem trigoni a h g per 1 p 6.</s> <s xml:id="echoid-s23340" xml:space="preserve"> Hac ergo triangula ſuperficie propoſita, & linea q e, poſsibile eſt per <lb/>29 p 6 ſuper datam lineam q e datę ſuperficiei trilaterę a h f ęquum parallelogrammum conſtituere, <lb/>quod addat ſuper cõpletionem datę lineæ q e ſuperficiem quadratã, dato quadrato a h ſimilem.</s> <s xml:id="echoid-s23341" xml:space="preserve"> Sit <lb/>ergo conſtituta, & parallelogrãmum ſit q m ęquale trigono a h f conſtitutum ſuper lineã q e, addẽs <lb/>ſuper cõpletionem datę lineę q e quadratũ e m, ſimile quadrato a h.</s> <s xml:id="echoid-s23342" xml:space="preserve"> Palã ergo, quòd illud, quod fit <lb/>ex ductu datę lineę q e, cum adiecta e z in ipſam adiectam lineam e z, uel eius ęqualem lineam z m, <lb/> <anchor type="figure" xlink:label="fig-0353-03a" xlink:href="fig-0353-03"/> eſt ęquale propoſito trigono a h f.</s> <s xml:id="echoid-s23343" xml:space="preserve"> Ergo & eius <lb/>ęquali, ſcilicet quadrato a h.</s> <s xml:id="echoid-s23344" xml:space="preserve"> Et hoc eſt propo <lb/>ſitum:</s> <s xml:id="echoid-s23345" xml:space="preserve"> quoniam linea e z eſt lineę q e taliter, ut <lb/>proponitur, adiuncta.</s> <s xml:id="echoid-s23346" xml:space="preserve"> Poteſt & idẽ aliter de-<lb/>monſtrari.</s> <s xml:id="echoid-s23347" xml:space="preserve"> Deſcribatur enim circulus, cuius <lb/>diameter ſit q e, & eius cẽtrum b, ducaturq́;</s> <s xml:id="echoid-s23348" xml:space="preserve"> li-<lb/>nea contingens circulum, ut contingit in pun <lb/>cto g per 17 p 3:</s> <s xml:id="echoid-s23349" xml:space="preserve"> & reſecetur ad ęqualitatem li-<lb/>neę a g:</s> <s xml:id="echoid-s23350" xml:space="preserve"> & ſit g a:</s> <s xml:id="echoid-s23351" xml:space="preserve"> & ab eius termino a ducatur li <lb/>nea per centrũ b ſecans peripheriam circuli in <lb/>punctis e & q.</s> <s xml:id="echoid-s23352" xml:space="preserve"> Quia ergo id, quod fit ex ductu <lb/>lineę q a in lineam a e, eſt æquale quadrato li-<lb/>neę a g per 36 p 3:</s> <s xml:id="echoid-s23353" xml:space="preserve"> patet, quòd lineę q e eſt adiecta linea e a, ut proponebatur.</s> <s xml:id="echoid-s23354" xml:space="preserve"/> </p> <div xml:id="echoid-div859" type="float" level="0" n="0"> <figure xlink:label="fig-0353-01" xlink:href="fig-0353-01a"> <variables xml:id="echoid-variables364" xml:space="preserve">h a g f</variables> </figure> <figure xlink:label="fig-0353-02" xlink:href="fig-0353-02a"> <variables xml:id="echoid-variables365" xml:space="preserve">a g q e m q e z</variables> </figure> <figure xlink:label="fig-0353-03" xlink:href="fig-0353-03a"> <variables xml:id="echoid-variables366" xml:space="preserve">q a a e g b e g q</variables> </figure> </div> </div> <div xml:id="echoid-div861" type="section" level="0" n="0"> <head xml:id="echoid-head702" xml:space="preserve" style="it">128. Sumpta circuli diametro, & ſumpto in circumferentia puncto æqualiter diſtante à ter-<lb/>minis diametri: poßibile eſt ab eodem puncto ad diametrũ eductã <lb/>extra circulum, ducere lineam rectam, quæ à circumferentia cir <lb/>culi extra circulum uſ ad concurſum cum diametro, ſit datæ li-<lb/>neæ æqualis.</head> <figure> <variables xml:id="echoid-variables367" xml:space="preserve">d <gap/> q g h e a z b</variables> </figure> <p> <s xml:id="echoid-s23355" xml:space="preserve">Eſto data linea q e:</s> <s xml:id="echoid-s23356" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s23357" xml:space="preserve"> g b diameter dati circuli, qui ſit a b g:</s> <s xml:id="echoid-s23358" xml:space="preserve"> & ſit <lb/>a punctus datus in circuli circũferentia æqualiter diſtans ab extre-<lb/>mis terminis diametri, qui ſunt g & b.</s> <s xml:id="echoid-s23359" xml:space="preserve"> Dico, quòd poſsibile eſt ab a <lb/>pũcto peripheriæ circuli duci lineã uſq;</s> <s xml:id="echoid-s23360" xml:space="preserve"> ad eductã diametrũ g b, quę <lb/>ſit ęqualis datę lineæ q e.</s> <s xml:id="echoid-s23361" xml:space="preserve"> Ducantur.</s> <s xml:id="echoid-s23362" xml:space="preserve"> n.</s> <s xml:id="echoid-s23363" xml:space="preserve"> duæ lineæ a b & a g:</s> <s xml:id="echoid-s23364" xml:space="preserve"> illæ ergo <lb/>neceſſariò erunt æquales ex hypotheſi, quoniã punctus a ęqualiter <lb/>diſtat à terminis diametri g & b:</s> <s xml:id="echoid-s23365" xml:space="preserve"> & adiũgatur lineæ q e linea talis, ut <lb/>illud, qđ fit ex ductu totius lineę cũ adiuncta in adiunctã, æquale ſit <lb/>quadrato lineæ a g per pręcedentẽ proximã:</s> <s xml:id="echoid-s23366" xml:space="preserve"> & ſit adiũcta e z.</s> <s xml:id="echoid-s23367" xml:space="preserve"> Cũ er-<lb/>go id, quod fit ex ductu q z in e z ſit æquale ei, qđ fit ex ductu lineę <lb/>a g in ſeipſam:</s> <s xml:id="echoid-s23368" xml:space="preserve"> erit linea q z maior ꝗ̃ linea a g, & linea e z minor illa.</s> <s xml:id="echoid-s23369" xml:space="preserve"> Si <lb/>enim linea e z fuerit maior, uel ęqualis lineę a g, tũc eſt impoſsibile, <lb/>utid, qđ fit ex ductu lineę q z in lineã e z, ſit ęquale quadrato lineę <lb/>a g:</s> <s xml:id="echoid-s23370" xml:space="preserve"> quoniã linea q z eſt maior ꝗ̃ linea e z, ut totũ parte.</s> <s xml:id="echoid-s23371" xml:space="preserve"> Si aũt linea e z <lb/>ſit minor ꝗ̃ linea a g, palã, quoniã linea q z eſt maior ꝗ̃ linea a g:</s> <s xml:id="echoid-s23372" xml:space="preserve"> ꝓdu-<lb/>catur ergo linea a g, donec fiat ęqualis lineę e q per 3 p 1:</s> <s xml:id="echoid-s23373" xml:space="preserve"> & ſit a g t.</s> <s xml:id="echoid-s23374" xml:space="preserve"> Poſito ergo pede circini ſuք pũctũ <lb/>a, fiat circulus ſecundũ quantitatẽ lineæ a g t, qui circulus ſecabit diametrũ b g eductam:</s> <s xml:id="echoid-s23375" xml:space="preserve"> ſecet ergo <lb/> <pb o="52" file="0354" n="354" rhead="VITELLONIS OPTICAE"/> ipſam in puncto d:</s> <s xml:id="echoid-s23376" xml:space="preserve"> & ducatur linea a d, quæ neceſſariò ſecabit circulũ:</s> <s xml:id="echoid-s23377" xml:space="preserve"> quoniã concurrit cũ diame-<lb/>tro:</s> <s xml:id="echoid-s23378" xml:space="preserve"> ſi enim non ſecet circulum, cõtingens erit & æquidiſtans diametro g b, nunquã cõcurrens cum <lb/>eadem:</s> <s xml:id="echoid-s23379" xml:space="preserve"> quia ex hypotheſi linea a g & a b ſunt æquales, & punctum a ęqualiter diſtat ab utriſq;</s> <s xml:id="echoid-s23380" xml:space="preserve"> termi <lb/>nis diametri, ſcilicet b & g.</s> <s xml:id="echoid-s23381" xml:space="preserve"> Secet ergo linea d a circulum a g b in puncto h:</s> <s xml:id="echoid-s23382" xml:space="preserve"> & ducatur linea g h.</s> <s xml:id="echoid-s23383" xml:space="preserve"> Palã <lb/>ergo, quòd cum ſuperficies a b g h ſit quadrangulum intra circulũ deſcriptum, quòd duo eius angu <lb/>li oppoſiti, ſcilicet a b g & a h g ualent duos rectos per 22 p 3:</s> <s xml:id="echoid-s23384" xml:space="preserve"> ſed a g b angulus æqualis eſt angulo a b <lb/>g per 5 p 1:</s> <s xml:id="echoid-s23385" xml:space="preserve"> angulus ergo a g b cũ angulo a h g ualet duos rectos.</s> <s xml:id="echoid-s23386" xml:space="preserve"> Cũ itaq;</s> <s xml:id="echoid-s23387" xml:space="preserve"> ք 13 p 1 angulus d g a cũ an-<lb/>gulo a g b ualeat duos rectos:</s> <s xml:id="echoid-s23388" xml:space="preserve"> palã, quia angulus a h g erit ęqualis angulo d g a:</s> <s xml:id="echoid-s23389" xml:space="preserve"> & angulus h a g cõmu <lb/>nis eſt totali triangulo a d g, & partiali trigono, qui eſt h a g:</s> <s xml:id="echoid-s23390" xml:space="preserve"> reſtat ergo per 32 p 1, ut angulus h d g ſit <lb/>ęqualis angulo h g a, & totalis triangulus d g a ęquiãgulus triangulo g h a.</s> <s xml:id="echoid-s23391" xml:space="preserve"> Ergo per 4 p 6 latera ipſo-<lb/>rũ æquos angulos reſpicientia ſunt proportionalia.</s> <s xml:id="echoid-s23392" xml:space="preserve"> Eſt ergo proportio lateris d a ad latus a g, ficut <lb/>lateris a g ad latus a h.</s> <s xml:id="echoid-s23393" xml:space="preserve"> Illud ergo qđ fit ex ductu lineę d a in lineã a h, eſt ęquale quadrato lineę a g ք <lb/>17 p 6:</s> <s xml:id="echoid-s23394" xml:space="preserve"> ſed linea d a eſt ęqualis lineæ a t per definitionem circuli.</s> <s xml:id="echoid-s23395" xml:space="preserve"> Ergo linea d a eſt ęqualis lineę q z, <lb/>quoniã linea t a ex pręmiſsis eſt ęqualis lineę q z.</s> <s xml:id="echoid-s23396" xml:space="preserve"> Quia uerò illud, quod fit ex ductu lineę d a in lineã <lb/>h a eſt ęquale quadrato lineę a g, quod ex pręmiſsis eſt ęquale ei, quod fit ex ductu lineę q z in lineã <lb/>e z:</s> <s xml:id="echoid-s23397" xml:space="preserve"> patet, qđ id, qđ fit ex ductu lineę a d in lineã h a, eſt ęquale ei, quod fit ex ductu lineę q z in lineã <lb/>e z:</s> <s xml:id="echoid-s23398" xml:space="preserve"> & linea d a eſt ęquális lineę q z:</s> <s xml:id="echoid-s23399" xml:space="preserve"> relin quitur ergo, ut linea a h ſit æqualis lineę e z.</s> <s xml:id="echoid-s23400" xml:space="preserve"> Erit ergo linea <lb/>d h æqualis ipſi lineę q e, quę eſt data linea:</s> <s xml:id="echoid-s23401" xml:space="preserve"> eſt autem à dato in peripheria circuli puncto a ad cõcur-<lb/>ſum diametri b g ſic producta.</s> <s xml:id="echoid-s23402" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s23403" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div862" type="section" level="0" n="0"> <head xml:id="echoid-head703" xml:space="preserve" style="it">129. Inter duas rectas angulariter cõiunctas à dato puncto rectãducere, cuius una partium <lb/>interiacens unã cõiunctarũ, & datũ punctũ, ſit cuicun datæ lineæ, & inſuper reliquæ ſuæ par <lb/>ti, datũ punctũ & alterã coniunctarum interiacenti æqualis. 4 theor. 2 conicorũ Apollonij.</head> <p> <s xml:id="echoid-s23404" xml:space="preserve">Exẽpli cauſa, ſit, ut duę lineę rectę in puncto uno angulariter coniungantur:</s> <s xml:id="echoid-s23405" xml:space="preserve"> quę ſint f k, & t k, cõ-<lb/> <anchor type="figure" xlink:label="fig-0354-01a" xlink:href="fig-0354-01"/> currentes in pũcto k, inter quas ſit datus pũctus m, & ſit <lb/>data linea m c:</s> <s xml:id="echoid-s23406" xml:space="preserve"> proponitur nobis, ut à puncto m ducatur <lb/>linea recta intra lineas f k & t k, ſecans illas in pũctis o & <lb/>lita, ut eius pars, quę eſt l m, ſit ęqualis datę lineæ m c, & <lb/>inſuper reliquę ſuę parti, quæ eſt m o.</s> <s xml:id="echoid-s23407" xml:space="preserve"> Ad hoc aũt ք lineas <lb/>rectas uel circulares demõſtrandũ, lõgus labor & multæ <lb/>diuerſitatis nobis incidit, & nõ fuit nobis hoc poſsibile <lb/>cõplere ք huiuſmodi lineas abſq;</s> <s xml:id="echoid-s23408" xml:space="preserve"> motu & imaginatione <lb/>mechanica, ita ut cũ lineę f k & t k datę ſint nobis indefi-<lb/>nitę, linea l o fixa in puncto m, imaginetur moueri, quo-<lb/>uſq;</s> <s xml:id="echoid-s23409" xml:space="preserve"> nobis accidat res quęſita.</s> <s xml:id="echoid-s23410" xml:space="preserve"> Hoc tñ Apollonius Pergę <lb/>us in libro ſuo de conicis elementis lιbro ſecũdo, ꝓpoſi-<lb/>tione quarta, ք ductionẽ ſectionis amblygonię à dato pũ <lb/>cto inter duas lineas aſymptotas, nullã earũ linearũ ſecã <lb/>tis demonſtrauit:</s> <s xml:id="echoid-s23411" xml:space="preserve"> cuius nos demonſtrationem, ut à mul-<lb/>tis ſui libri principijs pręambulis dependentẽ hic ſupponimus, & ipſa utimur ſicut demonſtrata.</s> <s xml:id="echoid-s23412" xml:space="preserve"/> </p> <div xml:id="echoid-div862" type="float" level="0" n="0"> <figure xlink:label="fig-0354-01" xlink:href="fig-0354-01a"> <variables xml:id="echoid-variables368" xml:space="preserve">k f t ſ c m o</variables> </figure> </div> </div> <div xml:id="echoid-div864" type="section" level="0" n="0"> <head xml:id="echoid-head704" xml:space="preserve" style="it">130. Sumpta circuli diametro, & ſumpto in circũfererẽtia puncto inæqualiter diſtante à ter-<lb/>minis diametri: poßibile eſt à ſumpto puncto ad eductã diametrũ lineã ducere, q̃, uel cui{us} pars <lb/>interiacẽs քipheriã et diametrũ ſit datæ lineæ æqualis. Alha. 30 n 5.</head> <figure> <variables xml:id="echoid-variables369" xml:space="preserve">q d n g e a b</variables> </figure> <p> <s xml:id="echoid-s23413" xml:space="preserve">Diſponantur omnia, ut in 128 huius, niſi quòd pũctus datus in cir-<lb/>cumferentia circuli, qui ſit a, inęqualiter diſtet à terminis diametri, ꝗ <lb/>ſint g & b:</s> <s xml:id="echoid-s23414" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s23415" xml:space="preserve"> lineę a b & a g inęquales:</s> <s xml:id="echoid-s23416" xml:space="preserve"> ideo quòd punctũ a inæ-<lb/>qualiter eſt diſtans à punctis g & b.</s> <s xml:id="echoid-s23417" xml:space="preserve"> Protrahatur ergo à pũcto g linea <lb/>ęquidiſtans lineę a b per 31 p 1, quę ſit g n, & ſumatur linea quęcunq;</s> <s xml:id="echoid-s23418" xml:space="preserve">, <lb/>utpote z t, & fiat ſuper punctum eius z angulus æqualis angulo a g d <lb/>per 23 p 1, qui ſit angulus t z f, ducta linea z f:</s> <s xml:id="echoid-s23419" xml:space="preserve"> & ducatur à puncto t li-<lb/>nea ęquidiſtans lineę z f, ut prius, quę ſit t m:</s> <s xml:id="echoid-s23420" xml:space="preserve"> & ex angulo t z f, ſecetur <lb/>angulus ęqualis angulo d g n ք 27 huius, qui ſit t z m, ducta linea z m, <lb/>quę ք z huius neceſſariò cõcurret cũ linea t m, cũ ſit ducta inter ęqui-<lb/>diſtantes:</s> <s xml:id="echoid-s23421" xml:space="preserve"> ſit ergo punctus concurſus m:</s> <s xml:id="echoid-s23422" xml:space="preserve"> reſtat ergo ut angulus m z f <lb/>ſit ęqualis angulo a g n.</s> <s xml:id="echoid-s23423" xml:space="preserve"> A pũcto itaq;</s> <s xml:id="echoid-s23424" xml:space="preserve"> t ducatur linea ęquidiſtãs lineę <lb/>z m, quę ſit t o hęc quoq;</s> <s xml:id="echoid-s23425" xml:space="preserve"> neceſſariò cõcurret cũ linea f z ք 2 huius:</s> <s xml:id="echoid-s23426" xml:space="preserve"> ſit <lb/>ergo earum cõcurſus in puncto k.</s> <s xml:id="echoid-s23427" xml:space="preserve"> Sumatur quoq;</s> <s xml:id="echoid-s23428" xml:space="preserve"> ք 3 huius linea, cu-<lb/>ius proportio ad lineã z t, ſit ſicut diameter g b ad lineã q e lineã datã:</s> <s xml:id="echoid-s23429" xml:space="preserve"> <lb/>& hęc ſit linea i.</s> <s xml:id="echoid-s23430" xml:space="preserve"> Deinde à pũcto m dato inter duas lineas k f & k o du <lb/>catur per pręmiſſam linea, quę ſit l c m o ſecans lineã l k in pũcto l, & <lb/>lineã k o in pũcto o, ita, ut eius pars c m ſit ęqualis datæ lineę i, & eius <lb/>pars l c ſit ęqualis lineę m o:</s> <s xml:id="echoid-s23431" xml:space="preserve"> & à puncto t ducatur linea t f ęquidiſtãs <lb/>lineę l o per 31 p 1:</s> <s xml:id="echoid-s23432" xml:space="preserve"> hęc quoq;</s> <s xml:id="echoid-s23433" xml:space="preserve"> per 29 huius ſecabitur à linea z m:</s> <s xml:id="echoid-s23434" xml:space="preserve"> ſit ergo punctus ſectionis y.</s> <s xml:id="echoid-s23435" xml:space="preserve"> Fiat er-<lb/>go ſupra punctum a terminũ lineę g a (punctũ ſcilicet, qui eſt in circumferẽtia circuli) angulus d a g <lb/> <pb o="53" file="0355" n="355" rhead="LIBER PRIMVS."/> æqualis angulo z f t per lineam a n d.</s> <s xml:id="echoid-s23436" xml:space="preserve"> Palàm autem, quòd hęc linea concurret cum producta diame-<lb/>tro g d.</s> <s xml:id="echoid-s23437" xml:space="preserve"> Cũ enim angulus d a g ſit ęqualis angulo z f t, & angulus a g n æqualis angulo f z m, & angu-<lb/> <anchor type="figure" xlink:label="fig-0355-01a" xlink:href="fig-0355-01"/> lus d g n eſt æqualis angulo t z m, totusq́;</s> <s xml:id="echoid-s23438" xml:space="preserve"> angulus a g d æqualis toti <lb/>angulo f z t, & cũ lineę f t & z t cõcurrãt:</s> <s xml:id="echoid-s23439" xml:space="preserve"> ergo & lineę a d & g d cõcur <lb/>rẽt:</s> <s xml:id="echoid-s23440" xml:space="preserve"> ergo linea a d aut cõtinget circulũ, aut ſecabit ipſum.</s> <s xml:id="echoid-s23441" xml:space="preserve"> Sit ergo li-<lb/>nea a d primò contingens circulum in puncto a.</s> <s xml:id="echoid-s23442" xml:space="preserve"> Cum ergo angulus <lb/>g a n ſit æqualis angulo z f t, & angulus a g n ſit ęqualis angulo f z y:</s> <s xml:id="echoid-s23443" xml:space="preserve"> <lb/>palàm per 32 p 1, quia angulus a n g erit ęqualis angulo z y f:</s> <s xml:id="echoid-s23444" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s23445" xml:space="preserve"> tri <lb/>angulus a g n ęquiangulus triangulo z f y:</s> <s xml:id="echoid-s23446" xml:space="preserve"> ergo eſt per 4 p 6 propor-<lb/>tio lineę a n ad lineam a g, ſicut lineę f y ad lineam f z.</s> <s xml:id="echoid-s23447" xml:space="preserve"> Similiter cum <lb/>angulus a g d ſit æqualis angulo f z t, & angulus g a d ęqualis angulo <lb/>z f t:</s> <s xml:id="echoid-s23448" xml:space="preserve"> erit per eandem triangulus a g d ſimilis triãgulo f z t:</s> <s xml:id="echoid-s23449" xml:space="preserve"> ergo ut pri <lb/>us, quæ eſt proportio lineæ a g ad lineam g d, eadem eſt lineę f z ad <lb/>lineam z t.</s> <s xml:id="echoid-s23450" xml:space="preserve"> Si ergo, quę eſt proportio lineę a n ad lineam a g, eadẽ eſt <lb/>lineę f y ad lineam f z, & quę eſt proportio lineæ a g ad lineam g d, ea <lb/>dem eſt lineę f z ad lineam z t:</s> <s xml:id="echoid-s23451" xml:space="preserve"> erit ergo per ęquam proportionalita-<lb/>tem per 22 p 5, ut quę eſt proportio lineę a n ad lineam g d, eadem ſit <lb/>lineę f y ad lineam z t.</s> <s xml:id="echoid-s23452" xml:space="preserve"> Quia uerò linea t m eſt ęquidiſtans lineę f l, & <lb/>linea f t ęquidiſtans lineę l m:</s> <s xml:id="echoid-s23453" xml:space="preserve"> erit ſuperficies l f t m ęquidiſtantibus <lb/>contenta lateribus:</s> <s xml:id="echoid-s23454" xml:space="preserve"> palã ergo per 34 p 1, quoniam linea f t eſt ęqualis <lb/>lineę l m.</s> <s xml:id="echoid-s23455" xml:space="preserve"> Quare erit linea f t ęqualis lineę c o, quoniam linea m o eſt <lb/>ęqualis ipſi l c per pręmiſſam:</s> <s xml:id="echoid-s23456" xml:space="preserve"> linea ergo c m addita utriq;</s> <s xml:id="echoid-s23457" xml:space="preserve">, adhuc e-<lb/>rũt æquales:</s> <s xml:id="echoid-s23458" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s23459" xml:space="preserve"> linea l m ęqualis lineæ c o:</s> <s xml:id="echoid-s23460" xml:space="preserve"> ſed linea m o eſt ęqualis lineæ y t per 34 p 1, & linea y m <lb/>eſt æqualis lineæ t o:</s> <s xml:id="echoid-s23461" xml:space="preserve"> reſtat ergo, ut linea f y ſit ęqualis lineę c m:</s> <s xml:id="echoid-s23462" xml:space="preserve"> ſed linea c m ex præmiſsis eſt ęqua-<lb/>lis lineæ i.</s> <s xml:id="echoid-s23463" xml:space="preserve"> Quare f y eſt æqualis i:</s> <s xml:id="echoid-s23464" xml:space="preserve"> eſt autem ex præmiſsis & per 5 huius proportio lineę i ad lineam <lb/>z t, ſicut diametri b g ad lineam e q:</s> <s xml:id="echoid-s23465" xml:space="preserve"> erit ergo per 7 p 5 proportio lineæ f y ad lineam t z, ſicut diame-<lb/>tri b g ad lineã e q.</s> <s xml:id="echoid-s23466" xml:space="preserve"> Quia uerò eſt proportio lineæ a n ad lineam g d, ſicut lineæ f y ad lineam z t:</s> <s xml:id="echoid-s23467" xml:space="preserve"> ergo <lb/>per 11 p 5 erit proportio lineæ a n ad lineam g d, ſicut lineæ g b ad lineam e q.</s> <s xml:id="echoid-s23468" xml:space="preserve"> Verùm angulus g a n eſt <lb/>ęqualis angulo g b a per 32 p 3:</s> <s xml:id="echoid-s23469" xml:space="preserve"> ſed angulus n g d eſt æqualis angulo g b a per 29 p 1, quia linea n g <lb/>ęquidiſtat lineę b a:</s> <s xml:id="echoid-s23470" xml:space="preserve"> igitur angulus n g d ęqualis eſt angulo n a g:</s> <s xml:id="echoid-s23471" xml:space="preserve"> & angulus n d g eſt communis am-<lb/>bobus trigonis n d g & a d g:</s> <s xml:id="echoid-s23472" xml:space="preserve"> ergo per 32 p 1 erit angulus d n g æqualis angulo d g a:</s> <s xml:id="echoid-s23473" xml:space="preserve"> ſunt ergo dicti <lb/>trianguli æquianguli:</s> <s xml:id="echoid-s23474" xml:space="preserve"> erit ergo per 4 p 6 proportio lineæ a d ad d g, ſicut lineæ g d ad n d.</s> <s xml:id="echoid-s23475" xml:space="preserve"> Ergo per <lb/>17 p 6 erit id, quod fit ex ductu lineę a d in d n ęquale quadrato g d:</s> <s xml:id="echoid-s23476" xml:space="preserve"> ſed id, quod fit ex ductu lineæ b d <lb/>& d g, per 36 p 3 eſt æquale quadrato d a:</s> <s xml:id="echoid-s23477" xml:space="preserve"> quadratum uerò lineæ d a eſt æquale ei, quod fit ex ductu <lb/>lineæ a d in d n, & a d in n a per 2 p 2:</s> <s xml:id="echoid-s23478" xml:space="preserve"> & id, quod fit ex ductu lineę b d in d g, eſt ęquale quadrato lineę <lb/>d g, & ei quod fit ex ductu b g in d g per 3 p 2.</s> <s xml:id="echoid-s23479" xml:space="preserve"> Ablatis ergo æqualibus hinc inde (quæ ſunt quadratũ <lb/>g d & rectangulum a d n) reſtatutid, quod fit ex ductu lineę a d in an, ſit ęquale ei, quod fit ex du-<lb/>ctu lineę b g in d g, eritq́ue per 16 p 6 proportio lineæ a n primæ ad lineam g d ſecundam, ſi-<lb/>cut lineę b g tertiæ ad lineam a d quartam:</s> <s xml:id="echoid-s23480" xml:space="preserve"> oſtenſum eſt autem ſuprà, quòd eſt proportio lineæ <lb/>a n ad lineam g d, ſicut lineæ b g ad lineam e q.</s> <s xml:id="echoid-s23481" xml:space="preserve"> Erit ergo per 9 p 5 linea e q æqualis lineæ <lb/>a d.</s> <s xml:id="echoid-s23482" xml:space="preserve"> Quod eſt propoſitum:</s> <s xml:id="echoid-s23483" xml:space="preserve"> quoniam ipſa linea a d eſt datæ lineæ æqualis:</s> <s xml:id="echoid-s23484" xml:space="preserve"> interiacet autem pe-<lb/>ripheriam circuli & eductam diametrum, eò quòd eſt contingens circulum.</s> <s xml:id="echoid-s23485" xml:space="preserve"> Quòd ſi linea a d <lb/> <anchor type="figure" xlink:label="fig-0355-02a" xlink:href="fig-0355-02"/> non ſit contingens, ſed ſecans circulum:</s> <s xml:id="echoid-s23486" xml:space="preserve"> aut igitur linea a g eſt <lb/>maior quàm linea a b:</s> <s xml:id="echoid-s23487" xml:space="preserve"> aut è contrario.</s> <s xml:id="echoid-s23488" xml:space="preserve"> Sit autem nunc linea a g <lb/>maior quàm linea b a:</s> <s xml:id="echoid-s23489" xml:space="preserve"> palàm, quia linea à puncto a ad diametrum <lb/>b g extra circulum ducta ſecabit circulum in arcu a g.</s> <s xml:id="echoid-s23490" xml:space="preserve"> Sit ergo, ut ſe <lb/>cet ipſum in puncto h:</s> <s xml:id="echoid-s23491" xml:space="preserve"> & ducatur linea h g.</s> <s xml:id="echoid-s23492" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s23493" xml:space="preserve">, cũ quadran <lb/>gulum a b g h ſit inſcriptum circulo, quia duo anguli a h g & a b g <lb/>per 22 p 3 ſunt ęquales duobus rectis.</s> <s xml:id="echoid-s23494" xml:space="preserve"> Ducatur quoque linea g n ę-<lb/>quidiſtans lineę b a:</s> <s xml:id="echoid-s23495" xml:space="preserve"> erit ergo per 29 p 1 angulus n g d ęqualis angu <lb/>lo g b a:</s> <s xml:id="echoid-s23496" xml:space="preserve"> ergo angulus n g d, & angulus a h g ſunt æquales duobus <lb/>rectis:</s> <s xml:id="echoid-s23497" xml:space="preserve"> ſed per 13 p 1 angulus n h g cũ angulo a h g ualet duos rectos:</s> <s xml:id="echoid-s23498" xml:space="preserve"> <lb/>ergo angulus n g d eſt ęqualis angulo n h g:</s> <s xml:id="echoid-s23499" xml:space="preserve"> angulus uerò n d g eſt <lb/>cõmunis ambobus trigonis g d n & h g d:</s> <s xml:id="echoid-s23500" xml:space="preserve"> erit ergo tertius angulus, <lb/>qui eſt d n g, ęqualis tertio, qui eſt d g h per 32 p 1.</s> <s xml:id="echoid-s23501" xml:space="preserve"> Ergo per 4 p 6 la-<lb/>tera æquos angulos reſpicientia ſunt proportionalia:</s> <s xml:id="echoid-s23502" xml:space="preserve"> eſt igitur ꝓ-<lb/>portio lineę h d ad lineam d g, ſicut lineæ d g ad lineam d n.</s> <s xml:id="echoid-s23503" xml:space="preserve"> Ergo ք <lb/>17 p 6 illud, quod fit ex ductu h d in d n eſt ęquale quadrato d g:</s> <s xml:id="echoid-s23504" xml:space="preserve"> & <lb/>illud, quod fit ex ductu a d in d h eſt ęquale ei, quod fit ex ductu b d <lb/>in d g per 36 p 3.</s> <s xml:id="echoid-s23505" xml:space="preserve"> Item illud, quod fit ex ductu a d in d h eſt ęquale ei, <lb/>quod fit ex ductu d h in d n, & d h in a n ք 1 p 2:</s> <s xml:id="echoid-s23506" xml:space="preserve"> illud uerò quod fit ex <lb/>ductu b d in d g eſt ęquale ei, quod fit ex ductu b g in g d, & quadra <lb/>to g d ք 3 p 2.</s> <s xml:id="echoid-s23507" xml:space="preserve"> A blatis igitur ęqualib.</s> <s xml:id="echoid-s23508" xml:space="preserve"> ab utriſq;</s> <s xml:id="echoid-s23509" xml:space="preserve"> (ſcilicet quadrato d g ex una parte, & illo, quod fit ex <lb/>ductu d h in d n ex altera) reſtat utillud, qđ fit ex ductu d h in a n, ſit ęquale ei, quod fit ex ductu b g <lb/> <pb o="54" file="0356" n="356" rhead="VITELLONIS OPTICAE"/> in d g:</s> <s xml:id="echoid-s23510" xml:space="preserve"> erit ergo per 16 p 6 proportio a n primi ad g d ſecundum, ſicut b g tertij ad d h quartũ:</s> <s xml:id="echoid-s23511" xml:space="preserve"> ſed pro <lb/>batum eſt in præcedentibus, quòd proportio lineæ a n ad lineam d g eſt, ſicut diameter b g ad lineã <lb/>e q.</s> <s xml:id="echoid-s23512" xml:space="preserve"> Igitur per 9 p 5 linea d h eſt æqualis lineę e q.</s> <s xml:id="echoid-s23513" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s23514" xml:space="preserve"> Si uerò linea a g ſit minor ꝗ̃ <lb/>linea a b, ſecabit linea d a circulum in arcu a b.</s> <s xml:id="echoid-s23515" xml:space="preserve"> Sit ergo ut ſecet ipſum in puncto h:</s> <s xml:id="echoid-s23516" xml:space="preserve"> & ducatur linea <lb/> <anchor type="figure" xlink:label="fig-0356-01a" xlink:href="fig-0356-01"/> g h & linea g n, æquidiſtans lineę b a.</s> <s xml:id="echoid-s23517" xml:space="preserve"> Palã ergo ք 29 p 1, quoniã an-<lb/>gulus n g d eſt ęqualis angulo a b g:</s> <s xml:id="echoid-s23518" xml:space="preserve"> ſed angulus a b g eſt ęqualis an-<lb/>gulo a h g per 27 p 3:</s> <s xml:id="echoid-s23519" xml:space="preserve"> quoniã ambo cadunt in arcũ g a, & ſunt ſuք cir <lb/>cumferentiã circuli:</s> <s xml:id="echoid-s23520" xml:space="preserve"> ergo angulus n g d eſt æqualis angulo a h g:</s> <s xml:id="echoid-s23521" xml:space="preserve"> & <lb/>angulus n d g cõmunis eſt ambobus trigonis, ſcilicet n d g & d h g:</s> <s xml:id="echoid-s23522" xml:space="preserve"> <lb/>eſt ergo tertius d n g ęqualis tertio, ſcilicet d g h ք 32 p 1.</s> <s xml:id="echoid-s23523" xml:space="preserve"> Ergo per 4 <lb/>p 6 erit proportio lineę h d ad lineam d g, ſicut lineæ d g ad lineam <lb/>d n:</s> <s xml:id="echoid-s23524" xml:space="preserve"> ergo per 17 p 6 illud, quod fit ex ductu h d in d n eſt ęquale qua-<lb/>drato lineę g d:</s> <s xml:id="echoid-s23525" xml:space="preserve"> ſed illud, qđ fit ex ductu b d in d g ք 36 p 3, eſt ęquale <lb/>ei quod fit ex ductu h d in d a:</s> <s xml:id="echoid-s23526" xml:space="preserve"> illud aũt, quod fit ex ductu h d in d a, <lb/>eſt ք 1 p 2 ęquale ei, qđ fit ex ductu lineę h d in d n, & lineę h d in n a:</s> <s xml:id="echoid-s23527" xml:space="preserve"> <lb/>illud uerò quod fit ex ductu lineę b d in d g, per 3 p 2 ualet illud, qđ <lb/>fit ex ductu lineæ b g in g d & quadratũ g d.</s> <s xml:id="echoid-s23528" xml:space="preserve"> Ablatis ergo ęqualibus <lb/>hinc inde, erit illud, quod fit ex ductu h d in n a ęquale ei, quod fit ex <lb/>ductu b g in g d:</s> <s xml:id="echoid-s23529" xml:space="preserve"> erit ergo, ut prius, ꝓportio lineæ a n ad lineam d g, <lb/>ſicut lineę b g ad lineã h d.</s> <s xml:id="echoid-s23530" xml:space="preserve"> Sed iã oſtenſum eſt ſuprà, quòd eſt ꝓpor <lb/>tio lineę a n ad lineã d g, ſicut lineę b g ad lineã e q.</s> <s xml:id="echoid-s23531" xml:space="preserve"> Igitur linea e q <lb/>eſt æqualis lineę h d per 9 p 5.</s> <s xml:id="echoid-s23532" xml:space="preserve"> Quod eſt propoſitum:</s> <s xml:id="echoid-s23533" xml:space="preserve"> quoniã à pun-<lb/>cto a dato ducta eſt linea ſecãs circulũ, cuius pars à pũcto ſectionis <lb/>uſque ad concurſum cum diametro producta, æqualis eſt datæ li-<lb/>neæ.</s> <s xml:id="echoid-s23534" xml:space="preserve"> Patet ergo quod proponebatur.</s> <s xml:id="echoid-s23535" xml:space="preserve"/> </p> <div xml:id="echoid-div864" type="float" level="0" n="0"> <figure xlink:label="fig-0355-01" xlink:href="fig-0355-01a"> <variables xml:id="echoid-variables370" xml:space="preserve">k t o z u y m f c l i</variables> </figure> <figure xlink:label="fig-0355-02" xlink:href="fig-0355-02a"> <variables xml:id="echoid-variables371" xml:space="preserve">q d n e g h a b</variables> </figure> <figure xlink:label="fig-0356-01" xlink:href="fig-0356-01a"> <variables xml:id="echoid-variables372" xml:space="preserve">d q n g a e b h</variables> </figure> </div> </div> <div xml:id="echoid-div866" type="section" level="0" n="0"> <head xml:id="echoid-head705" xml:space="preserve" style="it">131. Inter duas rectas ſe ſecantes ex unaparte à puncto dato hyperbolẽ, illas lineas nõ cõtingẽ <lb/>tem ducere, ex alia parte cõmunis puncti illarũ linearũ hyperbolẽ priori oppoſit ã deſignare. Ex <lb/>quo patet, quòd cũ fuerint duæ ſectiones oppoſitæ inter duas lineas, et producatur linea minima <lb/>ab una ſectione ad aliã: erit pars illi{us} lineæ interiacens unã ſectionũ, & reliquãlineam æqualis <lb/>ſuæ partialiam ſectionem, & reliquam lineam interiacenti. 4. 8 th. 2 conicorum Apollonij.</head> <p> <s xml:id="echoid-s23536" xml:space="preserve">Quod hic proponitur, demonſtratum eſt ab Apollonio in libro ſuo de conicis elementis:</s> <s xml:id="echoid-s23537" xml:space="preserve"> dicun-<lb/> <anchor type="figure" xlink:label="fig-0356-02a" xlink:href="fig-0356-02"/> tur aũt ſectiones am <lb/>blygonię ſiue hyper <lb/>bolæ oppoſitæ, qñ <lb/>gibboſitas unius i-<lb/>pſarũ ſequitur gib-<lb/>boſitatẽ alterius, ita <lb/>utillæ gibboſitates <lb/>ſe reſpiciãt, & amba <lb/>rum diametri ſintin <lb/>una linea recta.</s> <s xml:id="echoid-s23538" xml:space="preserve"> Ver <lb/>bi gratia:</s> <s xml:id="echoid-s23539" xml:space="preserve"> ſit, ut duæ <lb/>lineæ h l & z n ſecẽt <lb/>ſe in puncto x, & ex una parte ipſarum, ſcilicet ſub angulo h x z, uel ſub angulo h x n à dato puncto, <lb/>qui ſit t, ducatur ſectio amblygonia, quæ ſit t p, & ex altera parte ſub angulo n x l, uel ſub angulo z x <lb/>l ducatur ſectio illi opp oſita, quæ ſit c u, ita, quòd diametri quarumlibet oppoſitarum ambarum ſe-<lb/>ctionum illarũ ſint in una linea, quæ t c, à uertice unius ad uerticẽ alterius producta:</s> <s xml:id="echoid-s23540" xml:space="preserve"> quę neceſſariò <lb/>eſt minima omnium linearum inter illas duas ſectiones productarum.</s> <s xml:id="echoid-s23541" xml:space="preserve"> Et ex ijs declarauit Apollo-<lb/>nius illud, quod corollatiuè proponitur, ſcilicet, quòd ſi linea t c ſecet lineam h l in puncto f, & lineã <lb/>z n in puncto q, quòd linea t q erit ęqualis lineæ c f:</s> <s xml:id="echoid-s23542" xml:space="preserve"> & ſi linea t c pertrãſeat punctum x, erit linea t x <lb/>ęqualis lineę x c:</s> <s xml:id="echoid-s23543" xml:space="preserve"> & nos utimur hoc illo, ut per Apollonium demonſtrato, & propter conformitatẽ <lb/>portionis ſectionum reſpectu linearum ſe interſecantium.</s> <s xml:id="echoid-s23544" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s23545" xml:space="preserve"/> </p> <div xml:id="echoid-div866" type="float" level="0" n="0"> <figure xlink:label="fig-0356-02" xlink:href="fig-0356-02a"> <variables xml:id="echoid-variables373" xml:space="preserve">n u l c u x g t c m f q t k p h p z</variables> </figure> </div> </div> <div xml:id="echoid-div868" type="section" level="0" n="0"> <head xml:id="echoid-head706" xml:space="preserve" style="it">132. In uertice alteri{us} conicarum ſectionum poſito pede circini immobili, ſecundum quan-<lb/>titatem lineæ breuißimæ inter illas ſectiones ductæ, deſcriptus circulus ſectionem reliquam con-<lb/>tinget: ſecundum uerò maiorem, in duobus tantùm punctis reliquam ſecabit.</head> <p> <s xml:id="echoid-s23546" xml:space="preserve">Quod hic proponitur, facile eſt, & ſola indiget declaratione.</s> <s xml:id="echoid-s23547" xml:space="preserve"> Sint ut enim in præcedenti propoſi <lb/>tione duæ ſectiones conicæ oppoſitæ adinuicẽ, quę ſint t p & c u, inter quas linea minima uertices, <lb/>ſcilicet ambarum ſectionum continuans, ſit linea t c:</s> <s xml:id="echoid-s23548" xml:space="preserve"> & poſito in altero punctorum tuel c pede cir-<lb/>cini, utpote in puncto t, deſcribatur circulus ſecundum quantitatem diametri t c.</s> <s xml:id="echoid-s23549" xml:space="preserve"> Hic ergo cir-<lb/>culus, quia ſectionem c u non attingit niſi in puncto c, & omnes alię lineæ ducibiles interipſas ſe-<lb/>ctiones, ſunt maiores quã linea t c:</s> <s xml:id="echoid-s23550" xml:space="preserve"> ſunt ergo maiores ſemidiametro circuli:</s> <s xml:id="echoid-s23551" xml:space="preserve"> ſecabuntur ergo oẽs <lb/>per circulũ, nec attinget circulus alicubi ſectionem niſi in puncto c.</s> <s xml:id="echoid-s23552" xml:space="preserve"> Patet ergo primũ propoſitorũ.</s> <s xml:id="echoid-s23553" xml:space="preserve"> <lb/>Qđ ſi linea t c ſemidiameter circuli ſit maior ꝗ̃ linearũ minima, inter oppoſitas ſectiões ꝓductarũ, <lb/> <pb o="55" file="0357" n="357" rhead="LIBER PRIMVS."/> ut eſt t c:</s> <s xml:id="echoid-s23554" xml:space="preserve"> patet, quoniã illa minima linea intra ſuperficiem ſectionis producetur ad peripheriam cir <lb/>culi, ut in punctum m:</s> <s xml:id="echoid-s23555" xml:space="preserve"> aliqua ergo ſuperficies cõmunis erit circulo & ſectioni:</s> <s xml:id="echoid-s23556" xml:space="preserve"> circulus ergo & ſe-<lb/>ctio ſe ſecabunt.</s> <s xml:id="echoid-s23557" xml:space="preserve"> Hęc itaq;</s> <s xml:id="echoid-s23558" xml:space="preserve"> ſectio nõ erit niſi in duobus tantũ punctis g & k:</s> <s xml:id="echoid-s23559" xml:space="preserve"> quod per modum 10 p 3 <lb/>conuinci poteſt.</s> <s xml:id="echoid-s23560" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s23561" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div869" type="section" level="0" n="0"> <head xml:id="echoid-head707" xml:space="preserve" style="it">133. A pũcto dato in circuli circũferẽtia extra diametrũ: poßibile eſt ducere lineãք diametrũ <lb/>ad circũferentiã, ita, ut pars ei{us} interiacẽs diametrũ & reliquãpartẽ circũferẽtiæ, ſit æqualis <lb/>lineæ datæ eidẽ circulo inſcriptibili præmiſſo modo: ſed harum linearum æqualium ab eodẽ pun <lb/>cto dato in eodem circulo producibiles ſunt tantùm duæ. Alhazen 34 n 5.</head> <p> <s xml:id="echoid-s23562" xml:space="preserve">Eſto circulus a b g, cuius diameter ſit b g:</s> <s xml:id="echoid-s23563" xml:space="preserve"> & punctus datus in ſui circũferentia ſita:</s> <s xml:id="echoid-s23564" xml:space="preserve"> & ſit h z linea <lb/>data minor diametro b g, pręmiſſo modo poſsibilis inſcribi circulo.</s> <s xml:id="echoid-s23565" xml:space="preserve"> Dico, quòd â pũcto a poſsibile <lb/>eſt ducere lineã tranſeuntẽ per diametrũ b g, cuius pars interiacens diametrũ b g & circũferentiam <lb/>ſit ęqualis lineę datę, quę h z.</s> <s xml:id="echoid-s23566" xml:space="preserve"> Ducantur enim in circulo lineę b a & a g:</s> <s xml:id="echoid-s23567" xml:space="preserve"> & ſuper punctũ h lineę datę <lb/>h z fiat angulus ęqualis angulo a g b:</s> <s xml:id="echoid-s23568" xml:space="preserve"> qui ſit m h z, ducta linea m h, & ſuper idẽ punctũ h fiat angulus <lb/>æqualis angulo a b g, qui ſit l h z, ducta linea h l:</s> <s xml:id="echoid-s23569" xml:space="preserve"> & â puncto z ducatur linea æquidiſtans lineę h m, <lb/>quę ſit z n:</s> <s xml:id="echoid-s23570" xml:space="preserve"> quę quidẽ ſecabit lineã h l:</s> <s xml:id="echoid-s23571" xml:space="preserve"> ſit, ut ſecet ipſam in puncto x:</s> <s xml:id="echoid-s23572" xml:space="preserve"> & à pũcto z iterũ ducatur alia li <lb/>nea æquidiſtans lineę h l, quę ſit z t, ſecans lineã h m in puncto t:</s> <s xml:id="echoid-s23573" xml:space="preserve"> ſecabit autem per 2 huius:</s> <s xml:id="echoid-s23574" xml:space="preserve"> & à pun <lb/>cto t ducatur ſectio conica, quæ ſit t p, ſicut præmiſſum eſt in 131 huius.</s> <s xml:id="echoid-s23575" xml:space="preserve"> Hæc itaq;</s> <s xml:id="echoid-s23576" xml:space="preserve"> ſectio non contin <lb/>git aliquam linearũ z n & h l, inter quas ipſa iacet.</s> <s xml:id="echoid-s23577" xml:space="preserve"> Similiter fiat ſectio alia conica, iſti oppoſita, inter <lb/> <anchor type="figure" xlink:label="fig-0357-01a" xlink:href="fig-0357-01"/> eaſdem lineas ex parte alia, quę ſit c u:</s> <s xml:id="echoid-s23578" xml:space="preserve"> & inter il-<lb/>las ſectiones omnium linearum ductarum mini-<lb/>ma ducta à puncto t ad ſectionem c u, ſit linea t c.</s> <s xml:id="echoid-s23579" xml:space="preserve"> <lb/>Hæc ergo linea t c ſi fuerit æqualis diametro cir-<lb/>culi b g:</s> <s xml:id="echoid-s23580" xml:space="preserve"> circulus factus ſecundum ſemidiametrũ <lb/>t c (poſito pede circini in puncto t) palàm, quia <lb/>ſectionem c u cõtinget.</s> <s xml:id="echoid-s23581" xml:space="preserve"> Si uerò linea t c fuerit mi-<lb/>nor diametro b g:</s> <s xml:id="echoid-s23582" xml:space="preserve"> circulus factus modo prędicto <lb/>ſecundum quantitatem lineæ b g ſecabit ſectionẽ <lb/>c u in duobus punctis, ut patet per pręmiſſam.</s> <s xml:id="echoid-s23583" xml:space="preserve"> Sit <lb/>ergo nunc primùm linea t c ęqualis diametro b g.</s> <s xml:id="echoid-s23584" xml:space="preserve"> <lb/>Cum ergo linea t c ducatur ad ſectionem conicã, <lb/>quæ interiacet lineas h l & z n:</s> <s xml:id="echoid-s23585" xml:space="preserve"> neceſſariò ſecabit <lb/>linea t c illas ambas lineas:</s> <s xml:id="echoid-s23586" xml:space="preserve"> quas ſi in puncto x (ꝗ <lb/>eſt pũctus communis ſectionis illarum linearũ) <lb/>ſecuerit, erit linea t x æqualis lineæ x c:</s> <s xml:id="echoid-s23587" xml:space="preserve"> quòd ſi <lb/>ipſas in alijs punctis ſecuerit:</s> <s xml:id="echoid-s23588" xml:space="preserve"> ſecet ergo lineã z n <lb/> <anchor type="figure" xlink:label="fig-0357-02a" xlink:href="fig-0357-02"/> in puncto q, & lineam h l ín puncto f:</s> <s xml:id="echoid-s23589" xml:space="preserve"> & du-<lb/>catur à puncto z per 31 p 1 linea æquidiſtans <lb/>ipſi lineæ t c:</s> <s xml:id="echoid-s23590" xml:space="preserve"> quæ per 2 huius ſecabit lineas <lb/>h m & h l, ſicut etiam ſua ęquidiſtans t c:</s> <s xml:id="echoid-s23591" xml:space="preserve"> ſecet <lb/>ergo eas in punctis m & l:</s> <s xml:id="echoid-s23592" xml:space="preserve"> & ſit ipſa linea m z <lb/>l.</s> <s xml:id="echoid-s23593" xml:space="preserve"> Super diametri ergo g b terminum g per 23 <lb/>p 1 fiat angulus æqualis angulo h l m, qui ſit <lb/>angulus b g d:</s> <s xml:id="echoid-s23594" xml:space="preserve"> & ducantur duæ lineæ a d, d b.</s> <s xml:id="echoid-s23595" xml:space="preserve"> <lb/>Palàm ergo, cum angulus g a b ſit rectus per <lb/>31 p 3, quòd alij duo anguli trianguli g a b, ſcili <lb/>cet a g b & a b g ualent rectũ per 32 p 1:</s> <s xml:id="echoid-s23596" xml:space="preserve"> angu-<lb/>lus ergo l h m (qui æqualis eſt illis duobus angulis) eſt rectus:</s> <s xml:id="echoid-s23597" xml:space="preserve"> ergo æqualis angulo g d b:</s> <s xml:id="echoid-s23598" xml:space="preserve"> angulus <lb/>uerò h l m eſt æqualis angulo d g b:</s> <s xml:id="echoid-s23599" xml:space="preserve"> ergo per 32 p 1 angulus tertius unius, trigonorum g b d & h l m <lb/>erit ęqualis angulo tertio alterius, ſcilicet angulus h m l, angulo g b d:</s> <s xml:id="echoid-s23600" xml:space="preserve"> erit ergo per 4 p 6 proportio <lb/>lineæ g b ad b d, ſicut lineæ l m ad m h.</s> <s xml:id="echoid-s23601" xml:space="preserve"> Sit aũt pũctus, in quo linea a d ſecat diametrũ b g, punctus e.</s> <s xml:id="echoid-s23602" xml:space="preserve"> <lb/>Quia ergo ք 27 p 3 angulus a d b eſt æqualis angulo b g a:</s> <s xml:id="echoid-s23603" xml:space="preserve"> quia cadũt in eundẽ arcũ (qui a b) & an-<lb/>gulus b g a æqualis angulo m h z ex p̃miſsis:</s> <s xml:id="echoid-s23604" xml:space="preserve"> erit ergo angulus a d b æqualis angulo m h z:</s> <s xml:id="echoid-s23605" xml:space="preserve"> & patuit <lb/>prius, qđ angulus d b g eſt æqualis angulo h m z:</s> <s xml:id="echoid-s23606" xml:space="preserve"> erit ergo tertius angulus trianguli d e b per 32 p 1 <lb/>ęqualis tertio angulo trigoni m h z, ſcilicet angulus d e b angulo m z h.</s> <s xml:id="echoid-s23607" xml:space="preserve"> Quia ergo trigona d e b & m <lb/>z h ſunt æquiangula, erit per 4 p 6 proportio lineę b d ad d e, ſicut lineę m h ad h z.</s> <s xml:id="echoid-s23608" xml:space="preserve"> Oſtẽſum eſt aũt <lb/>ſuperius, qđ eſt ꝓportio lineę g b ad b d, ſicut lineę l m ad m h:</s> <s xml:id="echoid-s23609" xml:space="preserve"> ergo ք 22 p 5 erit ք ęquã ꝓportiona-<lb/>litatẽ ꝓportio lineę b g ad d e, ſicut lineę l m ad h z:</s> <s xml:id="echoid-s23610" xml:space="preserve"> ſed ſicut ք 131 huius declaratũ eſt, patet, qđ linea <lb/>q t eſt ęqualis lineę f c:</s> <s xml:id="echoid-s23611" xml:space="preserve"> ſed linea t q eſt ęqualis lineę m z ք 34 p 1, cũ parallelogrãmũ m t q z ſit ęquidi-<lb/>ſtãtiũ laterũ, ut patet ex pręmiſsis:</s> <s xml:id="echoid-s23612" xml:space="preserve"> eſt igitur linea m z ęqualis lineę f c:</s> <s xml:id="echoid-s23613" xml:space="preserve"> ſed ք 34 p 1 linea z l eſt æqua-<lb/>lis lineę t f.</s> <s xml:id="echoid-s23614" xml:space="preserve"> Eſt igitur totalis linea m l æqualis totali lineę t c:</s> <s xml:id="echoid-s23615" xml:space="preserve"> ergo ք 7 p 5 eſt ꝓportio lineę t c ad h z, <lb/>ſicut lineę l m ad h z.</s> <s xml:id="echoid-s23616" xml:space="preserve"> Eſt ergo ꝓportio lineæ g b ad lineã d e, ſicut lineę t c ad h z:</s> <s xml:id="echoid-s23617" xml:space="preserve"> & քmutatim.</s> <s xml:id="echoid-s23618" xml:space="preserve"> Cũ <lb/>ergo linea t c ſit æqualis lineę g b, erit linea e d æqualis ipſi h z datę lineę.</s> <s xml:id="echoid-s23619" xml:space="preserve"> Quod eſt propoſitũ.</s> <s xml:id="echoid-s23620" xml:space="preserve"> Si au <lb/>tem linea t c ſit minor diametro b g:</s> <s xml:id="echoid-s23621" xml:space="preserve"> producatur ultra ſectionem, donec ipſa ſit æqualis diametro <lb/> <pb o="56" file="0358" n="358" rhead="VITELLONIS OPTICAE"/> b g, & ſecundum quantitatem eius fiat circulus.</s> <s xml:id="echoid-s23622" xml:space="preserve"> Palàm per pręmiſſam, quòd ille ſecabit ſectionem <lb/>in punctis duobus, qui ſint c & u:</s> <s xml:id="echoid-s23623" xml:space="preserve"> à quibus lineæ ductę ad punctum t erunt æquales lineę b g per <lb/>definitionem circuli:</s> <s xml:id="echoid-s23624" xml:space="preserve"> & tunc à puncto z ducatur linea æquidiſtans alteri illarum, & item alia a qui-<lb/>diſtans alteri:</s> <s xml:id="echoid-s23625" xml:space="preserve"> & tunc erit ducere à puncto a per modum prędictum duas lineas e d æquales lineæ <lb/>datę:</s> <s xml:id="echoid-s23626" xml:space="preserve"> & erit idem penitus probandi modus, qui ſuprà.</s> <s xml:id="echoid-s23627" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s23628" xml:space="preserve"/> </p> <div xml:id="echoid-div869" type="float" level="0" n="0"> <figure xlink:label="fig-0357-01" xlink:href="fig-0357-01a"> <variables xml:id="echoid-variables374" xml:space="preserve">a g e b d</variables> </figure> <figure xlink:label="fig-0357-02" xlink:href="fig-0357-02a"> <variables xml:id="echoid-variables375" xml:space="preserve">h n t f x q c u p m z f</variables> </figure> </div> </div> <div xml:id="echoid-div871" type="section" level="0" n="0"> <head xml:id="echoid-head708" xml:space="preserve" style="it">134. Dato trigono orthogonio, & dato puncto in uno ſuorum laterum angulum rectum con-<lb/>tinentium: poßibile est ducere à puncto illo ad aliud laterum continentium angulum rectum <lb/>lineam ſecantem baſim it a, quòd pars ductæ lineæ interiacens punctum ſectionis, & latus, in <lb/>quo non est punctus datus, ſe habeat ad partem baſis, quæ est à ſectione ad latus, in quo eſt pun <lb/>ctus datus, ſicut data linea ad datam lineam. Alhazen 35 n 5.</head> <p> <s xml:id="echoid-s23629" xml:space="preserve">Eſto a b g triangulus datus, cuius angulus a b g ſit rectus:</s> <s xml:id="echoid-s23630" xml:space="preserve"> & in latere illius b g ſit pũctus datus, <lb/>qui ſit d, extra triangulum aut intra:</s> <s xml:id="echoid-s23631" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s23632" xml:space="preserve"> datę lineę duę e & z.</s> <s xml:id="echoid-s23633" xml:space="preserve"> Dico, quòd à puncto d poſsibile eſt <lb/>ducere lineam ſecantem baſim a g, & concurrentem cum latere a b, ita, quòd pars lineæ ſecãtis in-<lb/> <anchor type="figure" xlink:label="fig-0358-01a" xlink:href="fig-0358-01"/> teriacens latus a b & baſim a g, ſit eiuſdem proportionis ad partem <lb/>baſis a g, quę eſt ab illa linea uſq;</s> <s xml:id="echoid-s23634" xml:space="preserve"> ad punctum g, cuius eſt data linea <lb/>e ad datam lineam z.</s> <s xml:id="echoid-s23635" xml:space="preserve"> Sit enim primò pũctus d in ipſo trigono a b g:</s> <s xml:id="echoid-s23636" xml:space="preserve"> <lb/>& ducatur ab eo linea æquidiſtans lineę a b per 31 p 1, quę ſit d m:</s> <s xml:id="echoid-s23637" xml:space="preserve"> & <lb/>fiat circulus per tria puncta g, d, m per 5 p 4:</s> <s xml:id="echoid-s23638" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s23639" xml:space="preserve"> linea g m diame-<lb/>ter huius circuli per 31 p 3:</s> <s xml:id="echoid-s23640" xml:space="preserve"> ſubtenditur enim angulo recto per 29 p 1:</s> <s xml:id="echoid-s23641" xml:space="preserve"> <lb/>& protrahatur linea a d.</s> <s xml:id="echoid-s23642" xml:space="preserve"> Et quia per eandem 29 p 1 angulus g m d eſt <lb/>æqualis angulo g a b:</s> <s xml:id="echoid-s23643" xml:space="preserve"> palàm, quia angulus g m d erit maior angulo <lb/>g a d, cum angulus g a b ſit maior angulo g a d:</s> <s xml:id="echoid-s23644" xml:space="preserve"> ſecetur ergo ex angu-<lb/>lo g m d angulus æqualis angulo g a d per 27 huius, ducta linea m n <lb/>ad peripheriam circuli:</s> <s xml:id="echoid-s23645" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s23646" xml:space="preserve"> angulus d m n:</s> <s xml:id="echoid-s23647" xml:space="preserve"> quę autem eſt proportio <lb/>lineę e ad lineam z, eadẽ ſit per 3 huius proportio lineę a d ad lineã <lb/>h:</s> <s xml:id="echoid-s23648" xml:space="preserve"> & à puncton, qui eſt punctus in peripheria circuli, ducatur linea <lb/>ad diametrum g m, quę ſit n l, ſecans circulum in pũcto c, ita, ut eius <lb/>pars interiacens peripheriam circuli & diametrum, quę eſt c l, ſit æ-<lb/>qualis lineę datę h per 128 uel per 130 huius:</s> <s xml:id="echoid-s23649" xml:space="preserve"> & ducatur linea g c:</s> <s xml:id="echoid-s23650" xml:space="preserve"> & à <lb/>pũcto d ducatur linea ad punctũ c, quę cũ cadat inter duas lineas ę-<lb/>quidiſtantes, quę ſunt d m & b a, tenens angulum acutũ cum earum <lb/>altera, ut cũ m d, ſi producatur, neceſſariò concurret cũ reliqua per <lb/>2 huius:</s> <s xml:id="echoid-s23651" xml:space="preserve"> cõcurrat ergo in puncto q.</s> <s xml:id="echoid-s23652" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s23653" xml:space="preserve"> per 27 p 3 angulus g m <lb/>d eſt æqualis angulo g c d, & angulus g m d eſt ęqualis angulo g a b <lb/>per 29 p 1:</s> <s xml:id="echoid-s23654" xml:space="preserve"> palàm, quòd angulus g c d eſt ęqualis angulo g a b:</s> <s xml:id="echoid-s23655" xml:space="preserve"> ergo per 13 p 1 erit angulus g c q ęqua-<lb/>lis angulo b a l:</s> <s xml:id="echoid-s23656" xml:space="preserve"> ſed angulus b a l per 15 p 1 eſt ęqualis angulo g a q:</s> <s xml:id="echoid-s23657" xml:space="preserve"> angulus ergo g c q eſt æqualis an-<lb/> <anchor type="figure" xlink:label="fig-0358-02a" xlink:href="fig-0358-02"/> gulo g a q.</s> <s xml:id="echoid-s23658" xml:space="preserve"> Sit autẽ t punctus, in quo li <lb/>nea d q ſecat lineam a g:</s> <s xml:id="echoid-s23659" xml:space="preserve"> erit ergo per <lb/>15 p 1 angulus g t c æqualis angulo at <lb/>q.</s> <s xml:id="echoid-s23660" xml:space="preserve"> Quia ergo trigonorum a t q & t c g <lb/>duo anguli ſunt ęquales, erit & terti-<lb/>us tertio ęqualis:</s> <s xml:id="echoid-s23661" xml:space="preserve"> trianguli ergo a t q <lb/>& t c g ſunt æquianguli:</s> <s xml:id="echoid-s23662" xml:space="preserve"> ergo ք 4 p 6 <lb/>erit proportio lineę q t ad t g, ſicut li-<lb/>neę a t ad t c:</s> <s xml:id="echoid-s23663" xml:space="preserve"> uerùm angulus n m d ex <lb/>p̃miſsis eſt ęqualis angulo t a d.</s> <s xml:id="echoid-s23664" xml:space="preserve"> Quia <lb/>enim anguli g m d & t a b ſunt ęqua-<lb/>les:</s> <s xml:id="echoid-s23665" xml:space="preserve"> & anguli g m n & d a g ęquales:</s> <s xml:id="echoid-s23666" xml:space="preserve"> re-<lb/>linquitur n m a ęqualis angulo t a d:</s> <s xml:id="echoid-s23667" xml:space="preserve"> <lb/>ſed & angulus n c d ք 27 p 3 eſt ęqualis <lb/>angulo n m d:</s> <s xml:id="echoid-s23668" xml:space="preserve"> quare angulus n c d eſt <lb/>ęqualis angulo t a d:</s> <s xml:id="echoid-s23669" xml:space="preserve"> ergo ք 15 p 1 angu <lb/>lus t c l, qui eſt contra poſitus angulo <lb/>n c d, eſt æqualis angulo t a d.</s> <s xml:id="echoid-s23670" xml:space="preserve"> Quia er <lb/>go angulus t c l eſt cõmunis duobus <lb/>trigonis, ſcilicet trigono t c l & trigo-<lb/>no t a d, & anguli t c l & t a d ſunt ęquales:</s> <s xml:id="echoid-s23671" xml:space="preserve"> erũt ք 32 p 1 trigoni t c l & t a d ęquianguli:</s> <s xml:id="echoid-s23672" xml:space="preserve"> ergo ք 4 p 6 eſt <lb/>ꝓportio lineę t a ad lineã t c, ſicut lineæ a d ad lineã l c.</s> <s xml:id="echoid-s23673" xml:space="preserve"> Fuit aũt oſtẽ ſum ſuperius, qđ eſt ꝓportio li-<lb/>neę t q ad lineã t g, ſicut lineę a t ad lineã t c:</s> <s xml:id="echoid-s23674" xml:space="preserve"> ergo ք 11 p 5 erit proportio lineę a d ad l c, ſicut lineę q t <lb/>ad t g:</s> <s xml:id="echoid-s23675" xml:space="preserve"> ſed linea l c eſt æqualis lineę h, & proportio lineę a d, ad lineam h eſt, ſicut proportio lineæ <lb/>e a d z.</s> <s xml:id="echoid-s23676" xml:space="preserve"> Ergo ք 7 & 11 p 5 erit ꝓportio lineæ q t ad lineã t g, ſicut lineę e ad lineã z.</s> <s xml:id="echoid-s23677" xml:space="preserve"> Quod eſt ꝓpoſitũ.</s> <s xml:id="echoid-s23678" xml:space="preserve"> <lb/>Si uerò d pũctus datus ſit in latere trigoni, qđ eſt b g, extra triangulũ ꝓducto:</s> <s xml:id="echoid-s23679" xml:space="preserve"> ducatur prius à pun <lb/>cto d linea ęquidiſtãs lineę a b:</s> <s xml:id="echoid-s23680" xml:space="preserve"> & ſit d m:</s> <s xml:id="echoid-s23681" xml:space="preserve"> & ducatur linea a g, donec cõcurrat cũ linea d m in pũcto <lb/> <pb o="57" file="0359" n="359" rhead="LIBER I."/> m:</s> <s xml:id="echoid-s23682" xml:space="preserve"> & fiat, ut prius, circulus trãſiens ք tria pũcta g, d, m:</s> <s xml:id="echoid-s23683" xml:space="preserve"> erit ergo, ut prius, m g diameter iſtius circuli:</s> <s xml:id="echoid-s23684" xml:space="preserve"> <lb/>& ducatur linea a d:</s> <s xml:id="echoid-s23685" xml:space="preserve"> erit ꝗ dẽ angulus g a d maior angulo g m d ք 16 p 1:</s> <s xml:id="echoid-s23686" xml:space="preserve"> fiat ergo, ut prius, ſuք pũctũ <lb/>m lineæ d m angulus æqualis angulo g a d ք lineã m n, ꝗ ſit angulus d m n:</s> <s xml:id="echoid-s23687" xml:space="preserve"> & à puncto n, ꝗ ſit in cir-<lb/>cũferẽtia circuli, ducatur, ut prius, ք 128 uel ք 130 huius linea ad eductã diametrũ m g, cõcurrens cũ <lb/>ipſa in pũcto l, & ſecãs peripheriã circuli in pũcto c, ita, ut linea c l ſit ęqualis lineę h aſſumptę, ut pri <lb/>us, ſcilicet ut ք 3 huius ſit ꝓportio lineæ a d ad ipſam h, ſicut lineę datę e ad lineã datã z:</s> <s xml:id="echoid-s23688" xml:space="preserve"> & ducatur <lb/>linea d c fecans lineã a g in pũcto t, & lineã a b in puncto q.</s> <s xml:id="echoid-s23689" xml:space="preserve"> Cũ ergo angulus n m d, & angulus n c d <lb/>per 22 p 3 ſint æquales duobus rectis, & angulus n m d ſit æqualis angulo t a d ex præmiſsis:</s> <s xml:id="echoid-s23690" xml:space="preserve"> palàm <lb/>ex 13 p 1, quoniam erit angulus t c l æqualis angulo t a d:</s> <s xml:id="echoid-s23691" xml:space="preserve"> erunt ergo duo triangulit c l & t a d per 15 <lb/>& 32 p 1 æquianguli:</s> <s xml:id="echoid-s23692" xml:space="preserve"> erit ergo per 4 p 6 proportio lineæ d a ad lineam c l, ſicut lineæ t a ad lineã t c.</s> <s xml:id="echoid-s23693" xml:space="preserve"> <lb/>Cum autem per 27 p 3 duo anguli g c d & g m d ſint æquales, quoniam cadunt in eundem arcum, <lb/>qui eſt d g, angulus uerò t a q per 29 p 1 eſt æqualis angulo g m d:</s> <s xml:id="echoid-s23694" xml:space="preserve"> erit angulus t a q ęqualis angulo t <lb/>c g:</s> <s xml:id="echoid-s23695" xml:space="preserve"> ſed & anguli q t a & g t c ſunt ęquales ք 15 p 1:</s> <s xml:id="echoid-s23696" xml:space="preserve"> erũt ergo trigoni g t c & t a q ęquiãguli ք 32 p 1:</s> <s xml:id="echoid-s23697" xml:space="preserve"> erit <lb/>ergo per 4 p 6 proportio lineæ a t ad lineam t c, ſicut lineæ q t ad lineam t g.</s> <s xml:id="echoid-s23698" xml:space="preserve"> Eſt ergo ex præmiſsis <lb/>& per 11 p 5 proportio lineę a d ad lineam c l, quę eſt æqualis lιneæ h, ſicut lineæ q t ad lineam t g.</s> <s xml:id="echoid-s23699" xml:space="preserve"> Eſt <lb/>ergo per 11 p 5 proportio lineæ e ad lineam z, ſicut lineę q t ad lineam t g.</s> <s xml:id="echoid-s23700" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s23701" xml:space="preserve"/> </p> <div xml:id="echoid-div871" type="float" level="0" n="0"> <figure xlink:label="fig-0358-01" xlink:href="fig-0358-01a"> <variables xml:id="echoid-variables376" xml:space="preserve">q ſ a e z a h t d m c b d g n</variables> </figure> <figure xlink:label="fig-0358-02" xlink:href="fig-0358-02a"> <variables xml:id="echoid-variables377" xml:space="preserve">l d b q a a e z h d l g c e z h <gap/> t g c b q a d m n a m n d</variables> </figure> </div> </div> <div xml:id="echoid-div873" type="section" level="0" n="0"> <head xml:id="echoid-head709" xml:space="preserve" style="it">135. Datis duob{us} punctis, uno in circulo, alio extra circulum, uelutro extra circulum: poſ <lb/>ſibile eſt inuenire punctum in circumferentia dati circuli, ita, ut angulum contentum à lineis <lb/>à prædictis punctis ad punctum inuentum ductis diuidat per æqualia, linea in illo puncto cir-<lb/>culum contingens. Alhazen 36 n 5.</head> <p> <s xml:id="echoid-s23702" xml:space="preserve">Sunto duo pũcti dati, qui e & d, quorũ primò unus, ꝗ ſit e, ſit in circulo, & reliquus extra illũ:</s> <s xml:id="echoid-s23703" xml:space="preserve"> & <lb/>ſit datus circulus, cuius centrũ ſit g.</s> <s xml:id="echoid-s23704" xml:space="preserve"> Dico, quòd poſsibile eſt in peripheria circuli g inuenire pũctũ, <lb/>in quo linea cõtingens circulũ ducta, ſecet angulũ cõtentũ à lineis à pũctis d & e ad illũ punctũ du-<lb/>ctis per æqualia.</s> <s xml:id="echoid-s23705" xml:space="preserve"> Ducatur enim à pũcto e ad cẽtrũ g linea e g:</s> <s xml:id="echoid-s23706" xml:space="preserve"> & ꝓducatur uſq;</s> <s xml:id="echoid-s23707" xml:space="preserve"> ad circumferentiã, & <lb/>ſit e g s:</s> <s xml:id="echoid-s23708" xml:space="preserve"> deinde ducatur linea d g:</s> <s xml:id="echoid-s23709" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s23710" xml:space="preserve"> ex præmiſsis linea e g minor ꝗ̃ linea d g.</s> <s xml:id="echoid-s23711" xml:space="preserve"> Aſſumatur quoq;</s> <s xml:id="echoid-s23712" xml:space="preserve"> <lb/>linea m i, quę in puncto c taliter diuidatur, ut ꝓportio lineæ i c ad lineã c m ſit, ſicut lineę d g ad li-<lb/>neã g e ք 119 huius:</s> <s xml:id="echoid-s23713" xml:space="preserve"> diuidaturq́;</s> <s xml:id="echoid-s23714" xml:space="preserve"> linea m i ք æqualia in puncto n:</s> <s xml:id="echoid-s23715" xml:space="preserve"> à quo ſuper lineã m i ducatur per-<lb/>pẽdicularis ք 11 p 1, q̃ ſit n o:</s> <s xml:id="echoid-s23716" xml:space="preserve"> & ſuper punctũ m ք 23 p 1 fiat angulus ęqualis medietati anguli d g s di <lb/>uiſi per 9 p 1 ք æqualia:</s> <s xml:id="echoid-s23717" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s23718" xml:space="preserve"> linea m o.</s> <s xml:id="echoid-s23719" xml:space="preserve"> Palàm aũt, quòd angulus i m o erit minor recto, quoniã <lb/>angulus d g s eſt minor duob.</s> <s xml:id="echoid-s23720" xml:space="preserve"> rectis:</s> <s xml:id="echoid-s23721" xml:space="preserve"> ſed angulus o n m eſt rectus:</s> <s xml:id="echoid-s23722" xml:space="preserve"> igitur per 14 huius linea m o con-<lb/>curret cum linea n o:</s> <s xml:id="echoid-s23723" xml:space="preserve"> ſit autem punctus concurſus o:</s> <s xml:id="echoid-s23724" xml:space="preserve"> à puncto uerò c ducatur linea ad triangulum <lb/>m n o, quę ſit c k f, ita, ut proportio lineę k f ad lineã f m ſit, ſicut proportio lineæ e g ad lineam g s:</s> <s xml:id="echoid-s23725" xml:space="preserve"> <lb/>quod fieri poteſt per pręcedentẽ.</s> <s xml:id="echoid-s23726" xml:space="preserve"> Ducatur quoq;</s> <s xml:id="echoid-s23727" xml:space="preserve"> linea m k:</s> <s xml:id="echoid-s23728" xml:space="preserve"> & ſuper punctũ g terminũ lineę e g ք 23 <lb/>p 1 fiat angulus ęqualis angulo m f k, ք lineã uſq;</s> <s xml:id="echoid-s23729" xml:space="preserve"> ad circumferentiã productã:</s> <s xml:id="echoid-s23730" xml:space="preserve"> q̃ ſit a g:</s> <s xml:id="echoid-s23731" xml:space="preserve"> & ſit angu-<lb/>lus a g e:</s> <s xml:id="echoid-s23732" xml:space="preserve"> & ducantur duælineæ a g & a d.</s> <s xml:id="echoid-s23733" xml:space="preserve"> Dico, quòd a eſt quęſitus punctus.</s> <s xml:id="echoid-s23734" xml:space="preserve"> Ducatur enim linea <lb/> <anchor type="figure" xlink:label="fig-0359-01a" xlink:href="fig-0359-01"/> e a.</s> <s xml:id="echoid-s23735" xml:space="preserve"> Cum ergo ex pręmiſsis <lb/>angulus m f k ſit æqualis <lb/>angulo a g e, & ꝓportio li-<lb/>neę k f ad lineã f m, eſt, ſicut <lb/>ꝓportio lineæ e g ad lineã <lb/>g s:</s> <s xml:id="echoid-s23736" xml:space="preserve"> ergo ք 7 p 5 erit ꝓpor-<lb/>tio lineę k f ad lineã f m, ſi-<lb/>cut lineę e g ad lineam g a, <lb/>æqualem g s, quia ambę ex <lb/>cẽtro:</s> <s xml:id="echoid-s23737" xml:space="preserve"> erit triangulus a g e <lb/>ſimilis triangulo m f k ք 6 <lb/>p 6:</s> <s xml:id="echoid-s23738" xml:space="preserve"> igitur angulus f m k eſt <lb/>æqualis angulo e a g, & an-<lb/>gulus a e g æqualis angulo <lb/>m k f.</s> <s xml:id="echoid-s23739" xml:space="preserve"> Igitur à pũcto a duca <lb/>tur linea tenẽs cũ linea a e <lb/>angulũ æqualẽ angulo n m k:</s> <s xml:id="echoid-s23740" xml:space="preserve"> & ſit linea a z, q̃ neceſſariò cõcurret cũ linea e g:</s> <s xml:id="echoid-s23741" xml:space="preserve"> quoniã eſt proportio <lb/>e g ad g a, ſicut k f ad f m, & angulus g a z æqualis eſt angulo f m c:</s> <s xml:id="echoid-s23742" xml:space="preserve"> fuit enim prius angulus e a g ęqua <lb/>lis angulo f m k.</s> <s xml:id="echoid-s23743" xml:space="preserve"> Sicut ergo linea m o cõcurrit cũlinea k f in pũcto f, ſic cõcurret linea a z cũ linea g e.</s> <s xml:id="echoid-s23744" xml:space="preserve"> <lb/>Sit ergo cõcurfus in pũcto z:</s> <s xml:id="echoid-s23745" xml:space="preserve"> & ꝓducatur linea a z uſq;</s> <s xml:id="echoid-s23746" xml:space="preserve"> ad pũctũ q:</s> <s xml:id="echoid-s23747" xml:space="preserve"> donec linea a z ſe habeat ad lineã <lb/>q z, ſicut linea m c ad c i ք 3 huius:</s> <s xml:id="echoid-s23748" xml:space="preserve"> erit ergo ꝓportio lineę a z ad lineã q z, ſicut lineæ d g ad lineã g e:</s> <s xml:id="echoid-s23749" xml:space="preserve"> <lb/>& ducatur linea e q.</s> <s xml:id="echoid-s23750" xml:space="preserve"> Deinde à pũcto a ducatur linea æquidiſtãs lineę e q, q̃ ſit linea a t ք 31 p 1, & erit <lb/>angulus a q e æqualis angulo q a t ք 29 p 1, quoniã duo anguli z e a & e a t ſunt minores duobus re-<lb/>ctis, ideo, quia ք 29 p 1 anguli q e a & e a t ualẽt duos rectos:</s> <s xml:id="echoid-s23751" xml:space="preserve"> cõcurret linea a t neceſſariò cũ linea e z <lb/>ք 14 huius.</s> <s xml:id="echoid-s23752" xml:space="preserve"> Sit ergo pũctus cõcurſus t.</s> <s xml:id="echoid-s23753" xml:space="preserve"> Quia uerò angulus e a z eſt æqualis angulo n m k, ut ſuprà pa <lb/>tuit:</s> <s xml:id="echoid-s23754" xml:space="preserve"> ducta à pũcto e linea քpẽdiculari ſuք lineã a z ք 12 p 1, q̃ ſit e l, erũt trigoni a e l & n m k æquiã gu <lb/>li ք 32 p 1:</s> <s xml:id="echoid-s23755" xml:space="preserve"> erit ergo angulus a e l æqualis angulo m k n, & angul<emph style="sub">9</emph> a l e ęqualis angulo m n k, ꝗa uterq;</s> <s xml:id="echoid-s23756" xml:space="preserve"> <lb/>eſt rect<emph style="sub">9</emph>:</s> <s xml:id="echoid-s23757" xml:space="preserve"> ſed etιã angul<emph style="sub">9</emph> a e g eſt ex p̃miſsis ęqualis angulo m k f:</s> <s xml:id="echoid-s23758" xml:space="preserve"> reſtat ergo ք 13 p 1, ut angul<emph style="sub">9</emph> l e z ſit <lb/>ęqualis angulo n k c, & angul<emph style="sub">9</emph> e l z rectus eſt ęqualis angulo k n c recto:</s> <s xml:id="echoid-s23759" xml:space="preserve"> erit ergo ք 32 p 1 angul<emph style="sub">9</emph> e l z <lb/> <pb o="58" file="0360" n="360" rhead="VITELLONIS OPTICAE"/> æqualis angulo k c n.</s> <s xml:id="echoid-s23760" xml:space="preserve"> Igitur per 13 p 1 erit angulus e z q æqualis angulo k c i.</s> <s xml:id="echoid-s23761" xml:space="preserve"> Palàm ergo ex <lb/>præmiſsis, quòd triangulus a e g eſt æquiangulus triangulo f m k:</s> <s xml:id="echoid-s23762" xml:space="preserve"> & triangulus e a l æquiangulus <lb/>eſt triangulo k m n:</s> <s xml:id="echoid-s23763" xml:space="preserve"> & triangulus e l z æquiangulus triangulo k n c:</s> <s xml:id="echoid-s23764" xml:space="preserve"> & triangulus e a z æquiangulus <lb/>triangulo k m c.</s> <s xml:id="echoid-s23765" xml:space="preserve"> Eſt igitur per 4 p 6 proportio a z ad e z, ſicut m c ad c k:</s> <s xml:id="echoid-s23766" xml:space="preserve"> eſt autem propor-<lb/>tio q z ad z a, ſicut proportio i c ad c m, utpatet ex præmiſsis:</s> <s xml:id="echoid-s23767" xml:space="preserve"> erit ergo per 22 p 5 proportio q z ad <lb/>z e, ſicut i c ad c k:</s> <s xml:id="echoid-s23768" xml:space="preserve"> eſt ergo triangulus q z e per 6 p 6 æquiangulus triangulo i c k.</s> <s xml:id="echoid-s23769" xml:space="preserve"> Cum ergo triangu <lb/>lus e l z ſit æquiangulus triangulo k n c:</s> <s xml:id="echoid-s23770" xml:space="preserve"> erit totus triangulus q l e æquiangulus toti triangulo i k n:</s> <s xml:id="echoid-s23771" xml:space="preserve"> <lb/>eſt ergo per 4 p 6 proportio e l ad l q, ſicut k n ad n i:</s> <s xml:id="echoid-s23772" xml:space="preserve"> & ſimiliter eſt proportio a l ad l e, ſicut m n ad <lb/>n k:</s> <s xml:id="echoid-s23773" xml:space="preserve"> erit ergo per 22 p 5 proportio n m ad n i, ſicut a l ad l q:</s> <s xml:id="echoid-s23774" xml:space="preserve"> ſed linea n m eſt æqualis n i ex hypo-<lb/>theſi:</s> <s xml:id="echoid-s23775" xml:space="preserve"> ergo linea a l eſt æqualis lineæ l q:</s> <s xml:id="echoid-s23776" xml:space="preserve"> ergo per 4 p 1 linea e q erit æqualis lineæ e a:</s> <s xml:id="echoid-s23777" xml:space="preserve"> & angulus l q e <lb/>æqualis angulo l a e:</s> <s xml:id="echoid-s23778" xml:space="preserve"> ſed & angulus e q z per 29 p 1 eſt æqualis angulo t a l:</s> <s xml:id="echoid-s23779" xml:space="preserve"> angulus ergo e a l eſt æ-<lb/>qualis angulo t a l:</s> <s xml:id="echoid-s23780" xml:space="preserve"> quia angulus e q z eſt æqualis angulo t a l:</s> <s xml:id="echoid-s23781" xml:space="preserve"> & angulus e z q eſt æqualis angulo <lb/>a z t per 15 p 1:</s> <s xml:id="echoid-s23782" xml:space="preserve"> igitur tertius tertio:</s> <s xml:id="echoid-s23783" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s23784" xml:space="preserve"> triangulus z e q æquiangulus triangulo z a t.</s> <s xml:id="echoid-s23785" xml:space="preserve"> Eſt ergo per <lb/>4 p 6 proportio q z ad z a, ſicut e z ad z t, & ſicut e q ad a t:</s> <s xml:id="echoid-s23786" xml:space="preserve"> eſt autem ex præmiſsis linea e q æqualis <lb/>lineæ e a:</s> <s xml:id="echoid-s23787" xml:space="preserve"> ergo per 7 p 5 eſt proportio q z ad z a, ſicut a e ad a t:</s> <s xml:id="echoid-s23788" xml:space="preserve"> ſed q z ad z a eſt ex pręmiſsis, ſicut e g <lb/>ad g d.</s> <s xml:id="echoid-s23789" xml:space="preserve"> Igitur ք 11 p 5 eſt ꝓportio lineę a e ad a t, ſicut e g ad g d.</s> <s xml:id="echoid-s23790" xml:space="preserve"> Fiat aũt ſuper pũctũ a angulus ęqua-<lb/>lis angulo g a e:</s> <s xml:id="echoid-s23791" xml:space="preserve"> qui ſit u a g.</s> <s xml:id="echoid-s23792" xml:space="preserve"> ꝓ ducta linea a u, ſi poſsibile fuerit, uſq;</s> <s xml:id="echoid-s23793" xml:space="preserve"> ad lineã g s.</s> <s xml:id="echoid-s23794" xml:space="preserve"> Palàm ergo ex prę-<lb/>miſsis, quoniã angulus g a l eſt medietas anguli u a t:</s> <s xml:id="echoid-s23795" xml:space="preserve"> cũ enim angulus e a q ex pręmiſsis & ք 5 p 1, <lb/>ideo, quia lineę a e & e q ſunt æquales, ſit æqualis angulo a q e, qui per 29 p 1 eſt æqualis angulo <lb/>q a t:</s> <s xml:id="echoid-s23796" xml:space="preserve"> patet, quòd angulus e a l eſt æqualis angulo l a t:</s> <s xml:id="echoid-s23797" xml:space="preserve"> ſed angulus g a e eſt ęqualis angulo u a g.</s> <s xml:id="echoid-s23798" xml:space="preserve"> Eſt <lb/>ergo angulus g a l medietas anguli u a t:</s> <s xml:id="echoid-s23799" xml:space="preserve"> ſed angulus g a l cum ſit ex pręmiſsis ęqualis angulo f m c, <lb/>qui conſtitutus eſt ęqualis medietati anguli d g s, ęqualis eſt medietati anguli d g u.</s> <s xml:id="echoid-s23800" xml:space="preserve"> Angulus ergo <lb/>u a t eſt ęqualis angulo d g u:</s> <s xml:id="echoid-s23801" xml:space="preserve"> ſed anguli t a u & t u a ſunt minores duobus rectis argumento 32 p 1, <lb/>cum lineę a t & u t concurrant in puncto t.</s> <s xml:id="echoid-s23802" xml:space="preserve"> Quare duo anguli t u a, d g u ſunt minores duobus re-<lb/>ctis.</s> <s xml:id="echoid-s23803" xml:space="preserve"> Igitur linea a u concurret cum linea d g per 14 huius.</s> <s xml:id="echoid-s23804" xml:space="preserve"> Dico autem, quòd concurrent in puncto <lb/>d:</s> <s xml:id="echoid-s23805" xml:space="preserve"> efficiet enim linea u a producta ad lineam g d cum lineis u g & g d, triangulum ſimilem triangulo <lb/>a u t:</s> <s xml:id="echoid-s23806" xml:space="preserve"> quoniam iſti trigoni habent angulum a u g communem, & angulus t a u eſt æqualis angulo <lb/>d g u:</s> <s xml:id="echoid-s23807" xml:space="preserve"> erit ergo tertius tertio æqualis:</s> <s xml:id="echoid-s23808" xml:space="preserve"> ergo per 4 p 6 eſt proportio a u ad a t.</s> <s xml:id="echoid-s23809" xml:space="preserve"> ſicutu g ad lineam, quã <lb/>ſecat a u ex g d:</s> <s xml:id="echoid-s23810" xml:space="preserve"> & ꝓportio e a ad a u eſt, ſicut e g ad g u ꝓ 3 p 6:</s> <s xml:id="echoid-s23811" xml:space="preserve"> quia angulus u a g eſt ęqualis angulo <lb/>g a e.</s> <s xml:id="echoid-s23812" xml:space="preserve"> Cum ergo ex pręmiſsis eadem ſit proportio e a ad a t, quę e g ad g d:</s> <s xml:id="echoid-s23813" xml:space="preserve"> & proportio e a ad a t ſit <lb/>compoſita ex proportione e a ad a u, & a u ad a t:</s> <s xml:id="echoid-s23814" xml:space="preserve"> (quoniam per 13 huius proportio extremorum <lb/>componitur ſemper ex proportione cuiuſcunq;</s> <s xml:id="echoid-s23815" xml:space="preserve"> medię ad ambas extremas) erit proportio e g ad <lb/>g d cõpoſita ex eiſdẽ proportionibus.</s> <s xml:id="echoid-s23816" xml:space="preserve"> Quare erit cõpoſita ex proportione e g ad g u, & g u adlineã, <lb/>quã ſecat u a ex linea g d:</s> <s xml:id="echoid-s23817" xml:space="preserve"> ſed eſt cõpoſita ex proportionibus e g ad g u, & g u ad g d.</s> <s xml:id="echoid-s23818" xml:space="preserve"> Igitur linea, quã <lb/>ſecat a u ex g d, eſt linea g d.</s> <s xml:id="echoid-s23819" xml:space="preserve"> Ergo a u ſecat g d in pũcto d.</s> <s xml:id="echoid-s23820" xml:space="preserve"> Producatur ergo per 17 p 3 à pũcto a linea <lb/>cõtingens circulũ, quæ ſit a h:</s> <s xml:id="echoid-s23821" xml:space="preserve"> erit ergo angulus g a h rectus per 18 p 3:</s> <s xml:id="echoid-s23822" xml:space="preserve"> ſed angulus g a l eſt medietas <lb/>anguli d g u, ut patet ex præmiſsis.</s> <s xml:id="echoid-s23823" xml:space="preserve"> Igiturangulus l a h eſt medietas anguli d g e:</s> <s xml:id="echoid-s23824" xml:space="preserve"> ideo, quia anguli <lb/>d g u & d g e ualent duos rectos per 13 p 1, & angulus g a b eſt rectus.</s> <s xml:id="echoid-s23825" xml:space="preserve"> Sed cũ angulus t a u ſit æqualis <lb/>angulo d g u, erit angulus t a d æqualis angulo d g e per 13 p 1.</s> <s xml:id="echoid-s23826" xml:space="preserve"> Igitur angulus l a h eſt medietas angu <lb/>li t a d, & angulus e a l eſt medietas anguli e a t.</s> <s xml:id="echoid-s23827" xml:space="preserve"> Igitur angulus e a h eſt medietas anguli e a d.</s> <s xml:id="echoid-s23828" xml:space="preserve"> Quare <lb/>patet, quòd linea a h cõtingens circulũ, diuidit angulũ e a d per ęqualia:</s> <s xml:id="echoid-s23829" xml:space="preserve"> quod eſt propoſitũ.</s> <s xml:id="echoid-s23830" xml:space="preserve"> Cũ ue-<lb/>rò angulus u a g ſuper punctũ a terminũ lineæ g a factus, fuerit æqualis angulo g a e:</s> <s xml:id="echoid-s23831" xml:space="preserve"> tũc ſi linea a u <lb/> <anchor type="figure" xlink:label="fig-0360-01a" xlink:href="fig-0360-01"/> nõ cadit ſuper lineã <lb/>e s extra circulũ uel <lb/>intra circulum:</s> <s xml:id="echoid-s23832" xml:space="preserve"> pa-<lb/>lã, quia linea a u eſt <lb/>ęquidiſtãs lineæ e s:</s> <s xml:id="echoid-s23833" xml:space="preserve"> <lb/>quia in infinitũ pro-<lb/>tracta cum illa non <lb/>concurrit:</s> <s xml:id="echoid-s23834" xml:space="preserve"> erit quo-<lb/>que per 29 p 1 angu-<lb/>lus u a g æqualis an-<lb/>gulo a g e:</s> <s xml:id="echoid-s23835" xml:space="preserve"> ſed per <lb/>præmiſſa angulus g <lb/>a e eſt æqualis angulo u a g:</s> <s xml:id="echoid-s23836" xml:space="preserve"> ergo angulus g a e æqualis erit angulo a g e.</s> <s xml:id="echoid-s23837" xml:space="preserve"> Ergo per 6 p 1 in trigono a <lb/>g e latus a e eſt æquale lateri e g.</s> <s xml:id="echoid-s23838" xml:space="preserve"> Similiter angulus t a d erit æqualis angulo a t g per 29 p 1:</s> <s xml:id="echoid-s23839" xml:space="preserve"> ſunt e-<lb/>nim coalterni linearum æquidiſtantium ex hypotheſi:</s> <s xml:id="echoid-s23840" xml:space="preserve"> ſed iam oſtenſum eſt, quod angulus t a d eſt <lb/>æqualis angulo d g t.</s> <s xml:id="echoid-s23841" xml:space="preserve"> Igitur angulus a t g eſt ęqualis angulo d g t.</s> <s xml:id="echoid-s23842" xml:space="preserve"> Et ſimiliter duo anguli a d g & d g t <lb/>ſunt æquales per 29 p 1:</s> <s xml:id="echoid-s23843" xml:space="preserve"> ergo duo anguli a d g & a t g ſunt æquales.</s> <s xml:id="echoid-s23844" xml:space="preserve"> Sequitur ergo exijs, quòd linea, <lb/>quã ſecat a u ex linea g d, ſit æqualis lineæ a t:</s> <s xml:id="echoid-s23845" xml:space="preserve"> & iam præoſtenſum eſt, quòd linea e g eſt æqualis ipſi <lb/>a e, eſt ergo ք 7 p 5 ꝓportio lineæ e g ad lineã, quã ſecat a u ex d g, ſicut a e ad a t:</s> <s xml:id="echoid-s23846" xml:space="preserve"> ſed oſtẽſum eſt, qđ <lb/>a e ad a t eſt, ſicut e g ad g d.</s> <s xml:id="echoid-s23847" xml:space="preserve"> Igitur linea, quam ſecat a u ex g d, eſt g d.</s> <s xml:id="echoid-s23848" xml:space="preserve"> Et cũ ex pręmiſsis angulus t a d <lb/>ſit ęqualis angulo d g t:</s> <s xml:id="echoid-s23849" xml:space="preserve"> erit angulus l a h medietas anguli t a d, ut ſuprà patuit:</s> <s xml:id="echoid-s23850" xml:space="preserve"> & angulus e a l medie <lb/>tas anguli e a t.</s> <s xml:id="echoid-s23851" xml:space="preserve"> Erit ergo e a h medietas anguli e a d.</s> <s xml:id="echoid-s23852" xml:space="preserve"> Quod eſt ꝓpoſitũ.</s> <s xml:id="echoid-s23853" xml:space="preserve"> Eodẽq́;</s> <s xml:id="echoid-s23854" xml:space="preserve"> modo demõſtrãdũ, <lb/> <pb o="59" file="0361" n="361" rhead="LIBER PRIMVS."/> ſi ambo puncta e & d data ſint extra circulum.</s> <s xml:id="echoid-s23855" xml:space="preserve"> Patet ergo totum propoſitum.</s> <s xml:id="echoid-s23856" xml:space="preserve"/> </p> <div xml:id="echoid-div873" type="float" level="0" n="0"> <figure xlink:label="fig-0359-01" xlink:href="fig-0359-01a"> <variables xml:id="echoid-variables378" xml:space="preserve">d a h l z s u g e t q <lb/>o k s i c n m</variables> </figure> <figure xlink:label="fig-0360-01" xlink:href="fig-0360-01a"> <variables xml:id="echoid-variables379" xml:space="preserve">d a u m f t h z q c g s</variables> </figure> </div> </div> <div xml:id="echoid-div875" type="section" level="0" n="0"> <head xml:id="echoid-head710" xml:space="preserve" style="it">136. Dato circulo & in eo diametro, punctó extra circulum: poßibile eſt à dato pũcto ad dia <lb/>metrum ducere lineam, ſecantem circulum ſic, quòd pars ductæ lineæ interiacens circumferen <lb/>tiam & diametrum, ſit æqualis parti diametri interiacenti ipſam & centrũ. Alhazen 37 n 5.</head> <p> <s xml:id="echoid-s23857" xml:space="preserve">Eſto datus circulus, cuius centrum ſit g:</s> <s xml:id="echoid-s23858" xml:space="preserve"> & in eo data diameter ſit x g b:</s> <s xml:id="echoid-s23859" xml:space="preserve"> ſit quoq;</s> <s xml:id="echoid-s23860" xml:space="preserve"> punctus e pun-<lb/>ctus extra circulum.</s> <s xml:id="echoid-s23861" xml:space="preserve"> Dico, quòd poſsibile eſt duci à puncto e ad diametrum x g b lineam ſecantem <lb/>circulum ſecundum prædictum modum.</s> <s xml:id="echoid-s23862" xml:space="preserve"> Ducatur enim à puncto e perpendicularis ſuper diame-<lb/>trum x g b per 12 p 1, quæ ſit e c:</s> <s xml:id="echoid-s23863" xml:space="preserve"> & ſit exempli cauſſa, ut cadat illa perpendicularis ſuper ſemidiame-<lb/>trum b g, & ducatur linea e g:</s> <s xml:id="echoid-s23864" xml:space="preserve"> & aſſumatur linea q t æqualis lineę e c:</s> <s xml:id="echoid-s23865" xml:space="preserve"> & fiat per 33 p 3 ſuper lineam q t <lb/>portio circuli talis, ut quilibet angulus cadens in hanc portionem, ſit æqualis angulo e g b:</s> <s xml:id="echoid-s23866" xml:space="preserve"> & com-<lb/>pleatur circulus:</s> <s xml:id="echoid-s23867" xml:space="preserve"> & à medio puncto l, lineę q t, quod ſit ſuper ipſam q t ducatur perpendicularis per <lb/>10 & 11 p 1, & ducatur ex utraq;</s> <s xml:id="echoid-s23868" xml:space="preserve"> parte uſq;</s> <s xml:id="echoid-s23869" xml:space="preserve"> ád circumferentiam circuli:</s> <s xml:id="echoid-s23870" xml:space="preserve"> erit ergo ducta perpendicula <lb/>ris diameter circuli illius per 1 p 3:</s> <s xml:id="echoid-s23871" xml:space="preserve"> & à puncto q ducatur linea ad hanc diametrum, ſecans ipſam in <lb/>puncto f:</s> <s xml:id="echoid-s23872" xml:space="preserve"> & producatur uſq;</s> <s xml:id="echoid-s23873" xml:space="preserve"> ad p punctum circumferentiæ, ita, ut eius pars, quę f p, ſit æqualis me-<lb/>dietati lineę g b ſemidiametro dati circuli:</s> <s xml:id="echoid-s23874" xml:space="preserve"> quod fiet per 133 huius:</s> <s xml:id="echoid-s23875" xml:space="preserve"> & ducantur lineę p t & t f:</s> <s xml:id="echoid-s23876" xml:space="preserve"> & duca <lb/>tur à puncto p linea p u ęquidiſtans diametro, concurrens cum linea t f in puncto u (concurret au-<lb/>tem per 2 huius) & à puncto u ducatur linea æquidiſtans lineę q t, quę ſit u o, ſecans diametrum fl <lb/>in puncto m, & lineam p q in puncto o:</s> <s xml:id="echoid-s23877" xml:space="preserve"> & à puncto t ducatur perpendicularis ſuper lineam p q per <lb/>12 p 1, quę ſit t n:</s> <s xml:id="echoid-s23878" xml:space="preserve"> & à puncto t ducatur linea æquidiſtãs lineę p q per 31 p 1, quę ſit t s:</s> <s xml:id="echoid-s23879" xml:space="preserve"> & à puncto u du-<lb/>catur perpendicularis ſuper lineam p q, quę ſit u h.</s> <s xml:id="echoid-s23880" xml:space="preserve"> Dein de ex angulo b g e ſecetur angulus æqualis <lb/>angulo q p u per 27 huius, qui ſit b g d, ducta linea g d ad peripheriã circuli:</s> <s xml:id="echoid-s23881" xml:space="preserve"> & à puncto e ducatur li <lb/> <anchor type="figure" xlink:label="fig-0361-01a" xlink:href="fig-0361-01"/> <anchor type="figure" xlink:label="fig-0361-02a" xlink:href="fig-0361-02"/> nea e d z.</s> <s xml:id="echoid-s23882" xml:space="preserve"> Dico, quòd <lb/>linea d z eſt æqualis <lb/>parti diametri, q̃ eſt <lb/>z g, ſicut proponitur.</s> <s xml:id="echoid-s23883" xml:space="preserve"> <lb/>Ducatur enim à pun <lb/>cto d perpendicularis <lb/>ſuper lineam b g, quę <lb/>ſit d i:</s> <s xml:id="echoid-s23884" xml:space="preserve"> & ducatur à pũ <lb/>cto d linea contingens <lb/>circulũ per 17 p 3, quę <lb/>ſit d k.</s> <s xml:id="echoid-s23885" xml:space="preserve"> Palã itaq;</s> <s xml:id="echoid-s23886" xml:space="preserve"> (cũ <lb/>ex præmiſsis diame-<lb/>ter fi ſit perpendicularis ſuper lineam q t, & ſuper eius æquidiſtantem o u per 29 p 1, linea uerò p u <lb/>ſit æquidiſtans illi diametro) quòd angulus o u p erit rectus per eandem 29 p 1.</s> <s xml:id="echoid-s23887" xml:space="preserve"> Et cum linea o u di <lb/>uidatur per diametrum fl in partes æquales, & orthogonaliter per 29 p 1.</s> <s xml:id="echoid-s23888" xml:space="preserve"> 4 p 6 & 22 p 5, eò quòd li-<lb/>nea q t ſibi ęquidiſtans ſimiliter eſt diuiſa:</s> <s xml:id="echoid-s23889" xml:space="preserve"> erũt per 4 p 1 trianguli o f m & u f m ęquianguli:</s> <s xml:id="echoid-s23890" xml:space="preserve"> ergo per <lb/>4 p 6 cum latus f m ſit ęquale ſibijpſi, erit o m ęquale m u, & f o ęquale f u.</s> <s xml:id="echoid-s23891" xml:space="preserve"> Sed cum duo anguli p o u <lb/>& o p u ualeantunum rectum per 32 p 1, ideo quòd angulus p u o eſt rectus, ut patet ex pręmiſsis & <lb/>29 p 1, erit angulus f u p ęqualis angulo f p u:</s> <s xml:id="echoid-s23892" xml:space="preserve"> ideo, quia, ut pręmiſſum eſt, angulus f o u ęqualis eſt <lb/>angulo f u o:</s> <s xml:id="echoid-s23893" xml:space="preserve"> ſed angulus f p u cum angulo f o u ualet unum rectum, ut pręoſtenſum eſt:</s> <s xml:id="echoid-s23894" xml:space="preserve"> ergo angu-<lb/>lus f p u cum angulo f u o ualet unum rectum:</s> <s xml:id="echoid-s23895" xml:space="preserve"> eſt ergo angulus f u p æqualis angulo f p u, quia ſi ab <lb/>ęqualibus ęqualia demas, quę relin quuntur, & c.</s> <s xml:id="echoid-s23896" xml:space="preserve"> Ergo per 6 p 1 latus f p ęquale erit lateri f u:</s> <s xml:id="echoid-s23897" xml:space="preserve"> erit er-<lb/>go f p ęquale ipſi f o.</s> <s xml:id="echoid-s23898" xml:space="preserve"> Sic ergo erit linea p o ęqualis ſemidiametro g b, ergo & ipſi g d per definitionẽ <lb/>circuli:</s> <s xml:id="echoid-s23899" xml:space="preserve"> & ita erit per 7 p 5 proportio lineę e c, quę eſt ęqualis lineę q t, ad lineam g d, ſicut lineę q t <lb/>ad p o ęqualem g d.</s> <s xml:id="echoid-s23900" xml:space="preserve"> Sed cum angulus k d g ſit rectus per 18 p 3, ęqualis eſt ipſi angulo recto g i d, & <lb/>angulus i g d eſt communis:</s> <s xml:id="echoid-s23901" xml:space="preserve"> erit ergo per 32 p 1 triangulus i g d ęquiangulus triangulo k g d:</s> <s xml:id="echoid-s23902" xml:space="preserve"> erit er-<lb/>go per 4 p 6 proportio lineę g d ad d i, ſicut lineę g k ad k d:</s> <s xml:id="echoid-s23903" xml:space="preserve"> ſed angulus k g d eſt ęqualis angulo <lb/>q p u, & angulus g d k, qui rectus eſt per 18 p 3, eſt ęqualis angulo recto o u p:</s> <s xml:id="echoid-s23904" xml:space="preserve"> erit ergo per 32 p 1 ter-<lb/>tius tertio ęqualis, & triãgulus k d g ęquiangulus triangulo o u p:</s> <s xml:id="echoid-s23905" xml:space="preserve"> eſt ergo per 4 p 6 proportio lineę <lb/>g k ad k d, ſicut lineę o p ad o u.</s> <s xml:id="echoid-s23906" xml:space="preserve"> Et quoniã ex pręmiſsis eſt proportio lineę g k ad k d, ſicut lineę g d <lb/>ad d i:</s> <s xml:id="echoid-s23907" xml:space="preserve"> ergo per 11 p 5 eſt proportio lineę g d ad d i, ſicut lineę o p ad ou:</s> <s xml:id="echoid-s23908" xml:space="preserve"> fuit autem ex pręmiſsis pro-<lb/>portio lineę e c ad g d, ſicut lineę t q ad p o:</s> <s xml:id="echoid-s23909" xml:space="preserve"> ergo per 22 p 5 erit ꝓportio lineę e c ad d i, ſicut lineę q t <lb/>ad o u:</s> <s xml:id="echoid-s23910" xml:space="preserve"> ſed proportio q t ad o u eſt, ſicut t f ad f u per 29 p 1, & per 4 p 6, cum triangulus t f q ſit æqui-<lb/>angulus triangulo o f u.</s> <s xml:id="echoid-s23911" xml:space="preserve"> Verùm angulus u t s eſt æqualis angulo h f ù per 29 p 1, eſt enim coalternus <lb/>illi inter lineas ęquidiſtantes, quę ſunt h q & s t:</s> <s xml:id="echoid-s23912" xml:space="preserve"> ſed & angulus u s t eſt rectus ęqualis angulo f h u <lb/>recto, & angulus f u h æqualis eſt angulo s u t per 15 p 1:</s> <s xml:id="echoid-s23913" xml:space="preserve"> erit ergo triangulus u s t æquiangulus <lb/>triangulo h u f:</s> <s xml:id="echoid-s23914" xml:space="preserve"> ergo per 4 p 6 erit proportio lineæ t u ad u f, ſicut lineæ s u ad u h:</s> <s xml:id="echoid-s23915" xml:space="preserve"> ergo per 18 <lb/>p 5 erit cõiunctim proportio lineę t f ad f u, ſicut lineę s h ad h u:</s> <s xml:id="echoid-s23916" xml:space="preserve"> ſed linea t n ęqualis eſt lineę s h per <lb/>34 p 1:</s> <s xml:id="echoid-s23917" xml:space="preserve"> ergo per 7 p 5 erit proportio lineę t n ad lineã h u, ſicut lineę t f ad f u.</s> <s xml:id="echoid-s23918" xml:space="preserve"> Sed, ſicut patuit ex prę-<lb/>miſsis, quę eſt proportio lineę t f ad f u, eadem eſt lineę q t ad o u per 4 p 6.</s> <s xml:id="echoid-s23919" xml:space="preserve"> Ergo per 11 p 5 propor-<lb/>tio lineę q t ad o u eſt, ſicut lineę t n ad h u:</s> <s xml:id="echoid-s23920" xml:space="preserve"> ergo & proportio lineę e c ad d i eſt, ſicut lineę t n ad u h.</s> <s xml:id="echoid-s23921" xml:space="preserve"> <lb/>Sed cum angulus g i d ſit rectus, eſt ęqualis angulo p h u recto, & angulus i g d æqualis angulo h p u <lb/> <pb o="60" file="0362" n="362" rhead="VITELLONIS OPTICAE"/> expręmiſsis:</s> <s xml:id="echoid-s23922" xml:space="preserve"> erit ergo tertius tertio ęqualis ք 32 p 1:</s> <s xml:id="echoid-s23923" xml:space="preserve"> eſt ergo triangulus i g d ęquiangulus triangulo <lb/>h p u:</s> <s xml:id="echoid-s23924" xml:space="preserve"> eſt ergo ք 4 p 6 ꝓportio lineę i d ad d g, ſicut lineę h u ad u p:</s> <s xml:id="echoid-s23925" xml:space="preserve"> quare erit ք 22 p 5 ꝓportio lineę <lb/>e c ad g d, ſicut lineę t n ad u p.</s> <s xml:id="echoid-s23926" xml:space="preserve"> Sed cũ angulus c g e ſit æqualis angulo n p t ex hypotheſi, & angulus <lb/>g c e rectus, ęqualis angulo p n t:</s> <s xml:id="echoid-s23927" xml:space="preserve"> erit trigonorũn p t & g c e angulus reliquus reliquo ęqualis.</s> <s xml:id="echoid-s23928" xml:space="preserve"> Ergo <lb/>per 4 p 6 erit ꝓportio lineę e g ad e c, ſicut lineę p t ad n t:</s> <s xml:id="echoid-s23929" xml:space="preserve"> eſt igitur proportio lineę g e ad g d, ſicut <lb/>lineę p t ad u p per 22 p 5:</s> <s xml:id="echoid-s23930" xml:space="preserve"> ſed & angulus d g e æqualis eſt angulo u p t ex hypotheſi:</s> <s xml:id="echoid-s23931" xml:space="preserve"> quia enim angu <lb/>lus q p t eſt æqualis angulo b g e, & angulus q p u æqualis angulo b g d:</s> <s xml:id="echoid-s23932" xml:space="preserve"> remanet angulus u p t <lb/>æqualis angulo d g e.</s> <s xml:id="echoid-s23933" xml:space="preserve"> Igitur triangulus d g e eſt æquian gulus triangulo u p t per 6 p 6:</s> <s xml:id="echoid-s23934" xml:space="preserve"> ergo angulus <lb/>g d e ęqualis eſt angulo p u t:</s> <s xml:id="echoid-s23935" xml:space="preserve"> reſtat ergo per 13 p 1, ut angulus g d z ſit æqualis angulo f u p:</s> <s xml:id="echoid-s23936" xml:space="preserve"> ſed in tri <lb/>gonis g d z & p f u eſt angulus d g z æqualis angulo u p f:</s> <s xml:id="echoid-s23937" xml:space="preserve"> quare tertius tertio per 32 p 1:</s> <s xml:id="echoid-s23938" xml:space="preserve"> eſt ergo ք 4 <lb/>p 6 proportio lineę d z ad z g, ſicut lineę u f ad f p:</s> <s xml:id="echoid-s23939" xml:space="preserve"> ſed linea u f eſt ęqualis ipſi f p ex præmiſsis.</s> <s xml:id="echoid-s23940" xml:space="preserve"> Igi-<lb/>tur linea d z æqualis eſt ipſi z g.</s> <s xml:id="echoid-s23941" xml:space="preserve"> Quod eſt propoſitũ.</s> <s xml:id="echoid-s23942" xml:space="preserve"> Eſt aũtuniuerſalis hęc propoſitio ſiue intra cir <lb/>culũ ad aliquã partẽ diametri fiat ductio, ſiue ad ipſam peripheriã circuli, ita, ut lineę ductę pars in-<lb/>tra circulum fiat ęqualis ſemidiametro:</s> <s xml:id="echoid-s23943" xml:space="preserve"> ſiue fiat ductio ad aliquem punctum diametri extra circu-<lb/>lum ſic, quòd linea à puncto, quo tangit circuli peripheriam, ſit æqualis parti diametri, quam ab-<lb/>ſcindit.</s> <s xml:id="echoid-s23944" xml:space="preserve"> Patet ergo, quoniam hęc omnia eueniunt ſecundum quantitatem anguli k g d.</s> <s xml:id="echoid-s23945" xml:space="preserve"> Et hoc eſt <lb/>propoſitum.</s> <s xml:id="echoid-s23946" xml:space="preserve"/> </p> <div xml:id="echoid-div875" type="float" level="0" n="0"> <figure xlink:label="fig-0361-01" xlink:href="fig-0361-01a"> <variables xml:id="echoid-variables380" xml:space="preserve">p n f o m u q l c</variables> </figure> <figure xlink:label="fig-0361-02" xlink:href="fig-0361-02a"> <variables xml:id="echoid-variables381" xml:space="preserve">k b d z e i c g x</variables> </figure> </div> </div> <div xml:id="echoid-div877" type="section" level="0" n="0"> <head xml:id="echoid-head711" xml:space="preserve" style="it">137. Dato trigono orthogonio, dató aliquo puncto in maiore ſuorum laterum rectum an-<lb/>gulum continentium: poßibile eſt à dato puncto ducere lineam ad baſim ex alia ſui parte cum <lb/>reliquo latere concurrentem, quæ ſe habeat ad inferiorem partem abſciſſam baſis, ſicut linea <lb/>data ad lineam datam. Alhazen 38 n 5.</head> <p> <s xml:id="echoid-s23947" xml:space="preserve">Sint datę duę lineę, z minor & e maior:</s> <s xml:id="echoid-s23948" xml:space="preserve"> & ſit datum trigonum orthogonium a b g, cuius angulus <lb/>a b g ſit rectus, contentus à lineis g b & b a, & dato exempli cauſſa in g b latere maiore illius trigoni <lb/>puncto d.</s> <s xml:id="echoid-s23949" xml:space="preserve"> Dico, quòd poſsibile eſt à puncto d ad baſim g a ducere lineam ſecantẽ baſim a g in pun-<lb/>cto q, & ex alia ſui parte cum linea a b concurrentem in puncto t, ſic ut ipſa totalis linea t q habeat <lb/> <anchor type="figure" xlink:label="fig-0362-01a" xlink:href="fig-0362-01"/> proportionem ad lineam q g illam, quã habet <lb/>linea e ad lineã z.</s> <s xml:id="echoid-s23950" xml:space="preserve"> Ducatur enim à puncto d li <lb/>nea æquidiſtans lineæ d a per 31 p 1, quę ſit <lb/>d, m, & fiat circulus tranſiens per tria puncta <lb/>d, m g per 5 p 4.</s> <s xml:id="echoid-s23951" xml:space="preserve"> Et quoniã angulus g d m eſt <lb/>rectus per 29 p 1, quoniam angulus a b g eſt <lb/>rectus, erit linea m g diameter circuli per 31 <lb/>p 3:</s> <s xml:id="echoid-s23952" xml:space="preserve"> & ducatur linea d a.</s> <s xml:id="echoid-s23953" xml:space="preserve"> Sit quoq;</s> <s xml:id="echoid-s23954" xml:space="preserve"> h quædam <lb/>linea, ad quam ſe habeat linea d a, ſicut linea e <lb/>ad z per 3 huius.</s> <s xml:id="echoid-s23955" xml:space="preserve"> Et cum per 29 p 1 angulus <lb/>d, m, g ſit æqualis angulo b a g:</s> <s xml:id="echoid-s23956" xml:space="preserve"> ſecetur ex an-<lb/>gulo d m g angulus æqualis angulo d a g per <lb/>27 huius:</s> <s xml:id="echoid-s23957" xml:space="preserve"> & ſit angulus c m d:</s> <s xml:id="echoid-s23958" xml:space="preserve"> & ducatur m c, donec ſecet circumferentiam in puncto c:</s> <s xml:id="echoid-s23959" xml:space="preserve"> & à pun-<lb/>cto c ducatur linea ad diametrum m g, & uſque ad circumferentiam, quę ſit linea c n, ſecans diame <lb/>trum m g in puncto l taliter, quòd linea l n ſit æqualis lineę h datę per 133 huius:</s> <s xml:id="echoid-s23960" xml:space="preserve"> & ducatur linea n g, <lb/>& producatur d n linea concurrens cum linea a g in puncto q.</s> <s xml:id="echoid-s23961" xml:space="preserve"> Cum igitur angulus d m c ſit ęqualis <lb/>angulo d n c per 27 p 3:</s> <s xml:id="echoid-s23962" xml:space="preserve"> cadunt enim in eundem arcum, qui eſt d c:</s> <s xml:id="echoid-s23963" xml:space="preserve"> palàm, quia erit angulus q n l æ-<lb/>qualis angulo d a q:</s> <s xml:id="echoid-s23964" xml:space="preserve"> & angulus n q l eſt æqualis angulo d q a per 15 p 1:</s> <s xml:id="echoid-s23965" xml:space="preserve"> erit ergo per 32 p 1 triangulus <lb/>n q l ęquangulus triangulo d q a:</s> <s xml:id="echoid-s23966" xml:space="preserve"> igitur per 4 p 6 erit proportio lineę a q ad q n, ſicut lineę a d ad n l.</s> <s xml:id="echoid-s23967" xml:space="preserve"> <lb/>Sed cum angulus d m g ſit æqualis angulo d n g per 27 p 3:</s> <s xml:id="echoid-s23968" xml:space="preserve"> quia cadunt in eundem arcum d g:</s> <s xml:id="echoid-s23969" xml:space="preserve"> eſt au <lb/>tem per 29 p 1 angulus d m g ęqualis angulo t a g:</s> <s xml:id="echoid-s23970" xml:space="preserve"> patet, quia angulus q n g ęqualis angulo t a g.</s> <s xml:id="echoid-s23971" xml:space="preserve"> Sit <lb/>itaque t punctus, in quo linea d n concurrit cum a b:</s> <s xml:id="echoid-s23972" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s23973" xml:space="preserve"> per 15 p 1 angulus t q a ęqualis angulo n <lb/>q g:</s> <s xml:id="echoid-s23974" xml:space="preserve"> ergo per 32 p 1 erit triangulus t q a ęquiangulus triangulo g q n:</s> <s xml:id="echoid-s23975" xml:space="preserve"> erit ergo per 4 p 6 proportio li <lb/>neę a q ad lineam q n, ſicut lineę t q ad lineam q g:</s> <s xml:id="echoid-s23976" xml:space="preserve"> eſt igitur per 11 p 5 proportio lineę t q ad lineam <lb/>q g, ſicut lineę a d ad lineam n l:</s> <s xml:id="echoid-s23977" xml:space="preserve"> ſed linea n l eſt æqualis h aſſumptæ lineæ, & proportio lineæ a d ad <lb/>lineam h eſt, ſicut lineę e ad lineam z.</s> <s xml:id="echoid-s23978" xml:space="preserve"> Eſt ergo proportio lineę t q ad lineã q g, ſicut lineæ e ad line-<lb/>am z.</s> <s xml:id="echoid-s23979" xml:space="preserve"> Quod eſt propoſitũ.</s> <s xml:id="echoid-s23980" xml:space="preserve"> Et ſi contingat quòd à puncto c poſsint duci duæ lineæ ſimiles lineę c l n:</s> <s xml:id="echoid-s23981" xml:space="preserve"> <lb/>erit poſsibile à puncto d duci duas lineas ſimiles lineæ t q, ita ſcilicet, ut utriuſque ad partem, quam <lb/>ſecat ex baſi a g, ſit proportio, ſicut lineę e ad lineam z:</s> <s xml:id="echoid-s23982" xml:space="preserve"> & erit eadem demonſtratio.</s> <s xml:id="echoid-s23983" xml:space="preserve"> Plures autem <lb/>huiuſmodi lineas quàm duas nõ eſt poſsibile duci, ut patuit per 133 huius.</s> <s xml:id="echoid-s23984" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s23985" xml:space="preserve"> <lb/>Et licet hoc, quod hic proponitur, non uideatur penitus uniuerſale, quantum ad quælibet puncta <lb/>data, & quaslibet lineas datas, ad quarum proportionem fieri debeat ipſius baſis proportio:</s> <s xml:id="echoid-s23986" xml:space="preserve"> <lb/>nos tamen hoc propoſito theoremate non, niſi modo conuenienti <lb/>& poſsibili in ſequentibus utemur.</s> <s xml:id="echoid-s23987" xml:space="preserve"/> </p> <div xml:id="echoid-div877" type="float" level="0" n="0"> <figure xlink:label="fig-0362-01" xlink:href="fig-0362-01a"> <variables xml:id="echoid-variables382" xml:space="preserve">a a n m e z h q l b d g d t c</variables> </figure> </div> <pb o="61" file="0363" n="363"/> </div> <div xml:id="echoid-div879" type="section" level="0" n="0"> <head xml:id="echoid-head712" xml:space="preserve">VITELLONIS FI-<lb/>LII THVRINGORVM ET PO-<lb/>LONORVM OPTICAE LIBER SECVNDVS.</head> <p style="it"> <s xml:id="echoid-s23988" xml:space="preserve">VNiuerſalib{us} hui{us} ſcientiæ axiomatib{us} mathematicis præmißis:</s> <s xml:id="echoid-s23989" xml:space="preserve"> in hoc <lb/>ſecundo libro (ut promiſim{us}) uniuerſali actioni ſenſibilium formarum <lb/>quædã præambula naturalia præmittentes, de modo proiectionis luminis <lb/>per mediũ uni{us} diaphani, uel pluriũ ſuper diuerſas figuras corporum, & <lb/>de proiectione umbrarũ, & de figuratione lucis cadentis per fenestras aggredimur tra-<lb/>ctatum, ut de ijs, ſine quibus ſermonẽ uiſibilium formarũ aggredi conueniens non fuit, <lb/>prout in proceſſu postmodum patebit:</s> <s xml:id="echoid-s23990" xml:space="preserve"> quæ uerò præmittim{us}, ut nota ſenſui, ſunt iſta.</s> <s xml:id="echoid-s23991" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div880" type="section" level="0" n="0"> <head xml:id="echoid-head713" xml:space="preserve">DEFINITIONES.</head> <p> <s xml:id="echoid-s23992" xml:space="preserve">1.</s> <s xml:id="echoid-s23993" xml:space="preserve"> Corpus luminoſum, dicitur omne corpus, quod eſt ſui luminis diffuſiuũ.</s> <s xml:id="echoid-s23994" xml:space="preserve"> 2.</s> <s xml:id="echoid-s23995" xml:space="preserve"> Cor <lb/>pus diaphanum dιcitur omne corpus, per quod lumini patet tranſitus.</s> <s xml:id="echoid-s23996" xml:space="preserve"> 3.</s> <s xml:id="echoid-s23997" xml:space="preserve"> Corpus <lb/>umbroſum dicitur corpus, per quod lumini non patet tranſitus.</s> <s xml:id="echoid-s23998" xml:space="preserve"> 4.</s> <s xml:id="echoid-s23999" xml:space="preserve"> Lux prima dici-<lb/>turilla, quæ efficit ſecundã, ſicut lux intrans domũ per feneſtrã, & illuminãs domũ <lb/>reſiduã in loco, cui incidit, dicitur prima:</s> <s xml:id="echoid-s24000" xml:space="preserve"> in angulis uerò domus dicitur lux ſecun-<lb/>da.</s> <s xml:id="echoid-s24001" xml:space="preserve"> 5.</s> <s xml:id="echoid-s24002" xml:space="preserve"> Lux minima dicitur, quæ ſi diuidi intelligatur, nõ habebit amplius actũ lucis.</s> <s xml:id="echoid-s24003" xml:space="preserve"> <lb/>6.</s> <s xml:id="echoid-s24004" xml:space="preserve"> Radius dicitur linea luminoſa.</s> <s xml:id="echoid-s24005" xml:space="preserve"> 7.</s> <s xml:id="echoid-s24006" xml:space="preserve"> Linea radialis dicitur linea, per quam fit diffuſio <lb/>formarũ.</s> <s xml:id="echoid-s24007" xml:space="preserve"> 8.</s> <s xml:id="echoid-s24008" xml:space="preserve"> Linea refracta dicitur linea, cuius partes angulũ continẽt.</s> <s xml:id="echoid-s24009" xml:space="preserve"> 9.</s> <s xml:id="echoid-s24010" xml:space="preserve"> Pyramis ra-<lb/>dialis dicitur pyramis, cuius baſis eſt in ſuperficie corporis ſuã formã diffundentis, <lb/>& uertex in puncto alterius corporis cuiuſcunq;</s> <s xml:id="echoid-s24011" xml:space="preserve">. 10.</s> <s xml:id="echoid-s24012" xml:space="preserve"> Pyramis illuminatiõis dicitur <lb/>illa, cuius uertex eſt in pũcto corporis luminoſi, & baſis in ſuperficie rei illuminatę.</s> <s xml:id="echoid-s24013" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div881" type="section" level="0" n="0"> <head xml:id="echoid-head714" xml:space="preserve">PETITIONES.</head> <p> <s xml:id="echoid-s24014" xml:space="preserve">Petimus autẽ hæc, ut per ſe ſenſui nota:</s> <s xml:id="echoid-s24015" xml:space="preserve"> 1.</s> <s xml:id="echoid-s24016" xml:space="preserve"> Lucẽ cõpreſſam fortiorẽ eſſe luce diſ-<lb/>gregata.</s> <s xml:id="echoid-s24017" xml:space="preserve"> 2.</s> <s xml:id="echoid-s24018" xml:space="preserve"> Item lucem fortiorem uehementius illuminare, & lõgius ſe diffundere.</s> <s xml:id="echoid-s24019" xml:space="preserve"> <lb/>3.</s> <s xml:id="echoid-s24020" xml:space="preserve"> Item in abſentia luminis umbram fieri.</s> <s xml:id="echoid-s24021" xml:space="preserve"> 4.</s> <s xml:id="echoid-s24022" xml:space="preserve"> Item in allatione luminis umbram defi <lb/>cere.</s> <s xml:id="echoid-s24023" xml:space="preserve"> 5.</s> <s xml:id="echoid-s24024" xml:space="preserve"> Item aliquam umbram in ſui termino acui, & ad punctum terminari.</s> <s xml:id="echoid-s24025" xml:space="preserve"> 6.</s> <s xml:id="echoid-s24026" xml:space="preserve"> Item <lb/>lucẽ ad omnẽ poſitionis differentiam ęqualiter diffundi.</s> <s xml:id="echoid-s24027" xml:space="preserve"> 7.</s> <s xml:id="echoid-s24028" xml:space="preserve"> Item lucẽ res coloratas <lb/>pertrãſeuntẽ illarũ coloribus colorari, ut patet de luce trãſeunte uitreas feneſtras, <lb/>quę illorũ uitrorũ colorib.</s> <s xml:id="echoid-s24029" xml:space="preserve"> informat̃, ſecũ formas illorũ colorũ ſuper obiecta cor-<lb/>pora deferendo.</s> <s xml:id="echoid-s24030" xml:space="preserve"> 8.</s> <s xml:id="echoid-s24031" xml:space="preserve"> Itẽ quòd natura nihil fruſtra agit, ſicut nec deficit in neceſſarijs.</s> <s xml:id="echoid-s24032" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div882" type="section" level="0" n="0"> <head xml:id="echoid-head715" xml:space="preserve">THEOREMATA</head> <head xml:id="echoid-head716" xml:space="preserve" style="it">1. Radij quorumcun luminum & multiplic ationes formarum, ſecundum rectas lineas <lb/>protenduntur. Alhazen 2 n 7.</head> <p> <s xml:id="echoid-s24033" xml:space="preserve">HOc quod hic proponitur, non demonſtratione, ſed inſtrumentaliter poteſt declarari:</s> <s xml:id="echoid-s24034" xml:space="preserve"> <lb/>diuerſitas tamen antiquorũ ad hoc proban dũ pluribus & diuerſis uſa eſt inſtrumentis, <lb/>nos uerò utimuriſto, quod hic ſubſcribimus, quòd regularius huic ꝓpoſito credimus <lb/>cõuenire.</s> <s xml:id="echoid-s24035" xml:space="preserve"> Aſſumaturitaq;</s> <s xml:id="echoid-s24036" xml:space="preserve"> uas æneum rotundũ cõuenienter ſpiſſum, ad modum matris <lb/>aſtrolabij, cuius fundi latitudo ſit unius cubiti, uel maior, & altitudo oræ eius ſit æqua-<lb/>lis latitudini duorũ digitorũ perpẽdicularis ſuper baſim uaſis:</s> <s xml:id="echoid-s24037" xml:space="preserve"> & in medio dorſi huius uaſis ſit per-<lb/>pendiculariter erectũ aliquod corpus plurimũ rotundũ columnare, cuius longitudo ſit æqualis la <lb/>titudini trium digitorũ, latitudo uerò eιus ſit minor uno digito:</s> <s xml:id="echoid-s24038" xml:space="preserve"> & ponatur hoc uas ſecũdũ ſui pun-<lb/>cta media in tornatorio, & tornetur quouſq;</s> <s xml:id="echoid-s24039" xml:space="preserve"> peripheria eius ſit intrinſecus & extrinſecus ueræ ro-<lb/>tunditatis, & adæquentur planæ ſuperficies ipſius, & corpus columnare, quod eſt in medio dorſi, <lb/>fiat rotundũ.</s> <s xml:id="echoid-s24040" xml:space="preserve"> Signentur itaq;</s> <s xml:id="echoid-s24041" xml:space="preserve"> in interiori ſuperficie fundi huius uaſis duæ diametri orthogonaliter <lb/>ſe ſecantes, quæ ſint a b & c d:</s> <s xml:id="echoid-s24042" xml:space="preserve"> palàm, quoniam illę diametri tranſeunt per centrum circuli fun-<lb/>di, quod ſit e:</s> <s xml:id="echoid-s24043" xml:space="preserve"> deinde ſignetur in baſi oræ iſtius uaſis, quæ eſt circulus a c b d, in diſtantia extremita-<lb/>tis alterius diametrorum productarum, ut diametri a b, ſecundum latitudinem unius digiti pun-<lb/> <pb o="62" file="0364" n="364" rhead="VITELLONIS OPTICAE"/> ctum, quod ſit f:</s> <s xml:id="echoid-s24044" xml:space="preserve"> & ex hoc puncto tertia trahatur diameter per centrũ e, quę ſit f g:</s> <s xml:id="echoid-s24045" xml:space="preserve"> & à duob.</s> <s xml:id="echoid-s24046" xml:space="preserve"> termi-<lb/>nis iſtius diámetri f g ducãtur duę lineę in intrinſeca <lb/> <anchor type="figure" xlink:label="fig-0364-01a" xlink:href="fig-0364-01"/> ſuperficie orę uaſis:</s> <s xml:id="echoid-s24047" xml:space="preserve"> quę neceſſariò erunt perpẽdicu-<lb/>lares ſuper ſuperficiẽ fundi laminę, ideo, qđ ſuperfi-<lb/>cies orę, in qua perpẽdiculares iſtę ꝓducuntur, ſunt <lb/>erectę ſuper ſuperficiẽ fundi, ut patet ſuprà.</s> <s xml:id="echoid-s24048" xml:space="preserve"> Illę quo-<lb/>que perpendiculares ſint f h & g k:</s> <s xml:id="echoid-s24049" xml:space="preserve"> & in altera iſtarũ <lb/>linearũ, ut in f h, ſignentur tria puncta æquidiſtantia <lb/>ſecundũ quãtitatẽ medietatis grani hordei, quę ſint <lb/>l, m, n, quorũ primũ, qđ eſt l, ſit propinquius baſi ua-<lb/>ſis & ipſi puncto f, à quo diſtet per quantitatẽ medie <lb/>tatis grani hordei.</s> <s xml:id="echoid-s24050" xml:space="preserve"> Et deinde reducatur uas ad torna <lb/>torium, & ſignẽtur in ipſo tres circuli æquidiſtãtes, <lb/>tranſeuntes ք illa tria pũcta l, m, n:</s> <s xml:id="echoid-s24051" xml:space="preserve"> ք circuli diuident <lb/>lineã g k iſti diuiſę lineę, quę eſt f h, oppoſitã, ꝓpor-<lb/>tionaliter prius diuiſę per 17 p 11, ſintq́;</s> <s xml:id="echoid-s24052" xml:space="preserve"> diuiſiones li-<lb/>neę g k puncta o, p, q:</s> <s xml:id="echoid-s24053" xml:space="preserve"> & fient in in unoquoq;</s> <s xml:id="echoid-s24054" xml:space="preserve"> iſtorũ <lb/>triũ circulorũ duo pũcta oppoſita, q̃ ſunt extremita-<lb/>tes alicuius diametri illorũ circulorũ:</s> <s xml:id="echoid-s24055" xml:space="preserve"> ut pũcto diui-<lb/>ſionis lineę fh (qđ eſt punctũ l) opponitur in linea g k punctũ o, & fit linea l o diameter circuli æ-<lb/>quidiſtantis circulo a c b d:</s> <s xml:id="echoid-s24056" xml:space="preserve"> & ſimiliter linea m p fit diameter alterius circuli, & linea n q fit diame-<lb/>ter circulitertij.</s> <s xml:id="echoid-s24057" xml:space="preserve"> Diuidatur itaq;</s> <s xml:id="echoid-s24058" xml:space="preserve"> medius iſtorũ ctrculorũ in 360 partes, & ſi poſsibile fuerit, ք minu <lb/>ta:</s> <s xml:id="echoid-s24059" xml:space="preserve"> deinde ſuper lineã f h alterã duarũ linearũ perpẽdiculariũ, quę ſunt f h & g k, punctũ mediũ, qđ <lb/>eſt m, perforetur foramẽ rotundũ:</s> <s xml:id="echoid-s24060" xml:space="preserve"> & ſit medietas diametri foraminis ſecundũ quantitatẽ diſtantię <lb/>circulorũ, quę eſt linea m l:</s> <s xml:id="echoid-s24061" xml:space="preserve"> attinget ergo foramẽ illud ambos circulos extremos, & medius circulo <lb/>rũ diuidet circulũ foraminis ք æqualia, quoniã trãſit ք centrũ foraminis.</s> <s xml:id="echoid-s24062" xml:space="preserve"> Deinde accipiatur lamina <lb/>ænea plana aliquantulum ſpiſſa, & ſit eius ſpiſsitudo ſicut orę ipſius inſtrumẽti, & eius lõgitudo ſit <lb/>duorũ digitorũ, ſicut & ora uaſis, & eius latitudo ſit prope hoc, & ſit ęquidiſtantiũ ſuperficierũ:</s> <s xml:id="echoid-s24063" xml:space="preserve"> pla <lb/>neturq́;</s> <s xml:id="echoid-s24064" xml:space="preserve"> adeò, ut cõmunis ſectio ſuperficierũ ſuę latitudinis & ſpiſsitudinis ſit linea recta, quę ſit r s, <lb/>diui daturq́;</s> <s xml:id="echoid-s24065" xml:space="preserve"> in duo æqualia ք 10 p 1:</s> <s xml:id="echoid-s24066" xml:space="preserve"> & ab eius medio puncto, qđ ſit t, ducatur linea recta perpendi-<lb/>culariter ſuper ipſam lineã r s in ſuperficie latitudinis, quę ſit t u:</s> <s xml:id="echoid-s24067" xml:space="preserve"> & hęc, ut patet ex pręmiſsis & per <lb/>29 p 1, neceſſario ęquidiſtabit ambabus lineis lõgitudinis, diuidens ſuperficiẽ tabulę per ęqualia:</s> <s xml:id="echoid-s24068" xml:space="preserve"> & <lb/>in hac linea perpendiculari, quę eſt t u, à parte lineę r s, cui ſuperſtat, incipiendo, ſignentur tria pun <lb/>cta ęqualiter diſtantia ab inuicẽ ſecũdũ quãtitatẽ medietatis grani hordei, quę ſint x, y, z, & à medio <lb/>iſtorũ pũctorũ, quod eſt y, pforetur lamina foramine rotũdo:</s> <s xml:id="echoid-s24069" xml:space="preserve"> ſicq́;</s> <s xml:id="echoid-s24070" xml:space="preserve"> foraminis peripheria ad alia duo <lb/>puncta pertinget, eritq́;</s> <s xml:id="echoid-s24071" xml:space="preserve"> hoc foramen ęquale foramini l m n prius facto in <lb/> <anchor type="figure" xlink:label="fig-0364-02a" xlink:href="fig-0364-02"/> ora uaſis.</s> <s xml:id="echoid-s24072" xml:space="preserve"> Deinde in duo ęqualia diuidatur ſemidiameter uaſis fundi, quę <lb/>eſt f e, cuius extremitati in ora uaſis ſuperſtat una linearũ perpendiculariũ, <lb/>quę eſt f h:</s> <s xml:id="echoid-s24073" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s24074" xml:space="preserve"> punctus diuiſionis t:</s> <s xml:id="echoid-s24075" xml:space="preserve"> & ab hoc puncto medio t ducatur li-<lb/>nea perpen dicularis ſuper eandẽ diametrum, quę ſit r t s:</s> <s xml:id="echoid-s24076" xml:space="preserve"> deinde ponatur <lb/>baſis paruę laminę ſuper hãc lineã, donec linea, quę eſt differẽtia cõmunis <lb/>latitudinis & ꝓfunditatis laminę, quę eſt r t s, ſupponatur lineę iſti perpen <lb/>diculari ductę ſuper diametrũ, quę ſimiliter eſt r t s:</s> <s xml:id="echoid-s24077" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s24078" xml:space="preserve"> punctus diuidens <lb/>lineã laminę, quę eſt cõmunis differentia ſuperficierum latitudinis & pro-<lb/>funditatis, qui eſt punctus t, ſuperpoſitus puncto t, ſignato in linea f e ſemi <lb/>diametro uaſis:</s> <s xml:id="echoid-s24079" xml:space="preserve"> deinde cõſolidetur parua lamina fundo uaſis:</s> <s xml:id="echoid-s24080" xml:space="preserve"> erit quoq;</s> <s xml:id="echoid-s24081" xml:space="preserve"> tũc foramẽ <lb/> <anchor type="figure" xlink:label="fig-0364-03a" xlink:href="fig-0364-03"/> x y z, quod eſt in parua lamina, quę eſt r u s, directè oppoſitũ foramini l m n, qđ eſt in <lb/>uaſis ora:</s> <s xml:id="echoid-s24082" xml:space="preserve"> & erit linea recta, quę eſt m y, copulãs cẽtra iſtorũ foraminũ in ſuperficie cir <lb/>culi medij triũ circulorũ prius ſignatorũ, cuius diameter eſt linea m p:</s> <s xml:id="echoid-s24083" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s24084" xml:space="preserve"> linea m y <lb/>ęquidiſtans diametro uaſis, quę eſt f e.</s> <s xml:id="echoid-s24085" xml:space="preserve"> Deinde reſecetur ex ora uaſis pars interiacẽs <lb/>duas diametros orthogonaliter ſe ſecãtes, quę ſit pars quarta proximè ſequẽs quartã <lb/>illã, in qua eſt foramẽ, cui foramẽ laminę opponitur:</s> <s xml:id="echoid-s24086" xml:space="preserve"> & eſt in circulo a c b d, correſpõ-<lb/>dens arcui a d, & planetur locus ſectionis, donec fiat una ſuperficies cũ ſuperficie fun <lb/>di uaſis.</s> <s xml:id="echoid-s24087" xml:space="preserve"> Et ducta quarta circuli, quę ſit a d, ſecundũ quãtitatẽ circuli orę diuidatur in <lb/>90 grad.</s> <s xml:id="echoid-s24088" xml:space="preserve"> & diuidantur grad.</s> <s xml:id="echoid-s24089" xml:space="preserve"> in minuta:</s> <s xml:id="echoid-s24090" xml:space="preserve"> & iſti uaſi taliter informato & figurato, dein-<lb/>ceps damus nomẽ inſtrumẽti.</s> <s xml:id="echoid-s24091" xml:space="preserve"> Deinde accipiatur regula ęnea quadrãgula, cuius lõgi <lb/>tudo ſit unius cubiti, & ſint quatuor ſuperficies ipſam cõtinentes, latitudinis duorũ <lb/>digitorũ, & adęquẽtur ſuperficies eius, donec fiant ęquales rectãgulę.</s> <s xml:id="echoid-s24092" xml:space="preserve"> Deinde in me-<lb/>dio pũcto lõgitudinis regulę, & in medio alicuius illarũ ſuperficierũ fiat foramen ro-<lb/>tundũ, cuius amplitudo ſit capax corporis, qđ eſt in dorſo inſtrumẽti:</s> <s xml:id="echoid-s24093" xml:space="preserve"> & ſit foramen <lb/>perpẽdiculare ſuper ſuperficiẽ regulę trãſiens ad aliã partẽ ſuperficiei oppoſitę, fiatq́;</s> <s xml:id="echoid-s24094" xml:space="preserve"> <lb/>taliter, qđ reuoluatur in ipſo inſtrumentũ nõ leui reuolutione, ponaturq́;</s> <s xml:id="echoid-s24095" xml:space="preserve"> inſtrumen <lb/>tũ ſuper regulã immiſſo corpore, qđ eſt in eius dorſo in foramẽ regulę, donec ſuperfi <lb/>cies inſtrumẽti cõiungatur ſuքficiei regulę:</s> <s xml:id="echoid-s24096" xml:space="preserve"> erit q́;</s> <s xml:id="echoid-s24097" xml:space="preserve"> lõ git udo regulę ęqualis diametro <lb/> <pb o="63" file="0365" n="365" rhead="LIBER SECVNDVS."/> inſtrumenti:</s> <s xml:id="echoid-s24098" xml:space="preserve"> fiantq́;</s> <s xml:id="echoid-s24099" xml:space="preserve"> duæ pinnulæ latitudinis & ſpiſsitudinis regulæ, ſed lõgitudinis pluſquã unius <lb/>digiti, quę cõſolidẽtur ſuper extremitates regulę, ita, quòd ipſorũ præeminẽtia ſuper extremitates <lb/>regulæ ſit unius digiti, uel parũ plus, uel minus, & pinnulę illæ cõſolidatę ſint ſuper ſuperficiẽ regu <lb/>lę nõ perforatã.</s> <s xml:id="echoid-s24100" xml:space="preserve"> Et quia latitudo regulę eſt duorũ digitorũ, altitudo uerò corporis in dorſo inſtru-<lb/>mẽti eſt triũ digitorũ, ille tertius digitus, quo corpus pręeminet regulæ, perforetur, ſicut in aſtrola-<lb/>bio, & immittatur cuſpis cõtinens regulã cũ inſtrumẽto.</s> <s xml:id="echoid-s24101" xml:space="preserve"> Deinde aſſumatur alia regula ænea, cuius <lb/>latitudo ſit dupla ſuæ ſpiſs itudini, ſpiſsitudo uerò ſit æqualis diametro foraminis, qđ eſt in ora in-<lb/>ſtrumẽti, & lõgitudo eius ſit æqualis medietati cubiti, fiatq́;</s> <s xml:id="echoid-s24102" xml:space="preserve"> hæc regula recta & uera, & eius ſuperfi <lb/>cies æquales & æquidiſtãtes.</s> <s xml:id="echoid-s24103" xml:space="preserve"> Deinde ſecetur illa regula in una ſui parte obliquè, donec finis lõgitu <lb/>dinis eius cõtineat cũ termino latitudinis angulũ acutũ, ut facilius ualeat moueri.</s> <s xml:id="echoid-s24104" xml:space="preserve"> In parte uero al-<lb/>tera ſit finis latitudinis eius perpendicularis ſuper finẽ lõgitudinis.</s> <s xml:id="echoid-s24105" xml:space="preserve"> Deinde diuidatur linea eius la-<lb/>titudinis in duo æqualia, & à puncto ſectionis ducatur linea ęquidiſtans lineis lõgitudinis:</s> <s xml:id="echoid-s24106" xml:space="preserve"> quę erit <lb/>perpen dicularis ſuper lineã latitudinis per 29 p 1.</s> <s xml:id="echoid-s24107" xml:space="preserve"> Cũ itaq;</s> <s xml:id="echoid-s24108" xml:space="preserve"> hæc regula fuerit ſuperpoſita ſuperficiei <lb/>fundi inſtrumẽti taliter, ut eius ſpiſsitudo ſit orthogonaliter erecta ſuper fundũ inſtrumenti, & ſu-<lb/>perficies latitudinis applicetur ſuperficiei fundi ipſius inſtrumẽti:</s> <s xml:id="echoid-s24109" xml:space="preserve"> tũc erit eius ſuperior ſuperficies <lb/>in ſuperficie circuli medij triũ circulorũ in ora inſtrumenti protractorum, cuius diameter eſt linea <lb/>m p:</s> <s xml:id="echoid-s24110" xml:space="preserve"> ideo, quia ſpiſsitudo regulę eſt æqualis diametro foraminis, & diameter foraminis, quę eſt n l, <lb/>eſt æqualis lineæ perpẽdiculari exeunti à cẽtro foraminis ſuper ſuperficiẽ planã inſtrumẽti, quę eſt <lb/>linea m f, cui adiacet linea ſpiſsitudinis regulę, æqualis ipſi.</s> <s xml:id="echoid-s24111" xml:space="preserve"> Cũ itaq;</s> <s xml:id="echoid-s24112" xml:space="preserve"> propoſitã concluſionẽ experi-<lb/>mentaliter placuerit declarare, opponatur inſtrumẽtum pręmiſſum corpori ſolari, uel alteri corpo <lb/>ri luminoſo cuicunq;</s> <s xml:id="echoid-s24113" xml:space="preserve">, uel etiá candelæ, & applicetur cẽtrum foraminis inſtrumẽti, qđ eſt punctum <lb/>m, oppoſito corporis luminoſi, ſecundum qđ melius fuerit poſsibile, tranſibitq́;</s> <s xml:id="echoid-s24114" xml:space="preserve"> radius luminoſus <lb/>cẽtra amborũ oppoſitorũ foraminũ unius in ora inſtrumẽti, & alterius in tabella perforata exiſten-<lb/>tium, quę ſunt m & y:</s> <s xml:id="echoid-s24115" xml:space="preserve"> deſcribeturq́;</s> <s xml:id="echoid-s24116" xml:space="preserve"> circulus luminoſus in parte orę inſtrumẽti oppoſita foramini <lb/>l m n directè per diametrũ m p:</s> <s xml:id="echoid-s24117" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s24118" xml:space="preserve"> cẽtrum illius circuli luminoſi in puncto p:</s> <s xml:id="echoid-s24119" xml:space="preserve"> quod faciliter patê <lb/>re poteſt, ſi à puncto p ad utranq;</s> <s xml:id="echoid-s24120" xml:space="preserve"> partẽ peripheriæ circuli medij illorũ trium circulorũ ſecundũ gra <lb/>dus & minuta diuiſi, partes interiacẽtes luminoſi circuli peripheriã cõputentur:</s> <s xml:id="echoid-s24121" xml:space="preserve"> inuenientur enim <lb/>æquales numeri hinc inde.</s> <s xml:id="echoid-s24122" xml:space="preserve"> Eſt ergo punctũ p cẽtrum illius circuli luminoſi:</s> <s xml:id="echoid-s24123" xml:space="preserve"> linea itaq;</s> <s xml:id="echoid-s24124" xml:space="preserve"> m p, ſecun-<lb/>dum quã incidit radius, trãſiẽs per cẽtrum circuli utriuſq;</s> <s xml:id="echoid-s24125" xml:space="preserve"> foraminis, & per centrũ circuli luminoſi, <lb/>tota eſt in ſuperficie plana circuli medij illorũ trium circulorũ, & eſt diameter illius circuli.</s> <s xml:id="echoid-s24126" xml:space="preserve"> Eſt er-<lb/>go linea recta.</s> <s xml:id="echoid-s24127" xml:space="preserve"> Et ſi aliquod corpus forti colore medio coloratũ, ut uiride uel rubeum, ponatur ex-<lb/>tra foramen oræ inſtrumenti, ita, ut lumẽ ſolis uel alterius corporis tranſiẽs per illud corpus, poſt-<lb/>modũ incidat foraminibus inſtrumenti, & tranſeat per illa:</s> <s xml:id="echoid-s24128" xml:space="preserve"> tunc, ut patuit per 7 pręmiſſarũ ſuppoſi <lb/>tionũ, circa pũctũ p in ora inſtrumẽti deſcribetur circulus luminis colorati illo colore.</s> <s xml:id="echoid-s24129" xml:space="preserve"> Color ergo <lb/>mixtim cũ lumine diffundit formã ſuã ſecũdũ lineas rectas, ſicut & ipſũ lumẽ.</s> <s xml:id="echoid-s24130" xml:space="preserve"> Patet ergo, qđ radij <lb/>quorũcũq;</s> <s xml:id="echoid-s24131" xml:space="preserve"> luminũ & multiplicatiões formarũ ſecũdũ lineas rectas ꝓtendũtur.</s> <s xml:id="echoid-s24132" xml:space="preserve"> Et hoc eſt ꝓpoſitũ.</s> <s xml:id="echoid-s24133" xml:space="preserve"/> </p> <div xml:id="echoid-div882" type="float" level="0" n="0"> <figure xlink:label="fig-0364-01" xlink:href="fig-0364-01a"> <variables xml:id="echoid-variables383" xml:space="preserve">h n m l <gap/> a x r t s c e d z b g o p q k</variables> </figure> <figure xlink:label="fig-0364-02" xlink:href="fig-0364-02a"> <variables xml:id="echoid-variables384" xml:space="preserve">u g z y x r t s</variables> </figure> <figure xlink:label="fig-0364-03" xlink:href="fig-0364-03a"> <image file="0364-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0364-03"/> </figure> </div> </div> <div xml:id="echoid-div884" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables385" xml:space="preserve">a b c d</variables> </figure> <head xml:id="echoid-head717" xml:space="preserve" style="it">2. Lumen non impeditum, per totum ſibi proportionatum medium in inſtantineceſſa <lb/> rium eſt deferri.</head> <p> <s xml:id="echoid-s24134" xml:space="preserve">Sit linea proportionata delationi luminis fortioris, ut eſt in lumine ſolis mũdi diameter, <lb/>quę ſit linea a b c d, & ſit corpus fortiter luminoſum in puncto a.</s> <s xml:id="echoid-s24135" xml:space="preserve"> Si ergo dicatur, qđ lumẽ in <lb/>tẽpore defertur per lineã a b c d, & nõ in inſtãti:</s> <s xml:id="echoid-s24136" xml:space="preserve"> ergo in parte illius tẽporis defertur per lineã <lb/>a b, & in minimo tẽpore ſenſibili feretur ք minimã partẽ ſenſibilẽ lineę a b:</s> <s xml:id="echoid-s24137" xml:space="preserve"> quoniã ſi in tem-<lb/>pore ſenſibili ferretur per ſpatium inſenſibile, cõtingeret ſpatium ſenſibile ex inſenſibilibus <lb/>cõponi, ſicut tẽpus mẽſuratum poſt illud ſpatium cõpoſitum ex tẽporibus ſenſibilib.</s> <s xml:id="echoid-s24138" xml:space="preserve"> in ſuis <lb/>partibus:</s> <s xml:id="echoid-s24139" xml:space="preserve"> feretur ergo in tẽpore minimo ſenſibili per minimum ſpatiũ ſenſibile:</s> <s xml:id="echoid-s24140" xml:space="preserve"> ſed in eodẽ <lb/>tẽpore feretur per idẽ ſpatium forma luminoſi corporis debilioris illo corpore fortiori lumi <lb/>noſo:</s> <s xml:id="echoid-s24141" xml:space="preserve"> quoniã minimo ſpatio ſenſibili nõ eſt aliquod ſpatiũ ſenſibile minus:</s> <s xml:id="echoid-s24142" xml:space="preserve"> etiã minimo tem <lb/>pore ſenſibili nõ eſt aliquod ſenſibile tẽpus minus.</s> <s xml:id="echoid-s24143" xml:space="preserve"> Æ qualis ergo uirtutis erunt lumẽ fortius <lb/>& debilius:</s> <s xml:id="echoid-s24144" xml:space="preserve"> quod eſt impoſsibile, quoniã implicãtur cõtradictoria.</s> <s xml:id="echoid-s24145" xml:space="preserve"> Eſt ergo impoſsibile lu-<lb/>mẽ in tẽpore per proportionatum ſibi medium diffundi:</s> <s xml:id="echoid-s24146" xml:space="preserve"> neceſſe eſt ergo, qđ illa diffuſio fiat <lb/>in inſtãti.</s> <s xml:id="echoid-s24147" xml:space="preserve"> Quod eſt ꝓpoſitum.</s> <s xml:id="echoid-s24148" xml:space="preserve"> Ad hoc etiã aliquę deſeruiunt naturales rationes Ariſtotelis, <lb/>quas, qui uoluerit, percurrat, quia ſufficit nobis hoc unum inconueniens ſecutum.</s> <s xml:id="echoid-s24149" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div885" type="section" level="0" n="0"> <head xml:id="echoid-head718" xml:space="preserve" style="it">3. Omnis linea, qua peruenit lux à corpore luminoſo ad corp{us} oppoſitum, eſt linea na-<lb/>turalis ſenſibilis, latitudinem quandam habens, in qua est linea mathematica imagina-<lb/>biliter aſſumenda. Alhazen 16 n 4.</head> <p> <s xml:id="echoid-s24150" xml:space="preserve">Lux enim nõ procedit niſi à corpore, quoniã nõ eſt niſi in corpore:</s> <s xml:id="echoid-s24151" xml:space="preserve"> unde patet, quia in minima lu <lb/>ce, quę ſumi poteſt, eſt latitudo:</s> <s xml:id="echoid-s24152" xml:space="preserve"> quoniã minimã lucẽ dicimus, quæ ſi diuidatur, non habet am plius <lb/>actum lucis, quia nõ erit uiſibilis, ſed utraq;</s> <s xml:id="echoid-s24153" xml:space="preserve"> pars extinguetur, quia neutra pars eius erit lux, neque <lb/>apparebit ſenſui.</s> <s xml:id="echoid-s24154" xml:space="preserve"> Eſt ergo in linea radiali, ſecũdum quã fit diffuſio luminis, aliqua latitudo, propter <lb/>quã ineſt ei ſenſibilitas, & in medio illius lineæ eſt linea mathematica imaginabilis, cui oẽs aliæ li-<lb/>neæ mathematicæ in illa linea naturali ęquidiſtantes erunt.</s> <s xml:id="echoid-s24155" xml:space="preserve"> Et quoniã lux minima procedit ad mi-<lb/>nimã corporis partẽ, quã lux occupare poteſt:</s> <s xml:id="echoid-s24156" xml:space="preserve"> neceſſe eſt, quòd proceſſus eius ſit ſecundum lineam <lb/> <pb o="64" file="0366" n="366" rhead="VITELLONIS OPTICAE"/> mathematicã, quę eſt in medio lineę ſenſibilis, & ſecundum lineas extremas ęquidiſtãtes lineę me-<lb/>diæ:</s> <s xml:id="echoid-s24157" xml:space="preserve"> neq;</s> <s xml:id="echoid-s24158" xml:space="preserve"> cadit lux minima in punctum mathematicum corporis oppoſiti, ſed in punctum ſenſibilẽ <lb/>correſpondentẽ omnibus punctis mathematicis indiuiſibilibus, ad quos lineæ mathematicæ ipſi-<lb/>us lineæ ſenſibilis poſſunt terminari:</s> <s xml:id="echoid-s24159" xml:space="preserve"> & ob hoc utemur in demonſtrandis paſsionibus lucis figura-<lb/>tione linearum mathematicarum in proceſſu.</s> <s xml:id="echoid-s24160" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div886" type="section" level="0" n="0"> <head xml:id="echoid-head719" xml:space="preserve" style="it">4. Corpora diaphana ſunt apta penetrationi luminis & coloris ſine eſſentiali ſui tranſmuta <lb/>tione. Alhazen 28 n 1.</head> <p> <s xml:id="echoid-s24161" xml:space="preserve">Hęc enim corpora ꝓprietatẽ habẽt, ut nõ ꝓhibeant formas lucis & coloris ſe penetrare:</s> <s xml:id="echoid-s24162" xml:space="preserve"> attamẽ <lb/>nõ mutantur à lucibus uel coloribus, nec alterantur ab eis alteratione fixa:</s> <s xml:id="echoid-s24163" xml:space="preserve"> ſed fit per illa diffuſio lu <lb/>cis & coloris ſecundum lineas rectas per 1 huius:</s> <s xml:id="echoid-s24164" xml:space="preserve"> quarum aliquæ ſunt ęquidiſtãtes, aliquę ſecantes <lb/>ſe, & quæ dã diuerſi ſitus:</s> <s xml:id="echoid-s24165" xml:space="preserve"> & omnium iſtarum linearum diſtinctio fit per diſtinctum ſitum corporis <lb/>luminoſi, à quo fit diffuſio illius lucis uel coloris.</s> <s xml:id="echoid-s24166" xml:space="preserve"> Formæ itaq;</s> <s xml:id="echoid-s24167" xml:space="preserve"> lucis & coloris extẽſæ à corporibus <lb/>diuerſis in e o dẽ diaphano, extenduntur quęlibet ipſarum ſecundum lineam rectã, & pertranſeunt <lb/>ad corpora oppoſita.</s> <s xml:id="echoid-s24168" xml:space="preserve"> Corpus uero diaphanũ nõ tingitur per luces uel colores, ſed ſolùm penetra-<lb/>tur:</s> <s xml:id="echoid-s24169" xml:space="preserve"> neq;</s> <s xml:id="echoid-s24170" xml:space="preserve"> enim talia corpora propter luces & colores perdunt ſuas formas, neq;</s> <s xml:id="echoid-s24171" xml:space="preserve"> tinguntur per luces <lb/>& colores tinctura fixa:</s> <s xml:id="echoid-s24172" xml:space="preserve"> quia in eis non remanent formę lucis uel coloris poſt receſſum lucis uel co <lb/>loris ab ipſorum oppoſitione.</s> <s xml:id="echoid-s24173" xml:space="preserve"> Non ergo tranſmutantur illa corpora eſſentiali tranſmutatione per <lb/>luces & colores.</s> <s xml:id="echoid-s24174" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s24175" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div887" type="section" level="0" n="0"> <head xml:id="echoid-head720" xml:space="preserve" style="it">5. Luces & colores in corporib{us} diaphanis non admiſcentur adinuicem, ſed penetrant di-<lb/>ſtincti. Alhazen 29 n 1.</head> <p> <s xml:id="echoid-s24176" xml:space="preserve">Huius rei experimẽtaliter declarãdæ cauſſa, ponãtur in loco aliquo candelæ multæ localiter di-<lb/>ſtinctæ:</s> <s xml:id="echoid-s24177" xml:space="preserve"> & ſint oẽs oppoſitę uni foramini pertrãſeunti ad locũ obſcurum, & opponatur foramini in <lb/>loco obſcuro aliquod corpus non diaphanum.</s> <s xml:id="echoid-s24178" xml:space="preserve"> Luces itaq;</s> <s xml:id="echoid-s24179" xml:space="preserve"> cãdelarum apparent ſuper illud corpus <lb/>diſtinctè ſecundum numerum candelarũ, & quælibet illarum apparet oppoſita uni candelę ſecun-<lb/>dum lineã rectã tranſeuntẽ per foramẽ & per medium luminis candelæ:</s> <s xml:id="echoid-s24180" xml:space="preserve"> & ſi cooperiatur una cãde-<lb/>la, deſtruetur unum lumẽ oppoſitum illi cãdelæ tantùm, & diſcooperta cãdela, reuertitur lumẽ.</s> <s xml:id="echoid-s24181" xml:space="preserve"> Pa <lb/>làm itaq;</s> <s xml:id="echoid-s24182" xml:space="preserve">, qđ luces in medio foraminis, ubi ſe interſecãt oẽs uel plures in puncto uno, nõ admiſcen <lb/>tur in eodẽ puncto, ſed ſunt diſtinctæ per ſui ipſarum eſſentias:</s> <s xml:id="echoid-s24183" xml:space="preserve"> & ob hoc cum ulterius ꝓtẽduntur, <lb/>tunc ſecundum locorum, quibus incidũt, diuerſitatẽ localiter diſtinguuntur.</s> <s xml:id="echoid-s24184" xml:space="preserve"> Et quoniã luxres co-<lb/>loratas pertranſiẽs, illarum coloribus coloratur, ut ſuppoſitum eſt:</s> <s xml:id="echoid-s24185" xml:space="preserve"> palàm, ſi lumẽ penetrat diſtin-<lb/>ctum, & colores, qui feruntur cum lumine, penetrabunt diſtincti.</s> <s xml:id="echoid-s24186" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24187" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div888" type="section" level="0" n="0"> <head xml:id="echoid-head721" xml:space="preserve" style="it">6. Proportio uirtutis toti{us} corporis luminoſi ad totum corp{us} luminoſum eſt, ſicut determi-<lb/>natæ partis uirtutis ad partem corporis ſibi proportionalem.</head> <p> <s xml:id="echoid-s24188" xml:space="preserve">Sit corpus aliquod luminoſum a b.</s> <s xml:id="echoid-s24189" xml:space="preserve"> Dico, quòd ꝓportio uirtutis totius corporis a b ad totũ cor <lb/>pus a b eſt, ſicut proportio partis uirtutis, quæ <lb/> <anchor type="figure" xlink:label="fig-0366-01a" xlink:href="fig-0366-01"/> eſt a, ad partẽ corporis, quę eſt a.</s> <s xml:id="echoid-s24190" xml:space="preserve"> Si enim non <lb/>eſt iſtorum eadẽ ꝓportio:</s> <s xml:id="echoid-s24191" xml:space="preserve"> aut ergo maior, aut <lb/>minor:</s> <s xml:id="echoid-s24192" xml:space="preserve"> ſit primũ maior:</s> <s xml:id="echoid-s24193" xml:space="preserve"> & ſit uirtus totius cor <lb/>poris a b ſignata per lineã g d:</s> <s xml:id="echoid-s24194" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s24195" xml:space="preserve"> g uirtus partis corporis, quæ eſt a, & d ſit uirtus partis corporis, <lb/>quæ eſt b:</s> <s xml:id="echoid-s24196" xml:space="preserve"> quę eſt ergo proportio g ad a, eadẽ eſt ꝓportio d ad b:</s> <s xml:id="echoid-s24197" xml:space="preserve"> ergo per 18 p 5 erit cõiunctim g d <lb/>ad a b, ſicut g a d a.</s> <s xml:id="echoid-s24198" xml:space="preserve"> Si ergo ꝓportio g ad a eſt maior ꝓportione g d ad a b:</s> <s xml:id="echoid-s24199" xml:space="preserve"> erit quoq;</s> <s xml:id="echoid-s24200" xml:space="preserve"> maior ꝓportio <lb/>g d ad a b, ꝗ̃ g d ad a b:</s> <s xml:id="echoid-s24201" xml:space="preserve"> quod eſt impoſsibile:</s> <s xml:id="echoid-s24202" xml:space="preserve"> nõ enim poterunt eſſe unius rei ad aliã duę ꝓportiões, <lb/>quarum una ſit maior alia.</s> <s xml:id="echoid-s24203" xml:space="preserve"> Idẽ quoq;</s> <s xml:id="echoid-s24204" xml:space="preserve"> accidit impoſsibile danti, qđ minor ſit ꝓportio g partis uirtu <lb/>tis ad partẽ corporis, quę eſt a, ꝗ̃ g d uirtutis ad a b corpus.</s> <s xml:id="echoid-s24205" xml:space="preserve"> Si enim minor eſt proportio g ad a, ꝗ̃ g d <lb/>ad a b:</s> <s xml:id="echoid-s24206" xml:space="preserve"> & quę eſt g ad a, eadẽ eſt d ad b:</s> <s xml:id="echoid-s24207" xml:space="preserve"> erit ergo per 18 p 5 cõiunctim proportio totius uirtutis, quæ <lb/>eſt g d, ad corpus a b, minor proportione g d ad a b:</s> <s xml:id="echoid-s24208" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s24209" xml:space="preserve"> Eſt ergo proportio g ad a, <lb/>ſicut g d ad a b.</s> <s xml:id="echoid-s24210" xml:space="preserve"> Et hoc eſt propoſitum:</s> <s xml:id="echoid-s24211" xml:space="preserve"> & eſt uniuerſale, niſi fortè aliquid cõferat unio uirtuti:</s> <s xml:id="echoid-s24212" xml:space="preserve"> quo-<lb/>niam uirtus unita ſemper eſt fortior ſe ipſa diuiſa:</s> <s xml:id="echoid-s24213" xml:space="preserve"> unde tenet noſtra demonſtratio, quando partes <lb/>non diuiſæ à toto, agunt in ipſo toto non actualiter diſtinctę:</s> <s xml:id="echoid-s24214" xml:space="preserve"> cum enim diſtinctæ ſunt à toto, tunc <lb/>non ſunt partes:</s> <s xml:id="echoid-s24215" xml:space="preserve"> quia nomen partis, id quod dicit philoſophus, ſignat potentiam, non actum:</s> <s xml:id="echoid-s24216" xml:space="preserve"> & de <lb/>hoc completus in alijs ſermo fuit.</s> <s xml:id="echoid-s24217" xml:space="preserve"/> </p> <div xml:id="echoid-div888" type="float" level="0" n="0"> <figure xlink:label="fig-0366-01" xlink:href="fig-0366-01a"> <variables xml:id="echoid-variables386" xml:space="preserve">a b g d</variables> </figure> </div> </div> <div xml:id="echoid-div890" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables387" xml:space="preserve">a b g c d</variables> </figure> <head xml:id="echoid-head722" xml:space="preserve" style="it">7. Omnis corporis luminoſi intr anſmutabilis ſecun <lb/>dũ formã & ſitũ, in corp{us} aliud æquale et homogeneũ <lb/> idẽ immediatè uel per medium uniforme oppoſitũ, eſt ſemper actio æqualis & uniformis.</head> <p> <s xml:id="echoid-s24218" xml:space="preserve">Sit enim dati alicuius corporis luminoſi uirtus a:</s> <s xml:id="echoid-s24219" xml:space="preserve"> & ſit <lb/>corpus æquale & homogeneũ eidẽ oppoſitũ b g:</s> <s xml:id="echoid-s24220" xml:space="preserve"> & ſit im <lb/>preſsio uirtutis a in b g corpus ſignata ք c.</s> <s xml:id="echoid-s24221" xml:space="preserve"> Dico, quòd a <lb/>ſemper imprimit in corpus b g impreſsionẽ c, quę eſt ſem <lb/>per æqualis ſibijpſi & uniformis.</s> <s xml:id="echoid-s24222" xml:space="preserve"> Si enim detur, quòd a <lb/>quãdoq;</s> <s xml:id="echoid-s24223" xml:space="preserve"> imprimit in corpus b g impreſsionem, quę eſt c, <lb/>quãdoq;</s> <s xml:id="echoid-s24224" xml:space="preserve"> uerò nõ imprimit c, ſed aliud maius uel minus ipſo c, ut d:</s> <s xml:id="echoid-s24225" xml:space="preserve"> tũc cũ corpus obiectũ ſit homo <lb/> <pb o="65" file="0367" n="367" rhead="LIBER SECVNDVS."/> geneum & uniforme:</s> <s xml:id="echoid-s24226" xml:space="preserve"> erit diuerſitas impreſsionis nõ à corpore b g patiente, ſed à uirtute a diuerſifi <lb/>cata in ſe:</s> <s xml:id="echoid-s24227" xml:space="preserve"> hoc aũt eſt impoſsibile, cũ corpus luminoſum poſitum ſit intranſmutabile ſec undum for <lb/>mam & ſitum.</s> <s xml:id="echoid-s24228" xml:space="preserve"> Eſt ergo ipſius actio ſemper æqualis & uniformis in corpus eidẽ immediatè uel per <lb/>medium uniforme oppoſitum.</s> <s xml:id="echoid-s24229" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s24230" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div891" type="section" level="0" n="0"> <head xml:id="echoid-head723" xml:space="preserve" style="it">8. Neceſſe eſt terminum longitudinis cui{us}libet umbræ radium luminoſum eſſe.</head> <p> <s xml:id="echoid-s24231" xml:space="preserve">Quod hic ꝓponitur, ſatis patet ք p̃miſſa principia.</s> <s xml:id="echoid-s24232" xml:space="preserve"> Quoniã enim ք 3 ſuppoſitionẽ ſolũ in abſen-<lb/>tia luminis fit umbra, & ք 4 ſuppoſitionẽ in allatione luminis umbra deficit:</s> <s xml:id="echoid-s24233" xml:space="preserve"> tũc neceſſariò oportet <lb/>in tanto ſpatio umbrã cauſſari, in quãto lumẽ deficit:</s> <s xml:id="echoid-s24234" xml:space="preserve"> & ubi lumen accedit, ibi umbra deficit.</s> <s xml:id="echoid-s24235" xml:space="preserve"> T ermi <lb/>nus ergo lõgitudinis cuiuslibet umbrę cum ſit linea:</s> <s xml:id="echoid-s24236" xml:space="preserve"> patet, quòd oportet, ut illa linea ſit luminoſa.</s> <s xml:id="echoid-s24237" xml:space="preserve"> <lb/>Eſt ergo illa linea radius luminoſus per 6 definitionem.</s> <s xml:id="echoid-s24238" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24239" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div892" type="section" level="0" n="0"> <head xml:id="echoid-head724" xml:space="preserve" style="it">9. À<unsure/> terminis æquidiſt ãtiũ altitudinũ corporis luminoſi altioris, & corporis umbroſi baßioris <lb/>productæ lineæ cõcurrẽtes, ſunt ſuis altitudinib. proportionales. Ex quo patet, quòd eadẽ altitu-<lb/>do corporis umbroſi ex lumine baßiori longiorem proijcit umbram quàm ex lumine altiori.</head> <p> <s xml:id="echoid-s24240" xml:space="preserve">Sit altitudo corporis umbroſi cuiuſcũq;</s> <s xml:id="echoid-s24241" xml:space="preserve"> linea a b:</s> <s xml:id="echoid-s24242" xml:space="preserve"> & ſit altitudo alia illi æquidiſtãs ipſius corpo-<lb/>ris luminoſi, quæ ſit d e:</s> <s xml:id="echoid-s24243" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s24244" xml:space="preserve"> linea d e maior quàm li-<lb/> <anchor type="figure" xlink:label="fig-0367-01a" xlink:href="fig-0367-01"/> nea a b:</s> <s xml:id="echoid-s24245" xml:space="preserve"> ꝓducãturq́;</s> <s xml:id="echoid-s24246" xml:space="preserve"> lineæ e b & d a, quę ꝓtractæ con-<lb/>current ad aliquam partẽ in puncto g per 16 t 1 huius.</s> <s xml:id="echoid-s24247" xml:space="preserve"> <lb/>Dico, quòd erit ꝓportio lineę g b ad lineã g e, & lineę <lb/>g a ad lineã g d, ſicut lineæ a b ad lineã d e.</s> <s xml:id="echoid-s24248" xml:space="preserve"> Quia enim <lb/>linea b a æquidiſtat lineæ d e ex hypotheſi:</s> <s xml:id="echoid-s24249" xml:space="preserve"> palàm er-<lb/>go ք 29 p 1, quoniã angulus g b a eſt æqualis angulo g <lb/>e d, & angulus g a b æqualis angulo g d e:</s> <s xml:id="echoid-s24250" xml:space="preserve"> angulus <lb/>quoq;</s> <s xml:id="echoid-s24251" xml:space="preserve"> b g a cõmunis eſt ambobus trigonis d g e & a g <lb/>b:</s> <s xml:id="echoid-s24252" xml:space="preserve"> ergo ք 4 p 6 eſt ꝓportio lineæ g b ad lineã g e, ſicut <lb/>lineæ b a ad lineã e d:</s> <s xml:id="echoid-s24253" xml:space="preserve"> ergo ք 5 t 1 huius, erit è cõtrario <lb/>proportio lineę g e ad lineã b g, ſicut lineę e d ad lineã <lb/>a b.</s> <s xml:id="echoid-s24254" xml:space="preserve"> Palàm ergo eſt ꝓpoſitũ:</s> <s xml:id="echoid-s24255" xml:space="preserve"> quoniã eodem modo de-<lb/>monſtrari poteſt de lineis g a & g d.</s> <s xml:id="echoid-s24256" xml:space="preserve"> Et ex hoc patet, <lb/>quoniã eadem altitudo corporis umbroſi ex lumine <lb/>baſsiori lõgiorẽ proijcit umbram ꝗ̃ ex lumine altiori.</s> <s xml:id="echoid-s24257" xml:space="preserve"> <lb/>Eſto enim qđ aliquod corpus luminoſum ſit in pun-<lb/>cto h:</s> <s xml:id="echoid-s24258" xml:space="preserve"> cadatq́;</s> <s xml:id="echoid-s24259" xml:space="preserve"> radius h a in punctũ lineæ e g, qđ ſit k:</s> <s xml:id="echoid-s24260" xml:space="preserve"> <lb/>eritq́;</s> <s xml:id="echoid-s24261" xml:space="preserve"> ք pręmiſſum modũ ꝓportio e k ad b k, ſicut h e ad a b:</s> <s xml:id="echoid-s24262" xml:space="preserve"> ſed ք 8 p 5 ꝓportio h e ad a b eſt minor <lb/>ꝗ̃ d e ad a b:</s> <s xml:id="echoid-s24263" xml:space="preserve"> ſed ꝓportio d e ad a b eſt, ſicut ꝓportio e g ad b g, ut patuit:</s> <s xml:id="echoid-s24264" xml:space="preserve"> ergo ք 11 p 5 ꝓportio e k ad <lb/>b k eſt minor ꝗ̃ e g ad b g.</s> <s xml:id="echoid-s24265" xml:space="preserve"> Multũ ergo excreuit umbra b k reſpectu umbrę b g, ut patet ք 10 p 5 & per <lb/>4 t 1 huius.</s> <s xml:id="echoid-s24266" xml:space="preserve"> Et ex hoc accidit, quòd umbræ lunares ſemper ſunt lõgiores quàm umbrę ſolares:</s> <s xml:id="echoid-s24267" xml:space="preserve"> & ita <lb/>eſt de alijs corporibus luminoſis altioribus & baſsioribus quibuſcunq;</s> <s xml:id="echoid-s24268" xml:space="preserve">. Patet ergo propoſitum.</s> <s xml:id="echoid-s24269" xml:space="preserve"/> </p> <div xml:id="echoid-div892" type="float" level="0" n="0"> <figure xlink:label="fig-0367-01" xlink:href="fig-0367-01a"> <variables xml:id="echoid-variables388" xml:space="preserve">l h a e b g k</variables> </figure> </div> </div> <div xml:id="echoid-div894" type="section" level="0" n="0"> <head xml:id="echoid-head725" xml:space="preserve" style="it">10. Omnem r adium luminoſum per medium uni{us} diaphani trans uerticem alicui{us} corpo-<lb/>ris umbroſi protenſum, neceſſe est eſſe lineam unam rectam.</head> <p> <s xml:id="echoid-s24270" xml:space="preserve">Remaneat totalis diſpoſitio proximæ præcedẽtis, & ſit punctus g finis umbrę.</s> <s xml:id="echoid-s24271" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s24272" xml:space="preserve">, ut pa-<lb/>tet ք 8 huius, cuiuslibet umbrę terminus eſt radius <lb/> <anchor type="figure" xlink:label="fig-0367-02a" xlink:href="fig-0367-02"/> luminoſus:</s> <s xml:id="echoid-s24273" xml:space="preserve"> dico, quòd ille radius terminãs umbrã <lb/>eſt linea recta, ut eſt in propoſita figura linea d a g.</s> <s xml:id="echoid-s24274" xml:space="preserve"> Si <lb/>enim nõ eſt recta linea d a g, tũc cũ d a linea ſit recta <lb/>ք 1 huius, ideoq́;</s> <s xml:id="echoid-s24275" xml:space="preserve"> nullã habet cauſſam impedimẽti in <lb/>ꝓgreſſu, & linea a g ſimiliter eſt recta ք idẽ:</s> <s xml:id="echoid-s24276" xml:space="preserve"> cõiũgun <lb/>tur ergo lineę d a & a g angulariter in pũcto a:</s> <s xml:id="echoid-s24277" xml:space="preserve"> ſubtẽ-<lb/>datur ergo illi angulo, utcũq;</s> <s xml:id="echoid-s24278" xml:space="preserve"> cõtingat, baſis à pũctis <lb/>d & g:</s> <s xml:id="echoid-s24279" xml:space="preserve"> & ſit linea d u g recta:</s> <s xml:id="echoid-s24280" xml:space="preserve"> & ꝓtrahatur uel abſcin-<lb/>datur linea a b:</s> <s xml:id="echoid-s24281" xml:space="preserve"> trigonũ itaq;</s> <s xml:id="echoid-s24282" xml:space="preserve"> e d b g diuiditur ք lineã <lb/>b u æquidiſtãtẽ lineæ e d:</s> <s xml:id="echoid-s24283" xml:space="preserve"> ergo ք 29 p 1 erũt trigoni e <lb/>d g & b u g æquianguli:</s> <s xml:id="echoid-s24284" xml:space="preserve"> ergo ք 4 p 6 erit ꝓportio li-<lb/>neę g e ad lineã g b, ſicut lineæ e d ad lineã b u:</s> <s xml:id="echoid-s24285" xml:space="preserve"> ſed ք <lb/>proximã p̃miſſam eſt ꝓportio lineæ g e ad lineã g b, <lb/>ſicut lineæ d e ad lineã b a.</s> <s xml:id="echoid-s24286" xml:space="preserve"> Eſt ergo ք 11 p 5 eadẽ pro-<lb/>portio lineę d e ad ambas lineas b u & b a:</s> <s xml:id="echoid-s24287" xml:space="preserve"> qđ eſt cõ-<lb/>tra 8 p 5 & impoſsibile:</s> <s xml:id="echoid-s24288" xml:space="preserve"> ad minorẽ enim maior, & ad <lb/>maiorem minor eſt proportio:</s> <s xml:id="echoid-s24289" xml:space="preserve"> uel ſequetur maiorem lineam eſſe æqualem minori per 9 p 5:</s> <s xml:id="echoid-s24290" xml:space="preserve"> hoc au <lb/>tem eſt impoſsibile.</s> <s xml:id="echoid-s24291" xml:space="preserve"> Oportet ergo ut radius d a g ſit linea una recta.</s> <s xml:id="echoid-s24292" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s24293" xml:space="preserve"/> </p> <div xml:id="echoid-div894" type="float" level="0" n="0"> <figure xlink:label="fig-0367-02" xlink:href="fig-0367-02a"> <variables xml:id="echoid-variables389" xml:space="preserve">d u a u e b g</variables> </figure> </div> </div> <div xml:id="echoid-div896" type="section" level="0" n="0"> <head xml:id="echoid-head726" xml:space="preserve" style="it">11. Omnia corpora denſa non diaphana in partem luminoſo corpori aduerſam, umbrã proij-<lb/>ciunt uſ ad incidentiam radij per rei denſæ uerticem producti.</head> <p> <s xml:id="echoid-s24294" xml:space="preserve">Quia enim in corporibus dẽſis nõ diaphanis natura diaphanitatis & tranſparentiæ eſt impedita <lb/>ք admixtionẽ corporũ opacorũ terreorũ:</s> <s xml:id="echoid-s24295" xml:space="preserve"> ſunt enim omnia talia naturæ terre à dño:</s> <s xml:id="echoid-s24296" xml:space="preserve"> neceſſo eſt er-<lb/> <pb o="66" file="0368" n="368" rhead="VITELLONIS OPTICAE"/> go, ut trãſitũ luminis im pediãt:</s> <s xml:id="echoid-s24297" xml:space="preserve"> ergo ք 3 petitionẽ in abſentia luminis umbroſitatẽ efficiũt in ea par <lb/>te, in qua ք ipſas luminis acceſſus impeditur:</s> <s xml:id="echoid-s24298" xml:space="preserve"> hoc aũt <lb/> <anchor type="figure" xlink:label="fig-0368-01a" xlink:href="fig-0368-01"/> eſt in parte aduerſa corpori luminoſo.</s> <s xml:id="echoid-s24299" xml:space="preserve"> Sit aũt aliquod <lb/>taliũ umbroſorũ corporũ, cuius altitudo ab horizõte <lb/>ſit a b, & eius uertex a:</s> <s xml:id="echoid-s24300" xml:space="preserve"> & ſit corpus luminoſum altius <lb/>ꝗ̃ linea a b, cuius aliquis ſupremus punctus ſit d:</s> <s xml:id="echoid-s24301" xml:space="preserve"> radij <lb/>itaq;</s> <s xml:id="echoid-s24302" xml:space="preserve"> in tota linea a b incidẽtes, impediuntur à trãſitu <lb/>ꝓpter corporis opacitatẽ:</s> <s xml:id="echoid-s24303" xml:space="preserve"> cadat uerò radius d c pro xi <lb/>mus ſupra radiũ d a:</s> <s xml:id="echoid-s24304" xml:space="preserve"> hic ergo radius, ꝗ a nõ impeditur, <lb/>trãſit ultra corpus a b:</s> <s xml:id="echoid-s24305" xml:space="preserve"> in ſua ergo incidentia, q̃ ſit c, af-<lb/>fert lumen.</s> <s xml:id="echoid-s24306" xml:space="preserve"> Deficit ergo umbra.</s> <s xml:id="echoid-s24307" xml:space="preserve"> Et patet propoſitum.</s> <s xml:id="echoid-s24308" xml:space="preserve"/> </p> <div xml:id="echoid-div896" type="float" level="0" n="0"> <figure xlink:label="fig-0368-01" xlink:href="fig-0368-01a"> <variables xml:id="echoid-variables390" xml:space="preserve">d a b c</variables> </figure> </div> </div> <div xml:id="echoid-div898" type="section" level="0" n="0"> <head xml:id="echoid-head727" xml:space="preserve" style="it">12. Aequalium altitudinum corporum umbro-<lb/>ſorum, quod fuerit corpori luminoſo ſe altiori pro-<lb/>pinqui{us}, breuiorem facit umbram.</head> <p> <s xml:id="echoid-s24309" xml:space="preserve">Sit ſupremus pũctus corporis luminoſi g, qđ ſit al-<lb/>tius duob.</s> <s xml:id="echoid-s24310" xml:space="preserve"> corporibus umbroſis:</s> <s xml:id="echoid-s24311" xml:space="preserve"> cuius altitudo à ſup-<lb/>ficie horizontis ſit linea a g:</s> <s xml:id="echoid-s24312" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s24313" xml:space="preserve"> duorũ corporũ um <lb/>broſorũ æquales altitudines erectę ſuper lineã a b, ꝓ-<lb/>ductã in ipſa ſuperficie horizõtis, q̃ ſint d e & z h:</s> <s xml:id="echoid-s24314" xml:space="preserve"> qua-<lb/> <anchor type="figure" xlink:label="fig-0368-02a" xlink:href="fig-0368-02"/> rũ d e ſit ꝓpinquior corpori luminoſo a g, & z h remo <lb/>tior:</s> <s xml:id="echoid-s24315" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s24316" xml:space="preserve"> ք uerticẽ corporis d e radius g e t, ꝗ e-<lb/>rit line a una ք 10 huius:</s> <s xml:id="echoid-s24317" xml:space="preserve"> & ք uerticẽ corporis z h duca <lb/>tur radius g h b:</s> <s xml:id="echoid-s24318" xml:space="preserve"> erit itaq;</s> <s xml:id="echoid-s24319" xml:space="preserve"> ք p̃miſſam corporis d e um-<lb/>bra d e t:</s> <s xml:id="echoid-s24320" xml:space="preserve"> & corporis z h umbra z h b.</s> <s xml:id="echoid-s24321" xml:space="preserve"> Dico, qđ umbra <lb/>d e t eſt minor ꝗ̃ umbra z h b.</s> <s xml:id="echoid-s24322" xml:space="preserve"> Ducatur enim à pũcto h <lb/>linea æquidiſtãs lineæ e t ք 31 p 1, q̃ ſit h k:</s> <s xml:id="echoid-s24323" xml:space="preserve"> palãq́;</s> <s xml:id="echoid-s24324" xml:space="preserve"> ք 2 t 1 <lb/>huius, quoniã linea h k cõcurret cum linea a b, cũ qua <lb/>cõcurrit eius ęquidiſtãs, q̃ eſt linea e t.</s> <s xml:id="echoid-s24325" xml:space="preserve"> Et quoniã lineę <lb/>h b & e t cõcurrũt in pũcto g ſupremo pũcto coporis <lb/>luminoſi:</s> <s xml:id="echoid-s24326" xml:space="preserve"> cadet ergo punctũ k ք 2 & 14 t 1 huius inter <lb/>duo pũcta t & b.</s> <s xml:id="echoid-s24327" xml:space="preserve"> Copuletur ergo linea e h, q̃ ք 33 p 1 & <lb/>ex hypotheſi æqualιs & æquidιſtãs erit lineæ d z:</s> <s xml:id="echoid-s24328" xml:space="preserve"> ſed <lb/>per 34 p 1 lineę e h & t k ſunt æquales:</s> <s xml:id="echoid-s24329" xml:space="preserve"> lineę ergo t k & <lb/>d z ſunt ęquales.</s> <s xml:id="echoid-s24330" xml:space="preserve"> Addita ergo linea z t utriq;</s> <s xml:id="echoid-s24331" xml:space="preserve">, erit linea <lb/>d t æqualis lineæ z k:</s> <s xml:id="echoid-s24332" xml:space="preserve"> ergo per 1 p 6 umbra z h k eſt æqualis umbrę d e t:</s> <s xml:id="echoid-s24333" xml:space="preserve"> quoniam ſunt eiuſdem alti-<lb/>tudinis ex hypotheſi:</s> <s xml:id="echoid-s24334" xml:space="preserve"> ſed umbra z h k eſt minor quàm umbra z h b:</s> <s xml:id="echoid-s24335" xml:space="preserve"> quoniam eſt pars eius.</s> <s xml:id="echoid-s24336" xml:space="preserve"> Ergo & <lb/>umbra d e t eſt minor quàm umbra z h b.</s> <s xml:id="echoid-s24337" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24338" xml:space="preserve"/> </p> <div xml:id="echoid-div898" type="float" level="0" n="0"> <figure xlink:label="fig-0368-02" xlink:href="fig-0368-02a"> <variables xml:id="echoid-variables391" xml:space="preserve">g e h a d z t k b</variables> </figure> </div> </div> <div xml:id="echoid-div900" type="section" level="0" n="0"> <head xml:id="echoid-head728" xml:space="preserve" style="it">13. Vmbra lineæ rectæ perpendiculariter corpori luminoſo oppoſitæ, infixæ ſuperſiciei corpo-<lb/>ris denſi nulla eſt: eleuatæ uerò eſt linearis: apparet autem punctualis.</head> <p> <s xml:id="echoid-s24339" xml:space="preserve">Si enim ք ſuppoſitionẽ 3 in abſentia luminis fit umbra:</s> <s xml:id="echoid-s24340" xml:space="preserve"> tũc patet, qđ ſi lineã mathematicã natu-<lb/>ralis corporis ſuքficiei infixã accidat luminoſo corpori քpendiculariter offerri, nõ impedietur, niſi <lb/>unica linea radialis à trãſitu cũ alijs lineis radialibus, q̃ tráſeunt ad ſuքficiẽ illius corporis:</s> <s xml:id="echoid-s24341" xml:space="preserve"> nulla ue-<lb/>rò aliarũ linearũ radialiũ impeditur ꝓpter obiectũ illιus lineę:</s> <s xml:id="echoid-s24342" xml:space="preserve"> aliàs enim accideret duas uel plures <lb/>lineas radiales cũ una linea քpendiculari ipſis obiecta in uno pũcto cõcurrere:</s> <s xml:id="echoid-s24343" xml:space="preserve"> qđ eſt impoſsibile, <lb/>ꝗa indiuiſibilia in nullo ſe excedũt.</s> <s xml:id="echoid-s24344" xml:space="preserve"> Cũ aũt radius nõ ſit aliud ꝗ̃ linea luminoſa, ut patet ք 6 definiti <lb/>onẽ:</s> <s xml:id="echoid-s24345" xml:space="preserve"> palã, qđ radius ad modũ lineę incidit ſuքficiei corporis ſecũdũ pũctũ:</s> <s xml:id="echoid-s24346" xml:space="preserve"> ergo & impeditur ſecũ-<lb/>dũ pũctũ:</s> <s xml:id="echoid-s24347" xml:space="preserve"> ſed in allatione luminis umbra deficit ք 4 ſuppoſitionẽ.</s> <s xml:id="echoid-s24348" xml:space="preserve"> Quia ergo unicus radius eſt im-<lb/>peditus, & ille incidit ſecũdũ pũctũ:</s> <s xml:id="echoid-s24349" xml:space="preserve"> palã, qđ nõ manet aliqua umbra.</s> <s xml:id="echoid-s24350" xml:space="preserve"> Cũ uerò linea eleuatur fuper <lb/>dẽſi corporis ſuքficiẽ, ubicũq;</s> <s xml:id="echoid-s24351" xml:space="preserve"> ſub linea ponatur dẽſa ſuքficies, umbra inuenitur:</s> <s xml:id="echoid-s24352" xml:space="preserve"> & ſi ք diuerſa pun <lb/>cta fiat deſcẽſus, palã ꝗa umbra proijcitur linearis, eò qđ inter quælιbet duo pũcta eſt lineã mediã <lb/>ducere:</s> <s xml:id="echoid-s24353" xml:space="preserve"> apparet aũt ſemper punctualis in cõcurſu ſui cum ſuperficie corporis denſi:</s> <s xml:id="echoid-s24354" xml:space="preserve"> quia ibi ſolùm <lb/>cum umbra denſitatis ſuperficiei commiſcetur.</s> <s xml:id="echoid-s24355" xml:space="preserve"> Patet ergo illud, quod proponebatur.</s> <s xml:id="echoid-s24356" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div901" type="section" level="0" n="0"> <head xml:id="echoid-head729" xml:space="preserve" style="it">14. Vmbra ſuperficiei planæ cuiuſcun figuræ perpendicularis ſuper ſuperficiẽ corporis lumi <lb/>noſi, infixæ corpori denſo nulla eſt: eleuatæ uerò eſt ſuperficialis: ſed apparet linearis recta.</head> <p> <s xml:id="echoid-s24357" xml:space="preserve">Hoc patet ք p̃cedẽtẽ:</s> <s xml:id="echoid-s24358" xml:space="preserve"> ad quẽlibet enim pũctũ lineę terminãtis quãcunq;</s> <s xml:id="echoid-s24359" xml:space="preserve"> datã ſuperficiẽ corpori <lb/>luminoſo քpẽdiculariter oppoſitã, cõtingit ducere lineã քpẽdiculariter oppoſitã corpori lumino-<lb/>ſo.</s> <s xml:id="echoid-s24360" xml:space="preserve"> Vmbra ergo cuius libet illarũ linearũ, ſuքficie ꝓpoſita exiſtẽte infixa corpori denſo, nulla eſt:</s> <s xml:id="echoid-s24361" xml:space="preserve"> er-<lb/>go neq;</s> <s xml:id="echoid-s24362" xml:space="preserve"> umbra totius ſuքficiei fit aliqua.</s> <s xml:id="echoid-s24363" xml:space="preserve"> Eleuata uerò ſuperficie oppoſita ab illo dẽſo corpore, um-<lb/>bra cuiuslibet illarum linearũ ք præcedentẽ propoſitionẽ eſt punctualis:</s> <s xml:id="echoid-s24364" xml:space="preserve"> aggregata uerò talia pun <lb/>cta uidentur lineam conſtituere:</s> <s xml:id="echoid-s24365" xml:space="preserve"> apparet ergo umbra ſuperficiei taliter eleuatæ umbra linearis.</s> <s xml:id="echoid-s24366" xml:space="preserve"> Et <lb/>quoniam ſuperficies circulares ex ſuis diametris uel alijs perpendiculariter ſuper corpus lumino-<lb/>ſum productis, non accipiunt niſi puncta umbrarum, quæ ad lineam rectam inferius concurrunt, <lb/>quia impediunt tranſitum rectæ lineæ, fit ipſarum umbra linearis recta:</s> <s xml:id="echoid-s24367" xml:space="preserve"> non enim cauſſantur um-<lb/> <pb o="67" file="0369" n="369" rhead="LIBER SECVNDVS."/> bræ à figura quorumlibet obiectorum, niſi ſecundum quod tranſitus luminis impeditur.</s> <s xml:id="echoid-s24368" xml:space="preserve"> Cuiuſ-<lb/>cunque ergo figuræ fuerit propoſita ſuperficies, umbra apparẽs ſemper erit ſuperficialis:</s> <s xml:id="echoid-s24369" xml:space="preserve"> uidebitur <lb/>autem linearis propter præmiſſas cauſas.</s> <s xml:id="echoid-s24370" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24371" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div902" type="section" level="0" n="0"> <head xml:id="echoid-head730" xml:space="preserve" style="it">15. Omnis corporis denſi, cui{us} æqualis uel amplior eſt baſis, contrapoſita ſibi ſuperficie perẽ-<lb/>diculariter corpori luminoſo oppoſiti, infixi corpori denſo umbra nulla eſt: eleuati uerò eſt corpo-<lb/>ralis: uidetur autem ſuperficialis.</head> <p> <s xml:id="echoid-s24372" xml:space="preserve">Verbi gratia:</s> <s xml:id="echoid-s24373" xml:space="preserve"> ſit columna rotunda, uel aliud corpus, cuius baſis ſit æqualis uel amplior ſuperficie <lb/>illius eiuſdem corporis contrapoſita ipſi baſi, ſi ipſius corporis ſuperficies nõ terminetur ad unum <lb/>punctum, ut eſt in pyramide, quod infigatur ſuperficiei alicuius corporis ſolidi, & perpendiculari-<lb/>ter opponatur corpori luminoſo:</s> <s xml:id="echoid-s24374" xml:space="preserve"> dico, quòd uerũ eſt, quod proponitur.</s> <s xml:id="echoid-s24375" xml:space="preserve"> Si enim illud corpus ſit co-<lb/>lumna rotunda uel aliud corpus, cuius baſis ſit ęqualis ſuperficiei contrapoſitę baſis, & aduerſę cor <lb/>pori luminoſo, patet, quoniam radij luminoſi ex omni parte ſecundum lineas longitudinis perue-<lb/>niunt ad baſim:</s> <s xml:id="echoid-s24376" xml:space="preserve"> nulla ergo fit umbra.</s> <s xml:id="echoid-s24377" xml:space="preserve"> Et idem patet, ſi illud corpus ſit pyramidale:</s> <s xml:id="echoid-s24378" xml:space="preserve"> uel ſi baſis ſit ma-<lb/>ior ſibi contrapoſita ſuperficie aduerſa corpori luminoſo:</s> <s xml:id="echoid-s24379" xml:space="preserve"> tunc enim lumen nullatenus impeditur, <lb/>quod tamen accideret, ſi ſuperficies aduerſa corpori luminoſo, eſſet amplior ipſa baſi corporis um-<lb/>broſi:</s> <s xml:id="echoid-s24380" xml:space="preserve"> tunc enim impedito tranſitu luminis, cauſſaretur umbra.</s> <s xml:id="echoid-s24381" xml:space="preserve"> Sed quacunq;</s> <s xml:id="echoid-s24382" xml:space="preserve"> figura corporis exiſtẽ <lb/>te, ſi ipſum eleuetur ab alio corpore, cui fuit infixũ, apparebit umbra ſuperficialis:</s> <s xml:id="echoid-s24383" xml:space="preserve"> ſuperficies enim <lb/>ſecantes corpus, & perpendiculariter ſuperficiei corporis luminoſi incidentes, umbram conſtituũt <lb/>linearẽ per pręmiſſam.</s> <s xml:id="echoid-s24384" xml:space="preserve"> Et quia tota ſuperficies corporis oppoſita luminoſo corpori per tales ſuքfi-<lb/>cies exhauritur, lineę uerò tales cõiunctę ſuperficiẽ conſtituũt:</s> <s xml:id="echoid-s24385" xml:space="preserve"> palã, omnis corporis ſic diſpoſiti um <lb/>bram ſuperficialem apparere:</s> <s xml:id="echoid-s24386" xml:space="preserve"> erit autem illa umbra neceſſariò corporalis:</s> <s xml:id="echoid-s24387" xml:space="preserve"> quoniam erit dimenſio-<lb/>nata dimenſionibus corporis:</s> <s xml:id="echoid-s24388" xml:space="preserve"> quod poteſt declarari, ut prius.</s> <s xml:id="echoid-s24389" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24390" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div903" type="section" level="0" n="0"> <head xml:id="echoid-head731" xml:space="preserve" style="it">16. Longior radi{us} ad ſphæramuel circulum columnæ uelpyramidis rotundarum perueniẽs, <lb/>quaſi linea contingens eſt.</head> <p> <s xml:id="echoid-s24391" xml:space="preserve">Sit circulus magnus ſphæræ uel columnæ uel pyramidis rotundę, qui d g:</s> <s xml:id="echoid-s24392" xml:space="preserve"> cuius centrum ſit pun <lb/>ctum a, & diameter d g.</s> <s xml:id="echoid-s24393" xml:space="preserve"> Et quoniam lumen ad omnem differen <lb/> <anchor type="figure" xlink:label="fig-0369-01a" xlink:href="fig-0369-01"/> tiam poſitionis ſe diffundit, ſicut patet per 6 ſuppoſitionem:</s> <s xml:id="echoid-s24394" xml:space="preserve"> ſit <lb/>punctum corporis luminoſi z, cuius lumen ſe diffundat ſuper <lb/>circulum d g:</s> <s xml:id="echoid-s24395" xml:space="preserve"> ducaturq́ linea z a à pũcto corporis luminoſi ad <lb/>centrum illuminati circuli:</s> <s xml:id="echoid-s24396" xml:space="preserve"> & ſecundum diametrum a z deſcri-<lb/>batur circulus, ſecans circulum d g in punctis e & b:</s> <s xml:id="echoid-s24397" xml:space="preserve"> & copulen <lb/>tur radij z e, z b.</s> <s xml:id="echoid-s24398" xml:space="preserve"> Dico, quòd radij z e & z b ſunt contingentes <lb/>ſphæram, uel aliud aliorum corporum:</s> <s xml:id="echoid-s24399" xml:space="preserve"> & quòd nulli radij lon-<lb/>giores illis poſſunt ad illa corpora peruenire.</s> <s xml:id="echoid-s24400" xml:space="preserve"> Ducantur enim à <lb/>centro circuli d g (quod eſt punctum a) ad puncta ſectionum <lb/>b & e, lineę a e & a b.</s> <s xml:id="echoid-s24401" xml:space="preserve"> Palàm ergo per 31 p 3, quoniam duo angu-<lb/>li z e a & z b a ſunt recti:</s> <s xml:id="echoid-s24402" xml:space="preserve"> ergo per 16 p 3 patet, quòd lineæ z e & <lb/>z b contingunt circulum d g:</s> <s xml:id="echoid-s24403" xml:space="preserve"> productæ ergo non ſecabunt cir-<lb/>culum d g:</s> <s xml:id="echoid-s24404" xml:space="preserve"> ſuntitaq;</s> <s xml:id="echoid-s24405" xml:space="preserve"> lineæ z e & z b longiores lineę, quę à pun-<lb/>cto z ad illa corpora duci poſſunt.</s> <s xml:id="echoid-s24406" xml:space="preserve"> Si enim detur, quòd aliqui <lb/>longiores radij duci poſsint à puncto z ad illa corpora:</s> <s xml:id="echoid-s24407" xml:space="preserve"> patet <lb/>per 8 p 3, quòd illę non cadent in arcum e b:</s> <s xml:id="echoid-s24408" xml:space="preserve"> ipſæ ergo productę <lb/>ſecabunt lineas z e & z b prius, quàm perueniant ad arcus e d <lb/>uel b g:</s> <s xml:id="echoid-s24409" xml:space="preserve"> duę itaq;</s> <s xml:id="echoid-s24410" xml:space="preserve"> lineę rectę includent ſuperficiem:</s> <s xml:id="echoid-s24411" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s24412" xml:space="preserve"> Et hoc quidem non ſolùm <lb/>demonſtrabile eſt in corporibus illuminandis, ſed etiam per eundem modum demonſtrari poteſt <lb/>de corporibus luminoſis:</s> <s xml:id="echoid-s24413" xml:space="preserve"> quia & ab illis longior radius in obiecta corpora incidens, ipſa corpora l<gap/> <lb/>minoſa eſt contingens.</s> <s xml:id="echoid-s24414" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24415" xml:space="preserve"/> </p> <div xml:id="echoid-div903" type="float" level="0" n="0"> <figure xlink:label="fig-0369-01" xlink:href="fig-0369-01a"> <variables xml:id="echoid-variables392" xml:space="preserve">g a d b e z</variables> </figure> </div> </div> <div xml:id="echoid-div905" type="section" level="0" n="0"> <head xml:id="echoid-head732" xml:space="preserve" style="it">17. Impoßibile eſt, ut lumen egrediens à corpore luminoſo, egrediatur tantùm à centro corpo-<lb/>ris luminoſi. Ex quo patet, quòd neceſſe eſt à quolibet puncto ſuperficiei corporis luminoſi diffun-<lb/>di radios luminoſos.</head> <p> <s xml:id="echoid-s24416" xml:space="preserve">Si enim dicatur, quòd radij luminoſi tantùm <lb/> <anchor type="figure" xlink:label="fig-0369-02a" xlink:href="fig-0369-02"/> egrediuntur à centro corporis luminoſi:</s> <s xml:id="echoid-s24417" xml:space="preserve"> ſit cor <lb/>pus luminoſum circulus a b:</s> <s xml:id="echoid-s24418" xml:space="preserve"> cuius centrum g:</s> <s xml:id="echoid-s24419" xml:space="preserve"> <lb/>ſitq́;</s> <s xml:id="echoid-s24420" xml:space="preserve"> corpus illuminatum circulus d e:</s> <s xml:id="echoid-s24421" xml:space="preserve"> & à cen-<lb/>tro g corporis luminoſi egrediantur duo radij <lb/>longiſsimi, qui poſſunt ab illo puncto g corpo-<lb/>ri illuminando incidere, qui per præmiſſam e-<lb/>runt duæ lineæ contingentes fines corporis il-<lb/>luminati, quę ſint g d u, & g e z:</s> <s xml:id="echoid-s24422" xml:space="preserve"> & puncta con-<lb/>tactuũ, quæ ſint d & e, copulentur per lineã d e:</s> <s xml:id="echoid-s24423" xml:space="preserve"> <lb/>& ei æquidiſtanter ducatur linea u z per 31 p 1, eritq́ue pars corporis illuminati, ſuper quam cadit <lb/>lumen, pars d h e:</s> <s xml:id="echoid-s24424" xml:space="preserve"> & pars obſcura, ſuper quam non cadit lumen, quæ d c e.</s> <s xml:id="echoid-s24425" xml:space="preserve"> Et quia pars, ſupra quam <lb/> <pb o="68" file="0370" n="370" rhead="VITELLONIS OPTICAE"/> non cadit radius, non illuminatur:</s> <s xml:id="echoid-s24426" xml:space="preserve"> ergo pars contenta ſub terminis u d c e z eſt umbroſa, o bſcurans <lb/>lineas d e & u z ęquidiſtantes:</s> <s xml:id="echoid-s24427" xml:space="preserve"> ſunt itaq;</s> <s xml:id="echoid-s24428" xml:space="preserve"> per 29 p 1 trigoni u g z & d g e æquiãguli:</s> <s xml:id="echoid-s24429" xml:space="preserve"> quia angulus d g e <lb/>eſt communis ambobus trigonis.</s> <s xml:id="echoid-s24430" xml:space="preserve"> Eſt ergo per 4 p 6 proportio lineæ g e ad lineã g z, ſicut lineæ d e <lb/>ad lineam u z:</s> <s xml:id="echoid-s24431" xml:space="preserve"> ſed linea z g eſt maior quàm linea e g:</s> <s xml:id="echoid-s24432" xml:space="preserve"> ergo linea u z eſt maior quàm linea d e.</s> <s xml:id="echoid-s24433" xml:space="preserve"> Vmbra <lb/>ergo corporum omnium (cuiuſcunq;</s> <s xml:id="echoid-s24434" xml:space="preserve"> ſint proportionis ipſarum diametri ad diametros corporis lu <lb/>minoſi) ſemper eſt maior corpore umbroſo, & ſemper augmentantur ſecundum modum, quo elon <lb/>gantur ultra corpus umbroſum, cuius contrarium notum eſt ſenſui, Vnde fuit ſuppoſitum in princi <lb/>pio aliquam umbram in ſui termino acui, & ad punctum terminari.</s> <s xml:id="echoid-s24435" xml:space="preserve"> Palàm ergo eſt propoſitum.</s> <s xml:id="echoid-s24436" xml:space="preserve"> Et <lb/>cum lumen egrediatur à corpore luminoſo, & non ſolùm à centro, ut oſtendimus, manifeſtum eſt <lb/>corollarium:</s> <s xml:id="echoid-s24437" xml:space="preserve"> quoniam à quolibet puncto ſuperficiei corporis luminoſi neceſſe habet egredi ad cor <lb/>pora illuminanda:</s> <s xml:id="echoid-s24438" xml:space="preserve"> corpus enim luminoſum ſecũdum quodlibet ſui punctum unigeneum eſt:</s> <s xml:id="echoid-s24439" xml:space="preserve"> unde <lb/>qua ratione dabitur ab uno puncto ſuæ ſuperficiei lumen diffundi, eadem ratione dabitur de quoli <lb/>bet aliorum punctorum.</s> <s xml:id="echoid-s24440" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24441" xml:space="preserve"/> </p> <div xml:id="echoid-div905" type="float" level="0" n="0"> <figure xlink:label="fig-0369-02" xlink:href="fig-0369-02a"> <variables xml:id="echoid-variables393" xml:space="preserve">u d b <gap/> c h g z e a</variables> </figure> </div> </div> <div xml:id="echoid-div907" type="section" level="0" n="0"> <head xml:id="echoid-head733" xml:space="preserve" style="it">18. Impoßibile eſt, ut à ſuperficie corporis luminoſi egrediantur radij <lb/>ſolùm æquidiſtanter corpori illuminando incidentes.</head> <p> <s xml:id="echoid-s24442" xml:space="preserve">Si enim hoc dicatur eſſe neceſſarium, tunc ſequeretur euidens impoſ-<lb/> <anchor type="figure" xlink:label="fig-0370-01a" xlink:href="fig-0370-01"/> ſibile.</s> <s xml:id="echoid-s24443" xml:space="preserve"> Sit enim corpus luminoſum, cuius diameter a b:</s> <s xml:id="echoid-s24444" xml:space="preserve"> & corpus illumina <lb/>tum d g:</s> <s xml:id="echoid-s24445" xml:space="preserve"> & producantur à corpore luminoſo duo radij longiores, qui per <lb/>16 huius erunt duę lineę contingentes fines corporis g d:</s> <s xml:id="echoid-s24446" xml:space="preserve"> quę ſint a g u & <lb/>b d e, & ſint æquidiſtantes ex hypotheſi:</s> <s xml:id="echoid-s24447" xml:space="preserve"> pars quoq;</s> <s xml:id="echoid-s24448" xml:space="preserve"> illuminata, ſuper quã <lb/>cadit lumen, ſit g z d, & pars ſuper quã cadit umbra, ſit g h d.</s> <s xml:id="echoid-s24449" xml:space="preserve"> Vmbra ergo <lb/>continetur à duabus lineis e d & u g, quę ſunt æquidiſtantes.</s> <s xml:id="echoid-s24450" xml:space="preserve"> Si ergo uni-<lb/>cuiq;</s> <s xml:id="echoid-s24451" xml:space="preserve"> corpori illuminando correſpondeat æqualis ſibi pars corporis illu-<lb/>minantis (tunc enim ſolum ſecundum lineas æquidiſtantes radij incidẽt <lb/>per 33 p 1) patet ergo, quòd omnis umbra in omni ſui parte æqualis erit <lb/>ſuæ rei umbroſæ:</s> <s xml:id="echoid-s24452" xml:space="preserve"> igitur non augebitur umbra, neq;</s> <s xml:id="echoid-s24453" xml:space="preserve"> minuetur, ſed proten <lb/>detur ſemper in infinitum:</s> <s xml:id="echoid-s24454" xml:space="preserve"> quod eſt contra ſuppoſitionem:</s> <s xml:id="echoid-s24455" xml:space="preserve"> habet enim a-<lb/>liqua umbrarum terminum acutum:</s> <s xml:id="echoid-s24456" xml:space="preserve"> eſt ergo hoc impoſsibile:</s> <s xml:id="echoid-s24457" xml:space="preserve"> oppoſitum <lb/>eſt ergo neceſſarium.</s> <s xml:id="echoid-s24458" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s24459" xml:space="preserve"/> </p> <div xml:id="echoid-div907" type="float" level="0" n="0"> <figure xlink:label="fig-0370-01" xlink:href="fig-0370-01a"> <variables xml:id="echoid-variables394" xml:space="preserve">a b z z g d u h e</variables> </figure> </div> </div> <div xml:id="echoid-div909" type="section" level="0" n="0"> <head xml:id="echoid-head734" xml:space="preserve" style="it">19. Omnis punct{us} corporis luminoſi eam partem corporis umbroſi <lb/>illuminat, ad quã ab eodem pũcto rect{as} line{as} poßibile eſt produci. Ex <lb/>quo patet, quòd un{us} punct{us} luminoſi corporis non illuminat omne umbroſum corp{us}.</head> <p> <s xml:id="echoid-s24460" xml:space="preserve">Sunt enim corpora luminoſa unigenea in ſuis partibus:</s> <s xml:id="echoid-s24461" xml:space="preserve"> non ergo diuerſificatur effectus ſuarum <lb/>partium, neq;</s> <s xml:id="echoid-s24462" xml:space="preserve"> eſt poſsibile, ut ab una parte illuminent, & non ab alia:</s> <s xml:id="echoid-s24463" xml:space="preserve"> non tamen ab uno puncto cor <lb/>poris luminoſi ad quodlibet punctum umbroſi corporis poſſunt rectæ lineæ produci:</s> <s xml:id="echoid-s24464" xml:space="preserve"> & ob hoc u-<lb/>nus punctus non illuminat omnia, ſed illuminantur corpora umbroſa à diuerſis punctis corporis <lb/>luminoſi.</s> <s xml:id="echoid-s24465" xml:space="preserve"> Sit enim corpus luminoſum circulus a b:</s> <s xml:id="echoid-s24466" xml:space="preserve"> <lb/>quem contingat linea d g ſuper punctum a per 17 p 3:</s> <s xml:id="echoid-s24467" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0370-02a" xlink:href="fig-0370-02"/> ſitq́;</s> <s xml:id="echoid-s24468" xml:space="preserve"> corpus illuminatum concauum arcus e u, & ſecet <lb/>ipſum linea d g ſuper duo pũcta z & h.</s> <s xml:id="echoid-s24469" xml:space="preserve"> Dico, quòd poſ <lb/>ſibile eſt omnem arcum z h illuminari à puncto a cor-<lb/>poris luminoſi:</s> <s xml:id="echoid-s24470" xml:space="preserve"> quoniam, ut patet, poſsibile eſt, ut ab <lb/>omni puncto arcus z h ducatur linea recta ad punctũ <lb/>a:</s> <s xml:id="echoid-s24471" xml:space="preserve"> ſed ab arcu z e, & ab arcu h u aliquas lineas duci ad <lb/>punctum a eſt impoſsibile per 16 p 3:</s> <s xml:id="echoid-s24472" xml:space="preserve"> quoniam inter li <lb/>neam g d contin gentem circulum, & inter ipſum circu <lb/>lum a b aliquam lineam rectam intercipi eſt impoſsibi <lb/>le.</s> <s xml:id="echoid-s24473" xml:space="preserve"> Si ergo aliqua linea ab aliquo punctorum illorũ ar-<lb/>cuũ ducatur ad punctũ a, illa neceſſariò ſecabit circu-<lb/>lum, ſicut linea u a ſecat circulum a b in puncto t, pri-<lb/>uſquã perueniat ad pũctũ a.</s> <s xml:id="echoid-s24474" xml:space="preserve"> Et ſimiliter eſt de omnib.</s> <s xml:id="echoid-s24475" xml:space="preserve"> <lb/>lineis à quocunq;</s> <s xml:id="echoid-s24476" xml:space="preserve"> puncto arcuum u h & z e ad punctũ <lb/>a productis:</s> <s xml:id="echoid-s24477" xml:space="preserve"> oẽs enim ſecant circulũ a b in alio puncto <lb/>ab ipſo puncto a, priuſquã perueniãt ad punctũ a.</s> <s xml:id="echoid-s24478" xml:space="preserve"> Radius itaq;</s> <s xml:id="echoid-s24479" xml:space="preserve"> exiẽs à puncto a, nõ illuminat ambos <lb/>arcus u h & z e, ſed ſolũ arcũ h z:</s> <s xml:id="echoid-s24480" xml:space="preserve"> ſed illos arcus ab alijs punctis luminoſi corporis circuli a b, à quib.</s> <s xml:id="echoid-s24481" xml:space="preserve"> <lb/>ad eoſdem arcus rectę poſſunt produci lineę, nihil prohibet illuminari.</s> <s xml:id="echoid-s24482" xml:space="preserve"> Et ſimiliter eſt de alijs qui-<lb/>buſcunq;</s> <s xml:id="echoid-s24483" xml:space="preserve"> corporib illuminatis:</s> <s xml:id="echoid-s24484" xml:space="preserve"> quoniã ſi corpora cõcaua (de quibus plus uidetur, quòd poſsint ab <lb/>uno puncto illuminari) nõ illuminantur ab uno puncto corporis luminoſi:</s> <s xml:id="echoid-s24485" xml:space="preserve"> ergo multo minus cor-<lb/>pora recta plures planas ſuperficies habentia, uel corpora ſphærica, uel alia conuexa, poſſũnt ab u-<lb/>no puncto luminoſi corporis illuminari.</s> <s xml:id="echoid-s24486" xml:space="preserve"> Patet ergo propoſitum & eius corollarium.</s> <s xml:id="echoid-s24487" xml:space="preserve"/> </p> <div xml:id="echoid-div909" type="float" level="0" n="0"> <figure xlink:label="fig-0370-02" xlink:href="fig-0370-02a"> <variables xml:id="echoid-variables395" xml:space="preserve">g z a h d t e u b</variables> </figure> </div> </div> <div xml:id="echoid-div911" type="section" level="0" n="0"> <head xml:id="echoid-head735" xml:space="preserve" style="it">20. À puncto cui{us}libet corporis luminoſi lumen diffunditur ſecundum omnem rectam li-<lb/> <pb o="69" file="0371" n="371" rhead="LIBER SECVNDVS."/> neam, quæ ab illo puncto ad oppoſit am ſuperficiem duci poteſt: unicatantùm linea perpendicu-<lb/>lariter ſuperficiei obiecti corporis incidente. Ex quo patet, lucem cui{us}libet puncti corporis lumi <lb/>noſi ſecundum pyramidem illuminationis diffundi.</head> <p> <s xml:id="echoid-s24488" xml:space="preserve">Quòd enim lux cuiuslibet puncti corporis luminoſi diffun datur ſecundum omnem lineã duci-<lb/>bilem ab illo puncto ſuper ſuperficiem corporis obiecti, ad omnem poſitionis differentiã, hoc patet <lb/>per præmiſſam.</s> <s xml:id="echoid-s24489" xml:space="preserve"> Quòd autem unica tantũ linearum ab aliquo uno puncto corporis luminoſi produ <lb/>ctarũ ad ſuperficiẽ unam corporis oppoſiti ſit perpendicularis, hoc patet ex 20 t 1 huius.</s> <s xml:id="echoid-s24490" xml:space="preserve"> Vnica ergo <lb/>linea perpendiculariter incidit ſuperficiei ſibi oppoſitæ:</s> <s xml:id="echoid-s24491" xml:space="preserve"> omnes uerò aliæ lineæ ab eodem puncto <lb/>productæ incidunt obliquè.</s> <s xml:id="echoid-s24492" xml:space="preserve"> Patet ergo ex hoc, quòd cuiuslιbet puncti corporis luminoſi lumen ſe-<lb/>cundum pyramidem illuminationis diffunditur, cuius uertex eſt in puncto corporis luminoſi & ba <lb/>ſis in ſuperficie corporis obiecti:</s> <s xml:id="echoid-s24493" xml:space="preserve"> & hoc quidem inſtrumẽtaliter patet per 1 huius.</s> <s xml:id="echoid-s24494" xml:space="preserve"> Lumine enim trã <lb/>ſeunte foramen inſtrumenti, cuius cẽtrum eſt punctum m, & diffuſo ipſo in partem oppoſitam oræ <lb/>inſtrumenti ſecundum circulum, cuius centrum eſt punctum p:</s> <s xml:id="echoid-s24495" xml:space="preserve"> erit circulus p maior circulo m:</s> <s xml:id="echoid-s24496" xml:space="preserve"> qđ <lb/>ſenſibiliter poteſt uidèri, computatis hinc inde partibus in ora inſtrumenti, quę interiacẽt periphe-<lb/>rias illorum circulorum & centra.</s> <s xml:id="echoid-s24497" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24498" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div912" type="section" level="0" n="0"> <head xml:id="echoid-head736" xml:space="preserve" style="it">21. Corporis umbroſipars, cui à plurib{us} partib{us} corporis luminoſi lumen incidit, pl{us} illu-<lb/>minatur, quàm pars, cui à pauciorib. Ex quo patet, unumquod umbroſum circa radium ſibi <lb/>ṕerpendiculariter incidentem pl{us} ιlluminari.</head> <p> <s xml:id="echoid-s24499" xml:space="preserve">Sit corpus luminoſum circulus a b g:</s> <s xml:id="echoid-s24500" xml:space="preserve"> cuius centrum ſit punctũ d:</s> <s xml:id="echoid-s24501" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s24502" xml:space="preserve"> arcus ſui cõuexitate reſpi-<lb/>ciens corpus illuminandum (qui a b g) diuiſus per æqualia in puncto b:</s> <s xml:id="echoid-s24503" xml:space="preserve"> & ducatur linea z c contin-<lb/>gens circulum in puncto b per 17 p 3:</s> <s xml:id="echoid-s24504" xml:space="preserve"> & in puncto g contingat circulum linea i k, & in puncto a linea <lb/>t h:</s> <s xml:id="echoid-s24505" xml:space="preserve"> ſitq́ue corpus umbroſum arcus k z t i c h:</s> <s xml:id="echoid-s24506" xml:space="preserve"> ducatur <lb/>quoq;</s> <s xml:id="echoid-s24507" xml:space="preserve"> linea d b l à centro corporis luminoſi ad corpus <lb/>umbroſum:</s> <s xml:id="echoid-s24508" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s24509" xml:space="preserve"> hęc perpendicularis ſuper lineã c z, <lb/> <anchor type="figure" xlink:label="fig-0371-01a" xlink:href="fig-0371-01"/> contingẽtem circulũ in puncto b per 18 p 3:</s> <s xml:id="echoid-s24510" xml:space="preserve"> unaquęq;</s> <s xml:id="echoid-s24511" xml:space="preserve"> <lb/>igitur partium arcus h t illuminatur à puncto a corpo <lb/>ris luminoſi per 19 huius:</s> <s xml:id="echoid-s24512" xml:space="preserve"> punctus ergo lilluminatur <lb/>à puncto a.</s> <s xml:id="echoid-s24513" xml:space="preserve"> Similiterq́;</s> <s xml:id="echoid-s24514" xml:space="preserve"> arcus k i illuminatur à puncto <lb/>g:</s> <s xml:id="echoid-s24515" xml:space="preserve"> ergo & punctus l, totusq́;</s> <s xml:id="echoid-s24516" xml:space="preserve"> arcus z c illuminatur à pũ <lb/>cto b:</s> <s xml:id="echoid-s24517" xml:space="preserve"> ergo & punctus l:</s> <s xml:id="echoid-s24518" xml:space="preserve"> punctus itaq;</s> <s xml:id="echoid-s24519" xml:space="preserve"> l illuminatur à <lb/>tribus punctis corporis luminoſi, ſcilicet punctis a, b, <lb/>g, & totus arcus t i eſt communis illuminationi trium <lb/>punctorum a, b, g:</s> <s xml:id="echoid-s24520" xml:space="preserve"> arcus uerò c i eſt cõmunis duabus <lb/>tãtùm illuminationib.</s> <s xml:id="echoid-s24521" xml:space="preserve"> punctorum a & b:</s> <s xml:id="echoid-s24522" xml:space="preserve"> arcus quoq;</s> <s xml:id="echoid-s24523" xml:space="preserve"> <lb/>z t eſt ſimiliter cõmunis duabus tãtũ illuminationib.</s> <s xml:id="echoid-s24524" xml:space="preserve"> <lb/>punctorum b & g:</s> <s xml:id="echoid-s24525" xml:space="preserve"> quoniam eſt cõmunis arcubus z c <lb/>& k i ab illis duobus punctis illuminatis:</s> <s xml:id="echoid-s24526" xml:space="preserve"> arcus uerò <lb/>h c illuminatur tãtùm ab uno puncto a, & arcus z k ab <lb/>uno tantũ puncto g.</s> <s xml:id="echoid-s24527" xml:space="preserve"> Illuminatio ergo arcus ti triplicatum habet lumen, quod arcus z t & c i habent <lb/>duplum, & quod arcus c z & z k habent ſimplũ:</s> <s xml:id="echoid-s24528" xml:space="preserve"> magis ergo omnib.</s> <s xml:id="echoid-s24529" xml:space="preserve"> alijs arcubus illuminatur arcus <lb/>ti, qui eſt circa lineam perpendicularẽ, quæ eſt l d:</s> <s xml:id="echoid-s24530" xml:space="preserve"> & illuminatio duorum arcuũ z t & c i eſt ęqualis:</s> <s xml:id="echoid-s24531" xml:space="preserve"> <lb/>quoniam à totidem punctis corporis luminoſi illuminatur unus ut alius:</s> <s xml:id="echoid-s24532" xml:space="preserve"> ipſorũ uerò amborum il-<lb/>luminatio maior eſt illuminatione duorum arcuum c h & z k:</s> <s xml:id="echoid-s24533" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s24534" xml:space="preserve"> ſemper proportio exceſſus illu <lb/>minationis ſecundum numerum punctorum corporis illuminantis, reſpicientis partem corporis <lb/>illuminati.</s> <s xml:id="echoid-s24535" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s24536" xml:space="preserve"> exijs, quoniã ſemper id, quod eſt propinquius perpendiculari, fortius illumi <lb/>natur illo, quod eſt remotius ab eadem perpendiculari:</s> <s xml:id="echoid-s24537" xml:space="preserve"> ſuper ipſum namq;</s> <s xml:id="echoid-s24538" xml:space="preserve"> plus luminis cadit, quòd <lb/>à pluribus luminoſis partibus illuminatur.</s> <s xml:id="echoid-s24539" xml:space="preserve"> Quod enim nunc demonſtratum eſt in arcu k h, ſimiliter <lb/>accidit in alio corporum quocunq;</s> <s xml:id="echoid-s24540" xml:space="preserve">: exemplificauimus aũt iſtum in corpore concauo, quoniam il-<lb/>lud uidetur plus uniformiter debere illuminari.</s> <s xml:id="echoid-s24541" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24542" xml:space="preserve"/> </p> <div xml:id="echoid-div912" type="float" level="0" n="0"> <figure xlink:label="fig-0371-01" xlink:href="fig-0371-01a"> <variables xml:id="echoid-variables396" xml:space="preserve">t ſ <gap/> z b c <gap/> a k h d</variables> </figure> </div> </div> <div xml:id="echoid-div914" type="section" level="0" n="0"> <head xml:id="echoid-head737" xml:space="preserve" style="it">22. Omne corp{us} umbroſum puncto luminoſo propinqui{us}, illuminatur ab illo puncto forti{us} <lb/>corpore pl{us} diſtante.</head> <p> <s xml:id="echoid-s24543" xml:space="preserve">Sit corpus luminoſum in puncto a:</s> <s xml:id="echoid-s24544" xml:space="preserve"> & corpus illuminatum ſit apud lineã b g:</s> <s xml:id="echoid-s24545" xml:space="preserve"> & copulentur lineę <lb/>a b & a g.</s> <s xml:id="echoid-s24546" xml:space="preserve"> Virtus itaq;</s> <s xml:id="echoid-s24547" xml:space="preserve"> corporis a illuminans corpus b g, illuminat etiã aerẽ mediũ, qui continetur in <lb/>triangulo a b g:</s> <s xml:id="echoid-s24548" xml:space="preserve"> & ducatur linea d e ęquidiſtans lineę b g per 31 p 1:</s> <s xml:id="echoid-s24549" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s24550" xml:space="preserve"> linea b g propinquior corpo-<lb/>ri luminoſo in puncto a exiſtenti ꝗ̃ corpus d e.</s> <s xml:id="echoid-s24551" xml:space="preserve"> Dico, quòd corpus b g fortius illuminatur quã cor-<lb/>pus d e.</s> <s xml:id="echoid-s24552" xml:space="preserve"> Sit enim, ut radius a b cadatin punctum d, & radius a g in punctum e:</s> <s xml:id="echoid-s24553" xml:space="preserve"> & à puncto b ducatur <lb/>ſuper lineam b e linea perpendicularis, quę ſit b u:</s> <s xml:id="echoid-s24554" xml:space="preserve"> & à puncto g perpendicularis, quę ſit g z per 12 p <lb/>1.</s> <s xml:id="echoid-s24555" xml:space="preserve"> Erit ergo per 34 p 1 linea u z ęqualis lineę b g, & linea b u ęqualis lineæ g z.</s> <s xml:id="echoid-s24556" xml:space="preserve"> Ducãtur itaq;</s> <s xml:id="echoid-s24557" xml:space="preserve"> lineæ u a <lb/>& z a:</s> <s xml:id="echoid-s24558" xml:space="preserve"> hæ ergo ſecabunt lineam b g per 2 t 1 huius:</s> <s xml:id="echoid-s24559" xml:space="preserve"> ſecet ergo ipſam linea u a in puncto h, & linea z a <lb/>in puncto t.</s> <s xml:id="echoid-s24560" xml:space="preserve"> Quia ergo uirtus imprimens lumen in corpus b g eſt diffuſa per totum triangulũ a b g:</s> <s xml:id="echoid-s24561" xml:space="preserve"> <lb/>uirtus autem illuminans corpus u z æquale corpori b g, eſt diffuſa ſolùm per trigonum a h t:</s> <s xml:id="echoid-s24562" xml:space="preserve"> & <lb/> <pb o="70" file="0372" n="372" rhead="VITELLONIS OPTICAE"/> quia per 1 p 6 triangulus a b g eſt maior triãgulo a h t, quoniam baſis b g eſt maior baſi h t:</s> <s xml:id="echoid-s24563" xml:space="preserve"> plus itaq;</s> <s xml:id="echoid-s24564" xml:space="preserve"> <lb/>luminis diffuſum eſt in trigono a b g, quàm in trigono a h t:</s> <s xml:id="echoid-s24565" xml:space="preserve"> in quolibet enim iſtorum triangulorum <lb/>puncto eſt lumen ęqualiter diffuſum.</s> <s xml:id="echoid-s24566" xml:space="preserve"> Lumen ergo in-<lb/> <anchor type="figure" xlink:label="fig-0372-01a" xlink:href="fig-0372-01"/> cidens corpori exiſtenti in linea u z, illud corpus debi <lb/>lius illuminat quã corpus b g:</s> <s xml:id="echoid-s24567" xml:space="preserve"> quia paucius ſibi lumen <lb/>incidit:</s> <s xml:id="echoid-s24568" xml:space="preserve"> proportio enim uirtutis luminis incidentis li-<lb/>neę h t ad impreſsionẽ ſuã in corpus u z, eſt minor ꝓ-<lb/>portione uirtutis incidentis lineę b g ad impreſsionẽ <lb/>ſuam in corpus u z per 8 p 5:</s> <s xml:id="echoid-s24569" xml:space="preserve"> quoniam, ut patet ex prę <lb/>miſsis, lumen incidens lineę b g, eſt plus lumine inci-<lb/>dente lineę h t.</s> <s xml:id="echoid-s24570" xml:space="preserve"> Proportio uerò uirtutis incidẽtis lineę <lb/>h t ad impreſsionẽ ſuam in corpus u z, eſt ſicut propor <lb/>tio uirtutis incidentis lineæ b g ad impreſsionẽ ſuam <lb/>in corpus b g per 6 huius:</s> <s xml:id="echoid-s24571" xml:space="preserve"> ergo per 16 p 5 erit permuta <lb/>tim proportio uirtutis perueniẽtis ad lineã h t, ad uir-<lb/>tutẽ peruenientẽ ad lineã b g, ſicut impreſsionis factę <lb/>in corpus u z ad impreſsionẽ factã in corpus b g:</s> <s xml:id="echoid-s24572" xml:space="preserve"> ſed ք <lb/>p̃miſſa lumẽ քueniens ad lineã h t eſt debilius lumine <lb/>քueniẽte ad lineã b g.</s> <s xml:id="echoid-s24573" xml:space="preserve"> Ergo impreſsio քueniẽs à linea <lb/>h t in corpus u z, eſt debilior impreſsiõe քueniẽte à uirtute luminis incidẽtis lineę b g in corpus b g.</s> <s xml:id="echoid-s24574" xml:space="preserve"> <lb/>Corp<emph style="sub">9</emph> itaq;</s> <s xml:id="echoid-s24575" xml:space="preserve"> ꝓpinꝗus corpori luminoſo forti<emph style="sub">9</emph> illuminatur ꝗ̃ remotius ab eodẽ.</s> <s xml:id="echoid-s24576" xml:space="preserve"> Et hoc eſt ꝓpoſitũ.</s> <s xml:id="echoid-s24577" xml:space="preserve"/> </p> <div xml:id="echoid-div914" type="float" level="0" n="0"> <figure xlink:label="fig-0372-01" xlink:href="fig-0372-01a"> <variables xml:id="echoid-variables397" xml:space="preserve">a b h t g d u z e</variables> </figure> </div> </div> <div xml:id="echoid-div916" type="section" level="0" n="0"> <head xml:id="echoid-head738" xml:space="preserve" style="it">23. Puncto remotiori à corpore luminoſo incidunt radij à plurib. pun <lb/>ctis corporis luminoſi, quàm puncto propinquiori.</head> <figure> <variables xml:id="echoid-variables398" xml:space="preserve">g h c e f a b d</variables> </figure> <p> <s xml:id="echoid-s24578" xml:space="preserve">Sit corporis luminoſi circulus a b c, cuius centrum d:</s> <s xml:id="echoid-s24579" xml:space="preserve"> & ducatur perpen <lb/>dicularis d g, in qua ſignentur duo puncta g remotior, & h propinquior.</s> <s xml:id="echoid-s24580" xml:space="preserve"> Di <lb/>co, quòd puncto remotiori, qui eſt g, incidunt radij à plurib.</s> <s xml:id="echoid-s24581" xml:space="preserve"> punctis corpo <lb/>ris luminoſi, ꝗ̃ ipſi puncto h.</s> <s xml:id="echoid-s24582" xml:space="preserve"> Ducãtur enim radij longiſsimi à corpore lumi <lb/>noſo ad punctum g.</s> <s xml:id="echoid-s24583" xml:space="preserve"> Et ſimiliter ducantur radij lõgiſsimi à corpore lumino <lb/>ſo ad punctũ h:</s> <s xml:id="echoid-s24584" xml:space="preserve"> erunt itaq;</s> <s xml:id="echoid-s24585" xml:space="preserve"> per 16 huius illi radij cõtingentes ſphęrã.</s> <s xml:id="echoid-s24586" xml:space="preserve"> Contin <lb/>gant itaq;</s> <s xml:id="echoid-s24587" xml:space="preserve"> radij incidentes puncto g in punctis a & b, & radij incidentes pũ <lb/>cto h, contingant ſphæram in punctis e & f:</s> <s xml:id="echoid-s24588" xml:space="preserve"> palamq́;</s> <s xml:id="echoid-s24589" xml:space="preserve"> per 60 t 1 huius, quo-<lb/>niam puncta contingentiæ e & f cadent intra puncta a & b.</s> <s xml:id="echoid-s24590" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s24591" xml:space="preserve"> pun-<lb/>ctum h ſolum irradiatur à punctis arcus e c f, & non ab alijs:</s> <s xml:id="echoid-s24592" xml:space="preserve"> punctũ uerò g <lb/>irradiatur à punctis arcus a c b, qui eſt maior arcu e c f, patet propoſitũ:</s> <s xml:id="echoid-s24593" xml:space="preserve"> quo <lb/>niam punctũ g illuminabitur à ſuperficie corporis luminoſi, quã per ęqua-<lb/>lia diuidit arcus a c b:</s> <s xml:id="echoid-s24594" xml:space="preserve"> & punctum h illuminabitur à ſuperficie corporis lu-<lb/>minoſi, quã per ęqualia diuidit arcus e c f:</s> <s xml:id="echoid-s24595" xml:space="preserve"> tamen propter radiorum fortitu-<lb/>dinem, quæ conſequitur ipſorum breuitatem, fortius illuminabitur punctũ <lb/>h à paucioribus radijs, quã punctum g à plurib.</s> <s xml:id="echoid-s24596" xml:space="preserve"> multiplicitas enim luminis <lb/>in puncto remotiori eſt ex concurſu radiorum multorum obliquè inciden-<lb/>tium & debilium, ſed in puncto propinquiori fortificatur lux ex breuitate radij, ſecundum quam à <lb/>corpore luminoſo immittitur plus uirtutis.</s> <s xml:id="echoid-s24597" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div917" type="section" level="0" n="0"> <head xml:id="echoid-head739" xml:space="preserve" style="it">24. Omne corp{us} luminoſum min{us} ſpatium, à quo non egreditur, forti{us} illuminat quàm <lb/>ſpatium mai{us} illo.</head> <p> <s xml:id="echoid-s24598" xml:space="preserve">Quod hic proponitur, ſatis patet per exemplũ:</s> <s xml:id="echoid-s24599" xml:space="preserve"> u-<lb/> <anchor type="figure" xlink:label="fig-0372-03a" xlink:href="fig-0372-03"/> na enim candela paruam cameram fortius illuminat <lb/>quã domum uel cameram maiorem:</s> <s xml:id="echoid-s24600" xml:space="preserve"> poteſt tamen i-<lb/>dem figuraliter demonſtrari.</s> <s xml:id="echoid-s24601" xml:space="preserve"> Eſto enim, ut ſit pũctus <lb/>aliquis corporis luminoſi a:</s> <s xml:id="echoid-s24602" xml:space="preserve"> à quo per ſpatiũ magnũ, <lb/>in quo ſit linea b g, diffundantur radij a g, a b, a d:</s> <s xml:id="echoid-s24603" xml:space="preserve"> & ſit <lb/>radius a d perpendicularis ſuper lineam b g:</s> <s xml:id="echoid-s24604" xml:space="preserve"> illumi-<lb/>natur itaq;</s> <s xml:id="echoid-s24605" xml:space="preserve"> ſpatium totum b g ſecundum has lineas <lb/>à puncto a ſibi incidentes.</s> <s xml:id="echoid-s24606" xml:space="preserve"> Abſcindatur itaq;</s> <s xml:id="echoid-s24607" xml:space="preserve"> à linea <lb/>a b linea a e, ut placuerit, & â linea a g abſcindetur li-<lb/>nea a f æqualis lineæ a e:</s> <s xml:id="echoid-s24608" xml:space="preserve"> productaq́;</s> <s xml:id="echoid-s24609" xml:space="preserve"> linea e f ſecet li-<lb/>neam perpẽdicularẽ, quę eſt a d, in puncto h.</s> <s xml:id="echoid-s24610" xml:space="preserve"> Si ergo <lb/>in linea e h f terminetur ſpatium, ne lumen ultrà per-<lb/>tranſeat, erit illud ſpatium minus ſpatio terminato ք <lb/>lineam b d g per 2 p 6.</s> <s xml:id="echoid-s24611" xml:space="preserve"> Omnes autem radij peruenien <lb/>tes ad lineam b g, perueniunt ad lineam e f:</s> <s xml:id="echoid-s24612" xml:space="preserve"> plus ergo <lb/>aggregantur radij in ſpatio e f quã in ſpatio b g:</s> <s xml:id="echoid-s24613" xml:space="preserve"> fortiores ergo fiunt, cum ſint uirtutis plus unitæ:</s> <s xml:id="echoid-s24614" xml:space="preserve"> ma <lb/>gis ergo agunt quã in ſpatio b g, in quo ſunt diffuſiores.</s> <s xml:id="echoid-s24615" xml:space="preserve"> Plus ergo illuminatur ſpatium minus, cùm <lb/>ad eius terminos uirtus luminis terminatur, quàm ſpatium maius illo.</s> <s xml:id="echoid-s24616" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s24617" xml:space="preserve"/> </p> <div xml:id="echoid-div917" type="float" level="0" n="0"> <figure xlink:label="fig-0372-03" xlink:href="fig-0372-03a"> <variables xml:id="echoid-variables399" xml:space="preserve">a e h f b d g</variables> </figure> </div> <pb o="71" file="0373" n="373" rhead="LIBER SECVNDVS."/> </div> <div xml:id="echoid-div919" type="section" level="0" n="0"> <head xml:id="echoid-head740" xml:space="preserve" style="it">25. Omnis axis uel diameter corporis umbroſi non perpendiculariter reſpiciens ſuperficiem <lb/>corporis ſphærici luminoſi: alicui diametro illi{us} corporis æquidιſtat.</head> <p> <s xml:id="echoid-s24618" xml:space="preserve">Sit enim axis uel diameter corporis umbroſi linea a b, non perpendiculariter reſpiciens ſuperfi-<lb/>ciem corporis luminoſi ſphęrici, cuius centrũ ſit punctum c.</s> <s xml:id="echoid-s24619" xml:space="preserve"> Dico, quòd linea a b æquidιſtat alicui <lb/>diametrorũ corporis c.</s> <s xml:id="echoid-s24620" xml:space="preserve"> Ducatur enim linea a c à termino lineę a b ad centrum corporis luminoſi:</s> <s xml:id="echoid-s24621" xml:space="preserve"> & <lb/>ſuper punctum c terminum lineæ a c fiat an-<lb/> <anchor type="figure" xlink:label="fig-0373-01a" xlink:href="fig-0373-01"/> <anchor type="figure" xlink:label="fig-0373-02a" xlink:href="fig-0373-02"/> gulus æqualis angulo b a c per 23 p 1, qui ſit d <lb/>c a, producta linea d c taliter, ut anguli b a c <lb/>& a c d fiant coalterni:</s> <s xml:id="echoid-s24622" xml:space="preserve"> lineę ergo d c & a b ę-<lb/>quidiſtant adinuicem per 27 p 1.</s> <s xml:id="echoid-s24623" xml:space="preserve"> Et quoniam <lb/>linea c d eſt ducta à cẽtro corporis luminoſi:</s> <s xml:id="echoid-s24624" xml:space="preserve"> <lb/>patet, quòd ipſa eſt pars diametri ſphærici il-<lb/>lius corporis.</s> <s xml:id="echoid-s24625" xml:space="preserve"> Producta ergo diametro d c e, <lb/>patet, quòd ipſa æquidiſtat lineæ a b.</s> <s xml:id="echoid-s24626" xml:space="preserve"> Et hoc <lb/>eſt propoſitum.</s> <s xml:id="echoid-s24627" xml:space="preserve"/> </p> <div xml:id="echoid-div919" type="float" level="0" n="0"> <figure xlink:label="fig-0373-01" xlink:href="fig-0373-01a"> <variables xml:id="echoid-variables400" xml:space="preserve">e c d b a</variables> </figure> <figure xlink:label="fig-0373-02" xlink:href="fig-0373-02a"> <variables xml:id="echoid-variables401" xml:space="preserve">a g b e d u f z h</variables> </figure> </div> </div> <div xml:id="echoid-div921" type="section" level="0" n="0"> <head xml:id="echoid-head741" xml:space="preserve" style="it">26. Diametro corporis luminoſi ſphæri-<lb/>ci exiſtente æquali diametro corporis illu-<lb/>minãdi: tantũ ei{us} mediet{as} illuminatur: <lb/>& umbra fit æqualis rei in infinitum pro= <lb/>tenſa. Ariſtarch{us} Sami{us} in libro de ma <lb/>gnitudinib. & interuallis ſolis & lunæ.</head> <p> <s xml:id="echoid-s24628" xml:space="preserve">Eſto corporis illuminantis diameter a g:</s> <s xml:id="echoid-s24629" xml:space="preserve"> <lb/>cuius pars aſpiciens corpus illuminandũ ſit <lb/>a b g:</s> <s xml:id="echoid-s24630" xml:space="preserve"> diameter uerò corporis illuminandi ſit <lb/>d u ęqualis ex hypotheſi, & per pręmiſſam ęquidiſtãs diametro a g:</s> <s xml:id="echoid-s24631" xml:space="preserve"> & ſuperficies illuminata ſit d e u.</s> <s xml:id="echoid-s24632" xml:space="preserve"> <lb/>Dico, quòd d e u eſt medietas ſuperficiei corporis illuminandi.</s> <s xml:id="echoid-s24633" xml:space="preserve"> Ducantur enιm radij a d & g u.</s> <s xml:id="echoid-s24634" xml:space="preserve"> Quia <lb/>itaque diameter a g eſt æqualis & ęquidiſtans diametro d u ք hypotheſim & per pręmιſſam:</s> <s xml:id="echoid-s24635" xml:space="preserve"> palàm, <lb/>quòd radij a d & d u ſunt æquidiſtantes & æquales per 33 p 1:</s> <s xml:id="echoid-s24636" xml:space="preserve"> ergo in infinitum protracti nunquã cõ <lb/>current:</s> <s xml:id="echoid-s24637" xml:space="preserve"> non ergo illuminatur aliqua pars corporis d e u ultra diametrũ d u.</s> <s xml:id="echoid-s24638" xml:space="preserve"> Eius ergo corporis tan <lb/>tùm medietas illuminatur:</s> <s xml:id="echoid-s24639" xml:space="preserve"> protenditur enim umbra in infinitum æqualis diametri cum diametro <lb/>corporis:</s> <s xml:id="echoid-s24640" xml:space="preserve"> & eſt extenſa inter lineas d z & u h, & eſt linea z h ęqualis lineæ d u.</s> <s xml:id="echoid-s24641" xml:space="preserve"> Portio itaque arcus <lb/>d f u, quę eſt medietas totius ſuperficiei corporis d e u:</s> <s xml:id="echoid-s24642" xml:space="preserve"> & linea d z & u h continent umbram æqualẽ <lb/>rei umbroſæ, quæ protenditur in infinitum.</s> <s xml:id="echoid-s24643" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24644" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div922" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables402" xml:space="preserve">e d g b a</variables> </figure> <head xml:id="echoid-head742" xml:space="preserve" style="it">27. Diametro corporis luminoſi ſphærici existẽte maiore dia-<lb/> metro corporis ſphærici illuminandi: pl{us} medietate corporis il- luminatur: & baſis umbræ eſt minor magno circulo corporis il- luminati, concurrens ad punctum unũ retro corp{us}. Ariſtar- ch<emph style="sub">9</emph> Sami{us} in libro de magnitudinib. et interuallis ſolis et lunæ.</head> <p> <s xml:id="echoid-s24645" xml:space="preserve">Sit corpus luminoſum contentum circulo a b:</s> <s xml:id="echoid-s24646" xml:space="preserve"> & ſit corpus um-<lb/>broſum illuminandũ contentũ circulo g d:</s> <s xml:id="echoid-s24647" xml:space="preserve"> & ſit diameter circuli <lb/>a b maior diametro circuli g d:</s> <s xml:id="echoid-s24648" xml:space="preserve"> & ſint radij incidentes a g & b d:</s> <s xml:id="echoid-s24649" xml:space="preserve"> ij <lb/>ergo radij neceſſariò cõcurrent ultra corpus g d.</s> <s xml:id="echoid-s24650" xml:space="preserve"> Si enim nõ cõcur <lb/>rant, tunc ęquidiſtabunt:</s> <s xml:id="echoid-s24651" xml:space="preserve"> neceſſariũ ergo erit diametros a b & g d <lb/>eſſe æquales, quod eſt cõtra hypotheſim:</s> <s xml:id="echoid-s24652" xml:space="preserve"> cõcurrant itaq;</s> <s xml:id="echoid-s24653" xml:space="preserve"> in pũcto <lb/>e:</s> <s xml:id="echoid-s24654" xml:space="preserve"> patet ergo, quòd radij a g & b d nõ tranſeunt terminos diametri <lb/>circuli g d:</s> <s xml:id="echoid-s24655" xml:space="preserve"> ſi enim tranſeãt, palã, cũ illi radij per 16 huius circulum <lb/>g d contingant, quia anguli e g d & e d g erũt recti per 18 p 3.</s> <s xml:id="echoid-s24656" xml:space="preserve"> In triã <lb/>gulo ergo g d e ſunt duo anguli recti, quod eſt impoſsibile & con-<lb/>tra 32 p 1:</s> <s xml:id="echoid-s24657" xml:space="preserve"> palã ergo, quòd radij a e & b e nõ tranſeunt per terminos <lb/>diametri circuli g d, ſed ultra illos cõtingunt ſuperficiẽ corporis il <lb/>luminãdi:</s> <s xml:id="echoid-s24658" xml:space="preserve"> magis ergo medietate corporis illuminatur.</s> <s xml:id="echoid-s24659" xml:space="preserve"> Et quia mi <lb/>nor circulus illius ſphęrici corporis cõtinet umbram, patet, quòd <lb/>baſis umbræ minor eſt magno circulo corporis illuminati.</s> <s xml:id="echoid-s24660" xml:space="preserve"> Quod <lb/>eſt propoſitum.</s> <s xml:id="echoid-s24661" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div923" type="section" level="0" n="0"> <head xml:id="echoid-head743" xml:space="preserve" style="it">28. Diametro corporis luminoſi ſphærici exiſtẽte minore diame <lb/>tro corporis illuminãdi ſphærici: min{us} medietate illuminatur: <lb/>& eſt umbra multò maior corpore illuminato in infinitũ ꝓtẽſa.</head> <p> <s xml:id="echoid-s24662" xml:space="preserve">Sit corpus luminoſum, cuius maior circulus ſit d g:</s> <s xml:id="echoid-s24663" xml:space="preserve"> & corpus il-<lb/>luminãdum, cuius maior circulus ſit a b:</s> <s xml:id="echoid-s24664" xml:space="preserve"> & ſit diameter circuli d g <lb/>minor diametro circuli a b:</s> <s xml:id="echoid-s24665" xml:space="preserve"> concurrent itaque radij g a & b d ultra corpus luminoſum g d perpræ-<lb/> <pb o="72" file="0374" n="374" rhead="VITELLONIS OPTICAE"/> miſſam diametrorum proportionem:</s> <s xml:id="echoid-s24666" xml:space="preserve"> concurrant ergo in puncto e ultra diametrum corporis d g:</s> <s xml:id="echoid-s24667" xml:space="preserve"> ij <lb/>ergo radij non contingunt terminos diametri circuli a b:</s> <s xml:id="echoid-s24668" xml:space="preserve"> quia ſi ſic erunt, ut in pręmiſſa per 16 & 18 <lb/>p 3 trigoni a b e duo anguli recti:</s> <s xml:id="echoid-s24669" xml:space="preserve"> quod eſt impoſsibile:</s> <s xml:id="echoid-s24670" xml:space="preserve"> minus ergo medi<gap/>tate corporis a b illumina-<lb/>tur.</s> <s xml:id="echoid-s24671" xml:space="preserve"> Et quoniam magnus circulus corpo-<lb/> <anchor type="figure" xlink:label="fig-0374-01a" xlink:href="fig-0374-01"/> <anchor type="figure" xlink:label="fig-0374-02a" xlink:href="fig-0374-02"/> ris a b cadit intra umbram, & umbra ultra <lb/>illum protenſa ſemper dilatatur, cum per <lb/>14 t 1 huius radios g a & g b ad illam par-<lb/>tem cõcurrere ſit impoſsibile:</s> <s xml:id="echoid-s24672" xml:space="preserve"> patet, quòd <lb/>umbra extẽdetur in infinitum.</s> <s xml:id="echoid-s24673" xml:space="preserve"> Et hoc eſt <lb/>quod proponitur.</s> <s xml:id="echoid-s24674" xml:space="preserve"> Et per hęc pręmiſſa pe-<lb/>nitus ſimiliter in columnis & pyramidib.</s> <s xml:id="echoid-s24675" xml:space="preserve"> <lb/>poteſt demonſtrari:</s> <s xml:id="echoid-s24676" xml:space="preserve"> idem enim in illis eſt <lb/>demonſtrandi modus.</s> <s xml:id="echoid-s24677" xml:space="preserve"/> </p> <div xml:id="echoid-div923" type="float" level="0" n="0"> <figure xlink:label="fig-0374-01" xlink:href="fig-0374-01a"> <variables xml:id="echoid-variables403" xml:space="preserve">e d g c b a</variables> </figure> <figure xlink:label="fig-0374-02" xlink:href="fig-0374-02a"> <variables xml:id="echoid-variables404" xml:space="preserve">g e f d b c a</variables> </figure> </div> </div> <div xml:id="echoid-div925" type="section" level="0" n="0"> <head xml:id="echoid-head744" xml:space="preserve" style="it">29. Superficiem planam ſuper mediũ <lb/>umbræ erectam, corp{us} umbroſum & <lb/>corp{us} luminoſum, per æqualia diuide-<lb/>re eſt neceſſe.</head> <p> <s xml:id="echoid-s24678" xml:space="preserve">Sit corpus luminoſum a b, cuius cen-<lb/>trum c:</s> <s xml:id="echoid-s24679" xml:space="preserve"> & corpus umbroſum ſit d e, cuius <lb/>centrum f:</s> <s xml:id="echoid-s24680" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s24681" xml:space="preserve"> punctus in medio umbrę, <lb/>qui ſit g:</s> <s xml:id="echoid-s24682" xml:space="preserve"> & copuletur linea f g:</s> <s xml:id="echoid-s24683" xml:space="preserve"> cadet itaq;</s> <s xml:id="echoid-s24684" xml:space="preserve"> <lb/>linea f g in mediũ umbrę:</s> <s xml:id="echoid-s24685" xml:space="preserve"> ſuperficies itaq;</s> <s xml:id="echoid-s24686" xml:space="preserve"> <lb/>erecta ſuper medium umbræ, neceſſariò erit erecta ſuք lineam g f:</s> <s xml:id="echoid-s24687" xml:space="preserve"> tranſit ergo illa ſuperficies cen-<lb/>trum corporis umbroſi & centrum corporis luminoſi:</s> <s xml:id="echoid-s24688" xml:space="preserve"> neceſſariò ergo diuidet illa corpora per ęqua <lb/>lia per ea, quæ oſtenſa ſunt in principio huius.</s> <s xml:id="echoid-s24689" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24690" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div926" type="section" level="0" n="0"> <head xml:id="echoid-head745" xml:space="preserve" style="it">30. Superficiem planam corp{us} luminoſum & corp{us} umbroſum per æqualia diuidentem, ſu <lb/>per medium umbræ erigi eſt neceſſe. Ex quo patet, tot eſſe umbr{as} eiuſdẽ umbroſi corporis, quot <lb/>ipſum opponitur corporib{us} luminoſis.</head> <p> <s xml:id="echoid-s24691" xml:space="preserve">Sit corpus ſuper quod cadit lumen, quod cõtinetur à circulo a b, cuius centrũ eſt punctũ g:</s> <s xml:id="echoid-s24692" xml:space="preserve"> & ſit <lb/>unum corporũ luminoſorũ contentũ à circulo d e, cu-<lb/> <anchor type="figure" xlink:label="fig-0374-03a" xlink:href="fig-0374-03"/> ius centrũ eſt u:</s> <s xml:id="echoid-s24693" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s24694" xml:space="preserve"> aliud corpus luminoſum cõtẽtũ à <lb/>circulo z h, cuius cẽtrũ eſt c:</s> <s xml:id="echoid-s24695" xml:space="preserve"> uidebitur itaq;</s> <s xml:id="echoid-s24696" xml:space="preserve"> umbra op-<lb/>poſita luminoſo corpori d e, contenta à lineis a k, b l, cu <lb/>ius medius punctus ſit m.</s> <s xml:id="echoid-s24697" xml:space="preserve"> Cũ ergo aliqua ſuperficies di <lb/>uiſerit corpus luminoſum & corpus umbroſum per ę-<lb/>qualia:</s> <s xml:id="echoid-s24698" xml:space="preserve"> illa neceſſariò trãſibit ք lineã u g m:</s> <s xml:id="echoid-s24699" xml:space="preserve"> ſecabit ergo <lb/>per ęqualia ipſam umbrã:</s> <s xml:id="echoid-s24700" xml:space="preserve"> quιa perpẽdiculariter erecta <lb/>trãſit per ipſius corporis centrũ, quod eſt punctũ g.</s> <s xml:id="echoid-s24701" xml:space="preserve"> Si-<lb/>militer quoq;</s> <s xml:id="echoid-s24702" xml:space="preserve"> ſuքficies diuidẽs per ęqualia ambo cor-<lb/>pora z h, & a b tranſit per lineam c g ductã per centra il <lb/>lorũ corporum:</s> <s xml:id="echoid-s24703" xml:space="preserve"> ſed eadem pertranſit centrũ umbrę cõ <lb/>tentę ſub lineis a n & b s ſecundum punctũ medium i-<lb/>pſius, qui ſit q.</s> <s xml:id="echoid-s24704" xml:space="preserve"> Illa ergo ſuperficies diuidens corpora <lb/>z h & a b in duo media, diuidet etiã umbram per duo <lb/>ęqualia.</s> <s xml:id="echoid-s24705" xml:space="preserve"> Et quoniã ſuperficies planæ ſecantes corpora <lb/>umbroſa & luminoſa hinc inde ք æqualia ſunt diuiſæ:</s> <s xml:id="echoid-s24706" xml:space="preserve"> <lb/>patet quòd ſecundũ ipſas numerantur etiam & umbrę:</s> <s xml:id="echoid-s24707" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s24708" xml:space="preserve"> Vniuerſaliter enim <lb/>tot erunt umbrę eiuſdem umbroſi corporis, quotipſum opponitur corporibus luminoſis.</s> <s xml:id="echoid-s24709" xml:space="preserve"/> </p> <div xml:id="echoid-div926" type="float" level="0" n="0"> <figure xlink:label="fig-0374-03" xlink:href="fig-0374-03a"> <variables xml:id="echoid-variables405" xml:space="preserve">b d c n <gap/> a b e g k a b <gap/> m q l n</variables> </figure> </div> </div> <div xml:id="echoid-div928" type="section" level="0" n="0"> <head xml:id="echoid-head746" xml:space="preserve" style="it">31. Corporis umbroſi remotioris à corpore luminoſo umbra min{us} umbreſcit: propinquioris <lb/>uerò magis.</head> <p> <s xml:id="echoid-s24710" xml:space="preserve">Quoniam enim, ut patet per 22 huius, omne corpus umbroſum corpori luminoſo propinquius, <lb/>illuminatur fortius corpore plus diſtãte:</s> <s xml:id="echoid-s24711" xml:space="preserve"> patet, quòd umbra corporis propinquioris plus priuat lu-<lb/>minis:</s> <s xml:id="echoid-s24712" xml:space="preserve"> radij quoq;</s> <s xml:id="echoid-s24713" xml:space="preserve"> ipſam terminantes ſunt fortioris luminis:</s> <s xml:id="echoid-s24714" xml:space="preserve"> umbra ergo inter illos radios apparet <lb/>nigrior, & plus umbreſcit:</s> <s xml:id="echoid-s24715" xml:space="preserve"> quoniã radij terminantes illas umbras, ſunt plus luminoſi, propter quod <lb/>etiam plus apparent umbræ in pręſentia illorũ:</s> <s xml:id="echoid-s24716" xml:space="preserve"> Corporis uerò remotioris à corpore luminoſo um-<lb/>bra minus priuat luminis:</s> <s xml:id="echoid-s24717" xml:space="preserve"> radij quoque continentes ipſam umbram ſunt debilioris luminis:</s> <s xml:id="echoid-s24718" xml:space="preserve"> umbra <lb/>ergo inter illos radios apparet debilior:</s> <s xml:id="echoid-s24719" xml:space="preserve"> minus ergo umbreſcit.</s> <s xml:id="echoid-s24720" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24721" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div929" type="section" level="0" n="0"> <head xml:id="echoid-head747" xml:space="preserve" style="it">32. Omnis umbra multiplicata pl{us} umbreſcit.</head> <p> <s xml:id="echoid-s24722" xml:space="preserve">Eſto enim, ut ſit unũ corpus umbroſum obiectũ pluribus corporib.</s> <s xml:id="echoid-s24723" xml:space="preserve"> luminoſis:</s> <s xml:id="echoid-s24724" xml:space="preserve"> palã ergo per 30 <lb/>huius, quoniam tot erunt umbræ eiuſdem corporis umbroſi, quot ipſum opponitur corporib.</s> <s xml:id="echoid-s24725" xml:space="preserve"> lumi <lb/>noſis.</s> <s xml:id="echoid-s24726" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s24727" xml:space="preserve"> accidat, ut umbrę ſe interſecent:</s> <s xml:id="echoid-s24728" xml:space="preserve"> dico, quòd umbra multiplicata plus umbreſcit:</s> <s xml:id="echoid-s24729" xml:space="preserve"> quę-<lb/>libet enim umbrarum aufert aliquod lumen:</s> <s xml:id="echoid-s24730" xml:space="preserve"> multiplicata ergo umbra plura auferet lumina, quæ <lb/> <pb o="73" file="0375" n="375" rhead="LIBER SECVNDVS."/> remanẽt in alijs partibus medij, in quibus umbra nõ multiplicatur, ſed remanet ſimpliciter umbra.</s> <s xml:id="echoid-s24731" xml:space="preserve"> <lb/>Ergo illa ſimplex քfunditur aliquo lumine, qđ ad umbrã multiplicatã nõ քtingit.</s> <s xml:id="echoid-s24732" xml:space="preserve"> Multiplicata ergo <lb/>umbra plus umbreſcit:</s> <s xml:id="echoid-s24733" xml:space="preserve"> quoniã plurimo lumine priuatur locus illius umbræ.</s> <s xml:id="echoid-s24734" xml:space="preserve"> Patet ergo ꝓpoſitum.</s> <s xml:id="echoid-s24735" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div930" type="section" level="0" n="0"> <head xml:id="echoid-head748" xml:space="preserve" style="it">33. Duo corpora, quorum unum obumbrat reliquum ſecũdum ſui medium, in eadem ſuper-<lb/>ficie erecta ſuper corp{us} luminoſum conſiſtere neceſſe eſt: & ſi in eadem ſuperficie, propinqua <lb/>adinuicem conſiſtunt: unum reliquum ſecundum ſui medium obumbrabit.</head> <p> <s xml:id="echoid-s24736" xml:space="preserve">Hoc, quãtùm ad primam partem, patet per 30 huius:</s> <s xml:id="echoid-s24737" xml:space="preserve"> quoniam enim ſuperficies plana corpus lu-<lb/>minoſum & corpus umbroſum per æqualia diuidens eſt erecta ſuper ſuperficiem corporis lumi-<lb/>noſi, & ipſa erigitur ſuper medium umbræ rei umbroſæ:</s> <s xml:id="echoid-s24738" xml:space="preserve"> umbra uerò cadit ſuper lumẽ corporis ob-<lb/>umbrati:</s> <s xml:id="echoid-s24739" xml:space="preserve"> ergo oportet, quòd illud corpus obumbratum ſecundum ſui medium ſit in ſuperficie ere-<lb/>cta ſuper ſuperficiem corporis luminoſi.</s> <s xml:id="echoid-s24740" xml:space="preserve"> Ex hoc etiam patet ſecunda pars præſentis theorematis:</s> <s xml:id="echoid-s24741" xml:space="preserve"> <lb/>quoniam ſi duo corpora propinqua adinuicem ſecundũ ſui partes medias in eadẽ ſuperficie erecta <lb/>ſuper ſuperficiem luminoſi corporis conſiſtunt, unum reliquum obumbrabit:</s> <s xml:id="echoid-s24742" xml:space="preserve"> quoniam remotius à <lb/>lumine, quando fuerit propinquum illi, quod plus accedit ad lumen, cadet in umbra illius, quod eſt <lb/>propinquius lumini:</s> <s xml:id="echoid-s24743" xml:space="preserve"> ut quando idem radius tranſiens uerticem propinquioris, tranſit etiam uerti-<lb/>cem remotioris, uel punctum aliquod, quod ſit altius illo.</s> <s xml:id="echoid-s24744" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24745" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div931" type="section" level="0" n="0"> <head xml:id="echoid-head749" xml:space="preserve" style="it">34. Aequidiſtantia linearum radialium uel ipſarum concurſ{us} non eſt totaliter per ſe ex <lb/>natura radiorum, ſed ex proportione diametri corporis luminoſi ad diametros corporum um-<lb/>broſorum. Ex quo patet, quòd lumen diffunditur uniformiter per aerem circumſtantem.</head> <p> <s xml:id="echoid-s24746" xml:space="preserve">Hoc patet per 17 & 18 huius:</s> <s xml:id="echoid-s24747" xml:space="preserve"> & poteſt ſic exemplariter declarari.</s> <s xml:id="echoid-s24748" xml:space="preserve"> Sit enim corpus luminoſum cir-<lb/>culus a b:</s> <s xml:id="echoid-s24749" xml:space="preserve"> & una linearum radialium ab ipſa egredientium ſit linea a g, & alia linea b g, & cõcurrant <lb/>illæ in pũcto g:</s> <s xml:id="echoid-s24750" xml:space="preserve"> ſit item una linea e u, & alia d z:</s> <s xml:id="echoid-s24751" xml:space="preserve"> & ſint e u & d z æquidi-<lb/> <anchor type="figure" xlink:label="fig-0375-01a" xlink:href="fig-0375-01"/> ſtantes, ſitq́;</s> <s xml:id="echoid-s24752" xml:space="preserve"> corpus unum (cuius diameter ſit minor diametro corpo-<lb/>ris luminoſi) ſuper quod cadit lumen, poſitum inter duas a g & b g ſe <lb/>contingentes, cuius maior circulus ſit ti:</s> <s xml:id="echoid-s24753" xml:space="preserve"> & contingat ipſum linea b g <lb/>in puncto i, & linea a g in puncto t:</s> <s xml:id="echoid-s24754" xml:space="preserve"> & corpus aliud æquale corpori lu-<lb/>minoſo, ſuper quod cadit lumen, ſit poſitum inter duas lineas æquidi-<lb/>ſtantes e u & d z, illud corpus contingentes, cuius diameter ſit k l:</s> <s xml:id="echoid-s24755" xml:space="preserve"> con-<lb/>tingaturq́;</s> <s xml:id="echoid-s24756" xml:space="preserve"> à linea e u in puncto k, & à linea d z in puncto l.</s> <s xml:id="echoid-s24757" xml:space="preserve"> V mbra itaq;</s> <s xml:id="echoid-s24758" xml:space="preserve"> <lb/>proueniens à corpore t i, minuitur & terminatur, & fit pyramidalis per <lb/>27 huius, ideo quia radij contingentes corpus t i, qui ſunt a g, b g, con-<lb/>currunt in puncto g:</s> <s xml:id="echoid-s24759" xml:space="preserve"> umbra ergo corporis t i cõtinetur à duabus lineis <lb/>i g & t g, & ſuperficie corporis t i, quæ eſt à parte g.</s> <s xml:id="echoid-s24760" xml:space="preserve"> Vmbra ergo finitur <lb/>apud punctum g.</s> <s xml:id="echoid-s24761" xml:space="preserve"> Vmbra uerò corporis k l protenſa inter lineas æqui-<lb/>diſtantes l z & k u, ut patet per 26 huius, non terminatur ad aliquod <lb/>punctum:</s> <s xml:id="echoid-s24762" xml:space="preserve"> quoniam illæ lineæ continentes umbram in infinitum pro-<lb/>tractæ non concurrunt.</s> <s xml:id="echoid-s24763" xml:space="preserve"> Si uerò corpus ti motum extra lineas a g & b g <lb/>ponatur intra lineas e u & d z, concurrent lineæ e u & d z, & uariabitur <lb/>umbra ab ipſis prius contenta ſecundum diuerſitatem proportionis <lb/>diametrorum corporis t i, & corporis k l ad diametrum corporis lumi-<lb/>noſi.</s> <s xml:id="echoid-s24764" xml:space="preserve"> Et exhoc patet, quòd radij per ſe non ſunt lineæ neq;</s> <s xml:id="echoid-s24765" xml:space="preserve"> regulares, <lb/>neq;</s> <s xml:id="echoid-s24766" xml:space="preserve"> irregulares, neq;</s> <s xml:id="echoid-s24767" xml:space="preserve"> æquidiſtantes, neq;</s> <s xml:id="echoid-s24768" xml:space="preserve"> concurrentes:</s> <s xml:id="echoid-s24769" xml:space="preserve"> ſed accidit eis <lb/>lineatio per reſpectum ad corpora, quibus incidunt:</s> <s xml:id="echoid-s24770" xml:space="preserve"> & æquidiſtantia <lb/>& concurſus accidunt eis per proportionem diametrorum corporum <lb/>umbroſorum ad diametros corporis luminoſi.</s> <s xml:id="echoid-s24771" xml:space="preserve"> Diffunditur ergo lumẽ <lb/>uniformiter per totum aerem circumſtantem, ita, ut omnis punctus <lb/>aeris, à quo poſsibile eſt produci lineam rectam ad aliquod punctum <lb/>corporis luminoſi, illuminetur à lumine corporis luminoſi, ut patet <lb/>per 19 huius.</s> <s xml:id="echoid-s24772" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24773" xml:space="preserve"/> </p> <div xml:id="echoid-div931" type="float" level="0" n="0"> <figure xlink:label="fig-0375-01" xlink:href="fig-0375-01a"> <variables xml:id="echoid-variables406" xml:space="preserve">g <gap/> t b a d e l k z u</variables> </figure> </div> </div> <div xml:id="echoid-div933" type="section" level="0" n="0"> <head xml:id="echoid-head750" xml:space="preserve" style="it">35. Radij ab uno puncto luminoſi corporis procedentes, ſecũdum <lb/>linearum longitudinem ad æquidiſtantiam ſenſibilem pl{us} accedunt.</head> <p> <s xml:id="echoid-s24774" xml:space="preserve">Eſto, ut à puncto medio corporis luminoſi (quod ſit a) egrediantur radij a b & a g ęquales:</s> <s xml:id="echoid-s24775" xml:space="preserve"> copu-<lb/>letur quoq;</s> <s xml:id="echoid-s24776" xml:space="preserve"> baſis b g, & ducatur linea d e ſecans trigonum a b g citra medium ſui lateris a g æquidi-<lb/>ſtanter baſi b g per 10 & 31 p 1:</s> <s xml:id="echoid-s24777" xml:space="preserve"> protrahaturq́;</s> <s xml:id="echoid-s24778" xml:space="preserve"> à puncto a linea a z perpendiculariter ſuper baſim b g <lb/>per 12 p 1, quæ ſecet lineam d e in puncto u:</s> <s xml:id="echoid-s24779" xml:space="preserve"> diuidaturq́;</s> <s xml:id="echoid-s24780" xml:space="preserve"> linea e g in duo æqualia in puncto h per 10 <lb/>p 1, & linea d b in puncto t:</s> <s xml:id="echoid-s24781" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s24782" xml:space="preserve"> linea h t:</s> <s xml:id="echoid-s24783" xml:space="preserve"> linea ergo h t erit æquidiſtans baſi g b per 2 p 6:</s> <s xml:id="echoid-s24784" xml:space="preserve"> ſeca-<lb/>bit ergo lineam u z per 2 t 1 huius:</s> <s xml:id="echoid-s24785" xml:space="preserve"> ſit punctus ſectionis k.</s> <s xml:id="echoid-s24786" xml:space="preserve"> Ducãtur item à punctis e, d, h, t lineæ per-<lb/>pendiculares ſuper baſim b g:</s> <s xml:id="echoid-s24787" xml:space="preserve"> quæ ſint e l, d m, h n, t s:</s> <s xml:id="echoid-s24788" xml:space="preserve"> ſecabit quoq;</s> <s xml:id="echoid-s24789" xml:space="preserve"> perpendicularis e l lineam h t:</s> <s xml:id="echoid-s24790" xml:space="preserve"> <lb/>ſit punctus ſectionis q, & punctus ſectionis linearum d m & h t ſit f:</s> <s xml:id="echoid-s24791" xml:space="preserve"> erit ergo linea q f æqualis lineæ <lb/>e d per 34 p 1:</s> <s xml:id="echoid-s24792" xml:space="preserve"> patet ergo, quòd linea h t eſt maior quàm linea e d.</s> <s xml:id="echoid-s24793" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s24794" xml:space="preserve"> trigona a u e & e h q ſunt <lb/>æquiangula per 29 & 32 p 1:</s> <s xml:id="echoid-s24795" xml:space="preserve"> erunt per 4 p 6 latera ipſorum proportionalia.</s> <s xml:id="echoid-s24796" xml:space="preserve"> Quia ergo, ut patuit ſu-<lb/>prà, linea a e eſt maior quàm linea e h:</s> <s xml:id="echoid-s24797" xml:space="preserve"> erit ergo linea e u maior quàm linea h q:</s> <s xml:id="echoid-s24798" xml:space="preserve"> ſed linea h t eſt maior <lb/>quàm linea e d, ut præoſtenſum eſt:</s> <s xml:id="echoid-s24799" xml:space="preserve"> ergo per 9 t 1 huius maior eſt proportio lineæ e u ad lineam e d, <lb/> <pb o="74" file="0376" n="376" rhead="VITELLONIS OPTICAE"/> quàm lineæ h q ad lineam h t:</s> <s xml:id="echoid-s24800" xml:space="preserve"> eſt enim proportio lineæ e u ad lineam e d, ſicut lineæ h k ad lineam <gap/> <lb/>t per 4 p 6 & per 11 & 16 p 5:</s> <s xml:id="echoid-s24801" xml:space="preserve"> ſed linea h q eſt pars li-<lb/> <anchor type="figure" xlink:label="fig-0376-01a" xlink:href="fig-0376-01"/> neæ h k:</s> <s xml:id="echoid-s24802" xml:space="preserve"> ergo per 8 p 5 minor eſt proportio h q ad h t, <lb/>quàm h k ad h t.</s> <s xml:id="echoid-s24803" xml:space="preserve"> Minor eſt ergo proportio lineę h q ad <lb/>h t, quàm e u ad e d.</s> <s xml:id="echoid-s24804" xml:space="preserve"> Eodemq́;</s> <s xml:id="echoid-s24805" xml:space="preserve"> modo demonſtrãdum, <lb/>quod lineæ g n ad lineã g b minor eſt proportio, quã <lb/>lineæ h q ad lineã h t:</s> <s xml:id="echoid-s24806" xml:space="preserve"> exceſſus itaq;</s> <s xml:id="echoid-s24807" xml:space="preserve"> baſis g b ſuper ba-<lb/>ſim h t eſt minor exceſſu baſis h t ſuper baſim d e:</s> <s xml:id="echoid-s24808" xml:space="preserve"> & <lb/>quãtò baſes ſunt remotiores à puncto a corporis lu-<lb/>minoſi, tantò exceſſus remotiorum baſium ſuper ba-<lb/>ſes uiciniores plus minuẽtur.</s> <s xml:id="echoid-s24809" xml:space="preserve"> Palàm ergo, quia in re-<lb/>motiori diſtantia radij quaſi ad æquidiſtantiam plus <lb/>procedunt:</s> <s xml:id="echoid-s24810" xml:space="preserve"> & cũ quantitas exceſſus baſium ſit quan-<lb/>titatis non ſenſibilis:</s> <s xml:id="echoid-s24811" xml:space="preserve"> tunc lineæ radiales erunt quaſi <lb/>æquidiſtãtes.</s> <s xml:id="echoid-s24812" xml:space="preserve"> Quoniam enim linea b g ſenſibiliter nõ <lb/>excedit lineã h t:</s> <s xml:id="echoid-s24813" xml:space="preserve"> tunc erunt h g & t b radij quaſi ęqui-<lb/>diſtantes ſecũdum ſenſum.</s> <s xml:id="echoid-s24814" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s24815" xml:space="preserve"> Et <lb/>fortè ad iſtud multũ cooperatur proprietas radiorũ, <lb/>quę ſemper, ut poteſt, approximat ſuæ perpẽdiculari:</s> <s xml:id="echoid-s24816" xml:space="preserve"> <lb/>propter quod radij omniũ punctorũ totius corporis <lb/>luminoſi ſemper concurrunt in quolibet puncto cor-<lb/>poris illuminãdi:</s> <s xml:id="echoid-s24817" xml:space="preserve"> & ſic cõſtituunt pyramidẽ radialẽ.</s> <s xml:id="echoid-s24818" xml:space="preserve"/> </p> <div xml:id="echoid-div933" type="float" level="0" n="0"> <figure xlink:label="fig-0376-01" xlink:href="fig-0376-01a"> <variables xml:id="echoid-variables407" xml:space="preserve">a e u d h q k f t g n l z m s b</variables> </figure> </div> </div> <div xml:id="echoid-div935" type="section" level="0" n="0"> <head xml:id="echoid-head751" xml:space="preserve" style="it">36. Lumine incidente per feneſtram ſuper cor-<lb/>p{us} oppoſitum ſolidũ: erit luminis perimeter am-<lb/>plior perimetro feneſtræ.</head> <p> <s xml:id="echoid-s24819" xml:space="preserve">Eſto corpus luminoſum, cuius centrum a:</s> <s xml:id="echoid-s24820" xml:space="preserve"> & circulus magnus d e g:</s> <s xml:id="echoid-s24821" xml:space="preserve"> & ſit diameter feneſtræ b c:</s> <s xml:id="echoid-s24822" xml:space="preserve"> <lb/>ſitq́;</s> <s xml:id="echoid-s24823" xml:space="preserve"> linea t z in ſuperficie corporis ſolidi oppoſita lumini, cui incidit <lb/> <anchor type="figure" xlink:label="fig-0376-02a" xlink:href="fig-0376-02"/> radius:</s> <s xml:id="echoid-s24824" xml:space="preserve"> producantur quoq;</s> <s xml:id="echoid-s24825" xml:space="preserve"> lineæ radiales tangentes peripheriã fene-<lb/>ſtræ:</s> <s xml:id="echoid-s24826" xml:space="preserve"> quæ ſint e b & g c:</s> <s xml:id="echoid-s24827" xml:space="preserve"> hæ itaq;</s> <s xml:id="echoid-s24828" xml:space="preserve"> lineæ ſecabunt ſe in aliqua parte me-<lb/>dij:</s> <s xml:id="echoid-s24829" xml:space="preserve"> ſit pũctus cõmunis ſectionis f:</s> <s xml:id="echoid-s24830" xml:space="preserve"> & hæ lineæ productę incidãt ſuper-<lb/>ficiei corporis oppoſitæ lumini:</s> <s xml:id="echoid-s24831" xml:space="preserve"> cadatq́;</s> <s xml:id="echoid-s24832" xml:space="preserve"> linea e b in punctũ z, & linea <lb/>g c in punctũ t.</s> <s xml:id="echoid-s24833" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s24834" xml:space="preserve"> in trigono f t z, latus t z eſt maius latere b c:</s> <s xml:id="echoid-s24835" xml:space="preserve"> <lb/>quoniam trigonum f t z maius eſt trigono f c b.</s> <s xml:id="echoid-s24836" xml:space="preserve"> Et quoniã per omnem <lb/>punctum peripheriæ feneſtræ ſic incidũt radij ſe ſecãtes:</s> <s xml:id="echoid-s24837" xml:space="preserve"> ideo quòd à <lb/>quolibet pũcto corporis luminoſi in totam feneſtrã fit miſsio luminis <lb/>ք 20 huius:</s> <s xml:id="echoid-s24838" xml:space="preserve"> palàm, quoniã perimeter luminis incidẽtis corpori ſolido <lb/>oppoſito feneſtræ, eſt maior perimetro feneſtræ.</s> <s xml:id="echoid-s24839" xml:space="preserve"> Et hoc ꝓponebatur.</s> <s xml:id="echoid-s24840" xml:space="preserve"/> </p> <div xml:id="echoid-div935" type="float" level="0" n="0"> <figure xlink:label="fig-0376-02" xlink:href="fig-0376-02a"> <variables xml:id="echoid-variables408" xml:space="preserve">z t b c f g e a d</variables> </figure> </div> </div> <div xml:id="echoid-div937" type="section" level="0" n="0"> <head xml:id="echoid-head752" xml:space="preserve" style="it">37. Ad centrũ circularis for aminis radio à centro corporis lu-<lb/>minoſi perpẽdiculariter incidẽte: lumen in ſuperficie denſi corporis <lb/>æquidiſtante ſuperficiei for aminis eſt uerè circulare.</head> <p> <s xml:id="echoid-s24841" xml:space="preserve">Sit circulus foraminis a b g d, cuius cẽtrum e:</s> <s xml:id="echoid-s24842" xml:space="preserve"> cui ſit ęquidiſtans ſu-<lb/>perficies ſolidi corporis f h k l:</s> <s xml:id="echoid-s24843" xml:space="preserve"> & erigatur à centro e linea e z, perpendiculariter ſuper ſuperficiem <lb/>a b g d circuli:</s> <s xml:id="echoid-s24844" xml:space="preserve"> in quocunq;</s> <s xml:id="echoid-s24845" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s24846" xml:space="preserve"> puncto lineæ e z ſit <lb/> <anchor type="figure" xlink:label="fig-0376-03a" xlink:href="fig-0376-03"/> centrum corporis luminoſi, dico, quòd lumen inci-<lb/>dẽs ſuperficiei f h k l, eſt uerè circulare.</s> <s xml:id="echoid-s24847" xml:space="preserve"> Palàm enim <lb/>per 65 t 1 huius, quoniam omnes lineę z a, z b, z g, z d <lb/>ductæ à polo z ad circumferentiam, ſunt æquales, & <lb/>æquales angulos cõtinent cũ linea e z per 8 p 1.</s> <s xml:id="echoid-s24848" xml:space="preserve"> Pro-<lb/>ducatur itaq;</s> <s xml:id="echoid-s24849" xml:space="preserve"> linea z e ultra punctũ e ad ſuperficiem <lb/>æquidιſtãtem circulo foraminis, quæ eſt f h k l:</s> <s xml:id="echoid-s24850" xml:space="preserve"> inci-<lb/>detq́;</s> <s xml:id="echoid-s24851" xml:space="preserve"> perpendiculariter ſuper illam per 14 p 11:</s> <s xml:id="echoid-s24852" xml:space="preserve"> ſit ut <lb/>incidat in punctum m:</s> <s xml:id="echoid-s24853" xml:space="preserve"> producaturq́;</s> <s xml:id="echoid-s24854" xml:space="preserve"> linea z b ad ſu-<lb/>perficiem f h k l in punctum k, & linea z a in punctũ <lb/>f, & linea z d in punctum h, & linea z g in punctum l:</s> <s xml:id="echoid-s24855" xml:space="preserve"> <lb/>eruntq́;</s> <s xml:id="echoid-s24856" xml:space="preserve"> lineæ a f, b k, d h, g l per 25 t 1 huius æquales <lb/>propter æquidiſtantiam ſuperficierum & æqualita-<lb/>tom angulorum:</s> <s xml:id="echoid-s24857" xml:space="preserve"> tota ergo linea z f erit æqualis toti <lb/>lineæ z h:</s> <s xml:id="echoid-s24858" xml:space="preserve"> & z k æqualis lineæ z l.</s> <s xml:id="echoid-s24859" xml:space="preserve"> Ducantur quoq;</s> <s xml:id="echoid-s24860" xml:space="preserve"> li-<lb/>neæ f m, h m, k m, l m.</s> <s xml:id="echoid-s24861" xml:space="preserve"> In trigono itaq;</s> <s xml:id="echoid-s24862" xml:space="preserve"> f m z baſis f m <lb/>erit æqualis baſi h m trigoni h m z per 4 p 1:</s> <s xml:id="echoid-s24863" xml:space="preserve"> eodemq́;</s> <s xml:id="echoid-s24864" xml:space="preserve"> modo erit linea k m ęqualis lineæ h m, & linea <lb/>l m æqualis lineæ k m.</s> <s xml:id="echoid-s24865" xml:space="preserve"> Palàm ergo per 9 p 3, quoniam ſuperficies f h k l eſt circularis:</s> <s xml:id="echoid-s24866" xml:space="preserve"> & ipſa eſt, ad <lb/>quam terminantur radij luminis incidentis per feneſtram a b g d:</s> <s xml:id="echoid-s24867" xml:space="preserve"> quoniam de omnibus alijs lineis <lb/>eadem eſt demonſtratio.</s> <s xml:id="echoid-s24868" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24869" xml:space="preserve"/> </p> <div xml:id="echoid-div937" type="float" level="0" n="0"> <figure xlink:label="fig-0376-03" xlink:href="fig-0376-03a"> <variables xml:id="echoid-variables409" xml:space="preserve"><gap/> a z d h e b m g k l</variables> </figure> </div> <pb o="75" file="0377" n="377" rhead="LIBER SECVNDVS."/> </div> <div xml:id="echoid-div939" type="section" level="0" n="0"> <head xml:id="echoid-head753" xml:space="preserve" style="it">38. Per centrũ circularis foraminis radio luminoſo obliquè incidẽte ſuperficiei denſi corporis <lb/>ſubſtratæ ſuperficiei for aminis: lumẽ incidẽs erit figuræ ſectionis pyramidalis, cui{us} maior dia-<lb/>meter erit in ſuperficie erecta ſuper ſuperficiem feneſtræ, & ſuper ſuperficiẽ corporis ſubſtrati.</head> <p> <s xml:id="echoid-s24870" xml:space="preserve">Eſto foramen circulare a b c d, cuius centrum e:</s> <s xml:id="echoid-s24871" xml:space="preserve"> cui ſit ſuperficies æquidiſtans h m k l:</s> <s xml:id="echoid-s24872" xml:space="preserve"> & ſit f cen-<lb/>trum corporis luminoſi:</s> <s xml:id="echoid-s24873" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s24874" xml:space="preserve"> primò ut linea f e obliquè cadat ſuper ſuperficiem circuli a b c d:</s> <s xml:id="echoid-s24875" xml:space="preserve"> hæc <lb/>itaq;</s> <s xml:id="echoid-s24876" xml:space="preserve"> producta incidet ſuperficiei h m k l ſimiliter obliquè propter æquidiſtantiam ſuperficierũ, ar-<lb/>gumento 23 t 1 huius:</s> <s xml:id="echoid-s24877" xml:space="preserve"> incidat itaq;</s> <s xml:id="echoid-s24878" xml:space="preserve"> in punctum g:</s> <s xml:id="echoid-s24879" xml:space="preserve"> & ducatur linea a e b diameter circuli:</s> <s xml:id="echoid-s24880" xml:space="preserve"> ſit itaq;</s> <s xml:id="echoid-s24881" xml:space="preserve"> an-<lb/>gulus a e f acutus:</s> <s xml:id="echoid-s24882" xml:space="preserve"> erit ergo per 13 p 1 angulus b e f obtuſus:</s> <s xml:id="echoid-s24883" xml:space="preserve"> ducãtur ergo lineę f a, f b.</s> <s xml:id="echoid-s24884" xml:space="preserve"> Et quia quadra-<lb/>tum lineæ f a ualet minus duobus quadratis linearum e f & e a, per 13 p 2, & quadratum lineæ b f eſt <lb/>maius quadrato lineæ f e, & quadrato lineæ b e per <lb/> <anchor type="figure" xlink:label="fig-0377-01a" xlink:href="fig-0377-01"/> 12 p 2:</s> <s xml:id="echoid-s24885" xml:space="preserve"> quadratũ uerò lineæ b e æquale eſt quadrato <lb/>lineæ a e:</s> <s xml:id="echoid-s24886" xml:space="preserve"> quia ſunt æquales ſemidiametri:</s> <s xml:id="echoid-s24887" xml:space="preserve"> & quadra <lb/>tum lineæ f e eſt commune:</s> <s xml:id="echoid-s24888" xml:space="preserve"> patet, quòd quadratum <lb/>lineæ f b eſt maius quadrato lineæ f a:</s> <s xml:id="echoid-s24889" xml:space="preserve"> ergo linea f b <lb/>eſt maior quàm linea f a:</s> <s xml:id="echoid-s24890" xml:space="preserve"> productisq́;</s> <s xml:id="echoid-s24891" xml:space="preserve"> lineis f a & f b <lb/>ad ſuperficiem h m k l:</s> <s xml:id="echoid-s24892" xml:space="preserve"> ſi linea f a incidat ad pũctum <lb/>m, & linea f b ad punctum l:</s> <s xml:id="echoid-s24893" xml:space="preserve"> erit linea f l maior quàm <lb/>linea f m per eadem, quæ prius:</s> <s xml:id="echoid-s24894" xml:space="preserve"> copulatisq́;</s> <s xml:id="echoid-s24895" xml:space="preserve"> lineis l g <lb/>& m g ad punctum g, cui incidit radius trãſiens cen-<lb/>trum foraminis feneſtræ:</s> <s xml:id="echoid-s24896" xml:space="preserve"> erit quoq;</s> <s xml:id="echoid-s24897" xml:space="preserve"> per 2 p 6 & per <lb/>11 p 5 proportio lineæ l g ad lineam b e, ſicut lineæ g <lb/>m ad lineam e a:</s> <s xml:id="echoid-s24898" xml:space="preserve"> quoniã utrarumq;</s> <s xml:id="echoid-s24899" xml:space="preserve"> illarum propor-<lb/>tio eſt adinuicẽ, ſicut lineæ g f ad lineam f e:</s> <s xml:id="echoid-s24900" xml:space="preserve"> eſt ergo <lb/>per 16 p 5 proportio lineæ l g ad lineam m g, ſicut li-<lb/>neæ b e ad lineam e a:</s> <s xml:id="echoid-s24901" xml:space="preserve"> ſed linea b e eſt æqualis lineæ <lb/>e a:</s> <s xml:id="echoid-s24902" xml:space="preserve"> ergo linea l g eſt ęqualis lineæ g m.</s> <s xml:id="echoid-s24903" xml:space="preserve"> Ducatur tunc <lb/>c d diameter ſuper a b diametrum orthogonaliter, & continuentur lineæ f c, f d:</s> <s xml:id="echoid-s24904" xml:space="preserve"> producanturq́;</s> <s xml:id="echoid-s24905" xml:space="preserve"> ad <lb/>ſuperficiem h m k l in puncta h & k:</s> <s xml:id="echoid-s24906" xml:space="preserve"> & ducatur linea h g k.</s> <s xml:id="echoid-s24907" xml:space="preserve"> Et quoniam ſuperficies, in qua ſunt lineæ <lb/>f e & a b, ſola eſt erecta ſuper circulum feneſtræ, quoniam omnes alię ſuperficies, in quibus eſt linea <lb/>f e, incidunt illi ſuperficiei obliquè (ſic enim accipimus lineam a b) erit ergo ſuperficies a f b erecta <lb/>ſuper ſuperficiẽ circuli feneſtræ.</s> <s xml:id="echoid-s24908" xml:space="preserve"> Palàm ergo, quia angulus f e d eſt æqualis angulo f e c:</s> <s xml:id="echoid-s24909" xml:space="preserve"> eſt ergo per <lb/>4 p 1 linea f d æqualis lineæ f c:</s> <s xml:id="echoid-s24910" xml:space="preserve"> ergo, ut prius, erit linea h g æqualis k g, & linea f h æqualis lineæ f k, <lb/>& f g eſt communis:</s> <s xml:id="echoid-s24911" xml:space="preserve"> & quia linea h k eſt perpendicularis ſuper lineam m l, & ſuper lineam f g:</s> <s xml:id="echoid-s24912" xml:space="preserve"> palàm <lb/>per 4 p 11, quòd linea h g eſt perpendicularis ſuper ſuperficiem, in qua ſunt lineæ f g & m g:</s> <s xml:id="echoid-s24913" xml:space="preserve"> ergo per <lb/>18 p 11 erit ſuperficies h m k l erecta ſuper ſuperficiem f m g:</s> <s xml:id="echoid-s24914" xml:space="preserve"> ergo & ſuperficies f m g eſt erecta ſuper <lb/>ſuperficiem h m k l.</s> <s xml:id="echoid-s24915" xml:space="preserve"> Imaginetur ergo à puncto g termino axis, qui eſt f g, circumduci pyramidi illu-<lb/>minationis circulus per 102 t 1 huius:</s> <s xml:id="echoid-s24916" xml:space="preserve"> erit ergo per 100 & 89 t 1 huius axis f g erectus ſuper illum cir-<lb/>culum, & ipſe eſt obliquus ſuper ſuperficiẽ h m k l:</s> <s xml:id="echoid-s24917" xml:space="preserve"> erit ergo per 103 t 1 huius linea h m k l ſectio py-<lb/>ramidalis, cuius maior diameter erit in ſuperficie f m l erecta ſuper ſuperficiem h m k l.</s> <s xml:id="echoid-s24918" xml:space="preserve"> Patet ergo <lb/>propoſitum.</s> <s xml:id="echoid-s24919" xml:space="preserve"> Et ſi ſuperficies feneſtræ circularis ſit baſis pyramidis illuminationis, ita quòd cẽtrum <lb/>corporis luminoſi ſit polus circuli feneſtræ, & axis erectus ſit ſuper ſuperficiẽ feneſtræ, ſuperficies <lb/>uerò ſolidi corporis excipientis radios luminis, non fuerit æquidiſtans ſuperficiei feneſtræ:</s> <s xml:id="echoid-s24920" xml:space="preserve"> adhuc <lb/>erit figura luminis ſectio pyramidalis:</s> <s xml:id="echoid-s24921" xml:space="preserve"> quod eſt præmiſſo modo demonſtrandũ:</s> <s xml:id="echoid-s24922" xml:space="preserve"> ducta enim per 102 <lb/>t 1 huius à pũcto l termino longioris radij, qui eſt f l, ſuperficie æquidiſtante ſuperficiei feneſtræ:</s> <s xml:id="echoid-s24923" xml:space="preserve"> pa-<lb/>tet per 100 t 1 huius, quòd illa ſuperficies ſecabit pyramidẽ illuminationis ſecundũ circulum, qui ſit <lb/>1 p q.</s> <s xml:id="echoid-s24924" xml:space="preserve"> Ergo ſuperficies h m k l ſecat ipſam ſecundũ pyramidalem ſectionem.</s> <s xml:id="echoid-s24925" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24926" xml:space="preserve"/> </p> <div xml:id="echoid-div939" type="float" level="0" n="0"> <figure xlink:label="fig-0377-01" xlink:href="fig-0377-01a"> <variables xml:id="echoid-variables410" xml:space="preserve">p q m a f c e d h g k b <gap/></variables> </figure> </div> </div> <div xml:id="echoid-div941" type="section" level="0" n="0"> <head xml:id="echoid-head754" xml:space="preserve" style="it">39. Omne lumen per foramina angularia incidens rotundatur.</head> <p> <s xml:id="echoid-s24927" xml:space="preserve">Quod hic proponitur, patet per 35 huius.</s> <s xml:id="echoid-s24928" xml:space="preserve"> Quoniam enim omnes radij ab uno puncto luminoſi <lb/>corporis procedentes ſecundum linearum longitudinem ad æquidiſtantiam ſenſibilem plus acce-<lb/>dunt:</s> <s xml:id="echoid-s24929" xml:space="preserve"> patet, quòd radij ſecundum foraminum angularium diſpoſitionem ipſis angulis incidẽtes, ſe <lb/>applicant æquidiſtantiæ radij perpẽdiculariter uel circa ſuperficiei foraminis incidentis:</s> <s xml:id="echoid-s24930" xml:space="preserve"> retrahunt <lb/>ergo ſe ab angularitate, & ſic lumen ſuperficiei foramini obiectæ incidẽs incipit rotundari.</s> <s xml:id="echoid-s24931" xml:space="preserve"> Et quo-<lb/>niam, ut patet per 20 huius, à puncto cuiuslibet corporis luminoſi lumen diffunditur ſecundum o-<lb/>mnem lineam, quæ ab illo pũcto ad oppoſitam ſuperficiẽ duci poteſt:</s> <s xml:id="echoid-s24932" xml:space="preserve"> omnes enim illi radij in quo-<lb/>libet puncto medij concurrunt:</s> <s xml:id="echoid-s24933" xml:space="preserve"> patet, quòd ipſi in quolibet puncto ſe interſecent, & radij inferio-<lb/>rum punctorum ipſius corporis luminoſi in punctis linearũ feneſtrę alios radios ſuperiorum pun-<lb/>ctorum ſecant, & ultrà protenduntur:</s> <s xml:id="echoid-s24934" xml:space="preserve"> & ſic lumen huiuſmodi feneſtras pertranſiens rotundatur:</s> <s xml:id="echoid-s24935" xml:space="preserve"> <lb/>quod non adeò accideret, ſi ſolùm ab uno puncto luminoſi corporis egrederentur radij feneſtram <lb/>penetrantes.</s> <s xml:id="echoid-s24936" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s24937" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div942" type="section" level="0" n="0"> <head xml:id="echoid-head755" xml:space="preserve" style="it">40. Radio luminoſo medio puncto foraminis quadrati perpendiculariter incidente: lumen <lb/>ſuperficiei corporis æquidiſtantis ſuperficiei for aminis incidens, eſt quadr atum ad circulaxit a-<lb/>tem aliquam accedens.</head> <p> <s xml:id="echoid-s24938" xml:space="preserve">Sit centrum corporis luminoſi e:</s> <s xml:id="echoid-s24939" xml:space="preserve"> & foramẽ quadratum ſit a b c d:</s> <s xml:id="echoid-s24940" xml:space="preserve"> cuius puncto medio (qui ſit f) <lb/> <pb o="76" file="0378" n="378" rhead="VITELLONIS OPTICAE"/> incidat perpendiculariter radius e f:</s> <s xml:id="echoid-s24941" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s24942" xml:space="preserve"> ſuperficies corporis denſi æquidiſtans ſuperficiei forami-<lb/>nis, quæ eſt g h k l.</s> <s xml:id="echoid-s24943" xml:space="preserve"> Dico, quòd lumẽ incidens illi ſu-<lb/> <anchor type="figure" xlink:label="fig-0378-01a" xlink:href="fig-0378-01"/> perficiei, erit figuræ quadratæ:</s> <s xml:id="echoid-s24944" xml:space="preserve"> fiunt enim duæ pyra <lb/>mides unum uerticem habentes punctum e, quarũ <lb/>maioris baſis eſt g h k l, minoris uerò baſis eſt a b c <lb/>d, & earum baſes ſunt ęquidiſtantes:</s> <s xml:id="echoid-s24945" xml:space="preserve"> ſunt ergo ſimi-<lb/>les per 99 t 1 huius.</s> <s xml:id="echoid-s24946" xml:space="preserve"> Quia ergo baſis a b c d ex hypo-<lb/>theſi eſt quadrata, patet, quòd & baſis g h k l eſt qua <lb/>drata.</s> <s xml:id="echoid-s24947" xml:space="preserve"> Et eſt hoc propoſitum primum.</s> <s xml:id="echoid-s24948" xml:space="preserve"> Quoniã uerò <lb/>per 35 huius radij longiores ad aliquam ęquidiſtan-<lb/>tiam accedunt:</s> <s xml:id="echoid-s24949" xml:space="preserve"> accedit & hęc figura ad aliquam cir-<lb/>cularitatem, propter compreſsionem radiorum, uel <lb/>propter ipſorum interſectionem in punctis linearũ <lb/>terminãtium feneſtras, ut diximus in præmiſſa.</s> <s xml:id="echoid-s24950" xml:space="preserve"> Pa-<lb/>tet ergo propoſitum.</s> <s xml:id="echoid-s24951" xml:space="preserve"/> </p> <div xml:id="echoid-div942" type="float" level="0" n="0"> <figure xlink:label="fig-0378-01" xlink:href="fig-0378-01a"> <variables xml:id="echoid-variables411" xml:space="preserve">g h a b f e d c c k</variables> </figure> </div> </div> <div xml:id="echoid-div944" type="section" level="0" n="0"> <head xml:id="echoid-head756" xml:space="preserve" style="it">41. Per medium quadr ati foraminis radio ob-<lb/>liquè incidente ſuperficiei denſi corporis ſubſtratæ <lb/>ſuperficiei for aminis: lumen incidens erit figura <lb/>altera parte longior ſuis angulis æqualiter arcuatis.</head> <p> <s xml:id="echoid-s24952" xml:space="preserve">Eſto, ut in pręmiſſa, centrũ corporis luminoſi punctũ e:</s> <s xml:id="echoid-s24953" xml:space="preserve"> & peripheria quadrati foraminis a b c d, <lb/>cuius medio puncto, qui ſit e, obliquè incidat radius e f:</s> <s xml:id="echoid-s24954" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s24955" xml:space="preserve"> ſuքficies corporis dẽſi ſubſtrati illi fo-<lb/>ramini, quæ g h k l, cui ſimiliter obliquè incidat radius.</s> <s xml:id="echoid-s24956" xml:space="preserve"> Dico, quòd figura luminis in ſubſtrata ſuք-<lb/>ficie erit altera parte longior.</s> <s xml:id="echoid-s24957" xml:space="preserve"> Quoniam enim illæ ſuperficies non ſunt baſes pyramidum illumina-<lb/>tionis, ſed ſolùm ſecantes illas pyramides obliquè, patet per 99 t 1 huius, quoniam ambæ figuræ <lb/>a b c d & g h k l (ſiue earum ſuperficies æquidiſtent, ſiue non æquidiſtent) ſunt figuræ altera parte <lb/>longiores:</s> <s xml:id="echoid-s24958" xml:space="preserve"> quoniam illæ figuræ, quę ſecundum illa puncta, quibus axis e f propoſitis ſuperficiebus <lb/>obliquè incidit, pyramides terminant, ſunt ambæ quadratæ:</s> <s xml:id="echoid-s24959" xml:space="preserve"> reliquæ uerò obliquè ſecundum illa <lb/>pũcta axi incidẽtes, ſunt ambæ altera parte lõgiores.</s> <s xml:id="echoid-s24960" xml:space="preserve"> Patet ergo ꝓpoſitũ primũ.</s> <s xml:id="echoid-s24961" xml:space="preserve"> Et quoniã, ut patet <lb/>per 35 huius, radij longiores quaſi ad aliquã æquidiſtantiam accedũt, patet, quòd anguli illius figu-<lb/>ræ luminis aliqualiter arcuantur, ſicut & in duabus pręmiſsis declaratũ eſt.</s> <s xml:id="echoid-s24962" xml:space="preserve"> Et hoc eſt propoſitũ.</s> <s xml:id="echoid-s24963" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div945" type="section" level="0" n="0"> <head xml:id="echoid-head757" xml:space="preserve" style="it">42. Per medium ſecũdi diaphani denſioris primo radi{us} perpendicularis duct{us} à cẽtro cor-<lb/>poris luminoſi ſuper ſuperficiẽ obiecti corporis ſemper penetrat irrefract{us}. Alhazen 3 n 7.</head> <p> <s xml:id="echoid-s24964" xml:space="preserve">Huius propoſitionis probatio plus experientiæ inſtrumẽtorum innititur, quàm alteri demon-<lb/>ſtrationum.</s> <s xml:id="echoid-s24965" xml:space="preserve"> Cum ergo quis experiri uoluerit modum fractionis radiorum luminoſorum in medio <lb/>ſecũdι diaphani denſioris primo, ut in aqua, quæ eſt denſior aere:</s> <s xml:id="echoid-s24966" xml:space="preserve"> aſſumat uas rectarũ orarum qua-<lb/>liſcunq;</s> <s xml:id="echoid-s24967" xml:space="preserve"> uoluerit materiæ uel figuræ, dum tamẽ ſit altitudo orarum maior medietate cubiti, & dia-<lb/>meter latitudinis eius ſit non minor diametro inſtrumenti, quod faciendum præmiſimus in prima <lb/>huius:</s> <s xml:id="echoid-s24968" xml:space="preserve"> & planẽtur oræ illius uaſis, donec ſuperficies per eius oras tranſiens ſit æqualis plana:</s> <s xml:id="echoid-s24969" xml:space="preserve"> & po-<lb/>natur in fundo uaſis aliquod corpuſculum coloratum uiſibile, ut aliquod numiſma uel res picta di-<lb/>uerſi coloris:</s> <s xml:id="echoid-s24970" xml:space="preserve"> deinde impleatur uas aqua clara.</s> <s xml:id="echoid-s24971" xml:space="preserve"> Cùm ergo quieuerit motus aquæ, ſi aſpiciens uiſum <lb/>perpendiculariter proiecerit ſuper medium numiſmatis, uel picturæ:</s> <s xml:id="echoid-s24972" xml:space="preserve"> inueniet figuram & colorem <lb/>& ipſorum ſitum & partium ordinationem eo modo, quo ſunt ſecundũ ſe ordinata, ſi in aere uide-<lb/>rentur.</s> <s xml:id="echoid-s24973" xml:space="preserve"> Cõſideret ergo experimentator illũ ſui corporis ſitum, ſiue ſit ſtans ſiue ſedẽs, & ſui diſtan-<lb/>tiam à uaſe, & ſitum ipſius uaſis, & omnia circumſtãtia illam uiſionem.</s> <s xml:id="echoid-s24974" xml:space="preserve"> Ponatur itaq;</s> <s xml:id="echoid-s24975" xml:space="preserve"> uas iſtud ple-<lb/>num aqua clara in loco, in quo ſplendet ſol, & ſiſtatur uas taliter, ut ſuperficies circumferẽtiæ uaſis <lb/>ſit æquidiſtãs horizonti:</s> <s xml:id="echoid-s24976" xml:space="preserve"> hoc autẽ poteſt perpẽdi ex hoc, ſi ſuperficies aquę ſit æquidiſtãs periphe-<lb/>riæ uaſis.</s> <s xml:id="echoid-s24977" xml:space="preserve"> Deinde imponatur inſtrumentũ in hoc uas, ita quòd pinnulæ ſuper extre mitates regulæ <lb/>exiſtẽtes ſuperponãtur oræ uaſis ex utraq;</s> <s xml:id="echoid-s24978" xml:space="preserve"> parte:</s> <s xml:id="echoid-s24979" xml:space="preserve"> tũc ergo medietas inſtrumenti cũ tota regula erit <lb/>intra uas:</s> <s xml:id="echoid-s24980" xml:space="preserve"> deinde auferatur aqua, donec ſuperficies aquæ ſecet cẽtrum inſtrumẽti:</s> <s xml:id="echoid-s24981" xml:space="preserve"> & reuoluatur in-<lb/>ſtrumẽtum in circuitu uaſis, donec oræ ſuper aquã obumbrent alias ſub aqua:</s> <s xml:id="echoid-s24982" xml:space="preserve"> & tunc retẽta regula <lb/>cum altera manuum, reuoluatur inſtrumẽtum cum reliqua manu in circuitu ſui centri, donec lumẽ <lb/>ſolis pertranſeat foramẽ l m n, quod eſt in ora inſtrumẽti, & foramẽ laminæ quadratæ, & perueniat <lb/>ad ſuperficiẽ aquæ, quia lumẽ pertranſiens foramen rotundũ ampliatur ſemper per 36 huius.</s> <s xml:id="echoid-s24983" xml:space="preserve"> Siſta-<lb/>tur quoq;</s> <s xml:id="echoid-s24984" xml:space="preserve"> taliter inſtrumentum, utlumen cadens ſuper laminam ſecundi foraminis, quod eſt x y z, <lb/>ſitum habeat æqualẽ:</s> <s xml:id="echoid-s24985" xml:space="preserve"> & tunc experimẽtator reductis manibus ab inſtrumento, ſecundum omnem <lb/>ſitum & modum, quo prius aſpexit numiſma, inſpiciat ad fundũ aquæ ex parte quartæ inſtrumẽti, <lb/>cuius ora eſt abſciſſa, quæ eſt a d:</s> <s xml:id="echoid-s24986" xml:space="preserve"> inuenietq́;</s> <s xml:id="echoid-s24987" xml:space="preserve"> lumẽ pertranſiẽs ex duobus foraminibus ſuper ſuper-<lb/>ficiem oræ alterius, quæ eſt intra aquam, & lumen inter duos circulos extremos trium circulorum <lb/>æquidiſtanter ſignatorum, aut addens ſuper diſtantiam illorum circulorum modicùm:</s> <s xml:id="echoid-s24988" xml:space="preserve"> & erit addi-<lb/>tio æqualis ex duobus lateribus circulorum.</s> <s xml:id="echoid-s24989" xml:space="preserve"> Ex quo patet, quòd medium punctum huius luminis <lb/>cadit in aliquod punctum circumferẽtiæ medij circuli illorum trium circulorum, ut in punctum p.</s> <s xml:id="echoid-s24990" xml:space="preserve"> <lb/>Deinde acus ferrea uel lignũ minutum in interiori parte foraminis oræ inſtrumenti applicata per-<lb/>tranſeat medium foraminis diametraliter, & tunc inſpiciẽti, ut prius, uidebitur umbra acus in me-<lb/> <pb o="77" file="0379" n="379" rhead="LIBER SECVNDVS."/> dio lucis oppoſitæ, per 11 huius, diuidens eum per æqualia.</s> <s xml:id="echoid-s24991" xml:space="preserve"> Deinde retrahatur acus, donec acumen <lb/>eius ſit in medio foraminis, & erit umbra extremitatιs acus in medio lucis, quæ eſt in ſuքficie aquę, <lb/>& eius, quæ eſt intra aquam:</s> <s xml:id="echoid-s24992" xml:space="preserve"> & uniuerſaliter ſecundum quam proportionem acus peripheriam fo-<lb/>raminis ut chorda abſciderit, ſecundum eandem proportionem umbra acus peripheriam lucis in <lb/>ſuperficie aquæ & ſub aqua exiſtentis abſcindet:</s> <s xml:id="echoid-s24993" xml:space="preserve"> acu uerò penitus remota, lumẽ reuertetur.</s> <s xml:id="echoid-s24994" xml:space="preserve"> Palàm <lb/>ergo ex his, quòd punctus, qui eſt in medio lucis intra aquam exiſtẽtis, exit à<gap/>puncto medio lucis in <lb/>ſuperficie aquæ exiſtentis:</s> <s xml:id="echoid-s24995" xml:space="preserve"> & quòd punctus medius huius lucis exit à luce, quæ eſt in centro fora-<lb/>minis ſuperioris.</s> <s xml:id="echoid-s24996" xml:space="preserve"> Lux ergo, quæ peruenit ad cẽtrum lucis in ſuperficie aquæ exiſtentis, extenditur <lb/>ſecundum rectitudinem lineæ rectæ per duo puncta m & y, quę ſunt centra amborum foraminum, <lb/>tranſeuntis:</s> <s xml:id="echoid-s24997" xml:space="preserve"> & hęc linea eſt in ſuperficie medij circuli trium circulorũ:</s> <s xml:id="echoid-s24998" xml:space="preserve"> & eſt pars diametri illius cir-<lb/>culi, quæ eſt m p, cũ ſit æquidiſtans diametro circuli in baſi inſtrumenti exiſtẽtis, quæ eſt f e g.</s> <s xml:id="echoid-s24999" xml:space="preserve"> Pun-<lb/>ctus ergo, qui eſt in medio lucis, quæ eſt in ſuperficie aquæ, eſt in ſuperficie huius medij circuli:</s> <s xml:id="echoid-s25000" xml:space="preserve"> ſed <lb/>& punctus in medio lucis intra aquam exiſtentis, eſt in circumferentia medij circuli:</s> <s xml:id="echoid-s25001" xml:space="preserve"> hæc ergo duo <lb/>puncta erũt in ſuperficie medij circuli:</s> <s xml:id="echoid-s25002" xml:space="preserve"> ergo & tota illa linea erit in ſuperficie medij circuli per 1 p 11.</s> <s xml:id="echoid-s25003" xml:space="preserve"> <lb/>Quòd ſi lux, quę eſt in ſuperficie aquæ, non fuerit manifeſta:</s> <s xml:id="echoid-s25004" xml:space="preserve"> mittatur regula minor in aquam, & ſu-<lb/>perficies eius, in qua ſignata eſt linea, diuidens ſuperficiem eius latitudinis per æqualia, applicetur <lb/>ſuperficiei aquæ, ut fiat una ſuperficies cum illa, & alia eius ſuperficies applicetur ſuperficiei baſis <lb/>inſtrumẽti.</s> <s xml:id="echoid-s25005" xml:space="preserve"> Palàm ergo ex præmiſsis in 1 huius, quia linea, quæ eſt in ſuperficie regulæ, eſt in ſuper-<lb/>ficie medij circul<gap/> per m & y centra duorum foraminum tranſeuntis:</s> <s xml:id="echoid-s25006" xml:space="preserve"> apparebitq́;</s> <s xml:id="echoid-s25007" xml:space="preserve"> lux, quæ eſt in ſu-<lb/>perficie aquæ, ſuper ſuperficiem regulæ, & mediũ illius lucis ſuper lineam, quæ eſt in medio regu-<lb/>læ.</s> <s xml:id="echoid-s25008" xml:space="preserve"> Et ſi acus fuerit poſita ſuper medium foraminis ſuperioris, obumbrabitur linea, quæ eſt in me-<lb/>dio regulæ:</s> <s xml:id="echoid-s25009" xml:space="preserve"> & ſi acumen acus ponatur ſuper cẽtrum foraminis, cadet umbra acuminis acus in me-<lb/>dio lucis, quæ eſt ſuper regulam, & ablata acu redibit lumen.</s> <s xml:id="echoid-s25010" xml:space="preserve"> Sic ergo apparebit lumẽ cadens ſuper <lb/>ſuperficiem aquæ, apparitione manifeſta:</s> <s xml:id="echoid-s25011" xml:space="preserve"> & patebit, quòd lux incidens cẽtro foraminis ſuperioris, <lb/>ipſa eſt ſuper lineam tranſeuntem per centra duorum foraminum.</s> <s xml:id="echoid-s25012" xml:space="preserve"> Et quoniã ſuperficies aquę tran-<lb/>ſit centrum inſtrumenti, & ſuperficies regulæ eſt una cum ſuperficie aquæ:</s> <s xml:id="echoid-s25013" xml:space="preserve"> ſuperficies itaq;</s> <s xml:id="echoid-s25014" xml:space="preserve"> regulæ <lb/>tranſibit centrum inſtrumenti.</s> <s xml:id="echoid-s25015" xml:space="preserve"> Erit ergo remotio centri lucis à centro inſtrumenti, æqualis medie-<lb/>tati latitudinis regulæ, quæ eſt æqualis perpendiculari, cadenti à centro foraminis ſuper ſuperficiẽ <lb/>baſis inſtrumenti:</s> <s xml:id="echoid-s25016" xml:space="preserve"> erit ergo centrum lucis, quæ eſt in ſuperficie regulæ uel aquæ, cẽtrum medij cir-<lb/>culi.</s> <s xml:id="echoid-s25017" xml:space="preserve"> Reuoluatur ergo regula, donec angulus ipſius acutus tranſeat per centrũ inſtrumenti, & pars <lb/>inferior lineæ diuidentis angulũ eius per æqualia, ſit in centro luminis, quod eſt intra aquam:</s> <s xml:id="echoid-s25018" xml:space="preserve"> acui-<lb/>tas ergo ſuperior regulæ tranſibit centrum circuli medij:</s> <s xml:id="echoid-s25019" xml:space="preserve"> punctus ergo lineæ ſuperficiei ſuperioris <lb/>regulæ, qui eſt in ſuperficie aquæ, eſt centrũ medij circuli, & lucis, quæ eſt in ſuperficie aquæ:</s> <s xml:id="echoid-s25020" xml:space="preserve"> & erit <lb/>illa linea ſemidiameter circuli medij.</s> <s xml:id="echoid-s25021" xml:space="preserve"> Immittatur ergo acus longa in aquam ita, ut acumen ipſius ſit <lb/>in puncto anguli regulæ, ſecabitq́;</s> <s xml:id="echoid-s25022" xml:space="preserve"> umbra acus lucem, quæ eſt intra aquam, eritq́;</s> <s xml:id="echoid-s25023" xml:space="preserve"> umbra acuminis <lb/>acus ad finem regulæ, quæ eſt in medio lucis.</s> <s xml:id="echoid-s25024" xml:space="preserve"> Et ſi fixo acumine acus, moueatur acus:</s> <s xml:id="echoid-s25025" xml:space="preserve"> umbra acus <lb/>mutabit ſitum ad diuerſas partes lucis:</s> <s xml:id="echoid-s25026" xml:space="preserve"> umbra tamen acuminis non mutata à medio lucis:</s> <s xml:id="echoid-s25027" xml:space="preserve"> ablata <lb/>uerò totaliter acu, redibit lux totalis.</s> <s xml:id="echoid-s25028" xml:space="preserve"> Idem quoq;</s> <s xml:id="echoid-s25029" xml:space="preserve"> accidit, in quocunq;</s> <s xml:id="echoid-s25030" xml:space="preserve"> puncto lineæ, quæ eſt in ſu-<lb/>perficie regulæ, poſitum fuerit acumen acus.</s> <s xml:id="echoid-s25031" xml:space="preserve"> Ex quo patet, quòd lux exiſtens in aliquo puncto lu-<lb/>cis intra aquã, procedit à puncto ſibi ſimili in luce, quæ eſt in ſuperficie aquæ, & quòd à medio pun-<lb/>cto lucis, quæ ſuper aquam ad medium punctum lucis intra aquam protenditur radius ſecundũ li-<lb/>neam rectam, quæ eſt medium regulæ.</s> <s xml:id="echoid-s25032" xml:space="preserve"> Ex quo patet, quòd tranſitus lucis per corpus aquæ eſt ſe-<lb/>cundum lineas rectas per 1 p 11.</s> <s xml:id="echoid-s25033" xml:space="preserve"> Et hoc eſt, quod circa propoſitam propoſitionem experimentaliter <lb/>intendimus declarare.</s> <s xml:id="echoid-s25034" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div946" type="section" level="0" n="0"> <head xml:id="echoid-head758" xml:space="preserve" style="it">43. In medio ſecũdi diaphani, quod eſt denſi{us} primo diaphano, fit refr actio radiorum obli-<lb/>quorum ab anteriori ſuperficie diaphani ſecundi ad perpendicularem, exeuntem à puncto re-<lb/>fractionis ſuper ſuperficiem corporis ſecundi. Alhazen 4 n 7.</head> <p> <s xml:id="echoid-s25035" xml:space="preserve">Experimentaliter etiã & hoc propoſitũ theorema poteſt declarari.</s> <s xml:id="echoid-s25036" xml:space="preserve"> Oppoſito enim foramine ſu-<lb/>perioris inſtrumenti obliquè ipſi corpori ſolari, ita, ut radius obliquè incidat ad oram inſtrumenti <lb/>oppoſitã foramini, & perſcrutato per modũ, quo in præmiſſa, centro lucis, quę eſt intra aquã:</s> <s xml:id="echoid-s25037" xml:space="preserve"> ſigne-<lb/>tur illud per puncturam ferri duri in ſuperficie ipſa inſtrumẽti, & inuenietur illud centrũ non in li<gap/> <lb/>nea g k perpendiculariter erecta ſuper g terminũ diametri oppoſita lineæ f h, in qua eſt foramẽ oræ <lb/>inſtrumenti, ſed declinabit ab illa linea ad partem, in qua eſt ſol:</s> <s xml:id="echoid-s25038" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s25039" xml:space="preserve"> inter hoc centrũ lucis & pũ-<lb/>ctum p, (quod eſt communis differentia lineæ g k, perpendicularis ſuper terminũ diametri inſtru-<lb/>menti, & circũferentiæ circuli medij tranſeũtis per m & y cẽtra foraminũ) diſtantia ſenſibilis.</s> <s xml:id="echoid-s25040" xml:space="preserve"> Mit-<lb/>tatur itaq;</s> <s xml:id="echoid-s25041" xml:space="preserve"> regula in aquam, & applicetur ſuperficiei laminæ, ita, quòd terminus latior regulæ ſit ſu-<lb/>pra centrũ laminæ:</s> <s xml:id="echoid-s25042" xml:space="preserve"> & moueatur regula, quouſq;</s> <s xml:id="echoid-s25043" xml:space="preserve"> acuitas eius ſit perpendicularis ſuper ſuperficiem <lb/>aquæ, quo ad ſenſum:</s> <s xml:id="echoid-s25044" xml:space="preserve"> erit itaq;</s> <s xml:id="echoid-s25045" xml:space="preserve"> centrum lucis, quod eſt intra aquam, inter acumẽ regulæ, & lineam <lb/>g k perpendicularem ſuper f g diametrũ baſis inſtrumenti.</s> <s xml:id="echoid-s25046" xml:space="preserve"> Patet ergo ex hoc, quòd hęc refractio eſt <lb/>ad partẽ perpendicularis, exeuntis à loco refractionis perpẽdiculariter ſuper ſuperficiẽ aquæ.</s> <s xml:id="echoid-s25047" xml:space="preserve"> Hoc <lb/>ita inuento ſignetur in circũferentia circuli medij trium ſignatorum circulorũ ſuper punctũ extre-<lb/>mum perpendicularis, exeuntis à centro eiuſdem circuli perpendiculariter ſuper ſuperficiẽ aquæ, <lb/>ſignum fixũ per ferri duri puncturam.</s> <s xml:id="echoid-s25048" xml:space="preserve"> Et quia patuit per præmiſſam, quòd inſtrumento directè ſoli <lb/> <pb o="78" file="0380" n="380" rhead="VITELLONIS OPTICAE"/> oppoſito, & radio ſolis ſibi perpendiculariter incidente, lux, quæ peruenit ad cẽtrum lucis, quæ e<gap/> <lb/>intra aquam, eſt lux extenſa ſecundũ rectitudinem lineæ continuantis duo centra foraminum, quę <lb/>linea peruenit ad centrum medij circuli æquidiſtantis ſuperficiei baſis inſtrumenti, & eſt diameter <lb/>illius:</s> <s xml:id="echoid-s25049" xml:space="preserve"> ſi hæc linea fuerit imaginata extendi ſecundum rectitudinem intra aquam, donec perueniat <lb/>ad oram inſtrumenti:</s> <s xml:id="echoid-s25050" xml:space="preserve"> tunc erit totaliter æquidiſtans diametro inſtrumenti, & perueniet ad lineam <lb/>g k perpendicularem ſuper diametrum f g, in interiore parte oræ inſtrumenti ductam.</s> <s xml:id="echoid-s25051" xml:space="preserve"> Et quando <lb/>centrum lucis, quæ nunc eſt intra aquam, nõ eſt ſuper illam lineam perpendicularem in ora inſtru-<lb/>menti productam:</s> <s xml:id="echoid-s25052" xml:space="preserve"> tũc patet, quòd lux extenſa à medio lucis, quæ eſt in ſuperficie aquæ, non exten-<lb/>ditur ad medium lucis, quæ eſt intra aquam, ſecundum rectitudinem lineæ tranſeuntis per centra <lb/>duorum foraminum, ſed refringitur ab illa:</s> <s xml:id="echoid-s25053" xml:space="preserve"> declaratum eſt autem per 1 huius, quòd hæc lux exten-<lb/>ditur rectè à medio lucis, quæ eſt in ſuperficie aquæ, ad mediũ lucis, quæ eſt intra aquam.</s> <s xml:id="echoid-s25054" xml:space="preserve"> Eſt ergo <lb/>huius lucis refractio apud ſuperficiem aquæ.</s> <s xml:id="echoid-s25055" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s25056" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div947" type="section" level="0" n="0"> <head xml:id="echoid-head759" xml:space="preserve" style="it">44. Per medium ſecundi diaphani rarioris primo, radi{us} perpẽdiculariter incidens, à cen-<lb/>tro corporis luminoſi ſuper ſuperficiem corporis obiecti penetrat irrefr act{us}. Alhazen 6 n 7.</head> <p> <s xml:id="echoid-s25057" xml:space="preserve">Inſtrumentali ſimiliter experiẽtia propoſitum theorema poteſt declarari.</s> <s xml:id="echoid-s25058" xml:space="preserve"> Aſſumatur enim uitri <lb/>clari uel cryſtalli fruſtum figuræ cubicæ, longitudinis duplæ diametri foraminis oræ inſtrumenti:</s> <s xml:id="echoid-s25059" xml:space="preserve"> <lb/>& fiant planę ſuperficies eorum æquales & æquidiſtantes, & latera ipſorum ſint recta, & multũ po-<lb/>liantur:</s> <s xml:id="echoid-s25060" xml:space="preserve"> deinde ſignetur per ſculpturam ferri duri in medio baſis inſtrumenti linea recta, tranſiens <lb/>per centrum ipſius, quod eſt e, perpendiculariter ſuper ipſius diametrũ, quæ eſt f g, ſuper cuius ex-<lb/>tremitates ſunt in ora inſtrumenti productæ duę perpendiculares f h & g k:</s> <s xml:id="echoid-s25061" xml:space="preserve"> & producatur illa linea <lb/>in utranq;</s> <s xml:id="echoid-s25062" xml:space="preserve"> partem ſuperficiei circuli baſis, & ſit z e x.</s> <s xml:id="echoid-s25063" xml:space="preserve"> Ponatur itaq;</s> <s xml:id="echoid-s25064" xml:space="preserve"> unum uitrorum iſtorũ ſuper ſa-<lb/>perficiem baſis inſtrumenti, & applicetur unum laterum ſuorum perpendiculariter ductæ, quæ eſt <lb/>z e x, taliter, ut medium lateris uitri ſit uerè ſuper punctum e centrum inſtrumẽti:</s> <s xml:id="echoid-s25065" xml:space="preserve"> & ſit totum cor-<lb/>pus uitri ex parte foraminum, ſcilicet inter foramina oræ & tabulæ, & inter centrum inſtrumenti, <lb/>quod eſt e.</s> <s xml:id="echoid-s25066" xml:space="preserve"> Tranſit ergo dicta diameter inſtrumẽti (quæ eſt f g) per medium ſuperficiei uitri ſuper-<lb/>poſitæ baſi inſtrumenti.</s> <s xml:id="echoid-s25067" xml:space="preserve"> Applicetur itaq;</s> <s xml:id="echoid-s25068" xml:space="preserve"> uitrum baſi inſtrumẽti forti applicatione per bitumen fir-<lb/>mum, taliter tamen, quòd poſsit auferri, quando placuerit:</s> <s xml:id="echoid-s25069" xml:space="preserve"> deinde ponatur alterũ uitrum ultra pri-<lb/>mũ ſcilicet, ex eadẽ parte foraminũ:</s> <s xml:id="echoid-s25070" xml:space="preserve"> & applicetur aliqua ſuperficierũ eius ſuperficiei primi uitri, & <lb/>applicetur baſi inſtrumenti applicatione fixa:</s> <s xml:id="echoid-s25071" xml:space="preserve"> deinde tertiũ uitrum applicetur ſecundo, & adæque-<lb/>tur ſuperficies eius cum duabus ſuperficiebus laterum ſecundi uitri, & applicetur baſi inſtrumẽti, <lb/>& ſic fiat de pluribus uitris, quouſq;</s> <s xml:id="echoid-s25072" xml:space="preserve"> perueniant uitra ad aliam perpendicularem ſuper ſuperficiem <lb/>baſis inſtrumenti aut propè, ſcilicet uerſus punctum t.</s> <s xml:id="echoid-s25073" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s25074" xml:space="preserve"> uitra fuerint applicata ſuperficiei <lb/>baſis inſtrumenti ſecundum prædictum modum:</s> <s xml:id="echoid-s25075" xml:space="preserve"> palàm quoniam præmiſſa diameter inſtrumenti <lb/>(quæ eſt f g) tranſibit per medium omniũ ſuperficierum uitrorum ſuperpoſitorũ baſi inſtrumenti:</s> <s xml:id="echoid-s25076" xml:space="preserve"> <lb/>& altitudo omnium uitrorum eſt dupla diametro foraminis:</s> <s xml:id="echoid-s25077" xml:space="preserve"> diameter uerò foraminis eſt æqualis <lb/>perpendiculari m f exeunti à centro foraminis ſuper ſuperficiem baſis inſtrumenti, & ſuper diame-<lb/>trum eius f g:</s> <s xml:id="echoid-s25078" xml:space="preserve"> unaquæq;</s> <s xml:id="echoid-s25079" xml:space="preserve"> ergo perpendicularium, exeuntium à centris ſuperficierum uitrorum per-<lb/>pendicularium ſuper diametrũ baſis inſtrumenti, eſt æqualis lineæ m f, ſcilicet perpendiculari exe-<lb/>unti à centro foraminis ſuper ſuperficiem baſis inſtrumenti.</s> <s xml:id="echoid-s25080" xml:space="preserve"> Linea ergo, quæ tranſit centra ambo-<lb/>rum foraminum, tranſibit centra ſuperficierum uitrorum perpendicularium ſuper ſuperficiem ba-<lb/>ſis inſtrumenti.</s> <s xml:id="echoid-s25081" xml:space="preserve"> Accipiatur ergo regula ſubtilis, cuius formam præmiſimus:</s> <s xml:id="echoid-s25082" xml:space="preserve"> & erigatur ſuper oram <lb/>inſtrumenti in ſuperficie baſis inſtrumẽti:</s> <s xml:id="echoid-s25083" xml:space="preserve"> & ponatur ſuperficies regulæ, in qua ſignata eſt linea ex <lb/>parte primi uitri, quod eſt ſupra e cẽtrum baſis inſtrumenti:</s> <s xml:id="echoid-s25084" xml:space="preserve"> & ponatur regula prope uitrum, & ap-<lb/>plicetur taliter ut linea, quę eſt in ſuperficie regulæ, ſit in ſuperficie medij circuli, ſecabitq́;</s> <s xml:id="echoid-s25085" xml:space="preserve"> linea re-<lb/>cta, tranſiens per centra amborum foraminum, & per centra ſuperficierum uitrorum lineam latitu-<lb/>dinis regulæ perpendiculariter, & tranſibit ad punctum g.</s> <s xml:id="echoid-s25086" xml:space="preserve"> Tunc itaq;</s> <s xml:id="echoid-s25087" xml:space="preserve"> ponatur inſtrumentũ in uas <lb/>prædictum uacuum aqua, & ponatur uas in ſole directè oppoſitum centro ſolis, ut accipiat radium <lb/>perpẽdicularẽ:</s> <s xml:id="echoid-s25088" xml:space="preserve"> hoc aũt pteoſt fieri, ſi moueatur inſtrumentũ, quouſq;</s> <s xml:id="echoid-s25089" xml:space="preserve"> lux ſolis trãſeat per ambo fo-<lb/>ramina, & fiat apud ſecundũ foramẽ lux æqualis:</s> <s xml:id="echoid-s25090" xml:space="preserve"> & aſpiciatur ſuperficies regulæ oppoſita uitro, & <lb/>uidebitur lux exiens à duobus foraminibus ipſius inſtrumẽti, extenſa ſuper ſuperficiẽ ipſius regu-<lb/>læ:</s> <s xml:id="echoid-s25091" xml:space="preserve"> & illud umbroſum, quod circumdat lucẽ in ſuperficie regulæ, obumbrabitur per umbrã oræ in-<lb/>ſtrumẽti:</s> <s xml:id="echoid-s25092" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s25093" xml:space="preserve"> centrũ uiſus ipſius aſpiciẽtis ſuք lineã, quæ eſt in ſuքficie regulæ.</s> <s xml:id="echoid-s25094" xml:space="preserve"> Deinde acus ſub-<lb/>tilis ponatur ſuք ſuperius foramẽ, ita quòd extremitas acus ſit perpẽdicularis ſuper centrũ forami-<lb/>nis:</s> <s xml:id="echoid-s25095" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s25096" xml:space="preserve"> tũc umbra extremitatis acus ſuք centrũ lucis in linea, quæ eſt in ſuperficie regulæ.</s> <s xml:id="echoid-s25097" xml:space="preserve"> Tũc <lb/>itaq;</s> <s xml:id="echoid-s25098" xml:space="preserve"> ſignetur pũctus illius umbræ cũ incauſto ſubtiliter:</s> <s xml:id="echoid-s25099" xml:space="preserve"> & auferatur acus à ſuperiori foramine:</s> <s xml:id="echoid-s25100" xml:space="preserve"> & <lb/>eius extremitas ponatur ſuper centrũ inferioris foraminis:</s> <s xml:id="echoid-s25101" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s25102" xml:space="preserve"> iterũ umbra extremitatis acus <lb/>ſuper punctũ ſignatum in ſuperficie regulæ:</s> <s xml:id="echoid-s25103" xml:space="preserve"> ablata quoq;</s> <s xml:id="echoid-s25104" xml:space="preserve"> acu lux reuertitur.</s> <s xml:id="echoid-s25105" xml:space="preserve"> Ex quo patet, quoniã <lb/>lux, quæ eſt ſuք punctũ, quod eſt in ſuperficie regulæ, tranſit per cẽtra amborũ foraminum.</s> <s xml:id="echoid-s25106" xml:space="preserve"> Deinde <lb/>cũ incauſto ſignetur nota nigra in pũcto medio ſuperficiei uitri ex parte regulæ (poteſt aũt ille pũ-<lb/>ctus inueniri per 40 t 1 huius, quoniã ille pũctus eſt cõmunis ſectio duarũ diametrorũ ſuքficiei ui-<lb/>tri) & tũc intuens lucẽ, quæ eſt ſuper regulã, inueniet umbrã puncti, qui eſt in medio uitri ſuք pun-<lb/>ctum, quod eſt in ſuperficie regulæ.</s> <s xml:id="echoid-s25107" xml:space="preserve"> Patet ergo ex hoc, quoniã lux, quæ trãſit per centra duorũ fora-<lb/>minum, tranſit per punctum, quod eſt in medio uitri.</s> <s xml:id="echoid-s25108" xml:space="preserve"> Deinde euellatur uitrum primum, quod eſt <lb/> <pb o="79" file="0381" n="381" rhead="LIBER SECVNDVS."/> ſuper centrũ inſtrumenti, punctũ e:</s> <s xml:id="echoid-s25109" xml:space="preserve"> & in ſuperficie ſecundi uitri ſignetur punctũ medium, ut prius <lb/>factum eſt in ſuperficie uitri primi:</s> <s xml:id="echoid-s25110" xml:space="preserve"> & componatur inſtrumentũ ſecundò, & moueatur, quouſq;</s> <s xml:id="echoid-s25111" xml:space="preserve"> lux <lb/>tranſeat per duo foramina, peruenietq́;</s> <s xml:id="echoid-s25112" xml:space="preserve"> lux tranſiens per centra duorum foraminũ ad centrũ lucis, <lb/>quod eſt in ſuperficie regulę.</s> <s xml:id="echoid-s25113" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s25114" xml:space="preserve"> ex hoc, quòd lux pertranſiẽs centra duorũ foraminũ, tran-<lb/>ſit per punctum, quod eſt in medio ſuperficiei ſecundi uitri:</s> <s xml:id="echoid-s25115" xml:space="preserve"> & quòd lux, quę tranſit per centra duo-<lb/>rum foraminũ in prima experimentatione, tranſit etiã per punctũ, quod eſt in medio ſecundi uitri.</s> <s xml:id="echoid-s25116" xml:space="preserve"> <lb/>Extrahatur itaq;</s> <s xml:id="echoid-s25117" xml:space="preserve"> ſecũdũ uitrum, & opponatur tertiũ, & ſic de cęteris uſq;</s> <s xml:id="echoid-s25118" xml:space="preserve"> ad ultimũ.</s> <s xml:id="echoid-s25119" xml:space="preserve"> Et patet uniuer <lb/>ſaliter, quòd lux tranſiẽs per centra duorũ foraminũ, perueniens ad ſuperficiem regulæ, tranſit etiã <lb/>per centra ſuperficierũ uitrorũ omniũ, poſitorũ ſuper ſuperficiẽ laminę:</s> <s xml:id="echoid-s25120" xml:space="preserve"> & ſunt omnia centra ſuper-<lb/>ficierum uitrorũ omniũ in una linea recta cõtinuante centra duorũ foraminũ.</s> <s xml:id="echoid-s25121" xml:space="preserve"> Lux itaq;</s> <s xml:id="echoid-s25122" xml:space="preserve"> pertranſiẽs <lb/>centra foraminũ tam in corpore uitri ꝗ̃ extra corpus in aere, extenditur ſecundũ lineam rectã conti <lb/>nuantẽ centra duorũ foraminũ:</s> <s xml:id="echoid-s25123" xml:space="preserve"> & eſt illa linea m p, perpendicularis ſuper ſuperficies omniũ uitro-<lb/>rum oppoſitas foramini ք 14 p 11:</s> <s xml:id="echoid-s25124" xml:space="preserve"> illa enim linea m p eſt æquidiſtans lineę f g, diametro laminæ, quæ <lb/>eſt perpendicularis ſuper ſuperficiẽ uitrorũ, cum ſit perpendicularis ſuper differentiã cõmunem ſu <lb/>perficiei uitri, & ſuperficiei laminę.</s> <s xml:id="echoid-s25125" xml:space="preserve"> Et ſi omnibus uitris uel ipſorũ aliquo præmiſſo modo ſuper fun <lb/>dum inſtrumẽti diſpoſito, infundatur aqua uaſi uſq;</s> <s xml:id="echoid-s25126" xml:space="preserve"> ad concauũ ſuperficiei uitri:</s> <s xml:id="echoid-s25127" xml:space="preserve"> accidet tamen idẽ <lb/>quod prius, quoniã radius perpẽdicularis ſemper penetrat irrefractus.</s> <s xml:id="echoid-s25128" xml:space="preserve"> Itẽ ne putet aliquis, quòd re-<lb/>ctitudo radiorũ perpendiculariũ adiuuetur per cubicã figurã uitri:</s> <s xml:id="echoid-s25129" xml:space="preserve"> accipiatur medietas ſphærę ui-<lb/>treæ clarę uel cryſtallinæ, cuius ſemidiameter ſit minor diſtantia, quę eſt inter punctũ c & centrum <lb/>laminę, qđ eſt punctũ e:</s> <s xml:id="echoid-s25130" xml:space="preserve"> & inueniatur cẽtrũ baſis eius, ſuper qđ ſignetur linea ſubtilis cũ incauſto.</s> <s xml:id="echoid-s25131" xml:space="preserve"> <lb/>Deinde ex hac linea ex parte centri ſphærę ſeparetur linea æqualis l n diame-<lb/> <anchor type="figure" xlink:label="fig-0381-01a" xlink:href="fig-0381-01"/> metro foraminis orę inſtrumẽti:</s> <s xml:id="echoid-s25132" xml:space="preserve"> erit ergo hæc linea æqualis lineę m f, quę eſt <lb/>inter m centrũ foraminis, qđ eſt in ora inſtrumẽti, & ſuperficiẽ laminę:</s> <s xml:id="echoid-s25133" xml:space="preserve"> deinde <lb/>ſuք extremitatẽ huius lineę ſeparatę à diametro ꝓducatur perpẽdicularis ad <lb/>utramq;</s> <s xml:id="echoid-s25134" xml:space="preserve"> partẽ ſuperfici<gap/>i ſphæricę, qđ poteſt fieri per 11 p 1:</s> <s xml:id="echoid-s25135" xml:space="preserve"> & ſecetur ſphæra <lb/>uitrea ſecundũ illã lineã, planeturq́;</s> <s xml:id="echoid-s25136" xml:space="preserve"> ſuperficies uitri ſecti, donec ſit penitus æqualis, fiatq́;</s> <s xml:id="echoid-s25137" xml:space="preserve"> perpendi <lb/>culariter erecta ſuper ſuperficiem planã hemiſphęrij (quod per angulũ rectum corporeum poterit <lb/>menſurari) erit ergo tunc cõmunis differentia iſtius ſuperficiei erectę, & ſuperficiei baſis ſphærę li-<lb/>nea recta, ſuper quã erit perpendicularis linea prius à centro ſphærę producta:</s> <s xml:id="echoid-s25138" xml:space="preserve"> ergo etiã erit perpen <lb/>dicularis ſuper ſuperficiem erectã.</s> <s xml:id="echoid-s25139" xml:space="preserve"> Deinde in medio illius lineæ, quæ eſt cõmunis ſectio, fiat ſignum <lb/>cum incauſto:</s> <s xml:id="echoid-s25140" xml:space="preserve"> deinde uitrũ illud politũ optimè ſuper hanc ſuperficiem ſectã, ponatur ſuper ſuperfi-<lb/>ciem laminę inſtrumenti, ita quòd gibboſitas eius reſpiciat foramina, & mediũ lineę, quę eſt cõmu-<lb/>nis ſectio duarum ſuperficierum planarum uitri, applicetur centro laminæ, & figatur uitrum ſuper <lb/>laminã, ne cadat.</s> <s xml:id="echoid-s25141" xml:space="preserve"> Deinde ponatur regula ſubtilis ſuper ſuperficiem laminę inſtrumenti, ſicut in ex-<lb/>perimentatione uitrorũ cubicorum, ita quòd ſuperficies regulæ, in qua eſt linea recta latitudinis ſit <lb/>ex parte uitri, & prope illud.</s> <s xml:id="echoid-s25142" xml:space="preserve"> Deinde imponatur inſtrumentũ in uas prædictũ, & ponatur uas in ſole <lb/>uacuum aqua, & moueatur inſtrumentũ, donec lux ſolis tranſeat ambo foramina:</s> <s xml:id="echoid-s25143" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s25144" xml:space="preserve"> lux ſuper <lb/>ſuperficiem regulę.</s> <s xml:id="echoid-s25145" xml:space="preserve"> Deinde ponatur extremitas acus uel ſtili ferrei ſuper centrum ſuperioris fora-<lb/>minis:</s> <s xml:id="echoid-s25146" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s25147" xml:space="preserve"> umbra extremitatis acus ſuper centrum lucis:</s> <s xml:id="echoid-s25148" xml:space="preserve"> ablato quoq;</s> <s xml:id="echoid-s25149" xml:space="preserve"> ſtilo, reuertetur lumẽ ad <lb/>locum ſuũ.</s> <s xml:id="echoid-s25150" xml:space="preserve"> Idem quoq;</s> <s xml:id="echoid-s25151" xml:space="preserve"> accidit ponenti extremitatẽ acus ſuper centrum foraminis ſecundi.</s> <s xml:id="echoid-s25152" xml:space="preserve"> Deinde <lb/>ponatur extremitas acus ſuper centrũ ſphærę uitreę:</s> <s xml:id="echoid-s25153" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s25154" xml:space="preserve"> umbra extremitatis acus ſuper centrũ <lb/>lucis.</s> <s xml:id="echoid-s25155" xml:space="preserve"> Ex quo patet, quia lux tranſiens per centra duorũ foraminũ, tranſit etiã per centrũ ſphærę ui-<lb/>treę, & per mediũ ſuperficiei lucis, quę eſt in cõuexo uitri.</s> <s xml:id="echoid-s25156" xml:space="preserve"> Patet etiã ex his, qđ lux tranſiens in cor-<lb/>pus uitri, extenditur ſecundũ rectitudinẽ lineę tranſeuntis per centra duorũ foraminũ:</s> <s xml:id="echoid-s25157" xml:space="preserve"> & eſt illa li-<lb/>nea ſemidiameter ſphærę.</s> <s xml:id="echoid-s25158" xml:space="preserve"> Nam perpendicularis, exiens à centro baſis uitri ad laminam, eſt æqualis <lb/>diametro foraminis & lineę exeunti à centro foraminis perpendiculariter ad ſuperficiem laminę:</s> <s xml:id="echoid-s25159" xml:space="preserve"> & <lb/>quoniam hę duę perpendiculares cadunt ſuper diametrum laminę:</s> <s xml:id="echoid-s25160" xml:space="preserve"> palàm, quòd linea tranſiens per <lb/>centra duorũ foraminũ, cum extenditur in rectitudinẽ, peruenit ad centrum ſphærę uitreę:</s> <s xml:id="echoid-s25161" xml:space="preserve"> eſt ergo <lb/>in illa linea diameter huius ſphærę uitreę:</s> <s xml:id="echoid-s25162" xml:space="preserve"> eſt ergo perpẽdicularis ſuper ſuperficiẽ huius ſphęrę ք 72 <lb/>t 1 huius:</s> <s xml:id="echoid-s25163" xml:space="preserve"> quoniã enim trãſit centrũ ſphęrę, patet quòd ipſa eſt perpẽdicularis ſuper conuexã ſuper-<lb/>ficιẽ ſphærę, ſicut ſuperius patuit in uitris cubicis.</s> <s xml:id="echoid-s25164" xml:space="preserve"> Auferatur itaq;</s> <s xml:id="echoid-s25165" xml:space="preserve"> regula ſubtilis applicata ad ſuper <lb/>ficiem laminæ, & ponatur inſtrumentũ ſecundò in uas, ut prius, & moueatur quouſq;</s> <s xml:id="echoid-s25166" xml:space="preserve"> lux tranſeat <lb/>per duo foramina:</s> <s xml:id="echoid-s25167" xml:space="preserve"> inuenieturq́;</s> <s xml:id="echoid-s25168" xml:space="preserve"> lux ſuper oram inſtrumenti, & inuenietur centrũ lucis in puncto p, <lb/>quod eſt differentia cõmunis inter circumferentiã circuli medij, & lineam g k, perpendicularẽ in o-<lb/>ra inſtrumẽti:</s> <s xml:id="echoid-s25169" xml:space="preserve"> hoc eſt in extremitate diametri circulι medij, quæ eſt m p, tranſeuntis per centra duo-<lb/>rum foraminum m & y.</s> <s xml:id="echoid-s25170" xml:space="preserve"> Ex quo patet, quoniã lux tranſiens in corpus uitri, & perueniens ad centrũ <lb/>eius, prodiensq́;</s> <s xml:id="echoid-s25171" xml:space="preserve"> in corpus aeris, extenditur ſecundũ lineã, quę extendebatur in corpore uitri.</s> <s xml:id="echoid-s25172" xml:space="preserve"> Cum <lb/>enim linea recta tranſiens centra amborũ foraminũ, perpendicularis ſit ſuper ſuperficiẽ uitri:</s> <s xml:id="echoid-s25173" xml:space="preserve"> patet <lb/>quòd ipſa neceſſariò eſt perpendicularis ſuper ſuperficiẽ aeris tangentis uitri ſuperficiẽ.</s> <s xml:id="echoid-s25174" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s25175" xml:space="preserve"> ſi uaſi <lb/>infundatur aqua, remanente uitro in ſua poſitione, donec aqua ſuperfluat centro uitri:</s> <s xml:id="echoid-s25176" xml:space="preserve"> adhuc inue-<lb/>nietur centrũ lucis ſuper extremitatẽ diametri circuli medij:</s> <s xml:id="echoid-s25177" xml:space="preserve"> & ſi ſphæra uitrea tranſuertatur, ita ut <lb/>cõuexũ eius ſituetur ad ſecundũ foramẽ, & plana ſuperficies ad centrũ inſtrumẽti, ſcilicet punctũ e, <lb/>ſiue aqua ſuperfundatur, ſiue nõ, adhuc omnia alia accidẽt, quę in priori ſitu accidebãt:</s> <s xml:id="echoid-s25178" xml:space="preserve"> quoniã ſemք <lb/>radius trãſiens per cẽtra amborũ foraminũ, tranſibit etiã per centrũ ſphærę.</s> <s xml:id="echoid-s25179" xml:space="preserve"> Ex his omnibus ք uitra <lb/> <pb o="80" file="0382" n="382" rhead="VITELLONIS OPTICAE"/> cubica & ſphærica patet, quòd ſiue medium ſecundi diaphani fuerit denſius uel rarius, dum tamen <lb/>linea, per quam extenditur radius, fuerit perpendicularis ſuper ſuperficiem ſecũdi corporis, quòd <lb/>lux extenditur in ſecundo corpore ſecundum rectitudinẽ lineæ, per quam extendebatur in corpo-<lb/>re primo.</s> <s xml:id="echoid-s25180" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s25181" xml:space="preserve"> corpus enim uitri eſt denſioris diaphanitatis, quàm corpus aeris, <lb/>& etiam quàm corpus aquæ.</s> <s xml:id="echoid-s25182" xml:space="preserve"/> </p> <div xml:id="echoid-div947" type="float" level="0" n="0"> <figure xlink:label="fig-0381-01" xlink:href="fig-0381-01a"> <image file="0381-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0381-01"/> </figure> </div> </div> <div xml:id="echoid-div949" type="section" level="0" n="0"> <head xml:id="echoid-head760" xml:space="preserve" style="it">45. In medio ſecundi diaphani rarioris primo diaphano, fit refractio radiorum obliquè inci-<lb/>dentium à poſteriore ſuperficie ſecundi diaphani, à perpendiculari exeunte à puncto refractio-<lb/>nis ſuper ſuperficiem corporis ſecundi. Alhazen 7 n 7.</head> <p> <s xml:id="echoid-s25183" xml:space="preserve">Hoc quod nunc hic proponitur, eſt cõformiter prioribus per inſtrumentalẽ experientiã declaran <lb/>dum.</s> <s xml:id="echoid-s25184" xml:space="preserve"> Aſſumatur enim illud uitrũ ſphæricũ, quo iam in præcedenti ꝓximo theoremate uſi ſumus, & <lb/>ponatur ſuper laminã inſtrumẽti, ita qđ ſuperficies plana ipſius reſpiciat foramina, & quòd mediũ <lb/>lineæ rectę, quæ eſt in ipſo, ſit ſuper centrũ laminę, & linea, quæ eſt cõmunis ſectio ſuperficierũ pla-<lb/>narũ uitri, cadat obliquè ſuper diametrũ laminæ quacũq;</s> <s xml:id="echoid-s25185" xml:space="preserve"> obliquatiõe.</s> <s xml:id="echoid-s25186" xml:space="preserve"> Palàm ergo, quòd linea tran-<lb/>ſiens centra duorũ foraminũ, obliqua eſt ſuper ſuperficiẽ planã uitri.</s> <s xml:id="echoid-s25187" xml:space="preserve"> Cõiungatur itaq;</s> <s xml:id="echoid-s25188" xml:space="preserve"> uitrũ laminę <lb/>inſtrumẽti ſecundũ hunc ſitũ firmiter:</s> <s xml:id="echoid-s25189" xml:space="preserve"> & ponatur inſtrumentũ in uas, & uas in ſole, moueaturq́;</s> <s xml:id="echoid-s25190" xml:space="preserve"> in-<lb/>ſtrumentũ, donec lux tranſeat per duo foramina:</s> <s xml:id="echoid-s25191" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s25192" xml:space="preserve"> lux in interiori ora inſtrumenti:</s> <s xml:id="echoid-s25193" xml:space="preserve"> & centrũ <lb/>lucis erit in circunferẽtia medij circuli, ſed extra illũ punctũ p, qui eſt cõmunis differẽtia circũferen <lb/>tię medij circuli, & lineę ſtanti in ora inſtrumenti, quę eſt g k:</s> <s xml:id="echoid-s25194" xml:space="preserve"> & erit declinatio eius ad partẽ, in qua <lb/>eſt ſol:</s> <s xml:id="echoid-s25195" xml:space="preserve"> erit ergo ad partẽ perpendicularis exeuntis à loco refractiõis ſuper ſuperficiẽ ſphæricã uitri.</s> <s xml:id="echoid-s25196" xml:space="preserve"> <lb/>Et quoniã hæc lux extenditur in aere ſecundũ rectitudinẽ lineę tranſeuntis per centra duorũ fora-<lb/>minũ, ut patet per 1 huius:</s> <s xml:id="echoid-s25197" xml:space="preserve"> & hæc linea in hoc ſitu քuenit ad centrũ ſphærę uitreę:</s> <s xml:id="echoid-s25198" xml:space="preserve"> & eſt obliqua ſuք <lb/>ſuperficiẽ ſphærę planã:</s> <s xml:id="echoid-s25199" xml:space="preserve"> palã ergo, quia terminatio extenſiõis illius lucis eſt in centro uitri.</s> <s xml:id="echoid-s25200" xml:space="preserve"> Extendi <lb/>tur ergo lux in corpus uitri ſecundũ lineã rectã, exeuntẽ à cẽtro ſphærę ad circunferentiã, quę linea <lb/>cũ ſit dιameter, palàm per 72 t 1 huius, quoniã ipſa eſt perpẽdicularis ſuper ſphęricã ſuperficiẽ uitri:</s> <s xml:id="echoid-s25201" xml:space="preserve"> <lb/>ergo & ſuper concauã ſuperficiẽ aeris continentis ſphærã uitri:</s> <s xml:id="echoid-s25202" xml:space="preserve"> nõ ergo refringitur in aere ſecundo, <lb/>ſicutneq;</s> <s xml:id="echoid-s25203" xml:space="preserve"> in primo, ſed neq;</s> <s xml:id="echoid-s25204" xml:space="preserve"> refringitur in corpore uitri, nec in cõuexo ipſius:</s> <s xml:id="echoid-s25205" xml:space="preserve"> refringitur ergo apud <lb/>centrum uitri, quia fuit obliqua ſuper ſuperficiem eius planã, in qua eſt centrũ uitri.</s> <s xml:id="echoid-s25206" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s25207" xml:space="preserve"> ex <lb/>his experimentationibus illud, quod eſt etiã ſuperius declaratũ, ſcilicet quoniã lux, ſi fuerit exten-<lb/>ſa in corpore ſubtiliori obliquè incidens ſuperficiei corporis groſsioris, refringetur ab ipſo:</s> <s xml:id="echoid-s25208" xml:space="preserve"> & erit <lb/>eius refractio ad partẽ perpendicularis ſuper ſuperficiẽ ſphæricã corporis groſsioris, ſicut ք 43 hu-<lb/>ius patuit:</s> <s xml:id="echoid-s25209" xml:space="preserve"> ut ſi fiat refractio ex aere ad aquã, erit illa refractio ad partẽ perpẽdicularis exeũtis à loco <lb/>refractionis ſuper ſuperficiẽ aquæ, & nõ peruenit refractio ad perpendicularẽ.</s> <s xml:id="echoid-s25210" xml:space="preserve"> Quòd ſi uitrũ è con-<lb/>uerſo ſituetur, ſcilicet ut ſuperficies eius ſphęrica conuexa reſpiciat ſuperius foramẽ, & punctũ me-<lb/>diũ lineę (quę eſt cõmunis differentia ſuperficierũ planarũ) quod eſt centrũ ſphærę uitreę, ſit ſuper <lb/>centrũ inſtrumenti, cadatq́;</s> <s xml:id="echoid-s25211" xml:space="preserve"> hæc linea obliquè ſuper diametrũ laminæ:</s> <s xml:id="echoid-s25212" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s25213" xml:space="preserve"> in ipſa ſuperficie <lb/>laminæ à centro laminæ linea perpendicularis ſuper lineã, quæ eſt cõmunis ſectio illarum planarũ <lb/>ſuperficierũ, quę neceſſariò erit perpendicularis ſuper ſuperficiem planam uitri erectã ſuper ſuper-<lb/>ficiem laminæ:</s> <s xml:id="echoid-s25214" xml:space="preserve"> ponaturq́;</s> <s xml:id="echoid-s25215" xml:space="preserve"> inſtrumentũ in uaſe ſine aqua, & moueatur, quouſq;</s> <s xml:id="echoid-s25216" xml:space="preserve"> lux pertranſeat duo <lb/>foramina:</s> <s xml:id="echoid-s25217" xml:space="preserve"> cadet centrum lucis in circunferentia medij circuli extra punctum p, quod eſt differentia <lb/>cõmunis medij circuli, & lineæ g k perpendicularis ſuper ſuperficiẽ laminæ ducta in ora inſtrumen <lb/>ti, quod punctum p eſt extremitas diametri medij circuli, quæ eſt m p:</s> <s xml:id="echoid-s25218" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s25219" xml:space="preserve"> declinatio lucis ad par-<lb/>tem contrariam illi, in qua eſt perpendicularis educta à loco refractionis ſuper planam ſuperficiem <lb/>uitri.</s> <s xml:id="echoid-s25220" xml:space="preserve"> Hæc autẽ lux extenditur in uitro ſecundum rectitudinẽ lineæ tranſeuntis per centra duorum <lb/>foraminum:</s> <s xml:id="echoid-s25221" xml:space="preserve"> quoniã illa linea cum per centrum ſphæræ uitreæ tranſeat, eſt illa diameter ſphæræ ui-<lb/>treæ:</s> <s xml:id="echoid-s25222" xml:space="preserve"> fit itaq;</s> <s xml:id="echoid-s25223" xml:space="preserve"> refractio lucis apud centrum ſphæræ uitreæ:</s> <s xml:id="echoid-s25224" xml:space="preserve"> quoniam lux tranſiens centra amborum <lb/>foraminum fit obliqua ſuper ſuperficiem planam uitri, & ſuper ſuperficiem aeris contingentis ui-<lb/>trum.</s> <s xml:id="echoid-s25225" xml:space="preserve"> Et ſi aqua infundatur uaſi, quouſq;</s> <s xml:id="echoid-s25226" xml:space="preserve"> ſuperemineat centro inſtrumenti:</s> <s xml:id="echoid-s25227" xml:space="preserve"> cadet adhuc centrũ lu-<lb/>cis in circumferentia medij circuli extra extremitatem ſui diametri obliquè ad partem contrariam <lb/>illi parti, ſuper quam cadit perpendicularis.</s> <s xml:id="echoid-s25228" xml:space="preserve"> Et quoniã aer eſt ſubtilior quàm aqua, & aqua ſubtilior <lb/>uitro:</s> <s xml:id="echoid-s25229" xml:space="preserve"> maior fiet diſtantia centri lucis ab extremitate diametri medij circuli in aere, quàm in aqua.</s> <s xml:id="echoid-s25230" xml:space="preserve"> <lb/>Quòd ſi uitrum ponatur aliter in ſuperficie laminæ, ſcilicet ut linea, quæ eſt communis differentia <lb/>duarum ſuperficierũ planarum ipſius uitri, ſit ſuper lineam perpendiculariter diametrum laminæ <lb/>ſecantem, non tamen ſit eius medius punctus (qui eſt centrum ſphæræ uitreæ) ſuper centrum lami-<lb/>næ, & uertatur conuexum uitri ad foramina, & figatur regula ſubtilis ſuper ſuperficiem laminę ere-<lb/>cta ſuper oram eius, ſitq́;</s> <s xml:id="echoid-s25231" xml:space="preserve"> ſuperficies eius, in qua eſt linea, ex parte uitri:</s> <s xml:id="echoid-s25232" xml:space="preserve"> & terminus regulæ ſecet dia <lb/>metrum laminæ perpendiculariter:</s> <s xml:id="echoid-s25233" xml:space="preserve"> palàm, quia linea tranſiens per centra foraminum duorum, non <lb/>tranſit per centrũ ſphæræ, ſed per aliud punctum ſuperficiei planæ ipſius uitri:</s> <s xml:id="echoid-s25234" xml:space="preserve"> & erit obliqua ſuper <lb/>ſphæricam ſuperficiem per 72 t 1 huius.</s> <s xml:id="echoid-s25235" xml:space="preserve"> Ponatur itaq;</s> <s xml:id="echoid-s25236" xml:space="preserve"> inſtrumentum in uaſe, & uas in ſole, & mo-<lb/>ueatur inſtrumentum, quouſq;</s> <s xml:id="echoid-s25237" xml:space="preserve"> lux tranſeat per centra duorum foraminum:</s> <s xml:id="echoid-s25238" xml:space="preserve"> & non cadet lux di-<lb/>rectè ſuper ſuperficiem regulæ, neq;</s> <s xml:id="echoid-s25239" xml:space="preserve"> centrum lucis cadet in linea, quæ eſt in ſuperficie regulæ, ſed <lb/>declinabit obliquè extra lineam, quæ tranſit per centra duorum foraminum ad partem, in qua eſt <lb/>centrum uitri, hoc eſt ad partem contrariam perpendicularis, exeuntis à loco refractionis per-<lb/>pendiculariter ſuper ſuperficiem uitri ſphæricam:</s> <s xml:id="echoid-s25240" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s25241" xml:space="preserve"> linea pertranſiens centra duorum fo-<lb/> <pb o="81" file="0383" n="383" rhead="LIBER SECVNDVS."/> raminum perpendicularis ſuper ſuperficiem ultri planã per 8 p 11:</s> <s xml:id="echoid-s25242" xml:space="preserve"> quoniã illa linea eſt æquidiſtans <lb/>lineæ f g diametro laminæ, quæ ex hypotheſi eſt perpendicularis ſuper ſuperficiem planam uitri.</s> <s xml:id="echoid-s25243" xml:space="preserve"> Si <lb/>ergo lux tranſiret per centra duorum foraminũ, & extenderetur ſecundũ rectitudinem ad planã ui-<lb/>tri ſuperficiem:</s> <s xml:id="echoid-s25244" xml:space="preserve"> palàm, quòd tunc extenderetur ſecundũ rectitudinem in aere:</s> <s xml:id="echoid-s25245" xml:space="preserve"> ſed centrũ lucis, quæ <lb/>eſt in regula, cum nõ cadat in rectitudinẽ huius lineę:</s> <s xml:id="echoid-s25246" xml:space="preserve"> patet, quòd lux nõ extenditur in eius rectitu-<lb/>dine ad ſuperficiem planã uitri:</s> <s xml:id="echoid-s25247" xml:space="preserve"> eſt ergo lux refracta, ſed nõ refringitur in aere, neq;</s> <s xml:id="echoid-s25248" xml:space="preserve"> in corpore uitri.</s> <s xml:id="echoid-s25249" xml:space="preserve"> <lb/>Refringitur itaq;</s> <s xml:id="echoid-s25250" xml:space="preserve"> apud ſphæricã ſuperficiem uitri:</s> <s xml:id="echoid-s25251" xml:space="preserve"> incidit enim obliquè ſuper ſphæricã ſuperficiem, <lb/>quoniã linea tranſiens centra duorũ foraminũ, nõ tranſit per centrũ uitri:</s> <s xml:id="echoid-s25252" xml:space="preserve"> & hæc lux egrediẽs à pla-<lb/>na ſuperficie uitri, quoniã obliquè aeri incidit, plus refringitur.</s> <s xml:id="echoid-s25253" xml:space="preserve"> Quòd ſi uitrũ è cõtrario diſponatur, <lb/>ut eius ſuperficies plana opponatur foramini primò ſic, quòd cõmunis differentia ſit ſuper lineam <lb/>ſecantẽ diametrum laminæ perpendiculariter, & medius punctus illius lineæ ſit extra centrum la-<lb/>minæ:</s> <s xml:id="echoid-s25254" xml:space="preserve"> tunc ergo linea pertranſiens centra duorũ foraminum non tranſit per centrum uitri, ſed per <lb/>alium punctũ illius planę ſuperficiei, & eſt perpendicularis ſuper illam ſuperficiem.</s> <s xml:id="echoid-s25255" xml:space="preserve"> Moueatur itaq;</s> <s xml:id="echoid-s25256" xml:space="preserve"> <lb/>inſtrumentũ in ſole, donec lux tranſeat per ambo foramina:</s> <s xml:id="echoid-s25257" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s25258" xml:space="preserve"> centrum lucis, quę cadit in inte <lb/>riore parte oræ ipſius inſtrumenti in peripheria medij circuli, extra punctũ p, quod eſt extremitas <lb/>diametri medij circuli, quæ eſt linea m p, ſed declinabit ad partẽ, in qua eſt centrũ uitreæ ſphærę:</s> <s xml:id="echoid-s25259" xml:space="preserve"> & <lb/>linea, quę egreditur à centro huius ſphærę in imaginatione ad locum refractionis, eſt perpendicula <lb/>ris ſuper ſuperficiem huius ſphærę:</s> <s xml:id="echoid-s25260" xml:space="preserve"> eſt ergo perpendicularis ſuper ſuperficiem aeris contingentis <lb/>ſuperficiem ſphærę uitreæ.</s> <s xml:id="echoid-s25261" xml:space="preserve"> Hæc itaq;</s> <s xml:id="echoid-s25262" xml:space="preserve"> refractio eſt ad partem contrariã illi, in qua eſt perpendicula-<lb/>ris, exiens à loco refractionis ſuper ſuperficiem aeris cõtingentis ſphæram.</s> <s xml:id="echoid-s25263" xml:space="preserve"> Lux uerò tranſiens cen <lb/>tra<gap/>duorum foraminũ, pertranſit corpus uitri rectè, cũ ſit perpẽdicularis ſuper ſuperficiem planam <lb/>uitri:</s> <s xml:id="echoid-s25264" xml:space="preserve"> ſed non eſt perpendicularis ſuper ſuperficiẽ conuexam, cum non pertranſeat centrũ ſphæræ:</s> <s xml:id="echoid-s25265" xml:space="preserve"> <lb/>ergo etiam non eſt hæc lux perpendicularis ſuper ſuperficiem aeris contingentis conuexũ uitri:</s> <s xml:id="echoid-s25266" xml:space="preserve"> & <lb/>quia hæc lux refracta inuenitur:</s> <s xml:id="echoid-s25267" xml:space="preserve"> refringitur ergo apud cõuexam ſuperficiem ſphæræ uitreæ.</s> <s xml:id="echoid-s25268" xml:space="preserve"> Quòd <lb/>ſi a qua tunc infundatur uaſi infra centrum laminæ:</s> <s xml:id="echoid-s25269" xml:space="preserve"> inuenietur etiam lux refracta ad partem, in qua <lb/>eſt centrum uitri:</s> <s xml:id="echoid-s25270" xml:space="preserve"> hoc autem eſt ad partẽ contrariam illi, in quam cadit perpendicularis, exiẽs à loco <lb/>refractionis, quæ extenditur in corpore aeris, perpendicularis ſuper concauam ipſius aeris ſuperfi-<lb/>ciem conuexam uitri contingentem.</s> <s xml:id="echoid-s25271" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s25272" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div950" type="section" level="0" n="0"> <head xml:id="echoid-head761" xml:space="preserve" style="it">46. Omnem radium incidentem & refractum in eadem plana ſuperficie conſiſtere eſt neceſſe. <lb/>Alhazen 5 n 7.</head> <p> <s xml:id="echoid-s25273" xml:space="preserve">Sed & id, quod nunc proponitur, poteſt experimẽtaliter declarari.</s> <s xml:id="echoid-s25274" xml:space="preserve"> Quoniã enim omnibus diſpo <lb/>ſitis, ut in 43 huius, lux incidens centro lucis, quę eſt in ſuperficie aquæ, & à centro lucis exiſtentis <lb/>ſuper ſuperficiem aquæ, quod eſt centrum medij circuli, incidens centro lucis intra aquam exiſten-<lb/>tis, quod eſt in circumferentia circuli medij, tranſit per centra amborũ foraminũ, quæ ſimiliter ſunt <lb/>in ſuperficie medij circuli:</s> <s xml:id="echoid-s25275" xml:space="preserve"> palàm, quoniã linea, ſecundũ quã lumẽ incidit ſuperficiei aquę per mediũ <lb/>aerẽ, & ſecundũ quã refringitur in aquę medio, ſunt in eadem ſuperficie:</s> <s xml:id="echoid-s25276" xml:space="preserve"> quoniã utraq;</s> <s xml:id="echoid-s25277" xml:space="preserve"> ipſarũ eſt in <lb/>ſuperficie medij circuli trium aſsignatorũ circulorum.</s> <s xml:id="echoid-s25278" xml:space="preserve"> Inuenitur autẽ hæc refractio in radio ſolari, <lb/>quando radius ſolaris tranſiens per centra foraminum, fuerit obliquus ſuper aquæ ſuperficiem, nõ <lb/>quãdo fuerit perpendicularis:</s> <s xml:id="echoid-s25279" xml:space="preserve"> & propter obliquitatẽ ſitus inſtrumenti à centro ſphærę aquæ, nunꝗ̃ <lb/>fiet hæc linea radialis perpendicularis ſuper ſuperficiẽ aquæ, niſi ſol fuerit perpendiculariter ſuper <lb/>zenith capitis:</s> <s xml:id="echoid-s25280" xml:space="preserve"> ſole uerò ultra uel citra zenith capitum exiſtente, ſatis euidens eſt hæc experimenta <lb/>tio omni tẽpore.</s> <s xml:id="echoid-s25281" xml:space="preserve"> Patet ergo id, quod proponebatur.</s> <s xml:id="echoid-s25282" xml:space="preserve"> Et hanc ſuperficiẽ dicimus ſuperficiẽ refractio-<lb/>nis.</s> <s xml:id="echoid-s25283" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s25284" xml:space="preserve"> exijs omnibus quinq;</s> <s xml:id="echoid-s25285" xml:space="preserve"> præmiſsis propoſitionibus, quoniã omnis lux pertrãſit quæ-<lb/>cunq;</s> <s xml:id="echoid-s25286" xml:space="preserve"> corpora diaphana ſecundũ lineas rectas:</s> <s xml:id="echoid-s25287" xml:space="preserve"> & quãdiu lineę ſunt քpendiculares ſuք ſuperficies <lb/>corporũ, quantũcunq;</s> <s xml:id="echoid-s25288" xml:space="preserve"> etiã diuerſę ſint diaphanitatis, ſemper extẽditur ſecundũ rectitudinẽ eiuſdẽ <lb/>lineę, & nõ refringitur.</s> <s xml:id="echoid-s25289" xml:space="preserve"> In corpore uerò diuerſę diaphanitatis omnis lux ſuperficiei ſecũdi corporis <lb/>obliquè incidẽs, refringitur ſecundũ lineas rectas alias ab illis, ſecundũ quas incidebat primo cor-<lb/>pori:</s> <s xml:id="echoid-s25290" xml:space="preserve"> quę tamẽ lineę ſemper erũt in eadẽ ſuperficie plana, imaginatę ſecare utrunq;</s> <s xml:id="echoid-s25291" xml:space="preserve"> illorũ corporũ:</s> <s xml:id="echoid-s25292" xml:space="preserve"> <lb/>& hæc ſuperficies in inſpectiõe inſtrumẽti eſt medius circulus triũ circulorũ ſignatorũ in interiore <lb/>parte orę inſtrumẽti, cuius diameter eſt linea m p.</s> <s xml:id="echoid-s25293" xml:space="preserve"> Cũ uerò lux obliqua exiuerit à corpore ſubtiliori <lb/>ad groſsius:</s> <s xml:id="echoid-s25294" xml:space="preserve"> refringetur ad partẽ քpendicularis exeũtis à loco refractionis, quę eſt քpẽdicularis ſu-<lb/>per ſuperficiẽ groſsioris ſecũdi corporis:</s> <s xml:id="echoid-s25295" xml:space="preserve"> & cũ lux obliqua exiuit à corpore groſsiori ad ſubtilius, re <lb/>fringetur ad partẽ cõtrariã prędicto modo ductę ſuք ſuperficiẽ corporis ſecundi, ſcilicet ſubtilioris.</s> <s xml:id="echoid-s25296" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div951" type="section" level="0" n="0"> <head xml:id="echoid-head762" xml:space="preserve" style="it">47. Radio perpendiculari omne corp{us} diaphanũ penetrante, radi{us} obliquè incidens in me-<lb/>dio ſecũdi diaphani denſioris refringitur ad perpẽdicularẽ ductã à pũcto incidẽtiæ ſuper ſecundi <lb/>diaphani ſuperficiẽ: & in medio ſecundi diaphani rarioris refringitur ab eadẽ. Alhazen 8 n 7.</head> <p> <s xml:id="echoid-s25297" xml:space="preserve">Illud, quod particularibus experientijs hactenus inſtrumentaliter probatũ eſt, naturali demon-<lb/>ſtratiõe intendimus adiuuare.</s> <s xml:id="echoid-s25298" xml:space="preserve"> Omnes enim motus naturales, qui fiunt ſecundũ lineas perpendicu-<lb/>lares, ſunt fortiores, quoniã coadiuuãtur uirtute uniuerſali cœleſti ſecundũ lineã rectã breuiſsimã, <lb/>omni ſubiecto corpori influẽte.</s> <s xml:id="echoid-s25299" xml:space="preserve"> Impulſiones ꝓiectationũ factarũ perpendiculariter, ſunt fortiores <lb/>eis, quę fiunt obliquè:</s> <s xml:id="echoid-s25300" xml:space="preserve"> & ſimiliter percuſsiones, quę fiunt perpendiculariter, ſunt omnibus obliquis <lb/>percuſsionib.</s> <s xml:id="echoid-s25301" xml:space="preserve"> fortiores:</s> <s xml:id="echoid-s25302" xml:space="preserve"> & inter oẽs obliquas, fortiores ſunt illę, quæ plus accedũt ad perpẽdiculari-<lb/>tatẽ.</s> <s xml:id="echoid-s25303" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s25304" xml:space="preserve"> omnis corporis dẽſitas impedit tranſitũ luminis, neceſſe eſt lumẽ imaginari repelli à <lb/> <pb o="82" file="0384" n="384" rhead="VITELLONIS OPTICAE"/> tranſitu per reſiſtentiã corporis denſi, & plus per reſiſtentiã corporis denſioris:</s> <s xml:id="echoid-s25305" xml:space="preserve"> & per hanc reſiſten-<lb/>tiam qualitatis paſsiuæ, quæ eſt denſitas ad qualitatẽ actiuam, quæ eſt lumen, intelligimus quendã <lb/>modũ motionis luminis per medium corporũ reſiſtentiũ, quæ ſecundũ plus & minus capacia ſunt <lb/>impreſsionis luminaris, nõ quòd in tranſmutatione locali ipſius luminis ſit aliquis motus, ut patet <lb/>per 2 huius:</s> <s xml:id="echoid-s25306" xml:space="preserve"> ſed quia lumen in eodẽ inſtanti ſecundũ diuerſitatẽ mediorũ ſe plus comprimit uel dif-<lb/>fundit:</s> <s xml:id="echoid-s25307" xml:space="preserve"> & hoc uocamus hic motũ ipſius lucis.</s> <s xml:id="echoid-s25308" xml:space="preserve"> Omnis itaq;</s> <s xml:id="echoid-s25309" xml:space="preserve"> lux pertrãſiẽs corpus diaphanũ, motu ue <lb/>lociſsimo & inſenſibili pertrãſit:</s> <s xml:id="echoid-s25310" xml:space="preserve"> ſic tamẽ, quòd per magis diaphana uelocior fit motus ꝗ̃ per minus <lb/>diaphana.</s> <s xml:id="echoid-s25311" xml:space="preserve"> Omne enim corpus diaphanũ plus & minus reſiſtit penetrationi lucis, ſecundũ quod eſt <lb/>participans diaphanitate plus uel minus:</s> <s xml:id="echoid-s25312" xml:space="preserve"> groſsities enim corporũ reſiſtens eſt ſemper luminis pene <lb/>trationi.</s> <s xml:id="echoid-s25313" xml:space="preserve"> Cũ ergo lux pertranſiuerit corpus aliquod diaphanũ obliquè, & occurrerit corpori alij dia-<lb/>phano groſsiori:</s> <s xml:id="echoid-s25314" xml:space="preserve"> tũc corpus groſsius reſiſtit luci uehementius, ꝗ̃ prius corpus rarius reſiſtebat:</s> <s xml:id="echoid-s25315" xml:space="preserve"> ne-<lb/>ceſſe eſt ergo, quòd ꝓpter reſiſtentiã illius corporis denſioris motus lucis trãſmutetur:</s> <s xml:id="echoid-s25316" xml:space="preserve"> & ſi reſiſten <lb/>tia fuerit fortis, tunc motus ille ad partẽ contrariã refringetur:</s> <s xml:id="echoid-s25317" xml:space="preserve"> quia uerò nõ reſiſtit fortiter, ideo lu-<lb/>men nõ redibit in partẽ, ad quã mouebatur.</s> <s xml:id="echoid-s25318" xml:space="preserve"> Si uerò reſiſtentia fuerit debilis ꝓpter maiorẽ raritatẽ <lb/>corporis plus diaphani:</s> <s xml:id="echoid-s25319" xml:space="preserve"> tũc lux incidens nõ refringetur ad contrariã partẽ, nec poterit per illã lineã <lb/>ꝓcedere, per quã inceperat, ſed mutabitur in ſitu:</s> <s xml:id="echoid-s25320" xml:space="preserve"> cũ uerò քpendiculariter inciderit quibuslibet cor <lb/>poribus diaphanis & quantũcunq;</s> <s xml:id="echoid-s25321" xml:space="preserve"> diuerſæ diaphanitatis, nõ mutabitur, ſed directè omnia penetra <lb/>bit:</s> <s xml:id="echoid-s25322" xml:space="preserve"> quoniã perpendicularis fortior eſt omnibus, & obliqui uiciniores perpẽdiculari, ſunt fortiores <lb/>omnibus remotioribus.</s> <s xml:id="echoid-s25323" xml:space="preserve"> Cũ itaq;</s> <s xml:id="echoid-s25324" xml:space="preserve"> corpori diaphano groſsiori lux incidit obliquè, extenditur ſecun-<lb/>dum lineã rectam approximantẽ ad perpendicularẽ, exeuntẽ à puncto, in quo lux occurrit ſuperfi-<lb/>ciei corporis diaphani groſsi, productã ſuper ſuperficiẽ corporis groſsioris, ideo, quia facilimus mo <lb/>tuũ eſt ſecundũ lineam perpendicularẽ.</s> <s xml:id="echoid-s25325" xml:space="preserve"> Si ergo radius lucis inciderit ſuper lineã perpendicularẽ, <lb/>tranſibit rectè ꝓpter fortitudinẽ motus ſuper perpendicularẽ:</s> <s xml:id="echoid-s25326" xml:space="preserve"> & ſi radius inciderit obliquè, tunc nõ <lb/>poterit tranſire ꝓpter debilitatẽ motus ſuper lineas obliquas.</s> <s xml:id="echoid-s25327" xml:space="preserve"> Accidit ergo ut declinet ad partẽ ali-<lb/>quã, per quã facilior ſit tranſitus, ꝗ̃ per illam partẽ, ad quã per lineam incidentię mouebatur:</s> <s xml:id="echoid-s25328" xml:space="preserve"> facilior <lb/>aũt motuũ, & plus adiutus cœleſti influentia eſt ſupèr lineã perpendicularẽ:</s> <s xml:id="echoid-s25329" xml:space="preserve"> quod enim uicinius eſt <lb/>perpendiculari, facilioris eſt tranſitus, ꝗ̃ remotius ab illa.</s> <s xml:id="echoid-s25330" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s25331" xml:space="preserve"> ut à puncto a corporis luminoſi <lb/>incidant radij quàm plures per mediũ a b ſuper ſuperficiem alterius diaphani corporis, in qua ſit li-<lb/>nea b c d e:</s> <s xml:id="echoid-s25332" xml:space="preserve"> & ſit b f linea profunditatis illius corporis:</s> <s xml:id="echoid-s25333" xml:space="preserve"> & ſit <lb/> <anchor type="figure" xlink:label="fig-0384-01a" xlink:href="fig-0384-01"/> linea a b perpendicularis ſuper illam ſuperficiẽ.</s> <s xml:id="echoid-s25334" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s25335" xml:space="preserve"> <lb/>ſecundũ rationẽ præmiſſam fortitudinis perpendiculariũ, <lb/>& per experientias inſtrumẽtales ք 42 & 44 huius, quoniã <lb/>radius incidẽs ſecundũ lineã a b perpẽdiculariter, penetrat <lb/>totũ corpus b e f.</s> <s xml:id="echoid-s25336" xml:space="preserve"> Radius uerò incidens ſecundũ lineã a c, ſi <lb/>directè trãſeat corpus b e f:</s> <s xml:id="echoid-s25337" xml:space="preserve"> tunc nõ erit diuerſitas in diapha <lb/>nitate corporũ a b e & b e f:</s> <s xml:id="echoid-s25338" xml:space="preserve"> qđ eſt cõtra hypotheſim:</s> <s xml:id="echoid-s25339" xml:space="preserve"> linea <lb/>itaq;</s> <s xml:id="echoid-s25340" xml:space="preserve"> a c ꝓpter diuerſitatẽ reſiſtentiæ nõ erit linea cõtinua.</s> <s xml:id="echoid-s25341" xml:space="preserve"> <lb/>Sed ſi per corpus minus reſiſtẽs mouebatur liberè per lineã <lb/>a c, nõ poteſt in corpore plus uel minus reſiſtẽte per eandẽ <lb/>lineã moueri.</s> <s xml:id="echoid-s25342" xml:space="preserve"> Si ergo corpus b e f ſit denſius corpore a b e, <lb/>patet ex præmiſsis, quòd difficilior eſt trãſitus per illud.</s> <s xml:id="echoid-s25343" xml:space="preserve"> Si <lb/>itaq;</s> <s xml:id="echoid-s25344" xml:space="preserve"> linea a c refringitur à linea perpẽdiculari, ducta à pun-<lb/>cto c ſuper ſuperficiẽ corporis b c d e, quę ſit c g, debilitabi-<lb/>tur, nec ad aliquid perueniet effectus eius:</s> <s xml:id="echoid-s25345" xml:space="preserve"> fruſtra ergo inci <lb/>debat:</s> <s xml:id="echoid-s25346" xml:space="preserve"> natura aũt fruſtra nihil agit, ſicut in principio ſuppo-<lb/>ſitũ eſt:</s> <s xml:id="echoid-s25347" xml:space="preserve"> linea ergo a c (ut etiã oſtenſum eſt experimentaliter <lb/>ք 43 huius) refringitur neceſſariò ad partẽ perpendicularis <lb/>c g, ut fortificetur actio eius:</s> <s xml:id="echoid-s25348" xml:space="preserve"> ſimiliter quoq;</s> <s xml:id="echoid-s25349" xml:space="preserve"> eſt de radijs in-<lb/>cidentib.</s> <s xml:id="echoid-s25350" xml:space="preserve"> ſecundũ lineas a d & a e.</s> <s xml:id="echoid-s25351" xml:space="preserve"> Qđ ſi corpus, in cuius ſu <lb/>perficie eſt linea b c d e, fuerit diaphanitatis rarioris, ꝗ̃ ſit <lb/>corpus a b e, adhuc ꝓpter fortitudinẽ actionis, radius per-<lb/>pendicularis, ꝗ eſt a b, penetrat irrefractus, radius uerò ſe-<lb/>cundũ lineã a c tranſiens corpus denſius, & in puncto c inci <lb/>dens ſuperficiei corporis rarioris, nõ inuenit reſiſtentiã quã <lb/>prius.</s> <s xml:id="echoid-s25352" xml:space="preserve"> Et quia formarũ propriũ eſt ſemք ſe diffundere ſecundũ amplitudinẽ omnis capacis materię:</s> <s xml:id="echoid-s25353" xml:space="preserve"> <lb/>patet, quòd radius a c nõ ꝓcedit ſecundũ lineã a c:</s> <s xml:id="echoid-s25354" xml:space="preserve"> quia ſic diſpoſitio diaphanorũ corporũ ſecundũ <lb/>reſiſtentiã ad receptionẽ luminis eſſet uniformis, qđ eſt contra hypotheſim:</s> <s xml:id="echoid-s25355" xml:space="preserve"> refringitur ergo radius <lb/>a c, ſed nõ ad perpendicularẽ c g:</s> <s xml:id="echoid-s25356" xml:space="preserve"> quoniã illa refractio nõ fit propter reſiſtentiã materię, ſed ꝓpter ui <lb/>ctoriã formæ agentis ſuper materiã plus diſpoſitã ꝗ̃ prius:</s> <s xml:id="echoid-s25357" xml:space="preserve"> unde forma diffundit ſe uirtute ꝓpria ab <lb/>incepto ꝓgreſſu ſecundũ lineã a c, & ad partẽ cõtrariã ipſius perpendicularis c g, & eius æquidiſtan <lb/>tis, quę b f:</s> <s xml:id="echoid-s25358" xml:space="preserve"> & ſimiliter eſt de omnib.</s> <s xml:id="echoid-s25359" xml:space="preserve"> alijs obliquis radijs ut a d & a e.</s> <s xml:id="echoid-s25360" xml:space="preserve"> Motus itaq;</s> <s xml:id="echoid-s25361" xml:space="preserve"> radij incidentis ob-<lb/>liquè ſecũdũ lineã a c in corpore ſecũdi diaphani denſioris, qđ eſt b e f, cõponitur ex motu in partẽ <lb/>քpendicularis a b, trãſeuntis per corpus b e f, in quo eſt motus, & ex motu facto ſuper lineã c b, quæ <lb/>eſt perpẽdicularis ſuper lineã c g.</s> <s xml:id="echoid-s25362" xml:space="preserve"> Quoniã enim trãſitus perpẽdicularis eſt fortiſsimus & facillimus <lb/>motuũ, & denſitas corporis reſiſtit termino motus, ad quẽ intẽdebat, linea a c neceſſariò mouebitur <lb/> <pb o="83" file="0385" n="385" rhead="LIBER SECVNDVS."/> ad perpendicularem c g, exeuntem à puncto c, in quo radius a c occurrit ſuperficiei corporis den-<lb/>ſioris.</s> <s xml:id="echoid-s25363" xml:space="preserve"> Et quoniã illi motui reſiſtitur propter groſsiciem medij, & etiam propter naturã alterius m o-<lb/>tus, qui eſt ſuper lineam c b, qui propter reſiſtentiã medij non omnino dimittitur, ſed tantùm impe-<lb/>ditur:</s> <s xml:id="echoid-s25364" xml:space="preserve"> declinabit ergo lumen uerſus punctum b, ſemper approximans perpendiculari a b f:</s> <s xml:id="echoid-s25365" xml:space="preserve"> fit itaq;</s> <s xml:id="echoid-s25366" xml:space="preserve"> <lb/>in medio ſecundę diaphanitatis groſsiore medio primo, refractio radij a c ſecundũ lineam c l, pro-<lb/>pinquiorẽ perpendiculari c g, exeũti à puncto c, in quo occurrit corpori denſiori, quàm linea a c, per <lb/>quam incidebat ſuperficiei illius corporis, producta ultra punctum c a d punctũ h, propinqua fuerit <lb/>eidem perpendiculari eductæ ultra punctũ c ad punctum h, ita, ut angulus a c h ſit maior angulo l c <lb/>g:</s> <s xml:id="echoid-s25367" xml:space="preserve"> non concurret tamen cum perpendiculari b f uerſus punctum f, ſed uerſus punctum a per 2 t 1 hu <lb/>ius, quoniã concurrit cũ eius æquidiſtante linea c g in puncto c.</s> <s xml:id="echoid-s25368" xml:space="preserve"> Cum uerò radius a c exiuerit à cor-<lb/>pore groſsiore ad ſubtilius:</s> <s xml:id="echoid-s25369" xml:space="preserve"> tunc quia minus habet reſiſtentiæ, erit motus eius uelociter & ma gis <lb/>ſui diffuſiuus.</s> <s xml:id="echoid-s25370" xml:space="preserve"> Et quoniam reſiſtentia medij denſioris impellit ſemper lucem obliquam, ut coadune <lb/>tur ad perpendicularẽ lineam à puncto incidentiæ ſuper ſuperficiem illius corporis productã, quæ <lb/>eſt c g:</s> <s xml:id="echoid-s25371" xml:space="preserve"> patet, quòd in medio rarioris diaphani illa reſiſtentia erit minor quàm prima:</s> <s xml:id="echoid-s25372" xml:space="preserve"> fit ergo motus <lb/>lucis ad partem, à qua per reſiſtentiã repellebatur motus maior.</s> <s xml:id="echoid-s25373" xml:space="preserve"> Mouetur ergo lux in corpore dia-<lb/>phano rariore plus ad partem contrariã parti perpendicularis, ita, quòd angulus g c k ſit maior an-<lb/>gulo a c h:</s> <s xml:id="echoid-s25374" xml:space="preserve"> fit tamẽ ſemper motus lucis a c in refractione à corpore ſecundo rarioris diaphani quàm <lb/>primum, inter lineas c g & c e:</s> <s xml:id="echoid-s25375" xml:space="preserve"> quoniam <gap/>um angulus g c e ſit rectus, angulus g <gap/>k nunquam poteſt <lb/>fieri rectus.</s> <s xml:id="echoid-s25376" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s25377" xml:space="preserve"/> </p> <div xml:id="echoid-div951" type="float" level="0" n="0"> <figure xlink:label="fig-0384-01" xlink:href="fig-0384-01a"> <variables xml:id="echoid-variables412" xml:space="preserve">a h e d c b k q l g f</variables> </figure> </div> </div> <div xml:id="echoid-div953" type="section" level="0" n="0"> <head xml:id="echoid-head763" xml:space="preserve" style="it">48. À<unsure/> ſuperficie plana corporis diaphani omnium radiorum illi ſuperficiei incidentiũ, non <lb/>eſt poßibile fieri refractionem ad aliquod punctum unum.</head> <p> <s xml:id="echoid-s25378" xml:space="preserve">Quoniã enim, ut patet per præmiſſas, in omni corpore diaphano ſemper fit refractio uel ad ipſas <lb/>perpendiculares ductas à punctis incidentię radij ſuper ſuperficiem corporis diaphani, à qua fit re-<lb/>fractio:</s> <s xml:id="echoid-s25379" xml:space="preserve"> uel ab illis perpendicularibus (quomodocunq;</s> <s xml:id="echoid-s25380" xml:space="preserve"> autem hoc contingat) patet, cum illę perpen <lb/>diculares ſuper planam ſuperficiem ſint æquidiſtantes per 6 p 11, quoniam ſiue ad ipſas perpendicu <lb/>lares, ſiue ab ipſis fiat refractio:</s> <s xml:id="echoid-s25381" xml:space="preserve"> non eſt poſsibile, ut omnium radiorum illi planæ ſuperficiei inciden <lb/>tium refractio fiat ad punctum unum Patet ergo propoſitum.</s> <s xml:id="echoid-s25382" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div954" type="section" level="0" n="0"> <head xml:id="echoid-head764" xml:space="preserve" style="it">49. Nulla refractio tranſmutat ſitũ partiũ formæ refractæ, ſedſolũ auget uel minuit figurã.</head> <p> <s xml:id="echoid-s25383" xml:space="preserve">Quoniam enim, ut patet per 47 huius, omnis refractio fit in medio ſecundi diaphani, & in rario-<lb/>ri à perpendiculari, in denſiori uerò ad perpendicularẽ:</s> <s xml:id="echoid-s25384" xml:space="preserve"> palàm, quòd ſemper dexter radius remanet <lb/>dexter, & ſiniſter ſiniſter:</s> <s xml:id="echoid-s25385" xml:space="preserve"> & ſimiliter de alijs differentijs poſitionis.</s> <s xml:id="echoid-s25386" xml:space="preserve"> Situs ergo partium formæ refra <lb/>ctæ non mutantur, ſed ſemper permanẽt:</s> <s xml:id="echoid-s25387" xml:space="preserve"> modo aũt ſuo:</s> <s xml:id="echoid-s25388" xml:space="preserve"> cum à perpendiculari fit refractio, augetur <lb/>forma ſecundum dilatationem:</s> <s xml:id="echoid-s25389" xml:space="preserve"> & cum ad perpendicularem fit refractio, minuitur:</s> <s xml:id="echoid-s25390" xml:space="preserve"> quoniam anguli <lb/>ipſam continentes, anguſtantur.</s> <s xml:id="echoid-s25391" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s25392" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div955" type="section" level="0" n="0"> <head xml:id="echoid-head765" xml:space="preserve" style="it">50. In omni ſimili ſuperficie eiuſdem diaphani, radij ſecundum æquales angulos incidentes, <lb/>ſecundũ æquales angulos refringuntur: & ſi maiores ſunt anguli incidentiæ, maiores ſunt angu <lb/>li refractionum, & ſi minores, minores.</head> <p> <s xml:id="echoid-s25393" xml:space="preserve">Siue enim refractionis modus attendatur à parte <lb/> <anchor type="figure" xlink:label="fig-0385-01a" xlink:href="fig-0385-01"/> ſuperficierum corporũ, in quibus fit refractio:</s> <s xml:id="echoid-s25394" xml:space="preserve"> quo-<lb/>niam alia fit refractio à ſuperficie ſphærica, & alia à <lb/>plana:</s> <s xml:id="echoid-s25395" xml:space="preserve"> ſiue à parte diſpoſitionis diaphanorum:</s> <s xml:id="echoid-s25396" xml:space="preserve"> quo-<lb/>niam alia fit refractio à rariori diaphano, alia à den-<lb/>ſiori, ut patet per plures propoſitiones libri huius:</s> <s xml:id="echoid-s25397" xml:space="preserve"> <lb/>ſiue attendatur à parte angulorum incidentiæ, patet <lb/>ſemper quòd angulis incidentiæ exiſtentibus æqua <lb/>libus, ſecundum modum propoſitum nulla ſubeſt <lb/>cauſſa diuerſitatis modi refractionis.</s> <s xml:id="echoid-s25398" xml:space="preserve"> Fiet ergo ſem-<lb/>per refractio ſecundum angulos æquales.</s> <s xml:id="echoid-s25399" xml:space="preserve"> Et hoc eſt <lb/>propoſitũ primum.</s> <s xml:id="echoid-s25400" xml:space="preserve"> Et eſt huius exemplum, ut ſit <lb/>corpus ſphæricum diaphanum denſius ipſo aere, <lb/>in cuius ſuperficie ſit circulus a b c d e:</s> <s xml:id="echoid-s25401" xml:space="preserve"> cuius centrũ <lb/>ſit p:</s> <s xml:id="echoid-s25402" xml:space="preserve"> & à puncto f corporis luminoſi incidant lineæ <lb/>radiales, quæ ſint a f, b f, c f, d f, e f:</s> <s xml:id="echoid-s25403" xml:space="preserve"> incidatq́;</s> <s xml:id="echoid-s25404" xml:space="preserve"> radius f c <lb/>perpẽdiculariter, & alij obliquè:</s> <s xml:id="echoid-s25405" xml:space="preserve"> patet, quòd omnes <lb/>radij incidentes obliquè in ſuperficie illius corporis <lb/>diaphani, refringentur per 47 huius.</s> <s xml:id="echoid-s25406" xml:space="preserve"> Sit ergo exem-<lb/>pli cauſſa & breuitatis figuratiõis & denominatiõis <lb/>linearũ, ut oẽs illi radij refracti cõcurrãt in puncto g:</s> <s xml:id="echoid-s25407" xml:space="preserve"> <lb/>& ducãtur քpendiculariter ſuք ſuperficiẽ corporis li <lb/>neę, quę ſint p d q & p b r & p a x & p e z.</s> <s xml:id="echoid-s25408" xml:space="preserve"> Dico, quòd <lb/>ſi angulus incidẽtię (qui eſt f d q) ſit ęqualis angulo <lb/>f b r, quòd angulus g d p erit æqualis angulo g b p, ք pręmiſſa, ꝓpter uniformitatẽ omniũ prędictarũ <lb/> <pb o="84" file="0386" n="386" rhead="VITELLONIS OPTICAE"/> conditionum.</s> <s xml:id="echoid-s25409" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s25410" xml:space="preserve"> dico, quòd ſi angulus f d q ſit maior angulo f a x, quòd angulus p d g <lb/>erit maior angulo p a g.</s> <s xml:id="echoid-s25411" xml:space="preserve"> Fiat enim ſuper punctũ a terminũ lineę x a angulus æ qualis angulo f d q per <lb/>23 p 1, qui ſit angulus h a x:</s> <s xml:id="echoid-s25412" xml:space="preserve"> refringaturq́;</s> <s xml:id="echoid-s25413" xml:space="preserve"> radius h a in puncto a:</s> <s xml:id="echoid-s25414" xml:space="preserve"> concurratq́;</s> <s xml:id="echoid-s25415" xml:space="preserve"> cum linea f g in puncto <lb/>k:</s> <s xml:id="echoid-s25416" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s25417" xml:space="preserve"> per primam partem huius, angulus p a k æqualis angulo p d g:</s> <s xml:id="echoid-s25418" xml:space="preserve"> eſt autẽ angulus p a k maior <lb/>angulo p a g:</s> <s xml:id="echoid-s25419" xml:space="preserve"> non enim eſt æqualis, quoniam tunc ex præmilsis ſequeretur angulos incidentiæ eſſe <lb/>æquales, quod eſt contra hypotheſim, ſunt enim ſuppoſiti eſſe inęquales:</s> <s xml:id="echoid-s25420" xml:space="preserve"> ſed neq;</s> <s xml:id="echoid-s25421" xml:space="preserve"> minor:</s> <s xml:id="echoid-s25422" xml:space="preserve"> quoniã ſic <lb/>fieret refractio irregularis:</s> <s xml:id="echoid-s25423" xml:space="preserve"> quod eſt cõtra 43 & 45 huius:</s> <s xml:id="echoid-s25424" xml:space="preserve"> eſt ergo maior:</s> <s xml:id="echoid-s25425" xml:space="preserve"> ergo & angulus p d g eſt ma <lb/>ior p a g.</s> <s xml:id="echoid-s25426" xml:space="preserve"> Idẽ quoq;</s> <s xml:id="echoid-s25427" xml:space="preserve"> poteſt demõſtrari facilius, ut ſi angulus f e z fiat æqualis angulo f a x per 8 p 3, ut-<lb/>pote ſi arcus a c & c e aſſumantur æquales:</s> <s xml:id="echoid-s25428" xml:space="preserve"> tũc enim anguli p a g & p e g erunt per præmiſſa ęquales:</s> <s xml:id="echoid-s25429" xml:space="preserve"> <lb/>angulus uerò p d g minor eſt angulo p e g:</s> <s xml:id="echoid-s25430" xml:space="preserve"> quod patet etiã, ſi anguli refractiõis ponantur eſſe æqua-<lb/>les.</s> <s xml:id="echoid-s25431" xml:space="preserve"> De hac autem materia hic ſummariè loquimur, quoniã ipſam in 10 huius libro, ubilocum pro-<lb/>prium habet, perfectius perſequemur.</s> <s xml:id="echoid-s25432" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s25433" xml:space="preserve"/> </p> <div xml:id="echoid-div955" type="float" level="0" n="0"> <figure xlink:label="fig-0385-01" xlink:href="fig-0385-01a"> <variables xml:id="echoid-variables413" xml:space="preserve">f h r q x b c d z a e p g k</variables> </figure> </div> <figure> <variables xml:id="echoid-variables414" xml:space="preserve">a <gap/> b z d</variables> </figure> </div> <div xml:id="echoid-div957" type="section" level="0" n="0"> <head xml:id="echoid-head766" xml:space="preserve" style="it">51. Datam altitudinem per umbram quanta ſit cognoſcere ſo-<lb/>le apparente Euclides 18 theo. opticorum.</head> <p> <s xml:id="echoid-s25434" xml:space="preserve">Sit data altitudo a b, quam proponimus, quanta ſit cognoſcere ſo-<lb/>le apparẽte:</s> <s xml:id="echoid-s25435" xml:space="preserve"> & ſi illa altitudo eſt erecta ſuper ſuperficiem horizontis, <lb/>ducatur in illa ſuperficie linea b d perpendicularis ſuper terminum <lb/>altitudinis a b, qui ſit b:</s> <s xml:id="echoid-s25436" xml:space="preserve"> & incidat radius ſolaris per uerticem a b (qui <lb/>ſit a) ipſi pũcto d:</s> <s xml:id="echoid-s25437" xml:space="preserve"> & ſit a d:</s> <s xml:id="echoid-s25438" xml:space="preserve"> ergo per 11 huius, erit linea b d umbra altitu <lb/>dinis ipſius a b:</s> <s xml:id="echoid-s25439" xml:space="preserve"> erigaturq́;</s> <s xml:id="echoid-s25440" xml:space="preserve"> nota linea e z inter umbrã b d & radiũ a d <lb/>æquidiſtanter altitudini a b, ut ſi z e ſit baculus notæ quantitatis.</s> <s xml:id="echoid-s25441" xml:space="preserve"> <lb/>Erit ergo trigonus d z e per 29 p 1 æquiangulus trigono a b d:</s> <s xml:id="echoid-s25442" xml:space="preserve"> ergo <lb/>per 4 p 6, uel per 9 huius, erit proportio d z ad z e, ſicut d b ad b a:</s> <s xml:id="echoid-s25443" xml:space="preserve"> ſed <lb/>d z ad z e proportio eſt nota:</s> <s xml:id="echoid-s25444" xml:space="preserve"> quoniam cum z e ſit aſſumpta nota, po-<lb/>teſt & linea umbræ ſuæ, quæ eſt z d, modica menſuratione fieri nota:</s> <s xml:id="echoid-s25445" xml:space="preserve"> <lb/>ergo d b ad b a proportio eſt nota:</s> <s xml:id="echoid-s25446" xml:space="preserve"> ſed d b poteſt menſurando fieri no <lb/>ta.</s> <s xml:id="echoid-s25447" xml:space="preserve"> Ergo & a b erit nota.</s> <s xml:id="echoid-s25448" xml:space="preserve"> Quod eſt propoſitum, ut ſi linea a b ſit alti-<lb/>tudo alicuius turris uel parietis, qui ualeat adiri ad menſuranda ſpa-<lb/>tia umbrarum.</s> <s xml:id="echoid-s25449" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div958" type="section" level="0" n="0"> <head xml:id="echoid-head767" xml:space="preserve">VITELLONIS FILII</head> <head xml:id="echoid-head768" xml:space="preserve">THVRINGORVM ET PO-</head> <head xml:id="echoid-head769" xml:space="preserve">LONORVM OPTICAE LIBER TERTIVS.</head> <p style="it"> <s xml:id="echoid-s25450" xml:space="preserve">IN præmißis libris mathematicalia & naturalia principia præmiſim{us}, per <lb/>quæ, prout nostra poßibilit{as} fert, noſtri propoſiti conſequentia intendim{us} <lb/>declarare.</s> <s xml:id="echoid-s25451" xml:space="preserve"> Volentes autem formarum naturalium actiones ſub triplici ui-<lb/>dendi modo proſequi, ſcilicet illo, quι fit per ſimplicem uiſionem, & eo, qui <lb/>per reflexionem, & illo, qui per refractionẽ:</s> <s xml:id="echoid-s25452" xml:space="preserve"> in hoc tertio libro proſequimur modum ſim <lb/>plicis uiſionis, & diſpoſitionem propriam organi uiſiui.</s> <s xml:id="echoid-s25453" xml:space="preserve"> Supponim{us} autem hæc, quæ <lb/>ſequuntur, in locis alijs declarata, uel ut per ſe ipſa nota.</s> <s xml:id="echoid-s25454" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div959" type="section" level="0" n="0"> <head xml:id="echoid-head770" xml:space="preserve">PETITIONES.</head> <p> <s xml:id="echoid-s25455" xml:space="preserve">1.</s> <s xml:id="echoid-s25456" xml:space="preserve"> Viſionem non compleri, niſi apud peruentum formæ uiſibilis ad animam.</s> <s xml:id="echoid-s25457" xml:space="preserve"> 2 <lb/>Item quòd per ſe uiſibilia ſunt tantùm duo, ſcilιcet lux & color:</s> <s xml:id="echoid-s25458" xml:space="preserve"> quoniam lux <lb/>ex ſe ipſa uidetur:</s> <s xml:id="echoid-s25459" xml:space="preserve"> & ipſa eſt hypoſtaſis colorum:</s> <s xml:id="echoid-s25460" xml:space="preserve"> alia uerò per accidens uiſibilia <lb/>ſunt, utpote remotio, magnitudo, ſitus, corporeitas, figura, continuitas, ſepara-<lb/>tio uel diuiſio, numerus, motus, quies, aſperitas, lenitas, diaphanitas, denſitas, um-<lb/>bra, obſcuritas, pulchritudo, deformitas, cõſimilitudo & diuerſitas.</s> <s xml:id="echoid-s25461" xml:space="preserve"> Hæc enim non <lb/>ſolùm uiſu, ſed alijs ſenſibus comprehenduntur.</s> <s xml:id="echoid-s25462" xml:space="preserve"> 3.</s> <s xml:id="echoid-s25463" xml:space="preserve"> Item petimus lucem fortem <lb/>lædere uiſum diutius intuentem.</s> <s xml:id="echoid-s25464" xml:space="preserve"> 4.</s> <s xml:id="echoid-s25465" xml:space="preserve"> Item rem maioris quantitatis, quàm ſit o-<lb/>culus, oculo uideri.</s> <s xml:id="echoid-s25466" xml:space="preserve"> 5.</s> <s xml:id="echoid-s25467" xml:space="preserve"> Item rem uiſam ſecundum ſitum, figuram & ordinẽ ſua-<lb/>rum partium uideri.</s> <s xml:id="echoid-s25468" xml:space="preserve"> 6.</s> <s xml:id="echoid-s25469" xml:space="preserve"> Item uiſum ſimul diuerſa uiſibilia uidere.</s> <s xml:id="echoid-s25470" xml:space="preserve"> 7.</s> <s xml:id="echoid-s25471" xml:space="preserve"> Item ab <lb/>ambobus uiſibus ſimul unam rem uideri.</s> <s xml:id="echoid-s25472" xml:space="preserve"> 8 Itẽ quòd colornõ eſt motiuus ui-<lb/>ſus, niſi ſecundũ actũ lucidi.</s> <s xml:id="echoid-s25473" xml:space="preserve"> 9.</s> <s xml:id="echoid-s25474" xml:space="preserve"> Itẽ ſine contactu uiſionẽ nõ fieri, ſicut nec aliquã <lb/>actionẽ naturalẽ.</s> <s xml:id="echoid-s25475" xml:space="preserve"> 10.</s> <s xml:id="echoid-s25476" xml:space="preserve"> Item uirtutẽ uiſiuam finitã eſſe, & non extendi in infinitũ.</s> <s xml:id="echoid-s25477" xml:space="preserve"/> </p> <pb o="85" file="0387" n="387" rhead="LIBER TERTIVS."/> </div> <div xml:id="echoid-div960" type="section" level="0" n="0"> <head xml:id="echoid-head771" xml:space="preserve">THEOREMATA.</head> <head xml:id="echoid-head772" xml:space="preserve" style="it">1. Viſibili lucem actu non participante: ipſum impoßibile eſt uideri. Alhazen 39 n 1.</head> <p> <s xml:id="echoid-s25478" xml:space="preserve">Quæ enim, ut ſuppoſitum eſt, per ſe ſunt uiſibilia:</s> <s xml:id="echoid-s25479" xml:space="preserve"> ſunt lux & color:</s> <s xml:id="echoid-s25480" xml:space="preserve"> lux autẽ non eſt uiſibilis, præ-<lb/>terquam ex ſeipſa:</s> <s xml:id="echoid-s25481" xml:space="preserve"> & etiam lux cum ſit hypoſtaſis colorum, non eſt poſsibile colores uideri ſine lu-<lb/>ce:</s> <s xml:id="echoid-s25482" xml:space="preserve"> forma enim coloris eſt forma debilior, quàm ſit forma lucis:</s> <s xml:id="echoid-s25483" xml:space="preserve"> cum color ſit quædam lux incorpo-<lb/>rata corporibus mixtis.</s> <s xml:id="echoid-s25484" xml:space="preserve"> Viſus ergo nõ recipit formam coloris rei uiſæ, niſi ex luce admixta cum for-<lb/>ma coloris:</s> <s xml:id="echoid-s25485" xml:space="preserve"> & propter hoc alterantur colores multarum rerum apud uiſum per alterationem lucis <lb/>orientis ſuper ipſas:</s> <s xml:id="echoid-s25486" xml:space="preserve"> & ſi color, qui eſt per ſe uiſibilis, non eſt motiuus ipſius uiſus, niſi ſecundum a-<lb/>ctum lucidi:</s> <s xml:id="echoid-s25487" xml:space="preserve"> patet, quòd omni uiſibili actu lucem non participante ipſum impoſsibile eſt uideri.</s> <s xml:id="echoid-s25488" xml:space="preserve"> Pa-<lb/>tet ergo propoſitum.</s> <s xml:id="echoid-s25489" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div961" type="section" level="0" n="0"> <head xml:id="echoid-head773" xml:space="preserve" style="it">2. Inter quodlibet punctum ſuperficiei rei uiſibilis, & aliquod punctũ ſuperficiei uiſ{us} pro-<lb/>duci poſſe rect{as} line{as} eſt neceſſe, ut res actu uideatur. Ex quo patet, ſolùm in oppoſitione rei ui-<lb/>ſæ ad uiſum fieri uiſionem. Alhazen 21 n 1.</head> <p> <s xml:id="echoid-s25490" xml:space="preserve">Viſio enim ſiue fiat ex eo, quòd radij egrediuntur à uiſu ſuper puncta rei uiſæ, ſiue ex hoc, quòd <lb/>formę punctorum rei uiſę per lineas radiales perueniunt ad ſuperficiẽ organi uiſiui:</s> <s xml:id="echoid-s25491" xml:space="preserve"> ſemper neceſſe <lb/>eſt inter quodlibet punctum ſuperficiei rei uiſibilis, & aliquod punctum ſuperficiei uiſus produci <lb/>poſſe lineas rectas, ut res uideantur actu.</s> <s xml:id="echoid-s25492" xml:space="preserve"> Vnde cum hæ lineę ſecundũ quemcunq;</s> <s xml:id="echoid-s25493" xml:space="preserve"> propoſitũ modũ <lb/>produci poſſunt, fit uiſio:</s> <s xml:id="echoid-s25494" xml:space="preserve"> niſi fortè propter alterius impedimenti reſiſtentiã uiſus fuerit impeditus.</s> <s xml:id="echoid-s25495" xml:space="preserve"> <lb/>Cum itaq;</s> <s xml:id="echoid-s25496" xml:space="preserve"> uiſus fuerit oppoſitus rei uiſæ, uidebit ipſam:</s> <s xml:id="echoid-s25497" xml:space="preserve"> & cũ aufertur ab eius oppoſitione, non ſen <lb/>tiet ipſam, & cũ reuertetur ad oppoſitionẽ, reuertetur ſenſus:</s> <s xml:id="echoid-s25498" xml:space="preserve"> quoniã ab alijs partibus ꝗ̃ ab oppoſitis <lb/>directè non poteſt linea produci à punctis uiſibiliũ ad puncta ſuperficiei uiſus.</s> <s xml:id="echoid-s25499" xml:space="preserve"> Patet ergo ꝓpoſitũ.</s> <s xml:id="echoid-s25500" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div962" type="section" level="0" n="0"> <head xml:id="echoid-head774" xml:space="preserve" style="it">3. Organum uirtutis uiſiuæ neceſſe eſt ſphæricum eſſe. Alhazen 35 n 1.</head> <p> <s xml:id="echoid-s25501" xml:space="preserve">Si enim non ſit ſphæricum:</s> <s xml:id="echoid-s25502" xml:space="preserve"> dico, quòd non impeditur uiſio, utpote ſi ſit ſuperficiei planæ:</s> <s xml:id="echoid-s25503" xml:space="preserve"> tunc e-<lb/>nim nõ uidebit uno aſpectu, niſi ſibi ęquale.</s> <s xml:id="echoid-s25504" xml:space="preserve"> Siue enim radij egrediantur à uiſu ſuper rem uiſam, ſiue <lb/>formæ punctorum rei uiſæ per lineas radiales perueniant ad ſuperficiem organi uiſiui:</s> <s xml:id="echoid-s25505" xml:space="preserve"> patet, quòd <lb/>ſemper perpendiculares ſunt breuiores per 21 t 1 huius:</s> <s xml:id="echoid-s25506" xml:space="preserve"> unde res magis approximat uiſui ſecundum <lb/>illas, quoniam res uiſa directè ſecundum ipſas perpendiculares uidetur, nõ per aliquas lineas obli-<lb/>quas, quæ refringantur:</s> <s xml:id="echoid-s25507" xml:space="preserve"> quia ut patet per 48 t 2 huius, in corporibus planis nõ poteſt fieri refractio <lb/>formarum ad aliquod punctum unum:</s> <s xml:id="echoid-s25508" xml:space="preserve"> eò quòd in talibus nullus punctus eſt omnibus communis.</s> <s xml:id="echoid-s25509" xml:space="preserve"> <lb/>Sola ergo illa ab organo uiſiuo ſuperficiei planæ uideri poſſunt, quæ ſine refractione directè perue-<lb/>niunt ad ipſum:</s> <s xml:id="echoid-s25510" xml:space="preserve"> hæc autẽ ſunt ſecundum perpendiculares lineas peruenientia ad uiſum.</s> <s xml:id="echoid-s25511" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s25512" xml:space="preserve"> ſu <lb/>perficies plana uiſus, in qua ſit linea a b:</s> <s xml:id="echoid-s25513" xml:space="preserve"> & ſit in ſuperficie plana alicuius rei uiſę æquidiſtantis uiſui, <lb/>& lineæ a b, linea recta, quę c d e:</s> <s xml:id="echoid-s25514" xml:space="preserve"> & à pũcto c du <lb/> <anchor type="figure" xlink:label="fig-0387-01a" xlink:href="fig-0387-01"/> catur perpẽdicularis ſuper ſuperficiẽ uiſus per <lb/>11 p 11, quæ incidat in punctũ a, & ſit c a:</s> <s xml:id="echoid-s25515" xml:space="preserve"> & à pun <lb/>cto d ducatur ſimiliter ſuք ſuperficiẽ uiſus per-<lb/>pendicularis, quæ ſit d b.</s> <s xml:id="echoid-s25516" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s25517" xml:space="preserve"> lineæ a c & <lb/>b d ſint æquidiſtãtes & ęquales per 25 t 1 huius:</s> <s xml:id="echoid-s25518" xml:space="preserve"> <lb/>ergo per 33 p 1 linea a b æqualis erit lineæ c d.</s> <s xml:id="echoid-s25519" xml:space="preserve"> Et <lb/>quoniã linea a b æqualis eſt lineæ c d, ſed linea <lb/>c d e eſt maior quàm linea c d:</s> <s xml:id="echoid-s25520" xml:space="preserve"> ergo nõ uidetur <lb/>ſimul tota linea c d e:</s> <s xml:id="echoid-s25521" xml:space="preserve"> ꝗ a in hac diſpoſitione non <lb/>poteſt res uiſa excedere quantitatẽ ſuperficiei <lb/>uiſus.</s> <s xml:id="echoid-s25522" xml:space="preserve"> Et quoniã hoc eſt falſum & contra ſuppo-<lb/>ſitionem, quæ patet ſenſui:</s> <s xml:id="echoid-s25523" xml:space="preserve"> quoniam poſsibile eſt rem maiorem ipſo oculo uideri:</s> <s xml:id="echoid-s25524" xml:space="preserve"> palàm, quia non <lb/>eſt poſsibile, ut ſuperficies organi uiſiui ſit plana:</s> <s xml:id="echoid-s25525" xml:space="preserve"> ſed neq;</s> <s xml:id="echoid-s25526" xml:space="preserve"> alterius figuræ quàm ſphæricæ:</s> <s xml:id="echoid-s25527" xml:space="preserve"> quia ſem <lb/>per accident impoſsibilia inæqualitatis uiſionis.</s> <s xml:id="echoid-s25528" xml:space="preserve"> Neceſſariò ergo erit ſphærica ſuperficies organi ui <lb/>ſiui, in cuius centro fiat cõcurſus linearum radialium ex longè maiori magnitudine quàm ſit ipſum <lb/>organum uiſiuum.</s> <s xml:id="echoid-s25529" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s25530" xml:space="preserve"/> </p> <div xml:id="echoid-div962" type="float" level="0" n="0"> <figure xlink:label="fig-0387-01" xlink:href="fig-0387-01a"> <variables xml:id="echoid-variables415" xml:space="preserve">a b c d e</variables> </figure> </div> </div> <div xml:id="echoid-div964" type="section" level="0" n="0"> <head xml:id="echoid-head775" xml:space="preserve" style="it">4. Ocul{us} eſt organum uirtutis uiſiuæ ſphæricum, ex trib{us} humorib{us} & quatuor tunicis à <lb/>ſubstantia cerebri prodeuntib{us} ſphæricè ſe interſecantib{us} compoſitum. Alhazen 4 n 1.</head> <p> <s xml:id="echoid-s25531" xml:space="preserve">Quomodo ſit oculus uirtutis uiſiuę organum, negotio alterius partis philoſophiæ relinquimus:</s> <s xml:id="echoid-s25532" xml:space="preserve"> <lb/>quòd aũt ſit ſphæricus, neceſſariũ eſt per præcedentẽ propoſitionẽ:</s> <s xml:id="echoid-s25533" xml:space="preserve"> & etiã ex eo, quòd eſt naturę a-<lb/>queæ, cuius proprietas eſt ſemper rotundari, ut alibi eſt declaratũ.</s> <s xml:id="echoid-s25534" xml:space="preserve"> Quòd aũt ſit oculus ex tribus hu <lb/>moribus & quatuor tunicis cõpoſitus, diligens anatomizantiũ cura edocuit.</s> <s xml:id="echoid-s25535" xml:space="preserve"> Primus itaq;</s> <s xml:id="echoid-s25536" xml:space="preserve"> humorũ <lb/>iſtorũ dicitur cryſtallinus uel glacialis, qui propriè eſt organũ uirtutis uiſiuæ, & eſt in medio oculi <lb/>ſitus:</s> <s xml:id="echoid-s25537" xml:space="preserve"> eſtq́;</s> <s xml:id="echoid-s25538" xml:space="preserve"> ſphæra parua, alba, humida, humiditatis receptibilis formarũ uiſibiliũ, in qua eſt diapha-<lb/>nitas nõ intenſa ualde, cũ ſit in ea aliqua ſpiſsitudo:</s> <s xml:id="echoid-s25539" xml:space="preserve"> unde diaphanitas eius aſsimilatur diaphanitati <lb/>cryſtalli uel glaciei:</s> <s xml:id="echoid-s25540" xml:space="preserve"> & ob hoc dicitur humor cryſtallinus uel glacialis.</s> <s xml:id="echoid-s25541" xml:space="preserve"> Quia uerò eius humoris dia-<lb/>phanitas mutatur in ſui parte poſteriori uerſus cerebrũ, à qua parte totus oculus recipit nutrimẽtũ, <lb/>quod anteꝗ̃ perfectè uniatur humori cryſtallino (qui principaliter intenditur nutriri) nondũ plenè <lb/>in formis ſubſtantialibus & accidentalibus eidẽ aſsimilatũ, neceſſariò eſt alterius diaphanitatis ab <lb/> <pb o="86" file="0388" n="388" rhead="VITELLONIS OPTICAE"/> illo:</s> <s xml:id="echoid-s25542" xml:space="preserve"> & ob hoc dicitur alter humor:</s> <s xml:id="echoid-s25543" xml:space="preserve"> & uocatur uitreus:</s> <s xml:id="echoid-s25544" xml:space="preserve"> quia aſsimilatur uitro quaſ<gap/> fruſtato.</s> <s xml:id="echoid-s25545" xml:space="preserve"> Et quia <lb/>in omni, qđ nutritur, ſemper purũ ab impuro ſeparatur:</s> <s xml:id="echoid-s25546" xml:space="preserve"> illud, qđ ab humore cryſtallino nutrito, ut <lb/>ſuę puritati incõueniẽs, ſeparatur ad partẽ oppoſitã parti nutrimẽtali, hoc eſt, ad anterius cryſtallini <lb/>humoris ꝓfluit:</s> <s xml:id="echoid-s25547" xml:space="preserve"> & quia eſt diaphanũ, quoquo modo aſsimilatũ humori cryſtallino, nondũ tamẽ ſuæ <lb/>perfectę cõſiſtentię in denſitate, eò quòd eſt ſuperfluũ nutrimẽti corporis denſioris:</s> <s xml:id="echoid-s25548" xml:space="preserve"> patet, quòd ne-<lb/>ceſſariò eſt diaphanũ liquidũ:</s> <s xml:id="echoid-s25549" xml:space="preserve"> unde uocatus eſt humor albugineus, ꝗa ſimile eſt albumini oui in te-<lb/>nuitate & albedine & diaphanitate:</s> <s xml:id="echoid-s25550" xml:space="preserve"> eſt enim humor albus, clarus, tenuis, diaphanus:</s> <s xml:id="echoid-s25551" xml:space="preserve"> & hunc humo <lb/>rem ad partẽ anteriorẽ, ſicut uitreũ humorẽ ad partẽ poſteriorẽ pro cuſtodia humoris cryſtallini, ne <lb/>ab extrinſecis occaſionib.</s> <s xml:id="echoid-s25552" xml:space="preserve"> uel intrinſecis citius patiatur, & cadat ab officio organi uiſiui, naturę ſaga <lb/>citas deputauit.</s> <s xml:id="echoid-s25553" xml:space="preserve"> Cõtinet aũt primos duos humores, ſcilicet cryſtallinũ & uitreũ tela ualde tenuis & <lb/>ſubtilis, ſeparãs eos ab albugineo, & circundãs ambos eos, cuius etiã telę aliqua pars deſcendẽs per <lb/>mediũ ſeparat cryſtallinũ à uitreo:</s> <s xml:id="echoid-s25554" xml:space="preserve"> & hęc tela ꝓpter ſui ſubtilitatẽ tela aranea nominatur.</s> <s xml:id="echoid-s25555" xml:space="preserve"> Cũ aũt hu <lb/>mor albugineus ſit liquidus, per ſe nõ conſiſtens, neceſſariũ fuit ipſum per aliquod ſolidũ pro oculi <lb/>cuſtodia retineri:</s> <s xml:id="echoid-s25556" xml:space="preserve"> circũde dit ergo ipſum natura pelle uiſcoſa ſolida forti, nõ multũ diaphana, quę ſui <lb/>denſitate melius retineat, & ſui caliditate humorẽ albugineũ temperet, ne cryſtallinus cõgeletur, & <lb/>fiat inhabilis receptioni uiſibiliũ formarũ.</s> <s xml:id="echoid-s25557" xml:space="preserve"> Et ꝗa ꝓpter eius tunicę denſitatẽ & uiſcoſitatẽ formę uiſi <lb/>biles ad humorẽ cryſtallinũ undiq;</s> <s xml:id="echoid-s25558" xml:space="preserve"> tali tunica circundatũ nõ perueniſſent:</s> <s xml:id="echoid-s25559" xml:space="preserve"> ideo in anteriori parte o-<lb/>culi, ubi eſt locus receptionis formarũ uiſibiliũ, natura hanc tunicã intercîdit, factumq́;</s> <s xml:id="echoid-s25560" xml:space="preserve"> eſt foramen <lb/>rotundum, cuius diameter eſt quaſi æqualis lateri cubi inſcriptibilis intra illã ſphærã, uel lateri qua <lb/>drati inſcriptibilis circulo magno illius ſphærę:</s> <s xml:id="echoid-s25561" xml:space="preserve"> & eſt hoc foramen ideo rotundũ, ut ſit magis aptum <lb/>ſuſceptioni omniũ formarũ pertranſiẽs uſq;</s> <s xml:id="echoid-s25562" xml:space="preserve"> ad eiuſdẽ tunicę concauũ:</s> <s xml:id="echoid-s25563" xml:space="preserve"> & ob hoc hęc tunica dicta eſt <lb/>uuea, quia aſsimilatur uuæ in aſpectu:</s> <s xml:id="echoid-s25564" xml:space="preserve"> & eſt hæc tunica plurimũ nigra, ſæpe tamẽ uiridis, & quãdoq;</s> <s xml:id="echoid-s25565" xml:space="preserve"> <lb/>glauca:</s> <s xml:id="echoid-s25566" xml:space="preserve"> & corpus illius tunicę eſt tenue denſum nõ rarũ.</s> <s xml:id="echoid-s25567" xml:space="preserve"> Ne uerò humor albugineus effluat ex fora-<lb/>mine uueę, & ut nõ impediatur operatio uirtutis uiſiuę, neceſſariũ fuit naturę foramini uueæ ſuppo <lb/>nere uelamẽ diaphanũ ſolidũ ad modũ cornu albi clari:</s> <s xml:id="echoid-s25568" xml:space="preserve"> dictaq́;</s> <s xml:id="echoid-s25569" xml:space="preserve"> eſt hæc tunica cornea.</s> <s xml:id="echoid-s25570" xml:space="preserve"> Vbi uerò con <lb/>iungitur hęc tunica alijs partibus corporis circũpoſitis oculo, ibi ceſſat diaphanitas, fitq́;</s> <s xml:id="echoid-s25571" xml:space="preserve"> alterius di <lb/>ſpoſitiõis tunica ſolidior ꝗ̃ cornea nõ diaphana, cũ ipſa tamẽ cornea cõplẽs ſphærã unã, quę eſt ſphæ <lb/>ra totius oculi, & illius ſphærę poſterior pars nõ diaphana, ſed carnoſa fit alia tunica:</s> <s xml:id="echoid-s25572" xml:space="preserve"> & hæc dicitur <lb/>cõiunctiua uel conſolidatiua, quoniã cõiungit oculũ & cõſolidat ipſum cũ partibus corporis uicini.</s> <s xml:id="echoid-s25573" xml:space="preserve"> <lb/>Erit ergo tunica cornea humor albugineus & humor glacialis & humor uitreus ſe ad inuicẽ conſe-<lb/>quentes:</s> <s xml:id="echoid-s25574" xml:space="preserve"> & omnia iſta ſunt diaphana propter meliorẽ formarũ uiſibiliũ receptionẽ.</s> <s xml:id="echoid-s25575" xml:space="preserve"> À<unsure/> ſubſtantia ce <lb/>rebri prodeũt humores & tunicę oculi, quoniã ex anteriori parte cerebri à duabus partibus ipſius <lb/>creſcũt duo nerui optici, id eſt cõcaui cõſimiles habentes duas tunicas ortas à duab telis cerebri, & <lb/>procedunt ij nerui ad mediũ anterioris partis cerebri, ubi efficitur neruus unus opticus, qui in pro-<lb/>ceſſu iterũ diuiditur in duos neruos opticos cõſimiles & æquales, ꝗ tranſmutatis ſuis ſitibus, ita, ut <lb/>dexter fiat ſiniſter, & ſiniſter dexter, ſunt procedẽtes ad cõuexa duorũ oſsiũ concauorũ cõtinentiũ <lb/>oculos, quoniã in medijs iſtorũ duorũ oſsiũ cõcauorũ ſunt duo foramina æqualiter perforata, quæ <lb/>dicuntur foramina gyrationis neruorũ cõcauorũ:</s> <s xml:id="echoid-s25576" xml:space="preserve"> & quoniã illa duo foramina ſunt rotunda, pũctus <lb/>medius cuiuslibet illorũ foraminũ dicitur centrũ illius foraminis.</s> <s xml:id="echoid-s25577" xml:space="preserve"> Illi ergo nerui intrãt iſta duo fora <lb/>mina, & exeũt ad cõcauitatẽ duorũ oſsiũ prædictorũ, & illic dilatãtur & ampliãtur, & efficitur extre <lb/>mitas cuiuſq;</s> <s xml:id="echoid-s25578" xml:space="preserve"> ipſorũ quaſi inſtrumentũ ponendi uinũ in dolijs, hoc eſt ad modũ pyramidis rotundę <lb/>cõcauæ:</s> <s xml:id="echoid-s25579" xml:space="preserve"> & ꝗlibet oculorũ cõponitur ſuք unã extremitatẽ iſtius nerui, & cõſolidatur cũ ipſo.</s> <s xml:id="echoid-s25580" xml:space="preserve"> Cõſimi <lb/>liter & à tunicis iſtorũ neruorũ oriuntur tunicę oculorũ:</s> <s xml:id="echoid-s25581" xml:space="preserve"> nã tunica cornea oritur ex tunica extrinſe <lb/>ca duarũ tunicarũ iſtius nerui:</s> <s xml:id="echoid-s25582" xml:space="preserve"> & tunica uuea oritur ex tunica intrinſeca duarũ tunicarũ duorũ ner-<lb/>uorũ:</s> <s xml:id="echoid-s25583" xml:space="preserve"> intra iſtã tunicã uueã ordinatur humor cryſtallinus ſuք extremitatẽ cõcauitatis nerui median <lb/>te uitreo humore, ꝗ ambo ex medullari ſubſtãtia cerebri oriuntur:</s> <s xml:id="echoid-s25584" xml:space="preserve"> & inter humores iſtos & tunicã <lb/>uueã ex ſubtiliſsimis filis tunicę uueę cõtexitur tela aranea, quã alij uocãt tunicã retiuã, ꝗa eſt cõte-<lb/>xta ad modũ retis.</s> <s xml:id="echoid-s25585" xml:space="preserve"> Sphęricè ſe interſecãt humores & tunicę oculi:</s> <s xml:id="echoid-s25586" xml:space="preserve"> quia enim tunica uuea nõ քuenit <lb/>intra oculũ ad cõplemẽtũ ſphęrę, cũ, ſicut pręmiſsũ eſt, in anteriori ſui parte ſit foramẽ rotundũ, qđ <lb/>tegitur à cornea tunica:</s> <s xml:id="echoid-s25587" xml:space="preserve"> ſphæra ergo tunicę corneę neceſſariò interſecat ſphærã uueæ:</s> <s xml:id="echoid-s25588" xml:space="preserve"> & cõis ſectio <lb/>ſuarũ ſuperficierũ ſphęricarũ eſt circũferẽtia illius foraminis:</s> <s xml:id="echoid-s25589" xml:space="preserve"> & eſt linea circularis ք 80 t 1 huius.</s> <s xml:id="echoid-s25590" xml:space="preserve"> In <lb/>anteriori quoq;</s> <s xml:id="echoid-s25591" xml:space="preserve"> humoris cryſtallini ꝓpter meliorẽ formarũ receptionẽ eſt cõpreſsio ſuքficialis par <lb/>ua minoris curuitatis, ꝗ̃ ſit ſuքficies cornea cõtinẽs illã:</s> <s xml:id="echoid-s25592" xml:space="preserve"> ſphęricitas.</s> <s xml:id="echoid-s25593" xml:space="preserve"> n.</s> <s xml:id="echoid-s25594" xml:space="preserve"> ſuքficiei humoris cryſtallini <lb/>aſsimilatur cõpreſsiõi ſuքficiei lenticulę, ut patet cõſiderãtib.</s> <s xml:id="echoid-s25595" xml:space="preserve"> anatomiã oculi.</s> <s xml:id="echoid-s25596" xml:space="preserve"> Superficies ergo ante <lb/>rior ipſius eſt portio ſuperficiei maioris ſphęrę, ꝗ̃ ſit ſphęra uuea continẽs ipſam:</s> <s xml:id="echoid-s25597" xml:space="preserve"> & hęc cõpreſsio ę-<lb/>qualiter deflectitur ad oppoſitionẽ foraminis, qđ eſt in anteriori parte uueę:</s> <s xml:id="echoid-s25598" xml:space="preserve"> quia ſitus eius ab eo eſt <lb/>cõſimilis.</s> <s xml:id="echoid-s25599" xml:space="preserve"> Sicut aũt foramẽ rotundũ, quod eſt in anteriori parte uueę, eſt directè oppoſitũ extremi-<lb/>tati cõcauitatis nerui, ſuք quẽ collocatur oculus:</s> <s xml:id="echoid-s25600" xml:space="preserve"> ſic etiã in parte poſteriore cõcauitatis uueę eſt fo-<lb/>ramen rotundũ, quod eſt ſuper extremitatẽ cõcauitatis nerui:</s> <s xml:id="echoid-s25601" xml:space="preserve"> & foramẽ, quod eſt in anteriori uueę, <lb/>eſt oppoſitũ foramini concauitat<gap/>s nerui:</s> <s xml:id="echoid-s25602" xml:space="preserve"> quoniã neruus opticus interſecat tunicã coniunctiuã & <lb/>uueam, & penetrat omnes tunicas oculi uſq;</s> <s xml:id="echoid-s25603" xml:space="preserve"> ad ſphærã cryſtallinã, quæ pyramidẽ nerui interſecat, <lb/>ſicut & humer uitreus, qui in nerui optici pyramidali cõcauo collocatur:</s> <s xml:id="echoid-s25604" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s25605" xml:space="preserve"> cõmunis ſectio pyra <lb/>midis nerui optici, & ſphærę cryſtallinæ, eſt circulus per 110 t 1 huius:</s> <s xml:id="echoid-s25606" xml:space="preserve"> ſphæra itaq;</s> <s xml:id="echoid-s25607" xml:space="preserve"> glacialis eſt com-<lb/>poſita in extremitate cõcauitatis nerui optici, & in foramine poſteriori uueæ rotundo.</s> <s xml:id="echoid-s25608" xml:space="preserve"> Extremitas <lb/> <pb o="87" file="0389" n="389" rhead="LIBER TERTIVS."/> ergo nerui continet medium ſphærę glacialis:</s> <s xml:id="echoid-s25609" xml:space="preserve"> & eſt neruus ille concauus deferens in ſe ſpiritum ui <lb/>ſibilem à cerebro ad oculum, & per eius uenas paruas peruenit nutrimentũ ad oculum, & diffundi-<lb/>tur in illo per uias nutrimenti:</s> <s xml:id="echoid-s25610" xml:space="preserve"> & eſt in interſe-<lb/> <anchor type="figure" xlink:label="fig-0389-01a" xlink:href="fig-0389-01"/> ctiõe huius nerui in anteriori parte cerebri uir <lb/>tus uiſiua ſentiẽs & dijudicãs omne uiſibile:</s> <s xml:id="echoid-s25611" xml:space="preserve"> & <lb/>conſolidatur uuea cum glaciali in circulo conti <lb/>nente foramen rotundum in poſteriori uueæ.</s> <s xml:id="echoid-s25612" xml:space="preserve"> <lb/>Interſecãt quoq;</s> <s xml:id="echoid-s25613" xml:space="preserve"> ſe ſphæræ iſtæ duæ, ſcilicet gla <lb/>cialis & uitrea neceſſariò:</s> <s xml:id="echoid-s25614" xml:space="preserve"> cum cõuexum unius <lb/>obuiet cõuexo alterius:</s> <s xml:id="echoid-s25615" xml:space="preserve"> ſicut enim ſunt diuerſę <lb/>naturę & diaphanitatis, ſic ſunt portiões diuer-<lb/>ſarum ſphærarum ſe ſecantium:</s> <s xml:id="echoid-s25616" xml:space="preserve"> cõmunis itaq;</s> <s xml:id="echoid-s25617" xml:space="preserve"> <lb/>ſectio illarum ſphærarum eſt circulus per 80 t 1 <lb/>huius.</s> <s xml:id="echoid-s25618" xml:space="preserve"> Idem ergo circulus eſt baſis pyramidis <lb/>nerui optici, & interſectionis eiuſdem pyrami-<lb/>dis, & ſphærę cryſtallinæ, & conſolidationis u-<lb/>ueæ ſphæræ cum ſphæra cryſtallina, & fortè in-<lb/>terſectionis earundẽ ſphærarũ.</s> <s xml:id="echoid-s25619" xml:space="preserve"> Corpus uerò <lb/>conſolidatiuę continet partẽ pyramidalem ner <lb/>ui, quę eſt intra foramen oſsis, per quod tranſit <lb/>neruus, & intra circumferentiam ſphærę glacia <lb/>lis:</s> <s xml:id="echoid-s25620" xml:space="preserve"> & continet ſphærã uueam.</s> <s xml:id="echoid-s25621" xml:space="preserve"> Ex his itaq;</s> <s xml:id="echoid-s25622" xml:space="preserve"> pa-<lb/>tet humorẽ glacialem propriè eſſe organũ uir-<lb/>tutis uiſiuę:</s> <s xml:id="echoid-s25623" xml:space="preserve"> nam huius ſolius diaphanitas eſt rẽ <lb/>ceptibilis formarum uiſibiliũ:</s> <s xml:id="echoid-s25624" xml:space="preserve"> & eſt in medio o-<lb/>mnium & humorũ & tunicarũ collocatus:</s> <s xml:id="echoid-s25625" xml:space="preserve"> & ſi <lb/>alij cuicũq;</s> <s xml:id="echoid-s25626" xml:space="preserve"> tunicę uel humori accidat læſio, ſal <lb/>uo glaciali humore, ſemper auxilio medicinę re <lb/>cipit oculus curationem, & ſanatur ac reſtitui-<lb/>tur uiſus:</s> <s xml:id="echoid-s25627" xml:space="preserve"> ipſa uerò corrupta, corrumpitur uiſus <lb/>totus ſine ſpe reſtitutiõis per auxiliũ curę medi <lb/>cinalis.</s> <s xml:id="echoid-s25628" xml:space="preserve"> Eſt itaq;</s> <s xml:id="echoid-s25629" xml:space="preserve"> humor cryſtallinus uel glacia-<lb/>lis principaliter uirtutis uiſiuę organũ:</s> <s xml:id="echoid-s25630" xml:space="preserve"> propter <lb/>quod eſt diligentius conſeruatũ.</s> <s xml:id="echoid-s25631" xml:space="preserve"> Et cõſtituit na <lb/>tura duos oculos propter perfectionẽ bonita-<lb/>tis uiſionis, & complementũ eius.</s> <s xml:id="echoid-s25632" xml:space="preserve"> Sic ergo pa-<lb/>tet, quòd humores & tunicę oculi ſphæricè ſe <lb/>interſecant:</s> <s xml:id="echoid-s25633" xml:space="preserve"> & patet declaratio definitionis pro <lb/>poſitæ oculi ſecundũ omniũ eorũ experientiã <lb/>qui de ipſius anatomia hactenus ſcripſerũt.</s> <s xml:id="echoid-s25634" xml:space="preserve"> Hæc aũt omnia, quæ ſcilicet de cõpoſitione oculi in hac <lb/>quarta propoſitione huius tertij librι noſtræ perſpectiuæ ſunt præmiſſa:</s> <s xml:id="echoid-s25635" xml:space="preserve"> nunc ſummatim in figura <lb/>mathematica adiecta ſpectanda proponimus.</s> <s xml:id="echoid-s25636" xml:space="preserve"/> </p> <div xml:id="echoid-div964" type="float" level="0" n="0"> <figure xlink:label="fig-0389-01" xlink:href="fig-0389-01a"> <caption xml:id="echoid-caption1" xml:space="preserve">VERA OCVLI DESCRIPTIO <lb/>atq; effigies è recentiorib{us} anatomicis <lb/>libris deſumpta.</caption> </figure> </div> </div> <div xml:id="echoid-div966" type="section" level="0" n="0"> <head xml:id="echoid-head776" xml:space="preserve" style="it">5. Impoßιbile est uiſum reb{us} uiſis applicari per radios ab oculis egreſſos. Alhazen 23 n 1. <lb/>item 23 n 2.</head> <p> <s xml:id="echoid-s25637" xml:space="preserve">Si enim aliqui radij egrediuntur ab oculis, per quos uirtus uiſiua rebus extra cõiungitur:</s> <s xml:id="echoid-s25638" xml:space="preserve"> aut illi <lb/>radij ſunt corporei;</s> <s xml:id="echoid-s25639" xml:space="preserve"> uel incorporei.</s> <s xml:id="echoid-s25640" xml:space="preserve"> Si corporei, tũc cum uiſus uiderit ſtellas & cœlum:</s> <s xml:id="echoid-s25641" xml:space="preserve"> neceſſarium <lb/>eſt, ut à uiſu corporeũ exiẽs impleat totũ ſpacium uniuerſi, quod eſt inter uiſum & partẽ cœli uiſam <lb/>præter diminutionẽ ipſius oculi:</s> <s xml:id="echoid-s25642" xml:space="preserve"> quod & impoſsibile eſt fieri, & etiã tam citò fieri (ſubſtãtia & quan <lb/>titate oculi manente ſalua.</s> <s xml:id="echoid-s25643" xml:space="preserve">) Si uerò detur, quòd radij ſint incorporei, cum ſenſus nõ ſit niſi in re cor <lb/>porali:</s> <s xml:id="echoid-s25644" xml:space="preserve"> tunc ipſi radij nõ ſentirent rem uiſam:</s> <s xml:id="echoid-s25645" xml:space="preserve"> ergo nec oculus corporeus mediante hoc incorporeo <lb/>non ſentiente poterit ſentire:</s> <s xml:id="echoid-s25646" xml:space="preserve"> nec enim talia incorporea red dunt aliquid uiſui, quõ uiſus poſſet com <lb/>prehendere rem uiſam, cum uiſus non fiat, niſi per contactũ uiſus cum forma uiſa:</s> <s xml:id="echoid-s25647" xml:space="preserve"> quia ſine cõtactu <lb/>nõ fit actio.</s> <s xml:id="echoid-s25648" xml:space="preserve"> Radij ergo procedentes ab oculo ſi nihil reddunt uiſui:</s> <s xml:id="echoid-s25649" xml:space="preserve"> tunc non fiet per ipſos uiſio:</s> <s xml:id="echoid-s25650" xml:space="preserve"> ſi ue <lb/>rò aliquid reddunt uiſui, hæc erunt luces uel colores, quę per ſe uidentur, & quæ inter radios multi-<lb/>plicantur ad uiſum.</s> <s xml:id="echoid-s25651" xml:space="preserve"> Radij ergo nõ ſunt cauſſa applicationis uiſus cum rebus uiſis, ſed aliquid aliud, <lb/>quod ſe multiplicat ad uiſum, eſt per ſe cauſſa uiſiõis.</s> <s xml:id="echoid-s25652" xml:space="preserve"> Impoſsibile eſt ergo radios per ſe eſſe cauſſam <lb/>uiſionis, niſi fortè radij dicantur lineæ deſcriptę per puncta formarũ multiplicata à ſuperficiebus re <lb/>rum uiſarum ad uiſum:</s> <s xml:id="echoid-s25653" xml:space="preserve"> quoniam, ut patet per 2 huius, inter quodlibet punctum ſuperficiei rei uiſibi <lb/>lis, & aliquod punctum ſuperficiei uiſus neceſſe eſt poſſe produci lineas rectas, ut res actu uideatur:</s> <s xml:id="echoid-s25654" xml:space="preserve"> <lb/>tales uerò radij ab oculis non egrediuntur.</s> <s xml:id="echoid-s25655" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s25656" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div967" type="section" level="0" n="0"> <head xml:id="echoid-head777" xml:space="preserve" style="it">6. Viſio fit ex actione formæ uiſibilis in uiſum, & ex paßione uiſ{us} ab hac forma. Alhazen <lb/>1. 2. 3. 14 n 1.</head> <p> <s xml:id="echoid-s25657" xml:space="preserve">Formas uiſibiles agere in uiſum ex 2 & 3 ſuppoſitione patet:</s> <s xml:id="echoid-s25658" xml:space="preserve"> lęditur enim uiſus ex forti luce, ut in <lb/> <pb o="88" file="0390" n="390" rhead="VITELLONIS OPTICAE"/> aſpectu corporis ſolaris uel alterius lucis fortis, utlucis reflexæ ad oculum à corpore polito, uel ab <lb/>alio corpore ualde albo.</s> <s xml:id="echoid-s25659" xml:space="preserve"> In his enim debilitatur uiſus taliter, ut à ſua cadat operatione, quouſq;</s> <s xml:id="echoid-s25660" xml:space="preserve"> per <lb/>uirtutem intrinſecam naturalem fuerit reſtitutus.</s> <s xml:id="echoid-s25661" xml:space="preserve"> Sed & uiſus patitur à ſenſibilibus formis:</s> <s xml:id="echoid-s25662" xml:space="preserve">retinet <lb/>enim quandoq;</s> <s xml:id="echoid-s25663" xml:space="preserve"> in ſe fortes earũ impreſsiones.</s> <s xml:id="echoid-s25664" xml:space="preserve"> Viſus enim poſtquã diu inſpexerit fortem lucem uel <lb/>colorẽ, ſi poſtea aſpiciat locũ obſcurũ uel locũ debilis lucis:</s> <s xml:id="echoid-s25665" xml:space="preserve"> inueniet fortè illod uiſibile, quod prius <lb/>inſpexerat in ſe ipſo cũluce, colore, & figura ſua:</s> <s xml:id="echoid-s25666" xml:space="preserve"> & quandoq;</s> <s xml:id="echoid-s25667" xml:space="preserve"> color fortis impreſſus uiſui permiſce-<lb/>bitur coloribus rerũ uiſarũ in obſcuro, & uidebuntur resillæ alio colore mixto coloratę, ut fortè ui <lb/>ride uiſum facit res albas poſtea uiſas in loco obſcuriori mixtim uirides apparere:</s> <s xml:id="echoid-s25668" xml:space="preserve"> & ſi claudatur o-<lb/>culus, nihilominus occurret uιſui forma prius uiſa.</s> <s xml:id="echoid-s25669" xml:space="preserve"> Formæ ergo uiſibiles agunt in uiſum, & niſos pa <lb/>titur ab illis.</s> <s xml:id="echoid-s25670" xml:space="preserve"> Et quia uiſibilia per ſe ſunt lux & color, & lux eſt hypoſtaſis colorũ:</s> <s xml:id="echoid-s25671" xml:space="preserve"> lux aũt ſemper ſphę <lb/>ricè diffunditur ad omnẽ poſitionis differentiã:</s> <s xml:id="echoid-s25672" xml:space="preserve"> palàm ergo, ſic etiã colores diffundi.</s> <s xml:id="echoid-s25673" xml:space="preserve"> Cũ itaq;</s> <s xml:id="echoid-s25674" xml:space="preserve"> uiſus <lb/>opponitur alicui rei illuminatę uel coloratę, tunc multiplicatur lumẽ uel per ſe, uel cũ illo colore rei <lb/>oppoſitæ uiſui, & perueniẽs ad uiſus ſuperficiem & agit in uiſum, & uiſus patitur ab illo.</s> <s xml:id="echoid-s25675" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s25676" xml:space="preserve"> <lb/>lüx & color ueniunt ſimul ad ſuperficiem uiſus, & agunt in illũ, & uiſus patitur ab illis, & uirtus ani-<lb/>mæ propter unionẽ formarum uiſibilium cum ſuo organo fit cognoſcẽs:</s> <s xml:id="echoid-s25677" xml:space="preserve"> tunc fit uiſio propter præ-<lb/>ſentiam uiſibilium formarũ agentium in uiſum:</s> <s xml:id="echoid-s25678" xml:space="preserve"> & fit hæc actio & paſsio modo aliarum actionũ na-<lb/>turalium:</s> <s xml:id="echoid-s25679" xml:space="preserve"> quoniã totum agens agit in quodlibet paſsi punctũ, etiá in indiuiſibile, & totũ paſſum pa-<lb/>titur à quolibet puncto agẽtis.</s> <s xml:id="echoid-s25680" xml:space="preserve"> Forma ergo lucis & coloris, quę ſunt in aliquo punctorũ rei uiſibilis, <lb/>perueniunt ad totã ſuperficiem oculi:</s> <s xml:id="echoid-s25681" xml:space="preserve"> & formę omniũ punctorũ ſuperficiei rei uiſibilis perueniunt <lb/>ad punctum unum ſuperficiei oculi:</s> <s xml:id="echoid-s25682" xml:space="preserve"> & ſic fit actio & paſsio inter iſta.</s> <s xml:id="echoid-s25683" xml:space="preserve"> Non fit aũt actio formarum ui-<lb/>ſibilium in uiſum, niſi forma uiſibilis ſit potens ad agendũ & completæ hypoſtaſis ex luminis præ-<lb/>ſentia, & niſi mediũ extrinſecũ oculo & rei uiſibili fit lucidum actu, & niſi organũ uiſus fit receptiuũ <lb/>formarũ pertunicas medias, & humores diaphanos ſuæ propriæ diaphanitatis.</s> <s xml:id="echoid-s25684" xml:space="preserve"> Pars enim tunicæ <lb/>corneæ ſuperpoſita foramini uueę, quę primò aeri extrinſeco cõiungitur, & humor albugineus im-<lb/>plens foramẽ uueæ, ſi à propria ceciderit diaphanitate, utpote mutata qualitate ſibi propria uel im-<lb/>pedimento alio occurrente, uel etiã ipſe humor glacialis ſi per nimiam congelationẽ, uel alio modo <lb/>à formarũ receptione fuerit impeditus, nõ fit uiſio:</s> <s xml:id="echoid-s25685" xml:space="preserve"> quia forma ſenſibilis organo uiſiuo imprimi nõ <lb/>poteſt.</s> <s xml:id="echoid-s25686" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s25687" xml:space="preserve"> uiſibilis ueniens à re uiſa per medium lucidũ uſq;</s> <s xml:id="echoid-s25688" xml:space="preserve"> ad ſuperficiẽ uiſus, tranſit per <lb/>diaphanitatẽ tumcarũ uiſus, & peruenit ad uirtutẽ uiſiuam ex foramine, quod eſt in anteriori uueę, <lb/>& peruenit ad glacialem, & pertranſit in eo ſecundũ modum ſuæ diaphanitatis:</s> <s xml:id="echoid-s25689" xml:space="preserve"> & ob hoc natura o-<lb/>mnes tunicas oculi diaphanas ordinauit, ut à formis ſenſibilibus actũ lucidi habentιbus patiantur.</s> <s xml:id="echoid-s25690" xml:space="preserve"> <lb/>Viſus uerò licet patiatur à formis uiſibilibus:</s> <s xml:id="echoid-s25691" xml:space="preserve"> nõ tamen tingitur à forma lucis ucl coloris poſt receſ-<lb/>ſum præſentię corporis lucidi uel colorati, ſicut uniuerſaliter oſtendimus hanc paſsionẽ conuenire <lb/>omnicorpori diaphano per 4 t 2 huius:</s> <s xml:id="echoid-s25692" xml:space="preserve"> & licet quandoq;</s> <s xml:id="echoid-s25693" xml:space="preserve"> propter fortitudinẽ lucis & coloris fiat ali <lb/>qua impreſsio in ulſum, & alteratio ſecundum illas luces & colores:</s> <s xml:id="echoid-s25694" xml:space="preserve"> nõ tamen illę remanent in uiſu, <lb/>nιſi tempore modico:</s> <s xml:id="echoid-s25695" xml:space="preserve"> nõ eſt ergo talis alteratio fixa.</s> <s xml:id="echoid-s25696" xml:space="preserve"> Viſus itaq;</s> <s xml:id="echoid-s25697" xml:space="preserve"> non tingitur & coloribus & formis <lb/>lucis tinctura fixa, formis ſenſibilibus agentibus in uiſum.</s> <s xml:id="echoid-s25698" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s25699" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div968" type="section" level="0" n="0"> <head xml:id="echoid-head778" xml:space="preserve" style="it">7. Centrum ſphærætoti{us} oculi: & centrũ glacialis: & centrum ſuperficierum extrinſecæ & <lb/>intrinſecæ corneæ: & centrũ conuexæ ſuperficιei humoris albuginei neceſſe eſt idẽ eſſe. Ex quo pa <lb/>tet, quonia ſuperficies intrinſecæ corneæ ſuperficiei ſuæ extrinſecæ æquidiſtat. Alhazen 12 n 1.</head> <p> <s xml:id="echoid-s25700" xml:space="preserve">Reſumpta figura oculi, quam pręmiſimus in 4 huius:</s> <s xml:id="echoid-s25701" xml:space="preserve"> dico, quod uerum eſt, quòd hic proponitur, <lb/>quoniã punctũ a eſt cõmune centrum propoſitarũ ſphærarũ.</s> <s xml:id="echoid-s25702" xml:space="preserve"> Si enim detur, quod centrũ ſphærę to-<lb/>tius oculi (quod eſt punctũ a) non ſit centrum ſphærę glacialis, palàm per 75 t 1 huius, quoniã lineæ <lb/>rectę perpendiculares ſuper ſuperficiem ſphærę oculi, non ſunt perpendiculares ſuper ſuperficiem <lb/>ſphærę glacialis, niſi ſolùm illa, quæ tranſit per ambarum centra:</s> <s xml:id="echoid-s25703" xml:space="preserve"> cæterę uero omnes, quę erunt per-<lb/>pendiculares ſuper ſuperficiẽ uiſus, erunt declinantes ſuper ſuperficiem glacialis.</s> <s xml:id="echoid-s25704" xml:space="preserve"> Si ergo glacialis <lb/>comprehendat formas rerũ uiſarum ſecundũ incidentiã iſtarum linearũ, quę ſunt perpendiculares <lb/>ſuper ſuperficiem oculi, & obliquantur declinantes ſuper ſuperficiem glacialis:</s> <s xml:id="echoid-s25705" xml:space="preserve"> tunc neceſſariò gla-<lb/>cialis comprehendit oẽs formas rerum uiſibiliũ obliquatas, & declinantes à ſuo ſitu & figura, quam <lb/>habent extrà in ſuperficiebus rerũ uiſibiliũ, quod eſt contra 5 ſuppoſitionẽ præmiſſam in principio <lb/>huius libri.</s> <s xml:id="echoid-s25706" xml:space="preserve"> Et quoniã formę incidentes medio ſecundi diaphani denſioris ſecundũ lineas non per-<lb/>pendiculares refringuntur ad perpendicularẽ, ut patet per 47 t 2 huius:</s> <s xml:id="echoid-s25707" xml:space="preserve"> ſubſtantia uerò humorũ & <lb/>tunicarũ oculi denſior eſt aere circũſtante, & ſubſtantię diuerſę diaphanitatis inter ſe, ut patet per 4 <lb/>huius:</s> <s xml:id="echoid-s25708" xml:space="preserve"> palàm, quòd in ipſa ſuperficie glacialis fiet refractio alia quàm in ſuperficie corneæ:</s> <s xml:id="echoid-s25709" xml:space="preserve"> nõ diſtin <lb/>guet ergo glacialis aliquid in rebus uiſis propter refractionẽ formarũ in ſua ſuperficie factarũ:</s> <s xml:id="echoid-s25710" xml:space="preserve"> mani <lb/>ſeſtum eſt enim, quòd lineæ obliquè incidentes ſuperficiei uiſus, magis obliquãtur in ſuperficie gla <lb/>cialis:</s> <s xml:id="echoid-s25711" xml:space="preserve"> cum glacialis ſit alterius diaphanitatis à cornea uel albugineo humore:</s> <s xml:id="echoid-s25712" xml:space="preserve"> eſt enim in glaciali ali <lb/>qua diaphanitas, propter quã recipit formas, & aliqua ſpiſsitudo prohibens tranſitũ formarũ:</s> <s xml:id="echoid-s25713" xml:space="preserve"> & ob <lb/>hoc figuntur formę in eius ſuperficie & corpore.</s> <s xml:id="echoid-s25714" xml:space="preserve"> Nullam ergo formarũ uiſibilium cõprehendit gla-<lb/>cialis ſecundum eius ſitum, & figurã, quam habuit extra uiſum:</s> <s xml:id="echoid-s25715" xml:space="preserve"> hoc autẽ eſt impoſsibile:</s> <s xml:id="echoid-s25716" xml:space="preserve"> quoniã pa-<lb/>tet manifeſtè per 5 ſuppoſitionẽ, quòd glacialis cõprehendit formas rerũ uiſibilium ſecundũ ſitum <lb/>& figurã, quę habent in rebus extrà.</s> <s xml:id="echoid-s25717" xml:space="preserve"> Eſt ergo neceſſariũ, quòd lineę, quę ſunt perpendiculares ſuper <lb/>ſuperficiem oculi, ſint perpendiculares ſuper ſuperficiẽ glacialis:</s> <s xml:id="echoid-s25718" xml:space="preserve"> erunt ergo ſuperficies oculi, & gla <lb/> <pb o="89" file="0391" n="391" rhead="LIBER TERTIVS."/> cialis ſuperficies ſphærarum contentarum habentes idem centrum, & extremitates omnium linea-<lb/>rum imaginatarũ produci à quolibet puncto ſuperficiei rei uiſæ perpendiculariter ſuper ſuperficiẽ <lb/>oculi, cõcurrunt in hoc centro per 72 t 1 huius:</s> <s xml:id="echoid-s25719" xml:space="preserve"> & ſunt perpendiculares ſuper ſuperficiem glacialis <lb/>per 72 t 1 huius.</s> <s xml:id="echoid-s25720" xml:space="preserve"> Et quoniã ſuperficies corneæ anterius cõplet oculi ſuperficiem ſphæricã, & fit cum <lb/>illa una ſuperficies ſphærica:</s> <s xml:id="echoid-s25721" xml:space="preserve"> patet, quoniã centrum oculi eſt centrũ corneæ per definitionẽ ſphærę.</s> <s xml:id="echoid-s25722" xml:space="preserve"> <lb/>Patetitaq;</s> <s xml:id="echoid-s25723" xml:space="preserve">, quoniã centrum oculi, & centrum glacialis, & centrum corneę ſunt idem centrum.</s> <s xml:id="echoid-s25724" xml:space="preserve"> Quia <lb/>ergo centrum oculi (quod eſt centrum ſuperficiei exterioris ipſius corneæ, & centrum ſphærę gla-<lb/>cialis) ſunt unum cum centro totius oculi ex omnibus ſuis humoribus & telis cõſtante:</s> <s xml:id="echoid-s25725" xml:space="preserve"> conuenien <lb/>tius naturæ eſt, ut centrũ glacialis ſit ipſum centrum ſuperficiei interioris corneæ, ita quòd centra <lb/>omnium ſuperficierũ oppoſitarũ foramini uueæ ſit unum punctum cõmune, & ſuperficies concaua <lb/>corneæ ſphæræ fiat æquidiſtãs eius ſuperficiei conuexæ:</s> <s xml:id="echoid-s25726" xml:space="preserve"> ſic enim per 72 & 74 t 1 huius erunt omnes <lb/>lineæ exeuntes à centro ad ſuperficiem oculi perpendiculares ſuper omnes ſuperficies oppoſitas <lb/>foramini, & augebitur bonitas uiſionis:</s> <s xml:id="echoid-s25727" xml:space="preserve"> & erit totus oculus rotundus propter unitatẽ centri corneę <lb/>cum toto oculo.</s> <s xml:id="echoid-s25728" xml:space="preserve"> Et quoniã per 73 t 1 huius ſuperficies intrinſeca corneæ æquidiſtans eſt ſuperficiei <lb/>extrinſecę ipſius, cum ipſarum ambarum ſit idem centrum:</s> <s xml:id="echoid-s25729" xml:space="preserve"> humor uerò albugineus ſecundum eius <lb/>conuexum contingit concauum corneæ, ut præmiſſum eſt per experientiam anatomizantium in 4 <lb/>huius:</s> <s xml:id="echoid-s25730" xml:space="preserve"> ergo per 79 t 1 huius ſuperficies conuexa humoris albuginei erit pars ſuperficiei ſphæricæ <lb/>ſecundum eius conuexum ſuperficiem concauam ſphærę corneæ contingentis.</s> <s xml:id="echoid-s25731" xml:space="preserve"> Patet ergo per 73 t 1 <lb/>huius, quoniam conuexæ ſuperficiei humoris albuginei & concauę ſuperficiei corneę eſt idem cen <lb/>trum.</s> <s xml:id="echoid-s25732" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s25733" xml:space="preserve"> Et patet corollarium.</s> <s xml:id="echoid-s25734" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div969" type="section" level="0" n="0"> <head xml:id="echoid-head779" xml:space="preserve" style="it">8. Sphæram uueam neceſſe eſt toti oculo eccentricã eſſe, centruḿ ei{us} ad anteri{us} oculi pl{us} <lb/>acceder e: centrum uerò oculi ampli{us} profundari. Ex quo patet, centrum uueæ centris omnium <lb/>tunicarum & humorum anterioris partis oculi ampli{us} eleuari. Alhazen 8 n 1.</head> <p> <s xml:id="echoid-s25735" xml:space="preserve">Cum enim (ut patet per 4 huius, & per præcedentem) ſphęra cornea ſecundum eius ſuperficiem <lb/>manifeſtam ſit continua cum ſuperficie totius oculi, & pars ſphæræ ipſius, & totus oculus ſit ſphæ-<lb/>ra maior quàm ſphæra uuea:</s> <s xml:id="echoid-s25736" xml:space="preserve"> quoniam intra ſe cõtinet maximũ circulum ſphærę uueæ:</s> <s xml:id="echoid-s25737" xml:space="preserve"> patet per de-<lb/>finitionem ſphærarũ ſe intrinſecus interſecantiũ, quòd ſuperficies ſphærę corneę eſt maior ſuperfi-<lb/>cie ſphæræ uueæ:</s> <s xml:id="echoid-s25738" xml:space="preserve"> palàm itaq;</s> <s xml:id="echoid-s25739" xml:space="preserve"> ex definitione ſphærę maioris, quoniam ſemidiameter corneæ eſt ma <lb/>ior ſemidiametro uueæ.</s> <s xml:id="echoid-s25740" xml:space="preserve"> Et quia ſuperficies intrinſeca corneæ ſuperpoſita foramini uueę, eſt ſuper-<lb/>ficies concaua ſphærica æquidiſtans ſuperficiel manifeſtæ ipſius corneæ, eò quòd tota cornea eſt <lb/>æqualis ſpiſsitudinis, ut oſtenſum eſt in præcedenti, ideo quòd centrum ſuperficiei intrinſecæ cor-<lb/>neæ idem eſt cum centro ſuperficiei manifeſtæ conuexæ eiuſdem corneę<unsure/>:</s> <s xml:id="echoid-s25741" xml:space="preserve"> ſed ſuperficies concaua <lb/>corneæ ſecat ſuperficiem ſphærę uueæ ſuper circumferentiam foraminis, quod eſt in anteriori par-<lb/>te uueę, ut præmiſſum eſt in 4 huius, & declaratum per 80 t 1 huius:</s> <s xml:id="echoid-s25742" xml:space="preserve"> ergo per 84 t 1 huius centrum <lb/>ſphærę corneæ continentis ſphæram uueam neceſſe eſt remotius eſſe in profundo quàm centrum <lb/>ſphærę uueę.</s> <s xml:id="echoid-s25743" xml:space="preserve"> Patet ergo, quoniã ſphæram uueam neceſſe eſt toti oculo eccentricam eſſe, centrumq́;</s> <s xml:id="echoid-s25744" xml:space="preserve"> <lb/>eius ad anterius oculi plus accedere, centrum uerò oculi amplius profundari:</s> <s xml:id="echoid-s25745" xml:space="preserve"> quod eſt principale <lb/>propoſitũ.</s> <s xml:id="echoid-s25746" xml:space="preserve"> Et ex hoc etiam patet corollarium, quia cum ſphæra uueę non ſit in medio cõſolidatiuę, <lb/>ſed anterius ad partẽ ſuperficiei manifeſtę<unsure/> oculi, & cũ ſuperficies manifeſta ipſius oculi ſit pars ſphę <lb/>ræ maioris:</s> <s xml:id="echoid-s25747" xml:space="preserve"> palàm, ut præmiſſum eſt, quia centrum eius erit remotius in profundo centro uueæ.</s> <s xml:id="echoid-s25748" xml:space="preserve"> Ma <lb/>nifeſtum uerò oculi eſt ſuperficies ipſius corneæ extrinſeca cõuexa, cui æquidiſtat eiuſdem ſuper-<lb/>ficies intrinſeca concaua.</s> <s xml:id="echoid-s25749" xml:space="preserve"> Centrum ergo tam ſuperficiei concauæ quàm ſuperficiei conuexæ i-<lb/>pſius corneæ plus proſundatur in oculo quàm centrum uueæ.</s> <s xml:id="echoid-s25750" xml:space="preserve"> Et quia ſuperficies concaua corneæ <lb/>contingit ſuperficiem humoris albuginei, qui eſt in anteriori foraminis uueæ, & ſuperponitur <lb/>ei:</s> <s xml:id="echoid-s25751" xml:space="preserve"> patet ex præmiſſa, & per 79 t 1 huius, quoniam ſuperficies conuexa humoris albuginei eſt ſu-<lb/>perficies ſphærica, cuius centrum eſt centrum ſuperficiei ſibi ſuperpoſitæ.</s> <s xml:id="echoid-s25752" xml:space="preserve"> Superficies ergo conue-<lb/>xa corneæ, & ſuperficies concaua ipſius, & ſuperficies conuexa humoris albuginei, attingens con-<lb/>cauum corneę, cum ſint ſuperficies ſphæricę æquidiſtantium ſphærarũ, palàm per 73 t 1 huius, quia <lb/>centrum ipſarum omnium eſt unus punctus, qui amplius profundatur centro uueę.</s> <s xml:id="echoid-s25753" xml:space="preserve"> Et quia ſuperfi <lb/>cies anterioris glacialis eſt ſphærica concentrica totali oculo per præcedentẽ:</s> <s xml:id="echoid-s25754" xml:space="preserve"> & etiam quia ſuperfi <lb/>cies ſphærę glacialis cõuexa ſecat ſuperficiem ſphærę uueæ intrinſecus:</s> <s xml:id="echoid-s25755" xml:space="preserve"> patet per 84 t 1 huius, cum <lb/>ſuperficies glacialis ſit portio ſphærę maioris, quàm ſuperficies ſphærę uueę, quod amplius profun <lb/>datur centrum glacialis quàm centrum uueæ.</s> <s xml:id="echoid-s25756" xml:space="preserve"> Centrum itaq;</s> <s xml:id="echoid-s25757" xml:space="preserve"> uueæ centris omnium tunicarum & <lb/>humorum oculi, qui ſunt anterioris partis oculi ad partem aeris extrinſecam reſpicientes, amplius <lb/>eleuatur.</s> <s xml:id="echoid-s25758" xml:space="preserve"> Quod eſt totum propoſitum.</s> <s xml:id="echoid-s25759" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div970" type="section" level="0" n="0"> <head xml:id="echoid-head780" xml:space="preserve" style="it">9. Inter centrum oculi & centrum uueæ product a linea recta centrum circuli ſectionis uueæ, <lb/>& medium concauit atis nerui optici neceſſariò penetrabit. Alhazen 7 n 1.</head> <p> <s xml:id="echoid-s25760" xml:space="preserve">Oſten ſum eſt per 7 huius, idem eſſe centrum totius oculi & centrum corneæ:</s> <s xml:id="echoid-s25761" xml:space="preserve"> ſed linea, quæ con-<lb/>tinuat duo centra corneę & uueę (quæ in præmiſſa figura oculi in 4 huius eſt linea a f) hæc produ-<lb/>cta peruenit ad centrum circuli communis earũ ſectionis per 82 t 1 huius, ut in punctum f, centrum <lb/>circuli foraminis uueę, ſecundum cuius peripheriã illæ ſphærę ſe interſecant:</s> <s xml:id="echoid-s25762" xml:space="preserve"> ſuperficies enim con-<lb/>caua corneæ, & ſuperficies conuexa uueę ſunt duę ſuperficies ſphæricę ſecantes ſe ſecundum peri-<lb/> <pb o="90" file="0392" n="392" rhead="VITELLONIS OPTICAE"/> pheriam foraminis uueæ, ut patet per 4 huius:</s> <s xml:id="echoid-s25763" xml:space="preserve"> palàm quoq;</s> <s xml:id="echoid-s25764" xml:space="preserve"> per 86 t 1 huius, quòd eadem linea pro-<lb/>ducta peruenit ad duo media duarum ſuperficierum corneæ inter ſe æquidiſtantium ſuperpoſitarũ <lb/>illi foramini uueæ, cuius foraminis peripheria eſt circumferentia circuli ſectionis.</s> <s xml:id="echoid-s25765" xml:space="preserve"> Et quoniam fo-<lb/>ramen, quod eſt in anteriori uueæ, eſt directè oppoſitum foramini, quod eſt in poſteriori uueæ, <lb/>quod eſt extremitas concauitatis nerui:</s> <s xml:id="echoid-s25766" xml:space="preserve"> palàm per 111 t 1 huius, quoniam eadem linea producta me-<lb/>dium concauitatis nerui optici neceſſariò penetrabit:</s> <s xml:id="echoid-s25767" xml:space="preserve"> & hoc eſt centrum circuli baſis pyramidis ner <lb/>ui optici concaui.</s> <s xml:id="echoid-s25768" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s25769" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div971" type="section" level="0" n="0"> <head xml:id="echoid-head781" xml:space="preserve" style="it">10. Inter centra a ſphær arum glacialis & uueæ linea recta producta ad centrum circuli conſoli <lb/>dationis ſphær arum glacialis & uitreæ cum uuea neceſſario pertinget: & ſuper illi{us} circuli ſu-<lb/>perficiem erecta erit. Alhazen 9 n 1.</head> <p> <s xml:id="echoid-s25770" xml:space="preserve">Patuit ex præmiſsis in 4 huius, quoniã ſphæra glacialis interſecat intrinſecus ſphæram uueam:</s> <s xml:id="echoid-s25771" xml:space="preserve"> <lb/>linea ergo per centra iſtarum ſphærarum tranſiens 82 t 1 huius, erit perpendicularis ſuper centrum <lb/>circuli cõmunis ſectionis ipſarũ.</s> <s xml:id="echoid-s25772" xml:space="preserve"> Iſte uerò circulus ſectionis, aut eſt circulus diſtinguens finem con <lb/>ſolidationis harum ſphærarũ ad inuiẽc, aut æquidiſtans ei:</s> <s xml:id="echoid-s25773" xml:space="preserve"> ſuperficies enim, quæ eſt in nateriori par <lb/>te glacialis, oppoſita eſt foramini, quod eſt in anteriori parte uueæ, & ſitus eius ab eo eſt ſitus conſi-<lb/>milis, ut patuit in 4 huius:</s> <s xml:id="echoid-s25774" xml:space="preserve"> terminus ergo iſtius ſuperficiei, qui eſt circulus ſectiõis inter duas ſuper-<lb/>ficies ſphærę glacialis & uitreæ, aut eſt ipſe circulus conſolidationis iſtarũ ſphærarũ cum uuea, aut <lb/>æquidiſtãs ei.</s> <s xml:id="echoid-s25775" xml:space="preserve"> Si ergo circulus ſectionis inter duas ſuperficies, glacialis ſcilicet ſphæræ & uitreæ fue <lb/>rit ipſe circulus cõſolidationis ipſarũ cum uuea:</s> <s xml:id="echoid-s25776" xml:space="preserve"> iſte ergo circulus, eſt circulus ſectionis inter ſuper-<lb/>ficiẽ glaclalis & uueę:</s> <s xml:id="echoid-s25777" xml:space="preserve"> & tũc, ut prius, per 82 t 1 huius patet, ꝓpoſitũ.</s> <s xml:id="echoid-s25778" xml:space="preserve"> Quòd ſi circulus ſectionis inter <lb/>ſuperficiẽ ſphærę glacialis & ſuperficiẽ ſphærę uitreæ nõ fuerit ipſe circulus cõſolidationis ſphæra-<lb/>rum cryſtallinæ & uitreę cũ ſphæra uuea, ſed fuerit æquidiſtãs circulo cõſolidationis earũ cũ uuea:</s> <s xml:id="echoid-s25779" xml:space="preserve"> <lb/>tunc ſuperficies ſphærę glacialis ſi imaginetur extendi intellectu mathematico, ſuper id, quod for-<lb/>ma naturalis ſuæ ſphærę extenditur, ſecabit ſphæram uueę ſuper circulum æquidiſtantẽ iſti circulo <lb/>ſectionis ſphærę glacialis & uitreę:</s> <s xml:id="echoid-s25780" xml:space="preserve"> quoniã iſte circulus æqualem habet ſitum à circunferentia ſphæ <lb/>ræ uueæ:</s> <s xml:id="echoid-s25781" xml:space="preserve"> & quia iſte circulus eſt æquidiſtans circulo conſolidationis:</s> <s xml:id="echoid-s25782" xml:space="preserve"> erit neceſſario circulus ſectio <lb/>nis inter ſuperficiem glacialis & ſuperficiẽ uueæ, aut ipſe circulus conſolidationis, aut æquidiſtans <lb/>ei.</s> <s xml:id="echoid-s25783" xml:space="preserve"> Quòd ſi circulus iſte fuerit ipſe circulus conſolidationis, palàm per 82 t 1 huius, quia linea tran-<lb/>ſiens per centrum glacialis, & per centrum uueæ, tranſibit perpendiculariter per centrum iſtius cir <lb/>culi, eò quòd iſte circulus eſt circulus ſectiõis inter duas illas ſuperficies ſphæricas.</s> <s xml:id="echoid-s25784" xml:space="preserve"> Sed ſi iſte circu <lb/>lus fuerit æquidiſtans circulo conſolidationis, & eſt æquidiſtans circulo ſectionis inter ſuperficiem <lb/>glacialis & ſuperficiem uueę:</s> <s xml:id="echoid-s25785" xml:space="preserve"> eſt ergo cum circulo ſectionis inter ſuperficiem glacialis & uitreæ, in <lb/>ſuperficie una ſphærica, quæ eſt ſuperficies glacialis, & eſt æquidiſtans circulo dictę ſectionis.</s> <s xml:id="echoid-s25786" xml:space="preserve"> Sed <lb/>ſi in aliqua ſphæra duo circuli fuerint æquidiſtantes, linea tranſiens perpendiculariter centrum u-<lb/>nius, neceſſariò trãſibit perpẽdiculariter centrũ alterius, ut patet per 68 & 66 t 1 huius.</s> <s xml:id="echoid-s25787" xml:space="preserve"> Linea igitur <lb/>quæ tranſibit per<gap/>centrũ uueę & per centrũ glacialis tranſit per centrũ circuli conſolidationis ſphæ <lb/>rarum glacialis & uitreę cum uuea ſecun dum omnes diſpoſitiones ſphærarũ & illorum circulorũ:</s> <s xml:id="echoid-s25788" xml:space="preserve"> <lb/>eſt ergo illa linea erecta ſuper ſuperficiem illius circuli per 66 t 1 huius.</s> <s xml:id="echoid-s25789" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s25790" xml:space="preserve"> Sunt <lb/>tamen neceſſariò hi tres circuli circulus unus, quamuis etiam ſi fuerint diuerſi circuli, & æquidiſtan <lb/>tes, eadem propoſita omnibus occurrunt:</s> <s xml:id="echoid-s25791" xml:space="preserve"> ſecundum eundem enim circulum ſecant ſe glacialis & <lb/>uitrea, & ambę illę ſecant uueam, & conſolidantur ſecundũ eundem circulum cum illa:</s> <s xml:id="echoid-s25792" xml:space="preserve"> & eſt ille cir <lb/>culus baſis concauitatis nerui optici:</s> <s xml:id="echoid-s25793" xml:space="preserve"> & ſic ille unus circulus obtinet officium quatuor circulorum.</s> <s xml:id="echoid-s25794" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div972" type="section" level="0" n="0"> <head xml:id="echoid-head782" xml:space="preserve" style="it">11. Sphæram uitream neceſſe eſt ſphæræ glaciali eccentricã eſſe: centruḿ uitreæ ad anteri{us} <lb/>oculi pl{us} accedere. Alhazen 10 n 1.</head> <p> <s xml:id="echoid-s25795" xml:space="preserve">Quia enim ſuperficies ſphærę glacialis, & ſuperficies ſphærę uitreæ ſunt duæ ſuperficies ſphæri-<lb/>cæ ſecantes ſe:</s> <s xml:id="echoid-s25796" xml:space="preserve"> centrum ergo ſuperficiei anterioris reſpectu manifeſti oculi, eſt remotius in profun-<lb/>do, quàm centrum ſuperficiei poſterioris per 84 t 1 huius, poſterior uerò harum duarum eſt ſuperfi-<lb/>cies ipſius uitreæ, ut præoſtenſum eſt in 4 huius.</s> <s xml:id="echoid-s25797" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s25798" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div973" type="section" level="0" n="0"> <head xml:id="echoid-head783" xml:space="preserve" style="it">12. Lineam tranſeuntem centrum glacialis & uueæ, centrũ quo uitreæ, & medium concau<gap/> <lb/>tatis nerui optici neceſſarium eſt tranſire. Alhazen 11 n 1.</head> <p> <s xml:id="echoid-s25799" xml:space="preserve">Quia linea recta tranſiens centrum ſphæræ glacialis & uueę, producta ſuper centrum circuli con <lb/>ſolidationis glacialis cum uuea, perpendicularis eſt ſuper ſuperficiem circuli conſolidationis ſphæ <lb/>rarum glacialis & uitreæ cum uuea, ut patet per 10 huius.</s> <s xml:id="echoid-s25800" xml:space="preserve"> Huic autem circulo aut idem eſt circulus <lb/>interſectionis glacialis cum uitrea, aut æquidiſtans ei:</s> <s xml:id="echoid-s25801" xml:space="preserve"> quocunq;</s> <s xml:id="echoid-s25802" xml:space="preserve"> uerò iſtorũ modorũ exiſtente, ſem <lb/>per erit prædicta linea perpendicularis ſuper circulũ ſectionis ſphæræ glacialis cum uitrea:</s> <s xml:id="echoid-s25803" xml:space="preserve"> palàm <lb/>ergo per 83 t 1 huius, quoniam ipſa tranſit per centrum ſphæræ uitreæ.</s> <s xml:id="echoid-s25804" xml:space="preserve"> Quia ergo linea iſta tranſit <lb/>per centrum uitreæ, patet per 82 t 1 huius, quòd ipſa neceſſariò centrum circuli cõſolidationis per-<lb/>pendiculariter tranſibit.</s> <s xml:id="echoid-s25805" xml:space="preserve"> Extenditur ergo in medio cõcauitatis nerui optici, ſuper quẽ componitur <lb/>oculus:</s> <s xml:id="echoid-s25806" xml:space="preserve"> quoniã circulus conſolidationis eſt baſis, & extremitas cõcauitatis nerui optici, ut patet ex <lb/>4 huius.</s> <s xml:id="echoid-s25807" xml:space="preserve"> Quia uerò oſtenſum eſt ſuprà per 9 huius, quòd inter centrum oculi & centrum uueæ pro-<lb/>ducta linea centrum circuli ſectionis uueæ, & medium concauitatis nerui optici neceſſariò pe-<lb/> <pb o="91" file="0393" n="393" rhead="LIBER TERTIVS."/> netrat, cum ab eodem puncto, ut à medio nerui optici ſuper eandem ſuperfieiem plures perpendi-<lb/>culares non poſſunt produci, ut patet per 20 t1 huius:</s> <s xml:id="echoid-s25808" xml:space="preserve"> palàm quoniam linea eadem per cétrum cir-<lb/>culi ſectionis ſphæræ uueæ & glacialis, & centrum uueæ & centrum oculi, & ſphęræ glacialis & ui-<lb/>treæ, & per centrum circuli conſolidationis eſt tranſiens.</s> <s xml:id="echoid-s25809" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s25810" xml:space="preserve"> ex præmiſsis, quòd una & ea-<lb/>dem linea tranſit per inedium cócauitatis nerui optici & per duo media omnium tunicarum oppo-<lb/>ſitarum ſoramini uueæ:</s> <s xml:id="echoid-s25811" xml:space="preserve"> & eſt ipſa per 74 t 1 huius, perpendicularis ſuper ſuperficies omnium tuni-<lb/>carum oppoſitarum foramini uueæ:</s> <s xml:id="echoid-s25812" xml:space="preserve"> & eſt perpendicularis ſuper ſuperficiem foraminis uueę:</s> <s xml:id="echoid-s25813" xml:space="preserve"> & eſt <lb/>perpendicularis ſuper ſuperficiem circuli có ſolidationis:</s> <s xml:id="echoid-s25814" xml:space="preserve"> & extenditur in medio concauitatis ner-<lb/>ui optici, ſuper quẽ componitur oculus:</s> <s xml:id="echoid-s25815" xml:space="preserve"> & ipſa eſt axis totius oculi:</s> <s xml:id="echoid-s25816" xml:space="preserve"> qui in propoſita ſuperius figu-<lb/>ratione eſt in rectitudine literarum f a, extenſa per medium concauitatis nerui optici.</s> <s xml:id="echoid-s25817" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div974" type="section" level="0" n="0"> <head xml:id="echoid-head784" xml:space="preserve" style="it">13. Viſus non coprehendit res uiſas niſicorpore medio diaphano exiſtẽte. Alhaz. 22.41 n 1.</head> <p> <s xml:id="echoid-s25818" xml:space="preserve">Quia enim, ut patet per 6 huius, uiſio non eſt niſi ex a ctione formę uifibilis uenientis àre uiſa ad <lb/>uiſum:</s> <s xml:id="echoid-s25819" xml:space="preserve"> formæ uerò non extenduntur niſi in corporibus diaphanis conſimilis diaphanitatis, in qui-<lb/>bus fit lucis & formarum extenſio fecundum lineas rectas, ut patet per 1 t 2 huius.</s> <s xml:id="echoid-s25820" xml:space="preserve"> Cum ergo lineas <lb/>productas à rebus uiſibilibus ad uiſum nõ abſcindit aliquod corpus medium non diaphanum:</s> <s xml:id="echoid-s25821" xml:space="preserve"> túc <lb/>perueniunt formæ ad uiſum, & uiſio completur:</s> <s xml:id="echoid-s25822" xml:space="preserve"> quòd ſi aliquod corpus non diaphanum interue-<lb/>nerit, impeditur multiplicatio formæ ad uiſum.</s> <s xml:id="echoid-s25823" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s25824" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div975" type="section" level="0" n="0"> <head xml:id="echoid-head785" xml:space="preserve" style="it">14. Non fit uiſio corpore uiſibiliexiſtẽte ſimilis diaphanitatis cum medio. Alhazen 42 n 1.</head> <p> <s xml:id="echoid-s25825" xml:space="preserve">Si enim corpus uiſibile ſit diaphanum:</s> <s xml:id="echoid-s25826" xml:space="preserve"> tunc non eſt coloratum, nec eſt habens formam lucis, ſed <lb/>ſolum lucidi:</s> <s xml:id="echoid-s25827" xml:space="preserve"> ergo non uidetur, quoniam ut patet per 4 t 2 huius, luxnon figitur in corporibus dia-<lb/>phanis taliter, utipſa tingat, uel quòd eis præſtet actum uiſibilitatis.</s> <s xml:id="echoid-s25828" xml:space="preserve"> Cum ergo diaphanitas corpo-<lb/>ris uiſibilis fuerit ſimilis diaphanitati aerisιtune erit eius diſpoſitio ſicut diſpoſitio aeris, & non ap-<lb/>prehenditur à uiſu, ſicut nec aer.</s> <s xml:id="echoid-s25829" xml:space="preserve"> Et ſimiliter eſt de alio medio quocunq,:</s> <s xml:id="echoid-s25830" xml:space="preserve"> nullum enim talium uide-<lb/>tur, cum diaphanitas rei uiſæ non fuerit ſpiſsior corporis medij diaphanitate.</s> <s xml:id="echoid-s25831" xml:space="preserve"> Si uerò corpus uiſum <lb/>fuerit diaphanum, ſed minus quàm medium:</s> <s xml:id="echoid-s25832" xml:space="preserve"> ſicuti cryſtallus reſpectu aeris:</s> <s xml:id="echoid-s25833" xml:space="preserve"> tunc res uiſa, quoniam <lb/>habet aliquem colorem reſpectu ſuæ ſpiſsitudinis, uidebitur per mediũ aerem ueluti res colorata:</s> <s xml:id="echoid-s25834" xml:space="preserve"> <lb/>quoniam cum lux oritur ſuper ipſum, ſigetur in ipſo aliqua ſixione, ſcilicet ſecundum id, quod eſt <lb/>in ipſa de ſpiſsitudine, & pertranſibit in eo ſecundũ ſuam diaphanitatem:</s> <s xml:id="echoid-s25835" xml:space="preserve"> & erit in eo forma in aere <lb/>ſecundum colorem & lucem, quę ſuntin ſua ſuperſicie, & illa forma cum peruenerit ad uiſum, ope-<lb/>rabitur in uiſum, & ſentiet uiſus rem uiſam.</s> <s xml:id="echoid-s25836" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s25837" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div976" type="section" level="0" n="0"> <head xml:id="echoid-head786" xml:space="preserve" style="it">15. Inter uiſibile & oculi ſuperſiciẽ diſtantiam mediam neceſſariũ eſt eſſe. Alhazen 37 n 1.</head> <p> <s xml:id="echoid-s25838" xml:space="preserve">Non enim apprehendit uiſus rem uiſibilem, niſi quádo fuerit in ea aliqua lux media perιhuius:</s> <s xml:id="echoid-s25839" xml:space="preserve"> <lb/>hoc autem non eſt niſi per mediam diſtantiam.</s> <s xml:id="echoid-s25840" xml:space="preserve"> Quando ergo uiſibile fuerit ſuperpoſitum uiſui ſine <lb/>medio, tunc ipſum non uidetum:</s> <s xml:id="echoid-s25841" xml:space="preserve"> res enim per ſe luminoſæ non poſſunt immediatè ſuperficiei uiſus <lb/>applicari:</s> <s xml:id="echoid-s25842" xml:space="preserve"> talia enim ſunt, ut ſtellæ & ignis, quæ uiſui immediatè non poſſunt applicari:</s> <s xml:id="echoid-s25843" xml:space="preserve"> quoniam ex <lb/>eorum applicatione ſequeretur corruptio uidentis.</s> <s xml:id="echoid-s25844" xml:space="preserve"> Reliqua uerò corpora nõ luminoſa ſi uiſui ap-<lb/>plicentur, illa ſine lumine non uidebuntur requiritur ergo media diſtãtia inter illa corpora, & inter <lb/>ſuperficiem ipſius uiſus, in qua ſe diffundant corporum illorum formæ mediante luce.</s> <s xml:id="echoid-s25845" xml:space="preserve"> Et etiã cor-<lb/>poribus uiſibilibus ipſi uiſui immediatè applicatis:</s> <s xml:id="echoid-s25846" xml:space="preserve"> tunc corpus oculi ſecundum ſitum ſuum pro-<lb/>hibetur à uiſuali operatione.</s> <s xml:id="echoid-s25847" xml:space="preserve"> Quia enim uiſio non fit, niſi ex parte oppoſita foramini uueæ, ut patet <lb/>per 4 huius:</s> <s xml:id="echoid-s25848" xml:space="preserve"> ſi ergo uiſus comprehendat rem uiſibilem per immediatam applicationem:</s> <s xml:id="echoid-s25849" xml:space="preserve"> non com-<lb/>prehendetillam niſi ſecundum partem applicatam foramini uueæ, & nõ comprehendet reſiduum <lb/>rei uiſæ:</s> <s xml:id="echoid-s25850" xml:space="preserve"> & ſi imaginetur res uiſa moueri ſuper oculi ſuperſiciem quouſq;</s> <s xml:id="echoid-s25851" xml:space="preserve"> uiſus totã illam rem con-<lb/>tingat, non propter hoc erit iudicium per uiſum, ſed potius per tactum:</s> <s xml:id="echoid-s25852" xml:space="preserve"> nec enim ſic aget in uiſum <lb/>forma uiſibilis, quę eſt forma multiplicata extra rem ſenſibilem, ſed res ipſa.</s> <s xml:id="echoid-s25853" xml:space="preserve"> Non ergo erit uiſio, niſi <lb/>inter uiſibile & oculi ſuperficiem ſit aliqua media diſtantia.</s> <s xml:id="echoid-s25854" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s25855" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div977" type="section" level="0" n="0"> <head xml:id="echoid-head787" xml:space="preserve" style="it">16. Viſio non ſit ſine dolore & paßione à ſubſtãtia oculi abijciente, Ex quo patet, uiſum opor-<lb/>tere conuenientis diſpoſitionis in ſanitate eſſe ad hoc, ut completè exerceat uiſionem. Alhazen <lb/>26. item 1.2 n 1.</head> <p> <s xml:id="echoid-s25856" xml:space="preserve">Quoniam enim glacialis recipit formam lucis & coloris:</s> <s xml:id="echoid-s25857" xml:space="preserve"> & lux & color operantur in glacialem:</s> <s xml:id="echoid-s25858" xml:space="preserve"> <lb/>erit neceſſariò illa operatio non ſine dolore, quáuis quandoq;</s> <s xml:id="echoid-s25859" xml:space="preserve"> non ſentiatur ille dolor, ut cum non <lb/>eſt ualde fortis.</s> <s xml:id="echoid-s25860" xml:space="preserve"> Luces uerò fortes anguſtiant uiſum, & læduntipſum manifeſtè, ut patet in luce ſo-<lb/>lis, uel in luce reflexa à corporibus politis ad uiſum.</s> <s xml:id="echoid-s25861" xml:space="preserve"> Et quia operatio omnis lucis in uiſum eſt ex <lb/>uno genere, non diuerſificata niſi ſecundum magis & minus:</s> <s xml:id="echoid-s25862" xml:space="preserve"> & maior operatio cuiuslibet lucis in <lb/>uiſum eſt ex genere doloris, & non diuerſificatur in hoc niſi ſecundum magis & minus, ſic etiam <lb/>quòd quandoq;</s> <s xml:id="echoid-s25863" xml:space="preserve"> latet doloripſum ſenſum:</s> <s xml:id="echoid-s25864" xml:space="preserve"> ſemper tamen illa paſsio quantumcũq;</s> <s xml:id="echoid-s25865" xml:space="preserve"> inſenſibilis abij-<lb/>cit à ſubſtantia oculi.</s> <s xml:id="echoid-s25866" xml:space="preserve"> Exhoc ergo patet, quòd oportet uiſum conuenientis diſpoſitionis in ſanitare <lb/>eſſe ad hoc, ut completè exerceat uiſionem:</s> <s xml:id="echoid-s25867" xml:space="preserve"> quoniam ſemper comprehẽſio uiſibilium à uiſu eſt ſe-<lb/>cundum ſortitudinem uiſus:</s> <s xml:id="echoid-s25868" xml:space="preserve"> quia ſenſus uiſus oculorum diuerſificatur ſecundum uigorem & de-<lb/>bilitatem ipſorum:</s> <s xml:id="echoid-s25869" xml:space="preserve"> humidi enim oculi citius læduntur à lucibus & coloribus, & ſicci minus.</s> <s xml:id="echoid-s25870" xml:space="preserve"> Et hæc <lb/>uoluimus declarare.</s> <s xml:id="echoid-s25871" xml:space="preserve"/> </p> <pb o="92" file="0394" n="394" rhead="VITELLONIS OPTICAE"/> </div> <div xml:id="echoid-div978" type="section" level="0" n="0"> <head xml:id="echoid-head788" xml:space="preserve" style="it">17. Viſio diſtinctafit ſolùm ſecũdum perpendiculares lineas à punctis reiuiſa ad oculi ſuper-<lb/>ficiem productas. Ex quo patet, omnem formam uiſam ſic ordinari in oculi ſuperſicie, ſicui eſt <lb/>ordinatain ſuperficierei uiſæ. Alhazen 15.18 n 1.</head> <p> <s xml:id="echoid-s25872" xml:space="preserve">Licetenim, utoſtenſum eſt in 6 huius, tota forma rei uiſibilis agat in uiſum, & in quodlibet pun-<lb/>ctum ſuperficiei uiſus:</s> <s xml:id="echoid-s25873" xml:space="preserve"> quia tamen per 20 t1 huius forma tantùm unius puncti totius ſuperficiei rei <lb/>uiſæ oppoſitæ uiſui perpendiculariter incidit uni puncto ſuperficiei uiſus, & ſormæ omnium pun-<lb/>ctorum reſiduorum ſuperficiei rei uiſæ ueniunt ad illud idem punctũ ſuperficiei uiſus ſuper lineas <lb/>declinantes per 13 p 11, & in quoliber puncto ſuperficiei uiſus tranſeuntin eodem tempore formæ <lb/>omnium punctorum, quæ ſunt in ſuperficiebus omnium uiſibilium oppoſitorum uiſui in illo tem-<lb/>pore:</s> <s xml:id="echoid-s25874" xml:space="preserve"> quoniá ſuppoſitum eſt in principio huius 6 ſuppoſitione, uiſum ſimul diuerſa uiſibilia uidere:</s> <s xml:id="echoid-s25875" xml:space="preserve"> <lb/>ſola uerò forma puncti, quæ perpendiculariter incidit illi puncto ſuperficiei uiſus, per 47 t1 huius <lb/>tranſit rectè per diaphanitatem omnium tunicarum oculi:</s> <s xml:id="echoid-s25876" xml:space="preserve"> formæ uerò omnium aliorum punctorũ <lb/>refringuntur, & tranſeunt per diaphanitatem tunicarum uiſus ſecundum lineas declinantes ſuper <lb/>fuperficiem uiſus:</s> <s xml:id="echoid-s25877" xml:space="preserve"> & etiam ex quolibet puncto ſuperficiei glacialis erit una tantùm perpendicula-<lb/>ris ſuper ſuperficiem uiſus:</s> <s xml:id="echoid-s25878" xml:space="preserve"> quoniam cũ ſphæræ glacialis & totius oculi ſitidem centrũ, utpatet per <lb/>7 huius:</s> <s xml:id="echoid-s25879" xml:space="preserve"> quęcunq;</s> <s xml:id="echoid-s25880" xml:space="preserve"> linea fuerit perpendicularis ſuper ſuperficiẽ unius, & ſuper alterius ſuperficiem, <lb/>perpendicularis erit per 74 t1 huius:</s> <s xml:id="echoid-s25881" xml:space="preserve"> ſicut autem ex eodem puncto ſuperficiei ſphæræ glacialis ſe-<lb/>cundum ponentes radios egredi à uiſu, exeũt lineæ infinitæ ad ſuperficiẽ uiſus, quæ ſunt declinan <lb/>tes ſuper ſuperficiem uiſus:</s> <s xml:id="echoid-s25882" xml:space="preserve"> ſic à puncto aliquo ſuperficiei glacialis, ex quo exit perpendicularis ſu-<lb/>per ſuperficiem uiſus, & pertranſit foramen uueæ, exeũt lineæ aliæ infinitę tranſeuntes in foramen <lb/>uueæ, & peruenientes ad ſuperficiẽ uiſus declinantes.</s> <s xml:id="echoid-s25883" xml:space="preserve"> Et ſicut radij imaginati egredi à uiſibus quá-<lb/>do fuerint imaginati refringi ſecundũ modũ differẽtiæ diaphanitatis corneæ à diaphanitate aeris, <lb/>per 47 t 2 huius perueniunt ad diuerſa loca & ad puncta diuerſa in ſuperficiebus rerum uiſibilium <lb/>oppoſitarũ uiſui in uno tempore, & nulla iſtarum linearũ occurrit puncto, quod eſt apud extremi-<lb/>tatem perpendicularis.</s> <s xml:id="echoid-s25884" xml:space="preserve"> Sic etiam ſecundũ nos ponentes radios non egredi, ſed formas diffundi ad <lb/>uiſum, formæ punctorũ uiſibilium, quæ ſunt apud extremitates harum linearum, extenduntur ſe-<lb/>cundum rectitudinem harum linearũ, & perueniũt ad ſuperficiem uiſus, & per 47 t 2 huius refrin-<lb/>guntur ad idem punctũ ſuperficiei glacialis:</s> <s xml:id="echoid-s25885" xml:space="preserve"> ſolus autem punctus, qui eſt apud extremitatem per-<lb/>pendicularis, non refrin gitur, ſed ſemper extenditur ſecundú rectitudinem perpẽdicularis, & per-<lb/>tranſit ad illum punctũ glacialis.</s> <s xml:id="echoid-s25886" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s25887" xml:space="preserve"> glacialis ſecundũ lineas non perpendiculares ſentiat:</s> <s xml:id="echoid-s25888" xml:space="preserve"> tunc <lb/>puncti, qui ſunt in ſuperficiebus uiſibilium, nunquá ordinabuntur in ſenſu ſecundũ modum ordi-<lb/>nis ſui in ſuperficie rei uiſæ:</s> <s xml:id="echoid-s25889" xml:space="preserve"> quoniam in eodem puncto occurrũt formæ admixtæ ex multis formis <lb/>diuerſis, & ex coloribus diuerſis, & nõ diſtinguetur aliquid in illis:</s> <s xml:id="echoid-s25890" xml:space="preserve"> ſed ſi glacialis ſecundum lineas <lb/>perpendiculares tantùm ſentiet:</s> <s xml:id="echoid-s25891" xml:space="preserve"> tunc diſtin guentur in eo puncti, qui ſunt in ſuperficiebus uiſibi-<lb/>lium, nec erit differẽtia ſitus & ordinationis formarum uiſibilium in ſuperficie glacialis & in rebus <lb/>uiſibilibus, quæ ſunt extrà.</s> <s xml:id="echoid-s25892" xml:space="preserve"> Quoniam autem ſecũdum 5 ſuppoſitionem noſtram formæ uiſibilium <lb/>perueniunt ad uiſum ſub figuris, quas habẽt in rebus extrà:</s> <s xml:id="echoid-s25893" xml:space="preserve"> patet quòd ſecundũ ſolas perpendicu-<lb/>lares lineas fit uiſio:</s> <s xml:id="echoid-s25894" xml:space="preserve"> tunc enim ſolùm forma uiſa ſic ordinatur in oculi ſuperficie, ſicut eſt ordinata <lb/>in ſuperficie rei uiſæ.</s> <s xml:id="echoid-s25895" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s25896" xml:space="preserve"> Omnes itaq;</s> <s xml:id="echoid-s25897" xml:space="preserve"> lineæ diffuſionis quarumcunq;</s> <s xml:id="echoid-s25898" xml:space="preserve"> uiſarum <lb/>formarũ, quæ ſunt perpendiculares ſuper ſuperficies tunicarũ uiſus, continẽtur in pyramide, cuius <lb/>uertex eſt centrũ uiſus, & cuius baſis eſt circulus foraminis uueæ, uel pars ſuperficiei illius circuli:</s> <s xml:id="echoid-s25899" xml:space="preserve"> <lb/>& quantò magis exten ditur bæc pyramis, & remouetur à uiſu, tantò magis amplificatur:</s> <s xml:id="echoid-s25900" xml:space="preserve"> & omnes <lb/>formæ rerum cadentiũ intra illam pyramidem, extendũtur in rectitudinem linearũ radialiũ, & per-<lb/>tranſeunt tunicas oculorũ refractæ:</s> <s xml:id="echoid-s25901" xml:space="preserve"> & hác pyramidem dicimus pyramidem radialem.</s> <s xml:id="echoid-s25902" xml:space="preserve"> Formæ uerò <lb/>rerum uiſibiliũ, quæ ſunt extra hanc pyramidem, nun quam incidũt per aliquam illarũ linearũ per-<lb/>pendiculariũ, ſed fortè accidit ipſas extẽdi per lineas rectas, quæ ſunt inter ipſas & ſuperficiẽ uiſus <lb/>oppoſitam foramini uueæ, & illæ formę refringũtur à diaphanitate tunicarũ uiſus, & nó perueniũt <lb/>ordinatè ad uirtutem uiſiuam:</s> <s xml:id="echoid-s25903" xml:space="preserve"> unde non fit diſtincta uiſio ſecundũ illas:</s> <s xml:id="echoid-s25904" xml:space="preserve"> ueruntamẽ illas ſormas re-<lb/>fractas aliqualiter accidit uideri, ſed indiſtinctè, in cócurſu ſcilicet ipſarum cũ lineis perpendicula-<lb/>ribus à cẽtro oculi extra pyramidem radialem productis.</s> <s xml:id="echoid-s25905" xml:space="preserve"> Dicimus autem nũc ſuperficiem uiſus illá <lb/>partem ſuperficiei oculi, quæ eſt oppoſita ſuperficiei foraminis uueæ.</s> <s xml:id="echoid-s25906" xml:space="preserve"> Quòd autem uiſus compre-<lb/>hendat quádoq;</s> <s xml:id="echoid-s25907" xml:space="preserve"> illa, quæ ſunt extra pyramidẽ radialem, patet experimentaliter.</s> <s xml:id="echoid-s25908" xml:space="preserve"> Extremitas enim <lb/>acus uel ſtipulæ ſubtilis poſitæ in poſtremo oculi, utinter palpebras uel in parte lachrimali quie-<lb/>ſcente uiſu, uidebitur, cũ tamen illa extremitas ſit extra pyramidem radialẽ.</s> <s xml:id="echoid-s25909" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s25910" xml:space="preserve"> in eiſ-<lb/>dem locis circa oculũ erecto indice uel alio digito extra pyramidẽ radialem, quæ ualde ſubtilis eſt, <lb/>quoniam pyramidalitas eius nó eſt ampla:</s> <s xml:id="echoid-s25911" xml:space="preserve"> unde nihil ſui peruenit ad loca, quæ circũdant oculũ, ui-<lb/>debitur tamen ſuperficies ipſius indicis uel alterius digiti.</s> <s xml:id="echoid-s25912" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s25913" xml:space="preserve"> iſtorũ uiſibiliũ peruenit ad <lb/>ſuperficiẽ uiſus per lineas obliquas, quæ ſunt extra pyramidẽ radialẽ.</s> <s xml:id="echoid-s25914" xml:space="preserve"> Patet ergo, quòd formæ rerũ <lb/>taliter ſituatarũ reſpectu pyramidis radialis, perueniũt ad ſuperficiẽ uiſus ք refractionẽ factã in ſu-<lb/>perficie uiſus ab aere, ꝗ eſt rarioris diaphani, quá ſint tunicæ ipſius uiſus.</s> <s xml:id="echoid-s25915" xml:space="preserve"> Quòd aũt refractio fiat in <lb/>ſuքficie ipſi{us} uiſus formarũ obliquè uiſui incidẽtiũ, patet etiá in illis, quorũ formæ niſi ꝓhiberẽtur, <lb/>caderẽt intra pyramidẽ radialẽ.</s> <s xml:id="echoid-s25916" xml:space="preserve"> Si enim acus uel alia res ſubtilis minuta directè oppoſita foramini <lb/>uueæ interponatur uiſui & parieti albo:</s> <s xml:id="echoid-s25917" xml:space="preserve"> uidebitur tñ forma toti{us} parietis, cũ ſecũdũ ueritatẽ formæ <lb/> <pb o="93" file="0395" n="395" rhead="LIBER TERTIVS."/> partis parietis directè oppoſitæ acui & uiſui, directè nõ perueniat ad ſuperficiem ipſius uiſus, per-<lb/>uenit autem, ut patet, quoniam uidetur.</s> <s xml:id="echoid-s25918" xml:space="preserve"> Palàm ergo, quoniam peruenit per refractionem factam in <lb/>ſuperficie ipſius uiſus:</s> <s xml:id="echoid-s25919" xml:space="preserve"> omnia autem hæc uidẽtur indiſtinctè:</s> <s xml:id="echoid-s25920" xml:space="preserve"> unde reductis ipſis intra pyramidem <lb/>radialem, & ablato quolibet corpore interpoſito, uidebuntur illarum formæ diſtinctè & perſectius <lb/>quàm prius.</s> <s xml:id="echoid-s25921" xml:space="preserve"> Fit ergo uiſio diſtincta ſolùm ſecundum perpendiculares lineas à punctis rei uiſæ ad <lb/>oculi ſuperficiem productas:</s> <s xml:id="echoid-s25922" xml:space="preserve"> indiſtincta uerò uiſio fit per lineas non perpendiculares, & ita uiſio <lb/>indiſtincta coadiuuat diſtinctam.</s> <s xml:id="echoid-s25923" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div979" type="section" level="0" n="0"> <head xml:id="echoid-head789" xml:space="preserve" style="it">18. Omnium formarum uiſibilium diſtincta uiſio fit ſecundum pyramidem, cuius uertex eſt <lb/>in centro oculi, baſis uerò in ſuperficie rei uiſæ. Ex quo patet, omne quod uidetur, ſub angulo ui-<lb/>deri. Euclides 2 hypotbe.opt. Alhazen 19 n 1.</head> <p> <s xml:id="echoid-s25924" xml:space="preserve">Cum per 6 huius omnis uiſio fiat ex actione formę uiſibilis in uiſum:</s> <s xml:id="echoid-s25925" xml:space="preserve"> & quælibet pars formæ ui-<lb/>ſibilis & pũctus ſe multiplicet per medium extrinſecum ad oculi ſuperficiem totam:</s> <s xml:id="echoid-s25926" xml:space="preserve"> & tota ſuper-<lb/>ficies rei uiſæ ad unum punctũ oculi:</s> <s xml:id="echoid-s25927" xml:space="preserve"> quia tamen oculorũ tunicæ ſunt alterius diaphanitatis quàm <lb/>aer extrinſecus:</s> <s xml:id="echoid-s25928" xml:space="preserve"> ſolæ illæ lineæ formarum à ſuperficie rei uiſibilis ad ſuperficiẽ oculi productæ, quę <lb/>protractæ centrum oculi penetrant, cum ſint perpendiculares ſuper ſuperficiem oculi, non refrin-<lb/>guntur in medio diaphani ipſius corneæ, ut patet per 72 t 1 huius, & 47 t 2 huius, & per præ miſſam:</s> <s xml:id="echoid-s25929" xml:space="preserve"> <lb/>aliæ uerò lineæ omnes refringuntur, quia incidunt obliquè:</s> <s xml:id="echoid-s25930" xml:space="preserve"> unde nõ fit uiſio ſecundum illas.</s> <s xml:id="echoid-s25931" xml:space="preserve"> Quo-<lb/>niam autem ſolus glacialis propriè eſt organum uiſus, & non ſuperficies oculi, quæ eſt pars ſphęræ <lb/>corneæ:</s> <s xml:id="echoid-s25932" xml:space="preserve"> oportet neceſſariò ut lineæ, per quas debet fieri uiſio, perueniát ad glacialem.</s> <s xml:id="echoid-s25933" xml:space="preserve"> Et quia non <lb/>eſt poſsibile, ut uiſus comprehendat rem uiſam ſecundum ſuum eſſe, niſi quando apprehendit for-<lb/>mam unius puncti rei uiſæ ex uno tantùm puncto ſuæ ſuperficiei:</s> <s xml:id="echoid-s25934" xml:space="preserve"> quoniam, ut in præmiſſa oſtẽſum <lb/>eſt, omnis forma rei uiſæ ſic ordinatur in oculi ſuperficie, ſicut eſt ordinata in ſuperficie rei uiſæ.</s> <s xml:id="echoid-s25935" xml:space="preserve"> Nõ <lb/>eſt ergo poſsibile, ut glacialis comprehendat rem uiſam ſecundum ſuum eſſe, niſi quando compre-<lb/>hendit colorem uel ſormam unius puncti rei uiſæ ex uno tantùm puncto ſuperficiei uiſus uenien-<lb/>tem ad ſe.</s> <s xml:id="echoid-s25936" xml:space="preserve"> Et cum centrum oculi & centrum ſphæræ glacialis, ſicut patet per 7 huius, ſit idem pun-<lb/>ctum:</s> <s xml:id="echoid-s25937" xml:space="preserve"> neceſſe eſt, quòd omnes lineæ perpendiculariter productæ à punctis uiſibilium ſuper ſuper-<lb/>ficiem oculi diaphanam concurrant in centro glacialis:</s> <s xml:id="echoid-s25938" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s25939" xml:space="preserve"> quidẽ diametri in ſuperficiebus tu-<lb/>nicarum oculi perpendiculares ſuper ipſas tunicas oculi:</s> <s xml:id="echoid-s25940" xml:space="preserve"> eritq́ quælibet perpendicularis occurrẽs <lb/>ſuperficiei corneæ in puncto uno, & occurrens ſuperficiei glacialis in puncto uno:</s> <s xml:id="echoid-s25941" xml:space="preserve"> & una tantùm <lb/>perpendicularis tranſit per punctum aliquod glacialis à centro corneæ per ipſam ſuperficiem cor-<lb/>neæ ſuperpoſitam illi puncto glacialis, quæ ſit perpendicularis ſuperficiem rei uiſæ:</s> <s xml:id="echoid-s25942" xml:space="preserve"> quoniam <lb/>per 20 t 1 huius ab aliquo puncto ſuper ſuperficiem unam una tantùm perpendicularis duci poteſt.</s> <s xml:id="echoid-s25943" xml:space="preserve"> <lb/>Vnde cum ſuperficies rei uiſæ fuerit æ quidiſtãs ſuperficiei ipſius uiſus, erit per 23 t 1 huius illa linea <lb/>perpendicularis ſuper ſuperficiem uiſus & ſuper ſuperficiem rei uiſæ:</s> <s xml:id="echoid-s25944" xml:space="preserve"> aliæ uerò lineæ omnes ſunt <lb/>obliquæ ſuper ſuperficiem rei uiſæ, quamuis productæ ad centrum uiſus, fiant perpendiculares ſu-<lb/>per ſuperficiem uiſus, & ſuper ſuperficiem ipſius glacialis.</s> <s xml:id="echoid-s25945" xml:space="preserve"> Forma ergo cuiuslibet puncti ſuperficiei <lb/>rei uiſibilis mota ad uiſum ſecundum lineam unam perpendicularem productam ab eo ad ſuperfi-<lb/>ciem uiſus, occurrit ſuperficiei uiſus ſuper unum punctum, ſuper quem non occurrit ei aliqua for-<lb/>marum punctorum aliorum rei uiſibilis.</s> <s xml:id="echoid-s25946" xml:space="preserve"> Productis ergo a quolibet pũcto ſuperficiei rei uiſibilis ad <lb/>cẽtrum oculi lineis:</s> <s xml:id="echoid-s25947" xml:space="preserve"> palàm, quoniam iſtæ lineæ productæ in diuerſis punctis oculi, ſuperficiẽ ſphæ-<lb/>ricam oculi ſecabunt, & omnes in centrum oculi concurrent:</s> <s xml:id="echoid-s25948" xml:space="preserve"> quia omnes lineæ iſtæ continentur <lb/>quaſi in uno corpore continuo, quia à punctis quaſi continuis unius ſuperficiei rei uiſæ ad unum <lb/>punctum, qui eſt centrum oculi, terminantur.</s> <s xml:id="echoid-s25949" xml:space="preserve"> Palàm ergo, quoniam omnes iſtæ lineæ imaginandæ <lb/>ſuntin quadam pyramide uerticem habente in cẽtro oculi & baſim in ſuperficie rei uiſæ:</s> <s xml:id="echoid-s25950" xml:space="preserve"> erit enim <lb/>forma cuiuſcunq;</s> <s xml:id="echoid-s25951" xml:space="preserve"> puncti ſuperficiei rei uiſæ extenſa ſecundum rectitudinem lineæ, quæ eſt inter <lb/>illud punctum & uerticem pyramidis, qui eſt centrum uiſus:</s> <s xml:id="echoid-s25952" xml:space="preserve"> & omnes tunicarũ oculi & humorum <lb/>ſuperficies ſecant hanc pyramidem, quoniam formæ penetrant per illas:</s> <s xml:id="echoid-s25953" xml:space="preserve"> & ob hoc, quia ſuperficies <lb/>glacialis conuexa ſecat hác pyramidem quaſi æ quidiſtanter baſi, figuratur in illa ſuperficie glacia-<lb/>lis quaſi noua pyramis, cuius baſis eſt in ipſa ſuperficie glacialis & uertex, ubi prius, & baſes illarũ <lb/>pyramidum fiunt quaſi ſimiles, ut patet per 99 & 100 t 1 huius.</s> <s xml:id="echoid-s25954" xml:space="preserve"> Et ex hoc patet, omne quod uidetur, <lb/>ſub angulo uideri, quem continent lineæ radiales concurrentes in centro uiſus.</s> <s xml:id="echoid-s25955" xml:space="preserve"> Patet ergo propo-<lb/>ſitum.</s> <s xml:id="echoid-s25956" xml:space="preserve"> Linea itaq;</s> <s xml:id="echoid-s25957" xml:space="preserve"> recta tranſiens per omnia centra tunicarum uiſus ad locum gyrationis concaui <lb/>nerui, ſuper quem componitur oculus, quia illa, ut patet ex præmiſsis & 12 huius, tranſit per centrũ <lb/>uiſus & per centrum foraminis, quod eſt in anteriori uueæ, & per centrum ipſius uueæ exten ditur <lb/>in medio pyramidis radialis, dicatur axis pyramidis radialis:</s> <s xml:id="echoid-s25958" xml:space="preserve"> aliæ uerò lineæ huius pyramidis di-<lb/>cantur lineæ radiales.</s> <s xml:id="echoid-s25959" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div980" type="section" level="0" n="0"> <head xml:id="echoid-head790" xml:space="preserve" style="it">19. Corpus uiſibile oportet ut ſit alicuius quãtitatis reſpectu ſuperficiei uiſus, ad hoc, ut actu <lb/>uideatur. Alhazen 40 n 1.</head> <p> <s xml:id="echoid-s25960" xml:space="preserve">Iam enim oſtenſum eſt, quoniã uiſio ſemper fit per pyramidẽ, cuius conus eſt in centro oculi, & <lb/>baſis in ſup erficie rei uiſæ per præmiſſam:</s> <s xml:id="echoid-s25961" xml:space="preserve"> & quòd iſta pyramis diſtinguit ex ſuperficie mẽbri ſen-<lb/>tientis paruã partẽ, in qua ordinatur forma rei uiſæ, ut patet per 17 huius.</s> <s xml:id="echoid-s25962" xml:space="preserve"> In rebus ergo ualde par-<lb/>uis erit pyramis parua, & pars diſtincta per ipſam ex ſuperficie conuexa glacialis, quæ eſt primum <lb/>membrum ſentiens, erit quaſi punctus uel ualde parua:</s> <s xml:id="echoid-s25963" xml:space="preserve"> ſed membrũ ſentiens non ſentιt formã, niſi <lb/> <pb o="94" file="0396" n="396" rhead="VITELLONIS OPTICAE"/> quando pars ſuæ ſuperficiei, ad quam peruenit forma, fuerit quantitatis ſenſibilis, roſpectu totius <lb/>oculi, quoniã uirtutes ſenſus ſunt finitæ, & nõ extenduntur in infinitum:</s> <s xml:id="echoid-s25964" xml:space="preserve"> unde ſunt ſecundũ unum <lb/>aliquẽ terminum, ad quẽ peruenire poteſt uirtus ſenſitiua.</s> <s xml:id="echoid-s25965" xml:space="preserve"> Cum ergo pars membri ſentiẽtis, ad quá <lb/>peruenit forma, nõ eſt quantitatis ſenſibilis apud totum membrũ ſentiens:</s> <s xml:id="echoid-s25966" xml:space="preserve"> tunc nõ ſentit membrũ <lb/>actionem, quá agit ſorma rei uiſibilis in illa parte ꝓ pter paruitatẽ ipſius:</s> <s xml:id="echoid-s25967" xml:space="preserve"> quare nõ cõprehendit for-<lb/>mam rei tam paruę.</s> <s xml:id="echoid-s25968" xml:space="preserve"> Solæ itaq;</s> <s xml:id="echoid-s25969" xml:space="preserve"> res ſunt ſenſibiles actu, quarũ pyramides inter uiſum & centrũ uiſus <lb/>diſtinguunt ex ſuperficie glacialis partẽ aliquã ſenſibilis quátitatis, reſpectu totius ſuperficiei gla-<lb/>cialis:</s> <s xml:id="echoid-s25970" xml:space="preserve"> illæ ergo res oportet ut ſint alicuius quátitatis reſpectu ſuperficiei uiſus.</s> <s xml:id="echoid-s25971" xml:space="preserve"> Et hoc eſt ꝓpoſitũ.</s> <s xml:id="echoid-s25972" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div981" type="section" level="0" n="0"> <head xml:id="echoid-head791" xml:space="preserve" style="it">20. Viſio non completur, niſicum ordinatio formærecepta in ſuperficie glacialis, ad neruum <lb/>peruenerit communem. Alhazen 25 n 1.</head> <p> <s xml:id="echoid-s25973" xml:space="preserve">Quoniam enim, ut patet in 4 huius, in concurſu amborũ neruorum opticorum in anteriori par-<lb/>te cerebri conſtituta eſt uirtus uiſiua ſentiens & dijudicans omne uiſibile, propter quod in uno ui-<lb/>dente eſt unitas ſenſus uiſus, ob cuius unitatem ambobus uiſibus unam & eandẽ rem ſimul accidit <lb/>uideri:</s> <s xml:id="echoid-s25974" xml:space="preserve"> patet quòd uiſio nõ cõplebitur niſi cũ forma uiſibilis unietur uirtuti ſentiẽti, quę eſt in cõca-<lb/>uo cõmunis nerui:</s> <s xml:id="echoid-s25975" xml:space="preserve"> oportet enim cognoſcibile ſemper uniri ipſi cognoſcẽti.</s> <s xml:id="echoid-s25976" xml:space="preserve"> Quia uerò per 17 huius <lb/>formarum uiſibiliũ fit ordinatio in ipſius oculi ſuperficie, ſicut ordinatæ ſunt in ſuperficie rei uiſæ, <lb/>& ex 5 ſuppoſitione huius res uiſa ſecundum ſitum, figuram & or dinẽ ſuarum partium uidetur:</s> <s xml:id="echoid-s25977" xml:space="preserve"> ne-<lb/>ceſſe eſt ergo fieri ordinationem formæ in ipſo neruo communi ſecundum modum ordinationis, <lb/>quo eſtrecepta in ſuperficie glacialis, & aliter non complebitur uiſio.</s> <s xml:id="echoid-s25978" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s25979" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div982" type="section" level="0" n="0"> <head xml:id="echoid-head792" xml:space="preserve" style="it">21. Humorem uitreum alterius diaphanitatis à glaciali neceſſarium eſt eſſe. Alhaz. 2 n 2.</head> <p> <s xml:id="echoid-s25980" xml:space="preserve">Si enim diaphanitas iſtorum duorum, corporum glacialis ſcilicet humoris & uitrei, ſit conſimi-<lb/>lis:</s> <s xml:id="echoid-s25981" xml:space="preserve"> tunc (ut patet per 1 t 2 huius, & per 17 huius & per 72 t 1 huius) formę uiſibiles receptæ in ſuper-<lb/>ficie glacialis non refractæ ſecundum líneas radiales concurrent in cẽtro oculi propter conſimili-<lb/>tudinem diaphanitatis, & ibi ſe interſecãtes ulterius ſe diffundent.</s> <s xml:id="echoid-s25982" xml:space="preserve"> Quia uerò, ut patet per præmiſ-<lb/>ſam, uiſio non completur, niſi poſtquam ordinatio formæ, quæ recipitur in ſuperficie glacialis, per-<lb/>uenit ad neruum communem:</s> <s xml:id="echoid-s25983" xml:space="preserve"> ſitus autem partium formæ ſecũdum ſuum eſſe in ſuperficie glacia-<lb/>lis non poteſt peruenire ad neruum communem, niſi per extenſionem eius in cõcauo nerui;</s> <s xml:id="echoid-s25984" xml:space="preserve"> ſuper <lb/>quem componitur ſphæra glacialis, quia aliter eſt ipſum impoſsibile peruenire:</s> <s xml:id="echoid-s25985" xml:space="preserve"> forma uerò nõ po-<lb/>teſt extendi à ſuperficie glacialis ad concauum nerui communis ſecundum extenſionem linearum <lb/>rectarum, & cõſeruare ſitus ſuarum partium ſecundum ſuum eſſe, niſi natura alterius diaphani cla-<lb/>rioris ſibi occurrat, ante quá perueniat ad cẽtrum oculi:</s> <s xml:id="echoid-s25986" xml:space="preserve"> quoniá ſi non ſit medium alterius diaphani <lb/>cõmunis, iſtæ lineæ cõcurrent apud cẽtrum oculi, & efficietur quaſi unũ punctum.</s> <s xml:id="echoid-s25987" xml:space="preserve"> Et quia hoc cẽ-<lb/>trum oculi eſt ante locum unionis neruorum opticorum, patet per 91 t 1 huius, quòd ſi illę lineæ ul-<lb/>tra cẽtrum oculi debeát extẽdi, neceſſariò erit linearum illarum interſectio in cẽtro, & poſt cẽtrum <lb/>creabitur noua pyramis, cuius lineæ longitudinis ſecundum poſitionem & ſitum priori pyramidi <lb/>modo contrario ſe habebunt.</s> <s xml:id="echoid-s25988" xml:space="preserve"> Cõuertetur ergo totus ſitus figuræ rei uiſæ, quem habet in ſuperficie <lb/>rei uiſæ & in ſuperficie glacialis, taliter, ut illud, quod eſt in ſuperficie glaciali dextrú, fiat ſiniſtrum <lb/>apud ſenſum, & econtrariò, & ſuperius fiat inferius, & econtrariò:</s> <s xml:id="echoid-s25989" xml:space="preserve"> nec perueniet aliquid formæ di-<lb/>rectè ad neruum communem, niſi ſolum unum punctum, quod eſt in extremitate axis pyramidis.</s> <s xml:id="echoid-s25990" xml:space="preserve"> <lb/>Omnes ergo res ſecundum modum ſuo naturali ſitui contrarium uidentur:</s> <s xml:id="echoid-s25991" xml:space="preserve"> quod eſt contra 5 ſup-<lb/>poſitionem, & manifeſtè contra id, quod accidit in ſenſu.</s> <s xml:id="echoid-s25992" xml:space="preserve"> Patet ergo quod neceſſarium eſt, quod iſti <lb/>humores ſint diuerſæ diaphanitatis.</s> <s xml:id="echoid-s25993" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s25994" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div983" type="section" level="0" n="0"> <head xml:id="echoid-head793" xml:space="preserve" style="it">22. Superficiem communis ſectionis ſphæræ glacialis & uitreæ ad anterius cẽtro oculi ſitam <lb/>eſſe: humoreḿ uitreum & ſpiritum uiſibilem eiuſdẽ quaſi diaphanitatis, & utraplus dia-<lb/>phana humore glaciali neceſſe eſt eſſe. Alhazen 30 n 1. Item 4.5.6 n 2.</head> <p> <s xml:id="echoid-s25995" xml:space="preserve">Quoniam, ut patet per 20 huius, omnis forma rei uiſæ ſecundum ſitum, figuram & ordinem ſua.</s> <s xml:id="echoid-s25996" xml:space="preserve"> <lb/>rum partium peruenit ad neruum communem:</s> <s xml:id="echoid-s25997" xml:space="preserve"> palàm ſicut in præmiſſa oſtẽſum eſt, quòd neceſſa-<lb/>rium eſt, quòd fiat aliqua refractio ante peruentum formę ad centrum oculi:</s> <s xml:id="echoid-s25998" xml:space="preserve"> quia etiam ſi fiat refra-<lb/>ctio poſt centri tranſitum, erunt neceſſariò formæ cõuerſæ:</s> <s xml:id="echoid-s25999" xml:space="preserve"> quoniá & tunc per 91 t 1 huius erit mu-<lb/>tatus ſitus partium formę.</s> <s xml:id="echoid-s26000" xml:space="preserve"> Refractio uerò cum ſolum fiat ad perpendicularẽ, uel à perpẽdiculari, ut <lb/>patet per 47 t 2 huius:</s> <s xml:id="echoid-s26001" xml:space="preserve"> palàm, quia non tranſmutat ſitum partium, ſed ſolum auget uel minuit figu-<lb/>ram per 49 t 2 huius.</s> <s xml:id="echoid-s26002" xml:space="preserve"> Quia uerò glacialis, ad quem perueniunt formæ ſecundum rectitudinẽ, totus <lb/>eſt unius diaphani:</s> <s xml:id="echoid-s26003" xml:space="preserve"> refractio uero nõ fit niſi medio alterius diaphani:</s> <s xml:id="echoid-s26004" xml:space="preserve"> palàm, quia nõ poteſt fieri re-<lb/>fractio formarum niſi apud humorem uitreum, cuius corpus, ut in præcedẽti oſtenſum eſt, diuerſæ <lb/>eſt diaphanitatis à corpore glacialis.</s> <s xml:id="echoid-s26005" xml:space="preserve"> Hic ergo humor neceſſario antecedit cẽtrum oculi, ideo ut re-<lb/>fringantur formæ apud ipſum, priuſquã perueniãt ad ipſum cẽtrum oculi, quod eſt idẽ cẽtrum hu-<lb/>moris glacialis per 7 huius:</s> <s xml:id="echoid-s26006" xml:space="preserve"> quia aliàs enim in centro illo fieret cõcurſus omnium linearum radia-<lb/>liú per 72 t 1 huius:</s> <s xml:id="echoid-s26007" xml:space="preserve"> quia illæ lineæ ſunt omnes perpẽdiculares ſuper ſuperficiẽ glacialis:</s> <s xml:id="echoid-s26008" xml:space="preserve"> accideret <lb/>quoq;</s> <s xml:id="echoid-s26009" xml:space="preserve"> illis formis ulterius progrediẽtibus tranſmutatio ſecundum ſitum per 91 t 1 huius, ut pręmiſ-<lb/>ſum eſt:</s> <s xml:id="echoid-s26010" xml:space="preserve"> & ꝗa hoc eſt impoſsibile, patet ergo, quòd hum or uitreus antecedit cẽtrum glacialis.</s> <s xml:id="echoid-s26011" xml:space="preserve"> Quã-<lb/>uis itaq;</s> <s xml:id="echoid-s26012" xml:space="preserve"> glacialis, in quo eſt principiũ ſenſus, indigeat lineis radialib.</s> <s xml:id="echoid-s26013" xml:space="preserve"> extẽſis ſecũdũ rectitudinẽ, eò <lb/>quòd impoſsibile eſt, ut forma rei uiſę ſit ordinata in ſuքficie uiſus ꝓpter magnitudiné rei uiſę, & ք <lb/> <pb o="95" file="0397" n="397" rhead="LIBER TERTIVS."/> unitatem ſuperficiei corporis uiſus niſi per iſtas lineas, per quas completur comprehenſio rei uiſæ <lb/>ſecundum ſuum eſſe:</s> <s xml:id="echoid-s26014" xml:space="preserve"> peruentus tamen formarum ad ultimum ſentiens non indiget tantùm exten-<lb/>ſione formarum ſecũdum rectitudinem iſtarum linearum:</s> <s xml:id="echoid-s26015" xml:space="preserve"> quoniã receptio formarum in membro <lb/>ſentiente non eſt omnino ſunilis receptioni formarum in corpore diaphano:</s> <s xml:id="echoid-s26016" xml:space="preserve"> membrum enim ſen-<lb/>tiens recipit iſtas formas propter ſuam diaphanitatem, & ſentit eas propter eius uirtutem ſenſibi-<lb/>lem:</s> <s xml:id="echoid-s26017" xml:space="preserve"> & ſic recipit formas ſecundũ receptionem ſenſus, cum alia corpora diaphana recipiãt formas <lb/>tantùm ad repręſentandũ ipſas uiſui, non aũt ad ſentiendum.</s> <s xml:id="echoid-s26018" xml:space="preserve"> Qualitas ergo receptionis formarum <lb/>in humore uitreo ſecundum lineas refractas, eſt propter diuerſitatem ſuæ diaphanitatis à corpore <lb/>glacialis, & propter qualitatẽ receptionis ſenſibilis, quæ non eſt completa in humore glaciali.</s> <s xml:id="echoid-s26019" xml:space="preserve"> Sed <lb/>& corpus ſubtile, quod eſt in concauitate nerui inter humorẽ uitreum & neruum cõmunem, quod <lb/>corpus nominatur ſpiritus uiſibilis, quoniam in ipſo primò diſcurrunt ſpiritus uiſibiles, neceſſe eſt <lb/>diaphanum eſſe:</s> <s xml:id="echoid-s26020" xml:space="preserve"> quoniã ſormæ rerum uiſibilium quando perueniunt in corpus humoris uitrei, ex-<lb/>tenditur ſenſus ab illo in corpus ſentiens extenſum in concauo nerui continuati inter uiſum & an-<lb/>terius cerebri, & ſecundum extenſionem ſenſus extendũtur formæ ordinatæ ſecundũ ſuam diſpo-<lb/>ſitionem Patet ergo, quòd ordinatio partium corporis ſentientis ſormas, & ordinatio uirtutis ſen-<lb/>tientis æ qualiter eſt neceſſariò in corpore uitreo, & in omni corpore ſubtili extenſo in cõcauo ner-<lb/>ui.</s> <s xml:id="echoid-s26021" xml:space="preserve"> Cum enim ſorma peruenit ad aliquod punctum ſuperficiei uitreæ, extenditur directè, & non al-<lb/>teratur eius ſitus in concauitate nerui, in quo extenditur corpus ſentiens, & erunt formæ omnium <lb/>punctorum conſimilis ordinationis adinuicem.</s> <s xml:id="echoid-s26022" xml:space="preserve"> Corpus itaq;</s> <s xml:id="echoid-s26023" xml:space="preserve"> ſentiens, quod eſt in concauo nerui, <lb/>erit neceſſariò diaphanum propter receptionem formarũ uiſibilium:</s> <s xml:id="echoid-s26024" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s26025" xml:space="preserve"> diaphanitas eius quaſi <lb/>eadem cum diaphanitate humoris uitrei, ut nó obliquentur uel ſiant monſtruo ſæ formæ apud per-<lb/>uentum earum ad ultimam ſuperficiem uitrei ulcinantem corpori, quod eſt in concauo nerui.</s> <s xml:id="echoid-s26026" xml:space="preserve"> Per-<lb/>tranſeunt ergo ſormæ in iſto corpore ſubtili ratione diaphanitatis, & apparent uirtuti ſenſitiuæ ra-<lb/>tione ſpiſsitudinis eiuſdem corporis.</s> <s xml:id="echoid-s26027" xml:space="preserve"> Sentiens itaq;</s> <s xml:id="echoid-s26028" xml:space="preserve"> ultimum, quod eſt in neruo cómuni, compre-<lb/>henditlucem ex illuminatione corporis huius & colorem ex eius coloratione, quoniá horum ſor-<lb/>mæ tranſeunt & figuntur in ipſo.</s> <s xml:id="echoid-s26029" xml:space="preserve"> Fit autẽ refractio formarum apud humorẽ uitreum tam propter <lb/>diuerſitatẽ qualitatis receptionis ſenſus, quàm propter diuerſitatẽ diaphanitatis humoris glacialis <lb/>& uitrei.</s> <s xml:id="echoid-s26030" xml:space="preserve"> Et ſi diaphanitas ſuorũ corporum eſſet cõſimilis:</s> <s xml:id="echoid-s26031" xml:space="preserve"> eſſet forma extenſa in corpore uitreo ſe-<lb/>cundum rectitudinem linearũ radialiũ propter conſimilitudinẽ diaphanitatis, & eſſet refracta pro-<lb/>pter diuerſitatẽ qualitatis ſenſus inter hęc duo corpora:</s> <s xml:id="echoid-s26032" xml:space="preserve"> & ſic fieret ſorma aut móſtruoſa, aut eſſent <lb/>duæ formæ.</s> <s xml:id="echoid-s26033" xml:space="preserve"> Quádo uerò propter diaphanitatis diuerſitatẽ fit refractio, & diuerſitas qualitatis ſen-<lb/>ſus affirmat illá refractionẽ aut obliquationẽ:</s> <s xml:id="echoid-s26034" xml:space="preserve"> tunc erit forma poſt obliquationẽ refractionis, forma <lb/>una ordinata ſecũdum ſuarũ partium ſitum, figurá, & ordinẽ, qua haber ſorma in re extrà, & uirtus <lb/>ſenſitiua ſentit formam rei uiſæ ex toto corpore ſentiente extenſo à ſuperficie uiſus primò ſentiẽtis <lb/>& ſenſibiles formas recipiẽtis uſq;</s> <s xml:id="echoid-s26035" xml:space="preserve"> ad cõcauũ nerui cõmunis, quod eſt ultimũ corpus ſentiẽs:</s> <s xml:id="echoid-s26036" xml:space="preserve"> quo-<lb/>niam in ipſo cõſtituta eſt uirtus ſenſitiua.</s> <s xml:id="echoid-s26037" xml:space="preserve"> Sunt itaq;</s> <s xml:id="echoid-s26038" xml:space="preserve"> humor uitreus & corpus, quod eſt in cõcauita-<lb/>te nerui, eiuſdẽ quaſi diaphanitatis:</s> <s xml:id="echoid-s26039" xml:space="preserve"> quia inter ipſa nó fit refractio aliqua ſenſibilis diuerſa, ſed regu-<lb/>lariter per unitatẽ uirtutis ſenſitiuę ad unitatẽ ſimplicis extẽſionis formę poſt refractionẽ in ſuper-<lb/>ficie uitreæ.</s> <s xml:id="echoid-s26040" xml:space="preserve"> Et quoniá in ijs ambobus corporib.</s> <s xml:id="echoid-s26041" xml:space="preserve"> ſit ꝓ greſsio formarũ ultra centrũ oculi:</s> <s xml:id="echoid-s26042" xml:space="preserve"> patet, quòd <lb/>illa refractio facta eſt à քpẽdiculari erecta à pũcto refractionis ſuք ſuperficiẽ glacialis:</s> <s xml:id="echoid-s26043" xml:space="preserve"> utrũq;</s> <s xml:id="echoid-s26044" xml:space="preserve"> ergo <lb/>illorũ corporũ eſt plus diaphanũ corpore ipſius glacialis ք 45 uel 47 t 2 huius.</s> <s xml:id="echoid-s26045" xml:space="preserve"> Patet ergo ꝓpoſitũ.</s> <s xml:id="echoid-s26046" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div984" type="section" level="0" n="0"> <head xml:id="echoid-head794" xml:space="preserve" style="it">23. Superficiem communis ſectionis ſphæræ glacialis & uitreæneceße eſt planã eſſe: aut par-<lb/>tem ſphæræ maioris, quàm ſit ſphæra glacialis, & eccentricam ſuperficiei oculi. Alhazen 3 n 2.</head> <p> <s xml:id="echoid-s26047" xml:space="preserve">Iſtarum ſphærarum glacialis ſcilicet & uitreæ cõmunis ſectionis ſuperficies eſt neceſſariò plana, <lb/>aut talis, qualis proponitur:</s> <s xml:id="echoid-s26048" xml:space="preserve"> quoniá oportet ſuperficiẽ huius ſectionis eſſe ſimilis ordinationis, ita <lb/>quòd eius extremitates ordinẽtur in cõſimili & eadẽ diſtantia à cẽtro oculi, ut nõ appareãt formæ <lb/>monſtruoſæ poſt refractionẽ.</s> <s xml:id="echoid-s26049" xml:space="preserve"> Superficies cõſimilis ordinationis, aut eſt plana, aut eſt ſphærica:</s> <s xml:id="echoid-s26050" xml:space="preserve"> hæc <lb/>autẽ ſuperficies nõ poteſt eſſe ex ſphæra cõcentrica oculo:</s> <s xml:id="echoid-s26051" xml:space="preserve"> tũc enim eſſent lineæ radiales, quæ ſunt <lb/>perpẽdiculares ſuք ſuperficiẽ glacialis:</s> <s xml:id="echoid-s26052" xml:space="preserve"> perpẽdiculares etiá ſuper ipſam ex 74 t 1 huius:</s> <s xml:id="echoid-s26053" xml:space="preserve"> & nõ fieret <lb/>refractio ſormarũ, ſed cõcurrerent in cẽtro, & fierent formæ monſtruoſæ, ſicut per præmiſſam oſtẽ-<lb/>ſum eſt.</s> <s xml:id="echoid-s26054" xml:space="preserve"> Eſt ergo illa ſuperficies, ſi ſuerit pars ſphæræ, neceſſariò eccentrica oculo:</s> <s xml:id="echoid-s26055" xml:space="preserve"> ergo non poteſt <lb/>eſſe ex ſphæra minore quàm ſit ſphæra eccẽtrica oculo:</s> <s xml:id="echoid-s26056" xml:space="preserve"> quoniã ratione diuerſitatis cẽtri formæ cõ-<lb/>currerẽt ante peruentũ ſuum ad centrũ oculi:</s> <s xml:id="echoid-s26057" xml:space="preserve"> minoris enim ſphæræ min or eſt diameter quantũ eſt <lb/>de natura ſphæricitatis, & propter maiorẽ diaphanitatẽ ſphæræ uitreæ ſuper glacialẽ, quę oſtẽſa eſt <lb/>in præmiſſa, refringerẽtur formæ ab ipſa perpendiculari per 47 t 2 huius, ratione rarioris diaphani, <lb/>cui incidũt:</s> <s xml:id="echoid-s26058" xml:space="preserve"> ratione uerò ſphæræ minoris in ſuperficie cõmunis ſectionis frangerẽtur ad perpendi-<lb/>cularẽ.</s> <s xml:id="echoid-s26059" xml:space="preserve"> Sic ergo eſſicerẽtur formæ monſtruoſæ, quoniá procederẽt ad perpendicularẽ ratione ſuæ <lb/>perpẽdicularis ſuper ſuperficiẽ ſphæricá, quæ perpẽdiculares ſemper tranſeũt per centrũ per 72 t 1 <lb/>huius, & refringerẽtur à perpẽdiculari.</s> <s xml:id="echoid-s26060" xml:space="preserve"> Iſta ergo ſuperficies eſt aut plana aut ſphærica, utpote pars <lb/>ſphæræ alicuius bonæ quátitatis, ita quòd ſphæricitas eius cõueniat ordinationi ſecundũ propor-<lb/>tionẽ refractionis à perpẽdiculari, quæ fit ꝓpter naturã alterius diaphanitatis.</s> <s xml:id="echoid-s26061" xml:space="preserve"> Omnes ergo formæ <lb/>perueniẽtes in ſuperficiẽ glacialis, extendũtur per corpus glacialis ſecundũ rectitudinẽ linearũ ra-<lb/>dialiũ, quouſq;</s> <s xml:id="echoid-s26062" xml:space="preserve"> peruenerint ad iſtá ſuperficiẽ:</s> <s xml:id="echoid-s26063" xml:space="preserve"> tunc refringũtur apud ipſam ſecundũ lineas cõſimilis <lb/>ordinationis ſecátes lineas radiales.</s> <s xml:id="echoid-s26064" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s26065" xml:space="preserve"> perueniẽs in ali quod pũctum ſuperficiei glacialis, <lb/> <pb o="96" file="0398" n="398" rhead="VITELLONIS OPTICAE"/> ſemper extenditur ſuper ean dem incidentiam lineæ ad idem punctum ſuperficiei uiſus, & ad idem <lb/>punctum loci nerui communis:</s> <s xml:id="echoid-s26066" xml:space="preserve"> à quibuslibet ergo duobus punctis cõſimilis ſitus in reſpectu duo-<lb/>rum neruorum extenduntur duæ formæ ad idem punctum in neruo communi, donec fiat perfecta <lb/>unitas formarum.</s> <s xml:id="echoid-s26067" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div985" type="section" level="0" n="0"> <head xml:id="echoid-head795" xml:space="preserve" style="it">24. Inter omnes lineas pyramidis radialis neceſſe eſt ſolum axem tranſeuntem per centrum <lb/>for aminis uueæ ſuper ſuperficiem communem glacialis & uitreæ, & ſuper poſteriorem ſuper-<lb/>ficiem uitreæ perpendicularem eſſe. Alhazen 7 n 2.</head> <p> <s xml:id="echoid-s26068" xml:space="preserve">Axis enim hic, ſi non fuerit perpendicularis, ſed declinás ſuper aliquam iſtarum ſuperficierum.</s> <s xml:id="echoid-s26069" xml:space="preserve"> <lb/>accidet diuerſificatio ordinationis formarum peruenientium ad illam ſuperficiem, & mutabuntur <lb/>diſpoſitiones illarum formarum propter declinationem axis:</s> <s xml:id="echoid-s26070" xml:space="preserve"> ſolùm enim tũc, cum axis fuerit per-<lb/>pendicularis ſuper ſuperficiem glacialis, perueniet forma rei uiſæ in ſuperficiem glacialis ordinata <lb/>ſecũdum ordinem partium ſuperficiei rei uiſæ, & perueniet forma puncti, quod eſt apud extremi-<lb/>tatem axis in ſuperficie rei uiſæ, ad punctũ, quod eſt ſuper axem in ſuperficie glacialis, ut patet per <lb/>17 huius.</s> <s xml:id="echoid-s26071" xml:space="preserve"> Et quia axis radialis eſt perpendicularis ſuper ſuperficiem glacialis, palàm ex 18 p 11, quo-<lb/>niam omnes ſuperficies planæ exeuntes ab axe, & ſecantes ſuperficiem glacialis, erunt perpẽdicu-<lb/>lares ſuper iſtam ſuperficiem.</s> <s xml:id="echoid-s26072" xml:space="preserve"> Et quia ſuperficies humoris uitrei reſpiciens ipſam ſuperficiem gla-<lb/>cialis, quæ eſt cómunis ſectio ſphæræ glacialis & uitreæ, ut patet per præmiſſam, aut eſt ſuperficies <lb/>plana, aut ſphærica, & centrum eius non eſt centrum uiſus:</s> <s xml:id="echoid-s26073" xml:space="preserve"> ſi ergo axis radialis eſt declinans ſuper <lb/>iſtam ſuperficiem, & non eſt perpendicularis ſuper ipſam, non exibit ab axe ſuperficies plana per-<lb/>pendicularis ſuper iſtam ſuperficiem, niſi una tantũ ſuperficies, illa ſcilicet, quę tranſit per inæ qua-<lb/>litatem maximam angulorum, quæ patet per 30 t 1 huius:</s> <s xml:id="echoid-s26074" xml:space="preserve"> & omnes ſuperficies reſiduæ exeuntes ab <lb/>axe, erunt declinantes ſuper ipſam ſuperficiem uitreæ.</s> <s xml:id="echoid-s26075" xml:space="preserve"> Si enim duę ſuperficies uel plaures exeuntes <lb/>ab axe, ſunt perpendiculares ſuper dictam ſuperficiem, cum illæ ſuperficies de neceſsitate ſe inter-<lb/>ſecent, & ſua communis differentia ſit axis pyramidis radialis:</s> <s xml:id="echoid-s26076" xml:space="preserve"> erit per 19 p 11 axis perpendicularis <lb/>ſuper eandem ſuperficiem:</s> <s xml:id="echoid-s26077" xml:space="preserve"> datum autem fuit, quòd eſſet declinans.</s> <s xml:id="echoid-s26078" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s26079" xml:space="preserve"> centrum oculi punctũ <lb/>c:</s> <s xml:id="echoid-s26080" xml:space="preserve"> in ſuperficie quoq;</s> <s xml:id="echoid-s26081" xml:space="preserve"> oculi ſiue in ipſa ſuperficie glaciali, quæ per 7 huius & per 73 t 1 huius æ qui-<lb/>diſtat ſuperficiei ipſius oculi, ſit linea b a d:</s> <s xml:id="echoid-s26082" xml:space="preserve"> & <lb/> <anchor type="figure" xlink:label="fig-0398-01a" xlink:href="fig-0398-01"/> in ſuperficie humoris uitrei recipiente humo <lb/>rem glacialem ſit linea e g f:</s> <s xml:id="echoid-s26083" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s26084" xml:space="preserve"> axis pyrami-<lb/>dis radialis linea a c.</s> <s xml:id="echoid-s26085" xml:space="preserve"> Imaginemur ergo ſuper-<lb/>ficiem a b c d exeũtem ab axe, & erectam ſu-<lb/>per ſuperficiem glacialis tranſeuntẽ per cen-<lb/>trum oculi, quod eſtc:</s> <s xml:id="echoid-s26086" xml:space="preserve"> & hæc ſuperficies ere-<lb/>cta ſit etiá ſuper ſuperficiem humoris uitrei, <lb/>quæ eſt e g f:</s> <s xml:id="echoid-s26087" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s26088" xml:space="preserve"> communis ſectio huius ſu-<lb/>perficiei erectę a b c d cum ipſa ſuperficie gla-<lb/>cialis linea b a d:</s> <s xml:id="echoid-s26089" xml:space="preserve"> & ſint puncta b & d æ quali-<lb/>ter diſtantia à pũcto a, quòd ſit terminus axis <lb/>pyramidis uiſualis:</s> <s xml:id="echoid-s26090" xml:space="preserve"> & ſit cõmunis ſectio eius cum ſuperficie humoris uitrei linea e f:</s> <s xml:id="echoid-s26091" xml:space="preserve"> exeant quoq;</s> <s xml:id="echoid-s26092" xml:space="preserve"> <lb/>duæ lineæ à centro c, quæ ſint c b & c d:</s> <s xml:id="echoid-s26093" xml:space="preserve"> erunt ergo iſtæ duę lineæ c b & c d cum axe c a in ſuperficie <lb/>communi perpendiculari ſuper ſuper ficiem e g f per 1 p 11:</s> <s xml:id="echoid-s26094" xml:space="preserve"> quoniam omnia puncta c b d ſunt in illa <lb/>ſuperficie:</s> <s xml:id="echoid-s26095" xml:space="preserve"> eruntq́ ex hypotheſi duo anguli a c d & a c d ęquales:</s> <s xml:id="echoid-s26096" xml:space="preserve"> quod patet per 8 p 1, ſi illis arcubus <lb/>b a & a d ſubtendátur chordæ b a & d a:</s> <s xml:id="echoid-s26097" xml:space="preserve"> ſint quoq;</s> <s xml:id="echoid-s26098" xml:space="preserve"> lineæ c b & c d ſecantes lineam e f, quæ eſt com-<lb/>munis ſectio dictæ ſuperficiei erectę & ſuperficiei uitreæ ſuper duo pũcta f & e:</s> <s xml:id="echoid-s26099" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s26100" xml:space="preserve"> axis c a ean-<lb/>dem lineam e f ſuper punctum g.</s> <s xml:id="echoid-s26101" xml:space="preserve"> Si ergo ſuperficies, quæ eſt communis ſectio ſphæræ glacialis & <lb/>uitreæ, eſt plana:</s> <s xml:id="echoid-s26102" xml:space="preserve"> erit differẽtia communis, quæ eſt e g f, linea recta:</s> <s xml:id="echoid-s26103" xml:space="preserve"> & ſi axis a c fuerit declinans ſu-<lb/>per ſuperficiem uitreæ, & ipſa eſt in ſuperficie a b c d erecta ſuper ſuperficiem e g f:</s> <s xml:id="echoid-s26104" xml:space="preserve"> tunc neceſſariò <lb/>erit axis c a declinans ſuper lineam e f:</s> <s xml:id="echoid-s26105" xml:space="preserve"> erunt ergo anguli e g c & f g c inæquales:</s> <s xml:id="echoid-s26106" xml:space="preserve"> quoniam linea à <lb/>puncto g perpendiculariter producta ſuper lineam e g f e x 11 p 1 faciet angulos æquales cum linea <lb/>e f.</s> <s xml:id="echoid-s26107" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s26108" xml:space="preserve"> anguli e g c & f g c ſint inęquales:</s> <s xml:id="echoid-s26109" xml:space="preserve"> angulus quoq;</s> <s xml:id="echoid-s26110" xml:space="preserve"> c g f ſit exempli cauſſa, minor angulo <lb/>c g e, & duo anguli a c b & a c d ſint æ quales:</s> <s xml:id="echoid-s26111" xml:space="preserve"> erunt per 24 p 1, duæ lineæ e c & e f inęquales:</s> <s xml:id="echoid-s26112" xml:space="preserve"> eſt enim <lb/>linea e f breuior quàm linea e c:</s> <s xml:id="echoid-s26113" xml:space="preserve"> ſi enim illæ lineæ ſint æquales, cum anguli e c g & f c g ſint æquales, <lb/>& linea g c communis ambobus triangulis:</s> <s xml:id="echoid-s26114" xml:space="preserve"> erũt per 4 p 1 anguli e g c & f g c æ quales, quod eſt con-<lb/>tra datum:</s> <s xml:id="echoid-s26115" xml:space="preserve"> cum axis a c ſit declinans ſuper lineam e f.</s> <s xml:id="echoid-s26116" xml:space="preserve"> Sit ergo linea c g æ qualis lineæ c e, & ducatur <lb/>linea h g, quæ per 4 p 1 & ex præmiſsis eritæqualis lineæ e g:</s> <s xml:id="echoid-s26117" xml:space="preserve"> & à puncto g ducatur perpendicularis <lb/>g i ſuper lineam c h per 12 p 1.</s> <s xml:id="echoid-s26118" xml:space="preserve"> Ex penultima ergo primi latus g h oppoſitũ angulo recto in triangulo.</s> <s xml:id="echoid-s26119" xml:space="preserve"> <lb/>h i g, eſt maius latere g i:</s> <s xml:id="echoid-s26120" xml:space="preserve"> ergo per 19 p 1 erit linea g h maior quàm linea g f:</s> <s xml:id="echoid-s26121" xml:space="preserve"> cum enim angulus g f h ſit <lb/>extrinſecus angulo g i f recto:</s> <s xml:id="echoid-s26122" xml:space="preserve"> palàm, quòd an gulus g f h eſt obtuſus:</s> <s xml:id="echoid-s26123" xml:space="preserve"> eſtergo maior an gulorum tri-<lb/>goni f g h:</s> <s xml:id="echoid-s26124" xml:space="preserve"> ergo linea e g, quæ eſt æ qualis lineæ g h, maior eſt quàm linea g f:</s> <s xml:id="echoid-s26125" xml:space="preserve"> erunt ergo duo puncta <lb/>e & f diuerſæ diſtantiæ à puncto g:</s> <s xml:id="echoid-s26126" xml:space="preserve"> & iſta duo puncta e & f ſunt illa, ad quæ perueniunt formæ duo-<lb/>rum punctorum ſuperficiei glacialis, ſcilicet b & d, quæ ſunt æqualiter diſtantia ab axe.</s> <s xml:id="echoid-s26127" xml:space="preserve"> puncta itaq;</s> <s xml:id="echoid-s26128" xml:space="preserve"> <lb/>æqualiter diſtantia ab axe in ſuperficie glacialis, inæqualiter diſtant à puncto axis incidẽtis ſuper-<lb/>ficiei uitreæ:</s> <s xml:id="echoid-s26129" xml:space="preserve"> quod cum ita ſit, palàm, quia cum forma peruenerit â ſuperficie glacialis ad ſuperficiẽ <lb/> <pb o="97" file="0399" n="399" rhead="LIBER TERTIVS."/> humorĩs uitrei, erit ordinatio formæ non ſecundum eſſe, quod habet in ſuperficie glacialis, nec ſe-<lb/>cundum ſuum eſſe in ſuperficie rei uiſæ.</s> <s xml:id="echoid-s26130" xml:space="preserve"> Quando ergo axis fuerit declinãs ſuper ſuperficiẽ planam, <lb/>quę eſt cõmunis ſectio ſuperficiei glacialis & uitreæ:</s> <s xml:id="echoid-s26131" xml:space="preserve"> erit linea, quę eſt differentia cõmunis cuiusli-<lb/>bet ſuperficiei, exeuntis ab axe, erectæ ſuper ſuperficiẽ, & ſuperficiei ipſius uitreę, cõgitinẽs cũ <lb/>axe duos angulos in æquales, pręterquã in una tantùm ſuperficie, quæ ſecat ſecundum angulos re-<lb/>ctos ſuperficiem tranſeuntem per decliuitatem axis:</s> <s xml:id="echoid-s26132" xml:space="preserve"> quoniam huius tantũ ſuperficiei cõmunis dif-<lb/>ferentia continebit cum axe angulos rectos.</s> <s xml:id="echoid-s26133" xml:space="preserve"> Et cum duo anguli prædicti fuperint inæ quales, & angu <lb/>li apud centrum glacialis ęquales:</s> <s xml:id="echoid-s26134" xml:space="preserve"> erunt duę partes differentię cõmunis, quę eſt in ſuperficie uitrei, <lb/>inęquales.</s> <s xml:id="echoid-s26135" xml:space="preserve"> Formę ergo ſecundum iſta puncta, quæ ſunt in extremitatib.</s> <s xml:id="echoid-s26136" xml:space="preserve"> iſtarum differentiarũ perue <lb/>nientes ad ſuperficiem uitreę, erunt diuerſæ diſtantiæ à puncto axis, quod eſt in iſta ſuperficie:</s> <s xml:id="echoid-s26137" xml:space="preserve"> ſed <lb/>quia puncta iſtarum linearum in ſuperficie glacialis æ qualiter diſtant à puncto axis:</s> <s xml:id="echoid-s26138" xml:space="preserve"> in eadẽ ſuperfi <lb/>cie uidebuntur formæ non ſecundum ſuam ordinationem in ſuperficie glacialis & in rei uiſæ ſuքfi <lb/>cie.</s> <s xml:id="echoid-s26139" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s26140" xml:space="preserve"> demonſtrandum, ſi ſuperficies uitreę fuerit ſphærica, & fuerit axis declinans ſu <lb/>per ipſam:</s> <s xml:id="echoid-s26141" xml:space="preserve"> tunc enim axis non tranſibit per centrum uitreę, & tamen tranſibit per centrum glacia-<lb/>lis:</s> <s xml:id="echoid-s26142" xml:space="preserve"> lineę ergo quę exeunt à centro glacialis ad puncta, quorum diſtantia à puncto axis in ſuperficie <lb/>glacialis eſt æqualis, continent cũ axe apud centrum glacialis angulos ęqualies:</s> <s xml:id="echoid-s26143" xml:space="preserve"> & quia centrũ gla-<lb/>cialis non eſt centrũ uitreę, ut patet per 11 huius, diſtinguẽt iſtæ lineę ex ſuperficie uitreæ arcus inę-<lb/>quales.</s> <s xml:id="echoid-s26144" xml:space="preserve"> Cum enim linea e c, ut prędictum eſt, ſit <lb/> <anchor type="figure" xlink:label="fig-0399-01a" xlink:href="fig-0399-01"/> maior ꝗ̃ linea c f:</s> <s xml:id="echoid-s26145" xml:space="preserve"> ſit linea c h ę qualis lineę c e:</s> <s xml:id="echoid-s26146" xml:space="preserve"> & <lb/>protrahatur linea g h, ſuper quá deſcripta por-<lb/>tio circuli e g f, quæ ſit g h, erit ęqualis portioni <lb/>e g ք 24 p 3:</s> <s xml:id="echoid-s26147" xml:space="preserve"> ideo ꝗa chorda e g eſt æqualis chor-<lb/>dę g h per 4 p 1:</s> <s xml:id="echoid-s26148" xml:space="preserve"> producta ergo perpendiculari <lb/>g i, erit, ut prius, chorda g h maior ꝗ̃ chorda g f:</s> <s xml:id="echoid-s26149" xml:space="preserve"> <lb/>ergo arcus g h erit maior arcu g f per 28 p 3:</s> <s xml:id="echoid-s26150" xml:space="preserve"> er-<lb/>go & linea recta, quę eſt e g ęqualis lineę g h, e-<lb/>rit maior quàm linea g f recta:</s> <s xml:id="echoid-s26151" xml:space="preserve"> arcus ergo e g eſt <lb/>in ęqualis arcui g f per 28 p 3:</s> <s xml:id="echoid-s26152" xml:space="preserve"> nullę ergo lineę cõ <lb/>tinentes cum axe angulos rectos & exiſtentes cum linea a c in eadem ſuperficie, diſtinguunt ex ſu-<lb/>perficie uitreę duos arcus ęquales, niſi duę tantùm lineę, quæ ſunt in ſuperficie ſecante orthogona-<lb/>liter ſuperficiem erectam ſuper ſuperficiẽ uitreæ.</s> <s xml:id="echoid-s26153" xml:space="preserve"> Cũ ergo axis fuerit declinans ſuper ſuperficiẽ ui-<lb/>treę, formæ peruenientes ad ſuperficiẽ uitreę, erunt diuerſæ ordinationis, ſiue ſit ſuperficies uitreę <lb/>plana, ſiue ſphærica.</s> <s xml:id="echoid-s26154" xml:space="preserve"> Cũ uerò axis fuerit perpendicularis ſuper ſuperficiem uitrei, erit perpen dicula <lb/>ris ſuper omnes differentias quarumcunq;</s> <s xml:id="echoid-s26155" xml:space="preserve"> ſuperficierum planarum ductarum per lineá a c, & ſuքfi <lb/>ciei ipſius uitreę:</s> <s xml:id="echoid-s26156" xml:space="preserve"> & erunt quęlibet duę lineę, exeuntes à cẽtro glacialis, quod eſt unus pũctus axis, <lb/>continentes cũ axe angulos ęquales, & diſtinguentes ex differentia cõmuni, quę eſt in ſuperficie ui <lb/>treę, duas partes æquales, ſiue ſit ſuperficies illa plana, ſiue ſphęrica:</s> <s xml:id="echoid-s26157" xml:space="preserve"> & comprehenduntur formæ à <lb/>ſenſu ſecundũ ſuam ordinationem in ſuperficie glacialis & in ſuperficie rei uiſæ.</s> <s xml:id="echoid-s26158" xml:space="preserve"> Et quia talis eſt cõ-<lb/>prehenſio formarum, ut patet ex 5 ſuppoſitione:</s> <s xml:id="echoid-s26159" xml:space="preserve"> palã, quia ſemper axis pyramidis uiſualis eſt perpẽ <lb/>dicularis ſuper ſuperficiem humoris uitrei anteriorem & poſteriorem:</s> <s xml:id="echoid-s26160" xml:space="preserve"> quoniam eadẽ eſt cauſſa & <lb/>eodem modo demonſtrandum.</s> <s xml:id="echoid-s26161" xml:space="preserve"> Omnes uerò alię lineę erunt declinãtes ſuper has ſuperficies, quo-<lb/>niam procedunt, a c ſi ſecare poſsint axem ſuper centrum glacialis, & nulla ipſarum tranſit per cen-<lb/>trum uitreę, ſi fuerit ſphęrica, niſi axis tãtùm per 72 t 1 huius:</s> <s xml:id="echoid-s26162" xml:space="preserve"> quoniam ſolusille eſt perpendicularis <lb/>ſuperipſam.</s> <s xml:id="echoid-s26163" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s26164" xml:space="preserve"/> </p> <div xml:id="echoid-div985" type="float" level="0" n="0"> <figure xlink:label="fig-0398-01" xlink:href="fig-0398-01a"> <variables xml:id="echoid-variables416" xml:space="preserve">a b d h g e f i c</variables> </figure> <figure xlink:label="fig-0399-01" xlink:href="fig-0399-01a"> <variables xml:id="echoid-variables417" xml:space="preserve">a b d b g e f i c</variables> </figure> </div> </div> <div xml:id="echoid-div987" type="section" level="0" n="0"> <head xml:id="echoid-head796" xml:space="preserve" style="it">25. Motuoculi ſecundum ſe totum exiſtente poßibili: non eſt poßibile ſitum ſuarum partium <lb/>mutart. Alhazen 5. 13 n 1.</head> <p> <s xml:id="echoid-s26165" xml:space="preserve">Oſtenſum eſt in 4 huius, foramẽ eſſe in concauo oſsis, per quod tranſit neruus opticus:</s> <s xml:id="echoid-s26166" xml:space="preserve"> ſed inter <lb/>hoc foramen oſsis & inter circũferentiam glacialis cõiumctam cũ uuea, eſt ſpatium aliquantuiũ, & <lb/>neruus opticus extenditur in illo ſpatio exfine foraminis uſq;</s> <s xml:id="echoid-s26167" xml:space="preserve"> ad circũferentiã glacialis ſecundum <lb/>pyramidalitatem, & amplificatur quorſq;</s> <s xml:id="echoid-s26168" xml:space="preserve"> perueniat ad circumferentiam ſphærę glacialis, cum qua <lb/>conſolidatur.</s> <s xml:id="echoid-s26169" xml:space="preserve"> Cum ergo iſte neruus declinatur, erit eius declinatio apud foramen cõcauitatis ipſi-<lb/>us oſsis.</s> <s xml:id="echoid-s26170" xml:space="preserve"> Et quoniam cõcauitas oſsis continet totum oculum, declinato ſic neruo, etiam oculus mo <lb/>uebitur ſecundum ſe totũ in iſta cõcauitate:</s> <s xml:id="echoid-s26171" xml:space="preserve"> conſolidatiua enim, quę conſolidatur cũ eo, quod eſt in <lb/>an teriori oculi ex neruo & ex tunicis reſiduis ſemper eſt cuſtodiens ſitum eius:</s> <s xml:id="echoid-s26172" xml:space="preserve"> declinatio ergo ner <lb/>ui apud motum oculi non eſt niſi à poſteriore totius oculi:</s> <s xml:id="echoid-s26173" xml:space="preserve"> non eſt ergo poſsibile ſitum partiũ oculi <lb/>mutari, quoniam ut per 7 huius patuit, centrum ſuperficierũ tunicarum uiſus oppoſitarum forami-<lb/>ni uueę & corneę, eſt idẽ cũ centro oculi.</s> <s xml:id="echoid-s26174" xml:space="preserve"> Sicut ergo cum mouebitur oculus, non mutabitur centrũ <lb/>oculi, quoniam ſpnęra aliqua aliqualiter mota, nõ propter hoc mutatur fitus centri:</s> <s xml:id="echoid-s26175" xml:space="preserve"> ſic nec centrum <lb/>ſuperficierũ tunicarum oppoſitarũ formamini uueę mutatur:</s> <s xml:id="echoid-s26176" xml:space="preserve"> ergo neq;</s> <s xml:id="echoid-s26177" xml:space="preserve"> ſitus tunicarū oculi mutatur.</s> <s xml:id="echoid-s26178" xml:space="preserve"> <lb/>Quia enim linea tranſiens per centra omniũ tunicarũ & humorũ oculi, tranſit per medium cõcaui-<lb/>tatis nerui orthogonaliter erecta ſuք baſim pyramidis nerui, ut patet per 9 huius, & linea, quę trãſit <lb/>orthogonaliter per centrũ circuli baſis alicuius pyramidis, neceſſariò attingit uerticẽ pyramidis ք <lb/>89 t 1 huius:</s> <s xml:id="echoid-s26179" xml:space="preserve"> in pyramide uerò concaua nerui opticiuertex pyramidis moto oculo nõ mutatur:</s> <s xml:id="echoid-s26180" xml:space="preserve"> ne-<lb/> <pb o="98" file="0400" n="400" rhead="VITELLONIS OPTICAE"/> ceſſe eſt moto oculo ſecun dum ſe totũ, partes eius nullo modo mutari:</s> <s xml:id="echoid-s26181" xml:space="preserve"> quoniã linea, quæ trãſit per <lb/>centra illarum partium, tranſit per mediũ concauitatis nerui optici per 9 huius.</s> <s xml:id="echoid-s26182" xml:space="preserve"> Ex quo patet, quòd <lb/>partes oculi nullo modo mutantur.</s> <s xml:id="echoid-s26183" xml:space="preserve"> Declinatio enim partis pyramidalis nerui ſuք ſuperficiẽ circuli <lb/>cõſolidationis eſt ſemք declinatio cõſimilis:</s> <s xml:id="echoid-s26184" xml:space="preserve"> partes ergo oculi ſecundũ ſuum ſitũ non mutantur.</s> <s xml:id="echoid-s26185" xml:space="preserve"> Et <lb/>hoc eſt propoſitum.</s> <s xml:id="echoid-s26186" xml:space="preserve"> Et quoniã oculi ambo ſunt cõſimilis diſpoſitionis in ſuis tunicis & partib, & in <lb/>figuris ſuarũ tunicarũ, & in ſitu cuiuslibet bunicarum reſpectu totius oculi:</s> <s xml:id="echoid-s26187" xml:space="preserve"> patet, quòd nõ eſt diuer <lb/>ſitas inter illos, quò ad hoc, quod proponitur de ſuarũ partium ſitus mutatione, ipſis oculis motis:</s> <s xml:id="echoid-s26188" xml:space="preserve"> <lb/>ſitus enim linearum ambarum tranſeuntium per centra tunicarũ uiſus in utroq;</s> <s xml:id="echoid-s26189" xml:space="preserve"> oculorũ eſt ſemper <lb/>ſitus conſimilis in omnibus diſpoſitionibus oculorum.</s> <s xml:id="echoid-s26190" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s26191" xml:space="preserve"> illud, quod proponebatur.</s> <s xml:id="echoid-s26192" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div988" type="section" level="0" n="0"> <head xml:id="echoid-head797" xml:space="preserve" style="it">26. Vno oculo moto, neceſſe eſt alium eidem conformiter mooueri.</head> <p> <s xml:id="echoid-s26193" xml:space="preserve">Quoniam enim ſitus partium oculinon mutatur in utroq;</s> <s xml:id="echoid-s26194" xml:space="preserve"> oculorum, & motus unius oculi fit ք.</s> <s xml:id="echoid-s26195" xml:space="preserve"> <lb/>motum nerui optici in centro foraminis oſsis, motus uerò nerui partialis procedit à puncto nerui <lb/>communis:</s> <s xml:id="echoid-s26196" xml:space="preserve"> quoniam ſemper illud, quod mouetur in partib.</s> <s xml:id="echoid-s26197" xml:space="preserve"> aliarum, mouetur circa aliquod fixum:</s> <s xml:id="echoid-s26198" xml:space="preserve"> <lb/>motus itaq;</s> <s xml:id="echoid-s26199" xml:space="preserve"> nerui partialis incipit in puncto nerui cõmunis ambobus neruis opticis amborum ocu <lb/>lorum, in quo eſt uirtus animæ ſentientis & mouentis.</s> <s xml:id="echoid-s26200" xml:space="preserve"> Et quoniam illa uirtus eſt in diuiſibilis & u-<lb/>niformis & principium, quo primo mouet, eſt corpus naturale ſecundum ſui formam naturalẽ indi <lb/>uiſibilem:</s> <s xml:id="echoid-s26201" xml:space="preserve"> palàm, quòd mouendo unum oculum mouet & alterum:</s> <s xml:id="echoid-s26202" xml:space="preserve"> nec enim eſt maior ratio, qua u-<lb/>num oculum moueat, ꝗ̃ qua alterum:</s> <s xml:id="echoid-s26203" xml:space="preserve"> uno itaq;</s> <s xml:id="echoid-s26204" xml:space="preserve"> oculo moto, ambo oculi mouentur, & unus confor-<lb/>miter alteri mouetur:</s> <s xml:id="echoid-s26205" xml:space="preserve"> ut ſicut ab eodem principio motus amborum incipit, ſic ad eundem terminũ <lb/>terminentur ambo ιnotus, & ſicut ab uno indiuiſibili incipiũt, ſic ad unum diuiſibile terminentur.</s> <s xml:id="echoid-s26206" xml:space="preserve"> <lb/>Palàm eſt ergo illud, quod proponebatur.</s> <s xml:id="echoid-s26207" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div989" type="section" level="0" n="0"> <head xml:id="echoid-head798" xml:space="preserve" style="it">27. Duobus uiſibus uno uiſibili directè oppoſitis: neceſſe ect duas figur ari pyramides, quarum <lb/>communis baſis eſt ſuperficies reiuiſæ, & axis cuiuslibet tranſitper centrum for aminis uueæ, et <lb/>per centrum ſui uiſus.</head> <p> <s xml:id="echoid-s26208" xml:space="preserve">Quoniã enim, ut patet per 17 huius, ſitus partiũ ſuperficiei rei uiſæ peruenit ad ſuperficiẽ utriuſq;</s> <s xml:id="echoid-s26209" xml:space="preserve"> <lb/>uifus, & in illa figuratur ſecundum lineas perpendiculares ab omnib punctis fuperficiei rei uiſæ ad <lb/>oculi illius ſuperficiẽ productas, quarũ omnium cõcurſus ſecundũ puncta ſuarũ incidentiarũ reſpi <lb/>cit centrũ oculi, cuius ſuperficiei incidit, & demũ poſt refractionẽ quælibet illarũ figurarũ peruenit <lb/>ad mediũ punctum nerui communis:</s> <s xml:id="echoid-s26210" xml:space="preserve"> ambarũ itaq;</s> <s xml:id="echoid-s26211" xml:space="preserve"> illarum formarũ concurſus fit in puncto medio <lb/>nerui communis, cui incidunt.</s> <s xml:id="echoid-s26212" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s26213" xml:space="preserve"> centra duorũ uiſuum ſunt duo, palã, quia in uiſione eiuſ-<lb/>dem rei à duobus oculis duæ pyramides uiſuales modo propoſito figur antur.</s> <s xml:id="echoid-s26214" xml:space="preserve"> Superficies enim rei <lb/>uiſæ ſemper erit baſis utriuſq;</s> <s xml:id="echoid-s26215" xml:space="preserve"> pyramidis ab utroq;</s> <s xml:id="echoid-s26216" xml:space="preserve"> oculorũ prodeuntis, propter multiplicationem <lb/>formæ cuiuslibet puncti ſuperficiei rei uiſæ æqualiter ad uiſum, & axis cuiuslibet earum tranſit per <lb/>centra foraminis uueæ ad centrum ſui uiſus.</s> <s xml:id="echoid-s26217" xml:space="preserve"> Sicut enim uiſibile directè;</s> <s xml:id="echoid-s26218" xml:space="preserve"> opponitur uni uiſui, ſic di-<lb/>rectè opponitur & alteri, ex pypotheſi:</s> <s xml:id="echoid-s26219" xml:space="preserve"> & quoniam ambo uiſus æqualiter mouentur ad aliquid ui-<lb/>dendum, per pręmiſſam, patet, quòd ſemper in uiſione unius rei medium punctum ſuperficiei uiſus <lb/>oculi opponitur medio puncto ſuperficiei rei uiſæ, uel propinquo illi:</s> <s xml:id="echoid-s26220" xml:space="preserve"> medium autem punctũ ſuper <lb/>ficiei uiſus uel oculi eſt centrum foraminis uueę per 4 huius.</s> <s xml:id="echoid-s26221" xml:space="preserve"> Forma ergo illius puncti medij ſuperfi <lb/>ciei rei uiſę uel puncti propin qui illi, per centrum foraminis uueę peruenit ad centrum ſui uiſus.</s> <s xml:id="echoid-s26222" xml:space="preserve"> Et <lb/>hoc eſt propoſitum.</s> <s xml:id="echoid-s26223" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div990" type="section" level="0" n="0"> <head xml:id="echoid-head799" xml:space="preserve" style="it">28. Duobus exiſtentibus oculis, unius rei unam tantùm formam accidit. uideri. Alhazen <lb/>27 n 1. Item 9 n 3.</head> <p> <s xml:id="echoid-s26224" xml:space="preserve">Quoniam enim, ut prius pluries dictum eſt, forma recepta in ſuperficie glacialis, pertranſit cor-<lb/>pus glacialis, deinde extenditur per corpus ſubtile, quod eſt in neruo optico, & uenit ad anterius ce <lb/>rebri, in quo eſt ſentiens ultimum, quod eſt uirtus ſenſitiua, comprehendens ſenſibilia, cuius uirtu-<lb/>tis oculus eſt inſtrum entum, recipiens formas rerũ, & reddens eas ultimo ſentienti, ſic quòd apud <lb/>neruũ communem ambobus oculis, cuius nerui ſitus à duobus oculis eſt ſitus conſimilis, demum <lb/>completur uiſio:</s> <s xml:id="echoid-s26225" xml:space="preserve"> licet ergo duę formæ perueniant in duobus oculis ab una re uiſa;</s> <s xml:id="echoid-s26226" xml:space="preserve"> illæ tamen formę <lb/>ambę qũ perueniunt ad neruũ communem, con currunt & fiunt una forma, & per unionem harum <lb/>formarum comprehen dit ultimum ſentiens formam rei uiſæ, & ſic unius rei tantùm unam formam <lb/>accidit uideri:</s> <s xml:id="echoid-s26227" xml:space="preserve"> niſi fortè per aliquam occaſionem interuenientem accidat formas duobus oculis ac <lb/>ceptas non uniri, eò quòd non concurrunt in unionem amborum neruorũ opticorum:</s> <s xml:id="echoid-s26228" xml:space="preserve"> tunc enim <lb/>duas formas accidit uideri, ut cum aſpiciẽs mutauerit ſitum unius oculi ad anterius, & alius oculus <lb/>fuerit immotus.</s> <s xml:id="echoid-s26229" xml:space="preserve"> Quando uerò ſitus duorum oculorum fuerit naturalis, tunc quia ſitus ipſorũ ab u-<lb/>na re uiſa eſt ſitus conſimilis, peruenit forma ab una re uiſa in duo loca conſimilis ſitus, & cũ ſitus u-<lb/>nius oculorum fuerit declinans, tunc diuerſatur ſitus oculorum ab illa re uiſa, & ſic perueniunt duę <lb/>formę illius rei uiſæ diuerſi ſitus:</s> <s xml:id="echoid-s26230" xml:space="preserve"> ſed hoc non ineſt uiſui naturaliter, ſed ſolũ per uiolentiam, quã fa-<lb/>cit uoluntas uel naturalis debilitas conſuetudini naturæ.</s> <s xml:id="echoid-s26231" xml:space="preserve"> Quando itaq;</s> <s xml:id="echoid-s26232" xml:space="preserve"> ſitus oculorum ſuerit natu-<lb/>ralis, tunc ſemper ambobus uiſibus unius rei unam formam accidit uideri.</s> <s xml:id="echoid-s26233" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s26234" xml:space="preserve"> <lb/>Duæ ergo formæ uiſi puncti infiguntur in duobus medijs duarum ſuperficierum amborum uiſuũ, <lb/>& quilibet punctus alius formæ uiſæ infigetur in duobus locis conſimilis poſitionis in duobus ui-<lb/> <pb o="99" file="0401" n="401" rhead="LIBER TERTIVS."/> fibus:</s> <s xml:id="echoid-s26235" xml:space="preserve"> deinde formæ uiſæ perueniunt ad concauitatem communis nerui, & perueniũt duæ formę, <lb/>quę ſunt in puncto, quod eſt in duobus axibus illarũ duarum pyramidum ra dialiũ, ſecundum quas <lb/>fit uiſio, ad punctum, quod eſt in communi axe, & efficiuntur una forma:</s> <s xml:id="echoid-s26236" xml:space="preserve"> & quęlibet duę formę quę <lb/>ſunt in duobus punctis conſimilis poſitionis à duobus uiſib.</s> <s xml:id="echoid-s26237" xml:space="preserve"> peruenient ad idem punctum puncto-<lb/>rum circumſtantium punctum, qui eſt in axe communi.</s> <s xml:id="echoid-s26238" xml:space="preserve"> Sic ergo duæ formę totius rei uiſę ſuperpo <lb/>nuntur ſibi, & efficiuntur una forma:</s> <s xml:id="echoid-s26239" xml:space="preserve"> & ſic uiſum comprehenditur unum.</s> <s xml:id="echoid-s26240" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div991" type="section" level="0" n="0"> <head xml:id="echoid-head800" xml:space="preserve" style="it">29. Omnem punctum formæ incidentem ſuperficiebus uiſuum per axes radiales, ad centrum <lb/>foraminis gyrationis nerui concaui pertingere eſt neceſſe.</head> <p> <s xml:id="echoid-s26241" xml:space="preserve">Quoniam enim quilibet axium tranſit per centrũ foraminis uueę ad centrum uiſus, ut patet per <lb/>27 huius:</s> <s xml:id="echoid-s26242" xml:space="preserve"> ergo & pertranſit centrũ ipſius ſphęrę uueę per 8 huius:</s> <s xml:id="echoid-s26243" xml:space="preserve"> omnis uerò linea recta producta <lb/>inter centrum oculi & uueę, centũ circuli ſectionis uueę & medium punctum concauitatis nerui <lb/>neceſſariò penetrabit per 9 huius:</s> <s xml:id="echoid-s26244" xml:space="preserve"> palá ergo, cũ perpẽdicularis ſemքmaneat irrefracta per 47 t 2 hu <lb/>ius, quòd omnem punctum formę incidentem ſuperficieb, uiſuum per axes radiales ad centrum gy <lb/>rationis nerui communis pertingere eſt neceſſe:</s> <s xml:id="echoid-s26245" xml:space="preserve"> ab hoc autem puncto diffun ditur forma ad me-<lb/>dium punctum nerui communis:</s> <s xml:id="echoid-s26246" xml:space="preserve"> & quoniam medius punctus nerui communis eſt tantùm unus:</s> <s xml:id="echoid-s26247" xml:space="preserve"> <lb/>palàm, quia axes amborum uiſuum in uno puncto nerui communis ſemper concurrunt.</s> <s xml:id="echoid-s26248" xml:space="preserve"> Patet er-<lb/>go propoſitum.</s> <s xml:id="echoid-s26249" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div992" type="section" level="0" n="0"> <head xml:id="echoid-head801" xml:space="preserve" style="it">30. Si à terminis lineæ inter duo centra for aminum gyrationis neruorum concauorum pro-<lb/>ductæ duæ rectæ lineæ ad medium communis nerui producantur: neceße eſt in conſtituto trian-<lb/>gulo angulos ad baſim æquales eſſe. Ex quo patet, quòd lineæ illæ productæ ſunt æquales. Al-<lb/>hazen 6 n 3.</head> <p> <s xml:id="echoid-s26250" xml:space="preserve">Sint duo cẽtra foraminum gyrationis neruorum concauorũr & t, inter quę producatur linea r t:</s> <s xml:id="echoid-s26251" xml:space="preserve"> <lb/>ſitq́;</s> <s xml:id="echoid-s26252" xml:space="preserve"> medius punctus nerui communis a:</s> <s xml:id="echoid-s26253" xml:space="preserve"> & cõſtitua-<lb/> <anchor type="figure" xlink:label="fig-0401-01a" xlink:href="fig-0401-01"/> tur trian gulus r a t:</s> <s xml:id="echoid-s26254" xml:space="preserve"> dico, quòd angulus a r t eſt æqua <lb/>lis angulo a t r.</s> <s xml:id="echoid-s26255" xml:space="preserve"> Cum enim poſitio duorum neruorũ <lb/>in reſpectu concauitatis nerui communis ſit poſitio <lb/>conſimilis:</s> <s xml:id="echoid-s26256" xml:space="preserve"> quia concauitas nerui unius eſt omnino <lb/>ſimilis concauitati alterius per 4 huius:</s> <s xml:id="echoid-s26257" xml:space="preserve"> ergo & me-<lb/>dium concauitatis unius eſt ſimile medio concaui-<lb/>tatis alterius:</s> <s xml:id="echoid-s26258" xml:space="preserve"> unde axis nerui unius æqualis eſt axi <lb/>nerui alterius:</s> <s xml:id="echoid-s26259" xml:space="preserve"> ſed per eandem 4 huius poſitio duo-<lb/>rum neruorum in reſpectu duorum foraminium, eſt <lb/>poſitio conſimilis, in quorum neruorum medio fe-<lb/>rũtur lineę r a & t a, ut axes.</s> <s xml:id="echoid-s26260" xml:space="preserve"> Palàm ergo, quoniam po <lb/>ſitio duarum linearum r a & t a apud lineã r t eſt po-<lb/>ſitio conſimilis:</s> <s xml:id="echoid-s26261" xml:space="preserve"> hoc autem eſt impoſsibile, niſi angu <lb/>li a r t & a t r ſint æquales:</s> <s xml:id="echoid-s26262" xml:space="preserve"> quoniam ad inæqualita-<lb/>tem iſtorum angulorum ſequitur inæqualitas poſi-<lb/>tionis medij axis ipſorum neruorum concauorum:</s> <s xml:id="echoid-s26263" xml:space="preserve"> <lb/>& ex conſequenti ipſorum neruorum.</s> <s xml:id="echoid-s26264" xml:space="preserve"> Sunt ergo illi <lb/>anguli ad baſim æquales:</s> <s xml:id="echoid-s26265" xml:space="preserve"> ergo per 6 p 1 lineę illæ productę ſunt ęquales, ſcilicet linea a r lineæ a t.</s> <s xml:id="echoid-s26266" xml:space="preserve"> Pa <lb/>tet ergo propoſitum.</s> <s xml:id="echoid-s26267" xml:space="preserve"/> </p> <div xml:id="echoid-div992" type="float" level="0" n="0"> <figure xlink:label="fig-0401-01" xlink:href="fig-0401-01a"> <variables xml:id="echoid-variables418" xml:space="preserve">a r t</variables> </figure> </div> </div> <div xml:id="echoid-div994" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables419" xml:space="preserve">a r t b</variables> </figure> <head xml:id="echoid-head802" xml:space="preserve" style="it">31. Vnopunctorei uiſæ, ſuperficiebus amborum uiſuum perpen <lb/> diculariter incidente: neceſſe eſt axes radiales in centr is for ami- num gyrationis neruorum concauorum angulariter refringi.</head> <p> <s xml:id="echoid-s26268" xml:space="preserve">Quoniam enim, ut patet per 27 huius, quilibet illorũ axium per-<lb/>tranſit centrum foraminis uueæ & centrum oculi:</s> <s xml:id="echoid-s26269" xml:space="preserve"> motus autem cu-<lb/>iuslibet oculorum fit in centro foraminis gyrationis nerui optici:</s> <s xml:id="echoid-s26270" xml:space="preserve"> <lb/>patet, quoniam ſecundum motum oculorum uariantur axes illi ra-<lb/>diales, in quibus ſunt ſemper eædem ſemidiametri oculorum, quæ <lb/>ſcilicet ab ipſorum centris ad centra foraminum uueæ protendun-<lb/>tur:</s> <s xml:id="echoid-s26271" xml:space="preserve"> partes autem ſuperiores illorum axium, quibus à centris fora-<lb/>minum gyrationis neruorum concauorum formæ perueniunt ad <lb/>punctum medium nerui communis, ſemper manent ſecundum mo-<lb/>dum unum.</s> <s xml:id="echoid-s26272" xml:space="preserve"> Cum itaque aliæ partes illorum axium ſemper ſintim, <lb/>mobiles, & alię ſemper mobiles, cum per ipfas unus punctus uide-<lb/>tur:</s> <s xml:id="echoid-s26273" xml:space="preserve"> patet per 1 p 11, quoniam illę lineę nõ ſunt linea una:</s> <s xml:id="echoid-s26274" xml:space="preserve"> utpote ſi for <lb/>ma puncti b uideatur ſecundum ambos axes b r & b t, & ſicut factũ <lb/>eſt in præmiſſa, ducantur lineæ r a & t a ad medium punctum nerui <lb/>communis, qui ſit a, patet per 1 p 11, quoniam lineę b r & r a, non ſunt <lb/> <pb o="100" file="0402" n="402" rhead="VITELLONIS OPTICAE"/> linea una:</s> <s xml:id="echoid-s26275" xml:space="preserve"> eius enim partem in ſublimi, partẽ in plano accideret eſſe:</s> <s xml:id="echoid-s26276" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s26277" xml:space="preserve"> Pater er-<lb/>go, quoniam angulariter coniunguntur:</s> <s xml:id="echoid-s26278" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s26279" xml:space="preserve"> Et licet axes præmiſſo modo refrin-<lb/>gantur:</s> <s xml:id="echoid-s26280" xml:space="preserve"> formatio tamen pyramidum uiſualium fit, ac ſi axes integri ad uerticem peruenirẽt, neque <lb/>accidit uiſui aliqua diuerſitas exillo.</s> <s xml:id="echoid-s26281" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div995" type="section" level="0" n="0"> <head xml:id="echoid-head803" xml:space="preserve" style="it">32. Neceſſe eſt axes pyramidum uiſualium amborum uiſuum tranſeuntes per centra fora-<lb/>minum uueæ, ſemper coniungi in uno puncto ſuperficiei rei uiſæ, etiam motis uiſibus per ſuper-<lb/>ficiem reiuiſæ. Alhazen 2 n 3.</head> <p> <s xml:id="echoid-s26282" xml:space="preserve">Cum enim uidens intuebitur aliquam rem uiſam:</s> <s xml:id="echoid-s26283" xml:space="preserve"> tunc uterque uiſus erit in oppoſitione illius <lb/>rei uiſæ per 2 huius, & utraque pupillarum dirigetur ad illum uiſum directione æquali, propter ui-<lb/>ſuum æqualitatem per 4 huius.</s> <s xml:id="echoid-s26284" xml:space="preserve"> Sint ergo duo centra duorum uiſuum e & g:</s> <s xml:id="echoid-s26285" xml:space="preserve"> & ſit medius punctus <lb/>nerui communis punctus a:</s> <s xml:id="echoid-s26286" xml:space="preserve"> & ſit ſuperficies rei uiſæ b c d f, quę ſit exempi cauſſa ęquidiſtans lineę, <lb/>centra uiſuum connectenti, quæ ſit e g.</s> <s xml:id="echoid-s26287" xml:space="preserve"> Palàm ergo, quoniá à centris <lb/> <anchor type="figure" xlink:label="fig-0402-01a" xlink:href="fig-0402-01"/> uiſuum perpendiculares ſuper ipſam ſuperficiem b c d f productæ, <lb/>ſunt ęquidiſtantes per 6 p 11, quæ ſint e q & g x.</s> <s xml:id="echoid-s26288" xml:space="preserve"> In hac itaq;</s> <s xml:id="echoid-s26289" xml:space="preserve"> ſuperficie <lb/>b c d f ſignetur punctus, qui ſit u:</s> <s xml:id="echoid-s26290" xml:space="preserve"> dico, quòd propter ęqualitatem am <lb/>borum oculorum in omnibus ſuis diſpoſitionib.</s> <s xml:id="echoid-s26291" xml:space="preserve"> ſi alter uiſus ſuerit <lb/>motus ad uiden dum pũctum u, ſtatim etiam reliquus mouebitur ad <lb/>uidendum idem punctum u:</s> <s xml:id="echoid-s26292" xml:space="preserve"> ita quòd axes ambarum pyramidum ui-<lb/>ſualium tranſeuntes per centra foraminũ uueę coniungentur in pun <lb/>cto u, uno ipſorum ibi pertingente.</s> <s xml:id="echoid-s26293" xml:space="preserve"> Si enim uno illorum axium in ci-<lb/>dente in puncto u, alius incidit in alio puncto:</s> <s xml:id="echoid-s26294" xml:space="preserve"> ſit illud punctum z:</s> <s xml:id="echoid-s26295" xml:space="preserve"> e-<lb/>rũtq́;</s> <s xml:id="echoid-s26296" xml:space="preserve"> duo axes e u & g z, inter quorũ terminos linea z u producatur.</s> <s xml:id="echoid-s26297" xml:space="preserve"> <lb/>Et quoniam axes ſic protenſi à duobus uiſibus non concurrunt in a-<lb/>liquo punctorũ lineę z u, ſicut neq;</s> <s xml:id="echoid-s26298" xml:space="preserve"> concurrũt ſi ſolũ ſecundum per-<lb/>pendiculares lineas, quæ ſunt e q & g x, fiat uiſio:</s> <s xml:id="echoid-s26299" xml:space="preserve"> palàm, quòd nullũ <lb/>punctorum lineę z u uidebitur ambobus uiſibus, ſed tantùm uno:</s> <s xml:id="echoid-s26300" xml:space="preserve"> al <lb/>ter ergo oculorum uidetur ſuperfluere, cum unus oculorũ ſecundũ <lb/>ſui axem omnia puncta lineæ z u, poſsit imperceptibiliter tranſcur-<lb/>rere:</s> <s xml:id="echoid-s26301" xml:space="preserve"> conſtituit aũt natura duos oculos propter perfectionem boni-<lb/>tatis uiſionis & complementum eius, ut ipſorum uirtus unita ſit for <lb/>tior, ut patet per 4 huius.</s> <s xml:id="echoid-s26302" xml:space="preserve"> Si ergo axes uiſuales non concurrant in ali <lb/>quod pũctum unum lineę z u, ſequitur uel naturam ſuperfluere, uel <lb/>ipſam modo debiliori, quo poteſt, operari:</s> <s xml:id="echoid-s26303" xml:space="preserve"> quorum utrum q;</s> <s xml:id="echoid-s26304" xml:space="preserve"> eſt im-<lb/>poſsibile.</s> <s xml:id="echoid-s26305" xml:space="preserve"> Natura enim nihil agit fruſtra, nec deficit in neceſſarijs, ut <lb/>patet per 8 ſuppoſitionẽ 2 huius:</s> <s xml:id="echoid-s26306" xml:space="preserve"> accidit aũt hoc impoſsibile, ſi axes <lb/>ſolum incidant diuerſis punctis ſuperficiei uiſibilis:</s> <s xml:id="echoid-s26307" xml:space="preserve"> impoſsibile aũt <lb/>nunquam accideret, ſi incidant in idem punctum.</s> <s xml:id="echoid-s26308" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s26309" xml:space="preserve">, quo-<lb/>niã in idem punctum incidere axes pyramidum amborum uiſuum <lb/>ſemper eſt neceſſe:</s> <s xml:id="echoid-s26310" xml:space="preserve"> quoniam operatio amborum uiſuum eſt unifor-<lb/>mis.</s> <s xml:id="echoid-s26311" xml:space="preserve"> Cũ igitur uiſus fuerit motus ſuper rem uiſam:</s> <s xml:id="echoid-s26312" xml:space="preserve"> tunc uterq;</s> <s xml:id="echoid-s26313" xml:space="preserve"> uiſus <lb/>mouebitur ſuper illam, & axes congre gati in uno puncto ſuperficiei rei uiſæ, moto uiſu ambo mo-<lb/>uebuntur ſimul ad aliquod unum punctum ſuper ſuperficiem illius rei uiſæ:</s> <s xml:id="echoid-s26314" xml:space="preserve"> ambo enim oculi ſunt <lb/>ęquales in omnib.</s> <s xml:id="echoid-s26315" xml:space="preserve"> ſuis diſpoſitionibus:</s> <s xml:id="echoid-s26316" xml:space="preserve"> & eſt ambobus oculis unus neruus communis.</s> <s xml:id="echoid-s26317" xml:space="preserve"> Et quoniam <lb/>motus oculorum procedit ab una uirtute, neceſſe eſt uirtutem motam per unitatem nerui procede <lb/>re:</s> <s xml:id="echoid-s26318" xml:space="preserve"> hæc ergo moto uno oculo ambos oculos mouebit, ut patet per 26 huius.</s> <s xml:id="echoid-s26319" xml:space="preserve"> Actio itaq;</s> <s xml:id="echoid-s26320" xml:space="preserve"> & paſsio o-<lb/>culorum ſemper eſt ęqualis & conſimilis:</s> <s xml:id="echoid-s26321" xml:space="preserve"> & ſi alter uiſuum motus fuerit ad aliquid uidendũ, ſtatim <lb/>alter mouebitur ad hoc idem uidendum illo eodem motu:</s> <s xml:id="echoid-s26322" xml:space="preserve"> & ſi alter uiſuum quieſcat, reliquus qui-<lb/>eſcet.</s> <s xml:id="echoid-s26323" xml:space="preserve"> Impoſsibile eſt enim alterum uiſuũ moueri, & alterum quieſcere, niſi alter fuerit impeditus, ut <lb/>patet per 26 huius:</s> <s xml:id="echoid-s26324" xml:space="preserve"> & ſicut etiam declaratum eſt per 18 huius, ſuperficies rei uiſæ ſemper erit baſis <lb/>utriuſq;</s> <s xml:id="echoid-s26325" xml:space="preserve"> pyramidis ab utroq;</s> <s xml:id="echoid-s26326" xml:space="preserve"> oculorum prodeuntis:</s> <s xml:id="echoid-s26327" xml:space="preserve"> quoniam tunc poſitio pũcti, in quo ambo axes <lb/>ſunt cõiuncti, eſt poſitio conſimilis:</s> <s xml:id="echoid-s26328" xml:space="preserve"> quia eſt oppoſitio duobus medijs amborum uiſuum.</s> <s xml:id="echoid-s26329" xml:space="preserve"> Palàm er-<lb/>go propoſitũ:</s> <s xml:id="echoid-s26330" xml:space="preserve"> dicemusq́;</s> <s xml:id="echoid-s26331" xml:space="preserve"> punctũ cõcurſus amborũ axiũ in ſuperficie rei uiſæ punctũ cõiunctionis.</s> <s xml:id="echoid-s26332" xml:space="preserve"/> </p> <div xml:id="echoid-div995" type="float" level="0" n="0"> <figure xlink:label="fig-0402-01" xlink:href="fig-0402-01a"> <variables xml:id="echoid-variables420" xml:space="preserve">a e g b f z q x c u d</variables> </figure> </div> </div> <div xml:id="echoid-div997" type="section" level="0" n="0"> <head xml:id="echoid-head804" xml:space="preserve" style="it">33. Si à puncto medio nerui communis ad medium lineæ connect entis centra for aminum gy <lb/>rationis neruorum concauorum linea recta producatur: neceſſe eſt product ã ſuper diuiſam per-<lb/>pendicularem eſſe: & eam, puncto uiſo cum axibus incidente, trigonum ab axibus & diuiſa li-<lb/>nea contentum per æqualia diuidere. Alhazen 7 n 3.</head> <p> <s xml:id="echoid-s26333" xml:space="preserve">Quod hic proponitur, patet per præmiſſam & per 31 primi huius:</s> <s xml:id="echoid-s26334" xml:space="preserve"> ut autem particularius demon-<lb/>ſtretur, ſint omnia diſpoſita ut in 30 huius:</s> <s xml:id="echoid-s26335" xml:space="preserve"> & ſit linea r t, diuiſa per ęqualia in puncto s:</s> <s xml:id="echoid-s26336" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s26337" xml:space="preserve"> uiſibile <lb/>aliquod oppoſitum ambobus uiſibus, quod ſit d c, in cuius puncto medio, quod ſit b, concurrant ք <lb/>pręcedentem ipſi axes radiales, qui ſint r b & t b:</s> <s xml:id="echoid-s26338" xml:space="preserve"> & producatur à puncto a, quod eſt medius pũctus <lb/>concauitatis nerui, ad punctũ s linea a s.</s> <s xml:id="echoid-s26339" xml:space="preserve"> Dico, quòd linea a s eſt քpendicularis ſuper lineá r t.</s> <s xml:id="echoid-s26340" xml:space="preserve"> Quo-<lb/>niá enim angulus a r t eſt ęqualis angulo a t r, ք 30 huius, & linea a r eſt & qualis lineę a t:</s> <s xml:id="echoid-s26341" xml:space="preserve"> ſed linea a s <lb/> <pb o="101" file="0403" n="403" rhead="LIBER TERTIVS."/> eſt æqualis ſibi ipſi:</s> <s xml:id="echoid-s26342" xml:space="preserve"> ergo per 8 p 1 trigona ars & ats, ſunt ęquiangula:</s> <s xml:id="echoid-s26343" xml:space="preserve"> angulus ergo a s t eſt æqualis <lb/>angulo a s r:</s> <s xml:id="echoid-s26344" xml:space="preserve"> ergo per definitionem perpẽdicularis, linea a s eſt per-<lb/>pendicularis ſuper lineam r t.</s> <s xml:id="echoid-s26345" xml:space="preserve"> Producatur item linea a s uſq;</s> <s xml:id="echoid-s26346" xml:space="preserve"> ad pun <lb/> <anchor type="figure" xlink:label="fig-0403-01a" xlink:href="fig-0403-01"/> ctum coniunctionis amborum axium, quod ſit punctum b:</s> <s xml:id="echoid-s26347" xml:space="preserve"> dico, qđ <lb/>linea s b, diuidit per ęqualia trigonum r b t:</s> <s xml:id="echoid-s26348" xml:space="preserve"> hoc autem patet ex præ-<lb/>miſsis & ex 4 p 1:</s> <s xml:id="echoid-s26349" xml:space="preserve"> erιt enim trigonum partiale s r b æquale trigono <lb/>partiali s b t.</s> <s xml:id="echoid-s26350" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s26351" xml:space="preserve"> Et ex hoc patet, quoniam tota <lb/>linea a b cuicunq;</s> <s xml:id="echoid-s26352" xml:space="preserve"> puncto uiſo incidit, utcunq;</s> <s xml:id="echoid-s26353" xml:space="preserve"> tranſmutatis axibus, <lb/>non mutatur, ſed ſemper in medio eorum cõſiſtit:</s> <s xml:id="echoid-s26354" xml:space="preserve"> poſſumus ergo il-<lb/>lam nominare axem communem, quia ſemper ducitur ęqualiter ad <lb/>punctum coniunctionis amborum axium in ſuperficie rei uiſę à pũ <lb/>cto, qui eſt in medio concauitatis nerui, in quo duæ lineę extenſę in <lb/>duobus medijs concauitatum neruorum duorum ſe interſecãt:</s> <s xml:id="echoid-s26355" xml:space="preserve"> hic <lb/>uerò punctus ſemper eſt unus, non tranſmutabilis:</s> <s xml:id="echoid-s26356" xml:space="preserve"> & pũctus etiam <lb/>s ſemper eſt unus, non tranſmutabilis, per quem ſemper tranſit hęc <lb/>linea a b.</s> <s xml:id="echoid-s26357" xml:space="preserve"> Eſt ergo & ipſa ſemper intrãſmutabilis, licet alij axes trãſ-<lb/>mutentur quandoq;</s> <s xml:id="echoid-s26358" xml:space="preserve"> ab ipſo communi axe.</s> <s xml:id="echoid-s26359" xml:space="preserve"/> </p> <div xml:id="echoid-div997" type="float" level="0" n="0"> <figure xlink:label="fig-0403-01" xlink:href="fig-0403-01a"> <variables xml:id="echoid-variables421" xml:space="preserve">a r s t d b c</variables> </figure> </div> </div> <div xml:id="echoid-div999" type="section" level="0" n="0"> <head xml:id="echoid-head805" xml:space="preserve" style="it">34. Axe communi cum axibus radialibus puncto rei uiſæ in-<lb/>cidente: lineam copulantem centra for aminum gyrationis neruo <lb/>rum concauorum, & lineas ab his centris duct as ad nerui com-<lb/>munis medium & axem communem, amboś axes radiales in eadẽ ſuperficie conſiſtere eſt ne-<lb/>ceſſe. Alhazen 8 n 3.</head> <p> <s xml:id="echoid-s26360" xml:space="preserve">Sit diſpoſitio, quæ in proxima:</s> <s xml:id="echoid-s26361" xml:space="preserve"> dico, quòd lineam r t, & duas lineas r a & t a, & axem communem, <lb/>qui eſt a b, & duos axes radiales, ſcilicet r b & t b in eadem ſemper ſuperficie cõſiſtere oportet.</s> <s xml:id="echoid-s26362" xml:space="preserve"> Duo <lb/>enim axes t b & r b tranſeunt per centra r & t, per 29 huius:</s> <s xml:id="echoid-s26363" xml:space="preserve"> tranſeunt enim per centra foraminum <lb/>gyrationis duorum neruorum concauorum:</s> <s xml:id="echoid-s26364" xml:space="preserve"> & quia in pun cto coniunctionis concurrunt cum axe <lb/>communi ex hypotheſi:</s> <s xml:id="echoid-s26365" xml:space="preserve"> neceſſariò erunt cum axe communi in eadem ſuperficie per 2 p 11:</s> <s xml:id="echoid-s26366" xml:space="preserve"> ſed & li-<lb/>nea r t connectens centra foraminum gyrationis neruorum, ſecat hos duos axes radiales in punctis <lb/>r & t, & axem communem in puncto s:</s> <s xml:id="echoid-s26367" xml:space="preserve"> lineę quoq;</s> <s xml:id="echoid-s26368" xml:space="preserve"> r a & t a ſecant lineas r t & a b in punctis, in qui-<lb/>bus cum ipſis concurrunt:</s> <s xml:id="echoid-s26369" xml:space="preserve"> & quia omnes hæ lineę ſunt rectę, palàm per 1 p 11, quoniam quælibeti-<lb/>pſarum eſt in una ſuperficie.</s> <s xml:id="echoid-s26370" xml:space="preserve"> Patet ergo per 2 p 11, quoniam omnes ſunt in eadem ſuperficie.</s> <s xml:id="echoid-s26371" xml:space="preserve"> Et hoc <lb/>eſt propoſitum.</s> <s xml:id="echoid-s26372" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1000" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables422" xml:space="preserve">a r s t o p n q b</variables> </figure> <head xml:id="echoid-head806" xml:space="preserve" style="it">35. Neceſſe eſt axes radiales cum axe communi concurrẽtes <lb/> in puncto, cuius diſtantia à uiſu ſit multiplex lineæ connectenti centra oculorum: ſecundum ſui partes interiacentes punctum coniunctionis, & ſuperficies ipſorum uiſuum, æquales eſſe, ſu- perficiebuś amborum uiſuum, nec non ſuperficiei anteriori i- pſius uitreæ æqualiter incidere, & ſecundum angulos æquales. Alhazen 2 n 3.</head> <p> <s xml:id="echoid-s26373" xml:space="preserve">Sintitem, ut in 30 huius, duo centra duorum foraminũ gyratio <lb/>nis neruorum concauorum r & t.</s> <s xml:id="echoid-s26374" xml:space="preserve"> Quoniam ergo oculus mouetur <lb/>ſecundum totum nõ ſecundũ partem per 25 huius:</s> <s xml:id="echoid-s26375" xml:space="preserve"> palàm, quoniã <lb/>puncta r & t ſunt poſteriora oculo:</s> <s xml:id="echoid-s26376" xml:space="preserve"> figurentur ergo duo oculi qua-<lb/>ſi contingentes puncta r & t, circa centra o & p:</s> <s xml:id="echoid-s26377" xml:space="preserve"> & ab aliquo pun-<lb/>cto ſuperficiei rei uiſæ, quod ſit b, procedant axes ad centra uiſuũ:</s> <s xml:id="echoid-s26378" xml:space="preserve"> <lb/>& producantur ultrà ad puncta r & t.</s> <s xml:id="echoid-s26379" xml:space="preserve"> Palàm ita que, quoniam axes <lb/>r b & t b, tranſibunt totum uiſum:</s> <s xml:id="echoid-s26380" xml:space="preserve"> tranſeat ergo axis r b ſuperficiẽ <lb/>anteriorem ſui uiſus in puncto n:</s> <s xml:id="echoid-s26381" xml:space="preserve"> & axis t b, tranſeat anteriorem <lb/>ſuperficiem ſui uiſus in puncto q:</s> <s xml:id="echoid-s26382" xml:space="preserve"> & producatur linea n q:</s> <s xml:id="echoid-s26383" xml:space="preserve"> ſunt er-<lb/>go puncta q & n, puncta illa ſuperficierum uiſus, quibus infigitur <lb/>forma puncti coniunctionis axium, quod eſt b.</s> <s xml:id="echoid-s26384" xml:space="preserve"> Et quoniam axes <lb/>r b & t b ſunt æquales per præmiſſam:</s> <s xml:id="echoid-s26385" xml:space="preserve"> dico, quòd partes axium, <lb/>quæ ſunt b n & b q, ſunt æquales:</s> <s xml:id="echoid-s26386" xml:space="preserve"> & quòd incidunt uiſui ſecun-<lb/>dum angulos æquales.</s> <s xml:id="echoid-s26387" xml:space="preserve"> Cum enim lineę r n & t q ſint ęquales, quia <lb/>funt diametri æqualium oculorum ęqualiter à punctis r & t diſtan <lb/>tium, neceſſe eſt, ſi illæ ab æqualibus axibus abſcindantur, quòd <lb/>reſiduuum ſit æquale:</s> <s xml:id="echoid-s26388" xml:space="preserve"> erit ergo linea b n æqualis lineæ b q.</s> <s xml:id="echoid-s26389" xml:space="preserve"> Et quo-<lb/>niam linea n q æ quidiſtat lineę r t per 2 p 6:</s> <s xml:id="echoid-s26390" xml:space="preserve"> ideo quoniam latera <lb/>t b & r b, proportionaliter diuiduntur per lineã n q:</s> <s xml:id="echoid-s26391" xml:space="preserve"> ergo per 29 p 1 <lb/>erit angulus b n q ęqualis angulo b q n:</s> <s xml:id="echoid-s26392" xml:space="preserve"> angulus aũt b r t ęqualis eſt angulo b t r, quoniã linea b s diui <lb/> <pb o="102" file="0404" n="404" rhead="VITELLONIS OPTICAE"/> dit trigonum r t b per æqualia & baſim eius r t, ut patet per præmiſſam.</s> <s xml:id="echoid-s26393" xml:space="preserve"> Patet ergo, quoniam axes ra <lb/>diales ſuperficiebus uiſuum ęqualiter incidunt & ſecundum angulos ęquales.</s> <s xml:id="echoid-s26394" xml:space="preserve"> Et ſi incidant ſuperfi <lb/>ciebus uiſuum taliter, ut per centra uiſuũ tranſeant:</s> <s xml:id="echoid-s26395" xml:space="preserve"> palàm ergo, quoniam orthogonales ſunt ſuper <lb/>ſuperficies contingentes in punctis n & q:</s> <s xml:id="echoid-s26396" xml:space="preserve"> incidunt ergo ſuperficiebus uiſuum ęqualiter ſecundum <lb/>rectos angulos incidentes.</s> <s xml:id="echoid-s26397" xml:space="preserve"> Et propter hoc in omnium oculorum ordinatione, motu, uel quiete ſem <lb/>per duo axes eius ſunt æquales, aut non eſt in eis diuerſitas ſenſibilis, quę cauſſet aliquam diuerſita <lb/>tem uiſionis, maximè cũ res uiſa non fuerit ualde propinqua uiſui, ſed cũ diſtantia eius à uiſu fuerit <lb/>mediocris:</s> <s xml:id="echoid-s26398" xml:space="preserve"> cum enim res uiſa ualde uiſui approximauerit, ita ut linea, quę eſt inter duo centra ocu-<lb/>lorum, quę ſunt o & p, proportionem ęqualitatis uel exceſſus uel paruę diminutionis habuerit ad <lb/>axem radialem:</s> <s xml:id="echoid-s26399" xml:space="preserve"> tunc erunt axes ſenſibiliter inęquales, & facient angulos inęquales:</s> <s xml:id="echoid-s26400" xml:space="preserve"> aliàs uerò ſem-<lb/>per ſenſibiliter ęquales erunt:</s> <s xml:id="echoid-s26401" xml:space="preserve"> & conſtituent angulos ſenſibiliter ęquales:</s> <s xml:id="echoid-s26402" xml:space="preserve"> quia propter unitatem ui <lb/>ſuum, & uniformem receptionem formarum quodlibet punctum multiplicatur uniformiter ad u-<lb/>trunq;</s> <s xml:id="echoid-s26403" xml:space="preserve"> oculum:</s> <s xml:id="echoid-s26404" xml:space="preserve"> propter quod etiã omnes lineę ęqualiter diſtantes ab axibus faciunt angulos ęqua <lb/>les, & ipſę omnes ſenſibiliter ſunt ęquales.</s> <s xml:id="echoid-s26405" xml:space="preserve"> Eodẽ quoq;</s> <s xml:id="echoid-s26406" xml:space="preserve"> modo demonſtrari poteſt, quòd anguli, qui <lb/>per axes fiunt in ipſa ſuperficie uitreę, in qua fit refractio, ſunt ęquales.</s> <s xml:id="echoid-s26407" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s26408" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1001" type="section" level="0" n="0"> <head xml:id="echoid-head807" xml:space="preserve" style="it">36. Omnium linearum pyramidis radιalis obliquarum, plus uicinarum axirefractio fit ſecũ <lb/>dum angulos minores: remotiorum uerò ſecundum angulos maiores: æqualiter uerò diſtantium <lb/>ſecundum angulos œquales. Alhazen 9 n 2.</head> <p> <s xml:id="echoid-s26409" xml:space="preserve">Sit pyramis radialis, cuius uertex a:</s> <s xml:id="echoid-s26410" xml:space="preserve"> & diameter baſis, quæ per 18 huius, eſt ſuperficies rei uiſæ, ſit <lb/>b c d e f:</s> <s xml:id="echoid-s26411" xml:space="preserve"> axis uerò d a:</s> <s xml:id="echoid-s26412" xml:space="preserve"> & ſint lineę c a & e a lineę radiales obliquę, uicinę magis axi d a:</s> <s xml:id="echoid-s26413" xml:space="preserve"> & ſint b a & <lb/>f a remotiores.</s> <s xml:id="echoid-s26414" xml:space="preserve"> Dico, quòd lineę c a & e a ſecundũ mi <lb/> <anchor type="figure" xlink:label="fig-0404-01a" xlink:href="fig-0404-01"/> norem angulũ refringentur, & lineę b a & f a, ſecundũ <lb/>angulum maiorem.</s> <s xml:id="echoid-s26415" xml:space="preserve"> Intelligantur enim omnes iſtę li-<lb/>neę concurrere in puncto a, quod eſt uertex pyrami-<lb/>dis:</s> <s xml:id="echoid-s26416" xml:space="preserve"> & ſit in ſuperficie uitreę linea, cui incidũnt illæ li-<lb/>neę, g h i k l:</s> <s xml:id="echoid-s26417" xml:space="preserve"> hęc ergo linea erit recta uel curua circula <lb/>ris per 23 huius:</s> <s xml:id="echoid-s26418" xml:space="preserve"> ſit primũ recta:</s> <s xml:id="echoid-s26419" xml:space="preserve"> & incidat linea b ailli <lb/>lineę in puncto g:</s> <s xml:id="echoid-s26420" xml:space="preserve"> & linea c a in puncto h:</s> <s xml:id="echoid-s26421" xml:space="preserve"> & linea d a <lb/>axis in puncto i:</s> <s xml:id="echoid-s26422" xml:space="preserve"> & linea e a in puncto k:</s> <s xml:id="echoid-s26423" xml:space="preserve"> & linea f a in <lb/>puncto l.</s> <s xml:id="echoid-s26424" xml:space="preserve"> Quia ergo angulus g i a eſt rectus per pręce-<lb/>dentẽ:</s> <s xml:id="echoid-s26425" xml:space="preserve"> palá per 32 p 1, quòd angulus g h a eſt obtuſus:</s> <s xml:id="echoid-s26426" xml:space="preserve"> <lb/>ergo per 19 p 1, linea a g eſt maior quàm linea a h.</s> <s xml:id="echoid-s26427" xml:space="preserve"> Et <lb/>quia à puncto a exeunt duę lineę a c & a b, quę ſunt ad <lb/>baſim triãguli a g i, quę eſt g h i:</s> <s xml:id="echoid-s26428" xml:space="preserve"> angulus ergo a h i ma-<lb/>ior eſt angulo a g i per 16 p 1.</s> <s xml:id="echoid-s26429" xml:space="preserve"> Quia ergo angulus a h i <lb/>cũ angulo c h i ualet duos rectos per 13 p 1:</s> <s xml:id="echoid-s26430" xml:space="preserve"> & ſimiliter <lb/>angulus b g h cũ angulo a g h ualet duos rectos:</s> <s xml:id="echoid-s26431" xml:space="preserve"> palá, <lb/>quia angulus c h i minor eſt angulo b g i:</s> <s xml:id="echoid-s26432" xml:space="preserve"> ergo per 50 t <lb/>2 huius, angulus refractionis lineę c h eſt minor angu <lb/>lo refractionis lineę b g.</s> <s xml:id="echoid-s26433" xml:space="preserve"> Patet ergo, quòd linea c h re-<lb/>fringetur ſecundum minorem angulum, quàm linea <lb/>g b:</s> <s xml:id="echoid-s26434" xml:space="preserve"> & ſimiliter eſt de lineis e k & f l.</s> <s xml:id="echoid-s26435" xml:space="preserve"> Et quia lineæ æ-<lb/>qualiter diſtantes ab axe a d, ut ſunt exempli cauſſa li <lb/>neę a c & a e, ſecundum modum præmiſſum æquales <lb/>angulos faciunt in ſuperficie uitreę, qui ſunt c h i, & e k i:</s> <s xml:id="echoid-s26436" xml:space="preserve"> patet per 50 t 2 huius, quoniam anguli refra <lb/>ctionis ſunt æquales.</s> <s xml:id="echoid-s26437" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s26438" xml:space="preserve"> quoniam quando linea g h i k l fuerit linea circularis:</s> <s xml:id="echoid-s26439" xml:space="preserve"> <lb/>erit eodem modo dem onſtrandum per 50 t 2 huius.</s> <s xml:id="echoid-s26440" xml:space="preserve"/> </p> <div xml:id="echoid-div1001" type="float" level="0" n="0"> <figure xlink:label="fig-0404-01" xlink:href="fig-0404-01a"> <variables xml:id="echoid-variables423" xml:space="preserve">a g h i k l h i k b c d e f</variables> </figure> </div> </div> <div xml:id="echoid-div1003" type="section" level="0" n="0"> <head xml:id="echoid-head808" xml:space="preserve" style="it">37. Omnes formæ punctorum æqualiter circumſtantium puncta, quæ ſuperficiebus uiſuum <lb/>incidunt ſecundum axes radiales: ad punct a æqualiter circumſtantia medium punctum ner-<lb/>ui communis conſimiliter pertingunt.</head> <p> <s xml:id="echoid-s26441" xml:space="preserve">Diſponantur omnia alia, ut in 35 huius:</s> <s xml:id="echoid-s26442" xml:space="preserve"> ſignenturq́;</s> <s xml:id="echoid-s26443" xml:space="preserve"> in ſuperficie oculi, cuius centrum eſt punctũ <lb/>o, ex utraq;</s> <s xml:id="echoid-s26444" xml:space="preserve"> parte punctin duo puncta u & x:</s> <s xml:id="echoid-s26445" xml:space="preserve"> & in ſuperficie oculi, cuius centrum eſt punctum p, ſi-<lb/>gnentur ex utraq;</s> <s xml:id="echoid-s26446" xml:space="preserve"> parte puncti q duo puncta y & z:</s> <s xml:id="echoid-s26447" xml:space="preserve"> ſitq́ ſuperficies rei uiſę oppoſita uiſibus, in qua <lb/>ſit linea recta, quæ g b c, cuius punctus medius ſit b, & extremi puncti g & c:</s> <s xml:id="echoid-s26448" xml:space="preserve"> incidantq́;</s> <s xml:id="echoid-s26449" xml:space="preserve"> axes radia-<lb/>les, qui ſunt r b & t b, cum axe communi, qui ſit a b, ipſi puncto b, qui ſit punctus coniunctionis o-<lb/>mnium trium axium:</s> <s xml:id="echoid-s26450" xml:space="preserve"> protrahanturq́;</s> <s xml:id="echoid-s26451" xml:space="preserve"> à punctis u & x ſuperficiei uiſus, cuius centrum eſto, ad pun-<lb/>cta g & c, ſuperficiei rei uiſę, duæ lineę rectæ, quæ ſint u g & x c:</s> <s xml:id="echoid-s26452" xml:space="preserve"> & à punctis y & z ſuperficiei ui-<lb/>ſus, cuius centrum eſt p, protrahantur lineæ y g & z c.</s> <s xml:id="echoid-s26453" xml:space="preserve"> Dico, quòd formæ punctorum ſu-<lb/>perficiei rei uiſæ, quæ ſunt g & c, quæ in ſuperficie oculi o incidunt in punctis u & x, & in ſu-<lb/>perficie oculi p in punctisy & z, non perueniunt ad medium punctum nerui communis, quod <lb/>eſt a, ſed circunſtant ipſum punctum a, ſimilis diſpoſitionis, ut puncta c & g difpoſita ſunt ad <lb/>punctum b, in ipſa ſuperficie rei uiſæ taliter, ut punctus, qui eſt dexter ad punctum b, qui eſt <lb/> <pb o="103" file="0405" n="405" rhead="LIBER TERTIVS."/> punctus coniunctionis axiũ in ſuperficie rei uiſę, ſit dexter pertingẽs ad pũctũ a, & ſiniſter ipſi pun <lb/>cto b, fiat ſiniſter ipſi pũcto a:</s> <s xml:id="echoid-s26454" xml:space="preserve"> & ſic de alijs differẽtijs poſitionũ, ut quod eſt ſurſum ad pũctũ b, ſit ſur <lb/>ſum ad pũctũ a, & quod eſt deorſum ad pũctum b, deorſum fiat ad <lb/>punctũ a.</s> <s xml:id="echoid-s26455" xml:space="preserve"> Producatur enim in utroq;</s> <s xml:id="echoid-s26456" xml:space="preserve"> oculorum linea im, recta uel <lb/> <anchor type="figure" xlink:label="fig-0405-01a" xlink:href="fig-0405-01"/> curua, diſtinguẽs ſuperficiẽ uitreę à ſuperficie glacialis:</s> <s xml:id="echoid-s26457" xml:space="preserve"> & hęc li-<lb/>nea ſiue recta fuerit ſiue curua, quorũ alterũ eſt neceſſariũ per 23 <lb/>huius:</s> <s xml:id="echoid-s26458" xml:space="preserve"> ſemper tamẽ anguli incidẽtię erũt ęquales ք 35 huius, quo-<lb/>niã & eadẽ de illis eſt demõſtratio:</s> <s xml:id="echoid-s26459" xml:space="preserve"> ſed & anguli refractionis fiunt <lb/>ęquales per pręmiſſam, & ideo, quia propter conformitatẽ uiſuũ <lb/>& ęqualẽ diſtantiã pũctorũ g & c à pũcto b ex hypotheſi, ſequitur <lb/>trigona y g u & x c z eſſe ęquiangula:</s> <s xml:id="echoid-s26460" xml:space="preserve"> anguli ergo g y u & c x z, ſunt <lb/>æquales:</s> <s xml:id="echoid-s26461" xml:space="preserve"> ſed & figurę oculorũ ſunt penitus ſimiles, & diaphanitas <lb/>eſt cõformis:</s> <s xml:id="echoid-s26462" xml:space="preserve"> fiet ergo linearum c x & g y in ſuperficie refractionis <lb/>cõformis refractio:</s> <s xml:id="echoid-s26463" xml:space="preserve"> & fimiliter linearum g u & c z fiet cõformis re-<lb/>fractio & ſecundũ angulos æquales:</s> <s xml:id="echoid-s26464" xml:space="preserve"> quælibet ergo ipſarum refrin <lb/>getur æqualiter à perpẽdiculari:</s> <s xml:id="echoid-s26465" xml:space="preserve"> ſit ergo ut linea c x refringatur ad <lb/>punctum f, & linea g u ad punctũ h, quę ſunt puncta foraminis gy-<lb/>rationis nerui circa punctũ r:</s> <s xml:id="echoid-s26466" xml:space="preserve"> linea uero g y refringatur ad punctũ <lb/>l:</s> <s xml:id="echoid-s26467" xml:space="preserve"> & linea c z a d e punctum alterius foraminis, quod eſt circa pun-<lb/>ctum t.</s> <s xml:id="echoid-s26468" xml:space="preserve"> Et quoniam omnia puncta formarũ ſecundũ lineas rectas <lb/>breuiſsimas refringũtur à perpẽdiculari n r:</s> <s xml:id="echoid-s26469" xml:space="preserve"> palàm, quia non con-<lb/>currũt cum illa, ſed directè diffundẽtes ſe ad pũcta nerui cõmunis <lb/>ſimilẽ ſitũ & diſpoſitionẽ recipiunt eis, quę habẽt in ſuperficie rei <lb/>uiſæ, quę eſt baſis pyramidis uiſionis.</s> <s xml:id="echoid-s26470" xml:space="preserve"> Linea ergo x f, quæ uenit à <lb/>puncto c rei uiſæ, refringitur ad aliquod pũctũ nerui aliud à pũcto <lb/>a, quod ſit d:</s> <s xml:id="echoid-s26471" xml:space="preserve"> & linea u h, quę uenit à pũcto g rei uiſæ, refringitur ad <lb/>punctũ aliud à pũcto a, quod ſit k.</s> <s xml:id="echoid-s26472" xml:space="preserve"> Et quoniam unius diſpoſitionis <lb/>ſunt ambo uiſus, & oculorũ diſtantia eſt res modica, ut patet per 4 <lb/>huius:</s> <s xml:id="echoid-s26473" xml:space="preserve"> & lineę ad talia pũcta productę à uiſibus ambobus ſunt æ-<lb/>quales:</s> <s xml:id="echoid-s26474" xml:space="preserve"> & anguli incidẽtię ſunt ęquales per 35 huius, anguli quoq;</s> <s xml:id="echoid-s26475" xml:space="preserve"> <lb/>refractionis ſunt æ quales ք pręmiſſam:</s> <s xml:id="echoid-s26476" xml:space="preserve"> palàm, quia linea y l, quæ eſt forma pũcti g, refringetur ad <lb/>punctũ k, in quo cecidit forma eiuſdẽ pũcti g, ueniens per lineam u h:</s> <s xml:id="echoid-s26477" xml:space="preserve"> linea quoq;</s> <s xml:id="echoid-s26478" xml:space="preserve"> z e, quæ eſt forma <lb/>puncti c, refringetur ad pũctum d, in quo cadit eadẽ forma pũcti c, ueniens per lineam x f.</s> <s xml:id="echoid-s26479" xml:space="preserve"> Similiter <lb/>quoq;</s> <s xml:id="echoid-s26480" xml:space="preserve"> demõſtrandũ de quibuslibet duobus pũctis ſuperficiei rei uiſę, æqualiter diſtantibus à pun-<lb/>cto coniunctionis, quod eſt b.</s> <s xml:id="echoid-s26481" xml:space="preserve"> Omnes ergo formę pũctorum rei uiſę æqualiter circũſtantium pun-<lb/>cta, quę ſuperficiebus uiſuum incidũt ſecũdum axes radiales, ad pũcta æqualiter circũſtantia me-<lb/>dium pũctum nerui communis conſimiliter peringũt & ſeruatur figura & diſpoſitio totius ſuper-<lb/>ficiei rei uiſę in partibus ſuis, & in remotione à pũcto, quod eſt in axe, ſecũdum modũ diſtantiæ & <lb/>declinationis pũctorum, quorum formę illic recipiuntur à pũcto coniũctionis in ſuperficie rei ui-<lb/>ſæ ſecũdum diſpoſitionem angulorum refractionis in ſuperficie uitreæ:</s> <s xml:id="echoid-s26482" xml:space="preserve"> & duæ formę, quę infigun-<lb/>tur in duobus pũctis conſimilis poſitionis a pud ſuperficies duorũ uiſuum, perueniũt ad illũ eundẽ <lb/>pũctum concauitatis nerui communis, & ſuperponũtur ſibi in illo pũcto, & erũt una forma:</s> <s xml:id="echoid-s26483" xml:space="preserve"> lineæ <lb/>quoq;</s> <s xml:id="echoid-s26484" xml:space="preserve"> obliquè ſuperficiebus uiſuum incidẽtes, quę in ſuperficie ipſius uiſus refringũtur, ad eandẽ <lb/>ordinationem formę poſſunt peruenire.</s> <s xml:id="echoid-s26485" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s26486" xml:space="preserve"/> </p> <div xml:id="echoid-div1003" type="float" level="0" n="0"> <figure xlink:label="fig-0405-01" xlink:href="fig-0405-01a"> <variables xml:id="echoid-variables424" xml:space="preserve">k a d h r f s l t e i o m i p m x y u n q z g b c</variables> </figure> </div> </div> <div xml:id="echoid-div1005" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables425" xml:space="preserve">a e g b f c</variables> </figure> <head xml:id="echoid-head809" xml:space="preserve" style="it">38. Neceſſe eſt ambos axes radiales cum axe communi concur-<lb/> rentes in ſuperficie rei uiſæ, cum linea æquidiſtante line æ cõnecten ti centra oculorum, uelcum totali ſuperficie æquales hinc & inde angulos continere.</head> <p> <s xml:id="echoid-s26487" xml:space="preserve">Sunt enim ambo oculi ęqualis diſpoſitionis per 4 huius:</s> <s xml:id="echoid-s26488" xml:space="preserve"> patet e-<lb/>tiam ſenſui, quò d ſunt diſtantię modicę ab inuicem, & axis ſemper in <lb/>quolibet oculo una tantũ linea tranſiens per cẽtrũ foraminis uueę & <lb/>centra omniũ tunicarũ, ad cẽtrũ foraminis gyrationis nerui concaui <lb/>pertingẽs, ut patet ք 29 huius.</s> <s xml:id="echoid-s26489" xml:space="preserve"> Sit ergo, ut linea b f c ęquidiſtet lineę <lb/>e g, cõnectẽti cẽtra oculorũ e & g:</s> <s xml:id="echoid-s26490" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s26491" xml:space="preserve"> medius pũct{us} nerui cõmunis, <lb/>ꝗ a:</s> <s xml:id="echoid-s26492" xml:space="preserve"> & ſit ut forma pũcti ſuքficiei rei uiſę, qđ ſit f, ք axes f e & f g քue-<lb/>niat ad cẽtra oculorũ, quę ſunt e, g, connexa per lineã e g:</s> <s xml:id="echoid-s26493" xml:space="preserve"> pertingãtq́;</s> <s xml:id="echoid-s26494" xml:space="preserve"> <lb/>ad punctum a, quod ſit punctus medius nerui communis:</s> <s xml:id="echoid-s26495" xml:space="preserve"> & ſit axis <lb/>communis, qui a f, incidẽs ſuperficiei rei uiſę in puncto f ſecundũ an-<lb/>gulos rectos:</s> <s xml:id="echoid-s26496" xml:space="preserve"> quoniam ſuperficies, in qua ſunt omnes aſsignatę lineę <lb/>axium & pũcta per 34 huius, erecta eſt ſuper ſuperficiẽ rei uiſę, & axis <lb/>cõmunis incidit directè ք 33 huius, & ք 29 p 1:</s> <s xml:id="echoid-s26497" xml:space="preserve"> quoniã linea cõnectẽs <lb/>centra oculorũ, lineę r t connectẽti cẽtra foraminũ gyrationis nerui <lb/>concaui eſt ęquidiſtans:</s> <s xml:id="echoid-s26498" xml:space="preserve"> ergo & lineę uel ſuperficiei illi ęquidiſtanti <lb/> <pb o="104" file="0406" n="406" rhead="VITELLONIS OPTICAE"/> per 30 p 1.</s> <s xml:id="echoid-s26499" xml:space="preserve"> Quia ergo per 33 huius angulus a f e eſt æqualis angulo a f g:</s> <s xml:id="echoid-s26500" xml:space="preserve"> erit ergo reſiduũ duorũ re-<lb/>ctorum contentorum ab axe & linea b c, quę eſt communis ſectio ſuperficiei rei uiſæ, & ſuperficiei <lb/>axium inter ſe, hinc inde æquale.</s> <s xml:id="echoid-s26501" xml:space="preserve"> Axes ergo radiales incidunt ſuperficiei rei uiſæ ſecundum angu-<lb/>los ęquales.</s> <s xml:id="echoid-s26502" xml:space="preserve"> Et hoc eſt propoſitum:</s> <s xml:id="echoid-s26503" xml:space="preserve"> quoniam angulus e f b fit æqualis angulo g f c.</s> <s xml:id="echoid-s26504" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1006" type="section" level="0" n="0"> <head xml:id="echoid-head810" xml:space="preserve" style="it">39. A‘ puncto coniunctionis lineam æquidiſtantem lineæ connectenticentra oculorum in ſu <lb/>perficie rei uiſæ illi æquidiſtante protrahere.</head> <p> <s xml:id="echoid-s26505" xml:space="preserve">Sint centra duorum oculorũ puncta e & g:</s> <s xml:id="echoid-s26506" xml:space="preserve"> & ducatur linea e g:</s> <s xml:id="echoid-s26507" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s26508" xml:space="preserve"> ſuperficies rei uiſæ b c d f, à <lb/>cuius puncto dato, quod ſit a, linea æquidiſtans lineę <lb/>e g debeat produci.</s> <s xml:id="echoid-s26509" xml:space="preserve"> Diuidatur itaq;</s> <s xml:id="echoid-s26510" xml:space="preserve"> linea e g per ęqua <lb/> <anchor type="figure" xlink:label="fig-0406-01a" xlink:href="fig-0406-01"/> lia in puncto r per 10 p 1:</s> <s xml:id="echoid-s26511" xml:space="preserve"> & à puncto a ad punctum r <lb/>ducatur linea a r:</s> <s xml:id="echoid-s26512" xml:space="preserve"> & ducantur lineę e a & g a, quę ſint <lb/>axes uiſuales, concurrentes in puncto a ſuperficiei rei <lb/>uiſę:</s> <s xml:id="echoid-s26513" xml:space="preserve"> patet ergo, quoniã axis e a ęqualis eſt axi g a per <lb/>35 huius:</s> <s xml:id="echoid-s26514" xml:space="preserve"> & linea e r eſt ęqualis lineę g r, & linea r a cõ-<lb/>munis:</s> <s xml:id="echoid-s26515" xml:space="preserve"> erit ergo per 8 p 1 angulus e r a ęqualis angulo <lb/>g r a, & ambo recti:</s> <s xml:id="echoid-s26516" xml:space="preserve"> erit ergo linea a r perpendicularis <lb/>ſuper lineam e g per definitionem lineę perpendicu-<lb/>laris:</s> <s xml:id="echoid-s26517" xml:space="preserve"> & à centris uiſuum e & g ducantur æquidiſtan-<lb/>tes lineę r a, per 31 p 1, quæ ſint lineę e z & g y:</s> <s xml:id="echoid-s26518" xml:space="preserve"> hę ergo <lb/>inter ſe ſunt æquales & æquidiſtantes per 25 t 1 huius, <lb/>& ſunt in eadem ſuperficie per 1 t 1 huius.</s> <s xml:id="echoid-s26519" xml:space="preserve"> Et quia com <lb/>munis ſectio huius ſuperficiei & ſuperficiei rei uiſæ <lb/>tranſit per pũctum a:</s> <s xml:id="echoid-s26520" xml:space="preserve"> & eſt per 33 p 1 æquidiſtans lineę <lb/>e g:</s> <s xml:id="echoid-s26521" xml:space="preserve"> palàm, quòd ipſa linea z a y eſt linea, quę quęritur.</s> <s xml:id="echoid-s26522" xml:space="preserve"> <lb/>Eſt ergo factum id, quod proponebatur.</s> <s xml:id="echoid-s26523" xml:space="preserve"/> </p> <div xml:id="echoid-div1006" type="float" level="0" n="0"> <figure xlink:label="fig-0406-01" xlink:href="fig-0406-01a"> <variables xml:id="echoid-variables426" xml:space="preserve">c r g b f z a y c d</variables> </figure> </div> </div> <div xml:id="echoid-div1008" type="section" level="0" n="0"> <head xml:id="echoid-head811" xml:space="preserve" style="it">40. Omnes lineæ productæ ab ambobus uiſibus adidem punctum lineæ cum ambobus axibus <lb/>pyramidumidum radialium angulos rectos facientis, neceſſariò ſunt æquales. Alhazen 3 n 3.</head> <p> <s xml:id="echoid-s26524" xml:space="preserve">Verbi gratia:</s> <s xml:id="echoid-s26525" xml:space="preserve"> ſint, ut ſuprà in proxima pręcedente, centra duorum uiſuũ puncta e & g:</s> <s xml:id="echoid-s26526" xml:space="preserve"> & ſuperfi-<lb/>cies rei uiſę ſit b c d f:</s> <s xml:id="echoid-s26527" xml:space="preserve"> in cuius puncto a concurrant axes e a & g a:</s> <s xml:id="echoid-s26528" xml:space="preserve"> & à puncto a ad utranq;</s> <s xml:id="echoid-s26529" xml:space="preserve"> partem <lb/>producatur linea una, quæ ſit z a u, rectos angulos continens cum utroq;</s> <s xml:id="echoid-s26530" xml:space="preserve"> axium:</s> <s xml:id="echoid-s26531" xml:space="preserve"> producanturq́;</s> <s xml:id="echoid-s26532" xml:space="preserve"> à <lb/>centris uiſuum lineæ e u, g u, e z, g z.</s> <s xml:id="echoid-s26533" xml:space="preserve"> Dico, quòd lineę e u & g u ſunt æ quales inter ſe, & lineę e z & <lb/>g z ęquales inter ſe.</s> <s xml:id="echoid-s26534" xml:space="preserve"> Quoniam enim axes uiſuum æquales ſunt per 35 huius, palàm quòd axis e a eſt <lb/>ęqualis axi g a, & angulus e a u ęqualis angulo g a u:</s> <s xml:id="echoid-s26535" xml:space="preserve"> quoniam uterq;</s> <s xml:id="echoid-s26536" xml:space="preserve"> ipſorum eſt rectus ex hypothe <lb/>ſi:</s> <s xml:id="echoid-s26537" xml:space="preserve"> ſed linea a u eſt communis in triangulis e a u & g a u:</s> <s xml:id="echoid-s26538" xml:space="preserve"> erit ergo per 4 p 1 baſis e u æqualis baſi g u:</s> <s xml:id="echoid-s26539" xml:space="preserve"> <lb/>& ſimiliter erit baſis e z æqualis baſi g z:</s> <s xml:id="echoid-s26540" xml:space="preserve"> & eodẽ mo <lb/>do in punctis omnibus lineę z u accidit.</s> <s xml:id="echoid-s26541" xml:space="preserve"> Palàm ergo <lb/> <anchor type="figure" xlink:label="fig-0406-02a" xlink:href="fig-0406-02"/> eſt quod proponitur.</s> <s xml:id="echoid-s26542" xml:space="preserve"> Poteſt & hoc aliter demonſtra <lb/>ri:</s> <s xml:id="echoid-s26543" xml:space="preserve"> ducatur enim à pũcto a ſuperficiei rei uiſę, in quo <lb/>concurrunt axes, linea æquidiſtans lineę e g, quę eſt <lb/>inter duo centra oculorum, per pręcedentem:</s> <s xml:id="echoid-s26544" xml:space="preserve"> quæ <lb/>ſit linea k l:</s> <s xml:id="echoid-s26545" xml:space="preserve"> eritq́ illa linea k l in ſuperficie rei uiſę:</s> <s xml:id="echoid-s26546" xml:space="preserve"> <lb/>ducatur quoq;</s> <s xml:id="echoid-s26547" xml:space="preserve"> linea z a perpendicularis ſuper lineã <lb/>k l per 12 p 1.</s> <s xml:id="echoid-s26548" xml:space="preserve"> Item ducatur à puncto a linea orthogo-<lb/>naliter ſuper lineam e g, quę ſit linea a r:</s> <s xml:id="echoid-s26549" xml:space="preserve"> diuidetq́;</s> <s xml:id="echoid-s26550" xml:space="preserve"> li-<lb/>nea a r lineam e g per æqualia in pũcto r, per 31 t 1 hu <lb/>ius, & ex 35 huius, & ex 5 p 1:</s> <s xml:id="echoid-s26551" xml:space="preserve"> quoniam enim axes e a <lb/>& g a ſunt æquales, erunt anguli ad baſim ęquales, <lb/>& linea r a cõmunis ambobus trigonis e a r, & g a r, <lb/>anguliq́;</s> <s xml:id="echoid-s26552" xml:space="preserve"> ad punctũ r ſunt ęquales, quia recti:</s> <s xml:id="echoid-s26553" xml:space="preserve"> erit er-<lb/>go per 32 p 1, & per 4 p 6 linea e r ęqualis lineę r g:</s> <s xml:id="echoid-s26554" xml:space="preserve"> ꝓ-<lb/>ducaturq́;</s> <s xml:id="echoid-s26555" xml:space="preserve"> linea r z:</s> <s xml:id="echoid-s26556" xml:space="preserve"> erit ergo per 29 p 1 linear a per-<lb/>pendicularis ſuper lineam k a l.</s> <s xml:id="echoid-s26557" xml:space="preserve"> Et quoniã per 34 hu-<lb/>ius lineę e a, g a & r a ſunt in eadem ſuperficie, & linea z a eſt perpendicularis ſuper lineas e a & g a, <lb/>ut patet ex hypotheſi:</s> <s xml:id="echoid-s26558" xml:space="preserve"> ergo per 4 p 11 linea z a eſt perpendiculariter erecta ſuper illam ſuperficiẽ, in <lb/>qua ſunt lineę e a, g a:</s> <s xml:id="echoid-s26559" xml:space="preserve"> ergo & ſuper lineam r a.</s> <s xml:id="echoid-s26560" xml:space="preserve"> Itẽ per 4 p 11 linea k a erit perpẽdicularis ſuper ſuper-<lb/>ficiem r z a:</s> <s xml:id="echoid-s26561" xml:space="preserve"> erit ergo per 8 p 11 linea e r perpẽdicularis ſuper eandẽ ſuperficiem r z a:</s> <s xml:id="echoid-s26562" xml:space="preserve"> ex definitione <lb/>ergo lineę erectę ſuper ſuperficiem, erit linea e r perpendicularis ſuper lineam r z.</s> <s xml:id="echoid-s26563" xml:space="preserve"> Quia ergo duorũ <lb/>triangulorũ e r z & g r z duo anguli e r z, & g r z ſunt ęquales, quia recti, & linea e r ęqualis eſt lineę <lb/>r g, & latus r z commune:</s> <s xml:id="echoid-s26564" xml:space="preserve"> erit per 4 p 1 linea e z ęqualis lineę g z.</s> <s xml:id="echoid-s26565" xml:space="preserve"> Et eodẽ modo de quolibet aliorum <lb/>punctorum lineę z u demonſtrandum.</s> <s xml:id="echoid-s26566" xml:space="preserve"> Patet ergo prop oſitum.</s> <s xml:id="echoid-s26567" xml:space="preserve"/> </p> <div xml:id="echoid-div1008" type="float" level="0" n="0"> <figure xlink:label="fig-0406-02" xlink:href="fig-0406-02a"> <variables xml:id="echoid-variables427" xml:space="preserve">e r g b z f k m a n l c u d</variables> </figure> </div> </div> <div xml:id="echoid-div1010" type="section" level="0" n="0"> <head xml:id="echoid-head812" xml:space="preserve" style="it">41. Omnes lineæ productæ ab ambobus uiſibus adidem punctum lineæ cum ambobus axibus <lb/>angulos obliquos ſacientis, neceſſariò ſunt inæquales. Alhazen 4 n 3.</head> <p> <s xml:id="echoid-s26568" xml:space="preserve">Sit omnimoda diſpoſitio, ut ſuprà in pręcedente.</s> <s xml:id="echoid-s26569" xml:space="preserve"> Dico, quòd oẽs lineę ab ambobus uiſibus ad <lb/> <pb o="105" file="0407" n="407" rhead="LIBER TERTIVS"/> idem punctum extra lineam u z, quę ſola cũ ambobus axibus facitrectos, ſemper ſunt inæquales.</s> <s xml:id="echoid-s26570" xml:space="preserve"> <lb/>Signentur enim in linea k l, utcunq;</s> <s xml:id="echoid-s26571" xml:space="preserve"> ſecante lineam u z duo puncta à puncto a, prout placuerit, di-<lb/>ſtantia, quę ſint m & n:</s> <s xml:id="echoid-s26572" xml:space="preserve"> & ducantur lineę e m & e n.</s> <s xml:id="echoid-s26573" xml:space="preserve"> Dico, quod lineæ e m & g m ſunt in æquales, & li <lb/>neæ e n & g n inęquales.</s> <s xml:id="echoid-s26574" xml:space="preserve"> Ducatur enim à puncto r ad pũctum m linea, quæ ſit r m.</s> <s xml:id="echoid-s26575" xml:space="preserve"> Quoniam ergo <lb/>angulus e r a eſt rectus, ut patuit in præmiſſa:</s> <s xml:id="echoid-s26576" xml:space="preserve"> palàm, quia angulus e r m eſt minor recto:</s> <s xml:id="echoid-s26577" xml:space="preserve"> angulus er-<lb/>go g r m eſt maior recto per 13 p 1.</s> <s xml:id="echoid-s26578" xml:space="preserve"> In triangulis ergo g r m & e r m latus r m eſt commune, & linea e r <lb/>æqualis eſt lineę g r, & angulus g r m maior angulo e r m:</s> <s xml:id="echoid-s26579" xml:space="preserve"> ergo per 24 p 1 erit latus g m longius late-<lb/>re e m:</s> <s xml:id="echoid-s26580" xml:space="preserve"> & ſimiliter eſt de omnibus alijs punctis extra lineam u z argumentandum.</s> <s xml:id="echoid-s26581" xml:space="preserve"> Patet ergo pro-<lb/>poſitum.</s> <s xml:id="echoid-s26582" xml:space="preserve"> Iſta tamen inæ qualitas illarum linearum minus eſt ſenſibilis, cum puncta declinationis <lb/>fuerint propinqua puncto coniunctionis.</s> <s xml:id="echoid-s26583" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1011" type="section" level="0" n="0"> <head xml:id="echoid-head813" xml:space="preserve" style="it">42. Omnes lineæ ad puncta æquidiſtantia à puncto coniunctionis axium in linea cum ambo <lb/>bus axibus angulos obliquos faciente, ab alternis uiſibus productæ, neceſſariò ſunt æquales, & <lb/>æquales cum illis lineis angulos continentes. Alhazen 3 n 3.</head> <p> <s xml:id="echoid-s26584" xml:space="preserve">Sit omnis diſpoſitio ut ſuprà in duabus pręmiſsis, & fint m & n puncta in linea k l, angulos obli-<lb/>quos faciente cũ ambobus axibus, æqualiter diftan <lb/>tia à pũcto a, quod fit pũctum coniũctionis axium, <lb/> <anchor type="figure" xlink:label="fig-0407-01a" xlink:href="fig-0407-01"/> ita quòd linea m a ſit æqualis a n.</s> <s xml:id="echoid-s26585" xml:space="preserve"> Dico, quòd protra <lb/>ctæ lineę ab alternis uiſibus ut e n & g m, & e m & <lb/>g n ſunt æquales.</s> <s xml:id="echoid-s26586" xml:space="preserve"> Quoniã enim axis e a eſt æqualis <lb/>axi g a per 35 huius, & angulus incidentiæ axis e a, ꝗ <lb/>eſt angulus e a m, æqualis eſt angulo incidentię axis <lb/>g a, qui eſt angulus g a n:</s> <s xml:id="echoid-s26587" xml:space="preserve"> ideo quia anguli r a m & r a <lb/>n ſuntrecti:</s> <s xml:id="echoid-s26588" xml:space="preserve"> anguli quoq;</s> <s xml:id="echoid-s26589" xml:space="preserve"> r a e & r a g ſunt ęquales, ut <lb/>hæc patent ex prędemõſtratis in præmiſsis duabus <lb/>propoſitionibus:</s> <s xml:id="echoid-s26590" xml:space="preserve"> remanẽt ergo anguli e a m & g a n <lb/>æquales:</s> <s xml:id="echoid-s26591" xml:space="preserve"> ſed & axes e a & g a ſunt æquales, & linea <lb/>m a æqualis eſt lineæ n a ex hypotheſi:</s> <s xml:id="echoid-s26592" xml:space="preserve"> erit ergo li-<lb/>nea g n æqualis lineę e m per 4 p 1:</s> <s xml:id="echoid-s26593" xml:space="preserve"> & angulus g n a <lb/>ęqualis angulo e m a.</s> <s xml:id="echoid-s26594" xml:space="preserve"> Ergo in triangulis quoq;</s> <s xml:id="echoid-s26595" xml:space="preserve"> e m n <lb/>& g n m per 4 p 1 baſis e m æqualis eſt baſi g n.</s> <s xml:id="echoid-s26596" xml:space="preserve"> Et ſi-<lb/>militer demonſtrari poteſt in omnibus alijs pũctis <lb/>ſimilibus:</s> <s xml:id="echoid-s26597" xml:space="preserve"> lineæ enim g b & e f, g f & e b, & g k & e l, g l & e k, g c & e d, g d & e c, omnes, ut ſic nomi-<lb/>nantur, & ut ab alternis uiſibus ad puncta æqualiter à pũcto a diſtantia producũtur, neceſſariò ſunt <lb/>ęquales.</s> <s xml:id="echoid-s26598" xml:space="preserve"> Patet ergo propoſitum, quotcunq;</s> <s xml:id="echoid-s26599" xml:space="preserve"> etiam alijs lineis modo ſimili productis.</s> <s xml:id="echoid-s26600" xml:space="preserve"/> </p> <div xml:id="echoid-div1011" type="float" level="0" n="0"> <figure xlink:label="fig-0407-01" xlink:href="fig-0407-01a"> <variables xml:id="echoid-variables428" xml:space="preserve">e r g b f k m a n l c d</variables> </figure> </div> </div> <div xml:id="echoid-div1013" type="section" level="0" n="0"> <head xml:id="echoid-head814" xml:space="preserve" style="it">43. Secundum omnes lineas pyramidis radialis formarum fit certa comprehenſio à uiſu: <lb/>magis autem ſecundum lineas axi uiciniores: & maximè per axem centrum for aminis uueæ <lb/>tranſeuntem. Alhazen 8 n 2.</head> <p> <s xml:id="echoid-s26601" xml:space="preserve">Solus enim axis extenditur ſecundum rectitudinem, quouſque perueniat ad locum gyratio-<lb/>nis concaui nerui, & omnes aliæ lineæ obliquantur, ut patet per 24 huius:</s> <s xml:id="echoid-s26602" xml:space="preserve"> forma ergo rei uiſæ op-<lb/>poſitæ medio ſuperficiei uiſus, perueniet ad glacialem & uitreum ſecundum extenſionem uſq;</s> <s xml:id="echoid-s26603" xml:space="preserve"> ad <lb/>locum gyrationis nerui concaui:</s> <s xml:id="echoid-s26604" xml:space="preserve"> formę uerò, quæ ueniunt ſecundũ lineas alias, obliquantur.</s> <s xml:id="echoid-s26605" xml:space="preserve"> Et ꝗa <lb/>diſpoſitio formarũ obliquatarũ non eſt ſicut diſpoſitio formarũ extenſarũ rectè:</s> <s xml:id="echoid-s26606" xml:space="preserve"> quoniam obliqua <lb/>tio neceſſariò ipſas alterat aliqua alteratione in certitudine comprehenſionis:</s> <s xml:id="echoid-s26607" xml:space="preserve"> punctus ergo formæ <lb/>perueniens ad locũ gyrationis concaui nerui, qui extenditur ſecundũ rectitudinem axis, eſt magis <lb/>uerificatus omnibus punctis formarum.</s> <s xml:id="echoid-s26608" xml:space="preserve"> Et quia obliquatio linearũ uicinarum axi eſt minor, & re-<lb/>motiorum maior, eò quòd anguli, qui fiunt ex lineis, ſuper quas ueniunt formæ, & ex perpendicu-<lb/>laribus ſuper axem productis in ſuperficie obliquationis, linearũ uicinarũ axi ſunt acutiores, & re-<lb/>motiorũ minus acuti, ut patet per 36 huius:</s> <s xml:id="echoid-s26609" xml:space="preserve"> formę uerò, quarũ obliquatio eſt minor, magis manife-<lb/>ſtantur, quàm formę, quarũ obliquatio eſt maior:</s> <s xml:id="echoid-s26610" xml:space="preserve"> punctus ergo, qui eſt ſuper axem, perueniens ad <lb/>locũ gyrationis nerui concaui, eſt manifeſtior omnibus alijs pũctis, & certioris comprehenſionis:</s> <s xml:id="echoid-s26611" xml:space="preserve"> <lb/>& qui eſt propinquior illi, eſt manifeſtior remotiore ab illo:</s> <s xml:id="echoid-s26612" xml:space="preserve"> & ſimiliter eſt de forma perueniente in <lb/>neruũ communem, ex quo comprehendit uirtus ſenſitiua formas rerũ.</s> <s xml:id="echoid-s26613" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s26614" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1014" type="section" level="0" n="0"> <head xml:id="echoid-head815" xml:space="preserve" style="it">44. Puncto coniunctionis in axe communi exiſtente, certιßima fit uiſio: propinquè uerò illi <lb/>axi adhuc certa: remotius uerò minus certa. Alhazen 10 n 3.</head> <p> <s xml:id="echoid-s26615" xml:space="preserve">Sit linea connectens centra foraminũ uueę, quę a b:</s> <s xml:id="echoid-s26616" xml:space="preserve"> & ſit linea c e axis communis:</s> <s xml:id="echoid-s26617" xml:space="preserve"> pũctus quoq;</s> <s xml:id="echoid-s26618" xml:space="preserve"> <lb/>coniũctionis in ipſa linea c e ſit d, in quo concurrant axes a d & b d:</s> <s xml:id="echoid-s26619" xml:space="preserve"> & ſit medius pũctus concauita <lb/>tis nerui cõmunis punctũ h.</s> <s xml:id="echoid-s26620" xml:space="preserve"> Dico, quòd pũcto d exiſtente in linea c e, tũc certiſsima fit uiſio.</s> <s xml:id="echoid-s26621" xml:space="preserve"> Formę <lb/>enim uiſę perueniẽtes ad ſuperficieẽ uiſus ſunt tũc magis cõſimiles, eò qđ axibus cadẽtibus in cen-<lb/>tra foraminũ uueę, quę ſunt ſignata per puncta a & b, formę punctorũ circumſtantium punctum d, <lb/>diſtinctè & conſimiliter incidunt circa illa centra.</s> <s xml:id="echoid-s26622" xml:space="preserve"> Et quoniam axis communis, qui eſt e c, diuidit <lb/>lineam a b per æqualia in puncto c per 33 huius, & per 29 p 1, ideo quia linea connectens centra <lb/> <pb o="106" file="0408" n="408" rhead="VITELLONIS OPTICAE"/> foraminum uueæ eſt æquidiſtans lineæ r t, cónectenti centra foraminum gyrationis neruorũ con-<lb/>cauorum, ut patet ex pręmiſsis, & per 4 huius:</s> <s xml:id="echoid-s26623" xml:space="preserve"> unde per 31 t 1 huius <lb/> <anchor type="figure" xlink:label="fig-0408-01a" xlink:href="fig-0408-01"/> patet, quòd linea h c per æqualia diuiditlineam a b, & eſt perpendi <lb/>cularis ſuper illam:</s> <s xml:id="echoid-s26624" xml:space="preserve"> eſt ergo palàm per 4 p 1, quoniam axis a d eſt ę-<lb/>qualis axi b d, & angulus d a c æqualis angulo d b c:</s> <s xml:id="echoid-s26625" xml:space="preserve"> ſed & per 30 <lb/>huius anguli h a c & h b c ſunt æquales:</s> <s xml:id="echoid-s26626" xml:space="preserve"> & quoniam axis cõmunis, <lb/>qui eſt e c, pertingit ad h punctum mediũ concauitatis nerui com-<lb/>munis, ad quod formæ à punctis a & b diffunduntur:</s> <s xml:id="echoid-s26627" xml:space="preserve"> palàm per 26 <lb/>p 1, quoniam anguli c h a & c h b ſunt æquales.</s> <s xml:id="echoid-s26628" xml:space="preserve"> Idem quoq;</s> <s xml:id="echoid-s26629" xml:space="preserve"> accidit <lb/>in omnibus punctis, quibus incidunt lineæ radiales ipſis axibus <lb/>a d & b d propinquè, quæ ſunt æquales quaſi ad ſenſum, ut patet <lb/>per 40 huius:</s> <s xml:id="echoid-s26630" xml:space="preserve"> hæ enim lineæ radiales quaſi ęqualiter incidũt pun-<lb/>ctis æqualibus ſuperficiei nerui communis per 37 huius.</s> <s xml:id="echoid-s26631" xml:space="preserve"> Formæ i-<lb/>taq;</s> <s xml:id="echoid-s26632" xml:space="preserve"> pũctorum taliter uiſorum ſunt magis conſimiles:</s> <s xml:id="echoid-s26633" xml:space="preserve"> unde ſit tũc <lb/>uiſio certior.</s> <s xml:id="echoid-s26634" xml:space="preserve"> Sed cum punctus coniunctionis fuerit modicũ extra <lb/>communem axem, ut in puncto f, ſiue remotio illa fit ad partem ſi-<lb/>niſtram uel dextram, ſurſum uel deorſum, ſiue ad alias utcunq;</s> <s xml:id="echoid-s26635" xml:space="preserve">: tũc <lb/>adhuc duæ formæ, quę infiguntur duobus uiſibus, non multũ ha-<lb/>bent diuerſitatis:</s> <s xml:id="echoid-s26636" xml:space="preserve"> unde puncto formæ, cui duo axes infigũtur, ipſi <lb/>puncto h medio, ſcilicet puncto concauitatis nerui incidente, reſi-<lb/>dui puncti formæ rei uiſæ per lineas radiales uicinas axibus, 1 pſis <lb/>uiſibus incidentes, in concauitate nerui communis circa pũctum <lb/>h uniuntur, non tamen ſecũdum perfectionem prioris díſpoſitio-<lb/>nis:</s> <s xml:id="echoid-s26637" xml:space="preserve"> uidetur itaq;</s> <s xml:id="echoid-s26638" xml:space="preserve"> & tũc res certa uiſione, non tamen in gradu cer-<lb/>titudinis prioris.</s> <s xml:id="echoid-s26639" xml:space="preserve"> Cum uerò coniunctionis punctus fuerit remo-<lb/>tus extra communem axem, qui eſt c e, ut in puncto g, ad quamcun <lb/>que differentiam poſitionis hoc contingat:</s> <s xml:id="echoid-s26640" xml:space="preserve"> tũc adhuc punctus rei <lb/>uiſæ, in quo duo axes concurrunt, infigetur ipſi puncto h:</s> <s xml:id="echoid-s26641" xml:space="preserve"> ſed for-<lb/>mæ reſiduorum punctorum illius rei uiſæ infixæ in circuitu pun-<lb/>cti h, non recipient diſpoſitionem priorib.</s> <s xml:id="echoid-s26642" xml:space="preserve"> duabus ſimilẽ, neq;</s> <s xml:id="echoid-s26643" xml:space="preserve"> erit <lb/>illorum punctorum uiſio bene uerificata, ſed remanet minus certa.</s> <s xml:id="echoid-s26644" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s26645" xml:space="preserve"/> </p> <div xml:id="echoid-div1014" type="float" level="0" n="0"> <figure xlink:label="fig-0408-01" xlink:href="fig-0408-01a"> <variables xml:id="echoid-variables429" xml:space="preserve">h r s t a c b e g f d f y</variables> </figure> </div> </div> <div xml:id="echoid-div1016" type="section" level="0" n="0"> <head xml:id="echoid-head816" xml:space="preserve" style="it">45. Omne uiſum in puncto coniunctionis duorum axium uiſualium certius uidetur eo, quod <lb/>per radios axibus propinquos: & ſecundum remotionem ab axibus gradus certitudinis de-<lb/>creſcit. Ex quo patet, quòd puncta ſuperficiei rei uiſæ æqualiter diſt antia à puncto cõiunctionis, <lb/>ſimiliter uirtuti uiſiuæ offerentur. Alhazen 15 n 3.</head> <p> <s xml:id="echoid-s26646" xml:space="preserve">Quoniam enim, ut patet ք 43 huius, ſecũdũ oẽs lineas cuiuslibet pyramidis radialis fit certa com <lb/>prehẽſio formę uiſibilis à uiſu:</s> <s xml:id="echoid-s26647" xml:space="preserve"> magis aũt ſecũ lineas axi uiciniores, & maximè ք axem centrũ fo <lb/>raminis uueæ tranſeuntẽ:</s> <s xml:id="echoid-s26648" xml:space="preserve"> in pũcto aũt coniũctionis concurrũt duo axes ք 32 huius.</s> <s xml:id="echoid-s26649" xml:space="preserve"> Palàm ergo, cũ <lb/>uirtus duplicata ſit fortior ſui medietate, quòd in pũcto coniũctionis certior fit uiſio ſecũdũ totam <lb/>ſuperficiẽ rei uiſę, quę eſt baſis ambarũ pyramidũ uiſionis, & ſecũdũ proportionẽ dupliad duplũ, <lb/>quę eſt ſimpli ad ſimplũ.</s> <s xml:id="echoid-s26650" xml:space="preserve"> Secũdũ lineas uerò radiales, quæ ſunt propinquæ axibus, fit minus certa <lb/>uiſio quàm per axes:</s> <s xml:id="echoid-s26651" xml:space="preserve"> quoniam formę punctorũ peruenientes ad uirtutem ſenſitiuam, nõ perueni-<lb/>unt directè ad mediũ cõmunis nerui:</s> <s xml:id="echoid-s26652" xml:space="preserve"> unde non fit adeò perſectũ de illis iudiciũ, ut de formis perue <lb/>nientibus per ipſos axes.</s> <s xml:id="echoid-s26653" xml:space="preserve"> Secũdũ remotionem uerò illarũ linearũ ab axibus gradus certitudinis ui <lb/>ſionis decreſcit:</s> <s xml:id="echoid-s26654" xml:space="preserve"> quia cũ partes ſuperficiei rei uiſæ, quibus axes in cidũt, & partes illis proximæ ma-<lb/>nifeſtius uideantur per 43 huius:</s> <s xml:id="echoid-s26655" xml:space="preserve"> ſecũdũ partes remotiores illius ſuperficiei, quibus incidũt extre-<lb/>mæ lineæ longitudinis pyramidis radialis, eſt debiliſsima certitudo uiſionis:</s> <s xml:id="echoid-s26656" xml:space="preserve"> & ſecũdũ alias partes <lb/>medias fit media diſpoſitio certitudinis, ſecũdũ quod plus accedũt axibus, uel ſecũdũ quod ab illis <lb/>plus remouentur.</s> <s xml:id="echoid-s26657" xml:space="preserve"> Palàm ergo propoſitũ.</s> <s xml:id="echoid-s26658" xml:space="preserve"> Et per hoc patet corollariũ, quoniam in pũctis ſuperficiei <lb/>rei uiſæ æqualiter à pũcto coniũctionis diſtátibus eadẽ eſt ratio certitudinis uiſionis hinc & inde:</s> <s xml:id="echoid-s26659" xml:space="preserve"> <lb/>quoniam illarũ formæ æqualiter in ſuperficie ipſius uiſus, & ex conſequẽti in ſuperficie nerui com <lb/>munis ſemper figurantur.</s> <s xml:id="echoid-s26660" xml:space="preserve"> Patet ergo totum, quod proponebatur.</s> <s xml:id="echoid-s26661" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1017" type="section" level="0" n="0"> <head xml:id="echoid-head817" xml:space="preserve" style="it">46. Omne uiſum, in quo concurrunt duo axes uiſuales uel radij illis propinqui, uidetur ſem <lb/>per unum. Alhazen 14 n 3.</head> <p> <s xml:id="echoid-s26662" xml:space="preserve">Quoniam enim formę per axes radiales peruenientes ad uiſum, æqualiter incidunt uiſibus am-<lb/>bobus per 35 huius, & per 30 huius æqualiter perueniunt ad medium pũctum concauitatis nerui:</s> <s xml:id="echoid-s26663" xml:space="preserve"> <lb/>concurrunt ergo ambæ illæ formæ ad punctum unum, & una ipſarũ ſupponitur alteri, & fiunt for-<lb/>mauna.</s> <s xml:id="echoid-s26664" xml:space="preserve"> Et quoniam omnia uiſa nobis aſſueta ſemper ſunt oppoſita ambobus uiſibus, & ambo ui-<lb/>ſus aſpiciunt ad quodlibet illorum uiſibilium, propter quod duo axes duorũ uiſuum ſemper con-<lb/>currunt in uno puncto illorum uiſibilium per 32 huius:</s> <s xml:id="echoid-s26665" xml:space="preserve"> & poſitio radiorum reſiduorũ, qui circum-<lb/>incidunt communi pũcto ipſorũ, eſt poſitio conſimilis per 37 huius:</s> <s xml:id="echoid-s26666" xml:space="preserve"> maximè quando non differũt <lb/>in remotione à duobus axibus maxima differẽtia:</s> <s xml:id="echoid-s26667" xml:space="preserve"> propter hoc ergo quodlibet uiſorũ aſſuetorũ ui-<lb/>detur ambobus uiſib.</s> <s xml:id="echoid-s26668" xml:space="preserve"> unũ.</s> <s xml:id="echoid-s26669" xml:space="preserve"> Et quia, ut pręmiſſum eſt, patet per 37 huius, quoniá oẽs formę pũctorũ <lb/> <pb o="107" file="0409" n="409" rhead="LIBER TERTIVS."/> æqualiter circumſtantium puncta, quæ ſuperficiebus uiſuum incidunt ſecundum axes radiales, ad <lb/>puncta æqualiter circum ſtantia medium punctum nerui cõmunis conſimiliter pertingunt.</s> <s xml:id="echoid-s26670" xml:space="preserve"> Lineæ <lb/>uerò radiales propinquæ axibus uiſualibus, quia nõ multum obliquè incidũt uiſibus, ideo nõ mul-<lb/>tum obliquè refringuntur, quoniam ipſarum refractio eſt ſecundum angulos minores per 36 hu-<lb/>ius:</s> <s xml:id="echoid-s26671" xml:space="preserve"> directius ergo perueniunt ad cõcauitatẽ nerui:</s> <s xml:id="echoid-s26672" xml:space="preserve"> cõtingunt ergo ſe circa mediũ punctum cócaui <lb/>tatis nerui, & ſupponuntur ſibi adiouicem, ſiuntq́;</s> <s xml:id="echoid-s26673" xml:space="preserve"> forma una.</s> <s xml:id="echoid-s26674" xml:space="preserve"> Ethocproponebatur.</s> <s xml:id="echoid-s26675" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1018" type="section" level="0" n="0"> <head xml:id="echoid-head818" xml:space="preserve" style="it">47. Omne uiſum, in quo concurrit axis communis, & unus axium uiſualium, comprehen-<lb/>ditur ſemper unum.</head> <p> <s xml:id="echoid-s26676" xml:space="preserve">Axis enim cõmunis adiuuat certitudinem cõprehenſionis, & axis uiſualis unicus unam tantùm <lb/>formam regulariter diſpoſitam imprimit medio puncto nerui communis:</s> <s xml:id="echoid-s26677" xml:space="preserve"> uidetur ergo una tãtùm <lb/>forma, quia tunc nõ fit refractio alterius formæ ad aliquam partem nerui diſtinctam ſecundum par <lb/>tem uel ſecundum remotionem.</s> <s xml:id="echoid-s26678" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s26679" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1019" type="section" level="0" n="0"> <head xml:id="echoid-head819" xml:space="preserve" style="it">48. Nullum uiſorum ſimultotum æqualiter uidetur. Euclides in præfatione & 1 the. opti-<lb/>corum. Alhazen 16 n 3.</head> <p> <s xml:id="echoid-s26680" xml:space="preserve">Quoniam enim, ſiue aliquod uifum exiſtat in axe communi, ſiue extraillum, ſemper punctum e-<lb/>ius, cui incidunt axes uiſuales, certius uidetur, quàm puncta, quibus incidunt radij propinqui:</s> <s xml:id="echoid-s26681" xml:space="preserve"> & il <lb/>la puncta certius uidentur, quàm puncta, quibus incidunt radij remoti per 45 huius:</s> <s xml:id="echoid-s26682" xml:space="preserve"> pater, quòd <lb/>nullum uiſum totum ſimul æqualiter uidecur.</s> <s xml:id="echoid-s26683" xml:space="preserve"> Cum enim omnia puncta ipſius communiter per o-<lb/>mnes tres axes, uel ſaltem per duos uifuales motn oculi tranſcurſa fuerint:</s> <s xml:id="echoid-s26684" xml:space="preserve"> tũc ſolùm æqualiter eſt <lb/>totum uiſum:</s> <s xml:id="echoid-s26685" xml:space="preserve"> quoniam tunc forma cuiuslibet ſui puncti infigetur puncto medio concauitatis ner-<lb/>ui, & erit ſemper noua diſpoſitio totius form æ circa punctum illud:</s> <s xml:id="echoid-s26686" xml:space="preserve"> magis ergo æqualiter perpen-<lb/>detur tunc partium æqualitas adinuicem in omnibus diſpoſitionibus ſuis:</s> <s xml:id="echoid-s26687" xml:space="preserve"> tunc ergo totares ęqua <lb/>liter uidebitur:</s> <s xml:id="echoid-s26688" xml:space="preserve"> nullus autem motus eſt in inſtanti, ſed ſolùm in tempore:</s> <s xml:id="echoid-s26689" xml:space="preserve"> palàm ergo, quòd nullũ <lb/>uiſorum ſimul totum æqualiter uidebitur:</s> <s xml:id="echoid-s26690" xml:space="preserve"> ſed bene eſt poſsibile ipſum totum ſimul uideri inæ qua <lb/>liter:</s> <s xml:id="echoid-s26691" xml:space="preserve"> quoniam omnia puncta formæ oppoſitæ uifui, à quibus line æ rectæ poſſunt produciad ui-<lb/>ſum, ſimùl multiplicantur ad uiſum, quamuis ſecundum diuerſitatem angulorum diuerſimodè ſe-<lb/>cundum diuerſas partes uideantur:</s> <s xml:id="echoid-s26692" xml:space="preserve"> parua tamen corpora & propinquarum diametrorum æquali-<lb/>us uidentur, quàm corpora diametrorum maiorum:</s> <s xml:id="echoid-s26693" xml:space="preserve"> remotiores enim partes à puncto coniunctio-<lb/>nis non adeò bene certificantur, ut propinqua per 45 huius:</s> <s xml:id="echoid-s26694" xml:space="preserve"> & ſi uiſum fuerit unius coloris unifor-<lb/>me, minus accidic in eo in æqualιtatis, quàm ſi ruerit plurium colorum, autſi fuerit in ipſo lineatio, <lb/>aut pictura, aut aliæ ſubtiles intentiones:</s> <s xml:id="echoid-s26695" xml:space="preserve"> tunc enim forma extremorum erit magis dubitabilis, & <lb/>non bene certificata:</s> <s xml:id="echoid-s26696" xml:space="preserve"> hæ enim comprehenduntur per lineas radiales remotas ab axe.</s> <s xml:id="echoid-s26697" xml:space="preserve"> Patet ergo <lb/>propoſitum.</s> <s xml:id="echoid-s26698" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1020" type="section" level="0" n="0"> <head xml:id="echoid-head820" xml:space="preserve" style="it">49. Impoßibile ect plura ſimul æqualiter uideri.</head> <p> <s xml:id="echoid-s26699" xml:space="preserve">Quamuis enim uiſus quãdoq;</s> <s xml:id="echoid-s26700" xml:space="preserve"> eodem tempore opponitur multis uiſibilibus diuerſi coloris, in-<lb/>ter quorum quodlibet & uiſum produci poſſuntline æ rectæ in aere cõtinuato medio inter ea & ui <lb/>ſum, perueniantq́;</s> <s xml:id="echoid-s26701" xml:space="preserve"> formæ lucis & coloris, quę ſunt in rebus uiſis, ad ſuperficiem uiſus in eodẽ tem-<lb/>pore, & forma cuιuslibet ipſarum ad quamlibet partem ſuperficiei uiſus, propter earum directam <lb/>oppoſitionem:</s> <s xml:id="echoid-s26702" xml:space="preserve"> & licet uidens uideat in eodem tempore uiſibilia diuerſi coloris oppoſita uiſui, & <lb/>ſic in tota ſuperficie uiſus ſint multa lumina diuerſa & multi colores diuerſi, quorum quilibetim-<lb/>plet ſuperficiem uiſus ſibi oppoſitam, proutincidit perpendiculariter uel obliquè:</s> <s xml:id="echoid-s26703" xml:space="preserve"> tamen, ut patet <lb/>per 17 huius, non fit diſtincta uiſio, niſi ſolùm ſecundum perpendiculares lineas à punctis rei uiſæ <lb/>ad oculi ſuperficiem productas:</s> <s xml:id="echoid-s26704" xml:space="preserve"> & ſecundum bæc diſtinguuntur form æ ſecundum diſtinctionem <lb/>partium ſuperficiei uiſus, in quas ſolùm incidunt perpendiculares:</s> <s xml:id="echoid-s26705" xml:space="preserve"> & licetſic perueniant ad ſuper-<lb/>ficiem uiſus form æ admixtæ luminibus & coloribus diuerſis, uiſus tamen comprehendit omnes <lb/>form as ſecundum ipſarum proprietatem:</s> <s xml:id="echoid-s26706" xml:space="preserve"> non eſt ergo impoſsibile plura ſimul uidere, ſed in æqua-<lb/>liter & indiſtin ctè.</s> <s xml:id="echoid-s26707" xml:space="preserve"> Nam licet, ut patet per 17 huius, humor glacialis ſentiat formam unius rei fecun <lb/>dum ſuum eſſe, & figuram ordinatam in ſui ſuperficie ſecundum ordinem, quem habet in ſuperfi-<lb/>cie rei uiſæ:</s> <s xml:id="echoid-s26708" xml:space="preserve"> extrà tamen poterit etiam ſentire in illa diſpoſitione formas aliarum rerum uiſarũ, prę <lb/>ter illam rem uiſam ex pyramidibus diſting uentibus ex ſua ſuperficie alias huius rei partes:</s> <s xml:id="echoid-s26709" xml:space="preserve"> & pote <lb/>rit ſentire formam cu:</s> <s xml:id="echoid-s26710" xml:space="preserve"> uslibet illarum rerum uiſarum ſecũdum ſuum eſſe, & ſentire ſitus earum ad-<lb/>inuicem, non tamen æqualiter:</s> <s xml:id="echoid-s26711" xml:space="preserve"> ſed perfectius illud, quod uidet ſecundum pyramidem, cuius axis <lb/>incidit per centrum circuli uue æ ipſi centro uiſus, minus uerò perſectè alia, quorum pyramidum <lb/>axes incidunt ſecundum alia puncta ſuperficiei dicti circuli, ut patet per 43 huius:</s> <s xml:id="echoid-s26712" xml:space="preserve"> illorum enim o-<lb/>mnium axes ſunt longiores, etiamſi ab eadem diſtantia procedant.</s> <s xml:id="echoid-s26713" xml:space="preserve"> Aſpiciens itaque quando fuerit <lb/>oppoſitus multis rebus uiſibilibus, & uiſus eius fuerit quietus:</s> <s xml:id="echoid-s26714" xml:space="preserve"> inueniet rem oppoſitam medio ſui <lb/>uiſus manifeſtiorem illis, quæ ſunt à parte laterum illius medij, & quod eſt propin quius medio, e-<lb/>rit manifeſtius, & quod eſt remotius, erit minus manifeſtum, ut hæc omnia patent per 43 huius.</s> <s xml:id="echoid-s26715" xml:space="preserve"> E ſt <lb/>ergo impoſsibile, plura ſimul æqualiter uideri:</s> <s xml:id="echoid-s26716" xml:space="preserve"> quoniam impoſsibile eſt axem pyramidis radialis <lb/>tranſeuntem per centrum uueæ ſimul pluribus punctis, ne dum ſuperficiebus, incidere per 2011 <lb/>huius.</s> <s xml:id="echoid-s26717" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s26718" xml:space="preserve"/> </p> <pb o="108" file="0410" n="410" rhead="VITELLONIS OPTICAE"/> </div> <div xml:id="echoid-div1021" type="section" level="0" n="0"> <head xml:id="echoid-head821" xml:space="preserve" style="it">50. Interpoſitis ſibi diuerſis uiſibilibus, remotiorum quando ſecundum aliquìd uiſio impe-<lb/>ditur. Alhazen 5 n 3.</head> <p> <s xml:id="echoid-s26719" xml:space="preserve">Exempli cauſa, ſint duo puncta n & m centra duorũ uiſuũ, & ſitr punctũ cuiuſdam rei uiſę, quæ <lb/>ſit l o, remotior ab ambobus uiſibus quàm ſit res uiſa, quæ ſit b k c:</s> <s xml:id="echoid-s26720" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0410-01a" xlink:href="fig-0410-01"/> in cuius puncto k concurrát ambo axes uiſuales, qui ſint m k & n k:</s> <s xml:id="echoid-s26721" xml:space="preserve"> <lb/>ſitq́;</s> <s xml:id="echoid-s26722" xml:space="preserve"> punctum r taliter poſitũ, utipſum protractis axibus n k ad pun <lb/>ctum q, & m k ad punctũ h, intercipiatur inter axes, nihilq́;</s> <s xml:id="echoid-s26723" xml:space="preserve"> eius ca-<lb/>piatur per interpoſitionem rei uiſæ, quę eſt b c:</s> <s xml:id="echoid-s26724" xml:space="preserve"> ſit aũt uiſibile e d re-<lb/>motius quàm ſit ipſum b c, & propinquius pũcto r, inter duos axes <lb/>taliter diſpoſitum:</s> <s xml:id="echoid-s26725" xml:space="preserve"> ita quòd lineæ n b & m c protractæ, & cõcurren-<lb/>tes in ipſo p, aliquã partẽ eius intercipiát, quæ ſit f g:</s> <s xml:id="echoid-s26726" xml:space="preserve"> lineæ uerò m p <lb/>& n p interſecantes ſe in pucto p protractæ cõtingant peripheriã <lb/>corporis, in quo eſt punctum r, in punctis l & o:</s> <s xml:id="echoid-s26727" xml:space="preserve"> ſit uerò a quoddam <lb/>uiſum proximum uiſui, cadens inter axes m k & n k.</s> <s xml:id="echoid-s26728" xml:space="preserve"> Dico, quando <lb/>uiſus cõprehendit in eadem hora in ſimul formas uiſibilium, quæ <lb/>funt b c & e d & r, quòd quandoq;</s> <s xml:id="echoid-s26729" xml:space="preserve"> impeditur ſecũdum aliquid uiſio <lb/>ipſius e d:</s> <s xml:id="echoid-s26730" xml:space="preserve"> quoniã impeditur ſecundũ ſui partem, quæ eſt fg, quę cũ <lb/>ſit obumbrata uiſui per interpoſitionẽ uiſibilis, quod eſt b c:</s> <s xml:id="echoid-s26731" xml:space="preserve"> patet, <lb/>quòd forma illius partis nõ perueniet ad uiſum, neq;</s> <s xml:id="echoid-s26732" xml:space="preserve"> ſeruabitur in <lb/>neruo cómuni:</s> <s xml:id="echoid-s26733" xml:space="preserve"> forma uerò uiſibilis remotioris, quod eſt l o, in quo <lb/>eſt punctũ r, quoniá ipſum cadit inter lineas n b & m c, ſecantes ſe <lb/>in puncto p, quæ productę ultra punctum p, ſuis terminis l & o inci <lb/>dunt:</s> <s xml:id="echoid-s26734" xml:space="preserve"> patet quòd perueniet ad uiſum, nõ impediente uiſibili b c:</s> <s xml:id="echoid-s26735" xml:space="preserve"> ꝗa <lb/>tamen in nullo eius puncto cõcurrunt axes uiſuales, forma eius ui-<lb/>debitur inordinata ſecundũ ſitũ earundẽ partium ipſius formę, quę <lb/>ſi bi directè nõ ſuperponentur, ut oſtenſum fuit in 37 huius:</s> <s xml:id="echoid-s26736" xml:space="preserve"> ergo e-<lb/>runt inordinatæ ſecundũ remotionẽ à puncto medio nerui cõmu-<lb/>nis, quæ remotio erit hinc inde in æqualis, propter diuerſitatem in-<lb/>cidentiæ ip ſarũ linearũ, per quas adueniunt eadẽ puncta formarũ, <lb/>ut ſunt lineæ m l & n l, reſpectu formę puncti l, & lineæ m o & n o, <lb/>reſpectu form æ puncti o:</s> <s xml:id="echoid-s26737" xml:space="preserve"> pars tamen uniuerſi, qu æ attẽditur ſecun-<lb/>dum dextrã uel ſiniſtrã, ſurſum uel deorſum partium ipſius form æ nõ mutatur.</s> <s xml:id="echoid-s26738" xml:space="preserve"> Viſum enim b c cũ <lb/>ſit minus uiſo l o, in quo eſt punctũ r, quando in puncto k rei uiſæ b c cõiunguntur duo axes m k & <lb/>n k:</s> <s xml:id="echoid-s26739" xml:space="preserve"> tũc forma uiſi b c fit in duobus locis duorum uiſuum conſimilis poſitionis, & forma uiſi, quòd <lb/>eſt l o, diuerſificabitur ſecundum ſitum partiũ ſuæ formę, & ſecundum remotionẽ in æqualẽ à pun-<lb/>cto medio nerui cõmunis:</s> <s xml:id="echoid-s26740" xml:space="preserve"> quoniã eſt magna diuerſitas in angulis refractionis ſuarum partialium <lb/>formarum, ſicut & in angulis incidentiæ earundẽ, ut hoc patere poteſt per 36 huius:</s> <s xml:id="echoid-s26741" xml:space="preserve"> nõ tamen erit <lb/>errorin parte uniuerſi:</s> <s xml:id="echoid-s26742" xml:space="preserve"> quia form æ partiũ ſuo ordine diſponẽtur, ut ſuntin re, & res uidebitur una:</s> <s xml:id="echoid-s26743" xml:space="preserve"> <lb/>quòd nõ accidit in forma uiſi, ſcilicetipſius a, quod ꝓpinquius eſt uiſui, ſi ipſum paruę fuerit quan-<lb/>titatis, & nõ ſit in illorum corporũ poſitione differẽtia ſenſibilis, ita quòd corpus a cadat inter axes <lb/>m k & n k.</s> <s xml:id="echoid-s26744" xml:space="preserve"> Quando itaq;</s> <s xml:id="echoid-s26745" xml:space="preserve"> ambo uiſus ambas res uiſas, in quibus ſunt r & d e, cõprehendunt, & quan <lb/>do duo axes fixi ſunt in uiſo b c, ſecundũ loca nõ obumbrata inſtituuntur illarũ rerum uiſarũ d e & <lb/>l o formæ in duobus locis duorum uiſuũ, & fiunt cõſimilis poſitionis in parte uniuerſi, & nõ in re-<lb/>motione à puncto ιnedio nerui cõmunis, aut nõ oẽs partes earum erunt cõfimilis poſitionis in re-<lb/>motione à duobus axibus, nec forma earum erit certificata.</s> <s xml:id="echoid-s26746" xml:space="preserve"> De uiſo uerò a, quod eſt proximum <lb/>uiſibus, quoniã ipſum cadit inter axes m k & n k, & eſt propinquius uiſui, quia non figuntur in ipſo <lb/>axes, poteſt fieri poſitio eius, in reſpectu amborũ uiſuum, diuerſa in parte ipſius uniuerſi, ita, ut nec <lb/>uideatur ad ſiniſtrã nec ad dextrã, quoniã forma ipſius quantum eſt de ſe, ad nullã partium uniuerſi <lb/>ſecundum reſpectum puncti medij ipſius nerui concaui, cui axes uiſuales incidunt, ordinatur.</s> <s xml:id="echoid-s26747" xml:space="preserve"> Sic <lb/>ergo uiſu exiſtente fixo, interpoſitis ſibi diuerſis uiſibilibus, remotiorum quandoq;</s> <s xml:id="echoid-s26748" xml:space="preserve"> ſecundum ali-<lb/>quid uiſio impeditur, ut patet.</s> <s xml:id="echoid-s26749" xml:space="preserve"> Cum autem uiſus fuerint moti, & axes fuerint coniuncti in unoquo-<lb/>que uiſibiliũ cõprehenſorũ in ſimul:</s> <s xml:id="echoid-s26750" xml:space="preserve"> tunc formæ omniũ uiſibiliũ cõprehẽdentur ſimul in ambobus <lb/>uiſibus cõſimiles in parte & remotione, & cõprehendentur ſecundum modũ ſuę certitu dinis for-<lb/>mæ uniuſcuiuſq;</s> <s xml:id="echoid-s26751" xml:space="preserve"> uiſibiliũ.</s> <s xml:id="echoid-s26752" xml:space="preserve"> Huius aũtrei totius ratio eſt hæc, quia certitudo uiſionis fit ſecundum <lb/>axes, & uiſio fit per mulriplicationẽ formæ uiſibilis in uiſum, quę nonnunquã tunc per corpus ιn-<lb/>terpoſitum im peditur, cũ linea multiplicationis formæ aliã ſuperficiem corporis medij oppoſitam <lb/>uiſui ali qualiter attingit.</s> <s xml:id="echoid-s26753" xml:space="preserve"> Et hoc eſt quod uolebamus.</s> <s xml:id="echoid-s26754" xml:space="preserve"/> </p> <div xml:id="echoid-div1021" type="float" level="0" n="0"> <figure xlink:label="fig-0410-01" xlink:href="fig-0410-01a"> <variables xml:id="echoid-variables430" xml:space="preserve">n m a b k c e d f g p h q l r o</variables> </figure> </div> </div> <div xml:id="echoid-div1023" type="section" level="0" n="0"> <head xml:id="echoid-head822" xml:space="preserve" style="it">51. Omnis uiſio fit uelper aſpectum ſimplicẽ, uelper intuitionẽ diligentẽ. Alhazen 64 n 2.</head> <p> <s xml:id="echoid-s26755" xml:space="preserve">Aſpectum primũ ſimplicẽ dicimus illũ actum, quo primò ſimpliciter recipitur in oculi ſuperficio <lb/>forma rei uiſę:</s> <s xml:id="echoid-s26756" xml:space="preserve"> intuitionẽ uerò dicimus illũ actum, quo uiſus uerã cõprehenſionem formę rei dili-<lb/>genter perſpicien do perquirit, nõ cõtentus ſim plici receptione, ſed profunda indagine.</s> <s xml:id="echoid-s26757" xml:space="preserve"> Viſus itaq;</s> <s xml:id="echoid-s26758" xml:space="preserve"> <lb/>per aſpectum ſimplicẽ comprehendit intentiones manifeſtas, quę ſuntin rebus, nec certificat illas:</s> <s xml:id="echoid-s26759" xml:space="preserve"> <lb/> <pb o="109" file="0411" n="411" rhead="LIBER TERTIVS."/> per intuitionem uerò conſiderat oẽs intẽtiones partium formę uiſæ occultas aſpectui, & certificat <lb/>omnes diſpoſitiones illius formę uiſę.</s> <s xml:id="echoid-s26760" xml:space="preserve"> Et quia aſpectus ſimplex poteſt eſſe ſine intuitione, quamuis <lb/>intuitio non poſsit eſſe ſine ſimpliciaſpectu:</s> <s xml:id="echoid-s26761" xml:space="preserve"> patet, quòd omnis uiſio aut fit per unum iſtorum mo-<lb/>dorum, aut per alium.</s> <s xml:id="echoid-s26762" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s26763" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1024" type="section" level="0" n="0"> <head xml:id="echoid-head823" xml:space="preserve" style="it">52. Aſpectu ſimplici ſecundum totam pyramidem uiſualem exiſtente poßibili: intuitio fit <lb/>ſolùm ſecundum incidentiam axis pyr amidis uiſualis. Alhazen 65 n 2.</head> <p> <s xml:id="echoid-s26764" xml:space="preserve">Quoniam enim, ut patet per pręmiſſam, aſpectus ſimplex eſt folũ receptio formę ſenſibilis in ſu-<lb/>perficie uiſus:</s> <s xml:id="echoid-s26765" xml:space="preserve"> palàm, quòd ipſa fit ſecũdum totam pyramidẽ uiſualẽ:</s> <s xml:id="echoid-s26766" xml:space="preserve"> quęlibet enim perpẽdiculariũ <lb/>ſiue linearũ radialium illam pyramidẽ conſtituentium per 17 huius adducit aliquam formam pun-<lb/>cti ſuperficiei rei uiſibilis, quã tũc aſpicit uiſus:</s> <s xml:id="echoid-s26767" xml:space="preserve"> quia uerò intuitio certificat ueritatẽ formarũ com-<lb/>prehẽſarum:</s> <s xml:id="echoid-s26768" xml:space="preserve"> certificatio uerò omnium formarum uiſibilium plus fit per axes pyramidum uiſualiũ, <lb/>quàm per aliquã aliarum linearum illius pyramidis per 43 huius:</s> <s xml:id="echoid-s26769" xml:space="preserve"> patet, quòdintuitio fit ſolùm per <lb/>incidentiam illius axis.</s> <s xml:id="echoid-s26770" xml:space="preserve"> Cũ ergo uiſus fuerit fixus oppoſitus alicui rei uiſæ, quę fuerit alicuius quan <lb/>titatis:</s> <s xml:id="echoid-s26771" xml:space="preserve"> & illus, quod opponitur medio uiſus ex illa re uiſa, fuerit per axem uiſualẽ aut propeillum:</s> <s xml:id="echoid-s26772" xml:space="preserve"> <lb/>tunc erit ipſum, quod eſt in axe, uel quod approximat axi, manifeſtius reſiduis partibus rei uiſæ.</s> <s xml:id="echoid-s26773" xml:space="preserve"> Si <lb/>itaq;</s> <s xml:id="echoid-s26774" xml:space="preserve"> uidens uoluerit certificari de forma totali rei uiſæ, mouebit ambos uiſus, donec medium eius <lb/>opponatur cuilibet partium, uel punctorum ſuperficiei rei uiſę ſibi oppoſitę:</s> <s xml:id="echoid-s26775" xml:space="preserve"> & tunc, quia ambo <lb/>axes radiales per 32 huius incidẽt unicuiq;</s> <s xml:id="echoid-s26776" xml:space="preserve"> punctorum, fiet hoc modo intuitio completa totins for <lb/>mæ.</s> <s xml:id="echoid-s26777" xml:space="preserve"> Quando enim uiſus fuerit oppoſitus rei uiſę, tunc ſentiens comprehẽdet totam formam com-<lb/>prehẽſione qualicunq;</s> <s xml:id="echoid-s26778" xml:space="preserve"> per 43 huius, & partẽ, quę eſt apud extremum axis, comprehẽdet uera com-<lb/>prehẽſione:</s> <s xml:id="echoid-s26779" xml:space="preserve"> deinde mutatis axibus ad aliud punctum, tunc idẽ punctum uerius comprehẽdetur, & <lb/>tunc cum hoc tota forma prius comprehẽſa comprehẽdetur ſecundò, & etiam ille punctus, in quo <lb/>prius fixi fuerũt axes, & cum axes mutabũtur ad pũctum tertium, fiet tertiò cõprehẽſio totius for-<lb/>mę, & etiã illorum pũctorum, quibus prius axes incidebãt, & ita ſecũdum numerũ pũctorum, qui-<lb/>bus incidũt axes, numeratur comprehẽſio totius formę:</s> <s xml:id="echoid-s26780" xml:space="preserve"> ſemper tamẽ punctus, cui axes incidunt, <lb/>certius alijs punctis comprehẽdetur.</s> <s xml:id="echoid-s26781" xml:space="preserve"> Sic ergo intuens per motum axium cõprehẽdit certitudinem <lb/>cuiuslibet puncti rei uiſę, & inſuper reiterat frequẽtationẽ comprehẽſionis totius formæ ſecũdum <lb/>numerum punctorum, quibus incidunt ip ſi axes.</s> <s xml:id="echoid-s26782" xml:space="preserve"> Apparet ergo uiſui tunc omne id, quod poſsibile <lb/>eſt apparere in forma illius rei uiſę, & nõ certificabitur forma rei uiſę, niſi poſt motus uiſus ſecundũ <lb/>ſuos axes radiales ſuper oẽs partes uel puncta ſuperficiei rei uiſę:</s> <s xml:id="echoid-s26783" xml:space="preserve"> nec enim intẽtiones ſubtiles, quę <lb/>ſunt in re uiſa, apparẽt uiſui niſi per motum uiſus, & per tranſitum axis, aut radialium linearũ, quæ <lb/>ſunt prope ipſum, ſuper quamlibet partium rei uiſę.</s> <s xml:id="echoid-s26784" xml:space="preserve"> Et etiam ſires fuerit in fine paruitatis, & fuerit <lb/>oppoſita uiſui:</s> <s xml:id="echoid-s26785" xml:space="preserve"> non intuebitur illã uiſus intuitione perfecta, niſi donec moto uiſu axis radialis tran <lb/>ſiuerit per oẽs particulas uel puncta illius rei.</s> <s xml:id="echoid-s26786" xml:space="preserve"> Sic ergo fit ſolùm intuitio ſecundum axis pyramidis <lb/>radialis incidẽtiam, quamuis aſpectus ſimplex fiat ſecundum omnes lineas radiales totius pyrami-<lb/>dis uiſualis.</s> <s xml:id="echoid-s26787" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s26788" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1025" type="section" level="0" n="0"> <head xml:id="echoid-head824" xml:space="preserve" style="it">53. Axis radialis in toto motu ipſius oculi ſemper manet fixus in ſuo ſitu: quoniam ille mo-<lb/>tus oculi eſt in ſenſibilis uelocitatis. Alhazen 42 n 2.</head> <p> <s xml:id="echoid-s26789" xml:space="preserve">Motus enim axis ſuper partes rei uiſæ nõ eſt ք gyrationẽ axis à loco cẽtri ipſius uiſus, ſed ք mo-<lb/>tum eius ք ſe ſuper partes rei uiſæ:</s> <s xml:id="echoid-s26790" xml:space="preserve"> patet enim ք 25 & 32 huius, quòd linea axis extẽditur rectè uſq;</s> <s xml:id="echoid-s26791" xml:space="preserve"> <lb/>ad locum gyrationis nerui, ſuper quẽ componitur oculus:</s> <s xml:id="echoid-s26792" xml:space="preserve"> & quòd ſitus eius à uiſun nõ mutatur, ſed <lb/>cum totus oculus mouetur in oppoſitione rei uiſæ, & medium oculi, in quo eſt ſenſus uiſus, oppo-<lb/>nitur cuilibet partium rei uiſæ, tunc axis trãſit ք quamlibet partium rei uiſę:</s> <s xml:id="echoid-s26793" xml:space="preserve"> & ſecundum iſtum mo <lb/>dum tota forma cuiuslibet partis rei uiſæ extẽditur ad uiſum ſemper ſecundum rectitudinẽ axis:</s> <s xml:id="echoid-s26794" xml:space="preserve"> & <lb/>erit gyratio axis immutabilis à loco ſuo, reſpectu omnium partium & tunicarum oculi:</s> <s xml:id="echoid-s26795" xml:space="preserve"> ſed circum <lb/>gyrabitur axis in concauo oſsis cũ motu totius oculi.</s> <s xml:id="echoid-s26796" xml:space="preserve"> Et cũ uiſus uoluerit intueri rẽ uiſam, & incœ <lb/>perit intueri in extremitatẽ rei uiſæ:</s> <s xml:id="echoid-s26797" xml:space="preserve"> erit tunc extremũ axis ſuper extremitatẽ rei uiſæ, eritq́;</s> <s xml:id="echoid-s26798" xml:space="preserve"> in diſ-<lb/>poſitione maior pars totius rei uiſæ in parte ſuperficiei uiſus declinante aut obliqua ab axe ad aliã <lb/>partẽ, pręter partẽ, ſuper quam eſt axis:</s> <s xml:id="echoid-s26799" xml:space="preserve"> quoniã forma eius erit in medio uiſus & in loco axis, eritq́;</s> <s xml:id="echoid-s26800" xml:space="preserve"> <lb/>reſiduum formæ obliquum ad aliam partẽ ab axe:</s> <s xml:id="echoid-s26801" xml:space="preserve"> & cũ uiſus poſt illam diſpoſitionẽ mouebiturſu-<lb/>per aliquã diametrum rei uiſæ, trãsferetur axis ad partẽ ſequẽtẽ illã partẽ rei uiſæ, & erit forma pri-<lb/>mæ partis declinans ad locum alium oppoſitũ loco, ad quẽ mouetur axis, & nõ ceſſabit forma de-<lb/>clinare, quamdiu mouetur axis ſuperillam diametrum, quouſq;</s> <s xml:id="echoid-s26802" xml:space="preserve"> axis perueniat ad ultimũ illius dia <lb/>metri rei uiſæ, quę eſt pars alterius rei uiſæ:</s> <s xml:id="echoid-s26803" xml:space="preserve"> & fic erit forma totius rei uiſæ in iſta diſpoſitione obli-<lb/>qua uiſui & puncto oppoſito ipſi axi, etiam cui prius fuit obliqua, axe radiali in alijs punctis diuer-<lb/>ſis incidẽte, pręter quam ultima pars & extrema ipſius rei uiſæ, quę remanebitſuper axem & in me-<lb/>dio uiſus:</s> <s xml:id="echoid-s26804" xml:space="preserve"> & axis in iſto toto motu erit fixus in ſuo ſitu, quantum ad pertranſitum uniformẽ omni-<lb/>um tunicarum oculi.</s> <s xml:id="echoid-s26805" xml:space="preserve"> Patet ergo illud, quod proponebatur.</s> <s xml:id="echoid-s26806" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1026" type="section" level="0" n="0"> <head xml:id="echoid-head825" xml:space="preserve" style="it">54. Axis in motu intuitionis nunquam fit baſis anguli, quem reſpicit ſuperficies rei uiſæ, <lb/>ne ſemper ſecat angulum, quem reſpicit aliqua diametrorum rei uiſæ. Alhazen 43 n 2.</head> <p> <s xml:id="echoid-s26807" xml:space="preserve">Quia enim iam oſtẽſum eſt in præ cedẽte theoremate, quòd axis in toto motu oculi ad intuendũ <lb/> <anchor type="figure" xlink:label="fig-0411-01a" xlink:href="fig-0411-01"/> <pb o="110" file="0412" n="412" rhead="VITELLONIS OPTICAE"/> ſemper manet fixus:</s> <s xml:id="echoid-s26808" xml:space="preserve"> ſi ergo axis fieret baſis anguli, quẽ reſpicit ſuperficies rei uiſę:</s> <s xml:id="echoid-s26809" xml:space="preserve"> oporteret im mo <lb/>tas remanere lineas illũ angulũ cõtinẽtes, & moueri axem:</s> <s xml:id="echoid-s26810" xml:space="preserve"> hoc autẽ nõ eſſet poſsibile, niſi quando <lb/>axis moueretur per ſe, toto oculo quieſcẽte:</s> <s xml:id="echoid-s26811" xml:space="preserve"> & quia hoc eſt impoſsibile per præcedẽtẽ, totus enim <lb/>oculus mouetur apud intuitionẽ, & axis mouetur ք motũ eius, & moto axe, mouẽtur oẽs lineæ cõ-<lb/>tinẽtes angulum pyramidis, & tota pyramis uariato axe uariatur.</s> <s xml:id="echoid-s26812" xml:space="preserve"> Incidẽte enim axe radiali diuerſis <lb/>punctis ſuperficiei rei uiſę, licetidẽ remaneat uertex pyramidis, & etiam eadẽ baſis ſit:</s> <s xml:id="echoid-s26813" xml:space="preserve"> uariato ta-<lb/>mẽ axe, cauſſatur ſemper noua pyramis, quamuis uideatur ſemper una:</s> <s xml:id="echoid-s26814" xml:space="preserve"> ideo quia motus oculi eſt <lb/>inſenſibilis uelocitatis.</s> <s xml:id="echoid-s26815" xml:space="preserve"> Per hunc itaq;</s> <s xml:id="echoid-s26816" xml:space="preserve"> motum comprehendit uiſus quodlibet punctum ſuperficiei <lb/>rei uiſæ uiſui medio oppoſitum, in puncto ſcilicet axis, & per hunc motum mouetur forma rei uiſæ <lb/>ad ipſam ſuperficiẽ uiſus, & mutatur pars ſuperficiei uiſus, in qua prius fuit forma:</s> <s xml:id="echoid-s26817" xml:space="preserve"> quoniam forma <lb/>rei uiſæ apud motum axis erit in una parte ſuperficiei uiſus poſt aliam partẽ ſuperficiei uiſus.</s> <s xml:id="echoid-s26818" xml:space="preserve"> Quo-<lb/>tiens enim com prehẽderit uirtus ſentiens partẽ rei uiſæ, quæ eſt apud extremum axis, totiens com <lb/>prehẽdet cum hoc totam ſuperficiẽ rei uiſę, & comprehendettotam illam partẽ ſuperficiei uiſus, in <lb/>qua peruenit forma totius rei uiſæ, quę ſemper eſt alia & alia:</s> <s xml:id="echoid-s26819" xml:space="preserve"> quãdiu itaq;</s> <s xml:id="echoid-s26820" xml:space="preserve"> axis caditin aliquod pun <lb/>ctorum diametri rei uiſæ, non terminantium ipſam diametrum, tunc axis diuidit angulum, cui in <lb/>cẽtro uiſus ſubtẽditur illa diameter:</s> <s xml:id="echoid-s26821" xml:space="preserve"> ſed cum incidit ipſi termino diam etri, tunc ipſe axis fit una li-<lb/>nearum cõtinẽtium illum angulum.</s> <s xml:id="echoid-s26822" xml:space="preserve"> Non ergo ſecat ſemper illum angulum.</s> <s xml:id="echoid-s26823" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s26824" xml:space="preserve"/> </p> <div xml:id="echoid-div1026" type="float" level="0" n="0"> <figure xlink:label="fig-0411-01" xlink:href="fig-0411-01a"> <image file="0411-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0411-01"/> </figure> </div> </div> <div xml:id="echoid-div1028" type="section" level="0" n="0"> <head xml:id="echoid-head826" xml:space="preserve" style="it">55. Neceſſe eſt omnem uiſionem, quæ fit aſpectu ſimplici, fieri in ſtanti.</head> <p> <s xml:id="echoid-s26825" xml:space="preserve">Si enim fiat aſpectus fimplex in tempore, quantumcunq;</s> <s xml:id="echoid-s26826" xml:space="preserve"> paruum ſit illud tẽpus:</s> <s xml:id="echoid-s26827" xml:space="preserve"> erit ipſum pars <lb/>magni temporis:</s> <s xml:id="echoid-s26828" xml:space="preserve"> & quoniam non datur uiſio fieri in tempore, niſi per diſtantiam uiſibilis ab ipſo ui <lb/>ſu:</s> <s xml:id="echoid-s26829" xml:space="preserve"> palàm tunc, quòd ſecundum ſpatium diſtantiæ uiſibilis à uiſu multiplicabitur & tempus.</s> <s xml:id="echoid-s26830" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0412-01a" xlink:href="fig-0412-01"/> Producatur itaq;</s> <s xml:id="echoid-s26831" xml:space="preserve"> linea a b c d, & ſit uiſus ad punctum a, & aliquod uiſibile ſit apud punctũ b.</s> <s xml:id="echoid-s26832" xml:space="preserve"> <lb/>Cumitaq;</s> <s xml:id="echoid-s26833" xml:space="preserve">, ut dictum & declaratum eſt in 6 huius, forma puncti b multiplicatur ad uiſum, ſi <lb/>hoc fiat in tempore quocunq;</s> <s xml:id="echoid-s26834" xml:space="preserve">, etiam fortè imperceptibili:</s> <s xml:id="echoid-s26835" xml:space="preserve"> ſit aliud uiſibile in puncto c:</s> <s xml:id="echoid-s26836" xml:space="preserve"> & ſit <lb/>ſpatium a c multiplex ſpatio a b:</s> <s xml:id="echoid-s26837" xml:space="preserve"> erit ergo tempus, in quo forma punctic multiplicatur ad ui <lb/>ſum a, multiplex tempori, in quo ſorma puncti b multiplicatur ad uiſum a:</s> <s xml:id="echoid-s26838" xml:space="preserve"> & ſihoc tempus <lb/>nondũ ſit ſenſibile, ſit in ulteriori puncto uiſibile d remotius à uiſu a, quàm eſt ipſum c:</s> <s xml:id="echoid-s26839" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s26840" xml:space="preserve"> <lb/>ſpatium d a multiplex ſpatij c a:</s> <s xml:id="echoid-s26841" xml:space="preserve"> ergo erit ipſum magis multiplex ſpatij b a.</s> <s xml:id="echoid-s26842" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s26843" xml:space="preserve"> pun-<lb/>cti d multiplicabitur ad uiſum a in tempore multiplici tempori, in quo peruenit ad uiſum a <lb/>forma puncti c:</s> <s xml:id="echoid-s26844" xml:space="preserve"> ſed in pertranſitu formæ puncti d per ipſum ſpatiũ a d non requiritur in ipſa <lb/>operatione uiſiua plus temporis, quàm in ſpatio a b:</s> <s xml:id="echoid-s26845" xml:space="preserve"> apertis enim oculis æquè citò uidentur <lb/>remota & propinqua:</s> <s xml:id="echoid-s26846" xml:space="preserve"> neq;</s> <s xml:id="echoid-s26847" xml:space="preserve"> enim eſt ſenſibilis differẽtia temporis, quo uidetur res proxima, <lb/>aut aliqua ſtellarum ſixarum, cuius ferè diſtantia eſt ſecundum mundi ſemidiametrum, quæ <lb/>eſt maxima linearum naturalium entium.</s> <s xml:id="echoid-s26848" xml:space="preserve"> Impoſsibile eſt ergo uiſionẽ, quæ fit aſpectu ſim-<lb/>plici, fieri in tempore:</s> <s xml:id="echoid-s26849" xml:space="preserve"> ſed neceſſe eſt omnẽ huiuſmodi uiſionem, quantum ad aſpectum ſim-<lb/>plicẽ, fieri in inſtãti & ſubitò:</s> <s xml:id="echoid-s26850" xml:space="preserve"> eius itaq;</s> <s xml:id="echoid-s26851" xml:space="preserve"> principiũ nõ differt ab eius fine.</s> <s xml:id="echoid-s26852" xml:space="preserve"> Ethoc eſt ꝓpoſitũ.</s> <s xml:id="echoid-s26853" xml:space="preserve"/> </p> <div xml:id="echoid-div1028" type="float" level="0" n="0"> <figure xlink:label="fig-0412-01" xlink:href="fig-0412-01a"> <variables xml:id="echoid-variables431" xml:space="preserve">a b c d</variables> </figure> </div> </div> <div xml:id="echoid-div1030" type="section" level="0" n="0"> <head xml:id="echoid-head827" xml:space="preserve" style="it">56. Omnem intuitionem in tempore fieri eſt neceſſe: tempuś intuitionis intentio-<lb/>num uiſibilium diuerſatur ſecundum diuerſitatem intentionum formarum intuitarũ. <lb/>Alhazen 70. 74 n 2.</head> <p> <s xml:id="echoid-s26854" xml:space="preserve">Cum enim, ut patuit in 51 huius, intuitio ſit actus uirtutis uiſiuæ, quo uiſus ueram comprehen-<lb/>ſionem formæ rei uiſæ diligenter perſpiciendo perquirit, & ſemper in ipſa intuitione axes radiales <lb/>per omnia puncta ſuperficiei rei uiſæ moueantur, ut declaratum eſt per 52 huius.</s> <s xml:id="echoid-s26855" xml:space="preserve"> Cum ergo omnis <lb/>motus ſenſibilis fiat in tempore ſenſibili ideo, quia, ut alibi declarauimus, tempus eſt proportiona-<lb/>le motui:</s> <s xml:id="echoid-s26856" xml:space="preserve"> palàm, quia omnem intuitionẽ in tempore ſenſibili fieri eſt neceſſe.</s> <s xml:id="echoid-s26857" xml:space="preserve"> Tempus quoq;</s> <s xml:id="echoid-s26858" xml:space="preserve"> intui-<lb/>tionis diuerſatur ſecundum diuerſas intentiones formarum uiſibilium eorum, quæ quis intuetur, <lb/>cuius exemplum eſt:</s> <s xml:id="echoid-s26859" xml:space="preserve"> ut ſi uiſus comprehendat animallongũ multorũ paruorũ pedũ, quod mouea-<lb/>tur:</s> <s xml:id="echoid-s26860" xml:space="preserve"> tunc primò per modicam intuitionẽ comprehẽ dit motũ eius, & per motũ comprehẽdit ipſum <lb/>eſſe animal:</s> <s xml:id="echoid-s26861" xml:space="preserve"> dein de per modicã intuitionẽ in pedibus comprehẽdet ipſum eſſe multorum pedũ ex <lb/>comprehẽſione diſtantiæ inter pedes, non tamẽ cognoſcet numerũ ipſorum pedũ:</s> <s xml:id="echoid-s26862" xml:space="preserve"> & deinde dili-<lb/>gẽtius intuẽs cognoſcet numerum pedum pluri intuitione & maioris tẽporis conatu.</s> <s xml:id="echoid-s26863" xml:space="preserve"> Comprehẽ-<lb/>ſio ergo animalitatis eius erit in paruo tẽpore, & comprehenſio multitudinis pedum erit in tẽpore <lb/>maiore illo tempore priori, in quo cognitũ eſt ipſum eſſe animal:</s> <s xml:id="echoid-s26864" xml:space="preserve"> numerus aũt pedũ erit adhuc in <lb/>tẽpore maiori aliquo illorũ tẽporum:</s> <s xml:id="echoid-s26865" xml:space="preserve"> oportet enim uiſum intueri quẽlibetillorũ pedum, & nume-<lb/>rare illos:</s> <s xml:id="echoid-s26866" xml:space="preserve"> erit aũt quantitas tẽporis intuitionis pedũ ſecundum numerũ multitudinis uel paucita-<lb/>tis pedum:</s> <s xml:id="echoid-s26867" xml:space="preserve"> & hoc etiã patet per diuerſitatẽ aliarum uiſibilium intẽtionum.</s> <s xml:id="echoid-s26868" xml:space="preserve"> Tẽpus itaq;</s> <s xml:id="echoid-s26869" xml:space="preserve"> intuitionis <lb/>intentionum uiſibilium ſormarũ, quarum una eſt numerus, diuerſatur ſecundum diuerſitatem in-<lb/>tentionum formarum intuitarum.</s> <s xml:id="echoid-s26870" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s26871" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1031" type="section" level="0" n="0"> <head xml:id="echoid-head828" xml:space="preserve" style="it">57. Viſus non poteſt comprehendere ueram form am rei uiſæ primo aſpectu ſimplici, ſedpoct <lb/>diligentem intuitionem. Alhazen 76 n 2.</head> <p> <s xml:id="echoid-s26872" xml:space="preserve">Cum enim formæ uiſibilium ſint cõpoſitæ ex multis intentionibus particularibus, quibuſdam <lb/> <pb o="111" file="0413" n="413" rhead="LIBER TERTIVS."/> illarum exiſtentibus groſsis, primo aſpectui ſe offerentibus, quibuſdam uero ſubtilibus ualde, ut <lb/>ſunt lineationes minutæ & colores minutatim diſperſi, & ſimilia, quę primo aſpectui, qui eſt inſtan <lb/>tiuus per 55 huius, ſtatim ſe offerre non poſſunt:</s> <s xml:id="echoid-s26873" xml:space="preserve"> unde indigẽt tẽpore ut uideantur:</s> <s xml:id="echoid-s26874" xml:space="preserve"> poſt diligẽtem <lb/>ergo intuitum uidebuntur, & non prius.</s> <s xml:id="echoid-s26875" xml:space="preserve"> Viſus enim non comprehendit ueram formam rei uiſę, ni-<lb/>ſi per comprehenſionem omnium intentionum particularium, quæ ſunt in illa forma.</s> <s xml:id="echoid-s26876" xml:space="preserve"> Patet ergo, <lb/>quòd forma rei uiſæ, in qua ſubtiles ſunt intentiones, non comprehenditur à uiſu ſecundum ueri-<lb/>tatem ſui eſſe primo aſpectu, ſed poſt intuitionem diligentem.</s> <s xml:id="echoid-s26877" xml:space="preserve"> Et quoniam etiam in formis, in qui-<lb/>bus non ſunt ſubtiles intentiones, uiſus illarum carentiam à primo aſpectu dijudicare non poteſt:</s> <s xml:id="echoid-s26878" xml:space="preserve"> <lb/>ideo etiam tunc eſt opus intuitione:</s> <s xml:id="echoid-s26879" xml:space="preserve"> nec enim poteſt certificare ueritatẽ formæ, niſi poſt diligentẽ <lb/>intuitionẽ cuiuslibet partis illius formę rei uiſæ.</s> <s xml:id="echoid-s26880" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s26881" xml:space="preserve">, quia uiſus nũ quá poteſt cõprehẽdere <lb/>uerã formã rei uiſæ in primo aſpectu, ſed ſolùm poſt diligentẽ intuitionẽ.</s> <s xml:id="echoid-s26882" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s26883" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1032" type="section" level="0" n="0"> <head xml:id="echoid-head829" xml:space="preserve" style="it">58. Intuitus repetiti plus figunt & certificant formas ſenſibiles in anima remanentes. <lb/>Alhazen 66 n 2.</head> <p> <s xml:id="echoid-s26884" xml:space="preserve">Cum enim uiſus comprehendit aliquam rem uiſam, & fuerit certificata forma eius apud ſentien <lb/>tem:</s> <s xml:id="echoid-s26885" xml:space="preserve"> tunc forma illius rei uiſæ remanet in anima, & figitur in imaginatione ipſius uidentis, utin na-<lb/>turalibus animæ paſsionibus declaratum eſt:</s> <s xml:id="echoid-s26886" xml:space="preserve"> & ſi iterabitur comprehenſio rei uiſæ:</s> <s xml:id="echoid-s26887" xml:space="preserve"> tunc erit for-<lb/>ma eius magis fixa in anima quàm forma rei ſemel uiſæ:</s> <s xml:id="echoid-s26888" xml:space="preserve"> quia uiſus rarò comprehẽdit perfectè rem <lb/>ſemel uiſam, ſed ſemper exiteratione uiſionis peruenit forma denuò ad animam, & renouatur for-<lb/>ma prius uiſa apud animam:</s> <s xml:id="echoid-s26889" xml:space="preserve"> & ſi aliquid ex intẽtionibus illius formæ obliuioni traditum eſt, reſtau <lb/>ratur, & ſi prius uiſum non eſt, recuperatur.</s> <s xml:id="echoid-s26890" xml:space="preserve"> Anima autem per formam ſecundam rememoratur for <lb/>mæprimæ, & cum pluries iteratur euentus eiuſdẽ intẽtionis ſuper animam, erit anima magis re-<lb/>memorans illam intentionẽ:</s> <s xml:id="echoid-s26891" xml:space="preserve"> & ſic erit illa forma magis fixa in anima:</s> <s xml:id="echoid-s26892" xml:space="preserve"> ſed & magis certificata:</s> <s xml:id="echoid-s26893" xml:space="preserve"> quia <lb/>in prima uiſione, in qua forma rei uiſæ uenit ad animam, fortè anima non comprehẽdet omnes in-<lb/>tentiones, quæ ſunt in illa forma, neq;</s> <s xml:id="echoid-s26894" xml:space="preserve"> certificabitipſas:</s> <s xml:id="echoid-s26895" xml:space="preserve"> & cum forma redierit ſecundò, comprehen <lb/>det anima ex ea aliud, quod in prima uice non comprehendit:</s> <s xml:id="echoid-s26896" xml:space="preserve"> & quantò magis forma iterabitur ſu-<lb/>per animam, tantò magis manifeſtabitur ex ea, quod prius non apparebat:</s> <s xml:id="echoid-s26897" xml:space="preserve"> & cum anima compre-<lb/>henderit intentiones ſubtiliores formarum, magis certificabitur ſibi eſſe totius formæ, Patet ergo <lb/>ex his, quia intuitus repetiti erunt certiores, ut proponitur.</s> <s xml:id="echoid-s26898" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1033" type="section" level="0" n="0"> <head xml:id="echoid-head830" xml:space="preserve" style="it">59. Nullum uiſibilium comprehenditur ſolo ſenſu uiſus, niſi ſolùm luces & colores. Al-<lb/>hazen 18 n 2.</head> <p> <s xml:id="echoid-s26899" xml:space="preserve">Sola enim hęc cum ſint per ſe uiſibilia, ſicut in ſuppo ſitionibus huius libri præmiſſum eſt:</s> <s xml:id="echoid-s26900" xml:space="preserve"> patet, <lb/>quòd ipſa ſunt priora omnibus alijs uiſibilibus:</s> <s xml:id="echoid-s26901" xml:space="preserve"> unde ipſa ſine alijs offeruntur uiſui, ut ſine ſitu, figu <lb/>ra, & magnitudine, & ſimilibus:</s> <s xml:id="echoid-s26902" xml:space="preserve"> alia uerò non offeruntur uiſui ſine illis, uiſibili enim actu lucem nõ <lb/>participiante impoſsibile eſt illud uideri, ut patet per 1 huius:</s> <s xml:id="echoid-s26903" xml:space="preserve"> circa lucem ergo & colorem nõ fit a-<lb/>li qua alia operatio animæ niſi ſola ſenſatio uiſionis.</s> <s xml:id="echoid-s26904" xml:space="preserve"> Lux enim, quæ eſt in corpore illuminato, com-<lb/>prehenditur à uiſu ſecundum ſuum eſſe & per ſe exipſo ſenſu:</s> <s xml:id="echoid-s26905" xml:space="preserve"> lux uerò & color, quæ ſuntin corpo-<lb/>re colorato & illuminato, comprehenduntur à uiſu in ſimul & admixta:</s> <s xml:id="echoid-s26906" xml:space="preserve"> comprehenditur autem u-<lb/>trunq;</s> <s xml:id="echoid-s26907" xml:space="preserve"> illorum ſolo ſenſu uiſus:</s> <s xml:id="echoid-s26908" xml:space="preserve"> lux enim prima comprehenditur à uiſu exilluminatione corporis <lb/>ſentientis, quod eſt de ſubſtantia oculi, & color ex alteratione ſormæ eiuſdem corporis ſentientis <lb/>& eius coloratione cum admixtione lucis, quæ eſt hypoſtaſis coloris.</s> <s xml:id="echoid-s26909" xml:space="preserve"> Sicutenim ſentiens compre-<lb/>hendit in peruentu formæ lucis primæ ſolam lucem:</s> <s xml:id="echoid-s26910" xml:space="preserve"> ſic in peruentu formæ coloris comprehendit <lb/>lucem coloratam.</s> <s xml:id="echoid-s26911" xml:space="preserve"> Ergo hæc duo comprehenduntur ſolo ſenſu uiſus ſine alijs animæ potentijs & <lb/>operationibus, quod non accidit in aliquo aliorum inuiſibilium:</s> <s xml:id="echoid-s26912" xml:space="preserve"> quoniam illa quaſi plura à pluri-<lb/>bus ſenſibus ſentiuntur:</s> <s xml:id="echoid-s26913" xml:space="preserve"> & ſi aliqua ipſorum ſolo ſenſu uiſus ſentiantur, & non alijs ſenſibus parti-<lb/>cularibus:</s> <s xml:id="echoid-s26914" xml:space="preserve"> hoc accidit uel ex iſtorum aliqua participatione, uel iſtorum priuatione, ſicut eſt in dia-<lb/>phanitate & opacitate, tenebris & umbra, in quibus neceſſaria eſt ratio conferens hincinde, quæ <lb/>non eſt neceſſaria in comprehenſione lucis & coloris, Patet ergo propoſitum.</s> <s xml:id="echoid-s26915" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1034" type="section" level="0" n="0"> <head xml:id="echoid-head831" xml:space="preserve" style="it">60. Omne uiſibile aut comprehenditur à uiſu ſolo ſimpliciter: aut cum ratione & diſtin-<lb/>ctione. Alhazen 10 n 2.</head> <p> <s xml:id="echoid-s26916" xml:space="preserve">Vtenim patet per præ ce dentem, lucem & colorem per ſe ſimpliciter comprehendit ſolus uiſus:</s> <s xml:id="echoid-s26917" xml:space="preserve"> <lb/>ſunt tamen plura aliorum, quæ de numero uiſibilium ſunt ſuppoſita, quę uiſus quidẽ comprehen-<lb/>dit, non tamen ſimpliciter per ſeipſum, ſed alijs actionibus anιmę accedentibus:</s> <s xml:id="echoid-s26918" xml:space="preserve"> & ſunt plura talia <lb/>uiſibilia, quorum comprehẽſio non eſt puro ſenſu uiſus:</s> <s xml:id="echoid-s26919" xml:space="preserve"> quoniam uiſus quando comprehẽdit duo <lb/>in diuidua eiuſdem ſpeciei & formę eodem tempore:</s> <s xml:id="echoid-s26920" xml:space="preserve"> tunc comprehendet in diuidua, & comprehẽ-<lb/>det quòd ſunt ſimilia:</s> <s xml:id="echoid-s26921" xml:space="preserve"> ſed ſimilitudo duarum ſormarum non eſt ipſæ formæ ambę, neq;</s> <s xml:id="echoid-s26922" xml:space="preserve"> una ipſarũ, <lb/>ſed neq;</s> <s xml:id="echoid-s26923" xml:space="preserve"> forma tertia propria conſimilitudini, ſed eſt conuenientia illarum duarum formarum in <lb/>aliquo.</s> <s xml:id="echoid-s26924" xml:space="preserve"> Non ergo comprehẽdetur duarum formarum ſimilitudo, niſi ex comparatione unius ipſa-<lb/>rum ad alteram:</s> <s xml:id="echoid-s26925" xml:space="preserve"> Non fit ergo ſimilitudinis comprehenſio per ſolum uiſum, ſed ex potentia animæ, <lb/>quam dicimus rationem per actum ratiocinationis diuerſas formas uiſas ad inuicem comparantẽ.</s> <s xml:id="echoid-s26926" xml:space="preserve"> <lb/>Et etiam quando uiſus uidet duos colores albos, quorum unus eſt albior alio, comprehendet am-<lb/> <pb o="112" file="0414" n="414" rhead="VITELLONIS OPTICAE"/> borum albedinem, & quod alterum eſt fortioris albedinis:</s> <s xml:id="echoid-s26927" xml:space="preserve"> comprehendet ergo ſimilitudinem illo <lb/>rum duorum alborum in albedine, & diuerſitatem illorum in fortitudine & debilitate:</s> <s xml:id="echoid-s26928" xml:space="preserve"> diſtinctio <lb/>uerò inter illas duas albedines non eſt ipſe ſenſus albedinis:</s> <s xml:id="echoid-s26929" xml:space="preserve"> quoniam ſenſus albedinis eſt ex deal-<lb/>batione ſuperficiei uiſus, quę fit ab utraq;</s> <s xml:id="echoid-s26930" xml:space="preserve"> albedine:</s> <s xml:id="echoid-s26931" xml:space="preserve"> diſtinctio autem illarum albedinum fit propter <lb/>diuerſitatem actionis illarum duarum albedinum in ipſum.</s> <s xml:id="echoid-s26932" xml:space="preserve"> Non eſt ergo illa diſtinctio à ſo-<lb/>lo ſenſu, ſed eſt ab alia uirtute animæ, quam dicimus diſtinctiuam.</s> <s xml:id="echoid-s26933" xml:space="preserve"> Et ſimiliter eſt de comparatione <lb/>& diſtinctione aliarum ſenſibilium formarum:</s> <s xml:id="echoid-s26934" xml:space="preserve"> nihil enim illorum accipitur ſolo uiſu, ſed ratione & <lb/>uirtute diſtinctiua coadiuuantibus:</s> <s xml:id="echoid-s26935" xml:space="preserve"> uiſus enim per ſe non habet uirtutem diſtinguendi, ſed uirtus <lb/>diſtinctiua animæ diſtinguit omnia illa mediante uiſu.</s> <s xml:id="echoid-s26936" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s26937" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1035" type="section" level="0" n="0"> <head xml:id="echoid-head832" xml:space="preserve" style="it">61. Ex intentionibus formarum indiuidualium ſæpius intuitarum, remanet in anima <lb/>fixio & certificatio formæ uniuer ſalis, exiſtens uiſui principium cognoſcendi omnia indiuidua <lb/>eiuſdem ſpeciei. Alhazen 14. 67 n 2.</head> <p> <s xml:id="echoid-s26938" xml:space="preserve">Quia enim quodlibet uiſibilium indiuidualium habet formam & figuram, in quibus cõueniunt <lb/>omnia indiuidua illius ſpeciei, quæ diuerſantur ſolùm in intentionibus particularibus, comprehẽ-<lb/>ſis per ſenſum uiſus, & fortè erit in omnibus illis indiuiduis color unius modi, ut quaſi uniuerſali-<lb/>ter in indiuiduis auium, ut cygno, coruo, pica, & graculo, & ſimilibus, in quibus eſt uniſormitas co-<lb/>loris, conueniens toti ſpeciei, uelut in pluribus, quia iam uidimus coruum album & urſum album.</s> <s xml:id="echoid-s26939" xml:space="preserve"> <lb/>Si itaq;</s> <s xml:id="echoid-s26940" xml:space="preserve"> forma & figura & color & omnes intentiones, ex quibus componitur forma cuiuslibetin-<lb/>diuidui ſpeciei, eſt forma uniuerſalis totius ſpeciei:</s> <s xml:id="echoid-s26941" xml:space="preserve"> & uiſus comprehẽdit illam figuram & formam <lb/>& colorem & omnium illorum intentiones, quę conueniunt illi ſpeciei:</s> <s xml:id="echoid-s26942" xml:space="preserve"> tunc anima iudicabit illud <lb/>particulare uiſum eſſe indiuiduum illius ſpeciei:</s> <s xml:id="echoid-s26943" xml:space="preserve"> non tamen propter hoc cognoſcet unum in diui-<lb/>duum ab alio indiuiduo eiuſdem ſpeciei diſtinctum:</s> <s xml:id="echoid-s26944" xml:space="preserve"> donec comprehenderit etiam intẽtiones par-<lb/>ticulares, per quas diuerſantur indiuidua, & donec illæ quieuerint in anima & in ipſa uirtute imagi <lb/>natiua:</s> <s xml:id="echoid-s26945" xml:space="preserve"> tunc enim aliquo prius uiſorum indiuiduorum ipſi uiſui occurrente, per intentionem in di-<lb/>uiduorum illius ſpeciei, cuius forma eſt apud animam, iterabitur à uiſu intentio illius formæ uni-<lb/>uerſalis, quæ eſt illius ſpeciei, cum diuerſitate formarum particularium illorum indiuiduorum:</s> <s xml:id="echoid-s26946" xml:space="preserve"> & <lb/>cum illa forma uniuerſalis per intẽtionem alterius indiuidui eiuſdem ſpeciei comparabitur in ani-<lb/>ma:</s> <s xml:id="echoid-s26947" xml:space="preserve"> tũc figetur in anima, & quieſcet.</s> <s xml:id="echoid-s26948" xml:space="preserve"> Ex diuerſitate itaq;</s> <s xml:id="echoid-s26949" xml:space="preserve"> formarum particularium uenientiũ ad ui-<lb/>ſum cum formis uniuerſalibus, apud intuitionem comprehẽdet anima diuerſitem indiuiduorũ <lb/>eiuſdẽ ſpeciei, & per cõuenientiam accidentium uiſibilium in diuerſis indiuiduis, comprehendet <lb/>quòd forma, in qua conueniunt omnia indiuidua illius ſpeciei, eſt forma uniuerſalis illorũ omniũ.</s> <s xml:id="echoid-s26950" xml:space="preserve"> <lb/>Sic ergo remanet in anima forma uniuerſalis, & in eius uirtute imaginatiua:</s> <s xml:id="echoid-s26951" xml:space="preserve"> & eſt illa forma uiſui <lb/>principium cognoſcendi omnia indiuidua eiuſdem ſpeciei, quantum ad illud, quod eſt in ipſis ex <lb/>intentionibus uniuerſalibus indiuiduatum, & de intentionibus particularibus ſenſibilibus qui-<lb/>buſcunque.</s> <s xml:id="echoid-s26952" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s26953" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1036" type="section" level="0" n="0"> <head xml:id="echoid-head833" xml:space="preserve" style="it">62. Omnis uera comprehenſio formarum uiſibilium, aut est per ſolam intuitionem, aut <lb/>per intuitionem cum ſcientia præcedente. Alhazen 69 n 2.</head> <p> <s xml:id="echoid-s26954" xml:space="preserve">Comprehenſio uiſibilium ſola intuitione fit, quando comprehenduntur uiſibilia extranea, ut <lb/>quando uiſus comprehẽdit rem uiſam, quam antea non percepit nec in ſe nec in ſua ſpecie:</s> <s xml:id="echoid-s26955" xml:space="preserve"> per in-<lb/>tuitionem uerò diligentẽ acquirit omnes diſpoſitiones & formam eius ueram:</s> <s xml:id="echoid-s26956" xml:space="preserve"> non tamẽ cognoſcit <lb/>formam eius, quia ipſam antea non percepit, uel non recolit:</s> <s xml:id="echoid-s26957" xml:space="preserve"> ſic ergo comprehendetur illa forma <lb/>uera comprehenſione per ſolam intuitionem.</s> <s xml:id="echoid-s26958" xml:space="preserve"> Comprehenſio autem uera formarum uiſibilium a-<lb/>lia ab alia, quæ fit per ſolam intuitionem, quandoq;</s> <s xml:id="echoid-s26959" xml:space="preserve"> fit per intuitionem cum ſcientia præcedente, <lb/>ut quando uiſus comprehendit formã alicuius rei uiſæ, quã cõprehẽdit etiam antè, & cuius formę <lb/>intentio eſt apud animam auttota, aut aliqua pars illius:</s> <s xml:id="echoid-s26960" xml:space="preserve"> tũc enim uiſus ſtatim in aſpectu illius rei <lb/>cõprehendet eius formã:</s> <s xml:id="echoid-s26961" xml:space="preserve"> & deinde modica intuitione comprehendet totam formam eius, quę eſt <lb/>ſcientia uniuerſalis ſuæ ſpeciei, & cognoſcet formam uniuerſalem, quam comprehendet in illa re <lb/>uiſa apud comprehenſionẽ formæ in anima per rememorationem illius rei uiſæ ſpecialiter:</s> <s xml:id="echoid-s26962" xml:space="preserve"> & dein <lb/>de intuens intẽtiones reſiduas, quæ ſunt in illa re uiſa, certificabit particularem ſormam illius, ipſi <lb/>uiſo indiuiduo appropriatam:</s> <s xml:id="echoid-s26963" xml:space="preserve"> & ſi fuerit rememorans illius formę particulairs, ut prius per uiſum <lb/>comprehenſę, tunc cognoſcet illam formam indiuidualem.</s> <s xml:id="echoid-s26964" xml:space="preserve"> Et quia nulla res uiſa comprehenditur <lb/>uera comprehenſione, niſi aliquo iſtorum modorum.</s> <s xml:id="echoid-s26965" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s26966" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1037" type="section" level="0" n="0"> <head xml:id="echoid-head834" xml:space="preserve" style="it">63. Comprehenſio uiſualis per cognitionem ſemper fit per aliquem modum rationis confe-<lb/>rentis. Alhazen 11 n 2.</head> <p> <s xml:id="echoid-s26967" xml:space="preserve">Eſt enim cognitio comprehenſio conſimilitudinis duarũ formarũ, ſcilicet formæ, quam compre <lb/>hendit uiſus apud cognitionem, quando ſentit ſe cognoſcere rem, quam uidet, & formæ quieſcen-<lb/>tis in anima, prius cõprehenſæ:</s> <s xml:id="echoid-s26968" xml:space="preserve"> unde non fit uiſualis cognitio, niſi per rememorationem:</s> <s xml:id="echoid-s26969" xml:space="preserve"> quoniam <lb/>ſi nulla forma talis fuerit quieſcens apud animã & pręſens memoriæ, non cognoſcet uiſus rem ui-<lb/>ſam.</s> <s xml:id="echoid-s26970" xml:space="preserve"> Semper itaq;</s> <s xml:id="echoid-s26971" xml:space="preserve"> fit cognitio ex aſsimilatione formæ quieſcentis in anima ad formá poſtea uiſam <lb/>extrâ, ſiue forma quieſcens ſit forma ſpeciei uel indiuidui cognoſcendi.</s> <s xml:id="echoid-s26972" xml:space="preserve"> Viſus itaq;</s> <s xml:id="echoid-s26973" xml:space="preserve"> comprehendit <lb/> <pb o="113" file="0415" n="415" rhead="LIBER TERTIVS"/> multas res ք cognitionem:</s> <s xml:id="echoid-s26974" xml:space="preserve"> cognoſcit enim hominem eſſe hominem, & equũ eſſe equũ, & Socratẽ <lb/>eſſe Socratem:</s> <s xml:id="echoid-s26975" xml:space="preserve"> & cognoſcit animalia ſibi aſſueta, & arbores, & plantas, & lapides, quę prius uidit, & <lb/>cogn oſcit illis ſimilιa, & omnes intentiones ſibi aſſuetas in rebus uiſibilibus, & quantitates omniũ <lb/>rerum ſibi conſuetarũ, quę non cognoſcuntur ſolo uiſu per 59 huius:</s> <s xml:id="echoid-s26976" xml:space="preserve"> nec tamen cognoſcit uiſus o-<lb/>mne, quod uidit prius, niſi quando fuerit remem orans formæ prius uiſæ.</s> <s xml:id="echoid-s26977" xml:space="preserve"> Non eſt ergo cognitio ui-<lb/>ſualis comprehenſio ſolo ſenſu, ſed per rationẽ formam pręſentis rei uiſæ formę prius uiſæ & apud <lb/>ſe quieſcenti conferentem:</s> <s xml:id="echoid-s26978" xml:space="preserve"> nun quam enim poteſt fieri cognitio, niſi per comparationẽ formæ qui-<lb/>eſcentis in anima ad formã uiſam extra.</s> <s xml:id="echoid-s26979" xml:space="preserve"> Sic ergo patet, quoniá comprehẽſio uiſualis per cognitio-<lb/>nem ſemper ſit per aliquem modum rationis conferentis.</s> <s xml:id="echoid-s26980" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s26981" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1038" type="section" level="0" n="0"> <head xml:id="echoid-head835" xml:space="preserve" style="it">64. Omnem comprehenſionem uiſualem cognoſcitiuam in tempore fieriest neceſſe: ſedin mi <lb/>nori, quàm ſit tempus comprehenſionis per ſolam intuitionem. Alhazen 13. 71 n 2.</head> <p> <s xml:id="echoid-s26982" xml:space="preserve">Quoniam enim, ſicut in præce dente propoſitione pręmiſſum eſt, omnis cognitio fit per <lb/>intuitionem & formam in anima quieſcentem rememoratam & applicatam formæ nunc per dili-<lb/>gentem intuitum perſpectę:</s> <s xml:id="echoid-s26983" xml:space="preserve"> & quoniam omnis intuitio fit in tempore per 56 huius, & omnis reme <lb/>moratio formę prius uiſæ fit plurimũ in tẽpore, quoniam fit per diſcurſum animę per formas, quas <lb/>apud ſe habet in imaginatione, quæ ſi quęrenti animæ ſtatim occurreret, nõ eſſet rememoratio, ſed <lb/>continuata memoria.</s> <s xml:id="echoid-s26984" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s26985" xml:space="preserve"> ambo hęc, ſcilicet intuitio & rememoratio, uel ipſorum alterũ fit <lb/>in tempore:</s> <s xml:id="echoid-s26986" xml:space="preserve"> patet etiam, quòd omnis comprehẽſio uiſualis cognoſcitiua fit neceſſariò in tempore:</s> <s xml:id="echoid-s26987" xml:space="preserve"> <lb/>ſed in minori, quàm ſit tempus comprehenſionis per ſolam intuitionem:</s> <s xml:id="echoid-s26988" xml:space="preserve"> quoniam intẽtiones exi-<lb/>ſtentes in anima pręſentis memoriæ non indigent, ut cognoſcantur omnes intentiones, quę ſunt <lb/>in formis rerum cognitarum, ex quibus componuntur in rei ueritate, ſed ſufficit in comprehenſio-<lb/>ne eorum comprehenſio alicuius intẽtionis proprię illis.</s> <s xml:id="echoid-s26989" xml:space="preserve"> Cum ergo uirtus diſtinctiua comprehen-<lb/>derit in forma ueniente ad ipſam aliquam intentionem propriam illi formæ, erit rememorans pri-<lb/>mæ formæ, & cognoſcet omnes formas uenientes ad ipſam, quoniam omnis intentio appropriata <lb/>alicui formæ, eſt ſignans ſuper illas formas:</s> <s xml:id="echoid-s26990" xml:space="preserve"> ut quando uiſus intuens Socratem, comprehendit li-<lb/>neationem manus humanæ, ſtatim comprehendit quòd ſit homo, & antequam comprehendat li-<lb/>neationem ſuæ faciei uel partium aliarum.</s> <s xml:id="echoid-s26991" xml:space="preserve"> Ex comprehenſione ergo quarundam intentionũ, quæ <lb/>appropriantur ſormæ hominis, comprehendit quòd idem uiſibile ſit homo ſine in digentia cópre-<lb/>henſionis partium aliarum, quas comprehendit ſolùm per cognitionem pręcedentem ex formis <lb/>reſidentibus in anima, & per comprehenſionem alicuius intentionis propriæ illi indiuiduo, ut per <lb/>glaucitatem oculorum uel oris groſsiciem aut arcuitatẽ ſuperciliorum, aut ſimilibus comprehen-<lb/>dit totalis illius indiuidui intentiones:</s> <s xml:id="echoid-s26992" xml:space="preserve"> & ſimiliter cognoſcet equum per aliquam maculam in fron <lb/>te aut alibi in corpore:</s> <s xml:id="echoid-s26993" xml:space="preserve"> & ſcriptor ex quarun dam comprehenſione literarum cognoſcit omnes par <lb/>tes dictionis uel orationis, quam frequenter & continuè uidet.</s> <s xml:id="echoid-s26994" xml:space="preserve"> Et quoniam comprehenſio, quę ac-<lb/>quiritur tantùm per intuitionem, fit per conſiderationem omnium partium rei uiſæ, & omniũ in-<lb/>tentionum, quæ ſunt in ea:</s> <s xml:id="echoid-s26995" xml:space="preserve"> comprehenſio uerò per cognitionem fit per conſiderationem ſolũ qua-<lb/>rundam intentionum, quæ ſunt in illa forma:</s> <s xml:id="echoid-s26996" xml:space="preserve"> palàm, quòd uiſio, quę eſt per cognitionem, eſt in mi-<lb/>nori tem pore, quàm ſit uiſio per ſolam intuitionem:</s> <s xml:id="echoid-s26997" xml:space="preserve"> & propter hoc uiſus comprehẽdit uiſibilia aſ-<lb/>ſueta uelociter in paruo tẽpore quaſi latente ſenſum, & maximè illa, quę ſui primordio cognoſce-<lb/>re cõſueuit, uel cum quibus multo tẽore perſeuerauit.</s> <s xml:id="echoid-s26998" xml:space="preserve"> Patet ergo illud, quod proponebatur.</s> <s xml:id="echoid-s26999" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1039" type="section" level="0" n="0"> <head xml:id="echoid-head836" xml:space="preserve" style="it">65. Viſio per cognitionem præcedentem per modicam intuitionem non efficit certam formæ <lb/>rei comprehenſionem. Alhazen 75 n 2.</head> <p> <s xml:id="echoid-s27000" xml:space="preserve">Quoniam enim uiſio per cognitionem pręcedentem non eſt niſi circa totalitatem & uniuerſita-<lb/>tem rei uiſæ ſuperficialiter & in groſſo & per quæ dam exteriora ſigna illius rei uiſæ:</s> <s xml:id="echoid-s27001" xml:space="preserve"> & uirtus diſtin <lb/>ctiua comprehen dit intentiones particulares, quæ ſunt in illa re uiſa, ſecun dum modum, quo co-<lb/>gnouit res uiſas ex prima forma illius rei uiſæ in anima exiſtente:</s> <s xml:id="echoid-s27002" xml:space="preserve"> ſed omnes particulares intentio-<lb/>nes uiſibilium, quæ ſuntin rebus corruptibilibus, mutantur temporis mutatione:</s> <s xml:id="echoid-s27003" xml:space="preserve"> uiſus autem non <lb/>comprehendit mutationem intentionum rei uiſæ per formam prius habitam, cum mutatio fuerit <lb/>non manifeſta nec comprehenſibilis à uiſu primo aſpectu.</s> <s xml:id="echoid-s27004" xml:space="preserve"> Cognitio ergo præ cedens non efficit ue <lb/>ram rei cognitionem:</s> <s xml:id="echoid-s27005" xml:space="preserve"> utpote ſi in homine mundæ faciei prius cognito accidat poſtmodum macu-<lb/>la uel cicatrix in facie, quæ non ſit manifeſta:</s> <s xml:id="echoid-s27006" xml:space="preserve"> tum enim poſtea longo tempore uiſo illo homine, nõ <lb/>cognoſcet ipſum uidens ſecundum formam ſui, quam prius mem oriter ſeruauerat, nec tum com-<lb/>prehen det maculam uel cicatricem illam in facie illius, niſi poſt intuitionem diligentem factam in <lb/>illam maculam uel cicatricem:</s> <s xml:id="echoid-s27007" xml:space="preserve"> & tunc comprehendet formam eius ſecun dum ſuum eſſe.</s> <s xml:id="echoid-s27008" xml:space="preserve"> Et ſimili-<lb/>ter eſt ſi macula ſemper in facie ipſius cogniti fuerit, non tamen fuerit uiſui multum maniſeſta:</s> <s xml:id="echoid-s27009" xml:space="preserve"> tũc <lb/>enim licet habeat uidens apud ſe formam illius non maculatam, non tamen applicabit ipſam illius <lb/>faciei maculatæ, & non cognoſcet ipſum niſi poſt multam aliarũ intentionũ particularium intuitio <lb/>nem.</s> <s xml:id="echoid-s27010" xml:space="preserve"> Et ſimiliter eſt in alijs indiuiduis uiſibilium & intentionibus diuerſis ipſorum.</s> <s xml:id="echoid-s27011" xml:space="preserve"> In omnibus e-<lb/>nim ipſis uiſio per cognitionem præ cedentem per modicam intuitionem non <lb/>efficit certam formæ rei comprehenſionem.</s> <s xml:id="echoid-s27012" xml:space="preserve"> <lb/>Patet ergo propoſitum.</s> <s xml:id="echoid-s27013" xml:space="preserve"/> </p> <pb o="114" file="0416" n="416" rhead="VITELLONIS OPTICAE"/> </div> <div xml:id="echoid-div1040" type="section" level="0" n="0"> <head xml:id="echoid-head837" xml:space="preserve" style="it">66. Nullius entium quiddit as per ſe eſt uiſibilis, ſed per accidens, mediantibus intentionibus <lb/>ſenſibilibus, quæ per ſe uidentur. Alhazen 68 n 2.</head> <p> <s xml:id="echoid-s27014" xml:space="preserve">Quoniam enim, ut ſuppoſitum eſt in principio libri huius, uiſio nõ completur niſi apud peruen-<lb/>tum formarum uiſibilium ad animam, quę omnes ſunt de genere accidẽtis, ut patet in ipſarum ſin-<lb/>gulari enumeratione:</s> <s xml:id="echoid-s27015" xml:space="preserve"> palàm (cum nullius ſubſtátiæ quidditas ſit de genere accidẽtis) quòd nul-<lb/>la ipſarum per ſe eſt uiſibilis:</s> <s xml:id="echoid-s27016" xml:space="preserve"> per accidens autem quid ditas ſub ſtantiarum corporalium percipitur <lb/>à uiſu, ſcilicet per comprehẽſionem ſuarum intẽtionum uiſibiliũ, quę per ſe uidẽtur.</s> <s xml:id="echoid-s27017" xml:space="preserve"> Sic ergo quid-<lb/>ditas ſubſtantiæ non ſit niſi per cognitionem intrin ſecam animæ, quæ fit ex comparatione formæ <lb/>unius poſterius comprehẽſæ, ad formam aliam prius comprehẽſam quieſcẽtem in imaginatione.</s> <s xml:id="echoid-s27018" xml:space="preserve"> <lb/>Comprehenſio ergo quid ditatis ſubſtantiæ uiſæ, ut hominis uel canis uel alicuius alterius ſubſtan <lb/>tiæ, non eſt niſi ex comprehẽſione aſsimilationis formæ rei uiſæ ad aliquam formarum uniuerſaliũ <lb/>quieſcẽtium in anima & fixarum in imaginatione, quam uiſus antè comprehẽderat.</s> <s xml:id="echoid-s27019" xml:space="preserve"> Et quia uirtus <lb/>diſtin ctiua, quę eſt in anima, per quam anima rerum differẽtias dijudicat, ut hominem non eſſe ca-<lb/>nem & ecóuerſo, naturaliter aſsimilat ipſas formas uiſibilium nouiter ſcilicet uiſas, uiſibilibus for-<lb/>mis fixis in imaginatione.</s> <s xml:id="echoid-s27020" xml:space="preserve"> Cum ergo uiſus comprehẽderit ali quam rem uiſam, ſtatim uirtus diſtin <lb/>ctiua quærit eius ſimilem in formis exiſtentibus in imaginatione, & illa inuenta cognoſcit per illá, <lb/>rem uiſam, & comprehẽdit quidditatẽ eius:</s> <s xml:id="echoid-s27021" xml:space="preserve"> & ſi non inuenerit ex ſormis quieſcẽtibus in anima for <lb/>mam ſimilem formę illius rei uiſę, nõ cognoſcet illam rẽ uiſam, neq;</s> <s xml:id="echoid-s27022" xml:space="preserve"> comprehẽdet quid ditatẽ eius.</s> <s xml:id="echoid-s27023" xml:space="preserve"> <lb/>Sic ergo nulla quid ditas alicuius ſubſtantię comprehẽditur per ſe à uiſu, ſed peraccidẽs, ut propo-<lb/>nitur.</s> <s xml:id="echoid-s27024" xml:space="preserve"> Si enim aliqua talium quidditatum per ſe comprehenderetur à uiſu:</s> <s xml:id="echoid-s27025" xml:space="preserve"> ergo & omnis quiddi-<lb/>tas cuiuslibet uiſibilis ſubſtantiæ eſſet comprehẽſibilιs à uiſu, ſicut patetin lucibus & colorιbus, & <lb/>ſub ſtantię quantum ad ſenſum & ſenſibilem operationẽ exiſtentes indiuiſibiles per ſuas quiddita-<lb/>tes uiderẽtur, quod nõ eſt uerum:</s> <s xml:id="echoid-s27026" xml:space="preserve"> oportet enim ut corpus uiſibile ſit alicuius quantitatis reſpectu <lb/>ſuperficiei uiſus, ad hoc ut ipſum a ctu uideatur, ut patet per 19 huius.</s> <s xml:id="echoid-s27027" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s27028" xml:space="preserve"> patet de o-<lb/>mnibus alijs quorumcunque entium quid ditatibus:</s> <s xml:id="echoid-s27029" xml:space="preserve"> ſemper enim quidditas cuiuslibet compoſiti <lb/>compoſita eſt, & eius compoſitionem uiſus per ſe comprehendere non poteſt:</s> <s xml:id="echoid-s27030" xml:space="preserve"> & ſi uiſus aliquam <lb/>quid ditatem, ut eſt quidditas, cognoſceret:</s> <s xml:id="echoid-s27031" xml:space="preserve"> tunc uiſus omnem quidditatem cognoſceret, quarum <lb/>multæ tamen ſunt inuiſibiles, cum omnes ipſæ ſint per ſe intelligibiles:</s> <s xml:id="echoid-s27032" xml:space="preserve"> & cum hoc ſit impoſsibi-<lb/>le:</s> <s xml:id="echoid-s27033" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s27034" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1041" type="section" level="0" n="0"> <head xml:id="echoid-head838" xml:space="preserve" style="it">67. Primum quod comprehendit uirtus diſtinctiua ex intentionibus appropriatis formæ ui <lb/>ſibili, est quidditas lucis & coloris. Alhazen 17 n 2.</head> <p> <s xml:id="echoid-s27035" xml:space="preserve">Quamuis enim lux & color ſint per ſe ipſa & primò uiſibilia, ipſorũ tamẽ quidditates & differen <lb/>tiæ eſſentiales ſolo ſenſu uiſus comprehẽdi non poſſunt:</s> <s xml:id="echoid-s27036" xml:space="preserve"> quid ditas enim lucis nõ comprehenditur <lb/>ſolùm per uiſum, niſi cooperante uirtute animæ, quę eſt cognoſcitiua, quoniam uiſus cognoſcit lu <lb/>mẽſolis, & diſtinguit inter ipſum & lumẽ lunæ & lumẽ ignis per cognitionem prius factam & per <lb/>formam in anima reſeruatam:</s> <s xml:id="echoid-s27037" xml:space="preserve"> ſimiliter etiam quid ditas coloris non comprehen ditur à uirtute di-<lb/>ſtinctiua niſi per cognitionem, quando color rei uiſę fuerit ex colorbus aſſuetis.</s> <s xml:id="echoid-s27038" xml:space="preserve"> Illa autem cogni-<lb/>tio diſtinctiua fit ex comparatione formæ coloris nunc uiſi ad formas illi colori prius com-<lb/>prehenſas:</s> <s xml:id="echoid-s27039" xml:space="preserve"> non enim poteſt uiſus comprehendere colorẽ rubeum & quòd ſit rubeus, niſi quia co-<lb/>gnoſcit ipſum, quia in ipſa anima uidentis permanſit forma eius, ut prius uiſa.</s> <s xml:id="echoid-s27040" xml:space="preserve"> Si enim uiſus nun-<lb/>quam colorem rubeum antea uidiſſet, nũc ipſum uiſum cognoſcere nõ poſſet, ſed ipſum coloribus <lb/>illi propm quis ſibi cognitis aſsimilaret, ut quotidie facit in noua permixtione quorũlibet colorũ.</s> <s xml:id="echoid-s27041" xml:space="preserve"> <lb/>Cum itaq;</s> <s xml:id="echoid-s27042" xml:space="preserve"> uirtus diſtinctiua comprehen dit diuerſitatem lucis à colorum quidditate, quamuis forma, quã <lb/>ris.</s> <s xml:id="echoid-s27043" xml:space="preserve"> comprehen dit etiam diuerſitatem quid ditatis lucis à colorum quidditate, quamuis forma, quã <lb/>comprehendit uiſus, ſit a dmixta ex forma lucis & coloris, quæ ſunt in re uiſa.</s> <s xml:id="echoid-s27044" xml:space="preserve"> Et quoniam lux & co <lb/>lor ſunt prima uiſibilia, quorũ participatione & auxilio omnia alia uidentur:</s> <s xml:id="echoid-s27045" xml:space="preserve"> ideo neceſſe eſt ut pri <lb/>mum, quod comprehendit uirtus diſtinctiua ex intentionibus appropriatis ſormę uiſibili, ſit quid-<lb/>ditas lucis & coloris, ut ſicut illis primò & per ſe debetur uiſiua comprehenſio, ſic & illorũ quiddi-<lb/>tatibus debeatur per ſe & primò operatio uirtutis diſtinctiuæ, ut illis, quorum præſentia prius re-<lb/>lucetin organis uiſiuis, quæ omnia ſecundum plus & minus accedunt ad diaphanitatem.</s> <s xml:id="echoid-s27046" xml:space="preserve"> Patet er-<lb/>go propoſitum.</s> <s xml:id="echoid-s27047" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1042" type="section" level="0" n="0"> <head xml:id="echoid-head839" xml:space="preserve" style="it">68. Comprehenſio coloris, in eo, quod eſt color, eſt prior comprehenſione quidditatis coloris. <lb/>Ex quo patet, quòd prior eſt comprehenſio omnium uiſibilium in eo, quòd in ſuo genere uiſibilia <lb/>ſunt, quàm ſuarum ſpecialium quiddit atum. Alhazen 19 n 2.</head> <p> <s xml:id="echoid-s27048" xml:space="preserve">Viſus enim comprehẽdit colorem, & ſentit quòd eſt color, prius quàm ſentiat cuiuſmodiſit ille <lb/>color, ut patet in coloribus fortibus poſitis in loco non multum luminoſo.</s> <s xml:id="echoid-s27049" xml:space="preserve"> Ibi enim comprehendit <lb/>quidem uiſus colores indiſtinctè tantùm:</s> <s xml:id="echoid-s27050" xml:space="preserve"> diſtinguuntur autẽ per aduentum maioris lucis aut per <lb/>longam intuitionem.</s> <s xml:id="echoid-s27051" xml:space="preserve"> Primum ergo, quod comprehendit uiſus ex forma coloris, eſt mutatio mem-<lb/>brilentientis & coloratio eius:</s> <s xml:id="echoid-s27052" xml:space="preserve"> quoniam apud peruentum formæ in uiſum coloratur uiſus, qui ſen <lb/>tiens ſe coloratum ſtatim ſentit colorem:</s> <s xml:id="echoid-s27053" xml:space="preserve"> & deinde ex diſtinctione & comparatione ipſius ad colo-<lb/>res notos uiſui, comprehendit quidditatem coloris.</s> <s xml:id="echoid-s27054" xml:space="preserve"> Comprehenſio ergo coloris in eo, quod eſt <lb/>color, eſt ante comprehenſionẽ quidditatis ipſius coloris, quę fit non per ſolũ ſenſum uiſus, ſed per <lb/> <pb o="115" file="0417" n="417" rhead="LIBER TERTIVS."/> cognitionem, quando idem color prius fuit à uiſu comprehenſus, & forma eius eſt in memoria ani-<lb/>mæ conſeruata.</s> <s xml:id="echoid-s27055" xml:space="preserve"> Et ſi uiſus comprehendat colorem extraneum, quem nunquam uidit, tunc com-<lb/>prehendet quòd eſt color, & tamen neſciet cuiuſmodi ſit coloris, ſed comparando ipſum coloribus <lb/>alijs, aſsimilabit propinquiori colori ſimili ſibi, & fortè plures uidentes illum colorem ſimul in eo-<lb/>dem lumine, aſsimilabunt ipſum colorib.</s> <s xml:id="echoid-s27056" xml:space="preserve"> diuerſis, ut accidit in colore confecto ex diſſolutione cor-<lb/>poris commixti ex cupro & argento.</s> <s xml:id="echoid-s27057" xml:space="preserve"> Illum enim aliquis aſsimilabit uiriditati, quæ eſt ex cupro, & <lb/>aliquis lazulio colori, qui fit ex argento.</s> <s xml:id="echoid-s27058" xml:space="preserve"> Patet ergo per has experimentationes, quòd comprehen-<lb/>ſio coloris in eo, quòd eſt color, eſt prior comprehenſione quid ditatis coloris.</s> <s xml:id="echoid-s27059" xml:space="preserve"> Et quoniam color eſt <lb/>primũ uiſibile poſt lucem, patet, quòd prior eſt comprehenſio omnium uiſibilium in eo, quòd uiſi-<lb/>bilia ſunt, quàm ſuarum ſpecialium quidditatum:</s> <s xml:id="echoid-s27060" xml:space="preserve"> prius enim comprehenditur in ſenſu uiſus in ge-<lb/>nere ipſe ſitus, quàm aliqua ſpecies ſitus, & prius figura in genere, quàm aliqua ſpecialis figura:</s> <s xml:id="echoid-s27061" xml:space="preserve"> & ſi <lb/>contingat in uiſu abſolui ſpecialem, remanet tamen generalis, uel illa, quę eſt primi generis, uel illa, <lb/>quæ eſt generis ſecundi.</s> <s xml:id="echoid-s27062" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s27063" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1043" type="section" level="0" n="0"> <head xml:id="echoid-head840" xml:space="preserve" style="it">69. Diuerſarum intentio num uiſibilium per rationem & diſtinctionem fit comprehenſio ſi-<lb/>mul in inſtanti: ſimilium uerò in tempore. Alhazen 13. 15. 71 n 2.</head> <p> <s xml:id="echoid-s27064" xml:space="preserve">Figura enim & magnitudo, & diaphanitas, & plura ſimilia, quando comprehenduntur primo a-<lb/>ſpectu, qui ſemper fit in inſtanti temporis per 55 huius, ſtatim utſe uiſui præſentant, per rationem <lb/>& diſtinctionem, propter uelocitatem rationis in eodem inſtanti comprehenduntur, & omnes in-<lb/>tentiones, quæ ſunt in illis.</s> <s xml:id="echoid-s27065" xml:space="preserve"> Virtus enim diſtinctius non arguit per compoſitionem & ordinationẽ <lb/>propoſitionum ad formam ſyllogiſticam.</s> <s xml:id="echoid-s27066" xml:space="preserve"> Sicut ergo in intellectu, qui eſt habitus principiorum, in <lb/>actuali intelligentia propoſitionum uniuerſalium & per ſe manifeſtarum non indiget aliquanto tẽ-<lb/>pore, nec etiam indiget aliquanto tẽpore in apprehendẽdo concluſiones particulares ex illis, quo-<lb/>niam cum intellectu propoſitionis uniuerſalis ſimul accipit concluſionem, quæ immediatè ſequi-<lb/>tur exilla:</s> <s xml:id="echoid-s27067" xml:space="preserve"> ideo quia anima humana apta nata eſt ad arguen dum ſine difficultate & labore:</s> <s xml:id="echoid-s27068" xml:space="preserve"> unde e-<lb/>tiam non percipit homo, quòd comprehenſio, quæ fit per rationem & diſtinctionem, fiat per argu-<lb/>mentum, ſicut puerulus ex duobus pulchris diſtinguens & eligens pulchrius, non percipit quòd id <lb/>fiat per uiam argumentationis & conſiderationis eligendorum.</s> <s xml:id="echoid-s27069" xml:space="preserve"> Hoc itaque modo ſimili & confor-<lb/>mi, quatenus eſt poſsibile, fit omnium intẽtionum uiſibilium per rationem & diſtinctionem in in-<lb/>ſtanti comprehenſio.</s> <s xml:id="echoid-s27070" xml:space="preserve"> Diſtin ctio enim & argumentatio uirtutis diſtinctiuæ fit ſtatim uenientibus <lb/>formis intra medium nerui communis:</s> <s xml:id="echoid-s27071" xml:space="preserve"> quoniam totum corpus extenſum à ſuperficie primi oculi <lb/>recipiente formas uſque ad medium nerui communis, eſt ſentiens & diaphanum, & fit per ipſum <lb/>tranſitus intentionis formarum in inſtanti, cum ſtatim ultra oculi ſubſtantiam ſit ſpiritus uiſibilis <lb/>diaphanus, per quem uirtus ſenſitiua defertur ad totum diaphanum omnium humorum & tunica-<lb/>rum amborum oculorum:</s> <s xml:id="echoid-s27072" xml:space="preserve"> omnia enim diaphana illa illuminãtur à luce, & colorantur à colore uno <lb/>uel diuerſis ſecundum diuerſitatem colorum corporis ſenſati:</s> <s xml:id="echoid-s27073" xml:space="preserve"> & corpus, quod eſt in cõcauitate ner <lb/>ui communis, eſt ultimum corpus, ad quod perueniunt lux & color.</s> <s xml:id="echoid-s27074" xml:space="preserve"> Cum ergo extenditur forma <lb/>â ſuperficie prima membri ſentientis uſque ad medium nerui communis, quæ libet pars corporis <lb/>ſentientis ſentiet formam:</s> <s xml:id="echoid-s27075" xml:space="preserve"> & cum peruenerit in concauum nerui communis, tunc comprehendi-<lb/>tur ab ultimo ſentiente:</s> <s xml:id="echoid-s27076" xml:space="preserve"> & tunc fit diſtinctio formarum:</s> <s xml:id="echoid-s27077" xml:space="preserve"> non tamen inter a ctum diſtinctionis & a-<lb/>ctum primi aſpectus eſt differentia temporalis:</s> <s xml:id="echoid-s27078" xml:space="preserve"> quoniam ſicut lumen in uno inſtanti ſe multiplicat <lb/>per mundi diametrum propter corporis medij diaphanitatem:</s> <s xml:id="echoid-s27079" xml:space="preserve"> ſic etiam formæ ſenſibiles, ut oſten-<lb/>ſum eſt per 55 huius, in inſtanti pertingunt trans medium quodcun que corpus diaphanum ad me-<lb/>dium nerui communis, ubi per uirtutem animæ ſentiuntur, comprehenduntur.</s> <s xml:id="echoid-s27080" xml:space="preserve"> & diſtinguuntur.</s> <s xml:id="echoid-s27081" xml:space="preserve"> Et <lb/>quoniam uirtus animæ eſt indiuiſibilis, fit hoc totum ſimul in unico inſtanti.</s> <s xml:id="echoid-s27082" xml:space="preserve"> Quando uerò inten-<lb/>tiones uiſibilium ſunt ſimiles ualde, ut eſt uiriditas rutæ uiriditati mentę:</s> <s xml:id="echoid-s27083" xml:space="preserve"> tunc non fit ipſorum di-<lb/>ſtinctio in inſtanti illo, quo utraq;</s> <s xml:id="echoid-s27084" xml:space="preserve"> illarum uiriditatum comprehẽditur à uiſu, ſed poſt comparatio-<lb/>nem unius ad alteram factam:</s> <s xml:id="echoid-s27085" xml:space="preserve"> fit ergo in alio inſtanti, & ſic inter inſtans primi aſpectus ſimplicis & <lb/>inſtans diſtinctionis ex comparatione, neceſſarium eſt tempus medium aſſumi.</s> <s xml:id="echoid-s27086" xml:space="preserve"> Patet ergo illud, <lb/>quod proponebatur.</s> <s xml:id="echoid-s27087" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1044" type="section" level="0" n="0"> <head xml:id="echoid-head841" xml:space="preserve" style="it">70. Comprehenſionem quidditatis coloris in tempore fieri eſt neceſſe. Ex quo patet, quòd com-<lb/>prehenſio quidditatis omniũ ſimilium uiſibilium non fit niſi in tempore. Alhazen 20 n 2.</head> <p> <s xml:id="echoid-s27088" xml:space="preserve">Fit enim comprehenſio quidditatis coloris poſt comprehenſionem coloris in eo, quòd eſt co-<lb/>lor, ut patet per 68 huius.</s> <s xml:id="echoid-s27089" xml:space="preserve"> Et quoniam color in eo, quòd eſt color, non poteſt comprehendi per a-<lb/>ſpectum ſimplicem niſi in inſtanti per 55 huius:</s> <s xml:id="echoid-s27090" xml:space="preserve"> cum ergo comprehenſio quidditatis alicuius co-<lb/>loris ſit com poſita ex comprehenſione coloris in eo, quòd eſt color, & inſuper ex alia diſtinctiua <lb/>comparatione conſequente, per quam quid ditas unius coloris diſtinguitur à quidditate alterius <lb/>coloris:</s> <s xml:id="echoid-s27091" xml:space="preserve"> ideo quòd omnes colores mixti habent eſſentialem conuenientiam in actu & hypoſtaſi lu-<lb/>cis, & inſuper habent plures ipſorum adinuicem maximam conuenientiam in proximitate mixtio-<lb/>nis:</s> <s xml:id="echoid-s27092" xml:space="preserve"> palàm, quia illa diſtinctio quidditatis ipſorum colorum completur in alio inſtanti temporis, <lb/>quàm comprehendatur à uiſu, ſed inter quæ libet duo diſtantia eſt tempus medium.</s> <s xml:id="echoid-s27093" xml:space="preserve"> Quia ita que <lb/>cõprehenſio quidditatis coloris fit per diſtin ctionẽ unius coloris ab alio, palã per præmiſſam, quo- <pb o="116" file="0418" n="418" rhead="VITELLONIS OPTICAE"/> niam illa diſtinctio completur in tempore:</s> <s xml:id="echoid-s27094" xml:space="preserve"> ergo & comprehenſio quidditatis neceſſariò fit in tem-<lb/>pore.</s> <s xml:id="echoid-s27095" xml:space="preserve"> Viſus quoque non comprehendit quidditatem coloris, niſi per intuitionem:</s> <s xml:id="echoid-s27096" xml:space="preserve"> quoniam ſi co-<lb/>lor non fuerit in aliqua ſuperficie, ita ut ſibi poſsint infigi axes uiſuales in tempore ſenſibili, non cõ-<lb/>prehendit uiſus quidditatem colorum:</s> <s xml:id="echoid-s27097" xml:space="preserve"> unde in rebus uelociter motis nó diſtinguitur quidditas co <lb/>loris:</s> <s xml:id="echoid-s27098" xml:space="preserve"> ſed ſi plures in re uelociter mota ſint colores, uidebuntur omnes indiſtinctè unus permixtus <lb/>color, ut patet in pila diuerſi coloris uelociter mota per iactum fortem.</s> <s xml:id="echoid-s27099" xml:space="preserve"> Patet ergo, comprehenſio-<lb/>nem quidditatis ipſius coloris in tempore fieri eſt neceſſe:</s> <s xml:id="echoid-s27100" xml:space="preserve"> & ex hoc patet, quòd cóprehenſio quan-<lb/>titatis omuium formarum uiſibiliũ non fit, niſi in tempore.</s> <s xml:id="echoid-s27101" xml:space="preserve"> Si enim uiſus non comprehendit quid-<lb/>ditatem coloris, qui comprehẽditur ſolo ſenſu uiſus, niſi in tempore:</s> <s xml:id="echoid-s27102" xml:space="preserve"> palàm, quòd plus indiget tem <lb/>pore in intentionibus aliorum uiſibilium, quæ comprehenduntur plurimum diſtinctione & cogni <lb/>tione, Omnium itaque intentionum uiſibilium quidditatum comprehenſio fit in tempore, licet il-<lb/>lud tempus quandoque ſit ualde paruum.</s> <s xml:id="echoid-s27103" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s27104" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1045" type="section" level="0" n="0"> <head xml:id="echoid-head842" xml:space="preserve" style="it">71. Viſus in formis indiuidualibus minoritempore comprehendit intentiones ſpeciales quàm <lb/>indiuiduales. Alhazen 72 n 2.</head> <p> <s xml:id="echoid-s27105" xml:space="preserve">Quando enim uiſus comprehendit aliquod indiuiduum hominis, comprehendit ipſum eſſe ho-<lb/>minem prius, quàm comprehendit formam eius particularem:</s> <s xml:id="echoid-s27106" xml:space="preserve"> & ſortè per intentiones formæ ho-<lb/>minis, uel per aliqua conuenientia propria formæ hominis comprehendit ipſum eſſe hominem, <lb/>quamuis non comprehendat lineationem ſuæ faciei, utpote ex rectitudine corporis & ordinatione <lb/>membrorum corporis.</s> <s xml:id="echoid-s27107" xml:space="preserve"> Indiuidualitas autem rei uiſæ non comprehendetur niſi ex comprehenſio-<lb/>neintentionum particularium illi indiuiduo propriarum omnium aut quarundam:</s> <s xml:id="echoid-s27108" xml:space="preserve"> & hæc com-<lb/>prehendi non poſſunt niſi poſt comprehenſionem uniuerſalium intentionum, quæ ſunt ex gene-<lb/>re uel ſpecie illius indiuidui, omnium aut quarundam:</s> <s xml:id="echoid-s27109" xml:space="preserve"> ſed comprehenſio formæ partialis eſt <lb/>in minori tempore quàm formæ totius.</s> <s xml:id="echoid-s27110" xml:space="preserve"> Et quoniam indiuidualitas addit aliquid ſuper ſpeciali-<lb/>tatem, patet, quòd in diuidualitas eſt quaſi quædam totalitas reſpectu ſpecialitatis.</s> <s xml:id="echoid-s27111" xml:space="preserve"> Compre-<lb/>henſio ergo ſpecialitatis rei uiſæ eſt in minori tempore quàm comprehenſio indiuidualitatis.</s> <s xml:id="echoid-s27112" xml:space="preserve"> Et <lb/>hoc proponebatur.</s> <s xml:id="echoid-s27113" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1046" type="section" level="0" n="0"> <head xml:id="echoid-head843" xml:space="preserve" style="it">72. Intentiones ſpeciales & indiuiduales quorundam uiſibilium aſſuetorum minoritempore <lb/>alijs intentionibus ſpecialibus & indiuidualibus comprehenduntur. Alhazen 73 n 2.</head> <p> <s xml:id="echoid-s27114" xml:space="preserve">Quædam enim ſpecierum uiſibilium aſſuetorum non aſsimilantur alijs ſpeciebus, ut ſpecies ho-<lb/>minis, quæ propter corporis rectitudinem nulli aliorum animalium aſsimilatur:</s> <s xml:id="echoid-s27115" xml:space="preserve"> & quædam aſsimi <lb/>lantur alijs ſpeciebus, ut ſpecies equi, quæ aſsimilatur multis animalibus in tota forma.</s> <s xml:id="echoid-s27116" xml:space="preserve"> Tempus er <lb/>go, in quo uiſus comprehendit ſpeciem indiuidui hominis, & comprehendit ipſum eſſe hominem, <lb/>eſt minus tempore, in quo comprehendit equum eſſe equum, & maximè quando comprehendit <lb/>utrunque iſtorum in magna remotione:</s> <s xml:id="echoid-s27117" xml:space="preserve"> quoniam uiſus comprehendens indiuiduum hominis mo-<lb/>tum localiter, ſtatim comprehendet ipſum eſſe animal:</s> <s xml:id="echoid-s27118" xml:space="preserve"> ex motu & ex corporis erectione compre-<lb/>hendetipſum eſſe hominem:</s> <s xml:id="echoid-s27119" xml:space="preserve"> ſed licet per motum etiam poſsit comprehendere, quòd indiuiduum <lb/>equi ſit animal, & per numerum quatuor pedum comprehendatipſum eſſe beſtiam, non tamẽ pro-<lb/>pter hoc comprehendet ipſum eſſe equum:</s> <s xml:id="echoid-s27120" xml:space="preserve"> quoniam intentiones equinę, quæ ſunt à ſpatio remoto <lb/>uiſu perceptibiles, ſunt in pluribus quadrupedum, quæ aſsimilantur equo in pluribus eſſentiali-<lb/>bus & accidentalibus intentionibus, ut in mulo & alijs.</s> <s xml:id="echoid-s27121" xml:space="preserve"> Si itaque uiſus non comprehendit aliquam <lb/>intentionum propriarum equo, non comprehendet illum eſſe equum.</s> <s xml:id="echoid-s27122" xml:space="preserve"> Quia itaque tempus, in quo <lb/>comprehendit uiſus erectionem corporis hominis, non eſt ſicut tempus, in quo comprehendit for-<lb/>mam equi cum intentionibus particularibus, per quas diſtinguitur equus ab alijs beſtijs, ut eſt li-<lb/>neatio ſuæ faciei, & extenſio colli, & uelocitas motus, & paſſuum amplitudo:</s> <s xml:id="echoid-s27123" xml:space="preserve"> comprehenſio igitur <lb/>ſpeciei hominis eſt in minori tempore quàm comprehenſio ſpeciei equi:</s> <s xml:id="echoid-s27124" xml:space="preserve"> quamuis enim illa duo <lb/>tempora ſunt parua, tamen unum ipſorum ſecundum omnes diſpoſitiones eius eſtmaius altero:</s> <s xml:id="echoid-s27125" xml:space="preserve"> <lb/>& ſimiliter quia roſę hortenſi nullus alius flos aſsimilatur in forma ſuæ ſpeciei, uel etiam in inten-<lb/>tione ſuæ rubedinis, ideo uiſus in minori tempore comprehendit eius ſpeciem per rubedinem ro-<lb/>ſeaceum, quàm ſpeciem rutæ per eius uiriditatem, cui multæ herbarum aſsimilantur.</s> <s xml:id="echoid-s27126" xml:space="preserve"> Et uniuer-<lb/>ſaliter quidditates omnium ſpecierum, quæ poſſunt aſsimilari alijs, non adeò citò comprehen-<lb/>dunturà uiſu, ſicut quidditates omnium ſpecierum, quæ paucis uel nullis aſsimilantur.</s> <s xml:id="echoid-s27127" xml:space="preserve"> Et ſimi-<lb/>liter etiam eſt de indiuiduis:</s> <s xml:id="echoid-s27128" xml:space="preserve"> quoniam indiuiduum nulli alij aſsimilatum comprehenditur per mo-<lb/>dicam intuitionem & per ſigna:</s> <s xml:id="echoid-s27129" xml:space="preserve"> illud autem indiuiduum, quod aſsimilatur alij indiuiduo, opor-<lb/>tet quòd comprehendatur per multam intuitionem.</s> <s xml:id="echoid-s27130" xml:space="preserve"> Patet ergo illud, quod proponebatur.</s> <s xml:id="echoid-s27131" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1047" type="section" level="0" n="0"> <head xml:id="echoid-head844" xml:space="preserve" style="it">73. Virtus ſenſitiua comprehendit quantit atem anguli, quem in centro uiſus reſpicit ſuper-<lb/>ficies rei uiſæ ſolùm ex comprehenſione partis ſuperficiei uiſus, in qua figuratur forma rei uiſæ. <lb/>Alhazen 44 n 2.</head> <p> <s xml:id="echoid-s27132" xml:space="preserve">Quamuis enim ordo puræ matheſis ſit in hoc, ut per quantitatem angulorum ſciatur quantitas <lb/>partium ſuperficierum ſphæricarum illis angulis ſubtenſarum, eò quòd ſicut centrum eſt princi-<lb/>pium conſtitutionis totius ſphæræ:</s> <s xml:id="echoid-s27133" xml:space="preserve"> ſic partes angulorũ 8 ſolidorũ, qui ſunt circa centrũ ſphæræ, ut <lb/> <pb o="117" file="0419" n="419" rhead="LIBER QVARTVS."/> circa quodlibet uniuerſi pun ctum ſint principium diſtin ctiuũ omnis partis ſuperficiei ſphæræ per <lb/>87 t 1 huius:</s> <s xml:id="echoid-s27134" xml:space="preserve"> tamen in hac ſcientiæ ſenſibilis experientia, quę naturalium rerũ conditione permiſce <lb/>tur, uirtus ſenſitiua ex cõprehenſione partis ſuperficiei uiſus, in qua figuratur forma rei uiſæ, cõpre-<lb/>hendit à poſteriori uia ſenſib, competente quantitatem anguli, quem ιn centro uiſus reſpicic ſuper-<lb/>ficies pręfata.</s> <s xml:id="echoid-s27135" xml:space="preserve"> Senſus enim uiſus naturaliter comprehendit illam ſuperficiem, in qua figuratur ſor-<lb/>ma rei uiſæ per diſtin ctionem lucis & coloris, qui per le accidũt in illa parte ab alijs ſuperficieb.</s> <s xml:id="echoid-s27136" xml:space="preserve"> ui-<lb/>ſus diſtincta:</s> <s xml:id="echoid-s27137" xml:space="preserve"> & quando comprehendit quantitatem illius partis, tũc imaginatur angulos, quos re-<lb/>ſpiciũt illę partes, & comprehendit quantitates eorũ apud centrũ uiſus ſecũdũ quantitatem partiũ <lb/>ſuperſiciei uiſus illis angulis ſubtenſarũ:</s> <s xml:id="echoid-s27138" xml:space="preserve"> anguli autem tũc non certificantur, niſi per motum uiſus <lb/>reſpicientis ſuper diametros rei uiſę, aut ſuper ſpatiũ, cuius uiſus magnitudinem uult ſcιre.</s> <s xml:id="echoid-s27139" xml:space="preserve"> Patet er <lb/>go propoſitũ.</s> <s xml:id="echoid-s27140" xml:space="preserve"> Et licet lineę radiales in centro uiſus non concurrant, quoniam peruenit interſectio <lb/>axium uiſualiũ ad mediũ punctum nerui cõm unis, ut in pręcedentium theorem atum pluribus pa-<lb/>tuit:</s> <s xml:id="echoid-s27141" xml:space="preserve"> partes tamen ſuperficiei ipſius uiſus informantur ſecundum modũ, quo lineę radiales concur <lb/>rerent in centro ipſius uiſus, niſi ipſos refractio in medio ſecundi diaphani præueniret, ut patet per <lb/>22 huius:</s> <s xml:id="echoid-s27142" xml:space="preserve"> & hoc eſt notatu dignũ, quoniam nos in ſequentib.</s> <s xml:id="echoid-s27143" xml:space="preserve"> utemur centro uiſus, ac ſi lineę radia-<lb/>les in ipſo angulariter con currant:</s> <s xml:id="echoid-s27144" xml:space="preserve"> quia ſecundum hoc omnis uiſio informatur.</s> <s xml:id="echoid-s27145" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1048" type="section" level="0" n="0"> <head xml:id="echoid-head845" xml:space="preserve">VITELLONIS FI-<lb/>LII THVRINGORVM ET PO-<lb/>LONORVM OPTICAE LIBER QVARTVS.</head> <p style="it"> <s xml:id="echoid-s27146" xml:space="preserve"><emph style="sc">Tractavimvs</emph> in præmiſſo tertio libro deproprietatibus organi ui <lb/>ſiui, & de eſſentialibus modis uidendi:</s> <s xml:id="echoid-s27147" xml:space="preserve"> nunc aũt restat, ut in hoc quarto li <lb/>bro proſequamur proprietates omnium uiſibiliũ, quæ, ut in principio tertij <lb/>diximus, ſunt uigintiduo, quorum tantũ duo, ſcilicet lux & color ſunt per <lb/>ſe uiſibilia:</s> <s xml:id="echoid-s27148" xml:space="preserve"> alia uerò uidentur per accidens:</s> <s xml:id="echoid-s27149" xml:space="preserve"> uel quia pluribus alijs ſenſibus percipiuntur:</s> <s xml:id="echoid-s27150" xml:space="preserve"> <lb/>uel quia non uιdentur, niſi propter luces & colores, ut patet in ſingulis ipſorũ.</s> <s xml:id="echoid-s27151" xml:space="preserve"> Et quoniã <lb/>in præmiſſo tertio libro de uiſione lucis & coloris ſatis præmiſimus:</s> <s xml:id="echoid-s27152" xml:space="preserve"> ideo nũc alia 20 uiſi <lb/>bilia reſtant pertractãda.</s> <s xml:id="echoid-s27153" xml:space="preserve"> Hæc ita omnia, paßiones quo et deceptiones, quæ accidunt <lb/>uiſib.</s> <s xml:id="echoid-s27154" xml:space="preserve"> & potentijs intrinſecis animæ circa illa naturaliter uel mathematicè, prout natu-<lb/>rarei et poßibilitas noſtra fert, ſub modo demõſtratiõis ſuo ordine per curremus, unicui <lb/>ipſorũ ſuæ u<gap/>ſi<gap/>s modũ, et in ſe et in ſuis partib.</s> <s xml:id="echoid-s27155" xml:space="preserve"> præmittentes:</s> <s xml:id="echoid-s27156" xml:space="preserve"> deceptiones quo, quæ in <lb/>ipſo uel tantũ uirtuti uiſiuæ, uel etiam potentijs animæ intrinſecis, ut quæ uirtuti diſtin-<lb/>ctiuæ & ratiocinatiuæ accidũt, cũ ſtudio ſubiungemus:</s> <s xml:id="echoid-s27157" xml:space="preserve"> quæ aũt præmittimus, ſunt iſta.</s> <s xml:id="echoid-s27158" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1049" type="section" level="0" n="0"> <head xml:id="echoid-head846" xml:space="preserve">DEFINITIONES.</head> <p> <s xml:id="echoid-s27159" xml:space="preserve">1.</s> <s xml:id="echoid-s27160" xml:space="preserve"> Forma dicitur directè uiſibus incidere, à qua producta linea recta ſuper ſuper-<lb/>ficiem uiſus eſt perpendicularis, incidens ipſi centro foraminis uueæ.</s> <s xml:id="echoid-s27161" xml:space="preserve"> 2.</s> <s xml:id="echoid-s27162" xml:space="preserve"> Obliquè <lb/>uerò incidere dicitur, à qua producta recta dicto modo, nõ eſt perpẽdicularis.</s> <s xml:id="echoid-s27163" xml:space="preserve"> 3.</s> <s xml:id="echoid-s27164" xml:space="preserve"> Li <lb/>nea directè uiſui oppoſita dicitur illa, cuiaxis radialis perpendiculariter incidit ſe-<lb/>cundum aliquod eius punctum.</s> <s xml:id="echoid-s27165" xml:space="preserve"> 4.</s> <s xml:id="echoid-s27166" xml:space="preserve"> Linea obliquata ad uiſum dicitur, cui axis ra-<lb/>dialis ad nullum ſui punctum perpendiculariter poteſt incidere.</s> <s xml:id="echoid-s27167" xml:space="preserve"> 5.</s> <s xml:id="echoid-s27168" xml:space="preserve"> Superficies di-<lb/>rectè oppoſita dicitur, quando axis radialis perpendiculariter erigitur ſuper illam.</s> <s xml:id="echoid-s27169" xml:space="preserve"> <lb/>6.</s> <s xml:id="echoid-s27170" xml:space="preserve"> Superficies uerò obliquata ad uiſum dicitur, quando axis radιalis punctis illius <lb/>ſuperficiei incidit obliquè.</s> <s xml:id="echoid-s27171" xml:space="preserve"> 7.</s> <s xml:id="echoid-s27172" xml:space="preserve"> Complementũ directionis in oppoſitione uiſus eſt, <lb/>cum axis perpēdicularis incidit medio ſuperficiei, uel lineæ oppoſitæ uiſui:</s> <s xml:id="echoid-s27173" xml:space="preserve"> & quã-<lb/>tò magis punctus, cui incidit axis perpendiculariter, fuerit medio ſuperficiei aut li-<lb/>neæ propinquior, tantò erit ſuperficies uel linea maioris directionis in oppoſi-<lb/>tione.</s> <s xml:id="echoid-s27174" xml:space="preserve"> 8.</s> <s xml:id="echoid-s27175" xml:space="preserve"> Vera comprehenſio per uiſum, diciturilla, inter quã & ueritatem rei <lb/>uiſæ non eſt diuerſitas ſenſibilis omnino, reſpectu totius rei uiſæ.</s> <s xml:id="echoid-s27176" xml:space="preserve"> 9.</s> <s xml:id="echoid-s27177" xml:space="preserve"> Remo-<lb/>tio unius rei ab altera, eſt priuatio contactus interilla.</s> <s xml:id="echoid-s27178" xml:space="preserve"> 10.</s> <s xml:id="echoid-s27179" xml:space="preserve"> Conus dicitur pyra-<lb/> <pb o="118" file="0420" n="420" rhead="VITELLONIS OPTICAE"/> mis rotunda, uel uertex pyramidis cuiuſcunque rotundæ uellateratæ.</s> <s xml:id="echoid-s27180" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1050" type="section" level="0" n="0"> <head xml:id="echoid-head847" xml:space="preserve">PETITIONES.</head> <p> <s xml:id="echoid-s27181" xml:space="preserve">Petimus autem hæc.</s> <s xml:id="echoid-s27182" xml:space="preserve"> J.</s> <s xml:id="echoid-s27183" xml:space="preserve"> Sub eleuatioribus radijs uiſa eleuatiora apparere, ſub de-<lb/>cliuioribus uerò decliuiora:</s> <s xml:id="echoid-s27184" xml:space="preserve"> & ſimiliter ſub dexterioribus radijs uiſa dexteriora ap <lb/>parere, ſub ſiniſterioribus uerò ſiniſteriora.</s> <s xml:id="echoid-s27185" xml:space="preserve"> 2.</s> <s xml:id="echoid-s27186" xml:space="preserve"> Item ſub pluribus angulis uiſa per-<lb/>ſpicatius uideri.</s> <s xml:id="echoid-s27187" xml:space="preserve"> 3.</s> <s xml:id="echoid-s27188" xml:space="preserve"> Item omnes uiſus æqualis diſpoſitionis æquè ueloces eſſe.</s> <s xml:id="echoid-s27189" xml:space="preserve"> 4.</s> <s xml:id="echoid-s27190" xml:space="preserve"> <lb/>Item omne to tum uideri maius ſua parte.</s> <s xml:id="echoid-s27191" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1051" type="section" level="0" n="0"> <head xml:id="echoid-head848" xml:space="preserve">THE OREMATA</head> <head xml:id="echoid-head849" xml:space="preserve" style="it">1. Ex intemperata proportione circumſtantiarum formarum uiſibilium ad uiſum fit dece-<lb/>ptio in uiſu, non ſolùm ſecundũ ſe, ſed ſecundum uirtutẽ animæ diſtinctiuam. Alhazen 1 n 3.</head> <p> <s xml:id="echoid-s27192" xml:space="preserve">Exhis, quæ declarata ſunt in libro tertio, patet octo eſſe neceſſaria ad perfectam operationem <lb/>uiſus, quæ ſunt:</s> <s xml:id="echoid-s27193" xml:space="preserve"> lux per 1 th 3 huius.</s> <s xml:id="echoid-s27194" xml:space="preserve"> Item diſtantia uiſibilis à uiſu per 15 th 3 huius.</s> <s xml:id="echoid-s27195" xml:space="preserve"> Item ſitus oppoſi-<lb/>tionis ipſius uiſus per 2 th 3 huius:</s> <s xml:id="echoid-s27196" xml:space="preserve"> uel ſitus reſpectu axis cõmunis per 44 th 3 huius.</s> <s xml:id="echoid-s27197" xml:space="preserve"> Item magni-<lb/>tudo corporis per 19 th 3 huius.</s> <s xml:id="echoid-s27198" xml:space="preserve"> Item ſoliditas corporis uidendi per 14 th 3 huius.</s> <s xml:id="echoid-s27199" xml:space="preserve"> Item diaphani-<lb/>tas aeris per 13 th 3 huius.</s> <s xml:id="echoid-s27200" xml:space="preserve"> Item tempus conueniens intuitioni faciendę per 56 th 3 huius.</s> <s xml:id="echoid-s27201" xml:space="preserve"> Item fa-<lb/>nitas uiſus per 16 th 3 huius.</s> <s xml:id="echoid-s27202" xml:space="preserve"> Quodlιbet autem iſtorum latitudinem habet proportionatam ad rem <lb/>uiſam.</s> <s xml:id="echoid-s27203" xml:space="preserve"> Lux enim habet latitudinem, quoniam lux maxima impedit uiſum, & lux debilis non edu-<lb/>cit uiſibilia in actum agendi in uiſum:</s> <s xml:id="echoid-s27204" xml:space="preserve"> unde corpora minuta uel intentiones uiſibiles minutæ non <lb/>uidentur in luce debili:</s> <s xml:id="echoid-s27205" xml:space="preserve"> ſed eſt etiam latitudo in ea luce, quæ eſt magnitudini corporis proportiona <lb/>ta.</s> <s xml:id="echoid-s27206" xml:space="preserve"> Diſtantia quoq;</s> <s xml:id="echoid-s27207" xml:space="preserve"> uiſibilis à uiſu ſiue ipſius remotio latitudinem habet:</s> <s xml:id="echoid-s27208" xml:space="preserve"> corpus enim aliquod ab <lb/>aliqua diſtantia plenè comprehenditur, & ab alia non plenè:</s> <s xml:id="echoid-s27209" xml:space="preserve"> & inter illas diſtantias eſt latitudo ma-<lb/>gna, in qua fit plena comprehenſio corporis illius, & ſecundum quod maius fuerit corpus, maior e-<lb/>rit latitudo diſtantiæ ſpatij, ſecundum quam ipſum poterit uideri.</s> <s xml:id="echoid-s27210" xml:space="preserve"> Similiter cum magna fuerit de-<lb/>clinatio alicuius corporis à directione oppoſitionis ipſius uiſus, non comprehenduntur particulæ <lb/>uel notæ paruæ, quæ ſunt in ipſo, quę in parua declinatione corporis uiderentur:</s> <s xml:id="echoid-s27211" xml:space="preserve"> & eſt etiam inter <lb/>illas declinationes latitudo.</s> <s xml:id="echoid-s27212" xml:space="preserve"> Similiter corpus paruum ſitum extra axem communem uidebitur mul <lb/>tum elongatum & occultatum, & idem corpus ſitum circa axem communem uidebitur apertè:</s> <s xml:id="echoid-s27213" xml:space="preserve"> pa-<lb/>làm autem, quòd ſitus reſpectu axis communis habet latitudinem, quoniã habet habitudinẽ pro-<lb/>portionatam ad corporis magnitudinem & minutias ipſius.</s> <s xml:id="echoid-s27214" xml:space="preserve"> Magnitudo etiam corporis habet la-<lb/>titudinem:</s> <s xml:id="echoid-s27215" xml:space="preserve"> ſi enim partes rei uiſæ non fuerint proportionales totali magnitudini uiſæ, occultabun <lb/>tur uiſui:</s> <s xml:id="echoid-s27216" xml:space="preserve"> & ſi fuerint proportionales totali uiſę magnitudini, ſit tamen corpus totale modicum, ad-<lb/>huc non uidebuntur.</s> <s xml:id="echoid-s27217" xml:space="preserve"> unde in picturis modicis aliquas particulas non ſtatim percipimus uiſu, licet <lb/>proportionales ſint ſuis totis:</s> <s xml:id="echoid-s27218" xml:space="preserve"> latιtudo ergo magnitudinis rei uiſæ proportionata debet eſſe ad to-<lb/>tale corpus, cuius fuerit pars illa uiſa magnitudo.</s> <s xml:id="echoid-s27219" xml:space="preserve"> Soliditas quoque habet latitudinem proportio-<lb/>natam ad rem uiſam.</s> <s xml:id="echoid-s27220" xml:space="preserve"> Sienim in corpore aliquo color ualde acutus fuerit, licet ipſum ſit paucæ ſo-<lb/>liditatis:</s> <s xml:id="echoid-s27221" xml:space="preserve"> illud tamen corpus uideri poterit, quod non accideret maiori ſoliditate in illo corpore <lb/>exiſtente, quoniam fortè color propter reflexionem uehementem luminis impediret uiſum, quæ <lb/>reflexio fieret propter magnam corporis ſoliditatem:</s> <s xml:id="echoid-s27222" xml:space="preserve"> & ſi color fuerit obſcurus, tunc fortè accidet <lb/>minus ſolidum debilius uideri colore eius obſcuro exiſtente.</s> <s xml:id="echoid-s27223" xml:space="preserve"> Diaphanitas etiam aeris habet lati-<lb/>tudinem:</s> <s xml:id="echoid-s27224" xml:space="preserve"> quia per flammas & per ſumos non fit uiſio rerum minutarum, ſed ſortè groſſarum, ſicut <lb/>ſi per ipſa uideretur charta, non ſcriptura.</s> <s xml:id="echoid-s27225" xml:space="preserve"> Tempus etiam conueniens intuitioni facien dæ latitudi-<lb/>nem habet:</s> <s xml:id="echoid-s27226" xml:space="preserve"> quia corpus ſubitò uiſum pertranſiens, non comprehenditur à uiſu, & quandoque mo-<lb/>tus trochi non uidetur, quia eſt uelociſsimus in tempore ualde paruo.</s> <s xml:id="echoid-s27227" xml:space="preserve"> Sanitas etiam uiſus latitu-<lb/>dinem habet:</s> <s xml:id="echoid-s27228" xml:space="preserve"> in quibuſdam enim infirmitatibus minutiæ corporis, niſi abſcondantur, in minori ſpa <lb/>tio percipiuntur, & uiſus debiliores non uident illa, quæ occurrunt uiſibus fortioribus.</s> <s xml:id="echoid-s27229" xml:space="preserve"> Vniuer-<lb/>ſaliter ergo quilibet iſtorum modorum, in quo non uerificatur forma rei uiſæ, ſicut eſt in rei uerita-<lb/>te, eſt egreſſus à temperantia ad rem illam uidendam proportionata:</s> <s xml:id="echoid-s27230" xml:space="preserve"> & hęc omnia ſe alterutrum re-<lb/>ſpiciunt ſecundum conuenientes adinuicem proportiones:</s> <s xml:id="echoid-s27231" xml:space="preserve"> & quodlibet ipſorum ad alia octo con <lb/>uenientem oportet quòd habeat diſpoſitionem, quorum pertractationem relin quimus conſidera-<lb/>tioni animæ res propinquius intuentis.</s> <s xml:id="echoid-s27232" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1052" type="section" level="0" n="0"> <head xml:id="echoid-head850" xml:space="preserve" style="it">2. Impoßibile eſt uiſum unam intentionum uiſibilium per ſe ſolam comprehendere. Alha-<lb/>zen 63 n 2.</head> <p> <s xml:id="echoid-s27233" xml:space="preserve">Viſus enim perſe comprehendit formas uiſibilium, quæ ſunt corporales:</s> <s xml:id="echoid-s27234" xml:space="preserve"> omnes autem formæ <lb/>corporales ſunt compoſitę ex multis intentionibus uiſibilibus particularib.</s> <s xml:id="echoid-s27235" xml:space="preserve"> prædictis:</s> <s xml:id="echoid-s27236" xml:space="preserve"> ſicut magni <lb/>tudo nõ eſt ſine figura, & figura nõ eſt ſine ſitu:</s> <s xml:id="echoid-s27237" xml:space="preserve"> & hæc omnia nõ ſunt ſine colore, & color nõ eſt ſine <lb/>luce, & luxnon diffunditur niſi in corpore.</s> <s xml:id="echoid-s27238" xml:space="preserve"> Viſus itaq;</s> <s xml:id="echoid-s27239" xml:space="preserve"> non comprehendit aliquam iſtarum particu <lb/>lariũ intentionũ, niſi ex cõprehenſione formarũ uiſibiliũ cõpoſitarũ ex plurib.</s> <s xml:id="echoid-s27240" xml:space="preserve"> intentionibus parti-<lb/>cularibus, quarũ quãlibet ſimul cõprehendit uiſus.</s> <s xml:id="echoid-s27241" xml:space="preserve"> Et quoniã nulla intentionũ per ſe ſola cõplet ali <lb/> <pb o="119" file="0421" n="421" rhead="LIBER QVARTVS."/> quá formarum corporalium ſenſibiliũ:</s> <s xml:id="echoid-s27242" xml:space="preserve"> palá, quòd impoſsibile eſt uiſum cóprehendere aliquã illarũ <lb/>intentionum ſolam per ſe, ſed ſemper ſunt plures illarum intentionum ſimul in forma ſenſibili con-<lb/>gregatæ.</s> <s xml:id="echoid-s27243" xml:space="preserve"> Viſus ergo cõprehendit ſimul ſemper multas intentiones particulares, quę ſolũ diſtinguũ <lb/>tur anxilio uirtutιs diftinctiuæ per imaginationẽ:</s> <s xml:id="echoid-s27244" xml:space="preserve"> & ſic demum uiſus comprehendit intentionem <lb/>particularium quamlibet diſtinctam.</s> <s xml:id="echoid-s27245" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s27246" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1053" type="section" level="0" n="0"> <head xml:id="echoid-head851" xml:space="preserve" style="it">3. Non ſub quocun angulo res ſenſibiles uidentur.</head> <p> <s xml:id="echoid-s27247" xml:space="preserve">Quod omne quod uidetur, ſub angulo uideatur, patet per corollarium 18 t 3 huius:</s> <s xml:id="echoid-s27248" xml:space="preserve"> & etiam cum <lb/>per 19 th 3 huius, corpus uiſibile, oportet, ut ſit alicuius quantitatis reſpectu uiſus, ad hoc ut actu ui-<lb/>deatur:</s> <s xml:id="echoid-s27249" xml:space="preserve"> palàm ergo, quòd ſub angulo contingentiæ, qui eſt indiuiſibilis per 16 p 3, non erit poſsibile <lb/>aliquam rem uideri.</s> <s xml:id="echoid-s27250" xml:space="preserve"> Omnis enim angulus, ſub quo poteſt fieri uiſio, eſt diuiſibilis per axem pyrami <lb/>dis radialis ſuperficiei ipſius uiſus perpendiculariter incidentem:</s> <s xml:id="echoid-s27251" xml:space="preserve"> eò quòd omnis uiſio fit per pyra-<lb/>midem uiſualem, cuius baſis ſuperficies rei uiſę per 18 t 3 huius:</s> <s xml:id="echoid-s27252" xml:space="preserve"> uel ad minus ille angulus eft ſub illo <lb/>axe, & ſub alia linea longitudinis radialis pyramidis contentus, ut declaratum eſt in 54 th 3 huius:</s> <s xml:id="echoid-s27253" xml:space="preserve"> <lb/>eſt ergo rectilineus:</s> <s xml:id="echoid-s27254" xml:space="preserve"> eſt ergo diuiſibilis per 9 p 1.</s> <s xml:id="echoid-s27255" xml:space="preserve"> Et quoniam maximus angulorum, ſub quo fit uiſio, <lb/>eſt quaſi rectus, ideo, quòd diameter foraminis uueæ, quæ ſubtenditur illi angulo in centro uiſus, <lb/>eſt quaſi æ qualis lateri cubi inſcriptibilis ſphæræ uueę, uel lateri quadrati inſcriptibilis circulo ma-<lb/>gno illius ſphæræ, ut oſten dim us in 4 t 3 huius:</s> <s xml:id="echoid-s27256" xml:space="preserve"> illi autem lateri ſemper ſubtenditur angulus rectus <lb/>per 33 p 6:</s> <s xml:id="echoid-s27257" xml:space="preserve"> quoniam eius chorda eſt quarta circuli.</s> <s xml:id="echoid-s27258" xml:space="preserve"> Si ergo uiſio fieret ac ſi lineæ radiales in centro <lb/>uueę concurrerent:</s> <s xml:id="echoid-s27259" xml:space="preserve"> tunc maximus angulus, ſecun dum quem fit uiſio, eſſet quaſi angulus rectus ſo-<lb/>lidus, ita quòd pyramis uifualis maxima fieret rectangula, & ſemidiameter baſis illius pyramidis fie <lb/>ret æqualis axi:</s> <s xml:id="echoid-s27260" xml:space="preserve"> fit autem uiſio ac ſi lineę concurrant in centro uiſus, ut patet per 73 th 3 huius:</s> <s xml:id="echoid-s27261" xml:space="preserve"> cen-<lb/>trum uerò uiſus eſt remotius in profundo, quàm centrum uueę per 8 th 3 huius.</s> <s xml:id="echoid-s27262" xml:space="preserve"> Maior ergo angu-<lb/>lus, ſecundum quem fit uiſio, eſt minor recto, ſed non multùm minor, quia illorum centrorum, ſphę <lb/>ræ ſcilicet uueę & oculi, non eſt magna diſtantia:</s> <s xml:id="echoid-s27263" xml:space="preserve"> & fit axis maximæ pyramidis uiſualis maior ſemi-<lb/>diametro baſis eius, ſed non multò maior.</s> <s xml:id="echoid-s27264" xml:space="preserve"> Et hoc patet etiam experimento:</s> <s xml:id="echoid-s27265" xml:space="preserve"> quoniam ſi aliquis ſtet <lb/>in campo plano erectus, & aperiat oculum, ut amplius poteſt, tunc uidebit quaſi quartá circuli ma-<lb/>ioris ſphærę cœleſtis per Zenith capitis tranſeuntis:</s> <s xml:id="echoid-s27266" xml:space="preserve"> & per anguli huius diuiſionem fit uiſio partiũ <lb/>illius, & omnium rerum illis angulis ſubtenſarum, quouſq;</s> <s xml:id="echoid-s27267" xml:space="preserve"> perueniatur ad angulum minimum, qui <lb/>ſi diuideretur, non fieret uiſio ſecundum illum.</s> <s xml:id="echoid-s27268" xml:space="preserve"> Licet enim omnis angulus rectilineus mathemati-<lb/>cus ſit in inſinitum diuiſibilis:</s> <s xml:id="echoid-s27269" xml:space="preserve"> in angulis tamen naturalib.</s> <s xml:id="echoid-s27270" xml:space="preserve"> ſecundum quorum diſpoſitionem fit paſ-<lb/>ſio operationis ſenſibilis, oportet ut ſit ſtatus in diuiſione, quãdo minus ſenſibile illo non erit:</s> <s xml:id="echoid-s27271" xml:space="preserve"> neq;</s> <s xml:id="echoid-s27272" xml:space="preserve"> <lb/>ergo erit uiſio ſenſibilis ſecundũ illũ:</s> <s xml:id="echoid-s27273" xml:space="preserve"> ſed omnis uiſio eſt ſenſibilis, cũ ſit actio ſenſitiua:</s> <s xml:id="echoid-s27274" xml:space="preserve"> nulla ergo ui <lb/>ſio erit ſecũdum angulum minorem illo.</s> <s xml:id="echoid-s27275" xml:space="preserve"> Non ergo ſub quocunq;</s> <s xml:id="echoid-s27276" xml:space="preserve"> angulo res ſenſibiles uidentur:</s> <s xml:id="echoid-s27277" xml:space="preserve"> & <lb/>hoc intelligendũ eſt ſecun dum lineas radiales perpendiculariter ſuperficiebus uiſuũ incidentes, <lb/>nõ obliquè, ſecundum quas obliquas fit incerta uiſio, & confuſio formarum rerum uiſibilium in ui-<lb/>ſu, ut oſtendimus in 17 th 3 huius.</s> <s xml:id="echoid-s27278" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s27279" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1054" type="section" level="0" n="0"> <head xml:id="echoid-head852" xml:space="preserve" style="it">4. Forma lineæ perpendiculariter ſuperficiei uiſus oppoſitæ non uidetur: quoniam per ipſam <lb/>ſolùm fit distinctio punctualis: oppoſitæ uerò uiſui ſecundum longitudinem, ſecundum ſui for-<lb/>mam propriam uidetur.</head> <p> <s xml:id="echoid-s27280" xml:space="preserve">Eſto, ut uiſui, cuius centrum ſit d, perpendiculariter incidat linea a b, quæ ſit linea ſenſibilis, utpo <lb/>te corpus longum inſenſibilem habens latitudinem, <lb/>ut pilus, qui, licet ſit columna rotunda, uellaterata, ba <lb/> <anchor type="figure" xlink:label="fig-0421-01a" xlink:href="fig-0421-01"/> ſis tamen eius à uiſu percipi non poteſt:</s> <s xml:id="echoid-s27281" xml:space="preserve"> dico, quòd ta <lb/>le corpus taliter diſpoſitum non uidetur:</s> <s xml:id="echoid-s27282" xml:space="preserve"> eſt enim an <lb/>gulus in centro uiſus, cui ſubtenditur baſis eius dia-<lb/>metri penitus inſenſibilis, ſecundum quẽ non poteſt <lb/>fieri uiſio per præmiſſam.</s> <s xml:id="echoid-s27283" xml:space="preserve"> In formis tamen alijs uiſis <lb/>fiet per incidentiam formæ huiuſmodi corporis ali <lb/>qua diſtinctio pũctualis inſenſibilis:</s> <s xml:id="echoid-s27284" xml:space="preserve"> quoniam forma <lb/>puncti illius perpendiculariter incidentis ſe formis <lb/>punctorum circumſtantium aliarum formarum im-<lb/>miſcebit:</s> <s xml:id="echoid-s27285" xml:space="preserve"> & cum non ſit de genere illorum, neceſſariò <lb/>aliquam faciet diſtinctionem, ita, ut illorum corporũ <lb/>formæ actu, licet non multum ſenſibiliter diſtinguan <lb/>tur, nec ad naturam continuitatis unius lineæ pertin-<lb/>gant.</s> <s xml:id="echoid-s27286" xml:space="preserve"> Oppoſita uerò linea uiſui ſecundum longitudi-<lb/>nem, ſiue ſit poſitio directa uel obliqua, ſemper ipſa <lb/>ſecundũ ſui formã propriam uidebitur:</s> <s xml:id="echoid-s27287" xml:space="preserve"> quoniã tota <lb/>eius lõgitudo ſub angulo uno, & partes eius ſub angulis ſenſibilib.</s> <s xml:id="echoid-s27288" xml:space="preserve"> perueniẽt ad uiſum:</s> <s xml:id="echoid-s27289" xml:space="preserve"> ut ſi linea a b <lb/>c opponatur uiſui d ſecũ dũ ſuilõ gitudinẽ, & ſit diſtãtia cõueniens:</s> <s xml:id="echoid-s27290" xml:space="preserve"> tũc ipſa tota uidebitur ſub angu <lb/>lo a d c:</s> <s xml:id="echoid-s27291" xml:space="preserve"> & pars eius a b ſub angulo a d b, & pars eius b c ſub angulo b d c:</s> <s xml:id="echoid-s27292" xml:space="preserve"> & ſiue ſit recta uel curua, uel <lb/>irregularis, ſemք aliqua lõgitudo ſecũdũ latitudinẽ deſcribetur in oculi ſuքficie, ſecundũ qđ eſt in <lb/> <pb o="120" file="0422" n="422" rhead="VITELLONIS OPTICAE"/> ipſa linea, & per longitudinem ſenſibilem & latitudinem non ſenſatam uirtus diſtin ctiua formá li-<lb/>neę iudicabit, ut accidit in lineis naturalιbus, quę ſunt, ut quidam pili.</s> <s xml:id="echoid-s27293" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s27294" xml:space="preserve"/> </p> <div xml:id="echoid-div1054" type="float" level="0" n="0"> <figure xlink:label="fig-0421-01" xlink:href="fig-0421-01a"> <variables xml:id="echoid-variables432" xml:space="preserve">d a b a d b c</variables> </figure> </div> </div> <div xml:id="echoid-div1056" type="section" level="0" n="0"> <head xml:id="echoid-head853" xml:space="preserve" style="it">5. Superficiei oppoſitæ uiſui taliter, ut imaginata protrahi ſecet oculum per eius cẽtrum, una <lb/>tantum linea: oppoſitæ uerò uiſui ſecundum latitudinem forma propria uidetur.</head> <p> <s xml:id="echoid-s27295" xml:space="preserve">Oppoſita enim uiſui ſuperficie quacunq;</s> <s xml:id="echoid-s27296" xml:space="preserve"> per modum, quo proponitur, formæ omnium puncto-<lb/>rum perpendiculariter incident ſuperſiciei uiſus, & concurrent in centro.</s> <s xml:id="echoid-s27297" xml:space="preserve"> Et quoniã ſorma cuiusli <lb/>bet illorum punctorũ facit aliquam diſtinctionem in uiſu per pręcedentem:</s> <s xml:id="echoid-s27298" xml:space="preserve"> & omnia illa puncta ſe-<lb/>cundum longitudinem incidentia coniuncta cadunt in quadam linea:</s> <s xml:id="echoid-s27299" xml:space="preserve"> patet, quòd illius ſuperficiei <lb/>ſic diſpoſitę una tantùm linea uidetur.</s> <s xml:id="echoid-s27300" xml:space="preserve"> Oppoſita uerò illa ſuperficie ſecundum ſui longitudinem ui <lb/>ſui, forma cuiuslibet ſuę lineę uidetur ſecundum ſui formam propriam linearis per pręcedentẽ.</s> <s xml:id="echoid-s27301" xml:space="preserve"> To-<lb/>ta ergo ſuperficies ſecundum ſui formam propriam uidetur, quoniam ſemper uidebitur longitudo <lb/>& latitudo aliqua, ſiue illa ſuperficies ſit plana, ſiue concaua, uel conuexa:</s> <s xml:id="echoid-s27302" xml:space="preserve"> quia non eſt differentia in <lb/>illis, quantum ad propoſitam paſsionem.</s> <s xml:id="echoid-s27303" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s27304" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1057" type="section" level="0" n="0"> <head xml:id="echoid-head854" xml:space="preserve" style="it">6. Corporum uiſibus oppoſitorum ſolæ ſuperficies à ſolo uiſu comprehenduntur.</head> <p> <s xml:id="echoid-s27305" xml:space="preserve">Quia enim à ſolo uiſu corpora uidentur, ſecũdum quòd formę ipſorum uiſui ſe offerũt, & in eius <lb/>ſuperficie depinguntur, ut patet per 17 t 3 huius:</s> <s xml:id="echoid-s27306" xml:space="preserve"> formę uerò profunditatis corporum uiſib.</s> <s xml:id="echoid-s27307" xml:space="preserve"> non offe <lb/>runtur, ſed ſolùm ea, quibus ſecundum longum & latum lineę ductę à centro uiſus incidunt, ut pa-<lb/>tet per 2 t 3 huius:</s> <s xml:id="echoid-s27308" xml:space="preserve"> hęc aũt eſt diſpoſitio ſuperficialis.</s> <s xml:id="echoid-s27309" xml:space="preserve"> Corporum ergo uiſibus oppoſitorũ ſolę ſuper-<lb/>ficies à ſolo uiſu comprehenduntur:</s> <s xml:id="echoid-s27310" xml:space="preserve"> & ſi una ſit corporis ſuperficies, ſiue ſit illud corpus ſphæricũ <lb/>cócauum uel conuexum, una tantũ uidebitur ſuperficies:</s> <s xml:id="echoid-s27311" xml:space="preserve"> & ſi plures ſint corporis unius ſuperfici-<lb/>es, utin corporibus omnium planarum ſuperficierum & columnarum rotundarum, & pyramidum <lb/>& portionum ſphęricarum quarumcunq;</s> <s xml:id="echoid-s27312" xml:space="preserve">, ſemper non niſi plures ſuperficies uidebuntur, ac ſi non <lb/>eſſet corpus, ſed quędam ſuperficies ſic extenſa, ſine corporis medij incluſione, Patet ergo propoſi <lb/>tum.</s> <s xml:id="echoid-s27313" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s27314" xml:space="preserve"> paſsio in lineis uiſui accidens, deſcendit in ſuperficierum uiſionem, & paſsio in ſu-<lb/>perficiebus uiſui accidens deſcendit in corporũ uiſionem, ſola uerò corpora per ſe uideantur, quia <lb/>ſolùm corpora per ſe ſunt entia naturalia ſenſibilia, & ſuperficies & lineę in illis ſunt imaginabilia:</s> <s xml:id="echoid-s27315" xml:space="preserve"> <lb/>parcendum nobis eſt, ſi uiſuales paſsiones corporum proponimus per modũ paſsionum uiſualium <lb/>ſuperficierum uel linearum:</s> <s xml:id="echoid-s27316" xml:space="preserve"> quia quòd uiſibus in lineis accidit, corporum longitudini ſolùm uel la-<lb/>titudini ſolùm æſtimamus accidere, & quod ſuperficiebus accidit, corporum longitudini ſimul cũ <lb/>latitudine neceſſarium eſt euenire:</s> <s xml:id="echoid-s27317" xml:space="preserve"> unde ſecundum iſtorum conueniẽtiam ſuperficiebus uel lineis <lb/>nos poſterius utemur.</s> <s xml:id="echoid-s27318" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1058" type="section" level="0" n="0"> <head xml:id="echoid-head855" xml:space="preserve" style="it">7. Omnium æqualium uiſibilium quod à propinquiori uidetur, ſub maiori angulo uidetur: <lb/>quod uerò à remotiori, ſub minori. Euclides 5 th. opticorum.</head> <p> <s xml:id="echoid-s27319" xml:space="preserve">Sint duę magnitudines ęquales b c & d e:</s> <s xml:id="echoid-s27320" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s27321" xml:space="preserve"> centrum uiſus a:</s> <s xml:id="echoid-s27322" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s27323" xml:space="preserve"> b c propinquior uiſui a, quá <lb/>ipſa d e.</s> <s xml:id="echoid-s27324" xml:space="preserve"> Dico, quòd b c uidetur ſub maiori angulo quã <lb/> <anchor type="figure" xlink:label="fig-0422-01a" xlink:href="fig-0422-01"/> d e.</s> <s xml:id="echoid-s27325" xml:space="preserve"> Ducantur enim lineę a b & a c:</s> <s xml:id="echoid-s27326" xml:space="preserve"> & quoniam hę lineę <lb/>concurrunt in puncto a, palàm quòd non ęquidiſtant <lb/>per definitionem ęquidiſtantium llnearum:</s> <s xml:id="echoid-s27327" xml:space="preserve"> ſed neque <lb/>concurrent in aliquo alio puncto quá in a:</s> <s xml:id="echoid-s27328" xml:space="preserve"> quia ſic duę <lb/>rectę lineę ſuperficiem includerent, quod eſt impoſsi-<lb/>bile.</s> <s xml:id="echoid-s27329" xml:space="preserve"> Nunquam ergo concurrent alibi quã in puncto a:</s> <s xml:id="echoid-s27330" xml:space="preserve"> <lb/>protractę uerò ultra puncta b & c, ſemper ibunt in di-<lb/>ſtantiam:</s> <s xml:id="echoid-s27331" xml:space="preserve"> ergo nunquã tangent lineam d e, nec erit ui-<lb/>ſio aliquorum punctorũ lineę d e ſecundum illas per 2 <lb/>th:</s> <s xml:id="echoid-s27332" xml:space="preserve"> 3 huius.</s> <s xml:id="echoid-s27333" xml:space="preserve"> Si ergo extrema puncta lineę d e uideri de-<lb/>bent, hoc erit ſecũdum lineas cadentes intra lineas b a <lb/>& c a, quę ſint lineę a d & a e.</s> <s xml:id="echoid-s27334" xml:space="preserve"> Siue ergo magnitudines <lb/>b c & d e ęquidiſtent, ſiue non, ducta à puncto d ęquidi <lb/>ſtante & ęquali ipſi b c per 31 p 1, patet per 34 t 1 huius, <lb/>quoniam angulus b a c erit maior angulo d a e:</s> <s xml:id="echoid-s27335" xml:space="preserve"> lineæ <lb/>ergo a d & a e ſunt angulum b a c diuidentes.</s> <s xml:id="echoid-s27336" xml:space="preserve"> Quia ue <lb/>rò angulus partialis d a e eſt minor totali angulo b a c, <lb/>patetid, quod proponebatur.</s> <s xml:id="echoid-s27337" xml:space="preserve"> Et ſimiliter demonſtrandũ eſt, ſi linearũ b c & c d ęqualiũ ſit idem ter <lb/>minus, qui eſt c:</s> <s xml:id="echoid-s27338" xml:space="preserve"> uel ſi ſint adinuicem declinãtes:</s> <s xml:id="echoid-s27339" xml:space="preserve"> tũc enim idem accidit, quod prius.</s> <s xml:id="echoid-s27340" xml:space="preserve"> Totum tamen, <lb/>quod hic proponitur per 108 th:</s> <s xml:id="echoid-s27341" xml:space="preserve"> 1 huius, perfectius patet:</s> <s xml:id="echoid-s27342" xml:space="preserve"> remotioris enim uiſi axis pyramidis radia <lb/>lis, eſt lógior axe pyramidis radialis propinquioris uiſi:</s> <s xml:id="echoid-s27343" xml:space="preserve"> unde anguli ſolidi in uerticibus illarum py-<lb/>ramidum diuerſantur.</s> <s xml:id="echoid-s27344" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s27345" xml:space="preserve"/> </p> <div xml:id="echoid-div1058" type="float" level="0" n="0"> <figure xlink:label="fig-0422-01" xlink:href="fig-0422-01a"> <variables xml:id="echoid-variables433" xml:space="preserve">d e d b c a</variables> </figure> </div> </div> <div xml:id="echoid-div1060" type="section" level="0" n="0"> <head xml:id="echoid-head856" xml:space="preserve" style="it">8. Vnumquod uiſorũ longitudinem habet ſpatij, ultra quod non uidetur. Eucli. 3 th. optico.</head> <p> <s xml:id="echoid-s27346" xml:space="preserve">Sit centrũ oculi b:</s> <s xml:id="echoid-s27347" xml:space="preserve"> res autẽ d g ſit uiſa ſub minimo angulo uiſui determinato.</s> <s xml:id="echoid-s27348" xml:space="preserve"> Dico, quòd illa res, <lb/>quę eſt d g, in ulteriori ſpatio nõ uidebitur.</s> <s xml:id="echoid-s27349" xml:space="preserve"> Sit enim poſitũ g d in ſpatio ulteriori, in quo ſit pũctus k:</s> <s xml:id="echoid-s27350" xml:space="preserve"> <lb/> <pb o="121" file="0423" n="423" rhead="LIBER QVARTVS."/> ſi igitur g d uideturin pũctok, neceſſe eſt per præmiſſam ipſam ſub minori angulo uideri quàm ſub <lb/>illo minimo, qui eſt uiſui determinatus.</s> <s xml:id="echoid-s27351" xml:space="preserve"> Non autẽ ſub minori angulo <lb/> <anchor type="figure" xlink:label="fig-0423-01a" xlink:href="fig-0423-01"/> uiſibile potuit ad uiſum multiplicari:</s> <s xml:id="echoid-s27352" xml:space="preserve"> angulus enim multiplicationis <lb/>formarũ ad uiſum tam diu poteſt diminui, donec formæ punctorum <lb/>extremitatis rei uniantur, & fiant pũctus unus, nec res uidebitur niſi <lb/>punctualis, uel nullo modo uidebitur.</s> <s xml:id="echoid-s27353" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s27354" xml:space="preserve"/> </p> <div xml:id="echoid-div1060" type="float" level="0" n="0"> <figure xlink:label="fig-0423-01" xlink:href="fig-0423-01a"> <variables xml:id="echoid-variables434" xml:space="preserve">k d g b</variables> </figure> </div> </div> <div xml:id="echoid-div1062" type="section" level="0" n="0"> <head xml:id="echoid-head857" xml:space="preserve" style="it">9. Remotio rei uiſæ ab ipſo uiſu non eſt comprebenſibilis à ſolo <lb/>ſenſu uiſus, ſed auxilio uirtutis animæ cognoſcitiue & diſtinctiuæ. <lb/>Alhazen 24 n 2.</head> <p> <s xml:id="echoid-s27355" xml:space="preserve">Intentio enim remotionis inter duo corpora eſt priuatio cótactus <lb/>propter aliquod ſpatium inter illa duo corpora exiſtens:</s> <s xml:id="echoid-s27356" xml:space="preserve">nó compre-<lb/>henditur ergo remotio per ſe à uiſu, ſed auxilio uirtutis cognoſcitiuæ <lb/>& diſtinctiuæ cognoſcentis utrumq;</s> <s xml:id="echoid-s27357" xml:space="preserve"> extremorũ corporum & diſtin-<lb/>guentis inter illa:</s> <s xml:id="echoid-s27358" xml:space="preserve"> fit tamẽ talis comprehenſio non in tempore, ſed in <lb/>inſtanti:</s> <s xml:id="echoid-s27359" xml:space="preserve"> quieſcunt enim in anima intẽtiones ſenſibiles, per quas com <lb/>prehenditur remotio.</s> <s xml:id="echoid-s27360" xml:space="preserve"> Et quia illæ intentiones requieuerũt in anima <lb/>per tempora longiora, ideo propter nimiam frequentationem & ite-<lb/>rationem formarum illarum pluries in uiſu factam, nõ indiget uirtus <lb/>diſtinctiua nouis collationibus temporalibus apud comprehenſionẽ <lb/>illarum intentionũ, ſed ſtatim comprehendit remotionem ſimul cum <lb/>rei comprehenſione propter cognitionem antecedentem.</s> <s xml:id="echoid-s27361" xml:space="preserve"> Quia enim oculis apertis res oppoſita <lb/>uiſui ſtatim uidetur, & ſtatim clauſis oculis uelre ablata ab oppoſitione res non uidetur:</s> <s xml:id="echoid-s27362" xml:space="preserve"> concludit <lb/>ratio, quòdillud, quod accidit eſſe in uiſu apud aliquem certum ſitum, & non manet poſt eius abla-<lb/>tionem, non eſt fixum intra uiſum.</s> <s xml:id="echoid-s27363" xml:space="preserve"> Et quoniam forma ipſius, per quam uidetur, nó eſt intra uiſum:</s> <s xml:id="echoid-s27364" xml:space="preserve"> <lb/>eſt ergo ab extrinſeco, à corpore ſcilicet exiſtente extra uiſum, non contingente uiſum:</s> <s xml:id="echoid-s27365" xml:space="preserve"> eſt ergo in-<lb/>ter uiſum & illam rem uiſam remotio.</s> <s xml:id="echoid-s27366" xml:space="preserve"> Fit autem hæc argumentatio nô in tempore, ſed ſtatim ſimul <lb/>cum ſimplici aſpectu uιſionis:</s> <s xml:id="echoid-s27367" xml:space="preserve"> quoniam ex frequẽtia uiſionis cum hac argumentatione quieſcit in <lb/>anima uniuerſalis propoſitio, quam etiam anima non percipit apud ſe quieſcentem:</s> <s xml:id="echoid-s27368" xml:space="preserve"> & eſt, quòd o-<lb/>mnia uiſibilia ſunt extra uiſum, & quòd inter quamlibet rem uiſam & ipſum uiſum eſt remotio.</s> <s xml:id="echoid-s27369" xml:space="preserve"> Pa-<lb/>tet ergo propoſitum.</s> <s xml:id="echoid-s27370" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1063" type="section" level="0" n="0"> <head xml:id="echoid-head858" xml:space="preserve" style="it">10. Quantitas remotionis comprehenditur à uiſu auxilio uirtutis diſtinctiuæ, cum remotio <lb/>reſpicit corpora ordinatæ & continuata. Alhazen 24 n 2.</head> <p> <s xml:id="echoid-s27371" xml:space="preserve">Quantitas remotionis diuerſa eſt ab intẽtione remotionis in eo, quòd eſt remotio:</s> <s xml:id="echoid-s27372" xml:space="preserve"> quoniam in-<lb/>tentio remotionis facit priuationem contactus aliquorum duorum corporum propter ſpatium in-<lb/>ter illa duo corpora exiſtẽs:</s> <s xml:id="echoid-s27373" xml:space="preserve"> ſed quantitas remotionis eſt quantitas ſpatij inter illa duo corpora re-<lb/>mota exiſtentis.</s> <s xml:id="echoid-s27374" xml:space="preserve"> Nulla itaq;</s> <s xml:id="echoid-s27375" xml:space="preserve"> quantitas remotionis omnium uiſibilium comprehenditur per ſolum <lb/>ſenſum uiſus etiam cum auxilιo uirtutis diſtinctiuæ, niſi quantitas remotionis illorum uiſibilium, <lb/>quorum remotio reſpicit corpora ordinata & continuata, & quorum remotio eſt mediocris:</s> <s xml:id="echoid-s27376" xml:space="preserve"> tunc <lb/>enim cum uiſus comprehẽdit corpora ordinata & continuata, reſpicientia remotiones aliquorum <lb/>corporum, & certificat menſuras illorũ corporum:</s> <s xml:id="echoid-s27377" xml:space="preserve"> conſequenter quoq;</s> <s xml:id="echoid-s27378" xml:space="preserve"> certificat remotionis men <lb/>ſuram per menſuras illorum corporum & per quantitates ſpatiorum, quæ ſunt inter extremitates <lb/>corum:</s> <s xml:id="echoid-s27379" xml:space="preserve"> ſpatium enim, quod eſt inter duas extremitates uiſus & corporis, reſpicit rem otionem, quę <lb/>eſtinter uiſum & rem illam uiſam.</s> <s xml:id="echoid-s27380" xml:space="preserve"> Vnde cum uiſus comprehenderit menſuram illius ſpatij, com-<lb/>prehendet etiam menſuram remotionis rei uiſæ:</s> <s xml:id="echoid-s27381" xml:space="preserve"> & hoc fit certitudinaliter per corpora ordinata & <lb/>continuata in illo ſpatio exiſtentia & uerè comprehenſa, & cum remotio eſt mediocris.</s> <s xml:id="echoid-s27382" xml:space="preserve"> Dicimus <lb/>uerò corpora ordinata & continuata, quæ ſunt in aliqua linea quaſi recta diſpoſita, in æquali quaſi <lb/>ab inuicem diſtátia, ut ſunt arbores, uel montes, uel altæ turres, & ſimilia:</s> <s xml:id="echoid-s27383" xml:space="preserve"> per iſtorum enim nume-<lb/>rationem cum ipſorum diſtantia ab inuicem aliqualiter fuerit nota, & innoteſcit quantitas remo-<lb/>tionis eius, quòd ſecundum illam lineam à uiſibus eſt remotum.</s> <s xml:id="echoid-s27384" xml:space="preserve"> Mediocris uerò remotio eſt illa, in <lb/>qua non latet omnino quantitas rei ſenſibilis reſpectu quantitatis totius remotionis.</s> <s xml:id="echoid-s27385" xml:space="preserve"> Solum itaq;</s> <s xml:id="echoid-s27386" xml:space="preserve"> <lb/>illorum corporum remotio à uiſu comprehenditur uera comprehenſione, quorum remotio reſpi-<lb/>cit corpora ordinata & continuata;</s> <s xml:id="echoid-s27387" xml:space="preserve"> quorum corporum & ſpatiorum ipſa interiacentium quantitas <lb/>& menſura à uiſu poteſt comprehendi uera comprehenſione, & cum remotio eſt mediocris.</s> <s xml:id="echoid-s27388" xml:space="preserve"> Vnde <lb/>ſiue deficiat comprehenſio corporum cõtinuatorum & ordinatorum, ſiue deficiat mediocritas re-<lb/>motionis, nunquam comprehendetur remotio illorum corporum uera comprehẽſione, ſed ſolùm <lb/>ſecundum æſtimationem.</s> <s xml:id="echoid-s27389" xml:space="preserve"> Vnde uidens nubes in loco non montuoſo, æſtimabit nubes ualde pro-<lb/>pinquas cœlo:</s> <s xml:id="echoid-s27390" xml:space="preserve"> ſi autem nubes uideantur ſuper cacumina montiũ, uel ſub illis, tunc ſciet uiſus, quia <lb/>nubes ſunt propinquæ terræ.</s> <s xml:id="echoid-s27391" xml:space="preserve"> Cum ergo uiſus comprehendit uiſibilia, quorum remotionum quan-<lb/>titates non certificantur à uiſu:</s> <s xml:id="echoid-s27392" xml:space="preserve"> tunc uirtus diſtinctiua cognoſcit menſuras rem otionis eorum ſe-<lb/>cundum æſtimationem, non ſecundum certitudinem, & comparat remotionem earum ad remo-<lb/>tionem ſibi ſim lium ex uiſibilibus prius comprehenſis à uiſu.</s> <s xml:id="echoid-s27393" xml:space="preserve"> Quando itaq;</s> <s xml:id="echoid-s27394" xml:space="preserve"> uiſus comprehendit <lb/>aliquam rem uiſam remotam, ſtatim uirtus diſtinctiua comprehendit remotionem eius & menſu-<lb/>ram remotionis eius ſecundum quod poterit comprehendere, aut per certitudinem, aut per æſti-<lb/> <pb o="122" file="0424" n="424" rhead="VITELLONIS OPTICAE"/> mationem, & ſtatim remotio illius rei habebit in anima menſuram imaginatam.</s> <s xml:id="echoid-s27395" xml:space="preserve"> Corpora ucrò or-<lb/>dinata & continuata reſpicientia remotiones uiſibilium, ſunt utplurim um partes terræ & uiſibiliæ <lb/>affueta, quæ ſemper uel frequentius comprehenduntur à uiſu, ut quæ ſunt ſuper terræ ſuperficiem, <lb/>& corpus terrę interiacet illa corpora, ſicut etiam interiacetilla & corpus hominis aſpicientis:</s> <s xml:id="echoid-s27396" xml:space="preserve"> cor-<lb/>pus autem terræ in teriacens illa corpora, menſuratur à uiſu per numerum pedum, quoniam pes eſt <lb/>minima menſura conſueta hominibus ad menſurandum partes terræ propιnquas, per quas partes <lb/>terræ propin quas mẽſurantur partes terræ remotæ per uim diſtinctiuam animæ, propter frequen-<lb/>tationem comprehenſionis ſimilium partium illi parti terræ, quarum partium menſura quieſcit in <lb/>anima, ita, quòd etiam anima non percipit illarum partium quietem apud ſe ipſam.</s> <s xml:id="echoid-s27397" xml:space="preserve"> Peruenit autem <lb/>hæc menſura ad animam, quoniam quantitas ſpatiorum, quæ ſunt apud pedes hominum, compre-<lb/>henduntur à uiſu:</s> <s xml:id="echoid-s27398" xml:space="preserve"> menſurátur enim etiam ſine intentione per pedes hominum, quando frequen-<lb/>ter ambulant fuper illa ſpatia, ſicut etiam menſurantur per extêſiones brachiorum, & uirtus diſtin-<lb/>ctiua comprehen dit iſtam ueram menſurationem, & certificat ex ea quátitates partium terræ con-<lb/>tinuatarum cum corpore hominis uidentis:</s> <s xml:id="echoid-s27399" xml:space="preserve"> & hoc quieſcens in anima eſt principium menſuratio-<lb/>nis omnium remotionum ſecundum æſtimationem.</s> <s xml:id="echoid-s27400" xml:space="preserve"> Cum enim uiſus comprehen dit ſemper quan <lb/>titatem partium terræ ſibi uicinarum, remanet apud animam etiam quantitas linearum protenſa-<lb/>rum ab extremitatibus illarum partium terrę ad uiſum, & quantitas partis ſuperficiei membri ſen-<lb/>cientis, ad quam peruenit forma illarom partium terræ & per conſequens quantitates angulorum <lb/>peruenientium in centro uiſus, quos reſpiciunt illæ partes ſuperficierum uiſus per 73 th.</s> <s xml:id="echoid-s27401" xml:space="preserve"> 3 huius:</s> <s xml:id="echoid-s27402" xml:space="preserve"> <lb/>unde ſi homo erectus aſpexerit terram, quæ eſt ante pedes eius, tũclongitudo linearum radialium <lb/>erit quantitas lineæ erectionis, & ſuperducta ſuperiori palpebra uiſui, erit quaſi indiuiſibilis (ſicut <lb/>angulus contingentiæ) ille angulus, ſecũdum quem fit uiſio:</s> <s xml:id="echoid-s27403" xml:space="preserve"> & cum proſpexerit ulterius, augmen-<lb/>tabuntur lineæ radiales per 47 p 1, & eleuata ſuperiori palpebra, augebitur angulus, ita, ut cũ quan <lb/>titas ſpatij uiſi ad quantitatem ſemidiametri mundi acceſſerit, etiá quantitas anguli peruenit quaſi <lb/>ad rectum angulum, quoniam illi angulo ſubtendetur quarta circuli magniipſius ſphæræ cœleſtis <lb/>uiſæ.</s> <s xml:id="echoid-s27404" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s27405" xml:space="preserve"> hæ intentiones linearum & angulorum in anima quieuerint, fiunt principia com-<lb/>prehenſionis quantitatum remotionum quarumcunq;</s> <s xml:id="echoid-s27406" xml:space="preserve">: quoniam æ quales lineæ radiales & anguli <lb/>æſtimantur partibus æqualibus correſpondere, & utitur ijs uidens præteríntentionem compara-<lb/>tionis, & coadiuuatin hoc quantιtas angulorum & augmẽtatio ipſorum in longiori quantitate re-<lb/>ſpectu breuioris:</s> <s xml:id="echoid-s27407" xml:space="preserve"> & ſimiliter eſt in proportione longitudinis linearũ radialium, quam per ſe ſentit <lb/>uiſus auxilio uirtutis diſtin ctiuæ, perpendẽs quòd omne totum eſt maius ſua parte.</s> <s xml:id="echoid-s27408" xml:space="preserve"> Hoc itaq;</s> <s xml:id="echoid-s27409" xml:space="preserve"> mo-<lb/>do comprehendit uiſus auxilio uirtutis diſtinctiuę quantitatem remotionis rerum uiſarum ſecun-<lb/>dum lineas diſtantiarum ſuarum abinuicem & à uiſu, ſicut etiam uiſus quandoq;</s> <s xml:id="echoid-s27410" xml:space="preserve"> per uirtutem di-<lb/>ſtinctiuam comprehendit quantitates altitudinum alιquorum corporum eleuatorum ſuper ſuper-<lb/>ficiem terrę, ſicut turrium, parietum & montium, maximè cum remotio fuerit mediocris, uel etiam <lb/>altitudo, Cum autem remotio uel altitudo fuerit maxima:</s> <s xml:id="echoid-s27411" xml:space="preserve"> tunc partes paruæ, quæ ſunt in ultimo <lb/>ſpatij, non comprehenduntur à uiſu, nec dιſtιnguuntur per uirtutem diſtinctiuam, quoniam parua <lb/>quantitas in rem otione maxima latet uiſum:</s> <s xml:id="echoid-s27412" xml:space="preserve"> non enim facit angulum ſenſibilem apud centrum ui-<lb/>ſus, propter quod quantitas illorum nó certificatur per 3 huius.</s> <s xml:id="echoid-s27413" xml:space="preserve"> Nihil itaq;</s> <s xml:id="echoid-s27414" xml:space="preserve"> ex quantitatibus remo-<lb/>tionum uiſibilium certificatur, niſi per corpora ordinata & cõtinuata mediocris diſtantiæ ab inui-<lb/>cem & æqualis.</s> <s xml:id="echoid-s27415" xml:space="preserve"> Nulla quoq;</s> <s xml:id="echoid-s27416" xml:space="preserve"> remotio poteſt certificari, niſi cum uiſus aſsimilatrem otionẽ rei uiſæ <lb/>remotioni ſibi ſimili ex remotionibus aſſuetis & notis:</s> <s xml:id="echoid-s27417" xml:space="preserve"> remotio uerò mediocris, cuius quantitas <lb/>certificatur à uiſu, eſt rem otio, apud cuius ultimum non latet uiſum pars habẽs proportionem ſen-<lb/>ſibilem ad totam rem otionem:</s> <s xml:id="echoid-s27418" xml:space="preserve"> & cum uidens ſcit quantitatem anguli, ſecũdum quam uidetremo-<lb/>tionem certam cognitam ſibi:</s> <s xml:id="echoid-s27419" xml:space="preserve"> tunc ſecun dũ exceſſum uel diminutionem, uel æqualitatẽ, ad illum <lb/>angulum notum uirtus diſtinctiua iudicat remotiones ignotas, accipiendo ſecũdum quantitatem <lb/>anguli, quantitatem ipſius remotionis.</s> <s xml:id="echoid-s27420" xml:space="preserve"> Et etiã certificatur remotio per motum uiſus ſuper corpus <lb/>re piciens remotiones extremorũ alicuius ſuperficiei aut ſpatij.</s> <s xml:id="echoid-s27421" xml:space="preserve"> Generaliter autem forma rei uiſæ <lb/>& forma remotionis rei uiſæ, cuius remotio eſt mediocris, & reſpiciens corpora ordinata & conti-<lb/>nuata, perueniunt communiter in imaginatione ſimul apud intuitionem rei uiſæ, & uirtus diſtin-<lb/>ctiua illam dijudicat modo dicto.</s> <s xml:id="echoid-s27422" xml:space="preserve"> Pater ergo propoſitum.</s> <s xml:id="echoid-s27423" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1064" type="section" level="0" n="0"> <head xml:id="echoid-head859" xml:space="preserve" style="it">11. Aequalibus quantitatibus ex inæquali diſtantia uiſis: maior eſt proportio diſtantiæ ma-<lb/>ioris ad minorem, quàm maioris anguli, ſub quo fit uiſio, ad minorem. Euclides 8 th opt.</head> <p> <s xml:id="echoid-s27424" xml:space="preserve">Sint, exempli cauſſa, datæ æquales & æquidiſtantes magnitudines, quæ a b & g d:</s> <s xml:id="echoid-s27425" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s27426" xml:space="preserve"> cen-<lb/>trum uiſus punctum e:</s> <s xml:id="echoid-s27427" xml:space="preserve"> & ſit g d propinquior uiſui, a b uerò remotior:</s> <s xml:id="echoid-s27428" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s27429" xml:space="preserve"> illarum magnitudinum <lb/>una remota ab altera, & utraq;</s> <s xml:id="echoid-s27430" xml:space="preserve"> ipſarum ab ipſo centro uiſus ſenſibili remotione:</s> <s xml:id="echoid-s27431" xml:space="preserve"> ſtatuanturq́;</s> <s xml:id="echoid-s27432" xml:space="preserve"> tali-<lb/>ter, ut puncta b & d, quæ ſunt extremitates illarum duarum magnitudinum, ſint in uno axe pyra-<lb/>midis uiſualis:</s> <s xml:id="echoid-s27433" xml:space="preserve"> & ſecundum illum axem formæ illorum punctorum perueniát ad uiſum.</s> <s xml:id="echoid-s27434" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s27435" xml:space="preserve"> <lb/>puncta b & d ſecundum eandem lineam ad uiſum ſe multiplicent:</s> <s xml:id="echoid-s27436" xml:space="preserve"> palam quòd oportet puncta a & <lb/>g ſecundum diuerſas lineas, quę a e & g e, ad uiſum peruenire.</s> <s xml:id="echoid-s27437" xml:space="preserve"> Et quoniã, ut patet per 7 huius, ma-<lb/>gnitudo a b, quę eſt remotior à uiſu, ſub minori angulo uidetur, patet quòd linea e a ſecat angulum <lb/>g e d:</s> <s xml:id="echoid-s27438" xml:space="preserve"> ergo per 29 th.</s> <s xml:id="echoid-s27439" xml:space="preserve"> 1 huius ipſa ſecabit baſim g d:</s> <s xml:id="echoid-s27440" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s27441" xml:space="preserve"> punctus, in quo linea a e interſecat lineam g <lb/>d, punctus z:</s> <s xml:id="echoid-s27442" xml:space="preserve"> & centro exiſtente puncto e fiat arcus circuli ad quantitatem ſemidiametri e z:</s> <s xml:id="echoid-s27443" xml:space="preserve"> quine-<lb/> <pb o="123" file="0425" n="425" rhead="LIBER QVARTVS."/> ceſſariò ſecabit lineas e g & e b, cum linea e z, quæ eſt ſemidiameter, ſit minor illis ambabus lineis, <lb/>linea ſcilicet e b ex hypotheſi, & linea e g per 21 p 1:</s> <s xml:id="echoid-s27444" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0425-01a" xlink:href="fig-0425-01"/> ſecet ergo lineam e gin pũcto i, & lineã e b in pun-<lb/>cto t:</s> <s xml:id="echoid-s27445" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s27446" xml:space="preserve"> ille arcus i z t.</s> <s xml:id="echoid-s27447" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s27448" xml:space="preserve"> trigonum e g z <lb/>eſt maius ſectore e z i, & trigonũ e z d minus ſecto-<lb/>re e z t:</s> <s xml:id="echoid-s27449" xml:space="preserve"> ergo per 9 th.</s> <s xml:id="echoid-s27450" xml:space="preserve"> 1 huius trigonum e g z maiorẽ <lb/>habet proportionem ad trigonũ e z d, quàm ſector <lb/>e ziad ſectorẽ e z t:</s> <s xml:id="echoid-s27451" xml:space="preserve"> ergo per 11 th.</s> <s xml:id="echoid-s27452" xml:space="preserve"> 1 huius erit con-<lb/>iunctim maior proportio trigonie g d ad trigonum <lb/>e z d, quàm ſectoris eit ad ſectorẽ e zt:</s> <s xml:id="echoid-s27453" xml:space="preserve"> ſed propor-<lb/>tio e g d trigoni ad e z d trigonum per 1 p 6 eſt, ſicut <lb/>proportio lineæ g d ad lineam z d:</s> <s xml:id="echoid-s27454" xml:space="preserve"> ſed linea d g eſt <lb/>æqualis lineę a b ex hypotheſi:</s> <s xml:id="echoid-s27455" xml:space="preserve"> ergo per 7 p 5 linea-<lb/>rum g d & a b ad lineam d z eſt eadẽ proportio.</s> <s xml:id="echoid-s27456" xml:space="preserve"> Et <lb/>quoniam per 29 p 1 & ex hypotheſi trigona a e b & <lb/>e z d ſunt æquiangula, quia ambobus ipſis angulus <lb/>a e b eſt communis:</s> <s xml:id="echoid-s27457" xml:space="preserve"> eſt ergo per 4 p 6 proportio li-<lb/>neæ a b ad lineam d z, ſicut lineæ b e ad lineam e d:</s> <s xml:id="echoid-s27458" xml:space="preserve"> ergo per 11 p 5 erit proportio lineæ b e a d lineam <lb/>d e maior quàm proportio ſectoris e i t ad ſectorem e z t:</s> <s xml:id="echoid-s27459" xml:space="preserve"> ſed ſicut ſe habet ſector e i t a d ſectorem e <lb/>z t, ita ſe habet arcus it ad arcum z t:</s> <s xml:id="echoid-s27460" xml:space="preserve"> quod patet per 1 p 6, & nos hoc declarauimus in 35 th.</s> <s xml:id="echoid-s27461" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s27462" xml:space="preserve"> <lb/>eſt autem proportio arcus i t ad arcum z t, ſicut anguli i e t ad angulum z e t per 33 p 6.</s> <s xml:id="echoid-s27463" xml:space="preserve"> Eſt ergo ma-<lb/>ior proportio lineę b e ad lineam d e, quàm anguli i e t ad angulum z e t.</s> <s xml:id="echoid-s27464" xml:space="preserve"> Palàm ergo, quòd maior eſt <lb/>proportio diſtantiæ maioris ad diſtantiam minorem, quàm anguli maioris, ſub quo fit uiſio, ad an-<lb/>gulum minorem.</s> <s xml:id="echoid-s27465" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s27466" xml:space="preserve"> Illud enim, quod in æquidiſtantibus magnitudinibus de-<lb/>claratum eſt, in non æquidiſtantibus amplius patet:</s> <s xml:id="echoid-s27467" xml:space="preserve"> quoniam tunc uiſionis anguli minuuntur, ut <lb/>oſten dimus in 7 huius.</s> <s xml:id="echoid-s27468" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s27469" xml:space="preserve"/> </p> <div xml:id="echoid-div1064" type="float" level="0" n="0"> <figure xlink:label="fig-0425-01" xlink:href="fig-0425-01a"> <variables xml:id="echoid-variables435" xml:space="preserve">a g i z b t d e</variables> </figure> </div> </div> <div xml:id="echoid-div1066" type="section" level="0" n="0"> <head xml:id="echoid-head860" xml:space="preserve" style="it">12. Aequalitas remotionis extremorum lineæ uel ſuperficiei rei uiſæ à centro uiſus, dire-<lb/>ctionis comprehenſinis uiſiuæ eſt cauſſa, ſicut inæqualitas eadem corundem eſt cauſſa obliqua-<lb/>tionis. Alhazen 45 n 2.</head> <p> <s xml:id="echoid-s27470" xml:space="preserve">Aequalitas enim rem otionis extremorum lineæ uel ſuperficiei rei uiſæ cauſſat æqualitatem an-<lb/> <anchor type="figure" xlink:label="fig-0425-02a" xlink:href="fig-0425-02"/> gulorum ipſorum axium radialium illi lineæ uel ſuperficiei incidẽtium ſecundum media ipſorum <lb/>puncta.</s> <s xml:id="echoid-s27471" xml:space="preserve"> Vt ſi lineæ a b c extrema, quæ ſunt a & c, æqualiter diſtent à <lb/>centro uiſus, quod eſt d:</s> <s xml:id="echoid-s27472" xml:space="preserve"> & ducatur axis radialis, qui d b :</s> <s xml:id="echoid-s27473" xml:space="preserve"> & lineæ ra-<lb/>diales, quæ d a & d c:</s> <s xml:id="echoid-s27474" xml:space="preserve"> tunc patet ex hypotheſi, & per 8 p 1, quoniã an-<lb/>guli d b a & d b c ſunt æquales.</s> <s xml:id="echoid-s27475" xml:space="preserve"> Siuero extrema puncta, quæ ſunt a & <lb/>c, inæqualiter diſtent à centro d:</s> <s xml:id="echoid-s27476" xml:space="preserve"> tũc lineæ d a & d c fiuntinæquales:</s> <s xml:id="echoid-s27477" xml:space="preserve"> <lb/>& ſimiliter anguli d b a & d b c fiunt inæquales, & fit uiſio obliqua.</s> <s xml:id="echoid-s27478" xml:space="preserve"> Si <lb/>itaq;</s> <s xml:id="echoid-s27479" xml:space="preserve"> linea uel ſuperficies rei uiſę fuerit directè oppoſita uiſui:</s> <s xml:id="echoid-s27480" xml:space="preserve"> ſentiet <lb/>uiſus directionem eius ex ſenſu æqualitatis remotionum ſuarũ par-<lb/>tium ab axe uiſuali perpendiculariter illi lineæ uel ſuperficiei inci-<lb/>dente:</s> <s xml:id="echoid-s27481" xml:space="preserve"> quoniam tunc per definitionem lineæ uel ſuperficiei directè <lb/>uiſibus oppoſitæ, & per 38th 3 huius patet, quoniam ambo axes ra-<lb/>diales cõtinent hinc & inde angulos æquales.</s> <s xml:id="echoid-s27482" xml:space="preserve"> Et ſi ſuperficies rei ui-<lb/>ſæ fuerit obliqua:</s> <s xml:id="echoid-s27483" xml:space="preserve"> tũc ſentiet uiſus obliquationem eius ex ſenſu inæ-<lb/>qualitatis quantitatum remotionum extremorum eius, & etiam an-<lb/>gulorum eius:</s> <s xml:id="echoid-s27484" xml:space="preserve"> & ſic incipit latere quantitas magnitudinis eius uirtu <lb/>tem diſtinctiuam.</s> <s xml:id="echoid-s27485" xml:space="preserve"> Quoniam uirtus diſtinctiua comprehẽdit ex in æ-<lb/>qualitate remotionum diametrorum extrem orũ illius obliqui ſpa-<lb/>tij, obliquationem pyramidis continentis ipſum.</s> <s xml:id="echoid-s27486" xml:space="preserve"> Quare ſentit dimi-<lb/>nutionem magnitudinis baſis eius propter obliquationem:</s> <s xml:id="echoid-s27487" xml:space="preserve"> & non <lb/>conuenit ſecundum aſsimilationem quãtitas magnitudinis obliquè <lb/>uiſui oppoſitæ quantitati magnicudinis directè uιſui oppoſitæ niſi tunc, quando comparatio fue-<lb/>rit ad angulum ſolum:</s> <s xml:id="echoid-s27488" xml:space="preserve"> ſed ſi fiat comparatio ad angulum & ad longitudines linearum radialium in-<lb/>teriacentium uiſum & extrema rei uiſæ:</s> <s xml:id="echoid-s27489" xml:space="preserve"> tunc nullum erit dubium in diuerſitate quantitatum ma-<lb/>gnitudinis hinc inde:</s> <s xml:id="echoid-s27490" xml:space="preserve"> rem otiſsima enim remotionũ mediocrium, reſpectu rei uiſæ per obliquatio.</s> <s xml:id="echoid-s27491" xml:space="preserve"> <lb/>nem, eſt minor remotiſsima remotionum mediocrium, reſpectu illius eiuſdẽ rei uiſæ per directio-<lb/>nem.</s> <s xml:id="echoid-s27492" xml:space="preserve"> Rem otio uerò mediocris, reſpectu rei uiſæ, eſt, in qua nõ latet uiſum pars rei uiſæ proportio-<lb/>nem habẽs ſenſibilẽ ad totá rem uiſam.</s> <s xml:id="echoid-s27493" xml:space="preserve"> Tota itaq;</s> <s xml:id="echoid-s27494" xml:space="preserve"> res obliquata uiſui latet in remotione minori illa <lb/>remotiõe, in qua latet illa res uiſa in directiõe, & diminuitur quátitas eius in remotione minori illa <lb/>remotione, in qua minuitur quãtitas eius, quádo fuerit directè uiſui oppoſita.</s> <s xml:id="echoid-s27495" xml:space="preserve">Patet ergo ꝓpoſitũ.</s> <s xml:id="echoid-s27496" xml:space="preserve"/> </p> <div xml:id="echoid-div1066" type="float" level="0" n="0"> <figure xlink:label="fig-0425-02" xlink:href="fig-0425-02a"> <variables xml:id="echoid-variables436" xml:space="preserve">d n a b c c</variables> </figure> </div> </div> <div xml:id="echoid-div1068" type="section" level="0" n="0"> <head xml:id="echoid-head861" xml:space="preserve" style="it">13. Horizon uidetur quaſiperipheriæ terræcohærere: diſtantiæ tamẽ maioris apparet, quàm <lb/>zenith capitis uidentis.</head> <p> <s xml:id="echoid-s27497" xml:space="preserve">Quia enim inter horizontem (qui eſt circulus terminator uiſus ad cœli concauam ſuperficiem) <lb/> <pb o="124" file="0426" n="426" rhead="VITELLONIS OPTICAE"/> & inter extremã terræ peripheriam, quæ eſt ultima pars terrę uiſibilis, non comprehẽditur aliquod <lb/>ſpatium ſenſibile per uiſum, non poteſt uiſus illorũ certam rem otionẽ ad inuicem diſcernere:</s> <s xml:id="echoid-s27498" xml:space="preserve"> quo-<lb/>niam, ut patet per 10 huius, quantitas remotionis tũc ſolùm comprehenditur à uiſu auxilio uirtutis <lb/>diſtin ctiuæ, cum remotio reſpicit corpora cõtinuata & ordinata:</s> <s xml:id="echoid-s27499" xml:space="preserve"> & quia inter peripheriam terræ & <lb/>concauum cœli non ſunt huiuſmodi corpora:</s> <s xml:id="echoid-s27500" xml:space="preserve"> uidetur ergo horizon quaſi peripheriæ terræ cohæ-<lb/>rere.</s> <s xml:id="echoid-s27501" xml:space="preserve"> Diſtantia uerò peripheriæ horizontis à ſuo cẽtro (quod eſt centrũ uiſus) apparet ſenſibiliter <lb/>maior quàm diſtãtia zenith capitis uidẽtis, quod eſt polus horizõtis.</s> <s xml:id="echoid-s27502" xml:space="preserve"> Quia licet ſecundũ ueritatem.</s> <s xml:id="echoid-s27503" xml:space="preserve"> <lb/>illa quantitas diſtantiæ aut eadẽ ſit, aut inſenſibiliter maior (propter quod quaſi in omnibus aſtro-<lb/>nomicis cõſiderationibus, quæ per uiſum fiunt, centrũ uiſus ponitur centrũ mundi) apparet tamẽ <lb/>ſenſibiliter maior uiſui uirtute etiã diſtin ctiua ſic iudicãte:</s> <s xml:id="echoid-s27504" xml:space="preserve"> quod accidit propter latitudinẽ ſpatij ſu-<lb/>perficiei terræ, quod ſentitur inter uiſum & horizõta, cũ inter zenith capitis & terrá nihil percipia-<lb/>tur.</s> <s xml:id="echoid-s27505" xml:space="preserve"> Quia enim ex corporũ mediorũ ſenſibili diſtantia quãtitas remotionis cognoſcitur ք 10 huius, <lb/>neceſſe eſt, ubi maior ſẽſibilis quãtitas interiacere uidetur, maior diſtãtia iudicetur:</s> <s xml:id="echoid-s27506" xml:space="preserve"> multò ergo ma-<lb/>ior uidetur diſtãtia peripherię horizõtis ꝗ̃ diſtãtia zenith capitis uidẽtis:</s> <s xml:id="echoid-s27507" xml:space="preserve"> & ſimiliter eſt de qualibet <lb/>parte alia cœli uiſa:</s> <s xml:id="echoid-s27508" xml:space="preserve"> ꝓpter hoc, qđ uiſus in medio terræ latitudinẽ cõprehẽdit.</s> <s xml:id="echoid-s27509" xml:space="preserve"> Patet ergo ꝓpoſitũ.</s> <s xml:id="echoid-s27510" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1069" type="section" level="0" n="0"> <head xml:id="echoid-head862" xml:space="preserve" style="it">14. Locus rei uiſæ comprehenditur à uiſu ex remotione, & ex parte uniuerſi, & ex quanti-<lb/>tate remotionis, auxilio uirtutis diſtinctiuæ. Alhazen 22 n 2.</head> <p> <s xml:id="echoid-s27511" xml:space="preserve">Quia enim intentio remotionis non eſt ipſa quantitas remotionis:</s> <s xml:id="echoid-s27512" xml:space="preserve"> intentio enim remotionis eſt <lb/>priuatio contactus duorum corporum, & ex conſequenti comprehenſio cuiuſdam ſitus rerum ab <lb/>inuicem remotarum:</s> <s xml:id="echoid-s27513" xml:space="preserve"> comprehenſio uerò quantitatis rem otionis eſt comprehenſio quantitatis uel <lb/>magnitudinis ſpatij illa corpora interiacentis:</s> <s xml:id="echoid-s27514" xml:space="preserve"> palàm ergo, quòd comprehenſio loci rei uiſæ non eſt <lb/>comprehenſio remotionis cius.</s> <s xml:id="echoid-s27515" xml:space="preserve"> Conſiſtit autem comprehenſio loci rei uiſæ ex comprehẽſione lu-<lb/>cis & coloris rei & remotionis rei, & partis uniuerſi, in qua eſt res illa uiſa, reſpectu uidentis, & ex <lb/>comprehenſione quantitatis remotionis, quando hæc omnia ſimul comprehenduntur per uiam <lb/>cognitionis:</s> <s xml:id="echoid-s27516" xml:space="preserve"> & etiam quia, ut patet per 17 th.</s> <s xml:id="echoid-s27517" xml:space="preserve"> 3 huius, uiſio diſtincta fit ex peruẽtu formarum ſecun-<lb/>dum lineas perpendiculares ſuper ſuperficiem oculi incidentium ad ipſum uiſum.</s> <s xml:id="echoid-s27518" xml:space="preserve"> Cum ergo uiſus <lb/>ſenſerit formam ſic a duenientem, æſtimabit uirtus diſtinctiua rem uiſam eſſe apud extremitatem <lb/>illius lineæ, & ſecundum directionem illius lineæ comprehendet locum rei uiſæ.</s> <s xml:id="echoid-s27519" xml:space="preserve"> Locus ergo rei <lb/>uiſæ comprehenditur à ſentiente ex comprehẽſione ſitus rei uiſæ apud uiſionem per directionem <lb/>lineæ radialis ab illo loco ad uiſum.</s> <s xml:id="echoid-s27520" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s27521" xml:space="preserve"> forma rei uiſæ peruenit ad uiſum:</s> <s xml:id="echoid-s27522" xml:space="preserve"> tunc ſentiet uiſus <lb/>partem membri ſentientis, ad quam peruenit illa ſorma, & uirtus diſtinctiua comprehendet ſtatim <lb/>locum rei uiſæ per directiònem lineæ radialis ab illo loco:</s> <s xml:id="echoid-s27523" xml:space="preserve"> & quoniam intẽtio remotionis eſt quie-<lb/>ſcens in anima ipſa:</s> <s xml:id="echoid-s27524" xml:space="preserve"> ergo comprehendet locum & remotionem in ſimulin comprehenſione formæ <lb/>ab ipſo uiſu.</s> <s xml:id="echoid-s27525" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s27526" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1070" type="section" level="0" n="0"> <head xml:id="echoid-head863" xml:space="preserve" style="it">15. Aequalium uiſibilium inæqualiter à uiſu diſtantium æquali intuitu uiſorum, propin-<lb/>quioris certior eſt uiſio. Euclides 2 the. opt. Alhazen 40 n 2.</head> <p> <s xml:id="echoid-s27527" xml:space="preserve">Sit centrum uiſus b:</s> <s xml:id="echoid-s27528" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s27529" xml:space="preserve"> duo uiſibilia g d & k linæ qualiter diſtantia à centro uiſus b, quæ nunc <lb/>exempli cauſſa, ponantur æ quidiſtantia inter ſe (quoniam ſi ſint ſe contingentia uel ſecantia, patet <lb/>quòd ipſain puncto contactus uel ſectionis æqualiter diſtant à puncto b:</s> <s xml:id="echoid-s27530" xml:space="preserve"> de alijs uerò ipſorũ pun-<lb/>ctis eadem eſt demonſtratio, quæ de ipſis æquidiſtantibus, ipſorum partibus uariatis ſecũdum ap-<lb/>proximationem uel remotionem à uiſu, quantum ad modum certitudinis uiſionis.</s> <s xml:id="echoid-s27531" xml:space="preserve">) Ponãtur itaq;</s> <s xml:id="echoid-s27532" xml:space="preserve"> <lb/>g d & k l æ quidiſtare:</s> <s xml:id="echoid-s27533" xml:space="preserve"> & ſit g d propin quius uiſui:</s> <s xml:id="echoid-s27534" xml:space="preserve"> per-<lb/> <anchor type="figure" xlink:label="fig-0426-01a" xlink:href="fig-0426-01"/> ueniantq́;</s> <s xml:id="echoid-s27535" xml:space="preserve"> ad uiſum formæ punctorum terminalium <lb/>per lineas d b, g b, k b, l b:</s> <s xml:id="echoid-s27536" xml:space="preserve"> fientq́;</s> <s xml:id="echoid-s27537" xml:space="preserve"> trigoni b g d & b k l:</s> <s xml:id="echoid-s27538" xml:space="preserve"> <lb/>ducanturq́;</s> <s xml:id="echoid-s27539" xml:space="preserve"> lineæ l d & k g, quæ per 33 p 1 erunt æqui-<lb/>diſtantes & æquales.</s> <s xml:id="echoid-s27540" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s27541" xml:space="preserve"> punctil multiplicás <lb/>ſe ad uiſum b, nõ tranſibit punctũd, neq;</s> <s xml:id="echoid-s27542" xml:space="preserve"> forma pun-<lb/>cti k punctum g:</s> <s xml:id="echoid-s27543" xml:space="preserve"> quoniam ſi ſic, eſſet linea k g b linea <lb/>una, & linea l d b linea una:</s> <s xml:id="echoid-s27544" xml:space="preserve"> ergo lineæ k g & l d con-<lb/>current in puncto b, quæ ſunt æquidiſtantes:</s> <s xml:id="echoid-s27545" xml:space="preserve"> hoc au-<lb/>tem impoſsibile.</s> <s xml:id="echoid-s27546" xml:space="preserve"> Sed neq;</s> <s xml:id="echoid-s27547" xml:space="preserve"> fient formarum punctorũ <lb/>k & l multiplicationes ad uiſum b, extra aliquod pũ-<lb/>ctum lineæ g d:</s> <s xml:id="echoid-s27548" xml:space="preserve"> quia tunc, cum in trigono l k b cadat <lb/>linea d g æquidiſtanter lineæ k l, palàm per 2 p 6, quo-<lb/>niam erit linea g d minor quàm linea k l:</s> <s xml:id="echoid-s27549" xml:space="preserve"> poſita autẽ <lb/>eſt ęqualis illi:</s> <s xml:id="echoid-s27550" xml:space="preserve"> palàm ergo, quoniã lineæ k b & l b per-<lb/>tranſeunt aliqua pũcta lineæ g d:</s> <s xml:id="echoid-s27551" xml:space="preserve"> erit ergo aliqua pars <lb/>lineæ g d intra pyramidẽ uiſionis, quæ b k l.</s> <s xml:id="echoid-s27552" xml:space="preserve"> Sub quo-<lb/>cunq;</s> <s xml:id="echoid-s27553" xml:space="preserve"> ergo angulo uidetur k l, ſub eodẽ uidetur & aliquid ipſius g d, & nõ econuerſo:</s> <s xml:id="echoid-s27554" xml:space="preserve"> quoniá ut pa-<lb/>tet per 34 th.</s> <s xml:id="echoid-s27555" xml:space="preserve"> 1 huius, uel ք 7 huius, angulus g d b eſt maior angulo k b l.</s> <s xml:id="echoid-s27556" xml:space="preserve"> Quicquid ergo uirtutis ui-<lb/>fiuæ applicatur ipſi k l, applicatur etiam ipſi g d, & non econuerſo:</s> <s xml:id="echoid-s27557" xml:space="preserve"> fortius autem patet illud per 108 <lb/>th.</s> <s xml:id="echoid-s27558" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s27559" xml:space="preserve"> Sub plurib.</s> <s xml:id="echoid-s27560" xml:space="preserve"> ergo uiſibus & angulis uidetur g d quàm k l:</s> <s xml:id="echoid-s27561" xml:space="preserve"> ergo perſpicatius uidetur per 5 <lb/>ſuppoſitionẽ præmiſſam in principio libri huius.</s> <s xml:id="echoid-s27562" xml:space="preserve"> Ipſius ergo certior eſt uiſio.</s> <s xml:id="echoid-s27563" xml:space="preserve"> Et hoc eſt propoſitũ.</s> <s xml:id="echoid-s27564" xml:space="preserve"/> </p> <div xml:id="echoid-div1070" type="float" level="0" n="0"> <figure xlink:label="fig-0426-01" xlink:href="fig-0426-01a"> <variables xml:id="echoid-variables437" xml:space="preserve">c k d g b</variables> </figure> </div> <pb o="125" file="0427" n="427" rhead="LIBER QVARTVS."/> </div> <div xml:id="echoid-div1072" type="section" level="0" n="0"> <head xml:id="echoid-head864" xml:space="preserve" style="it">16.Viſioni uirtutis diſtinctiuæ error accidit in remotionis uiſione ex intẽperata diſpoſitio-<lb/>ne octo circumſtantiarum cuiuslibet rei uiſæ. Alhazen 23. 34. 45. 52. 58. 64. 66. 69 n 3.</head> <p> <s xml:id="echoid-s27565" xml:space="preserve">Accidit enim uirtuti diſtinctiuæ in uiſione remotionis ex intemperata lucis diſpoſitione error <lb/>in remotione rerum uiſarum.</s> <s xml:id="echoid-s27566" xml:space="preserve"> Exiſtẽte enim remotione temperata, non multum certa & debili lu-<lb/>ce, ſi fiat hominum uel alιarum rerum talis diſpoſitio, ut unus poſt alium ſit poſitus:</s> <s xml:id="echoid-s27567" xml:space="preserve"> tunc de nocte <lb/>uel in crepuſculis, & maximè uno uiſo adhibito, uidebuntur illi homines uel res aliæ ſibi quaſi co-<lb/>hærere, quia propter lucis debilitatem non comprehenditur diſtantia inter illa:</s> <s xml:id="echoid-s27568" xml:space="preserve"> & ſi illi homines <lb/>ad eandem partem moueantur æquali motu, ſemper ſimul moueri putabuntur, & non perpende-<lb/>tur diſtantia inter illa, ſed uidebuntur quaſi res una.</s> <s xml:id="echoid-s27569" xml:space="preserve"> Similiter etiam ex nimia diſtantia uirtuti di-<lb/>ſtin ctiuæ accidit error in rerum uiſarum remotione ab inuicem.</s> <s xml:id="echoid-s27570" xml:space="preserve"> Cum enim quis arbores ualde re-<lb/>motas inſpexerit, licet illæ plurimum diſtent inter ſe, uidebuntur tamen quaſi coniunctæ uel quaſi <lb/>propinquæ ad inuicem:</s> <s xml:id="echoid-s27571" xml:space="preserve"> & ita ſtellæ cœli aliquæ reputantur quaſi coniunctæ, licet plurimum à ſe <lb/>diſtent in ueritate:</s> <s xml:id="echoid-s27572" xml:space="preserve"> propter egreſſum etiam diſtantiæ à temperãtia ſtellæ uagantes æſtimantur fer-<lb/>ri in eadem ſuperficie cum ſtellis fixis, licet plurimum diſtent ab illis.</s> <s xml:id="echoid-s27573" xml:space="preserve"> Ex intemperata etiam diſpo-<lb/>ſitione ſitus in oppoſitione rei uiſibilis ad uiſum error accidit in remotionis uiſione:</s> <s xml:id="echoid-s27574" xml:space="preserve"> ut ſi uideantur <lb/>duo corpora, quorum unum ſit retro alterum, ita quòd anterius cooperiat partẽ poſterioris, & alia <lb/>pars emιneat, nec inter ea fuerint aliqua corpora uiſa, & ſit remotio temperata non multum certa:</s> <s xml:id="echoid-s27575" xml:space="preserve"> <lb/>tunc non plenè æſtimabitur menſura longitudinis unius ad alterum, & fortè iudicabit uiſus ipſa <lb/>eſſe ſibi ualde propinqua:</s> <s xml:id="echoid-s27576" xml:space="preserve"> & eſt hic error ex ſola ſitus oppoſitionis intemperãtia, quoniam ſi unum <lb/>non occultaret partem alterius, ſed utrunq;</s> <s xml:id="echoid-s27577" xml:space="preserve"> totum exponeretur uiſui, ita ut eſſet ſenſibilis diuerſi-<lb/>tas inter illa, tunc diſcerneretur diſtãtia unius ab alio:</s> <s xml:id="echoid-s27578" xml:space="preserve"> & ita patet, quòd ille error eſt propter intem-<lb/>perantiam ſitus, quoniam ſolo ſitu ad temperantiam reducto non accideret error talis.</s> <s xml:id="echoid-s27579" xml:space="preserve"> Ex intem-<lb/>perantia etiam di poſitionis quãtitatis error accidit in uiſione remotionis:</s> <s xml:id="echoid-s27580" xml:space="preserve"> unde ſi ſint duo corpo-<lb/>ra æqualiter à uiſu diſtantia ſecũdum temperatam remotionem non multum certam, quorũ unum <lb/>ſit lòngè maius alio, æſtιmabitur maius propιn quius uiſui, quia certius uidebitur:</s> <s xml:id="echoid-s27581" xml:space="preserve"> & ſic propter <lb/>quantitatem erit deceptio in rem otione, quoniam æ què rem otorum unum uidetur remotius alte-<lb/>ro.</s> <s xml:id="echoid-s27582" xml:space="preserve"> Ex intemperata quoq;</s> <s xml:id="echoid-s27583" xml:space="preserve"> ſoliditate corporũ accidit error uiſui in remotionis uiſione:</s> <s xml:id="echoid-s27584" xml:space="preserve"> ſi enim cor-<lb/>pus fuerit ualde rarum minimæ ſoliditatis, ſicut eſt cryſtallus pura, & ſit retro ipſum corpus ualde <lb/>coloratum lucidum:</s> <s xml:id="echoid-s27585" xml:space="preserve"> tunc non plenè comprehenditur cryſtallus, ſed quaſi non eſſet intermedia, <lb/>comprehendetur corpus per ipſam:</s> <s xml:id="echoid-s27586" xml:space="preserve"> & accidit error in comprehenſione cryſtalli propter remotio-<lb/>nem cryſtalli à uiſu.</s> <s xml:id="echoid-s27587" xml:space="preserve"> Exintemperãtia etiam diaphanitatis error accidit uiſui in remotionis uiſione:</s> <s xml:id="echoid-s27588" xml:space="preserve"> <lb/>ſi enim ſuerit aer nubiloſus, ſicut accidit plerunq;</s> <s xml:id="echoid-s27589" xml:space="preserve"> in crepuſculis:</s> <s xml:id="echoid-s27590" xml:space="preserve"> tunc res aliqua, ut turris oppoſita <lb/>uiſui in longitudine temperata, æſtimabitur à uiſu plus elongata quàm ſit ſecundũ ueritatem:</s> <s xml:id="echoid-s27591" xml:space="preserve"> quia <lb/>tunc propter denſitatem aeris non comprehenditur quantitas terræ interiacens uiſum & rem ui-<lb/>ſam, per quam accipitur menſura elongationis turris:</s> <s xml:id="echoid-s27592" xml:space="preserve"> fitq́;</s> <s xml:id="echoid-s27593" xml:space="preserve"> erroris cauſa ex ipſa intemperantia dia-<lb/>phanitatis aeris.</s> <s xml:id="echoid-s27594" xml:space="preserve"> Ex intemperantia etiã temporis fit error uiſui in remotione:</s> <s xml:id="echoid-s27595" xml:space="preserve"> ſi enim intueatur quis <lb/>aliquod remotum à turri a ta, quod ſtatim uiſui ſurripiatur:</s> <s xml:id="echoid-s27596" xml:space="preserve"> tũc uirtus diſtinctiua non poterit ple-<lb/>nè diſcernere inter remotionem illius à turri, & iudica bit fortè aut minus remotum à turri, aut ma-<lb/>gis, quàm fuerit in rei ueritate:</s> <s xml:id="echoid-s27597" xml:space="preserve"> quoniam in tam modico tempore non percipitur à uidente quanti-<lb/>tas terræ interiacens turrim & aliam rem uiſam, ſecundum quam per 10 huius perpenditur menſu-<lb/>ra remotionis illorum ab inuicem:</s> <s xml:id="echoid-s27598" xml:space="preserve"> nec enim in tam breui tempore potuit axis uiſualis quãtitatem <lb/>terræ intermediam per diligentem intuitum tranſcurrere:</s> <s xml:id="echoid-s27599" xml:space="preserve"> unde illam non plenè comprehendit:</s> <s xml:id="echoid-s27600" xml:space="preserve"> & <lb/>ſic ex breuitate tem poris fit error in rem otione.</s> <s xml:id="echoid-s27601" xml:space="preserve"> Exintem perantia etiam debilitatis uiſus error ac-<lb/>cidit uiſui in remotione:</s> <s xml:id="echoid-s27602" xml:space="preserve"> ſi enim opponantur uiſui duo corpora, quorum unum, quod eſt remotius <lb/>à uiſu, ſit coloris fortis, & alterum, quod eſt propinquius, ſit coloris debilis:</s> <s xml:id="echoid-s27603" xml:space="preserve"> tunc debilitas uiſus in-<lb/>certam faciet collationem:</s> <s xml:id="echoid-s27604" xml:space="preserve"> & quia apud fortes uiſus expertum eſt, & patet per præcedentem, quòd <lb/>corpus uiſui propinquius eſt maioris certitudinis:</s> <s xml:id="echoid-s27605" xml:space="preserve"> æſtimabit uiſus debilis illud, quod eſt certius, <lb/>eſſe propinquius:</s> <s xml:id="echoid-s27606" xml:space="preserve"> & ſic quia fortior color à uiſu debili melius percipitur, iudicabit uiſibile fortiori <lb/>colore coloratum propinquius ſibi, licet ſit remotius ſecundum ueritatem:</s> <s xml:id="echoid-s27607" xml:space="preserve"> & fit error in remotio-<lb/>ne ex uiſus debilitate.</s> <s xml:id="echoid-s27608" xml:space="preserve"> Et etiam quia ab oculis groſſa humiditate infectis fit reflexio formarú, ſicut <lb/>ctiam à ſpeculis, cum ab uno uiſuum facta reflexio peruenit ad alterum propter groſsitudinem ae-<lb/>ris extrinſecam, uidebit uiſus debilis formam ſibi propinquam, quæ eſt forma rei remotæ ſcilicet.</s> <s xml:id="echoid-s27609" xml:space="preserve"> <lb/>Sic ergo uiſioni uirtutis diſtinctiuæ error accidit in remotione ex intéperata diſpoſitione circum-<lb/>ſtantiarum quarumlibet rei uiſæ, quæ ſunt tantum octo, ut patuit per 1 th.</s> <s xml:id="echoid-s27610" xml:space="preserve"> huius, quarum euentum <lb/>percurrimus his exemplis & experimentationibus per ſenotis.</s> <s xml:id="echoid-s27611" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s27612" xml:space="preserve"> propoſitum.</s> <s xml:id="echoid-s27613" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1073" type="section" level="0" n="0"> <head xml:id="echoid-head865" xml:space="preserve" style="it">17. Magnitudo rei uiſæ comprehenditur à uiſu ſecundum magnitudιnem partis ſuperficiei <lb/>uiſus, ad quam peruenit forma rei, & anguli ſolidi, qui fit in centro uiſus. Alhazen 37 n 2.</head> <p> <s xml:id="echoid-s27614" xml:space="preserve">Pars enim ſuperficiei uiſus, ad quam peruenit forma rei uiſæ, per angulum uerticis pyramidis <lb/>radiàlis, ſecundum quam per 18th.</s> <s xml:id="echoid-s27615" xml:space="preserve"> 3 huius fit rei obiectæ uiſio, qui eſt apud centrum uiſus, ſemper <lb/>menſuratur.</s> <s xml:id="echoid-s27616" xml:space="preserve"> Quamuis enim uirtus ſenſitiua comprehendat quantitatem illius anguli ex compre-<lb/>henſione partis ſuperficiei uiſus, in qua figuratur forma rei uiſæ, ut patet per 73 th.</s> <s xml:id="echoid-s27617" xml:space="preserve"> 3 huius:</s> <s xml:id="echoid-s27618" xml:space="preserve"> propriè <lb/>tamen angulus eſt per ſe cauſſa menſurationis illius ſuperficiei:</s> <s xml:id="echoid-s27619" xml:space="preserve"> eſt enim ſemper proportio illius <lb/> <pb o="126" file="0428" n="428" rhead="VITELLONIS OPTICAE"/> partis ſuperficiei oculi ad totam ſphęricam ſuperficiem oculi, ſicut illius anguli ad octo angulos re-<lb/>ctos ſolidos per 87 th.</s> <s xml:id="echoid-s27620" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s27621" xml:space="preserve"> Cú enim pyramidis radialis baſis ſemper ſit in ſuperficie rei uiſæ per <lb/>18 th.</s> <s xml:id="echoid-s27622" xml:space="preserve"> 3 huius, ſecatur tamen ipſa pyramis quaſi æquidiſtanter ſuæ baſi per ſuperficiem ipſius uiſus, <lb/>& ſic unus angulus fit ambabus pyramidibus communis, radiali uidelicet totali & eius parti re-<lb/>ſectæ per ipſam ſuperficiem oculi:</s> <s xml:id="echoid-s27623" xml:space="preserve"> magnitudo itaq;</s> <s xml:id="echoid-s27624" xml:space="preserve"> partis ſuperficiei uiſus, ad quam peruenit for-<lb/>ma rei, & angulus, quem continet pyramis radialis, continens illam partem ſuperficiei uiſus, ſunt <lb/>ambo radix comprehenſionis magnitudinis rei uiſæ.</s> <s xml:id="echoid-s27625" xml:space="preserve"> Quamuis autem & hic angulus & hæc pars <lb/>ſuperficiei uiſus diuerſificentur ſecundum diuerſitatem remotionis:</s> <s xml:id="echoid-s27626" xml:space="preserve"> quantò enim magis elonga-<lb/>tur res, tantò magis ille angulus minorabitur per 106 th.</s> <s xml:id="echoid-s27627" xml:space="preserve"> 1 huius, quia pyramis radialis fit ftrictior, <lb/>& quaſi una pyramidum radialium, quæ eſt rei uiſæ remotioris, infcribitur pyramidi radiali, quę eſt <lb/>rei uiſæ propin quioris:</s> <s xml:id="echoid-s27628" xml:space="preserve"> angulus ergo in cẽtro uiſus fit acutior, & pars ſuperficiei uiſus correſpon-<lb/>dens illi angulo fit minor, & quantò plus approximat res uiſui, tantò plus ampliatur magnitudo.</s> <s xml:id="echoid-s27629" xml:space="preserve"> <lb/>Semper tamen magnitudo rei uiſæ comprehenditur à uiſu ſecundum magnitudinem partis præ-<lb/>miſſæ ſuperficiei uiſus, & anguli illius ſolidi, qui fit in centro uiſus.</s> <s xml:id="echoid-s27630" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s27631" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1074" type="section" level="0" n="0"> <head xml:id="echoid-head866" xml:space="preserve" style="it">18. Magnitudines omnes comprehenſæ à uiſu ſecundum oppoſitionem, ſunt quantitates ſu-<lb/>perficierum uiſibilium & partium illarum ſuperficierum: nec non ſuorum terminorum & ſpa-<lb/>tiorum inter uiſibilia diſtinctorum. Alhazen 41 n 2.</head> <p> <s xml:id="echoid-s27632" xml:space="preserve">Quantitas enim totius corporis rei uiſæ non comprehenditur à uiſu:</s> <s xml:id="echoid-s27633" xml:space="preserve"> quoniam uiſus non com-<lb/>prehendit totam ſuperficiem corporis, ſed ſolum illud, quod ſibi opponitur ex ſuperficie corporis <lb/>aut ex ſuperficiebus eius, quamuis corpus ſit paruum:</s> <s xml:id="echoid-s27634" xml:space="preserve"> utpote illud, inter quod & aliquam partem <lb/>ſuperficiei uiſus duci poſsint lineę rectæ per 2 th.</s> <s xml:id="echoid-s27635" xml:space="preserve"> 3 huius.</s> <s xml:id="echoid-s27636" xml:space="preserve"> Sic ergo uiſus comprehendit ſolam rei ſu-<lb/>perficiem:</s> <s xml:id="echoid-s27637" xml:space="preserve"> & ſi uiſus comprehenderit corporeitatem corporis:</s> <s xml:id="echoid-s27638" xml:space="preserve"> non propter hoc cõprehendet quan <lb/>titatem eius, ſed tantùm figuram corporeitatis:</s> <s xml:id="echoid-s27639" xml:space="preserve"> quòd ſi fortaſſe corpus fuerit motum aut uiſus mo-<lb/>tus, ita quòd uiſus comprehendet totam corporis ſuperficiem:</s> <s xml:id="echoid-s27640" xml:space="preserve"> tunc uirtus diſtinctiua comprehen-<lb/>det quantitates corporeitatis eius alia operatione quàm uiſa ſit apud uiſionem:</s> <s xml:id="echoid-s27641" xml:space="preserve"> & ſimiliter eſt de <lb/>partibus corporis.</s> <s xml:id="echoid-s27642" xml:space="preserve"> Quantitates ergo, quas uiſus comprehendit per oppoſitionem, non ſunt, nifi <lb/>quantitates ſuperficierum & linearum terminantium illas ſuperficies uel ipſas menſurantium ſe-<lb/>cundum longum uel ſecundum latum.</s> <s xml:id="echoid-s27643" xml:space="preserve"> Et quoniam comprehenſis diuerſorum corporum ſuperfi-<lb/>ciebus diuerſis & ipſarum terminis, neceſſariò cõprehenditur diſtantia inter illa corpora per com-<lb/>prehenſiones partium ſuperficiei uiſus nó coloratarum colore uiſorum corporum, ſed interiacen-<lb/>tium partes ſuperficiei uiſus coloratas coloribus illorum corporũ, nec ſunt plures magnitudines, <lb/>quæ uiſu comprehendantur:</s> <s xml:id="echoid-s27644" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s27645" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1075" type="section" level="0" n="0"> <head xml:id="echoid-head867" xml:space="preserve" style="it">19. Omnia uiſa ſub eodem angulo, quorum diſtantia ab inuicem non perpenditur, æqualia <lb/>uidentur. Euclides 7 hypotheſi opticorum.</head> <p> <s xml:id="echoid-s27646" xml:space="preserve">Sit uiſus centrum punctum a:</s> <s xml:id="echoid-s27647" xml:space="preserve"> & ſit res uiſa linea b g:</s> <s xml:id="echoid-s27648" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s27649" xml:space="preserve"> lineæ, ſecundum quas puncta g & b <lb/>perueniunt ad uiſum, g a & b a:</s> <s xml:id="echoid-s27650" xml:space="preserve"> uidetur itaq;</s> <s xml:id="echoid-s27651" xml:space="preserve"> linea b g ſub angulo g a b:</s> <s xml:id="echoid-s27652" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s27653" xml:space="preserve"> alia res, quæ eſt d e, ca-<lb/>dens inter eaſdem lineas g a & b a, ita ut ipſa uideatur <lb/> <anchor type="figure" xlink:label="fig-0428-01a" xlink:href="fig-0428-01"/> ſub eodem angulo g a b:</s> <s xml:id="echoid-s27654" xml:space="preserve"> dico, quòd lineæ g b & d e ui-<lb/>debuntur æquales, ſi lineæ d b & e g non perpendan <lb/>tur à uiſu.</s> <s xml:id="echoid-s27655" xml:space="preserve"> Quia enim uiſus a comprehendit duo pun-<lb/>cta d & b ſuper unam lineam, quæ eſt a b, & duo pun-<lb/>cta e & g ſuper unam lineam, quæ eſt a g:</s> <s xml:id="echoid-s27656" xml:space="preserve"> non ergo ui-<lb/>det aliquem terminum alicuius duarum quantitatum <lb/>b g & d e egredi ab alia, ſed uidet fines extremitatum <lb/>æquales.</s> <s xml:id="echoid-s27657" xml:space="preserve"> Et quia nó perpendit quantitatem linearum <lb/>d b & e g eſſe aliquam, apparet uiſui punctus d ſuper <lb/>punctum b, & punctus e ſuper punctum g:</s> <s xml:id="echoid-s27658" xml:space="preserve"> eorum ue-<lb/>rò, quorum alterum alteri ſuperpoſitum non excedit <lb/>reliquum, nec exceditur ab illo, illa ſunt ad inuicem <lb/>æqualia:</s> <s xml:id="echoid-s27659" xml:space="preserve"> duæ ergo lineæ d e & b g uidentur æquales:</s> <s xml:id="echoid-s27660" xml:space="preserve"> <lb/>quoniam ſecundum iudicium uiſus una ipſarũ aliam <lb/>cooperit, neq;</s> <s xml:id="echoid-s27661" xml:space="preserve"> extremitates unius ſuperant alterius <lb/>extremitates.</s> <s xml:id="echoid-s27662" xml:space="preserve"> Et per hũc modum in noctibus aliqua-<lb/>liter lucidis, ut cum luna lucet de ſub nubibus, uel in <lb/>horis crepuſcularibus, ſi accidat hominẽ uel aliud aliquid cum alta arbore uel turri ſub eodem an-<lb/>gulo uideri:</s> <s xml:id="echoid-s27663" xml:space="preserve"> iudicabitur homo uel res alia fortè altitudinis ipſius arboris uel turris:</s> <s xml:id="echoid-s27664" xml:space="preserve"> & fit propter <lb/>hoc multa deceptio in uiſu.</s> <s xml:id="echoid-s27665" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s27666" xml:space="preserve"> propoſitum.</s> <s xml:id="echoid-s27667" xml:space="preserve"/> </p> <div xml:id="echoid-div1075" type="float" level="0" n="0"> <figure xlink:label="fig-0428-01" xlink:href="fig-0428-01a"> <variables xml:id="echoid-variables438" xml:space="preserve">b d g e a</variables> </figure> </div> </div> <div xml:id="echoid-div1077" type="section" level="0" n="0"> <head xml:id="echoid-head868" xml:space="preserve" style="it">20. Omne quod ſub maiori angulo uidetur, maius uidetur, & quod ſub minori minus: ex quo <lb/>patet idem ſub maiori angulo uiſum apparere maius ſe ipſo ſub minori angulo uiſo: & uniuer-<lb/>ſaliter ſecundum proportioncm anguli fit proportio quantitatis rei directè uel ſub eadem obli-<lb/>quitate uiſæ. Euclides 5 & 6 hypotheſi opt.</head> <pb o="127" file="0429" n="429" rhead="LIBER QVARTVS."/> <p> <s xml:id="echoid-s27668" xml:space="preserve">Eſto centrum uiſus in puncto a:</s> <s xml:id="echoid-s27669" xml:space="preserve"> & ſit res, quę f e, uiſa ſub angulo f a e:</s> <s xml:id="echoid-s27670" xml:space="preserve"> productis quoq;</s> <s xml:id="echoid-s27671" xml:space="preserve"> lineis a f & <lb/>a e, producatur inter ipſas linea g b æ quidiſtanter lineæ f e:</s> <s xml:id="echoid-s27672" xml:space="preserve"> uidebitur ergo linea g b ſub angulo f a e, <lb/>quam fortè accidet uideri eſſe æqualem lineæ f e per præmiſſam, ut ſi lineas g f & b e non contingat <lb/>uideri, ſed uiſis lineis g f & b e, uidetur minor, quia eſt ſecundum ueritatem per 4 p 6 linea g b mi-<lb/>nor, quàm ſit linea f e, cum linea a g ſit minor quàm linea a f ex hypotheſi.</s> <s xml:id="echoid-s27673" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s27674" xml:space="preserve"> à puncto <lb/>e linea æquidiſtans lineæ a g per 31 p 1, quæ ſecet protractam lineam g b in puncto d:</s> <s xml:id="echoid-s27675" xml:space="preserve"> erit ergo per <lb/>34 p 1 linea g d æqualis lineæ f e:</s> <s xml:id="echoid-s27676" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s27677" xml:space="preserve"> linea a d, ſecans protractam lineam e f in puncto h:</s> <s xml:id="echoid-s27678" xml:space="preserve"> e <lb/>ritq́;</s> <s xml:id="echoid-s27679" xml:space="preserve"> linea h f maior quàm linea e f:</s> <s xml:id="echoid-s27680" xml:space="preserve"> & angulus f a h <lb/> <anchor type="figure" xlink:label="fig-0429-01a" xlink:href="fig-0429-01"/> eſt maior angulo f a e per 29 th.</s> <s xml:id="echoid-s27681" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s27682" xml:space="preserve"> Et quoniam <lb/>angulus f a e eſt pars anguli f a h, linea uerò f h uide-<lb/>tur maior quàm linea e f, & linea d g uidetur maior <lb/>quàm linea b g:</s> <s xml:id="echoid-s27683" xml:space="preserve"> quia uiſus partem à toto dijudicat:</s> <s xml:id="echoid-s27684" xml:space="preserve"> <lb/>quod ergo ſub minori angulo uidetur, minus uide-<lb/>tur:</s> <s xml:id="echoid-s27685" xml:space="preserve"> ſed & quandoq;</s> <s xml:id="echoid-s27686" xml:space="preserve"> f e per præcedentem uidetur æ-<lb/>qualis lineæ g b:</s> <s xml:id="echoid-s27687" xml:space="preserve"> ergo poteſt uideri linea e f minor <lb/>quàm linea g d, quæ eſt æqualis lineæ f e, ut patet ex <lb/>pręmiſsis:</s> <s xml:id="echoid-s27688" xml:space="preserve"> quod ergo ſub maiori angulo uidetur, ma-<lb/>ius uidetur, & quod uidetur ſub minori, uidetur mi-<lb/>nus.</s> <s xml:id="echoid-s27689" xml:space="preserve"> Conus itaq;</s> <s xml:id="echoid-s27690" xml:space="preserve"> pyramidis uiſualis, quæ eſt f a e, ſe-<lb/>cundum quam uidetur res remotior, quæ eſt f e, mi-<lb/>nor & acutior eſt quàm conus pyramidis g a d.</s> <s xml:id="echoid-s27691" xml:space="preserve"> Et <lb/>quoniam ſuperficies oculi ſecat ambas iſtas pyrami-<lb/>des, cum ipſarum ambarum conus ſit quaſi in centro <lb/>oculi per 18 th.</s> <s xml:id="echoid-s27692" xml:space="preserve"> 3 huius, neceſſe eſt ergo baſim pyrami <lb/>dis abſciſſæ à pyramide f a e minorem eſſe baſi pyramidis abſciſſæ à totali pyramide g a d per 109 <lb/>th.</s> <s xml:id="echoid-s27693" xml:space="preserve"> 1 huius, cum illæ duæ a bſciſſæ pyramides æ qualis ſint altitudinis:</s> <s xml:id="echoid-s27694" xml:space="preserve"> quoniam linea producta à cen <lb/>tro foraminis gyrationis nerui concaui ad ſuperficiem oculi extrinſecam eſt axis ambarum illa-<lb/>rum pyramidum abſciſſarum.</s> <s xml:id="echoid-s27695" xml:space="preserve"> Pars ergo ſuperficiei uiſus ibi figurata per formam rei uiſæ, quæ eſt <lb/>g d, eſt maior quàm pars eiuſdem ſuperficiei figurata per formam rei uiſæ, quæ eſt f e:</s> <s xml:id="echoid-s27696" xml:space="preserve"> uidetur <lb/>ergo linea g d maior quàm linea fe.</s> <s xml:id="echoid-s27697" xml:space="preserve"> Et quoniam ſecundum quantitatem illarum partium ſuperfi-<lb/>ciei ipſius uiſus uirtus ſenſitiua comprehendit angulum, quem lineæ radiales continentin centro <lb/>per 73 th.</s> <s xml:id="echoid-s27698" xml:space="preserve"> 3 huius:</s> <s xml:id="echoid-s27699" xml:space="preserve"> patet quòd rei, quæ uidetur maior, correſpondet angulus maior, & rei, quæ uide-<lb/>tur minor, correſpondet angulus minor:</s> <s xml:id="echoid-s27700" xml:space="preserve"> quoniam ſecundum quod forma rei uiſæ recipitur in ſu-<lb/>perficie organi uiſiui, ſecundum hoc accipitur quantitas anguli, ſub quo fit uiſio, & ſecundum hoc <lb/>idem etiam fit iudicium quantitatis rei uiſæ.</s> <s xml:id="echoid-s27701" xml:space="preserve"> Omnis ergo res ſub maiori angulo uiſa maior uidetur <lb/>ſe ipſa uiſa ſub angulo minori.</s> <s xml:id="echoid-s27702" xml:space="preserve"> Et uniuerſaliter in rebus directè uiſis ſecundum excrem entum angu <lb/>li fit excrementum quantitatis rei uiſæ:</s> <s xml:id="echoid-s27703" xml:space="preserve"> unde ſub duplo angulo uiſum duplum uidetur, & ſub triplo <lb/>triplum, & ſic ſecundum proportionem angulorum.</s> <s xml:id="echoid-s27704" xml:space="preserve"> In obliquè tamen uiſis, uel in his, quorum u-<lb/>num uidetur directè, & aliud obliquè, non ſic.</s> <s xml:id="echoid-s27705" xml:space="preserve"> Si enim trigonum a e f fit orthogonium, ita ut <lb/>eius angulus a e f ſit rectus, diuidaturq́;</s> <s xml:id="echoid-s27706" xml:space="preserve"> angulus f a e per æ qualia, producta linea a k, ſecante lineam <lb/>f e in puncto k:</s> <s xml:id="echoid-s27707" xml:space="preserve"> non propter hoc diuidetur linea e f per æqualia in puncto k:</s> <s xml:id="echoid-s27708" xml:space="preserve"> quoniã, ut patet per 35 <lb/>th.</s> <s xml:id="echoid-s27709" xml:space="preserve"> 1 huius, minor eſt proportio anguli f a k ad angulũ k a e, quàm lineæ f k ad lineam k e:</s> <s xml:id="echoid-s27710" xml:space="preserve"> & ſic ſecun-<lb/>dum proportionem anguli ad angulum, non ſemper fit proportio quantitatis uiſæ ad quantitatem <lb/>uiſam:</s> <s xml:id="echoid-s27711" xml:space="preserve"> neq;</s> <s xml:id="echoid-s27712" xml:space="preserve"> enim talia uiſa ſecundum eandem uidentur diſpoſitionem & ſitum reſpectu ipſius ui-<lb/>ſus.</s> <s xml:id="echoid-s27713" xml:space="preserve"> In conformibus autem uiſibilibus ſecundum diſtantiam & ſitum & alia accidentia, quæ requi-<lb/>runtur ad conditionem & circunſtantiam uidendi, quæ patent per 1 the.</s> <s xml:id="echoid-s27714" xml:space="preserve"> huius, ſemper ſecundum <lb/>proportionem anguli uidetur proportionaliter quantitas rei uiſæ:</s> <s xml:id="echoid-s27715" xml:space="preserve"> unde etiam illud, quod ſub mi-<lb/>nimo angulo uidetur, minimum uidetur, & quod ſub nullo uel inſenſibili angulo peruenit ad uiſus <lb/>ſuperficiem, nullo modo uldetur, ut patet per 19 th.</s> <s xml:id="echoid-s27716" xml:space="preserve"> 3 huius.</s> <s xml:id="echoid-s27717" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s27718" xml:space="preserve"/> </p> <div xml:id="echoid-div1077" type="float" level="0" n="0"> <figure xlink:label="fig-0429-01" xlink:href="fig-0429-01a"> <variables xml:id="echoid-variables439" xml:space="preserve">f g k e b a h d</variables> </figure> </div> </div> <div xml:id="echoid-div1079" type="section" level="0" n="0"> <head xml:id="echoid-head869" xml:space="preserve" style="it">21. Parallelæ lineæ ſecundum remotiores à uiſu partes quaſi concurrere uidentur: nunquam <lb/>tamen uidebuntur concurrentes. Euclides 6 the. opt.</head> <p> <s xml:id="echoid-s27719" xml:space="preserve">Vniuerſale eſt quod proponitur, uiſu quocunq;</s> <s xml:id="echoid-s27720" xml:space="preserve"> modo ſe habente ad illas lineas parallelas:</s> <s xml:id="echoid-s27721" xml:space="preserve"> ſiue <lb/>enim uiſus ſit in illarum ſuperficie, ſiue ſupra illam, ſiue ſub illa, ſemper eadem paſsio uiſui accidit.</s> <s xml:id="echoid-s27722" xml:space="preserve"> <lb/>Sit ergo primò uiſus in illarum ſuperficie, & ſint duæ parallelæ lineæ a b & g d:</s> <s xml:id="echoid-s27723" xml:space="preserve"> hæ ergo per 1 th.</s> <s xml:id="echoid-s27724" xml:space="preserve"> 1 <lb/>huius neceſſariò erunt in eadem ſuperficie:</s> <s xml:id="echoid-s27725" xml:space="preserve"> ſit ergo in ipſarum ſuperficie uiſus, qui ſit e, uel prope <lb/>illam.</s> <s xml:id="echoid-s27726" xml:space="preserve"> Dico, quòd ſuperficiei interiacentis lineas a b & g d, in æqualis apparebit latitudo:</s> <s xml:id="echoid-s27727" xml:space="preserve"> & quòd <lb/>pars ſui propinquior uiſui apparebit latior, quàm pars eius à uiſu remotior, & ita lineæ a b & g d <lb/>quaſi concurrere uidebuntur.</s> <s xml:id="echoid-s27728" xml:space="preserve"> Signentur enim puncta æquidiſtanter & ſimiliter in lineis a b & g <lb/>d, quæ ſint in linea a b puncta z & t, & in linea d g pucta l & k:</s> <s xml:id="echoid-s27729" xml:space="preserve"> & coniungantur illa puncta, & pun-<lb/>cta terminalia ductis lineis b d, z l, t k, a g:</s> <s xml:id="echoid-s27730" xml:space="preserve"> quæ omnes erunt æ quidiſtantes ex hypotheſi & per 33 p 1:</s> <s xml:id="echoid-s27731" xml:space="preserve"> <lb/>& producantur lineæ e b, e z, e t, e a:</s> <s xml:id="echoid-s27732" xml:space="preserve"> e d, e l, e k, e g.</s> <s xml:id="echoid-s27733" xml:space="preserve"> Et quoniam angulus b e d maior eſt angulo z e <lb/>l.</s> <s xml:id="echoid-s27734" xml:space="preserve"> ſicut totum parte (quod patet per 34 theo.</s> <s xml:id="echoid-s27735" xml:space="preserve"> 1 huius) palàm per præmiſſam quia maior uidebitur <lb/> <pb o="128" file="0430" n="430" rhead="VITELLONIS OPTICAE"/> linea b d quàm linea z l:</s> <s xml:id="echoid-s27736" xml:space="preserve"> & eodem modo maior uidebitur linea z l quàm linea t k, maiorq́;</s> <s xml:id="echoid-s27737" xml:space="preserve"> uidebitur <lb/>linea t k quàm linea a g.</s> <s xml:id="echoid-s27738" xml:space="preserve"> Et quia ſic diminuuntur in uiſu lineæ latitu-<lb/>dinis:</s> <s xml:id="echoid-s27739" xml:space="preserve"> palàm;</s> <s xml:id="echoid-s27740" xml:space="preserve"> quòd ſuperficies interiacens lineas minor uidebitur:</s> <s xml:id="echoid-s27741" xml:space="preserve"> li-<lb/> <anchor type="figure" xlink:label="fig-0430-01a" xlink:href="fig-0430-01"/> neæ ergo a b & g d quaſi concurrere uidebuntur:</s> <s xml:id="echoid-s27742" xml:space="preserve"> nunquam tamen ui <lb/>debuntur concurrentes, quia ſemper lineæ latitudinis ſub aliquo an-<lb/>gulo uidentur, cui in termino uiſionis ſubtenditur baſis cuiuſcunq;</s> <s xml:id="echoid-s27743" xml:space="preserve"> <lb/>ruerit paruitatis:</s> <s xml:id="echoid-s27744" xml:space="preserve"> nunquam ergo uidebuntur concurrentes.</s> <s xml:id="echoid-s27745" xml:space="preserve"> Si uerò <lb/>uiſui, qui ſit a, parallelæ ſubiaceant, quæ ſint lineæ l g & x e, ità quòd <lb/>uiſus ſit erectus ſuper ſuperficiem horizontis, & lineæ illæ ſint in ſu-<lb/>perficie ipſius horizontis, adhuc illæ lineæ ſecundum remotiores à <lb/>uiſu partes quaſi concurrere uidebuntur.</s> <s xml:id="echoid-s27746" xml:space="preserve"> Dimittatur enim à uiſu a <lb/>perpendicularis ſuper ſuperficiem horizontis per 11 p 11, quæ ſit a b:</s> <s xml:id="echoid-s27747" xml:space="preserve"> <lb/>ſintq́;</s> <s xml:id="echoid-s27748" xml:space="preserve">, ut prius, lineæ l x, k n, t m parallelæ.</s> <s xml:id="echoid-s27749" xml:space="preserve"> Dico, quoniam adhuc inæ-<lb/>qualis latitudinis apparet ſuperficies interiacens lineas l g & x e:</s> <s xml:id="echoid-s27750" xml:space="preserve"> & <lb/>partes linearum remotiores à uiſu quaſi concurrere uidentur.</s> <s xml:id="echoid-s27751" xml:space="preserve"> Duca-<lb/>rur enim linea à puncto b perpendiculariter ſuper lineam l x, quę ſit <lb/>b r:</s> <s xml:id="echoid-s27752" xml:space="preserve"> eruntq́, lineæ b r & l x in eadem ſuperficie per 2 p 11:</s> <s xml:id="echoid-s27753" xml:space="preserve"> & produca-<lb/>tur linea b r ſuper lineam g e in punctum f:</s> <s xml:id="echoid-s27754" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s27755" xml:space="preserve"> lineam k n in pun-<lb/>cto p, & lineam t m in puncto c:</s> <s xml:id="echoid-s27756" xml:space="preserve"> & ducantur lineæ l a, k a, t a, x a, n a, m <lb/>a:</s> <s xml:id="echoid-s27757" xml:space="preserve"> ſimiliter etiá ducantur lineæ a r, a p, a c.</s> <s xml:id="echoid-s27758" xml:space="preserve"> Quoniã itaq;</s> <s xml:id="echoid-s27759" xml:space="preserve"> angulus a b r <lb/>eſt rectus, palàm quòd ſuperficies a b c erecta eſt ſuper ſuperficiẽ l x <lb/>e g, & earum communis ſectio eſt linea b f per 19 th.</s> <s xml:id="echoid-s27760" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s27761" xml:space="preserve"> quoniam illa lineà b f eſt in ambabus <lb/>illis ſuperficiebus.</s> <s xml:id="echoid-s27762" xml:space="preserve"> Quia ergo linea a r producta eſt in ſuperficie a b c, <lb/> <anchor type="figure" xlink:label="fig-0430-02a" xlink:href="fig-0430-02"/> & ſimiliter lineæ a p & a f:</s> <s xml:id="echoid-s27763" xml:space="preserve"> palàm per definitionem, quoniam anguli a <lb/>r x & a p n & a c m ſunt recti:</s> <s xml:id="echoid-s27764" xml:space="preserve"> & ita illi trigoni, qui ſunt a b r, & a b p, & <lb/>a b c ſunt orthogonij:</s> <s xml:id="echoid-s27765" xml:space="preserve"> ſed linèà p n eſt æ qualis lineæ r x ex hypothe-<lb/>ſi, & per 34 p 1.</s> <s xml:id="echoid-s27766" xml:space="preserve"> Quia uerò angulus a b r eſt rectus, erit angulus a <lb/>r b acutus per 32 p 1:</s> <s xml:id="echoid-s27767" xml:space="preserve"> ergo per 13 p 1 angulus a r p eſt obtuſus:</s> <s xml:id="echoid-s27768" xml:space="preserve"> li-<lb/>nea ergo a p maior eſt quàm linea a r per 19 p 1:</s> <s xml:id="echoid-s27769" xml:space="preserve"> angulus ergo r a x per <lb/>34 the.</s> <s xml:id="echoid-s27770" xml:space="preserve"> 1 huius maior eſt angulo p a n:</s> <s xml:id="echoid-s27771" xml:space="preserve"> maior ergo uidebitur linea r x <lb/>quàm linea p n, per præmiſſam:</s> <s xml:id="echoid-s27772" xml:space="preserve"> ſimiliterq́;</s> <s xml:id="echoid-s27773" xml:space="preserve"> maior uidebitur linea lr <lb/>quàm linea k p:</s> <s xml:id="echoid-s27774" xml:space="preserve"> quoniam eadem eſt demonſtratio:</s> <s xml:id="echoid-s27775" xml:space="preserve"> eſt enim linea lr <lb/>æqualis lineę k p per principium:</s> <s xml:id="echoid-s27776" xml:space="preserve"> ſi ab ęqualibus, &c.</s> <s xml:id="echoid-s27777" xml:space="preserve"> Tota ergo linea <lb/>l x uidebitur maior quàm tota linea k n:</s> <s xml:id="echoid-s27778" xml:space="preserve"> eodemq́;</s> <s xml:id="echoid-s27779" xml:space="preserve"> modo tota linea k <lb/>n uidebitur maior quàm tota linea t m.</s> <s xml:id="echoid-s27780" xml:space="preserve"> Superficiei ergo l x g e partes <lb/>remotiores uiſui uidebuntur ſtrictiores:</s> <s xml:id="echoid-s27781" xml:space="preserve"> lineæ ergo l g & x e uidebun <lb/>tur quaſi concurrere:</s> <s xml:id="echoid-s27782" xml:space="preserve"> nó tamen uidebuntur unquam concurrentes, <lb/>quia ſemper ſub angulo aliquo uidebuntur.</s> <s xml:id="echoid-s27783" xml:space="preserve"> Et eodem penitus modo <lb/>demonſtrandum, ſi lineæ parallelæ uiſæ ſint uiſu ſuperiores, ut ſi uiſu <lb/>inferius exiſtente lineæ ipſæ paralellæ ſint in aliqua ſuperficie ſuper <lb/>uiſum, ut accidit in tectis domuum, & ſimilibus, uiſu exiſtente infe-<lb/>rius.</s> <s xml:id="echoid-s27784" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s27785" xml:space="preserve"/> </p> <div xml:id="echoid-div1079" type="float" level="0" n="0"> <figure xlink:label="fig-0430-01" xlink:href="fig-0430-01a"> <variables xml:id="echoid-variables440" xml:space="preserve">g a h c l z d b e</variables> </figure> <figure xlink:label="fig-0430-02" xlink:href="fig-0430-02a"> <variables xml:id="echoid-variables441" xml:space="preserve">g f e t c m k p n l r x b a</variables> </figure> </div> </div> <div xml:id="echoid-div1081" type="section" level="0" n="0"> <head xml:id="echoid-head870" xml:space="preserve" style="it">22. Lineis pluribus æqualiter ab inuicem æquidiſtantibus, obiectis uiſui: diſtantia remotiorũ <lb/>minor uiſui apparet. Euclides 4 theo. opt.</head> <p> <s xml:id="echoid-s27786" xml:space="preserve">Eſto, utin præmiſſa, uiſus, cuius centrum ſit a, erectus in aere ſe-<lb/> <anchor type="figure" xlink:label="fig-0430-03a" xlink:href="fig-0430-03"/> cundum erectionem uidentis:</s> <s xml:id="echoid-s27787" xml:space="preserve"> in ſuperficie quoq;</s> <s xml:id="echoid-s27788" xml:space="preserve"> horizontis ſubia-<lb/>ceant uiſui lineæ æquales & æquidiſtantes, & ſecundum æqualem di <lb/>ſtantiam ab inuicem diſtantes, quæ ſint l x, k n, t m, g e, hoc ordine po <lb/>ſitæ ut linea l x ſit uiſui propinquior, aliæ uerò ſuæ nominationis or-<lb/>dine ſint remotiores à uiſu.</s> <s xml:id="echoid-s27789" xml:space="preserve"> Dico, quòd linearum k n & t m diſtantia <lb/>minor uidebitur quàm linearum l x & k n.</s> <s xml:id="echoid-s27790" xml:space="preserve"> Cum enim iſtæ lineæ ſint <lb/>æquales & æquidiſtantes, quæ ſunt l x, k n, & t m:</s> <s xml:id="echoid-s27791" xml:space="preserve"> copulatis ipſarũ ter <lb/>minis per lineas l g & x e:</s> <s xml:id="echoid-s27792" xml:space="preserve"> erit per 30 & 33 p 1, linea l g æqualis lineæ x <lb/>e:</s> <s xml:id="echoid-s27793" xml:space="preserve"> & ducatur, ut in proxima præcedente, linea a b perpendicularis ſu <lb/>per ſuperficiẽ l x g e:</s> <s xml:id="echoid-s27794" xml:space="preserve"> & facta demonſtratione, ut in illa, ſequetur angu <lb/>lum r a p eſſe maiorẽ angulo p a c.</s> <s xml:id="echoid-s27795" xml:space="preserve"> Facilius tamen patet hoc per 35 th.</s> <s xml:id="echoid-s27796" xml:space="preserve"> 1 <lb/>huius:</s> <s xml:id="echoid-s27797" xml:space="preserve"> quoniã in trigono orthogonio a b f partes æquales fũt abſciſſę <lb/>ab uno laterũ rectũ angulũ cõtinentiũ, quę r p, & p c, & c f:</s> <s xml:id="echoid-s27798" xml:space="preserve"> eſt ergo an <lb/>gulus r a p maior angulo p a c ք 10 p 5:</s> <s xml:id="echoid-s27799" xml:space="preserve"> linea ergo r p ք 20 huius uide-<lb/>bitur maior ꝗ̃ linea p c, & linea ṕ c maior ꝗ̃ linea c f.</s> <s xml:id="echoid-s27800" xml:space="preserve"> Remotior ergo <lb/>iſtarũ diſtantiarũ, quęſunt r p, & p c, & c f, minor apparet uiſui per 20 <lb/>huius.</s> <s xml:id="echoid-s27801" xml:space="preserve"> Et hoc eſt propoſitũ.</s> <s xml:id="echoid-s27802" xml:space="preserve"> Et uniuerſaliter in omni uiſus diſpoſitióe <lb/>ad datàs parallelas poteſt hocidem, ut in præcedenti, demonſtrari.</s> <s xml:id="echoid-s27803" xml:space="preserve"/> </p> <div xml:id="echoid-div1081" type="float" level="0" n="0"> <figure xlink:label="fig-0430-03" xlink:href="fig-0430-03a"> <variables xml:id="echoid-variables442" xml:space="preserve">g f e t c m k p n l r x b a</variables> </figure> </div> <pb o="129" file="0431" n="431" rhead="LIBER QVARTVS."/> </div> <div xml:id="echoid-div1083" type="section" level="0" n="0"> <head xml:id="echoid-head871" xml:space="preserve" style="it">23. Aequalium partium eiuſdem uiſibilis lineæ connectenti centra for aminum gyrationis <lb/>neruo rum concauorum æquidiſtantis, remotior à uiſu minor uidetur. Euclides 4 theor. opt.</head> <p> <s xml:id="echoid-s27804" xml:space="preserve">Sit linea r t connectens centra foraminum gyrationis neruorum concauorum:</s> <s xml:id="echoid-s27805" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s27806" xml:space="preserve"> æquales <lb/>partes eiuſdem uiſibilis ſuper lineam æquidiſtantem lineæ r t collocatæ:</s> <s xml:id="echoid-s27807" xml:space="preserve"> quæ ſint a b, b g, g d, d f:</s> <s xml:id="echoid-s27808" xml:space="preserve"> tra-<lb/>haturq́;</s> <s xml:id="echoid-s27809" xml:space="preserve"> perpendicularis a e, in qua ſit centrum oculi e.</s> <s xml:id="echoid-s27810" xml:space="preserve"> Dico, quòd maior apparebit pars a b quàm b <lb/>g, & b g quàm g d, & g d quàm d f.</s> <s xml:id="echoid-s27811" xml:space="preserve"> Cum enim perpẽdi-<lb/> <anchor type="figure" xlink:label="fig-0431-01a" xlink:href="fig-0431-01"/> cularis e a ſit breuior omnibus lineis ducibilibus à <lb/>puncto e ad lineam a d, ut omnibus lineis e b, e g, e d, <lb/>quod per 47 p 1 palàm eſt:</s> <s xml:id="echoid-s27812" xml:space="preserve"> manifeſtum eſt ergo, quo-<lb/>niam pars a b eſt propinquior uiſui omnibus illis par-<lb/>tibus, quæ ſunt b g & g d & d f.</s> <s xml:id="echoid-s27813" xml:space="preserve"> Ducantur enim lineæ, <lb/>per quas accedunt formæ punctorum ad uiſum, quæ <lb/>ſint b e, g e, d e, f e:</s> <s xml:id="echoid-s27814" xml:space="preserve"> & ducatur per 31 p 1 linea b z æquidi <lb/>ſtans lineæ g e.</s> <s xml:id="echoid-s27815" xml:space="preserve"> Quia igitur in trigono a e g linea b z æ-<lb/>quidiſtat lateri e g:</s> <s xml:id="echoid-s27816" xml:space="preserve"> palàm per 2 p 6, quoniam eſt pro-<lb/>portio lineæ a z ad lineã z e, ſicut lineæ a b ad lineam b <lb/>g:</s> <s xml:id="echoid-s27817" xml:space="preserve"> ſed linea a b æqualis eſt lineæ b g ex hypotheſi:</s> <s xml:id="echoid-s27818" xml:space="preserve"> er-<lb/>go linea a z eſt æqualis lineæ z e:</s> <s xml:id="echoid-s27819" xml:space="preserve"> ſed per 47 p 1 linea z <lb/>b eſt maior quàm linea z a:</s> <s xml:id="echoid-s27820" xml:space="preserve"> ergo linea b z eſt maior <lb/>quàm linea z e:</s> <s xml:id="echoid-s27821" xml:space="preserve"> angulus ergo z e b per 18 p 1 maior eſt <lb/>angulo z b e:</s> <s xml:id="echoid-s27822" xml:space="preserve"> ſed angulus z b e per 29 p 1 æqualis eſt <lb/>angulo b e g, quia ſunt coalterni inter lineas æquidi-<lb/>ſtantes, quæ ſunt z b & e g:</s> <s xml:id="echoid-s27823" xml:space="preserve"> ergo angulus a e b maior <lb/>eſt angulo b e g.</s> <s xml:id="echoid-s27824" xml:space="preserve"> Ergo per 20 huius maius uidebitur a b quam b g:</s> <s xml:id="echoid-s27825" xml:space="preserve"> ſub maiori enim angulo uidebi-<lb/>tur.</s> <s xml:id="echoid-s27826" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s27827" xml:space="preserve"> ducta à puncto g linea æquidiſtante lineæ e d, eadem eſt demonſtratio.</s> <s xml:id="echoid-s27828" xml:space="preserve"> Idem <lb/>quoq;</s> <s xml:id="echoid-s27829" xml:space="preserve"> accidit, ſi lineæ e a, e b, e g, e d, e f non ſunt in una linea naturali, dum tamẽ linea mathematica <lb/>inter ipſas imaginata æquidiſtet lineæ g e.</s> <s xml:id="echoid-s27830" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s27831" xml:space="preserve"/> </p> <div xml:id="echoid-div1083" type="float" level="0" n="0"> <figure xlink:label="fig-0431-01" xlink:href="fig-0431-01a"> <variables xml:id="echoid-variables443" xml:space="preserve">f r d g b a z z z e c</variables> </figure> </div> </div> <div xml:id="echoid-div1085" type="section" level="0" n="0"> <head xml:id="echoid-head872" xml:space="preserve" style="it">24. Aequalium diuerſorum uiſibilium ſecundum eandem rectam lineam æquidiſtantem li <lb/>neæ connectenti centra for aminum gyrationis neruorum concauorum uiſuiobiectorum, quod <lb/>propinquius est uiſui, apparet maius. Euclides 7 theo. opt.</head> <p> <s xml:id="echoid-s27832" xml:space="preserve">Sint duo uiſibilia diſcontinuata diuerſa, ſed æqualia a b & g d, oppoſita uiſui ſecundum lineam <lb/>a d:</s> <s xml:id="echoid-s27833" xml:space="preserve"> quæ ſit æquidiſtans lineæ r t, connectenti centra foraminum gyrationis neruorum concauo-<lb/>rum:</s> <s xml:id="echoid-s27834" xml:space="preserve"> & ſint in æqualiter diſtantes à centro uiſus, quod ſit e:</s> <s xml:id="echoid-s27835" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s27836" xml:space="preserve"> lineæ à terminis uiſibilium <lb/>ad centrum uiſus, quæ ſint e d & e a:</s> <s xml:id="echoid-s27837" xml:space="preserve"> & ſit linea e a maior quâm linea d e.</s> <s xml:id="echoid-s27838" xml:space="preserve"> Dico, quòd g d apparebit <lb/>uiſui maius quàm a b.</s> <s xml:id="echoid-s27839" xml:space="preserve"> Producantur enim lineæ e g & <lb/>e b:</s> <s xml:id="echoid-s27840" xml:space="preserve"> & circa trigonum a e d deſcribatur circulus per 5 <lb/>p 4:</s> <s xml:id="echoid-s27841" xml:space="preserve"> & producatur linea e g ad circumferentiã in pun <lb/>ctum l, & linea e b in punctum z:</s> <s xml:id="echoid-s27842" xml:space="preserve"> & à puncto g duca <lb/>tur perpendicularis ſuper a d per 11 p 1, quę protracta <lb/> <anchor type="figure" xlink:label="fig-0431-02a" xlink:href="fig-0431-02"/> ad circumferentiam ſit g k:</s> <s xml:id="echoid-s27843" xml:space="preserve"> & à puncto b ducàtur li-<lb/>nea b c æquidiſtans lineæ g k:</s> <s xml:id="echoid-s27844" xml:space="preserve"> erit ergo per 29 p 1 li-<lb/>nea b c perpendicularis ſuper lineam a d:</s> <s xml:id="echoid-s27845" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s27846" xml:space="preserve"> pe-<lb/>ripheriam circuli in puncto c.</s> <s xml:id="echoid-s27847" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s27848" xml:space="preserve"> à terminis <lb/>lineæ a d intra circulum collocatæ æquales partes <lb/>ſunt reſectæ, quæ ſunt a b & g d, quoniam illæ ſunt æ-<lb/>quales ex hypotheſi:</s> <s xml:id="echoid-s27849" xml:space="preserve"> & à punctis ſectionum ſunt duę <lb/>lineæ perpendiculares ſuper lineam d a productæ <lb/>ad peripheriam illius circuli, quæ ſunt g k & b c:</s> <s xml:id="echoid-s27850" xml:space="preserve"> e-<lb/>rit ergo per 45 th.</s> <s xml:id="echoid-s27851" xml:space="preserve"> 1 huius linea b c æqualis lineæ g <lb/>k:</s> <s xml:id="echoid-s27852" xml:space="preserve"> ſed & linea a b eſt æqualis lineæ g d ex hypotheſi, <lb/>& angulus a b c æqualis eſt angulo k g d, quia uterq;</s> <s xml:id="echoid-s27853" xml:space="preserve"> <lb/>rectus:</s> <s xml:id="echoid-s27854" xml:space="preserve"> ergo chorda k d æqualis eſt chordæ c a per 4 <lb/>p 1:</s> <s xml:id="echoid-s27855" xml:space="preserve"> ergo per 28 p 3 arcus d k æqualis eſt arcui c a:</s> <s xml:id="echoid-s27856" xml:space="preserve"> ſed arcus c a eſt maior arcu z a:</s> <s xml:id="echoid-s27857" xml:space="preserve"> ergo & arcus k d ma <lb/>ior eſt arcu z a:</s> <s xml:id="echoid-s27858" xml:space="preserve"> arcus uerò l d maior eſt arcu k d:</s> <s xml:id="echoid-s27859" xml:space="preserve"> ergo multò maior eſt arcus l d arcu z a:</s> <s xml:id="echoid-s27860" xml:space="preserve"> ſed in arcum <lb/>z a cadit angulus a e z, & in arcũ l d cadit angulus l e d:</s> <s xml:id="echoid-s27861" xml:space="preserve"> ergo per 33 p 6 angulus l e d maior eſt angulo <lb/>z e a:</s> <s xml:id="echoid-s27862" xml:space="preserve"> ſed ſub angulo a e z uidetur linea a b, & ſub angulo l e d uidetur linea g d:</s> <s xml:id="echoid-s27863" xml:space="preserve"> maior ergo apparet <lb/>uiſui linea g d, quàm linea a b per 20 huius.</s> <s xml:id="echoid-s27864" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s27865" xml:space="preserve"/> </p> <div xml:id="echoid-div1085" type="float" level="0" n="0"> <figure xlink:label="fig-0431-02" xlink:href="fig-0431-02a"> <variables xml:id="echoid-variables444" xml:space="preserve">a z c b l k g d e f</variables> </figure> </div> </div> <div xml:id="echoid-div1087" type="section" level="0" n="0"> <head xml:id="echoid-head873" xml:space="preserve" style="it">25. Aequalium & æquidistantium magnitudinum inæqualiter à uiſu distantium pro-<lb/>pinquior ſemper maior uidetur: non tamen proportionaliter ſuis distantijs uidetur. Euclides <lb/>5 theo. opticorum.</head> <pb o="130" file="0432" n="432" rhead="VITELLONIS OPTICAE"/> <p> <s xml:id="echoid-s27866" xml:space="preserve">Sint duæ magnitudines uiſæ a b & g d inæqualiter diſtantes ab oculo:</s> <s xml:id="echoid-s27867" xml:space="preserve"> cuius centrum ſit e, ſitq́;</s> <s xml:id="echoid-s27868" xml:space="preserve"> ui <lb/>ſui propinquior g d quàm a b.</s> <s xml:id="echoid-s27869" xml:space="preserve"> Dico, quòd maior apparebit g d quàm a b.</s> <s xml:id="echoid-s27870" xml:space="preserve"> Producantur enim lineæ <lb/>e a, e b, e d, e g:</s> <s xml:id="echoid-s27871" xml:space="preserve"> uidebiturq́;</s> <s xml:id="echoid-s27872" xml:space="preserve"> g d ſub angulo g e d, qui eſt maior angulo a e b, ut parte ſua per 34 th.</s> <s xml:id="echoid-s27873" xml:space="preserve"> 1 <lb/>huius.</s> <s xml:id="echoid-s27874" xml:space="preserve"> Patet ergo per 20 huius, quia linea g d uidebitur maior quàm <lb/> <anchor type="figure" xlink:label="fig-0432-01a" xlink:href="fig-0432-01"/> linea a b.</s> <s xml:id="echoid-s27875" xml:space="preserve"> Et hoc eodem modo demonſtrandum, ſiue centrum uiſus <lb/>& res uiſæ ſint in eadem altitudine, ſiue in diuerſis:</s> <s xml:id="echoid-s27876" xml:space="preserve"> ut ſi uiſus ſit al-<lb/>tior rebus uiſis, uel etiam econtrà.</s> <s xml:id="echoid-s27877" xml:space="preserve"> Non tamen uidentur hæc propor <lb/>tionaliter ſuis diſtantijs, uidelicet ut proportio g d maioris ſecundũ <lb/>apparentiam ad a b minorem ſecundum apparentiam ſit, ſicut b e di <lb/>ſtantiæ maioris ad d e diſtantiam minorem:</s> <s xml:id="echoid-s27878" xml:space="preserve"> quoniam, ut patet per 11 <lb/>huius, maior eſt proportio b e diſtantiæ maioris ad d e diſtantiam mi <lb/>norem, quàm anguli g e d maioris ad angulum a e b minorem.</s> <s xml:id="echoid-s27879" xml:space="preserve"> Sed <lb/>quantùm angulus g e d eſt maior angulo a e b, tantò linea g d uide-<lb/>tur maior quàm linea a b, ut diximus in 20 huius, quoniam illa uiſi-<lb/>bilia conformiter ordinantur ad uiſum.</s> <s xml:id="echoid-s27880" xml:space="preserve"> Non uidentur ergo lineæ g <lb/>d & a b proportinaliter ſuis diſtantijs:</s> <s xml:id="echoid-s27881" xml:space="preserve"> quoniam diſtantiarum maior <lb/>eſt proportio.</s> <s xml:id="echoid-s27882" xml:space="preserve"> Ethoc eſt propoſitum.</s> <s xml:id="echoid-s27883" xml:space="preserve"/> </p> <div xml:id="echoid-div1087" type="float" level="0" n="0"> <figure xlink:label="fig-0432-01" xlink:href="fig-0432-01a"> <variables xml:id="echoid-variables445" xml:space="preserve">b a d g e</variables> </figure> </div> </div> <div xml:id="echoid-div1089" type="section" level="0" n="0"> <head xml:id="echoid-head874" xml:space="preserve" style="it">26. Omne uiſibile obliquatum à uiſu, minus uidetur ſe ipſo, ſe-<lb/>cundum proximum ſui terminum directè uiſui oppoſito.</head> <p> <s xml:id="echoid-s27884" xml:space="preserve">Sit enim linea connectens centra oculorum r t:</s> <s xml:id="echoid-s27885" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s27886" xml:space="preserve"> centrum ui-<lb/>ſus a:</s> <s xml:id="echoid-s27887" xml:space="preserve"> & ſit uiſibile obliquatum à uiſu, b c:</s> <s xml:id="echoid-s27888" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s27889" xml:space="preserve"> lineæ a b & a c:</s> <s xml:id="echoid-s27890" xml:space="preserve"> <lb/>& à puncto c, qui ſit terminus rei uiſæ proximus uiſui, ducatur li-<lb/>nea c d, æqualis lineæ c b, & æquidiſtans lineæ r t, connectenti centra oculorum, quod fieri poteſt <lb/>per 39 th.</s> <s xml:id="echoid-s27891" xml:space="preserve"> 3 huius:</s> <s xml:id="echoid-s27892" xml:space="preserve"> illa ergo dιrectè uiſui opponetur per <lb/>1 definitionem huius:</s> <s xml:id="echoid-s27893" xml:space="preserve"> ducatur quoq;</s> <s xml:id="echoid-s27894" xml:space="preserve"> linea a d.</s> <s xml:id="echoid-s27895" xml:space="preserve"> Et quo-<lb/>niam per 7 huius linea c d ſub maiori angulo uidetur <lb/> <anchor type="figure" xlink:label="fig-0432-02a" xlink:href="fig-0432-02"/> quàm linea c b:</s> <s xml:id="echoid-s27896" xml:space="preserve"> patet per 20 huius, quoniam minor ui-<lb/>detur linea c b obliquata quàm ſua æqualis, quæ eſt li-<lb/>nea c d, directè uiſui oppoſita ſecundum proximum <lb/>terminum ipſius lineæ c b, quo uiſui plus appropin-<lb/>quat, qui eſt punctus c.</s> <s xml:id="echoid-s27897" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s27898" xml:space="preserve"/> </p> <div xml:id="echoid-div1089" type="float" level="0" n="0"> <figure xlink:label="fig-0432-02" xlink:href="fig-0432-02a"> <variables xml:id="echoid-variables446" xml:space="preserve">b d r c a t</variables> </figure> </div> </div> <div xml:id="echoid-div1091" type="section" level="0" n="0"> <head xml:id="echoid-head875" xml:space="preserve" style="it">27. Vera rerum quantitas non comprehendi-<lb/>tur à uiſu, niſi auxilio uirtutis diſtinctiuæ. Alha-<lb/>zen 36. 38 n 2.</head> <p> <s xml:id="echoid-s27899" xml:space="preserve">Quoniam enim, ut patet ex præmiſsis, anguli, qui <lb/>formantur in centro uiſus, & partes ſuperficierum ui-<lb/>ſus, ſecundum quas fit comprehenſio magnitudinis <lb/>rei uiſæ, ſemper diuerſantur ſecundum approximatio <lb/>nem & remotionẽ eiuſdem rei, & ſecundum eandem <lb/>directionem uel obliquationem ſe habentis ad uiſum <lb/>& ad axes radiales:</s> <s xml:id="echoid-s27900" xml:space="preserve"> uirtus ergo diſtinctiua diſtinguens quantitaẽ ueram rei uiſæ, non conſiderabit <lb/>ſolum angulum uel ſolam remotionẽ:</s> <s xml:id="echoid-s27901" xml:space="preserve"> quoniam neutrum illorum per ſe ſufficit, ſed conſiderabit an-<lb/>gulum & remotionẽ ſimul.</s> <s xml:id="echoid-s27902" xml:space="preserve"> Quantitates ergo ueræ ipſorum uiſibilium non comprehendentur niſi <lb/>per diſtinctionem & comparationem:</s> <s xml:id="echoid-s27903" xml:space="preserve"> hæc autem comparatio erit ſimul:</s> <s xml:id="echoid-s27904" xml:space="preserve"> & erit ipſius baſis pyrami-<lb/>dis radialis (quæ per 18 th.</s> <s xml:id="echoid-s27905" xml:space="preserve"> 3 huius eſt ſuperficies rei uiſæ) ad angulum pyramidis, & ad quantitatem <lb/>longitudinis axis pyramidis, quæ eſt linea remotionis rei uiſæ à uiſu.</s> <s xml:id="echoid-s27906" xml:space="preserve"> Conſideratio uerò uirtutis di-<lb/>ſtinctiuę ipſius ſuperficiei eſt ſemper in parte colorata ſuperficiei uiſus, angulo dicto correſponden <lb/>ti, cum conſideratione rem otionis ipſius rei uiſæ à ſuperficie uiſus:</s> <s xml:id="echoid-s27907" xml:space="preserve"> quoniam quantitas illius par-<lb/>tis coloratæ ſuperficiei uiſus ſemper eſt ſecundum quantitatẽ illius anguli per 73 th 3 huius.</s> <s xml:id="echoid-s27908" xml:space="preserve"> Nó eſt <lb/>autem in illa conſideratione uirtutis diſtinctiuæ inter remotionem rei uiſæ à ſuperficie uiſus & re-<lb/>motionẽ eius à centro uiſus diuerſitas ſenſibilis.</s> <s xml:id="echoid-s27909" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s27910" xml:space="preserve"> uiſus comprehendit lineas pyramidis <lb/>radialis perpendiculariter ſibi incidentes:</s> <s xml:id="echoid-s27911" xml:space="preserve"> tunc uirtus diſtinctiua imaginabitur quantitatem exten <lb/>ſionis, ſecundum quantitatẽ extenſionis iſtarum linearum à centro uiſus uſq;</s> <s xml:id="echoid-s27912" xml:space="preserve"> ad terminos rei uiſæ:</s> <s xml:id="echoid-s27913" xml:space="preserve"> <lb/>& quando cũ hoc comprehenderit quantitatẽ remotionis rei uiſæ per 10 huius:</s> <s xml:id="echoid-s27914" xml:space="preserve"> tunc imaginabitur <lb/>quantitatẽ longitudinũ iſtarum linearũ & quantitatẽ ſpatiorũ, quæ ſunt inter ipſarũ extremitates, <lb/>quæ ſpatia ſunt diametri ipſius rei uiſæ.</s> <s xml:id="echoid-s27915" xml:space="preserve"> Quando ergo uirtus diſtinctiua imaginabitur quantitatẽ <lb/>anguli, & quantitatẽ partis ſuperficiei uiſus, correſpondentis illi angulo, & quantitatẽ longitudinis <lb/>linearum radialium, & quantitatem ſitus ipſarum adinuicem, & quantitatem ſpatiorum, quæ ſunt <lb/>inter extremitates earum:</s> <s xml:id="echoid-s27916" xml:space="preserve"> tunc ipſa comprehendet quantitatem rei uiſæ ſecundum ſuum eſſe:</s> <s xml:id="echoid-s27917" xml:space="preserve"> quo-<lb/>niam tunc nihil eorum, quibus comprehenditur magnitudo rei uiſæ, remanet incomprehenſum.</s> <s xml:id="echoid-s27918" xml:space="preserve"> <lb/>Hæc eſt ita que qualitas comprehenſionis magnitudinis rerum uiſarum, & fit plurimum propter <lb/> <pb o="131" file="0433" n="433" rhead="LIBER QVARTVS."/> affuetudinem uiſus in diſtinctione remotionum uiſibilium:</s> <s xml:id="echoid-s27919" xml:space="preserve"> qui quando ſenſerit formam & remo-<lb/>tionem rei uiſæ, ſtatim imaginabitur quantitatem loci & quantitatem remotionis, & ex ijs compre-<lb/>hendet magnitudinem rei uiſæ.</s> <s xml:id="echoid-s27920" xml:space="preserve"> Patet ergo illud, quod proponebatur.</s> <s xml:id="echoid-s27921" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1092" type="section" level="0" n="0"> <head xml:id="echoid-head876" xml:space="preserve" style="it">28. In magnitudinis uiſione uirtuti diſtinctiua error accidit ex intemperata diſpoſitione o-<lb/>cto circunſtantiarum cuiuslibet reiuiſa Alhazen 26. 37. 47. 54. 59. 64. 66. 69 n 3.</head> <p> <s xml:id="echoid-s27922" xml:space="preserve">Exintemperata enim lucis diſpoſitione, ut de nocte uel in crepuſculis cum lux eſt dubia, inſpe-<lb/>cto homine & uiſo nemore aut pariete remotis ab illo homine, cum latuerit hominem uidentem di <lb/>ſtantia inter hominem & nemus aut parietem uiſum, quamuis illa diſtantia ſecundum ueritatem ſit <lb/>plurima:</s> <s xml:id="echoid-s27923" xml:space="preserve"> tunc uidebitur propinquitas hominis ad nemus uel ad parietem:</s> <s xml:id="echoid-s27924" xml:space="preserve"> & ſi accidit, utidem ra-<lb/>dius pertingens ad caputhominis perueniat ad cacumen nemoris:</s> <s xml:id="echoid-s27925" xml:space="preserve"> tunc per 19 huius uidebuntur <lb/>homo & nemus aut paries eιuſdem altitudinis:</s> <s xml:id="echoid-s27926" xml:space="preserve"> quooiá ſub eodem angulo uidentur:</s> <s xml:id="echoid-s27927" xml:space="preserve"> & forſitan ho-<lb/>mo uidebitur maioris altitudinis ipſo nemore:</s> <s xml:id="echoid-s27928" xml:space="preserve"> ut ſi radius tranſiens caput hominis ad nemoris uel <lb/>parietis altitudinem non pertingat.</s> <s xml:id="echoid-s27929" xml:space="preserve"> Et huius ſimile accidit iuxta ciuitatem Vratislauiæ apud ne-<lb/>mus uillæ Boret:</s> <s xml:id="echoid-s27930" xml:space="preserve"> uiſt ſunt enim homines ibi in crepuſculis altiores nemore illo alto:</s> <s xml:id="echoid-s27931" xml:space="preserve"> & uiſus eſt lu-<lb/>pus iuxta lignum & caſtrum Poloniæ æqualis altitudinis ipſi nemori:</s> <s xml:id="echoid-s27932" xml:space="preserve"> ſed hoc accidit in horis cre-<lb/>puſcularibus:</s> <s xml:id="echoid-s27933" xml:space="preserve"> cum lux eſt dubia:</s> <s xml:id="echoid-s27934" xml:space="preserve"> & æſtimata ſunt illa uiſa fuiſſe phantaſmata à uidentibus.</s> <s xml:id="echoid-s27935" xml:space="preserve"> Non ac-<lb/>cideret autem aliquid talium, luce exiſtente in temperamento, quoniam tunc diſtantia hominis à <lb/>nemore diſceroeretur, & altitudo uniuſcuiuſq;</s> <s xml:id="echoid-s27936" xml:space="preserve"> ſecundum terminum ιpſius apparentem menſura-<lb/>retur.</s> <s xml:id="echoid-s27937" xml:space="preserve"> Similιter etiam ex coloris debιlitate accidit error in uiſione magnitudinis:</s> <s xml:id="echoid-s27938" xml:space="preserve"> quoniam ſi in a-<lb/>liquo loco ſtatuatur aliquod corpus fortis coloris, non latebit uiſum:</s> <s xml:id="echoid-s27939" xml:space="preserve"> quòd ſi in eodem loco pona-<lb/>tur eorpus æquale priorι, ſed coloris debilis, non uidebitur illud corpus.</s> <s xml:id="echoid-s27940" xml:space="preserve"> Sic etiam accidιt error iſte <lb/>ex colorum identitate in corpore medio & in re uiſa.</s> <s xml:id="echoid-s27941" xml:space="preserve"> Vnde corpus album in loco aliquo poſitum <lb/>effuſa aliqua albedine in ſuperficie terræ interiacentis uiſum & rẽ uiſam, nõ uidebitur:</s> <s xml:id="echoid-s27942" xml:space="preserve"> remota uerò <lb/>albedine ſpatij interiacẽtis:</s> <s xml:id="echoid-s27943" xml:space="preserve"> ſtatim forma illius albi corporιs cóprehendetut.</s> <s xml:id="echoid-s27944" xml:space="preserve"> Fit ergo tũc occultatio <lb/>ex conuenientia colorum:</s> <s xml:id="echoid-s27945" xml:space="preserve"> quoniam ſi loco illius albi corporis ponatur corpus æquale ſibi alterius <lb/>coloris, bene uidebitur ipſum trans medium dealbatum.</s> <s xml:id="echoid-s27946" xml:space="preserve"> Exintemperara etiam longitudinis di-<lb/>ſtantia fit error in magnitudinis uiſione:</s> <s xml:id="echoid-s27947" xml:space="preserve"> quoniam tunc uidebitur res multò minor quàm ſit in ue-<lb/>ritate per 24 th.</s> <s xml:id="echoid-s27948" xml:space="preserve"> huius:</s> <s xml:id="echoid-s27949" xml:space="preserve"> tunc enim etiam partes eiuſdem rei improportionales ſuo tori abſcondun-<lb/>tur uiſui, quia non poſſunt in tanta diſtantia uideri per 23 th.</s> <s xml:id="echoid-s27950" xml:space="preserve"> huius:</s> <s xml:id="echoid-s27951" xml:space="preserve"> & fit minor totalis rei apparen-<lb/>tia:</s> <s xml:id="echoid-s27952" xml:space="preserve"> quoniam plura inſenſibiliter abſcondita faciuntrei ſenſibilem ablationem, quæ non fieret in di-<lb/>ſtantia temperata.</s> <s xml:id="echoid-s27953" xml:space="preserve"> Intemperata etiam approximatio errorem inducit in uiſione magnitudinis:</s> <s xml:id="echoid-s27954" xml:space="preserve"> <lb/>quoniam corpus approximatum oculo, uidetur maioris quantitatis quàm ſit reuera:</s> <s xml:id="echoid-s27955" xml:space="preserve"> quoniã pro-<lb/>pter magnitudinem anguli corpus uidetur maius, ut prius propter paruitatẽ anguli corpus uiſum <lb/>eſt minus:</s> <s xml:id="echoid-s27956" xml:space="preserve"> & patent hæc per 20 th.</s> <s xml:id="echoid-s27957" xml:space="preserve"> huius:</s> <s xml:id="echoid-s27958" xml:space="preserve"> ſecundum quantitatem enim ampliorem anguli pyramidà <lb/>lis amplior ſuperficies uiſus informatur, ut patet per 87 th 1 huius:</s> <s xml:id="echoid-s27959" xml:space="preserve"> unde ſecũdum quantitatẽ illius <lb/>anguli & elongationem corporis fit æſtimatio quantitatιs rei uiſæ, ut præ miſſum eſt in præcedente <lb/>propoſitione:</s> <s xml:id="echoid-s27960" xml:space="preserve"> nec enim longitudo dιſtantiæ rei ad interiora uidentis penetrat;</s> <s xml:id="echoid-s27961" xml:space="preserve"> cum pars capitis in-<lb/>terior non ſit capax totius quantitatis radialium linearú, nec poteſt certitudinaliter menſurari:</s> <s xml:id="echoid-s27962" xml:space="preserve"> & <lb/>propter hoc rei quantitas refertur ad anguli capacitatem & notam longitudinem.</s> <s xml:id="echoid-s27963" xml:space="preserve"> Vera autem re-<lb/>inotio corporis atten ditur ſecundum lineam à centro u ſus ad ſuperſiciem rei procedentem, reſpe-<lb/>ctu cuius lineæ ſemidiameter oculi incipit eſſe inſenſibilis:</s> <s xml:id="echoid-s27964" xml:space="preserve"> unde non facit aliquem ſenſibilem erro-<lb/>rem in longitudinis illius æſtimatione:</s> <s xml:id="echoid-s27965" xml:space="preserve"> ſed corpore approximato uiſui ultra illam diſtantiam, tune <lb/>fit ſemidiameter oculi proportionalis diſtantιæ corporis proportione ſenſibili:</s> <s xml:id="echoid-s27966" xml:space="preserve"> erit enim aliquan-<lb/>do maior, aliquando æ qualis, aliquan do minor proportione modica, ut fortè ſubdupla uel ſubtri-<lb/>pla, uel huiuſmodi:</s> <s xml:id="echoid-s27967" xml:space="preserve"> unde in tali propinquitate rei uiſæ, magnitudo angul pytamid pytamidalis, & ſenſibilis <lb/>minoricas longitudinis æſtimatæ, reſpectu ueræ inducunt ſenſibilem apparentiam maioritatis in <lb/>corpore.</s> <s xml:id="echoid-s27968" xml:space="preserve"> Exinordinata etiam ſitus oppoſitione fit error in magnitudinis uiſione:</s> <s xml:id="echoid-s27969" xml:space="preserve"> cum enim ali-<lb/>quis in alto exiſtens uidet ſub illa altitudine aliqua exiſtentia inter ſe æqualia, quorum eſt unum <lb/>poſt aliud in ordine diſpoſitum:</s> <s xml:id="echoid-s27970" xml:space="preserve"> tunc enim per 25 huius iudicabitur poſtremum, quod eſt uidenti <lb/>propin quius, altius omnibus alijs uel maius:</s> <s xml:id="echoid-s27971" xml:space="preserve"> ut uigil ſtans in turris alicuius eminentia, uidens ho-<lb/>mines uel arbores æquales, inæ qualiter à ſe diſtantes, propinquiorem ſibi æſtimat altiorem.</s> <s xml:id="echoid-s27972" xml:space="preserve"> Ex <lb/>intemperata etiam quantitatis rei uiſæ diſpoſitione accidit error in magnitudinis uiſione:</s> <s xml:id="echoid-s27973" xml:space="preserve"> propoſi-<lb/>tis enim uiſui duobus corporibus, quorum unum ſit modicum maius alio, aut in ſola longitudine, <lb/>aut in latitudine, aut in utraq;</s> <s xml:id="echoid-s27974" xml:space="preserve"> ipſarum:</s> <s xml:id="echoid-s27975" xml:space="preserve"> forſitan illa indicabuntur æqualia in omni dimenſione, <lb/>quoniam paruitas ill us exceſſus non ſentitur propter ſui paruitatem:</s> <s xml:id="echoid-s27976" xml:space="preserve"> non enim excedit fines tem-<lb/>perantiæ reſpectu ipſius uiſus.</s> <s xml:id="echoid-s27977" xml:space="preserve"> Exintemperata etiam ſoliditate fit error in uiſione magnitudinis:</s> <s xml:id="echoid-s27978" xml:space="preserve"> <lb/>in cryſtallo enim angulata, extrema angulorum, quia parum ſolida ſunt, quandoq;</s> <s xml:id="echoid-s27979" xml:space="preserve"> non uidentur, <lb/>cum corporis ſolidi anguli uideri poſſent.</s> <s xml:id="echoid-s27980" xml:space="preserve"> Exintemperantia etiam raritatis in uiſione magnitu-<lb/>dinis error accidit:</s> <s xml:id="echoid-s27981" xml:space="preserve"> quoniam in aere nubiloſo obſcuro, ut in horis crepuſcularibus plurimum acci-<lb/>dit, quòd corpus uiſum maius apparet quàm in aere temperato, ut nos infrà declarabimus, cum tra <lb/>ctatum de ijs, quæ uidentur per medium ſecundi diaphani faciemus.</s> <s xml:id="echoid-s27982" xml:space="preserve"> Ex intemperantia eriam <lb/>temporis fit error in uιſione quantiatis:</s> <s xml:id="echoid-s27983" xml:space="preserve"> cum enim ardens titio ſæpius per aliquod ſpatium ue-<lb/>lociter mouetur, apparet totum ſpatium ignitum:</s> <s xml:id="echoid-s27984" xml:space="preserve"> quia non perpenditur quantitas temporis, <lb/> <pb o="132" file="0434" n="434" rhead="VITELLONIS OPTICAE"/> propter uelocitatem motus titionis:</s> <s xml:id="echoid-s27985" xml:space="preserve"> & ſic ignis paruus æſtimatur maior propter ſui motus tempe-<lb/>ris breuitatem.</s> <s xml:id="echoid-s27986" xml:space="preserve"> Exintemperantia & uiſus debilitate in magnitudinis uiſione error accidit:</s> <s xml:id="echoid-s27987" xml:space="preserve"> quia e-<lb/>tiam res fortè parua nullo modo uidetur:</s> <s xml:id="echoid-s27988" xml:space="preserve"> ut patet in ſenibus, qui non poſſunt diſcernore literam mi <lb/>nutam.</s> <s xml:id="echoid-s27989" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s27990" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1093" type="section" level="0" n="0"> <head xml:id="echoid-head877" xml:space="preserve" style="it">29. Viſio comprehendit omnem ſitum per comprehenſionem debitæ remotionis in ipſis rebus <lb/>ſituatis. Alhazen 26 n 2.</head> <p> <s xml:id="echoid-s27991" xml:space="preserve">Siue enim nomen ſitus dicat totius rei uiſę, ſiue partium eius oppoſitionem ad uiſum ſecundum <lb/>directionem uel obliquationem:</s> <s xml:id="echoid-s27992" xml:space="preserve"> ſiue dicat ordinationem ſuperficierum rei uiſæ, uel partium eius <lb/>apud ſuperficiem ipſius uiſus, ut cum res uiſa eſt multarũ ſuperficierum apparentium uiſui:</s> <s xml:id="echoid-s27993" xml:space="preserve"> ſiue no <lb/>men ſitus dicat ſituationem linearum, quæ ſuntipſarum ſuperficierum uiſibilium:</s> <s xml:id="echoid-s27994" xml:space="preserve"> ſiue dicat ſitum.</s> <s xml:id="echoid-s27995" xml:space="preserve"> <lb/>ſpatiorũ, quæ ſunt inter quælibet duo uiſibilia ſimul comprehenſa à uiſu:</s> <s xml:id="echoid-s27996" xml:space="preserve"> ſemper accepto ſitu ſecun <lb/>dum quemcunq;</s> <s xml:id="echoid-s27997" xml:space="preserve"> iſtorum modorum, hæc omnia & ſingula comprehendit uiſus, ut hæc ſunt diſp oſi-<lb/>ta in corporibus lucidis uel coloratis, ut in per ſe uiſibilibus & in illis ſundata:</s> <s xml:id="echoid-s27998" xml:space="preserve"> & ſemper cóprehen-<lb/>dit quemlibet modũ ſitus, cóprehenſa remotione à uiſu uel inter ſe, quæ debentur ipſis totis uel par <lb/>tibus ſituatis.</s> <s xml:id="echoid-s27999" xml:space="preserve"> Pater ergo propoſitũ:</s> <s xml:id="echoid-s28000" xml:space="preserve"> quoniá hos modos particulariter in ſequentibus proſequemur.</s> <s xml:id="echoid-s28001" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1094" type="section" level="0" n="0"> <head xml:id="echoid-head878" xml:space="preserve" style="it">30. Situs oppoſitionis rei uiſa & partium eius ad uiſum, comprehenditur à ſenſu uiſus auxi <lb/>lio uirtutis diſtinctiua. Alhazen 27 n 2.</head> <p> <s xml:id="echoid-s28002" xml:space="preserve">Cum enim ſitus cuiuslibet habentis ſitum ad aliud, componatur ex remotione illorum duorum <lb/>ad inuicem:</s> <s xml:id="echoid-s28003" xml:space="preserve"> palàm, quòd oppoſitio rei uiſæ ad uiſum, quæ quidem ſitus eſt, componitur ex remotio <lb/>ne rei uiſæ à uiſu, & ex parte uniuerſi, in qua eſt res uiſa reſpectu uiſus.</s> <s xml:id="echoid-s28004" xml:space="preserve"> Comprehenſio autem remo-<lb/>tionis rei uiſæeſt ab ipſa uirtute diſtinctiua per intentionem quieſcentem in anima, ut oſtenſum eſt <lb/>per 9 & 10 the.</s> <s xml:id="echoid-s28005" xml:space="preserve"> huius.</s> <s xml:id="echoid-s28006" xml:space="preserve"> Cum ergo uirtus diſtinctiua comprehendet locum rei uiſæ & ſuam remotio-<lb/>nem:</s> <s xml:id="echoid-s28007" xml:space="preserve"> tunc in ſimul cum illis comprehendet rei oppoſitionem:</s> <s xml:id="echoid-s28008" xml:space="preserve"> uerus autem locus rei uiſæ compre-<lb/>henditur ex ſitu ipſius uiſus, & ex ſitu ipſius rei uiſæ apud uiſionem, quoniam uiſus nõ comprehen <lb/>dit rem uiſam niſi ex oppoſitione.</s> <s xml:id="echoid-s28009" xml:space="preserve"> Diſtιnguet ergo uirtus diſtinctiua inter locum obliquum uiſui, & <lb/>locum propinquum ei:</s> <s xml:id="echoid-s28010" xml:space="preserve"> uirtus enim diſtinctiua comprehendit omnia loca rerum locatarũ per com-<lb/>prehenſionem remotionis & partis uniuerſi, ad quam eſt illa remotio, ut patuit per 14 huius:</s> <s xml:id="echoid-s28011" xml:space="preserve"> unde <lb/>etiam comprehendet locum oppoſitum uiſui apud comprehenſionẽ rei uiſæ.</s> <s xml:id="echoid-s28012" xml:space="preserve"> Et quoniam uiſu abla <lb/>to ab illa re uiſa, deſtruitur uiſio illius rei, tunc uirtus diſtinctiua comprehendit, quòd res uiſa non <lb/>eſt, niſi in parte oppoſita uiſui apud uiſionem illius rei uiſæ:</s> <s xml:id="echoid-s28013" xml:space="preserve"> & ſecundum hunc modum diſtinguun-<lb/>turloca uiſibilium, quoniam uiſibilia diſtincta non diſtinguuntur à uiſu niſi ex diſtinctione loco-<lb/>rum diſtinctorum in ſuperficie membri ſentientis, ad quod perueniunt formæ uiſibilium diſtincto-<lb/>rum.</s> <s xml:id="echoid-s28014" xml:space="preserve"> Sicutitaq;</s> <s xml:id="echoid-s28015" xml:space="preserve"> loca uocum & ſonorum comprehenduntur à ſenſu auditus:</s> <s xml:id="echoid-s28016" xml:space="preserve"> & deinde mediante au <lb/>ditu à uirtute diſtinctiua:</s> <s xml:id="echoid-s28017" xml:space="preserve"> ita loca uiſibilium comprehenduntur mediante uiſu à uirtute diſtinctiua.</s> <s xml:id="echoid-s28018" xml:space="preserve"> <lb/>Cum enim forma rei uiſę peruenerit in ſuperſiciem uiſus, ſentiet uirtus uidens locum membri ſen-<lb/>tientis, ad quem peruenit illa forma, & ex rectitudine lineæ perpendiculariter incidentis illi loco <lb/>comprehendet uirtus diſtinctiua locum rei uiſæ:</s> <s xml:id="echoid-s28019" xml:space="preserve"> & quia intentio remotionis eſt quieſcenns apudi-<lb/>pſam animam, ipſa ergo comprehendet locum rei uiſæ, & remotionem eius in ſimul apud compre-<lb/>henſionem formæ à uiſu ſentiente.</s> <s xml:id="echoid-s28020" xml:space="preserve"> In peruentu ergo formæ uiſę ad uiſum comprehendit uiſus lu <lb/>cem & colorem rei uiſæ, & partem ſuperficiei uiſus, quæ illuminatur & coloratur ab iſta forma, & <lb/>uirtus diſtinctiua comprehendit locum & remotionem rei uiſæ, & per conſequens oppoſitionem <lb/>ipſius totius rei uiſæ & omnium partium eius adinuicem in ſuo toto, & omnnium iſtorum compre-<lb/>henſio ſit ſimul.</s> <s xml:id="echoid-s28021" xml:space="preserve"> Situs ergo oppoſitionis rei uiſæ & partium eius ad uiſum comprehenditur à ſenſu <lb/>uiſus auxilio uirtutis diſtinctiuæ.</s> <s xml:id="echoid-s28022" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s28023" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1095" type="section" level="0" n="0"> <head xml:id="echoid-head879" xml:space="preserve" style="it">31. Viſus comprehendit directionem & obliquationem line arum, ſuperſicierum, & ſpatio-<lb/>rum ex comprehenſione diuerſitate remotionum ſuarum extremit atum, auxilio uirtutis diſtin <lb/>ctiua. Alhazen 28 n 2.</head> <p> <s xml:id="echoid-s28024" xml:space="preserve">Cum enim axes radiales ſecant lineas, uel ſuperſicies, uel ſpatia, ut ſuper illa perpendiculariter e-<lb/>recti:</s> <s xml:id="echoid-s28025" xml:space="preserve"> tunc uiſus comprehendit ſuperſiciem rei uiſæ, & remotiones extremitatum eius æ quales ex <lb/>utraq;</s> <s xml:id="echoid-s28026" xml:space="preserve"> parte axis erecti:</s> <s xml:id="echoid-s28027" xml:space="preserve"> & tunc comprehendit illam ſuperſiciem eſſe directè uiſui oppoſitam, & iu-<lb/>dicabit uirtus diſtinctiua ſuperficiem illam directè oppoſitam uiſui.</s> <s xml:id="echoid-s28028" xml:space="preserve"> Cum autem uiſus comprehen-<lb/>derit remotionem extremitatum ſuperſiciei uiſæ diuerſam, & à puncto coniunctionis axium extra <lb/>lineam, in quam incidunt axes perpendiculariter, non inuenit in tota ſuperſicie ſibi oppoſita duo <lb/>puncta æ qualis remotionis à ſuperficie uiſus:</s> <s xml:id="echoid-s28029" xml:space="preserve"> tunc comprehendet illam ſuperſiciem obliquatam in <lb/>eius oppoſitione, & uirtus diſtinctiua iudicabitipſam obliquatam.</s> <s xml:id="echoid-s28030" xml:space="preserve"> Et ſimiliter eſt de ſuibus linea-<lb/>rum & ſpatiorum cadentium inter res plures uiſas ſimul:</s> <s xml:id="echoid-s28031" xml:space="preserve"> ipſorum enim directionem & obliquatio-<lb/>nem iudicabit uiſus auxilio uirtutis diſtinctiuæ.</s> <s xml:id="echoid-s28032" xml:space="preserve"> Et iſta æqualitas directionis & diuerſitas obliqua-<lb/>tionis multotiens comprehenditur à ſentiente per ſolam æſtimationem & per ſigna:</s> <s xml:id="echoid-s28033" xml:space="preserve"> in maxima e-<lb/>nim diſtantia uel remotione comprehendetur ſuperſicies uel linea uel ſpatium, quod eſt obliqua-<lb/>tum, quaſi ſit directum, quando ſcilicet non perfectè comprehenditur diuerſitas, quæ eſt inter <lb/>remotiones extremitatũ eius:</s> <s xml:id="echoid-s28034" xml:space="preserve"> unde ad hoc, quòd uiſus bene hoc comprehendat, oportet ut talium <lb/> <pb o="133" file="0435" n="435" rhead="LIBER QVARTVS."/> uiſibilium ſit diſtantia mediocris:</s> <s xml:id="echoid-s28035" xml:space="preserve"> quia etiam in magna diſtantia parum obliquata uidentur, ut peni <lb/>tus directa.</s> <s xml:id="echoid-s28036" xml:space="preserve"> Et licet ſecundum modum prędictum ſuperficies aliqua, uel linea, uel ſpatium uiſui ſint <lb/>directè oppoſita:</s> <s xml:id="echoid-s28037" xml:space="preserve"> nulla tamen pars illius ſuperficiei, lineæ, uel ſpatij per ſe directè opponitur uiſui:</s> <s xml:id="echoid-s28038" xml:space="preserve"> <lb/>quoniam axes radiales ubicunq;</s> <s xml:id="echoid-s28039" xml:space="preserve"> extra unum punctum perpendicularitatis incidant, ſemper inci-<lb/>dunt obliquè, & ſecundum angulos in æ quales per 20 th.</s> <s xml:id="echoid-s28040" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s28041" xml:space="preserve"> Si autem ſuperficies, lineę, uel ſpa-<lb/>tia æquidiſtent axibus uiſualibus, nec ſecentur ab illis, opponantur autem uiſui:</s> <s xml:id="echoid-s28042" xml:space="preserve"> tunc etiam ſitus i-<lb/>pſorum in directione & obliquatione comprehenditur à uiſu per remotionem ſuarum extremita-<lb/>tum:</s> <s xml:id="echoid-s28043" xml:space="preserve"> & poteſt fieri proportio iſtorum ad ſuperficies, lineas, uel ſpatia, quæ ſecant axes radiales, qui-<lb/>bus axibus ipſa æquidiſtant.</s> <s xml:id="echoid-s28044" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s28045" xml:space="preserve"> illud, quod proponebatur.</s> <s xml:id="echoid-s28046" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1096" type="section" level="0" n="0"> <head xml:id="echoid-head880" xml:space="preserve" style="it">32. Situs partium & ſitus terminorum ſuperficiei rei uiſa aut ſitus ſuperficierum eius adin <lb/>uicem: & ſitus plurium uiſibilium ſimul uiſorum ex comprebenſione diuerſitatis in remotione <lb/>& ordinatione formarum peruenientium ad uiſum, comprehenditur à uiſu auxilio uirtutis di <lb/>ctinctiua. Alhazen 30 n 2.</head> <p> <s xml:id="echoid-s28047" xml:space="preserve">Quoniam enim forma cuiuslibet partis ſuperficiei rei uiſæ peruenit ad aliquam partem ſuperfi-<lb/>ciei uiſus, ad quam peruenit forma totius rei uiſæ:</s> <s xml:id="echoid-s28048" xml:space="preserve"> unde cum ſuperficies rei uiſæ fuerit diuerſorum <lb/>colorum diſtinctorum:</s> <s xml:id="echoid-s28049" xml:space="preserve"> tunc erit forma perueniens in uiſum, diuerſorum colorum, & erunt partes <lb/>eius diſtinctæ ſecundum diſtinctionem partium ſuperficiei rei uiſæ.</s> <s xml:id="echoid-s28050" xml:space="preserve"> Tunc itaq;</s> <s xml:id="echoid-s28051" xml:space="preserve"> uiſus ſentiet quam-<lb/>libet partem formæ uiſæ exſenſu colorum illarum partium & lucis, quæ eſt in eis, & ſentit loca for-<lb/>marum partium in ſuperficie uiſus ex ſenſu colorum partium illarum & lucis earũ:</s> <s xml:id="echoid-s28052" xml:space="preserve"> & uirtus diſtin-<lb/>ctiua cõprehendit ordinationẽ illorum colorum ex cõprehenſione diuerſitatis partiũ formę, & ex <lb/>comprehenſione differentiarũ ipſarũ partiũ:</s> <s xml:id="echoid-s28053" xml:space="preserve"> & ſic cõprehendit aliquid cõtiguũ & aliquid ſeparatũ.</s> <s xml:id="echoid-s28054" xml:space="preserve"> <lb/>Similiter etiã eſt de ipſis uiſibilibus contiguis uel diſiunctis.</s> <s xml:id="echoid-s28055" xml:space="preserve"> Situs uerò partium rei uiſæ adinuicẽ <lb/>ſecundũ acceſsionem & remotionẽ, uel ſecundũ præeminentiá unius ipſarũ ſuper alterá, & profun <lb/>dationẽ unius ipſarũ ſub altera, cõprehenditur à uiſu ex cõprehenſione quantitatis remotionis par <lb/>tium ſecundũ magis & minus.</s> <s xml:id="echoid-s28056" xml:space="preserve"> Termini aũt ſuperficiei rei uiſæ aut ſuperficierũ eius, quę ſunt lineæ <lb/>ipſas ſuperficies terminantes, & ordinatio ipſorũ, cõprehenditur à uiſu per comprehenſionẽ partis <lb/>ſuperficiei eius, in quã peruenit color ipſius ſuperficiei rei uiſę per illos terminos uel lineas termina <lb/>tæ, & lux eius, & per comprehenſionem terminorum illius partis & ordinationis illius partis, auxi-<lb/>lio uirtutis diſtin ctiuæ.</s> <s xml:id="echoid-s28057" xml:space="preserve"> Et quoniam omnia propoſita ſecundum hunc modum comprehenduntur:</s> <s xml:id="echoid-s28058" xml:space="preserve"> <lb/>patetergo illud, quod proponebatur.</s> <s xml:id="echoid-s28059" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1097" type="section" level="0" n="0"> <head xml:id="echoid-head881" xml:space="preserve" style="it">33. Omnis linea uel ſuperficies reì uiſa directè uiſibus ueluiſui oppoſita, perfectius uidetur <lb/>quàm obliquata: & ſecundũ quantitatẽ obliquationis fit imperfectio uiſionis. Alhazen 17 n 3.</head> <p> <s xml:id="echoid-s28060" xml:space="preserve">Eſto centrum uiſus a:</s> <s xml:id="echoid-s28061" xml:space="preserve"> & ſit, exempli gratia, ſuperficies plana rei uiſæ directè uiſibus oppoſitæ, in <lb/>qua ſit linea b c d e f:</s> <s xml:id="echoid-s28062" xml:space="preserve"> & ſint b c, c d, d e, e f partes illius lineæ æquales uel inæquales:</s> <s xml:id="echoid-s28063" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s28064" xml:space="preserve"> ſuperficies <lb/>obliquata uiſibus, in qua ſit linea f g <lb/> <anchor type="figure" xlink:label="fig-0435-01a" xlink:href="fig-0435-01"/> h i k:</s> <s xml:id="echoid-s28065" xml:space="preserve"> & ſit taliter, ut obliquatio illius <lb/>ſuperficiei incipiat à puncto f:</s> <s xml:id="echoid-s28066" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s28067" xml:space="preserve"> li <lb/>nea a d perpendicularis ſuper lineã <lb/>b f:</s> <s xml:id="echoid-s28068" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s28069" xml:space="preserve"> à centro uiſus lineæ <lb/>a f, a e, a d, a c, a b:</s> <s xml:id="echoid-s28070" xml:space="preserve"> quæ omnes produ-<lb/>cantur ad ſuperficiem obliquatam:</s> <s xml:id="echoid-s28071" xml:space="preserve"> <lb/>incidatq́;</s> <s xml:id="echoid-s28072" xml:space="preserve"> linea a e in punctum g, & li <lb/>nea a d in punctum h, & linea a c in <lb/>punctum i, & linea a b in punctum k.</s> <s xml:id="echoid-s28073" xml:space="preserve"> <lb/>Et quia per 13 p 1 angulus h d f eſt re-<lb/>ctus, quia angulus a d f eſt rectus ex <lb/>hypotheſi:</s> <s xml:id="echoid-s28074" xml:space="preserve"> palàm ergo ք 47 p 1, quo-<lb/>niam linea f h eſt maior quàm linea f <lb/>d.</s> <s xml:id="echoid-s28075" xml:space="preserve"> Et ſi à puncto g ducatur linea ęqui <lb/>diſtans lineę f d per 31 p 1, quę ſit g in:</s> <s xml:id="echoid-s28076" xml:space="preserve"> <lb/>erit per 29 p 1, & 4 p 6, & 47 p 1, linea <lb/>g h maior quàm linea e d:</s> <s xml:id="echoid-s28077" xml:space="preserve"> & ſimiliter <lb/>fiet de omnibus punctis inter punctaf & h datis.</s> <s xml:id="echoid-s28078" xml:space="preserve"> Item à puncto h ducatur linea æ quidiſtans lineę <lb/>d c:</s> <s xml:id="echoid-s28079" xml:space="preserve"> quæ ſit h n.</s> <s xml:id="echoid-s28080" xml:space="preserve"> Et quoniam per 32 p 1 angulus a c d eſt acutus, erit per 13 p 1 angulus i c d obtuſus:</s> <s xml:id="echoid-s28081" xml:space="preserve"> er-<lb/>go per 29 p 1 angulus in h eſt obtuſus:</s> <s xml:id="echoid-s28082" xml:space="preserve"> ergo per 19 p 1 & per 4 p 6 linea h i eſt maior quàm d c.</s> <s xml:id="echoid-s28083" xml:space="preserve"> Eodẽ <lb/>inodo fit de omnibus punctis lineæ h k.</s> <s xml:id="echoid-s28084" xml:space="preserve"> Patet ergo, quòd eidem angulo, qui fit in centro uiſus, ſem <lb/>per ſubten duntur maiores partes lineæ obliquatæ, quàm lineæ directè oppoſitæ uiſui.</s> <s xml:id="echoid-s28085" xml:space="preserve"> Partes itaq;</s> <s xml:id="echoid-s28086" xml:space="preserve"> <lb/>ſuperficiei rei uiſæ directè uiſui uel uiſibus oppoſitę, æqualiter diſtantes à puncto axis, uel à puncto <lb/>coniunctionis, ſimiliter uirtuti uiſiuę offeruntur per 45 th.</s> <s xml:id="echoid-s28087" xml:space="preserve"> 3 huius, propter quod perfectius tota illa <lb/>ſuperficies uidetur, & oẽs ſubtiles intentiones, quę ſunt in ipſa:</s> <s xml:id="echoid-s28088" xml:space="preserve"> ſuperficies uerò obliquata uiſibus <lb/>acquirit formam dubitabilem, ſiue per unũ uiſum uideatur, ſiue per ambos:</s> <s xml:id="echoid-s28089" xml:space="preserve"> & ſiue illa forma per a-<lb/>xes perueniat ad uiſum, ſiue extra axes:</s> <s xml:id="echoid-s28090" xml:space="preserve"> & etiã ſi diſtantia ſit mediocris ipſius ſuperficiei obliquatæ <lb/> <pb o="134" file="0436" n="436" rhead="VITELLONIS OPTICAE"/> à uiſu:</s> <s xml:id="echoid-s28091" xml:space="preserve"> partes enim ſuperficiei illius æquales partibus ſuperficiei directè uiſui oppofitæ, ut patet ex <lb/>prædemonſtratis, ſub minori angulo uidentur, quàm ſi eſſent directè uiſibus oppoſitæ:</s> <s xml:id="echoid-s28092" xml:space="preserve"> quia lineæ <lb/>ſuarum extremitatum à centro uiſus productæ, minoribus angulis ſubtenduntur.</s> <s xml:id="echoid-s28093" xml:space="preserve"> Sic ergo totales <lb/>illæ ſuperficies inſtituuntur in ſuperficiebus uiſus, quaſi congregatæ propter ſuam obliquationem:</s> <s xml:id="echoid-s28094" xml:space="preserve"> <lb/>angulus enim, quem ſubtendit ſuperficies ipſius uiſus, quæ eſt informata forma ſuperficiei obliqua <lb/>tæ, eſt paruus & ſenſibιliter minor eo, quem faceret eadem ſuperficies uiſibus oppoſita directè, uel <lb/>ſuperficies aliqua alia æqualis ſuperficiei obliquatæ.</s> <s xml:id="echoid-s28095" xml:space="preserve"> Quia ergo ipſa ſuperficies uiſus informata ex <lb/>illa obliquata ſuperficie eſt minor, & partes paruæ illius ſuperficiei obliquatæ incidunt angulis <lb/>quaſi inſenſibilibus, propter maximam obliquationem:</s> <s xml:id="echoid-s28096" xml:space="preserve"> ideo de neceſsitate illa ſuperficies obliqua-<lb/>ta uidetur minus perfectè.</s> <s xml:id="echoid-s28097" xml:space="preserve"> Cum enim parua ſuperficies fuerit multũ obliquata:</s> <s xml:id="echoid-s28098" xml:space="preserve"> tunc duæ lineæ ex-<lb/>euntes à centro uiſus ad extrema illius partis, fient quaſi linea una:</s> <s xml:id="echoid-s28099" xml:space="preserve"> quapropter ſentiens non com <lb/>prehendet angulum contentum interillas, neq;</s> <s xml:id="echoid-s28100" xml:space="preserve"> partem, quam diſtinguunt ex ſuperficie uiſus.</s> <s xml:id="echoid-s28101" xml:space="preserve"> To <lb/>ta ergo ſuperficies obliquata uiſui multũ amittit ſenſibilitatis:</s> <s xml:id="echoid-s28102" xml:space="preserve"> quia ſi ιn ipſa fuerint ſubtiles aliquę <lb/>intentiones, non comprehendentur à uiſu, propter latitantiã ſuarum partium paruarum.</s> <s xml:id="echoid-s28103" xml:space="preserve"> Et quoniã <lb/>ſuperficiebus plus obliquatis plus accidit propoſitæ paſsionis:</s> <s xml:id="echoid-s28104" xml:space="preserve"> ideo ſecundum quantitatẽ obliqua-<lb/>tionis fit imperfectio uiſionis.</s> <s xml:id="echoid-s28105" xml:space="preserve"> Patet ergo illud, quod proponebatur.</s> <s xml:id="echoid-s28106" xml:space="preserve"/> </p> <div xml:id="echoid-div1097" type="float" level="0" n="0"> <figure xlink:label="fig-0435-01" xlink:href="fig-0435-01a"> <variables xml:id="echoid-variables447" xml:space="preserve">f g e h m d a i n c k o b</variables> </figure> </div> </div> <div xml:id="echoid-div1099" type="section" level="0" n="0"> <head xml:id="echoid-head882" xml:space="preserve" style="it">34. Exceſſu remotionis nimio exiſtente: res à uiſibus obliquata quando uidetur directè op-<lb/>poſita. Alhazen 29 n 2.</head> <p> <s xml:id="echoid-s28107" xml:space="preserve">Quoniam enim, ut patet per 10 huius, quantitas remotionis attenditur ſecundum quantitatem <lb/>diametrorum rei uiſæ:</s> <s xml:id="echoid-s28108" xml:space="preserve"> ideo & nimietas exceſſus remotionis attenditur ſecundum quantitatem dia <lb/>metrorum rei uiſæ.</s> <s xml:id="echoid-s28109" xml:space="preserve"> Quæ enim magno uiſibili non eſt <lb/>nimia diſtantia à uiſu, hæc minori uiſibili eſt nimia:</s> <s xml:id="echoid-s28110" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0436-01a" xlink:href="fig-0436-01"/> quoniam nõ eodem modo in eadem diſtantia maius <lb/>& minus percipiuntur à uiſu, ut patet per 7 & 20 hu-<lb/>ius.</s> <s xml:id="echoid-s28111" xml:space="preserve"> Sititaq;</s> <s xml:id="echoid-s28112" xml:space="preserve"> centrũ uiſus a:</s> <s xml:id="echoid-s28113" xml:space="preserve"> & res uiſa obliquata, quæ <lb/>b c:</s> <s xml:id="echoid-s28114" xml:space="preserve"> cuius alter terminorum, qui ſit b, propinquior ſit <lb/>uiſui:</s> <s xml:id="echoid-s28115" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s28116" xml:space="preserve"> illa res uiſa ſub angulo b a c:</s> <s xml:id="echoid-s28117" xml:space="preserve"> erit ergo argu <lb/>mento 26 & 20 huius angulus b a c minor, quàm ſi i-<lb/>pſa res uiſa (quæ b c) à proximo ſui termino ad uiſum <lb/>(qui eſt a) directè uideretur:</s> <s xml:id="echoid-s28118" xml:space="preserve"> ſed per 11 huius in omni <lb/>bus uiſis maior eſt proportio diſtantiæ maioris ad di <lb/>ſtantiam minoris, quàm ſit anguli maioris ad angulũ <lb/>minorem:</s> <s xml:id="echoid-s28119" xml:space="preserve"> in nimia aũt remotione diſtantiarum pro-<lb/>portio diſtantiæ maioris unius extremorum rei uiſę, <lb/>utin propoſito ipſius c ad diſtantiã minorẽ alterius <lb/>extremorũ, ut ipſius b, eſt differentia inſenſibilis, ut <lb/>lineæ a c longioris ad lineã a b breuiorẽ:</s> <s xml:id="echoid-s28120" xml:space="preserve"> ergo multò <lb/>magis inſenſibilis eſt differentia ipſorum angulorũ.</s> <s xml:id="echoid-s28121" xml:space="preserve"> <lb/>Videbitur ergo b c in maxima remotione quaſi directè uiſibus oppoſita, cum ſit obliquata.</s> <s xml:id="echoid-s28122" xml:space="preserve"> Ethoe <lb/>eſt propoſitum.</s> <s xml:id="echoid-s28123" xml:space="preserve"/> </p> <div xml:id="echoid-div1099" type="float" level="0" n="0"> <figure xlink:label="fig-0436-01" xlink:href="fig-0436-01a"> <variables xml:id="echoid-variables448" xml:space="preserve">c c b a</variables> </figure> </div> </div> <div xml:id="echoid-div1101" type="section" level="0" n="0"> <head xml:id="echoid-head883" xml:space="preserve" style="it">35. Omne uiſum exiſtens extra communem axem, in uno tantùm axe uiſuali: uelper radios <lb/>propinquos axi: uel etiam per propinquos ambobus axibus uiſualibus comprebenſum, uidetur <lb/>axi communi approximare plus eius ſitu uero.</head> <p> <s xml:id="echoid-s28124" xml:space="preserve">Axis enim radialis, ut patet ք 37 th.</s> <s xml:id="echoid-s28125" xml:space="preserve"> 3 huius, ſemper defert punctũ, cui incidit ad punctũ medium <lb/>nerui cõmunis, cui ſemper in hæret terminus axis cõmunis.</s> <s xml:id="echoid-s28126" xml:space="preserve"> Cum ergo uiſus comprehendit rem ui-<lb/>ſam ſecundũ quod eſt, & inſtituitur forma in concauitate cõmunis nerui in uno loco, & continua ſi-<lb/>bi adinuicem ſecundum continuationẽ rei uiſæ, & punctus rei uiſæ, qui eſt ſuper radialem axem, li-<lb/>cet non fuerit ſuper axem cõmunem, uidetur tamen in loco propinquiori axi communi, quàm ſit in <lb/>ſuo uero loco:</s> <s xml:id="echoid-s28127" xml:space="preserve"> tunc puncta reſidua etiam uidentur in loco propinquiori axi communi, quàm ſint in <lb/>ſuo uero loco:</s> <s xml:id="echoid-s28128" xml:space="preserve"> quia ſunt continuata cum parte, quę eſt apud extremum axis:</s> <s xml:id="echoid-s28129" xml:space="preserve"> & ſi axes amborum ui-<lb/>ſuum concurrerintin aliqua re uiſa extra axem communem:</s> <s xml:id="echoid-s28130" xml:space="preserve"> uidebitur tunc illa res in loco propin-<lb/>quiori communi axi, quàm ſit in ſuo loco uero.</s> <s xml:id="echoid-s28131" xml:space="preserve"> Hoc tamen rarò accidit, quia cum axes uiſuales con <lb/>currerint in aliquo uiſo:</s> <s xml:id="echoid-s28132" xml:space="preserve"> tunc ut plurimũ axis communis tranſibit per illud uiſum:</s> <s xml:id="echoid-s28133" xml:space="preserve"> quia rarò axes <lb/>amborum uiſuum concurrunt in aliquo uiſo extra axem communem, niſi per laborem aut impedi-<lb/>mentum cogens uiſum ad hoc:</s> <s xml:id="echoid-s28134" xml:space="preserve"> unde hæc diſpoſitio non eſt uiſibus aſſueta, quia ſi eſſet talis diſpoſi-<lb/>tio uiſibus multum aſſueta:</s> <s xml:id="echoid-s28135" xml:space="preserve"> tunc ipſa accideret in omni uiſione uel in pluribus:</s> <s xml:id="echoid-s28136" xml:space="preserve"> quod tamen non eſt <lb/>uerum.</s> <s xml:id="echoid-s28137" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s28138" xml:space="preserve"> propoſitum.</s> <s xml:id="echoid-s28139" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1102" type="section" level="0" n="0"> <head xml:id="echoid-head884" xml:space="preserve" style="it">36. Omnium uiſibilium ſecundum ſui longitudinem ante oculos extenſorum: quæ ſunt à de-<lb/>xtris in ſiniſtram, & quæ in ſiniſtris, ad dextr am educi uidentur partem Euclides 12. th opt.</head> <p> <s xml:id="echoid-s28140" xml:space="preserve">Sint duo uiſibilia ſecundũ ſui longitudinẽ ante oculos extenſa, quæ exẽpli cauſſa ſint ęquidiſtan-<lb/>tia:</s> <s xml:id="echoid-s28141" xml:space="preserve"> & ſint a b & d g:</s> <s xml:id="echoid-s28142" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s28143" xml:space="preserve"> centrũ uiſus e:</s> <s xml:id="echoid-s28144" xml:space="preserve"> ducãturq́;</s> <s xml:id="echoid-s28145" xml:space="preserve"> lineæ ad puncta illorũ uiſibiſiũ:</s> <s xml:id="echoid-s28146" xml:space="preserve"> in dexteriore qui-<lb/>dẽ parte, quę ſit a b, ducãtur lineę e b, e c, e k, e a:</s> <s xml:id="echoid-s28147" xml:space="preserve"> & in ſiniſteriore, quę ſit g d, ducãtur lineæ e d, e z, e i, <lb/> <pb o="135" file="0437" n="437" rhead="LIBER QVARTVS."/> e g.</s> <s xml:id="echoid-s28148" xml:space="preserve"> Dico, quòd lineę e z, e i, e g uidentur quaſi in partẽ ſiniſtrã productæ, & lineæ e c, e k, e a uidentur <lb/>quaſi protractæ in partẽ dextrã.</s> <s xml:id="echoid-s28149" xml:space="preserve"> Sit enim linea e d perpendicularis ſuper lineã d g, & linea e b per-<lb/>pendicularis ſuper lineam b a:</s> <s xml:id="echoid-s28150" xml:space="preserve"> erit ergo per 19 p 1 linea e d breuior omnibus lineis e z, e i, e g:</s> <s xml:id="echoid-s28151" xml:space="preserve"> & linea <lb/>e b breuior omnibus lineis e c, e k, e a.</s> <s xml:id="echoid-s28152" xml:space="preserve"> Lineæ ergo e d & e b minimam <lb/>à uiſu denotabunt diſtantiam linearũ g d & a b:</s> <s xml:id="echoid-s28153" xml:space="preserve"> ſecundum illas ergo <lb/> <anchor type="figure" xlink:label="fig-0437-01a" xlink:href="fig-0437-01"/> lineas perſectior ſit uiſio partiũ rerum uiſarum, quibus incidunt per <lb/>23 huius:</s> <s xml:id="echoid-s28154" xml:space="preserve"> linea ergo e d apparebit dexterior omnibus lineis ſuo uiſi-<lb/>bili incidentibus, & linea e b ſiniſterior omnibus lineis ſuo uiſi-<lb/>incidentibus:</s> <s xml:id="echoid-s28155" xml:space="preserve"> illis quoq;</s> <s xml:id="echoid-s28156" xml:space="preserve"> lineis propin quis incidentes mutabunt ſi-<lb/>tus diſpoſitionem ſecundum receſſum ab illis lineis:</s> <s xml:id="echoid-s28157" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s28158" xml:space="preserve"> linea e z <lb/>dexterior quàm linea e i, & linea e i dexterior quàm linea e g.</s> <s xml:id="echoid-s28159" xml:space="preserve"> Palàm <lb/>ergo, quoniam linea e g uidetur in ſiniſtra à linea e i, & linea e i ſimi-<lb/>liter uidetur in ſiniſtra à linea e z.</s> <s xml:id="echoid-s28160" xml:space="preserve"> Eodem quoq;</s> <s xml:id="echoid-s28161" xml:space="preserve"> modo uidebitur li-<lb/>nea e a in dextram educi à linea e k, & linea e k uidetur in dextram e-<lb/>duci à linea e c:</s> <s xml:id="echoid-s28162" xml:space="preserve"> punctum ergo z plus approximat ad ſiniſtram quàm <lb/>punctum d, & punctum i plus quàm punctum z, & punctum g plus <lb/>quàm punctum i.</s> <s xml:id="echoid-s28163" xml:space="preserve"> Tota ergo linea d g uidetur ſiniſtrari, & tota linea <lb/>b a uidetur dextrari:</s> <s xml:id="echoid-s28164" xml:space="preserve"> quoniam puncto b exiſtente ſiniſtro, punctum c <lb/>uidetur plus dextrum illo, & item punctum k plus dextrum puncto <lb/>c, & punctum a plus dextrum puncto k.</s> <s xml:id="echoid-s28165" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s28166" xml:space="preserve"> quo-<lb/>niam ſimiliter eſt in quibuslibet alijs punctis demonſtrandum:</s> <s xml:id="echoid-s28167" xml:space="preserve"> quæ <lb/>enim ſub dexterioribus radijs uidentur, dexteriora apparent, & quæ <lb/>ſub ſiniſterioribus ſiniſteriora, ut patet per 1 ſuppoſitionem hu-<lb/>ius.</s> <s xml:id="echoid-s28168" xml:space="preserve"> Hæc autem omnia ideo accidunt, quia lineæ parallelæ ſecundum remotiores ſui â uiſu partes <lb/>concurrere uidentur per 21 huius.</s> <s xml:id="echoid-s28169" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s28170" xml:space="preserve"/> </p> <div xml:id="echoid-div1102" type="float" level="0" n="0"> <figure xlink:label="fig-0437-01" xlink:href="fig-0437-01a"> <variables xml:id="echoid-variables449" xml:space="preserve">g a i k z c d e b</variables> </figure> </div> </div> <div xml:id="echoid-div1104" type="section" level="0" n="0"> <head xml:id="echoid-head885" xml:space="preserve" style="it">37. Superſicierum ſub oculo iacentium, remotiores à uiſu, altiores uidentur. Euclides 10 <lb/>theo. opticorum.</head> <p> <s xml:id="echoid-s28171" xml:space="preserve">Sit centrum uiſus a in altiori ſitu collocatum, quàm ſuperſicies rei uiſæ, in qua ſint lineæ b e, e d, <lb/>d g:</s> <s xml:id="echoid-s28172" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s28173" xml:space="preserve"> lineę a b, a e, a d, a g:</s> <s xml:id="echoid-s28174" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s28175" xml:space="preserve"> cauſſa exempli ſitus talis, ut linea a b ſit perpendicularis ſu <lb/>per lineam b g, in qua collocantur lineæ b e, e d, d g:</s> <s xml:id="echoid-s28176" xml:space="preserve"> quoniã in alijs ſitibus maior eſt diuerſitas.</s> <s xml:id="echoid-s28177" xml:space="preserve"> Dico <lb/>quòd linea d g altior uidetur quàm linea d e, & linea <lb/> <anchor type="figure" xlink:label="fig-0437-02a" xlink:href="fig-0437-02"/> d e altior quàm linea b e.</s> <s xml:id="echoid-s28178" xml:space="preserve"> Sumatur enim in linea b e, <lb/>punctus z, à quo ducatur per 11 p 1 linea z i perpendi-<lb/>cularis ſuper lineam b e.</s> <s xml:id="echoid-s28179" xml:space="preserve"> Quoniam ergo punctorum <lb/>formæ g, d, e procedentes ad uiſum, primò pertran-<lb/>ſeunt lineam z i, quàm perueniant ad punctum a cen <lb/>trũ uiſus:</s> <s xml:id="echoid-s28180" xml:space="preserve"> ſit, ut linea g a ſecet lineam z i in puncto i, <lb/>& linea d a in puncto t, & & linea e a in pũcto k.</s> <s xml:id="echoid-s28181" xml:space="preserve"> Quia <lb/>ergo punctus i eleuatior eſt puncto t, & punctus t <lb/>puncto k:</s> <s xml:id="echoid-s28182" xml:space="preserve"> ideo quòd linea a t maior eſt quàm linea a <lb/>i, & linea a k maior ꝗ̃ linea a t per 19 p 1:</s> <s xml:id="echoid-s28183" xml:space="preserve"> & in linea, in <lb/>qua eſt punctum i, eſt etiam punctum g, & in linea, <lb/>in qua eſt punctum t, eſt etiam punctũ d, & in linea, <lb/>in qua eſt punctum k, eſt etiam punctum e:</s> <s xml:id="echoid-s28184" xml:space="preserve"> per com-<lb/>prehenſionem uerò punctorum d & g uidetur linea <lb/>d g:</s> <s xml:id="echoid-s28185" xml:space="preserve"> & per puncta e & d uidetur linea e d:</s> <s xml:id="echoid-s28186" xml:space="preserve"> palàm, quo <lb/>niam linea g d eleuatior apparebit quàm linea d e:</s> <s xml:id="echoid-s28187" xml:space="preserve"> & <lb/>ſimiliter d e apparebit eleuatior quàm linea b e.</s> <s xml:id="echoid-s28188" xml:space="preserve"> Cuius enim pũcti forma multiplicando ſe ad uiſum <lb/>magis eleuatur, hoc altius apparet uiſui per 1 ſuppoſitionẽ huius, quia in altiori ſitu offertur uiſui, <lb/>& ſecundum illum modũ ſiguratur in ſuperſicie uiſus.</s> <s xml:id="echoid-s28189" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s28190" xml:space="preserve"> Et patet ex hoc, quòd <lb/>multum exaltato uiſu ſuperficies planæ iacentes longè à uiſu concauæ uidebuntur:</s> <s xml:id="echoid-s28191" xml:space="preserve"> tendunt enim <lb/>formæ talium punctorum ad uiſum per modum circumferentiæ circa centrũ uiſus propter æquali-<lb/>tatẽ uirtutis uiſiuę.</s> <s xml:id="echoid-s28192" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s28193" xml:space="preserve"/> </p> <div xml:id="echoid-div1104" type="float" level="0" n="0"> <figure xlink:label="fig-0437-02" xlink:href="fig-0437-02a"> <variables xml:id="echoid-variables450" xml:space="preserve">a i t k b z e d g</variables> </figure> </div> </div> <div xml:id="echoid-div1106" type="section" level="0" n="0"> <head xml:id="echoid-head886" xml:space="preserve" style="it">38. Superſicierum uiſui ſuperiacentium remotiores à uiſu decliuiores uidentur. Euclides <lb/>11 theo. opticorum.</head> <p> <s xml:id="echoid-s28194" xml:space="preserve">Sit centrum uiſus punctus a in inferiori ſitu collocatum, quàm ſuperficies rei uiſæ, in qua ſint li-<lb/>neæ b e, e d, d g:</s> <s xml:id="echoid-s28195" xml:space="preserve"> & ducantur, ſicut in præcedenti, lineæ a b, a e, a d, a g:</s> <s xml:id="echoid-s28196" xml:space="preserve"> quarum a b ſit perpendicularis <lb/>ſuper ſuperſiciem ſuppoſitam uiſui.</s> <s xml:id="echoid-s28197" xml:space="preserve"> Dico, quòd linea g d apparebit decliuior quàm linea d e, & li-<lb/>nea d e decliuior quàm linea b e.</s> <s xml:id="echoid-s28198" xml:space="preserve"> Ducatur enim, utin præcedente, linea z i æquidiſtans lineæ a b, <lb/>ſecans lineam g a in puncto i, & lineam d a in puncto c, & lineam e a in puncto k:</s> <s xml:id="echoid-s28199" xml:space="preserve"> ergo per ea, quæ in <lb/>præcedenti diximus, forma puncti g decliuior uidebitur quàm forma puncti d, & forma puncti d <lb/> <pb o="136" file="0438" n="438" rhead="VITELLONIS OPTICAE"/> decliuior quàm forma punctie, & forma puncti e de-<lb/>cliuior quàm forma puncti b:</s> <s xml:id="echoid-s28200" xml:space="preserve"> ſed per formas puncto-<lb/>rum g & d forma lineæ g d occurrit uiſui, & per for-<lb/> <anchor type="figure" xlink:label="fig-0438-01a" xlink:href="fig-0438-01"/> mas punctorum d & e uidebitur forma lineæ d e, & <lb/>per formas punctorum e & b uidebitur forma lineæ <lb/>e b.</s> <s xml:id="echoid-s28201" xml:space="preserve"> Quoniam itaq;</s> <s xml:id="echoid-s28202" xml:space="preserve">, ut oſtendimus in præmiſſa, linea <lb/>a c eſt maior quàm linea a i, & linea a k maior quàm li-<lb/>nea a c:</s> <s xml:id="echoid-s28203" xml:space="preserve"> & ſecundum harum linearum diſpoſitionem <lb/>fit formarum illorum punctorum uiſio.</s> <s xml:id="echoid-s28204" xml:space="preserve"> Palàm ergo, <lb/>quoniá centro uiſus & ipſo uiſibili ſic diſpoſitis, remo <lb/>tiora à uiſu, decliuiora uiſui occurrunt, quàm propin-<lb/>quiora.</s> <s xml:id="echoid-s28205" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s28206" xml:space="preserve"/> </p> <div xml:id="echoid-div1106" type="float" level="0" n="0"> <figure xlink:label="fig-0438-01" xlink:href="fig-0438-01a"> <variables xml:id="echoid-variables451" xml:space="preserve">b z e d g k c i a</variables> </figure> </div> </div> <div xml:id="echoid-div1108" type="section" level="0" n="0"> <head xml:id="echoid-head887" xml:space="preserve" style="it">39. Aequalium magnitudinum ſub eodem uiſu <lb/>erect arum, remotiores altiores apparent. Euclides <lb/>13 tbeo. opticorum.</head> <p> <s xml:id="echoid-s28207" xml:space="preserve">Sit centrum uiſus punctum i:</s> <s xml:id="echoid-s28208" xml:space="preserve"> & ſint uiſæ æquales <lb/>magnitudines, quæ ſub ipſo uiſu ſint erectæ, quæ ſint <lb/>a b, g d, e z:</s> <s xml:id="echoid-s28209" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s28210" xml:space="preserve"> a b remotior à uiſu, & deinde g d, & <lb/>deinde e z:</s> <s xml:id="echoid-s28211" xml:space="preserve"> & ſit centrum oculi punctum i eleuatius exiſtẽs illis magnitudinibus:</s> <s xml:id="echoid-s28212" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s28213" xml:space="preserve"> lineæ <lb/>i a, i g, i e.</s> <s xml:id="echoid-s28214" xml:space="preserve"> Dico, quòd magnitudinum illarum a b ap-<lb/>paret altiour quàm g d, & g d altior quàm e z.</s> <s xml:id="echoid-s28215" xml:space="preserve"> Quoniã <lb/>enim linea i a eſt eleuation quàm linea i g, & linea i <lb/>g eleuatior quàm linea i e, & in linea, cui incidunt li-<lb/>neæ i a, i g, i e ſunt pũcta a, g, e, & per 37 huius uidẽtur <lb/>puncta remotiora uiſui altiora:</s> <s xml:id="echoid-s28216" xml:space="preserve"> pũcta uerò a, g, e ſunt <lb/> <anchor type="figure" xlink:label="fig-0438-02a" xlink:href="fig-0438-02"/> in magnitudinibus a b, g d, e z:</s> <s xml:id="echoid-s28217" xml:space="preserve"> ergo magnitudo a b <lb/>apparet eleuatior quàm ipſa magnitudo g d, & ma-<lb/>gnitudo g d apparet altior quàm ipſa e z.</s> <s xml:id="echoid-s28218" xml:space="preserve"> Quod eſt <lb/>propoſitũ.</s> <s xml:id="echoid-s28219" xml:space="preserve"> Et quia de qualibet magnitudine longio-<lb/>ri poteſt abſcindi æ qualis breuiori:</s> <s xml:id="echoid-s28220" xml:space="preserve"> ideo in omnibus <lb/>magnitudinibus ſubiacentibus uiſui præſens tenet <lb/>demonſtratio:</s> <s xml:id="echoid-s28221" xml:space="preserve"> quoniam ſemper remotiores uiden-<lb/>tur altiores, quàm ſint ſecundum ueritatem.</s> <s xml:id="echoid-s28222" xml:space="preserve"/> </p> <div xml:id="echoid-div1108" type="float" level="0" n="0"> <figure xlink:label="fig-0438-02" xlink:href="fig-0438-02a"> <variables xml:id="echoid-variables452" xml:space="preserve">i a g e b d z</variables> </figure> </div> </div> <div xml:id="echoid-div1110" type="section" level="0" n="0"> <head xml:id="echoid-head888" xml:space="preserve" style="it">40. Aequalium magnitudinum uiſui ſuperere <lb/>ctarum remotiores decliuiores apparent. Euclides <lb/>14 theo. opt.</head> <p> <s xml:id="echoid-s28223" xml:space="preserve">Eſto, ſicut in præcedenti, centrum uiſus punctum i:</s> <s xml:id="echoid-s28224" xml:space="preserve"> & ſint æquales magnitudines, quæ a b, g d, e <lb/>z, erectæ ſuperſtantes uiſui:</s> <s xml:id="echoid-s28225" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s28226" xml:space="preserve"> a b remotior uiſui quàm aliæ, & e z propinquior.</s> <s xml:id="echoid-s28227" xml:space="preserve"> Dico, quòd <lb/>magnitudo a b apparet decliuior quàm g d, & ma-<lb/> <anchor type="figure" xlink:label="fig-0438-03a" xlink:href="fig-0438-03"/> gnitudo g d decliuior quàm e z.</s> <s xml:id="echoid-s28228" xml:space="preserve"> Ducantur enim, ut <lb/>in præmiſſa, lineę i b, i d, i z.</s> <s xml:id="echoid-s28229" xml:space="preserve"> Quoniam ergo, ſicut pa-<lb/>tet per 38 huius, forma ueniẽs per lineam i b, eſt de-<lb/>cliuiori modo uiſui incidens, quàm forma ueniens <lb/>per lineam id, & forma uiſui adueniens per lineam <lb/>i d, decliuiori modo incidit, quàm forma ueniẽs per <lb/>lineam i z:</s> <s xml:id="echoid-s28230" xml:space="preserve"> ſed in linea, cui incidunt lineæ i z, i d, i b, <lb/>ſunt puncta z, d, b, quę puncta ſunt in magnitudini-<lb/>bus a b, g d, e z.</s> <s xml:id="echoid-s28231" xml:space="preserve"> Palàm ergo, quoniam iſtarũ magni-<lb/>tudinũ illa, quę eſt a b, decliuior apparet quàm g d, <lb/>& g d quàm e z.</s> <s xml:id="echoid-s28232" xml:space="preserve"> Et hoc eſt propoſitũ.</s> <s xml:id="echoid-s28233" xml:space="preserve"> Eſt autem uni <lb/>uerſale illo modo, quo diximus in præcedenti.</s> <s xml:id="echoid-s28234" xml:space="preserve"/> </p> <div xml:id="echoid-div1110" type="float" level="0" n="0"> <figure xlink:label="fig-0438-03" xlink:href="fig-0438-03a"> <variables xml:id="echoid-variables453" xml:space="preserve">a g e b d z i</variables> </figure> </div> </div> <div xml:id="echoid-div1112" type="section" level="0" n="0"> <head xml:id="echoid-head889" xml:space="preserve" style="it">41. Altioris magnitudinis uiſibilis per uerti <lb/>cem inferioris aſpectæ, accedente & recedente ui-<lb/>ſu ſecundum lineam uertici inferioris perpendi-<lb/>culariter incidentem: ſemper idem erit exceſſus, <lb/>non uidebitur autem idem. Euclides 17 th. opt.</head> <p> <s xml:id="echoid-s28235" xml:space="preserve">Sint duæ uiſæ magnitudines inæquales a b maior, & g d minot:</s> <s xml:id="echoid-s28236" xml:space="preserve"> quarum uertices ſint a & g:</s> <s xml:id="echoid-s28237" xml:space="preserve"> & ſit <lb/>centrum uiſus punctum e:</s> <s xml:id="echoid-s28238" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s28239" xml:space="preserve"> linea g e perpendicularis ſuper lineam g d, ſecans lineam a b <lb/>in puncto z.</s> <s xml:id="echoid-s28240" xml:space="preserve"> Dico, quòd oculo accedente & recedente ſecundum lineam g e, ſemperidem uidebi-<lb/>tur exceſſus lineæ a b ſuper lineam g d, qui exceſſus eſt linea z a.</s> <s xml:id="echoid-s28241" xml:space="preserve"> Accedat enim uiſus ad punctum i, <lb/>propinquius puncto g quàm punctum e, uel remoueatur ad aliud punctum f, remotius quàm pun-<lb/>ctum e:</s> <s xml:id="echoid-s28242" xml:space="preserve"> ſemper autem perpendiculariter non incidet forma alicuius punctorum lineæ g d ipſi uiſui, <lb/> <pb o="137" file="0439" n="439" rhead="LIBER QVARTVS."/> niſi ſola forma puncti z, in quam cadit perpendiculariter e z:</s> <s xml:id="echoid-s28243" xml:space="preserve"> quoniam per 20 th.</s> <s xml:id="echoid-s28244" xml:space="preserve"> 1 huius duas lineas <lb/>eidem ſuperficiei ab eodem puncto ductas perpen-<lb/> <anchor type="figure" xlink:label="fig-0439-01a" xlink:href="fig-0439-01"/> diculariter inſiſtere eſt impoſsibile:</s> <s xml:id="echoid-s28245" xml:space="preserve"> palàm ergo pro <lb/>poſitum.</s> <s xml:id="echoid-s28246" xml:space="preserve"> Videbitur tamen linea a z minui uel augu-<lb/>mentari ſecun dum diuerſitatẽ angulorum, ſub qui-<lb/>bus fiet uiſio per 20 huius.</s> <s xml:id="echoid-s28247" xml:space="preserve"> Et eſt, ut patet ex pręmiſ-<lb/>ſis, & per 21 p 1, angulus a i z maior angulo a e z, & <lb/>angulus a e z maior angulo a f z:</s> <s xml:id="echoid-s28248" xml:space="preserve"> ſecundum hoc au-<lb/>tem diuerſificatur in uiſu quãtitas lineæ a z:</s> <s xml:id="echoid-s28249" xml:space="preserve"> ſemper <lb/>tamen illius lineæ a z eadem eſt quantitas in ſeipſa.</s> <s xml:id="echoid-s28250" xml:space="preserve"> <lb/>Et hoc eſt propoſitum.</s> <s xml:id="echoid-s28251" xml:space="preserve"/> </p> <div xml:id="echoid-div1112" type="float" level="0" n="0"> <figure xlink:label="fig-0439-01" xlink:href="fig-0439-01a"> <variables xml:id="echoid-variables454" xml:space="preserve">a z g i e f b d</variables> </figure> </div> </div> <div xml:id="echoid-div1114" type="section" level="0" n="0"> <head xml:id="echoid-head890" xml:space="preserve" style="it">42. Altioris uiſibilis per uerticem inferioris <lb/>aſpecti, accedente uiſu ſecundum lineam exceſſui <lb/>altioris perpendiculariter incidentẽ: maior pars <lb/>altioris uidetur, recedente uerò uiſu ſecundũ ean-<lb/>dem lineam minor pars altioris uidetur: ſecundũ <lb/>aliam uerò lineam accedente uel recedente uiſu, <lb/>accidit econuerſo. Euclides 16 the. opt.</head> <p> <s xml:id="echoid-s28252" xml:space="preserve">Sint, ut in præmiſſa, duæ in æquales magnitudines, quæ a b & g d, quarum maior ſit a b:</s> <s xml:id="echoid-s28253" xml:space="preserve"> & ſit cen-<lb/>trum uiſus in puncto e poſitũ in linea e a, perpendiculariter incidente puncto a, qui ſit altior termi-<lb/>nus lineę a b:</s> <s xml:id="echoid-s28254" xml:space="preserve"> ambę ergo magnitudines tam a b quàm g d ſubiacebunt uiſui, cum uertex altioris (qui <lb/>eſt a) ſit in perpen diculari ducta à centro uiſus ad magnitudinem altiorem:</s> <s xml:id="echoid-s28255" xml:space="preserve"> ſint enim magnitudines <lb/>a b & g d taliter erectæ, ut punctum a ſit altius, quàm punctum g, per-<lb/>ueniatq́;</s> <s xml:id="echoid-s28256" xml:space="preserve"> forma alicuius punctorũ lineæ a b, quod ſit z, per uerticem <lb/> <anchor type="figure" xlink:label="fig-0439-02a" xlink:href="fig-0439-02"/> lineę d g, qui ſit g, ad uiſum e:</s> <s xml:id="echoid-s28257" xml:space="preserve"> & ſit linea, ſecundũ quã aduenit illa for-<lb/>ma, linea z e.</s> <s xml:id="echoid-s28258" xml:space="preserve"> Sub linea itaq;</s> <s xml:id="echoid-s28259" xml:space="preserve"> z e uidetur linea z a, pars magnitudinis <lb/>a b, & tota magnitudo d g, remanetq́;</s> <s xml:id="echoid-s28260" xml:space="preserve"> pars lineæ a b, quæ non uidetur <lb/>per uerticem g:</s> <s xml:id="echoid-s28261" xml:space="preserve"> & hęc eſt linea z b.</s> <s xml:id="echoid-s28262" xml:space="preserve"> Accedat autem uiſus propinquius <lb/>ad punctum a, ut fiat in eadem linea in puncto i.</s> <s xml:id="echoid-s28263" xml:space="preserve"> Palàm quoq;</s> <s xml:id="echoid-s28264" xml:space="preserve">, quia in <lb/>hoc ſitu aliquis punctus lineæ a b inferior puncto z peruenit ad ui-<lb/>ſum, qui ſit punctus t:</s> <s xml:id="echoid-s28265" xml:space="preserve"> & ducatur linea ti per uerticem g ad uiſum:</s> <s xml:id="echoid-s28266" xml:space="preserve"> ſub <lb/>linea ergo it uidetur pars magnitudinis a b, quæ eſt t a, & tota magni <lb/>tudo g d, remanetq́;</s> <s xml:id="echoid-s28267" xml:space="preserve"> pars lineæ a b, quæ eſt a t, uiſa, Et quoniam linea <lb/>a teſt maior quàm linea z a, quæ uidebatur uiſu exiſtente remotiore:</s> <s xml:id="echoid-s28268" xml:space="preserve"> <lb/>neceſſarium autem eſt lineamt a fieri maiorẽ quàm ſit linea z a:</s> <s xml:id="echoid-s28269" xml:space="preserve"> ideo <lb/>quòd angulus ait eſt maior angulo a e z per 16 p 1:</s> <s xml:id="echoid-s28270" xml:space="preserve"> illud ergo, quod ui <lb/>detur ſub angulo ait, eſt maius illo, quod uidetur ſub angulo a e z ք <lb/>20 huius:</s> <s xml:id="echoid-s28271" xml:space="preserve"> linea ergo a t maior uidebitur:</s> <s xml:id="echoid-s28272" xml:space="preserve"> & per 19 p 1 maior eſt quàm <lb/>linea a z.</s> <s xml:id="echoid-s28273" xml:space="preserve"> Et quando linea, in qua e centrum uiſus, perpendiculariter <lb/>incidit cuicunq;</s> <s xml:id="echoid-s28274" xml:space="preserve"> puncto exceſſus lineæ a b ſuper lineam g d, eadem <lb/>eſt demonſtratio.</s> <s xml:id="echoid-s28275" xml:space="preserve"> Palàm ergo, quòd accedente uiſu ſuperapparens <lb/>pars lineæ a b ſemper fit maior, recedente uerò uiſu fit minor.</s> <s xml:id="echoid-s28276" xml:space="preserve"> Et hoc <lb/>eſt propoſitum primum.</s> <s xml:id="echoid-s28277" xml:space="preserve"> Secundum aliam uerò lineam, quæ ſit perpendicularis ſuper lineam a b, <lb/> <anchor type="figure" xlink:label="fig-0439-03a" xlink:href="fig-0439-03"/> non tamen incidat in punctum a, uel in aliquod punctum exceſſus, <lb/>ſed in aliquod aliud punctum lineæ a b, baſsius toto exceſſu lineæ a <lb/>b ſuper lineam g d, ut in punctum f:</s> <s xml:id="echoid-s28278" xml:space="preserve"> uiſu accedente uel recedente ac-<lb/>cidit ecóuerſo.</s> <s xml:id="echoid-s28279" xml:space="preserve"> Nam accedente uiſu, totius magnitudinis a b minus <lb/>uidetur per uerticem g, & recedente uiſu, magis:</s> <s xml:id="echoid-s28280" xml:space="preserve"> exiſtente enim uiſu <lb/>in pũcto e, multiplicabitur ad uiſum forma lineę z a, accedente uerò <lb/>uiſu in punctumi, & ductis lineis e g z & i g t, patet, quòd illæ lineæ <lb/>ſecabunt ſe in puncto g, & non perueniet ad uiſum forma alicuius <lb/>punctorum lineæ z t, ſed ſolùm forma lineæ t a, quę eſt neceſſariò mi <lb/>nor quàm linea z a.</s> <s xml:id="echoid-s28281" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s28282" xml:space="preserve"/> </p> <div xml:id="echoid-div1114" type="float" level="0" n="0"> <figure xlink:label="fig-0439-02" xlink:href="fig-0439-02a"> <variables xml:id="echoid-variables455" xml:space="preserve">a i e f i e z g z t i b d</variables> </figure> <figure xlink:label="fig-0439-03" xlink:href="fig-0439-03a"> <variables xml:id="echoid-variables456" xml:space="preserve">a t z g s i e b d</variables> </figure> </div> </div> <div xml:id="echoid-div1116" type="section" level="0" n="0"> <head xml:id="echoid-head891" xml:space="preserve" style="it">43. Inæqualium uiſibilium uerticibus in eadem linea æquidi-<lb/>ctante horiz onti existentibus: pars inferior longioris uiſa per ba-<lb/>ſim breuioris accedente uiſu ſecundum lineã exceſſui longiouis per <lb/>pendiculariter in cidentem, maior pars longioris unidebitur: rece-<lb/>dente uerò uiſu ſecũdũ eandẽ lineã minor pars altioris uidebitur: <lb/>ſecundũ aliam uerò lineam accidit econuerſo. Euclides 15 th. opt.</head> <p> <s xml:id="echoid-s28283" xml:space="preserve">Hæc non differt in hypotheſi à præmiſſa, niſi quòd in illa uiſibilia <lb/>ſunt ſubiacẽtia uiſui, in hac uerò ſunt ſuperſtantia.</s> <s xml:id="echoid-s28284" xml:space="preserve"> Sint ergo inæqua <lb/>les quantitates a b & g d:</s> <s xml:id="echoid-s28285" xml:space="preserve"> quarũ maior fit a b:</s> <s xml:id="echoid-s28286" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s28287" xml:space="preserve"> uertices illarum <lb/> <pb o="138" file="0440" n="440" rhead="VITELLONIS OPTICAE"/> quantitatum b & d:</s> <s xml:id="echoid-s28288" xml:space="preserve"> & fit linea b d æ quidiſtans horizonti:</s> <s xml:id="echoid-s28289" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s28290" xml:space="preserve"> centrum uiſus in puncto e:</s> <s xml:id="echoid-s28291" xml:space="preserve"> multipli-<lb/>ceturq́;</s> <s xml:id="echoid-s28292" xml:space="preserve"> forma alicuius puncti lineæ a b, ut z, per baſim g ad uiſum e:</s> <s xml:id="echoid-s28293" xml:space="preserve"> ſiatq́;</s> <s xml:id="echoid-s28294" xml:space="preserve"> linea z g e:</s> <s xml:id="echoid-s28295" xml:space="preserve"> ſub linea ergo <lb/>z e continentur z a & g d:</s> <s xml:id="echoid-s28296" xml:space="preserve"> & b z non apparet uiſui, propter interpoſi-<lb/> <anchor type="figure" xlink:label="fig-0440-01a" xlink:href="fig-0440-01"/> tionem ipſius g d:</s> <s xml:id="echoid-s28297" xml:space="preserve"> inferior uerò ipſius pars decliuior apparet per 40 <lb/>huius, remanetq́;</s> <s xml:id="echoid-s28298" xml:space="preserve"> a z pars lineæ a b apparens uiſui ultra lineam g d.</s> <s xml:id="echoid-s28299" xml:space="preserve"> <lb/>Accedat ergo uiſus, & ſit in puncto i propin quiori ad punctum a, in <lb/>eadẽlinea perpen diculari ſuper lineã a b, quæ ſit e f:</s> <s xml:id="echoid-s28300" xml:space="preserve"> hæc enim æ qui-<lb/>diſtat uerticibus ipſorum uiſorum, qui ſunt b & d:</s> <s xml:id="echoid-s28301" xml:space="preserve"> multiplicabiturq́;</s> <s xml:id="echoid-s28302" xml:space="preserve"> <lb/>forma alicuius puncti lineæ a b per punctum g ad uiſum exiſtentem <lb/>in puncto i:</s> <s xml:id="echoid-s28303" xml:space="preserve"> ſit ille punctus t:</s> <s xml:id="echoid-s28304" xml:space="preserve"> & ducatur linea t g i:</s> <s xml:id="echoid-s28305" xml:space="preserve"> ſub linea ergo t g i <lb/>cótinentur magnitudines g d & t a:</s> <s xml:id="echoid-s28306" xml:space="preserve"> ſub linea uerò e z cõtinentur ma-<lb/>gnitudines z a & g d.</s> <s xml:id="echoid-s28307" xml:space="preserve"> Et quoniam linea t z a maior eſt quàm linea z a, <lb/>cum angulus tifper 16 p 1 ſit maior angulo z e ſiergo per 20 huius li-<lb/>neat fuiſa ſub angulo tif maior eſt quàm linea z f, uiſa ſub angulo z <lb/>e f.</s> <s xml:id="echoid-s28308" xml:space="preserve"> Et non ſolùm apparebit uiſui maior:</s> <s xml:id="echoid-s28309" xml:space="preserve"> imò & erit maior.</s> <s xml:id="echoid-s28310" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s28311" xml:space="preserve"> <lb/>ambabus lineis t f & z f communis eſt linea f a:</s> <s xml:id="echoid-s28312" xml:space="preserve"> pater, quòd tota linea <lb/>t a erit maior quàm linea z a.</s> <s xml:id="echoid-s28313" xml:space="preserve"> Et hoc eſt primum propoſitorum.</s> <s xml:id="echoid-s28314" xml:space="preserve"> Siue <lb/>rò uiſus accedat non ſecundum lineam e f, ſed fiat in punctoi, extra <lb/>illam lineam e f, in alia linea e f perpendiculariter incidente lineæ <lb/>a b, non in aliquod punctorum exceſſus a b ſuper d g:</s> <s xml:id="echoid-s28315" xml:space="preserve"> dico, quòd acci <lb/>det econuerſo:</s> <s xml:id="echoid-s28316" xml:space="preserve"> erit enim linea t a minor quàm linea z a.</s> <s xml:id="echoid-s28317" xml:space="preserve"> Ducantur e-<lb/>nim lineæ t g i, & a i, & i z:</s> <s xml:id="echoid-s28318" xml:space="preserve"> palàm quoq;</s> <s xml:id="echoid-s28319" xml:space="preserve"> per 32 p 1 quoniam angulus a <lb/>i t eft minor angulo a i z:</s> <s xml:id="echoid-s28320" xml:space="preserve"> ideo quia angulus a z i minor eſt angulo a t i <lb/>per 21 p 1, & angulus t a i communis:</s> <s xml:id="echoid-s28321" xml:space="preserve"> uiſum ergo à puncto i ſub angulo a i t eſt minus uiſo ſub angulo <lb/>aiz:</s> <s xml:id="echoid-s28322" xml:space="preserve"> linea ergo z a eſt maior quàm linea t a:</s> <s xml:id="echoid-s28323" xml:space="preserve"> & uidetur maior.</s> <s xml:id="echoid-s28324" xml:space="preserve"> Et hoc accidit, cum centrum uiſus col-<lb/>locatur ſupra lineam primã e f, & altius quàm illa:</s> <s xml:id="echoid-s28325" xml:space="preserve"> ſi uerò ipſum collocetur inferius, quàm linea pri-<lb/>ma e f:</s> <s xml:id="echoid-s28326" xml:space="preserve"> tunc accidit econuerſo.</s> <s xml:id="echoid-s28327" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s28328" xml:space="preserve"/> </p> <div xml:id="echoid-div1116" type="float" level="0" n="0"> <figure xlink:label="fig-0440-01" xlink:href="fig-0440-01a"> <variables xml:id="echoid-variables457" xml:space="preserve">b d f i e t z g f z i r t a</variables> </figure> </div> </div> <div xml:id="echoid-div1118" type="section" level="0" n="0"> <head xml:id="echoid-head892" xml:space="preserve" style="it">44. In ſitus uiſione uirtutidiſtinctiuæ error accidit ex intemper ata diſpoſitione octo circun-<lb/>ſtantiarum cuiuslibet rei uiſæ. Alhazen 24. 35. 46 53. 58. 64. 66. 69 n 3.</head> <p> <s xml:id="echoid-s28329" xml:space="preserve">Exintemperantia enim lucis uirtuti diſtinctiuæ error accidit in uiſione ſitus:</s> <s xml:id="echoid-s28330" xml:space="preserve"> ut ſi in nocte non <lb/>obſcura aliquid modicè declinet à uiſu:</s> <s xml:id="echoid-s28331" xml:space="preserve"> tunc æſtimabitur in eo ſitus rectitudo propter debilitatem <lb/>lucis egreſſam à temperamento.</s> <s xml:id="echoid-s28332" xml:space="preserve"> Nimia etiam remotio in uiſione ſitus errorẽ inducit:</s> <s xml:id="echoid-s28333" xml:space="preserve"> unde res ui-<lb/>ſibilis ualde remota à uiſu & obliquata uiſui, uidebitur directè oppoſita per 34 huius.</s> <s xml:id="echoid-s28334" xml:space="preserve"> Intemperátia <lb/>etiam ſitus errorẽ facit in ſitus uiſione:</s> <s xml:id="echoid-s28335" xml:space="preserve"> cadente enim axe uiſuali in corpus ſecundum temperatã di-<lb/>ſtantiã uiſui oppoſitũ, & ſum pto alio corpore multũ elongato ab axe, & declinato modicũ ſuper li-<lb/>neam imaginatã, ſuper quã cadit axis radialis perpendiculariter:</s> <s xml:id="echoid-s28336" xml:space="preserve"> tunc uiſus non cõprehendit corpo <lb/>ris illius declinati onẽ propter ſitum à temperamento egreſſum:</s> <s xml:id="echoid-s28337" xml:space="preserve"> quoniã non fit plena cõprehenſio <lb/>corporũ longè ab axe poſitorũ per 45 th.</s> <s xml:id="echoid-s28338" xml:space="preserve"> 3 huius:</s> <s xml:id="echoid-s28339" xml:space="preserve"> & ita propter hunc errorẽ res obliquè uiſibus op-<lb/>poſita, iudicabitur oppoſita directè.</s> <s xml:id="echoid-s28340" xml:space="preserve"> Intemperãtia etiã magnitudinis in uiſione ſitus eſſicit errorẽ:</s> <s xml:id="echoid-s28341" xml:space="preserve"> <lb/>quoniã granũ ſinapis ſi fuerit ab oculis declinatũ, uidetur tamẽ ac ſi eſſet directè oppoſitũ:</s> <s xml:id="echoid-s28342" xml:space="preserve"> quia eius <lb/>declinatio propter paruitatẽ corporis non poteſt comprehẽdi:</s> <s xml:id="echoid-s28343" xml:space="preserve"> nec enim eſt ſenſibilis declinatio hu <lb/>ius grani ab axe cõmuni orthogonaliter ſuper uiſibilia cadente, ſecundũ quã diſcernitur obliquatio <lb/>rerum uiſarũ reſpectu uiſus:</s> <s xml:id="echoid-s28344" xml:space="preserve"> quoniã nõ plenè diſcernitur diftãtia inter hunc axem & extremitates <lb/>grani, quæ eſt quaſi minima linea omniũ linearũ ſenſibiliũ.</s> <s xml:id="echoid-s28345" xml:space="preserve"> Ex intẽperata etiã ſoliditate error acci-<lb/>dit uiſui in ſitu:</s> <s xml:id="echoid-s28346" xml:space="preserve"> quoniã ſi corporis rari ſitus, reſpectu uiſus, fuerit declinatus, occultabitur eius decli <lb/>natio, & ſi fortè uidebitur directè opponi:</s> <s xml:id="echoid-s28347" xml:space="preserve"> una enim extremitatũ illius corporis eiuſdẽ diſtantiæ re-<lb/>putabitur cũ alia, cũ tamen ſint diuerſæ:</s> <s xml:id="echoid-s28348" xml:space="preserve"> & accidit hoc propter nimiã raritatẽ non terminantẽ certi-<lb/>tudinaliter uiſibilẽ operationẽ, & inducentẽ incertitudinẽ in quãtitate anguli, ſub quo fit uiſio.</s> <s xml:id="echoid-s28349" xml:space="preserve"> In-<lb/>temperata etiã diaphanitas efſicit errorẽ uiſui in ſitu:</s> <s xml:id="echoid-s28350" xml:space="preserve"> ſi enim corpus uiſum ſub parua obliquatione <lb/>obijciatur uiſui in aere denſo obſcuro, ſicut accidit in horis cre puſcularibus, occultabitur declina-<lb/>tio, quę pateret in aere lucido claro:</s> <s xml:id="echoid-s28351" xml:space="preserve"> fit ergo error in ſitu oppoſitiõis corporis ad uiſum.</s> <s xml:id="echoid-s28352" xml:space="preserve"> Exintẽpera <lb/>ta etiã quantitate tẽporis fit error uiſui in ſitu:</s> <s xml:id="echoid-s28353" xml:space="preserve"> ut cũ aliquid occurrit uiſui ſubitò, qđ ſtatim recedit:</s> <s xml:id="echoid-s28354" xml:space="preserve"> <lb/>hoc enim fortè directè uiſui oppoſitũ reputabitur obliquatũ, uel ecõuerſo, ſi fuerit obliquatũ uiſui, <lb/>fortè reputabitur rectũ.</s> <s xml:id="echoid-s28355" xml:space="preserve"> Ex diſpoſitiõe etiã uiſus in ſanitate fit error uiſui in ſitu:</s> <s xml:id="echoid-s28356" xml:space="preserve"> ut ſi ab aliquãta di <lb/>ſtãtia licet tẽperata corpus aliqđ in oppoſitiõe uiſus modicũ obliquetur:</s> <s xml:id="echoid-s28357" xml:space="preserve"> tũc enim uiſu exiſtẽte de-<lb/>bili, nõ ſentietur obliquatio, cũ tamẽ ſit obliquatio ſecũdũ uerũ.</s> <s xml:id="echoid-s28358" xml:space="preserve"> Sic ergo in ſitus uiſiõe uirtuti diſtin <lb/>ctiuę error accidit ex intẽperata diſpoſitiõe octo circũſtantiarũ cuiuslibet rei uiſæ, ut ꝓponebatur.</s> <s xml:id="echoid-s28359" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1119" type="section" level="0" n="0"> <head xml:id="echoid-head893" xml:space="preserve" style="it">45. Figura circularis ſuperſicieirei uiſæ cõprehenditur à uiſu ex circularitate formæ in ſuper-<lb/>ficie oculi deſcriptæ. Alhazen 32 n 2.</head> <p> <s xml:id="echoid-s28360" xml:space="preserve">Quoniã enim formę rerũ deſcribuntur in oculi ſuperficie, ſicut ſunt in rebus extrà per 17 th.</s> <s xml:id="echoid-s28361" xml:space="preserve"> 3 hu-<lb/>ius, & formæ ſecundũ figurã, qua deſcribuntur in oculi ſuperficie, ſic perueniũt ad neruũ cõmunẽ, <lb/>& circa eius punctũ mediũ figurantur, prout patet ք 37 th.</s> <s xml:id="echoid-s28362" xml:space="preserve"> 3 huius, & ibi cõprehenduntur ab anima <lb/>ſecundũ ſui diſpoſitionẽ:</s> <s xml:id="echoid-s28363" xml:space="preserve"> tũ c patet, quòd forma circularis ſuperficiei rei uiſæ cõprehenditur à uiſu <lb/> <pb o="139" file="0441" n="441" rhead="LIBER QVARTVS."/> ex circularitate formę in ſuperficie oculi deſcriptæ:</s> <s xml:id="echoid-s28364" xml:space="preserve"> & ſimiliter comprehen ditur circularitas cuiuſ-<lb/>libet partium ſuperficiei reι uiſæ.</s> <s xml:id="echoid-s28365" xml:space="preserve"> Certificatur autem hæc uiſio, cum uidens mouerit axes radiales <lb/>ambos uel ſaltem unum per totam circum ferentiam rei uiſæ aut partis eius:</s> <s xml:id="echoid-s28366" xml:space="preserve"> ſic enim ex certifica-<lb/>tione ſituum terminorum formæ comprehen det figuram ſuperficiei circularem ex conſimilitudi-<lb/>ne uel diſsimilitu dine partium, & ex comprehẽſione æqualιtatis uel in æqualitatis remotionis par-<lb/>tium rei uiſæ ab inuicem, uel æ qualitatis uel linæ qualitatis eleuationum, partium rei uiſæ ad inui-<lb/>cem.</s> <s xml:id="echoid-s28367" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s28368" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1120" type="section" level="0" n="0"> <head xml:id="echoid-head894" xml:space="preserve" style="it">46. Figura rectilinea comprehenditur à uiſu ex ſuorum terminorum comprehenſione.</head> <p> <s xml:id="echoid-s28369" xml:space="preserve">Quoniam enim figura eſt, quæ termino uel terminis continetur:</s> <s xml:id="echoid-s28370" xml:space="preserve"> termini autem figurarum ſunt <lb/>lineæ, quæ comprehenduntur uiſu non decepto ſecũdum ipſarum ſituationem in ſuperficie oculi, <lb/>ficut eſt ipſarum ſituatio in ſuperficie rei uiſæ.</s> <s xml:id="echoid-s28371" xml:space="preserve"> Palàm ergo, quoniam ipſarum comprehenſio à uiſu <lb/>eſt comprehenſio figuræ in ipſis contentæ, cuius ſunt termini illi.</s> <s xml:id="echoid-s28372" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s28373" xml:space="preserve"> Sed in his <lb/>omnibus uiſus requirit diſtantiam mediocrem & alias circumſtantias uiſui debitas, ne fortè fiat <lb/>deceptio in ipſo uiſu.</s> <s xml:id="echoid-s28374" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1121" type="section" level="0" n="0"> <head xml:id="echoid-head895" xml:space="preserve" style="it">47. Planicies ſuperſiciei ſecũdum mediocrem diſtantiam directè uiſui oppoſitæ comprehen-<lb/>ditur ex comprehenſione æqualitatis remotionis partium, & conſimilitudinis ordinationis <lb/>ipſarum. Alhazen 35 n z.</head> <p> <s xml:id="echoid-s28375" xml:space="preserve">Sit ſuperficies plana a b c d:</s> <s xml:id="echoid-s28376" xml:space="preserve"> & ſit centrum uiſus e:</s> <s xml:id="echoid-s28377" xml:space="preserve"> à quo ducatur ſuper datam ſuperficiẽ perpen.</s> <s xml:id="echoid-s28378" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0441-01a" xlink:href="fig-0441-01"/> dicularis e f.</s> <s xml:id="echoid-s28379" xml:space="preserve"> Et quoniam ſup erficies illa eſt directè uiſui oppoſita, <lb/>ſic quòd perpendicularis incidat in medium punctum illius ſuper-<lb/>ficiei:</s> <s xml:id="echoid-s28380" xml:space="preserve"> producantur quoq;</s> <s xml:id="echoid-s28381" xml:space="preserve"> ad puncta æqualiter à puncto f diſtantia, <lb/>qu æ ſunt a, b, c, d lineæ e a, e b, e c, e d:</s> <s xml:id="echoid-s28382" xml:space="preserve"> & continuentur lineæ f a, f b, f <lb/>c, f d:</s> <s xml:id="echoid-s28383" xml:space="preserve"> quæ omnes erunt æquales propter æqualem ipſarum diſtan-<lb/>tiam à puncto f.</s> <s xml:id="echoid-s28384" xml:space="preserve"> Cum ergo omnes illæ lineæ f a, f b, f c, f d per defini-<lb/>tionem lineæ ſuper ſuperficiem erectæ fint perpendiculares ſuper <lb/>lineam e f:</s> <s xml:id="echoid-s28385" xml:space="preserve"> patet per 4 p 1, quoniam lineæ e a, e b, e c, e d ſunt æqua-<lb/>les:</s> <s xml:id="echoid-s28386" xml:space="preserve"> ſup erficies itaq;</s> <s xml:id="echoid-s28387" xml:space="preserve"> a b c d ſecun dum illos eius terminos æ qualiter <lb/>diſtat à uiſu.</s> <s xml:id="echoid-s28388" xml:space="preserve"> Sed & alijs lineis ad puncta alia æqualiter diſtantia à <lb/>puncto f, à centro uiſus productis, illarum omnium ad inuicem ex <lb/>præmiſsis concluditur æqualitas.</s> <s xml:id="echoid-s28389" xml:space="preserve"> Tota ergo ſuperficies ſecundum <lb/>omnes ſui partes æ qualiter diſtantes ex omni parte à puncto f con-<lb/>fimiliter peruenit ad uiſum.</s> <s xml:id="echoid-s28390" xml:space="preserve"> Tota itaq;</s> <s xml:id="echoid-s28391" xml:space="preserve"> ſuperficies uidebitur plana <lb/>ex comprehenſione æqualitatis remotionis partium & conſimili-<lb/>tudinis ordinationis ipſarum.</s> <s xml:id="echoid-s28392" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s28393" xml:space="preserve"> Sed & ſi axes <lb/>radiales non incidant ad medium, nihilominus per eadem demon-<lb/>ſtrandum:</s> <s xml:id="echoid-s28394" xml:space="preserve"> ſemp er enim termini cuiuslibet partiũ ſuperficiei erunt <lb/>lineæ rectæ.</s> <s xml:id="echoid-s28395" xml:space="preserve"> Superficies ergo eſt plana.</s> <s xml:id="echoid-s28396" xml:space="preserve"/> </p> <div xml:id="echoid-div1121" type="float" level="0" n="0"> <figure xlink:label="fig-0441-01" xlink:href="fig-0441-01a"> <variables xml:id="echoid-variables458" xml:space="preserve">e a b f c d</variables> </figure> </div> </div> <div xml:id="echoid-div1123" type="section" level="0" n="0"> <head xml:id="echoid-head896" xml:space="preserve" style="it">48. Conuexitas ſuperficiei comprehenditur à uiſu ex propin-<lb/>quit ate partium mediarum, & æquali remotione partium extremarum. Alhazen 33 n 2.</head> <p> <s xml:id="echoid-s28397" xml:space="preserve">Cum enim ſuperficies conuexa directè uiſui opponitur ſecundum mediocrem diſtantiam:</s> <s xml:id="echoid-s28398" xml:space="preserve"> tunc <lb/>cum omnis regularis ſup erficies conuexa ſit pars alicuius ſphæræ uel columnæ rotundæ uel pyra-<lb/>midis rotundæ per 118th.</s> <s xml:id="echoid-s28399" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s28400" xml:space="preserve"> ſi ſuperficies illa oppoſita uiſui ſit pars ſphæricæ ſuperficiei, & fi à <lb/>centro uiſus ad centrum ſphæræ linea recta ducatur, aliæq́;</s> <s xml:id="echoid-s28401" xml:space="preserve"> præter centrum lineæ plurimæ produ-<lb/>cantur, patet per 72 th.</s> <s xml:id="echoid-s28402" xml:space="preserve"> 1 huius, quòd ſola illa, quæ centrum tranſit, eſt perpendicularis ſuper ſphęræ <lb/>ſuperficiem:</s> <s xml:id="echoid-s28403" xml:space="preserve"> aliæ uerò omnes lineæ à centro uiſus ad illam ſphæricam ſuperficiem productæ, ſunt <lb/>ſuper illam ſuperficiẽ incidentes obliquè.</s> <s xml:id="echoid-s28404" xml:space="preserve"> Erit ergo per 8 p 3 pars perpendicularis interiacens cen-<lb/>trum uiſus & ſup erficiem ſphæricam omnium aliarum linearum breuiſsima:</s> <s xml:id="echoid-s28405" xml:space="preserve"> ergo ſecundum illam <lb/>fit maxima approximatio ad uiſum, & omnes circuli ſecundum punctum, cui incidit illa perpendi-<lb/>cularis, in ſup erficie ſphæræ deſcripti, erunt uiſui proximiores ſecundum illa puncta, & ſecun dum <lb/>alias lineas obliquè incidẽtes, erunt uiſui remotiores:</s> <s xml:id="echoid-s28406" xml:space="preserve"> quia omnes lineæ perpendiculari lineæ pro-<lb/>pinquiores modo dicto, ſunt minores remotioribus:</s> <s xml:id="echoid-s28407" xml:space="preserve"> quoniam per prænominatam 8 p 3 omnes li-<lb/>neæ à centro uiſus ad peripherias maiorum circulorũ productæ ſunt longiores lineis propinquio-<lb/>ribus ipſi perpendiculari.</s> <s xml:id="echoid-s28408" xml:space="preserve"> Ex comprehenſione ergo propin quitatis partium mediarum in illa ſu-<lb/>perficie, & remotione aliarum partium, quæ ſunt in terminis, apparet maior eleuatio partium me-<lb/>diarum quàm extremarum:</s> <s xml:id="echoid-s28409" xml:space="preserve"> & ex inæqualitate eleuationis partium ſup erficiei uidetur gibbofitas;</s> <s xml:id="echoid-s28410" xml:space="preserve"> <lb/>quæ eſt cauſſa conuexitatis.</s> <s xml:id="echoid-s28411" xml:space="preserve"> Et quoniam in omnl puncto ſuperficiei ſphæricæ ſecant ſe circuli ma-<lb/>gni tranſeuntes per centrum illius ſphæræ, & omnes lineæ, quę lineæ breuiſsimæ utrinq;</s> <s xml:id="echoid-s28412" xml:space="preserve"> æquè ap-<lb/>propinquant, ſunt æquales:</s> <s xml:id="echoid-s28413" xml:space="preserve"> ideo ſecundum æ qualem diſtantiam à perpendiculari fit æqualitas o-<lb/>mnium linearum ad ſphæræ ſuperficiem à centro uiſus productarũ, & apparet deflexio gibbofita-<lb/>tis æqualis ſecundũ omnem differentiam poſitionis in ſphæricis ſuperficiebus, maximè cũ directè <lb/>uiſibus opponũtur.</s> <s xml:id="echoid-s28414" xml:space="preserve"> Si uerò ſup erficies cõuexa oppoſita uiſui fuerit pars ſuperficiei columnaris aut <lb/>pyramidalis rotundarum:</s> <s xml:id="echoid-s28415" xml:space="preserve"> tunc fit eadẽ dem onftratio productis lineis perpendicularibus à centro <lb/> <pb o="140" file="0442" n="442" rhead="VITELLONIS OPTICAE"/> uiſus ad centrum circuli baſis, & omnium circulorum æquidiſtantium baſi:</s> <s xml:id="echoid-s28416" xml:space="preserve"> alijs quoq;</s> <s xml:id="echoid-s28417" xml:space="preserve"> lineis pluri-<lb/>bus ab eodem cẽtro uiſus non perpendiculariter per eoſdem circulos productis, complebitur de-<lb/>monſtratio ut prius.</s> <s xml:id="echoid-s28418" xml:space="preserve"> Et ſi illæ ſuperficies quomodocunq;</s> <s xml:id="echoid-s28419" xml:space="preserve"> obliquatæ ſint ad uiſum, nihilominus per <lb/>eadem eſt demonſtrandum:</s> <s xml:id="echoid-s28420" xml:space="preserve"> ſiue enim gibboſitas ſit inferius, ſiue ſuperius, ſiue à dextris, ſiue à ſini-<lb/>ſtris, ſemper partium in æqualis diſtantia propoſitum cõcludet:</s> <s xml:id="echoid-s28421" xml:space="preserve"> & de irregularibus conuexitatibus <lb/>per eadem fit comprehenſio in uiſu.</s> <s xml:id="echoid-s28422" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s28423" xml:space="preserve"> Vniuerſaliter enim conuexitas com-<lb/>prehenditur à uiſu ex propin quitate partiũ mediarum, & æquali remotione partium extremarum.</s> <s xml:id="echoid-s28424" xml:space="preserve"> <lb/>Patet ergo quod proponebatur.</s> <s xml:id="echoid-s28425" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1124" type="section" level="0" n="0"> <head xml:id="echoid-head897" xml:space="preserve" style="it">49. Concauit as ſuperficiei comprehẽditur à uiſu ex remotione partium mediarũ, & æquali <lb/>appropinquatione partium extremarum. Alhazen 34 n 2.</head> <p> <s xml:id="echoid-s28426" xml:space="preserve">Per eadem, quæ in præcedenti, demonſtrandum, & ſimiliter per omnem ſuperficiem tranſcur-<lb/>rendum.</s> <s xml:id="echoid-s28427" xml:space="preserve"> Semper enim per 8 p 3 linea à centro uiſus ad centrum ſphæræ uel circuli producta, quia <lb/>continet diametrum, eſt omnium longiſsima, & ſibi propinquiores ſunt cæteris remotioribus ma-<lb/>iores, & omnes æqualiter ab illa diſtantes ſunt æquales.</s> <s xml:id="echoid-s28428" xml:space="preserve"> Ergo termini illius ſuperficiei uidebuntur <lb/>arcuales, & tota ſuperficies uidebitur concaua.</s> <s xml:id="echoid-s28429" xml:space="preserve"> Et ſi illæ ſuperficies fint obliquatæ uiſibus, ſiue ar-<lb/>cualitas terminorum ſit ſuperius, ſiue inferius, ſiue à dextris, ſiue à ſiniſtris, ſemper per eandem de-<lb/>monſtrandum.</s> <s xml:id="echoid-s28430" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s28431" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1125" type="section" level="0" n="0"> <head xml:id="echoid-head898" xml:space="preserve" style="it">50. Centro for aminis uueæ & circumferentia circuli in eadẽ ſuperficie exiſtẽtibus: circum-<lb/>ferentia ad aliquam rectitudinem accedere uidetur. Euclides in præfat. & 22 the. opt.</head> <p> <s xml:id="echoid-s28432" xml:space="preserve">Eſto foraminis uueæ centrum a, in eadem exiſtens ſuperficie cum circumferentia circuli uiſi:</s> <s xml:id="echoid-s28433" xml:space="preserve"> ita <lb/> <anchor type="figure" xlink:label="fig-0442-01a" xlink:href="fig-0442-01"/> quòd plana ſuperficies circuli imaginata produci ſecet ſphærã oculi <lb/>trans centrum:</s> <s xml:id="echoid-s28434" xml:space="preserve"> illius quoq;</s> <s xml:id="echoid-s28435" xml:space="preserve"> circumferentia circuli ſit g b:</s> <s xml:id="echoid-s28436" xml:space="preserve"> & eius cen-<lb/>trum k:</s> <s xml:id="echoid-s28437" xml:space="preserve"> & à punctis illius circumferentiæ ducantur lineæ plurimæ ad <lb/>uiſum a:</s> <s xml:id="echoid-s28438" xml:space="preserve"> quæ ſint b a, d a, e a, z a, i a, c a, g a:</s> <s xml:id="echoid-s28439" xml:space="preserve"> ſecũdum quas lineas formæ <lb/>illorum punctorum accedunt ad uiſum.</s> <s xml:id="echoid-s28440" xml:space="preserve"> Dico, quoniam arcus b g ap-<lb/>paret uiſui linea recta.</s> <s xml:id="echoid-s28441" xml:space="preserve"> Ducãtur enim à centro illius circuli lineæ k b, <lb/>k d, k e, k z, k i, k c, k g.</s> <s xml:id="echoid-s28442" xml:space="preserve"> Quoniam ergo linea k b uidetur ſub angulo k a <lb/>b, & linea k d ſub angulo k a d, qui minor eſt angulo k a b, quoniam <lb/>pars eius eſt:</s> <s xml:id="echoid-s28443" xml:space="preserve"> ergo per 20 huius palàm eſt, quia maior uidebitur linea <lb/>k b quàm k d, quoniam ſub maiori angulo uidetur:</s> <s xml:id="echoid-s28444" xml:space="preserve"> & ſimiliter uide-<lb/>bitur linea k d maior quàm k e, & k e maior quàm k z:</s> <s xml:id="echoid-s28445" xml:space="preserve"> & eodem modo <lb/>uidebitur k g maior quàm k e, & k c maior quàm k i, & k i maior quàm <lb/>k z.</s> <s xml:id="echoid-s28446" xml:space="preserve"> Punctus quoq;</s> <s xml:id="echoid-s28447" xml:space="preserve"> z inter omnes datos punctos, quoniã cadit in per-<lb/>pendiculari a k, propinquior uidebitur centro k quàm punctus e, & <lb/>punctus e propinquior quàm punctus d, & punctus d propinquior <lb/>quàm punctus b.</s> <s xml:id="echoid-s28448" xml:space="preserve"> In apparentia ergo uiſui aliquid tollitur de curui-<lb/>tate arcus z b.</s> <s xml:id="echoid-s28449" xml:space="preserve"> Et ſimiliter eſt de arcu z g.</s> <s xml:id="echoid-s28450" xml:space="preserve"> Accedere ergo uidetur ad <lb/>rectitudinem arcus g b.</s> <s xml:id="echoid-s28451" xml:space="preserve"> Cum enim per 8 p 3 linea a z fit omnium bre-<lb/>uiſsima, & linea a e breuior ſit quàm linea a d, & a d breuior quàm a <lb/>b:</s> <s xml:id="echoid-s28452" xml:space="preserve"> patet quòd in uiſu aliquid remanet curuitatis apprehẽſæ & ſic non <lb/>uidebitur tota peripheria linea recta, ſed ad rectitudinem aliqualiter accedens.</s> <s xml:id="echoid-s28453" xml:space="preserve"> Patet ergo propoſi-<lb/> <anchor type="figure" xlink:label="fig-0442-02a" xlink:href="fig-0442-02"/> tum.</s> <s xml:id="echoid-s28454" xml:space="preserve"> Ethoc idẽ accidet cõuexis & concauis partibus peripheriæ cir-<lb/>culi uiſui oppoſitis.</s> <s xml:id="echoid-s28455" xml:space="preserve"> Quia ſi à puncto z ducatur aliqua perpendicula-<lb/>ris ſuper lineam a z:</s> <s xml:id="echoid-s28456" xml:space="preserve"> tunc non eſt differentia magna uiſui inter arcum <lb/>& lineam contingentem, cum per maius ſpatium uiſio fit, propè uerò <lb/>exiſtente uiſu, maior percipitur conuexitas uel concauitas:</s> <s xml:id="echoid-s28457" xml:space="preserve"> & magis <lb/>apparet.</s> <s xml:id="echoid-s28458" xml:space="preserve"> Quòd ſi centrum oculi & circulus non ſint in eadem ſuper-<lb/>ficie:</s> <s xml:id="echoid-s28459" xml:space="preserve"> tunc circum ferentia circuli uidebitur curua:</s> <s xml:id="echoid-s28460" xml:space="preserve"> quoniam tũc ſitus <lb/>partiũ lineę circularis ſecundũ ſuũ ſitũ & eſſe propriũ peruenit ad ui-<lb/>ſum, & depingitur ſecundũ ſuã curuitatẽ in ſuperficie illius, licet quã-<lb/>doq;</s> <s xml:id="echoid-s28461" xml:space="preserve"> forma ſphærica illius curuitatis ſecundũ aliquid ſui uarietur.</s> <s xml:id="echoid-s28462" xml:space="preserve"/> </p> <div xml:id="echoid-div1125" type="float" level="0" n="0"> <figure xlink:label="fig-0442-01" xlink:href="fig-0442-01a"> <variables xml:id="echoid-variables459" xml:space="preserve">k g c i z e d b a</variables> </figure> <figure xlink:label="fig-0442-02" xlink:href="fig-0442-02a"> <variables xml:id="echoid-variables460" xml:space="preserve">b e c d a</variables> </figure> </div> </div> <div xml:id="echoid-div1127" type="section" level="0" n="0"> <head xml:id="echoid-head899" xml:space="preserve" style="it">51. Circulo centró for aminis uueæ in eadem ſuperficie exiſten-<lb/>tibus: minus ſemicirculo uidetur.</head> <p> <s xml:id="echoid-s28463" xml:space="preserve">Sit centrum foraminis uueæ, quod ſit punctum a:</s> <s xml:id="echoid-s28464" xml:space="preserve"> & circulus b c d, <lb/>cuius diameter b e, in eadem ſuperficie plana exiſtentia:</s> <s xml:id="echoid-s28465" xml:space="preserve"> uideaturq́;</s> <s xml:id="echoid-s28466" xml:space="preserve"> <lb/>arcus b c d:</s> <s xml:id="echoid-s28467" xml:space="preserve"> dico, quòd minus ſemicirculo uidebitur.</s> <s xml:id="echoid-s28468" xml:space="preserve"> Si enim arcus b <lb/>c d, qui uidetur, ſit ſemicirculus, neceſſe eſt lineas a b & a e ſuper ter-<lb/>minos diametri b e incidere:</s> <s xml:id="echoid-s28469" xml:space="preserve"> aliter enim ſemicirculus non uidebitur:</s> <s xml:id="echoid-s28470" xml:space="preserve"> <lb/>quia ſola diameter eſt, quæ diuidit circulũ per æqualia per 17 defin.</s> <s xml:id="echoid-s28471" xml:space="preserve"> 1.</s> <s xml:id="echoid-s28472" xml:space="preserve"> <lb/>Ergo lineæ a b & a e ſemper contingent circulum, quoniam à termi-<lb/>nis diametri producuntur.</s> <s xml:id="echoid-s28473" xml:space="preserve"> Palàm ergo per 18 p 3, quoniam utraq;</s> <s xml:id="echoid-s28474" xml:space="preserve"> cum diametro b e angulũ rectum <lb/>contin ebit:</s> <s xml:id="echoid-s28475" xml:space="preserve"> triangulus itaq;</s> <s xml:id="echoid-s28476" xml:space="preserve"> a b e habebit duos angulos rectos, & tertium angulum:</s> <s xml:id="echoid-s28477" xml:space="preserve"> quod eſt cõtra <lb/>32 p 1, & impoſsibile.</s> <s xml:id="echoid-s28478" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s28479" xml:space="preserve"/> </p> <pb o="141" file="0443" n="443" rhead="LIBER QVARTVS."/> </div> <div xml:id="echoid-div1128" type="section" level="0" n="0"> <head xml:id="echoid-head900" xml:space="preserve" style="it">52. Centro foraminis uueæ exiſtente in circumferentia uel in centro circuli: totalis circu-<lb/>lus uidetur.</head> <p> <s xml:id="echoid-s28480" xml:space="preserve">Eſto centrum ſoraminis uueæ punctum a in circumferentia circuli <lb/>d b:</s> <s xml:id="echoid-s28481" xml:space="preserve"> dico, quòd totus circulus d b uidebitur.</s> <s xml:id="echoid-s28482" xml:space="preserve"> Nec enim eſt punctus in <lb/> <anchor type="figure" xlink:label="fig-0443-01a" xlink:href="fig-0443-01"/> toto circulo, à quo ad quemlibet punctum datum in circumferentia <lb/>duci linea recta non poſsit.</s> <s xml:id="echoid-s28483" xml:space="preserve"> Et quia, ut oſtenſum eſt per 2 th.</s> <s xml:id="echoid-s28484" xml:space="preserve"> 3 huius, <lb/>poſsibilile eſtſolum illud uideri, inter cuius quodlibet pũctum & ali <lb/>quod punctum ſuperficiei uiſus produci lineas rectas eſt poſsibile:</s> <s xml:id="echoid-s28485" xml:space="preserve"> <lb/>formæ ergo omnium punctorum circuli pertingere poſſunt ad uiſum <lb/>nullo extrinſeco corpore impediente.</s> <s xml:id="echoid-s28486" xml:space="preserve"> Totalis ergo circulus ſecundũ <lb/>omnia ſua puncta uideri poterit cẽtro foraminis uueæ in illius circuli <lb/>circumferentia collocata.</s> <s xml:id="echoid-s28487" xml:space="preserve"> Et quoniam centro foraminis uueæ in cen-<lb/>tro circuli exiſtente, adhuc omnes lineæ ducibiles à punctis circum-<lb/>ferentiæ ad centrum, ad ipſum uiſum perueniunt:</s> <s xml:id="echoid-s28488" xml:space="preserve"> patet, quia fiet uiſio <lb/>ſecundum lineas, quæ à punctis circum ferentiæ ducũtur ad centrum <lb/>uiſus per 17 th.</s> <s xml:id="echoid-s28489" xml:space="preserve"> 3 huius.</s> <s xml:id="echoid-s28490" xml:space="preserve"> Ethoc eſt propoſitum.</s> <s xml:id="echoid-s28491" xml:space="preserve"/> </p> <div xml:id="echoid-div1128" type="float" level="0" n="0"> <figure xlink:label="fig-0443-01" xlink:href="fig-0443-01a"> <variables xml:id="echoid-variables461" xml:space="preserve">b a a d</variables> </figure> </div> </div> <div xml:id="echoid-div1130" type="section" level="0" n="0"> <head xml:id="echoid-head901" xml:space="preserve" style="it">53. Exiſtente cẽtro oculi in linea à centro circuli ſuper ſuperficiem circuli erecta, aut in ter-<lb/>mino lineæ obliquè ſuperficiei circuli inſiſtẽtis æqualis ſemidiametro: omnes diametri in eodem <lb/>circulo productæ æquales uiſui apparebunt. Euclides 35. 36 th. 0pt.</head> <p> <s xml:id="echoid-s28492" xml:space="preserve">Eſto circulus d e g:</s> <s xml:id="echoid-s28493" xml:space="preserve"> cuius centrum ſit punctus a:</s> <s xml:id="echoid-s28494" xml:space="preserve"> erigaturq́ linea a b perpendiculariter ſuper cir-<lb/>culi ſuperficiem:</s> <s xml:id="echoid-s28495" xml:space="preserve"> & ducãtur diametri e z & d g:</s> <s xml:id="echoid-s28496" xml:space="preserve"> ponaturq́ centrũ oculi in linea a b in puncto b.</s> <s xml:id="echoid-s28497" xml:space="preserve"> Di-<lb/>co, quòd omnes diametri ductæ trãs ſuperficiem circuli, ut e z & d g, æ quales adinuicem uidebun-<lb/>tur.</s> <s xml:id="echoid-s28498" xml:space="preserve"> Ducantur enim à centro uiſus line æ b e, b z, b d, b g.</s> <s xml:id="echoid-s28499" xml:space="preserve"> Quoniam ergo linea z a æqualis eſt lineæ a <lb/> <anchor type="figure" xlink:label="fig-0443-02a" xlink:href="fig-0443-02"/> g, & linea b a cõmunis ambobus trigonis a b g & a b z, anguli quoq;</s> <s xml:id="echoid-s28500" xml:space="preserve"> <lb/>ad centrum a ſunt æ quales, quia recti:</s> <s xml:id="echoid-s28501" xml:space="preserve"> palàm per 4 p 1, quoniam linea <lb/>b g eſt æ qualis lineæ b z, & angulus a b z eſt æqualis angulo a b:</s> <s xml:id="echoid-s28502" xml:space="preserve"> g:</s> <s xml:id="echoid-s28503" xml:space="preserve"> & <lb/>eodem modo erit angulus a b d æqualis angulo a b e, & omnes an-<lb/>guliad centrum uiſus inter ſe ſunt æquales.</s> <s xml:id="echoid-s28504" xml:space="preserve"> Ergo per 19 uel 20 huius <lb/>omnes ſemidiametri æquales apparent:</s> <s xml:id="echoid-s28505" xml:space="preserve"> imò & ipſi diametri:</s> <s xml:id="echoid-s28506" xml:space="preserve"> ſub æ-<lb/>qualibus enim angulis omnia uidẽtur, & totales diametri & partes.</s> <s xml:id="echoid-s28507" xml:space="preserve"> <lb/>Sed & omnes lineæ æquidiſtantes alteri diametrorum, uidentur mi-<lb/>nores diametris, & remotiores minores propinquioribus:</s> <s xml:id="echoid-s28508" xml:space="preserve"> quod pa-<lb/>tet ducta linea ſh æquidiſtante diametro d g, cuius medio pũcto, qui <lb/>ſit k, incidat linea b k:</s> <s xml:id="echoid-s28509" xml:space="preserve"> & copulentur lineæ b f, & b h, & a k:</s> <s xml:id="echoid-s28510" xml:space="preserve"> eritq́ linea <lb/>a k per 3 p 3 per pẽdicularis ſuper lineam fh, quoniam ueniens à cen-<lb/>tro diuidit ipſam per æqualia in puncto k.</s> <s xml:id="echoid-s28511" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s28512" xml:space="preserve"> in trigonis b a g <lb/>& b k h anguli b a g & b k h ſunt recti, ut b a g ex hypotheſi, & b k h <lb/>per 22 th.</s> <s xml:id="echoid-s28513" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s28514" xml:space="preserve"> linea uerò b k eſt maior quàm linea b a, & linea a g <lb/>eſt maior quàm linea k h:</s> <s xml:id="echoid-s28515" xml:space="preserve"> ergo per 37 th.</s> <s xml:id="echoid-s28516" xml:space="preserve"> 1 huius angulus b h k eſt ma-<lb/>ior angulo b g a:</s> <s xml:id="echoid-s28517" xml:space="preserve"> ſimiliter quoq;</s> <s xml:id="echoid-s28518" xml:space="preserve"> angulus b f h erit maior angulo b d a.</s> <s xml:id="echoid-s28519" xml:space="preserve"> <lb/>In trigonis ergo d b g & f b h erit per 32 p 1 angulus d b g maior angu-<lb/>lo f b h:</s> <s xml:id="echoid-s28520" xml:space="preserve"> diameter ergo d g uidebitur maior, quàm linea fh per 20 hu-<lb/>ius.</s> <s xml:id="echoid-s28521" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s28522" xml:space="preserve"> eſt de omnibus alijs lineis æquidiſtantibus dia <lb/>metro, reſpectu ipſius diametri, & ad inuicem demonſtrãdum.</s> <s xml:id="echoid-s28523" xml:space="preserve"> Quælibet ergo minor uidebitur mi-<lb/> <anchor type="figure" xlink:label="fig-0443-03a" xlink:href="fig-0443-03"/> nor:</s> <s xml:id="echoid-s28524" xml:space="preserve"> & ita totus circulus uidebitur propriæ ſuæ figurę.</s> <s xml:id="echoid-s28525" xml:space="preserve"> Et hoc eſt pro-<lb/>poſitum primum.</s> <s xml:id="echoid-s28526" xml:space="preserve"> Si uerò linea a b non ſit erecta ſuper circuli ſuperfi-<lb/>ciem, ſed obliquè inſiſtens, ſit tamen æqualis ſemidiametro circuli, ad <lb/>huc diametri d g & z e uidebuntur æquales, centro uiſus in puncto b <lb/>exiſtente.</s> <s xml:id="echoid-s28527" xml:space="preserve"> Gum enim ex hypotheſi z a ſemidiameter ſit æqualis lineæ <lb/>a b, & ſemidiameter a e æqualis ſit eidem:</s> <s xml:id="echoid-s28528" xml:space="preserve"> palàm quoniam lineæ a b, a <lb/>e, a z ſunt æquales.</s> <s xml:id="echoid-s28529" xml:space="preserve"> Si ergo ſuper punctum a ad quantitatem ſemidia-<lb/>metri e a circulus deſcribatur in ſuperficie, in qua ſunt lineæ a e, a z, a <lb/>b:</s> <s xml:id="echoid-s28530" xml:space="preserve"> palàm quia tranſibit per punctum b:</s> <s xml:id="echoid-s28531" xml:space="preserve"> ergo per 31 p 3 angulus e b z eſt <lb/>rectus:</s> <s xml:id="echoid-s28532" xml:space="preserve"> ſimiliter quoq;</s> <s xml:id="echoid-s28533" xml:space="preserve"> oſtendetur angulum g b d eſſe rectum.</s> <s xml:id="echoid-s28534" xml:space="preserve"> Et quia <lb/>omnes anguli recti ſunt æquales, & ſub æqualibus angulis uiſa æqua-<lb/>lia apparẽt ք 19 uel 20 huius:</s> <s xml:id="echoid-s28535" xml:space="preserve"> palàm quia omnes diametri illius circu-<lb/>li quotcũq;</s> <s xml:id="echoid-s28536" xml:space="preserve"> ducãtur, ęquales apparebũt, ſicut diameter e z ipſi diame-<lb/>tro g d:</s> <s xml:id="echoid-s28537" xml:space="preserve"> qđ eſt ꝓpoſitũ ſecundũ.</s> <s xml:id="echoid-s28538" xml:space="preserve"> Patet ergo totũ, quod ꝓponebatur.</s> <s xml:id="echoid-s28539" xml:space="preserve"/> </p> <div xml:id="echoid-div1130" type="float" level="0" n="0"> <figure xlink:label="fig-0443-02" xlink:href="fig-0443-02a"> <variables xml:id="echoid-variables462" xml:space="preserve">b e d a g f k h v</variables> </figure> <figure xlink:label="fig-0443-03" xlink:href="fig-0443-03a"> <variables xml:id="echoid-variables463" xml:space="preserve">b g e a z d</variables> </figure> </div> </div> <div xml:id="echoid-div1132" type="section" level="0" n="0"> <head xml:id="echoid-head902" xml:space="preserve" style="it">54. Centro oculi exiſtente in termino lineæ maioris uel minoris ſemidiametro circuli (cuius <lb/>ſuperſiciei in centro obliquè eſt inſiſtens) æquales angulos cum diuerſis ſemidiametris continen-<lb/>tis: illæ diametri eiuſdem circuli æquales apparebunt. Euclides ſecunda parte 30 & 38 th. opt.</head> <p> <s xml:id="echoid-s28540" xml:space="preserve">Sit circulus b g d e, cuius centrũ a:</s> <s xml:id="echoid-s28541" xml:space="preserve"> & ſit centrũ uiſus z:</s> <s xml:id="echoid-s28542" xml:space="preserve"> ſitq́ linea a z nõ erecta, ſed obliquè incidẽs <lb/> <pb o="142" file="0444" n="444" rhead="VITELLONIS OPTICAE"/> ſuperficiei circuli maior uel minor ſemidiametro d a:</s> <s xml:id="echoid-s28543" xml:space="preserve"> ſit tamẽ angulus d a z æqualis angulo g a z, & <lb/>angulus e a z æqualis angulo b a z.</s> <s xml:id="echoid-s28544" xml:space="preserve"> Dico, quòd adhuc diametri d b & e g uidebuntur æquales:</s> <s xml:id="echoid-s28545" xml:space="preserve"> quo-<lb/> <anchor type="figure" xlink:label="fig-0444-01a" xlink:href="fig-0444-01"/> niam enim linea d a eſt ęqualis a g, & linea z a communis duobus trigo-<lb/>nis z a g, & z a d:</s> <s xml:id="echoid-s28546" xml:space="preserve"> eſt quoq;</s> <s xml:id="echoid-s28547" xml:space="preserve"> ex hypotheſi angulus d a z æqualis angulo g <lb/>a z:</s> <s xml:id="echoid-s28548" xml:space="preserve"> erit per 4 p 1 linea z d æqualis lineæ z g, & angulus d z a æqualis an-<lb/>gulo g z a:</s> <s xml:id="echoid-s28549" xml:space="preserve"> ergo per 19 uel 20 huius baſis d a uidebitur æqualis g a baſi.</s> <s xml:id="echoid-s28550" xml:space="preserve"> <lb/>Similiter quoq;</s> <s xml:id="echoid-s28551" xml:space="preserve"> per eadem demonſtrabitur angulus e z a æqualis angu-<lb/>lo b z a:</s> <s xml:id="echoid-s28552" xml:space="preserve"> & per pręmiſſa uidebitur linea e a ęqualis lineæ b a, & angulus a <lb/>z g æqualis eſt angulo a z d, & angulus e z a æqualis angulo a z g:</s> <s xml:id="echoid-s28553" xml:space="preserve"> ideo <lb/>accidit ut totalis angulus d z b totali angulo e z g ſit æqualis.</s> <s xml:id="echoid-s28554" xml:space="preserve"> Videbitur <lb/>ergo, ut ſuprà patuit, diameter d b æqualis diametro e g.</s> <s xml:id="echoid-s28555" xml:space="preserve"> Quod eſt pro-<lb/>poſitum.</s> <s xml:id="echoid-s28556" xml:space="preserve"> Poſsibile eſt autem hoc in quibuſdam diametris accidere, non <lb/>autem in omnibus diametris circuli taliter uiſui oppoſiti:</s> <s xml:id="echoid-s28557" xml:space="preserve"> nõ ergo opor-<lb/>tet quòd omnes diametri illius circuli uideantur æquales:</s> <s xml:id="echoid-s28558" xml:space="preserve"> non enim <lb/>illæ diametri uidebuntur æquales, cum quibus linea z a facit angulos <lb/>in æquales.</s> <s xml:id="echoid-s28559" xml:space="preserve"/> </p> <div xml:id="echoid-div1132" type="float" level="0" n="0"> <figure xlink:label="fig-0444-01" xlink:href="fig-0444-01a"> <variables xml:id="echoid-variables464" xml:space="preserve">e z b a d g z</variables> </figure> </div> </div> <div xml:id="echoid-div1134" type="section" level="0" n="0"> <head xml:id="echoid-head903" xml:space="preserve" style="it">55. Sirect a linea à centro circuli centro oculi incidens, non eri-<lb/>gatur ſuper ſuperficiem circuli, ne æquales angulos contineat cum <lb/>diametris, ſit́ maior ſemidiametro: diametri illius circuliinæqua-<lb/>les apparebunt: totuś circulus uidebitur ſectio columnaris: cuius <lb/>maxima eſt diameter illa, cui perpendiculariter incidit linea radia-<lb/>lis. Euclides 37. 39 th. opt.</head> <p> <s xml:id="echoid-s28560" xml:space="preserve">Eſto circulus a g b d:</s> <s xml:id="echoid-s28561" xml:space="preserve"> cuius centrum z:</s> <s xml:id="echoid-s28562" xml:space="preserve"> & ducantur diametri a b & g d, ſe ad inuicem orthogona <lb/>liter ſecantes:</s> <s xml:id="echoid-s28563" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s28564" xml:space="preserve"> centrum oculi e:</s> <s xml:id="echoid-s28565" xml:space="preserve"> à quo ducatur linea e z ad centrum circuli, diametro quidem d <lb/>g ſecundum angulum rectum perpendiculariter incidens, diametro uerò a b obliquè, ut acciderit:</s> <s xml:id="echoid-s28566" xml:space="preserve"> <lb/>non erit ergo linea e z erecta ſuper ſuperficiem circuli:</s> <s xml:id="echoid-s28567" xml:space="preserve"> ſitq́ linea e z maior ſemidiametro circuli.</s> <s xml:id="echoid-s28568" xml:space="preserve"> <lb/>Dico, quòd diametri a b & g d uidebuntur in æquales:</s> <s xml:id="echoid-s28569" xml:space="preserve"> & g d maxima quidem, a b uerò minima:</s> <s xml:id="echoid-s28570" xml:space="preserve"> & <lb/>quòd totus circulus uidebitur altera parte longior, ueluti ſectio columnaris:</s> <s xml:id="echoid-s28571" xml:space="preserve"> & quòd omnis dia-<lb/>meter circuli, quæ ceciderit propior minimæ, uidebitur minor remotiore ab illa:</s> <s xml:id="echoid-s28572" xml:space="preserve"> & duæ tãtùm dia-<lb/>metri apparebunt æquales, ut illæ, quæ æqualiter diſtant ab utraq;</s> <s xml:id="echoid-s28573" xml:space="preserve"> parte à minima diametro, quæ <lb/>eſt a b.</s> <s xml:id="echoid-s28574" xml:space="preserve"> Quoniam enim diameter g d eſt perpendicularis ſuper diametrum a b, & ſuper lineam z e, <lb/>palàm per 4 p 11 quoniam linea g z eſt perpendicularis ſuper ſuperficiem, in qua ſunt lineæ e z & a <lb/>z, uel a b:</s> <s xml:id="echoid-s28575" xml:space="preserve"> ergo per 18 p 11 erit circulus propoſitus orthogonalis ſuper ſuperficiem e a z:</s> <s xml:id="echoid-s28576" xml:space="preserve"> ergo & e a z <lb/>ſuperficies erecta erit ſuper circulum.</s> <s xml:id="echoid-s28577" xml:space="preserve"> Ducatur ergo à puncto e ſuper ſuperficiem circuli a b g d <lb/> <anchor type="figure" xlink:label="fig-0444-02a" xlink:href="fig-0444-02"/> perpendicularis per 11 p 11:</s> <s xml:id="echoid-s28578" xml:space="preserve"> hæc itaque per præmiſſa ne-<lb/>ceſſariò cadet in communem ſectionem illarum ſuper-<lb/>ficierum, quæ eſt a b:</s> <s xml:id="echoid-s28579" xml:space="preserve"> cadat ergo, & ſit e k:</s> <s xml:id="echoid-s28580" xml:space="preserve"> & ducantur li-<lb/>neæ e a, e b, e d, e g:</s> <s xml:id="echoid-s28581" xml:space="preserve"> producaturq́ diameter circuli alia, <lb/>quæ ſit s z p, conſtituens cum diametro g z d angulum <lb/>p z d æqualem angulo g z s per 15 p 1:</s> <s xml:id="echoid-s28582" xml:space="preserve"> ducatur quoque <lb/>alia diameter, quæ ſit i z t:</s> <s xml:id="echoid-s28583" xml:space="preserve"> ita ut anguli g z s & i z g ſint <lb/>æquales.</s> <s xml:id="echoid-s28584" xml:space="preserve"> Quia itaque à puncto e in aere dato ſuper ſub-<lb/>ſtratam planam ſuperficiem circuli, qui eſt a b g d, du-<lb/>cuntur duæ lineæ, una perpendiculariter, quæ eſt e k, & <lb/>alia obliquè, quæ eſt e z, & inter puncta incidentiæ, quæ <lb/>ſunt k & z, copulatur linea z k in ipſa ſuperſicie:</s> <s xml:id="echoid-s28585" xml:space="preserve"> patet <lb/>per 39 th.</s> <s xml:id="echoid-s28586" xml:space="preserve"> 1 huius, quoniam angulus e z k minimus eſt o-<lb/>mnium angulorum ſub linea e z obliquè incidente, & <lb/>ſemidiametro z i uel z p, uel quacunq;</s> <s xml:id="echoid-s28587" xml:space="preserve"> alia diametro con <lb/>tentorum:</s> <s xml:id="echoid-s28588" xml:space="preserve"> & omnis angulus iſtorum angulorum pro-<lb/>pinquior angulo e z k eſt minor remotiore:</s> <s xml:id="echoid-s28589" xml:space="preserve"> duo quo que <lb/>anguli ex utraque parte æqualiter angulo e z k appro-<lb/>ximantes, ut ſunt anguli i z k, & p z k inter ſe ſunt æqua-<lb/>les.</s> <s xml:id="echoid-s28590" xml:space="preserve"> Copulentur quoq;</s> <s xml:id="echoid-s28591" xml:space="preserve"> lineæ e i, e s, e p, e t.</s> <s xml:id="echoid-s28592" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s28593" xml:space="preserve"> ab <lb/>angulis duorũ trigonorũ d e g & t e i, ad medietates ſua-<lb/>rũ baſiũ æqualiũ in trigono d e g linea e z perpẽdicula-<lb/>riter incidit, & in trigono tei obliquè, eſtq́;</s> <s xml:id="echoid-s28594" xml:space="preserve"> linea e z ma <lb/>ior medietate utriuſq;</s> <s xml:id="echoid-s28595" xml:space="preserve"> illarũ baſium g d, & i t, ut patet ex <lb/>hypotheſi:</s> <s xml:id="echoid-s28596" xml:space="preserve"> ergo ք 49 th.</s> <s xml:id="echoid-s28597" xml:space="preserve"> 1 huius erit angulus d e g maior <lb/>angulo t e i:</s> <s xml:id="echoid-s28598" xml:space="preserve"> ergo ք 20 huius diameter d g uidebitur ma-<lb/>ior diametro i t.</s> <s xml:id="echoid-s28599" xml:space="preserve"> Et quoniã, ut oſtẽſum eſt ք 39 th.</s> <s xml:id="echoid-s28600" xml:space="preserve"> 1 huius, <lb/>angulus e z i eſt maior angulo e z a, ambabus uerò baſib.</s> <s xml:id="echoid-s28601" xml:space="preserve"> trigonorũ t e i & a e b, quæ ſunt i t & a b, ad <lb/> <pb o="143" file="0445" n="445" rhead="LIBER QVARTVS."/> medium punctum, quod eſt z, linea e z incidit obliquè:</s> <s xml:id="echoid-s28602" xml:space="preserve"> erit per 51 th.</s> <s xml:id="echoid-s28603" xml:space="preserve"> 1 huius angulus t e i maior an-<lb/>gulo a e b:</s> <s xml:id="echoid-s28604" xml:space="preserve"> ergo per 20 huius diameter it uidebitur maior diametro a b.</s> <s xml:id="echoid-s28605" xml:space="preserve"> Et ſic per præmiſſa de quali-<lb/>bet aliarum diametrorum, reſpectu diametri a b, eſt demonſtrandum.</s> <s xml:id="echoid-s28606" xml:space="preserve"> Omnium itaq;</s> <s xml:id="echoid-s28607" xml:space="preserve"> diametrorum <lb/>circuli propoſiti g d uidetur maxima, & a b minima:</s> <s xml:id="echoid-s28608" xml:space="preserve"> & propinquiores diametro g d uidẽtur maio-<lb/>res, & propinquiores diametro a b uidentur minores:</s> <s xml:id="echoid-s28609" xml:space="preserve"> duæ quoq;</s> <s xml:id="echoid-s28610" xml:space="preserve"> diametri æqualiter hinc inde di-<lb/>ſtantes, uidentur æquales, ut ſuntit & s p per præmiſſam:</s> <s xml:id="echoid-s28611" xml:space="preserve"> quoniam propter æqualitatem angulo-<lb/>rum aliquorum, qui ſunt e z i & e z p per 39 th.</s> <s xml:id="echoid-s28612" xml:space="preserve"> 1 huius, anguli t e i & s e p fiunt æquales per 4 p 1 To-<lb/>tus ergo circulus uidetur altera parte longior, ueluti ſectιo colũnaris.</s> <s xml:id="echoid-s28613" xml:space="preserve"> Sed & ſuppoſitis ijs, quæ per <lb/>39 th 1 huius declarata ſunt, poteſt reliquum aliter demonſtrari.</s> <s xml:id="echoid-s28614" xml:space="preserve"> Extra hanc enim figuram protra-<lb/>hatur lineal m æqualis diametro d g per 3 p 1, & diuidatur linea l m per æqualia in puncto n per 10 p <lb/>1:</s> <s xml:id="echoid-s28615" xml:space="preserve"> & à puncto n ducatur linea n x perpendiculariter ſuper lineam l m per 11 p 1, & reſecetur linea n x <lb/>ad æqualitatem lineæ z e, quæ eſt ex hypoth eſi maior quàm linea n m, æqualis ſemidiametro z g, ut <lb/>patet ex præmiſsis:</s> <s xml:id="echoid-s28616" xml:space="preserve"> ductisq́ lineis l x & m x, compleatur trigonum l m x:</s> <s xml:id="echoid-s28617" xml:space="preserve"> & per 5 p 4 circumſcrιba-<lb/> <anchor type="figure" xlink:label="fig-0445-01a" xlink:href="fig-0445-01"/> tur ei portio circuli, quę ſit l m x:</s> <s xml:id="echoid-s28618" xml:space="preserve"> eſt itaq;</s> <s xml:id="echoid-s28619" xml:space="preserve"> illa portio <lb/>circuli l m x maior ſemιcirculo, ideo quia linea n x <lb/>eſt maior utraq;</s> <s xml:id="echoid-s28620" xml:space="preserve"> linearum n m & n l.</s> <s xml:id="echoid-s28621" xml:space="preserve"> Et quoniã tri-<lb/>gonorum g z e & l n x latus g z eſt æquale lateri n l, <lb/>& latus z e æquale lateri n x, & angulus g z e æqua-<lb/>lis angulo l n x, quoniã, ut patet ex pręmiſsis, uterq;</s> <s xml:id="echoid-s28622" xml:space="preserve"> <lb/>ipſorũ eſt rectus:</s> <s xml:id="echoid-s28623" xml:space="preserve"> erit per 4 p 1 baſis ge æqualis baſi <lb/>l x:</s> <s xml:id="echoid-s28624" xml:space="preserve"> & ſimiliter iterata demonſtratione in trigonis d <lb/>z e & n x m:</s> <s xml:id="echoid-s28625" xml:space="preserve"> erit linea d e æqualis lineæ m x:</s> <s xml:id="echoid-s28626" xml:space="preserve"> & erit <lb/>totus angulus l x m æqualis totali angulo g e d.</s> <s xml:id="echoid-s28627" xml:space="preserve"> Fiat <lb/>quoq;</s> <s xml:id="echoid-s28628" xml:space="preserve"> ſuper punctum n terminũ lineæ l n per 23 p 1 <lb/>angulus æqualis angulo i z e:</s> <s xml:id="echoid-s28629" xml:space="preserve"> & ſit angulus l n o:</s> <s xml:id="echoid-s28630" xml:space="preserve"> <lb/>fiatq́ per 3 p 1 linea n o æqualis lineæ e z:</s> <s xml:id="echoid-s28631" xml:space="preserve"> & ducãtur <lb/>lineæ l o & m o:</s> <s xml:id="echoid-s28632" xml:space="preserve"> deſcribaturq́;</s> <s xml:id="echoid-s28633" xml:space="preserve">, ut ſuprà, circa trigo <lb/>num l o m portio circuli, quæ ſit l o m:</s> <s xml:id="echoid-s28634" xml:space="preserve"> erit quoq;</s> <s xml:id="echoid-s28635" xml:space="preserve"> ſe-<lb/>cundum præmiſſum probandi modum angulus lo <lb/>m æqualis angulo i e t.</s> <s xml:id="echoid-s28636" xml:space="preserve"> Item, ut prius, per 23 p 1 con-<lb/>ſtituatur ſuper punctum n terminum lineæ l n angulus l n p æqualis angulo a z e:</s> <s xml:id="echoid-s28637" xml:space="preserve"> & fiat linea n p <lb/>æqualis lineæ e z:</s> <s xml:id="echoid-s28638" xml:space="preserve"> & ducantur lineæ l p & p m:</s> <s xml:id="echoid-s28639" xml:space="preserve"> & cιrca trigonum l p m delcrιbatur portio circuli, ut <lb/>prius, quæ ſit l p m:</s> <s xml:id="echoid-s28640" xml:space="preserve"> erit quoq;</s> <s xml:id="echoid-s28641" xml:space="preserve"> modo præmiſſo angulus l p m æqualis angulo a e b:</s> <s xml:id="echoid-s28642" xml:space="preserve"> ducaturq́ linea à <lb/>puncto lad punctum ſectionis, ubi linea m o ſecat circumferentiã portionis circuli, quæ l x m, quæ <lb/>linea ſit l q.</s> <s xml:id="echoid-s28643" xml:space="preserve"> Et quia per 27 p 3 angulus l q m æqualis eſt angulo l x m, cadũt enim in eundem arcum, <lb/>quem chordat linea l m:</s> <s xml:id="echoid-s28644" xml:space="preserve"> angulus uerò l q m maior eſt angulo l o m per 16 p 1, patet quia angulus l x <lb/>m maior eſt angulo l o m:</s> <s xml:id="echoid-s28645" xml:space="preserve"> angulus uerò l x m æqualis eſt angulo g e d, & angulus l o m æqualis eſt <lb/>angulo i e t:</s> <s xml:id="echoid-s28646" xml:space="preserve"> palàm ergo, quoniá angulus g e d maior eſt angulo l e t.</s> <s xml:id="echoid-s28647" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s28648" xml:space="preserve"> ducta linea l r <lb/>ad punctum ſectionis, in quo linea m p ſecat arcum l o m:</s> <s xml:id="echoid-s28649" xml:space="preserve"> palàm ut prius, quoniã angulus l o m ma-<lb/>ior eſt angulo l p m:</s> <s xml:id="echoid-s28650" xml:space="preserve"> & quoniá angulus l p m eſt æqualis angulo a e b:</s> <s xml:id="echoid-s28651" xml:space="preserve"> erit angulus i e t maior.</s> <s xml:id="echoid-s28652" xml:space="preserve"> angulo <lb/>a e b:</s> <s xml:id="echoid-s28653" xml:space="preserve"> ergo per 20 huius maior apparebιt uiſui in puncto e poſito diameter g d, quàm diameterit, & <lb/>diameter i t maior diametro a b.</s> <s xml:id="echoid-s28654" xml:space="preserve"> Et quoniam de omnibus diametris cadentibus in arcum i a eadem <lb/>eſt demonſtratio, reſpectu diametri a b:</s> <s xml:id="echoid-s28655" xml:space="preserve"> patet quòd omnibus illis maior uidebitur diameter g d, & <lb/>minor uidebitur diameter a b.</s> <s xml:id="echoid-s28656" xml:space="preserve"> Omnium itaq;</s> <s xml:id="echoid-s28657" xml:space="preserve"> diametrorum cõcurrentium cum linea e z in puncto <lb/>z diameter a b uidetur minima, & g d maxima:</s> <s xml:id="echoid-s28658" xml:space="preserve"> diameter uerò media diuidẽs angulũ a z g per ęqua-<lb/>lia, modo medio uidebitur inter diametros g d & a b.</s> <s xml:id="echoid-s28659" xml:space="preserve"> Et quia per præmiſſa angulus i e t æqualis eſt <lb/>angulo s e p, palàm quia diametri i t & s p ęquales uidebuntur, quoniam ſunt à diametris g d & a b <lb/>æqualiter diſtantes, ut pater per præmiſſam & per 15 p 1.</s> <s xml:id="echoid-s28660" xml:space="preserve"> Hoc ergo eſt propoſitum.</s> <s xml:id="echoid-s28661" xml:space="preserve"/> </p> <div xml:id="echoid-div1134" type="float" level="0" n="0"> <figure xlink:label="fig-0444-02" xlink:href="fig-0444-02a"> <variables xml:id="echoid-variables465" xml:space="preserve">k e a i p g z d s b t</variables> </figure> <figure xlink:label="fig-0445-01" xlink:href="fig-0445-01a"> <variables xml:id="echoid-variables466" xml:space="preserve">p r o q x l n m</variables> </figure> </div> </div> <div xml:id="echoid-div1136" type="section" level="0" n="0"> <head xml:id="echoid-head904" xml:space="preserve" style="it">56. Silinea recta à centro circuli centro uiſus incidens, non erigatur ſuper ſuperficiem cir-<lb/>culi, ne æquales angulos contineat cum diametris, ſit́ minor diametro: diametri illius circu-<lb/>li inæquales appærebunt: totuś circulus uidebitur ſectio columnaris, cuius maxιma diameter <lb/>eſt illa, cui oblιquè incidιt linea radialis. Euclides 37. 39 th. opt.</head> <p> <s xml:id="echoid-s28662" xml:space="preserve">Eſto circulus a b g d:</s> <s xml:id="echoid-s28663" xml:space="preserve"> cuius centrum e:</s> <s xml:id="echoid-s28664" xml:space="preserve"> & ducantur duæ dιametri a g & b d ſe inuicem ad rectos <lb/>angulos ſecantes in centro e:</s> <s xml:id="echoid-s28665" xml:space="preserve"> & ducatur linea e z, quæ neque ſit erecta ſuper ſuperficiem circuli da-<lb/>ti, nec angulos æquales continens cum diametris a g & b d:</s> <s xml:id="echoid-s28666" xml:space="preserve"> & ſit minor ſemidiametro continens <lb/>angulos rectos cum diametro ga, & in æquales cum diametro d b.</s> <s xml:id="echoid-s28667" xml:space="preserve"> Dico, quòd diametri propoſiti <lb/>circuli apparebuntin æquales;</s> <s xml:id="echoid-s28668" xml:space="preserve">: & quòd totus circulus uidebitur ſectio columnaris, cuius dιame-<lb/>ter g a apparebit omnium minima, & diameter d b maxima:</s> <s xml:id="echoid-s28669" xml:space="preserve"> dιametri uerò æqualiter ab iſtis am-<lb/>babus diametris diſtantes, æquales apparebunt oculo in puncto z exiſtente, ut ſunt diametri h <lb/>p & s r.</s> <s xml:id="echoid-s28670" xml:space="preserve"> Quia enim angulus z e g eſt rectus:</s> <s xml:id="echoid-s28671" xml:space="preserve"> ducantur lineæ z g, z d, z a, z b:</s> <s xml:id="echoid-s28672" xml:space="preserve"> & ducantur ad <lb/>diametrum h p lineæ z h, z p:</s> <s xml:id="echoid-s28673" xml:space="preserve"> & ad diametrum s r lineæ z s & z r, & omnibus alijs, ut in præ-<lb/>miſſa, diſpoſitis, ſcilicet ducta linea z k ſuper diametrum g a, cui perpendiculariter incidit li-<lb/>nea z e.</s> <s xml:id="echoid-s28674" xml:space="preserve"> Per 39 itaque th.</s> <s xml:id="echoid-s28675" xml:space="preserve"> 1 huius patet, quòd angulus z e k eſt minimus omnium angulorum <lb/> <pb o="144" file="0446" n="446" rhead="VITELLONIS OPTICAE"/> illorum:</s> <s xml:id="echoid-s28676" xml:space="preserve"> & omnis angulus illi propinquior eſt minor remotiore.</s> <s xml:id="echoid-s28677" xml:space="preserve"> Quia uerò ab angulo trigoni g z a <lb/> <anchor type="figure" xlink:label="fig-0446-01a" xlink:href="fig-0446-01"/> deſcendit linea z e ad medium baſis, quæ eſt a g, per-<lb/>pendiculariter, & ab angulo trigonih z p deſcẽdit ea-<lb/>dem linea z e obliquè ad medium baſis h p:</s> <s xml:id="echoid-s28678" xml:space="preserve"> eſtq́ linea <lb/>z e minor medietate utriuſq;</s> <s xml:id="echoid-s28679" xml:space="preserve"> illarum baſium æqualiũ, <lb/>ut patet ex hypotheſi:</s> <s xml:id="echoid-s28680" xml:space="preserve"> palàm per 50 th.</s> <s xml:id="echoid-s28681" xml:space="preserve"> 1 huius, quoniã <lb/>angulus g z a eſt minor angulo h z p:</s> <s xml:id="echoid-s28682" xml:space="preserve"> item per 51 th.</s> <s xml:id="echoid-s28683" xml:space="preserve"> 1 <lb/>huius angulus h z p eſt minor angulo d z b.</s> <s xml:id="echoid-s28684" xml:space="preserve"> Similiter <lb/>quoq;</s> <s xml:id="echoid-s28685" xml:space="preserve"> de quibuſcunq;</s> <s xml:id="echoid-s28686" xml:space="preserve"> diametris medijs demonſtran-<lb/>dum.</s> <s xml:id="echoid-s28687" xml:space="preserve"> Patet ergo per 20 huius, quoniam omnium illa-<lb/>rum diametrorum a guidetur minima, & d b maxima, <lb/>& mediæ medio modo ſe habentes, ſecundum quod <lb/>plus approximant hinc & inde.</s> <s xml:id="echoid-s28688" xml:space="preserve"> Duæ quoq;</s> <s xml:id="echoid-s28689" xml:space="preserve"> diametri <lb/>æqualiter diſtantes ab extremis, uidentur ęquales per <lb/>54 huius.</s> <s xml:id="echoid-s28690" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s28691" xml:space="preserve"> Sed & ſuppoſitis ijs, <lb/>quæ per 39 th.</s> <s xml:id="echoid-s28692" xml:space="preserve"> 1 huius declarata ſunt, poteſt reliquum <lb/>aliter demonſtrari:</s> <s xml:id="echoid-s28693" xml:space="preserve"> Aſſumatur, ut in præmiſſa, linea k l <lb/>æqualis diametro g d:</s> <s xml:id="echoid-s28694" xml:space="preserve"> & diuidatur in duo æqualia in <lb/>puncto m:</s> <s xml:id="echoid-s28695" xml:space="preserve"> & producatur à puncto m perpendiculari-<lb/>ter linea m o æqualis lineæ e z:</s> <s xml:id="echoid-s28696" xml:space="preserve"> erit ergo linea m o ex hypotheſi minor ſemidiametro g e, & minor <lb/> <anchor type="figure" xlink:label="fig-0446-02a" xlink:href="fig-0446-02"/> linea k m:</s> <s xml:id="echoid-s28697" xml:space="preserve"> & ducãtur lineę k o & l o.</s> <s xml:id="echoid-s28698" xml:space="preserve"> Trigono quoq;</s> <s xml:id="echoid-s28699" xml:space="preserve"> <lb/>k o l circumſcribatur circuli portio per 5 p 4, quę ſit <lb/>k o l:</s> <s xml:id="echoid-s28700" xml:space="preserve"> eſt autem illa portio minor ſemicirculo:</s> <s xml:id="echoid-s28701" xml:space="preserve"> quia <lb/>linea m o eſt minor ſemidiametro:</s> <s xml:id="echoid-s28702" xml:space="preserve"> eritq́ per 4 & 8 <lb/>pιangulus k o l æqualis angulo g z a.</s> <s xml:id="echoid-s28703" xml:space="preserve"> Sititem per <lb/>23 p 1 angulo p e z æqualis angulus k m x:</s> <s xml:id="echoid-s28704" xml:space="preserve"> & ſit linea <lb/>x m æqualis lineæ e z:</s> <s xml:id="echoid-s28705" xml:space="preserve"> ductisq́ lineis k x & l x, cir-<lb/>cumſcribatur trigono k x l portio circuli k x l:</s> <s xml:id="echoid-s28706" xml:space="preserve"> & <lb/>erit modo præmiſſo angulus k x l æqualis angulo h <lb/>z p.</s> <s xml:id="echoid-s28707" xml:space="preserve"> Item ſit angulus k m q æqualis angulo a e z:</s> <s xml:id="echoid-s28708" xml:space="preserve"> & ſit linea m q æqualis e z:</s> <s xml:id="echoid-s28709" xml:space="preserve"> ductisq́ lineis k q & l q, <lb/>ut prius, deſcribatur portio circuli k q l:</s> <s xml:id="echoid-s28710" xml:space="preserve"> & êrit angulus k q l æqualis angulo d z b.</s> <s xml:id="echoid-s28711" xml:space="preserve"> Et quia ut in præ-<lb/>miſſa patuit, erit angulus k o l minor angulo k x l, & angulus k x l minor angulo k q l:</s> <s xml:id="echoid-s28712" xml:space="preserve"> erit angulus g <lb/>z a minorangulo h z p, & angulus h z p minor angulo d z b.</s> <s xml:id="echoid-s28713" xml:space="preserve"> Apparebit ergo diameter d b maior <lb/>quàm diameter h p, & h p maior quàm g d.</s> <s xml:id="echoid-s28714" xml:space="preserve"> Diameter uerò h p & e i æqualiter condiſtans (quæ s r) <lb/>à diametro g a, æquales apparebunt per 54 huius.</s> <s xml:id="echoid-s28715" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s28716" xml:space="preserve"/> </p> <div xml:id="echoid-div1136" type="float" level="0" n="0"> <figure xlink:label="fig-0446-01" xlink:href="fig-0446-01a"> <variables xml:id="echoid-variables467" xml:space="preserve">d s z p g k e d h b r</variables> </figure> <figure xlink:label="fig-0446-02" xlink:href="fig-0446-02a"> <variables xml:id="echoid-variables468" xml:space="preserve">o x k q m l</variables> </figure> </div> </div> <div xml:id="echoid-div1138" type="section" level="0" n="0"> <head xml:id="echoid-head905" xml:space="preserve" style="it">57. Centro uiſus exiſtente in linea erecta ſuper ſuperficiem quadrati in pũcto interſectionis <lb/>duorũ diagoniorũ: latera quadrati æqualia apparent, & diametri æquales. Euclides 59 th. opt.</head> <p> <s xml:id="echoid-s28717" xml:space="preserve">Sittetragonus a b g d:</s> <s xml:id="echoid-s28718" xml:space="preserve"> & protrahátur in ipſo diagonij a g, b d:</s> <s xml:id="echoid-s28719" xml:space="preserve"> & earum interſectio ſit e:</s> <s xml:id="echoid-s28720" xml:space="preserve"> erigatur <lb/>e z ſuper ſuperficiem tetragoni per 12 p 11:</s> <s xml:id="echoid-s28721" xml:space="preserve"> ponatur̀q́ oculus in aliquo <lb/> <anchor type="figure" xlink:label="fig-0446-03a" xlink:href="fig-0446-03"/> puncto lineæ e z, ut in z:</s> <s xml:id="echoid-s28722" xml:space="preserve"> & ducátur lineæ z a, z b, z d, z g.</s> <s xml:id="echoid-s28723" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s28724" xml:space="preserve"> per <lb/>40 th.</s> <s xml:id="echoid-s28725" xml:space="preserve"> 1 huius medietates diagoniorum inter ſe ſunt æquales, ut d e & <lb/>g e, & linea e z eſt communis duobus trigonis d z e & g z e, & anguli <lb/>circa e ſunt recti per definitionem lineæ ſuper ſuperficiem erectæ:</s> <s xml:id="echoid-s28726" xml:space="preserve"> erit <lb/>per 4 p 1 baſis z g æ qualis baſi z d, & angulus e z g ęqualis angulo e z d:</s> <s xml:id="echoid-s28727" xml:space="preserve"> <lb/>uidebitur ita q;</s> <s xml:id="echoid-s28728" xml:space="preserve"> linea d e æ qualis line æ g e per 20 huius.</s> <s xml:id="echoid-s28729" xml:space="preserve"> Et ſimiliter per <lb/>eadem, quia angulus a z e eſt ęqualis angulo b z e, uidebitur ergo linea <lb/>a e æ qualis line æ b e:</s> <s xml:id="echoid-s28730" xml:space="preserve"> tota quoq;</s> <s xml:id="echoid-s28731" xml:space="preserve"> linea d b apparebit æ qualis toti lineæ <lb/>a g.</s> <s xml:id="echoid-s28732" xml:space="preserve"> Et quoniá linea g z eſt æ qualis lineæ b z, & linea a z æqualis lineæ <lb/>d z, & linea a b eſt æ qualis ipſi g d:</s> <s xml:id="echoid-s28733" xml:space="preserve"> quoniam ſuntlatera eiuſdem qua-<lb/>drati, & ſic tria latera unius trigoni ſunt æ qualia tribus lateribus alte-<lb/>rius:</s> <s xml:id="echoid-s28734" xml:space="preserve"> ergo per 8 p 1 anguli æ qualibus lateribus contenti ſunt æ quales:</s> <s xml:id="echoid-s28735" xml:space="preserve"> <lb/>omnia itaq;</s> <s xml:id="echoid-s28736" xml:space="preserve"> latera ipſius quadrati hoc modo æ qualia apparebunt.</s> <s xml:id="echoid-s28737" xml:space="preserve"> Et <lb/>hoc eſt propoſitum:</s> <s xml:id="echoid-s28738" xml:space="preserve"> quoniam in omni puncto lineæ e z eadem eſt de-<lb/>monſtratio, concludendo ſem per per 20 huius.</s> <s xml:id="echoid-s28739" xml:space="preserve"/> </p> <div xml:id="echoid-div1138" type="float" level="0" n="0"> <figure xlink:label="fig-0446-03" xlink:href="fig-0446-03a"> <variables xml:id="echoid-variables469" xml:space="preserve">z a b e g d</variables> </figure> </div> </div> <div xml:id="echoid-div1140" type="section" level="0" n="0"> <head xml:id="echoid-head906" xml:space="preserve" style="it">58. Sirect a linea maior uel minor medietate diagonij quadrati, <lb/>à medio puncto centro uiſus incidens, obliquata ſuper eius ſuperfi-<lb/>ciem, æquales angulos contineat cum diuerſis medietatibus diago-<lb/>niorum: diagonij illius quadrati apparebunt æquales.</head> <p> <s xml:id="echoid-s28740" xml:space="preserve">Sit quadratum a b c d:</s> <s xml:id="echoid-s28741" xml:space="preserve"> cuius medius punctus inueniatur per 40 th.</s> <s xml:id="echoid-s28742" xml:space="preserve"> 1 huius, quod ſit e:</s> <s xml:id="echoid-s28743" xml:space="preserve"> & ducãtur <lb/>diagonija e b & c e d:</s> <s xml:id="echoid-s28744" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s28745" xml:space="preserve"> cẽtrum uiſus f:</s> <s xml:id="echoid-s28746" xml:space="preserve"> & linea fe ſit maior quàm linea e a medietate diagonij, uel <lb/>minor illa:</s> <s xml:id="echoid-s28747" xml:space="preserve"> ſit quoq;</s> <s xml:id="echoid-s28748" xml:space="preserve"> linea f e obliquata ſuper ſuperficiem quadrati, ſit tamen angulus f e a æqualis <lb/>angulo f e c.</s> <s xml:id="echoid-s28749" xml:space="preserve"> Dico, quòd adhuc diagonij ipſius quadrati ęquales apparebunt.</s> <s xml:id="echoid-s28750" xml:space="preserve"> Circa pũctuιn enim e <lb/>deſcribatur circulus ad quantitatem ſemidiametri e a:</s> <s xml:id="echoid-s28751" xml:space="preserve"> palàm ergo (cum omnes medietates diago-<lb/> <pb o="145" file="0447" n="447" rhead="LIBER QVARTVS."/> niorum ſint ęquales per 40 th.</s> <s xml:id="echoid-s28752" xml:space="preserve"> 1 huius) quoniam per 9 p 3 circulus iſte circũſcribetur totali quadra-<lb/> <anchor type="figure" xlink:label="fig-0447-01a" xlink:href="fig-0447-01"/> to, omnes terminos diagoniorũ attingens:</s> <s xml:id="echoid-s28753" xml:space="preserve"> erunt er-<lb/>go diagonij quadrati diametri deſcripti circuli.</s> <s xml:id="echoid-s28754" xml:space="preserve"> Sed <lb/>manifeſtum eſt per 54 huius, quoniam diametri cir-<lb/>culorum in hac diſpoſitione omnes uidẽtur ęquales:</s> <s xml:id="echoid-s28755" xml:space="preserve"> <lb/>ergo & diagonij quadrati, cum ſint eędem cũ illis.</s> <s xml:id="echoid-s28756" xml:space="preserve"> Et <lb/>hoc eſt propoſitum.</s> <s xml:id="echoid-s28757" xml:space="preserve"> Idem quoq;</s> <s xml:id="echoid-s28758" xml:space="preserve"> accidit in omnibus <lb/>figuris polygonijs cuiuſc unq;</s> <s xml:id="echoid-s28759" xml:space="preserve"> formæ:</s> <s xml:id="echoid-s28760" xml:space="preserve"> & per eadẽ ue <lb/>ſimilia demonſtrandum.</s> <s xml:id="echoid-s28761" xml:space="preserve"/> </p> <div xml:id="echoid-div1140" type="float" level="0" n="0"> <figure xlink:label="fig-0447-01" xlink:href="fig-0447-01a"> <variables xml:id="echoid-variables470" xml:space="preserve">a f c e d d b f</variables> </figure> </div> </div> <div xml:id="echoid-div1142" type="section" level="0" n="0"> <head xml:id="echoid-head907" xml:space="preserve" style="it">59. Linea recta ad punctum medium ſuperficie i <lb/>quadratæ obliquè à centro uiſus incidente, & in æ-<lb/>quales angulos cum diagonijs continente, ſiue ma-<lb/>ior ſiue minor ſemidiagonio fuerit: ſemper diago-<lb/>nij quadrati inæquales apparebunt. Euclides 61 <lb/>th. opticorum.</head> <p> <s xml:id="echoid-s28762" xml:space="preserve">Remaneat diſpoſitio proximę preęcedentis:</s> <s xml:id="echoid-s28763" xml:space="preserve"> conti <lb/>neatq́;</s> <s xml:id="echoid-s28764" xml:space="preserve"> linea ſe inęquales angulos cum diagonijs, ita <lb/>quòd angulus ſe a ſit inęqualis angulo f e c:</s> <s xml:id="echoid-s28765" xml:space="preserve"> & circunducatur circulus quadrato circa centrum e, ut <lb/>prius:</s> <s xml:id="echoid-s28766" xml:space="preserve"> & ſi linea fe fuerit maior ſemidiagonio a e, concludetur per 55 huius diametros circuli (qui <lb/>ſunt diagonij propoſiti quadrati) inęquales uideri.</s> <s xml:id="echoid-s28767" xml:space="preserve"> Quòd ſi linea fe fuerit minor ſemidiagonio a e:</s> <s xml:id="echoid-s28768" xml:space="preserve"> <lb/>tunc ſimiliter per 56 huius cõuincetur diagonios quadrati inęquales uideri.</s> <s xml:id="echoid-s28769" xml:space="preserve"> Diuerſitas tamẽ iſtarũ <lb/>inęqualitatum fit ſecundum modum illic in circulis propoſitum, ſecundum diuerfitatem angulorũ <lb/>incidentię hinc inde.</s> <s xml:id="echoid-s28770" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s28771" xml:space="preserve"> Et eodem modo poteſt de alijs figuris, ut de quadran-<lb/>gulo altera parte longiore, & de hexagonis, octogonis, & uniuerſaliter de omnibus polygonijs pa-<lb/>rium angulorum faciliter demonſtrari, quòd ipſorum diagonij quandoq;</s> <s xml:id="echoid-s28772" xml:space="preserve"> ęquales uidentur, & quan <lb/>doq;</s> <s xml:id="echoid-s28773" xml:space="preserve"> inæquales:</s> <s xml:id="echoid-s28774" xml:space="preserve"> nec in talibus duximus immorandum, quia quilibet huius ſcientiæ perſcrutator <lb/>hoc faciliter comprehendet.</s> <s xml:id="echoid-s28775" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1143" type="section" level="0" n="0"> <head xml:id="echoid-head908" xml:space="preserve" style="it">60. Centro for aminis uueæ in puncto medio ſuperficiei cuiuſcun figuræ recti lineæ exiſtente, <lb/>ſemper figur a ſecundum ſui formam propriam uiſui occurret.</head> <p> <s xml:id="echoid-s28776" xml:space="preserve">Verbi gratia ſit figura data, exempli cauſſa, quadrata:</s> <s xml:id="echoid-s28777" xml:space="preserve"> & inueniatur pũctus medius per 40 th.</s> <s xml:id="echoid-s28778" xml:space="preserve"> 1 hu <lb/>ius, in quo ponatur centrum foraminis uueę:</s> <s xml:id="echoid-s28779" xml:space="preserve"> & hoc eſt, ut ſuperponatur oculus illi puncto.</s> <s xml:id="echoid-s28780" xml:space="preserve"> Et quo-<lb/>niam ab illo puncto ad omnem punctum laterum & angulorum poſſunt duci lineę ęquales uel pro-<lb/>portionales ijs, quę in ipſa ſuperficie:</s> <s xml:id="echoid-s28781" xml:space="preserve"> patet, quòd forma cuius libetillorum punctorum uidebitur:</s> <s xml:id="echoid-s28782" xml:space="preserve"> & <lb/>propter ęqualitatem linearũ radialium ad eas, quę in ſuperficie, lineas, figurabitur figura in oculi ſu <lb/>perficie, ſicut eſt extrà in ſuperficie rei uiſę.</s> <s xml:id="echoid-s28783" xml:space="preserve"> Patet ergo, quòd totalis forma & figura illius ſuperficiei <lb/>uidebitur, ſicut eſt propria illi figuratio, cuiuſcunq;</s> <s xml:id="echoid-s28784" xml:space="preserve"> ſit figurę.</s> <s xml:id="echoid-s28785" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s28786" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1144" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables471" xml:space="preserve">e b a d c</variables> </figure> <head xml:id="echoid-head909" xml:space="preserve" style="it">61. Figura quadr at a uno ſolo latere directè uiſui oppoſito, è di-<lb/> ſtantia uiſa alter a parte longior uidetur.</head> <p> <s xml:id="echoid-s28787" xml:space="preserve">Sit enim figura quadrata a b c d:</s> <s xml:id="echoid-s28788" xml:space="preserve"> & centrum uiſus e:</s> <s xml:id="echoid-s28789" xml:space="preserve"> & latus qua-<lb/>drati, quod ſit a b, opponatur uiſui directè:</s> <s xml:id="echoid-s28790" xml:space="preserve"> palàm ergo, quoniam alia <lb/>uiſui opponentur obliquè:</s> <s xml:id="echoid-s28791" xml:space="preserve"> fed per 26 huius quantitas obliquè uiſui <lb/>oppoſita uidetur minor, quoniam ſub minori angulo uidetur:</s> <s xml:id="echoid-s28792" xml:space="preserve"> dire-<lb/>ctè uerò uiſui oppoſita uidetur ſuę proprię quãtitatis, quàm obliquè <lb/>uiſa:</s> <s xml:id="echoid-s28793" xml:space="preserve"> ſub maiori enim angulo uidentur omnia directè uiſibus oppoſi <lb/>ta, ꝗ̃ ſibi æqualia, quæ opponuntur uiſibus obliquè.</s> <s xml:id="echoid-s28794" xml:space="preserve"> Tota ergo figura <lb/>quadrata uidebitur altera parte longior.</s> <s xml:id="echoid-s28795" xml:space="preserve"> Superficies uerò quadrata <lb/>è diſtantia uiſa altera parte longior uidetur, ut proponitur:</s> <s xml:id="echoid-s28796" xml:space="preserve"> ſed & eſt <lb/>poſsibile, altera parte longior appareat uiſui eſſe quadrata, ut ſi latus <lb/>eius breuius directè opponatur uiſui & lõgius obliquè:</s> <s xml:id="echoid-s28797" xml:space="preserve"> tũc enim po <lb/>teſt fieri propter diſpoſitionẽ obliquitatis, ut longius latus appareat <lb/>æquale breuiori.</s> <s xml:id="echoid-s28798" xml:space="preserve"> Multa quoq;</s> <s xml:id="echoid-s28799" xml:space="preserve"> ſimilia accidunt ex hac radice, utpote <lb/>irregularitas in quibuslibet polygonijs figuris æquilateris & ęquian <lb/>gulis.</s> <s xml:id="echoid-s28800" xml:space="preserve"> In alijs quoq;</s> <s xml:id="echoid-s28801" xml:space="preserve"> accidit ſuę formę diuerſitas in uiſiõe, quę omnia <lb/>relinquimus diligentię particulariter perquirentis:</s> <s xml:id="echoid-s28802" xml:space="preserve"> ſufficit enim no-<lb/>bis hoc uniuerſaliter propoſitum in radice.</s> <s xml:id="echoid-s28803" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1145" type="section" level="0" n="0"> <head xml:id="echoid-head910" xml:space="preserve" style="it">62. Si quadr atum, cuius latus non ſit excedens diſtantiam oculorum, uiſibus propius appo-<lb/>natur: uidebitur alter a parte longius: & latera uiſibus obuiantia ex parte uiſuum concurre-<lb/>re uidebuntur.</head> <p> <s xml:id="echoid-s28804" xml:space="preserve">Sit quadratum a b c d, utin præmiſſa, cuius latus a b non ſit excedens quantitatem lineæ conne-<lb/>ctentis centra oculorum, hoc eſt diſtantiam oculorum:</s> <s xml:id="echoid-s28805" xml:space="preserve"> & applicetur uiſibus, ut propius poteſt, ſe-<lb/> <pb o="146" file="0448" n="448" rhead="VITELLONIS OPTICAE"/> cundum latus ſuũ a b:</s> <s xml:id="echoid-s28806" xml:space="preserve"> dico, quòd uidebitur altera parte longius.</s> <s xml:id="echoid-s28807" xml:space="preserve"> Latera enim eius duo, ſcilicet a c & <lb/>b d directè ſubij ciuntur uiſui, quoniam quo dlibet illorum laterum imaginatum extendi ſecundum <lb/>ſuum continuum & directum, penetrat centrum uiſus, cui directè ſubijcitur:</s> <s xml:id="echoid-s28808" xml:space="preserve"> & ſic forma eius dire-<lb/>ctè depingitur in ſuperficie ipſius uiſus, & latus c d directè opp onitur uiſui:</s> <s xml:id="echoid-s28809" xml:space="preserve"> uidebũtur ergo illa ſuæ <lb/>proprię quantitatis per 26 huius:</s> <s xml:id="echoid-s28810" xml:space="preserve"> latus uerò a b uidetur obliquè, quoniam cadit intra axes uiſuales, <lb/>nec ſuper ipſum erigitur aliquis axium uiſualium:</s> <s xml:id="echoid-s28811" xml:space="preserve"> uidetur ergo minus per eandem 26 huius.</s> <s xml:id="echoid-s28812" xml:space="preserve"> Totũ <lb/>ergo quadratum a b c d uidetur altera parte longius, & lineæ c a & d b, quę ſunt latera illius quadra-<lb/>ti uiſibus obuiantia, uidebuntur plus diſtare ſecundum lineam c d, quã ſecundum lineam a b:</s> <s xml:id="echoid-s28813" xml:space="preserve"> uiden <lb/>tur ergo concurrere uerſus partem uiſus.</s> <s xml:id="echoid-s28814" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s28815" xml:space="preserve"> Et eadem paſsio accidit figurę qua-<lb/>drangulæ altera parte longiori, nec eſt differentia quò ad illam:</s> <s xml:id="echoid-s28816" xml:space="preserve"> quod etiam per eadem poteſt demõ <lb/>ſtrari.</s> <s xml:id="echoid-s28817" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s28818" xml:space="preserve"> Et quoniam figura corporalis quæ dam figura eſt, licet uiſio corporei-<lb/>tatis ſit alia â uiſione figurę, quomodo uirtuti diſtinctiuæ error in uiſione figuræ accidat, duximus <lb/>in poſterius differendum.</s> <s xml:id="echoid-s28819" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1146" type="section" level="0" n="0"> <head xml:id="echoid-head911" xml:space="preserve" style="it">63. Corporeitas comprehenditur à uiſu, in quibuſdam corporibus per ſe, & in quibuſdã auxi-<lb/>lio uirtutis iudicatiuæ. Alhazen 31 n 2.</head> <p> <s xml:id="echoid-s28820" xml:space="preserve">Cum enim corporeitas ſit extenſio corporis ſecundum trinã dimẽſionem:</s> <s xml:id="echoid-s28821" xml:space="preserve"> dico, quòd ipſa quan-<lb/>doq;</s> <s xml:id="echoid-s28822" xml:space="preserve"> comprehenditur in quibuſdam corporibus à uiſu per ſe:</s> <s xml:id="echoid-s28823" xml:space="preserve"> quæ dam enim corpora continentur <lb/>à ſuperficiebus planis ſecantibus ſe rectè uel obliquè adinuicem:</s> <s xml:id="echoid-s28824" xml:space="preserve"> & quædam à ſuperficieb.</s> <s xml:id="echoid-s28825" xml:space="preserve"> cõcauis <lb/>& conuexis:</s> <s xml:id="echoid-s28826" xml:space="preserve"> & quædam à ſuperficie bus cõuexis & planis:</s> <s xml:id="echoid-s28827" xml:space="preserve"> & quędã à ſuperficieb.</s> <s xml:id="echoid-s28828" xml:space="preserve"> concauis & pla-<lb/>nis:</s> <s xml:id="echoid-s28829" xml:space="preserve"> & quędam à diuerſis ſuperficiebus conuexis, cocauis & planis ſe interſecantibus:</s> <s xml:id="echoid-s28830" xml:space="preserve"> & quæ dã cõ-<lb/>tinentur ab una ſola ſuperficie rotunda.</s> <s xml:id="echoid-s28831" xml:space="preserve"> Corpus itaq;</s> <s xml:id="echoid-s28832" xml:space="preserve"> contentum à ſuperficiebus ſecantib.</s> <s xml:id="echoid-s28833" xml:space="preserve"> ſe, cuius <lb/>una ſuperficies eſt plana:</s> <s xml:id="echoid-s28834" xml:space="preserve"> quando ſuperficies eius fuerit oppoſita uiſui ſecundum directã oppoſitio <lb/>nem ſiue obliquatam, ita tamen, quòd cõmunis ſectio duarum ſuperficierum uideatur, & quòd am-<lb/>bę ſuperficies ſe ſecantes occurrant ſimul uiſui:</s> <s xml:id="echoid-s28835" xml:space="preserve"> tunc extenſio corporis ſecundum longitudinem & <lb/>latitudinem, & ſecundum proſunditatẽ à uiſu comprehendetur.</s> <s xml:id="echoid-s28836" xml:space="preserve"> Sic ergo corporeitas comprehen-<lb/>detur.</s> <s xml:id="echoid-s28837" xml:space="preserve"> Corpora quoq;</s> <s xml:id="echoid-s28838" xml:space="preserve">, quorum ſuperficies eſt conuexa, ſiue ſit una, ſiue multæ, cum opponuntur ui <lb/>ſui ſecundum directionem uel obliquationem, erunt remotiores partiũ eius à uiſu inæ quales, & e-<lb/>rit mediũ cõuexi eius propin quius extremitatibus uiſus per 8 p 3:</s> <s xml:id="echoid-s28839" xml:space="preserve"> reliquę uerò partes eius erunt à <lb/>uiſu remotiores, qua comprehenſione ſentiet uiſus corporeitatem:</s> <s xml:id="echoid-s28840" xml:space="preserve"> quoniam cõprehendet profun-<lb/>ditatem partium plus remotarum à ſe reſpectu partium propinquiorum ſibi:</s> <s xml:id="echoid-s28841" xml:space="preserve"> & cum hoc comprehẽ <lb/>det longitudinem & latitudinem dimenſionum illorum corporum.</s> <s xml:id="echoid-s28842" xml:space="preserve"> Corporis quoq;</s> <s xml:id="echoid-s28843" xml:space="preserve"> concaui conca <lb/>uitas percipi poteſt à uiſu ſecun dum mediocrem diſtantiam:</s> <s xml:id="echoid-s28844" xml:space="preserve"> tunc enim, quia medium eius maxime <lb/>elõgatur à uiſu per 8 p 3, ut prius:</s> <s xml:id="echoid-s28845" xml:space="preserve"> profunditas illius corporis cõprehẽditur à uiſu propter maiorem <lb/>diſtantiam unius partis reſpectu aliarum:</s> <s xml:id="echoid-s28846" xml:space="preserve"> ſed ex conſequenti lõgitudo & latitudo patent.</s> <s xml:id="echoid-s28847" xml:space="preserve"> Quòd ſi <lb/>plures ſunt in ipſo ſuperficies ſe ſecãtes, quarũ communes ſectiones ſe ad uiſum offerant, corporei-<lb/>tas ipſorum cõprehenditur à uiſu cum ſentitur obliquitas illarum ſuperficierum.</s> <s xml:id="echoid-s28848" xml:space="preserve"> In ijs aũt omnib.</s> <s xml:id="echoid-s28849" xml:space="preserve"> <lb/>attendenda eſt mediocritas diſtantię, quoniam in maximis remotionib.</s> <s xml:id="echoid-s28850" xml:space="preserve"> eſt ſecus:</s> <s xml:id="echoid-s28851" xml:space="preserve"> tunc enim per ui-<lb/>ſum nudum non comprehenditur corpus propter uiſionem ſuperficiei, ſed auxilio uirtutis animæ <lb/>ſuperioris:</s> <s xml:id="echoid-s28852" xml:space="preserve"> eſt enim principium quieſcens in anima ex conſuetudine uiſionum:</s> <s xml:id="echoid-s28853" xml:space="preserve"> & eſt tale, quòd ni-<lb/>hil uidetur niſi corpus.</s> <s xml:id="echoid-s28854" xml:space="preserve"> Vnde quando uiſus uidet aliquam uiſibilem ſuperficiem, ſtatim uirtus iudi-<lb/>catiua animæ dicet, quòd uidens uidet corpus, quamuis non comprehendat uiſus extenſionem e-<lb/>ius in profundum.</s> <s xml:id="echoid-s28855" xml:space="preserve"> Nam latitudinem & longitudinem per ſe comprehendet uiſus per comprehen-<lb/>ſionem ſuperficiei cuiuſcunque per 17 th.</s> <s xml:id="echoid-s28856" xml:space="preserve"> 3 huius:</s> <s xml:id="echoid-s28857" xml:space="preserve"> non autem comprehendet ſemper corporum <lb/>profunditatem, quę eſt tertia dimenſio ipſorum, niſi auxilio uirtutis ſuperioris ipſius animę.</s> <s xml:id="echoid-s28858" xml:space="preserve"> Patet <lb/>ergo propoſitum.</s> <s xml:id="echoid-s28859" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1147" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables472" xml:space="preserve">g a d e b z</variables> </figure> <head xml:id="echoid-head912" xml:space="preserve" style="it">64. Longior linea ab aliquo puncto ſuperficiei conuexæ <lb/> ſphæricæ ad uiſum accedens, eſt linea contingens cir culum magnum illius ſphæræ.</head> <p> <s xml:id="echoid-s28860" xml:space="preserve">Eſto data ſphæra d g:</s> <s xml:id="echoid-s28861" xml:space="preserve"> cuius centrum ſit a:</s> <s xml:id="echoid-s28862" xml:space="preserve"> circulus eius ma <lb/>gnus d g e b:</s> <s xml:id="echoid-s28863" xml:space="preserve"> quę ſphęra ſit uiſa ab oculo, cuius centrũ ſit pũ-<lb/>ctum z:</s> <s xml:id="echoid-s28864" xml:space="preserve"> & ſuper lineam diſtantiæ centri ſphęrę, quod eſt a, & <lb/>centri oculi, quod eſt z, poſitam pro diametro, quæ ſit a z, fi-<lb/>guretur circulus a b e z:</s> <s xml:id="echoid-s28865" xml:space="preserve"> & ducantur ad ſectiones circulorũ <lb/>iſtorum lineę z b & z e.</s> <s xml:id="echoid-s28866" xml:space="preserve"> Dico, quòd hę lineę cõtingunt circu-<lb/>lum d g e b, qui eſt circulus magnus ꝓpoſitę ſphęrę:</s> <s xml:id="echoid-s28867" xml:space="preserve"> & quòd <lb/>ipſæ ſunt lõgiores omnibus alijs lineis ducibilib.</s> <s xml:id="echoid-s28868" xml:space="preserve"> à quibuſ-<lb/>cunq;</s> <s xml:id="echoid-s28869" xml:space="preserve"> punctis ſuperficiei ſphærę ad centrum uiſus.</s> <s xml:id="echoid-s28870" xml:space="preserve"> Ducan-<lb/>tur enim à centro ſphærę, quod eſta, duę lineę ad terminos <lb/>linearum z e & z b, quę facient cum eis angulos rectos:</s> <s xml:id="echoid-s28871" xml:space="preserve"> fient <lb/>enim anguli a e z & a b z recti per 31 p 3, quia uterq;</s> <s xml:id="echoid-s28872" xml:space="preserve"> illorum <lb/>cadit in ſemicirculo:</s> <s xml:id="echoid-s28873" xml:space="preserve"> ergo per 16 p 3 illæ duę lineę z e & z b <lb/>ſunt contingentes circulum d g e b:</s> <s xml:id="echoid-s28874" xml:space="preserve"> protractæ ergo circulũ <lb/> <pb o="147" file="0449" n="449" rhead="LIBER QVARTVS."/> non ſecabunt.</s> <s xml:id="echoid-s28875" xml:space="preserve"> Si uerò dicatur, quòd illę cõtingentes nõ ſunt longiſsimę, quę perueniunt à punctis <lb/>ſuperficiei ſphærę uiſę ad centrum uiſus z:</s> <s xml:id="echoid-s28876" xml:space="preserve"> ſint alię longiores.</s> <s xml:id="echoid-s28877" xml:space="preserve"> Et quia, ut patet ex præmiſsis, ſi linea <lb/>z b protrahatur, ipſa non ſecabit circulum, quem contingit per 16 p 3:</s> <s xml:id="echoid-s28878" xml:space="preserve"> ergo ſi à pũcto z centro uiſus <lb/>in ſuperficie, in qua ſunt lineę z e & z b, protrahatur linea longior quã ſit linea z b uſq;</s> <s xml:id="echoid-s28879" xml:space="preserve"> ad circulum:</s> <s xml:id="echoid-s28880" xml:space="preserve"> <lb/>palàm ergo, quia iſta recta cum linea z b ſup erficiem includet:</s> <s xml:id="echoid-s28881" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s28882" xml:space="preserve"> Illæ ergo duæ <lb/>lineę contingentes circulum, ſunt omnibus alijs lineis longiores.</s> <s xml:id="echoid-s28883" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s28884" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1148" type="section" level="0" n="0"> <head xml:id="echoid-head913" xml:space="preserve" style="it">65. Sphæræ à remotiſsimo uiſæ ſuperficies cõuexa uel cõcaua uidetur plana. Euclides 25 th. opt.</head> <p> <s xml:id="echoid-s28885" xml:space="preserve">Sit ſphæra, cuius centrum ſit a:</s> <s xml:id="echoid-s28886" xml:space="preserve"> & in ea circulus magnus b c d:</s> <s xml:id="echoid-s28887" xml:space="preserve"> & ſit centrum uiſus e:</s> <s xml:id="echoid-s28888" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s28889" xml:space="preserve"> li <lb/>neę e a, e b, e c, e d:</s> <s xml:id="echoid-s28890" xml:space="preserve"> palamq́;</s> <s xml:id="echoid-s28891" xml:space="preserve"> per 50 huius, quoniam forma arcus b c d ipſi <lb/> <anchor type="figure" xlink:label="fig-0449-01a" xlink:href="fig-0449-01"/> uiſui e à remotiori incidentiæ arcus b c d, accedit ad rectitudinem:</s> <s xml:id="echoid-s28892" xml:space="preserve"> & i-<lb/>dem eſt de alijs arcubus quibuſcunq;</s> <s xml:id="echoid-s28893" xml:space="preserve"> uiſus incidit in tota data ſphæra.</s> <s xml:id="echoid-s28894" xml:space="preserve"> <lb/>Totalis ergo portio conuexæ ſuperficiei ſphęrę, cui uiſus incidit, uide-<lb/>tur plana:</s> <s xml:id="echoid-s28895" xml:space="preserve"> & ſicut arcus circulorum in ſuperficie ipſius deſcriptibilium <lb/>accedunt ad rectitudinẽ linearum, ſic totalis ſphærę ſuperficies ad pla-<lb/>niciem accedit.</s> <s xml:id="echoid-s28896" xml:space="preserve"> Et per eadem poteſt fieri dem onſtratio de concaua ſu-<lb/>perficie ipſius ſphærę.</s> <s xml:id="echoid-s28897" xml:space="preserve"> Cum enim nulla partium rei uiſę plus altera di-<lb/>ſtare uidetur, neceſſe eſt unius diſpoſitionis apparere totam ſuperficiẽ <lb/>rei uiſæ.</s> <s xml:id="echoid-s28898" xml:space="preserve"> Cum itaque totum conuexum corpus uel concauum in remo <lb/>tione maxima ſuerit à uiſu:</s> <s xml:id="echoid-s28899" xml:space="preserve"> tunc uiſus non comprehend det concauitatẽ <lb/>uel conuexitatem, ſed comprehendet ipſum quaſi planũ:</s> <s xml:id="echoid-s28900" xml:space="preserve"> quia ſitus par <lb/>tium ſuperficiei ſuę adinuicem nõ comprehẽduntur à uiſu in aliqua di-<lb/>uerſitate, ſed ſecundum continuitatem ęqualem perueniunt ad uiſum, <lb/>& in ipſius uiſus ſuperficie ſecundum diuerſitatẽ ſitus figurantur:</s> <s xml:id="echoid-s28901" xml:space="preserve"> unde <lb/>plana iudicantur, & plana uidebitur totalis ſuperficies rei uiſæ.</s> <s xml:id="echoid-s28902" xml:space="preserve"> Et o b <lb/>hoc figuræ ſuperficierum ſolis & lunę uidentur planæ:</s> <s xml:id="echoid-s28903" xml:space="preserve"> ſemidiametri e-<lb/>nim ipſorum ad lineam ſuæ diſtantię, quę à centro uiſus ad ipſorum ſo-<lb/>lis & lunę centra ducitur, non habet aliquã ſenſibilem proportionẽ:</s> <s xml:id="echoid-s28904" xml:space="preserve"> un <lb/>de nihil aufert à quantitate lineę à centro uiſus productę contingente <lb/>ſphæras illas per præmiſſam.</s> <s xml:id="echoid-s28905" xml:space="preserve"> Longior enim linea ab aliquo puncto ſuperficiei conuexę ipſius ſphę <lb/>rę ad uiſum accedens, eſt linea circulum magnum illius ſphęrę contingens:</s> <s xml:id="echoid-s28906" xml:space="preserve"> & illæ lineę omnes ſunt <lb/>ęquales inter ſe per 58 th.</s> <s xml:id="echoid-s28907" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s28908" xml:space="preserve"> Et quoniam ſenſibiliter non excedunt lineam à centro uiſus ſuper <lb/>ſuperficies illaram ſphęrarum productas:</s> <s xml:id="echoid-s28909" xml:space="preserve"> ideo omnes illæ lineę uidentur quaſi ęquales ipſis perpẽ-<lb/>dicularibus, quę tranſeunt centra illorum corporum à centro uiſus productę, & arcus interiacentes <lb/>rectitudini accedunt:</s> <s xml:id="echoid-s28910" xml:space="preserve"> unde totales ſuperficies uidẽtur planę.</s> <s xml:id="echoid-s28911" xml:space="preserve"> Et hoc idem propter eandem cauſſam <lb/>accidit in omnibus alijs ſtellis, quę propter remotionem maximã quaſi quędam ſuperficies paruo-<lb/>rum circulorum uidentur.</s> <s xml:id="echoid-s28912" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s28913" xml:space="preserve"/> </p> <div xml:id="echoid-div1148" type="float" level="0" n="0"> <figure xlink:label="fig-0449-01" xlink:href="fig-0449-01a"> <variables xml:id="echoid-variables473" xml:space="preserve">e c d b a</variables> </figure> </div> </div> <div xml:id="echoid-div1150" type="section" level="0" n="0"> <head xml:id="echoid-head914" xml:space="preserve" style="it">66. Sphæricæ ſuperficiei conuxæ illuminatæ uno oculo uiſæ, ſemper minus hemiſphærio appa <lb/>ret: & pars eius uiſa circulo continetur. Euclides 23 th. opt.</head> <p> <s xml:id="echoid-s28914" xml:space="preserve">Sit ſphærę uiſę centrũ a:</s> <s xml:id="echoid-s28915" xml:space="preserve"> & ſit centrum uiſus b:</s> <s xml:id="echoid-s28916" xml:space="preserve"> producaturq́;</s> <s xml:id="echoid-s28917" xml:space="preserve"> linea a b:</s> <s xml:id="echoid-s28918" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s28919" xml:space="preserve">, ut ſuperficies plana <lb/>tranſiens punctum b, ſecet ſphęram:</s> <s xml:id="echoid-s28920" xml:space="preserve"> erit ergo per 69 th.</s> <s xml:id="echoid-s28921" xml:space="preserve"> 1 huius <lb/>communis ſectio illius ſuperficiei & ſphærę circulus:</s> <s xml:id="echoid-s28922" xml:space="preserve"> ſit ille cir-<lb/> <anchor type="figure" xlink:label="fig-0449-02a" xlink:href="fig-0449-02"/> culus g d:</s> <s xml:id="echoid-s28923" xml:space="preserve"> & ſuper diametrum a b, quæ interiacet centrum uiſus <lb/>& centrum ſphærę uiſæ, deſcribatur circulus, qui ſit a g d b:</s> <s xml:id="echoid-s28924" xml:space="preserve"> & <lb/>producãtur lineæ g b, d b, a g, a d.</s> <s xml:id="echoid-s28925" xml:space="preserve"> Quia ergo arcus a g b eſt ſemi-<lb/>circulus, palàm per 31 p 3, quia angulus a g b eſt rectus:</s> <s xml:id="echoid-s28926" xml:space="preserve"> ſimiliter <lb/>autem & angulus a d b eſt rectus:</s> <s xml:id="echoid-s28927" xml:space="preserve"> ergo lineæ b g & b d ſunt con-<lb/>tingentes cιrculum per 16 p 3.</s> <s xml:id="echoid-s28928" xml:space="preserve"> Copuletur itaq;</s> <s xml:id="echoid-s28929" xml:space="preserve"> linea g d ducta ք <lb/>puncta contactuum, quã ſecabit linea b a per æ qualia per 58 th.</s> <s xml:id="echoid-s28930" xml:space="preserve"> <lb/>1 huius:</s> <s xml:id="echoid-s28931" xml:space="preserve"> ſit ergo punctus ſectionis k:</s> <s xml:id="echoid-s28932" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s28933" xml:space="preserve"> per 4 p 1 trigona g <lb/>k b & d k b æquiangula:</s> <s xml:id="echoid-s28934" xml:space="preserve"> patet & hoc per 3 p 3.</s> <s xml:id="echoid-s28935" xml:space="preserve"> Ducatur quoque <lb/>per centrum a linea it æquidiſtanter lineæ g d per 31 p 1:</s> <s xml:id="echoid-s28936" xml:space="preserve"> erit er-<lb/>go per 29 p 1 linea a b perpendicularis ſuper lineam it, cum ipſa <lb/>ſit perpendicularis ſuper lineam g d ęquidiſtantem lineę it:</s> <s xml:id="echoid-s28937" xml:space="preserve"> er <lb/>go per 16 p 3 erit linea i a contingens circulum a g b d:</s> <s xml:id="echoid-s28938" xml:space="preserve"> & ipſa eſt <lb/>diameter circuli d g:</s> <s xml:id="echoid-s28939" xml:space="preserve"> arcus ergo d g, qui uidetur, minor eſt ſemi-<lb/>circulo, prout etiam patet per 51 huius.</s> <s xml:id="echoid-s28940" xml:space="preserve"> Trigonus itaq;</s> <s xml:id="echoid-s28941" xml:space="preserve"> b g k, ma <lb/>nente fixo latere b k, intelligatur circũduci, quouſq;</s> <s xml:id="echoid-s28942" xml:space="preserve"> redeat ad <lb/>locum unde cœpit:</s> <s xml:id="echoid-s28943" xml:space="preserve"> & palàm, quoniam linea b g contingens circulum d g, unumquodq;</s> <s xml:id="echoid-s28944" xml:space="preserve"> punctũ ſu-<lb/>perficiei ſphęrę, cui ipſa circũducitur, continget, & linea k g motu ſuo faciet circuli ſectionem, fietq́;</s> <s xml:id="echoid-s28945" xml:space="preserve"> <lb/>pyramis, cuius uertex erit punctum b, quod eſt centrum uiſus, baſisq́;</s> <s xml:id="echoid-s28946" xml:space="preserve"> eius erit circulus per motum <lb/>lineę k g factus:</s> <s xml:id="echoid-s28947" xml:space="preserve"> pars ergo uiſa ſub circulo continetur.</s> <s xml:id="echoid-s28948" xml:space="preserve"> Palàm quo que, quoniam uidetur minus hemi <lb/>ſphærio:</s> <s xml:id="echoid-s28949" xml:space="preserve"> eſt enim, ut præmiſſum eſt, ſphæræ uiſæ diameterit, & linea g d illi ęquidiſtans minor dia-<lb/> <pb o="148" file="0450" n="450" rhead="VITELLONIS OPTICAE"/> metro:</s> <s xml:id="echoid-s28950" xml:space="preserve"> eſt autem linea g d diameter baſis pyramidis uiſionis:</s> <s xml:id="echoid-s28951" xml:space="preserve"> minus ergo hemiſphærio uidetur.</s> <s xml:id="echoid-s28952" xml:space="preserve"> <lb/>Quod eſt prop oſitum.</s> <s xml:id="echoid-s28953" xml:space="preserve"/> </p> <div xml:id="echoid-div1150" type="float" level="0" n="0"> <figure xlink:label="fig-0449-02" xlink:href="fig-0449-02a"> <variables xml:id="echoid-variables474" xml:space="preserve">i a t g k d b</variables> </figure> </div> </div> <div xml:id="echoid-div1152" type="section" level="0" n="0"> <head xml:id="echoid-head915" xml:space="preserve" style="it">67. Viſu ſphæræ illuminatæ conuexæ approximante, minus ſuperficiei ſphæræ uidetur: appa-<lb/>ret autem quaſi magis uideatur. Euclides 24 th. opt.</head> <p> <s xml:id="echoid-s28954" xml:space="preserve">Eſto, ut in præ miſſa, ſphæra, cuius centrum a:</s> <s xml:id="echoid-s28955" xml:space="preserve"> ſit quoq;</s> <s xml:id="echoid-s28956" xml:space="preserve"> centrum uiſus b:</s> <s xml:id="echoid-s28957" xml:space="preserve"> & ducatur linea a b:</s> <s xml:id="echoid-s28958" xml:space="preserve"> & cir <lb/>ca diam etrum a b deſcribatur circulus g b d:</s> <s xml:id="echoid-s28959" xml:space="preserve"> & ducatur à pũcto <lb/> <anchor type="figure" xlink:label="fig-0450-01a" xlink:href="fig-0450-01"/> a linea e a z perpendiculariter ſuper lineam a b per 11 p 1.</s> <s xml:id="echoid-s28960" xml:space="preserve"> Et quia <lb/>lineæ a b & e z ſunt in una ſuperficie per 2 p 11:</s> <s xml:id="echoid-s28961" xml:space="preserve"> intelligatur hæc ſu <lb/>perficies plana ſecare ſphæram:</s> <s xml:id="echoid-s28962" xml:space="preserve"> ipſa autem per 69 th.</s> <s xml:id="echoid-s28963" xml:space="preserve"> 1 huius ſe-<lb/>cabit ſphæram ſecundum circulum, qui ſit g e z d:</s> <s xml:id="echoid-s28964" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s28965" xml:space="preserve"> puncta <lb/>ſectionis duorum propoſitorum circulorum, quę g & d:</s> <s xml:id="echoid-s28966" xml:space="preserve"> & ducan <lb/>tur lineæ g a, d a, b g, b d:</s> <s xml:id="echoid-s28967" xml:space="preserve"> & patet per modum proximæ præcedẽ, <lb/>tis, quoniam lineę b g & b d contingunt ſphæram, & uidetur ab <lb/>oculo exiſtente in puncto b pars ſphæræ g d.</s> <s xml:id="echoid-s28968" xml:space="preserve"> Sit ergo, ut appro-<lb/>pinquet oculus ſphęrę, & fiat in pũcto c:</s> <s xml:id="echoid-s28969" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s28970" xml:space="preserve"> c a, circa quã, <lb/>ut diametrum, deſcribatur circulus a k c l:</s> <s xml:id="echoid-s28971" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s28972" xml:space="preserve"> lineæ c k, <lb/>c l, a k, a l:</s> <s xml:id="echoid-s28973" xml:space="preserve"> ergo ք pręmiſſam uidebitur ab oculo exiſtẽte in pũcto <lb/>c, pars ſphærę, quę eſt k l, quæ minor eſt parte ſphærę g d uiſæ ab <lb/>oculo exiſtente in puncto b:</s> <s xml:id="echoid-s28974" xml:space="preserve"> quoniã arcus cadẽs inter puncta cõ <lb/>tingentię linearum c k & c l, quę per 64 huius contingunt ſphę.</s> <s xml:id="echoid-s28975" xml:space="preserve"> <lb/>ram, minor eſt arcu g d, qui cadit inter puncta contingentiæ li-<lb/>nearũ b g & b d:</s> <s xml:id="echoid-s28976" xml:space="preserve"> quod patet per 60 th.</s> <s xml:id="echoid-s28977" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s28978" xml:space="preserve"> Palàm ergo quo-<lb/>niam appropinquante oculo ipſi ſphęrę, minus ſuperficiei ſphę <lb/>ricę uidetur.</s> <s xml:id="echoid-s28979" xml:space="preserve"> Quia uerò, ut patet per 60 th.</s> <s xml:id="echoid-s28980" xml:space="preserve"> 1 huius, lineę g b & c k concurrunt, ſi producantur uerſus <lb/>punctum g:</s> <s xml:id="echoid-s28981" xml:space="preserve"> palàm per 16 p 1, quoniam angulus k c a maior eſt angulo g b a:</s> <s xml:id="echoid-s28982" xml:space="preserve"> ſimiliter angulus a clma <lb/>ior eſt angulo a b d:</s> <s xml:id="echoid-s28983" xml:space="preserve"> totus ergo angulus k c l eſt maior toto angulo g b d.</s> <s xml:id="echoid-s28984" xml:space="preserve"> Pars ergo ſphęrę, in qua eſt <lb/>arcus k l, ſub maiori angulo uidebitur, quã pars ſphęrę, in qua eſt arcus g d.</s> <s xml:id="echoid-s28985" xml:space="preserve"> Apparet ergo per 20 hu-<lb/>ius maior uiſui pars ſphęrę, quę eſt k l, quàm pars eius, quæ eſt g d.</s> <s xml:id="echoid-s28986" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s28987" xml:space="preserve"/> </p> <div xml:id="echoid-div1152" type="float" level="0" n="0"> <figure xlink:label="fig-0450-01" xlink:href="fig-0450-01a"> <variables xml:id="echoid-variables475" xml:space="preserve">e a z g k l d c b</variables> </figure> </div> </div> <div xml:id="echoid-div1154" type="section" level="0" n="0"> <head xml:id="echoid-head916" xml:space="preserve" style="it">68. Diametro ſphæræ illuminatæ conuexæ, lineæ connectentic entra amborum oculorumæ-<lb/>quali exiſtente: hemiſphærium eſt, quod ambobus uiſibus uidetur. <lb/>Euclides 26 th. opt.</head> <figure> <variables xml:id="echoid-variables476" xml:space="preserve">g a b e z d</variables> </figure> <p> <s xml:id="echoid-s28988" xml:space="preserve">Sphæræ datę ſit centrum a:</s> <s xml:id="echoid-s28989" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s28990" xml:space="preserve"> circulus eius maior, cuius diame <lb/>ter ſti b g:</s> <s xml:id="echoid-s28991" xml:space="preserve"> quę ex hypotheſi ſit ęqualis diſtantiæ oculorum, hoc eſt <lb/>lineę connectenti centra uiſuum amborum, qui ſint e & d.</s> <s xml:id="echoid-s28992" xml:space="preserve"> Ducan-<lb/>tur quoq;</s> <s xml:id="echoid-s28993" xml:space="preserve"> à punctis b & g perpendiculares b d & g e, quę fiant ęqua-<lb/>les per 3 p 1:</s> <s xml:id="echoid-s28994" xml:space="preserve"> & copuletur linea d e:</s> <s xml:id="echoid-s28995" xml:space="preserve"> quæ per 33 p 1 & ex hypotheſi erit <lb/>æqualis & ęquidιſtans lineæ g b.</s> <s xml:id="echoid-s28996" xml:space="preserve"> Ducatur quo que perpendicularis <lb/>à puncto a centro ſphęrę ſuper lineam g b per 11 p 1:</s> <s xml:id="echoid-s28997" xml:space="preserve"> quę producta ad <lb/>lineam d e ſecet ipſam in puncto z.</s> <s xml:id="echoid-s28998" xml:space="preserve"> Palàm ergo per 29 p 1, quoniam <lb/>linea a z eſt per pendicularis ſuper lineam e d, & per 28 p 1 erit linea <lb/>a z ęquidiſtãs lineę g e:</s> <s xml:id="echoid-s28999" xml:space="preserve"> ergo per 33 p 1 patet, quòd linea e d diuiditur <lb/>per æqualia in puncto z, quia, ut patet ex hypotheſi, oculi ſunt in <lb/>punctis d & e:</s> <s xml:id="echoid-s29000" xml:space="preserve"> dico, quòd hemiſphęrium eſt quod uidetur.</s> <s xml:id="echoid-s29001" xml:space="preserve"> Manen-<lb/>te enim fixa linea a z, circumuoluatur parallelogrãmum a b z d, do-<lb/>nec redeat ad locum, unde incœpit:</s> <s xml:id="echoid-s29002" xml:space="preserve"> linea ergo a b mota deſcribet cir <lb/>culum ęqualem circulo g b, cuius ipſa eſt ſemidiam eter:</s> <s xml:id="echoid-s29003" xml:space="preserve"> eſt autẽ cir-<lb/>culus magnus ſphęrę datæ circulus g d:</s> <s xml:id="echoid-s29004" xml:space="preserve"> ergo per motũ lineę a b de-<lb/>ſcribitur circulus magnus:</s> <s xml:id="echoid-s29005" xml:space="preserve"> hic autem ſphęram diuidit in duo ęqua-<lb/>lia.</s> <s xml:id="echoid-s29006" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s29007" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1155" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables477" xml:space="preserve">u e f c a h d b g</variables> </figure> <head xml:id="echoid-head917" xml:space="preserve" style="it">69. Linea connectens centra amborum oculorum, ſimaior diametro ſphæræ illuminatæ con-<lb/>uexæ fuerit: plus hemiſphærio eſt, quod ambo-<lb/> bus uiſibus uidetur. Euclides 27 th. opt.</head> <p> <s xml:id="echoid-s29008" xml:space="preserve">Sit ſphæra data, cuius centrum a:</s> <s xml:id="echoid-s29009" xml:space="preserve"> & eius circu <lb/>lus magnus ſit e c d i:</s> <s xml:id="echoid-s29010" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s29011" xml:space="preserve"> centra amborum o-<lb/>culorum b & g:</s> <s xml:id="echoid-s29012" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s29013" xml:space="preserve"> linea b g producta maior dia <lb/>metro datę ſphęræ & eius circuli magni.</s> <s xml:id="echoid-s29014" xml:space="preserve"> Dico, <lb/>quòd ambobus uiſibus maius hemiſphęrio ui-<lb/>debitur.</s> <s xml:id="echoid-s29015" xml:space="preserve"> Ducantur enim à centris oculorum li-<lb/>neæ b e & g d contingentes circulum e d ci per <lb/>17 p 3:</s> <s xml:id="echoid-s29016" xml:space="preserve"> contingantq́;</s> <s xml:id="echoid-s29017" xml:space="preserve"> in punctis e & d:</s> <s xml:id="echoid-s29018" xml:space="preserve"> & ducatur <lb/>à puncto a diameter ſphęrę ęquidiſtãs lineę b g <lb/> <pb o="149" file="0451" n="451" rhead="LIBER QVARTVS"/> per 31 p 1.</s> <s xml:id="echoid-s29019" xml:space="preserve"> Et quia diameter ſphęrę ex hypotheſi eſt minor quàm linea b g, palàm quoniam lineæ b e <lb/>& g d ultra diametrum fh concurrent per 16 th.</s> <s xml:id="echoid-s29020" xml:space="preserve"> 1 huius concurrant ergo in puncto z.</s> <s xml:id="echoid-s29021" xml:space="preserve"> Quia ergo ab <lb/>uno puncto z ducuntur duę lineę contingentes circulum, ſcilicet e z & z d:</s> <s xml:id="echoid-s29022" xml:space="preserve"> palàm, quia portio cir-<lb/>culi, quæ eſt e c d eſt minor ſemicirculo per 58 th.</s> <s xml:id="echoid-s29023" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s29024" xml:space="preserve"> ergo portio eiuſdem circuli reliqua, quæ <lb/>eſt e i d eſt m aior ſemicirculo:</s> <s xml:id="echoid-s29025" xml:space="preserve"> hęc autem portio eſt illa, quę uidetur.</s> <s xml:id="echoid-s29026" xml:space="preserve"> Et quia idem eſt de omnib.</s> <s xml:id="echoid-s29027" xml:space="preserve"> cir-<lb/>culis magnis in tota ſphęra ſignatis:</s> <s xml:id="echoid-s29028" xml:space="preserve"> palàm, quia maius hemiſphęrio eſt, qđ de ſuperficie ſphęrica, <lb/>hypotheſi tali exiſtente, uidetur.</s> <s xml:id="echoid-s29029" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s29030" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1156" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables478" xml:space="preserve">f a h b y i d e z</variables> </figure> <head xml:id="echoid-head918" xml:space="preserve" style="it">70. Linea connectens centra amborum uiſuum, ſi diametro ſphæ <lb/> ræ conuexæ minor fuerit: minus hemiſphærio eſt, quod uidetur. Eu- clides 28 th. opt.</head> <p> <s xml:id="echoid-s29031" xml:space="preserve">Sit ſphęra data, cuius centrum a:</s> <s xml:id="echoid-s29032" xml:space="preserve"> & circuli eius magni diameter ſit <lb/>f h:</s> <s xml:id="echoid-s29033" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s29034" xml:space="preserve"> centra oculorum d & e:</s> <s xml:id="echoid-s29035" xml:space="preserve"> & producatur linea d e, connectens <lb/>centra oculorum minor exiſtens diametro ſ h:</s> <s xml:id="echoid-s29036" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s29037" xml:space="preserve"> lineæ illũ <lb/>circulum cõtingentes, quę ſint d b & e g.</s> <s xml:id="echoid-s29038" xml:space="preserve"> Dico, quòd minus hemiſphę <lb/>rio eſt illud, quod uidetur.</s> <s xml:id="echoid-s29039" xml:space="preserve"> Protrahantur enim lineę b d & g e.</s> <s xml:id="echoid-s29040" xml:space="preserve"> Et quo-<lb/>niam linea d e, eſt minor diametro f h, palàm per 16 th.</s> <s xml:id="echoid-s29041" xml:space="preserve"> 1 huius, quoniã <lb/>lineæ b d & g e, concurrent ultra ambos uiſus:</s> <s xml:id="echoid-s29042" xml:space="preserve"> ſit ergo concurſus pun <lb/>ctus z Palàm per 58 th.</s> <s xml:id="echoid-s29043" xml:space="preserve"> 1 huius, quoniam cum à puncto z ducãtur duę <lb/>lineæ unum circulum contingentes, quæ ſunt z b & z g, quòd arcus b <lb/>i g eſt minor ſemicirculo:</s> <s xml:id="echoid-s29044" xml:space="preserve"> minus ergo ſemicirculo b g uidetur ſub ocu <lb/>lis d & e.</s> <s xml:id="echoid-s29045" xml:space="preserve"> Ergo, ut prius, minus hemiſphærio uidebitur ſub oculis d & <lb/>e.</s> <s xml:id="echoid-s29046" xml:space="preserve"> Et hoc eſt, quod proponebatur.</s> <s xml:id="echoid-s29047" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1157" type="section" level="0" n="0"> <head xml:id="echoid-head919" xml:space="preserve" style="it">71. Centro for aminis uueæ in ſuperficie ſphæræ concauæ illumina <lb/>tæ exiſtente, tota ſphæræ intrinſeca ſuperficies uidetur. Alha-<lb/>zen 44 n 4.</head> <p> <s xml:id="echoid-s29048" xml:space="preserve">Eſto centrum ſoraminis uueæ punctus a:</s> <s xml:id="echoid-s29049" xml:space="preserve"> & ſit ſphæra data, cuius maior circulus ſit b a g tranſiẽs <lb/>per centrum a.</s> <s xml:id="echoid-s29050" xml:space="preserve"> Patet ergo per 52 huius, quoniam ſic uiſu diſpoſito to-<lb/>tus circulus b a g poterit uideri.</s> <s xml:id="echoid-s29051" xml:space="preserve"> Et quia plurimi circuli magni ſphęræ <lb/> <anchor type="figure" xlink:label="fig-0451-02a" xlink:href="fig-0451-02"/> ſe ſecant ſuper polos ſphęrę, quilibet autem punctus ſphęræ eſt polus <lb/>ſphæræ:</s> <s xml:id="echoid-s29052" xml:space="preserve"> palàm, quia omnes circuli magni ſphærę datę, qui per omnia <lb/>puncta ſuperficiei ſphęrę imagin ari poſſunt, tranſeuntes ſe interſeca-<lb/>bunt ſuper punctum a:</s> <s xml:id="echoid-s29053" xml:space="preserve"> erit ergo punctum a, quod eſt centrum ſorami <lb/>nis ipſius uueæ in quolibet illorum magnorum circulorũ:</s> <s xml:id="echoid-s29054" xml:space="preserve"> omnes aũt <lb/>illi circuli magni ſphæræ totam ſphæræ ſuperficiem euacuant:</s> <s xml:id="echoid-s29055" xml:space="preserve"> quia <lb/>non eſt dare punctum in ſphærę ſuperficie, quem aliquis circulus ma-<lb/>gnus non tranſeat.</s> <s xml:id="echoid-s29056" xml:space="preserve"> Viſu ergo taliter diſpoſito, tota concaua ſphærę ſu <lb/>perficies uidebitur.</s> <s xml:id="echoid-s29057" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s29058" xml:space="preserve"/> </p> <div xml:id="echoid-div1157" type="float" level="0" n="0"> <figure xlink:label="fig-0451-02" xlink:href="fig-0451-02a"> <variables xml:id="echoid-variables479" xml:space="preserve">a b g</variables> </figure> </div> </div> <div xml:id="echoid-div1159" type="section" level="0" n="0"> <head xml:id="echoid-head920" xml:space="preserve" style="it">72. Centro for aminis uueæ intra ſphæræ concauæ illuminatæ ſuperficiem, uel extra illam exi <lb/>ſtente, portio circularis ſphæræ uidebitur, cui incidunt æquales lineæ à centro uiſus ductæ: erit́ <lb/>uiſum quando hemiſphærium: quando mairo portio: quando minor. Alhazen 44 n 4.</head> <p> <s xml:id="echoid-s29059" xml:space="preserve">Eſto centrum foraminis uueę punctum a, & ſit ſphęra concaua, cuius circulus magnus ſit b c d:</s> <s xml:id="echoid-s29060" xml:space="preserve"> & <lb/>centrum ſphærę ſit punctum e.</s> <s xml:id="echoid-s29061" xml:space="preserve"> Si ergo centrum uiſus ſuerit in puncto e centro ſphæræ, quod eſt e-<lb/>tiam centrum circuli magni, qui eſt b c d, per definitionem circuli magni:</s> <s xml:id="echoid-s29062" xml:space="preserve"> tunc manifeſtum eſt per <lb/>52 huius, quòd totus circulus b c d uidebitur:</s> <s xml:id="echoid-s29063" xml:space="preserve"> ſed & per eandẽ 52 hu-<lb/>ius, omnes alij circuli ſubiecti hemiſphærij æquidiſtantes circulo b <lb/> <anchor type="figure" xlink:label="fig-0451-03a" xlink:href="fig-0451-03"/> c d uidebuntur, quoniam omnium illorum polus erit centrum <lb/>uιſus:</s> <s xml:id="echoid-s29064" xml:space="preserve"> omnes quoq;</s> <s xml:id="echoid-s29065" xml:space="preserve"> lineę rectæ ductę à polo ad peripheriam ſui cir-<lb/>culi ſunt æquales per 65 th.</s> <s xml:id="echoid-s29066" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s29067" xml:space="preserve"> & quoniam hi omnes circu-<lb/>li totum hemiſphęrium exhauriunt:</s> <s xml:id="echoid-s29068" xml:space="preserve"> patet, quòd in hoc ſitu exiſten-<lb/>te uiſu, totum hemiſphęrium uidebitur.</s> <s xml:id="echoid-s29069" xml:space="preserve"> Quòd ſi punctum a, cen-<lb/>trum foraminis uueæ ſit ſub centro ſphæræ, quod eſt punctum e, <lb/>tunc per eadem minus hemiſphęrio uidebitur:</s> <s xml:id="echoid-s29070" xml:space="preserve"> ſi ſit ſupra centrum e, <lb/>ſiue ſit intra ſphęram, ſiue extra:</s> <s xml:id="echoid-s29071" xml:space="preserve"> tũc ſimiliter per 2 th.</s> <s xml:id="echoid-s29072" xml:space="preserve"> 3 huius, omnes <lb/>circuli, ad quorum circum ferentias poſſunt produci lineę rectę, ui-<lb/>debuntur:</s> <s xml:id="echoid-s29073" xml:space="preserve"> maius ergo hemiſphęrio uidebitur.</s> <s xml:id="echoid-s29074" xml:space="preserve"> Et ſi linea à centro ui-<lb/>ſus ad ſuperficem ſphæræ ducta, obliquè incidat ſuperficiei ipſius <lb/>ſphęrę:</s> <s xml:id="echoid-s29075" xml:space="preserve"> tunc palàm, quòd etiam ſuperficiebus multorum circulorũ obliquè incidet:</s> <s xml:id="echoid-s29076" xml:space="preserve"> & poteſt acci-<lb/>dere, quòd tota figura ſphærę uidebitur inęqualis, ſuorum circulorum peripherijs quibuſdam ten <lb/>dentibus ad figuram ſectionis columnaris per 55 & 56 huius.</s> <s xml:id="echoid-s29077" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s29078" xml:space="preserve"/> </p> <div xml:id="echoid-div1159" type="float" level="0" n="0"> <figure xlink:label="fig-0451-03" xlink:href="fig-0451-03a"> <variables xml:id="echoid-variables480" xml:space="preserve">b c a e d</variables> </figure> </div> <pb o="150" file="0452" n="452" rhead="VITELLONIS OPTICAE"/> </div> <div xml:id="echoid-div1161" type="section" level="0" n="0"> <head xml:id="echoid-head921" xml:space="preserve" style="it">73. Viſu hemiſphærio concauo appropinquante, minus ſuperficiei ſphæræ uidebitur: apparet <lb/>autem plus uideri.</head> <p> <s xml:id="echoid-s29079" xml:space="preserve">Hęc poteſt demonſtrari, ſicut & 67 huius, de ſphæra cõuexa eſt demonſtrata:</s> <s xml:id="echoid-s29080" xml:space="preserve"> eſt enim per omnia <lb/>idem hinc inde demonſtrandi modus.</s> <s xml:id="echoid-s29081" xml:space="preserve"> Vnde hic ſphæra concaua figuretur, ut illic conuexa, & ſub <lb/>eiſdem literis conſignetur figuratio totalis, & per eadem concludetur.</s> <s xml:id="echoid-s29082" xml:space="preserve"> Et hæc quidem de uiſione <lb/>ſphærarum dicta ſunt, ſuperficie bus ipſarum oppoſitis uiſui totaliter exiſtentibus luminoſis per ſe, <lb/>uel illuminatis aliun de:</s> <s xml:id="echoid-s29083" xml:space="preserve"> quoniam hoc non exiſtente, licet in ſphærarum ſuperficiebus permaneat <lb/>dictorum modorum uiſibilitas, non tam en actu uidebuntur, niſi luminis interuentu, ut patet per 1 <lb/>th.</s> <s xml:id="echoid-s29084" xml:space="preserve"> 3 huius, & ſecundum diuerſitatem lumin oſitatis in partibus ſuperficiei ſphærarum, quæ uiden-<lb/>tur, nouæ paſsiones uiſibus generantur, quales ſunt hæ, quas nunc intendimus explicare.</s> <s xml:id="echoid-s29085" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1162" type="section" level="0" n="0"> <head xml:id="echoid-head922" xml:space="preserve" style="it">74. Diametro ſphæræ uiſæ illuminatæ maiore diſtantia oculorum exiſtente, & diametro ſphæ <lb/>ræ illuminantis eidem æquali uel maiore, circuló baſis pyr amidis uiſionis æquidiſtante circulo <lb/>baſis pyr amidis illuminationis uel ipſum intrinſecus contingente: tota ſuperficies baſis pyrami-<lb/>dis uiſionis illuminata uiſibus occurrit: uidetur autem in maiori diſtantia quaſi plana.</head> <p> <s xml:id="echoid-s29086" xml:space="preserve">Patet enim per 26 uel 27 th.</s> <s xml:id="echoid-s29087" xml:space="preserve"> 2 huius, quoniam tanta exiſtente quantitate diametrorum iſtorum <lb/>corporum, ut proponitur:</s> <s xml:id="echoid-s29088" xml:space="preserve"> tunc baſis pyramidis illuminationis aut eſt circulus magnus ſphæræ illu <lb/>minatæ, aut æquidiſtans ei.</s> <s xml:id="echoid-s29089" xml:space="preserve"> Circulus autem, qui eſt baſis pyramidis uiſionis, ut patet per 70 huius, <lb/>ſemper eſt minor circulo magno ſphęrę uiſę, quoniam, ut patet ex hypotheſi, diameter ſphęræ uiſæ <lb/>eſt maior quàm diſtantia oculorum.</s> <s xml:id="echoid-s29090" xml:space="preserve"> Si ergo circũferentia circuli minoris ſit ęquidiſtans circum-<lb/>ferentiæ circuli maioris:</s> <s xml:id="echoid-s29091" xml:space="preserve"> tunc per 68 th.</s> <s xml:id="echoid-s29092" xml:space="preserve"> 1 huius, centra duorum illorum circulorum in eadem ſphæ-<lb/>rę diametro conſiſtunt, & tota baſis pyramidis uiſionis occurrit uiſibus, quia tota eſt illuminata:</s> <s xml:id="echoid-s29093" xml:space="preserve"> ui-<lb/>detur autem ſuperficies plana per 65 huius.</s> <s xml:id="echoid-s29094" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s29095" xml:space="preserve"> Sed etiam ſi centra iſtorum circu <lb/>lorum uſq;</s> <s xml:id="echoid-s29096" xml:space="preserve"> ad punctum contactus circumferentiarum mutentur, quandiu unus circulus alium non <lb/>ſecat, ſemper tota baſis pyramidis uiſionis uidetur illuminata:</s> <s xml:id="echoid-s29097" xml:space="preserve"> & lumen in ſphæræ uiſę ſuperficie ui <lb/>detur ſemper circulare, & tota baſis pyramidis illuminata:</s> <s xml:id="echoid-s29098" xml:space="preserve"> plus tamen tenebreſcit baſis pyramidis <lb/>uiſionis ad illam partem, ubi fit contactus illorum circulorum per 21 th 3 huius.</s> <s xml:id="echoid-s29099" xml:space="preserve"> Patet ergo propo-<lb/>ſitum.</s> <s xml:id="echoid-s29100" xml:space="preserve"> Et quod hic de duobus oculis oſtenſum eſt, euidentius patet, ſi uiſio tantùm uno fiat ocu-<lb/>lo, per 66 huius.</s> <s xml:id="echoid-s29101" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1163" type="section" level="0" n="0"> <head xml:id="echoid-head923" xml:space="preserve" style="it">75. Si diametro ſphæræ uiſæ illuminatæ maiore diſtantia oculorũ exiſtente, diametró ſphæ-<lb/>ræ illuminantis eidem æquali uel maiore, baſis pyramidis uiſionis inter ſecet baſim pyramidis il-<lb/>luminationis, it a ut ambo centra baſium ſint ſub ſuperficie communis ſectionis: erit illa commu-<lb/>nis ſectio pars ſuperficiei ſphæricæ irregularis: uidebituŕ ſuperficies plana gibberoſa, ut duabus <lb/>curuis lineis inæqualis quantitatis & curuit atis contenta.</head> <p> <s xml:id="echoid-s29102" xml:space="preserve">Imaginentur enim centra baſium (quę per pręcedentem in eadem diametro ſphęrę uiſę fore diſ-<lb/>ponuntur) tantùm ab inuicem elongari, ut circuli baſium ſe ſecent quantumcunq;</s> <s xml:id="echoid-s29103" xml:space="preserve">, dum tamen cẽ-<lb/>tra ambarum baſium ſub ſuperficie, quæ eſt communis ambabus illis baſibus, remaneant:</s> <s xml:id="echoid-s29104" xml:space="preserve"> tunc illa <lb/>communis ſectio erit pars ſuperficiei ſphęricæ figurę irregularis:</s> <s xml:id="echoid-s29105" xml:space="preserve"> quoniam, ut patet per 26 uel per <lb/>27 th.</s> <s xml:id="echoid-s29106" xml:space="preserve"> 2 huius, & ex 70 huius, & ut oſtenſum eſt in præmiſſa proxima, arcus circuli baſis pyramidis <lb/>illuminationis eſt maior arcu circuli baſis pyramidis uiſionis:</s> <s xml:id="echoid-s29107" xml:space="preserve"> & ſi illius ſuperficiei acciperetur pun <lb/>ctus medius, lineæ ab illo puncto ad peripherias arcuum ductę, eſſent inęquales.</s> <s xml:id="echoid-s29108" xml:space="preserve"> Videtur autem ſu-<lb/>perficies illa eſſe plana per 65 huius:</s> <s xml:id="echoid-s29109" xml:space="preserve"> & erit gibberoſa, ut duabus præmiſsis curuis lineis in æqualis <lb/>quantitatis & curuitatis contenta:</s> <s xml:id="echoid-s29110" xml:space="preserve"> quoniã arcus circuli pyramidis uiſionis eſt curuior & maior por <lb/>tio ſuę circumferentię, quàm arcus circuli baſis pyramidis ιlluminationis ſit portio ſuę circumferẽ-<lb/>tię.</s> <s xml:id="echoid-s29111" xml:space="preserve"> Quod accidit propter inęqualitatem circulorum.</s> <s xml:id="echoid-s29112" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s29113" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1164" type="section" level="0" n="0"> <head xml:id="echoid-head924" xml:space="preserve" style="it">76. Baſi pyramidis uiſionis ſphæræ interſecante baſim pyramidis illuminationis, ita quòd <lb/>ipſorum axes angulum rectum contineant: communis earum ſectio est quarta ſuperficiei <lb/>ſphæricæ: uidetur autem in maiori diftantia plana ſuperficies una recta linea & ſemicircu-<lb/>lo contenta.</head> <p> <s xml:id="echoid-s29114" xml:space="preserve">Quòd illuminatio cuiuslibet ſphæræ fiat ſecundum pyramidem, cuius baſis in ſuperficie ſphærę <lb/>illuminatę eſt circulus, hoc patet per 26 & 27 & 28 th.</s> <s xml:id="echoid-s29115" xml:space="preserve"> 2 huius:</s> <s xml:id="echoid-s29116" xml:space="preserve"> quòd etiam baſis pyramidis uiſionis <lb/>omnis ſphęrę ſit circulus, patet per 66 & 68 & 69 & & 70 huius.</s> <s xml:id="echoid-s29117" xml:space="preserve"> Et quoniam axes iſtarum pyramidũ <lb/>ex hypotheſi producti ad inuicem angulũ rectũ continent:</s> <s xml:id="echoid-s29118" xml:space="preserve"> tunc patet per 33 p 6, quòd ab illorũ axiũ <lb/>cõcurſus puncto ſecũdũquantitatẽ ſemidiametri ſphæræ uiſę circũducto circulo, interiacebit quar <lb/>ta circuli inter axes.</s> <s xml:id="echoid-s29119" xml:space="preserve"> Et quoniã uterq;</s> <s xml:id="echoid-s29120" xml:space="preserve"> axiũ eſt per pendicularis ſuper ſuperficiẽ ſphæræ illuminatæ <lb/>uiſæ, palã per 111 th.</s> <s xml:id="echoid-s29121" xml:space="preserve"> 1 huius, quòd uterq;</s> <s xml:id="echoid-s29122" xml:space="preserve"> axiũ tranſibit per centrum illius ſphęrę:</s> <s xml:id="echoid-s29123" xml:space="preserve"> punctus itaq;</s> <s xml:id="echoid-s29124" xml:space="preserve"> inter-<lb/>ſectionis axium eſt in cẽtro illius ſphæræ:</s> <s xml:id="echoid-s29125" xml:space="preserve"> & ſolũ ille punctus, qui eſt centrũ ſphærę, ambobus axib.</s> <s xml:id="echoid-s29126" xml:space="preserve"> <lb/>erit cõmunis.</s> <s xml:id="echoid-s29127" xml:space="preserve"> Axibus itaq;</s> <s xml:id="echoid-s29128" xml:space="preserve"> interiacet quarta magni circuli ſphæræ ęqualiter diſtãtis à duobus pun-<lb/>ctis duarũ interſectionũ circulorũ baſis pyramidis illuminationis & baſis pyramidis uiſionis:</s> <s xml:id="echoid-s29129" xml:space="preserve"> cõmu <lb/> <pb o="151" file="0453" n="453" rhead="LIBER QVARTVS."/> nis itaq́;</s> <s xml:id="echoid-s29130" xml:space="preserve"> ſectio iſtarum duarum baſiũ eſt quarta ſuperficiei ſphę;</s> <s xml:id="echoid-s29131" xml:space="preserve">. Et quoniá tota ſuperficies ſphę;</s> <s xml:id="echoid-s29132" xml:space="preserve"> <lb/>rica in maiori diſtantia uidetur plana ſuperficies per 65 huius:</s> <s xml:id="echoid-s29133" xml:space="preserve"> palàm & hãc ſuperficiem ſphęricam <lb/>planá à maiori diſtantia uideri:</s> <s xml:id="echoid-s29134" xml:space="preserve"> axis enim pyramidis uiſionis caditin ſuperficie circuli baſis pyrami <lb/>dis illuminationis, propter erectionem ſui ſuper axem illius pyramidis, quod patetper 4 p 11.</s> <s xml:id="echoid-s29135" xml:space="preserve"> Pa-<lb/>làm ergo cum centrũ uiſus ſit in uertice axis pyramidis uiſionis, quoniam circulus baſis pyramidis <lb/>illuminationies eſt in eadem ſuperficie cũ centro uiſus:</s> <s xml:id="echoid-s29136" xml:space="preserve"> palà ergo per 50 huius quoniá ipſe uide-<lb/>tur linea recta.</s> <s xml:id="echoid-s29137" xml:space="preserve"> Semicirculus uerò baſis illuminationis, quia non eſt in eadé ſuperficie cũ centro ui-<lb/>ſus, uidetur circularis.</s> <s xml:id="echoid-s29138" xml:space="preserve"> Sic ergo illa ſuperficies communis ſectionis uidetur ſuperficies plana, una li <lb/>nea recta & alia curua contenta.</s> <s xml:id="echoid-s29139" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s29140" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1165" type="section" level="0" n="0"> <head xml:id="echoid-head925" xml:space="preserve" style="it">77. Baſi pyramidis uiſionis ſphæræ interſecante baſim pyramidis illuminationis, earum <lb/>communis ſectio, cui neutrius axis incidit, ect portio minor quarta parte ſuperficiei ſphæ-<lb/>ricæ: uidetur autem plana ſuperficies duobus quaſiæqualibus circunferentiarum baſium ar-<lb/>cubus contenta.</head> <p> <s xml:id="echoid-s29141" xml:space="preserve">Quia enim, ut in proxima præ miſſum eſt, omnis illuminatio ſphærę fit ſecundũ pyramidẽ, cuíus <lb/>baſis eſt circulus, ut patet per plures propoſitiones ſecũdi huius, & ſimiliter baſis pyramidis uiſio-<lb/>nis eſt circulus per 66 huius:</s> <s xml:id="echoid-s29142" xml:space="preserve"> palàm ſi iſti circuli, qui ſunt baſes pyramidũ, ſe non ſecent, ut quia ipſi <lb/>ſiti ſuntin oppoſitis quaſi partibus ſuperficiei ſphærę, cuius una pars eſt illuminata uel aliàs uiſa, <lb/>nec incidentia luminis, quæ ſic ſuperficiei ſphærę incidit, aliqualiter à uiſu perpen detur, utpote ſi <lb/>globum ligneum uel cereum, cuius diameter ſit maior diſtantia oculorum, oculis & lumini directè <lb/>interponas, reuoluto aũt globo ita ut lumẽ ſuperficiei ſphęriæ ipſius globi in cidens aliqualiter ap <lb/>pareat, tunc uidebitur ipſius ſuperficiei globi illuminata pars, quã recipit circũſerentiam baſis pyra-<lb/>midis uiſionis.</s> <s xml:id="echoid-s29143" xml:space="preserve"> Et quoniam illa pars uiſa, ut illuminata eſt, terminatur per circũſerentiam baſis py-<lb/>ramidis illuminationis:</s> <s xml:id="echoid-s29144" xml:space="preserve"> patet quòdilla uiſa portio ſphærę eſt minor quarta parte ſuperficiei ſphæ-<lb/>rę.</s> <s xml:id="echoid-s29145" xml:space="preserve"> Cum enim neutrius pyramidũ axis incidat ſuperficiei cómunis ſectionis, ut patet ex hypotheſi:</s> <s xml:id="echoid-s29146" xml:space="preserve"> <lb/>palàm per 33 p 6, quia arcus diuidẽs illã ſuperficiẽ,æ qualiter diſtãs à duobus punctis interſectionũ <lb/>circulorũ dictarũ baſium, diuidens totã ſphę & illã cõmunem ſectionis ſuperficiẽ per æ qualia, eſt <lb/>minor quarta circuli.</s> <s xml:id="echoid-s29147" xml:space="preserve"> Quoniam enim angulus ei ſubtenſus eſt minor recto, patet quòd arcus ille eſt <lb/>minor quarta circuli:</s> <s xml:id="echoid-s29148" xml:space="preserve"> & ipſa uiſa ſuperficies uidetur plana per 65 huius.</s> <s xml:id="echoid-s29149" xml:space="preserve"> Et quia nullus illorũ circu-<lb/>lorum uel arcuũ directè uiſibus opponitur:</s> <s xml:id="echoid-s29150" xml:space="preserve"> quiblibet illorũ in ſua uidetur curuitate, quoniam forma <lb/>punctorum cuiuslibet illorum arcuum ſecundũ ſitũ ſuum peruenit ad uiſum.</s> <s xml:id="echoid-s29151" xml:space="preserve"> Illa ergo portio com-<lb/>munis ſectionis baſium dictarum pyramidum uidetur quaſi duo bus æ qualibus arcubus contenta <lb/>propter inſenſibilitatem in æ qualitatis, maximè cũ à remotiori ſpatio fituiſio per 50 huius.</s> <s xml:id="echoid-s29152" xml:space="preserve"> Certũ <lb/>tamẽ eſt per 27 th.</s> <s xml:id="echoid-s29153" xml:space="preserve"> 2 huius, & per 70 huius, quia arcus baſis pyramidis illuminationis eſt pars maio <lb/>tis circuli, quàm arcus baſis pyramidis uiſionis:</s> <s xml:id="echoid-s29154" xml:space="preserve"> quoniã diameter ſphærę corporis illuminantis eſt <lb/>maior diametro ſphęrę illuminatæ, & diſtantia oculorũ minor illa.</s> <s xml:id="echoid-s29155" xml:space="preserve"> Pater ergo propoſitum.</s> <s xml:id="echoid-s29156" xml:space="preserve"> Ex his <lb/>itaq;</s> <s xml:id="echoid-s29157" xml:space="preserve"> quatuor theorematibus patet, quare forma lunæ ſit in receſſu à coniunctioe nouacularis.</s> <s xml:id="echoid-s29158" xml:space="preserve"> In <lb/>tempore enim coniunctionis luna non uidetur, inſi fiat eclipſis ſolis, ita quòd radij ſolis penetran-<lb/>tes diaphanitatem corporis lunæ propter differentiã denſitatis corporis lunaris ad diaphanitatem <lb/>partium ſuæ ſphęræ uicinarum, & peruenientes ad uiſum faciant corpus ſphæricum lunę uiſibile:</s> <s xml:id="echoid-s29159" xml:space="preserve"> <lb/>tunc enim uidetur luna ſecundum ſui figuram diſtinctè:</s> <s xml:id="echoid-s29160" xml:space="preserve"> ſed proprio lumine priuata.</s> <s xml:id="echoid-s29161" xml:space="preserve"> In alijs autem <lb/>coniunctionibus quia radij perpendiculariter incidentes corporilunę, aut ualde obliquè aut nul-<lb/>lo modo peruenient ad uiſum:</s> <s xml:id="echoid-s29162" xml:space="preserve"> tunc corpus lunę non uidetur, eò quòd baſis pyramidis uifionis in-<lb/>ciditin partem oppoſitam baſi pyramidis illuminationis, nec ſecat una illarum baſium aliam.</s> <s xml:id="echoid-s29163" xml:space="preserve"> Cum <lb/>autem luna recedente à ſole, iſtę baſes ſe incipiuntinterſecare:</s> <s xml:id="echoid-s29164" xml:space="preserve"> tũcipſorum communis ſectio (quę <lb/>eſt portio ſuperficiei ſphærici corporis lunę) uidetur, & propter magnitudinem diſtantię uidetur <lb/>illa portio ſphęrę quaſi plana ſuperficies duabus curuis lineis ſecundunm eius conuexum & conca-<lb/>uum contenta, quę uidentur æ quales propter remotionem:</s> <s xml:id="echoid-s29165" xml:space="preserve"> non ſunt autem æ quales, ſed ſemperil <lb/>la, quę eſt in conuexo, quia eſt arcus circuli baſis pyramidis illuminationis, eſt pars maioris circuli, <lb/>quàm illa, quę eſt in concauo, quę eſt arcus circul baſis pyramidis uiſionis.</s> <s xml:id="echoid-s29166" xml:space="preserve"> Et quoniam axis pyra-<lb/>midis illuminationis ſemper eſt perpendicularis ſuper corpus ſolis, ut patet per 111 th.</s> <s xml:id="echoid-s29167" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s29168" xml:space="preserve"> ideo <lb/>ſemper conuexum lunæ eſt auerſum ſoli, & cornua uidentur ſemper reſpicere ad ſolem.</s> <s xml:id="echoid-s29169" xml:space="preserve"> Vnde illo-<lb/>rum ſitus ſemper uariatur ſecundum ſitum ſolis, & ſecundum latitudinẽ motus lunæ.</s> <s xml:id="echoid-s29170" xml:space="preserve"> Et durat ſem <lb/>per in luna hæc figura.</s> <s xml:id="echoid-s29171" xml:space="preserve"> quouſq;</s> <s xml:id="echoid-s29172" xml:space="preserve"> axes pyramidum ſecant ſe ad angulos rectos per 76 huius:</s> <s xml:id="echoid-s29173" xml:space="preserve"> tunce-<lb/>nim luna uidebitur in quadratura, quoniam quarta part ſuę ſphęrę interiacens peripherias dicta-<lb/>rum baſium uidebitur:</s> <s xml:id="echoid-s29174" xml:space="preserve"> & in prima quadratura & in ſecun da ſemper arcus illluminationis, quia <lb/>directè uiſibus opponitur, uidebitur linearecta, & arcus pyramidis illuminationis ſemper curuus.</s> <s xml:id="echoid-s29175" xml:space="preserve"> <lb/>Mutato autem hoc ſitu, tunc centra baſium ambarum pyramidum ſunt in ſuperficie communis ſe-<lb/>ctionis:</s> <s xml:id="echoid-s29176" xml:space="preserve"> uidebitur ergo luna gibberoſa & planę ſuperficiei per 65 huius:</s> <s xml:id="echoid-s29177" xml:space="preserve"> & hoc durabit, quouſque <lb/>circuli baſium intrinſecus ſe contingant, tunc enim luna uidetur plena.</s> <s xml:id="echoid-s29178" xml:space="preserve"> Et quando centra circulo-<lb/>rum dictarum baſium ſibi ad inuicem ſuperponentur, ita ut ambo fiant in linea una, ut quando illi <lb/>circuli baſiunt ęquidiſtantes in eadem ſuperficie ſphęrę lunę, ut patet per 68 th.</s> <s xml:id="echoid-s29179" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s29180" xml:space="preserve"> tunc erit ue-<lb/>ra lunę impletio, & limen ex omni parte circunſertur ęauale:</s> <s xml:id="echoid-s29181" xml:space="preserve"> & deinde luna mota uſque ad con-<lb/>cauum circulorum ipſarum baſium, uidetur ſemper plena, tamen aliquantum obfuſcatur lumen <lb/> <pb o="152" file="0454" n="454" rhead="VITELLONIS OPTICAE"/> approximans tenebroſitati:</s> <s xml:id="echoid-s29182" xml:space="preserve"> & ſic procedit luna in figuris eidem diſtantiæ competentibus ab oppo <lb/>ſitione ad coniunctionem, ſicut à coniumctione ad oppoſitionem.</s> <s xml:id="echoid-s29183" xml:space="preserve"> Ethoc quidem in luna propter <lb/>eius propinquitatẽ ad uiſus noſtros euidentius apparet:</s> <s xml:id="echoid-s29184" xml:space="preserve"> in alijs tamen ominibus ſtellis ſuum lumẽ <lb/>& actualitatem ſuiluminis à ſole uel ab alijs ſtellis accipientibus, neceſſe eſt eaſdem figuras exprę-<lb/>miſsis tribus theorematibus prouenire.</s> <s xml:id="echoid-s29185" xml:space="preserve"> Et ſecundum hoc cœleſtium influentiarum aſpectus & mo <lb/>di diuerſificantur:</s> <s xml:id="echoid-s29186" xml:space="preserve"> non apparet aũt hoc uiſibiliter in ſtellis alijs à luna, propter ipſarum magnam re <lb/>motionem à uiſu, ratione cuius accidit error uiſui, ut patet per 16 huius.</s> <s xml:id="echoid-s29187" xml:space="preserve"> Videntur itaq;</s> <s xml:id="echoid-s29188" xml:space="preserve"> omnes aliæ <lb/>ſtellæ, præter lunam ſemper rotundæ propter ſui remotionem à uiſibus, propter quod etiam ignis <lb/>remotus à uiſibus uidetur rotundus.</s> <s xml:id="echoid-s29189" xml:space="preserve"> Videntur aũt ſtellæ eædem maximè plenæ quádoq;</s> <s xml:id="echoid-s29190" xml:space="preserve"> maiores <lb/>quandoq;</s> <s xml:id="echoid-s29191" xml:space="preserve"> minores, quodnos eidé cauſſæ paucitati ſcilicent ſuæ illuminationis uel multitudini cre-<lb/>dimus expræmiſsis adſcribendum.</s> <s xml:id="echoid-s29192" xml:space="preserve"> De his tamen ſuo loco ſermo erit, ad præſens uerò nobis ſuffi-<lb/>ciat ex pręmiſsis propoſitionibus demonſtrationem præſentibus attuliſle:</s> <s xml:id="echoid-s29193" xml:space="preserve"> ſiue enim ſtellarum dia <lb/>metri ſint omnes ad inuicem æ quales, ſiue una ipſarum ſit maior altera:</s> <s xml:id="echoid-s29194" xml:space="preserve"> ſemper tamen pater, quòd <lb/>omnis diameter cuiuſcunq;</s> <s xml:id="echoid-s29195" xml:space="preserve"> ſtelię eſt maior quàm ſit diſtantia oculorum cuiuſcũq;</s> <s xml:id="echoid-s29196" xml:space="preserve"> uidentis:</s> <s xml:id="echoid-s29197" xml:space="preserve"> & ſic <lb/>hãc paſsionem uiſibus in ipſarum illuminatione accidere eſt neceſſe, quamuis illam diſtinctè non <lb/>comprehendat uiſus.</s> <s xml:id="echoid-s29198" xml:space="preserve"> Et hoc quidem & ante nos dixit arabs Meſſahala, ſed ſuper hoc nullam at-<lb/>tulit demonſtrationem.</s> <s xml:id="echoid-s29199" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1166" type="section" level="0" n="0"> <head xml:id="echoid-head926" xml:space="preserve" style="it">78. Columnærotundæ uel cylindri conuexi ſub uno oculo uiſi, minus medietate curuæ ſuper-<lb/>ficiei uidetur. Euclides 29 th. opt.</head> <p> <s xml:id="echoid-s29200" xml:space="preserve">Eſto columna rotunda, cuius una baſis ſit circulus gb:</s> <s xml:id="echoid-s29201" xml:space="preserve"> & eius diameter f h:</s> <s xml:id="echoid-s29202" xml:space="preserve"> & centrum a:</s> <s xml:id="echoid-s29203" xml:space="preserve"> ſitq́ in <lb/>ſuperficie illius circuli centrũ oculi punctũd:</s> <s xml:id="echoid-s29204" xml:space="preserve"> & producatur linea d a, copulans centrũ uiſus cũ cen <lb/>tro circuli baſis columnę:</s> <s xml:id="echoid-s29205" xml:space="preserve"> & ducantur lineæ d b & d g:</s> <s xml:id="echoid-s29206" xml:space="preserve"> quę contingant circulũg b per17 p 3:</s> <s xml:id="echoid-s29207" xml:space="preserve"> & pro-<lb/>ducantur à punctis g & b duę lineæ longitudinis colum <lb/> <anchor type="figure" xlink:label="fig-0454-01a" xlink:href="fig-0454-01"/> nę per 101th.</s> <s xml:id="echoid-s29208" xml:space="preserve"> 1 huius, quę ſint b e & g z:</s> <s xml:id="echoid-s29209" xml:space="preserve"> & erunt illę lineę <lb/>orthogonaliter ſuper baſim g b erectę per 92th.</s> <s xml:id="echoid-s29210" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s29211" xml:space="preserve"> <lb/>ſitq́, ut per lineas b e & b d unatranſeat ſuperficies pla-<lb/>na, & per lineas g d & g z alia ſuperficies plana.</s> <s xml:id="echoid-s29212" xml:space="preserve"> Neutra <lb/>ergo iſtarum ſuperficierum ſecat columnam:</s> <s xml:id="echoid-s29213" xml:space="preserve"> quoniam <lb/>lineę d b & d g ſunt contingentes circulũ baſis, & lineæ <lb/>b e & g z ſunt lineę;</s> <s xml:id="echoid-s29214" xml:space="preserve"> longitudinis in ſuperficie columnæ <lb/>non ſecantes illá:</s> <s xml:id="echoid-s29215" xml:space="preserve"> ſunt ergo illę ſuperficies ipſam colum <lb/>nam contingentes.</s> <s xml:id="echoid-s29216" xml:space="preserve"> Iſtarum quoq;</s> <s xml:id="echoid-s29217" xml:space="preserve"> ſuperficierum contin <lb/>gentium columnã (quia ambæ tranſeunt centra uiſus, <lb/>ut patet expręmiſsis, & ipſarum communis ſectio eſt li-<lb/>nea recta per 3 p 11) interſectio fit in quadá linea tranſe-<lb/>unte centrum uiſus æquidiſtanter axi columnę:</s> <s xml:id="echoid-s29218" xml:space="preserve"> & hoc, <lb/>quod inter ipſas de ſuperficie colũnę intercipitur, hoc <lb/>ſolũ uidetur.</s> <s xml:id="echoid-s29219" xml:space="preserve"> Quia uerò lineę longitudinis b e & g z ſunt <lb/>æ quidiſtantes per 6 p 11, palâm per 33 p 1, quoniam chor <lb/>dę arcuum baſium inter ipſas cadentes, quę ſunt g b & <lb/>z e, ſuntęquales:</s> <s xml:id="echoid-s29220" xml:space="preserve"> ergo per 28 p 3, arcus illis chordis cor-<lb/>reſpondentes eruntęquales.</s> <s xml:id="echoid-s29221" xml:space="preserve"> Portiones itaq;</s> <s xml:id="echoid-s29222" xml:space="preserve"> circulorũ <lb/>ipſarum baſium interceptę inter has lineas lógitudinis <lb/>colũnę b e & g z, & omniũ circulorum ęquidiſtantiũ ba <lb/>ſibus, ſunt ę quales portioni circuli g b:</s> <s xml:id="echoid-s29223" xml:space="preserve"> eſt autẽ hęc mi-<lb/>nor ſemicirculo per 51 huius:</s> <s xml:id="echoid-s29224" xml:space="preserve"> ergo & oẽs portiões alio-<lb/>rũ circulorũ ſunt minores ſuis ſemicirculis.</s> <s xml:id="echoid-s29225" xml:space="preserve"> Videbitur <lb/>ergo minus medietate colũnę.</s> <s xml:id="echoid-s29226" xml:space="preserve"> Quod eſt propoſitũ.</s> <s xml:id="echoid-s29227" xml:space="preserve"> Idẽ <lb/>quoq;</s> <s xml:id="echoid-s29228" xml:space="preserve"> accideret in columnis lateratis, niſi quòd anguli <lb/>quandoq;</s> <s xml:id="echoid-s29229" xml:space="preserve"> impediunt, quòdoq;</s> <s xml:id="echoid-s29230" xml:space="preserve"> iuuant uiſionis quátita-<lb/>té, quorũ uiſionis modũ propter infinitatem numerorũ omittimus:</s> <s xml:id="echoid-s29231" xml:space="preserve"> quia radice pręſenti ſuppoſita <lb/>diligens inueſtigator multa particularia concludet.</s> <s xml:id="echoid-s29232" xml:space="preserve"/> </p> <div xml:id="echoid-div1166" type="float" level="0" n="0"> <figure xlink:label="fig-0454-01" xlink:href="fig-0454-01a"> <variables xml:id="echoid-variables481" xml:space="preserve">d b g h a f e z h f</variables> </figure> </div> </div> <div xml:id="echoid-div1168" type="section" level="0" n="0"> <head xml:id="echoid-head927" xml:space="preserve" style="it">79. Linea connectens centra amborum uiſuum ſiæqualis diametro baſis cylindrifuerit, ſe-<lb/>micylindri conuexum uidebitur: ſi maior, mainus: ſi minor, minus.</head> <p> <s xml:id="echoid-s29233" xml:space="preserve">Eſto circulus baſis cylindri, cuius centrum ſit punctum a:</s> <s xml:id="echoid-s29234" xml:space="preserve"> punctus uerò extrà ſignatus ſit z:</s> <s xml:id="echoid-s29235" xml:space="preserve"> <lb/>& ducatur linea a z:</s> <s xml:id="echoid-s29236" xml:space="preserve"> & producatur à puncto a diameter g d orthogonaliter ſuper lineam z a <lb/>per 11 p 1:</s> <s xml:id="echoid-s29237" xml:space="preserve"> & deſcribatur ſuper lineam a z, ut ſuper diametrum, circulus a b z e:</s> <s xml:id="echoid-s29238" xml:space="preserve"> & producan-<lb/>tur lineę a b, b z, a e, e z:</s> <s xml:id="echoid-s29239" xml:space="preserve"> duę itaque lineę, quę z e & z b, contingunt circulum b e d g per <lb/>31 & 16 p 3.</s> <s xml:id="echoid-s29240" xml:space="preserve"> Producantur ergo à punctis b & e per 101 th.</s> <s xml:id="echoid-s29241" xml:space="preserve"> 1 huius duę lineę longitudinis:</s> <s xml:id="echoid-s29242" xml:space="preserve"> quę <lb/>erunt perpendiculares ſuper lineas a e, a b per 92 th.</s> <s xml:id="echoid-s29243" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s29244" xml:space="preserve"> ideo quòd ſunt erectę ſuper ba-<lb/>fim.</s> <s xml:id="echoid-s29245" xml:space="preserve"> Superficies quoque ductę ſupper lineas z e & z b, & per lineas longitudinum ſibi conter-<lb/> <pb o="153" file="0455" n="455" rhead="LIBER QVARTVS."/> minales ſecabunt ſe in linea per centrum commune amborum uiſuum, quod eſt in medio puncto <lb/>nterſectionis nerui concaui, ducta æquidiſtanter axi columnę, quando linea connectens cétraam <lb/>borũ uiſuũ fuerit minor diametro baſis colũnę:</s> <s xml:id="echoid-s29246" xml:space="preserve"> quę ſi maior fuerit, <lb/> <anchor type="figure" xlink:label="fig-0455-01a" xlink:href="fig-0455-01"/> illæ diametri cócurrét ad partẽ oppoſitã in aliqua linea ſuperficiei <lb/>ductæ per lineam ductam per centrum cómun æquidiſtanter axi, <lb/>& per ipſum axem.</s> <s xml:id="echoid-s29247" xml:space="preserve"> Si uerò fuerint diametri baſis columnæ uiſæ & <lb/>linea cónectenscentra oculorum æquales:</s> <s xml:id="echoid-s29248" xml:space="preserve"> tunc lineę longitudinis <lb/>ductæ cadunt ſuper terminos diametri æquidiſtantis centris ocu-<lb/>lorum, & ſuperficies productæ nunauã concurrent.</s> <s xml:id="echoid-s29249" xml:space="preserve"> Superficies au <lb/>tẽ columnæ inter has ſuperficies columnã cótingentes intercepta <lb/>eſt portio ſuperficiei columnę, quę uidetur:</s> <s xml:id="echoid-s29250" xml:space="preserve"> ſunt aũt omnes portio <lb/>nes circulorũ interceptæ inter eas, æquales portioni baſis interce-<lb/>ptæ.</s> <s xml:id="echoid-s29251" xml:space="preserve"> Si ergo illa ſuerit ſemicirculus, medietas cylindri uidebitur:</s> <s xml:id="echoid-s29252" xml:space="preserve"> ſi <lb/>minor ſemicirculo, ut eſt in propoſito arcus b e:</s> <s xml:id="echoid-s29253" xml:space="preserve"> tũc minus ſemicy-<lb/>lindro uidebitur:</s> <s xml:id="echoid-s29254" xml:space="preserve"> ſi maior, maius:</s> <s xml:id="echoid-s29255" xml:space="preserve"> horum autem omnium deducti o <lb/>eſt euidens expræmiſsis pluries repetitis.</s> <s xml:id="echoid-s29256" xml:space="preserve"> Patet ergo propoſitu m.</s> <s xml:id="echoid-s29257" xml:space="preserve"/> </p> <div xml:id="echoid-div1168" type="float" level="0" n="0"> <figure xlink:label="fig-0455-01" xlink:href="fig-0455-01a"> <variables xml:id="echoid-variables482" xml:space="preserve">g a d h e z</variables> </figure> </div> </div> <div xml:id="echoid-div1170" type="section" level="0" n="0"> <head xml:id="echoid-head928" xml:space="preserve" style="it">80. Viſu appropinquante cylindro conuexo, minus curuæ ſu-<lb/>perficiei uidebitur: apparet autem ac ſi magis uideatur. Eucli-<lb/>des 30th. opt.</head> <p> <s xml:id="echoid-s29258" xml:space="preserve">Sit cylindri baſis circulus b g:</s> <s xml:id="echoid-s29259" xml:space="preserve"> cuius centrũ ſit a:</s> <s xml:id="echoid-s29260" xml:space="preserve"> & diameter f h:</s> <s xml:id="echoid-s29261" xml:space="preserve"> <lb/>oculi uerò cẽtrum ſit in puncto e:</s> <s xml:id="echoid-s29262" xml:space="preserve"> & ducatur linea e a inter illa cen-<lb/>tra:</s> <s xml:id="echoid-s29263" xml:space="preserve"> & ducantur line æ e b & e g circulũ cõtingentes per 17 p 3:</s> <s xml:id="echoid-s29264" xml:space="preserve">& du <lb/>cantur à punctis b & g per 101 th.</s> <s xml:id="echoid-s29265" xml:space="preserve"> 1 huius lineæ longitudinis cylin-<lb/>dri, quæ ſint b i & g z.</s> <s xml:id="echoid-s29266" xml:space="preserve"> Videtur itaq;</s> <s xml:id="echoid-s29267" xml:space="preserve"> per modũ pręmiſſarũ ſub oculo <lb/>exiſtente in puncto e, ſuperficies cylindri i b g z:</s> <s xml:id="echoid-s29268" xml:space="preserve"> quæ minor eſt ſe-<lb/>micylindro per 78 huius.</s> <s xml:id="echoid-s29269" xml:space="preserve"> Appropinquet ergo uiſus columnæ:</s> <s xml:id="echoid-s29270" xml:space="preserve"> & <lb/>ſit in puncto t:</s> <s xml:id="echoid-s29271" xml:space="preserve">& ducantur lineæ cótingentes baſim columnæ, quæ <lb/>ſint t k & t l:</s> <s xml:id="echoid-s29272" xml:space="preserve"> & à punctis k & l ducantur lineę longitu dinis cylindri, <lb/>quę ſintl n & k m.</s> <s xml:id="echoid-s29273" xml:space="preserve"> Videbitur ergo ſub uiſu exiſtente in puncto t, ſu <lb/>perficies cylindri, quę eſt l n k m, quę minor eſt ſuքficie i b g z uiſa <lb/>in puncto e:</s> <s xml:id="echoid-s29274" xml:space="preserve"> cuius declaratio eſt ſimilis declarartioni factę in 67 hu-<lb/> <anchor type="figure" xlink:label="fig-0455-02a" xlink:href="fig-0455-02"/> ius.</s> <s xml:id="echoid-s29275" xml:space="preserve"> Appropinquante ergo uiſu ad cylin drum, minus ipſius ſuperfr <lb/>ciei uidetur:</s> <s xml:id="echoid-s29276" xml:space="preserve">apparet aũt ac ſi magis uideatur:</s> <s xml:id="echoid-s29277" xml:space="preserve"> quoniam per 60 th.</s> <s xml:id="echoid-s29278" xml:space="preserve"> 1 <lb/>huius, & per 21 p 1 angulus l t k maior eſt angulo b e g:</s> <s xml:id="echoid-s29279" xml:space="preserve"> concurrunt <lb/>enim lineæ t k & e g uerſus pũctũ g.</s> <s xml:id="echoid-s29280" xml:space="preserve"> Pater ergo ꝓpoſitũ ք 20 huius.</s> <s xml:id="echoid-s29281" xml:space="preserve"/> </p> <div xml:id="echoid-div1170" type="float" level="0" n="0"> <figure xlink:label="fig-0455-02" xlink:href="fig-0455-02a"> <variables xml:id="echoid-variables483" xml:space="preserve">f a h b l k g t i n m z e</variables> </figure> </div> </div> <div xml:id="echoid-div1172" type="section" level="0" n="0"> <head xml:id="echoid-head929" xml:space="preserve" style="it">81. Axe unius tantũ uiſus cẽtro baſis colũnæ rotundæ uelia <lb/>teratæ cuiuſcun incidente: uelſi diſtantia oculorũ æqualis, uel <lb/>minor fuerit diametro baſis cylindri obiexctæ directè uiſui: ſola <lb/>baſis uidetur: quæ ſi maior baſi ſuerit, totus uidebitur cylindrus, <lb/>baſiremotiore duntaxat excepta.</head> <p> <s xml:id="echoid-s29282" xml:space="preserve">Cum enim uno oculo fiat uiſio, & axis incidat centro circuli ba.</s> <s xml:id="echoid-s29283" xml:space="preserve"> <lb/>ſis columnę rotundę uel lateratę:</s> <s xml:id="echoid-s29284" xml:space="preserve"> tunc quia oẽs lineę longito dinis <lb/>ſunt perpendiculares ſuper baſim, ut patet per 92 th.</s> <s xml:id="echoid-s29285" xml:space="preserve"> 1 huius, nõ ui-<lb/>debitur forma puncti alicuius illarũ lin earũ, niſi ſolus pũctus com-<lb/>munis lineę longitudinis & peripherię ſuperficiei baſis:</s> <s xml:id="echoid-s29286" xml:space="preserve"> uidebitur <lb/>ergo ſola baſis.</s> <s xml:id="echoid-s29287" xml:space="preserve"> Etidem eſt ſi uiſio fiat ambobus uiſibus, ſi tamẽ di-<lb/>ſtantia oculorum, queę eſt linea connectens cẽtra oculorum, fuerit <lb/>æqualis uel minor diametro baſis:</s> <s xml:id="echoid-s29288" xml:space="preserve"> tunc enim, ut pater per 4 huius, <lb/>nullalinearum longitudinis columnę perueniet ad ambos uiſus, <lb/>niſi ſolùm, ut prius oſtenſum eſt, punctus, qui eſt communis ſectio <lb/>alicuius illarũlinearũ & peripherię ipſus baſis.</s> <s xml:id="echoid-s29289" xml:space="preserve"> Siuerò maior fue-<lb/>rit diſtantia oculorum ipſa diametro baſis:</s> <s xml:id="echoid-s29290" xml:space="preserve"> tunc omnes loneę longi <lb/>tudinis columnę perueniẽt ad ambos uiſus:</s> <s xml:id="echoid-s29291" xml:space="preserve">& uidebitur tota con <lb/>uexitas uiſę columnę, & baſis ſuperior uicinior uiſibus:</s> <s xml:id="echoid-s29292" xml:space="preserve"> in ferior ue <lb/>rò baſis nõ uidetur:</s> <s xml:id="echoid-s29293" xml:space="preserve"> quia nullus eius punctus peruenit ad uiſum, ni <lb/>ſi peripherię ſuę cũ lineis longitud <lb/>nis columnę, quę ad illam peri <lb/>pheriam terminãtur.</s> <s xml:id="echoid-s29294" xml:space="preserve"> Quòd ſi uno tantũ oculo uiſione ſacta.</s> <s xml:id="echoid-s29295" xml:space="preserve"> axis ceciderit extra centrum baſis:</s> <s xml:id="echoid-s29296" xml:space="preserve"> ui-<lb/>debitur aliqua pars linearum longitudinis totius columnæ:</s> <s xml:id="echoid-s29297" xml:space="preserve"> quoniã tunc peripheria baſis ſecat py-<lb/>ramidem uiſionis.</s> <s xml:id="echoid-s29298" xml:space="preserve"> Patet ergo illud, quod proponebatur.</s> <s xml:id="echoid-s29299" xml:space="preserve"> Eſt autẽ poſsibile, ut uiſu obliquè baſi co-<lb/>lumnę incidente, tota columna, & ſi regularis ſit, uideatur eius baſis altera parte longior, & tota co <lb/>lumna figuræ irregolaris per 55 uel 56 hui <lb/>us.</s> <s xml:id="echoid-s29300" xml:space="preserve"> Et hoc eſi nota tu dignum.</s> <s xml:id="echoid-s29301" xml:space="preserve"/> </p> <pb o="154" file="0456" n="456" rhead="VITELLONIS OPTICAE"/> </div> <div xml:id="echoid-div1173" type="section" level="0" n="0"> <head xml:id="echoid-head930" xml:space="preserve" style="it">82. Vnius tantùm uiſus axe, centro columnaris ſectionis (quæ eſt baſis abſidis columnaris ro <lb/>tundæ) incidente: totailla baſis & parts linearum longitudinis abſidis uidentur.</head> <p> <s xml:id="echoid-s29302" xml:space="preserve">Sit enim aliqua columna rotun da taliter abſciſſa, ut axis non ſit perpendicularis erectus ſuper <lb/>baſim:</s> <s xml:id="echoid-s29303" xml:space="preserve"> palàm ergo per 103 th.</s> <s xml:id="echoid-s29304" xml:space="preserve"> 1 huius, quòd baſis hæc eſt ſectio, quę dicitur colũnaris uel ſectio oxy-<lb/>gonia:</s> <s xml:id="echoid-s29305" xml:space="preserve"> & ipſa pars columnæ abſciſſa dicitur abſis.</s> <s xml:id="echoid-s29306" xml:space="preserve"> Dico, quòd ſi axis uiſualis incidat centro illius ba <lb/>ſis, quòd pars linearum longitudinis abſidis, illa ſcilicet, quæ in decliuiori parte approximat, uide-<lb/>bitur uno etiam uiſu.</s> <s xml:id="echoid-s29307" xml:space="preserve"> Huius autẽ cauſſa eſt obliquatio baſis, quæ ſub minoria angulo uidetur per 26 <lb/>huius:</s> <s xml:id="echoid-s29308" xml:space="preserve"> propter quod etiam uidentur formæ punctorum linearum longitudinis illius obliquitatis <lb/>remotiori parti adiacentium, cum reſidui anguli perueniunt ad uiſum:</s> <s xml:id="echoid-s29309" xml:space="preserve"> quod nõ accideret, ſi illa ba-<lb/>ſis poſſet directè uiſui opponi:</s> <s xml:id="echoid-s29310" xml:space="preserve">hoc autem impoſsibile ſine linearum longitudinis abſidis uiſione.</s> <s xml:id="echoid-s29311" xml:space="preserve"> <lb/>Patet ergo propoſitum.</s> <s xml:id="echoid-s29312" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1174" type="section" level="0" n="0"> <head xml:id="echoid-head931" xml:space="preserve" style="it">83. Centro for aminis uueæ in ſuperficie illuminata concaua columnæ cuiuſcunæ exiſtente: <lb/>ſemper columnæ tota concauit as uidetur: in alijs autem partinum columnarum concauarum ui <lb/>ſionibus idem accidit, quod ſphærarum concauitati.</head> <p> <s xml:id="echoid-s29313" xml:space="preserve">Diſpoſito enim uiſu ſecũdũ propoſitũ modũ, reſpectu cuiuslibet colũnę;</s> <s xml:id="echoid-s29314" xml:space="preserve">cócauæ, formæ omniũ <lb/>punctorũ linearũ lõgitudinis, quas ſecat ſuperficies ſoraminis uueæ, tũ oẽs perueniunt ad uiſum:</s> <s xml:id="echoid-s29315" xml:space="preserve"> <lb/>ideo quòd ad centrũ illius foraminis ſecundũ lineas rectas pertingunt:</s> <s xml:id="echoid-s29316" xml:space="preserve"> & ſuperficiẽ o culi cõtingit <lb/>tantùm una in illo centro:</s> <s xml:id="echoid-s29317" xml:space="preserve"> aliæ ueròipſam contingunt in punctis diuerſis circuli foraminis.</s> <s xml:id="echoid-s29318" xml:space="preserve"> Vide-<lb/>buntur ergo oẽs per 2 th.</s> <s xml:id="echoid-s29319" xml:space="preserve"> 3 huius.</s> <s xml:id="echoid-s29320" xml:space="preserve"> Et quoniã formæ omuiũ aliarũ linearũ longitudinũ, & oẽs puncti <lb/>baſium directè uel obliquè perueniunt ad uiſum:</s> <s xml:id="echoid-s29321" xml:space="preserve"> palã, quia tota colũnę cócauitas uidetur ſecundũ <lb/>omnia puncta ſuæ ſuperficiei.</s> <s xml:id="echoid-s29322" xml:space="preserve"> Sed fortè accidet figuræ uiſæ irregularitas propter aliquarũ ſuarum <lb/>partiũ obliquarionẽ ad uiſum per 55 uel 56 huius.</s> <s xml:id="echoid-s29323" xml:space="preserve"> In alijs quoq;</s> <s xml:id="echoid-s29324" xml:space="preserve"> uiſionibus partiũ columnarũ con-<lb/>cauarũ idẽ accidit, quod in ſphęris cõcauis:</s> <s xml:id="echoid-s29325" xml:space="preserve"> quoniã uiſu poſito in pũcto medio quadranguli termi-<lb/>nantis ſemicylindrum, ille totaliter uidebuitur per 60 huius.</s> <s xml:id="echoid-s29326" xml:space="preserve"> Sed & quodilbet punctorũ ſuperficiei <lb/>concauæ & baſium uiſibus occurrit.</s> <s xml:id="echoid-s29327" xml:space="preserve"> Etrecedente uiſu ab illo puncto, ſemper uidebitur portio co-<lb/>lumnæ minor uel maior ſemicylindro.</s> <s xml:id="echoid-s29328" xml:space="preserve"> Pater ergo propoſitum.</s> <s xml:id="echoid-s29329" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1175" type="section" level="0" n="0"> <head xml:id="echoid-head932" xml:space="preserve" style="it">84. Pyramidis rotundæ baſi in eadem ſuperficie cum centro unius oculorum exiſtente: minus <lb/>medietate ſuperficiei conuexæ pyramidis uidetur. Euchlides 31th. opt.</head> <p> <s xml:id="echoid-s29330" xml:space="preserve">Sit pyramis rotunda, cuius baſis ſit circulus, qui b g:</s> <s xml:id="echoid-s29331" xml:space="preserve"> cuius diameter fh:</s> <s xml:id="echoid-s29332" xml:space="preserve"> centrum k:</s> <s xml:id="echoid-s29333" xml:space="preserve"> uertex uerò <lb/>illius pyramidis ſit punctũa:</s> <s xml:id="echoid-s29334" xml:space="preserve"> & ſit centrũ uiſus d:</s> <s xml:id="echoid-s29335" xml:space="preserve"> & ducantur lineæ <lb/> <anchor type="figure" xlink:label="fig-0456-01a" xlink:href="fig-0456-01"/> d b & d g contingentes circulũ b g per 17 p 3:</s> <s xml:id="echoid-s29336" xml:space="preserve"> eſt ergo per 58 th.</s> <s xml:id="echoid-s29337" xml:space="preserve"> 1 hu <lb/>ius arcus b g minor ſemicirculo.</s> <s xml:id="echoid-s29338" xml:space="preserve"> Ducátur quoq:</s> <s xml:id="echoid-s29339" xml:space="preserve"> à uertice a pyrami <lb/>dis per101th.</s> <s xml:id="echoid-s29340" xml:space="preserve"> 1 huius lineæ longitudinis, quæſint a b & a g.</s> <s xml:id="echoid-s29341" xml:space="preserve"> Palàm <lb/>itaq;</s> <s xml:id="echoid-s29342" xml:space="preserve"> ad modũ eorũ.</s> <s xml:id="echoid-s29343" xml:space="preserve"> quæ demonſtrauimus in columinis, auoniã ſu-<lb/>perficies intercepta lineis a b & a g, ſola uidetur.</s> <s xml:id="echoid-s29344" xml:space="preserve"> Et quoniam hæ li-<lb/>neæ ex omnibus circulis ęquidiſtátibus baſi pyramidis partes ſimi <lb/>les reſecant, & intra ſe illas cõtinẽt, & cũ per 58 th.</s> <s xml:id="echoid-s29345" xml:space="preserve"> 1 huius arcus b g <lb/>ſit minor ſemicirculo:</s> <s xml:id="echoid-s29346" xml:space="preserve"> erunt neceſſariò arcus omnium aliorũ <lb/>lorũ minores ſemicirculis ſuis:</s> <s xml:id="echoid-s29347" xml:space="preserve"> ergo portio uiſa minor erit hemico <lb/>nio:</s> <s xml:id="echoid-s29348" xml:space="preserve">quoniam ſicut tota conuexa ſuperficies pyramidis toti baſire-<lb/>ſpondet:</s> <s xml:id="echoid-s29349" xml:space="preserve"> ſic pars proportionalis ad totá conuexam ſuperficiẽ parti <lb/>proportionali baſis ad totã baſim:</s> <s xml:id="echoid-s29350" xml:space="preserve"> quoniam lineæ lõgitudinis pro-<lb/>ductæ à uertice ad peripheriã baſis, ſicut diuidũ conicã ſuperficiẽ:</s> <s xml:id="echoid-s29351" xml:space="preserve"> <lb/>ſic lineæ à terminis illarũ linearũ ad centrũ baſis pyramidis produ <lb/>ctæ diuiduntipſam.</s> <s xml:id="echoid-s29352" xml:space="preserve"> Et poteſt hoc conuinci argumẽto 5 p 12 Eucli <lb/>dis.</s> <s xml:id="echoid-s29353" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s29354" xml:space="preserve"/> </p> <div xml:id="echoid-div1175" type="float" level="0" n="0"> <figure xlink:label="fig-0456-01" xlink:href="fig-0456-01a"> <variables xml:id="echoid-variables484" xml:space="preserve">a f k h b g d</variables> </figure> </div> </div> <div xml:id="echoid-div1177" type="section" level="0" n="0"> <head xml:id="echoid-head933" xml:space="preserve" style="it">85. Cẽtris amborũ uiſuũ in eadẽ ſuperficie cũ baſiconiexiſten <lb/>tibus, ſilinea cõnectens cẽtra uiſuũ æqualis fucrit diametro ba-<lb/>ſis, hemiconium uidebitur: ſi maior, maius: ſi minor, minus.</head> <p> <s xml:id="echoid-s29355" xml:space="preserve">Diſpoſitione ordinata ad conũ, quæ in 79 huius ad columnam, <lb/>hoc ſolo adiecto, quòd centra uiſuũ ſint ſolũ in eadẽ ſuperficie cũ <lb/>baſi pyramidis, & non eleuentur ſecundũ lineam axi coni æquidi <lb/>ſtantem, ſicut poteſt fieri in columna:</s> <s xml:id="echoid-s29356" xml:space="preserve"> ſi enim uiſus in lineaæ quidi-<lb/>ſtante axi columnæ eleuetur, idem accidit, quod eo in baſi exiſten-<lb/>te:</s> <s xml:id="echoid-s29357" xml:space="preserve"> quia in columna ſufficit, etiá ſi ſint in ſuperficie baſi æ quidiſtan-<lb/>ti.</s> <s xml:id="echoid-s29358" xml:space="preserve"> Patet ergo, quod hic proponitur, & eſt idem demonſtrandi mo-<lb/>dus.</s> <s xml:id="echoid-s29359" xml:space="preserve"> Vnde fruſtra eſt membranas denuò occupare.</s> <s xml:id="echoid-s29360" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1178" type="section" level="0" n="0"> <head xml:id="echoid-head934" xml:space="preserve" style="it">86. Appropinquãte centro uiſus in ſuperficie baſis coni: minus conicæ ſuperficieiuidebitur: <lb/>apparet autem plus uideri. Euclides 32 th. opt.</head> <p> <s xml:id="echoid-s29361" xml:space="preserve">Sit circulus a b baſis coni:</s> <s xml:id="echoid-s29362" xml:space="preserve"> cuius cẽtrum l:</s> <s xml:id="echoid-s29363" xml:space="preserve">& ſit uertex coni punctum g:</s> <s xml:id="echoid-s29364" xml:space="preserve"> cẽtrum quoq;</s> <s xml:id="echoid-s29365" xml:space="preserve"> oculiſit d:</s> <s xml:id="echoid-s29366" xml:space="preserve"> <lb/> <pb o="155" file="0457" n="457" rhead="LIBER QVARTVS."/> ducatur linea d lad centrum uiſus à centro baſis pyramidis:</s> <s xml:id="echoid-s29367" xml:space="preserve"> & ducanturlineæ d b & d a contingen <lb/>tes circulũ, qui eſt baſis coni, in pũctis b & a:</s> <s xml:id="echoid-s29368" xml:space="preserve"> & ducãtur à uertice pyra <lb/> <anchor type="figure" xlink:label="fig-0457-01a" xlink:href="fig-0457-01"/> midis lineæ lõgitudinis coni, quæ ſint g a & g b:</s> <s xml:id="echoid-s29369" xml:space="preserve"> ergo p̀er ea, quæ pri-<lb/>us in pręcedẽtibus dicta ſunt, ſuperficies g a b uidetur ſub oculo d:</s> <s xml:id="echoid-s29370" xml:space="preserve"> & <lb/>eſt minorhemiconio.</s> <s xml:id="echoid-s29371" xml:space="preserve"> Appropinquet aũt oculus, & fiat in pũcto e:</s> <s xml:id="echoid-s29372" xml:space="preserve"> du <lb/>canturq́;</s> <s xml:id="echoid-s29373" xml:space="preserve"> lineæ e z, e i cõtingentes circulũ, qui eſt baſis coni:</s> <s xml:id="echoid-s29374" xml:space="preserve"> & à uerti <lb/>ce coni cõtinuẽtur lineæ g z & g i.</s> <s xml:id="echoid-s29375" xml:space="preserve"> Videbitur itaq;</s> <s xml:id="echoid-s29376" xml:space="preserve"> ab uno oculo exi-<lb/>ſtente in puncto e portio ſuperficiei conicæ, quæ eſt g z i minor por-<lb/>tione g a b.</s> <s xml:id="echoid-s29377" xml:space="preserve"> Videtur autẽ apparere maior portiõe g a b propter maio-<lb/>ritatẽ anguli z e i ſupra angulum a d b.</s> <s xml:id="echoid-s29378" xml:space="preserve"> Ethoc eſt propoſitum.</s> <s xml:id="echoid-s29379" xml:space="preserve"/> </p> <div xml:id="echoid-div1178" type="float" level="0" n="0"> <figure xlink:label="fig-0457-01" xlink:href="fig-0457-01a"> <variables xml:id="echoid-variables485" xml:space="preserve">g l a z i b e d</variables> </figure> </div> </div> <div xml:id="echoid-div1180" type="section" level="0" n="0"> <head xml:id="echoid-head935" xml:space="preserve" style="it">87. Lineis à centro uiſus ad baſim coni cõtingenter ductis, & à <lb/>punctis contactuum ductis lineis logitudinis coni: ſi in cõmuni ſe-<lb/>ctione ſuperficierum per eaſdem line as & per cẽtrum oculi produ-<lb/>ctarum uiſus cono appropin quet: eadẽ portio ſuperficiei conicæ ui-<lb/>debitur, quæ prius, & eiuſdem quantitatis apparebit. Eucli-des 33th. opt.</head> <p> <s xml:id="echoid-s29380" xml:space="preserve">Eſto conus, cuius baſis ſit circulus b z g:</s> <s xml:id="echoid-s29381" xml:space="preserve"> & uertex eius punctũ a:</s> <s xml:id="echoid-s29382" xml:space="preserve"> <lb/>axis quoq;</s> <s xml:id="echoid-s29383" xml:space="preserve"> ſit a h:</s> <s xml:id="echoid-s29384" xml:space="preserve"> centrumq́;</s> <s xml:id="echoid-s29385" xml:space="preserve"> oculi ſit d:</s> <s xml:id="echoid-s29386" xml:space="preserve"> & ducantur per 17 p 3 lineæ à <lb/>centro uilus d contingentes circulũ b z g, quæ ſint d z & d g.</s> <s xml:id="echoid-s29387" xml:space="preserve"> Et quo-<lb/>niam hoc fit ex hypotheſi:</s> <s xml:id="echoid-s29388" xml:space="preserve"> tũc patet per 16 p 3 & 2 p 11, quoniã centrũ <lb/>uiſus eſt in ſuperficie baſis coni uiſi.</s> <s xml:id="echoid-s29389" xml:space="preserve"> Et ducátur à punctis contactuũ <lb/>z & g duæ lineæ longitudinis per coni uerticẽ punctũ a, quæ ſint z a <lb/>& g a:</s> <s xml:id="echoid-s29390" xml:space="preserve"> quod fiet per 101 th.</s> <s xml:id="echoid-s29391" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s29392" xml:space="preserve"> & à centro uiſus puncto d ad uerti-<lb/>cem coni punctũ a ducatur linea d a:</s> <s xml:id="echoid-s29393" xml:space="preserve"> & ducátur duæ ſuperficies, una <lb/>per lineas d g & g a, alia uerò per lineas d z & z a.</s> <s xml:id="echoid-s29394" xml:space="preserve"> Et quoniá eę ſuper-<lb/>ficies cõcurrũtin centro uiſus d & in uertice conia:</s> <s xml:id="echoid-s29395" xml:space="preserve"> erit ipſarũ com-<lb/>munis ſectio linea a d per 1 p 11 & per 19 th.</s> <s xml:id="echoid-s29396" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s29397" xml:space="preserve"> Dico, quòd ſi ocu.</s> <s xml:id="echoid-s29398" xml:space="preserve"> <lb/>lus appropinquet cono ſecundum lineam d a:</s> <s xml:id="echoid-s29399" xml:space="preserve"> non uidebitur maior <lb/>conicæ ſuperficiei portio nũc quàm prius, oculo in puncto d exiſtente.</s> <s xml:id="echoid-s29400" xml:space="preserve"> Sit enim, ut approximando <lb/>ipſrcono perueniat in punctum e lineæ d a:</s> <s xml:id="echoid-s29401" xml:space="preserve"> & ducantur à puncto e lineę æquidiſtantes lineis d g & <lb/>d z a d ſuperficiẽ coni uiſam:</s> <s xml:id="echoid-s29402" xml:space="preserve"> hę eruntergo neceſſariò <lb/>cõtingẽtes aliquẽ circulũ coni ęquidiſtátẽ baſi b z g:</s> <s xml:id="echoid-s29403" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0457-02a" xlink:href="fig-0457-02"/> ergo neceſſariò cadent in aliqua puncta linearum a z <lb/>& a g:</s> <s xml:id="echoid-s29404" xml:space="preserve"> ideo quòd illæ ſecant proportionaliter baſim <lb/>coni, & oẽs circulos ei æ quidiſtátes:</s> <s xml:id="echoid-s29405" xml:space="preserve"> quoniá ſecundũ <lb/>lineas illas terminatur uiſus, & ſecundũ illas ſuperfi <lb/>cies contingẽtes terminatur uiſio circulorũ.</s> <s xml:id="echoid-s29406" xml:space="preserve"> Si enim <lb/>dicatur, quòd illæ lineę contingentes aliquẽ dictorũ <lb/>circulorũ ductæ à puncto e, cadant extra lineas a z & <lb/>a g, cũ lineæ à pũcto e in lineas a z & a g ductæ termi-<lb/>nentuiſum, & ſimiliter illæ cõtingentes terminẽt ui-<lb/>ſum:</s> <s xml:id="echoid-s29407" xml:space="preserve"> ſequetur uel lineas radiales eſſe refractas in me-<lb/>dio unius diaphani:</s> <s xml:id="echoid-s29408" xml:space="preserve"> quod eſt cõtra ea, quæ demõſtra <lb/>ta ſuntper 44 & ſequẽtes ſecũdi huius:</s> <s xml:id="echoid-s29409" xml:space="preserve"> uel ſequetur <lb/>lineas radiales eſſe curuas:</s> <s xml:id="echoid-s29410" xml:space="preserve"> quod eſt cõtra 1 th.</s> <s xml:id="echoid-s29411" xml:space="preserve"> 2 hu-<lb/>ius:</s> <s xml:id="echoid-s29412" xml:space="preserve"> uel ſequetur duas rectas lineas ſuperficiẽ inclu-<lb/>dere:</s> <s xml:id="echoid-s29413" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s29414" xml:space="preserve"> Cadent ergo dictę lineæ <lb/>pertingentes ad ſuperficiẽ conicã ductæ à puncto e <lb/>ιn lineas a z & a g:</s> <s xml:id="echoid-s29415" xml:space="preserve"> cadant ita q;</s> <s xml:id="echoid-s29416" xml:space="preserve"> in ipſarũ duo puncta, <lb/>quę ſinti & c, & ſint lineę e i & e c.</s> <s xml:id="echoid-s29417" xml:space="preserve"> Quia ergo angulus <lb/>c e ι eſt æ qualis angulo g d z per 10 p 11, ſicut & anguli <lb/>cõtenti ſub lineis c i & g z, quoniã oẽs illi anguli con-<lb/>tinentur ſub lineis æquidiſtantibus angulariter con-<lb/>runctis, patet per 20 huius uerum eſſe quod proponi <lb/>tur.</s> <s xml:id="echoid-s29418" xml:space="preserve"> Et quia ubicunq;</s> <s xml:id="echoid-s29419" xml:space="preserve"> uiſus in linea d a ponitur, ſemper anguli ad uiſum ſunt æ quales per 10 p 11, pa-<lb/>làm ergo eſt propoſitum.</s> <s xml:id="echoid-s29420" xml:space="preserve"> Et hocidem ſuo modo in ambobus poteſt uiſibus demonſtrari.</s> <s xml:id="echoid-s29421" xml:space="preserve"/> </p> <div xml:id="echoid-div1180" type="float" level="0" n="0"> <figure xlink:label="fig-0457-02" xlink:href="fig-0457-02a"> <variables xml:id="echoid-variables486" xml:space="preserve">a e c d z h b g</variables> </figure> </div> </div> <div xml:id="echoid-div1182" type="section" level="0" n="0"> <head xml:id="echoid-head936" xml:space="preserve" style="it">88. Eleuato uiſu, reſpectu ſuperficiei conicæ: maius erit, quod uidetur, uidebitur autem mi-<lb/>nus uideri: depreſſo uerò uiſu, minus erit quod uidebitur, ſed apparebit maius prius uiſo. Eu-<lb/>clides 34th. optico.</head> <p> <s xml:id="echoid-s29422" xml:space="preserve">Eſto conus, cuius baſis circulus b g:</s> <s xml:id="echoid-s29423" xml:space="preserve"> & uertex punctus a:</s> <s xml:id="echoid-s29424" xml:space="preserve"> & ducantur lineæ longitudinis, quæ <lb/> <pb o="156" file="0458" n="458" rhead="VITELLONIS OPTICAE"/> fint a b & a g:</s> <s xml:id="echoid-s29425" xml:space="preserve"> & ducatur linea b g:</s> <s xml:id="echoid-s29426" xml:space="preserve"> & producatur uſq;</s> <s xml:id="echoid-s29427" xml:space="preserve"> ad punctum l:</s> <s xml:id="echoid-s29428" xml:space="preserve"> & à puncto t, quod ſitinferius <lb/>puncto a uertice coni, ducatur linea æquidiſtãs lineę <lb/> <anchor type="figure" xlink:label="fig-0458-01a" xlink:href="fig-0458-01"/> a b per 31 p 1, quæ producta uerſus lineam b l, ſecetil-<lb/>lam in pũcto p:</s> <s xml:id="echoid-s29429" xml:space="preserve"> & ſit aliquis pũctus eius inſerior pun <lb/>cto t pũctus k:</s> <s xml:id="echoid-s29430" xml:space="preserve"> & ſit illa linea t k p.</s> <s xml:id="echoid-s29431" xml:space="preserve"> Dico, quòd oculo <lb/>poſito ſuper pũctum t, qui eſt eleuatior pũcto k:</s> <s xml:id="echoid-s29432" xml:space="preserve"> pars <lb/>ſuperficiei conicę uiſa, maior quidem erit, minor aũt <lb/>uidebitur, quàm uideatur oculo exiſtẽte in pũcto k.</s> <s xml:id="echoid-s29433" xml:space="preserve"> <lb/>Ducátur enim lineæ a k & a t:</s> <s xml:id="echoid-s29434" xml:space="preserve"> & producatur linea a t, <lb/>donec cõcurrat cum linea b l:</s> <s xml:id="echoid-s29435" xml:space="preserve"> cõcurrent aũt per con-<lb/>uerſam 2 p 6.</s> <s xml:id="echoid-s29436" xml:space="preserve"> Quoniã enim linea t p eſt minor quàm <lb/>linea a b, ut patet ex præmiſsis, & illæ lineæ æquidi-<lb/>ſtant, patet quòd lineæ a t & b l cõcurrẽt:</s> <s xml:id="echoid-s29437" xml:space="preserve"> ſit ergo pun <lb/>ctus cócurſus i:</s> <s xml:id="echoid-s29438" xml:space="preserve"> & ſimiliter lineæ a k & b l concurrẽt:</s> <s xml:id="echoid-s29439" xml:space="preserve"> <lb/>ſitq́;</s> <s xml:id="echoid-s29440" xml:space="preserve"> pũctus concurſus l.</s> <s xml:id="echoid-s29441" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s29442" xml:space="preserve"> quia magis ui-<lb/>debitur de cono ſuper punctũ i, quàm ſuper pũctum <lb/>l per 86 huius:</s> <s xml:id="echoid-s29443" xml:space="preserve"> ꝓpinquior enim eſt ipſi cono pũctus <lb/>l, quàm pũctus i.</s> <s xml:id="echoid-s29444" xml:space="preserve"> Quod autẽ de ſuperficie conica ui-<lb/>detur, oculo exiſtente in pũcto i, idem per præceden <lb/>tem proximam uidetur cẽtro uiſus exiſtẽte per totã <lb/>lineam i a, utpote in pũcto t:</s> <s xml:id="echoid-s29445" xml:space="preserve"> & illud, quod uidetur ui <lb/>ſu exiſtẽte in pũcto l, uidetur in quolibet pũcto lineę <lb/>l a exiſtẽte uiſu:</s> <s xml:id="echoid-s29446" xml:space="preserve"> ergo & in pũcto k.</s> <s xml:id="echoid-s29447" xml:space="preserve"> Sed quod uidetur <lb/>â pũcto i maius eſt eo, quod uidetur à puncto l, & mi <lb/>nus eſſe uidetur per 86 huius:</s> <s xml:id="echoid-s29448" xml:space="preserve"> ergo illud, quod uide-<lb/>tur à pũcto t maius eſt illo, quod uidetur à pũcto k, & minus uidetur eſſe.</s> <s xml:id="echoid-s29449" xml:space="preserve"> Ethoc eſt quod proponi <lb/>tur.</s> <s xml:id="echoid-s29450" xml:space="preserve"> Ethocidẽ etiam ſuo modo de ambobus uiſibus poteſt demonſtrari.</s> <s xml:id="echoid-s29451" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s29452" xml:space="preserve"/> </p> <div xml:id="echoid-div1182" type="float" level="0" n="0"> <figure xlink:label="fig-0458-01" xlink:href="fig-0458-01a"> <variables xml:id="echoid-variables487" xml:space="preserve">a t k g h b p l i</variables> </figure> </div> </div> <div xml:id="echoid-div1184" type="section" level="0" n="0"> <head xml:id="echoid-head937" xml:space="preserve" style="it">89. Linea à centro uiſus ad uerticem coni duct a perpendiculari existẽte ſuper axem: ſuper-<lb/>ficiei conicæ medietas uidetur. Alhazen 36 n 4.</head> <p> <s xml:id="echoid-s29453" xml:space="preserve">Verbi gratia ſit pyramis a c n:</s> <s xml:id="echoid-s29454" xml:space="preserve"> cuius axis a d, & uertex a:</s> <s xml:id="echoid-s29455" xml:space="preserve"> palàm ergo per 89 th.</s> <s xml:id="echoid-s29456" xml:space="preserve"> 1 huius, quòd pun <lb/>ctum d eſt centrũ circuli baſis ipſius coni:</s> <s xml:id="echoid-s29457" xml:space="preserve"> ſitq̀;</s> <s xml:id="echoid-s29458" xml:space="preserve"> centrũ uiſus b:</s> <s xml:id="echoid-s29459" xml:space="preserve"> & ducatur linea b a faciens angulum <lb/>b a d rectũ.</s> <s xml:id="echoid-s29460" xml:space="preserve"> Dico, quòd conicæ ſuperficiei a c n medietas uidebitur.</s> <s xml:id="echoid-s29461" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0458-02a" xlink:href="fig-0458-02"/> Secet enim aliqua ſuperficies conum a c n æquidiſtáter baſi c n:</s> <s xml:id="echoid-s29462" xml:space="preserve"> hæc <lb/>ergo per 100 th.</s> <s xml:id="echoid-s29463" xml:space="preserve"> 1 huius ſecabit ipſam ſecũdum circulũ, qui ſit f g:</s> <s xml:id="echoid-s29464" xml:space="preserve"> & <lb/>eius cẽtrum, quod ſit pũctum l, erit in aliquo puncto axis a d:</s> <s xml:id="echoid-s29465" xml:space="preserve"> ſecetq́:</s> <s xml:id="echoid-s29466" xml:space="preserve"> <lb/>ſuperficies plana pyramidẽ per axem a d, & per cẽtrũ uiſus d:</s> <s xml:id="echoid-s29467" xml:space="preserve"> illa er-<lb/>go ſuperficies ſecabit circulum f g:</s> <s xml:id="echoid-s29468" xml:space="preserve"> linea quoq;</s> <s xml:id="echoid-s29469" xml:space="preserve"> cõmunis huic ſuper-<lb/>ficiei & circulo f g erit orthogonalis ſuper axem:</s> <s xml:id="echoid-s29470" xml:space="preserve"> quoniã axis eſt ere-<lb/>ctus ſuper ſuperficiẽ circuli, & tráſibit cẽtrũ circuli.</s> <s xml:id="echoid-s29471" xml:space="preserve"> Sit quoq;</s> <s xml:id="echoid-s29472" xml:space="preserve"> illa li-<lb/>nea k l:</s> <s xml:id="echoid-s29473" xml:space="preserve"> quę erit ք 28 p 1 æquidiſtás lineæ b a, & eſt cũilla in eadẽ ſu-<lb/>perficie.</s> <s xml:id="echoid-s29474" xml:space="preserve"> Ducatur quoq;</s> <s xml:id="echoid-s29475" xml:space="preserve"> ք cẽtrum circuli diameter f l g orthogona-<lb/>lis ſuper lineá k l ք 11 p 1:</s> <s xml:id="echoid-s29476" xml:space="preserve"> & à terminis huius diametri protrahantur <lb/>duę lineæ cótingentes circulũ per 17 p 3, quę ſint f e & g h:</s> <s xml:id="echoid-s29477" xml:space="preserve"> & ab eiſdẽ <lb/>pũctis g & h ducãtur duę lineæ lógitudinis ad uerticẽ coni ք 101 th.</s> <s xml:id="echoid-s29478" xml:space="preserve"> 1 <lb/>huius, quę ſint f a & g a:</s> <s xml:id="echoid-s29479" xml:space="preserve"> duę ergo ſuperficies planę, in quarũ una ſunt <lb/>lineę f e & f a, & in quarũ altera ſunt lineæ g h & g a, palàm quoniam <lb/>cõtingẽt pyramidẽ ſecũdũ lineas lõgitudinis, quę ſunt f a & g a ք 95 <lb/>th.</s> <s xml:id="echoid-s29480" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s29481" xml:space="preserve"> Et quoniã linea k l æquidiſtat lineæ b a, & lineis cotingen <lb/>tibus circulũ, quę ſunt f e & g h, ut patet per 16 p 3, & per 28 p 1:</s> <s xml:id="echoid-s29482" xml:space="preserve"> erunt <lb/>per 9 p 11 lineæ f e & g h æquidiſtãtes lineæ b a:</s> <s xml:id="echoid-s29483" xml:space="preserve"> quęlibet ergo ipſarũ <lb/>eſt in eadẽ ſuperficie cũ illa per 1 th.</s> <s xml:id="echoid-s29484" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s29485" xml:space="preserve"> Illę ergo duæ ſuperficies <lb/>neceſſariò ſecabũt ſe ſuper lineã b a per 19 th.</s> <s xml:id="echoid-s29486" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s29487" xml:space="preserve"> utraq;</s> <s xml:id="echoid-s29488" xml:space="preserve"> ergo ſuperficierũ pyramidẽ propoſitã <lb/>in terminis diametri unius ſuorũ circulorũ cótigentiũ trãſit per cẽtrum uiſus.</s> <s xml:id="echoid-s29489" xml:space="preserve"> Quod ergo ſuperfi-<lb/>ciei conicę inter illas ſuperficies cadit, apparet uiſui:</s> <s xml:id="echoid-s29490" xml:space="preserve"> eſt aũt hæc medietas pyramidis, quoniá illas li <lb/>neas contingentes interiacet medietas circuli.</s> <s xml:id="echoid-s29491" xml:space="preserve"> In hoc ergo ſitu medietas ſuperficiei conicæ uide-<lb/>tur.</s> <s xml:id="echoid-s29492" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s29493" xml:space="preserve"/> </p> <div xml:id="echoid-div1184" type="float" level="0" n="0"> <figure xlink:label="fig-0458-02" xlink:href="fig-0458-02a"> <variables xml:id="echoid-variables488" xml:space="preserve">b a f j g e k y n d c</variables> </figure> </div> </div> <div xml:id="echoid-div1186" type="section" level="0" n="0"> <head xml:id="echoid-head938" xml:space="preserve" style="it">90. Linea à centro uiſus ad uerticem coni duct a angulũ obtuſum cũ axetenente, nec tamen <lb/>cum aliqua line arum longitudinis coni unita: uidetur ſnperficiei conicæ pars maior medietate. <lb/>Alhazen 37 n 4.</head> <p> <s xml:id="echoid-s29494" xml:space="preserve">Sit pyramis b i m:</s> <s xml:id="echoid-s29495" xml:space="preserve"> cuius axis b d:</s> <s xml:id="echoid-s29496" xml:space="preserve"> uertex b:</s> <s xml:id="echoid-s29497" xml:space="preserve"> palamq́;</s> <s xml:id="echoid-s29498" xml:space="preserve"> per 89 th.</s> <s xml:id="echoid-s29499" xml:space="preserve"> 1 huius, quòd cẽtrũ circuli baſis eſt <lb/>punctũ d:</s> <s xml:id="echoid-s29500" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s29501" xml:space="preserve"> punctũ a centrũ uiſus:</s> <s xml:id="echoid-s29502" xml:space="preserve"> & ducta linea a b, fiat angulus a b d obtuſus, ita tamẽ, ut linea <lb/>a b nõ fiat una linea cũ aliqua linearũ lõgitudinis coni, ſed ſecet eas utcũq;</s> <s xml:id="echoid-s29503" xml:space="preserve"> poſsibile eſt productas <lb/>oẽs:</s> <s xml:id="echoid-s29504" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s29505" xml:space="preserve"> tũc uiſus altior uertice pyramidis:</s> <s xml:id="echoid-s29506" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s29507" xml:space="preserve">, ut in pręcedẽte, circulus e h æquidiſtás baſipy-<lb/> <pb o="157" file="0459" n="459" rhead="LIBER QVARTVS."/> ramidis, quæ eſt i m:</s> <s xml:id="echoid-s29508" xml:space="preserve"> & linea communis huic ſuperficiei & circulo, (in quo eſt centrũ uiſus punctũ <lb/>a, & axis coni, qui eſt b d) ſit linea e h:</s> <s xml:id="echoid-s29509" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s29510" xml:space="preserve"> linea e h perpendicula-<lb/>ris ſuper axem b d:</s> <s xml:id="echoid-s29511" xml:space="preserve"> & producatur linea e h extra pyramidem, donec <lb/> <anchor type="figure" xlink:label="fig-0459-01a" xlink:href="fig-0459-01"/> concurrat cum linea b a, producta ultra punctum b:</s> <s xml:id="echoid-s29512" xml:space="preserve"> cócurret autem <lb/>ք 14 th.</s> <s xml:id="echoid-s29513" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s29514" xml:space="preserve"> ideo, quia angulus a b d eſt obtuſus ex hypotheſi, & <lb/>angulus d b h eſt acutus per 32 p 1, & linea e h eſt perpendicularis ſu-<lb/>per axem b d.</s> <s xml:id="echoid-s29515" xml:space="preserve"> Sit ergo concurſus punctus g:</s> <s xml:id="echoid-s29516" xml:space="preserve"> & â puncto g producan-<lb/>tur duæ lineę g f & g r circulũ e h cõtingentes per 17 p 3:</s> <s xml:id="echoid-s29517" xml:space="preserve"> contingãtq́;</s> <s xml:id="echoid-s29518" xml:space="preserve"> <lb/>circulũ in duobus punctis f & r:</s> <s xml:id="echoid-s29519" xml:space="preserve"> & ab ijs punctis per 101 th.</s> <s xml:id="echoid-s29520" xml:space="preserve"> 1 huius ꝓ-<lb/>ducantur lineæ longitudinis ad uerticem coni punctũ b, quę ſint f b <lb/>& r b:</s> <s xml:id="echoid-s29521" xml:space="preserve"> ſuperficies ergo illæ, in quibus ſunt lineæ g f & f b, & lineæ g r <lb/>& r b cõtingunt pyramidem, & in utraq;</s> <s xml:id="echoid-s29522" xml:space="preserve"> iſtarum ſuperficierum erit <lb/>uertex pyramidis punctus b, & punctus g, in quo concurrit linea a b <lb/>cum linea e h:</s> <s xml:id="echoid-s29523" xml:space="preserve"> ergo linea a b g per 1 p 11 & 19 th.</s> <s xml:id="echoid-s29524" xml:space="preserve"> 1 huius eſt in utraq;</s> <s xml:id="echoid-s29525" xml:space="preserve"> il-<lb/>larum ſuperficierum:</s> <s xml:id="echoid-s29526" xml:space="preserve"> ergo utraq;</s> <s xml:id="echoid-s29527" xml:space="preserve"> ſuperficies tran ſit per punctũ a cen <lb/>trum uiſus.</s> <s xml:id="echoid-s29528" xml:space="preserve"> Et quoniam per 58 th.</s> <s xml:id="echoid-s29529" xml:space="preserve"> 1 huius duæ lineæ g f & g r inclu-<lb/>dunt minorem partem circuli:</s> <s xml:id="echoid-s29530" xml:space="preserve"> quoniam arcus circuli interiacẽs pun <lb/>cta contingentię duarum linearum ab eodem puncto productarum, <lb/>eſt minor ſemicirculo:</s> <s xml:id="echoid-s29531" xml:space="preserve"> tunc patet quòd illæ duę ſuperficies includũt <lb/>minorem partẽ ſuperficiei conicę quàm ſit medietas:</s> <s xml:id="echoid-s29532" xml:space="preserve"> reſiduũ ergo il <lb/>lius ſuperficiei eſt maius medietate:</s> <s xml:id="echoid-s29533" xml:space="preserve"> hoc autẽ uidetur à uiſu taliter, <lb/>ut proponitur, collocato.</s> <s xml:id="echoid-s29534" xml:space="preserve"> Pars ergo ſuperficiei conicæ maior medietate taliter uidetur.</s> <s xml:id="echoid-s29535" xml:space="preserve"> Ethoc eſt <lb/>propoſitum.</s> <s xml:id="echoid-s29536" xml:space="preserve"> Ambobus uero uiſibus adhuc uidetur magis.</s> <s xml:id="echoid-s29537" xml:space="preserve"/> </p> <div xml:id="echoid-div1186" type="float" level="0" n="0"> <figure xlink:label="fig-0459-01" xlink:href="fig-0459-01a"> <variables xml:id="echoid-variables489" xml:space="preserve">a g e c f h g r i d m</variables> </figure> </div> </div> <div xml:id="echoid-div1188" type="section" level="0" n="0"> <head xml:id="echoid-head939" xml:space="preserve" style="it">91. Cum linea longitudinis coni producta ultra uerticem cum centro uiſus concurrerit, nihil <lb/>uiſum totius ſuperficiei conicæ latebit: niſi linea longitudinis illa ſola. Alhazen 38 n 4.</head> <p> <s xml:id="echoid-s29538" xml:space="preserve">Sit pyramis, cuius uertex ſit punctũ b:</s> <s xml:id="echoid-s29539" xml:space="preserve"> & linea longitudinis ſitq́;</s> <s xml:id="echoid-s29540" xml:space="preserve"> centrum uiſus punctũ a:</s> <s xml:id="echoid-s29541" xml:space="preserve"> <lb/>& linea c b producta ultra punctũ b concurrat cũ cẽtro uiſus puncto a.</s> <s xml:id="echoid-s29542" xml:space="preserve"> Dico, quòd non latebit ui-<lb/>ſum totius huius ſuperficiei conicæ pars aliqua, præter quandã lineam intellectualẽ, quæ eſt ipſa li <lb/>nealongitudinis b c.</s> <s xml:id="echoid-s29543" xml:space="preserve"> Omnis enim ſuperficies, in qua eſt linea à centro uiſus ad aliquem punctum <lb/>axis ducta, ſecabit pyramidẽ, excepta tantũ illa ſuperficie, in qua eſt linea a b c:</s> <s xml:id="echoid-s29544" xml:space="preserve"> hæc enim contingit <lb/>pyramidem ſecundum lineam b c per 95 th.</s> <s xml:id="echoid-s29545" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s29546" xml:space="preserve"> Et quoniam illud, quod ſub ſuperficie contin-<lb/>gente pyramidem & tranſeunte centrum uiſus continetur, occurrit uiſui per 17 th.</s> <s xml:id="echoid-s29547" xml:space="preserve"> 3 huius:</s> <s xml:id="echoid-s29548" xml:space="preserve"> ſormæ <lb/>enim omnium punctorũ ſuperficiei illius conicæ in ſuperficie uiſus de-<lb/>pinguntur:</s> <s xml:id="echoid-s29549" xml:space="preserve"> palàm ergo quoniã tota ſuperficies conica uidetur, excepta <lb/> <anchor type="figure" xlink:label="fig-0459-02a" xlink:href="fig-0459-02"/> ſola linea intellectuali, quæ eſt b c.</s> <s xml:id="echoid-s29550" xml:space="preserve"> Dato enim quocunq;</s> <s xml:id="echoid-s29551" xml:space="preserve"> puncto ſuperfi-<lb/>ciei pyramidalis extra lineam b c:</s> <s xml:id="echoid-s29552" xml:space="preserve"> dico, quòd illud uidebitur.</s> <s xml:id="echoid-s29553" xml:space="preserve"> Sit enim il-<lb/>lud punctũ h:</s> <s xml:id="echoid-s29554" xml:space="preserve"> & ducatur ad ipſum à centro uiſus a linea a h:</s> <s xml:id="echoid-s29555" xml:space="preserve"> & ab illo eo-<lb/>dẽ per 101 th.</s> <s xml:id="echoid-s29556" xml:space="preserve"> 1 huius ducatur linea longitudinis, quæ ſit h b:</s> <s xml:id="echoid-s29557" xml:space="preserve"> fietq́;</s> <s xml:id="echoid-s29558" xml:space="preserve"> trian-<lb/>gulus h b a, qui neceſſariò eritin aliqua ſuperficie pyramidẽ ſecãte, per <lb/>tranſeunte centrũ uiſus a:</s> <s xml:id="echoid-s29559" xml:space="preserve"> ex lineis aũt illius ſuperficiei non cadunt, niſi <lb/>duæ in ſuperficiẽ conicã pyramidis, ſcilicet linea lõgitudinis b h, & linea <lb/>oppoſita lineæ b h in alia parte pyramidis:</s> <s xml:id="echoid-s29560" xml:space="preserve"> quoniã, ut patet ք 90 th.</s> <s xml:id="echoid-s29561" xml:space="preserve"> 1 hu-<lb/>ius, planę ſuperficiei ſecantis conũ trans axem & ſuperficiei conicæ cõ-<lb/>munis ſection eſt trigonũ duabus lineis longitudinis pyramidis & diame <lb/>tro baſis contentũ:</s> <s xml:id="echoid-s29562" xml:space="preserve"> linea uerò a h ſecat lineam b h in pũcto h, & linea c b <lb/>ſecat eandem b h in puncto b per 91 th.</s> <s xml:id="echoid-s29563" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s29564" xml:space="preserve"> lineæ ergo a h nulla linea <lb/>cõcurret à uertice pyramidis niſi in puncto a:</s> <s xml:id="echoid-s29565" xml:space="preserve"> nec enim ad aliquod pun-<lb/>ctũ mediũ lineę a h à uertice b ductæ lineę incident:</s> <s xml:id="echoid-s29566" xml:space="preserve"> nõ occultabitur er-<lb/>go pũctus h ab alìquo alio pũcto, quò minus perueniat ad centrũ uiſus <lb/>a.</s> <s xml:id="echoid-s29567" xml:space="preserve"> Occurrit ergo punctus h uiſui, cũ inter ipſum & uiſum nõ accidat ſoli-<lb/>dicorporis interpoſitio.</s> <s xml:id="echoid-s29568" xml:space="preserve"> Eadẽ quoq;</s> <s xml:id="echoid-s29569" xml:space="preserve"> eſt probatio de quolibet alio dato <lb/>puncto ſuperficiei pyramidis:</s> <s xml:id="echoid-s29570" xml:space="preserve"> in linea uerò b c, quę perpendicularis eſt <lb/>ſuper ſuperficiẽ uiſus per 72 th.</s> <s xml:id="echoid-s29571" xml:space="preserve"> 1 huius, folũ tantũ punctũ poſsibile eſt uideri, ut oſtẽſum eſt in 4 hu <lb/>ius:</s> <s xml:id="echoid-s29572" xml:space="preserve"> omnia uerò alia puncta lineæ b c neceſſariò occultãtur.</s> <s xml:id="echoid-s29573" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s29574" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s29575" xml:space="preserve"> ex ijs, <lb/>quoniã in hoc ſitu nulla ſuperficierũ pyramidũ contingẽtiũ peruenit ad cẽtrũ uiſus, pręter illã, quę <lb/>in linea b c longitudinis centrũ uiſus tranſeuntis pyramidẽ cõtingit:</s> <s xml:id="echoid-s29576" xml:space="preserve"> & oẽs ſuperficies aliæ conum <lb/>contingentes ſecant lineam productã à centro ad ipſam pyramidẽ inter uerticẽ coni & cẽtrũ uiſus.</s> <s xml:id="echoid-s29577" xml:space="preserve"/> </p> <div xml:id="echoid-div1188" type="float" level="0" n="0"> <figure xlink:label="fig-0459-02" xlink:href="fig-0459-02a"> <variables xml:id="echoid-variables490" xml:space="preserve">a b h c</variables> </figure> </div> </div> <div xml:id="echoid-div1190" type="section" level="0" n="0"> <head xml:id="echoid-head940" xml:space="preserve" style="it">92. Axe pyramidis cum centro uiſus uerſus uerticem concurrente: tota conica ſuperficies <lb/>uno oculo uidetur. Alhazen 39 n 4.</head> <p> <s xml:id="echoid-s29578" xml:space="preserve">Eſto data pyramis, cuius axis b c:</s> <s xml:id="echoid-s29579" xml:space="preserve"> uertex quoq;</s> <s xml:id="echoid-s29580" xml:space="preserve"> punctus b:</s> <s xml:id="echoid-s29581" xml:space="preserve"> & ſit uiſus centrũ pũctũ a:</s> <s xml:id="echoid-s29582" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s29583" xml:space="preserve">, ut axis <lb/>b c ꝓductus currat in punctũ a.</s> <s xml:id="echoid-s29584" xml:space="preserve"> Dico, quòd in hoc ſitu oculi tota conica ſuperficies pyramidis oc-<lb/>currit uni uiſui:</s> <s xml:id="echoid-s29585" xml:space="preserve"> nullus enim punctus ſuperficiei conicæ totins pyramidis uiſui occultatur.</s> <s xml:id="echoid-s29586" xml:space="preserve"> Dato e-<lb/>nim quocũq;</s> <s xml:id="echoid-s29587" xml:space="preserve"> puncto, ſit ille l:</s> <s xml:id="echoid-s29588" xml:space="preserve"> & ducatur ad ipſum à cẽtro uiſus a linea a l:</s> <s xml:id="echoid-s29589" xml:space="preserve"> & ab ipſo pũcto l ducatur <lb/> <pb o="158" file="0460" n="460" rhead="VITELLONIS OPTICAE"/> per 101 th.</s> <s xml:id="echoid-s29590" xml:space="preserve"> 1 huius linea longitudinis pyramidis, quæ ſit l b:</s> <s xml:id="echoid-s29591" xml:space="preserve"> fietq́;</s> <s xml:id="echoid-s29592" xml:space="preserve"> trigonũ l b a, quod neceſſariõ erit <lb/>in ſuperficie pyramidẽ ſecante, ideo quòd linea a e ducta à cẽtro uiſus in-<lb/>tratin ipſam pyramidẽ, ſecãs ipſam, & ipſa eſt in dicta ſuperficie per 1 p 11, <lb/> <anchor type="figure" xlink:label="fig-0460-01a" xlink:href="fig-0460-01"/> quoniã linea a b eſt in illa ſuperficie:</s> <s xml:id="echoid-s29593" xml:space="preserve"> linea uerò a l ſecat lineá b l in puncto <lb/>l:</s> <s xml:id="echoid-s29594" xml:space="preserve"> ex lineis uerò ſuperficiei, in qua ſunt duæ lineæ a l & b l, nó ſunt, niſi duę <lb/>tantũ lineæ in ſuperficie pyramidis, ſcilicet linea lõgitudinis, quæ eſt b l, <lb/>& linea alia longitudinis illi oppoſita, quæ ſit b k, ut patet per 90 th.</s> <s xml:id="echoid-s29595" xml:space="preserve"> 1 hu-<lb/>ius:</s> <s xml:id="echoid-s29596" xml:space="preserve"> hęc ergo linea b k producta ultra punctũ b, cũ ſit in eadẽ ſuperficie cũ <lb/>lineis a b & b l, neceſſariò ſecabit angulũ a b l:</s> <s xml:id="echoid-s29597" xml:space="preserve"> ergo per 29 th.</s> <s xml:id="echoid-s29598" xml:space="preserve"> 1 huius ipſa <lb/>ſecabit & baſim a l:</s> <s xml:id="echoid-s29599" xml:space="preserve"> ſit ergo ut ſecet illã in puncto d.</s> <s xml:id="echoid-s29600" xml:space="preserve"> Et quia linea a l ſecat <lb/>duas lineas k b & l b, quæ ſolæ ex lineis ſuperficiei pyramidẽ ſecãtis ſunt <lb/>in pyramidis ſuperficie, ſecat enim linea a l lineá k b extra pyramidem in <lb/>pũcto d, & lineam l b in ſuperficie pyramidis, in pũcto l:</s> <s xml:id="echoid-s29601" xml:space="preserve"> producta ergo li-<lb/>nea a k in infinitũ, nõ concurret cũ aliqua illarũ linearũ:</s> <s xml:id="echoid-s29602" xml:space="preserve"> nõ interponetur <lb/>ergo ſolidũ punctũ, quod eſt k inter uiſum & pũctũ l:</s> <s xml:id="echoid-s29603" xml:space="preserve"> ſed nec aliquod alio <lb/>rũ punctorũ ipſius pyramidis:</s> <s xml:id="echoid-s29604" xml:space="preserve"> quoniã nullũ ipſorũ caditin illa ſuperficie.</s> <s xml:id="echoid-s29605" xml:space="preserve"> <lb/>Nõ occultabitur ergo tũc uiſui exiſtẽti in pũcto a datũ punctũ l:</s> <s xml:id="echoid-s29606" xml:space="preserve"> cũinter <lb/>ipſum & cẽtrũ uiſus nõ accidat aliqua ſolidi corporis interpoſitio.</s> <s xml:id="echoid-s29607" xml:space="preserve"> Et ea-<lb/>dẽ eſt demonſtratio de quolibet dato pũcto in tota ſuperficie pyramidis.</s> <s xml:id="echoid-s29608" xml:space="preserve"> <lb/>Patet ergo propoſitũ.</s> <s xml:id="echoid-s29609" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s29610" xml:space="preserve"> ex his, quoniã in hoc ſitu nulla ſuperfi <lb/>cierũ cõtingentiũ pyramidẽ tranſit per centrũ uiſus, ſed quælibetipſarũ <lb/>ſecabit lineam à centro uiſus per uerticem conum intrantem inter centrum uiſus & pyramidem, <lb/>ſcilicet in uertice ipſius axis, ut patetintuenti.</s> <s xml:id="echoid-s29611" xml:space="preserve"/> </p> <div xml:id="echoid-div1190" type="float" level="0" n="0"> <figure xlink:label="fig-0460-01" xlink:href="fig-0460-01a"> <variables xml:id="echoid-variables491" xml:space="preserve">a d b k j c</variables> </figure> </div> </div> <div xml:id="echoid-div1192" type="section" level="0" n="0"> <head xml:id="echoid-head941" xml:space="preserve" style="it">93. Omnes lineæ uel ſuperficies, inter lineas uel ſuperficies cõtingentes colũnã uel pyramidẽ <lb/>rotũdãſuքficiẽ uiſam terminãtes àcẽtro uiſus ꝓductæ, colũnã uel pyramidẽ neceſſariò ſecabũt.</head> <p> <s xml:id="echoid-s29612" xml:space="preserve">Verbi gratia, ſint duæ lineæ lõgitudinis columnę uel pyramidis terminantes uiſam ſuperficiem, <lb/>quę ſint a b & c d.</s> <s xml:id="echoid-s29613" xml:space="preserve"> Dico, quòd ſi à centro uiſus (quod <lb/>eſte) ducatur linea e finter lineas illas a b & c d, quo <lb/> <anchor type="figure" xlink:label="fig-0460-02a" xlink:href="fig-0460-02"/> niam linea e f ſecabit propoſitã columnã uel pyramic <lb/>dẽ.</s> <s xml:id="echoid-s29614" xml:space="preserve"> Tranſeat enim ſuperficies plana columnã uel py-<lb/>ramidẽ ſecans ipſam in puncto f æquidiſtanter baſi:</s> <s xml:id="echoid-s29615" xml:space="preserve"> <lb/>eritq́;</s> <s xml:id="echoid-s29616" xml:space="preserve"> per 100 th.</s> <s xml:id="echoid-s29617" xml:space="preserve"> 1 huius cõmunis ſectio circulus, qui <lb/>ſit g f h:</s> <s xml:id="echoid-s29618" xml:space="preserve"> qui ſecet lineas lõgitudinis colũnę uel pyra-<lb/>midis, eam ſcilicet, quę a b, in pũcto g, & eam, quę eſt <lb/>c d, in puncto h:</s> <s xml:id="echoid-s29619" xml:space="preserve"> & ducantur à pũcto e per 17 p 3 duæ <lb/>lineę cõtingẽtes illũ circulũ, quę ſint e g & e h:</s> <s xml:id="echoid-s29620" xml:space="preserve"> palàm <lb/>aũt per 57 th.</s> <s xml:id="echoid-s29621" xml:space="preserve"> 1 huius quoniá linea e fin eadẽ ſuperfi-<lb/>cie cũlineis illis exiftẽs, ſecat circulũ g fh:</s> <s xml:id="echoid-s29622" xml:space="preserve"> ergo ſeca <lb/>bit columná uel pyramidẽ, quæ per eundẽ circulum <lb/>ſecatur.</s> <s xml:id="echoid-s29623" xml:space="preserve"> Idẽ quoq;</s> <s xml:id="echoid-s29624" xml:space="preserve"> accidit ſi per ſectionem lineæ lon <lb/>gitudinis hoc placuerit demonſtrari, & in idem re-<lb/>dit.</s> <s xml:id="echoid-s29625" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s29626" xml:space="preserve"/> </p> <div xml:id="echoid-div1192" type="float" level="0" n="0"> <figure xlink:label="fig-0460-02" xlink:href="fig-0460-02a"> <variables xml:id="echoid-variables492" xml:space="preserve">c k a f y g e d g</variables> </figure> </div> </div> <div xml:id="echoid-div1194" type="section" level="0" n="0"> <head xml:id="echoid-head942" xml:space="preserve" style="it">94. Pluribus planis ſuperficiebus centrum uiſus <lb/>tranſeuntibus ſecundũ lineas longitudinis partis <lb/>ſuperficiei uiſæ columnã uel pyramidẽ conuexam <lb/>ſecantibus: ſolã ſuperficiẽ axem columnæ pertran-<lb/>ſeuntẽ, ſuperficiẽ colũnarẽ uel pyramidalẽ uiſam <lb/>per æqualia diuidere: & econuerſo ſuperficiẽ per æ-<lb/>qualia illam uiſam ſuperficiem diuidentem, axem tranſire eſt neceſſe.</head> <p> <s xml:id="echoid-s29627" xml:space="preserve">Sit colũna cõuexa, cuius ſuperficies uiſa ſit e d f g:</s> <s xml:id="echoid-s29628" xml:space="preserve"> & axis eius ſit h i:</s> <s xml:id="echoid-s29629" xml:space="preserve"> & ſit centrũ uiſus punctũ a:</s> <s xml:id="echoid-s29630" xml:space="preserve"> <lb/>ſintq́;</s> <s xml:id="echoid-s29631" xml:space="preserve"> lineę longitudinis colũnæ, continẽtes uiſam ſuperficiẽ, quæ e d & f g.</s> <s xml:id="echoid-s29632" xml:space="preserve"> Imaginẽtur quoq;</s> <s xml:id="echoid-s29633" xml:space="preserve"> mul-<lb/>tę planæ ſuperficies tranſeuntes centrũ uiſus a, & ſecantes e d f g uiſam ſuperficiẽ columnæ.</s> <s xml:id="echoid-s29634" xml:space="preserve"> Dico, <lb/>quòd ſola illa, quæ pertrãſit axem h i, ipſam uiſam ſuperficiẽ ք ęqualia diuidit, & nulla aliarũ:</s> <s xml:id="echoid-s29635" xml:space="preserve"> ſola e-<lb/>nim hæc erecta eſt ſuper cõuexam ſuperficiẽ colũn æ:</s> <s xml:id="echoid-s29636" xml:space="preserve"> quoniã cõmunis ſectio illius ſuperficiei ſecan <lb/>tis, & ſuperficiei colũnæ eſt rectangulũ ſub duabus lineis lõgitudinis colũnæ & duabus diametris <lb/>baſium cõtentũ, ut patet ք 93 th.</s> <s xml:id="echoid-s29637" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s29638" xml:space="preserve"> ergo cõmunis ſectio illius ſuperficiei & uiſę ſuperficiei con <lb/>uexæ ipſius colũnæ ſit linea lõgitudinis colũnæ, quæ m o:</s> <s xml:id="echoid-s29639" xml:space="preserve"> & imaginetur ſuperficies plana cõtingẽs <lb/>columnã ſecũdũ lineã longitudinis m o ք 95 th.</s> <s xml:id="echoid-s29640" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s29641" xml:space="preserve"> erũt ergo illa cõtingẽs ſuperficies & ſuper-<lb/>ficies ſecás per axem erectæ ad inuicẽ per 97 th.</s> <s xml:id="echoid-s29642" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s29643" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s29644" xml:space="preserve"> in linea m o ſignetur punctũ p:</s> <s xml:id="echoid-s29645" xml:space="preserve"> & in <lb/>ſuperficie cótingente ducatur linea t p s:</s> <s xml:id="echoid-s29646" xml:space="preserve"> tunc palàm quòd linea t p s cõtinget quẽdã circulũ ſuper-<lb/>ficiei colũnæ æ quidiſtantẽ baſibus.</s> <s xml:id="echoid-s29647" xml:space="preserve"> qui ſit b q:</s> <s xml:id="echoid-s29648" xml:space="preserve"> & eius centrũ ſit u:</s> <s xml:id="echoid-s29649" xml:space="preserve"> ducãturq́;</s> <s xml:id="echoid-s29650" xml:space="preserve"> per 17 p 3 lineę a b & a q <lb/> <pb o="159" file="0461" n="461" rhead="LIBER QVARTVS"/> à centro uiſus circulũ b q cõtingentes:</s> <s xml:id="echoid-s29651" xml:space="preserve"> erũt ergo illæ lineæ æquales per 58 th.</s> <s xml:id="echoid-s29652" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s29653" xml:space="preserve"> ſecentq́;</s> <s xml:id="echoid-s29654" xml:space="preserve"> lineã <lb/>illã circulũ contingentẽ, quæ eſt t p s, in punctis t & s:</s> <s xml:id="echoid-s29655" xml:space="preserve"> & ducatur linea a p:</s> <s xml:id="echoid-s29656" xml:space="preserve"> quæ producta, ut patet ք <lb/>18 p 3 pertinget ad axem in pũctũ u centrũ circuli:</s> <s xml:id="echoid-s29657" xml:space="preserve"> & <lb/>ducãtur intra columnã lineæ b u & q u ſemidiametri <lb/> <anchor type="figure" xlink:label="fig-0461-01a" xlink:href="fig-0461-01"/> circuli b q.</s> <s xml:id="echoid-s29658" xml:space="preserve"> Trigona ita q;</s> <s xml:id="echoid-s29659" xml:space="preserve"> a b u & a q u ſunt ęquilate-<lb/>ra:</s> <s xml:id="echoid-s29660" xml:space="preserve"> ergo per 8 p 1 ſunt ęquiangula:</s> <s xml:id="echoid-s29661" xml:space="preserve"> angulus ergo u a b <lb/>eſt æqualis angulo u a q:</s> <s xml:id="echoid-s29662" xml:space="preserve"> ſed in trigono at p angulus <lb/>a p t eſt æqualis angulo a p s trigoni a p s per defini-<lb/>tionẽ lineę ſuper ſuperficiẽ erectę:</s> <s xml:id="echoid-s29663" xml:space="preserve"> ergo per 32 p 1 an-<lb/>gulus a t p eſt æqualis angulo a s p:</s> <s xml:id="echoid-s29664" xml:space="preserve"> ergo per 6 p 1 eſt <lb/>linea a t æqualis lineæ a s.</s> <s xml:id="echoid-s29665" xml:space="preserve"> Et quia lineę a b & a q ſunt <lb/>ęquales, ut ſuprà patuit:</s> <s xml:id="echoid-s29666" xml:space="preserve"> ablatis ergo hinc inde lineis <lb/>a t & a s, remanebit linea t q æqualis lineæ s b:</s> <s xml:id="echoid-s29667" xml:space="preserve"> ſed li-<lb/>nea t q eſt æqualis lineæ t p per 58 t 1 huius:</s> <s xml:id="echoid-s29668" xml:space="preserve"> quoniã à <lb/>puncto t ductæ ſunt duæ lineę circulũ contingẽtes, <lb/>quæ ſunt lineę t q & t p:</s> <s xml:id="echoid-s29669" xml:space="preserve"> ſimiliter quoq;</s> <s xml:id="echoid-s29670" xml:space="preserve"> fit linea s b ę-<lb/>qualis lineæ s p.</s> <s xml:id="echoid-s29671" xml:space="preserve"> Cũ ergo per 13 p 1 anguli b s p & q t p <lb/>ſint æquales, erit per 4 p 1 chorda p b ęqualis chordę <lb/>p q:</s> <s xml:id="echoid-s29672" xml:space="preserve"> ergo per 28 p 3 erit arcus p b æqualis arcui p q.</s> <s xml:id="echoid-s29673" xml:space="preserve"> Et <lb/>quoniam idẽ accidit in baſibus columnę, & in quoli-<lb/>bet aliorũ circulorũ æquidiſtãte baſibus:</s> <s xml:id="echoid-s29674" xml:space="preserve"> patet ergo <lb/>propoſitũ primũ, ſcilicet quòd ſuperficies plana ſe <lb/>cans columná per axem & tranſiens cẽtrũ uiſus, ſe-<lb/>cat ſuperficiẽ uiſam per æqualia.</s> <s xml:id="echoid-s29675" xml:space="preserve"> Et quoniã oẽs aliæ <lb/>ſuperficies declinantes ab axe obliquè incidunt ſu-<lb/>perficiei contingenti columnã in media linea ſuperficiei uiſæ ipſius columnæ, quæ eſt linea m o, pa <lb/>tet quòd nulla ipſarũ illã ſuperficiẽ uiſam per æqualia ſecat.</s> <s xml:id="echoid-s29676" xml:space="preserve"> Sed etiã ſuperficies, quę uiſam partem <lb/>ſuperficiei columnę per ęqualia ſecat, neceſſariò tranſit per axem.</s> <s xml:id="echoid-s29677" xml:space="preserve"> Sit enim diſpoſitio, quæ prius, & <lb/>ducantur oẽs lineæ priores:</s> <s xml:id="echoid-s29678" xml:space="preserve"> erit ergo linea m o, cui illa ſup erficies incidit, diuidẽs ſuperficiẽ uiſam <lb/>per æqualia:</s> <s xml:id="echoid-s29679" xml:space="preserve"> & ipſa eſt cõmunis ſectio ſuperficierũ ſecantis & cõtingentis:</s> <s xml:id="echoid-s29680" xml:space="preserve"> erit itaq;</s> <s xml:id="echoid-s29681" xml:space="preserve"> per 61 th.</s> <s xml:id="echoid-s29682" xml:space="preserve"> 1 hu <lb/>ius linea p t ęqualis lineę p s:</s> <s xml:id="echoid-s29683" xml:space="preserve"> ſed linea p t eſt ęqualis lineæ t q per 58 th.</s> <s xml:id="echoid-s29684" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s29685" xml:space="preserve"> & ſimiliter linea p s <lb/>æqualis ipſi lineę s b:</s> <s xml:id="echoid-s29686" xml:space="preserve"> relinquitur ergo linea a t æqualis eſſe lineę a s.</s> <s xml:id="echoid-s29687" xml:space="preserve"> Et quoniã in illis trigonis a p s <lb/>& a p t linea a p eſt cõmunis ambobus ipſis:</s> <s xml:id="echoid-s29688" xml:space="preserve"> erit ergo per 8 p 1 angulus a p t æqualis angulo a p s:</s> <s xml:id="echoid-s29689" xml:space="preserve"> u-<lb/>terq;</s> <s xml:id="echoid-s29690" xml:space="preserve"> ergo illorũ angulorũ eſt rectus, & linea a p eſt perpẽdicularis ſuper lineã t p s:</s> <s xml:id="echoid-s29691" xml:space="preserve"> linea ergo a p cũ <lb/>æquales angulos cõtineat cũ linea m o:</s> <s xml:id="echoid-s29692" xml:space="preserve"> palã per definitionẽ quoniã ipſa eſt erecta ſuper ſuperficiẽ <lb/>contingentẽ columnã in linea m o:</s> <s xml:id="echoid-s29693" xml:space="preserve"> ergo per 18 p 11 ſuperficies, in qua eſt linea a p ſecans columnam, <lb/>erecta eſt ſuper ſuperficiem ipſam contingentem columnam ſecundũ lineam m o.</s> <s xml:id="echoid-s29694" xml:space="preserve"> Ergo per 97 th.</s> <s xml:id="echoid-s29695" xml:space="preserve"> 1 <lb/>huius patet quòd ipſa tranſit per illius columnę axem.</s> <s xml:id="echoid-s29696" xml:space="preserve"> Et penitus eodem modo eſt in rotundis py-<lb/>ramidibus demonſtrandum.</s> <s xml:id="echoid-s29697" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s29698" xml:space="preserve"/> </p> <div xml:id="echoid-div1194" type="float" level="0" n="0"> <figure xlink:label="fig-0461-01" xlink:href="fig-0461-01a"> <variables xml:id="echoid-variables493" xml:space="preserve">o a g i e s p t b u q m d f h</variables> </figure> </div> </div> <div xml:id="echoid-div1196" type="section" level="0" n="0"> <head xml:id="echoid-head943" xml:space="preserve" style="it">95. Rect angulæ magnitudines à maiori diſtantia uiſæ circulares apparẽt. Euclides 9 th. opt.</head> <p> <s xml:id="echoid-s29699" xml:space="preserve">Sit magnitudo rectangula uiſa ex magna diſtantia, quæ ſit b g d z.</s> <s xml:id="echoid-s29700" xml:space="preserve"> Quoniã ergo unumquodq;</s> <s xml:id="echoid-s29701" xml:space="preserve"> ui-<lb/>ſorum habet longitudinẽ diſtantię, qua facta non fiet uiſio, ut patet <lb/> <anchor type="figure" xlink:label="fig-0461-02a" xlink:href="fig-0461-02"/> per 8 huius:</s> <s xml:id="echoid-s29702" xml:space="preserve"> corpus uerò angulare circa angulum eſt minus, quàm <lb/>circa alìas ſui partes:</s> <s xml:id="echoid-s29703" xml:space="preserve"> eſt ergo neceſſe prius deficere uiſui corpus <lb/>circa angulũ g quàm circa puncta remotiora, quæ ſunt d, z:</s> <s xml:id="echoid-s29704" xml:space="preserve"> & ſimi-<lb/>liter accidet in unoquoq;</s> <s xml:id="echoid-s29705" xml:space="preserve"> aliorum angulorum.</s> <s xml:id="echoid-s29706" xml:space="preserve"> Tota ergo periphe-<lb/>ria corporis quãtũ ad prominentiã angulorũ propter ſui diſtantiã <lb/>à uiſu nõ apparebit.</s> <s xml:id="echoid-s29707" xml:space="preserve"> Videtur itaq;</s> <s xml:id="echoid-s29708" xml:space="preserve"> uiſui corpus rectangulũ eſſe ſigu-<lb/>ræ circularis:</s> <s xml:id="echoid-s29709" xml:space="preserve"> ut turris quadrata uidebitur rotunda.</s> <s xml:id="echoid-s29710" xml:space="preserve"> Quando ita q;</s> <s xml:id="echoid-s29711" xml:space="preserve"> <lb/>uiſus comprehendit quadratum aut polygonium à remoto, cõpre <lb/>hendet illud rotundũ, ſi ſuerit æqualium diametrorũ:</s> <s xml:id="echoid-s29712" xml:space="preserve"> aut compre <lb/>hendet ipſum oblongũ figurę teretis, ſi fueritinęqualiũ diametro-<lb/>rum, ut eſt figura altera parte longior:</s> <s xml:id="echoid-s29713" xml:space="preserve"> ut plurimũ ſunt quadrangu-<lb/>læ turres, quæ cũ à remoto uidentur, apparent teretis figuræ:</s> <s xml:id="echoid-s29714" xml:space="preserve"> nec enim exceſſus radiorũ ab angulis <lb/>ſuperficiei quadratæ prodeuntium ad uiſum ſuper longitudinem radiorum prodeuntium à lateri-<lb/>bus planis eſt proportionalis reſpectu diſtantię totius corporis à uiſu aliqua proportione ſenſibili:</s> <s xml:id="echoid-s29715" xml:space="preserve"> <lb/>unde propter inſenſibilitatẽ exceſſus oẽs radij æſtimantur eſſe æquales:</s> <s xml:id="echoid-s29716" xml:space="preserve"> magis aũt hoc ſolet accide <lb/>re in alijs polygonis figuris:</s> <s xml:id="echoid-s29717" xml:space="preserve"> oxygona enim corpora plurimũ ex aliqua magna diſtãtia uiſa uidẽtur <lb/>rotunda:</s> <s xml:id="echoid-s29718" xml:space="preserve"> & eſt hoc quaſi per eadem pręmiſsis demonſtrandum.</s> <s xml:id="echoid-s29719" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s29720" xml:space="preserve"/> </p> <div xml:id="echoid-div1196" type="float" level="0" n="0"> <figure xlink:label="fig-0461-02" xlink:href="fig-0461-02a"> <variables xml:id="echoid-variables494" xml:space="preserve">g z d g</variables> </figure> </div> </div> <div xml:id="echoid-div1198" type="section" level="0" n="0"> <head xml:id="echoid-head944" xml:space="preserve" style="it">96. Curruum rotæ uel lapidum molarium figuræ quando circulares, quando oblongæ ap-<lb/>parent. Euclides 40 th. opt.</head> <p> <s xml:id="echoid-s29721" xml:space="preserve">Quod ſuprà per 55 & 56 huius cõcluſum eſt de figuris ſuperficialibus:</s> <s xml:id="echoid-s29722" xml:space="preserve"> hic proponimus ſimiliter <lb/> <pb o="160" file="0462" n="462" rhead="VITELLONIS OPTICAE"/> decorporalibus figuris, paſsiones proprias ipſarum ſuperficierũ illis corporibus, quorũ ſunt lpſæ <lb/>ſuperficies, applicãtes.</s> <s xml:id="echoid-s29723" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s29724" xml:space="preserve"> rota a b g d:</s> <s xml:id="echoid-s29725" xml:space="preserve"> cuius diametri ſint b a & <lb/>g d ſecantes ſe orthogonaliter ſuper cẽtrũ e:</s> <s xml:id="echoid-s29726" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s29727" xml:space="preserve"> oculus in ſuperficie <lb/> <anchor type="figure" xlink:label="fig-0462-01a" xlink:href="fig-0462-01"/> circuli uel circa.</s> <s xml:id="echoid-s29728" xml:space="preserve"> Si ergo linea, quę cadit à centro oculi ſuper centrum <lb/>rotæ (quod eſt punctum e) obliquè incidat ſuperficiei ipſius rotæ, <lb/>ita ut non ſit perpendicularis ſuper rotæ ſuperficiem, nec æqualis ſe-<lb/>midiametro:</s> <s xml:id="echoid-s29729" xml:space="preserve"> dico quòd diametri rotę inæquales apparebunt, & una <lb/>quidem maxima, alia uerò minima:</s> <s xml:id="echoid-s29730" xml:space="preserve"> aliæ uerò omnes, quę ſunt mediæ <lb/>inter maximã & minimã, propinquiores minimę ſunt minores remo <lb/>tioribus ab illa:</s> <s xml:id="echoid-s29731" xml:space="preserve"> quælibet aũt duę æqualiter diſtantes ab altera diame <lb/>trorum, æquales apparebunt.</s> <s xml:id="echoid-s29732" xml:space="preserve"> Rotæ ergo oblongę, ut ſectio columna <lb/>ris uel conica oxygonia uidentur.</s> <s xml:id="echoid-s29733" xml:space="preserve"> Etidẽ accidit in figuris lapidũ mo <lb/>lariũ, & omnibus alijs quibuſcũq;</s> <s xml:id="echoid-s29734" xml:space="preserve"> figuris.</s> <s xml:id="echoid-s29735" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s29736" xml:space="preserve"/> </p> <div xml:id="echoid-div1198" type="float" level="0" n="0"> <figure xlink:label="fig-0462-01" xlink:href="fig-0462-01a"> <variables xml:id="echoid-variables495" xml:space="preserve">b g c d a</variables> </figure> </div> </div> <div xml:id="echoid-div1200" type="section" level="0" n="0"> <head xml:id="echoid-head945" xml:space="preserve" style="it">97. In figuræ uiſione uirtuti diſtinctiuæ error accidit ex intemper at a diſpoſitione octo circum <lb/>ſtantiarum cuiuslibet rei uiſæ. Alhazen 25. 36. 47. 54. 59. 64. 66. 69 n 3.</head> <p> <s xml:id="echoid-s29737" xml:space="preserve">Ex intemperata enim lucis diſpoſitione figura polygonia æquilatera uidebitur de nocte circula <lb/>ris uel ſphærica:</s> <s xml:id="echoid-s29738" xml:space="preserve"> quoniam lux nimis debilis occultat angulos:</s> <s xml:id="echoid-s29739" xml:space="preserve"> & etiã ſphęra ſub luce ualde debili ui <lb/>ſa, æſtimatur ſuperficiei planę, quia propter lucis debilitatẽ occultatur uiſui partiũ pręeminẽtia in <lb/>ſuperficie ipſius ſphærę.</s> <s xml:id="echoid-s29740" xml:space="preserve"> Exintẽperata etiã longitudine diſtantiæ figura quadrata quandoq;</s> <s xml:id="echoid-s29741" xml:space="preserve"> uide-<lb/>tur rotunda ſphęrica:</s> <s xml:id="echoid-s29742" xml:space="preserve"> & etiã figura quadrata quandoq;</s> <s xml:id="echoid-s29743" xml:space="preserve"> apparet uiſui altera parte longior, ut patet <lb/>ք 59 huiu:</s> <s xml:id="echoid-s29744" xml:space="preserve"> squãdo etiã propter remotionẽ nimiam obliquatio alterius lateris quadrati nõ ſentitur.</s> <s xml:id="echoid-s29745" xml:space="preserve"> <lb/>tunc propter ipſam remotionẽ quadratũ altera parte lõgius uidetur, ut patet per 62 huius.</s> <s xml:id="echoid-s29746" xml:space="preserve"> Accidit <lb/>etiã error uiſioni figuræ ex longitudinis immoderatione:</s> <s xml:id="echoid-s29747" xml:space="preserve"> figura enim multorũ laterũ æqualiũ op-<lb/>poſita uiſui directè, in magna diſtantia uidetur circularis rotunda, quia anguli eius ſunt uiſui imper <lb/>ceptibiles, quod patet per 95 huius:</s> <s xml:id="echoid-s29748" xml:space="preserve"> & linea curua æſtimatur recta per 50 huius:</s> <s xml:id="echoid-s29749" xml:space="preserve"> & figura ſphærica <lb/>uidetur plana per 65 huius.</s> <s xml:id="echoid-s29750" xml:space="preserve"> Ex inordinatione etiã ſitus error accidit in figurę uiſione.</s> <s xml:id="echoid-s29751" xml:space="preserve"> Si enim cor-<lb/>pus circulare, ut ſcutella, ab axe elongetur, & modicũ ſuper lineam, cui axis perpendiculariter inci <lb/>dit, obliquetur, uidebuntur eius diametri inæquales per 96 huius:</s> <s xml:id="echoid-s29752" xml:space="preserve"> & figura circularis per 55 & 56 <lb/>huius uidebitur ſectionis oxygonię uel columnaris figurę:</s> <s xml:id="echoid-s29753" xml:space="preserve"> & ſimiliter propter æqualitatẽ oppoſi-<lb/>tionis unius laterum ad uiſum figura quadrata æſtimabitur altera parte lõgior per 61 huius.</s> <s xml:id="echoid-s29754" xml:space="preserve"> Ex in-<lb/>temperantia etiã quantitatis uel magnitudinis accidit error uiſioni figurarum.</s> <s xml:id="echoid-s29755" xml:space="preserve"> Cum enim ſuperfi-<lb/>cies uiſa fuerit multũm parua, ſi ſuerint in ea anguli, occultabuntur uiſui:</s> <s xml:id="echoid-s29756" xml:space="preserve"> unde fortè forma eius an-<lb/>gularis æſtimabitur rotunda, ſphęrica, aut columnaris.</s> <s xml:id="echoid-s29757" xml:space="preserve"> Et ſi fuerint in eius ſuperficie aliquę pręemi <lb/>nentię, latebunt uiſum, & æſtimabitur eorũ ſuperficies plana, ut hęc patere poſſunt in atomis ſolis.</s> <s xml:id="echoid-s29758" xml:space="preserve"> <lb/>quarum certa figura nõ comprehẽditur, quoniã anguli ipſarum uiſui à minori diſtãtia occultãtur.</s> <s xml:id="echoid-s29759" xml:space="preserve"> <lb/>ut patet per 8 huius.</s> <s xml:id="echoid-s29760" xml:space="preserve"> Ex intẽperata etiã ſoliditate accidit error uiſioni figurarũ.</s> <s xml:id="echoid-s29761" xml:space="preserve"> Si enim corpus ſue <lb/>ritminus ſolidum, in quo fuerint anguli, illi fortè occultabuntur uidenti, & angularis forma puta-<lb/>bitur ſphęrica, ſortè & ſphęricitas illorũ corporum uidebitur plana.</s> <s xml:id="echoid-s29762" xml:space="preserve"> Intemperata quoq;</s> <s xml:id="echoid-s29763" xml:space="preserve"> diaphani-<lb/>tas in unſione figurarum errorem in ducit:</s> <s xml:id="echoid-s29764" xml:space="preserve"> quoniã exiſtente aere nubiloſo, obſcuro, ut in crepuſcu-<lb/>lis, ſi in corpore illo fuerint anguli, fortè apparebit ſphęricitas:</s> <s xml:id="echoid-s29765" xml:space="preserve"> & ſi in ipſo fuerit ſphęricitas, appare <lb/>bit ſorrè planities:</s> <s xml:id="echoid-s29766" xml:space="preserve"> quoniã medium nõ eſt taliter diſp oſitum, ut per ipſum poſsit fieri cõpleta uiſio, <lb/>ad quã requiritur lumen, ut patet per 1 th.</s> <s xml:id="echoid-s29767" xml:space="preserve"> 2 huius.</s> <s xml:id="echoid-s29768" xml:space="preserve"> Breuitas etiã temporis errorẽ uiſibus in uiſione <lb/>figurarum adducit:</s> <s xml:id="echoid-s29769" xml:space="preserve"> modica enim gibboſitas in re ſubitò uiſa latet uiſum, & æſtimatur planities:</s> <s xml:id="echoid-s29770" xml:space="preserve"> & <lb/>ſi fuerint res figurę angularis ſubitò uiſæ, ſortè ſphæricę apparebunt.</s> <s xml:id="echoid-s29771" xml:space="preserve"> Viſus quoq;</s> <s xml:id="echoid-s29772" xml:space="preserve"> debilitas errorẽ <lb/>cauſſat in figurarum uiſione:</s> <s xml:id="echoid-s29773" xml:space="preserve"> modicus enim gibbus, & multiplex angulus debilem latent uiſum:</s> <s xml:id="echoid-s29774" xml:space="preserve"> & <lb/>uidentur res ſphę planæ, & angulares ſphericę:</s> <s xml:id="echoid-s29775" xml:space="preserve"> ſic ergo patet propoſitum in omnibus circum-<lb/>ſtantijs uiſibilium.</s> <s xml:id="echoid-s29776" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s29777" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1201" type="section" level="0" n="0"> <head xml:id="echoid-head946" xml:space="preserve" style="it">98. In uiſione corporeitatis erroures accidentes uirtuti diſtinctiuæ ex intemperata diſpoſi-<lb/>tione octo circumſtantiarum cuiuslibet rei uiſæ, ſunt ijdem illis, qui in ſitus & figuræ accidune <lb/>uiſione. Alhazen 25 n 3.</head> <p> <s xml:id="echoid-s29778" xml:space="preserve">Corporeitas enim, ut patet in 63 huius, à uiſu comprehenditur ex comprehenſione figurarum, <lb/>quas faciunt ſuperficies corpus continentes:</s> <s xml:id="echoid-s29779" xml:space="preserve"> eſt ergo eadem hinc inde erroris cauſſa:</s> <s xml:id="echoid-s29780" xml:space="preserve"> & omnis er-<lb/>ror, qui poteſt accidere uiſui in non comprehenſione uerę corporeitatis, uel in erronea cõprehen-<lb/>fione, accidit ex errore proueniente circa ſpecies figurarum:</s> <s xml:id="echoid-s29781" xml:space="preserve"> ut ſi ſuperficies ſphęrica conuexa uel <lb/>concaua ęſtimetur plana per 65 huius:</s> <s xml:id="echoid-s29782" xml:space="preserve"> quia in corporibus maximę remotionis à uiſu non compre-<lb/>hendit uiſus corporeitatem, quando non comprehendit obliquationem ſuperficierum.</s> <s xml:id="echoid-s29783" xml:space="preserve"> Et hoc to-<lb/>tum accidit propter deceptionem circa figuras factam:</s> <s xml:id="echoid-s29784" xml:space="preserve"> non enim cõprehendit tunc uiſus ſitus par-<lb/>tium illarum ſuperficierum ad inuicem, qui ſitus eſſicit figuram:</s> <s xml:id="echoid-s29785" xml:space="preserve"> unde cum certitudinaliter cõpre-<lb/>henditur figura, certitudinaliter comprehenditur corporeitas:</s> <s xml:id="echoid-s29786" xml:space="preserve"> & cum comprehenditur figura in-<lb/>diſcinctè, comprehenditur etiam corporeitas indiſtinctè.</s> <s xml:id="echoid-s29787" xml:space="preserve"> Et hoc accidit in omnibus modis, quibus <lb/>error accidit in uiſionibus figurarum, Et quia ſitus eſt cauſſa figurarum, ideo etiam errores acciden <lb/> <pb o="161" file="0463" n="463" rhead="LIBER QVARTVS."/> tes ſitui, accidunt & corporeitati:</s> <s xml:id="echoid-s29788" xml:space="preserve"> quia enim corporeitas includitur ſub figura & ſitu, ideo errorem <lb/>corporeitatis gerit error in ſe ſitus & figuræ.</s> <s xml:id="echoid-s29789" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1202" type="section" level="0" n="0"> <head xml:id="echoid-head947" xml:space="preserve" style="it">99. Diftinctio uiſibilium comprehenditur à uiſu ex diſtinctione formarum ipſarum uiſibili-<lb/>um in diuerſis ſuperſiciei uiſus partibus impreſſarum. Alhazen 46 n 2.</head> <p> <s xml:id="echoid-s29790" xml:space="preserve">Diſtinctio, quę eſt inter quęlibet duo corpora, aut eſt exluce:</s> <s xml:id="echoid-s29791" xml:space="preserve"> aut ex colore actum lucidi haben-<lb/>te:</s> <s xml:id="echoid-s29792" xml:space="preserve"> aut ex obſcurιtate:</s> <s xml:id="echoid-s29793" xml:space="preserve"> hęc enim ſunt princιpium diſtinctionis formarum in ſuperficie uiſus:</s> <s xml:id="echoid-s29794" xml:space="preserve"> quo-<lb/>niam hæc per ſe perueniunt in partem ſuperficiei uiſus.</s> <s xml:id="echoid-s29795" xml:space="preserve"> Quandoq;</s> <s xml:id="echoid-s29796" xml:space="preserve"> autem lux & coloruel obſcuri-<lb/>tas ſunt in ιpſis formis, quę diſtinguuntur:</s> <s xml:id="echoid-s29797" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s29798" xml:space="preserve"> uerò lux & color uel obſcuritas diſtinguentia <lb/>formas in ipſa ſuperficie uiſus, ſunt in corporibus medijs ſecundum ſιtum diſtinguentibus corpo-<lb/>ra, quorum formę diſtinguuntur in uiſu:</s> <s xml:id="echoid-s29799" xml:space="preserve"> & tunc ſi uiſus non ſen ſerit quòd lux, color aut obſcuritas, <lb/>quę eſt in loco diſtinctionis, non eſt in corpore continuato cum utroq;</s> <s xml:id="echoid-s29800" xml:space="preserve"> corporum, quę ſunt in eius <lb/>lateribus:</s> <s xml:id="echoid-s29801" xml:space="preserve"> tunc non ſentiet diſtinctionem duorum corporum.</s> <s xml:id="echoid-s29802" xml:space="preserve"> Et etiam quandoq;</s> <s xml:id="echoid-s29803" xml:space="preserve"> fit diſtinctio uiſi-<lb/>bilium ex hoc, quia non eſt poſsibile plura uiſιbilia æ qualiter uideri per 49 th.</s> <s xml:id="echoid-s29804" xml:space="preserve">3 huius.</s> <s xml:id="echoid-s29805" xml:space="preserve"> Aut enim ſu <lb/>perficies cuiuslibet illorum corporum eſt obliqua ad ſuperficiem uiſus in loco diſtin ctionis, & eſt <lb/>in æ qualis obliquitatis:</s> <s xml:id="echoid-s29806" xml:space="preserve"> aut unius ipſorum forma eſt uiſui obliquè, alterius uerò forma eſt uiſui dire-<lb/>ctè oppoſita, maniſeſtior uiſui quàm alia, quæ eſt uiſui obliquè oppoſita, uel quę ſibi opponitur <lb/>plus obliquè:</s> <s xml:id="echoid-s29807" xml:space="preserve"> & ſecundum hoc comprehendet uiſus diſtinctionem uiſibilium formarum:</s> <s xml:id="echoid-s29808" xml:space="preserve"> ſiue ipſo-<lb/>rum diſtinctio ſecundum ſpatium interiacens ſit ampla ſiue ſtricta, dum tamen ſit ſenſibilis, reſpe-<lb/>ctu remotionis corporum uiſorum & reſpectu quantitatis corporũ diſtinctorum:</s> <s xml:id="echoid-s29809" xml:space="preserve"> quia fortè quan-<lb/>doque diſtinctio formarum eſt quantitatis unius capilli:</s> <s xml:id="echoid-s29810" xml:space="preserve"> & illud diminutum non affert diſtantiam <lb/>ſenſibilem in uiſu.</s> <s xml:id="echoid-s29811" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s29812" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1203" type="section" level="0" n="0"> <head xml:id="echoid-head948" xml:space="preserve" style="it">100. Continuitas uiſibilium comprehenditur à uiſu ex diſtantiæ priuatione. Alha-<lb/>zen 47 n 2.</head> <p> <s xml:id="echoid-s29813" xml:space="preserve">Cum enim uiſus non ſenſeritin corpore aliquam diſtantiam, comprehendet ipſum eſſe conti-<lb/>nuum:</s> <s xml:id="echoid-s29814" xml:space="preserve"> & ſi in corpore fuerit diſtantia occulta non comprehenſa à uiſu:</s> <s xml:id="echoid-s29815" xml:space="preserve"> comprehendet uiſus illud <lb/>corpus eſſe continuum, & diſcernet inter continuationem & continuationem ex com prehenſione <lb/>aggregationis duorum terminorum duorum corporum.</s> <s xml:id="echoid-s29816" xml:space="preserve"> Si ergo ſentiẽs non ſenſerit, quòd utrũq;</s> <s xml:id="echoid-s29817" xml:space="preserve"> <lb/>duorum corporum contiguorum eſt diuerſum ab altero & diſtinctum ab eo:</s> <s xml:id="echoid-s29818" xml:space="preserve"> tunc non ſentiet con-<lb/>tiguationem, ſed indicabit eſſe inter illa uiſa perfectam continuationem & totius ſuperficiei uiſæ <lb/>perſectam unitatem, quæ eſt continuitas.</s> <s xml:id="echoid-s29819" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s29820" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1204" type="section" level="0" n="0"> <head xml:id="echoid-head949" xml:space="preserve" style="it">101. Numerus comprehenditur à uiſu per hoc, quòd unum uiſibilium comprehenditur ab al <lb/>terodiſtinctum. Alhazen 48 n 2.</head> <p> <s xml:id="echoid-s29821" xml:space="preserve">Quando enim uiſus comprehendit in una hora multa uiſibilia in ſimul diſtincta, & illorum di-<lb/>ſtinctionem, comprehendet quòd quodlibet ipſorum eſt ab altero diuiſum.</s> <s xml:id="echoid-s29822" xml:space="preserve"> Comprehendit ergo <lb/>multitudinem:</s> <s xml:id="echoid-s29823" xml:space="preserve"> & tunc uirtus diſtinctiua comprehendet numerum ex multitudine ill orum, & ſi eſt <lb/>par uel impar, & medietatem paris numeri & quamlibet ipſorũ unitatem:</s> <s xml:id="echoid-s29824" xml:space="preserve"> & per hũc modũ omniũ <lb/>rerum uiſarum numerum comprehendit & mathematicum & naturalem.</s> <s xml:id="echoid-s29825" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s29826" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1205" type="section" level="0" n="0"> <head xml:id="echoid-head950" xml:space="preserve" style="it">102. Omnis forma uiſibus obliquè incidens ſemper apparet ultra locum formæ directè inci-<lb/>dentis. Ex quo patet quòdformæ ambobus uiſibus ſecundum æqualitatem angulorum obliqui-<lb/>us incidentes plurimum à ſe diſtant.</head> <p> <s xml:id="echoid-s29827" xml:space="preserve">Quod hic proponitur, ſatis patet.</s> <s xml:id="echoid-s29828" xml:space="preserve"> Quando enim linea radialis ſuperficiei uiſus obliquè incidit:</s> <s xml:id="echoid-s29829" xml:space="preserve"> <lb/>tunc ipſa per 47 th.</s> <s xml:id="echoid-s29830" xml:space="preserve"> 2 huius refringitur à ſuperficie oculi, & ad concauum nerui peruenit plus obli-<lb/>què:</s> <s xml:id="echoid-s29831" xml:space="preserve"> quoniam tunc ſecundum angulum incidentiæ formatur quantitas anguli refractionis per 36 <lb/>th.</s> <s xml:id="echoid-s29832" xml:space="preserve"> 3 huius.</s> <s xml:id="echoid-s29833" xml:space="preserve"> Palàm ergo quoniam illa linea obliquè ſuperficiei ipſius uiſus incidens, propter ſuæ in-<lb/>cidentiæ obliquitatem & anguli acuitatem facit angulum ſuæ refractionis acutum:</s> <s xml:id="echoid-s29834" xml:space="preserve"> unde tunc li-<lb/>nea refractionis interſecat lineam directè incidentem, & à ſuperficie oculi æ qualiter refractam:</s> <s xml:id="echoid-s29835" xml:space="preserve"> & <lb/>ſic forma obliqua uidetur ultra formam rectè uiſam.</s> <s xml:id="echoid-s29836" xml:space="preserve"> Et ſi ambę formę obliquè incidant ſecundum <lb/>eundem ſuę obliquitatis modum, ita, ut utrobique ſit ęqualitas angulorum incidentię & refractio-<lb/>nis:</s> <s xml:id="echoid-s29837" xml:space="preserve"> tunc forma oculo dextro incidens, ſecans lineam, per quam directè incidens ad medium pun-<lb/>ctum concauitatis nerui perueniſſet, ſit ſiniſtra ab illa, & ſorma oculo ſiniſtro obliquè incidens, re-<lb/>ſpectu illius medij puncti concauitatis nerui, fit dextra:</s> <s xml:id="echoid-s29838" xml:space="preserve"> & ſic quandoq;</s> <s xml:id="echoid-s29839" xml:space="preserve"> acciditillas formas à ſe plu <lb/>rimum diſtare.</s> <s xml:id="echoid-s29840" xml:space="preserve"> Et quoniam quęlibet ipſarum offertur uirtuti ſenſitiuæ, quoniam ſecundum luces <lb/>& colores, quę ſunt in ipſa forma, quę eſt extra, depingitur ipſa forma in ſuperficie organi membri <lb/>ſentientis in duobus locis ſecundum numerum oculorum, quibus incidit, & à quorum ſuperficie <lb/>refringitur:</s> <s xml:id="echoid-s29841" xml:space="preserve"> forma uerò directè in cidens ad unum ſecundum omnes eius partes ordinatur locum <lb/>conſimiliter, ut patet per 37 th.</s> <s xml:id="echoid-s29842" xml:space="preserve"> 3 huius.</s> <s xml:id="echoid-s29843" xml:space="preserve"> Forma ergo obliquè incidens ſemper apparet <lb/>ultra locum formæ dirctè incidentis.</s> <s xml:id="echoid-s29844" xml:space="preserve"> Patet ergo propoſitum, <lb/>& eius corollarium.</s> <s xml:id="echoid-s29845" xml:space="preserve"/> </p> <pb o="162" file="0464" n="464" rhead="VITELLONIS OPTICAE"/> </div> <div xml:id="echoid-div1206" type="section" level="0" n="0"> <head xml:id="echoid-head951" xml:space="preserve" style="it">103. Omne uiſum, quod directè opponitur medio unius uiſus, & in reſpectu ad reliquum ni-<lb/>ſum eſt obliquum: ſemper uidetur duo. Alhazen 13 n 3.</head> <p> <s xml:id="echoid-s29846" xml:space="preserve">Nam formapuncti, quæ directè incidit medio alterius uiſuum, per uenit ad punctum mediũ coa <lb/>cauitatis nerui, ur patet per 29 th.</s> <s xml:id="echoid-s29847" xml:space="preserve"> 3 huius, quoniam forma illius puncti incidit uiſui ſecundũ axem <lb/>pyramidis radialis:</s> <s xml:id="echoid-s29848" xml:space="preserve"> ſorma uerò puncti obliquè inciden-<lb/>tis in medio ſuperficiei alterius uiſus uenit ad punctum <lb/> <anchor type="figure" xlink:label="fig-0464-01a" xlink:href="fig-0464-01"/> aliud quàm ad medium punctũ concauitatis ipſius ner-<lb/>ui, ſecundau obliquationem puncti ſuperficiei uiſus:</s> <s xml:id="echoid-s29849" xml:space="preserve"> & <lb/>ſic non concurrunt illę formę in eodẽ pũcto medio con <lb/>cauitatis nerui.</s> <s xml:id="echoid-s29850" xml:space="preserve"> Verbi gratia, ſint centra duorum uiſuum <lb/>a & b:</s> <s xml:id="echoid-s29851" xml:space="preserve"> ſit linea e f quoddam uiſum directè oppoſitũ cen-<lb/>tro uiſus a:</s> <s xml:id="echoid-s29852" xml:space="preserve"> ſit autem ipſa linea e f obliquè oppoſita uiſui, <lb/>cuius centrum eſt punctum b.</s> <s xml:id="echoid-s29853" xml:space="preserve"> Quia ergo forma lineæ e f <lb/>directè peruenit ad medium cõcauitatis nerui commu-<lb/>nis per 29 th.</s> <s xml:id="echoid-s29854" xml:space="preserve"> 3 huius:</s> <s xml:id="echoid-s29855" xml:space="preserve"> palàm quòd forma eius circa illum <lb/>punctum medium concauitatis nerui ſecũdum omnes ſitus ſuarum partium ordinatur per 37 th.</s> <s xml:id="echoid-s29856" xml:space="preserve"> 3 <lb/>huius.</s> <s xml:id="echoid-s29857" xml:space="preserve"> Quia uerò forma eiuſdem lineę e f tota obliquè in cidit ſuperſiciei uiſus b:</s> <s xml:id="echoid-s29858" xml:space="preserve"> palàm per ea, quæ <lb/>declarata ſunt in 37 th.</s> <s xml:id="echoid-s29859" xml:space="preserve"> 3 huius.</s> <s xml:id="echoid-s29860" xml:space="preserve"> quòd forma eius non peruenit ad punctũ medium concauitatis ner-<lb/>ui, ſed ad aliquod ipſius punctum aliud:</s> <s xml:id="echoid-s29861" xml:space="preserve"> non ſuperponetur ergo priori formę, ſed remanebit diſtin-<lb/>cta ab illa.</s> <s xml:id="echoid-s29862" xml:space="preserve"> Apparebunt ergo duę formę, quoniam in duobus locis ipſius membri ſentientis offertur <lb/>forma ipſius uiſibilis ipſi uirtuti ſentiẽti:</s> <s xml:id="echoid-s29863" xml:space="preserve"> & ſic iudicat illas eſſe duas, & nõ unã.</s> <s xml:id="echoid-s29864" xml:space="preserve"> Patet ergo ꝓpoſitũ.</s> <s xml:id="echoid-s29865" xml:space="preserve"/> </p> <div xml:id="echoid-div1206" type="float" level="0" n="0"> <figure xlink:label="fig-0464-01" xlink:href="fig-0464-01a"> <variables xml:id="echoid-variables496" xml:space="preserve">e f a b</variables> </figure> </div> </div> <div xml:id="echoid-div1208" type="section" level="0" n="0"> <head xml:id="echoid-head952" xml:space="preserve" style="it">104. Omnis forma rei uiſæ intra axes radiales conſtitutæ, obliquè ambobus uiſibus occurrit: <lb/>unde ſemper uidetur duo. Alhazen 11 n 3.</head> <p> <s xml:id="echoid-s29866" xml:space="preserve">Verbi gratia, ſint centra duorũ uiſuũ a & b:</s> <s xml:id="echoid-s29867" xml:space="preserve"> & concurrant axes uiſuales in puncto c:</s> <s xml:id="echoid-s29868" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s29869" xml:space="preserve"> axis cõ-<lb/>munis d c:</s> <s xml:id="echoid-s29870" xml:space="preserve"> & ſit res intra axes uiſa, quę e.</s> <s xml:id="echoid-s29871" xml:space="preserve"> Dico, quòd forma rei uiſę, quę eſte, ſemper obliquè occur-<lb/>rit ambobus uiſibus:</s> <s xml:id="echoid-s29872" xml:space="preserve"> unde ſem per uidebitur eſſe duę.</s> <s xml:id="echoid-s29873" xml:space="preserve"> Quòd autẽ ob-<lb/>liquè ſemper incidat ambobus uiſibus, patet:</s> <s xml:id="echoid-s29874" xml:space="preserve"> cũ enim à puncto c du-<lb/> <anchor type="figure" xlink:label="fig-0464-02a" xlink:href="fig-0464-02"/> cta ſit linea c a perpendiculariter ſuper centrum foraminis uueæ o cu-<lb/>li, cuius centrum eſt punctum a, ut patet per 24 th.</s> <s xml:id="echoid-s29875" xml:space="preserve"> 3 huius, & cũ linea <lb/>c b ducta ſit perpendiculariter ſuper centrum foraminis uueæ o culi, <lb/>suius centrũ eſt punctum b:</s> <s xml:id="echoid-s29876" xml:space="preserve"> palàm per 13 p 11 quoniam ab aliquo pun-<lb/>cto ſuperficiei rei uiſę, quę eſt e, a d dicta centra foraminum peropendi-<lb/>culares aliæ duci non poſſunt:</s> <s xml:id="echoid-s29877" xml:space="preserve"> omnes ergo lineæ à ſuperficie corpo-<lb/>ris e ad ſup erficiem uiſuum productę, ſunt obliquę per 24 th.</s> <s xml:id="echoid-s29878" xml:space="preserve"> 3 huius:</s> <s xml:id="echoid-s29879" xml:space="preserve"> <lb/>non ergo propter reſractionem concurrent in puncto medio conca-<lb/>uitatis nerui, ſed ultra, & plurimùm à ſe diſtabũt per 102 huius.</s> <s xml:id="echoid-s29880" xml:space="preserve"> Vide-<lb/>buntur ergo ſemper dua per pręcedentem.</s> <s xml:id="echoid-s29881" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s29882" xml:space="preserve"> axes duarum <lb/>pyramidum uiſualium concurrant in aliquo puncto rei uiſę, & duo a-<lb/>lij radij obliqui comperehendand aliud uiſum propin quius duobus ui <lb/>ſibus aut remotius intra axes:</s> <s xml:id="echoid-s29883" xml:space="preserve"> tunc poſitio eius apud duos uiſus erit <lb/>diuerſa in parte:</s> <s xml:id="echoid-s29884" xml:space="preserve"> nam illud uiſum erit dextrũ uni axium uiſualiũ, & ſi-<lb/>niſtrum alteri ipſorũ:</s> <s xml:id="echoid-s29885" xml:space="preserve"> radij quoq;</s> <s xml:id="echoid-s29886" xml:space="preserve"> exeuntes ab ipſa re taliter uiſa ad al-<lb/>terũ uiſum, erũt dexcri ab axe, & ad reliquũ uiſum exeuntes, erũt ſiniſtri ab illius axe:</s> <s xml:id="echoid-s29887" xml:space="preserve"> & ſic poſitio <lb/>eius apud duos uiſus erit diuerſa in parte:</s> <s xml:id="echoid-s29888" xml:space="preserve"> & forma unius uiſi incidit duobus uiſibus in duobus lo-<lb/>cis diuerſè poſitis, & peruenit ad loca diuerſa concauitatis communis nerui à duobus lateribus ſui <lb/>puncti medij, & partes illius formę non ſuperponuntur ſibi:</s> <s xml:id="echoid-s29889" xml:space="preserve"> erunt ergo duæ formæ.</s> <s xml:id="echoid-s29890" xml:space="preserve"> Et ita ſemper <lb/>forma rei taliter ad uiſum diſpoſitę uidentur duę formæ, & res ipſa uiſa uidetur ſemper duo.</s> <s xml:id="echoid-s29891" xml:space="preserve"> Quod <lb/>eſt propoſitum.</s> <s xml:id="echoid-s29892" xml:space="preserve"/> </p> <div xml:id="echoid-div1208" type="float" level="0" n="0"> <figure xlink:label="fig-0464-02" xlink:href="fig-0464-02a"> <variables xml:id="echoid-variables497" xml:space="preserve">c e e e a d b</variables> </figure> </div> </div> <div xml:id="echoid-div1210" type="section" level="0" n="0"> <head xml:id="echoid-head953" xml:space="preserve" style="it">105. Lineæ rectæ uicinæ uiſibus in ſuperficie axis communis erectæ ſuper trigonum axiũ ra-<lb/>dialium puncto coniunctionis incidente, ſolum illud punctum uidebitur unum: omnia uerò alia <lb/>dictæ lineæ punct a uidebuntur duo, & æqualiter à puncto coniunctionis declinantia, ac ſi duæ <lb/>lineæ ſeinterſecent in puncto coniunctionis.</head> <p> <s xml:id="echoid-s29893" xml:space="preserve">Sit centrum uiſus ſiniſtri punctum a, dextri uerò punctum b:</s> <s xml:id="echoid-s29894" xml:space="preserve"> & ſit linea recta h z:</s> <s xml:id="echoid-s29895" xml:space="preserve"> quæ ſecundum <lb/>medium pũctum naſi ambobus uiſibus interpoſita extendatur taliter, ut in aliquo pũcto ſuo ſigna-<lb/>to, quod ſit q, concurrant axes uiſuales:</s> <s xml:id="echoid-s29896" xml:space="preserve"> erit ergo q punctum coniũctionis amborum axium uiſua-<lb/>lium.</s> <s xml:id="echoid-s29897" xml:space="preserve"> Et quoniam ipſum punctum eſt in linea h z, quę ſic extenditur inter ambos axes radiales:</s> <s xml:id="echoid-s29898" xml:space="preserve"> <lb/>tunc palàm eſt, quòd ipſa eſt in ſuperficie, in qua eſt axis communis, erecta ſuper baſim trigoni <lb/>b q a per 33 th.</s> <s xml:id="echoid-s29899" xml:space="preserve"> 3 huius.</s> <s xml:id="echoid-s29900" xml:space="preserve"> Dico ergo, quòd ubicunque punctus coniunctionis, qui eſt q lineæ h z, obli-<lb/>què incidit uiſibus, hoc eſt ambobus axibus b q, & a q, uel eorum alteri, angulosrectos non conti-<lb/>nentibus cum linea h z, ſolus punctus q uidebitur unus, ut eſt:</s> <s xml:id="echoid-s29901" xml:space="preserve"> quoniam forma eius ſolius peram-<lb/>bos axes radiales peruenit ad medium punctum concauitatis nerui:</s> <s xml:id="echoid-s29902" xml:space="preserve"> & ſic forma una uidetur rei <lb/> <pb o="163" file="0465" n="465" rhead="LIBER QVARTVS."/> unius, ut hoc patere poteſt per 46 & 47 th.</s> <s xml:id="echoid-s29903" xml:space="preserve"> 3 huius:</s> <s xml:id="echoid-s29904" xml:space="preserve"> reliqua uerò puncta omnia lineæ h z uidenturę-<lb/>qualiter à puncto coniunctionis declinantia, ac ſi duæ lineæ ſe interſecent in puncto cõiunctionis <lb/>quod eſt q:</s> <s xml:id="echoid-s29905" xml:space="preserve"> quia radij diuerſi ab illis punctis perueni.</s> <s xml:id="echoid-s29906" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0465-01a" xlink:href="fig-0465-01"/> entes ad ambos uiſus & ſiniſtrantur & dextrantur.</s> <s xml:id="echoid-s29907" xml:space="preserve"> <lb/>Omnes enim radij exeuntes ab alijs pũctis lineæ h q <lb/>ad uiſum dextrum ex parte axis h q, fiunt ſiniſtri ab <lb/>axe a q, & peruenientes ad ſiniſtrum uiſum ex parte <lb/>axis h q, fiunt dextri ab axe b q:</s> <s xml:id="echoid-s29908" xml:space="preserve"> perueniunt enim ad <lb/>ſuperficiem uiſus ex una parte ſemidiametri ſorami-<lb/>nis, quæ à centro uueæ reſpicit axem communem:</s> <s xml:id="echoid-s29909" xml:space="preserve"> & <lb/>radij peruenientes à punctis lineæ q z, ad uiſum de-<lb/>xtrum, fiunt item ſiniſtri ab axe aq, & peruenientes <lb/>ad uiſum ſiniſtrum, fiunt dextri:</s> <s xml:id="echoid-s29910" xml:space="preserve"> perueniunt enim u-<lb/>trique radij ad ſuperficiem uiſus ex parte ſemidiame <lb/>tri cum priori ſemidiametro, diametrum totam illius <lb/>foraminis uueæ complente.</s> <s xml:id="echoid-s29911" xml:space="preserve"> Et quoniam ambo ocu-<lb/>li ſunt in omnibus diſpoſitionibus ęquales per 4 th.</s> <s xml:id="echoid-s29912" xml:space="preserve"> 3 <lb/>huius:</s> <s xml:id="echoid-s29913" xml:space="preserve"> palàm, quòd anguli utriuſq;</s> <s xml:id="echoid-s29914" xml:space="preserve"> axium & iſtarum <lb/>ſemidiametrorum ſunt æ quales circa centrum utri-<lb/>uſque circuli foraminis.</s> <s xml:id="echoid-s29915" xml:space="preserve"> Anguli quoq;</s> <s xml:id="echoid-s29916" xml:space="preserve"> c q z, & d q z, <lb/>propter eadem ſunt æ quales.</s> <s xml:id="echoid-s29917" xml:space="preserve"> Ducta itaq;</s> <s xml:id="echoid-s29918" xml:space="preserve"> linea à puncto z æ quidiſtante lineæ a b per 31 p 1, quæ ſit c <lb/>z d:</s> <s xml:id="echoid-s29919" xml:space="preserve"> producatur linea a q in punctum d, & linea b q in punctum c:</s> <s xml:id="echoid-s29920" xml:space="preserve"> patet, quòd ſecundum illas lineas <lb/>fit uiſio illarum formarum.</s> <s xml:id="echoid-s29921" xml:space="preserve"> Quoniam enim anguli, ſecundum quos fit obliquatio uiſionis, qui ſunt <lb/>c q z, & d q z ſunt æ quales:</s> <s xml:id="echoid-s29922" xml:space="preserve"> ergo ք 13.</s> <s xml:id="echoid-s29923" xml:space="preserve"> 15.</s> <s xml:id="echoid-s29924" xml:space="preserve"> & 14 p 1 lineæ uiſuales, quę, exẽpli cauſſa, ſint lineę b q, & q c, <lb/>cõiunctæ ſunt linea una:</s> <s xml:id="echoid-s29925" xml:space="preserve"> & ſimiliter eſt de lineis a q, & q d.</s> <s xml:id="echoid-s29926" xml:space="preserve"> videtur aũt linea una radialis duæ lineę <lb/>propter diuerſitatem incidentiæ formę illius puncti ambobus uiſibus, quæ obliquatio fit quaſi per <lb/>modum duarum linearum ſeſecantium circa punctum q:</s> <s xml:id="echoid-s29927" xml:space="preserve"> forma enim ſecundum axes radiales uiſt-<lb/>bus incidens ad medium punctum concauitatis nerui pertingit, & formæ obliquè incidentes, circa <lb/>ipſum ſe ſecantes figurátur.</s> <s xml:id="echoid-s29928" xml:space="preserve"> Remotiones enim duarum quarumlibet linearum radialium ab aliquo <lb/>puncto lineæ h z ad ambos axes peruenientium, ſemper erunt in duabus partibus diuerſis:</s> <s xml:id="echoid-s29929" xml:space="preserve"> quapro <lb/>pter duæ formæ cuiuslibet puncti eius incident duobus punctis concauitatis nerui communis à <lb/>duobus lateribus pũcti medij, ut oſtẽdimus in præmiſsis.</s> <s xml:id="echoid-s29930" xml:space="preserve"> Pater ergo propoſitum.</s> <s xml:id="echoid-s29931" xml:space="preserve"> Patet etiam quòd <lb/>mutato pũcto coniunctionis, linearum interſectarum quantitas mutatur:</s> <s xml:id="echoid-s29932" xml:space="preserve"> ſempertamen ex utraq:</s> <s xml:id="echoid-s29933" xml:space="preserve"> <lb/>parte ſectionis partes linearum ſuntæ quales:</s> <s xml:id="echoid-s29934" xml:space="preserve"> & ſecundum approxiamationem ad uiſus anguli me-<lb/>dij, ut ſunt a q b, & c q d fiunt maiores, & ſecundum elongationem à uiſu fiunt minores, quouſq;</s> <s xml:id="echoid-s29935" xml:space="preserve"> cir <lb/>ca axes radiales pyramides deſcribuntur, quarum baſis eſt tota ſuperficies reiuiſæ:</s> <s xml:id="echoid-s29936" xml:space="preserve"> & horum pro-<lb/>batio experimentalis accidit, ſi uiſibus modo dicto diſpoſitis unusipſorum claudatur, alterq́ue a-<lb/>pertus reſeruerur, ſic uices mutando quantùm placet.</s> <s xml:id="echoid-s29937" xml:space="preserve"/> </p> <div xml:id="echoid-div1210" type="float" level="0" n="0"> <figure xlink:label="fig-0465-01" xlink:href="fig-0465-01a"> <variables xml:id="echoid-variables498" xml:space="preserve">c c z d d q q a h b</variables> </figure> </div> </div> <div xml:id="echoid-div1212" type="section" level="0" n="0"> <head xml:id="echoid-head954" xml:space="preserve" style="it">106. Si à puncto coniunctionis linea inter duas perpendiculares productas à terminis lineæ <lb/>connectentis centra uiſuum, eidem æ qualis & æ quidiſtans fuerit producta: forma cuiuslibet <lb/>punctiproductæ lineæ aut rei ſuper ipſam exiſtentis, & formarei exiſtentis ſuper alteram per-<lb/>pendicularium in puncto propinquo prædictæ lineæ, uidebitur tant ùm una: exiſtentis autem <lb/>in eadem perpendiculari remotæ à producta linea uidebitur ſem-<lb/>per duæ.</head> <figure> <variables xml:id="echoid-variables499" xml:space="preserve">d c f r t q k b a</variables> </figure> <p> <s xml:id="echoid-s29938" xml:space="preserve">Sint centra duotum uiſuũ a & b:</s> <s xml:id="echoid-s29939" xml:space="preserve"> linea ergo connectens centra eſt <lb/>a b:</s> <s xml:id="echoid-s29940" xml:space="preserve"> & ab illius terminis erigantur perpendiculares a c & b d per 11 p <lb/>1:</s> <s xml:id="echoid-s29941" xml:space="preserve"> & ſit punctus coniunctionis q:</s> <s xml:id="echoid-s29942" xml:space="preserve"> erunt ergo axes uiſuales a q & b q:</s> <s xml:id="echoid-s29943" xml:space="preserve"> <lb/>à punctus.</s> <s xml:id="echoid-s29944" xml:space="preserve"> uerò q per 31 p 1 ducatur linea k q t æ quidiſitás lineæ a b.</s> <s xml:id="echoid-s29945" xml:space="preserve"> Di <lb/>co, quòd forma cuiuslibet puncti lineæ k t, aut rei ſuper ipſam exi-<lb/>ſtentis, ſemper uidebitur una:</s> <s xml:id="echoid-s29946" xml:space="preserve"> & ſi in aliqua perpendicularium a c & <lb/>b d, in puncto propinquo lineæ k t, utin puncto r, ſit res uiſa:</s> <s xml:id="echoid-s29947" xml:space="preserve"> adhuc <lb/>uidebitur eius ſorma una.</s> <s xml:id="echoid-s29948" xml:space="preserve"> Quod ſi fuerit in puncto ualde remoto, <lb/>ut in puncto f, tunc uidebitur una res ibi exiſtens eſſe duæ.</s> <s xml:id="echoid-s29949" xml:space="preserve"> Ducan-<lb/>tur enim à puncto b lineæ b k, b r, b f.</s> <s xml:id="echoid-s29950" xml:space="preserve"> Palàm ergo per 47 & 19 p 1, <lb/>quoniam linea b k eſt maior quàm linea b t:</s> <s xml:id="echoid-s29951" xml:space="preserve"> ſed linea k q eſt æ qualis <lb/>lineæ q t ex hypotheſi:</s> <s xml:id="echoid-s29952" xml:space="preserve"> ergo per 35 th.</s> <s xml:id="echoid-s29953" xml:space="preserve"> 1 huius angulus t b q eſt ma-<lb/>iorangulo q b k:</s> <s xml:id="echoid-s29954" xml:space="preserve"> eſt enim in trigono orthogonio, quod eſt t b k pro-<lb/>ducta linea b q ab angulo t b k:</s> <s xml:id="echoid-s29955" xml:space="preserve"> ergo proportio anguli q b k ad an-<lb/>gulũ t b q minor, quã partis baſis, quæ eſt q k, ad p artem baſis, quæ <lb/>eſt q t:</s> <s xml:id="echoid-s29956" xml:space="preserve"> ſed partes illę baſis ad inuicẽ ſunt ęquales:</s> <s xml:id="echoid-s29957" xml:space="preserve"> ergo angulus t b q <lb/>eſt maior angulo q b k per 10 p 5:</s> <s xml:id="echoid-s29958" xml:space="preserve"> ſed ք 4 p 1 angulus t b q eſt ęqualιs <lb/>angulo k a q:</s> <s xml:id="echoid-s29959" xml:space="preserve">angulus ergo k a q eſt maior angulo k b q:</s> <s xml:id="echoid-s29960" xml:space="preserve"> ergo ք argumentũ 1 petitionis factę in prin-<lb/> <pb o="164" file="0466" n="466" rhead="VITELLONIS OPTICAE"/> cipijs primi libri huius remotio lineæ a k ab axe a q eſt maior quá remotio lineæ b k ab axe b q.</s> <s xml:id="echoid-s29961" xml:space="preserve"> Dif-<lb/>ferentia tamen inter has duas remotiones eſt modica:</s> <s xml:id="echoid-s29962" xml:space="preserve"> quoniam differẽtia inter duos augulos k a q, <lb/>& k b q eſt modica.</s> <s xml:id="echoid-s29963" xml:space="preserve"> Forma ergo puncti k non multum obliquabitur ab axibus uiſualibus, qui ſunt <lb/>b q, & a q.</s> <s xml:id="echoid-s29964" xml:space="preserve"> Non ergo uidebitur illius puncti k forma niſi una, quoniam forma eius non multùm elon <lb/>gatur à puncto medio cócauitatis nerui.</s> <s xml:id="echoid-s29965" xml:space="preserve"> Et quoniá corpore aliquo exiſtente in pũctor, patet, quòd <lb/>radij exeuntes ad ipſum, ſunt b r & a r:</s> <s xml:id="echoid-s29966" xml:space="preserve"> & quia etiam duo anguli r a q & r b q nõ multũ differunt, quo <lb/>niam angulus k b r, qui eſt illorum angulorum differentia, ut patet, non habet ſenſibilẽ quantitatẽ, <lb/>quando punctus r fuerit ual de propinquus puncto k:</s> <s xml:id="echoid-s29967" xml:space="preserve"> forma ergo puncti r adhuc non uidebitur niſi <lb/>una, Siuerò corpus aliquod, cuius ſorma ſe offert uiſui, exiſtat in aliquo puncto lineę perpendicu-<lb/>laris ſuper ſuperficiem uiſus, quæ eſta c, remoto ualde à puncto k, ut eſt punctum ſ:</s> <s xml:id="echoid-s29968" xml:space="preserve"> tunc quia angu-<lb/>li f b q & f a q ſunt diuerſi maxiama diuerſitate, ideo, quòd angulus f b k, qui eſt illorum angulorum <lb/>differentia, eſt ſenſibilis quantitatis:</s> <s xml:id="echoid-s29969" xml:space="preserve"> tunc corpus, quod eſt apud punctum f, uidebitur duo, quando <lb/>duo axes concurrunt in puncto q:</s> <s xml:id="echoid-s29970" xml:space="preserve"> forma enim puncti fobliquè incidit ſuperficiei uiſus b:</s> <s xml:id="echoid-s29971" xml:space="preserve"> unde nõ <lb/>peruenit ad medium punctum concauitatis nerui, ut patet per 102 huius, ſed apparet ultra illud:</s> <s xml:id="echoid-s29972" xml:space="preserve"> ſic <lb/>ergo numeratur forma illius punctif.</s> <s xml:id="echoid-s29973" xml:space="preserve"> Exhocitaq;</s> <s xml:id="echoid-s29974" xml:space="preserve"> patet, quòd uiſum, in quo concurrunt duo axes, <lb/>ſemper uidetur unum, ſicut etiam patuit per 46 th.</s> <s xml:id="echoid-s29975" xml:space="preserve"> 3 huius, & quòd unumquodq;</s> <s xml:id="echoid-s29976" xml:space="preserve"> uiſorum, in quo <lb/>concurrunt radij conſimilis poſitionis, inter quos non eſt magna diſtantia ab ambobus axibus, ui-<lb/>detur etiam unum:</s> <s xml:id="echoid-s29977" xml:space="preserve"> illud uerò uiſum, in quo concurrunt radij multùm diſtantes ab axibus, uidetur <lb/>duo:</s> <s xml:id="echoid-s29978" xml:space="preserve"> propterea quòd ipſum uni uiſuũ incidit directè & alteri ualde obliquè:</s> <s xml:id="echoid-s29979" xml:space="preserve">uel ſi ambobus uiſibus <lb/>incidit obliquè, & una illarum obliquitatum eſt ſenſibiliter maior quá altera.</s> <s xml:id="echoid-s29980" xml:space="preserve"> Videtur ergo talis res <lb/>duæ per 104 huius.</s> <s xml:id="echoid-s29981" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s29982" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1213" type="section" level="0" n="0"> <head xml:id="echoid-head955" xml:space="preserve" style="it">107. Puncto coniunctionis cadente in angulũ trigoni, cui ſubtenſa baſis ſit æqualis line æ con-<lb/>nectenti centra oculorum, ſecundum terminos ſuæ baſis applicati centris amborum uiſnum: <lb/>quodlibet duorum laterum trigoni duas formas uiſuirepræſentat.</head> <p> <s xml:id="echoid-s29983" xml:space="preserve">Sint centra amborum uiſuum a & b:</s> <s xml:id="echoid-s29984" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s29985" xml:space="preserve"> trigonum a b q applicatum uiſibus taliter ut proponi-<lb/>tur:</s> <s xml:id="echoid-s29986" xml:space="preserve"> uel ſit ita, ut trigoni a b q baſis a b ſit baſsior centris oculorum, in-<lb/> <anchor type="figure" xlink:label="fig-0466-01a" xlink:href="fig-0466-01"/> cidantq́;</s> <s xml:id="echoid-s29987" xml:space="preserve"> axes uiſuales in punctum q, qui ſit punctus coniunctionis:</s> <s xml:id="echoid-s29988" xml:space="preserve"> & <lb/>axis communis ſit h q.</s> <s xml:id="echoid-s29989" xml:space="preserve"> Dico quòd laterum trigoni, quæ ſunt a q & b q, <lb/>unumquodq;</s> <s xml:id="echoid-s29990" xml:space="preserve"> duas formas uidenti præſentabit.</s> <s xml:id="echoid-s29991" xml:space="preserve"> Quoniam enim utra-<lb/>que formarum linearum a q & b q, utriq;</s> <s xml:id="echoid-s29992" xml:space="preserve"> uiſui ſe offert directè & obli <lb/>què, ut linea dextra, quæ eſt a q, dextro uiſui (qui eſt a) ſe offert dire-<lb/>ctè, quoniam omnes radij à quolibet ſuorum punctorũ exeuntes inci-<lb/>duntin cẽtrum foraminis uueæ per 24 th.</s> <s xml:id="echoid-s29993" xml:space="preserve"> 3 huius:</s> <s xml:id="echoid-s29994" xml:space="preserve"> & linea ſiniſtra, quę <lb/>eſt b q, incidit obliquè uiſui dextro, qui eſt a:</s> <s xml:id="echoid-s29995" xml:space="preserve"> & ecõuerſo linea b q ſini-<lb/>ſtro uiſui (qui eſt b) directè incidit, & linea a q eidem uiſui ſiniſtro, qui <lb/>eſt b, incidit obliquè, ut hęc omnia patent per 24 th.</s> <s xml:id="echoid-s29996" xml:space="preserve"> 3 huius.</s> <s xml:id="echoid-s29997" xml:space="preserve"> Forma i-<lb/>taque obliquè incidens dextro uiſui declinat ultra latus ſiniſtrum, cu-<lb/>ius ipſa eſt forma, & fit ſiniſtra ab axe:</s> <s xml:id="echoid-s29998" xml:space="preserve"> & forma obliquè incidens ſini-<lb/>ſtro uiſui, declinat ad latus dextrum, cuius ipſa eſt forma, & ſit dextra <lb/>ab axe:</s> <s xml:id="echoid-s29999" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s30000" xml:space="preserve"> laterum trigoni omnia puncta in apparentia uiſuum <lb/>duplicata:</s> <s xml:id="echoid-s30001" xml:space="preserve"> præter ſolum punctum q, qui eſt punctus coniunctionis:</s> <s xml:id="echoid-s30002" xml:space="preserve"> & <lb/>eſt ratio huius apparitionis eadẽ illi in præcedenti theoremate declarata.</s> <s xml:id="echoid-s30003" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s30004" xml:space="preserve"/> </p> <div xml:id="echoid-div1213" type="float" level="0" n="0"> <figure xlink:label="fig-0466-01" xlink:href="fig-0466-01a"> <variables xml:id="echoid-variables500" xml:space="preserve">q b h a</variables> </figure> </div> </div> <div xml:id="echoid-div1215" type="section" level="0" n="0"> <head xml:id="echoid-head956" xml:space="preserve" style="it">108. Vnam rem nonnunquam uideri duas experiment aliter declaratur. Alhazen 12 n 3.</head> <p> <s xml:id="echoid-s30005" xml:space="preserve">Aſſumatur tabula lignea planarum ſuperficierum, cuius lineæ longitudinis æ quidiſtantes & æ-<lb/>quales ſint a c, & b d:</s> <s xml:id="echoid-s30006" xml:space="preserve"> & ſint unius cubiti:</s> <s xml:id="echoid-s30007" xml:space="preserve"> latitudinis uerò ipſius lineæ æ quales & ęquidiftãtes:</s> <s xml:id="echoid-s30008" xml:space="preserve"> ſintq́:</s> <s xml:id="echoid-s30009" xml:space="preserve"> <lb/>a b, & c d:</s> <s xml:id="echoid-s30010" xml:space="preserve"> & ſint quatuor digitorum, orthogonaliter ſuper lineas longitudinis erectę:</s> <s xml:id="echoid-s30011" xml:space="preserve"> ducanturq́ue <lb/>duæ diagonij, quæ ſint a d, & b c, ſecantes ſe in puncto q:</s> <s xml:id="echoid-s30012" xml:space="preserve"> & à puncto q, quod per 40 th.</s> <s xml:id="echoid-s30013" xml:space="preserve"> 1 huius eſt <lb/>medius punctus ſuperficiei totius tabulæ a b c d, ducatur ad utrum que latus longitudinis linea ę-<lb/>quidiſtans lineis latitudinis per 31 p 1, quæ ſit k q t:</s> <s xml:id="echoid-s30014" xml:space="preserve"> & ab eodem puncto q ducatur linea h q z æ quidi-<lb/>ſtans lineis longitudinis a c, & b d:</s> <s xml:id="echoid-s30015" xml:space="preserve"> & intingantur omnes iſtę lineę b c, a d, t k, h z tincturis lucidis di <lb/>uerſorum colorum, ut bene appareant:</s> <s xml:id="echoid-s30016" xml:space="preserve"> ſed ramen duę diagonij, quæ ſunt a d, & b c, ſint unius colo-<lb/>ris:</s> <s xml:id="echoid-s30017" xml:space="preserve"> & ſuper punctum h interiorem terminum lineæ z h in medio latitudinis ipſius tabulæ cauetur <lb/>tabula quaſi pyramidaliter, ut ibi poſsit intrare cornu naſi:</s> <s xml:id="echoid-s30018" xml:space="preserve"> ita ut cum tabula ſuperponitur ſuperio-<lb/>ri parti ipſius naſi, tangant duo anguli tabulæ ferè duo media ſuperficierum duorum uiſuum:</s> <s xml:id="echoid-s30019" xml:space="preserve"> & ſit <lb/>hæc concauitas m h n.</s> <s xml:id="echoid-s30020" xml:space="preserve"> Fiant itaque de cera tria corpuſcula columnaria, & ſint diuerſorum colorum:</s> <s xml:id="echoid-s30021" xml:space="preserve"> <lb/>quæ ſint e, g, p:</s> <s xml:id="echoid-s30022" xml:space="preserve"> & erigantur iſtæ columnæ ſuper ſuperficiem tabulæ in linea k q t, ita, quòd corpus g <lb/>ſit ſuք punctũ q, & corpus p ſuper punctũ k, & corpus e ſuper punctũ t:</s> <s xml:id="echoid-s30023" xml:space="preserve"> & applicẽtur illa corpora fir <lb/>miter ipſi tabulę, ita quòd nõ cadãt, & tũc applicetur tabula uiſib.</s> <s xml:id="echoid-s30024" xml:space="preserve"> ut ſuprà pręmiſsũ eſt.</s> <s xml:id="echoid-s30025" xml:space="preserve"> Deinde ex-<lb/>perimẽtator inſpiciat forti intuitu corp{us} g, qđ eſt in pũcto q medio pũcto tabulę:</s> <s xml:id="echoid-s30026" xml:space="preserve"> tũc ergo duo axes <lb/>amborũ uiſuũ cõcurrent in aliquo pũcto ſuքſiciei corporis g, & ſuքponentur dua bus diagonijs ta-<lb/>bulę, quę ſunt b q, & a q, aut erunt æ quidiſtátes illis, & axis cõmunis ſuperponetur lineæ h q.</s> <s xml:id="echoid-s30027" xml:space="preserve"> Et ſi in <lb/> <pb o="163" file="0467" n="467" rhead="LIBER QVARTVS."/> hac diſpoſitione intueantur ambo uiſus omnia, quæ ſunt in ſuperficie tabulæ & corpora & lineas-<lb/>inuenietur forma uniuſcuiuſq;</s> <s xml:id="echoid-s30028" xml:space="preserve"> corporũ, quę ſunt <lb/>e, g, p, forma una, & tota forma lineæ k q t erit una:</s> <s xml:id="echoid-s30029" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0467-01a" xlink:href="fig-0467-01"/> linea uerò h z extenſa in longitudine tabulæ appa-<lb/>rebit lineæ duæ, ſecantes ſe ſuper punctum q, uel ſu <lb/>per quodcunq;</s> <s xml:id="echoid-s30030" xml:space="preserve"> aliud punctũ cõcurrant radij uiſua <lb/>les:</s> <s xml:id="echoid-s30031" xml:space="preserve"> & etiam quælibet duarũ diagoniorũ, quę ſunt <lb/>b c & a d, apparebit duplicata, ita ut uideátur qua-<lb/>tuor diagonij:</s> <s xml:id="echoid-s30032" xml:space="preserve"> angulus uerò a q b apparebit am-<lb/>plior quá ſit ſecundum ueritatem.</s> <s xml:id="echoid-s30033" xml:space="preserve"> Et ſi alter uiſuũ <lb/>claudatur, uidebũtur duę tantũ diagnoij:</s> <s xml:id="echoid-s30034" xml:space="preserve"> & diago-<lb/>nius remota à medio ſequetur uiſum coopertum.</s> <s xml:id="echoid-s30035" xml:space="preserve"> <lb/>Ex quo patet, quòd duæ diagonij, quę uidentur re <lb/>motæ, ſunt illę, quarũ utraq;</s> <s xml:id="echoid-s30036" xml:space="preserve"> uidetur uiſu obliquo:</s> <s xml:id="echoid-s30037" xml:space="preserve"> <lb/>& propter hoc cõprehenditur per radios remotos <lb/>ab axe dextros & ſiniſtros:</s> <s xml:id="echoid-s30038" xml:space="preserve"> unde inſtituũtur in cõ-<lb/>cauitate nerui cõmunis ab inuicẽ remotæ:</s> <s xml:id="echoid-s30039" xml:space="preserve"> infigun <lb/>tur enim in duabus partib, contrarijs reſpectu pun <lb/>cti medij nerui cõmunis, & in partib.</s> <s xml:id="echoid-s30040" xml:space="preserve"> remotis ab il <lb/>lo pũcto:</s> <s xml:id="echoid-s30041" xml:space="preserve"> unde illæ duę diagonij habent duas for-<lb/>mas propinquas ſibi, & duas remotas à ſeinuicẽ.</s> <s xml:id="echoid-s30042" xml:space="preserve"> <lb/>Deinde experimentator figat axes uiſuales ſuper <lb/>aliquod corporum, quę ſunt e & p, quę ſunt ſuper <lb/>pũcta t & k extrema lineę t q k:</s> <s xml:id="echoid-s30043" xml:space="preserve"> tunc enim appare-<lb/>bũt omnia numero, quo prius.</s> <s xml:id="echoid-s30044" xml:space="preserve"> Quòd ſi corpora e <lb/>& p auſerantur à locis ſuis, & ponátur in linea h z <lb/>ęquidiſtanter à pũcto q:</s> <s xml:id="echoid-s30045" xml:space="preserve">& ſit corpus e uicinius uiſibus in puncto l circa pũctum q:</s> <s xml:id="echoid-s30046" xml:space="preserve"> & corpus p ſitre-<lb/>motius à uiſu in pũcto s, ultra punctum q, & applicata tabula ipſis uiſibus, ſigantur axes uiſuales ſu-<lb/>per corpus g, quod eſt in puncto q medio:</s> <s xml:id="echoid-s30047" xml:space="preserve"> tunc unũquodq;</s> <s xml:id="echoid-s30048" xml:space="preserve"> corporum e & p apparebit duo, & appa <lb/>rebũt ambo illa corpora, quatuor corpora obliquè à medio corpore g, duo ſcilicet in dextro, & duo <lb/>in ſiniſtro:</s> <s xml:id="echoid-s30049" xml:space="preserve"> & uidebuntur ſuper duas lineas, quãuis ſecũdum ueritatem ſint ſuper lineã unam, & ap-<lb/>parebũt quęlibet duo illorum quatuor eorporum ſuper alterã ill arum duarum linearum.</s> <s xml:id="echoid-s30050" xml:space="preserve"> Idẽ quoq:</s> <s xml:id="echoid-s30051" xml:space="preserve"> <lb/>accidit ſi corpora e & p ponantur ſuper alterá duarũ diagoniorũ ſecundũ omnem modum, quo po <lb/>ſita fuerint ſuper lineá h z:</s> <s xml:id="echoid-s30052" xml:space="preserve"> taliter ut æ quidiſtent corporig, & unũ ſit propin quius uiſui quã alterũ:</s> <s xml:id="echoid-s30053" xml:space="preserve"> <lb/>quia tunc utraq;</s> <s xml:id="echoid-s30054" xml:space="preserve"> diagoniorum apparebit duę:</s> <s xml:id="echoid-s30055" xml:space="preserve"> unde ſuper utramq;</s> <s xml:id="echoid-s30056" xml:space="preserve"> linearũ, quę ſunt unius diagonij, <lb/>duo apparebunt corpora, unũ in parte ipſius uiſus, & aliud ultra corpus g poſitũ in medio illorum <lb/>duorũ corporũ.</s> <s xml:id="echoid-s30057" xml:space="preserve"> Et ſimiliter ſi corpora e & p ponantur ſuper ambas diagonios, unũ ſuper unam, & <lb/>aliud ſuper aliã, & ambo in parte uiſus:</s> <s xml:id="echoid-s30058" xml:space="preserve"> tũc enιm apparebũt quatuor corpora, duo ꝓpinqua, & duo <lb/>remota.</s> <s xml:id="echoid-s30059" xml:space="preserve"> Deinde auferantur duo corpora e & p à tabula, & ponatur alterũ ipſorum ſuper marginem <lb/>tabulæ in linea a c ultra punctum k, & tamẽ ualde uicinè illi pũcto k, & ſit ſuper punctũr:</s> <s xml:id="echoid-s30060" xml:space="preserve"> & tunc ap-<lb/>plicata tabula uiſibus, ditigantur adhuc axes ad corpus g poſitũ in medio:</s> <s xml:id="echoid-s30061" xml:space="preserve"> & tunc apparebit forma <lb/>punctie tan tũ una.</s> <s xml:id="echoid-s30062" xml:space="preserve"> Quòd ſi corpus ein eadem linea a c ponatur ſuper punctũ f remotius à pũcro k, <lb/>quàm ſit punctum 1:</s> <s xml:id="echoid-s30063" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s30064" xml:space="preserve"> puncti ſ à puncto k diſtátia ſenſibilis:</s> <s xml:id="echoid-s30065" xml:space="preserve"> & ſic directis axib.</s> <s xml:id="echoid-s30066" xml:space="preserve"> uiſualibus ad cor <lb/>pus g medium, apparebit forma corporis e duplicata.</s> <s xml:id="echoid-s30067" xml:space="preserve"> Idẽ quoq;</s> <s xml:id="echoid-s30068" xml:space="preserve"> accidit, ſi ambo axes uiſuales ſecun <lb/>dũ iſtá diſpoſitionem dirigantur ad quodcunq;</s> <s xml:id="echoid-s30069" xml:space="preserve"> punctũ lineę t k:</s> <s xml:id="echoid-s30070" xml:space="preserve"> ſem per enim tũc corpus e poſitum <lb/>in pũcto ſuidebitur eſſe duo.</s> <s xml:id="echoid-s30071" xml:space="preserve"> Hęc uerò, quę præmiſſa ſunt, omnia per 105 huius & propoſitiones ſe-<lb/>quẽtes declarantur, ut patet intuẽti.</s> <s xml:id="echoid-s30072" xml:space="preserve"> Quòd ſi experimentator direxerit axes uiſuales ad punctũ ali-<lb/>quem tabulæ extra lineam k t:</s> <s xml:id="echoid-s30073" xml:space="preserve"> tunc ipſum corpus g, poſitũ in medio ſuperficiei tabulæ in puncto q, <lb/>uidebitur duo:</s> <s xml:id="echoid-s30074" xml:space="preserve"> & ſi corpus eponatur in pũcto t, & corpus p in pũcto k:</s> <s xml:id="echoid-s30075" xml:space="preserve"> tunc trun q;</s> <s xml:id="echoid-s30076" xml:space="preserve"> ip ſorũ uidebi-<lb/>tur duo.</s> <s xml:id="echoid-s30077" xml:space="preserve"> Sed redeuntib.</s> <s xml:id="echoid-s30078" xml:space="preserve"> axibus uiſualibus ſuper punctũ q, aut ſuper aliquod punctũ lineę t k:</s> <s xml:id="echoid-s30079" xml:space="preserve"> tũc re <lb/>uertetur prior diſpoſitio.</s> <s xml:id="echoid-s30080" xml:space="preserve"> Deinde accipiat experimẽtator tres ſchedulas pergameni patuas & ęqua <lb/>les, & inſctibat oẽs ipſas una ſcripturptura maniſeſta ęqualis quãtitatis:</s> <s xml:id="echoid-s30081" xml:space="preserve"> & ponat unã ipſarũ in medio p̃.</s> <s xml:id="echoid-s30082" xml:space="preserve"> <lb/>miſſæ tabulæ in puncto q, & alteram ipſarũ ſuper punctũ k, ſigendo cũ cera, ut ſtent erectè, & appli-<lb/>cata tabula ipſis uiſibus, ut prius, intueatur ſchedulã poſitã ſuper punctũ q, & cõprehendet eius ſcri <lb/>pturá certa cóprehenſione:</s> <s xml:id="echoid-s30083" xml:space="preserve"> & ſimiliter ſcriptureã ſchedulę poſitę in pun cto k cõprehendet:</s> <s xml:id="echoid-s30084" xml:space="preserve"> ſed nõ ita <lb/>perſectè ut ſcripturã ſchedulę poſitę in puncto q, licet ſint illę ſeripturæ cóſimiles in figura, forma & <lb/>quãtitate.</s> <s xml:id="echoid-s30085" xml:space="preserve"> Deinde aſſumatur tertia ſchedula, & ponatur quaſι in medio pũcto lineę e z, & manu ք-<lb/>tracta ſecundũ rectitudinem lineę k c, teneatur ultra tabulá in ſitu & poſitione duarũ aliarum ſche-<lb/>dularũ:</s> <s xml:id="echoid-s30086" xml:space="preserve"> tunc enim fixis ambobus axibus uiſuum in ſchedula poſita in pũcto q, & tũc uiſa tertia ſche <lb/>dula, uidebitur forma ſcripturæ ſuę dubitabilis & in diſtincta:</s> <s xml:id="echoid-s30087" xml:space="preserve"> & ſi ſchedula pũcti k depoſita ſchedu-<lb/>latertia ponatur penes primã, quę eſt in puncto q:</s> <s xml:id="echoid-s30088" xml:space="preserve">tũc ambę ſchedulę cõprehendentur in ſuis ſcri-<lb/>pturis ę qualiter diſpoſitę, n e c erit differentia ſenſibilis inter illas:</s> <s xml:id="echoid-s30089" xml:space="preserve"> & ſi tertia ſchedula moueatur pla.</s> <s xml:id="echoid-s30090" xml:space="preserve"> <lb/>nè ſuք lineã q k, axibus ill orũ uiſuũ cadentib.</s> <s xml:id="echoid-s30091" xml:space="preserve"> in pũctũ q:</s> <s xml:id="echoid-s30092" xml:space="preserve"> uidebitur tũc diminui diſtinctio ſcripturæ <lb/>ſchedulę motę ſecundum diſtantiam, quę ſit per motum, donec perueniat ad punctum k:</s> <s xml:id="echoid-s30093" xml:space="preserve"> & tune <lb/> <pb o="166" file="0468" n="468" rhead="VITELLONIS OPTICAE"/> paulatim à puncto k extra rabulam moueatur ſecundum lineam latitudinis a k protèſam:</s> <s xml:id="echoid-s30094" xml:space="preserve"> tunc ſem-<lb/>per minuetur ſcripturæ diſtinctio, ita quòd tandem nulla erit diſcretio ipſius.</s> <s xml:id="echoid-s30095" xml:space="preserve"> Peractisq́;</s> <s xml:id="echoid-s30096" xml:space="preserve"> circa lin eá <lb/>c d eiſdem, quæcum his ſchedulis facta ſuntcirca lineam k t, eadem tuncuiſibus apparent, quęp ri-<lb/>us, ſeruata diſtantiæ proportione:</s> <s xml:id="echoid-s30097" xml:space="preserve"> & etiam ſi elongenturultra longitudinem tabulæ.</s> <s xml:id="echoid-s30098" xml:space="preserve"> Quæ itaq ue <lb/>ex his paſsionibus ambobus uiſibus accidunt, plus accident uni uiſuum, ſi alter fuerit coopert us.</s> <s xml:id="echoid-s30099" xml:space="preserve"> <lb/>Deinde aſſumatur ſchedula quatuor digitorum quadrata, in qua punctus medius ſignetur per 40 <lb/>th.</s> <s xml:id="echoid-s30100" xml:space="preserve"> 1 huius, & alia ſchedula ſcribatur ſcriptura aliqua diſtincta, & erigatur hęc ſchedula ſuper line am <lb/>k t, & dirigatur uiſus ad medium illius ſchedulæ:</s> <s xml:id="echoid-s30101" xml:space="preserve"> tunc enim uidebitur ſcriptura bene diſtin cta, ſed <lb/>ſcriptura, quæ eſt circa medium ſchedulæ, uidebitur diſtinctior, quàm quæ in extremis.</s> <s xml:id="echoid-s30102" xml:space="preserve"> Dein de <lb/>parum obliquetur ſchedula ſuper lineam t k, in puncto q:</s> <s xml:id="echoid-s30103" xml:space="preserve"> & tunc axibus uiſuum cadentibus ſuper <lb/>medium punctum ſchedulæ, inuenietur ſchedula minus diſtincta quàm prius, cum ſchedula fuerit <lb/>fuper lineam k t:</s> <s xml:id="echoid-s30104" xml:space="preserve"> & ſi ſchedula plus obliquabitur, indiſtinctior uidebitur ſcriptura, & quantò ma-<lb/>gis obliquabitur ſchedula, tantò magis latebit utrun que uiſum uel alterum ipſa ſcriptura.</s> <s xml:id="echoid-s30105" xml:space="preserve"> Et ſi ſche <lb/>dula ſecundum alterum ſuorum extremorum ponatur in puncto q, & erigatur ſuper ſuperficiem <lb/>tabulæ ſecundum lineam k q:</s> <s xml:id="echoid-s30106" xml:space="preserve"> tunc patet, quòd medietas ſchedulæ cadet extra tabulam.</s> <s xml:id="echoid-s30107" xml:space="preserve"> Viſu ita-<lb/>que cadente in pun ctum q, tunc uidebitur ſcriptura circa punctum q diftinctior, minus autem ſe-<lb/>cundum partes remotiores abillo:</s> <s xml:id="echoid-s30108" xml:space="preserve"> & ſiobliquetur ſchedula ſuper lineam q k, apparebit latentior <lb/>ſcriptura ſecundum quantitatem obliquationis & diſtantię à puncto q:</s> <s xml:id="echoid-s30109" xml:space="preserve"> & ſi ſchedula ponatur ſuper <lb/>lineam c d, tunc uifibus directis ad medium punctum ſchedulæ, erit litera legibiliter diſtincta:</s> <s xml:id="echoid-s30110" xml:space="preserve"> & ſi <lb/>obliquetur ſchcdula ſuper punctũ z:</s> <s xml:id="echoid-s30111" xml:space="preserve"> tunc erit ſcriptura latentior quàm prius.</s> <s xml:id="echoid-s30112" xml:space="preserve"> Et uniuerſaliter pera-<lb/>cto circa lineam c d, quod prius actum eſt circa lineam t k, idem accidet in diſtinctione ſcripturæ <lb/>proportionaliterilli ſpatio diſtantiæ:</s> <s xml:id="echoid-s30113" xml:space="preserve"> & etiam ſi elongetur ſchedula ultra longitudinem tabulæ.</s> <s xml:id="echoid-s30114" xml:space="preserve"> <lb/>Quod autem accidit ambobus uiſibus in hac experimentatione, etiam accidit uni uiſuum, altero <lb/>cooperto.</s> <s xml:id="echoid-s30115" xml:space="preserve"> Patet ergo ex his experimentationibus exemplum eorum, quę per plura theoremata <lb/>proponuntur:</s> <s xml:id="echoid-s30116" xml:space="preserve"> & patet manifeſtè, quòd pluribus modis accidit unam rem uideri duas.</s> <s xml:id="echoid-s30117" xml:space="preserve"> patetergo <lb/>propoſitum.</s> <s xml:id="echoid-s30118" xml:space="preserve"/> </p> <div xml:id="echoid-div1215" type="float" level="0" n="0"> <figure xlink:label="fig-0467-01" xlink:href="fig-0467-01a"> <variables xml:id="echoid-variables501" xml:space="preserve">d z e s f r f q k e g p l h b n m x</variables> </figure> </div> </div> <div xml:id="echoid-div1217" type="section" level="0" n="0"> <head xml:id="echoid-head957" xml:space="preserve" style="it">109. In uiſione diuiſionis, continuationis & numeri error accidit uirtuti diſtinctiuæ ex in-<lb/>temperata diſpoſitioneocto circumſtantiarum cuiuslibet rei uiſe. Alhazen 27. 38. 48. 55. <lb/>59. 64. 67. 69 n 3.</head> <p> <s xml:id="echoid-s30119" xml:space="preserve">Exlucis enim debilitate error accidit in præmiſſorum uiſione:</s> <s xml:id="echoid-s30120" xml:space="preserve"> quia ſi de nocte uideatur tabula, <lb/>in qua ſint linearum obſcurarum protractiones, uidens illas putabit fortè diuiſiones eſſe uel ſciſſu-<lb/>ras:</s> <s xml:id="echoid-s30121" xml:space="preserve"> & ita continuum etiam putabitur diuiſum, & partes eiuſdem continui plura putabuntur ut di-<lb/>uiſa, cum ta men tabula ſit continua & tantùm una.</s> <s xml:id="echoid-s30122" xml:space="preserve"> Similiter exiſtente uiſu in ſorti luce reflexa, ſi <lb/>ipſi uiſui adhibeantur corpora modicùm diſtantia, apparebunt continua & unum, propter rexflexio <lb/>nem lucis factæ ab illis corporibus, quę non permittit eorum diſtantiam diſcerni.</s> <s xml:id="echoid-s30123" xml:space="preserve"> Ex intemperatata <lb/>etiam diftantia fit error in præmiſſorum uiſione.</s> <s xml:id="echoid-s30124" xml:space="preserve"> Pariete enim aliquo à longè uiſo, ſi in parte eius <lb/>fuerit color tenebroſus:</s> <s xml:id="echoid-s30125" xml:space="preserve"> ſortè putabitur facta eſſe diuifio illius parietis ſecundum ſpatium illius co-<lb/>loris.</s> <s xml:id="echoid-s30126" xml:space="preserve"> Similiter etiá ſi propc parieté illum creſcat altitudo herbarum, ut cóſueuitin talibus creſcere <lb/>hedera:</s> <s xml:id="echoid-s30127" xml:space="preserve"> uidebitur fortè paries ſecundum hederæſpatium diuiſus.</s> <s xml:id="echoid-s30128" xml:space="preserve"> Et fimiliter luce ſolis ſuper uiſum <lb/>album partietem ſplendente, ſi ſortis umbra aliqua lu cem parietis diuiſerit, æſtimabitur paries diui-<lb/>ſus:</s> <s xml:id="echoid-s30129" xml:space="preserve"> & ita his modis omnibus & etiam pluribus alijs hoc poteſt accidere, ut continuum æſtimetur <lb/>diuiſum:</s> <s xml:id="echoid-s30130" xml:space="preserve"> & ex conſequenti unum plura.</s> <s xml:id="echoid-s30131" xml:space="preserve"> Sed & quandoq;</s> <s xml:id="echoid-s30132" xml:space="preserve"> ipſa ſecundum ueritatem diuifa æſtiman-<lb/>tur continua, & plura ęſtimantur unum.</s> <s xml:id="echoid-s30133" xml:space="preserve"> Corpora enim à longè uiſa in colore ſimilia, & adinuicem <lb/>propinqua, creuntur continua, & propter hoc tabulę parietis uel ſcamni apparent quandoq;</s> <s xml:id="echoid-s30134" xml:space="preserve"> con-<lb/>tinuę, cũ modica diuiſione ab inuicem ſunt diuiſæ:</s> <s xml:id="echoid-s30135" xml:space="preserve"> & ſic diuiſa æſtimantur propter remotionẽ à ui-<lb/>ſu eſſe continua, & plura ęſtimantur unũ.</s> <s xml:id="echoid-s30136" xml:space="preserve"> Exinordinato etiam ſitu oppoſitionis oritur error in prę-<lb/>miſſorum uiſione:</s> <s xml:id="echoid-s30137" xml:space="preserve"> ſi enim alicuius corporis magna fuerit à uiſu obliquatio, in quo ſuerint pũcta ſen <lb/>ſibilia nigra uel ualde tenebroſa:</s> <s xml:id="echoid-s30138" xml:space="preserve"> illa quædá diuiſiones putabuntur, & inter partes illis punctis con <lb/>ſines, iudicabitur diuiſio & pluralitas, licet in eis ſit cótinuitatis unio:</s> <s xml:id="echoid-s30139" xml:space="preserve"> & ſi in hoc corpore ſuerint li <lb/>neę tenebroſę ſenſibiles, iudicabũtur partes eius cótinuales diuiſę, cũ ſint cótinuę, & plures, cũ ſint <lb/>unum.</s> <s xml:id="echoid-s30140" xml:space="preserve"> Similiter eſt ex obliquatione ſitus plurium parietum ad uiſum, quorum unus eſt ordinatè <lb/>poſtalium modicùm diſtans ab illo, ita quòd uno aſpectu uideri ualeant, fortè occultabitur ui-<lb/>denti ſpatium, quod eſt inter illos parieters, & putabuntur continui & unus, cum ſint diuerſi & plu-<lb/>res.</s> <s xml:id="echoid-s30141" xml:space="preserve"> Qualiter autem propter ſitum eius erret in numero, ſatis patet per propoſitionem præmiſſam.</s> <s xml:id="echoid-s30142" xml:space="preserve"> <lb/>Ex intemperata etiam magnitudine error accidit in uiſione pręmiſſorum:</s> <s xml:id="echoid-s30143" xml:space="preserve"> adhærente enim capillo <lb/>uaſi uitreo, apparebit uitrum ſiſſum:</s> <s xml:id="echoid-s30144" xml:space="preserve"> quod ideo accidit, quia capilli paruitas non ſentitur eſſe cor-<lb/>pus.</s> <s xml:id="echoid-s30145" xml:space="preserve"> Si enim iaceret ſuper uas uitreum calamus aut corpus aliud ſenſibile, non propter hoc ſen-<lb/>tiretur uitrum eſſe ſiſſum.</s> <s xml:id="echoid-s30146" xml:space="preserve"> Similiter etiam accidit error in continutiata:</s> <s xml:id="echoid-s30147" xml:space="preserve"> ſi enim ſolia pergameni te-<lb/>nuis æqualis altitudinis, ita quòd in eadem plana ſuperſicie conſtitutam, & bene compreſſa ſint, & ui <lb/>dens ignoret eſſe folia, iudicabit ipſa eſſe continua, & unam ſuperficiem ipſorum:</s> <s xml:id="echoid-s30148" xml:space="preserve"> huius autem er-<lb/>roris cauſſa eſt paruitas quantitatis ſpatij & aeris, ſecundum quod ſe illa folia contingunt, & ſic etiá <lb/>numerus inducit errorem.</s> <s xml:id="echoid-s30149" xml:space="preserve"> Exintemperantia quoq;</s> <s xml:id="echoid-s30150" xml:space="preserve"> ſoliditatis ſit error in præmiſſorum uiſione:</s> <s xml:id="echoid-s30151" xml:space="preserve"> in <lb/>corpore enim magnę raritatis, utin cryſtallo pura, ſi in aliqua parte ſupficiei ſuę fuerit linea nigra, <lb/> <pb o="167" file="0469" n="469" rhead="LIBER QVARTVS."/> apparebit totum corpus ſiſſum ſecundũ locum, in quem cadit illa linea, & ita æſtimatur uitrũ diſcó-<lb/>tinuum & plura:</s> <s xml:id="echoid-s30152" xml:space="preserve"> & hoc accidit propter perſpicuitatem, quæ accidit ex deſectu ſoliditatis.</s> <s xml:id="echoid-s30153" xml:space="preserve"> Et ſi duo <lb/>corpora talia fuerint modicùm à ſe diſtantia, reputabuntur continua & unum.</s> <s xml:id="echoid-s30154" xml:space="preserve"> Ex intemperantia e-<lb/>tiam raritatis accidit error in præmiſſorum uiſione idem, qui ex defectu ſoliditatis, augmentatus ta <lb/>men propter exceſſum raritatis.</s> <s xml:id="echoid-s30155" xml:space="preserve"> Expaucitate etiam temporis accidit error in præmifforum uiſio-<lb/>ne.</s> <s xml:id="echoid-s30156" xml:space="preserve"> Si enim corpus, in quo ſit linea nigra, fubitò à uiſu diuertatur, putabitur illa linea eſſe partium di <lb/>uiſio:</s> <s xml:id="echoid-s30157" xml:space="preserve"> & ſi corpora contigua aut ualde propin qua ſubitò uideantur, æſtimabuntur cõtinua, ſicut ac-<lb/>cidit in tabulis ſcamnorum ſubitò inſpectis, & ſit error in continuitate & numero.</s> <s xml:id="echoid-s30158" xml:space="preserve"> Ex intemperan-<lb/>tia etiam debilitatis uiſus error accidit in uiſione præmiſſorum, & ſecundum modos temporis bre-<lb/>uitate accidentes:</s> <s xml:id="echoid-s30159" xml:space="preserve"> quod enim ſano uiſui accidit in temporis breuitate, debili accidit in maiori tem-<lb/>pore, & ſortè ſemper durante uiſus debilitate:</s> <s xml:id="echoid-s30160" xml:space="preserve"> & etiam ſtrabo uel debilis in uno oculo unum quan-<lb/>doq;</s> <s xml:id="echoid-s30161" xml:space="preserve"> iudicat duo:</s> <s xml:id="echoid-s30162" xml:space="preserve"> tunc enim res uiſa habet diuer ſitatem ſitus reſpectu talium duorũ oculorum, quę <lb/>diuerſitas facit ut unum uideatur duo, etiam per duos oculos ſanos & æqualis ordinationis, utſa-<lb/>tis demonſtratum eſt ex præmiſsis.</s> <s xml:id="echoid-s30163" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s30164" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1218" type="section" level="0" n="0"> <head xml:id="echoid-head958" xml:space="preserve" style="it">110. Motus comprehenditur à uiſu ex comprehenſione rei mote ſecundum diuerſos ſui ſitus <lb/>in inſtantibus diuerſis, inter quæ ſenſibile cadit tempus. Alhazen 49 n 2.</head> <p> <s xml:id="echoid-s30165" xml:space="preserve">Quoniam enim moueri eſt aliter ſe habere nunc, quàm prius:</s> <s xml:id="echoid-s30166" xml:space="preserve"> palàm quòd facilitas huius com-<lb/>prehenſionis motus ſit ex comparatione rei motæ uiſæ ad aliud uiſibile quieſcens non motũ.</s> <s xml:id="echoid-s30167" xml:space="preserve"> Quã-<lb/>do enim comprehenditure ſitus unius rei mobilis, reſpectu alterius rei uiſibilis, tunc etiam compre-<lb/>henditur diuerſitas ſitus eius reſpectu illius uiſibilis, & tune comprehenditur motus.</s> <s xml:id="echoid-s30168" xml:space="preserve"> Semperitaq;</s> <s xml:id="echoid-s30169" xml:space="preserve"> <lb/>motus comprehenditur à uiſu aut ex comprehenſione diuerſitatis & mutationis ſitus rei uiſæ mo-<lb/>tæ, reſpectu alterius uiſibilis, quod eſt remotius aut propinquius uiſui, ipſo tamen uiſu in parte alte <lb/>ra exiſtente in ſuo loco:</s> <s xml:id="echoid-s30170" xml:space="preserve"> aut comprehenditur motus experimentatione ſitus alicuius partis, uel par <lb/>tium rei uiſę motæ, reſpe ctuillius uiſibilis non ſecun dum ſe totum modti:</s> <s xml:id="echoid-s30171" xml:space="preserve"> & hoc modo comprehen-<lb/>dit uiſus motum circularem.</s> <s xml:id="echoid-s30172" xml:space="preserve"> Similiter etiam accidit motum à uiſu comprehendi, ſi res uiſa mota <lb/>ad multa immota uiſibilia comparetur.</s> <s xml:id="echoid-s30173" xml:space="preserve"> Cum enim uiſus fuerit quietus, & res uiſa mota ad ipſum ui <lb/>ſum uel à uiſu:</s> <s xml:id="echoid-s30174" xml:space="preserve"> tunc uiſus ſentiens diuerſam locationem corporis moti, ſentiet motum:</s> <s xml:id="echoid-s30175" xml:space="preserve"> aut enim <lb/>mobile tunc elongabitur aut appropinquabit uiſui per motum, Et quia, ut patet per 9 huius, elon-<lb/>gatio aut appropin quatio à uiſu ſentitur, palàm quia motus tunc ſentitur.</s> <s xml:id="echoid-s30176" xml:space="preserve"> Quòd ſi mobile mouetur <lb/>tantùm circa uiſum circulariter, tunc cum ſuperſicies uiſiua oculinon ſittota ſphærica, ut patet per <lb/>4th.</s> <s xml:id="echoid-s30177" xml:space="preserve"> 3 huius, quoniam ſola ſuperficies foraminis uueæ eſt uiſiua, & non aliæ partes ſuperſiciei ocu-<lb/>li:</s> <s xml:id="echoid-s30178" xml:space="preserve"> aliqua itaque re mota circa uiſum, neceſſariò mutabitur ſitus partis oppoſitæ uiſui, & cum illa <lb/>pars rei uiſę motæ ſuerit mutata, ſentiet uiſus mutationem eius:</s> <s xml:id="echoid-s30179" xml:space="preserve"> & ſic uiſu exiſtente in ſuo loco ſen-<lb/>tiet uiſus motum rei uiſæ.</s> <s xml:id="echoid-s30180" xml:space="preserve"> Et ſi ipſe uiſus moueatur, comprehendet tamẽ motum ſecundum quem-<lb/>libet iſtorum modorum, ut cum uiſus ſentit diuerſitatem ſitus rei uiſæ motæ, ſentiẽdo quòd illa di-<lb/>uerſitas non eſt propter motum ipſius uiſus:</s> <s xml:id="echoid-s30181" xml:space="preserve"> ſed tamen quando ipſe uiſus & etiá res uiſa ambo mo-<lb/>uentur, adhuc diſcernit uiſus motum:</s> <s xml:id="echoid-s30182" xml:space="preserve"> quoniam diſtinguit inter diuerſitatem uiſus, quæ accidit rei <lb/>uiſæ motę propter motum ipſius rei, uel propter motum ipſius uiſus, quoniam moto uiſu ſentiun-<lb/>tur etiam formæ corporum exiſtentium non motæ, nec ſemper iudicat uiſus rem uiſam moueri <lb/>propter ſui ipſius motum, niſi fortè perueniat in uiſum forma rei uiſæ motæ.</s> <s xml:id="echoid-s30183" xml:space="preserve"> Et quoniam motus <lb/>omnis eſt in tempore, non comprehendit uiſus motum niſi in tempore:</s> <s xml:id="echoid-s30184" xml:space="preserve"> diuerſitas enim ſitus par-<lb/>tium rei uiſę non poteſt comprehendi inſi ad minus in duobus inſtantibus:</s> <s xml:id="echoid-s30185" xml:space="preserve"> & quia inter quælibet <lb/>duo inſtantia cadit tempus medium:</s> <s xml:id="echoid-s30186" xml:space="preserve"> palàm quòd inter illa duo inſtantia cadit tempus medium:</s> <s xml:id="echoid-s30187" xml:space="preserve"> & <lb/>quoniam uirtus uiſiua eſt uirtus ſenſitiua, oportet tempus ab ipſa comprehenſum eſſe ſen ſibile.</s> <s xml:id="echoid-s30188" xml:space="preserve"> Et <lb/>hoc proponebatur.</s> <s xml:id="echoid-s30189" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1219" type="section" level="0" n="0"> <head xml:id="echoid-head959" xml:space="preserve" style="it">111. Zualit as motus comprehenditur à uiſu ex comprehenſione ſpatij, ſuper quod mouetur res <lb/>ipſauiſa. Alhazen 50 n 2.</head> <p> <s xml:id="echoid-s30190" xml:space="preserve">Siue enim motus ſit ſurſum uel deorſum, uel etiam ſuperipſam ſuperficiem horizontis uel æqui-<lb/>diſtantem illi, ſiue etiam nó ſit motus rectus, ſed ſit tortuo ſus uel circularis:</s> <s xml:id="echoid-s30191" xml:space="preserve"> ſemper qualitas motus <lb/>comprehenditur à uiſu ex comprehenſione ſpatij, ſuper quod mouetur res ipſa uiſa.</s> <s xml:id="echoid-s30192" xml:space="preserve"> Qualitas enim <lb/>motus recti comprehenditur ex comprehenſione ſpatij, ſuper quod mouetur res uiſa ſecundum ſe <lb/>totam motu recto, & tunc uiſus certiſicat qualitatẽ motus per certificationem figuræ ſpatij directi, <lb/>ſuper quod fit motus in ſuperficie horizontis, autin ſuperficie æquidiſtante ei, aut in linea perpen <lb/>diculari uel obliqua ſuper ſuperficiem horizontis.</s> <s xml:id="echoid-s30193" xml:space="preserve"> Similiter quoque qualitas aliorum motuum us <lb/>tortuoſi & circularis comprehenditur à uiſu ex comprehenſione ſpatij tortuoſi uel etiam circula-<lb/>ris, in ſuperficie horizontis, aut æ quidiſtante ipſi, aut erecta ſuper ipſam:</s> <s xml:id="echoid-s30194" xml:space="preserve"> motum enim compoſi-<lb/>tum excirculari & recto uiſus comprehendet ex comprehenſione ſpatij tortuoſi, ſuper quod fit mo <lb/>tus.</s> <s xml:id="echoid-s30195" xml:space="preserve"> Comprehendit etiam uiſus diuerſitatem & æqualitatem motuum ſecundum uelocitatem & <lb/>tarditatem ex comprehenſione ſpatiorum, ſuper quæ mouentur uiſibilia mota, & cognitione tem-<lb/>poris, in quo ſiunt illi motus.</s> <s xml:id="echoid-s30196" xml:space="preserve"> Cum enim uiſus ſentit quòd unum ſpatium pertranſitum ab uno <lb/>mobili in aliquo tempore, eſt maius alio ſpatio pertranſito ab alio mobli in eodem tempore:</s> <s xml:id="echoid-s30197" xml:space="preserve"> uel <lb/>cum uiſus ſenſerit æqualitatem duorum ſpatiorum cum inæ qualitate temporum duorum motnũ:</s> <s xml:id="echoid-s30198" xml:space="preserve"> <lb/> <pb o="168" file="0470" n="470" rhead="VITELLONIS OPTICAE"/> tunc enim ſtatim auxilio uirtutis animæ diſtinctiuę & cognoſcitiuę ſentiet uelocitatẽ unius mobi-<lb/>lis ſuper alterum & duorum motuũ inę qualitatem.</s> <s xml:id="echoid-s30199" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s30200" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1220" type="section" level="0" n="0"> <head xml:id="echoid-head960" xml:space="preserve" style="it">112. Zuies comprehenditur à uiſu ex comprehenſione rei uiſæ in eodem loco & ſitu tempore <lb/>ſenſibili permanente. Alhazen 52 n 2.</head> <p> <s xml:id="echoid-s30201" xml:space="preserve">Cum enim uiſus compreh enderit rem uiſam in eodem loco, & ſecundum eundem ſitum in duo-<lb/>bus inſtantibus diuerſis, inter quæ cadit medium tempus ſenſibile:</s> <s xml:id="echoid-s30202" xml:space="preserve"> tunc comprehendet rem in illo <lb/>tempore non fuiſſe motam per 110 huius:</s> <s xml:id="echoid-s30203" xml:space="preserve"> quoniam ſi illa res in illo tempore fuit mota mutatus eſt <lb/>ſitus eius:</s> <s xml:id="echoid-s30204" xml:space="preserve"> comprehẽdet ergo illam rem quieſcentem.</s> <s xml:id="echoid-s30205" xml:space="preserve"> Comprehenditur aũt ſitus rei uiſæ quieſcẽtis <lb/>non mutatus reſpectu alterius rei uel aliarum rerum uiſarum, & etiam reſpectu ipſius uiſus.</s> <s xml:id="echoid-s30206" xml:space="preserve"> Secun-<lb/>dum hunc ergo modum ſit comprehenſio quietis uiſorum corporum à uiſu.</s> <s xml:id="echoid-s30207" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s30208" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1221" type="section" level="0" n="0"> <head xml:id="echoid-head961" xml:space="preserve" style="it">113. Eſt locus, in quo oculo manente & tranſpoſita re uiſa, res ſemper æqualis apparet. Eucli-<lb/>des 44 the. opt.</head> <p> <s xml:id="echoid-s30209" xml:space="preserve">Sit res uiſa b g:</s> <s xml:id="echoid-s30210" xml:space="preserve"> & ſit centrum uiſus in puncto a:</s> <s xml:id="echoid-s30211" xml:space="preserve"> & accedant formę punctorum b & g ad uiſum a <lb/>ſecundum lineas b a & g a:</s> <s xml:id="echoid-s30212" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s30213" xml:space="preserve"> trigonum a b g, Dico, quòd eſt locus, in quo non mutato centro ui-<lb/> <anchor type="figure" xlink:label="fig-0470-01a" xlink:href="fig-0470-01"/> ſus à puncto a, & tranſpoſita magnitudine b g, ſemper eiuſdẽ <lb/>quantitatis uidebitur magnitudo b g.</s> <s xml:id="echoid-s30214" xml:space="preserve"> Trigono enim a b g cir-<lb/>cunſcribatur circulus per 5 p 4:</s> <s xml:id="echoid-s30215" xml:space="preserve"> & ſuper punctum g terminum <lb/>lineę a g cõſtiotuatur angulus æqualis angulo a g b ք 23 p 1, qui <lb/>ſit a g d.</s> <s xml:id="echoid-s30216" xml:space="preserve"> & producta linea g d ad peripheriá circuli copulentur <lb/>lineæ a b & a d:</s> <s xml:id="echoid-s30217" xml:space="preserve"> erit q́;</s> <s xml:id="echoid-s30218" xml:space="preserve"> per 26 p 3 arcus ad ęqualis arcui b a:</s> <s xml:id="echoid-s30219" xml:space="preserve"> ergo <lb/>per 29 p 3 eſt chorda a b ęqualis chordę a d:</s> <s xml:id="echoid-s30220" xml:space="preserve"> & arcus g d, qui eſt.</s> <s xml:id="echoid-s30221" xml:space="preserve"> <lb/>reſiduus ſemicirculi, eſt ęqualis arcui b g:</s> <s xml:id="echoid-s30222" xml:space="preserve"> chorda quo q;</s> <s xml:id="echoid-s30223" xml:space="preserve"> g d e-<lb/>rit ęqualis chordæ b g per 29 p 3:</s> <s xml:id="echoid-s30224" xml:space="preserve"> ergo per 8 p 1, uel per 27 p 3 e-<lb/>rit angulus b a g æqualis angulo d a g, quoniam illi anguli ca-<lb/>dunt in æquales arcus, qui ſunt d g & b g.</s> <s xml:id="echoid-s30225" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s30226" xml:space="preserve"> lineæ b g <lb/>& d g ęquales ſub æ qualibus angulis, qui ſunt d a g & b a g, hinc & inde uidentur:</s> <s xml:id="echoid-s30227" xml:space="preserve"> palàm quoniam <lb/>illæ lineę ęquales uiſui apparent per 20 huius.</s> <s xml:id="echoid-s30228" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s30229" xml:space="preserve"> Idẽ quo q;</s> <s xml:id="echoid-s30230" xml:space="preserve"> contingeret, ſi cen <lb/>tro oculi in centro circuli manente ſixo, res uiſa ſuք circuli peripheriã moueatur:</s> <s xml:id="echoid-s30231" xml:space="preserve"> tunc enim uiſibili <lb/>tranſmutato res uiſa ſemper uidebitur ęqualis uiſui nó tranſinutato:</s> <s xml:id="echoid-s30232" xml:space="preserve"> quoniam ſub eodẽ ſemper an-<lb/>gulo uidebitur, ut poteſt patere ſecundum præmiſſum modum.</s> <s xml:id="echoid-s30233" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s30234" xml:space="preserve"/> </p> <div xml:id="echoid-div1221" type="float" level="0" n="0"> <figure xlink:label="fig-0470-01" xlink:href="fig-0470-01a"> <variables xml:id="echoid-variables502" xml:space="preserve">b s a d</variables> </figure> </div> </div> <div xml:id="echoid-div1223" type="section" level="0" n="0"> <head xml:id="echoid-head962" xml:space="preserve" style="it">114. Eſt locus, in quo oculo trãſmutato re uiſa non mota, ſem <lb/>per res uiſa æqualis apparet. Euclides 45 th. opt.</head> <figure> <variables xml:id="echoid-variables503" xml:space="preserve">z d e b g</variables> </figure> <p> <s xml:id="echoid-s30235" xml:space="preserve">Sitres uiſa b g:</s> <s xml:id="echoid-s30236" xml:space="preserve"> & ſit oculus in puncto z dato in aere, ut contin-<lb/>git:</s> <s xml:id="echoid-s30237" xml:space="preserve"> & ducantur à terminis rei uiſæ lineæ b z & g z:</s> <s xml:id="echoid-s30238" xml:space="preserve"> & cirumſcri-<lb/>batur trigono b z g, circulus per 5 p 4, ut in præmiſſa:</s> <s xml:id="echoid-s30239" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s30240" xml:space="preserve"> ille cir-<lb/>culus z d g b:</s> <s xml:id="echoid-s30241" xml:space="preserve"> & mutetur centrum oculi à puncto z in punctum d:</s> <s xml:id="echoid-s30242" xml:space="preserve"> <lb/>& ducantur lineę b d & g d:</s> <s xml:id="echoid-s30243" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s30244" xml:space="preserve"> per 27 p 3 angulus b z g ęqualis <lb/>angulo b d g:</s> <s xml:id="echoid-s30245" xml:space="preserve"> ergo per 20 huius in utroq;</s> <s xml:id="echoid-s30246" xml:space="preserve"> ſitu magnitudo b g ſem-<lb/>per uidebitur æqualis.</s> <s xml:id="echoid-s30247" xml:space="preserve"> Idem quoq;</s> <s xml:id="echoid-s30248" xml:space="preserve"> accidit uiſui per omnia pũcta <lb/>arcus b z g tranſmutato.</s> <s xml:id="echoid-s30249" xml:space="preserve"> Ethoc eſt propoſitum.</s> <s xml:id="echoid-s30250" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1224" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables504" xml:space="preserve">a e b g</variables> </figure> <head xml:id="echoid-head963" xml:space="preserve" style="it">115. Zuantitas erecta ſuper aliquam planam ſuperficiem, in qua ſit centrum uiſus, mota ſe-<lb/> cundum circuli peripheriam pro centro habentis centrum oculi, ſemper æqualis uidetur. Ideḿ accidit ſecundum lineam à centro circulirerectam centrooculi ſuper circuli ſuperficiem eleuato. Eucli des 41. 42 th. opt.</head> <p> <s xml:id="echoid-s30251" xml:space="preserve">Eſto a b aliqua magnitudo uiſa erecta ſuper quamcunq;</s> <s xml:id="echoid-s30252" xml:space="preserve"> ſuperficiẽ <lb/>planam datam:</s> <s xml:id="echoid-s30253" xml:space="preserve"> in qua ſit centrum uiſus, quod ſit g:</s> <s xml:id="echoid-s30254" xml:space="preserve"> & ducatur ab alte <lb/>ro terminorum rei uiſæ ad centrum uiſus linea g b:</s> <s xml:id="echoid-s30255" xml:space="preserve"> & ſecundũ quan-<lb/>titatem lineę g b, centro exiſtente in puncto g, deſcribatur circulus.</s> <s xml:id="echoid-s30256" xml:space="preserve"> <lb/>Dico, quòd ſi ſuper illius circuli peripheriam moueatur magnitudo <lb/>erecta, quę eſt a b, quòd ſemper uidebitur ęqualis oculo ipſo in pun-<lb/>cto g exiſtente.</s> <s xml:id="echoid-s30257" xml:space="preserve"> Quia enim linea a b eſt erecta ſuper ſuperficiem, palá <lb/>per definitionẽ, quia ſemper facit angulum a b g rectũ, & ſemper an-<lb/>gulũ ęqualem cũ linea g b, utcunque contingit, ducta linea a b:</s> <s xml:id="echoid-s30258" xml:space="preserve"> ſed & <lb/>linea g b ſemper eſtæqualis ſibijpſi, cum ſit diameter circuli, & li-<lb/>nea a b ſemper eſt æqualis ſib jpſi:</s> <s xml:id="echoid-s30259" xml:space="preserve"> ducatur itaque linea a g:</s> <s xml:id="echoid-s30260" xml:space="preserve"> pa-<lb/>lamq́ue quòd per totam circuli peripheriam angulus a g b eſt æ-<lb/>qualis ſibijpſi:</s> <s xml:id="echoid-s30261" xml:space="preserve"> ergo per 20 huius magnitudo a b ſemper uidebi-<lb/>tur æqualis:</s> <s xml:id="echoid-s30262" xml:space="preserve"> quod eſt primum propoſitorum.</s> <s xml:id="echoid-s30263" xml:space="preserve"> Ducatur itaque li-<lb/>nea g e à centro oculi erecta ſuper ſuperficiem circuli:</s> <s xml:id="echoid-s30264" xml:space="preserve"> erit ergo <lb/> <pb o="169" file="0471" n="471" rhead="LIBER QVARTVS."/> linea g e æquidiſtans line æ a b per 6 p 11, & centrum uiſus eleuetur ſuper ſuperficiem circuli ſecun-<lb/>dum aliquod punctum lineæ g e, quod ſit e, in quo figatur uiſus.</s> <s xml:id="echoid-s30265" xml:space="preserve"> Dico, quòd adhuc magnitudo a b <lb/>mota ſuper circuli peripheriam æquidiſtanter lineæ g e, ſemper uidebitur æ qualis.</s> <s xml:id="echoid-s30266" xml:space="preserve"> Productis enim <lb/>lineis a e & b e:</s> <s xml:id="echoid-s30267" xml:space="preserve"> patet per 4 p 1 quoniam angulus a e b ſemper eſt æqualis ſibijpſi.</s> <s xml:id="echoid-s30268" xml:space="preserve"> Cum enim an gu-<lb/>lus b g e ſit ſemper æqualis ſibiipſi, erit baſis b e ſibijpſi ſemper æqualis, & angulus e b g æqualis <lb/>ſibijpſi:</s> <s xml:id="echoid-s30269" xml:space="preserve"> ergo etiá angulus a e b eſt ſemper æqualis ſibijpſi:</s> <s xml:id="echoid-s30270" xml:space="preserve"> ergo & baſis a e, & angulus a e b erit ſem-<lb/>per æqualis ſibijpſi:</s> <s xml:id="echoid-s30271" xml:space="preserve"> ergo per 20 huius linea a b ſemper uidebitur æqualis ſibijpſi:</s> <s xml:id="echoid-s30272" xml:space="preserve"> patet ergo ſecun-<lb/>dum propoſitorum.</s> <s xml:id="echoid-s30273" xml:space="preserve"> Et hoc eſt totum, quod proponebatur.</s> <s xml:id="echoid-s30274" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1225" type="section" level="0" n="0"> <head xml:id="echoid-head964" xml:space="preserve" style="it">116. Zuantitas obliquè incidens ſuperficiei planæ, in qua eſt centrum uiſus, uniformiter <lb/>mota ſecundum circuli peripheriam, cuius centrum eſt centrum uiſus, ſemper æqualis uidebi-<lb/>tur: ipſa uerò exiſtente æquali ſemidiametro illius circuli, mota quo ſecundum ſui ſitus æqui-<lb/>diſtantiam per illius circuli peripheriam quando æqualis, quando minor, quando maior <lb/>uiſui apparebit. Euclides 43 th. opt.</head> <p> <s xml:id="echoid-s30275" xml:space="preserve">Sit circulus a d:</s> <s xml:id="echoid-s30276" xml:space="preserve"> cuius centrum ſit punctum e:</s> <s xml:id="echoid-s30277" xml:space="preserve"> & in eius peripheria ſumatur punctum d:</s> <s xml:id="echoid-s30278" xml:space="preserve"> ſit quoq;</s> <s xml:id="echoid-s30279" xml:space="preserve"> <lb/>linea d z obliquè incidens ſuperſiciei circuli:</s> <s xml:id="echoid-s30280" xml:space="preserve"> & ſit centrum oculi in puncto e centro circuli.</s> <s xml:id="echoid-s30281" xml:space="preserve"> Dico, <lb/>quò d ſi linea d z in circuli peripheria tranſponatur uniſormiter, ita ut cum ſemidiametris illius cir-<lb/>culi ſemper æqualem contineat angulum, quòd ipſa ſemper æqualis apparet:</s> <s xml:id="echoid-s30282" xml:space="preserve"> hoc autẽ poteſt euin-<lb/>ci per 4 p 1, ut in præcedente:</s> <s xml:id="echoid-s30283" xml:space="preserve"> eſt enim angulus d e z ſemper æqualis ſibijpſi:</s> <s xml:id="echoid-s30284" xml:space="preserve"> ergo & res ſemper ui-<lb/>d etur æqualis per 20 huius.</s> <s xml:id="echoid-s30285" xml:space="preserve"> Et hoc eſt propoſitum primum.</s> <s xml:id="echoid-s30286" xml:space="preserve"> Rurſum ſit centrum uiſus in puncto e <lb/>centro circuli a d, cuius ſuperſiciei obliquè incidat linea d z, quæ ſit æqualis ſemidiametro d e:</s> <s xml:id="echoid-s30287" xml:space="preserve"> mo-<lb/>ueaturq́;</s> <s xml:id="echoid-s30288" xml:space="preserve"> per circuli illius peripheriam ſecundũ ſui primi ſitus æquidiſtantiam, ſitq́;</s> <s xml:id="echoid-s30289" xml:space="preserve"> exempli cauſſa <lb/> <anchor type="figure" xlink:label="fig-0471-01a" xlink:href="fig-0471-01"/> angulus z d e acutus.</s> <s xml:id="echoid-s30290" xml:space="preserve"> Dico, quòd aliquando appare-<lb/>bitlinea mota, quæ d z, æqualis ſuæ propriæ quanti-<lb/>tati, utpote ſemidiamtro circuli, aliquãdo maior, ali-<lb/>quando minor.</s> <s xml:id="echoid-s30291" xml:space="preserve"> Ducatur enim à centro circuli e linea <lb/>e g æquidiſtans lineæ d z per 31 p 1, quæ fiat æqualis <lb/>eidem per 3 p 1:</s> <s xml:id="echoid-s30292" xml:space="preserve"> ducatur quoq;</s> <s xml:id="echoid-s30293" xml:space="preserve"> à puncto g perpendi-<lb/>cularis ſuper circuli ſuperficiem per 11 p 11, quæ ſit g i:</s> <s xml:id="echoid-s30294" xml:space="preserve"> <lb/>& ducatur à centro circuli linea e i:</s> <s xml:id="echoid-s30295" xml:space="preserve"> quæ producatur <lb/>ad peripheriam circuli in pũctum a:</s> <s xml:id="echoid-s30296" xml:space="preserve"> & à puncto a du-<lb/>catur linea æquidiſtans lineæ e g per 31 p 1, quę ſit a b:</s> <s xml:id="echoid-s30297" xml:space="preserve"> <lb/>quæ reſectur per 3 p 1 æqualis lineæ d z:</s> <s xml:id="echoid-s30298" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s30299" xml:space="preserve"> linea <lb/>a b æquidiſtans etiá lineæ d z per 30 p 1 uel per 9 p 11.</s> <s xml:id="echoid-s30300" xml:space="preserve"> <lb/>Et quoniam linea g e, ut patet exhypotheſi, eſt obli-<lb/>qua ſuper ſuperficiem circuli a d, & à pũcto g in aere <lb/>dato ad ſubſtratam planam ſuperficiem incidit linea <lb/>g i perpendiculariter, & linea g e obliquè:</s> <s xml:id="echoid-s30301" xml:space="preserve"> tunc patet <lb/>per 39 th.</s> <s xml:id="echoid-s30302" xml:space="preserve"> 1 huius, quòniam angulus g ea minimus eſt <lb/>omnium angulorũ ſub illa linea obliqua g e, & qua-<lb/>cunq;</s> <s xml:id="echoid-s30303" xml:space="preserve"> linea in ſubſtrata ſuperficie circuli a d protracta contento:</s> <s xml:id="echoid-s30304" xml:space="preserve"> & omnis angulus illi propin quior <lb/>eſt minor remotiore:</s> <s xml:id="echoid-s30305" xml:space="preserve"> & duo anguli ex utra q;</s> <s xml:id="echoid-s30306" xml:space="preserve"> parte illi æqualiter approximantes ſunt inter ſe æqua <lb/>les.</s> <s xml:id="echoid-s30307" xml:space="preserve"> Dico itaq;</s> <s xml:id="echoid-s30308" xml:space="preserve">, quoniam linea a b omnium linearum æqualium lineæ d z tranſpoſitarum ſecundum <lb/>peripheriam circuli minima apparebit.</s> <s xml:id="echoid-s30309" xml:space="preserve"> Ducantur enim lineæ g z, g b, e b, e z, e d.</s> <s xml:id="echoid-s30310" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s30311" xml:space="preserve"> linea g e <lb/>eſt æ quidiſtás lineæ a b & æqualis ut patet per 34 p 1, quoniã linea g b eſt æqualis lineæ e a & æqui-<lb/>diſtans eidem:</s> <s xml:id="echoid-s30312" xml:space="preserve"> ſunt ergo duæ ſuperficies parallelogrammæ, quæ e g b a & e d z g.</s> <s xml:id="echoid-s30313" xml:space="preserve"> Quia uerò angu-<lb/>lus g e a eſt acutus, ut pateter ex præmiſsis, propter obliquationem lineæ g e ſuper ſuperficiẽ circuli <lb/>a d:</s> <s xml:id="echoid-s30314" xml:space="preserve"> erit ergo angulus g e d obtuſus per 13 p 1:</s> <s xml:id="echoid-s30315" xml:space="preserve"> quoniam enim, ut patet per 39 th.</s> <s xml:id="echoid-s30316" xml:space="preserve"> 1 huius, angulus g e a <lb/>eſt minimus omnium angulorum contentorum ſub quacunq;</s> <s xml:id="echoid-s30317" xml:space="preserve"> linea in ſuperficie circuli ducta ad <lb/>punctum e, & ſub linea g e:</s> <s xml:id="echoid-s30318" xml:space="preserve"> eſt ergo angulus g e a minor quàm angulus g e d:</s> <s xml:id="echoid-s30319" xml:space="preserve"> ſed cũ linea e z ſit dia-<lb/>gonius parallelogrammi e d z g:</s> <s xml:id="echoid-s30320" xml:space="preserve"> palàm quòd angulus d e z eſt medietas g e d anguli ք 4 p 1:</s> <s xml:id="echoid-s30321" xml:space="preserve"> & ſimi-<lb/>liter angulus b e a eſt medietas anguli g e a:</s> <s xml:id="echoid-s30322" xml:space="preserve"> angulus itaq;</s> <s xml:id="echoid-s30323" xml:space="preserve"> d e z eſt maior angulo b e a.</s> <s xml:id="echoid-s30324" xml:space="preserve"> Ergo per 20 <lb/>huius quantitás lineæ b a ininor uidebitur quàm quantitas lineæ z d.</s> <s xml:id="echoid-s30325" xml:space="preserve"> Et per præmiſſa cum angulus <lb/>g e a ſit minimus omnium angulorum, qui cõtinentur ſub linea g e & aliqua linea in ſuperficie cir-<lb/>culi a d producta:</s> <s xml:id="echoid-s30326" xml:space="preserve"> palàm quia medietas anguli g e a erit minor medietate cuiuslibet aliorum angu-<lb/>lorum.</s> <s xml:id="echoid-s30327" xml:space="preserve"> Quantitas ergo lineæ a b uidebitur omnium aliarum ſibi æqualium quantitate minima.</s> <s xml:id="echoid-s30328" xml:space="preserve"> Et <lb/>quoniam angulus z e d eſt maximus omnium illorum aliorum angulorum:</s> <s xml:id="echoid-s30329" xml:space="preserve"> uidebitur ergo quanti-<lb/>tas z d maxima:</s> <s xml:id="echoid-s30330" xml:space="preserve"> mediæ uerò modo medio uidebuntur, & quantitates in circuli peripheria æ quali-<lb/>ter diſtantes ab utraq;</s> <s xml:id="echoid-s30331" xml:space="preserve"> quantitatũ, quæ a b & d z, ad inuicé uidebũtur ęquales.</s> <s xml:id="echoid-s30332" xml:space="preserve"> Et hoc eſt propoſitũ.</s> <s xml:id="echoid-s30333" xml:space="preserve"/> </p> <div xml:id="echoid-div1225" type="float" level="0" n="0"> <figure xlink:label="fig-0471-01" xlink:href="fig-0471-01a"> <variables xml:id="echoid-variables505" xml:space="preserve">z b g z a i e d</variables> </figure> </div> </div> <div xml:id="echoid-div1227" type="section" level="0" n="0"> <head xml:id="echoid-head965" xml:space="preserve" style="it">117. Re uiſa ſuper ſuperſiciem planam erecta ſixta manente, & centro oculi ſecundum circuli <lb/>peripheriam moto circa punctum, in quo resuiſa ſuperſiciei coniungitur: res ſemper æqualis ui-<lb/>ſui apparebit: quod non accidit centro uiſus moto ſuper peripheriam oxygonie ſectionis.</head> <pb o="170" file="0472" n="472" rhead="VITELLONIS OPTICAE"/> <p> <s xml:id="echoid-s30334" xml:space="preserve">Sit a b magnitudo erecta ſuper ſuperficiem planam, tangens ipſam in puncto b:</s> <s xml:id="echoid-s30335" xml:space="preserve">ſit q́, centrũ ocu <lb/> <anchor type="figure" xlink:label="fig-0472-01a" xlink:href="fig-0472-01"/> li in puncto g in eadem ſuperficie:</s> <s xml:id="echoid-s30336" xml:space="preserve"> & centro quidem exiſtente pun-<lb/>cto b, ſecundum ſpatium b g lineæ deſcirbatur circulus, qui ſit g d.</s> <s xml:id="echoid-s30337" xml:space="preserve"> <lb/>Dico, quòd ſi tranſponatur centrum oculi à puncto g, ſuper totam <lb/>circuli g d peripheriá:</s> <s xml:id="echoid-s30338" xml:space="preserve"> apparebit uiſui linea a b ſemper ęqualis.</s> <s xml:id="echoid-s30339" xml:space="preserve"> Quo-<lb/>niam enim angulus a b g eſt ſemper rectus per definitionem lineæ <lb/>ſuper ſuperficiẽ erectæ:</s> <s xml:id="echoid-s30340" xml:space="preserve"> palàm quia omnes anguli a g b per 4 p 1 ſunt <lb/>ubiq;</s> <s xml:id="echoid-s30341" xml:space="preserve"> æquales:</s> <s xml:id="echoid-s30342" xml:space="preserve"> ergo per 20 huius res uiſa, quæ a b, ſemper uidebitur <lb/>æqualis.</s> <s xml:id="echoid-s30343" xml:space="preserve"> Ethoc eſt propoſitum primum.</s> <s xml:id="echoid-s30344" xml:space="preserve"> Non accidit autem hoc cen <lb/>tro uiſus moto ſuper peripheriam oxygoniæ ſectionis:</s> <s xml:id="echoid-s30345" xml:space="preserve"> quoniá tunc <lb/>quantitas rei apparet inæ qualis, quę ſuper ipſius ſectionis punctum <lb/>medium eſt erecta:</s> <s xml:id="echoid-s30346" xml:space="preserve"> quoniam ſectio oxygonia habet ſemidiametros <lb/>in æquales, & omnes lineæ à centro uſq;</s> <s xml:id="echoid-s30347" xml:space="preserve"> ad circumſerentiam ductæ <lb/>ſunt inæquales:</s> <s xml:id="echoid-s30348" xml:space="preserve"> appropinquantes enim ſemidiametro maiori ſunt <lb/>maiores, & approximátes ſemidiametro minori ſunt minores:</s> <s xml:id="echoid-s30349" xml:space="preserve"> con-<lb/>trarium ergo neceſſariò accidit ei, quod oculo moto ſecundum cir-<lb/>culi peripheriam accidebat:</s> <s xml:id="echoid-s30350" xml:space="preserve"> quod patet per 7 & per 20 huius.</s> <s xml:id="echoid-s30351" xml:space="preserve"> Patet <lb/>ergo totum, quod proponebatur.</s> <s xml:id="echoid-s30352" xml:space="preserve"/> </p> <div xml:id="echoid-div1227" type="float" level="0" n="0"> <figure xlink:label="fig-0472-01" xlink:href="fig-0472-01a"> <variables xml:id="echoid-variables506" xml:space="preserve">a g b d</variables> </figure> </div> </div> <div xml:id="echoid-div1229" type="section" level="0" n="0"> <head xml:id="echoid-head966" xml:space="preserve" style="it">118. Re uiſa ſix a manente, oculo uerò moto ſecundum lineam <lb/>rectam obliquè incidentẽ quantitatirei uiſæ: illa quãtitas quan-<lb/>do æqualis, quando inæqualis uiſui apparet. Euclides 46 th. opt.</head> <p> <s xml:id="echoid-s30353" xml:space="preserve">Sitres uiſa, quæ a b:</s> <s xml:id="echoid-s30354" xml:space="preserve"> & ſit centrum uiſus punctum e:</s> <s xml:id="echoid-s30355" xml:space="preserve"> incidatq́;</s> <s xml:id="echoid-s30356" xml:space="preserve"> linea e g obliquè lineæ a b:</s> <s xml:id="echoid-s30357" xml:space="preserve"> produ-<lb/>catur enim linea a b in punctum g, donec concurrat cum linea e g:</s> <s xml:id="echoid-s30358" xml:space="preserve"> & item producatur linea e g in <lb/>continuum & directum ultra punctum e ad punctũ d:</s> <s xml:id="echoid-s30359" xml:space="preserve"> ſitilla linea indeſinita d e g.</s> <s xml:id="echoid-s30360" xml:space="preserve"> Dico, quòd ocu-<lb/>lo tranſmutato ſecũdum lineam d g, quandoq;</s> <s xml:id="echoid-s30361" xml:space="preserve"> linea a b uidetur minor:</s> <s xml:id="echoid-s30362" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s30363" xml:space="preserve"> maior:</s> <s xml:id="echoid-s30364" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s30365" xml:space="preserve"> <lb/>æ qualis.</s> <s xml:id="echoid-s30366" xml:space="preserve"> Sumatur enim per 13 p 6 inter duas lineas b g & a g linea medio loco propotionalis, quæ <lb/>ſit, exempli cauſſa, linea e g:</s> <s xml:id="echoid-s30367" xml:space="preserve"> hoc autem eſt poſibile per reſectionem lineæ d g per 3 p 1:</s> <s xml:id="echoid-s30368" xml:space="preserve"> ponaturq́;</s> <s xml:id="echoid-s30369" xml:space="preserve"> <lb/>cẽtrum oculi in puncto eiproudcaturq́;</s> <s xml:id="echoid-s30370" xml:space="preserve"> linea e b.</s> <s xml:id="echoid-s30371" xml:space="preserve"> & producatur in ſuperficie trigoni b e g à puncto <lb/>b linea perpendicularis ſuper lineam b a, quæ ſit b d:</s> <s xml:id="echoid-s30372" xml:space="preserve"> quæ per 14 th.</s> <s xml:id="echoid-s30373" xml:space="preserve"> 1 huius concurret cum linea e g:</s> <s xml:id="echoid-s30374" xml:space="preserve"> <lb/>ideo quòd angulus e g b eſt acutus, & angulus g b d rectus:</s> <s xml:id="echoid-s30375" xml:space="preserve"> cõcurrantitaq;</s> <s xml:id="echoid-s30376" xml:space="preserve"> in puncto d.</s> <s xml:id="echoid-s30377" xml:space="preserve"> Dico, quod <lb/>moto uiſu pertotam lineam e d, ſemper uiſum b a inæ quale apparet.</s> <s xml:id="echoid-s30378" xml:space="preserve"> Ducantur enim lineæ a e, a d:</s> <s xml:id="echoid-s30379" xml:space="preserve"> <lb/>& deſcribatur per 5 p 4 circa a e b trigonum porcio circuli, quæ ſimiliter ſit a e b.</s> <s xml:id="echoid-s30380" xml:space="preserve"> Et quoniam illud, <lb/> <anchor type="figure" xlink:label="fig-0472-02a" xlink:href="fig-0472-02"/> quod ſit ex ductu lineæ b g in <lb/>lineam a g, ut patet per 17 p 6 & <lb/>ex præmiſsis, eſt æ quale qua-<lb/>drato lineæ e g:</s> <s xml:id="echoid-s30381" xml:space="preserve"> pater per 37 p 3 <lb/>quoniam linea g è eſt contin-<lb/>gens circulum b e a in puncto <lb/>e.</s> <s xml:id="echoid-s30382" xml:space="preserve"> Et à termino quo q;</s> <s xml:id="echoid-s30383" xml:space="preserve"> a lineæ g <lb/>a ducaturlinea a z per 23 p 1 ita, <lb/>ut fiat angulus g a z ęqualis an-<lb/>gulo g d b:</s> <s xml:id="echoid-s30384" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s30385" xml:space="preserve"> punctum z <lb/>inlineam d g inter puncta e & <lb/>g per 29th.</s> <s xml:id="echoid-s30386" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s30387" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s30388" xml:space="preserve"> b a z d <lb/>quadrilaterum inſcriptibile cir <lb/>culo per 22 p 3:</s> <s xml:id="echoid-s30389" xml:space="preserve"> quilibet enim <lb/>eius duo anguli ex aduerſo collocati ualét duos rectos:</s> <s xml:id="echoid-s30390" xml:space="preserve"> angulus enim d z a per 32 p 1 ualet angulum <lb/>z g a, & angulum z a g:</s> <s xml:id="echoid-s30391" xml:space="preserve"> ſed angulus z a g, ut patet ex præmiſsis, eſt æ qualis an gulo g d b:</s> <s xml:id="echoid-s30392" xml:space="preserve"> ſed angulus <lb/>d b g rectus cum angulis b d g & d g b ualet duos rectos per 32 p 1:</s> <s xml:id="echoid-s30393" xml:space="preserve"> angulus itaq;</s> <s xml:id="echoid-s30394" xml:space="preserve"> d z a cum angulo d <lb/>b g ualet duos rectos:</s> <s xml:id="echoid-s30395" xml:space="preserve"> ſed omnes anguli quadranguli cuiuſcunq;</s> <s xml:id="echoid-s30396" xml:space="preserve"> ualẽt quatuor rectos:</s> <s xml:id="echoid-s30397" xml:space="preserve"> quia quod-<lb/>libetillorum eſt diuiſibile in duos triangulos, quorum cuiuslibet anguli ualent duos rectos:</s> <s xml:id="echoid-s30398" xml:space="preserve"> ergo <lb/>anguli z d b & z a b ualẽt duos rectos:</s> <s xml:id="echoid-s30399" xml:space="preserve"> eſt ergo quadrilaterum z d b a circulo inſcriptibile.</s> <s xml:id="echoid-s30400" xml:space="preserve"> Circum-<lb/>ſcribatur ergo ei circulus per 22 p 3 & per 9 p 4:</s> <s xml:id="echoid-s30401" xml:space="preserve"> & ſit circumſcripta portio circuli, quæ ſit b d z a:</s> <s xml:id="echoid-s30402" xml:space="preserve"> du-<lb/>caturq́;</s> <s xml:id="echoid-s30403" xml:space="preserve"> linea b z ſecans arcum e a in punctot:</s> <s xml:id="echoid-s30404" xml:space="preserve"> ſecabit enim ipſum ideo, quia, ut patet ex præmiſsis, <lb/>punctum z cadit inter puncta e & g:</s> <s xml:id="echoid-s30405" xml:space="preserve"> & ducatur linea t a:</s> <s xml:id="echoid-s30406" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s30407" xml:space="preserve"> per 16 p 1 angulus a t b extrinſecus <lb/>maior angulo a z b intrinſeco:</s> <s xml:id="echoid-s30408" xml:space="preserve"> ſed angulus a t b eſt æ qualis angulo a e b per 27 p 3, quoniam cadunt <lb/>in eundem arcum, qui eſt b a, portionis circuli minoris, qui b e a:</s> <s xml:id="echoid-s30409" xml:space="preserve"> angulus itaq;</s> <s xml:id="echoid-s30410" xml:space="preserve"> a e b maior eſt angu-<lb/>lo a z b:</s> <s xml:id="echoid-s30411" xml:space="preserve"> angulus uerò a z b æqualis eſt angulo a d b per eandem 27 p 3:</s> <s xml:id="echoid-s30412" xml:space="preserve"> quoniam ambo illi anguli ca-<lb/>dunt in eundem arcum, qui eſt a b circuli maioris, qui eſt b d z a:</s> <s xml:id="echoid-s30413" xml:space="preserve"> angulus itaq;</s> <s xml:id="echoid-s30414" xml:space="preserve"> a e b maior eſt angu-<lb/>lo a d b.</s> <s xml:id="echoid-s30415" xml:space="preserve"> Centro uerò uiſus exiſtente in puncto d, uidetur linea a b ſub angulo a d b:</s> <s xml:id="echoid-s30416" xml:space="preserve"> ipſo autem exi-<lb/>ſtente in puncto e uidetur ſub angulo a e b:</s> <s xml:id="echoid-s30417" xml:space="preserve"> maior itaq;</s> <s xml:id="echoid-s30418" xml:space="preserve"> uidebitur in puncto e, quàm in puncto d per <lb/>20 huius.</s> <s xml:id="echoid-s30419" xml:space="preserve"> Mutato ergo oculo ſecundum puncta lineæ e d, ſemper in æ qualis uidebitur magnitudo <lb/>b a:</s> <s xml:id="echoid-s30420" xml:space="preserve"> quoniam ſemper minor ſe ipſa:</s> <s xml:id="echoid-s30421" xml:space="preserve"> & quantò plus accedit ad punctum d, tantò uidebitur minor:</s> <s xml:id="echoid-s30422" xml:space="preserve"> & <lb/> <pb o="171" file="0473" n="473" rhead="LIBER QVARTVS."/> quantò plus appropinquat puncto e, tantò apparet maior:</s> <s xml:id="echoid-s30423" xml:space="preserve"> eodemq́;</s> <s xml:id="echoid-s30424" xml:space="preserve"> modo uiſu mutato ſuper pun-<lb/>cta lineæ e g, in æqualis uidebitur linea a b, & minor quàm ſuper pũctum e:</s> <s xml:id="echoid-s30425" xml:space="preserve"> quoniam linearum du-<lb/>ctarum per punctum aliquod lineæ e g à terminis lineæ a b, ſemper angulus erit minor angulo b e <lb/>a:</s> <s xml:id="echoid-s30426" xml:space="preserve"> quoniam angulus à lineis ad circumferentiam arcus e a ductis per 21 p 1 maior erit illo conſtituto <lb/>ſuper aliquod punctorum lineæ e g, per lineam trans idem punctum arcus ab altero terminorum <lb/>lineæ a b productam, & per lineam à reliquo eius termino copulatam.</s> <s xml:id="echoid-s30427" xml:space="preserve"> Quilibet autem angulorum <lb/>cõſtitutorum ſuper aliquod punctorũ arcus e a, per lineas à terminis lineæ a b productas eſt æqua-<lb/>lis angulo b e a per 27 p 3:</s> <s xml:id="echoid-s30428" xml:space="preserve"> ergo per 20 huius linea a b maior uidebitur centro uiſus exiſtente in pun-<lb/>cto e quàm ipſo exiſtente in aliquo punctorum lineæ e g.</s> <s xml:id="echoid-s30429" xml:space="preserve"> Semper quoq;</s> <s xml:id="echoid-s30430" xml:space="preserve"> minor apparebit ſecundũ <lb/>quod plus appropinquat puncto g:</s> <s xml:id="echoid-s30431" xml:space="preserve"> ita quòd centro uiſus exiſtente in puncto g, non uidebitur niſi <lb/>unicus eius punctus, qui eſt a, ut patet per 4 huius.</s> <s xml:id="echoid-s30432" xml:space="preserve"> Maior autem ſemper apparebit ſecũdum quod <lb/>appropinquat ad punctum e:</s> <s xml:id="echoid-s30433" xml:space="preserve"> ad punctum uerò z apparebit ſicut ad punctum d, æqualis ſibi:</s> <s xml:id="echoid-s30434" xml:space="preserve"> ideo <lb/>quòd anguli b d a & b z a per 27 p 3, ut ſuprà patuit, ſunt æquales.</s> <s xml:id="echoid-s30435" xml:space="preserve"> Et quoniam, utiam oſtendimus, <lb/>uiſu exiſtente in puncto g, non uidebitur linea a b, imò tota linea g b, niſi punctus:</s> <s xml:id="echoid-s30436" xml:space="preserve"> palàm quòd in-<lb/>ter puncta g & z modica fit additio.</s> <s xml:id="echoid-s30437" xml:space="preserve"> Semper ergo uidebitur linea a b inæ qualis:</s> <s xml:id="echoid-s30438" xml:space="preserve"> in ęquidiſtantia ue-<lb/>rò à punctis d & z, uidebitur etiam æqualitas propter æqualitatem angulorum proueniẽtium hinc <lb/>& inde.</s> <s xml:id="echoid-s30439" xml:space="preserve"> Quòd ſi linea e g non ex parte puncti a, ſed ex parte puncti b cõcurrat cum linea a b, eadem <lb/>eſt demóſtratio.</s> <s xml:id="echoid-s30440" xml:space="preserve"> Sit enim, ut fiat <lb/> <anchor type="figure" xlink:label="fig-0473-01a" xlink:href="fig-0473-01"/> cócurſus, ſicut prius, in puncto <lb/>g:</s> <s xml:id="echoid-s30441" xml:space="preserve"> & ſit linea g e medio loco pro <lb/>portionalis inter lineas a g & g <lb/>b:</s> <s xml:id="echoid-s30442" xml:space="preserve"> & copulatis lineis e a & e b, <lb/>trigono a e b circũſcribatur por-<lb/>tio circuli, quæ fit, ut prius, b e a:</s> <s xml:id="echoid-s30443" xml:space="preserve"> <lb/>& ducantur lineæ d b & d a:</s> <s xml:id="echoid-s30444" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s30445" xml:space="preserve"> <lb/>cẽtrum oculi ſuper punctum d:</s> <s xml:id="echoid-s30446" xml:space="preserve"> <lb/>& ad punctum, in quo linea a d <lb/>interſecat circumferentiam cir-<lb/>culi b e a, qui ſit z, ducatur linea <lb/>b z.</s> <s xml:id="echoid-s30447" xml:space="preserve"> Et quia aǹgulus b z a eſt maior angulo b d a per 16 p 1, & angulus b e a æ qualis eſt angulo b z a <lb/>per 27 p 3, quoniam cadunt in eundem arcum a b:</s> <s xml:id="echoid-s30448" xml:space="preserve"> palàm quia angulus b e a maior eſt angulo b d a.</s> <s xml:id="echoid-s30449" xml:space="preserve"> <lb/>Viſus itaq;</s> <s xml:id="echoid-s30450" xml:space="preserve"> centro exiſtẽte ſuper punctum e maior apparebit linea b a per 20 huius quàm ipſo exi-<lb/>ſtente in puncto d:</s> <s xml:id="echoid-s30451" xml:space="preserve"> in punctis uerò d & z apparebit linea a b æ qualis:</s> <s xml:id="echoid-s30452" xml:space="preserve"> & omnia alia accidũt, ut prius <lb/>declaratum eſt.</s> <s xml:id="echoid-s30453" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s30454" xml:space="preserve"/> </p> <div xml:id="echoid-div1229" type="float" level="0" n="0"> <figure xlink:label="fig-0472-02" xlink:href="fig-0472-02a"> <variables xml:id="echoid-variables507" xml:space="preserve">d e t z b a g</variables> </figure> <figure xlink:label="fig-0473-01" xlink:href="fig-0473-01a"> <variables xml:id="echoid-variables508" xml:space="preserve">d e z a b g</variables> </figure> </div> </div> <div xml:id="echoid-div1231" type="section" level="0" n="0"> <head xml:id="echoid-head967" xml:space="preserve" style="it">119. Re uiſa fixa manente, uiſu autem moto ſecundum lineam æquidiſtantem rei uiſæ: eius <lb/>quantit as quando æqualis, quando inæqualis uidetur. Euclides 47 th. opt.</head> <p> <s xml:id="echoid-s30455" xml:space="preserve">Eſto uiſa magnitudo, quæ fixa & immota permanens ſit a b:</s> <s xml:id="echoid-s30456" xml:space="preserve"> diuidaturq́;</s> <s xml:id="echoid-s30457" xml:space="preserve"> per æqualia in puncto e:</s> <s xml:id="echoid-s30458" xml:space="preserve"> <lb/>& erigatur ſuper ipſam perpendiculariter linea e z per 11 p 1:</s> <s xml:id="echoid-s30459" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s30460" xml:space="preserve"> centrum oculi in puncto z:</s> <s xml:id="echoid-s30461" xml:space="preserve"> ducan-<lb/>turq́;</s> <s xml:id="echoid-s30462" xml:space="preserve"> lineæ z a & z b, ita ut compleatur trigonũ a z b:</s> <s xml:id="echoid-s30463" xml:space="preserve"> & deſcribatur circa a z b trigonum portio cir-<lb/>culi a z b per 5 p 4:</s> <s xml:id="echoid-s30464" xml:space="preserve"> ducaturq́, linea z d parallela lineæ b a per 31 p 1:</s> <s xml:id="echoid-s30465" xml:space="preserve"> moueaturq́;</s> <s xml:id="echoid-s30466" xml:space="preserve"> centrũ oculi in pun-<lb/>ctum d:</s> <s xml:id="echoid-s30467" xml:space="preserve"> & ducantur lineæ d a & d b;</s> <s xml:id="echoid-s30468" xml:space="preserve"> & ad punctum, in quo linea d b ſecat circulum, quod ſit l, duca-<lb/>tur linea a l.</s> <s xml:id="echoid-s30469" xml:space="preserve"> Palàm ergo per 16 p 1 quoniam angulus a l b maior eſt angulo a d b:</s> <s xml:id="echoid-s30470" xml:space="preserve"> ſed per 27 p 3 angu-<lb/>lus a z b eſt ęqualis a l b:</s> <s xml:id="echoid-s30471" xml:space="preserve"> eſt ergo angulus a z b maior angulo a d b:</s> <s xml:id="echoid-s30472" xml:space="preserve"> maior ergo uidebitur magnitudo <lb/>a b, centro oculi exiſtẽte in puncto z quàm in puncto d, ut patet per 20 huius.</s> <s xml:id="echoid-s30473" xml:space="preserve"> Et ſi linea z g ſit ęqua-<lb/>lis lineæ z d, æqualis uidebitur linea a b in punctis d & g:</s> <s xml:id="echoid-s30474" xml:space="preserve"> hoc enim cõcluditur per 34 & 4 p 1, ductis <lb/>lineis g b & g a:</s> <s xml:id="echoid-s30475" xml:space="preserve"> angulus enim b g a æqualis eſt angulo b d a:</s> <s xml:id="echoid-s30476" xml:space="preserve"> & ſimiliter patethoc in alijs punctis <lb/>æqualiter diſtantibus à punctis d & g:</s> <s xml:id="echoid-s30477" xml:space="preserve"> ergo per 20 huius in talibus punctis uidebitur linea b a ſem-<lb/> <anchor type="figure" xlink:label="fig-0473-02a" xlink:href="fig-0473-02"/> per ſibijpſi æ qualis.</s> <s xml:id="echoid-s30478" xml:space="preserve"> Si uerò li-<lb/>nea z h ſit minor quàm linea z <lb/>d:</s> <s xml:id="echoid-s30479" xml:space="preserve"> tunc ducátur lineæ b h & a h:</s> <s xml:id="echoid-s30480" xml:space="preserve"> <lb/>& ꝓducatur linea a b ultra pũ-<lb/>ctum b ad punctum q.</s> <s xml:id="echoid-s30481" xml:space="preserve"> Quoniá <lb/>itaq;</s> <s xml:id="echoid-s30482" xml:space="preserve"> angulus z e b eſt rectus, pa <lb/>tet per 32 p 1 quoniam angulus <lb/>z b e eſt acutus:</s> <s xml:id="echoid-s30483" xml:space="preserve"> erit ergo per 13 <lb/>p 1 angulus q b z obtuſus:</s> <s xml:id="echoid-s30484" xml:space="preserve"> ergo <lb/>per 29 p 1 angulus h z b eſt ob-<lb/>tuſus:</s> <s xml:id="echoid-s30485" xml:space="preserve"> ergo per 16 p 1 angulus g <lb/>h b eſt obtuſus:</s> <s xml:id="echoid-s30486" xml:space="preserve"> linea ergo b g <lb/>eſt maior quàm linea b h per 19 <lb/>p 1.</s> <s xml:id="echoid-s30487" xml:space="preserve"> Quia uerò ք 4 p 1 & ex hy-<lb/>potheſi patet quòd angul{us} z b <lb/>a eſt æ qualis angulo z a b:</s> <s xml:id="echoid-s30488" xml:space="preserve"> angulus ergo b a h eſt maior angulo h b a:</s> <s xml:id="echoid-s30489" xml:space="preserve"> ergo per 19 p 1 linea b h eſt ma-<lb/>ior quàm linea a h:</s> <s xml:id="echoid-s30490" xml:space="preserve"> ergo & linea b g eſt maior quàm linea a h.</s> <s xml:id="echoid-s30491" xml:space="preserve"> Et quoniam lineæ b g & a h ſe interſe-<lb/> <pb o="172" file="0474" n="474" rhead="VITELLONIS OPTICAE"/> cant, ſit punctus ſectionis p.</s> <s xml:id="echoid-s30492" xml:space="preserve"> Et quoniam per 37 p 1 trigonum b g a eſt, ablato <lb/>ab ambobus communi trigono b p a:</s> <s xml:id="echoid-s30493" xml:space="preserve"> remanebit trigonum b h p æ quale trigono a p g:</s> <s xml:id="echoid-s30494" xml:space="preserve"> ſed per 15 p 1 <lb/>angulus a p g eſt æ qualis angulo b p h:</s> <s xml:id="echoid-s30495" xml:space="preserve"> ergo per 15 p 6 & 16 p 5 erit proportio line æ a p ad lineam b <lb/>p, ſicut lineæ h p ad lineam g p:</s> <s xml:id="echoid-s30496" xml:space="preserve"> ergo per 15 p 5 erit proportio totius lineæ a h ad totam lineam b g, <lb/>ſicut lineæ a p ad lineam b p:</s> <s xml:id="echoid-s30497" xml:space="preserve"> ſed linea a h eſt minor quàm lineam b g, <lb/>a p eſt minor quàm linea b p:</s> <s xml:id="echoid-s30498" xml:space="preserve"> linea ergo b p eſt maior quàm linea a p.</s> <s xml:id="echoid-s30499" xml:space="preserve"> Quæ eſt ergo proportio lineæ <lb/>b p ad lineam a p, eadem ſit lineæ a p ad lineam p o per 3th.</s> <s xml:id="echoid-s30500" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s30501" xml:space="preserve"> erit ergo ex præmiſsis linea p o <lb/>minor quàm linea p b:</s> <s xml:id="echoid-s30502" xml:space="preserve"> abſcindatur ergo linea p o à linea p b per 3 p 1, & ducatur linea h o.</s> <s xml:id="echoid-s30503" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s30504" xml:space="preserve"> <lb/>per 11 p 5 & ex præmiſsis eſt proportio lineæ a p ad lineam p o, ſicut lineæ h p ad lineam p g, & an-<lb/>gulus h p o eſt æqualis angulo a p g per 15 p 1, palàm per 6 p 6 quoniam trigona h p o & g p a ſunt ad <lb/>inuicem æ quiangula:</s> <s xml:id="echoid-s30505" xml:space="preserve"> eſt ergo angulus o h p æ qualis angulo a g p.</s> <s xml:id="echoid-s30506" xml:space="preserve"> Et quoniam linea h o diuidit ba-<lb/>ſim b p trigoni b h p, pater per 29 th.</s> <s xml:id="echoid-s30507" xml:space="preserve"> 1 huius quoniam ipſa linea h o diuidit etiam angulum b h p:</s> <s xml:id="echoid-s30508" xml:space="preserve"> eſt <lb/>ergo angulus b h a maior angulo o h p:</s> <s xml:id="echoid-s30509" xml:space="preserve"> ergo & eius æ quali, ſcilicet angulo b g a.</s> <s xml:id="echoid-s30510" xml:space="preserve"> Quantitas ergo li-<lb/>neæ b a per 20 huius maior uidebitur, centro uiſus exiſtente in puncto h quàm in puncto g:</s> <s xml:id="echoid-s30511" xml:space="preserve"> minor <lb/>autem quàm in puncto z.</s> <s xml:id="echoid-s30512" xml:space="preserve"> Sit enim punctus, in quo linea b h ſecat circulum b z a, punctus x:</s> <s xml:id="echoid-s30513" xml:space="preserve"> & duca-<lb/>tur linea a x:</s> <s xml:id="echoid-s30514" xml:space="preserve"> patet quoq;</s> <s xml:id="echoid-s30515" xml:space="preserve"> per 16 p 1 & per 27 p 3 quoniá angulus b z a eſt maior angulo b h a.</s> <s xml:id="echoid-s30516" xml:space="preserve"> Et quo-<lb/>niam quibuſcunq;</s> <s xml:id="echoid-s30517" xml:space="preserve"> punctis lineæ d z uellineæ z g datis, ſiue linea d z ſit maior quàm linea z g, ſiue <lb/>minor:</s> <s xml:id="echoid-s30518" xml:space="preserve"> ſemper eodem modo poteſt demõſtrari:</s> <s xml:id="echoid-s30519" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s30520" xml:space="preserve"> Angulus enim b z a fit ma-<lb/>ximus omniũ illorũ angulorũ:</s> <s xml:id="echoid-s30521" xml:space="preserve"> & ei propinquiores fiunt remotioribus maiores:</s> <s xml:id="echoid-s30522" xml:space="preserve"> & æ qualiter ab illo <lb/>diſtãtes fiũt æ quales:</s> <s xml:id="echoid-s30523" xml:space="preserve"> & ſecundũ illorũ angulorũ quãtitates per 20 huius mutatur quátitas rei uiſæ.</s> <s xml:id="echoid-s30524" xml:space="preserve"/> </p> <div xml:id="echoid-div1231" type="float" level="0" n="0"> <figure xlink:label="fig-0473-02" xlink:href="fig-0473-02a"> <variables xml:id="echoid-variables509" xml:space="preserve">d z y g x p l o q b e a</variables> </figure> </div> </div> <div xml:id="echoid-div1233" type="section" level="0" n="0"> <head xml:id="echoid-head968" xml:space="preserve" style="it">120. Sunt loca, in quibus oculo trãſpoſito, æquales magnitudines communiter loca quædam <lb/>directè occupantes, quando æquales, quando inæquales apparent.</head> <p> <s xml:id="echoid-s30525" xml:space="preserve">Cõmuniter dicuntur magnitudines occupare loca ſua, quãdo una applicatur alteri taliter, quòd <lb/>nihil cadit medium inter ipſas, neq;</s> <s xml:id="echoid-s30526" xml:space="preserve"> ſecundum rectam lineam æ qualiter utriq;</s> <s xml:id="echoid-s30527" xml:space="preserve"> magnitudinũ con-<lb/>iunctam, neq;</s> <s xml:id="echoid-s30528" xml:space="preserve"> ſecundum lineam alteri illarum magnitudinum angulariter incidẽtem.</s> <s xml:id="echoid-s30529" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s30530" xml:space="preserve"> cen-<lb/>trum oculi in puncto d:</s> <s xml:id="echoid-s30531" xml:space="preserve"> & ſint uiſæ magnitudines æ quales, quæ a b & b g, communiter occupantes <lb/>locum b:</s> <s xml:id="echoid-s30532" xml:space="preserve"> & à puncto b ſuper ambas illas magnitudines ducatur linea perpendicularis, quæ ſit b z:</s> <s xml:id="echoid-s30533" xml:space="preserve"> <lb/>ſitq́;</s> <s xml:id="echoid-s30534" xml:space="preserve"> oculus diſpoſitus in tali ſitu, ut linea z b protracta ultra punctum b, concurrat cum puncto, in <lb/>quo eſt centrum uiſus.</s> <s xml:id="echoid-s30535" xml:space="preserve"> Et quoniam in quocunq;</s> <s xml:id="echoid-s30536" xml:space="preserve"> puncto lineæ d z poſito centro uiſus, erunt ſemper <lb/> <anchor type="figure" xlink:label="fig-0474-01a" xlink:href="fig-0474-01"/> per 4 p 1 anguli b d g & b d a in centro uiſus æ quales:</s> <s xml:id="echoid-s30537" xml:space="preserve"> <lb/>manifeſtũ ergo per 20 huius quoniã ſecundũ quem-<lb/>cunq;</s> <s xml:id="echoid-s30538" xml:space="preserve"> punctum lineæ d z poſito centro uiſus d, ſem-<lb/>per magnitudines b g & a b æ quales apparebũt.</s> <s xml:id="echoid-s30539" xml:space="preserve"> Trãſ <lb/>ponatur autem oculus:</s> <s xml:id="echoid-s30540" xml:space="preserve"> & ſit extra lineam d z in pun-<lb/>cto e:</s> <s xml:id="echoid-s30541" xml:space="preserve"> dico quoniam magnitudines a b & b g in æqua-<lb/>les apparẽt:</s> <s xml:id="echoid-s30542" xml:space="preserve"> ꝓducátur enim lineæ e a, e b, e g:</s> <s xml:id="echoid-s30543" xml:space="preserve"> & deſcri-<lb/>batur circa a e g trigonum circulus, qui ſit a e d g, per <lb/>5 p 4, & adijciatur line æ e b linea recta b i, attingens <lb/>in parte oppoſita puncto e circumferentiá.</s> <s xml:id="echoid-s30544" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s30545" xml:space="preserve"> <lb/>arcus a z eſt æ qualis arcui z g per 33 p 6 propter recti-<lb/>tudinem angulorum ad punctum b, ſiue punctum ſit <lb/>cẽtrum deſcripti circuli, ſiue non:</s> <s xml:id="echoid-s30546" xml:space="preserve"> ſemper enim ex hy.</s> <s xml:id="echoid-s30547" xml:space="preserve"> <lb/>potheſi, & per 3 p 3 & 4 p 1 & per 28 p 3 erit arcus a i <lb/>maior arcu i g.</s> <s xml:id="echoid-s30548" xml:space="preserve"> Palàm ergo item per 33 p 6, quoniam <lb/>angulus a e i maior eſt angulo i e g:</s> <s xml:id="echoid-s30549" xml:space="preserve"> ſed ſub angulo a e <lb/>i uidetur magnitudo a b ab oculo exiſtẽte centraliter <lb/>in puncto e, & ſub angulo i e g uidetur magnitudo b g:</s> <s xml:id="echoid-s30550" xml:space="preserve"> apparet ergo a b maior quâm b g, oculo tali-<lb/>ter diſp oſito, ut patet per 20 huius.</s> <s xml:id="echoid-s30551" xml:space="preserve"> Palàm etiam per 118 huius quòd ſi oculus tranſmutetur ſecun-<lb/>dum lineam e i illis magnitudinibus obliquè incidentem, ſemper uiſæ magnitudines a b & b g ap-<lb/>parent in æquales:</s> <s xml:id="echoid-s30552" xml:space="preserve"> & quantò propinquius ad punctum b, tantò apparẽt maiores per 16 p 1 & per 20 <lb/>huius:</s> <s xml:id="echoid-s30553" xml:space="preserve"> quoniã ſemper angulus extrinſecus maior fit angulo intrinſeco ſibi oppoſito.</s> <s xml:id="echoid-s30554" xml:space="preserve"> Si ergo ſuper <lb/>circuli circumferentiam centrum uiſus moueri intelligatur:</s> <s xml:id="echoid-s30555" xml:space="preserve"> ſemper in æquales apparent magnitu-<lb/>dines a b & b g:</s> <s xml:id="echoid-s30556" xml:space="preserve"> & ſi oculus extra circulum ponatur nõ exiſtens in directo lineæ d z, a d huc in æqua <lb/>les apparent magnitudines a b & b g.</s> <s xml:id="echoid-s30557" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s30558" xml:space="preserve"/> </p> <div xml:id="echoid-div1233" type="float" level="0" n="0"> <figure xlink:label="fig-0474-01" xlink:href="fig-0474-01a"> <variables xml:id="echoid-variables510" xml:space="preserve">d e e a b g a b g z i i</variables> </figure> </div> </div> <div xml:id="echoid-div1235" type="section" level="0" n="0"> <head xml:id="echoid-head969" xml:space="preserve" style="it">121. Sunt loca, in quibus poſito uiſu, æquales magnitudines communiter loca quædam obli-<lb/>què occupantes, quando æquales, quando inæquales apparent.</head> <p> <s xml:id="echoid-s30559" xml:space="preserve">Eſto cẽtrum uiſus in puncto z:</s> <s xml:id="echoid-s30560" xml:space="preserve"> & ſint du æ magnitudines æ quales uiſæ:</s> <s xml:id="echoid-s30561" xml:space="preserve"> quæ g d & g b, quæ com.</s> <s xml:id="echoid-s30562" xml:space="preserve"> <lb/>muniter locum unum occupent nullo medio corpore interpoſito:</s> <s xml:id="echoid-s30563" xml:space="preserve"> obliquè tamen coniungãtur ſe-<lb/>cundum angulum, qui ſit d g b:</s> <s xml:id="echoid-s30564" xml:space="preserve"> hunc ergo angulũ per æqualia diuidat linea g z per 9 p 1 Dico quòd <lb/>in quocunq;</s> <s xml:id="echoid-s30565" xml:space="preserve"> puncto line æ z g cadat oculus, ſemper æquales uidebuntur magnitudines b g & g d.</s> <s xml:id="echoid-s30566" xml:space="preserve"> <lb/>Poteſt autem hoc conuinci per 4 p 1 & per 20 huius:</s> <s xml:id="echoid-s30567" xml:space="preserve"> ſemper enim angulus g z b eſt æ qualis angulo <lb/>g z d.</s> <s xml:id="echoid-s30568" xml:space="preserve"> Idem quoq;</s> <s xml:id="echoid-s30569" xml:space="preserve"> accidit, ſi ſuper utranq;</s> <s xml:id="echoid-s30570" xml:space="preserve"> illarum linearum b g & g d ſemicirculus deſcribatur:</s> <s xml:id="echoid-s30571" xml:space="preserve"> & à <lb/>puncto ſectionis illorum ſemicirculorum, qui ſit z, ducantur lineæ z b & z d, z g:</s> <s xml:id="echoid-s30572" xml:space="preserve"> tunc enim, quia <lb/> <pb o="173" file="0475" n="475" rhead="LIBER QVARTVS."/> uterq;</s> <s xml:id="echoid-s30573" xml:space="preserve"> angulorum b z g & d z g eritrectus per 31 p 3:</s> <s xml:id="echoid-s30574" xml:space="preserve"> pater ergo per 20 huius propoſitum.</s> <s xml:id="echoid-s30575" xml:space="preserve"> Idẽ quoq;</s> <s xml:id="echoid-s30576" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0475-01a" xlink:href="fig-0475-01"/> <anchor type="figure" xlink:label="fig-0475-02a" xlink:href="fig-0475-02"/> accidit ſi ultra pun-<lb/>ctum ſectionis ſemi-<lb/>circulorum linea g z <lb/>ꝓducatur:</s> <s xml:id="echoid-s30577" xml:space="preserve"> & in eius <lb/>pũcto z centrũ oculi <lb/>ponatur.</s> <s xml:id="echoid-s30578" xml:space="preserve"> Sed eſt etiá <lb/>locus, in quo illę ma-<lb/>gnitudines datæ æ-<lb/>quales, quæ ſunt b g <lb/>& g d, uiſui in æqua-<lb/>les apparẽt:</s> <s xml:id="echoid-s30579" xml:space="preserve"> ad quem <lb/>inuenlendum, circa <lb/>lineam g b ſemicirculus deſcribatur, qui ſit b z g, & circa lineam g d portio maior ſemicirculo, quæ <lb/> <anchor type="figure" xlink:label="fig-0475-03a" xlink:href="fig-0475-03"/> ſit g z d.</s> <s xml:id="echoid-s30580" xml:space="preserve"> Poſsibile quoque eſt hoc ſuper g d deſcribere <lb/>portionem circuli capientem angulum dato acuto an-<lb/>gulo æqualẽ per 33 p 3:</s> <s xml:id="echoid-s30581" xml:space="preserve"> ſed illa portio maior eſt ſemicir-<lb/>culo per 31 p 3:</s> <s xml:id="echoid-s30582" xml:space="preserve"> ſit ergo deſcripta, & ſit g z d:</s> <s xml:id="echoid-s30583" xml:space="preserve"> & ducantur <lb/>lineæ b z & g z & d z:</s> <s xml:id="echoid-s30584" xml:space="preserve"> angulus itaq;</s> <s xml:id="echoid-s30585" xml:space="preserve"> b z g eſt rectus per 31 <lb/>p 3, & angulus g z d acutus per eandem 31:</s> <s xml:id="echoid-s30586" xml:space="preserve"> ſed ſub maio-<lb/>ri angulo uiſa maiora apparent per 20 huius.</s> <s xml:id="echoid-s30587" xml:space="preserve"> Eſtitaque <lb/>locus, in quo magnitudines æquales inæquales appa-<lb/>rent:</s> <s xml:id="echoid-s30588" xml:space="preserve"> ut punctus ſectionis portionis maioris ſemicircu-<lb/>lo conſtitutæ ſuper unam magnitudinum, & ſemicirculi ſuper alteram conſtituti.</s> <s xml:id="echoid-s30589" xml:space="preserve"> Et hoc eſt, quod <lb/>proponitur.</s> <s xml:id="echoid-s30590" xml:space="preserve"/> </p> <div xml:id="echoid-div1235" type="float" level="0" n="0"> <figure xlink:label="fig-0475-01" xlink:href="fig-0475-01a"> <variables xml:id="echoid-variables511" xml:space="preserve">z z d b g</variables> </figure> <figure xlink:label="fig-0475-02" xlink:href="fig-0475-02a"> <variables xml:id="echoid-variables512" xml:space="preserve">z d z b g</variables> </figure> <figure xlink:label="fig-0475-03" xlink:href="fig-0475-03a"> <variables xml:id="echoid-variables513" xml:space="preserve">z d b g</variables> </figure> </div> </div> <div xml:id="echoid-div1237" type="section" level="0" n="0"> <head xml:id="echoid-head970" xml:space="preserve" style="it">122. Eſt locus, in quo inæquales magnitudines communiter loca quædam obliquè occupan-<lb/>tes, quando inæquales, quando æquales apparent. Euclides 49 th opt.</head> <p> <s xml:id="echoid-s30591" xml:space="preserve">Sit, ut in præcedente, centrum uiſus in puncto z:</s> <s xml:id="echoid-s30592" xml:space="preserve"> & ſint duæ magnitudines quarum maior b g, <lb/>minor uerò g d, coniunctæ ſecundum angulum d g b:</s> <s xml:id="echoid-s30593" xml:space="preserve"> qui diuidatur per 9 p 1 per æ qualia, ducta li-<lb/> <anchor type="figure" xlink:label="fig-0475-04a" xlink:href="fig-0475-04"/> <anchor type="figure" xlink:label="fig-0475-05a" xlink:href="fig-0475-05"/> nea g z.</s> <s xml:id="echoid-s30594" xml:space="preserve"> Dico, quòd <lb/>oculo exiſtente ſuper <lb/>quodcunq;</s> <s xml:id="echoid-s30595" xml:space="preserve"> pũctum li-<lb/>neæ z g, ſemper magni <lb/>tudines b g & g d uide <lb/>buntur inæquales:</s> <s xml:id="echoid-s30596" xml:space="preserve"> & <lb/>b g maior, Ductis e-<lb/>nim lineis b z & d z, an <lb/>guli ad pũctum z fiunt <lb/>in æquales, & maior, <lb/>cui maior baſis ſubten <lb/>ditur per 25 p 1.</s> <s xml:id="echoid-s30597" xml:space="preserve"> Quoniá <lb/>ſi detur quod illi anguli ſint ęquales:</s> <s xml:id="echoid-s30598" xml:space="preserve"> erũt trigoni b z g & g z d ęquianguli & æquilateri, quod eſt cõ-<lb/>tra hypotheſim:</s> <s xml:id="echoid-s30599" xml:space="preserve"> palàm ergo quòd illi anguli erunt inæquales:</s> <s xml:id="echoid-s30600" xml:space="preserve"> uidebũtur itaq;</s> <s xml:id="echoid-s30601" xml:space="preserve"> per 20 huius illæ ma-<lb/>gnitudines in æquales:</s> <s xml:id="echoid-s30602" xml:space="preserve"> & maior uidebitur ipſa g b, quoniam ſub maiori angulo uidebitur.</s> <s xml:id="echoid-s30603" xml:space="preserve"> Sed & <lb/>quandoq;</s> <s xml:id="echoid-s30604" xml:space="preserve"> illæ magnitudines uidentur æ quales.</s> <s xml:id="echoid-s30605" xml:space="preserve"> Deſcribatur enim, ſicut in præmiſſa, circa lineam b <lb/>g maiorem ipſarum portio maior ſemicirculo, quæ ſit b z g:</s> <s xml:id="echoid-s30606" xml:space="preserve"> & ducantur lineæ b z & z g:</s> <s xml:id="echoid-s30607" xml:space="preserve"> & circum-<lb/>ſcribatur lineæ g d minori portio ſimilis portioni b z g, hoc eſt angulum æ qualem angulo b z g ca-<lb/>piens:</s> <s xml:id="echoid-s30608" xml:space="preserve"> ſit quoq;</s> <s xml:id="echoid-s30609" xml:space="preserve"> communis punctus iſtarum ſectionum punctus z:</s> <s xml:id="echoid-s30610" xml:space="preserve"> & ducantur lineæ z b, & z g, z <lb/>d.</s> <s xml:id="echoid-s30611" xml:space="preserve"> Quia ita q;</s> <s xml:id="echoid-s30612" xml:space="preserve"> angulus d z g eſt æ qualis angulo b z g, quoniã in ſimiles cadunt portiones.</s> <s xml:id="echoid-s30613" xml:space="preserve"> Oculi itaq;</s> <s xml:id="echoid-s30614" xml:space="preserve"> <lb/>centro poſito in puncto z, qui eſt punctus communis ſectionis illarum portionum, magnitudines <lb/>b g & g d æ quales apparent.</s> <s xml:id="echoid-s30615" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s30616" xml:space="preserve"/> </p> <div xml:id="echoid-div1237" type="float" level="0" n="0"> <figure xlink:label="fig-0475-04" xlink:href="fig-0475-04a"> <variables xml:id="echoid-variables514" xml:space="preserve">z z b g</variables> </figure> <figure xlink:label="fig-0475-05" xlink:href="fig-0475-05a"> <variables xml:id="echoid-variables515" xml:space="preserve">z z d b g</variables> </figure> </div> </div> <div xml:id="echoid-div1239" type="section" level="0" n="0"> <head xml:id="echoid-head971" xml:space="preserve" style="it">123. Sunt loca, in quibus centro uiſus poſito, æquales magnitudines erectæ ſuper ſubiacẽtem <lb/>planam ſuperſiciem, quando æquales, quando inæquales apparent. Euclides 48 th. opt.</head> <p> <s xml:id="echoid-s30617" xml:space="preserve">Sint duæ magnitudines a b, & g d æ quales & erectæ ſuper ſubiacẽtem ipſis planam ſuperficiem:</s> <s xml:id="echoid-s30618" xml:space="preserve"> <lb/>dico quòd eſt locus ubi poſito cẽtro uiſus, magnitudines a b & g d apparẽt æquales.</s> <s xml:id="echoid-s30619" xml:space="preserve"> Ducatur enim <lb/>inter ipſas in ſubiecta plana ſuperſicie linea recta, quæ ſit b d:</s> <s xml:id="echoid-s30620" xml:space="preserve"> quæ diuidatur in duo æ qualia in pun-<lb/>cto e per 10 p 1:</s> <s xml:id="echoid-s30621" xml:space="preserve"> & à puncto e protrahatur perpendiculariter linea e z ſuper lineam b d in eadem ſu-<lb/>perficie per 11 p 1.</s> <s xml:id="echoid-s30622" xml:space="preserve"> Dico quòd ſuper lineam è z perpendicularem ſuper lineam d b exiſtente centro <lb/>uiſus, ſemper magnitudines a b, & g d æ quales apparebunt.</s> <s xml:id="echoid-s30623" xml:space="preserve"> Sit enim o culus in puncto z:</s> <s xml:id="echoid-s30624" xml:space="preserve"> & ducan-<lb/>tur lineæ z a, z b, z g, z d.</s> <s xml:id="echoid-s30625" xml:space="preserve"> Quoniã ergo trigonorũ b e z, & d e z latus b e eſt æ quale lateri d e, & latus e <lb/>z eſt commune, anguli uerò z e b, & z e d ſunt æ quales, quia recti:</s> <s xml:id="echoid-s30626" xml:space="preserve"> palàm per 4 p 1 quoniam linea z b <lb/>eſt æ qualis lineæ z d:</s> <s xml:id="echoid-s30627" xml:space="preserve"> ſed & linea a b eſt ęqualis lineæ g d per hypotheſim, & anguli g d z & a b z ſunt <lb/>recti per deſinitionem lineæ ſuper ſuperficiem erectæ erit ergo per 4 p 1 linea z a æ qualis lineæ z g, <lb/> <pb o="174" file="0476" n="476" rhead="VITELLONIS OPTICAE"/> & reliqui anguli reliquis angulis.</s> <s xml:id="echoid-s30628" xml:space="preserve"> Angulus ergo a z b æ qualis eſt angulo g z d:</s> <s xml:id="echoid-s30629" xml:space="preserve"> ergo per 20 huius æ.</s> <s xml:id="echoid-s30630" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0476-01a" xlink:href="fig-0476-01"/> quales apparent magnitudines a b, & g d.</s> <s xml:id="echoid-s30631" xml:space="preserve"> Dico etiam <lb/>quòd quandoq;</s> <s xml:id="echoid-s30632" xml:space="preserve"> inæquales apparent ipſæ magnitudi-<lb/>nes a b, & g d.</s> <s xml:id="echoid-s30633" xml:space="preserve"> Remanẽte enim præmiſſa diſpoſitione in <lb/>eadem ſubſtrata ſuperſicie tranſmutetur centrum ocu-<lb/>li extra lineam e z:</s> <s xml:id="echoid-s30634" xml:space="preserve"> & fiat in puncto i:</s> <s xml:id="echoid-s30635" xml:space="preserve"> & ducatur linea i e <lb/>ad medium punctum line æ b d:</s> <s xml:id="echoid-s30636" xml:space="preserve"> & ducantur lineę i a, i b, <lb/>i g, i d:</s> <s xml:id="echoid-s30637" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s30638" xml:space="preserve"> per 24 p 1 linea i b maior quã linea i d:</s> <s xml:id="echoid-s30639" xml:space="preserve"> ideo <lb/>quòd angulus b e i eſt maior angulo d e i, æquis inter ſe <lb/>lateribus contento:</s> <s xml:id="echoid-s30640" xml:space="preserve"> abſcin datur ergo à linea i b æ qualis <lb/>lineæ i d per 3 p 1:</s> <s xml:id="echoid-s30641" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s30642" xml:space="preserve"> linea b t æ qualis lineæ i d, & du-<lb/>catur linea a t.</s> <s xml:id="echoid-s30643" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s30644" xml:space="preserve"> per definitionem lineæ ſuper <lb/>ſuperficiem erectæ anguli i b a & i d g ſunt ęquales, quia <lb/>recti:</s> <s xml:id="echoid-s30645" xml:space="preserve"> erit ergo per 4 p 1 angulus b t a æqualis angulo g i <lb/>d:</s> <s xml:id="echoid-s30646" xml:space="preserve"> ſed angulus b t a per 16 p 1 eſt maior angulo b i a, quia <lb/>eſt extrinſecus trigono a t i:</s> <s xml:id="echoid-s30647" xml:space="preserve"> angulus ergo g i d maior eſt <lb/>angulo b i a:</s> <s xml:id="echoid-s30648" xml:space="preserve"> ergo per 20 huius uiſu exiſtente in puncto <lb/>i, maior apparet linea d g quàm linea a b:</s> <s xml:id="echoid-s30649" xml:space="preserve"> & eodẽ modo <lb/>de quolibet puncto extra lineã z e dato eſt demonſtran <lb/>dum.</s> <s xml:id="echoid-s30650" xml:space="preserve"> Variantur autem magnitudines in uiſu ſecũdum <lb/>approximationem uel elongationem ab altero uiſibi-<lb/>lium.</s> <s xml:id="echoid-s30651" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s30652" xml:space="preserve"/> </p> <div xml:id="echoid-div1239" type="float" level="0" n="0"> <figure xlink:label="fig-0476-01" xlink:href="fig-0476-01a"> <variables xml:id="echoid-variables516" xml:space="preserve">a g b e d t z a</variables> </figure> </div> </div> <div xml:id="echoid-div1241" type="section" level="0" n="0"> <head xml:id="echoid-head972" xml:space="preserve" style="it">124. Sunt loca, in quibus centro uiſus poſito, in ea-<lb/>dem ſuperficie æqualia latera rectanguli quandoque <lb/>æqualia, quando inæqualia uidentur.</head> <p> <s xml:id="echoid-s30653" xml:space="preserve">Sit rectangulũ a b g d, cuius duo latera a b & g d ſint <lb/>æ qualia.</s> <s xml:id="echoid-s30654" xml:space="preserve"> Dico quòd ſunt loca, in quibus cẽtro uiſus po-<lb/>ſito, illa duo latera uidebũtur æqualia.</s> <s xml:id="echoid-s30655" xml:space="preserve"> Circumſcirbatur <lb/>enim illi rectangulo per 40th.</s> <s xml:id="echoid-s30656" xml:space="preserve"> 1 huius, & per 9 p 3 circulus, in cuius alterius arcuum (qui ſunt b d & <lb/>a g) quocunq;</s> <s xml:id="echoid-s30657" xml:space="preserve"> puncto ponatur cétrum uiſus:</s> <s xml:id="echoid-s30658" xml:space="preserve"> ſit autẽ, exempli cauſſa, poſitus in pũcto inedio arcus <lb/>b d, qui ſit o:</s> <s xml:id="echoid-s30659" xml:space="preserve"> & copulentur lineæ, quæ o a, o g, o b, o d.</s> <s xml:id="echoid-s30660" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s30661" xml:space="preserve"> latera a b & d g ſunt æ qualia:</s> <s xml:id="echoid-s30662" xml:space="preserve"> erunt <lb/>per 28 p 3 arcus a b, & d g æ quales:</s> <s xml:id="echoid-s30663" xml:space="preserve"> ergo per 27 p 3 erũt anguli a o b & g o d æ quales:</s> <s xml:id="echoid-s30664" xml:space="preserve"> ergo per 20 hu-<lb/>ius latera a b & d g uidentur æqualia uiſu exiſtente in puncto o.</s> <s xml:id="echoid-s30665" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s30666" xml:space="preserve"> demonſtrandũ eſt <lb/>de quolibet puncto amborum arcuum b d, & a g:</s> <s xml:id="echoid-s30667" xml:space="preserve"> ſemperenim centro uiſus in quorumcunq;</s> <s xml:id="echoid-s30668" xml:space="preserve"> illorũ <lb/>punctorũ exiſtente, uidẽtur a b & g d magnitudines æ quales.</s> <s xml:id="echoid-s30669" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s30670" xml:space="preserve"> fi linea b d diui datur <lb/> <anchor type="figure" xlink:label="fig-0476-02a" xlink:href="fig-0476-02"/> per æqualia in puncto ſper 10 p 1:</s> <s xml:id="echoid-s30671" xml:space="preserve"> & in puncto f ponatur <lb/>centrũ uiſus:</s> <s xml:id="echoid-s30672" xml:space="preserve"> tuncitem per 4 p 1 & 20 huius lineæ a b & <lb/>g d uidebuntur æquales.</s> <s xml:id="echoid-s30673" xml:space="preserve"> Et ſi à pũcto f ducatur per 11 p 1 <lb/>linea perpendicularis ſuper lineam b d, quę ſit f z, ſecans <lb/>peripheriam circuli in puncto o:</s> <s xml:id="echoid-s30674" xml:space="preserve"> tunc a dhuc ſecun dum <lb/>præmiſſa, in quocũq;</s> <s xml:id="echoid-s30675" xml:space="preserve"> pũcto lineæ f z ponatur centrũ ui-<lb/>ſus, ſemper per 4 p 1 & 20 huius dictę lineæ a b & g d ap-<lb/>parebunt ęquales.</s> <s xml:id="echoid-s30676" xml:space="preserve"> Quòd ſi centrũ oculi ſit extra circulũ <lb/>a b g d, ut in pũcto e, quod ſit, exẽpli cauſſa, propinquius <lb/>lineæ d g, quá ipſi b a:</s> <s xml:id="echoid-s30677" xml:space="preserve"> dico quòd uidebitur linea a b ma-<lb/>ior quàm linea g d.</s> <s xml:id="echoid-s30678" xml:space="preserve"> Protrahantur enim lineæ e a, e g, e b, <lb/>e d:</s> <s xml:id="echoid-s30679" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s30680" xml:space="preserve"> linea e a peripheriá circuli in pũcto t, & linea <lb/>e g in puncto r:</s> <s xml:id="echoid-s30681" xml:space="preserve"> & copulentur lineæ b t & d r.</s> <s xml:id="echoid-s30682" xml:space="preserve"> Et quoniã, <lb/>ut ſuprà patuit, lineæ a b & g d ſunt æquales ex hypo-<lb/>theſi:</s> <s xml:id="echoid-s30683" xml:space="preserve"> ergo per 28 p 3 erit arcus a b æquales arcui g d:</s> <s xml:id="echoid-s30684" xml:space="preserve"> erũt <lb/>ergo per 27 p 3 anguli a t b & g r d æquales propter duo-<lb/>rum arcuũ æqualitatẽ:</s> <s xml:id="echoid-s30685" xml:space="preserve"> ergo per 13 p 1 anguli b t e & d r e <lb/>ſunt æquales.</s> <s xml:id="echoid-s30686" xml:space="preserve"> Quia uerò arcus b t eſt maior arcu d r, pro <lb/>pter maiorẽ propinquitatẽ puncti e ad lineá d g:</s> <s xml:id="echoid-s30687" xml:space="preserve"> erit er-<lb/>go per 29 p 3 latus b t maius latere r d:</s> <s xml:id="echoid-s30688" xml:space="preserve"> linea uerò e t eſt <lb/>minor quàm linea r e:</s> <s xml:id="echoid-s30689" xml:space="preserve"> quod patet ex 17 p 6 & 36 p 3 pro-<lb/>tracta prius à pũcto e per 17 p 3 linea e q circulũ contin-<lb/>gente in pũcto q.</s> <s xml:id="echoid-s30690" xml:space="preserve"> Tunc ergo, cũlinea a e ſit maior quàm <lb/>linea e g ex hypotheſi:</s> <s xml:id="echoid-s30691" xml:space="preserve"> patet etiã per 8 & 10 p 5 lineam er <lb/>eſſe maiorẽ linea e t.</s> <s xml:id="echoid-s30692" xml:space="preserve"> Quia ergo linea b t eſt maior quàm <lb/>linea r d, & linea e t eſt minor quàm linea e r:</s> <s xml:id="echoid-s30693" xml:space="preserve"> fiat per 3 <lb/>th.</s> <s xml:id="echoid-s30694" xml:space="preserve"> 1 huius, ut quæ eſt proportio lineæ b t ad lineam t e, eadẽ ſit lineæ r d ad aliquã lineam quartam:</s> <s xml:id="echoid-s30695" xml:space="preserve"> <lb/>quæ neceſſario, ut patet ex præmiſsis, erit minor quá linea r e.</s> <s xml:id="echoid-s30696" xml:space="preserve"> Abſcindatur ergo ք 3 p 1 æ qualis illi à <lb/>linea r e, quę ſit r p:</s> <s xml:id="echoid-s30697" xml:space="preserve"> copuletur quoq;</s> <s xml:id="echoid-s30698" xml:space="preserve"> linea p d.</s> <s xml:id="echoid-s30699" xml:space="preserve"> Ergo per 6 p 6 trigona b t e & r d p æ quiangula erũt, <lb/> <pb o="175" file="0477" n="477" rhead="LIBER QVARTVS."/> eritq́;</s> <s xml:id="echoid-s30700" xml:space="preserve"> angulus r p d æqualis angulo b e t:</s> <s xml:id="echoid-s30701" xml:space="preserve"> ſed per 16 p 1 angulus r p d maior eſt angulo p e d:</s> <s xml:id="echoid-s30702" xml:space="preserve"> angu-<lb/>lus ergo a e b eſt maior angulo g e d:</s> <s xml:id="echoid-s30703" xml:space="preserve"> ergo per 20 huius uidebitur linea a b maior quàm linea g d.</s> <s xml:id="echoid-s30704" xml:space="preserve"> Si <lb/>autem centrum oculi conſiſtat intra circulum:</s> <s xml:id="echoid-s30705" xml:space="preserve"> tunc immutetur figura, ſitq́;</s> <s xml:id="echoid-s30706" xml:space="preserve">, ut prius, circulus à b d g <lb/>d circumſcriptus rectangulo a b g d, cuius latus b d diuidatur per æqualia in puncto f:</s> <s xml:id="echoid-s30707" xml:space="preserve"> & ducatur à <lb/>puncto f ad peripheriam circuli perpendicularis ſu-<lb/> <anchor type="figure" xlink:label="fig-0477-01a" xlink:href="fig-0477-01"/> per lineam b d:</s> <s xml:id="echoid-s30708" xml:space="preserve"> quę ſit z f:</s> <s xml:id="echoid-s30709" xml:space="preserve"> conſiſtatq́;</s> <s xml:id="echoid-s30710" xml:space="preserve"> centrum uiſus in-<lb/>tra portionem z f d, ut in puncto o:</s> <s xml:id="echoid-s30711" xml:space="preserve"> dico quòd linea g d <lb/>apparebit maior quàm linea a b.</s> <s xml:id="echoid-s30712" xml:space="preserve"> Sit enim centrum il-<lb/>lius circuli punctũ e:</s> <s xml:id="echoid-s30713" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s30714" xml:space="preserve"> lineæ o a, o b, o g, o d:</s> <s xml:id="echoid-s30715" xml:space="preserve"> <lb/>producaturq́;</s> <s xml:id="echoid-s30716" xml:space="preserve"> linea a o uſq;</s> <s xml:id="echoid-s30717" xml:space="preserve"> in punctum circumferen-<lb/>tiæ, quod ſit s, & linea g o uſ q;</s> <s xml:id="echoid-s30718" xml:space="preserve"> in punctum q, & linea <lb/>e o uſq;</s> <s xml:id="echoid-s30719" xml:space="preserve"> in punctum i:</s> <s xml:id="echoid-s30720" xml:space="preserve"> & copulentur lineæ q d & g b.</s> <s xml:id="echoid-s30721" xml:space="preserve"> <lb/>Cum itaq;</s> <s xml:id="echoid-s30722" xml:space="preserve"> linea a s ſit maior quàm linea g q per 7 p 3:</s> <s xml:id="echoid-s30723" xml:space="preserve"> <lb/>propter hoc quòd punctus o, in quo eſt centrum ui-<lb/>ſus, datus eſt in portione z f d propinquior lineæ d g <lb/>quàm lineæ a b, & propinquior puncto g ꝗ̃ puncto a:</s> <s xml:id="echoid-s30724" xml:space="preserve"> li <lb/>nea quoq;</s> <s xml:id="echoid-s30725" xml:space="preserve"> a s eſt propinquior centro e quàm linea g q:</s> <s xml:id="echoid-s30726" xml:space="preserve"> <lb/>eſt ergo portio circuli & arcus a s maior portione cir-<lb/>culi & arcu q g:</s> <s xml:id="echoid-s30727" xml:space="preserve"> ſed, ut patet ex præmiſsis, arcus a b æ-<lb/>qualis eſt arcui g d per 28 p 3 & ex hypotheſi:</s> <s xml:id="echoid-s30728" xml:space="preserve"> ablatis er <lb/>go hinc & inde arcubus æqualibus, remanebit arcus b <lb/>s maior arcu q d:</s> <s xml:id="echoid-s30729" xml:space="preserve"> ergo per 29 p 3 erit chorda b s maior <lb/>quàm chorda q d:</s> <s xml:id="echoid-s30730" xml:space="preserve"> ſed per 7 p 3 linea o s eſt minor quàm <lb/>linea o q, cum linea o s ſit propinquior diametro e i quàm linea o q, ut patet ex præmiſsis.</s> <s xml:id="echoid-s30731" xml:space="preserve"> Quoniam <lb/>ergo anguli b s a & g q d per 27 p 3 ſunt æquales, quoniam cadunt in arcus æquales:</s> <s xml:id="echoid-s30732" xml:space="preserve"> in trigonis <lb/>quo q;</s> <s xml:id="echoid-s30733" xml:space="preserve"> b o s & d o q latus b s eſt maius latere q d, & latus q o maius latere s o, ut patet ex præmiſsis:</s> <s xml:id="echoid-s30734" xml:space="preserve"> <lb/>& h æ c latera hinc & inde continent angulos æ quales:</s> <s xml:id="echoid-s30735" xml:space="preserve"> tunc per modum, quo in præmiſsis ſuperius <lb/>uſi ſumus, patet quòd angulus b o s maior eſt angulo q o d:</s> <s xml:id="echoid-s30736" xml:space="preserve"> ergo per 13 p 1 angulus b o a eſt minor <lb/>angulo g o d:</s> <s xml:id="echoid-s30737" xml:space="preserve"> ergo per 20 huius uidebitur linea g d maior quàm linea a b, centro oculi exiſtente in <lb/>puncto o.</s> <s xml:id="echoid-s30738" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s30739" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s30740" xml:space="preserve"> ſi centrum uiſus fuerit portione z f b, uidebitur <lb/>linea a b maior quàm linea d g.</s> <s xml:id="echoid-s30741" xml:space="preserve"> Hæc ergo latera rectanguli quandoq;</s> <s xml:id="echoid-s30742" xml:space="preserve"> uidentur æqualia, quandoq;</s> <s xml:id="echoid-s30743" xml:space="preserve"> <lb/>in æqualia in diuerſis locis centro uiſus poſito.</s> <s xml:id="echoid-s30744" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s30745" xml:space="preserve"/> </p> <div xml:id="echoid-div1241" type="float" level="0" n="0"> <figure xlink:label="fig-0476-02" xlink:href="fig-0476-02a"> <variables xml:id="echoid-variables517" xml:space="preserve">a g q b f d t o r p z c</variables> </figure> <figure xlink:label="fig-0477-01" xlink:href="fig-0477-01a"> <variables xml:id="echoid-variables518" xml:space="preserve">a i g e b f d o z q s</variables> </figure> </div> </div> <div xml:id="echoid-div1243" type="section" level="0" n="0"> <head xml:id="echoid-head973" xml:space="preserve" style="it">125. Sunt loca, in quibus oculo poſito, inæquales magnitudines in idem compoſitæ, æquales u-<lb/>tri inæqualium apparent. Euclides 50 th. opt.</head> <p> <s xml:id="echoid-s30746" xml:space="preserve">Sit duarum magnitudinum datarum b g maior, & d g minor:</s> <s xml:id="echoid-s30747" xml:space="preserve"> & circa utranq;</s> <s xml:id="echoid-s30748" xml:space="preserve"> ſemicirculus deſcri <lb/>batur, ut circa lineam d g ſemicirculus d z g, & circa lineam b g ſemicirculus g k b:</s> <s xml:id="echoid-s30749" xml:space="preserve"> & tertius ſemicir-<lb/> <anchor type="figure" xlink:label="fig-0477-02a" xlink:href="fig-0477-02"/> culus deſcribatur circa totam lineam d b, qui ſit d a b.</s> <s xml:id="echoid-s30750" xml:space="preserve"> <lb/>Ductis itaq;</s> <s xml:id="echoid-s30751" xml:space="preserve"> lineis d a & b a, palàm quia productæ lineę <lb/>ſecant minores ſemicirculos:</s> <s xml:id="echoid-s30752" xml:space="preserve"> ſecet ergo linea a b ſemi-<lb/>circulum g k b in puncto k, & linea d a ſem:</s> <s xml:id="echoid-s30753" xml:space="preserve"> circulum <lb/>d z g in puncto z:</s> <s xml:id="echoid-s30754" xml:space="preserve"> & ducátur lineæ z g & k g.</s> <s xml:id="echoid-s30755" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s30756" xml:space="preserve"> <lb/>per 31 p 3, quoniam anguli d z g & g k b & d a b omnes <lb/>ſunt æquales:</s> <s xml:id="echoid-s30757" xml:space="preserve"> quia recti.</s> <s xml:id="echoid-s30758" xml:space="preserve"> Oculi itaq;</s> <s xml:id="echoid-s30759" xml:space="preserve"> centro ſecundum <lb/>puncta k, a, z tranſmutato, uidebitur linea b g æqualies <lb/>lineæ g d, & linea d b æqualis alteri datarum, & linea d <lb/>g æqualies a mbabus lineis d g & b g.</s> <s xml:id="echoid-s30760" xml:space="preserve"> Et idem accidit cen <lb/>tro oculi ſecundum quæcunq;</s> <s xml:id="echoid-s30761" xml:space="preserve"> puncta form arum ſemicirculorũ tranſmutato.</s> <s xml:id="echoid-s30762" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s30763" xml:space="preserve"/> </p> <div xml:id="echoid-div1243" type="float" level="0" n="0"> <figure xlink:label="fig-0477-02" xlink:href="fig-0477-02a"> <variables xml:id="echoid-variables519" xml:space="preserve">a k z b g d</variables> </figure> </div> </div> <div xml:id="echoid-div1245" type="section" level="0" n="0"> <head xml:id="echoid-head974" xml:space="preserve" style="it">126. Poßibile eſt inueniri loca, à quibus æqualis magnitudo apparet medietas, uelquarta <lb/>pars: & uniuerſaliter in eaproportione, ſecundum quampropoſitus angulus diuidetur. Eucli-<lb/>des 51 theo. opt.</head> <figure> <variables xml:id="echoid-variables520" xml:space="preserve">e k h a f b g</variables> </figure> <p> <s xml:id="echoid-s30764" xml:space="preserve">Sint duæ magnitudines a b & g b æquales:</s> <s xml:id="echoid-s30765" xml:space="preserve"> & <lb/>circa a b deſcribatur ſemicirculus, qui ſit a k b:</s> <s xml:id="echoid-s30766" xml:space="preserve"> qui <lb/>per 30 p 3 diuidatur per æqualia in pũcto k, ductis <lb/>lineis a k & b k:</s> <s xml:id="echoid-s30767" xml:space="preserve"> palãq́;</s> <s xml:id="echoid-s30768" xml:space="preserve"> ք 31 p 3 quoniã angulus a k <lb/>b eſt rectus:</s> <s xml:id="echoid-s30769" xml:space="preserve"> diuidaturq́;</s> <s xml:id="echoid-s30770" xml:space="preserve"> angulus a k b per æqualia <lb/>per 9 p 1 ducta linea k ſ:</s> <s xml:id="echoid-s30771" xml:space="preserve"> quę per 33 p 6 neceſſariò e-<lb/>rit perpendicularis ſuper diametrũ a b, & incidet <lb/>centro ſemicirculi:</s> <s xml:id="echoid-s30772" xml:space="preserve"> ideo quia arcus ſemicirculi di-<lb/>uiſus eſt per ęqualia in puncto k:</s> <s xml:id="echoid-s30773" xml:space="preserve"> & per 33 p 3 ſupra <lb/>lineam b g deſcribatur portio circuli capiens an-<lb/>gulũ æqualẽ angulo a k f.</s> <s xml:id="echoid-s30774" xml:space="preserve"> Et quoniã angulus a k f <lb/>eſt acutus.</s> <s xml:id="echoid-s30775" xml:space="preserve"> angulus enim a k b, ꝗ eſt rectus, eſt du-<lb/> <pb o="176" file="0478" n="478" rhead="VITELLONIS OPTICAE"/> plus angulo a k f:</s> <s xml:id="echoid-s30776" xml:space="preserve"> erit ergo illa deſcripta portio maior ſemicirculo per 31 p 3, quæ ſit b e g:</s> <s xml:id="echoid-s30777" xml:space="preserve"> eritq́ angu <lb/>lus a k b duplus angulo b e g:</s> <s xml:id="echoid-s30778" xml:space="preserve"> cadatq́;</s> <s xml:id="echoid-s30779" xml:space="preserve"> punctus e in medio arcus b e g.</s> <s xml:id="echoid-s30780" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s30781" xml:space="preserve"> lineæ a b & b g ui-<lb/>dentur directè uiſui oppoſitæ, cum uiſus centrum eſt in punctis k & e:</s> <s xml:id="echoid-s30782" xml:space="preserve"> uidebitur ergo per 20 huius <lb/>linea b a in puncto k dupla lineæ b g uiſæ in puncto e.</s> <s xml:id="echoid-s30783" xml:space="preserve"> Et quoniam omnes anguli in una portione <lb/>circuli ſuper arcum conſiſtentes ſunt æquales per 21 p 3, palàm quòd accidit ſimiliter ſuper omnia <lb/>puncta illorum arcuum ſemicirculi ſcilicet præmiſsi, qui a k b, & portionis b e g, à quibus ductæ li-<lb/>neæ continent æquales angulos cũ diametro, ita ut obliquitas uiſionis hincinde ſit ſemper eadẽ.</s> <s xml:id="echoid-s30784" xml:space="preserve"> Vi <lb/>ſu ita q;</s> <s xml:id="echoid-s30785" xml:space="preserve"> exiſtente in puncto communis ſectionis ipſarum, qui ſit punctus h:</s> <s xml:id="echoid-s30786" xml:space="preserve"> tunc eodem intuitu <lb/>uidebitur linea a b quaſi dupla lineæ b g.</s> <s xml:id="echoid-s30787" xml:space="preserve"> Et eodem modo diuerſificatur rerum æqualium apparen-<lb/>tia, diuiſo angulo per alium numerum quemcunq;</s> <s xml:id="echoid-s30788" xml:space="preserve">. Generale enim eſt hoc, data magnitudine & an-<lb/>gulo diuidere angulum ſecundum aliquam proportionẽ per 27 th.</s> <s xml:id="echoid-s30789" xml:space="preserve"> 1 huius, & circa magnitudinem <lb/>deſcribere portionem circuli capientem angulum alicui diuidentium æqualem:</s> <s xml:id="echoid-s30790" xml:space="preserve"> & ſemper poſito <lb/>centro uiſus ad illum angulum uidebitur apparentia magnitudinis uariari ſecundum illum.</s> <s xml:id="echoid-s30791" xml:space="preserve"> Hoc <lb/>eſt ergo propoſitum.</s> <s xml:id="echoid-s30792" xml:space="preserve"> In hoc tamen non modicum effectum habet longitudo diſtantiæ ſecundum <lb/>rectam lineam protenſæ à puncto concurſus linearum illum angulum continentium:</s> <s xml:id="echoid-s30793" xml:space="preserve"> quoniam in <lb/>omnibus uiſis ex inæquali diſtantia, maior eſtproportio diſtantię maioris ad minorẽ, quàm anguli <lb/>ad angulum, ut patet per 11 huius.</s> <s xml:id="echoid-s30794" xml:space="preserve"> Idem quoq;</s> <s xml:id="echoid-s30795" xml:space="preserve"> accidit, ſi angulus a k b ſecundum aliam proportionẽ <lb/>fuerit diuiſus, & ei æqualis in portione circuli ſuper lineam b g conſtituatur angulus:</s> <s xml:id="echoid-s30796" xml:space="preserve"> & eadem eſt <lb/>demonſtratio.</s> <s xml:id="echoid-s30797" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s30798" xml:space="preserve"> propoſitum.</s> <s xml:id="echoid-s30799" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1246" type="section" level="0" n="0"> <head xml:id="echoid-head975" xml:space="preserve" style="it">127. Sunt loca, in quibus poſito uiſu, eadẽ magnitudo quandog totius ſuæ quantitatis, quan-<lb/>do medietatis, quando quartæ, uel ſecundum datam proportionem uidetur.</head> <figure> <variables xml:id="echoid-variables521" xml:space="preserve">e d c g a b</variables> </figure> <p> <s xml:id="echoid-s30800" xml:space="preserve">Eſto a b magnitudo uiſa:</s> <s xml:id="echoid-s30801" xml:space="preserve"> dico quòd ipſa (tranſmutato centro ui-<lb/>ſus ad diuerſa puncta) quandoque apparet ſuæ propriæ quantita-<lb/>tis, quandoq;</s> <s xml:id="echoid-s30802" xml:space="preserve"> in alia quacunq;</s> <s xml:id="echoid-s30803" xml:space="preserve"> proportione.</s> <s xml:id="echoid-s30804" xml:space="preserve"> Deſcribatur enim cir-<lb/>ca lineam a b circulus a e b, ita quòd linea a b non ſit diameter il-<lb/>lius circuli:</s> <s xml:id="echoid-s30805" xml:space="preserve"> quod poteſt fieri ſumpta pro diametro circuli aliqua li <lb/>nea maiore quàm ſit linea a b.</s> <s xml:id="echoid-s30806" xml:space="preserve"> Sit itaque centrum illius circuli pun-<lb/>ctum g:</s> <s xml:id="echoid-s30807" xml:space="preserve"> & ducantur lineæ a g, b g, a e, b e.</s> <s xml:id="echoid-s30808" xml:space="preserve"> Palàm ergo per 20 p 3 <lb/>quoniam angulus a g b duplus eſt angulo a e b.</s> <s xml:id="echoid-s30809" xml:space="preserve"> Oculi itaque cen-<lb/>tro exiſtente in centro circuli g, linea a b apparebit duplo maior <lb/>quàm appareat centro oculi exiſtente in arcu a e b per 20 huius:</s> <s xml:id="echoid-s30810" xml:space="preserve"> <lb/>quoniam omnes anguli contenti ſub lineis ab iſtis punctis ad pun-<lb/>cta a, b ductis ſunt æquales per 21 p 3:</s> <s xml:id="echoid-s30811" xml:space="preserve"> & cuilibet illorum duplus eſt <lb/>angulus, quiad centrum g, per 20 p 3.</s> <s xml:id="echoid-s30812" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s30813" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1247" type="section" level="0" n="0"> <head xml:id="echoid-head976" xml:space="preserve" style="it">128. Oculo, ei, quod uidetur, propius accedente: uidebitur rei uiſæ quantitas augmentari. Eu-<lb/>clides 55. theo. opt.</head> <figure> <variables xml:id="echoid-variables522" xml:space="preserve">g b d z</variables> </figure> <p> <s xml:id="echoid-s30814" xml:space="preserve">Sit linea uiſa b g:</s> <s xml:id="echoid-s30815" xml:space="preserve"> & ſit oculus in puncto z:</s> <s xml:id="echoid-s30816" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s30817" xml:space="preserve"> lineæ z b & <lb/>z g:</s> <s xml:id="echoid-s30818" xml:space="preserve"> & accedat oculus propius lineæ b g:</s> <s xml:id="echoid-s30819" xml:space="preserve"> & ſit ſuper d punctum:</s> <s xml:id="echoid-s30820" xml:space="preserve"> intel-<lb/>ligimus autem hic acceſsionem ſecundum lineam rectam perpendi-<lb/>cularem ſuper magnitudinem uiſam.</s> <s xml:id="echoid-s30821" xml:space="preserve"> Ducantur ergo lineæ b d & g d.</s> <s xml:id="echoid-s30822" xml:space="preserve"> <lb/>Et quia per 21 p 1 angulus b d g eſt maior angulo b z g:</s> <s xml:id="echoid-s30823" xml:space="preserve"> res autem <lb/>ſub maiori angulo uiſa maior uidetur per 20 huius.</s> <s xml:id="echoid-s30824" xml:space="preserve"> Videbitur er-<lb/>go augmentata quantitas lineæ b g, oculo ſuper d exiſtente, reſpe-<lb/>ctu eius, quod fuit, exiſtente centro uiſus in puncto z.</s> <s xml:id="echoid-s30825" xml:space="preserve"> Et hoc eſt <lb/>propoſitum.</s> <s xml:id="echoid-s30826" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1248" type="section" level="0" n="0"> <head xml:id="echoid-head977" xml:space="preserve" style="it">129. Augmentatæ magnitudines uidebuntur oculo appropin-<lb/>quare. Euclides 58 th. opt.</head> <p> <s xml:id="echoid-s30827" xml:space="preserve">Sit magnitudo a b, quæ uidetur:</s> <s xml:id="echoid-s30828" xml:space="preserve"> & centrum oculi ſit in puncto g:</s> <s xml:id="echoid-s30829" xml:space="preserve"> & <lb/> <anchor type="figure" xlink:label="fig-0478-03a" xlink:href="fig-0478-03"/> ducantur lineæ g a & g b:</s> <s xml:id="echoid-s30830" xml:space="preserve"> & augmentetur b a magnitu-<lb/>do ita, ut fiat magnitudo b d maior quàm b a:</s> <s xml:id="echoid-s30831" xml:space="preserve"> & ducatur <lb/>linea d g.</s> <s xml:id="echoid-s30832" xml:space="preserve"> Quia ergo angulus b g d maior eſt angulo b <lb/>g a, ut patet per 29 th.</s> <s xml:id="echoid-s30833" xml:space="preserve"> 1 huius, quia eſt maior, ſicuttotũ <lb/>ſua parte:</s> <s xml:id="echoid-s30834" xml:space="preserve"> palàm per 20 huius quoniá maior apparet ma <lb/>gnitudo b d quàm b a:</s> <s xml:id="echoid-s30835" xml:space="preserve"> maiora uerò ſe ipſis prius uiſis ui <lb/>dentur omnia poſtmodũ aucta:</s> <s xml:id="echoid-s30836" xml:space="preserve"> in eo uerò quòd maio-<lb/>ra ſunt, ſub maiori angulo uidẽtur.</s> <s xml:id="echoid-s30837" xml:space="preserve"> Et quoniá tale uiſum <lb/>uidetur idẽ ei, qđ prius uiſú eſt, & ęſtimatur ęquale ſibi <lb/>ipſi:</s> <s xml:id="echoid-s30838" xml:space="preserve"> omnium aũt æqualiũ, qđ à ꝓpinquiori uidetur, ſub <lb/>maiori angulo uidetur, ut patet per 7 huius:</s> <s xml:id="echoid-s30839" xml:space="preserve"> uirtus ergo <lb/> <pb o="177" file="0479" n="479" rhead="LIBER QVARTVS."/> diſtinctiua animæ ſentiens angulum, ſub quo fit uiſio, augmentari, & ęſtimans rem eadem, iudicat <lb/>ſe illam à propinquiori uidere.</s> <s xml:id="echoid-s30840" xml:space="preserve"> Omnes ergo auctæ magnitudines uidentur oculo appropinquare.</s> <s xml:id="echoid-s30841" xml:space="preserve"> <lb/>Et hoc eſt propoſitum.</s> <s xml:id="echoid-s30842" xml:space="preserve"/> </p> <div xml:id="echoid-div1248" type="float" level="0" n="0"> <figure xlink:label="fig-0478-03" xlink:href="fig-0478-03a"> <variables xml:id="echoid-variables523" xml:space="preserve">d a b g</variables> </figure> </div> </div> <div xml:id="echoid-div1250" type="section" level="0" n="0"> <head xml:id="echoid-head978" xml:space="preserve" style="it">130. Omnes magnitudines in eadem ſuperficie iacentes, extremis ſuis non in directo ſuo me-<lb/>dio existentibus, totalem ſuam figuram quando concauã, quando ueròfaciunt conuexam: <lb/>Euclides 59. theo. opticorum.</head> <figure> <variables xml:id="echoid-variables524" xml:space="preserve">b g d k</variables> </figure> <figure> <variables xml:id="echoid-variables525" xml:space="preserve">g d b k</variables> </figure> <p> <s xml:id="echoid-s30843" xml:space="preserve">Verbi gratia uideatur magnitudo g b d iacens in aliqua ſuperficie:</s> <s xml:id="echoid-s30844" xml:space="preserve"> <lb/>& eius punctum medium, quod eſt b, non ſit in directo ſuorum extre-<lb/>morum, ſed extra illa:</s> <s xml:id="echoid-s30845" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s30846" xml:space="preserve"> oculus in puncto k:</s> <s xml:id="echoid-s30847" xml:space="preserve"> & ducantur lineæ k g <lb/>& k b & k d.</s> <s xml:id="echoid-s30848" xml:space="preserve"> Videbitur itaq;</s> <s xml:id="echoid-s30849" xml:space="preserve"> tota figura g b d concaua, ſi eius medius <lb/>punctus ſit remotior à uiſu.</s> <s xml:id="echoid-s30850" xml:space="preserve"> Accedat uerò medius punctus rei uiſæ, <lb/>quod eſt b, àd uiſum:</s> <s xml:id="echoid-s30851" xml:space="preserve"> & fiat propinquior oculo:</s> <s xml:id="echoid-s30852" xml:space="preserve"> dico quòd uidebitur <lb/>tota magnitudo conuexa:</s> <s xml:id="echoid-s30853" xml:space="preserve"> uidet enim uiſus ſimul puncta media & ex-<lb/>trema, quorum formæ ſecundum ipſorum ſitum & diſtantiam de-<lb/>ſcribuntur in ſuperficie uiſus:</s> <s xml:id="echoid-s30854" xml:space="preserve"> & accidit uiſui paſsio, quæ accidit ex <lb/>ſuperſiciebus concauis & conuexis.</s> <s xml:id="echoid-s30855" xml:space="preserve"> Apparent ergo illa concaua & <lb/>conuexa ſecundum diuerſitatem ſitus ſui puncti medij.</s> <s xml:id="echoid-s30856" xml:space="preserve"> Et hoc eſt <lb/>propoſitum.</s> <s xml:id="echoid-s30857" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1251" type="section" level="0" n="0"> <head xml:id="echoid-head979" xml:space="preserve" style="it">131. Omnium mobilium æqueuelocium ſecundum eandem lineã <lb/>motorũ, ultra punctum cõiunctionis axium uiſualium proximum <lb/>uiſui exiſtentium, remotior a uidentur tardius moueri.</head> <p> <s xml:id="echoid-s30858" xml:space="preserve">Sint duo mobilia b & c, quæ moueantur æque uelociter:</s> <s xml:id="echoid-s30859" xml:space="preserve"> & ſit centrum uiſus a:</s> <s xml:id="echoid-s30860" xml:space="preserve"> & ſit, ut mobilia <lb/>b & c ſint ſuper lineam a g:</s> <s xml:id="echoid-s30861" xml:space="preserve"> & ſit b remotius à uiſu quàm c.</s> <s xml:id="echoid-s30862" xml:space="preserve"> Quia ergo linea a b eſt maior quàm linea <lb/> <anchor type="figure" xlink:label="fig-0479-03a" xlink:href="fig-0479-03"/> a c:</s> <s xml:id="echoid-s30863" xml:space="preserve"> palàm per 7 huius quoniam ſecund ũlineam a b ſub minori angu-<lb/>lo fit uiſio quàm ſecundum lineam a c.</s> <s xml:id="echoid-s30864" xml:space="preserve"> Viſio ergo, quæ fit in puncto <lb/>b, minus erit certa, quàm quæ fit in puncto c:</s> <s xml:id="echoid-s30865" xml:space="preserve"> & ſimiliter per eandem <lb/>7 huius ſub minori angulo uidetur ſpatium, quod in aliquo tempo-<lb/>re pertranſit mobile b, quàm illud ſpatium, quod in eodem tempo-<lb/>re pertranſit mobile c.</s> <s xml:id="echoid-s30866" xml:space="preserve"> Motus ergo mobilis b non comprehenditut <lb/>tam perfectè, ut motus mobilis c:</s> <s xml:id="echoid-s30867" xml:space="preserve"> uidebitur ergo b tardius moueri, <lb/>quia ſub minore angulo uidetur mobile b, quàm mobile c.</s> <s xml:id="echoid-s30868" xml:space="preserve"> Et ſimili-<lb/>ter ſpatium, quod pertranſit mobile b, ſub minori angulo uidebitur <lb/>quàm ſpatium, per quod in eodem tempore tranſit mobile c.</s> <s xml:id="echoid-s30869" xml:space="preserve"> Minus <lb/>ergo uidebitur ſpatium, per quod motum eſt mobile b, ſpatio, quod <lb/>pertranſit mobile c per 20 huius.</s> <s xml:id="echoid-s30870" xml:space="preserve"> Et ſi h æc mobilia ambo ſint in linea <lb/>obliqua ad uiſum extra axem, ut linea a d:</s> <s xml:id="echoid-s30871" xml:space="preserve"> tunc ambo minus uidebun <lb/>tur moueri ſuis ueris motibus:</s> <s xml:id="echoid-s30872" xml:space="preserve"> minus autem adhuc uidebitur moue-<lb/>ri b, quod eſt remotius à uiſu, quàm ipſum c.</s> <s xml:id="echoid-s30873" xml:space="preserve"> Quòd ſi ambobus ipſis <lb/>exiſtentibus in uno axe uiſuali, & aliquod ipſorum fuerit intra con <lb/>curſum axium propinquiſsimum uiſui, illud propinquius penitus obliquè uidebitur, ut per mul-<lb/>tas præcedentium patuit:</s> <s xml:id="echoid-s30874" xml:space="preserve"> unde æſtimabitur tardius moueri, licetipſum ſit propin quius uiſui.</s> <s xml:id="echoid-s30875" xml:space="preserve"> Patec <lb/>ergo propoſitum.</s> <s xml:id="echoid-s30876" xml:space="preserve"/> </p> <div xml:id="echoid-div1251" type="float" level="0" n="0"> <figure xlink:label="fig-0479-03" xlink:href="fig-0479-03a"> <variables xml:id="echoid-variables526" xml:space="preserve">d g b b c c a</variables> </figure> </div> </div> <div xml:id="echoid-div1253" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables527" xml:space="preserve">a d b e z</variables> </figure> <head xml:id="echoid-head980" xml:space="preserve" style="it">132. Omnium mobilium æqueuelocium ſuper lineas æquidistan-<lb/> tes non proximas uiſui motorum, remotior a uidentur tardius mo ueri. Euclides 56 theo. opt.</head> <p> <s xml:id="echoid-s30877" xml:space="preserve">Sint duo mobilia a & b æqueuelociter mota ſuper duas lineas æ-<lb/>quidiſtantes & æquales, quæſint a d & b e, quarum remotior à uiſu <lb/>ſit a d:</s> <s xml:id="echoid-s30878" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s30879" xml:space="preserve"> centrum uiſus punctum z:</s> <s xml:id="echoid-s30880" xml:space="preserve"> à quo ducantur lineæ z a, z b, <lb/>z d, z e.</s> <s xml:id="echoid-s30881" xml:space="preserve"> Dico quòd mobile a, quod eſt uiſui remotius, uidebitur fieri <lb/>tardius quàm mobile b, quod eſt propinquius:</s> <s xml:id="echoid-s30882" xml:space="preserve"> quia per 7 & 20 hu-<lb/>ius linea a d uidebitur minor quàm linea b c, cum tamen ſint æqua-<lb/>les.</s> <s xml:id="echoid-s30883" xml:space="preserve"> Mobile ergo a, quod in æquali tempore æquales partes lineæ a <lb/>d abſcindit, uidetur tardius moueri quàm mobile b, quod in eo-<lb/>dem tempore proportionaliter diuiſioni lineæ a d, maiores partes li-<lb/>neæ b e abſcindere uidetur, quamuis, ut patet ex hypotheſi, illæ par-<lb/>tes hinc & inde ſint æquales.</s> <s xml:id="echoid-s30884" xml:space="preserve"> Apparet ergo uelocius moueri mobi-<lb/>le b, quàm mobile a remotius uiſui.</s> <s xml:id="echoid-s30885" xml:space="preserve"> Quando enim mobile b perue-<lb/>nit ad punctum e:</s> <s xml:id="echoid-s30886" xml:space="preserve"> tunc mobile a peruenit ad punctum d, qui uide-<lb/>tur eſſe retro punctum e:</s> <s xml:id="echoid-s30887" xml:space="preserve"> & ita uidetur mobile a præpoſteratum mobili b:</s> <s xml:id="echoid-s30888" xml:space="preserve"> quia linea b e uidetur <lb/> <pb o="178" file="0480" n="480" rhead="VITELLONIS OPTICAE"/> maior quàm linea a d.</s> <s xml:id="echoid-s30889" xml:space="preserve"> Mobile ergo a ęſtimatur tardius moueri quàm mobile b.</s> <s xml:id="echoid-s30890" xml:space="preserve"> Quod eſt propoſitũ.</s> <s xml:id="echoid-s30891" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1254" type="section" level="0" n="0"> <head xml:id="echoid-head981" xml:space="preserve" style="it">133. Oculo fixo exiſtente, & axe uiſuali æqualiter tranſmutato, remotior a uiſorum æqualiter <lb/>diſtantium à priori ſitu axis, poſteriorari uidentur. Euclides 57. theo. opt.</head> <p> <s xml:id="echoid-s30892" xml:space="preserve">Sint duo uiſibilia a & g exiſtentia in duabus lineis æqualibus, quęſint a b & g d:</s> <s xml:id="echoid-s30893" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s30894" xml:space="preserve"> centrum ui-<lb/> <anchor type="figure" xlink:label="fig-0480-01a" xlink:href="fig-0480-01"/> ſus e:</s> <s xml:id="echoid-s30895" xml:space="preserve"> & ſit, ut axis uiſualis tranſeat ex puncto d ad punctum b:</s> <s xml:id="echoid-s30896" xml:space="preserve"> erit er-<lb/>go punctum b remotius à uiſu, quàm ſit punctum d.</s> <s xml:id="echoid-s30897" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s30898" xml:space="preserve"> per 7 <lb/>huius quoniam linea a b remotior à uiſu ſub minori angulo uidetur, <lb/>quàm ſua æ qualis, quæ eſt g d, propinquior uiſui.</s> <s xml:id="echoid-s30899" xml:space="preserve"> Angulus ergo d e g <lb/>eſt maior angulo b e a:</s> <s xml:id="echoid-s30900" xml:space="preserve"> ergo ք 20 huius linea g d uidetur maior quàm <lb/>linea a b.</s> <s xml:id="echoid-s30901" xml:space="preserve"> Manente itaq;</s> <s xml:id="echoid-s30902" xml:space="preserve"> oculo fixo in pũcto e, & axe uiſuali moto per <lb/>ſpatium totum, in quo ſunt uiſibilia a & g, pertranſit axis propter mi-<lb/>noritatem anguli b e a, reſpectu anguli d e g, citius uiſibile a, quàm ui-<lb/>ſibile g.</s> <s xml:id="echoid-s30903" xml:space="preserve"> Videtur ergo uiſibile a fieri poſterius uiſibili g:</s> <s xml:id="echoid-s30904" xml:space="preserve"> quoniam uiſo <lb/>g uidebitur a retrò illud.</s> <s xml:id="echoid-s30905" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s30906" xml:space="preserve"/> </p> <div xml:id="echoid-div1254" type="float" level="0" n="0"> <figure xlink:label="fig-0480-01" xlink:href="fig-0480-01a"> <variables xml:id="echoid-variables528" xml:space="preserve">b a d g e</variables> </figure> </div> </div> <div xml:id="echoid-div1256" type="section" level="0" n="0"> <head xml:id="echoid-head982" xml:space="preserve" style="it">134. Mobilium ſecundũ lineam, cui perpendiculariter inſiſtunt, <lb/>æquidistantem lineæ ab oculo ductæ, æqualiter ad ductam ab oculo <lb/>lineam motorum: illud, quod remotius à centro uiſus eſt, antecede-<lb/>dere, propinquius uerò ſequi uidetur: tranſitu uerò facto ad aliam <lb/>partem lineæ ab oculo ductæ, remotius quidẽ ſubſequi, propinquius <lb/>uerò antecedere uidetur. Euclides 52 th. opt.</head> <p> <s xml:id="echoid-s30907" xml:space="preserve">Sint æquali uelocitate mota tria mobilia, ſcilicet b g, d z, k a ſuper lineam, quæ ſit g a, cui orthogo <lb/>naliter inſiſtant ſecundũ puncta g, z, a:</s> <s xml:id="echoid-s30908" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s30909" xml:space="preserve"> mobile b g remotius à centro uiſus, quod ſit punctũ m:</s> <s xml:id="echoid-s30910" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0480-02a" xlink:href="fig-0480-02"/> & ſit mobile a k uiſui ꝓpinquius:</s> <s xml:id="echoid-s30911" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s30912" xml:space="preserve"> à uiſu à pun <lb/>cto ſcilicet m per 31 p 1 linea parallela lineę g a, quæ ſit m <lb/>l:</s> <s xml:id="echoid-s30913" xml:space="preserve"> & ducantur lineæ m g, m z, m a:</s> <s xml:id="echoid-s30914" xml:space="preserve"> producanturq́;</s> <s xml:id="echoid-s30915" xml:space="preserve"> lineæ k <lb/>a, d z, b g ad lineá m l:</s> <s xml:id="echoid-s30916" xml:space="preserve"> incidatq́;</s> <s xml:id="echoid-s30917" xml:space="preserve"> linea k a lineæ m l in pun <lb/>ctum f, & linea d z in punctũ e, & linea b g in punctũ l.</s> <s xml:id="echoid-s30918" xml:space="preserve"> Et <lb/>quoniam lineæ g a & m l ſunt parallelæ:</s> <s xml:id="echoid-s30919" xml:space="preserve"> palàm per 21 hu-<lb/>ius quoniá ad partẽ l concurrere uidentur:</s> <s xml:id="echoid-s30920" xml:space="preserve"> propinquior <lb/>igitur uidebitur g ad punctũ l, quàm z ad punctũ e, uel a <lb/>ad punctũ f.</s> <s xml:id="echoid-s30921" xml:space="preserve"> Videtur igitur præcedens b g, ſubſequés ue <lb/>rò d z, & ultimũ ipſorum k a.</s> <s xml:id="echoid-s30922" xml:space="preserve"> Protrahatur itaq;</s> <s xml:id="echoid-s30923" xml:space="preserve"> linea g a <lb/>ultra punctum a ad punctum q, & copuletur linea q m.</s> <s xml:id="echoid-s30924" xml:space="preserve"> <lb/>Quia ergo per 16 p 1 angulus m a q eſt maior angulo m z <lb/>a, & angulus m z a eſt maior angulo m g z, palàm quòd li <lb/>nea m g magis approximare uidetur ad punctũ g, quàm <lb/>linea m z ad punctum z, uel linea m a ad punctũ a:</s> <s xml:id="echoid-s30925" xml:space="preserve"> quo-<lb/>niá anguli extrinſeci maiores ſunt intrinſecis.</s> <s xml:id="echoid-s30926" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s30927" xml:space="preserve"> mo-<lb/>bile b g, quod eſt remotius, uidebitur præcedere mobi-<lb/>lia d z & k a (accedentibus ſecundum lineam rectá, quę <lb/>eſt g a, ad lineam m l æqueuelociter ipſis mobilibus k a, <lb/>d z, b g) mobile uerò k a, quod eſt poſtremum, uidetur <lb/>ſubſequi:</s> <s xml:id="echoid-s30928" xml:space="preserve"> quia magis uidetur à linea m l elongari.</s> <s xml:id="echoid-s30929" xml:space="preserve"> Ethoc <lb/>durabit, quouſq;</s> <s xml:id="echoid-s30930" xml:space="preserve"> linea g a ſuperponatur lineæ m l:</s> <s xml:id="echoid-s30931" xml:space="preserve"> tunc <lb/>ſecun dum lineam rectam m l mobile k a propinquius ui <lb/>ſui uidetur quàm alia, & maius per 7 & 20 huius.</s> <s xml:id="echoid-s30932" xml:space="preserve"> Facto <lb/>autem tranſitu ultra lineam m l, ita ut mobilia, quæ fue-<lb/>runt prius dextra uiſui, fiant ſiniſtra, uel econtrariò:</s> <s xml:id="echoid-s30933" xml:space="preserve"> tunc <lb/>mobile remotius uiſui uidebitur ſequi, & propinquius præcedere propter eandem cauſſam, quam <lb/>præmiſimus.</s> <s xml:id="echoid-s30934" xml:space="preserve"> Et ut hoc exemplariter pateat, ſit ut mobile b g, quod eſt remotius à centro uiſus m, <lb/>pertranſita linea m l, perueniat ad locum lineæ n x, & mobile d z ad locum lineæ p r, & mobile k a, <lb/>quod eſt propinquius uiſui, perueniat ad locum lineæ s t.</s> <s xml:id="echoid-s30935" xml:space="preserve"> Ducantur quoq;</s> <s xml:id="echoid-s30936" xml:space="preserve"> à centro uiſus ad puncta <lb/>n, p, s lineę m n, m p, m s.</s> <s xml:id="echoid-s30937" xml:space="preserve"> Videbitur ergo mobile n x ſubſequi duo alia mobilia, ideo, quòd, ſicut præ-<lb/>miſſum eſt, linea n x magis approximat ad punctum l, quàm linea p r ad punctum e, uel quàm linea <lb/>s t ad punctũ f.</s> <s xml:id="echoid-s30938" xml:space="preserve"> Igitur mobile b g, quod fuerat prius præcedens, cum peruenerit ad lineam n x, uide-<lb/>bitur ſequi:</s> <s xml:id="echoid-s30939" xml:space="preserve"> & linea a k, quæ fuerat prius ſubſequẽs, ſuper lineam s t uidebitur præcedere.</s> <s xml:id="echoid-s30940" xml:space="preserve"> Et ſic iſto-<lb/>rum mobilium mutato ſitu, motus uidebitur diuerſus.</s> <s xml:id="echoid-s30941" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s30942" xml:space="preserve"/> </p> <div xml:id="echoid-div1256" type="float" level="0" n="0"> <figure xlink:label="fig-0480-02" xlink:href="fig-0480-02a"> <variables xml:id="echoid-variables529" xml:space="preserve">b g l n x d z e p r k a f s t q q m</variables> </figure> </div> </div> <div xml:id="echoid-div1258" type="section" level="0" n="0"> <head xml:id="echoid-head983" xml:space="preserve" style="it">135. Pluribus mobilibus non æquè uelociter ad eandem partem motis, ad quam mouetur & <lb/>uiſus, æqueuelocia uiſui, quieſcere: tardior a uerò cõtrà moueri: & celeriora antecedere uidebun-<lb/>cur. Euclides 53 th. opt.</head> <pb o="179" file="0481" n="481" rhead="LIBER QVARTVS."/> <p> <s xml:id="echoid-s30943" xml:space="preserve">Sint tria mobilia b, c, d:</s> <s xml:id="echoid-s30944" xml:space="preserve"> & ſit centrum oculi punctũ a:</s> <s xml:id="echoid-s30945" xml:space="preserve"> ſit autem inter hæc mobilia, b tardiſsimum, <lb/>& c æqueuelox uiſui, d uerò ſit uelocius quàm c:</s> <s xml:id="echoid-s30946" xml:space="preserve"> & omnia moueantur ad eandem partem uniuerſi:</s> <s xml:id="echoid-s30947" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0481-01a" xlink:href="fig-0481-01"/> à centro quoq;</s> <s xml:id="echoid-s30948" xml:space="preserve"> uiſus a ducantur lineæ a b, a c, a d.</s> <s xml:id="echoid-s30949" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s30950" xml:space="preserve"> motus fuerit <lb/>oculus a:</s> <s xml:id="echoid-s30951" xml:space="preserve"> tunc mobile c, quod eſt æqueuelox oculo, æqualiter motum eſt <lb/>cum oculo:</s> <s xml:id="echoid-s30952" xml:space="preserve"> non ergo mutat ſitum reſpectu oculi:</s> <s xml:id="echoid-s30953" xml:space="preserve"> ergo per 112 huius ipſum <lb/>quieſcere uidebitur.</s> <s xml:id="echoid-s30954" xml:space="preserve"> Mobile uerò b, quia eſt tardiſsimum, patet quòd mo <lb/>to uiſu ipſum eſt pertranſitum per motum uelociorẽ ipſius uiſus:</s> <s xml:id="echoid-s30955" xml:space="preserve"> & quia <lb/>mobile c uidetur quieſcere, & mobile b ſemper magis & magis remoue-<lb/>tur à mobili c, propter exceſſum uelocitatis mobilis c ſuper mobile b:</s> <s xml:id="echoid-s30956" xml:space="preserve"> ui-<lb/>detur ergo mobile b ad partem contrariam moueri.</s> <s xml:id="echoid-s30957" xml:space="preserve"> Mobile uerò d, quia <lb/>uelociſsimum eſt, præcedit mobile c, & ipſum uiſum:</s> <s xml:id="echoid-s30958" xml:space="preserve"> & ſemper fit plus di <lb/>ſtans à uiſu.</s> <s xml:id="echoid-s30959" xml:space="preserve"> Videtur ergo præcedere.</s> <s xml:id="echoid-s30960" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s30961" xml:space="preserve"> propoſitum.</s> <s xml:id="echoid-s30962" xml:space="preserve"/> </p> <div xml:id="echoid-div1258" type="float" level="0" n="0"> <figure xlink:label="fig-0481-01" xlink:href="fig-0481-01a"> <variables xml:id="echoid-variables530" xml:space="preserve">d c b a</variables> </figure> </div> </div> <div xml:id="echoid-div1260" type="section" level="0" n="0"> <head xml:id="echoid-head984" xml:space="preserve" style="it">136. Si aliquibus mobilibus æqueuelociter motis uiſis apparet ali-<lb/>quid immotum: illud uidebitur adpartem contrariam alijs mobilibus <lb/>moueri. Euclides 54 theo. opt.</head> <p> <s xml:id="echoid-s30963" xml:space="preserve">Sint enim duo mobilia b & d, quęmoueantur æqueuelociter ad unam <lb/>partem quamcunq,:</s> <s xml:id="echoid-s30964" xml:space="preserve"> & ſit c aliquid non motum:</s> <s xml:id="echoid-s30965" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s30966" xml:space="preserve"> centrum uiſus a:</s> <s xml:id="echoid-s30967" xml:space="preserve"> & ducantur à centro uiſus li-<lb/>neæ a b, a c, a d.</s> <s xml:id="echoid-s30968" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s30969" xml:space="preserve"> mobile b mouetur ad aliquem terminum:</s> <s xml:id="echoid-s30970" xml:space="preserve"> palàm quoniam ipſum fit pro-<lb/> <anchor type="figure" xlink:label="fig-0481-02a" xlink:href="fig-0481-02"/> pinquius ad illum quàm corpus c, quod non mouetur:</s> <s xml:id="echoid-s30971" xml:space="preserve"> ſed & mobile <lb/>d æqueuelociter motum eſt mobili b:</s> <s xml:id="echoid-s30972" xml:space="preserve"> uidentur ergo mobilia b & d nõ <lb/>mutare ſitum adinuicem:</s> <s xml:id="echoid-s30973" xml:space="preserve"> corpus uerò c mutat ſitum reſpectu illorum <lb/>amborum mobilium:</s> <s xml:id="echoid-s30974" xml:space="preserve"> uidetur ergo c ad partem illis contrariam moue <lb/>ri, quod patet ք 110 huius.</s> <s xml:id="echoid-s30975" xml:space="preserve"> Et hoc eſt ꝓpoſitũ.</s> <s xml:id="echoid-s30976" xml:space="preserve"> Et ex hoc apparet, qua-<lb/>re motis uelociter nubibus luna uiſa uidetur ad partem contrariam <lb/>moueri.</s> <s xml:id="echoid-s30977" xml:space="preserve"> Quia enim partes nubium æqueuelociter mouentur, ut b & <lb/>d:</s> <s xml:id="echoid-s30978" xml:space="preserve"> lunæ uerò motus proprius à uiſu propter remotionẽ in paruo tem-<lb/>pore non percipitur, ideo uidetur luna, ut mobile c, ad partem con-<lb/>triam moueri.</s> <s xml:id="echoid-s30979" xml:space="preserve"/> </p> <div xml:id="echoid-div1260" type="float" level="0" n="0"> <figure xlink:label="fig-0481-02" xlink:href="fig-0481-02a"> <variables xml:id="echoid-variables531" xml:space="preserve">c d b a</variables> </figure> </div> </div> <div xml:id="echoid-div1262" type="section" level="0" n="0"> <head xml:id="echoid-head985" xml:space="preserve" style="it">137 Puncta ſignata in re circulariter mota, uidentur circuli: & <lb/>lineæ ſuperficies rotundæ.</head> <p> <s xml:id="echoid-s30980" xml:space="preserve">Cum enim talia mobilia ſic ſignata mouentur circulariter, quodli-<lb/>bet ſuorum punctorum motu ſuo deſcribit circulum:</s> <s xml:id="echoid-s30981" xml:space="preserve"> quoniã quodli-<lb/>bet punctum non figitur in eodem loco tempore ſenſibili, ſed in par-<lb/>uo tempore circumgyrat totam circumferentiam, ſuper quam uolui-<lb/>tur:</s> <s xml:id="echoid-s30982" xml:space="preserve"> peruenit ergo tunc forma puncti ſignati in ſuperficiem uiſus per modum circumferentiæ circu <lb/>li.</s> <s xml:id="echoid-s30983" xml:space="preserve"> Quoniam enim motus circularis eſt totus unus, non diuidens tempus:</s> <s xml:id="echoid-s30984" xml:space="preserve"> non poteſt uiſus compre-<lb/>hendere formam puncti ſignati niſi ſecundum circumſerentiam circuli:</s> <s xml:id="echoid-s30985" xml:space="preserve"> in minimo enim tempore <lb/>comprehendit colorem illius puncti circumgyratũ:</s> <s xml:id="echoid-s30986" xml:space="preserve"> & ſi plura ſunt puncta ſecun dũ ordinem unius <lb/>ſub altero ſignata, plures uidebuntur circuli ſubalternatim & ordinatè cõtenti.</s> <s xml:id="echoid-s30987" xml:space="preserve"> Ethic eſt ludus pue <lb/>rorum in trochis ſuper planas ſuperficies circulariter exagitatis:</s> <s xml:id="echoid-s30988" xml:space="preserve"> quoniã quando trochus fuerit cir-<lb/>cumgyratus motu forti, & aſpexerit quis ipſum, ſi unus eſt punctus in ipſo ſignatus, uidebitur circu <lb/>lus:</s> <s xml:id="echoid-s30989" xml:space="preserve"> & ſi plura ſunt pũcta ab inuicẽ diſtãtia, uidebuntur plures circuli ęquidiſtantes, & circa idẽ cen <lb/>trum:</s> <s xml:id="echoid-s30990" xml:space="preserve"> & uidebit uiſus differentiã colorum cuiuslibet illorũ circulorũ.</s> <s xml:id="echoid-s30991" xml:space="preserve"> Et ſi plura puncta diuerſorũ <lb/>colorũ ſibi ad inuicẽ approximantur, cóprehẽdet uiſus oẽs illorum punctorũ colores quaſi unũ co-<lb/>lorẽ, diuerſum ab omnibus coloribus, qui ſunt in illis punctis, quaſi ſit color cõpoſitus ex omnibus <lb/>coloribus illorũ punctorũ, & no cõprehendet lineationẽ neq;</s> <s xml:id="echoid-s30992" xml:space="preserve"> diuerſitatẽ colorũ.</s> <s xml:id="echoid-s30993" xml:space="preserve"> Et ſi motus fuerit <lb/>ualde ſortis, cõprehendet uiſus illud corpus motũ, quaſi quieſcẽs & circulariter figuratũ:</s> <s xml:id="echoid-s30994" xml:space="preserve"> ideo quòd <lb/>nullũ illius corporis pũctũ figitur in loco tẽpore ſenſibili, ſed in minimo tẽpore gyratur tota circũfe <lb/>rentia, ſup qua reuoluitur.</s> <s xml:id="echoid-s30995" xml:space="preserve"> Et ſimiliter mota linea uidebitur ſecũdũ lineę lõgitudinẽ latitudo cuiuſ-<lb/>dam ſuքſiciei rotundę deſcripta in ſuperfic e ipſius uiſus:</s> <s xml:id="echoid-s30996" xml:space="preserve"> & ſi linea illa fuerit colorata:</s> <s xml:id="echoid-s30997" xml:space="preserve"> tunc propter <lb/>motus uelocitatẽ, motus facit totã ſuperficiẽ rotundá apparere coloratam.</s> <s xml:id="echoid-s30998" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s30999" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1263" type="section" level="0" n="0"> <head xml:id="echoid-head986" xml:space="preserve" style="it">138. In motus & quietis uiſione error accidit uirtuti diſtinctiue ex intemperata diſpoſitione <lb/>octo circumſtantiarum cuiuslibet rei uiſæ. Alhazen 28. 39. 49. 55. 60. 65. 67. 70 n 3.</head> <p> <s xml:id="echoid-s31000" xml:space="preserve">Ex intemperata enim luce accidit error in uiſione motus & quietis.</s> <s xml:id="echoid-s31001" xml:space="preserve"> Si enim de nocte cõprehen-<lb/>derit uiſus hominẽ ante aliquod nemus, fortè occultabitur ei diſtantia hominis ad nemus.</s> <s xml:id="echoid-s31002" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s31003" xml:space="preserve"> <lb/>uidens moueatur uerſus hominẽ uiſum, quantò magis ad illũ acceſſerit, tantò diſtantiá illam cer-<lb/>tius uidebit:</s> <s xml:id="echoid-s31004" xml:space="preserve"> unde cum prius ſimul unà eũ nemore appareret ei homo uiſus, & quantò ad eum plus <lb/>accedit, tantò plus uidetur à nemore remotus:</s> <s xml:id="echoid-s31005" xml:space="preserve"> & certũ eſt ei nemus immotũ remanere:</s> <s xml:id="echoid-s31006" xml:space="preserve"> æſtimabit <lb/>ergo hominẽ ad partem contrariã nemoris incedere, licet ueritas ſit ipſum hominem uiſum immo-<lb/>tum & quietũ eſſe.</s> <s xml:id="echoid-s31007" xml:space="preserve"> Et etiá ſi homo de nocte uiſus non plenè cõprehenditur, qui modicũ moueatur, <lb/>nõ diſcernetur motus eius, & uidebitur quieſcens:</s> <s xml:id="echoid-s31008" xml:space="preserve"> hi aũt errores non acciderent in temperata luce.</s> <s xml:id="echoid-s31009" xml:space="preserve"> <lb/> <pb o="180" file="0482" n="482" rhead="VITELLONIS OPTICAE"/> Exintemperata etiam remotione error accidit in uiſione motus & quietis.</s> <s xml:id="echoid-s31010" xml:space="preserve"> Si quis enim ad partẽ, in <lb/>qua lunam aut ſolem aut ſtellã aliquã uiderit, moueatur, cum poſt plurimũ motum lunã ante ſe uide <lb/>rit elongatã nõ minus ꝗ̃ in principio ſui motus, æſtimat ipſam lunã ad eandem partẽ ſecum moueri, <lb/>& ab eo recedere, & ob hoc elongatiões durare:</s> <s xml:id="echoid-s31011" xml:space="preserve"> & euenit hoc etiã in luna ad partẽ contrariã prope-<lb/>rante.</s> <s xml:id="echoid-s31012" xml:space="preserve"> Acciditq́ hic errorideo, quia notũ eſt homini quòd in his naturis inferioribus exiſtentibus <lb/>duobus corporibus, quorũ unum moueatur in partem aliquã, ſi tunc permanſerit identitas ſitus re-<lb/>ſpectu alterius corporis, tunc neceſſe eſt etiã aliud corpus in eandẽ partẽ æ quali motu fuiſſe motũ:</s> <s xml:id="echoid-s31013" xml:space="preserve"> <lb/>hoc tamen non oporter ſic æſtimari in luna uel ſtellis, quoniã magnitudo uiæ, quã peragit quis mo-<lb/>tu ſuo, non eſt proportionalis magnitudini corporis lunæ uel alterius ſtellæ:</s> <s xml:id="echoid-s31014" xml:space="preserve"> ergo neq;</s> <s xml:id="echoid-s31015" xml:space="preserve"> exceſſus po <lb/>ſtremę propin quitatis ad ſtellam ſuper primã propinquitatẽ eſt ſenſibilis, reſpectu totalis remotio-<lb/>nis.</s> <s xml:id="echoid-s31016" xml:space="preserve"> Idem etiã error accidit in motu nubium:</s> <s xml:id="echoid-s31017" xml:space="preserve"> creditur enim uelociſsimus eſſe motus lunæ, quia par-<lb/>tes nubiũ, per quas uidetur luna, ſubitò mutantur, & luna nunc cũ his partibus nubiũ, nunc cum il-<lb/>lis uidetur eſſe ſita:</s> <s xml:id="echoid-s31018" xml:space="preserve"> & quia luna eſt corpus luminoſum uiſibilius quàm nubes, æſtimatur luna moue <lb/>ri motu, quo ſecundũ ueritatẽ nõ mouetur.</s> <s xml:id="echoid-s31019" xml:space="preserve"> Similiter etiá accidit error in quiete:</s> <s xml:id="echoid-s31020" xml:space="preserve"> aliquis enim à lon-<lb/>gè uiſus non ueloci motu motus, quieſcere uidetur:</s> <s xml:id="echoid-s31021" xml:space="preserve">& propter hoc planetas credimus immotos, li-<lb/>cet uelociter moueãtur.</s> <s xml:id="echoid-s31022" xml:space="preserve"> Vię enim, quas incedunt in tẽpore paruo, nõ ſunt perceptibiles uiſui à tãta <lb/>remotione:</s> <s xml:id="echoid-s31023" xml:space="preserve"> unde durante ſitu ipſorum, reſpectu uidentis identitate quieſcere putãtur.</s> <s xml:id="echoid-s31024" xml:space="preserve"> Similiter e-<lb/>tiam accidit hic error, ſi in eadem linea uiſuali uel axe corpus aliquod uiſum uel à uiſu moueatur.</s> <s xml:id="echoid-s31025" xml:space="preserve"> <lb/>Tunc enim niſi motus eius fuerit ualde fortis, putabitur immotũ:</s> <s xml:id="echoid-s31026" xml:space="preserve"> quia non percipitur an partes uel <lb/>ipſum totũ ſe aliter habeat nũc ꝗ̃ prius:</s> <s xml:id="echoid-s31027" xml:space="preserve"> uia enim, qua incedit, eſt imperceptibilis à tanta remotione.</s> <s xml:id="echoid-s31028" xml:space="preserve"> <lb/>Exintemperata etiã ſitus oppoſitionis obliquitate accidit error uirtuti diſtin ctiuę in pręmiſſorũ ui <lb/>ſione:</s> <s xml:id="echoid-s31029" xml:space="preserve"> unde aliquo uelociter nauigáte in flumine, & obliquè inſpiciẽte arbores in ripa fluminis:</s> <s xml:id="echoid-s31030" xml:space="preserve"> tũc <lb/>arbores ab axe uiſuali multum elongatas æſtimabit moueri, illæ uerò arbores, quibus axis uiſualis <lb/>incidet, quieſcere uidebuntur.</s> <s xml:id="echoid-s31031" xml:space="preserve"> Similiter rota aliqua mota, ut molendini obliquè uiſa uidetur quie-<lb/>ſcere.</s> <s xml:id="echoid-s31032" xml:space="preserve"> Eſt autem hic error propter ſolam obliquationem ſitus rei ad uiſum, quoniá talis rota direcè <lb/>intuita moueri uidetur.</s> <s xml:id="echoid-s31033" xml:space="preserve"> Exintemperata etiá magnitudine accidit error in uiſione præmiſſorum.</s> <s xml:id="echoid-s31034" xml:space="preserve"> Si <lb/>enim moueantur duo, quorum unum ſit paululũ uelocius alio, putabit uidens eſſe æ qualem ipſorũ <lb/>motum, cum inſenſibile ſit unius motus ſuper alium excrementum, & ſimiliter quantitas ex-<lb/>ceſſus uiæ, quam tranſit alius, imperceptibilis eſt uiſui:</s> <s xml:id="echoid-s31035" xml:space="preserve"> unde iudicatur æ qualitas motuũ & uiarum:</s> <s xml:id="echoid-s31036" xml:space="preserve"> <lb/>& ſimiliter res parua mota fortè æſtimabitur non moueri, etiam ſi diſtantia à uiſu fuerit tẽperata.</s> <s xml:id="echoid-s31037" xml:space="preserve"> Ex <lb/>intemperata etiam raritate accidit error in præmiſsis.</s> <s xml:id="echoid-s31038" xml:space="preserve"> Si enim in aere nubiloſo obſcuro duo corpo-<lb/>ra moueantur, quorũ unum alio paululum uelocius moueatur:</s> <s xml:id="echoid-s31039" xml:space="preserve"> iudica buntur forſitan æ quales ipſo-<lb/>rum motus, cum propter intemperiem diaphanitatis aeris diſcerni nõ poſsit motus unius ad motũ <lb/>alterius exceſſus:</s> <s xml:id="echoid-s31040" xml:space="preserve"> nec enim tunc percipitur à uiſu exceſſus uiæ pertranſitæ ab uno, à uia pertranſitæ <lb/>ab alio.</s> <s xml:id="echoid-s31041" xml:space="preserve"> Similiter etiam in tali aere à longitudine media, non tamen parua, ſi quis uideat aquã fluen-<lb/>tem, aut iudicabit eam immotá, aut ſi fuerit fortis eius fluxus, æſtimabitur minus mota quàm mo-<lb/>neatur.</s> <s xml:id="echoid-s31042" xml:space="preserve"> Exintemperata etiam temporis diſpoſitione ſit maximus error in uiſione motus & quietis, <lb/>quæ perſe tempore menſurátur.</s> <s xml:id="echoid-s31043" xml:space="preserve"> Cum enim duorũ mobilium unũ paulò uelocius alio mouebitur:</s> <s xml:id="echoid-s31044" xml:space="preserve"> <lb/>tunc motus in tẽpore modico cõprehenſi æ quales iudicabuntur:</s> <s xml:id="echoid-s31045" xml:space="preserve"> quia nõ eſt tam ſubitò cóprehenſi <lb/>bilis ipſorũ exceſſus:</s> <s xml:id="echoid-s31046" xml:space="preserve"> & ſi aliquid tardè moueatur, hoc in tẽpore modico inſpectũ nõ uidebitur mo-<lb/>ueri:</s> <s xml:id="echoid-s31047" xml:space="preserve"> quoniá uia, per quá mouetur in modico tẽpore, eſt imperceptibilis uiſui propter ſui paruitatẽ:</s> <s xml:id="echoid-s31048" xml:space="preserve"> <lb/>ſed & uelociſsimè motum circulariter & in eodem loco manens, ut trochus, non æſtimatur moue-<lb/>ri:</s> <s xml:id="echoid-s31049" xml:space="preserve"> locus enim trochi non mutatur, & partes uelociſsimè redeunt ad priorem ſitum.</s> <s xml:id="echoid-s31050" xml:space="preserve"> Ex intemperan <lb/>tia etiam diſpoſitionis uiſus accidit error uiſioni præmiſſorum.</s> <s xml:id="echoid-s31051" xml:space="preserve"> Cum enim quis ſæpius in circuitu <lb/>fuerit reuolutus, & pòſt quieſcit:</s> <s xml:id="echoid-s31052" xml:space="preserve"> tunc putat quòd uicini parietes moueantur:</s> <s xml:id="echoid-s31053" xml:space="preserve"> ideo quia ſpiritus uiſi <lb/>biles interius moti diſcurrunt ex motu corporis ipſius facto, nec ſtatim quieſcente corpore exterio <lb/>ri ſpiritus intrinſecus moti quieſcunt, eò quòd leuiores corpore groſſo, ſuntillo mobiliores, & mi <lb/>nor uirtus animæ mouet illos, illi autem moti formas motas uirtuti diſtinctiuæ repręſentant:</s> <s xml:id="echoid-s31054" xml:space="preserve"> uiden <lb/>tur ergo omnia moueri, quorum formæ motis ſpiritibus uirtuti animæ offeruntur etiam poſt quie-<lb/>tem ipſius uidentis.</s> <s xml:id="echoid-s31055" xml:space="preserve"> Ethuius ſimile eſt etiam in alijs motis:</s> <s xml:id="echoid-s31056" xml:space="preserve"> trochus enim diu poſt quietem manus <lb/>motricis mouetur, & non quieſcit, quouſq;</s> <s xml:id="echoid-s31057" xml:space="preserve"> uirtus influxa ſibi deſinit mouere.</s> <s xml:id="echoid-s31058" xml:space="preserve"> Eſt etiam quædam <lb/>corporis & oculorum infirmitas, in qua uidentur omnia circumuolui.</s> <s xml:id="echoid-s31059" xml:space="preserve"> Si etiam corpus ſimilium <lb/>partiũ uoluatur tardè, ut accidit in quibuſdam rotis horologiorũ:</s> <s xml:id="echoid-s31060" xml:space="preserve"> tunc uiſus debilis non percipiet <lb/>motũ eius, neq;</s> <s xml:id="echoid-s31061" xml:space="preserve"> etiam ſanus uiſus percipiet motum parui temporis.</s> <s xml:id="echoid-s31062" xml:space="preserve"> Si uerò ſit corpus diſsimilium <lb/>partium, ut in rotis molendini:</s> <s xml:id="echoid-s31063" xml:space="preserve"> tunc fortè etiam uiſus debilis comprehendet motũ, niſi ualde feſti-<lb/>na fuerit rotæ reuolutio:</s> <s xml:id="echoid-s31064" xml:space="preserve"> quia propter uelocitatem motus fortè diſsimilitudo partium rotæ non po <lb/>terit comprehendi.</s> <s xml:id="echoid-s31065" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s31066" xml:space="preserve"> illud, quod proponebatur.</s> <s xml:id="echoid-s31067" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1264" type="section" level="0" n="0"> <head xml:id="echoid-head987" xml:space="preserve" style="it">139. Alperitas comprehenditur à uiſu ex cõprehenſione lucis ſuperficiei corporis aſperi inci-<lb/>dentis, per quã comprehenditur diuerſitas ſituũ partium ſuperficiei corporis. Alhazen 53 n 2.</head> <p> <s xml:id="echoid-s31068" xml:space="preserve">Cum aſperitas ſit diuerſitas ſitus partiũ ſuperficiei corporis, palàm per 11 th.</s> <s xml:id="echoid-s31069" xml:space="preserve"> 2 huius, quòd partes <lb/>præeminentes umbram faciunt, quando luxinciderit ſuperficiei illius corporis:</s> <s xml:id="echoid-s31070" xml:space="preserve"> partes ergo præe-<lb/>minentes erunt manifeſtæ luci & diſcoopertæ, & in partes profundas perueniũt umbræ, permiſcen <lb/>tes lucem illis partibus incidentem.</s> <s xml:id="echoid-s31071" xml:space="preserve"> Diuerſificabitur ergo forma lucis in ſuperficie illius corporis, <lb/> <pb o="181" file="0483" n="483" rhead="LIBER QVARTVS."/> quod non accidit in ſuperficie plana:</s> <s xml:id="echoid-s31072" xml:space="preserve"> eius enim partes ſunt conſimilis ſitus, & fit forma lucis in o-<lb/>mnibus ſuis partibus conſimilis.</s> <s xml:id="echoid-s31073" xml:space="preserve"> Viſus itaq;</s> <s xml:id="echoid-s31074" xml:space="preserve"> cognoſcit formam lucis in ſuperficiebus aſperis & pla-<lb/>nis diuerſam, propter frequentationem uiſionis ſuperficierum aſperarum & planarum:</s> <s xml:id="echoid-s31075" xml:space="preserve"> & ſecun-<lb/>dum hoc dijudicat aſperitatem ſuperficierum uel planitiem in corporibus aſperis quibuſcũq;</s> <s xml:id="echoid-s31076" xml:space="preserve">. Sed <lb/>ſi ſuperficiei aſperæ partes fuerint ualde præeminentes, poteſt etiam uiſus comprehendere præ e-<lb/>minentiam illarum partium ex comprehenſione diſtantiæ, quę eſt inter partes:</s> <s xml:id="echoid-s31077" xml:space="preserve"> & ſic ex comprehen <lb/>ſione diuerſitatis ſitus partium ſuperficiei corporis aſperi comprehendet etiam aſperitatem illius:</s> <s xml:id="echoid-s31078" xml:space="preserve"> <lb/>& erit etiam lux in illa aſperitate maximę diuerſitatis, quoniam maioribus umbris diſtinctè permi-<lb/>ſcetur, & ex diuerſitate formæ lucis uidebitur diſtantia partium, & diuerſitas ſitus earum:</s> <s xml:id="echoid-s31079" xml:space="preserve"> & ex hoc <lb/>uidebitur corporis aſperitas.</s> <s xml:id="echoid-s31080" xml:space="preserve"> Quòd ſi præeminentiæ partium ſuperficiei rei uiſæ fuerint paruæ ual-<lb/>de, non comprehendet uiſus illam aſperitatẽ corporis, niſi cum multa appropinquatione intuitus.</s> <s xml:id="echoid-s31081" xml:space="preserve"> <lb/>Sic ergo per diuerſitatem lucis ſuperficiebus corporum aſperorum incidentis, & ex cõſequenti per <lb/>eomprehenſionem diuerſitatis ſituum partium ſuperficiei corporis, aſperitas comprehenditur à ui <lb/>ſu.</s> <s xml:id="echoid-s31082" xml:space="preserve"> Pater ergo propoſitum.</s> <s xml:id="echoid-s31083" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1265" type="section" level="0" n="0"> <head xml:id="echoid-head988" xml:space="preserve" style="it">140. Lenitas ſiue planities comprehenditur à uiſu comprehenſione lucis ſuperficiei lenis cor-<lb/>poris incidentis, tum etiam per ſuarum partium omnimodam æqualitatem. Alhazen 54 n 2.</head> <p> <s xml:id="echoid-s31084" xml:space="preserve">Quia enim lenitas eſt æ qualitas ſitus partium ſuperficiei, patet quòd partes corporis lenis ſunt <lb/>conſimilis ſitus:</s> <s xml:id="echoid-s31085" xml:space="preserve"> lux ergo illis corporibus incidens fit conſimilis & nullis umbris permixta:</s> <s xml:id="echoid-s31086" xml:space="preserve"> unde e-<lb/>tiam corporis terſitudo ſiue politio, quæ eſt quædam lenitas uel planities, comprehenditur à uifu <lb/>ex ſcintillatione lucis in ſuperficie illius corporis, & ex ſitu, ſecundum quẽ reflectitur lux ad uiſum, <lb/>uel ad aliud corpus obiectum.</s> <s xml:id="echoid-s31087" xml:space="preserve"> Comprehendit etiam uiſus quandoq;</s> <s xml:id="echoid-s31088" xml:space="preserve"> planitiem per intuitum dili-<lb/>gentem, per quem comprehendit partium ſuperficiei uiſæ æqualitatem:</s> <s xml:id="echoid-s31089" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s31090" xml:space="preserve"> etiam compre-<lb/>hendit ipſam planitiem ſuperpoſito uiſu in una parte illius ſuperficiei uiſæ:</s> <s xml:id="echoid-s31091" xml:space="preserve"> & cum formæ partium <lb/>extremarũ illius ſuperficiei, quæ ſunt remotiores à uiſu, ſecundum lineas rectas perueniunt ad ui-<lb/>ſum in ipſa ſuperficie productas:</s> <s xml:id="echoid-s31092" xml:space="preserve"> tunc uiſus ſic ipſius ſuperficiei planitiem comprehendit.</s> <s xml:id="echoid-s31093" xml:space="preserve"> Patet <lb/>ergo propoſitum.</s> <s xml:id="echoid-s31094" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1266" type="section" level="0" n="0"> <head xml:id="echoid-head989" xml:space="preserve" style="it">141. In aſperitatis & lenitatis uiſione error accidit uirtuti diſtinctiuæ ex intemperata diſpo <lb/>ſitione octo circunſtantiarum cuiuslibet rei uiſæ. Alhazen 29. 40. 50. 56. 61. 65. 68. 71. n 3.</head> <p> <s xml:id="echoid-s31095" xml:space="preserve">Ex debilitate enim lucis error accidit uniſioniaſperitatis & lenitatis:</s> <s xml:id="echoid-s31096" xml:space="preserve"> quia de nocte uiſa aſperitas <lb/>fortè iudicabitur lenitas aut econuerſo, ſecundum qualitatem rei uiſæ.</s> <s xml:id="echoid-s31097" xml:space="preserve"> Et etiam cum à capillis ni-<lb/>gris lotis fit lucis reflexio, æſtimantur illi capilli ſummè plani, cum ſint ſecundum ueritatem aſpe-<lb/>ri, eò quòd eſt in eis diuerſitas & diſtantia innumeroſa.</s> <s xml:id="echoid-s31098" xml:space="preserve"> Superflua etiam longitudo diſtantiæ erro-<lb/>rem in gerit uiſioni aſperitatis & lenitatis:</s> <s xml:id="echoid-s31099" xml:space="preserve"> unde in pictis capillis uel pilis alicuius pictæ imaginis <lb/>propter longitudinem diſtantiæ æſtimatur aſperitas:</s> <s xml:id="echoid-s31100" xml:space="preserve"> ideo quia ſenſus conſueuit accipere aſperita-<lb/>tem in capillis ueris:</s> <s xml:id="echoid-s31101" xml:space="preserve"> & idem accidit in rugis ueſtium depictarum, quæ propter diſtantiam uiden-<lb/>tur replicatæ, cum ſint in una ſuperficie conſtitutæ.</s> <s xml:id="echoid-s31102" xml:space="preserve"> Similiter etiam ſi à magna diſtantia opponatur <lb/>uiſui corpus, in quo eſt modica aſperitas, putabitur lenitas:</s> <s xml:id="echoid-s31103" xml:space="preserve"> quia à tali diſtantia non poteſt diſcerni <lb/>diuerſitas partium aut proiectio umbrę partium eminentium ſuper depreſſas:</s> <s xml:id="echoid-s31104" xml:space="preserve"> unde iudicatur in eo <lb/>lenitas.</s> <s xml:id="echoid-s31105" xml:space="preserve"> Exintemperantia etiam ſitus fit error in uiſione aſperitatis & lenitatis.</s> <s xml:id="echoid-s31106" xml:space="preserve"> Si enim à capillis <lb/>depictis alicuius pictæ imaginis fiat obliqua reflexio lucis, utpote uiſu non exiſtente in loco refle-<lb/>xionis, fiet comprehenſio aſperitatis capillorum, cum non ſit niſi lenitas in illis:</s> <s xml:id="echoid-s31107" xml:space="preserve">hoc autem non ac-<lb/>cideret uiſui directè lucem reflexam excipienti:</s> <s xml:id="echoid-s31108" xml:space="preserve"> quia tunc uera lenitas appareret.</s> <s xml:id="echoid-s31109" xml:space="preserve"> Cum etiam cor-<lb/>pus aliquod, in quo eſt modica aſperitas, obliquatũ fuerit ab axe uiſuali:</s> <s xml:id="echoid-s31110" xml:space="preserve"> tunc apparebit lene:</s> <s xml:id="echoid-s31111" xml:space="preserve"> quod <lb/>ſi directè uiſui opponeretur, ſua aſperitas uiſui ſe offerret.</s> <s xml:id="echoid-s31112" xml:space="preserve"> Ex intemperantia etiam magnitudinis <lb/>error accidit uiſioni præ miſſorum:</s> <s xml:id="echoid-s31113" xml:space="preserve"> cum enim occurrerit uiſui res multum parua, uidebitur fortè le-<lb/>nitas, ubi eſt aſperitas, aut econuerſo:</s> <s xml:id="echoid-s31114" xml:space="preserve"> non enim comprehenditur præeminentia partium aliarum <lb/>ſuper alias propter nimiam corporis paruitatem.</s> <s xml:id="echoid-s31115" xml:space="preserve"> Ex ſoliditatis etiam intemperantia error acci-<lb/>dit uiſioni præmiſſorũ.</s> <s xml:id="echoid-s31116" xml:space="preserve"> Si enim in corpore multũ raro fuerit aſperitas nõ magna, putabitur fortè le-<lb/>nitas:</s> <s xml:id="echoid-s31117" xml:space="preserve"> & ſi totum fuerit lene, & trans ipſum uideatur corpus aſperum aut diuerſorum colorum:</s> <s xml:id="echoid-s31118" xml:space="preserve"> æſti-<lb/>mabitur hoc corpus, quod eſt rarum & lene, eſſe aſperum:</s> <s xml:id="echoid-s31119" xml:space="preserve"> & erit error in aſperitate & lenitate.</s> <s xml:id="echoid-s31120" xml:space="preserve"> Ex <lb/>intemperantia etiam raritatis error accidit uiſioni præ miſſorum:</s> <s xml:id="echoid-s31121" xml:space="preserve"> quia in aere nubiloſo obſcuro ui-<lb/>debitur corpus aſperum eſſe lene, propter latentes aſperitatis cauſſas, & uiſa re polita, cum non di-<lb/>ſcernitur reflexio ab ea, æſtimabitur fortè aſpera.</s> <s xml:id="echoid-s31122" xml:space="preserve"> Ex paruitate etiam temporis fit error in uiſione <lb/>præmiſſorum:</s> <s xml:id="echoid-s31123" xml:space="preserve"> cum enim ſubitò uidetur aliquod aſperum, æſtimabitur lene, & ſi lene uiſum fuerit ſu <lb/>bitò, non poterit diſcernilenitas aut aſperitas:</s> <s xml:id="echoid-s31124" xml:space="preserve"> unde ſub dubio fit error.</s> <s xml:id="echoid-s31125" xml:space="preserve"> Ex uiſus etiam debilitate fit <lb/>error in uiſione præmiſſorũ:</s> <s xml:id="echoid-s31126" xml:space="preserve"> quia uiſus debilis reputabit corpus modicè aſperũ fortè lene, uel econ-<lb/>uerſo, ſi in formis corporis aſperi & lenis fuerit diſsimilitudo.</s> <s xml:id="echoid-s31127" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s31128" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1267" type="section" level="0" n="0"> <head xml:id="echoid-head990" xml:space="preserve" style="it">142. Diaphanitas cõprehenditur à uiſu ex comprehenſione formæ cõrporis ultra corpus dia-<lb/>phanum exiſtentis. Alhazen 55 n 2.</head> <p> <s xml:id="echoid-s31129" xml:space="preserve">Quòd diaphanitas comprehendatur modo propoſito, ſatis patet:</s> <s xml:id="echoid-s31130" xml:space="preserve"> dicimus enim, ut in principio 2 <lb/> <pb o="182" file="0484" n="484" rhead="VITELLONIS OPTICAE"/> huius præmiſimus, illa corpora diaphana, quæ ſunt peruia uiſui ad alia corpora uideneda.</s> <s xml:id="echoid-s31131" xml:space="preserve"> Corpus <lb/>itaq;</s> <s xml:id="echoid-s31132" xml:space="preserve"> diaphanum perſe non uidetur, ut patet per 14 t 3 huius, niſi in ipſo ſit aliqua ſpiſitudo, reſpe-<lb/>ctu diaphanitatis aeris interiacentis uiſum, ut eſt cryſtallus & beryllus, & ſimilia denfa diaphana:</s> <s xml:id="echoid-s31133" xml:space="preserve"> <lb/>ſed etiam illorum diaphanitas à uiſu non comprehenditur, niſi ex comprehenſione formæ corpo-<lb/>ris exiſtentis ultra illa uel in circuitu ipſorum, quorum luxuel color per media illa diaphana perue-<lb/>nit ad uiſum.</s> <s xml:id="echoid-s31134" xml:space="preserve"> Cum ergo uiſus comprehendit, quòd forma lucis uel coloris comprehenſi à ſe eſt ſo-<lb/>lùm corporis ultra corpus diaphanum exiſtentis:</s> <s xml:id="echoid-s31135" xml:space="preserve"> tunc ſentiet diaphanitatem corporis diaphani.</s> <s xml:id="echoid-s31136" xml:space="preserve"> <lb/>Quòd ſi corpus diaphanum fuerit debilis diaphanitatis, utpote maioris ſpiſsitudinis quàm alia dia <lb/>phana, & corpora ultra ipſum exiſtentia fuerint debilis lucis uel coloris:</s> <s xml:id="echoid-s31137" xml:space="preserve"> tunc diaphanitas eius uix <lb/>comprehenditur à uiſu, niſi apponatur forti luci:</s> <s xml:id="echoid-s31138" xml:space="preserve"> tunc enim poteſt eius diaphanitas melius compre <lb/>hendi:</s> <s xml:id="echoid-s31139" xml:space="preserve"> propter applicationem autem proximam corporum ualde ſpiſſorum talibus corporibus dia <lb/>phanis, ipſorum comprehenſio à uiſu, quantùm ad partem applicationis, penitus impeditur, ut pa-<lb/>tet de hyaſpide in auro.</s> <s xml:id="echoid-s31140" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s31141" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1268" type="section" level="0" n="0"> <head xml:id="echoid-head991" xml:space="preserve" style="it">143. Spißitudo ſiue denſitas comprehenditur à uiſuex priuatione diaphanitatis. Alha-<lb/>zen 56 n 2.</head> <p> <s xml:id="echoid-s31142" xml:space="preserve">Cum enim uiſus comprehendit corpus aliquod, & non ſentiet in ipſo aliquam diaphanitatem, <lb/>ſtatim arguet ipſius ſpiſsitudinem:</s> <s xml:id="echoid-s31143" xml:space="preserve"> quia cum ſtatim ad illud corpus terminatur operatio uiſiua, nec <lb/>aliquid penetrat per illud, nec uiſus exercetur ad uidendum ultra ipſum formas aliorũ corporum:</s> <s xml:id="echoid-s31144" xml:space="preserve"> <lb/>tunc iudicat uiſus ipſum eſſe ſpiſſum ſiue denſum & partium compactarum:</s> <s xml:id="echoid-s31145" xml:space="preserve"> & ſic comprehenditur <lb/>ſpiſsitudo uel denſitas à uiſu ex priuatione diaphanitatis.</s> <s xml:id="echoid-s31146" xml:space="preserve"> Quod proponebatur.</s> <s xml:id="echoid-s31147" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1269" type="section" level="0" n="0"> <head xml:id="echoid-head992" xml:space="preserve" style="it">144. In raritatis & ſolidit atis uiſione error accidit uirtuti diſtinctiuæ ex intemper at a diſpo-<lb/>ſitione octo circunſtantiarum cuiuslibet rei uiſæ. Alhazen 30. 41. 50. 56. 61. 65. 68. 71 n 3.</head> <p> <s xml:id="echoid-s31148" xml:space="preserve">Ex lucis enim debilitate, ut de nocte, uidebitur corporis multum rari minor eſſe raritas:</s> <s xml:id="echoid-s31149" xml:space="preserve"> quia <lb/>cum trans ipſum non plena fiat comprehenſio formæ corporis ſolidi, æſtimabitur remiſsio rarita-<lb/>tis uiam tranſitus formarum prohibere, & corpus modicè rarum etiam tunc iudicabitur ſolidum.</s> <s xml:id="echoid-s31150" xml:space="preserve"> <lb/>Exintemperantia etiam remotionis fit error in uiſione præmiſſorum:</s> <s xml:id="echoid-s31151" xml:space="preserve"> cum enim circa oculum eri-<lb/>gitur acus, aut aliquid aliud multum ſubtile, licet illud appareat uiſui maius, quàm ſit, tamen nihil <lb/>occultatur ei de oppoſito pariete aut alio corpore:</s> <s xml:id="echoid-s31152" xml:space="preserve"> unde quia raritas non perpenditur, niſi quòd <lb/>retro corpora rara alia corpora uidentur, ut patet per 142 huius:</s> <s xml:id="echoid-s31153" xml:space="preserve"> æſtimabitur diaphanitas eſſe in a-<lb/>cu, aut in alio corpore, cum retro ipſum totus paries uideatur, quod tamen accidit ideo, quia remo-<lb/>tio tam modica, reſpectu occultationis acus eſt immoderata.</s> <s xml:id="echoid-s31154" xml:space="preserve"> Similiter etiam ſi quis à longè intuea-<lb/>tur corpus rarum, retro quod non ſit aliquod corpus coloratum aut tenebroſum, non reputabitur <lb/>illud corpus rarum, ſed ſolidum:</s> <s xml:id="echoid-s31155" xml:space="preserve"> quia retro ipſum non percipitur aliud corpus:</s> <s xml:id="echoid-s31156" xml:space="preserve"> quæ eſt proprietas <lb/>corporum rarorum.</s> <s xml:id="echoid-s31157" xml:space="preserve"> Exintemperata etiam ſitus diſpoſitione accidit error in prædictorũ uiſione.</s> <s xml:id="echoid-s31158" xml:space="preserve"> <lb/>Si enim deſcẽderit lux declinata in uitrum plenum uino, & lateat uiſum tranſitus lucis per uitrum, <lb/>& ſit magna declinatio lucis illius à radijs incidentibus, lateat quoq;</s> <s xml:id="echoid-s31159" xml:space="preserve"> uidentem uinum eſſe in uaſe <lb/>uitreo:</s> <s xml:id="echoid-s31160" xml:space="preserve"> tunc æſtimabitur à uidente uinum eſſe corpus ſolidum, ſcilicet uinum cum uaſe uitreo:</s> <s xml:id="echoid-s31161" xml:space="preserve"> & <lb/>non accideret hic error in tranſitu lucis per uas uitreum directè oppoſitum.</s> <s xml:id="echoid-s31162" xml:space="preserve"> Ex intemperata e-<lb/>tiam magnitudine accidit error in uiſione præmiſſorum.</s> <s xml:id="echoid-s31163" xml:space="preserve"> Si quis enim intueatur corpus ualde par-<lb/>uum politum, ut ab eo lux poſsit reflecti, & ſit ſimile margaritæ:</s> <s xml:id="echoid-s31164" xml:space="preserve"> iudicabit ipſum uiſus eſſe rarum <lb/>cum ſit denſum:</s> <s xml:id="echoid-s31165" xml:space="preserve"> ſimiliter uiſo corpore raro multum paruo, quia poſt ipſum non fit corporis ſoli-<lb/>di comprehenſio, aſsimilabitur ſolido.</s> <s xml:id="echoid-s31166" xml:space="preserve"> Exintemperata etiam ſoliditate fit error in uiſione præ-<lb/>miſſorum.</s> <s xml:id="echoid-s31167" xml:space="preserve"> Si enim retro corpus ualde rarum ſit aliquod corpus non multum rarum & colore forti <lb/>coloratum:</s> <s xml:id="echoid-s31168" xml:space="preserve"> tunc apparebit primum non multum rarum, ſed aſsimilabitur eius raritas poſterioris <lb/>corporis raritati:</s> <s xml:id="echoid-s31169" xml:space="preserve"> ut uitrum alij uitro ſuperpoſitum non apparet ita rarum, ſicut apparet adhibito <lb/>uiſu ſibi ſoli:</s> <s xml:id="echoid-s31170" xml:space="preserve"> unde fit error in raritate.</s> <s xml:id="echoid-s31171" xml:space="preserve"> Si autem poſt corpus rarum ponatur ualde propin què cor-<lb/>pus ſolidum:</s> <s xml:id="echoid-s31172" xml:space="preserve"> tunc primum iudicabitur ſolidum:</s> <s xml:id="echoid-s31173" xml:space="preserve"> & fit error in ſoliditate.</s> <s xml:id="echoid-s31174" xml:space="preserve"> Si etiam uas uitreum ual <lb/>de rarum contineat uinum, cum poſt illud non percipiatur lux aut corpus aliud:</s> <s xml:id="echoid-s31175" xml:space="preserve"> iudicabitur ſortè <lb/>uinum ipſum cum uaſe uitreo eſſe unũ corpus ſolidum.</s> <s xml:id="echoid-s31176" xml:space="preserve"> Idem etiã accidit error in uiſione præmiſ-<lb/>ſorum ex paucitate raritatis.</s> <s xml:id="echoid-s31177" xml:space="preserve"> In aere enim nubiloſo obſcuro corpus rarum apparebit minus rarum, <lb/>& fortè putabitur ſolidum:</s> <s xml:id="echoid-s31178" xml:space="preserve"> & ita fit error in ſoliditate & raritate.</s> <s xml:id="echoid-s31179" xml:space="preserve"> Ex paruitate etiam temporis fic <lb/>error in uiſione præ miſſorum:</s> <s xml:id="echoid-s31180" xml:space="preserve"> luce enim declinata ſuper corpus remiſſè rarum, ipſo quoq;</s> <s xml:id="echoid-s31181" xml:space="preserve"> deſcen-<lb/>dente ſubitò per uiſum, cum non percipiatur declinatio lucis, putabitur forſitan, quod illud ſit rarú <lb/>in fine raritatis, cui ſi in tempore maiori fiat intuitus, percipietur ab ipſo uiſu declinationem lucis <lb/>eſſe cauſſam apparentiæ maioris raritatis in corpore remiſſè rarò.</s> <s xml:id="echoid-s31182" xml:space="preserve"> Si quis etiam inſtanter intueatur <lb/>corpus rarum, & poſt ipſum non diſcernat lucis tranſitum, putabitipſum eſſe ſolidum.</s> <s xml:id="echoid-s31183" xml:space="preserve"> Debilitas <lb/>etiam uiſus errorem inuehit uiſioni præ miſſorum:</s> <s xml:id="echoid-s31184" xml:space="preserve"> cum enim fuerit in corpore raro ſoliditas pauca, <lb/>æſtimabitur à uiſu debili illa ſoliditas maior quàm uera:</s> <s xml:id="echoid-s31185" xml:space="preserve"> & cum fuerit in corpore raro color fortis, <lb/>aut poſt ipſum, aut raritas modica, putabitur illud corpus uiſui debili eſſe ſolidum.</s> <s xml:id="echoid-s31186" xml:space="preserve"> Patet ergo uni-<lb/>uerſaliter in omnibus illud, quod proponebatur.</s> <s xml:id="echoid-s31187" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1270" type="section" level="0" n="0"> <head xml:id="echoid-head993" xml:space="preserve" style="it">145. Vmbra comprehenditur à uiſu ex priuatione alicuius lucis luce altera præſente. Al-<lb/>hazen 57 n 2.</head> <pb o="183" file="0485" n="485" rhead="LIBER QVARTVS."/> <p> <s xml:id="echoid-s31188" xml:space="preserve">Eſtenim umbra priuatio cuiuſdam lucis, exiſtente actu præſentia lucis alterius in loco umbro-<lb/>ſo.</s> <s xml:id="echoid-s31189" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s31190" xml:space="preserve"> ſenſerit uiſus corpus uicinum umbræ maioris illuminationis, & fortioris quàm cor-<lb/>pus exiſtensin loco umbroſo:</s> <s xml:id="echoid-s31191" xml:space="preserve">tunc ſentiet obumbrationem illius loci & priuationem lucis inciden <lb/>tis corporibus uicinis ipſi.</s> <s xml:id="echoid-s31192" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s31193" xml:space="preserve"> uiſus ſenſerit aliquam lucem in aliquo loco, qui careat luce ſo-<lb/>lis prima, quæ proijcitur ſecundum directionem radiorum, percipiet tamen ſecundam, quæ fit ex <lb/>diffuſione lucis primæ:</s> <s xml:id="echoid-s31194" xml:space="preserve"> ut cum in domum unicam habentem feneſtram radius ſolis incidit, totam <lb/>domum ſui diffuſione illuminantis:</s> <s xml:id="echoid-s31195" xml:space="preserve"> tunc uiſus extra locum radij exiſtens ſentiet obumbrationem <lb/>loci, & priuationem à prima luce ſolis, quæ eſt in radio, uel ab alia luce forti:</s> <s xml:id="echoid-s31196" xml:space="preserve"> & fortè uiſus quando-<lb/>que ſtatim ſentiet corpus umbroſum, quandoq;</s> <s xml:id="echoid-s31197" xml:space="preserve"> non niſi per diligentem intuitionem, & quandoq;</s> <s xml:id="echoid-s31198" xml:space="preserve"> <lb/>uidebit umbram multiplicatam ſecundum diuerſarum lucium priuationem, ſemper aliqua luce re-<lb/>manente, ex cuius actualitate uiſus poſsit ſuam actionem ad alia exercere.</s> <s xml:id="echoid-s31199" xml:space="preserve"> Vniuerſaliter itaq;</s> <s xml:id="echoid-s31200" xml:space="preserve"> ſecun <lb/>dum omnes modos umbrarum, quos præmiſimus, poſſunt uideri umbræ.</s> <s xml:id="echoid-s31201" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s31202" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1271" type="section" level="0" n="0"> <head xml:id="echoid-head994" xml:space="preserve" style="it">146. Obſcuritas comprehenditur à uiſu ex omnimoda priuatione lucis. Alhazen 58 n 2.</head> <p> <s xml:id="echoid-s31203" xml:space="preserve">Cum uiſus comprehendit aliquem locum & nullam lucem in illo:</s> <s xml:id="echoid-s31204" xml:space="preserve"> tunc ſentiet eius obſcuritatẽ, <lb/>licet fortè illa obſcuritas ab umbris cauſſetur, ut in carcere tetro de die propter umbras denſorum <lb/>parietum uidetur obſcuritas:</s> <s xml:id="echoid-s31205" xml:space="preserve"> & nox obſcura eſt ex umbra terræ.</s> <s xml:id="echoid-s31206" xml:space="preserve"> Eſt ergo obſcuritas umbra magna, <lb/>cuius terminus ad aliquid lucidum pertingere non ſentitur:</s> <s xml:id="echoid-s31207" xml:space="preserve"> ſicut etiam umbra eſt obſcuritas parua <lb/>habens aliquem actum lucis, & ad aliquod lucidum terminata.</s> <s xml:id="echoid-s31208" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s31209" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1272" type="section" level="0" n="0"> <head xml:id="echoid-head995" xml:space="preserve" style="it">147. In umbræ & obſcuritatis uiſione error accidit uirtuti diſtinctiuæ ex intemper at a diſpo <lb/>ſitione octo circumſtantiarum cuiuslibet rei uiſæ. Alhazen 31. 42. 50. 56. 62. 65. 68. 71 n 3.</head> <p> <s xml:id="echoid-s31210" xml:space="preserve">Ex intemperata lucis diſpoſitione error accidit in uiſione umbræ & obſcuritatis.</s> <s xml:id="echoid-s31211" xml:space="preserve"> Si enim in pa-<lb/>riete albo fuerint partes obſcuræ, & cadat ſuper parietem album lux candelæ:</s> <s xml:id="echoid-s31212" xml:space="preserve">poteſt accidere quòd <lb/>uidens illam obſcuritatem, iudicabit ipſam eſſe umbram, & forſan uidebitur quod procedat appa-<lb/>rens umbra à pariete uicino.</s> <s xml:id="echoid-s31213" xml:space="preserve"> Et ſi fuerit in parte parietis nigredo multùm intenſa, æſtimabitur for-<lb/>tè uacuitas foraminis præbens iter egredientibus tenebris:</s> <s xml:id="echoid-s31214" xml:space="preserve"> & ſi tota ſuperficies parietis ſit deni-<lb/>grata intenſa nigredine, forſan totus paries æſtimabitur quædam obſcuritas tenebrarum, ſicut ac-<lb/>cidit in pariete cooperto fuligine fumorum uiſo ſub debili luce.</s> <s xml:id="echoid-s31215" xml:space="preserve"> Ex ſuperfluitate etiam remotio-<lb/>nis error accidit in uiſione umbræ & obſcuritatis.</s> <s xml:id="echoid-s31216" xml:space="preserve"> Si enima à maxima diſtantia opponatur uiſui cor-<lb/>pus album, in quo ſit aliqua pars tenebroſa, luce ſolis ſuper corpus illud deſcendente:</s> <s xml:id="echoid-s31217" xml:space="preserve"> apparebit <lb/>umbra in parte corporis tenebroſa:</s> <s xml:id="echoid-s31218" xml:space="preserve"> & ſi tunc uideatur corpus aliud iuxta illud primum:</s> <s xml:id="echoid-s31219" xml:space="preserve"> æſtimabi-<lb/>tur quòd umbra apparens proijciatur ab illo alio corpore ſuper primum.</s> <s xml:id="echoid-s31220" xml:space="preserve"> Sic ergo propter exceſ-<lb/>ſum diſtantiæ fit error in uiſione umbræ.</s> <s xml:id="echoid-s31221" xml:space="preserve"> Si etiam à longè uideatur corpus album, in quo ſint mul-<lb/>tæ partes nigræ, æſtimabuntur fortaſsis in parte illa tenebræ credetur enim aliquod corpus album <lb/>ſecundum ſui partes nigras perforatum, per quæ fiat egreſsio tenebrarum exiſtentium retro cor-<lb/>pus album:</s> <s xml:id="echoid-s31222" xml:space="preserve"> hoc autem non accideret in temperata remotione.</s> <s xml:id="echoid-s31223" xml:space="preserve"> Exinordinatione etiam ſitus oppo-<lb/>ſitionis accidit error in uiſione præmiſſorum, ſicut & ex intemperata remotione:</s> <s xml:id="echoid-s31224" xml:space="preserve"> corpore enim a-<lb/>liquo elongato, ſi fuerit in eo pars tenebroſa, putabitur fortaſsis umbra:</s> <s xml:id="echoid-s31225" xml:space="preserve"> & ſi corpus aliquod fuerit <lb/>circa illud primum poſitum, æſtimabitur umbra proijci ab illo ſecundo corpore ſuper primum:</s> <s xml:id="echoid-s31226" xml:space="preserve"> & <lb/>ſi in corpore illo fuerit pars multum nigra, æſtimabitur fortè in loco illo cuiuſdam foraminis per-<lb/>foratio, per quam egrediantur tenebræ exiſtentes retro corpus album:</s> <s xml:id="echoid-s31227" xml:space="preserve"> hoc autem non accideret in <lb/>corpore approximante directioni oppoſitionis.</s> <s xml:id="echoid-s31228" xml:space="preserve"> Ex paruitate etiam quantitatis rei uiſæ accidit er-<lb/>ror in uiſione præmiſſorum.</s> <s xml:id="echoid-s31229" xml:space="preserve"> Si enim in pariete albo uiſui oppoſito fuerit punctorum non ualde ni-<lb/>grorum diſtinctio, adhibita luce ſolis directè in parietem cadente uel propè:</s> <s xml:id="echoid-s31230" xml:space="preserve"> æſtimabuntur à uiden <lb/>te ſingula puncta illa ſingula eſſe foramina, in quibus fit umbra, cum lux non penetret ea, ſicut ſolet <lb/>accidere luce ſuper ſuperficiem foraminum multorum cadente:</s> <s xml:id="echoid-s31231" xml:space="preserve"> & fit error umbræ ex ſola puncto-<lb/>rum paruitate:</s> <s xml:id="echoid-s31232" xml:space="preserve"> quòd ſi illa puncta ſunt maximę nigritudinis, tunc æſtimabuntur eſſe foramina par-<lb/>ua, per quæ tranſeant tenebræ:</s> <s xml:id="echoid-s31233" xml:space="preserve"> & ſic etiam ſola illorum punctorum paruitas eſt cauſſa apparitionis <lb/>tenebrarum.</s> <s xml:id="echoid-s31234" xml:space="preserve"> Ex intemperata etiam ſoliditate, utpote propter defectum ſoliditatis fit error in <lb/>umbræ & obſcuritatis uiſione.</s> <s xml:id="echoid-s31235" xml:space="preserve"> Luce enim ſolis in domum per foramen aliquod deſcendente, & ſu-<lb/>per feneſtram uitream cadente, ſi domus illa fuerit umbroſa:</s> <s xml:id="echoid-s31236" xml:space="preserve"> apparebit ſuper feneſtram illam um-<lb/>bra, licet in ueritate lux ſuper ipſam inciderit, quæ quidem lux comprehenderetur, ſi ſolidum eſſet <lb/>feneſtrę corpus:</s> <s xml:id="echoid-s31237" xml:space="preserve"> quoniam tunc lux non penetraret, & ita ſuper ſolidum corpus lux apparet:</s> <s xml:id="echoid-s31238" xml:space="preserve"> fit ergo <lb/>error in umbra propter defectum ſoliditatis.</s> <s xml:id="echoid-s31239" xml:space="preserve"> Similiter etiam fit error in uiſione tenebrarum ſiue <lb/>obſcuritatis ex indiſpoſitione ſoliditatis:</s> <s xml:id="echoid-s31240" xml:space="preserve"> quia luce ſolis in aquam fluminis directè non deſcen-<lb/>dente aut in mare, ſicut accidit in hora matuatina & ueſpertina, ſi fuerit magna claritas in a qua, ap-<lb/>parebit tenebroſa, & quantò fuerit clarior, tantò apparebit tenebroſior:</s> <s xml:id="echoid-s31241" xml:space="preserve"> & accidit hoc, quoniá pars <lb/>aquæ ſuperior umbram proijcit ſuper proximam partem aquę inferiorem, & illa proxima ſuper a-<lb/>liam proximam inferiorem, & ita per ſingulas partes ſemper ſuperior proijcit umbram ſuper in-<lb/>feriorem uſq;</s> <s xml:id="echoid-s31242" xml:space="preserve"> ad fundum aquæ:</s> <s xml:id="echoid-s31243" xml:space="preserve"> & licet ſingularum partium umbra in ſe ſit modica, plures tamen <lb/>umbræ coniunctæ unam faciunt maximam umbram, ſicut palàm eſt in colore uini accidere.</s> <s xml:id="echoid-s31244" xml:space="preserve"> In mo-<lb/>dica enim quantitate uini color eſt debilis, & in multa quantitate uini licet totum uinum ſit homo-<lb/> <pb o="184" file="0486" n="486" rhead="VITELLONIS OPTICAE"/> geneum in ſubſtantia & colore, fit fortior idem color.</s> <s xml:id="echoid-s31245" xml:space="preserve"> Cauſſa autem, quare in mari umbra ſuis <lb/>partibus ſuperioribus ſuper inferiores iacentibus, uideantur eſſe tenebræ in maris claritate, hæc <lb/>eſt:</s> <s xml:id="echoid-s31246" xml:space="preserve"> quoniam intenſa ipſius clarltas eſt ſignum intenſæ raritatis, quæ formis uiſibilibus maiorem <lb/>concedit penetrationem:</s> <s xml:id="echoid-s31247" xml:space="preserve">unde fit maior diffuſio formarum plurium maris partium umbram facien <lb/>tium, quarum umbrarum aggre gatarum perceptio inducit ſimilitudinem tenebrarum.</s> <s xml:id="echoid-s31248" xml:space="preserve"> Si uerò ma-<lb/>re fuerit turbulentum, propter diminutam raritatem penetrabunt formæ partium paucæ perue-<lb/>nientes ad uiſum, & comprehendetur modica aquæ pars, quę licet faciat umbrã, tamen cum ipſa ſit <lb/>modica, erit umbra remiſſa, & uincet color illius partis umbram.</s> <s xml:id="echoid-s31249" xml:space="preserve"> In turbida enim aqua aliquis co-<lb/>lor partium aquæ apparet, & in clara nullus:</s> <s xml:id="echoid-s31250" xml:space="preserve"> unde & propter apparentiorem turbidum colorem, <lb/>& propter umbræ partis apparentis remiſsionem non comprehenduntur in aqua tenebræ:</s> <s xml:id="echoid-s31251" xml:space="preserve"> & in-<lb/>de cum fuerit turbida, apparebit colorata, & cum eſt clara, apparebit tenebroſa.</s> <s xml:id="echoid-s31252" xml:space="preserve"> Solis autem radio <lb/>cadente directè ſuper maris ſuperficiem, cum ei propter raritatem eius pateat tranſitus, abijciuntur <lb/>omnes tenebrę & umbræ apparentia.</s> <s xml:id="echoid-s31253" xml:space="preserve"> Ex defectu itaq;</s> <s xml:id="echoid-s31254" xml:space="preserve"> ſoliditatis cauſſantur & umbra & tenebræ:</s> <s xml:id="echoid-s31255" xml:space="preserve"> <lb/>quia per corpus perfectè ſolidum non fit tranſitus luminis, & per corpus perfectæ raritatis fiet tran <lb/>ſitus luminis ſine umbra.</s> <s xml:id="echoid-s31256" xml:space="preserve"> Ex intemperantia etiam raritatis accidit error in uiſione præ miſſorum.</s> <s xml:id="echoid-s31257" xml:space="preserve"> <lb/>Si ultra aerem nubiloſum uel tenebroſum, utin crepuſculis, uideatur corpus album, in quo ſint par <lb/>ticulæ rotundę nigræ:</s> <s xml:id="echoid-s31258" xml:space="preserve"> tunc luce ignis in corpus illud cadente, ita ut non mutetur tota diſpoſitio ae-<lb/>ris illius, apparebit in locis illis umbra, aut fortè reputabuntur foramina præſtantia uiam tene-<lb/>bris, quæ ſunt retro illud corpus ad uiſum pertingentes:</s> <s xml:id="echoid-s31259" xml:space="preserve"> ſic ergo propter corporis intemperatam <lb/>raritatem accidet error in uiſione umbrę & obſcuritatis.</s> <s xml:id="echoid-s31260" xml:space="preserve"> Ex paruitate etiã temporis accidit error in <lb/>uiſione præmiſſorum.</s> <s xml:id="echoid-s31261" xml:space="preserve"> Si enim in albo pariete ſint partes ſubnigræ, deſcendente ſuper ipſum parie-<lb/>tem luce ignis:</s> <s xml:id="echoid-s31262" xml:space="preserve"> illæ partes nigræ ſubitò uiſæ putabuntur eſſe umbræ.</s> <s xml:id="echoid-s31263" xml:space="preserve"> Si uerò nigredo illarum par-<lb/>tium fuerit intenſa, tunc æſtimabuntur foramina tenebris plena.</s> <s xml:id="echoid-s31264" xml:space="preserve"> Ex uiſus etiam debilitate error ac-<lb/>cidit uiſioni præmiſſorum.</s> <s xml:id="echoid-s31265" xml:space="preserve"> In pariete enim albo maculæ ſubnigrę, deſcendente luce ſuper ipſas, ap-<lb/>parent debili uiſui eſſe umbræ:</s> <s xml:id="echoid-s31266" xml:space="preserve"> & ſi fuerint multum nigræ, apparebunt eſſe foramina, per quæ tene-<lb/>bræ exlocis, quæ ſunt retro illum album parietem, perueniant ad uiſum.</s> <s xml:id="echoid-s31267" xml:space="preserve"> In omnibus ergo præmiſ-<lb/>ſis octo uiſibilium circumſtantijs patet quod proponebatur.</s> <s xml:id="echoid-s31268" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1273" type="section" level="0" n="0"> <head xml:id="echoid-head996" xml:space="preserve" style="it">148. Pulchritudo comprehenditur à uiſu ex comprehenſione ſimplici formarum uiſibilium <lb/>placentium animæ, uel coniunctione plurium uiſibilium intentionum, habentium ad inuicem <lb/>proportionem debitam formæ uiſæ. Alhazen 59 n 2.</head> <p> <s xml:id="echoid-s31269" xml:space="preserve">Fit enim placentia animæ, quæ pulchritudo dicitur, quandoq;</s> <s xml:id="echoid-s31270" xml:space="preserve"> ex comprehenſione ſimplici uiſi-<lb/>bilium formarum, ut patet per omnes ſpecies uiſibilium diſcurrendo:</s> <s xml:id="echoid-s31271" xml:space="preserve"> ut enim exemplariter dica-<lb/>mus, & alia per hoc accipiantur:</s> <s xml:id="echoid-s31272" xml:space="preserve"> lux, quæ eſt primum uiſibile, facit pulchritudinem:</s> <s xml:id="echoid-s31273" xml:space="preserve"> unde uidentur <lb/>pulchra ſol & luna & ſtellæ propter lucem ſolam.</s> <s xml:id="echoid-s31274" xml:space="preserve"> Color etiam facit pulchritudinem, ſicut color ui-<lb/>ridis & roſeus, & alij colores ſcintillantes formam ſibi appropiati luminis uiſui diffundentes.</s> <s xml:id="echoid-s31275" xml:space="preserve"> Re-<lb/>motio quoq;</s> <s xml:id="echoid-s31276" xml:space="preserve"> & approximatio faciunt pulchritudinem in uiſu:</s> <s xml:id="echoid-s31277" xml:space="preserve"> ιn quibuſdam enim formis pulchris <lb/>ſunt maculæ turpes paruæ & rugoſæ, diſplicentes animæ uidenti, quæ propter remotionem latent <lb/>uiſum, & forma placita animæ ex illa remotione peruenit ad uiſum.</s> <s xml:id="echoid-s31278" xml:space="preserve"> In multis quoq;</s> <s xml:id="echoid-s31279" xml:space="preserve"> formis pul-<lb/>chris ſunt intentiones paruæ ſubtiles cooperantes pulchritudini formarum, ſicut eſt lineatio de-<lb/>cens & ordinatio partium uenuſta, quæ tantùm in propinquitate ad uiſum apparent, & faciunt for-<lb/>mam uiſui pulchram apparere.</s> <s xml:id="echoid-s31280" xml:space="preserve"> Magnitudo etiam facit pulchritudinem in uiſu:</s> <s xml:id="echoid-s31281" xml:space="preserve"> & propter hoc lu-<lb/>na apparet pulchrior alijs ſtellis, quia uidetur maior, & ſtellæ maiores pulchriores mínoribus, ut <lb/>maximè patet in illis ſtellis, quæ ſunt magnitudinis primæ uel ſecundæ.</s> <s xml:id="echoid-s31282" xml:space="preserve"> Situs quoq;</s> <s xml:id="echoid-s31283" xml:space="preserve"> facit pulchri-<lb/>tudinem in uiſu:</s> <s xml:id="echoid-s31284" xml:space="preserve"> quoniam plures intentiones pulchræ non uidentur pulchræ, niſi per ordinatio-<lb/>nem partium & ſituum:</s> <s xml:id="echoid-s31285" xml:space="preserve"> unde ſcriptura & pictura, omnesq́;</s> <s xml:id="echoid-s31286" xml:space="preserve"> intentiones uiſibiles ordinatæ & per-<lb/>mutatæ non apparent pulchræ niſi percompetentem ſibi ſitum:</s> <s xml:id="echoid-s31287" xml:space="preserve"> quamuis enim figuræ literarum <lb/>ſint omnes per ſe bene diſpoſitæ & pulchræ, ſi tamen una ipſarum eſt magna & alia parua, non <lb/>iudicabit uiſus pulchras ſcripturas, quæ ſunt ex illis.</s> <s xml:id="echoid-s31288" xml:space="preserve"> Figura etiam facit pulcritudinem:</s> <s xml:id="echoid-s31289" xml:space="preserve"> unde ar-<lb/>tificiata bene figurata uidentur pulchra, magis autem opera naturæ:</s> <s xml:id="echoid-s31290" xml:space="preserve"> unde oculi hominis cum <lb/>ſint figuræ amygdalaris & oblongæ, uidentur pulchri, rotundi uerò oculi uidentur penitus defor-<lb/>mes.</s> <s xml:id="echoid-s31291" xml:space="preserve"> Corporeitas etiam facit pulchritudinem in uiſu:</s> <s xml:id="echoid-s31292" xml:space="preserve"> unde uidetur pulchrum corpus ſphæra & co-<lb/>lumna rotunda & bene quadratum corpus.</s> <s xml:id="echoid-s31293" xml:space="preserve"> Continuatio quoq;</s> <s xml:id="echoid-s31294" xml:space="preserve"> facit pulchritudinem in uiſu:</s> <s xml:id="echoid-s31295" xml:space="preserve"> un-<lb/>de ſpatia uiridia continua placent uiſui, & plantæ ſpiſſæ uirides:</s> <s xml:id="echoid-s31296" xml:space="preserve"> quia quæ accedunt continuitati, <lb/>ſunt pulchriores eiſdem diſperſis.</s> <s xml:id="echoid-s31297" xml:space="preserve"> Diuiſio etiam facit pulchritudinem in uiſu:</s> <s xml:id="echoid-s31298" xml:space="preserve"> unde ſtellæ ſepa-<lb/>ratæ & diſtinctæ ſunt pulchriores ſtellis approximatis nimis ad inuicem, ut ſtellæ galaxiæ & cande <lb/>læ diſtinctæ ſunt pulchriores magno adunato igne.</s> <s xml:id="echoid-s31299" xml:space="preserve"> Numerus etiam facit pulchritudinem in ui-<lb/>ſu:</s> <s xml:id="echoid-s31300" xml:space="preserve"> & propter hoc loca cœli multarum ſtellarum diſtinctarum ſunt pulchriora locis paucarum ſtel-<lb/>larum, & plures candelæ ſunt pulchriores paucis.</s> <s xml:id="echoid-s31301" xml:space="preserve"> Motus quoq;</s> <s xml:id="echoid-s31302" xml:space="preserve"> & quies faciunt in uiſu pulchritu-<lb/>dinem:</s> <s xml:id="echoid-s31303" xml:space="preserve"> motus enim hominis in ſermone & ſeparatione eius facit pulchritudinem:</s> <s xml:id="echoid-s31304" xml:space="preserve"> & propter hoc <lb/>apparet pulchra grauitas in loquendo & taciturnitas diſtinguens ordinatè uerba.</s> <s xml:id="echoid-s31305" xml:space="preserve"> Aſperitas etiam <lb/>facit pulchritudinem:</s> <s xml:id="echoid-s31306" xml:space="preserve"> uilloſitas enim pannorum catenatorum & aliorum placet uiſui.</s> <s xml:id="echoid-s31307" xml:space="preserve"> Planities <lb/>quoq;</s> <s xml:id="echoid-s31308" xml:space="preserve"> uiſui pulchritudinem facit:</s> <s xml:id="echoid-s31309" xml:space="preserve"> quia planities pannorum ſericorum & ſi etiam ad politionem <lb/> <pb o="185" file="0487" n="487" rhead="LIBER QVARTVS."/> ſiue terſionem accedant, placet animæ & eſt pulchrum uiſui.</s> <s xml:id="echoid-s31310" xml:space="preserve"> Diaphanitas etiam facit pulchritudi-<lb/>nem apparere:</s> <s xml:id="echoid-s31311" xml:space="preserve"> quia per ipſam uidentur de nocte res micantes, ut patet de aere ſereno, per quem in <lb/>nocte uidentur ſtellæ, quod non accidit in aere condenſato propter uapores.</s> <s xml:id="echoid-s31312" xml:space="preserve"> Spiſsitudo etiam fa-<lb/>cit pulchritudinem:</s> <s xml:id="echoid-s31313" xml:space="preserve"> quoniam lux & color & figura & lineatio & omne pulchrum uιſibile compre-<lb/>henduntur à uiſu propter terminationem corporum, quibus inſunt, quæ terminatio à ſpifsitudine <lb/>cauſſatur.</s> <s xml:id="echoid-s31314" xml:space="preserve"> Et umbra facit apparere pulchritudinem:</s> <s xml:id="echoid-s31315" xml:space="preserve"> quoniam in multis formis uiſibilium ſunt ma-<lb/>culæ ſubtiles reddentes ipſas turpes cum fuerint in luce, quæ in umbra uel luce debili uiſum ſunt <lb/>latentes.</s> <s xml:id="echoid-s31316" xml:space="preserve"> Tortuoſitas quoq;</s> <s xml:id="echoid-s31317" xml:space="preserve">, quæ eſt in plumis auium, ut pauonum & aliarum, quia facit umbras, <lb/>facit apparere pulchritudinẽ uiſui propter umbram, quæ in ſui admixtione cum lumine cauſſat ua-<lb/>rios colores, qui tamen non apparent in umbra uel in luce debili.</s> <s xml:id="echoid-s31318" xml:space="preserve"> Obſcuritas etiam facit pulchri-<lb/>tudinem apparere uiſui:</s> <s xml:id="echoid-s31319" xml:space="preserve"> quoniam ſtellæ non uidentur niſi in obſcuro.</s> <s xml:id="echoid-s31320" xml:space="preserve"> Similitudo etiam pulchritu-<lb/>dinem facit:</s> <s xml:id="echoid-s31321" xml:space="preserve"> quoniam membra eiuſdem animalis, ut Socratis, non apparent pulchra, niſi quando <lb/>fuerint conſimilia:</s> <s xml:id="echoid-s31322" xml:space="preserve"> unde oculi, quorum unus eſt rotundus & alter oblongus, non ſunt pulchri, uel ſi <lb/>unus maior fuerit altero, uel unus niger & alter uiridis, uel ſi una gena fuerit profunda & altera pro-<lb/>minens:</s> <s xml:id="echoid-s31323" xml:space="preserve"> erit enim tota facies non pulchra, quando eius partes congeneæ non fuerint conſimiles.</s> <s xml:id="echoid-s31324" xml:space="preserve"> <lb/>Diuerſitas etiam facit pulchritudinem:</s> <s xml:id="echoid-s31325" xml:space="preserve"> quoniam diuerſæ partes uniuerſi ornant & pulchrum fa-<lb/>ciunt uniuerſum, & diuerſæ partes animalium animalia:</s> <s xml:id="echoid-s31326" xml:space="preserve"> eandem quoq;</s> <s xml:id="echoid-s31327" xml:space="preserve"> manum ornat diuerſitas <lb/>digitorum, omnis enim pulchritudo membrorum eſt ex diuerſitate figurarum partium ipſarum.</s> <s xml:id="echoid-s31328" xml:space="preserve"> <lb/>Sic ergo pulchritudo comprehenditur à uiſu ex comprehenſione ſimplici formarum uiſibilium pla <lb/>centium animæ:</s> <s xml:id="echoid-s31329" xml:space="preserve"> quæ libet tamen iſtarum uiſibilium intentionum non facit pulchritudinem in qua-<lb/>libet forma, in qua uenit illa intentio ad uiſum:</s> <s xml:id="echoid-s31330" xml:space="preserve"> quælibet enim figura non facit pulchritudinem in <lb/>qualibet formarum, & ſimiliter de alijs omnibus intentionibus particularibus uiſibilium quorum-<lb/>cunq;</s> <s xml:id="echoid-s31331" xml:space="preserve">. Exconiunctione quoq;</s> <s xml:id="echoid-s31332" xml:space="preserve"> plurium ιntentionum formarum uifibilium adinuicem, & non ſo-<lb/>lum ex ipſis intentionibus uiſibilium fit pulchritudo in uiſu, ut colores ſcintillantes & pictura ſimi-<lb/>liter proportionati ſunt pulchriores coloribus & picturis carentibus ordinatione conſimili:</s> <s xml:id="echoid-s31333" xml:space="preserve"> & ſimi-<lb/>liter eſt in uultu humano:</s> <s xml:id="echoid-s31334" xml:space="preserve"> rotunditas enim faciei cum tenuitate & ſubtilitate coloris eſt pulchrior <lb/>quàm unum ſine altero, & mediocris paruitas oris cum gracilitate labiorum proportionali eſt pul-<lb/>chrior paruitate oris cum groſsitudine labiorum.</s> <s xml:id="echoid-s31335" xml:space="preserve"> In multis itaq;</s> <s xml:id="echoid-s31336" xml:space="preserve"> formis uiſibilium coniunctio, quæ <lb/>eſt in formis diuerſis, facit modum pulchritudinis, quem non facit una illarum intentionum per ſe.</s> <s xml:id="echoid-s31337" xml:space="preserve"> <lb/>Facit autem proportionalitas partium debita alicui formæ naturali uel artificiali in coniunctione <lb/>intentionum ſenſibilium pulchritudinem magis, quàm aliqua intentionum particularium:</s> <s xml:id="echoid-s31338" xml:space="preserve"> omnes <lb/>enim pulchritudines, quas faciunt intentiones ſenſibiles ex ipſarũ coniunctione adinuicem, conſi-<lb/>ſtunt in proportionalitate debita formis, quas perficiunt ſub modo illius coniunctionis.</s> <s xml:id="echoid-s31339" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s31340" xml:space="preserve"> <lb/>uiſus comprehendit aliquam rem uiſam, in qua eſt aliqua intentio particularis, faciens per ſe pul-<lb/>chritudinem:</s> <s xml:id="echoid-s31341" xml:space="preserve"> tunc peruenit forma ill us intentionis poſt intuitum ad uirtutem ſentientem, & com-<lb/>prehendet uirtus diſtinctiua pulchritudinem rei uiſæ, in qua eſt illa intentio:</s> <s xml:id="echoid-s31342" xml:space="preserve"> & ſic coniunctio di-<lb/>uerſarum intentionum fit cauſſans pulchritudinem, cum peruenerit illa coniunctio ad ſentientem:</s> <s xml:id="echoid-s31343" xml:space="preserve"> <lb/>tunc uirtus diſtinctiua comparabit illas intentiones ad inuicem, & tunc comprehendet pulchritu-<lb/>dinem rei uiſæ compoſitæ exillarum intentionum coniunctione, quæ ſunt in ea.</s> <s xml:id="echoid-s31344" xml:space="preserve"> Et hi ſunt modi, pe <lb/>nes quos accipitur à uiſu omnium formarum ſenſibilium pulchritudo:</s> <s xml:id="echoid-s31345" xml:space="preserve"> in pluribus tamen iſtorum <lb/>conſuetudo facit pulchritudinem:</s> <s xml:id="echoid-s31346" xml:space="preserve"> unde unaquæq;</s> <s xml:id="echoid-s31347" xml:space="preserve"> gens hominum approbat ſuæ conſuetudinis for <lb/>mam, ſicutillud, quod per ſe æſtimat pulchrum in fine pulchritudinis:</s> <s xml:id="echoid-s31348" xml:space="preserve"> alios enim colores & propor <lb/>tiones partium corporis humani & picturarũ approbat Maurus, & alios Danus, & inter hæc extre-<lb/>ma & ipſis proxima Germanus approbat medios colores & corporis proceritates & mores:</s> <s xml:id="echoid-s31349" xml:space="preserve"> & ſicut <lb/>unicuiq;</s> <s xml:id="echoid-s31350" xml:space="preserve"> ſuus proprius mos eſt, ſic & propria æſtimatio pulchritudinis accidit unicuiq;</s> <s xml:id="echoid-s31351" xml:space="preserve">. De his er-<lb/>go topicè & figuraliter ſit dictum.</s> <s xml:id="echoid-s31352" xml:space="preserve"> Et patet quod proponebatur.</s> <s xml:id="echoid-s31353" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1274" type="section" level="0" n="0"> <head xml:id="echoid-head997" xml:space="preserve" style="it">149. Turpitudo comprehenditur à uiſu, cum intentiones ſenſibiles ne per ſe, ne ex coniun <lb/>ctione ipſarum adinuicem aliquam pulchritudinem ſunt cauſſantes. Alhazen 60 n 2.</head> <p> <s xml:id="echoid-s31354" xml:space="preserve">Turpitudo formarum eſt priuatio pulchritudinis in eis:</s> <s xml:id="echoid-s31355" xml:space="preserve"> iam autem præmiſſum eſt, quò inten-<lb/>tiones non faciunt pulchritudinem in omnibus formis, ſed in quibuſdam tantum.</s> <s xml:id="echoid-s31356" xml:space="preserve"> Formæ itaq;</s> <s xml:id="echoid-s31357" xml:space="preserve">, in <lb/>quibus non faciunt intentiones particulares aliquam pulchritudinẽ neq;</s> <s xml:id="echoid-s31358" xml:space="preserve"> per ſe neq;</s> <s xml:id="echoid-s31359" xml:space="preserve"> per ſuam con-<lb/>iunctionem, ut illa, in quibus non eſt aliqua conſueta proportionalitas inter ipſorum partes, carent <lb/>omni pulchritudine:</s> <s xml:id="echoid-s31360" xml:space="preserve"> & ſic ſunt turpes:</s> <s xml:id="echoid-s31361" xml:space="preserve"> & ſi quandoq;</s> <s xml:id="echoid-s31362" xml:space="preserve"> accidat in eadem forma congregari intentio-<lb/>nes pulchras & turpes:</s> <s xml:id="echoid-s31363" xml:space="preserve"> tunc uiſus comprehendit pulchritudinem ex pulchro, & turpitudinem ex <lb/>turpi, auxilio uirtutis diſtinctiuæ, quando fuerit intuens intentiones, quæ ſunt in illa forma.</s> <s xml:id="echoid-s31364" xml:space="preserve"> Patet <lb/>ergo quomodo à uiſu comprehenditur turpitudo:</s> <s xml:id="echoid-s31365" xml:space="preserve"> ſed etiam in hoc plurimum coadiuuat conſuetu-<lb/>do, propter quam nonnunquam accidit uni uideri turpe, quod uidetur alteri perpulchrum.</s> <s xml:id="echoid-s31366" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1275" type="section" level="0" n="0"> <head xml:id="echoid-head998" xml:space="preserve" style="it">150 In pulchritudinis & deformitatis uiſione uirtuti diſtinctiuæ error accidit ex intempera-<lb/>ta diſpoſitione octo circumstantiarũ cuiuslibet reiuiſæ. Alhazen 32. 43. 51. 57. 63. 65. 68. 71 n 3.</head> <p> <s xml:id="echoid-s31367" xml:space="preserve">Ex paruitate enim lucis error accidit uiſioni pulchritudinis & deformitatis:</s> <s xml:id="echoid-s31368" xml:space="preserve"> de nocte enim ui-<lb/>detur facies formoſa, licet in ea ſint maculæ, ſicut lentigines uel ſicut cicatrices puſtularum.</s> <s xml:id="echoid-s31369" xml:space="preserve"> Et <lb/> <pb o="186" file="0488" n="488" rhead="VITELLONIS OPTICAE"/> ſi fuerintin reuiſa picturæ ſubtiles rem perfectius decorantes, cum illæ in nocte uiſum lateant, ui-<lb/>detur res deformis.</s> <s xml:id="echoid-s31370" xml:space="preserve"> Remotio etiam excedens modum, eſt cauſſa erroris uiſionis præmiſſorum.</s> <s xml:id="echoid-s31371" xml:space="preserve"> <lb/>Cum enim à longè reſpicitur res aliqua, ſi fuerint in ea maculæ paruæ ipſam deformantes, illas ex <lb/>diſtantia accidit occultari, & iudicabitur res formoſa:</s> <s xml:id="echoid-s31372" xml:space="preserve"> & ſi à magna diſtantia uideatur res, in qua <lb/>ſunt picturæ minutæ, in quibus conſiftit pulchritudo illius rei, illa res iudicabitur deformis:</s> <s xml:id="echoid-s31373" xml:space="preserve"> quo-<lb/>niam uirtus diſtinctiua iudicat res ſecundum quod apparent.</s> <s xml:id="echoid-s31374" xml:space="preserve"> Exinordinatione etiam ſitus oppo-<lb/>ſitionis accidit error uiſioni pręmiſſorum.</s> <s xml:id="echoid-s31375" xml:space="preserve"> Cum enim corpus aliquod remotum fuerit ab axe uiſua-<lb/>li, in quo ſunt maculæ minutæ deformantes rem:</s> <s xml:id="echoid-s31376" xml:space="preserve"> tunc nonnunquam maculæ illæ occultabuntur <lb/>propter obliquationem reſpectu axis uiſualis:</s> <s xml:id="echoid-s31377" xml:space="preserve"> & ob hoc facies lentiginoſa obliquè uiſa uidetur pul-<lb/>chra:</s> <s xml:id="echoid-s31378" xml:space="preserve"> unde etiam accidit, quòd cum luna obliquè aſpicitur, latent umbroſæ maculæ ipſius, & tunc <lb/>pulchrior uidetur:</s> <s xml:id="echoid-s31379" xml:space="preserve"> ſi autem in corpore aliquo uiſo fuerint picturæ ſubtiles rem decorantes, illæ pi-<lb/>cturæ obliquatæ ad uiſum, latebunt ipſum, & adiudicabitur pulchritudo deformitati.</s> <s xml:id="echoid-s31380" xml:space="preserve"> Ex paruitate <lb/>ctiam magnitudinis accidit error uiſioni præmiſſorum in exemplis præmiſsis:</s> <s xml:id="echoid-s31381" xml:space="preserve"> com propter ſolam <lb/>fui paruitatem aliqua minuta ipſas res uiſibiles deformantia uel decorantia non uidentur.</s> <s xml:id="echoid-s31382" xml:space="preserve"> Exde-<lb/>fectu etiam ſoliditatis fit error in uiſione præ miſſorum.</s> <s xml:id="echoid-s31383" xml:space="preserve"> Sienim in uafe uitreo multùm raro ſint ali-<lb/>quæ paruæ particulæ uel menſurationes ipſi decorem inferentes, & imponatur uaſi illi uinum tur-<lb/>bidum & turpe uel feculentum:</s> <s xml:id="echoid-s31384" xml:space="preserve"> tunc occultabuntur illæ decoris cauſſæ, & iudicabitur uas defor-<lb/>me:</s> <s xml:id="echoid-s31385" xml:space="preserve"> & ſi uas tale deformant aliquæ particulæ, & imponatur ei uinum clarum lucidum coloris for-<lb/>moſi, placidi, occultabuntur illæ cauſſæ turpitudinis, & apparebit uas pulchrum.</s> <s xml:id="echoid-s31386" xml:space="preserve"> Ex intemperantia <lb/>etiam raritatis error accidit uiſioni præmiſſorum, cum propter aerem obſcurum nubiloſum cauſſæ <lb/>pulchritudinis uel deformitatis non uidentur.</s> <s xml:id="echoid-s31387" xml:space="preserve"> Extemporis quoq;</s> <s xml:id="echoid-s31388" xml:space="preserve"> breuitate error accidit uiſioni <lb/>præmiſſorum:</s> <s xml:id="echoid-s31389" xml:space="preserve"> quoniam in paruo tempore non ſunt comprehenſibiles minutæ cauſſæ pulchritudi-<lb/>nis uel deformitatis:</s> <s xml:id="echoid-s31390" xml:space="preserve"> ſicut accidit cum aliquis inſpiciens per foramen uiderit aliquam faciem:</s> <s xml:id="echoid-s31391" xml:space="preserve">tunc <lb/>enim aliquando deformem iudicat eſſe pulchram, & aliquando econuerſo:</s> <s xml:id="echoid-s31392" xml:space="preserve"> & idem accidit mota re <lb/>uiſa ſubitò, remanente oculo non moto.</s> <s xml:id="echoid-s31393" xml:space="preserve"> Ex uiſus etiam debilitate error accidit uiſioni præmiſſorũ:</s> <s xml:id="echoid-s31394" xml:space="preserve"> <lb/>minuta enim, quæ ſunt cauſſa pulchritudinis uel deformitatis, uiſus debilis non uidet:</s> <s xml:id="echoid-s31395" xml:space="preserve"> unde modo <lb/>contrario iudicat unum quodq;</s> <s xml:id="echoid-s31396" xml:space="preserve"> iſtorum.</s> <s xml:id="echoid-s31397" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s31398" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1276" type="section" level="0" n="0"> <head xml:id="echoid-head999" xml:space="preserve" style="it">151. Conſimilitudo comprehenditur à uiſu ex conuenientia formarum comprehenſarũ ad in-<lb/>uicem. Alhazen 61 n 2.</head> <p> <s xml:id="echoid-s31399" xml:space="preserve">Eſt enim conſimilitudo æqualitas duarum formarum aut duarum intentionum in re, in qua <lb/>ſunt conſimiles.</s> <s xml:id="echoid-s31400" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s31401" xml:space="preserve"> uiſus comprehenderit duas formas aut duas intentiones conſimiles in <lb/>ſimul, comprehendet conſimilitudinem illarum ex comprehenſione cuiuslibet illarũ duarum for-<lb/>marum & ſuarum intentionum ex comparatione alterius illarum ad alteram.</s> <s xml:id="echoid-s31402" xml:space="preserve"> Viſus itaq;</s> <s xml:id="echoid-s31403" xml:space="preserve"> compre-<lb/>hendet conſimilitudinem in formis & intentionibus conſimilibus ex comprehenſione cuiuslibet <lb/>formarum intentionum ſecundum ſuum eſſe, & ex comprehenſione illarum ad inuicem.</s> <s xml:id="echoid-s31404" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1277" type="section" level="0" n="0"> <head xml:id="echoid-head1000" xml:space="preserve" style="it">152. Diuerſitas comprehenditur à uiſuex priuatione conſimilitudinis in formis ſenſibilibus <lb/>comprehenſis. Alhazen 62 n 2.</head> <p> <s xml:id="echoid-s31405" xml:space="preserve">Cum enim diuerſitas, ut hic accipitur, non ſit aliud, quàm differentia form arũ ſenſibilium com-<lb/>prehenſarum à uiſu, hæc diuerſitas comprehenditur à uiſu in formis diuerſis ex comprehenſione <lb/>cuiuslibet illarum formarum diuerſarum, & ex comparatione alterius illarum ad alterã, & ex com-<lb/>prehenſione priuationis conſimilitudinis in eis.</s> <s xml:id="echoid-s31406" xml:space="preserve"> Diuerſitas ergo comprehenditur per ſenſum uiſus <lb/>ex comprehenſione cuiuslibet formarum & intentionum per ſe, & ex comparatione ipſarum adin-<lb/>uicem, & ex ſenſu priuationis conſimilitudinis ab ipſo ſentiente.</s> <s xml:id="echoid-s31407" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1278" type="section" level="0" n="0"> <head xml:id="echoid-head1001" xml:space="preserve" style="it">153. In ſimilitudinis & diuerſitatis uiſione error accidit uirtuti diſtinctiuæ ex intempera-<lb/>ta diſpoſitione octo circumſtantiarum cuiuslibet rei uiſæ. Alhazen 33. 44. 51. 57. 63. 65. <lb/>68. 71. n 3.</head> <p> <s xml:id="echoid-s31408" xml:space="preserve">Ex paucitate enim lucis error accidit in uiſione ſimilitudinis & diuerſitatis corporum eiuſdem <lb/>coloris ſecundum ſpeciem, uel eiuſdem figuræ ſecundum ſpeciem, in quibus partialis diuerſi-<lb/>tas per latentia ſigna diſtincta eſt:</s> <s xml:id="echoid-s31409" xml:space="preserve">tunc enim illa in luce debili non uidentur:</s> <s xml:id="echoid-s31410" xml:space="preserve"> & ob hoc inter illa cor <lb/>pora omnimoda iudicabitur ſimilitudo.</s> <s xml:id="echoid-s31411" xml:space="preserve"> Et ſi aliqua corpora ſolùm propter aliqua minuta ſignai-<lb/>pſis communia participent ſimilitudine:</s> <s xml:id="echoid-s31412" xml:space="preserve"> tunc propter lucis debilitatem illis cauſsis conſimilitudi-<lb/>nis non perceptis, iudicabitur diuerſitas totalis, quod non accideret in luce temperata.</s> <s xml:id="echoid-s31413" xml:space="preserve"> Ex ſuper-<lb/>flua etiam elongatione accidit error in præmiſſorum uiſione, ut patet in præmiſsis exemplis.</s> <s xml:id="echoid-s31414" xml:space="preserve"> Minu <lb/>tæ enim cauſſæ ſimilitudinis uel diſsimilitudinis à magna remotione non uidentur per 8 huius.</s> <s xml:id="echoid-s31415" xml:space="preserve"> Et <lb/>ſimiliter etià eiſdem error accidit ex ſitus nimia obliquatione, quæ res paruas non ſinit comprehen <lb/>di à uiſu per 26 huius.</s> <s xml:id="echoid-s31416" xml:space="preserve"> Accidit etiam error in præmiſſorũ uiſione, propter cauſſarũ ſimilitudinis uel <lb/>diſsimilitudinis paruitatẽ, propter quã, cæteris exiſtentibus cõuenienter uiſui diſpoſitis, huiuſmo-<lb/>dinõ uidentur.</s> <s xml:id="echoid-s31417" xml:space="preserve"> Ex defectu etiã ſoliditatis error accidit uiſioni præmiſſorũ.</s> <s xml:id="echoid-s31418" xml:space="preserve"> Sienim duo uaſa multũ <lb/>rara cõueniãt in ſpecie, figura & raritate, ſed diſcrepẽt in aliqua ſuarũ partiũ diſpoſitiõe:</s> <s xml:id="echoid-s31419" xml:space="preserve"> tũc uino e-<lb/>iuſdẽ coloris & claritatis ambob.</s> <s xml:id="echoid-s31420" xml:space="preserve"> repletis latebũt cauſſę diuerſitatis, & reputabũtur omnino ſimilia.</s> <s xml:id="echoid-s31421" xml:space="preserve"> <lb/> <pb o="187" file="0489" n="489" rhead="LIBER QVARTVS."/> Et ſi differant ſpecie, figura & raritate, ſed ſolùm in aliquibus partialibus formulis cõueniant:</s> <s xml:id="echoid-s31422" xml:space="preserve"> tunc <lb/>uino ſimili plena putabuntur omnino ſimilia:</s> <s xml:id="echoid-s31423" xml:space="preserve"> qui error accidit propter defectum ipſorum ſolidita-<lb/>tis:</s> <s xml:id="echoid-s31424" xml:space="preserve"> quia cũ ſint peruia, ideo res per ipſa uiſa ſimilitudinis uel diſsimilitudinis aufert cauſſas.</s> <s xml:id="echoid-s31425" xml:space="preserve"> Exin-<lb/>temperantia etiam raritatis accidit error in uiſione præmiſſorum:</s> <s xml:id="echoid-s31426" xml:space="preserve">in aere enim nubiloſo & obſcuro <lb/>minutæ cauſſæ ſimilitudinis uel diſsimilitudinis non uidẽtur.</s> <s xml:id="echoid-s31427" xml:space="preserve"> Ex temporis etiã breuitate præmiſ-<lb/>ſorum uiſioni error accidit:</s> <s xml:id="echoid-s31428" xml:space="preserve"> quoniam particulares ſimilitudinis uel diſsimilitudinis cauſſæ paruiſ-<lb/>ſimo tempore inſpectæ latent uiſum.</s> <s xml:id="echoid-s31429" xml:space="preserve"> Debilitas etiam uiſus errorem illorum uiſioni adducit, quia <lb/>minutas ipſorum ſcilicet ſimilitudinis uel diſsimilitudinis cauſſas uiſus debilis perſpicere non po-<lb/>teſt.</s> <s xml:id="echoid-s31430" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s31431" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1279" type="section" level="0" n="0"> <head xml:id="echoid-head1002" xml:space="preserve" style="it">154. Virtuti diſtinctiuæ error quando accidit ex cauſſarum plurium aggregatione, qua-<lb/>rum nulla per ſe ad errorem ſufficit cauſſandum. Alhazen 72 n 3.</head> <p> <s xml:id="echoid-s31432" xml:space="preserve">Quandoq;</s> <s xml:id="echoid-s31433" xml:space="preserve"> enim duæ intemperantiæ circumſtantiarum octo omnium uiſibilium concurrunt in <lb/>uno uiſibili, & faciũt errorem in uiſu, licet neutra ipſarum per ſe ſufficeret ad cauſſandum errorem.</s> <s xml:id="echoid-s31434" xml:space="preserve"> <lb/>Si enim moueatur aliquid à magna diſtantia motu tardo, illud ſubitò uiſum uidebitur nõ motum, <lb/>& motus ille poſſet percipi in diſtantia temperata etiam ſubito uiſu, uel etiam poſſet percipi in illa <lb/>remota diſtantia per intuitum diligentem tempore conuenienti.</s> <s xml:id="echoid-s31435" xml:space="preserve"> Sed illis duabus cauſsis erroris <lb/>concurrentibus, tunc errabit uirtus diſtinctiua, & uidebitur res immota.</s> <s xml:id="echoid-s31436" xml:space="preserve"> Sed etiam quandoq;</s> <s xml:id="echoid-s31437" xml:space="preserve"> con-<lb/>currunt intemperantiæ plures ad unum errorem cauſſandum, quam nulla illarum per ſe cauſſaret.</s> <s xml:id="echoid-s31438" xml:space="preserve"> <lb/>Si enim à magna diſtantia ſub debili luce in tempore modico opponatur uiſui debili corpus diuer-<lb/>ſorum colorum motum tardo motu:</s> <s xml:id="echoid-s31439" xml:space="preserve">tũc fortè uidebitur quieſcere:</s> <s xml:id="echoid-s31440" xml:space="preserve">ſed motus eius qualibet illarum <lb/>cauſſarum aliqua deficiente percipi fortè poſſet:</s> <s xml:id="echoid-s31441" xml:space="preserve"> & fortè quandoq;</s> <s xml:id="echoid-s31442" xml:space="preserve"> intemperátiæ omnium circum-<lb/>ſtantiarum corporum uiſibilium cõcurruntad unum errorem cauſſandum, uel quandoq;</s> <s xml:id="echoid-s31443" xml:space="preserve"> plurium <lb/>illarum, & ſecundum diuerſas combinationes, quæ plus experientiam quàm rationem reſpiciunt <lb/>ſecundum omnem ſui diuerſitatem:</s> <s xml:id="echoid-s31444" xml:space="preserve"> unde de his ſic eſſe ſufficit exemplatum.</s> <s xml:id="echoid-s31445" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1280" type="section" level="0" n="0"> <head xml:id="echoid-head1003" xml:space="preserve" style="it">155. Error accidit uiſuiuia ſcientiæ per inconueniẽtem applicationem formæ, quæ eſt in ani-<lb/>ma alicuirei uiſæ, in intemperantia cuiuslibet octo circumſtantiaru reiuiſæ. Alhazen 21 n 3.</head> <p> <s xml:id="echoid-s31446" xml:space="preserve">Cum enim res alia aut alterius ſpeciei uiſui apparet quàm ſit in rei ueritate:</s> <s xml:id="echoid-s31447" xml:space="preserve"> tunc fit error uia <lb/>ſcientiæ in uiſu:</s> <s xml:id="echoid-s31448" xml:space="preserve"> quoniam forma quieſcens in anima inconuenienter alteri rei applicatur, cui non <lb/>conuenit:</s> <s xml:id="echoid-s31449" xml:space="preserve"> & hoc accidit propter intemperantiam cuiuslibet octo circumſtantiarum rerum uiſibi-<lb/>lium.</s> <s xml:id="echoid-s31450" xml:space="preserve"> Propter defectum enim lucis fit plurimus error in rerum cognitione, ut hoc euidenter per ſe <lb/>patet.</s> <s xml:id="echoid-s31451" xml:space="preserve"> Debilitas enim lucis nimia errorem infert formæ uiſæ:</s> <s xml:id="echoid-s31452" xml:space="preserve"> unde accidit error in crepuſculis in <lb/>omnibus uiſis:</s> <s xml:id="echoid-s31453" xml:space="preserve"> unde etiam noctiluca uidẽtur lucere in tenebris, quorum forma non eſt lumen, nec <lb/>etiam ſcintillans color:</s> <s xml:id="echoid-s31454" xml:space="preserve"> quæ omnia non acciderent in luce temperata.</s> <s xml:id="echoid-s31455" xml:space="preserve"> Etpropter diſtantiam etiam <lb/>nimiam uiſibilis à uiſu accidit hominem notum quandoq;</s> <s xml:id="echoid-s31456" xml:space="preserve"> pro extraneo reputari, & econtrario, uel <lb/>etiam notum unum pro alio noto, ut Socratem pro Platone, aut econtrario:</s> <s xml:id="echoid-s31457" xml:space="preserve"> & quandoq;</s> <s xml:id="echoid-s31458" xml:space="preserve"> aliquis <lb/>uidens equum, putat ſe uidere aſinum.</s> <s xml:id="echoid-s31459" xml:space="preserve"> Et uniuerſaliter fit error ſcientiæ, uel à ſpecie ad ſpeciem, uel <lb/>ab indiuiduo ad indiuiduum eiuſdem ſpeciei:</s> <s xml:id="echoid-s31460" xml:space="preserve"> uel ab indiuiduo ſpeciei unius ad indiuiduũ ſpeciei <lb/>alterius, ut cum equus Petri æſtimatur mulus Martini.</s> <s xml:id="echoid-s31461" xml:space="preserve"> Et quandoq;</s> <s xml:id="echoid-s31462" xml:space="preserve"> quis uidens ignem remotum <lb/>longè in aere, putat ſe ſtellam uidere:</s> <s xml:id="echoid-s31463" xml:space="preserve"> hæc enim omnia ſi propè eſſent, uiderentur ſine errore.</s> <s xml:id="echoid-s31464" xml:space="preserve"> Situs <lb/>etiam oppoſitionis errorẽ inducit:</s> <s xml:id="echoid-s31465" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s31466" xml:space="preserve"> enim Petrus remotus ab axe uiſuali, putabitur Mar-<lb/>tinus, & quandoq;</s> <s xml:id="echoid-s31467" xml:space="preserve"> equus uiſus putabitur eſſe aſinus, quæ ſi directè uiſui opponãtur, error penitùs <lb/>ceſſabit.</s> <s xml:id="echoid-s31468" xml:space="preserve"> Quantitas etiam extra temperantiam exiſtens errorem facit uiſui & ſcientiæ, ut cum gra-<lb/>num ſinapis creditur eſſe granum naſturtij.</s> <s xml:id="echoid-s31469" xml:space="preserve"> Soliditas etiã eſt cauſſa huius erroris:</s> <s xml:id="echoid-s31470" xml:space="preserve"> unde cryſtallus, <lb/>quia parum eſt ſolida, creditur color eius eſſe color rubini, ſuppoſito ſibi tali colore & uiſu in op-<lb/>poſito exiſtente.</s> <s xml:id="echoid-s31471" xml:space="preserve"> Diaphanitas etiam nimis diminuta huius erroris eſt cauſſa:</s> <s xml:id="echoid-s31472" xml:space="preserve"> uitro enim colorato <lb/>uiſui & rei uiſæ coloratæ interpoſito, æſtimabitur color corporis oppoſiti mixtus ex colore pro-<lb/>prio & colore uitri:</s> <s xml:id="echoid-s31473" xml:space="preserve"> & ſi oculis & rebus uiſis interponatur pannus multùm rarus, apparebit color <lb/>corporis mixtus, non quòd ſecũdum ueritatem partes coloris rei per foramina pannitranſentes <lb/>cum coloribus filorum miſceantur, ſed quia pũcta coloris rei uiſæ & filorum ſine diſtantia ſenſibili <lb/>propè adinuicem in uiſus ſuperficie ſituantur:</s> <s xml:id="echoid-s31474" xml:space="preserve"> unde illi colores diuerſi uidentur punctualiter ad-<lb/>inuicem coniumcti, propter quod apparet uiſui unus color ex illis ambobus coloribus mixtus:</s> <s xml:id="echoid-s31475" xml:space="preserve"> un-<lb/>de ſi magna ſint panni foramina, diſcernentur colores & panni & rei uiſæ ſine aliqua mixtura.</s> <s xml:id="echoid-s31476" xml:space="preserve"> Et <lb/>ex hoc accidit quòd uiſo colore alicuius corporis per pannum laneũ, uidebitur mixtura colorum <lb/>plurimùm conſonans colori filorum:</s> <s xml:id="echoid-s31477" xml:space="preserve"> quia foramina panni lanei ſunt ſtricta, quæ pilis multis colo-<lb/>ratis conteguntur:</s> <s xml:id="echoid-s31478" xml:space="preserve"> & etiam cum ioculatores faciunt ſub pannis ſe circumdantibus imagines li-<lb/>gneas pictas moueri:</s> <s xml:id="echoid-s31479" xml:space="preserve"> tunc ſimilitudines illarum imaginum inſpicienti per pannum lineum ſubti-<lb/>lem, ſicut ſolet fieri, apparebunt aues uel alia animalia illis formis conuenientia:</s> <s xml:id="echoid-s31480" xml:space="preserve"> & hoc propter de-<lb/>fectum diaphanitatis medij, quia in aere præter pannum aliud uidetur.</s> <s xml:id="echoid-s31481" xml:space="preserve"> Temporis etiam intem-<lb/>perantia huius erroris eſt cauſſa.</s> <s xml:id="echoid-s31482" xml:space="preserve"> Si quis enim per foramen reſpiciat aliquod corpus tranſiens ue-<lb/>loci motu, & non plenè acquirat formam corporis, uel ſi quis ſubitò aliquid uideat, quod ſtatim <lb/>à uiſu recedat, errabit in indruiduo illius formæ:</s> <s xml:id="echoid-s31483" xml:space="preserve"> unde forſan eſt error in ſpecie uel in indiuiduo <lb/>uel in utroque:</s> <s xml:id="echoid-s31484" xml:space="preserve"> forſan enim æſtimabit equum fuiſſe mulum, uel Petrum Martinum, uel equum <lb/> <pb o="188" file="0490" n="490" rhead="VITELLONIS OPTICAE"/> Petri fuiſſe múlum Martini.</s> <s xml:id="echoid-s31485" xml:space="preserve"> Debilitas quoq;</s> <s xml:id="echoid-s31486" xml:space="preserve"> uiſus huius erroris eſt cauſſa:</s> <s xml:id="echoid-s31487" xml:space="preserve">læſus enim uiſus à colo-<lb/>re forti, cui incidit lumen forte, iudicat omnem colorem uiſum illius coloris, uel alterius coloris ex <lb/>illis duobus mixti:</s> <s xml:id="echoid-s31488" xml:space="preserve"> & etiam propter oculorum ægritudinẽ aliquando equus apparet aſinus, & So-<lb/>crates uidetur Plato.</s> <s xml:id="echoid-s31489" xml:space="preserve"> Et ſimiliter in alijs uiſibilibus errabit uiſus propter ſolam intemperãtiam ſuæ <lb/>æqualis diſpoſitionis nullo alio impedimento accedente.</s> <s xml:id="echoid-s31490" xml:space="preserve"> Sic ergo errores ſcientiæ accidunt uiſui <lb/>ſecundum ſingulas intemperãtias 8 circumſtantiarum rei uiſæ, ut patet.</s> <s xml:id="echoid-s31491" xml:space="preserve"> His autem & eorum ſimi-<lb/>libus non duximus multum inſiſtendum, quia hæc, quæ diximus, ſufficiunt pro talium omnium <lb/>radice.</s> <s xml:id="echoid-s31492" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s31493" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1281" type="section" level="0" n="0"> <head xml:id="echoid-head1004" xml:space="preserve" style="it">156. In ſolo uiſu error quando accidit propter intemperãtiam cuiuslibet octo circumſtan-<lb/>tiarum rerum per ipſum propriè uiſarum. Alhazen 20 n 3.</head> <p> <s xml:id="echoid-s31494" xml:space="preserve">Quia enim, ut patet per principium 3 huius, lux & color ſunt per ſe obiectum uiſus, palàm quòd <lb/>eiſoli non poteſt error accidere niſi in luce & colore.</s> <s xml:id="echoid-s31495" xml:space="preserve"> Accidit autem uiſui in illis error propter ipſo-<lb/>rum intemperantiam in fortitudine, ut lux fortis non permittit alia uiſibilia uideri, & color fortis <lb/>facit res alias quaſcunq;</s> <s xml:id="echoid-s31496" xml:space="preserve"> in colore ſibi ſimiles uideri, cum tamen illorum color ſit diuerſus.</s> <s xml:id="echoid-s31497" xml:space="preserve"> Et ſimi-<lb/>liter eſt in lucis & coloris debilitate.</s> <s xml:id="echoid-s31498" xml:space="preserve"> Si enim corpus, in quo ſit multa colorum diuerſitas, occurrat <lb/>uiſui ſub luce multùm debili, ut ueſtis diuerſi coloris, apparebit unius coloris.</s> <s xml:id="echoid-s31499" xml:space="preserve"> Et ſi color ſit ualde <lb/>debilis, etiam in luce temperata non uidebitur, & ſic lux extra temperantiam facit uiſui deceptio-<lb/>nem ſecundum utrunq;</s> <s xml:id="echoid-s31500" xml:space="preserve"> extremorum.</s> <s xml:id="echoid-s31501" xml:space="preserve"> Diſtantia etiã uiſibilium errorem inducit uiſui:</s> <s xml:id="echoid-s31502" xml:space="preserve"> quia propter <lb/>improportionatam diſtantiam res colorum diuerſorum minuratim ipſis aſperſa, uidebitur unius <lb/>coloris.</s> <s xml:id="echoid-s31503" xml:space="preserve"> Situs etiam oppoſitionis ſenſum errare facit:</s> <s xml:id="echoid-s31504" xml:space="preserve"> quia cum corpus uiſum fuerit multùm obli-<lb/>quatum, occultabuntur propter ſui obliquationem ipſi uiſui minutæ eius particulæ:</s> <s xml:id="echoid-s31505" xml:space="preserve"> & ſi fuerit in <lb/>partibus minutis colorum diuerſitas, apparebit in totali corpore:</s> <s xml:id="echoid-s31506" xml:space="preserve"> & ſi corpus redieritad directam <lb/>oppoſitionem, illorum colorum diuerſitas apparebit, niſit fortè elongatio partium colorati corpo-<lb/>ris ab axe uiſuali fuerit nimis magna.</s> <s xml:id="echoid-s31507" xml:space="preserve"> Magnitudo etiam uiſui errorem inducit:</s> <s xml:id="echoid-s31508" xml:space="preserve"> quia etiam luce & <lb/>diſtantia, & ſitu uiſioni conuenientibus, colores paruarum partium corporis, diuerſi coloris eua-<lb/>dunt uiſum, & uidetur res unius coloris:</s> <s xml:id="echoid-s31509" xml:space="preserve"> quod non fieret, ſi paruitas partium temperamentum non <lb/>exiret.</s> <s xml:id="echoid-s31510" xml:space="preserve"> Soliditas etiam eſt cauſſa deceptionis uiſus, ſi nimis remiſſa fuerit:</s> <s xml:id="echoid-s31511" xml:space="preserve"> unde cryſtallus uidetur <lb/>colorata colore rei ſibi ſuppoſitæ propter ſuæ ſoliditatis paruitatem:</s> <s xml:id="echoid-s31512" xml:space="preserve"> quod non accideret, ſi cryſtal-<lb/>lus plus ſolida eſſet.</s> <s xml:id="echoid-s31513" xml:space="preserve"> Ex diaphanitate etiam error accidit uiſui:</s> <s xml:id="echoid-s31514" xml:space="preserve"> quia propter interp oſitionem flam-<lb/>mæ inter uiſum & rem uiſam, etiam ſi illa res uiſa fortis ſit coloris, uidebitur illud corpus tenebro-<lb/>fum propter ſolam carentiam diaphanitatis in medio.</s> <s xml:id="echoid-s31515" xml:space="preserve"> Tempus etiam eſt cauſſa erroris:</s> <s xml:id="echoid-s31516" xml:space="preserve"> quia ſi ſubi-<lb/>tò ſuper corpus diuerſorum colorum fiat uiſus directio, apparebit illud corpus coloris unius, do-<lb/>nec per diligentem intuitum diſcernatur.</s> <s xml:id="echoid-s31517" xml:space="preserve"> Debilitas etiam uiſus errorem prætendit in uiſione præ-<lb/>miſſorum:</s> <s xml:id="echoid-s31518" xml:space="preserve"> luce enim forti in uiſum agẽte, læditur uiſus ſtatim, & ad colorem alicuius corporis con-<lb/>uerſus ipſum colorem tenebroſum recipit, donec poſt aliquod tempus læſio receſſerit.</s> <s xml:id="echoid-s31519" xml:space="preserve"> Similiter <lb/>etiam cum adeſt oculis infirmitas, occulta bitur uiſui colorum uarietas:</s> <s xml:id="echoid-s31520" xml:space="preserve"> & ſic fit error in talibus ex <lb/>ſola uiſus qualitate à temperamento recedente.</s> <s xml:id="echoid-s31521" xml:space="preserve"> Patet ergo quòd ſecundum omnes circumſtantias <lb/>rerum uiſibilium in ſolo uiſu fieri deceptionem eſt poſsibile.</s> <s xml:id="echoid-s31522" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s31523" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1282" type="section" level="0" n="0"> <head xml:id="echoid-head1005" xml:space="preserve" style="it">157. Fulgidum mixtum nigro, ſiue per nigrum medium, uiſui colorem præſentat puniceum.</head> <p> <s xml:id="echoid-s31524" xml:space="preserve">Huius declaratio eſt ex ſenſibilιbus naturalibus experientijs:</s> <s xml:id="echoid-s31525" xml:space="preserve"> uidemus enim quòd in ſpeculis <lb/>benè terſis fulgidis res fulgida uiſui præſentatur in ſui fulgore:</s> <s xml:id="echoid-s31526" xml:space="preserve"> quòd ſi ſpeculum fulgidum nõ fue-<lb/>rit, tunc forma fulgidi permixta nigro colore ſpeculi præſentatur uiſui, non intentione ſui fulgoris, <lb/>ſed quaſi aliquantulum denigrata, & ita rubea ſiue punicea apparet.</s> <s xml:id="echoid-s31527" xml:space="preserve"> Vniuerſale enim eſt, ut in prin-<lb/>cipio 2 huius ſuppoſitum eſt, quòd rerum ualde coloratarum colores lumenq́ue ipſius medij co-<lb/>lori permixta ferátur ad uiſum, ut ſi per uitrum coloratum aliqua res uideatur, quòd color rei ui-<lb/>ſæ ex colore proprio & colore uitri permixtus uiſui præſentetur:</s> <s xml:id="echoid-s31528" xml:space="preserve"> & horum multas experientias <lb/>planè poterit quis uidere.</s> <s xml:id="echoid-s31529" xml:space="preserve"> Euenit etiam humidos oculos habentibus, quòd forma albi fulgidi per <lb/>infectos humores & tunicas oculi ad centrum oculi perueniens, in medium colorem uiſus iudicio <lb/>permutatur, & apparet oculo coloris punicei phantaſia.</s> <s xml:id="echoid-s31530" xml:space="preserve"> Et etiam uidemus uiridium lignorũ flam-<lb/>mam rubeam appropinquare puniceo colori:</s> <s xml:id="echoid-s31531" xml:space="preserve"> quia ignis fulgidus & albus exiſtens per fumum ni-<lb/>grum propter groſsitiem materiæ, & humiditatem aqueam, quę illi fumo miſcetur, puniceus uide-<lb/>tur.</s> <s xml:id="echoid-s31532" xml:space="preserve"> Per caliginem quoq;</s> <s xml:id="echoid-s31533" xml:space="preserve"> & fumum nigrum uidetur ſol non fulgidus ſed puniceus, quando talem <lb/>fumum uel caliginem ſoli & uiſibus accidit interponi:</s> <s xml:id="echoid-s31534" xml:space="preserve"> & hoc idem in alijs ſtellis poterit perpendi.</s> <s xml:id="echoid-s31535" xml:space="preserve"> <lb/>Item circuli, qui circa candelas uidentur, propter groſsitiem aeris & nigredinem purpurei uiden-<lb/>tur:</s> <s xml:id="echoid-s31536" xml:space="preserve"> quoniam aer ingroſſatus à natura lucidi aliqualiter impeditur, & propter admixtionẽ umbræ <lb/>nigredine permiſceri uidetur, uel alio medio colore ſecundum diſpoſitionem luminis & admixtæ <lb/>umbræ.</s> <s xml:id="echoid-s31537" xml:space="preserve"> Et ad hoc etiam plenius declarandum diligẽs inquiſitor plures experientias poterit appli-<lb/>care.</s> <s xml:id="echoid-s31538" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s31539" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1283" type="section" level="0" n="0"> <head xml:id="echoid-head1006" xml:space="preserve" style="it">158. Viſum protenſum longè debiliorem fieri patens eſt.</head> <p> <s xml:id="echoid-s31540" xml:space="preserve">Non enim uiſus uidet ſimiliter de longè poſita, quemadmodum propè exiſtẽtia.</s> <s xml:id="echoid-s31541" xml:space="preserve"> Si enim uidea-<lb/>tur de longè corpus foraminoſum, cuius ſint parua foramina, totũ uidetur continuum:</s> <s xml:id="echoid-s31542" xml:space="preserve"> unde ſi ali-<lb/> <pb o="189" file="0491" n="491" rhead="LIBER QVINTVS."/> quis uaporem roridum de longè uideat, totum ipſum fore unum corpus continuum uiſus indica-<lb/>bit:</s> <s xml:id="echoid-s31543" xml:space="preserve"> quin etiam uiſus recta curua, rotunda quadrata ex remotione iudicat, ſicut eſt in præmiſsis hu-<lb/>ius libritheorematibus declaratum.</s> <s xml:id="echoid-s31544" xml:space="preserve"> Et ſi uiſus pannum coloratum, in quo eſt minuta colorum di-<lb/>uerſorum conſperſio, ad quos proportionata partium elongatio ſit intemperata ipſi uiſui, diutius <lb/>etiam aſpexerit:</s> <s xml:id="echoid-s31545" xml:space="preserve"> apparebit pannus ille unius coloris tantùm, quoniam extra temperantiam eſt lon-<lb/>gitudo, reſpectu partialium colorum, licet omnia alia conueniantin debita temperantia, reſpectu <lb/>uiſus.</s> <s xml:id="echoid-s31546" xml:space="preserve"> Quia ergo uiſibilem rei circum ſtantiam uiſus protenſus nõ perſpicit, palàm quia debilitatur <lb/>ex protenſione ſui ad uiſibile, ſiue ex remotione uiſibilis ab ipſo.</s> <s xml:id="echoid-s31547" xml:space="preserve"> Et hoc eſt, quod proponebatur.</s> <s xml:id="echoid-s31548" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1284" type="section" level="0" n="0"> <head xml:id="echoid-head1007" xml:space="preserve" style="it">159. Nigredinis in re non nigra apparitio ex uiſus prouenit defectione.</head> <p> <s xml:id="echoid-s31549" xml:space="preserve">Experientia ſimiliter comprobatur, quod hic proponitur, auxilio pręcedentis.</s> <s xml:id="echoid-s31550" xml:space="preserve"> Quia enim uiſum <lb/>protenſum longè debiliorem fieri patens eſt, ut præmiſſum eſt:</s> <s xml:id="echoid-s31551" xml:space="preserve"> ideo accιdit quòd ea, quæ longè ui-<lb/>dentur, propter uiſus debilitationem omnia nigriora apparent, ſicut etiã corpora remotiora & mi-<lb/>nora & planiora quàm ſint, uiſibus apparent:</s> <s xml:id="echoid-s31552" xml:space="preserve"> quoniam eminentiæ ſuarum partium aſperitates & <lb/>tumores in ipſis facientes non uidentur.</s> <s xml:id="echoid-s31553" xml:space="preserve"> Similiter etiam, quæ in ſpeculis uidentur, quia propter re-<lb/>flexionem ipſorum diſtantia augetur, ideo propter remotionem, quæ accidit uiſui, talia nigriora <lb/>uidentur experimentanti.</s> <s xml:id="echoid-s31554" xml:space="preserve"> Quantò enim magis ex remotione etiã rei albæ immoto ſpeculo diſtan-<lb/>tia à ſuperficie ſpeculi augmẽtatur, tantò magis color ille albus uiſui ad nigre dinem accedit:</s> <s xml:id="echoid-s31555" xml:space="preserve"> unde <lb/>etiam nubes apparentes in aqua nigriores uidentur quàm in loco ſuo, uiſu in eodem loco exiſtẽte, <lb/>quoniam reflexio facta in aqua auget diſtantiam:</s> <s xml:id="echoid-s31556" xml:space="preserve"> nihil autem differt aliquid multum diſtans uiſui <lb/>apparere, aut uiſum per multam diſtantiam uiſionem rei complere:</s> <s xml:id="echoid-s31557" xml:space="preserve"> ſemper enim fit iudicium uir-<lb/>tutis uiſiuæ, ſecundum quod forma eſt in uiſus organo recepta.</s> <s xml:id="echoid-s31558" xml:space="preserve"> Neq;</s> <s xml:id="echoid-s31559" xml:space="preserve"> latebit hic experimentantem, <lb/>quia quando clara nubes fuerit uicina ſoli, tunc alicui aſpicienti ad nubem, nubes nõ uidebitur niſi <lb/>alba:</s> <s xml:id="echoid-s31560" xml:space="preserve"> ſed ſi reflectatur ab aqua, & eam uiſus in aqua uideat:</s> <s xml:id="echoid-s31561" xml:space="preserve"> tunc illa nubes alba aliquem colorem ex <lb/>medijs coloribus uiſui præſentabit, ut puniceum, purpureum, uiridẽ, & lazulium:</s> <s xml:id="echoid-s31562" xml:space="preserve"> unde ſicut uiſus <lb/>colorem nigrum per reflexionem uidet eſſe nigriorem, ſic & colorem album uidet minus album <lb/>propter reflexionem.</s> <s xml:id="echoid-s31563" xml:space="preserve"> Nubem itaq;</s> <s xml:id="echoid-s31564" xml:space="preserve"> albam exiſtentem uidet uiſus propter diſtantiã ampliorem, quę <lb/>fit per reflexionem, in ſuo colore nigram, & ſimilem priuationi & negationi propter uiſus protenſi <lb/>debilitatem.</s> <s xml:id="echoid-s31565" xml:space="preserve"> Et quoniam coloratio nubis fit ex impreſsione luminis ab aliquo corpore luminoſo, <lb/>poteſt concludi ex præmiſsis, quòd in omni corpore, cui lumen uel color ex corpore luminoſo im-<lb/>primitur, eandem cauſſam & effectum participem habebit.</s> <s xml:id="echoid-s31566" xml:space="preserve"> Ethoc eſt, quod proponebatur.</s> <s xml:id="echoid-s31567" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1285" type="section" level="0" n="0"> <head xml:id="echoid-head1008" xml:space="preserve">VITELLONIS FI-<lb/>LII THVRINGORVM ET PO-<lb/>LONORVM OPTICAE LIBER QVINTVS.</head> <p style="it"> <s xml:id="echoid-s31568" xml:space="preserve"><emph style="sc">Expeditis</emph> aliqualiter his, quæ ſimplici & directæ uiſioni neceſſaria <lb/>exiſtere, & eius deceptionibus accidere uiſa ſunt reſtatnũc ut conuenien-<lb/>ter eum modum uiſionis, qui fit per reflexionem à politis corporibus, quæ <lb/>ſpecula dicimus, proſequẽtes, de omni reflexionis modo à quibuſcun ſpe-<lb/>culis ex quiſitius pertractemus.</s> <s xml:id="echoid-s31569" xml:space="preserve"> Primò ita in præſenti quinto huius ſciẽtiæ libro præmit-<lb/>temus quælibet illorum, quæ æſtimamus cõmunia omnibus ſpeculis:</s> <s xml:id="echoid-s31570" xml:space="preserve"> & deinde adiun-<lb/>gemus paßiones, quæ accidunt rebus & uiſui à ſolis fpeculis planis, quorum ſpeculorum <lb/>forma ſimplicior eſt formis omniũ aliorum ſpeculorum:</s> <s xml:id="echoid-s31571" xml:space="preserve"> propter quod & ſpeculorũ pla-<lb/>norum paßiones quibuſdam alijs ſpeculis ſunt cõmunes, ut patebit in libris ſequentibus, <lb/>quibus aliorum ſpeculorum paßiones proprias reſeruamus.</s> <s xml:id="echoid-s31572" xml:space="preserve"> Veruntamen ſicut in princi-<lb/>pio huius ſcientiæ diximus, non intelligimus in hoctractatu per ſpecula corpora tantùm <lb/>formata & polita per artificiüm, ſed etiam ipſa corpora naturalia, à quorũ ſuperficie-<lb/>bus fit eadem reflexio, quæ & à corporum artificialium ſuperficiebus accidit.</s> <s xml:id="echoid-s31573" xml:space="preserve"> Nec in-<lb/>telligimus, quòd ſolum hæc reflexio fiat ad uiſus animalium, ſed etiam ipſis uiſibus non <lb/>præſentibus fit reflexio formarũ, & accidit uiſibus, ſi inlocis reflexarũ formarum diſ-<lb/>ponantur, quòd fiat reflexio ad ipſos:</s> <s xml:id="echoid-s31574" xml:space="preserve"> quod manifeſtè patet per hæc, quia non in omni <lb/>loco fit reflexio ad quemcunq;</s> <s xml:id="echoid-s31575" xml:space="preserve"> uiſum à ſpeculo quocu;</s> <s xml:id="echoid-s31576" xml:space="preserve">. Eſt tamẽ in receptione harum <lb/>formarũ reflexarũ in uiſibus aliqua proprietas, & maximè in illis reflexionũ modis, in <lb/> <pb o="190" file="0492" n="492" rhead="VITELLONIS OPTICAE"/> quibus fit aliqua deceptio in uiſu.</s> <s xml:id="echoid-s31577" xml:space="preserve"> Quamuis autem, ut in proæmio buius ſcientiæ dixi-<lb/>mus, idem immittatur in contrarium & in ſenſum:</s> <s xml:id="echoid-s31578" xml:space="preserve"> quoniam unius rei una & eadem <lb/>forma ſemper diffunditur per medium, propter quod eadem forma reflectitur à ſuper-<lb/>ficiebus ſpeculorum, quæ etiam in modo ſimplicis uiſionis directè uiſibus occurrit:</s> <s xml:id="echoid-s31579" xml:space="preserve"> non <lb/>poteſt tamen in reflexione facta à ſuperficiebus ſpeculorum quorumcun comprehendi <lb/>ueritas formæ, ſicut comprehenditur in uiſione ſimplici directa.</s> <s xml:id="echoid-s31580" xml:space="preserve"> In reflexionibus enim à <lb/>quibuſcun ſpeculis factis apparet forma rei ut plurimum præ oculis, ipſis uiſibus quaſi <lb/>oppoſita, cum tamen ſecundum ueritatem illis non opponatur.</s> <s xml:id="echoid-s31581" xml:space="preserve"> Lux quo & color cor-<lb/>poris uiſi ſemper miſcentur cum colore ſpeculi, à quo fit reflexio, quam mixturam in re-<lb/>flexionibus uiſus percipit, & nõ ueram lucem uel uerum rei uiſæ colorem.</s> <s xml:id="echoid-s31582" xml:space="preserve"> Omnis quo <lb/>reflexio, ut nos inferius perfectius declarabimus, debilitat luces & colores:</s> <s xml:id="echoid-s31583" xml:space="preserve"> unde in o-<lb/>mnireflexione latet uiſum ueritas lucis & coloris, plus quàm in directa ſimplici uiſio-<lb/>ne.</s> <s xml:id="echoid-s31584" xml:space="preserve"> Quæ uerò ad hunc uiſionis modum, quæ fit per reflexionem à quibuſcun, & à pla-<lb/>nis maximè ſpeculis, præmittimus, ſunt iſta.</s> <s xml:id="echoid-s31585" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1286" type="section" level="0" n="0"> <head xml:id="echoid-head1009" xml:space="preserve">DEFINITIONES.</head> <p> <s xml:id="echoid-s31586" xml:space="preserve">1.</s> <s xml:id="echoid-s31587" xml:space="preserve"> Politio corporum eſt cõtinuitas partium ſuperficiei politi corporis ſine ſen-<lb/>ſibilitate pororum uel diuiſionis.</s> <s xml:id="echoid-s31588" xml:space="preserve"> 2.</s> <s xml:id="echoid-s31589" xml:space="preserve"> Speculum dicitur omne corpus politũ ope-<lb/>re artis uelnaturæ.</s> <s xml:id="echoid-s31590" xml:space="preserve"> 3.</s> <s xml:id="echoid-s31591" xml:space="preserve"> Linea incidentiæ diciturilla, ſecundum quam forma rei in-<lb/>cidit ſuperficiei ſpeculi.</s> <s xml:id="echoid-s31592" xml:space="preserve"> 4.</s> <s xml:id="echoid-s31593" xml:space="preserve"> Linea reflexionis diciturilla, ſecundum quam forma <lb/>reuerberata, propter ſoliditatem ſpeculi, quam penetrare nõ poteſt, reflectitur ad <lb/>uiſum.</s> <s xml:id="echoid-s31594" xml:space="preserve"> 5.</s> <s xml:id="echoid-s31595" xml:space="preserve"> Punctus incidentiæ dicitur ille punctus, in quo linea incidentiæ incidit <lb/>ſuperficiei ſpeculi:</s> <s xml:id="echoid-s31596" xml:space="preserve"> & idem eſt punctus reflexionis, quoniam formarum reflexio ad <lb/>uiſum ſemper fit à puncto incidentiæ.</s> <s xml:id="echoid-s31597" xml:space="preserve"> 6.</s> <s xml:id="echoid-s31598" xml:space="preserve"> Perpendicularis ſuper ſuperficiem ſpe-<lb/>culi, à quo fit reflexio, dicitur linea orthonogaliter erecta à puncto incidentiæ ſu-<lb/>per ſuperficiẽ ſpeculi illius, à quo ſit reflexio, ſi illa ſuperficies ſit plana:</s> <s xml:id="echoid-s31599" xml:space="preserve"> quòd ſi illa <lb/>ſuperficies ſit conuexa uel concaua:</s> <s xml:id="echoid-s31600" xml:space="preserve"> tunc dicitur perpendicularis ſuperipſam, quæ <lb/>eſt perpendicularis ſuper ſuperficiem planam, illam ſuperficiẽ conuexam uel con-<lb/>cauam in puncto incidentiæ contingẽtem.</s> <s xml:id="echoid-s31601" xml:space="preserve"> 7.</s> <s xml:id="echoid-s31602" xml:space="preserve"> Superficies reflexionis dicitur ſu-<lb/>perficies continens lineam incidentiæ & reflexionis, & perpẽdicularem à puncto <lb/>contingentiæ productam ſuperipſam ſpeculi ſuperficiem, uel ſuper ſuperficiem <lb/>ipſam contingentem.</s> <s xml:id="echoid-s31603" xml:space="preserve"> 8.</s> <s xml:id="echoid-s31604" xml:space="preserve"> Cathetus incidẽtiæ dicitur linea perpendiculariter ere-<lb/>cta ſuper ſuperficiem planam ſpeculi, aut ſuper lineam rectam contingẽtem com-<lb/>munem ſectionem ſuperficiei reflexionis, & ſuperficiei ſpeculi conuexi uel conca-<lb/>ui, ducta à puncto, à quo incipit incidentia, ut à cẽtro uiſus, uel ab alio pũcto quo-<lb/>cunq;</s> <s xml:id="echoid-s31605" xml:space="preserve">, cuius forma à ſpeculo reflectitur ad uiſum.</s> <s xml:id="echoid-s31606" xml:space="preserve"> 9.</s> <s xml:id="echoid-s31607" xml:space="preserve"> Cathetus reflexionis dicitur <lb/>linea erecta ſuper illam eandem ſuperficiem uel lineam à puncto, ad quem termi-<lb/>natur ipſa linea reflexionis, ut à centro uiſus uel ab alio puncto, ad quem reflexio <lb/>terminatur.</s> <s xml:id="echoid-s31608" xml:space="preserve"> 10.</s> <s xml:id="echoid-s31609" xml:space="preserve"> Superficies incidentiæ dicitur ſuperficies contenta à linea rei ui-<lb/>ſæ, & à cathetis incidentiæ terminorum illius lineæ.</s> <s xml:id="echoid-s31610" xml:space="preserve"> 11.</s> <s xml:id="echoid-s31611" xml:space="preserve"> Angulus incidentiæ dici-<lb/>tur angulus, quem in ſuperficie reflexionis continet linea incidentiæ, cū linea, quę <lb/>eſt communis ſectio ſuperficiei reflexionis, & ſuperficiei ipſius ſpeculi, uel ſuperfi-<lb/>ciei ſpeculum in puncto reflexionis contingentis.</s> <s xml:id="echoid-s31612" xml:space="preserve"> 12.</s> <s xml:id="echoid-s31613" xml:space="preserve"> Angulus reflexionis dici-<lb/>tur angulus, quem in ſuperficie reflexionis continet linea reflexionis cum dicta <lb/>communi ſectione.</s> <s xml:id="echoid-s31614" xml:space="preserve"> 13.</s> <s xml:id="echoid-s31615" xml:space="preserve"> Imago dicitur forma in ſpeculo cõprehenſa.</s> <s xml:id="echoid-s31616" xml:space="preserve"> 14.</s> <s xml:id="echoid-s31617" xml:space="preserve"> Locus <lb/>imaginis dicitur locus uiſionis illius formæ, ſcilicetlocus, in quo uidetur forma.</s> <s xml:id="echoid-s31618" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1287" type="section" level="0" n="0"> <head xml:id="echoid-head1010" xml:space="preserve">PETITIONES.</head> <p> <s xml:id="echoid-s31619" xml:space="preserve">Supponimus autem hæc.</s> <s xml:id="echoid-s31620" xml:space="preserve"> 1.</s> <s xml:id="echoid-s31621" xml:space="preserve"> Rei elongatæ & approximatæ ſpeculo, extrema <lb/>quandoq;</s> <s xml:id="echoid-s31622" xml:space="preserve"> uideri.</s> <s xml:id="echoid-s31623" xml:space="preserve"> 2.</s> <s xml:id="echoid-s31624" xml:space="preserve"> Item quòd uniformis ſituatio puncti rei uiſæ reſpectu ſuper-<lb/>ficiei cuiuſcunq;</s> <s xml:id="echoid-s31625" xml:space="preserve"> ſpeculi, à qua eius forma reflectitur, fit ſolum ſecundum cathe-<lb/>tum ſuæ incidentiæ.</s> <s xml:id="echoid-s31626" xml:space="preserve"/> </p> <pb o="191" file="0493" n="493" rhead="LIBER QVINTVS."/> </div> <div xml:id="echoid-div1288" type="section" level="0" n="0"> <head xml:id="echoid-head1011" xml:space="preserve">THE OREMATA</head> <head xml:id="echoid-head1012" xml:space="preserve" style="it">1. Corporum terſorum politorum, cuiuſcun figuræ ſint, ſuperficies à quolibet ſuorum pun-<lb/>ctorum luces, colores, & formas rerum oppoſitarum reflectunt ſecundum rectitudinem linea-<lb/>rum. Euclides 2 hypothe. catoptr. Ptolemæus 1 & 3 the. 1 catoptr. Alhazen 2 n 4.</head> <p> <s xml:id="echoid-s31627" xml:space="preserve">Quoniam enim, ut patuit per 1 th.</s> <s xml:id="echoid-s31628" xml:space="preserve"> 2 huius, forma lucis à corpore luminoſo ſemper ſecundum <lb/>lineam rectam diffunditur in omne corpus ei oppoſitum, & ſimiliter forma colorata habentis <lb/>actum luminis.</s> <s xml:id="echoid-s31629" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s31630" xml:space="preserve"> hæc incidunt alicui corpori terſo polito:</s> <s xml:id="echoid-s31631" xml:space="preserve"> quia in tali corpore non patet <lb/>tranſitus lumini uel colori propter talis corporis denſitatem & priuationem diaphanitatis, cũ ſint <lb/>planarum ſuperficierũ, in quibus nulla eſt aſperitas, ſemper ab illis fit luminis & coloris & forma-<lb/>rum reflexio:</s> <s xml:id="echoid-s31632" xml:space="preserve"> & ob hoc oppofito ſpeculo lumini forti obliquè incidenti, manifeſtè fit ad parietem <lb/>uicinum luminis reflexio & coloris, ſi color fuerit coniumctus lumini, & uidebitur lumen reflexum <lb/>incidens parieti cum colore:</s> <s xml:id="echoid-s31633" xml:space="preserve"> & moto ſpeculo radius reflexus mouebitur mutans locum, & ablato <lb/>ſpeculo lumen reflexum aufertur:</s> <s xml:id="echoid-s31634" xml:space="preserve"> & ſi à loco, cui incidit radius luminoſus, manus uel aliud corpus <lb/>mundum uel politum ſecundum lineam rectam ducatur ad ſuperficiem corporis, à qua fit reflexio:</s> <s xml:id="echoid-s31635" xml:space="preserve"> <lb/>patens erit quoniam ſecundum rectitudinem linearum reflexio eſt facta, quoniam ipſi experimen-<lb/>tanti ſecundum lineam rectam ad corpus, à quo fit reflexio, redeunti, ſemper reflexionem luminis <lb/>accidit uideri.</s> <s xml:id="echoid-s31636" xml:space="preserve"> In omni itaq;</s> <s xml:id="echoid-s31637" xml:space="preserve"> polita ſuperficie cuiuſcunq;</s> <s xml:id="echoid-s31638" xml:space="preserve"> ſit figuræ, à quolibet ſuorum pũctorum fit <lb/>reflexio ſecundum rectitudinem linearum:</s> <s xml:id="echoid-s31639" xml:space="preserve"> caditenim in quo dlibet puctum corporis politi lux à <lb/>quolibet puncto corporis luminoſi.</s> <s xml:id="echoid-s31640" xml:space="preserve"> Vnde ſicut oſtenſum eſt in 20 th.</s> <s xml:id="echoid-s31641" xml:space="preserve"> 2 huius ſuper quodlibet pun-<lb/>ctum corporis politi fit pyramis, cuius uertex eſt in pũcto corporis politi, & baſis in ſuperficie cor-<lb/>poris luminoſi:</s> <s xml:id="echoid-s31642" xml:space="preserve"> & à quolibet puncto luminoſi corporis procedit pyramis, cuius uertex eſt in pun-<lb/>cto corporis luminoſi, & baſis in ſuperficie corporis politi.</s> <s xml:id="echoid-s31643" xml:space="preserve"> Et ſi à corpore luminoſo procedit lux <lb/>ad corpus politum ſecundum lineas æquidiſtantes, illæ lineæ quaſi columnam continentes termi-<lb/>nantur ad baſes pyramidum præmiſſarum.</s> <s xml:id="echoid-s31644" xml:space="preserve"> Per quaſcunq;</s> <s xml:id="echoid-s31645" xml:space="preserve"> autem lineas lumen corpori polito inci-<lb/>dit, ſecundum illarum proprietatem reflectitur, ſiue ſint perpendiculares ſiue obliquæ:</s> <s xml:id="echoid-s31646" xml:space="preserve"> patet ergo <lb/>propoſitum.</s> <s xml:id="echoid-s31647" xml:space="preserve"> Fit autẽ à corporibus politis reflexio lucis, non autem à corporibus non politis, aſpe-<lb/>ris:</s> <s xml:id="echoid-s31648" xml:space="preserve"> quoniam in illis ſunt pori & foueæ, quas ſubintrat lumen, & redit in ſe permixtum cum umbra <lb/>illorum corporum:</s> <s xml:id="echoid-s31649" xml:space="preserve">unde non fit reflexio ſenſibilis ab illis.</s> <s xml:id="echoid-s31650" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1289" type="section" level="0" n="0"> <head xml:id="echoid-head1013" xml:space="preserve" style="it">2. Ab omni corpore colorato præſente luce, color ad corpus oppoſitum politum mixtim cum <lb/>lumine mittitur: & quando totaliter, quando partialiter reflectitur ab illo, ſicut & ipſum <lb/>lumen. Ptolemæus 3 th. 1 catoptr. Alhazen 3 n 4.</head> <p> <s xml:id="echoid-s31651" xml:space="preserve">Quòd hic proponitur, experimẽtaliter declaratur.</s> <s xml:id="echoid-s31652" xml:space="preserve"> Sit enim, ut intra domum unius tantùm fene-<lb/>ſtræ deſcendat lux ſolis ſuper corpus multum coloratũ forti colore:</s> <s xml:id="echoid-s31653" xml:space="preserve"> & ponatur in oppoſitione con-<lb/>tra ipſum ſpeculum argenteum, & iterum cõtra ſpeculum ponatur uas concauum ad modum ſcy-<lb/>phi, quod ſit interius album, uel in quo ponatur corpus album, & aptetur taliter ut lux reflexa inci-<lb/>dat ſuper illud corpus album.</s> <s xml:id="echoid-s31654" xml:space="preserve"> Apparebit itaq, ſuper faciem albi corporis color illius corporis, in <lb/>quod primò fit deſcenſus lucis.</s> <s xml:id="echoid-s31655" xml:space="preserve"> Color itaq;</s> <s xml:id="echoid-s31656" xml:space="preserve"> mixtim cum luce reflectitur:</s> <s xml:id="echoid-s31657" xml:space="preserve"> ergo etiam mixtim cũlu-<lb/>mine incidit corpori polito:</s> <s xml:id="echoid-s31658" xml:space="preserve"> quod corpus politum ſi denſum & durum fuerit, color cum luce tota-<lb/>liter ab ipſo reflectitur, ita ut non coloret corpus politum.</s> <s xml:id="echoid-s31659" xml:space="preserve"> Si uerò corpus politum ſit rarum & luci-<lb/>dum actu, ſicut ſunt aqua & uitrum, & ſimilia:</s> <s xml:id="echoid-s31660" xml:space="preserve"> tunc refle ctũtur ab illo colores & luces, & penetrant <lb/>in illud:</s> <s xml:id="echoid-s31661" xml:space="preserve"> quod patet per hoc, quòd forma reflexionis ab his corporibus eſt debilioris lucis & colo-<lb/>ris, quàm ab alijs corporibus denſioribus, quàm ſint illa:</s> <s xml:id="echoid-s31662" xml:space="preserve"> & etiam circa aliquod punctum ſub iſtis <lb/>corporibus, uel in iſtis uidentur formæ lucis & coloris incidẽtes ſuperiori ſuperficiei iſtorum cor-<lb/>porum.</s> <s xml:id="echoid-s31663" xml:space="preserve"> Patet ergo illud, quod proponebatur.</s> <s xml:id="echoid-s31664" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1290" type="section" level="0" n="0"> <head xml:id="echoid-head1014" xml:space="preserve" style="it">3. Omnis reflexio debilitat luces & colores: & uniuerſaliter omnes formas. Alhazẽ 4 n 4.</head> <p> <s xml:id="echoid-s31665" xml:space="preserve">Quoniam enim lux continua fortior eſt luce diſgregata per 1 petitionem 2 huius, & quantò lux <lb/>ab ortu ſuo plus elongatur, tantò plus debilitatur per 22 th.</s> <s xml:id="echoid-s31666" xml:space="preserve"> 2 huius:</s> <s xml:id="echoid-s31667" xml:space="preserve"> patet quòd cum ſecundum ali-<lb/>quem punctum corporis luminoſi procedit lux ad ſuperficiem corporis politi in modum pyrami-<lb/>dis, quòd quantò magis elongatur à puncto illo, tantò maior eſt eius debilitatio, & propter elonga-<lb/>tionem ab ortu lucis, & propter diſgregationẽ.</s> <s xml:id="echoid-s31668" xml:space="preserve"> Lux uerò reflexa ab aliquo polito corpore plus de-<lb/>bilitatur, tum propter elongationem à loco reflexionis & diſgregationẽ, tum propter ipſam refle-<lb/>xionem.</s> <s xml:id="echoid-s31669" xml:space="preserve"> Luces quoq;</s> <s xml:id="echoid-s31670" xml:space="preserve"> ſecundum lineas æquidiſtantes politis corporibus incidẽtes, ſunt debiliores <lb/>quàm luces obliquè incidentes, quoniã minus aggregantur.</s> <s xml:id="echoid-s31671" xml:space="preserve"> Colorum quoq;</s> <s xml:id="echoid-s31672" xml:space="preserve"> reflexio quamuis fiat <lb/>ab omni corpore polito, ſicut & lucis, ut patet per 1 huius:</s> <s xml:id="echoid-s31673" xml:space="preserve"> non tamen eſt multum ſenſibilis, propter <lb/>debilitationem, quę fit ex reflexione, & propter admixtionẽ coloris ipſius ſpeculi conformis colo-<lb/>rum reflexorum, niſi fortè à ſpeculo argẽteo fiat reflexio.</s> <s xml:id="echoid-s31674" xml:space="preserve"> In ferreo enim ſpeculo color apparet de-<lb/>bilior, quoniam color ferri mixtus cum luce reflexa, & ipſo colore reflexo debilitat ipſum colorem <lb/>reflexum.</s> <s xml:id="echoid-s31675" xml:space="preserve"> Omnes itaq;</s> <s xml:id="echoid-s31676" xml:space="preserve"> reflexiones colorum optimè experiri poſſunt in domo unici foraminis, cui <lb/>foramini albus paries opponitur.</s> <s xml:id="echoid-s31677" xml:space="preserve"> Tunc enim in radio ſolis poſito ſpeculo argenteo, & ipſi ſpeculo <lb/>& parieti interpoſita re aliqua colorata:</s> <s xml:id="echoid-s31678" xml:space="preserve"> erit reflexio coloris ad parietẽ album ſenſibilis.</s> <s xml:id="echoid-s31679" xml:space="preserve"> Idem quoq:</s> <s xml:id="echoid-s31680" xml:space="preserve"> <lb/> <pb o="192" file="0494" n="494" rhead="VITELLONIS OPTICAE"/> accidit ſi in radio incidentiæ ipſius ſpeculi ponatur corpus diaphanum coloratum, per quod tran-<lb/>ſeat radius, incidens ipſi ſpeculo, utpote ſi ante feneſtram ponatur uitrum coloratum, uel ſi modo <lb/>ſimili, ut experimentanti uidebitur, diſponatur.</s> <s xml:id="echoid-s31681" xml:space="preserve"> Cadẽte itaq;</s> <s xml:id="echoid-s31682" xml:space="preserve"> luce forti ſuper ſpeculum argenteum <lb/>& ipſa reflexa ſuper parietem album, notabiliter uidebitur lux parietis debilior quàm ſpeculi.</s> <s xml:id="echoid-s31683" xml:space="preserve"> Re-<lb/>flexio ergo lucem debilitat.</s> <s xml:id="echoid-s31684" xml:space="preserve"> Et eodẽ modo color reflexus eſt debilior colore, à quo fit reflexio.</s> <s xml:id="echoid-s31685" xml:space="preserve"> Pa-<lb/>làm ergo quòd reflexio debilitat luces & colores, ſed colores magis quàm luces.</s> <s xml:id="echoid-s31686" xml:space="preserve"> Colores enim de-<lb/>biliori modo incidunt quàm luces:</s> <s xml:id="echoid-s31687" xml:space="preserve"> unde etiam in reflexione facilius debilitantur.</s> <s xml:id="echoid-s31688" xml:space="preserve"> Color enim de-<lb/>bilis cum ad ſpeculũ peruenerit, miſcetur colori ſpeculi & immutatur propter illius admixtionem:</s> <s xml:id="echoid-s31689" xml:space="preserve"> <lb/>quare color reflexus apparet debilis & tenebroſus:</s> <s xml:id="echoid-s31690" xml:space="preserve"> & uniuerſaliter formæ reflexæ ſunt debiliores <lb/>quàm ſint in loco, à quo reflectuntur.</s> <s xml:id="echoid-s31691" xml:space="preserve"> Sic ergo patet quòd omnis reflexio eſt cauſſa debilitatis.</s> <s xml:id="echoid-s31692" xml:space="preserve"> Nam <lb/>& hoc patet ſenſibiliter in luce:</s> <s xml:id="echoid-s31693" xml:space="preserve"> licet enim lux directa & lux reflexa ęqualiter diſtent ab ortu ſuo, ta-<lb/>men debilior eſt lux reflexa.</s> <s xml:id="echoid-s31694" xml:space="preserve"> Opponatur enim in aere radio ſolis intrãti per feneſtram domum ali-<lb/>quam, in qua unica eſt feneſtra, ſpeculum minus foramine, ita ut lux reſidua foraminis, quę non in-<lb/>cidit in ſpeculo, cadat in terram ſuper corpus album:</s> <s xml:id="echoid-s31695" xml:space="preserve"> & lux à ſpeculo reflexa cadat ſimiliter ſuper <lb/>corpus album eleuatum à terra, hoc obſeruato, ut ſit eadem diſtantia corporis eleuati & iacentis à <lb/>centro foraminis feneſtræ:</s> <s xml:id="echoid-s31696" xml:space="preserve"> uidebitur itaq;</s> <s xml:id="echoid-s31697" xml:space="preserve"> ſuper corpus album eleuatum, ad quod fit reflexio, lux <lb/>minor, quàm ſuper corpus iacens:</s> <s xml:id="echoid-s31698" xml:space="preserve"> cuius minoritatis ſola reflexio eſt cauſſa.</s> <s xml:id="echoid-s31699" xml:space="preserve"> Etidem poteſt in colo-<lb/>rum reflexione faciliter demonſtrari, & eodem modo.</s> <s xml:id="echoid-s31700" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s31701" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1291" type="section" level="0" n="0"> <head xml:id="echoid-head1015" xml:space="preserve" style="it">4. Omnis lux reflexa, etſi debilior ſit luce prima, eſt tamen fortior quàm lux ſecunda, æqua-<lb/>liter ab origine diſtantibus ambabus: & idem eſt in colore. Alhazen 5 n 4.</head> <p> <s xml:id="echoid-s31702" xml:space="preserve">Luce enim reflexa cadente in aliquod corpus, ſi aliud ſimile corpus ponatur extra locum refle-<lb/>xionis, & ſit cum illo eiuſdem elongationis à ſpeculo:</s> <s xml:id="echoid-s31703" xml:space="preserve"> uidebitur ſuper ipſum corpus ſecunda lux <lb/>minor, quàm in illo, quod eſt poſitum in loco reflexionis.</s> <s xml:id="echoid-s31704" xml:space="preserve"> Sit enim, quòd in directo foraminis, per <lb/>quod radiusdomum aliquam ingreditur, ponatur ſpeculum in terra, ſuſcipiens totam lucem radij <lb/>incidentis per illam feneſtram, quam lucem ſuperius in principio 2 libri huius ſcientiæ diximus lu-<lb/>cem primam:</s> <s xml:id="echoid-s31705" xml:space="preserve"> tunc enim fiet palàm, quòd erit lux fortior ſuper corpus in loco reflexionis poſitum, <lb/>quàm ſuper aliud corpus ſimile poſitum extra illum locum tantundem à ſpeculo elongatum.</s> <s xml:id="echoid-s31706" xml:space="preserve"> Et <lb/>idem accidit ſi ſuperficies ſpeculi non ſuſcipiat radium directè, ſed obliquè.</s> <s xml:id="echoid-s31707" xml:space="preserve"> Idem etiam patet in co-<lb/>loribus:</s> <s xml:id="echoid-s31708" xml:space="preserve"> quoniam facta reflexione coloris à ſpeculo argenteo, corpus album poſitum in loco refle-<lb/>xionis plurimum recipit coloris:</s> <s xml:id="echoid-s31709" xml:space="preserve"> aliud uerò corpus æquè album exiſtens extra locum reflexionis, <lb/>& in eadem diſtantia à ſpeculo, apparet quidem coloratum, ſed debilius ualde quàm corpus poſi-<lb/>tum in loco reflexionis:</s> <s xml:id="echoid-s31710" xml:space="preserve"> & ſi ferreum fuerit ſpeculum fortè in corpore, quod eſt in loco reflexionis, <lb/>modicus uidebitur color, extra uerò locum reflexionis in corpore æquè albo, quaſi nullus appare-<lb/>bit color.</s> <s xml:id="echoid-s31711" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s31712" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1292" type="section" level="0" n="0"> <head xml:id="echoid-head1016" xml:space="preserve" style="it">5. Natura agit in omnibus ſecundum lineas breuiores. Euclides in præfatione opticorum. <lb/>Ptolemæus 1 th. 1 catoptr.</head> <p> <s xml:id="echoid-s31713" xml:space="preserve">Hoc uniuerſaliter patet in omnibus operibus naturæ.</s> <s xml:id="echoid-s31714" xml:space="preserve"> Omnes enim motus naturales ſic fiũt:</s> <s xml:id="echoid-s31715" xml:space="preserve"> de-<lb/>ſcendunt enim grauia perpendiculariter ſuper ſuperficiem horizontis.</s> <s xml:id="echoid-s31716" xml:space="preserve"> Sagittæ etiam emiſſæ uio-<lb/>lenter ab arcubus ſeruntur linea breuiori ſecundum angulum ſuæ emiſsionis:</s> <s xml:id="echoid-s31717" xml:space="preserve"> per breuiorem enim <lb/>lineam ab eodem termino in eundem terminum uelociter eſt motus.</s> <s xml:id="echoid-s31718" xml:space="preserve"> Et quia, ut in principio 2 libri <lb/>huius ſcientiæ ſuppoſitum eſt, natura nihil agit fruſtra, neq;</s> <s xml:id="echoid-s31719" xml:space="preserve"> deficit in neceſſarijs:</s> <s xml:id="echoid-s31720" xml:space="preserve"> palàm quòd ne-<lb/>ceſſariò agit ſecundum lineas breuiores.</s> <s xml:id="echoid-s31721" xml:space="preserve"> Si enim poſsit operatio nem intentam complere per mo-<lb/>tum uel actionem per lineam a b, & agat per <lb/>lineam a b c:</s> <s xml:id="echoid-s31722" xml:space="preserve"> omnis actio, quam facit in linea <lb/>b c eſt fruſtra, quoniam cõſecuta eſt finem in <lb/> <anchor type="figure" xlink:label="fig-0494-01a" xlink:href="fig-0494-01"/> puncto b:</s> <s xml:id="echoid-s31723" xml:space="preserve"> non ergo agit ſecundum aliquod punctum lineæ b c.</s> <s xml:id="echoid-s31724" xml:space="preserve"> Et hoc idem per multa naturalia <lb/>exempla patere poteſt.</s> <s xml:id="echoid-s31725" xml:space="preserve"> Vnde & animalia, quorum motrix eſt anima, ſecundum breuiorem lineam <lb/>mouentur ad terminũ, ut patet in rectitudine filorum aranearum, ex quibus texunt telas ſuas, quæ <lb/>telæ etſi nonnunquã inueniantur circulares, ſunt tamen ex rectis filis & in ſtamine, & in ſubtelari <lb/>contextæ propter lineæ breuitatem.</s> <s xml:id="echoid-s31726" xml:space="preserve"> Idẽ quoq;</s> <s xml:id="echoid-s31727" xml:space="preserve"> patet in canibus, qui omiſsis duobus lateribus tri-<lb/>goni currunt per tertium, ac ſi naturaliter informati nouerint, quod duo latera trigoni maiora ſint <lb/>tertio, quòd homines geometres edocet 20 p 1.</s> <s xml:id="echoid-s31728" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s31729" xml:space="preserve"> propoſitum, prout poſsibile nobis fuit.</s> <s xml:id="echoid-s31730" xml:space="preserve"/> </p> <div xml:id="echoid-div1292" type="float" level="0" n="0"> <figure xlink:label="fig-0494-01" xlink:href="fig-0494-01a"> <variables xml:id="echoid-variables532" xml:space="preserve">a b c</variables> </figure> </div> </div> <div xml:id="echoid-div1294" type="section" level="0" n="0"> <head xml:id="echoid-head1017" xml:space="preserve" style="it">6. Omnis reflexio luminis & coloris fit ſecundum lineas ſenſibiles latitudinem habentes. <lb/>Alhazen 16 n 4.</head> <p> <s xml:id="echoid-s31731" xml:space="preserve">Secundum enim tales lineas fit lucis incidẽtia, etiam lucis minimæ ſuper corpus politum, ut pa-<lb/>tet per 3 th.</s> <s xml:id="echoid-s31732" xml:space="preserve"> 2 huius.</s> <s xml:id="echoid-s31733" xml:space="preserve"> Latitudo itaq;</s> <s xml:id="echoid-s31734" xml:space="preserve"> lineæ reflexionis eſt æqualis latitudini lineæ incidentiæ:</s> <s xml:id="echoid-s31735" xml:space="preserve"> & linea <lb/>mathematica, quę eſt linea media totius lineæ reflexionis, eundem habet ſitum in loco reflexionis, <lb/>quem habet linea mathematica, quæ eſt linea media lineæ incidentiæ ſenſibilis in loco incidentiæ:</s> <s xml:id="echoid-s31736" xml:space="preserve"> <lb/>& ſimiliter quæliber aliarum linearum mathematicarum in linea ſenſibili reflexionis eundem reti-<lb/>net ſitum, quem ſua compar in linea incidẽtiæ ſenſibili:</s> <s xml:id="echoid-s31737" xml:space="preserve"> & ob hoc lineis mathematicis pro ipſis ſen-<lb/>ſibilibus non inconueniens eſt uti in tractatibus reflexionum.</s> <s xml:id="echoid-s31738" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s31739" xml:space="preserve"/> </p> <pb o="193" file="0495" n="495" rhead="LIBER QVINTVS."/> </div> <div xml:id="echoid-div1295" type="section" level="0" n="0"> <head xml:id="echoid-head1018" xml:space="preserve" style="it">7. In reflexionibus factis à quibuſcun ſpeculis, fit deceptio propter intem perantiãlucis: uel <lb/>propter diuerſitatem ſitus:uel propter remotionem puncti, cuius forma reflect itur:uel etiãcen-<lb/>tri ipſius uiſus à ſuperficie cuiuslibet ſpeculorum. Alhazen 3 n 6.</head> <p> <s xml:id="echoid-s31740" xml:space="preserve">Vniuerſaliter enim quibuſcunq;</s> <s xml:id="echoid-s31741" xml:space="preserve"> modis contingit decipi uiſum circa intentiones uiſibilium per <lb/>ſi mplicem uiſionem uiſorum:</s> <s xml:id="echoid-s31742" xml:space="preserve"> eiſdem etiam modis contingit uiſum decipi in uiſione, quæ fit per re-<lb/>flexionem:</s> <s xml:id="echoid-s31743" xml:space="preserve"> quoniam & hæc uiſio eſt quædam uiſio, in qua forma lucis & colorum & aliarum inten-<lb/>tionum uiſibilium ipſi uirtuti diſtinctiuæ præſentantur.</s> <s xml:id="echoid-s31744" xml:space="preserve"> Et hoc, ut patuit per 1 th.</s> <s xml:id="echoid-s31745" xml:space="preserve"> 4 huius, & multis <lb/>illius theorematibus, accidit octo modis.</s> <s xml:id="echoid-s31746" xml:space="preserve"> Plurimum tamen & manifeſtius fit hoc in ſpeculis:</s> <s xml:id="echoid-s31747" xml:space="preserve"> uel <lb/>propter debilitatem lucis:</s> <s xml:id="echoid-s31748" xml:space="preserve"> uel propter diuerſitatem ſitus, propter quam lineas reflexionũ remoueri <lb/>accidit ab axibus uiſualibus:</s> <s xml:id="echoid-s31749" xml:space="preserve">uel propter remotionem puncti rei uiſæ, cuius forma reflectitur à ſu-<lb/>perficie ipſius ſpeculi:</s> <s xml:id="echoid-s31750" xml:space="preserve">uel etiam propter remotionem ipſius centri uiſus, ad quod remota fit refle-<lb/>xio à ſuperficie ipſius ſpeculi.</s> <s xml:id="echoid-s31751" xml:space="preserve"> In alijs uerò quinq;</s> <s xml:id="echoid-s31752" xml:space="preserve"> modis licet ſimiliter cauſſetur error in uiſione for <lb/>marum reflexarum à quibuſcunq;</s> <s xml:id="echoid-s31753" xml:space="preserve"> ſpenculis ad uiſum, non eſt tamẽ ille error tam ſenſibilis, ut in iſtis <lb/>modis propoſitis:</s> <s xml:id="echoid-s31754" xml:space="preserve"> nec tamen fit totalis excuſatio ab illis.</s> <s xml:id="echoid-s31755" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s31756" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1296" type="section" level="0" n="0"> <head xml:id="echoid-head1019" xml:space="preserve" style="it">8. Specula, à quibus regularis fit reflexio, ſunt tantùm ſeptem.</head> <p> <s xml:id="echoid-s31757" xml:space="preserve">Quoniam enim regularis reflexio non poteſt fieri niſi à corporibus regularibus:</s> <s xml:id="echoid-s31758" xml:space="preserve"> corpora uerò re <lb/>gularia non poſſunt eſſe niſi corpora ut plurimum planarum ſuperficierum uel unius ſuperficiei cõ <lb/>cauæ uel conuexæ.</s> <s xml:id="echoid-s31759" xml:space="preserve"> Sicut autem patet ſenſui, licet corporum planorum ſpecies ſecundum figuras & <lb/>numerum angulorum uarientur:</s> <s xml:id="echoid-s31760" xml:space="preserve"> quantùm tamen ad naturam reflexionis, in omnibus illis eſt iden <lb/>titas ſuperficiei planæ:</s> <s xml:id="echoid-s31761" xml:space="preserve"> nec enim in ipſis, quo ad hæc, uariatio inuenitur:</s> <s xml:id="echoid-s31762" xml:space="preserve"> ut autem patet per 138 th.</s> <s xml:id="echoid-s31763" xml:space="preserve"> 1 <lb/>huius, omnis ſuperficies conuexa uel concaua regularis aut eſt pars ſuperficiei ſphæræ, aut colu-<lb/>mnæ, aut pyramidis rotundæ.</s> <s xml:id="echoid-s31764" xml:space="preserve"> Sic ergo habentur in uniuerſo ſeptem ſpecula:</s> <s xml:id="echoid-s31765" xml:space="preserve"> quorum unũ eſt pla-<lb/>num cuiuſcunq;</s> <s xml:id="echoid-s31766" xml:space="preserve"> figuræ:</s> <s xml:id="echoid-s31767" xml:space="preserve"> & tria ſunt conuexa, ſphærica, columnaria & pyramidalia:</s> <s xml:id="echoid-s31768" xml:space="preserve"> & tria ſunt con-<lb/>caua, ſphęrica, columnaria & pyramidalia:</s> <s xml:id="echoid-s31769" xml:space="preserve"> nec eſt poſsibile plura eſſe ſpecula, à quibus regularis fiat <lb/>reflexio.</s> <s xml:id="echoid-s31770" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s31771" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1297" type="section" level="0" n="0"> <head xml:id="echoid-head1020" xml:space="preserve" style="it">9. Inſtrumentum conſtituimus, in quo modi omnium reflexionum à quibuſcun regularib. <lb/>ſpeculis inſtrumentaliter declarantur. Alhazen 7 n 4.</head> <p> <s xml:id="echoid-s31772" xml:space="preserve">Aſſumatur ſemicirculus æneus cõuenientis ſpiſsitudinis, utpote medietatis grani hordei uel cir <lb/>ca illud, & conuenientis quantitatis:</s> <s xml:id="echoid-s31773" xml:space="preserve"> qui ſit a <lb/>b c, cuius diameter ſit a c, & eius centrum d:</s> <s xml:id="echoid-s31774" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0495-01a" xlink:href="fig-0495-01"/> producaturq́;</s> <s xml:id="echoid-s31775" xml:space="preserve"> linea d b perpendiculariter ſu-<lb/>per diametrum a c per 11 p 1:</s> <s xml:id="echoid-s31776" xml:space="preserve"> eſt ergo d b ſemi-<lb/>diameter circuli diuidens ſemicirculum per <lb/>æqualia per 33 p 6.</s> <s xml:id="echoid-s31777" xml:space="preserve"> Abſcin datur itaq;</s> <s xml:id="echoid-s31778" xml:space="preserve"> ex linea <lb/>d b ſuperius ſexta pars ipſius per 9 p 6, quę ſit <lb/>b e:</s> <s xml:id="echoid-s31779" xml:space="preserve"> & ſecundum quantitatem lineæ e d à cen <lb/>tro d fiat ſemicirculus, qui ſit f e g.</s> <s xml:id="echoid-s31780" xml:space="preserve"> Arcus ita-<lb/>que b c diuidatur in partes, quot libuerit, ſe-<lb/>cundum puncta h, i, k:</s> <s xml:id="echoid-s31781" xml:space="preserve"> & arcus b a in totidem <lb/>partes diuidatur ſecundum puncta l, m, n:</s> <s xml:id="echoid-s31782" xml:space="preserve"> ita <lb/>quòd arcus l b fiat ęqualis arcui b h, & arcus m l arcui h i, & arcus n m arcui i k, per 23 p 1 & 26 p 3, ꝓ-<lb/>ductis lineis d h, d i, d k, d l, d m, d n.</s> <s xml:id="echoid-s31783" xml:space="preserve"> Deinde iterũ à ſemidiametro b d inferius abſcindatur ſexta pars <lb/>ipſius, quæ ſit d o:</s> <s xml:id="echoid-s31784" xml:space="preserve"> & à pũcto o ducatur linea ęquidiſtans diametro ſemicirculi, quæ eſt a c, per 31 p 1:</s> <s xml:id="echoid-s31785" xml:space="preserve"> <lb/>quę ſit p o q:</s> <s xml:id="echoid-s31786" xml:space="preserve"> hanc itaq;</s> <s xml:id="echoid-s31787" xml:space="preserve"> interſe cabũt omnes lineæ ad partes diuiſionis à centro d productę.</s> <s xml:id="echoid-s31788" xml:space="preserve"> Punctus <lb/>ergo, in quo linea d n ipſam interfecat, ſit r, & in quo linea d k ipſam interſecat, ſit s:</s> <s xml:id="echoid-s31789" xml:space="preserve"> & pũcta, in quib.</s> <s xml:id="echoid-s31790" xml:space="preserve"> <lb/>ipſam ſecat ſemicirculus f e g, ſint t & u.</s> <s xml:id="echoid-s31791" xml:space="preserve"> Deinde à totali ſemicirculo abſcindatur pars d a p r exuna <lb/>parte, & ex alia pars d c q s:</s> <s xml:id="echoid-s31792" xml:space="preserve"> & planentur optimè ſuperficies:</s> <s xml:id="echoid-s31793" xml:space="preserve"> & acuatur d centrum aſſumpti ſemicir-<lb/>culi quaſi punctus, ita ut ipſum punctũ d maneat in eadẽ ſuperficie ſemicirculi cũ lineis productis.</s> <s xml:id="echoid-s31794" xml:space="preserve"> <lb/>Nos aũt quantitatẽ lineæ b e, quę eſt ſexta pars ſemidiametri d b, deinceps digitum appellamus:</s> <s xml:id="echoid-s31795" xml:space="preserve"> eſt <lb/>ergo diameter a c duodecim digitorũ.</s> <s xml:id="echoid-s31796" xml:space="preserve"> Deinde aſſumatur tabula lignea quadrata plana, cuius latus <lb/>fit 14 præmiſſorum digitorum, excedens diametrum a c duobus digitis:</s> <s xml:id="echoid-s31797" xml:space="preserve"> & ſpiſsitudo eius ſit 7 digi-<lb/>torum:</s> <s xml:id="echoid-s31798" xml:space="preserve"> & in hac tabula ſignetur punctus medius ք 40 th.</s> <s xml:id="echoid-s31799" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s31800" xml:space="preserve"> & ſuper ipſum fiat circulus ſecun-<lb/>dum quantitatem lateris tabulę:</s> <s xml:id="echoid-s31801" xml:space="preserve"> hic ergo excedet circulum a b c quãtitate unius digiti ex omni par <lb/>te:</s> <s xml:id="echoid-s31802" xml:space="preserve"> quoniam eius diameter in duobus digitis excedit diametrum a c.</s> <s xml:id="echoid-s31803" xml:space="preserve"> Fiat iterum ſuperidem centrũ <lb/>tabulæ ligneæ circulus ęqualis circulo f e g:</s> <s xml:id="echoid-s31804" xml:space="preserve"> diuidaturq́;</s> <s xml:id="echoid-s31805" xml:space="preserve"> circulus tabulę ligneę proportionaliter ſe-<lb/>micirculo æneo, qui eſt a b c, ita ut prima pars circuli lignei reſpondeat primæ, & ſecunda ſecundæ, <lb/>& ſic deinceps:</s> <s xml:id="echoid-s31806" xml:space="preserve"> & à centro tabulæ ligneæ ducantur ad puncta diuiſionis lineę rectę:</s> <s xml:id="echoid-s31807" xml:space="preserve"> & rotũdetur ta-<lb/>bula lignea extrinſecus ſecundum circulum maiorem:</s> <s xml:id="echoid-s31808" xml:space="preserve"> & excidatur pars interior tabulæ minori cir-<lb/>culo contenta:</s> <s xml:id="echoid-s31809" xml:space="preserve"> remanebitq́;</s> <s xml:id="echoid-s31810" xml:space="preserve"> quędam armilla lignea, cuius latitudo eſt duorum digitorũ, diameter <lb/>exterioris circuli 14:</s> <s xml:id="echoid-s31811" xml:space="preserve"> interioris circuli 10:</s> <s xml:id="echoid-s31812" xml:space="preserve"> & totius armillæ profunditas uel altitudo erit 7 digitorũ:</s> <s xml:id="echoid-s31813" xml:space="preserve"> <lb/>cuius ſuperficies curuę optimè planentur ad modum columnæ rotundæ:</s> <s xml:id="echoid-s31814" xml:space="preserve"> remanebuntq́;</s> <s xml:id="echoid-s31815" xml:space="preserve"> in ſuperfi-<lb/> <pb o="194" file="0496" n="496" rhead="VITELLONIS OPTICAE"/> cie plana illius armillæ, lineæ diuidentes circulum ſecundum diuiſionem ſemicirculi a b c.</s> <s xml:id="echoid-s31816" xml:space="preserve"> A‘capi-<lb/>tibus itaque illarum linearum producantur lineę in ſuperficie conuexa altitudinis armillæ, perpen <lb/>diculares ſuper <lb/> <anchor type="figure" xlink:label="fig-0496-01a" xlink:href="fig-0496-01"/> planam ſuperfi <lb/>ciem latitudinis <lb/>ipſius.</s> <s xml:id="echoid-s31817" xml:space="preserve"> Ponatur <lb/>enim pes circini <lb/>ſuper terminum <lb/>lineę diuidentis <lb/>circulum:</s> <s xml:id="echoid-s31818" xml:space="preserve"> & fiat <lb/>ſemicirculus in <lb/>ſuperficie cõue-<lb/>xa armillæ, qui <lb/>diuidatur per ę <lb/>qualia per 30 p 3:</s> <s xml:id="echoid-s31819" xml:space="preserve"> <lb/>& producatur a <lb/>puncto ad pun <lb/>ctũ linea:</s> <s xml:id="echoid-s31820" xml:space="preserve"> palãq́;</s> <s xml:id="echoid-s31821" xml:space="preserve"> <lb/>ք 105 th.</s> <s xml:id="echoid-s31822" xml:space="preserve"> 1 huius <lb/>quoniam illa li-<lb/>nea eſt perpẽdi-<lb/>cularis ſuper ſu-<lb/>perficiem latitu <lb/>dinis, quæ pars <lb/>eſt baſis colũnę:</s> <s xml:id="echoid-s31823" xml:space="preserve"> <lb/>& eodem modo <lb/>à terminis illarũ <lb/>diuidentiũ pro-<lb/>ducantur perpẽ <lb/>diculares in ſu-<lb/>perficie armillæ <lb/>concaua.</s> <s xml:id="echoid-s31824" xml:space="preserve"> In qua <lb/>etiam ſuperficie <lb/>ex parte planæ ſuperficiei non druiſæ ſum atur altitudo duorum digitorum:</s> <s xml:id="echoid-s31825" xml:space="preserve"> & in perpedicularibus <lb/>lineis omnibus in illa ſuperficie productis fiant ſigna:</s> <s xml:id="echoid-s31826" xml:space="preserve"> & ſecundum ſigna illa fiat circulus ęquidiſtãs <lb/>planę ſuperficiei armillæ, immiſſa tabella acuta, quantitatis circulif e g, uel alio modo, prout conue <lb/>nientius poſsit fieri:</s> <s xml:id="echoid-s31827" xml:space="preserve"> & ſecundum quantitatem medietatis grani hordei fiant item alia ſigna intra il-<lb/>los duos digitos:</s> <s xml:id="echoid-s31828" xml:space="preserve"> & circũducatur circulus æ quidiſtans priori circulo ſecundum quantitatẽ pręmiſ-<lb/>ſam medietatis grani hordei:</s> <s xml:id="echoid-s31829" xml:space="preserve"> & ſub hoc ſecundo circulo intra altitudinem duorũ illorum digitorũ, <lb/>ſecundum profunditatem ſemicirculi ænei a b c ſignentur alia puncta in prædictis perpendicula-<lb/>ribus, & iterum fiat circulus ſecundum illa puncta:</s> <s xml:id="echoid-s31830" xml:space="preserve"> & excepto per aliqua inſtrumẽta illo corpore li-<lb/>gneo inter hos duos ſecũdos circulos exiſtẽte, fiat concauitas unius digiti profunda:</s> <s xml:id="echoid-s31831" xml:space="preserve"> & coaptetur <lb/>huic cõcauitati ænea ſemicirculi portio, quę eſt p b q, quæ intrabit concauitatem uſq;</s> <s xml:id="echoid-s31832" xml:space="preserve"> ad portionem <lb/>minoris circuli, quę eſt t e u:</s> <s xml:id="echoid-s31833" xml:space="preserve">ideo quòd diſtantia iſtorum duorum arcuum eſt unius digiti, & eadem <lb/>eſt profunditas concauitatis factæ in tabula lignea.</s> <s xml:id="echoid-s31834" xml:space="preserve"> Flat autem taliter, ut ſuperficies circuli f e g diui <lb/>ſa per lineam à centro d ad circumferentiam producta, ſit ad partem ſuperficiei armillæ diuiſæ.</s> <s xml:id="echoid-s31835" xml:space="preserve"> Li-<lb/>neę itaq;</s> <s xml:id="echoid-s31836" xml:space="preserve"> perpẽdiculares ductæ in concaua ſuperficie armillæ tangent lineas diuiſionis circuli f e g, <lb/>& cadent perpendiculariter ſuper ſuperficiem circuli f e g.</s> <s xml:id="echoid-s31837" xml:space="preserve"> Item in cõuexa ſuperficie armillæ ex par <lb/>te ſup erficiei non diuiſæ ſignetur punctus in qualibet perpendicularium productarum ſecundum <lb/>diſtantiam duorũ digitorum ab ipſa plana ſuperficie non diuiſa:</s> <s xml:id="echoid-s31838" xml:space="preserve"> & poſito pede circini ſuper quodli-<lb/>bet punctorum ſignatorum, fiant circuli, quorũ cuiuslibet diameter ſit ęqualis quantitati grani hor <lb/>dei:</s> <s xml:id="echoid-s31839" xml:space="preserve"> & ſecundum illorum circulorum quantitatem fiant foramina columnaria rotunda:</s> <s xml:id="echoid-s31840" xml:space="preserve"> & in aliquo <lb/>ipſorũ coaptetur baculus ligneus, qui cum tranſierit ad interiorem concauitatem armillæ, tãget ſe-<lb/>micirculi f e g ſuperficiem:</s> <s xml:id="echoid-s31841" xml:space="preserve"> quoniam, ut patet ex præmiſsis, centrum cuiuslibet illorum circulorum <lb/>paruorum erit in circũferentia circuli prius ſignati in ſuperficie concaua armillę, à quo diſtat ſuper <lb/>ficies circuli ęnei, qui eſt f e g, ſecundum quantitatem medietatis grani hordei.</s> <s xml:id="echoid-s31842" xml:space="preserve"> Deinde ſumatur alia <lb/>tabula lignea quadrata, cuius diameter ſit æqualis diametro armillæ ligneæ:</s> <s xml:id="echoid-s31843" xml:space="preserve"> & perquiſito puncto <lb/>medio ipſius per 40 th.</s> <s xml:id="echoid-s31844" xml:space="preserve"> 1 huius, ab illo puncto medio circunducatur circulus ad quantitatem ſemi-<lb/>diametri d e:</s> <s xml:id="echoid-s31845" xml:space="preserve"> & hic circulus erit ęqualis circulo f e g & baſi concauitatis armillę.</s> <s xml:id="echoid-s31846" xml:space="preserve"> Item ſuper centrum <lb/>huius circuli fiat quadratum, cuius latera ſint quatuor digitorum lateribus ſuis æqualiter diſtanti-<lb/>bus à lateribus huius tabulæ ligneę:</s> <s xml:id="echoid-s31847" xml:space="preserve"> quod poteſt fieri per 41 th.</s> <s xml:id="echoid-s31848" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s31849" xml:space="preserve"> & fodiatur hoc quadratum <lb/>ad profunditatem unius digiti:</s> <s xml:id="echoid-s31850" xml:space="preserve"> & planentur omnes ſuperficies concauitatis ſuæ, ut fiant rectangu-<lb/>læ, & fundus eius fiat planus.</s> <s xml:id="echoid-s31851" xml:space="preserve"> Deinde huic tabulæ coaptetur immobiliter baſis armillæ, ita ut circu-<lb/>lus minor huius tabulę applicetur concauitati armillæ.</s> <s xml:id="echoid-s31852" xml:space="preserve"> Deinde fiat columna ferrea concaua aliquã <lb/> <pb o="195" file="0497" n="497" rhead="LIBER QVINTVS"/> tulum ſpiſſa, cuius baſis diameter ſit æqualis quantitati grani hordei, ſicut diametri foraminum:</s> <s xml:id="echoid-s31853" xml:space="preserve"> & <lb/>ponatur illa columna in prius factis foraminib.</s> <s xml:id="echoid-s31854" xml:space="preserve"> quę, cũ peruenerit ad concauum armillæ, continget <lb/>lineas in circulo f e g productas.</s> <s xml:id="echoid-s31855" xml:space="preserve"> Fiat aũt in capite columnæ quodcunq;</s> <s xml:id="echoid-s31856" xml:space="preserve"> artificium, non permittens <lb/>columnam intrare, niſi ad locum determιnatum:</s> <s xml:id="echoid-s31857" xml:space="preserve"> & ut firmius ſtare poſsit, modicum cerę ſibi circũ-<lb/>ponatur:</s> <s xml:id="echoid-s31858" xml:space="preserve"> & ſit tantæ longitudinis columna, ut procedens ſuper ſuperficiem circuli f e g, contingere <lb/>poſsitlatus quadrati concaui in tabula lignea, quod eſt ęquidiſtans lineę r s, ductę in ſuperficie cir-<lb/>culi ænei.</s> <s xml:id="echoid-s31859" xml:space="preserve"> Deinde fiant ſeptem regulæ ligneę planę ęquidiſtantium ſuperficierũ orthogonalium, æ-<lb/>quales & penitus ſimiles, quarum longitudo ſit digitorum ſex:</s> <s xml:id="echoid-s31860" xml:space="preserve"> latitudo quatuor:</s> <s xml:id="echoid-s31861" xml:space="preserve"> & ſpiſsitudo con-<lb/>ueniens, ut inferius neceſsitas ipſius finis edocebit:</s> <s xml:id="echoid-s31862" xml:space="preserve"> & una ipſarum ada-<lb/>ptetur quadrato cõcauo, ita ut orthogonaliter cadat ſuper fundum qua-<lb/> <anchor type="figure" xlink:label="fig-0497-01a" xlink:href="fig-0497-01"/> drati concaui, & ut faciliter intret ſine compreſsione:</s> <s xml:id="echoid-s31863" xml:space="preserve"> ducaturq́ taliter ut <lb/>punctus d centrum ſcilicet circuli a b c contingat unam ſuperficierum la <lb/>titudinis regulæ:</s> <s xml:id="echoid-s31864" xml:space="preserve"> & in puncto contactus fiat ſignum in regula, quod ſit x:</s> <s xml:id="echoid-s31865" xml:space="preserve"> <lb/>& à puncto ſignato x producatur in extremitates regulę linea ęquidiſtãs <lb/>longioribus lateribus regulę, quę ſit b x p.</s> <s xml:id="echoid-s31866" xml:space="preserve"> Et palàm quoniam illa erit li <lb/>nea longitudinis regulę.</s> <s xml:id="echoid-s31867" xml:space="preserve"> Deinde in longiori parte illius lineę à puncto x <lb/>ſignato ſumatur altitudo medij grani hordei:</s> <s xml:id="echoid-s31868" xml:space="preserve"> & fiat ibi punctum z:</s> <s xml:id="echoid-s31869" xml:space="preserve"> erit i <lb/>taq;</s> <s xml:id="echoid-s31870" xml:space="preserve"> z medius punctus longitudinis regulæ, centrisq́ foraminum oppo-<lb/>ſitus directè:</s> <s xml:id="echoid-s31871" xml:space="preserve"> centra enim foraminum altiora ſunt ſuperficie circuli a b c <lb/>in quantitate medij grani hordei, & diſtant à baſi armillę per duos digi-<lb/>tos:</s> <s xml:id="echoid-s31872" xml:space="preserve"> punctus ergo z diſtat ab eadẽ baſi per duos digitos, & regula in qua-<lb/>drato concauo per digitum unum.</s> <s xml:id="echoid-s31873" xml:space="preserve"> Et quia ab extremitate regulę uſq;</s> <s xml:id="echoid-s31874" xml:space="preserve"> ad <lb/>punctum z ſunt digiti tres, longitudo quoq;</s> <s xml:id="echoid-s31875" xml:space="preserve"> regulę eſt tantùm ſex digito <lb/>rum:</s> <s xml:id="echoid-s31876" xml:space="preserve"> patet, quòd punctum z eſt medium longitudinis regulę.</s> <s xml:id="echoid-s31877" xml:space="preserve"> Ducaturi-<lb/>taq;</s> <s xml:id="echoid-s31878" xml:space="preserve"> per punctum z lineę ęquidiſtans lineis extremitatum latitudinis re-<lb/>gulæ, quę ſit c q:</s> <s xml:id="echoid-s31879" xml:space="preserve"> eſt itaque linea longitudinis regulæ, quę eſt b p, diuiſa ք <lb/>ęqualia in puncto z:</s> <s xml:id="echoid-s31880" xml:space="preserve"> cuius item medietates, quę ſunt b z & z p, diuidãtur <lb/>per ęqualia in punctis k & y, ſemper ductis lineιs latitudinis à pũctis ſe-<lb/>ctionis k & y perpendiculariter ſuper lineam longitudinis b p & ęquidiſtanter lineæ c q.</s> <s xml:id="echoid-s31881" xml:space="preserve"> Sic ergo e-<lb/>rit linea b p, & conſequenter tota regula diuiſa in quatuor partes ęquales.</s> <s xml:id="echoid-s31882" xml:space="preserve"> Et hoc modo omnes aliæ <lb/>ſex regulæ diuidantur, & ſignentur:</s> <s xml:id="echoid-s31883" xml:space="preserve"> & ſic factum eſt, quod proponebatur.</s> <s xml:id="echoid-s31884" xml:space="preserve"/> </p> <div xml:id="echoid-div1297" type="float" level="0" n="0"> <figure xlink:label="fig-0495-01" xlink:href="fig-0495-01a"> <variables xml:id="echoid-variables533" xml:space="preserve">b a p n m l h i k q c f t e u g r o s d</variables> </figure> <figure xlink:label="fig-0496-01" xlink:href="fig-0496-01a"> <variables xml:id="echoid-variables534" xml:space="preserve">n m l b h i k</variables> </figure> <figure xlink:label="fig-0497-01" xlink:href="fig-0497-01a"> <variables xml:id="echoid-variables535" xml:space="preserve">p k c z q x y b</variables> </figure> </div> </div> <div xml:id="echoid-div1299" type="section" level="0" n="0"> <head xml:id="echoid-head1021" xml:space="preserve" style="it">10. In ſpeculis planis radij obliquè incidentis fit ad aliam partem reflexio: ſempeŕ angulum <lb/>incidentiæ æqualem eße angulo reflexionis experimentaliter comprobatur. Euclides 1 the. ca-<lb/>toptr. Ptolemæus 4 th. 1 catoptricorum. Alhazen 10 n 4.</head> <p> <s xml:id="echoid-s31885" xml:space="preserve">Fiat itaq;</s> <s xml:id="echoid-s31886" xml:space="preserve"> ex ferro mundo ſpeculum planum circularis figurę:</s> <s xml:id="echoid-s31887" xml:space="preserve"> cuius diameter modo præmiſſo ſit <lb/>trium digitorum:</s> <s xml:id="echoid-s31888" xml:space="preserve"> & concauetur regula præmiſſa ſecundum centrum z, qui eſt medιus punctus regu <lb/>læ circulariter ad quantitatem diametri ſpeculi:</s> <s xml:id="echoid-s31889" xml:space="preserve"> & profundetur ſecundum ſpiſsitudinem ipſius ſpe <lb/>culi, apteturq́;</s> <s xml:id="echoid-s31890" xml:space="preserve"> taliter, ut una fiat ſuperficies ſpeculi & regulę:</s> <s xml:id="echoid-s31891" xml:space="preserve"> & ut centrum circuli rotunditatis ſpe-<lb/>culi directè ſuperponatur puncto z.</s> <s xml:id="echoid-s31892" xml:space="preserve"> Linea itaq;</s> <s xml:id="echoid-s31893" xml:space="preserve"> c q diuidens latiorem ſuperficiem regulæ per duo <lb/>æqualia, diuidet etiam ſuperficiem ſpeculi per duo ęqualia:</s> <s xml:id="echoid-s31894" xml:space="preserve"> & in hoc experimentantis diligentia cõ <lb/>ſiſtat.</s> <s xml:id="echoid-s31895" xml:space="preserve"> Immittatur itaq;</s> <s xml:id="echoid-s31896" xml:space="preserve"> ligneę armillæ hęc regula, donec centrum d, quod eſt acumen tabulę æneæ, <lb/>cadat ſuper ſpeculũ:</s> <s xml:id="echoid-s31897" xml:space="preserve"> & tunc illa regula ſit cũ ſpeculo in figura quadrato concauo per aliquod artifi-<lb/>cium appodiata, ne uacillet, ſed ſtet firma.</s> <s xml:id="echoid-s31898" xml:space="preserve"> Deinde bene obturentur omnia foramina inſtrumenti, <lb/>pręter unum, quod obliquè ſuper regulę ſuperficiem declinet:</s> <s xml:id="echoid-s31899" xml:space="preserve"> & ſit exempli cauſſa, foramen corre-<lb/>ſpõdens lineęd l in circulo a b c ęneo:</s> <s xml:id="echoid-s31900" xml:space="preserve"> & hoc foramen apertum adhibeatur radιo ſolis, & melius eſt-<lb/>radio ſolis per ſeneſtram domus intrãti.</s> <s xml:id="echoid-s31901" xml:space="preserve"> Radius itaq;</s> <s xml:id="echoid-s31902" xml:space="preserve"> ſpeculo plano incidẽs uidebitur reflecti ad fo <lb/>ramen aliud correſpondens lineę d h in circulo a b c ęneo:</s> <s xml:id="echoid-s31903" xml:space="preserve"> & ſi foramen illud puncti h aperiatur, & <lb/>foramen prius opertum, quod fuit punctil, obſtruatur, reflectitur itẽ radius in illud foramẽ cooper-<lb/>tum.</s> <s xml:id="echoid-s31904" xml:space="preserve"> Angulus autem b d l eſt ęqualis angulo b d h, ut patet ex hypotheſi in pręmiſſa:</s> <s xml:id="echoid-s31905" xml:space="preserve"> ergo angulus <lb/>l d a eſt ęqualis angulo h d c:</s> <s xml:id="echoid-s31906" xml:space="preserve"> quoniã totus angulus b d a eſt ęqualis toti angulo b d c:</s> <s xml:id="echoid-s31907" xml:space="preserve"> quia uterq;</s> <s xml:id="echoid-s31908" xml:space="preserve"> eſt <lb/>rectus.</s> <s xml:id="echoid-s31909" xml:space="preserve"> Si etiam imponatur foramini aperto columna ferrea concaua, de qua pręmiſimus:</s> <s xml:id="echoid-s31910" xml:space="preserve"> deſcẽdet <lb/>lux per columnę cõcauitatẽ ad ſpeculũ, & reflectetur in foramine reſpiciens ęqualem angulum, ut <lb/>prius.</s> <s xml:id="echoid-s31911" xml:space="preserve"> Et ſi ad ſecundum foramen columna transferatur, reflectetur radius ad primũ:</s> <s xml:id="echoid-s31912" xml:space="preserve"> ſemper tñ erit <lb/>debilior lux per columnã deſcendens, quã ſine columna per ipſum foramen deſcẽdens.</s> <s xml:id="echoid-s31913" xml:space="preserve"> Et idem eſt <lb/>experimentandi modus, ſi aliquod foraminum cum cera obſtruatur, & circa centrũ eius cum ſtilo <lb/>ferreo fiat modicum foramẽ:</s> <s xml:id="echoid-s31914" xml:space="preserve"> tunc enim lumen reflectetur in ſimile ſpatiũ paruũ circa centrũ forami <lb/>nis alterius, illud primũ in anguli æqualitate reſpicientis.</s> <s xml:id="echoid-s31915" xml:space="preserve"> Et ſi concauitas columnę ferreę concaua <lb/>obturata fuerit, facto foramine primo ſecundum centrum ſuæ baſis, deſcendet lux per axẽ colũnæ, <lb/>& ad centrum alterius foraminis reflectetur, ſemper, ęqualitate angulorum in omnibus obſerua-<lb/>ta.</s> <s xml:id="echoid-s31916" xml:space="preserve"> Et ſi aptetur inſtrumentũ taliter, ut lux intret per duo foramina, reflectetur ſimiliter ք alia duo il <lb/>lis ſimilia:</s> <s xml:id="echoid-s31917" xml:space="preserve"> ſemք enim declinatio linearũ reflexionis eſt ęqualis declinationi linearum incidẽtiæ.</s> <s xml:id="echoid-s31918" xml:space="preserve"> Et <lb/>quoniã linea b x p(quę eſt linea media lõgitudinis regulę) eſt orthogonalis ſuք lineã latitudinis re <lb/>gulę inferiorẽ ęquidiſtantẽ lineę c q:</s> <s xml:id="echoid-s31919" xml:space="preserve"> quoniá illa eſt cõmunis ſectio ſuperficiei regulę & ſuքficiei fun <lb/> <pb o="196" file="0498" n="498" rhead="VITELLONIS OPTICAE"/> di quadrati concaui æ quidiſtãtis ſuperficiei a b c circuli ænei:</s> <s xml:id="echoid-s31920" xml:space="preserve"> & linea media ſuperficiei fundi ęqui-<lb/>diſtatlineæ d b (quę eſt media diameter circuli) & quia linea, quę eſt cõmunis ſectio ſemicirculi a b <lb/>c & ſuperficiei regule, in qua eſt linea latitudinis regulę, eſt ſquidiſtans cõmuni ſectioni ſuperficiei <lb/>fundi & regulę per 28 p 1, quoniam linea b x p cadit perpendiculariter ſuper ambasillas lineas lati-<lb/>tudinis regulę:</s> <s xml:id="echoid-s31921" xml:space="preserve"> & quoniã linea b x p eſt erecta ſuper ſuperficiem fundi:</s> <s xml:id="echoid-s31922" xml:space="preserve"> palã per 23 th.</s> <s xml:id="echoid-s31923" xml:space="preserve"> 1 huius quoniã <lb/>linea b x p eſt perpendicularis ſuper ſuperficiem circuli a b c ęquidiſtantẽ fuperficiei fundi tabulæ.</s> <s xml:id="echoid-s31924" xml:space="preserve"> <lb/>Ergo per definitionem lineę ſuper ſuperficiẽ erectę, diameter d b eſt perpẽdicularis ſuք lineã b x p, <lb/>cum ſecent ſe in puncto d:</s> <s xml:id="echoid-s31925" xml:space="preserve"> eſtergo linea d b erecta ſuper ſupficiem ſpeculi plani, & ſuper eius circu-<lb/>li diametrum:</s> <s xml:id="echoid-s31926" xml:space="preserve"> quia ſuperficies circuli a b c eſt ęquidiſtãs ſuperficiei circuli trãſeuntis per cẽtra fora <lb/>minum:</s> <s xml:id="echoid-s31927" xml:space="preserve"> quoniã diftantia omnium centrorũ foraminum à ſuperficie circuli a b c eſt eadem, ſcilicet <lb/>medietas quantitatis granihordei.</s> <s xml:id="echoid-s31928" xml:space="preserve"> Superficies uerò tranſiens centra omniũ ſoraminum ſecat co-<lb/>lumnã ferreã per axem:</s> <s xml:id="echoid-s31929" xml:space="preserve"> eſt ergo axis colũnę in illa ſuperficie, Et quia columna ſerrea in ſuo deſcẽſu <lb/>tangit aliquã linearum in ſuperficie circuli a b c, à cẽtro d a d circumferentiã productarum, utpote li <lb/>neã d b, uellineã d m, uel aliquã aliã illarum linearum:</s> <s xml:id="echoid-s31930" xml:space="preserve"> palã per præmiſſa, quia axis columnæ ęquidi <lb/>ſtatilli lineę, quę tangitur per lineã lõgitudinis columnę.</s> <s xml:id="echoid-s31931" xml:space="preserve"> Et quoniã per quodcunq;</s> <s xml:id="echoid-s31932" xml:space="preserve"> foraminũ colu-<lb/>mna deſendente, ſemք axis eius caditin linea b x p & in punctũ z:</s> <s xml:id="echoid-s31933" xml:space="preserve"> linea uerò z b ſemper eſt perpen <lb/>dicularis ſuper ſuperficiẽ a b c:</s> <s xml:id="echoid-s31934" xml:space="preserve"> linea quoq;</s> <s xml:id="echoid-s31935" xml:space="preserve"> à puncto z ipſius regulę protracta ad centrũ foraminis.</s> <s xml:id="echoid-s31936" xml:space="preserve"> <lb/>quod eſt contingens punctum n, eft ęquidiſtans lineę d n, & ſimiliter de alijs centris foraminũ & pũ <lb/>ctis m, l, h, i, k ſignatis in circumferentia a b c:</s> <s xml:id="echoid-s31937" xml:space="preserve"> oẽs enim ſemidiametri foraminũ ſunt ęquales & ęqui <lb/>diſtãtes lineę z b ք 25 th.</s> <s xml:id="echoid-s31938" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s31939" xml:space="preserve"> ſunt enim oẽs ſemidiametri foraminum perpẽdiculares ſuք ſuperfi <lb/>ciẽ circuli a b c:</s> <s xml:id="echoid-s31940" xml:space="preserve"> quoniã ſunt partes lineeę lõgitudinis armillæ.</s> <s xml:id="echoid-s31941" xml:space="preserve"> Lineę itaq;</s> <s xml:id="echoid-s31942" xml:space="preserve"> l d & d h ſunt ęquidiſtantes <lb/>duabus lineis imaginatis duci à puncto regulę, quod eſtz, ad centrum duorũ foraminum cõtingen <lb/>tium puncta l & h per 33 p 1:</s> <s xml:id="echoid-s31943" xml:space="preserve"> ergo per 10 p 11 anguli ab illis lineis in ſuperficiebus ęquidiſtantibus cõ <lb/>tenti ſunt æquales.</s> <s xml:id="echoid-s31944" xml:space="preserve"> Et ſi à puncto z ducatur linea ad centrum medij foraminis, eritipſa per præmiſſa <lb/>ęquidiſtans lineę d b, diuidens angulum linearum ſecum cõcurrentium per æqualia, ſicut linea d b <lb/>diuidit angulum l d h per æqualia.</s> <s xml:id="echoid-s31945" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s31946" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1300" type="section" level="0" n="0"> <head xml:id="echoid-head1022" xml:space="preserve" style="it">11. In ſpeculis planis radium perpendiculariter incidentem reflec̃ti in ſe ipſum inſtrument æ-<lb/>liter declar atur. Euclides 2 the. catoptr. Alhazen 11 n 4.</head> <p> <s xml:id="echoid-s31947" xml:space="preserve">Remanente enim omni diſpoſitione inſtrumenti, ut prius:</s> <s xml:id="echoid-s31948" xml:space="preserve"> & regula, in qua ſitum eſt ſpeculũ pla-<lb/>num, erecta ſuper fundum quadrati concaui, quod eſt in tabula lignea, quæ eſt baſis inſtrumẽti, ob-<lb/>turentur omnia ſoramina, præter medium, cui reſpondet ſemidiameter d b circuli a b c:</s> <s xml:id="echoid-s31949" xml:space="preserve"> & fiat bacu <lb/>lus columnaris ad quantitatem foraminis, cuius extremitas acuaturita, ut remaneat ſolus punctus, <lb/>qui eſt terminus axis eius, qui immittatur per foramẽ ad ſpeculum:</s> <s xml:id="echoid-s31950" xml:space="preserve"> ſigneturq́;</s> <s xml:id="echoid-s31951" xml:space="preserve"> in cauſto punctus, in <lb/>quẽ ceciderit.</s> <s xml:id="echoid-s31952" xml:space="preserve"> Deinde extracto baculo opponaturforamen apertum radio:</s> <s xml:id="echoid-s31953" xml:space="preserve"> cadetq́ radius ſuper pũ <lb/>ctum ſignatum, & circa ipſum efficiet circulum.</s> <s xml:id="echoid-s31954" xml:space="preserve"> Signeturitaq;</s> <s xml:id="echoid-s31955" xml:space="preserve"> in fine huius lucis circularis punctũ, <lb/>& ſecundum quantitatem lineę interiacentis puncta ſignata fiat circulus, qui erit maior circulo fo, <lb/>raminis per 36 th.</s> <s xml:id="echoid-s31956" xml:space="preserve"> 2 huius:</s> <s xml:id="echoid-s31957" xml:space="preserve"> quoniam ſemper proceſſus lucis per foramen ingredientis eſt in modum <lb/>pyramidis:</s> <s xml:id="echoid-s31958" xml:space="preserve"> in nullo aũt aliorum foraminum neq;</s> <s xml:id="echoid-s31959" xml:space="preserve"> in aliqua parte cõcauitatis armillæ uidebitur lux <lb/>reflexa.</s> <s xml:id="echoid-s31960" xml:space="preserve"> Palàm ergo quòd lux deſcendens per axẽ, reflectitur per eundem, & ſecundum illius refle-<lb/>xionem ordinatur totaliter reflexio luminis incidentis.</s> <s xml:id="echoid-s31961" xml:space="preserve"> Quãuis aũt uideatur lux circularis circa ba <lb/>fim interiorẽ foraminis maior luce incidente uel radio, & quãuis illa lux uideatur major ipſius lucis <lb/>interioris circulo, palãq́ ſit illã lucẽ apparere per reflexionẽ:</s> <s xml:id="echoid-s31962" xml:space="preserve"> non tñ accidit hoc ք reflexionẽ rad j ք-<lb/>pendiculariter incidentis, qui eſt axis illius pyramidis luminoſæ:</s> <s xml:id="echoid-s31963" xml:space="preserve"> ſed accidithoc propter reflexionẽ <lb/>aliorum radiorũ pyramidis obliquè ſpeculo incidentiũ, qui etiã ſecundum modũ ſuæ obliquitatis <lb/>ad partes oppoſitas, & nõ in ſe refle ctuntur:</s> <s xml:id="echoid-s31964" xml:space="preserve"> quod pater, ſi obturetur ք cerã utraq;</s> <s xml:id="echoid-s31965" xml:space="preserve"> baſis foraminis, fa <lb/>cto modico foramine ſecundũ axẽ:</s> <s xml:id="echoid-s31966" xml:space="preserve"> tunc enim radio ſolis ք uiã tantũ axis deſcendẽte, nõ apparebit <lb/>lux reflexa circularis circa interiorẽ baſim foraminis.</s> <s xml:id="echoid-s31967" xml:space="preserve"> Patet ergo quòd non procedebat illa lux cir-<lb/>cularis ex reflexa luce axis, ſed ex reflexione lucis obliquè incidentis ipſi ſpeculo.</s> <s xml:id="echoid-s31968" xml:space="preserve"> Quòd ſi regula, in <lb/>qua ſitum eſt dictum ſpeculum planum, aliquãtum retrorſum inclinetur:</s> <s xml:id="echoid-s31969" xml:space="preserve"> tunc palã eſt quòd radius <lb/>per medium foramen incidens non cadit perpendiculariter ſuper ſuperficiem ſpeculi, uidebiturq́;</s> <s xml:id="echoid-s31970" xml:space="preserve"> <lb/>lux reflexa à medio foramine remota ſecundum modum declinationis ſpeculi:</s> <s xml:id="echoid-s31971" xml:space="preserve"> ſemper tñ centrum <lb/>lucis cadet ſuper lineã ductam in cõcaua ſuperficie armillę perpendicularẽ ſuper ſuperficiẽ a b c cir <lb/>culi ęnei, & deſcendentẽ per centrum baſis foraminis medij:</s> <s xml:id="echoid-s31972" xml:space="preserve"> hoc enim ſecat ſemper lucẽ circularẽ <lb/>reflexã, & diuidit circulum eius per medium.</s> <s xml:id="echoid-s31973" xml:space="preserve"> Et ſi regula ad latus dextrũ uel ſiniſtrum declinetur, <lb/>ſemper radius ſecundũ hoc obliquabitur:</s> <s xml:id="echoid-s31974" xml:space="preserve"> regula uerò ad rectitudinẽ redeunte, reuertetur lucis re-<lb/>flexio ad interiorẽ baſim foraminis, ut prius.</s> <s xml:id="echoid-s31975" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s31976" xml:space="preserve"> ſemper enim in ſpeculis planis <lb/>radius perpendiculariter incidens reflectitur in ſe ipſum:</s> <s xml:id="echoid-s31977" xml:space="preserve"> ſed in radijs obliquè incidentib.</s> <s xml:id="echoid-s31978" xml:space="preserve"> angulus <lb/>incidentiæ fit æqualis angulo reflexionis, ut patet per præmiſſam.</s> <s xml:id="echoid-s31979" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1301" type="section" level="0" n="0"> <head xml:id="echoid-head1023" xml:space="preserve" style="it">12. In ſphæricis conuexis ſpeculis radio incidente & reflexo, ſemper angulus incidentiæ eſt æ-<lb/>qualis angulo reflexionis. Ex quo patet quia radius perpendicularis reflectitur in ſe ipſum. Eu-<lb/>clides I the. catoptr. Ptolemæus 5 th. 1 catoptr. Alhazen 12 n 4.</head> <p> <s xml:id="echoid-s31980" xml:space="preserve">Fiat ex ſerro mundo ſpeculum ſphæricum cõuexum hoc modo.</s> <s xml:id="echoid-s31981" xml:space="preserve"> Deſcribatur circulus maximus <lb/> <pb o="197" file="0499" n="499" rhead="LIBER QVINTVS."/> ſphæræ, cuius diameter ſit ſex digitorum aſſumptorum prius:</s> <s xml:id="echoid-s31982" xml:space="preserve"> & inſcribatur ei linea æqualis ſemi-<lb/>diametro per 1 p 4:</s> <s xml:id="echoid-s31983" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s31984" xml:space="preserve"> erit chorda trium digitorum.</s> <s xml:id="echoid-s31985" xml:space="preserve"> Ducatur quoq;</s> <s xml:id="echoid-s31986" xml:space="preserve"> à centro ſphærę ſemidiameter <lb/>perpendiculariter ſuper illam chordam per 12 p 1:</s> <s xml:id="echoid-s31987" xml:space="preserve"> & producatur ad arcum, cadetq́;</s> <s xml:id="echoid-s31988" xml:space="preserve"> in medium arcus <lb/>punctum per 4 p 1, & per 28 p 3:</s> <s xml:id="echoid-s31989" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s31990" xml:space="preserve"> ſinus uerſus minor medio digito.</s> <s xml:id="echoid-s31991" xml:space="preserve"> Abſcindaturitaq;</s> <s xml:id="echoid-s31992" xml:space="preserve"> illa minor <lb/>portio circuli, & ſecundum illius quantitatẽ & cõcauitatẽ fabricetur ſpeculum, quod limetur & po <lb/>liatur planiſsimè extrinſecus:</s> <s xml:id="echoid-s31993" xml:space="preserve"> aſſumaturq́ regula lignea ſimilis penitus prius ſumptę in omni linea <lb/>tione & creatione:</s> <s xml:id="echoid-s31994" xml:space="preserve"> & facta cõcauitate in linea ad modum ſpeculi, applicetur ſpeculum regulęita, ut <lb/>medium punctũ conuexi ſpeculi cadat ſuք z medium punctum regulę:</s> <s xml:id="echoid-s31995" xml:space="preserve"> & ſit in ſuperficie ipſius re-<lb/>gulę, quod poteſt ſciri per applicationẽ alterius regulę uel amuſsis, ut placuerit.</s> <s xml:id="echoid-s31996" xml:space="preserve"> Erigatur quoq;</s> <s xml:id="echoid-s31997" xml:space="preserve"> re-<lb/>gula cum ſpeculo orthogonaliter ſuper fundum quadrati, ut in ſpeculis planis, & operatione priori <lb/>repetita, & luce per foramen obliquum uel medium defcendente fiat reflexio, ut prius.</s> <s xml:id="echoid-s31998" xml:space="preserve"> Et ſimiliter <lb/>fiet, ſiregula declinetur.</s> <s xml:id="echoid-s31999" xml:space="preserve"> Semper enim luces per diuerſas lineas obliquas ſpeculo ſphęrico cõuexo <lb/>incidẽtes, per diuerſas lineas obliquas reflectuntur:</s> <s xml:id="echoid-s32000" xml:space="preserve"> & quę fecun dum perpendiculares lineas ſpe-<lb/>culo luces incidunt, reflectuntur in ſe ipſas, & ſemper angulus incidentiæqualis angulo refle-<lb/>xionis.</s> <s xml:id="echoid-s32001" xml:space="preserve"> Quod proponebatur.</s> <s xml:id="echoid-s32002" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1302" type="section" level="0" n="0"> <head xml:id="echoid-head1024" xml:space="preserve" style="it">13. In ſphæricis concauis ſpeculis radio incidente & reflexo, ſemper angulus incidentiæ eſt æ-<lb/>qualis angulo reflexionis. Euclides 1 the. catoptr. Alhazen 12 n 4.</head> <p> <s xml:id="echoid-s32003" xml:space="preserve">Fiat ſpeculum ſphęricũ ut fuprà:</s> <s xml:id="echoid-s32004" xml:space="preserve"> & ſecundum conuexã portionẽ illius circuli limetur & poliatur <lb/>planiſsimè intrinfecus:</s> <s xml:id="echoid-s32005" xml:space="preserve"> & aſſumatur alia regula lignea ſimilis priori, & coaptetur ei ſpeculũ taliter, <lb/>ut circulus baſis ſpeculi fit in ſuperficie regulæ:</s> <s xml:id="echoid-s32006" xml:space="preserve"> & centrũ illius circuli cadat in punctum z:</s> <s xml:id="echoid-s32007" xml:space="preserve"> & linea <lb/>c q, quæ diuidit ſuperficiem regulæ per ęqualia, continuetur diametro baſis ſpeculi, & fiat iftorũ di-<lb/>ligens in quiſitio per artificium, quod induſtriæ experimentantis cõmittimus.</s> <s xml:id="echoid-s32008" xml:space="preserve"> Immittaturq́;</s> <s xml:id="echoid-s32009" xml:space="preserve"> regula <lb/>cũ ſpeculo ipſi inſtrumento, ut prius, & fiat operatio ſimilis omnino priori, ſic tamen ut femper pũ-<lb/>ctus d, qui eſt centrũ ſemicirculi ęnei, cadat ſuper medium punctum ſpeculi:</s> <s xml:id="echoid-s32010" xml:space="preserve"> hoc enim eſt ſem per in <lb/>omnib.</s> <s xml:id="echoid-s32011" xml:space="preserve"> ſpeculis cõuexis & cõcauis obſeruandũ:</s> <s xml:id="echoid-s32012" xml:space="preserve"> declarabiturq́;</s> <s xml:id="echoid-s32013" xml:space="preserve"> angulorũ incidentię & reflexionis <lb/>ęqualitas, ut prius, tã in radijs obliquè incidẽtib.</s> <s xml:id="echoid-s32014" xml:space="preserve"> ꝗ̃ in ipſo radio perpẽdiculari.</s> <s xml:id="echoid-s32015" xml:space="preserve"> Patet ergo ꝓpoſitũ.</s> <s xml:id="echoid-s32016" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1303" type="section" level="0" n="0"> <head xml:id="echoid-head1025" xml:space="preserve" style="it">14. In columnaribus conuexis ſpeculis radio incidente & reſlexo, ſemper angulus in cidentiæ <lb/>eſt æqualis angulo reflexionis. Euclides 1 th. catoptr. Alhazen 12 n 4.</head> <p> <s xml:id="echoid-s32017" xml:space="preserve">Sumatur enim columna rotunda, quæ ſit altitudinis trium digitorum:</s> <s xml:id="echoid-s32018" xml:space="preserve"> & cuius baſis circuli dia-<lb/>meter ſit ſex digitorum:</s> <s xml:id="echoid-s32019" xml:space="preserve"> & reſecetur portio circuli baſis illius columnæ, ut prius in ſpeculis ſphæri-<lb/>cis:</s> <s xml:id="echoid-s32020" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s32021" xml:space="preserve"> ex ferro mundo portio columnę, cuius baſis ſit illa portio ciculi, & altitudo ipſius trium <lb/>digitorum:</s> <s xml:id="echoid-s32022" xml:space="preserve"> & ſecundum concauitatem illius formetur cõuexitas illius portionis:</s> <s xml:id="echoid-s32023" xml:space="preserve"> fiantq́;</s> <s xml:id="echoid-s32024" xml:space="preserve"> omnes li-<lb/>neę lõgitudinis eius perpendiculares ſuper utraſq;</s> <s xml:id="echoid-s32025" xml:space="preserve"> baſes:</s> <s xml:id="echoid-s32026" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s32027" xml:space="preserve"> ſinus uerfus bafis minor medietate <lb/>unius digiti.</s> <s xml:id="echoid-s32028" xml:space="preserve"> Hoc itaq;</s> <s xml:id="echoid-s32029" xml:space="preserve"> ſpeculum optimè politum ſui conuexo applicetur uni regularum ſimili prio <lb/>ribus, ita ut medius punctus eius cadat ſuper medium punctũ regulæ, quieſtz, & ita ut linea longi-<lb/>tudinis diuidens ipſius conuexam ſuperficiẽ per ęqualia, ſit in ſuperficie regulæ:</s> <s xml:id="echoid-s32030" xml:space="preserve"> & applicetur ei ſe-<lb/>cundum lineã longitudinis eius, quę eſt b p:</s> <s xml:id="echoid-s32031" xml:space="preserve"> & hoc fieri poterit, ſiutriuſq;</s> <s xml:id="echoid-s32032" xml:space="preserve"> baſis arcus per ęqualia di <lb/>uidatur, & puncta media ſignata lineę b p applicẽtur.</s> <s xml:id="echoid-s32033" xml:space="preserve"> Immittaturitaq;</s> <s xml:id="echoid-s32034" xml:space="preserve"> regula cũ ſpeculo ipſi inſtru-<lb/>mento, ut prius, & fiat operatio ſimilis priori:</s> <s xml:id="echoid-s32035" xml:space="preserve"> demõſtrabiturq́;</s> <s xml:id="echoid-s32036" xml:space="preserve"> angulorũ incidentię & reflexionis ę-<lb/>qualitas, ut ſuprà:</s> <s xml:id="echoid-s32037" xml:space="preserve"> nec eſt in aliquo à paſsiõe ſpeculorũ planorũ in his ſpeculis diuerfitas, niſi in hoc:</s> <s xml:id="echoid-s32038" xml:space="preserve"> <lb/>quòd ſi radio per foramen medium incidente, regula hæc obliquetur ſecundum partẽ dextram uel <lb/>finiftram:</s> <s xml:id="echoid-s32039" xml:space="preserve"> apparebit inde lux reflecti ſuper idem medium foramen & medium lucis ſuper medium <lb/>foraminis, quę lux in ſpeculis alijs obliquatur.</s> <s xml:id="echoid-s32040" xml:space="preserve"> Quoniã enim in ſpeculis columnarib.</s> <s xml:id="echoid-s32041" xml:space="preserve"> radins perpen <lb/>diculariter incidens uni lineæ longitudinis, perpendiculariter unicuiq;</s> <s xml:id="echoid-s32042" xml:space="preserve"> aliarum ſibi oppofitarũ in-<lb/>cidit:</s> <s xml:id="echoid-s32043" xml:space="preserve"> propter hoc in omnibus ipſis accidit uniformitas reflexionis:</s> <s xml:id="echoid-s32044" xml:space="preserve"> & ſemperradius perpendicula-<lb/>ris reflectitur in ſeipſum.</s> <s xml:id="echoid-s32045" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s32046" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1304" type="section" level="0" n="0"> <head xml:id="echoid-head1026" xml:space="preserve" style="it">15. In pyramidalibus conuexis ſpeculis radio incidente & reflexo, ſemper angulus incidetiæ <lb/>eſt æqualis angulo reflexionis. Euclides 1 the. catoptr. Alhazen 12 n 4.</head> <p> <s xml:id="echoid-s32047" xml:space="preserve">Fiat ex ferro puro ſpeculum pyramidale, cuius baſis ſit ęqualis baſi ſpeculi columnaris:</s> <s xml:id="echoid-s32048" xml:space="preserve"> erit ergo <lb/>chorda illius baſis trium digitorum:</s> <s xml:id="echoid-s32049" xml:space="preserve"> & ſinus uerſus minor medietate unius digiti.</s> <s xml:id="echoid-s32050" xml:space="preserve"> Sit aũt linea lõgi-<lb/>tudinis ſpeculi quatuor digitorum & dimidij, & hoc optimè exterius politum applicetur uni ſimi-<lb/>lium regularum taliter concauatæ, ut medius punctus eius ſit ſuper punctum z medium punctũ re-<lb/>gulæ:</s> <s xml:id="echoid-s32051" xml:space="preserve"> & ut acumen eius ſit in termino lineę b p:</s> <s xml:id="echoid-s32052" xml:space="preserve"> & linea diuidens portionem pyramidalẽ per ęqua <lb/>lia, quę ſcilicet â uertice pyramidis ad mediũ punctũ arcus baſis producitur, ſit in ſuperficie regulę.</s> <s xml:id="echoid-s32053" xml:space="preserve"> <lb/>Immiſſa quoq;</s> <s xml:id="echoid-s32054" xml:space="preserve"> regula cũ ſpeculo in inſtrumentũ, fiat operatio, ut prius:</s> <s xml:id="echoid-s32055" xml:space="preserve"> & accidũt omnia, quęin ſpe <lb/>culis columnarib.</s> <s xml:id="echoid-s32056" xml:space="preserve"> conuexis accidebãt, Eſt ergo & in ipſis angulus incidentię æqualis angulo refle-<lb/>xionis:</s> <s xml:id="echoid-s32057" xml:space="preserve"> & radius քpẽdicularis ſemք reflectitur in ſeipſum, ut patuit in p̃miſsis.</s> <s xml:id="echoid-s32058" xml:space="preserve"> Pater ergo ꝓpoſitũ.</s> <s xml:id="echoid-s32059" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1305" type="section" level="0" n="0"> <head xml:id="echoid-head1027" xml:space="preserve" style="it">16. In columnaribus concauis ſpeculis radio incidente & reflexo, ſemper angulus incidentiæ <lb/>eſt æqualis angulo reflexionis. Euclides 1 the. catoptr. Alhazen 12 n 4.</head> <p> <s xml:id="echoid-s32060" xml:space="preserve">Fiat ferreum ſpeculũ columnare cõcauum, cuius concauitas ſit omnino æqualis prioris colũna-<lb/> <pb o="198" file="0500" n="500" rhead="VITELLONIS OPTICAE"/> ris ſpeculi conuexitati:</s> <s xml:id="echoid-s32061" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s32062" xml:space="preserve"> optimè ſecundum concauitatẽ arcus portionis baſis interius politũ:</s> <s xml:id="echoid-s32063" xml:space="preserve"> & <lb/>hoc applicetur uniregularum ſimilium concauatæ, ut prius, taliter, ut chordæ arcus utriuſq;</s> <s xml:id="echoid-s32064" xml:space="preserve"> baſis <lb/>cum extremis lineis lõgitudinis ſint in ſuperficie regulę:</s> <s xml:id="echoid-s32065" xml:space="preserve"> & fiat operatio, ut prius:</s> <s xml:id="echoid-s32066" xml:space="preserve"> accidentq́;</s> <s xml:id="echoid-s32067" xml:space="preserve"> omnia, <lb/>quæ in ſpeculis columnaribus conuexis accidebant.</s> <s xml:id="echoid-s32068" xml:space="preserve"> Et per hoc patet propoſitum.</s> <s xml:id="echoid-s32069" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1306" type="section" level="0" n="0"> <head xml:id="echoid-head1028" xml:space="preserve" style="it">17. In pyramidalibus concauis ſpeculis radio incidente & reflexo, ſemper angulus incidentiæ <lb/>eſt æqualis angulo reflexianis. Euclides 1 the. catoptr. Alhazen 12 n 4.</head> <p> <s xml:id="echoid-s32070" xml:space="preserve">Fiat ſpeculum ferreum pyramidale concauum, cuius concauitas ſit omnino æqualis præmiſsi cõ <lb/>uexi pyramidalis ſpeculi conuexitati:</s> <s xml:id="echoid-s32071" xml:space="preserve"> & poliatur interius:</s> <s xml:id="echoid-s32072" xml:space="preserve"> appliceturq́;</s> <s xml:id="echoid-s32073" xml:space="preserve"> uni regularum ſimilium ta-<lb/>liter, ut medius punctus eius ſit ſuper punctum z, & ut acumẽ eius ſit directè in linea b p, & ut chor-<lb/>da arcus ipſius baſis ſit in ſuperficie regulæ, Cum aũt linea longitudinis portionis pyramidalis ſpe-<lb/>culi ſit quatuor digitorum & dimidij, reſtat ex longitudine regulæ digitus & dimidius tam in ſpecu <lb/>lo concauo quàm in conuexo.</s> <s xml:id="echoid-s32074" xml:space="preserve"> Immiſſa quoq;</s> <s xml:id="echoid-s32075" xml:space="preserve"> regula cum ſpeculo in inſtrumentum, fiat operatio, <lb/>ut prius:</s> <s xml:id="echoid-s32076" xml:space="preserve"> accidentq́;</s> <s xml:id="echoid-s32077" xml:space="preserve"> omnia, quę in ſpeculis pyramidalibus conuexis accidebantin reflexioneradio <lb/>rum obliquè incidentiũ ad an gulos ęquales:</s> <s xml:id="echoid-s32078" xml:space="preserve"> & in reflexione radiorum perpendiculariũ in ſe ipſos.</s> <s xml:id="echoid-s32079" xml:space="preserve"> <lb/>Patet ergo propoſitũ.</s> <s xml:id="echoid-s32080" xml:space="preserve"> Palã itaq;</s> <s xml:id="echoid-s32081" xml:space="preserve"> ex præmiſsis, quoniã in omni reflexione à quibuſcunq;</s> <s xml:id="echoid-s32082" xml:space="preserve"> ſpeculis po <lb/>litis regularibus (ut ſunt hæc ſeptem ſpecula) ſemperradius ſecundum lineam rectam perpendicu <lb/>lariter incidens ſecundum eandem rectam perpendicularem reflectitur:</s> <s xml:id="echoid-s32083" xml:space="preserve"> & quòd radius ſecundum <lb/>lineã rectã obliquè incidens, ſecundũ aliã lineã obliquã reflectitur, ita tamẽ quòdangulus inciden-<lb/>tiæ eſt ſemper ęqualis angulo reflexionis:</s> <s xml:id="echoid-s32084" xml:space="preserve"> unde hoc per rationabilẽ ſenſus experiẽtiã inuẽto, ſem ք <lb/>utuniuerſaliprincipio, deinceps in omnib.</s> <s xml:id="echoid-s32085" xml:space="preserve"> his ſpeculis utemur:</s> <s xml:id="echoid-s32086" xml:space="preserve"> & licet hoc, ut quidẽ huius ſcientiæ <lb/>principium, ſit experimentaliter declaratum:</s> <s xml:id="echoid-s32087" xml:space="preserve"> poteſttamen etiam per aliquẽ demonſtrationis mo-<lb/>dum ad ipſius ſciẽtiam perueniri:</s> <s xml:id="echoid-s32088" xml:space="preserve"> unde nos ipſum, prout diligentius poterimus, tentabimus demõ-<lb/>ſtrare:</s> <s xml:id="echoid-s32089" xml:space="preserve"> propter quod duo ſequentia theorem ata duximus præmittenda.</s> <s xml:id="echoid-s32090" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1307" type="section" level="0" n="0"> <head xml:id="echoid-head1029" xml:space="preserve" style="it">18. Omnis res uiſa per ſpeculum quodcun, ſub breuiſsimis lineis comprehenditur à uiſu. Pto <lb/>lemaus 4 th. 1 catoptr.</head> <p> <s xml:id="echoid-s32091" xml:space="preserve">Sit ſpeculum, in cuius ſuperficie ſit linea recta uel curua, quæ ſit a c b:</s> <s xml:id="echoid-s32092" xml:space="preserve"> rei quoq;</s> <s xml:id="echoid-s32093" xml:space="preserve"> uiſæ punctus ſit <lb/>d:</s> <s xml:id="echoid-s32094" xml:space="preserve"> & centrum oculi ſit f:</s> <s xml:id="echoid-s32095" xml:space="preserve"> & punctus d uideatur reflexus à puncto ſpeculi c.</s> <s xml:id="echoid-s32096" xml:space="preserve"> Dico quòd lineę f c & d c, <lb/>ſunt breuiores omnibus lineis protractis à punctis d & fad quælibet alia puncta ſpeculi.</s> <s xml:id="echoid-s32097" xml:space="preserve"> Ducantur <lb/> <anchor type="figure" xlink:label="fig-0500-01a" xlink:href="fig-0500-01"/> enim à puncto alio ſuperficiei ſpeculi (quod ſit e) lineæ <lb/>e d & e f, quæ non ſint breuiores quàm lineæ c d & c f, <lb/>neque æquales illis, ſed longiores.</s> <s xml:id="echoid-s32098" xml:space="preserve"> Quia ergo, ut patet ք <lb/>5 huius, natura in omnibus agit ſecundũ lineas breuio-<lb/>res:</s> <s xml:id="echoid-s32099" xml:space="preserve"> multiplicatio uerò formarum ad ſuperficies ſpecu <lb/>lorum eft naturalis:</s> <s xml:id="echoid-s32100" xml:space="preserve"> quoniam fit opere naturæ, ſicut & <lb/>omnis alia diffuſio formarum, ut in philoſophia natura <lb/>licapitulo de naturali a ctione oſtendimus:</s> <s xml:id="echoid-s32101" xml:space="preserve"> & ſimiliter <lb/>reflexio formarum à ſuperficieb.</s> <s xml:id="echoid-s32102" xml:space="preserve"> ſpeculorum ad uiſum <lb/>eſt purè naturalis, quoniam ſit ab opere naturæ, & com-<lb/>pletur per actionem animæ, ſicut & omnis alia uiſio, ut <lb/>patet per totum quartum huius noſtræ ſcientiæ librum:</s> <s xml:id="echoid-s32103" xml:space="preserve"> eſt autem anima tanquam natura anima-<lb/>lium.</s> <s xml:id="echoid-s32104" xml:space="preserve"> Patet ergo quòd hæc diffuſio formæ & reflexio & comprehenfio, quæ fit ſecundum ipſam, eſt <lb/>uerè naturalis:</s> <s xml:id="echoid-s32105" xml:space="preserve"> fiet ergo ſecundum lineas breuiores:</s> <s xml:id="echoid-s32106" xml:space="preserve"> quod eſt propoſitum:</s> <s xml:id="echoid-s32107" xml:space="preserve"> fruſtra enim fieret ſecun <lb/>dum lineas longiores, cum poſsit melius & certius fieri ſecundum lineas breuiores.</s> <s xml:id="echoid-s32108" xml:space="preserve"/> </p> <div xml:id="echoid-div1307" type="float" level="0" n="0"> <figure xlink:label="fig-0500-01" xlink:href="fig-0500-01a"> <variables xml:id="echoid-variables536" xml:space="preserve">f d a c e b</variables> </figure> </div> </div> <div xml:id="echoid-div1309" type="section" level="0" n="0"> <head xml:id="echoid-head1030" xml:space="preserve" style="it">19. Lineæ incidentiæ & reflexionis, continentes angulos æquales cum perpẽdiculari à puncto <lb/>ſui concurſus ſuper ſuperficiem ſpeculi plani uel cõuexiextracta, ſunt breuiores omnib. lineis ab <lb/>eiſdem termin is ſuper eandem ſuperficiem ſpeculi productis, continentιb. angulos inæquales cũ <lb/>perpendicularibus à punctis ſui concurſus extractis.</head> <p> <s xml:id="echoid-s32109" xml:space="preserve">Quod hic proponitur, faciliter per 17 & 18 th.</s> <s xml:id="echoid-s32110" xml:space="preserve"> 1 huius poteſt demõſtrari:</s> <s xml:id="echoid-s32111" xml:space="preserve"> ſed quia aliter eſtidẽ de-<lb/>monſtrabile, ſit res uiſa quęcunq;</s> <s xml:id="echoid-s32112" xml:space="preserve">, in qua ſit punctus c:</s> <s xml:id="echoid-s32113" xml:space="preserve"> & ſit ſpeculum planum, in cuius fuperficie ſit <lb/>linea h d e:</s> <s xml:id="echoid-s32114" xml:space="preserve"> ſit autem nunc, exempli cauſſa, ſpeculum datum planum:</s> <s xml:id="echoid-s32115" xml:space="preserve"> erit ergo linea h d e linea recta:</s> <s xml:id="echoid-s32116" xml:space="preserve"> <lb/>lineæ quoq;</s> <s xml:id="echoid-s32117" xml:space="preserve"> continentes angulos ęquales cum linea h d e, fint c d & d f.</s> <s xml:id="echoid-s32118" xml:space="preserve"> Aut ergo centrum oculi erit <lb/>in eadem linea æquidiſtante lineæ h d e, in qua eſt c punctus rei uiſæ, aut n on.</s> <s xml:id="echoid-s32119" xml:space="preserve"> Si ſit:</s> <s xml:id="echoid-s32120" xml:space="preserve"> eſto itaq;</s> <s xml:id="echoid-s32121" xml:space="preserve"> pun-<lb/>ctum oculi f:</s> <s xml:id="echoid-s32122" xml:space="preserve"> & protrahatur linea c f:</s> <s xml:id="echoid-s32123" xml:space="preserve"> & extrahatur à puncto d perpendicularis ſuper ſpeculi ſuperfi-<lb/>ciem per 12 p 11, quæ protracta, quia ſecat angulum c d f, patet per 29 th.</s> <s xml:id="echoid-s32124" xml:space="preserve"> huius, quoniã ipſa ſecabit li <lb/>neã c f:</s> <s xml:id="echoid-s32125" xml:space="preserve"> eſt enim in eadẽ ſuքficie cũ illa:</s> <s xml:id="echoid-s32126" xml:space="preserve"> hęc ergo perpẽdicularis ꝓducta ad lineã c f, ſit d g:</s> <s xml:id="echoid-s32127" xml:space="preserve"> erit ergo li <lb/>nea d g perpẽdicularis ſuper lineã c f ęquidiftantẽ lineæ d e per 29 p 1.</s> <s xml:id="echoid-s32128" xml:space="preserve"> Quia ergo c d h angulus eſt ę-<lb/>qualis f d e angulo, dẽptis illis angulis ęqualib.</s> <s xml:id="echoid-s32129" xml:space="preserve"> à duobus rectis, qui ſunt g d h & g d e, erũt anguli reſi <lb/>dui ęquales:</s> <s xml:id="echoid-s32130" xml:space="preserve"> eſt ergo angulus c d g ęqualis angulo g d f.</s> <s xml:id="echoid-s32131" xml:space="preserve"> Et quoniã trigonorũ c d g & f d g ambo angu <lb/>li, qui ſunt ad pũctũ g, ſunt recti:</s> <s xml:id="echoid-s32132" xml:space="preserve"> palam ք 32 p 1, quoniã anguli d c g & d f g ſunt ęquales.</s> <s xml:id="echoid-s32133" xml:space="preserve"> Sunt itaq;</s> <s xml:id="echoid-s32134" xml:space="preserve"> tri <lb/>goni c d g & f d g ęquiãguli:</s> <s xml:id="echoid-s32135" xml:space="preserve"> latera ergo ęquos angulos reſpiciẽtia ſunt ꝓportionalia ք 4 p 6:</s> <s xml:id="echoid-s32136" xml:space="preserve"> & quo-<lb/>niam latus d g ęquale eſt ſibijpſi, erit latus f d æquale lateri c d:</s> <s xml:id="echoid-s32137" xml:space="preserve"> ductisq́;</s> <s xml:id="echoid-s32138" xml:space="preserve"> lineis f e & c e ſuper pun-<lb/> <pb o="199" file="0501" n="501" rhead="LIBER QVINTVS."/> ctum e punctum lineæ d e, quæ ut patet ex pręmiſsis, eſt æ quidiſtans lineæ e f, patet quòd linea <lb/> <anchor type="figure" xlink:label="fig-0501-01a" xlink:href="fig-0501-01"/> c e eſt maior quàm linea f e per <lb/>19 p 1:</s> <s xml:id="echoid-s32139" xml:space="preserve"> eſt enim angulus c f e ma-<lb/>ior angulo g f d, & angulus f c e <lb/>eſt minor angulo g c d:</s> <s xml:id="echoid-s32140" xml:space="preserve"> reſtat <lb/>ergo ut angulus c f e ſit maior <lb/>angulo f c e, & quòd linea c e <lb/>fit maior quàm linea f e, Et quia <lb/>ſuper eandem baſim, quæ c f, & <lb/>inter lineas æquidiſtantes, quæ <lb/>ſunt d e & c f, collocatur trigo-<lb/>num c f d, cuius latera c d & d <lb/>f ſunt æqualia, & trigonum c f e, <lb/>cuius latera c e & f e ſunt inæ-<lb/>qualia, ut patet ex præmiſsis:</s> <s xml:id="echoid-s32141" xml:space="preserve"> di-<lb/>co quòd latera c d & d f ambo <lb/>ſimul ſumpta ſunt minora ambo <lb/>bus lateribus c e & f e ſimul <lb/>ſumptis.</s> <s xml:id="echoid-s32142" xml:space="preserve"> Producatur enim linea <lb/>c d ultra punctum d in continuum & directum ad punctum l, ita ut linea d l ſit æ qualis, lineæ d f:</s> <s xml:id="echoid-s32143" xml:space="preserve"> <lb/>ſed & linea c e, quæ eſt longius latus trigoni c f e, producatur ultra punctum e ad punctum m.</s> <s xml:id="echoid-s32144" xml:space="preserve"> <lb/>donec linea e m fiat æqualis lineæ e f:</s> <s xml:id="echoid-s32145" xml:space="preserve"> & copuletur linea l m & linea e l.</s> <s xml:id="echoid-s32146" xml:space="preserve"> Et quia angulus f d e eſt <lb/>æqualis angulo d f c per 29 p 1, & angulus d f c eſt æqualis angulo d c f, ut patet ex pręmiſsis:</s> <s xml:id="echoid-s32147" xml:space="preserve"> <lb/>angulus uerò e d l æqualis eſt angulo f c d per 29 p 1:</s> <s xml:id="echoid-s32148" xml:space="preserve"> erit ergo angulus f d e ęqualis angulo e d l:</s> <s xml:id="echoid-s32149" xml:space="preserve"> <lb/>ſed linea d l eſt æqualis lineę d f, & linea d e eſt ambobus trigonis (quę ſunt f d e & e d l) com-<lb/>munis:</s> <s xml:id="echoid-s32150" xml:space="preserve"> ergo per 4 p 1, eſt linea f e æqualis lineæ l e:</s> <s xml:id="echoid-s32151" xml:space="preserve"> ergo & lineæ e m:</s> <s xml:id="echoid-s32152" xml:space="preserve"> ergo per 5 p 1 anguli e l m <lb/>& e m l ſunt æquales:</s> <s xml:id="echoid-s32153" xml:space="preserve"> totalis ergo angulus c l m eſt maior angulo c m l:</s> <s xml:id="echoid-s32154" xml:space="preserve"> ergo per 19 p 1 linea c m <lb/>eſt maior quàm linea c l.</s> <s xml:id="echoid-s32155" xml:space="preserve"> Duo ergo latera c e & e f pariter accepta maiora ſunt duobus lateribus <lb/>c d & d f pariter acceptis.</s> <s xml:id="echoid-s32156" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s32157" xml:space="preserve"> Si autem uiſus & res uiſa non ſint in eadem li-<lb/>nea æquidiſtante lineę h e:</s> <s xml:id="echoid-s32158" xml:space="preserve"> ſit punctus reiuiſæ, ut prius, c:</s> <s xml:id="echoid-s32159" xml:space="preserve"> & centrum uiſus ſit b:</s> <s xml:id="echoid-s32160" xml:space="preserve"> & ducatur linea <lb/>b a æ quidiſtans lineę h d e, quę eſt in ſpeculi ſuperficie:</s> <s xml:id="echoid-s32161" xml:space="preserve"> & producatur linea d c ad punctum a:</s> <s xml:id="echoid-s32162" xml:space="preserve"> & <lb/>protrahantur lineę c d, b d, c e, a e, e b:</s> <s xml:id="echoid-s32163" xml:space="preserve"> & ſint lineę continentes æquales angulos cum linea d e, quæ <lb/>c d & d b:</s> <s xml:id="echoid-s32164" xml:space="preserve"> in ęquales uerò angulos contineant c e & b e:</s> <s xml:id="echoid-s32165" xml:space="preserve"> erunt ergo, ut ſuprà, lineę a d & b d ęqua-<lb/>les, producta perpendiculari d k à puncto d.</s> <s xml:id="echoid-s32166" xml:space="preserve"> Comparato ergo trigono a d b ad trigonum a e b:</s> <s xml:id="echoid-s32167" xml:space="preserve"> erũt <lb/>lineę a d & d b minores quàm lineę a e & e b, ut patet ſecundum pręmiſſa, Cum enim lineę a d & d b <lb/>ſint æquales per 2 p 6, 18 p 5 & corollarium 4 p 5:</s> <s xml:id="echoid-s32168" xml:space="preserve"> ideo quia lineę c d & d f ſunt æquales:</s> <s xml:id="echoid-s32169" xml:space="preserve"> lineę uerò <lb/>a e & b e ſuntinęquales:</s> <s xml:id="echoid-s32170" xml:space="preserve"> erũt duo latera a e & b e ſimul iuncta maiora duobus lateribus a d & d b ſi-<lb/>mul iunctis:</s> <s xml:id="echoid-s32171" xml:space="preserve"> ergo cum a c & c e duo latera trigoni a c e per 20 p 1 ſintlongiora latere a e:</s> <s xml:id="echoid-s32172" xml:space="preserve"> eruntiſtę <lb/>tres lineę a c, c e, e b longiores duabus lineis, quę ſunt a d & d b:</s> <s xml:id="echoid-s32173" xml:space="preserve"> ergo dempta hincinde ipſa a c com <lb/>muni, remanèbunt lineę c e & e b maiores quàm lineę c d & d b.</s> <s xml:id="echoid-s32174" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s32175" xml:space="preserve"> Eteodẽ mo-<lb/>do poteſt demonſtrari in quibuſcunq;</s> <s xml:id="echoid-s32176" xml:space="preserve"> alijs ſpeculis conuexis.</s> <s xml:id="echoid-s32177" xml:space="preserve"> Sit ergo ſpeculum non planum, cu-<lb/>iuſcunq;</s> <s xml:id="echoid-s32178" xml:space="preserve"> figurę conuexę placuerit, & ſit nunc, exemplicauſſa, ſphęricum conuexum, quia idem ac-<lb/> <anchor type="figure" xlink:label="fig-0501-02a" xlink:href="fig-0501-02"/> ciditin alijs:</s> <s xml:id="echoid-s32179" xml:space="preserve"> & ſit h a b:</s> <s xml:id="echoid-s32180" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s32181" xml:space="preserve"> centrum uiſus g:</s> <s xml:id="echoid-s32182" xml:space="preserve"> & pun-<lb/>ctum uiſum d;</s> <s xml:id="echoid-s32183" xml:space="preserve"> & lineę g a & d a ęquales angulos con-<lb/>tineant cum linea circulum contingente in puncto a, <lb/>quę ſit e f, ita ut angulus e a g ſit ęqualis angulo f a d:</s> <s xml:id="echoid-s32184" xml:space="preserve"> <lb/>incidantq́;</s> <s xml:id="echoid-s32185" xml:space="preserve"> lineę g b & d b in punctum alium ſpeculi, <lb/>qui ſit b, ita ut inęquales angulos contineant cum li-<lb/>nea contingente ſpeculum in puncto b.</s> <s xml:id="echoid-s32186" xml:space="preserve"> Dico, quòd li <lb/>neę g a & a d ſunt minores lineis g b & d b.</s> <s xml:id="echoid-s32187" xml:space="preserve"> Quoniam <lb/>enim angulus contingẽtię, qui eſt h a e ęqualis eſt an-<lb/>gulo b a f, uterq;</s> <s xml:id="echoid-s32188" xml:space="preserve"> enim eſt minimus acutorum per 16 <lb/>p 3:</s> <s xml:id="echoid-s32189" xml:space="preserve"> angulus uerò e a g eſt ęqualis angulo f a d:</s> <s xml:id="echoid-s32190" xml:space="preserve"> ſit pun <lb/>ctus, in quo linea g b ſecat lineam contingentem, quę <lb/>eſt e f, punctus z:</s> <s xml:id="echoid-s32191" xml:space="preserve"> & ducatur linea d z:</s> <s xml:id="echoid-s32192" xml:space="preserve"> palàm per 16 p 1, <lb/>quoniam angulus e a g eſt maior angulo e z g:</s> <s xml:id="echoid-s32193" xml:space="preserve"> ergo <lb/>angulus d a z eſt maior angulo g z a:</s> <s xml:id="echoid-s32194" xml:space="preserve"> ſed angulus d z f <lb/>eſt maior angulo d a z:</s> <s xml:id="echoid-s32195" xml:space="preserve"> ergo angulus f z d eſt maior <lb/>angulo g z a:</s> <s xml:id="echoid-s32196" xml:space="preserve"> ergo per 17 th.</s> <s xml:id="echoid-s32197" xml:space="preserve"> 1 huius duę lineę g a & d a <lb/>ſunt minores duabus lineis g z & d z:</s> <s xml:id="echoid-s32198" xml:space="preserve"> ſed lineę g z & d z ſunt minores lineis g b & d b:</s> <s xml:id="echoid-s32199" xml:space="preserve"> quoniã linea <lb/>g b eſt maior quàm linea g z, ut totum parte:</s> <s xml:id="echoid-s32200" xml:space="preserve"> linea uerò d b eſt maior quàm linea d z per 8 p 3.</s> <s xml:id="echoid-s32201" xml:space="preserve"> Patet <lb/>ergo propoſitum uniuerſaliter in ſuperficiebus quorumlibet ſpeculorum cōuexorum.</s> <s xml:id="echoid-s32202" xml:space="preserve"> Hoc aũt idẽ <lb/>ut prędiximus, poteſt per 17 uel per 18 th.</s> <s xml:id="echoid-s32203" xml:space="preserve"> 1 huius facilius demōſtrari:</s> <s xml:id="echoid-s32204" xml:space="preserve"> quoniã in illis oſtẽdimus, qđ <lb/>lineę rectę continentes angulos ęquales cum linea, cui ad unum punctum incidũt, ſunt breuiores <lb/> <pb o="200" file="0502" n="502" rhead="VITELLONIS OPTICAE"/> omnibus lineis ab eiſdem terminis ſuper eandem lineam ad unũ punctum alium productis, Ethoc <lb/>propoſuimus per 17 th.</s> <s xml:id="echoid-s32205" xml:space="preserve"> 1 huius in lineis rectis, per 18 eiuſdẽ primi in lineis conuexis.</s> <s xml:id="echoid-s32206" xml:space="preserve"/> </p> <div xml:id="echoid-div1309" type="float" level="0" n="0"> <figure xlink:label="fig-0501-01" xlink:href="fig-0501-01a"> <variables xml:id="echoid-variables537" xml:space="preserve">a k b c g f b d e l m</variables> </figure> <figure xlink:label="fig-0501-02" xlink:href="fig-0501-02a"> <variables xml:id="echoid-variables538" xml:space="preserve">d g f o a b h</variables> </figure> </div> </div> <div xml:id="echoid-div1311" type="section" level="0" n="0"> <head xml:id="echoid-head1031" xml:space="preserve" style="it">20. In omnireflexione à quibuſcun ſpeculis facta, ſemper angulus incidẽtiæ eſt aqualis an-<lb/>gulo reflexionis: ex quo patet, quòd linearum inæqualit as natur am reflexionis non immutat. <lb/>Euclides 1 th. catoptr. Ptolemæus 4. 5 th. 1 catoptr. Alhazen. 10. 18 n 4.</head> <p> <s xml:id="echoid-s32207" xml:space="preserve">Quoniam enim, ut patet per 18 huius, omnis res uiſa per quodcunq;</s> <s xml:id="echoid-s32208" xml:space="preserve"> ſpeculũ planũ uel cõuexum <lb/>uel concauũ, ſub breuiſsimis lineis comprehẽditur:</s> <s xml:id="echoid-s32209" xml:space="preserve"> lineę uerò ab eiſdẽ punctis, utpote â pũcto rei <lb/>uiſę, & cẽtro uiſus ad ſuperficiẽ cuiuſcũq;</s> <s xml:id="echoid-s32210" xml:space="preserve"> ſpeculi productę breuiſsimę ſunt, quę continẽt angulos <lb/>æquales, & cũ lineis contingẽtibus ſuperficies ſpeculorũ, & cũ perpẽdicularibus à pũctis fui cõcur <lb/>ſus productis ſuper ſuperficies ſpeculorũ, ut patet ք pręmiſſam:</s> <s xml:id="echoid-s32211" xml:space="preserve"> angulus uerò, quẽ facit linea â pun <lb/>cto rei uiſę producta, eſt angulus incidentię, & angulus, quẽ facit linea ab illo pũcto ad centrũ uiſus <lb/>producta, eſt angulus reflexionis:</s> <s xml:id="echoid-s32212" xml:space="preserve"> patet ergo quòd angulus incidẽtię <lb/> <anchor type="figure" xlink:label="fig-0502-01a" xlink:href="fig-0502-01"/> ſemper eſt æqualis angulo reflexionis, à quocunq;</s> <s xml:id="echoid-s32213" xml:space="preserve"> ſpeculo plano uel <lb/>cõuexo fiat reflexio.</s> <s xml:id="echoid-s32214" xml:space="preserve"> Sed & idẽ patẽt in cõcauis ſpeculis quibuſcunq;</s> <s xml:id="echoid-s32215" xml:space="preserve">. <lb/>Sit enim aliquod ſpeculũ cõuexũ, in quo ſit circulus e b d:</s> <s xml:id="echoid-s32216" xml:space="preserve"> quẽ in pun <lb/>cto b cõtingat extrinſecus circulus a b c:</s> <s xml:id="echoid-s32217" xml:space="preserve"> & ducatur à pũcto b linea f b <lb/>g ambos circulos cõtingẽs in pũcto b:</s> <s xml:id="echoid-s32218" xml:space="preserve"> & ſit pũctus rei uiſęh, cuius for-<lb/>ma à pũcto b ſpeculi cõuexi reflectatur ad uiſum exiſtentẽ in pũcto k:</s> <s xml:id="echoid-s32219" xml:space="preserve"> <lb/>eritq́;</s> <s xml:id="echoid-s32220" xml:space="preserve"> ք p̃miſſangulus h b f æqualis angulo k b g:</s> <s xml:id="echoid-s32221" xml:space="preserve"> ſed & angulus a b f <lb/>eſt ęqualis angulo c b g per 16 p 3, quoniã ſunt anguli cõtingẽtię:</s> <s xml:id="echoid-s32222" xml:space="preserve"> relin <lb/>quitur ergo angulus h b a, qui eſt angulus incidẽtię in ſpeculo cõcauo <lb/>a b c, æ qualis angulo k b c, qui eſt angulus reflexionis.</s> <s xml:id="echoid-s32223" xml:space="preserve"> Patet ergo pro-<lb/>poſitũ.</s> <s xml:id="echoid-s32224" xml:space="preserve"> Vniuerſaliter enim in omnibus ſpeculis cõcauis hęc demõſtra <lb/>tio poteſt coaptari.</s> <s xml:id="echoid-s32225" xml:space="preserve"> Eſt aũt etiã hoc rationabile.</s> <s xml:id="echoid-s32226" xml:space="preserve"> Si enim linea inciden-<lb/>tię, quę ſit, exempli cauſſa, a b, lineã rectã c b d protractã in ſuperficie <lb/>plani ſpeculi, uel contingentẽ ſuperficiẽ conuexam uel concauã alicu <lb/>ius ſpeculi ſine reflexione penetraret in puncto b uſq;</s> <s xml:id="echoid-s32227" xml:space="preserve"> ad punctũ e:</s> <s xml:id="echoid-s32228" xml:space="preserve"> pa <lb/>làm per 15 p 1, quòd angulus incidẽtię a b c fieret ęqualis angulo e b d.</s> <s xml:id="echoid-s32229" xml:space="preserve"> <lb/>Si ergo fiat reflexio ſecũdum lineam b f:</s> <s xml:id="echoid-s32230" xml:space="preserve"> conuenientius <lb/> <anchor type="figure" xlink:label="fig-0502-02a" xlink:href="fig-0502-02"/> eſt, utfiat ſecũdum angulum ęqualem illi contrapofito <lb/>quàm ſecũdum aliquem aliũ angulũ, ita utangulus f b <lb/>d fiat ęqualis angulo e b d, & angulo a b c.</s> <s xml:id="echoid-s32231" xml:space="preserve"> Si enim pun-<lb/>ctis c & d exiſtẽtibus immotis, linea c d imaginetur re-<lb/>uolui:</s> <s xml:id="echoid-s32232" xml:space="preserve"> tũc enim linea e b propter ęqualitatem angulorũ <lb/>e b d & d b f cadet ſuper lineã b f:</s> <s xml:id="echoid-s32233" xml:space="preserve"> & hocuidetur impor-<lb/>tare nomẽ reflexionis.</s> <s xml:id="echoid-s32234" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s32235" xml:space="preserve"> Patet etiã <lb/>ex hoc corollariũ:</s> <s xml:id="echoid-s32236" xml:space="preserve"> linearũ enim inęqualitas, quia nõ im-<lb/>mutat angulorũ quantitatem:</s> <s xml:id="echoid-s32237" xml:space="preserve"> ergo neq;</s> <s xml:id="echoid-s32238" xml:space="preserve"> naturã reflexio <lb/>nis.</s> <s xml:id="echoid-s32239" xml:space="preserve"> Vnde pũcta eiuſdem lineę remotiora à pũcto refle-<lb/>xionis, poſſunt reflecti ad uiſum, ſicut pũcta eiuſdem lineę propinquiora puncto reflexionis.</s> <s xml:id="echoid-s32240" xml:space="preserve"> Vni-<lb/>uerſaliter enim omnia puncta eiuſdem lineę ſecundum ęqualem angulum reflecti poſſunt.</s> <s xml:id="echoid-s32241" xml:space="preserve"> Ethoc <lb/>proponebatur.</s> <s xml:id="echoid-s32242" xml:space="preserve"/> </p> <div xml:id="echoid-div1311" type="float" level="0" n="0"> <figure xlink:label="fig-0502-01" xlink:href="fig-0502-01a"> <variables xml:id="echoid-variables539" xml:space="preserve">e d f b g h k a c</variables> </figure> <figure xlink:label="fig-0502-02" xlink:href="fig-0502-02a"> <variables xml:id="echoid-variables540" xml:space="preserve">a f c b d e</variables> </figure> </div> </div> <div xml:id="echoid-div1313" type="section" level="0" n="0"> <head xml:id="echoid-head1032" xml:space="preserve" style="it">21. Omnis formæ ſecundum lineam perpendicularem ſuper ſuperficiem cuiuſcũ ſpeculi in-<lb/>cidentis, reflexio fit ſecundum lineam eandem. Euclides 2 th. catoptr. Alhazen 11 n 4.</head> <figure> <variables xml:id="echoid-variables541" xml:space="preserve">e a b d c</variables> </figure> <p> <s xml:id="echoid-s32243" xml:space="preserve">Verbi gratia, eſto, ut forma pũctia ſuperficiei ſpeculi b d cincidat <lb/>ſecundum lineam a d perpendicularem ſuper ſuperficiem b d c:</s> <s xml:id="echoid-s32244" xml:space="preserve"> dico <lb/>quòd reflexio formę puncti a erit ſecundum eandem lineam d a.</s> <s xml:id="echoid-s32245" xml:space="preserve"> Da-<lb/>to enim quòd ſecundum aliam lineam fiat reflexio:</s> <s xml:id="echoid-s32246" xml:space="preserve"> tunc, cum an-<lb/>gulus incidentiæ ſemper ſit ęqualis angulo reflexionis, ut patet per <lb/>præmiſſam:</s> <s xml:id="echoid-s32247" xml:space="preserve"> & in propoſito angulus incidentiæ ſit rectus:</s> <s xml:id="echoid-s32248" xml:space="preserve"> infiniti <lb/>quoque fint anguli recti ordinatim ſuper punctum d, nec ſit de-<lb/>clinatio formę plus ad unum punctum ſuperficiei b c, quàm ad ali-<lb/>ud:</s> <s xml:id="echoid-s32249" xml:space="preserve"> ęqualiter enim ſe habet linea a d, quę eſt linea incidentię, ad pun-<lb/>ctum b, & ad punctum c, & ad omnia alia puncta ſuperficiei b c, Sic <lb/>ergo erunt infinitę reflexiones ad infinita puncta ſuperficiei b c:</s> <s xml:id="echoid-s32250" xml:space="preserve"> quia <lb/>qua ratione ad unam differentiam poſitionis fieret reflexio:</s> <s xml:id="echoid-s32251" xml:space="preserve"> eadem <lb/>ratione fieret ad aliam & ad om nem:</s> <s xml:id="echoid-s32252" xml:space="preserve"> quod eſt inconueniens.</s> <s xml:id="echoid-s32253" xml:space="preserve"> Dabi-<lb/>tur ergo neceſſariò quòd fiat reflexio ſuper unam & eandẽ lineã a d, <lb/>ſecundũ quã incidentia fiebat.</s> <s xml:id="echoid-s32254" xml:space="preserve"> Perpẽdiculares ergo uel nõ reflectun-<lb/>tur, uel redeunt in ſe ipſas, & fortificatur actio talium formarũ.</s> <s xml:id="echoid-s32255" xml:space="preserve"> Si tamẽ dicatur qđ perpẽdicularis <lb/> <pb o="201" file="0503" n="503" rhead="LIBER QVINTVS."/> incidens per aliam lineam reflectitur:</s> <s xml:id="echoid-s32256" xml:space="preserve"> ſit, ut reflectatur per lineam d e:</s> <s xml:id="echoid-s32257" xml:space="preserve"> tunc ergo cum angulus inci-<lb/>dentiæ, ut patuit per præmiſſam, ſemper ſit æ qualis angulo reflexionis, erit angulus a d c æ qualis <lb/>angulo a d e:</s> <s xml:id="echoid-s32258" xml:space="preserve"> ſed angulus a d c æqualis eſt angulo a d b per hypotheſim:</s> <s xml:id="echoid-s32259" xml:space="preserve"> erit ergo angulus a d e ęqua <lb/>lis angulo a d b, pars ſuo toti:</s> <s xml:id="echoid-s32260" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s32261" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s32262" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1314" type="section" level="0" n="0"> <head xml:id="echoid-head1033" xml:space="preserve" style="it">22. Inter punct a formæ ſuperficiei cuiuſcun ſpeculi incidentis & ſpeculi oppoſiti ſuperfici-<lb/>em, neceſſe eſt infinitas pyramides figurari, conos & baſes binc inde mutuas babentes. Alha-<lb/>zen 14 n 4.</head> <p> <s xml:id="echoid-s32263" xml:space="preserve">Declaratum eſt enim per 1 huius, quoniam à quolibet pũcto corporis oppoſiti procedit lux uel <lb/>color ad quodlibet punctum ſpeculi:</s> <s xml:id="echoid-s32264" xml:space="preserve"> oẽs enim lineæ ductæ à quolibet puncto corporis recidunt <lb/>in unum punctum ſpeculi:</s> <s xml:id="echoid-s32265" xml:space="preserve"> & forma unius puncti corporis incidit omnibus punctis ſuperficiei to-<lb/>tius ſpeculi:</s> <s xml:id="echoid-s32266" xml:space="preserve"> eò quò ad omnẽ poſitionis differẽtiam fit diffuſio formarum.</s> <s xml:id="echoid-s32267" xml:space="preserve"> Tota ergo forma corpo <lb/>ris erit in unoquoq;</s> <s xml:id="echoid-s32268" xml:space="preserve"> puncto ſpeculi:</s> <s xml:id="echoid-s32269" xml:space="preserve"> & forma cuiuslibet puncti corporis in tota ſpeculi fuperficie.</s> <s xml:id="echoid-s32270" xml:space="preserve"> <lb/>Quot ergo ſunt puncta in ſuperficie ſpeculi, tot ſunt pyramides ad totam ſuperficiẽ formæ corpo-<lb/>ris terminatæ, quę ſuperficies ſit baſis omnium illarum pyramidum:</s> <s xml:id="echoid-s32271" xml:space="preserve"> & quot ſunt puncta in totali ſu <lb/>perficie corporis, cuius forma incidit ſpeculo, tot ſiunt pyramides ad totam ſuperficiẽ ſpeculi ter-<lb/>minatæ, quę fit baſis omniũ illarũ pyramidũ.</s> <s xml:id="echoid-s32272" xml:space="preserve"> Et ſunt oẽs iſtæ pyramides cõtinuę propter continui-<lb/>tatẽ punctorum in dictis ſuperficiebus exiſtentium potẽtia, nó actu:</s> <s xml:id="echoid-s32273" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s32274" xml:space="preserve"> axis cuiuslibet harũ py-<lb/>ramidũ punctus, ſecũdum quẽ ſpeculo incidit, pũctus medius totius formę ſpeculo incidẽtis:</s> <s xml:id="echoid-s32275" xml:space="preserve"> quo-<lb/>niam ab illo incidunt ſecundum æ qualem diſtantiam omnes punctialij circumſtantes æqualiter <lb/>medium punctum formæ.</s> <s xml:id="echoid-s32276" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s32277" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1315" type="section" level="0" n="0"> <head xml:id="echoid-head1034" xml:space="preserve" style="it">23. Impoßibile est uideri imagines in quibuſcun ſpeculis propter reflexionem radiorum ui-<lb/>ſualium à ſpeculis ad res uiſas: ſed ſolùm propter reflexionem formarum à ſpeculis ad uiſum-<lb/>Alhazen 20 n 4.</head> <p> <s xml:id="echoid-s32278" xml:space="preserve">Sienim radij uiſuales refle cterẽtur à ſpeculo ad res, quarũ uiſus accipit imagines, referrẽtq́;</s> <s xml:id="echoid-s32279" xml:space="preserve"> ipſas <lb/>formas à ſpeculis ad uiſum:</s> <s xml:id="echoid-s32280" xml:space="preserve">tũc quælibet imago uideretur in loco ſuæ rei, cuius eſt imago:</s> <s xml:id="echoid-s32281" xml:space="preserve"> quod eſt <lb/>contra ſenſum Et quia, ut præoſtẽſum eſt per 2 huius, ab omni corpore colorato præſente luce co-<lb/>lor ad corpus oppoſitũ politũ mittitur mixtim cũ lumine, & quãdoq;</s> <s xml:id="echoid-s32282" xml:space="preserve"> totaliter, quãdoq;</s> <s xml:id="echoid-s32283" xml:space="preserve"> partialiter <lb/>reflectitur ab illo:</s> <s xml:id="echoid-s32284" xml:space="preserve"> tũc ſi radij uiſuales incidẽtes ſpeculis reflecterẽtur ab illis ad res ipſas, & deferrẽt <lb/>ſecum formas:</s> <s xml:id="echoid-s32285" xml:space="preserve"> accideret quòd duæ uiderẽtur imagines uniuſcuiuſq;</s> <s xml:id="echoid-s32286" xml:space="preserve"> rei, quarũ unam offerret uiſui <lb/>ipſe uiſualis radius reflexus, & aliá ipſe radius ſormæ rei incidẽs ſpeculo, in quo formæ rerũ impri-<lb/>muntur, & reflexus à ſpeculo ad uiſum:</s> <s xml:id="echoid-s32287" xml:space="preserve"> quod totũ eſt impoſsibile ſenſui.</s> <s xml:id="echoid-s32288" xml:space="preserve"> Sed ad huius oppoſitum <lb/>quidá antiquorũ demonſtrationẽ attulit, quã & nos utindifferẽtẽ uigoratá fortius præſenti ꝓpoſi <lb/>to applicamus.</s> <s xml:id="echoid-s32289" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s32290" xml:space="preserve">, exempli cauſſa, ſpeculũ planũ erectũ ſuper ſuperficiẽ horizontis orthogo-<lb/>naliter:</s> <s xml:id="echoid-s32291" xml:space="preserve"> in quo ſit linea diuidẽs ſuperficiẽ ſpeculi ք æqualia, quę ſit a b:</s> <s xml:id="echoid-s32292" xml:space="preserve"> & ſit cẽtrũ uiſus g:</s> <s xml:id="echoid-s32293" xml:space="preserve"> à quo du-<lb/>catur linea g t perpẽdicularis ſuper ſuperficiẽ ſpeculi ք 11 p 11.</s> <s xml:id="echoid-s32294" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s32295" xml:space="preserve">, ut linea g t cadat ſuper lineã <lb/>a b in pũctũ t:</s> <s xml:id="echoid-s32296" xml:space="preserve"> erit ergo linea g t perpẽdicularis ſuper lineã a b.</s> <s xml:id="echoid-s32297" xml:space="preserve"> Et ducantur à pũcto g lineæ g a & g b <lb/> <anchor type="figure" xlink:label="fig-0503-01a" xlink:href="fig-0503-01"/> ęquales:</s> <s xml:id="echoid-s32298" xml:space="preserve"> erunt ergo per 5 p 1 anguli <lb/>g a b & g b a æ quales:</s> <s xml:id="echoid-s32299" xml:space="preserve"> & anguli ad <lb/>pũctum t ſunt recti:</s> <s xml:id="echoid-s32300" xml:space="preserve"> ergo per 26 p 1, <lb/>& per hypotheſim erit linea at æ-<lb/>qualis lineæ b t.</s> <s xml:id="echoid-s32301" xml:space="preserve"> Producãtur itaq;</s> <s xml:id="echoid-s32302" xml:space="preserve"> li <lb/>neæ g a & g b ultra ſpeculũ ad pun-<lb/>cta d & e:</s> <s xml:id="echoid-s32303" xml:space="preserve"> ιta ut lineæ g a d & g b e <lb/>fint ęquales:</s> <s xml:id="echoid-s32304" xml:space="preserve"> & cóiũgatur linea d e:</s> <s xml:id="echoid-s32305" xml:space="preserve"> <lb/>producaturq́;</s> <s xml:id="echoid-s32306" xml:space="preserve"> linea g tad lineá d e:</s> <s xml:id="echoid-s32307" xml:space="preserve"> <lb/>& incidat illi in pũcto h.</s> <s xml:id="echoid-s32308" xml:space="preserve"> Erit ergo ք <lb/>præmiſſa & 26 p 1 linea d h æ qualis <lb/>lineæ h e:</s> <s xml:id="echoid-s32309" xml:space="preserve"> ergo ք 8 p 1, & per defini-<lb/>tionẽ perpẽdicularis, anguli ad pũ-<lb/>ctũ h ſuntrecti:</s> <s xml:id="echoid-s32310" xml:space="preserve"> ergo ք 28 p 1 line æ <lb/>d h & a t ſunt æquidiſtátes, & lineę <lb/>h e & t b æquidiltátes:</s> <s xml:id="echoid-s32311" xml:space="preserve"> ꝓducaturq́;</s> <s xml:id="echoid-s32312" xml:space="preserve"> <lb/>linea t g ultra uiſum g, donec linea <lb/>ti ſit æqualis line æ t h:</s> <s xml:id="echoid-s32313" xml:space="preserve"> & ducãtur à pũcto i lineę i u & i z æ quidiſtáter lineæ a b:</s> <s xml:id="echoid-s32314" xml:space="preserve"> & ſit linea u z ęqua-<lb/>lis line æ d e:</s> <s xml:id="echoid-s32315" xml:space="preserve"> & ducátur lineę u a & z b.</s> <s xml:id="echoid-s32316" xml:space="preserve"> Quia ergo linea t i eſt æqualis ipſi lineę t h, & linea u z æ qua-<lb/>lis lineę d e, & linea a b ęqualis eſt ſibi ipſi, erit ſuperficies a b z u ęqualis ſuperficiei a b d e.</s> <s xml:id="echoid-s32317" xml:space="preserve"> Superpo <lb/>fita enim nec excedetnec excedetur.</s> <s xml:id="echoid-s32318" xml:space="preserve"> Linea ergo u a eſt ęqualis lineę a d, & z b eſt ęqualis ipſi lineæ <lb/>b e:</s> <s xml:id="echoid-s32319" xml:space="preserve"> & angulus a u z ęqualis eſt angulo a d e, & angulus u z b eſt æqualis angulo d e b:</s> <s xml:id="echoid-s32320" xml:space="preserve"> & angulus d <lb/>a b ęqualis angulo u a b:</s> <s xml:id="echoid-s32321" xml:space="preserve"> radius ergo g a per 20 huius reflectetur ad pũctum u.</s> <s xml:id="echoid-s32322" xml:space="preserve"> Si enim producatur li <lb/>nea a b ultra pũctũ a ad pũctũr, & ultra pũctũ b ad pũctũ l palã ex pręmiſsis & ք 13 p 1 quia linea a r <lb/>diuidet angulũ u a d ք duo ęqualia:</s> <s xml:id="echoid-s32323" xml:space="preserve"> erit ergo angulus r a u ęqualis angulo r a d:</s> <s xml:id="echoid-s32324" xml:space="preserve"> & ſimiliter erit angu-<lb/>lus z b l ęqualis angulo e b l:</s> <s xml:id="echoid-s32325" xml:space="preserve"> ſed angulus r a d eſt ęqualis angulo g a b, & angulus l b e ęqualis angu-<lb/> <pb o="202" file="0504" n="504" rhead="VITELLONIS OPTICAE"/> lo g b a ք 15 p 1:</s> <s xml:id="echoid-s32326" xml:space="preserve"> ergo angulus r a u eſt ęqualis angulo g a b, & angulus l b z ęqualis angulo g b a:</s> <s xml:id="echoid-s32327" xml:space="preserve"> ergo <lb/>ք 20 huι{us} duo radij g a & g b reflectẽtur à duob.</s> <s xml:id="echoid-s32328" xml:space="preserve"> pũctis a & b ad duo pũcta u & z.</s> <s xml:id="echoid-s32329" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s32330" xml:space="preserve"> cêtrũ uiſus, <lb/>quod eſt g, appropinquet ſuperficiei ſpeculi, & lineæ a b, ut ſi perueniat in punctũ f:</s> <s xml:id="echoid-s32331" xml:space="preserve"> tũc quia angu-<lb/>lus incidẽtię, qui eſt f a t minor eſt angulo incidẽtiæ, ꝗ eſt g a t, erit per 20 huius angulus reflexiõis, <lb/>qui ſit q a r, minor angulo prioris reflexionis, qui eſt u a r:</s> <s xml:id="echoid-s32332" xml:space="preserve"> & erit angulus q a r maior angulo u a g, & <lb/>linea q i maior linea u i.</s> <s xml:id="echoid-s32333" xml:space="preserve"> Approximante ergo uiſu ſuperficiei ſpeculi, non uidebuntur extremitates <lb/>rei prius uiſæ, quæ ſunt u & z, ſecundum extremitates ſpeculi, quæ ſunt a & b.</s> <s xml:id="echoid-s32334" xml:space="preserve"> Sed & uiſu perſiſten <lb/>tein puncto g, & linea u z approximante ſpeculo uſq;</s> <s xml:id="echoid-s32335" xml:space="preserve"> ad punctum x (quod ſit punctum line æ i h) <lb/>non uidebuntur extremitates lineæ u z, quæ ſunt u & z, ſed ſolùm aliqua pũcta ipfius, in quibus ra-<lb/>dius g a uiſualis reflexus à ſuperficie ſpeculi ſecat u z, quæ ſint pũcta m & n:</s> <s xml:id="echoid-s32336" xml:space="preserve"> erit enim linean m mi-<lb/>nor quàm linea u z:</s> <s xml:id="echoid-s32337" xml:space="preserve"> quod patet per 34 p 1, ductis lineis æquidiſtantibus & perpendicularibus, quæ <lb/>ſint n o & m p.</s> <s xml:id="echoid-s32338" xml:space="preserve"> Et ſi linea u z elongata fuerit à ſuperficie ſpeculi, nullum eius punctum uidebitur ſe-<lb/>cũdum radios a u & b z:</s> <s xml:id="echoid-s32339" xml:space="preserve"> quia alij radij uiſuales à punctis extremis ipſius ſpeculi, qui ſunt a & b, non <lb/>reflectuntur ad aliquod punctum lineæ u z, ſed ultra illa:</s> <s xml:id="echoid-s32340" xml:space="preserve"> quod patet per 34 p 1, copulatis lineis æ-<lb/>quidiſtantibus, quę ſint u u & zz.</s> <s xml:id="echoid-s32341" xml:space="preserve"> Non uidebitur ergo in tali diſpoſitione reſpectu ſpeculi aliquis <lb/>punctorum lineæ u z:</s> <s xml:id="echoid-s32342" xml:space="preserve"> quod eſt contra experientiam & ſenſum.</s> <s xml:id="echoid-s32343" xml:space="preserve"> Acciditenim extrema rei approxi-<lb/>matæ & elongatæ in ſpeculo quandoque uideri, ut ſuppoſitum eſt in huius libri principio.</s> <s xml:id="echoid-s32344" xml:space="preserve"> Et fi-<lb/>cut hoc patet in ſpeculis planis:</s> <s xml:id="echoid-s32345" xml:space="preserve"> ſic etiam patet in alijs ſpeculis quibuſcunque:</s> <s xml:id="echoid-s32346" xml:space="preserve"> quoniam de omni-<lb/>bus eadẽ eſt demóſtratio.</s> <s xml:id="echoid-s32347" xml:space="preserve"> Patet ergo ꝓpoſitũ, aut ad minus ex his nõ cõcluditur oppoſitum ipſius.</s> <s xml:id="echoid-s32348" xml:space="preserve"/> </p> <div xml:id="echoid-div1315" type="float" level="0" n="0"> <figure xlink:label="fig-0503-01" xlink:href="fig-0503-01a"> <variables xml:id="echoid-variables542" xml:space="preserve">d r u q n o a h t x i b f g e l z m p</variables> </figure> </div> </div> <div xml:id="echoid-div1317" type="section" level="0" n="0"> <head xml:id="echoid-head1035" xml:space="preserve" style="it">24. Comprehenſionem formarum uiſibilium in ſpeculo ſola efficit reflexio, qua ad uiſum: un-<lb/>de ſecundum diſpoſitionem linearũ reflexionis uiſus neceſſariò informatur. Alhazen 21 n 4.</head> <p> <s xml:id="echoid-s32349" xml:space="preserve">Quòd enim radij ab oculo non exeant, qui redeuntes ad uiſum referant ſecum formas uiſibili-<lb/>um, hoc oſtenſum eſt per pręmiſſam:</s> <s xml:id="echoid-s32350" xml:space="preserve"> quòd autem forma ſenſibilis non informet ipſum ſpeculum, <lb/>ſicut forma naturalis ſuam materiam, hoc patet ex hoc:</s> <s xml:id="echoid-s32351" xml:space="preserve"> quòd non in omni differentia poſitionis ui-<lb/>dentur formæ in ſpeculis quibuſcunq;</s> <s xml:id="echoid-s32352" xml:space="preserve">. Intuens enim aliquis accedens ad ſpeculum fixum, uidet <lb/>formam, quam prius non uidit, & recedens à loco uiſionis formæ prius in ſpeculo fixo uiſæ, non <lb/>amplius uidet illam:</s> <s xml:id="echoid-s32353" xml:space="preserve"> & uiſa parte ſpeculi, non propter hocuidetur pars formarum in ſpeculo ap-<lb/>parétium, ſed in eodẽ pũcto ſpeculi diuerſi aſpiciẽtes uidere poſſunt formas diuerſas & diſtinctas, <lb/>quę tamẽ, ut quidam actus completiui, eandẽ partem ſpeculinon poſſunt ſimul informare.</s> <s xml:id="echoid-s32354" xml:space="preserve"> Vide-<lb/>tur etiam in ſpeculis forma rei, quæ ſecũdum lineam rectam non poteſt multiplicari ad uifum:</s> <s xml:id="echoid-s32355" xml:space="preserve"> mul <lb/>ta quoq;</s> <s xml:id="echoid-s32356" xml:space="preserve"> alia accidũt, quorum ratio poſterior eſt, magnam tamẽ impoſsibilitatem demonſtrat.</s> <s xml:id="echoid-s32357" xml:space="preserve"> Pa-<lb/>làm itaq;</s> <s xml:id="echoid-s32358" xml:space="preserve"> formas à ſpeculo non procedere, ut in ſpeculo exiſtẽtes & multiplicantes ſe ad uiſum, ſed <lb/>ut incidẽtes ip ſis ſpeculis à rebus ſormatis & à ſpeculis ad uiſum reflecti.</s> <s xml:id="echoid-s32359" xml:space="preserve"> Secũdum diſpoſitionem <lb/>ergo linearum reflexionis uiſus neceſſariò inſormatur:</s> <s xml:id="echoid-s32360" xml:space="preserve"> quando quidẽ uiſus uerè rem aliam non ui-<lb/>det, niſi cuius formam comprehẽdit à ſpeculo reflexam.</s> <s xml:id="echoid-s32361" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s32362" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1318" type="section" level="0" n="0"> <head xml:id="echoid-head1036" xml:space="preserve" style="it">25. In omni reflexione à quocun ſpeculo facta, ſuperficiem reflexionis ſuper illius ſpeculi ſu-<lb/>perficiem, uel ſuper ſuperficiem illud ſpeculum in puncto reflexionis contingentem, erectam eſſe <lb/>eſt neceſſe. Alhazen 13 n 4.</head> <p> <s xml:id="echoid-s32363" xml:space="preserve">Quoniam enim ſi lux uel forma alicui ſpeculo ſecũdum perpẽdicularem lineam incidit, illa ſe-<lb/>cundum eandem reflectitur per 21 huius:</s> <s xml:id="echoid-s32364" xml:space="preserve"> palàm quòd tũc ſit incidentia & reflexio ſecundum ean-<lb/>dem lineam:</s> <s xml:id="echoid-s32365" xml:space="preserve"> & ſuperſiciem reflexionis neceſſe eſt eſſe erectam ſuper ſuperficiem ipſius ſpeculi per <lb/>18 p 11.</s> <s xml:id="echoid-s32366" xml:space="preserve"> Si uerò lux uel forma ſecũdum lineas obliquas incidit ſuperficiei ſpeculi cuiuſcũq;</s> <s xml:id="echoid-s32367" xml:space="preserve">, tũc angu <lb/>lus incidẽtiæ, quẽ facit linea incidentię cum perpẽdiculari, ſemper eſt æ qualis angulo reflexionis, <lb/>quem continet linea reflexionis cum eadẽ perpẽdiculari, ut patet per 20 huius:</s> <s xml:id="echoid-s32368" xml:space="preserve"> utraq;</s> <s xml:id="echoid-s32369" xml:space="preserve"> ergo ipſarũ <lb/>eſt in eadẽ ſuperficie cú linea perpẽdiculari per 2 p 11:</s> <s xml:id="echoid-s32370" xml:space="preserve"> ergo & ipſæ ambę ſuntin eadẽ ſuperficie, quę <lb/>eſt, ut patet per definitionẽ, ſuperficies reflexiõis.</s> <s xml:id="echoid-s32371" xml:space="preserve"> Eft ergo per 18 p 11 illa ſuperficies erecta ſuper ſu-<lb/>perficiẽ ſpeculi, uel ſuper ſuperficiẽ ſpeculum contingẽtem in puncto reflexionis.</s> <s xml:id="echoid-s32372" xml:space="preserve"> Ethoc exempla <lb/>riter patet in ſuperficie circuli ſecantis armillam inſtrumẽti in 9 huius pręmiſsi æquidiſtanter baſi-<lb/>bus ſuis per omnia cétra foraminum, & æ quidiſtantis ſuperficiei circuli ænei, qui eſt a b c.</s> <s xml:id="echoid-s32373" xml:space="preserve"> Radio e-<lb/>nim per foramẽ medium incidẽte & ſpeculo declinato ſecundum regulam eadẽ eſt demonſtratio, <lb/>quę in radijs obliquè incidẽtibus:</s> <s xml:id="echoid-s32374" xml:space="preserve"> reflectitur enim tunc ſemper radius ad lineam longitudinis ar-<lb/>millæ, quæ tunc non æquidiſtat line æ b z p, quę eſt linea longitudinis regulæ.</s> <s xml:id="echoid-s32375" xml:space="preserve"> Et quoniam fit tunc <lb/>reflexio à puncto z, cui incidit axis columnæ rotundę, uel radij perpẽdiculariter ſuper lineam c q, <lb/>quæ eſt communis ſectio ſuperſiciei regulæ & ſuperſiciei circuli tranſeuntis per cẽtra foraminum, <lb/>& huic axi æ quidiſtat linea d b ſemidiameter circuli a b c:</s> <s xml:id="echoid-s32376" xml:space="preserve"> ſunt ergo in eadẽ ſuperficie per 1 th.</s> <s xml:id="echoid-s32377" xml:space="preserve"> 1 hu-<lb/>ius.</s> <s xml:id="echoid-s32378" xml:space="preserve"> Sed linea d b eſt perpẽdicularis ſuper lineam latitudinis regulæ, quæ eſt communis ſectio ſu-<lb/>perficiei regulæ & circuli a b c:</s> <s xml:id="echoid-s32379" xml:space="preserve"> ergo per definition ẽ ſuperficiei ſuper ſuperficiẽ erectæ, ſuperficies, <lb/>in qua ſunt axis columnæ ferreæ uel radij incidẽtis & linea b d, eſt erecta ſuper ſuperficiem regulæ <lb/>uel ſpeculi:</s> <s xml:id="echoid-s32380" xml:space="preserve"> & in hac ſuperficie eſt linea perpédicularis, quæ eſt linea altitudinis armillæ, tranſiens <lb/>per punctum b, & per cẽtrum foraminis medij, in quá lineam fit reflexio lucis axis pyramidis ra-<lb/>dialis.</s> <s xml:id="echoid-s32381" xml:space="preserve"> Patet ergo propoſitum etiam in unoquoq;</s> <s xml:id="echoid-s32382" xml:space="preserve"> ſpeculorum:</s> <s xml:id="echoid-s32383" xml:space="preserve"> quoniam ad omne ſpeculum h æ c <lb/>demonſtratio ſe extendit, ut patuit ex pręmiſsis.</s> <s xml:id="echoid-s32384" xml:space="preserve"/> </p> <pb o="203" file="0505" n="505" rhead="LIBER QVINTVS."/> </div> <div xml:id="echoid-div1319" type="section" level="0" n="0"> <head xml:id="echoid-head1037" xml:space="preserve" style="it">26. In omni reflexione à cuiuſcun ſpeculi ſuperficie, linea recta per aqualia diuidens angu-<lb/>lum contentum ſub lineis incidentiæ & reflexionis, ſuper lineam, quæ eſt cõmunis ſectio ſu-<lb/>perficiei reflexionis & ſpeculi, uel ſuperficiei in puncto incidentiæ ſpeculum contingentis, neceſ-<lb/>ſariò perpendicularis exiſtit: ex quo patet illam lineam erectã eſſe ſuper ſuperficiem in illo pun-<lb/>cto ſpeculum contingentem.</head> <p> <s xml:id="echoid-s32385" xml:space="preserve">Sit enim, ut forma puncti a incidat ſuperficiei alicuius ſpeculi ſecundũ punctũ b:</s> <s xml:id="echoid-s32386" xml:space="preserve"> & refle ctatur in <lb/>punctum c:</s> <s xml:id="echoid-s32387" xml:space="preserve"> eſt itaq;</s> <s xml:id="echoid-s32388" xml:space="preserve"> linea incidẽtiæ linea a b, & linea reflexionis linea b c, quę ſunt in una ſuperficie <lb/>erecta ſuք ſuperficiẽ ſpeculi ք præmiſſam:</s> <s xml:id="echoid-s32389" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s32390" xml:space="preserve"> aliqua ſuperficies plana cõtingens ſpeculũ ſecundũ <lb/> <anchor type="figure" xlink:label="fig-0505-01a" xlink:href="fig-0505-01"/> punctũ b:</s> <s xml:id="echoid-s32391" xml:space="preserve"> cõmunis aút ſectio huius ſuperficiei & ſuperficiei reflexio <lb/>nis ſit linea d b e:</s> <s xml:id="echoid-s32392" xml:space="preserve"> angulũ uerò a b c diuidatlinea b f per ęqualia.</s> <s xml:id="echoid-s32393" xml:space="preserve"> Dico, <lb/>quòd linea f b eſt neceſſarò perpendicularis ſuper lineã d b e.</s> <s xml:id="echoid-s32394" xml:space="preserve"> Quia e-<lb/>nim angulus d b a eſt ęqualis angulo e b c per 20 huius:</s> <s xml:id="echoid-s32395" xml:space="preserve"> angulus enim <lb/>incidentiæ d b a eſt æqualis angulo reflexionis, qui eſt e b c:</s> <s xml:id="echoid-s32396" xml:space="preserve"> & quia <lb/>angulus a b f eſt æqualis angulo f b c e x hypotheſi:</s> <s xml:id="echoid-s32397" xml:space="preserve"> palàm quòd totus <lb/>angulus f b d eſt æqualis toti angulo f b e:</s> <s xml:id="echoid-s32398" xml:space="preserve"> eſt ergo linea f b perpendi-<lb/>cularis ſuper lineam d e per deſinitionem line æ perpendicularis:</s> <s xml:id="echoid-s32399" xml:space="preserve"> & <lb/>hoc ſi linea d b e ſit linea recta:</s> <s xml:id="echoid-s32400" xml:space="preserve"> quę ſi fuerit curua, ſit, ut g h linea recta <lb/>ipſam contingatin puncto b per 17 p 3.</s> <s xml:id="echoid-s32401" xml:space="preserve"> Et quia anguli contingentiæ <lb/>g b d & h b e ſunt æquales:</s> <s xml:id="echoid-s32402" xml:space="preserve"> relinquitur quòd angulι f b g & f b h ſint ę-<lb/>quales, & erit item linea f b perpẽdicularis ſuper lineam g h, & ſuper <lb/>lineam d e.</s> <s xml:id="echoid-s32403" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s32404" xml:space="preserve"> linea f b ſit ducta in ſuperficie reflexionis, quæ <lb/>ex pręmiſſa eſt arecta ſuper ſuperficiẽ ſpeculi, uel ſuper ſuperficiem, <lb/>ſpeculũ in puncto incidẽtiæ contingentẽ, & cũipſa ſit ſuper ipſarum <lb/>communẽ ſectionẽ perpẽdicularis:</s> <s xml:id="echoid-s32405" xml:space="preserve"> patet quòd linea fb eſt erecta ſuper ſuperficiẽ ſpeculum in illo <lb/>pũcto contingentẽ continet enim cũ omnibus lineis in illa ſuperficie productis angulos æquales.</s> <s xml:id="echoid-s32406" xml:space="preserve"> <lb/>Et quoniam eodem modo poteſt fieri declaratio in ſectionibus:</s> <s xml:id="echoid-s32407" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s32408" xml:space="preserve"/> </p> <div xml:id="echoid-div1319" type="float" level="0" n="0"> <figure xlink:label="fig-0505-01" xlink:href="fig-0505-01a"> <variables xml:id="echoid-variables543" xml:space="preserve">a f c g b h d e</variables> </figure> </div> </div> <div xml:id="echoid-div1321" type="section" level="0" n="0"> <head xml:id="echoid-head1038" xml:space="preserve" style="it">27. In omni ſuperficie reflexionis à ſpeculis quibuſcun, centrum uiſus: & punctũ forma ui-<lb/>ſæ: & punctum reflexionis: & terminum perpendicularis & cathetiutriuſ, conſiſtere eſt neceſ <lb/>ſe: ex quo patet lineam perpendicularem à puncto reflexionis ductam, omnibus ſuperficiebus re <lb/>flexionis illi puncto incidentibus communem eſſe. Alhazen 23 n 4.</head> <p> <s xml:id="echoid-s32409" xml:space="preserve">Oſtenſum eſt per 25 huius quoniam in omni reflexione à quocunq;</s> <s xml:id="echoid-s32410" xml:space="preserve"> ſpeculo facta, ſemper ſuper-<lb/>ficies reflexionis (in qua ſunt lineæ reflexionis, incidẽtiæ & perpẽdicularis ſuper ſuperficiem ſpe-<lb/>culi ducta à puncto reflexionis) erecta eſt ſuper ſuperficiẽ ſpeculi, à quo fit reflexio.</s> <s xml:id="echoid-s32411" xml:space="preserve"> Cum aũt linea <lb/>incidentiæ incipiat à puncto formæ comprehẽſæ, & terminetur in pũctum reflexionis, & linea re-<lb/>flexionis incipiat à pũcto reflexionis, & terminetur ad cẽtrũ:</s> <s xml:id="echoid-s32412" xml:space="preserve"> palàm quò hæ tria pũcta ſunt <lb/>in eadẽ ſuperficie.</s> <s xml:id="echoid-s32413" xml:space="preserve"> Sed cũ perpẽdicularis ſit erecta ſuper ſuperficiem ſpeculi, ſuper quá per 25 huius <lb/>& ſuperficies reflexionis eſt erecta, quoniá & in illa ſuperficie eſt tota perpẽdicularis:</s> <s xml:id="echoid-s32414" xml:space="preserve"> cũ enim ipſa <lb/>perpẽdicularis in pũcto reflexionis ſecet lineas incidẽtię & reflexionis, cũ quibus ipſa ex definitio-<lb/>ne eſt in eadẽ ſuperficie:</s> <s xml:id="echoid-s32415" xml:space="preserve"> ergo per 1 p 11 terminus perpendicularis ſuperior neceſſariò erit in eadem <lb/>ſuperficie cum punctis prædictis.</s> <s xml:id="echoid-s32416" xml:space="preserve"> Si enim illa perpendicularis ad punctũ aliũ extra ſuperficiem re-<lb/>flexionis terminetur:</s> <s xml:id="echoid-s32417" xml:space="preserve"> patet quòd illa per pẽdicularis in alia erit ſuperficie, qđ eſt contra definitio-<lb/> <anchor type="figure" xlink:label="fig-0505-02a" xlink:href="fig-0505-02"/> nem ſuperficiei reflexionis:</s> <s xml:id="echoid-s32418" xml:space="preserve"> & etiam, ſi ipſa in alia fuerit ſuperficie, e-<lb/>rit rectus minor recto, quod eſt impoſsibile.</s> <s xml:id="echoid-s32419" xml:space="preserve"> Linea enim à puncto re <lb/>flexionis producta in ipfa ſuperficie reflexionis erecta ſuper ſuperfi-<lb/>ciem ſpeculi, cum linea in ſuperficie ſpeculi ab eodem puncto pro-<lb/>ducta continet angulum rectum, & perpendicularis ſimiliter.</s> <s xml:id="echoid-s32420" xml:space="preserve"> Si er-<lb/>go illæ duæ lineę ad diuerſa puncta terminantur, fit rectus maior re-<lb/>cto.</s> <s xml:id="echoid-s32421" xml:space="preserve"> Sed per eundem modum patet id, quod proponitur de cathe-<lb/>tis.</s> <s xml:id="echoid-s32422" xml:space="preserve"> Et quoniam omnes fuperficies reflexionis, quę tranſeunt eun-<lb/>dem punctum reflexionis, & aliquem punctum formæ comprehen-<lb/>fum, licet ad diuerſa centra uiſuum terminentur, ſemper tranſeunt <lb/>eundem terminum perpendicularis, quoniam omnes ſunt erectæ ſu <lb/>per ſuperficiem ſpeculi, uel ſuper ſuperficiem ſpeculum in pũcto re-<lb/>flexionis contingẽtem:</s> <s xml:id="echoid-s32423" xml:space="preserve"> palàm quoniam omnes ſecant ſein perpen-<lb/>diculari.</s> <s xml:id="echoid-s32424" xml:space="preserve"> Eſt ergo perpẽdicularis eis omnibus cõmunis.</s> <s xml:id="echoid-s32425" xml:space="preserve"> Sed & hoc <lb/>figuraliter eſt declarandum.</s> <s xml:id="echoid-s32426" xml:space="preserve"> Sit enim ſuperficies ſpeculi cuiuſcunq;</s> <s xml:id="echoid-s32427" xml:space="preserve"> <lb/>a c b:</s> <s xml:id="echoid-s32428" xml:space="preserve"> in cuius punctum c incidat radius à puncto rei uiſæ, quod ſit f, <lb/>per lineam f c:</s> <s xml:id="echoid-s32429" xml:space="preserve"> & reflectatur ad centrum uiſus, quod ſit e, per lineam <lb/>c e:</s> <s xml:id="echoid-s32430" xml:space="preserve"> extrahatur quoq;</s> <s xml:id="echoid-s32431" xml:space="preserve"> քpẽdicularis ſuper ſuքficiẽ ſpeculi, quę eſt b c a, à púcto c, quę ſit c d.</s> <s xml:id="echoid-s32432" xml:space="preserve"> ք 12 p 11.</s> <s xml:id="echoid-s32433" xml:space="preserve"> <lb/>Intelligatur quoq;</s> <s xml:id="echoid-s32434" xml:space="preserve"> à púcto e perpẽdicularis protrahi ſuper ſuperficiẽ b c a, aut ei cõtinuã per 11 p 11, <lb/>quę ſit e a:</s> <s xml:id="echoid-s32435" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s32436" xml:space="preserve"> linea e a æ quidiſtans lineę d c per 6 p 11, quoniam ambæ ſunt orthogonales ſuper <lb/> <pb o="204" file="0506" n="506" rhead="VITELLONIS OPTICAE"/> eandem ſuperficiem ſpeculi, quę eſt b a.</s> <s xml:id="echoid-s32437" xml:space="preserve"> Et quoniam lineæ d c & e a ſunt æquidiſtantes:</s> <s xml:id="echoid-s32438" xml:space="preserve"> palàm per <lb/>1 th.</s> <s xml:id="echoid-s32439" xml:space="preserve"> 1 huius quia ſunt in eadẽ plana ſuperficie:</s> <s xml:id="echoid-s32440" xml:space="preserve"> & linea recta a b cũ utraq;</s> <s xml:id="echoid-s32441" xml:space="preserve"> illarũ linearum, ſcilicet d c <lb/>& e a continebit angulum rectú, & erit in eadẽ ſuperficie cũ utraq;</s> <s xml:id="echoid-s32442" xml:space="preserve"> ipſarú per 2 p 11:</s> <s xml:id="echoid-s32443" xml:space="preserve"> & linea e c tene-<lb/>bit cũ his ambabus lineis, quæ ſunt e a & d c, angulos acutos propter diuiſionẽ angulorũ rectorũ.</s> <s xml:id="echoid-s32444" xml:space="preserve"> <lb/>Et quoniá linea incidentiæ & reflexionis cũ perpẽ diculari d c ſunt in eadẽ ſuperficie, & linea e c re-<lb/>cta copulat extremitates linearũ e a & d c, erit ipſa per 2 p 11 in eadẽ ſuperficie cũ dictis perpendicu <lb/>laribus.</s> <s xml:id="echoid-s32445" xml:space="preserve"> Omnes ergo lineæ, quæ ſunt e a, e c, d c, f c ſunt in una & eadẽ ſuperficie.</s> <s xml:id="echoid-s32446" xml:space="preserve"> Quatuor ergo prę-<lb/>miffa puncta ſunt in eadẽ ſuperficie reflexionis.</s> <s xml:id="echoid-s32447" xml:space="preserve"> Et hoc proponebatur:</s> <s xml:id="echoid-s32448" xml:space="preserve"> quoniam inſpecto quocũq;</s> <s xml:id="echoid-s32449" xml:space="preserve"> <lb/>alio puncto corporis uiſi uel ſpeculi, ſemper accidit idẽ ſitus linearũ radialium cũ ipſis perpendicu <lb/>laribus.</s> <s xml:id="echoid-s32450" xml:space="preserve"> Et ſimiliter patet de utriſq;</s> <s xml:id="echoid-s32451" xml:space="preserve"> cathetis & incidentiæ & reflexionis per 1 p 11.</s> <s xml:id="echoid-s32452" xml:space="preserve"> Patete ergo propo-<lb/>ſitum.</s> <s xml:id="echoid-s32453" xml:space="preserve"> Et ex hoc patet concluſio corollaria ſatis manifeſtè.</s> <s xml:id="echoid-s32454" xml:space="preserve"/> </p> <div xml:id="echoid-div1321" type="float" level="0" n="0"> <figure xlink:label="fig-0505-02" xlink:href="fig-0505-02a"> <variables xml:id="echoid-variables544" xml:space="preserve">e d f a c b</variables> </figure> </div> </div> <div xml:id="echoid-div1323" type="section" level="0" n="0"> <head xml:id="echoid-head1039" xml:space="preserve" style="it">28. Omnem pun ctum reflexionis formæ puncti obliquè ſpeculo incidentis, inter cathetũ in-<lb/>cidentia & reflexionis in ſuperficie ſpeculi conſiſtere eſt neceſſe.</head> <p> <s xml:id="echoid-s32455" xml:space="preserve">Sit ſuperficies cuiuſcunq;</s> <s xml:id="echoid-s32456" xml:space="preserve"> ſpeculi, in quo cõmunis ſectio ſuperficiei reflexionis & ſuperficiei ſpe <lb/>culi ſit linea a b c, recta uel curua:</s> <s xml:id="echoid-s32457" xml:space="preserve"> & ſit punctus rei uiſæ, qui d:</s> <s xml:id="echoid-s32458" xml:space="preserve"> & centrũ uiſus púctũ e:</s> <s xml:id="echoid-s32459" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s32460" xml:space="preserve"> cathetus <lb/>incidentiæ, quę d a, & cathetus reflexionis, quæ e b.</s> <s xml:id="echoid-s32461" xml:space="preserve"> Dico quòd omnem púctum reflexionis formæ <lb/>puncti d ad centrum uiſus e, inter pũcta ſuperficiei ſpeculi a & b conſiſtere eſt neceſſe.</s> <s xml:id="echoid-s32462" xml:space="preserve"> Si enim de-<lb/> <anchor type="figure" xlink:label="fig-0506-01a" xlink:href="fig-0506-01"/> <anchor type="figure" xlink:label="fig-0506-02a" xlink:href="fig-0506-02"/> tur quòd in ipſis pun <lb/>ctis a uel b fiat refle-<lb/>xio formę púcti d ad <lb/>uiſum e:</s> <s xml:id="echoid-s32463" xml:space="preserve"> ſit ergo, ut fi <lb/>at à puncto ſpeculi, <lb/>qđ eſt a:</s> <s xml:id="echoid-s32464" xml:space="preserve"> & ducatur <lb/>linea a e:</s> <s xml:id="echoid-s32465" xml:space="preserve"> tunc cum li-<lb/>nea d a ſit perpẽdicu <lb/>laris, & linea a e non <lb/>fit perpẽdicularis, & <lb/>per 20 huius angu-<lb/>lus incidẽtiæ ſit ęqua <lb/>lis angulo reflexiõis:</s> <s xml:id="echoid-s32466" xml:space="preserve"> <lb/>erit ergo angulus e a <lb/>b rectus, ſed & angu-<lb/>lus e b a eſt rectus:</s> <s xml:id="echoid-s32467" xml:space="preserve"> tri <lb/>goni ergo e a b duo <lb/>anguli ſunt recti:</s> <s xml:id="echoid-s32468" xml:space="preserve"> qđ <lb/>eſt impoſsibile.</s> <s xml:id="echoid-s32469" xml:space="preserve"> Similiterq́;</s> <s xml:id="echoid-s32470" xml:space="preserve"> deducendum ſi detur reflexionẽ fieri <lb/>à puncto b, quoniã idé accidit impoſsibile.</s> <s xml:id="echoid-s32471" xml:space="preserve"> Nó fit ergo reflexio ab <lb/>aliquo pũctorũ a uel b, quibus incidũt catheti.</s> <s xml:id="echoid-s32472" xml:space="preserve"> Sed neq;</s> <s xml:id="echoid-s32473" xml:space="preserve"> ab aliquo <lb/>punctorũ lineæ a b c, extra puncta a & b:</s> <s xml:id="echoid-s32474" xml:space="preserve"> ſit enim, ut forma puncti d reflectatur ad uiſum e à puncto <lb/>ſpeculi c:</s> <s xml:id="echoid-s32475" xml:space="preserve"> ductis ergo lineis d c & c e, diuidatur angulus d c e per æ qualia per 9 p 1:</s> <s xml:id="echoid-s32476" xml:space="preserve"> & ducatur linea <lb/>c f ſecás lineam b e in pũcto f:</s> <s xml:id="echoid-s32477" xml:space="preserve"> erit ergo ք pręmiſſam linea c f perpẽdicularis ſuper lineam a c:</s> <s xml:id="echoid-s32478" xml:space="preserve"> trigo-<lb/>ni ergo b f c duo anguli ſunt recti:</s> <s xml:id="echoid-s32479" xml:space="preserve"> quod eſt impoſsibile, ut prius.</s> <s xml:id="echoid-s32480" xml:space="preserve"> Et eodẽmodo deducẽdũ, ſi detur <lb/>fieri reflexio ab aliquo puncto linea a b c, ultra punctũ a, ut à pũcto g, ducta linea g h angulum d g e <lb/>per æqualia diuidẽte.</s> <s xml:id="echoid-s32481" xml:space="preserve"> Patet ergo quòd ſolùm inter pũcta a & b fiet reflexio ab aliquo pũctorum li-<lb/>neæ a b, uidelicet inter cathetũ incidétiæ & cathetum reflexionis.</s> <s xml:id="echoid-s32482" xml:space="preserve"> Quod eſt propoſitum in ſpecu-<lb/>lis planis:</s> <s xml:id="echoid-s32483" xml:space="preserve"> & patet uniuerſaliter in omnibus reflexionibus à ſpeculis quibuſcunq;</s> <s xml:id="echoid-s32484" xml:space="preserve">: quia danti op-<lb/>poſitum eadem impoſsibilia ſequentur, ducta chorda arcus interiacentis data puncta reflexionum <lb/>& cathetorum productarum, & ductis lineis contingentibus in illis punctis ipſas ſuperficies ſpe-<lb/>culorum, uel lineas, quę ſunt communes ſectiones ipſorum ſpeculorum & ſuperficierum reflexio-<lb/>nis.</s> <s xml:id="echoid-s32485" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s32486" xml:space="preserve"/> </p> <div xml:id="echoid-div1323" type="float" level="0" n="0"> <figure xlink:label="fig-0506-01" xlink:href="fig-0506-01a"> <variables xml:id="echoid-variables545" xml:space="preserve">d e h f g a b l</variables> </figure> <figure xlink:label="fig-0506-02" xlink:href="fig-0506-02a"> <variables xml:id="echoid-variables546" xml:space="preserve">h f a b g c</variables> </figure> </div> </div> <div xml:id="echoid-div1325" type="section" level="0" n="0"> <head xml:id="echoid-head1040" xml:space="preserve" style="it">29. Impoßibile eſt ſimul duo puncta eiuſdem rei uiſæ ab eodem puncto cuiuſcun. ſpeculi re-<lb/>flectiadidem centrum uiſus: uel à duob. punctis ſpeculorum planorũ uel conuexorum formam <lb/>unius puncti. Alhazen 51 n 4.</head> <p> <s xml:id="echoid-s32487" xml:space="preserve">Quòd enim puncto a licuius formę perpendiculariter ſuperficiei ſpeculi incidẽte, aliam lineam <lb/>ab alio puncto rei eiuſdẽ, uel alterius perpẽdiculariter duci ſuper eandem ſuperficiem ad idẽ pun-<lb/>ctum fit impoſsibile, patet per 13 p 11:</s> <s xml:id="echoid-s32488" xml:space="preserve"> quòd autem perpendicularis reflectatur in ſe ipſam patet per <lb/>21 huius:</s> <s xml:id="echoid-s32489" xml:space="preserve"> impoſsibile eſt ergo duo puncta eiuſdem formæ uiſæ ab eodem puncto ſpeculi ad idẽ cen <lb/>trum uiſus reflecti perpẽdiculariter.</s> <s xml:id="echoid-s32490" xml:space="preserve"> Sed neq;</s> <s xml:id="echoid-s32491" xml:space="preserve"> eſt hoc poſsibile fieri, linea incidẽtiæ obliqua exiſten <lb/>te.</s> <s xml:id="echoid-s32492" xml:space="preserve"> Omnis enim punctus cuiuslibet formę incidit ſpeculo, & reflectitur ad uiſum ſecũdũ lineas bre <lb/>uiores per 18 huius:</s> <s xml:id="echoid-s32493" xml:space="preserve"> & omnis talis reflexio ad uiſum & ipſarũ formarũ cõprehẽſio fit ſecũ diſpo-<lb/>fitionem linearũ reflexarũ per 24 huius:</s> <s xml:id="echoid-s32494" xml:space="preserve"> illæ ergo duę formę ſi ad unũ pũctũ, quod eſt centrũ oculi.</s> <s xml:id="echoid-s32495" xml:space="preserve"> <lb/> <pb o="205" file="0507" n="507" rhead="LIBER QVINTVS."/> incidunt, & ab uno puncto reflectuntur:</s> <s xml:id="echoid-s32496" xml:space="preserve"> tuncilla duo puncta, à quibus ſuarum formarum fit inci, <lb/>dentia, quia non perueniunt ad uiſum niſi ſe cundum lineas incidentiæ, quæ a b uno puncto reflexę <lb/>perueniút ad uiſum, uidebũtur unus pũctus:</s> <s xml:id="echoid-s32497" xml:space="preserve"> & ſic erit confuſio formarú in uiſu.</s> <s xml:id="echoid-s32498" xml:space="preserve"> Si enim lineæ inci-<lb/>dentiæ formarum diuerſorum pũctorum non diuerſificant pũcta reflexionis, ſed incidũt eidẽ pun-<lb/>cto:</s> <s xml:id="echoid-s32499" xml:space="preserve"> palàm quòd aut aliqua forma tota, aut plura pũcta illius formę poſunt uni pũcto incidere, & <lb/>in unum pũctum reflecti, qui eſt cẽ trum uiſus:</s> <s xml:id="echoid-s32500" xml:space="preserve"> & uidebitur tota forma unus pũctus.</s> <s xml:id="echoid-s32501" xml:space="preserve"> Item ſi detur li-<lb/>neas incidétiæ & reflexionis propter angulorũ ſuorum diuerſitatem ſomper diuerſas eſſe:</s> <s xml:id="echoid-s32502" xml:space="preserve"> ſicut er-<lb/>go ſunt duę lineę incidẽtiæ, quę à diuerſis pũctis formæ incidũt eidem pũcto ſpeculi:</s> <s xml:id="echoid-s32503" xml:space="preserve"> ſic fient duæ <lb/>lineæ reflexionis, quę ad idem cẽtrum uiſus terminantur:</s> <s xml:id="echoid-s32504" xml:space="preserve"> ut ſi à duobus pũctis formę incidẽtis ſpe <lb/> <anchor type="figure" xlink:label="fig-0507-01a" xlink:href="fig-0507-01"/> <anchor type="figure" xlink:label="fig-0507-02a" xlink:href="fig-0507-02"/> culo, quæ ſunt a & b, incidant eidẽ pun <lb/>cto ſpeculi, qui ſit c, quę lineæ a c & b c:</s> <s xml:id="echoid-s32505" xml:space="preserve"> <lb/>& ab illo reflectantur ad idẽ cétrum ui-<lb/>ſus, quod ſit d:</s> <s xml:id="echoid-s32506" xml:space="preserve"> ſequetur adhuc ſi ab uno <lb/>pũcto reflexionis c, diuerſę formę pun-<lb/>ctorum a & b ad centrũ uiſus d perue-<lb/>niant, duas lineas rectas, quę ſunt c d, <lb/>fuperficiem includere:</s> <s xml:id="echoid-s32507" xml:space="preserve"> quod eſt impoſ-<lb/>ſibile.</s> <s xml:id="echoid-s32508" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s32509" xml:space="preserve"> Sed neq;</s> <s xml:id="echoid-s32510" xml:space="preserve"> à <lb/>duobus punctis alicuius ſpeculi plani <lb/>uel cõuexi ad idẽ cẽtrũ uiſus ſimul poſ-<lb/>ſibile eſtidẽ pũctum formę reflecti.</s> <s xml:id="echoid-s32511" xml:space="preserve"> Sit <lb/>enim, ſi poſsibile eſt, ut forma puncti a <lb/>reflectatur ad cẽtrum uiſus b à duobus <lb/>pũctis ſpeculi plani uel conuexi cuiuſ-<lb/>cũq;</s> <s xml:id="echoid-s32512" xml:space="preserve">, qui ſint c & d, ſignati ſuper lineá, <lb/>quę eſt communis ſectio ſuperficiei re-<lb/>flexionis & ſpeculi uel ſuperficiei contingentis ſpeculum conuexum, quę ſit e f.</s> <s xml:id="echoid-s32513" xml:space="preserve"> Cum ergo per 24 <lb/>huius ſecundum diſpoſitionem linearum reflexionis, uiſus ſemper informetur:</s> <s xml:id="echoid-s32514" xml:space="preserve"> tũc forma pũctia, <lb/>quę eſt indiuifibilis, occurret uiſui, ut forma lineæ c d, quę eſt diuiſibilis linea.</s> <s xml:id="echoid-s32515" xml:space="preserve"> Nó ergo occurret ui-<lb/>fui, niſi tantũ unus pũctus formę reflexę ab uno pũcto ſpeculi:</s> <s xml:id="echoid-s32516" xml:space="preserve"> neq;</s> <s xml:id="echoid-s32517" xml:space="preserve"> unũ punctum formę à duobus <lb/>punctis ſpeculi plani uel conuexi poſsibile eſt reflecti.</s> <s xml:id="echoid-s32518" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s32519" xml:space="preserve"/> </p> <div xml:id="echoid-div1325" type="float" level="0" n="0"> <figure xlink:label="fig-0507-01" xlink:href="fig-0507-01a"> <variables xml:id="echoid-variables547" xml:space="preserve">a b d c</variables> </figure> <figure xlink:label="fig-0507-02" xlink:href="fig-0507-02a"> <variables xml:id="echoid-variables548" xml:space="preserve">a b e c d v</variables> </figure> </div> </div> <div xml:id="echoid-div1327" type="section" level="0" n="0"> <head xml:id="echoid-head1041" xml:space="preserve" style="it">30. Abuno puncto ſuperficiei ſpeculi cuiuſcun formam unius punctireiuiſæ ad duos uiſus <lb/>non eſt poßibile reflecti. Alhazen 51 n 4.</head> <p> <s xml:id="echoid-s32520" xml:space="preserve">Linea enim reflexionis ad unum uifum procedens quia cum perpendiculari erecta à puncto re-<lb/>flexionis ſuper ſuperficiem ſpeculi angulũ tenet ęqualẽ angulo, quẽtenet linea incidẽtię cum eadé <lb/>perpẽdiculari, ut patet per 20 huius:</s> <s xml:id="echoid-s32521" xml:space="preserve"> palàm quòd non poteſt in eadẽ ſuperficie alia linea ſumi, quæ <lb/>æqualẽ angulũ efficiat cũ ducta perpẽdiculari:</s> <s xml:id="echoid-s32522" xml:space="preserve"> unde ab hoc pũcto nõreflectetur forma eiuſdẽ pun-<lb/>cti ad uiſum aliũ.</s> <s xml:id="echoid-s32523" xml:space="preserve"> Oportet igitur, ut à diuerſis pũctis ſpeculi cuiuſcunq;</s> <s xml:id="echoid-s32524" xml:space="preserve"> fiat ad uiſus diuerſos refle-<lb/>xio.</s> <s xml:id="echoid-s32525" xml:space="preserve"> Et quoniam duo tãtùm ſunt uiſus, oportet ad minus, ut à duobus pũctis ſuperficiei ſpeculi cu-<lb/>inſcũq;</s> <s xml:id="echoid-s32526" xml:space="preserve"> fiat reflexio formę unius punctirei uifæ ad ambos uiſus.</s> <s xml:id="echoid-s32527" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s32528" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1328" type="section" level="0" n="0"> <head xml:id="echoid-head1042" xml:space="preserve" style="it">31. Ab uno puncto reflexionis cuiuſun ſpeculi ad diuerſos uiſus poßibile eſt formas pun-<lb/>ctorum plurium reflecti: & à diuerſis unam. Alhazen 51 n 4.</head> <p> <s xml:id="echoid-s32529" xml:space="preserve">Quamuis enim, ut patet per 29 huius, ſolùm formę unius pũcti incidẽtis ab uno tantùm puncto <lb/>ſpeculi reflexio fimul ſit poſsibilis ad unum centrum uiſus:</s> <s xml:id="echoid-s32530" xml:space="preserve"> eſt tamẽ poſsibile fieri ſimul ad diuer-<lb/>ſos uiſus ab uno puncto ſpeculi diuerſorum pũctorũ formę incidẽtis reflexionẽ:</s> <s xml:id="echoid-s32531" xml:space="preserve"> quoniã illa pũcta <lb/>ſecundum angulos diuerſos incidũt, & ſecundũ diuerſos reflectũtur:</s> <s xml:id="echoid-s32532" xml:space="preserve"> ergo ad pũcta diuerſa termi-<lb/>nantur lineę reflexę, in quibus diuerſi uiſus cadẽtes pũcta diuerſarũ formarũ comprehẽdẽt ab uno <lb/>puncto ſpeculi ad diuerſos uiſus reflexa.</s> <s xml:id="echoid-s32533" xml:space="preserve"> Et ſi unus uiſus motus fuerit, & ſitum uariauerit, ſpeculo <lb/>exiftẽte immoto:</s> <s xml:id="echoid-s32534" xml:space="preserve"> tũc etiam ſecũdum ſitus ſui diuerſitatem ab eodẽ pũcto ſpeculi ad ipſum pũcta di <lb/>uerſerũ formarũ reflectentur, ſemper tamen complebitur pyramis reliquarũ formarũ.</s> <s xml:id="echoid-s32535" xml:space="preserve"> Sed & unus <lb/>uiſus motus, uel diuerſi uiſus eandẽ formã uidebũt à diuerſis pũctis ſpeculi reflexam:</s> <s xml:id="echoid-s32536" xml:space="preserve"> quia quilibet <lb/>pũctus formę incidẽtis totali ſuperficiei ſpeculi incidens ad aliquá partem oppoſitã reflectitur, & <lb/>ſecũdum modum, quo in 22 & 24 huius proponitur, patet quòd formarú pyramides diuerſantur.</s> <s xml:id="echoid-s32537" xml:space="preserve"> <lb/>Et quia diuerſis uiſibus diuerſi axes pyramidum incidunt, qui ſunt eiuſdẽ formæ, accidit ut à diuer <lb/>ſis uiſibus una forma à diuerfis punctis ſuperficiei ſpeculi reflexa uideatur.</s> <s xml:id="echoid-s32538" xml:space="preserve"> Etidem accidit etiam <lb/>eidem uiſui moto, quando ſpeculum permanet immotum.</s> <s xml:id="echoid-s32539" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s32540" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1329" type="section" level="0" n="0"> <head xml:id="echoid-head1043" xml:space="preserve" style="it">32. A‘ centro oculi duct a perpendiculari ſuper ſuperficiem cuiuſcun ſpeculi plani uel con-<lb/>uexi, non eſt poßibile aliquem punctum ductæ line æ reflecti ad uiſum, niſieum ſolùm, in quo du-<lb/>ct a perpendicularis ſuperficiem oculi interſecat: & ab eo ſolo puncto, in quo duct a perpendicula <lb/>ris incidit ipſius ſpeculi ſuperficiei. Alhazen 13 n 5.</head> <p> <s xml:id="echoid-s32541" xml:space="preserve">Sit centrũ uiſus punctũ a:</s> <s xml:id="echoid-s32542" xml:space="preserve"> & ſit linea, quæ eſt cõmunis ſectio ſuperficiei reflexionis & ſuperficiei <lb/> <pb o="206" file="0508" n="508" rhead="VITELLONIS OPTICAE"/> ſpeculi cuiuſcunq;</s> <s xml:id="echoid-s32543" xml:space="preserve"> plani uel conuexi, & ſit nũc, exempli cauſſa, ſpeculi plani dati linea b g:</s> <s xml:id="echoid-s32544" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s32545" xml:space="preserve"> per-<lb/>pendicularis ducta à pũcto a ſuper lineam b g linea a g:</s> <s xml:id="echoid-s32546" xml:space="preserve"> ſit quo q;</s> <s xml:id="echoid-s32547" xml:space="preserve">, ut linea a g ſecet ſuperficiem ſphę-<lb/>ricá cóuexam oculi in puncto d.</s> <s xml:id="echoid-s32548" xml:space="preserve"> Dico quòd in tota perpendicularia g quantumcũq;</s> <s xml:id="echoid-s32549" xml:space="preserve"> protracta non <lb/>eſt punctus, qui reflectatur ab hoc ſpeculo ad centrũ uiſus a, niſi ſolus punctus d.</s> <s xml:id="echoid-s32550" xml:space="preserve"> Si enim alius pun <lb/>ctus dictę perpẽdicularis ad uiſum reflectitur pręter punctũ d:</s> <s xml:id="echoid-s32551" xml:space="preserve"> aut ille pũctus eſt ultra centrũ uiſus <lb/>a:</s> <s xml:id="echoid-s32552" xml:space="preserve"> aut ſub uiſu.</s> <s xml:id="echoid-s32553" xml:space="preserve"> Si ultra uiſum, ſit ille pũctus h:</s> <s xml:id="echoid-s32554" xml:space="preserve"> palã ergo quòd non perueniet forma eius ad ſpeculũ <lb/>ſuper perpẽdicularẽ h a, propter ſolidi corporis in, erpoſitionẽ, qđ eſt ultra uiſum in capite uiden-<lb/>tis.</s> <s xml:id="echoid-s32555" xml:space="preserve"> Nó reflectetur ergo forma pũcti h ſuper քpen-<lb/> <anchor type="figure" xlink:label="fig-0508-01a" xlink:href="fig-0508-01"/> dicularẽ h g.</s> <s xml:id="echoid-s32556" xml:space="preserve"> Si uerò dicatur qđ ab alio pũcto ſpe-<lb/>culi pręter pũctum g, poteſt reflecti forma pũcti h <lb/>a d uiſum a:</s> <s xml:id="echoid-s32557" xml:space="preserve"> ſit illud pũctum b:</s> <s xml:id="echoid-s32558" xml:space="preserve"> & ſit linea incidẽtiæ <lb/>h b:</s> <s xml:id="echoid-s32559" xml:space="preserve"> & linea reflexionis h a:</s> <s xml:id="echoid-s32560" xml:space="preserve"> diuidaturq́;</s> <s xml:id="echoid-s32561" xml:space="preserve"> angulus h <lb/>b a ք æqualia ք lineã b t ductã ad perpendicularẽ <lb/>h g auxilio 9 p 1:</s> <s xml:id="echoid-s32562" xml:space="preserve"> erit ergo ք 26 huius linea b t per-<lb/>pẽdicularis ſuper lineã b g:</s> <s xml:id="echoid-s32563" xml:space="preserve"> ſed linea t g eſt perpen <lb/>dicularis ſuper eandẽ lineá h g.</s> <s xml:id="echoid-s32564" xml:space="preserve"> Ab eodẽ ergo pun <lb/>cto t eſt ducere duas perpẽdiculares ſuper lineam <lb/>b g, & ſuper ipſam ſuperficiẽ ſpeculi:</s> <s xml:id="echoid-s32565" xml:space="preserve"> q đ eſt impoſ <lb/>ſibile.</s> <s xml:id="echoid-s32566" xml:space="preserve"> Sequetur enim trigoni a b g duos angulos <lb/>eſſerectos, ſcilicet angulos t g b & t b g:</s> <s xml:id="echoid-s32567" xml:space="preserve"> & ab eodẽ <lb/>pũcto plures ducerẽtur քpẽdiculares lineæ ſuper <lb/>eandẽ ſuperficiẽ, qđ eſt cõtra 20 th.</s> <s xml:id="echoid-s32568" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s32569" xml:space="preserve"> Nulla <lb/>ergo forma pũctorũ lineę h d poteſt reflecti ad ui-<lb/>ſum, niſi ſolum pũctũ d:</s> <s xml:id="echoid-s32570" xml:space="preserve"> quoniã de omnibus alijs <lb/>punctis eodẽ modo eſt demóſtrandũ.</s> <s xml:id="echoid-s32571" xml:space="preserve"> Neq;</s> <s xml:id="echoid-s32572" xml:space="preserve"> enim <lb/>poteſt dici quòd aliqua forma alicuius pũcti ſum-<lb/>pti inter pũcta a & d reflectatur ad uiſum, niſi per <lb/>lineá քpendicularẽ d a:</s> <s xml:id="echoid-s32573" xml:space="preserve"> quoniã pũcti inter centrũ <lb/>uiſus & ſuperficiẽ eius poſiti ſunt ualde rari:</s> <s xml:id="echoid-s32574" xml:space="preserve"> unde <lb/>nõ mittitur alicuius ipſorũ forma in uiſum, neque <lb/>ab aliquo ſpeculo refle ctitur, ut ſentiatur.</s> <s xml:id="echoid-s32575" xml:space="preserve"> Sed ne-<lb/>que forma alicuius pũctorum lineæ d g poteſt re-<lb/>flecti ad uiſum a à pũcto ſpeculi g, ut forma pũcti f:</s> <s xml:id="echoid-s32576" xml:space="preserve"> quoniá ſi illud pũctũ d ſolidi corporis fuerit, pa-<lb/>tet quòd ipſum impediet reflexionẽ ad uiſum ք lineã d g:</s> <s xml:id="echoid-s32577" xml:space="preserve"> quia propter ſoliditatẽ ipſius forma pun-<lb/>cti fnõ poterit tranſire & ad uiſum քuenire:</s> <s xml:id="echoid-s32578" xml:space="preserve"> & ſi fueritrarũ, adhuc forma reflexa à ſpeculo miſcebi-<lb/>tur ei, & adhęrebit ſibi, neq;</s> <s xml:id="echoid-s32579" xml:space="preserve"> քueniet ad uiſum.</s> <s xml:id="echoid-s32580" xml:space="preserve"> Sed neq;</s> <s xml:id="echoid-s32581" xml:space="preserve"> poteſt forma alicuius illorũ pũctorũ refle-<lb/>cti à pũcto alio ſpeculi ꝗ̃ à pũcto k:</s> <s xml:id="echoid-s32582" xml:space="preserve"> quoniã ductis lineis f k & a k, & diuiſo angulo a k f <lb/>ք ęqualia ք lineã k l, ſequeturidẽ impoſsibile, qđ prius, ſcilicet lineas l k & l g perpẽdiculares eſſe ſu <lb/>per ſuperficiẽ ſpeculi, uel ſuper ſuքficiẽ ſpeculũ cõtingentẽ:</s> <s xml:id="echoid-s32583" xml:space="preserve"> qđ eſt cótra 20 th.</s> <s xml:id="echoid-s32584" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s32585" xml:space="preserve"> Omniũ ita q;</s> <s xml:id="echoid-s32586" xml:space="preserve"> <lb/>pũctorũ lineæ h g n õ reflectitur aliquis ad uiſum a niſi ſolũ pũctũ d.</s> <s xml:id="echoid-s32587" xml:space="preserve"> Et quoniã quodlibet pũctũ to-<lb/>tius uiſibilis, in quo eſt linea h g, pręter pũctum d, in ſuperficie uiſus impreſſum opponitur ſpeculo <lb/>nõ ad angulũ rectũ:</s> <s xml:id="echoid-s32588" xml:space="preserve"> quoniã omnia pũcta circũſtátia pũctũ d cócurrũt in cẽtro uiſus a, & faciunt co-<lb/>nũ pyramidis, cuius baſis eſt in ſuperficie ſpeculi circa axẽ a g:</s> <s xml:id="echoid-s32589" xml:space="preserve"> uidebútur formæ omniú illorũ pun-<lb/>ctorú ſuper perpédiculares ab eis ad ſuperficiẽ ſpeculi ductas.</s> <s xml:id="echoid-s32590" xml:space="preserve"> Patet ergo ꝓpoſitũ:</s> <s xml:id="echoid-s32591" xml:space="preserve"> quoniá in ſpecu <lb/>lis cóuexis linea h g eſt ſemper perpẽdicularis ſuper ſuperficiẽ ſpeculi, nec ab aliquo ſuorũ puncto-<lb/>rũ ſuper ſpeculi ſuperficiẽ alia perpendicularis duci poteſt ք 20 th.</s> <s xml:id="echoid-s32592" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s32593" xml:space="preserve"> ita tamẽ quòd hæc, quæ <lb/>pręmiſſa ſunt, in uno tantũ uiſu intelligátur in omnib.</s> <s xml:id="echoid-s32594" xml:space="preserve"> ſpeculis planis & quibuſcũq;</s> <s xml:id="echoid-s32595" xml:space="preserve"> conuexis, ſicut <lb/>ꝓpoſitio proponit:</s> <s xml:id="echoid-s32596" xml:space="preserve"> quoniã forma eiuſdẽ pũcti rei uiſæ ad ambos uiſus reflexa, ſi uni uiſuũ perpen-<lb/>diculariter incidat, poteſt alij uiſui obliquè incidere ſecundũ lineã reflexionis obliquè à ſuperficie <lb/>ſpeculi ad cétrum uiſus procedétem:</s> <s xml:id="echoid-s32597" xml:space="preserve"> & uidebitur idẽ pũctus rei uiſę à duobus uiſibus ſecũdum di-<lb/>uerſum modum ſuæ reflexionis:</s> <s xml:id="echoid-s32598" xml:space="preserve"> in ſpeculis uerò concauis quibuſcunq;</s> <s xml:id="echoid-s32599" xml:space="preserve"> eſt ſecus.</s> <s xml:id="echoid-s32600" xml:space="preserve"/> </p> <div xml:id="echoid-div1329" type="float" level="0" n="0"> <figure xlink:label="fig-0508-01" xlink:href="fig-0508-01a"> <variables xml:id="echoid-variables549" xml:space="preserve">h t a d l y g k j e</variables> </figure> </div> <figure> <variables xml:id="echoid-variables550" xml:space="preserve">e c a d b</variables> </figure> </div> <div xml:id="echoid-div1331" type="section" level="0" n="0"> <head xml:id="echoid-head1044" xml:space="preserve" style="it">33. Impoßibile eſt formã obliquè ſpeculo incidentẽ ſecundum li-<lb/>neam ſuæ incidentiæ aduiſum reſlecti, uelex parte ſui anguli mi-<lb/>noris. Euclides 3 th. catoptr.</head> <p> <s xml:id="echoid-s32601" xml:space="preserve">Eſto ut ſpeculo a d b incidat forma pũcti c obliquè in puncto d, ita <lb/>ut angulus c d b ſit maior angulo c d a.</s> <s xml:id="echoid-s32602" xml:space="preserve"> Dico quòd forma pũcti c ſecun <lb/>dum lineam c d non reflectetur in ſe ipſam propter inęqualitatem an-<lb/>gulorum:</s> <s xml:id="echoid-s32603" xml:space="preserve"> cum ſemper angulus incidentiæ ſit æ qualis angulo reflexio <lb/>nis per 20 huius:</s> <s xml:id="echoid-s32604" xml:space="preserve"> ſed neq;</s> <s xml:id="echoid-s32605" xml:space="preserve"> ex parte ſui anguli minoris, qui eſt c d a.</s> <s xml:id="echoid-s32606" xml:space="preserve"> Fiat <lb/>enim, utreflectatur ſecundum lineam d e diuidentem angulum c d a:</s> <s xml:id="echoid-s32607" xml:space="preserve"> <lb/>erit ergo angulus c d b æ qualis angulo e d a:</s> <s xml:id="echoid-s32608" xml:space="preserve"> ſed angulus c d b maior <lb/>eſt angulo d a c per hypotheſin:</s> <s xml:id="echoid-s32609" xml:space="preserve"> erit ergo angulus e d a maior angulo <lb/>c d a, pars ſuo toto:</s> <s xml:id="echoid-s32610" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s32611" xml:space="preserve"> Semper ergo ſecũdũ angulũ <lb/>maiorẽ, ꝗ in ꝓpoſito eſt angulus c d b fiet reflexio, Et hoc eſt ꝓpoſitũ.</s> <s xml:id="echoid-s32612" xml:space="preserve"/> </p> <pb o="207" file="0509" n="509" rhead="LIBER QVINTVS."/> </div> <div xml:id="echoid-div1332" type="section" level="0" n="0"> <head xml:id="echoid-head1045" xml:space="preserve" style="it">34. Inomni ſpeculo formarum punctorum mediorũ cuiuslibet rei uiſæ reflexio fit inter pun-<lb/>cta reflexionum formarum punctorum extremorum eiuſdem rei uiſæ.</head> <p> <s xml:id="echoid-s32613" xml:space="preserve">Sit res uiſa per reflexionẽ à quocunq;</s> <s xml:id="echoid-s32614" xml:space="preserve"> ſpeculo, quæ a b c:</s> <s xml:id="echoid-s32615" xml:space="preserve"> cuius extrema puncta ſint a & c:</s> <s xml:id="echoid-s32616" xml:space="preserve"> aliquis <lb/>uerò mediorum punctorum lineæ a b c ſit punctus b:</s> <s xml:id="echoid-s32617" xml:space="preserve"> & ſit ſuperficies illius ſpeculi, ſiue plana, ſiue <lb/>conuexa, uel concaua fuerit, in qua ſit communis ſectio ſuperficiei reflexionis & ſpeculi linea d e f:</s> <s xml:id="echoid-s32618" xml:space="preserve"> <lb/>& ſit centrum uiſus punctum g:</s> <s xml:id="echoid-s32619" xml:space="preserve"> reflectaturq́;</s> <s xml:id="echoid-s32620" xml:space="preserve"> <lb/>forma puncti a ad uiſum g â puncto ſpeculi, <lb/> <anchor type="figure" xlink:label="fig-0509-01a" xlink:href="fig-0509-01"/> quod ſit d:</s> <s xml:id="echoid-s32621" xml:space="preserve"> & forma puncti c à puncto ſpecu-<lb/>li, quod ſit f:</s> <s xml:id="echoid-s32622" xml:space="preserve"> & forma puncti b, qui ſit aliquis <lb/>mediorum punctorum lineæ a b c, reflecta-<lb/>tur ad uiſum à puncto ſpeculi e.</s> <s xml:id="echoid-s32623" xml:space="preserve"> Dico, quòd <lb/>punctus e neceſſariò cadit inter puncta d & f, <lb/>quæ ſunt puncta reflexionum formarũ p un-<lb/>ctorum extremorũ a & c.</s> <s xml:id="echoid-s32624" xml:space="preserve"> Si enim cadat pun-<lb/>ctum e extra puncta d & f:</s> <s xml:id="echoid-s32625" xml:space="preserve"> linea ergo b e, quæ <lb/>eſt linea incidentiæ formæ puncti b, ſecabit <lb/>aliquam linearum, quæ ſunt a d & c f:</s> <s xml:id="echoid-s32626" xml:space="preserve"> quam-<lb/>cunq;</s> <s xml:id="echoid-s32627" xml:space="preserve"> uerò illa ſecuerit, ſit punctum ſectionis h.</s> <s xml:id="echoid-s32628" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s32629" xml:space="preserve"> quòd forma puncti h reflectetur ad ui-<lb/>ſum g à duobus punctis ſpeculi, quæ ſunt e & f, uel e & d:</s> <s xml:id="echoid-s32630" xml:space="preserve"> quod in ſpeculis planis & conuexis patet <lb/>eſſe impoſsibile per 29 huius.</s> <s xml:id="echoid-s32631" xml:space="preserve"> In ſpeculis quoq;</s> <s xml:id="echoid-s32632" xml:space="preserve"> concauis duplicabuntur puncti reflexionum illis <lb/>ſpeculis cõuenientium:</s> <s xml:id="echoid-s32633" xml:space="preserve"> nulla quo q;</s> <s xml:id="echoid-s32634" xml:space="preserve"> forma in aliquo ſpeculorum ſecũdum ſitum & ordinationem <lb/>propriam ſuarum partium uidebitur:</s> <s xml:id="echoid-s32635" xml:space="preserve"> quod totum eſt impoſsibile.</s> <s xml:id="echoid-s32636" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s32637" xml:space="preserve"/> </p> <div xml:id="echoid-div1332" type="float" level="0" n="0"> <figure xlink:label="fig-0509-01" xlink:href="fig-0509-01a"> <variables xml:id="echoid-variables551" xml:space="preserve">a b c h h g e d e f e</variables> </figure> </div> </div> <div xml:id="echoid-div1334" type="section" level="0" n="0"> <head xml:id="echoid-head1046" xml:space="preserve" style="it">35. Figura ſuperficiei corporis incidentis & ſpeculi, ſitú ſimilibus exiſtentibus, erit in omni <lb/>ſpeculo complementum formæ corporis & figuræ. Alhazen 22 n 4.</head> <p> <s xml:id="echoid-s32638" xml:space="preserve">Cum enim figura ſpeculi & corporis eſt eadem & ſitus idem:</s> <s xml:id="echoid-s32639" xml:space="preserve"> ut ſi utraq;</s> <s xml:id="echoid-s32640" xml:space="preserve"> illarum figurarũ ſit pla-<lb/>na & æquidiſtent:</s> <s xml:id="echoid-s32641" xml:space="preserve"> tũc forma puncti primi ſuperficiei uiſi corporis incidit puncto primo ſpeculi, & <lb/>forma puncti ſecundi puncto ſecundo, & ſic de omnibus alijs punctis ſe reſpicientibus.</s> <s xml:id="echoid-s32642" xml:space="preserve"> Sic ergo in <lb/>ſuperficie ſpeculi fit totalis figura ſuperficiei corporis uiſi:</s> <s xml:id="echoid-s32643" xml:space="preserve"> quod non accidit in ſpeculo alterius fi-<lb/>guræ.</s> <s xml:id="echoid-s32644" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s32645" xml:space="preserve"> ſumpta quacunq;</s> <s xml:id="echoid-s32646" xml:space="preserve"> ſpeculi parte, cuius figura ſit ſimilis figuræ corporis, & ſi-<lb/>tus æquidiſtans:</s> <s xml:id="echoid-s32647" xml:space="preserve"> erit ſemper complementũ figuræ corporis in ea.</s> <s xml:id="echoid-s32648" xml:space="preserve"> Et cum infinitæ ſint tales ſpeculi <lb/>partes, palàm quòd infinitæ erũt formæ corporis ſpeculo incidentes, quę ſemper ad diuerſa pũcta <lb/>reflectuntur, ex quibus formam corporis uiſus diuerſi in eodẽ ſpeculo comprehendunt.</s> <s xml:id="echoid-s32649" xml:space="preserve"> Hocitaq;</s> <s xml:id="echoid-s32650" xml:space="preserve"> <lb/>accidit in omnibus ſpeculis:</s> <s xml:id="echoid-s32651" xml:space="preserve"> ſed maximè euidens eſt in planis.</s> <s xml:id="echoid-s32652" xml:space="preserve"> Cum enim quolibet puncto ſuperfi-<lb/>ciei planæ ſuperficiei ſpeculi plani incidente, figura partium circũſtantium ſit ſimilis ordinationis <lb/>& ſitus, accidit ex omnibus pũctis ſimilis reflexio & ſimul & in eodem modo:</s> <s xml:id="echoid-s32653" xml:space="preserve"> & ſic fit complemen-<lb/>tum in ſpeculo formæ corporis & figuræ.</s> <s xml:id="echoid-s32654" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s32655" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1335" type="section" level="0" n="0"> <head xml:id="echoid-head1047" xml:space="preserve" style="it">36. In ſpeculis quibuſcun unumquod punctorum conſpecto-<lb/>rũ in catheto ſuæ incidentiæ uidetur. Euclides 16.17.18.th. catoptr. <lb/>Alhazen 9 n 5.</head> <figure> <variables xml:id="echoid-variables552" xml:space="preserve">a b g d</variables> </figure> <p> <s xml:id="echoid-s32656" xml:space="preserve">Sit ſpeculum quodcũq;</s> <s xml:id="echoid-s32657" xml:space="preserve">: & ſit nunc, exempli cauſſa, planum:</s> <s xml:id="echoid-s32658" xml:space="preserve"> quod <lb/>ſit g d, punctusq́;</s> <s xml:id="echoid-s32659" xml:space="preserve"> uiſus ſit a:</s> <s xml:id="echoid-s32660" xml:space="preserve"> & centrum oculi ſit b:</s> <s xml:id="echoid-s32661" xml:space="preserve"> & ducatur à pũcto <lb/>rei uiſæ, quod eſt a, cathetus incidentiæ, quæ ſit a g.</s> <s xml:id="echoid-s32662" xml:space="preserve"> Dico, quòd ima-<lb/>go puncti a ſemper uidetur in linea a g:</s> <s xml:id="echoid-s32663" xml:space="preserve"> ſuppoſitum enim eſt in prin-<lb/>cipio huius libri 2 ſuppoſitione quòd uniformis ſituatio puncti rei <lb/>uiſæ reſpectu ſuperficiei cuiu ſcunq;</s> <s xml:id="echoid-s32664" xml:space="preserve"> ſpeculi, à qua eius forma reflecti <lb/>tur, fit ſolùm ſecundum cathetum ſuæ incidentiæ:</s> <s xml:id="echoid-s32665" xml:space="preserve"> forma autem, quæ <lb/>in ſpeculo uidetur, eſt imago rei uiſæ, ut patet per definitionem:</s> <s xml:id="echoid-s32666" xml:space="preserve"> ne-<lb/>ceſſe eſt ergo imaginem illam uideri ſecundum ſituationem unifor-<lb/>mem ipſius puncti rei uiſæ ad ſpeculum:</s> <s xml:id="echoid-s32667" xml:space="preserve"> quoniam aliàs non uidere-<lb/>tur illa forma per modũ imaginis.</s> <s xml:id="echoid-s32668" xml:space="preserve"> Videbitur ergo neceſſariò in ipſa <lb/>catheto incidentiæ ſuæ.</s> <s xml:id="echoid-s32669" xml:space="preserve"> Quod eſt propoſitum:</s> <s xml:id="echoid-s32670" xml:space="preserve"> in alijs enim ſpeculis <lb/>eſt eodem modo declarandum.</s> <s xml:id="echoid-s32671" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1336" type="section" level="0" n="0"> <head xml:id="echoid-head1048" xml:space="preserve" style="it">37. Locum imaginis rei uiſæ in ſpeculis quibuſcun in puncto concurſus lineæ reflexionis cũ <lb/>catheto incidentiæ neceſſe eſt eſſe. Alhazen 2.4.6.7.8 n 5.</head> <p> <s xml:id="echoid-s32672" xml:space="preserve">Huius exemplum eſt:</s> <s xml:id="echoid-s32673" xml:space="preserve"> ſi pyramis orthogonia erigatur perpendiculariter ſuper ſuperficiem ſpecu <lb/>li cuiuſcunq;</s> <s xml:id="echoid-s32674" xml:space="preserve">: tunc enim apparebit uiſui alia pyramis continua, tenens ſe cum pyramide extrinſeca <lb/>quaſi ad modũ rhombi:</s> <s xml:id="echoid-s32675" xml:space="preserve"> & uidebuntur harũ pyramidũ uertices quaſi uniformiter diſtantes à ſuper-<lb/>ficie ſpeculi.</s> <s xml:id="echoid-s32676" xml:space="preserve"> Et ſi linea recta imaginetur duci à uertice unius pyramidis ad uerticẽ alterius:</s> <s xml:id="echoid-s32677" xml:space="preserve"> palàm <lb/>quoniam ipſa erit perpẽdicularis ſuper baſim uiſæ pyramidis, & ita ſuper ſuperficiem ſpeculi, cum <lb/>eadem ſit ſuperficies ſpeculi & baſis uiſæ pyramidis, ut in ſpeculis planis, uel baſis uiſæ pyramidis <lb/> <pb o="208" file="0510" n="510" rhead="VITELLONIS OPTICAE"/> æ quidiſtet ſuperficiei ſpeculum contingenti, ut in ſpeculis conuexis, quorum ſpeculorum ſuperfi-<lb/>cies ipſa baſis uiſæ pyramidis eſt contingens, uel æ quidiſtans ſuperficiei contingenti ſuperficiẽ ſpe <lb/>culi, ut in ſpeculis concauis, in quibus baſis pyramidis erectæ ſuper ſpeculum æquidiſtat ſuperfi-<lb/>siei planæ ſpeculum contingenti:</s> <s xml:id="echoid-s32678" xml:space="preserve"> uertex itaq;</s> <s xml:id="echoid-s32679" xml:space="preserve"> pyramidis ſemper uidebitur in linea perpendicula-<lb/>ri ab eo educta ad ſpeculum.</s> <s xml:id="echoid-s32680" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s32681" xml:space="preserve"> à quocunq;</s> <s xml:id="echoid-s32682" xml:space="preserve"> puncto pyramidis ducatur linea æ quidi-<lb/>ſtanter axi, ſemper incidet ad punctũ ſimile ſibi reſpiciens ipſum in alia pyramide:</s> <s xml:id="echoid-s32683" xml:space="preserve"> & erit linea pro-<lb/>ducta per 8 p 11 ſemper orthogonalis ſuper baſes dictarum pyramidum, & ſuper ſuperficiẽ ſpeculi, <lb/>uel ſuper ſuperficiẽ ſpeculũ contingentẽ.</s> <s xml:id="echoid-s32684" xml:space="preserve"> Imago ergo cuiuslibet pũctorũ pyramidis ſic ſpeculo op <lb/>poſitæ cadit in perpẽdiculari intellecta duci à puncto illo ſuper ſuperficiem ſpeculi.</s> <s xml:id="echoid-s32685" xml:space="preserve"> Sed quicunq;</s> <s xml:id="echoid-s32686" xml:space="preserve"> <lb/>punctus corporis opponatur ſpeculo, neceſſe eſt imaginari pyramidem orthogonalem ſuper ſu-<lb/>perficiem ſpeculi aut ei continuam, uel ſuper ſuperficiem ipſum ſpeculum contingẽtem, uel ſuper-<lb/>ficiei contingenti æ quidiſtantem, ut patet per 22 huius, cuius pyramidis uertex eſt punctus ille ui <lb/>ſus, & baſis eius ſuperficies ſpeculi aut ſuperficies eicontinua.</s> <s xml:id="echoid-s32687" xml:space="preserve"> Et conuenit ut imaginetur alia pyra <lb/>mis oppoſita illi, cum illa quaſi complens rhombum, quarum utriuſq;</s> <s xml:id="echoid-s32688" xml:space="preserve"> eſt baſis uel eadem, uel una <lb/>baſium eſt alteri æquidiſtans, & perpendicularis à uertice unius ad uerticẽ alterius ducta erit per-<lb/>pendicularis ſuper ſpeculi ſuperficiem.</s> <s xml:id="echoid-s32689" xml:space="preserve"> Et quia imago cuiuslibet puncti ſpeculo oppoſiti cadit in li <lb/>neam perpendicularem ductam ab illo puncto ad ſpeculi ſuperficiẽ aut ei continuam:</s> <s xml:id="echoid-s32690" xml:space="preserve"> patet quòd <lb/>locus imaginis eſt in linea illa perpendiculari, ut etiam patuit per præmiſſam.</s> <s xml:id="echoid-s32691" xml:space="preserve"> Sed quia in ſpeculis <lb/>quibuſcunq;</s> <s xml:id="echoid-s32692" xml:space="preserve"> non accidit comprehenſio formarum niſi perlineas reflexionum, ut pater per 24 hu-<lb/>ius:</s> <s xml:id="echoid-s32693" xml:space="preserve"> palàm etiá quia imago cuiuslibet uiſi puncti cadit in lineam reflexionis:</s> <s xml:id="echoid-s32694" xml:space="preserve"> & quia quælibet taliũ <lb/>linearum eſt recta:</s> <s xml:id="echoid-s32695" xml:space="preserve"> imago ergo cuiuslibet puncti formæ reflexæ cadit in punctum ſectionis perpen <lb/>dicularis & lineæ reflexionis.</s> <s xml:id="echoid-s32696" xml:space="preserve"> Videtur ergo quandoq;</s> <s xml:id="echoid-s32697" xml:space="preserve"> citra ſuperficiem ſpeculi, ut cum talium linea.</s> <s xml:id="echoid-s32698" xml:space="preserve"> <lb/>rum interſectio uidelicet lineæ reflexionis & catheti incidentiæ non poteſt fieri niſi ſub ſuperficie <lb/>ſpeculi.</s> <s xml:id="echoid-s32699" xml:space="preserve"> Concurrit autem linea reflexionis protracta cum catheto incidentiæ.</s> <s xml:id="echoid-s32700" xml:space="preserve"> Quia enim linea re-<lb/>flexionis concurrit cum linea perpendiculari educta à puncto reflexionis ſuper ipſam ſpeculi ſu-<lb/>perficiem, ut patet ex pręmiſsis:</s> <s xml:id="echoid-s32701" xml:space="preserve"> ſed in ſpeculis planis illa perpendicularis æ quidiſtat catheto inci-<lb/>dentiæ per 6 p 11:</s> <s xml:id="echoid-s32702" xml:space="preserve"> ſunt enim ambæ ſuper ſpeculi ſuperficiẽ perpẽdiculares:</s> <s xml:id="echoid-s32703" xml:space="preserve"> manifeſtũ ergo per 2 th.</s> <s xml:id="echoid-s32704" xml:space="preserve"> 1 <lb/>huius, quia in illis ſpeculis linea reflexionis concurrit cum catheto incidentiæ.</s> <s xml:id="echoid-s32705" xml:space="preserve"> In alijs autẽ ſpecu-<lb/>lis eſt hoc magis manifeſtùm, quoniá in pluribus illis cathetus incidentiæ concurrit cũ perpendi-<lb/>culari ducta à puncto reflexionis ſuper ſuperficiẽ ſpeculi:</s> <s xml:id="echoid-s32706" xml:space="preserve"> de ſingulis tamen ſpeculis hoc in ſequen-<lb/>tibus demonſtratur:</s> <s xml:id="echoid-s32707" xml:space="preserve"> & in iſtarum linearum concurſu uidetur imago.</s> <s xml:id="echoid-s32708" xml:space="preserve"> Eſt ergo ibi locus imaginis, ut <lb/>proponebatur.</s> <s xml:id="echoid-s32709" xml:space="preserve"> Hoc aũt eſt neceſſe ideo, quia cum medium diſtantiæ inter punctũ uiſu comprehen.</s> <s xml:id="echoid-s32710" xml:space="preserve"> <lb/>ſum & ſpeculi ſuperficiem non ſit uacuum, fit reflexio formæ corporis med jad uiſum, ſicut & pun.</s> <s xml:id="echoid-s32711" xml:space="preserve"> <lb/>cti corporis, ad quod intendit uiſus:</s> <s xml:id="echoid-s32712" xml:space="preserve"> nec eſt differentia reflexionis formę corporis medij à reflexio <lb/>ne formæ puncti intenti, niſi ſicut alicuius formæ unius totius corporis continui, cuius ſolùm pars <lb/>modica intenditur uideri:</s> <s xml:id="echoid-s32713" xml:space="preserve"> ut ſi foramen acus intendatur uideri in ſpeculo & form a illius multipl ì-<lb/>cetur ad uiſum:</s> <s xml:id="echoid-s32714" xml:space="preserve"> nihilominus ordinaturin ſpeculo tota form a acus.</s> <s xml:id="echoid-s32715" xml:space="preserve"> Et quoniam formæ cadentes in <lb/>uiſibus & ſpeculis quibuſcunq;</s> <s xml:id="echoid-s32716" xml:space="preserve"> regularibus, retinẽt eſſentialem ordinẽ ſuarũ partiũ & figuartũ, ut <lb/>patet per 34 huius:</s> <s xml:id="echoid-s32717" xml:space="preserve"> ideo neceſſe eſt puncta formarũ incidentiũ ſpeculis quãdoq;</s> <s xml:id="echoid-s32718" xml:space="preserve"> in quadam diſtan-<lb/>tia uideri, ut quando diſtant pũcta rei extrà, & quando linea reflexionis & cathetus concuty ũt ſub <lb/>ſpeculi ſuperficie uel inter uiſum & ſpeculũ, & nõ in ipſa ſuperficie ſpeculi uel retro uiſum, in qui-<lb/>bus omnibus eſt eadẽ uniuerſalis cauſſa, quæ præmiſſa eſt, differens ſolùm ſecundũ uarios modos <lb/>reflexionum.</s> <s xml:id="echoid-s32719" xml:space="preserve"> Accidit enim rebus ſecundum quod formę ipſarũ diffund duntur per mediú ad ſuper-<lb/>ficiem ſpeculi, in formis ſuis ſpecificis differre, cũ ſenſibiliter non ferantur ad ſpeculum, niſi lux & <lb/>color & figura & ſimilia, quę non faciunt differẽtiam ſpecificá in rebus, ut in ligno & lapide.</s> <s xml:id="echoid-s32720" xml:space="preserve"> quam-<lb/>uis uirtus diſtin ctiua per accidẽtium cognitionẽ ſpecificam accipiat differẽtiam, ſcilicet per appli-<lb/>cationẽ illorũ accidentiũ ad propria ſubiecta, quę uiſibus directè uidentibus ſub talibus accidenti-<lb/>bus occurrunt.</s> <s xml:id="echoid-s32721" xml:space="preserve"> Sicut ergo unius corporis naturalis continui partium formæ feruntur ad ſpeculi ſu <lb/>perficiem, & ſeruata forma totali & figura, accidit neceſſariò partes remotiores à ſpeculſi ſuperficie <lb/>remotiores uideri, ne forma & figura rerum uiſarũ confundantur:</s> <s xml:id="echoid-s32722" xml:space="preserve"> ſic accidit neceſſariò de rebus ui <lb/>ſis per mediũ aerem, ut pręordinata forma aeris in ſitu ſuo, reſpectu formę rei per mediũ aerem ui-<lb/>ſæ, omnium ſuorũ pũctorũ formę uideantur:</s> <s xml:id="echoid-s32723" xml:space="preserve"> aliàs enim figura & forma rerũ multi plicatarũ ad ſpe-<lb/>culi ſuperficiẽ confunderentur.</s> <s xml:id="echoid-s32724" xml:space="preserve"> Et hæc mihi uiſa eſt eſſe cauſſa rei per alios multis ambagibus per-<lb/>quiſitę.</s> <s xml:id="echoid-s32725" xml:space="preserve"> Videtur itaq;</s> <s xml:id="echoid-s32726" xml:space="preserve"> res neceſſariò in perpendiculari, quoniam, ut patet per 21 th.</s> <s xml:id="echoid-s32727" xml:space="preserve"> 1 huius, hæc eſt <lb/>breuiſsima eius diſtantia à ſuperficie ſpeculi, à qua fit reflexio ad uiſum, aut à ſuperficie ei cõtimua:</s> <s xml:id="echoid-s32728" xml:space="preserve"> <lb/>& ſecũdum hanc ſit rei uiſæ, reſpectu ſpeculi, uniformis diſpoſitio:</s> <s xml:id="echoid-s32729" xml:space="preserve"> & ex hoc forma rei nomen acci-<lb/>pit imaginis, ut diximus in præ miſſa.</s> <s xml:id="echoid-s32730" xml:space="preserve"> Licet ergo forma rei ſecũdum aliam lineam reflectatur ad ui-<lb/>ſum:</s> <s xml:id="echoid-s32731" xml:space="preserve"> iudiciũ tamen uirtutis uiſiuę fit ſecũdum lineam breuiſsimam, ſecũdum quam incidit forma <lb/>uiſa ſuperficiei ipſius ſpeculi aut ei continuæ, propter conuenientẽ ordinationẽ formarũ in ſpecu-<lb/>li ſuperſicie & in uiſu, & propter certiorem cognitionem ſuæ proprię quantitatis.</s> <s xml:id="echoid-s32732" xml:space="preserve"> Cum enim neceſ <lb/>ſe ſit imaginẽ eſſe in linea reflexionis, ſi uideretur citra cathetum propinquior ad uiſum, uideretur <lb/>major:</s> <s xml:id="echoid-s32733" xml:space="preserve"> ſi ultra cathetum, uideretur minor, ut à remotiori uiſa:</s> <s xml:id="echoid-s32734" xml:space="preserve"> in catheto uerò quantũ permittit figu <lb/>ra ſpeculi & uiſus diſtantia, ſecundum ſui propriam quantitatem uidetur.</s> <s xml:id="echoid-s32735" xml:space="preserve"> Eſt ergo neceſſariũ ipſam <lb/>uideri in puncto concurſus lineæ reflexionis cũ catheto incidentiæ.</s> <s xml:id="echoid-s32736" xml:space="preserve"> Viſus enim cũ per reflexionẽ <lb/> <pb o="209" file="0511" n="511" rhead="LIBER QVINTVS."/> formas comprehendit, non animaaduertit quòd hęc comprehenſio fiat per reflexionẽ:</s> <s xml:id="echoid-s32737" xml:space="preserve"> quoniam re-<lb/>flexio, ut ſuprà in proœmio huius ſcientiæ diximus, non accidit ex proprietate uiſus:</s> <s xml:id="echoid-s32738" xml:space="preserve"> uiſu enim re-<lb/>moto, nihilominus fit reflexio à ſpeculis, quoniam forma corporalis non minus incidit ſuperficie-<lb/>bus ſpeculorum:</s> <s xml:id="echoid-s32739" xml:space="preserve"> ſed quoniam inuenit tranſeundi reſiſtentiam ex ſoliditate corporis ſpecularis, re-<lb/>flectitur ab illis:</s> <s xml:id="echoid-s32740" xml:space="preserve"> & ſi contingat uiſum eſſe in loco, in quo fit linearum reflexarum aggregatio, com-<lb/>prehendet uiſus, illas formas in capitibus illarũ:</s> <s xml:id="echoid-s32741" xml:space="preserve"> & eſt quælibet formarum reflexarũ à quo-<lb/>cunq;</s> <s xml:id="echoid-s32742" xml:space="preserve"> ſpeculo in illo ſpeculo tanquam non adueniens;</s> <s xml:id="echoid-s32743" xml:space="preserve"> ſed ac ſi naturalis eſſet forma ſpeculi:</s> <s xml:id="echoid-s32744" xml:space="preserve"> cum <lb/>tamen non ſit aliquid eſſentiæ ipſius ſpeculi.</s> <s xml:id="echoid-s32745" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s32746" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1337" type="section" level="0" n="0"> <head xml:id="echoid-head1049" xml:space="preserve" style="it">38. Formam omnis rei uiſe comprehenſe per reflexionem à ſuperficie alicuius ſpeculi: figuræ <lb/>ſuperficiei illius ſpeculi eſt neceſſarium aliqualiter aßimilari. Alhazen 37 n 6.</head> <p> <s xml:id="echoid-s32747" xml:space="preserve">Quoniam enim, ut patet per præmiſſam, locus imaginis cuiuſcunq;</s> <s xml:id="echoid-s32748" xml:space="preserve"> puncti formę uiſæ eſt in con <lb/>curſu lineæ reflexionis cum catheto incidentię:</s> <s xml:id="echoid-s32749" xml:space="preserve"> harum aũt linearũ concurſus diuerſificatur ſecũdũ <lb/>figurã ſuperficierũ ſpeculorũ, à quibus fit reflexio:</s> <s xml:id="echoid-s32750" xml:space="preserve"> quoniã ſecũdũ illius figurę diſpoſitionẽ fit diuer <lb/>ſitas concurſus catheti incidẽtiæ & perpẽdicularis ductæ à pũcto formę incidẽtis ſuper ſuperficiẽ <lb/>ſpeculi, uel ſuper ſuperficiẽ contingentẽ ſpeculũ in pũcto reflexionis ſuperficiei ſpeculi, à qua fit re <lb/>flexio ad uiſum:</s> <s xml:id="echoid-s32751" xml:space="preserve"> quarũ perpendiculariũ cõcurſus diuerſificat concurſum linearũ reflexionis cũ ca-<lb/>theto incidẽtiæ, in quo cõcurſu eſt locus imaginũ, ut declaratũ eſt in præmiſſa.</s> <s xml:id="echoid-s32752" xml:space="preserve"> Habet itaq;</s> <s xml:id="echoid-s32753" xml:space="preserve"> ſuperfi-<lb/>cies ſpeculi, à qua fit reflexio, aliquã dignitatẽ in formatione imaginũ uiſarũ, quę ab ipſis reflectun-<lb/>tur:</s> <s xml:id="echoid-s32754" xml:space="preserve"> non tamen fit ſemper hęc aſsimilatio ſecũdũ totã diſpoſitionẽ formarũ, niſi cũ loca imaginũ ca <lb/>dũt in ipſis ſuperficiebus ſpeculorũ non intra ſpecula uel extra ipſa:</s> <s xml:id="echoid-s32755" xml:space="preserve"> ſed & tũc ſecũdũ aliquid aſsi-<lb/>milantiur formæ uiſæ ipſis formis uel figuris ſpeculorũ:</s> <s xml:id="echoid-s32756" xml:space="preserve"> quoniã in ſpeculis pyramidalibus apparẽt <lb/>formæ aliqualiter pyramidales:</s> <s xml:id="echoid-s32757" xml:space="preserve"> & ſic aliqualiter accidit in alijs ſpeculis.</s> <s xml:id="echoid-s32758" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s32759" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1338" type="section" level="0" n="0"> <head xml:id="echoid-head1050" xml:space="preserve" style="it">39. Diuiſa cuiuſcun ſpeculi ſuperficie, accidit formam unius puncti rei uiſæ numero illarũ <lb/>partium numer ari.</head> <p> <s xml:id="echoid-s32760" xml:space="preserve">Hoc, quòd hic proponitur, uerum eſt, quando per diuiſionẽ ſuperficiei alicuius ſpeculi ſenſibilis <lb/>accidit diuerſitas ordinis & ſitus partialium ſuperficierũ & in ſe, & reſpectu ipſius uiſus:</s> <s xml:id="echoid-s32761" xml:space="preserve"> ut plurimũ <lb/>accidit in ſpeculis uitreis plumbatis, quę per diuiſionẽ ab unitate ſuperficiei defacili recedũt:</s> <s xml:id="echoid-s32762" xml:space="preserve"> quod <lb/>non accidit in alijs ſpeculis tam faciliter.</s> <s xml:id="echoid-s32763" xml:space="preserve"> Quãdo itaq;</s> <s xml:id="echoid-s32764" xml:space="preserve"> aliorũ ſpeculorũ ſuperficies propter diuiſio-<lb/>nẽ in ipſis factam ab unitate ſuperficiei ſecũdum ſitũ & ordinẽ pręmiſſo modo recedũt:</s> <s xml:id="echoid-s32765" xml:space="preserve"> accidit for-<lb/>má unius pũcti rei uiſę numero illarũ partium numerari.</s> <s xml:id="echoid-s32766" xml:space="preserve"> Tũc enim diuerſę fiunt catheti incidentię <lb/>formæ eiuſdem pũcti rei uiſæ, reſpectu illarũ diuerſarum partialiũ ſuperficierũ, & ſimiliter diuerſa <lb/>fiunt pũcta reflexionũ & diuerſæ reflexionũ lineæ ad centrũ eiuſdẽ uiſus.</s> <s xml:id="echoid-s32767" xml:space="preserve"> Et quia locus cuiuslibet <lb/>imaginis ſemper fit in pũcto cócurſus lineæ reflexionis cum catheto incidentiæ, ut patet per 37 hu <lb/>ius:</s> <s xml:id="echoid-s32768" xml:space="preserve"> ideo patet quòd ſecundum numerum iſtarum linearum, & ſui concurſus formæ eiuſdem pun-<lb/>cti imagines numerantur.</s> <s xml:id="echoid-s32769" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s32770" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1339" type="section" level="0" n="0"> <head xml:id="echoid-head1051" xml:space="preserve" style="it">40. In omnis ſpeculi ſuperficie fit formarum reflexio in longitudine & latitudine ſecundum <lb/>modum polituræ.</head> <p> <s xml:id="echoid-s32771" xml:space="preserve">Quod hic proponitur, exemplariter patet in ſpeculis quibuſcunq;</s> <s xml:id="echoid-s32772" xml:space="preserve"> artificio politis.</s> <s xml:id="echoid-s32773" xml:space="preserve"> Si enim for-<lb/>biantur in longũ, ut accidit in ſuperficiebus enſium:</s> <s xml:id="echoid-s32774" xml:space="preserve"> tũc facies intuentis uidebitur oblong a reſpe-<lb/>ctu ſuæ proprię diſpoſitionis:</s> <s xml:id="echoid-s32775" xml:space="preserve"> & ſi forbiantur aliquę ſuperficies ſecũdũ ipſarũ latitudinẽ:</s> <s xml:id="echoid-s32776" xml:space="preserve"> tũc imago <lb/>faciei illa intuentis uidebitur latior quàm ſit eius proprietas uera ſecundũ illam diſpoſitionem:</s> <s xml:id="echoid-s32777" xml:space="preserve"> & <lb/>quandoq;</s> <s xml:id="echoid-s32778" xml:space="preserve"> uidebitur imago tranſuerſalis propter tranſuerſalitatẽ forbitionis.</s> <s xml:id="echoid-s32779" xml:space="preserve"> In omnibus uerò his <lb/>cauſſa eſt unitio maior ſuperficierũ ipſarũ corporũ politorũ, à quib.</s> <s xml:id="echoid-s32780" xml:space="preserve"> & â quarũ partibus cófluitre-<lb/>flexio ad unionẽ formarũ reflexarũ, quæ ſecundum illud perueniunt ad uiſum.</s> <s xml:id="echoid-s32781" xml:space="preserve"> Etenim, ut in prin-<lb/>cipijs huius libri:</s> <s xml:id="echoid-s32782" xml:space="preserve"> 1.</s> <s xml:id="echoid-s32783" xml:space="preserve"> definitióe diximus, politio eſt cõtinuitas partium ſuperficiei politi corporis ſi-<lb/>ne ſenſibilitate pororũ uel diuiſionis:</s> <s xml:id="echoid-s32784" xml:space="preserve"> unde cũ ad aliquã differentiá poſitionis illi pori complanan-<lb/>tur:</s> <s xml:id="echoid-s32785" xml:space="preserve">neceſſe eſt ſecundũ illá differentiá formas pluribus punctis illis incidentes in unitatem formæ <lb/>cõfluere & uniri, & ſecũdum illũ modũ formam uiſam ſecũdum reflexionẽ augmẽtari & uideri ma <lb/>iorem:</s> <s xml:id="echoid-s32786" xml:space="preserve"> ſecũdum alias uerò poſitionũ differentias neceſſe eſt ipſam uideri ſuę diſpoſitionis proprię, <lb/>uel circa illá.</s> <s xml:id="echoid-s32787" xml:space="preserve">Et ſic accidit quædá monſtruoſitas in imaginibus formarũ taliter uiſarũ:</s> <s xml:id="echoid-s32788" xml:space="preserve"> quia ipſarum <lb/>reflexio eſt inæqualis hinc inde:</s> <s xml:id="echoid-s32789" xml:space="preserve"> & fit irregularis ſecũdum illud.</s> <s xml:id="echoid-s32790" xml:space="preserve"> Vt itaq;</s> <s xml:id="echoid-s32791" xml:space="preserve"> à corporibus arte politis re <lb/>flexio fiat regularis & conueniens diſpoſitioni formarũ reflexarũ:</s> <s xml:id="echoid-s32792" xml:space="preserve"> neceſſe eſt ipſorũ ſuperficies for <lb/>biari ſecundum modum circularẽ non in longum nec in latum uel tranſuerſum, ne ſecundum illos <lb/>modos formarum propria diſpoſitio difformetur.</s> <s xml:id="echoid-s32793" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s32794" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1340" type="section" level="0" n="0"> <head xml:id="echoid-head1052" xml:space="preserve" style="it">41. In omni ſpeculo accidit eandem imaginem à duobus uiſibus quando uideriduas.</head> <p> <s xml:id="echoid-s32795" xml:space="preserve">Huius rei euétus accidit uiſui in unius imaginis uiſione à quocũq;</s> <s xml:id="echoid-s32796" xml:space="preserve"> ſpeculorũ reflexæ, ſicut & idẽ <lb/>error ſibi accidit in ſimplici rerũ uiſione, cũ e ædẽ cauſſę concurrũt uel illarũ aliqua, quas declaraui <lb/>mus in 103,104,105,106 & 107 th.</s> <s xml:id="echoid-s32797" xml:space="preserve">4 huius:</s> <s xml:id="echoid-s32798" xml:space="preserve"> utpote cũ eiuſdẽ rei forma ab e o dẽ ſpeculo reflexa uni ui <lb/>fuũ offertur directè, & alteri obliquè:</s> <s xml:id="echoid-s32799" xml:space="preserve"> uel cũ forma reflexa cõſtituta intra axes radiales ambob.</s> <s xml:id="echoid-s32800" xml:space="preserve"> uiſi-<lb/>bus occurrit obliquè.</s> <s xml:id="echoid-s32801" xml:space="preserve"> Quibuſcunq;</s> <s xml:id="echoid-s32802" xml:space="preserve"> enim modis accidit formam eiuſdem rei uideri duas, eiſdé mo <lb/>dis poſsibile eſt imaginem illius formæ uideri duas, ſi ſecundum modum ſuæ uiſionis ad uiſum ab <lb/> <anchor type="figure" xlink:label="fig-0511-01a" xlink:href="fig-0511-01"/> <pb o="210" file="0512" n="512" rhead="VITELLONIS OPTICAE"/> aliquo ſpeculo reflectatur.</s> <s xml:id="echoid-s32803" xml:space="preserve"> Et propterea talibus nõ oportet aliter immorari, quàm ut in ſimpliciui-<lb/>ſione dictum eſt:</s> <s xml:id="echoid-s32804" xml:space="preserve"> non enim accidit illud propter diuerſitatem punctorum reflexionis formæ eiuſdẽ <lb/>puncti ad ambos uiſus:</s> <s xml:id="echoid-s32805" xml:space="preserve"> quoniã illa diuerſitas aut nulla eſt, aut non eſt ſenſibilis:</s> <s xml:id="echoid-s32806" xml:space="preserve"> unde nullum ſenſi-<lb/>bilem inducit uiſibus errorem, ſed ambo uiſus ſecundum illum bene perueniunt ad uiſionem uni-<lb/>tatis eiuſdem formę, ut poſterius declarabitur:</s> <s xml:id="echoid-s32807" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s32808" xml:space="preserve"/> </p> <div xml:id="echoid-div1340" type="float" level="0" n="0"> <figure xlink:label="fig-0511-01" xlink:href="fig-0511-01a"> <image file="0511-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0511-01"/> </figure> </div> </div> <div xml:id="echoid-div1342" type="section" level="0" n="0"> <head xml:id="echoid-head1053" xml:space="preserve" style="it">42. Imago rei uiſæ motæ in omni ſpeculo moueri uidetur.</head> <p> <s xml:id="echoid-s32809" xml:space="preserve">Huius cauſſa non eſt alia, niſi uniformitas reflexionis à quolibet puncto ſpeculi, ſuper quod fit <lb/>motus.</s> <s xml:id="echoid-s32810" xml:space="preserve"> Et quia omnia puncta rei uiſæ à diuerſis, quàm prius, punctis reflectũtur, efficitur noua ima <lb/>go totius rei uiſæ, ſecundum quod per eius motum puncta, à quibus facta eſt reflexio, permutátur.</s> <s xml:id="echoid-s32811" xml:space="preserve"> <lb/>Videtur itaq;</s> <s xml:id="echoid-s32812" xml:space="preserve"> forma moueri, licet ſecũdum ueritatẽ nõ moueatur, ſed potius noua imago mutato <lb/>ſitu rei uiſæ generetur.</s> <s xml:id="echoid-s32813" xml:space="preserve"> Hoc aũt accidit propter continuitatem punctorum reflexionis in ſuperficie <lb/>ſpeculorum.</s> <s xml:id="echoid-s32814" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s32815" xml:space="preserve"> His itaq;</s> <s xml:id="echoid-s32816" xml:space="preserve"> communibus omnium ſpeculorum paſsionibus prę-<lb/>miſsis:</s> <s xml:id="echoid-s32817" xml:space="preserve"> reſtat ut ad planorum ſpeculorum paſsiones proprias calamum conuertamus.</s> <s xml:id="echoid-s32818" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1343" type="section" level="0" n="0"> <head xml:id="echoid-head1054" xml:space="preserve" style="it">43. In omni reflexione à ſpeculis planis facta, lineæ incidentiæ & reflexionis proportionales <lb/>ſunt cathetis à punctis ſuorum terminorum demißis, & ipſis baſibus in ſpeculorũ ſuperficie in-<lb/>teriectis. Euclides 3 hypotheſi catoptr.</head> <p> <s xml:id="echoid-s32819" xml:space="preserve">Sit ſpeculum planũ, in cuius ſuperficie ſit linea d c e:</s> <s xml:id="echoid-s32820" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0512-01a" xlink:href="fig-0512-01"/> & ſit linea incidentiæ a c:</s> <s xml:id="echoid-s32821" xml:space="preserve"> reflexionis uerò c b:</s> <s xml:id="echoid-s32822" xml:space="preserve"> & ducan <lb/>tur catheti a d incidẽtiæ & reflexionis b e.</s> <s xml:id="echoid-s32823" xml:space="preserve"> Dico quòd <lb/>quæ eſt proportio a d ad b e, eadẽ eſt a c ad b c & d c ad <lb/>c e.</s> <s xml:id="echoid-s32824" xml:space="preserve"> Quoniá enim in trigono a d c angulus rectus, quia <lb/>d c, eſt æ qualis angulo, qui b e c, recto:</s> <s xml:id="echoid-s32825" xml:space="preserve"> & angulus a c d, <lb/>qui eſt angulus incidentiæ, eſt per 20 huius ęqualis an-<lb/>gulo b c e, qui eſt angulus reflexiõis:</s> <s xml:id="echoid-s32826" xml:space="preserve"> erit neceſſariò an-<lb/>gulus d a c trigoni a d c æ qualis angulo c b e trigoni b e <lb/>c per 32 p 1:</s> <s xml:id="echoid-s32827" xml:space="preserve"> ergo per 4 p 6 latera iſtorũ trigonorũ ęqua-<lb/>les angulos reſpicientia ſunt proportionalia:</s> <s xml:id="echoid-s32828" xml:space="preserve"> quæ eſt <lb/>ergo proportio lineæ a d ad lineam b e, eadẽ eſt proportio lineę d c ad e c.</s> <s xml:id="echoid-s32829" xml:space="preserve"> Et quo-<lb/>niam ſemper manet eadem proportio reſultans ex æqualitate angulorum:</s> <s xml:id="echoid-s32830" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s32831" xml:space="preserve"/> </p> <div xml:id="echoid-div1343" type="float" level="0" n="0"> <figure xlink:label="fig-0512-01" xlink:href="fig-0512-01a"> <variables xml:id="echoid-variables553" xml:space="preserve">a b d c e</variables> </figure> </div> </div> <div xml:id="echoid-div1345" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables554" xml:space="preserve">a f b d e</variables> </figure> <head xml:id="echoid-head1055" xml:space="preserve" style="it">44. Forma punctirei uiſæ ſuperſiciei plani ſpeculi incidente: lo <lb/> cum, in quo uiſu conſtituto, ad ipſum fiat reflexio, inuenire.</head> <p> <s xml:id="echoid-s32832" xml:space="preserve">Eſto punctus, cuius forma ſpeculo plano incidat a:</s> <s xml:id="echoid-s32833" xml:space="preserve"> & ſit linea b c d <lb/>communis ſectio ſuperficiei reflexionis & ſpeculi ducta in ſuperficie <lb/>ſpeculi:</s> <s xml:id="echoid-s32834" xml:space="preserve"> incidatq́;</s> <s xml:id="echoid-s32835" xml:space="preserve"> punctus a ſpeculo ſecundum punctum c:</s> <s xml:id="echoid-s32836" xml:space="preserve"> & duca-<lb/>tur linea incidentiæ, quæ a c:</s> <s xml:id="echoid-s32837" xml:space="preserve"> & à puncto a ducatur linea a b perpen-<lb/>dicularis ſuper lineam b c d per 12 p 1:</s> <s xml:id="echoid-s32838" xml:space="preserve"> & producatur uſque ad pun-<lb/>ctum e, donec per 3 p 1 linea b e fiat æ qualis ipſi a b:</s> <s xml:id="echoid-s32839" xml:space="preserve"> & continuetur li-<lb/>nea e c:</s> <s xml:id="echoid-s32840" xml:space="preserve"> quæ ꝓdacatur ultra c ad punctum f.</s> <s xml:id="echoid-s32841" xml:space="preserve"> Dico quòd uiſu exiſtẽte <lb/>in quocũq;</s> <s xml:id="echoid-s32842" xml:space="preserve"> puncto lineæ c f, ſemper fiet reflexio ad ipſum, & uidebit <lb/>formá pũctia.</s> <s xml:id="echoid-s32843" xml:space="preserve"> Copuletur enim linea a c:</s> <s xml:id="echoid-s32844" xml:space="preserve"> eritq́ue angulus a b c æqua <lb/>lis angulo c b e, quia, ut patet ex pręmiſsis, ambo illi anguli ſunt recti.</s> <s xml:id="echoid-s32845" xml:space="preserve"> <lb/>Quoniam ergo per 4 p 1 cũ ex hypotheſi linea b e ſit æ qualis ipſi a b, <lb/>& latus b c cómune, trigona a b c & c b e ſint æquiangula:</s> <s xml:id="echoid-s32846" xml:space="preserve"> erit angu-<lb/>lus a c b æ qualis angulo b c e:</s> <s xml:id="echoid-s32847" xml:space="preserve"> ſed per 15 p 1 angulus f c d eſt æqualis <lb/>angulo b c e:</s> <s xml:id="echoid-s32848" xml:space="preserve"> ergo angulus f c d eſt æqualis angulo a c b:</s> <s xml:id="echoid-s32849" xml:space="preserve"> ergo per 20 <lb/>huius, cum linea a c ſit linea incidentiæ, erit c f linea reflexionis.</s> <s xml:id="echoid-s32850" xml:space="preserve"> Vifu <lb/>ergo in illa poſito, fiet reflexio ad uiſum.</s> <s xml:id="echoid-s32851" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s32852" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1346" type="section" level="0" n="0"> <head xml:id="echoid-head1056" xml:space="preserve" style="it">45. Forma puncti à ſpeculo plano non reflectitur ad eundem ui-<lb/>ſum, niſi ab uno puncto tantùm. Alhazen 14 n 5.</head> <p> <s xml:id="echoid-s32853" xml:space="preserve">Eſto centrum uiſus a:</s> <s xml:id="echoid-s32854" xml:space="preserve"> & punctum uiſum b:</s> <s xml:id="echoid-s32855" xml:space="preserve"> & ſit z h ſuperficies ſpeculi plani.</s> <s xml:id="echoid-s32856" xml:space="preserve"> Dico quòd ab uno <lb/>tantùm puncto ſuperficiei z h reflectitur forma puncti b ad uiſum a.</s> <s xml:id="echoid-s32857" xml:space="preserve"> Si enim à duobus pũctis ſit poſ <lb/>ſibile illá reflecti, ſint illa duo pũcta d & e:</s> <s xml:id="echoid-s32858" xml:space="preserve"> & ducatur linea à centro uiſus puncto a ad punctũ uiſum <lb/>b:</s> <s xml:id="echoid-s32859" xml:space="preserve"> quę ſit a b.</s> <s xml:id="echoid-s32860" xml:space="preserve">Linea itaq;</s> <s xml:id="echoid-s32861" xml:space="preserve"> a b ꝓtracta ultra alterũ punctorũ, quę ſunt b uel a, aut concurrit cũ ſuperfi-<lb/>cie ſpeculi, aut æquidiſtat.</s> <s xml:id="echoid-s32862" xml:space="preserve"> Si cõcurrit ſiue ſit քpẽdicularis ſuper ſuperficiẽ ſpeculi, à quo fit reflexio, <lb/>ſiue non, ſemper ipſa erit neceſſariò in una ſola ſuperficie reflexionis.</s> <s xml:id="echoid-s32863" xml:space="preserve"> Si enim ipſa ſit perpendicula-<lb/>ris ſuper ſuperficiẽ ſpeculi:</s> <s xml:id="echoid-s32864" xml:space="preserve"> tunc patet quòd ipſa eſt in una ſuperficie reflexionis per 27 huius:</s> <s xml:id="echoid-s32865" xml:space="preserve"> quo-<lb/>niam ipſa reflectitur in ſe ipſam per 21 huius.</s> <s xml:id="echoid-s32866" xml:space="preserve"> Si uerò linea a b ſuper ſuperficiẽ ſpeculi non ſit perpen <lb/>dicularis, cum ſit linea recta extenſa inter duo puncta extrema, quę am bo per 25 huius neceſſariò <lb/>ſunt in una ſuperficie reflexionis erecta ſuper ſuperficiẽ ſpeculi, erit etiam linea a b in una ſola tali <lb/>ſuperficie:</s> <s xml:id="echoid-s32867" xml:space="preserve"> quoniam ſi in duabus talibus ſuperficiebus fuerit, tunc ipſa erit communis ſectio dua-<lb/>bus illis ſuperficiebus orthogonalibus ſuper ſuperficiem ſpeculi per 19 th.</s> <s xml:id="echoid-s32868" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s32869" xml:space="preserve"> unde ſumpto in <lb/> <anchor type="figure" xlink:label="fig-0512-03a" xlink:href="fig-0512-03"/> <pb o="211" file="0513" n="513" rhead="LIBER QVINTVS."/> ea puncto & ducta ab illo puncto linea in altera ſuperficierum ſuper lineam cómmunem huic ſuperfi <lb/>ciei & ſuperficiei ſpeculi, erit hęc linea erecta ſuper ſuperficiem ſpeculi per definitionem ſuperficiei <lb/>ſuper ſuperficiem erectæ:</s> <s xml:id="echoid-s32870" xml:space="preserve"> & ſimiliter ab eodem pun-<lb/>cto ducatur linea in alia ſuperficie ſuper lineam com <lb/> <anchor type="figure" xlink:label="fig-0513-01a" xlink:href="fig-0513-01"/> munem ei & ſuperficiei ſpeculi, & erit iterum hæc li-<lb/>nea orthogonalis ſuper ſuperficiem ſpeculi.</s> <s xml:id="echoid-s32871" xml:space="preserve"> Ab eo-<lb/>dem ergo puncto contingeret ducere duas perpen <lb/>diculares ſuper eãdem ſuperficiem ſpeculi:</s> <s xml:id="echoid-s32872" xml:space="preserve"> q uod eſt <lb/>impoſsibile & contra 20 th.</s> <s xml:id="echoid-s32873" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s32874" xml:space="preserve"> Ergo li nea b a e-<lb/>ritin una ſola ſuperficie reflexionis, erecta ſuper ſu-<lb/>perficiem ſpeculi plani:</s> <s xml:id="echoid-s32875" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s32876" xml:space="preserve"> tria puncta a, e, b in <lb/>eadem ſuperficie reflexion is per 1 p 11:</s> <s xml:id="echoid-s32877" xml:space="preserve"> & erunt lineę <lb/>a e & e d & e b per 25 huius in illa ſuperficie reflex io <lb/>nis, in qua eſt linea a b:</s> <s xml:id="echoid-s32878" xml:space="preserve"> & ſimiliter lineę e d & d b & <lb/>d a.</s> <s xml:id="echoid-s32879" xml:space="preserve"> Quare lineę e a & e b erunt in eadem ſuperficie <lb/>cum lineis d a & d b per 2 p 11.</s> <s xml:id="echoid-s32880" xml:space="preserve"> Sed angulus a e h eſt <lb/>maior angulo a d e per 16 p 1, extrinſecus enim eſt-<lb/>maior intrinſeco:</s> <s xml:id="echoid-s32881" xml:space="preserve"> ſed per 20 huius angulus inciden-<lb/>tiæ, qui eſt a e h, eſt æ qualis angulo reflexionis, qui <lb/>eſt b e d:</s> <s xml:id="echoid-s32882" xml:space="preserve"> & angulus a d e eſt æ qualis angulo b d z:</s> <s xml:id="echoid-s32883" xml:space="preserve"> an <lb/>gulus ergo d e b maior eſt angulo a d e:</s> <s xml:id="echoid-s32884" xml:space="preserve"> ergo & ipſius ęquali, ſcilicet angulo b d z, quod eſt contra 16 <lb/>p 1, extrinſecus enim, qui eſt b d z, maior eſt intrιnſeco, qui eſt b e d:</s> <s xml:id="echoid-s32885" xml:space="preserve"> ergo & angulus a d h maior eſt <lb/>angulo b e d:</s> <s xml:id="echoid-s32886" xml:space="preserve"> & ſic idem angulus eodem angulo erit maior & minor, quod eſt impoſsibile.</s> <s xml:id="echoid-s32887" xml:space="preserve"> A ſolo er <lb/>go puncto ſpeculi plani fit reflexio formæ puncti b ad uiſum a.</s> <s xml:id="echoid-s32888" xml:space="preserve"> Si uero linea a b ſit perpendicularis <lb/>ſuper ſuperficiem ſpeculi plani, patet per 32 huius, quòd unus tantùm punctus reflectitur ſecundú <lb/>ipſam ad uiſum, & ab uno ſolo ſpeculi puncto.</s> <s xml:id="echoid-s32889" xml:space="preserve"> Quòd ſi linea a b non concurrat cum aliqua linearũ <lb/>protractarum in ſuperficie ſpeculi, ſed ſit ęquidiftans alicui illarum:</s> <s xml:id="echoid-s32890" xml:space="preserve"> ergo per 9 p 11 ipſa erit ęquidi-<lb/>ſtans cuilibet æquidiſtanti illi lineę in ſpeculi ſuperficie productę.</s> <s xml:id="echoid-s32891" xml:space="preserve"> Sit ergo ęquidiſtans lineę h z:</s> <s xml:id="echoid-s32892" xml:space="preserve"> e-<lb/>runt quoq;</s> <s xml:id="echoid-s32893" xml:space="preserve"> per 1 th.</s> <s xml:id="echoid-s32894" xml:space="preserve"> 1 huius lineę a b & h z in eadem ſuperficie:</s> <s xml:id="echoid-s32895" xml:space="preserve"> fiat ergo deductio, ut prius, quoniam <lb/>intrinſecus angulus erit maior extrinſeco:</s> <s xml:id="echoid-s32896" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s32897" xml:space="preserve"> Ergo & illud, ex quo ſequebatur.</s> <s xml:id="echoid-s32898" xml:space="preserve"> <lb/>Patert ergo, quod proponebatur.</s> <s xml:id="echoid-s32899" xml:space="preserve"/> </p> <div xml:id="echoid-div1346" type="float" level="0" n="0"> <figure xlink:label="fig-0512-03" xlink:href="fig-0512-03a"> <image file="0512-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0512-03"/> </figure> <figure xlink:label="fig-0513-01" xlink:href="fig-0513-01a"> <variables xml:id="echoid-variables555" xml:space="preserve">a b h e d z</variables> </figure> </div> </div> <div xml:id="echoid-div1348" type="section" level="0" n="0"> <head xml:id="echoid-head1057" xml:space="preserve" style="it">46. In ſpeculis planis dati puncti uiſi ad cẽtrum uiſus datum, punctum reſlexionis inuenire. <lb/>Alhazen 12 n 5.</head> <p> <s xml:id="echoid-s32900" xml:space="preserve">Sit ſpeculum planum, in cuius ſuperficie ſit linea a g:</s> <s xml:id="echoid-s32901" xml:space="preserve"> & ſit centrum uiſus b:</s> <s xml:id="echoid-s32902" xml:space="preserve"> punctusq́;</s> <s xml:id="echoid-s32903" xml:space="preserve"> rei uiſæ ſit <lb/>d:</s> <s xml:id="echoid-s32904" xml:space="preserve"> & ducantur catheti a d & g b perpendiculariter ſuper <lb/>ſuperficiem ſpeculi per 11 p 11:</s> <s xml:id="echoid-s32905" xml:space="preserve"> diuidaturq́;</s> <s xml:id="echoid-s32906" xml:space="preserve"> linea a g in pũ <lb/> <anchor type="figure" xlink:label="fig-0513-02a" xlink:href="fig-0513-02"/> cto h, ita ut ſit proportio lineæ a h ad lineam h g, ſicut li-<lb/>neæ a d ad lineam g b per 119 th.</s> <s xml:id="echoid-s32907" xml:space="preserve"> I huius.</s> <s xml:id="echoid-s32908" xml:space="preserve"> Dico itaq;</s> <s xml:id="echoid-s32909" xml:space="preserve"> quòd <lb/>forma puncti d reflectetur ad uiſum b à pũcto ſpeculi h.</s> <s xml:id="echoid-s32910" xml:space="preserve"> <lb/>Ducãtur enim lineę d h & b h.</s> <s xml:id="echoid-s32911" xml:space="preserve"> Palã itaq;</s> <s xml:id="echoid-s32912" xml:space="preserve"> p 6 p 6 & ex hy-<lb/>potheſi quoniam triangulus d h a eſt æ quiangulus trian <lb/>gulo h g b:</s> <s xml:id="echoid-s32913" xml:space="preserve"> angulus enim h a d eſt æqualis angulo h g b, <lb/>quia ſunt amborecti, & eſt proportio lineę a d ad lineá <lb/>g b, ſicut lineę a h ad lineam h g:</s> <s xml:id="echoid-s32914" xml:space="preserve">angulus itaque a h d eſt <lb/>æqualis angulo g h b.</s> <s xml:id="echoid-s32915" xml:space="preserve"> A puncto itaq;</s> <s xml:id="echoid-s32916" xml:space="preserve"> ſpeculi, quod eſt <lb/>h, refle ctetur forma puncti d ad uiſum b per 20 huius:</s> <s xml:id="echoid-s32917" xml:space="preserve"> angulus enim incidentiæ eſt æqualis angulo <lb/>reflexionis, Si autem punctus h obſtruatur per alιquod ſuperpoſitum, utpote per ceram uel per pi-<lb/>cem aut ſimile:</s> <s xml:id="echoid-s32918" xml:space="preserve"> nulla uidebitur imago puncti d, centro ipſius uiſus, quod eſt b, diſpoſito ſecundum <lb/>præmiſſum modum:</s> <s xml:id="echoid-s32919" xml:space="preserve"> quoniam à puncto alio impoſsibile eſt fieri reflexionem per præmiſſam:</s> <s xml:id="echoid-s32920" xml:space="preserve"> acci-<lb/>dit enim à puncto alio uariari proportionem, & angulos incidentiæ & reflexionis fieriinæ quales.</s> <s xml:id="echoid-s32921" xml:space="preserve"> <lb/>Patet ergo propoſitum.</s> <s xml:id="echoid-s32922" xml:space="preserve"/> </p> <div xml:id="echoid-div1348" type="float" level="0" n="0"> <figure xlink:label="fig-0513-02" xlink:href="fig-0513-02a"> <variables xml:id="echoid-variables556" xml:space="preserve">d b a h s</variables> </figure> </div> </div> <div xml:id="echoid-div1350" type="section" level="0" n="0"> <head xml:id="echoid-head1058" xml:space="preserve" style="it">47. Lineæ reflexionis formæ eiuſàem puncti à diuerſis punctis ſpeculi plani non ſunt æquidi-<lb/>ſtantes: attamen in centro unius uiſus non concurrunt. Ex quo patet quòd unus uiſus uidere nõ <lb/>poteſt idolum eiuſdem formæ à diuerſis punctis eiuſdem plani ſpeculi reflexum. Euclides 4 the. <lb/>catoptr. Ptolemæus 7 th. 1 catoptr.</head> <p> <s xml:id="echoid-s32923" xml:space="preserve">Eſto ſpeculum planum, in cuius ſuperficie ſit linea a b c d:</s> <s xml:id="echoid-s32924" xml:space="preserve"> cuius duobus punctis b & c à puncto <lb/>rei uiſæ, quod ſite, incidant lineę e b & e c:</s> <s xml:id="echoid-s32925" xml:space="preserve"> & ſit centrum uiſus g:</s> <s xml:id="echoid-s32926" xml:space="preserve"> & reflectatur linea e b ſecundum li <lb/>neam b f, & linea e c ſecundum lineam c g.</s> <s xml:id="echoid-s32927" xml:space="preserve"> Dico quòd lineæ c g & b ſ non ſunt æ quidiſtantes, nec tñ <lb/>concurrent in centro unius uiſus, quãuis etiã ſint in eadẽ ſuperficie:</s> <s xml:id="echoid-s32928" xml:space="preserve"> angulus enim incidétię, qui eſt <lb/>e c d, eſt ęqualis angulo reflexiõis, qui eſt g c a:</s> <s xml:id="echoid-s32929" xml:space="preserve"> & angulus c b d eſt æqualis angulo f b a, utpater per <lb/>20 huius, Quia ergo trigoni e b c latus b c protrahitur ad punctum d:</s> <s xml:id="echoid-s32930" xml:space="preserve"> erit per 16 p 1 angulus e c d ex-<lb/>trinſecus maior angulo intrinſeco, qui eſt e b d:</s> <s xml:id="echoid-s32931" xml:space="preserve"> palá ergo per 20 huius quia & angulus g c a maior <lb/> <pb o="212" file="0514" n="514" rhead="VITELLONIS OPTICAE"/> eſt angulo f b a:</s> <s xml:id="echoid-s32932" xml:space="preserve"> ergo per 14th.</s> <s xml:id="echoid-s32933" xml:space="preserve"> 1 huius lineæ g c & b f non ſunt æquidiſtantes:</s> <s xml:id="echoid-s32934" xml:space="preserve"> angulus enim extrin-<lb/>ſecus maior eſt intrinſeco cadente linea a d ſuper ambas <lb/>lineas g c & b f:</s> <s xml:id="echoid-s32935" xml:space="preserve"> ſed neq;</s> <s xml:id="echoid-s32936" xml:space="preserve"> concurrentin centro unius ui-<lb/>ſus.</s> <s xml:id="echoid-s32937" xml:space="preserve"> Dato enim quòd concurrant in centro uiſus, quod <lb/> <anchor type="figure" xlink:label="fig-0514-01a" xlink:href="fig-0514-01"/> ſit f, & linea e c reflectatur ad uiſum ſ ſecundum lineam <lb/>c f:</s> <s xml:id="echoid-s32938" xml:space="preserve"> tunc quia per 20 huius angulus incidentię, qui eſt f b <lb/>a, æqualis eſt angulo reflexionis, qui eſt e b d, & angulus <lb/>e c d æ qualis angulo b c f:</s> <s xml:id="echoid-s32939" xml:space="preserve"> ſed angulus fb a maior eſt an-<lb/>gulo f c b per 16 p 1:</s> <s xml:id="echoid-s32940" xml:space="preserve"> ergo & angulus e b c intrinſecus ma-<lb/>ior eſt angulo e c d extrinſeco:</s> <s xml:id="echoid-s32941" xml:space="preserve"> quod eſt contra ean-<lb/>dem 16 p 1, & impoſsibile.</s> <s xml:id="echoid-s32942" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s32943" xml:space="preserve"> Et ex <lb/>hoc patet planè totum corollarium.</s> <s xml:id="echoid-s32944" xml:space="preserve"> Si enim lineæ refle-<lb/>xionis formæ eiuſdem puncti non poſſunt in centro unius uiſus concurrere:</s> <s xml:id="echoid-s32945" xml:space="preserve"> tunc eſt manifeſtum, <lb/>quòd unus uiſus non poteſtidolum eiuſdem formæ uidere reflexum à diuerſis punctis ſuperficiei <lb/>eiuſdem ſpeculi plani, Quod eſt totum propoſitum.</s> <s xml:id="echoid-s32946" xml:space="preserve"/> </p> <div xml:id="echoid-div1350" type="float" level="0" n="0"> <figure xlink:label="fig-0514-01" xlink:href="fig-0514-01a"> <variables xml:id="echoid-variables557" xml:space="preserve">f g e a b c d</variables> </figure> </div> </div> <div xml:id="echoid-div1352" type="section" level="0" n="0"> <head xml:id="echoid-head1059" xml:space="preserve" style="it">48. In ſpeculis planis forma puncti adcentrum uiſus reflexa, locum imagin is inuenire.</head> <p> <s xml:id="echoid-s32947" xml:space="preserve">Eſto ſpeculum planum, in cuius ſuperficie ſit linea a b c:</s> <s xml:id="echoid-s32948" xml:space="preserve"> ſit quoq;</s> <s xml:id="echoid-s32949" xml:space="preserve">, ut form a puncti rei uiſæ, quod <lb/> <anchor type="figure" xlink:label="fig-0514-02a" xlink:href="fig-0514-02"/> ſit d, reflectatur ad centrum uiſus, quod ſit e, à puncto ſpe-<lb/>culi b:</s> <s xml:id="echoid-s32950" xml:space="preserve"> & ducatur linea incidentiæ, quæ ſit d b, & lineare-<lb/>flexionis, quę ſit b e.</s> <s xml:id="echoid-s32951" xml:space="preserve"> Dico quòd eſt poſsibile inueniri locũ <lb/>imaginis, in quo uidetur forma puncti d.</s> <s xml:id="echoid-s32952" xml:space="preserve"> Quoniam enim <lb/>per 27 huius puncta d, b, e ſunt in eadem ſuperficie:</s> <s xml:id="echoid-s32953" xml:space="preserve"> patet <lb/>per 1 & 2 p 11 quoniam linea a b c eſt cum lineis d b & b e <lb/>in eadẽ ſuperficie.</s> <s xml:id="echoid-s32954" xml:space="preserve"> Imaginetur ergo extendi linea a b c in <lb/>continuum, quouſq;</s> <s xml:id="echoid-s32955" xml:space="preserve"> a puncto e ſuper ipſam producatur <lb/>per 12 p 1 linea perpendicularis, quæ ſit e c, & ei ęquidiſtás <lb/>a puncto d, quæ ſit d a, per 31 p 1.</s> <s xml:id="echoid-s32956" xml:space="preserve"> Quia ita que linea e b con-<lb/>currit cum linea e c in puncto e, palá per 2th.</s> <s xml:id="echoid-s32957" xml:space="preserve"> I huius quo-<lb/>niam ipſa concurret cum linea d a producta:</s> <s xml:id="echoid-s32958" xml:space="preserve"> ſit concurſus punctus f.</s> <s xml:id="echoid-s32959" xml:space="preserve"> Dico per 37 huius quoniá pun <lb/>ctus f eſt locus imaginis formæ puncti d.</s> <s xml:id="echoid-s32960" xml:space="preserve"> Pater ergo propoſitum.</s> <s xml:id="echoid-s32961" xml:space="preserve"/> </p> <div xml:id="echoid-div1352" type="float" level="0" n="0"> <figure xlink:label="fig-0514-02" xlink:href="fig-0514-02a"> <variables xml:id="echoid-variables558" xml:space="preserve">e d c b a f</variables> </figure> </div> </div> <div xml:id="echoid-div1354" type="section" level="0" n="0"> <head xml:id="echoid-head1060" xml:space="preserve" style="it">49. Eadem eſt diſtantia loci imaginis à ſuperficie ſpeculi plani ſub ſpeculo, quæ eſt punctiuiſi <lb/>ab eadem ſuperficie ſupra ſpeculum planũ exiſtentis. Euclides 19th. catoptr. Alhazen 11 n 5.</head> <p> <s xml:id="echoid-s32962" xml:space="preserve">Sit punctus rei uiſæ a:</s> <s xml:id="echoid-s32963" xml:space="preserve"> & ſit centrum uiſus b:</s> <s xml:id="echoid-s32964" xml:space="preserve"> & ſit c d e linea communis ſuperficiei reflexionis & <lb/>ſuperficiei ſpeculi plani:</s> <s xml:id="echoid-s32965" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s32966" xml:space="preserve"> d punctus reflexionis:</s> <s xml:id="echoid-s32967" xml:space="preserve"> & à puncto d ducatur linea d f perpendiculari-<lb/>ter ſuper lineam c d e per 11 p 1, uel ſuper totam ſuperficiem ſpeculi plani per 12 p 11:</s> <s xml:id="echoid-s32968" xml:space="preserve"> & à puncto a du <lb/>catur perpẽdicularis ſuper ſuperficiem ſpeculi per 11 p 11, quæ ſit a c, <lb/>quæ producatur ultra ſpeculum:</s> <s xml:id="echoid-s32969" xml:space="preserve"> & ducatur linea incidentiæ, quæ <lb/> <anchor type="figure" xlink:label="fig-0514-03a" xlink:href="fig-0514-03"/> ſit a d, & linea reflexionis, quę ſit b d.</s> <s xml:id="echoid-s32970" xml:space="preserve"> Pater ergo per 27 huius quo-<lb/>niam lineę a d, f d, b d ſuntin ſuperficie reflexionis.</s> <s xml:id="echoid-s32971" xml:space="preserve"> Et cum linea f d <lb/>ſit ęquidiſtans lineę a cper 28 p 1, uel per 6 p 11, & linea b d cócurrat <lb/>cum linea f d in puncto d, patet per 2 th.</s> <s xml:id="echoid-s32972" xml:space="preserve"> 1 huius quia linea b d protra <lb/>cta concurret cum linea a c protracta:</s> <s xml:id="echoid-s32973" xml:space="preserve"> concurrat ergo in puncto g.</s> <s xml:id="echoid-s32974" xml:space="preserve"> <lb/>Dico quòd linea g c eſt æqualis lineæ a c.</s> <s xml:id="echoid-s32975" xml:space="preserve"> Quoniam enim angulus <lb/>b d e eſt æqualis angulo a d c per 20 huius, ſunt enim anguli incidẽ-<lb/>tiæ & reflexionis:</s> <s xml:id="echoid-s32976" xml:space="preserve"> ſed angulus b d e eſt æqualis angulo c d g per 15 p <lb/>1, quoniam ſunt anguli contra ſe poſiti:</s> <s xml:id="echoid-s32977" xml:space="preserve"> angulus ergo a d c eſt ęqua-<lb/>lis angulo c d g:</s> <s xml:id="echoid-s32978" xml:space="preserve"> angulus uerò a c d eſt ęqualis angulo d c g, quon iá <lb/>uterque eſt rectus:</s> <s xml:id="echoid-s32979" xml:space="preserve"> erit ergo per 32 p 1 angulus c a d trigoni c a d æ-<lb/>qualis angulo c g d trigoni c g d:</s> <s xml:id="echoid-s32980" xml:space="preserve"> erunt ergo per 4 p 6 latera æ qu o s <lb/>angulos continentia proportionalia:</s> <s xml:id="echoid-s32981" xml:space="preserve"> ſed latus c d æ quale eſt ſibi i-<lb/>pſi:</s> <s xml:id="echoid-s32982" xml:space="preserve"> erunt ergo cætera latera æquos angulos reſpicientia inter ſe æ-<lb/>qualia, ut a c ipſi c g, & a d ipſi a g.</s> <s xml:id="echoid-s32983" xml:space="preserve"> Quia ergo in puncto g eſt locus i-<lb/>maginis per 37 huius, & linea c g eſt ęqualis ipſi a c:</s> <s xml:id="echoid-s32984" xml:space="preserve"> pater ergo pro-<lb/>poſitum.</s> <s xml:id="echoid-s32985" xml:space="preserve"> Si ergo è perpendiculari ultra ſuperficiem ſpeculi imagi-<lb/>netur linea c g æ qualis lineæ a c reſecari:</s> <s xml:id="echoid-s32986" xml:space="preserve"> ſemper erit in puncto g lo-<lb/>cus imaginis tantùm diſtans à ſuperficie ſpeculi plani ſub ſpeculo, <lb/>quantùm punctus rei uiſæ, cuius forma uidetur in ſpeculo, diſtat ab eadem ſuperficie ſpeculi ſupra <lb/>ſpeculum.</s> <s xml:id="echoid-s32987" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s32988" xml:space="preserve"/> </p> <div xml:id="echoid-div1354" type="float" level="0" n="0"> <figure xlink:label="fig-0514-03" xlink:href="fig-0514-03a"> <variables xml:id="echoid-variables559" xml:space="preserve">a f b c d e g</variables> </figure> </div> </div> <div xml:id="echoid-div1356" type="section" level="0" n="0"> <head xml:id="echoid-head1061" xml:space="preserve" style="it">50. In omni reflexione à ſpeculis planis facta, linea à centro uiſus ad locum imaginis produ-<lb/>cta, æqualis eſt lineæ incidentiæ & reflexionis ſimuliunctis.</head> <p> <s xml:id="echoid-s32989" xml:space="preserve">Eſto in ſpeculo plano linea a b c:</s> <s xml:id="echoid-s32990" xml:space="preserve"> & ſit centrum uiſus d:</s> <s xml:id="echoid-s32991" xml:space="preserve"> & punctus rei uiſæ ſit e:</s> <s xml:id="echoid-s32992" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s32993" xml:space="preserve"> reflexio for-<lb/> <pb o="213" file="0515" n="515" rhead="LIBER QVINTVS."/> mæ puncti e ad uiſum d à puncto ſpeculi plani, quod ſit b:</s> <s xml:id="echoid-s32994" xml:space="preserve"> erit ergo linea incidentiæ, quæ e b, & li-<lb/> <anchor type="figure" xlink:label="fig-0515-01a" xlink:href="fig-0515-01"/> nea reflexionis, quę b d:</s> <s xml:id="echoid-s32995" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s32996" xml:space="preserve"> locus imaginis punctus g:</s> <s xml:id="echoid-s32997" xml:space="preserve"> <lb/>hic ergo per 37 huius eritin concurſu lineæ reflexionis <lb/>d b cum catheto incidẽtiæ.</s> <s xml:id="echoid-s32998" xml:space="preserve"> Sit ergo, ut cathetus e g pro-<lb/>ducta ſecet lineam a c in pũcto f.</s> <s xml:id="echoid-s32999" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s33000" xml:space="preserve"> angulus inci <lb/>dentię, qui eſt e b f, eſt ęqualis angulo reflexiois, qui eſt <lb/>a b d, per 20 huius:</s> <s xml:id="echoid-s33001" xml:space="preserve"> & angulus g b f æqualis a b d per 15 p <lb/>1:</s> <s xml:id="echoid-s33002" xml:space="preserve"> eſt ergo angulus g b f æ qualis angulo e b f:</s> <s xml:id="echoid-s33003" xml:space="preserve"> ſed & angu <lb/>lus e f b æ qualis eſt angulo g f b, quia amborecti:</s> <s xml:id="echoid-s33004" xml:space="preserve"> ergo <lb/>per 32 p 1 trigoni b g f & b e f ſunt æ quianguli:</s> <s xml:id="echoid-s33005" xml:space="preserve"> ergo per <lb/>4 p 6 latera illorum æ quos angulos continentia ſunt <lb/>proportionalia:</s> <s xml:id="echoid-s33006" xml:space="preserve"> ſed latus b f eſt æquale ſibi ipſi:</s> <s xml:id="echoid-s33007" xml:space="preserve"> ergo g b <lb/>eſt æquale ipſi b e.</s> <s xml:id="echoid-s33008" xml:space="preserve"> Ergo linea d g à centro uiſus ad locum imaginis g producta, eſt ęqualis ambabus <lb/>lineis d b & b e ſimul acceptis.</s> <s xml:id="echoid-s33009" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s33010" xml:space="preserve"/> </p> <div xml:id="echoid-div1356" type="float" level="0" n="0"> <figure xlink:label="fig-0515-01" xlink:href="fig-0515-01a"> <variables xml:id="echoid-variables560" xml:space="preserve">d e a b f c g</variables> </figure> </div> </div> <div xml:id="echoid-div1358" type="section" level="0" n="0"> <head xml:id="echoid-head1062" xml:space="preserve" style="it">51. In ſpeculo plano ab utro uiſu uno puncto comprehenſo, idem erit imaginis locus uiſib. am <lb/>bobus: ex quo patet quòd una ſola imago utri uiſuioccurrit. Alhazen 15 n 5.</head> <p> <s xml:id="echoid-s33011" xml:space="preserve">Sint duo uiſus b & g:</s> <s xml:id="echoid-s33012" xml:space="preserve"> & ſit a punctus rei uiſæ:</s> <s xml:id="echoid-s33013" xml:space="preserve"> & ſit q d z e linea in ſuperficie ſpeculi plani ducta:</s> <s xml:id="echoid-s33014" xml:space="preserve"> <lb/>ſitq́;</s> <s xml:id="echoid-s33015" xml:space="preserve"> linea a d perpendicularis ducta à puncto a ſuper ſuperficiem ſpeculi, Et quia per 30 huius a b u-<lb/>no puncto ſpeculi propoſiti ad ambos uiſus non poteſt fieri reflexio, ſed ad minus à duobus:</s> <s xml:id="echoid-s33016" xml:space="preserve"> ſint i-<lb/>taq;</s> <s xml:id="echoid-s33017" xml:space="preserve"> illa duo puncta t & z:</s> <s xml:id="echoid-s33018" xml:space="preserve"> & ducátur lineæ b t, a t, a z, z g Palàm ergo per 25 huius quia linea b t & a t <lb/>& a d ſunt in eadẽ ſuperficie reflexionis, erecta ſuper <lb/>ſuperficiem ſpeculi:</s> <s xml:id="echoid-s33019" xml:space="preserve"> & ſimiliter lineę a d, a z, z g ſunt <lb/> <anchor type="figure" xlink:label="fig-0515-02a" xlink:href="fig-0515-02"/> in eadem ſuperficie:</s> <s xml:id="echoid-s33020" xml:space="preserve"> & linea d t eſt communis ſectio <lb/>ſuperficiei reflexionis, quæ eſt a d t b, & ſuperficiei i-<lb/>pſius ſpeculi:</s> <s xml:id="echoid-s33021" xml:space="preserve"> & linea d z eſt communis ſectio ſuperfi <lb/>ciei reflexionis, quæ eſt a d z g, & ſuperſiciei ſpeculi <lb/>per 19 th.</s> <s xml:id="echoid-s33022" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s33023" xml:space="preserve"> Si ergo ambæ lineę reflexionis, quę <lb/>ſunt b t & g z, fuerint in eadem ſuperficie erecta ſu-<lb/>per ſuperficiem ſpeculi:</s> <s xml:id="echoid-s33024" xml:space="preserve"> palàm quia linea t d z erit li-<lb/>nea una recta:</s> <s xml:id="echoid-s33025" xml:space="preserve"> ideo quia communis ſectio ſuperfi-<lb/>ciei ſpeculi, & ſuperficiei cuiuſcunque ſuper ipſam <lb/>erectæ eſt linea una recta per 3 p 11:</s> <s xml:id="echoid-s33026" xml:space="preserve"> tunc ergo & per-<lb/>pendicularis a d, quæ eſt inter duas lineas illas refle-<lb/>xionis, quę t b & g z, aut erit in eadem ſuperficie cum <lb/>illis, aut extra illas in alia ſuperficie:</s> <s xml:id="echoid-s33027" xml:space="preserve"> quodcunq;</s> <s xml:id="echoid-s33028" xml:space="preserve"> iſto-<lb/>rum fuerit, ſemper linea reflexionis, quę b t, protra-<lb/>cta ſecabit ex perpendiculari, quę eſt a d, ultra ſpecu <lb/>lum protracta partem æ qualẽ ipſi a d per 49 huius, quę ſit d h:</s> <s xml:id="echoid-s33029" xml:space="preserve"> quoniá ſemper lineæ b t & a d ſunt in <lb/>aliqua eadem ſuperficie per 27 huius, ut præmiſſum eſt.</s> <s xml:id="echoid-s33030" xml:space="preserve"> Et ſimiliter per 49 huius linea g z protracta <lb/> <anchor type="figure" xlink:label="fig-0515-03a" xlink:href="fig-0515-03"/> <anchor type="figure" xlink:label="fig-0515-04a" xlink:href="fig-0515-04"/> ultra ſpeculum ſeca-<lb/>bit ex protracta ca-<lb/>theto ad lineá ęqua-<lb/>lem ipſi lineę a d:</s> <s xml:id="echoid-s33031" xml:space="preserve"> ſe-<lb/>cabit ergo ipſam in <lb/>puncto h.</s> <s xml:id="echoid-s33032" xml:space="preserve"> Imago er-<lb/>go puncti a in eodé <lb/>puncto perpendicu <lb/>laris, quod eſt h, քci <lb/>pietur ab utroq;</s> <s xml:id="echoid-s33033" xml:space="preserve"> ui-<lb/>ſu, & idem erit ima-<lb/>ginis locus.</s> <s xml:id="echoid-s33034" xml:space="preserve"> Vna er-<lb/>go tátũ erit imago, <lb/>& in uno eodẽq́;</s> <s xml:id="echoid-s33035" xml:space="preserve"> lo-<lb/>co uidebitur ab am-<lb/>bobus uiſib.</s> <s xml:id="echoid-s33036" xml:space="preserve"> in quo <lb/>puncto uno tantùm <lb/>uiſu perciperetur.</s> <s xml:id="echoid-s33037" xml:space="preserve"> <lb/>Si uerò puncta t & znon fuerint in eadem ſuperficie reflexionis, ad-<lb/>huc eadẽ facta deductione una tantũ imago uidebitur, & unus tantũ <lb/>erit imaginis locus, ut prius.</s> <s xml:id="echoid-s33038" xml:space="preserve"> Sẽper.</s> <s xml:id="echoid-s33039" xml:space="preserve"> n.</s> <s xml:id="echoid-s33040" xml:space="preserve"> utraq;</s> <s xml:id="echoid-s33041" xml:space="preserve"> linea reflexiõis ſecabit ex քpẽdiculari ꝓtracta partẽ æ-<lb/>qualẽ ipſi a d:</s> <s xml:id="echoid-s33042" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s33043" xml:space="preserve"> ſectio ambarũ linearũ reflexionis cũ illa ꝗpẽdiculari in eodẽ puncto h, ꝗ per 37 <lb/>huius, erit ſemք imaginis locus.</s> <s xml:id="echoid-s33044" xml:space="preserve"> Et hoc eſt ꝓpoſitũ:</s> <s xml:id="echoid-s33045" xml:space="preserve"> quoniã ſi cẽtra amborũ uiſuũ, quę ſunt b & g, fue <lb/>rint ex eadẽ parte rei uiſę, quę eſt a, ſemper eodẽ modo eſt demonſtrandũ:</s> <s xml:id="echoid-s33046" xml:space="preserve"> cõcurrent enim lineę re-<lb/> <pb o="214" file="0516" n="516" rhead="VITELLONIS OPTICAE"/> flexionum cum catheto in eodem puncto:</s> <s xml:id="echoid-s33047" xml:space="preserve"> & erit idẽ imaginis locus, & eadem imago uiſib occuret</s> </p> <div xml:id="echoid-div1358" type="float" level="0" n="0"> <figure xlink:label="fig-0515-02" xlink:href="fig-0515-02a"> <variables xml:id="echoid-variables561" xml:space="preserve">b g a q t d z e h</variables> </figure> <figure xlink:label="fig-0515-03" xlink:href="fig-0515-03a"> <variables xml:id="echoid-variables562" xml:space="preserve">a g b e d z t q h</variables> </figure> <figure xlink:label="fig-0515-04" xlink:href="fig-0515-04a"> <variables xml:id="echoid-variables563" xml:space="preserve">b a g t z d h</variables> </figure> </div> </div> <div xml:id="echoid-div1360" type="section" level="0" n="0"> <head xml:id="echoid-head1063" xml:space="preserve" style="it">52. In ſpeculis planis figurarei uiſæ & ſitus partium ſecundum quantitatem longitudini & <lb/>latitudinis non mutatur. Ex quo patet, quòdimago cuius libet rei uiſæ in ſpeculo plano æqualis <lb/>eſt formæ rei extrà. Euclides 19 th. catoptr. Alhazen 2 n 6.</head> <p> <s xml:id="echoid-s33048" xml:space="preserve">Sit ſpeculum planum, in quo ſectio communis ſuperficiei illius ſpeculi & ſuperficiei reflexionis <lb/>ſit linea a b:</s> <s xml:id="echoid-s33049" xml:space="preserve"> & duo puncta extrema alicuius rei uiſæ ſint f & l:</s> <s xml:id="echoid-s33050" xml:space="preserve">erigaturq́;</s> <s xml:id="echoid-s33051" xml:space="preserve"> cathetus perpendiculariter <lb/> <anchor type="figure" xlink:label="fig-0516-01a" xlink:href="fig-0516-01"/> ſuper ſuperficiem ſpeculi à puncto l, quæ ſit l h:</s> <s xml:id="echoid-s33052" xml:space="preserve"> & à pun-<lb/>cto f cathetus, quę ſit f z:</s> <s xml:id="echoid-s33053" xml:space="preserve"> & erunt z & h duo puncta in ſu <lb/>perficie reflexionis per 27 huius:</s> <s xml:id="echoid-s33054" xml:space="preserve"> producanturq́;</s> <s xml:id="echoid-s33055" xml:space="preserve"> taliter <lb/>ſub ſpeculum, ut linea h g ſit æ qualis ipſi l h, & linea z d <lb/>æqualis ipſi f z:</s> <s xml:id="echoid-s33056" xml:space="preserve"> ſit quoq, centrum uiſus e:</s> <s xml:id="echoid-s33057" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s33058" xml:space="preserve"> per <lb/>11 p 11 à puncto e cathetus ſuper ſpeculũ, quæ ſit e b.</s> <s xml:id="echoid-s33059" xml:space="preserve"> Palã <lb/>itaq;</s> <s xml:id="echoid-s33060" xml:space="preserve"> ex 28 huius quoniam forma puncti l reflectitur ad <lb/>uiſum e ab aliquo puncto ſpeculi lineæ h b:</s> <s xml:id="echoid-s33061" xml:space="preserve"> & locus ima <lb/>ginis ſuæ per 49 huius eſt punctum g, tantũ diſtans à ſu-<lb/>perficie ſpeculi ultra ſpeculum, quátum punctus l ſupra <lb/>ſpeculum.</s> <s xml:id="echoid-s33062" xml:space="preserve"> Similiter forma puncti freflectitur ad uiſum e <lb/>ab aliquo puncto lineæ z b:</s> <s xml:id="echoid-s33063" xml:space="preserve"> & locus imaginis eſt punctũ <lb/>d.</s> <s xml:id="echoid-s33064" xml:space="preserve"> Ducta quoq;</s> <s xml:id="echoid-s33065" xml:space="preserve"> linea f l, & linea d g:</s> <s xml:id="echoid-s33066" xml:space="preserve"> palá quia quodcũq;</s> <s xml:id="echoid-s33067" xml:space="preserve"> <lb/>punctum lineæ f l reflectitur ad uiſum e, ſimiliter locus <lb/>imaginis ſuę eſt tantùm diſtans à ſuperficie ſpeculi ultra <lb/>ſpeculum, quantùm ille punctus eſt ſupra ſpeculũ.</s> <s xml:id="echoid-s33068" xml:space="preserve"> Qui-<lb/>libet ergo punctus lineæ f l tantùm uidetur diſtare ſub <lb/>ſpeculo, quantùm ip ſe punctus a ſuperficie ſpeculi ſupra <lb/>ſpeculum.</s> <s xml:id="echoid-s33069" xml:space="preserve"> Si ergo linea f l fuerit recta, erit linea d grecta:</s> <s xml:id="echoid-s33070" xml:space="preserve"> ſi linea f l fuerit arcus circuli, erit quo q;</s> <s xml:id="echoid-s33071" xml:space="preserve"> li-<lb/>nead g arcus circuli, & ſemper eiuſdem curuitatis & diſpoſitionis.</s> <s xml:id="echoid-s33072" xml:space="preserve"> Linea ergo f l ſemper apparebit <lb/>eiuſdem quan titatis & figuræ, cuius eſt extra ſpeculum.</s> <s xml:id="echoid-s33073" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s33074" xml:space="preserve"> Supponendum ta-<lb/>men eſt, ut tale ſpeculum planum ſit æ qualiter politum:</s> <s xml:id="echoid-s33075" xml:space="preserve"> quoniam ſi ad longitudinem uellatitudi-<lb/>nem nimis declinet politio, declinabit & forma ſecundum idem per 40 huius:</s> <s xml:id="echoid-s33076" xml:space="preserve"> neceritin longitudi <lb/>ne & latitudine debitus ordo formæ.</s> <s xml:id="echoid-s33077" xml:space="preserve"/> </p> <div xml:id="echoid-div1360" type="float" level="0" n="0"> <figure xlink:label="fig-0516-01" xlink:href="fig-0516-01a"> <variables xml:id="echoid-variables564" xml:space="preserve">f l e a z b h d g</variables> </figure> </div> </div> <div xml:id="echoid-div1362" type="section" level="0" n="0"> <head xml:id="echoid-head1064" xml:space="preserve" style="it">53. Altitudines & profunditates à planis ſpeculis reuerſæuidentur, cum ſpeculorum ſuperfi <lb/>ciebus perpendiculariter inſiſtunt. Euclides 7th. catoptr.</head> <p> <s xml:id="echoid-s33078" xml:space="preserve">Eſto altitudo uiſa, quæ a b c e:</s> <s xml:id="echoid-s33079" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s33080" xml:space="preserve"> centrum uerò communis ſuperficiei reflexionis <lb/>& ſuperficiei ſpeculi plani ſit <lb/>e f g h i:</s> <s xml:id="echoid-s33081" xml:space="preserve"> incidatq́;</s> <s xml:id="echoid-s33082" xml:space="preserve"> forma pun <lb/> <anchor type="figure" xlink:label="fig-0516-02a" xlink:href="fig-0516-02"/> <anchor type="figure" xlink:label="fig-0516-03a" xlink:href="fig-0516-03"/> cti a ſecundum lineam a h, & <lb/>reflectatur ſecundum lineá <lb/>h d:</s> <s xml:id="echoid-s33083" xml:space="preserve"> & forma pũcti b incidat <lb/>ſecundum lineã b g, & refle-<lb/>ctatur ſecundum lineá g d:</s> <s xml:id="echoid-s33084" xml:space="preserve"> <lb/>& forma puncti c incidat ſe-<lb/>cundum lineam c f, & refle-<lb/>ctatur ſecundum lineam f d.</s> <s xml:id="echoid-s33085" xml:space="preserve"> <lb/>Dico, quòd altitudo e a uide <lb/>bitur reuerſa.</s> <s xml:id="echoid-s33086" xml:space="preserve"> Protracta e-<lb/>nim linea e a, quę perpendi-<lb/>cularis eſt ſuper lineam e i, <lb/>ſub ſpeculum, & protractis <lb/>omnib.</s> <s xml:id="echoid-s33087" xml:space="preserve"> lineis reflexionis ad <lb/>concurſum eum protracta linea a e ultra punctũ e:</s> <s xml:id="echoid-s33088" xml:space="preserve"> incidat linea d h in punctum m:</s> <s xml:id="echoid-s33089" xml:space="preserve"> & linea d g in pũ <lb/>ctum l:</s> <s xml:id="echoid-s33090" xml:space="preserve"> & linea d f in punctum k.</s> <s xml:id="echoid-s33091" xml:space="preserve"> Palàm per pręmiſſam quoniam linea k e ęqualis eſt ipſi lineę e c, & <lb/>leipſi e b, & m e ęqualis ipſi e a.</s> <s xml:id="echoid-s33092" xml:space="preserve"> Pũcta ergo altitudinis e a propinquiora ſuperficiei ſpeculi ſuperius <lb/>exiſtentia, propin quiora uidebuntur eidem ſub ſpeculo inferius, & puncta remotiora ſuperficiei <lb/>ſpeculi ſuperius, remotiora uidebuntur ſub ſpeculo inferius.</s> <s xml:id="echoid-s33093" xml:space="preserve"> Videbitur ergo altitudo reuerſa ſub <lb/>ſpeculo:</s> <s xml:id="echoid-s33094" xml:space="preserve"> quod enim eſt ſuperius in altitudine, uidebitur inferius, quoniá ſub maiori diſtantia à uiſu <lb/>uidetur:</s> <s xml:id="echoid-s33095" xml:space="preserve"> & quod eſt inferius in altitudine, uidebitur ſuperius, quoniá propinquius uiſui uidetur.</s> <s xml:id="echoid-s33096" xml:space="preserve"> Et <lb/>eodem modo demonſtrandũ, ſi linea a b c ſit linea profunditatis alicuius rei.</s> <s xml:id="echoid-s33097" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s33098" xml:space="preserve"/> </p> <div xml:id="echoid-div1362" type="float" level="0" n="0"> <figure xlink:label="fig-0516-02" xlink:href="fig-0516-02a"> <variables xml:id="echoid-variables565" xml:space="preserve">a d b c d f g h i k l m</variables> </figure> <figure xlink:label="fig-0516-03" xlink:href="fig-0516-03a"> <variables xml:id="echoid-variables566" xml:space="preserve">m c k i h g f e b c d a</variables> </figure> </div> </div> <div xml:id="echoid-div1364" type="section" level="0" n="0"> <head xml:id="echoid-head1065" xml:space="preserve" style="it">54. Obliquæ longitudines à planis ſpeculis uidentur, quemadmodum ſe habent. Euclides <gap/> <lb/>the. catoptr.</head> <p> <s xml:id="echoid-s33099" xml:space="preserve">Sit d e longitudo obliquè diſtans à ſuperficie plani ſpeculi, ita ut punctum eius, quod eſt e, ſit re-<lb/>motius ab ipſa ſuperficie ſpeculi:</s> <s xml:id="echoid-s33100" xml:space="preserve"> cõmunis quoq;</s> <s xml:id="echoid-s33101" xml:space="preserve"> ſectio ſuperficiei reflexionis & ſuperficiei ſpeculi <lb/> <pb o="215" file="0517" n="517" rhead="LIBER QVINTVS"/> ſit linea k q a g:</s> <s xml:id="echoid-s33102" xml:space="preserve"> cẽtrumq́;</s> <s xml:id="echoid-s33103" xml:space="preserve"> uiſus ſit punctus b:</s> <s xml:id="echoid-s33104" xml:space="preserve"> & incidat forma puncti d ipſi ſpeculo ſecundum lineá <lb/> <anchor type="figure" xlink:label="fig-0517-01a" xlink:href="fig-0517-01"/> d a, & reflectatur ſecundum lineá a b ad centrũ ui-<lb/>ſus:</s> <s xml:id="echoid-s33105" xml:space="preserve"> & incidat forma puncti e ſecũdum lineam eg, <lb/>& reflectatur ad uiſum ſecundum lineam g b:</s> <s xml:id="echoid-s33106" xml:space="preserve"> pro-<lb/>trahaturq́;</s> <s xml:id="echoid-s33107" xml:space="preserve"> cathetus d k perpendiculariter, & linea <lb/>reflexionis, quę eſt b a, donec concurrantin pun-<lb/>cto m:</s> <s xml:id="echoid-s33108" xml:space="preserve"> & protrahatur cathetus e q perpendiculari-<lb/>ter, donec concurrat cũ linea b g in puncto l:</s> <s xml:id="echoid-s33109" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s33110" xml:space="preserve"> <lb/>per 49 huius linea d k æqualis lineæ k m, & linea <lb/>e q æ qualis lineę q l.</s> <s xml:id="echoid-s33111" xml:space="preserve"> Et quoniam lógitudo d e obli <lb/>què ſe habet ad ſuperficiem ſpeculi, (etenim pun <lb/>ctum e remotius eſt à ſpeculo ꝗ̃ punctum d) erit li <lb/>nea e q longior quã linea d k:</s> <s xml:id="echoid-s33112" xml:space="preserve"> ergo & linea q l lon-<lb/>gior quá linea k m.</s> <s xml:id="echoid-s33113" xml:space="preserve"> Punctũ ergo illius obliquę ma-<lb/>gnitudinis, quod eſt remotius ſuper ſuperficiẽ ſpe <lb/>culi, hoc ſimiliter ſub ſuperfιcie ſpeculi à remotio-<lb/>ri uidetur:</s> <s xml:id="echoid-s33114" xml:space="preserve"> & quòd ſuperius propin quius eſt ſpe-<lb/>culo, hoc ſub ſpeculo etiá uidetur eſſe in loco pro-<lb/>pinquiori.</s> <s xml:id="echoid-s33115" xml:space="preserve"> Videntur ergo tales magnitudines quẽadmodũ ſe habẽt.</s> <s xml:id="echoid-s33116" xml:space="preserve"> Et hoc eſt, quod ꝓponebatur.</s> <s xml:id="echoid-s33117" xml:space="preserve"/> </p> <div xml:id="echoid-div1364" type="float" level="0" n="0"> <figure xlink:label="fig-0517-01" xlink:href="fig-0517-01a"> <variables xml:id="echoid-variables567" xml:space="preserve">d e b k q a g m l</variables> </figure> </div> </div> <div xml:id="echoid-div1366" type="section" level="0" n="0"> <head xml:id="echoid-head1066" xml:space="preserve" style="it">55. In ſpeculis planis dextra apparent ſiniſtra, & ſiniſtra dextra. Euclides 19 th. catoptr.</head> <p> <s xml:id="echoid-s33118" xml:space="preserve">Eſto ſpeculum planum g s t:</s> <s xml:id="echoid-s33119" xml:space="preserve"> & uiſa res ſit d b:</s> <s xml:id="echoid-s33120" xml:space="preserve"> ſint quoq;</s> <s xml:id="echoid-s33121" xml:space="preserve"> lineæ incidentię d g & b s:</s> <s xml:id="echoid-s33122" xml:space="preserve"> & ſit centrum <lb/>uiſus p:</s> <s xml:id="echoid-s33123" xml:space="preserve">lineæ quoq;</s> <s xml:id="echoid-s33124" xml:space="preserve"> reflexionis ſint p g & p s:</s> <s xml:id="echoid-s33125" xml:space="preserve"> & ſit, ut linea reflexionis, quæ eſt p g, concurrat cũ ca-<lb/>theto incidentiæ, quæ d h in puncto f:</s> <s xml:id="echoid-s33126" xml:space="preserve"> & linea reflexio <lb/> <anchor type="figure" xlink:label="fig-0517-02a" xlink:href="fig-0517-02"/> nis, quę eſt p s, concurrat cum catheto b tin puncto e:</s> <s xml:id="echoid-s33127" xml:space="preserve"> <lb/>producaturq́;</s> <s xml:id="echoid-s33128" xml:space="preserve"> linea fe, quæ eſt per 52 huius imago rei <lb/>uiſæ, quæ d b:</s> <s xml:id="echoid-s33129" xml:space="preserve"> apparebunt ergo dextra ſiniſtra, & ſini-<lb/>ſtra dextra.</s> <s xml:id="echoid-s33130" xml:space="preserve"> Quoniam enim per 33 huius ſemper ad an <lb/>gulum maiorem angulo incidentiæ ſit reflexio, & it a <lb/>ad partem oppoſitam parti incidétiæ:</s> <s xml:id="echoid-s33131" xml:space="preserve"> patet quòd de-<lb/>xtrum rei uiſæ ſemper uidebitur ſub linea reflexionis <lb/>magis ſiniſtra, & ſiniſtrum ſub linea reflexionis magis <lb/>dextra:</s> <s xml:id="echoid-s33132" xml:space="preserve"> illa enim linea reflexionis, quę plus eſt dextra, <lb/>cadet ſuper dextram parté imaginis, & ſiniſtra cadet <lb/>ſuper ſiniſtram.</s> <s xml:id="echoid-s33133" xml:space="preserve"> Sic ergo dextrum rei apparet ſub ſini-<lb/>ſtro imaginis, & econuerſo:</s> <s xml:id="echoid-s33134" xml:space="preserve"> quoniam imago rei uide-<lb/>tur ſe habere ad rem, ſicut homo ſtans erecta facie con <lb/>tra aliquem alium:</s> <s xml:id="echoid-s33135" xml:space="preserve"> tunc enim pars ſiniſtra opponitur <lb/>dextræ, & dextra ſiniſtræ:</s> <s xml:id="echoid-s33136" xml:space="preserve"> quia ſemper cum aliquis ho <lb/>mo alij opponitur, contrarius eſt eis oppoſitis adinui <lb/>cem ſitus:</s> <s xml:id="echoid-s33137" xml:space="preserve">ad eandem enim poſitionis differentiam eſt <lb/>dextrũ unius ſiniſtrum alterius, & econuerſo:</s> <s xml:id="echoid-s33138" xml:space="preserve"> & ſic quod eſt rei uiſę dextrũ, fit ſuę imaginis ſiniſtrũ:</s> <s xml:id="echoid-s33139" xml:space="preserve"> <lb/>& quod eſt rei uiſæ ſiniſtrum, in imagine dextrum erit ſecundum uiſum.</s> <s xml:id="echoid-s33140" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s33141" xml:space="preserve"/> </p> <div xml:id="echoid-div1366" type="float" level="0" n="0"> <figure xlink:label="fig-0517-02" xlink:href="fig-0517-02a"> <variables xml:id="echoid-variables568" xml:space="preserve">b d p t h s g e f</variables> </figure> </div> </div> <div xml:id="echoid-div1368" type="section" level="0" n="0"> <head xml:id="echoid-head1067" xml:space="preserve" style="it">56. Poſsibile eſt ſpeculum planum taliter ſiſti, ut intuens propria imagine non uiſa, uideat i-<lb/>maginem rei alterius non uiſæ, Ptolemæus 9 th. 2 catoptr.</head> <p> <s xml:id="echoid-s33142" xml:space="preserve">Sit a b lignũ horizonti perpẽdiculariter infixũ, uel ſuperficiei ſibi ęquidiſtanti, uel aliter quomo-<lb/>docun q;</s> <s xml:id="echoid-s33143" xml:space="preserve"> diſpoſitę, quæ ſit b g:</s> <s xml:id="echoid-s33144" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s33145" xml:space="preserve"> ſpeculum planũ, in quo ſit linea d b:</s> <s xml:id="echoid-s33146" xml:space="preserve"> & ſit quadratum.</s> <s xml:id="echoid-s33147" xml:space="preserve"> Et quia li-<lb/>gnum a b eſt perpendiculariter erectum ſuper g b ſuperficiem, ducatur linea g b, ut cótingit;</s> <s xml:id="echoid-s33148" xml:space="preserve"> palàm <lb/>ergo quòd angulus a b g eſt rectus:</s> <s xml:id="echoid-s33149" xml:space="preserve"> diuidatur ergo ille angulus rectus in tres partes æquales per 28 <lb/>th.</s> <s xml:id="echoid-s33150" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s33151" xml:space="preserve"> inclineturq́;</s> <s xml:id="echoid-s33152" xml:space="preserve">ſpeculum d b taliter à ligno a b, ut angulus d b a ſit tertia pars unius recti, ꝗ <lb/>eſt a b g:</s> <s xml:id="echoid-s33153" xml:space="preserve"> erit ergo angulus d b g duę tertiæ partes unius recti.</s> <s xml:id="echoid-s33154" xml:space="preserve"> In hoc autem conſiſtit bonitas opera <lb/>tionis mechanicæ & utilior effectus:</s> <s xml:id="echoid-s33155" xml:space="preserve"> quęcunq;</s> <s xml:id="echoid-s33156" xml:space="preserve"> tamen alia pars recti anguli abſcindatur, ad idé per-<lb/>uenit demonſtratio, ut patet.</s> <s xml:id="echoid-s33157" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s33158" xml:space="preserve"> angulus a d b tertia pars unius recti:</s> <s xml:id="echoid-s33159" xml:space="preserve"> & producatur linea ſpe-<lb/>culi, quę eſt b d, ultra punctum d in continuum & directum uſq;</s> <s xml:id="echoid-s33160" xml:space="preserve"> ad punctum, quod ſit e.</s> <s xml:id="echoid-s33161" xml:space="preserve"> Et quoniá <lb/>linea g b eſt perpendicularis ſuper lineam a b, cum linea quoq;</s> <s xml:id="echoid-s33162" xml:space="preserve"> ſpeculi, quæ eſt d b, continet angulũ <lb/>acutum:</s> <s xml:id="echoid-s33163" xml:space="preserve"> tunc à puncto g, quod ſit in ſuperficie orthogonaliter erecta ſuper ſpeculi ſuperficiem, du-<lb/>catur linea perpendicularis ſuper lineam b e per 12 p 1, quę ſit g e:</s> <s xml:id="echoid-s33164" xml:space="preserve"> angulus igitur b e g erit rectus.</s> <s xml:id="echoid-s33165" xml:space="preserve"> Sit <lb/>itaque locus ipſius uiſus punctũ g, à quo ad pun ctum d protrahatur linea g d:</s> <s xml:id="echoid-s33166" xml:space="preserve"> à puncto quoq;</s> <s xml:id="echoid-s33167" xml:space="preserve"> d ꝓ-<lb/>ducatur linea cadens ſuꝗ lineam b g, quę in cidat in punctum z, ita ut angulus z d g ſit æqualis angu <lb/>lo e g d conſtituto ſuper terminum lineę g d per 23 p 1:</s> <s xml:id="echoid-s33168" xml:space="preserve"> erit ergo linea z d ęquidiſtans lineę g e per 27 <lb/>p 1:</s> <s xml:id="echoid-s33169" xml:space="preserve"> ergo ք 8 p 11:</s> <s xml:id="echoid-s33170" xml:space="preserve"> erit linea z d erecta perpẽdiculariter ſuper ſuperficiẽ ſpeculi, & perpendicularis ſu-<lb/>per cõmunem ſectionẽ ſuperſiciei reflexionis & ſpeculi, quę eſt b d:</s> <s xml:id="echoid-s33171" xml:space="preserve"> angulus ergo z d b eſt rectus æ-<lb/>qualis angulo g e d ex præmiſsis, & etiam per 29 p 1.</s> <s xml:id="echoid-s33172" xml:space="preserve"> A puncto quoque z ducatur linea z h per-<lb/>pendicularis ſuper ſuperficiem g b per 11 p.</s> <s xml:id="echoid-s33173" xml:space="preserve"> 11:</s> <s xml:id="echoid-s33174" xml:space="preserve"> & ſuper punctum d terminum lineæ z d conſti-<lb/> <pb o="216" file="0518" n="518" rhead="VITELLONIS OPTICAE"/> tuatur angulus æ qualis angulo g d z, qui ſit angulus z d i.</s> <s xml:id="echoid-s33175" xml:space="preserve"> Et quoniã ք 2 th.</s> <s xml:id="echoid-s33176" xml:space="preserve"> 1 huius cõcurret linea di <lb/>cũ linea zh:</s> <s xml:id="echoid-s33177" xml:space="preserve"> ideo quia linea d i pro ducta ultra punctũ d, cõcurret cũ linea a b, ut patet expræmiſsis, <lb/>& per 14 th.</s> <s xml:id="echoid-s33178" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s33179" xml:space="preserve"> ſit ergo linea-<lb/> <anchor type="figure" xlink:label="fig-0518-01a" xlink:href="fig-0518-01"/> rũ di & z h concurſus in puncto i:</s> <s xml:id="echoid-s33180" xml:space="preserve"> <lb/>& à puncto i ducatur linea ęquidi <lb/>ſtans lineę b d per 31 p 1, quę ſit li-<lb/>neait:</s> <s xml:id="echoid-s33181" xml:space="preserve"> & à puncto b extrahatur ք-<lb/>pendicularis ſuper ſuperficiẽ ſpe-<lb/>culi per 12 p 11, quę fit b q:</s> <s xml:id="echoid-s33182" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s33183" xml:space="preserve"> ք-<lb/>28 p 1 linea b q æquidiſtans lineæ <lb/>g e:</s> <s xml:id="echoid-s33184" xml:space="preserve"> ergo per 8 p 11 linea b q, ſicut & <lb/>linea g e, erecta eſt perpendiculari <lb/>ter ſuք ſuperficiẽ ſpeculi, quæ eſt <lb/>d b.</s> <s xml:id="echoid-s33185" xml:space="preserve"> Super punctũ ergo b terminũ <lb/>lineę q b conſtituatur angulus æ-<lb/>qualis angulo g b q, qui ſit q b t:</s> <s xml:id="echoid-s33186" xml:space="preserve"> cõ <lb/>curret ergo linea b t cum linea æ-<lb/>quidiſtanter ducta lineę d b à pun <lb/>ctoi, quæ eſt linea it, per 2 th.</s> <s xml:id="echoid-s33187" xml:space="preserve"> 1 hu-<lb/>ius:</s> <s xml:id="echoid-s33188" xml:space="preserve"> ſit concurſus punctus t:</s> <s xml:id="echoid-s33189" xml:space="preserve"> & cõ-<lb/>pleatur tabula i t.</s> <s xml:id="echoid-s33190" xml:space="preserve"> Depingatur ita-<lb/>quein tabula, in qua eſt linea it, i-<lb/>mago quæcunq;</s> <s xml:id="echoid-s33191" xml:space="preserve"> placuerit:</s> <s xml:id="echoid-s33192" xml:space="preserve"> & ponatur tabula depictæ imaginis in loco lineęit, ſecundum medium <lb/>lineæ tabulæ correſpondens lineæ zi:</s> <s xml:id="echoid-s33193" xml:space="preserve"> & perforetur ſuperficies g b ſecundum lineam z b, ita ut for-<lb/>ma picturæ poſsit uenire ad ſpeculum d b.</s> <s xml:id="echoid-s33194" xml:space="preserve"> Cũ itaq;</s> <s xml:id="echoid-s33195" xml:space="preserve"> centrum uiſus fuerit in puncto g, uidebit intuẽs <lb/>formam imaginis depictę in tabula it, propriam uerò non uidebit imaginẽ:</s> <s xml:id="echoid-s33196" xml:space="preserve"> cuius hęc eſt demõſtra-<lb/>tio.</s> <s xml:id="echoid-s33197" xml:space="preserve"> Quia enim angulus g e b eſtrectus, patet per 16 p 1 quoniã angulus g d b eſt obtuſus:</s> <s xml:id="echoid-s33198" xml:space="preserve"> & ſimiliter <lb/>omniũ punctorũ formæ uel faciei ipſius uidentis incidentium ſpeculo d b, anguli ſunt obtuſi ք eã-<lb/>dem 16.</s> <s xml:id="echoid-s33199" xml:space="preserve"> Quia uerò anguli incidentiæ ſemper ſunt æquales angulis reflexionis per 20 huius:</s> <s xml:id="echoid-s33200" xml:space="preserve"> palã ք <lb/>13 p 1 quoniã nunꝗ̃ erit reflexio formæ ipſius uidentis ad centrum uiſus, ſed ſemper ad puncta, quæ <lb/>ſunt ſub uiſu, quod patet per 33 huius.</s> <s xml:id="echoid-s33201" xml:space="preserve"> Nũquã ergo uidebit quis exiſtens ſecundũ centrũ uiſus in pũ <lb/>cto g propriam imaginẽ in ſpeculo plano taliter ordinato ſecundũ ſitum.</s> <s xml:id="echoid-s33202" xml:space="preserve"> Et ſi uiſus elongetur à fpe <lb/>culo ſecundũ quodcunq;</s> <s xml:id="echoid-s33203" xml:space="preserve"> punctũ ultra punctũ g, utpote ad punctum f:</s> <s xml:id="echoid-s33204" xml:space="preserve"> palàm quoniã angulus f e b <lb/>eſt maior recto:</s> <s xml:id="echoid-s33205" xml:space="preserve"> ſed & angulus f d b eſt maior angulo f e b per 16 p 1:</s> <s xml:id="echoid-s33206" xml:space="preserve"> nunquá ergo fiet reflexio ad pũ-<lb/>ctum f, ſed ſemper ad alium punctum ſub linea.</s> <s xml:id="echoid-s33207" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s33208" xml:space="preserve"> accedente uiſu ad ſpeculũ ſecundũ-<lb/>quodcun q;</s> <s xml:id="echoid-s33209" xml:space="preserve"> punctum lineę g z, pręterꝗ̃ ſecundum ipſum punctúz, nunꝗ̃ uidebit uidens ſui ipſius i-<lb/>maginem:</s> <s xml:id="echoid-s33210" xml:space="preserve"> ſola enim perpendicularis, quæ eſt linea z d, ut patet expræmiſsis, per 21 huius reflectitur <lb/>in ſe ipſam:</s> <s xml:id="echoid-s33211" xml:space="preserve"> & ita in puncto z conſtituto centro uiſus uidebit intuens formá ſui ipſius oculi à ſpecu-<lb/>lo plano taliter diſpoſito reflexá, non aũt aliã partem faciei:</s> <s xml:id="echoid-s33212" xml:space="preserve"> quoniá ſola perpendicularis, quæ eſt li-<lb/>nea unica, reflectitur in ſe ipſam:</s> <s xml:id="echoid-s33213" xml:space="preserve"> & ita ſolius illius puncti fit reflexio, non aũt punctorũ aliorũ.</s> <s xml:id="echoid-s33214" xml:space="preserve"> Si er-<lb/>go uiſus à puncto g appropinquet ſpeculo ſecundũ punctũ k cadentẽ inter puncta g & z:</s> <s xml:id="echoid-s33215" xml:space="preserve"> ſi à pũcto <lb/>k ducatur linea ad punctũ d, quę ſit k d:</s> <s xml:id="echoid-s33216" xml:space="preserve"> palã per 14 th 1 huius, & ex pręmiſsis quòd lineę d k & e g cõ <lb/>currãt ultra lineã g k:</s> <s xml:id="echoid-s33217" xml:space="preserve"> ſola enim linea d z æ quidiſtat lineæ e g:</s> <s xml:id="echoid-s33218" xml:space="preserve"> angulus uerò g e d eſtrectus, & angu-<lb/>lus z d b rectus:</s> <s xml:id="echoid-s33219" xml:space="preserve"> ergo angulus k d b eſt obtuſus:</s> <s xml:id="echoid-s33220" xml:space="preserve"> fiet ergo reflexio ad alium pũctũ ſub pũcto k.</s> <s xml:id="echoid-s33221" xml:space="preserve"> A pũ-<lb/>cto uerò z, ut prædictũ eſt, fiet reflexio in ipſum punctum z:</s> <s xml:id="echoid-s33222" xml:space="preserve"> ideo quia linea z d ęquidiſtãs lineæ g e, <lb/>eſt perpẽdicularis ſuք lineam d b per 29 p 1, & ex hypotheſi.</s> <s xml:id="echoid-s33223" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s33224" xml:space="preserve"> poſito uiſu in quocũ q;</s> <s xml:id="echoid-s33225" xml:space="preserve"> <lb/>puncto lineę z b (quoniá à quolibet punctorũ illorũ eſt ducere perpendicularẽ ſuper ſuperficiẽ ſpe-<lb/>culi, uel ſuper lineá d b) reflectitur illarum quælibetin ſe ipſam ք 21 huius.</s> <s xml:id="echoid-s33226" xml:space="preserve"> Palã itaq;</s> <s xml:id="echoid-s33227" xml:space="preserve"> quoniã cõſtitu <lb/>to uiſu in linea g z, non uidebit intuens imaginẽ ſui ipſius:</s> <s xml:id="echoid-s33228" xml:space="preserve"> quia, ut dictum eſt, ſola perpendicularis <lb/>ſecundum unicum punctũ reflectitur ad uiſum, non aũt alia puncta formæ.</s> <s xml:id="echoid-s33229" xml:space="preserve"> Quia uerò angulus i d z <lb/>eſt æ qualis angulo z d g, & linea z d eſt perpendicularis ſuper ſuperficiẽ ſpeculi d b:</s> <s xml:id="echoid-s33230" xml:space="preserve"> ergo per 20 hu-<lb/>ius forma puncti i à puncto ſpeculi d reflectitur ad uiſum in puncto g exiſtentem.</s> <s xml:id="echoid-s33231" xml:space="preserve"> Et quia angulus <lb/>t b q eſt æ qualis angulo g b q, ut patet expræmiſsis, & linea b q perpendicularis eſt ſuper ſuperficiẽ <lb/>ſpeculi:</s> <s xml:id="echoid-s33232" xml:space="preserve"> palàm per 20 huius quoniam forma puncti t à puncto ſpeculi b reflectitur ad uiſum in pun <lb/>cto g:</s> <s xml:id="echoid-s33233" xml:space="preserve"> ergo per 34 huius forma totius lineæ i t reflectitur à ſpeculo d b ad uiſum in puncto g.</s> <s xml:id="echoid-s33234" xml:space="preserve"> Non ui <lb/>debitur autem ipſa tabula depicta i t, quoniã eſt ſub ſuperficie, cui ſuperſtat ſpeculũ & uiſus.</s> <s xml:id="echoid-s33235" xml:space="preserve"> Poteſt <lb/>aũt ſic fieri ut ſecundum longitudinẽ lineæ z b ſit factus murus ſuper terram ad altitudinem uiden-<lb/>tium, qui interius ſit concauus, ſuperius uerſus ſpeculum apertus:</s> <s xml:id="echoid-s33236" xml:space="preserve"> & in illo muro deponatur tabu-<lb/>la picta, quæ eſt i t, ęquidiſtanter ſpeculo b d, & ſit uiſus in diſtantia à ſpeculo ſecundum ſitum pun-<lb/>cti g, & ſit prohibitus per aliquod inedium, ne poſsit propius accedere:</s> <s xml:id="echoid-s33237" xml:space="preserve"> tunc enim omnes formæ pũ <lb/>ctorũ depictæ imaginis incidẽt uiſui.</s> <s xml:id="echoid-s33238" xml:space="preserve"> Diſponatur ergo taliter ք ingeniũ, ut tabula depicta nullo mo <lb/>do uideatur:</s> <s xml:id="echoid-s33239" xml:space="preserve"> & ſit ſpeculũ ſitũ uerſus lumen, ita utaer circa ipſum ſit luminoſus:</s> <s xml:id="echoid-s33240" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s33241" xml:space="preserve"> tabula depicta <lb/>ſimiliter lumẽ habens:</s> <s xml:id="echoid-s33242" xml:space="preserve"> quia aliter in tenebris latens non poſſet uideri:</s> <s xml:id="echoid-s33243" xml:space="preserve"> mediãte enim lumine formã <lb/>fuã multiplicat ք medium, & peruenit ad ſpeculum, & reflectitur ad uiſum.</s> <s xml:id="echoid-s33244" xml:space="preserve"> Palã ergo propoſitum.</s> <s xml:id="echoid-s33245" xml:space="preserve"/> </p> <div xml:id="echoid-div1368" type="float" level="0" n="0"> <figure xlink:label="fig-0518-01" xlink:href="fig-0518-01a"> <variables xml:id="echoid-variables569" xml:space="preserve">e a d f g k z b i h q t</variables> </figure> </div> <pb o="217" file="0519" n="519" rhead="LIBER QVINTVS."/> </div> <div xml:id="echoid-div1370" type="section" level="0" n="0"> <head xml:id="echoid-head1068" xml:space="preserve" style="it">57. Poßibile est ſpeculum unum planum in camera propriataliter ſiſti, ut in ipſo uideantur <lb/>ea, quæ geruntur in domo alia uel in uicis & plateis. Ptolemæus 7 th. 2 catoptr.</head> <p> <s xml:id="echoid-s33246" xml:space="preserve">Sit in camera uidentis locus aliquis, in quo exiſtente uiſu placet uidere per ſpeculum planum o-<lb/>mne illud, quod alibiagitur:</s> <s xml:id="echoid-s33247" xml:space="preserve"> quilocus cameræ, in quo ſiſtitur centrũ uiſus, ſit ſignatus puncto a:</s> <s xml:id="echoid-s33248" xml:space="preserve"> & <lb/>ſit locus, in quo eſt uoluntas aliquid uidendi, quod in <lb/>illo loco agitur, ſignaturs puncto b:</s> <s xml:id="echoid-s33249" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s33250" xml:space="preserve"> rima ſiue fene <lb/> <anchor type="figure" xlink:label="fig-0519-01a" xlink:href="fig-0519-01"/> ſtra in camera uidentis oppoſita loco b, quę ſit g:</s> <s xml:id="echoid-s33251" xml:space="preserve"> & du <lb/>catur linea b g:</s> <s xml:id="echoid-s33252" xml:space="preserve"> & producatur in continuũ & directũ <lb/>intra cameram ad aliquem punctum, qui ſit d:</s> <s xml:id="echoid-s33253" xml:space="preserve"> quod to <lb/>tum poteſt fieri per aſtrolabium ſiue quadrantẽ uel a-<lb/>liud inſtrumentum certificationis uiſuum:</s> <s xml:id="echoid-s33254" xml:space="preserve"> uiſo enim <lb/>puncto b, reuoluatur uiſus fixo inſtrumento, & cadat <lb/>uiſus per eaſdem pinnulas immotas in punctũ came-<lb/>ræ d.</s> <s xml:id="echoid-s33255" xml:space="preserve"> Ducantur ergo lineę d a & g a:</s> <s xml:id="echoid-s33256" xml:space="preserve"> & diuidatur linea <lb/>g a per 119 th.</s> <s xml:id="echoid-s33257" xml:space="preserve"> 1 huius in puncto e, ita ut ſit proportio li <lb/>neæ a e ad lineã e g, ſicut lineæ a d ad lineam d g:</s> <s xml:id="echoid-s33258" xml:space="preserve"> quæ <lb/>ambę per inſtrumẽti acceptionẽ ſunt notæ:</s> <s xml:id="echoid-s33259" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s33260" xml:space="preserve"> <lb/>linea e d:</s> <s xml:id="echoid-s33261" xml:space="preserve"> diuidet ergo per 3 p 6 linea d e angulum a d <lb/>g per æ qualia.</s> <s xml:id="echoid-s33262" xml:space="preserve"> Ponatur itaq;</s> <s xml:id="echoid-s33263" xml:space="preserve"> ſpeculũ perpẽdiculariter <lb/>erectũ ſuper lineã d e in puncto d per cõuerſam 11 p 11, <lb/>in quo ſpeculo ſit linea f h.</s> <s xml:id="echoid-s33264" xml:space="preserve"> A puncto itaq;</s> <s xml:id="echoid-s33265" xml:space="preserve"> ſpeculi d re <lb/>flectetur forma puncti g ad uiſum a per 20 huius:</s> <s xml:id="echoid-s33266" xml:space="preserve"> ergo <lb/>& forma puncti b per eandẽ 20 huius:</s> <s xml:id="echoid-s33267" xml:space="preserve"> diſtantia enim ſecundũ eandem lineam naturã reflexionis nó <lb/>immutat.</s> <s xml:id="echoid-s33268" xml:space="preserve"> Videbit itaq;</s> <s xml:id="echoid-s33269" xml:space="preserve"> uiſus ſecundũ eius centrũ in puncto cameræ, quod eſta, exiſtens, omne, qđ <lb/>erit, & quod agetur in loco b, ſiue ſit domus alia ſiue uicus ſiue platea.</s> <s xml:id="echoid-s33270" xml:space="preserve"> Et hoc eſt, quod ꝓ ponebatur.</s> <s xml:id="echoid-s33271" xml:space="preserve"/> </p> <div xml:id="echoid-div1370" type="float" level="0" n="0"> <figure xlink:label="fig-0519-01" xlink:href="fig-0519-01a"> <variables xml:id="echoid-variables570" xml:space="preserve">a f h e d g b</variables> </figure> </div> </div> <div xml:id="echoid-div1372" type="section" level="0" n="0"> <head xml:id="echoid-head1069" xml:space="preserve" style="it">58. Poßibile eſt ſpeculum ex ſpeculis planis compoſitum conſtrui, in quo uideantur ſolius aſpi-<lb/>cientis plures imagines ad modum chorearum. Ptolemæus 6 th. 2 catoptr.</head> <p> <s xml:id="echoid-s33272" xml:space="preserve">Aſſumatur arcus circuli a z, cuius centrum ſit h:</s> <s xml:id="echoid-s33273" xml:space="preserve"> & quoniam arcus a z indefinitus aſſumitur, eſto, <lb/>ut ipſe exempli cauſſa, diuiſus ſit in quinq;</s> <s xml:id="echoid-s33274" xml:space="preserve"> partes æ quales, uel quotcũq;</s> <s xml:id="echoid-s33275" xml:space="preserve"> quis uoluerit, partes, ita ut <lb/>arcui a b ſint æ quales arcus b g, g d, d e, e z:</s> <s xml:id="echoid-s33276" xml:space="preserve"> & ducantur chordę a b, b g, g d, d e, e z, quæ omnes erunt <lb/>æquales per 29 p 3:</s> <s xml:id="echoid-s33277" xml:space="preserve"> & à centro h ducantur lineę h a, h b, h g, h d, h e, h z:</s> <s xml:id="echoid-s33278" xml:space="preserve"> & ablatis arcubus ſuper chor <lb/>das a b & b g & alijs, erigantur ſpecula plana quadrangula parallelogramma, ita ut eorum latera a i <lb/>b k, g l, d m, e n, z x ſint ęquidiſtátia:</s> <s xml:id="echoid-s33279" xml:space="preserve"> & ſint ſpecula con <lb/>tinua ad inuicem taliter, ut latera eorũ, quæ ſunt b k, <lb/>g l, d m, e n ſint cõmunia:</s> <s xml:id="echoid-s33280" xml:space="preserve"> ſint aũt ſpecula adinuicem <lb/> <anchor type="figure" xlink:label="fig-0519-02a" xlink:href="fig-0519-02"/> taliter cóp oſita, ut anguli contenti à lineis a i & i k:</s> <s xml:id="echoid-s33281" xml:space="preserve"> b <lb/>k, & k l:</s> <s xml:id="echoid-s33282" xml:space="preserve"> g l, & l m:</s> <s xml:id="echoid-s33283" xml:space="preserve"> d m, & m n:</s> <s xml:id="echoid-s33284" xml:space="preserve"> e n, & n x ſint æ quales an <lb/>gulis contentis à lineis h a & a b:</s> <s xml:id="echoid-s33285" xml:space="preserve"> h b & b g:</s> <s xml:id="echoid-s33286" xml:space="preserve"> h g & g d:</s> <s xml:id="echoid-s33287" xml:space="preserve"> h <lb/>e & e z:</s> <s xml:id="echoid-s33288" xml:space="preserve"> ſintq́ ſuperficies inſiſtétes lineis a b, b g, g d, d <lb/>e, e z uerſę inferius, & ſuppoſitæ ſuperficiebus alijs ſu <lb/>perius eleuatis, in quibus ſunt lineæ i k, k l, l m, m n, n <lb/>x:</s> <s xml:id="echoid-s33289" xml:space="preserve"> & ſint ſuperficies ſuperiores inferioribus æquidi-<lb/>ſtantes:</s> <s xml:id="echoid-s33290" xml:space="preserve"> hęc enim omnia ſpecula taliter diſpoſita aſpe <lb/>ctum uniformẽ habebunt ad uiſum exiſtenté in cen-<lb/>tro h.</s> <s xml:id="echoid-s33291" xml:space="preserve"> Quonιam enim lineæ h a, h b, h g, h d, h e, h z du-<lb/>cuntur à centro h ad puncta cōmunia chordis & arcu <lb/>bus, patet per 18 p 3 quoniá omnes ſunt perpendicu-<lb/>lares ſuper lineas circulum a z in illis purctis contin-<lb/>gentes:</s> <s xml:id="echoid-s33292" xml:space="preserve"> ergo per 21 huius omnes illæ lineæ reflectun-<lb/>tur in ſe ipſas:</s> <s xml:id="echoid-s33293" xml:space="preserve"> erit ergo diſtin ctio imaginũ ſecundum <lb/>illas:</s> <s xml:id="echoid-s33294" xml:space="preserve"> ſed & perpendiculares, quæ à puncto h ducuntur ſuper ſuperficies ſpeculorum planorũ, quæ <lb/>per 20 th.</s> <s xml:id="echoid-s33295" xml:space="preserve"> 1 huius ſolùm numerantur numero ſuperficierum ſpeculorũ:</s> <s xml:id="echoid-s33296" xml:space="preserve"> & circa omnes illas fit uni-<lb/>formis reflexio ad uiſum:</s> <s xml:id="echoid-s33297" xml:space="preserve"> numerabuntur ergo imagines numero ſpeculorum, quorum numero & <lb/>loca imaginum numerantur:</s> <s xml:id="echoid-s33298" xml:space="preserve"> ideo quia à puncto h productæ perpendiculares non concurrunt ul-<lb/>tra ſpecula, cum omnes in puncto h concurrant:</s> <s xml:id="echoid-s33299" xml:space="preserve"> eſt autem locus cuiuſq;</s> <s xml:id="echoid-s33300" xml:space="preserve"> imaginis in concurſu ca-<lb/>theti cum linea reflexionis per 37 huius.</s> <s xml:id="echoid-s33301" xml:space="preserve"> Et cum hæc ſpecula uniformiter reſpiciant uiſum in pun <lb/>cto h:</s> <s xml:id="echoid-s33302" xml:space="preserve"> patet quòd qua ratione reflexio fit ab uno ipſorum ad uiſum, eadem ratione fit reflexio à quo-<lb/>libet aliorum:</s> <s xml:id="echoid-s33303" xml:space="preserve"> & ſic reflexionum lineæ numerantur numero cathetorum.</s> <s xml:id="echoid-s33304" xml:space="preserve"> Plures ergo uidebuntur <lb/>imagines diſp oſitę adinuicem numero & ordine ſpeculorum.</s> <s xml:id="echoid-s33305" xml:space="preserve"> Quia uerò ſpecula reſpiciunt uiſum, <lb/>ut ſui centrum, ad modũ arcus circuli, & imagines ipſius uidentis reſpicient uidentem ad modum <lb/>chorearũ.</s> <s xml:id="echoid-s33306" xml:space="preserve"> Quod eſt propoſitũ.</s> <s xml:id="echoid-s33307" xml:space="preserve"> Poſſunt & per hoc ſpeculũ uariato ſitu plures elici imaginum ſitua-<lb/>tiones, quod experimentantis induſtriæ cenſuimus relinquendũ, ut ſi ſpeculũ a b ſecundũ baſim ai <lb/>fituetur æquidiſtãs ſuperficiei horizontis, uel ſecundũ alios modos diuerſos, ut libuerit, diuerſetur.</s> <s xml:id="echoid-s33308" xml:space="preserve"/> </p> <div xml:id="echoid-div1372" type="float" level="0" n="0"> <figure xlink:label="fig-0519-02" xlink:href="fig-0519-02a"> <variables xml:id="echoid-variables571" xml:space="preserve">i k j m n x b g d e a y i</variables> </figure> </div> <pb o="218" file="0520" n="520" rhead="VITELLONIS OPTICAE"/> </div> <div xml:id="echoid-div1374" type="section" level="0" n="0"> <head xml:id="echoid-head1070" xml:space="preserve" style="it">59. Poßbile eſt ſpeculum ex ſpeculis planis compoſitum conſtrui, in quo aſpiciens ſuam uideat <lb/>imaginem uolantem. Ptolemæus 6 th. 2. catoptr.</head> <p> <s xml:id="echoid-s33309" xml:space="preserve">Aſſumatur trigonum iſoſceles rectangulum, quod ſit b a g:</s> <s xml:id="echoid-s33310" xml:space="preserve"> & ſit angulus eius, qui b a g, rectus:</s> <s xml:id="echoid-s33311" xml:space="preserve"> & <lb/>linea b g ſecetur in duo æqualia in puncto c:</s> <s xml:id="echoid-s33312" xml:space="preserve"> & ducatur linea a c:</s> <s xml:id="echoid-s33313" xml:space="preserve"> & ſuper lineam a g ponatur ſpecu-<lb/>lum planum, quod ſit z h:</s> <s xml:id="echoid-s33314" xml:space="preserve"> & ſuper lineam b a ponatur aliud ſpeculum planũ, quod lit d e:</s> <s xml:id="echoid-s33315" xml:space="preserve"> & ſit uiſus <lb/>intuentis in linea a c, reſpiciẽs in quodcunq;</s> <s xml:id="echoid-s33316" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0520-01a" xlink:href="fig-0520-01"/> illorum ſpeculorum uoluerit, ut in z h:</s> <s xml:id="echoid-s33317" xml:space="preserve"> & alte <lb/>rum ſpeculum, quod eſt e d, iaceatin plana ſu <lb/>perficie, ſuper quod ſtat intuẽs:</s> <s xml:id="echoid-s33318" xml:space="preserve"> & accedat & <lb/>recedat intuẽs, donec calcanei ſui forma per <lb/>ueniat ad ſpeculum e d:</s> <s xml:id="echoid-s33319" xml:space="preserve"> dico quòd reuerbera <lb/>bitur in aliud ſpeculum, quod eſt z h, in quo <lb/>aſpiciens putabit propriam imaginem uola-<lb/>re:</s> <s xml:id="echoid-s33320" xml:space="preserve"> quoniam uidebit ipſam eleuatam ſecun-<lb/>dum ſe totam in aere, cum tamen ipſe aſpi-<lb/>ciens ſtet ſuper ſuperficiem terrę uel alterius <lb/>rei, in qua eſt ſpeculum e d:</s> <s xml:id="echoid-s33321" xml:space="preserve"> quoniam forma <lb/>calcanei incidens inferiori ſpeculo, quod eſt e d, reflectetur ad ſuperius ſpeculum, & in illo figura-<lb/>bitur tota forma intuentis.</s> <s xml:id="echoid-s33322" xml:space="preserve"> Et ſi intuens mouerit ſe aliqualiter, ita tamen, ut nó muterur ſitus reſpe-<lb/>ctu reflexionũ, quæ fiunt à ſpeculo:</s> <s xml:id="echoid-s33323" xml:space="preserve"> moueri uidebitur imago in aere per 42 huius:</s> <s xml:id="echoid-s33324" xml:space="preserve"> & ſic uidebit aſpi-<lb/>ciens ſuam imaginem uolantẽ.</s> <s xml:id="echoid-s33325" xml:space="preserve"> Quod proponitur.</s> <s xml:id="echoid-s33326" xml:space="preserve"> Et circa hoc plura alia diligentia artificis perqui-<lb/>ret.</s> <s xml:id="echoid-s33327" xml:space="preserve"> Vt aũt idem propoſitũ & aliter melius pateat figuraliter demonſtratũ:</s> <s xml:id="echoid-s33328" xml:space="preserve"> ſit orthogonium trigonũ <lb/>a b c, cuius angulus b a c ſit rectus:</s> <s xml:id="echoid-s33329" xml:space="preserve"> & in cuius latere a b ſituetur ſpeculum planũ, cuius media linea <lb/>ſit d e:</s> <s xml:id="echoid-s33330" xml:space="preserve"> cuius punctus d ſit propinquior puncto b, quàm punctus e:</s> <s xml:id="echoid-s33331" xml:space="preserve"> & ſit trigonũ a b c ſecundũ eius la <lb/>rus a b poſitum in ſuperficie horizontis uel alia quacunq;</s> <s xml:id="echoid-s33332" xml:space="preserve"> ſuperficie, ſuper quã eleuata ſit ſtatura in-<lb/>tuentis, cuius plantæ pedis ſtent in puncto g aliqualiter eleuato ſuper lineam a b:</s> <s xml:id="echoid-s33333" xml:space="preserve"> & ducatur linea g <lb/>d:</s> <s xml:id="echoid-s33334" xml:space="preserve"> & ſuper punctũ d terminũ lineæ b d fiat per 23 p 1 angulus æqualis angulo g d b, qui ſit h d a, pro-<lb/>ducta linea d h ad lineam a c:</s> <s xml:id="echoid-s33335" xml:space="preserve"> & ſuper punctum h terminum lineæ c h fiatangulus d h k æqualis an-<lb/>gulo d h a, producta linea h k ad lineam b c:</s> <s xml:id="echoid-s33336" xml:space="preserve"> poſitoq́;</s> <s xml:id="echoid-s33337" xml:space="preserve"> <lb/>centro uiſus in puncto k:</s> <s xml:id="echoid-s33338" xml:space="preserve"> pater ex præmiſsis & per <lb/>20 huius, quoniam forma puncti g à puncto h refle-<lb/> <anchor type="figure" xlink:label="fig-0520-02a" xlink:href="fig-0520-02"/> ctetur ad uiſum, ſi punctum h ſuerit punctum ſpecu <lb/>li alicuius:</s> <s xml:id="echoid-s33339" xml:space="preserve"> inuentoq́;</s> <s xml:id="echoid-s33340" xml:space="preserve"> per 46 huius in ſpeculo d e <lb/>puncto reflexionis formæ puncti m, quod ſit in uer-<lb/>tice uidẽtis:</s> <s xml:id="echoid-s33341" xml:space="preserve"> ſit formæ puncti illius punctus reflexio <lb/>nis e:</s> <s xml:id="echoid-s33342" xml:space="preserve"> & ducatur linea m e:</s> <s xml:id="echoid-s33343" xml:space="preserve"> & angulo m e d ſuper pun <lb/>ctum e terminum lineę m e per 23 p 1 fiat æqualis an-<lb/>gulus, qui ſit a e l;</s> <s xml:id="echoid-s33344" xml:space="preserve"> producta linea el ad lineam a c:</s> <s xml:id="echoid-s33345" xml:space="preserve"> & <lb/>inter puncta a & h ſituertur ſpeculum, quod ſit l h, <lb/>ita quòd puncta l & h ſint in ſuperficie illius ſpeculi, <lb/>& ſimiliter punctum a.</s> <s xml:id="echoid-s33346" xml:space="preserve"> Et quoniam forma puncti m <lb/>à puncto ſpeculi d e (quod eſt e) reflectitur ad totam <lb/>ſuperficiem ſpeculi 1 h per 22 huius, & ab illo puncto <lb/>ſpeculi 1 h, in quo anguli e l a, & h l k ſunt æquales, <lb/>(quodcunq;</s> <s xml:id="echoid-s33347" xml:space="preserve"> enim ſuerit illud punctú, ſemper ipſum <lb/>dicatur punctuml) fiet reflexio ad uiſum k.</s> <s xml:id="echoid-s33348" xml:space="preserve"> Quoniá <lb/>enim, ut patet per 26 huius, anguli k l c, & k h c ſunt <lb/>acuti, patet per 14 th.</s> <s xml:id="echoid-s33349" xml:space="preserve"> 1 huius quoniam illæ lineę con <lb/>current, ſitq́;</s> <s xml:id="echoid-s33350" xml:space="preserve"> punctus concurſus k.</s> <s xml:id="echoid-s33351" xml:space="preserve"> Palàm ergo per <lb/>34 huius quòd tota imago aſpicientis, quæ eſt linea <lb/>g m, à ſuperficie ſpeculi e d reflectitur ad ſpeculum <lb/>l h, & à ſuperficie ſpeculi l h reflectitur ad uiſum exi-<lb/>ſtentem in puncto k.</s> <s xml:id="echoid-s33352" xml:space="preserve"> Et quoniam, ut pater per 37 huius, locus imaginis formæ uniuſcuiuſq;</s> <s xml:id="echoid-s33353" xml:space="preserve"> puncti <lb/>eſt in concurſu catheti ſuæ incidentiæ cum linea ſuæ reflexionis:</s> <s xml:id="echoid-s33354" xml:space="preserve"> producatur itaq;</s> <s xml:id="echoid-s33355" xml:space="preserve"> à puncto ſpeculi <lb/>d e, à quo fit reflexio formæ puncti g, quod eſt d, per 11 p 11 linea perpendicularis ſuper ſpeculi a h ſu <lb/>perficiem:</s> <s xml:id="echoid-s33356" xml:space="preserve"> & patet, cum exhypotheſi angulus d a h ſit rectus, quòd illa perpendicularis eſt linea d a.</s> <s xml:id="echoid-s33357" xml:space="preserve"> <lb/>Similiter quoq;</s> <s xml:id="echoid-s33358" xml:space="preserve"> perpendicularis à puncto reflexionis formæ punctim, quod eſt ſpeculi de punctũ <lb/>e, ducta ſuper ſuperficiem ſpeculi a h, eſt eadem linea, quæ e a:</s> <s xml:id="echoid-s33359" xml:space="preserve"> hæc itaq;</s> <s xml:id="echoid-s33360" xml:space="preserve"> linea eſt cathetus inciden-<lb/>tiæ formarum punctorum g & m reflexorum à punctis d & e ad ſpeculum l h.</s> <s xml:id="echoid-s33361" xml:space="preserve"> Et quoniam, ut præ-<lb/>miſſum eſt, patet per 26 huius, quòd anguli k h c & k l c ſunt acuti, quoniá linea angulum d h k uel <lb/>e l k per æ qualia diuidens, eſt perpendicularis ſuper lineam l h:</s> <s xml:id="echoid-s33362" xml:space="preserve"> angulus uerò d a h eſt rectus:</s> <s xml:id="echoid-s33363" xml:space="preserve"> ergo <lb/>per 14 th.</s> <s xml:id="echoid-s33364" xml:space="preserve"> 1 huius linea d e a concurret cum ambabus lineis k l & k h:</s> <s xml:id="echoid-s33365" xml:space="preserve"> ſit ergo, ut punctus concurſus <lb/>linearum d a & k h ſit n:</s> <s xml:id="echoid-s33366" xml:space="preserve"> & punctus concurſus linearũ e a & k l ſit o.</s> <s xml:id="echoid-s33367" xml:space="preserve"> Erit ergo linea o n imago formæ <lb/> <pb o="219" file="0521" n="521" rhead="LIBER QVINTVS."/> totius lineæ m g:</s> <s xml:id="echoid-s33368" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s33369" xml:space="preserve"> punctum, quod eſt imago formæ puncti g, plantarum ſcilicet ipſius intuen-<lb/>tis altius in aere quàm punctum o, quod eſt imago formæ puncti m, uerticis ipſius uidentis.</s> <s xml:id="echoid-s33370" xml:space="preserve"> Vide-<lb/>bit ergo ex puncto k intuens ſpeculum l h, ſuam imaginem in aere uolantem:</s> <s xml:id="echoid-s33371" xml:space="preserve"> quoniam uidebit pe-<lb/>des altius in aere quàm ipſum caput, collocatos.</s> <s xml:id="echoid-s33372" xml:space="preserve"> Per eadem quoq;</s> <s xml:id="echoid-s33373" xml:space="preserve"> demonſtrandum, ſi trigonum <lb/>a b c fuerit oxygonium, niſi quòd imago intuentis aliam recipit ſitus diſpoſitionem:</s> <s xml:id="echoid-s33374" xml:space="preserve"> catheti enim <lb/>incidentiæ aliter ſuperficiei ſpeculi incidunt quàm <lb/> <anchor type="figure" xlink:label="fig-0521-01a" xlink:href="fig-0521-01"/> prius:</s> <s xml:id="echoid-s33375" xml:space="preserve"> ſemper tamen trigono a b c exiſtente orthogo <lb/>nio uel oxygonio uidebitur imago intuentis uolans <lb/>ſub ſpeculo.</s> <s xml:id="echoid-s33376" xml:space="preserve"> Quòd ſi trigonum a b c fuerit ambly-<lb/>gonium, poſsibile eſt fieri, ut imago ſit uolans in aere <lb/>retro uiſum:</s> <s xml:id="echoid-s33377" xml:space="preserve"> quoniã ut patet per 14 th.</s> <s xml:id="echoid-s33378" xml:space="preserve"> 1 huius, catheti <lb/>incidentiæ & lineæ reflexionum concurrent retro <lb/>centrum uiſus.</s> <s xml:id="echoid-s33379" xml:space="preserve"> Non uidebitur autẽ talis imago, quo-<lb/>niam ſemper fugiet abſcõſa ab ipſo uiſu, niſi fortè ab <lb/> <anchor type="figure" xlink:label="fig-0521-02a" xlink:href="fig-0521-02"/> alio ſpeculo tertio ad uiſum poſſet fieri reflexio.</s> <s xml:id="echoid-s33380" xml:space="preserve"> Patet ergo illud, quod proponebatur:</s> <s xml:id="echoid-s33381" xml:space="preserve"> & hoc:</s> <s xml:id="echoid-s33382" xml:space="preserve"> uiſu <lb/>ſolùm reſpiciente in ſpeculum a h, non in ſpeculũ d e.</s> <s xml:id="echoid-s33383" xml:space="preserve"> Et hæc quidem demonſtrata ſunt, ac ſi à pun-<lb/>ctis primarum reflexionum, quæ ſunt d & e, ducantur catheti incidentiæ:</s> <s xml:id="echoid-s33384" xml:space="preserve"> quæ ſi imaginentur à lo-<lb/>cis primarum imaginum duci, multò fortius ſecundæ imagines, quæ uidentur in ſpeculo a h, uide-<lb/>buntur eſſe diſpoſitæ, ut uolantes.</s> <s xml:id="echoid-s33385" xml:space="preserve"/> </p> <div xml:id="echoid-div1374" type="float" level="0" n="0"> <figure xlink:label="fig-0520-01" xlink:href="fig-0520-01a"> <variables xml:id="echoid-variables572" xml:space="preserve">a N d y e g c b</variables> </figure> <figure xlink:label="fig-0520-02" xlink:href="fig-0520-02a"> <variables xml:id="echoid-variables573" xml:space="preserve">n o a e l y d y c m g k</variables> </figure> <figure xlink:label="fig-0521-01" xlink:href="fig-0521-01a"> <variables xml:id="echoid-variables574" xml:space="preserve">a n o e j d y g c m g k</variables> </figure> <figure xlink:label="fig-0521-02" xlink:href="fig-0521-02a"> <variables xml:id="echoid-variables575" xml:space="preserve">a e j d y g c k m n o g</variables> </figure> </div> </div> <div xml:id="echoid-div1376" type="section" level="0" n="0"> <head xml:id="echoid-head1071" xml:space="preserve" style="it">60. Per duo ueltria ſpecula plana orthogonaliter ad inuicẽ diſpoſita, poſsibile eſt eiuſdem pun <lb/>cti imaginem uideri. Euclides 13 th. catoptr.</head> <p> <s xml:id="echoid-s33386" xml:space="preserve">Sit uiſibile aliquod, in quo ſit punctum a:</s> <s xml:id="echoid-s33387" xml:space="preserve"> & ſit centrum uiſus punctũ b:</s> <s xml:id="echoid-s33388" xml:space="preserve"> & ſint tria ſpecula plana <lb/>g d, d e & e z orthogonaliter ad inuicẽ diſpoſita:</s> <s xml:id="echoid-s33389" xml:space="preserve"> ducatur quoq;</s> <s xml:id="echoid-s33390" xml:space="preserve"> à puncto a linea a z perpendiculari <lb/>ter ſuper ſuperficiem ſpeculi e z per 11 p 11:</s> <s xml:id="echoid-s33391" xml:space="preserve"> & producatur linea a z in continuũ:</s> <s xml:id="echoid-s33392" xml:space="preserve"> abſcindaturq́;</s> <s xml:id="echoid-s33393" xml:space="preserve"> in pun <lb/>cto c taliter per 3 p 1, ut linea z c ſit <lb/> <anchor type="figure" xlink:label="fig-0521-03a" xlink:href="fig-0521-03"/> æqualis lineę a z:</s> <s xml:id="echoid-s33394" xml:space="preserve"> & à puncto b, qđ <lb/>eſt centrum uiſus, ducatur linea b <lb/>g perpendiculariter ſuper ſpeculũ <lb/>d g:</s> <s xml:id="echoid-s33395" xml:space="preserve"> & producatur taliter, ut linea g <lb/>s ſit æ qualis lineæ b g.</s> <s xml:id="echoid-s33396" xml:space="preserve"> A puncto <lb/>quoq;</s> <s xml:id="echoid-s33397" xml:space="preserve"> c ducatur perpendicularis <lb/>ſuper ſuperficiem ſpeculi d e, quæ <lb/>ſit c k:</s> <s xml:id="echoid-s33398" xml:space="preserve"> & producatur ultra punctũ <lb/>k a d punctũ l, quouſq;</s> <s xml:id="echoid-s33399" xml:space="preserve"> linea c k ſit <lb/>æ qualis lineæ k l:</s> <s xml:id="echoid-s33400" xml:space="preserve"> & à puncto l du-<lb/>catur linea ad punctũ s, ſecans ſpe-<lb/>culum d e in puncto m, & ſpeculũ <lb/>d g in puncto f:</s> <s xml:id="echoid-s33401" xml:space="preserve"> & à puncto m duca <lb/>tur ad punctum c linea m c ſecans <lb/>ſpeculum e z in puncto r:</s> <s xml:id="echoid-s33402" xml:space="preserve"> & ducan <lb/>tur lineæ a r & b f.</s> <s xml:id="echoid-s33403" xml:space="preserve"> Quia ergo linea <lb/>b g eſt æ qualis lineę g s, & linea g f <lb/>cōmunis ambobus trigonis s g f & g f b, & angulus b g f æqualis eſt angulo s g f, quia ambo illi an-<lb/>guli ſunt recti:</s> <s xml:id="echoid-s33404" xml:space="preserve"> erit per 4 p 1 linea b f æ qualis lineæ s f, & angulus g f b æ qualis angulo g f s, & angu-<lb/>lus f b g æqualis angulo f s g:</s> <s xml:id="echoid-s33405" xml:space="preserve"> ſed angulus s f g eſt æ qualis angulo d f m per 15 p 1:</s> <s xml:id="echoid-s33406" xml:space="preserve"> ergo angulus d f m <lb/>æ qualis eſt angulo g f b.</s> <s xml:id="echoid-s33407" xml:space="preserve"> Poteſt ergo ք 20 huius forma puncti m reflecti ad uiſum b.</s> <s xml:id="echoid-s33408" xml:space="preserve"> Quia uerò linea <lb/>c k eſt æqualis lineæ k l, & linea k m cõmunis eſt ambob.</s> <s xml:id="echoid-s33409" xml:space="preserve"> trigonis c k m & l m k:</s> <s xml:id="echoid-s33410" xml:space="preserve"> angulus quoq;</s> <s xml:id="echoid-s33411" xml:space="preserve"> l k m <lb/>æqualis eſt angulo m k c, quia ambo recti:</s> <s xml:id="echoid-s33412" xml:space="preserve"> erit per 4 p 1 linea l m æ qualis lineæ m c, & angulus l m k <lb/> <pb o="220" file="0522" n="522" rhead="VITELLONIS OPTICAE"/> æ qualis angulo k m c:</s> <s xml:id="echoid-s33413" xml:space="preserve"> ergo angulus d m f eſt æ qualis angulo k m c:</s> <s xml:id="echoid-s33414" xml:space="preserve"> quoniam per 15 p 1 ipſe eſt æ qua <lb/>lis angulo l m k:</s> <s xml:id="echoid-s33415" xml:space="preserve"> ergo per 20 huius forma puncti r poteſt reflecti à puncto m ad punctũ f:</s> <s xml:id="echoid-s33416" xml:space="preserve"> & à puncto <lb/>f ad punctum b centrum uiſus.</s> <s xml:id="echoid-s33417" xml:space="preserve"> Per duo ergo ſpecula, quæ ſunt d e & d g, uidetur forma puncti r re-<lb/>flexa ad idem centrum uiſus, quod eſt b.</s> <s xml:id="echoid-s33418" xml:space="preserve"> Et quia linea a z eſt æ qualis lineę z c, & linea z r communis <lb/>eſt ambobus trigonis a r z & z r c:</s> <s xml:id="echoid-s33419" xml:space="preserve"> angulus quoq;</s> <s xml:id="echoid-s33420" xml:space="preserve"> a z r eſt æqualis angulo r z c, quia ambo recti <lb/>ſunt:</s> <s xml:id="echoid-s33421" xml:space="preserve"> erit angulus a r z per 4 p 1 æ qualis angulo z r c:</s> <s xml:id="echoid-s33422" xml:space="preserve"> ergo per 15 p 1 angulus m r e eſt æ qualis angulo <lb/>a r z.</s> <s xml:id="echoid-s33423" xml:space="preserve"> Forma ergo puncti a reflectitur à puncto r ſpeculi z e ad punctum m ſpeculi d e, & à puncto m <lb/>ad punctum f ſpeculi d g, & à puncto f ad centrum uiſus b.</s> <s xml:id="echoid-s33424" xml:space="preserve"> A tribus ergo ſpeculis uidetur forma & <lb/>imago eiuſdem puncti a.</s> <s xml:id="echoid-s33425" xml:space="preserve"> Quod eſt propoſitũ:</s> <s xml:id="echoid-s33426" xml:space="preserve"> & hoc accidit uiſu ſolùm reſpiciente in ſpeculum d g.</s> <s xml:id="echoid-s33427" xml:space="preserve"/> </p> <div xml:id="echoid-div1376" type="float" level="0" n="0"> <figure xlink:label="fig-0521-03" xlink:href="fig-0521-03a"> <variables xml:id="echoid-variables576" xml:space="preserve">l k e m d r f c z a b g s</variables> </figure> </div> </div> <div xml:id="echoid-div1378" type="section" level="0" n="0"> <head xml:id="echoid-head1072" xml:space="preserve" style="it">61. Poßibile eſt per quotcun quis uoluerit plana ſpecula ſecundum diſpoſitionem polygonij <lb/>æquilateri & æquianguli ad inuicem diſpoſita, eiuſdem puncti imaginem uideri. Euclides 14 <lb/>th. catoptr. Ptolemæus 8 th. 2 catoptr.</head> <p> <s xml:id="echoid-s33428" xml:space="preserve">Sit centrum uiſus punctum a:</s> <s xml:id="echoid-s33429" xml:space="preserve"> & punctum rei uiſæ ſit b:</s> <s xml:id="echoid-s33430" xml:space="preserve"> & ducatur linea a b:</s> <s xml:id="echoid-s33431" xml:space="preserve"> & ſecundum quanti-<lb/>tatem lineæ a b deſcribatur polygonium æ quilaterum & æ quiangulũ, quotcunq;</s> <s xml:id="echoid-s33432" xml:space="preserve"> laterũ uiſum fue-<lb/>rit ordinari.</s> <s xml:id="echoid-s33433" xml:space="preserve"> Sit autem nunc, exempli cauſſa, polygonium a e d g b pentagonũ:</s> <s xml:id="echoid-s33434" xml:space="preserve"> cui circunſcribatur <lb/>circulus per 14 p 4:</s> <s xml:id="echoid-s33435" xml:space="preserve"> & ducantur lineæ ad centrum circuli, quod ſit c, ab angul:</s> <s xml:id="echoid-s33436" xml:space="preserve"> s polygonij, quæ ſint <lb/>a c, e c, d c, g c, b c:</s> <s xml:id="echoid-s33437" xml:space="preserve"> palàm itaq;</s> <s xml:id="echoid-s33438" xml:space="preserve"> quoniam omnes illæ li-<lb/> <anchor type="figure" xlink:label="fig-0522-01a" xlink:href="fig-0522-01"/> neæ ſunt æquales per definitionem circuli:</s> <s xml:id="echoid-s33439" xml:space="preserve"> anguli er <lb/>go ad baſes oẽs ſunt æ quales per 5 & 8 p 1:</s> <s xml:id="echoid-s33440" xml:space="preserve"> & in con-<lb/>curſu quorumliber dictorum laterum ponatur ſpecu <lb/>lum planum, præter quàm in punctis a & b, ut ad pun <lb/>cta e, d, g:</s> <s xml:id="echoid-s33441" xml:space="preserve"> uel ſi fuerit polygoniũ plurium laterum, po <lb/>nantur plura:</s> <s xml:id="echoid-s33442" xml:space="preserve"> & erigantur omnia orthogonaliter ſu-<lb/>per lineas ad centrum circuli productas, ut ſunt hæ li <lb/>neæ d c & g c:</s> <s xml:id="echoid-s33443" xml:space="preserve"> quod fier per 11 p 11:</s> <s xml:id="echoid-s33444" xml:space="preserve"> ita ut ſpeculum f h <lb/>ſuper lineam g c ſit perpendiculariter in ſiſtens:</s> <s xml:id="echoid-s33445" xml:space="preserve"> ad u-<lb/>num uerò angulum ſit punctum rei uiſæ, & ad alium <lb/>ſibi proximum ſit centrum uiſus, ut ſunt hic puncta <lb/>a & b.</s> <s xml:id="echoid-s33446" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s33447" xml:space="preserve"> angulus c g f eſt æ qualis angulo h g <lb/>c;</s> <s xml:id="echoid-s33448" xml:space="preserve"> quia ambo ſunt recti per 18 p 3:</s> <s xml:id="echoid-s33449" xml:space="preserve"> ſed & angulus c g b <lb/>eſt ęqualis angulo c g d, ut patet per pręmiſſa & per 8 <lb/>p 1:</s> <s xml:id="echoid-s33450" xml:space="preserve"> angulus ergo b g r æ qualis eſt angulo d g h.</s> <s xml:id="echoid-s33451" xml:space="preserve"> Ergo <lb/>forma puncti b à puncto g ſpeculi f h reflectitur ad <lb/>punctum ſpeculi proximi, quod eſt ad punctũ d:</s> <s xml:id="echoid-s33452" xml:space="preserve"> per æ quales enim angulos fit omnis reflexio, ut pa <lb/>tet per 20 huius.</s> <s xml:id="echoid-s33453" xml:space="preserve"> Et quoniã omnes anguli illi præmiſsis duobus angulis ſimiles inter ſe, ſunt æ qua-<lb/>les:</s> <s xml:id="echoid-s33454" xml:space="preserve"> palàm quia fit reflexio à puncto d ad punctũ e, & à puncto e ad punctũ a, quod eſt centrũ uiſus.</s> <s xml:id="echoid-s33455" xml:space="preserve"> <lb/>Viſus itaq;</s> <s xml:id="echoid-s33456" xml:space="preserve"> exiſtens in puncto a, & intuens ſolum ſpeculum, cuius eſt punctũ e, uidebit formam b, <lb/>quæ immediatè nõ reflecteretur ad ipſum à puncto ſpeculi e, reflexam mediantibus ſpeculis g & d.</s> <s xml:id="echoid-s33457" xml:space="preserve"> <lb/>Quod eſt propoſitum.</s> <s xml:id="echoid-s33458" xml:space="preserve"> Quòd ſi centrum uiſus ſit in puncto c, quod eſt centrũ circuli, cuius periphe <lb/>riam contingunt omnia ſpecula in angulis polygoniorũ conſtituta:</s> <s xml:id="echoid-s33459" xml:space="preserve"> palàm quòd forma puncti c ab <lb/>omnibus pũctis reflectitur in ſe ipſam:</s> <s xml:id="echoid-s33460" xml:space="preserve"> quoniã omnes lineę, quæ ſunt c a, c b, c g, c d, c e ſunt perpen-<lb/>diculares ſuper ſpeculo rũ ſuperficies:</s> <s xml:id="echoid-s33461" xml:space="preserve"> reflectuntur ergo in ſe ipſas ad punctũ c per 21 huius.</s> <s xml:id="echoid-s33462" xml:space="preserve"> Palàm <lb/>ergo eſt propoſitũ.</s> <s xml:id="echoid-s33463" xml:space="preserve"> Et ſi plurima ordinantur hoc modo ſpecula, de omnibus eſt eadẽ demonſtratio <lb/>& idem modus circum ſcribendi circulũ alteri polygonio, qui & pentagono.</s> <s xml:id="echoid-s33464" xml:space="preserve"> Per hęc itaq;</s> <s xml:id="echoid-s33465" xml:space="preserve"> duo theo <lb/>remata patet quòd rei, quę nõ uidetur, imago poteſt in ſpeculo uideri:</s> <s xml:id="echoid-s33466" xml:space="preserve"> ut ſi res taliter diſponatur ad <lb/>primũ ſpeculum, quòd ad ipſum uiſus pertingere non poſsit:</s> <s xml:id="echoid-s33467" xml:space="preserve"> hoc autem faciliter accidit cogitanti.</s> <s xml:id="echoid-s33468" xml:space="preserve"/> </p> <div xml:id="echoid-div1378" type="float" level="0" n="0"> <figure xlink:label="fig-0522-01" xlink:href="fig-0522-01a"> <variables xml:id="echoid-variables577" xml:space="preserve">p y e g c f a b</variables> </figure> </div> </div> <div xml:id="echoid-div1380" type="section" level="0" n="0"> <head xml:id="echoid-head1073" xml:space="preserve" style="it">62. A‘ pluribus ſpeculis planis poßibile eſt formã rei per ſe uiſæ, uelrei non uiſæ reflecti ad ui <lb/>ſum, it a ut diſtantia imaginis à centro uiſus ſit æqualis omnibus lineis incidentiæ & ipſi lineæ <lb/>reflexionis.</head> <p> <s xml:id="echoid-s33469" xml:space="preserve">Sit centrũ uiſus in puncto a:</s> <s xml:id="echoid-s33470" xml:space="preserve"> & punctus rei uiſę b:</s> <s xml:id="echoid-s33471" xml:space="preserve"> & inter illos punctos, ſi placet, exempli cauſſa, <lb/>ſit aliqua magnitudo tegens unũ illorum punctorum ab altero, ut paries uel aliud aliquid, quod ſit <lb/>p g:</s> <s xml:id="echoid-s33472" xml:space="preserve"> & à punctis a & b ad oppoſita ipſis loca ducantur lineæ æquidiſtantes per 31 p 1, quæ ſint a d & b <lb/>e:</s> <s xml:id="echoid-s33473" xml:space="preserve"> & copuletur linea d e:</s> <s xml:id="echoid-s33474" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s33475" xml:space="preserve">, exempli cauſſa, lineæ b e & a d perpendiculares ſuper lineam e d:</s> <s xml:id="echoid-s33476" xml:space="preserve"> & <lb/>diuidatur angulus a d e per æ qualia per 9 p 1 ducta linéa d z:</s> <s xml:id="echoid-s33477" xml:space="preserve"> & ſimiliter diuidatur angulus b e d per <lb/>æqualia per lineã e h:</s> <s xml:id="echoid-s33478" xml:space="preserve"> & ſuper punctũ d terminũ lineæ z d erigatur perpendiculariter linea k d c per <lb/>11 p 1:</s> <s xml:id="echoid-s33479" xml:space="preserve"> & ſimiliter ſuper punctũ e terminũ lineæ h e erigatur perpendiculariter linea l e m:</s> <s xml:id="echoid-s33480" xml:space="preserve"> & his dua-<lb/>bus lineis k d c & l e m imaginentur ſuperponi duo plana ſpecula.</s> <s xml:id="echoid-s33481" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s33482" xml:space="preserve"> puncti b incidet ſpe-<lb/>culo plano, quod eſt m e l, in puncto e, & reflectetur in punctũ d per 20 huius:</s> <s xml:id="echoid-s33483" xml:space="preserve"> quia anguli b e m & d <lb/>e l ſunt æ quales:</s> <s xml:id="echoid-s33484" xml:space="preserve"> anguli enim h e l & h e m ſunt æquales, quia recti:</s> <s xml:id="echoid-s33485" xml:space="preserve"> ſed & anguli h e d & h e b ſunt æ-<lb/>quales ex præmiſsis.</s> <s xml:id="echoid-s33486" xml:space="preserve"> Item forma incidens ſpeculo k d c ab eius puncto d reflectetur ad punctum a, <lb/>quod eſt centrũ uiſus per 20 huius:</s> <s xml:id="echoid-s33487" xml:space="preserve"> quoniã, ut ſuprà patuit, anguli e d z & z d a ſunt æ quales.</s> <s xml:id="echoid-s33488" xml:space="preserve"> Vide-<lb/>bitur ergo ſorma puncti b per uiſum exiſtentẽ in puncto a, cum tamen res, in qua eſt punctũ b, non <lb/>ſit uiſibilis per ſe ipſam.</s> <s xml:id="echoid-s33489" xml:space="preserve"> Linea quoq;</s> <s xml:id="echoid-s33490" xml:space="preserve"> reflexionis ad uiſum, quæ eſt d a, eſt ſemper una, quãuis lineæ <lb/> <pb o="221" file="0523" n="523" rhead="LIBER QVINTVS."/> incidentiarũ ſecundum numerum talium ſpeculorum numerentur.</s> <s xml:id="echoid-s33491" xml:space="preserve"> Et ſi à puncto rei uiſę, quod eſt <lb/>b, ducatur per 11 p 11 linea perpendicularis ſuper ſuperficiem ſpeculi, quę ſit b m, ſecans lineam e l m <lb/>in puncto m:</s> <s xml:id="echoid-s33492" xml:space="preserve"> erit angulus b m <lb/>c rectus:</s> <s xml:id="echoid-s33493" xml:space="preserve"> ergo per 32 p 1 erit an-<lb/> <anchor type="figure" xlink:label="fig-0523-01a" xlink:href="fig-0523-01"/> gulus e b m acutus.</s> <s xml:id="echoid-s33494" xml:space="preserve"> Cum ergo <lb/>angulus b e d ſit rectus:</s> <s xml:id="echoid-s33495" xml:space="preserve"> palàm <lb/>per 14 th.</s> <s xml:id="echoid-s33496" xml:space="preserve"> 1 huius quia lineæ b <lb/>m & d e concurrent:</s> <s xml:id="echoid-s33497" xml:space="preserve"> ſit concur <lb/>ſus ipſarum in puncto n.</s> <s xml:id="echoid-s33498" xml:space="preserve"> Quia <lb/>itaq;</s> <s xml:id="echoid-s33499" xml:space="preserve"> linea m e l cadens ſuper li <lb/>neas e h & b n facit angulum e <lb/>m b intrinſecũ æ qualem angu <lb/>lo l e h extrinſeco:</s> <s xml:id="echoid-s33500" xml:space="preserve"> patet per 28 <lb/>p 1 quoniá lineæ b n & e h ſunt <lb/>ęquidiſtantes.</s> <s xml:id="echoid-s33501" xml:space="preserve"> Ergo angulus d <lb/>e h extrinſecus eſt æqualis an-<lb/>gulo e n b intrinſeco per 29 p <lb/>1, & angulus e b n eſt æqualis <lb/>angulo b e h:</s> <s xml:id="echoid-s33502" xml:space="preserve"> quia ſunt coalter <lb/>ni:</s> <s xml:id="echoid-s33503" xml:space="preserve"> ſed angulus b e h eſt æqualis angulo h e d, utpatet ex præmiſsis, diuiſus eſt enim angulus b e d <lb/>per æqualia per lineam h e:</s> <s xml:id="echoid-s33504" xml:space="preserve"> erit ergo angulus e b n æqualis angulo e n b:</s> <s xml:id="echoid-s33505" xml:space="preserve"> ergo per 6 p 1 lineę n e & b c <lb/>ſunt æ quales:</s> <s xml:id="echoid-s33506" xml:space="preserve"> eſt aũt per 37 huius punctũn locus imaginis formę puncti b reflexi ad uiſum exiſten-<lb/>tem in puncto d à ſpeculi m e l puncto e.</s> <s xml:id="echoid-s33507" xml:space="preserve"> Item à puncto n ducatur linea perpendicularis ſuper lineá <lb/>c d k per 12 p 1:</s> <s xml:id="echoid-s33508" xml:space="preserve"> quę ſit n k.</s> <s xml:id="echoid-s33509" xml:space="preserve"> Patet ergo, ut prius, per 32 p 1 quòd angulus d n k eſt acutus:</s> <s xml:id="echoid-s33510" xml:space="preserve"> ſed angulus n <lb/>d a eſt rectus:</s> <s xml:id="echoid-s33511" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s33512" xml:space="preserve"> 1 huius lineę n k & a d productæ concurrent:</s> <s xml:id="echoid-s33513" xml:space="preserve"> ſit punctus concurſus s.</s> <s xml:id="echoid-s33514" xml:space="preserve"> <lb/>Quia itaq;</s> <s xml:id="echoid-s33515" xml:space="preserve"> linea d k cadens ſuper lineas z d & n s facit angulũ z d c extrinſecũ æ qualem angulo n k <lb/>d intrinſeco, uterq;</s> <s xml:id="echoid-s33516" xml:space="preserve"> enim illorũ angulorũ eſt rectus:</s> <s xml:id="echoid-s33517" xml:space="preserve"> patet ergo per 28 p 1 quòd lineę n s & z d æqui-<lb/>diſtant:</s> <s xml:id="echoid-s33518" xml:space="preserve"> ergo per 29 p 1 eſt angulus z d a extrinſecus æqualis angulo n s d intrinſeco:</s> <s xml:id="echoid-s33519" xml:space="preserve"> ſed & anguli s <lb/>n d & n d z ſunt æquales, quia coalterni:</s> <s xml:id="echoid-s33520" xml:space="preserve"> & anguli n d z & z d a ſunt æquales, ut patet ex præmiſsis:</s> <s xml:id="echoid-s33521" xml:space="preserve"> <lb/>angulus enim n d a diuiditur per æqualia per lineã z d:</s> <s xml:id="echoid-s33522" xml:space="preserve"> angulus ergo d n s eſt æ qualis angulo d s n:</s> <s xml:id="echoid-s33523" xml:space="preserve"> <lb/>ergo per 6 p 1 duę lineę d s & d n ſunt æquales.</s> <s xml:id="echoid-s33524" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s33525" xml:space="preserve"> linea e n eſt æqualis lineę e b:</s> <s xml:id="echoid-s33526" xml:space="preserve"> erit linea d <lb/>n æqualis duabus lineis d e & e b:</s> <s xml:id="echoid-s33527" xml:space="preserve"> ergo linea d s eſt æqualis illis eiſdẽ duabus lineis d e & e b.</s> <s xml:id="echoid-s33528" xml:space="preserve"> Et ꝗa <lb/>per 37 huius punctũ s eſt locus imaginis formę puncti n reflexę à puncto ſpeculi k d c, quod eſt d, ad <lb/>uiſum exiſtentẽ in puncto a:</s> <s xml:id="echoid-s33529" xml:space="preserve"> patet quòd linea a s, quę eſt diſtantia imaginis à centro uiſus eſt æqua <lb/>lis duabus lineis incidẽtię, quę ſunt b e & e d, & inſuper lineę reflexionis, quę eſt d a.</s> <s xml:id="echoid-s33530" xml:space="preserve"> Et hoc eſt pro-<lb/>poſitum:</s> <s xml:id="echoid-s33531" xml:space="preserve"> quoniam ſi à pluribus ſpeculis fiat reflexio, eodem penitus modo erit demonſtrandum.</s> <s xml:id="echoid-s33532" xml:space="preserve"/> </p> <div xml:id="echoid-div1380" type="float" level="0" n="0"> <figure xlink:label="fig-0523-01" xlink:href="fig-0523-01a"> <variables xml:id="echoid-variables578" xml:space="preserve">c s d a z p g k l h e b m n</variables> </figure> </div> </div> <div xml:id="echoid-div1382" type="section" level="0" n="0"> <head xml:id="echoid-head1074" xml:space="preserve" style="it">63. Reflexione à pluribus ſpeculis planis ad eundẽ uiſum facta, ab imparibus quidẽ dextra <lb/>apparẽt ſiniſtra, & ſiniſtra dextra: à paribus uerò dextra apparent dextra, & ſiniſtra ſiniſtra: <lb/>& diſtantia imaginis à uiſu conſtabit ex quantitate omnium linearum incidentiæ & lineæ re-<lb/>flexionis. Ptolemæus 3 th. 2 cattoptr.</head> <p> <s xml:id="echoid-s33533" xml:space="preserve">Sιt centrũ uiſus a:</s> <s xml:id="echoid-s33534" xml:space="preserve"> & linea rei uiſę ſit b g:</s> <s xml:id="echoid-s33535" xml:space="preserve"> & ſi placet, ſit inter centrũ uiſus & rem uiſam aliqđ cor-<lb/>pus denſum ſimplicẽ prohibens uiſionẽ, ut paries uel aliquid ſimile, quod ſit d:</s> <s xml:id="echoid-s33536" xml:space="preserve"> fiatq́ reflexio ex tri <lb/>bus ſpeculis, quę ſunt e z & h c & k l:</s> <s xml:id="echoid-s33537" xml:space="preserve"> reflectaturq́;</s> <s xml:id="echoid-s33538" xml:space="preserve"> forma lineę b g per hæc tria ſpecula ad uiſum exi <lb/>ſtentẽ in puncto a:</s> <s xml:id="echoid-s33539" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s33540" xml:space="preserve">, ut punctus b lineę b g incidat ſpeculo k l in pũcto k, & ſpeculo h cin puncto <lb/>h, & ſpeculo e z in puncto e:</s> <s xml:id="echoid-s33541" xml:space="preserve"> reflectaturq́;</s> <s xml:id="echoid-s33542" xml:space="preserve"> ad uiſum a ſecundũ lineam e a.</s> <s xml:id="echoid-s33543" xml:space="preserve"> Et fimiliter forma puncti g <lb/>incidat ſpeculo k lin pũcto l, & ſpeculo h cin pũcto c, & ſpeculo e z in puncto z:</s> <s xml:id="echoid-s33544" xml:space="preserve"> & reflectatur ad ui-<lb/>ſum ſecundũ lineá z a.</s> <s xml:id="echoid-s33545" xml:space="preserve"> Et ducantur hę lineę incidentię & reflexionis, quę erunt b k, k h, h e, e a:</s> <s xml:id="echoid-s33546" xml:space="preserve"> & g l, <lb/>l c, c z, z a:</s> <s xml:id="echoid-s33547" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s33548" xml:space="preserve"> locus imaginis formę puncti b in primo ſpeculo, qđ eſt k l, punctũ t:</s> <s xml:id="echoid-s33549" xml:space="preserve"> & locus imagi-<lb/>nis formę puncti g in pri-<lb/> <anchor type="figure" xlink:label="fig-0523-02a" xlink:href="fig-0523-02"/> mo ſpeculo ſit pũctũ q:</s> <s xml:id="echoid-s33550" xml:space="preserve"> & <lb/>ducatur linea t q:</s> <s xml:id="echoid-s33551" xml:space="preserve"> quę per <lb/>49 huius erit æqualis li-<lb/>neę b g.</s> <s xml:id="echoid-s33552" xml:space="preserve"> In ſecundo uerò <lb/>ſpeculo, qđ eſt h c, linea i-<lb/>maginis ſit s o.</s> <s xml:id="echoid-s33553" xml:space="preserve"> In tertio ue <lb/>rò ſpeculo, qđ eſt e z, linea <lb/>imaginis ſit m n.</s> <s xml:id="echoid-s33554" xml:space="preserve"> Pater ita-<lb/>que quoniam in quolibet <lb/>iſtorum ſpeculorum táta <lb/>eſt diſtantia imaginis ſub <lb/>ſpeculo à ſuperficie ſpecu <lb/>li, quanta eſt diſtantia for-<lb/>mę, quę reflectitur à ſpeculo, à ſuperficie ipſius ſpeculi per 49 huius:</s> <s xml:id="echoid-s33555" xml:space="preserve"> linea ergo k b, quę eſt diſtantia <lb/> <pb o="222" file="0524" n="524" rhead="VITELLONIS OPTICAE"/> puncti rei uiſę à ſuperficie ſpeculi extra ſpeculum, eſt æqualis lineę k t, quę eſt diſtantia imaginis à <lb/>ſpeculo ſub illo:</s> <s xml:id="echoid-s33556" xml:space="preserve"> & linea g l eſt æqualis lineę l q:</s> <s xml:id="echoid-s33557" xml:space="preserve"> itẽ linea t h, quę eſt diſtantia formæ uiſæ à ſuperficie <lb/>ſpeculι h c, eſt æqualis lineæ h s, quę eſt diſtátia loci imaginis ſub eodẽ ſpeculo:</s> <s xml:id="echoid-s33558" xml:space="preserve"> & linea q c eſt æ qua <lb/>lis lineę c o:</s> <s xml:id="echoid-s33559" xml:space="preserve"> linea quoq;</s> <s xml:id="echoid-s33560" xml:space="preserve"> s e, quæ eſt diſtãtia formæ reflexę à ſpeculo z e eſt æ qualis lineę e m, quę eſt <lb/>diſtantia formæ ab eodẽ ſpeculo ſub illo:</s> <s xml:id="echoid-s33561" xml:space="preserve"> & ſimiliter linea o z eſt æ qualis lineæ z n.</s> <s xml:id="echoid-s33562" xml:space="preserve"> Et quoniá, ut pa <lb/>tet per 37 huius, locus imaginis uniuſcuiuſq;</s> <s xml:id="echoid-s33563" xml:space="preserve"> formæ puncti uiſi eſt in puncto concurſus catheti ſuæ <lb/>incidentiæ cũ linea reflexióis:</s> <s xml:id="echoid-s33564" xml:space="preserve"> & in ſpeculis planis imago ſemper eſt æqualis rei uiſæ ք 52 huius:</s> <s xml:id="echoid-s33565" xml:space="preserve"> pa <lb/>tet quòd uiſus exiſtens in puncto a, cõprehendet imaginẽ formæ lineę b g in loco lineę m n æqualé <lb/>ipſi rei uiſæ:</s> <s xml:id="echoid-s33566" xml:space="preserve"> & eius diſtantia à uiſu, quę eſt ſecundũ lineas a m & a n, eſt æ qualis omnibus lineis inci <lb/>dentię:</s> <s xml:id="echoid-s33567" xml:space="preserve"> quoniá linea a m eſt æqualis lineæ reflexionis, quę eſt e a, & lineę m e, quæ eſt æqualis lineæ <lb/>e s, quę ſecundũ præmiſſa eſt æqualis lineæ incidentię, quæ eſt e h, & lineę h s, æquali lineę t h, quæ <lb/>eſt æqualis lineę k h, & lineę t k, quę linea t k eſt æ qualis lineę k b.</s> <s xml:id="echoid-s33568" xml:space="preserve"> Et ſimiliter linea a n eſt æqualis li <lb/>neę reflexiõis, quæ eſt a z, & omnibus lineis incidentię, ut iam patuit.</s> <s xml:id="echoid-s33569" xml:space="preserve"> Et quoniã, ut patet per 55 hu-<lb/>ius in ſpeculis planis dextra apparent ſiniſtra & ſiniſtra dextra:</s> <s xml:id="echoid-s33570" xml:space="preserve"> palàm quòd in ſpeculo prιmo reſpe <lb/>ctu rei uiſibilis, quod eſt ſpeculũ l k, fit imago formę rei b g uiſę, quę eſt imago t q, traſmutata modo <lb/>dicto:</s> <s xml:id="echoid-s33571" xml:space="preserve"> ſed & eadem imago reflexa à ſecundo ſpeculo, quod eſt h c, mutat dextrum in ſiniſtrũ & ſini-<lb/>ſtrum in dextrum.</s> <s xml:id="echoid-s33572" xml:space="preserve"> Redιt ergo in ſpeculo numeri paris diſpoſitio partiũ imaginis ad diſpoſitionem <lb/>partiũ ipſius rei uiſæ.</s> <s xml:id="echoid-s33573" xml:space="preserve"> Et quia in ſpeculo tertio, quod eſt e z, imago ſecunda, quę eſt s o, mutat ſitum <lb/>partiũ ſuarum:</s> <s xml:id="echoid-s33574" xml:space="preserve"> patet quòd imaginis m n ſitus eſt alius à diſpoſitione formę rei, quę eſt b g.</s> <s xml:id="echoid-s33575" xml:space="preserve"> In fpecu-<lb/>lis itaq;</s> <s xml:id="echoid-s33576" xml:space="preserve"> numeri paris fit imago ſimilis rei ſecundum dextrum & ſiniſtrum, & in ſpeculis imparibus <lb/>tranſmutaturr.</s> <s xml:id="echoid-s33577" xml:space="preserve"> Et ſic uniuerſaliter quotcunq;</s> <s xml:id="echoid-s33578" xml:space="preserve"> ſpeculis paribus uel imparibus pofitis, fecundum hæc <lb/>imaginum diſpoſitio uariatur ſecundum dextrum & ſiniſtrum.</s> <s xml:id="echoid-s33579" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s33580" xml:space="preserve"/> </p> <div xml:id="echoid-div1382" type="float" level="0" n="0"> <figure xlink:label="fig-0523-02" xlink:href="fig-0523-02a"> <variables xml:id="echoid-variables579" xml:space="preserve">s b g o k h t q l c d m e a n z</variables> </figure> </div> </div> <div xml:id="echoid-div1384" type="section" level="0" n="0"> <head xml:id="echoid-head1075" xml:space="preserve" style="it">64. Duo ſpecula plana rectangula & æqualia poßibile eſt ſic ſiſti, ut intuens in uno ſpeculorũ <lb/>ſuam imaginem uideat uenientem, & in altero recedentem. Ptolemæus 4 th. 2 catoptr.</head> <p> <s xml:id="echoid-s33581" xml:space="preserve">Sint duo ſpecula plana rectangula & æqualia, cuiuſcũq;</s> <s xml:id="echoid-s33582" xml:space="preserve"> placuerit quantitatis ſuorũ laterũ, dum <lb/>tamen latera unius ſint æqualia lateribus alterius:</s> <s xml:id="echoid-s33583" xml:space="preserve"> & ſint latera eiuſdẽ ſpeculi inter ſe proportiona-<lb/>lia, ita ut longitudo ſit dupla latitudini eiuſdé ſpeculi:</s> <s xml:id="echoid-s33584" xml:space="preserve"> aſſumaturq́, linea, cuius longitudo ſit multò <lb/>maior uno latere illorũ ſpeculorũ:</s> <s xml:id="echoid-s33585" xml:space="preserve"> & ſit, exempli cauſſa, quatuor cubitorũ, quæ ſit a b:</s> <s xml:id="echoid-s33586" xml:space="preserve"> & ſecetur ex <lb/>ea portio æqualis quartæ parti unius lateris longitudιnιs ſpeculi per 3 p 1, quę ſit a g:</s> <s xml:id="echoid-s33587" xml:space="preserve"> & diuidatur li-<lb/>nea g b in duo æ qualia in puncto d:</s> <s xml:id="echoid-s33588" xml:space="preserve"> & à pũcto d ducatur linea perpendiculariter ſuper lineá a b per <lb/>11 p 1:</s> <s xml:id="echoid-s33589" xml:space="preserve"> producaturq́;</s> <s xml:id="echoid-s33590" xml:space="preserve"> in continuũ & directum:</s> <s xml:id="echoid-s33591" xml:space="preserve"> & abſcindatur ab ipſa linea æ qualis altitudini ſpeculi, <lb/>quę ſit linea d z:</s> <s xml:id="echoid-s33592" xml:space="preserve"> & à puncto b ducatur linea æqualis & æquidiſtans lineę d z, quę ſit b c:</s> <s xml:id="echoid-s33593" xml:space="preserve"> & produca-<lb/>tur linea c z orthogonaliter ſuper lineam b c, quę erit ęqualis lineę b d per 33 p 1:</s> <s xml:id="echoid-s33594" xml:space="preserve"> & producatur linea <lb/>c z in continuũ & directũ:</s> <s xml:id="echoid-s33595" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s33596" xml:space="preserve"> à puncto g linea g e æquidiſtans & æqualis lineę d z:</s> <s xml:id="echoid-s33597" xml:space="preserve"> erit ergo <lb/>linea g e per 30 p 1 æ-<lb/> <anchor type="figure" xlink:label="fig-0524-01a" xlink:href="fig-0524-01"/> qualis & ęquidιſtans <lb/>lineæ b c.</s> <s xml:id="echoid-s33598" xml:space="preserve"> Et ſuք pun <lb/>ctum e centrum exi-<lb/>ſtens deſcribatur por <lb/>tio circuli ſecundum <lb/>modum quantitatis <lb/>placitę, quę ſit r i:</s> <s xml:id="echoid-s33599" xml:space="preserve"> di-<lb/>uidaturq́;</s> <s xml:id="echoid-s33600" xml:space="preserve"> arcus r i ք <lb/>æ qualia per 30 p 3 in <lb/>puncto l:</s> <s xml:id="echoid-s33601" xml:space="preserve"> & ducatur <lb/>linea l e:</s> <s xml:id="echoid-s33602" xml:space="preserve"> & à puncto e <lb/>ducatur una linea perpendicularis ſuper lineá l e, quę ſit e m:</s> <s xml:id="echoid-s33603" xml:space="preserve"> & itẽ alia, quę ſit e n, quę tamen lineæ <lb/>adinoicem coniunctę ſunt linea una per 14 p 1:</s> <s xml:id="echoid-s33604" xml:space="preserve"> & ſit linea m e æqualis lineę n e:</s> <s xml:id="echoid-s33605" xml:space="preserve"> & tota linea m n ſit <lb/>æqualis longitudini ſpeculi.</s> <s xml:id="echoid-s33606" xml:space="preserve"> Si ergo duoru ſpeculorũ planorum rectangulorũ & ęqualiũ angularis <lb/>coniunctio fiat ſuper lineam m n.</s> <s xml:id="echoid-s33607" xml:space="preserve"> tunc diuident lineę m e & n e ſuperficies illorũ coniunctorũ ſpe-<lb/>culorum per æ qualia:</s> <s xml:id="echoid-s33608" xml:space="preserve"> patetq́;</s> <s xml:id="echoid-s33609" xml:space="preserve"> quòd illa ſpecula non erunt in una plana ſuperficie diſpoſita:</s> <s xml:id="echoid-s33610" xml:space="preserve"> perpen <lb/>diculares ergo à centro uiſus ſuper illa ſpecula ductæ, quæ ſunt catheti incidentię formæ ipſius ui-<lb/>dentis, ſunt diuerſæ Poſito ergo centro u ſus in pũcto d, & motis ſpeculis ſuper lineam l e fixam:</s> <s xml:id="echoid-s33611" xml:space="preserve"> ui <lb/>debit homo ſeipſum ſuper unũ duorum ſpeculorum uenienté & in altero recedentẽ:</s> <s xml:id="echoid-s33612" xml:space="preserve"> eſt enim lon-<lb/>gitudo amborum illorũ ſpeculorú, quę eſt linea m n, quaſi dupla latitudinis unius ipſotũ:</s> <s xml:id="echoid-s33613" xml:space="preserve"> & ſic pun <lb/>ctum eſt quaſi mediũ ſuperficiei amborũ illorum ſpeculorũ:</s> <s xml:id="echoid-s33614" xml:space="preserve"> unde circa ipſum æqualior fit motus.</s> <s xml:id="echoid-s33615" xml:space="preserve"> <lb/>Et ſi hæc ſpecula fuerint taliter ordinata, ut claudantur & aperiantur, & angulos inter ſe exiftentes <lb/>uarient, cum reuoluentur:</s> <s xml:id="echoid-s33616" xml:space="preserve"> multa deformitas efficitur imagιnũ unius etiam rei:</s> <s xml:id="echoid-s33617" xml:space="preserve"> anguli tamen taliter <lb/>ſint diſpoſiti, ut ab uno ſpeculo in aliud fieri poſsit reflexio:</s> <s xml:id="echoid-s33618" xml:space="preserve"> nec æſtimamus hæc demonſtratione <lb/>alia ꝗ̃ in his, quæ præmiſſę ſunt in ſimplicibus planis ſpeculis, indigere:</s> <s xml:id="echoid-s33619" xml:space="preserve"> & hæc practicę artificũ du-<lb/>cimus cõmittenda:</s> <s xml:id="echoid-s33620" xml:space="preserve"> quia & hæc, quæ præmiſimus, plus habilitatem operιs mechanici reſpiciunt, ꝗ̃ <lb/>firmitudinẽ demonſtrationis:</s> <s xml:id="echoid-s33621" xml:space="preserve"> fuit enim iſtud diligens inuentio antiquorũ, cui poteſt addere & de-<lb/>mere ille, qui diligenter perſpexerit ea, quę demonſtrationis neceſsitate cõſcripſimus in hoc libro.</s> <s xml:id="echoid-s33622" xml:space="preserve"/> </p> <div xml:id="echoid-div1384" type="float" level="0" n="0"> <figure xlink:label="fig-0524-01" xlink:href="fig-0524-01a"> <variables xml:id="echoid-variables580" xml:space="preserve">n e i z c m l r a g d c</variables> </figure> </div> <pb o="223" file="0525" n="525" rhead="LIBER SEXTVS."/> </div> <div xml:id="echoid-div1386" type="section" level="0" n="0"> <head xml:id="echoid-head1076" xml:space="preserve" style="it">65. Abuno ſpeculo plano ſoli oppoſito ignem eſt impoßibile accẽdi: à pluribus uerò poſsibile.</head> <p> <s xml:id="echoid-s33623" xml:space="preserve">Hoc enim euidens eſt:</s> <s xml:id="echoid-s33624" xml:space="preserve"> quia ignis non accenditur niſi per aggregationem plurium radiorum:</s> <s xml:id="echoid-s33625" xml:space="preserve"> li-<lb/>neæ uerò reflexionis à ſpeculorum planorum diuerſis punctis productæ nõ concurrunt, ut per 47 <lb/>huius demonſtratum eſt:</s> <s xml:id="echoid-s33626" xml:space="preserve"> in nullo ergo puncto conueniunt illi radij reflexi ad generationem ignis <lb/>poſsibilis in materia combuſtibili quacunq;</s> <s xml:id="echoid-s33627" xml:space="preserve"> Patet ergo primum propoſitorum.</s> <s xml:id="echoid-s33628" xml:space="preserve"> Iam autẽ dixit An-<lb/>themius neſcio qua ductus experientia, quòd ſolùm uiginti quatuor reflexi radij concurrentes in <lb/>uno puncto materiæ inflammabilis, ignem in illa accendant:</s> <s xml:id="echoid-s33629" xml:space="preserve"> & coniunxit ſeptem ſpecula plana he-<lb/>xagona colligatione ſtabili fixa, ſcilicet ſex extrema circa unum, quod ſtatuit in medio illorum, & <lb/>uniebantur illa ſpecula in quibuslibet angulis hexagoni:</s> <s xml:id="echoid-s33630" xml:space="preserve"> ideo quia figuræ hexagonæ replent locũ <lb/>ſuperficialẽ:</s> <s xml:id="echoid-s33631" xml:space="preserve"> ualent enim tres anguli hexagoni quatuor rectos.</s> <s xml:id="echoid-s33632" xml:space="preserve"> Et dixit Anthemius quòd ad quam-<lb/>cunq;</s> <s xml:id="echoid-s33633" xml:space="preserve"> diſtantiam ſic ignis potuit accendi:</s> <s xml:id="echoid-s33634" xml:space="preserve"> quæ ſi ad complendá unam planam ſuperficiem coniun-<lb/>xerat, non poterat, ut ex præmiſsis patére poteſt, intentionem ſuam aliter conſequi, quàm ſicut ex <lb/>uno ſpeculo plano:</s> <s xml:id="echoid-s33635" xml:space="preserve"> quoniam, ut prædictum eſt, tres ſuperficies hexagonæ replent punctum unum:</s> <s xml:id="echoid-s33636" xml:space="preserve"> <lb/>quia angulus quilibet hexagoni ualet duas tertias duorum rectorum, & tres anguli hexagoni ualẽt <lb/>quatuor rectos:</s> <s xml:id="echoid-s33637" xml:space="preserve"> concurrentes ergo tales tres anguli nullum uacuum dimittunt:</s> <s xml:id="echoid-s33638" xml:space="preserve"> nihil eſt ergo quod <lb/>punctum ſui concurſus diſtinguat à natura planę ſuperficiei & unius.</s> <s xml:id="echoid-s33639" xml:space="preserve"> Quòd ſi ijdem hexagoni tali-<lb/>ter adinuicem inclinentur, ut ab una ſphæra fiant circumſcriptibiles:</s> <s xml:id="echoid-s33640" xml:space="preserve"> tunc ad centrum illius ſphæ-<lb/>ræ fiet reflexio omniũ radiorum perpendiculariter ab uno puncto illis ſuperficiebus incidentium;</s> <s xml:id="echoid-s33641" xml:space="preserve"><unsure/> <lb/>& augebitur uigor caliditatis:</s> <s xml:id="echoid-s33642" xml:space="preserve"> unde tale ſpeculum melius poſſet ex trigonis quàm hexagonis com-<lb/>poni:</s> <s xml:id="echoid-s33643" xml:space="preserve"> quoniam numero ſuperficierum numerabuntur radij, & uirtus augebitur caloris:</s> <s xml:id="echoid-s33644" xml:space="preserve"> hęc tamen;</s> <s xml:id="echoid-s33645" xml:space="preserve"> <lb/>quia facilia ſunt, non duximus proſequenda ipſa, relinquentes artificis induſtrij animæ.</s> <s xml:id="echoid-s33646" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1387" type="section" level="0" n="0"> <head xml:id="echoid-head1077" xml:space="preserve">VITELLONIS FI-<lb/>LII THVRINGORVM ET PO-<lb/>LONORVM OPTICAE LIBER SEXTVS.</head> <p style="it"> <s xml:id="echoid-s33647" xml:space="preserve"><emph style="sc">Lativs</emph>, quoad potuimus, ſpeculorũ planorũ paßionibus percurſis:</s> <s xml:id="echoid-s33648" xml:space="preserve"> ſuper-<lb/>eſt nũc ut ad aliorum ſpeculorum paßiones proprias diuertamus.</s> <s xml:id="echoid-s33649" xml:space="preserve"> Et quia <lb/>ſpecula conuexa ſunt ſimpliciora concauis:</s> <s xml:id="echoid-s33650" xml:space="preserve"> quoniã quædam paßionũ ſpe-<lb/>culorum conuexorum deſcendunt in concaua, ut in illa, quorum paßiones <lb/>propriè diuerſimodè uariantur:</s> <s xml:id="echoid-s33651" xml:space="preserve"> cõuenit ut prιmò tractatum ſpeculorum conuexorum <lb/>alijs præmittamus.</s> <s xml:id="echoid-s33652" xml:space="preserve"> Sed quia inter ſpecula conuexa (quorum quædam ſunt ſphærica, <lb/>quædam columnaria, quædã pyramidalia) ipſa ſpecula ſphærιca ſunt alijs ſimpliciora:</s> <s xml:id="echoid-s33653" xml:space="preserve"> <lb/>paßiones enim & cauſſæ reflexionum ſpeculorũ ſphæricorum conuexorum deſcendunt <lb/>in ſpecula columnaria & pyramidalia cõuexa, cum in illis ab aliquibus punctis ſuorũ <lb/>circulorum accidit fieri reflexionem, 4icut & paßiones ſpeculorũ planorum deſcendunt <lb/>in eadem ſpecula columnaria & pyramidalia, quãdo ab aliquo puncto alicuius linea-<lb/>rum longitudinis illorum ſpeculorum ad uiſum fit reflexio;</s> <s xml:id="echoid-s33654" xml:space="preserve"> poſt tractatum ergo plano-<lb/>rum ſpeculorũ de ſpeculis ſphæricis conuexis, ut de ſimplicιoribus omnιbus aijs, & con-<lb/>cauis ſpeculιs proſequi dignum uiſum eſt.</s> <s xml:id="echoid-s33655" xml:space="preserve"> Quæ itaq;</s> <s xml:id="echoid-s33656" xml:space="preserve"> ad ſpeculorũ ſphæricorum proprias <lb/>paßiones proſequendas præmittimus, ſunt iſta.</s> <s xml:id="echoid-s33657" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1388" type="section" level="0" n="0"> <head xml:id="echoid-head1078" xml:space="preserve">DEFINITIONES.</head> <p> <s xml:id="echoid-s33658" xml:space="preserve">1.</s> <s xml:id="echoid-s33659" xml:space="preserve"> Maius ſpeculum ſphęricum conuexũ uel concauũ dicimus, cuius ſphęræ dia-<lb/>meter eſt maior:</s> <s xml:id="echoid-s33660" xml:space="preserve"> & minus, cuius minor.</s> <s xml:id="echoid-s33661" xml:space="preserve"> 2.</s> <s xml:id="echoid-s33662" xml:space="preserve"> Diametrum ſpeculi ſphærici dicimus <lb/>diametrum ſphæræ, cuius portio eſt ſpeculũ.</s> <s xml:id="echoid-s33663" xml:space="preserve"> 3.</s> <s xml:id="echoid-s33664" xml:space="preserve"> Centrũ ſpeculi dicimus centrum <lb/>ſphæræ, cuius portio eſt ſpeculũ 4.</s> <s xml:id="echoid-s33665" xml:space="preserve"> Diametrum uiſualẽ dicimus lineam à centro <lb/>uiſus per centrũ ſpeculi ſphærici tranſeuntẽ:</s> <s xml:id="echoid-s33666" xml:space="preserve"> & eadẽ dicitur cathetus reflexionis.</s> <s xml:id="echoid-s33667" xml:space="preserve"> <lb/>5.</s> <s xml:id="echoid-s33668" xml:space="preserve"> Lineã rectā æquidiſtare ſpeculo ſphærico conuexo dicimus, quæ ſecundū eius <lb/>punctũ mediū æquidiſtat lineæ aliquem arcū circuli magni illius ſpeculi ſecũdum <lb/>mediũ eius punctũ contingenti.</s> <s xml:id="echoid-s33669" xml:space="preserve"> 6.</s> <s xml:id="echoid-s33670" xml:space="preserve"> Finis contingentiæ dicitur punctus ubi altera <lb/>cathetorũ ſecat lineam in puncto reflexionis ſpeculum contin gẽtem.</s> <s xml:id="echoid-s33671" xml:space="preserve"> 7.</s> <s xml:id="echoid-s33672" xml:space="preserve"> Metam <lb/>locorum imaginũ dicimus punctũ uel lineam, ultra quam imagines nõ uidentur.</s> <s xml:id="echoid-s33673" xml:space="preserve"/> </p> <pb o="224" file="0526" n="526" rhead="VITELLONIS OPTICAE"/> </div> <div xml:id="echoid-div1389" type="section" level="0" n="0"> <head xml:id="echoid-head1079" xml:space="preserve">THEOREMATA</head> <head xml:id="echoid-head1080" xml:space="preserve" style="it">1. Communem ſectionem ſuperſiciei reflexionis & ſuperficiei ſpeculi ſphærici conuexineceſſe <lb/>eſt circulum magnum uel arcum circuli magni ſphæræ eſſe: ex quo patet quòdomnis ſuperficies <lb/>reflexionis diuidit ſphæram ſpeculi per æqualia.</head> <p> <s xml:id="echoid-s33674" xml:space="preserve">Quoniam enim, ut patet in principio 5 huius, 7 definitione, ſuperficies reflexionis dicitur ſuper-<lb/>ficies continens lineam incidentiæ & lineam reflexionis & perpendicularem à puncto contingen-<lb/>tiæ productam ſuper ſuperficiem ſphæricum ſpeculum in puncto incidentiæ contingentem, quæ <lb/>omnes lineæ rectæ ſunt:</s> <s xml:id="echoid-s33675" xml:space="preserve"> patet quòd ſuperficies reflexionis eſt ſuperficies plana.</s> <s xml:id="echoid-s33676" xml:space="preserve"> Omne autem ſpe-<lb/>culum ſphæricum conuexum, aut ſphæra eſt, aut pars ſphæræ, ut patet per 8 th.</s> <s xml:id="echoid-s33677" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s33678" xml:space="preserve"> ergo per 69 <lb/>th.</s> <s xml:id="echoid-s33679" xml:space="preserve"> 1 huius ſi ſuperficies reflexionis ſecet ſpeculum, ipſorum communis ſectio neceſſariò erit circu-<lb/>lus uel pars circuli.</s> <s xml:id="echoid-s33680" xml:space="preserve"> Et quoniam perpendiculares ſuper ſuperficies ſphæras contingentes, neceſſa-<lb/>riò tranſeunt per centrum ſphæræ, ut oſtẽdi poteſt per 72 th.</s> <s xml:id="echoid-s33681" xml:space="preserve"> 1 huius, & per definitionem lineæ per-<lb/>pendicularis ſuper ſuperficiem ſphæræ poſitam in principio huius:</s> <s xml:id="echoid-s33682" xml:space="preserve"> patet quòd omnis ſuperficies <lb/>reflexionis tranſit centrum ſpeculi:</s> <s xml:id="echoid-s33683" xml:space="preserve"> eſt ergo illa communis ſectio circulus magnus uel arcus circuli <lb/>magni ſphæræ illius ſpeculi per definitionem circuli magni.</s> <s xml:id="echoid-s33684" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s33685" xml:space="preserve"> Patet etiam co-<lb/>rollarium:</s> <s xml:id="echoid-s33686" xml:space="preserve"> quia cum omnis ſuperſicies reflexionis tranſeat per centrũ ſpeculi:</s> <s xml:id="echoid-s33687" xml:space="preserve"> patet manifeſtè quo-<lb/>niam ipſa diuidit ſphæram ſpeculi per æqualia.</s> <s xml:id="echoid-s33688" xml:space="preserve"> Ethoc proponebatur.</s> <s xml:id="echoid-s33689" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1390" type="section" level="0" n="0"> <head xml:id="echoid-head1081" xml:space="preserve" style="it">2. A centro uiſus ad ſuperficiem ſpeculi ſphærici conuexi ducta contingẽs, circa fixam ui-<lb/>ſualem diametrum æqualiter mota portionem ſuperficiei ſpeculi determinat, à cuius pũctis fiet <lb/>formarum reflexio ad uiſum. Alhazen 25 n 4.</head> <p> <s xml:id="echoid-s33690" xml:space="preserve">Sit centrum uiſus punctus a:</s> <s xml:id="echoid-s33691" xml:space="preserve"> & cõmunis ſectio ſuperficiei reflexionis & ſuperficiei ſpecúli ſphæ-<lb/>rici conuexi ſit circulus b c d k:</s> <s xml:id="echoid-s33692" xml:space="preserve"> cuius centrum ſit e:</s> <s xml:id="echoid-s33693" xml:space="preserve"> & à puncto a ducatur per 17 p 3 linea contingẽs <lb/>circulum in puncto d, quæ ſit a d:</s> <s xml:id="echoid-s33694" xml:space="preserve"> ducatur & diameter uiſualis, quæ ſit a e, ſecãs peripheriam circuli <lb/>b c d in puncto c.</s> <s xml:id="echoid-s33695" xml:space="preserve"> Dico, quòd fi diametro a e manente fixa linea contingens, quæ eſt a d, imaginetur <lb/>æqualiter moueri ſuper peripheriam ſpeculi, ſeruans ſemper æqualitatem anguli e a d, quouſq;</s> <s xml:id="echoid-s33696" xml:space="preserve"> re-<lb/>deat ad locum, unde exiuit:</s> <s xml:id="echoid-s33697" xml:space="preserve"> quòd ipſa motu ſuo ſecundum punctum d deſcribet circulum determi-<lb/>nantem portionem ſpeculi ſphærici conuexi, à qua fit reflexio omniũ formarum ad uiſum exiſten-<lb/>tem in puncto a, ab illa parte alia ſuperficiei ſpeculi, à qua non fit reflexio.</s> <s xml:id="echoid-s33698" xml:space="preserve"> Producatur enim linea a <lb/>d ultra punctum contingentiæ d ad punctum f:</s> <s xml:id="echoid-s33699" xml:space="preserve"> & duca-<lb/> <anchor type="figure" xlink:label="fig-0526-01a" xlink:href="fig-0526-01"/> tur linea e d:</s> <s xml:id="echoid-s33700" xml:space="preserve"> quę producatur extra ſpeculum ultra pun-<lb/>ctum d uſq;</s> <s xml:id="echoid-s33701" xml:space="preserve"> ad punctum g.</s> <s xml:id="echoid-s33702" xml:space="preserve"> Erunt ergo per 18 p 3 anguli <lb/>omnes ad punctum d recti:</s> <s xml:id="echoid-s33703" xml:space="preserve"> omnes ergo puncti in linea <lb/>d f conftituti uidebuntur directè:</s> <s xml:id="echoid-s33704" xml:space="preserve"> ideo quia linea a f ma-<lb/>nens una non reflectitur à puncto d.</s> <s xml:id="echoid-s33705" xml:space="preserve"> Quia tamen eadem <lb/>linea contingit ſpeculum, incipiunt puncta lineæ d f ali-<lb/>quid participare naturæ reflexionis:</s> <s xml:id="echoid-s33706" xml:space="preserve"> unde uidebuntur à <lb/>puncto d reflecti ſecũ dum lineam d a ad uiſum a per 20 <lb/>th.</s> <s xml:id="echoid-s33707" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s33708" xml:space="preserve"> quoniam angulus incidentiæ, qui eſt f d g, eſt <lb/>æqualis angulo reflexionis, qui eſt g d a.</s> <s xml:id="echoid-s33709" xml:space="preserve"> Dico etiá quòd <lb/>à nullo puncto arcus d k b poteſt fieri reflexio ad uiſum <lb/>a.</s> <s xml:id="echoid-s33710" xml:space="preserve"> Si enim ſit hoc poſsibile, eſto quòd à puncto h arcus d <lb/>k b fiat reflexio formæ alicuius puncti ad uiſum exiſten-<lb/>tem in puncto a:</s> <s xml:id="echoid-s33711" xml:space="preserve"> & ducatur linea reflexionis ad uiſum a, <lb/>quæ ſit h a:</s> <s xml:id="echoid-s33712" xml:space="preserve"> h æc ergo non poteſt tranſire ſolidum corpus <lb/>ſpeculi, ſcilicet arcum circuli b c d ſecádo:</s> <s xml:id="echoid-s33713" xml:space="preserve"> tranſibit ergo <lb/>extra circulum.</s> <s xml:id="echoid-s33714" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s33715" xml:space="preserve"> angulus contingentię, qui eſt <lb/>h d f eſt indiuiſibilis per 16 p 3, patet quòd illa linea refle-<lb/>xionis, quę eſt h a, non tranſibit punctum d:</s> <s xml:id="echoid-s33716" xml:space="preserve"> ſecabit ergo <lb/>lineam d g:</s> <s xml:id="echoid-s33717" xml:space="preserve"> ſit, ut ſecet ipſam in púcto l.</s> <s xml:id="echoid-s33718" xml:space="preserve"> Et quia linea re-<lb/>flexionis, quæ eſt h a, nó ſecat angulum h d f:</s> <s xml:id="echoid-s33719" xml:space="preserve"> palàm, cum <lb/>non ſecet arcum h d, quòd ſecat lineam d f:</s> <s xml:id="echoid-s33720" xml:space="preserve"> ſit, ut ſecet <lb/>ipſam in puncto m.</s> <s xml:id="echoid-s33721" xml:space="preserve"> Si ergo linea h m à puncto m perue-<lb/>niat ad punctum a:</s> <s xml:id="echoid-s33722" xml:space="preserve"> patet quòd duæ rectæ, quæ ſunt m l a <lb/>& m d a includũt ſuperficiem, quod eſt impoſsibile.</s> <s xml:id="echoid-s33723" xml:space="preserve"> Vel <lb/>deducatur ſic:</s> <s xml:id="echoid-s33724" xml:space="preserve"> trigoni d l m angulus m d l eſt rectus:</s> <s xml:id="echoid-s33725" xml:space="preserve"> ergo angulus d l m per 32 p 1 eſt acutus:</s> <s xml:id="echoid-s33726" xml:space="preserve"> ergo per <lb/>13 p 1 angulus a l d eſt obtuſus:</s> <s xml:id="echoid-s33727" xml:space="preserve"> ſed angulus a d l eſt rectus, quia angulus a d e eſt rectus:</s> <s xml:id="echoid-s33728" xml:space="preserve"> ergo per 14 <lb/>th.</s> <s xml:id="echoid-s33729" xml:space="preserve"> 1 huius, cum linea e g cadat ſuper ambas lineas a d & h a, & faciat angulos prędicto modo diſpo-<lb/>ſitos:</s> <s xml:id="echoid-s33730" xml:space="preserve"> patet quòd lineæ h l a & d a ad illam partem concurrent, ad quam ſunt anguli minores.</s> <s xml:id="echoid-s33731" xml:space="preserve"> Non <lb/>ergo reflectetur forma aliqua à puncto h ad punctum a, quod eſt oppoſitum dati.</s> <s xml:id="echoid-s33732" xml:space="preserve"> Patet ergo propo-<lb/>ſitum:</s> <s xml:id="echoid-s33733" xml:space="preserve"> quoniam quocunq;</s> <s xml:id="echoid-s33734" xml:space="preserve"> puncto arcus d k b dato, eodem modo poteſt fieri deductio.</s> <s xml:id="echoid-s33735" xml:space="preserve"/> </p> <div xml:id="echoid-div1390" type="float" level="0" n="0"> <figure xlink:label="fig-0526-01" xlink:href="fig-0526-01a"> <variables xml:id="echoid-variables581" xml:space="preserve">a b c d l g e m h k f</variables> </figure> </div> <pb o="225" file="0527" n="527" rhead="LIBER SEXTVS."/> </div> <div xml:id="echoid-div1392" type="section" level="0" n="0"> <head xml:id="echoid-head1082" xml:space="preserve" style="it">3. Oppoſito uiſui ſpeculo ſphærico conuexo, it a ut uiſus non ſit in ſuperficie illius ſpeculi aut <lb/>ſuperficie ei continua: erit communis ſectio baſis pyramidis uiſionis & ſuperficiei ſpeculi circu-<lb/>lus minor magno circulo ſphæram ſpeculi per æqualia ſecante. Alhazen 24 n 4.</head> <p> <s xml:id="echoid-s33736" xml:space="preserve">Opponatur uiſui ſpeculum ſphęricum taliter ut uiſus non ſit in ſuperficie illius ſpeculi aut in ſu-<lb/>perficie ei continua:</s> <s xml:id="echoid-s33737" xml:space="preserve"> dico quòd pars ſpeculi à uiſu comprehenſa erit pars ſphæræ circulo incluſa, <lb/>quem efficit motu ſuo radius contingens ſuperficiem ſphęræ.</s> <s xml:id="echoid-s33738" xml:space="preserve"> Quia enim, ut patet per 16 th.</s> <s xml:id="echoid-s33739" xml:space="preserve"> 2 huius, <lb/>longior radius ad ſphæræ ſuperficiem pertingens, quaſi linea ſpeculum contingẽs eſt:</s> <s xml:id="echoid-s33740" xml:space="preserve"> ſi ille radius <lb/>imaginetur per gyrum moueri attingendo ſphæram, donec redeat ad pũctum primum, à quo ſum-<lb/>pſit motus principium:</s> <s xml:id="echoid-s33741" xml:space="preserve"> palàm per præmiſſam quia punctus contingentiæ in ſphæræ ſuperficie cir-<lb/>culum deſcribet.</s> <s xml:id="echoid-s33742" xml:space="preserve"> Hic uerò circulus minor erit circulo magno illius ſphæræ.</s> <s xml:id="echoid-s33743" xml:space="preserve"> Quoniam ſi intelligan-<lb/>tur ſuperficies ſecantes ſe ſuper diametrum ſphæræ tranſeuntes polos prædicti circuli & ſphæram <lb/>per æqualia ſecantes:</s> <s xml:id="echoid-s33744" xml:space="preserve"> patet quòd omnes illi circuli contingentes lineas habent illas, quę ſunt lineæ <lb/>longitudinis pyramidis uiſionis:</s> <s xml:id="echoid-s33745" xml:space="preserve"> ergo per 58 th.</s> <s xml:id="echoid-s33746" xml:space="preserve"> 1 huius quilibet arcuum interiacentium ipſi ſuper-<lb/>ficiei ſphæræ, & his ſuperficiebus planis ſecantibus ſphęram, erit minor ſemicirculo circuli magni.</s> <s xml:id="echoid-s33747" xml:space="preserve"> <lb/>Verbi gratia, ſit per 69 th.</s> <s xml:id="echoid-s33748" xml:space="preserve"> 1 huius circulus, qui eſt communis ſectio ſuperficiei ſphæræ & ſuperficiei <lb/>planæ tranſeuntis per uiſum a extra ſphæram exiſtentẽ, & per cen-<lb/>trum ſphæræ, quod ſit b, circulus c s d:</s> <s xml:id="echoid-s33749" xml:space="preserve"> cuius centrum ſit b:</s> <s xml:id="echoid-s33750" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s33751" xml:space="preserve"> po-<lb/> <anchor type="figure" xlink:label="fig-0527-01a" xlink:href="fig-0527-01"/> lus circuli intellecti, ſecundum quem baſis pyramidis uiſionis ſecat <lb/>ſuperficiem ſpeculi pũctus s:</s> <s xml:id="echoid-s33752" xml:space="preserve"> producaturq́;</s> <s xml:id="echoid-s33753" xml:space="preserve"> b s ſemidiameter ad ui-<lb/>ſum a:</s> <s xml:id="echoid-s33754" xml:space="preserve"> & ſit linea b s a:</s> <s xml:id="echoid-s33755" xml:space="preserve"> & à puncto a centro uiſus ducatur linea con-<lb/>tingens circulum, quæ ſit a c:</s> <s xml:id="echoid-s33756" xml:space="preserve"> & à puncto contingétiæ, qui eſt c, du-<lb/>catur ad centrum b linea c b.</s> <s xml:id="echoid-s33757" xml:space="preserve"> Dico quòd arcus c s eſt minor quàm <lb/>quarta circuli magni.</s> <s xml:id="echoid-s33758" xml:space="preserve"> Angulus enim b c a eſt rectus per 18 p 3:</s> <s xml:id="echoid-s33759" xml:space="preserve"> angu-<lb/>lus ergo c b a eſt acutus:</s> <s xml:id="echoid-s33760" xml:space="preserve"> quia non poſſunt eſſe duo recti in eodem <lb/>trigono a b c per 32 p 1:</s> <s xml:id="echoid-s33761" xml:space="preserve"> hunc autem angulum in centro exiſtentem <lb/>reſpicit arcus c s:</s> <s xml:id="echoid-s33762" xml:space="preserve"> palàm ergo per 33 p 6 quoniã ipſe minor eſt quàm <lb/>quarta circuli.</s> <s xml:id="echoid-s33763" xml:space="preserve"> Et quia idem accidit in omnibus pũctis imaginato-<lb/>rum circulorum, manifeſtum quoniá quilibet arcuum illorum cir-<lb/>culorũ eſt minor quàm quarta circuli magni.</s> <s xml:id="echoid-s33764" xml:space="preserve"> Ergo circulus termi-<lb/>nans uiſum eſt minor circulo magno ſphæræ propoſitæ.</s> <s xml:id="echoid-s33765" xml:space="preserve"> Et hoc eſt <lb/>quod proponebatur.</s> <s xml:id="echoid-s33766" xml:space="preserve"> Tenet autẽ hæc demõſtratio in uno uiſu tan-<lb/>tùm, uel in ambobus uiſibus, dum modò diameter ſpeculi ſphærici <lb/>ſit maior quàm diſtantia oculorum:</s> <s xml:id="echoid-s33767" xml:space="preserve"> quoniã iſtis exiſtẽtibus æqua-<lb/>libus circulus maior ſphæræ erit circulus propoſitæ ſectionis, & <lb/>medietas ſphæræ uidebitur.</s> <s xml:id="echoid-s33768" xml:space="preserve"> Si uerò diſtantia oculorum ſit maior <lb/>diametro ſpeculi, plus medietate ſphæræ uidebitur:</s> <s xml:id="echoid-s33769" xml:space="preserve"> & erit communis ſectio circulus minor, ut hæe <lb/>ſunt demonſtrata in 4 huius.</s> <s xml:id="echoid-s33770" xml:space="preserve"/> </p> <div xml:id="echoid-div1392" type="float" level="0" n="0"> <figure xlink:label="fig-0527-01" xlink:href="fig-0527-01a"> <variables xml:id="echoid-variables582" xml:space="preserve">a s d b c</variables> </figure> </div> </div> <div xml:id="echoid-div1394" type="section" level="0" n="0"> <head xml:id="echoid-head1083" xml:space="preserve" style="it">4. In ſpeculis ſphæricis conuexis ſecundũ acceſſum uiſuum <lb/>ad ſpecula, circulorum uiſum terminantium quantitas mi-<lb/>nuitur, ad receſſum uerò augetur.</head> <figure> <variables xml:id="echoid-variables583" xml:space="preserve">f g h c b e d k a</variables> </figure> <p> <s xml:id="echoid-s33771" xml:space="preserve">Eſto enim ſpeculum ſphæricum conuexum, cuius centrum <lb/>b:</s> <s xml:id="echoid-s33772" xml:space="preserve"> & ſit centrum uiſus a:</s> <s xml:id="echoid-s33773" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s33774" xml:space="preserve"> circulus terminás uiſum in ſuper-<lb/>ficie ſpeculi, quic g h e.</s> <s xml:id="echoid-s33775" xml:space="preserve"> Dico quòd ſecũdum acceſſum & receſ-<lb/>ſum uiſuum à ſpeculis, illorum circulorum quantitas mutatur:</s> <s xml:id="echoid-s33776" xml:space="preserve"> <lb/>diminuitur enim ſecundum acceſſum, & augetur ſecũdum re-<lb/>ceſſum.</s> <s xml:id="echoid-s33777" xml:space="preserve"> Sit enim cõmunis ſectio ſuperficiei reflexionis & ſpe-<lb/>culi circulus c d e f:</s> <s xml:id="echoid-s33778" xml:space="preserve"> cuius arcus c d e ſit erectus ſuper circulum <lb/>c g h e, uiſam partem ſpeculi continentem:</s> <s xml:id="echoid-s33779" xml:space="preserve"> ſitq́ ipſius arcus c d <lb/>e medius pũctus d:</s> <s xml:id="echoid-s33780" xml:space="preserve"> & ducátur lineę a c, a d b, c b, a e:</s> <s xml:id="echoid-s33781" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s33782" xml:space="preserve"> per 18 <lb/>p 3 angulus a c b rectus:</s> <s xml:id="echoid-s33783" xml:space="preserve"> accedat ergo uiſus ſecundũ lineá a b ad <lb/>punctum k.</s> <s xml:id="echoid-s33784" xml:space="preserve"> Si ergo uiſus terminatur ad eundem circulum c g h <lb/>e, ut prius:</s> <s xml:id="echoid-s33785" xml:space="preserve"> ducatur linea k c.</s> <s xml:id="echoid-s33786" xml:space="preserve"> Et quoniam per 16 th.</s> <s xml:id="echoid-s33787" xml:space="preserve"> 2 huius lon-<lb/>gior radius à uiſu ad ſphærá pertingens quaſi linea contingens <lb/>eſt:</s> <s xml:id="echoid-s33788" xml:space="preserve"> patet per 18 p 3 quoniam angulus k c b eſt rectus:</s> <s xml:id="echoid-s33789" xml:space="preserve"> ſed & an-<lb/>gulus a c b fuit rectus:</s> <s xml:id="echoid-s33790" xml:space="preserve"> eſt ergo rectus minor recto:</s> <s xml:id="echoid-s33791" xml:space="preserve"> quod eſt im-<lb/>poſsibile.</s> <s xml:id="echoid-s33792" xml:space="preserve"> Exiſtẽte ergo uiſu in puncto k, non terminabitur ui-<lb/>ſio ad circulum c g h e, ſed ad aliquem circulum ipſo circulo c g <lb/>h e minorem.</s> <s xml:id="echoid-s33793" xml:space="preserve"> Quia enim inter duas lineas contingentes circu-<lb/>lum, quæ ſunt a c & a e, ab uno puncto a ductas, à puncto k du-<lb/>cuntur aliæ duæ lineæ eundem circulum contingentes:</s> <s xml:id="echoid-s33794" xml:space="preserve"> palàm <lb/>ergo per 60 th.</s> <s xml:id="echoid-s33795" xml:space="preserve"> 1 huius quòd puncta contingentiæ interiorum <lb/>cadent intra puncta contingentiæ exteriorum.</s> <s xml:id="echoid-s33796" xml:space="preserve"> Minorem ergo <lb/>arcum circuli comprehendent lineæ propinquiores quàm remotiores.</s> <s xml:id="echoid-s33797" xml:space="preserve"> Patet ergo propſitum.</s> <s xml:id="echoid-s33798" xml:space="preserve"/> </p> <pb o="226" file="0528" n="528" rhead="VITELLONIS OPTICAE"/> </div> <div xml:id="echoid-div1395" type="section" level="0" n="0"> <head xml:id="echoid-head1084" xml:space="preserve" style="it">5. A quolibet puncto ſuperficiei ſpeculi ſphærici conuexi oppoſitæ uiſui poteſt fieri reflexie <lb/>aduiſum. Alhazen 25 n 4.</head> <p> <s xml:id="echoid-s33799" xml:space="preserve">Eſto diſpoſitio eadem, quæ in 3 huius:</s> <s xml:id="echoid-s33800" xml:space="preserve"> dico quòd à quolibet puncto portionis oppoſitæ uiſui, ut <lb/>à quolibet puncto arcus c s, & omnium ſibi ſimilium arcuum poteſt fieri reflexio ad uiſum.</s> <s xml:id="echoid-s33801" xml:space="preserve"> Signe-<lb/>tur enim aliquis punctus arcus c s, qui ſit d:</s> <s xml:id="echoid-s33802" xml:space="preserve"> & ducatur ſemidiameter d b.</s> <s xml:id="echoid-s33803" xml:space="preserve"> Palàm per 72 th 1 huius <lb/>quoniam linea d b eſt perpendicularis ſuper ſuperficiem planam contingentem ſpeculum in pun-<lb/>cto d.</s> <s xml:id="echoid-s33804" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s33805" xml:space="preserve"> forma puncti rei uiſæ puncto d inciderit, palàm per 27 th.</s> <s xml:id="echoid-s33806" xml:space="preserve"> 5 huius quia linea refle-<lb/>xionis erit in eadem ſuperficie cum ſemidiametro d b & cum catheto a b ortho gonaliter cadente <lb/>ſuper ſuperficiem ſpeculi, eò quòd tranſeat per centrum eius b.</s> <s xml:id="echoid-s33807" xml:space="preserve"> Et ducatur à puncto d linea contin-<lb/>gens circulum c d s per 17 p 3, quæ ſit linea h d k:</s> <s xml:id="echoid-s33808" xml:space="preserve"> erit ergo per 18 p <lb/>3 angulus b d krectus:</s> <s xml:id="echoid-s33809" xml:space="preserve"> erit ergo trigoni d b a angulus a d b o btu-<lb/> <anchor type="figure" xlink:label="fig-0528-01a" xlink:href="fig-0528-01"/> ſus.</s> <s xml:id="echoid-s33810" xml:space="preserve"> Si ergo producatur linea b d extra ſphæram a d punctũ f:</s> <s xml:id="echoid-s33811" xml:space="preserve"> erit <lb/>per 13 p 1 angulus fd a acutus:</s> <s xml:id="echoid-s33812" xml:space="preserve"> ideo quòd angulus b d a ſit obtu-<lb/>ſus, ut patet ex præmiſsis per 13 p 1:</s> <s xml:id="echoid-s33813" xml:space="preserve"> & etiã ex hoc, quia cum linea <lb/>a d cadatintra lineam a cſpeculum contingentem:</s> <s xml:id="echoid-s33814" xml:space="preserve"> palàm per 57 <lb/>th.</s> <s xml:id="echoid-s33815" xml:space="preserve"> 1 huius quia linea a d producta ſecabit ſphæram ſpeculi:</s> <s xml:id="echoid-s33816" xml:space="preserve"> & ſu-<lb/>perficies contingens ſphæram in puncto d, in qua ſint lineæ h k, <lb/>d g, decliuior erit quàm linea a d, ſecabitq́;</s> <s xml:id="echoid-s33817" xml:space="preserve"> lineam a b.</s> <s xml:id="echoid-s33818" xml:space="preserve"> Et quia ſe-<lb/>midiameter b d eſt perpẽdicularis ſuper ſuperficiem h k d g ſpe-<lb/>culum in puncto d contingentem, erunt anguli f d k & f d h recti:</s> <s xml:id="echoid-s33819" xml:space="preserve"> <lb/>ergo etiam erit angulus b d k rectus:</s> <s xml:id="echoid-s33820" xml:space="preserve"> angulus quoq;</s> <s xml:id="echoid-s33821" xml:space="preserve"> b d a maior <lb/>recto, & angulus f d a minor recto.</s> <s xml:id="echoid-s33822" xml:space="preserve"> Refecato ergo ab angulo re-<lb/>cto, quieſt f d h, angulum acutum æqualẽ angulo f d a per 27 th.</s> <s xml:id="echoid-s33823" xml:space="preserve"> 1 <lb/>huius, qui ſit m d f:</s> <s xml:id="echoid-s33824" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s33825" xml:space="preserve"> lineæ continentes hos angulos in ea-<lb/>dem ſuperficie.</s> <s xml:id="echoid-s33826" xml:space="preserve"> Punctus ergo rei uiſæ exiſtens in linea m d, & ſu-<lb/>perficiei ſpecul incidens ad punctum d, reflectetur ad uiſum per <lb/>lineam d a per 11 uel 20 th.</s> <s xml:id="echoid-s33827" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s33828" xml:space="preserve"> continent enim lineæ m d & a <lb/>dangulos æquales cum perpendiculari b f:</s> <s xml:id="echoid-s33829" xml:space="preserve"> & lineæ illæ inciden-<lb/>tiæ & reflexionis, ut oſtenſum fuit per 25 th.</s> <s xml:id="echoid-s33830" xml:space="preserve"> 5 huius, erunt in ea-<lb/>dem ſuperficie, quæ erit ſuperficies reflexionis erecta ſuper ſu-<lb/>perficiem ſphęram ſpeculi in puncto d contingentem.</s> <s xml:id="echoid-s33831" xml:space="preserve"> Et eodem <lb/>modo demonſtrabitur de quolibetpũcto arcus c s, & cuiusslibet <lb/>arcus ſui ſimilis:</s> <s xml:id="echoid-s33832" xml:space="preserve"> hoc eſt de tota portione ſpeculi uiſui oppoſita:</s> <s xml:id="echoid-s33833" xml:space="preserve"> <lb/>quoniam de quolibet dato puncto poteſteodem modo demon-<lb/>ſtrari.</s> <s xml:id="echoid-s33834" xml:space="preserve"> Patet ergo quoniam à quolibet puncto ſuperficiei ſpeculi ſphærici conuexi oppoſitæ uiſui <lb/>poteſt fieri reflexio ad uiſum, ſicut proponebatur.</s> <s xml:id="echoid-s33835" xml:space="preserve"/> </p> <div xml:id="echoid-div1395" type="float" level="0" n="0"> <figure xlink:label="fig-0528-01" xlink:href="fig-0528-01a"> <variables xml:id="echoid-variables584" xml:space="preserve">a k f s d m b g c h</variables> </figure> </div> </div> <div xml:id="echoid-div1397" type="section" level="0" n="0"> <head xml:id="echoid-head1085" xml:space="preserve" style="it">6. In omni ſuperficie reflexionis à ſpeculis ſphæricis conuexis, centrum uiſus: & centrum ſpe-<lb/>culi: punctum reflexionis: & punctum reflexum cõſiſtere eſt neceſſe: ex quo patet lineam à cen-<lb/>tro uiſus ad centrum ſpeculi productam omnibus ſuperficiebus ſectionum ſecundũ diuerſa pun-<lb/>cta ſpecula huiuſmodi ſecantium communem eſſe. Alhazen 23. 25 n 4.</head> <p> <s xml:id="echoid-s33836" xml:space="preserve">Hoc patet per 25 th.</s> <s xml:id="echoid-s33837" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s33838" xml:space="preserve"> In omni enim ſuperficie reflexionis neceſſariò ſunt linea incidentiæ <lb/>& linea reflexionis:</s> <s xml:id="echoid-s33839" xml:space="preserve"> hæ autem lineæ continent tria puncta:</s> <s xml:id="echoid-s33840" xml:space="preserve"> punctum reflexum, & punctum refle-<lb/>xionis, & centrum uiſus.</s> <s xml:id="echoid-s33841" xml:space="preserve"> Et quia quælibet illarum ſuperficierum eſt erecta ſuper ſuperficiem ſpe-<lb/>culi, à quo fit reflexio:</s> <s xml:id="echoid-s33842" xml:space="preserve"> erunt lineæ in ipſa productæ, quæ ſunt erectæ ſuper ſuperficiem ſpeculi, cen-<lb/>trum ſpeculi tranſeuntes per 72 th.</s> <s xml:id="echoid-s33843" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s33844" xml:space="preserve"> manifeſtum ergo quia quælibet illarum ſuperficierum <lb/>tranſit centrum ſphæræ.</s> <s xml:id="echoid-s33845" xml:space="preserve"> In qualibet ergo ſuperficierũ reflexionis ſunt præn ominata quatuor pun-<lb/>cta:</s> <s xml:id="echoid-s33846" xml:space="preserve"> centrum uiſus:</s> <s xml:id="echoid-s33847" xml:space="preserve"> cẽtrum ſpeculi:</s> <s xml:id="echoid-s33848" xml:space="preserve"> punctum reflexionis:</s> <s xml:id="echoid-s33849" xml:space="preserve"> punctum reflexum.</s> <s xml:id="echoid-s33850" xml:space="preserve"> Ex his patet, quia cum <lb/>ſuperficierum planarum ſe interſecátium communis ſectio ſit linea recta, ut patet per 3 p 11, iſtarum <lb/>ſuperficierum neceſſariò communis ſectio erit linea à cẽtro uiſus ad centrũ ſpeculi producta:</s> <s xml:id="echoid-s33851" xml:space="preserve"> quo-<lb/>niam alijs duobus punctis uariatis ſecundũ numerum ſuperficierum reflexionis, hæc duo puncta <lb/>ſcilicet centrũ uiſus & cẽtrum ſpeculi in talibus ſuperficiebus ſemper manẽt.</s> <s xml:id="echoid-s33852" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s33853" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1398" type="section" level="0" n="0"> <head xml:id="echoid-head1086" xml:space="preserve" style="it">7. Omnis linea reflexionis (præter lineas contingẽtes) ſecat circulum (qui eſt communis ſe-<lb/>ctio ſuperficiei reflexionis, & ſuperficiei ſpeculi ſphærici conuexi) in duobus tantùm punctis: in <lb/>puncto uidelicet reflexionis & in puncto alio portionis ſuperficiei ſpeculinon apparentis.</head> <p> <s xml:id="echoid-s33854" xml:space="preserve">Sit communis ſectio ſuperficiei ſpeculi ſphærici conuexi & ſuperficiei reflexionis circulus a b c <lb/>d:</s> <s xml:id="echoid-s33855" xml:space="preserve"> cuius centrum ſit punctum g:</s> <s xml:id="echoid-s33856" xml:space="preserve"> & ſit centrum uiſus e:</s> <s xml:id="echoid-s33857" xml:space="preserve"> à quo ducantur lineæ contingẽtes illum cir-<lb/>culum, quæ ſint e a & e c.</s> <s xml:id="echoid-s33858" xml:space="preserve"> Palàm ergo per 2 huius quoniam à toto arcu a b c fit reflexio ad uiſum.</s> <s xml:id="echoid-s33859" xml:space="preserve"> <lb/>Sitergo ut à puncto b, quod eſt inter puncta a & c, fiat reflexio ad uiſum e:</s> <s xml:id="echoid-s33860" xml:space="preserve"> & ſit linea refle-<lb/>xionis b e.</s> <s xml:id="echoid-s33861" xml:space="preserve"> Dico quòd linea e b producta ultra punctum b ſecabit circulum a b c in aliquo puncto <lb/>arcus ſpeculi non apparentis, quod ſit d.</s> <s xml:id="echoid-s33862" xml:space="preserve"> Ducatur enim diameter uiſualis e fg h diuidens circulum <lb/>per æqualia in duos ſemicirculos, qui ſunt f c h, & f a h:</s> <s xml:id="echoid-s33863" xml:space="preserve"> oſtenſum eſt autem per 57 th.</s> <s xml:id="echoid-s33864" xml:space="preserve"> 1 huius <lb/> <pb o="227" file="0529" n="529" rhead="LIBER SEXTVS."/> quoniam ab uno puncto ad datum ſemicirculum tantùm unam lineam contingentem duci eſt poſ-<lb/>ſibile:</s> <s xml:id="echoid-s33865" xml:space="preserve"> & cooſtenſum ibieſt quòd omnis linea ab eodem puncto ſub <lb/>linea contingente ducta, ſecat ſemicirculum in puncto uno ſupra <lb/> <anchor type="figure" xlink:label="fig-0529-01a" xlink:href="fig-0529-01"/> punctum contingentiæ & in alio ſub ipſo.</s> <s xml:id="echoid-s33866" xml:space="preserve"> Patet ergo, cum à puncto <lb/>e ducatur linea e c circulum contingens, & ab eodem puncto e du-<lb/>catur ſub linea contingente linea e b, quoniam linea e b ſecat ſemi-<lb/>circulum f c h in uno puncto ſupra illum punctum contingẽtiæ, qui <lb/>ſit d, & in alío puncto b ſub illo puncto c, qui eſt terminus portionis <lb/>arcus apparehtis uiſui.</s> <s xml:id="echoid-s33867" xml:space="preserve"> Punctus ergo d cadit in portione c d a non <lb/>apparente uiſui.</s> <s xml:id="echoid-s33868" xml:space="preserve"> Quod eſt propoſitũ.</s> <s xml:id="echoid-s33869" xml:space="preserve"> Eodẽ ergo modo de quolibet <lb/>puncto arcus a f poteſt demõſtrari.</s> <s xml:id="echoid-s33870" xml:space="preserve"> Patet ergo, quod proponebatur.</s> <s xml:id="echoid-s33871" xml:space="preserve"/> </p> <div xml:id="echoid-div1398" type="float" level="0" n="0"> <figure xlink:label="fig-0529-01" xlink:href="fig-0529-01a"> <variables xml:id="echoid-variables585" xml:space="preserve">d h g c a b f e</variables> </figure> </div> </div> <div xml:id="echoid-div1400" type="section" level="0" n="0"> <head xml:id="echoid-head1087" xml:space="preserve" style="it">8. In omni reflexione à ſpeculis ſphæricis couexis, linea à cen-<lb/>tro ſpeculi ad punctũ reflexionis ducta, diuidit angulum à lineis <lb/>incidẽtiæ & reflexionis cõtentũ per duo æqualia. Alhaz. 13 n 4.</head> <p> <s xml:id="echoid-s33872" xml:space="preserve">Sit centrum uiſus a:</s> <s xml:id="echoid-s33873" xml:space="preserve"> & punctus rei uiſæ per reflexionem à ſpecu-<lb/>lo propoſito ſit b:</s> <s xml:id="echoid-s33874" xml:space="preserve"> ſitq́ cõmunis ſectio ſuperficiei reflexionis & ſpe-<lb/>culi circulus c d e:</s> <s xml:id="echoid-s33875" xml:space="preserve"> cuius cẽtrum ſit f:</s> <s xml:id="echoid-s33876" xml:space="preserve"> & reflectatur forma puncti b ad <lb/>uiſum a à pũcto ſpeculi d:</s> <s xml:id="echoid-s33877" xml:space="preserve"> & ducatur linea d f.</s> <s xml:id="echoid-s33878" xml:space="preserve"> Dico quòd linea f d ꝓ <lb/>ducta extra circulum ad punctum g, diuidit angulũ a d b per æqua-<lb/>lia:</s> <s xml:id="echoid-s33879" xml:space="preserve"> ita ut angulus a d g ſit æ qualis angulo g d b.</s> <s xml:id="echoid-s33880" xml:space="preserve"> Ducatur enim linea <lb/>contingens circulum c d e in puncto d per 17 p 3, quæ ſit h k:</s> <s xml:id="echoid-s33881" xml:space="preserve"> erunt ergo per 18 p 3 anguli f d k & f d h <lb/>recti:</s> <s xml:id="echoid-s33882" xml:space="preserve"> ergo per 13 p 1 anguli g d k & g d h ſunt recti & æquales:</s> <s xml:id="echoid-s33883" xml:space="preserve"> ſed an-<lb/> <anchor type="figure" xlink:label="fig-0529-02a" xlink:href="fig-0529-02"/> gulus b d k cũ ſit angulus incidẽtiæ, eſt per 20 th.</s> <s xml:id="echoid-s33884" xml:space="preserve"> 5 huius æqualis an <lb/>gulo a d h, qui eſt angulus reflexionis:</s> <s xml:id="echoid-s33885" xml:space="preserve"> remanet ergo angulus a d g <lb/>ęqualis angulo g d b Linea ergo f d producta à cẽtro ſpeculi ad pun-<lb/>ctum reflexionis, quod eſt d, diuidit angulum a d b per æqualia.</s> <s xml:id="echoid-s33886" xml:space="preserve"> Pa-<lb/>tet ergo propoſitum.</s> <s xml:id="echoid-s33887" xml:space="preserve"/> </p> <div xml:id="echoid-div1400" type="float" level="0" n="0"> <figure xlink:label="fig-0529-02" xlink:href="fig-0529-02a"> <variables xml:id="echoid-variables586" xml:space="preserve">b g a k d h e f c</variables> </figure> </div> </div> <div xml:id="echoid-div1402" type="section" level="0" n="0"> <head xml:id="echoid-head1088" xml:space="preserve" style="it">9. In conuexis ſpeculis ſphæricis omnem lineam reflexionis cum <lb/>catheto incidentiæ ab eodem puncto ad centrũ ſpeculi productam, <lb/>concurrere eſt neceſſe. Alhazen 8 n 5.</head> <p> <s xml:id="echoid-s33888" xml:space="preserve">Eſto communis ſectio ſuperficiei reflexionis & conuexi ſpeculí <lb/>ſphærici circulus g d:</s> <s xml:id="echoid-s33889" xml:space="preserve"> cuius centrum ſit z:</s> <s xml:id="echoid-s33890" xml:space="preserve"> & ſit centrum uiſus pun-<lb/>ctum b:</s> <s xml:id="echoid-s33891" xml:space="preserve"> punctusq́;</s> <s xml:id="echoid-s33892" xml:space="preserve"> rei uiſæ ſit a:</s> <s xml:id="echoid-s33893" xml:space="preserve"> reflectaturq́ forma puncti a ad cen-<lb/>trũ uiſus b à puncto ſpeculi d:</s> <s xml:id="echoid-s33894" xml:space="preserve"> & ſit linea reflexionís d b:</s> <s xml:id="echoid-s33895" xml:space="preserve"> linea quoq;</s> <s xml:id="echoid-s33896" xml:space="preserve"> <lb/>incidentiæ ſit a d, Ducatur itaq;</s> <s xml:id="echoid-s33897" xml:space="preserve"> linea à puncto dato a ad centrũ ſpe-<lb/>culiz, quæ ſit cathetus a z, ſecans ſuperficiem ſpeculi in puncto g:</s> <s xml:id="echoid-s33898" xml:space="preserve"> & <lb/>copuletur linea d z:</s> <s xml:id="echoid-s33899" xml:space="preserve"> & producatur b d intra ſpeculum, donec cõcurrat cum linea a z (concurret au-<lb/>tem per 29 th.</s> <s xml:id="echoid-s33900" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s33901" xml:space="preserve"> quoniã enim linea b d producta ſecat angulum <lb/>a d z, ut patet per præcedentem & per 15 p 1:</s> <s xml:id="echoid-s33902" xml:space="preserve"> ergo ſecabit & baſim a z) <lb/> <anchor type="figure" xlink:label="fig-0529-03a" xlink:href="fig-0529-03"/> ſititaq;</s> <s xml:id="echoid-s33903" xml:space="preserve"> punctus concurſus e:</s> <s xml:id="echoid-s33904" xml:space="preserve"> eſt autem linea a z cathetus incidentiæ <lb/>punctia, ut patet per definitionem catheti, & per 72 th.</s> <s xml:id="echoid-s33905" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s33906" xml:space="preserve"> Patet <lb/>ergo propoſitum, quoniam linea reflexionis concurrit cum catheto <lb/>incidentiæ.</s> <s xml:id="echoid-s33907" xml:space="preserve"> Quod autem hic de concurſu lineæ incidentiæ cum ca-<lb/>theto incidentiæ demonſtrauimus, hoc adiunximus propter 37 th.</s> <s xml:id="echoid-s33908" xml:space="preserve"> 5 <lb/>huius:</s> <s xml:id="echoid-s33909" xml:space="preserve"> ſecũdum enim utramq;</s> <s xml:id="echoid-s33910" xml:space="preserve"> illarum linearum eſt neceſſarium fieri <lb/>uiſionem:</s> <s xml:id="echoid-s33911" xml:space="preserve"> quoniam ſecũdum lineam reflexionis forma reflectitur ad <lb/>uiſum, & ſecundum cathetum incidẽtiæ reſpicit res ipſum ſpeculum, <lb/>à cuius ſuperſicie ſorma rei uiſæ reflectitur ad uiſum.</s> <s xml:id="echoid-s33912" xml:space="preserve"/> </p> <div xml:id="echoid-div1402" type="float" level="0" n="0"> <figure xlink:label="fig-0529-03" xlink:href="fig-0529-03a"> <variables xml:id="echoid-variables587" xml:space="preserve">b a d g e z</variables> </figure> </div> </div> <div xml:id="echoid-div1404" type="section" level="0" n="0"> <head xml:id="echoid-head1089" xml:space="preserve" style="it">10. Centro uiſus poſito in catheto incidẽtiæ ſuper ſpeculum ſphæ-<lb/>ricum conuexum incidente: ab uno tantùm puncto ſpeculi fiet re-<lb/>ſlexio: & uidebitur imago in ſuperficie ſpeculi in ipſo ſcilicet puncto <lb/>reflexionis: niſi fortè propter continuitatem ſui cum punctis alijs <lb/>formæ uiſæ ad alium locum imaginis pertrahatur. Alhazen 19 n 5.</head> <p> <s xml:id="echoid-s33913" xml:space="preserve">Often ſum eſt per 32 th.</s> <s xml:id="echoid-s33914" xml:space="preserve"> 5 huius quòd omnis perpendicularis reflectitur in ſeipſam:</s> <s xml:id="echoid-s33915" xml:space="preserve"> nunc autem <lb/>oſten demus, quod hic proponitur.</s> <s xml:id="echoid-s33916" xml:space="preserve"> Sit ergo g centrum uiſus:</s> <s xml:id="echoid-s33917" xml:space="preserve"> & d centrum ſpeculi propoſiti:</s> <s xml:id="echoid-s33918" xml:space="preserve"> ſitq́ g <lb/>k z d cathetus in cidẽtíæ, ducta à centro uiſus ad ſpeculum, ſecans ſuperficiem oculi in puncto k, & <lb/>incidens ſuperficiei ſpeculi in puncto z.</s> <s xml:id="echoid-s33919" xml:space="preserve"> Dico quod ſolius puncti k forma reflectitur ad uiſum:</s> <s xml:id="echoid-s33920" xml:space="preserve"> quo-<lb/>niam de alijs punctis lineæ d g quibuſcunq;</s> <s xml:id="echoid-s33921" xml:space="preserve"> datis, quantùm ad ip ſorum reflexionem, eodem modo <lb/>demonſtrandum, ut in 32 th.</s> <s xml:id="echoid-s33922" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s33923" xml:space="preserve"> Sed neq;</s> <s xml:id="echoid-s33924" xml:space="preserve"> aliquod punctum huius lineę reflectitur ab alio pun-<lb/>cto ſpeculi.</s> <s xml:id="echoid-s33925" xml:space="preserve"> Dato enim quòd ab alio puncto fiat reflexio:</s> <s xml:id="echoid-s33926" xml:space="preserve"> ſit illud aliud punctum a:</s> <s xml:id="echoid-s33927" xml:space="preserve"> & ducatur li-<lb/>nea g a, quæ ſit linea reflexionis:</s> <s xml:id="echoid-s33928" xml:space="preserve"> ducatur quoque linea incidentiæ ad punctum a ab aliquo puncto <lb/> <pb o="228" file="0530" n="530" rhead="VITELLONIS OPTICAE"/> lineæ g d, cuius forma à puncto a reflectitur, qui ſit x:</s> <s xml:id="echoid-s33929" xml:space="preserve"> hæc ergo linea x a cõtinebit angulum cum li-<lb/>nea g a, qui ſit x a g:</s> <s xml:id="echoid-s33930" xml:space="preserve"> & ducatur diameter d a:</s> <s xml:id="echoid-s33931" xml:space="preserve"> hęc ergo extra circulum <lb/> <anchor type="figure" xlink:label="fig-0530-01a" xlink:href="fig-0530-01"/> producta neceſſariò diuidet angulum x a g per æqualia per 8 huius:</s> <s xml:id="echoid-s33932" xml:space="preserve"> <lb/>eò quod ueniens à cẽtro ſpeculi & ad iſtum punctum reflexionis eſt-<lb/>perpendicularis ſuper lpſum:</s> <s xml:id="echoid-s33933" xml:space="preserve"> concurret ergo diameter d a cum per-<lb/>pendiculari g d inter punctum x reflexum & punctum g cẽtrum ui-<lb/>fus.</s> <s xml:id="echoid-s33934" xml:space="preserve"> Sic ergo duæ lineæ rectæ, quæ ſunt x d & d a, in duobus punctis <lb/>concurrent, & ſuperficiem continebunt:</s> <s xml:id="echoid-s33935" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s33936" xml:space="preserve"> Patet <lb/>ergo propoſitum:</s> <s xml:id="echoid-s33937" xml:space="preserve"> quoniam ab uno tantùm puncto ſpeculi reflexio-<lb/>nem fieri eſtneceſſe:</s> <s xml:id="echoid-s33938" xml:space="preserve"> ergo & una tãtùm uidebitur imago.</s> <s xml:id="echoid-s33939" xml:space="preserve"> Et quia lo-<lb/>cum ipſius nulla linearũ interſectio determinat, ut patet per 37 th.</s> <s xml:id="echoid-s33940" xml:space="preserve"> 5 <lb/>huius, palàm quòd illa imago uidetur in proprio loco ſuo:</s> <s xml:id="echoid-s33941" xml:space="preserve"> hoc autẽ <lb/>eſt in ſuperficie ipſius ſpeculi in puncto ſcilicetreflexionis:</s> <s xml:id="echoid-s33942" xml:space="preserve"> niſi fortè <lb/>propter cotinuitatem ſui cum punctis alijs formæ naturalis uiſæ ad <lb/>locum alium imaginis pertrahatur.</s> <s xml:id="echoid-s33943" xml:space="preserve"> Pater ergo propoſitum.</s> <s xml:id="echoid-s33944" xml:space="preserve"/> </p> <div xml:id="echoid-div1404" type="float" level="0" n="0"> <figure xlink:label="fig-0530-01" xlink:href="fig-0530-01a"> <variables xml:id="echoid-variables588" xml:space="preserve">x e g k z a d</variables> </figure> </div> </div> <div xml:id="echoid-div1406" type="section" level="0" n="0"> <head xml:id="echoid-head1090" xml:space="preserve" style="it">11. Locum imaginis uiſæ in ſpeculis ſphæricis conuexis in con-<lb/>curſu lineæ reflexionis cum catheto inctdentiæ neceſſe eft eſſe: ex <lb/>quo patet, quòd in omnireflexione ab his ſpeculis facta, ſemper <lb/>imago totius rei uiſæ continetur in aliqua linea inter loca imagi-<lb/>num ſuorum extremorum punctorum producta: patet etiã quòd <lb/>in his ſpeculis poßibile eſt locum imaginis inueniri. Euclides 17 th. catoptr. Alhazen 3. 16 n 5.</head> <p> <s xml:id="echoid-s33945" xml:space="preserve">Quòd linea reflexionis concurrat cum catheto incidentiæ, patet per 9 huius:</s> <s xml:id="echoid-s33946" xml:space="preserve"> poteſt & idem de-<lb/>monſtrari aliter.</s> <s xml:id="echoid-s33947" xml:space="preserve"> Sit enim punctus rei uiſæ a:</s> <s xml:id="echoid-s33948" xml:space="preserve"> cẽtrum oculi b:</s> <s xml:id="echoid-s33949" xml:space="preserve"> punctus reflexionis g:</s> <s xml:id="echoid-s33950" xml:space="preserve"> centrum ſpeculi <lb/>n.</s> <s xml:id="echoid-s33951" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s33952" xml:space="preserve"> per 25.</s> <s xml:id="echoid-s33953" xml:space="preserve"> th.</s> <s xml:id="echoid-s33954" xml:space="preserve"> 5 huius quòd a g linea incidentiæ, g b linea reflexionis ſunt in eadem ſuper-<lb/>ficie erecta ſuper ſuperficiem ſpeculum in puncto g contingentem.</s> <s xml:id="echoid-s33955" xml:space="preserve"> Linea itaq;</s> <s xml:id="echoid-s33956" xml:space="preserve"> communis ſuperfi-<lb/>ciei reflexionis & ſuperficiei ſpeculi ſit circulus z g q:</s> <s xml:id="echoid-s33957" xml:space="preserve"> & linea cõmunis ſuperficiei contingenti ſpe-<lb/>culum in puncto g & ſuperficiei reflexionis ſit linea e g p:</s> <s xml:id="echoid-s33958" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s33959" xml:space="preserve"> linea h g perpendicularis ſuper <lb/>lineam e g p per 11 p 1.</s> <s xml:id="echoid-s33960" xml:space="preserve"> Et patet per 19 p 3 quòd linea h g producta per-<lb/> <anchor type="figure" xlink:label="fig-0530-02a" xlink:href="fig-0530-02"/> tinget ad centrum circuli z g q:</s> <s xml:id="echoid-s33961" xml:space="preserve"> qui cum ſit circulus magnus, ut patet <lb/>per 1 huius:</s> <s xml:id="echoid-s33962" xml:space="preserve"> palàm quòd centrũ eius eſt cẽtrum ipſius ſpeculi.</s> <s xml:id="echoid-s33963" xml:space="preserve"> Tran-<lb/>ſit ergo linea h g producta ultra punctum g per cẽtrum ſpeculi, quod <lb/>eſt n:</s> <s xml:id="echoid-s33964" xml:space="preserve"> aliter enim linea à centro ſpeculi ad pũctum g ducta erit etiam <lb/>perpendicularis ſuper lineam p g e, & linea h g producta eſt perpen-<lb/>dicularis ſuper eandem:</s> <s xml:id="echoid-s33965" xml:space="preserve"> ab eodem ergo puncto ad eũdem punctum <lb/>lineæ rectæ continget ducere duas perpendiculares ſuper unam li-<lb/>neam:</s> <s xml:id="echoid-s33966" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s33967" xml:space="preserve"> Pertinget ergo linea h g ad punctum n.</s> <s xml:id="echoid-s33968" xml:space="preserve"> <lb/>Ducatur ergo linea a n à puncto uiſo ad centrum ſpeculi:</s> <s xml:id="echoid-s33969" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s33970" xml:space="preserve"> linea <lb/>a n per 72 th.</s> <s xml:id="echoid-s33971" xml:space="preserve"> 1 huius perpendicularis ſuper ſuperficiem ſpeculi:</s> <s xml:id="echoid-s33972" xml:space="preserve"> ergo <lb/>& ſuper ſuperficiem contingentẽ ſpeculum in puncto illo, per quem <lb/>tranſit.</s> <s xml:id="echoid-s33973" xml:space="preserve"> Et quia inter duas lineas h g & p g angulum rectum cõtinen-<lb/>tes cadit linea b g:</s> <s xml:id="echoid-s33974" xml:space="preserve"> palàm quia ipſa non contingit circulum z g q:</s> <s xml:id="echoid-s33975" xml:space="preserve"> ipſa <lb/>ergo producta ſecat circulum:</s> <s xml:id="echoid-s33976" xml:space="preserve"> concurret ergo cũ linea a n:</s> <s xml:id="echoid-s33977" xml:space="preserve"> ſit, ut con-<lb/>currat in puncto d.</s> <s xml:id="echoid-s33978" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s33979" xml:space="preserve">, ut patet per 6 huius, punctum a, cuius <lb/>forma à puncto ſpeculi g reflectitur, & centrum ſpeculi, quod eſt n, <lb/>neceſſario ſint in eadem ſuperficie:</s> <s xml:id="echoid-s33980" xml:space="preserve"> erit ergo per 1 p 11 linea a n in ea-<lb/>dem ſuperficie cum linea b g.</s> <s xml:id="echoid-s33981" xml:space="preserve"> Palàm ergo per 37 th.</s> <s xml:id="echoid-s33982" xml:space="preserve"> 5 huius quia pun-<lb/>ctus d erit locus imaginis:</s> <s xml:id="echoid-s33983" xml:space="preserve"> quoniam ipſe eſt pũctus communis lineæ <lb/>reflexionis, in qua neceſſariò eſt forma, & lineæ a n, quæ eſt cathetus incidentiæ formæ puncti a, ſe-<lb/>cundum quam, ut ſecun dum lineam breuiorem, neceſſariò uidetur forma.</s> <s xml:id="echoid-s33984" xml:space="preserve"> Patet ergo principaliter <lb/>propoſitũ per 37 th.</s> <s xml:id="echoid-s33985" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s33986" xml:space="preserve"> Et per hoc patet corollarium, quòd in omni reflexione à ſpeculis ſphę-<lb/>ricis conuexis facta, ſemper imago totius rei uiſæ continetur in aliqua linea inter loca imaginum <lb/>ſuorum extremorum punctorum producta:</s> <s xml:id="echoid-s33987" xml:space="preserve"> quoniam catheti incidẽtiæ punctorum mediorum ca-<lb/>dunt ſemper inter cathetos incidentiæ punctorum extremorũ:</s> <s xml:id="echoid-s33988" xml:space="preserve"> nec enim catheti incidentiæ ab ali-<lb/>quo illorum punctorum extremorum productæ ad centrum ſpeculi, ſecare poſſunt aliquam cathe-<lb/>tum incidentiæ punctorum mediorum.</s> <s xml:id="echoid-s33989" xml:space="preserve"> Patet etiam quòd in his ſpeculis cuiuſcunq;</s> <s xml:id="echoid-s33990" xml:space="preserve"> puncti rei uiſæ <lb/>poſsibile eſt locum imaginis inueniri:</s> <s xml:id="echoid-s33991" xml:space="preserve"> producta enim linea recta à puncto quocunq;</s> <s xml:id="echoid-s33992" xml:space="preserve"> uiſo per refle-<lb/>xionem ad centrum ſpeculi, & producta linea reflexionis ad cõcurſum cum illa:</s> <s xml:id="echoid-s33993" xml:space="preserve"> erit punctus com <lb/>munis ſectionis illarum linearum ſemper locus imaginis.</s> <s xml:id="echoid-s33994" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s33995" xml:space="preserve"/> </p> <div xml:id="echoid-div1406" type="float" level="0" n="0"> <figure xlink:label="fig-0530-02" xlink:href="fig-0530-02a"> <variables xml:id="echoid-variables589" xml:space="preserve">a h b e g p d z n q</variables> </figure> </div> </div> <div xml:id="echoid-div1408" type="section" level="0" n="0"> <head xml:id="echoid-head1091" xml:space="preserve" style="it">12. Cathetum incidentiæ linea reflexionis à circulo (qui eſt communis ſectio ſuperficiei refle-<lb/>xionis & ſpeculi ſphærici conuexi) ſecante, & à puncto reflexionis duct a recta illum circulum <lb/>contingente, quæ ſecet cathetum: erit totius catheti proportio ad inferiorem partẽ ſui reſectam <lb/> <pb o="229" file="0531" n="531" rhead="LIBER SEXTVS."/> uerſus centrum ſicut partis extrinſecus reſectæ per contingentem ad eam partem, quæ utraſ <lb/>interiacet ſectiones. Alhazen 18 n 5.</head> <p> <s xml:id="echoid-s33996" xml:space="preserve">Maneat diſpoſitio figuræ præcedentis:</s> <s xml:id="echoid-s33997" xml:space="preserve"> dico quòd proportio totius lineæ a n ad lineam n d eſt, <lb/>ſicut proportio lineæ a e ad e d.</s> <s xml:id="echoid-s33998" xml:space="preserve"> Quia enim angulus b g h æ qualis eſt angulo h g a per 8 huius:</s> <s xml:id="echoid-s33999" xml:space="preserve"> angu-<lb/>lus uerò b g h æ qualis eſt angulo d g n per 15 p 1, quia ſunt anguli cõ-<lb/> <anchor type="figure" xlink:label="fig-0531-01a" xlink:href="fig-0531-01"/> tra ſe poſiti:</s> <s xml:id="echoid-s34000" xml:space="preserve"> patet quòd angulus h g à æ qualis eſt angulo d g n.</s> <s xml:id="echoid-s34001" xml:space="preserve"> Et <lb/>quia anguli n gle & h g e ſunt recti per 18 p 3:</s> <s xml:id="echoid-s34002" xml:space="preserve"> ideo quod linea e g eſt <lb/>perpendicularis ſuper lineam h g n:</s> <s xml:id="echoid-s34003" xml:space="preserve"> patet quòd æ qualibus angulis <lb/>ab his hinc inde demptis, erunt anguli a g e & d g e æquales.</s> <s xml:id="echoid-s34004" xml:space="preserve"> E t quia <lb/>in trigono a g d linea g e angulum a g d per æ qualia ſecat:</s> <s xml:id="echoid-s34005" xml:space="preserve"> palà m ex 3 <lb/>p 6 quia proportio lineæ a e ad lineã e d eſt;</s> <s xml:id="echoid-s34006" xml:space="preserve"> ſicut lineæ a g ad lineam <lb/>d g.</s> <s xml:id="echoid-s34007" xml:space="preserve"> Protrahatur itaq;</s> <s xml:id="echoid-s34008" xml:space="preserve"> à puncto a linea æ quidiſtans lineæ d g p er 31 p 1 <lb/>concurrens cum linea h n in puncto h:</s> <s xml:id="echoid-s34009" xml:space="preserve"> quæ ſit h a (cõcurrent autem <lb/>illæ lineæ per 2 th.</s> <s xml:id="echoid-s34010" xml:space="preserve"> 1 huius) erit ergo per 29 p 1 angulus n g d æ qualis <lb/>angulo g h a:</s> <s xml:id="echoid-s34011" xml:space="preserve"> ſed ex præmiſsis patet quòd angulus n g d æqualis eſt <lb/>angulo a g h:</s> <s xml:id="echoid-s34012" xml:space="preserve"> eſt ergo angulus a g h æ qualis angulo a h g:</s> <s xml:id="echoid-s34013" xml:space="preserve"> ergo per 6 <lb/>p 1 erit latus a g æquale lateri a h:</s> <s xml:id="echoid-s34014" xml:space="preserve"> ergo per 7 p 5 erit proportio lineæ <lb/>a g ad g d, ſicut lineæ a h ad g d:</s> <s xml:id="echoid-s34015" xml:space="preserve"> ſed proportio lineę a h ad g d eſt ſicut <lb/>proportio lineæ a n ad d n per 29 p 1 & per 4 p 6:</s> <s xml:id="echoid-s34016" xml:space="preserve"> quia ergo quæ, eſt <lb/>proportio lineæ a h ad d g, eadem eſt lineæ a n ad d n:</s> <s xml:id="echoid-s34017" xml:space="preserve"> proportio ue-<lb/>rò lineæ a h uel a g ad d g, ut patet ex pręmiſsis, eſt, ſicut proportio li-<lb/>neæ a e ad e d:</s> <s xml:id="echoid-s34018" xml:space="preserve"> ergo per 11 p 5 eſt proportio lineæ a n ad n d, ſicut li-<lb/>neæ a e ad e d.</s> <s xml:id="echoid-s34019" xml:space="preserve"> Quod eſt propoſitum:</s> <s xml:id="echoid-s34020" xml:space="preserve"> quoniam linea e d utraſq;</s> <s xml:id="echoid-s34021" xml:space="preserve"> in-<lb/>teriacet ſectiones.</s> <s xml:id="echoid-s34022" xml:space="preserve"/> </p> <div xml:id="echoid-div1408" type="float" level="0" n="0"> <figure xlink:label="fig-0531-01" xlink:href="fig-0531-01a"> <variables xml:id="echoid-variables590" xml:space="preserve">y g a e g p d z n q</variables> </figure> </div> </div> <div xml:id="echoid-div1410" type="section" level="0" n="0"> <head xml:id="echoid-head1092" xml:space="preserve" style="it">13. In omni ſpeculo ſphærico conuexo linea recta interiacens centrum ſpeculi; & locum imæ. <lb/>ginis, maior eſt rect a interiacente locum imaginis & punctum reflexionis. Alhazen 17 n 5.</head> <p> <s xml:id="echoid-s34023" xml:space="preserve">Sit diſpoſitio quemadmodum in præcedẽte:</s> <s xml:id="echoid-s34024" xml:space="preserve"> dico quòd linea n d eſt maior quàm linea d g.</s> <s xml:id="echoid-s34025" xml:space="preserve"> Secet <lb/>enim linea p g e lineam a n in puncto e:</s> <s xml:id="echoid-s34026" xml:space="preserve"> palàm quòd punctum e di-<lb/> <anchor type="figure" xlink:label="fig-0531-02a" xlink:href="fig-0531-02"/> citur ſinis contingentiæ, ut patet ex principijs libri huius 6 defini <lb/>tione.</s> <s xml:id="echoid-s34027" xml:space="preserve"> Et quia per pręcedentem eſt proportio lineæ a n ad lineam n <lb/>d, ſicut lineæ a e ad lineam e d:</s> <s xml:id="echoid-s34028" xml:space="preserve"> proportio uerò lineæ a e ad e d per 3 <lb/>p 6, eſt ſicut proportio lineę a g ad g d:</s> <s xml:id="echoid-s34029" xml:space="preserve"> quoniá, ut præoſtẽſum eſt, li-<lb/>nea e g diuidit angulum a g d per æ qualia:</s> <s xml:id="echoid-s34030" xml:space="preserve"> eſt ergo proportio lineæ <lb/>a n ad n d, ſicut lineæ a g ad lineam g d per 11 p 5:</s> <s xml:id="echoid-s34031" xml:space="preserve"> ergo per 16 p 5 erit <lb/>permutatim proportio lineæ a n ad a g ſicut lineę d n ad d g:</s> <s xml:id="echoid-s34032" xml:space="preserve"> ſed per <lb/>19 p 1 linea a n eſt maior quàm a g:</s> <s xml:id="echoid-s34033" xml:space="preserve"> ideo quòd angulus a g n eſt obtu-<lb/>ſus, cum ſit maior angulo n g e recto:</s> <s xml:id="echoid-s34034" xml:space="preserve"> ergo linea n d eſt maior quàm <lb/>linea d g.</s> <s xml:id="echoid-s34035" xml:space="preserve"> Et quia per 11 huius punctus d eſt locus imaginis:</s> <s xml:id="echoid-s34036" xml:space="preserve"> patet <lb/>quòd linea n d interiacens centrum ſpeculi & locum imaginis eſt <lb/>maior linea d g interiacente locum imaginis & punctum reflexio-<lb/>nis, quod eſt g.</s> <s xml:id="echoid-s34037" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s34038" xml:space="preserve"/> </p> <div xml:id="echoid-div1410" type="float" level="0" n="0"> <figure xlink:label="fig-0531-02" xlink:href="fig-0531-02a"> <variables xml:id="echoid-variables591" xml:space="preserve">a h b e g p f d z n q</variables> </figure> </div> </div> <div xml:id="echoid-div1412" type="section" level="0" n="0"> <head xml:id="echoid-head1093" xml:space="preserve" style="it">14. Ducta catheto incidentiæ ad centrum circuli, qui eſt com-<lb/>munis ſectio ſuperficiei reflexionis & ſuperficiei ſpeculi ſphærici <lb/>conuexi: ducta quo & linea in puncto reflexionis eundem cir-<lb/>culum contingente: pars catheti interiacens finem contingentiæ <lb/>& circumferetiam circuli ſemidiametro eiuſdem circuli eſt mi-<lb/>nor. Alhazen 18 n 5.</head> <p> <s xml:id="echoid-s34039" xml:space="preserve">Remaneat omnino diſpoſitio, quæ ſuprà.</s> <s xml:id="echoid-s34040" xml:space="preserve"> Et quia punctus e eſt finis contingentiæ:</s> <s xml:id="echoid-s34041" xml:space="preserve"> interſecet li-<lb/>nea a n circumferẽtiam circuli in puncto f.</s> <s xml:id="echoid-s34042" xml:space="preserve"> Dιco quòd linea e f eſt minor ſemidiametro circuli, quæ <lb/>eſt f n.</s> <s xml:id="echoid-s34043" xml:space="preserve"> Quoniam enimm, ut patet ex pręmiſsis in proximo theoremate, proportio lineæ a g ad g d eſt, <lb/>ſicut proportlo lineæ a e ad e d:</s> <s xml:id="echoid-s34044" xml:space="preserve"> & proportio lineæ a n ad d n eſt, ſicut lineæ a d ad d g:</s> <s xml:id="echoid-s34045" xml:space="preserve"> igitur per 11 p <lb/>5 erit proportio lineæ a n ad d n, ſicut lineæ a e ad e d:</s> <s xml:id="echoid-s34046" xml:space="preserve"> ergo per 16 p 5 erit perinutatim proportio li-<lb/>neæ a n ad a e, ſicut d n ad d e:</s> <s xml:id="echoid-s34047" xml:space="preserve"> ſed linea a n eſt maior quàm linea a e, quoniam totum eſt maius ſua <lb/>parte:</s> <s xml:id="echoid-s34048" xml:space="preserve"> ergo linea d n eſt maior quàm linea d e:</s> <s xml:id="echoid-s34049" xml:space="preserve"> erit ergo linea d n multò maior quàm linea f e, quę eſt <lb/>pars ipſius d e:</s> <s xml:id="echoid-s34050" xml:space="preserve"> multò magis ergo linea n f erit maior quàm linea f e.</s> <s xml:id="echoid-s34051" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s34052" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1413" type="section" level="0" n="0"> <head xml:id="echoid-head1094" xml:space="preserve" style="it">15. Lineæ reflexionis formæ eiuſdem punctià diuerſis punctis ſpeculi ſphærici conuexi non <lb/>ſunt æquidiſtantes: attamen in centro unius uiſus non concurrunt. Ex quo patet quòd unus ui-<lb/>ſus non poteſt uidere idolum eiuſdem formæ reflexum à diuerſis punctis eiuſdem ſpeculi ſphæri-<lb/>ci conuexi. Euclides 4 th. catoptr. Ptolemæus 8 th. 1 catoptr.</head> <pb o="230" file="0532" n="532" rhead="VITELLONIS OPTICAE"/> <p> <s xml:id="echoid-s34053" xml:space="preserve">Eſto centrum uiſus b:</s> <s xml:id="echoid-s34054" xml:space="preserve"> & punctus rei uiſæ ſit e:</s> <s xml:id="echoid-s34055" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s34056" xml:space="preserve"> cõmunis ſectio ſuperficiei reflexionis & ſpe-<lb/>culi ſphærici conuexi circulus a g:</s> <s xml:id="echoid-s34057" xml:space="preserve"> incidatq́;</s> <s xml:id="echoid-s34058" xml:space="preserve"> punctus e diuerſis punctis ſpeculi in circulo a g:</s> <s xml:id="echoid-s34059" xml:space="preserve">quæ <lb/>ſint a & g.</s> <s xml:id="echoid-s34060" xml:space="preserve"> Dico quòd duæ lineæ reflexionis b a & b g <lb/>non ſuntæquidiſtante s:</s> <s xml:id="echoid-s34061" xml:space="preserve"> attamen in unius centro ui-<lb/> <anchor type="figure" xlink:label="fig-0532-01a" xlink:href="fig-0532-01"/> ſus non concurrent.</s> <s xml:id="echoid-s34062" xml:space="preserve"> Dato enim quòd concurrant in <lb/>puncto b:</s> <s xml:id="echoid-s34063" xml:space="preserve"> ducatur intra circulum chorda arcus a g:</s> <s xml:id="echoid-s34064" xml:space="preserve"> <lb/>quæ ſit recta a g:</s> <s xml:id="echoid-s34065" xml:space="preserve"> & producatur extra circulum uſq;</s> <s xml:id="echoid-s34066" xml:space="preserve"> <lb/>ad punctũ f e x parte a, & ex parte g uſq;</s> <s xml:id="echoid-s34067" xml:space="preserve"> ad punctum <lb/>n.</s> <s xml:id="echoid-s34068" xml:space="preserve"> Et quia per 20 th.</s> <s xml:id="echoid-s34069" xml:space="preserve"> 5 huius angulus e g n eſt æqualis <lb/>angulo b g a:</s> <s xml:id="echoid-s34070" xml:space="preserve"> ſed angulus e g n maior eſt angulo e a g <lb/>per 16 p 1:</s> <s xml:id="echoid-s34071" xml:space="preserve"> ergo angulus b g a maior eſt angulo e a g:</s> <s xml:id="echoid-s34072" xml:space="preserve"> <lb/>ſed angulus b a f maior eſt angulo b g a per 16 p 1:</s> <s xml:id="echoid-s34073" xml:space="preserve"> er-<lb/>go angulus b a f eſt maior angulo e a g.</s> <s xml:id="echoid-s34074" xml:space="preserve"> Non ergo re-<lb/>flectitur form a puncti e ad uiſum exiſtẽtem in pun-<lb/>cto b à puncto ſpeculi a per 20 th.</s> <s xml:id="echoid-s34075" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s34076" xml:space="preserve"> Et tamen <lb/>quia angulus b a f non eſt æqualis angulo b g a, ſed <lb/>minor:</s> <s xml:id="echoid-s34077" xml:space="preserve"> ideo quia per 16 p 1 angulus e g n eſt maior <lb/>angulo e a g:</s> <s xml:id="echoid-s34078" xml:space="preserve"> ergo per 20 th.</s> <s xml:id="echoid-s34079" xml:space="preserve"> 5 huius, & ex hypotheſi <lb/>erit angulus b g a maior angulo b a f Palàm ergo per <lb/>14 th.</s> <s xml:id="echoid-s34080" xml:space="preserve"> 1 huius quia duæ lineæ b a & b g non ſunt æquidiſtantes:</s> <s xml:id="echoid-s34081" xml:space="preserve"> ſed ut patet ex præmiſsis, ipſæ nun-<lb/>quam concurrent in puncto b, in quo eſt centrum uiſus.</s> <s xml:id="echoid-s34082" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s34083" xml:space="preserve"> Et per hoc patet <lb/>quòd unus uiſus non poteſt uidere idolum eiuſdem formæ à diuerſis punctis talium ſpeculorum <lb/>reflexum.</s> <s xml:id="echoid-s34084" xml:space="preserve"> Quod proponebatur.</s> <s xml:id="echoid-s34085" xml:space="preserve"/> </p> <div xml:id="echoid-div1413" type="float" level="0" n="0"> <figure xlink:label="fig-0532-01" xlink:href="fig-0532-01a"> <variables xml:id="echoid-variables592" xml:space="preserve">b e n a f g</variables> </figure> </div> </div> <div xml:id="echoid-div1415" type="section" level="0" n="0"> <head xml:id="echoid-head1095" xml:space="preserve" style="it">16. A ſuperficie ſpeculi ſphærici conuexi non poteſt forma alicuius puncti ad uiſum unum, <lb/>niſi à ſolo puncto reflecti: & una ſola imago uiſui occurrit. Alhazen 29 n 5.</head> <p> <s xml:id="echoid-s34086" xml:space="preserve">Quoniam enim per 10 huius patet quòd forma perpendiculariter huiuſmodi ſpeculo incidens, <lb/>centro uiſus in illa perpendiculari exiſtente, ab uno tantùm puncto reflectitur ad uiſum:</s> <s xml:id="echoid-s34087" xml:space="preserve"> non opor-<lb/>tet nos nunc propoſitũ niſi de lineis obliquè his ſpeculis ſphæricis conuexis incidentibus demon-<lb/>ſtrare.</s> <s xml:id="echoid-s34088" xml:space="preserve"> Sit ergo punctum uiſum b:</s> <s xml:id="echoid-s34089" xml:space="preserve"> & centrum uiſus à:</s> <s xml:id="echoid-s34090" xml:space="preserve"> & non ſit punctum a in perpendiculari ducta <lb/>à re uiſa a d centrum ſpeculi, quod ſit n.</s> <s xml:id="echoid-s34091" xml:space="preserve"> Dico quòd ſorma puncti b reflectitur ad a centrum uiſus ab <lb/>uno ſolo puncto ſpeculi:</s> <s xml:id="echoid-s34092" xml:space="preserve"> & una ſola imago uiſui occurrit.</s> <s xml:id="echoid-s34093" xml:space="preserve"> Palàm enim per 5 huius quòd uiſibili, in <lb/>quo eſt punctum b, modo conuenienti oppoſito ipſi ſpeculo, ab aliquo puncto ſuperficiei ſpeculi <lb/>poteſt reflecti forma puncti b ad uiſum a.</s> <s xml:id="echoid-s34094" xml:space="preserve"> Sit illud punctum reflexionis g:</s> <s xml:id="echoid-s34095" xml:space="preserve"> & ducantur lineæ b g & a <lb/>g:</s> <s xml:id="echoid-s34096" xml:space="preserve"> & ducatur cathetus incidentiæ, quę ſit b n, ſecans ſuperficiem ſpeculi in puncto l:</s> <s xml:id="echoid-s34097" xml:space="preserve"> & ſit a n diame-<lb/>ter uiſualis, ſecans ſuperficiem ſpeculi in puncto r.</s> <s xml:id="echoid-s34098" xml:space="preserve"> Sint quoq;</s> <s xml:id="echoid-s34099" xml:space="preserve"> pũcta d & e termini portionis ſuper-<lb/>ficiei ſpeculi uiſui oppoſitæ:</s> <s xml:id="echoid-s34100" xml:space="preserve"> producaturq́;</s> <s xml:id="echoid-s34101" xml:space="preserve"> linea reflexionis a g:</s> <s xml:id="echoid-s34102" xml:space="preserve"> quæ producta ultra punctum g ſe-<lb/>cabit per 9 huius perpẽdicularem b n:</s> <s xml:id="echoid-s34103" xml:space="preserve"> ſecet ergo illam in puncto q, qui punctus q, ut patet per 11 hu-<lb/>ius, eſt locus imaginis.</s> <s xml:id="echoid-s34104" xml:space="preserve"> Palàm itaque per 6 huius <lb/>quia puncta a, n, b ſunt in eadem ſuperficie ortho-<lb/> <anchor type="figure" xlink:label="fig-0532-02a" xlink:href="fig-0532-02"/> gonali ſuper ſuperficiem ſpeculi.</s> <s xml:id="echoid-s34105" xml:space="preserve"> Et quia ſuperfi-<lb/>cierum erectarum ſuper ſphæram ſpeculi, in qui-<lb/>bus ſunt puncta b & n, nulla extẽdi poteſt ad pun-<lb/>ctum a, quod eſt centrũ uiſus, niſi una tantùm:</s> <s xml:id="echoid-s34106" xml:space="preserve"> quo <lb/>niam punctus a eſt in diuiſibilis, qui ad ſuperficies <lb/>ſe circa ipſum uel lineam, in qua eſt, non ſecantes <lb/>communis eſſe non poteſt:</s> <s xml:id="echoid-s34107" xml:space="preserve"> tũc palàm quia puncti <lb/>a & b ſunt tantùm in una ſuperficie erecta ſuper <lb/>ſphæram ſpeculi, & non in pluribus.</s> <s xml:id="echoid-s34108" xml:space="preserve"> Non ergo fiet <lb/>reflexio pũcti b ad uiſum a, niſi in circulo ſphæræ, <lb/>qui eſt cõmunis ſectio ſuperficiei ſpeculi & ſuper-<lb/>ficiei a n b.</s> <s xml:id="echoid-s34109" xml:space="preserve"> Sit ergo hic circulus d g e.</s> <s xml:id="echoid-s34110" xml:space="preserve"> Dico quòd à <lb/>nullo puncto huius circuli d g e, præterquam à ſo-<lb/>lo punctò, quod propoſitum eſt eſſe g, ſiet reflexio <lb/>formæ puncti b ad a cẽtrum uiſus.</s> <s xml:id="echoid-s34111" xml:space="preserve"> Si enim ſit poſ-<lb/>ſibile fieri reflexionem ab alio puncto circuli d g e, <lb/>quàm à puncto g:</s> <s xml:id="echoid-s34112" xml:space="preserve"> ſit ille datus punctus l, in quo ca-<lb/>thetus incidẽtiæ, quę eſt b n, ſecat ſuperficiem ſpe-<lb/>culi.</s> <s xml:id="echoid-s34113" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s34114" xml:space="preserve"> linea b n ſit perpendicularis ſuper <lb/>ſuperficiem ſpeculi, & linea a l nõ ſit perpendicu-<lb/>laris ſuper illam, quia non tranſit centrum ſpeculi, <lb/>quod eſt n:</s> <s xml:id="echoid-s34115" xml:space="preserve"> & forma ſecundum lineam perpendi-<lb/>cularem ueniens, neceſſariò ſecundum perpendi-<lb/>cularem reflectatur quoniam ſemper angulus incidẽtiæ eſt æqualis angulo reflexionis:</s> <s xml:id="echoid-s34116" xml:space="preserve"> palàm quia <lb/>non reflectitur forma puncti b ad uiſum a à pũcto l.</s> <s xml:id="echoid-s34117" xml:space="preserve"> Palàm etiã quòd non reflectetur ab aliquo pun-<lb/> <pb o="231" file="0533" n="533" rhead="LIBER SEXTVS."/> cto arcus le:</s> <s xml:id="echoid-s34118" xml:space="preserve"> hocenim eſt impoſsibile:</s> <s xml:id="echoid-s34119" xml:space="preserve"> quia ad quodcunq;</s> <s xml:id="echoid-s34120" xml:space="preserve"> punctũ illius arcus ducatur linea à pun-<lb/>ctob, tenebit cum linea contingẽte circulum ín puncto illo, angulũ obtuſum expartee.</s> <s xml:id="echoid-s34121" xml:space="preserve"> Ideo enim <lb/>quòd angulus contentus ſub diametro circuli, & linea in illo puncto circulum contingente eſt re-<lb/>ctus per 18 p 3, & illa ſemidiameter educta non peruenit ad punctum b, quoniam ibi peruenit ſemi-<lb/>diameter n l:</s> <s xml:id="echoid-s34122" xml:space="preserve"> erit ergo angulus contentus ſub linea ducta à puncto b, & ſub illa linea contingente <lb/>exparte punctib, neceſſariò obtuſus:</s> <s xml:id="echoid-s34123" xml:space="preserve"> & linea ducta à puncto a tenebit cum illa linea contingente <lb/>in puncto dato angulum acutum uerſus l:</s> <s xml:id="echoid-s34124" xml:space="preserve"> linea enim à centro ſpeculi ad punctum illum cõtingen-<lb/>tiæ perueniens tenebit cum linea contingente circulum in illo puncto angulum rectum per 18 p 3:</s> <s xml:id="echoid-s34125" xml:space="preserve"> <lb/>â puncto uerò a linea ueniens cum eadem contingente tenebit angulum minorem recto ex parte <lb/>puncti k hęc enim contingens̀ à puncto a ducinon poteſt, quod patet per 57 th.</s> <s xml:id="echoid-s34126" xml:space="preserve"> 1 huius, quoniam li-<lb/>nea a e ſuperficiem ſpeculi eſt contingens exhypotheſi:</s> <s xml:id="echoid-s34127" xml:space="preserve"> propter hoc, quia lineæ a e & b d continent <lb/>arcum circuli d g e uiſui apparentem, qui per 2 huius à ſuperficie ſpeculi non apparente uiſui per li-<lb/>neas contingentes determinatur, Quare ſi ab illo puncto ſieret reflexio, tunc per 20 th.</s> <s xml:id="echoid-s34128" xml:space="preserve"> 5 huius ac-<lb/>cideret, quòd eſſet angulus acutus æ qualis obtuſo:</s> <s xml:id="echoid-s34129" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s34130" xml:space="preserve"> Non ergo fiet reflexio ab <lb/>aliquo puncto arcus l e.</s> <s xml:id="echoid-s34131" xml:space="preserve"> Sed etiam à nullo puncto arcus gl poteſt in hac diſpoſitione fieri reflexio.</s> <s xml:id="echoid-s34132" xml:space="preserve"> <lb/>Sit enim, ſi poſsibile eſt, ut fiat à puncto z:</s> <s xml:id="echoid-s34133" xml:space="preserve"> & ducatur linea a z o, ſecans cathetum incidẽtiæ, quæ eſt <lb/>b n, in puncto o:</s> <s xml:id="echoid-s34134" xml:space="preserve"> & ducatur linea contingens circulum in puncto z:</s> <s xml:id="echoid-s34135" xml:space="preserve"> hæc ergo contingens neceſſariò <lb/>cadet inter lineas b g & b l, quoniam punctus z eſt inter punct a g & l.</s> <s xml:id="echoid-s34136" xml:space="preserve"> Sit ergo illa contingens linea <lb/>z m:</s> <s xml:id="echoid-s34137" xml:space="preserve"> & ſit g flinea contingens circulum in puncto g:</s> <s xml:id="echoid-s34138" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s34139" xml:space="preserve"> linea z m cathetum incidentiæ in pun-<lb/>cto m:</s> <s xml:id="echoid-s34140" xml:space="preserve"> & linea g fin puncto f.</s> <s xml:id="echoid-s34141" xml:space="preserve"> Palàm ergo per 12 huius quòd proportio lineæ b n ad lineam n q eſt.</s> <s xml:id="echoid-s34142" xml:space="preserve"> <lb/>ſicut lineæ b f ad lineam ſ q:</s> <s xml:id="echoid-s34143" xml:space="preserve"> & ſimiliter erit proportio lineæ b n ad n o, ſicut proportio lineæ b m ad <lb/>m o:</s> <s xml:id="echoid-s34144" xml:space="preserve"> ſed quia linea o n maior eſt quàm linea q n, quoniam totum maius eſt ſua parte:</s> <s xml:id="echoid-s34145" xml:space="preserve"> erit per 8 p 5 li-<lb/>neæ b n ad n q maior proportio quàm ad lineam n o:</s> <s xml:id="echoid-s34146" xml:space="preserve"> maior ergo proportio eſt lineæ b f ad ſ q quàm <lb/>lineæ b m ad m o:</s> <s xml:id="echoid-s34147" xml:space="preserve"> quod eſt impoſsibile, & contra 9 th.</s> <s xml:id="echoid-s34148" xml:space="preserve"> 1 huius, cum linea b f ſit minor quàm linea b <lb/>m, & f q ſit maior quàm m o:</s> <s xml:id="echoid-s34149" xml:space="preserve"> reſtat ergo ut à puncto z non fiat reflexio.</s> <s xml:id="echoid-s34150" xml:space="preserve"> Sed neq;</s> <s xml:id="echoid-s34151" xml:space="preserve"> ab aliquo alio pun-<lb/>cto arcus g l:</s> <s xml:id="echoid-s34152" xml:space="preserve"> quoniam dato quocunq;</s> <s xml:id="echoid-s34153" xml:space="preserve"> puncto alio à puncto z, poteſt fieri deductio præmiſſo modo.</s> <s xml:id="echoid-s34154" xml:space="preserve"> <lb/>Similiter quoq;</s> <s xml:id="echoid-s34155" xml:space="preserve"> nec ab aliquo puncto arcus g d fiet reflexio.</s> <s xml:id="echoid-s34156" xml:space="preserve"> Si enim fiat ab aliquo, ſit iſtud t:</s> <s xml:id="echoid-s34157" xml:space="preserve"> & du-<lb/>catur linea b t, & linea a th ſecans cathetum b n in puncto h:</s> <s xml:id="echoid-s34158" xml:space="preserve"> & ducatur contingens circulũ in pun-<lb/>cto t, quæ ſit t p, ſecans cathetum b n in puncto p.</s> <s xml:id="echoid-s34159" xml:space="preserve"> Erit ergo per 12 huius proportio lineæ b n ad n h.</s> <s xml:id="echoid-s34160" xml:space="preserve"> <lb/>ſicut lineæ b p ad p h, & lineæ b n ad n q eſt ſicut lineæ b f ad ſ q:</s> <s xml:id="echoid-s34161" xml:space="preserve"> ſed maior eſt proportio lineæ b n ad <lb/>n h, quàm lineæ b n ad n q per 8 p 5:</s> <s xml:id="echoid-s34162" xml:space="preserve"> maior eſt ergo proportio lineæ b p ad p h, quàm lineæ b f ad f q:</s> <s xml:id="echoid-s34163" xml:space="preserve"> <lb/>quod eſt impoſsibile, & contra 9 th.</s> <s xml:id="echoid-s34164" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34165" xml:space="preserve"> maioris enim ad minorem maior eſt proportio, quàm <lb/>minoris ad maiorem per eandem 9 th.</s> <s xml:id="echoid-s34166" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34167" xml:space="preserve"> eſt enim linea b f maior quàm b p, & p h maior quàm <lb/>f q.</s> <s xml:id="echoid-s34168" xml:space="preserve"> Palàm ergò quòd à nullo pũcto arcus g d fiet reflexio formæ pũcti b ad uiſum a.</s> <s xml:id="echoid-s34169" xml:space="preserve"> Quodlibet ergo <lb/>punctũ formæ uiſæ ab uno ſolo puncto ſpeculi conuexi ſphærici ad uiſum reflectitur:</s> <s xml:id="echoid-s34170" xml:space="preserve"> una ſola ergo <lb/>erit linea reflexionis cuiuslibet puncti uiſi:</s> <s xml:id="echoid-s34171" xml:space="preserve"> ſed eſt etiam unica cathetus incidẽtiæ per 20 th.</s> <s xml:id="echoid-s34172" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34173" xml:space="preserve"> <lb/>unicus ergo punctus eſt, in quo illæ lineæ rectæ ſe ſecant, qui eſt locus imaginis, ut patet per 11 hu-<lb/>ius.</s> <s xml:id="echoid-s34174" xml:space="preserve"> Vnius ergo puncti eius eſt unica imago.</s> <s xml:id="echoid-s34175" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s34176" xml:space="preserve"/> </p> <div xml:id="echoid-div1415" type="float" level="0" n="0"> <figure xlink:label="fig-0532-02" xlink:href="fig-0532-02a"> <variables xml:id="echoid-variables593" xml:space="preserve">b k a p f m j z s t r e o q h n d</variables> </figure> </div> </div> <div xml:id="echoid-div1417" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables594" xml:space="preserve">b k u a p e l g t q n d</variables> </figure> <head xml:id="echoid-head1096" xml:space="preserve" style="it">17. In una catheto incidẽtiæ ſuperficiei ſpeculi ſphærici conuexi ſumptis duobus punctis, que <lb/>rum formæ à ſuperficie ſpeculi ſint reflexibiles <lb/>ad unum uiſum: erit punctus reflexionis puncti <lb/> propinquioris centro ſpeculi remotior à centro uiſus, quàm puncti remotioris ab eodem centro ſpeculi ſit ab ipſo centro uiſus. Alhazen 30 n 5.</head> <p> <s xml:id="echoid-s34177" xml:space="preserve">Remanẽte diſpoſitione, quæ in pręcedente, ſint <lb/>in catheto incidẽtiæ, quæ eſt n b, duo pũcti ſigna-<lb/>ti, qui ſint p & b:</s> <s xml:id="echoid-s34178" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s34179" xml:space="preserve"> punctus p propinquior cen-<lb/>tro ſpeculi puncto ſcilicent n, cẽtro circuli d g e, qui <lb/>eſt communis ſectio ſuperficiei reflexionis & ſu-<lb/>perficiei ſpeculi dati:</s> <s xml:id="echoid-s34180" xml:space="preserve"> & ſit punctus b remotior ab <lb/>eodem centro:</s> <s xml:id="echoid-s34181" xml:space="preserve"> & ſit a cẽtrum uiſus:</s> <s xml:id="echoid-s34182" xml:space="preserve"> & ſit locus re-<lb/>flexionis puncti b punctus g.</s> <s xml:id="echoid-s34183" xml:space="preserve"> Dico quòd punctus <lb/>reflexionis formę punctip remotior eſt à cẽtro ui-<lb/>ſus, qui eſt punctus à, quàm g, qui eſt punctus re-<lb/>flexionis formæ puncti b.</s> <s xml:id="echoid-s34184" xml:space="preserve"> Ducantur enim à pun-<lb/>cto a duę lineæ contingentes circulum, & portio-<lb/>nem circuli oppoſitam uiſui continẽes per 2 hu-<lb/>ius, quæ ſint a e & a d:</s> <s xml:id="echoid-s34185" xml:space="preserve"> & ſit punctus, in quo cathe-<lb/>tus b n ſecat circulum propoſitum, punctusl.</s> <s xml:id="echoid-s34186" xml:space="preserve"> Pa-<lb/>làm ergo quòd forma puncti p non reflectitur à <lb/>puncto l ad punctum a:</s> <s xml:id="echoid-s34187" xml:space="preserve"> quoniam ſola perpendicularis uiſualis reflectitur in ſeipſam per 10 huius:</s> <s xml:id="echoid-s34188" xml:space="preserve"> <lb/>neq;</s> <s xml:id="echoid-s34189" xml:space="preserve"> reflectitur forma puncti p à puncto g:</s> <s xml:id="echoid-s34190" xml:space="preserve"> quoniam ab illo reflectitur forma puncti b, ut patet per <lb/> <pb o="232" file="0534" n="534" rhead="VITELLONIS OPTICAE"/> præmiſſam:</s> <s xml:id="echoid-s34191" xml:space="preserve"> ſed neceſſe eſt ut refle ctatur ab aliquo puncto arcus g l inter puncta g & l.</s> <s xml:id="echoid-s34192" xml:space="preserve"> Sienim de-<lb/>tur quòd ab aliquo puncto arcus g d fiat reflexio formæ puncti p ad uiſum:</s> <s xml:id="echoid-s34193" xml:space="preserve"> ſit illud punctum t:</s> <s xml:id="echoid-s34194" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s34195" xml:space="preserve"> <lb/>p tlinea incidentiæ formæ puncti p:</s> <s xml:id="echoid-s34196" xml:space="preserve"> ducatur itaq;</s> <s xml:id="echoid-s34197" xml:space="preserve"> ad punctum t perpendiculàris n t u:</s> <s xml:id="echoid-s34198" xml:space="preserve"> hæc ergo per <lb/>8 huius neceſſariò diuidit angulum p t a per æqualia.</s> <s xml:id="echoid-s34199" xml:space="preserve"> Ducatur quoq;</s> <s xml:id="echoid-s34200" xml:space="preserve"> ad punctum g perpendicula-<lb/>ris n g k:</s> <s xml:id="echoid-s34201" xml:space="preserve"> palàm ergo per 21 p 1 quòd angulus n t a maior eſt angulo n g a:</s> <s xml:id="echoid-s34202" xml:space="preserve"> angulus ergo u t a (qui per <lb/>13 p 1 eſt reſiduum duorum rectorum ſuper angulum n t a) eſt minor angulo k g a, qui eſt reſiduum <lb/>duorum rectorum ſuper angulum n g a:</s> <s xml:id="echoid-s34203" xml:space="preserve"> ſed angulus k g a per 8 huius æ qualis eſt angulo b g k:</s> <s xml:id="echoid-s34204" xml:space="preserve"> an-<lb/>gulus ergo u t a eſt minor angulo b g k:</s> <s xml:id="echoid-s34205" xml:space="preserve"> angulus ergo p t u (qui per 8 huius eſt æqualis angulouta) <lb/>minor eſt angulo b g k:</s> <s xml:id="echoid-s34206" xml:space="preserve"> ſed angulus p t u ualet angulum p n t, & angulum t p n per 32 p 1, & angulus <lb/>b g k ualet angulum g b n, & angulum g n b per eandem 32 p 1:</s> <s xml:id="echoid-s34207" xml:space="preserve"> erũt ergo duo angulitnp & t p n mi-<lb/>nores duobus angulis g b n & g n b:</s> <s xml:id="echoid-s34208" xml:space="preserve"> quod eſt impoſsibile:</s> <s xml:id="echoid-s34209" xml:space="preserve"> cum angulus pnt contineat angulum b <lb/>n g, tanquam partem ſui, & angulus t p n ſit maior angulo g b n per 16 p 1.</s> <s xml:id="echoid-s34210" xml:space="preserve"> Palàm ergo quòd punctus <lb/>p non reflectitur niſi ab aliquo puncto arcus g linteriacente puncta g & l Et quoniam inter puncta <lb/>g & l punctus g eſt propinquior puncto a, qui eſt centrum uiſus:</s> <s xml:id="echoid-s34211" xml:space="preserve"> patet quòd omne punctum arcus <lb/>g l aliud à puncto g, eſt remotius à centro uiſus a, quàm punctumg, quod eſt punctum reflexionis <lb/>formæ puncti b.</s> <s xml:id="echoid-s34212" xml:space="preserve"> Punctum ergo reflexionis formæ puncti propinquioris centro ſpeculi, eſt remo-<lb/>tius à centro uiſus, quàm punctus reflexionis formæ puncti remotioris à centro ſpeculi.</s> <s xml:id="echoid-s34213" xml:space="preserve"> Quod <lb/>eſt propoſitum.</s> <s xml:id="echoid-s34214" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1418" type="section" level="0" n="0"> <head xml:id="echoid-head1097" xml:space="preserve" style="it">18. Formæ omnium punctorum æqualiter diſtantium à centro ſpeculi ſphærici conuexi, ſe-<lb/>cundum æquales angulos ſub cathet is incidentiæ & diametris uiſualibus in centro ſpeculi con-<lb/>tentos reflectuntur ad uiſus.</head> <p> <s xml:id="echoid-s34215" xml:space="preserve">Sit communis ſectio ſuperficiei reflexionis & ſuperficiei ſpeculi ſphærici conuexi circulus a b c:</s> <s xml:id="echoid-s34216" xml:space="preserve"> <lb/>cuius centrum ſit d:</s> <s xml:id="echoid-s34217" xml:space="preserve"> patetq́;</s> <s xml:id="echoid-s34218" xml:space="preserve"> per 1 huius quoniam punctum d eſt centrum ſpeculi:</s> <s xml:id="echoid-s34219" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s34220" xml:space="preserve"> duo puncta <lb/>e & f æqualiter diſtantia à centro ſpeculi, quod eſt d:</s> <s xml:id="echoid-s34221" xml:space="preserve"> <lb/>erunt ergo lineæ e d & f d æ quales.</s> <s xml:id="echoid-s34222" xml:space="preserve"> Dico quòd neceſ-<lb/>ſarium eſt formas illorum punctorum reflecti ad ui-<lb/> <anchor type="figure" xlink:label="fig-0534-01a" xlink:href="fig-0534-01"/> ſum ſecundum angulos æ quales:</s> <s xml:id="echoid-s34223" xml:space="preserve"> ut ſi forma puncti e <lb/>reflectatur ad uiſum exiſtentem in puncto g à puncto <lb/>ſpeculi h:</s> <s xml:id="echoid-s34224" xml:space="preserve"> & forma punctif, quæ per præmiſſam non <lb/>poteſt reflecti ad uiſum g à pũcto h, reflectatur ad ui-<lb/>ſum exiſtentem in puncto k à puncto l:</s> <s xml:id="echoid-s34225" xml:space="preserve"> & ducantur li-<lb/>neæ g d & k d:</s> <s xml:id="echoid-s34226" xml:space="preserve"> dico quòd angulus e d g eſt æ qualis an-<lb/>gulo f d k.</s> <s xml:id="echoid-s34227" xml:space="preserve"> Sit enim, ut cathetus incidẽtiæ, quæ eſt e d, <lb/>ſecet circulum in pũcto a:</s> <s xml:id="echoid-s34228" xml:space="preserve"> & cathetus f d in puncto b:</s> <s xml:id="echoid-s34229" xml:space="preserve"> <lb/>& diameter uiſualis g d ſecet circulum in puncto c, & <lb/>diameter k d in puncto m.</s> <s xml:id="echoid-s34230" xml:space="preserve"> Quia itaque lineæ e d & f d <lb/>ſunt æ quales, patet per præmiſſam, quoniam puncta <lb/>reflexionis, quę ſunt h & l, æ qualiter diſtant à uiſibus, <lb/>ad quos reflectuntur, ut quantùm diſtat h punctus re-<lb/>flexionis à puncto c, in quo diameter uiſualis g d ſecat <lb/>circulum, tantùm diſtat punctus reflexionis, qui eſt l, à puncto m, in quo diameter uiſualis, quæ eſt <lb/>k d, ſecat circulum:</s> <s xml:id="echoid-s34231" xml:space="preserve"> quoniam punctus reflexionis formæ puncti minus diſtantis à centro ſpeculi fit <lb/>per præ miſſam remotior à centro uiſus, & plus diſtantis propinquior.</s> <s xml:id="echoid-s34232" xml:space="preserve"> Ergo in illis, quæ æqualiter <lb/>diſtant, erit æ qualitas diſtantiæ à uiſibus, ad quos reflectuntur.</s> <s xml:id="echoid-s34233" xml:space="preserve"> Nec eſt in hoc diuerſitas, ſiue aliqua <lb/>puncta ſint in diuerſis cathetis incidentiæ, uel in una:</s> <s xml:id="echoid-s34234" xml:space="preserve"> ſemper enim punctorũ æ qualiter diſtantium <lb/>à centro eiuſdem ſpeculi, eadem eſt habitudo & ratio reflexionis:</s> <s xml:id="echoid-s34235" xml:space="preserve"> arcus ergo h c eſt æ qualis arcui <lb/>l m:</s> <s xml:id="echoid-s34236" xml:space="preserve"> & eadem ratione eſt arcus a h æ qualis arcui b l.</s> <s xml:id="echoid-s34237" xml:space="preserve"> Quoniã ergo per 33 p 6 peripheria circuli (ſicut <lb/>& per 87 th.</s> <s xml:id="echoid-s34238" xml:space="preserve"> 1 huius tota ſuperficies ſpeculi) æ qualiter ſe habet ad centrum:</s> <s xml:id="echoid-s34239" xml:space="preserve"> & puncta e & f æ quali-<lb/>ter diſtant ab eodem centro:</s> <s xml:id="echoid-s34240" xml:space="preserve"> totus ergo arcus a c eſt æ qualis toti arcui b m:</s> <s xml:id="echoid-s34241" xml:space="preserve"> ergo per 27 p 3 angulus <lb/>e d g eſt æ qualis angulo f d k.</s> <s xml:id="echoid-s34242" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s34243" xml:space="preserve"/> </p> <div xml:id="echoid-div1418" type="float" level="0" n="0"> <figure xlink:label="fig-0534-01" xlink:href="fig-0534-01a"> <variables xml:id="echoid-variables595" xml:space="preserve">k g e f a m h l c g d</variables> </figure> </div> </div> <div xml:id="echoid-div1420" type="section" level="0" n="0"> <head xml:id="echoid-head1098" xml:space="preserve" style="it">19. Impoſsibile eſt duo puncta æqualis diſtantiæ à centro ſpeculi ſphærici conuexi, ex eadem <lb/>parte diametri uiſualis exiſtentia, ab arcu (qui eſt communis ſectio ſuperficiei reflexionis & ſu-<lb/>perficiei ſpeculi) adeundem uiſum reflecti.</head> <p> <s xml:id="echoid-s34244" xml:space="preserve">Sit communis ſectio ſuperficiei reflexionis & ſpeculi ſphærici conuexi circulus a b c, cuius cen-<lb/>trum ſit punctum d:</s> <s xml:id="echoid-s34245" xml:space="preserve"> & ſint duo puncta æ qualiter diſtantia à centro ſpeculi, quæ ſinte & f:</s> <s xml:id="echoid-s34246" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s34247" xml:space="preserve"> cen-<lb/>trum uiſus in puncto g, in eadem ſuperficie cum punctis e & f, & exuna parte ipſorum:</s> <s xml:id="echoid-s34248" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s34249" xml:space="preserve"> pun-<lb/>ctum e remotius à puncto g quàm puuctum f.</s> <s xml:id="echoid-s34250" xml:space="preserve"> Dico quòd illa duo puncta e & fnon eſt poſsibile re-<lb/>flecti ad unum uiſum exiſtentem in puncto g.</s> <s xml:id="echoid-s34251" xml:space="preserve"> Ducantur enim lineæ e d, f d, g d:</s> <s xml:id="echoid-s34252" xml:space="preserve"> pater itaq;</s> <s xml:id="echoid-s34253" xml:space="preserve"> ex hypo-<lb/>theſi quòd angulus e d g eſt maior angulo f d g, ſicut totum ſua parte:</s> <s xml:id="echoid-s34254" xml:space="preserve"> fiat itaq;</s> <s xml:id="echoid-s34255" xml:space="preserve"> ſuper punctum d ter-<lb/>minum lineæ f d angulus æ qualis angulo e d g per 23 p 1, qui ſit f d h.</s> <s xml:id="echoid-s34256" xml:space="preserve"> Palàm ergo per præcedentem, <lb/> <pb o="233" file="0535" n="535" rhead="LIBER SEXTVS."/> quoniam forma puncti frefle ctetur ad punctum h, quod erit ultra punctum g:</s> <s xml:id="echoid-s34257" xml:space="preserve"> nõ ergo ad punctum <lb/>g per 15 huius.</s> <s xml:id="echoid-s34258" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s34259" xml:space="preserve"> Si enim <lb/>detur, ut reflectatur ad pũctum g, erit per præ-<lb/>miſſam angulus partialis, qui f d g, æqualis an-<lb/>gulo e d g:</s> <s xml:id="echoid-s34260" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s34261" xml:space="preserve"/> </p> <figure> <variables xml:id="echoid-variables596" xml:space="preserve">y a e p g c f g</variables> </figure> </div> <div xml:id="echoid-div1421" type="section" level="0" n="0"> <head xml:id="echoid-head1099" xml:space="preserve" style="it">20. Puncto rei uiſæ & centro uiſus æqua-<lb/>liter à ſuperficie ſpeculi ſphærici conuexi di-<lb/>ſtantibus, punctum reflexionis inuenire. <lb/>Alhazen 31 n 5.</head> <p> <s xml:id="echoid-s34262" xml:space="preserve">Eſto b pũctus rei uiſæ:</s> <s xml:id="echoid-s34263" xml:space="preserve"> & ſit a cẽtrum uiſus:</s> <s xml:id="echoid-s34264" xml:space="preserve"> <lb/>ſit quoq;</s> <s xml:id="echoid-s34265" xml:space="preserve"> dati ſpeculi conuexi ſphærici cẽtrum <lb/>c:</s> <s xml:id="echoid-s34266" xml:space="preserve"> & ſit circulus (qui eſt cõmunis ſectio ſuper-<lb/>ficierum reflexionis & ſpeculi) quie f g:</s> <s xml:id="echoid-s34267" xml:space="preserve"> & du-<lb/>cantur catheti b c & a c, ſecantes circulum in punctis f & g.</s> <s xml:id="echoid-s34268" xml:space="preserve"> Quia ergo propter æ qualitatem altitu-<lb/>dinis puncti rei uiſæ cum centro uiſus, iſtæ duæ lineæ b c & a c ſunt <lb/> <anchor type="figure" xlink:label="fig-0535-02a" xlink:href="fig-0535-02"/> æ quales, cum manifeſtum ſit per ea, quæ patuerũt in demonſtratio-<lb/>ne 17 huius, quoniam ab aliquo puncto arcus f g interiacentis cathe-<lb/>tos incidentię & reflexionis neceſſariò fiet reflexio:</s> <s xml:id="echoid-s34269" xml:space="preserve"> ſecetur itaq;</s> <s xml:id="echoid-s34270" xml:space="preserve"> per <lb/>9 p 1 angulus a c b per æ qualia per lineam c d, ſecantem arcum f g in <lb/>puncto e.</s> <s xml:id="echoid-s34271" xml:space="preserve"> Patet quoq;</s> <s xml:id="echoid-s34272" xml:space="preserve"> per 26 p 3 quoniam arcus f e eſt æ qualis arcui <lb/>e g:</s> <s xml:id="echoid-s34273" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s34274" xml:space="preserve"> linea c d per pendicularis ſuper lineam circulũ contingen-<lb/>tem in pũcto e per 18 p 3.</s> <s xml:id="echoid-s34275" xml:space="preserve"> Ducantur ergo ad punctũ e duæ lineæ a e & <lb/>b e:</s> <s xml:id="echoid-s34276" xml:space="preserve"> erũtq́;</s> <s xml:id="echoid-s34277" xml:space="preserve"> duo trianguli a e c & b e c per 4 p 1 & ex hypotheſi ęquian-<lb/>guli & æ quilateri:</s> <s xml:id="echoid-s34278" xml:space="preserve"> angulus ergo a e d æqualis erit angulo d e b:</s> <s xml:id="echoid-s34279" xml:space="preserve"> erit <lb/>ergo per 8 huius punctus e, qui eſt medius pũctus arcus f g, punctus <lb/>reflexionis formę pũcti b ad uiſum a.</s> <s xml:id="echoid-s34280" xml:space="preserve"> Ethoceſt propoſitũ.</s> <s xml:id="echoid-s34281" xml:space="preserve"> Si uerò li-<lb/>neæ b c & a c fuerint inæ quales, fiat in ipſis æ qualitas longioris, ut ſi <lb/>linea b c ſit lõgior quàm a c, cũ f c ſit ęqualis c g;</s> <s xml:id="echoid-s34282" xml:space="preserve"> quia ſunt ſemidiame-<lb/>tri eiuſdẽ circuli:</s> <s xml:id="echoid-s34283" xml:space="preserve"> reſecetur linea b f ad æ qualitatẽ lineæ a g in pũcto <lb/>h:</s> <s xml:id="echoid-s34284" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s34285" xml:space="preserve"> f h æ qualis ipſi a g:</s> <s xml:id="echoid-s34286" xml:space="preserve"> palàm ergo per pręmiſſa quoniã forma pũ-<lb/>cti h reflectitur ad uiſum a à puncto e.</s> <s xml:id="echoid-s34287" xml:space="preserve"> Puncta uerò uiciniora centro <lb/>c, quia per 17 huius ſunt in puncto ſuæ reflexionis magis diſtantia à <lb/>puncto, quod eſt centrum uiſus, nec poſſunt cadere in punctum e:</s> <s xml:id="echoid-s34288" xml:space="preserve"> <lb/>palàm quia reflectunturà punctis arcus e f, & ſecundum elongatio-<lb/>nem ſui à centro circuli c, erit punctorum ipſorum reflexionis approximatio ad centrum uiſus ſe:</s> <s xml:id="echoid-s34289" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0535-03a" xlink:href="fig-0535-03"/> cundum puncta ſuæ reflexionis.</s> <s xml:id="echoid-s34290" xml:space="preserve"> Remotiora uerò pũcta, ut illa, quæ <lb/>ſunt ſupra punctum h, ſcilicet puncta m & b, erunt ſecundum pun-<lb/>cta ſuę reflexionis propinquiora centro uiſus quàm punctum e:</s> <s xml:id="echoid-s34291" xml:space="preserve"> ca-<lb/>dent ergo in arcum e g, & ſecundum approximationem ſui ad cen-<lb/>trum circuli c, erit punctorum reflexionis maior elongatio à centro <lb/>uiſus b.</s> <s xml:id="echoid-s34292" xml:space="preserve"> Hoc autem licet ſic in groſſo ſcientiam afferat:</s> <s xml:id="echoid-s34293" xml:space="preserve"> eſt tamen ſe-<lb/>cundum ſingulorum punctorum reflexionis à punctis ſingulis ſu-<lb/>perficiei ſpeculi diligentius perſcrutandum.</s> <s xml:id="echoid-s34294" xml:space="preserve"/> </p> <div xml:id="echoid-div1421" type="float" level="0" n="0"> <figure xlink:label="fig-0535-02" xlink:href="fig-0535-02a"> <variables xml:id="echoid-variables597" xml:space="preserve">g d a f e g c</variables> </figure> <figure xlink:label="fig-0535-03" xlink:href="fig-0535-03a"> <variables xml:id="echoid-variables598" xml:space="preserve">b d m h a f e g c</variables> </figure> </div> </div> <div xml:id="echoid-div1423" type="section" level="0" n="0"> <head xml:id="echoid-head1100" xml:space="preserve" style="it">21. Si angulus contentus ſub linea incidentiæ à puncto rei uiſæ <lb/>obliquè duct a ad pũctum aliquem ſuperficiei ſpeculi ſphærici con-<lb/>uexi, & linea à centro ſpeculi ad eundem punctum duct a nõ fue-<lb/>rit maior recto, impoßibile eſt fieri reflexionem perfectam ad ali-<lb/>quem uiſum ſecundum illum punctum. Alhazen 40 n 5.</head> <p> <s xml:id="echoid-s34295" xml:space="preserve">Eſto a centrum uiſus:</s> <s xml:id="echoid-s34296" xml:space="preserve"> & b punctus rei uiſæ:</s> <s xml:id="echoid-s34297" xml:space="preserve"> ſit quoq;</s> <s xml:id="echoid-s34298" xml:space="preserve"> punctum g <lb/>centrum ſpeculi ſphærici conuexi:</s> <s xml:id="echoid-s34299" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s34300" xml:space="preserve"> communis ſectio ſuperficiei <lb/>reflexionis & ſpeculi circulus, cuius centrum erit punctũ g per 1 hu-<lb/>ius:</s> <s xml:id="echoid-s34301" xml:space="preserve"> ſit quoq;</s> <s xml:id="echoid-s34302" xml:space="preserve"> d punctus aliquis reflexionis:</s> <s xml:id="echoid-s34303" xml:space="preserve"> & ducantur lineæ g d, b <lb/>d & a d, quæ neceſſariòt erũt in ſuperſicie reflexionis per 6 huius, uel <lb/>per 27 th.</s> <s xml:id="echoid-s34304" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s34305" xml:space="preserve"> Dico quòd ſi à puncto d debet fieri reflexio, neceſ-<lb/>ſe eſt angulum b d g eſſe maiorem recto:</s> <s xml:id="echoid-s34306" xml:space="preserve"> quia ſi non ſit maior recto, nunquam fiet ab illo puncto re-<lb/>flexio.</s> <s xml:id="echoid-s34307" xml:space="preserve"> Sienim angulus b d g non eſt maior recto:</s> <s xml:id="echoid-s34308" xml:space="preserve"> aut ergo eſt rectus, aut minor recto.</s> <s xml:id="echoid-s34309" xml:space="preserve"> Si dicatur <lb/>quòd ipſe ſit rectus:</s> <s xml:id="echoid-s34310" xml:space="preserve"> ergo per 16 p 3 linea b d contingit circulum in puncto d:</s> <s xml:id="echoid-s34311" xml:space="preserve"> ſed per 20 th.</s> <s xml:id="echoid-s34312" xml:space="preserve"> 5 huius <lb/>angulus incidentiæ eſt æ qualis angulo reflexionis:</s> <s xml:id="echoid-s34313" xml:space="preserve"> ergo & angulus a d g erit rectus & contingens <lb/>circulum in puncto d:</s> <s xml:id="echoid-s34314" xml:space="preserve"> ergo per 14 p 1 duę lineæ b d & d a coniunctæ in puncto d ſunt linea una.</s> <s xml:id="echoid-s34315" xml:space="preserve"> Non <lb/>ergo fit reflexio ſecundum perfectam naturá reflexionis formæ puncti b à púcto ſpeculi d ad uiſum <lb/>exiſtentem in puncto a, ſed fit ſimpliciter uiſio ſecũdum lineam a d b, quod eſt contra hypotheſim:</s> <s xml:id="echoid-s34316" xml:space="preserve"> <lb/>quoniam pũctum d eſt poſitum eſſe punctum reflexionis.</s> <s xml:id="echoid-s34317" xml:space="preserve"> Si uerò angulus b d g dicature eſſe minor <lb/>recto:</s> <s xml:id="echoid-s34318" xml:space="preserve"> tunc â puncto d ducatur linea circulum contingens in puncto d per 17 p 3;</s> <s xml:id="echoid-s34319" xml:space="preserve"> quæ producatur <lb/> <pb o="234" file="0536" n="536" rhead="VITELLONIS OPTICAE"/> ad partem lineæ d b, & ſit d e:</s> <s xml:id="echoid-s34320" xml:space="preserve"> erit ergo per 18 p 3 angulus g d e rectus.</s> <s xml:id="echoid-s34321" xml:space="preserve"> Et quoniam angulus b d g eſt <lb/>datus minor recto:</s> <s xml:id="echoid-s34322" xml:space="preserve"> eſt ergo angulus b d g minor angu-<lb/>lo e d g.</s> <s xml:id="echoid-s34323" xml:space="preserve"> Et quoniam lineam b d, quæ eſt linea inciden-<lb/> <anchor type="figure" xlink:label="fig-0536-01a" xlink:href="fig-0536-01"/> tiæ formæ puncti b, extra ſpeculum cadere eſt neceſſe:</s> <s xml:id="echoid-s34324" xml:space="preserve"> <lb/>erit ergo neceſſarium peripſam diuidi angulũ contin-<lb/>gentiæ lineæ d e:</s> <s xml:id="echoid-s34325" xml:space="preserve"> quod eſt impoſsibile, & contra 16 p 3.</s> <s xml:id="echoid-s34326" xml:space="preserve"> <lb/>Non eſt ergo poſsibile angulum b d g eſſe minorem re-<lb/>cto, ſed neq;</s> <s xml:id="echoid-s34327" xml:space="preserve"> æqualem:</s> <s xml:id="echoid-s34328" xml:space="preserve"> neceſſarium ergo eſt ipſum eſſe <lb/>maiorem recto, & hoc proponebatur.</s> <s xml:id="echoid-s34329" xml:space="preserve"/> </p> <div xml:id="echoid-div1423" type="float" level="0" n="0"> <figure xlink:label="fig-0536-01" xlink:href="fig-0536-01a"> <variables xml:id="echoid-variables599" xml:space="preserve">a b d a e b b g</variables> </figure> </div> </div> <div xml:id="echoid-div1425" type="section" level="0" n="0"> <head xml:id="echoid-head1101" xml:space="preserve" style="it">22. Puncto rei uiſæ dato plus diſtante à cẽtro ſpẽ <lb/>culi ſphærici conuexi quàm centrum õculi: poßibile <lb/>eſt in ſuperſicie ſpeculi inuenire certum pũctum refle-<lb/>xionis formæ dati puncti ad datum centrum uiſus. Alhazen 39 n. 5.</head> <p> <s xml:id="echoid-s34330" xml:space="preserve">Eſto punctum a centrum uiſus:</s> <s xml:id="echoid-s34331" xml:space="preserve"> & ſit b datus punctus rei uiſæ:</s> <s xml:id="echoid-s34332" xml:space="preserve"> ſitq́ue g centrum ſpeculi ſphærici <lb/>conuexi:</s> <s xml:id="echoid-s34333" xml:space="preserve"> ducanturq́ue lineæ a g & b g:</s> <s xml:id="echoid-s34334" xml:space="preserve"> ſitq́ue exempli cauſſa, linea b g maior quàm linea a g, ideo <lb/>ut punctus b plus diſtet à centro ſpeculi g quàm centrum uiſus a.</s> <s xml:id="echoid-s34335" xml:space="preserve"> Et quoniam lineæ a g & b g ſunt <lb/>in ſuperficie reflexionis per 25 th.</s> <s xml:id="echoid-s34336" xml:space="preserve"> 5 huius, ſit communis ſectio ſuperficiei reflexionis & ſpeculi cir-<lb/>culus, cuius centrumg.</s> <s xml:id="echoid-s34337" xml:space="preserve"> Dico quòd in hoc circulo poſsibile eſt inuenire punctum reflexionis, à quo <lb/>refle ctitur forma puncti b ad uiſum a.</s> <s xml:id="echoid-s34338" xml:space="preserve"> Diuidatur enim angulus b g a per æ qualia per 9 p 1, ducta li-<lb/>nea e g ſecante peripheriam circuli in punctou.</s> <s xml:id="echoid-s34339" xml:space="preserve"> Sumatur quoque alia linea, quæ ſit m k:</s> <s xml:id="echoid-s34340" xml:space="preserve"> & diuida-<lb/>tur in puncto ftaliter, ut eius pars f m ſe habeat ad fk, ſicut linea b g ad lineam g a per 119 th.</s> <s xml:id="echoid-s34341" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34342" xml:space="preserve"> <lb/>& diuidatur linea m k per æ qualia in puncto o per 10 p 1:</s> <s xml:id="echoid-s34343" xml:space="preserve"> & à puncto o educatur perpendicularis <lb/>indefinita ſuper lineam m k per 11 p 1, quæ ſit o c:</s> <s xml:id="echoid-s34344" xml:space="preserve"> & ducatur à puncto k linea ad lineam co, tenens <lb/>cum ipſa linea c o angulum æ qualem angulo e g b, quæ ſit k c:</s> <s xml:id="echoid-s34345" xml:space="preserve"> eſt autem poſsibile hoc fieri.</s> <s xml:id="echoid-s34346" xml:space="preserve"> Cum <lb/>enim linea o c fuerit accepta indefinita, & linea g e indefinita, ducatur per 12 p 1 à puncto b perpen-<lb/>dicularis ſuper lineam g e, quæ ſit b e:</s> <s xml:id="echoid-s34347" xml:space="preserve"> eritq́ue angulus b e g æ qualis angulo c o k, quia uterque re-<lb/>ctus:</s> <s xml:id="echoid-s34348" xml:space="preserve"> ſuper punctum ergo kterminum lineæ o k ſiat per 23 p 1 angulus o k c æ qualis angulo e b g, <lb/>producta linea k c, quæ per 14 th.</s> <s xml:id="echoid-s34349" xml:space="preserve"> 1 huius neceſſariò concurret cum linea o c:</s> <s xml:id="echoid-s34350" xml:space="preserve"> quoniam cum angu-<lb/>lus k o c ſit rectus, patet quòd angulus o k c, qui eſt æ qualis angulo e b d, eſt acutus:</s> <s xml:id="echoid-s34351" xml:space="preserve"> palàm per 32 p 1 <lb/>quoniam angulus o c k eſt æ qualis angulo b g e.</s> <s xml:id="echoid-s34352" xml:space="preserve"> Quia ergo trigonum k o c eſt orthogonium, in cu-<lb/>ius latere o k eſt datus punctus f, tunc per 137 th.</s> <s xml:id="echoid-s34353" xml:space="preserve"> 1 huius à dato puncto f ducatur linea ad baſim tri-<lb/>gonick, quæ ſit fp:</s> <s xml:id="echoid-s34354" xml:space="preserve"> & concurrat cum producto latere c o in punctos, ita ut proportio lineæ s p ad <lb/>p k ſit, ſicut lineæ b g ad ſemidiametrum circuli, cuius centrum eſt punctum g:</s> <s xml:id="echoid-s34355" xml:space="preserve"> quæ ſit g u.</s> <s xml:id="echoid-s34356" xml:space="preserve"> Ex angu-<lb/>lo quoq;</s> <s xml:id="echoid-s34357" xml:space="preserve"> b g a ſecetur angulus æ qualis angulo f p k per 27 th.</s> <s xml:id="echoid-s34358" xml:space="preserve"> 1 huius, qui ſit b g d:</s> <s xml:id="echoid-s34359" xml:space="preserve"> hoc autem eſt poſ-<lb/>ſibile propter hoc, quia angulus p c s eſt æ qualis medietati anguli b g a:</s> <s xml:id="echoid-s34360" xml:space="preserve"> eſt autem angulus p c s ma-<lb/>ior angulo c s p per 18 p 1:</s> <s xml:id="echoid-s34361" xml:space="preserve"> quoniam ſic oportet duci lineam s p, ut linea s p fiat maior quàm linea c p, <lb/>ad quæſitum propoſitum inueniendum:</s> <s xml:id="echoid-s34362" xml:space="preserve"> aliàs enim non poſſet per lineam m k punctus quæren-<lb/>dæ reflexionis inueniri, ſed oporteret aliam lineam aſſumi:</s> <s xml:id="echoid-s34363" xml:space="preserve"> eſt ergo angulus f p k minor angulo b <lb/>g a per 32 p 1:</s> <s xml:id="echoid-s34364" xml:space="preserve"> & ducantur li-<lb/>neæ k s & b d.</s> <s xml:id="echoid-s34365" xml:space="preserve"> Quia ergo <lb/> <anchor type="figure" xlink:label="fig-0536-02a" xlink:href="fig-0536-02"/> <anchor type="figure" xlink:label="fig-0536-03a" xlink:href="fig-0536-03"/> proportio lineæ s p ad p k <lb/>eſt, ſicut lineæ b g ad ſemi-<lb/>diametrum g d, & anguli his <lb/>lineis proportionalibus con <lb/>tenti ſunt ęquales:</s> <s xml:id="echoid-s34366" xml:space="preserve"> erunt per <lb/>6 p 6 trianguli s p k & b g d <lb/>æquianguli:</s> <s xml:id="echoid-s34367" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s34368" xml:space="preserve"> angulus <lb/>s k p æ qualis angulo b d g.</s> <s xml:id="echoid-s34369" xml:space="preserve"> <lb/>Sed fortè ſecundũ quod pro <lb/>ponitur in 133 th.</s> <s xml:id="echoid-s34370" xml:space="preserve"> 1 huius, & <lb/>declaraturin 137 th.</s> <s xml:id="echoid-s34371" xml:space="preserve"> 1 huius, <lb/>poſsibile eſt à puncto f duci <lb/>lineam aliam ad lineam c k <lb/>ſimilem lineæ s p:</s> <s xml:id="echoid-s34372" xml:space="preserve"> ut ſi duca-<lb/>tur hoc modo linea y fr, ſe <lb/>cans lineam c s in puncto y, & lineam ck in puncto r talîter, ut proponitur, ſcilicet ut ſit eius pro-<lb/>portio ad r k partem lineæ, quam ſecabit ex linea c k, ſicut lineæ s p ad p k:</s> <s xml:id="echoid-s34373" xml:space="preserve"> & tunc à puncto k <lb/>ad lineam o s ducatur linea k y alia quàm linea s k, aliumq́ue cum linea c k angulum continens <lb/>maiorem uel minorem angulo c k s, qui ſit angulus c k y.</s> <s xml:id="echoid-s34374" xml:space="preserve"> Si ergo maior angulus ex his non <lb/>fuerit maior recto, non erit inuenire punctum reflexionis, ut patet per præmiſſam:</s> <s xml:id="echoid-s34375" xml:space="preserve"> quoniam & <lb/>tunc angulus contentus ſub linea reflexionis & ſemidiametro ſpeculi non erit maior recto.</s> <s xml:id="echoid-s34376" xml:space="preserve"> Si <lb/> <pb o="235" file="0537" n="537" rhead="LIBER SEXTVS."/> uerò aliquis illorum angulorum fuerit maior recto, erit poſsibile fieri reflexionem, & purictum <lb/>eius inueniri.</s> <s xml:id="echoid-s34377" xml:space="preserve"> Sit igitur primò angulus c k s maior recto, eritq́;</s> <s xml:id="echoid-s34378" xml:space="preserve"> poſsibile inueniri punctum re-<lb/>flexionis.</s> <s xml:id="echoid-s34379" xml:space="preserve"> Palàm enim ſi angulus c k s eſt maior recto, quòd eius æ qualis b d g eſt maior recto:</s> <s xml:id="echoid-s34380" xml:space="preserve"> <lb/>ducatur itaq;</s> <s xml:id="echoid-s34381" xml:space="preserve"> à puncto d linea contingens circulum per 17 p 3, quæ ſit n d y:</s> <s xml:id="echoid-s34382" xml:space="preserve"> cuius punctus n cadat <lb/>in lineam b g per 14 th.</s> <s xml:id="echoid-s34383" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s34384" xml:space="preserve"> Et cum angulus p k o ſit minor recto per 32 p 1, ideo quia angulus <lb/>c o k eſt rectus, ut patet ex præmiſsis:</s> <s xml:id="echoid-s34385" xml:space="preserve"> ſecetur ergo ex angulo b d g æ qualis angulo p k o per 27 <lb/>th.</s> <s xml:id="echoid-s34386" xml:space="preserve"> 1 huius, qui ſit angulus q d g, ducta linea d q ſecante lineam b g in puncto q.</s> <s xml:id="echoid-s34387" xml:space="preserve"> Cum igitur angu-<lb/>lus s p k ſit æ qualis angulo d g q, & angulus p k ſ æ qualis angulo q d g, erit per 32 p 1 triangulus <lb/>f p k æ quianguuls triangulo q g d:</s> <s xml:id="echoid-s34388" xml:space="preserve"> erit ergo angulus p f k æ qualis angulo d q g:</s> <s xml:id="echoid-s34389" xml:space="preserve"> ergo per 13 p 1 erit an <lb/>gulus d q b æ qualis angulo k f s.</s> <s xml:id="echoid-s34390" xml:space="preserve"> Et quia angulus b d q eſt æ qualis angulo f k s:</s> <s xml:id="echoid-s34391" xml:space="preserve"> ideo quia cum totus <lb/>angulus b d g ſit æ qualis toti angulo c k s, & angulus q d g ſit æ qualis angulo p k f:</s> <s xml:id="echoid-s34392" xml:space="preserve"> reſtat ut angu-<lb/>lus b d q æ qualis ſit angulo f k s:</s> <s xml:id="echoid-s34393" xml:space="preserve"> ergo per 32 p 1 angulorum duorum illorum trigonorum b d q & f <lb/>k s erit tertius tertio æ qualis, ſcilicetangulus d b q angulo k s f:</s> <s xml:id="echoid-s34394" xml:space="preserve"> trianguli ergo b d q & f k s ſunt <lb/>per 4 p 6 ſimiles.</s> <s xml:id="echoid-s34395" xml:space="preserve"> Producatur autem linea q d extra circulum:</s> <s xml:id="echoid-s34396" xml:space="preserve"> & à puncto b ducatur perpendicula-<lb/>ris ſuperipſam:</s> <s xml:id="echoid-s34397" xml:space="preserve"> quæ ſit b z:</s> <s xml:id="echoid-s34398" xml:space="preserve"> erit ergo angulus b q z per 13 p 1 æ qualis angulo s f o, & angulus b z q <lb/>rectus æ qualis eſt angulo s o f recto:</s> <s xml:id="echoid-s34399" xml:space="preserve"> erit ergo per præmiſſa triangulus b q z ſimilis triangulo s f o.</s> <s xml:id="echoid-s34400" xml:space="preserve"> <lb/>Producatur ergo linea d z ultra punctum z uſq;</s> <s xml:id="echoid-s34401" xml:space="preserve"> ad punctum i, ita quòd linea z i ſit æ qualis lineæ z <lb/>d per 3 p 1:</s> <s xml:id="echoid-s34402" xml:space="preserve"> palàm ergo ex ſimilitudine triangulorum quoniam proportio lineę z q ad q b eſt, ſicut li-<lb/>neæ of ad f s:</s> <s xml:id="echoid-s34403" xml:space="preserve"> & proportio lineæ b q ad q d eſt ſicut lineæ fs ad f k:</s> <s xml:id="echoid-s34404" xml:space="preserve"> erit ergo per 22 p 5, proportio <lb/>lineæ z q ad q d, ſicut o f ad f k:</s> <s xml:id="echoid-s34405" xml:space="preserve"> ergo per 18 p 5 erit coniunctim proportio lineæ z d ad q d, ſicut <lb/>lineæ o k ad f k:</s> <s xml:id="echoid-s34406" xml:space="preserve"> ergo per 15 p 5 erit proportio lineæ i d ad lineam q d, ſicut m k ad f k:</s> <s xml:id="echoid-s34407" xml:space="preserve"> eſt enim <lb/>linea i d dupla ad lineam d z, ſicut linea m k dupla ad lineam o k:</s> <s xml:id="echoid-s34408" xml:space="preserve"> ergo per 17 p 5 erit diui-<lb/>ſim proportio i q ad q d, ſicut m f ad f k:</s> <s xml:id="echoid-s34409" xml:space="preserve"> eſt autem ex præmiſsis proportio m f ad f k, ſicut g b <lb/>ad g a:</s> <s xml:id="echoid-s34410" xml:space="preserve"> ergo per 11 p 5 erit proportio i q ad q d, ſicut b g ad g a:</s> <s xml:id="echoid-s34411" xml:space="preserve"> quoniam accepta eſt proportio <lb/>m ſ ad f k, ſicut b g ad g a.</s> <s xml:id="echoid-s34412" xml:space="preserve"> Ducatur itaque linea b i:</s> <s xml:id="echoid-s34413" xml:space="preserve"> cui à puncto d ducatur æ quidiſtans d l per 31 <lb/>p 1:</s> <s xml:id="echoid-s34414" xml:space="preserve"> & producatur linea b g donec concurrat cum linea d l in puncto l:</s> <s xml:id="echoid-s34415" xml:space="preserve"> concurrent autem illæ li-<lb/>neæ per 2 th.</s> <s xml:id="echoid-s34416" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34417" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s34418" xml:space="preserve"> per 15 & per 29 p 1, & 4 p 6 triangulus l d q ſimilis trian gulo b q i:</s> <s xml:id="echoid-s34419" xml:space="preserve"> & e-<lb/>rit proportio q i ad q d, ſicut bi ad d l.</s> <s xml:id="echoid-s34420" xml:space="preserve"> Et cum linea i z ſitæ qualis lineæ z d, & linea b z perpendicu-<lb/>laris ſit ſuper lineá d i, ut patet ex pręiniſsis:</s> <s xml:id="echoid-s34421" xml:space="preserve"> erit per 4 p 1 linea b d æ qualis b i:</s> <s xml:id="echoid-s34422" xml:space="preserve"> erit ergo proportio li <lb/>neæ b d ad d l per 7 p 5 ſicut lineæ b i ad d l:</s> <s xml:id="echoid-s34423" xml:space="preserve"> eſt ergo proportio lineæ b d ad d l, ſicut lineæ i q ad q <lb/>d:</s> <s xml:id="echoid-s34424" xml:space="preserve"> ergo per 11 p 5, ſicut lineæ b g ad g a.</s> <s xml:id="echoid-s34425" xml:space="preserve"> Ducatur autem à puncto d linea, quæ ſit d h, æ qualem te-<lb/>nens angulum cum linea d l angulo b g a per 23 p 1:</s> <s xml:id="echoid-s34426" xml:space="preserve"> qui ſit angulus h d l:</s> <s xml:id="echoid-s34427" xml:space="preserve"> cadatq́;</s> <s xml:id="echoid-s34428" xml:space="preserve"> punctus h in linea <lb/>b g.</s> <s xml:id="echoid-s34429" xml:space="preserve"> Cum ergo lineæ h l & d l concurrantin puncto l:</s> <s xml:id="echoid-s34430" xml:space="preserve"> erunt duo anguli l h d & l d h minores duobus <lb/>rectis per 32 p 1 uel per 14 th.</s> <s xml:id="echoid-s34431" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34432" xml:space="preserve"> ergo duo anguli a g h & d h g, qui ſunt æ quales iſtis, ut patet <lb/>ex præ miſsis, ſunt minores duobus rectis:</s> <s xml:id="echoid-s34433" xml:space="preserve"> quare linea h d cõcurret cũ linea g a per 14 th.</s> <s xml:id="echoid-s34434" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s34435" xml:space="preserve"> Di <lb/>co quòd concurret in puncto a.</s> <s xml:id="echoid-s34436" xml:space="preserve"> Palàm enim quòd angulus g d n eſtrectus per 18 p 3:</s> <s xml:id="echoid-s34437" xml:space="preserve"> ſed per 32 p 1 cũ <lb/>trigoni o k c angulus c o k ſit rectus, & duo anguli o c k & c k o ſint & qualestecto:</s> <s xml:id="echoid-s34438" xml:space="preserve"> eſt angulus g d <lb/>n æ qualis illis duobus angulis o k c & o c k, & angulus o k e, ut patet expræmiſsis, æqualis eſt an-<lb/>gulo g d q:</s> <s xml:id="echoid-s34439" xml:space="preserve"> reſtat ergo, ut angulus q d n ſit æqualis angulo o c k, qui, ut pater ex præ miſsis, æqua.</s> <s xml:id="echoid-s34440" xml:space="preserve"> <lb/>lis eſt angulo b g e, ſcilicet medietati anguli b g a:</s> <s xml:id="echoid-s34441" xml:space="preserve"> eſt ergo angulus q d n medietas anguli b g a, <lb/>& ita medietas anguli h d l:</s> <s xml:id="echoid-s34442" xml:space="preserve"> ſed angulus q d b eſt medietas anguli b d l per 3 p 6:</s> <s xml:id="echoid-s34443" xml:space="preserve"> quoniam eſt pro-<lb/>portio lineæ b q ad q l, ſicut lineæ b d ad d l:</s> <s xml:id="echoid-s34444" xml:space="preserve"> cum, ſicut ſuprà oſtenſum eſt, triangulus d q l ſimilis ſit <lb/>triangulo b q i, & linea b d æqualis ſit lineæ b i, ut patet ex præ miſsis:</s> <s xml:id="echoid-s34445" xml:space="preserve"> reſtat igitur ut angulus b <lb/>d n ſit medietas anguli h d b:</s> <s xml:id="echoid-s34446" xml:space="preserve"> & ita angulus b d n eritæ qualis angulo n d h.</s> <s xml:id="echoid-s34447" xml:space="preserve"> Cum enim angulus b <lb/>d q ſit æ qualis angulo q d l, patet quòd angulus b d h excedit angulum h d l in duplo anguli q d h:</s> <s xml:id="echoid-s34448" xml:space="preserve"> <lb/>eſt ergo angulus b d n æ qualis angulo n d h.</s> <s xml:id="echoid-s34449" xml:space="preserve"> Producatur itaq;</s> <s xml:id="echoid-s34450" xml:space="preserve"> linea g d ultra punctum d ad pun-<lb/>ctum f.</s> <s xml:id="echoid-s34451" xml:space="preserve"> Et quia anguli f d n & g d n ſunt recti:</s> <s xml:id="echoid-s34452" xml:space="preserve"> reſtat ut angulus b d f ſit æ qualis angulo h d g:</s> <s xml:id="echoid-s34453" xml:space="preserve"> duca-<lb/>tur ergo per 31 p 1 linea h t æquidiſtans lineæ b d, cuius punctus t cadat in lineam d g.</s> <s xml:id="echoid-s34454" xml:space="preserve"> Palàm ergo <lb/>per 29 p 1 quòd angulus b d f eſt æ qualis angulo h t d:</s> <s xml:id="echoid-s34455" xml:space="preserve"> ſed & angulus b d f æ qualis eſt angu-<lb/>lo h d g:</s> <s xml:id="echoid-s34456" xml:space="preserve"> ergo per 6 p 1 linea h t eſt æ qualis lineę h d:</s> <s xml:id="echoid-s34457" xml:space="preserve"> ſed eſt proportio lineæ b d ad h t ſicut lineæ b <lb/>g ad g h per 29 p 1 & per 4 p 6:</s> <s xml:id="echoid-s34458" xml:space="preserve"> cum lineę b d & h t ſuntæquidiſtantes:</s> <s xml:id="echoid-s34459" xml:space="preserve"> eſt ergo per 7 p 5 proportio <lb/>lineæ b d ad d h, ſicut lineæ b g ad g h:</s> <s xml:id="echoid-s34460" xml:space="preserve"> ſed ex præmiſsis patet quòd linea h d producta ultra pun-<lb/>ctum d concurret cum linea g a, & fiet per 32 p 1 triangulus ſimilis triangulo h d l, cum habeant <lb/>angulum l h d communem, & angulus h d l ſit ex præmiſsis æqualis angulo h g a:</s> <s xml:id="echoid-s34461" xml:space="preserve"> igitur per 4 p 6 <lb/>eſt proportio lineę h d ad lineam d l, ſicut lineę h g ad lineam, quam ſecat linea h d exlinea a g:</s> <s xml:id="echoid-s34462" xml:space="preserve"> & <lb/>proportio lineę b d ad d l per 13 th.</s> <s xml:id="echoid-s34463" xml:space="preserve"> 1 huius conſtat exproportione lineę b d ad d h, & lineę d h ad l d:</s> <s xml:id="echoid-s34464" xml:space="preserve"> <lb/>igitur, ut patet ex præmiſsis, proportio lineę b d ad lineam d l conſtat ex proportione lineę b g ad g <lb/>h, & lineę g h ad lineã, quã h d ſecat ex g a:</s> <s xml:id="echoid-s34465" xml:space="preserve"> ſed proportio b d ad d l, ut patuit ſuperius, eſt ſicut b g a d <lb/>g a:</s> <s xml:id="echoid-s34466" xml:space="preserve"> ergo proportio b g ad g a cõſtat ex proportionibus b g ad g h, & ipſius g h ad lineá, quá ſecat h <lb/>dex g a:</s> <s xml:id="echoid-s34467" xml:space="preserve"> cõſtat aũt proportio lineę b g ad lineã g a per 13 th.</s> <s xml:id="echoid-s34468" xml:space="preserve"> 1 huius ex proportiõe lineę b g ad g h, & <lb/>lineę g h ad g a:</s> <s xml:id="echoid-s34469" xml:space="preserve"> igitur g a eſt linea, quã ſecat h d ex linea a g:</s> <s xml:id="echoid-s34470" xml:space="preserve"> & ita linea h d concurrit cum a g in pun-<lb/>cto a.</s> <s xml:id="echoid-s34471" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s34472" xml:space="preserve">, ut patet ex præmiſsis, angulus b d f eſt æqualis angulo h d g, & angulus h d g æ qua <lb/>lis eſt angulo f d a ſibi contrapoſito per 15 p 1:</s> <s xml:id="echoid-s34473" xml:space="preserve"> patet quòd angulus b d f ęqualis eſt angulo f da.</s> <s xml:id="echoid-s34474" xml:space="preserve"> Il-<lb/>lud ergo punctum d eſt punctus reflexionis per 8 huius:</s> <s xml:id="echoid-s34475" xml:space="preserve"> quoniam in ipſo angulus incidentię fic <pb o="236" file="0538" n="538" rhead="VITELLONIS OPTICAE"/> æqualis angulo reflexionis.</s> <s xml:id="echoid-s34476" xml:space="preserve"> Quod eſt propoſitum, quando angulus cks eſt maior recto.</s> <s xml:id="echoid-s34477" xml:space="preserve"> Quòd ſi <lb/>neuter angulorũ, qui ſunt c k s & c k y fuerit maior recto:</s> <s xml:id="echoid-s34478" xml:space="preserve"> dico quòd non fiet reflexio ab aliquo pun <lb/>cto ſpeculi ad uiſum.</s> <s xml:id="echoid-s34479" xml:space="preserve"> Sienim dicatur quòd hoc ſit poſsibile:</s> <s xml:id="echoid-s34480" xml:space="preserve"> ſit ergo punctus reflexionis d, ductis <lb/>lineis a d, b d, a g, b g, d g.</s> <s xml:id="echoid-s34481" xml:space="preserve"> Et quia fit reflexio à puncto ſpeculid, pater per præmiſſam, quòd oportet <lb/>angulum b d g eſſe maiorem recto:</s> <s xml:id="echoid-s34482" xml:space="preserve"> non ergo fiet reflexio ab his ſpeculis ſecundum diſpoſitionem <lb/>talem figuræ, ut angulorum c k s & c k y quilibet ſit maior recto.</s> <s xml:id="echoid-s34483" xml:space="preserve"> Sed & idem aliter demonſtran-<lb/>dum.</s> <s xml:id="echoid-s34484" xml:space="preserve"> Producaturitaq;</s> <s xml:id="echoid-s34485" xml:space="preserve"> linea a dintra circulum uſq;</s> <s xml:id="echoid-s34486" xml:space="preserve"> ad h pun ctum lineæ g b:</s> <s xml:id="echoid-s34487" xml:space="preserve"> & producatur linea d l <lb/>intra circulum taliter, ut fiat angulus l d h æqualis angulo a g b per 23 p 1:</s> <s xml:id="echoid-s34488" xml:space="preserve"> protracta quoq;</s> <s xml:id="echoid-s34489" xml:space="preserve"> linea b g, <lb/>quouſq;</s> <s xml:id="echoid-s34490" xml:space="preserve"> cócurrat cum linea d l in puncto l:</s> <s xml:id="echoid-s34491" xml:space="preserve"> concurret aũt per 14 th.</s> <s xml:id="echoid-s34492" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34493" xml:space="preserve"> quoniam angulus g d l <lb/>eſt minor recto per 42 th.</s> <s xml:id="echoid-s34494" xml:space="preserve"> 1 huius, & angulus d g b, ut patet ք 3 huius, & ք 33 p 6, eſt etiá ininor recto:</s> <s xml:id="echoid-s34495" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0538-01a" xlink:href="fig-0538-01"/> & ducatur linea cótingés <lb/>circulum in puncto d, quę <lb/>ſit n d y:</s> <s xml:id="echoid-s34496" xml:space="preserve"> & à puncto d pro <lb/>tracta linea d q fecante li-<lb/>neam g b in puncto q, fiat <lb/>angulus q d n ęqualis me-<lb/>dietati anguli a g b ք 9 & <lb/>23 p 1:</s> <s xml:id="echoid-s34497" xml:space="preserve"> palã ergo quòd trian <lb/>gulus h d l æquiangulus <lb/>cſt triangulo h g a.</s> <s xml:id="echoid-s34498" xml:space="preserve"> Quia e-<lb/>nim angulus h d l æqualis <lb/>eſt angulo h g a, & angulus <lb/>a h g eſt cõmunis, erit per <lb/>32 p 1 tertius tertio æqua-<lb/>lis:</s> <s xml:id="echoid-s34499" xml:space="preserve"> ergo per 4 p 6 erit pro-<lb/>portio lineæ h d ad d l, ſi-<lb/>cut lineæ h g ad g a.</s> <s xml:id="echoid-s34500" xml:space="preserve"> Duca-<lb/>tur itaq;</s> <s xml:id="echoid-s34501" xml:space="preserve"> à puncto h per 31 p 1 linea æquidiſtans lineæ b d, quæ ſit h t:</s> <s xml:id="echoid-s34502" xml:space="preserve"> erit ergo per 29 p 1 & per 4 p 6 <lb/>proportio lineę b d ad th, ſicut lineæ b g ad g h.</s> <s xml:id="echoid-s34503" xml:space="preserve"> Quia uero ex hypotheſi forma puncti b reflectitur <lb/>ad uiſum a à puncto ſpeculi d, ducatur linea g d extra circulum ad punctum e:</s> <s xml:id="echoid-s34504" xml:space="preserve"> erit quoq;</s> <s xml:id="echoid-s34505" xml:space="preserve"> per 8 hu-<lb/>ius angulus e d b æqualis angulo e d a:</s> <s xml:id="echoid-s34506" xml:space="preserve"> ergo per 15 & 29 p 1 erit angulus d t h æqualis angulo h d t:</s> <s xml:id="echoid-s34507" xml:space="preserve"> <lb/>ergo per 6 p 1 erit linea d h æqualis lineæ h t.</s> <s xml:id="echoid-s34508" xml:space="preserve"> Quia ergo, ut patet per 4 p 6, cum linea t h ſit æquidi-<lb/>ſtans lineæ d b, erit proportio b g ad g h, ſicut b d ad h t:</s> <s xml:id="echoid-s34509" xml:space="preserve"> ſed linea t h æqualis eſt ipſi d h:</s> <s xml:id="echoid-s34510" xml:space="preserve"> eſt ergo per <lb/>7 p 5 proportio b d ad d h, ſicut b g ad g h:</s> <s xml:id="echoid-s34511" xml:space="preserve"> fuit autem proportio h d ad d l, ſicut h g ad g a:</s> <s xml:id="echoid-s34512" xml:space="preserve"> ergo per <lb/>22 p 5 erit proportio b d ad d l, ſicut b g ad g a:</s> <s xml:id="echoid-s34513" xml:space="preserve"> ſed cũ angulus b d e ſit æ qualis angulo h d g per præ-<lb/>miſſa, & angulus n d e æqualis angulo n d g, quia uterq;</s> <s xml:id="echoid-s34514" xml:space="preserve"> rectus:</s> <s xml:id="echoid-s34515" xml:space="preserve"> relinquitur angulus b d n æqualis <lb/>angulo n d h:</s> <s xml:id="echoid-s34516" xml:space="preserve"> eſt ergo angulus h d n medietas anguli b d h:</s> <s xml:id="echoid-s34517" xml:space="preserve"> ſed angulus n d q eſt medietas anguli a g <lb/>b ex præmiſsis:</s> <s xml:id="echoid-s34518" xml:space="preserve"> ergo & eſt medietas anguli h d l, qui eſt æqualis angulo a g b:</s> <s xml:id="echoid-s34519" xml:space="preserve"> igitur angulus b d q <lb/>eſt medietas anguli b d l:</s> <s xml:id="echoid-s34520" xml:space="preserve"> eſt ergo angulus b d q æqualis angulo q d l:</s> <s xml:id="echoid-s34521" xml:space="preserve"> ergo per 3 p 6 in trigono b d l <lb/>erit ꝓportio b q ad q l, ſicut b d ad d l.</s> <s xml:id="echoid-s34522" xml:space="preserve"> Ducatur quoq;</s> <s xml:id="echoid-s34523" xml:space="preserve"> à puncto b per 31 p 1 linea æquidiſtãs lineę d l, <lb/>quæ ſit b i:</s> <s xml:id="echoid-s34524" xml:space="preserve"> & concurrat linea d q cum linea b i in puncto i:</s> <s xml:id="echoid-s34525" xml:space="preserve"> concurret autem per 2 th.</s> <s xml:id="echoid-s34526" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34527" xml:space="preserve"> & diui-<lb/>datur linea d i per æqualia in puncto z per 10 p 1:</s> <s xml:id="echoid-s34528" xml:space="preserve"> & ducatur linea b z.</s> <s xml:id="echoid-s34529" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s34530" xml:space="preserve"> per 15 & 29 & 32 <lb/>p 1 quoniam trigona b q i & q d l ſunt æquiangula:</s> <s xml:id="echoid-s34531" xml:space="preserve"> ergo per 4 p 6 erit proportio lineæ b q ad <lb/>q l, ſicut lineæ b i ad d l:</s> <s xml:id="echoid-s34532" xml:space="preserve"> fuit autem ex præmiſsis proportio b q ad q l, ſicut b d ad d l:</s> <s xml:id="echoid-s34533" xml:space="preserve"> ergo per 11 p 5 <lb/>eſt proportio b i ad d l, ſicuti b d ad d l:</s> <s xml:id="echoid-s34534" xml:space="preserve"> ergo per 9 p 5 lineæ b i & b d ſunt æquales:</s> <s xml:id="echoid-s34535" xml:space="preserve"> ergo per 31 th.</s> <s xml:id="echoid-s34536" xml:space="preserve"> 1 hu <lb/>ius linea b z eſt perpendicularis ſuper lineam d i:</s> <s xml:id="echoid-s34537" xml:space="preserve"> eſt autẽ, ſicut ex præmiſsis patet, proportio i q ad <lb/>q d, ſicut m f ad ſ k:</s> <s xml:id="echoid-s34538" xml:space="preserve"> ergo per 18 p 5 erit cóiunctim proportio i d ad d q, ſicut m k ad ſ k:</s> <s xml:id="echoid-s34539" xml:space="preserve"> & erit per 15 p <lb/>5 proportio d z ad q d, ſicut o kad f k:</s> <s xml:id="echoid-s34540" xml:space="preserve"> ergo per 17 p 5 erit proportio z q ad q d, ſicut o f ad f k.</s> <s xml:id="echoid-s34541" xml:space="preserve"> Produ-<lb/>catur quoq;</s> <s xml:id="echoid-s34542" xml:space="preserve"> linea b z intra ſpeculum, donec concurrat cum linea e g:</s> <s xml:id="echoid-s34543" xml:space="preserve"> concurret autem per 14 <lb/>th.</s> <s xml:id="echoid-s34544" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34545" xml:space="preserve"> cum angulus d z b ſit rectus, ut præoſtenſum eſt, & angulus z d g ſit minor recto, qui <lb/>eſt angulus n d g:</s> <s xml:id="echoid-s34546" xml:space="preserve"> ſit ergo punctum concurſus x.</s> <s xml:id="echoid-s34547" xml:space="preserve"> Palàm àutem ex præmiſsis, quoniam eſt propor-<lb/>tio lineæ b g ad g d, ſicut lineæ s p ad p k.</s> <s xml:id="echoid-s34548" xml:space="preserve"> Cum ergo angulus c k s dicatur non eſſe maior recto:</s> <s xml:id="echoid-s34549" xml:space="preserve"> fiat <lb/>ſuper punctum k lineæ c k angulus maior recto:</s> <s xml:id="echoid-s34550" xml:space="preserve"> hoc autem eſt poſsibile fieri:</s> <s xml:id="echoid-s34551" xml:space="preserve"> quia cum ſicut pa-<lb/>tet ex præmiſsis, angulus q d n ſit æqualis medietati anguli a g b, & eidem æqualis conſtitutus <lb/>ſit angulus k c o, neceſſe eſt quòd angulus q d n ſit æqualιs angulo k c o:</s> <s xml:id="echoid-s34552" xml:space="preserve"> erit ergo, ut patet ex præ-<lb/>miſsis, angulus q d g æqualis angulo c k o, quod patet ut prius.</s> <s xml:id="echoid-s34553" xml:space="preserve"> Cum enim trigonum c k o ſit or-<lb/>thogonium, palàm quòd duo anguli k c o & c k o ualent unum rectum per 32 p 1:</s> <s xml:id="echoid-s34554" xml:space="preserve"> ſunt ergo æ-<lb/>quales angulo n d g.</s> <s xml:id="echoid-s34555" xml:space="preserve"> Et quia angulus k c o eſt æquàlis angulo n d q:</s> <s xml:id="echoid-s34556" xml:space="preserve"> relinquitur angulus c k o æ-<lb/>qualis angulo q d g.</s> <s xml:id="echoid-s34557" xml:space="preserve"> Fiat ergo ſuper punctum k lineæ f k angulus æqualis angulo b d q:</s> <s xml:id="echoid-s34558" xml:space="preserve"> & pona-<lb/>tur quòd linea tenens hunc angulum, concurrat cum linea c o in puncto s:</s> <s xml:id="echoid-s34559" xml:space="preserve"> & ducatur linea s p <lb/>tranſiens per punctum f, quæ ſit alia à priori linea s ſ p.</s> <s xml:id="echoid-s34560" xml:space="preserve"> Dico quòd iſtius lineæ s p ad lineam <lb/>p k partem lineæ c k erit proportio, ſicut lineæ b g ad g d.</s> <s xml:id="echoid-s34561" xml:space="preserve"> Cum enim angulus b z d ſit re-<lb/>ctus, æqualis angulo s o k:</s> <s xml:id="echoid-s34562" xml:space="preserve"> erit triangulus b z d ex præmiſsis ſimilis triangulo s o k:</s> <s xml:id="echoid-s34563" xml:space="preserve"> eſt ergo <lb/>proportio lineæ b z ad b d, ſicut lineæ o s ad lineam s k, & lineæ b z ad z d, ſicut lineæ s o ad <lb/> <pb o="237" file="0539" n="539" rhead="LIBER SEXTVS."/> o k:</s> <s xml:id="echoid-s34564" xml:space="preserve"> fuit autem oſtenſum prius, quia eſt proportio lineæ z q ad q d, ſicut lineæ o f ad f k:</s> <s xml:id="echoid-s34565" xml:space="preserve"> ergo per 5 <lb/>th.</s> <s xml:id="echoid-s34566" xml:space="preserve"> 1 huius erit econtrario proportio lineæ q d ad z q, ſicut ſ kad o f:</s> <s xml:id="echoid-s34567" xml:space="preserve"> ergo per 18 p 5 eſt proportio <lb/>totius lineæ z d ad z q, ſicut totius lineæ o k ad o f:</s> <s xml:id="echoid-s34568" xml:space="preserve"> ergo per 22 p 5 erit z b ad z q, ſicut s o ad of:</s> <s xml:id="echoid-s34569" xml:space="preserve"> er-<lb/>go per 6 p 6 trigona z q b & of s ſunt æquiangula:</s> <s xml:id="echoid-s34570" xml:space="preserve"> angulus ergo z b q eſt æqualis angulo o s f:</s> <s xml:id="echoid-s34571" xml:space="preserve"> re-<lb/>manet ergo angulus q b d æqualis angulo f s k:</s> <s xml:id="echoid-s34572" xml:space="preserve"> ſed & angulus f k s factus fuit æqualis angulo b d q, <lb/>& angulus p k f æqualis eſt angulo q d g:</s> <s xml:id="echoid-s34573" xml:space="preserve"> totus ergo angulus s k p æqualis eſt angulo b d g:</s> <s xml:id="echoid-s34574" xml:space="preserve"> ergo per <lb/>32 p 1, & ex 4 p 6 erit triangulus b d g ſimilis triangulo s p k:</s> <s xml:id="echoid-s34575" xml:space="preserve"> & totus triangulus b d x ſimilis totali <lb/>triangulo c k s:</s> <s xml:id="echoid-s34576" xml:space="preserve"> eſtigitur proportio lineæ s p ad p k, ſicut b g ad g d.</s> <s xml:id="echoid-s34577" xml:space="preserve"> Conſtituto ergo ſuper centrum <lb/>g angulo, æquali angulo iſti s p k, & ducta ſemidiametro circuli, quæ ſit g u, patet ſecundum præ-<lb/>miſſum modum, quoniam punctum u erit punctum reflexionis.</s> <s xml:id="echoid-s34578" xml:space="preserve"> Et quia, ut patet per 16 p 1, & ex <lb/>præmiſsis, prior angulus s p k eſt maior præſenti angulo s p k, quoniam eſt extrinſecus:</s> <s xml:id="echoid-s34579" xml:space="preserve"> palàm <lb/>quòd à duobus punctis ſpeculi, quæ ſunt d & u, fiet reflexio:</s> <s xml:id="echoid-s34580" xml:space="preserve"> quod eſt contra 16 huius.</s> <s xml:id="echoid-s34581" xml:space="preserve"> Non ergo po <lb/>teſt angulus s p k unquam eſſe non maior recto, ſi ſecundum ipſum debeat fieri puncti reflexionis <lb/>inuentio:</s> <s xml:id="echoid-s34582" xml:space="preserve"> quia ſecundum talem diſpoſitionem collocatis puncto rei uiſæ & centro uiſus, non eſt <lb/>poſsibile fieri reflexionem.</s> <s xml:id="echoid-s34583" xml:space="preserve"> Item impoſsibile eſt quòd duo anguli conſtituti ſuper lineam m o ſint <lb/>uterq;</s> <s xml:id="echoid-s34584" xml:space="preserve"> maior recto.</s> <s xml:id="echoid-s34585" xml:space="preserve"> Si enim uterq;</s> <s xml:id="echoid-s34586" xml:space="preserve"> talium maior fuerit recto, cum ſuper g centrum circuli propoſiti <lb/>fiat angulus æqualis angulo s k m, ſiet ſuper idem centrum angulus alius diuerſus ab iſto:</s> <s xml:id="echoid-s34587" xml:space="preserve"> quem ef-<lb/>ficiet ſuper k m alia linea ſimilis priori lineæ s k:</s> <s xml:id="echoid-s34588" xml:space="preserve"> & ita â puncto d & ab alio puncto illius circuli fiet <lb/>reflexio formæ eiuſdem puncti ad uiſum eundem:</s> <s xml:id="echoid-s34589" xml:space="preserve"> quod eſt contra 16 huius.</s> <s xml:id="echoid-s34590" xml:space="preserve"> Oportet ergo ut tan-<lb/>tùm unus illorum angulorum ſit maior recto, non ambo maiores uel ambo minores recto.</s> <s xml:id="echoid-s34591" xml:space="preserve"> Patet <lb/>ergo propoſitum.</s> <s xml:id="echoid-s34592" xml:space="preserve"/> </p> <div xml:id="echoid-div1425" type="float" level="0" n="0"> <figure xlink:label="fig-0536-02" xlink:href="fig-0536-02a"> <variables xml:id="echoid-variables600" xml:space="preserve">c p r m o f k y s</variables> </figure> <figure xlink:label="fig-0536-03" xlink:href="fig-0536-03a"> <variables xml:id="echoid-variables601" xml:space="preserve">b f e m h a d a c z q t i g j</variables> </figure> <figure xlink:label="fig-0538-01" xlink:href="fig-0538-01a"> <variables xml:id="echoid-variables602" xml:space="preserve">c p p s p s b e n h d a k z q t j g x</variables> </figure> </div> </div> <div xml:id="echoid-div1427" type="section" level="0" n="0"> <head xml:id="echoid-head1102" xml:space="preserve" style="it">23. Super unam cathetum incidentiæ ſuper ficiei ſpeculi ſphærici conuexi, uelſuper diuerſas <lb/>aduiſum, ad quem fit reflexio, conſimiliter ſe habentes, datis duobus punctis, quorum formæ à <lb/>ſuperficie ſpeculi ſint reflexibiles ad uiſum: erit locus imaginis puncti centro ſpeculi propinquio <lb/>ris remotior à centro ſpeculi, & remotioris propinquior.</head> <p> <s xml:id="echoid-s34593" xml:space="preserve">Sit circulus (qui eſt cómunis ſectio ſuperficiei reflexionis & ſuperficiei ſpeculi ſphærici cóuexi,) <lb/>a b c:</s> <s xml:id="echoid-s34594" xml:space="preserve"> cuius centrum d:</s> <s xml:id="echoid-s34595" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s34596" xml:space="preserve"> centrum uiſus e:</s> <s xml:id="echoid-s34597" xml:space="preserve"> & cathetus incidentiæ ſit d f g:</s> <s xml:id="echoid-s34598" xml:space="preserve"> in qua ſint duo puncta <lb/>f & g, quorum formæ ſint reflexibiles ad uiſum:</s> <s xml:id="echoid-s34599" xml:space="preserve"> & ſit punctum f propinquius centro ſpeculi, & pun <lb/>ctum g remotius:</s> <s xml:id="echoid-s34600" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s34601" xml:space="preserve"> eadem cathetus circulum a b cin puncto c.</s> <s xml:id="echoid-s34602" xml:space="preserve"> Dico quòd locus imaginis for <lb/>mæ puncti f remotior eſt à centro ſpeculi, quod eſt d, quàm locus imaginis formæ puncti g.</s> <s xml:id="echoid-s34603" xml:space="preserve"> Quo-<lb/>niam enim, ut patet per hypotheſim, quælibet for-<lb/> <anchor type="figure" xlink:label="fig-0539-01a" xlink:href="fig-0539-01"/> marum iſtorum punctorum ab aliquo puncto ſpe-<lb/>culi reflectitur ad uiſum:</s> <s xml:id="echoid-s34604" xml:space="preserve"> patet cum illa puncta ſint <lb/>in eadem catheto incidentiæ conſiſtentia, quòd <lb/>centrum uiſus e eſt cum ambobus illis punctis in <lb/>eadem ſuperficie reflexionis per 6 huius:</s> <s xml:id="echoid-s34605" xml:space="preserve"> fiet ergo <lb/>reflexio cuiuslibetillorum punctorum ad uiſum e <lb/>ab aliquo puncto circuli a b c.</s> <s xml:id="echoid-s34606" xml:space="preserve"> Sit ergo, ut forma <lb/>puncti g reflectatur à puncto a, & forma puncti f à <lb/>puncto b:</s> <s xml:id="echoid-s34607" xml:space="preserve"> erit ergo per 17 huius punctus b remo-<lb/>tior à centro uiſus e quàm punctus a.</s> <s xml:id="echoid-s34608" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s34609" xml:space="preserve"> <lb/>diameter uiſualis, quæ e d:</s> <s xml:id="echoid-s34610" xml:space="preserve"> & ducantur lineæ inci-<lb/>dentiæ, quæ ſint g a & f b:</s> <s xml:id="echoid-s34611" xml:space="preserve"> & lineæ reflexionis, quæ <lb/>ſint a e & b e:</s> <s xml:id="echoid-s34612" xml:space="preserve"> quæ productæ intra circulum ſeca-<lb/>bunt cathetum d f g ք 9 huius.</s> <s xml:id="echoid-s34613" xml:space="preserve"> Et quoniam concur <lb/>runt cum diametro uiſuali, quæ eſt e d:</s> <s xml:id="echoid-s34614" xml:space="preserve"> ſit ergo, ut <lb/>linea e a ſecet cathetum g d in puncto h, & linea e b <lb/>in puncto k.</s> <s xml:id="echoid-s34615" xml:space="preserve"> Erit ergo punctum h locus imaginis <lb/>formæ puncti g, & punctum k locus imaginis formæ puncti f per 11 huius.</s> <s xml:id="echoid-s34616" xml:space="preserve"> Quoniam uerò pun-<lb/>ctum h eſt propinquius centro d quàm punctum k per 29 th.</s> <s xml:id="echoid-s34617" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34618" xml:space="preserve"> quia enim linea h e ſecat an-<lb/>gulum d e k, palàm quia ipſa ſecabit baſim illi ſubtenſam, quæ eſt d k:</s> <s xml:id="echoid-s34619" xml:space="preserve"> eſt ergo punctum h propin-<lb/>quius centro ſpeculi, quod eſt d, quàm punctum k.</s> <s xml:id="echoid-s34620" xml:space="preserve"> Et quoniam, ut patet ſecundum hunc modum, <lb/>omnes lineæ ductæ à centro uiſus, quod eſt e, per quæcunq;</s> <s xml:id="echoid-s34621" xml:space="preserve"> puncta arcus a c, intermedia puncto-<lb/>rum a & c ad cathetum d g, cadunt in puncta ſemidiametri d c à centro remotiora quàm punctum <lb/>h, patet propoſitum.</s> <s xml:id="echoid-s34622" xml:space="preserve"> Et ex hoc etiam patet quòd quantò puncta lineæ c g ſunt propinquiora cen-<lb/>tro d, tantò loca ſuarum imaginum ſunt magis elongata à centro ſpeculi, quod eſt d.</s> <s xml:id="echoid-s34623" xml:space="preserve"> Et quoniam <lb/>omnes catheti incidentiæ concurrunt in centro ſpeculi:</s> <s xml:id="echoid-s34624" xml:space="preserve"> palàm quòd de punctis diuerſarum cathe-<lb/>torum ad uiſum, ad quem fit reflexio, conſimiliter ſe habentium, eadem eſt demonſtratio, quæ de <lb/>punctis eiuſdem catheti:</s> <s xml:id="echoid-s34625" xml:space="preserve"> quoniá unicuiq;</s> <s xml:id="echoid-s34626" xml:space="preserve"> punctorũ in una ſimili catheto ſignatorũ, pũctus ſimilis, <lb/>qui ſit eiuſdem diſtantię à centro ſpeculi, in catheto alia reſpondet:</s> <s xml:id="echoid-s34627" xml:space="preserve"> & illorũ quorumcunq;</s> <s xml:id="echoid-s34628" xml:space="preserve"> puncto-<lb/>rum (quia conſimiliter reſpiciunt uiſum) loca imaginum reſpectu centri ſpeculi confimiliter ordi-<lb/>nantur.</s> <s xml:id="echoid-s34629" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s34630" xml:space="preserve"/> </p> <div xml:id="echoid-div1427" type="float" level="0" n="0"> <figure xlink:label="fig-0539-01" xlink:href="fig-0539-01a"> <variables xml:id="echoid-variables603" xml:space="preserve">e g a b f c k h d</variables> </figure> </div> <pb o="238" file="0540" n="540" rhead="VITELLONIS OPTICAE"/> </div> <div xml:id="echoid-div1429" type="section" level="0" n="0"> <head xml:id="echoid-head1103" xml:space="preserve" style="it">24. Si ab aliquo puncto ſpeculi ſphærici conuexi linea reflexionis producta circulum (qui eſt <lb/>communis ſectio ſuperficiei reflexionis & ſpeculi) taliter ſecuerit, quòd lineæ productæ pars, quæ <lb/>eſt intra circulum, ſit æqualis ſemidiametro circuli: locus uiſæ imaginis ſemper erit intra conue-<lb/>xuæ ſpeculi. Alhazen 20 n 5.</head> <p> <s xml:id="echoid-s34631" xml:space="preserve">Eſto centrũ uiſus g:</s> <s xml:id="echoid-s34632" xml:space="preserve"> & centrũ ſpeculi ſphærici conuexi ſit punctú d:</s> <s xml:id="echoid-s34633" xml:space="preserve"> ſitq́:</s> <s xml:id="echoid-s34634" xml:space="preserve"> cómunis ſectio ſuperfi-<lb/>ciei reflexionis & ſpeculi circulus a b k à centro quoq;</s> <s xml:id="echoid-s34635" xml:space="preserve"> uiſus puncto g ducantur per 17 p 3 duæ lineę <lb/>contingentes circulũ a b k, quę ſint g a & g b:</s> <s xml:id="echoid-s34636" xml:space="preserve"> eritq́ ք 2 huius circuli a b k portio a b apparẽs uiſui:</s> <s xml:id="echoid-s34637" xml:space="preserve"> & <lb/>centrũ eius ſit punctũ d.</s> <s xml:id="echoid-s34638" xml:space="preserve"> Quoniá aũt uiſus & ſpecula mu <lb/> <anchor type="figure" xlink:label="fig-0540-01a" xlink:href="fig-0540-01"/> tantlocum:</s> <s xml:id="echoid-s34639" xml:space="preserve"> ſit talis facta diſpoſitio uiſus ad ſpeculum, ut <lb/>à puncto g centro uiſus ductæ lineæ ſecantis circulum a <lb/>b k, pars intra circulum, quæ eſt chorda arcus circuli, qui <lb/>h k, ſit æqualis ſemidiametro illius circuli:</s> <s xml:id="echoid-s34640" xml:space="preserve"> & ſit illa linea <lb/>g h k, cuius pars h k intra circulum ſit ęqualis ſemidiame <lb/>tro d k.</s> <s xml:id="echoid-s34641" xml:space="preserve"> Hoc aũt poſsibile eſt fieri, ſi ք 1 p 4 inſcribatur cir <lb/>culo a b k linea h k æqualisſemidiam etro illius circuli:</s> <s xml:id="echoid-s34642" xml:space="preserve"> & <lb/>in illa linea k h producta extra circulum ponatur centrú <lb/>uiſus.</s> <s xml:id="echoid-s34643" xml:space="preserve"> Dico quòd locus imaginis reflexę à puncto h ſem <lb/>per eſt intra conuexam ſuperficiem ſpeculi.</s> <s xml:id="echoid-s34644" xml:space="preserve"> Producatur <lb/>enim à puncto h ſuք lineã cõtingentẽ circulũ in pũcto h <lb/>քpendicularis, quę ſit h m:</s> <s xml:id="echoid-s34645" xml:space="preserve"> hæc ergo producta in circulũ <lb/>tranſit per centrum d per 19 p 3.</s> <s xml:id="echoid-s34646" xml:space="preserve"> Dico quòd cum forma a-<lb/>licuius rei uiſæ reflectatur à puncto h, locus imaginis ſuę <lb/>erit ſemperintra conuexũ ſpeculi.</s> <s xml:id="echoid-s34647" xml:space="preserve"> Ducaturenim à pun-<lb/>cto h linea conſtituens ſuper punctum h terminum li-<lb/>neæ h m angulum æqualem angulo g h m per 23 p 1, qui <lb/>ſit p h m, producta linea h p:</s> <s xml:id="echoid-s34648" xml:space="preserve"> reflectẽturergo per 20 th.</s> <s xml:id="echoid-s34649" xml:space="preserve"> 5 <lb/>puncta huius lineę h p ad uiſum g à puncto ſpeculi h:</s> <s xml:id="echoid-s34650" xml:space="preserve"> nec <lb/>alterius lineæ puncta à puncto h ad uiſum poterunt re-<lb/>flecti.</s> <s xml:id="echoid-s34651" xml:space="preserve"> Sumatur ergo alιquod eius punctum, quod ſit p:</s> <s xml:id="echoid-s34652" xml:space="preserve"> & <lb/>ducatur linea ab ipſo ad centrum ſpeculi, quæ ſit p d:</s> <s xml:id="echoid-s34653" xml:space="preserve"> erit <lb/>quoq, per 1 huius, & per 72 th 1 huius linea p d perpendi <lb/>cularis ſuper ſuperficiem contingentẽ ſpeculum in pun-<lb/>cto, quo ipſa linea p d ſecat circum ſerentiam circuli a b <lb/>k:</s> <s xml:id="echoid-s34654" xml:space="preserve"> copuletur quoq;</s> <s xml:id="echoid-s34655" xml:space="preserve"> linea d k.</s> <s xml:id="echoid-s34656" xml:space="preserve"> Et quia angulus p h m inci-<lb/>dentiæ eſt æqualis angulo m h g reflexionis, ut pater ex præmiſsis, angulus u erò g h m per 15 p 1 æ-<lb/>qualis eſt angulo k h d:</s> <s xml:id="echoid-s34657" xml:space="preserve"> angulus igitur p h m eſt æqualis angulo k h d:</s> <s xml:id="echoid-s34658" xml:space="preserve"> ſed angulus k h d æqualιs eſt <lb/>angulo h d k per 5 p 1, ideo quia latus h k ex hypotheſi æquale eſt ſemidiametro d k:</s> <s xml:id="echoid-s34659" xml:space="preserve"> angulus ergo <lb/>p h m eſt æqualis angulo h d k.</s> <s xml:id="echoid-s34660" xml:space="preserve"> Quia ergo linea m d cadens ſuper lineas h p & d k facit angulum <lb/>extrinſecum, qui eſt m h p, æqualem angulo intrinſeco, qui eſt m d k:</s> <s xml:id="echoid-s34661" xml:space="preserve"> linea ergo h p per 28 p 1 æqui-<lb/>diſtat lineæ d k:</s> <s xml:id="echoid-s34662" xml:space="preserve"> lineæ ergo h p & d k in infinitum protractæ nunquam concurrent.</s> <s xml:id="echoid-s34663" xml:space="preserve"> Et linea p d, quæ <lb/>eſt cathetus incidentiæ ſormæ puncti p, uel quæcunq;</s> <s xml:id="echoid-s34664" xml:space="preserve"> alia linea ducta à quocunq;</s> <s xml:id="echoid-s34665" xml:space="preserve"> puncto lineæ h <lb/>p ad centrum d, ſemper inter puncta h & k interſecabit lineam h k interiacentem lineas æquidiſtan <lb/>tes, quæ ſunt k d & h p, ut patet per 29 th.</s> <s xml:id="echoid-s34666" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34667" xml:space="preserve"> diuidunt enim omnes illæ catheti angulum h d k:</s> <s xml:id="echoid-s34668" xml:space="preserve"> <lb/>ergo & ſecabunt baſim h k:</s> <s xml:id="echoid-s34669" xml:space="preserve"> quælibet enim illarum cathetorum incidentiæ ſemper ducitur ad cen-<lb/>trum ſpeculi, ut ad punctum d.</s> <s xml:id="echoid-s34670" xml:space="preserve"> Quodcunq;</s> <s xml:id="echoid-s34671" xml:space="preserve"> ergo punctum ſumatur in linea p h:</s> <s xml:id="echoid-s34672" xml:space="preserve"> ſemper linea ducta <lb/>ab illo puncto ad punctum d ſecabit lineam reflexionis, quæ eſt g h k intra cõuexum ſpeculi:</s> <s xml:id="echoid-s34673" xml:space="preserve"> quo-<lb/>niam ſemper cathetus incidentiæ producta ad centrum ſpeculi perpendicularis eſt ſuper ſuperfi-<lb/>ciem ſpeculi, ſicut nunc eſt p d.</s> <s xml:id="echoid-s34674" xml:space="preserve"> Imago ergo cuiuſcũq;</s> <s xml:id="echoid-s34675" xml:space="preserve"> puncti lineæ p h per 11 huius apparebit intra <lb/>conuexum ſpeculi.</s> <s xml:id="echoid-s34676" xml:space="preserve"> Ethoc proponebatur.</s> <s xml:id="echoid-s34677" xml:space="preserve"/> </p> <div xml:id="echoid-div1429" type="float" level="0" n="0"> <figure xlink:label="fig-0540-01" xlink:href="fig-0540-01a"> <variables xml:id="echoid-variables604" xml:space="preserve">g m h z p b d a k</variables> </figure> </div> </div> <div xml:id="echoid-div1431" type="section" level="0" n="0"> <head xml:id="echoid-head1104" xml:space="preserve" style="it">25. A‘ quocuń puncto arcus circuli (quieſt communis ſectio ſuperficiei reflexionis & ſpe-<lb/>culi ſphærici conuexi) interiacentis puncta, in quibus caιhetus reflexionis & linea reflexionis, <lb/>(cuius pars intra circulum est æqualis ſemidiametro circuli) ſecant circulu, fiat reflexio: locus <lb/>uiſæ imagin is ſemper erit intra ſpeculum. Alhazen 21 n 5.</head> <p> <s xml:id="echoid-s34678" xml:space="preserve">Sit diſpoſitio, quæ in præmiſſa, ita ut linea reflexionis, quæ g h k ſecet circulũ a b k, taliter ut eius <lb/>pars intra circulum, quę eſt h k, ſit æqualis ſemidiametro circuli:</s> <s xml:id="echoid-s34679" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s34680" xml:space="preserve"> cathetus reflexionis à <lb/>uiſu ad centrũ ſpeculi:</s> <s xml:id="echoid-s34681" xml:space="preserve"> quæ ſit g d, ſecans circulum a b k in puncto z Dico quòd à quocunq;</s> <s xml:id="echoid-s34682" xml:space="preserve"> puncto <lb/>arcus h z fiat reflexio, ſemper erit locus imaginis intra ſpeculum.</s> <s xml:id="echoid-s34683" xml:space="preserve"> Sit enim ita, ut à puncto illius ar-<lb/>cus h z (quod ſit i) fiat reflexio:</s> <s xml:id="echoid-s34684" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s34685" xml:space="preserve"> à puncto g centro uiſus ad punctũ i linea ſecans circulũ <lb/>ſuper punctũ i, quę ſit g i s:</s> <s xml:id="echoid-s34686" xml:space="preserve"> & & ducatur ſuper ſuperſiciem ſpeculi linea perpendicularis à puncto i:</s> <s xml:id="echoid-s34687" xml:space="preserve"> <lb/>quod fiet per 72 th.</s> <s xml:id="echoid-s34688" xml:space="preserve"> 1 huius, ſi à centro ſpeculi puncto d producatur linea, quæ ſit d i t:</s> <s xml:id="echoid-s34689" xml:space="preserve"> ſuper cuius <lb/> <pb o="239" file="0541" n="541" rhead="LIBER SEXTVS."/> punctum i fiat angulus æqualis angulo ti g per 23 p 1, qui ſit p it.</s> <s xml:id="echoid-s34690" xml:space="preserve"> Palàm ergo quòd ſolùm puncta li-<lb/>neæ p i reflectuntur à puncto i ad uiſum g per 20 th.</s> <s xml:id="echoid-s34691" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s34692" xml:space="preserve"> Palàm etiam per 15 p 3 quòd linea i s <lb/>maior eſt quàm linea h k:</s> <s xml:id="echoid-s34693" xml:space="preserve"> ergo linea is eſt maior ſemi-<lb/> <anchor type="figure" xlink:label="fig-0541-01a" xlink:href="fig-0541-01"/> diametro s d.</s> <s xml:id="echoid-s34694" xml:space="preserve"> In trigono ergo s i d angulus s d i eſt ma-<lb/>ior angulo s i d per 18 p 1:</s> <s xml:id="echoid-s34695" xml:space="preserve"> ergo per 15 p 1 angulus s d i eſt <lb/>maior angulo ti g:</s> <s xml:id="echoid-s34696" xml:space="preserve"> eſt ergo angulus s d i maior angulo t <lb/>i p, ꝗ ex præmiſsis eſt æqualis angulo ti g:</s> <s xml:id="echoid-s34697" xml:space="preserve"> ergo ք 14 th.</s> <s xml:id="echoid-s34698" xml:space="preserve"> 1 <lb/>huius lineæ p i & d s non ſunt æquidiſtantes:</s> <s xml:id="echoid-s34699" xml:space="preserve"> in infini-<lb/>tum tamen protractæ ex parte ſuorum punctorum p & <lb/>s nunquam concurrent, ſed ex ſuis partibus i & d pro-<lb/>tractæ concurrent.</s> <s xml:id="echoid-s34700" xml:space="preserve"> A quocunq;</s> <s xml:id="echoid-s34701" xml:space="preserve"> ergo puncto lineæ p i <lb/>ad centrum d ducatur cathetus incidentiæ, illa ſecabit <lb/>lineam g i s, quæ eſt linea reflexionis, intra conuexum <lb/>ſpeculi:</s> <s xml:id="echoid-s34702" xml:space="preserve"> & omnis linea ducta à quocunq;</s> <s xml:id="echoid-s34703" xml:space="preserve"> puncto lineæ <lb/>p i ad punctum d, erit perpendicularis ſuper ſpeculi ſu-<lb/>perficiem per 72 th.</s> <s xml:id="echoid-s34704" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34705" xml:space="preserve"> ergo ipſa eſt cathetus inci-<lb/>dentiæ, ſicut nunc eſt linea p d.</s> <s xml:id="echoid-s34706" xml:space="preserve"> Et cum locus imaginis <lb/>ſit in concurſu catheti incidentiæ, & lineæ reflexionis <lb/>per 11 huius:</s> <s xml:id="echoid-s34707" xml:space="preserve"> palàm quia locus imaginis cuiuſcunq;</s> <s xml:id="echoid-s34708" xml:space="preserve"> pun <lb/>cti lineę p i ſemper erit intra conuexum ſpeculi.</s> <s xml:id="echoid-s34709" xml:space="preserve"> Et quo-<lb/>niam dato quocunq;</s> <s xml:id="echoid-s34710" xml:space="preserve"> puncto arcus h z, ſemper eadem <lb/>eſt demonſtratio:</s> <s xml:id="echoid-s34711" xml:space="preserve"> maniſeſtum ergo quòd omnium ima <lb/>ginũ arcus h z proprius locus erit intra ſpeculũ.</s> <s xml:id="echoid-s34712" xml:space="preserve"> Quod <lb/>eſt propoſitum.</s> <s xml:id="echoid-s34713" xml:space="preserve"/> </p> <div xml:id="echoid-div1431" type="float" level="0" n="0"> <figure xlink:label="fig-0541-01" xlink:href="fig-0541-01a"> <variables xml:id="echoid-variables605" xml:space="preserve">t g p b h i z a d k s</variables> </figure> </div> </div> <div xml:id="echoid-div1433" type="section" level="0" n="0"> <head xml:id="echoid-head1105" xml:space="preserve" style="it">26. A‘ quocũ pũcto arcus circuli (qui eſt cõmu-<lb/>nis ſectio ſuperficiei reflexiõis & ſpeculi ſphærici cõue-<lb/>xi) interiacentis punctũ, in quo linea reflexionis, cu-<lb/>ius pars intra circulũ eſt æqualis ſemidiametro circu <lb/>li, ſecat circulum, & punctum proximũ, in quo linea <lb/>ducta à centro uiſus contingit circulũ, fiat reflexio: locus uiſæ imaginis quandog erit intra ſpe-<lb/>culum: quando in ſuperficie conuex a ſpeculi: & quando extra ſpeculum. Alhazen 22 n 5.</head> <p> <s xml:id="echoid-s34714" xml:space="preserve">Remaneat totalis diſpoſitio figuræ, quæ in præcedente & in 24 huius, in hoc ſcilicet ut linea re-<lb/>flexionis, quæ g h k, ſecet circulum a b k, cuius centrum eſt punctum d, taliter, ut eius pars intra cir <lb/>culum, quę eſt h k, ſit æqualis ſemidiametro d z:</s> <s xml:id="echoid-s34715" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0541-02a" xlink:href="fig-0541-02"/> & lineæ g a & g b ſint contin gentes circulum a <lb/>b k in punct s a & b:</s> <s xml:id="echoid-s34716" xml:space="preserve"> & ſit pũctus b propin quior <lb/>puncto h.</s> <s xml:id="echoid-s34717" xml:space="preserve"> Dico quòd à quocunq;</s> <s xml:id="echoid-s34718" xml:space="preserve"> puncto arcus <lb/>h b fiat reflexio:</s> <s xml:id="echoid-s34719" xml:space="preserve"> erit locus uiſæ imaginis quan-<lb/>doq;</s> <s xml:id="echoid-s34720" xml:space="preserve"> intra ſpeculum:</s> <s xml:id="echoid-s34721" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s34722" xml:space="preserve"> in ſuperficie ſpe <lb/>culi:</s> <s xml:id="echoid-s34723" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s34724" xml:space="preserve"> extra ſpeculam.</s> <s xml:id="echoid-s34725" xml:space="preserve"> Sumatur enim <lb/>aliquod punctũ arcus h b, à quo fiat reflexio ad <lb/>uiſum g:</s> <s xml:id="echoid-s34726" xml:space="preserve"> & illud punctum reflexionis ſit n:</s> <s xml:id="echoid-s34727" xml:space="preserve"> & du <lb/>catur linea reflexionis ſecás circulum, quæ du-<lb/>cta trans circulum, ſit g n q:</s> <s xml:id="echoid-s34728" xml:space="preserve"> & ducatur à centro <lb/>d ſemidiameter d q:</s> <s xml:id="echoid-s34729" xml:space="preserve"> & ad punctum reflexionis <lb/>ducatur perpendicularis d n f:</s> <s xml:id="echoid-s34730" xml:space="preserve"> & producatur, ut <lb/>in præmiſsis, linea n e continens cum catheto d <lb/>n fangulum æqualem angulo fn g:</s> <s xml:id="echoid-s34731" xml:space="preserve"> qui ſit angu-<lb/>lus fne.</s> <s xml:id="echoid-s34732" xml:space="preserve"> Et quòniam linea n q per 15 p 3 minor <lb/>eſt ꝗ̃ linea h k:</s> <s xml:id="echoid-s34733" xml:space="preserve"> palàm quia linea n q eſt minor ſe-<lb/>midiametro q d.</s> <s xml:id="echoid-s34734" xml:space="preserve"> Quoniam enim linea h k eſt æ-<lb/>qualis ipſi q d ex hypotheſi:</s> <s xml:id="echoid-s34735" xml:space="preserve"> erit ergo linea q n <lb/>minor ꝗ̃ linea q d:</s> <s xml:id="echoid-s34736" xml:space="preserve"> angulus ergo q d n trigoni q d <lb/>n eſt minorangulo d n q ք 18 p 1:</s> <s xml:id="echoid-s34737" xml:space="preserve"> ergo ք 15 p eiuſ <lb/>dẽ angulus q d n minor eſt angulo g n f:</s> <s xml:id="echoid-s34738" xml:space="preserve"> ergo & <lb/>ſuo æquali, qui eſt e n f.</s> <s xml:id="echoid-s34739" xml:space="preserve"> Igitur lineę d q & n e con <lb/>current ad partem minorum angulorum per 14 <lb/>th.</s> <s xml:id="echoid-s34740" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34741" xml:space="preserve"> ſit ergo concurſus earum in puncto <lb/>e.</s> <s xml:id="echoid-s34742" xml:space="preserve"> Palàm autem, ut in præmiſsis, quia linea e q d eſt perpendicularis ſuper ſuperficiẽ ſpeculi per 72 <lb/>th.</s> <s xml:id="echoid-s34743" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34744" xml:space="preserve"> eſt ergo linea e d cathetus incidentiæ formę puncti è:</s> <s xml:id="echoid-s34745" xml:space="preserve"> & ſecat lineam g n q, quæ eſt linea <lb/>reflexionis in puncto q, qui eſt punctus ſuperficiei ſpeculi.</s> <s xml:id="echoid-s34746" xml:space="preserve"> Imago ergo puncti e, quando fuerit refle <lb/>xio facta à puncto arcus h b (quod eſt n) uidebitur in puncto q, quod eſt in ſuperficie conuexa <lb/>ſpeculi.</s> <s xml:id="echoid-s34747" xml:space="preserve"> Et quoniam linea reflexionis, quæ eſt g q, peripheriam arcus b k in unico tantùm pun-<lb/> <pb o="240" file="0542" n="542" rhead="VITELLONIS OPTICAE"/> cto interſecat, ut patet per 7 huius:</s> <s xml:id="echoid-s34748" xml:space="preserve"> palàm quia non accidit uideri imaginem formæ alicuius <lb/>punctorum lineæ n e in ipſa ſuperficie ſpeculi, niſi ſolùm in illo uno puncto, in quo ad ipſum ducta <lb/>cathetus ſecat lineam reflexionis in ipſa ſuperficie ſpeculi, ut eſt in propoſito cathetus puncti e.</s> <s xml:id="echoid-s34749" xml:space="preserve"> Si <lb/>uerò in linea e n ſumatur punctum ultra e, quod ſit punctum r:</s> <s xml:id="echoid-s34750" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s34751" xml:space="preserve"> cathetus incidentiæ ducta ab il <lb/>lo puncto r ad centrum ſpeculi, quæ ſitr d, ſecans lineam reflexionis, quæ eſt g n q productam ultra <lb/>punctum q, in puncto l:</s> <s xml:id="echoid-s34752" xml:space="preserve"> tunc erit ſectio extra ſuperficiem ſpeculi.</s> <s xml:id="echoid-s34753" xml:space="preserve"> Quare imago puncti cuiuslibet <lb/>lineæ n e ultra punctum e ſumpti uidebitur extra ſuperficiem ſpeculi ſecundum diſtantiam puncti <lb/>incidentis, & ſemper, ut pater per 11 huius, erit locus imaginis in puncto ſectionis linearum catheti <lb/>& reflexiõis:</s> <s xml:id="echoid-s34754" xml:space="preserve"> ut formæ puncti r locus imaginis eſt nuncin puncto l, qui eſt cómunis ſectio præmiſ-<lb/>ſarum linearum.</s> <s xml:id="echoid-s34755" xml:space="preserve"> Si uero in linea e n inter puncta n & e ſumatur aliquod punctum, ut c, cathetus ab <lb/>eo ducta ad ſpeculi cèntrũ, ſecabit lineam reflexionis, quę g n q, intra ſpeculum:</s> <s xml:id="echoid-s34756" xml:space="preserve"> ſecabit enim ipſam <lb/>in puncto aliquo èorum, quę ſunt inter puncta n & q.</s> <s xml:id="echoid-s34757" xml:space="preserve"> Imago ergo cuiuslibet puncti lineæ e n inter <lb/>puncta e & n ſumpti uidebitur intra ſpeculum.</s> <s xml:id="echoid-s34758" xml:space="preserve"> Et ſimiliter in quolibet alio puncto arcus b h pote-<lb/>ritidem & eodem modo de diuerſis punctis linearũ incidentię demonſtrari, & hoc eſt propoſitum.</s> <s xml:id="echoid-s34759" xml:space="preserve"> <lb/>Sicutitaq;</s> <s xml:id="echoid-s34760" xml:space="preserve"> in arcu z b dèmonſtrauimus in præmiſsis tribus theorematibus:</s> <s xml:id="echoid-s34761" xml:space="preserve"> ſic etiá figuratione ad-<lb/>hibita in arcu z a poterit dem onſtrari:</s> <s xml:id="echoid-s34762" xml:space="preserve"> quoniá eſt omnimoda ſimilitudo hinc in de:</s> <s xml:id="echoid-s34763" xml:space="preserve"> & idem eſt de o-<lb/>mnibus circulis ſpèculi ſphærici conuexi, circulo a b k ſimilibus.</s> <s xml:id="echoid-s34764" xml:space="preserve"> Si enim perpendiculari g z d ma-<lb/>nente fixa, linea g h ſecundũ æqualitatem anguli d g h imàginetur moueri quouſq;</s> <s xml:id="echoid-s34765" xml:space="preserve"> redeat ad locũ <lb/>ſuum, unde moueri incepit:</s> <s xml:id="echoid-s34766" xml:space="preserve"> tunc linea g h mota ſecabit ex tota ſpeculi cõuexa ſuperficie motu ſuo <lb/>portionẽ ſuperficiei:</s> <s xml:id="echoid-s34767" xml:space="preserve"> & imago formæ cuiuslibet puncti reflexi ab aliquo punctorũ huius portionis <lb/>uidebitur ſemper intra ſpeculũ.</s> <s xml:id="echoid-s34768" xml:space="preserve"> Si uerò fixa manente diametro g z d, linea cõtingens circulum a <lb/>b k, quę eſt g b, moueatur, quouſq;</s> <s xml:id="echoid-s34769" xml:space="preserve"> ad locum, unde exiuit, redeat, ſecabit ex ſphæra portionem ma-<lb/>iorẽ:</s> <s xml:id="echoid-s34770" xml:space="preserve"> & facta reflexione formæ cuiuslibet puncti à quibuſcũq;</s> <s xml:id="echoid-s34771" xml:space="preserve"> punctis ſuperficiei ſpeculi deſcriptæ <lb/>per arcũ h b, uel à punctis arcuũ illi ſimilium:</s> <s xml:id="echoid-s34772" xml:space="preserve"> tunc catheto incidentiæ ſecante lineam reflexionis in <lb/>ipſa ſuperficie ſpeculi, ſemper locus imaginis formæ puncti illius erit in ipſa ſuperficie ſpeculi:</s> <s xml:id="echoid-s34773" xml:space="preserve"> ſed <lb/>aliorum punctorũ in illa eadem linea exiſtentiũ quorundam locus imaginis eſt intrà ſpeculú, quo-<lb/>rundam extra ſpeculũ, ſecundum quod catheti ab illis punctis ad cèntrũ ſpeculi productæ ſecant li <lb/>neas ſuarum reflexionũ.</s> <s xml:id="echoid-s34774" xml:space="preserve"> Et quoniá ſitus centri uiſus, uel ſuperficiei ſpeculi, uel etiam ipſius rei uiſæ <lb/>poteſt multipliciter uariàri:</s> <s xml:id="echoid-s34775" xml:space="preserve"> hoc experimèntanti relin quimus, ut ſpeculorũ ſphæricorum conuexo <lb/>rum, quorũ uſus ut plurimũ apud homines noſtrę habitabilis eſt cõmunis (quoniá uitra, quę ſpecu <lb/>lantur, modo ſphærico diffundente ſe, artificũ ſpiritu exufflantur) quamcũq;</s> <s xml:id="echoid-s34776" xml:space="preserve"> portionẽ quis taliter <lb/>collocet, ut quandoq;</s> <s xml:id="echoid-s34777" xml:space="preserve"> imago puncti uiſi appareat intra ſpeculũ, hoc eſt ultra ſuperficiẽ ipſius, quan <lb/>doq;</s> <s xml:id="echoid-s34778" xml:space="preserve"> in ipſa ſuperficie ſpeculi:</s> <s xml:id="echoid-s34779" xml:space="preserve"> & quandoq;</s> <s xml:id="echoid-s34780" xml:space="preserve"> extra ſuperficiem ſpeculi, ita quòd ſuperficies ſpeculi <lb/>non ſit media inter imaginem, quæ uidetur, & oculum uidentis, ſed ad latus extrà uideatur:</s> <s xml:id="echoid-s34781" xml:space="preserve"> & hoc <lb/>iam pluries experimentantibus euenit.</s> <s xml:id="echoid-s34782" xml:space="preserve"> Vndè & periſta pater, quòd ſpeculum ſphæricum conue-<lb/>xum centrumq;</s> <s xml:id="echoid-s34783" xml:space="preserve"> uiſus, & res uiſa ſic ſiſti poſſent, ut imago extra ſpeculum in aere àppareat:</s> <s xml:id="echoid-s34784" xml:space="preserve"> quod re <lb/>linquimus artificio perquirentis.</s> <s xml:id="echoid-s34785" xml:space="preserve"/> </p> <div xml:id="echoid-div1433" type="float" level="0" n="0"> <figure xlink:label="fig-0541-02" xlink:href="fig-0541-02a"> <variables xml:id="echoid-variables606" xml:space="preserve">g a z h n d b c q k f e r</variables> </figure> </div> </div> <div xml:id="echoid-div1435" type="section" level="0" n="0"> <head xml:id="echoid-head1106" xml:space="preserve" style="it">27. Omnis diameter ſpeculi ſphærici conuexi, in quam locus imaginis cadit, in ipſa ſuperficie <lb/>ſpeculi aut extra ſpeculum: portioni ſphær æ ſpeculi nõ apparenti uiſuineceſſariò applicatur. Ex <lb/>quo patet, quòd ipſa eſt demißior qualibet linearum contingentium à centro uiſus ad ſpeculi ſu-<lb/>perficiem productarum.</head> <p> <s xml:id="echoid-s34786" xml:space="preserve">Quod hic proponitur, patet per præmiſſas, reſumpta figuratione præcedétis.</s> <s xml:id="echoid-s34787" xml:space="preserve"> Et quia, ut patet, à <lb/>quolibet puncto arcus a b poteſt fieri reflexio:</s> <s xml:id="echoid-s34788" xml:space="preserve"> omnis quoq;</s> <s xml:id="echoid-s34789" xml:space="preserve"> linea reflexionis, quoniã à centro uiſus <lb/>ſub linea à centro uiſus producta circulum contingente, ducitur, patet per 57 th.</s> <s xml:id="echoid-s34790" xml:space="preserve"> 1 huius quoniam <lb/>ipſa ſecat circulum.</s> <s xml:id="echoid-s34791" xml:space="preserve"> Et quandocunq;</s> <s xml:id="echoid-s34792" xml:space="preserve"> locus imaginis fuerit in ipſa ſpeculi ſuperficie uel extra, patet <lb/>quòd hoc non poteſt accidere in diametris ſpeculi applicatis arcui a b:</s> <s xml:id="echoid-s34793" xml:space="preserve"> non enim poteſt in illis dia-<lb/>metris locus imaginis eſſe in ipſa ſpeculi ſuperficie:</s> <s xml:id="echoid-s34794" xml:space="preserve"> quoniam catheti incidentię & lineæ reflexio-<lb/>num illorum punctorum in illis punctis concurrere non poſſunt.</s> <s xml:id="echoid-s34795" xml:space="preserve"> Sed neq;</s> <s xml:id="echoid-s34796" xml:space="preserve"> extra ſpeculorũ ſuperfi-<lb/>cies poteſt in illis diametris eſſe locus reflexionis:</s> <s xml:id="echoid-s34797" xml:space="preserve"> quoniam lineæ reflexionum ad partem illam ex-<lb/>tra ſpeculũ non cócurrent.</s> <s xml:id="echoid-s34798" xml:space="preserve"> Omnes ergo diametros ſpeculi cuiuſcunq;</s> <s xml:id="echoid-s34799" xml:space="preserve"> ſphærici conuexi, in quibus <lb/>loca imaginũ ſunt in ipſa ſuperficie ſpeculi, uel extra ſpeculum, neceſſariò applicantur portioni ſpe <lb/>culi non apparenti uiſui.</s> <s xml:id="echoid-s34800" xml:space="preserve"> Et quoniã portio ſpeculi apparens & non apparens per lineas contingen <lb/>tes à centro uiſus ad ſpeculi ſuperficiem ductas determinatur, ut patet per 2 huius:</s> <s xml:id="echoid-s34801" xml:space="preserve"> ideo manifeſtũ <lb/>eſt propoſitũ corollarium.</s> <s xml:id="echoid-s34802" xml:space="preserve"> Quælibet enim diametrorũ, in qua eſt locus imaginis in ipſa ſuperflcie <lb/>ſpeculi aut extra ſpeculum, oportet ut ſit demiſsior qualibet linearũ cõtingentiũ à centro uiſus ad <lb/>ſpeculi ſuperficiem productarum.</s> <s xml:id="echoid-s34803" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s34804" xml:space="preserve"> Poteſt aũt diameter, in qua apparet locus <lb/>imaginis intra ſpeculum, eſſe uel altior uel demiſsior illa contingente, ut patet ex his, quæ ſunt in <lb/>præmiſsis demonſtrata.</s> <s xml:id="echoid-s34805" xml:space="preserve"> Reſtar autem, ut nos deinceps loca imaginum certius determinemus.</s> <s xml:id="echoid-s34806" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1436" type="section" level="0" n="0"> <head xml:id="echoid-head1107" xml:space="preserve" style="it">28. Ad diametrum ſpeculi ſphærici conuexi ducta linea reflexionis ſecante ſpeculum, ita ut <lb/>pars ductæ lineæ interiacẽs ſuperficiem ſpeculi & diametrũ, ſit æqualis parti diametri interia-<lb/>centi punctum ſectionis & centrum ſpeculi: in illa parte diametri non eſt locus alicuius imagi-<lb/>nis, ſed eſt imaginum met a, ſicut & in illo puncto ſectionis. Alhazen 23 n 5.</head> <p> <s xml:id="echoid-s34807" xml:space="preserve">Eſto circulus communis ſectionis ſuperficiei reflexionis & ſuperficiei ſpeculi ſphærici cõuexi, <lb/> <pb o="241" file="0543" n="543" rhead="LIBER SEXTVS."/> qui a b ſ e g:</s> <s xml:id="echoid-s34808" xml:space="preserve"> & ſit punctum h centrum uiſus, punctum quoq;</s> <s xml:id="echoid-s34809" xml:space="preserve"> d centrum ſpeculi:</s> <s xml:id="echoid-s34810" xml:space="preserve"> & ſit d e ſemidiame <lb/>ter ſpeculi, quę neceſſariò eſt perpendicularis ſuper ſuperficiem ſpeculi per 72 th.</s> <s xml:id="echoid-s34811" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34812" xml:space="preserve"> & ſit linea <lb/>z h linea reflexionis, ſecans ſuperficiem conuexam ſpeculi ſuper punctum f:</s> <s xml:id="echoid-s34813" xml:space="preserve"> & concurrens cum e d <lb/>ſemidiametro ſpeculi ſuper punctum z.</s> <s xml:id="echoid-s34814" xml:space="preserve"> Sit quoq;</s> <s xml:id="echoid-s34815" xml:space="preserve"> linea z fęqualis lineę z d:</s> <s xml:id="echoid-s34816" xml:space="preserve"> quod poteſt fieri per 136 <lb/>th.</s> <s xml:id="echoid-s34817" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s34818" xml:space="preserve"> Dico quòd in linea z d non eſt locus alicuius imaginis.</s> <s xml:id="echoid-s34819" xml:space="preserve"> Neque enim pũctus z poteſt eſſe <lb/>locus alicuius imaginis, niſi ſolũ alicuius punctorum lineę e d protractę:</s> <s xml:id="echoid-s34820" xml:space="preserve"> quia ut patet per 11 huius, <lb/>locus imaginis formæ cuiuſque puncti ſemper eſt ſuper cathetum ſuę incidentiæ:</s> <s xml:id="echoid-s34821" xml:space="preserve"> & hęc eſt in ſpe-<lb/>culis ſphęricis conuexis in linea ab illo puncto ad centrum ſphæræ ducta.</s> <s xml:id="echoid-s34822" xml:space="preserve"> Quòd uerò punctus z nó <lb/>ſit locus alicuius imaginis punctorum lineę e d, patet.</s> <s xml:id="echoid-s34823" xml:space="preserve"> Ducatur enim <lb/> <anchor type="figure" xlink:label="fig-0543-01a" xlink:href="fig-0543-01"/> perpendicularis à centro d ſuper punctum f, quę producta extra cir-<lb/>culum ſit d f n:</s> <s xml:id="echoid-s34824" xml:space="preserve"> & ſuper ductam perpendicularem fiat in puncto fan-<lb/>gulus ęqualis angulo n f h per 23 p 1, qui ſit q f n:</s> <s xml:id="echoid-s34825" xml:space="preserve"> eſt ergo per 15 p 1 an-<lb/>gulus q f n ęqualis angulo z f d:</s> <s xml:id="echoid-s34826" xml:space="preserve"> ſed cum z d & z f lineę ex hypotheſi <lb/>ſint æquales:</s> <s xml:id="echoid-s34827" xml:space="preserve"> erit per 5 p 1 angulus z d f æqualis angulo z f d:</s> <s xml:id="echoid-s34828" xml:space="preserve"> ergo & <lb/>angulus q f n æqualis eſt angulo z d f:</s> <s xml:id="echoid-s34829" xml:space="preserve"> ergo per 28 p 1 lineę z d & q f <lb/>ſunt adinuicem æ quidiſtantes:</s> <s xml:id="echoid-s34830" xml:space="preserve"> in infinitum ergo protractę nunquá <lb/>concurrent.</s> <s xml:id="echoid-s34831" xml:space="preserve"> Nullius ergo puncti lineæ e d quantumcunq;</s> <s xml:id="echoid-s34832" xml:space="preserve"> protractę <lb/>forma mouebitur ad punctum f per lineam incidentię q f:</s> <s xml:id="echoid-s34833" xml:space="preserve"> ſed nõ po-<lb/>reſt eſſe locus alicuius imaginis in pũcto z, niſi moueatur ad punctũ <lb/>f forma per lineam q f:</s> <s xml:id="echoid-s34834" xml:space="preserve"> aliàs enim linea f h non fieret linea reflexio-<lb/>nis, in cuius interſectione cum diametro d e eſt pũctus z.</s> <s xml:id="echoid-s34835" xml:space="preserve"> Non eſt er-<lb/>go punctus z locus alicuius imaginis punctorum lineę e d:</s> <s xml:id="echoid-s34836" xml:space="preserve"> ergo nec <lb/>alicuius alterius imaginis formę cuiuſcunq;</s> <s xml:id="echoid-s34837" xml:space="preserve"> puncti extra lineam d e.</s> <s xml:id="echoid-s34838" xml:space="preserve"> <lb/>Et eadem erit demonſtratio quantacunq;</s> <s xml:id="echoid-s34839" xml:space="preserve"> ſumpta diametro e d.</s> <s xml:id="echoid-s34840" xml:space="preserve"> Sed <lb/>& nullus alius punctus lineæ z d præter z, poteſt eſſe locus alicuius <lb/>imaginis.</s> <s xml:id="echoid-s34841" xml:space="preserve"> Dato enim quòd punctus p poſsit eſſe locus alicuius ima-<lb/>ginis;</s> <s xml:id="echoid-s34842" xml:space="preserve"> ducatur linea h p ſecans conuexam ſuperficiem ſpeculi in pun <lb/>cto b:</s> <s xml:id="echoid-s34843" xml:space="preserve"> & ducatur perpendicularis d b m:</s> <s xml:id="echoid-s34844" xml:space="preserve"> & ut ſuprà, angulo m b h fiat <lb/>æqualis angulus ſuper punctũ b, quit b m.</s> <s xml:id="echoid-s34845" xml:space="preserve"> Palàm ergo, ut prius, quòd angulus t b m eſt æqualis an-<lb/>gulo p b d:</s> <s xml:id="echoid-s34846" xml:space="preserve"> ſed angulus d p b per 16 p 1 eſt maior angulo p z h, cum ſit ei extrinſecus in trigono p z h:</s> <s xml:id="echoid-s34847" xml:space="preserve"> <lb/>igitur duo alij anguli trigoni p d b lunt minores duobus alijs angulis trigoni d z f:</s> <s xml:id="echoid-s34848" xml:space="preserve"> ſed angulus p d b <lb/>eſt maior angulo z d f, eo quò d totum maius eſt ſua parte:</s> <s xml:id="echoid-s34849" xml:space="preserve"> & etiam patet hoc per 29 th.</s> <s xml:id="echoid-s34850" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s34851" xml:space="preserve"> Se-<lb/>quitur ergo ut angulus d b p ſit minor angulo d f z:</s> <s xml:id="echoid-s34852" xml:space="preserve"> angulus uerò d f z eſt æqualis angulo z d f, ut pri <lb/>us patuit:</s> <s xml:id="echoid-s34853" xml:space="preserve"> angulus ergo d b p minor eſt angulo z d f:</s> <s xml:id="echoid-s34854" xml:space="preserve"> multò ergo minor eſt angulus d b p angulo p d <lb/>b:</s> <s xml:id="echoid-s34855" xml:space="preserve"> angulus itaq;</s> <s xml:id="echoid-s34856" xml:space="preserve"> t b m minor eſt angulo p d b:</s> <s xml:id="echoid-s34857" xml:space="preserve"> lineę igiturt b & e d per 14 th.</s> <s xml:id="echoid-s34858" xml:space="preserve"> 1 huius nunquá concur-<lb/>rentad partem, à qua poſſet fieri reflexio.</s> <s xml:id="echoid-s34859" xml:space="preserve"> Nulla ergo forma incidens puncto b reflectetur ad uiſum <lb/>h, ita ut locus imaginis fiat in puncto p.</s> <s xml:id="echoid-s34860" xml:space="preserve"> Similiter neque imago alicuius alterius puncti ſe offeret ui-<lb/>ſui ſuper aliquem punctum lineę z d.</s> <s xml:id="echoid-s34861" xml:space="preserve"> Tota ergo linea z d erit ſemper uacua imaginibus:</s> <s xml:id="echoid-s34862" xml:space="preserve"> nec un quá <lb/>erit locus imaginum in ipſa.</s> <s xml:id="echoid-s34863" xml:space="preserve"> Et ſimiliter poteſt de qualibet alia diametro propoſiti ſpeculi demon-<lb/>ſtrari hypotheſi ſeruata.</s> <s xml:id="echoid-s34864" xml:space="preserve"> Patet etiam ex præmiſsis quoniam linea z d eſt meta imaginum.</s> <s xml:id="echoid-s34865" xml:space="preserve"> Quoniam <lb/>ſi linea f z fuerit maior quá linea z d, nulla unquá apparebit imago:</s> <s xml:id="echoid-s34866" xml:space="preserve"> quoniá angulus z d f per 18 p 1 e-<lb/>rit maior angulo d f z:</s> <s xml:id="echoid-s34867" xml:space="preserve"> ergo & angulo n f h per 15 p 1:</s> <s xml:id="echoid-s34868" xml:space="preserve"> ergo & angulo q f n per 8 huius.</s> <s xml:id="echoid-s34869" xml:space="preserve"> Lineę ergo e d <lb/>& q f per 14 th.</s> <s xml:id="echoid-s34870" xml:space="preserve"> 1 huius non conncurrent ad partem punctorum e & q, ſed ad partẽ punctorum d & f:</s> <s xml:id="echoid-s34871" xml:space="preserve"> <lb/>non ergo aliqua poterit apparere imago in puncto z:</s> <s xml:id="echoid-s34872" xml:space="preserve"> ergo nec in aliquo punctorum lineę z d.</s> <s xml:id="echoid-s34873" xml:space="preserve"> Quòd <lb/>ſi linea ſ z ſit minor quã linea z d:</s> <s xml:id="echoid-s34874" xml:space="preserve"> tunc ſecundũ pręmiſſum modũ erit angulus z d f minor angulo q <lb/>f n:</s> <s xml:id="echoid-s34875" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s34876" xml:space="preserve"> 1 huius lineę e d & q f concurrent ad partẽ punctorũ e & q:</s> <s xml:id="echoid-s34877" xml:space="preserve"> & ab illo pũcto poteſt <lb/>alicuius punctorum lineę e d fieri reflexio ad uiſum:</s> <s xml:id="echoid-s34878" xml:space="preserve"> & locus imaginis erit per 11 huius in puncto z:</s> <s xml:id="echoid-s34879" xml:space="preserve"> <lb/>& erit linea z d locus imaginis ſecundum omnẽ ſuum punctũ, quouſq;</s> <s xml:id="echoid-s34880" xml:space="preserve"> linea incidentię reſpectu dia <lb/>metri recipiat propoſitam diuiſionem.</s> <s xml:id="echoid-s34881" xml:space="preserve"> Patet ergo quòd cum linea z d eſt ęqualis lineæ z f, quòd li-<lb/>nea z d eſt meta imaginum ultra quã nulla, & citra quã omnis uidetur imago.</s> <s xml:id="echoid-s34882" xml:space="preserve"> Et ſimiliter punctus z <lb/>eſt meta imaginum:</s> <s xml:id="echoid-s34883" xml:space="preserve"> quoniam, ut patet ex pręmiſsis, omnis linea incidentię à quocunq;</s> <s xml:id="echoid-s34884" xml:space="preserve"> puncto ſpe <lb/>culi ad uiſum h inter puncta z & d ducta, eſt maior quá linea, quę perillã reſecatur exlinea z d:</s> <s xml:id="echoid-s34885" xml:space="preserve"> quo-<lb/>niam iſta eſt maior quá linea z f, per 15 p 3:</s> <s xml:id="echoid-s34886" xml:space="preserve"> eſt ergo etiam maior quá linea z d exhypotheſi, ut patet <lb/>de linea b p, quæ eſt maior quá linea p d, uel linea z d:</s> <s xml:id="echoid-s34887" xml:space="preserve"> omnisq̀;</s> <s xml:id="echoid-s34888" xml:space="preserve"> linea inter pũcta z & e ad uiſum h du <lb/>cta interiacens peripheriam circuli & diametrum, eſt minor quã linea f z:</s> <s xml:id="echoid-s34889" xml:space="preserve"> ergo & minor quã linea <lb/>z d:</s> <s xml:id="echoid-s34890" xml:space="preserve"> ergo eſt etiam minor quã linea, quã ipſa reſecat ex ſemidiametro d e.</s> <s xml:id="echoid-s34891" xml:space="preserve"> Sunt ergo, ut patet ք præ-<lb/>miſſa, in linea z e loca imaginum, præter quá in puncto z:</s> <s xml:id="echoid-s34892" xml:space="preserve"> in linea uerò z d non ſunt aliqua loca ima-<lb/>ginum.</s> <s xml:id="echoid-s34893" xml:space="preserve"> Et ſic patet quòd punctus z eſt meta imaginum:</s> <s xml:id="echoid-s34894" xml:space="preserve"> nec eſt differentia an punctus z cadat intra <lb/>circulum:</s> <s xml:id="echoid-s34895" xml:space="preserve"> an extra:</s> <s xml:id="echoid-s34896" xml:space="preserve"> an in ipſa ſuperficie ſpeculi:</s> <s xml:id="echoid-s34897" xml:space="preserve"> quia ſemper ubicun que acciderit lineam z d ęqua-<lb/>lem fieri parti lineę reflexionis interiacenti punctum reflexionis & punctum z:</s> <s xml:id="echoid-s34898" xml:space="preserve"> erit ſemper in pun-<lb/>cto z meta imaginum:</s> <s xml:id="echoid-s34899" xml:space="preserve"> & ſimiliter eſt de tota linea z d.</s> <s xml:id="echoid-s34900" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s34901" xml:space="preserve"/> </p> <div xml:id="echoid-div1436" type="float" level="0" n="0"> <figure xlink:label="fig-0543-01" xlink:href="fig-0543-01a"> <variables xml:id="echoid-variables607" xml:space="preserve">m t n q h b f e z p d a g</variables> </figure> </div> </div> <div xml:id="echoid-div1438" type="section" level="0" n="0"> <head xml:id="echoid-head1108" xml:space="preserve" style="it">29. Aßignata meta imaginum in quacunque diametro inter line as contingentes à uiſu ad <lb/>ſpeculum ſphæricum conuexum ductas, præter uiſualem diametrum: in punctis tantùm datæ <lb/> <pb o="242" file="0544" n="544" rhead="VITELLONIS OPTICAE"/> diametri, inter ſuperficiem ſphæræ & punctum, quieſt imaginum meta, exiſtentibus ſunt loca <lb/>imaginum illius diametri. Alhazen 24 n 5.</head> <p> <s xml:id="echoid-s34902" xml:space="preserve">Sit b centrum uiſus:</s> <s xml:id="echoid-s34903" xml:space="preserve"> & ſint b z & b e lineæ ſpeculum ſphęricum conuexum contingentes in pun <lb/>ctis z & e:</s> <s xml:id="echoid-s34904" xml:space="preserve"> & ſit a centrũ ſpeculi:</s> <s xml:id="echoid-s34905" xml:space="preserve"> & b h a diameter uiſualis:</s> <s xml:id="echoid-s34906" xml:space="preserve"> & ſit a g d diameter alia, in qua meta ima-<lb/>ginum aſsignata ſitin puncto t per præcedentem, & per 136 th.</s> <s xml:id="echoid-s34907" xml:space="preserve"> 1 hu-<lb/> <anchor type="figure" xlink:label="fig-0544-01a" xlink:href="fig-0544-01"/> ius:</s> <s xml:id="echoid-s34908" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s34909" xml:space="preserve"> linea a d ſuperficiem ſpeculi in puncto g.</s> <s xml:id="echoid-s34910" xml:space="preserve"> Dico quòd ſo-<lb/>lùm in punctis lineæ t g, quæ ſunt inter puncta g & t, ſuntloca imagi <lb/>num diametri d g a.</s> <s xml:id="echoid-s34911" xml:space="preserve"> Quòd enim imagines illæ non cadant in punctũ <lb/>g, qui eſt in ſuperficie ſpeculi:</s> <s xml:id="echoid-s34912" xml:space="preserve"> uel quòd non cadant extra ſuperficiẽ <lb/>ſpeculi, palàm per 27 huius:</s> <s xml:id="echoid-s34913" xml:space="preserve"> oportet enim ſemper diametrum, in qua <lb/>locus imaginis eſt in ſuperficie ſpeculi aut extra, demiſsiorem eſſe <lb/>puncto contingentiæ:</s> <s xml:id="echoid-s34914" xml:space="preserve"> diameter uerò a d eſt inter lineas contingen-<lb/>tes:</s> <s xml:id="echoid-s34915" xml:space="preserve"> nec ergo in ſuperficie ſpeculi, nec extra ſphæram ipſius appare-<lb/>bit imago ſecundum illam diametrum.</s> <s xml:id="echoid-s34916" xml:space="preserve"> Sed quòd quilibet punctus <lb/>inter puncta g & t ſumptus ſit locus imaginis, patet.</s> <s xml:id="echoid-s34917" xml:space="preserve"> Detur enim ali-<lb/>quod punctum lineæ g t:</s> <s xml:id="echoid-s34918" xml:space="preserve"> quod ſit q:</s> <s xml:id="echoid-s34919" xml:space="preserve"> & ducatur linea à uiſu ad illum <lb/>punctum, quæ ſit b q, ſecans ſuperficiem ſpeculi in puncto p:</s> <s xml:id="echoid-s34920" xml:space="preserve"> & duca <lb/>tur perpend:</s> <s xml:id="echoid-s34921" xml:space="preserve"> cularis a p l:</s> <s xml:id="echoid-s34922" xml:space="preserve"> & ſecundum ſæpius præmiſſa angulo l p b <lb/>fiat per 23 p 1 angulus æqualis, qui ſit d p l:</s> <s xml:id="echoid-s34923" xml:space="preserve"> & ducatur linea b t ſecans <lb/>ſuperficiem ſpeculi in puncto f.</s> <s xml:id="echoid-s34924" xml:space="preserve"> Ducatur quoq;</s> <s xml:id="echoid-s34925" xml:space="preserve"> perpendicularis a f.</s> <s xml:id="echoid-s34926" xml:space="preserve"> <lb/>Triangulus itaque a p b continet triangulum a f b:</s> <s xml:id="echoid-s34927" xml:space="preserve"> angulus ergo a f b <lb/>maior eſt angulo a p b per 21 p 1:</s> <s xml:id="echoid-s34928" xml:space="preserve"> ſed angulus a ft cum angulo a f b ua <lb/>let duos rectos, & angulus a p q cum angulo a p b ualet duos rectos <lb/>per 13 p 1.</s> <s xml:id="echoid-s34929" xml:space="preserve"> Palàm ergo quia angulus a ft minor eſt angulo a p q:</s> <s xml:id="echoid-s34930" xml:space="preserve"> ſed an <lb/>gulus a f t eſt ęqualis angulo f a t per 5 p 1, quoniam latus ft eſt ęqua-<lb/>lelateri t a per 136 th.</s> <s xml:id="echoid-s34931" xml:space="preserve"> 1 huius, & exhypotheſi:</s> <s xml:id="echoid-s34932" xml:space="preserve"> angulus ergo a p q maior eſt angulo ſ a t:</s> <s xml:id="echoid-s34933" xml:space="preserve"> quare etiam <lb/>erit maior angulo p a q, qui eſt pars anguli f a t.</s> <s xml:id="echoid-s34934" xml:space="preserve"> Et quia anguli a p q & l p b ſunt ęquales ք 15 p 1 ſunt <lb/>enim contra ſe poſiti:</s> <s xml:id="echoid-s34935" xml:space="preserve"> erit angulus l p b maior angulo p a q:</s> <s xml:id="echoid-s34936" xml:space="preserve"> eſt ergo per 8 huius angulus d p l maior <lb/>angulo p a q.</s> <s xml:id="echoid-s34937" xml:space="preserve"> Patet igitur quod lineę p d & a q concurrent per 14 th.</s> <s xml:id="echoid-s34938" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34939" xml:space="preserve"> ſit ergo d punctus con-<lb/>curſus ipſarum.</s> <s xml:id="echoid-s34940" xml:space="preserve"> Forma igitur puncti d reflectetur ad uiſum in punctum b à puncto ſuperficiei ſpecu <lb/>li, quod eſt p, perlineam p b:</s> <s xml:id="echoid-s34941" xml:space="preserve"> & locus imaginis ſuę eſt punctum q per 11 huius.</s> <s xml:id="echoid-s34942" xml:space="preserve"> Eadem quoq;</s> <s xml:id="echoid-s34943" xml:space="preserve"> eſt de-<lb/>monſtratio ſumpto quocunq;</s> <s xml:id="echoid-s34944" xml:space="preserve"> puncto inter g & t.</s> <s xml:id="echoid-s34945" xml:space="preserve"> In diametro uerò b h a (quę eſt diameter uiſualis) <lb/>non eſt aliquis locus imaginis, niſi ut proponit 10 huius.</s> <s xml:id="echoid-s34946" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s34947" xml:space="preserve"/> </p> <div xml:id="echoid-div1438" type="float" level="0" n="0"> <figure xlink:label="fig-0544-01" xlink:href="fig-0544-01a"> <variables xml:id="echoid-variables608" xml:space="preserve">b l z h f p d g q t e a</variables> </figure> </div> </div> <div xml:id="echoid-div1440" type="section" level="0" n="0"> <head xml:id="echoid-head1109" xml:space="preserve" style="it">30. Linea reflexionis, circulum (qui eſt communis ſectio ſuperficiei reflexionis & ſpeculi ſphæ <lb/>rici conuexi) taliter ſecante, quòd pars lineæ productæ intra circulum ſit æqualis ſemidiametro <lb/>ſpeculi: pars diametri in termino huius lineæ ſecantis ſpeculum, interiacens punctum ſectionis <lb/>ſpeculi, & punctum ſectionis ſui cum linea contingenter à uiſu ductæ ad ſpeculum, eſt locus ima-<lb/>ginum punctorum illius diametri: & nullus punctus alius diametri eiuſdem: erit́ locus ima-<lb/>ginis ſemper extra ſpeculum. Alhazen 25 n 5.</head> <p> <s xml:id="echoid-s34948" xml:space="preserve">Sint a c & a g lineę contingentes circulum, qui eſt communis ſectio ſuperficiei reflexionis & ſu-<lb/>perficiei ſpeculi ſphęrici conuexi, cuius centrum ſit punctum b:</s> <s xml:id="echoid-s34949" xml:space="preserve"> ſit quoq;</s> <s xml:id="echoid-s34950" xml:space="preserve"> in puncto a cẽtrum uiſus:</s> <s xml:id="echoid-s34951" xml:space="preserve"> <lb/>ſitq́;</s> <s xml:id="echoid-s34952" xml:space="preserve"> linea a d b z diameter uiſualis, ſecãs <lb/> <anchor type="figure" xlink:label="fig-0544-02a" xlink:href="fig-0544-02"/> ſuperficiem ſpeculi in punctis d & z:</s> <s xml:id="echoid-s34953" xml:space="preserve"> pro-<lb/>trahaturq́;</s> <s xml:id="echoid-s34954" xml:space="preserve"> à centro ſpeculi b ad punctum <lb/>contingẽtię g linea b g.</s> <s xml:id="echoid-s34955" xml:space="preserve"> Palàm ergo per 59 <lb/>th.</s> <s xml:id="echoid-s34956" xml:space="preserve"> 1 huius quòd arcus d g eſt minor quarta <lb/>circuli:</s> <s xml:id="echoid-s34957" xml:space="preserve"> arcus ergo g z eſt maior quarta cir <lb/>culi:</s> <s xml:id="echoid-s34958" xml:space="preserve"> ergo per 33 p 6 patet quòd angulus z <lb/>b g eſt maior recto.</s> <s xml:id="echoid-s34959" xml:space="preserve"> Hoc etiã patet ſic.</s> <s xml:id="echoid-s34960" xml:space="preserve"> Cũ <lb/>enim in triãgulo b a g angulus a g b ſit re-<lb/>ctus per 18 p 3, erit angulus g b a minor re-<lb/>cto:</s> <s xml:id="echoid-s34961" xml:space="preserve"> palàm ergo per 13 p 1 quòd angulus z <lb/>b g eſt maior recto.</s> <s xml:id="echoid-s34962" xml:space="preserve"> Abſcindatur ergo ab i-<lb/>pſo angulus h b g rectus per 23 p 1:</s> <s xml:id="echoid-s34963" xml:space="preserve"> erit per <lb/>28 p 1 linea h b ęquidiſtans lineę contingẽ <lb/>ti circulum, quæ eſt a g.</s> <s xml:id="echoid-s34964" xml:space="preserve"> Palàm ergo quo-<lb/>niam lineæ b h & a g productæ nunquam <lb/>concurrent:</s> <s xml:id="echoid-s34965" xml:space="preserve"> & quælibet diameter cadens <lb/>in arcum h g inter puncta h & g, cõcurret <lb/>cum linea a g producta per 2 uel 14 th.</s> <s xml:id="echoid-s34966" xml:space="preserve"> 1 hu <lb/>ius, quoniam angulum acutum continebit cum linea b h.</s> <s xml:id="echoid-s34967" xml:space="preserve"> Ducatur ergo à puncto a linea ſecans ſpe-<lb/>culum, quæ ſit a m o, ita quòd chorda m o ſit æqualis ſemidiametro ſpeculi, quæ ſit b o:</s> <s xml:id="echoid-s34968" xml:space="preserve"> hoc autem <lb/> <pb o="243" file="0545" n="545" rhead="LIBER SEXTVS."/> poſsibile eſt fieri per 136 th.</s> <s xml:id="echoid-s34969" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s34970" xml:space="preserve"> eritq;</s> <s xml:id="echoid-s34971" xml:space="preserve"> linea b o, & punctum o meta imaginum per 28 huius:</s> <s xml:id="echoid-s34972" xml:space="preserve"> con <lb/>curratq́;</s> <s xml:id="echoid-s34973" xml:space="preserve"> diameter b o cum linea a g in puncto t.</s> <s xml:id="echoid-s34974" xml:space="preserve"> Dico quòd in quolibet puncto lineæ t o eſt locus i-<lb/>maginis:</s> <s xml:id="echoid-s34975" xml:space="preserve"> & quòd in nullo alio puncto diametri t b eſt locus alicuius imaginis:</s> <s xml:id="echoid-s34976" xml:space="preserve"> & ſunt puncta o & t <lb/>metæ locorũ imaginũ, punctum o in ſuperficie ſpeculi, & punctũ t extra ſpeculum:</s> <s xml:id="echoid-s34977" xml:space="preserve"> ſolũ enim in his <lb/>duobus punctis concurret diameter b o cũ lineis reflexionis, quæ ſunt a m & a g.</s> <s xml:id="echoid-s34978" xml:space="preserve"> Sumatur enim ali <lb/>quod punctũ lineę t o, quod ſit k:</s> <s xml:id="echoid-s34979" xml:space="preserve"> & ducatur linea a n k, ſecans cõuexam ſuperficiẽ ſpeculi in puncto <lb/>n:</s> <s xml:id="echoid-s34980" xml:space="preserve"> & ducatur քpẽdicularis b n x:</s> <s xml:id="echoid-s34981" xml:space="preserve"> & angulo a n x fiat ęqualis angulus ſuք punctũ n, utin alijs p̃miſsis:</s> <s xml:id="echoid-s34982" xml:space="preserve"> <lb/>& producatur linea n f taliter, ut angulus x n f ſit æqualis angulo a n x per 23 p 1:</s> <s xml:id="echoid-s34983" xml:space="preserve"> protrahaturq́;</s> <s xml:id="echoid-s34984" xml:space="preserve"> per-<lb/>pendicularis b t ad lineam n f in punctum f:</s> <s xml:id="echoid-s34985" xml:space="preserve"> punctum enim concurſus, quicunq;</s> <s xml:id="echoid-s34986" xml:space="preserve"> fuerit, uocabimus <lb/>f:</s> <s xml:id="echoid-s34987" xml:space="preserve"> palàm uerò per 14 th.</s> <s xml:id="echoid-s34988" xml:space="preserve"> 1 huius quoniã concurrent.</s> <s xml:id="echoid-s34989" xml:space="preserve"> Linea itaq;</s> <s xml:id="echoid-s34990" xml:space="preserve"> n fnõ cadet inter puncta circuli, quæ <lb/>ſunt h & g:</s> <s xml:id="echoid-s34991" xml:space="preserve"> non enim ſecat ſpeculum:</s> <s xml:id="echoid-s34992" xml:space="preserve"> neq;</s> <s xml:id="echoid-s34993" xml:space="preserve"> ſecat lineam ipſum ſpeculum contingentem in puncto g, <lb/>quę eſt a g t, niſi in uno puncto, quod eſt extra ſuperficiem ſpeculi ſupra punctum g.</s> <s xml:id="echoid-s34994" xml:space="preserve"> Siaũt daretur <lb/>quòd linea n f caderet inter puncta h & g:</s> <s xml:id="echoid-s34995" xml:space="preserve"> oporteret ut uel ſecaret ſuperficiem ſpeculi uel lineam a g <lb/>in duobus punctis:</s> <s xml:id="echoid-s34996" xml:space="preserve"> in uno infra punctum g, & in alio ſupra punctum g, ubi fit reflexio ad uiſum exi-<lb/>ſtentem in puncto a:</s> <s xml:id="echoid-s34997" xml:space="preserve"> & ſic duæ lineę rectę ſuperficiem includerent:</s> <s xml:id="echoid-s34998" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s34999" xml:space="preserve"> Forma er <lb/>go puncti f mouebitur per lineam f n ad punctum n, & reflectetur ad a per lineam a n:</s> <s xml:id="echoid-s35000" xml:space="preserve"> apparebitq́;</s> <s xml:id="echoid-s35001" xml:space="preserve"> i-<lb/>mago eius in puncto k, in concurſu catheti incidentiæ, quæ eſt f b, cum linea reflexionis, quę eſt a k <lb/>extra ſpeculi ſuperficiem.</s> <s xml:id="echoid-s35002" xml:space="preserve"> Et eodem modo de omnibus punctis lineę o t eſt demonſtrãdum:</s> <s xml:id="echoid-s35003" xml:space="preserve"> & ima-<lb/>gines omnium uidentur extra ſpeculum.</s> <s xml:id="echoid-s35004" xml:space="preserve"> Et quoniam à puncto m nulla poteſt fieri reflexio formæ <lb/>alicuius punctorũ lineæ b f:</s> <s xml:id="echoid-s35005" xml:space="preserve"> quoniam omnes lineę reflexionum à puncto m ad punctum a factarum <lb/>ęquidiſtat diametro b f:</s> <s xml:id="echoid-s35006" xml:space="preserve"> quod patet, ſi ducatur քpendicularis b m, quę producatur uſque ad punctũ <lb/>q:</s> <s xml:id="echoid-s35007" xml:space="preserve"> & fiat angulus p m q æqualis angulo q m a.</s> <s xml:id="echoid-s35008" xml:space="preserve"> Tunc enim, quia anguli b m o & m b o ſunt ęquales ex <lb/>hypotheſi, & per 5 p 1:</s> <s xml:id="echoid-s35009" xml:space="preserve"> erunt, ſicut oſtẽdimus in 28 huius, anguli p m q & m b o ęquales:</s> <s xml:id="echoid-s35010" xml:space="preserve"> ergo per 28 p <lb/>1 lineę m p & b f ęquidiſtant:</s> <s xml:id="echoid-s35011" xml:space="preserve"> non ergo concurrunt:</s> <s xml:id="echoid-s35012" xml:space="preserve"> nec unquam fiet reflexio formæ alicuius puncti <lb/>diametri b f à puncto ſpeculi m:</s> <s xml:id="echoid-s35013" xml:space="preserve"> punctum ergo o non erit locus alicuius imaginis punctorũ diame-<lb/>tri b f.</s> <s xml:id="echoid-s35014" xml:space="preserve"> Omnia ergo illa loca ſunt extra ſpeculũ in linea t o:</s> <s xml:id="echoid-s35015" xml:space="preserve"> ita quòd puncta t & o ſunt loca imaginũ.</s> <s xml:id="echoid-s35016" xml:space="preserve"> <lb/>Patet ergo propoſitum:</s> <s xml:id="echoid-s35017" xml:space="preserve"> ita tamen ut punctum t accipiatur ut ſimpliciter uiſum, & ut reflexum, pro-<lb/>ut diximus in 2 huius:</s> <s xml:id="echoid-s35018" xml:space="preserve"> quoniam ipſum cadit in linea contingente.</s> <s xml:id="echoid-s35019" xml:space="preserve"/> </p> <div xml:id="echoid-div1440" type="float" level="0" n="0"> <figure xlink:label="fig-0544-02" xlink:href="fig-0544-02a"> <variables xml:id="echoid-variables609" xml:space="preserve">a c d b m q n x g p o k t f z h</variables> </figure> </div> </div> <div xml:id="echoid-div1442" type="section" level="0" n="0"> <head xml:id="echoid-head1110" xml:space="preserve" style="it">31. Catheto incidentiæ ſecante quemcun punctum arcus circuli (qui eſt communis ſectio <lb/>ſuperficiei reflexionis & ſpeculi ſphærici conuexi) interiacentis punctum contingentiæ lineæ <lb/>à centro uiſus ductæ, & punctum, quo linea reflexionis (cuius pars intra circulum eſt æqualis <lb/>ſemidiametro circuli) ſecat arcum circuli non apparentem uiſui: erũt locorum imaginum plu-<lb/>raintra ſpeculi conuexam ſuperficiem: unum tantũ in ipſa ſuperficie, & plurima extra ipſam. <lb/>Alhazen 26 n 5.</head> <p> <s xml:id="echoid-s35020" xml:space="preserve">Diſponantur omnia, ut in præhabita demõſtratione:</s> <s xml:id="echoid-s35021" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s35022" xml:space="preserve"> linea a m o circulum taliter, ut linea <lb/> <anchor type="figure" xlink:label="fig-0545-01a" xlink:href="fig-0545-01"/> m o ſit æqualis ſemidiametro ſpeculi:</s> <s xml:id="echoid-s35023" xml:space="preserve"> & linea a g t <lb/>contingat ſpeculum in puncto g.</s> <s xml:id="echoid-s35024" xml:space="preserve"> Dico quòd in ar-<lb/>cu g o erunt loca imaginum, ut proponitur.</s> <s xml:id="echoid-s35025" xml:space="preserve"> Suma <lb/>tur ergo punctus illius arcus g o, qui ſitl:</s> <s xml:id="echoid-s35026" xml:space="preserve"> & protra <lb/>hatur à cẽtro ſpeculi diameter b l, uſquequò ſecet <lb/>lineam contingentem circulum in puncto g, quæ <lb/>eſt a t:</s> <s xml:id="echoid-s35027" xml:space="preserve"> ſecabit aũt per 14 th.</s> <s xml:id="echoid-s35028" xml:space="preserve"> 1 huius, & ք ea, quæ de-<lb/>clarata ſunt in proxima præcedente.</s> <s xml:id="echoid-s35029" xml:space="preserve"> Sit ergo pun-<lb/>ctus ſectionis e:</s> <s xml:id="echoid-s35030" xml:space="preserve"> & producatur linea a l ſecá s appa-<lb/>rentem ſuperficiem ſpeculi in pũcto r:</s> <s xml:id="echoid-s35031" xml:space="preserve"> & palàm ex <lb/>15 p 3 quoniã linea l r minor eſt quã linea m o.</s> <s xml:id="echoid-s35032" xml:space="preserve"> Cũ <lb/>ergo exhypotheſi linea m o ſit ęqualis ſemidiame <lb/>tro b l, patet quòd linea r l minor eſt ſemidiametro <lb/>b l.</s> <s xml:id="echoid-s35033" xml:space="preserve"> Si ergo per 136 th.</s> <s xml:id="echoid-s35034" xml:space="preserve"> 1 huius à puncto a ducatur li <lb/>nea ad diametrum b l, cuius pars interiacẽs circu-<lb/>lum & diametrum ſit ęqualis parti diametri inter-<lb/>iacenti punctum huius ſectionis & centrum circu <lb/>li b:</s> <s xml:id="echoid-s35035" xml:space="preserve"> hæc linea reflexionis cadet inter puncta b & l.</s> <s xml:id="echoid-s35036" xml:space="preserve"> <lb/>Quia ſi detur, quòd cadat inter punctal & e:</s> <s xml:id="echoid-s35037" xml:space="preserve"> erit li <lb/>near l maior quá linea l b:</s> <s xml:id="echoid-s35038" xml:space="preserve"> omnis enim linea inter-<lb/>iacens centrum circuli, & illam partem lineę refle <lb/>xionis illi parti diametri æqualem, erit maior illa <lb/>parte diametri, ſicut in commento 28 huius per 15 <lb/>p 3 oſtẽdimus de linea b p, quę eſt maior quã linea <lb/>f z, æ quali parti diametri z d, ut ibi patet:</s> <s xml:id="echoid-s35039" xml:space="preserve"> eſt aũt li-<lb/>near l minor quàm linea b l:</s> <s xml:id="echoid-s35040" xml:space="preserve"> quoniam per 15 p 3 linear l eſt minor quã linea m o, quę ex hypotheſi eſt <lb/>æqualis ipſil b.</s> <s xml:id="echoid-s35041" xml:space="preserve"> Non ergo cadit illa linea inter puncta l & e, ſed neque in punctum l, propter eandem <lb/> <pb o="244" file="0546" n="546" rhead="VITELLONIS OPTICAE"/> cauſſam:</s> <s xml:id="echoid-s35042" xml:space="preserve"> cadit ergo inter puncta b & l.</s> <s xml:id="echoid-s35043" xml:space="preserve"> Sit ergo punctus, in quem cadit illa linea, punctus i:</s> <s xml:id="echoid-s35044" xml:space="preserve"> & duca-<lb/>tur linea a i ſecans portionem apparentẽ ſpeculi in puncto u:</s> <s xml:id="echoid-s35045" xml:space="preserve"> cuius pars u i ſit æqualis parti diame-<lb/>tri, quæ eſt b i.</s> <s xml:id="echoid-s35046" xml:space="preserve"> Dico ergo quòd in quolibet pũcto inter e & i ſumpto eſt locus imaginis:</s> <s xml:id="echoid-s35047" xml:space="preserve"> & ſunt pun <lb/>cta e & i metæ imaginum.</s> <s xml:id="echoid-s35048" xml:space="preserve"> Sumatur enim aliquod punctum lineę l e, quod ſit f:</s> <s xml:id="echoid-s35049" xml:space="preserve"> & ducatur linea f a ſe-<lb/>cans apparentem portionem ſpeculi in puncto h:</s> <s xml:id="echoid-s35050" xml:space="preserve"> & ducatur à centro ſpeculi perpendicularis, quæ <lb/>ſit b h k:</s> <s xml:id="echoid-s35051" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s35052" xml:space="preserve"> per 23 p 1 ſuper punctum h terminum lineæ k h angulus ęqualis angulo a h k, qui ſit k <lb/>h y:</s> <s xml:id="echoid-s35053" xml:space="preserve"> palamq́;</s> <s xml:id="echoid-s35054" xml:space="preserve"> ex præmiſsis in præcedente, quoniam lineę b e & h y productæ concurrent per 14 th.</s> <s xml:id="echoid-s35055" xml:space="preserve"> 1 <lb/>huius:</s> <s xml:id="echoid-s35056" xml:space="preserve"> ſit punctus concurſus y.</s> <s xml:id="echoid-s35057" xml:space="preserve"> Et quoniam linea h y cadit extra ſpeculum:</s> <s xml:id="echoid-s35058" xml:space="preserve"> forma ergo puncti y mo <lb/>uebitur per lineam y h a d ſpeculum:</s> <s xml:id="echoid-s35059" xml:space="preserve"> reflectetur quoq;</s> <s xml:id="echoid-s35060" xml:space="preserve"> à puncto ſpeculi, quod eſt h, ad uiſum exiſten <lb/>tem in puncto a:</s> <s xml:id="echoid-s35061" xml:space="preserve"> apparebitq́;</s> <s xml:id="echoid-s35062" xml:space="preserve"> imago eius in puncto f, in concurſu catheti incidentię, quæ eſt b f, cum <lb/>linea reflexionis, quæ eſt a h, extra ſpeculi ſuperficiem.</s> <s xml:id="echoid-s35063" xml:space="preserve"> Et eodem modo eſt de omnib.</s> <s xml:id="echoid-s35064" xml:space="preserve"> punctis lineæ <lb/>l e demonſtrandum.</s> <s xml:id="echoid-s35065" xml:space="preserve"> Imagines enim formarum omnium illorum punctorum uidentur extra ſpecu-<lb/>lum, excepto ſolo l, in quo diameter b l ſecat ſpeculi ſuperficiem:</s> <s xml:id="echoid-s35066" xml:space="preserve"> quoniam in illo puncto locus ima <lb/>ginis eſt in ſuperficie ſpeculi:</s> <s xml:id="echoid-s35067" xml:space="preserve"> ideo quòd in ſuperficie eius ſe interſecat linea reflexionis, quæ eſt a l, <lb/>cum catheto incidentiæ, quæ eſt b y:</s> <s xml:id="echoid-s35068" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s35069" xml:space="preserve"> punctum, cuius formæ imago uidetur in pũcto l, reflexa <lb/>à punctor, conſiſtens in diametro b y producta ultra punctum l, ut patet per 27 huius:</s> <s xml:id="echoid-s35070" xml:space="preserve"> ſed, ut patet ք <lb/>30 huius, omnes formę punctorum cadentium in diametro b y, ultra punctum reflexum à punctor.</s> <s xml:id="echoid-s35071" xml:space="preserve"> <lb/>reflectuntur ab aliquo puncto arcus r u, & loca imaginum omnium illorũ punctorum ſuntin linea <lb/>il:</s> <s xml:id="echoid-s35072" xml:space="preserve"> ideo, quia, ut patet ex præmiſsis, punctum i eſt meta imaginum, ultra quod punctum nunquá ap-<lb/>paret aliqua imaginum uiſu exiſtente in puncto a, & ſpeculo ſic diſpoſito, ut patet ex hypotheſi.</s> <s xml:id="echoid-s35073" xml:space="preserve"> Pa-<lb/>làm ergo quòd in quolibet puncto lineę e i ſumpto inter puncta e & i, eſt locus imaginis formæ ali-<lb/>cuius punctorum diametri b e eductæ ultra punctũ e.</s> <s xml:id="echoid-s35074" xml:space="preserve"> Quędá ergo imagines in diametro e b ſortiun <lb/>tur loca intra ſpeculũ, quędá extra ſpeculum, & una ſola in ſuperficie ſpeculi, ſcilicet in puncto l.</s> <s xml:id="echoid-s35075" xml:space="preserve"> Et <lb/>eodem modo in quolibet puncto arcus o g poterit demonſtrari diametris data puncta arcus o g trá <lb/>ſeuntibus & ſuperficiem ſpeculi ſecantibus, prout demonſtrationum neceſsitas requirit.</s> <s xml:id="echoid-s35076" xml:space="preserve"/> </p> <div xml:id="echoid-div1442" type="float" level="0" n="0"> <figure xlink:label="fig-0545-01" xlink:href="fig-0545-01a"> <variables xml:id="echoid-variables610" xml:space="preserve">a d k u m r h b g i l e f o z t y</variables> </figure> </div> </div> <div xml:id="echoid-div1444" type="section" level="0" n="0"> <head xml:id="echoid-head1111" xml:space="preserve" style="it">32. In quemcun punctum arcus circuli (quieſt cõmunis ſectio ſuperficiei reflexionis & ſpe-<lb/>culi ſphærici conuexi) interiacentis punctũ, in quo linea reflexionis (cuius pars intra circulũ) <lb/>eſt æqualis ſemidiametro circuli, in portione nõ apparente ſecat circulum, & punctum diſtantẽ <lb/>à puncto contingentiæ per quartam eiuſdem circuli cathetus, incidenιiæ ceciderit: locus imagi <lb/>nis ſemper erit extra ſpeculum. Alhazen 27 n 5.</head> <p> <s xml:id="echoid-s35077" xml:space="preserve">Diſponantur omnia, ut in pręcedentib ita ut linea a m o ſic ſecet circulum ſpeculi, ut linea m o ſit <lb/>æqualis ſemidiametro ſpeculi:</s> <s xml:id="echoid-s35078" xml:space="preserve"> & ſit, utin 30 huius, angulus h b g rectus:</s> <s xml:id="echoid-s35079" xml:space="preserve"> & linea a g p contingat ſpe-<lb/>culum in puncto g.</s> <s xml:id="echoid-s35080" xml:space="preserve"> Dico quòd arcui o h cathetis incidentiæ occurrentib.</s> <s xml:id="echoid-s35081" xml:space="preserve"> locusimaginis erit ſemp <lb/>extra ſpeculum.</s> <s xml:id="echoid-s35082" xml:space="preserve"> Ducatur enim per aliquod punctorum arcus o h dia <lb/> <anchor type="figure" xlink:label="fig-0546-01a" xlink:href="fig-0546-01"/> meter b q:</s> <s xml:id="echoid-s35083" xml:space="preserve"> quæ concurrat cum contingente a g p in puncto p:</s> <s xml:id="echoid-s35084" xml:space="preserve"> & duca <lb/>tur à centro uiſus linea a u q, ſecans ſuperius in portione uiſui appa-<lb/>rente ſpeculum in puncto u.</s> <s xml:id="echoid-s35085" xml:space="preserve"> Et quia, ut prius patuit, linea m o eſt æ-<lb/>qualis lineę b q, & linea u q eſt maior quã linea m o per 15 p 3:</s> <s xml:id="echoid-s35086" xml:space="preserve"> ergo li-<lb/>neau q eſt maior quàm linea q b.</s> <s xml:id="echoid-s35087" xml:space="preserve"> Linea quoq;</s> <s xml:id="echoid-s35088" xml:space="preserve"> ducta à circum ferentia <lb/>ad diametrum b p, quæ eſt æqualis parti diametri p b, interiacenti i-<lb/>pſam & centrum ſpeculi, non cadetinter puncta q & b.</s> <s xml:id="echoid-s35089" xml:space="preserve"> Si en m hoc <lb/>ſit poſsibile, tunc, ut prius, erit linea u q minor quàm linea q b:</s> <s xml:id="echoid-s35090" xml:space="preserve"> quo-<lb/>niam ſi linea illa caderet in punctum q, eſſet eius pars intra circumfe <lb/>rentiam maior quá linea u q per 15 p 3:</s> <s xml:id="echoid-s35091" xml:space="preserve"> reſtat ergo ut linea æqualis ca <lb/>dat inter p & q.</s> <s xml:id="echoid-s35092" xml:space="preserve"> Quòd enim non cadatin punctum p, palàm per hoc, <lb/>quia angulusp g b eſt rectus:</s> <s xml:id="echoid-s35093" xml:space="preserve"> eſt ergo per 19 p 1 in trigono p b g latus <lb/>p b maius latere p g.</s> <s xml:id="echoid-s35094" xml:space="preserve"> Cadat itaque linea taliter ducta citra p:</s> <s xml:id="echoid-s35095" xml:space="preserve"> & ſit pũ <lb/>ctus, in quem cadit, s:</s> <s xml:id="echoid-s35096" xml:space="preserve"> erit ergo per 28 huius punctus s meta loco-<lb/>rum imaginum:</s> <s xml:id="echoid-s35097" xml:space="preserve"> & quilibet punctus inter pũcta p & s erit locus ima-<lb/>ginum:</s> <s xml:id="echoid-s35098" xml:space="preserve"> & eſt eadem demonſtratio, quæ in ſuperioribus ſcilicet 30 & <lb/>31 huius:</s> <s xml:id="echoid-s35099" xml:space="preserve"> in quolibet quoque puncto arcus h o eſt eadem demonſtra <lb/>tio.</s> <s xml:id="echoid-s35100" xml:space="preserve"> Ex his ergo præmiſsis propoſitionibus palàm eſt, quia imagi-<lb/>nes diametrorum arcus h o omnes ſunt extra ſuperficiem ſpeculi:</s> <s xml:id="echoid-s35101" xml:space="preserve"> i-<lb/>maginũ uerò diametri f y, ut in 31 huius, una ſola eſt in ſuperficie ſpe-<lb/>culi, ut illa, quę eſt in puncto l:</s> <s xml:id="echoid-s35102" xml:space="preserve"> aliæ uerò ſunt intra ſuperficiem ſpecu <lb/>li, ut quæ cadunt in parte diametri, quæ eſt i l:</s> <s xml:id="echoid-s35103" xml:space="preserve"> alię uerò omnes ſunt extra ſpeculum, ut quę cadunt <lb/>in linea l e.</s> <s xml:id="echoid-s35104" xml:space="preserve"> Omnium quo que imaginum diametrorum arcus o g quędam ſunt intra ſuperficiem ſpe <lb/>culi:</s> <s xml:id="echoid-s35105" xml:space="preserve"> quædam extra ipſam:</s> <s xml:id="echoid-s35106" xml:space="preserve"> quędam in ipſa ſuperficie ſpeculi conuexa, ut ibidem in præmiſſa conclu <lb/>ſum eſt.</s> <s xml:id="echoid-s35107" xml:space="preserve"> Patetitaq;</s> <s xml:id="echoid-s35108" xml:space="preserve">, quod proponebatur.</s> <s xml:id="echoid-s35109" xml:space="preserve"/> </p> <div xml:id="echoid-div1444" type="float" level="0" n="0"> <figure xlink:label="fig-0546-01" xlink:href="fig-0546-01a"> <variables xml:id="echoid-variables611" xml:space="preserve">a d u m b s o t q z h s x</variables> </figure> </div> </div> <div xml:id="echoid-div1446" type="section" level="0" n="0"> <head xml:id="echoid-head1112" xml:space="preserve" style="it">33. In arcum circuli (communis ſectionis ſuperficiei reflexionis & ſuperficiei ſpeculi ſphæri-<lb/>ci conuexi) interiacentem punctum, ubi diameter uiſualis & punctum diſtans à puncto contin <lb/> <pb o="245" file="0547" n="547" rhead="LIBER SEXTVS."/> gentiæ per quartam circuli inferius ſecant circulum, non poteſt cadere cathetus incide ntiæ, in <lb/>qua aliquis locus imaginis occurrat. Alhazen 28 n 5.</head> <p> <s xml:id="echoid-s35110" xml:space="preserve">Omnibus alijs diſp oſitis, ut in proxima ſuperiori figura:</s> <s xml:id="echoid-s35111" xml:space="preserve"> dico quòd in arcum h z non poteſt cade <lb/>re aliqua diameter, in qua ſit locus alicuius imaginis.</s> <s xml:id="echoid-s35112" xml:space="preserve"> Quoniam enim linea contingẽs, quæ eſt a g p, <lb/>ęquidiſtat diametro b h per 28 p 1:</s> <s xml:id="echoid-s35113" xml:space="preserve"> tunc patet quòd uerſus punctũ p <lb/> <anchor type="figure" xlink:label="fig-0547-01a" xlink:href="fig-0547-01"/> nulla diameter cadens in arcum z h, concurrit cum linea contingen <lb/>te, quæ eſt a p:</s> <s xml:id="echoid-s35114" xml:space="preserve"> & à quocunq;</s> <s xml:id="echoid-s35115" xml:space="preserve"> puncto talium diametrorũ ducatur li-<lb/>nea ad ſuperficiem ſpeculi conuexam, cadit in portionem nõ ap pa-<lb/>rentem ipſius ſpeculi, utpote in portionem circuli, quæ eſt g z c:</s> <s xml:id="echoid-s35116" xml:space="preserve"> & <lb/>nulla ipſarum cadit in portionem circuli g d c uiſui oppoſitam, niſi <lb/>ſecando ſphæram ſpeculi.</s> <s xml:id="echoid-s35117" xml:space="preserve"> Nulla ergo forma puncti alicuius talium <lb/>diametrorum ueniet ad portionem uiſui apparentem uel ad uiſum.</s> <s xml:id="echoid-s35118" xml:space="preserve"> <lb/>Omnia aũt iſta, quæ in ſemicirculo d g z, & in eius arcubus in præ-<lb/>miſsis ſex theorematib.</s> <s xml:id="echoid-s35119" xml:space="preserve"> declarata ſunt, in arcubus quoq;</s> <s xml:id="echoid-s35120" xml:space="preserve"> ſemicircu-<lb/>li d c z ſimiliter poſſunt demonſtrari, ut in arcubus ſemicirculi d g z.</s> <s xml:id="echoid-s35121" xml:space="preserve"> <lb/>Similibus enim acceptis utrinq;</s> <s xml:id="echoid-s35122" xml:space="preserve"> diſpoſitionib.</s> <s xml:id="echoid-s35123" xml:space="preserve"> arcuum, & ſimilibus <lb/>factis protractionibus linearũ, eædem in omnibus occurrẽt paſsio-<lb/>nes:</s> <s xml:id="echoid-s35124" xml:space="preserve"> & idem eſt demonſtrandi modus.</s> <s xml:id="echoid-s35125" xml:space="preserve"> Et ſimiliter etiam quod nunc <lb/>declaratur in circulo c d g z, poteſt in unoquoq;</s> <s xml:id="echoid-s35126" xml:space="preserve"> circulorũ, qui ſunt <lb/>communes ſectiones ſuperficierum reflexionis & ſuperficierũ con <lb/>uexi ſpeculi ſphærici declarari.</s> <s xml:id="echoid-s35127" xml:space="preserve"> Vnde omnes paſsiones probatę ſe-<lb/>cundum quoſcunq;</s> <s xml:id="echoid-s35128" xml:space="preserve"> punctos circuli d g z c, in completis circulis ac-<lb/>cidunt per totam ſpeculi ſuperficiem:</s> <s xml:id="echoid-s35129" xml:space="preserve"> ſicut ſi punctus g, uel alius pũ <lb/>ctus ſignatus moueatur per ſphærę ſuperficiem, & circulum deſcri <lb/>bat.</s> <s xml:id="echoid-s35130" xml:space="preserve"> Paſsiones uerò arcuum circuli d g z c perueniuntin quadam lata ſuperficie contenta ſub ter-<lb/>minis æquidiſtantium circulorum per totam ſphæram ſpeculi:</s> <s xml:id="echoid-s35131" xml:space="preserve"> ſicut ſi arcus aliquis æquidiſtans po <lb/>lo, motus ſpeculi aliquam ſuperficiem diſtinguat, ut patet intuẽti.</s> <s xml:id="echoid-s35132" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s35133" xml:space="preserve"> linea b h moueatur, eodẽ <lb/>manente angulo h b z:</s> <s xml:id="echoid-s35134" xml:space="preserve"> ſignabit ipſa motu ſuo ſecundum punctum z portionẽ ſphæræ:</s> <s xml:id="echoid-s35135" xml:space="preserve"> in cuius dia-<lb/>metris nullus erit imaginis locus.</s> <s xml:id="echoid-s35136" xml:space="preserve"> Et ſi linea b h immota exiſtente, moueatur arcus o h, deſcribetur <lb/>portio ſphæræ, cuius omnes imagines in diametro b o, uel alia protracta exiſtentes, ſunt extra ſpe-<lb/>culum:</s> <s xml:id="echoid-s35137" xml:space="preserve"> moto uerò arcu o g, fiet portio ſpeculi, cuius diametrorum quædam imagines ſuntin ſuper-<lb/>ficie ſpeculi:</s> <s xml:id="echoid-s35138" xml:space="preserve"> quædam extra:</s> <s xml:id="echoid-s35139" xml:space="preserve"> & quædam intra ſpeculum.</s> <s xml:id="echoid-s35140" xml:space="preserve"> Verùm uiſus non ſemper comprehendit, <lb/>quæ imagines ſint in ſuperficie ſpeculi, uel quæ ſint extra:</s> <s xml:id="echoid-s35141" xml:space="preserve"> nec certificatur in iſtorum cõprehenſio-<lb/>ne, niſi in tantum, quia ſentit, quòd ſunt ultra portionem ſphæræ apparentem.</s> <s xml:id="echoid-s35142" xml:space="preserve"> Sic ergo expræmiſ-<lb/>ſis ſextheorematib.</s> <s xml:id="echoid-s35143" xml:space="preserve"> patet in propoſitis ſpeculis loca imaginum eſſe determinata, ſecundum quod <lb/>imagines horum ſpeculorum uni tantùm uiſui offeruntur.</s> <s xml:id="echoid-s35144" xml:space="preserve"/> </p> <div xml:id="echoid-div1446" type="float" level="0" n="0"> <figure xlink:label="fig-0547-01" xlink:href="fig-0547-01a"> <variables xml:id="echoid-variables612" xml:space="preserve">a d c u m b g o t q p n z h</variables> </figure> </div> </div> <div xml:id="echoid-div1448" type="section" level="0" n="0"> <head xml:id="echoid-head1113" xml:space="preserve" style="it">34. Ambobus uiſibus à duobus punctis reflexionis ſuperficiei ſpeculi ſphærici conuexiforma <lb/>unius punctioccurrẽte: unicus imaginis eſt locus: & imago tantũ unica uidetur. Alha. 41 n 5.</head> <p> <s xml:id="echoid-s35145" xml:space="preserve">Sint centra duorum uiſuum a & b:</s> <s xml:id="echoid-s35146" xml:space="preserve"> & punctus uiſus ſit c:</s> <s xml:id="echoid-s35147" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s35148" xml:space="preserve"> d centrum circuli magni, qui eſt ſe-<lb/>cans ambos circulos, qui ſunt communes ſectiones ſuperficierum ambarũ reflexionis & ſpeculi, à <lb/>cuius punctis fit reflexio, & cuius portio apparens uiſui ſit e f:</s> <s xml:id="echoid-s35149" xml:space="preserve"> ſitq́ue <lb/> <anchor type="figure" xlink:label="fig-0547-02a" xlink:href="fig-0547-02"/> punctus reflexionis formæ puncti c ad uiſum a, punctus g:</s> <s xml:id="echoid-s35150" xml:space="preserve"> & pũctus <lb/>reflexionis formę puncti c ad uiſum b ſit punctus h:</s> <s xml:id="echoid-s35151" xml:space="preserve"> & ducatur cathe <lb/>tus incidentiæ à puncto c ad centrum ſpeculi, quæ ſit c d, ſecans cir-<lb/>culum in puncto o:</s> <s xml:id="echoid-s35152" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s35153" xml:space="preserve"> linea reflexionis, quæ eſt a g, producta i-<lb/>pſam cathetum c d in puncto k, & linea b h in puncto i:</s> <s xml:id="echoid-s35154" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s35155" xml:space="preserve"> primò <lb/>uiſus ambo æqualiter diſtantes à centro ſpeculi d:</s> <s xml:id="echoid-s35156" xml:space="preserve"> & à puncto rei ui-<lb/>ſę, quod eſt c.</s> <s xml:id="echoid-s35157" xml:space="preserve"> Dico quòd ambobus uiſibus a & b, formę puncti uiſi c, <lb/>licet duo ſint reflexionum puncta, quæ g & h, una tantùm imago ui-<lb/>detur:</s> <s xml:id="echoid-s35158" xml:space="preserve"> quia unicus eſt imaginis locus.</s> <s xml:id="echoid-s35159" xml:space="preserve"> Ducantur enim lineæ a d & <lb/>b d à centris amborum uiſuum ad centrum ſphæræ ſecantes ſpecu-<lb/>lum in punctis l & m.</s> <s xml:id="echoid-s35160" xml:space="preserve"> Et palàm quoniam illæ lineæ ſunt ęquales, ocu <lb/>lis enim æqualiter diſtantib.</s> <s xml:id="echoid-s35161" xml:space="preserve"> à centro ſpeculi, quod eſt d, palàm quòd <lb/>linea a b continuans centra oculorum cum ambabus lineis a d & b d <lb/>continet angulos ęquales argumento 30 th.</s> <s xml:id="echoid-s35162" xml:space="preserve"> 3 huius:</s> <s xml:id="echoid-s35163" xml:space="preserve"> ergo per 6 p 1 li-<lb/>neę a d & b d ſunt æquales.</s> <s xml:id="echoid-s35164" xml:space="preserve"> Si ergo ſitus puncti c reſpectu utriuſque <lb/>uiſus a & b ſit idem, ita ut linea a c ſit æqualis lineę b c:</s> <s xml:id="echoid-s35165" xml:space="preserve"> tunc patet per <lb/>8 p 1 quòd utraq;</s> <s xml:id="echoid-s35166" xml:space="preserve"> diametrorum uiſualium ſcilicet a d & b d cum ca-<lb/>theto c d continet angulos ęquales:</s> <s xml:id="echoid-s35167" xml:space="preserve"> ergo per 26 p 3 arcus ſpeculi l o <lb/>& m o ſunt æquales.</s> <s xml:id="echoid-s35168" xml:space="preserve"> Quia enim a d & b d diametri uiſuales ſecantex <lb/>circulis communibus ſuperficiebus ſpeculi & reflexionis arcus, & continent angulos æquales cú <lb/>catheto c d in cẽtro d:</s> <s xml:id="echoid-s35169" xml:space="preserve"> palã per 26 p 3 quia illi arcus lineas c d & b d ex una parte, & ex alia lineas c d <lb/> <pb o="246" file="0548" n="548" rhead="VITELLONIS OPTICAE"/> & a d interiacentes duo puncta reflexionis, quæ ſunt h & g, & punctum o, ſunt æ quales per 26 p 3:</s> <s xml:id="echoid-s35170" xml:space="preserve"> <lb/>quoniam perpendiculares ductæ à centro ad puncta reflexionum, quæ ſunt d g p & d h q, cum linea <lb/>c d continent angulos æquales.</s> <s xml:id="echoid-s35171" xml:space="preserve"> Et quia arcus h o & g o ſunt ęquales, & ſemidiametri d h & d g ęqua <lb/>les:</s> <s xml:id="echoid-s35172" xml:space="preserve"> erunt etiam lineæ reflexionum, quę ſunt h b & g a, ęquales per 4 p 1:</s> <s xml:id="echoid-s35173" xml:space="preserve"> quoniam ad uiſus ęqualiter <lb/>diſtantes à centro ſpeculi ſecundum æ quales angulos ſunt incidentes:</s> <s xml:id="echoid-s35174" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s35175" xml:space="preserve"> ſimiliter lineæ g c & <lb/>h c æquales:</s> <s xml:id="echoid-s35176" xml:space="preserve"> lineæ uerò b h & a g neceſſariò ſe ſecant:</s> <s xml:id="echoid-s35177" xml:space="preserve"> quoniam cum anguli ſint minores duobus re-<lb/>ctis, palàm per 14 th.</s> <s xml:id="echoid-s35178" xml:space="preserve"> 1 huius quia lineæ b h & a g in aliquo puncto neceſſe habent cõcurrere.</s> <s xml:id="echoid-s35179" xml:space="preserve"> Et quia <lb/>anguli reflexionis ad ambos uiſus propter æqualem diſtantiam amborum uiſuum à puncto rei ui-<lb/>ſę, & à centro ſpeculi, ſunt ęquales:</s> <s xml:id="echoid-s35180" xml:space="preserve"> erunt & anguli c g a & c h b inter ſe ęquales:</s> <s xml:id="echoid-s35181" xml:space="preserve"> palã ergo per 13 & 32 <lb/>p 1 quia trigonum g c k eſt ęquiangulum trigono h c i, & linea c h eſt ęqualis ipſi lineę c g:</s> <s xml:id="echoid-s35182" xml:space="preserve"> erit ergo <lb/>per 4 p 6 linea h i ęqualis lineæ g k, & linea c k æqualis ipſi lineę ci:</s> <s xml:id="echoid-s35183" xml:space="preserve"> puncta ergo k & i ſunt punctus <lb/>unus.</s> <s xml:id="echoid-s35184" xml:space="preserve"> Superidem ergo punctum catheti c d erit ſectio ambarum linearum reflexionis, quę ſunt a g <lb/>& b h, cum catheto incidentiæ quę eſt c d:</s> <s xml:id="echoid-s35185" xml:space="preserve"> & in hoc puncto utriq;</s> <s xml:id="echoid-s35186" xml:space="preserve"> uiſui apparebit imago.</s> <s xml:id="echoid-s35187" xml:space="preserve"> Videbitur <lb/>ergo una ſola imago:</s> <s xml:id="echoid-s35188" xml:space="preserve"> quia unus & idẽ imaginis locus erit.</s> <s xml:id="echoid-s35189" xml:space="preserve"> Quòd ſi uiſus non æqualiter diſtent à ſpe-<lb/>culo uel à re uiſa:</s> <s xml:id="echoid-s35190" xml:space="preserve"> adhuc tamen unica uidebitur imago.</s> <s xml:id="echoid-s35191" xml:space="preserve"> Licet enim imago puncti uiſi cadat in diuer <lb/>ſis punctis perpendicularis:</s> <s xml:id="echoid-s35192" xml:space="preserve"> hoc tamen eſt imperceptibile, quia diſtantia illorum punctorum eſt im <lb/>perceptibilis.</s> <s xml:id="echoid-s35193" xml:space="preserve"> Imago ergo cuiuſcunq;</s> <s xml:id="echoid-s35194" xml:space="preserve"> puncti à quocunq;</s> <s xml:id="echoid-s35195" xml:space="preserve"> uideatur oculo, ſemper ſeruat identitatem <lb/>partis:</s> <s xml:id="echoid-s35196" xml:space="preserve"> & ob hoc apparet unitas imaginis.</s> <s xml:id="echoid-s35197" xml:space="preserve"> Remotio enim puncti uiſi ab uno uiſu modicò eſt maior <lb/>ꝗ̃ ab alio:</s> <s xml:id="echoid-s35198" xml:space="preserve"> & ob hocloca imaginum ſunt imperceptibiliter remota:</s> <s xml:id="echoid-s35199" xml:space="preserve"> & ob hoc apparent ſimul:</s> <s xml:id="echoid-s35200" xml:space="preserve"> quoniã <lb/>ex illis fit una imago compacta:</s> <s xml:id="echoid-s35201" xml:space="preserve"> quia loca imaginis non totaliter à ſe diſtant, licet partialiter aliquã-<lb/>tulnm diſtent.</s> <s xml:id="echoid-s35202" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s35203" xml:space="preserve"> Poteſt tamen quandoq;</s> <s xml:id="echoid-s35204" xml:space="preserve"> & hoc accidere, ut ſi forma reflexa <lb/>ualde obliquè incidat alteri uiſuũ:</s> <s xml:id="echoid-s35205" xml:space="preserve"> quòd ꝓpter obliquitatẽ una forma uideatur duę:</s> <s xml:id="echoid-s35206" xml:space="preserve"> ut cũ in una ſu-<lb/>perſicie reflexionis ſunt centra amborũ uiſuũ:</s> <s xml:id="echoid-s35207" xml:space="preserve"> tũc enim præmiſsi anguli in cẽtro ſpeculi fiuntinæ-<lb/>quales, & accidit uideri duas ſormas, ſicut & nos in ſimplici modo uidẽdi dixim{us} in quarto libro hu <lb/>ius, capitulis de uiſiõe numerali:</s> <s xml:id="echoid-s35208" xml:space="preserve"> ſed hoc euenit ut rarò, & nos de hoc aliquid dixim{us} in 7 th.</s> <s xml:id="echoid-s35209" xml:space="preserve"> 5 hui{us}.</s> <s xml:id="echoid-s35210" xml:space="preserve"/> </p> <div xml:id="echoid-div1448" type="float" level="0" n="0"> <figure xlink:label="fig-0547-02" xlink:href="fig-0547-02a"> <variables xml:id="echoid-variables613" xml:space="preserve">a b c p q l m g h o i k d e f</variables> </figure> </div> </div> <div xml:id="echoid-div1450" type="section" level="0" n="0"> <head xml:id="echoid-head1114" xml:space="preserve" style="it">35. In ſpeculo ſphærico conuexo eſt ordinatio punctorum imaginum in ambobus uiſibus, ſicut <lb/>ordinatio punctorum rei uiſæ. Alhazen 42 n 5. Item 4 n 6.</head> <p> <s xml:id="echoid-s35211" xml:space="preserve">Ducantur à terminis lineę, quę eſt in reuiſa, duę catheti ad cẽtrum ſpeculi.</s> <s xml:id="echoid-s35212" xml:space="preserve"> Palàm ergo quòd tũc <lb/>erit triangulus, in quo continebuntur omnes imagines omniũ punctorum illius lineę:</s> <s xml:id="echoid-s35213" xml:space="preserve"> & ſi in illa li-<lb/>nea ſit punctus non eiuſdem ſitus reſpectu amborum uiſuũ:</s> <s xml:id="echoid-s35214" xml:space="preserve"> imago puncti remotioris ab illo erit in <lb/>diametro remotiori ab eius diametro, & propinquioris in propin quiori:</s> <s xml:id="echoid-s35215" xml:space="preserve"> quoniã ſemper imago cu-<lb/>iuslibet puncti rei uiſæ uidebitur in cóncurſu lineæ reflexionis cum catheto incidentię ducta ab il-<lb/>lo puncto ad cẽtrum ſpeculi, ut patet per 11 huius.</s> <s xml:id="echoid-s35216" xml:space="preserve"> Sic ergo obſeruabitur ſitus partium in imaginib.</s> <s xml:id="echoid-s35217" xml:space="preserve"> <lb/>ſicut fuerit ſitus in pũctis uiſis.</s> <s xml:id="echoid-s35218" xml:space="preserve"> Sumpta uerò linea, in qua eſt punctũ eiuſdẽ ſitus, quodlibet punctũ <lb/>illius lineę eiuſdem erit ſitus reſpectu oculorũ.</s> <s xml:id="echoid-s35219" xml:space="preserve"> Si aũt ſumatur linea, quę angulũ, quẽ continent duę <lb/>lineę à centris oculorum ad punctum uiſum productę, diuidit per æqualia:</s> <s xml:id="echoid-s35220" xml:space="preserve"> ſitus cuiuslibet puncti <lb/>illius lineę quantumcunq;</s> <s xml:id="echoid-s35221" xml:space="preserve"> productę eſt ſitus cõſimilis utriq;</s> <s xml:id="echoid-s35222" xml:space="preserve"> uiſui ſicut uni.</s> <s xml:id="echoid-s35223" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s35224" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1451" type="section" level="0" n="0"> <head xml:id="echoid-head1115" xml:space="preserve" style="it">36. In quibuſdam ſitibus poßibile eſt à ſpeculis <lb/>ſphæricis conuexis, plurib. uiſibus rem apparere <lb/>unicam unaḿ imaginem habentem.</head> <figure> <variables xml:id="echoid-variables614" xml:space="preserve">c e p g a o b h k d f q</variables> </figure> <p> <s xml:id="echoid-s35225" xml:space="preserve">Sit cõmunis ſectio ſuperficiei reflexionis & ſpe-<lb/>culi ſphęrici conuexi circulus a b:</s> <s xml:id="echoid-s35226" xml:space="preserve"> cuius centrũ ſit <lb/>d:</s> <s xml:id="echoid-s35227" xml:space="preserve"> & ſit punctum c punctum rei uiſę:</s> <s xml:id="echoid-s35228" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s35229" xml:space="preserve"> li-<lb/>nea c d à puncto uiſo in centrũ d, ſecans ſpeculi pe-<lb/>ripheriam in puncto o:</s> <s xml:id="echoid-s35230" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s35231" xml:space="preserve"> arcus a o ęqualis arcui <lb/>o b:</s> <s xml:id="echoid-s35232" xml:space="preserve"> & ducãtur lineę c a & c b, quę per 8 p 3 & ex hy-<lb/>potheſi erũt ęquales.</s> <s xml:id="echoid-s35233" xml:space="preserve"> Et à puncto a ducatur linea f <lb/>a e contingens circulum per 17 p 3, & à puncto b li-<lb/>nea p b q:</s> <s xml:id="echoid-s35234" xml:space="preserve"> & ducaturlinea a b.</s> <s xml:id="echoid-s35235" xml:space="preserve"> Patet ergo per 5 p 1 <lb/>quoniam anguli c a b & c b a ſunt ęquales:</s> <s xml:id="echoid-s35236" xml:space="preserve"> ſed & an <lb/>guli o a b & o b a linea curua & recta contenti ſunt <lb/>æquales per 43 th.</s> <s xml:id="echoid-s35237" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s35238" xml:space="preserve"> ſed & anguli contingen <lb/>tię o a e & o b p per 16 p 3 ſunt æquales:</s> <s xml:id="echoid-s35239" xml:space="preserve"> relinquitur <lb/>ergo angulus c a e æqualis angulo c b p.</s> <s xml:id="echoid-s35240" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s35241" xml:space="preserve"> ſuper <lb/>punctũ a terminum lineę c a conſtituatur angulus <lb/>æqualis angulo c a e per 23 p 1, qui ſit g a c, & ſuper <lb/>b terminũ lineę c b conſtituatur angulus æqualis <lb/>angulo p b c, qui ſit h b c:</s> <s xml:id="echoid-s35242" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s35243" xml:space="preserve"> angulus h b c ęqua <lb/>lis angulo g a c.</s> <s xml:id="echoid-s35244" xml:space="preserve"> Poſitis itaq;</s> <s xml:id="echoid-s35245" xml:space="preserve"> uiſib.</s> <s xml:id="echoid-s35246" xml:space="preserve"> in pũctis g & h:</s> <s xml:id="echoid-s35247" xml:space="preserve"> <lb/>palã per 20 th.</s> <s xml:id="echoid-s35248" xml:space="preserve"> 1 huius quoniam forma puncti c re-<lb/>flectitur ad ambos uiſus exiſtentes in punctis g & <lb/>h:</s> <s xml:id="echoid-s35249" xml:space="preserve"> ad punctũ quidem g à puncto a, ad punctũ quoq;</s> <s xml:id="echoid-s35250" xml:space="preserve"> h à puncto b.</s> <s xml:id="echoid-s35251" xml:space="preserve"> Producatur quoq;</s> <s xml:id="echoid-s35252" xml:space="preserve"> ultra punctũ a <lb/>linea g a ad lineã c d, quę cõcurret cũ illa ք 14 th.</s> <s xml:id="echoid-s35253" xml:space="preserve"> 1 huius, ideo quia anguli g a c & a c d ſunt minores <lb/> <pb o="247" file="0549" n="549" rhead="LIBER SEXTVS."/> duobus rectis:</s> <s xml:id="echoid-s35254" xml:space="preserve"> cõcurrãtitaq;</s> <s xml:id="echoid-s35255" xml:space="preserve"> in pũcto k:</s> <s xml:id="echoid-s35256" xml:space="preserve"> & ꝓducatur linea h b ad lineã c d:</s> <s xml:id="echoid-s35257" xml:space="preserve"> quę ſimiliter cõcurret ք <lb/>pręmiffa, & in eodẽ pũcto k.</s> <s xml:id="echoid-s35258" xml:space="preserve"> Quia enim, ut patet ex pręmiſsis, linea a c eſt æqualis lineæ c b, & a d æ-<lb/>qualis ipſi b d, quia ſemidiametri, & linea c d cõmunis eſt ambobus trigonis a c d & b c d:</s> <s xml:id="echoid-s35259" xml:space="preserve"> erũt angu <lb/>li a c d & d c b æquales per 8 p 1, & angulus g a c, ut patet expręmiſsis, eſt ęqualis angulo h b c:</s> <s xml:id="echoid-s35260" xml:space="preserve"> ſed & <lb/>angulus p b c oſtẽſus fuit æqualis eſſe angulo e a c:</s> <s xml:id="echoid-s35261" xml:space="preserve"> eſt ergo angulus h b q æqualis angulo g a f per 13 <lb/>p 1:</s> <s xml:id="echoid-s35262" xml:space="preserve"> ſed angulus e a k eſt æqualis angulo g a f, & angulus p b k æqualis angulo h b q ք 15 p 1:</s> <s xml:id="echoid-s35263" xml:space="preserve"> ergo an-<lb/>gulus e a k æqualis eſt angulo p b k:</s> <s xml:id="echoid-s35264" xml:space="preserve"> erit ergo totalis angulus c a k æqualis totali angulo c b k:</s> <s xml:id="echoid-s35265" xml:space="preserve"> ergo <lb/>per 32 p 1 trianguli c a k & c b k ſunt æquianguli:</s> <s xml:id="echoid-s35266" xml:space="preserve"> ergo per 4 p 6 cũ a c fit æqualis ipſi b c, erit latus a k <lb/>æquale lateri b k:</s> <s xml:id="echoid-s35267" xml:space="preserve"> cõcurrent ergo in uno puncto k:</s> <s xml:id="echoid-s35268" xml:space="preserve"> quoniã latus c k eſt in ambobus trigonis æquale <lb/>ſibijpſi.</s> <s xml:id="echoid-s35269" xml:space="preserve"> Sed pũctus k eſt locus imaginis pũcti c:</s> <s xml:id="echoid-s35270" xml:space="preserve"> erit ergo ambobus uiſibus idẽ locus imaginis.</s> <s xml:id="echoid-s35271" xml:space="preserve"> Siue <lb/>ergo propriã faciẽ aſpicientes uideant, ſiue res alias à loco pũcti c à pũctis a & b reflexas ad uiſus in <lb/>pũctis g & h exiſtentes, idẽ accidit utrobiq;</s> <s xml:id="echoid-s35272" xml:space="preserve">. Idem quoq;</s> <s xml:id="echoid-s35273" xml:space="preserve"> accidit in toto circulo tranſeunte pũcta b <lb/>& a:</s> <s xml:id="echoid-s35274" xml:space="preserve"> quoniã in quolibet pũcto illius circuli modo prædicto diſpoſitis uiſibus eadem eſt demonſtra.</s> <s xml:id="echoid-s35275" xml:space="preserve"> <lb/>tio.</s> <s xml:id="echoid-s35276" xml:space="preserve"> Palàm ergo propoſitũ.</s> <s xml:id="echoid-s35277" xml:space="preserve"> Si aũt anguli reflexionũ ſint diuerſi:</s> <s xml:id="echoid-s35278" xml:space="preserve"> tũc res una diuerſis uiſibus in locis <lb/>uidebitur diuerſis, & plura idola obtinebit.</s> <s xml:id="echoid-s35279" xml:space="preserve"> Et hoc eſt notandũ, & ſatis patuit ք pręmiſſa:</s> <s xml:id="echoid-s35280" xml:space="preserve"> quia illæ <lb/>reflexionũ lineę in diuerſis pũctis diametri ſpeculi concurrunt:</s> <s xml:id="echoid-s35281" xml:space="preserve"> & ob hoc loca imaginũ conſtituũt <lb/>diuerſa, ut patet per 11 huius.</s> <s xml:id="echoid-s35282" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s35283" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1452" type="section" level="0" n="0"> <head xml:id="echoid-head1116" xml:space="preserve" style="it">37. In ſpeculis ſphæricis conuexis minor eſt diſtantia imaginis à ſpeculi ſuperficie, quàm ipſius <lb/>rei extra. Euclides 20 th. catoptr.</head> <p> <s xml:id="echoid-s35284" xml:space="preserve">Eſto circulus (qui eſt cõmunis ſectio ſuperficiei reflexionis & ſpeculi ſphęrici cõuexi) q h k r:</s> <s xml:id="echoid-s35285" xml:space="preserve"> cu <lb/>ius cẽtrũ z:</s> <s xml:id="echoid-s35286" xml:space="preserve"> & linea uiſa obliquè incidẽs ſpeculo ſit e f:</s> <s xml:id="echoid-s35287" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s35288" xml:space="preserve"> centrũ uiſus b:</s> <s xml:id="echoid-s35289" xml:space="preserve"> & reflectatur pũctus e à <lb/>pũcto ſpeculi h ad uiſum b, & f à pũcto q:</s> <s xml:id="echoid-s35290" xml:space="preserve"> ducãturq́;</s> <s xml:id="echoid-s35291" xml:space="preserve"> lineę e h, h b, f q, q b:</s> <s xml:id="echoid-s35292" xml:space="preserve"> & ducãtur քpendiculariter <lb/>ſuper ſuperficiẽ ſpeculi catheti e z, f z:</s> <s xml:id="echoid-s35293" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s35294" xml:space="preserve"> linea e z circulũ ſpeculi in pũcto r:</s> <s xml:id="echoid-s35295" xml:space="preserve"> & f z in pũcto k:</s> <s xml:id="echoid-s35296" xml:space="preserve"> & <lb/>b h producta intra ſpeculũ, ſecet e z in pũcto a:</s> <s xml:id="echoid-s35297" xml:space="preserve"> & b q <lb/> <anchor type="figure" xlink:label="fig-0549-01a" xlink:href="fig-0549-01"/> ſecet f z in pũcto g:</s> <s xml:id="echoid-s35298" xml:space="preserve"> & producatur linea a g:</s> <s xml:id="echoid-s35299" xml:space="preserve"> quę per 11 <lb/>huius erit imago lineæ e f:</s> <s xml:id="echoid-s35300" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s35301" xml:space="preserve"> à pũcto h linea <lb/>circulũ cõtingens ք 17 p 3, quę ſit h t:</s> <s xml:id="echoid-s35302" xml:space="preserve"> & hæc ꝓducta <lb/>ſecet lineã e z in pũcto t:</s> <s xml:id="echoid-s35303" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s35304" xml:space="preserve"> punctus t finis cõtin-<lb/>gentię lineę h t:</s> <s xml:id="echoid-s35305" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s35306" xml:space="preserve"> linea t h ꝓducta ultra h, lineã <lb/>b g in pũcto l:</s> <s xml:id="echoid-s35307" xml:space="preserve"> & à pũcto t ducatur perpẽdicularis ſu <lb/>per lineã e z ք 11 p 1, quę producta ſecet e h lineam in <lb/>pũcto d, & ſit t d.</s> <s xml:id="echoid-s35308" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s35309" xml:space="preserve"> angulus b h l eſt æqualis <lb/>angulo e h t ք 20 th.</s> <s xml:id="echoid-s35310" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s35311" xml:space="preserve"> ſed & angulus t h a ęqua <lb/>lis eſt angulo b h l ք 15 p 1:</s> <s xml:id="echoid-s35312" xml:space="preserve"> ergo angulus e h t eſt ęqua <lb/>lis angulo t h a:</s> <s xml:id="echoid-s35313" xml:space="preserve"> ergo ք 3 p 6 erit proportio lineæ e h <lb/>ad h a, ſicut lineę e t ad lineã t a:</s> <s xml:id="echoid-s35314" xml:space="preserve"> ſed linea e h eſt ma-<lb/>ior ꝗ̃ linea h a:</s> <s xml:id="echoid-s35315" xml:space="preserve"> ergo & linea e t eſt maior ꝗ̃ t a.</s> <s xml:id="echoid-s35316" xml:space="preserve"> Quòd <lb/>aũt linea e h ſit maior ꝗ̃ linea h a, patet.</s> <s xml:id="echoid-s35317" xml:space="preserve"> Cũ enim an-<lb/>gulus e t d ſit rectus:</s> <s xml:id="echoid-s35318" xml:space="preserve"> erit angulus e t h maior recto:</s> <s xml:id="echoid-s35319" xml:space="preserve"> <lb/>eſt ergo ք 13 p 1 angulus e t h maior angulo a t h:</s> <s xml:id="echoid-s35320" xml:space="preserve"> ſed <lb/>& angulus e t h maior eſt angulo e h t per 32 p 1:</s> <s xml:id="echoid-s35321" xml:space="preserve"> ſed angulus e h t eſt æqualis angulo a h t, ut patet ex <lb/>pręmiſsis.</s> <s xml:id="echoid-s35322" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s35323" xml:space="preserve"> anguli trigoni e t h oẽs ſimul ſumpti, ſunt æquales angulis trigoni a t h omni-<lb/>bus ſimul ſumptis ք 32 p 1:</s> <s xml:id="echoid-s35324" xml:space="preserve"> relin quitur ergo angulus t a h trigoni t h a maior angulo t e h trigoni h e <lb/>t.</s> <s xml:id="echoid-s35325" xml:space="preserve"> In trigono itaq;</s> <s xml:id="echoid-s35326" xml:space="preserve"> a e h angulus e a h maior eſt angulo a e h:</s> <s xml:id="echoid-s35327" xml:space="preserve"> ergo in trigono e a h latus e h maius eſt la <lb/>tere h a ք 19 p 1:</s> <s xml:id="echoid-s35328" xml:space="preserve"> maior eſt ergo linea e t ꝗ̃ linea t a:</s> <s xml:id="echoid-s35329" xml:space="preserve"> multò magis ergo linea e r eſt maior ꝗ̃ linea r a:</s> <s xml:id="echoid-s35330" xml:space="preserve"> ſed <lb/>linea r a eſt diſtãtia imaginis pũcti a à ſuperficie ſpeculi intra ſpeculũ:</s> <s xml:id="echoid-s35331" xml:space="preserve"> & linea e r eſt diſtãtia pũcti ui <lb/>ſi, ꝗ eſt e, à ſuքficie ſpeculi extra ſpeculũ.</s> <s xml:id="echoid-s35332" xml:space="preserve"> Et ſi à pũcto q ducatur linea cõtingẽs eirculũ, q̃ ꝓducta ad <lb/>cathetũ f z ſecet ipſam in pũcto m:</s> <s xml:id="echoid-s35333" xml:space="preserve"> & à pũcto m ducatur ք pendicularis ſuper f z, q̃ producta ad f q ſit <lb/>m n:</s> <s xml:id="echoid-s35334" xml:space="preserve"> patebit ſimiliter quoniá linea f k eſt maior ꝗ̃ linea k g.</s> <s xml:id="echoid-s35335" xml:space="preserve"> Hoc eſt ergo propoſitũ:</s> <s xml:id="echoid-s35336" xml:space="preserve"> quoniã ſi à me-<lb/>dijs pũctis lineę e f ducantur lineę, ſicut ab extremis, patebit idẽ in omnibus imaginibus ipforum, <lb/>quę per 11 huius cadunt omnes in lineam a g.</s> <s xml:id="echoid-s35337" xml:space="preserve"> Patet ergo hoc, quod proponebatur.</s> <s xml:id="echoid-s35338" xml:space="preserve"/> </p> <div xml:id="echoid-div1452" type="float" level="0" n="0"> <figure xlink:label="fig-0549-01" xlink:href="fig-0549-01a"> <variables xml:id="echoid-variables615" xml:space="preserve">f b e d t m n k h q r a g z</variables> </figure> </div> </div> <div xml:id="echoid-div1454" type="section" level="0" n="0"> <head xml:id="echoid-head1117" xml:space="preserve" style="it">38. Re conſpecta à tali longitudine, quòd eius certa quantitas uiſu comprehendi non poßιt: <lb/>nonnunquã uidebitur imago reiuiſæ in ſpeculo ſphærico cõuexo æqualis: quando maior quàm <lb/>forma per ſe uiſui occurrens. Alhazen 6 n 6.</head> <p> <s xml:id="echoid-s35339" xml:space="preserve">Sit a centrum ſpeculi ſphęrici conuexi:</s> <s xml:id="echoid-s35340" xml:space="preserve"> & circulus (qui eſt communis ſectio ſuperficiei reflexio <lb/>nis & ſuperficiei ſpeculi) ſit e d b:</s> <s xml:id="echoid-s35341" xml:space="preserve"> & ſit e d diameter illius circuli:</s> <s xml:id="echoid-s35342" xml:space="preserve"> & educatur dιameter e d ultra d <lb/>uſque a d z taliter, ut illud, quod fit ex ductu e z in z d ſit ęquale quadrato a d ſemidiametri per 127 <lb/>th.</s> <s xml:id="echoid-s35343" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s35344" xml:space="preserve"> ac ſi e d & a d ſint duę lineę datę.</s> <s xml:id="echoid-s35345" xml:space="preserve"> Diuidaturq́;</s> <s xml:id="echoid-s35346" xml:space="preserve"> linea z d per ęqualia in puncto h per 10 p 1:</s> <s xml:id="echoid-s35347" xml:space="preserve"> <lb/>eritigitur a h medietas lineę e z:</s> <s xml:id="echoid-s35348" xml:space="preserve"> ergo per 1 p 6 illud, quod fit ex ductu a h in d z, eſt ęquale medie-<lb/>tati quadrati lineę a d.</s> <s xml:id="echoid-s35349" xml:space="preserve"> Ergo per eandem 1 p 6 illud, quod fit ex ductu a h in h d ęquale eſt quartę <lb/>parti quadrati a d.</s> <s xml:id="echoid-s35350" xml:space="preserve"> Et quia illud, quod fit ex ductu a h in h d maius eſt quadrato h d per 3 p 2:</s> <s xml:id="echoid-s35351" xml:space="preserve"> ſit il-<lb/>lud, quod fit ex ductu a h in t h ęquale quadrato h d:</s> <s xml:id="echoid-s35352" xml:space="preserve"> erit ergo h t minor quàm h d.</s> <s xml:id="echoid-s35353" xml:space="preserve"> Fiat ergo cir-<lb/>culus ſecundum quantitatem lineę a h:</s> <s xml:id="echoid-s35354" xml:space="preserve"> qui neceſſariò ęquidiſtabit circulo priori:</s> <s xml:id="echoid-s35355" xml:space="preserve"> quoniam ipſo-<lb/> <pb o="248" file="0550" n="550" rhead="VITELLONIS OPTICAE"/> rum eſt idem centrum punctum a, & ipſorum ſemidiametri ſunt inęquales:</s> <s xml:id="echoid-s35356" xml:space="preserve"> & à puncto h ducatur <lb/>chorda ęqualis medietati lineę h d per 1 p 4, quę fit h q:</s> <s xml:id="echoid-s35357" xml:space="preserve"> & producantur lineę q a, q t:</s> <s xml:id="echoid-s35358" xml:space="preserve"> & ſuper pun-<lb/>tum q lineę h q fiat angulus ęqualis angulo q a h per 23 p 1, qui ſit h q n, ducta linea q n ſuper lineam <lb/>a h.</s> <s xml:id="echoid-s35359" xml:space="preserve"> Et quoniam trianguli h q a angulus q a h ęqualis eſt angulo h q n trigoni h q n, & angulus a <lb/>h q utrique communis, erit tertius tertio ęqualis per 32 p 1, ſcilicet angulus a q h angulo h n q:</s> <s xml:id="echoid-s35360" xml:space="preserve"> er-<lb/>go per 4 p 6 erit proportio h a ad q h, ſicut q h ad h n:</s> <s xml:id="echoid-s35361" xml:space="preserve"> ergo per 17 p 6 illud, quod fit ex ductu a h in <lb/>h n ęquale erit quadrato h q:</s> <s xml:id="echoid-s35362" xml:space="preserve"> ſed quadratum h q eſt quarta pars quadrati h d per 4 p 2:</s> <s xml:id="echoid-s35363" xml:space="preserve"> eſt enim h q <lb/>medietas lineę h d:</s> <s xml:id="echoid-s35364" xml:space="preserve"> ductus ergo a h in h n eſt ęqualis quartę parti quadrati d h:</s> <s xml:id="echoid-s35365" xml:space="preserve"> ergo & quartę par-<lb/>ti ductus a h in h t:</s> <s xml:id="echoid-s35366" xml:space="preserve"> eſt ergo linea h n ęqualis quartę parti lineę h t per 1 p 6:</s> <s xml:id="echoid-s35367" xml:space="preserve"> cadit ergo punctum n <lb/>inter puncta h & t:</s> <s xml:id="echoid-s35368" xml:space="preserve"> remanetq́;</s> <s xml:id="echoid-s35369" xml:space="preserve"> linea t n tres quartę lineę h t:</s> <s xml:id="echoid-s35370" xml:space="preserve"> reſtat ergo, ut ductus h t in t n ſit tres <lb/>quartę quadrati h t per 2 p 2:</s> <s xml:id="echoid-s35371" xml:space="preserve"> ſed & per 1 p 6 erit ductus lineę a h <lb/> <anchor type="figure" xlink:label="fig-0550-01a" xlink:href="fig-0550-01"/> in t n tres quartę quadrati h d.</s> <s xml:id="echoid-s35372" xml:space="preserve"> Quoniam autem angulus a q h <lb/>eſt acutus per 42 th.</s> <s xml:id="echoid-s35373" xml:space="preserve"> 1 huius, & ipſe eſt ęqualis angulo q h a per 5 <lb/>p 1, quoniam latera a h & a q ſunt ęqualia:</s> <s xml:id="echoid-s35374" xml:space="preserve"> patet ergo quia angu-<lb/>lus q h a eſt ęqualis angulo h n q in minori triangulo:</s> <s xml:id="echoid-s35375" xml:space="preserve"> ergo per <lb/>6 p 1 latus n q eſt ęquale lateri h q:</s> <s xml:id="echoid-s35376" xml:space="preserve"> & angulus h n q eſt acutus:</s> <s xml:id="echoid-s35377" xml:space="preserve"> <lb/>ergo per 13 p 1 angulus q n t eſt obtuſus:</s> <s xml:id="echoid-s35378" xml:space="preserve"> ergo quadratum lineę <lb/>t q amplius eſt quadrato lineę q n, & quadrato lineę t n, in illo, <lb/>quod fit ex ductu t n in n h per 12 p 2.</s> <s xml:id="echoid-s35379" xml:space="preserve"> Si enim à puncto q du-<lb/>catur perpendicularis ſuper h n:</s> <s xml:id="echoid-s35380" xml:space="preserve"> palàm per 31 th 1 huius, cum la-<lb/>tera q h & q n ſint ęqualia, quòd ipſa cadet in medio puncto li-<lb/>neę h n:</s> <s xml:id="echoid-s35381" xml:space="preserve"> ex prima uerò 2 ductus n t in h n ęquipollet illi, quod fit <lb/>ex ductu t n in medietatem h n bis:</s> <s xml:id="echoid-s35382" xml:space="preserve"> ſed ductus t n in n h cum <lb/>quadrato n t ęqualis eſt ductui h t in t n per 3 p 2:</s> <s xml:id="echoid-s35383" xml:space="preserve"> igitur ductus <lb/>h t in t n eſt exceſſus quadrati lineę t q ſupra quadratum lineę <lb/>n q:</s> <s xml:id="echoid-s35384" xml:space="preserve"> ergo & upra quadratum h q, cum h q ſit æqualis ipſi n q.</s> <s xml:id="echoid-s35385" xml:space="preserve"> <lb/>Quia uerò quadratũ t q eſt maius quadrato h q, & linea t q erit <lb/>maior linea h q:</s> <s xml:id="echoid-s35386" xml:space="preserve"> ſit ergo per 3 th.</s> <s xml:id="echoid-s35387" xml:space="preserve"> 1 huius ꝓportio a i ad a h, ſicut t q ad q h.</s> <s xml:id="echoid-s35388" xml:space="preserve"> Quia ergo linea q t eſt ma <lb/>ior ꝗ̃ linea q h, erit linea a i maior ꝗ̃ linea a h:</s> <s xml:id="echoid-s35389" xml:space="preserve"> erit quoq;</s> <s xml:id="echoid-s35390" xml:space="preserve"> ք 20 p 6 ꝓportio quadrati lineę a i ad quadra <lb/>tũ lineę a h, ſicut quadrati lineę t q ad quadratũ lineę h q:</s> <s xml:id="echoid-s35391" xml:space="preserve"> quoniã ſicut ſimpli ad ſimplũ, ſic dupli ad <lb/>duplum:</s> <s xml:id="echoid-s35392" xml:space="preserve"> proportio uerò quadratorum dupla eſt proportioni laterũ ex 20 p 6:</s> <s xml:id="echoid-s35393" xml:space="preserve"> erit ergo per 17 p 5 <lb/>exceſſus quadrati a i ſupra quadratum a h ad quadratum a h, ſicut ductus h t in t n ad quadratum <lb/>q h.</s> <s xml:id="echoid-s35394" xml:space="preserve"> Et quoniam ex 4 p 2 & ex pręmiſsis quadratum lineæ q h quater ſumptum efficit quadratum <lb/>lineæ h d, & ductus h t in n t quater ſumptus efficit triplum quadrati h t:</s> <s xml:id="echoid-s35395" xml:space="preserve"> ideo quòd ductus h t in tn <lb/>eſt tres quartę quadrati h t ut præmiſſum eſt, quater uerò tria ſunt 12, in quibus tria integra conti-<lb/>nentur:</s> <s xml:id="echoid-s35396" xml:space="preserve"> erit ergo per 15 p 5 ductus h t in t n ad quadratum q h, ſicut tripli quadrati h t ad quadra-<lb/>tum h d.</s> <s xml:id="echoid-s35397" xml:space="preserve"> Sit autem h o linea tripla ad lineam h t:</s> <s xml:id="echoid-s35398" xml:space="preserve"> erit ergo per 1 p 6 ductus o h in t h triplus quadrati <lb/>h t:</s> <s xml:id="echoid-s35399" xml:space="preserve"> ſed quoniam ductus a h in h t eſt æqualis quadrato h d, erit per 17 p 6 proportio h a ad h d, ſicut <lb/>h d ad h t:</s> <s xml:id="echoid-s35400" xml:space="preserve"> erit ergo h t ad h a, ſicut quadrati h t ad quadratum h d ex corollarij;</s> <s xml:id="echoid-s35401" xml:space="preserve"> 20 p 6 & 4 p 5.</s> <s xml:id="echoid-s35402" xml:space="preserve"> Ve-<lb/>rùm proportio lineæ o h ad lineam h a eſt, ſicut ductus o h in h t ad ductũ a h in h t ex 1 p 6:</s> <s xml:id="echoid-s35403" xml:space="preserve"> & ita per <lb/>11 p 5 eſt proportio lineæ o h ad lineam h a, ſicut tripli quadrati h t ad quadratum h d:</s> <s xml:id="echoid-s35404" xml:space="preserve"> ſed hæc erat <lb/>proportio exceſſus quadrati lineæ a i ſupra quadratum lineæ a h ad quadratum a h:</s> <s xml:id="echoid-s35405" xml:space="preserve"> eſt ergo con-<lb/>iunctim per 18 p 5 proportio lineæ o a ad lineam h a, ſicut quadrati lineæ a i, ad quadratum a h:</s> <s xml:id="echoid-s35406" xml:space="preserve"> ex-<lb/>ceſſus enim quadrati a i ſupra quadratum a h cum quadrato h a efficit quadratum a i:</s> <s xml:id="echoid-s35407" xml:space="preserve"> igitur ex 20 <lb/>p 6 erit linea i a medio loco proportionalis inter lineas o a & h a:</s> <s xml:id="echoid-s35408" xml:space="preserve"> eſt enim ut in corollario 20 p 6 <lb/>proponitur, trium linearum continuè proportionalium proportio primæ ad tertiam, ſicut quadra-<lb/>ti conſtituti ſuper primam ad quadratum conſtitutum ſuper ſecundam:</s> <s xml:id="echoid-s35409" xml:space="preserve"> igitur proportio lineæ <lb/>o a ad i a eſt ficut lineæ i a ad h a:</s> <s xml:id="echoid-s35410" xml:space="preserve"> erit ergo per 19 p 5 eadem proportio reſidui ad reſiduum, ſcilicet <lb/>o i ad i h.</s> <s xml:id="echoid-s35411" xml:space="preserve"> Cum itaque i a ſit maior quàm a h:</s> <s xml:id="echoid-s35412" xml:space="preserve"> erit o i maior quàm i h:</s> <s xml:id="echoid-s35413" xml:space="preserve"> ergo linea i h eſt minor medie-<lb/>tate lineæ o h.</s> <s xml:id="echoid-s35414" xml:space="preserve"> Item, ut prius oſtenſum eſt, ductus lineæ a h in lineam h d eſt æqualis quartæ parti <lb/>quadrati lineæ a d:</s> <s xml:id="echoid-s35415" xml:space="preserve"> ſed linea a d eſt minor quàm a h:</s> <s xml:id="echoid-s35416" xml:space="preserve"> ductus ergo a d in h d eſt minor quarta parte <lb/>quadrati lineæ a d:</s> <s xml:id="echoid-s35417" xml:space="preserve"> linea ergo h d eſt minor quarta parte lineæ a d.</s> <s xml:id="echoid-s35418" xml:space="preserve"> Quoniam ſi eſſet linea h d æqua-<lb/>lis quartæ parti lineæ a d:</s> <s xml:id="echoid-s35419" xml:space="preserve"> tunc per 1 p 6 ductus a d in h d eſſet æqualis quartæ parti quadrati lineæ <lb/>a d, cum ambo ſint altitudinis lineæ a d:</s> <s xml:id="echoid-s35420" xml:space="preserve"> eſt ergo linea h d minor quinta parte lineæ a h.</s> <s xml:id="echoid-s35421" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s35422" xml:space="preserve"> <lb/>linea a h ſit maior quàm quintupla lineæ h d:</s> <s xml:id="echoid-s35423" xml:space="preserve"> ductus uerò lineæ a h in lineam h t ſit æqualis qua-<lb/>drato lineæ h d, ut pater ex pręmiſsis:</s> <s xml:id="echoid-s35424" xml:space="preserve"> erit per 17 p 6 linea h d maior quàm quintupla lineæ h t:</s> <s xml:id="echoid-s35425" xml:space="preserve"> quo-<lb/>niam quæ eſt proportio lineæ a h ad lineam h d, eadem eſt proportio lineæ h d ad lineam h t:</s> <s xml:id="echoid-s35426" xml:space="preserve"> eſt <lb/>ergo h t minor quinta parte lineæ h d, & h d eſt minor quinta parte lineæ a h:</s> <s xml:id="echoid-s35427" xml:space="preserve"> ergo h t eſt minor 25 <lb/>parte lineæ a h.</s> <s xml:id="echoid-s35428" xml:space="preserve"> Eſt autem ex præmiſsis proportio lineæ o i ad lineam i h, ſicut lineæ i a ad h a:</s> <s xml:id="echoid-s35429" xml:space="preserve"> ergo <lb/>per 18 p 5 erit coniunctim proportio lineæ o h ad lineam i h, ſicut lineę i a cum linea a h ad lineã a h:</s> <s xml:id="echoid-s35430" xml:space="preserve"> <lb/>ergo per 15 p 5 erit proportio tertiæ partis primę lineę ad ſecundam, ſicut tertiæ partis ipſius ter-<lb/>tiæ lineæ ad quartam.</s> <s xml:id="echoid-s35431" xml:space="preserve"> Quia uerò linea h o affumpta eſt tripla lineæ h t:</s> <s xml:id="echoid-s35432" xml:space="preserve"> patet quòd linea h t eſt ter-<lb/>tia pars lineę o h:</s> <s xml:id="echoid-s35433" xml:space="preserve"> eſt ergo proportio lineæ h t ad lineam i h, ſicut tertiæ partis lineæ i a cum tertia <lb/>parte lineę a h ad lineam a h.</s> <s xml:id="echoid-s35434" xml:space="preserve"> Eſt igitur proportio lineę h t ad i a, ſicut duarum tertiarum lineæ a h <lb/> <pb o="249" file="0551" n="551" rhead="LIBER SEXTVS."/> cum una tertia lineę i h ad lineam a h.</s> <s xml:id="echoid-s35435" xml:space="preserve"> Quia enim linea a h bis accipitur, ſemel per ſeipſam & ſemel <lb/>in linea i h:</s> <s xml:id="echoid-s35436" xml:space="preserve"> ergo & eius tertia bis accipitur:</s> <s xml:id="echoid-s35437" xml:space="preserve"> linea uerò i h accipitur ſemel in linea a i:</s> <s xml:id="echoid-s35438" xml:space="preserve"> unde & eius <lb/>tertia eſt tantùm ſemel accipienda.</s> <s xml:id="echoid-s35439" xml:space="preserve"> Quia uerò linea o i eſt maior quàm linea i h, ut ſuprà patuit, & <lb/>linea i h eſt minor medietate lineæ o h:</s> <s xml:id="echoid-s35440" xml:space="preserve"> ergo tertia pars lineę i h erit minor ſexta parte lineæ o h <lb/>per 15 p 5.</s> <s xml:id="echoid-s35441" xml:space="preserve"> Sed cum linea h t ſit tertia pars lineæ o h:</s> <s xml:id="echoid-s35442" xml:space="preserve"> ergo medietas lineę h t eſt æqualis ſextæ parti <lb/>lineæ o h:</s> <s xml:id="echoid-s35443" xml:space="preserve"> eſt ergo tertia pars lineę i h minor medietate lineæ h t:</s> <s xml:id="echoid-s35444" xml:space="preserve"> ergo duę tertię lineę a h cum mi-<lb/>nore parte lineę quàm ſit medietas lineę h t, habent proportionem ad lineam a h illam, quam ha-<lb/>bet linea h t ad lineam i h:</s> <s xml:id="echoid-s35445" xml:space="preserve"> ergo econtrario per 5 th.</s> <s xml:id="echoid-s35446" xml:space="preserve"> 1 huius erit proportio lineę i h ad lineam h t, ſi-<lb/>cut lineę a h ad duas ſui tertias, cum linea minore medietate lineę h t:</s> <s xml:id="echoid-s35447" xml:space="preserve"> eſt autem linea h t, ut patet <lb/>per præmiſſa, minor 25 parte lineæ a h, & eius medietas minor eſt medietate 25 partis lineæ a h:</s> <s xml:id="echoid-s35448" xml:space="preserve"> ſed <lb/>linea a h in 25 partes diuiſa, duæ eius tertiæ cum medietate 25 partis non efficiunt 18 partes ipſius:</s> <s xml:id="echoid-s35449" xml:space="preserve"> <lb/>quoniam duę tertię de 24 ſunt 16, & remanet unum, cuius duę tertię cum illo, quod eſt minus di-<lb/>midio, fortè eſt plus quàm unum integrum, minus autem quàm duo integra.</s> <s xml:id="echoid-s35450" xml:space="preserve"> Igitur proportio lineę <lb/>i h ad lineam h t eſt maior quàm 25 ad 18 per 8 p 5.</s> <s xml:id="echoid-s35451" xml:space="preserve"> Item cum linea h t ſit minor 25 parte lineæ a h:</s> <s xml:id="echoid-s35452" xml:space="preserve"> e-<lb/>rit linea a t maior 24 partibus illarum partium, quarum linea a h eſt 25.</s> <s xml:id="echoid-s35453" xml:space="preserve"> Sed linea i h eſt minor me-<lb/>dietate lineę o h:</s> <s xml:id="echoid-s35454" xml:space="preserve"> eſt autem o h tripla ipſi h t:</s> <s xml:id="echoid-s35455" xml:space="preserve"> ergo linea o h eſt minor una & dimidia partium ex par <lb/>tibus, quarum a h eſt 25:</s> <s xml:id="echoid-s35456" xml:space="preserve"> ergo multò magis linea i h eſt minor una parte & dimidia illarum 25 parti-<lb/>um lineę a h:</s> <s xml:id="echoid-s35457" xml:space="preserve"> eſt ergo proportio lineæ a i ad lineam a t, ſicut lineæ minoris quàm 26 partes & dimi-<lb/>dię ad lineam maiorem quàm 24 partes partium earundem.</s> <s xml:id="echoid-s35458" xml:space="preserve"> Eſt ergo proportio lineę a i ad lineam <lb/>a t minor proportione 26 & dimidię ad 24 ք 8 p 5.</s> <s xml:id="echoid-s35459" xml:space="preserve"> Proportio uerò lineę i h ad lineam h t eſt maior <lb/>quàm 24 partium ad 18:</s> <s xml:id="echoid-s35460" xml:space="preserve"> quoniam ex pręmiſsis ipſa eſt maior quàm 25 partium ad 18.</s> <s xml:id="echoid-s35461" xml:space="preserve"> Igitur propor <lb/>tio lineę i h ad lineam h t eſt maior, quàm proportio lineę i a ad lineam a t:</s> <s xml:id="echoid-s35462" xml:space="preserve"> quoniam minor eſt pro-<lb/>portio 26 & dimidię ad 24, quàm 24 ad 18, quæ eſt ſeſquitertia.</s> <s xml:id="echoid-s35463" xml:space="preserve"> Sit quoq;</s> <s xml:id="echoid-s35464" xml:space="preserve"> per 3 th.</s> <s xml:id="echoid-s35465" xml:space="preserve"> 1 huius proportio <lb/>lineæ i m ad lineam m t, ſicut lineæ i a ad lineam a t.</s> <s xml:id="echoid-s35466" xml:space="preserve"> Eſt ergo maior proportio lineæ i h ad lineã h t, <lb/>quàm lineę i m ad lineam m t:</s> <s xml:id="echoid-s35467" xml:space="preserve"> cadit ergo punctus m inter puncta i & h:</s> <s xml:id="echoid-s35468" xml:space="preserve"> linea ergo m t eſt maior quá <lb/>h m:</s> <s xml:id="echoid-s35469" xml:space="preserve"> ergo per 8 p 5 maior eſt porportio i m ad h m, quàm ad m t:</s> <s xml:id="echoid-s35470" xml:space="preserve"> ergo maior eſt proportio i m ad <lb/>m h, quàm lineæ i a ad a t:</s> <s xml:id="echoid-s35471" xml:space="preserve"> ergo maior proportio i m ad m h, quàm i a ad a h:</s> <s xml:id="echoid-s35472" xml:space="preserve"> quoniam per 8 p 5 ma-<lb/>ior eſt proportio i a ad a t, quàm ad a h, cum a t ſit minor quàm a h.</s> <s xml:id="echoid-s35473" xml:space="preserve"> Sit ergo per 3 th.</s> <s xml:id="echoid-s35474" xml:space="preserve"> 1 huius propor-<lb/>tio lineæ i l ad l h, ſicut lineę i a ad a h:</s> <s xml:id="echoid-s35475" xml:space="preserve"> cadet ergo, ut prius, punctus l inter duo puncta m & i:</s> <s xml:id="echoid-s35476" xml:space="preserve"> quod <lb/>poteſt oſtendi, ſicut prius.</s> <s xml:id="echoid-s35477" xml:space="preserve"> Et his ſic pręmiſsis innouabimus figuram.</s> <s xml:id="echoid-s35478" xml:space="preserve"> Fiat itaque omnimoda diſ-<lb/>poſitio, ut in pręmiſſa figuratione, & in demonſtratione ulterius procedatur.</s> <s xml:id="echoid-s35479" xml:space="preserve"> A punctis itaq;</s> <s xml:id="echoid-s35480" xml:space="preserve"> l & <lb/>m ducantur duę lineæ contingentes circulum d b e per 17 p 3, quę ſint l b & m g:</s> <s xml:id="echoid-s35481" xml:space="preserve"> & copulentur lineę <lb/>i b, h b, i g, t g, a b, a g:</s> <s xml:id="echoid-s35482" xml:space="preserve"> & educantur lineæ a b, a g ad circulum exteriorem, quælibet in punctum z.</s> <s xml:id="echoid-s35483" xml:space="preserve"> <lb/>Quia itaque ex pręmiſsis eſt proportio lineę i l ad lineam l h, ſicut catheti i a ad ſui partem a h:</s> <s xml:id="echoid-s35484" xml:space="preserve"> pa-<lb/>tet per 12 huius quoniam punctus h eſt locus imagi-<lb/>nisformę puncti i reflexę à puncto ſpeculi, quod eſt b:</s> <s xml:id="echoid-s35485" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0551-01a" xlink:href="fig-0551-01"/> quia danti oppoſitum accidit contrarium proportio-<lb/>nis prędemonſtratę lineę i a ad lineam a h:</s> <s xml:id="echoid-s35486" xml:space="preserve"> erit enim <lb/>tunc proportio lineæ i a ad lineam ductam ad locum <lb/>imaginis à puncto a, ſicut lineę i l ad lineam ductam à <lb/>puncto l ad locũ imaginis.</s> <s xml:id="echoid-s35487" xml:space="preserve"> Et quia, ut pręoſtẽſum eſt, <lb/>ꝓportio lineę i l ad lineã h l eſt, ſicut lineę i a ad lineá h <lb/>a:</s> <s xml:id="echoid-s35488" xml:space="preserve"> erit ergo pũctus h locus imaginis:</s> <s xml:id="echoid-s35489" xml:space="preserve"> erit quoq;</s> <s xml:id="echoid-s35490" xml:space="preserve"> angul{us} <lb/>i b z cõtẽtus ſub linea incidẽtię i b, & ſub քpendicula-<lb/>ri a b z ducta à cẽtro ſpeculi ad pũctũ reflexionis, ęqua <lb/>lis angulo h b a, quẽ cõtinet linea reflexionis cũ eadẽ <lb/>քpẽdiculari a b z:</s> <s xml:id="echoid-s35491" xml:space="preserve"> quoniã, ut patet ք 9 huius, illa linea <lb/>reflexionis cõcurrit cũ catheto incidẽtię, quę eſt a i:</s> <s xml:id="echoid-s35492" xml:space="preserve"> u-<lb/>terq;</s> <s xml:id="echoid-s35493" xml:space="preserve"> enim illorũ angulorũ eſt æ qualis cuidã angulo <lb/>reflexiõis, ꝗ, exempli cauſſa, ſit z b x, ita ut cẽtrũ uiſus <lb/>ſit in pũcto x, uel in aliquo puncto illius lineę:</s> <s xml:id="echoid-s35494" xml:space="preserve"> angulo <lb/>itaq;</s> <s xml:id="echoid-s35495" xml:space="preserve"> z b x æquatur angulus i b z ք 20 th.</s> <s xml:id="echoid-s35496" xml:space="preserve"> 5 huius, ք qđ <lb/>oſtẽditur qđ angulus incidẽtię eſt æqualis angulo re-<lb/>flexiõis:</s> <s xml:id="echoid-s35497" xml:space="preserve"> & angulus h b a ęquatur angulo x b z ք 15 p 1.</s> <s xml:id="echoid-s35498" xml:space="preserve"> <lb/>Et ſimiliter cũ punctus h ſit locus imaginis, & linea l b <lb/>ſit cõtingẽs circulũ in pũcto b:</s> <s xml:id="echoid-s35499" xml:space="preserve"> erũt anguli l b z & a b l <lb/>recti per 18 p 3:</s> <s xml:id="echoid-s35500" xml:space="preserve"> ſed angulus i b z eſt æqualis angulo h <lb/>b a:</s> <s xml:id="echoid-s35501" xml:space="preserve"> relinquitur ergo angulus i b l ęqualis angulo l b h.</s> <s xml:id="echoid-s35502" xml:space="preserve"> <lb/>Similiter quoq;</s> <s xml:id="echoid-s35503" xml:space="preserve"> erit angulus i g z æqualis angulo t g a.</s> <s xml:id="echoid-s35504" xml:space="preserve"> Et cũ linea m g ſit cõtingens circulũ in pun-<lb/>cto g, & perpendicularis ſuper ſemidiametrum a g:</s> <s xml:id="echoid-s35505" xml:space="preserve"> erit ſecundum pręmiſſa angulus i g m æqualis <lb/>angulo m g t:</s> <s xml:id="echoid-s35506" xml:space="preserve"> eſt enim ſecundum pręmiſſa pũctus t locus imaginis formę pũcti i reflexę à pũcto ſpe <lb/>culi, quod eſt g.</s> <s xml:id="echoid-s35507" xml:space="preserve"> Item ducatur à puncto h ad lineam a b per 31 p 1 linea æquidiſtans lineę i b, quæ <lb/>ſit h p:</s> <s xml:id="echoid-s35508" xml:space="preserve"> & â puncto t ducatur ſuper lineam a g ęquidiſtans lineę i g, quę ſit t r:</s> <s xml:id="echoid-s35509" xml:space="preserve"> erit ergo per 29 p 1 an-<lb/>gulus i b z ęqualis angulo h p b:</s> <s xml:id="echoid-s35510" xml:space="preserve"> ſed angulus i b z ex pręmiſsis eſt ęqualis angulo h b a:</s> <s xml:id="echoid-s35511" xml:space="preserve"> <lb/> <pb o="250" file="0552" n="552" rhead="VITELLONIS OPTICAE"/> duo ergo anguli h b a & h p b ſunt ęquales:</s> <s xml:id="echoid-s35512" xml:space="preserve"> ergo per 6 p 1 duo latera h b & h p ſunt ęqualia:</s> <s xml:id="echoid-s35513" xml:space="preserve"> & ſi-<lb/>militer ſequitur, quòd duo latera t g & t r ſunt æqualia.</s> <s xml:id="echoid-s35514" xml:space="preserve"> Quia itaque in trigono h p b duo anguli h <lb/>p b & h b p ſunt æquales:</s> <s xml:id="echoid-s35515" xml:space="preserve"> patet per 32 p 1 quoniam uterque ipſorum eſt acutus:</s> <s xml:id="echoid-s35516" xml:space="preserve"> angulus s ergo h p a <lb/>eſt obtuſus:</s> <s xml:id="echoid-s35517" xml:space="preserve"> ergo per 19 p 1 in trigono h a p latus a h eſt maius latere h p:</s> <s xml:id="echoid-s35518" xml:space="preserve"> ergo & linea a h eſt maior <lb/>quàm linea h b:</s> <s xml:id="echoid-s35519" xml:space="preserve"> & ſimiliter erit linea a t maior quàm linea t g.</s> <s xml:id="echoid-s35520" xml:space="preserve"> Amplius quoniam linea h p eſt æ-<lb/>quidiſtãs lineæ i b:</s> <s xml:id="echoid-s35521" xml:space="preserve"> erit per 29 p 1 & per 4 p 6 proportio lineæ a i ad lineã a h, ſicut lineæ a b ad lineã <lb/>a p.</s> <s xml:id="echoid-s35522" xml:space="preserve"> Et ſimiliter cũ linea t r ſit æquidiſtans lineæ i g:</s> <s xml:id="echoid-s35523" xml:space="preserve"> erit proportio lineę a i ad lineã a t, ſicut lineę a g <lb/>ad lineá a r:</s> <s xml:id="echoid-s35524" xml:space="preserve"> ergo erit econtrario per 5 th.</s> <s xml:id="echoid-s35525" xml:space="preserve"> 1 huius proportio lineę a h ad lineã a i, ſicut lineæ a p ad li-<lb/>neã a b:</s> <s xml:id="echoid-s35526" xml:space="preserve"> ſed linea a g eſt æqualis lineæ a b per definitionẽ circuli:</s> <s xml:id="echoid-s35527" xml:space="preserve"> ergo per 7 p 5 ea dẽ eſt proportio li <lb/>nearum a g & a b ad lineam a r:</s> <s xml:id="echoid-s35528" xml:space="preserve"> eſt ergo proportio lineæ a i ad lineam a t, ſicut a b ad a r.</s> <s xml:id="echoid-s35529" xml:space="preserve"> Ablatis er-<lb/>go hinc inde eiſdem medijs, quæ ſunt a i & a b, erit per 22 p 5 proportio lineæ a h ad lineam a t, ſicut <lb/>lineæ a p ad lineam a r.</s> <s xml:id="echoid-s35530" xml:space="preserve"> Verùm cum angulus h p a ſit obtuſus:</s> <s xml:id="echoid-s35531" xml:space="preserve"> palàm per 12 p 2 quia quadratum li-<lb/>neæ a h excedet ambo quadrata linearum h p & a p in eo, quod fit bis ex ductu lineæ a p in lineam <lb/>ductam à puncto p uſque ad locum perpendicularis ductę à puncto h ſuper lineam a p:</s> <s xml:id="echoid-s35532" xml:space="preserve"> ſed perpen-<lb/>dicularis ducta à puncto h ſuper lineam a p productam, neceſſariò cadet in medio lineę p b per 31 <lb/>th.</s> <s xml:id="echoid-s35533" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s35534" xml:space="preserve"> quoniam lineæ h b & h p ſunt æquales:</s> <s xml:id="echoid-s35535" xml:space="preserve"> ergo per 1 p 2 quadratum lineę a h excedit am-<lb/>bo quadrata linearum h p & a p in eo, quod fit ex ductu lineæ a p in lineam p b:</s> <s xml:id="echoid-s35536" xml:space="preserve"> ſed per 3 p 2 il-<lb/>lud, quod fit ex ductu lineę a b in lineam a p, eſt æquale ei, quod fit ex ductu lineæ a p in lineam <lb/>p b & quadrato lineæ a p.</s> <s xml:id="echoid-s35537" xml:space="preserve"> Quadratum ergo lineę a h excedit quadratum lineę h p in eo, quod <lb/>fit ex ductu lineæ a b in lineam a p.</s> <s xml:id="echoid-s35538" xml:space="preserve"> Eodem quoque modo demonſtrandum, quòd quadratum li-<lb/>neę a t excedit quadratum lineę t r in eo, quod fit ex ductu unius linearum a g uel a b in a r:</s> <s xml:id="echoid-s35539" xml:space="preserve"> cum li-<lb/>nea a g ſit æqualis ipſi a b.</s> <s xml:id="echoid-s35540" xml:space="preserve"> Ducatur ergo linea a b in ambas lineas a p & a r, & prouenient duo prę-<lb/>miſsi exceſſus, quorum alterius ad alterum proportio per 1 p 6 eſt ſicut lineę a p ad lineam a r, cum <lb/>ipſorum ſit eadem altitudo, quę eſt lineę a b.</s> <s xml:id="echoid-s35541" xml:space="preserve"> Eſt autem ex pręmiſsis proportio lineę a p ad line-<lb/>am a r, ſicut lineę a h ad lineam a t:</s> <s xml:id="echoid-s35542" xml:space="preserve"> erit ergo proportio exceſſus quadrati a h ſupra quadratum <lb/>h p ad exceſſum quadrati a t ſupra quadratum t r, ſicut lineæ a h ad lineam a t.</s> <s xml:id="echoid-s35543" xml:space="preserve"> Et cum h p ſit <lb/>æqualis ipſi h b, & t r ſit æqualis ipſi t g:</s> <s xml:id="echoid-s35544" xml:space="preserve"> erit proportio exceſſus quadrati a h ſupra quadratum <lb/>h b ad exceſſum quadrati a t ſupra quadratum t g, ſicut lineę a h ad lineam a t.</s> <s xml:id="echoid-s35545" xml:space="preserve"> Quia uerò per <lb/>36 p 3 illud, quod fit ex ductu lineæ e h in h d, eſt æquale quadrato lineę contingentis, ductę à <lb/>puncto h ad circulum d b e, quę per 60 th.</s> <s xml:id="echoid-s35546" xml:space="preserve"> 1 huius, & per 8 p 5 erit minor quàm linea h b:</s> <s xml:id="echoid-s35547" xml:space="preserve"> illud er-<lb/>go, quod fit ex ductu lineę e h in lineam h d eſt minus quadrato lineę h b:</s> <s xml:id="echoid-s35548" xml:space="preserve"> patet ergo quòd il-<lb/>lud, quod fit ex ductu a h in b d, minus eſt quadrato h b.</s> <s xml:id="echoid-s35549" xml:space="preserve"> Fiat ergo per 127 th.</s> <s xml:id="echoid-s35550" xml:space="preserve"> 1 huius ut illud, quod <lb/>fit ex ductu a h in h u maiorem linea h d, ęquale ſit quadrato lineę h b.</s> <s xml:id="echoid-s35551" xml:space="preserve"> Et quoniam linea a h eſt <lb/>maior quàm linea h b, erit quoque a h maior quàm h u:</s> <s xml:id="echoid-s35552" xml:space="preserve"> abſcindatur ergo h u à linea a h per 3 <lb/>p 1 in puncto u:</s> <s xml:id="echoid-s35553" xml:space="preserve"> patetitaque per 2 p 2 quia quadratum lineæ a h eſt æquale ei, quod fit ex ductu li-<lb/>neæ a h in h u, & in a u:</s> <s xml:id="echoid-s35554" xml:space="preserve"> illud ergo, quod fit ex ductu a h in a u, eſt exceſſus quadrati a h ſupra <lb/>quadratum h b.</s> <s xml:id="echoid-s35555" xml:space="preserve"> Eſt ergo proportio lineæ a h ad lineam a t, ſicut eius, quod fit ex ductu a h in <lb/>a u ad exceſſum quadrati a t ſupra quadratum t g.</s> <s xml:id="echoid-s35556" xml:space="preserve"> Si itaque duæ lineæ a h & a t ducantur in li-<lb/>neam a u:</s> <s xml:id="echoid-s35557" xml:space="preserve"> erit per 1 p 6 proportio eius, quod fit ex ductu a h in a u ad illud, quod fit ex ductu <lb/>a t in a u, ſicut lineæ a h ad lineam a t:</s> <s xml:id="echoid-s35558" xml:space="preserve"> ergo per 9 p 5 illud, quod fit ex ductu lineæ a t in a u, eſt <lb/>æquale exceſſui quadrati a t ſupra quadratum t g:</s> <s xml:id="echoid-s35559" xml:space="preserve"> ſed per 2 p 2 quadratum lineæ a t eſt æquale <lb/>ei, quod fit ex ductu a t in a u, & a t in t u:</s> <s xml:id="echoid-s35560" xml:space="preserve"> eſt ergo illud, quod fit ex ductu a t in t u æquale <lb/>quadrato t g.</s> <s xml:id="echoid-s35561" xml:space="preserve"> Palàm ergo quoniam ductus lineæ a h in h u eſt æqualis quadrato h b, & ductus <lb/>a t in t u eſt æqualis quadrato t g.</s> <s xml:id="echoid-s35562" xml:space="preserve"> Item arcus b g diuidatur per æqualia in puncto o per 30 p 3:</s> <s xml:id="echoid-s35563" xml:space="preserve"> <lb/>ducaturq́;</s> <s xml:id="echoid-s35564" xml:space="preserve"> linea a o:</s> <s xml:id="echoid-s35565" xml:space="preserve"> & à punctis b & o & g ducantur tres perpendiculares ſuper lineam a h per <lb/>12 p 1 ſcilicet b f, o y, g k:</s> <s xml:id="echoid-s35566" xml:space="preserve"> & à puncto g ducatur linea æquidiſtans lineæ a h per 31 p 1, quæ ſit g s:</s> <s xml:id="echoid-s35567" xml:space="preserve"> <lb/>& à puncto b ducatur perpendicularis ſuper lineam a g, quæ ſit b c:</s> <s xml:id="echoid-s35568" xml:space="preserve"> & hæc quidem b c ſi pro-<lb/>duceretur ad peripheriam circuli, diuideret ipſam linea a g in duo æqualia per 3 p 3:</s> <s xml:id="echoid-s35569" xml:space="preserve"> & ſimiliter <lb/>diuideret arcum, cuius chorda eſſet producta b c, per æqualia in puncto g:</s> <s xml:id="echoid-s35570" xml:space="preserve"> & ita ſecaretur alius <lb/>arcus æqualis arcui b g:</s> <s xml:id="echoid-s35571" xml:space="preserve"> quoniam in illum arcum caderet angulus c b g:</s> <s xml:id="echoid-s35572" xml:space="preserve"> & ita angulus c b g eſt <lb/>medietas anguli, qui ſuper centrum a caderet in illum arcum per 20 p 3:</s> <s xml:id="echoid-s35573" xml:space="preserve"> ſed ille angulus per 27 p 3 <lb/>eſt æqualis angulo g a b:</s> <s xml:id="echoid-s35574" xml:space="preserve"> quoniam cadunt in arcus æquales ſuper centrum a:</s> <s xml:id="echoid-s35575" xml:space="preserve"> igitur angulus c b g <lb/>eſt medietas anguli g a b:</s> <s xml:id="echoid-s35576" xml:space="preserve"> eſt ergo per 27 p 3 angulus c b g æqualis angulo o a g.</s> <s xml:id="echoid-s35577" xml:space="preserve"> Duo autem an-<lb/>guli b s g & b c g ſunt recti:</s> <s xml:id="echoid-s35578" xml:space="preserve"> ergo per 31 p 3 ſi imaginetur circulus, cuius diameter ſit b g, tranſi-<lb/>ens per punctum s:</s> <s xml:id="echoid-s35579" xml:space="preserve"> ille neceſſariò tranſibit per punctum c:</s> <s xml:id="echoid-s35580" xml:space="preserve"> & fiet arcus c s, in quem cadent duo <lb/>anguli c b s & c g s:</s> <s xml:id="echoid-s35581" xml:space="preserve"> ergo hi duo anguli per 27 p 3 ſunt æquales:</s> <s xml:id="echoid-s35582" xml:space="preserve"> ſed angulus g a y æqualis eſt an-<lb/>gulo c g s per 29 p 1, quoniam lineę g s & a y ęquidiſtant:</s> <s xml:id="echoid-s35583" xml:space="preserve"> eſt ergo angulus g a y æqualis an-<lb/>gulo c b s:</s> <s xml:id="echoid-s35584" xml:space="preserve"> ut autem prius oſtenſum eſt, angulus c b g eſt ęqualis angulo o a g:</s> <s xml:id="echoid-s35585" xml:space="preserve"> ergo totalis an-<lb/>gulus o a y ęqualis totali angulo g b s:</s> <s xml:id="echoid-s35586" xml:space="preserve"> ſed anguli a y o & g s b ſunt recti:</s> <s xml:id="echoid-s35587" xml:space="preserve"> eſt ergo trigonum <lb/>y a o ęquiangulum trigono g b s:</s> <s xml:id="echoid-s35588" xml:space="preserve"> ergo per quartam pr.</s> <s xml:id="echoid-s35589" xml:space="preserve"> ſexti eſt proportio lineę g b ad lineam b s, <lb/>ficut lineę o a ad lineam a y, & proportio g b ad g s, ſicut a o ad o y.</s> <s xml:id="echoid-s35590" xml:space="preserve"> Itẽ quia angulus a h b eſt acutus <lb/>per quadrageſimumſecundum th.</s> <s xml:id="echoid-s35591" xml:space="preserve"> primi huius, palàm per decimamtertiam pr.</s> <s xml:id="echoid-s35592" xml:space="preserve"> ſecundi, quia qua-<lb/>dratum lineę a b minus eſt ambobus quadratis linearum a h & h b in eo, quod fit ex ductu lineę a <lb/>h in lineam h f bis:</s> <s xml:id="echoid-s35593" xml:space="preserve"> igitur quadratum lineę a h cum quadrato lineę h b, maius eſt quadrato lineę a b, <lb/> <pb o="251" file="0553" n="553" rhead="LIBER SEXTVS."/> uel quadrato eius æqualis, quę eſt a d, in eo, quod fit ex ductu lineę a h in lineam h f bis:</s> <s xml:id="echoid-s35594" xml:space="preserve"> ſed il-<lb/>lud, quod fit ex ductu a h in h f bis eſt per 1 p 2 ęquale ei, quod ſit ex ductu a h in h d bis, & ex du-<lb/>ctu a h in d f bis:</s> <s xml:id="echoid-s35595" xml:space="preserve"> illud autem, quod fit ex ductu a h in h d bis, cum quadrato lineę a d, eſt ęquale <lb/>quadrato lineæ a h cum quadrato lineę h d per 7 p 2:</s> <s xml:id="echoid-s35596" xml:space="preserve"> quadratũ ergo lineę a d cũ eo, quod fit ex du-<lb/>ctu a h in h d bis, quia eſt commune utrobiq;</s> <s xml:id="echoid-s35597" xml:space="preserve">, auſeratur:</s> <s xml:id="echoid-s35598" xml:space="preserve"> remanet ergo quadratũ lineę d h, quod cũ <lb/>eo, quod fit ex ductu lineę a h in f d bis, æquale quadrato lineę h b.</s> <s xml:id="echoid-s35599" xml:space="preserve"> Sed ex præmiſsis patet, quò il-<lb/>lud, quod fit ex ductu a h in h t, eſt æquale quadrato h d, & illud quod fit ex ductu a h in h u eſt æqua <lb/>le quadrato h b:</s> <s xml:id="echoid-s35600" xml:space="preserve">erit ergo ductus a h in h u æqualis ductui a h in h t ſemel & bis in d f:</s> <s xml:id="echoid-s35601" xml:space="preserve"> ablato ergo du <lb/>ctu a h in h t, qui communis ponitur utrobiq;</s> <s xml:id="echoid-s35602" xml:space="preserve">:relinquitur, ut illud, quod fit ex ductu a h in tu ſemel <lb/>ſit æquale ei, quod fit ex ductu a h in d f bis.</s> <s xml:id="echoid-s35603" xml:space="preserve"> Ergo per 1 p 6 erit linea tu dupla lineæ d f.</s> <s xml:id="echoid-s35604" xml:space="preserve"> Item cú an-<lb/>gulus a t g ſit acutus, erit ſecundum prędictum modum quadratum lineę a t cum quadrato lineę t g <lb/>æquale quadrato lineę a d, & ei quod fit ex ductu a t in t k bis, & ita ei, quod fit ex ductu a t in d t bis <lb/>& in d k bis:</s> <s xml:id="echoid-s35605" xml:space="preserve">remanebitq́;</s> <s xml:id="echoid-s35606" xml:space="preserve"> ut prius, quadratum lineę t g æquale quadrato lineæ t d, & ei, quod fit ex <lb/>ductu a t in d k bis.</s> <s xml:id="echoid-s35607" xml:space="preserve"> Sit autem per 10 p 6 ut quę eſt proportio a t ad t d, eadem ſit ipſius t d ad t æ:</s> <s xml:id="echoid-s35608" xml:space="preserve"> er-<lb/>go per 17 p 6 illud, quod fit ex ductu a t in t æ eſt æ quale quadrato t d:</s> <s xml:id="echoid-s35609" xml:space="preserve"> ſed ex pręmiſsis illud, quod <lb/>fit ex ductu a t in tu, eſt æquale quadrato t g:</s> <s xml:id="echoid-s35610" xml:space="preserve"> ablato ergo utrobiq;</s> <s xml:id="echoid-s35611" xml:space="preserve"> eo, quod fit ex ductu a t in t æ, re-<lb/>ſtat, ut illud, quod fit ex ductu a t in æ u ſemel, ſit æquale ei, quod fit ex ductu a t in d k bis:</s> <s xml:id="echoid-s35612" xml:space="preserve">igitur per <lb/>1 p 6 linea æ u eſt dupla lineę d k:</s> <s xml:id="echoid-s35613" xml:space="preserve"> ſed iam oſten ſum eſt quòd t u eſt dupla ipſi d f:</s> <s xml:id="echoid-s35614" xml:space="preserve"> reſtat ergo ut linea <lb/>æ t ſit dupla lineę k f.</s> <s xml:id="echoid-s35615" xml:space="preserve"> Item quia ex pręmiſsis illud, quod fit ex ductu a h in h t eſt æquale quadrato h <lb/>d:</s> <s xml:id="echoid-s35616" xml:space="preserve"> ergo per 17 p 6 erit proportio a h ad h d, ſicuth d ad h t:</s> <s xml:id="echoid-s35617" xml:space="preserve">eſt ergo proportio lineę a h ad h t propor-<lb/>tio duplicata lineæ a h ad h d:</s> <s xml:id="echoid-s35618" xml:space="preserve"> & ſimiliter per eandem rationem proportio at ad t æ eſt duplicata <lb/>proportio a t ad t d:</s> <s xml:id="echoid-s35619" xml:space="preserve">ſed maior eſt proportio a t ad t d, quàm a h ad h d per 4 th.</s> <s xml:id="echoid-s35620" xml:space="preserve"> 1 huius, quoniam e-<lb/>iuſdem lineę, quę t h, prioribus antecedenti & conſequenti ſit additio:</s> <s xml:id="echoid-s35621" xml:space="preserve">ergo maior eſt proportio li-<lb/>neæ a t ad lineę t æ, quàm lineę a h ad lineam at:</s> <s xml:id="echoid-s35622" xml:space="preserve"> ergo per 10 th.</s> <s xml:id="echoid-s35623" xml:space="preserve"> 1 huius erit permutatim maior pro-<lb/>portio lineę a t ad lineam a h, quâm lineę t æ ad lineam h t:</s> <s xml:id="echoid-s35624" xml:space="preserve"> ſed a h eſt maior quàm a t, quoniam to-<lb/>tum eſt maius parte:</s> <s xml:id="echoid-s35625" xml:space="preserve"> ergo h t eſt maior quàm t æ:</s> <s xml:id="echoid-s35626" xml:space="preserve"> ſed t æ eſt dupla ad f k, ut patuit ſuperius:</s> <s xml:id="echoid-s35627" xml:space="preserve"> ergo h t <lb/>eſt magis quàm dupla ad f k.</s> <s xml:id="echoid-s35628" xml:space="preserve"> Item, ut ſuprà demonſtratum eſt, proportio b g ad g s eſt, ſicut o a ad <lb/>o y:</s> <s xml:id="echoid-s35629" xml:space="preserve"> ergo permutatim per 16 p 5 erit proportio b g ad o a, ſicut g s ad o y:</s> <s xml:id="echoid-s35630" xml:space="preserve"> ſed o a eſt æqualis ipſi b a <lb/>per circuli definitionem, & g s eſt æqualis ipſi f k per 34 p 1:</s> <s xml:id="echoid-s35631" xml:space="preserve"> erit ergo per 7 p 5 proportio b g ad b a, <lb/>ficut f k ad o y.</s> <s xml:id="echoid-s35632" xml:space="preserve"> Item quia, ut prius quaſi in principio patuit, linea i h eſt minor medietate lineę <lb/>o h, & linea o h eſt tripla lineę h t:</s> <s xml:id="echoid-s35633" xml:space="preserve"> erit ergo linea i h minor quàm linea h t, & quàm ipſius medie-<lb/>tas:</s> <s xml:id="echoid-s35634" xml:space="preserve"> ſed linea h t eſt minor quinta parte lineę h d, ut prius declaratum eſt, ergo linea i h eſt mi-<lb/>nor quàm linea t d:</s> <s xml:id="echoid-s35635" xml:space="preserve"> ſed linea n d eſt maior quam t d:</s> <s xml:id="echoid-s35636" xml:space="preserve"> ergo i h eſt multò minor quàm n d:</s> <s xml:id="echoid-s35637" xml:space="preserve"> eſt au-<lb/>tem m i minor quàm i h:</s> <s xml:id="echoid-s35638" xml:space="preserve"> ergo m i eſt multo minor quàm n d:</s> <s xml:id="echoid-s35639" xml:space="preserve"> & quoniam z h eſt æqualis ipſi h d, <lb/>ut pręmiſſum eſt:</s> <s xml:id="echoid-s35640" xml:space="preserve"> patet quòd punctum i cadet inter duo puncta h & z:</s> <s xml:id="echoid-s35641" xml:space="preserve"> ergo & punctum m cadit in-<lb/>ter duo puncta h & z.</s> <s xml:id="echoid-s35642" xml:space="preserve"> Item illud, quod fit ex ductu e z in z d ſuppoſitum eſt æquale eſſe quadrato <lb/>ſemidiametri a d:</s> <s xml:id="echoid-s35643" xml:space="preserve"> igitur illud, quod fit ex ductu e m in m d eſt minus quadrato a d:</s> <s xml:id="echoid-s35644" xml:space="preserve"> eſt autem id, <lb/>quod fit ex ductu e m in m d, æquale quadraro lineę contingentis circulum, quę m g, per 36 p 3:</s> <s xml:id="echoid-s35645" xml:space="preserve"> <lb/>quadratum ergo lineę m g eſt minus quadrato lineę a d:</s> <s xml:id="echoid-s35646" xml:space="preserve"> ergo linea a d eſt maior quàm linea m g-<lb/>Igitur linea m g eſt minor quàm linea a g, quę eſt æqualis ipſi lineę a d, cum ſint ſemidiametri eiuſ-<lb/>dem circuli.</s> <s xml:id="echoid-s35647" xml:space="preserve"> Et quia duo trigonia g m & m g k habent unum angulum a m g communem:</s> <s xml:id="echoid-s35648" xml:space="preserve"> ſed & <lb/>angulus a g m eſt rectus per 18 p 3, & angulus m k g eſt rectus per definitionem perpendicularis:</s> <s xml:id="echoid-s35649" xml:space="preserve"> <lb/>ergo per 32 p 1 illi trigoni ſunt æquianguli:</s> <s xml:id="echoid-s35650" xml:space="preserve"> ergo per 4 p 6 eſt proportio m k ad k g, ſicut m g ad g a:</s> <s xml:id="echoid-s35651" xml:space="preserve"> <lb/>ſed m g eſt minor quàm a g, utiam patuit:</s> <s xml:id="echoid-s35652" xml:space="preserve"> ergo m k eſt minor quàm k g:</s> <s xml:id="echoid-s35653" xml:space="preserve"> ſed k g eſt minor quàm <lb/>o y per 15 p 3:</s> <s xml:id="echoid-s35654" xml:space="preserve"> & h d eſt minor quàm m k:</s> <s xml:id="echoid-s35655" xml:space="preserve"> erit ergo h d minor quàm k g:</s> <s xml:id="echoid-s35656" xml:space="preserve"> erit ergo h d minor <lb/>quàm o y.</s> <s xml:id="echoid-s35657" xml:space="preserve"> Et quia per pręmiſſa & per 17 p 6 eſt proportio a h ad h d, ſicut h d ad ht:</s> <s xml:id="echoid-s35658" xml:space="preserve"> cum ita-<lb/>que linea h q ſit medietas lineę h d:</s> <s xml:id="echoid-s35659" xml:space="preserve"> erit per 15 p 5 proportio lineę a h ad lineam h q, ficut li-<lb/>neę h d ad medietatem lineę h t:</s> <s xml:id="echoid-s35660" xml:space="preserve"> patuit autem ſuprà quòd linea h t eſt magis quàm dupla lineę <lb/>k f:</s> <s xml:id="echoid-s35661" xml:space="preserve"> & linea h d eſt minor quàm linea o y:</s> <s xml:id="echoid-s35662" xml:space="preserve"> eſt ergo maior proportio medietatis lineę h t ad line-<lb/>am h d, quàm lineę f k ad lineam o y per 9 th.</s> <s xml:id="echoid-s35663" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s35664" xml:space="preserve"> eſt ergo per 11 p 5, & per 5 th.</s> <s xml:id="echoid-s35665" xml:space="preserve"> 1 huius pro-<lb/>portio q h ad a h maior quàm f k ad o y.</s> <s xml:id="echoid-s35666" xml:space="preserve"> Item linea a q ſecat circulum e b d:</s> <s xml:id="echoid-s35667" xml:space="preserve"> ſit punctus ſectionis œ:</s> <s xml:id="echoid-s35668" xml:space="preserve"> <lb/>& ducatur chorda d œ, quę propter æquidiſtantiam arcuum h q, d œ, erit æquidiſtans chordę h q <lb/>per 43 th.</s> <s xml:id="echoid-s35669" xml:space="preserve"> 1 huius, & per 28 p 1:</s> <s xml:id="echoid-s35670" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s35671" xml:space="preserve"> per 29 p 1, & per 4 p 6 proportio h q ad a h, ſicut d œ ad a d:</s> <s xml:id="echoid-s35672" xml:space="preserve"> <lb/>ſed proportio h q ad h a eſt maior quàm f k ad o y:</s> <s xml:id="echoid-s35673" xml:space="preserve"> erit ergo proportio d œ ad d a, maior quàm fk <lb/>ad o y:</s> <s xml:id="echoid-s35674" xml:space="preserve"> eſt autem ex pręmiſsis f k ad o y, ſicut g b ad a b:</s> <s xml:id="echoid-s35675" xml:space="preserve"> eſt ergo maior proportio œ d ad d a, quàm <lb/>b g ad b a:</s> <s xml:id="echoid-s35676" xml:space="preserve"> ſed d a eſt æqualis ipſi b a, quia ſemidiametri:</s> <s xml:id="echoid-s35677" xml:space="preserve"> ergo per 10 p 5 chorda œ d eſt maior quàm <lb/>chorda b g:</s> <s xml:id="echoid-s35678" xml:space="preserve"> ergo per 28 p 3 erit arcus d œ maior arcu b g.</s> <s xml:id="echoid-s35679" xml:space="preserve"> Producatur item linea a q extra circu-<lb/>lum ad punctum s, donec per 3 p 1 fiat a s æ qualis lineæ a i:</s> <s xml:id="echoid-s35680" xml:space="preserve"> & copuletur lineæ s i, quæ per 7 p 5, <lb/>& per 2 p 6 erit æquidiſtans lineæ h q:</s> <s xml:id="echoid-s35681" xml:space="preserve"> ergo per 29 p 1 & per 4 p 6 erit proportio s i ad h q, ſicut <lb/>i a ad a h:</s> <s xml:id="echoid-s35682" xml:space="preserve"> eſt autem præoſtenſum quòd eſt proportio i a ad a h, ſicut t q ad q h:</s> <s xml:id="echoid-s35683" xml:space="preserve"> ergo per 9 p 5 <lb/>linea s i eſt æqualis lineæ t q:</s> <s xml:id="echoid-s35684" xml:space="preserve"> cum ipſarum ambarum ad lineam q h eadem ſit proportio, quę li-<lb/>neę i a ad lineam a b.</s> <s xml:id="echoid-s35685" xml:space="preserve"> Quia uerò numerus aſſumendarum linearum excedit multipliciter nume-<lb/>rum literarum latinarum, ne fortè fiat intricatio in omnibus ipſarum linearum, mutetur figu-<lb/>ra.</s> <s xml:id="echoid-s35686" xml:space="preserve"> Et quoniam linea nouiter aſſumpta, quę eſt a s, poſita eſt ęqualis lineę a i, fiat circulus ſuper <lb/> <pb o="252" file="0554" n="554" rhead="VITELLONIS OPTICAE"/> centrum a ſecundum ipſarum quantitatẽ, & loco s ponatur litera n:</s> <s xml:id="echoid-s35687" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s35688" xml:space="preserve"> circulus d g b ſimilis prio-<lb/>ri circulo, qui d b e, & producátur lineæ a b & a g uſq;</s> <s xml:id="echoid-s35689" xml:space="preserve"> ad circulũ exteriorẽ in puncta c & r:</s> <s xml:id="echoid-s35690" xml:space="preserve"> & ſint li-<lb/>neæ a b c & a g r, permutẽturq́;</s> <s xml:id="echoid-s35691" xml:space="preserve"> lineæ a i & a s, ita ut linea a d i ſit loco lineę a œ s, & loco lineę a d i ſit <lb/>linea a f n:</s> <s xml:id="echoid-s35692" xml:space="preserve"> ponaturq́;</s> <s xml:id="echoid-s35693" xml:space="preserve"> loco literę s litera n, & loco literę œ ponatur f:</s> <s xml:id="echoid-s35694" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s35695" xml:space="preserve"> ut pręoſtenſum eſt, arcus <lb/>d f maior arcu g b.</s> <s xml:id="echoid-s35696" xml:space="preserve"> Sit ergo arcus b m æqualis arcui d f, quod fiet per 33 p 6, ſi prius per 23 p 1 ſuper a <lb/>terminum lineę a b fiat angulus æ qualis angulo d a f, qui ſit b a m:</s> <s xml:id="echoid-s35697" xml:space="preserve"> producatur quoque linea a m ad <lb/>exteriorem perιpheriam in pũctum u:</s> <s xml:id="echoid-s35698" xml:space="preserve"> & ſit a m u:</s> <s xml:id="echoid-s35699" xml:space="preserve"> ducantur etiã lineę i b, i g, i m, n m, q m:</s> <s xml:id="echoid-s35700" xml:space="preserve"> quæ pro-<lb/>ducatur uſq;</s> <s xml:id="echoid-s35701" xml:space="preserve"> ad exteriorẽ circulũ:</s> <s xml:id="echoid-s35702" xml:space="preserve"> & cadat in pũctum z:</s> <s xml:id="echoid-s35703" xml:space="preserve"> & ducantur lineę z a, z g.</s> <s xml:id="echoid-s35704" xml:space="preserve"> Cũ itaq;</s> <s xml:id="echoid-s35705" xml:space="preserve"> arcus b m <lb/>ſit æqualis arcui d f, addito cõmuni arcu d m, erit arcus m fęqualis arcui d b:</s> <s xml:id="echoid-s35706" xml:space="preserve"> ergo per 27 p 3 erit an-<lb/>gulus n a m æqualis angulo i a b.</s> <s xml:id="echoid-s35707" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s35708" xml:space="preserve"> trigonorum n a m, i a b duo latera unius ſunt æqualia <lb/>duobus lateribus alterius, & angulus angulo:</s> <s xml:id="echoid-s35709" xml:space="preserve"> ergo per 4 p 1 erit linea n m æ qualis lineæ i b, & angu <lb/> <anchor type="figure" xlink:label="fig-0554-01a" xlink:href="fig-0554-01"/> lus n m a æqualis angulo i b a:</s> <s xml:id="echoid-s35710" xml:space="preserve"> remanet <lb/>ergo ք 13 p 1 angulus n m u æ qualis an-<lb/>gulo i b c.</s> <s xml:id="echoid-s35711" xml:space="preserve"> Et cũ in pręmiſſa proxima fi-<lb/>guratione linea a h fuerit poſita æqualis <lb/>ipſi lineæ a q:</s> <s xml:id="echoid-s35712" xml:space="preserve"> erunt trigonorum q a m & <lb/>a h b duo latera a q & a m æqualia duo-<lb/>bus lateribus a h & a b, & angulus q a m <lb/>eſt æqualis angulo h a b:</s> <s xml:id="echoid-s35713" xml:space="preserve"> erit ergo per 4 <lb/>p 1 linea q m æqualis lineę h b:</s> <s xml:id="echoid-s35714" xml:space="preserve"> & angu-<lb/>lus q m a æqualis angulo h b a:</s> <s xml:id="echoid-s35715" xml:space="preserve"> remanet <lb/>ergo angulus q m n ęqualis angulo h b <lb/>i:</s> <s xml:id="echoid-s35716" xml:space="preserve"> & angulus q m u æ qualis angulo h b c <lb/>per 13 p 1.</s> <s xml:id="echoid-s35717" xml:space="preserve"> Et quia lineæ a n & ai ſunt æ-<lb/>quales ք definitionẽ circuli, & linea a q <lb/>eſt æqualis ipſi a h ex hypotheſi:</s> <s xml:id="echoid-s35718" xml:space="preserve"> rema-<lb/>netlinea n q æqualis lineę i h.</s> <s xml:id="echoid-s35719" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s35720" xml:space="preserve"> <lb/>angulus n m u eſt ęqualis angulo i b c, <lb/>& angulus i b c, ut præoſtẽſum eſt, ęqua <lb/>lis eſt angulo h b a:</s> <s xml:id="echoid-s35721" xml:space="preserve"> angulus uerò h b a <lb/>eſt æqualis angulo q m a:</s> <s xml:id="echoid-s35722" xml:space="preserve"> erit angulus n m u æqualis angulo q m a.</s> <s xml:id="echoid-s35723" xml:space="preserve"> Patet etiam quòd linea m z tota <lb/>eſt extra circulum:</s> <s xml:id="echoid-s35724" xml:space="preserve"> quia cum linea contingens circulum ducta à puncto b cadat inter puncta i & h, <lb/>ut pręoſtendimus:</s> <s xml:id="echoid-s35725" xml:space="preserve"> & quia eſt eadẽ remotio puncti b à puncto h, quę puncti m à puncto q:</s> <s xml:id="echoid-s35726" xml:space="preserve"> quoniam <lb/>oſtenſum eſt, quod linea b h eſt ęqualis lineę q m, & linea i h eſt æqualis lineę n q:</s> <s xml:id="echoid-s35727" xml:space="preserve"> patet quod cõtin-<lb/>gens ducta à puncto m cadet inter puncta n & q.</s> <s xml:id="echoid-s35728" xml:space="preserve"> Igitur cũ linea q m cadat ſub linea cõtingente, pa-<lb/>tet per 16 p 3 quoniã ipſa ſecat circulũ:</s> <s xml:id="echoid-s35729" xml:space="preserve"> eſt ergo tota linea m z extra circulũ:</s> <s xml:id="echoid-s35730" xml:space="preserve"> quoniã linea q m z poſi <lb/>ta eſt eſſe linea una recta:</s> <s xml:id="echoid-s35731" xml:space="preserve"> propter qđ etiã erit per 15 p 1 angulus q m a æqualιs angulo u m z:</s> <s xml:id="echoid-s35732" xml:space="preserve"> ſed an-<lb/>gulus n m u oſtenſus eſt eſſe æqualis angulo q m a:</s> <s xml:id="echoid-s35733" xml:space="preserve"> erit ergo angulus n m u æqualis angulo u m z:</s> <s xml:id="echoid-s35734" xml:space="preserve"> <lb/>ergo per 8 huius forma puncti n reflectitur à puncto ſpeculi m ad uιſum exiſtentem in puncto z:</s> <s xml:id="echoid-s35735" xml:space="preserve"> & <lb/>erit per 11 huius locus imaginis punctus q.</s> <s xml:id="echoid-s35736" xml:space="preserve"> Itẽ quia angulus n m u eſt æqualis angulo u m z:</s> <s xml:id="echoid-s35737" xml:space="preserve">erũt per <lb/>ſuppoſitionẽ 1 huius lineę n m, z m ęqualiter diſtãtes à diametro a u:</s> <s xml:id="echoid-s35738" xml:space="preserve"> ergo per 7 p 3 ipſę ſunt æqua-<lb/>les.</s> <s xml:id="echoid-s35739" xml:space="preserve"> Ducantur itaq;</s> <s xml:id="echoid-s35740" xml:space="preserve"> lineę n u & z u, quę per 4 p 1 perunt æquales, cõmuni exiſtente linea m u ambo-<lb/>bus trigonis n m u, & z m u:</s> <s xml:id="echoid-s35741" xml:space="preserve">ergo ք 28 p 3 arcus n u eſt ęqualis arcui u z:</s> <s xml:id="echoid-s35742" xml:space="preserve"> ergo per 27 p 3 angulus n a u <lb/>eſt æqualis angulo u a z.</s> <s xml:id="echoid-s35743" xml:space="preserve"> Sed ex pręmiſsis patet qđ angulus n a u eſt æqualis angulo i a c:</s> <s xml:id="echoid-s35744" xml:space="preserve"> erit ergo <lb/>angulus i a c ęqualis angulo u a z.</s> <s xml:id="echoid-s35745" xml:space="preserve"> Angulus uerò b a g aut erit ęqualis angulo g a m, aut minor, aut <lb/>maior:</s> <s xml:id="echoid-s35746" xml:space="preserve"> ſit primò ęqualis.</s> <s xml:id="echoid-s35747" xml:space="preserve"> Siigitur ab angulo i a b ſubtrahatur angulus b a g, & ab angulo z a u angu-<lb/>lus g a m, remanebit angulus i a g ęqualis angulo z a g:</s> <s xml:id="echoid-s35748" xml:space="preserve"> & quia duo latera i a & a g ſunt ęqualia duo-<lb/>bus lateribus z a & ag:</s> <s xml:id="echoid-s35749" xml:space="preserve"> ergo per 4 p 1 erit linea i g ęqualis lineę z g, & angulus i g a ęqualis angulo <lb/>z g a:</s> <s xml:id="echoid-s35750" xml:space="preserve"> ergo per 13 p 1 angulus i g r eſt ęqualis angulo z g r.</s> <s xml:id="echoid-s35751" xml:space="preserve"> Fiat itaq;</s> <s xml:id="echoid-s35752" xml:space="preserve"> ſuper g terminum lineę a g angu-<lb/>lus ęqualis angulo i g r per 23 p 1, qui ſit angulus t g a, ducta linea g t ſuper lineam i a:</s> <s xml:id="echoid-s35753" xml:space="preserve">erit ergo angu-<lb/>lus t g a ęqualis angulo z g r.</s> <s xml:id="echoid-s35754" xml:space="preserve"> Si igitur linea t g producatur ad peripheriam circuli:</s> <s xml:id="echoid-s35755" xml:space="preserve"> palàm per 15 p 1 <lb/>quoniam ipſa perueniet ad punctum z:</s> <s xml:id="echoid-s35756" xml:space="preserve"> lineę enim z g & t g coniunctę in puncto g fiunt linea una <lb/>per 14 p 1:</s> <s xml:id="echoid-s35757" xml:space="preserve"> eſt ergo t g z linea una recta.</s> <s xml:id="echoid-s35758" xml:space="preserve"> Forma ergo puncti i reflectitur à puncto ſpeculi g ad uiſum <lb/>exiſtentem in puncto z:</s> <s xml:id="echoid-s35759" xml:space="preserve"> & locus imaginis eius eſt punctum t.</s> <s xml:id="echoid-s35760" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s35761" xml:space="preserve"> quoniã ad uiſum exiſten-<lb/>tem in puncto z reflectuntur formę duorum punctorum n & i à duobus pũctis ſpeculi ſphęrici con <lb/>uexi, quę ſunt m & g:</s> <s xml:id="echoid-s35762" xml:space="preserve"> & loca imaginum ſunt puncta t & q.</s> <s xml:id="echoid-s35763" xml:space="preserve"> Igitur per 11 huius linea t q erit imago to-<lb/>tius lineę in:</s> <s xml:id="echoid-s35764" xml:space="preserve"> probatum eſt autem ſuprà quòd linea t q eſt ęqualis lineę n i:</s> <s xml:id="echoid-s35765" xml:space="preserve"> palàm ergo quoniam ac-<lb/>cidit in his ſpeculis imaginem eſſe ęqualem rei uiſę.</s> <s xml:id="echoid-s35766" xml:space="preserve"> Quod eſt unum propoſitorum.</s> <s xml:id="echoid-s35767" xml:space="preserve"> quòd ſi angu-<lb/>lus b a g fuerit maior angulo g a m:</s> <s xml:id="echoid-s35768" xml:space="preserve"> abſtrahatur b a g ab angulo i ab, & angulus g a m ab angulo z a u <lb/>ęqualis angulo i a b:</s> <s xml:id="echoid-s35769" xml:space="preserve"> remanebit ergo angulus z a g maior angulo i a g.</s> <s xml:id="echoid-s35770" xml:space="preserve"> Sit ergo angulus k a g ęqualis <lb/>angulo i a g per 23 p 1, ducta linea à centro ad circumferentiam in pũctum k, & copuletur linea k g:</s> <s xml:id="echoid-s35771" xml:space="preserve"> <lb/>erit quoque angulus k a g minor angulo z a g:</s> <s xml:id="echoid-s35772" xml:space="preserve"> punctum ergo k erit altius puncto z, & punctũ m eſt <lb/>altius puncto g:</s> <s xml:id="echoid-s35773" xml:space="preserve"> linea ergo k g ſecabit lineam z m.</s> <s xml:id="echoid-s35774" xml:space="preserve"> Sit, ut ſecet ipſam in puncto l:</s> <s xml:id="echoid-s35775" xml:space="preserve"> & producatur k g ſu <lb/>per lineam i a in punctum t:</s> <s xml:id="echoid-s35776" xml:space="preserve"> fiat quo q;</s> <s xml:id="echoid-s35777" xml:space="preserve"> deductio, ut ſtatim in proxima linea t g.</s> <s xml:id="echoid-s35778" xml:space="preserve"> Palàm ergo quod ui-<lb/> <pb o="253" file="0555" n="555" rhead="LIBER SEXTVS."/> ſu exiſtente in puncto l, reflectetur ad ipſum forma puncti n à puncto m:</s> <s xml:id="echoid-s35779" xml:space="preserve"> & locus imaginis erit q:</s> <s xml:id="echoid-s35780" xml:space="preserve"> & <lb/> <anchor type="figure" xlink:label="fig-0555-01a" xlink:href="fig-0555-01"/> ſimiliter ad ipſum reflectetur forma <lb/>puncti i à puncto g, & locus imaginis <lb/>erit t ſecundum priorem probationẽ:</s> <s xml:id="echoid-s35781" xml:space="preserve"> <lb/>erit quoque linea t q imago lineæ n i, <lb/>quæ eſt ęqualis ipſi, ut ſupra oſtenſum <lb/>eſt:</s> <s xml:id="echoid-s35782" xml:space="preserve"> & ſic ſequitur idem propoſitũ qđ <lb/>prius.</s> <s xml:id="echoid-s35783" xml:space="preserve"> Si uerò angulus b a g fuerit mi-<lb/>nor angulo g a m, erit, ut ſupra, angu-<lb/>lus z a g minor angulo i a g.</s> <s xml:id="echoid-s35784" xml:space="preserve"> Sit ergo <lb/>angulus o a g ducta linea a o ad peri-<lb/>pheriam circuli æqualis angulo i a g:</s> <s xml:id="echoid-s35785" xml:space="preserve"> <lb/>erit ergo angulus o a g maior angulo <lb/>z a g:</s> <s xml:id="echoid-s35786" xml:space="preserve"> eſt ergo punctũ o inferius pũcto <lb/>z:</s> <s xml:id="echoid-s35787" xml:space="preserve"> & producatur linea o g, quę incidat <lb/>lineæ i a in puncto t.</s> <s xml:id="echoid-s35788" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s35789" xml:space="preserve"> quòd <lb/>forma pũcti i reflectitur ad uiſum exi-<lb/>ſtentẽ in puncto o à puncto ſpeculi g.</s> <s xml:id="echoid-s35790" xml:space="preserve"> <lb/>Linea itaq;</s> <s xml:id="echoid-s35791" xml:space="preserve"> o g aut ſecabit lineã z m q <lb/>extra circulũ ſpeculi, aut non:</s> <s xml:id="echoid-s35792" xml:space="preserve"> ſi ſit poſsibile, ſecet ipſam extra circulũ.</s> <s xml:id="echoid-s35793" xml:space="preserve"> Si in puncto ſectionis fuerit <lb/>uiſus, reflectentur ad ipſum duæ formę punctorũ n & i à pũctis ſpeculi m & g, & loca imaginũ erũt <lb/>puncta q & t:</s> <s xml:id="echoid-s35794" xml:space="preserve"> & tota linea q t imago totius lineæ n i, & erit per præmiſſa æqualis ei:</s> <s xml:id="echoid-s35795" xml:space="preserve"> patet itaq;</s> <s xml:id="echoid-s35796" xml:space="preserve"> hoc <lb/>quod prius:</s> <s xml:id="echoid-s35797" xml:space="preserve"> quoniã imago rei uidebitur in hoc ſitu æqualis ipſi rei.</s> <s xml:id="echoid-s35798" xml:space="preserve"> Si forte linea o g ſecet lineam z <lb/>m q intra circulum ſpeculi:</s> <s xml:id="echoid-s35799" xml:space="preserve"> tunc nõ poteſt accedere probatio pręmiſſa, ſed extra totalẽ hanc ſuper-<lb/>ficiem eſt poſsibile inueniri punctũ, in quo poſito uiſu reflectantur ad ipſum formę duorũ puncto-<lb/>rũ n & i à duobus punctis ſpeculi, & ipſorũ imagines erunt puncta q & t.</s> <s xml:id="echoid-s35800" xml:space="preserve"> Quoniã enim, ut patet ex <lb/>prius pręoſtenſis, angulus n a z eſt duplus angulo i a b:</s> <s xml:id="echoid-s35801" xml:space="preserve"> quoniã eſt duplus angulo n a u æquali angu <lb/>lo i a b, ut patet ex pręmiſsis:</s> <s xml:id="echoid-s35802" xml:space="preserve"> & angulus i a o eſt duplus angulo i a g:</s> <s xml:id="echoid-s35803" xml:space="preserve"> eſt aũt angulus i a b maior angu <lb/>lo i a gin angulo g a b.</s> <s xml:id="echoid-s35804" xml:space="preserve"> Et quia angulus g a b eſt ex hypotheſi minor angulo m a g:</s> <s xml:id="echoid-s35805" xml:space="preserve"> patet quòd angu-<lb/>lus g a b eſt minor medietate anguli m a b:</s> <s xml:id="echoid-s35806" xml:space="preserve"> totus uero angulus m a b eſt per 33 p 6 æqualis angulo n <lb/>a i, quoniã arcus d f eſt æqualis arcui m b:</s> <s xml:id="echoid-s35807" xml:space="preserve"> ergo angulus g a b eſt minor medietate anguli n a i:</s> <s xml:id="echoid-s35808" xml:space="preserve"> angu-<lb/>lus ergo n a z excedens angulũ i a o in duplo anguli g a b, nõ excedet ipſum in angulo maiori quàm <lb/>ſit angulus n a i:</s> <s xml:id="echoid-s35809" xml:space="preserve"> duo ergo anguli n a i & n a z ſunt maiores tertio, qui eſt i a o:</s> <s xml:id="echoid-s35810" xml:space="preserve"> & duo anguli n a z & <lb/>i a o ſunt maiores tertio, qui eſt n a i:</s> <s xml:id="echoid-s35811" xml:space="preserve"> & duo anguli i a o & n a i ſunt maiores tertio, qui eſt n a z:</s> <s xml:id="echoid-s35812" xml:space="preserve"> ſunt <lb/>ergo iſti tres anguli n a i, n a z, & i a o, quorũ quilibet duo ſunt maiores tertio, oẽs aũt tres ſimul 4 <lb/>rectis ſunt minores:</s> <s xml:id="echoid-s35813" xml:space="preserve">quoniã angulos, qui ſuper centrũ a 4 rectis ſunt æquales, ipſos impoſsibile eſt <lb/> <anchor type="figure" xlink:label="fig-0555-02a" xlink:href="fig-0555-02"/> euacuare, ut patet.</s> <s xml:id="echoid-s35814" xml:space="preserve"> Igitur per 23 p 11 p oſsibi <lb/>le eſt ex illis fieri unũ angulũ ſolidũ:</s> <s xml:id="echoid-s35815" xml:space="preserve"> fiat er <lb/>go ille ſuper centrũ a ք eandẽ 23 p 11:</s> <s xml:id="echoid-s35816" xml:space="preserve"> & ſit li <lb/>nea s a eleuata ſuper ſuperficiem circuli in <lb/>puncto a taliter, ut angulus i a s ſit æqualis <lb/>angulo i a o, & angulus n a s ſit æqualis an-<lb/>gulo n a z, angulus uerò n a i maneat, ut eſt <lb/>in ſuperficie circuli immotus.</s> <s xml:id="echoid-s35817" xml:space="preserve"> Fiat itaq;</s> <s xml:id="echoid-s35818" xml:space="preserve"> li-<lb/>nea a s æqualis a licui linearũ a n, uel a i, uel <lb/>a o, quæ oẽs ſunt æquales, quia ſunt ſemi-<lb/>diametri eiuſdẽ circuli:</s> <s xml:id="echoid-s35819" xml:space="preserve"> & ꝓducátur lineæ <lb/>t s, q s.</s> <s xml:id="echoid-s35820" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s35821" xml:space="preserve"> angulus tas eſt æqualis <lb/>angulo t a o, ut patet ex pręmiſsis, & duo <lb/>latera t a & a o ſunt æ qualia duobus lateri-<lb/>bus t a & a s, & angulus ta o eſt ęqualis an-<lb/>gulo t a s, ut patet ex pręmiſsis:</s> <s xml:id="echoid-s35822" xml:space="preserve"> erit ք 4 p 1 <lb/>baſis t s æ qualis baſi t o, & totus triangu-<lb/>19 toti triãgulo:</s> <s xml:id="echoid-s35823" xml:space="preserve"> erit ergo angulus o t a uel <lb/>g t a ęqualis angulo s t a.</s> <s xml:id="echoid-s35824" xml:space="preserve"> Similiter quoque <lb/>angulus q a s eſt æqualis angulo q a z, & duo latera duob.</s> <s xml:id="echoid-s35825" xml:space="preserve"> laterib.</s> <s xml:id="echoid-s35826" xml:space="preserve"> erit ergo, ut prius, angulus z q a ꝗ <lb/>eſt m q a, ęqualis angulo s q a.</s> <s xml:id="echoid-s35827" xml:space="preserve"> Diuidatur itaq;</s> <s xml:id="echoid-s35828" xml:space="preserve"> angulus t a s ք æ qualia ք lineã a y ex 9 p 1:</s> <s xml:id="echoid-s35829" xml:space="preserve"> & ſit y pun <lb/>ct{us}, in quo linea diuidẽs angulũ, ſecat lineã t s:</s> <s xml:id="echoid-s35830" xml:space="preserve"> palã cũ angulus i a g ſit medietas anguli i a o, ut patet <lb/>ex p̃miſsis, erit angulus t a g ęqualis angulo t a y:</s> <s xml:id="echoid-s35831" xml:space="preserve"> ſed & angulus g t a oſtẽſus eſt ęqualis angulo y t a.</s> <s xml:id="echoid-s35832" xml:space="preserve"> <lb/>Et ꝗa duob.</s> <s xml:id="echoid-s35833" xml:space="preserve"> trigonis y t a & g t a latus t a eſt cõmune, erit ք 26 p 1 trigonus y t a ęqualis trigono g t a:</s> <s xml:id="echoid-s35834" xml:space="preserve"> <lb/>quoniã latus t y erit æquale lateri t g, & latus a y æquale lateri a g:</s> <s xml:id="echoid-s35835" xml:space="preserve">erit ergo pũctus y in ſuperficie ſpe <lb/>culi, ſicut & punctũ g:</s> <s xml:id="echoid-s35836" xml:space="preserve"> cũ ambo æqualiter diſtẽt à cẽtro ſpeculi, q đ eſt a.</s> <s xml:id="echoid-s35837" xml:space="preserve"> Et quia angulus t a g eſt æ-<lb/>qualis angulo t a y, erit angulus i a g æqualis angulo i a y, & latera lateribus ſunt æqualia:</s> <s xml:id="echoid-s35838" xml:space="preserve"> quoniã i a <lb/>eſt commune, & a y eſt æquale ipſi a g:</s> <s xml:id="echoid-s35839" xml:space="preserve"> ergo ք 4 p 1 erit angulus a g i æqualis angulo a y i, & linea i y <lb/> <pb o="254" file="0556" n="556" rhead="VITELLONIS OPTICAE"/> producta erit æqualis lineæ i g.</s> <s xml:id="echoid-s35840" xml:space="preserve"> Et producatur a y extra ſpeculũ uſq;</s> <s xml:id="echoid-s35841" xml:space="preserve"> ad punctũ p:</s> <s xml:id="echoid-s35842" xml:space="preserve"> reſtat ergo angu-<lb/>lus i g r ęqualis angulo i y p.</s> <s xml:id="echoid-s35843" xml:space="preserve"> Verùm cũ linea t s ſit ęqualis lineæ t o, ut ſuprà patuit, & t y ęqualis ipſi-<lb/>t g:</s> <s xml:id="echoid-s35844" xml:space="preserve"> reſtat linea g o æ qualis lineæ y s:</s> <s xml:id="echoid-s35845" xml:space="preserve"> duo ergo latera a y & y s funt æqualia duobus lateribus a g, & <lb/>g o, & baſis a s eſt æqualis baſi a o:</s> <s xml:id="echoid-s35846" xml:space="preserve"> ergo per 8 p 1 trigonorũ a y s, a g o anguli æquis lateribus conten <lb/>ti ſunt æquales:</s> <s xml:id="echoid-s35847" xml:space="preserve"> angulus ergo a y s eſt æqualis angulo a g o:</s> <s xml:id="echoid-s35848" xml:space="preserve"> reſtat ergo per 13 p 1 angulus s y p æqua <lb/>lis angulo o g r:</s> <s xml:id="echoid-s35849" xml:space="preserve">igitur duo anguhi g r & o g r æquales ſunt duobus angulis i y p, s y p.</s> <s xml:id="echoid-s35850" xml:space="preserve"> Verùm linea <lb/>a s ſecat ſuperficiẽ cõuexã ſpeculi:</s> <s xml:id="echoid-s35851" xml:space="preserve"> ſit pũctus ſectiõis e:</s> <s xml:id="echoid-s35852" xml:space="preserve"> tria ergo pũcta, quęſunte, y, d ſunt in ſuperfi <lb/>cie cõuexa ſpeculi:</s> <s xml:id="echoid-s35853" xml:space="preserve"> lineæ ergo a centro ſpeculi, quod eſt a, ad illa tria puncta productæ ſunt æqua-<lb/>les.</s> <s xml:id="echoid-s35854" xml:space="preserve"> Quia uerò trigonũ t a s eſt per 2 p 11 totũ in eadẽ ſuperficie:</s> <s xml:id="echoid-s35855" xml:space="preserve"> patet quòd iſta tria pũcta d, y, e, quę <lb/>ſunt in lateribus illius trigoni, ſunt in eadẽ ſuperficie:</s> <s xml:id="echoid-s35856" xml:space="preserve"> ergo linea e y d eſt per 9 p 3 arcus circuli ma-<lb/>gni ſphæræ ſpeculi, cuius centrũ eſt a centrũ ſpeculi:</s> <s xml:id="echoid-s35857" xml:space="preserve">eſt aũt in ſuperficie reflexionis cõmunis ſectio <lb/>ſuքpficiei ſpeculi & reflexionis t s p ք 1 huius:</s> <s xml:id="echoid-s35858" xml:space="preserve"> ergo forma pũcti i reflectitur ad uiſum exiſtẽtẽ in pun <lb/>cto s à pũcto ſpeculi y:</s> <s xml:id="echoid-s35859" xml:space="preserve"> & locus imaginis eſt pũctũ t.</s> <s xml:id="echoid-s35860" xml:space="preserve"> Similιter diuiſo angulo n a s per ęqualia ք lineã <lb/>a x ductã ſuper q s in punctũ x, & productã extra ſpeculi ſuperficiẽ in punctũ œ, demõſtrabitur prę <lb/>dicto modo, quia linea q x erιt æqualis lineæ q m, & linea a x æqualis lineæ a m, & linea x s æqualis <lb/>lineæ m z:</s> <s xml:id="echoid-s35861" xml:space="preserve"> & duo anguli n x œ, & s x œ erunt æquales duob.</s> <s xml:id="echoid-s35862" xml:space="preserve"> angulis n m u, & z m u:</s> <s xml:id="echoid-s35863" xml:space="preserve"> & ita forma pun <lb/>cti n reflectetur ad uiſum exiſtentẽ in pũcto s à pũcto ſpeculi x:</s> <s xml:id="echoid-s35864" xml:space="preserve"> & locus imaginis eſt punctũ q:</s> <s xml:id="echoid-s35865" xml:space="preserve"> & ita <lb/>ut prius, formæ duorũ punctorũ n & ireflectuntur à duobus pũctis ſpeculi x & y ad uiſum exiſten-<lb/>tẽ in puncto s:</s> <s xml:id="echoid-s35866" xml:space="preserve"> & erit linea t q imago lineæ i n:</s> <s xml:id="echoid-s35867" xml:space="preserve"> eſt aũt linea t q æqualis lineæ in.</s> <s xml:id="echoid-s35868" xml:space="preserve"> Patet ergo propoſi-<lb/>tũ, ut prius.</s> <s xml:id="echoid-s35869" xml:space="preserve"> Itẽ ſi à pũcto i ducatur perpendicularis ſuper lineã n a, illa cadet inter puncta n & q, nõ <lb/>extra punctũ n:</s> <s xml:id="echoid-s35870" xml:space="preserve">quia cũ per 42 th.</s> <s xml:id="echoid-s35871" xml:space="preserve"> 1 huius angulus in a ſit acutus, ſi caderet extra puncũ n, fieret acu <lb/>tus extrinſecus recto, & ita maior per 16 p 1:</s> <s xml:id="echoid-s35872" xml:space="preserve"> quod eſt impoſsibile:</s> <s xml:id="echoid-s35873" xml:space="preserve"> cadet ergo illa perpẽdicularis ci-<lb/>tra punctũ n:</s> <s xml:id="echoid-s35874" xml:space="preserve"> faciet ergo illa perpendicularis angulũ rectũ ſuper lineã n q, quẽ reſpiciet linea in:</s> <s xml:id="echoid-s35875" xml:space="preserve"> er-<lb/>go ք 19 p 1 erit linea in maior illa perpẽdiculari:</s> <s xml:id="echoid-s35876" xml:space="preserve"> ergo illa perpẽdicularis erit minor quàm linea t q, <lb/>quę eſt æqualis lineę in.</s> <s xml:id="echoid-s35877" xml:space="preserve"> Pũctus itaq;</s> <s xml:id="echoid-s35878" xml:space="preserve"> lineę n q, in quẽcadit illa perpendicularis, qui fit k, refle ctitur <lb/>ad uiſum in puncto s exiſtentẽ ab aliquo puncto ſpeculi:</s> <s xml:id="echoid-s35879" xml:space="preserve"> & locus imaginis ſuæ erit in linea n a per <lb/>11 huius:</s> <s xml:id="echoid-s35880" xml:space="preserve"> erit aũt remotior à cẽtro ſpeculi, quod eſt a, ultra punctũ q, quàm ſit ipſum punctũ q, ut pa-<lb/>tet per 17 huius.</s> <s xml:id="echoid-s35881" xml:space="preserve"> Quantò enim remotiora ſunt puncta, quorũ formę reflectuntur à ſpeculis ſphæri-<lb/>cis cõuexis, tantò locaimaginũ magis accedunt ad centrũ ſpeculi:</s> <s xml:id="echoid-s35882" xml:space="preserve"> ſed punctus i illius perpendicu-<lb/>laris refle ctitur ad uiſum à pũcto ſpeculi y:</s> <s xml:id="echoid-s35883" xml:space="preserve">& locus ſuę imaginis eſt punctũ t.</s> <s xml:id="echoid-s35884" xml:space="preserve"> Quæcunq;</s> <s xml:id="echoid-s35885" xml:space="preserve"> uerò linea <lb/>ducitur à pũcto t ad aliquod punctũ lineæ n q ultra q, propius ad punctũ n, ut linea t k, illa cũ oppo <lb/>natur angulo obtuſo, ut patet, erit per 19 p 1 maior quàm linea t q:</s> <s xml:id="echoid-s35886" xml:space="preserve"> ergo etiam erit maior quàm linea <lb/>in, quæ eſt maior illa perpendiculari, cuius imago uiſui occurrit.</s> <s xml:id="echoid-s35887" xml:space="preserve"> Patet ergo quòd imago illius per-<lb/>pendicularis erit maior ipſa perpendiculari.</s> <s xml:id="echoid-s35888" xml:space="preserve"> Et idẽ accidit, quæcunq;</s> <s xml:id="echoid-s35889" xml:space="preserve"> linea ducatur à puncto i ad li <lb/>neam n q, inter illã perpendicularẽ i k & lineã in:</s> <s xml:id="echoid-s35890" xml:space="preserve"> erit enim ſemperlinea in maior illa linea per 47 <lb/>uel 19 p 1:</s> <s xml:id="echoid-s35891" xml:space="preserve"> & imago illius lineę ſemper erit maior quàm linea q t:</s> <s xml:id="echoid-s35892" xml:space="preserve"> & ita ſemper eritimago ipſius ma-<lb/>ior quàm ipſa.</s> <s xml:id="echoid-s35893" xml:space="preserve"> Quod eſt propoſitũ.</s> <s xml:id="echoid-s35894" xml:space="preserve"> Poſſunt aũt hęc clarius patefieri.</s> <s xml:id="echoid-s35895" xml:space="preserve"> Quia enim forma puncti n re-<lb/>flectitur ad uiſum exiſtentẽ in pũcto z à puncto ſpeculi m:</s> <s xml:id="echoid-s35896" xml:space="preserve"> & locus imaginis eſt punctũ q:</s> <s xml:id="echoid-s35897" xml:space="preserve"> patet qđ <lb/>linea reflexionis, quę eſt z m q, ſecat circulũ:</s> <s xml:id="echoid-s35898" xml:space="preserve"> ſit punctũ ſectionis z:</s> <s xml:id="echoid-s35899" xml:space="preserve"> patet ergo quòd contingens du <lb/>cta à puncto z ad circulũ, qui eſt cõmunis ſectio ſuperficiei reflexionis & ſpeculi, non poteſt cadere <lb/>in punctũm:</s> <s xml:id="echoid-s35900" xml:space="preserve"> quia per 21 huius angulus a m z oportet quò ſit maior recto, quod eſſet contra 18 p 3, <lb/>ſi linea z m eſſet circulũ contingens:</s> <s xml:id="echoid-s35901" xml:space="preserve"> neq;</s> <s xml:id="echoid-s35902" xml:space="preserve"> poteſt cadere in punctũ z, quia ibi ſecat & non contingit:</s> <s xml:id="echoid-s35903" xml:space="preserve"> <lb/>cadet ergo in aliquod punctũ arcus m e, & ꝓducta ad lineã n a, cadet altius quàm punctũ q:</s> <s xml:id="echoid-s35904" xml:space="preserve"> quoniã <lb/>punctus, in quẽ cadit, dicitur finis cõtingẽtiæ, qui ſit n:</s> <s xml:id="echoid-s35905" xml:space="preserve"> & eſt meta imaginũ, ut patet ք 7 definitionẽ:</s> <s xml:id="echoid-s35906" xml:space="preserve"> <lb/>huius & puncta ſub illo pũcto, ꝗ eſt meta imaginũ exiſtentia, non poterunt reflecti ad uiſum, ſupe-<lb/>riora uero illo poterunt reflecti.</s> <s xml:id="echoid-s35907" xml:space="preserve"> Igitur perpendicularis ducta à puncto i ſuper lineam n q, ſi cecide-<lb/>rit altius puncto n, qui eſt meta imaginũ, poteſt reflecti ad uiſum pũctus ille lineæ n q, in quẽ ipſa ք-<lb/>pendicularis cadit:</s> <s xml:id="echoid-s35908" xml:space="preserve"> & erit, ut pręmiſſum eſt, imago perpendicularis maior ipſa perpẽdiculari.</s> <s xml:id="echoid-s35909" xml:space="preserve"> Si ue-<lb/>rò perpẽdicularis cadatin ipſum punctum n, qui eſt meta imaginum, uel inferius illo:</s> <s xml:id="echoid-s35910" xml:space="preserve"> tunc forma <lb/>pũcti, in quẽ cadit perpẽdicularis, nõ reflectetur:</s> <s xml:id="echoid-s35911" xml:space="preserve">quare nulla erit imago ipſius քpẽdicularis:</s> <s xml:id="echoid-s35912" xml:space="preserve"> uerun <lb/>tamẽ quando n finis cõtingentiæ eſt inferior quàm linea i n, & plus ad centrũ:</s> <s xml:id="echoid-s35913" xml:space="preserve"> erunt inter pũctum, <lb/>qui eſt finis contingentiæ, & punctũ n infinita pũcta, quorũ quodlibet reflectitur ad uiſum:</s> <s xml:id="echoid-s35914" xml:space="preserve"> & ima-<lb/>go cuiuslibet erit ſuper lineã n q:</s> <s xml:id="echoid-s35915" xml:space="preserve"> & cuiuslibet lineæ ductę à pũcto i ad quodlibet illorũ, erit imago <lb/>maior illa linea, cuius eſt imago.</s> <s xml:id="echoid-s35916" xml:space="preserve"> Patet ergo propoſitũ longis ambagibus certius per quiſitum.</s> <s xml:id="echoid-s35917" xml:space="preserve"/> </p> <div xml:id="echoid-div1454" type="float" level="0" n="0"> <figure xlink:label="fig-0550-01" xlink:href="fig-0550-01a"> <variables xml:id="echoid-variables616" xml:space="preserve">o z i l h m n q t d a b e</variables> </figure> <figure xlink:label="fig-0551-01" xlink:href="fig-0551-01a"> <variables xml:id="echoid-variables617" xml:space="preserve">o z i l s m h n q t d z a <gap/> k c g y <gap/> f r s b z u a d x x e</variables> </figure> <figure xlink:label="fig-0554-01" xlink:href="fig-0554-01a"> <variables xml:id="echoid-variables618" xml:space="preserve">u r c z i h n m g b q f a</variables> </figure> <figure xlink:label="fig-0555-01" xlink:href="fig-0555-01a"> <variables xml:id="echoid-variables619" xml:space="preserve">i u r k c z l n d t m g b q f a</variables> </figure> <figure xlink:label="fig-0555-02" xlink:href="fig-0555-02a"> <variables xml:id="echoid-variables620" xml:space="preserve">r c u z i o n k q t d b m g b p g f a x e s æ</variables> </figure> </div> </div> <div xml:id="echoid-div1456" type="section" level="0" n="0"> <head xml:id="echoid-head1118" xml:space="preserve" style="it">39. In omni diſtãtia, qua certa quãtitasrei à uiſu potect cõprehẽdi, imago cuiuslibet rei uiſæ in <lb/>ſpeculo ſphærico cõuexo minor uidetur ꝗ̃ forma rei extra. Eucl. 21 th. catoptr. Alhazen 5 n 6.</head> <p> <s xml:id="echoid-s35918" xml:space="preserve">Sit a b linea uiſa:</s> <s xml:id="echoid-s35919" xml:space="preserve"> & ſit z x arcus circuli, qui eſt communis ſectio ſuperficiei reflexionis ſpeculi <lb/>ſphærici conuexi, cuius centrum d:</s> <s xml:id="echoid-s35920" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s35921" xml:space="preserve"> e centrum uiſus:</s> <s xml:id="echoid-s35922" xml:space="preserve"> & reflectatur forma puncti a ad uiſum e à <lb/>puncto reflexionis h arcus z x:</s> <s xml:id="echoid-s35923" xml:space="preserve"> & forma puncti b à puncto n:</s> <s xml:id="echoid-s35924" xml:space="preserve"> intelligaturq́;</s> <s xml:id="echoid-s35925" xml:space="preserve"> linea a b produci intra <lb/>ſpeculum.</s> <s xml:id="echoid-s35926" xml:space="preserve"> Aut er go ipſa tranſit centrum ſpeculi:</s> <s xml:id="echoid-s35927" xml:space="preserve"> aut non.</s> <s xml:id="echoid-s35928" xml:space="preserve"> Sit autem primò, quòd tranſeat:</s> <s xml:id="echoid-s35929" xml:space="preserve"> & duca-<lb/>tur linea a b d:</s> <s xml:id="echoid-s35930" xml:space="preserve"> ducatur quoq;</s> <s xml:id="echoid-s35931" xml:space="preserve"> à puncto n linea contingens circulũ, quæ ſit n l:</s> <s xml:id="echoid-s35932" xml:space="preserve"> & à puncto h ducatur <lb/>contingens, quæ h m:</s> <s xml:id="echoid-s35933" xml:space="preserve"> & ducantur lineæ incidentię & reflexionis, quę ſint b n, e n, a h, e h:</s> <s xml:id="echoid-s35934" xml:space="preserve"> producan <lb/> <pb o="255" file="0557" n="557" rhead="LIBER SEXTVS."/> turq́;</s> <s xml:id="echoid-s35935" xml:space="preserve"> lineæ reflexionis e h & e n, donec cadan t in perpendicularem a d:</s> <s xml:id="echoid-s35936" xml:space="preserve"> & incidat linea e h in pun-<lb/> <anchor type="figure" xlink:label="fig-0557-01a" xlink:href="fig-0557-01"/> ctum t, & linea e n in punctum q.</s> <s xml:id="echoid-s35937" xml:space="preserve"> Palàm ergo per 11 huius quoniam t <lb/>eſt locus imaginis formæ puncti a:</s> <s xml:id="echoid-s35938" xml:space="preserve"> & q eſt locus imaginis formę pun <lb/>cti b.</s> <s xml:id="echoid-s35939" xml:space="preserve"> Dico quòd linea a b eſt maior quàm linea q t.</s> <s xml:id="echoid-s35940" xml:space="preserve"> Patet enim ex12 <lb/>huius quia proportio a d ad d t eſt, ſicut a m ad m t.</s> <s xml:id="echoid-s35941" xml:space="preserve"> Similiter per ean-<lb/>dem proportio b d ad d q eſt ſicut proportio b l ad l q:</s> <s xml:id="echoid-s35942" xml:space="preserve"> ſed a d eſt ma-<lb/>ior quàm b d, & d t eſt minor quàm d q:</s> <s xml:id="echoid-s35943" xml:space="preserve">ergo per 9 th.</s> <s xml:id="echoid-s35944" xml:space="preserve"> 1 huius maior e-<lb/>rit proportio a d ad dt, quàm b d ad d q:</s> <s xml:id="echoid-s35945" xml:space="preserve"> ergo per 11 p 5 maior erit ꝓ-<lb/>portio a m ad m t, quàm b l ad q l.</s> <s xml:id="echoid-s35946" xml:space="preserve"> Secetur ergo linea a m, in puncto f <lb/>per 3 th.</s> <s xml:id="echoid-s35947" xml:space="preserve"> 1 huius, ita ut proportio f m ad m t ſit ſicut b l ad l q:</s> <s xml:id="echoid-s35948" xml:space="preserve"> & ita cum <lb/>m t ſit maior quàm l q:</s> <s xml:id="echoid-s35949" xml:space="preserve"> erit per 14 p 5 f m maior quàm b l:</s> <s xml:id="echoid-s35950" xml:space="preserve"> ergo ք 8 p 5 <lb/>erit f m ad t m maior proportio quàm b l ad t m:</s> <s xml:id="echoid-s35951" xml:space="preserve"> erit ergo minor pro-<lb/>portio b l ad m t, quàm b l ad l q:</s> <s xml:id="echoid-s35952" xml:space="preserve"> & multò magis erit minor propor-<lb/>tio b m ad m t, quam b l ad q l.</s> <s xml:id="echoid-s35953" xml:space="preserve"> Secetur ergo m t in puncto k taliter, ut <lb/>proportio b m ad m k ſit ſicut b l ad l q.</s> <s xml:id="echoid-s35954" xml:space="preserve"> Palàm ergo pernaturam pro-<lb/>portionis, & per 8 p 5 quoniam punctus k neceſlariò cadetinter pun <lb/>cta m & q:</s> <s xml:id="echoid-s35955" xml:space="preserve"> linea enim l q minor eſt quàm m q, & linea b l eſt maior ꝗ̃ <lb/>linea b m.</s> <s xml:id="echoid-s35956" xml:space="preserve"> Cum igitur ſit proportio f m ad m t, ſicut b l ad l q, & ſicut <lb/>b m ad m k:</s> <s xml:id="echoid-s35957" xml:space="preserve"> erit per 19 p 5 proportio f b ad k t, ſicut b l ad l q:</s> <s xml:id="echoid-s35958" xml:space="preserve"> ſed b l eſt <lb/>maior quàm l q:</s> <s xml:id="echoid-s35959" xml:space="preserve"> ergo f b eſt maior quàm k t:</s> <s xml:id="echoid-s35960" xml:space="preserve"> ſed f b eſt minor quàm <lb/>a b, & k t eſt maior quàm q t.</s> <s xml:id="echoid-s35961" xml:space="preserve"> Si ergo f b eſt maior quàm k t:</s> <s xml:id="echoid-s35962" xml:space="preserve"> ergo mul-<lb/>tò fortius a b eſt maior quàm q t.</s> <s xml:id="echoid-s35963" xml:space="preserve"> Et hoc eſt propo ſitum.</s> <s xml:id="echoid-s35964" xml:space="preserve"> Si uerò linea a b producta nó perueniat ad <lb/> <anchor type="figure" xlink:label="fig-0557-02a" xlink:href="fig-0557-02"/> <anchor type="figure" xlink:label="fig-0557-03a" xlink:href="fig-0557-03"/> cẽtrum d:</s> <s xml:id="echoid-s35965" xml:space="preserve"> <lb/>ducatur à <lb/>puncto a li <lb/>nea ad cen <lb/>trũ d, quæ <lb/>ſit a d:</s> <s xml:id="echoid-s35966" xml:space="preserve"> & à <lb/>pũcto b du <lb/>catur b d:</s> <s xml:id="echoid-s35967" xml:space="preserve"> <lb/>& locus i-<lb/>maginis a <lb/>ſit pũctus <lb/>g:</s> <s xml:id="echoid-s35968" xml:space="preserve"> locus i-<lb/>maginis b <lb/>ſit pũctus <lb/>p:</s> <s xml:id="echoid-s35969" xml:space="preserve"> & duca-<lb/>tur linea p <lb/>g:</s> <s xml:id="echoid-s35970" xml:space="preserve"> erit ergo <lb/>linea p g i-<lb/>magolineę <lb/>a b.</s> <s xml:id="echoid-s35971" xml:space="preserve"> Dico quia a b eſt maior ꝗ̃ p g.</s> <s xml:id="echoid-s35972" xml:space="preserve"> Aut enim p g eſt æqui diſtãs lineæ a b:</s> <s xml:id="echoid-s35973" xml:space="preserve"> aut nõ.</s> <s xml:id="echoid-s35974" xml:space="preserve"> Si fuerit æquidiſtãs, <lb/>palàm quia p g eſt minor ꝗ̃ a g per 29 p 1 & per 4 p 6:</s> <s xml:id="echoid-s35975" xml:space="preserve"> cũ ſit proportio a b ad p g, ſicut a d ad d g, & a d <lb/>ſit maior quàm d g, erit a b maior quàm p g.</s> <s xml:id="echoid-s35976" xml:space="preserve"> Si uerò linea p g non ſit æquidiſtãs ip ſi a b, producatur <lb/>uſq;</s> <s xml:id="echoid-s35977" xml:space="preserve"> quo concurrat cum a b:</s> <s xml:id="echoid-s35978" xml:space="preserve"> & ſit punctus concurſus z:</s> <s xml:id="echoid-s35979" xml:space="preserve"> & à puncto p ducatur æquidiſtans a b, quæ <lb/>ſit p h:</s> <s xml:id="echoid-s35980" xml:space="preserve"> angulus ergo p g h ſi ſit rectus uel maior recto, erit per 19 p 1 latus p h maius latere p g:</s> <s xml:id="echoid-s35981" xml:space="preserve"> ſed p h <lb/>eſt minus quàm a b per 29 p 1.</s> <s xml:id="echoid-s35982" xml:space="preserve"> 4 p 6:</s> <s xml:id="echoid-s35983" xml:space="preserve"> ergo p g eſt minus quàm a b.</s> <s xml:id="echoid-s35984" xml:space="preserve"> Si angulus p g h fuerit acutus, ma-<lb/>ior tamen angulo p h g, adhuc ſequitur idem quod prius.</s> <s xml:id="echoid-s35985" xml:space="preserve"> Quòd autem angulus p g h ſit minor an-<lb/>gulo p h g, hoc non poteſt accidere, niſi cum tanta fuerit rei à ſpeculo diſtantia, quòd illa diſtantia <lb/>ip ſi etiam uiſui uideretur minor quàm ſit ſecundum ueritatem:</s> <s xml:id="echoid-s35986" xml:space="preserve"> tunc aũt poteſt imago uideri maior <lb/>quàm forma per ſe uiſui occurrens, ut patet per pręmiſſam.</s> <s xml:id="echoid-s35987" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s35988" xml:space="preserve"/> </p> <div xml:id="echoid-div1456" type="float" level="0" n="0"> <figure xlink:label="fig-0557-01" xlink:href="fig-0557-01a"> <variables xml:id="echoid-variables621" xml:space="preserve">a f b m k q n e t h d v z</variables> </figure> <figure xlink:label="fig-0557-02" xlink:href="fig-0557-02a"> <variables xml:id="echoid-variables622" xml:space="preserve">b a e p g d</variables> </figure> <figure xlink:label="fig-0557-03" xlink:href="fig-0557-03a"> <variables xml:id="echoid-variables623" xml:space="preserve">a b h z h p g e d</variables> </figure> </div> </div> <div xml:id="echoid-div1458" type="section" level="0" n="0"> <head xml:id="echoid-head1119" xml:space="preserve" style="it">40. In minorib. ſpeculis ſphæricis couexis eiuſdẽ rei apparẽtidola minora. Eucl. 22 th. catoptr.</head> <p> <s xml:id="echoid-s35989" xml:space="preserve">Sint duo ſpecula ſphærica conuexa ſuper idem centrum t collocata, exempli cauſſa, quorum ma <lb/>ioris circulus communis ſibi & ſuperficiei reflexionis ſit a g, minoris uero ſit e i:</s> <s xml:id="echoid-s35990" xml:space="preserve"> fiat quoq;</s> <s xml:id="echoid-s35991" xml:space="preserve"> reflexio <lb/>formæ alicuius uiſibilis, ut ipſius h d, ab utroq;</s> <s xml:id="echoid-s35992" xml:space="preserve"> illorũ ſpeculorũ, ita ut forma puncti d reflectatur à <lb/>puncto g circuli ſpeculi maioris, ſcilicet ipſius a g, ad uiſum, qui ſit b.</s> <s xml:id="echoid-s35993" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s35994" xml:space="preserve"> idem uiſibile d reflecta <lb/>tur ad uiſum b ab aliquo puncto circuli e i ſpeculi minoris, ut à puncto o:</s> <s xml:id="echoid-s35995" xml:space="preserve">nõ eſt poſsibile ut linea re <lb/>flexionis, quę ſit o b, cadat in punctũ g ſpeculi circuli maioris.</s> <s xml:id="echoid-s35996" xml:space="preserve"> Detur enim, ut cadat in punctũg, & <lb/>reflectatur ad uiſum b:</s> <s xml:id="echoid-s35997" xml:space="preserve"> & ducatur linea d g, ut prius.</s> <s xml:id="echoid-s35998" xml:space="preserve"> Manifeſtũ itaq;</s> <s xml:id="echoid-s35999" xml:space="preserve"> ք 8 huius quoniã linea à centro <lb/>ſpeculι t ad punctũ g producta diuidit angulũ d g b ք duo æqualia:</s> <s xml:id="echoid-s36000" xml:space="preserve"> quę producta ſit t g q.</s> <s xml:id="echoid-s36001" xml:space="preserve"> Et quoniã <lb/>forma puncti d incidit pũcto ſpeculi minoris, quod eſt o:</s> <s xml:id="echoid-s36002" xml:space="preserve"> ducatur linea t o à cẽtro ſpeculi:</s> <s xml:id="echoid-s36003" xml:space="preserve"> hæc ergo <lb/>diuidet angulum d o b per æqualia:</s> <s xml:id="echoid-s36004" xml:space="preserve"> & producta fit t o p.</s> <s xml:id="echoid-s36005" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s36006" xml:space="preserve"> angulus d g b extrinſecus eſt ex <lb/>hypotheſi angulo d o b in trιgono d o g:</s> <s xml:id="echoid-s36007" xml:space="preserve"> palàm per 16 p 1 quoniam ipſe eſt maior illo:</s> <s xml:id="echoid-s36008" xml:space="preserve"> ergo medietas <lb/>anguli d g b eſt maior medietate anguli d o b:</s> <s xml:id="echoid-s36009" xml:space="preserve"> & ita angul{us} q g b maior eſt angulo p o g:</s> <s xml:id="echoid-s36010" xml:space="preserve"> ſed angul{us} o <lb/> <pb o="256" file="0558" n="558" rhead="VITELLONIS OPTICAE"/> gteſt ęqualis angulo q g b ք 15 p 1:</s> <s xml:id="echoid-s36011" xml:space="preserve"> ergo angul{us} p o g extrinſecus erit ęqualis angulo o g tintrinſeco <lb/> <anchor type="figure" xlink:label="fig-0558-01a" xlink:href="fig-0558-01"/> in trigono t o g:</s> <s xml:id="echoid-s36012" xml:space="preserve"> quod eſt contra 16 <lb/>p 1 & impoſsibile:</s> <s xml:id="echoid-s36013" xml:space="preserve"> non ergo tranſi-<lb/>bit linea reflexionis o b punctũ g.</s> <s xml:id="echoid-s36014" xml:space="preserve"> <lb/>Sed neq;</s> <s xml:id="echoid-s36015" xml:space="preserve"> ultra pũctũ g uerſus pun-<lb/>ctum a ad aliquod aliud punctũ ſpe <lb/>culi maioris incidere poteſt.</s> <s xml:id="echoid-s36016" xml:space="preserve"> Si e-<lb/>nim hoc ſit poſsibile:</s> <s xml:id="echoid-s36017" xml:space="preserve"> ſit, ut ad pun-<lb/>ctũ r incidens reflectatur linea d o <lb/>ad b:</s> <s xml:id="echoid-s36018" xml:space="preserve">palã autẽ per 17 huius (cum a <lb/>punctus lineæ d a cadat in ſuperfi-<lb/>cie ſpeculi, & reflectatur ab illo pũ-<lb/>cto, cui incidit, & punctum d refle-<lb/>ctatur à puncto g) quia quodlibet <lb/>punctorũ lineę d a reflectitur ab ali <lb/>quo punctorũ arcus a g & fiũt pro-<lb/>pinquiora centro ſpeculi, quod eſt <lb/>t:</s> <s xml:id="echoid-s36019" xml:space="preserve">quia reflectuntur à puncto remo-<lb/>tiori à centro uiſus, quod eſt b.</s> <s xml:id="echoid-s36020" xml:space="preserve"> Ali-<lb/>quod ergo pũctorũ lineæ d a refle ctetur à pũcto rad b:</s> <s xml:id="echoid-s36021" xml:space="preserve">ſit illud m:</s> <s xml:id="echoid-s36022" xml:space="preserve"> & accidet idẽ impoſsibile, q đ pri <lb/>us, ductis lineis m r, r b, t r.</s> <s xml:id="echoid-s36023" xml:space="preserve"> Vel ſi forma pũcti d reflectitur à puncto ſpeculi maioris, q đ eſt g:</s> <s xml:id="echoid-s36024" xml:space="preserve"> & ité ք <lb/>reflexionẽ à pũcto ſpeculi minoris, q đ eſt o, incidit pũcto ſpeculi maioris, q đ eſt r:</s> <s xml:id="echoid-s36025" xml:space="preserve"> à duob.</s> <s xml:id="echoid-s36026" xml:space="preserve"> ergo pu-<lb/>ctis maioris ſpeculi, quæ ſunt g & r, reflectitur forma unius pũcti ad uiſum b:</s> <s xml:id="echoid-s36027" xml:space="preserve">coincidũt ergo radij à <lb/>duob.</s> <s xml:id="echoid-s36028" xml:space="preserve"> pũctis huius ſpeculi reflexi, q đ eſt contra 15 huius, & impoſsibile.</s> <s xml:id="echoid-s36029" xml:space="preserve"> Nõ cadet ergo radius refle <lb/>xionis à pũcto o ſpeculi minoris in aliqđ pũctũ arcus a g ſpeculi maioris, à quo fit reflexio formarũ <lb/>pũctorũ lineæ a d, ſed directè քuenit ad uiſum in pũctũ b, trãs aliquẽ pũctorũ.</s> <s xml:id="echoid-s36030" xml:space="preserve"> arcus circuli ſpeculi <lb/>maioris, citra pũctũ g.</s> <s xml:id="echoid-s36031" xml:space="preserve"> Similiterq́;</s> <s xml:id="echoid-s36032" xml:space="preserve"> ſit, ut pũctus h lineæ d h ex alia parte uiſus b, ꝗ̃ ſit pũctũ d, reflecta <lb/>tur ad uiſum b ab aliquo puncto ſpeculi maioris, qđ ſit f:</s> <s xml:id="echoid-s36033" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s36034" xml:space="preserve"> f per 17 huius ex alia parte puncti g:</s> <s xml:id="echoid-s36035" xml:space="preserve"> <lb/>reflectaturq́;</s> <s xml:id="echoid-s36036" xml:space="preserve"> forma pũcti h à pũcto i minoris ſpeculi ad pũctũ b:</s> <s xml:id="echoid-s36037" xml:space="preserve"> fiet quoq;</s> <s xml:id="echoid-s36038" xml:space="preserve"> reflexio à pũcto i ad b ſi-<lb/>militer, ut prius.</s> <s xml:id="echoid-s36039" xml:space="preserve"> Quia ergo angulus g b f, ſub quo apparetidolũ in maiori ſpeculo, eſt maior ꝗ̃ angu <lb/>lus o b i, patet ք 20 th.</s> <s xml:id="echoid-s36040" xml:space="preserve"> 4 huius quoniá in maiori ſpeculo maius apparetidolũ ꝗ̃ in minori:</s> <s xml:id="echoid-s36041" xml:space="preserve"> formæ e-<lb/>nim magis coanguftátur circa cẽtra minorũ ſpeculorũ, ꝗ̃ circa cẽtra maiorũ:</s> <s xml:id="echoid-s36042" xml:space="preserve"> unde fiunt ſemper ma-<lb/>iores in ſpeculis maiorib.</s> <s xml:id="echoid-s36043" xml:space="preserve"> Vniuerſaliter aũt in omni ſitu ꝓ portionato rerũ ad ſpecula poteſt patere <lb/>propoſitũ per 46 th.</s> <s xml:id="echoid-s36044" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s36045" xml:space="preserve"> quoniá partes diametri circuli maioris ſunt maiores & minoris mino-<lb/>res:</s> <s xml:id="echoid-s36046" xml:space="preserve"> & fiunt ex cóſequenti imagines maiores & minores, ut patet per 11 huius.</s> <s xml:id="echoid-s36047" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s36048" xml:space="preserve"/> </p> <div xml:id="echoid-div1458" type="float" level="0" n="0"> <figure xlink:label="fig-0558-01" xlink:href="fig-0558-01a"> <variables xml:id="echoid-variables624" xml:space="preserve">a h m p u q b a r g f e o i t</variables> </figure> </div> </div> <div xml:id="echoid-div1460" type="section" level="0" n="0"> <head xml:id="echoid-head1120" xml:space="preserve" style="it">41. In eodem ſpeculo ſphærico conuexo, centro uiſus immoto exiſtente: imago rei approxima-<lb/>tæ ſuperficiei ſpeculi uidetur maior, & ſecundum eandem lineam elong at æ minor.</head> <p> <s xml:id="echoid-s36049" xml:space="preserve">Quoniam enim, ut patet per 11 huius, imagines punctorum rei uiſæ uidentur in cathetis ſuæ inci <lb/>dentiæ, & imagines rerum uiſarum inter cathetos incidentię ſuorum terminorum:</s> <s xml:id="echoid-s36050" xml:space="preserve"> catheti uerò <lb/>punctorũ terminalium rei à ſpeculi ſuperficie elõgatæ continentangulum minorẽ, & approxima-<lb/>tæ maiorem per 34 th.</s> <s xml:id="echoid-s36051" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s36052" xml:space="preserve"> linea enim æ qualis & æ quidiſtans baſi trigoni uicinior angulo ſupre <lb/>mo, maiori angulo ſubtenditur.</s> <s xml:id="echoid-s36053" xml:space="preserve"> Et quoniam mutata re ſecun dum locum, mutatur ipſius imago in <lb/>omni ſpeculo, ut patet per 42 th.</s> <s xml:id="echoid-s36054" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s36055" xml:space="preserve"> patet quòd imago rei elongatę fit minor:</s> <s xml:id="echoid-s36056" xml:space="preserve"> unde & uide-<lb/>tur minor:</s> <s xml:id="echoid-s36057" xml:space="preserve"> & approximatæ ſuperficiei ſpeculi fit maior:</s> <s xml:id="echoid-s36058" xml:space="preserve"> unde & uidetur maior:</s> <s xml:id="echoid-s36059" xml:space="preserve"> quoniam ſecun-<lb/>dum pręmiſſa in proxima pręcedente uidetur ſub maior:</s> <s xml:id="echoid-s36060" xml:space="preserve"> angulo contento in centro uiſus ſub li-<lb/>neis reflexionum ipſorum punctorum terminalium illius rei, ut patere poteſt per 34 th, 1 huius, & <lb/>per 23 huius.</s> <s xml:id="echoid-s36061" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s36062" xml:space="preserve"> Et per hæc & per præmiſſam poteſt patere, quoniam ſi ſit pro-<lb/>portio elongationis rei uiſæ à ſuperficie ſpeculi maioris ad elongationẽ à ſuperficie ſpeculi mino-<lb/>ris, ſicut exceſſus imaginum, quæ proueniunt in illis ſpeculis excedentes ſe ſecundum proportio-<lb/>nẽ diametrorum ſpeculorum:</s> <s xml:id="echoid-s36063" xml:space="preserve"> poſsibile eſt in ſpeculo maiori plus elongato à re uiſa, & in ſpeculo <lb/>minori plus approximato eidẽ rei, ęqualé imaginem uideri eiuſdem rei, quæ aliàs in ſpeculo maio-<lb/>ri appareret maior, & in ſpeculo minori minor, ut patet per pręmiſſam.</s> <s xml:id="echoid-s36064" xml:space="preserve"> Et hoc eſt notatu dignum.</s> <s xml:id="echoid-s36065" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1461" type="section" level="0" n="0"> <head xml:id="echoid-head1121" xml:space="preserve" style="it">42. In ſpeculo cõue xo ſphærico dextr a rei uiſæ apparẽt ſiniſtra, et ſiniſtra dextra. Euc. 20 th. catop.</head> <p> <s xml:id="echoid-s36066" xml:space="preserve">Hæc non requirit aliam dem onſtrationem ab illa, quæ ſimilem paſsionem declarat in ſpeculis <lb/>planis:</s> <s xml:id="echoid-s36067" xml:space="preserve">un de eodem modo demonſtrandum:</s> <s xml:id="echoid-s36068" xml:space="preserve">nec aliter oportet immorari.</s> <s xml:id="echoid-s36069" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1462" type="section" level="0" n="0"> <head xml:id="echoid-head1122" xml:space="preserve" style="it">43. Altitudines & profunditates perpendiculariter incidentes ſpeculis ſphæricis conuexis, <lb/>reuerſæ apparent. Euclides 8 th. catoptr.</head> <p> <s xml:id="echoid-s36070" xml:space="preserve">Eſto ſpeculum ſphæricum cõuexum a d g:</s> <s xml:id="echoid-s36071" xml:space="preserve">cuius centrũ m:</s> <s xml:id="echoid-s36072" xml:space="preserve"> incidatq́;</s> <s xml:id="echoid-s36073" xml:space="preserve"> ſuperficiei ſpeculi perpen-<lb/>diculariter altitudo, quæ ſit e a, cuius altius punctum ſit e:</s> <s xml:id="echoid-s36074" xml:space="preserve"> & ſit centrum uiſus b:</s> <s xml:id="echoid-s36075" xml:space="preserve"> reflectaturq́;</s> <s xml:id="echoid-s36076" xml:space="preserve"> pun-<lb/>ctus a à puncto ſpeculi, qui ſit a:</s> <s xml:id="echoid-s36077" xml:space="preserve"> & ſit linea reflexionis, quæ a b:</s> <s xml:id="echoid-s36078" xml:space="preserve"> refle ctatur quoq;</s> <s xml:id="echoid-s36079" xml:space="preserve"> forma puncti alti-<lb/>tudinis e à puncto ſpeculi g:</s> <s xml:id="echoid-s36080" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s36081" xml:space="preserve"> linea reflexionis g b:</s> <s xml:id="echoid-s36082" xml:space="preserve"> & alter punctus lineæ e a (qui ſit t) inferior <lb/>pũcto e, reflectatur ad uiſum b à puncto ſpeculi d:</s> <s xml:id="echoid-s36083" xml:space="preserve"> & ſit linea reflexionis d b.</s> <s xml:id="echoid-s36084" xml:space="preserve"> Producatur ita q;</s> <s xml:id="echoid-s36085" xml:space="preserve"> linea <lb/>altitudinis e a ultra punctum a:</s> <s xml:id="echoid-s36086" xml:space="preserve"> palam q́;</s> <s xml:id="echoid-s36087" xml:space="preserve"> ex hypotheſi, & per 72 th.</s> <s xml:id="echoid-s36088" xml:space="preserve"> 1 huius quoniá ipſa tranſibit cen <lb/> <pb o="257" file="0559" n="559" rhead="LIBER SEXTVS."/> trũ m:</s> <s xml:id="echoid-s36089" xml:space="preserve"> & ꝓducatur linea reflexionis b g intra ſpeculũ.</s> <s xml:id="echoid-s36090" xml:space="preserve"> Et ꝗa lineæ e a & b g ſuntin eadẽ ſuքficie re <lb/> <anchor type="figure" xlink:label="fig-0559-01a" xlink:href="fig-0559-01"/> <anchor type="figure" xlink:label="fig-0559-02a" xlink:href="fig-0559-02"/> flexiõis ք 27 th.</s> <s xml:id="echoid-s36091" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s36092" xml:space="preserve"> palá <lb/>cũ nõ ſint ęquidiſtátes, ut pa <lb/>tet per 9 huius, quia concur-<lb/>rent:</s> <s xml:id="echoid-s36093" xml:space="preserve"> cócurrant itaq;</s> <s xml:id="echoid-s36094" xml:space="preserve"> in pun-<lb/>cto h:</s> <s xml:id="echoid-s36095" xml:space="preserve"> ſed & b d linea reflexio <lb/>nis cócurrat cũ linea e a pro-<lb/>ducta, in puncto f.</s> <s xml:id="echoid-s36096" xml:space="preserve"> Et quoniã <lb/>per 11 huius pũcta h & f ſunt <lb/>loca imaginũ pũctorũ e & t:</s> <s xml:id="echoid-s36097" xml:space="preserve"> <lb/>palá quòd linea h f eſt imago <lb/>lineæ e t:</s> <s xml:id="echoid-s36098" xml:space="preserve"> ſimiliter quoq;</s> <s xml:id="echoid-s36099" xml:space="preserve"> de <lb/>alijs punctis lineæ e a demon <lb/>ſtrádũ.</s> <s xml:id="echoid-s36100" xml:space="preserve"> Eritq́;</s> <s xml:id="echoid-s36101" xml:space="preserve"> imago lineę e a <lb/>linea a h:</s> <s xml:id="echoid-s36102" xml:space="preserve"> reuerſa ergo uide-<lb/>tur altitudo:</s> <s xml:id="echoid-s36103" xml:space="preserve"> quod enim ſu-<lb/>premũ eſt, uidetur infimũ, & <lb/>ecõuerſo Patet enim ք 23 huius quoniá ſuper uná cathetũ incidẽtię ſignatis duob.</s> <s xml:id="echoid-s36104" xml:space="preserve"> pũctis, eritlocus <lb/>imaginis pũcti à cẽtro ſpeculi ꝓpinquioris, remotior à cẽtro ſpeculi, & remotioris propin quior:</s> <s xml:id="echoid-s36105" xml:space="preserve"> re <lb/>motior itaq;</s> <s xml:id="echoid-s36106" xml:space="preserve"> uidebitur à cẽtro m imago pũcti t, q̃ eſt f, ꝗ̃ imago pũcti e, q̃ eſt h.</s> <s xml:id="echoid-s36107" xml:space="preserve"> Palã itaq;</s> <s xml:id="echoid-s36108" xml:space="preserve"> eſt ꝓpoſitũ <lb/>primũ.</s> <s xml:id="echoid-s36109" xml:space="preserve"> Et eodẽ modo eſt de ꝓfunditatib.</s> <s xml:id="echoid-s36110" xml:space="preserve"> demõſtrãdũ:</s> <s xml:id="echoid-s36111" xml:space="preserve">infimũ.</s> <s xml:id="echoid-s36112" xml:space="preserve">n.</s> <s xml:id="echoid-s36113" xml:space="preserve">pũctũ reflectitur ad pũctũ imaginis <lb/>ſupremũ, & ecõuerſo.</s> <s xml:id="echoid-s36114" xml:space="preserve"> Media quoq;</s> <s xml:id="echoid-s36115" xml:space="preserve"> pũcta modo medio reuerſè diſponũtur.</s> <s xml:id="echoid-s36116" xml:space="preserve"> Propoſitũ aũt eſt hoc.</s> <s xml:id="echoid-s36117" xml:space="preserve"/> </p> <div xml:id="echoid-div1462" type="float" level="0" n="0"> <figure xlink:label="fig-0559-01" xlink:href="fig-0559-01a"> <variables xml:id="echoid-variables625" xml:space="preserve">e b t d g a f h m</variables> </figure> <figure xlink:label="fig-0559-02" xlink:href="fig-0559-02a"> <variables xml:id="echoid-variables626" xml:space="preserve">m h s g d a t b e</variables> </figure> </div> </div> <div xml:id="echoid-div1464" type="section" level="0" n="0"> <head xml:id="echoid-head1123" xml:space="preserve" style="it">44. Obliquarum longitudinum idola à conuexis ſpeculis reflexa apparent ſuæpropriæ diſpo-<lb/>ſitionis. Euclides 10 th. catoptr.</head> <p> <s xml:id="echoid-s36118" xml:space="preserve">Eſto longitudo d e obliquè incidens ſpeculo ſphærico conuexo, quod ſit a g:</s> <s xml:id="echoid-s36119" xml:space="preserve"> & eius centrũ f:</s> <s xml:id="echoid-s36120" xml:space="preserve"> & <lb/> <anchor type="figure" xlink:label="fig-0559-03a" xlink:href="fig-0559-03"/> ſit altius pũctũ d quàm e pũctũ à ſuperficie ſpeculi da <lb/>ti:</s> <s xml:id="echoid-s36121" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s36122" xml:space="preserve"> centrũ oculi b:</s> <s xml:id="echoid-s36123" xml:space="preserve"> & reflectatur punctus d ad ui-<lb/>ſum b à pũcto ſpeculi a, & pũctus e à pũcto g.</s> <s xml:id="echoid-s36124" xml:space="preserve"> Et à pun <lb/>cto d ducatur perpendicularis ſuper ſuperficiẽ ſpecu-<lb/>li, quæ per 72 th.</s> <s xml:id="echoid-s36125" xml:space="preserve"> 1 huius neceſſariò tranſibit centrum <lb/>ſpeculi, quod eſt f:</s> <s xml:id="echoid-s36126" xml:space="preserve"> quæ ſit d f:</s> <s xml:id="echoid-s36127" xml:space="preserve"> & ſimiliter ducatur ca-<lb/>thetus e f:</s> <s xml:id="echoid-s36128" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s36129" xml:space="preserve"> lineæ reflexionum b a & b g:</s> <s xml:id="echoid-s36130" xml:space="preserve"> & <lb/>producãtur intra ſpeculũ:</s> <s xml:id="echoid-s36131" xml:space="preserve"> cõcurratq́;</s> <s xml:id="echoid-s36132" xml:space="preserve"> b a cũ d fin pun <lb/>cto h, & b g cũ e fin pũcto k:</s> <s xml:id="echoid-s36133" xml:space="preserve"> & ducatur linea h k, eritq́;</s> <s xml:id="echoid-s36134" xml:space="preserve"> <lb/>ք 11 huius linea h k imago lineæ d e:</s> <s xml:id="echoid-s36135" xml:space="preserve"> eſt autẽ linea k h <lb/>obliquè ſe habẽs ad uiſum b, ſicut linea d e ad ſpeculũ.</s> <s xml:id="echoid-s36136" xml:space="preserve"> <lb/>Quoniã ք 23 hui{us} pũcti e, qđ eſt ꝓpinquius cẽtro ſpe-<lb/>culi, imago, q̃ eſt k, remotior fit à cẽtro ſpeculi f:</s> <s xml:id="echoid-s36137" xml:space="preserve"> & pun <lb/>ctũ h, q đ eſt imago pũcti d remotioris à cẽtro ſpeculi, <lb/>fit ꝓpinquius cẽtro ſpeculi:</s> <s xml:id="echoid-s36138" xml:space="preserve"> q đ patet ք hoc:</s> <s xml:id="echoid-s36139" xml:space="preserve"> quoniã ali <lb/>cuius pũcti catheti d f tãtũ diſtantis à pũcto f, quantũ <lb/>pũctũ e:</s> <s xml:id="echoid-s36140" xml:space="preserve"> locus imaginis eſt remotior à cẽtro f, ꝗ̃ locus <lb/>imaginis pũcti d ք 23 huius:</s> <s xml:id="echoid-s36141" xml:space="preserve"> eſt itaq;</s> <s xml:id="echoid-s36142" xml:space="preserve"> h remotius à con <lb/>uexa ſuքficie ſpeculi apparẽs, & pũctũ k propinquius eidẽ ſuperficiei.</s> <s xml:id="echoid-s36143" xml:space="preserve"> Sic aũt & pũctus d fuit remo <lb/>tior à ſuperficie ſpeculi, & pũctus e ꝓpinquior.</s> <s xml:id="echoid-s36144" xml:space="preserve"> Patet ergo ꝓpoſitũ, quoniá obliquę lõgitudines ap <lb/>parẽt illius diſtantiæ à ſuperficie ſpeculi, cuius ſunt ſecundum ueritatẽ in ſua propria diſpoſitione.</s> <s xml:id="echoid-s36145" xml:space="preserve"/> </p> <div xml:id="echoid-div1464" type="float" level="0" n="0"> <figure xlink:label="fig-0559-03" xlink:href="fig-0559-03a"> <variables xml:id="echoid-variables627" xml:space="preserve">d b e g a k h f</variables> </figure> </div> </div> <div xml:id="echoid-div1466" type="section" level="0" n="0"> <head xml:id="echoid-head1124" xml:space="preserve" style="it">45. Duobus punctis rei uiſæ æqualiter diſtantibus à centro ſpeculi ſphæriciconuexi, & inæ-<lb/>qualiter à centro uiſus in eadẽ ſuperficie uel diuerſis:erunt imago & finis cõtingentiæ punctire <lb/>motioris à centro uiſus remotiora à centro ſpeculi, quàm imago & finis cõtingentiæ puncti pro-<lb/>pinquioris:ex quo patet quòd punctorũ æqualiter diſtantiũ à centro ſpeculi & à centro uiſus, <lb/>imagines à centro ſpeculi æqualiter diſtabunt. Alhazen 7 n 6.</head> <p> <s xml:id="echoid-s36146" xml:space="preserve">Sintt & d duo pũcta æ qualiter à puncto g cẽtro ſpeculi remota:</s> <s xml:id="echoid-s36147" xml:space="preserve"> & ſit e cẽtrũ uiſus:</s> <s xml:id="echoid-s36148" xml:space="preserve"> & ſit cõmunis <lb/>ſectio ſuperficiei reflexionis & ſpeculi ſphærici conuexi circulus a b:</s> <s xml:id="echoid-s36149" xml:space="preserve"> cuius cẽtrũ erit pũctũ g ք 1 hu <lb/>ius.</s> <s xml:id="echoid-s36150" xml:space="preserve"> Sitq́;</s> <s xml:id="echoid-s36151" xml:space="preserve"> pũctũ d ꝓpinquius uiſui, ꝗ eſt e, ꝗ̃ pũctũ t:</s> <s xml:id="echoid-s36152" xml:space="preserve"> & ducãtur duæ catheti incidẽtiæ à pũctis t & d <lb/>ad cẽtrũ circuli g, q̃ ſint t g & d g:</s> <s xml:id="echoid-s36153" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s36154" xml:space="preserve"> cathetust g ſuքficiẽ ſpeculi in pũcto b:</s> <s xml:id="echoid-s36155" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s36156" xml:space="preserve"> angulo e g d ſu <lb/>per lineã t g ęqualis angulus, ք ſit t g z:</s> <s xml:id="echoid-s36157" xml:space="preserve"> & angulo e g t ęqualis angulus, ꝗ ſit t g h ք 23 p 1:</s> <s xml:id="echoid-s36158" xml:space="preserve"> lecetq́;</s> <s xml:id="echoid-s36159" xml:space="preserve"> linea <lb/>h g circulũ in pũcto a:</s> <s xml:id="echoid-s36160" xml:space="preserve"> & ſumatur ք 20 uel 22 huius in circulo pũctũ, à quo forma pũcti t reflectatur <lb/>ad pũctũ z:</s> <s xml:id="echoid-s36161" xml:space="preserve"> qđ ſit pũctũ q.</s> <s xml:id="echoid-s36162" xml:space="preserve"> Palã e rgo qđ forma pũctitnõ reflectitur ad pũctũ h ab aliquo pũcto arc{us} <lb/>b q:</s> <s xml:id="echoid-s36163" xml:space="preserve"> nõ.</s> <s xml:id="echoid-s36164" xml:space="preserve">n.</s> <s xml:id="echoid-s36165" xml:space="preserve">à pũcto b:</s> <s xml:id="echoid-s36166" xml:space="preserve">quoniã cũille ſit in catheto incidẽtię, palã ք 10 huius ꝗa reflectitur in ſeipſum & <lb/>nõ ad pũctũ h.</s> <s xml:id="echoid-s36167" xml:space="preserve"> Sed neq;</s> <s xml:id="echoid-s36168" xml:space="preserve"> à pũcto q:</s> <s xml:id="echoid-s36169" xml:space="preserve"> quoniá ab illo forma pũcti t refle ctitur ad pũctũ z.</s> <s xml:id="echoid-s36170" xml:space="preserve"> Quocũq;</s> <s xml:id="echoid-s36171" xml:space="preserve"> uerò <lb/>pũcto ſumpto in arcu b q, linea à pũcto h ad illud pũctũ ducta ſecabit lineá q z:</s> <s xml:id="echoid-s36172" xml:space="preserve"> igitur ad illud pũctũ <lb/>ſectιõis reflectitur form a pũcti t à pũcto aliquo arcus b q, & ad idẽ pũct{us} ſectiõis reflectitur à pũcto <lb/>q:</s> <s xml:id="echoid-s36173" xml:space="preserve">ergo forma puncti treflectitur à duob.</s> <s xml:id="echoid-s36174" xml:space="preserve"> punctis ſuքficiei ſpeculi ad unũ punctũ:</s> <s xml:id="echoid-s36175" xml:space="preserve"> qđ eſt impoſsibi-<lb/> <pb o="258" file="0560" n="560" rhead="VITELLONIS OPTICAE"/> le, & contra 16 huius.</s> <s xml:id="echoid-s36176" xml:space="preserve"> Reſtat ergo ut form a puncti treflectatur ad punctũ h ab aliquo puncto arcus <lb/> <anchor type="figure" xlink:label="fig-0560-01a" xlink:href="fig-0560-01"/> q a Sit iſlud pũctũ m:</s> <s xml:id="echoid-s36177" xml:space="preserve"> & à puncto m ducat ur linea <lb/>cõtingẽs circulũ ք 17 p 3:</s> <s xml:id="echoid-s36178" xml:space="preserve"> & ꝓducatur uſq;</s> <s xml:id="echoid-s36179" xml:space="preserve"> ad ca <lb/>thetũ g t:</s> <s xml:id="echoid-s36180" xml:space="preserve"> & ſit m n:</s> <s xml:id="echoid-s36181" xml:space="preserve">eritq́;</s> <s xml:id="echoid-s36182" xml:space="preserve"> pũctus n finis cõtingẽtię <lb/>puncti t reſpectu puncti h:</s> <s xml:id="echoid-s36183" xml:space="preserve"> & à puncto q ducatur <lb/>linea cõtingẽs circulũ, quę ꝓducta ad cathetũ t g, <lb/>ſit q o:</s> <s xml:id="echoid-s36184" xml:space="preserve"> hęc ergo neceſſariò cadet ſub linea n m per <lb/>60 th.</s> <s xml:id="echoid-s36185" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s36186" xml:space="preserve"> & ꝓducatur linea z q donec cadat <lb/>ſuper cathetũ g t in pũcto p (cadet aũt ք 9 huius) <lb/>& erit ք 11 huius punctus p locus imaginis form æ <lb/>puncti t:</s> <s xml:id="echoid-s36187" xml:space="preserve"> erit quo q;</s> <s xml:id="echoid-s36188" xml:space="preserve"> ք 12 huius ꝓportio g t ad p g, <lb/>ſicut t o ad o p:</s> <s xml:id="echoid-s36189" xml:space="preserve"> ergo ք 16 p 5 erit permutatim ꝓ-<lb/>portio g t a d t o, ſicut g p ad p o:</s> <s xml:id="echoid-s36190" xml:space="preserve"> ſed maior eſt pro-<lb/>portio g t ad t n, quã ad t o ք 8 p 5:</s> <s xml:id="echoid-s36191" xml:space="preserve"> cũ t n ſit minor <lb/>ꝗ̃ t o, ut patet ex pręmiſsis:</s> <s xml:id="echoid-s36192" xml:space="preserve"> maior ergo erit ꝓpor-<lb/>tio g t ad t n, ꝗ̃ g p ad p o:</s> <s xml:id="echoid-s36193" xml:space="preserve"> eſt autẽ ք 8 p 5 maior ꝓ-<lb/>portio g p ad p o, ꝗ̃ ad p n:</s> <s xml:id="echoid-s36194" xml:space="preserve"> ergo multò maior eſt <lb/>ꝓportio t g ad t n, ꝗ̃ g p ad p n:</s> <s xml:id="echoid-s36195" xml:space="preserve"> quoniã p o minor <lb/>eſt ꝗ̃ p n.</s> <s xml:id="echoid-s36196" xml:space="preserve"> Diuidatur ergo ք 119 th.</s> <s xml:id="echoid-s36197" xml:space="preserve"> 1 huius linea g n <lb/>in puncto 1 taliter, ut ſit ꝓportio t g ad t n, ſicut g l <lb/>ad l n:</s> <s xml:id="echoid-s36198" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s36199" xml:space="preserve"> g l maior ꝗ̃ g p, nõ ęqualis n eq;</s> <s xml:id="echoid-s36200" xml:space="preserve"> minor <lb/>ք 8 p 5:</s> <s xml:id="echoid-s36201" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s36202" xml:space="preserve"> ք 16 p 5 ꝓportio t g ad g l, ſicutt n ad <lb/>l n:</s> <s xml:id="echoid-s36203" xml:space="preserve"> ergo ք cõuerſam 12 huius erit punctũ llocus i-<lb/>maginis puncti h.</s> <s xml:id="echoid-s36204" xml:space="preserve"> Sint ergo lineę h g, e g, z g æ qua <lb/>les inter ſe:</s> <s xml:id="echoid-s36205" xml:space="preserve"> & g f ſit æqualis g p, & g s æqualis lineæ g o.</s> <s xml:id="echoid-s36206" xml:space="preserve"> Cũigitur angulus e g d ſit æ qualis angulo t <lb/>g z:</s> <s xml:id="echoid-s36207" xml:space="preserve"> erit ք 1 ſuppoſ.</s> <s xml:id="echoid-s36208" xml:space="preserve"> 1 huius remotio pũcti d à puncto e, ſicut remotio puncti z à puncto t.</s> <s xml:id="echoid-s36209" xml:space="preserve"> Quoniá cũ <lb/>pũcta d & t ſint eiuſdẽ diſtãtiæ à cẽtro ſpeculi, qđ eſt g:</s> <s xml:id="echoid-s36210" xml:space="preserve"> erũt lineæ d g & t g æ quales:</s> <s xml:id="echoid-s36211" xml:space="preserve">erit ergo per 23 <lb/>huius imago formę pũcti d reſpectu uiſus e tãtũ eleuata in catheto g d, quãtũ imago pũcti t eleuata <lb/>eſt, reſpectu pũcti z in catheto g t:</s> <s xml:id="echoid-s36212" xml:space="preserve">erit ergo locus imaginis formę pũcti d in pũcto f, ſicut locus ima-<lb/>ginis formæ pũcti t eſt in pũcto p:</s> <s xml:id="echoid-s36213" xml:space="preserve"> cũ lineæ g f & g p ſint æ quales.</s> <s xml:id="echoid-s36214" xml:space="preserve"> Et ſimiliter finis cõtingẽtię pũcti <lb/>d, reſpectu pũcti e erit eiuſdẽ altitudinis, cuius eſt finis cõtingẽtię pũcti t, reſpectu pũcti z:</s> <s xml:id="echoid-s36215" xml:space="preserve"> erit ergo <lb/>ſecũdũ p̃miſſa finis cõtingẽtiæ pũcti d in pũcto s.</s> <s xml:id="echoid-s36216" xml:space="preserve"> Verũ ꝗ a angulus e g t æ qualis eſt angulo t g h, & li <lb/>nea h g æ qualis eſt lineę e g.</s> <s xml:id="echoid-s36217" xml:space="preserve"> erit ք 33 p 6 ꝓpter æ qualitatẽ angulorũ, æ qualitas arcuũ interiacẽtium <lb/>cathetũ t g & lineas h g & e g:</s> <s xml:id="echoid-s36218" xml:space="preserve">erit ergo ք p̃miſſa pũctus llocus imaginis puncti t, reſpectu e, ſicut eſt <lb/>reſpectu h:</s> <s xml:id="echoid-s36219" xml:space="preserve"> & erit pũct{us} n finis cõtingẽtię, reſpectu pũctie, ſicut eſt reſpectu pũcti h.</s> <s xml:id="echoid-s36220" xml:space="preserve"> Imago ergo pun <lb/>cti remotioris ab e cẽtro uiſus remotior eſt à cẽtro ſpeculi ꝗ̃ imago pũcti ꝓpincuioris:</s> <s xml:id="echoid-s36221" xml:space="preserve"> & finis cõtin <lb/>gétiæ pũcti remotioris remotior eſt ab eodẽ cẽtro ꝗ̃ finis cótingétiæ ꝓpιnquioris.</s> <s xml:id="echoid-s36222" xml:space="preserve"> Et hoc eſt ꝓpoſi <lb/>tũ.</s> <s xml:id="echoid-s36223" xml:space="preserve"> Ex quo patet qđ ſi pũcta uiſain ſpeculo ſphærico cóuexo ęqualiter diſtẽt à cẽtro ſpeculi, & à cẽ-<lb/>tro uiſus, qđ imagines ipſorũ à cẽtro ſpeculι æ qualiter diſtabũt:</s> <s xml:id="echoid-s36224" xml:space="preserve">nec enim, ut patet ex p̃miſsis, fit di <lb/>uerſitas in locis imaginũ, cũ fines cõtingẽtiarũ ſemper ſint æ qualiter à cẽtro ſpeculi diſtãtes, ſecun-<lb/>dum quos accidit diſtantia imaginum à centro ſpeculi, quod eſt g.</s> <s xml:id="echoid-s36225" xml:space="preserve"> Patet ergo, quod ꝓponebatur.</s> <s xml:id="echoid-s36226" xml:space="preserve"/> </p> <div xml:id="echoid-div1466" type="float" level="0" n="0"> <figure xlink:label="fig-0560-01" xlink:href="fig-0560-01a"> <variables xml:id="echoid-variables628" xml:space="preserve">d t e z h s n o b l q m a f p g</variables> </figure> </div> </div> <div xml:id="echoid-div1468" type="section" level="0" n="0"> <head xml:id="echoid-head1125" xml:space="preserve" style="it">46. Imago arcus concentrici ſpeculo ſphærico conuexo (diametro uiſuali erecta ſuper ſuperfi-<lb/>ciem incidentiæ) uidetur curua, & ſemper æquidiſtans arcui, cuius eſt imago. Alhaz. 11 n 6.</head> <figure> <variables xml:id="echoid-variables629" xml:space="preserve">b e a d h t z l m q g</variables> </figure> <p> <s xml:id="echoid-s36227" xml:space="preserve">Eſto a b arcus oppoſitus ſpeculo ſphęrico cóuexo:</s> <s xml:id="echoid-s36228" xml:space="preserve"> in quo cómunis <lb/>ſectio ſuperficiei reflexionis & ſpeculi ſit circulus h t z:</s> <s xml:id="echoid-s36229" xml:space="preserve"> & ſit g cẽtrũ <lb/>illius arcus a b, & ſimiliter centrũ ſpeculi:</s> <s xml:id="echoid-s36230" xml:space="preserve"> quoniã ex hypotheſi arcus <lb/>uiſus & ſpeculũ ſunt cõcẽtrica:</s> <s xml:id="echoid-s36231" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s36232" xml:space="preserve"> d cẽtrũ uiſus:</s> <s xml:id="echoid-s36233" xml:space="preserve"> & ducátur lineę d <lb/>g a g, b g:</s> <s xml:id="echoid-s36234" xml:space="preserve"> & ſumatur in arcu a b pũctus e quocũq;</s> <s xml:id="echoid-s36235" xml:space="preserve"> modo, & ducatur <lb/>linea e g:</s> <s xml:id="echoid-s36236" xml:space="preserve"> erit ita q;</s> <s xml:id="echoid-s36237" xml:space="preserve"> ſuքficies a g b ſuperficies incidẽtiæ, in qua erit li-<lb/>nea e g:</s> <s xml:id="echoid-s36238" xml:space="preserve"> & linea d g eſt diameter uiſualis, q̃ ex hypotheſi eſt erecta ſu-<lb/>per ſuքficiẽ a g b:</s> <s xml:id="echoid-s36239" xml:space="preserve"> erũt ergo ք definitionẽ lineę ſuք ſuքficiẽ erectę an <lb/>guli d g a, d g b, d g e recti & oẽs æ quales:</s> <s xml:id="echoid-s36240" xml:space="preserve">ſed & latera laterib.</s> <s xml:id="echoid-s36241" xml:space="preserve"> ęqualia <lb/>ſunt, quoniã d g eſt ęquale ſibijpſi, & alia latera ſunt æ qualia ք defini <lb/>tionẽ circuli:</s> <s xml:id="echoid-s36242" xml:space="preserve"> ergo ք 4 p 1 baſes illorũ triangulorũ ſunt æ quales.</s> <s xml:id="echoid-s36243" xml:space="preserve"> Oẽs <lb/>ergo pũcti arcus a b eiuſdẽ diſtãtiæ ſunt à cẽtro uiſus:</s> <s xml:id="echoid-s36244" xml:space="preserve"> quare imagi-<lb/>nes omniũ illorũ pũctorũ eiuſdẽ diſtãtię erũt à cẽtro ſpeculi ք corol <lb/>lariũ p̃miſſę Sitq́;</s> <s xml:id="echoid-s36245" xml:space="preserve"> q m limago arcus a e b:</s> <s xml:id="echoid-s36246" xml:space="preserve"> erit igitur linea g q ęqualis <lb/>lineis g m & g l:</s> <s xml:id="echoid-s36247" xml:space="preserve"> quare ք 9 p 3 linea q m l erit arcus circuli, cuius cẽtrũ <lb/>erit pũctũ g:</s> <s xml:id="echoid-s36248" xml:space="preserve"> erit ergo cõuexitas ipſius reſpectu cẽtri g, nõ reſpectu ſu <lb/>քficiei cõuexæ ſpeculi ſiue loci reflexiõis.</s> <s xml:id="echoid-s36249" xml:space="preserve"> Et quoniã curuitas arcus <lb/>a b reſpicit cõuexitatẽ ſuքficiei ſpeculi, ut cõcẽtrica ipſi ex hypotheſi, patet qđ idẽ arcus eſt cõcen-<lb/>tricus ſuę imagini:</s> <s xml:id="echoid-s36250" xml:space="preserve"> ergo per 73 th.</s> <s xml:id="echoid-s36251" xml:space="preserve"> 1 huius patet q đ imago ęquidiſtat arcui uiſo:</s> <s xml:id="echoid-s36252" xml:space="preserve"> quoniam eſt ſemper <lb/>in ſuperficie incidentiæ:</s> <s xml:id="echoid-s36253" xml:space="preserve"> eſt enim ſemperimago cuiuslibet puncti in catheto ſuæ in cidentiæ per 11 <lb/>huius:</s> <s xml:id="echoid-s36254" xml:space="preserve"> omnes autem catheti illæ ſunt in ſuperficie in cidẽtiæ.</s> <s xml:id="echoid-s36255" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s36256" xml:space="preserve"/> </p> <pb o="259" file="0561" n="561" rhead="LIBER SEXTVS."/> </div> <div xml:id="echoid-div1469" type="section" level="0" n="0"> <head xml:id="echoid-head1126" xml:space="preserve" style="it">47. Imago arcus concentrici ſpeculo ſphærico conuexo (diametro uiſuali ſuperficiei inciden-<lb/>tiæ obli q u è incidente) uidetur curua, non æquidiſtans arcui, cuius eſt imago, niſiperpendicula-<lb/>riduct a à uiſu ſuper aliquem punctum uiſi arcus incidente. Alhazen 12 n 6.</head> <p> <s xml:id="echoid-s36257" xml:space="preserve">Diſponãtur omnia, ut in p̃cedẽre theoremate, niſi qđ diameter uiſualis, q̃ eſt d g, nõ ſit erecta, ſed <lb/>obliquè incidẽs ſuքficiei a b g.</s> <s xml:id="echoid-s36258" xml:space="preserve"> Dico qđ imago arcus a b uidetur curua.</s> <s xml:id="echoid-s36259" xml:space="preserve"> Ducatur enim ք pendicula-<lb/>ris à pũcto d ſuք hac ſuքficiẽ ք 11 p 11.</s> <s xml:id="echoid-s36260" xml:space="preserve"> Cũ itaq;</s> <s xml:id="echoid-s36261" xml:space="preserve"> illa քpendicularis ſit minor omnib l neis ductis à pũ-<lb/>cto d a d hác ſuքficiẽ ք 21 t.</s> <s xml:id="echoid-s36262" xml:space="preserve"> 1 huius, erit angulus rectus, qu cõtinet hęc ք pẽdicularis uerſus pũctũ g, <lb/>minor quolibet angulo uerſus punctũ g imaginato, quẽ cótinet alia linea à pũcto d ad ſuperficiẽ illã <lb/>ducta ք 16 p 1:</s> <s xml:id="echoid-s36263" xml:space="preserve"> & linea à pũcto d ad ſuքficiẽ illá ducta, quãtò remotior erit à քpẽdiculari, tátò maior <lb/>erit & maiorẽ angulũ cõtinebit uerſus g:</s> <s xml:id="echoid-s36264" xml:space="preserve"> ꝗ a minorẽ cõtinet uerſus ք pẽdicularẽ ք 21 p 1.</s> <s xml:id="echoid-s36265" xml:space="preserve"> Si ergo hęc <lb/>քpendicularis nõ cadat in arcũ a e b, ſed ultra ipſum:</s> <s xml:id="echoid-s36266" xml:space="preserve"> tũc erũt oẽs line æ ductę à pũcto d ad hũc arcũ <lb/>declinatæ in partẽ unã, & remotiores maiores & maiorẽ angulum cõtinentes uerſus pũctũ g, ꝗ̃ ꝓ-<lb/>pinquiores ք pendiculari.</s> <s xml:id="echoid-s36267" xml:space="preserve"> Si ergo ſumãtur tria pũcta in arcu a b, q̃ ſint e, c, b:</s> <s xml:id="echoid-s36268" xml:space="preserve"> & finis cõtigẽtiæ pũcti <lb/>b ſit l:</s> <s xml:id="echoid-s36269" xml:space="preserve"> & finis cõtingẽtiæ pũcti c ſit m:</s> <s xml:id="echoid-s36270" xml:space="preserve"> palã ք 45 huius, ꝗa ex eo, qđ pũctũ c eſt ꝓpinquius urſui d ꝗ̃ <lb/>pũctus b:</s> <s xml:id="echoid-s36271" xml:space="preserve">erit pũctus m ꝓpinquior cẽtro g ꝗ̃ pũctus l:</s> <s xml:id="echoid-s36272" xml:space="preserve">ſunt aũt lineæ g b & g c ęquales exhypotheſi, <lb/>& ք definitionẽ circuli:</s> <s xml:id="echoid-s36273" xml:space="preserve">eſt ergo linea c m maior ꝗ̃ b l.</s> <s xml:id="echoid-s36274" xml:space="preserve"> Sit aũt q imago pũcti c, & ſit timago pũcti b:</s> <s xml:id="echoid-s36275" xml:space="preserve"> & <lb/>ducatur linea q t:</s> <s xml:id="echoid-s36276" xml:space="preserve"> & ducãtur lineæ c b & m l:</s> <s xml:id="echoid-s36277" xml:space="preserve">q̃ ꝗ dẽ ꝓductę cõcurrẽt.</s> <s xml:id="echoid-s36278" xml:space="preserve"> Quia ſi à pũcto m ducatur linea <lb/>ęquidiſtans lineæ c b illa ſecabit exlinea g b lineam ęqualem ipſim c per 2 p 6:</s> <s xml:id="echoid-s36279" xml:space="preserve">eſt autem c m maior <lb/>quàm b l:</s> <s xml:id="echoid-s36280" xml:space="preserve">concurrant ergo lineæ c b & m lin puncto o.</s> <s xml:id="echoid-s36281" xml:space="preserve"> Et quoniam per 12 huius proportio eſt lineæ <lb/> <anchor type="figure" xlink:label="fig-0561-01a" xlink:href="fig-0561-01"/> g c ad g q, ſicut lineæ c m ad m q:</s> <s xml:id="echoid-s36282" xml:space="preserve"> <lb/>erit ք 16 p 5 permutatim propor-<lb/>tio g c ad c m, ſicut g q ad q m:</s> <s xml:id="echoid-s36283" xml:space="preserve"> & <lb/>ſimiliter erit g b ad b l, ſicut g t ad <lb/>t l:</s> <s xml:id="echoid-s36284" xml:space="preserve">ergo per 124 th.</s> <s xml:id="echoid-s36285" xml:space="preserve"> 1 huius, cumli <lb/>neæ g c & g b angulariter cõiun-<lb/>ctæ ſint proportionaliter diuiſæ, <lb/>& à punctis ſectionum ducantur <lb/>lineæ cõcurrẽtes, quæ c o & m o:</s> <s xml:id="echoid-s36286" xml:space="preserve"> <lb/>palàm quòd linea q t concurret <lb/>cum lineis c b, m l:</s> <s xml:id="echoid-s36287" xml:space="preserve"> & erit ipſarum <lb/>concurſus in puncto o.</s> <s xml:id="echoid-s36288" xml:space="preserve"> Finis ue-<lb/>rò contingẽtiæ puncti e ſit n.</s> <s xml:id="echoid-s36289" xml:space="preserve"> Et <lb/>quoniam pũctus n per 45 huius <lb/>demiſsior eſt puncto m:</s> <s xml:id="echoid-s36290" xml:space="preserve"> erit, ut <lb/>prius, e n lιnea maior quã linea <lb/>c m.</s> <s xml:id="echoid-s36291" xml:space="preserve"> Productis ergo lineis e c & <lb/>n m, patet, ut prius, quòd concur <lb/>rent:</s> <s xml:id="echoid-s36292" xml:space="preserve"> ſit ergo punctus concurſus p:</s> <s xml:id="echoid-s36293" xml:space="preserve"> & ducatur linea q p, & procedat donec ſecet lineam e g in pun-<lb/>cto f:</s> <s xml:id="echoid-s36294" xml:space="preserve"> & producatur linea o q uſq;</s> <s xml:id="echoid-s36295" xml:space="preserve"> ad lineam e g, quã ſecet in puncto k.</s> <s xml:id="echoid-s36296" xml:space="preserve"> Palã quoq;</s> <s xml:id="echoid-s36297" xml:space="preserve"> propter hoc, quòd <lb/>punctus n eſt demiſsior puncto m, quia punctum k erit ſuperius quã punctum f, & linea g k maior <lb/>erit quá g f:</s> <s xml:id="echoid-s36298" xml:space="preserve"> patet autẽ per 123 th.</s> <s xml:id="echoid-s36299" xml:space="preserve"> 1 huius quoniam proportio lineæ g e ad e n eſt ſicut lineæ g f ad fn:</s> <s xml:id="echoid-s36300" xml:space="preserve"> <lb/>ſed finis contingentiæ eſt punctus n:</s> <s xml:id="echoid-s36301" xml:space="preserve">locus ergo imaginis erit punctus f per 12 huius.</s> <s xml:id="echoid-s36302" xml:space="preserve"> Igitur linea f q <lb/>t erit imago arcus circuli e c b:</s> <s xml:id="echoid-s36303" xml:space="preserve"> & erit linea curua, nõ recta, utpote arcus illis tribus punctis per 5 p 4 <lb/>circũſcriptus.</s> <s xml:id="echoid-s36304" xml:space="preserve"> Nõ erit aũt ille arcus æ ꝗ diſtans arcui ſpeculi neq;</s> <s xml:id="echoid-s36305" xml:space="preserve"> arcui uiſo:</s> <s xml:id="echoid-s36306" xml:space="preserve">quoniã, ut patet, lineę t b <lb/>& q c & f e ſunt inæ quales, propter q đ remanẽt lineę g t, g q & g finęquales.</s> <s xml:id="echoid-s36307" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s36308" xml:space="preserve"> demõ-<lb/>ſtrãdũ ſi perpẽdicularis ducta à puncto d, cadat ex alia parte arcus a b citra ipſum:</s> <s xml:id="echoid-s36309" xml:space="preserve"> tunc enim ſimilis <lb/>erit probatio.</s> <s xml:id="echoid-s36310" xml:space="preserve"> Patet ergo propoſitum primum.</s> <s xml:id="echoid-s36311" xml:space="preserve"> Si uerò perpendicularis ducta à puncto d ſuper ſuքfi <lb/>ciem incidentię cadat in medio arcus a b:</s> <s xml:id="echoid-s36312" xml:space="preserve">lineę à puncto d ex diuerſis partibus ad arcum ductę ęqua <lb/>liter diſtantes à perpendiculari, erunt æ quales, & ęquales angulos continentes uerſus punctum g:</s> <s xml:id="echoid-s36313" xml:space="preserve"> <lb/>& imagines ipſarum æ qualiter diſtabunt à centro g:</s> <s xml:id="echoid-s36314" xml:space="preserve"> & fines contingentiarum ſimiliter.</s> <s xml:id="echoid-s36315" xml:space="preserve"> Imago itaq;</s> <s xml:id="echoid-s36316" xml:space="preserve"> <lb/>æ quidiſtabit arcui a b, & arcui ſpeculi:</s> <s xml:id="echoid-s36317" xml:space="preserve"> quoniam imago figurabitur ſuper centrum ſpeculi, quod eſt <lb/>g:</s> <s xml:id="echoid-s36318" xml:space="preserve"> & erit illi concẽtrica per 73 th.</s> <s xml:id="echoid-s36319" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s36320" xml:space="preserve"> Poteſt quoq;</s> <s xml:id="echoid-s36321" xml:space="preserve"> probari predicto modo de utraq;</s> <s xml:id="echoid-s36322" xml:space="preserve"> parte arcus <lb/>per ſe, ſecundum quod diuiditur à perpen diculari:</s> <s xml:id="echoid-s36323" xml:space="preserve"> quòd eius imago ſit linea curua modo prædicto <lb/>æ quidiſtans arcui uiſo propter æ qualitatem linearum à centro ſpeculi & arcus uiſi ad loca imagi-<lb/>nũ productarũ.</s> <s xml:id="echoid-s36324" xml:space="preserve"> Quod eſt propoſitũ:</s> <s xml:id="echoid-s36325" xml:space="preserve"> de imagine enim arcus a e poteſt ſecũdũ præmiſſa idem patere.</s> <s xml:id="echoid-s36326" xml:space="preserve"/> </p> <div xml:id="echoid-div1469" type="float" level="0" n="0"> <figure xlink:label="fig-0561-01" xlink:href="fig-0561-01a"> <variables xml:id="echoid-variables630" xml:space="preserve">p o b c e l m t n a q k f d g</variables> </figure> </div> </div> <div xml:id="echoid-div1471" type="section" level="0" n="0"> <head xml:id="echoid-head1127" xml:space="preserve" style="it">48. Imago arcus eccentrici circulo (qui eſt cõmunis ſectio ſuperficiei incidẽtiæ & ſpeculi ſphæ <lb/>rici conuexi) ſecundũ mediũ eius punctum propinquior is cẽtro ſpeculi (uiſu exiſtẽte extra ſuքfi <lb/>ciẽ incidentiæ) uidetur rnaιoris curuit at is qua arcus etdẽ circulo ſpeculi æքdiſtãtis. Alha. 3 n 6.</head> <p> <s xml:id="echoid-s36327" xml:space="preserve">Eſto arcus uiſus b e a:</s> <s xml:id="echoid-s36328" xml:space="preserve">circulus q́;</s> <s xml:id="echoid-s36329" xml:space="preserve"> communis ſuperficiei reflexionis & ſpeculi ſit h z:</s> <s xml:id="echoid-s36330" xml:space="preserve"> cuius centrũ <lb/>ſit g:</s> <s xml:id="echoid-s36331" xml:space="preserve">ſitq́;</s> <s xml:id="echoid-s36332" xml:space="preserve"> arcus b e a eccentricus arcui h z:</s> <s xml:id="echoid-s36333" xml:space="preserve"> ſint tamẽ iſti arcus in eadẽ ſuperficie:</s> <s xml:id="echoid-s36334" xml:space="preserve"> & ſit e medius pun-<lb/>ctus arcus b e a ꝓpinquior cẽtro g:</s> <s xml:id="echoid-s36335" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s36336" xml:space="preserve"> uiſus extra ſuperficiẽ incidẽtiæ.</s> <s xml:id="echoid-s36337" xml:space="preserve"> Dico quòd imago arcus b a <lb/>erit curua, & maioris curuitatis ꝗ̃ imago alterius arcus cõcentrici ipſi ſpeculo.</s> <s xml:id="echoid-s36338" xml:space="preserve"> Ducatur enim linea à <lb/>cetro ſpeculi, q đ eſt g, ad cẽtrũ arcus b e a, q đ ſit f:</s> <s xml:id="echoid-s36339" xml:space="preserve"> ꝓ ductaq́ linea g e, palã ք 7 p 3 quoniã ipſa eſt bre-<lb/> <pb o="260" file="0562" n="562" rhead="VITELLONIS OPTICAE"/> uior omnibus lineis à cẽtro g ad arcũ a e b productis.</s> <s xml:id="echoid-s36340" xml:space="preserve"> Et quoniã arcus b e eſt æ qualis arcui e a, palã <lb/> <anchor type="figure" xlink:label="fig-0562-01a" xlink:href="fig-0562-01"/> per 7 p 3 quoniam linea g a ęqualis eſt lineę g b:</s> <s xml:id="echoid-s36341" xml:space="preserve"> ductis q́;</s> <s xml:id="echoid-s36342" xml:space="preserve"> lineis g a, <lb/>g b, ſecũdum ipſarũ quantitatem deſcribatur arcus â centro g:</s> <s xml:id="echoid-s36343" xml:space="preserve"> pa-<lb/>lamq́;</s> <s xml:id="echoid-s36344" xml:space="preserve"> per pręmiſſa, quoniam arcus deſcriptus ſecundum ſui pun-<lb/>ctum medium magis diſtabit ab arcu h z, quàm arcus b e a, Sit ergo <lb/>deſcriptus arcus b d a:</s> <s xml:id="echoid-s36345" xml:space="preserve"> & ducatur linea g d ad medium punctum il-<lb/>lius arcus, quę erit ęqualis g b:</s> <s xml:id="echoid-s36346" xml:space="preserve">excedit ergo arcus b d a arcum b e a.</s> <s xml:id="echoid-s36347" xml:space="preserve"> <lb/>Manifeſtum aũt ex præcedentib.</s> <s xml:id="echoid-s36348" xml:space="preserve"> quia imago arcus b d a eſt curua <lb/>uiſu qualitercunq;</s> <s xml:id="echoid-s36349" xml:space="preserve"> ſe habente ad ſuperficiem reflexionis.</s> <s xml:id="echoid-s36350" xml:space="preserve"> Puncta er <lb/>go cõmunia iſtis duobus arcubus, quę ſunt a & b, habebunt imagi-<lb/>nes ſuas ſitas uniformiter priorib.</s> <s xml:id="echoid-s36351" xml:space="preserve"> ſed cum punctũ d ſit remotius à <lb/>centro g quã punctum e:</s> <s xml:id="echoid-s36352" xml:space="preserve">eius imago erit propinquior centro ſpecu <lb/>li quá imago puncti e per 23 th.</s> <s xml:id="echoid-s36353" xml:space="preserve"> huius:</s> <s xml:id="echoid-s36354" xml:space="preserve"> & ita cuiuslibet pũcti arcus <lb/>b d a imago eſt propinquior centro imagine puncti ſibi correſpon <lb/>dentis in arcu b e a.</s> <s xml:id="echoid-s36355" xml:space="preserve"> Quare uidebitur imago arcus a e b curuior ima <lb/>gine arcus a d b.</s> <s xml:id="echoid-s36356" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s36357" xml:space="preserve"> Et ſecundum hunc <lb/>modũ in alijs ſitibus arcuũ & ſpeculorũ poteſt fieri demonſtratio, <lb/>quando uiſus non fuerit in ſuperficie incidentiæ, ſed extra illam.</s> <s xml:id="echoid-s36358" xml:space="preserve"/> </p> <div xml:id="echoid-div1471" type="float" level="0" n="0"> <figure xlink:label="fig-0562-01" xlink:href="fig-0562-01a"> <variables xml:id="echoid-variables631" xml:space="preserve">b d e a h l z g f</variables> </figure> </div> </div> <div xml:id="echoid-div1473" type="section" level="0" n="0"> <head xml:id="echoid-head1128" xml:space="preserve" style="it">49. In ſpeculis ſphæricis cõexis, uiſu nõ exiſtẽte in ſuperficie li <lb/>neæ rectæ æꝗdiſtãtis ſpeculo, imago uidetur curua. Alha. 14 n 6.</head> <p> <s xml:id="echoid-s36359" xml:space="preserve">Sit linea recta uiſa a b:</s> <s xml:id="echoid-s36360" xml:space="preserve"> & ſit ſpeculi ſphærici conuexi centrum g:</s> <s xml:id="echoid-s36361" xml:space="preserve"> erit ergo ſuperficies incidentiæ <lb/> <anchor type="figure" xlink:label="fig-0562-02a" xlink:href="fig-0562-02"/> a g b:</s> <s xml:id="echoid-s36362" xml:space="preserve"> extra quã ſit centrum uiſus, quod ſit d:</s> <s xml:id="echoid-s36363" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s36364" xml:space="preserve"> linea a b æ quidiſtãs <lb/>ſpeculo:</s> <s xml:id="echoid-s36365" xml:space="preserve"> hoc eſt lineæ contingenti arcum eirculi (qui eſt communis <lb/>ſectio ſuperficiei incidentiæ & ſuperficiei ſpeculi) ſecundum mediũ <lb/>punctum illius arcus.</s> <s xml:id="echoid-s36366" xml:space="preserve"> Dico quò d imago lineæ rectæ a b curua uide-<lb/>bitur.</s> <s xml:id="echoid-s36367" xml:space="preserve"> Ducantur enim lineę rectę:</s> <s xml:id="echoid-s36368" xml:space="preserve">d g à centro uiſus ad centrum ſpecu <lb/>li, & g b, g a à centro ſpeculi ad terminos lineæ a b.</s> <s xml:id="echoid-s36369" xml:space="preserve"> Hæ autem lineæ <lb/>a g & b g, cum linea a b æ quidiſtet ſpeculo, palàm quò d ſunt ęquales.</s> <s xml:id="echoid-s36370" xml:space="preserve"> <lb/>Fiat ergo circulus concentricus ſpeculo ſecundum quantitatem illa-<lb/>rum linearum, qui ſit a e b:</s> <s xml:id="echoid-s36371" xml:space="preserve"> cadet ergo linea a b intra illum circulum:</s> <s xml:id="echoid-s36372" xml:space="preserve"> <lb/>eritq́;</s> <s xml:id="echoid-s36373" xml:space="preserve"> per 46 uel 47 huius imago arcus a e b curua.</s> <s xml:id="echoid-s36374" xml:space="preserve"> Sit ergo imago ar <lb/>cus a e b arcus z th, ita quòd imago pũcti a ſit z, & imago pũcti e ſit t, <lb/>& imago puncti b ſit h:</s> <s xml:id="echoid-s36375" xml:space="preserve"> & ducatur linea g e ſecans rectá a b in pũcto f.</s> <s xml:id="echoid-s36376" xml:space="preserve"> <lb/>Palá ergo quòd punctus e eſt in eadẽ linea cũ puncto f, ſed remotior <lb/>à centro g:</s> <s xml:id="echoid-s36377" xml:space="preserve">erit ergo per 23 huius imago puncti e propinquior centro <lb/>ſpeculi, quàm imago puncti f:</s> <s xml:id="echoid-s36378" xml:space="preserve"> communiũ uerò punctorũ, quæ ſunt <lb/>a & b, imagines ſunt e ædem.</s> <s xml:id="echoid-s36379" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s36380" xml:space="preserve"> punctus m imago puncti f:</s> <s xml:id="echoid-s36381" xml:space="preserve"> erit <lb/>ergo z m h imago a b lineæ rectæ.</s> <s xml:id="echoid-s36382" xml:space="preserve"> Patet autem quòd linea z m h eſt li-<lb/>nea curua, cum linea z t h ſit recta:</s> <s xml:id="echoid-s36383" xml:space="preserve"> & omnium punctorum lineæ re-<lb/>ctę, quæ a f, loca imaginum ordinentur ſecundum conuenientem ſi-<lb/>bi proportionẽ inter puncta z & m, reſpectu arcus z m, & omniũ pun <lb/>ctorũ lineę b floca imaginũ ordinentur ſecundũ conuenientẽ ſibi proportionẽ inter puncta h & m <lb/>reſpectu arcus h m.</s> <s xml:id="echoid-s36384" xml:space="preserve"> Patet ergo propoſitũ:</s> <s xml:id="echoid-s36385" xml:space="preserve"> reſe ctisq́ue lineis a f & b f ęqualiter, eadẽ eſt demõſtratio.</s> <s xml:id="echoid-s36386" xml:space="preserve"/> </p> <div xml:id="echoid-div1473" type="float" level="0" n="0"> <figure xlink:label="fig-0562-02" xlink:href="fig-0562-02a"> <variables xml:id="echoid-variables632" xml:space="preserve">b e a f d h m t z g</variables> </figure> </div> <figure> <variables xml:id="echoid-variables633" xml:space="preserve">q e b a d h m z g</variables> </figure> </div> <div xml:id="echoid-div1475" type="section" level="0" n="0"> <head xml:id="echoid-head1129" xml:space="preserve" style="it">50. Lineæ rectæ non æquidiſtantis ſpeculo, quæ producta non cõ-<lb/>tingeret, uel ſecaret ſuքficiẽ ſpeculi ſphærici cõuexi (uiſu nõ exiſten <lb/>te in ſuperficie incidentiæ) imago uidetur curua. Alhaz. 15 n 6.</head> <p> <s xml:id="echoid-s36387" xml:space="preserve">Diſponantur omnia, ut in præcedente, niſi quòd linea a b non æ-<lb/>quidiſtet ſpeculo, nec contingat, nec ſecet ſpeculum, ſed tantùm ob-<lb/>liquetur ſuper ipſum.</s> <s xml:id="echoid-s36388" xml:space="preserve"> Palàm ergo quòd lineę g b & g a productę ſunt <lb/>inæquales.</s> <s xml:id="echoid-s36389" xml:space="preserve"> Sit ergo a g maior quàm g b:</s> <s xml:id="echoid-s36390" xml:space="preserve"> & fiat circulus ſuper centrũ <lb/>g ad quantitatem lineę a g maioris, qui ſit a e q:</s> <s xml:id="echoid-s36391" xml:space="preserve"> & ducatur g b ultra b, <lb/>uſquequò cadat in circulum, in punctũ e.</s> <s xml:id="echoid-s36392" xml:space="preserve"> Patet autem ex 46 uel 47 <lb/>huius quoniá imago arcus a e eſt curua:</s> <s xml:id="echoid-s36393" xml:space="preserve"> punctus aũt imaginis a ſit z.</s> <s xml:id="echoid-s36394" xml:space="preserve"> <lb/>pũctus uerò imaginis e ſit m:</s> <s xml:id="echoid-s36395" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s36396" xml:space="preserve"> z m imago arcus a e.</s> <s xml:id="echoid-s36397" xml:space="preserve"> Et quoniã i-<lb/>mago pũcti b eſt remotior à cẽtro imagine pũcti e per 23 huius:</s> <s xml:id="echoid-s36398" xml:space="preserve"> patet <lb/>quòd erit imago lineę a b curua:</s> <s xml:id="echoid-s36399" xml:space="preserve"> qđ etiã per pũcta media arcus a e & <lb/>lineę a b faciliter poterit oſtendi.</s> <s xml:id="echoid-s36400" xml:space="preserve"> Patet ergo propoſitũ:</s> <s xml:id="echoid-s36401" xml:space="preserve"> reſectaq́ue li-<lb/>nea a b ex quacũq;</s> <s xml:id="echoid-s36402" xml:space="preserve"> ſui parte ſemper eadẽ eſt demõſtratio, quæ prius.</s> <s xml:id="echoid-s36403" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1476" type="section" level="0" n="0"> <head xml:id="echoid-head1130" xml:space="preserve" style="it">51. Imago lineæ rectæ, quæ product a contingeret ſpeculum ſphæ <lb/>ricum conuexum (uiſu non exiſtente in ſuperficie incidẽtiæ) ſem <lb/>per uidetur curua. Alhazen 16 n 6.</head> <p> <s xml:id="echoid-s36404" xml:space="preserve">Sit diſpofitio, quæ prius, ita tamen ut linea a b producta, contingat ſpeculum in puncto e:</s> <s xml:id="echoid-s36405" xml:space="preserve"> & du-<lb/>cantur à centro ſpeculi, quod ſit g, lineæ g b & g a:</s> <s xml:id="echoid-s36406" xml:space="preserve">ſitq́, utſuperficies incidentiæ, quæ ſit a b g ſecet <lb/> <pb o="261" file="0563" n="563" rhead="LIBER SEXTVS."/> ſpeculum in arcu e h z:</s> <s xml:id="echoid-s36407" xml:space="preserve"> & ſit d cẽtrũ uiſus:</s> <s xml:id="echoid-s36408" xml:space="preserve"> ſitq́ ſectio communis ſuperficiei reflexionis (in qua ſunt <lb/>lineæ g a & g d) & ſuperſiciei ſpeculi, arcus z p.</s> <s xml:id="echoid-s36409" xml:space="preserve"> Communis uerò ſectio ſuperficiei reflexionis (in qua <lb/>ſunt lineæ g h & g d) & ſuperficiei ſpeculi, ſit arcus h p.</s> <s xml:id="echoid-s36410" xml:space="preserve"> Palàm ergo per ea, quæ demõſtrata ſunt in 16 <lb/>huius, quòd forma pũcti b reflectitur ad uiſum d ab aliquo pũcto arcus h p.</s> <s xml:id="echoid-s36411" xml:space="preserve"> Si ergo à pũcto illo du-<lb/>catur linea cõtingẽs arcũ h p, illa ſecabit lineã b g:</s> <s xml:id="echoid-s36412" xml:space="preserve"> & finis cõtingẽtiæ erit pũctus illius ſectionis.</s> <s xml:id="echoid-s36413" xml:space="preserve"> Sit <lb/>pũctus ille m.</s> <s xml:id="echoid-s36414" xml:space="preserve"> Palã etiã quòd ſià pũcto m ducatur linea cõtingẽs arcũ e h.</s> <s xml:id="echoid-s36415" xml:space="preserve"> qđilla cadet citra punctũ <lb/>e ք 60 th.</s> <s xml:id="echoid-s36416" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s36417" xml:space="preserve"> quoniã linea a b producta eſt cõtingens circulũ in pũctus b eſt altior pũ <lb/>cto m.</s> <s xml:id="echoid-s36418" xml:space="preserve"> Cadat ergo contingẽs à pũcto m ducta in pũctũ f:</s> <s xml:id="echoid-s36419" xml:space="preserve"> & hęc cõtingẽs producta in cotinuũ & dire <lb/>ctum per 60 th.</s> <s xml:id="echoid-s36420" xml:space="preserve"> 1 huius ſecabit lineam a e:</s> <s xml:id="echoid-s36421" xml:space="preserve"> ergo ſecet in pũcto t:</s> <s xml:id="echoid-s36422" xml:space="preserve"> & ex alia parte ſecabιt lineam g a per <lb/>14 th.</s> <s xml:id="echoid-s36423" xml:space="preserve"> 1 huius, cum illæ omnes lineę ſint in una ſuperficie:</s> <s xml:id="echoid-s36424" xml:space="preserve"> ſecet ergo ipſam in puncto c.</s> <s xml:id="echoid-s36425" xml:space="preserve"> Fiat quoque <lb/> <anchor type="figure" xlink:label="fig-0563-01a" xlink:href="fig-0563-01"/> ſuper g terminum lineę b g angulus æqualis angulo <lb/>b g d per 23 p 1, qui ſit angulus b g s, cadente puncto s <lb/>in peripheriam circuli:</s> <s xml:id="echoid-s36426" xml:space="preserve"> & producatur linea g s ad æ-<lb/>qualitatem lineæ g d, quæ ſit g l.</s> <s xml:id="echoid-s36427" xml:space="preserve"> Erit ergo per 26 p 3 <lb/>arcus s h æ qualis arcui h p.</s> <s xml:id="echoid-s36428" xml:space="preserve"> sicut ergo reflectitur ſor-<lb/>ma puncti b ad uiſum in puncto d ab aliquo puncto <lb/>arcus h p:</s> <s xml:id="echoid-s36429" xml:space="preserve"> ſicreflectetur ad punctum l ab aliquo pun-<lb/>cto arcus h s:</s> <s xml:id="echoid-s36430" xml:space="preserve"> & erit reflexio à puncto f, ſicut in arcu <lb/>h p fit reflexio à puncto, à quo ducitur contingens ad <lb/>punctum m:</s> <s xml:id="echoid-s36431" xml:space="preserve"> quoniam illi arcus neceſſariò ſunt ęqua-<lb/>les, ut patet per 58 th.</s> <s xml:id="echoid-s36432" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s36433" xml:space="preserve"> Et quoniam à puncto m <lb/>uenit utraque illarum linearum contingentium:</s> <s xml:id="echoid-s36434" xml:space="preserve"> palã <lb/>quòd ipſæ ambæ ſunt æ quales per 58 th.</s> <s xml:id="echoid-s36435" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s36436" xml:space="preserve"> Du <lb/>cantur ergo lineę b f & l f.</s> <s xml:id="echoid-s36437" xml:space="preserve"> Similiter quo q;</s> <s xml:id="echoid-s36438" xml:space="preserve"> forma pun <lb/>cti a reflectitur per 16 huius ad uiſum d ab aliquo pun <lb/>cto arcus z p.</s> <s xml:id="echoid-s36439" xml:space="preserve"> Verùm in triangulo curuilineo h z p duo arcus h z & h p ſunt maiores tertio per 28 p 3 <lb/>& per 20 p 1:</s> <s xml:id="echoid-s36440" xml:space="preserve"> ſed arcus h p eſt æqualis arcui h s:</s> <s xml:id="echoid-s36441" xml:space="preserve"> igitur arcus z p eſt minor arcu z s.</s> <s xml:id="echoid-s36442" xml:space="preserve"> Reſcindatur ergo <lb/>arcus z s ad æ qualitatem arcus z p (quod poteſt fieri auxilio 34 p 3) ſit ergo factum in puncto y:</s> <s xml:id="echoid-s36443" xml:space="preserve"> & <lb/>ducatur linea g y:</s> <s xml:id="echoid-s36444" xml:space="preserve"> quæ producta ad æqualitatem lineæ g s, ſecabit neceſſario lineam f l:</s> <s xml:id="echoid-s36445" xml:space="preserve"> ideo quia <lb/>linea g d eſt æqualis lineæ g l.</s> <s xml:id="echoid-s36446" xml:space="preserve"> Quia itaque linea illa ſecat angulum l g z:</s> <s xml:id="echoid-s36447" xml:space="preserve"> ergo ſecabit etiam baſim ei <lb/>ſubtenſam per 29 th.</s> <s xml:id="echoid-s36448" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s36449" xml:space="preserve"> Secet ergo in puncto x:</s> <s xml:id="echoid-s36450" xml:space="preserve"> & ſit linea g y k ęqualis lineæ g d.</s> <s xml:id="echoid-s36451" xml:space="preserve"> Palam ergo <lb/>quoniam ſicut ſorma puncti a reflectitur ad uiſum d ab aliquo puncto arcus z p:</s> <s xml:id="echoid-s36452" xml:space="preserve"> ſimiliter eadem for <lb/>ma puncti a refle ctitur ad k ab aliquo puncto arcus.</s> <s xml:id="echoid-s36453" xml:space="preserve"> z y.</s> <s xml:id="echoid-s36454" xml:space="preserve"> Sed non reflectetur a ad k, niſi ab aliquo pun <lb/>cto.</s> <s xml:id="echoid-s36455" xml:space="preserve"> quod eſt citra punctum fex parte puncti z.</s> <s xml:id="echoid-s36456" xml:space="preserve"> Si enim dicatur quòd a puncto f uel ab alio puncto <lb/>arcus f y reflectitur forma puncti a ad punctum k:</s> <s xml:id="echoid-s36457" xml:space="preserve"> ſit, ut fiat illa reflexio à puncto f:</s> <s xml:id="echoid-s36458" xml:space="preserve"> palàm ergo quòd <lb/>tunclinea ducta à puncto a ad punctum reflexionis f, ſecabit in aliquo puncto lineam b f:</s> <s xml:id="echoid-s36459" xml:space="preserve"> quia linea <lb/>contingens circulum in puncto e tranſit per punctum b.</s> <s xml:id="echoid-s36460" xml:space="preserve"> Ad illud ergo punctum communis ſectio-<lb/>nis illarum linearum a f & b f reflectetur punctus k, & ad idem punctum à puncto freflectetur pun-<lb/>ctus l:</s> <s xml:id="echoid-s36461" xml:space="preserve"> & ita duo puncta in his ſpeculis reflectentur ad idem punctum ab eodem puncto f & exea-<lb/>dem parte diametri uiſualis, quod eſt contra 16 huius.</s> <s xml:id="echoid-s36462" xml:space="preserve"> Sed neque ab alio puncto arcus f y:</s> <s xml:id="echoid-s36463" xml:space="preserve"> quoniã <lb/>tunc, ut prius, linea ducta à puncto a ad punctũ reflexionis, ſecabit lineã b f:</s> <s xml:id="echoid-s36464" xml:space="preserve"> ſit punctũ ſectionis u.</s> <s xml:id="echoid-s36465" xml:space="preserve"> <lb/>Ad illud ergo punctũ ſectionis u reflectetur ſorma pũcti k & forma puncti l:</s> <s xml:id="echoid-s36466" xml:space="preserve"> & ita duo pũcta eiuſdẽ <lb/>diſtãtię à centro propoſiti ſpeculi, quod eſt pũctũ g (quoniã ambæ l g, k g ſunt ęquales ipſi g d exhy <lb/>potheſi) reflectentur ad idem centrum uiſus ex eadem parte diametri uiſualis, quæ a b illo puncto <lb/>ſectionis lineę b f, quod eſtu, eſt ducibilis ad punctum g centrũ ſpeculi, Erit ergo per 18 huius angu <lb/>lus l g u æqualis angulo k g u, totum ſuæ parti:</s> <s xml:id="echoid-s36467" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s36468" xml:space="preserve"> Non ergo refle ctitur forma pun <lb/>cti a ad punctum k ab aliquo puncto arcus f y:</s> <s xml:id="echoid-s36469" xml:space="preserve"> reſtat ergo, ut punctus a refle ctatur ad punctum k ab <lb/>aliquo puncto arcus z f alio, quàm ſit punctum f.</s> <s xml:id="echoid-s36470" xml:space="preserve"> Si igitur ab illo puncto ducatur linea contingens <lb/>circulum, illa producta neceſſariò ſecabit lineã a z:</s> <s xml:id="echoid-s36471" xml:space="preserve"> & cadetinter puncta z & cper 60 th.</s> <s xml:id="echoid-s36472" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s36473" xml:space="preserve"> i-<lb/>deo quòd punctus ſ reſpectu diametri g a demiſsior eſt quolibet puncto arcus z f:</s> <s xml:id="echoid-s36474" xml:space="preserve"> & ita linea con-<lb/>tingens à puncto f, quæeſt f c, altior eſt alijs contingentibus à punctis arcus z f ductis.</s> <s xml:id="echoid-s36475" xml:space="preserve"> Cadat ergo <lb/>contingens illa in punctum n:</s> <s xml:id="echoid-s36476" xml:space="preserve"> & ducatur linea m n:</s> <s xml:id="echoid-s36477" xml:space="preserve"> quæ quidẽ linea cum tranſeat per acumen trian <lb/>guli b m t, & producta diuidat angulum b m t per 15 p 1.</s> <s xml:id="echoid-s36478" xml:space="preserve"> quoniã & ipſa diuidit angulũ g m c, ut patet <lb/>expræmiſsis.</s> <s xml:id="echoid-s36479" xml:space="preserve"> Quia ergo diuidit b m t, ergo neceſſariò ſecabit baſim b t per 29 th.</s> <s xml:id="echoid-s36480" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s36481" xml:space="preserve"> Secet ergo <lb/>ipſam in puncto q:</s> <s xml:id="echoid-s36482" xml:space="preserve"> & ducatur linea g q:</s> <s xml:id="echoid-s36483" xml:space="preserve"> ſit autẽ i imago puncti a:</s> <s xml:id="echoid-s36484" xml:space="preserve"> & ſit o imago puncti b:</s> <s xml:id="echoid-s36485" xml:space="preserve"> & r ſit imago <lb/>puncti q.</s> <s xml:id="echoid-s36486" xml:space="preserve"> Palàm autem ex 45 huius, cum punctum b ſit propinquius puncto g centro ſpeculi quàm <lb/>punctum a, quod erit imago puncti b remotior a puncto g quã i imago puncti a.</s> <s xml:id="echoid-s36487" xml:space="preserve"> Ducatur ergo linea <lb/>o i, quæ per 11 huius erit imago lineæ a b.</s> <s xml:id="echoid-s36488" xml:space="preserve"> Palàm etiam per 12 huius & per 16 p 5 quòd proportio a g <lb/>ad a n eſt, ſicut g i ad i n, & proportio b g ad b m per eadem eſt ſicut g o ad o m.</s> <s xml:id="echoid-s36489" xml:space="preserve"> Cum ergo lineę a g & <lb/>b g diuidantur ſecundum proportionem ſimilem, utraq:</s> <s xml:id="echoid-s36490" xml:space="preserve"> ipſarum in duobus punctis, & à punctis di <lb/>uiſionum ducantur lineæ, quarum duæ ſcilicet g q & m n concurrant ad idẽ punctum q, tertia (quę <lb/>eſt i o) neceſſariò concurret ad idem punctum per 124 th.</s> <s xml:id="echoid-s36491" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s36492" xml:space="preserve"> Linea ergo i o producta cadet ſu-<lb/>per punctum q:</s> <s xml:id="echoid-s36493" xml:space="preserve"> eſt ergo linea io q linea recta.</s> <s xml:id="echoid-s36494" xml:space="preserve"> Igitur linea i o r non erit recta:</s> <s xml:id="echoid-s36495" xml:space="preserve"> ſed linea i o r eſt imago <lb/>lineæ a q.</s> <s xml:id="echoid-s36496" xml:space="preserve"> Quare palã quòd imago lineę a q erit curua.</s> <s xml:id="echoid-s36497" xml:space="preserve"> Poſito aũt b loco pũcti q, & alio pũcto lineæ a <lb/> <pb o="262" file="0564" n="564" rhead="VITELLONIS OPTICAE"/> b poſito loco pũctib, eodẽ modo penitus ꝓbatur, qđ imago lineæ a b eſt curua.</s> <s xml:id="echoid-s36498" xml:space="preserve"> Et hoc eſt ꝓpoſitũ.</s> <s xml:id="echoid-s36499" xml:space="preserve"/> </p> <div xml:id="echoid-div1476" type="float" level="0" n="0"> <figure xlink:label="fig-0563-01" xlink:href="fig-0563-01a"> <variables xml:id="echoid-variables634" xml:space="preserve">l k x s y e t q b a f r g o h m u i z n c p d</variables> </figure> </div> </div> <div xml:id="echoid-div1478" type="section" level="0" n="0"> <head xml:id="echoid-head1131" xml:space="preserve" style="it">52. Imago lineæ rectæ, quæ producta ſecaret circulum (qui eſt communis ſectio ſuperficiei inci <lb/>dentiæ, & ſuperſiciei ſpeculi ſphærici conuexi) non tamen per centrum, uiſu non exiſtente in ſu-<lb/>perſicie incidentiæ, uidetur curua, Alhazen 17 n 6.</head> <p> <s xml:id="echoid-s36500" xml:space="preserve">Manente priori diſpoſitione, ſit, ut linea a b producta, circulũ e h z (qui eſt cõmunis ſectio ſuperfi <lb/>ciei incidentiæ & ſpeculi) ſecet in pũcto e:</s> <s xml:id="echoid-s36501" xml:space="preserve"> & punctus reflexionis formæ puncti b ad punctum l ſit <lb/>punctum f:</s> <s xml:id="echoid-s36502" xml:space="preserve"> & ſit m finis contingentiæ lineę contingentis circulum e h z in pũcto f productæ ad li-<lb/> <anchor type="figure" xlink:label="fig-0564-01a" xlink:href="fig-0564-01"/> neam b g.</s> <s xml:id="echoid-s36503" xml:space="preserve"> Reflectetur itaque b ad a ab aliquo pũcto <lb/>arcus h p, ſicut in præcedente propoſitione pręmiſ-<lb/>ſum eſt.</s> <s xml:id="echoid-s36504" xml:space="preserve"> Arcus quoq;</s> <s xml:id="echoid-s36505" xml:space="preserve"> ab illo puncto reflexionis uſq;</s> <s xml:id="echoid-s36506" xml:space="preserve"> <lb/>ad pũctum h, aut eſt æ qualis arcui h e, aut maior, aut <lb/>minor.</s> <s xml:id="echoid-s36507" xml:space="preserve"> Si ęqualis, cum per præmiſſa in pręcedente <lb/>arcus ille ſit ęqualis arcui h f:</s> <s xml:id="echoid-s36508" xml:space="preserve"> ideo quia à puncto m <lb/>productę lineæ contingentes pertingunt ad arcus <lb/>æquales per 59 th, 1 huius.</s> <s xml:id="echoid-s36509" xml:space="preserve"> Sit ergo q punctus ipſius <lb/>circuli, in quem cadet contingens ducta à puncto <lb/>m ex parte e.</s> <s xml:id="echoid-s36510" xml:space="preserve"> Igitur linea a e tranſit per punctum q:</s> <s xml:id="echoid-s36511" xml:space="preserve"> <lb/>& ita linea m q ſecat lineam a e trãs punctũ e:</s> <s xml:id="echoid-s36512" xml:space="preserve"> quo-<lb/>niam utrunq;</s> <s xml:id="echoid-s36513" xml:space="preserve"> punctorum e & q eſt in peripheria cir <lb/>culi, & eſt punctum unum.</s> <s xml:id="echoid-s36514" xml:space="preserve"> Si uero arcus ille ſit mi-<lb/>nor arcu h e:</s> <s xml:id="echoid-s36515" xml:space="preserve"> ſecabit linea q m lineam e a ultra pun-<lb/>ctũ q:</s> <s xml:id="echoid-s36516" xml:space="preserve"> ſit, ut ſecet ipſam in puncto t, ut eſſiciatur triã <lb/>gulus ducta linea e q.</s> <s xml:id="echoid-s36517" xml:space="preserve"> Si uerò arcus ille fuerit maior <lb/>arcu h e:</s> <s xml:id="echoid-s36518" xml:space="preserve"> ſecabit linea m q lineam a e citra punctum <lb/>q:</s> <s xml:id="echoid-s36519" xml:space="preserve"> quodcunque iſtorum acciderit:</s> <s xml:id="echoid-s36520" xml:space="preserve"> iteretur probatio præmiſſę:</s> <s xml:id="echoid-s36521" xml:space="preserve"> & eodem modo penitus probabitur.</s> <s xml:id="echoid-s36522" xml:space="preserve"> <lb/>quòd imago lineę a b eſt curua.</s> <s xml:id="echoid-s36523" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s36524" xml:space="preserve"/> </p> <div xml:id="echoid-div1478" type="float" level="0" n="0"> <figure xlink:label="fig-0564-01" xlink:href="fig-0564-01a"> <variables xml:id="echoid-variables635" xml:space="preserve">l k x b a s y g p d</variables> </figure> </div> <figure> <variables xml:id="echoid-variables636" xml:space="preserve">d e g h a b z</variables> </figure> <figure> <variables xml:id="echoid-variables637" xml:space="preserve">d e g h b a z</variables> </figure> </div> <div xml:id="echoid-div1480" type="section" level="0" n="0"> <head xml:id="echoid-head1132" xml:space="preserve" style="it">53. Imago lineæ rectæ, quæ producta tranſiret centrum circuli <lb/>(quieſt cõmunis ſectio ſuperficiei incidentiæ & ſpeculi ſphærici con-<lb/>uexi) centro uiſus exiſtente in eadem ſuperficie, uel extra illam, nõ <lb/>tamen in illa linea, ſemper uidetur recta, Alhazen 18 n 6.</head> <p> <s xml:id="echoid-s36525" xml:space="preserve">Diſponãtur omnia, utin pręcedentib.</s> <s xml:id="echoid-s36526" xml:space="preserve"> niſi quòd hactenus locuti ſu <lb/>mus de paſsionibus harum linearum, uiſu non exiſtente in ſuperficie <lb/>incidentię, & nunc uiſum ſupponimus quandoq;</s> <s xml:id="echoid-s36527" xml:space="preserve"> eſſe in ſuperficie in <lb/>cidentię, qui ſit, ut prius, in pũcto d:</s> <s xml:id="echoid-s36528" xml:space="preserve"> & ducatur linea g d:</s> <s xml:id="echoid-s36529" xml:space="preserve"> concurratq́ <lb/>linea a b protracta cum circulo e h z, tranſiens ipſius centrum g.</s> <s xml:id="echoid-s36530" xml:space="preserve"> Palá <lb/>ergo quòd angulus illarum linearũ a g, & d g cadet ſuper g centrũ ſpe <lb/>culi:</s> <s xml:id="echoid-s36531" xml:space="preserve"> uidebiturq́;</s> <s xml:id="echoid-s36532" xml:space="preserve"> imago lineę a b una linea recta.</s> <s xml:id="echoid-s36533" xml:space="preserve"> Imago enim cuiusli <lb/>bet puncti illius lineę a b, cum ipſa ſit in catheto ſuæ incidentię diſpo <lb/>ſita, apparebit in ipſa linea a b producta ad centrum g per 11 huius:</s> <s xml:id="echoid-s36534" xml:space="preserve"> e-<lb/>rit ergo imago illius totius lineę recta, ſicut & ipſalinea a b producta, <lb/>eſt linea recta.</s> <s xml:id="echoid-s36535" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s36536" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1481" type="section" level="0" n="0"> <head xml:id="echoid-head1133" xml:space="preserve" style="it">54. Lineæ rectæ declinatæ à centro circuli (quieſt communis ſe-<lb/>ctio ſuperficiei incidentiæ & ſpeculi ſphærici conuexi) centro uiſus <lb/>exiſtente in eadem ſuperficie incidentiæ, ita quòd declinatio lineæ <lb/>ſit adpartem aliam à uiſu, & ſit tangens ſuperficiem ſpeculi, tan-<lb/>tùm imago unius puncti uidetur. Alhazen 19 n 6.</head> <p> <s xml:id="echoid-s36537" xml:space="preserve">Ordinentur omnia, ut prius in 51 huius:</s> <s xml:id="echoid-s36538" xml:space="preserve"> & ſit linea a b declinata ſu-<lb/>per circulũ e h z, ita qđ nõ attingat centrũ eius:</s> <s xml:id="echoid-s36539" xml:space="preserve"> ſitq́ uiſus d in ſuperfi <lb/>cie incidẽtię:</s> <s xml:id="echoid-s36540" xml:space="preserve"> & ſit declinatio lineę ad partẽ aliã ab illa, in qua eſt uiſus:</s> <s xml:id="echoid-s36541" xml:space="preserve"> <lb/>ut ſi uiſus ſit in parte dextra, declinet pũctũ a ad ſiniſtrã, uel ecõtrario:</s> <s xml:id="echoid-s36542" xml:space="preserve"> <lb/>& linea pertingat ad ſuperficiẽ ſpeculi:</s> <s xml:id="echoid-s36543" xml:space="preserve"> dico quòd tantũ unius pũcti <lb/>lineę a b imago uidebitur.</s> <s xml:id="echoid-s36544" xml:space="preserve"> Sumatur enim per auxiliũ 16 huius pũctus <lb/>circuli, à quo reflecti poſsit aliquid ad uiſum, ꝗ ſit h:</s> <s xml:id="echoid-s36545" xml:space="preserve"> & ſumatur aliqua <lb/>linea reſlexionis punctorũ a b lineæ declinatæ, ut pũcti b:</s> <s xml:id="echoid-s36546" xml:space="preserve"> & illa cadet <lb/>forſitan ſuper hanclineã reflexionis d h:</s> <s xml:id="echoid-s36547" xml:space="preserve"> quod ſi fuerit, nõ uidebitur <lb/>quidẽ imago lineæ huius declinatæ, qeę a b, niſi ſecundũ ſolũ illũ pun <lb/>ctũ b:</s> <s xml:id="echoid-s36548" xml:space="preserve"> quod patet ducta catheto incidentiæ à pũcto a, qui ſit a g:</s> <s xml:id="echoid-s36549" xml:space="preserve"> tunc <lb/>enim arcus interiacens punctũ h, à quo reflectitur forma puncti b, & <lb/>punctum ſectionis circuli e h z per cathetũ a g (quod ſit z) continet <lb/>omnia puncta reflexionis formarũ punctorum lineæ a b, ut oſtenſum <lb/>eſt in propoſitione 50 huius.</s> <s xml:id="echoid-s36550" xml:space="preserve"> Producta ergo à cẽtro uiſus ad cen-<lb/>trum ſpeculi linea, quæ ſit d g, ſecans circulum e h z in puncto e:</s> <s xml:id="echoid-s36551" xml:space="preserve"> ſi ſumatur in arcu circuli <lb/> <pb o="263" file="0565" n="565" rhead="LIBER SEXTVS."/> quieh, eitra hanc lineam d h punctus, à quo reflectitur ad uiſum aliquis punctus lineę declinatę a b:</s> <s xml:id="echoid-s36552" xml:space="preserve"> <lb/>ſed ille punctus reflectitur à puncto alιquo arcus h z prius aſsignati, qui eſt terminus lineæ ſuæ refle <lb/>xiõis:</s> <s xml:id="echoid-s36553" xml:space="preserve"> cum linea ſuę reflexionis ſit ultra lineam reflexionis formæ puncti b:</s> <s xml:id="echoid-s36554" xml:space="preserve"> & ita illæ punctus lineæ <lb/>declinatæ reflectitur ad eundem uiſum à duobus punctis arcus ſpeculi:</s> <s xml:id="echoid-s36555" xml:space="preserve"> quod eſt impoſsibile, & cõ <lb/>tra 16 huius.</s> <s xml:id="echoid-s36556" xml:space="preserve"> Non ergo reflectitur ad uiſum ab aliquo puncto arcus e h interiacentis lineam d g, & cõ <lb/>punctum reflexionis formæ puncti b, quiarcus non impeditur per lineam interpoſitam uiſui & ſpe <lb/>culo.</s> <s xml:id="echoid-s36557" xml:space="preserve"> Item ſi aliquis punctorum lineæ a b, præter punctum b, reflecteretur ad uiſum ab aliquo pun-<lb/>cto arcus e h interiacentis lineam d g & punctum reflexionis formæ pũcti b, cum illa puncta omnia <lb/>ſint in eadem ſuperſicie incidentię, ſicut & centrum uiſus:</s> <s xml:id="echoid-s36558" xml:space="preserve"> tunc patet per 1 p 11 quòd omnes lineæ re <lb/>flexionum ſunt in eadem ſuperficie:</s> <s xml:id="echoid-s36559" xml:space="preserve"> linea ergo incidentiæ illius puncti ſecaret lιneam incidentiæ <lb/>formæ puncti b.</s> <s xml:id="echoid-s36560" xml:space="preserve"> Forma ergo puncti illius ſectionis reflecteretur ad eundem uiſum d à duobus pun-<lb/>ctis, ſcilicet à puncto h puncto reflexionis formæ puncti b, & ab alio puncto dato:</s> <s xml:id="echoid-s36561" xml:space="preserve"> quod totũ eſt im-<lb/>poſsibile, & contra 16 huius.</s> <s xml:id="echoid-s36562" xml:space="preserve"> Non ergo reflectitur aliquis punctorum lineę a b, pręter punctũ b, ad <lb/>uiſum d ab aliquo puncto arcus e h diſcooperti.</s> <s xml:id="echoid-s36563" xml:space="preserve"> Licet autem reflectatur quilibet punctus lineę a b <lb/>ab aliquo puncto arcus h z prius ſumpti, non tamen uidebitur, cũ ſit in linea reflexionis, quæ occul-<lb/>tatur uiſui per pręcedentia puncta lineę ſolidę:</s> <s xml:id="echoid-s36564" xml:space="preserve"> & ita linea adiacens lineę reflexionis formæ pun <lb/>cti b non uidetur, uiſu ſic diſpoſito, ut pręmiſſum eſt.</s> <s xml:id="echoid-s36565" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s36566" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1482" type="section" level="0" n="0"> <head xml:id="echoid-head1134" xml:space="preserve" style="it">55. Lineæ rectæ declinatæ à cẽtro circuli (quieſt cõmunis ſectio ſuperficiei incidẽtiæ & ſpeculi <lb/>ſphærici cõuexi) cẽtro uiſus exiſtẽte in eadẽ ſuperficie incidentiæ, it a ꝗ declinatio lineæ ſit ad par <lb/>tẽ uiſus, ſiue ſit tangens ſuperficiẽ ſpeculι ſiue non, nullius puncti imago uidetur. Alhaz. 20 n 6.</head> <p> <s xml:id="echoid-s36567" xml:space="preserve">Sit diſpoſitio, quę ſuprà:</s> <s xml:id="echoid-s36568" xml:space="preserve"> & ſumatur a b linea declinata, ut proponitur:</s> <s xml:id="echoid-s36569" xml:space="preserve"> & eius declinatio ſit ex par <lb/> <anchor type="figure" xlink:label="fig-0565-01a" xlink:href="fig-0565-01"/> te uiſus d:</s> <s xml:id="echoid-s36570" xml:space="preserve"> dico quòd nullus punctus illius lineę uidebitur.</s> <s xml:id="echoid-s36571" xml:space="preserve"> Deture-<lb/>nim, quòd aliquis punctorũ illius lineę poſsit reflecti ab aliquo pun <lb/>cto arcus interiacentis lineam reflexionis, non impeditam per cor-<lb/>pus lineę interiacentis uiſum & ſpeculum & lineam d g, à centro ui-<lb/>ſus ductam ad centrũ ſpeculi:</s> <s xml:id="echoid-s36572" xml:space="preserve"> & ducatur linea ab illo puncto ad pun <lb/>ctum arcus ſumptum:</s> <s xml:id="echoid-s36573" xml:space="preserve"> hęcitaque ſecabit lineam reflexionis:</s> <s xml:id="echoid-s36574" xml:space="preserve"> & pun <lb/>ctus ſectionis reflectitur ad uiſum à duobus punctis ſpeculi:</s> <s xml:id="echoid-s36575" xml:space="preserve"> quod <lb/>eſt impoſsibile.</s> <s xml:id="echoid-s36576" xml:space="preserve"> Si uerò dicatur quòd punctus ſumptus in linea a b <lb/>reflectitur à puncto arcus circuli, qui eſt ſub illa linea a b, hoc erit im <lb/>poſsibile:</s> <s xml:id="echoid-s36577" xml:space="preserve"> quia totus ille arcus occultatur per lineam interpoſitam <lb/>uiſui & ſpeculo, abſcindentem omnes lineas reflexionum ſuorum <lb/>punctorũ.</s> <s xml:id="echoid-s36578" xml:space="preserve"> Et præterea ſecundũ hanc diſpoſitionẽ uiſus eſt ex parte <lb/>anguli minoris lineę obliquè ſpeculo incidentis:</s> <s xml:id="echoid-s36579" xml:space="preserve"> reflexio uero ſolũ <lb/>fit ex parte anguli maioris, ut patet per 33 th.</s> <s xml:id="echoid-s36580" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s36581" xml:space="preserve"> Non eſt ergo <lb/>poſsibile aliquod punctorũ illius lineę reflecti ad uiſumſic ſituatum.</s> <s xml:id="echoid-s36582" xml:space="preserve"> <lb/>Nullius ergo pũcti illius lineę a b imago uidetur.</s> <s xml:id="echoid-s36583" xml:space="preserve"> Quod eſt ꝓpoſitũ</s> </p> <div xml:id="echoid-div1482" type="float" level="0" n="0"> <figure xlink:label="fig-0565-01" xlink:href="fig-0565-01a"> <variables xml:id="echoid-variables638" xml:space="preserve">a d b b g</variables> </figure> </div> </div> <div xml:id="echoid-div1484" type="section" level="0" n="0"> <head xml:id="echoid-head1135" xml:space="preserve" style="it">56. Lineæ rectæ obliquæ, non tangentis ſuperficiem ſpeculi ſphæ <lb/>rici conuexi uiſu exiſtente in ſuperficie incidentiæ, ita quòd obli-<lb/>quatio lineæ ſit ad partem aliam à uiſu: modicùm imaginis uide-<lb/>tur: & erit imago ſemper curua. Alhazen 21 n 6.</head> <p> <s xml:id="echoid-s36584" xml:space="preserve">Diſponantur omnia, utin præcedentibus:</s> <s xml:id="echoid-s36585" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s36586" xml:space="preserve"> linea a b obliquata ſuper ſuperficiem ſpeculi:</s> <s xml:id="echoid-s36587" xml:space="preserve"> ita <lb/> <anchor type="figure" xlink:label="fig-0565-02a" xlink:href="fig-0565-02"/> quòd producta centrum eius non tranſeat nec tangat ſuperſiciem <lb/>ſpeculi:</s> <s xml:id="echoid-s36588" xml:space="preserve"> ſed diſtet punctus b aliqualiter ab illa in aere exiſtens:</s> <s xml:id="echoid-s36589" xml:space="preserve"> ſitq́ue <lb/>uiſus d in ſuperſicie incidentiæ illius lineæ a b.</s> <s xml:id="echoid-s36590" xml:space="preserve"> Dico quòd modicùm <lb/>imaginis lineæ a b uiſui occurret.</s> <s xml:id="echoid-s36591" xml:space="preserve"> Ducatur enim linea d b ſuper ſu-<lb/>perficiem ſpeculi, incidens in punctum c circuli e h z, (qui eſt com-<lb/>munis ſectio ſuperſiciei incidentię & ſuperſiciei ſpeculi:</s> <s xml:id="echoid-s36592" xml:space="preserve">) A‘puncto <lb/>quoque c ducatur linea contingens circulum per 17 p 3, quæ ſit l c m:</s> <s xml:id="echoid-s36593" xml:space="preserve"> <lb/>& ſuper c terminum lineæ m c fiat angulus æ qualis angulo d c l, per <lb/>lineam c f ſecantem lineam a b in puncto f:</s> <s xml:id="echoid-s36594" xml:space="preserve"> & à puncto f ducatur ca-<lb/>thetus f g ad centrum ſpeculi:</s> <s xml:id="echoid-s36595" xml:space="preserve"> & ducatur cathetus b g.</s> <s xml:id="echoid-s36596" xml:space="preserve"> Palàm itaque <lb/>quòd ſorma puncti freflectitur ad uiſum d à puncto c per 20 th.</s> <s xml:id="echoid-s36597" xml:space="preserve"> 5 hu-<lb/>ius:</s> <s xml:id="echoid-s36598" xml:space="preserve"> eritq́ue locus imaginis in linea f g:</s> <s xml:id="echoid-s36599" xml:space="preserve"> ſimiliterq́ue ſorma puncti b, <lb/>cum non habeat aliquod obſtaculum, reflectetur ad uiſum ab aliquo <lb/>puncto ſpeculi:</s> <s xml:id="echoid-s36600" xml:space="preserve"> & locus imaginis erit in linea b g per 11 huius.</s> <s xml:id="echoid-s36601" xml:space="preserve"> Et <lb/>quia propter interpoſitionem lineæ ſolidæ, quæ f b, alia puncta li-<lb/>neæ a b non poſſunt reflecti ad uiſum, niſi puncta lineæ b f, quo-<lb/>rum omnium imago cadit in linea ducta à punctis ſectionum linea-<lb/>rum reflexionum punctorum b & f, & cathetorum b g & f g:</s> <s xml:id="echoid-s36602" xml:space="preserve"> (quæ <lb/>eſt res modica) patet quòd imaginis lineæ a b pars modica uide.</s> <s xml:id="echoid-s36603" xml:space="preserve"> <lb/>tur.</s> <s xml:id="echoid-s36604" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s36605" xml:space="preserve"> Augetur tamen illa quantitas imaginis ſecundum quod centrum <lb/> <pb o="264" file="0566" n="566" rhead="VITELLONIS OPTICAE"/> uiſus in eadem ſuperficie declinat plus ad ſuperficiem ſpeculi.</s> <s xml:id="echoid-s36606" xml:space="preserve"> Vnde ſi uiſus perueniat inter ſuperfi <lb/>ciem ſpeculi & punctum b:</s> <s xml:id="echoid-s36607" xml:space="preserve"> totius lineę a b uidebitur imago.</s> <s xml:id="echoid-s36608" xml:space="preserve"> Tunc enim cadit hæc linea a b inter li-<lb/>neam reflexionis formæ punctia, & inter productam cathetum puncti a ultra lineam a g.</s> <s xml:id="echoid-s36609" xml:space="preserve"> Etſi tali-<lb/>ter ſituetur hęclinea a b, ut cadat inter lineam reflexionis d c & inter lineam per punctum reflexio-<lb/>nis puncti b tranſeuntem ad centrum ſpeculi, poterit uideri imago totius lineæ.</s> <s xml:id="echoid-s36610" xml:space="preserve"> Videbitur autem <lb/>imago totius lineę a b uel partis eius ſemper curua:</s> <s xml:id="echoid-s36611" xml:space="preserve"> quod poteſt oſtendi per modum 50 huius:</s> <s xml:id="echoid-s36612" xml:space="preserve"> & mi <lb/>nuitur curuitas imaginis huius lineę, ſecundum quod magis acceſſerit ad lineam tranſeuntem ad <lb/>centrum per punctum reflexionis formæ puncti b.</s> <s xml:id="echoid-s36613" xml:space="preserve"> Vniuerſaliter uerò quicquid interpoſitum uiſui <lb/>& ſpeculo impedit peruentum formarum punctorum ſpeculi ad uiſum:</s> <s xml:id="echoid-s36614" xml:space="preserve"> illius imago non uidebitur <lb/>in his ſpeculis.</s> <s xml:id="echoid-s36615" xml:space="preserve"> Hæc autem, qu& hic propoſita ſunt, intelligenda ſunt de lineis occurrentibus uiſui <lb/>in arcu circuli, qui apparet uiſui, utpote in arcu, qui interiacet duas contingentes ductas à centro ui <lb/>ſus ad ſpeculum:</s> <s xml:id="echoid-s36616" xml:space="preserve"> quoniam ille ſolum opponitur uiſui per 5 huius:</s> <s xml:id="echoid-s36617" xml:space="preserve"> linearum uerò concurrentium <lb/>cum ſpeculo in parte circuli occulta uiſui, aliqua poteſt eſſe æquidiſtans lineę contingenti, & illa nõ <lb/>uidebitur:</s> <s xml:id="echoid-s36618" xml:space="preserve"> ſimiliter & cõterminalis illi æquidiſtanti, quæ cadit ſub æ quidiſtante, penitus occultabi-<lb/>tur uiſui:</s> <s xml:id="echoid-s36619" xml:space="preserve"> ſed linea conterminalis ęquidiſtanti cadens ſuper ipſam ex parte illa, poterit uideri.</s> <s xml:id="echoid-s36620" xml:space="preserve"> Et hęc <lb/>experimentantium induſtriæ ex præhabitis principijs relinquimus demonſtranda:</s> <s xml:id="echoid-s36621" xml:space="preserve"> erunt tamẽ hoc <lb/>modo uiſarum linearum rectarum imagines ſemper curuę.</s> <s xml:id="echoid-s36622" xml:space="preserve"/> </p> <div xml:id="echoid-div1484" type="float" level="0" n="0"> <figure xlink:label="fig-0565-02" xlink:href="fig-0565-02a"> <variables xml:id="echoid-variables639" xml:space="preserve">a d f b l m e h c z g</variables> </figure> </div> </div> <div xml:id="echoid-div1486" type="section" level="0" n="0"> <head xml:id="echoid-head1136" xml:space="preserve" style="it">57. Viſu exiſtente inſuperficie incidentiæ lineæ rectæ, non concurrentis cum ſuperficie ſpeculi <lb/>ſphærici conuexi, ſed æquidiſtantis lineæ interiacenti centrum ſpeculi & uiſus, uel concurrentis <lb/>cum illa extra ſpeculum ex parte uiſus: imago uidebitur curua. Alhazen 22 n 6.</head> <p> <s xml:id="echoid-s36623" xml:space="preserve">Sit d centrum uiſus:</s> <s xml:id="echoid-s36624" xml:space="preserve"> & g centrum ſpeculi:</s> <s xml:id="echoid-s36625" xml:space="preserve"> & h e ſit linea uiſa:</s> <s xml:id="echoid-s36626" xml:space="preserve"> quę quidem linea non concurrat cũ <lb/>circulo, qui eſt communis ſectio ſuperficiei incidentiæ & ſpeculi, ſed ſit æ quidiſtans lineæ d g, uel <lb/>ſecet eam ex parte d.</s> <s xml:id="echoid-s36627" xml:space="preserve"> Sit quoque a b circulus, qui eſt communis ſectio ſuperficiei incidentiæ uel re-<lb/>flexionis, in qua ſuntlineę d g & h e, & ſuperficiei ſpeculi propoſiti:</s> <s xml:id="echoid-s36628" xml:space="preserve"> & producatur linea h g, in qua <lb/>ſit punctus z imago puncti h.</s> <s xml:id="echoid-s36629" xml:space="preserve"> Punctus quoq;</s> <s xml:id="echoid-s36630" xml:space="preserve"> circuli, à quo reflectitur forma puncti h ad uiſum d, ſit <lb/>b:</s> <s xml:id="echoid-s36631" xml:space="preserve"> ducaturq́ue à puncto b linea circulum contingens, quæ ſecet lineam h g ſuper punctum t:</s> <s xml:id="echoid-s36632" xml:space="preserve"> eritq́ <lb/>punctus t finis contingentie.</s> <s xml:id="echoid-s36633" xml:space="preserve"> Ducatur etiam linea g b, quæ producta neceſſariò cõcurret cum linea <lb/>h e.</s> <s xml:id="echoid-s36634" xml:space="preserve"> Sienim h e fuerit ęquidiſtans d g, concurret quidem per 2 th.</s> <s xml:id="echoid-s36635" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s36636" xml:space="preserve"> ſi uerò d g concurrat cum <lb/>h e, multò fortius g b concurret cum eadem per 29 th.</s> <s xml:id="echoid-s36637" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s36638" xml:space="preserve"> Concurſus quoque ille aut erit in li-<lb/>nea h e, aut ultra hanc lineam:</s> <s xml:id="echoid-s36639" xml:space="preserve"> ſi ultra, concurrat in puncto m.</s> <s xml:id="echoid-s36640" xml:space="preserve"> Ducatur quoque linea m g:</s> <s xml:id="echoid-s36641" xml:space="preserve"> quæ erit <lb/>cathetus incidentię puncti m:</s> <s xml:id="echoid-s36642" xml:space="preserve"> & imago puncti m ſit q.</s> <s xml:id="echoid-s36643" xml:space="preserve"> Imaginata quo que linea à pũcto reflexionis <lb/>formæ puncti m ad lineam g m producta:</s> <s xml:id="echoid-s36644" xml:space="preserve"> finis contingentiæ ſit punctus s:</s> <s xml:id="echoid-s36645" xml:space="preserve"> & ducatur linea z q copu <lb/>lanslocaimaginum:</s> <s xml:id="echoid-s36646" xml:space="preserve"> ſimiliter ducatur lineat s cupulans fines cõtingentiarum.</s> <s xml:id="echoid-s36647" xml:space="preserve"> Sit quoque, ut linea <lb/>d g ſecet circulum a b in puncto a:</s> <s xml:id="echoid-s36648" xml:space="preserve"> & producatur à puncto a linea contingens circulum, quæ ſit a u.</s> <s xml:id="echoid-s36649" xml:space="preserve"> <lb/>Palàm itaque quoniam arcus a b eſt minor quarta circuli, cum uiſus d uideat ex circulo minus me-<lb/>dietate per 3 huius:</s> <s xml:id="echoid-s36650" xml:space="preserve"> quare angulus a g b eſt acutus per 33 p 6, & angulus u a g eſt rectus per 18 p 3:</s> <s xml:id="echoid-s36651" xml:space="preserve"> igi-<lb/>tur linea a u concurret cumlinea g b per 14 th.</s> <s xml:id="echoid-s36652" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s36653" xml:space="preserve"> concurrat ergo in puncto u.</s> <s xml:id="echoid-s36654" xml:space="preserve"> Dico quia pun-<lb/> <anchor type="figure" xlink:label="fig-0566-01a" xlink:href="fig-0566-01"/> ctus u cadet ultra punctũ s.</s> <s xml:id="echoid-s36655" xml:space="preserve"> Quia cumper 16 huius <lb/>punctus m reſlectatur ab aliquo puncto arcus a b:</s> <s xml:id="echoid-s36656" xml:space="preserve"> <lb/>& punctus a ſit demiſsior illo puncto reflexionis <lb/>formę puncti m:</s> <s xml:id="echoid-s36657" xml:space="preserve"> erit finis contingentiæ lineæ du-<lb/>ctæ à puncto a contingentis circulum, altior fine <lb/>contingentiæ illius puncti per 60 th.</s> <s xml:id="echoid-s36658" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s36659" xml:space="preserve"> & ita <lb/>erit punctus s demiſsior puncto u.</s> <s xml:id="echoid-s36660" xml:space="preserve"> Protrahatur er <lb/>go linea t s, donec concurrat cumlinea u a:</s> <s xml:id="echoid-s36661" xml:space="preserve"> cõcur <lb/>ret autem per 14 th.</s> <s xml:id="echoid-s36662" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s36663" xml:space="preserve"> & ſit concurſus in pun <lb/>cto k:</s> <s xml:id="echoid-s36664" xml:space="preserve"> & ducatur linea g k:</s> <s xml:id="echoid-s36665" xml:space="preserve"> quæ producta concur-<lb/>ret cum h m per 2 uelper 29 th.</s> <s xml:id="echoid-s36666" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s36667" xml:space="preserve"> ſit concur-<lb/>ſus in puncto c.</s> <s xml:id="echoid-s36668" xml:space="preserve"> Punctus itaque c reflectitur ad ui <lb/>ſum d ab aliquo puncto arcus a b:</s> <s xml:id="echoid-s36669" xml:space="preserve"> quòd patet per <lb/>e a, quæ demonſtrata ſuntin 16 huius:</s> <s xml:id="echoid-s36670" xml:space="preserve"> ſit ille pun-<lb/>ctus ſ:</s> <s xml:id="echoid-s36671" xml:space="preserve"> à quo ducatur linea contingens ſpeculum <lb/>uſque ad cathetum g c:</s> <s xml:id="echoid-s36672" xml:space="preserve"> quæ quidem erit demiſ-<lb/>ſior quàm linea a k:</s> <s xml:id="echoid-s36673" xml:space="preserve"> & ſit fo, ſecans lineam g c in <lb/>puncto o:</s> <s xml:id="echoid-s36674" xml:space="preserve"> qui ſit finis contingentiæ:</s> <s xml:id="echoid-s36675" xml:space="preserve"> ergo per 60 <lb/>th.</s> <s xml:id="echoid-s36676" xml:space="preserve"> 1 huius erit punctus o demiſsior puncto k:</s> <s xml:id="echoid-s36677" xml:space="preserve"> ſunt <lb/>enim puncta k & o fines contingentiarum.</s> <s xml:id="echoid-s36678" xml:space="preserve"> Producatur quoque linea d f, uſquequò cadat ſuper g e <lb/>cathetum:</s> <s xml:id="echoid-s36679" xml:space="preserve"> cadet aũtper 9 huius:</s> <s xml:id="echoid-s36680" xml:space="preserve"> ſit ergo, ut cadat in punctũ r:</s> <s xml:id="echoid-s36681" xml:space="preserve"> & producatur linea z q uſq;</s> <s xml:id="echoid-s36682" xml:space="preserve"> ad lineam <lb/>g c:</s> <s xml:id="echoid-s36683" xml:space="preserve"> & cadat in punctum 1.</s> <s xml:id="echoid-s36684" xml:space="preserve"> Dico quoniam punctum l eſt altius quã punctũ r.</s> <s xml:id="echoid-s36685" xml:space="preserve"> Lineę enim h c & t k, & <lb/>z l aut ſunt ęquidiſtãtes aut cõcurrũt.</s> <s xml:id="echoid-s36686" xml:space="preserve"> Sint primò ęquidiſtãtes:</s> <s xml:id="echoid-s36687" xml:space="preserve"> cũ ergo hæ lineę ęquidiſtãtes ſecent <lb/>lineam c g, ſuper tria puncta c, k, l, & ſecent utranque linearum m g & h g:</s> <s xml:id="echoid-s36688" xml:space="preserve"> & cum ſit proportio lineę <lb/>h g ad h t, ſicut lineæ g z ad z t per 12 huius, & per 16 p 5:</s> <s xml:id="echoid-s36689" xml:space="preserve"> & ſimiliter cum ſit proportio lineæ m g <lb/> <pb o="265" file="0567" n="567" rhead="LIBER SEXTVS."/> ad m s, ſicut g q ad s:</s> <s xml:id="echoid-s36690" xml:space="preserve"> erit eadẽ proportio g c ad c k, quæ g l ad l k per 122 th.</s> <s xml:id="echoid-s36691" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s36692" xml:space="preserve"> ſed palàm per 11 <lb/>huius quoniam r eſt imago puncti c:</s> <s xml:id="echoid-s36693" xml:space="preserve"> linea enim d ſ eſt linea reflexionis, concurrens cum catheto c <lb/>g in puncto r:</s> <s xml:id="echoid-s36694" xml:space="preserve"> & o eſt finis contingentię:</s> <s xml:id="echoid-s36695" xml:space="preserve"> eſt ergo <lb/>per 12 huius, & per 16 p 5 proportio g c ad c o, <lb/> <anchor type="figure" xlink:label="fig-0567-01a" xlink:href="fig-0567-01"/> <anchor type="figure" xlink:label="fig-0567-02a" xlink:href="fig-0567-02"/> ſicut g r ad r o:</s> <s xml:id="echoid-s36696" xml:space="preserve"> led maior eſt proportio g c ad <lb/>c k, quàm g c ad co per 8 p 5:</s> <s xml:id="echoid-s36697" xml:space="preserve"> & ita erit maior <lb/>proportio g l ad l k, quàm g r ad r o:</s> <s xml:id="echoid-s36698" xml:space="preserve"> ergo <lb/>econtrario conuerfim per 6 th.</s> <s xml:id="echoid-s36699" xml:space="preserve"> 1 huius erit mi-<lb/>nor proportio l kad g l, quàm r o ad g r:</s> <s xml:id="echoid-s36700" xml:space="preserve"> eſt ergo <lb/>maior proportio o r ad g r, quàm k l ad g l:</s> <s xml:id="echoid-s36701" xml:space="preserve"> ergo <lb/>coniunctim per 11 th.</s> <s xml:id="echoid-s36702" xml:space="preserve"> 1 huius maior proportio <lb/>eſt o g ad r g, quàm k g ad l g:</s> <s xml:id="echoid-s36703" xml:space="preserve"> ſed k g eſt <lb/>maior quàm o g:</s> <s xml:id="echoid-s36704" xml:space="preserve"> ergo per 14 p 5 l g eſt maior <lb/>quàm r g.</s> <s xml:id="echoid-s36705" xml:space="preserve"> Eſt ergo punctus r demiſsior puncto <lb/>l:</s> <s xml:id="echoid-s36706" xml:space="preserve"> ſed z q l eſt linea recta:</s> <s xml:id="echoid-s36707" xml:space="preserve"> ergo linea z q r eſt linea <lb/>curua:</s> <s xml:id="echoid-s36708" xml:space="preserve"> ergo imago lineę h c eſt curua.</s> <s xml:id="echoid-s36709" xml:space="preserve"> Poſito er-<lb/>go alio puncto lineæ h e loco puncti m, & pun-<lb/>cto e loco puncti c:</s> <s xml:id="echoid-s36710" xml:space="preserve"> erit probare quod imago li-<lb/>neæ h e rectę ſit curua.</s> <s xml:id="echoid-s36711" xml:space="preserve"> Si uerò lineæ h m, t s, & <lb/>z q non ſunt æ quidiſtantes, concurrant ergo:</s> <s xml:id="echoid-s36712" xml:space="preserve"> & <lb/>erit cõcurſus aut ex parte d, aut ex parte h:</s> <s xml:id="echoid-s36713" xml:space="preserve"> ſit ex <lb/>parte d, & concurrant in pun-<lb/>cto c:</s> <s xml:id="echoid-s36714" xml:space="preserve"> erit ergo per 53 huius z q l <lb/>linea recta:</s> <s xml:id="echoid-s36715" xml:space="preserve"> quare z q r erit linea <lb/>curua:</s> <s xml:id="echoid-s36716" xml:space="preserve"> eſt ergo imago lineæ h <lb/>e rectæ curua, demonſtratione <lb/>completa, ut prius.</s> <s xml:id="echoid-s36717" xml:space="preserve"> Hoc ergo <lb/>eſt pro poſitum.</s> <s xml:id="echoid-s36718" xml:space="preserve"/> </p> <div xml:id="echoid-div1486" type="float" level="0" n="0"> <figure xlink:label="fig-0566-01" xlink:href="fig-0566-01a"> <variables xml:id="echoid-variables640" xml:space="preserve">h e m c c u s k b c z q l r f g a d</variables> </figure> <figure xlink:label="fig-0567-01" xlink:href="fig-0567-01a"> <variables xml:id="echoid-variables641" xml:space="preserve">h e m c s t z q b o r f d g a</variables> </figure> <figure xlink:label="fig-0567-02" xlink:href="fig-0567-02a"> <variables xml:id="echoid-variables642" xml:space="preserve">i h e m o c u z s b o k q l r f d o g a</variables> </figure> </div> </div> <div xml:id="echoid-div1488" type="section" level="0" n="0"> <head xml:id="echoid-head1137" xml:space="preserve" style="it">58. Omnis arcus circuli, in <lb/>cuius ſuperſicie incidentiæ fue <lb/>rit centrum uiſus, imago ſen-<lb/>ſibiliter apparens intra ſpecu-<lb/>lum ſphæricum conuexum ui-<lb/>detur ſemper curua. Alha-<lb/>zen 23 n 6.</head> <p> <s xml:id="echoid-s36719" xml:space="preserve">Sit arcus uiſus a b:</s> <s xml:id="echoid-s36720" xml:space="preserve"> & ſit cen-<lb/>trum ſpeculi punctum g:</s> <s xml:id="echoid-s36721" xml:space="preserve"> & cen <lb/>trum uiſus punctum d:</s> <s xml:id="echoid-s36722" xml:space="preserve"> ſiuq́;</s> <s xml:id="echoid-s36723" xml:space="preserve"> hoc centrum uiſus in ſuperficie incidentiæ, quæ eſt a b g Dico quòd <lb/>imago arcus a b uidetur ſemper curua, quando ſenſibιlιter intra ſpeculum uidetur.</s> <s xml:id="echoid-s36724" xml:space="preserve"> Ducatur enim <lb/> <anchor type="figure" xlink:label="fig-0567-03a" xlink:href="fig-0567-03"/> chorda a b:</s> <s xml:id="echoid-s36725" xml:space="preserve"> palamq́;</s> <s xml:id="echoid-s36726" xml:space="preserve"> ex præmιſsis propoſitιonιbus, quoniam ima-<lb/>go chordæ a b ſecundum omnem ſui ſitum, reſpectu ſpeculi uide-<lb/>tur ſemper curua:</s> <s xml:id="echoid-s36727" xml:space="preserve"> niſi ſolùm tunc, quando ipſa ſit in catheto inci-<lb/>dentiæ unius ſuæ extremitatis:</s> <s xml:id="echoid-s36728" xml:space="preserve"> ut cum ipſa eſt perpendicularis ſu <lb/>per ſpeculi ſuperficiem pertranſiens eius centrú:</s> <s xml:id="echoid-s36729" xml:space="preserve"> tunc enim ipſius <lb/>imago uidetur recta, ut patet per 53 huius.</s> <s xml:id="echoid-s36730" xml:space="preserve"> Arcum uero a b eſſe in <lb/>catheto incidentiæ ſuarum extremitatũ eſt impoſsibile:</s> <s xml:id="echoid-s36731" xml:space="preserve"> cum qui-<lb/>libet ſuorum punctorum diuerſam habeat incidentiæ cathetum.</s> <s xml:id="echoid-s36732" xml:space="preserve"> <lb/>Ergo nunquam uidebitur imago arcus taliter diſpoſiti in linea re-<lb/>cta:</s> <s xml:id="echoid-s36733" xml:space="preserve"> quoniam ſemper loca imaginum diuerſorum punctorum in <lb/>diuerſis ſunt cathetis.</s> <s xml:id="echoid-s36734" xml:space="preserve"> Curuitas uerò imaginum poteſt faciliter <lb/>concludi ſecundum modum, quo in præcedentibus in lineis re-<lb/>ctis uſi ſumus:</s> <s xml:id="echoid-s36735" xml:space="preserve"> & coadiuuabit ad hæc 45 huius.</s> <s xml:id="echoid-s36736" xml:space="preserve"> Patet ergo pro-<lb/>poſitum.</s> <s xml:id="echoid-s36737" xml:space="preserve"/> </p> <div xml:id="echoid-div1488" type="float" level="0" n="0"> <figure xlink:label="fig-0567-03" xlink:href="fig-0567-03a"> <variables xml:id="echoid-variables643" xml:space="preserve">a b d g</variables> </figure> </div> </div> <div xml:id="echoid-div1490" type="section" level="0" n="0"> <head xml:id="echoid-head1138" xml:space="preserve" style="it">59. Conuexitas imaginum quorumlibet arcuum, cum locus <lb/>ipſarũ eſt intra ſpeculum ſphæricum cõue xum uelextra ipſum, <lb/>conuexit ati arcuum fit contraria ſecundum ſitum.</head> <p> <s xml:id="echoid-s36738" xml:space="preserve">Eſto quòd arcus a b reſpiciat ſecundum ſui concauum uel conuexum centrum ſpeculi ſphærici <lb/>conuexi, quod ſit punctum g.</s> <s xml:id="echoid-s36739" xml:space="preserve"> Dico quòd conuexitas ipſius imaginis erit cõtraria ſecundum ſitum <lb/>conuexitati ipſius ſpeculi, quando imagoto taliter eſt intra ſpeculũ, ueltotaliter extra, uelſecun-<lb/> <pb o="266" file="0568" n="568" rhead="VITELLONIS OPTICAE"/> dum partem intra, ſecundum partem extra, & ſecundum partem in ipſa ſuperficie ſpeculi Loca e-<lb/>nim imaginum punctorum remotiorum à ſuperficie ſpeculi fiunt <lb/> <anchor type="figure" xlink:label="fig-0568-01a" xlink:href="fig-0568-01"/> propinquiora centro ſpeculi, & loca punctorum propinquiorum <lb/>ſuperficiei ſpeculi fiunt remotiora à centro ſpeculi, ut patet per 23 <lb/>huius.</s> <s xml:id="echoid-s36740" xml:space="preserve"> Et quia imagines accipiunt continuitatem ſitus ſuarum par-<lb/>tium à continuitate rerum, quarum ipſę ſuntimagines:</s> <s xml:id="echoid-s36741" xml:space="preserve"> patet quòd <lb/>conuexitas ipſarum imaginum conuexitati ipſorũ uiſorum arcuum <lb/>fit contraria ſecundum ſitum, prout etiam oſtendimus in 46 huius.</s> <s xml:id="echoid-s36742" xml:space="preserve"> <lb/>Patet ergo propoſitum.</s> <s xml:id="echoid-s36743" xml:space="preserve"/> </p> <div xml:id="echoid-div1490" type="float" level="0" n="0"> <figure xlink:label="fig-0568-01" xlink:href="fig-0568-01a"> <variables xml:id="echoid-variables644" xml:space="preserve">a b g</variables> </figure> </div> </div> <div xml:id="echoid-div1492" type="section" level="0" n="0"> <head xml:id="echoid-head1139" xml:space="preserve" style="it">60. Imaginum curuarum eiuſdem arcus uiſi remotioris à cen-<lb/>tro ſpeculi ſphærici conuexi curuior uidetur.</head> <p> <s xml:id="echoid-s36744" xml:space="preserve">Sit a b arcus uiſus, cuius punctus medius ſit e:</s> <s xml:id="echoid-s36745" xml:space="preserve"> & cuius arcus ima-<lb/>go ſit curua:</s> <s xml:id="echoid-s36746" xml:space="preserve"> & eius chorda ſit a b linea recta:</s> <s xml:id="echoid-s36747" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s36748" xml:space="preserve"> centrum ſpeculi <lb/>g.</s> <s xml:id="echoid-s36749" xml:space="preserve"> Dico quòd accedente linea a b ad ſpeculum, imago eius fit mino-<lb/>ris curuitatis, & recedente ipſa fit maioris.</s> <s xml:id="echoid-s36750" xml:space="preserve"> Ducantur enim catheti <lb/>a g & b g, in quibus erunt loca imaginum punctorum a & b per 11 hu <lb/>ius.</s> <s xml:id="echoid-s36751" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s36752" xml:space="preserve"> accedente linea recta a b ad ſuperficiem ſpeculi, angulus a g b fit maior, & receden-<lb/>te ipſa angulus a g b fit minor per 34 th.</s> <s xml:id="echoid-s36753" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s36754" xml:space="preserve"> imago uerò pũcti <lb/> <anchor type="figure" xlink:label="fig-0568-02a" xlink:href="fig-0568-02"/> e plus elongati à centro ſpeculi fit propinquior centro ſpeculi, & <lb/>imago eiuſdem approximantis ſpeculo fit remotior à centro per <lb/>23 th.</s> <s xml:id="echoid-s36755" xml:space="preserve"> huius:</s> <s xml:id="echoid-s36756" xml:space="preserve"> extrema uerò puncta illius imaginis ſemper ſunt in <lb/>cathetis a g & a b:</s> <s xml:id="echoid-s36757" xml:space="preserve"> patet ergo quòd imago arcus a b remotioris à <lb/>centró ſpeculi plus coanguſtatur, & approximati plus amplia-<lb/>tur:</s> <s xml:id="echoid-s36758" xml:space="preserve"> & ſecundũ hoc ipſius curuitatis modus uariatur modo propo <lb/>fito.</s> <s xml:id="echoid-s36759" xml:space="preserve"> Quoniam ipſius remotioris à centro ſpeculi imago fit cur-<lb/>uior, & propinquioris fit minus curua:</s> <s xml:id="echoid-s36760" xml:space="preserve"> quoniam ipſa ſemper fit <lb/>pars circuli maioris in acceſſu ad centrũ ſpeculi, & fit pars circuli <lb/>minoris in receſſu à cẽtro:</s> <s xml:id="echoid-s36761" xml:space="preserve"> & ſecundũ quantitatẽ acceſſus illius & <lb/>receſſus uariatur quãtitas dictarũ imaginũ.</s> <s xml:id="echoid-s36762" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s36763" xml:space="preserve"/> </p> <div xml:id="echoid-div1492" type="float" level="0" n="0"> <figure xlink:label="fig-0568-02" xlink:href="fig-0568-02a"> <variables xml:id="echoid-variables645" xml:space="preserve">b i d B t k e a c z g</variables> </figure> </div> </div> <div xml:id="echoid-div1494" type="section" level="0" n="0"> <head xml:id="echoid-head1140" xml:space="preserve" style="it">61. Omnis imago in ſuperficie ſpeculi ſphærici conuexi uiſui <lb/>occurrens, ſemper apparet conuexa. Euclides 23. th. catoptr.</head> <p> <s xml:id="echoid-s36764" xml:space="preserve">Eſto ſpeculum ſphæricum conuexum a g:</s> <s xml:id="echoid-s36765" xml:space="preserve"> & ſit centrum uiſus <lb/>e:</s> <s xml:id="echoid-s36766" xml:space="preserve"> & ſit linea recta uel curua uiſa, quæ d k:</s> <s xml:id="echoid-s36767" xml:space="preserve"> in qua ſignentur puncta <lb/>b & i, ſitq́;</s> <s xml:id="echoid-s36768" xml:space="preserve"> utloca imaginum iſtorum punctorum ſint in ſuperficie <lb/>ipſius ſpeculi, lineis incidentiæ exiſtentibus ipſis, quæ d a, b c, i z, <lb/>& k g:</s> <s xml:id="echoid-s36769" xml:space="preserve"> lineis quoq;</s> <s xml:id="echoid-s36770" xml:space="preserve"> reflexionis exiſtẽtibus a e, c e, z e, & g e.</s> <s xml:id="echoid-s36771" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s36772" xml:space="preserve"> <lb/>aliqua illarũ linearũ reflexionis ſit perpẽdicularis ſuper ſuperficiẽ <lb/>ſpeculi:</s> <s xml:id="echoid-s36773" xml:space="preserve"> palàm per 72 th.</s> <s xml:id="echoid-s36774" xml:space="preserve"> 1 huius quoniã ipſa trãſibit centrũ ſpeculi:</s> <s xml:id="echoid-s36775" xml:space="preserve"> ergo ք <lb/>8 p 3 uel per 21 th.</s> <s xml:id="echoid-s36776" xml:space="preserve"> 1 huius illa erit breuiſsima omniũ linearũ illarũ reflexio <lb/>nis, & illi propinquiores ſunt remotioribus breuiores.</s> <s xml:id="echoid-s36777" xml:space="preserve"> Patet ergo quoniã <lb/> <anchor type="figure" xlink:label="fig-0568-03a" xlink:href="fig-0568-03"/> illa imago uidetur curua:</s> <s xml:id="echoid-s36778" xml:space="preserve"> quoniam aliqua pars ipſius propinquior eſt ui-<lb/>ſui, & aliqua remotior.</s> <s xml:id="echoid-s36779" xml:space="preserve"> Idem quoq;</s> <s xml:id="echoid-s36780" xml:space="preserve"> accidit, ſi nulla illarum linearum refle-<lb/>xionis ſit perpendicularis ſuper ſpeculi ſuperficiem:</s> <s xml:id="echoid-s36781" xml:space="preserve"> quoniam ducta per-<lb/>pendiculari linea à puncto e ſuper ſuperficiem ſpeculi per 11 p 11:</s> <s xml:id="echoid-s36782" xml:space="preserve"> palàm <lb/>quòd omnes lineæ reflexionis illi perpendiculari propinquiores remo-<lb/>tioribus ſunt breuiores:</s> <s xml:id="echoid-s36783" xml:space="preserve"> & ſic item imago lineæ rectæ uel curuæ, quæ eſt <lb/>d k, occurrens uiſui in ſuperficie ſpeculi, uidetur ſemper curua.</s> <s xml:id="echoid-s36784" xml:space="preserve"> Et quoniã <lb/>eodem modo eſt demonſtrandum de qualibet imagine apparente in ſu-<lb/>perficie ſpeculi:</s> <s xml:id="echoid-s36785" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s36786" xml:space="preserve"/> </p> <div xml:id="echoid-div1494" type="float" level="0" n="0"> <figure xlink:label="fig-0568-03" xlink:href="fig-0568-03a"> <variables xml:id="echoid-variables646" xml:space="preserve">e a b a e b g</variables> </figure> </div> </div> <div xml:id="echoid-div1496" type="section" level="0" n="0"> <head xml:id="echoid-head1141" xml:space="preserve" style="it">62. Imago lineæ curuæ ſecundũ eius concauit atẽ reſpiciẽtis ſuperficiẽ <lb/>ſpeculi ſphærici conuexi, nonnunquã uidetur recta. Alhazen 23 n 6.</head> <p> <s xml:id="echoid-s36787" xml:space="preserve">Sit linea curua a b c, oppoſita ſpeculo ſphærico cõuexo ſecundũ ſui par <lb/>tem concauã.</s> <s xml:id="echoid-s36788" xml:space="preserve"> Dico qđ nõnunꝗ̃ imago ipſius poteſt uideri li-<lb/> <anchor type="figure" xlink:label="fig-0568-04a" xlink:href="fig-0568-04"/> nea recta.</s> <s xml:id="echoid-s36789" xml:space="preserve"> Ducatur enim eius chorda recta linea, quę ſit a c:</s> <s xml:id="echoid-s36790" xml:space="preserve"> pa <lb/>lamq́;</s> <s xml:id="echoid-s36791" xml:space="preserve"> per plures præmiſſarũ propoſitionum huius, quoniá in <lb/>aliquo ſitu imago ipſius lineę rectæ uidetur curua curua curuitate <lb/>reſpiciẽte centrũ ſpeculi.</s> <s xml:id="echoid-s36792" xml:space="preserve"> Quia ergo extremitates lineę curuę <lb/>a b c, quę ſunt a & c, uidẽtur in extremitatib.</s> <s xml:id="echoid-s36793" xml:space="preserve"> imaginis lineę re <lb/>ctę a c:</s> <s xml:id="echoid-s36794" xml:space="preserve"> imaginetur ipſi curuę imagini lineę rectę a c ſubtendi <lb/>chorda intra ſpeculũ.</s> <s xml:id="echoid-s36795" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s36796" xml:space="preserve"> hoc acciderit, quòd eſt poſsibi-<lb/>le, ut curuitas ipſius arcus (ꝗ eſt a b) ſit ſimilis curuitati imagi-<lb/>nis ipſius chordæ, ita qđ eius ſinus uerſi hinc inde ſint ſimi-<lb/> <pb o="267" file="0569" n="569" rhead="LIBER SEXTVS."/> les:</s> <s xml:id="echoid-s36797" xml:space="preserve"> palá ք 23 & ք 45 huius quòd imago lineę curuę, quę a b c, erit linea recta, ſubtẽſa per modũ chor <lb/>d æ ipſi imagini curuatæ.</s> <s xml:id="echoid-s36798" xml:space="preserve"> Videbitur ergo linea recta imago ipſius curuæ lineę a b c.</s> <s xml:id="echoid-s36799" xml:space="preserve"> Quod eſt propo <lb/>ſitũ.</s> <s xml:id="echoid-s36800" xml:space="preserve"> Patet hoc etiá aliter.</s> <s xml:id="echoid-s36801" xml:space="preserve"> Quia enim, ut in pręmiſſa proxima dictũ eſt, omnis imago in ſuperficie ſpe <lb/>culi ſphærici cõuexi uiſui occurrẽs, ſemper uidetur cũuexa, centrũ ſpeculi reſpiciens ſecundũ eius <lb/>concauitatẽ, & eiuſdem arcus imago cadens intra ſpeculum reſpicit centrum ſpeculi ſecundum ſui <lb/>concauum.</s> <s xml:id="echoid-s36802" xml:space="preserve"> Cum ergo non eatur ab extremo in extremũ ſine medio in huiuſmodi reflexionibus & <lb/>ſuperficiebus partium eiuſdem imaginis:</s> <s xml:id="echoid-s36803" xml:space="preserve"> palàm quòd illa imago in aliquo ſitu habeat diſpoſitionẽ <lb/>rectitudinis.</s> <s xml:id="echoid-s36804" xml:space="preserve"> Et quia omnia loca imaginũ punctorum illius arcus cadunt in unam lineam rectam, <lb/>quem ſitum tamen & uiſus & rei uiſæ & ſpeculi perquirere eſſet longum & inutile:</s> <s xml:id="echoid-s36805" xml:space="preserve"> patebit tamen <lb/>ſimpliciter expræmiſsis uia illud perquirere uolenti.</s> <s xml:id="echoid-s36806" xml:space="preserve"> Per hunc itaq;</s> <s xml:id="echoid-s36807" xml:space="preserve"> modum accidit circulum quan <lb/>doq;</s> <s xml:id="echoid-s36808" xml:space="preserve"> uideri ad modũ ſemicirculi & diametri, & ex portione circuli fit portio reuerſa, ita quòd ima-<lb/>go rectæ lineæ fit curua, & curuæ lineę fit recta:</s> <s xml:id="echoid-s36809" xml:space="preserve"> & quandoq;</s> <s xml:id="echoid-s36810" xml:space="preserve"> ambæ uidentur curuæ ad eandem par <lb/>tem, ſi curuitas arcus uiſi ſit minor curuitate imaginis ſuę chordæ:</s> <s xml:id="echoid-s36811" xml:space="preserve"> & quandoq;</s> <s xml:id="echoid-s36812" xml:space="preserve"> ad partes diuerſas, <lb/>ſicut interſectione duorum circulorũ inæqualium ſuperficies incluſa:</s> <s xml:id="echoid-s36813" xml:space="preserve"> & harum imaginum eſt mul-<lb/>ta diuerſitas, quá ex præmiſsis principijs diligenti ſolertiæ relinquimus exquirendam.</s> <s xml:id="echoid-s36814" xml:space="preserve"> In his itaq;</s> <s xml:id="echoid-s36815" xml:space="preserve"> <lb/>ſpeculis imago lineę rectę apparet curua, & lineę curuę imago ſemper uidetur curua:</s> <s xml:id="echoid-s36816" xml:space="preserve"> & quandoq;</s> <s xml:id="echoid-s36817" xml:space="preserve"> <lb/>apparet uiſui recta.</s> <s xml:id="echoid-s36818" xml:space="preserve"> Et quod oſtendimus de lineis, accidit etiam in ipſis ſuperficiebus planis conca-<lb/>uis & conuexis per lineas, quæ inſunt illis ſuperficiebus:</s> <s xml:id="echoid-s36819" xml:space="preserve"> & idem penitus eſt in lineis longitudinis <lb/>& latitudinis ipſarum.</s> <s xml:id="echoid-s36820" xml:space="preserve"> Si autem proponatur uiſui in his ſpeculis corpus curuũ longum, modicum <lb/>habens latitudinis:</s> <s xml:id="echoid-s36821" xml:space="preserve"> apparebit illius corporis curuitas manifeſtè, cũ ipſa diſcerni poſsit per ea, quæ <lb/>ſunt ſupra corpus, aut circa illud aut infra:</s> <s xml:id="echoid-s36822" xml:space="preserve"> non enim bene diſcernitur curuitas non magna, quando <lb/>occultæ fuerint extremitates longitudinis & latitudinis:</s> <s xml:id="echoid-s36823" xml:space="preserve"> unde in corpore conuexitatis modicę, & <lb/>quantitatis magnæ non bene diſcernitur eius conuexitas, licet imago ipſius ſit conuexa, cum non <lb/>appareant termini corporis in longitudine uellatitudine, qui termini coadiuuantnõ modicè com-<lb/>prehenſionem conuexitatis.</s> <s xml:id="echoid-s36824" xml:space="preserve"/> </p> <div xml:id="echoid-div1496" type="float" level="0" n="0"> <figure xlink:label="fig-0568-04" xlink:href="fig-0568-04a"> <variables xml:id="echoid-variables647" xml:space="preserve">a b c</variables> </figure> </div> </div> <div xml:id="echoid-div1498" type="section" level="0" n="0"> <head xml:id="echoid-head1142" xml:space="preserve" style="it">63. A ſuperficie ſpeculi ſphærici conuexi ex diuerſis ſuperficiebus ſphærarum compoſita, for <lb/>m æ reflexæ monſtruoſæ imaginis uidentur.</head> <p> <s xml:id="echoid-s36825" xml:space="preserve">Quia enim diuerſarum ſphæricarum ſuperficierũ diuerſa ſunt centra, & locus imaginis cuiuſq;</s> <s xml:id="echoid-s36826" xml:space="preserve"> <lb/>puncti in ſpeculis ſphæricis cóuexis per 11 huius eſt in catheto ſuæ incidentię, ducta à puncto uiſo <lb/>ad centrum ſpeculi:</s> <s xml:id="echoid-s36827" xml:space="preserve"> hæc aũt centra diuerſificantur in huiuſmodi ſpeculis irregularibus:</s> <s xml:id="echoid-s36828" xml:space="preserve"> patet ergo <lb/>quòd formę diuerſorum punctorum in partes diuerſas protrahuntur.</s> <s xml:id="echoid-s36829" xml:space="preserve"> Et quoniam à toata ſuperficie <lb/>fit reflexio, & puncta reflexa ſecundum loca diuerſificantur, non ſecundum eundem ſitum:</s> <s xml:id="echoid-s36830" xml:space="preserve"> patet <lb/>quòd imago tota, quæ ex locis talium punctorum aggregatur & unitur, ſuarum partiũ recipit inor-<lb/>dinatũ ſitum.</s> <s xml:id="echoid-s36831" xml:space="preserve"> Videtur ergo imago in talibus ſpeculis monſtruoſa:</s> <s xml:id="echoid-s36832" xml:space="preserve"> & ſit extenſio uniformis aliqua-<lb/>rum ſuarum partium ſecundum uniformem extenſionem illarum ſuperficierũ, & aliarum partium <lb/>fit deformitas ab alijs.</s> <s xml:id="echoid-s36833" xml:space="preserve"> Vnde quædá imaginis partes trahuntur in longũ, quædam in latum, quædã <lb/>in tranſuerſum, ſecundum quod partes aliquę ſuperficiei ſpeculi reſpiciunt diuerſa centra diuerſa-<lb/>rum ſphærarum.</s> <s xml:id="echoid-s36834" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s36835" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1499" type="section" level="0" n="0"> <head xml:id="echoid-head1143" xml:space="preserve" style="it">64. Poßibile eſt per plura, quotcun quis uoluerit, conuexa ſphærica ſpecula eiuſdem puncti <lb/>imaginem uideri. Euclides 15 th. catoptr.</head> <p> <s xml:id="echoid-s36836" xml:space="preserve">Fiat hic diſpoſitio, quæ in 61 th.</s> <s xml:id="echoid-s36837" xml:space="preserve"> 5 huius de ſpeculis planis dicta eſt:</s> <s xml:id="echoid-s36838" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s36839" xml:space="preserve"> a centrum uiſus:</s> <s xml:id="echoid-s36840" xml:space="preserve"> & pun-<lb/>ctus uiſus b:</s> <s xml:id="echoid-s36841" xml:space="preserve"> & deſcribatur, exempli cauſſa, polygonium æ quilaterũ <lb/> <anchor type="figure" xlink:label="fig-0569-01a" xlink:href="fig-0569-01"/> & æ quiangulũ, quod ſit a b g d e:</s> <s xml:id="echoid-s36842" xml:space="preserve"> & ad puncta g, d, e ſint ſpecula ſphæ <lb/>rica conuexa contingentia puncta angulorũ ęqualium:</s> <s xml:id="echoid-s36843" xml:space="preserve"> & imaginen <lb/>tur lineę contingentes ſpecula in eiſdem punctis, ut in puncto g, li-<lb/>nea l k.</s> <s xml:id="echoid-s36844" xml:space="preserve"> Et quoniam angulus b g k eſt æqualis angulo d g l:</s> <s xml:id="echoid-s36845" xml:space="preserve"> palàm per <lb/>20 th.</s> <s xml:id="echoid-s36846" xml:space="preserve"> 5 huius quoniam forma puncti b reflectetur à puncto g ad pun <lb/>ctum d:</s> <s xml:id="echoid-s36847" xml:space="preserve"> & eadem ratione à puncto d ad punctum e, & à puncto e ad <lb/>punctum a.</s> <s xml:id="echoid-s36848" xml:space="preserve"> Hoc autem eſt, quod proponebatur.</s> <s xml:id="echoid-s36849" xml:space="preserve"/> </p> <div xml:id="echoid-div1499" type="float" level="0" n="0"> <figure xlink:label="fig-0569-01" xlink:href="fig-0569-01a"> <variables xml:id="echoid-variables648" xml:space="preserve">e d j g k a b</variables> </figure> </div> </div> <div xml:id="echoid-div1501" type="section" level="0" n="0"> <head xml:id="echoid-head1144" xml:space="preserve" style="it">65. A‘ ſuperficie unius ſpeculi ſphærici conuexiignem impoßi-<lb/>bile eſt accendi: ex plurium tamen compoſitione poßibile.</head> <p> <s xml:id="echoid-s36850" xml:space="preserve">Quoniam enim, ut oſtenſum eſt in 15 huius, lineę reflexionis for-<lb/>mæ eiuſdem puncti à diuerſis punctis eiuſdem ſpeculi ſphærici conuexi non ſunt æquidiſtantes, <lb/>attamen in centro unius uiſus non concurrunt:</s> <s xml:id="echoid-s36851" xml:space="preserve"> ergo neq;</s> <s xml:id="echoid-s36852" xml:space="preserve"> radij ſolares uel alij, ſuperficiei huius ſpe <lb/>culi incidentes in aliquo unquam puncto poſſunt cócurrere, ſed diſperguntur in ipſo medio.</s> <s xml:id="echoid-s36853" xml:space="preserve"> Non <lb/>ergo illi radij aggregati unquá corpus aliquo modo quodcunq;</s> <s xml:id="echoid-s36854" xml:space="preserve"> etiá ipſum ſit combuſtibile, poſſunt <lb/>incendere, ut reflectuntur à ſuperficie ſperculi unius:</s> <s xml:id="echoid-s36855" xml:space="preserve"> ex plurium tamen ſpeculorum compoſitione <lb/>poſſet aliquid huiuſmodi effici, ita ut à quolibet illorum ſpeculorũ uno puncto reflecteretur unus <lb/>radius ad unum punctum, cum aliorum ſpeculorum radijs concurrens:</s> <s xml:id="echoid-s36856" xml:space="preserve"> & ſic ſortificaretur actio ra <lb/>diorum in illo puncto, & ſecundum numerum ſpeculorum ſieret numerus radiorum, & unio uel <lb/>aggregatio uirtutis.</s> <s xml:id="echoid-s36857" xml:space="preserve"> Hæc autem ſpeculorum compoſitio plus eſſet difficilis quàm utilis:</s> <s xml:id="echoid-s36858" xml:space="preserve"> unde tali <lb/>operi nos non dignum credimus inſiſti.</s> <s xml:id="echoid-s36859" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s36860" xml:space="preserve"> propoſitum.</s> <s xml:id="echoid-s36861" xml:space="preserve"/> </p> <pb o="268" file="0570" n="570"/> </div> <div xml:id="echoid-div1502" type="section" level="0" n="0"> <head xml:id="echoid-head1145" xml:space="preserve">VITELLONIS FILII <lb/>THVRINGORVM ET PO-<lb/>LONORVM OPTICAE LIBER SEPTIMVS.</head> <p style="it"> <s xml:id="echoid-s36862" xml:space="preserve">O Rdinis realis ſeries nos admonet, ut qui planorũ ſpeculorũ & ſphæricorum <lb/>conuexorũ paßiones proprias, prout potuimus, trãſcurrimus:</s> <s xml:id="echoid-s36863" xml:space="preserve"> nunc ad ſpecu <lb/>lorũ columnariũ & pyramιdaliũ proprietates diuertamus.</s> <s xml:id="echoid-s36864" xml:space="preserve"> Sunt enim ſpe <lb/>culorũ iſtorũ aliquæ paßiones ex paßionibus præmiſſorũ ſpeculorũ conξtan <lb/>tes uel cõpoſitæ, ſicut & figuræ iſtorum ſpeculorũ ex figuris illorum præmiſſorũ ſpeculo-<lb/>rum aliqualiter cõponuntur.</s> <s xml:id="echoid-s36865" xml:space="preserve"> Speculũ enim columnare cum ſit pars columnæ rotundæ <lb/>(ſicut in 8, 14 & 15 th.</s> <s xml:id="echoid-s36866" xml:space="preserve"> 5 huius declarauimus.</s> <s xml:id="echoid-s36867" xml:space="preserve">) palàm ex præmißis in primo libro hu-<lb/>ius ſcientiæ, & in principijs 11 Euclidis quoniam fit ex tranſitu quadrilateri rectangu-<lb/>li, quod uno ſuorum laterum fixo, motis alijs, circumducitur, quouſ redeat ad locũ, un <lb/>de motus accepit principiũ.</s> <s xml:id="echoid-s36868" xml:space="preserve"> Speculũ quo pyramidale cauſſatur ex motu trigoni rectan <lb/>guli, cuius unũ laterũ rectum angulum continentiũ figitur, & alia duo modo præmiſſo, <lb/>quouſq;</s> <s xml:id="echoid-s36869" xml:space="preserve"> ad locũ, unde moueri cæperunt, circumducuntur.</s> <s xml:id="echoid-s36870" xml:space="preserve"> Vtrumq;</s> <s xml:id="echoid-s36871" xml:space="preserve"> ergo iſtorum ſpecu-<lb/>lorum, quia ex motu linearũ rectarum ortũ habet, palàm quia rectarũ linearũ paßio-<lb/>nes proprias non euadit.</s> <s xml:id="echoid-s36872" xml:space="preserve"> In quantum uerò illæ lineæ cauſſant circulorũ figuras, cũ circu-<lb/>lariter circumferuntur:</s> <s xml:id="echoid-s36873" xml:space="preserve"> in tantũ bæc ſpecula paßiones circulares, hoc eſt ſphæricas, qua-<lb/>rum origo eſt circulus, cõmuniter conſequuntur:</s> <s xml:id="echoid-s36874" xml:space="preserve"> & hoc maximè in ſpeculis columnari-<lb/>bus euidentius apparet, prout manifeſtabimus in proceſſu.</s> <s xml:id="echoid-s36875" xml:space="preserve"> Propriè uerò istorũ ſpeculorũ <lb/>paßiones, ut illæ, quæ ſecundum oxygonias ſectiones accidunt, quæſolis his ſpeculis, ſiue <lb/>ſint conuexa, ſiue concaua, conueniunt, ex quadam cõmuni naura linearum rectarum <lb/>& motus accidunt in illis:</s> <s xml:id="echoid-s36876" xml:space="preserve"> hæc ergo ſpecula posteriorẽ ordinem recipiunt ad plana ſpe-<lb/>cula & ſphærica conuexa.</s> <s xml:id="echoid-s36877" xml:space="preserve"> Prius uerò de his ſpeculis columnaribus & pyramidalibus <lb/>conuexis proſequemur, quàm de quibuſcun concauis & ſphæricis, propter ſimplicita-<lb/>tem paßionum ſpeculorum conuexorum reſpectu concauorum, ut illarum, quæ in alias <lb/>deſcendunt, Quæ uerò præmittimus, ſunt iſta.</s> <s xml:id="echoid-s36878" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1503" type="section" level="0" n="0"> <head xml:id="echoid-head1146" xml:space="preserve">DEFINITIONES.</head> <p> <s xml:id="echoid-s36879" xml:space="preserve">1.</s> <s xml:id="echoid-s36880" xml:space="preserve"> Maius ſpeculum columnare uel pyramidale conuexum uel concauũ dicimus, <lb/>quod eſt pars maioris columnæ uel pyramidis:</s> <s xml:id="echoid-s36881" xml:space="preserve"> & minus, quod eſt pars minoris.</s> <s xml:id="echoid-s36882" xml:space="preserve"> 2.</s> <s xml:id="echoid-s36883" xml:space="preserve"> <lb/>Axem ſpeculi columnaris uel pyramidalis dicimus axem illius columnæ uel pyra-<lb/>midis, cuius pars ſpeculum exiſtit.</s> <s xml:id="echoid-s36884" xml:space="preserve"> 3.</s> <s xml:id="echoid-s36885" xml:space="preserve"> Baſes ſpeculorũ propoſitorũ dìcimus baſcs <lb/>ſuarum columnarum uel pyramidum quarumcunq;</s> <s xml:id="echoid-s36886" xml:space="preserve">. 4.</s> <s xml:id="echoid-s36887" xml:space="preserve"> Diametrũ uiſualem dici <lb/>mus lineam à centro uiſus perpendicularem ſuper ſuperficiem ſpeculi & ad axem <lb/>productam:</s> <s xml:id="echoid-s36888" xml:space="preserve"> & eadẽ dicitur cathetus reflexionis.</s> <s xml:id="echoid-s36889" xml:space="preserve"> 5.</s> <s xml:id="echoid-s36890" xml:space="preserve"> Cathetus incidentię dicitur, ut <lb/>prius, linea pep endicularis ducta à puncto rei uiſæ ſuper lineam, quæ eſt cõmunis <lb/>ſectio ſuperficiei reflexionis & ſpeculi:</s> <s xml:id="echoid-s36891" xml:space="preserve"> utpote ſuper lineam rectã, quæ eſt linea lon <lb/>gitudinis ſpeculi, uel ſuper circulũ, uel ſuper oxygoniã ſectionẽ, ſecundum quod <lb/>ab aliqua iſtarum linearum reflexio procedít.</s> <s xml:id="echoid-s36892" xml:space="preserve"> 6.</s> <s xml:id="echoid-s36893" xml:space="preserve"> Finis contingentiæ dicitur pun-<lb/>ctus, in quo altera cathetorum ſecat lineam in puncto reflexionis ſpeculum ſecun <lb/>dum circulum uel ſectionem oxygoniam contingentem.</s> <s xml:id="echoid-s36894" xml:space="preserve"> 7.</s> <s xml:id="echoid-s36895" xml:space="preserve"> Metam locorum di-<lb/>cimus, ut in ſpeculis ſphęricis, punctũ uel lineam, ultra quã imagines nõ uidentur.</s> <s xml:id="echoid-s36896" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1504" type="section" level="0" n="0"> <head xml:id="echoid-head1147" xml:space="preserve">THEOREMATA</head> <head xml:id="echoid-head1148" xml:space="preserve" style="it">1. Oppoſito uiſui ſpeculo colũnari uel pyramidali conuexo orthogonaliter erecto, ita ut uiſus <lb/>non ſit in ſuperficie ſpeculi, aut ei continua: linea recta à centrc uiſus ducta, cum axe ſpeculi in <lb/> <pb o="269" file="0571" n="571" rhead="LIBER SEPTIMVS."/> uertice acutum angulum tenente: à parte ſuperficiei ſpeculi interiacente ſuperficies contingen-<lb/>tes ductas à centro uiſus ad ſpeculi ſuperſiciem ſolùm fit reflexio ad uiſum. Alhazen 35 n 4.</head> <p> <s xml:id="echoid-s36897" xml:space="preserve">Hoc, quod hic proponitur, uniuerſaliter cõuenit ſpeculo columnari conuexo, ſiue ſecundũ an-<lb/>gulum rectũ ſiue ſecundũ acutum ſibi incidat linea uiſualis:</s> <s xml:id="echoid-s36898" xml:space="preserve"> ſemper enim ſicut per 78 th.</s> <s xml:id="echoid-s36899" xml:space="preserve"> 4 huius o-<lb/>ſtenſum eſt, minus medietate ſuperficiei columnaris uiſui occurrit, & abilla ſolùm fit reflexio ad ui <lb/>ſum.</s> <s xml:id="echoid-s36900" xml:space="preserve"> Hæc aũt ſuperſicies ſpeculi columnaris contenta eſt duabus ſu <lb/> <anchor type="figure" xlink:label="fig-0571-01a" xlink:href="fig-0571-01"/> perſiciebus à cẽtro uiſus productis, ſecundũ lineã lõgitudinis cõtin-<lb/>gentibus columnã.</s> <s xml:id="echoid-s36901" xml:space="preserve"> Et quoniã huius paſsionis idẽ eſt demon ſtrandi <lb/>modus in utroq:</s> <s xml:id="echoid-s36902" xml:space="preserve"> ꝓpoſitorũ ſpeculorũ:</s> <s xml:id="echoid-s36903" xml:space="preserve"> difficilius uerò in pyramida-<lb/>libus, ſufficit, exẽpli cauſſa, propoſitũ in ſpeculis pyramidalibus de-<lb/>monſtrari.</s> <s xml:id="echoid-s36904" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s36905" xml:space="preserve"> ſpeculum pyramidale conuexum, cuius axis ſit a <lb/>d:</s> <s xml:id="echoid-s36906" xml:space="preserve"> & uertex a:</s> <s xml:id="echoid-s36907" xml:space="preserve"> diameter baſis c n:</s> <s xml:id="echoid-s36908" xml:space="preserve"> centrũ baſis d:</s> <s xml:id="echoid-s36909" xml:space="preserve"> & ſit hæc pyramis e-<lb/>recta ſuper ſuperficiem horizontis, ita quòd non inclinetur ſuper il-<lb/>lam:</s> <s xml:id="echoid-s36910" xml:space="preserve"> & ſit centrũ uiſus b:</s> <s xml:id="echoid-s36911" xml:space="preserve"> cõcurratq́;</s> <s xml:id="echoid-s36912" xml:space="preserve"> linea b a à uiſus centro ad uerti-<lb/>cem ſpeculi producta cum axe datæ pyramidis, continens cum ipſo <lb/>angulum acutũ, qui eſt d a b.</s> <s xml:id="echoid-s36913" xml:space="preserve"> Dico quòd ſolùm à parte ſuperficiei co-<lb/>nicæ huius pyramidis, quæ interiacet ſuperficies contingentes du-<lb/>ctas à centro uiſus ad eandẽ ſuperficiem, fit reflexio ad uiſum.</s> <s xml:id="echoid-s36914" xml:space="preserve"> Imagi <lb/>nemur enim ſuperficiẽ à centro uiſus prodeuntẽ, quæ ſecet pyrami-<lb/>dem orthogonaliter per axem:</s> <s xml:id="echoid-s36915" xml:space="preserve"> & palàm per 100 th.</s> <s xml:id="echoid-s36916" xml:space="preserve"> 1 huius quoniam <lb/>cõmunis ſectio illius ſuperficiei, & ſuperficiei pyramidis erit circu-<lb/>lus æquidiſtans baſi pyramidis.</s> <s xml:id="echoid-s36917" xml:space="preserve"> Sit ergo ille circulus f g:</s> <s xml:id="echoid-s36918" xml:space="preserve"> & à centro <lb/>uiſus ducantur duæ lineæ b f & b g illum circulum contingentes per <lb/>17 p 3:</s> <s xml:id="echoid-s36919" xml:space="preserve"> & per 101 th.</s> <s xml:id="echoid-s36920" xml:space="preserve"> 1 huius ducantur à punctis f & g duę lineę longitu <lb/>dinis pyramidis, quę ſint c f a, & n g a.</s> <s xml:id="echoid-s36921" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s36922" xml:space="preserve"> quoniã ſuperficies, <lb/>in qua ſunt lineę c f a & linea b f continget pyramidem.</s> <s xml:id="echoid-s36923" xml:space="preserve"> Si enim dicatur quòd ſecet illam, & nó con <lb/>tingit:</s> <s xml:id="echoid-s36924" xml:space="preserve"> palàm quoniá linea b f, quæ eſt in illa ſuperficie, ſecabit circulũ f g, & non cõtinget:</s> <s xml:id="echoid-s36925" xml:space="preserve"> ducta au-<lb/>tem eſt ad contingentiam:</s> <s xml:id="echoid-s36926" xml:space="preserve"> ſecare igitur eſt impoſsibile.</s> <s xml:id="echoid-s36927" xml:space="preserve"> Superficies ergo illa pyramidem cõtinget.</s> <s xml:id="echoid-s36928" xml:space="preserve"> <lb/>Et ſimiliter oſtendendum eſt de ſuperficie, in qua ſunt lineę n g a & b g, quòd & illa pyramidem con <lb/>tingat.</s> <s xml:id="echoid-s36929" xml:space="preserve"> Superficies ergo pyramidis interiacẽs has duas ſuperficies contingentes, uiſui occurret, & <lb/>ſolùm ab hac fiet reflexio ad uiſum:</s> <s xml:id="echoid-s36930" xml:space="preserve"> quia, ut per 16 th.</s> <s xml:id="echoid-s36931" xml:space="preserve"> 2 huius oſtẽſum eſt, longior radius ad circulũ <lb/>columnæ uel pyramidis rotundarũ perueniens, quaſi linea contingens eſt.</s> <s xml:id="echoid-s36932" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s36933" xml:space="preserve"> <lb/>quoniam in ſpeculo columnari eſt ſimiliter demonſtrandum.</s> <s xml:id="echoid-s36934" xml:space="preserve"/> </p> <div xml:id="echoid-div1504" type="float" level="0" n="0"> <figure xlink:label="fig-0571-01" xlink:href="fig-0571-01a"> <variables xml:id="echoid-variables649" xml:space="preserve">a b f g c d n</variables> </figure> </div> </div> <div xml:id="echoid-div1506" type="section" level="0" n="0"> <head xml:id="echoid-head1149" xml:space="preserve" style="it">2. Si à centro oculi ad lineas, quæ ſunt termini ſuperficierũ ſpe-<lb/>culorum columnarium uel pyramidalium conuexorum apparen-<lb/>lium uiſui, duæ ſuperficies reflexionis producantur: neceſſe eſt per <lb/>ipſas ambas ſpeculum contingi. Alhazen 26 n 4.</head> <p> <s xml:id="echoid-s36935" xml:space="preserve">Verbi gratia, ſint conuexo ſpeculo columnari, quod ſit d f e g, duæ <lb/> <anchor type="figure" xlink:label="fig-0571-02a" xlink:href="fig-0571-02"/> lineæ longitudinis, quæ ſint d e & f g:</s> <s xml:id="echoid-s36936" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s36937" xml:space="preserve"> illæ lineæ termini ſuperfi <lb/>ciei columnæ ſpeculi apparentis uiſui, ut patet ex præmiſſa & per 78 <lb/>th.</s> <s xml:id="echoid-s36938" xml:space="preserve"> 4 huius:</s> <s xml:id="echoid-s36939" xml:space="preserve"> & ſit centrũ uiſus a:</s> <s xml:id="echoid-s36940" xml:space="preserve"> productisq;</s> <s xml:id="echoid-s36941" xml:space="preserve"> lineis a d, a f, a g, a e:</s> <s xml:id="echoid-s36942" xml:space="preserve"> erunt <lb/>ſuperficies trigonę a d e, & a f g.</s> <s xml:id="echoid-s36943" xml:space="preserve"> Drco quòd illæ ſuperficies contingẽt <lb/>columná.</s> <s xml:id="echoid-s36944" xml:space="preserve"> Si enim dicatur qđ altera ipſarũ ſecat columná, ut ſuperfi-<lb/>cies a d e, planũ eſt quòd illa ſectio erit ſuper lineá longitudinis d e, <lb/>in quá cadit illa ſuperficies:</s> <s xml:id="echoid-s36945" xml:space="preserve"> & ſimiliter erit procedendú ſi ſuperficies <lb/>a f g ſecet columná:</s> <s xml:id="echoid-s36946" xml:space="preserve"> & ſit ſectio ſuper lineã f g.</s> <s xml:id="echoid-s36947" xml:space="preserve"> Sit ergo, ut ſuperficies <lb/>plana pertranſiẽs centrũ uiſus, ſecet columná ęquidiſtanter baſibus:</s> <s xml:id="echoid-s36948" xml:space="preserve"> <lb/>eritq́;</s> <s xml:id="echoid-s36949" xml:space="preserve"> per 100 th.</s> <s xml:id="echoid-s36950" xml:space="preserve"> 1 huius ſectio cõmunis illi ſuperficiei & ſpeculi cir-<lb/>culus, qui ſit b c:</s> <s xml:id="echoid-s36951" xml:space="preserve"> hic ergo tranſit per duas lineas longitudinis d e, & f <lb/>g:</s> <s xml:id="echoid-s36952" xml:space="preserve"> ducantur ergo lineæ a b & a c ad hunc circulum.</s> <s xml:id="echoid-s36953" xml:space="preserve"> Hæ ergo cum ſint <lb/>in illis ſuperficiebus ſecantibus ſuperficiem columnę, ſecabunt cir-<lb/>culum b c:</s> <s xml:id="echoid-s36954" xml:space="preserve"> minus ergo uidebitur de arcu b c, quàm ſit illud, quod ſub <lb/>lineis circulũ b c contingentibus à centro uiſus puncto ſcilicet a du-<lb/>ctis continetur, quod eſt contra ea, quæ declarata ſunt in 51 th.</s> <s xml:id="echoid-s36955" xml:space="preserve"> 4 hu-<lb/>ius:</s> <s xml:id="echoid-s36956" xml:space="preserve"> & ſimiliter de baſibus columnæ declarandum.</s> <s xml:id="echoid-s36957" xml:space="preserve"> Non erunt ergo il <lb/>Iæ ſuperficies productæ ad terminos ſuperficiei columnæ a pparen-<lb/>tis uiſui, ſed citra illas:</s> <s xml:id="echoid-s36958" xml:space="preserve"> quod eſt contra hypotheſim.</s> <s xml:id="echoid-s36959" xml:space="preserve"> Eodem modo <lb/>quoq;</s> <s xml:id="echoid-s36960" xml:space="preserve"> eſt de ſpeculis pyramidalibus demonſtrandũ:</s> <s xml:id="echoid-s36961" xml:space="preserve"> & ſequitur idẽ <lb/>impoſsibile, quod prius, per 84 th.</s> <s xml:id="echoid-s36962" xml:space="preserve"> 4 huius:</s> <s xml:id="echoid-s36963" xml:space="preserve"> quod eſt contra hypothe <lb/>ſim.</s> <s xml:id="echoid-s36964" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s36965" xml:space="preserve"/> </p> <div xml:id="echoid-div1506" type="float" level="0" n="0"> <figure xlink:label="fig-0571-02" xlink:href="fig-0571-02a"> <variables xml:id="echoid-variables650" xml:space="preserve">a e g c b d h f</variables> </figure> </div> </div> <div xml:id="echoid-div1508" type="section" level="0" n="0"> <head xml:id="echoid-head1150" xml:space="preserve" style="it">3. Cõmunis ſectio omnium ſuperficierum à uiſu productarũ, contingentiũ ſpeculũ columna-<lb/>re cõuexum, eſt linea tranſiens centrum uiſus æquidiſtanter axi illius ſpeculi. Alhazen 26 n 4.</head> <pb o="270" file="0572" n="572" rhead="VITELLONIS OPTICAE"/> <p> <s xml:id="echoid-s36966" xml:space="preserve">Quod hic proponitur, patet.</s> <s xml:id="echoid-s36967" xml:space="preserve"> Eſto enim axis ſpeculi columnaris conuexi h k i:</s> <s xml:id="echoid-s36968" xml:space="preserve"> & baſis ſuperior c<gap/> <lb/>lumnæ circulus f d:</s> <s xml:id="echoid-s36969" xml:space="preserve"> cuius centrum ſit h:</s> <s xml:id="echoid-s36970" xml:space="preserve"> & interior baſis circulus g e:</s> <s xml:id="echoid-s36971" xml:space="preserve"> cuius centrum i:</s> <s xml:id="echoid-s36972" xml:space="preserve"> & communis <lb/>ſectio alicuius ſuperficiei reflexiõis & ſuperficiei ſpeculi columna-<lb/> <anchor type="figure" xlink:label="fig-0572-01a" xlink:href="fig-0572-01"/> ris ſit circulus b l:</s> <s xml:id="echoid-s36973" xml:space="preserve"> cuius centrum k.</s> <s xml:id="echoid-s36974" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s36975" xml:space="preserve"> axis h i, qui orthogo <lb/>nalis eſt ſuper baſes, ut patet per 92 th.</s> <s xml:id="echoid-s36976" xml:space="preserve"> 1 huius, ſit etiam orthogona <lb/>lis ſuper circulũ b l per 100 & per 23 th.</s> <s xml:id="echoid-s36977" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s36978" xml:space="preserve"> & per eadem ſint li-<lb/>neæ longitudinis columnæ d e & f g orthogonales ſuper circulum <lb/>b l.</s> <s xml:id="echoid-s36979" xml:space="preserve"> Superficies ergo contingentes columnã ſecundum illas lineas <lb/>d e & f g, erectæ erunt ſuper circulum b l per 18 p 11:</s> <s xml:id="echoid-s36980" xml:space="preserve"> ergo & ſuper ſu <lb/>perficiem reflexionis, ſecantẽ columná ſecundũ illum circulum b <lb/>l:</s> <s xml:id="echoid-s36981" xml:space="preserve"> ergo per 19 p 11 cõmunis ſectio illarũ ſuperficierum contingentiũ <lb/>columnã orthogonalis erit ſuper illam ſuperficiẽ reflexionis.</s> <s xml:id="echoid-s36982" xml:space="preserve"> Ergo <lb/>per 6 p 11 illarũ ſuperficierũ cõmunis ſectio æquidiſtans erit axi co <lb/>lumnæ, qui ſuper eandem ſuperficiem eſt orthogonaliter erectus.</s> <s xml:id="echoid-s36983" xml:space="preserve"> <lb/>Secant aũt illæ ſuperficies ſe in centro uiſus:</s> <s xml:id="echoid-s36984" xml:space="preserve"> quoniam centrum ui <lb/>ſus in omnibus illis exiſtit, ut patet ex hypotheſi de ſuperficiebus <lb/>planis ſpeculum propoſitum contingentibus, & de ſuperficie refle <lb/>xionis ex 27 th.</s> <s xml:id="echoid-s36985" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s36986" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s36987" xml:space="preserve"/> </p> <div xml:id="echoid-div1508" type="float" level="0" n="0"> <figure xlink:label="fig-0572-01" xlink:href="fig-0572-01a"> <variables xml:id="echoid-variables651" xml:space="preserve">m f h d b k j p q g i e a</variables> </figure> </div> </div> <div xml:id="echoid-div1510" type="section" level="0" n="0"> <head xml:id="echoid-head1151" xml:space="preserve" style="it">4. Ad quodcũ punctũ ſignatũ in ſugnatũ in ſuperficie apparẽte ſpecu <lb/>li colũnaris uel pyramidalis cõuexi à centro uiſus ducatur linea <lb/>rect a: illa product a neceſſariò ſpeculũ ſecabit. Alhazen 27 h 4.</head> <p> <s xml:id="echoid-s36988" xml:space="preserve">Sit diſpoſitio omnimoda præmiſſæ:</s> <s xml:id="echoid-s36989" xml:space="preserve"> ſigneturq́;</s> <s xml:id="echoid-s36990" xml:space="preserve"> in apparẽte uiſui <lb/>portione ſpeculi, quę eſt e d f g, punctus q:</s> <s xml:id="echoid-s36991" xml:space="preserve"> & producatur linea a q.</s> <s xml:id="echoid-s36992" xml:space="preserve"> <lb/>Dico quò d linea a q ꝓducta neceffariò ſpeculũ ſecabit.</s> <s xml:id="echoid-s36993" xml:space="preserve"> Produca-<lb/>tur enim à puncto q linea longitudinis columnæ, quæ fit q m, ք 101 <lb/>th.</s> <s xml:id="echoid-s36994" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s36995" xml:space="preserve"> hæc itaq;</s> <s xml:id="echoid-s36996" xml:space="preserve"> linea erit æquidiſtans ambabus lineis longi-<lb/>tudinis d e & f g per 92 th.</s> <s xml:id="echoid-s36997" xml:space="preserve"> 1 huius, 6 p 11 & 30 p 1.</s> <s xml:id="echoid-s36998" xml:space="preserve"> Sit quoq;</s> <s xml:id="echoid-s36999" xml:space="preserve"> ut fuper-<lb/>ficies aliqua reflexionis ſecet columná ultra punctũ q ſecundũ cir-<lb/>culum b l per 100 th.</s> <s xml:id="echoid-s37000" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s37001" xml:space="preserve"> Linea ergo q m neceſſariò tráſibit per <lb/>circulũ ſectionis, qui eſt b l, ſecás ipſum in pũcto:</s> <s xml:id="echoid-s37002" xml:space="preserve"> ſit ergo illud pun <lb/>ctũ p:</s> <s xml:id="echoid-s37003" xml:space="preserve"> ducaturq́ linea a p.</s> <s xml:id="echoid-s37004" xml:space="preserve"> Hæc ergo, quia caditinter lineas à centro uiſus a ad circulũ b l ꝓductas il <lb/>lum cõtingentes, quæ ſunt a b & a l, palã quia ſecabit circulum.</s> <s xml:id="echoid-s37005" xml:space="preserve"> Ergo etiã ſuperficies à centro uiſus <lb/>ad ſpeculi ſuperficiem protenſa, in qua ſunt lineę a p & a q, ſecabit ſpeculũ:</s> <s xml:id="echoid-s37006" xml:space="preserve"> quia illa ſuperficies ſeca <lb/>bit ſuperficiẽ columnaris ſpeculi ſecundũ lineam longitudinis, quæ eſt m q.</s> <s xml:id="echoid-s37007" xml:space="preserve"> Palàm ergo quoniam <lb/>linea a q ꝓducta ſecabit ſpeculũ:</s> <s xml:id="echoid-s37008" xml:space="preserve"> eodẽq́;</s> <s xml:id="echoid-s37009" xml:space="preserve"> modo patet de quolibet a-<lb/> <anchor type="figure" xlink:label="fig-0572-02a" xlink:href="fig-0572-02"/> lio dato puncto.</s> <s xml:id="echoid-s37010" xml:space="preserve"> In ſpeculis quoq;</s> <s xml:id="echoid-s37011" xml:space="preserve"> pyramidalibus cõuexis eodem <lb/>modo demonſtrandũ, ducta linea à uertice pyramidis ad punctum <lb/>quẽcunq;</s> <s xml:id="echoid-s37012" xml:space="preserve"> in illius ſpeculi ſuperficie datũ.</s> <s xml:id="echoid-s37013" xml:space="preserve"> Palàm eſt ergo ꝓpoſitũ.</s> <s xml:id="echoid-s37014" xml:space="preserve"/> </p> <div xml:id="echoid-div1510" type="float" level="0" n="0"> <figure xlink:label="fig-0572-02" xlink:href="fig-0572-02a"> <variables xml:id="echoid-variables652" xml:space="preserve">m f y d z b j s n p t r o g i e a</variables> </figure> </div> </div> <div xml:id="echoid-div1512" type="section" level="0" n="0"> <head xml:id="echoid-head1152" xml:space="preserve" style="it">5. Omnis ſuperficies plana in aliqua linea lõgitudinis ſuperfi-<lb/>ciei apparentis uiſui ſpeculi colũnaris uel pyramidalis conuexi, <lb/>contingens ſpeculũ, ſecat ſuperficies à uiſu productas, quæ cõtin-<lb/>gunt portionis apparentis extremitates: omneś illæ ſuperficies <lb/>inter uiſum & ſpeculi ſuperficiẽ extenduntur. Alhazen 27 n 4.</head> <p> <s xml:id="echoid-s37015" xml:space="preserve">Maneat ſuperior diſpoſitio:</s> <s xml:id="echoid-s37016" xml:space="preserve"> cõtingatq́;</s> <s xml:id="echoid-s37017" xml:space="preserve"> aliqua ſuքficies plana ſu-<lb/>perficiẽ apparentẽ ſpeculi ſecundũ lineã lõgitudinis, quę eſt m o, ք <lb/>95 th.</s> <s xml:id="echoid-s37018" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s37019" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s37020" xml:space="preserve"> ſuքficies reflexiõis, quę ſit a b l:</s> <s xml:id="echoid-s37021" xml:space="preserve"> & in ea ꝓ <lb/>ducatur linea cõtingẽs circulũ b l in pũcto p, quę ſit s p t.</s> <s xml:id="echoid-s37022" xml:space="preserve"> Palã ergo <lb/>q đ linea s p t ſecabit lineas a b & a l.</s> <s xml:id="echoid-s37023" xml:space="preserve"> Ducatur enim linea p l.</s> <s xml:id="echoid-s37024" xml:space="preserve"> Quia <lb/>ergo linea s p t ſecat angulũ a p l, patet ք 29 th.</s> <s xml:id="echoid-s37025" xml:space="preserve"> 1 huius quoniá ipſa <lb/>fecabit lineã a l.</s> <s xml:id="echoid-s37026" xml:space="preserve"> Similiter ducta linea p b, patet q đ linea s p ſecabit <lb/>lineã a b:</s> <s xml:id="echoid-s37027" xml:space="preserve"> palá ergo quoniá lineę a l & p t cõcurrent.</s> <s xml:id="echoid-s37028" xml:space="preserve"> Sed linea p t eſt <lb/>in ſuքficie cõtingẽte columná ſecundũ lineã lõgitudinis m o:</s> <s xml:id="echoid-s37029" xml:space="preserve"> linea <lb/>uerò a l eſt in ſuքficie cõtingẽte columnã ſecũdũ lineá lõgitudinis <lb/>d e, quę eſt extremitas portionis apparẽtis.</s> <s xml:id="echoid-s37030" xml:space="preserve"> Patet ergo ꝓpoſitũ pri <lb/>mum.</s> <s xml:id="echoid-s37031" xml:space="preserve"> Sed & oẽs tales ſuքficies, qualis eſt ſuքficies, in qua eſt linea <lb/>s t, inter uiſum & ſpeculi ſuքficiẽ extenduntur.</s> <s xml:id="echoid-s37032" xml:space="preserve"> Et de ſpeculi quidẽ <lb/>ſuperficie patet, cũ ſint illę ſuperficies cõtingentes ipſam ſpeculi ſu <lb/>perficiẽ, & nõ ſecantés illá:</s> <s xml:id="echoid-s37033" xml:space="preserve"> ſed & patet de cetro uiſus.</s> <s xml:id="echoid-s37034" xml:space="preserve"> Sit enim pun <lb/>ctum n proximũ punctũ ſignabile ſub puncto b, in arcu l b:</s> <s xml:id="echoid-s37035" xml:space="preserve"> & ima-<lb/>ginetur aliqua ſuperficies cõtingens ſuքficiẽ colũnę in linea lõgitu <lb/>dinis, in qua ſit punctus n:</s> <s xml:id="echoid-s37036" xml:space="preserve"> hæc ergo neceſſariò ſecabit ſuքficiẽ refle <lb/>xionis, quę eſt a b l:</s> <s xml:id="echoid-s37037" xml:space="preserve"> quoniá eſt orthogonalis ſuper illã per 18 p 11.</s> <s xml:id="echoid-s37038" xml:space="preserve"> Sit <lb/>itaq;</s> <s xml:id="echoid-s37039" xml:space="preserve"> ſuperficiei reflexióis, quę a b l, & dictę ſuperficiei cómunis ſectio linea recta, quę ſit n r.</s> <s xml:id="echoid-s37040" xml:space="preserve"> Palàm <lb/>ergo ք pręmiſſa quoniá linea n r cõtingit circulũ b n in pũcto n:</s> <s xml:id="echoid-s37041" xml:space="preserve"> ſed punctũ n demiſsius eſt pũcto b:</s> <s xml:id="echoid-s37042" xml:space="preserve"> <lb/> <pb o="271" file="0573" n="573" rhead="LIBER SEPTIMVS."/> ergo cõtingẽs linea, quę n r, erit demiſsior linea cõtingẽte, quę eſt a b, ք 60 th.</s> <s xml:id="echoid-s37043" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s37044" xml:space="preserve"> Nõ ergo per-<lb/>tinget linea n r ad punctũ a centrũ uiſus.</s> <s xml:id="echoid-s37045" xml:space="preserve"> Eodẽ modo demõſtrandũ in alijs ꝗ buſcũq;</s> <s xml:id="echoid-s37046" xml:space="preserve"> ſuperficiebus <lb/>taliter cõtingentibus ſuperficiem apparentem ſpeculi columnaris.</s> <s xml:id="echoid-s37047" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s37048" xml:space="preserve"> dem onſtrandũ <lb/>eſt de ſuperficiebus contingentibus ſpecula pyramidalia quæcunq;</s> <s xml:id="echoid-s37049" xml:space="preserve">. Patet ergo propoſitum.</s> <s xml:id="echoid-s37050" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1513" type="section" level="0" n="0"> <head xml:id="echoid-head1153" xml:space="preserve" style="it">6. Omnis ſuperficies reflexionis, in qua ſunt linea contingens baſim ſpeculi columnaris uel <lb/>pyramidalis conuexi & linea longitudinis eiuſdem ſpeculi: idem ſpeculum ſecũdum lineam ſuæ <lb/>longitudinis neceſſariò eſt contingens.</head> <p> <s xml:id="echoid-s37051" xml:space="preserve">Hoc patet per modum 2 huius:</s> <s xml:id="echoid-s37052" xml:space="preserve"> quoniam eadem huius & illius eſt demõſtratio.</s> <s xml:id="echoid-s37053" xml:space="preserve"> Sit enim, reſum-<lb/>pta figura præ cedenti, ſuperficies reflexionis g a f, in qua ſit linea z f contingens columnam uel py-<lb/>ramidem in puncto f, & linea longitudinis columnæ uel pyramidis, quę ſit g f.</s> <s xml:id="echoid-s37054" xml:space="preserve"> Dico quòd illa ſuper-<lb/>ficies reflexionis continget columnam uel pyramidem.</s> <s xml:id="echoid-s37055" xml:space="preserve"> Si enim de-<lb/> <anchor type="figure" xlink:label="fig-0573-01a" xlink:href="fig-0573-01"/> tur quòd illa ſuperficies columnam uel pyramidẽ ſpeculi ſecet:</s> <s xml:id="echoid-s37056" xml:space="preserve"> tunc <lb/>& linea z f b a ſim illius ſpeculi ſecabit:</s> <s xml:id="echoid-s37057" xml:space="preserve"> quod eſt contra hypotheſim.</s> <s xml:id="echoid-s37058" xml:space="preserve"> <lb/>Palàm ergo propoſitum.</s> <s xml:id="echoid-s37059" xml:space="preserve"/> </p> <div xml:id="echoid-div1513" type="float" level="0" n="0"> <figure xlink:label="fig-0573-01" xlink:href="fig-0573-01a"> <variables xml:id="echoid-variables653" xml:space="preserve">y f d y g i e c</variables> </figure> </div> </div> <div xml:id="echoid-div1515" type="section" level="0" n="0"> <head xml:id="echoid-head1154" xml:space="preserve" style="it">7. Oppoſito uiſui ſpeculo columnari uel pyramidali cõuexo, ita <lb/>ut centrum uiſus non ſit in ſuperficie columnæ uel pyramidis, & <lb/>punctus rei uiſæ ſit cum uiſu in eadem ſuperficie ſpeculum ſecun-<lb/>dum axem ſecante: communis ſectio ſuperficiei reflexionis & ſu-<lb/>perficiei apparentis ſpeculi erit linea longitudinis ſpeculi: & ſi illa <lb/>communis ſectio ſit linea longitudinis, ſuperficies reflexion is ſecat <lb/>ſpeculum per axem. Alhazen 29 n 4.</head> <p> <s xml:id="echoid-s37060" xml:space="preserve">Sit ſpeculum columnare cõuexum, cuius axis ſit h i:</s> <s xml:id="echoid-s37061" xml:space="preserve"> cuius ſuper-<lb/>ficies apparens uiſui ſit e d f g:</s> <s xml:id="echoid-s37062" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s37063" xml:space="preserve"> a cẽtrum uiſus, & b punctum ui-<lb/>ſum:</s> <s xml:id="echoid-s37064" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s37065" xml:space="preserve"> ſuperficies reflexionis (in qua per 27 th.</s> <s xml:id="echoid-s37066" xml:space="preserve"> 5 huius neceſ <lb/>ſariò ſunt puncta a & b) ipſum ſpeculum ſecundum axem h i.</s> <s xml:id="echoid-s37067" xml:space="preserve"> Dico <lb/>quod communis ſectio illius ſuperficiei reflexionis & ſuperficiei e d <lb/>f g, eſt linea longitudinis ſpeculi.</s> <s xml:id="echoid-s37068" xml:space="preserve"> Quoniam enim per 93 th.</s> <s xml:id="echoid-s37069" xml:space="preserve"> 1 huius <lb/>cõmunis ſectio illius ſuperficiei planę & ſuperficiei totius column æ <lb/>ſpeculi eſt quadrangulum rectangulum ſub duabus lineis longitu-<lb/>dinis & duabus diametris baſium columnę contentum, cum ſuper-<lb/>ficies reflexionis tranſeat per centrum uiſus, cui directè in ſpeculo <lb/>opponitur ſuperficies apparens uiſui, per 1 huius:</s> <s xml:id="echoid-s37070" xml:space="preserve"> patet quòd com-<lb/>munis ſectio illarum duarum ſuperficierum erit linea una longitu-<lb/>dinis, quæ eſt unum latus illius rectanguli, quod eſt communis ſectio illius ſuperficiei planæ & ſu-<lb/>perficiei totius columnæ.</s> <s xml:id="echoid-s37071" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s37072" xml:space="preserve"> patet per 90 th.</s> <s xml:id="echoid-s37073" xml:space="preserve"> 1 huius de ſpeculo pyramidali:</s> <s xml:id="echoid-s37074" xml:space="preserve"> quoniam <lb/>communis ſectio ſuperficiei reflexionis & ſuperficiei conicæ ſpecu-<lb/> <anchor type="figure" xlink:label="fig-0573-02a" xlink:href="fig-0573-02"/> li uiſui apparentis, eſt unum latus illius trigoni (qui eſt communis <lb/>ſectio huius planæ ſuperficiei & totius ſuperficiei ipſius pyramidis <lb/>ſpeculi) quod eſt una linearum longitudinis pyramidis.</s> <s xml:id="echoid-s37075" xml:space="preserve"> Patet ergo <lb/>propoſitum.</s> <s xml:id="echoid-s37076" xml:space="preserve"/> </p> <div xml:id="echoid-div1515" type="float" level="0" n="0"> <figure xlink:label="fig-0573-02" xlink:href="fig-0573-02a"> <variables xml:id="echoid-variables654" xml:space="preserve">m f y d o g i c a</variables> </figure> </div> </div> <div xml:id="echoid-div1517" type="section" level="0" n="0"> <head xml:id="echoid-head1155" xml:space="preserve" style="it">8. Omnium ſuperficierum planarum ſuperficiem ſpeculi colu-<lb/>mnaris uel pyramidalis conuexi contingentium unica ſuper ſu-<lb/>perficiem reftexionis ſpeculum ſecundũ axem ſecãtem eſt erecta, <lb/>ut quæ ſecundum communem ſectionem illius ſuperficiei & ſpecu-<lb/>li, lineam ſcilicet longitudinis, ſuperficiem apparẽtem ſpeculi per <lb/>æqualia diuidentem, ſpeculum eſt contingens.</head> <p> <s xml:id="echoid-s37077" xml:space="preserve">Sit ſpeculum columnare conuexum, cuius apparens uiſui ſuper-<lb/>ficies ſit e d f g:</s> <s xml:id="echoid-s37078" xml:space="preserve"> & axis h i:</s> <s xml:id="echoid-s37079" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s37080" xml:space="preserve"> centrum uiſus punctum a:</s> <s xml:id="echoid-s37081" xml:space="preserve"> & commu-<lb/>nis ſectio ſuperficiei reflexionis ſpeculum ſecundũ axem ſecantis & <lb/>ſpeculi ſit linea longitudinis, quæ m o, per æqualia diuidens ſuper-<lb/>ficiem e d f g:</s> <s xml:id="echoid-s37082" xml:space="preserve"> cõtingantq́;</s> <s xml:id="echoid-s37083" xml:space="preserve"> ſuperficiẽ ſpeculi ſuperficies planæ quot-<lb/>cunq;</s> <s xml:id="echoid-s37084" xml:space="preserve"> Dico quòd unica illa, quæ ſecũdum lineam longitudinis m o <lb/>ſpeculum contingit, erecta eſt ſuper illam ſuperficiem reflexionis:</s> <s xml:id="echoid-s37085" xml:space="preserve"> & <lb/>quòd omnes aliæ ſuper ipſam ſunt obliquatæ.</s> <s xml:id="echoid-s37086" xml:space="preserve"> Vtenim patet per 92 <lb/>th.</s> <s xml:id="echoid-s37087" xml:space="preserve"> 1 huius linea m o rectos eſt angulos cõtinens cum ſemidiametris <lb/>baſium columnæ, & ſimiliter cum ſemidiametris omnium circulo-<lb/>rum baſibus illis æ quidiſtantium, ſecantium columnam, ut patet <lb/>per 100 & per 23 th.</s> <s xml:id="echoid-s37088" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s37089" xml:space="preserve"> palàm quoq;</s> <s xml:id="echoid-s37090" xml:space="preserve"> per 96 th.</s> <s xml:id="echoid-s37091" xml:space="preserve"> 1 huius quoniam <lb/>omnes perpendiculares, quæ intra columnam ducibiles ſunt ſu-<lb/>per ipſam ſuperficiem contingentem ſpeculum, neceſſariò tranſ-<lb/>cunt per axem ſpeculi:</s> <s xml:id="echoid-s37092" xml:space="preserve"> omnes uerò illæ perpendiculares cadunt in <lb/> <pb o="272" file="0574" n="574" rhead="VITELLONIS OPTICAE"/> ſuperficie ſpeculum ſecundum axem ſecante, Ergo per definitionem illa ſuperficies contingẽs eſt <lb/>erectaſuper ſuperficiẽ illam reflexionis.</s> <s xml:id="echoid-s37093" xml:space="preserve"> Omnes ergo aliæ ſuperficies dictã ſuperficiẽ ſpeculi ſecun-<lb/>dum alias lineas longitudinum contingẽtes, ſuper illam ſuperficiem reflexionis ſunt obliquæ.</s> <s xml:id="echoid-s37094" xml:space="preserve"> Ali-<lb/>ter enim cum illæ ſuperficies contingẽtes ſe neceſſariò interſecent:</s> <s xml:id="echoid-s37095" xml:space="preserve"> ſi ab aliquo puncto lineæ (quæ <lb/>per 3 p 11 eſt communis ſectio illarum ſuperficierum) duæ lineæ in illis ſuperfi ciebus contin genti-<lb/>bus ad ſuperficiem reflexionis perducantur, qũarum extremitates in ipſa ſupe rſicie reflexionis per <lb/>lineam tertiam coniungantur:</s> <s xml:id="echoid-s37096" xml:space="preserve"> erũt procreati illius trigoni duo an-<lb/> <anchor type="figure" xlink:label="fig-0574-01a" xlink:href="fig-0574-01"/> guli recti:</s> <s xml:id="echoid-s37097" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s37098" xml:space="preserve"> Non eſt ergo aliqua aliarum ſuper-<lb/>ficierum ſpeculum contingẽtium ſuper illam ſuperficiem reflexio-<lb/>nis erecta:</s> <s xml:id="echoid-s37099" xml:space="preserve"> niſi unica in illa communi ſectione ſpeculum contingẽs.</s> <s xml:id="echoid-s37100" xml:space="preserve"> <lb/>Et eodẽ modo in ſpeculis pyramidalibus poteſt demonſtratio for-<lb/>mari.</s> <s xml:id="echoid-s37101" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s37102" xml:space="preserve"/> </p> <div xml:id="echoid-div1517" type="float" level="0" n="0"> <figure xlink:label="fig-0574-01" xlink:href="fig-0574-01a"> <variables xml:id="echoid-variables655" xml:space="preserve">f y d z c g i e a g</variables> </figure> </div> </div> <div xml:id="echoid-div1519" type="section" level="0" n="0"> <head xml:id="echoid-head1156" xml:space="preserve" style="it">9. Oppoſito uiſui ſpeculo columnari conuexo, ita ut uiſus non <lb/>ſit in ipſa ſuperficie columnæ, & punctus rei uiſæ ſit cum uiſu in <lb/>eadem ſuperficie æquidiſtanti baſibus columnæ: communis ſectio <lb/>ſuperficiei reflexionis & ſpeculi erit circulus equidiſtans baſibus <lb/>columnæ. Alhazen 30 n 4.</head> <p> <s xml:id="echoid-s37103" xml:space="preserve">Eſto columnare ſpeculum cõuexum, cuius axis ſit h i:</s> <s xml:id="echoid-s37104" xml:space="preserve"> & baſis ſu-<lb/>perior circulus f d:</s> <s xml:id="echoid-s37105" xml:space="preserve"> inferior baſis circulus g e:</s> <s xml:id="echoid-s37106" xml:space="preserve"> & ſit centrũ uiſus pũ-<lb/>ctum a:</s> <s xml:id="echoid-s37107" xml:space="preserve"> & punctum rei uiſæ ſit b:</s> <s xml:id="echoid-s37108" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s37109" xml:space="preserve"> ſpeculum directè uiſui oppo-<lb/>ſitum, ut proponitur.</s> <s xml:id="echoid-s37110" xml:space="preserve"> Dico quòd ſuperficies reflexionis (quæ ſit a b <lb/>c z) ſecabit ſuperficiem propoſiti ſpeculi taliter, quòd communis <lb/>ſectio, quę ſit c z, erit circulus æquidiſtãs baſibus ſpeculi.</s> <s xml:id="echoid-s37111" xml:space="preserve"> Hoc enim <lb/>patet ex hypotheſi, & per 100 th.</s> <s xml:id="echoid-s37112" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s37113" xml:space="preserve"> uel etiam hoc modo.</s> <s xml:id="echoid-s37114" xml:space="preserve"> Du-<lb/>cantur enim duæ lineæ productæ à uiſu contingentes ſpeculũ, quæ <lb/>ſint a z & a c:</s> <s xml:id="echoid-s37115" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s37116" xml:space="preserve"> z & c puncta contingentiæ oppoſita adinuicem <lb/>in eadem ſuperficie:</s> <s xml:id="echoid-s37117" xml:space="preserve"> & ab utroq;</s> <s xml:id="echoid-s37118" xml:space="preserve"> illorum pũctorum ducantur lineæ <lb/>ſecundum longitudinem columnæ, quæ ſint d c e & f z g.</s> <s xml:id="echoid-s37119" xml:space="preserve"> Et quoniã <lb/>linea d c eſt æ qualis lineæ f z, & linea c e æ qualis lineæ z g ex hypo-<lb/>theſi & per 25 th.</s> <s xml:id="echoid-s37120" xml:space="preserve"> 1 huius, propter æ quidiſtantiam baſium ſpeculi & <lb/>ſuperficiei reflexionis:</s> <s xml:id="echoid-s37121" xml:space="preserve"> palàm quia linea z c (quæ eſt communis ſe-<lb/>ctio ſuperficiei reflexionis & ſuperficiei & ſpeculi) ęquidiſtabit ar-<lb/>cubus baſium, qui ſunt d f & g e.</s> <s xml:id="echoid-s37122" xml:space="preserve"> Ductis enim rectis lineis d f, c z, g <lb/>e, erunt illæ lineæ rectæ æquidiſtantes per 33 p 1:</s> <s xml:id="echoid-s37123" xml:space="preserve"> ergo & hæ curuæ, quæ in eiſdẽ ſunt ſuperficiebus, <lb/>erunt æquidiſtantes:</s> <s xml:id="echoid-s37124" xml:space="preserve"> & ſunt circulares:</s> <s xml:id="echoid-s37125" xml:space="preserve"> quoniam ſunt æ quidiſtantes in eadẽ ſuperficie columnari.</s> <s xml:id="echoid-s37126" xml:space="preserve"> <lb/>Patet ergo propoſitum.</s> <s xml:id="echoid-s37127" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1520" type="section" level="0" n="0"> <head xml:id="echoid-head1157" xml:space="preserve" style="it">10. Oppoſito uiſui ſpeculo columnari uel pyramidali conuexo, <lb/>it a ut uiſus non ſit in ſuperficie colũnæ uel pyramidis, ſuperficie <lb/>reflexionis obliquè axi ſpeculi incidente: communis ſectio ſuper-<lb/>ficiei reflexionis & ſpeculi erit oxygonia ſectio. Alhazen 31 n 4.</head> <figure> <variables xml:id="echoid-variables656" xml:space="preserve">f h d g i e b a</variables> </figure> <p> <s xml:id="echoid-s37128" xml:space="preserve">Eſto, ut in præmiſsis, ſpeculum columnare uel pyramidale con-<lb/>uexum, cuius axis ſit linea h i:</s> <s xml:id="echoid-s37129" xml:space="preserve"> & ſuperficies eius apparens uiſui ſit <lb/>e d f g:</s> <s xml:id="echoid-s37130" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s37131" xml:space="preserve"> centrum uiſus punctum a:</s> <s xml:id="echoid-s37132" xml:space="preserve"> & punctus rei uiſæ b:</s> <s xml:id="echoid-s37133" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s37134" xml:space="preserve"> <lb/>ſuperficies reflexionis ſpeculum obliquè trans axem, ſcilicet non <lb/>æ quidiſtanter baſibus columnæ.</s> <s xml:id="echoid-s37135" xml:space="preserve"> Dico quòd communis ſectio ſu-<lb/>perficiei reflexionis & ſuperficiei ſpeculi uiſui apparentis eſt pars <lb/>oxygoniæ ſectionis.</s> <s xml:id="echoid-s37136" xml:space="preserve"> Quoniam enim per 103 th.</s> <s xml:id="echoid-s37137" xml:space="preserve"> 1 huius patet quòd <lb/>omnis ſuperficiei ſecantis columnam uel pyramidem trans axem <lb/>non æquidiſtãter baſibus & ſuperficiei totius pyramidis uel colu-<lb/>m n æ communem ſectionem circulũ eſſe eſt impoſsibile, uel etiam <lb/>lineam longitudinis per 7 huius, cum talis ſuperficies plana nõ ſe-<lb/>cet pyramidem uel columnam ſecundum axis longitudinem:</s> <s xml:id="echoid-s37138" xml:space="preserve"> pa-<lb/>tet quòd communis ſectio ſuperficiei reflexionis (quæ plana eſt) <lb/>& partis ſuperficiei ſpeculi pyramidalis uel columnaris oppoſitæ <lb/>uiſui non poterit eſſe arcus circuli, neq;</s> <s xml:id="echoid-s37139" xml:space="preserve"> linea longitudinis.</s> <s xml:id="echoid-s37140" xml:space="preserve"> Erit er-<lb/>go pars ſectionis oxygonię:</s> <s xml:id="echoid-s37141" xml:space="preserve"> quia totam talem ſectionem totius ſu-<lb/>perficiei pyramidalis uel columnaris, & ſuperficiei p lanæ ſecantis <lb/>pyramidem uel columnam diximus oxygoniam ſectionem in 98 <lb/>th.</s> <s xml:id="echoid-s37142" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s37143" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s37144" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1521" type="section" level="0" n="0"> <head xml:id="echoid-head1158" xml:space="preserve" style="it">11. Communi ſectione ſuperficiei reflexionis & ſpeculi colu-<lb/>mnaris circulo exiſtente: omnes ſuperficies planæ ſpeculum con-<lb/>tingentes, ſuper ſuperficiem reflexionis ſunt erectæ.</head> <pb o="273" file="0575" n="575" rhead="LIBER SEPTIMVS."/> <p> <s xml:id="echoid-s37145" xml:space="preserve">Remaneat diſpoſitio, quæ præceſsit in 9 huius.</s> <s xml:id="echoid-s37146" xml:space="preserve"> Et quia per 95 th.</s> <s xml:id="echoid-s37147" xml:space="preserve"> 1 huius omnes planæ ſuperfi-<lb/>cies columnam contingentes, ſecundum lineam longitudinis contingunt, patet per 92 th.</s> <s xml:id="echoid-s37148" xml:space="preserve"> 1 huius, <lb/>cum omnes lineæ longitudinis rectos angulos cum ſemidiametris baſium contineant, quoniam <lb/>omnes ſuper illas baſes ſunt erectæ.</s> <s xml:id="echoid-s37149" xml:space="preserve"> Ergo per 100 & 23 th.</s> <s xml:id="echoid-s37150" xml:space="preserve"> 1 huius illæ lineæ omnes ſunt erectæ ſu-<lb/>per circulum æquidiſtantem baſibus columnæ.</s> <s xml:id="echoid-s37151" xml:space="preserve"> Hic autem eſt circulus (qui eſt communis ſectio <lb/>ſuperficiei reflexionis & ſpeculi per 9 huius.</s> <s xml:id="echoid-s37152" xml:space="preserve">) Ergo per definitionem ſuperficierum erectarum ſu-<lb/>per ſuperficies, omnes illæ ſuperficies contingentes columnam, ſuper præfatam ſuperficiem refle-<lb/>xionis eriguntur.</s> <s xml:id="echoid-s37153" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s37154" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1522" type="section" level="0" n="0"> <head xml:id="echoid-head1159" xml:space="preserve" style="it">12. Communem ſectionem ſuperficiei reflexionis & ſpeculi pyramidalis conuexi circulum <lb/>impoßibile eſt eſſe. Alhazen 41 n 4. Item 50 n 5.</head> <p> <s xml:id="echoid-s37155" xml:space="preserve">Sit pyramidale ſpeculum conuexum a b c:</s> <s xml:id="echoid-s37156" xml:space="preserve"> cuius uertex a:</s> <s xml:id="echoid-s37157" xml:space="preserve"> diameter baſis b c:</s> <s xml:id="echoid-s37158" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s37159" xml:space="preserve"> axis ſpeculi li-<lb/>nea a d:</s> <s xml:id="echoid-s37160" xml:space="preserve"> eſt ergo per 89 th.</s> <s xml:id="echoid-s37161" xml:space="preserve"> 1 huius punctum d centrum baſis:</s> <s xml:id="echoid-s37162" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s37163" xml:space="preserve"> centrum uiſus e:</s> <s xml:id="echoid-s37164" xml:space="preserve"> & punctus rei ui-<lb/>ſæ ſit f.</s> <s xml:id="echoid-s37165" xml:space="preserve"> Dico quod forma puncti f non poteſt reflecti ad uiſum e ab aliquo puncto ſpeculi propoſiti, <lb/>ita ut communis ſectio ſuperficiei reflexionis & ſpeculi ſit circulus.</s> <s xml:id="echoid-s37166" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0575-01a" xlink:href="fig-0575-01"/> Si enim hoc ſit poſsibile:</s> <s xml:id="echoid-s37167" xml:space="preserve"> eſto quòd reflectatur forma puncti f ad ui-<lb/>ſum e à puncto ſpeculi g:</s> <s xml:id="echoid-s37168" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s37169" xml:space="preserve"> circulus g h communis ſectio ſuper-<lb/>ficiei reflexionis & ſpeculi:</s> <s xml:id="echoid-s37170" xml:space="preserve"> cuius centrum ſit k:</s> <s xml:id="echoid-s37171" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s37172" xml:space="preserve"> per 100 th.</s> <s xml:id="echoid-s37173" xml:space="preserve"> 1 <lb/>huius circulus g h æ quidiſtans baſi b c.</s> <s xml:id="echoid-s37174" xml:space="preserve"> Producatur ergo à puncto g <lb/>extra ſpeculum linea g m perpendiculariter ſuper ſuperficiem con-<lb/>tingentem pyramidem in puncto g per 12 p 11.</s> <s xml:id="echoid-s37175" xml:space="preserve"> Quia uerò ſuperficies <lb/>baſis non eſt orthogonalis ſuper ſuperficiem contingentem pyra-<lb/>midem in puncto g:</s> <s xml:id="echoid-s37176" xml:space="preserve"> ideo quòd omnis ſuperficies contingens pyra-<lb/>midem ſecundum lineam longitudinis eſt contingens, ut patet per <lb/>95 th.</s> <s xml:id="echoid-s37177" xml:space="preserve"> 1 huius, & linea longitudinis obliquè ſuperſtat ſuperficiei ba-<lb/>ſis:</s> <s xml:id="echoid-s37178" xml:space="preserve"> palàm quòd ſuperficies circuli h g ę quidiſtantis baſi, non eſt or-<lb/>thogonalis ſuper ſuperficiem ſpeculum contingentem in puncto g.</s> <s xml:id="echoid-s37179" xml:space="preserve"> <lb/>Producta ergo linea perpendiculari, quæ eſt g m, intra pyramidem:</s> <s xml:id="echoid-s37180" xml:space="preserve"> <lb/>palàm quòd ipſa non pertinget ad centrum circuli, quod eſt k, ſed <lb/>cadet ſub illo in alio puncto axis, qui ſit punctus n:</s> <s xml:id="echoid-s37181" xml:space="preserve"> & continebit li-<lb/>nea m g n acutum angulum cum axe uerſus punctum uerticis, ſcili-<lb/>cet angulum g n a, qui neceſſariò eſt acutus per 32 p 1, ideo quò d an-<lb/>gulus g k n eſt rectus per 29 p 1:</s> <s xml:id="echoid-s37182" xml:space="preserve"> cum angulus a d c ſit rectus.</s> <s xml:id="echoid-s37183" xml:space="preserve"> Et quo-<lb/>niam, ut patet per 27 th.</s> <s xml:id="echoid-s37184" xml:space="preserve"> 5 huius, punctum m, qui eſt terminus lineæ <lb/>perpendicularis ſuper ſuperficiem ſpeculi (quæ perpendicularis eſt linea n g m) in ſuperficie refle-<lb/>xionis conſiſtere eſt neceſſe:</s> <s xml:id="echoid-s37185" xml:space="preserve"> linea ergo h k g non eſt in illa ſuperficie.</s> <s xml:id="echoid-s37186" xml:space="preserve"> Palàm ergo quòd formę pun-<lb/>cti f ad uiſum e non fiet reflexio à puncto ſpeculi g, ut à puncto circuli.</s> <s xml:id="echoid-s37187" xml:space="preserve"> Si enim fieret reflexio à pun-<lb/>cto g, ut à puncto circuli g h:</s> <s xml:id="echoid-s37188" xml:space="preserve"> oporteret neceſſariò ſuperficiem circuli g h perpendicularem eſſe ſu-<lb/>per ſuperficiem planam contingentem ſpeculum in puncto g, & perpendicularem m g produci ad <lb/>centrum circuli k:</s> <s xml:id="echoid-s37189" xml:space="preserve"> quod eſt impoſsibile per præmiſſa.</s> <s xml:id="echoid-s37190" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s37191" xml:space="preserve"/> </p> <div xml:id="echoid-div1522" type="float" level="0" n="0"> <figure xlink:label="fig-0575-01" xlink:href="fig-0575-01a"> <variables xml:id="echoid-variables657" xml:space="preserve">a f m h k g n e b d e</variables> </figure> </div> </div> <div xml:id="echoid-div1524" type="section" level="0" n="0"> <head xml:id="echoid-head1160" xml:space="preserve" style="it">13. Oppoſito uiſui ſpeculo pyramidali conuexo, it a ut uiſus non ſit in ſuperficie pyramidis aut <lb/>ei continua, punctuś rei uiſæ ſit cum centro uiſus in eadem ſuperficie æquidiſtante baſi pyra-<lb/>midis: impoßibile eſt reflexionem fieri ad uiſum.</head> <p> <s xml:id="echoid-s37192" xml:space="preserve">Exiſtente enim tali diſpoſitione centri uiſus & puncti rei uiſæ, reſpectu ſpeculi pyramidalis con-<lb/>uexi, ut proponitur:</s> <s xml:id="echoid-s37193" xml:space="preserve"> palàm per 100 th.</s> <s xml:id="echoid-s37194" xml:space="preserve"> 1 huius, cum ſuperficies reflexionis ſit ſuperficies plana, quia <lb/>communis ſectio ſui & ſuperficiei conicæ ſpeculi eſt circulus.</s> <s xml:id="echoid-s37195" xml:space="preserve"> Patet ergo propoſitum per præmiſ-<lb/>ſam.</s> <s xml:id="echoid-s37196" xml:space="preserve"> Eſt enim in illa oſtenſum impoſsibile eſſe, ut communis ſectio ſuperficiei reflexionis & ſpeculi <lb/>pyramidalis conuexi ſit circulus.</s> <s xml:id="echoid-s37197" xml:space="preserve"> Quia ſi ſectio illa communis eſſet circulus, eſſet ipſa per 100 th.</s> <s xml:id="echoid-s37198" xml:space="preserve"> 1 <lb/>huius æquidiſtãs baſi ſpeculi, & eſſet ſuperficies illius circuli in ſuperficie reflexionis.</s> <s xml:id="echoid-s37199" xml:space="preserve"> Et quia axis <lb/>a d eſt perpendicularis ſuper illum circulum per 23 th.</s> <s xml:id="echoid-s37200" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s37201" xml:space="preserve"> erunt lineæ longitudinis pyramidis <lb/>declinatæ ſuper illum circulum angulos acutos continentes cum diametris baſis:</s> <s xml:id="echoid-s37202" xml:space="preserve"> & ita eſſent illæ <lb/>lineæ obliquæ ſuper ſuperficiem reflexionis.</s> <s xml:id="echoid-s37203" xml:space="preserve"> Ergo in illa ſuperficie non poſſet duci perpendicula-<lb/>ris ſuper lineam longitudinis:</s> <s xml:id="echoid-s37204" xml:space="preserve"> ſed per 27 th.</s> <s xml:id="echoid-s37205" xml:space="preserve"> 5 huius perpendicularis ducta ſuper ſuperficiem con-<lb/>tingentem ſpeculum ſecundum punctum reflexionis, eſt in ſuperficie reflexionis & perpendicula-<lb/>ris ſuper lineam longitudinis:</s> <s xml:id="echoid-s37206" xml:space="preserve"> cum quælibet ſuperficies contingẽs pyramidem, contingat illam ſe-<lb/>cundum lineam longitudinis.</s> <s xml:id="echoid-s37207" xml:space="preserve"> Ergo nunquam fiet reflexio ad uiſum in h o c ſitu ſormæ alicuius pũ-<lb/>ctorum rei uiſæ, ſuperficie reflexionis ſpeculum pyramidale, ut pyramidale, contingente.</s> <s xml:id="echoid-s37208" xml:space="preserve"> Si uerò <lb/>ſuperficies, in qua eſt linea contingens ſpeculi circulum, ſecundum aliquod punctum illius circuli <lb/>ſecet ſuperficiem ſpeculi:</s> <s xml:id="echoid-s37209" xml:space="preserve"> tunc eſt poſsibile ab his ſpeculis, & ab illo puncto circuli reflexionem fie-<lb/>ri, non ut à ſpeculis pyramidalibus, ſed in quantum ipſorum cõuexa ſuperficies communicat cum <lb/>ſpeculis ſphæricis uel columnaribus conuexis, quorum paſsiones declarauimus in præmiſsis:</s> <s xml:id="echoid-s37210" xml:space="preserve"> nec <lb/>tunc hæc paſsio ad proprietatem ſpeculorum pyramidalium accedit.</s> <s xml:id="echoid-s37211" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s37212" xml:space="preserve"/> </p> <pb o="274" file="0576" n="576" rhead="VITELLONIS OPTICAE"/> </div> <div xml:id="echoid-div1525" type="section" level="0" n="0"> <head xml:id="echoid-head1161" xml:space="preserve" style="it">14. Superficierum reflexionis (quarum communis ſectio cum ſuperficie ſpeculi pyramidalis <lb/>eſt linea recta) ſecundum diuerſas uiſus ſituationes quando ſolùm unam, quando plurimas <lb/>ad eundem uiſum poßibile eſt applicari.</head> <p> <s xml:id="echoid-s37213" xml:space="preserve">Quocunq;</s> <s xml:id="echoid-s37214" xml:space="preserve"> enim modo uiſu taliter diſpoſito, ut minus medietate ſuperficiei conicæ pyramidis <lb/>uideatur per 84 th.</s> <s xml:id="echoid-s37215" xml:space="preserve"> 4 huius:</s> <s xml:id="echoid-s37216" xml:space="preserve"> tunc ſolùm unica ſuperficies reflexionis tranſit peruiſum, cuius com-<lb/>munis ſectio cũ ſuperficie pyramidis ſit linea longitudinis:</s> <s xml:id="echoid-s37217" xml:space="preserve"> quoniam unica tunc tranſibit per axem <lb/>pyramidis.</s> <s xml:id="echoid-s37218" xml:space="preserve"> Oſtenſum eſt enim per 7 huius quoniam in omni ſuperficie reflexionis factæ à ſpeculis <lb/>pyramidalibus (quãdo communis ſectio ſuperficiei reflexionis & ſpeculi ſuerit linea longitudinis <lb/>ſpeculi) neceſſe eſt eſſe axem ſpeculi.</s> <s xml:id="echoid-s37219" xml:space="preserve"> Taliter uerò diſpoſito uiſu, ut tota pyramis uideatur per 92 <lb/>th.</s> <s xml:id="echoid-s37220" xml:space="preserve"> 4 huius, nõ ſolùm plures, ſed etiam inſinitæ ſuperficies reflexionum (quarum communis ſectio <lb/>eſt linea longitudinis) ut proponitur, poſſunt ad oculum applicari:</s> <s xml:id="echoid-s37221" xml:space="preserve"> quoniam tunc centrum uiſus <lb/>omnibus lineis longitudinis totius ſpeculi eſt commune:</s> <s xml:id="echoid-s37222" xml:space="preserve"> & omnes ſe æ qualiter habent ad uiſum.</s> <s xml:id="echoid-s37223" xml:space="preserve"> <lb/>Cum enim radius uiſualis continuus fuerit axi pyramidis:</s> <s xml:id="echoid-s37224" xml:space="preserve"> tota pyramis uidetur per 92 th.</s> <s xml:id="echoid-s37225" xml:space="preserve"> 4 huius.</s> <s xml:id="echoid-s37226" xml:space="preserve"> <lb/>In qualibet ergo ſuperficie reflexionis ſit totus axis & linea perpendicularis ſuper ſpeculi ſuperfi-<lb/>ciem, a d axem tranſiens à puncto reflexionis:</s> <s xml:id="echoid-s37227" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s37228" xml:space="preserve"> cuiuslibet ſuperficiei reflexionis, & ſuperficiei <lb/>pyramidalis ſpeculi ſectio linea longitudinis in hoc ſitu:</s> <s xml:id="echoid-s37229" xml:space="preserve"> quoniam quælibet ſuperficies, in qua eſt <lb/>totus axis, communem habet lineam longitudinis illius pyramidis cum ſuperficie pyramidis per <lb/>90 th.</s> <s xml:id="echoid-s37230" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s37231" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s37232" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1526" type="section" level="0" n="0"> <head xml:id="echoid-head1162" xml:space="preserve" style="it">15. Omnis ſuperficies reflexionis (cuius communis ſectio & ſuperſiciei ſpeculi columnaris uel <lb/>pyramidalis conuexi eſt linea longitudinis ſpeculi) per æqualiæ <lb/>diuidit ſuperficiem ſpeculi apparentem.</head> <figure> <variables xml:id="echoid-variables658" xml:space="preserve">p y d m g x j s p t g i e o a</variables> </figure> <p> <s xml:id="echoid-s37233" xml:space="preserve">Eſto ſpeculum columnare conuexũ, cuius apparens ſuperficies <lb/>uiſui ſit e d f g:</s> <s xml:id="echoid-s37234" xml:space="preserve"> & axis h i:</s> <s xml:id="echoid-s37235" xml:space="preserve"> & ſit cẽtrum uiſus a, ut prius in præmiſsis.</s> <s xml:id="echoid-s37236" xml:space="preserve"> <lb/>Patet itaq;</s> <s xml:id="echoid-s37237" xml:space="preserve"> per 7 huius quoniã ſuperficies reflexionis taliter ſecans <lb/>ſpeculum columnare uel pyramidale, ſecat ipſum ſecundum axis h <lb/>i longitudinem.</s> <s xml:id="echoid-s37238" xml:space="preserve"> Sit autem linea longitudinis, ſecundum quam illa <lb/>ſuperficies reflexionis ſecat ſpeculum, linea m o.</s> <s xml:id="echoid-s37239" xml:space="preserve"> Dico quòd linea <lb/>m o per æqualia diuidit ſuperficiem ſpeculi e d f g uiſui apparẽtem.</s> <s xml:id="echoid-s37240" xml:space="preserve"> <lb/>Patet enim per 25 th.</s> <s xml:id="echoid-s37241" xml:space="preserve"> 5 huius quòd illa ſuperficies reflexionis eſt <lb/>orthogonalis ſuper ſuperficiem contingentem columnam in linea <lb/>m o.</s> <s xml:id="echoid-s37242" xml:space="preserve"> Si ergo in linea m o ſignetur punctum p:</s> <s xml:id="echoid-s37243" xml:space="preserve"> & ducatur linea a p:</s> <s xml:id="echoid-s37244" xml:space="preserve"> & <lb/>â puncto p ducàtur linea t p s in ſuperficie ſpeculum contingente, <lb/>taliter ut linea s p t contingat quendam circulum columnæ æqui-<lb/>diſtantem baſibus, qui ſit b l:</s> <s xml:id="echoid-s37245" xml:space="preserve"> erit linea a p perpendicularis ſuper li-<lb/>neam t p s:</s> <s xml:id="echoid-s37246" xml:space="preserve"> quoniam ducitur in ſuperficie ſuper illam ſuperficiem <lb/>erecta:</s> <s xml:id="echoid-s37247" xml:space="preserve"> ergo per 19 p 3 linea a p producta tranſit centrum circuli b l, <lb/>quod ſit x.</s> <s xml:id="echoid-s37248" xml:space="preserve"> Ducanturq́;</s> <s xml:id="echoid-s37249" xml:space="preserve"> lineæ a b & a l, quæ ſunt æ quales per 58 th.</s> <s xml:id="echoid-s37250" xml:space="preserve"> 1 <lb/>huius:</s> <s xml:id="echoid-s37251" xml:space="preserve"> copulentur quoq;</s> <s xml:id="echoid-s37252" xml:space="preserve"> ſemidiametri x b & x l.</s> <s xml:id="echoid-s37253" xml:space="preserve"> Erunt ergo trigoni <lb/>a b x & a l x æ quianguli per 8 p 1:</s> <s xml:id="echoid-s37254" xml:space="preserve"> & erit angulus p a l æ qualis angu-<lb/>l o p a b:</s> <s xml:id="echoid-s37255" xml:space="preserve"> ergo per 58 th.</s> <s xml:id="echoid-s37256" xml:space="preserve"> 1 huius linea a p diuiditarcũ l p b per æ qua-<lb/>lia in puncto p:</s> <s xml:id="echoid-s37257" xml:space="preserve"> ſed arcus l p b eſt æ quidiſtans baſibus columnæ.</s> <s xml:id="echoid-s37258" xml:space="preserve"> Li-<lb/>n e æ quoq;</s> <s xml:id="echoid-s37259" xml:space="preserve"> rectæ terminantes ſuperficiem ſpeculi uiſui apparẽtem <lb/>æ quidiſtant lineę m o:</s> <s xml:id="echoid-s37260" xml:space="preserve"> quod patet per 92 th.</s> <s xml:id="echoid-s37261" xml:space="preserve"> 1 huius, & per 28 p 1.</s> <s xml:id="echoid-s37262" xml:space="preserve"> Li-<lb/>nea ita q;</s> <s xml:id="echoid-s37263" xml:space="preserve"> m o diuidet per æ qualia baſes columnæ:</s> <s xml:id="echoid-s37264" xml:space="preserve"> eſt autem linea <lb/>m o in ſuperficie reflexionis.</s> <s xml:id="echoid-s37265" xml:space="preserve"> Palàm ergo quòd illa ſuperficies refle-<lb/>xionis diuidit ſuperficiem ſpeculi apparentem uiſui per æqualia.</s> <s xml:id="echoid-s37266" xml:space="preserve"> <lb/>Et quoniam in ſpeculo pyramidali ſiue unica ſine plurimæ ſint illæ <lb/>ſuperficies reflexionis, ut patet per præmiſſam, ſemper eadẽ eſt demõſtratio.</s> <s xml:id="echoid-s37267" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s37268" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1527" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables659" xml:space="preserve">g y f l r k h p a c l d</variables> </figure> <head xml:id="echoid-head1163" xml:space="preserve" style="it">16. Omnium ſuperficierum reflexionum ab eodem ſpeculo columnari cõuexo ad eundem ui-<lb/> ſum factarum unicà eſt, cuius cõmunis ſe- ctio & ſuperficiei ſpe- culieſt linea lõgitudi- nis illius ſpeculi. Al- hazen 29 n 4.</head> <p> <s xml:id="echoid-s37269" xml:space="preserve">Sit diſpoſitio figuræ <lb/>eadem, quæ in præce-<lb/>dente.</s> <s xml:id="echoid-s37270" xml:space="preserve"> Et quia nunquã <lb/>cõmunis ſectio ſuper-<lb/>ficiei reflexionis & ſpe <lb/>culi propoſiti eſt linea longitudinis ſpeculi, niſi ſolùm ſuperficie reflexionis columnam per axem <lb/>ſecante per 7 huius:</s> <s xml:id="echoid-s37271" xml:space="preserve"> in hoc autem ſitu ſuperficies reflexionis (quæ eſt a h i) ſecat ſuperficiem e d f g <lb/> <pb o="275" file="0577" n="577" rhead="LIBER SEPTIMVS."/> apparentem uiſui per duo æqualia, ut patet per præmiſſam huius, & ſuperficies tranſiens per axem <lb/>hi, eſt unica:</s> <s xml:id="echoid-s37272" xml:space="preserve"> patet quòd huius ſolius & ſuperficiei ſpeculi communis ſectio eſt linea longitudinis <lb/>ſpeculi.</s> <s xml:id="echoid-s37273" xml:space="preserve"> Si autem dicatur quòd & alia ſuperficies reflexionis eſt, cuius communis ſectio & ſuperfi-<lb/>ciei ſpeculi eſt linea longitudinis ſpeculi:</s> <s xml:id="echoid-s37274" xml:space="preserve"> ergo per 7 huius illa ſuperficies ſecat ſpeculum ſecũdum <lb/>axem h i.</s> <s xml:id="echoid-s37275" xml:space="preserve"> Ducatur ergo in illa ſuperficie linea à centro uiſus ad axem h i, quæ ſit a r k:</s> <s xml:id="echoid-s37276" xml:space="preserve"> & ducatur in <lb/>propoſita ſuperficie reflexionis ſuperficiẽ apparentẽ ſpeculi per æ qualia ſecante, linea a p k.</s> <s xml:id="echoid-s37277" xml:space="preserve"> Palàm <lb/>ergo quòd iſtæ duę rectæ includẽt ſuperficiẽ:</s> <s xml:id="echoid-s37278" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s37279" xml:space="preserve"> Patet ergo ꝓpoſitũ.</s> <s xml:id="echoid-s37280" xml:space="preserve"> Vnica enim <lb/>poteſt imaginari ſuperficies, in qua ſintaxis colũnæ & centrũ uiſus & pũctus rei uiſæ, & nõ plures.</s> <s xml:id="echoid-s37281" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1528" type="section" level="0" n="0"> <head xml:id="echoid-head1164" xml:space="preserve" style="it">17. Omnium ſuperſicierum reflexionum ab eodem ſperculo columnari cõuexo ad eundem ui-<lb/>ſum factarum unica eſt, cuius communis ſectio & ſuperficiei ſpeculi eſt circulus æquidiſtans ba-<lb/>ſibus columnæ. Alhazen 30 n 4.</head> <p> <s xml:id="echoid-s37282" xml:space="preserve">Sit diſpoſitio, quæ ſuprà, ita ut cõmunis ſectio ſuperficiei reflexionis & ſpeculi columnaris con-<lb/>uexi ſit circulus.</s> <s xml:id="echoid-s37283" xml:space="preserve"> Quia ergo in omni tali ſuperficie reflexionis linea <lb/> <anchor type="figure" xlink:label="fig-0577-01a" xlink:href="fig-0577-01"/> perpendiculariter erecta ſuper ſuperficiem contingẽtem ſpeculum <lb/>in puncto reflexionis, eſt diameter circuli baſibus columnę æquidi-<lb/>ſtantis:</s> <s xml:id="echoid-s37284" xml:space="preserve"> & non poteſt eſſe in ſuperficie columnæ, niſi unus circulus <lb/>æ quidiſtans baſibus columnæ, qui cum centro uiſus ſit in eadẽ ſu <lb/>perficie:</s> <s xml:id="echoid-s37285" xml:space="preserve"> palàm quia omnium ſuperficierum reflexionum ab eodem <lb/>ſpeculo columnari cõuexo ad eundẽ uiſum factarũ unica eſt, cuius <lb/>communis ſectio & ſuperficiei ſpeculi eſt circulus æquidiſtãs baſi-<lb/>bus columnę.</s> <s xml:id="echoid-s37286" xml:space="preserve"> Si enim dicatur quòd ſint plures:</s> <s xml:id="echoid-s37287" xml:space="preserve"> ſit communis ſectio <lb/>unius illarum ſuperficierũ & ſuperficiei ſpeculi linea circularis, quę <lb/>ſit b p t:</s> <s xml:id="echoid-s37288" xml:space="preserve"> alterius uerò x y z:</s> <s xml:id="echoid-s37289" xml:space="preserve"> puncta quoq;</s> <s xml:id="echoid-s37290" xml:space="preserve">, in quibus axi columnę in <lb/>cidunt centra illorum circulorum ſint k & r:</s> <s xml:id="echoid-s37291" xml:space="preserve"> & producantur lineæ a <lb/>k & a r à cẽtro uiſus ad illa puncta.</s> <s xml:id="echoid-s37292" xml:space="preserve"> Palàm ergo propter æ quidiſtan-<lb/>tiam baſium ad iſtas, quoniã in trigono a k r duo anguli ad baſim k r <lb/>ſunt recti:</s> <s xml:id="echoid-s37293" xml:space="preserve"> linea enim k r, cũ ſit pars lineæ h i axis columnæ, ſicut eſt <lb/>e recta ſuper baſes colũnæ per 92 th.</s> <s xml:id="echoid-s37294" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s37295" xml:space="preserve"> ita & ſuper ſuperficies <lb/>circulorum illis baſibus æquidiſtãtium per 23 th.</s> <s xml:id="echoid-s37296" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s37297" xml:space="preserve"> Ergo & ſu-<lb/>per diametros illorũ circulorum eſt perpendicularis:</s> <s xml:id="echoid-s37298" xml:space="preserve"> ſunt autẽ illæ <lb/>diametri in lineis a k & a r.</s> <s xml:id="echoid-s37299" xml:space="preserve"> Lineà ergo k r eſt perpendicularis ſuper <lb/>ambas lineas a k & a r:</s> <s xml:id="echoid-s37300" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s37301" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s37302" xml:space="preserve"/> </p> <div xml:id="echoid-div1528" type="float" level="0" n="0"> <figure xlink:label="fig-0577-01" xlink:href="fig-0577-01a"> <variables xml:id="echoid-variables660" xml:space="preserve">f y d b k t p x r z y g i e a</variables> </figure> </div> </div> <div xml:id="echoid-div1530" type="section" level="0" n="0"> <head xml:id="echoid-head1165" xml:space="preserve" style="it">18. Superficierum reflexionis (quarum communis ſectio cum <lb/>ſuperficie ſpeculi colũnaris uel pyramidalis conuexi eſt ſectio oxy-<lb/>gonia) plures ab eadem portione apparẽte ſpeculi ad eundem ui-<lb/>ſum eſt poſsibile applicari. Alhazen 31. n 4.</head> <p> <s xml:id="echoid-s37303" xml:space="preserve">Fiat ordinatio figuræ, quæ ſuprà in 15 huius:</s> <s xml:id="echoid-s37304" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s37305" xml:space="preserve"> cõmunis ſectio <lb/>ſuperficiei reflexionis tranſeuntis per axem h i, linea m o:</s> <s xml:id="echoid-s37306" xml:space="preserve"> & cõmu-<lb/>nis ſectio ſuperficiei reflexionis æ quidiſtantis axibus columnę cir-<lb/>culus b p l.</s> <s xml:id="echoid-s37307" xml:space="preserve"> Palàm ex præhabitis, quon am ab omnibus punctis ſuperficiei columnaris m p b & m p <lb/>l poteſt fieri reflexio ad uiſum a ſecundũ partes ſectionis columnaris.</s> <s xml:id="echoid-s37308" xml:space="preserve"> Quia enim ad quodlibet illo-<lb/>rum punctorum poteſt aliquis punctus rerum uiſarum incidere:</s> <s xml:id="echoid-s37309" xml:space="preserve"> patet quòd à quolibet illorũ pun-<lb/>ctorum fieri poteſt reflexio ad uiſum per 1 th.</s> <s xml:id="echoid-s37310" xml:space="preserve"> huius.</s> <s xml:id="echoid-s37311" xml:space="preserve"> Manifeſtum eſt ergo quòd partes illarũ ſectio-<lb/>num columnarium uel pyramidalium poſſunt eſſe infinitæ, quarum quælibet ſecundum eandem <lb/>lineam perpendicularem ſuper axem ſecat columnam uel pyramidem ſpeculi, ut patet per 104 th.</s> <s xml:id="echoid-s37312" xml:space="preserve"> 1 <lb/>huius.</s> <s xml:id="echoid-s37313" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s37314" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1531" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables661" xml:space="preserve">d q f l y o m e d</variables> </figure> <head xml:id="echoid-head1166" xml:space="preserve" style="it">19. Linea longitudinis exiſtente cõmuni ſectione ſuperficiei reflexionis & ſpeculi columna-<lb/> ris uel pyramidalis cõ- uexi: à quocun pun- ctorum illius lineæ fiat reflexio ad uiſum, ſem- per fit in eadẽ ſuperfi- cie. Alhaz 32. 42 n 4.</head> <p> <s xml:id="echoid-s37315" xml:space="preserve">Signata, ut in pręmiſ-<lb/>ſa 15 huius, ſuperficie re <lb/>flexionis tali, ut propo-<lb/>nitur, quæ ſecet ſuper-<lb/>ficiem ſpeculi ſecũdum <lb/>lineam m o.</s> <s xml:id="echoid-s37316" xml:space="preserve"> Dico quòd à quocunq;</s> <s xml:id="echoid-s37317" xml:space="preserve"> puncto illius lineæ fiat reflexio ad uiſum:</s> <s xml:id="echoid-s37318" xml:space="preserve"> ſemper omnes lineæ <lb/>reflexionis erunt in eadem ſuperficie a m o.</s> <s xml:id="echoid-s37319" xml:space="preserve"> Quoniam enim in ſuperficie a m o eſt per 7 huius <lb/>axis h i:</s> <s xml:id="echoid-s37320" xml:space="preserve"> & unica ſuperficies contingens ſpeculum in illa linea m o, erecta eſt ſuper ſuperficiem <lb/> <pb o="276" file="0578" n="578" rhead="VITELLONIS OPTICAE"/> reflexionis, ut patet per 8 huius:</s> <s xml:id="echoid-s37321" xml:space="preserve"> palàm quia quocunq;</s> <s xml:id="echoid-s37322" xml:space="preserve"> pũcto in illa linea m o ſumpto, perpendicu-<lb/>laris ab eo ad axem h i ducta, ſemper erit in eadẽ ſuperficie cũ axe h i:</s> <s xml:id="echoid-s37323" xml:space="preserve"> & erit illa linea orthogonalis <lb/>ſuper ſuperficiem contingẽtem ſuperficiem columnæ ſecundũ illam lineam m o:</s> <s xml:id="echoid-s37324" xml:space="preserve"> quia per 18 p 3 illa <lb/>linea à puncto contactus ad centrũ circuli ducta eſt perpendicularis ſuper lineã, contingentem cir-<lb/>culum ductã in ſuperficie columnã contingente.</s> <s xml:id="echoid-s37325" xml:space="preserve"> Superficies ergo m o h i eſt erecta ſuper ſuperficiẽ <lb/>in linea m o ſpeculum contin gentem:</s> <s xml:id="echoid-s37326" xml:space="preserve"> ſed centrũ uiſus eſt in ſuperficie orthogonali ſuper eandẽ ſu-<lb/>perficiem:</s> <s xml:id="echoid-s37327" xml:space="preserve"> quoniã in ſuperficie una eſt cẽtrum uiſus & linea m o & <lb/> <anchor type="figure" xlink:label="fig-0578-01a" xlink:href="fig-0578-01"/> axis ſpeculi h i, ut patet per præmiſſa:</s> <s xml:id="echoid-s37328" xml:space="preserve"> una ſola autem ſuperficies eſt <lb/>orthogonalis ſuper illam ſuperficiem contingentem ſecundum li-<lb/>neam m o:</s> <s xml:id="echoid-s37329" xml:space="preserve"> quoniam dato oppoſito, contingeret duas lineas ſuper <lb/>pũctum unum ad ſuperficiem unam orthogonaliter inſiſtere, quod <lb/>eſt impoſsibile per 13 p 11.</s> <s xml:id="echoid-s37330" xml:space="preserve"> Omnes ergo reflexiones à punctis lineæ <lb/>m o factæ ſunt in una & eadem ſuperficie.</s> <s xml:id="echoid-s37331" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s37332" xml:space="preserve"/> </p> <div xml:id="echoid-div1531" type="float" level="0" n="0"> <figure xlink:label="fig-0578-01" xlink:href="fig-0578-01a"> <variables xml:id="echoid-variables662" xml:space="preserve">f h d m b k t p g i e o a</variables> </figure> </div> </div> <div xml:id="echoid-div1533" type="section" level="0" n="0"> <head xml:id="echoid-head1167" xml:space="preserve" style="it">20. Sectione communi ſuperficiei reflexionis & ſpeculi colu-<lb/>mnaris conuexi, exiſtẽte circulo: à quocun puncto illius circuli <lb/>fiat reflexio, ſemper fit in eadem ſuperficie. Alhazen 32 n 4.</head> <p> <s xml:id="echoid-s37333" xml:space="preserve">Fiat figuratio utin 17 huius:</s> <s xml:id="echoid-s37334" xml:space="preserve"> & ſignetur quodcũq;</s> <s xml:id="echoid-s37335" xml:space="preserve"> punctum pla-<lb/>cuerit in circulo b p t:</s> <s xml:id="echoid-s37336" xml:space="preserve"> palàm quoniam ſemper ſemidiameter illius <lb/>circuli ducta à puncto k centro illius circuli b p t erit perpẽdicula-<lb/>ris ſuper ſuperficiem contingentem ſpeculum in illo puncto refle-<lb/>xionis dato:</s> <s xml:id="echoid-s37337" xml:space="preserve"> erit ergo quælibet talium perpendicularium producta <lb/>extrà ſuper ſuperficiem contingentem columnam in eadem ſuper-<lb/>ficie conſiſtens tota per 1 p 11:</s> <s xml:id="echoid-s37338" xml:space="preserve"> eſt autem illa ſuperficies educta extra <lb/>colum nam ſuperficies reflexionis.</s> <s xml:id="echoid-s37339" xml:space="preserve"> Quia ergo quæ libet talium per-<lb/>pendicularium eſt in ſuperficie illius circuli, & pũctum uiſus, quod <lb/>eſt a, ſimiliter eſt in ead em ſuperficie.</s> <s xml:id="echoid-s37340" xml:space="preserve"> In hac ergo ſola ſuperficie erit <lb/>reflexio cuiuſcunq;</s> <s xml:id="echoid-s37341" xml:space="preserve"> puncti rei uiſæ facta à quolibet punctorum to-<lb/>tius illius circuli uel portionis ſuæ uiſæ.</s> <s xml:id="echoid-s37342" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s37343" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1534" type="section" level="0" n="0"> <head xml:id="echoid-head1168" xml:space="preserve" style="it">21. Omnis perpendicularis à puncto reflexionis ſuper ſpeculi <lb/>columnaris conuexam ſuperficiem erecta, producta intra ſpecu-<lb/>lum eſt diameter cir culi æquidiſtantis baſibus columnæ: & econ-<lb/>uerſo. Alhazen 34 n 4.</head> <p> <s xml:id="echoid-s37344" xml:space="preserve">Sit diſpoſitio figuræ, ut prius:</s> <s xml:id="echoid-s37345" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s37346" xml:space="preserve"> punctum reflexionis p, ſiue <lb/>communis ſectio ſuperficiei reflexionis & ſpeculi ſit linea longitudinis uel circulus uel ſectio colu-<lb/>mnaris:</s> <s xml:id="echoid-s37347" xml:space="preserve"> & à puncto p ducatur linea perpendicularis ſuper ſuperfi-<lb/> <anchor type="figure" xlink:label="fig-0578-02a" xlink:href="fig-0578-02"/> ciem contin gentem ſpeculum in eodem puncto p:</s> <s xml:id="echoid-s37348" xml:space="preserve"> quæ ſit p q.</s> <s xml:id="echoid-s37349" xml:space="preserve"> Dico <lb/>ſi linea p q intelligatur produci intra ſpeculum, quòd ipſa cadet in <lb/>punctum k, quod eſt centrum circuli b p l:</s> <s xml:id="echoid-s37350" xml:space="preserve"> & erit diameter illius cir-<lb/>culi.</s> <s xml:id="echoid-s37351" xml:space="preserve"> Quia ſi detur, quòd non:</s> <s xml:id="echoid-s37352" xml:space="preserve"> cum conſtet per 18 p 3 diametrum k p <lb/>perpendicularem eſſe ſuper lineam s t contingentem circulum b p l <lb/>in puncto p, & ex conſequen ti ſuper ſuperficiem in illo puncto con-<lb/>tingentem columnam, in qua per 6 huius eſt linea s t:</s> <s xml:id="echoid-s37353" xml:space="preserve"> cum etiam li-<lb/>nea q p ſit perpendicularis ſuper eandem lineam & ſuperficiem in <lb/>eodem puncto ſpeculum contingentem:</s> <s xml:id="echoid-s37354" xml:space="preserve"> palàm quòd erunt hæ duæ <lb/>perpendiculares q p & k p coniunctæ in puncto p linea una per 14 <lb/>p 1:</s> <s xml:id="echoid-s37355" xml:space="preserve"> ambæ enim illæ lineæ exeunt ab uno puncto p lineæ s p, & con-<lb/>tinet quęlibet ipſarum angulum rectum cum eadem:</s> <s xml:id="echoid-s37356" xml:space="preserve"> & danti oppo-<lb/>ſitum etiam accidit ex eodem puncto p ſuperficiei contingẽtis duas <lb/>erigi perpendiculares ſuper illam ſuperficiem, quod eſt contra 13 p <lb/>11.</s> <s xml:id="echoid-s37357" xml:space="preserve"> Producta enim diametro k p extra ſpeculum, ſi ipſa non pertin-<lb/>gat ad punctum q:</s> <s xml:id="echoid-s37358" xml:space="preserve"> ſit, ut ipſa pertingat ad pũctum z extra ſpeculum <lb/>ſuper ſuperficiem contingentem:</s> <s xml:id="echoid-s37359" xml:space="preserve"> accidet ergo ipſam p z & perpen <lb/>dicularem q p ſuper eandem ſuperficiem ad idem punctum p pro-<lb/>ductas perpendiculares eſſe:</s> <s xml:id="echoid-s37360" xml:space="preserve"> quod eſt impoßsibile.</s> <s xml:id="echoid-s37361" xml:space="preserve"> Patet ergo pro-<lb/>poſitum primum.</s> <s xml:id="echoid-s37362" xml:space="preserve"> Conuerſa quoq;</s> <s xml:id="echoid-s37363" xml:space="preserve"> patet per eundem modum.</s> <s xml:id="echoid-s37364" xml:space="preserve"/> </p> <div xml:id="echoid-div1534" type="float" level="0" n="0"> <figure xlink:label="fig-0578-02" xlink:href="fig-0578-02a"> <variables xml:id="echoid-variables663" xml:space="preserve">f h d m u b k l s p t g i e o q z</variables> </figure> </div> </div> <div xml:id="echoid-div1536" type="section" level="0" n="0"> <head xml:id="echoid-head1169" xml:space="preserve" style="it">22. Superficiei reflexionis & ſpeculi columnaris conuexi com-<lb/>muni ſectione quacun linea exiſtente: formæ eiuſdem puncti rei <lb/>uiſæ non fit reflexio ad uiſum eundem, niſi ab uno tantùm illius <lb/>ſectionis puncto. Alhazen 33 n 4.</head> <p> <s xml:id="echoid-s37365" xml:space="preserve">Communi enim ſectione ſuperficiei reflexionis & ſpeculorum <lb/>propoſitorum exiſtente linea recta per 7 huius:</s> <s xml:id="echoid-s37366" xml:space="preserve"> tunc non fiet refle-<lb/> <pb o="277" file="0579" n="579" rhead="LIBER SEPTIMVS."/> xio, niſi ab uno tantùm puncto illius lineæ, ſicut de ſpeculis planis oſtenſum eſt per 45 th.</s> <s xml:id="echoid-s37367" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s37368" xml:space="preserve"> <lb/>Si uerò communis ſectio ſuperficiei reflexionis & ſpeculi columnaris fuerit circulus, ut patet per 9 <lb/>huius:</s> <s xml:id="echoid-s37369" xml:space="preserve"> tunc ab uno tantùm puncto illius circulifiet reflexio, quemadmo dum in ſpeculis ſphæricis <lb/>conuexis oſtenſum eſt per 16 th.</s> <s xml:id="echoid-s37370" xml:space="preserve"> 6 huius.</s> <s xml:id="echoid-s37371" xml:space="preserve"> Si uerò illa communis ſectio fuerit oxygonia, ut patet per <lb/>10 huius:</s> <s xml:id="echoid-s37372" xml:space="preserve"> tunc eſt hoc propoſitum in ſpeculis propoſitis ſpecialiter demonſtrandum.</s> <s xml:id="echoid-s37373" xml:space="preserve"> Fiat ergo diſ-<lb/>poſitio figuræ, ut in præmiſſa proxima:</s> <s xml:id="echoid-s37374" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s37375" xml:space="preserve"> pars colũnaris ſectionis linea, quæ eſt p u.</s> <s xml:id="echoid-s37376" xml:space="preserve"> Dico quòd <lb/>ab uno tantùm puncto lineæ p u fiet reflexio ad uiſum in illa ſuperficie.</s> <s xml:id="echoid-s37377" xml:space="preserve"> Dato enim quocumq;</s> <s xml:id="echoid-s37378" xml:space="preserve"> pun-<lb/>cto alio, palàm quoniam perpendicularis ab illo puncto reflexionis erecta ſuper ſuperficiem colu-<lb/>mnæ, orthogonalis eſt ſuper lineam longitudinis columnæ perillum punctum tranſeuntis:</s> <s xml:id="echoid-s37379" xml:space="preserve"> quare <lb/>& ſuper axem perpendicularis erit per 29 p 1:</s> <s xml:id="echoid-s37380" xml:space="preserve"> & erit illa perpendicularis, diameter circuli æquidi-<lb/>ſtantis baſibus ſpeculi per præmiſſam.</s> <s xml:id="echoid-s37381" xml:space="preserve"> Et ſuperficies reflexionis & circulus ille ſecant ſe, & linea eis <lb/>communis eſt diameter illius circuli per 104 th.</s> <s xml:id="echoid-s37382" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s37383" xml:space="preserve"> & diameter illa eſt perpendicularis ſuper <lb/>ſuperficiem ſpeculum in illo puncto contingentem:</s> <s xml:id="echoid-s37384" xml:space="preserve"> & ſuperficies reflexionis eſt ſecãs illam lineam <lb/>longitudinis columnæ, ſuper quam fit contingentia:</s> <s xml:id="echoid-s37385" xml:space="preserve"> & eſt declinata ſuper eam:</s> <s xml:id="echoid-s37386" xml:space="preserve"> ergo & ſuper axem <lb/>erit illa ſuperficies reflexionis declinata.</s> <s xml:id="echoid-s37387" xml:space="preserve"> Sed in ſuperficie plana ſuper aliquã lineam declinata (ut <lb/>ſpecualiter patet de ſectione oxygonia per 112 th.</s> <s xml:id="echoid-s37388" xml:space="preserve"> 1 huius) non poteſt intelligi niſi una linea ortho-<lb/>gonaliter cadens in ipſam lineam uelin ipſum axem:</s> <s xml:id="echoid-s37389" xml:space="preserve"> quoniam linea terminans illam ſuperficiem, <lb/>in uno tantùm puncto ſecat illam lineam, ſuper quam ſuperficies declinatur:</s> <s xml:id="echoid-s37390" xml:space="preserve"> ab uno itaq;</s> <s xml:id="echoid-s37391" xml:space="preserve"> puncto <lb/>tantùm illius ſectionis fiet reflexio.</s> <s xml:id="echoid-s37392" xml:space="preserve"> Si enim à duobus punctis illius ſectionis daretur fieri reflexio <lb/>ad eundem uiſum:</s> <s xml:id="echoid-s37393" xml:space="preserve"> ſequeretur quòd in eadem ſuperficie illius reflexionis eſſent duę lineæ illius ſu-<lb/>perficiei orthogonales ſuper axem columnæ:</s> <s xml:id="echoid-s37394" xml:space="preserve"> quod eſſe non poteſt, cum illa ſuperficies ſit declina-<lb/>ta ſuper ipſum axem.</s> <s xml:id="echoid-s37395" xml:space="preserve"> Perpendicularis enlm ducta à puncto reflexionis, cadit in circulum æquidi-<lb/>ſtantem baſibus columnæ in punctum axis:</s> <s xml:id="echoid-s37396" xml:space="preserve"> & eſt communis ſectio ſuperficiei circuli & huius ſu-<lb/>perficiei reflexionis per 104 th.</s> <s xml:id="echoid-s37397" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s37398" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s37399" xml:space="preserve"> fieret reflexio etiam ab alio puncto:</s> <s xml:id="echoid-s37400" xml:space="preserve"> tunc itẽ perpen-<lb/>dicularis ducta à puncto illo reflexionis, eſſet per proximam propoſitionem diameter alterius cir-<lb/>culi illi primo circulo æquidiſtantis, & caderet in punctum axis, in quod nõ cadit ſuperficies refle-<lb/>xionis.</s> <s xml:id="echoid-s37401" xml:space="preserve"> In omnibus ergo his reflexionum ſuperficiebus ab uno tantum puncto lineæ communis fit <lb/>reflexio in eadem ſuperficie, reſpectu eiuſdem uiſus:</s> <s xml:id="echoid-s37402" xml:space="preserve"> quamuis reſpectu duorum uiſuum poſsit fieri <lb/>reflexio à duobus púctis ſuperficiei ſpeculi, ut à duobus diametri circuli terminis, quæ eſt perpen-<lb/>dicularis ſuper ipſam ſectionem:</s> <s xml:id="echoid-s37403" xml:space="preserve"> ita tamen ſi diameter illa ſit æqualis diftantiæ oculorum, uel mi-<lb/>nor, non aliter:</s> <s xml:id="echoid-s37404" xml:space="preserve"> ad unum uerò uiſum hæc fieri non poteſt:</s> <s xml:id="echoid-s37405" xml:space="preserve"> quoniã ab illo ſemper uidetur minus me-<lb/>dietate columnæ ſpeculi per 78 th.</s> <s xml:id="echoid-s37406" xml:space="preserve"> 4 huius.</s> <s xml:id="echoid-s37407" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s37408" xml:space="preserve"> quod nos demũ particularius <lb/>proſequemur, oſtendẽtes quòd in his ſpeculis quacunq;</s> <s xml:id="echoid-s37409" xml:space="preserve"> linea communi ſectione ſuperficiei refle-<lb/>xionis & ſpeculi exiftentc, ab uno tantum puncto totius ſpeculi fiet reflexio ad uiſum.</s> <s xml:id="echoid-s37410" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1537" type="section" level="0" n="0"> <head xml:id="echoid-head1170" xml:space="preserve" style="it">23. Linea uiſa non exiſtente in eadem ſuperficie, in qua eſt centrum uiſus & axis ſpeculi co-<lb/>lumnaris uel pyramidalis cõuexi, ſi linea uiſa reſpectu baſis ſpeculi ſuerit altior uel baßior cen-<lb/>tro uiſus, ſiue reflexio fiat à linea longitudinis ſpeculi ſiue à circulo: ſemper fiet ſecundum oxy-<lb/>gonias ſectiones ſuperficiem ſpeculi ſecundum pun-<lb/>ct a illarum linearum continua ſecantes.</head> <figure> <variables xml:id="echoid-variables664" xml:space="preserve">a d s f h f h b e g c b c</variables> </figure> <p> <s xml:id="echoid-s37411" xml:space="preserve">Sit linea uiſa ſiue ſit recta ſiue curua, quæ b c:</s> <s xml:id="echoid-s37412" xml:space="preserve"> & ſit <lb/>centrum uiſus a:</s> <s xml:id="echoid-s37413" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s37414" xml:space="preserve"> axis ſpeculi columnaris uel py-<lb/>ramidalis cóuexi d e:</s> <s xml:id="echoid-s37415" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s37416" xml:space="preserve"> lineæ a d & a e con-<lb/>tinentes cum axe d e trigonum a d e, in cuius ſuperfi-<lb/>cie non ſit linea b c, ſed extra illã;</s> <s xml:id="echoid-s37417" xml:space="preserve"> ſiue ſecet trigonum <lb/>a d e, ſiue non.</s> <s xml:id="echoid-s37418" xml:space="preserve"> Secet ipſum:</s> <s xml:id="echoid-s37419" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s37420" xml:space="preserve"> lineæ b c reflexio ad <lb/>uiſum a à ſuperficie ſpeculi propoſiti.</s> <s xml:id="echoid-s37421" xml:space="preserve"> Palàm autem <lb/>quòd ab uno puncto ſpeculi tota linea b c ad uiſum a <lb/>reflecti non poteſt per 29 th.</s> <s xml:id="echoid-s37422" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s37423" xml:space="preserve"> Dico quòd ſi li-<lb/>nea b c reflectatur ad uiſum a à linea lõgitudinis ſpe-<lb/>culi, quæ ſit s g (ut ſi linea b c æquidiſtet axi d e, & ſu-<lb/>perficies, in qua eſt linea b c, ſecet ſpeculum trans <lb/>axem orthogonaliter ſuper baſim ſpeculi:</s> <s xml:id="echoid-s37424" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s37425" xml:space="preserve"> ſu-<lb/>perficiem, in qua ſunt centrũ uiſus & axis ſpeculi, qui <lb/>eſt d e, ita quòd communis ſectio illarum ſuperficie-<lb/>rum ſit axis d e) fiet tamen reflexiò ad uiſum ſecun-<lb/>dum oxygonias ſectiones, quamuis fiat à linea lon-<lb/>gitudinis ſpeculi, quæ eſt s g.</s> <s xml:id="echoid-s37426" xml:space="preserve"> Palàm enim per 27 th.</s> <s xml:id="echoid-s37427" xml:space="preserve"> 5 <lb/>huius quoniam in omni ſuperficie reflexionis opor-<lb/>tet ut ſit cẽtrum uiſus, & punctus, cuius forma refle-<lb/>ctitur ad uiſum, & punctus ſpeculi, qui eſt punctus <lb/>reflexionis.</s> <s xml:id="echoid-s37428" xml:space="preserve"> Sit ergo, ut punctus b reflectatur ad uiſum a à puncto ſpeculi f:</s> <s xml:id="echoid-s37429" xml:space="preserve"> & punctus c à puncto h.</s> <s xml:id="echoid-s37430" xml:space="preserve"> <lb/>Et ducantur lineæ a f, b f, a h, c h.</s> <s xml:id="echoid-s37431" xml:space="preserve"> Quià itaq;</s> <s xml:id="echoid-s37432" xml:space="preserve"> punctus b lineæ b c non eſt in ſuperficie a d e ex hypo-<lb/>theſi:</s> <s xml:id="echoid-s37433" xml:space="preserve"> patet quòd ſuperficies ſuæ reflexionis, quæ eſt a f b, ſecat ſuperficiem a d e ſuper punctum a, <lb/> <pb o="278" file="0580" n="580" rhead="VITELLONIS OPTICAE"/> & ſuper punctum ſpeculi f:</s> <s xml:id="echoid-s37434" xml:space="preserve"> ſecat ergo ipſam ſecundum lineam a f:</s> <s xml:id="echoid-s37435" xml:space="preserve"> & ſecat ſpeculum trans axem d e.</s> <s xml:id="echoid-s37436" xml:space="preserve"> <lb/>non autem æquidiſtat baſi ex hypotheſi:</s> <s xml:id="echoid-s37437" xml:space="preserve"> quoniam illa linea uiſa, quæ b c, non eſt in ſuperficie a d e, <lb/>ſed extra illam.</s> <s xml:id="echoid-s37438" xml:space="preserve"> Superficies ergo b f a, quæ eſt ſuperficies reflexionis, tranſuerſaliter ſecat axem d e:</s> <s xml:id="echoid-s37439" xml:space="preserve"> <lb/>quoniam linea uiſa eſt altior uel baſsior centro uiſus ex hypotheſi.</s> <s xml:id="echoid-s37440" xml:space="preserve"> Communis ergo ſection ſuperfi-<lb/>ciei reflexionis & ſpeculi per 10 huius eſt oxygonia ſectio.</s> <s xml:id="echoid-s37441" xml:space="preserve"> Similiterq́;</s> <s xml:id="echoid-s37442" xml:space="preserve"> eſt de puncto c, & quolibet <lb/>medio puncto lineæ b c.</s> <s xml:id="echoid-s37443" xml:space="preserve"> Licet itaq;</s> <s xml:id="echoid-s37444" xml:space="preserve"> omnia puncta lineæ b c reflectantur ad centrum uiſus a à linea <lb/>longitudinis ſpeculi:</s> <s xml:id="echoid-s37445" xml:space="preserve"> cuiuslibet tamen puncti reflexio ad uiſum fiet ſecundum oxygoniam ſectio-<lb/>nem.</s> <s xml:id="echoid-s37446" xml:space="preserve"> Similiterq́;</s> <s xml:id="echoid-s37447" xml:space="preserve"> demonſtrandum, ſi ſuperficies incidẽtiæ lineæ b c orthogonaliter ſecet axem ſpe-<lb/>culi, & ſuperficiem a d e:</s> <s xml:id="echoid-s37448" xml:space="preserve"> tunc enim communis ſectio ſuperficiei incidentiæ lineæ b c & ſuperficiei <lb/>ſpeculi fiet circulus æquidiſtans baſi ſpeculi per 100 th.</s> <s xml:id="echoid-s37449" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s37450" xml:space="preserve"> Vnde ſi fiat reflexio ad uiſum, fiet ab <lb/>arcu circuli æquidiſtantis baſi ſpeculi:</s> <s xml:id="echoid-s37451" xml:space="preserve"> quælibet tamen ſuperficies reflexionis tranſiens centrum <lb/>uiſus ſecabit obliquè axem ſpeculi ſecũdum aliquod punctum illius arcus.</s> <s xml:id="echoid-s37452" xml:space="preserve"> Licet itaq;</s> <s xml:id="echoid-s37453" xml:space="preserve"> omnia pun-<lb/>cta lineæ b c reflectantur ad uiſum a ab arcu circuli ſpeculi:</s> <s xml:id="echoid-s37454" xml:space="preserve"> fit tamen cniuslibet puncti illius lineæ <lb/>reflexio ſecundum oxygoniam ſectionem.</s> <s xml:id="echoid-s37455" xml:space="preserve"> Si tamẽ aliquis punctorum lineæ b c ſuerit cum centro <lb/>uiſus in eadem ſuperficie æquidiſtanter baſi ſpeculum ſecante:</s> <s xml:id="echoid-s37456" xml:space="preserve"> illius ſolius reflexio fiet ſecundum <lb/>circulum, aliorum uerò omnium punctorum reflexio fiet ſecundũ oxygonias ſectiones:</s> <s xml:id="echoid-s37457" xml:space="preserve"> & ſic pun-<lb/>cta illius lineæ diuerſas afferent uiſui paſsiones.</s> <s xml:id="echoid-s37458" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s37459" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1538" type="section" level="0" n="0"> <head xml:id="echoid-head1171" xml:space="preserve" style="it">24. In omni ſuperficie reſlexionis à ſpeculis columnaribus uel pyramidalibus conuexis, cen-<lb/>trum uiſus: punctum uiſum: punctum reflexionis: punctum axis, in quem cadit perpendicula-<lb/>ris ducta à puncto reflexionis ſuper ſuperficiem ſpeculi, conſiſtere eſt neceſſe. Alhaz. 23. 34 n 4.</head> <p> <s xml:id="echoid-s37460" xml:space="preserve">Quòd centrum uiſus:</s> <s xml:id="echoid-s37461" xml:space="preserve"> & punctum reflexionis:</s> <s xml:id="echoid-s37462" xml:space="preserve"> & punctum reflexum ſintin ſuperficie reflexionis <lb/>patet per 27 th.</s> <s xml:id="echoid-s37463" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s37464" xml:space="preserve"> In omni enim ſuperficie reflexionis neceſſariò ſunt linea incidentiæ & re-<lb/>flexionis, quæ continent tria puncta prædicta.</s> <s xml:id="echoid-s37465" xml:space="preserve"> Et ſi ſuperſicies reflexionis ſecet ſpeculum ſecũdum <lb/>lineam ſuæ longitudinis:</s> <s xml:id="echoid-s37466" xml:space="preserve"> palàm per 7 huius quòd totus axis & punctum, in quod cadit perpendi-<lb/>cularis à puncto reflexionis ducta, ſunt in hac ſuperficie.</s> <s xml:id="echoid-s37467" xml:space="preserve"> Si uero communis ſectio ſuperficiei refle-<lb/>xionis & ſpeculi ſit circulus:</s> <s xml:id="echoid-s37468" xml:space="preserve"> palàm quia centrum illius circuli, qui eſt punctus axis, ad quem per 21 <lb/>huius omnes perpendiculares à puncto reflexionis totius circuli productæ concurrunt, eſt in ſu-<lb/>perficie reflexionis:</s> <s xml:id="echoid-s37469" xml:space="preserve"> quoniam tunc totus circulus eſt in ſuperficie reflexionis.</s> <s xml:id="echoid-s37470" xml:space="preserve"> Siautem communis <lb/>ſectio ſuperficiei reflexionis & ſpeculi ſit ſectio oxygonia:</s> <s xml:id="echoid-s37471" xml:space="preserve"> palàm per 10 huius quia hæc ſectio de-<lb/>clinis eſt ſuper axem columnæ, interſecans axem in puncto, cui incidit perpendicularis producta à <lb/>puncto reflexionis ſuper ſuperficiem contingentem columnam in pũcto ſectionis.</s> <s xml:id="echoid-s37472" xml:space="preserve"> Patet ergo pro-<lb/>poſitum ſecundum omnem diuerſitatem dictarum ſectionum.</s> <s xml:id="echoid-s37473" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1539" type="section" level="0" n="0"> <head xml:id="echoid-head1172" xml:space="preserve" style="it">25. In ſuperficie apparente ſpeculi columnaris conuexi ſiue communis ſectio ſuperficiei refle-<lb/>xionis & ſpeculi ſit linea longitudinis ſpeculi, ſiue circulus, ſiue oxygonia ſectio: à quolibet pun-<lb/>cto poteſt fieri reflexio aduiſum. Alhazen 28 n 4.</head> <p> <s xml:id="echoid-s37474" xml:space="preserve">Signentur termini apparẽtis portionis colũnæ, ut prius:</s> <s xml:id="echoid-s37475" xml:space="preserve"> & ſit illa portio d e f g:</s> <s xml:id="echoid-s37476" xml:space="preserve"> & ſit p punctus <lb/>datus in ſuperficie illa apparẽte:</s> <s xml:id="echoid-s37477" xml:space="preserve"> ſitq́:</s> <s xml:id="echoid-s37478" xml:space="preserve"> x punctus rei uiſæ.</s> <s xml:id="echoid-s37479" xml:space="preserve"> Dico quòd à puncto p poteſt fieri reflexio <lb/>formæ puncti x ad centrum uiſus, quod ſit a.</s> <s xml:id="echoid-s37480" xml:space="preserve"> Sit enim primò, ut ſuperficies reflexionis (in qua ſunt <lb/>punctus uiſus, qui eſt x, & centrum uiſus a, & punctus, à quo ſit reflexio, qui eſt p) ſecet columnam <lb/>ſpeculi ſecũdum axem h k i:</s> <s xml:id="echoid-s37481" xml:space="preserve"> erit ergo per 7 huius communis ſectio illius ſuperficiei & ſpeculi linea <lb/>longitudinis columnæ, quæ ſit m p n.</s> <s xml:id="echoid-s37482" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s37483" xml:space="preserve"> linea x p:</s> <s xml:id="echoid-s37484" xml:space="preserve"> & à puncto p erigatur linea perpendi-<lb/>cularis ſuper lineam m n per 11 p 1, quæ ſit p z:</s> <s xml:id="echoid-s37485" xml:space="preserve"> & ſuper punctum p terminum lineæ z p ſiat angulus <lb/>æqualis angulo x p z, qui ſit z p q.</s> <s xml:id="echoid-s37486" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s37487" xml:space="preserve"> centrum uiſus, quod eſt a, ſuerit in linea p q, palàm per 20 <lb/>th.</s> <s xml:id="echoid-s37488" xml:space="preserve"> 5 huius, cum angulus incidentiæ ſit æqualis angulo reflexionis, quoniam à puncto p fiet reflexio <lb/>formæ puncti x ad uiſum a exiſtentem in linea p q.</s> <s xml:id="echoid-s37489" xml:space="preserve"> Quòd ſi ſuperſicies reflexionis ſecet columnam <lb/>ſpeculi æquidiftanter baſibus:</s> <s xml:id="echoid-s37490" xml:space="preserve"> palàm quia communis ſectio erit circulus per 9 huius;</s> <s xml:id="echoid-s37491" xml:space="preserve"> fietq́;</s> <s xml:id="echoid-s37492" xml:space="preserve"> iterum <lb/>à puncto p reflexio ad uiſum.</s> <s xml:id="echoid-s37493" xml:space="preserve"> Ducatur enim per 102 th.</s> <s xml:id="echoid-s37494" xml:space="preserve"> 1 huius circulus æquidiſtans baſibus colu-<lb/>mnæ, tranſiens per punctum p, qui ſit b p l:</s> <s xml:id="echoid-s37495" xml:space="preserve"> cuius centrum ſit k:</s> <s xml:id="echoid-s37496" xml:space="preserve"> in cuius ſuperficie extenſa extra ſpe-<lb/>culum ſi fuerit punctum uiſum, & ducatur linea x p, quæ producta ſi tranſeat centrum circuli k:</s> <s xml:id="echoid-s37497" xml:space="preserve"> pa-<lb/>làm, cum axis columnæ h k i ſit orthogonalis ſuper ſuperficiem Illius circuli, ſicut & ſuper baſes co-<lb/>lumnæ per 100 & 23 th.</s> <s xml:id="echoid-s37498" xml:space="preserve"> 1 huius, quoniam & ipſe axis h k i orthogonalis erit ſuper lineam x p:</s> <s xml:id="echoid-s37499" xml:space="preserve"> ergo & <lb/>linea longitudinis columnæ (quæ eſt m p) erit orthogonalis ſuper lineam x p per 29 p 1.</s> <s xml:id="echoid-s37500" xml:space="preserve"> Reflectetur <lb/>ergo per 21 th.</s> <s xml:id="echoid-s37501" xml:space="preserve"> 5 huius linea x p in ſeipſam, & in ea exiſtente uiſu ſorma pũcti x uiſui occurret.</s> <s xml:id="echoid-s37502" xml:space="preserve"> Si ue-<lb/>rò linea x p producta non tranſeat centrum circuli k, ſed obliquetur ab illo:</s> <s xml:id="echoid-s37503" xml:space="preserve"> tunc copuletur ſemi-<lb/>diameter, quæ k p, quæ, ut patet ex pręmiſsis, erit orthogonalis ſuper axẽ h i:</s> <s xml:id="echoid-s37504" xml:space="preserve"> erit ergo linea k p per-<lb/>pendicularis ſuper lineam longitudinis, quæ eſt m p per 29 p 1:</s> <s xml:id="echoid-s37505" xml:space="preserve"> erit ergo k p perpendicularis ſuper <lb/>ſuperficiem contingentem columnam ſecundũ lineam longitudinis m p:</s> <s xml:id="echoid-s37506" xml:space="preserve"> in qua ducatur linea con-<lb/>tingens circulum b p line puncto p, quæ ſit s p t:</s> <s xml:id="echoid-s37507" xml:space="preserve"> educaturq́;</s> <s xml:id="echoid-s37508" xml:space="preserve"> linea k p perpendiculariter ſuper illam <lb/>ſuperfioiem in punctum u:</s> <s xml:id="echoid-s37509" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s37510" xml:space="preserve">, ut prius, centrum uiſus, quod eſt a, in linea q p in eadem ſuperficie <lb/>eirculi.</s> <s xml:id="echoid-s37511" xml:space="preserve"> Et quoniam in illa ſuperficie circulum contingente eſt linea s t, erit angulus k p t rectus:</s> <s xml:id="echoid-s37512" xml:space="preserve"> <lb/>ergo & angulus s p u eſt rectus per 15 p 1.</s> <s xml:id="echoid-s37513" xml:space="preserve"> Palàm ergo quia angulus a p s eſt minor recto:</s> <s xml:id="echoid-s37514" xml:space="preserve"> ergo eſt <lb/> <pb o="279" file="0581" n="581" rhead="LIBER SEPTIMVS."/> acutus:</s> <s xml:id="echoid-s37515" xml:space="preserve"> ergo per 13 p 1 angulus a p t eſt obtuſus:</s> <s xml:id="echoid-s37516" xml:space="preserve"> reſcindatur ergo ab angulo u p t recto angulus <lb/>æqualis angulo a p u per 27 th.</s> <s xml:id="echoid-s37517" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s37518" xml:space="preserve"> Si ergo linea x <lb/> <anchor type="figure" xlink:label="fig-0581-01a" xlink:href="fig-0581-01"/> p illum angulum contineat:</s> <s xml:id="echoid-s37519" xml:space="preserve"> palàm per 20 th.</s> <s xml:id="echoid-s37520" xml:space="preserve"> 5 huius <lb/>quoniam à pũcto p reflectetur forma pũcti x ad pun-<lb/>ctum a centrum uiſus.</s> <s xml:id="echoid-s37521" xml:space="preserve"> Quòd ſi linea x p illum angulũ <lb/>non contineat:</s> <s xml:id="echoid-s37522" xml:space="preserve"> tunc, ut prius, ſuper punctum p termi-<lb/>num lineę u p fiat angulus ęqualis angulo x p u per 23 <lb/>p 1.</s> <s xml:id="echoid-s37523" xml:space="preserve"> In linea quoq;</s> <s xml:id="echoid-s37524" xml:space="preserve"> illum angulum continente poſito <lb/>centro uiſus a, patet propoſitum, ut prius.</s> <s xml:id="echoid-s37525" xml:space="preserve"> Et quoniã <lb/>perpendicularis k p u eſt cum puncto a in eadem ſu-<lb/>perficie per præmiſſam, erit linea a p in eadem ſuper-<lb/>ficie cum linea x p:</s> <s xml:id="echoid-s37526" xml:space="preserve"> & erit hæc ſuperficies ipſa ſuperfi-<lb/>cies reflexionis & orthogonalis ſuper ſuperficiẽ ſpe-<lb/>culum contingentem ſecundũ lineam m n:</s> <s xml:id="echoid-s37527" xml:space="preserve"> quoniam <lb/>perpẽdicularis p u (quæ eſt in ſuperficie reflexionis) <lb/>erecta eſt ſuper ſuperficiem ſecundũ lineam m n ſpe-<lb/>culum contingentem:</s> <s xml:id="echoid-s37528" xml:space="preserve"> & eſt in ea circulus b p l æqui-<lb/>diſtans baſibus columnæ.</s> <s xml:id="echoid-s37529" xml:space="preserve"> Et ſimiliter poteſt demon-<lb/>ſtrari de alijs punctis datis in dicta ſuperficie ſpeculi.</s> <s xml:id="echoid-s37530" xml:space="preserve"> <lb/>Idem quoq;</s> <s xml:id="echoid-s37531" xml:space="preserve"> patetſi cõmunis ſectio ſuperficiei refle-<lb/>xionis & ſpeculi colũnaris ſuerit ſectio oxygonia per <lb/>10 huius:</s> <s xml:id="echoid-s37532" xml:space="preserve"> quoniam, ut oſtendimus in 21 huius, patet <lb/>quò d ſemper perpẽdicularis ducta à pũcto reflexio-<lb/>nis cadit in aliquod punctum axis, & eſt ſemidiame-<lb/>ter circuli cuiuſdam ſecãtis ſuperficiem ſpeculi ęqui-<lb/>diſtanter baſibus columnæ:</s> <s xml:id="echoid-s37533" xml:space="preserve"> ductaq́;</s> <s xml:id="echoid-s37534" xml:space="preserve"> linea in puncto dato ſpeculum ſecundum oxygoniam ſectio-<lb/>nem contingente, & producta illa perpendiculari, ſi punctus rei uiſæ & centrũ uiſus cadant in ean-<lb/>dem perpendicularem, uel in lineas in eadem ſuperficie cum perpendiculari exiſtentes, & æquales <lb/>angulos cum ipſa continentes:</s> <s xml:id="echoid-s37535" xml:space="preserve"> fiet ſecundum præmiſſa reflexio ad uiſum.</s> <s xml:id="echoid-s37536" xml:space="preserve"> Patet ergo uniuerſaliter <lb/>propoſitum in omni ſectione communi ſuperficiei reflexionis & ſuperficiei ſpeculi columnaris.</s> <s xml:id="echoid-s37537" xml:space="preserve"/> </p> <div xml:id="echoid-div1539" type="float" level="0" n="0"> <figure xlink:label="fig-0581-01" xlink:href="fig-0581-01a"> <variables xml:id="echoid-variables665" xml:space="preserve">x f h d m b k l t z s p u g i e n a q</variables> </figure> </div> </div> <div xml:id="echoid-div1541" type="section" level="0" n="0"> <head xml:id="echoid-head1173" xml:space="preserve" style="it">26. Superficiei reflexionis & ſpeculi columnaris conuexi communi ſectione linea longitudi-<lb/>nis ſpeculi exiſtente: formæ eiuſdem punctirei uiſæ ab uno tantùm puncto totius ſuperficiei ſpe-<lb/>culi ad unum uiſum fit reflexio. Alhazen 46 n 5.</head> <p> <s xml:id="echoid-s37538" xml:space="preserve">Eſto ſpeculum columnare conuexum, cuius axis ſit c d:</s> <s xml:id="echoid-s37539" xml:space="preserve"> ſitq̃;</s> <s xml:id="echoid-s37540" xml:space="preserve"> ſuperficies reflexionis a b g, ita ut <lb/>forma puncti b reflectatur ad a cẽtrum circuli à puncto g ſuperficiei ſpeculi:</s> <s xml:id="echoid-s37541" xml:space="preserve"> & ſit communis ſectio <lb/>ſuperficierum iſtarum linea f g n, quæ eſt linea longitudinis ſpeculi.</s> <s xml:id="echoid-s37542" xml:space="preserve"> Dico quòd forma puncti b non <lb/>poteſt reflecti ad centrum uiſus a ab alio pun-<lb/>cto ſpeculi quàm à puncto g.</s> <s xml:id="echoid-s37543" xml:space="preserve"> Ducatur enim à <lb/> <anchor type="figure" xlink:label="fig-0581-02a" xlink:href="fig-0581-02"/> puncto g perpendicularis ſuper ſuperficiem <lb/>contingentem columnam ſecundum lineam <lb/>f g n per 12 p 11:</s> <s xml:id="echoid-s37544" xml:space="preserve"> quæ ſit linea g q, ſecans lineam <lb/>a b productam inter punctum uiſum & cen-<lb/>trũ uiſus in puncto q.</s> <s xml:id="echoid-s37545" xml:space="preserve"> Palàm per 21 huius quo-<lb/>niam hæc linea g q producta intra ſpeculum <lb/>ſecat ipſum trans axem c d:</s> <s xml:id="echoid-s37546" xml:space="preserve"> ſecet ergo in pun-<lb/>cto e.</s> <s xml:id="echoid-s37547" xml:space="preserve"> Et quia linea longitudinis, quę eſt f n, eſt <lb/>in ſuperficie reflexionis:</s> <s xml:id="echoid-s37548" xml:space="preserve"> palàm quoniam axis <lb/>c d erit in eadem per 7 huius:</s> <s xml:id="echoid-s37549" xml:space="preserve"> ergo & pũctum <lb/>e erit in illa ſuperficie.</s> <s xml:id="echoid-s37550" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s37551" xml:space="preserve"> una ſola ſuperficies poſsit intelligi, in qua ſunt ſimul omnia pũcta <lb/>a, b, g & e, & lineæ f n & c d:</s> <s xml:id="echoid-s37552" xml:space="preserve"> palàm quòd à ſuperficie totius ſpeculi non poteſt reflecti forma puncti <lb/>b ad a centrum uiſus, niſi à linea longitudinis f n:</s> <s xml:id="echoid-s37553" xml:space="preserve"> ſed per 45 th.</s> <s xml:id="echoid-s37554" xml:space="preserve"> 5 huius oſtenſum eſt quòd in ſpecu-<lb/>lis planis ab uno ſolo puncto fit unius puncti reflexio ad uiſum:</s> <s xml:id="echoid-s37555" xml:space="preserve"> ergo & in his ſpeculis non poteſt <lb/>fieri reflexio ab alio pũcto quàm a b uno ſolo puncto ſcilicet lineæ f n.</s> <s xml:id="echoid-s37556" xml:space="preserve"> Forma ergo puncti b reflecti-<lb/>tur ad uiſum a ab uno ſolo puncto ſuperficiei totius ſpeculi.</s> <s xml:id="echoid-s37557" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s37558" xml:space="preserve"/> </p> <div xml:id="echoid-div1541" type="float" level="0" n="0"> <figure xlink:label="fig-0581-02" xlink:href="fig-0581-02a"> <variables xml:id="echoid-variables666" xml:space="preserve">a q k b f g l n c e i d h</variables> </figure> </div> </div> <div xml:id="echoid-div1543" type="section" level="0" n="0"> <head xml:id="echoid-head1174" xml:space="preserve" style="it">27. Superficiei reflexionis & ſpeculi columnaris conuexi communi ſectione exiſtente circu-<lb/>lo baſibus ſpeculi æquidiſtante: ab uno ſolo puncto ſuperficieitotius ſpeculi formæ eiuſdem puncte <lb/>reiuiſæ fit reflexio aduiſum. Alhazen 46 n 5.</head> <p> <s xml:id="echoid-s37559" xml:space="preserve">Sit diſpoſitio, quæ in præcedente, palamq́;</s> <s xml:id="echoid-s37560" xml:space="preserve"> per 17 huius quoniam hác hypotheſi exiſtẽte, ſuper-<lb/>ficies reflexionis a b g erit æquidiſtans baſibus columnæ:</s> <s xml:id="echoid-s37561" xml:space="preserve"> circulus quoq;</s> <s xml:id="echoid-s37562" xml:space="preserve">, qui eſt communis ſectio <lb/>ſuperficiei a b g & columnæ, cuius axis eſt c d, qui eſt æquidiſtans baſibus columnæ, ſit g h:</s> <s xml:id="echoid-s37563" xml:space="preserve"> cuius <lb/>centrum ſit punctum e.</s> <s xml:id="echoid-s37564" xml:space="preserve"> Dico quòd à circulo g h (qui eſt communis ſectio ſuperficiei a b g & ſuper-<lb/>ficiei ſpeculi) non poteſt fieri reflexio formæ b ad a uiſuin, niſi ab uno tantùm pũcto g.</s> <s xml:id="echoid-s37565" xml:space="preserve"> Patuit enim <lb/>per 16 th.</s> <s xml:id="echoid-s37566" xml:space="preserve"> 6 huius quia in ſpeculis ſphæricis conuexis à circulo, ſuper quem fit reflexio, non poteſt <lb/> <pb o="280" file="0582" n="582" rhead="VITELLONIS OPTICAE"/> fieri reflexio, niſi ab uno tantùm puncto:</s> <s xml:id="echoid-s37567" xml:space="preserve"> ergo nec in iſtis ſpeculis columnaribus fiet reflexio ſormæ <lb/>unius puncti rei uiſæ and uiſum, niſi ab uno tantùm puncto, quod ſit g.</s> <s xml:id="echoid-s37568" xml:space="preserve"> Siuerò detur, quòd ab alio <lb/>puncto ſpeculi huius (ut à pũctol) ſimiliter fiat reflexio, ſicut à puncto g:</s> <s xml:id="echoid-s37569" xml:space="preserve"> producatur à puncto da-<lb/>to linea l k per 12 p 11 perpendicularis ſuper ſuperficiem columnæ:</s> <s xml:id="echoid-s37570" xml:space="preserve"> hæc ergo producta cadet or-<lb/>thogonaliter ſuper axem c d per 21 huius:</s> <s xml:id="echoid-s37571" xml:space="preserve"> cadat in punctum axis, quod ſiti.</s> <s xml:id="echoid-s37572" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s37573" xml:space="preserve"> linea l <lb/>k, ut patet ex præmiſsis, ſecabit lineam a b productam inter punctum rei uiſæ & centrum uiſus:</s> <s xml:id="echoid-s37574" xml:space="preserve"> ſe-<lb/>cetq́;</s> <s xml:id="echoid-s37575" xml:space="preserve"> ipſam in puncto k:</s> <s xml:id="echoid-s37576" xml:space="preserve"> quod ſiue ſuerit idem cum puncto q, ſiue aliud à puncto q, ducatur ſem-<lb/>per linea k e ad centrum circuli g h:</s> <s xml:id="echoid-s37577" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s37578" xml:space="preserve"> linea k e orthogonalis ſuper axem c d:</s> <s xml:id="echoid-s37579" xml:space="preserve"> quoniam eſt in <lb/>ſuperficie reflexionis orthogonaliter axem c d ſecante.</s> <s xml:id="echoid-s37580" xml:space="preserve"> Duæ ergo lineæ k e & k i cum linea e i, par-<lb/>te axis continent triangulum, cuius duo anguli ſunt recti:</s> <s xml:id="echoid-s37581" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s37582" xml:space="preserve"> Palàm ergo quòd <lb/>in tali diſpoſitione non reflectitur forma p uncti b ad uiſum a, ab aliquo puncto ſuperficiei totius <lb/>ſpeculi alio, quàm à puncto g.</s> <s xml:id="echoid-s37583" xml:space="preserve"> Ethoc eſt propoſitum.</s> <s xml:id="echoid-s37584" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1544" type="section" level="0" n="0"> <head xml:id="echoid-head1175" xml:space="preserve" style="it">28. Superficiei reflexionis & ſpeculi columnaris conuexi communi ſectione exiſtente oxy-<lb/>gonia: formæ eiuſdem puncti rei uiſæ ab uno ſolo puncto totius ſuperficiei ſpeculifit reflexio ad <lb/>uiſum. Alhazen 47 n 5.</head> <p> <s xml:id="echoid-s37585" xml:space="preserve">Sit ſuperficies reflexionis a b g:</s> <s xml:id="echoid-s37586" xml:space="preserve"> cuius communis ſectio cum ſuperficie ſpeculi columnaris ſit o-<lb/>xygonia ſectio, tranſiens in ſuperficie ſpeculi punctum g:</s> <s xml:id="echoid-s37587" xml:space="preserve"> & ſit b punctus rei uiſæ:</s> <s xml:id="echoid-s37588" xml:space="preserve"> & a centrum ui-<lb/>ſus:</s> <s xml:id="echoid-s37589" xml:space="preserve"> & g punctus reflexionis.</s> <s xml:id="echoid-s37590" xml:space="preserve"> Dico quoniam forma puncti b non reflectitur ad centrum uiſus a ab <lb/>aliquo puncto totius ſuperficiei ſpeculi, niſi à puncto g.</s> <s xml:id="echoid-s37591" xml:space="preserve"> Ducatur enim à puncto a ſuperficies æqui-<lb/>diſtans baſibus columnæ, ſecans ſpeculum ſecundum circulum, qui ſit e z i:</s> <s xml:id="echoid-s37592" xml:space="preserve"> quod ſic fiet.</s> <s xml:id="echoid-s37593" xml:space="preserve"> Producta <lb/>enim à puncto a linea perpendiculari ſuper axem columnę per 12 p 1:</s> <s xml:id="echoid-s37594" xml:space="preserve"> erit hęc linea perpendicularis <lb/>erecta ſuper ſuperficiem columnæ:</s> <s xml:id="echoid-s37595" xml:space="preserve"> quia erit perpendicularis ſuper lineam longitudinis columnæ, <lb/>cui ipſa incidit per 29 p 1.</s> <s xml:id="echoid-s37596" xml:space="preserve"> Ducatur item ab eodẽ puncto axis, quod ſit q, alia linea rectum continens <lb/>angulum cum axe, quæ ſit linea q e.</s> <s xml:id="echoid-s37597" xml:space="preserve"> Ergo per 18 p 11 patet quoniam ſuperficies plana lineas illas a q <lb/>& q e imaginata pertranſire, ſuper ſuperficiem ſpeculi erit orthogonaliter erecta.</s> <s xml:id="echoid-s37598" xml:space="preserve"> Et quoniam per 4 <lb/>p 11 axis ſpeculi erectus eſt ſuper illã ſuperficiem, patet per 14 p 11 & per 92 th.</s> <s xml:id="echoid-s37599" xml:space="preserve"> 1 huius quoniam illa <lb/>ſuperficies æquidiſtat baſibus ſpeculi:</s> <s xml:id="echoid-s37600" xml:space="preserve"> ergo per 100 th.</s> <s xml:id="echoid-s37601" xml:space="preserve"> 1 huius, cum ipſa ſecet ſuperficiem columnæ <lb/>æquidiſtanter baſibus:</s> <s xml:id="echoid-s37602" xml:space="preserve"> patet quòd ipſa ſecat ſecundum circulũ, qui ſit e zi, cuius centrum erit pun-<lb/>ctum q.</s> <s xml:id="echoid-s37603" xml:space="preserve"> Et eodem modo à puncto g ducatur ſuperficies æquidiſtans baſibus ſpeculi, quæ ſecet ſpe-<lb/>culum ſecundum circulum s g p:</s> <s xml:id="echoid-s37604" xml:space="preserve"> cuius centrum ſit t:</s> <s xml:id="echoid-s37605" xml:space="preserve"> & in illo circulo ducatur ab axe linea ad pun-<lb/>ctum g, quæ ſit t g:</s> <s xml:id="echoid-s37606" xml:space="preserve"> & hæc per 21 huius erit perpendicularis ſuper ſuperficiem contingentem colu-<lb/>mnam in linea lõgitudinis, <lb/> <anchor type="figure" xlink:label="fig-0582-01a" xlink:href="fig-0582-01"/> in qua eſt punctus g.</s> <s xml:id="echoid-s37607" xml:space="preserve"> Linea <lb/>quoq;</s> <s xml:id="echoid-s37608" xml:space="preserve"> t g producta concur-<lb/>rat cum linea a b in puncto <lb/>k:</s> <s xml:id="echoid-s37609" xml:space="preserve"> cõcurret autem per 29 th.</s> <s xml:id="echoid-s37610" xml:space="preserve"> <lb/>1 huius:</s> <s xml:id="echoid-s37611" xml:space="preserve"> ideo ꝗa diuidit an-<lb/>gulum a g b, & puncta g, a, b <lb/>ſunt in eadem ſuperficie re-<lb/>flexionis per 24 huius.</s> <s xml:id="echoid-s37612" xml:space="preserve"> Du-<lb/>catur etiam à pũcto g linea <lb/>longitudinis ſpeculi per 101 <lb/>th.</s> <s xml:id="echoid-s37613" xml:space="preserve"> 1 huius, quæ ſit g z, cadẽs <lb/>inter duas ſectiones æqui-<lb/>diſtãtes baſibus ſpeculi nũc <lb/>ductas:</s> <s xml:id="echoid-s37614" xml:space="preserve"> & erit per 25 th.</s> <s xml:id="echoid-s37615" xml:space="preserve"> 1 <lb/>huius pars axis æqualis li-<lb/>neæ g z, linea t q:</s> <s xml:id="echoid-s37616" xml:space="preserve"> & à puncto b rei uiſæ ducatur linea perpendicularis ſuper ſuperficiem, ſecantem <lb/>ſpeculum ſecundum circulum e zi per 11 p 11:</s> <s xml:id="echoid-s37617" xml:space="preserve"> quæ ſit b h:</s> <s xml:id="echoid-s37618" xml:space="preserve"> & ducãtur duæ lineæ a z & h z:</s> <s xml:id="echoid-s37619" xml:space="preserve"> & ducatur <lb/>à puncto z in ſuperficie illa ad axẽ ſpeculi linea z q:</s> <s xml:id="echoid-s37620" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s37621" xml:space="preserve"> hæc linea z q perpendicularis ſuper axem <lb/>q t per 21 huius, ſicut & ſuperficies e z i, in qua protrahitur:</s> <s xml:id="echoid-s37622" xml:space="preserve"> & erit per eandem 21 huius linea z q per-<lb/>pendicularis ſuper ſuperficiem contingẽtem ſpeculum in puncto z.</s> <s xml:id="echoid-s37623" xml:space="preserve"> Quia ergo linea q z educta ex-<lb/>tra ſpeculi ſuperficiem neceſſariò diuidit angulum h z a, eò quò d â concurſu linearum h z & a z or-<lb/>thogonaliter producitur ſuper ſuperficiem contingẽtem, cui ſuperficiei lineæ a z & h z obliquè in-<lb/>cidunt:</s> <s xml:id="echoid-s37624" xml:space="preserve"> palàm per 29 th.</s> <s xml:id="echoid-s37625" xml:space="preserve"> 1 huius quia producta linea z q concurret cum linea a k, quæ ſubtenditur <lb/>angulo a z h:</s> <s xml:id="echoid-s37626" xml:space="preserve"> concurrat ergo in puncto l.</s> <s xml:id="echoid-s37627" xml:space="preserve"> Dico quoniã forma puncti h lineæ b h reflectitur ad uiſum <lb/>a à puncto ſpeculi z.</s> <s xml:id="echoid-s37628" xml:space="preserve"> Ducatur enim à puncto a linea æquidiſtans k g lineæ, quę ſit a m:</s> <s xml:id="echoid-s37629" xml:space="preserve"> hæc utiq;</s> <s xml:id="echoid-s37630" xml:space="preserve"> per <lb/>2 th.</s> <s xml:id="echoid-s37631" xml:space="preserve"> 1 huius concurret cum linea b g, cum qua ſua æquidiſtans concurrit:</s> <s xml:id="echoid-s37632" xml:space="preserve"> ſunt enim lineæ a b, b g, k <lb/>g omnes in eadem ſuperficie reflexionis:</s> <s xml:id="echoid-s37633" xml:space="preserve"> ſit ergo punctus cõcurſus linearum b g & a m punctus m.</s> <s xml:id="echoid-s37634" xml:space="preserve"> <lb/>Palàm quoq;</s> <s xml:id="echoid-s37635" xml:space="preserve"> per 6 p 11 quoniam linea g z æ quidiſtat lineæ b h, cum utraq;</s> <s xml:id="echoid-s37636" xml:space="preserve"> ipſarum ſit orthogonalis <lb/>ſuper ſuperficiem e z i æquidiſtantem baſibus columnæ:</s> <s xml:id="echoid-s37637" xml:space="preserve"> eſt ergo per 7 p 11 linea b g m in eadem ſu-<lb/>perficie, cum ſecet illas duas lineas æquidiſtãtes.</s> <s xml:id="echoid-s37638" xml:space="preserve"> In ſuperficie ergo reflexionis (quæ eſt a b g) ſunt <lb/> <pb o="281" file="0583" n="583" rhead="LIBER SEPTIMVS."/> tria punctam, z, h.</s> <s xml:id="echoid-s37639" xml:space="preserve"> Itẽ quia linea a m eſt æ quidiſtans lineæ k g, ſed & linea z l eſt æquidiſtans line æ k <lb/>g per 33 p 1:</s> <s xml:id="echoid-s37640" xml:space="preserve"> ſunt enim lineæ g z & t q æ quales & æ quidiſtãtes, ut patert ex præmiſsis, & linea t g pro-<lb/>ducitur in punctum k:</s> <s xml:id="echoid-s37641" xml:space="preserve"> & linea q z in punctũ l:</s> <s xml:id="echoid-s37642" xml:space="preserve"> erit ergo per 30 p 1 linea l z æ quidiſtãs lineæ a m.</s> <s xml:id="echoid-s37643" xml:space="preserve"> Sunt <lb/>ergo per 2 th.</s> <s xml:id="echoid-s37644" xml:space="preserve"> 1 huius lineæ l z & am in eadẽ ſuperſicie:</s> <s xml:id="echoid-s37645" xml:space="preserve"> & in eadem eſt linea h a per 7 p 11.</s> <s xml:id="echoid-s37646" xml:space="preserve"> Igitur tria <lb/>puncta m, z, h ſunt in eadem ſuperficie, in qua ſunt lineæ l z & a m & h a, quæ eſt ſuperficies h l z m.</s> <s xml:id="echoid-s37647" xml:space="preserve"> <lb/>Sed iam patuit ſuprà quòd ſunt in ſuperficie m b h:</s> <s xml:id="echoid-s37648" xml:space="preserve"> igitur ſuntin linea cõmuni illis duabus ſuper-<lb/>ficiebus:</s> <s xml:id="echoid-s37649" xml:space="preserve"> ergo per 3 p 11 linea h z m eſt linea recta.</s> <s xml:id="echoid-s37650" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s37651" xml:space="preserve"> punctus g ſit punctus reſlexionis exhy-<lb/>potheſi:</s> <s xml:id="echoid-s37652" xml:space="preserve"> erit per 20 th.</s> <s xml:id="echoid-s37653" xml:space="preserve"> 5 huius angulus a g k æqualis angulo k g b:</s> <s xml:id="echoid-s37654" xml:space="preserve"> ſed angulus k g b per 29 p 1 eſt <lb/>æqualis angulo a m g, cum ſit extrinſecus ad illum, & linea k g æquidiſtet lineæ a m:</s> <s xml:id="echoid-s37655" xml:space="preserve"> ſed & angulus <lb/>a g k eſt æ qualis angulo m a g per eandem 29 p 1, quia eſt illi-coalternus:</s> <s xml:id="echoid-s37656" xml:space="preserve"> ergo anguli a m g & m a g <lb/>ſunt æ quales:</s> <s xml:id="echoid-s37657" xml:space="preserve"> ergo per 6 p 1 duæ lineæ a g & m g ſunt æquales, quia uerò linea g z eſt erecta ſuper <lb/>ſuperficiem a h z, ut patet ex præmifsis:</s> <s xml:id="echoid-s37658" xml:space="preserve"> erit linea g z ortho gonalis ſuper quamlibet lineam ſuper-<lb/>ficiei a h z, ductam à puncto z:</s> <s xml:id="echoid-s37659" xml:space="preserve"> ergo erit perpẽdicularis ſuper lineam z m:</s> <s xml:id="echoid-s37660" xml:space="preserve"> angulus ergo m z g erit re-<lb/>ctus:</s> <s xml:id="echoid-s37661" xml:space="preserve"> erit quoq;</s> <s xml:id="echoid-s37662" xml:space="preserve"> per 47 p 1 quadratum lineæ m g æ quale quadratis duobus linearũ m g & g z:</s> <s xml:id="echoid-s37663" xml:space="preserve"> & ſimi <lb/>liter quadratum lineæ a g eſt æquale quadratis linearum a z & g z:</s> <s xml:id="echoid-s37664" xml:space="preserve"> ſed-quadratum lineæ m g æ qua-<lb/>le eſt quadrato lineæ a g:</s> <s xml:id="echoid-s37665" xml:space="preserve"> quoniá lineæ m g & a g ſunt æ quales:</s> <s xml:id="echoid-s37666" xml:space="preserve"> ablato ergo utrobiq;</s> <s xml:id="echoid-s37667" xml:space="preserve"> quadrato com-<lb/>muni, quod eſt quadratum lineæ g z:</s> <s xml:id="echoid-s37668" xml:space="preserve"> relinquitur quadratum lineæ m z æquale quadrato lineæ a z:</s> <s xml:id="echoid-s37669" xml:space="preserve"> <lb/>eſt igitur linea m z æ qualis lineæ a z:</s> <s xml:id="echoid-s37670" xml:space="preserve"> ergo per 5 p 1 angulus a m z eſt ęqualis angulo z a m:</s> <s xml:id="echoid-s37671" xml:space="preserve"> ſed per 29 <lb/>p 1 angulus l z h extrinſecus æqualis eſt angulo a m zintrinſeco, & angulus a m z eſt æ qualis angu-<lb/>10 l z a per eandem 29 p 1, quia illi anguli ſunt coalterni:</s> <s xml:id="echoid-s37672" xml:space="preserve"> ergo angulus a z l eſt æqualis angulo l z h.</s> <s xml:id="echoid-s37673" xml:space="preserve"> <lb/>Forma ergo puncti h incidens ſpeculo in puncto z reflectitur ad a centrum uiſus à puncto ſpeculi, <lb/>quod eſt z, ut patet per 20th.</s> <s xml:id="echoid-s37674" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s37675" xml:space="preserve"> Siuerò dicatur quòd ab alio puncto quàm à puncto g poteſt <lb/>forma puncti b reflecti ad uiſum a illud aliud punctum aut erit in linea longitudinis, quę eſt g z, aut <lb/>in alia.</s> <s xml:id="echoid-s37676" xml:space="preserve"> Si eſt in linea g z, ducatur à dato puncto lineæ g z, quod ſit d, linea perpendicularis ſuperli-<lb/>neam g z:</s> <s xml:id="echoid-s37677" xml:space="preserve"> quæ a d utramq;</s> <s xml:id="echoid-s37678" xml:space="preserve"> partem producta ſit linea o d f:</s> <s xml:id="echoid-s37679" xml:space="preserve"> & copulentur lineæ a d & b d.</s> <s xml:id="echoid-s37680" xml:space="preserve"> Linea itaq;</s> <s xml:id="echoid-s37681" xml:space="preserve"> <lb/>o d f per 29 th.</s> <s xml:id="echoid-s37682" xml:space="preserve"> 1 huius neceſſariò ſecabit lineam a b:</s> <s xml:id="echoid-s37683" xml:space="preserve"> & erit & quidiſtans lineæ a m per 28 p 1:</s> <s xml:id="echoid-s37684" xml:space="preserve"> & linea <lb/>ducta à puncto b a d illud punctum d, neceſſariò cõcurret cum linea a m per 2th.</s> <s xml:id="echoid-s37685" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s37686" xml:space="preserve"> & erit pun-<lb/>ctus d & punctus m in eadem ſuperficie:</s> <s xml:id="echoid-s37687" xml:space="preserve"> quoniam lineæ d f & a m, cum ſint æ quidiſtantes, ſunt in <lb/>eadem ſuperficie per 1 th.</s> <s xml:id="echoid-s37688" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s37689" xml:space="preserve"> Linea ergo b d aut cad et ſuper punctum m, aut ſuper aliud pun-<lb/>ctum lineæ a m.</s> <s xml:id="echoid-s37690" xml:space="preserve"> Si cadat ſuper punctum m, erit ducere à puncto b ad pũctum m duas rectas lineas, <lb/>ut lineam b g m, & lineam b d m:</s> <s xml:id="echoid-s37691" xml:space="preserve"> quod eſt impoſsibile:</s> <s xml:id="echoid-s37692" xml:space="preserve"> quoniam tunc duæ rectæ lineæ ſuperficiem <lb/>includerent.</s> <s xml:id="echoid-s37693" xml:space="preserve"> Si uerò ad aliud punctum lineæ a m, quàm ad punctũ m, incidat linea b d:</s> <s xml:id="echoid-s37694" xml:space="preserve"> ſit illud pun-<lb/>ctum n:</s> <s xml:id="echoid-s37695" xml:space="preserve"> & ducatur à puncto n linea n z a d punctum z:</s> <s xml:id="echoid-s37696" xml:space="preserve"> & poteſt probari, quòd hæc linea n z cum li-<lb/>nea h z facit lineam rectam, ſicut prius probatum eſt de linea m z.</s> <s xml:id="echoid-s37697" xml:space="preserve"> Quoniam enim puncta n, z, h ſunt <lb/>in duabus planis ſuperficiebus:</s> <s xml:id="echoid-s37698" xml:space="preserve"> ergo ſunt in illarum communi ſectione:</s> <s xml:id="echoid-s37699" xml:space="preserve"> ergo per 3 p 11 erit linea h z <lb/>n linea recta:</s> <s xml:id="echoid-s37700" xml:space="preserve"> & ita à puncto h erit ducere duas lineas rectas per punctum z tranſeuntes, & in diuer-<lb/>ſa puncta line æ a m cadentes:</s> <s xml:id="echoid-s37701" xml:space="preserve"> quod eſt impoſsibile per 1 p 11.</s> <s xml:id="echoid-s37702" xml:space="preserve"> Palàm ergo quòd à nullo puncto lineæ <lb/>g z poteſt forma puncti b reflecti ad uiſum a, niſi à ſolo puncto g.</s> <s xml:id="echoid-s37703" xml:space="preserve"> Si dicatur quòd extra hanc lineam <lb/>ſumpto puncto in ſuperficie ſpeculi ab illo poſsit reſle cti forma puncti b ad a uiſum:</s> <s xml:id="echoid-s37704" xml:space="preserve"> ducatur ſuper <lb/>illud punctum ſpeculi linea longitudinis ſpeculi per 101 th.</s> <s xml:id="echoid-s37705" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s37706" xml:space="preserve"> & à puncto circuli e z i, in quem <lb/>cadit hæc linea, probabitur forma puncti h reflecti ad uiſum a ſecundum prædictam probationem:</s> <s xml:id="echoid-s37707" xml:space="preserve"> <lb/>ſed iam probatum eſt quòd forma puncti h à puncto ſpeculi z reflectitur ad uiſum a:</s> <s xml:id="echoid-s37708" xml:space="preserve"> & ita formæ <lb/>eiuſdem puncti h a d eundem uiſum a à punctis duobus unius circuli fiet reflexio, quod eſt contra <lb/>16 th.</s> <s xml:id="echoid-s37709" xml:space="preserve"> 6 huius, & impoſsibile.</s> <s xml:id="echoid-s37710" xml:space="preserve"> Supereſt ergo, ut à ſolo pũcto ſpeculi propoſiti reflectatur forma pun-<lb/>cti b ad uiſum a.</s> <s xml:id="echoid-s37711" xml:space="preserve"> Palàm enim, quia ſi communis ſectio ſuperficiei reflexionis & ſpeculi columnaris <lb/>fuerit oxygonia ſectio, quia tunc nõ fiet reflexio, niſi ab uno tantùm puncto:</s> <s xml:id="echoid-s37712" xml:space="preserve"> quoniam, ut patet per <lb/>24 huius in omni ſuperficie reflexionis factæ a b his ſpeculis de neceſsitate oportet, ut ſit punctus <lb/>axis, in quem cadit perpen dicularis ducta à puncto reflexionis, quæ orthogonalis eſt ſuper lineam <lb/>longitudinis ſpeculi per punctum illũ tranſeuntem:</s> <s xml:id="echoid-s37713" xml:space="preserve"> ergo & ſuper axem ſpeculi per 28 p 1:</s> <s xml:id="echoid-s37714" xml:space="preserve"> quoniam <lb/>linea longitudinis columnæ & axis ſemper æ quidiſtant per 92 th.</s> <s xml:id="echoid-s37715" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s37716" xml:space="preserve"> Eſt autem illa perpendi-<lb/>cularis communi ſectioni oxygoniæ, à cuius pũcto fiet reflexio, & cuidam circulo æ quidiftanti ba-<lb/>ſibus ſpeculi per 104 th.</s> <s xml:id="echoid-s37717" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s37718" xml:space="preserve"> eſt ergo ſemidiameter illius circuli.</s> <s xml:id="echoid-s37719" xml:space="preserve"> Superficies itaq;</s> <s xml:id="echoid-s37720" xml:space="preserve"> reflexionis, & <lb/>ille circulus ſecant ſe in illa perpendiculari ſemidiametro circuli ſuper peripheriam circuli per 21 <lb/>huius:</s> <s xml:id="echoid-s37721" xml:space="preserve"> & ſupèrficies reflexionis, in qua eſt illa ſectio oxygonia, eſt declinata ſuper ſuperficiem cir-<lb/>culi, & ſuper illam ſemidiametrum, quæ eſt perpendicularis à puncto reflexionis ducta:</s> <s xml:id="echoid-s37722" xml:space="preserve"> ſuper ali-<lb/>quam uerò ſuperficiem declinatam ſuper axem columnę non poteſt intelligi, niſi una tantùm linea <lb/>perpendiculariter cadens ſuper axem per 112 th.</s> <s xml:id="echoid-s37723" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s37724" xml:space="preserve"> Si uerò ab eadem oxygonia ſectione ſieret <lb/>a duobus punctis reflexio:</s> <s xml:id="echoid-s37725" xml:space="preserve"> eſſet neceſſarium, ut in illa ſectio nis ſuperſicie poſſent duci duę perpen <lb/>diculares ſuper axem ſpeculi:</s> <s xml:id="echoid-s37726" xml:space="preserve"> quod eſt impoſsibile:</s> <s xml:id="echoid-s37727" xml:space="preserve"> cum unus uiſus ſemper uideat minus medieta-<lb/>te columnæ.</s> <s xml:id="echoid-s37728" xml:space="preserve"> Et ſimiliter patet per 79th.</s> <s xml:id="echoid-s37729" xml:space="preserve"> 4 huius quòd duo uiſus uidẽ minus medietate columnę, <lb/>quando diameter baſis columnæ maior eſt quàm diſtantia oculorum:</s> <s xml:id="echoid-s37730" xml:space="preserve"> hoc autem planius declara-<lb/>tum eſt in 22 huius.</s> <s xml:id="echoid-s37731" xml:space="preserve"> Patert itaq;</s> <s xml:id="echoid-s37732" xml:space="preserve"> propoſitum.</s> <s xml:id="echoid-s37733" xml:space="preserve"/> </p> <div xml:id="echoid-div1544" type="float" level="0" n="0"> <figure xlink:label="fig-0582-01" xlink:href="fig-0582-01a"> <variables xml:id="echoid-variables667" xml:space="preserve">a l f k b h d z g n q e o t s m i p</variables> </figure> </div> </div> <div xml:id="echoid-div1546" type="section" level="0" n="0"> <head xml:id="echoid-head1176" xml:space="preserve" style="it">29. Oxygonia ſectione exiſtente communi ſuperſiciei reflexionis & ſpeculi columnaris con-<lb/> <pb o="282" file="0584" n="584" rhead="VITELLONIS OPTICAE"/> uexi: dati punctiuiſi ad datum centrum uiſus punctum reflexionis inuenire. Alhazen 48 n 9.</head> <p> <s xml:id="echoid-s37734" xml:space="preserve">Communi ſectione ſuperficiei reflexionis & ſpeculi propoſiti exiſtente linea longitudinis ſpe-<lb/>culi, punctus reflexionis poterit faciliter inueniri, ſicut in ſpeculis planis per 46 th.</s> <s xml:id="echoid-s37735" xml:space="preserve"> 5 huius oſten-<lb/>ſum eſt.</s> <s xml:id="echoid-s37736" xml:space="preserve"> Siuerò illa communis ſectio fuerit circulus:</s> <s xml:id="echoid-s37737" xml:space="preserve"> tunc punctus reflexionis poterit faciliter in-<lb/>ueniri, ſicut in ſpeculis ſphæricis conuexis oſtenſum eſt per 20 uel 22 th.</s> <s xml:id="echoid-s37738" xml:space="preserve"> 6 huius.</s> <s xml:id="echoid-s37739" xml:space="preserve"> Si autem illa com-<lb/>munis ſectio ſit oxygonia ſectio, qualis proponitur:</s> <s xml:id="echoid-s37740" xml:space="preserve"> ſit rei uiſæ datus punctus b, qui reflectatur ab <lb/>aliquo puncto ſectionis oxygoniæ ad a centrum uiſus.</s> <s xml:id="echoid-s37741" xml:space="preserve"> Dico quòd poſsibile eſt inueniri punctum <lb/>reflexionis.</s> <s xml:id="echoid-s37742" xml:space="preserve"> Ducatur enim à puncto a, utin præcedente propoſitione, ſuperficies æquidiftans baſi-<lb/>bus columnæ:</s> <s xml:id="echoid-s37743" xml:space="preserve"> quæ ſecabit columnam ſuper circulum, qui ſit e zi:</s> <s xml:id="echoid-s37744" xml:space="preserve"> & ducatur à puncto b perpendi-<lb/>cularis ſuper hanc ſuperficiem per 11 p 11, qu æ ſit b h, & per 20 uel 22 t 6 huius, ſicut in ſpeculis ſphę-<lb/>ricis conuexis oſtenſum eſt, inueniatur in hac ſuperficie punctus, àquo reflectitur forma puncti h <lb/>ad uiſum a, qui ſit pũctus z:</s> <s xml:id="echoid-s37745" xml:space="preserve"> & à puncto z per 101 th.</s> <s xml:id="echoid-s37746" xml:space="preserve"> 1 huius ducatur linea longitudinis, quæ ſit z g:</s> <s xml:id="echoid-s37747" xml:space="preserve"> & <lb/>ducatur linea h a:</s> <s xml:id="echoid-s37748" xml:space="preserve"> & à pũ cto z ducatur perpendicularis ſuper lineam h a per 12 p 1, quæ ſit z l:</s> <s xml:id="echoid-s37749" xml:space="preserve"> & huic <lb/>ducatur æquidiſtans à puncto a per 31 p 1, qu æ ſit a m:</s> <s xml:id="echoid-s37750" xml:space="preserve"> & linea h z producatur uſque quò concurrat <lb/>cum linea a m:</s> <s xml:id="echoid-s37751" xml:space="preserve"> & ſit cõcurſus in puncto m:</s> <s xml:id="echoid-s37752" xml:space="preserve"> & à puncto m ducatur linea ad punctum b, quæ neceſſa-<lb/>riò ſecabit lineam z g, cum ſit in eadem ſuperficie cum illa:</s> <s xml:id="echoid-s37753" xml:space="preserve"> quoniam cum linea b h ſit æquidiſtans <lb/>lineæ g z per 6 p 11, eò quòd amb æ lineæ b h & g z ſunt perpendiculares ſuper eandem ſuperficiem <lb/>e zi æ quidiſtantem baſibus column æ erit ergo linea h m in ſuperficie illarum per 7 p 11:</s> <s xml:id="echoid-s37754" xml:space="preserve"> & ita linea <lb/>m b erit in eadem ſuperficie:</s> <s xml:id="echoid-s37755" xml:space="preserve"> quæ ſi ſecuerit lineam z g in puncto g:</s> <s xml:id="echoid-s37756" xml:space="preserve"> palàm ex his, quę in præcedẽte <lb/>propoſitione præmiſſa ſunt, quòd punctus g erit punctus reflexionis formæ puncti b ad a uiſum.</s> <s xml:id="echoid-s37757" xml:space="preserve"> <lb/>Hæc omnia pluraq́;</s> <s xml:id="echoid-s37758" xml:space="preserve"> alia patent per ea, quæ dicta ſunt in præcedente demõſtratione.</s> <s xml:id="echoid-s37759" xml:space="preserve"> Et hoc eſt pro-<lb/>poſitum:</s> <s xml:id="echoid-s37760" xml:space="preserve"> quoniam ſecundum hunc modum cuiuslibet dati punctiad datum uiſum punctus refle-<lb/>xionis poterit inueniri.</s> <s xml:id="echoid-s37761" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1547" type="section" level="0" n="0"> <head xml:id="echoid-head1177" xml:space="preserve" style="it">30. Linea rectæ æquidiſt antis axi ſpeculi columnaris conuexi, uiſu non exiſtente in eadem <lb/>ſuperficie, reflexio fit à linea longitudinis ſpeculi ad uiſum. Alhazen 26 n 6.</head> <p> <s xml:id="echoid-s37762" xml:space="preserve">Eſto axis ſpeculi columnaris conuexi linea 3 k:</s> <s xml:id="echoid-s37763" xml:space="preserve"> & ſit linea uiſa axi æquidiſtans, quæ t h:</s> <s xml:id="echoid-s37764" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s37765" xml:space="preserve"> cen-<lb/>trum uiſus e extra ſuperſiciem t h z k.</s> <s xml:id="echoid-s37766" xml:space="preserve"> Dico quòd formà lineæ t h reflectitur ad uiſum e à linea lon-<lb/>gitudinis ſpeculi, quæ eſt communis fectio ſuperficiei t h z k, & ſuperficiel ſpeculi.</s> <s xml:id="echoid-s37767" xml:space="preserve"> Et quia uiſus e <lb/>non eſt in ſuperficie t h z k:</s> <s xml:id="echoid-s37768" xml:space="preserve"> ſit ſuperficies per ipſum uiſum tranſiens, ſecas columnam ſpeculi æqui-<lb/>diſtanter baſibus:</s> <s xml:id="echoid-s37769" xml:space="preserve"> eritq́, hæc ſuperficies ſecans columnam ſecundum circulum per 100 th 1 huius:</s> <s xml:id="echoid-s37770" xml:space="preserve"> <lb/>qui circulus ſit b f.</s> <s xml:id="echoid-s37771" xml:space="preserve"> Palàm ergo cum linea h t ex hypotheſi æquidiſtet axi z k, quòd aliquis eius pun-<lb/>ctus reflectitur ad uiſum e ab aliquo puncto circuli b f:</s> <s xml:id="echoid-s37772" xml:space="preserve"> ſit ergo hoc à pũcto b.</s> <s xml:id="echoid-s37773" xml:space="preserve"> Punctus quoq;</s> <s xml:id="echoid-s37774" xml:space="preserve"> lineæ <lb/>th, qui reflectitur ad uiſum e à puncto ſpeculi b, ſit q:</s> <s xml:id="echoid-s37775" xml:space="preserve"> & ducantur lineæ q b, e b, q e:</s> <s xml:id="echoid-s37776" xml:space="preserve"> & ducatur per <lb/>101th.</s> <s xml:id="echoid-s37777" xml:space="preserve"> 1 huius à puncto b linea longitudinis columnæ quæ ſit a b g:</s> <s xml:id="echoid-s37778" xml:space="preserve"> & ducatur à puncto b perpen-<lb/>dicularis cadens ſuper axem z k in punctum 1:</s> <s xml:id="echoid-s37779" xml:space="preserve"> quæ producta ad lineam q e, ſecabit ipſam per 2 th.</s> <s xml:id="echoid-s37780" xml:space="preserve"> 1 <lb/>huius:</s> <s xml:id="echoid-s37781" xml:space="preserve"> quoniam illæ duæ lineæ æquidiſtant, ut patet ex præmiſsis.</s> <s xml:id="echoid-s37782" xml:space="preserve"> Et quoniam ſuperſicies e q b eſt <lb/>ſuperficies reflexionis:</s> <s xml:id="echoid-s37783" xml:space="preserve"> patet quòd punctum b cum linea e q eſt in eadem ſuperficie.</s> <s xml:id="echoid-s37784" xml:space="preserve"> Secet ergo li-<lb/>nea b l producta ipſam lineam q e in puncto m:</s> <s xml:id="echoid-s37785" xml:space="preserve"> & ſit lineam l:</s> <s xml:id="echoid-s37786" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s37787" xml:space="preserve"> à puncto e linea æquidi-<lb/>ſtans lineæ m l per 31 p 1, quæſit e o:</s> <s xml:id="echoid-s37788" xml:space="preserve"> & produ-<lb/> <anchor type="figure" xlink:label="fig-0584-01a" xlink:href="fig-0584-01"/> catur linea q b ultra punctũ b:</s> <s xml:id="echoid-s37789" xml:space="preserve"> quæ quia corl-<lb/>currit cũ linea m l:</s> <s xml:id="echoid-s37790" xml:space="preserve"> palàm per 2th.</s> <s xml:id="echoid-s37791" xml:space="preserve"> 1 huius quià <lb/>ipſa cõcurret cum eius æquidiſtante, quæ eſt <lb/>lineà e o:</s> <s xml:id="echoid-s37792" xml:space="preserve"> ſit ergo punctus cõcurſus o.</s> <s xml:id="echoid-s37793" xml:space="preserve"> Palàm <lb/>aũt per 20 th.</s> <s xml:id="echoid-s37794" xml:space="preserve"> 5 huius quoniam angulus inci-<lb/>dentiæ, qui eſt q b g, eſt æ qualis angulo refle-<lb/>xionis, qui eſt e b atanguli uerò m b g & m b a <lb/>funt æquales, quia recti:</s> <s xml:id="echoid-s37795" xml:space="preserve"> relin quitur ergo an-<lb/>gulus q b mæ qualis angulo reliquo, qui eſt e <lb/>b m:</s> <s xml:id="echoid-s37796" xml:space="preserve"> ſed per 29 p 21 angulus q b m eſt æ qualis <lb/>angulo b o e:</s> <s xml:id="echoid-s37797" xml:space="preserve"> quoniam extrinſecus intrinſe-<lb/>co eſt æ qualis:</s> <s xml:id="echoid-s37798" xml:space="preserve"> ſed & angulus m b e æqualis <lb/>eſt angulo b e o:</s> <s xml:id="echoid-s37799" xml:space="preserve"> quia coalternus eſt:</s> <s xml:id="echoid-s37800" xml:space="preserve"> ergo an-<lb/>gulus b o e æ qualis angulo b e o:</s> <s xml:id="echoid-s37801" xml:space="preserve"> ergo per 6 p <lb/>1 in trigono b e o latus b e eſt æ quale lateri b o.</s> <s xml:id="echoid-s37802" xml:space="preserve"> Sumatur autẽ & alius pũctus in linea th, qui ſit pun-<lb/>ctus t:</s> <s xml:id="echoid-s37803" xml:space="preserve"> & ducatur linea t o.</s> <s xml:id="echoid-s37804" xml:space="preserve"> Quia ergo linea th æquidiſtat lineæ longitudinis ſpeculi, quæ eſt a g per <lb/>30 p 1:</s> <s xml:id="echoid-s37805" xml:space="preserve"> ideòd quòd utraq;</s> <s xml:id="echoid-s37806" xml:space="preserve"> illarum eſt æquidiſtans axi z k:</s> <s xml:id="echoid-s37807" xml:space="preserve"> palàm ergo per 1 th.</s> <s xml:id="echoid-s37808" xml:space="preserve"> 1 huius quòd lineæ th & <lb/>a g ſunt in eadem ſuperficie, cum etiam linea t h & z k axis ſint in eadem ſuperficie.</s> <s xml:id="echoid-s37809" xml:space="preserve"> Ergo per 7 p 11 <lb/>linea q b o ſecans illas lineas æ quidiſtantes, quę ſunt t h & a g, eſt cum illis in eadem ſuperficie:</s> <s xml:id="echoid-s37810" xml:space="preserve"> & <lb/>fimiliter linea t o eſt in eadem ſuperficie cum illis per 1 p 11:</s> <s xml:id="echoid-s37811" xml:space="preserve"> ſunt enim puncta t & o in dicta ſuper-<lb/>ficie:</s> <s xml:id="echoid-s37812" xml:space="preserve"> ſecabit ergo linea t o lineam a g:</s> <s xml:id="echoid-s37813" xml:space="preserve"> ſit punctus ſectionis g:</s> <s xml:id="echoid-s37814" xml:space="preserve"> & ducantur lineæ e g & e t.</s> <s xml:id="echoid-s37815" xml:space="preserve"> Quia itaq:</s> <s xml:id="echoid-s37816" xml:space="preserve"> <lb/>a g, quę eſt linea longitudinis ſpeculis, eſt perpendicularis ſuper ſuperficiem circuli b f per 8 p 11:</s> <s xml:id="echoid-s37817" xml:space="preserve"> <lb/>ideo quòd axis z k, cui æquidiſtat linea a g, perpendicularis eſt ſuper eandem circuli ſuper-<lb/>ficiem per 23 th.</s> <s xml:id="echoid-s37818" xml:space="preserve"> 1 huius, cum ipſa ſit perpendicularis ſuper baſim columnæ per 92 th:</s> <s xml:id="echoid-s37819" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s37820" xml:space="preserve"> <lb/> <pb o="283" file="0585" n="585" rhead="LIBER SEPTIMVS."/> Superficies añt circuli b f eſt pars ſuքficiei e o b f:</s> <s xml:id="echoid-s37821" xml:space="preserve"> hæc enim ſuperficies ſecat columnã æ quidiſtáter <lb/>baſi, ut patet ex præmiſsis:</s> <s xml:id="echoid-s37822" xml:space="preserve"> ergo per definitionẽ lineæ ſuper ſuքficiẽ erectę angulus g b o eſt rectus, <lb/>& angulus g b e rectus:</s> <s xml:id="echoid-s37823" xml:space="preserve"> ergo per 47 p 1 quadratũ lineæ g o ualet ambo quadrata linearũ g b & b o:</s> <s xml:id="echoid-s37824" xml:space="preserve"> & <lb/>quadratũ lineę g e ualet ambo quadrata linearũ g b & b e.</s> <s xml:id="echoid-s37825" xml:space="preserve"> Et quoniá oſten ſum eſt quòd lineæ b e & <lb/>b o ſunt æquales, erunt etiá ipſaruin quadrata æqualia, & quadratũ b g utriq;</s> <s xml:id="echoid-s37826" xml:space="preserve"> eſt commune:</s> <s xml:id="echoid-s37827" xml:space="preserve"> erit er-<lb/>go quadra ũ lineæ g e æquale quadrato lineę g o:</s> <s xml:id="echoid-s37828" xml:space="preserve"> erit igitur per 6 p 1 in trigono e g o linea g e æ qua <lb/>lis lineæ g o:</s> <s xml:id="echoid-s37829" xml:space="preserve"> ergo per 5 p 1 erit angulus g e o æ qualis angulo g o e.</s> <s xml:id="echoid-s37830" xml:space="preserve"> A puncto itaq:</s> <s xml:id="echoid-s37831" xml:space="preserve"> g ducatur perpen <lb/>dicularis ſuper axem ſpeculi, qui eſt z k, per 12 p 1, quæ ſit linea g z:</s> <s xml:id="echoid-s37832" xml:space="preserve"> & h æc produæa ultra punctum g <lb/>ad lineam t e, ſit z g n:</s> <s xml:id="echoid-s37833" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s37834" xml:space="preserve"> linea z n æ quidiſtans lineę l m per 28 p 1:</s> <s xml:id="echoid-s37835" xml:space="preserve"> quoniam lineæ n z & m lambę <lb/>ſunt perpendiculares ſuper axem z k:</s> <s xml:id="echoid-s37836" xml:space="preserve"> ſed & linea e o æ quidiſtat lineæ m l, ut patet ex præmiſis.</s> <s xml:id="echoid-s37837" xml:space="preserve"> Li-<lb/>nea ergo z n æ quidiſtat lineæ e o per 30 p 1.</s> <s xml:id="echoid-s37838" xml:space="preserve"> Erit ergo per 29 p 1 angulus t g n exiſtens extrinſecus, æ-<lb/>qualis angulo g o e intrinſeco:</s> <s xml:id="echoid-s37839" xml:space="preserve"> & angulus n g e æ qualis.</s> <s xml:id="echoid-s37840" xml:space="preserve"> angulo g e o, quia ſunt coalterni:</s> <s xml:id="echoid-s37841" xml:space="preserve"> ſed angu-<lb/>lus g e o oſtenſus eſt eſſe æ qualis angulo g o e:</s> <s xml:id="echoid-s37842" xml:space="preserve"> ergo angulus t g n eſt æqualis angulo n g e.</s> <s xml:id="echoid-s37843" xml:space="preserve"> Cum er-<lb/>go linea t g o & linea n g z ſint in in eadem ſuperficie, in qua eſt punctus g:</s> <s xml:id="echoid-s37844" xml:space="preserve"> puncta ergo o, g, t erunt <lb/>in eadem ſuperficie:</s> <s xml:id="echoid-s37845" xml:space="preserve"> ergo in eadem ſuperficie ſunt lineæ e g, o g, t g per 1 p 11.</s> <s xml:id="echoid-s37846" xml:space="preserve"> Forma ergo puncti t re <lb/>flectitur ad uiſum e à puncto ſpeculi g, ut patet per 20 th.</s> <s xml:id="echoid-s37847" xml:space="preserve"> 5 huius, propter æ qualitatem angulorum <lb/>r g n & n g e.</s> <s xml:id="echoid-s37848" xml:space="preserve"> Sumpto autem in linea t h puncto h eiuſdem diſtantiæ à puncto q, & à céntro uiſus e, <lb/>cuius eſt punctus t:</s> <s xml:id="echoid-s37849" xml:space="preserve"> & ducta linea h o, tranſibit hæc per lineam longitudinis ſpeculi, quæ eſt a g:</s> <s xml:id="echoid-s37850" xml:space="preserve"> ſit <lb/>punctum tranſitus a:</s> <s xml:id="echoid-s37851" xml:space="preserve"> & ducta à puncto a linea perpendiculari ſuper axem z k, quæ ſit a d, & quæ <lb/>producta ad lineam h e, ſit d r, & ducta linea e a, patebit, ſicut prius, quia duo anguli a b e & a b o <lb/>ſuntrecti, & latera a e & a o ſunt æ qualia:</s> <s xml:id="echoid-s37852" xml:space="preserve"> ſiuntq́, ut prius, duo anguli h a r & e a r æ quales Forma <lb/>ergo punctih, utſuprà patuit, reflectitur ad uiſum e à puncto ſpeculi a.</s> <s xml:id="echoid-s37853" xml:space="preserve"> Similiter quoque ſumpto <lb/>quocũq;</s> <s xml:id="echoid-s37854" xml:space="preserve"> uncto lineæ t h, erit probare, quòd ille punctus reflectitur ad e ab aliquo puncto longitu-<lb/>dinis ſpeculi, quę eſt a g.</s> <s xml:id="echoid-s37855" xml:space="preserve"> T ota ergo linea t h reflectitur ab una linea longitudinis ſpeculi, quę eſt a g, <lb/>ad uiſum e:</s> <s xml:id="echoid-s37856" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s37857" xml:space="preserve"> Eſt tamen notandum, quòd in hac diſpoſitione figuræ punctum <lb/>q lineæ th eſt medius punctus illius lineæ & eſt in eadem ſuperficie cum centro uiſus e:</s> <s xml:id="echoid-s37858" xml:space="preserve"> propter <lb/>quod puncta t & h æqualiter diſtant à uiſu:</s> <s xml:id="echoid-s37859" xml:space="preserve"> & ſimiliter puncta reflexionum, quæ ſunt g & a:</s> <s xml:id="echoid-s37860" xml:space="preserve"> pro-<lb/>pter quod pater, quod lineæ g b & b a ſunt æquales:</s> <s xml:id="echoid-s37861" xml:space="preserve"> & tota diſpoſitio figuræ fit ſecundum illa.</s> <s xml:id="echoid-s37862" xml:space="preserve"> <lb/>Quòd ſi uiſus ſit inferior tota linea th:</s> <s xml:id="echoid-s37863" xml:space="preserve"> notandum quòd fit reflexio à linea a g, prout ſecat plurimas <lb/>oxygonias ſectiones, ut patet per 23 huius:</s> <s xml:id="echoid-s37864" xml:space="preserve"> aliàs uerò quandoq;</s> <s xml:id="echoid-s37865" xml:space="preserve"> ab aliquo puncto circuli neceſſe <lb/>eſt fieri reflexionem.</s> <s xml:id="echoid-s37866" xml:space="preserve"/> </p> <div xml:id="echoid-div1547" type="float" level="0" n="0"> <figure xlink:label="fig-0584-01" xlink:href="fig-0584-01a"> <variables xml:id="echoid-variables668" xml:space="preserve">t n q g z m b l f h r a d e k o</variables> </figure> </div> </div> <div xml:id="echoid-div1549" type="section" level="0" n="0"> <head xml:id="echoid-head1178" xml:space="preserve" style="it">31. Linea longitudinis exiſtente communi ſectione ſuperficiei reflexionis & ſpeculi pyr amida <lb/>lis conuexi: à quolibet puncto ſuperficiei ſpeculi apparent is uiſui poteſt fieri reflexio ad uiſum. <lb/>Alhazen 40 n 4.</head> <p> <s xml:id="echoid-s37867" xml:space="preserve">Eſto ſpeculum pyramidale conuexũ b x p:</s> <s xml:id="echoid-s37868" xml:space="preserve"> cuius uertex ſit b:</s> <s xml:id="echoid-s37869" xml:space="preserve"> & diameter baſis x p:</s> <s xml:id="echoid-s37870" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s37871" xml:space="preserve"> centrũ <lb/>baſis q:</s> <s xml:id="echoid-s37872" xml:space="preserve"> erit ergo linea b q axis ipſius ſpeculi.</s> <s xml:id="echoid-s37873" xml:space="preserve"> Sit quoq;</s> <s xml:id="echoid-s37874" xml:space="preserve"> quicunq;</s> <s xml:id="echoid-s37875" xml:space="preserve"> datus punctus in ipſius ſuperficie <lb/>apparente, punctus g:</s> <s xml:id="echoid-s37876" xml:space="preserve"> & ſit centrũ uiſus a:</s> <s xml:id="echoid-s37877" xml:space="preserve"> & punctus rei uiſæ ſit n.</s> <s xml:id="echoid-s37878" xml:space="preserve"> Dico quòd forma puncti n refle-<lb/>cti poteſt à puncto g ad uiſum a, ſi ſuerit in ſitu cõuenienti reflexioni, Circunducatur enim per 102 <lb/>th.</s> <s xml:id="echoid-s37879" xml:space="preserve"> 1 huius à puncto g circulus pyramidi ſpeculi æ quidiſtans baſi x p:</s> <s xml:id="echoid-s37880" xml:space="preserve"> cuius centrũ ſit d, & cuius dia-<lb/>meter ſit g c:</s> <s xml:id="echoid-s37881" xml:space="preserve"> ſemidiameter g d, quę neceſſariò erit perpendicularis ſuper axẽ b q per 29 p 1:</s> <s xml:id="echoid-s37882" xml:space="preserve"> eò quòd <lb/>x q ſemidiameter baſis ſpeculi eſt perpendicularis ſuper eundem axem b q, ſicut & alia ſemidiame-<lb/>ter baſis in eadẽ ſuperficie exiſtés cũ diametro g c æ quidiſtat illi:</s> <s xml:id="echoid-s37883" xml:space="preserve"> eſt <lb/> <anchor type="figure" xlink:label="fig-0585-01a" xlink:href="fig-0585-01"/> enim axis b q perpendicularis ſuper ſuperficies amborum circulorũ <lb/>x p & g e per 23 th.</s> <s xml:id="echoid-s37884" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s37885" xml:space="preserve"> & producatur linea g b à dato puncto g ad <lb/>uerticé pyramidis b.</s> <s xml:id="echoid-s37886" xml:space="preserve"> Palàm ergo per 32 p 1 quoniã angulus g b d eſt <lb/>a cutus:</s> <s xml:id="echoid-s37887" xml:space="preserve"> cũ angulus b d g ſit rectus.</s> <s xml:id="echoid-s37888" xml:space="preserve"> In ſuperficie quoq;</s> <s xml:id="echoid-s37889" xml:space="preserve"> trigoni g b d <lb/>ſit linea reflexionis, quæ eſt a g per 7 huius, & ex hypotheſi erunt li-<lb/>neæ reflexionis a g & longitudinis b g & axis b d q in eadem ſuperfi <lb/>cie.</s> <s xml:id="echoid-s37890" xml:space="preserve"> Et quoniam angulus b g d eſt acutus, fiat per 23 p 1 angulus b g r <lb/>rectus, producta linea g rad axem:</s> <s xml:id="echoid-s37891" xml:space="preserve"> eritq́ r g linea perpendicularis ſu <lb/>per lineam longitudinis, quæ eſt b x:</s> <s xml:id="echoid-s37892" xml:space="preserve"> eritq́ g r linea in eadem ſuperfi <lb/>cie cum alijs lateribus trigoni b g r per 2 p 11.</s> <s xml:id="echoid-s37893" xml:space="preserve"> A puncto quoq;</s> <s xml:id="echoid-s37894" xml:space="preserve"> g duca <lb/>catur linea contingens circulũ per 17 p 3, quæ ſit linea l g s:</s> <s xml:id="echoid-s37895" xml:space="preserve"> eritq́ per <lb/>18 p 3 linea l g s perpendicularis ſuper diametrũ g c:</s> <s xml:id="echoid-s37896" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s37897" xml:space="preserve"> alia <lb/>diameter circuli g c perpen dicularis ſuper diametrũ g c:</s> <s xml:id="echoid-s37898" xml:space="preserve"> quæ extra-<lb/>hatur à pũ cto d per 11 p 1:</s> <s xml:id="echoid-s37899" xml:space="preserve"> & ſit f k:</s> <s xml:id="echoid-s37900" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s37901" xml:space="preserve">, ſicut prius, diameter f k per-<lb/>pendicularis ſuper axem b q:</s> <s xml:id="echoid-s37902" xml:space="preserve"> erit ergo per 4 p 11 diameter f k perpen <lb/>dicularis ſuper ſuperficiem, in qua ſunt lineæ g c & b q:</s> <s xml:id="echoid-s37903" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s37904" xml:space="preserve"> diame <lb/>ter f k æquidiſtans lineę contingenti circulum, quę eſt l g s, per 18 p 3 <lb/>& per 28 p 1:</s> <s xml:id="echoid-s37905" xml:space="preserve"> ergo per 8 p 11 linea contingens circulũ g c, quæ eſt s g l, <lb/>perpédicularis eſt ſuք ſuperficiẽ, in qua ſunt diameter g c & axis b q:</s> <s xml:id="echoid-s37906" xml:space="preserve"> <lb/>ergo ք definition ẽ lineę erectę angulus l g r eſt rectus.</s> <s xml:id="echoid-s37907" xml:space="preserve"> Si ergo imaginemur ſuperficiẽ contingentẽ <lb/>pyramidẽ, in qua ſit linea l g s cõtingens circulũ b c:</s> <s xml:id="echoid-s37908" xml:space="preserve"> palã quoniã linea r g erecta eſt ſuper illã ſuper-<lb/> <pb o="284" file="0586" n="586" rhead="VITELLONIS OPTICAE"/> ficiem Si ergo linea reflexionis, quæ eſt a g, tranſiens pyramidem, fiat una linea cũ linea g r, erit ipfa <lb/>orthogonalis ſuper ſuperficiem contingentẽ ſpeculũ in puncto g:</s> <s xml:id="echoid-s37909" xml:space="preserve"> fiet ergo per 21 th.</s> <s xml:id="echoid-s37910" xml:space="preserve"> 5 huius formæ <lb/>fecundũ illam lineã ſuperficiei ſpeculi incidentis reflexio per eandẽ.</s> <s xml:id="echoid-s37911" xml:space="preserve"> Et ſi punctus n ſit in illa linea, <lb/>poterit forma eius reflecti ad uiſum a à puncto ſpeculi g per lineã a g.</s> <s xml:id="echoid-s37912" xml:space="preserve"> Siuerò linea a g non fiat una <lb/>linea cum linea g r:</s> <s xml:id="echoid-s37913" xml:space="preserve"> palàm per cõuerſam 14 p 1 quod angulus a g l eſt minorrecto uel maior:</s> <s xml:id="echoid-s37914" xml:space="preserve"> quoniã <lb/>ſi erit rectus, tunc lineę a g & g r ambæ coniunctæ ſunt linea una per eandẽ 14 p 1.</s> <s xml:id="echoid-s37915" xml:space="preserve"> Sit ergo angulus <lb/>a g l acutus:</s> <s xml:id="echoid-s37916" xml:space="preserve"> & productatur linea r g in continuũ & directũ uſq;</s> <s xml:id="echoid-s37917" xml:space="preserve"> ad punctũ u:</s> <s xml:id="echoid-s37918" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s37919" xml:space="preserve"> linea u g perpen-<lb/>dicularis ſuper ſuperficiem contingenté ſpeculum in puncto g:</s> <s xml:id="echoid-s37920" xml:space="preserve"> & erit angulus u g l rectus per 15 p 1:</s> <s xml:id="echoid-s37921" xml:space="preserve"> <lb/>erit ergo angulus u g a acutus.</s> <s xml:id="echoid-s37922" xml:space="preserve"> Ducatur ergo in eadẽ ſuperficie linea g h æ q u alẽ continẽs angulum <lb/>cum lineá ù g angulo u g a per 23 p 1.</s> <s xml:id="echoid-s37923" xml:space="preserve"> Si ergo punctus rei uiſę, qui poſitus eſt eſſe n, fuerit in linea h g:</s> <s xml:id="echoid-s37924" xml:space="preserve"> <lb/>palàm per 20 th.</s> <s xml:id="echoid-s37925" xml:space="preserve"> 5 huius quoniã poſsibile eſt à puncto g fieri reflexionẽ ad uiſum a:</s> <s xml:id="echoid-s37926" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s37927" xml:space="preserve"> linea inci-<lb/>dentiæ, quę eſt n g cũ linea reflexiõis, quę eſt g a, in eadẽ ſuperficie orthogonali ſuper ſuperficiẽ, cõ-<lb/>tingentẽ pyramidem in puncto reflexionis, quod eſt g:</s> <s xml:id="echoid-s37928" xml:space="preserve"> reflecteturq́;</s> <s xml:id="echoid-s37929" xml:space="preserve"> forma puncti rei uiſæ ſecundũ <lb/>punctum n a d uiſum, qui eſt in puncto a, à puncto ſpeculi, quod eſt g.</s> <s xml:id="echoid-s37930" xml:space="preserve"> Et eodem modo de quolibet <lb/>alio dato puncto ſuperficiei ſpeculi demonſtrandum.</s> <s xml:id="echoid-s37931" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s37932" xml:space="preserve"/> </p> <div xml:id="echoid-div1549" type="float" level="0" n="0"> <figure xlink:label="fig-0585-01" xlink:href="fig-0585-01a"> <variables xml:id="echoid-variables669" xml:space="preserve">b l a u f d c h n g r k s x q p</variables> </figure> </div> </div> <div xml:id="echoid-div1551" type="section" level="0" n="0"> <head xml:id="echoid-head1179" xml:space="preserve" style="it">32. Dato puncto ſpeculi pyramidalis couexi, à quo fiat reflexio dati puncti rei uiſæ ad datum <lb/>centrum uiſus à puncto oxygoniæ ſectionis, uel à linea longitudinis ſpeculi: poßibile eſt loca inue <lb/>niri, in quibus centro uiſus & puncto reiuiſæ collocatis, ſiat reflexio ad uiſum ab eodẽ dato pun <lb/>cto ſpeculi, prout eſt punctus circuli æquidiſtantis baſi. Alhazen 52 n 5.</head> <p> <s xml:id="echoid-s37933" xml:space="preserve">Sit a centrum uiſus:</s> <s xml:id="echoid-s37934" xml:space="preserve"> b punctus rei uiſæ:</s> <s xml:id="echoid-s37935" xml:space="preserve"> & ſit g punctus reflexionis ſuperficiei ſpeculi pyramida-<lb/>lis conuexi, cuius uertex ſit e.</s> <s xml:id="echoid-s37936" xml:space="preserve"> Dico quòd poſsibile eſt inueniri id, quod proponitur.</s> <s xml:id="echoid-s37937" xml:space="preserve"> Ducatur enim, <lb/>prout docuimus in 28 huius, ſuper punctpum g ſuperficies æ quidiſtans baſi, ſecans pyramidem ſu-<lb/>per circulum baſi æ quidiſtantem per 100 th.</s> <s xml:id="echoid-s37938" xml:space="preserve"> 1 huius, quæ ſit p g:</s> <s xml:id="echoid-s37939" xml:space="preserve"> cuius centrum ſit 1:</s> <s xml:id="echoid-s37940" xml:space="preserve"> & ducantur li-<lb/>neę a g, b g, a b:</s> <s xml:id="echoid-s37941" xml:space="preserve"> & à puncto g ducatur ad centrum circuli linea g t:</s> <s xml:id="echoid-s37942" xml:space="preserve"> & à uertice pyramidis, qui eſt pun <lb/>ctus e, ducatur axis e t.</s> <s xml:id="echoid-s37943" xml:space="preserve"> Et quoniam ſuperficies reflexionis ſemper eſt erecta ſuper ſupe.</s> <s xml:id="echoid-s37944" xml:space="preserve"> ficiem ſper-<lb/>culum in puncto reflexionis cõtingentem, ut patet per 8 huius, uel per 25 th.</s> <s xml:id="echoid-s37945" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s37946" xml:space="preserve"> ducatur in ſu-<lb/>perficie reflexionis linea perpendicularis ſuper ſuperficiem, contingentem ſpeculum in puncto re <lb/>flexionis, quod eſt g, quæ ſit h g:</s> <s xml:id="echoid-s37947" xml:space="preserve"> & palàm per 26 th.</s> <s xml:id="echoid-s37948" xml:space="preserve"> 5 huius quoniam hæc diuidit an gulum a g b per <lb/>æqualia:</s> <s xml:id="echoid-s37949" xml:space="preserve"> ipſa ergo producta ſecabit lineam a b per 29 th.</s> <s xml:id="echoid-s37950" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s37951" xml:space="preserve"> ſit ergo, ut ſecet eam in puncto z.</s> <s xml:id="echoid-s37952" xml:space="preserve"> <lb/>Ducatur quoq;</s> <s xml:id="echoid-s37953" xml:space="preserve"> à puncto e uertice pyramidis linea longi <lb/> <anchor type="figure" xlink:label="fig-0586-01a" xlink:href="fig-0586-01"/> tudinis ſpeculi, quæ ſit e g:</s> <s xml:id="echoid-s37954" xml:space="preserve"> & huic lineę e g ducatur æqui <lb/>diſtans à puncto á centro uiſus, quę neceſſariò ſecabit ſu <lb/>perficiem circuli p g:</s> <s xml:id="echoid-s37955" xml:space="preserve"> ſecet ergo ipſam in puncto n:</s> <s xml:id="echoid-s37956" xml:space="preserve"> & ſit a <lb/>n.</s> <s xml:id="echoid-s37957" xml:space="preserve"> Et ſimiliter à puncto b ducàtur linea æ quidiſtans ei-<lb/>dem lineæ e g, quæ ſit b m, ſecans ſuperficiem circuli g p <lb/>in puncto m.</s> <s xml:id="echoid-s37958" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s37959" xml:space="preserve"> ambæ lineæ a n & b m æ quidi-<lb/>ſtant eidem lineæ longitudinis ſpeculi, quæ eſt e g:</s> <s xml:id="echoid-s37960" xml:space="preserve"> patet <lb/>per 30 p 1 quia ipſæ adinuicẽ æ quidiſtãt, ſcilicet lineæ a n <lb/>& b m.</s> <s xml:id="echoid-s37961" xml:space="preserve"> A puncto ergo n ducatur per 31 p 1 linea æ quidi-<lb/>ſtans ſemidiametro circuli, quæ eſt g t, ſitq́l illa æ quidi-<lb/>ftans linea n f:</s> <s xml:id="echoid-s37962" xml:space="preserve"> & ducantur lineę n g, m g, n m.</s> <s xml:id="echoid-s37963" xml:space="preserve"> palàm itaq;</s> <s xml:id="echoid-s37964" xml:space="preserve"> <lb/>per 29 th.</s> <s xml:id="echoid-s37965" xml:space="preserve"> 1 huius quia linea t g producta ſecabit lineam <lb/>n m:</s> <s xml:id="echoid-s37966" xml:space="preserve"> ideo, quia ſecat angulum m g n:</s> <s xml:id="echoid-s37967" xml:space="preserve"> eſt enim trãſuerſim <lb/>ducta in eadem ſuperficie:</s> <s xml:id="echoid-s37968" xml:space="preserve"> & etiam lineæ n f & g t ſunt æ-<lb/>quidiſtantes:</s> <s xml:id="echoid-s37969" xml:space="preserve"> ſed linea n m ſecat lineam n f:</s> <s xml:id="echoid-s37970" xml:space="preserve"> ergo & ipſa <lb/>ſecabit per 2 th.</s> <s xml:id="echoid-s37971" xml:space="preserve"> 1 huius lineam g t:</s> <s xml:id="echoid-s37972" xml:space="preserve"> ſecet ergo in puncto q.</s> <s xml:id="echoid-s37973" xml:space="preserve"> <lb/>Palàm etiam per 2 th.</s> <s xml:id="echoid-s37974" xml:space="preserve"> 1 huius quòd linea m g producta <lb/>ſecabit lineam n f, cum ſecet lineam g t æ quidiſtantem <lb/>ipſi n f:</s> <s xml:id="echoid-s37975" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s37976" xml:space="preserve"> punctus ſectionis f:</s> <s xml:id="echoid-s37977" xml:space="preserve"> à puncto a ducatur li-<lb/>nea æ quidiſtans lineę perpendiculari ſuper ſuperficiem, <lb/>contingentem ſpeculum in puncto g, quæ eſt linea h z:</s> <s xml:id="echoid-s37978" xml:space="preserve"> & <lb/>ſit illa æ quidiſtans linea al.</s> <s xml:id="echoid-s37979" xml:space="preserve"> Palàm ergo per 2 th.</s> <s xml:id="echoid-s37980" xml:space="preserve"> 1 huius <lb/>quòd linea b g concurret cum linea a l:</s> <s xml:id="echoid-s37981" xml:space="preserve"> quia ſecat eius æ-<lb/>quidiſtantem lineam h z:</s> <s xml:id="echoid-s37982" xml:space="preserve"> ſit ergo punctus concurſus l.</s> <s xml:id="echoid-s37983" xml:space="preserve"> <lb/>Ducatur quoq;</s> <s xml:id="echoid-s37984" xml:space="preserve"> linea, quæ eſt ſectio communis ſuperfi-<lb/>ciei contingenti ſpeculum in puncto g, & ſuperficiei cir-<lb/>culi p g, quæ ſit linea g o.</s> <s xml:id="echoid-s37985" xml:space="preserve"> Palàm quòd linea g o erit orthogonalis ſuper ſemidiametrum circuli, quæ <lb/>eſt g t per 18 p 3:</s> <s xml:id="echoid-s37986" xml:space="preserve"> ideo quia linea g o eſt contingens circulum p g:</s> <s xml:id="echoid-s37987" xml:space="preserve"> quoniam ipſa ducta eſt in ſuperfi-<lb/>cie plana contingente ſpeculum in puncto g.</s> <s xml:id="echoid-s37988" xml:space="preserve"> Et quoniam lineæ n f & g t æquidiſtant:</s> <s xml:id="echoid-s37989" xml:space="preserve"> erit per <lb/>29 p 1 linea g o orthogonalis ſuper lineam n f æquidiftantem lineæ g t.</s> <s xml:id="echoid-s37990" xml:space="preserve"> Sumatur etiam linea, <lb/>quæ eſt communis ſectio ſuperficiei reflexionis & ſuperficiei contingentis ſpeculum in pun-<lb/>cto g, quæ ſit linea g d:</s> <s xml:id="echoid-s37991" xml:space="preserve"> quæ quidem, cumfecet lineam g h in puncto g, palàm per 2 th.</s> <s xml:id="echoid-s37992" xml:space="preserve"> 1 huius quia <lb/> <pb o="285" file="0587" n="587" rhead="LIBER SEPTIMVS."/> ipſa ſecabit lineam a l æ quidiftantem lineæ g h:</s> <s xml:id="echoid-s37993" xml:space="preserve"> ſit ergo punctus ſectionis d:</s> <s xml:id="echoid-s37994" xml:space="preserve"> & erit linea g d perpen <lb/>dicularis ſuper lineam a l per 29 p 1:</s> <s xml:id="echoid-s37995" xml:space="preserve"> eſt enim linea g d perpendicularis ſuper lineam g h.</s> <s xml:id="echoid-s37996" xml:space="preserve"> Quia cũ li-<lb/>nea h g ſit perpendicularis ſuper ſuperficiem cõtingentẽ circulum in puncto g:</s> <s xml:id="echoid-s37997" xml:space="preserve"> erit neceſſariò per-<lb/>peridicularis ſuper lineã g d productã ab eodẽ puncto in illa ſuperficie per definitionẽ lineæ ſuper <lb/>ſuperficiem erectæ.</s> <s xml:id="echoid-s37998" xml:space="preserve"> Palàm aũt ex prædictis, quoniá linea n f eſt;</s> <s xml:id="echoid-s37999" xml:space="preserve"> æ quidiftans ſemidiametro circuli p <lb/>g, quę eſt g t:</s> <s xml:id="echoid-s38000" xml:space="preserve"> ſimiliter quoq;</s> <s xml:id="echoid-s38001" xml:space="preserve"> linea a l eſt ęquidiſtans lineę g h:</s> <s xml:id="echoid-s38002" xml:space="preserve"> igitur per 15 p 11 ſuperficies, in qua ſunt <lb/>lineę n & al(quę ꝓductę ultra pũcta l & f, neceſſariòcócurret ք 14 th.</s> <s xml:id="echoid-s38003" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38004" xml:space="preserve"> quonã anguli f n a & <lb/>l a f, <gap/>tet, ſunt minores duobus rectis) eſt æquidiſtás ſuperficiei g t h:</s> <s xml:id="echoid-s38005" xml:space="preserve"> ſed & linea e g æ quidiſtat <lb/>lineę b in, ut patet ex præmiſsis:</s> <s xml:id="echoid-s38006" xml:space="preserve"> ergo per 1 th.</s> <s xml:id="echoid-s38007" xml:space="preserve"> 1 huius ipſæ ſunt in eadẽ ſuperficie ſecante prædictas <lb/>duas ſuperficies æ quidiſtantes:</s> <s xml:id="echoid-s38008" xml:space="preserve"> unã ipſarum ſuper lineam e g:</s> <s xml:id="echoid-s38009" xml:space="preserve"> aliã uerò ſuper lineã fl.</s> <s xml:id="echoid-s38010" xml:space="preserve"> Ergo per 16 p <lb/>11 cõmunes ipſarum ſectiones erunt æ quidiſtantes:</s> <s xml:id="echoid-s38011" xml:space="preserve"> erit ergo linea f l æ quidiſtãs lineę e g:</s> <s xml:id="echoid-s38012" xml:space="preserve"> ſed linea <lb/>a n eſt æ quidiſtãs lineæ e g, ut patet ex præmiſsis;</s> <s xml:id="echoid-s38013" xml:space="preserve"> ergo per 30 p 1 erit linea f l æ quidiſtans lineæ a n.</s> <s xml:id="echoid-s38014" xml:space="preserve"> <lb/>Verũ ſuperficies contingens ſpeculũ in puncto g ſecat eaſdẽ ſuperficies æ quidiſtantes, quę ſunt g t <lb/>h & n f a l:</s> <s xml:id="echoid-s38015" xml:space="preserve"> uná earum ſuper lineã e g, ſecundum quá ipſa eſt ſpeculũ contingens:</s> <s xml:id="echoid-s38016" xml:space="preserve"> & aliam ipſarũ ſuք <lb/>lineam o d:</s> <s xml:id="echoid-s38017" xml:space="preserve"> ergo per 16 p 11 linea o d æ quidiſtat lineæ e g:</s> <s xml:id="echoid-s38018" xml:space="preserve"> igitur per 30 p 1 erit linea o d æ quidiſtãs li-<lb/>neis a n, & l f æ quidiſtantibus lineæ e g.</s> <s xml:id="echoid-s38019" xml:space="preserve"> Et quia lineę n f & al, inter quas ducuntur lineæ n a, o d, f l, <lb/>ſunt in eadẽ ſuperficie per 2 p 11:</s> <s xml:id="echoid-s38020" xml:space="preserve"> patet quòd lineę a n, o d, f l ſunt in eadẽ ſuperficie, Ducatur itaq;</s> <s xml:id="echoid-s38021" xml:space="preserve"> à <lb/>puncto f linea æquidiſtans lineę l a per 31 p 1 ſecans lineã o d in puncto k, & lineam a n in puncto i:</s> <s xml:id="echoid-s38022" xml:space="preserve"> <lb/>eritq́;</s> <s xml:id="echoid-s38023" xml:space="preserve"> linea f i æ qualis lineę l a per 34 p 1:</s> <s xml:id="echoid-s38024" xml:space="preserve"> & ſimiliter erit linea f k æ qualis lineæ l d, & k i æ qualis ipſi <lb/>d a.</s> <s xml:id="echoid-s38025" xml:space="preserve"> Eſtaũt per 2 p 6 proportio i k ad k f, ſicut n o ad o f:</s> <s xml:id="echoid-s38026" xml:space="preserve"> ergo per 7 p 5 erit proportio lineæ a d ad li-<lb/>neam d l, ſicut lineæ n o ad lineã o f.</s> <s xml:id="echoid-s38027" xml:space="preserve"> Et quoniã ex præmiſsis angulus b g z eſt æqualis angulo a g z:</s> <s xml:id="echoid-s38028" xml:space="preserve"> <lb/>quoniá linea g z diuidit angulũ a g b per æqualia per 26 th.</s> <s xml:id="echoid-s38029" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s38030" xml:space="preserve"> ſed angulus b g z eſt æqualis an-<lb/>gulo g l a por 29 p 1, extrinſecus enim intrinſeco eſtæ qualis:</s> <s xml:id="echoid-s38031" xml:space="preserve"> & lineæ h z & a l ſunt æquidiſtantes:</s> <s xml:id="echoid-s38032" xml:space="preserve"> ſi-<lb/>militer angulus z g z per eandẽ 29 p 1 æqualis eſt angulo g a l, quia coalternus:</s> <s xml:id="echoid-s38033" xml:space="preserve"> angulus ergo g l ã æ-<lb/>qualis eſt angulo g a l:</s> <s xml:id="echoid-s38034" xml:space="preserve"> ergo per 6 p 1 lineæ g a & g l ſunt æ qualies:</s> <s xml:id="echoid-s38035" xml:space="preserve"> & linea g d eſt perpendicularis ſu-<lb/>perlineã al, ut patet ex præmiſisis:</s> <s xml:id="echoid-s38036" xml:space="preserve"> trigonũ ergo a g l diuiſum eſt in duos trigonos æquiangulos & <lb/>ſimiles ք 31 th.</s> <s xml:id="echoid-s38037" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38038" xml:space="preserve"> eſt ergo ꝓportio lineę a d ad lineã d l, ſicut lineę g a ad lineá g l:</s> <s xml:id="echoid-s38039" xml:space="preserve"> ſed linea a g, ut <lb/>patet ex præmiſsis, eſt æ qualis lineę g l:</s> <s xml:id="echoid-s38040" xml:space="preserve"> eſt ergo linea a d æ qualis lineę d l:</s> <s xml:id="echoid-s38041" xml:space="preserve"> ergo & linea n o eſt æqua <lb/>lis lineę o f:</s> <s xml:id="echoid-s38042" xml:space="preserve"> & linea g o eſt per 29 p 1 perpendicularis ſuper lineán f:</s> <s xml:id="echoid-s38043" xml:space="preserve"> quoniá linea g o eſt perpendicu <lb/>laris ſuper lineã g t, ut patet ex præmiſsis per 18 p 3, & lineæ g t & n f æ quidiſtant, ut præmiſſum eſt.</s> <s xml:id="echoid-s38044" xml:space="preserve"> <lb/>Quia itaq;</s> <s xml:id="echoid-s38045" xml:space="preserve"> angulus g o f eſt æ qualis angulo g o n, & linea o f æ qualis lineæ o n, & linea g o cõmounis:</s> <s xml:id="echoid-s38046" xml:space="preserve"> <lb/>erit ergo per 4 p 1 angulus o f g æqualis angulo o n g:</s> <s xml:id="echoid-s38047" xml:space="preserve"> ſed angulus q g m æ qualis eſt angulo o f g per <lb/>29 p 1;</s> <s xml:id="echoid-s38048" xml:space="preserve"> cum ſit ei extrinſecus, & angulus q g n æ qualis eſt angulo o n g, cum ſit ei conalternus, & lineę <lb/>t q & n f æ quidiſtent, ut patet ex præmiſsis:</s> <s xml:id="echoid-s38049" xml:space="preserve"> erit ergo q g m angulus æ qualis angulo q g n:</s> <s xml:id="echoid-s38050" xml:space="preserve"> ergo per <lb/>20 th.</s> <s xml:id="echoid-s38051" xml:space="preserve"> 5 huius à puncto g circuli p g poteſt ſorma punctim reflecti ad uiſum exiſtentẽ in puncto n:</s> <s xml:id="echoid-s38052" xml:space="preserve"> <lb/>non tamen quòd ſecundũ circulum ſiat reflexio ab his ſpeculis pyramidalibus conuexis, ſed ſic ſei-<lb/>licet quòd punctus g cõmunicat circulo, qui eſt ſectio ſphærę uel columnæ intra ſpeculũ pyramida <lb/>le imaginatæ:</s> <s xml:id="echoid-s38053" xml:space="preserve"> quoniã ſuperficies contingens circulum p g eſt erecta ſuper ſuperficiem reflexionis:</s> <s xml:id="echoid-s38054" xml:space="preserve"> <lb/>propter quod neceſſe habet pyramidem ſpeculi in ſui parte ampliore, ut in ea, quę eſt uerſus baſim, <lb/>ſecare ſecundù æquidiſtantiã axis pyramidis ſpeculi:</s> <s xml:id="echoid-s38055" xml:space="preserve"> & ſic ſuperficies reflexionis (in qua ſunt cen-<lb/>trum uiſus, & punctus rei, & circulus p g) erecta eſt ſuper illam ſuperficiem contingentẽ:</s> <s xml:id="echoid-s38056" xml:space="preserve"> & puncta <lb/>n & m ſe reſpiciunt in ſuperficie illius circulis ſecundũ angulos æquales contentos cum diametro <lb/>ipſius.</s> <s xml:id="echoid-s38057" xml:space="preserve"> Collocato ergo centro uiſus in puncto n & puncto rei uiſæ in puncto m uel econtuerſo:</s> <s xml:id="echoid-s38058" xml:space="preserve"> refle <lb/>ctetur ſemper forma ad centrum uiſus corpore ſpeculi pyramidalis nõ præſtante impedimentum:</s> <s xml:id="echoid-s38059" xml:space="preserve"> <lb/>ut ſi fortè lineæ a n & b m cadant in ipſo circulo baſis, & propter corpus pyramidis ſpeculi non ua-<lb/>leat à puncto g ad uiſum aliquid reflecti.</s> <s xml:id="echoid-s38060" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s38061" xml:space="preserve"/> </p> <div xml:id="echoid-div1551" type="float" level="0" n="0"> <figure xlink:label="fig-0586-01" xlink:href="fig-0586-01a"> <variables xml:id="echoid-variables670" xml:space="preserve">l d a e p t m f k h i g a q o n b</variables> </figure> </div> </div> <div xml:id="echoid-div1553" type="section" level="0" n="0"> <head xml:id="echoid-head1180" xml:space="preserve" style="it">33. Communi ſectione ſuperficieireflexionis & ſpeculipyramidalis conuexi exiſtente linea <lb/>longitudinis ſpeculi, ab uno tantùm puncto ſuperficiei ſpeculi fit formæ unius punctirei uiſæ re-<lb/>flexio ad uiſum. Alhazen 53 n 5.</head> <p> <s xml:id="echoid-s38062" xml:space="preserve">Sit omnino diſpoſitio, quæ eſt in proxima præcedente:</s> <s xml:id="echoid-s38063" xml:space="preserve"> & reflectatur forma puncti b ad uiſum <lb/>exiſtentẽ in puncto a, à puncto ſpeculi pyramidalis cõuexi, quod ſit g:</s> <s xml:id="echoid-s38064" xml:space="preserve"> ita quòd cõmunis ſectio ſu-<lb/>perficiel reflexiõis & ſpeculi ſit linea lõgitudinis ſpeculi, quæ eſt e g.</s> <s xml:id="echoid-s38065" xml:space="preserve"> dico quòd forma pũcti b refle <lb/>ctitur ad uiſum a à ſolo puncto ſuperficiei ſpeculi, qđ eſt;</s> <s xml:id="echoid-s38066" xml:space="preserve"> g.</s> <s xml:id="echoid-s38067" xml:space="preserve"> Si enim dicatur quòd poteſt reflecti ab <lb/>alio puncto ſuperficiei ſpeculi:</s> <s xml:id="echoid-s38068" xml:space="preserve"> tunc illud punctũ aliud aut erit in linea longitudinis ſpeculi, quę eſt <lb/>e g, aut n õ.</s> <s xml:id="echoid-s38069" xml:space="preserve"> Si ſit in linea longitudinis ſpeculi, quæ eſt e g:</s> <s xml:id="echoid-s38070" xml:space="preserve"> ſit e g:</s> <s xml:id="echoid-s38071" xml:space="preserve"> ſit illud punctũ x:</s> <s xml:id="echoid-s38072" xml:space="preserve"> & ab eo ducatur perpen <lb/>dicularis ſuք ſuperficiẽ cõtingenté ſpeculũ in illo pũcto ք 12 p 11:</s> <s xml:id="echoid-s38073" xml:space="preserve"> hęc ergo քpẽdicularis ſit x z:</s> <s xml:id="echoid-s38074" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s38075" xml:space="preserve"> <lb/>linea x z per 6 p 11 æ quidiſtans lineæ z g, quæ prius ducta eſt perpendicularis ſuper eandem ſuper-<lb/>ficiem:</s> <s xml:id="echoid-s38076" xml:space="preserve"> cum punctũ g & x ſint in eadem linea longitudinis, ſecundum quã ſuperficies illa pyramidẽ <lb/>contingit.</s> <s xml:id="echoid-s38077" xml:space="preserve"> Et quia lìnea h z & a l ſunt æquidiſtantes, ut patet per illa, quæ dicta ſunt in præmiſſa:</s> <s xml:id="echoid-s38078" xml:space="preserve"> e-<lb/>rit ergo per 30 p 1 illa perpendicularis x z æquidiſtans lineæ a l.</s> <s xml:id="echoid-s38079" xml:space="preserve"> Et quia linea x z ſicut & linea z h, <lb/>eſt in ſperficier reflexionis, quę per 8 huius uel 25 th.</s> <s xml:id="echoid-s38080" xml:space="preserve"> 5 huius eſt erecta ſuper ſuperficiem cõtingen-<lb/>tem ſpeculum in linea e g:</s> <s xml:id="echoid-s38081" xml:space="preserve"> erit ergo per 1 th.</s> <s xml:id="echoid-s38082" xml:space="preserve"> 1 huius linea a l in ſuperficie reflexionis huius lineæ <lb/>perpendicularis, quę eſt x z & erit ſimiliter in ſuperficie reflexionis lineę perpendicularis, quę eſt z <lb/>g.</s> <s xml:id="echoid-s38083" xml:space="preserve"> Igitur illæ duæ ſuperficies reflexionis ſecát ſe ſuper lineá al per 19 th.</s> <s xml:id="echoid-s38084" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38085" xml:space="preserve"> ſed ſecant ſe etiã ſu-<lb/> <pb o="286" file="0588" n="588" rhead="VITELLONIS OPTICAE"/> per punctũ b:</s> <s xml:id="echoid-s38086" xml:space="preserve"> quoniá illud eſt, quod reflectitur per utranq;</s> <s xml:id="echoid-s38087" xml:space="preserve">: hoc aũt eſt impoſsibile:</s> <s xml:id="echoid-s38088" xml:space="preserve"> quoniá punctú <lb/>b nó eſt in linea al.</s> <s xml:id="echoid-s38089" xml:space="preserve"> Oſten ſum eſt enim prius lineam fl ęquidiftantẽ eſſe lineę b m:</s> <s xml:id="echoid-s38090" xml:space="preserve"> quę duę lineę uel <lb/>concurrerẽt ſi punctú b eſſet in linea a l:</s> <s xml:id="echoid-s38091" xml:space="preserve"> uel ſequeretur puncta m & n cadere ex una parte lineæ g q-<lb/>Non ergo fieret reflexio punctorũ m & n adinuicé à puncto g, quod eſt cótra demonſtrata in præ-<lb/>miſſa.</s> <s xml:id="echoid-s38092" xml:space="preserve"> Reſtat ergo, ut à nullo puncto lineę longitudinis, quę e g, præterquá à puncto g, forma puncti <lb/>b poſsit reflecti ad centrũ uiſus exiſtens in puncto a.</s> <s xml:id="echoid-s38093" xml:space="preserve"> Si <lb/> <anchor type="figure" xlink:label="fig-0588-01a" xlink:href="fig-0588-01"/> aũt poſsibile eſt, ut reflectatur forma puncti b ad uiſum <lb/>a ab aliquo puncto ſpeculi extra lineá longitudinis g e, <lb/>fit ille punctus u:</s> <s xml:id="echoid-s38094" xml:space="preserve"> & per 101 th.</s> <s xml:id="echoid-s38095" xml:space="preserve"> 1 huius ducatur linea lon-<lb/>gitudinis ſpeculi, quæ ſit linea e u c:</s> <s xml:id="echoid-s38096" xml:space="preserve"> quæ in puncto c ſe-<lb/>cet peripheriá circuli g p.</s> <s xml:id="echoid-s38097" xml:space="preserve"> Et ſumatur ſuperſicies ęquidi <lb/>ſtans baſi tranſiens per punctũ u:</s> <s xml:id="echoid-s38098" xml:space="preserve"> palàm ergo per 8 p 11 <lb/>quoniá linea a n ſecat hanc ſuperficiem:</s> <s xml:id="echoid-s38099" xml:space="preserve"> ideo, quia linea <lb/>e g, cui æquidiſtat linea a n, ſecat eandẽ ſuperficiẽ:</s> <s xml:id="echoid-s38100" xml:space="preserve"> ſunt <lb/>aút per 1 th.</s> <s xml:id="echoid-s38101" xml:space="preserve"> 1 huius lineę a n & e g in eadẽ ſuperficie, cũ <lb/>ſint æ quidiftantes:</s> <s xml:id="echoid-s38102" xml:space="preserve"> ſit ergout linea a n ſecet illá ſuperfi-<lb/>ciem in puncto y.</s> <s xml:id="echoid-s38103" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s38104" xml:space="preserve"> linea b m æ quidiſtás <lb/>lineæ e g, ſecabit eandẽ ſuperficiẽ:</s> <s xml:id="echoid-s38105" xml:space="preserve"> ſit quoq;</s> <s xml:id="echoid-s38106" xml:space="preserve"> punctus ſe-<lb/>ctionis k:</s> <s xml:id="echoid-s38107" xml:space="preserve"> & ducátur lineę k u, y u, a k.</s> <s xml:id="echoid-s38108" xml:space="preserve"> Et cum illa ſuperfi <lb/>cies per 100 th.</s> <s xml:id="echoid-s38109" xml:space="preserve"> 1 huius ſecet pyramidẽ ſecundũ circulũ, <lb/>tranſeuntẽ per punctũ u, ducatur à pũcto u linea ad cen <lb/>trum huius circuli, quę ſit r u:</s> <s xml:id="echoid-s38110" xml:space="preserve"> & producatur extra ſpecu <lb/>lum:</s> <s xml:id="echoid-s38111" xml:space="preserve"> & ſit item u r:</s> <s xml:id="echoid-s38112" xml:space="preserve"> & à uertice pyramidis ſpeculi pũcto <lb/>ſcilicet e ducantur lineæ e k, e y, quæ neceſſariò ſecabũt <lb/>ſuperficié circuli p g:</s> <s xml:id="echoid-s38113" xml:space="preserve"> & ſint puncta ſectionũ i & s:</s> <s xml:id="echoid-s38114" xml:space="preserve"> & du-<lb/>cantur lineę i c & s c.</s> <s xml:id="echoid-s38115" xml:space="preserve"> Sicut ergo per præcedentem pro-<lb/>batum eſt de forma puncti m, quod non impediente <lb/>pyramide poteſt reflecti ad uiſum exiſtentem in pun-<lb/>cto n à puncto ſpeculi g:</s> <s xml:id="echoid-s38116" xml:space="preserve"> eodẽ modo probari poteſt de <lb/>puncto k, quod reflectetur ad uiſum exiſtentẽ in pũcto <lb/>y à puncto ſpeculi u:</s> <s xml:id="echoid-s38117" xml:space="preserve"> angulus ergo r u y erit æqualis an-<lb/>gulo r u k.</s> <s xml:id="echoid-s38118" xml:space="preserve"> Et quoniã linea b h æquidiſtat lineę e g, & li-<lb/>nea cõmunis ſuperficiei b g e k & ſuperficiei circuli p g eſt linea m g per 19 th.</s> <s xml:id="echoid-s38119" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38120" xml:space="preserve"> quoniá linea <lb/>m g eſt in utraq;</s> <s xml:id="echoid-s38121" xml:space="preserve"> illarũ ſuperficierũ:</s> <s xml:id="echoid-s38122" xml:space="preserve"> patet quòd linea e k, cum ſit in hac ſuperficie b g e k, & ſecet ſu-<lb/>perficié circuli p g, cadet ſuper lineam communẽ, quę eſt m g:</s> <s xml:id="echoid-s38123" xml:space="preserve"> cadit aũt in punctũ illius ſuperficiei, <lb/>quod eſt s, ut peręmiſſum eſt:</s> <s xml:id="echoid-s38124" xml:space="preserve"> quoniã linea e k s eſt linea una:</s> <s xml:id="echoid-s38125" xml:space="preserve"> erit igitur linea s m g linea recta.</s> <s xml:id="echoid-s38126" xml:space="preserve"> Eodẽ <lb/>modo cũ ſuperficies n y e g ſecet ſuperficiẽ circuli p g ſuper lineá n g, linea e y cõcurret cũ linea n g <lb/>in puncto i per modũ præmiſſum:</s> <s xml:id="echoid-s38127" xml:space="preserve"> ergo linea i n g eſt una linea recta.</s> <s xml:id="echoid-s38128" xml:space="preserve"> Palàm etiá quòd ſuperficies i c <lb/>e ſecabit ſuperficiem circuli p g ſuper lineã i c:</s> <s xml:id="echoid-s38129" xml:space="preserve"> ſecat aũt ſuperficiẽ huic ſuperficiei ęquidiftantẽ, quę <lb/>tranſit per punctũ u ſuper lineam y u:</s> <s xml:id="echoid-s38130" xml:space="preserve"> ergo per 16 p 11 linea i c æquidiſtat lineæ y u.</s> <s xml:id="echoid-s38131" xml:space="preserve"> Similiter ſuperfi <lb/>cies s c e ſecat ſuperficies illas æ quidiſtantes, ſcilicet ſuperficies g p & u y ſuper duas lineas s c & k <lb/>u:</s> <s xml:id="echoid-s38132" xml:space="preserve"> ergo per eandẽ 16 p 11 lineæ s c & k u ſunt æquidiſtátes.</s> <s xml:id="echoid-s38133" xml:space="preserve"> Similiter ſi ſumatur ſuperficies ſecans ſpe <lb/>culũ ſuper lineá longitudinis, quę eſt e c, in qua ſuperficie ſunt puncta r & u:</s> <s xml:id="echoid-s38134" xml:space="preserve"> ſunt enim puncta r, u, <lb/>c, Min eadẽ ſuperficie:</s> <s xml:id="echoid-s38135" xml:space="preserve"> cũ puncta r, u, t, & aliquis punctus lineę s g ſint in eadẽ ſuperficie:</s> <s xml:id="echoid-s38136" xml:space="preserve"> quia eadẽ <lb/>eſt demonſtratio dato alio quocunq;</s> <s xml:id="echoid-s38137" xml:space="preserve"> puncto lineę c M:</s> <s xml:id="echoid-s38138" xml:space="preserve"> ſemper enim ſuperficies hoc modo ſecans <lb/>ſpeculũ ſecundũ lineá e c, ſecabit illas ſuperficies æquidiſtantes ſuper duas lineas M c & r u:</s> <s xml:id="echoid-s38139" xml:space="preserve"> igitur, <lb/>ut prius per 16 p 11 illę duę M c & r u ſunt æ quidiftantes:</s> <s xml:id="echoid-s38140" xml:space="preserve"> igitur per 10 p 11 angulus s c M æqua-<lb/>lis eſt angulo k u r:</s> <s xml:id="echoid-s38141" xml:space="preserve"> & angulus M c i æqualis angulo r u y:</s> <s xml:id="echoid-s38142" xml:space="preserve"> ſed iam patuit quòd angulus k u r æqualis <lb/>eſt angulo r u y:</s> <s xml:id="echoid-s38143" xml:space="preserve"> ergo angulus s c M æqualis eſt angulo M c i.</s> <s xml:id="echoid-s38144" xml:space="preserve"> Quare forma pũcti s poteſt reflecti a d <lb/>uiſum exiſtentẽ in puncto i à puncto ſpeculic, nó impediente corpore pyramidis ſpeculi.</s> <s xml:id="echoid-s38145" xml:space="preserve"> Sed iam <lb/>probatũ eſt per præmiſſa, quòd forma puncti m refle cti poteſt ad uiſum exiſtentẽ in puncto i à pun <lb/>cto g circuli p g:</s> <s xml:id="echoid-s38146" xml:space="preserve"> quoniá poteſt reflecti ad punctũ n:</s> <s xml:id="echoid-s38147" xml:space="preserve"> & puncta n & i ſunt in eadẽ linea recta cõſiſten-<lb/>tia, ut præ oftenſum eſt.</s> <s xml:id="echoid-s38148" xml:space="preserve"> Poterit ergo forma.</s> <s xml:id="echoid-s38149" xml:space="preserve"> puncti m à pũcto ſpeculi g reflecti ad uiſum exiſtentẽ in <lb/>puncto i:</s> <s xml:id="echoid-s38150" xml:space="preserve"> & ita punctũ s, qđ eſt in linea s m g, poteſt reflecti ad uiſum exiſtentẽ in puncto i à puncto <lb/>g.</s> <s xml:id="echoid-s38151" xml:space="preserve"> Igitur forma pũcti s reflectitur ad uiſum in pũcto i à duobus punctis circuli p g, qđ eſt impoſsibi-<lb/>le, & cõtra 16 p 6 huius, & contra 27 huius.</s> <s xml:id="echoid-s38152" xml:space="preserve"> Reſtat ergo, ut primũ ſit impoſsibile:</s> <s xml:id="echoid-s38153" xml:space="preserve"> ſcilicet quòd forma <lb/>puncti b reflecti poſsit ad uiſum exiſtentẽ in puncto a ab aliquo alio puncto ſpeculi, quàm à puncto <lb/>g.</s> <s xml:id="echoid-s38154" xml:space="preserve"> Ab uno ſolo ergo puncto fiet reflexio formæ eiuſdẽ puncti cómuni ſectione ſuperficiei reflexio-<lb/>nis & ſpeculi pyramidalis conuexi exiſtente linea longitudinis ſpeculi.</s> <s xml:id="echoid-s38155" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s38156" xml:space="preserve"/> </p> <div xml:id="echoid-div1553" type="float" level="0" n="0"> <figure xlink:label="fig-0588-01" xlink:href="fig-0588-01a"> <variables xml:id="echoid-variables671" xml:space="preserve">l d a e f z x y t u p r k o h y x m n q m i b c</variables> </figure> </div> </div> <div xml:id="echoid-div1555" type="section" level="0" n="0"> <head xml:id="echoid-head1181" xml:space="preserve" style="it">34. Cõmuni ſectione ſuperficiei reflexionis & ſpeculi pyramidalis conuexiexiſtente oxygo-<lb/>nia ſectione: à quolibet puncto ſuperficiei ſpeculi apparentis uiſuipoteſt reflexio aduiſum: <lb/>& ab uno uel à duobus punctis tantùm. Alhazen 43 n 4.</head> <p> <s xml:id="echoid-s38157" xml:space="preserve">Efto ſpeculum pyramidale conuexum f k s:</s> <s xml:id="echoid-s38158" xml:space="preserve"> cuius uertex f:</s> <s xml:id="echoid-s38159" xml:space="preserve"> diameter baſis k s:</s> <s xml:id="echoid-s38160" xml:space="preserve"> centrumq́;</s> <s xml:id="echoid-s38161" xml:space="preserve"> baſis n:</s> <s xml:id="echoid-s38162" xml:space="preserve"> <lb/> <pb o="287" file="0589" n="589" rhead="LIBER SEPTIMVS."/> erit ergo axis ſpeculi linea f n:</s> <s xml:id="echoid-s38163" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s38164" xml:space="preserve"> centrum uiſus punctus a.</s> <s xml:id="echoid-s38165" xml:space="preserve"> Dico quòd cómuni ſectione ſuperfi-<lb/>ciei reflexionis & ſpeculi exiſtente ſectione oxygonia, quæ ſit b i:</s> <s xml:id="echoid-s38166" xml:space="preserve"> poſsibile eſt à quolibet puncto <lb/>ſpeculi propoſiti fieri reflexionẽ alicuius puncti uiſi ad punctum a, quod eſt centrũ uiſus.</s> <s xml:id="echoid-s38167" xml:space="preserve"> Sit enim <lb/>punctus b datus in ſuperficie ſpeculi, de quo dubitatur utrũ ab eo poſsit fierl reflexio formæ alicu-<lb/>ius puncti rei uiſæ ad centrum uiſus, quod eſt a.</s> <s xml:id="echoid-s38168" xml:space="preserve"> Ducatur ergo à puncto b linea longitudinis pyra-<lb/>midis ſpeculi per 101 th.</s> <s xml:id="echoid-s38169" xml:space="preserve"> 1 huius, quæ ſit b f:</s> <s xml:id="echoid-s38170" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s38171" xml:space="preserve"> à puncto b perpendicularis ſuper illá lineam <lb/>longitu dinis extra ſpeculum, quæ ſit b g:</s> <s xml:id="echoid-s38172" xml:space="preserve"> & ſuper punctum b terminum lineæ b g fiat per 23 p 1 an-<lb/>gulus æqualis angulo a b g, qui ſit g b p, ducta linea b p in eadem ſuperficie reſlexionis:</s> <s xml:id="echoid-s38173" xml:space="preserve"> patetq́;</s> <s xml:id="echoid-s38174" xml:space="preserve"> per <lb/>20 th 5 huius quia omnis punctus rei uiſæ exiſtens in linea b p reflectetur ad uiſum in punctum a:</s> <s xml:id="echoid-s38175" xml:space="preserve"> <lb/>ſed & à ſolo puncto b uel duobus tantùm fiet reflexio ad uiſum exiſtentẽ in puncto a.</s> <s xml:id="echoid-s38176" xml:space="preserve"> Palám enim <lb/>per 96 th 1 huius quòd ſi perpendicularis g b producatur intra pyramidem:</s> <s xml:id="echoid-s38177" xml:space="preserve"> quoniá concurret cum <lb/>axe f n:</s> <s xml:id="echoid-s38178" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s38179" xml:space="preserve"> punctus concurſus c.</s> <s xml:id="echoid-s38180" xml:space="preserve"> Palàm ergo quoniam angulus g c f cum ſit in ſuperficie ſectionis, <lb/>uerſus uerticem pyramidis eſt acutus per 32 p 1, quoniá in trigono b c f angulus c b f eſt rectus.</s> <s xml:id="echoid-s38181" xml:space="preserve"> Cir-<lb/>cunducatur ergo per 102 th.</s> <s xml:id="echoid-s38182" xml:space="preserve"> 1 huius à puncto reflexionis, quod eſt <lb/> <anchor type="figure" xlink:label="fig-0589-01a" xlink:href="fig-0589-01"/> b, circulus ſpeculo pyramidali:</s> <s xml:id="echoid-s38183" xml:space="preserve"> cuius diameter ſit b d:</s> <s xml:id="echoid-s38184" xml:space="preserve"> & eius cen-<lb/>trum e, ſecás axem f n in puncto e.</s> <s xml:id="echoid-s38185" xml:space="preserve"> Et quia ille circulus per 100 th.</s> <s xml:id="echoid-s38186" xml:space="preserve"> 1 <lb/>huius eſt æquidiſtans baſſ ſpeculi, palàm quia perpendicularis g c <lb/>acutum angulum tenens cum axe fn, declinata erit ſuper circuli il-<lb/>lius ſuperficiem:</s> <s xml:id="echoid-s38187" xml:space="preserve"> quia linea æquidiſtans lineæ g c ſi produceretur à <lb/>puncto n centro baſis ſpeculi, patet quòd declinata eſt ſuper baſim <lb/>pyramidis, ut ſit linea n q.</s> <s xml:id="echoid-s38188" xml:space="preserve"> Producta ergo linea c d à puncto axis c <lb/>ad circuli peripherlam, cum angulus b e c ſit æ qualis angulo d e c, <lb/>quoniam uterq;</s> <s xml:id="echoid-s38189" xml:space="preserve"> ipſorum eſt rectus:</s> <s xml:id="echoid-s38190" xml:space="preserve"> omnes enim anguli contenti <lb/>ſub ſemidiametris circuli & axe f e ſunt æ quales, & lineæ à centro <lb/>ad circumferentiam æ quales, e c uerò linea eſt communis:</s> <s xml:id="echoid-s38191" xml:space="preserve"> palàm <lb/>per 4 p 1 quoniam latus b c æquale eſt lateric d:</s> <s xml:id="echoid-s38192" xml:space="preserve"> & omnes anguli fa <lb/>ctorum trigonorum ſunt æquales:</s> <s xml:id="echoid-s38193" xml:space="preserve"> quia idem eſt de omnibus lineis <lb/>à puncto c ad circuli b d circumferentiam prdoductis per 65 th.</s> <s xml:id="echoid-s38194" xml:space="preserve"> 1 hu <lb/>ius:</s> <s xml:id="echoid-s38195" xml:space="preserve"> quoniam punctus c eſt polus circuli b d.</s> <s xml:id="echoid-s38196" xml:space="preserve"> Fiet ergo noua pyra-<lb/>mis, cuius baſis eſt circulus b d, uertex c, & axis c e.</s> <s xml:id="echoid-s38197" xml:space="preserve"> Superficies er-<lb/>go reflexionis ſecans ſpeculum ſecundum oxygoniam ſectionem:</s> <s xml:id="echoid-s38198" xml:space="preserve"> <lb/>aut continget hanc pyramidem c b d:</s> <s xml:id="echoid-s38199" xml:space="preserve"> aut ſecabit.</s> <s xml:id="echoid-s38200" xml:space="preserve"> Si contingat, di-<lb/>co quòd à ſolo puncto b, quod eſt punctus reflexionis, tantùm fiet <lb/>reflexio ſecundum illam ſuperficiem eandem.</s> <s xml:id="echoid-s38201" xml:space="preserve"> Palàm enim quòd ſuperficies reflexionis contingat <lb/>pyramidem ſuper lineam longitudinis illius pyramidis per 95 th.</s> <s xml:id="echoid-s38202" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38203" xml:space="preserve"> hæc autem erit linea b c, <lb/>in qua eſt punctũ b, à quo ducitur linea b c perpédicularis ſuper ſuperficiẽ ſpeculi, & linea reflexio-<lb/>nis b a.</s> <s xml:id="echoid-s38204" xml:space="preserve"> A puncto quoq, f, quod eſt uertex pyramidis ſpeculi, ducátur lineæ plures ad ſectionẽ oxy-<lb/>goniã, quæ eſt cõmunis ſection ſuperficiei reflexióis & pyramidis ſpeculi, quæ eſt f k s.</s> <s xml:id="echoid-s38205" xml:space="preserve"> Omnes itaq;</s> <s xml:id="echoid-s38206" xml:space="preserve"> <lb/>illæ lineę prius cadent in ſuperficiẽ circuli b d, quę eſt baſis pyramidis intellectæ, ꝗ̃ cadant in ipſam <lb/>ſectionẽ, præter uná ſolam, quę cadetin punctũ reflexiõis b, quę eſt linea f b.</s> <s xml:id="echoid-s38207" xml:space="preserve"> A folo itaq;</s> <s xml:id="echoid-s38208" xml:space="preserve"> puncto b <lb/>fiet reflexio a d uiſum.</s> <s xml:id="echoid-s38209" xml:space="preserve"> Si enim detur quod ab alio puncto dictę ſectiõis oxygonię, ut à puncto i, fiat <lb/>ad uiſum a reflexio:</s> <s xml:id="echoid-s38210" xml:space="preserve"> tunc linea ab illo punctio i ad punctũ c, quod eſt uertex pyramidis intelletę, du <lb/>cta, quę ſit i c:</s> <s xml:id="echoid-s38211" xml:space="preserve"> erit, ut prius, perpendicularis ſuper ſuperficiem ſpeculi per 96 th.</s> <s xml:id="echoid-s38212" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s38213" xml:space="preserve"> Cum enim <lb/>illa perpen dicularis neceſſariò ſit in ſuperficie reflexióis, in qua eſt ſectio:</s> <s xml:id="echoid-s38214" xml:space="preserve"> oportet quòd ipſa cadat <lb/>in punctũ c.</s> <s xml:id="echoid-s38215" xml:space="preserve"> Ergo erit perpendicularis ſuper lineam lógitudinis pyramidis ſpeculi per illud punctũ <lb/>i tranſeuntẽ, quæ ſit fil.</s> <s xml:id="echoid-s38216" xml:space="preserve"> Sit quoq;</s> <s xml:id="echoid-s38217" xml:space="preserve"> punctus, in quo linea f i ſecat circulũ d b, pun ctus r.</s> <s xml:id="echoid-s38218" xml:space="preserve"> Patet aũt per <lb/>præmiſſa & per 96 th.</s> <s xml:id="echoid-s38219" xml:space="preserve"> 1 huius quoniã linea c rà uertice pyramidis intellectę ducta ad illá lineá lon-<lb/>gitudinis necſſariò eſt perpendicularis ſuper illá, ſicut linea c b eſt perpendicularis ſuper lineá lon <lb/>gitudinis ſpeculi, quæ eſt f b:</s> <s xml:id="echoid-s38220" xml:space="preserve"> quoniã, ut patet per 89 th.</s> <s xml:id="echoid-s38221" xml:space="preserve"> 1 huius anguli omniũ linearũ longitudinis <lb/>cum ſemidiametro baſis & cũ axe ad uerticẽ ſunt æ quales:</s> <s xml:id="echoid-s38222" xml:space="preserve"> erunt ergo in triangulo cir duo anguli <lb/>recti:</s> <s xml:id="echoid-s38223" xml:space="preserve"> quod eſt impoſsibile & cótra 32 p 1.</s> <s xml:id="echoid-s38224" xml:space="preserve"> Non ergo fiet reflexio a b alio puncto ſect.</s> <s xml:id="echoid-s38225" xml:space="preserve"> onis oxygoniæ, <lb/>quę eſt b i, ꝗ̃ à puncto b, ſuperficie reflexionis pyramidem c b d contingente.</s> <s xml:id="echoid-s38226" xml:space="preserve"> Quòd fi ſuperficies re <lb/>flexionis ſecet pyramidẽ c b d, palàm per 104 th.</s> <s xml:id="echoid-s38227" xml:space="preserve"> 1 huius quoniã ſecabit circulũ b d, ꝗ eſt baſis eiuſdẽ <lb/>pyramidis, in duobus tantùm punctis.</s> <s xml:id="echoid-s38228" xml:space="preserve"> Dico ergo quòd in his ſolis duobus punctis poteſt fieri refle <lb/>xio ad uiſum à tota data oxygonia ſectione.</s> <s xml:id="echoid-s38229" xml:space="preserve"> Quoniá enim a b utroq;</s> <s xml:id="echoid-s38230" xml:space="preserve"> iſtorũ punctorũ linea ducta ad <lb/>punctũ c, qđ eſt uertex pyramidis c b d, eſt perpendicularis ſuper lineá longitudinis ſpeculi tranſ <lb/>euntẽ per illum punctũ, ut patet ex præmiſsis:</s> <s xml:id="echoid-s38231" xml:space="preserve"> ab illis ergo duobus punctis poteſt fieri reflexio ad <lb/>uiſum a, prout modo præmiſſo demonſtrari poteſt.</s> <s xml:id="echoid-s38232" xml:space="preserve"> Quòd ſi dentur puncta alia illius ſectionis oxy-<lb/>goniæ, à quibus dicatur poſſe fieri reflexio:</s> <s xml:id="echoid-s38233" xml:space="preserve"> tunc ſemper linea à dato puncto, quod ſit h, ducta ad <lb/>punctum c uerticem imaginatæ pyramidis, tenebit angulum rectum cum linea longitudinis ſpe-<lb/>culi per illum dictum punctum tranſeuntem:</s> <s xml:id="echoid-s38234" xml:space="preserve"> & fiet angulus extrinſecus æqualis intrinſeco ſi-<lb/>bi oppoſito, quod eſt contra 16 p 1:</s> <s xml:id="echoid-s38235" xml:space="preserve"> aut duo anguli trianguli fient recti, quod eſt contra 32 p 1, <lb/>ut prius.</s> <s xml:id="echoid-s38236" xml:space="preserve"> Linea enim à puncto c ad communem ſectionem eiuſdem lineæ longitudinis & circuli <lb/> <pb o="288" file="0590" n="590" rhead="VITELLONIS OPTICAE"/> b d ducta tenebit cum linea longitudinis angulum rectum.</s> <s xml:id="echoid-s38237" xml:space="preserve"> Si uerò angulus f h c ſit acutus, ideo <lb/>quòd angulus c r h ſit rectus:</s> <s xml:id="echoid-s38238" xml:space="preserve"> palàm per 13 p 1 quòd angulus c h l eſt obtuſus.</s> <s xml:id="echoid-s38239" xml:space="preserve"> Omnes enim lineæ du <lb/>ctę à puncto c, quod eſt uertex pyramidis intellectæ, quæ eſt c b d, ad <lb/> <anchor type="figure" xlink:label="fig-0590-01a" xlink:href="fig-0590-01"/> puncta ſectionis, quę interiacent uerticem ſpeculi & peripheriam cir-<lb/>culi b d, facient angulos obtuſos cum lineis longitudinis uerſus uerti-<lb/>cem pyramidis, qui eſt f:</s> <s xml:id="echoid-s38240" xml:space="preserve"> & omnes lineæ, quę ducuntur à puncto c ad <lb/>puncta interiacentia circulum b d & baſim ſpeculi k s, facient cum li-<lb/>neis longitudinis angulos acutos uerſus uerticem ſpeculi, qui eſt f, & <lb/>obtuſos ex parte baſis.</s> <s xml:id="echoid-s38241" xml:space="preserve"> A nullo ergo omnium illorum punctorum <lb/>fiet reflexio, ſed à ſolis punctis circuli b d:</s> <s xml:id="echoid-s38242" xml:space="preserve"> ſed neq;</s> <s xml:id="echoid-s38243" xml:space="preserve"> ab illis poteſt fieri <lb/>reflexio per 25 huius, niſi in ſectione oxygonia ceciderint:</s> <s xml:id="echoid-s38244" xml:space="preserve"> hoc autem <lb/>non eſt poſsibile, ut præmiſſum eſt, niſi in uno tantùm puncto ſectione <lb/>oxygonia imaginatam pyramidem contingente, uel tantùm in duobus <lb/>punctis dicta ſectione baſim pyramidis imaginatæ ſecante:</s> <s xml:id="echoid-s38245" xml:space="preserve"> nec enim <lb/>poſſunt hæc per modos alios uariari.</s> <s xml:id="echoid-s38246" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s38247" xml:space="preserve"/> </p> <div xml:id="echoid-div1555" type="float" level="0" n="0"> <figure xlink:label="fig-0589-01" xlink:href="fig-0589-01a"> <variables xml:id="echoid-variables672" xml:space="preserve">f a d e r c b y i h p s l n f q</variables> </figure> <figure xlink:label="fig-0590-01" xlink:href="fig-0590-01a"> <variables xml:id="echoid-variables673" xml:space="preserve">f a d e r c b g h p l s n k</variables> </figure> </div> </div> <div xml:id="echoid-div1557" type="section" level="0" n="0"> <head xml:id="echoid-head1182" xml:space="preserve" style="it">35. Dato ſpeculo pyramidali conuexo, centró uiſus & puncto <lb/>rei uiſe exiſtentibus inter ſuperficiem æquidiſtanter baſi ſpeculum <lb/>in uertice contingentem & inter ipſam baſim: poßile eſt inueni-<lb/>ri punctum reflexionis. Alhazen 54 n 5.</head> <p> <s xml:id="echoid-s38248" xml:space="preserve">Eſto datum ſpeculum pyramidale, cuius uertex ſit punctus g:</s> <s xml:id="echoid-s38249" xml:space="preserve"> & fiat <lb/>ſuper ipſum uerticem ſuperficies æquidiſtans baſi pyramidis, quę ſit m n g:</s> <s xml:id="echoid-s38250" xml:space="preserve"> quod fiet ductis à pun-<lb/>cto g uertice ſpeculi tribus lineis perpendicularibus ſuper axem ſpeculi per 11 p 1, & imaginata pla <lb/>na ſuperficie inter illas lineas extenſa:</s> <s xml:id="echoid-s38251" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s38252" xml:space="preserve"> a punctus rei uiſæ & b centrũ uiſus:</s> <s xml:id="echoid-s38253" xml:space="preserve"> quæ ſint ambo ſub <lb/>illa ſuperficie m n g, inter ipſam ſcilicet & baſim ſpeculi:</s> <s xml:id="echoid-s38254" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s38255" xml:space="preserve"> exempli cauſſa, punctũ b propinquius <lb/>uertici ſpeculi g quàm punctum a:</s> <s xml:id="echoid-s38256" xml:space="preserve"> quoniam ſi poſitum fuerit eſſe econuerſo, ſemper eadem eſt de-<lb/>monſtratio.</s> <s xml:id="echoid-s38257" xml:space="preserve"> Dico quòd eſt poſsibile punctum reflexionis inueniri.</s> <s xml:id="echoid-s38258" xml:space="preserve"> Ducatur enim à puncto a, qui <lb/>eſt punctus rei uiſæ, ſuperficies ſecans pyramidem æ quidiſtanter baſi, ut prius:</s> <s xml:id="echoid-s38259" xml:space="preserve"> & ducatur à uertice <lb/>ſpeculi, qui eſt punctum g, linea ad punctum b, quod eſt centrum uiſus, quæ ſit g b.</s> <s xml:id="echoid-s38260" xml:space="preserve"> Hæc itaq;</s> <s xml:id="echoid-s38261" xml:space="preserve"> linea <lb/>producta cadet in ſuperficiem à puncto a rei uiſæ ductam æquidiſtanter baſi pyramidis:</s> <s xml:id="echoid-s38262" xml:space="preserve"> cum illa li-<lb/>nea g b ſit inter ſuperficies æ quidiſtantes ducta à uertice axis ambas illas ſuperficies tranſeuntis:</s> <s xml:id="echoid-s38263" xml:space="preserve"> <lb/>punctus ergo, in quem cadit hæc linea g b, ſit punctus h.</s> <s xml:id="echoid-s38264" xml:space="preserve"> Ergo per modum demonſtrandi, quo uſi <lb/>ſumus in 32 huius, demonſtrari poteſt quoniam forma puncti a reflectetur ad uiſum exiſtentem in <lb/>puncto h ab aliquo puncto circuli, quem efficit ſuperficies ſecans pyramidem, ducta à punctis a & <lb/>h:</s> <s xml:id="echoid-s38265" xml:space="preserve"> cuius circuli centrum ſit punctum axis ſpeculi, quod <lb/> <anchor type="figure" xlink:label="fig-0590-02a" xlink:href="fig-0590-02"/> eſt t:</s> <s xml:id="echoid-s38266" xml:space="preserve"> & ſit punctus reflexionis inuentus in illo circulo <lb/>punctus e:</s> <s xml:id="echoid-s38267" xml:space="preserve"> & ducatur inter a punctum rei uiſæ & cen-<lb/>trum uiſus b linea a b:</s> <s xml:id="echoid-s38268" xml:space="preserve"> & linea longitudinis ſpeculi, quę <lb/>fit g e:</s> <s xml:id="echoid-s38269" xml:space="preserve"> & axis pyramidis ſpeculi ſit g t:</s> <s xml:id="echoid-s38270" xml:space="preserve"> & ducatur à pun-<lb/>cto e linea ad centrum ſui circuli, quæ ſit e t:</s> <s xml:id="echoid-s38271" xml:space="preserve"> hæc enim ca <lb/>det ſuper axẽ g t perpendiculariter per 100 & per 89 th.</s> <s xml:id="echoid-s38272" xml:space="preserve"> 1 <lb/>huius, uel per 21 huius:</s> <s xml:id="echoid-s38273" xml:space="preserve"> & etiam ideo quòd axis g t cum <lb/>ſit perpendicularis ſuper baſim pyramidis ſpeculi & etiã <lb/>erectus ſuper ſuperficiem circuli æquidiftantis illi baſi <lb/>per 23 th.</s> <s xml:id="echoid-s38274" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38275" xml:space="preserve"> eſt ergo per definitionem lineę ſuper ſu <lb/>perficiem erectæ axis g t perpendicularis ſuper ſemidia-<lb/>metrum e t, & erit linea e t erecta ſuper lineam contin-<lb/>gentem illum circulũ in puncto e per 18 p 3:</s> <s xml:id="echoid-s38276" xml:space="preserve"> & hæc linea <lb/>t e producta extra circulum ductis lineis h e & a e, ſeca-<lb/>bit angulum a b eis contentum per æ qualia, ſcilicet angu <lb/>lum h e a per 26 th.</s> <s xml:id="echoid-s38277" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s38278" xml:space="preserve"> ergo per 29 th.</s> <s xml:id="echoid-s38279" xml:space="preserve"> 1 huius eadem <lb/>linea t e producta lineam h a ductá ſecabit:</s> <s xml:id="echoid-s38280" xml:space="preserve"> cum ſit cum <lb/>illa in eadẽ ſuperficie reflexionis, ut patet per 24 huius:</s> <s xml:id="echoid-s38281" xml:space="preserve"> <lb/>ſit ergo linearum t e & h a punctus ſectionis r.</s> <s xml:id="echoid-s38282" xml:space="preserve"> Et quia li-<lb/>neæ g e & e t efficiunt ſuperficiem ſecantem lineam a b:</s> <s xml:id="echoid-s38283" xml:space="preserve"> <lb/>ſit punctus ſectionis ſ:</s> <s xml:id="echoid-s38284" xml:space="preserve"> & ab illo puncto f ducatur per 12 p <lb/>1 linea perpendicularis ſuper lineá longitudinis g e, quæ <lb/>fit f q:</s> <s xml:id="echoid-s38285" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s38286" xml:space="preserve"> linea f q per definitionem lineæ ſuper ſuper-<lb/>ficiem erectæ, perpendicularis ſuper ſuperficiem contin <lb/>gentem pyramidem ſuper lineam g e.</s> <s xml:id="echoid-s38287" xml:space="preserve"> Deinde à puncto <lb/>a ducatur linea æ quidiſtás lineæ f q, quæ fit linea a l:</s> <s xml:id="echoid-s38288" xml:space="preserve"> pro-<lb/>ducaturq́;</s> <s xml:id="echoid-s38289" xml:space="preserve"> linea f q, donec concurrat cum axe g t in puncto k.</s> <s xml:id="echoid-s38290" xml:space="preserve"> Ducatur item à puncto a linea æquidi <lb/>ſtans line æ r t, quæ ſit a s:</s> <s xml:id="echoid-s38291" xml:space="preserve"> & ducatur à puncto e linea, quæ ſit communis ſectio ſuperficiei reflexio-<lb/> <pb o="289" file="0591" n="591" rhead="LIBER SEPTIMVS."/> nis, quæ eſt a e h, & ſuperficiei contingentis pyramidem ſpeculi in linea longitudinis, quæ eſt g e:</s> <s xml:id="echoid-s38292" xml:space="preserve"> & <lb/>fit hæc linea e o:</s> <s xml:id="echoid-s38293" xml:space="preserve"> quæ cum ſit perpendicularis ſuper ſemidiametrum circuli, quæ eſt e t, ut pater per <lb/>18 p 3:</s> <s xml:id="echoid-s38294" xml:space="preserve"> contingit enim linea e o circulum, cuius eſt centrum punctum t:</s> <s xml:id="echoid-s38295" xml:space="preserve"> palàm quòd ipſa eſt perpen-<lb/>dicularis ſuper lineam e r:</s> <s xml:id="echoid-s38296" xml:space="preserve"> ergo per 29 p 1 erit linea e o perpendicularis ſuper lineam a s:</s> <s xml:id="echoid-s38297" xml:space="preserve"> quoniam li <lb/>nea a s æquidiſtat lineæ t r, ut patet ex præmiſsis.</s> <s xml:id="echoid-s38298" xml:space="preserve"> Ducatur quoque linea b q, quæ producta neceſſa-<lb/>riò concurret cum linea a l per 2 th.</s> <s xml:id="echoid-s38299" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38300" xml:space="preserve"> quia concurrit cum eius æ quidiſtante, ſcilicet linea f q:</s> <s xml:id="echoid-s38301" xml:space="preserve"> <lb/>fit punctus concurſus l:</s> <s xml:id="echoid-s38302" xml:space="preserve"> & ducatur à puncto q linea, quæ eſt communis ſectio ſuperficiei contingen <lb/>tis ſpeculum ſecundum lineam longitudinis g e & ſuperficiei a b l:</s> <s xml:id="echoid-s38303" xml:space="preserve"> quæ fit q p:</s> <s xml:id="echoid-s38304" xml:space="preserve"> quæ per 2 th.</s> <s xml:id="echoid-s38305" xml:space="preserve"> 1 huius <lb/>ſecabit lineã a l:</s> <s xml:id="echoid-s38306" xml:space="preserve"> quia ſecat eius æ quidiſtãtẽ, quę eſt f k:</s> <s xml:id="echoid-s38307" xml:space="preserve"> ſit pũctus ſectionis p:</s> <s xml:id="echoid-s38308" xml:space="preserve"> producaturq́;</s> <s xml:id="echoid-s38309" xml:space="preserve"> linea h e, <lb/>donec concurrat cum linea a s:</s> <s xml:id="echoid-s38310" xml:space="preserve"> concurret autem per 2 th.</s> <s xml:id="echoid-s38311" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38312" xml:space="preserve"> ſit punctus concurſus s:</s> <s xml:id="echoid-s38313" xml:space="preserve"> & ducan-<lb/>tur duę lineęl s & p o.</s> <s xml:id="echoid-s38314" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s38315" xml:space="preserve"> linea r t eſt perpendicularis ſuper axem g t, & linea f k acutum an-<lb/>gulum continet cum axe g t, angulus enim f k g per 32 p 1 eſt acutus:</s> <s xml:id="echoid-s38316" xml:space="preserve"> ideo quia angulus f q g, ut patet <lb/>ex pręmiſsis, eſt rectus:</s> <s xml:id="echoid-s38317" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s38318" xml:space="preserve"> 1 huius lineæ r t & f k concurrent in a liquo puncto ultra axẽ <lb/>g t:</s> <s xml:id="echoid-s38319" xml:space="preserve"> ſed & illarũ ęquidiſtãtes lineę, quæ ſunt a l & a s cõcurrunt in pũcto a:</s> <s xml:id="echoid-s38320" xml:space="preserve"> ſuntq́;</s> <s xml:id="echoid-s38321" xml:space="preserve"> in alia ſuperficie, ꝗ̃ <lb/>lineę r t & f k, quę ſunt in ſuքficie g e k ք 1 p 11:</s> <s xml:id="echoid-s38322" xml:space="preserve"> palá ergo quoniá ſuքficies a l s eſt ę quidiftás ſuքficiel <lb/>g e kք 15 p 11.</s> <s xml:id="echoid-s38323" xml:space="preserve"> Lineę quoq;</s> <s xml:id="echoid-s38324" xml:space="preserve"> q e & p o ſunt in ſuքficie cótingtẽte ſpeculũ in linea lõgitudinis g e, & ſecá <lb/>te illas duas ſuperficies ęquidiſtantes ſuք duas lineas, quę ſunt q e & p o:</s> <s xml:id="echoid-s38325" xml:space="preserve"> igitur linea q e æ quidiftat <lb/>lineę p o per 16 p 11.</s> <s xml:id="echoid-s38326" xml:space="preserve"> Et quia linea h e producta concurrit cum linea a s in puncto s:</s> <s xml:id="echoid-s38327" xml:space="preserve"> erit ergo linea e s <lb/>in ſuperficie h e g per 1 p 11, & in eadem ſuperficie eſt linea b l:</s> <s xml:id="echoid-s38328" xml:space="preserve"> & hæc ſuperficies ſecat prædictas fu-<lb/>perficies ęquidiſtantes, quæ ſunta l g & g e k, in duabus lineis e q & l s:</s> <s xml:id="echoid-s38329" xml:space="preserve"> igitur per 16 p 11 linea e q eſt <lb/>æquidiſtans lineæ l s:</s> <s xml:id="echoid-s38330" xml:space="preserve"> ergo per 30 p 1 linea p o, quæ eſt æquidiſtans lineæ q s, ut ſuprà patuit, erit æ-<lb/>quidiſtans ipſi lineę l s.</s> <s xml:id="echoid-s38331" xml:space="preserve"> Erit ergo per 2 p 6 proportio lineę a o ad lineam o s, ſicut line ę a p ad lineá <lb/>p l:</s> <s xml:id="echoid-s38332" xml:space="preserve"> ſed quoniam per 20 th.</s> <s xml:id="echoid-s38333" xml:space="preserve"> 5 huius angulus h e r eſt æqualis angulo r e a, & angulus h e r æ qualis an-<lb/>gulo e s a per 29 p 1:</s> <s xml:id="echoid-s38334" xml:space="preserve"> quoniam extrinſecus intrinſeco eſt æ qualis:</s> <s xml:id="echoid-s38335" xml:space="preserve"> & angulus e a s æ qualis eſt angulo <lb/>r e a, quia coalternus:</s> <s xml:id="echoid-s38336" xml:space="preserve"> palàm quia angulus e s a eſt ęqualis angulo e a s:</s> <s xml:id="echoid-s38337" xml:space="preserve"> ergo per 6 p 1 erit linea e a <lb/>ęqualis lineę e s, quia linea e o eſt perpendicularis ſuper lineam a s, erũt per 31 th.</s> <s xml:id="echoid-s38338" xml:space="preserve"> 1 huius trigonia e <lb/>o & s e o ſimiles:</s> <s xml:id="echoid-s38339" xml:space="preserve"> ergo per 1 definit.</s> <s xml:id="echoid-s38340" xml:space="preserve"> 6 ipſorum latera æquos angulos reſpicientia ſunt proportio-<lb/>nalia.</s> <s xml:id="echoid-s38341" xml:space="preserve"> Sed expræmiſsis patet quòd latus a e eſt æ quale lateri e s:</s> <s xml:id="echoid-s38342" xml:space="preserve"> ergo & latus a o erit æ quale lateri <lb/>o s:</s> <s xml:id="echoid-s38343" xml:space="preserve"> ergo & linea a p eſt ęqualis ipſi lineę p l:</s> <s xml:id="echoid-s38344" xml:space="preserve"> & linea p q eſt per 29 p 1 perpendicularis ſuper lineam <lb/>a l:</s> <s xml:id="echoid-s38345" xml:space="preserve"> cum ipſa ſit perpendicularis ſuper lineam f k ęquidiſtantem lineę a l.</s> <s xml:id="echoid-s38346" xml:space="preserve"> in trigonis ergo q p a & q p <lb/>l anguli ad p ſunt æquales, quia recti, & latus l p eſt æ quale lateri p a, latusq́;</s> <s xml:id="echoid-s38347" xml:space="preserve"> p q ambobus trigonis <lb/>q p l & q p a eſt commune:</s> <s xml:id="echoid-s38348" xml:space="preserve"> ergo per 4 p 1 erit linea a q ęqualis lineę q l, & angulus q l a æqualis eſt an <lb/>gulo q a l:</s> <s xml:id="echoid-s38349" xml:space="preserve"> ſed angulus q l a ęqualis eſt angulo b q f per 29 p 1, cum ſit ei extrinſecus:</s> <s xml:id="echoid-s38350" xml:space="preserve"> & angulus q a l <lb/>ęqualis eſt angulo a q f, cum ſit ei coalternus:</s> <s xml:id="echoid-s38351" xml:space="preserve"> erit ergo angulus b q f æqualis angulo a q f.</s> <s xml:id="echoid-s38352" xml:space="preserve"> Igitur per <lb/>20 th.</s> <s xml:id="echoid-s38353" xml:space="preserve"> 5 huius forma puncti a reflectitur ad uiſum b à puncto ſpeculi q.</s> <s xml:id="echoid-s38354" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s38355" xml:space="preserve"/> </p> <div xml:id="echoid-div1557" type="float" level="0" n="0"> <figure xlink:label="fig-0590-02" xlink:href="fig-0590-02a"> <variables xml:id="echoid-variables674" xml:space="preserve">g m n b f q k l t s e p o h r a</variables> </figure> </div> </div> <div xml:id="echoid-div1559" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables675" xml:space="preserve">g m q n t e b r a</variables> </figure> <head xml:id="echoid-head1183" xml:space="preserve" style="it">36. Dato ſpeculo pyramidali cõuexo, centró uiſus <lb/> & puncto rei uiſæ exiſtentibus in ſuperficie ſpeculum æquidiſtanter baſi in uertice contingente: poßibile ect inueniri punctum reflexionis. Alhazen 55 n 5.</head> <p> <s xml:id="echoid-s38356" xml:space="preserve">Fiat diſpoſitio ut proximę pręcedẽtis:</s> <s xml:id="echoid-s38357" xml:space="preserve"> ſitq́ue uertex <lb/>ſpeculi pyramidalis punctus g:</s> <s xml:id="echoid-s38358" xml:space="preserve"> in quo ipſum contin-<lb/>gat ſuperficies plana, quę ſit m n g, ęquidiſtans baſi ipſi-<lb/>us:</s> <s xml:id="echoid-s38359" xml:space="preserve"> & ſint centrum uiſus & punctus rei uiſę in ſuperficie <lb/>m n g, ita quòd unum ſit in puncto m, aliud in puncto n.</s> <s xml:id="echoid-s38360" xml:space="preserve"> <lb/>Dico quòd poſsibile eſt punctum reflexionis inueniri.</s> <s xml:id="echoid-s38361" xml:space="preserve"> <lb/>Ducantur enim lineę m g, n g, m n:</s> <s xml:id="echoid-s38362" xml:space="preserve"> & diuidatur angulus <lb/>m g n per ęqualia per lineam q g.</s> <s xml:id="echoid-s38363" xml:space="preserve"> Palàm ergo per 20 th.</s> <s xml:id="echoid-s38364" xml:space="preserve"> <lb/>5 huius quoniam forma puncti n à puncto fpeculi g re-<lb/>flectitur ad uiſum m.</s> <s xml:id="echoid-s38365" xml:space="preserve"> Palàm etiá quòd linea m g & axis <lb/>pyramidis ſpeculi, qui ſit g t, ſunt in ſuperficie ſecante <lb/>pyramidem ſuper lineam longitudinis pyramidis, quæ <lb/>fit g e.</s> <s xml:id="echoid-s38366" xml:space="preserve"> Et à puncto q duncatur perpendicularis ſuper <lb/>hanclineam longitudinis, quę eſt g e, per 12 p 1, quę ſit <lb/>q e:</s> <s xml:id="echoid-s38367" xml:space="preserve"> & ſuper punctum e ducatur ſuperficies æquidiftans <lb/>baſi ſpeculi:</s> <s xml:id="echoid-s38368" xml:space="preserve"> quę ſecabit pyramidem ſecundum circulũ, <lb/>per 100 th.</s> <s xml:id="echoid-s38369" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s38370" xml:space="preserve"> Linea uerò communis ſuperficiei q <lb/>e g & huic circulo ſit linea e t.</s> <s xml:id="echoid-s38371" xml:space="preserve"> Palàm ergo quoniam hęc <lb/>linea cadet ſuper axem ſpeculi in centro circuli, quod <lb/>fit t.</s> <s xml:id="echoid-s38372" xml:space="preserve"> Deinde à puncto m centro uiſus ducatur linea æ-<lb/>quidiſtans lineę longitudinis ſpeculi, quæ eſt e g per <lb/>31 p 1:</s> <s xml:id="echoid-s38373" xml:space="preserve"> quę producta in ſuperficiem illius circuli, ca-<lb/>dat in punctum b:</s> <s xml:id="echoid-s38374" xml:space="preserve"> & ſimiliter à puncto n, qui eſt punctu s rei uiſæ, ducaturlinea ęquidiftás lineę g e, <lb/> <pb o="290" file="0592" n="592" rhead="VITELLONIS OPTICAE"/> quæ producta in dictam ſuperficiem, cadatin punctum a:</s> <s xml:id="echoid-s38375" xml:space="preserve"> & ducatur linea b a in ſuperficie plana ſe-<lb/>cante ſpeculum ſecundum prædictum circulum:</s> <s xml:id="echoid-s38376" xml:space="preserve"> & producatur linea te extra ſpeculum, quæ ſeca-<lb/>bit neceſſariò lineam b a per 29 th.</s> <s xml:id="echoid-s38377" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38378" xml:space="preserve"> cum illæ ambæ lineæ in eadem ſint ſuperficie circuli:</s> <s xml:id="echoid-s38379" xml:space="preserve"> ſe-<lb/>cet ergo ipſam in punctor.</s> <s xml:id="echoid-s38380" xml:space="preserve"> Quia uerò linea m b æ quidiſtat lineę e g:</s> <s xml:id="echoid-s38381" xml:space="preserve"> palàm per 1 th.</s> <s xml:id="echoid-s38382" xml:space="preserve"> 1 huius quia eſt <lb/>cum ipſa in eadem ſuperficie:</s> <s xml:id="echoid-s38383" xml:space="preserve"> quę ſuperficies ſecat ſuperficiem m n g, & ſuperficiem b e a ſuper duas <lb/>lineas m g & b e:</s> <s xml:id="echoid-s38384" xml:space="preserve"> ſuperficies uerò m g n & b e a ſunt æquidiftantes per 24 th.</s> <s xml:id="echoid-s38385" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38386" xml:space="preserve"> quoniam ipſæ <lb/>ambæ æquidiſtant baſi ſpeculi:</s> <s xml:id="echoid-s38387" xml:space="preserve"> ergo per 16 11 linea m g eſt æquidiſtans lineæ b e.</s> <s xml:id="echoid-s38388" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s38389" xml:space="preserve"> <lb/>lineę a n & g e ſunt in ſuperficie ſecante illas æquidiſtantes ſuperficies ſuper lineas n g & e a:</s> <s xml:id="echoid-s38390" xml:space="preserve"> igitur <lb/>per 16 p 11 linea n g æquiſtat lineæ a e.</s> <s xml:id="echoid-s38391" xml:space="preserve"> Similiter ſuperficies q g e ſecat eaſdem ſuperficies ęquidiſtan <lb/>tes ſecundum duas lineas r e & q g:</s> <s xml:id="echoid-s38392" xml:space="preserve"> igitur, ut prius, lineę r e & q g æ quidiftant.</s> <s xml:id="echoid-s38393" xml:space="preserve"> Igitur duæ lineæ q g <lb/>& m g æ quidiſtant duabus lineis b e & r e:</s> <s xml:id="echoid-s38394" xml:space="preserve"> ergo per 10 p 11 angulus m g q eſt æ qualis angulo b e r:</s> <s xml:id="echoid-s38395" xml:space="preserve"> & <lb/>angulus q g n eadẽ ratione eſt ęqualis an gulo r e a.</s> <s xml:id="echoid-s38396" xml:space="preserve"> Ergo per 20 th.</s> <s xml:id="echoid-s38397" xml:space="preserve"> 5 huius forma puncti a poteſt re-<lb/>flecti ad uiſum b à puncto ſpeculi e.</s> <s xml:id="echoid-s38398" xml:space="preserve"> Si ergo à puncto a ducatur linea ę quidiſtãs ductę lineę q e, & a-<lb/>lia ęquidiſtans lineę r e:</s> <s xml:id="echoid-s38399" xml:space="preserve"> & copulentur lineę m e & n e, & producatur linea m e, donec concurrat cũ <lb/>linea ę quidiſtante lineę q e ducta à puncto a, & ducantur lineę communes, ut in proxima pręceden <lb/>te, & iteretur probatio, ut in illa:</s> <s xml:id="echoid-s38400" xml:space="preserve"> patebit quoniam forma punctin poteſt reflecti ad uiſum m à pun-<lb/>cto ſpeculie.</s> <s xml:id="echoid-s38401" xml:space="preserve"> Igitur punctus e erit punctus reflexionis.</s> <s xml:id="echoid-s38402" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s38403" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1560" type="section" level="0" n="0"> <head xml:id="echoid-head1184" xml:space="preserve" style="it">37. Dato ſpeculo pyramidali conuexo, & centro uiſus & puncto rei uiſæ exiſtentibus ultra <lb/>ſuperficiem æquidiſtanter baſi ſpeculum in uertice contingentem: poßibile eſt punctum reflexio-<lb/>nis inueniri. Alhazen 56 n 5.</head> <p> <s xml:id="echoid-s38404" xml:space="preserve">Sit diſpoſitio, quę prius:</s> <s xml:id="echoid-s38405" xml:space="preserve"> & ſit b centrum uiſus:</s> <s xml:id="echoid-s38406" xml:space="preserve"> & a punctus rei uiſæ ultra ſuperficiem m g n ſpecu <lb/>lum in puncto g uertice pyramidis contin gentem.</s> <s xml:id="echoid-s38407" xml:space="preserve"> Dico quòd eſt poſsibile inueniri punctum refle-<lb/>xionis.</s> <s xml:id="echoid-s38408" xml:space="preserve"> Fiat enim pyramis huic oppoſita:</s> <s xml:id="echoid-s38409" xml:space="preserve"> & eſt hoc per 91th.</s> <s xml:id="echoid-s38410" xml:space="preserve"> 1 huius poſsibile, lineis omnibus lon-<lb/>gitudinis pyramidis ſpeculi imaginatis protrahi ultra ipſarum cómunem ſectionem, quę fit in uer-<lb/>tice g:</s> <s xml:id="echoid-s38411" xml:space="preserve"> eritq́ue baſis huius pyramidis ęquidiſtans baſi pyramidis primæ.</s> <s xml:id="echoid-s38412" xml:space="preserve"> Ducatur itaque à puncto a <lb/>(qui eſt punctus rei uiſæ) ſuperficies ſecans hanc ſecundam pyramidem æquidiſtanter baſibus uni <lb/>us & alterius pyramidum.</s> <s xml:id="echoid-s38413" xml:space="preserve"> Et quoniam illæ baſes ad inuicem æquidiſtant:</s> <s xml:id="echoid-s38414" xml:space="preserve"> palàm per 23 & 24 th.</s> <s xml:id="echoid-s38415" xml:space="preserve"> 1 <lb/>huius quoniam illa ſuperficies ęquidiſtat ambabus baſibus pyramidum:</s> <s xml:id="echoid-s38416" xml:space="preserve"> palàm autem per 100 th.</s> <s xml:id="echoid-s38417" xml:space="preserve"> 1 <lb/>huius quoniam illa ſuperficies ſecabit pyramidem illam ſecundam ſecundum circulum, qui ſit y z.</s> <s xml:id="echoid-s38418" xml:space="preserve"> <lb/>Centrum ita que uiſus (quod eſt b) aut erit in hac ſuperficie pyramidem ſecante, autnon.</s> <s xml:id="echoid-s38419" xml:space="preserve"> Si fuerit <lb/>in illa ſuperficie, fiat ductio linearum ab ipſo puncto b, & compleatur demonſtratio ſicut in 35 hu-<lb/>ius, quantùm ad hoc quòd fiat reflexio formę puncti a ad centrum uiſus b ab aliquo puncto ſecun-<lb/>dę pyramidis, quod ſit z:</s> <s xml:id="echoid-s38420" xml:space="preserve"> quo habito, compleatur demonſtratio, ut infrà ſtatim patebit.</s> <s xml:id="echoid-s38421" xml:space="preserve"> Quòd ſi pun <lb/>ctus b (qui eſt centrum uiſus) non fuerit in illa ſuper-<lb/> <anchor type="figure" xlink:label="fig-0592-01a" xlink:href="fig-0592-01"/> ficie:</s> <s xml:id="echoid-s38422" xml:space="preserve"> ducatur à puncto g uertice ipſius ſpeculi ad cen-<lb/>trum uiſus, quod eſt b.</s> <s xml:id="echoid-s38423" xml:space="preserve"> linea g b:</s> <s xml:id="echoid-s38424" xml:space="preserve"> quę producatur uſque-<lb/>quò concurrat cum hac ſuperficie circuli y z:</s> <s xml:id="echoid-s38425" xml:space="preserve"> & ſit con-<lb/>curſus in puncto d.</s> <s xml:id="echoid-s38426" xml:space="preserve"> Palàm itaque quòd forma puncti a <lb/>reflectitur ad uiſum exiſtentẽ in puncto d ab aliquo pun <lb/>cto circuli y z arcus ſui interioris, ut patuit per 32 huius.</s> <s xml:id="echoid-s38427" xml:space="preserve"> <lb/>Sit ergo ille punctus z:</s> <s xml:id="echoid-s38428" xml:space="preserve"> & ducantur lineæ a z, d z, a d:</s> <s xml:id="echoid-s38429" xml:space="preserve"> an-<lb/>gulum quoque a z d diuidat linea p z per æqualia:</s> <s xml:id="echoid-s38430" xml:space="preserve"> ca-<lb/>detq́ue punctus p in linea a d:</s> <s xml:id="echoid-s38431" xml:space="preserve"> & ducatur linea a b:</s> <s xml:id="echoid-s38432" xml:space="preserve"> & à <lb/>puncto z ducatur linea z g per 101th.</s> <s xml:id="echoid-s38433" xml:space="preserve"> 1 huius, quæ ſit li-<lb/>nea lõgitudinis ſecundæ pyramidis.</s> <s xml:id="echoid-s38434" xml:space="preserve"> Palàm quoque per <lb/>91 th.</s> <s xml:id="echoid-s38435" xml:space="preserve"> 1 huius quoniam eadem linea producta trans uer-<lb/>ticem pyramidis ſpeculi, erit linea longitudinis primæ <lb/>pyramidis ipſius ſpeculi:</s> <s xml:id="echoid-s38436" xml:space="preserve"> quæ ſit linea z g e.</s> <s xml:id="echoid-s38437" xml:space="preserve"> Palàm ergo <lb/>quoniam ſuperficies p z e ſecabit lineam a b:</s> <s xml:id="echoid-s38438" xml:space="preserve"> ſecet ergo <lb/>ipſam in puncto q:</s> <s xml:id="echoid-s38439" xml:space="preserve"> & à puncto q per 12 p 1 ducatur linea <lb/>perpendicularis ſuper lineam g e:</s> <s xml:id="echoid-s38440" xml:space="preserve"> & cadat in punctum <lb/>e:</s> <s xml:id="echoid-s38441" xml:space="preserve"> & erit linea q e perpendicularis ſuper ſuperficiem cõ-<lb/>tingentem pyramidem ſecundum lineam g e:</s> <s xml:id="echoid-s38442" xml:space="preserve"> quoniam <lb/>linea q e eſt perpendicularis ſuper curuam ſuperficiem <lb/>pyramidis, ut patet.</s> <s xml:id="echoid-s38443" xml:space="preserve"> Super punctum quoque e fiat per <lb/>102 th.</s> <s xml:id="echoid-s38444" xml:space="preserve"> 1 huius ſuperficies ęquidiſtans baſi, quę ſit f e h, <lb/>& ducatur à puncto b centro uiſus linea æquidiſtans li-<lb/>neę z e longitudinis ſpeculi:</s> <s xml:id="echoid-s38445" xml:space="preserve"> quæ ſit b h, concurrens cũ <lb/>ſuperficie illa f e h in puncto h:</s> <s xml:id="echoid-s38446" xml:space="preserve"> & eidem lineæ z e du-<lb/>catur à puncto a rei uiſę linea ę quidiſtans, quę ſit a f, ſe-<lb/>cans ſuperficiem f e h in puncto, qui eſt ſ.</s> <s xml:id="echoid-s38447" xml:space="preserve"> Palàm itaque per 1 th.</s> <s xml:id="echoid-s38448" xml:space="preserve"> 1 huius, cum linea b h ſit æquidi-<lb/>ſtans lineę z e, quoniam illę lineę ſunt in eadem ſuperficie:</s> <s xml:id="echoid-s38449" xml:space="preserve"> ſed & puncta b & d ſunt in eadem linea:</s> <s xml:id="echoid-s38450" xml:space="preserve"> <lb/> <pb o="291" file="0593" n="593" rhead="LIBER SEPTIMVS."/> quia per 1 p 11 lineę d z & h e ſunt in eadem ſuperficie, quę ſecat ſuperficies illas ęquidiſtãtes, ſcilicet <lb/>y z & f e h ſuք duas lineas d z & h e.</s> <s xml:id="echoid-s38451" xml:space="preserve"> Igitur ք 16 p 11 illæ duę lineę d z & h e ſunt ęquidiſtãtes.</s> <s xml:id="echoid-s38452" xml:space="preserve"> Et ſimi <lb/>liter quoniam ſuperficies ducta per punctum a ſecat pyramidem ſecundam ęquidiſtãter ambabus <lb/>baſibus præmiſſarum pyramidum, ſpeculi ſcilicet, & pyramidis imaginatę ſecundum circulum y z, <lb/>& ſuperficies ducta per lineam f e, quæ eſt ſuperficies f e h, ſecat pyramidem ſpeculi ſecundum cir-<lb/>culum æquidiſtantem baſi ſpeculi:</s> <s xml:id="echoid-s38453" xml:space="preserve"> patet quòd ſuperficies, in qua ſunt lineę a z & f e, ſunt ęquidiſtan <lb/>tes per 24 th.</s> <s xml:id="echoid-s38454" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38455" xml:space="preserve"> lineę ergo a z & f e ſunt ęquidiſtantes.</s> <s xml:id="echoid-s38456" xml:space="preserve"> Patet ergo quòd duę lineę d z & a z æ-<lb/>quidiſtant duabus lineis h e & f e:</s> <s xml:id="echoid-s38457" xml:space="preserve"> ergo per 10 p 11 angulus d z a eſt ęqualis angulo h e ſ.</s> <s xml:id="echoid-s38458" xml:space="preserve"> Copuletur <lb/>quoque linea h f.</s> <s xml:id="echoid-s38459" xml:space="preserve"> Et quoniam linea p z eſt diuidens per ęqualia angulum d z a:</s> <s xml:id="echoid-s38460" xml:space="preserve"> erit ipſa per 26 th.</s> <s xml:id="echoid-s38461" xml:space="preserve"> 5 <lb/>huius perpendicularis ſuper lineam, circulum y z contingentem in pũcto z:</s> <s xml:id="echoid-s38462" xml:space="preserve"> ergo per 19 p 3 linea p z <lb/>producta tranſibit centrum circuli y z.</s> <s xml:id="echoid-s38463" xml:space="preserve"> Superficies ergo p z e ſecat ſpeculum trans axem:</s> <s xml:id="echoid-s38464" xml:space="preserve"> ſecat ergo <lb/>circulum ductum per punctum e tranſeuntem.</s> <s xml:id="echoid-s38465" xml:space="preserve"> Sit ergo communis ſectio ſuperficiei p z e & illius <lb/>circuli linea r e.</s> <s xml:id="echoid-s38466" xml:space="preserve"> Sicut ergo linea s p z tranſit centrum circuli y z:</s> <s xml:id="echoid-s38467" xml:space="preserve"> ſimiliter linea r e diuidens angulũ <lb/>h e f tranſibit centrum alterius circuli, ſuper quem ſuperficies f e h ſecat pyramidem ſpeculi ęqui-<lb/>diſtanter baſi.</s> <s xml:id="echoid-s38468" xml:space="preserve"> Et quia ſuperficies, in qua ſunt duę lineę p z & e r, ſecat illas duas ſuperficies ęquidi-<lb/>ſtantes ſuper duas lineas p z & r e:</s> <s xml:id="echoid-s38469" xml:space="preserve"> igitur per 16 p 11 lineę p z & r e ſunt æquidiſtantes.</s> <s xml:id="echoid-s38470" xml:space="preserve"> Duę ergo li-<lb/>neę a z & z p ſunt ęquidiſtantes duabus lineis f e & e r:</s> <s xml:id="echoid-s38471" xml:space="preserve"> ergo per 10 p 11 angulus a z p ęqualis eſt an-<lb/>gulo f e r.</s> <s xml:id="echoid-s38472" xml:space="preserve"> Similiter & angulus d z p eſt ęqualis angulo r e i:</s> <s xml:id="echoid-s38473" xml:space="preserve"> quoniam ſicut totus angulus d z a eſt ę-<lb/>qualis toti h e f, ſic medietas medietati:</s> <s xml:id="echoid-s38474" xml:space="preserve"> ergo angulus f e r ęqualis eſt angulo h e r.</s> <s xml:id="echoid-s38475" xml:space="preserve"> Patet ergo per <lb/>20 th.</s> <s xml:id="echoid-s38476" xml:space="preserve"> 5 huius quoniam forma puncti ſ reflectitur ad uiſum exiſtentem in puncto h à puncto ſpeculi <lb/>e.</s> <s xml:id="echoid-s38477" xml:space="preserve"> Ergo ſi à puncto f protrahatur linea ęquidiſtans lineę q e, & alia linea ęquidiſtans lineę r e, & lineę <lb/>aliæ communes, ut in 35 huius, reiterata demonſtratione illius:</s> <s xml:id="echoid-s38478" xml:space="preserve"> patebit quoniam forma puncti a re-<lb/>flectitur ad uiſum b à puncto ſpeculi e.</s> <s xml:id="echoid-s38479" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s38480" xml:space="preserve"> Quòd ſi à puncto q non poſsit duci li <lb/>nea perpendicularis ſuper lineam g e, nulla ſiet reflexio formæ puncti a ad uiſum b in tali diſpoſitio <lb/>ne conſtitutum:</s> <s xml:id="echoid-s38481" xml:space="preserve"> aliàs autem ſemper fiet reflexio, ut præ oſtenſum eſt:</s> <s xml:id="echoid-s38482" xml:space="preserve"> & patet per 14 huius, & per <lb/>90 th.</s> <s xml:id="echoid-s38483" xml:space="preserve"> 4 huius.</s> <s xml:id="echoid-s38484" xml:space="preserve"/> </p> <div xml:id="echoid-div1560" type="float" level="0" n="0"> <figure xlink:label="fig-0592-01" xlink:href="fig-0592-01a"> <variables xml:id="echoid-variables676" xml:space="preserve">z y v p d q b m n g t e f r h</variables> </figure> </div> </div> <div xml:id="echoid-div1562" type="section" level="0" n="0"> <head xml:id="echoid-head1185" xml:space="preserve" style="it">38. Dato ſpeculo pyramidali conuexo, punctó rei uiſæ exiſtente ſub ſuperficie ſpeculũ æqui-<lb/>diſtanter baſi in uertice contingente, & centro uiſus in eadem ſuperficie: poßibile est punctum <lb/>reflexionis inueniri. Alhazen 57 n 5.</head> <p> <s xml:id="echoid-s38485" xml:space="preserve">Permaneat prior diſpoſitio pręmiſſarum:</s> <s xml:id="echoid-s38486" xml:space="preserve"> & ſit a punctus rei uiſę, qui ſit ſub ſuperficie m n g con-<lb/>tingente pyramidem ſpeculi in uertice g ęquidiſtanter baſi:</s> <s xml:id="echoid-s38487" xml:space="preserve"> & ſit centrum uiſus in illa ſuperficie.</s> <s xml:id="echoid-s38488" xml:space="preserve"> Di <lb/>co quòd adhuc poſsibile eſt inueniri punctum reflexionis.</s> <s xml:id="echoid-s38489" xml:space="preserve"> Sit enim centrum uiſus in puncto m ſu-<lb/>perficiei m g n, quę poſita eſt ſuperficies contingens ſpeculũ in puncto uerticis g ęquidiſtanter baſi <lb/>ſpeculi:</s> <s xml:id="echoid-s38490" xml:space="preserve"> & à puncto a rei uiſę ducatur ſuperficies ęquidiſtans baſi pyramidis:</s> <s xml:id="echoid-s38491" xml:space="preserve"> quę per 100 th.</s> <s xml:id="echoid-s38492" xml:space="preserve"> 1 huius <lb/>ſecabit pyramidem ſuper circulum, qui ſit d e k:</s> <s xml:id="echoid-s38493" xml:space="preserve"> cuius centrum ſit punctum t, & ducatur axis ſpecu <lb/>li, qui ſit g t:</s> <s xml:id="echoid-s38494" xml:space="preserve"> & à puncto m centro uiſus ducatur ad a punctum rei uiſę linea m a:</s> <s xml:id="echoid-s38495" xml:space="preserve"> & linea perpendicu <lb/>laris ſuper dictam ſuperficiem circuli, quę ſit m h:</s> <s xml:id="echoid-s38496" xml:space="preserve"> & à puncto had cẽ <lb/>trum circuli ducatur linea h t:</s> <s xml:id="echoid-s38497" xml:space="preserve"> & à puncto rei uiſę (quι eſt a) ducatur <lb/> <anchor type="figure" xlink:label="fig-0593-01a" xlink:href="fig-0593-01"/> ad lineam h t linea a e q intra circulum, ſecãs peripheriam circuli in <lb/>puncto e, & producta taliter ut pars ductę lineę intra circulũ, quę eſt <lb/>e q, ſit æqualis lineę q t ſcilicet parti diametri interiacenti punctum <lb/>ſectionis & centrum:</s> <s xml:id="echoid-s38498" xml:space="preserve"> quod poteſt fieri per 136 th.</s> <s xml:id="echoid-s38499" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38500" xml:space="preserve"> & ducatur <lb/>linea t e i:</s> <s xml:id="echoid-s38501" xml:space="preserve"> & à puncto h ducatur in eadem ſuperficie ſpeculum ſecan <lb/>te ſecundũ circulũ d e k, linea ęquidiſtans & æqualis lineę t e, quæ ſit <lb/>h b:</s> <s xml:id="echoid-s38502" xml:space="preserve"> & ducãtur lineę m b, & b e, & g e:</s> <s xml:id="echoid-s38503" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s38504" xml:space="preserve"> g e linea lõgitudinis ſpe-<lb/>culi.</s> <s xml:id="echoid-s38505" xml:space="preserve"> Palàm quoniã ſuperficies g t e ſecans ſpeculum trans axẽ, ſecat <lb/>etiam lineam a m:</s> <s xml:id="echoid-s38506" xml:space="preserve"> ſit ergo punctus ſectionis f:</s> <s xml:id="echoid-s38507" xml:space="preserve"> & ducatur à puncto f <lb/>perpendicularis ſuper lineá longitudinis ſpeculi, quæ eſt g e, cadens <lb/>in punctum o:</s> <s xml:id="echoid-s38508" xml:space="preserve"> & producatur ad axem g t:</s> <s xml:id="echoid-s38509" xml:space="preserve"> & ſit f o p ſecans axem g t <lb/>in puncto p:</s> <s xml:id="echoid-s38510" xml:space="preserve"> & ducantur lineæ m o & a o.</s> <s xml:id="echoid-s38511" xml:space="preserve"> Dico quoniam punctus o, <lb/>(qui eſt punctus ſuperficiei ſpeculi:</s> <s xml:id="echoid-s38512" xml:space="preserve"> cum ſit in linea ſuę longitudinis, <lb/>quæ eſt g e) eſt punctus reflexionis formæ punctia, ad centrum ui-<lb/>ſus punctum m.</s> <s xml:id="echoid-s38513" xml:space="preserve"> Palàm enim ex pręmiſsis, quoniam linea h b eſt æ-<lb/>qualis & æquidiſtans lineę t e:</s> <s xml:id="echoid-s38514" xml:space="preserve"> igitur per 33 p 1 erit linea h t ęqualis & <lb/>ęquidiſtans lineę b e:</s> <s xml:id="echoid-s38515" xml:space="preserve"> ſed linea m h eſt ęqualis & ęquidiſtans axi g t <lb/>per 25 th.</s> <s xml:id="echoid-s38516" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38517" xml:space="preserve"> eò quòd ipſę ſunt lineæ ęquidiſtantes inter ſuper-<lb/>ficies ęquidiſtantes productæ:</s> <s xml:id="echoid-s38518" xml:space="preserve"> ergo per 33 p 1 linea h t æquidiſtat lineę m g:</s> <s xml:id="echoid-s38519" xml:space="preserve"> ergo per 30 p 1 linea m g <lb/>ęquidiſtat lineę b e:</s> <s xml:id="echoid-s38520" xml:space="preserve"> & eſt ęqualis illi.</s> <s xml:id="echoid-s38521" xml:space="preserve"> Palàm etiam quòd angulus q t e eſt ęqualis angulo q e t per 5 <lb/>p 1:</s> <s xml:id="echoid-s38522" xml:space="preserve"> ideo quia lineæ e q & q t, ut patet ex præmiſsis, ſunt æquales:</s> <s xml:id="echoid-s38523" xml:space="preserve"> ſed angulus q e t æqualis eſt angu <lb/>lo a e i per 15 p 1:</s> <s xml:id="echoid-s38524" xml:space="preserve"> angulus ergo q t e eſt æqualis angulo a e i:</s> <s xml:id="echoid-s38525" xml:space="preserve"> ſed angulus q t e per 29 p 1 eſt ęqualis an-<lb/>gulo i e b:</s> <s xml:id="echoid-s38526" xml:space="preserve"> propter hoc quòd lineę e b & t h æquidiſtant:</s> <s xml:id="echoid-s38527" xml:space="preserve"> ergo angulus j e b eſt ęqualis angulo i e a.</s> <s xml:id="echoid-s38528" xml:space="preserve"> <lb/>Patet ergo per 20 th.</s> <s xml:id="echoid-s38529" xml:space="preserve"> 5 huius quoniam forma puncti a reflectitur ad uiſum exiſtentem in puncto b <lb/>à puncto ſpeculi e.</s> <s xml:id="echoid-s38530" xml:space="preserve"> Et cum linea b m ęquidiſtans ſit lineę g e, ſi à puncto a ducatur linea ęquidiſtans <lb/> <pb o="292" file="0594" n="594" rhead="VITELLONIS OPTICAE"/> lineæ f o p, & linea æquidiſtans lineę it:</s> <s xml:id="echoid-s38531" xml:space="preserve"> & iteretur figura ſuperà dicta 35 huius, & probatio eiuſdem:</s> <s xml:id="echoid-s38532" xml:space="preserve"> <lb/>palàm quia ſorma punctι a reflectetur ad cẽtrum uiſus exiſtens in puncto m à puncto ſpeculi o.</s> <s xml:id="echoid-s38533" xml:space="preserve"> Qđ <lb/>eſt propoſitum:</s> <s xml:id="echoid-s38534" xml:space="preserve"> nec refert quemadmodum demonſtraui hoc in ſequenti proxιma:</s> <s xml:id="echoid-s38535" xml:space="preserve"> ſiue punctum rei <lb/>uiſæ, ſiue centrum uiſus ſιt in ſuperficie m g n:</s> <s xml:id="echoid-s38536" xml:space="preserve"> quoniam idem eſt modus & ratio reflexionis hinc <lb/>& inde.</s> <s xml:id="echoid-s38537" xml:space="preserve"/> </p> <div xml:id="echoid-div1562" type="float" level="0" n="0"> <figure xlink:label="fig-0593-01" xlink:href="fig-0593-01a"> <variables xml:id="echoid-variables677" xml:space="preserve">m n g f p i b a h e q t d k</variables> </figure> </div> </div> <div xml:id="echoid-div1564" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables678" xml:space="preserve">y z m q p a n y t e f r h</variables> </figure> <head xml:id="echoid-head1186" xml:space="preserve" style="it">39. Dato ſpeculo pyramidali conuexo, punctó rei uiſæ exiſten <lb/> te ultra ſuperficiem ſpeculum æquidιſt anter baſι ιn uertice contin gentem, & centro uιſus in eadem ſuperficie: poßibile est punctum reflexionis ιnueniri. Alhazen 58 n 5.</head> <p> <s xml:id="echoid-s38538" xml:space="preserve">Remanente diſpoſitione figuræ pręcedentis:</s> <s xml:id="echoid-s38539" xml:space="preserve"> ſit centrum uiſus in <lb/>puncto m ſuperficιei m g n:</s> <s xml:id="echoid-s38540" xml:space="preserve"> & ſit a punctus rei uiſę ultra illam ſuper-<lb/>ficiem:</s> <s xml:id="echoid-s38541" xml:space="preserve"> fiatq́ue pyramiś alia huic oppoſita:</s> <s xml:id="echoid-s38542" xml:space="preserve"> & fiat ſuper punctum a <lb/>ſuperficies ęquidiſtans baſi huius pyramidis:</s> <s xml:id="echoid-s38543" xml:space="preserve"> & per proximam prę-<lb/>cedentem inueniatur in circulo huius ſuperficiei punctus reflexιo-<lb/>nis ex punctis interioribus:</s> <s xml:id="echoid-s38544" xml:space="preserve"> & ducatur à puncto ιllo lιnea ad pun-<lb/>ctum g:</s> <s xml:id="echoid-s38545" xml:space="preserve"> & producatur taliter in ſuperficie ipſius, ut ipſa fiat linea lõ-<lb/>gitudιnis pyramidis ipſius ſpeculι:</s> <s xml:id="echoid-s38546" xml:space="preserve"> inuenιeturq́;</s> <s xml:id="echoid-s38547" xml:space="preserve"> punctus reflexio-<lb/>nis ſecundum ea, quę præmiſimus in 37 huius:</s> <s xml:id="echoid-s38548" xml:space="preserve"> eiusq́;</s> <s xml:id="echoid-s38549" xml:space="preserve"> ιdem proban-<lb/>di modus penitus, quι prius in eadem 37.</s> <s xml:id="echoid-s38550" xml:space="preserve"> Et hoc eſt prop oſitum.</s> <s xml:id="echoid-s38551" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1565" type="section" level="0" n="0"> <head xml:id="echoid-head1187" xml:space="preserve" style="it">40. Dato ſpeculo pyramidali conuexo, punctó rei uiſæ exiſten <lb/>te ſub ſuperficie pyramidem æquidiſt anter baſi in uertice contin-<lb/>gente, & centro uiſus ſuper eandem, uel econuerſo: poßibile est <lb/>punctum reflexionis inueniri. Alhazen 59 n 5.</head> <p> <s xml:id="echoid-s38552" xml:space="preserve">Diſpoſitione priori remanente:</s> <s xml:id="echoid-s38553" xml:space="preserve"> ſit punctus a rei uiſæ ſub ſuperficie m n g:</s> <s xml:id="echoid-s38554" xml:space="preserve"> & punctus b centrum <lb/>uiſus ultra eandem ſuperficiem ſpeculum in uertice g contingentem:</s> <s xml:id="echoid-s38555" xml:space="preserve"> uel econuerſo a punctus rei <lb/>uiſæ ſit ultra ſuperficιem m n g, & b centrum uiſus ſub ſuperficie m n g.</s> <s xml:id="echoid-s38556" xml:space="preserve"> Dico quòd adhuc poſsibile <lb/>eſt punctum reflexionis inueniri.</s> <s xml:id="echoid-s38557" xml:space="preserve"> Sit enim, exempli gratia, punctum a ſub ſuperficie m n g, & b ul-<lb/>tra ιllam:</s> <s xml:id="echoid-s38558" xml:space="preserve"> ducaturq́ue à puncto a ſuperficies ęquidiſtãs <lb/>baſi ſpeculi ſecans per 100 th.</s> <s xml:id="echoid-s38559" xml:space="preserve"> 1 huius pyramidem ſpecu <lb/> <anchor type="figure" xlink:label="fig-0594-02a" xlink:href="fig-0594-02"/> li ſuper circulum, qui ſit d e:</s> <s xml:id="echoid-s38560" xml:space="preserve"> cuius centrum ſit t:</s> <s xml:id="echoid-s38561" xml:space="preserve"> & duca <lb/>tur axis ſpeculi, qui ſit g t:</s> <s xml:id="echoid-s38562" xml:space="preserve"> & ducatur linea b g à puncto <lb/>ulteriori, in quo eſt centrum uiſus, ad uerticem pyrami <lb/>dis:</s> <s xml:id="echoid-s38563" xml:space="preserve"> quæ producta concurret neceſſariò cum ſuperficie <lb/>a e d:</s> <s xml:id="echoid-s38564" xml:space="preserve"> quoniam concurrit cum axe ſuper ipſam erecto.</s> <s xml:id="echoid-s38565" xml:space="preserve"> <lb/>Sit concurſus punctus k:</s> <s xml:id="echoid-s38566" xml:space="preserve"> & in circulo d e inueniatur <lb/>per 135 th.</s> <s xml:id="echoid-s38567" xml:space="preserve"> 1 huius pũctus, qui ſit e, ita ut linea circulum <lb/>contingens à puncto e ducta, quę ſit e s, dιuidat per æ-<lb/>qualia angulum, quem continent ductæ lineę k e & a e:</s> <s xml:id="echoid-s38568" xml:space="preserve"> <lb/>copulenturq́;</s> <s xml:id="echoid-s38569" xml:space="preserve"> lineę longitudinis, quæ ſint g e & g d:</s> <s xml:id="echoid-s38570" xml:space="preserve"> & <lb/>à puncto b ducatur linea æquidiſtans lineę g e:</s> <s xml:id="echoid-s38571" xml:space="preserve"> quę ne-<lb/>ceſſariò concurret cum linea k e concurrente cum eius <lb/>æquidiſtante quæ eſt g e, per 2 th.</s> <s xml:id="echoid-s38572" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38573" xml:space="preserve"> ſit concurſus <lb/>in puncto h.</s> <s xml:id="echoid-s38574" xml:space="preserve"> Palàm itaque per 1 p 11 quia punctus b eſt <lb/>in ſuperficie g e k:</s> <s xml:id="echoid-s38575" xml:space="preserve"> quoniam eſt in linea k g b, quę ducta <lb/>eſt in illa ſuperficιe, & linea b h eſt in eadem ſuperficie <lb/>per 1 th.</s> <s xml:id="echoid-s38576" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38577" xml:space="preserve"> quoniam ipſa linea b h eſt ęquidiſtans <lb/>lineę g e:</s> <s xml:id="echoid-s38578" xml:space="preserve"> & ducatur linea t e i à centro circuli t per pun-<lb/>ctum contactus e.</s> <s xml:id="echoid-s38579" xml:space="preserve"> Palàm itaque quoniam ſuperficies g <lb/>t e ſecans ſpeculum trans axem g t, ſecat etiam lineam <lb/>b a.</s> <s xml:id="echoid-s38580" xml:space="preserve"> Secet ergo ipſam in puncto u:</s> <s xml:id="echoid-s38581" xml:space="preserve"> & à puncto u duca-<lb/>tur perpẽdicularis ſuper ſuperficiẽ contingentẽ ſpecu-<lb/>lũ ſecundum lineã longitudinis ſpeculi, quę eſt g e:</s> <s xml:id="echoid-s38582" xml:space="preserve"> hęc <lb/>enim ſuperficies continget circulum d e in puncto e, <lb/>quæ linea ſit o p u, ſecans ſuperficiem ſpeculi in pun-<lb/>cto o, & axem g t in pũcto p:</s> <s xml:id="echoid-s38583" xml:space="preserve"> & ducãtur lineæ a o & b o.</s> <s xml:id="echoid-s38584" xml:space="preserve"> <lb/>Cum itaque, ut patet ex præmiſsis, angulus a e s ſit ęqualis angulo s e k, & cum angulus i e s ſit re-<lb/>ctus per 18 p 3, & angulus s e t rectus:</s> <s xml:id="echoid-s38585" xml:space="preserve"> palàm quòd angulus i e a eſt æqualis angulo t e k:</s> <s xml:id="echoid-s38586" xml:space="preserve"> ſed & angu-<lb/>lus t e k ęqualis eſt angulo i e h ք 15 p 1:</s> <s xml:id="echoid-s38587" xml:space="preserve"> ergo angulus a e i eſt ęqualis angulo i e h.</s> <s xml:id="echoid-s38588" xml:space="preserve"> Poteſt ergo forma <lb/>pũcti a reflecti ad uiſum exiſtentẽ in pũcto h à puncto ſpeculi, qđ eſt e, ք 20 th.</s> <s xml:id="echoid-s38589" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s38590" xml:space="preserve"> Si ergo à pun <lb/>cto a ducatur linea æquidiſtans lineæ u p, & linea æquidiſtãs lineæ i t.</s> <s xml:id="echoid-s38591" xml:space="preserve"> & iteretur probatio 35 huius:</s> <s xml:id="echoid-s38592" xml:space="preserve"> <lb/> <pb o="293" file="0595" n="595" rhead="LIBER SEPTIMVS."/> palàm quoniam forma puncti a reflectetur à puncto ſpeculi, quod eſto, punctum lineę g e, ad uiſum <lb/>exiſtentem in puncto b.</s> <s xml:id="echoid-s38593" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s38594" xml:space="preserve"> Et quoniam ſemper eſt eodem modeo demonſtran-<lb/>dum quodcunque punctorum a uel b fuerit ex quacũ que altera parte ſuperficiei m n g:</s> <s xml:id="echoid-s38595" xml:space="preserve"> patet illud, <lb/>quod proponebatur.</s> <s xml:id="echoid-s38596" xml:space="preserve"> Et imaginandum eſt ita, quòd in figura ſolida punctum b cadat in lineam e g, <lb/>quod in plano non potuimus taliter figurare.</s> <s xml:id="echoid-s38597" xml:space="preserve"> Palàm itaque ex præmiſsis ſex theorematibus, cũ nõ <lb/>ſit poſsibile alio modo ſe habere punctum rei uiſæ ſecundum ſitum reflexibilitatis à ſpeculis pyra-<lb/>midalibus conuexis ad centra uiſus, niſi modis propoſitis:</s> <s xml:id="echoid-s38598" xml:space="preserve"> quoniam aut ambo erunt ſub ſuperficie <lb/>m n g:</s> <s xml:id="echoid-s38599" xml:space="preserve"> aut ambo ultra illam:</s> <s xml:id="echoid-s38600" xml:space="preserve"> aut ambo in illa:</s> <s xml:id="echoid-s38601" xml:space="preserve"> aut unum in illa, aliud ſub illa uel ultra illam:</s> <s xml:id="echoid-s38602" xml:space="preserve"> aut unum <lb/>ſub illa, aliud ultra illam:</s> <s xml:id="echoid-s38603" xml:space="preserve"> & omnibus his modis reflexionis punctum eſt inuentum.</s> <s xml:id="echoid-s38604" xml:space="preserve"> Vniuer ſaliter er <lb/>go in tota ſuperficie ſpeculi pyramidalis conuexi quocunq;</s> <s xml:id="echoid-s38605" xml:space="preserve"> modo ſe habente rei uiſibilis pũcto ad <lb/>centrum uiſus, punctum reflexionis eſt poſsibile inueniri:</s> <s xml:id="echoid-s38606" xml:space="preserve"> quod principaliter quærebatur.</s> <s xml:id="echoid-s38607" xml:space="preserve"/> </p> <div xml:id="echoid-div1565" type="float" level="0" n="0"> <figure xlink:label="fig-0594-02" xlink:href="fig-0594-02a"> <variables xml:id="echoid-variables679" xml:space="preserve">a s t d k e h i o p u g m n b</variables> </figure> </div> </div> <div xml:id="echoid-div1567" type="section" level="0" n="0"> <head xml:id="echoid-head1188" xml:space="preserve" style="it">41. Speculo pyramidali conuexo ſuper ipſius baſim erecto: poßibile eſt rectam lineam rei uiſæ <lb/>& centrum uiſus ſic ſiſti, ut ab una linea longitudinis ſpeculi fiat formarum omnium puncto-<lb/>rum illius lineæ reflexio ad uiſum. Alhazen 31 n 6.</head> <p> <s xml:id="echoid-s38608" xml:space="preserve">Sit ſpeculum pyramidale conuexum, cuius uertex ſit a:</s> <s xml:id="echoid-s38609" xml:space="preserve"> axis uerò a h:</s> <s xml:id="echoid-s38610" xml:space="preserve"> linea longitudinis a z:</s> <s xml:id="echoid-s38611" xml:space="preserve"> & à <lb/>puncto z ducatur linea perpendicularis ſuper ſuperficiem contingentem ſpeculum in linea longi-<lb/>tudinis:</s> <s xml:id="echoid-s38612" xml:space="preserve"> quæ producta neceſſariò concurret cum axe a h per 96 th.</s> <s xml:id="echoid-s38613" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38614" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s38615" xml:space="preserve"> linea h z t ſecans axẽ <lb/>a h in puncto h:</s> <s xml:id="echoid-s38616" xml:space="preserve"> & eius punctus t ſit extra ſuperficiem ſpeculi:</s> <s xml:id="echoid-s38617" xml:space="preserve"> & erit angulus a z h rectus:</s> <s xml:id="echoid-s38618" xml:space="preserve"> ergo per <lb/>32 p 1 angulus a h z eſt acutus.</s> <s xml:id="echoid-s38619" xml:space="preserve"> Ducatur quo que à puncto a uertice ſpeculi linea extra pyramidem <lb/>ultra ſuperficiem contingentem pyramidem in linea a z, continens angulum acutum cum ſpeculi <lb/>axe, qui eſt a h, & cũ linea lõgitudinis a z, quæ ſit a n:</s> <s xml:id="echoid-s38620" xml:space="preserve"> lineę quoq;</s> <s xml:id="echoid-s38621" xml:space="preserve"> a h, a z, a n non ſint in eadem ſuperfi <lb/>cie, ſed in diuerſis:</s> <s xml:id="echoid-s38622" xml:space="preserve"> & in ſuperficie h a n à puncto h ducatur linea, cum axe continens angulum æqua <lb/>lem angulo a h z:</s> <s xml:id="echoid-s38623" xml:space="preserve"> quæ linea concurret cum linea a n per 14 th.</s> <s xml:id="echoid-s38624" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38625" xml:space="preserve"> cum anguli h a n & a h z ſint a-<lb/>cuti, ut patet ex præmiſsis:</s> <s xml:id="echoid-s38626" xml:space="preserve"> concurrant ergo in puncto o:</s> <s xml:id="echoid-s38627" xml:space="preserve"> & ſit linea h o:</s> <s xml:id="echoid-s38628" xml:space="preserve"> & facto ſuper punctum z cir <lb/>culo æquidiſtante baſit per 102 th.</s> <s xml:id="echoid-s38629" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38630" xml:space="preserve"> palàm quoniam linea h o tranſibit ſuperficiem illius circu <lb/>li:</s> <s xml:id="echoid-s38631" xml:space="preserve"> ſicut etiam linea h z t tranſit per ſuperficiem eiuſdem circuli.</s> <s xml:id="echoid-s38632" xml:space="preserve"> Fit enim punctus h polus illius cir-<lb/>culi:</s> <s xml:id="echoid-s38633" xml:space="preserve"> ideo quòd ſemidiameter illius circuli cum axe a h continet angulum rectum, & anguli a h z & a <lb/>h o ſunt acuti, ut patet ex præmiſsis.</s> <s xml:id="echoid-s38634" xml:space="preserve"> Secet itaq;</s> <s xml:id="echoid-s38635" xml:space="preserve"> linea h z t ſuperficiem illius circuli in puncto z:</s> <s xml:id="echoid-s38636" xml:space="preserve"> & li-<lb/>nea h o in puncto u:</s> <s xml:id="echoid-s38637" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s38638" xml:space="preserve"> linea longitudinis ſpeculi, quæ ſit a u s:</s> <s xml:id="echoid-s38639" xml:space="preserve"> ducatur quoq;</s> <s xml:id="echoid-s38640" xml:space="preserve"> linea o z:</s> <s xml:id="echoid-s38641" xml:space="preserve"> quæ <lb/>producatur uſque ad punctum f.</s> <s xml:id="echoid-s38642" xml:space="preserve"> Et quoniam linea o z eſt ultra ſuperficiem contingentem pyrami-<lb/>dem in linea a z, cum linea h z ſit <lb/>perpendicularis ſuper illam ſu-<lb/> <anchor type="figure" xlink:label="fig-0595-01a" xlink:href="fig-0595-01"/> perficiem:</s> <s xml:id="echoid-s38643" xml:space="preserve"> palàm quia angulus o <lb/>z h erit maior recto, cum angulus <lb/>a z h ſit rectus:</s> <s xml:id="echoid-s38644" xml:space="preserve"> igitur per 13 p 1 an-<lb/>gulus f z h eſt minor recto.</s> <s xml:id="echoid-s38645" xml:space="preserve"> A <lb/>puncto ergo z ducatur linea con-<lb/>tingens circulum per 17 p 3, quæ <lb/>ſit z m:</s> <s xml:id="echoid-s38646" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s38647" xml:space="preserve"> linea z m in ſuper-<lb/>ficie contingente ſpeculum ſecũ-<lb/>dum lineam longitudinis, quę eſt <lb/>a z:</s> <s xml:id="echoid-s38648" xml:space="preserve"> eſt ergo linea h z perpendicu <lb/>laris ſuper lineam m z:</s> <s xml:id="echoid-s38649" xml:space="preserve"> & à pun-<lb/>cto f ducatur linea perpendιcula <lb/>ris ſuper lineam a z per 12 p 1, quę <lb/>ſit linea f e, concurrens cum linea <lb/>a z producta in puncto e:</s> <s xml:id="echoid-s38650" xml:space="preserve"> quę li-<lb/>nea f e producta concurret cum <lb/>linea a n per 14 th.</s> <s xml:id="echoid-s38651" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38652" xml:space="preserve"> quia <lb/>cum angulus a e f ſit rectus, angulus e a n eſt acutus:</s> <s xml:id="echoid-s38653" xml:space="preserve"> concurrant ergo in puncto n:</s> <s xml:id="echoid-s38654" xml:space="preserve"> & à puncto e du-<lb/>catur linea ęquidiſtans lineæ t h:</s> <s xml:id="echoid-s38655" xml:space="preserve"> quæ ſit e q, per 31 p 1:</s> <s xml:id="echoid-s38656" xml:space="preserve"> item q́ue ab eodem puncto e ducatur linea æ-<lb/>quidiſtans lineę m z, quæ ſit e l.</s> <s xml:id="echoid-s38657" xml:space="preserve"> Palàm autem quod linea m z eſt perpendicularis ſuper lineam a e <lb/>per 22 th.</s> <s xml:id="echoid-s38658" xml:space="preserve"> 1 huius, quoniam ipſa eſt perpẽdicularis ſuper lineam t h, ut ſuper diametrum circuli, quẽ <lb/>ipſa eſt contingens in puncto z.</s> <s xml:id="echoid-s38659" xml:space="preserve"> Igitur linea l e, cum ipſa ſit æquidiſtans lineæ m z, eſt per 29 p 1 per-<lb/>pendicularis ſuper lineam a e.</s> <s xml:id="echoid-s38660" xml:space="preserve"> Sunt quoque lineæ m z & l e in eadem ſuperficie per 1 th.</s> <s xml:id="echoid-s38661" xml:space="preserve"> 1 huius, cum <lb/>ipſæ ſint æquidiſtantes:</s> <s xml:id="echoid-s38662" xml:space="preserve"> producaturq́;</s> <s xml:id="echoid-s38663" xml:space="preserve"> linea q e ultra punctum e:</s> <s xml:id="echoid-s38664" xml:space="preserve"> & hæc per 2 th.</s> <s xml:id="echoid-s38665" xml:space="preserve"> 1 huius ſecabit axẽ <lb/>a h, cum ipſa ſit in eadem ſuperficie cum linea h t per 1 th.</s> <s xml:id="echoid-s38666" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38667" xml:space="preserve"> ſecet ergo axem in puncto d:</s> <s xml:id="echoid-s38668" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s38669" xml:space="preserve"> <lb/>angulus h d q acutus æqualis angulo a h t per 29 p 1.</s> <s xml:id="echoid-s38670" xml:space="preserve"> Fiat quoque ſuperficies l e d q ſecans pyrami-<lb/>dem:</s> <s xml:id="echoid-s38671" xml:space="preserve"> erit ergo illius ſuperficiei & ſuperficiei pyramidis communis ſectio oxygonia per 103 th.</s> <s xml:id="echoid-s38672" xml:space="preserve"> 1 hu-<lb/>ius.</s> <s xml:id="echoid-s38673" xml:space="preserve"> Cum ergo linea a e ſit perpendicularιs ſuper lineam f n, & ſuper lineam d q, & ſuper lineam l e:</s> <s xml:id="echoid-s38674" xml:space="preserve"> <lb/>patet per definitionem lineæ erectę ſuper ſuperficiem, quoniam linea longitudinis pyramidis, quæ <lb/>eſt a e, erecta eſt ſuper ſuperficiem illius ſectionis oxygoniæ, quæ eſt l e d q.</s> <s xml:id="echoid-s38675" xml:space="preserve"> Et quia linea a e eſt per-<lb/>pendicularis ſuper lineam f e n:</s> <s xml:id="echoid-s38676" xml:space="preserve"> erit ergo linea f n in ſuperficie illa ſecante pyramidem ſecundũ illã <lb/> <pb o="294" file="0596" n="596" rhead="VITELLONIS OPTICAE"/> ſectionem:</s> <s xml:id="echoid-s38677" xml:space="preserve"> fiat ergo, ut in illa ſuperficie ſectionis à puncto f ducatur linea f p per 31 p 1 æquidiſtãs li-<lb/>neę e q:</s> <s xml:id="echoid-s38678" xml:space="preserve"> ergo per 9 p 11 erit linea f p æquidiſtans lineæ z t.</s> <s xml:id="echoid-s38679" xml:space="preserve"> Verùm cum angulus o z t ſit acutus:</s> <s xml:id="echoid-s38680" xml:space="preserve"> ideo <lb/>quod angulus o z h eſt obtuſus:</s> <s xml:id="echoid-s38681" xml:space="preserve"> erit per 13 p 1 angulus t z f obtuſus.</s> <s xml:id="echoid-s38682" xml:space="preserve"> Ducatur ita que à puncto z linea <lb/>faciens cum linea t z angulum æqualem angulo o z t:</s> <s xml:id="echoid-s38683" xml:space="preserve"> quę quidem linea producta neceſſariò ſecabit <lb/>lineam f p per 2 th.</s> <s xml:id="echoid-s38684" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38685" xml:space="preserve"> cum linea f p ſit æquidiſtans lineę z t.</s> <s xml:id="echoid-s38686" xml:space="preserve"> Secet ergo ipſam in puncto p:</s> <s xml:id="echoid-s38687" xml:space="preserve"> & du <lb/>catur linea p e:</s> <s xml:id="echoid-s38688" xml:space="preserve"> quæ per 1 p 11 erit in ſuperficie l e d q:</s> <s xml:id="echoid-s38689" xml:space="preserve"> erit ergo angulus a e p rectus, ut patet ex præ-<lb/>miſsis & per definitionem lineæ ſuper ſuperficiem erectæ.</s> <s xml:id="echoid-s38690" xml:space="preserve"> Cum ergo lineæ p z & o z, ut patet ex præ <lb/>miſsis, in eadem ſint ſuperficie pyramidem ſecante, & angulus o z t æqualis ſit angulo t z p:</s> <s xml:id="echoid-s38691" xml:space="preserve"> palàm <lb/>per 20 th.</s> <s xml:id="echoid-s38692" xml:space="preserve"> 5 huius quia forma puncti o reflectitur ad uiſum exiſtentem in puncto p à puncto ſpeculi <lb/>z.</s> <s xml:id="echoid-s38693" xml:space="preserve"> Verùm quia angulus o z t per 29 p 1 eſt æqualis angulo z f p, quia eſt extrinſecus illi:</s> <s xml:id="echoid-s38694" xml:space="preserve"> & angulus h <lb/>z f æqualis eſt angulo o z t per 15 p 1:</s> <s xml:id="echoid-s38695" xml:space="preserve"> ſed angulus z p f æqualis eſt angulo p z t per 29 p 1, quia eſt co-<lb/>alternus:</s> <s xml:id="echoid-s38696" xml:space="preserve"> palàm quia angulus z f p ęqualis eſt angulo z p f:</s> <s xml:id="echoid-s38697" xml:space="preserve"> ergo per 6 p 1 latus z f ęquale eſt lateri z p.</s> <s xml:id="echoid-s38698" xml:space="preserve"> <lb/>Et quia angulus f e z eſt rectus:</s> <s xml:id="echoid-s38699" xml:space="preserve"> ideo quia linea a e eſt perpendicularis ſuper lineã f n:</s> <s xml:id="echoid-s38700" xml:space="preserve"> palàm per 47 <lb/>p 1 quia quadratum lineæ f z ualet ambo quadrata linearum e f & e z:</s> <s xml:id="echoid-s38701" xml:space="preserve"> ſed eadem ratione quadratũ <lb/>lineę z p ualet ambo quadrata linearum e z & e p:</s> <s xml:id="echoid-s38702" xml:space="preserve"> quoniam, ut patet ex præmiſsis, angulus p e z eſt <lb/>rectus:</s> <s xml:id="echoid-s38703" xml:space="preserve"> quadratum uerò lineę p z eſt ęquale quadrato lineę z f:</s> <s xml:id="echoid-s38704" xml:space="preserve"> quoniam, ut patet ex præmiſsis, lineę <lb/>z f & z p ſunt ęquales:</s> <s xml:id="echoid-s38705" xml:space="preserve"> illa ergo duo quadrata hinc inde ſunt ęqualia:</s> <s xml:id="echoid-s38706" xml:space="preserve"> ergo ablato communi quadra-<lb/>to lineæ z e, remanet quadratum lineę e p æquale quadrato lineę e f:</s> <s xml:id="echoid-s38707" xml:space="preserve"> igitur latus f e æquale eſt lateri <lb/>p e:</s> <s xml:id="echoid-s38708" xml:space="preserve"> ergo per 5 p 1 angulus e p f eſt æqualis angulo e f p.</s> <s xml:id="echoid-s38709" xml:space="preserve"> Sed angulus n e q eſt æqualis angulo e f p ք <lb/>29 p 1, quoniam extrinſecus eſt illi:</s> <s xml:id="echoid-s38710" xml:space="preserve"> & angulus q e p æqualis angulo e p f, quia coalternus eſt illi:</s> <s xml:id="echoid-s38711" xml:space="preserve"> an-<lb/>gulus ergo n e q & q e p ſunt æquales:</s> <s xml:id="echoid-s38712" xml:space="preserve"> qui cum ſint in eadem ſuperficie, quæ eſt p e n:</s> <s xml:id="echoid-s38713" xml:space="preserve"> palàm per 20 <lb/>th.</s> <s xml:id="echoid-s38714" xml:space="preserve"> 5 huius quoniam forma puncti n reflectitur ad uiſum exiſtentem in puncto p à puncto ſpeculi, <lb/>quod eſt e.</s> <s xml:id="echoid-s38715" xml:space="preserve"> Similiter ſi ducatur à puncto f quæcunq;</s> <s xml:id="echoid-s38716" xml:space="preserve"> linea ad aliquod punctum lineę z e, & produca <lb/>tur uſque ad lineam o n:</s> <s xml:id="echoid-s38717" xml:space="preserve"> ſemper probabitur de puncto lineę o n, in quem cadit producta linea, quòd <lb/>ipſe reflectetur ad punctum p à puncto aliquo lineę z e, quem ſecat illa linea.</s> <s xml:id="echoid-s38718" xml:space="preserve"> Simili modo & omniũ <lb/>huiuſmodi linearum probatio ſumet initium à linea per pendiculari, quæ eſt f e, & à parte lineæ e z, <lb/>quæ erit communis omnib.</s> <s xml:id="echoid-s38719" xml:space="preserve"> illis triangulis:</s> <s xml:id="echoid-s38720" xml:space="preserve"> & ita quo dlibet punctũ lineę a o n reflectitur ad uiſum <lb/>exiſtentẽ in puncto p ab aliquo pũcto lineę z e:</s> <s xml:id="echoid-s38721" xml:space="preserve"> quia de omnib.</s> <s xml:id="echoid-s38722" xml:space="preserve"> eſt eadem demõſtratio:</s> <s xml:id="echoid-s38723" xml:space="preserve"> quod etiam <lb/>patet per 34 th.</s> <s xml:id="echoid-s38724" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s38725" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s38726" xml:space="preserve"> quęcunq;</s> <s xml:id="echoid-s38727" xml:space="preserve"> linea recta cuiuſcũq;</s> <s xml:id="echoid-s38728" xml:space="preserve"> rei uiſæ ponatur in loco lineę a o n, & <lb/>centrũ uiſus ſiſtatur in puncto p:</s> <s xml:id="echoid-s38729" xml:space="preserve"> ſemper fiet reflexio ad uiſum ab aliquo punctorum lineę a z e, quę <lb/>eſt linea lõgitudinis ſpeculi:</s> <s xml:id="echoid-s38730" xml:space="preserve"> & hoc proponebatur faciendum.</s> <s xml:id="echoid-s38731" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s38732" xml:space="preserve"/> </p> <div xml:id="echoid-div1567" type="float" level="0" n="0"> <figure xlink:label="fig-0595-01" xlink:href="fig-0595-01a"> <variables xml:id="echoid-variables680" xml:space="preserve">a o u m h z t s b c n d l e q f p</variables> </figure> </div> </div> <div xml:id="echoid-div1569" type="section" level="0" n="0"> <head xml:id="echoid-head1189" xml:space="preserve" style="it">42. Cum ſuperficiei reflexionis & ſpeculi columnaris uel pyramidalis conuexi communis ſe-<lb/>ctio ſuerit linea longitudinis: erunt loca imaginum & diſtantia ipſarum à uiſibus, quæ & in ſpe <lb/>culis planis. Alhazen 43. 49 n 5.</head> <p> <s xml:id="echoid-s38733" xml:space="preserve">Quando cauſſa in diuerſis ſubiectis uniuocatur, & paſsio uniuocabitur:</s> <s xml:id="echoid-s38734" xml:space="preserve"> ob hoc non repetimus <lb/>illa hic, quæ in ſpeculis planis dicta ſunt in quinto libro huius ſcientiæ.</s> <s xml:id="echoid-s38735" xml:space="preserve"> Quia enim utrobiq;</s> <s xml:id="echoid-s38736" xml:space="preserve"> in pla-<lb/>nis ſcilicet, & propoſitis ſpeculis lineę incidentiæ & reflexionis incidunt & reflectuntur à lineis re-<lb/>ctis:</s> <s xml:id="echoid-s38737" xml:space="preserve"> erit utrobiq;</s> <s xml:id="echoid-s38738" xml:space="preserve"> locus imaginis in perpendiculari à puncto uiſo ducta ſuper ſuperficiem ſpeculi, tá <lb/>tùm diſtans à ſuperficie ſpeculi, quantùm punctus rei uiſæ diſtat ab eadem ſpeculi ſuperficie:</s> <s xml:id="echoid-s38739" xml:space="preserve"> ideo <lb/>quia ſemper imago rei uiſæ uidetur in cõcurſu lineę reflexionis cum catheto incidẽtiæ in omnibus <lb/>his ſpeculis, ut patet per 37 th.</s> <s xml:id="echoid-s38740" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s38741" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s38742" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1570" type="section" level="0" n="0"> <head xml:id="echoid-head1190" xml:space="preserve" style="it">43. Cum ſuperficiei reflexionis & ſpeculi columnaris conuexi cõmunis ſectio fuerit circulus: <lb/>erunt punct a reflexionũ & loca imaginũ, quæ & in ſpeculis ſphæricis conuexis. Alha. 43 n 5.</head> <p> <s xml:id="echoid-s38743" xml:space="preserve">Erit enim aliquando locus imaginis intra ſpeculum columnare conuexum:</s> <s xml:id="echoid-s38744" xml:space="preserve"> aliquando in ſuperfi <lb/>cie ſpeculi:</s> <s xml:id="echoid-s38745" xml:space="preserve"> aliquando extra ſpeculũ, ſecundum modum quo cathetus incidentiæ & linea reflexio-<lb/>nis in diuerſis punctis concurrunt:</s> <s xml:id="echoid-s38746" xml:space="preserve"> cuius qui cauſſam & demõſtrationem quæſierit, recurrat ad ea, <lb/>quæ in ſexto huius ſcientiæ libro de ſpeculis ſphæricis conuexis demonſtrata ſunt:</s> <s xml:id="echoid-s38747" xml:space="preserve"> nam eadem pe-<lb/>nitus eſt ratio hincinde:</s> <s xml:id="echoid-s38748" xml:space="preserve"> quia & fines contingentiarum & metæ imaginum & loca, & eædem pro-<lb/>portiones linearum ſunt in illis ſpeculis & in iſtis.</s> <s xml:id="echoid-s38749" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s38750" xml:space="preserve"> per illa propoſitum:</s> <s xml:id="echoid-s38751" xml:space="preserve"> nec uiſum eſt no-<lb/>bis dignum in his amplius immorari.</s> <s xml:id="echoid-s38752" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1571" type="section" level="0" n="0"> <head xml:id="echoid-head1191" xml:space="preserve" style="it">44. A puncto ſectionis columnaris, cui incidit cathetus incidentiæ ad perpendicularẽ du-<lb/>ctam à puncto reflexionis ſuper ſuperficiem ſpeculi columnaris conuexi, ducta recta ad axem cõ <lb/>tinente angulum acutum cum eadem: erit concurſus catheti incidentiæ cum illa perpendιcula-<lb/>ri ſub axe. Alhazen 24 n 6.</head> <p> <s xml:id="echoid-s38753" xml:space="preserve">Hoc, quòd hic proponitur demonſtrandum, patet per 114 th.</s> <s xml:id="echoid-s38754" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38755" xml:space="preserve"> ut autem huic noſtro propo <lb/>ſito concluſio mathematica ſenſibiliter applicetur, eandẽ demonſtrationẽ duximus iterandam.</s> <s xml:id="echoid-s38756" xml:space="preserve"> Sit <lb/>ergo a e b c columnaris ſectio:</s> <s xml:id="echoid-s38757" xml:space="preserve"> & ſit e datus punctus, cui incidat cathetus incidentiæ formæ puncti <lb/>n:</s> <s xml:id="echoid-s38758" xml:space="preserve"> qui ſit punctus rei uiſæ:</s> <s xml:id="echoid-s38759" xml:space="preserve"> & b ſit punctus reflexionis, à quo ducta ſit linea b d perpendicularis ſuք <lb/>axem ſpeculi, qui ſit h k:</s> <s xml:id="echoid-s38760" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s38761" xml:space="preserve"> cathetus incidentię ducta à puncto n, qui eſt punctus rei uiſæ, ipſum <lb/>ſpeculũ ſecundũ punctũ propoſitæ ſectionis, qui eſt e:</s> <s xml:id="echoid-s38762" xml:space="preserve"> dico uerum eſſe, quod proponitur.</s> <s xml:id="echoid-s38763" xml:space="preserve"> Ducatur <lb/> <pb o="295" file="0597" n="597" rhead="LIBER SEPTIMVS."/> enim linea e d:</s> <s xml:id="echoid-s38764" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s38765" xml:space="preserve"> ita, ut fiat e d b angulus acutus:</s> <s xml:id="echoid-s38766" xml:space="preserve"> ſit ergo q e l linea contingens ſectionem in <lb/>puncto e:</s> <s xml:id="echoid-s38767" xml:space="preserve"> & ſuper punctum ſectιonis b fiat circulus æquidιſtans baſibus ſpeculi per 102 th.</s> <s xml:id="echoid-s38768" xml:space="preserve"> 1 huius, <lb/>qui ſit b t o:</s> <s xml:id="echoid-s38769" xml:space="preserve"> cuius centrũ ſit d:</s> <s xml:id="echoid-s38770" xml:space="preserve"> & ducatur à pũcto e linea longitudinis ſpeculi per 101 th.</s> <s xml:id="echoid-s38771" xml:space="preserve"> 1 huius, quæ <lb/>ſit e t.</s> <s xml:id="echoid-s38772" xml:space="preserve"> A puncto quoq;</s> <s xml:id="echoid-s38773" xml:space="preserve"> d per 11 p 1 ducatur linea d g per pendicularis ſuper lineam b d in ipſa circu-<lb/>li ſuperficie.</s> <s xml:id="echoid-s38774" xml:space="preserve"> Palàm ergo quod ſuperficies h d g (cum per axem h k tranſeat, qui per 92 th.</s> <s xml:id="echoid-s38775" xml:space="preserve"> 1 huius <lb/>eſt erectus ſuper circuli ſuperficiem) perpendicularis eſt ſuper eandem circuli ſuperficiem per <lb/>18 p 11.</s> <s xml:id="echoid-s38776" xml:space="preserve"> Superficies uerò contingens ſpeculum in puncto b, erit æquidiſtans ſuperficiei h d g ſpecu-<lb/>lum ſecanti.</s> <s xml:id="echoid-s38777" xml:space="preserve"> Ideo enim quia linea longitudinis ſpeculi ducta à puncto b eſt æquidiſtans axi h k, & li <lb/>nea cιrculum b t o contιngens ſuper punctum b, eſt æquidiſtans lineæ g d per 29 p 1:</s> <s xml:id="echoid-s38778" xml:space="preserve"> angulus enim <lb/>g d b eſt rectus, ut patet ex pręmiſsis, & angulus contentus ſub linea d b & ſub linea contingente <lb/>cιrculum in pũcto b eſt rectus per 18 p 3:</s> <s xml:id="echoid-s38779" xml:space="preserve"> ergo <lb/> <anchor type="figure" xlink:label="fig-0597-01a" xlink:href="fig-0597-01"/> illæ ſuperficies æquidiſtant per 14 p 11.</s> <s xml:id="echoid-s38780" xml:space="preserve"> Igitur <lb/>ſuperficies, in qua ſunt lineæ l e & e t, non eſt <lb/>æquidιſtans ſuperficiei h d g:</s> <s xml:id="echoid-s38781" xml:space="preserve"> quod patet per <lb/>24 th.</s> <s xml:id="echoid-s38782" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38783" xml:space="preserve"> quoniam ſuperficies contin-<lb/>gens ſectionem oxygoniam in puncto b non <lb/>eſt æquidiſtans ſuperficiei contingenti ean-<lb/>dem ſectionem in puncto e, in quo ſunt linea <lb/>l e q contingens ſectionem, & linea longitu-<lb/>dinis, quæ eſt e t.</s> <s xml:id="echoid-s38784" xml:space="preserve"> Angulus enim e d b, ut pa-<lb/>tet ex hypotheſi, eſt acutus:</s> <s xml:id="echoid-s38785" xml:space="preserve"> ſuperficies ergo <lb/>h d g non æquidiſtat ſuperficiei l e t:</s> <s xml:id="echoid-s38786" xml:space="preserve"> ergo con <lb/>curret cum illa:</s> <s xml:id="echoid-s38787" xml:space="preserve"> concurrat ergo in linea l g & ducatur linea g t:</s> <s xml:id="echoid-s38788" xml:space="preserve"> quæ neceſſariò erit contingens <lb/>circulum b t o:</s> <s xml:id="echoid-s38789" xml:space="preserve"> cum ſuperficies, in quá ducitur linea g t, ipſum ſpeculum ſit contingens.</s> <s xml:id="echoid-s38790" xml:space="preserve"> Ducta <lb/>autem linea t d, erit angulus g t d rectus per 18 p 3:</s> <s xml:id="echoid-s38791" xml:space="preserve"> quoniam linea t d eſt diameter circuli & li-<lb/>nea g t contingit ιllum cιrculum in puncto t.</s> <s xml:id="echoid-s38792" xml:space="preserve"> Fiat quo que, ut prius, ſuper e punctum ſectionis <lb/>circulus æquidiſtans baſibus ſpeculi, qui ſit e s z p:</s> <s xml:id="echoid-s38793" xml:space="preserve"> & centrum huius circuli ſit punctus axιs, qui <lb/>k:</s> <s xml:id="echoid-s38794" xml:space="preserve"> & ducatur linea k e:</s> <s xml:id="echoid-s38795" xml:space="preserve"> & ducatur etιam linea d l:</s> <s xml:id="echoid-s38796" xml:space="preserve"> quæ quidem ſecabit ſuperficιem circuli e s p:</s> <s xml:id="echoid-s38797" xml:space="preserve"> <lb/>ſecet ergo illam in puncto f Quia ιtaque punctum d eſt in ſuperficie ſectιonis per 24 huius:</s> <s xml:id="echoid-s38798" xml:space="preserve"> cum <lb/>ipſa ſectionis ſuperficies ſit ſuperficies reflexionis, & punctum l, quod eſt punctum lineæ contin-<lb/>gentιs ſectιonem, eſtιn eadem ſuperficie ſectionis:</s> <s xml:id="echoid-s38799" xml:space="preserve"> ergo per 1 p 11 tota linea d l eſt in ſuperficie ſe-<lb/>ctionis:</s> <s xml:id="echoid-s38800" xml:space="preserve"> punctum ergo f eſt in ſuperficie ſectionis:</s> <s xml:id="echoid-s38801" xml:space="preserve"> ſed ipſum eſt in ſuperficie circuli e z p:</s> <s xml:id="echoid-s38802" xml:space="preserve"> eſt er-<lb/>goin communi ſectione illarum ſuperficierũ, circuli ſcilicet & ſectionis:</s> <s xml:id="echoid-s38803" xml:space="preserve"> ſed & punctum e eſt in am <lb/>babus eιſdem ſuperficιebus:</s> <s xml:id="echoid-s38804" xml:space="preserve"> ergo item per 1 p 11 linea e f ducta erit in ambabus illis ſuperficiebus:</s> <s xml:id="echoid-s38805" xml:space="preserve"> <lb/>ergo per 19 th.</s> <s xml:id="echoid-s38806" xml:space="preserve"> 1 huius ſecundum lineam e f ſecant ſe ſuperficies ſectionis & circuli e z p.</s> <s xml:id="echoid-s38807" xml:space="preserve"> Duca-<lb/>tur ιtaque lιnea k f:</s> <s xml:id="echoid-s38808" xml:space="preserve"> & a puncto f ducatur perpendicularιs ſuper ſuperficiem circuli b t o per 11 <lb/>p 11, quæ ſit f m:</s> <s xml:id="echoid-s38809" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s38810" xml:space="preserve"> punctus m in linea d g, ut patet:</s> <s xml:id="echoid-s38811" xml:space="preserve"> & ducatur linea t m.</s> <s xml:id="echoid-s38812" xml:space="preserve"> Palàm quoniam <lb/>linea k d æquidιſtans & æqualis eſt lineæ f m per 25 th.</s> <s xml:id="echoid-s38813" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38814" xml:space="preserve"> ſunt enim lineæ k d & f m am-<lb/>bæ perpendιculares ſuper ſuperficiem circuli b t o:</s> <s xml:id="echoid-s38815" xml:space="preserve"> quia illi circuli æquidιſtant per 24 th.</s> <s xml:id="echoid-s38816" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38817" xml:space="preserve"> <lb/>utraque enιm ipſarum æquidιſtat baſibus columnæ per 100 th.</s> <s xml:id="echoid-s38818" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s38819" xml:space="preserve"> Quoniam ergo linea f m <lb/>eſt æqualιs & æquidiſtans lineæ d k, quæ eſt pars axis:</s> <s xml:id="echoid-s38820" xml:space="preserve"> ergo per 33 p 1 linea k f æqualis & æ-<lb/>quιdiſtans eſt lineæ d m:</s> <s xml:id="echoid-s38821" xml:space="preserve"> & ſimiliter erit linea f m æqualis & æquidiſtans lineæ longitudinis, <lb/>quę eſt e t, per 33 p 1:</s> <s xml:id="echoid-s38822" xml:space="preserve"> quoniam linea e t eſt æqualis & æquidiſtans axi k d per 92 th.</s> <s xml:id="echoid-s38823" xml:space="preserve"> 1 huius, <lb/>cum ſιt lιnea longitudinis ſpeculi:</s> <s xml:id="echoid-s38824" xml:space="preserve"> & erit, ut prius, linea k e æqualιs & æquidiſtans lineæ d t, & <lb/>linea e f æqualis eſt & æquidiſtans lineæ t m per eandem 33 p 1.</s> <s xml:id="echoid-s38825" xml:space="preserve"> Verùm etiam ſuperficies k d l g <lb/>(quιa tranſit axem columnæ, & angulus g d b eſt rectus) orthogonalis eſt ſuper ſuperficiem ſe-<lb/>ctionis oxygoniæ, quę eſt a e b c per definitionem ſuperficiei erectę:</s> <s xml:id="echoid-s38826" xml:space="preserve"> & eadem ſuperficies k d l g <lb/>orthogonalιs eſt ſuper ſuperficiem circuli e s p:</s> <s xml:id="echoid-s38827" xml:space="preserve"> quoniam illa ſuperficies k d l tranſiens per axem, <lb/>per 18 p 11 erecta eſt ſuper baſes columnæ:</s> <s xml:id="echoid-s38828" xml:space="preserve"> ergo & ſuper ſuperficiem circuli e s p, æquιdiſtantis <lb/>baſibus erecta eſt eadem ſuperficies k d l.</s> <s xml:id="echoid-s38829" xml:space="preserve"> Quia itaque dicta ſuperficies k d l eſt erecta ſuper ſu-<lb/>perficiem ſectionis oxygonię & circuli e s p:</s> <s xml:id="echoid-s38830" xml:space="preserve"> eſt ergo orthogonalιs ſuper lineam communem di-<lb/>ctę ſectιonιs & circuli (quę eſt linea e f) per 19 p 11 Et quia linea e f eſt erecta ſuper ſuperficiem <lb/>k d l, in qua ducta eſt lιnea k f:</s> <s xml:id="echoid-s38831" xml:space="preserve"> igitur per definitionem lineę ſuper ſuperficiem erectę angulus <lb/>e f k eſt rectus:</s> <s xml:id="echoid-s38832" xml:space="preserve"> ergo & angulus t m d eſt rectus per 10 p 11:</s> <s xml:id="echoid-s38833" xml:space="preserve"> latera enim illos angulos continen-<lb/>tia in æquιdιſtantιbus circulorum ſuperficiebus protracta æqualia ſunt & æquidiſtantia, ut pa-<lb/>tet ex pręmiſsis.</s> <s xml:id="echoid-s38834" xml:space="preserve"> Cum ergo angulus d m t ſit rectus, & angulus g t d ſit rectus per 18 p 3:</s> <s xml:id="echoid-s38835" xml:space="preserve"> in trigono <lb/>ergo orthogonιo d t g ducta eſt ab angulo ad baſim perpendicularis t m:</s> <s xml:id="echoid-s38836" xml:space="preserve"> ergo per 8 & 17 p 6 <lb/>illud, quod ſit ex ductu lιneæ d m in g m eſt æquale quadrato lineæ m t.</s> <s xml:id="echoid-s38837" xml:space="preserve"> Et quoniam linea g t <lb/>contingit cιrculum b t o, cum ſit in ſuperficιe contingente ducta ad punctum contingentię, <lb/>quod eſt t:</s> <s xml:id="echoid-s38838" xml:space="preserve"> palàm quòd linea l g eſt ęquidiſtans axi k d.</s> <s xml:id="echoid-s38839" xml:space="preserve"> Quoniam enim ſuperficies lecun-<lb/>dum lineam longιtudinis ſpeculum contingentes ſunt erectę ſuper baſium columnę ſuperfi-<lb/>cιes:</s> <s xml:id="echoid-s38840" xml:space="preserve"> ergo per 19 p 11 earum communis ſectio, quę in propoſito eſt linea l g, ſuper eaſdem ſu-<lb/>perficιes baſium perpendιcularis erιt:</s> <s xml:id="echoid-s38841" xml:space="preserve"> ęquidiſtabit ergo axi h k per 6 p 11:</s> <s xml:id="echoid-s38842" xml:space="preserve"> ergo etiam ęquidi-<lb/>ſtabιt lineæ f m per 30 p 1.</s> <s xml:id="echoid-s38843" xml:space="preserve"> Quia ergo in trigono l g d linea f m æquidiſtat baſi l g:</s> <s xml:id="echoid-s38844" xml:space="preserve"> patet per 2 p 6 <lb/> <pb o="296" file="0598" n="598" rhead="VITELLONIS OPTICAE"/> quoniam ſecat alia latera illius trigoni proportionaliter.</s> <s xml:id="echoid-s38845" xml:space="preserve"> Eſt ergo proportio lineæ d f ad f l, ſi-<lb/>cut lineæ d m ad m g:</s> <s xml:id="echoid-s38846" xml:space="preserve"> ergo permutatim per 16 p 5 erit proportio lineæ d f ad d m, ſicut lineæ <lb/>f l ad m g:</s> <s xml:id="echoid-s38847" xml:space="preserve"> ſed linea d f maior eſt quàm linea d m per 19 p 1:</s> <s xml:id="echoid-s38848" xml:space="preserve"> quoniam in trigono f d m angulus <lb/>f m d eſt rectus per præmiſſa uel 8 p 11:</s> <s xml:id="echoid-s38849" xml:space="preserve"> ergo & linea f l eſt maior quàm linea m g:</s> <s xml:id="echoid-s38850" xml:space="preserve"> ergo illud, quod <lb/>fit ex ductu lineæ f d in fl maius eſt illo, quod fit ex ductu lineæ d m in m g:</s> <s xml:id="echoid-s38851" xml:space="preserve"> ergo & quadrato lineæ <lb/>t m:</s> <s xml:id="echoid-s38852" xml:space="preserve"> ſed linea t m eſt æqualis lineæ e f, ut patet ex præmiſsis:</s> <s xml:id="echoid-s38853" xml:space="preserve"> ergo illud, quod fit ex ductu lineę d fin <lb/>l f maius eſt quadrato lineæ e f.</s> <s xml:id="echoid-s38854" xml:space="preserve"> Eſt ergo in trigono d e l angulus l e d maior recto per 30 th.</s> <s xml:id="echoid-s38855" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s38856" xml:space="preserve"> <lb/>Quia ſi eſſet rectus, tunc cum linea e f ſit perpendicularis ſuper lineam d l:</s> <s xml:id="echoid-s38857" xml:space="preserve"> eſſet per 8 & per 17 p 6 il-<lb/>lud, quod fit ex ductu lineæ d f in f l, æquale quadrato lineæ e f.</s> <s xml:id="echoid-s38858" xml:space="preserve"> Reftat ergo ut linea perpendicularis <lb/>ſuper lineam contingentem ſectionem a e b c, quæ eſt linea q l, ducta à puncto e, cadat ſub linea e d, <lb/>non perueniens in punctum d.</s> <s xml:id="echoid-s38859" xml:space="preserve"> Sit ergo illa perpendicularis linea e u.</s> <s xml:id="echoid-s38860" xml:space="preserve"> Et quia angulus e d b eſt acu-<lb/>tus, & angulus d e u acutus:</s> <s xml:id="echoid-s38861" xml:space="preserve"> quoniam angulus u e q eſt rectus:</s> <s xml:id="echoid-s38862" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s38863" xml:space="preserve"> 1 huius lineæ e u & b d <lb/>productæ concurrent in puncto aliquo ſub axe h k:</s> <s xml:id="echoid-s38864" xml:space="preserve"> & ſub concurſu lineæ e d cum linea b d:</s> <s xml:id="echoid-s38865" xml:space="preserve"> quod <lb/>eſt euidens.</s> <s xml:id="echoid-s38866" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s38867" xml:space="preserve"/> </p> <div xml:id="echoid-div1571" type="float" level="0" n="0"> <figure xlink:label="fig-0597-01" xlink:href="fig-0597-01a"> <variables xml:id="echoid-variables681" xml:space="preserve">n q t e l g o f m k d h c a s u b p z</variables> </figure> </div> </div> <div xml:id="echoid-div1573" type="section" level="0" n="0"> <head xml:id="echoid-head1192" xml:space="preserve" style="it">45. Perpendicularẽ duct ã à puncto reflexionis ſectionis pyramidalis ſuper ſuperficiẽ ſpeculi <lb/>pyramidalis cõuexi, cũ catheto incidentiæ puncto remotiori à uertice ſpeculi, quàm ſit punctus <lb/>reflexionis, incidente, ſub axe ſpeculi cõcurrere eſt neceſſe: dũ tamẽ linea à pũcto catheti inciden <lb/>tiæ duct a ad perpendicularẽ, ſuper axem angulum contine at acutũ. Alhaz. 30 n 6.</head> <p> <s xml:id="echoid-s38868" xml:space="preserve">Hæc quoq;</s> <s xml:id="echoid-s38869" xml:space="preserve"> propoſitio patet per 113 th.</s> <s xml:id="echoid-s38870" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38871" xml:space="preserve"> ut autem iam facilius pyramidalibus ſpeculis ap-<lb/>plicetur:</s> <s xml:id="echoid-s38872" xml:space="preserve"> ſit ſpeculum pyramidale conuexum a b g:</s> <s xml:id="echoid-s38873" xml:space="preserve"> cuius uertex ſit a:</s> <s xml:id="echoid-s38874" xml:space="preserve"> & axis a k:</s> <s xml:id="echoid-s38875" xml:space="preserve"> cadatq́;</s> <s xml:id="echoid-s38876" xml:space="preserve"> ιn ipſum ſe-<lb/>ctio oxygonia:</s> <s xml:id="echoid-s38877" xml:space="preserve"> à cuius circumferentia formę pũctorum lineę uiſæ reflectátur ad uiſum, quę ſit e f z:</s> <s xml:id="echoid-s38878" xml:space="preserve"> <lb/>punctum quoq;</s> <s xml:id="echoid-s38879" xml:space="preserve"> reflexionis ſit e:</s> <s xml:id="echoid-s38880" xml:space="preserve"> & ſit linea e d exiens à puncto e (quod eſt punctum reflexionis) <lb/>perpendicularis ſuper ſuperficiẽ contingentẽ ſpeculũ:</s> <s xml:id="echoid-s38881" xml:space="preserve"> quæ producta in ſuperficie ſectionis concur <lb/>ret quidẽ cũ axe a k per 14 th.</s> <s xml:id="echoid-s38882" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38883" xml:space="preserve"> angulus enim e a k eſt acutus, & angulus a e d eſt rectus:</s> <s xml:id="echoid-s38884" xml:space="preserve"> cõcur <lb/>rat ergo in puncto d:</s> <s xml:id="echoid-s38885" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s38886" xml:space="preserve"> cathetus incidentiæ formæ puncti alicuius reflexi à pũcto ſpeculi e z:</s> <s xml:id="echoid-s38887" xml:space="preserve"> quę <lb/>ſit h z.</s> <s xml:id="echoid-s38888" xml:space="preserve"> Dico quòd cathetus h z concurret cum perpendiculari e d ultra punctum d ſub axe ſpecu-<lb/>li.</s> <s xml:id="echoid-s38889" xml:space="preserve"> Ducatur enim linea t z q, quæ contingat ſectionem e f z in puncto z, dum tamẽ ſit punctũ z remo-<lb/>tius à puncto a uertice ſpeculi, quàm ſit punctũ e:</s> <s xml:id="echoid-s38890" xml:space="preserve"> ducta quoq;</s> <s xml:id="echoid-s38891" xml:space="preserve"> linea z d angulũ acutũ cõtineat cum <lb/>perpendiculari e d ſuper ipſum axem ſpeculi, in quẽ cadit punctum d.</s> <s xml:id="echoid-s38892" xml:space="preserve"> Trãſeat quoq;</s> <s xml:id="echoid-s38893" xml:space="preserve"> ſuper punctũ <lb/>z ſuperficies æquidiſtans baſi ſpeculi, quæ ſecando ſpeculum faciat circulũ r z g per 100 th.</s> <s xml:id="echoid-s38894" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38895" xml:space="preserve"> <lb/>iſte ergo circulus ſecat ſectionẽ e f z in duobus tantũ locis per 104 th.</s> <s xml:id="echoid-s38896" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38897" xml:space="preserve"> quoniã circulus eſt ք-<lb/>pendicularis ſuper axem a d, & ſectio eſt obliqua ſuper eundem axem:</s> <s xml:id="echoid-s38898" xml:space="preserve"> & ducantur lineæ a z & a e.</s> <s xml:id="echoid-s38899" xml:space="preserve"> <lb/>Linea quoq;</s> <s xml:id="echoid-s38900" xml:space="preserve"> a e, quæ ex hypotheſi eſt breuior quàm linea a z (ideo quòd punctum z remotius eſt à <lb/>uertice pyramidis quàm punctum e, protrahatur ultra punctum e) donec cõcurrat cum circumfe-<lb/>rentia circuli r z g:</s> <s xml:id="echoid-s38901" xml:space="preserve"> & ſit concurſus punctus o:</s> <s xml:id="echoid-s38902" xml:space="preserve"> ergo punctus o eſt remotior à puncto a uertice ſpecu-<lb/>li, quàm ſit pũctus e:</s> <s xml:id="echoid-s38903" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s38904" xml:space="preserve"> linea a o æqualis lineę a z ք <lb/> <anchor type="figure" xlink:label="fig-0598-01a" xlink:href="fig-0598-01"/> 89 th.</s> <s xml:id="echoid-s38905" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38906" xml:space="preserve"> ideo quia ambæ à uertice pyramidis du <lb/>cuntur ad circuli circum ferentiam.</s> <s xml:id="echoid-s38907" xml:space="preserve"> Cum ergo exierit <lb/>â puncto o perpendicularis ſuper ſuperficiem contin <lb/>gentem ſpeculum ſecundum lineam a o:</s> <s xml:id="echoid-s38908" xml:space="preserve"> concurret il-<lb/>la linea cum axe a k ultra punctum d (cui prius data <lb/>eſt incidere perpendicularis e d) per 2 th.</s> <s xml:id="echoid-s38909" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38910" xml:space="preserve"> erit <lb/>enim linea illa æquidiſtans lineę e d per 6 p 11:</s> <s xml:id="echoid-s38911" xml:space="preserve"> ſit ergo <lb/>punctus cõcurſus k:</s> <s xml:id="echoid-s38912" xml:space="preserve"> ducantur ergo lineæ k z & d z.</s> <s xml:id="echoid-s38913" xml:space="preserve"> Et <lb/>quia linea k z eſt ęqualis lineę k o per 65 th.</s> <s xml:id="echoid-s38914" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38915" xml:space="preserve"> eſt <lb/>enim k polus circuli:</s> <s xml:id="echoid-s38916" xml:space="preserve"> ſed & linea a o eſt æqualis lineæ <lb/>a z per 89 th.</s> <s xml:id="echoid-s38917" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38918" xml:space="preserve"> cum ſint lineæ lõgitudinis unius <lb/>pyramidis, & linea a k cõmunis eſt ambobus illis tri-<lb/>gonis:</s> <s xml:id="echoid-s38919" xml:space="preserve"> erũt ergo ք 8 p 1 trianguli a o k & a z k æ quian-<lb/>guli:</s> <s xml:id="echoid-s38920" xml:space="preserve"> ſed angulus a o k eſt rectus:</s> <s xml:id="echoid-s38921" xml:space="preserve"> ergo & angulus a z k <lb/>eſt rectus:</s> <s xml:id="echoid-s38922" xml:space="preserve"> eſt ergo linea k z perpẽdicularis ſuper lineã <lb/>lõgitudinis ſpeculi a z, q̃ eſt in ſuքficie cõtingẽte ſpe-<lb/>culũ:</s> <s xml:id="echoid-s38923" xml:space="preserve"> eſt ergo linea k z erecta ſuper ſuperficiẽ contin-<lb/>gentẽ ſpeculũ ſecũ dũ lineã a z:</s> <s xml:id="echoid-s38924" xml:space="preserve"> ergo ք 18 p 11 & ſuper-<lb/>ficies z k o eſt erecta ſuper illã ſuperficiẽ contingentẽ.</s> <s xml:id="echoid-s38925" xml:space="preserve"> <lb/>Et quia à puncto z ducta eſt linea cõtingẽs ſectionẽ, q̃ <lb/>eſt t z q:</s> <s xml:id="echoid-s38926" xml:space="preserve"> cũ ergo, ut patet, linea k z ſit erecta ſuper ſuք-<lb/>ficiẽ ſpeculum contingentẽ ſecundum lineam a z, & <lb/>cõmunis ſectio ſuperficiei ſectionis & illius ſuperficiei ſpeculũ contingentis ſit linea t z q cõtingẽs <lb/>ſectionẽ:</s> <s xml:id="echoid-s38927" xml:space="preserve"> erit linea k z քpẽdicularis ſuper lineã t z q:</s> <s xml:id="echoid-s38928" xml:space="preserve"> erit ergo angulus k z q rect{us} ք definitionẽ lineæ <lb/>ſuք ſuքficiẽ cõtingẽtẽ erectę.</s> <s xml:id="echoid-s38929" xml:space="preserve"> Et ꝗa, ut patet ex p̃miſsis, angul{us} k z q eſt rectus:</s> <s xml:id="echoid-s38930" xml:space="preserve"> trigonũ quoq;</s> <s xml:id="echoid-s38931" xml:space="preserve"> a z k <lb/>erectũ eſt ſuper ſuperficiẽ ſpeculũ ſecũdũ lineã a z cõtingẽtẽ:</s> <s xml:id="echoid-s38932" xml:space="preserve"> & linea k z eſt ſimiliter perpẽdicularis <lb/> <pb o="297" file="0599" n="599" rhead="LIBER SEPTIMVS."/> ſuper hanc ſuperficiẽ cõtingentẽ.</s> <s xml:id="echoid-s38933" xml:space="preserve"> Extrahamus ergo à puncto z cõmunẽ ſectionẽ ſuperficiei circuli <lb/>r z g & ſuperficiei pyramidẽ ſecũdum lineã a z contingentis:</s> <s xml:id="echoid-s38934" xml:space="preserve"> hęc aũt per 3 p 11 eſt linea recta:</s> <s xml:id="echoid-s38935" xml:space="preserve"> ſit ergo <lb/>hæc linea z y:</s> <s xml:id="echoid-s38936" xml:space="preserve"> & palàm per pręmiſſa, quòd linea z y cõtingit circulũ r z g:</s> <s xml:id="echoid-s38937" xml:space="preserve"> ſit quoq;</s> <s xml:id="echoid-s38938" xml:space="preserve"> cẽtrum huius cir-<lb/>culi c:</s> <s xml:id="echoid-s38939" xml:space="preserve"> & producatur linea c z:</s> <s xml:id="echoid-s38940" xml:space="preserve"> angulus ergo c z y eſt rectus per 18 p 3:</s> <s xml:id="echoid-s38941" xml:space="preserve"> & ducatur à puncto c, quod eſt <lb/>centrũ circuli r z g, linea cõtinens cũ linea z c angulũ rectũ per 11 p 1:</s> <s xml:id="echoid-s38942" xml:space="preserve"> & ſit linea c r:</s> <s xml:id="echoid-s38943" xml:space="preserve"> linea ergo c r eſt <lb/>æ quidiſtans lineę z y per 28 p 1:</s> <s xml:id="echoid-s38944" xml:space="preserve"> linea uerò c r eſt perpendicularis ſuper ſuperficiem a z c per 4 p 11:</s> <s xml:id="echoid-s38945" xml:space="preserve"> <lb/>ideo quia angulus z c r eſt rectus expręmiſsis, & angulus z c a eſt rectus:</s> <s xml:id="echoid-s38946" xml:space="preserve"> ideo quia axis a c eſt per-<lb/>pendicularis ſuper ſuperficiem circuli r z g per 89 th.</s> <s xml:id="echoid-s38947" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38948" xml:space="preserve"> & quia etiam axis eſt per pendicularis <lb/>ſuper baſim pyramidis, cui circulus æquidiſtat:</s> <s xml:id="echoid-s38949" xml:space="preserve"> ergo & axis erit erectus ſuper circulum per 23 th.</s> <s xml:id="echoid-s38950" xml:space="preserve"> 1 <lb/>huius:</s> <s xml:id="echoid-s38951" xml:space="preserve"> linea ergo z y æ quidiſtans lineæ c r, eſt perpendicularis ſuper ſuperficiẽ a z c per 8 p 11:</s> <s xml:id="echoid-s38952" xml:space="preserve"> er-<lb/>go linea z q contingens ſectionem eſt obliqua ſuper ſuperficiem a z c:</s> <s xml:id="echoid-s38953" xml:space="preserve"> ergo & ſuper lineam c z.</s> <s xml:id="echoid-s38954" xml:space="preserve"> Pro-<lb/>ducatur ergo à puncto z in ſectionis ſuperficie extra ipſam ſectionis peripheriam linea recta conti-<lb/>nens cum linea t q angulum rectum per 11 p 1:</s> <s xml:id="echoid-s38955" xml:space="preserve"> quæ ſit z h.</s> <s xml:id="echoid-s38956" xml:space="preserve"> Et quia punctus d per 24 huius eſt in ſu-<lb/>perficie ſectionis in aliquo puncto axis:</s> <s xml:id="echoid-s38957" xml:space="preserve"> palàm quòd ipſum aliud eſt à pũcto k, qui eſt punctus axis <lb/>inferior puncto d extra ſuperficiẽ ſectionis:</s> <s xml:id="echoid-s38958" xml:space="preserve"> ſed pũctus z eſt in ipſius ſuperficie:</s> <s xml:id="echoid-s38959" xml:space="preserve"> patet ergo quoniã <lb/>linea k z eſt extra ſuperficiẽ ſectionis.</s> <s xml:id="echoid-s38960" xml:space="preserve"> Linea ergo k z ſecat lineã z h, nec cõtinuatur cũ ipſa:</s> <s xml:id="echoid-s38961" xml:space="preserve"> quoniã <lb/>linea z h eſt in ipſa ſuperficie ſectiõis, & linea k z eſt extra illã.</s> <s xml:id="echoid-s38962" xml:space="preserve"> Et quoniã lineæ k z & h z ſecant ſe in <lb/>pũcto z:</s> <s xml:id="echoid-s38963" xml:space="preserve"> patet quòd ipſę ſunt in aliqua ſuperficie una per 2 p 11:</s> <s xml:id="echoid-s38964" xml:space="preserve"> ſint ergo lineę z k & z h in alia ſuper <lb/>ficie pręter ſuperficiẽ ſectionis, quę ſecet ſuperficiẽ ſectiõis ſuper lineã p z h in ambabus iſtis ſuper <lb/>ficiebus exiſtentẽ per 19 th.</s> <s xml:id="echoid-s38965" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s38966" xml:space="preserve"> & ſit z p eadẽ linea cũ z h, quę eſt producta in ſuperficie ſectio-<lb/>nis.</s> <s xml:id="echoid-s38967" xml:space="preserve"> Linea uerò d z, quę eſt in ſuperficie ſectionis, eſt extra ſuperficiem, in qua ſunt lineæ k z & z h:</s> <s xml:id="echoid-s38968" xml:space="preserve"> <lb/>ſed linea z k continet cum linea z q angulum rectum:</s> <s xml:id="echoid-s38969" xml:space="preserve"> ideo quia, ut prædictũ eſt, linea k z eſt perpen <lb/>dicularis ſuper ſuperficiem contingentem pyramidem, quę tranſit lineas a z & z q:</s> <s xml:id="echoid-s38970" xml:space="preserve"> & ſuperficies <lb/>k z h ſecat ſuperficiem d z h ſuper lineam illis duabus ſuperficiebus communem per 19 th.</s> <s xml:id="echoid-s38971" xml:space="preserve"> 1 huius, <lb/>quæ eſt h z.</s> <s xml:id="echoid-s38972" xml:space="preserve"> Verùm linea d z eſt in ſuperficie ſectionis, ut ſuprà patuit, & ſecatur à linea k z in pun-<lb/>cto z, & pũcta t & q ſunt à lateribus ſuperficiei k z p h:</s> <s xml:id="echoid-s38973" xml:space="preserve"> ergo ſuperficies h z k ſecat ſuperficiem d z q:</s> <s xml:id="echoid-s38974" xml:space="preserve"> <lb/>differentia ergo communis ſuperficierum h z k & d z q eſt in ſuperficie h z k:</s> <s xml:id="echoid-s38975" xml:space="preserve"> eſt quoq;</s> <s xml:id="echoid-s38976" xml:space="preserve"> illa commu-<lb/>nis ſectio linea recta per 3 p 11:</s> <s xml:id="echoid-s38977" xml:space="preserve"> continet ergo illa linea cum linea z q angulum rectum.</s> <s xml:id="echoid-s38978" xml:space="preserve"> Nam linea z q <lb/>cum ſit perpendicularis ſuper lineam z h, & ſuper lineam z k:</s> <s xml:id="echoid-s38979" xml:space="preserve"> patet per 4 p 11 quoniam ipſa eſt ere-<lb/>cta ſuper ſuperficiẽ h z k:</s> <s xml:id="echoid-s38980" xml:space="preserve"> ergo & ſuper lineam z p.</s> <s xml:id="echoid-s38981" xml:space="preserve"> Et quoniam ſuperficies h z k ſecat ſuperficiem d <lb/>z q, & declinatio ſuperficiei h z k à ſuperficie ſectiõis, cuius pars eſt ſuperficies d z q, fit ex parte ſe-<lb/>midiametri z c, erit linea (quę eſt differentia communis his duabus ſuperficiebus) media inter <lb/>duas lineas q z & z d:</s> <s xml:id="echoid-s38982" xml:space="preserve"> ergo angulus q z d eſt obtuſus:</s> <s xml:id="echoid-s38983" xml:space="preserve"> & linea h z eſt in ſuperficie, in qua ſunt lineæ <lb/>d z & z q, quę eſt ſuperficies ſectionis, & continet cum linea z q angulum rectum:</s> <s xml:id="echoid-s38984" xml:space="preserve"> linea ergo z h ꝓ-<lb/>ducta intra ſectionem ultra punctum z, ſecabit angulum d z q:</s> <s xml:id="echoid-s38985" xml:space="preserve"> & linea h z concurret cum linea e d <lb/>ſub pũcto d puncto axis per 14 th.</s> <s xml:id="echoid-s38986" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s38987" xml:space="preserve"> Angulus enim z d e eſt acutus ex hypotheſi, & angulus d <lb/>z p acutus.</s> <s xml:id="echoid-s38988" xml:space="preserve"> Cathetus itaq;</s> <s xml:id="echoid-s38989" xml:space="preserve"> incidentię, quę eſt h z, cum perpendiculari e d, quę ducitur à puncto re-<lb/>flexionis ſuper ſuperficiem ſpeculum contingentem, concurret ſub axe:</s> <s xml:id="echoid-s38990" xml:space="preserve"> & ſub puncto ipſius axis, <lb/>qui eſt d:</s> <s xml:id="echoid-s38991" xml:space="preserve"> ſit itaq;</s> <s xml:id="echoid-s38992" xml:space="preserve"> punctum concurſus p.</s> <s xml:id="echoid-s38993" xml:space="preserve"> Ethoc eſt propoſitum.</s> <s xml:id="echoid-s38994" xml:space="preserve"/> </p> <div xml:id="echoid-div1573" type="float" level="0" n="0"> <figure xlink:label="fig-0598-01" xlink:href="fig-0598-01a"> <variables xml:id="echoid-variables682" xml:space="preserve">a e t b o f z h d g y k p b q</variables> </figure> </div> <figure> <variables xml:id="echoid-variables683" xml:space="preserve">a h l z x m o k e q d y p f b g</variables> </figure> </div> <div xml:id="echoid-div1575" type="section" level="0" n="0"> <head xml:id="echoid-head1193" xml:space="preserve" style="it">46. Perpendicularem ductam à puncto re-<lb/>flexionis ſectionis pyramidalis ſuper ſuperficiem <lb/>ſpeculi pyramidalis conuexi, cum catheto inci-<lb/>dentιæ pũcto propinquiori uertici ſpeculi, quàm <lb/>ſit punctus reflexionis incidente, ſub axe ſpecu-<lb/>li concurrere eſt neceſſe: altioris quoque puncti <lb/>cathetus cum eadem porpendiculari concur-<lb/>ret remotius ſub axe: dum tamen linea à pun-<lb/>cto ſuperiori cum perpendiculari ducta à pun-<lb/>cto inferiori ſuper axem angulum contine at a-<lb/>cutum.</head> <p> <s xml:id="echoid-s38995" xml:space="preserve">Sit, ut in præmiſſa, ſpeculum pyramidale <lb/>conuexum a b g:</s> <s xml:id="echoid-s38996" xml:space="preserve"> cuius uertex ſit a:</s> <s xml:id="echoid-s38997" xml:space="preserve"> & axis a d:</s> <s xml:id="echoid-s38998" xml:space="preserve"> <lb/>ſitq́;</s> <s xml:id="echoid-s38999" xml:space="preserve"> in ipſo ſectio pyramidalis, quæ e f z:</s> <s xml:id="echoid-s39000" xml:space="preserve"> pun-<lb/>ctum quoque reflexionis ſit e:</s> <s xml:id="echoid-s39001" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s39002" xml:space="preserve"> linea e d per-<lb/>pendicularis ſuper ſuperficiem ſpeculi, concur-<lb/>rens cum axe a k in puncto d in ſuperficie ſectio-<lb/>nis:</s> <s xml:id="echoid-s39003" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s39004" xml:space="preserve"> cathetus incidentiæ formæ puncti ali-<lb/>cuius reflexi à puncto e, quæ ſit h z:</s> <s xml:id="echoid-s39005" xml:space="preserve"> cuius pun-<lb/>ctum z ſit propinquius uertici ſpeculi quàm pun-<lb/>ctum e:</s> <s xml:id="echoid-s39006" xml:space="preserve"> ita tamen quòd linea z d cum linea e d in <lb/>puncto d contineat angulum acutum.</s> <s xml:id="echoid-s39007" xml:space="preserve"> Dico quòd uerũ eſt, quod proponitur.</s> <s xml:id="echoid-s39008" xml:space="preserve"> Circũducatur enim à <lb/> <pb o="298" file="0600" n="600" rhead="VITELLONIS OPTICAE"/> puncto zipſi ſpeculo circulus per 102 th.</s> <s xml:id="echoid-s39009" xml:space="preserve"> 1 huius, qui ſit z m l:</s> <s xml:id="echoid-s39010" xml:space="preserve"> & ducantur lineę a z & a e.</s> <s xml:id="echoid-s39011" xml:space="preserve"> Linea quoq;</s> <s xml:id="echoid-s39012" xml:space="preserve"> <lb/>a e ex hypotheſi eſt longior quàm linea a z:</s> <s xml:id="echoid-s39013" xml:space="preserve"> patet ergo per 100 & 89 th.</s> <s xml:id="echoid-s39014" xml:space="preserve"> 1 huius quoniã abſcinditur <lb/>per ſuperficiem circuli z m l:</s> <s xml:id="echoid-s39015" xml:space="preserve"> ideo quia punctũ z propinquius eſt uertici pyramidis, qui eſt a, ꝗ̃ pun-<lb/>ctum e.</s> <s xml:id="echoid-s39016" xml:space="preserve"> Sit ergo, ut abſcindatur in puncto o:</s> <s xml:id="echoid-s39017" xml:space="preserve"> eſt ergo punctum o propinquius uertici ipſius ſpeculi, <lb/>quàm e punctũ:</s> <s xml:id="echoid-s39018" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s39019" xml:space="preserve"> linea a o æqualis lineæ a z per 89 th.</s> <s xml:id="echoid-s39020" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s39021" xml:space="preserve"> Cum ergo exierit à pũcto o per-<lb/>pendicularis ſuper lineam a o, quæ ſit o k, ſecans axem a d in puncto k:</s> <s xml:id="echoid-s39022" xml:space="preserve"> erit per 28 p 1 linea o k æ qui-<lb/>diſtans lineæ e d.</s> <s xml:id="echoid-s39023" xml:space="preserve"> Ducantur ergo lineæ k z & d z.</s> <s xml:id="echoid-s39024" xml:space="preserve"> Et quia linea k z eſt æqualis lineæ k o per 65 <lb/>th.</s> <s xml:id="echoid-s39025" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39026" xml:space="preserve"> eſt enim pũctus k polus circuli z m l:</s> <s xml:id="echoid-s39027" xml:space="preserve"> ſed & linea a o eſt æqualis lineæ a z ք 89 th.</s> <s xml:id="echoid-s39028" xml:space="preserve"> 1 huius, <lb/>& linea a k eſt cõmunis ambobus illis trigonis:</s> <s xml:id="echoid-s39029" xml:space="preserve"> erunt ergo per 8 p 1 trigoni a o k & a z k ęquianguli:</s> <s xml:id="echoid-s39030" xml:space="preserve"> <lb/>ſed angulus a o k eſt rectus, quia o k perpendicularis ducta eſt ſuper lineam a o:</s> <s xml:id="echoid-s39031" xml:space="preserve"> uel etiã per 29 p 1:</s> <s xml:id="echoid-s39032" xml:space="preserve"> <lb/>ideo quia angulus a e d eſt rectus, & lineæ e d & o k æquidiſtant:</s> <s xml:id="echoid-s39033" xml:space="preserve"> ergo & angulus a z k eſt rectus:</s> <s xml:id="echoid-s39034" xml:space="preserve"> eſt <lb/>ergo linea k z perpendicularis ſuper lineã lõgitudinis ſpeculi a z, quæ eſt in ſuperficie contingente <lb/>ſpeculũ:</s> <s xml:id="echoid-s39035" xml:space="preserve"> eſt ergo linea k z erecta ſuper ſuperficiẽ cõtingentẽ ſpeculũ ſecundũ lineã a z.</s> <s xml:id="echoid-s39036" xml:space="preserve"> Ducta quoq;</s> <s xml:id="echoid-s39037" xml:space="preserve"> <lb/>â puncto z linea cõtingente ſectionẽ in puncto z, quæ ſit t z q, perficiatur demonſtratio, ut in proxi-<lb/>ma præmiſſa:</s> <s xml:id="echoid-s39038" xml:space="preserve"> patetq́;</s> <s xml:id="echoid-s39039" xml:space="preserve"> propoſitũ nunc, ut prius.</s> <s xml:id="echoid-s39040" xml:space="preserve"> Cadet enim punctus p, qui ſit cõmunis ſectio cathe-<lb/>ti incidẽtiæ ductæ à pũcto z cũ քpendiculari e d, ſub axe a d & ſub pũcto d.</s> <s xml:id="echoid-s39041" xml:space="preserve"> Et ſi in peripheria ipſius <lb/>ſectionis ſignetur pũctus propinquior uertici, ꝗ̃ ſit punctũ z, qui ſit punctus x:</s> <s xml:id="echoid-s39042" xml:space="preserve"> ab eo quoq;</s> <s xml:id="echoid-s39043" xml:space="preserve"> ducatur <lb/>cathetus incidentiæ, quæ ſit x y:</s> <s xml:id="echoid-s39044" xml:space="preserve"> quæ eodẽ modo, ſi angulus x d e fuerit acutus, demõſtrabitur con-<lb/>currere cum perpendiculari e d ſub axe a d:</s> <s xml:id="echoid-s39045" xml:space="preserve"> ſit concurſus in puncto y.</s> <s xml:id="echoid-s39046" xml:space="preserve"> Dico quòd punctus y remo-<lb/>tior erit ſub axe a d quàm punctũ p:</s> <s xml:id="echoid-s39047" xml:space="preserve"> non enim ſecabit linea x y angulũ a z p, neq;</s> <s xml:id="echoid-s39048" xml:space="preserve"> lineam z p:</s> <s xml:id="echoid-s39049" xml:space="preserve"> quoniã <lb/>cathetus ducta à puncto altiori ulterius protenditur ſub axem:</s> <s xml:id="echoid-s39050" xml:space="preserve"> & cathetus angulum rectum conti-<lb/>nens cum perpendiculari e d concurret cum illa in puncto axis d.</s> <s xml:id="echoid-s39051" xml:space="preserve"> Reliquæ uerò catheti harum me-<lb/>diæ, à quarum punctis incidentiæ ductæ lineæ ad punctum d, angulos continent acutos cum per-<lb/>pendiculari e d, non ſecabunt lineam d p Patet ergo propoſitum.</s> <s xml:id="echoid-s39052" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1576" type="section" level="0" n="0"> <head xml:id="echoid-head1194" xml:space="preserve" style="it">47. Cathetum incidentiæ linea reflexionis intra ſectionem oxygoniam ſecante, & à puncto <lb/>reflexionis duct a contingente, quæ ſecet cathetum: erit totius catheti proportio ad partẽ ſui re-<lb/>ſectam intra ſectionem oxygoniam, ſicut partis extrinſecus reſectæ adeam, quæutraſ interia <lb/>cet ſectiones. Alhazen 44 n 5.</head> <p> <s xml:id="echoid-s39053" xml:space="preserve">Eſto a b c ſection oxygonia:</s> <s xml:id="echoid-s39054" xml:space="preserve"> cuius punctus b ſit punctus reflexionis:</s> <s xml:id="echoid-s39055" xml:space="preserve"> & ſit e punctus rei uiſæ:</s> <s xml:id="echoid-s39056" xml:space="preserve"> d cen <lb/>trum uiſus:</s> <s xml:id="echoid-s39057" xml:space="preserve"> à puncto quoque reflexionis, quod eſt b, ducatur linea perpendicularis ſuper ſuperfici-<lb/>em contingentem ſpeculum in puncto b, quæ ſit g b q, ducta intra ſpeculum propoſitum in punctũ <lb/>q:</s> <s xml:id="echoid-s39058" xml:space="preserve"> & ducatur à puncto e linea e k perpendicularis ſuper ipſam ſectionẽ, aut ſuper lineã ſectionẽ con <lb/>tingentem, ut fuerit poſsibile:</s> <s xml:id="echoid-s39059" xml:space="preserve"> ducatur quoque linea contingens ſpeculũ in puncto b, quæ ſit t b u, <lb/>& alia contingens ſectionem in puncto k.</s> <s xml:id="echoid-s39060" xml:space="preserve"> Duæ ita que perpendiculares, quæ ſunt g b q & e k con-<lb/>current intra ſectionem ſub axe ſpeculi per tres præcedentes:</s> <s xml:id="echoid-s39061" xml:space="preserve"> ſit ergo punctus concurſus illarum <lb/>perpendicularium punctum q.</s> <s xml:id="echoid-s39062" xml:space="preserve"> Sed & hoc in propoſito aliter declarandum.</s> <s xml:id="echoid-s39063" xml:space="preserve"> Ducantur enim lineæ <lb/>e b, d b, k b:</s> <s xml:id="echoid-s39064" xml:space="preserve"> palàm per 29 th.</s> <s xml:id="echoid-s39065" xml:space="preserve"> 1 huius, & ex præmiſsis quoniam linea k m cadet intra ſuperficiem e <lb/>k b, & linea b t cader intra eandem ſuperficiem:</s> <s xml:id="echoid-s39066" xml:space="preserve"> igitur linea b t ſecabit lineam e k:</s> <s xml:id="echoid-s39067" xml:space="preserve"> ſit, ut ſecet ipſam <lb/>in puncto t:</s> <s xml:id="echoid-s39068" xml:space="preserve"> & linea k m ſecabit lineam b e:</s> <s xml:id="echoid-s39069" xml:space="preserve"> & ſit, ut ſecet ipſam in puncto m.</s> <s xml:id="echoid-s39070" xml:space="preserve"> Cum ergo angulus e <lb/>k m ſit rectus, ut patet ex præmiſsis:</s> <s xml:id="echoid-s39071" xml:space="preserve"> palàm quòd angulus e k b maior eſt recto:</s> <s xml:id="echoid-s39072" xml:space="preserve"> & ſimiliter quia an-<lb/>gulus g b t eſt rectus, erit angulus g b k maior recto:</s> <s xml:id="echoid-s39073" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0600-01a" xlink:href="fig-0600-01"/> palàm ergo per 14 th.</s> <s xml:id="echoid-s39074" xml:space="preserve"> 1 huius quoniam duæ perpen-<lb/>diculares g b & e k concurrent in aliquo puncto <lb/>ſuperficiei ſectionis, cũ ſint in eadem ſuperficie:</s> <s xml:id="echoid-s39075" xml:space="preserve"> ſit, <lb/>ut prius, earum concurſus in pũcto q:</s> <s xml:id="echoid-s39076" xml:space="preserve"> ſimiliter quo-<lb/>que angulus d b k eſt maior angulo recto, qui eſt <lb/>g b t, ut patet ex præmiſsis:</s> <s xml:id="echoid-s39077" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s39078" xml:space="preserve"> 1 hu-<lb/>ius lineæ d b & e k concurrent:</s> <s xml:id="echoid-s39079" xml:space="preserve"> ſit ipſarum con-<lb/>curſus punctus h:</s> <s xml:id="echoid-s39080" xml:space="preserve"> igitur per 37 th.</s> <s xml:id="echoid-s39081" xml:space="preserve"> 1 huius punctus h <lb/>eſt locus imaginis formæ puncti e.</s> <s xml:id="echoid-s39082" xml:space="preserve"> Dico itaq;</s> <s xml:id="echoid-s39083" xml:space="preserve"> quòd <lb/>erit proportio lineæ e q, quæ eſt cathetus inciden-<lb/>tiæ formæ puncti e, ad lineam q h, ſicut lineæ e t ad <lb/>lineam t h.</s> <s xml:id="echoid-s39084" xml:space="preserve"> Quia enim lineæ e k & b e concurrunt in <lb/>puncto e:</s> <s xml:id="echoid-s39085" xml:space="preserve"> ducatur à puncto h linea h f æ quidiſtans <lb/>lineæ e b per 31 p 1.</s> <s xml:id="echoid-s39086" xml:space="preserve"> Et quoniam angulus e b t eſt per <lb/>20 th.</s> <s xml:id="echoid-s39087" xml:space="preserve"> 5 huius æqualis angulo d b u, & per 15 p 1 an-<lb/>gulus d b u eſt æqualis angulo t b h:</s> <s xml:id="echoid-s39088" xml:space="preserve"> palàm quòd <lb/>angulus e b t erit æqualis angulo t b h.</s> <s xml:id="echoid-s39089" xml:space="preserve"> Reſtater-<lb/>go, ut angulus e b g ſit æqualis angulo h b q:</s> <s xml:id="echoid-s39090" xml:space="preserve"> ideo quia anguli t b q & t b g ſunt recti & æquales.</s> <s xml:id="echoid-s39091" xml:space="preserve"> <lb/>Cum igitur linea t b diuidat angulum e b h per æqualia:</s> <s xml:id="echoid-s39092" xml:space="preserve"> erit per 3 p 6 proportio lineæ e t ad t h, <lb/>ſicut lineæ e b a d b h:</s> <s xml:id="echoid-s39093" xml:space="preserve"> ſed per 29 p 1 angulus e b g eſt æqualis angulo h f b:</s> <s xml:id="echoid-s39094" xml:space="preserve"> angulus ergo h f b eſt æ-<lb/>qualis angulo h b f, quoniã ut præoſtenſum eſt, angulus e b g eſt æqualis angulo h b f:</s> <s xml:id="echoid-s39095" xml:space="preserve"> ergo per 6 p 1 <lb/> <pb o="299" file="0601" n="601" rhead="LIBER SEPTIMVS."/> linea h b eſt æqualis lineæ h f:</s> <s xml:id="echoid-s39096" xml:space="preserve"> ergo per 7 p 5 proportio lineæ e b ad lineã h f eſt, ſicut ad lineã h b:</s> <s xml:id="echoid-s39097" xml:space="preserve"> eſt <lb/>aũt proportio lineæ e b ad h f, ſicut lineæ e q ad q h ք 4 p 6:</s> <s xml:id="echoid-s39098" xml:space="preserve"> quia per 29 p 1 trigona e q b & h q f ſunt <lb/>æquiangula:</s> <s xml:id="echoid-s39099" xml:space="preserve"> erit ergo proportio lineæ e b ad h b, ſicut lineæ e q ad q h.</s> <s xml:id="echoid-s39100" xml:space="preserve"> Erit ergo per 11 p 5 proportio <lb/>lineæ e q ad lineam q h, ſicut lineæ e t ad lineam th.</s> <s xml:id="echoid-s39101" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s39102" xml:space="preserve"/> </p> <div xml:id="echoid-div1576" type="float" level="0" n="0"> <figure xlink:label="fig-0600-01" xlink:href="fig-0600-01a"> <variables xml:id="echoid-variables684" xml:space="preserve">e g d t m b u k h f q a c</variables> </figure> </div> </div> <div xml:id="echoid-div1578" type="section" level="0" n="0"> <head xml:id="echoid-head1195" xml:space="preserve" style="it">48. In omni ſpeculo columnari uel pyramidali uel pyramidali conuexo, communi ſectione ſuperficiei reflexio <lb/>nis & ſpeculi oxygonia exiſtente: linea rect a interiacens punctum concurſus duarum præmiſſa <lb/>rum perpendicularium & locum imaginis, maior eſt linea rect a interiacente locum imaginis <lb/>& punctum reflexionis. Alhazen 44 n 5.</head> <p> <s xml:id="echoid-s39103" xml:space="preserve">Sit omnimoda diſpoſitio & probatio, ut in præcedente proxima.</s> <s xml:id="echoid-s39104" xml:space="preserve"> Et quia eſt proportio lineæ e q <lb/>ad lineam q h, ſicut lineæ e b ad lineam h f per 4 p 6 & 29 p 1:</s> <s xml:id="echoid-s39105" xml:space="preserve"> & proportio lineæ e b ad h f eſt, ſicut li <lb/>neæ e b ad lineam h b per 6 p 1 & 7 p 5:</s> <s xml:id="echoid-s39106" xml:space="preserve"> erit proportio lineæ e b ad lineam b h, ſicut lineæ e q ad lineã <lb/>q h per 11 p 5:</s> <s xml:id="echoid-s39107" xml:space="preserve"> ergo permutatim per 16 p 5 & corollarium 4 p 5 erit proportio lineæ e q ad e b, ſicut <lb/>q h ad h b:</s> <s xml:id="echoid-s39108" xml:space="preserve"> ſed linea e q maior eſt quàm linea e b per 19 p 1, eò quòd angulus e b q maior eſt recto, ut <lb/>patet ex pręmiſsis, quia angulus t b q eſt rectus:</s> <s xml:id="echoid-s39109" xml:space="preserve"> ergo linea q h eſt maior ꝗ̃ linea h b.</s> <s xml:id="echoid-s39110" xml:space="preserve"> Quod eſt ꝓpo-<lb/>ſitum:</s> <s xml:id="echoid-s39111" xml:space="preserve"> eſt enim punctus q ille punctus, in quo cõcurrunt duę perpẽdiculares g b q & e k, quę eſt ca-<lb/>thetus incidentiæ:</s> <s xml:id="echoid-s39112" xml:space="preserve"> & punctus h eſt locus imaginis formæ puncti e:</s> <s xml:id="echoid-s39113" xml:space="preserve"> & punctus b eſt pũctus reflexio <lb/>nis formæ puncti e ad centrum uiſus exiſtentis in puncto d.</s> <s xml:id="echoid-s39114" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1579" type="section" level="0" n="0"> <head xml:id="echoid-head1196" xml:space="preserve" style="it">49. Communi ſectione ſuperficiei reflexionis & ſpeculi columnaris uel pyramidalis conuexi <lb/>exiſtente oxygonia, formá rei uiſæ obliquè ſpeculo incidente: locus imaginum formarum uiſo-<lb/>rum punctorum quando erit in ſuperficie ſpeculi: quando intra ſpeculum: & quãdo extra <lb/>ipſum. Alhazen 45. 51 n 5.</head> <p> <s xml:id="echoid-s39115" xml:space="preserve">Quod hic proponitur, locum habet, cum punctus rei uiſæ non fuerit in diametro uiſuali perpen <lb/>diculari ſuper ſuperficiem ſpeculi:</s> <s xml:id="echoid-s39116" xml:space="preserve"> tunc enim unius ſolius forma puncti ſuper lineam perpendicu-<lb/>larem accedit ad ſpeculum, & ſecundum eandem lineam refle ctetur ad uiſum, utpote punctus ipſi-<lb/>us perpẽdicularis lineæ, qui eſt in ſuperficie oculi uidentis.</s> <s xml:id="echoid-s39117" xml:space="preserve"> Punctus enim ultra ſuperficiem oculi <lb/>ſumptus non poteſt reflecti ſuper hanc perpendicularem:</s> <s xml:id="echoid-s39118" xml:space="preserve"> quia non poteſt accedere ad ſpeculum ſu <lb/>per lineam perpendicularem, propter rationem aſsignatam in 32 th.</s> <s xml:id="echoid-s39119" xml:space="preserve"> 5 & 10 th.</s> <s xml:id="echoid-s39120" xml:space="preserve"> 6 huius.</s> <s xml:id="echoid-s39121" xml:space="preserve"> Et ſimiliter <lb/>non poteſt reflecti forma illius puncti ad uiſum ab alio puncto ſpeculi, quàm à puncto illo, cui inci-<lb/>dit ipſa perpendicularis.</s> <s xml:id="echoid-s39122" xml:space="preserve"> Si enim daretur hoc poſſe fieri:</s> <s xml:id="echoid-s39123" xml:space="preserve"> tunc accideret duas perpendiculares du-<lb/>ctas à ſuperficie ſpeculi concurrere in centro eiuſdẽ uiſus, quod eſſet contra 6 p 11 & contra 20 th.</s> <s xml:id="echoid-s39124" xml:space="preserve"> 1 <lb/>huius:</s> <s xml:id="echoid-s39125" xml:space="preserve"> & duo anguli trianguli fierent recti:</s> <s xml:id="echoid-s39126" xml:space="preserve"> quod eſſet contra 32 p 1, & impoſsibile.</s> <s xml:id="echoid-s39127" xml:space="preserve"> In tali ergo ſitu <lb/>perpendicularis reflectitur tantũ in ſeipſam.</s> <s xml:id="echoid-s39128" xml:space="preserve"> Sit au <lb/> <anchor type="figure" xlink:label="fig-0601-01a" xlink:href="fig-0601-01"/> tem nunc, ut forma rei uiſæ incidat ſuperficiei ſpe-<lb/>culi non perpendiculariter, ſed obliquè:</s> <s xml:id="echoid-s39129" xml:space="preserve"> & eſto, ut <lb/>ſuperficies reflexionis ſecet ſpeculum columnare <lb/>conuexum, & communis eorum ſectio ſit oxygo-<lb/>nia ſectio, quæ a b g:</s> <s xml:id="echoid-s39130" xml:space="preserve"> ad cuius punctum a ſumatur li <lb/>nea contingens ſectionem, quæ ſit e a t:</s> <s xml:id="echoid-s39131" xml:space="preserve"> & ducatur <lb/>perpendicularis à puncto a per 11 p 1 ſuper lineam <lb/>e t intra ſectionem, quæ ſit a d:</s> <s xml:id="echoid-s39132" xml:space="preserve"> cadatq́;</s> <s xml:id="echoid-s39133" xml:space="preserve"> punctus d <lb/>intra ſectionẽ.</s> <s xml:id="echoid-s39134" xml:space="preserve"> Palàm ergo per 115 th.</s> <s xml:id="echoid-s39135" xml:space="preserve"> 1 huius quòd <lb/>linea d a diuidit ſectionem in duas partes, in qua-<lb/>rum utraq;</s> <s xml:id="echoid-s39136" xml:space="preserve"> eſt punctus unicus, in quo pũcto linea <lb/>ſectionem contingens, erit æ quidiſtans lineæ d a:</s> <s xml:id="echoid-s39137" xml:space="preserve"> <lb/>ſit ergo citra unum illorum punctorum alius, qui <lb/>ſit punctus g, cuius puncti contingens concurrat <lb/>cum linea d a in puncto h extra ſectionem:</s> <s xml:id="echoid-s39138" xml:space="preserve"> & duca <lb/>tur linea perpendicularis ſuper hanc lineam cõtin <lb/>gentẽ (quę eſt g h) per 11 p 1, quę perpendicularis <lb/>ſit g q, ſecãs lineã aliã cõtingẽtẽ, quę eſt e a t, in pũcto t:</s> <s xml:id="echoid-s39139" xml:space="preserve"> erit ergo punctũ t finis cõtingẽtię per 6 de-<lb/>finitionẽ 6 huius:</s> <s xml:id="echoid-s39140" xml:space="preserve"> & hæc quidem perpendicularis (quæ g q) neceſſariò concurret cum linea h d <lb/>ք 14 th.</s> <s xml:id="echoid-s39141" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39142" xml:space="preserve"> ideo qđ angulus q g h eſt rectus, & angulus g h d acutus:</s> <s xml:id="echoid-s39143" xml:space="preserve"> ſit ergo in pũcto d ipſarũ <lb/>cõcurſus:</s> <s xml:id="echoid-s39144" xml:space="preserve"> & ducatur linea g a:</s> <s xml:id="echoid-s39145" xml:space="preserve"> quæ producatur extra ſectionẽ uſq;</s> <s xml:id="echoid-s39146" xml:space="preserve"> ad punctũ p:</s> <s xml:id="echoid-s39147" xml:space="preserve"> & ducatur linea q a.</s> <s xml:id="echoid-s39148" xml:space="preserve"> <lb/>Igitur angulus q a h aut eſt æqualis angulo h a p:</s> <s xml:id="echoid-s39149" xml:space="preserve"> aut maior:</s> <s xml:id="echoid-s39150" xml:space="preserve"> aut minor.</s> <s xml:id="echoid-s39151" xml:space="preserve"> Si ſit æqualis:</s> <s xml:id="echoid-s39152" xml:space="preserve"> incidet ergo <lb/>forma puncti q ſpeculo in pũcto a, & reflectetur ad cẽtrũ uiſus exiſtẽs in pũcto p per 20 th.</s> <s xml:id="echoid-s39153" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s39154" xml:space="preserve"> <lb/>& locus imaginis erit pũctus g, ꝗ eſt pũctus ſectionis oxygoniæ & ſuքficiei colũnę ſpeculi ք 37 th.</s> <s xml:id="echoid-s39155" xml:space="preserve"> 5 <lb/>huius:</s> <s xml:id="echoid-s39156" xml:space="preserve"> quoniã in illo pũcto cõcurrit cathetus incidẽtiæ ducta à pũcto rei uiſæ, quę eſt q, ſuper lineã <lb/>cõtingentẽ ſectionẽ in pũcto g, cũ linea reflexionis, quę eſt p a.</s> <s xml:id="echoid-s39157" xml:space="preserve"> Et quia pũctus g eſt in ſuperficie ſpe <lb/>culi:</s> <s xml:id="echoid-s39158" xml:space="preserve"> patet qđ tũc uidebitur imago formæ pũcti q in ſuքficie ſpeculi.</s> <s xml:id="echoid-s39159" xml:space="preserve"> Si uerò in linea g q ſupra pun-<lb/>ctũ q ſumatur alius pũctus, ut f, & ducatur linea f a:</s> <s xml:id="echoid-s39160" xml:space="preserve"> erit quidẽ angulus f a h minor angulo h a p:</s> <s xml:id="echoid-s39161" xml:space="preserve"> eſt <lb/>enim angulus f a h minor angulo q a h, ꝗ eſt æqualis angulo h a p:</s> <s xml:id="echoid-s39162" xml:space="preserve"> fiat ergo angulo f a h ſuper a termi <lb/>nũ lineę h a æqualis angulus, ꝗ ſit h a n ք 23 p 1:</s> <s xml:id="echoid-s39163" xml:space="preserve"> & ꝓducatur linea n a intra ſectionẽ:</s> <s xml:id="echoid-s39164" xml:space="preserve"> cõcurretq́;</s> <s xml:id="echoid-s39165" xml:space="preserve"> cum <lb/> <pb o="300" file="0602" n="602" rhead="VITELLONIS OPTICAE"/> catheto f q g d:</s> <s xml:id="echoid-s39166" xml:space="preserve"> & ſit pũctus cõcurſus k.</s> <s xml:id="echoid-s39167" xml:space="preserve"> Palàm ergo per 20 th.</s> <s xml:id="echoid-s39168" xml:space="preserve"> 5 huius quòd forma pũcti f reflectitur <lb/>à pũcto ſpeculi, qđ eſt a, ad uiſum exiſtentẽ in pũcto n:</s> <s xml:id="echoid-s39169" xml:space="preserve"> & locus imaginis formæ pũcti f erit in pun-<lb/>cto k:</s> <s xml:id="echoid-s39170" xml:space="preserve"> & imagines omniũ punctorũ lineæ q f, quæ ſunt ultra punctũ q, erunt intra columnã ſpeculi, <lb/>ut patet ք 34 th.</s> <s xml:id="echoid-s39171" xml:space="preserve"> 5 huius, & ex pręmiſsis.</s> <s xml:id="echoid-s39172" xml:space="preserve"> Si uerò inter punctũ q & pũctũ t (qui eſt finis cõtingẽtiæ) <lb/>ponatur punctus aliquis, utr:</s> <s xml:id="echoid-s39173" xml:space="preserve"> erit angulus r a h maior angulo q a h, ergo & angulo h a p:</s> <s xml:id="echoid-s39174" xml:space="preserve"> fiat ergo ei <lb/>æqualis angulus, qui ſit h a m.</s> <s xml:id="echoid-s39175" xml:space="preserve"> Palàm quò d linea m a producta cadet ſuper lineam g q extra ſectio-<lb/>nẽ.</s> <s xml:id="echoid-s39176" xml:space="preserve"> Ideo enim, quia linea p a continens cum linea a h angulum p a h æqualem angulo q a h, cadit in <lb/>ipſam ſectionẽ in punctum g:</s> <s xml:id="echoid-s39177" xml:space="preserve"> patet quia linea m a ſecabit lineam g q extra ſectionẽ:</s> <s xml:id="echoid-s39178" xml:space="preserve"> ſit q́;</s> <s xml:id="echoid-s39179" xml:space="preserve">, ut cadat in <lb/>punctum o:</s> <s xml:id="echoid-s39180" xml:space="preserve"> erit ergo per 37 th.</s> <s xml:id="echoid-s39181" xml:space="preserve"> 5 huius imago formæ puncti r in puncto o:</s> <s xml:id="echoid-s39182" xml:space="preserve"> & omnium punctorum li <lb/>neæ r q, excepto puncto q, imagines erunt extra ſpeculum, inter puncta o & g.</s> <s xml:id="echoid-s39183" xml:space="preserve"> Si autem angulus q <lb/>a h fuerit minor angulo h a p:</s> <s xml:id="echoid-s39184" xml:space="preserve"> ſecetur ex angulo h a p angulus h a n æqualis angulo q a h per 27 th.</s> <s xml:id="echoid-s39185" xml:space="preserve"> 1 <lb/>huius.</s> <s xml:id="echoid-s39186" xml:space="preserve"> Palàm ergo, ut prius, quòd formæ puncti q imago erit in puncto k:</s> <s xml:id="echoid-s39187" xml:space="preserve"> & omniũ ſuperiorũ pun-<lb/>ctorum lineæ q fimagines erunt intra ſectionem.</s> <s xml:id="echoid-s39188" xml:space="preserve"> Si uerò punctus r ſumatur inferior puncto q, ita <lb/>ut angulus r a h ſit æqualis angulo h a p:</s> <s xml:id="echoid-s39189" xml:space="preserve"> tunc erit imago formę puncti r in ſectionis puncto g, quod <lb/>eſt in ſuperficie ſpeculi:</s> <s xml:id="echoid-s39190" xml:space="preserve"> & omnium punctorum inter r & q imagines erũt intra ſpeculum:</s> <s xml:id="echoid-s39191" xml:space="preserve"> & omniũ <lb/>punctorum inter puncta r & t imagines erunt extra ſpeculi ſuperficiem.</s> <s xml:id="echoid-s39192" xml:space="preserve"> Si uerò angulus q a h fuerit <lb/>maior angulo h a p:</s> <s xml:id="echoid-s39193" xml:space="preserve"> fiat angulus h a m æqualis angulo q a h:</s> <s xml:id="echoid-s39194" xml:space="preserve"> palam q́;</s> <s xml:id="echoid-s39195" xml:space="preserve"> quòd linea m a producta ſeca-<lb/>bit ſectionem:</s> <s xml:id="echoid-s39196" xml:space="preserve"> linea enim e a t eſt contingens ſectionem in puncto a, propter quod linea m a produ <lb/>cta neceſſariò ſectionem ſecabit:</s> <s xml:id="echoid-s39197" xml:space="preserve"> ſecet ergo in puncto b:</s> <s xml:id="echoid-s39198" xml:space="preserve"> & ducatur linea contingens ſectionem in <lb/>puncto b, quæ concurrat cum linea d h in puncto l:</s> <s xml:id="echoid-s39199" xml:space="preserve"> cõcurret autem per 14 th.</s> <s xml:id="echoid-s39200" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39201" xml:space="preserve"> angulus enim <lb/>d b l eſt rectus, & angulus l d b acutus, ducta linea d b:</s> <s xml:id="echoid-s39202" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s39203" xml:space="preserve"> angulus d l b acutus per 32 p 1:</s> <s xml:id="echoid-s39204" xml:space="preserve"> cum an-<lb/>gulus d b l ſit rectus:</s> <s xml:id="echoid-s39205" xml:space="preserve"> eſt ergo per 13 p 1 angulus h l b obtuſus:</s> <s xml:id="echoid-s39206" xml:space="preserve"> linea ergo l b concurret cum linea h d, <lb/>ut patet per 60 th.</s> <s xml:id="echoid-s39207" xml:space="preserve"> 1 huius, ex parte punctorum b & g:</s> <s xml:id="echoid-s39208" xml:space="preserve"> quia quantùm ad hoc eadem ratio eſt in circu <lb/>lis & in ſectionibus:</s> <s xml:id="echoid-s39209" xml:space="preserve"> facietq́;</s> <s xml:id="echoid-s39210" xml:space="preserve"> cum ipſa angulum acutum.</s> <s xml:id="echoid-s39211" xml:space="preserve"> Ducatur ergo perpendicularis ſuper line-<lb/>am l b à puncto b per 11 p 1, quæ ſit b s:</s> <s xml:id="echoid-s39212" xml:space="preserve"> hęc ergo cõiuncta cum linea d b fiet linea una per 14 p 1:</s> <s xml:id="echoid-s39213" xml:space="preserve"> quo-<lb/>niam utraq;</s> <s xml:id="echoid-s39214" xml:space="preserve"> ipſarum cum linea l b in eodem puncto, qui eſt b, continet angulum rectum:</s> <s xml:id="echoid-s39215" xml:space="preserve"> & linea b s <lb/>ſecabit lineam h g:</s> <s xml:id="echoid-s39216" xml:space="preserve"> ſit, ut ſecet ipſam in puncto x.</s> <s xml:id="echoid-s39217" xml:space="preserve"> Et quoniam linea l b protracta concurrit cum li-<lb/>nea h d, & angulus s b l eſt rectus:</s> <s xml:id="echoid-s39218" xml:space="preserve"> patet quòd linea b s cum linea h g ex parte puncti h continet an-<lb/>gulum acutum per 14 th.</s> <s xml:id="echoid-s39219" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39220" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s39221" xml:space="preserve"> angulus s x h acutus:</s> <s xml:id="echoid-s39222" xml:space="preserve"> ergo & angulus g x b illi contrapoſitus <lb/>ſimiliter eſt acutus per 15 p 1.</s> <s xml:id="echoid-s39223" xml:space="preserve"> Quia uerò linea h g ſecat lineam q a, ſit pũctus ſectionis u.</s> <s xml:id="echoid-s39224" xml:space="preserve"> Et quoniã <lb/>angulus h g d eſt rectus, & linea q a concurrit cum linea d g in puncto q:</s> <s xml:id="echoid-s39225" xml:space="preserve"> quoniam omnes hæ lineæ <lb/>ſunt in una ſuperficie:</s> <s xml:id="echoid-s39226" xml:space="preserve"> palàm per 14 th.</s> <s xml:id="echoid-s39227" xml:space="preserve"> 1 huius quòd linea h g cum linea q a continet angulum acu-<lb/>tum ſuper punctum u, qui eſt angulus h u a.</s> <s xml:id="echoid-s39228" xml:space="preserve"> Quia ergo angulus s x h eſt acutus, & angulus q u g con <lb/>trapoſitus angulo h u a per 15 p 1 eſt acutus:</s> <s xml:id="echoid-s39229" xml:space="preserve"> patet per 14 th.</s> <s xml:id="echoid-s39230" xml:space="preserve"> 1 huius quòd lineæ s b & q u cõcurrunt:</s> <s xml:id="echoid-s39231" xml:space="preserve"> <lb/>ſit ergo concurſus ipſarum in puncto z.</s> <s xml:id="echoid-s39232" xml:space="preserve"> Forma itaque puncti z mouebitur ad ſpeculum per lineam <lb/>z a, & reflectetur per lineam a m ad uiſum exiſtentem in puncto m:</s> <s xml:id="echoid-s39233" xml:space="preserve"> & locus imaginis erit punctus <lb/>b:</s> <s xml:id="echoid-s39234" xml:space="preserve"> & loca omnium imaginum punctorum lineæ z s ultra punctum z erũt intra ſectionem:</s> <s xml:id="echoid-s39235" xml:space="preserve"> & omniũ <lb/>punctorum lineæ z b, quæ ſunt citra z, loca imaginum erunt extra ſectionem.</s> <s xml:id="echoid-s39236" xml:space="preserve"> Quod eſt propoſitũ.</s> <s xml:id="echoid-s39237" xml:space="preserve"/> </p> <div xml:id="echoid-div1579" type="float" level="0" n="0"> <figure xlink:label="fig-0601-01" xlink:href="fig-0601-01a"> <variables xml:id="echoid-variables685" xml:space="preserve">s f n h q x r p l z u t m a b o g e k d</variables> </figure> </div> </div> <div xml:id="echoid-div1581" type="section" level="0" n="0"> <head xml:id="echoid-head1197" xml:space="preserve" style="it">50. Lineæ rectæ æquidiſtantis axi ſpeculi columnaris conuexi, centró uiſus exiſtente inea-<lb/>dem ſuperficie, reflexionem poſsibile eſt fieri à tota linea longitudinis ſpeculi ad uiſum: imagó <lb/>eius uidebitur recta, æqualis rei uiſæ. Alhazen 25 n 6.</head> <p> <s xml:id="echoid-s39238" xml:space="preserve">Eſto ſpeculum columnare, ut in 30 huius:</s> <s xml:id="echoid-s39239" xml:space="preserve"> cuius axi z k æquidiſtet linea recta, quæ ſit t h:</s> <s xml:id="echoid-s39240" xml:space="preserve"> erit ergo <lb/>per 30 p 1 & per 92 th.</s> <s xml:id="echoid-s39241" xml:space="preserve"> 1 huius linea t h æquidi <lb/> <anchor type="figure" xlink:label="fig-0602-01a" xlink:href="fig-0602-01"/> ſtans lineę longitudinis ſpeculi columnaris, <lb/>quæ exiſtens in eadem ſuperficie t h z k, ſit li-<lb/>nea a g.</s> <s xml:id="echoid-s39242" xml:space="preserve"> Dico quòd ſi uiſus (cuius centrum <lb/>ſit e) fuerit in eadem ſuperficie t h z k cum <lb/>linea t h, & cum axe z k:</s> <s xml:id="echoid-s39243" xml:space="preserve"> poſsibile eſt ut o-<lb/>mnia puncta lineæ t h reflectantur ad ui-<lb/>ſum e:</s> <s xml:id="echoid-s39244" xml:space="preserve"> quoniam per 30 huius poſsibile eſt, <lb/>ut puncta reflexionis omnium punctorum <lb/>lineæ t h ſint in linea longitudinis colu-<lb/>mnę, quæ eſt g a:</s> <s xml:id="echoid-s39245" xml:space="preserve"> quia illa linea ſuperficiei re-<lb/>flexionis, in qua ſunt uiſus e, & axis z k, & <lb/>linea t h, & ſuperficiei columnæ eſt com-<lb/>munis, ut patet per 93 th.</s> <s xml:id="echoid-s39246" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s39247" xml:space="preserve"> Videbitur <lb/>ergo imago formæ lineæ t h recta:</s> <s xml:id="echoid-s39248" xml:space="preserve"> ideo quia <lb/>quælibet perpendicularis ducta à puncto lineæ t h, erit in eadem ſuperficie cum uiſu & axe:</s> <s xml:id="echoid-s39249" xml:space="preserve"> & pro-<lb/>babuntur loca imaginum punctorum lineæ t h eſſe ſecundum lineam rectam diſpoſita, <lb/>ſicut in ſpeculis planis per 52 th.</s> <s xml:id="echoid-s39250" xml:space="preserve"> 5 huius extitit probatum de lineis rectis <lb/>uiſis.</s> <s xml:id="echoid-s39251" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s39252" xml:space="preserve"/> </p> <div xml:id="echoid-div1581" type="float" level="0" n="0"> <figure xlink:label="fig-0602-01" xlink:href="fig-0602-01a"> <variables xml:id="echoid-variables686" xml:space="preserve">t n q g z m b f f h r a d e k o</variables> </figure> </div> <pb o="301" file="0603" n="603" rhead="LIBER SEPTIMVS."/> </div> <div xml:id="echoid-div1583" type="section" level="0" n="0"> <head xml:id="echoid-head1198" xml:space="preserve" style="it">51. Lineærectæ æquidiſtantes axi ſpeculi columnaris conuexi, uiſu non exiſtente in eadẽ ſu-<lb/>perficie, imago curua uidetur modicæ curuitatis, & minor re uiſa. Alhazen 27 n 6.</head> <p> <s xml:id="echoid-s39253" xml:space="preserve">Sit diſpoſitio, quæ prius in 30 huius:</s> <s xml:id="echoid-s39254" xml:space="preserve"> reflectaturq́;</s> <s xml:id="echoid-s39255" xml:space="preserve"> forma lineæ t h à linea longitudinis ſpeculi, <lb/>quę ſit a g.</s> <s xml:id="echoid-s39256" xml:space="preserve"> Dico quòd imago lineæ th uidebitur aliquando curua:</s> <s xml:id="echoid-s39257" xml:space="preserve"> forma enim puncti eius, quod <lb/>eſt q, ut ſuprà patuit in 30 huius, reflectitur ad uiſum e à puncto ſpeculi b, qui eſt punctus circuli b f:</s> <s xml:id="echoid-s39258" xml:space="preserve"> <lb/>linea ergo à puncto q ducta ad centrũ circuli b f, quod eſt l, quæ erit q l, ipſa eſt cathetus incidentiæ <lb/>formæ puncti q:</s> <s xml:id="echoid-s39259" xml:space="preserve"> quoniam, ut patet per 18 p 3, linea q l eſt perpendicularis ſuper lineam contingentẽ <lb/>circulum b f, cuius peripheria eſt communis ſectio ſuperficiei reflexionis & ſpeculi:</s> <s xml:id="echoid-s39260" xml:space="preserve"> hæc quoq;</s> <s xml:id="echoid-s39261" xml:space="preserve"> ca-<lb/>thetus q l, ut patet, concurret cum perpẽdiculari producta à puncto b, quod eſt punctum reflexio-<lb/>nis, ſuper ipſam ſuperficiem ſpeculi ſuper axẽ z k:</s> <s xml:id="echoid-s39262" xml:space="preserve"> & erit concurſus in puncto axis l, ſcilicet in cẽtro <lb/>circuli b f per 96 th.</s> <s xml:id="echoid-s39263" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s39264" xml:space="preserve"> Cõcurrat ergo linea q l cũ linea m lin puncto axis l:</s> <s xml:id="echoid-s39265" xml:space="preserve"> producatur quoq;</s> <s xml:id="echoid-s39266" xml:space="preserve"> <lb/>linea reflexionis, quæ eſt e b, quouſq;</s> <s xml:id="echoid-s39267" xml:space="preserve"> cõcurrat cum catheto q l:</s> <s xml:id="echoid-s39268" xml:space="preserve"> & ſit punctus concurſus c:</s> <s xml:id="echoid-s39269" xml:space="preserve"> uidebi-<lb/>tur ergo per 37th.</s> <s xml:id="echoid-s39270" xml:space="preserve"> 5 huius imago formę puncti q in puncto c:</s> <s xml:id="echoid-s39271" xml:space="preserve"> & eſt punctus c per 1 p 11 in ſuperficie, <lb/>in qua ſunt linea q h, & axis z k, & linea longitudinis a g.</s> <s xml:id="echoid-s39272" xml:space="preserve"> Itẽ forma puncti t lineę t h reflectitur à pun <lb/>cto ſpeculi g, qui per 10 huius eſt punctus ſectionis oxygonię, cum punctus t ſit altior centro uiſus, <lb/>quod eſt e, nec ipſi ſint in eadẽ ſuperficie.</s> <s xml:id="echoid-s39273" xml:space="preserve"> Eſt autẽ à puncto t, unam tantũ ducere perpẽdicularẽ ſu-<lb/>peripſam oxygoniam ſectionẽ, quæ eſt communis ſectio ſuperficiei reflexionis & ſpeculi, uel ſuper <lb/>lineam contingẽtẽ ſpeculum in puncto aliquo oxygoniæ ſectionis:</s> <s xml:id="echoid-s39274" xml:space="preserve"> per 12 p 1 ſit ducta:</s> <s xml:id="echoid-s39275" xml:space="preserve"> hæc ergo per <lb/>114 th.</s> <s xml:id="echoid-s39276" xml:space="preserve"> 1 huius uel per 44 huius concurret cũ perpẽdiculari ducta à puncto eiuſdẽ ſectiõis, quod eſt <lb/> <anchor type="figure" xlink:label="fig-0603-01a" xlink:href="fig-0603-01"/> g, ſuper <lb/>axẽ z k, <lb/>quæ eſt <lb/>linea n g <lb/>z:</s> <s xml:id="echoid-s39277" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s39278" xml:space="preserve"> <lb/>concur-<lb/>fus ſub <lb/>axe, hoc <lb/>eſt ſub <lb/>pũcto z, <lb/>ꝗ eſt cõ-<lb/>curſus ք <lb/>pendicu <lb/>laris n z, <lb/>& axis z <lb/>k:</s> <s xml:id="echoid-s39279" xml:space="preserve"> quo-<lb/>niam du <lb/>cta linea t z, erit angulus t z n acutus:</s> <s xml:id="echoid-s39280" xml:space="preserve"> ideo quòd angulus n z y eſt rectus, axe k z producto ultra pun <lb/>ctum z ad punctũ y.</s> <s xml:id="echoid-s39281" xml:space="preserve"> Producatur itaq;</s> <s xml:id="echoid-s39282" xml:space="preserve"> linea n z ultra pũctum z ad pũctum x:</s> <s xml:id="echoid-s39283" xml:space="preserve"> & ducatur à pũcto t li-<lb/>nea concurrens cum linea n z producta ultra pũctum z in pũcto x:</s> <s xml:id="echoid-s39284" xml:space="preserve"> concurret autem per 14 th.</s> <s xml:id="echoid-s39285" xml:space="preserve"> 1 hu-<lb/>ius:</s> <s xml:id="echoid-s39286" xml:space="preserve"> ideo quia angulus x n t eſt rectus, uel acutus, & angulus x t n acutus:</s> <s xml:id="echoid-s39287" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s39288" xml:space="preserve"> linea t x axẽ k z in <lb/>pũcto y:</s> <s xml:id="echoid-s39289" xml:space="preserve"> & producatur linea e g ultra pũctum g, donec concurrat cum linea t x:</s> <s xml:id="echoid-s39290" xml:space="preserve"> concurrẽt autẽ per <lb/>29 th.</s> <s xml:id="echoid-s39291" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39292" xml:space="preserve"> linea enim e g producta ſecat angulum t g x:</s> <s xml:id="echoid-s39293" xml:space="preserve"> ergo & baſim t x:</s> <s xml:id="echoid-s39294" xml:space="preserve"> quoniam illę lineę ſunt <lb/>in eadem ſuperficie, ut patet:</s> <s xml:id="echoid-s39295" xml:space="preserve"> ſit ipſarum ſectio in pũcto i:</s> <s xml:id="echoid-s39296" xml:space="preserve"> erit ergo punctus i locus imaginis formæ <lb/>puncti t per 37 th.</s> <s xml:id="echoid-s39297" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s39298" xml:space="preserve"> Similiter ducta à puncto h lineę th, linea, quę ſit orthogonalis ſuper lineã <lb/>contingentem ſpeculum in aliquo pũcto ſectionis oxygonię, à qua reflectitur forma pũcti h ad ui-<lb/>ſum e per 10 huius, illa concurret cum perpendiculari d a r ſub pũcto d, qui eſt pũctus axis per 114 <lb/>th.</s> <s xml:id="echoid-s39299" xml:space="preserve"> 1 huius uel per 44 huius:</s> <s xml:id="echoid-s39300" xml:space="preserve"> concurrat ergo in pũcto p:</s> <s xml:id="echoid-s39301" xml:space="preserve"> & ducatur linea e a ultra punctum a, donec <lb/>concurrat cum linea h p:</s> <s xml:id="echoid-s39302" xml:space="preserve"> & ſit ſecũdum pręmiſſos modos punctus concurſus s:</s> <s xml:id="echoid-s39303" xml:space="preserve"> erit quoq;</s> <s xml:id="echoid-s39304" xml:space="preserve">, ut prius, <lb/>punctus s imago puncti h.</s> <s xml:id="echoid-s39305" xml:space="preserve"> Ducatur quoq;</s> <s xml:id="echoid-s39306" xml:space="preserve"> linea s i:</s> <s xml:id="echoid-s39307" xml:space="preserve"> palàm ergo cum linea t i concurrat in puncto x <lb/>cum perpendiculari n z, quę eſt æquidiſtans lineæ e o, quòd eadem concurret cum linea e o per 2 <lb/>th.</s> <s xml:id="echoid-s39308" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39309" xml:space="preserve"> concurrat ergo in puncto u:</s> <s xml:id="echoid-s39310" xml:space="preserve"> ſimiliter linea h s cum concurrat cum perpẽdiculari d r, quę <lb/>eſt æquidiſtans lineę e o, concurret cum linea e o per 2 th.</s> <s xml:id="echoid-s39311" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s39312" xml:space="preserve"> Sed quoniam ſitus puncti t lineæ <lb/>th reſpectu pũctie, quod eſt centrum uiſus, idem eſt cum ſitu puncti h, & eadem diſtantia à uiſu:</s> <s xml:id="echoid-s39313" xml:space="preserve"> <lb/>quoniam linea t h æquidiſtat axi z k, & ſimiliter puncta t & h æqualiter diſtant à pũcto q, &, ut patet <lb/>ex pręmiſsis in 30 huius, ſitus puncti t & puncti h ad punctum o eſt idem, & punctorum i & s, reſpe-<lb/>ctu puncti o eſt etiam idem ſitus, ut patet ex pręmiſsis in pręſente demonſtratione:</s> <s xml:id="echoid-s39314" xml:space="preserve"> ergo per 1 p 11 e-<lb/>rit linearum ti & h s reſpectu lineæ e o idem ſitus.</s> <s xml:id="echoid-s39315" xml:space="preserve"> Lineę ergo t i & h s concurrent ſuper idem pun-<lb/>ctum lineę e o:</s> <s xml:id="echoid-s39316" xml:space="preserve"> concurrant ergo in pũcto u:</s> <s xml:id="echoid-s39317" xml:space="preserve"> erit ergo t u h triangulus, & in ſuperficie huius triangu-<lb/>li erit linea i s:</s> <s xml:id="echoid-s39318" xml:space="preserve"> axis autem ſpeculi, qui eſt z k, nõ eſt in hac ſuperficie:</s> <s xml:id="echoid-s39319" xml:space="preserve"> uerùm linea t h eſt in eadem ſu-<lb/>perficie cum axe, ut patet ex hypotheſi & per 1 th.</s> <s xml:id="echoid-s39320" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39321" xml:space="preserve"> ergo ſuperficies illa ſecat ſuperſiciem tri-<lb/>anguli t u h ſuper lineam communem, quæ eſt e h, nõ ſuper aliam.</s> <s xml:id="echoid-s39322" xml:space="preserve"> Cum ergo punctus t ſit in ſuper-<lb/>ficie lineę t h, & ſimiliter axis z k ſit in eadem ſuperficie, & punctus c non ſit in linea t h:</s> <s xml:id="echoid-s39323" xml:space="preserve"> ergo non eſt <lb/>in ſuperficie trianguli t u h:</s> <s xml:id="echoid-s39324" xml:space="preserve"> & duo puncta i & s ſuntin ſuperficie illius triaguli:</s> <s xml:id="echoid-s39325" xml:space="preserve"> linea ergo i c s erit <lb/> <pb o="302" file="0604" n="604" rhead="VITELLONIS OPTICAE"/> curua per 1 p 11.</s> <s xml:id="echoid-s39326" xml:space="preserve"> Et quia ipſa eſt imago lineæ t h:</s> <s xml:id="echoid-s39327" xml:space="preserve"> palàm quòd imago lineæ rectæ, quæ eſt t h, eſt cur-<lb/>ua:</s> <s xml:id="echoid-s39328" xml:space="preserve"> quod eſt primum propoſitum.</s> <s xml:id="echoid-s39329" xml:space="preserve"> Sed eius curuitas modica eſt:</s> <s xml:id="echoid-s39330" xml:space="preserve"> quia perpendicularis ducta à pun-<lb/>cto c a d lineam i s, ad punctum ſcilicet ſectionis lineæ i s, & ſuperficiei cirucli eſt ualde parua:</s> <s xml:id="echoid-s39331" xml:space="preserve"> ſed <lb/>quantò maior ſuerit linea uiſa, quę eſt t h æquidiſtans lineæ longitudinis ſpeculi, tantò imago eius.</s> <s xml:id="echoid-s39332" xml:space="preserve"> <lb/>erit minus curua:</s> <s xml:id="echoid-s39333" xml:space="preserve"> & quantò minor fuerit linea th, tantò curuitas erit maior.</s> <s xml:id="echoid-s39334" xml:space="preserve"> Et quoniam linea i c mi <lb/>nor eſt quàm linea t q, & linea s c minor quàm linea h q:</s> <s xml:id="echoid-s39335" xml:space="preserve"> quoniam linea i s, à quo modicùm declinat <lb/>linea i c s, cadit inter lineas t u & h u concurrentes in puncto u, & eſt quaſi æquidiſtans lineæ t h, ſi-<lb/>cut & axi k z:</s> <s xml:id="echoid-s39336" xml:space="preserve"> patet ergo quòd linea imaginis (quæ eſt i c s) minor eſt reuiſa, in qua eſt linea t h:</s> <s xml:id="echoid-s39337" xml:space="preserve"> & <lb/>hoc eſt ſecundum propoſitum.</s> <s xml:id="echoid-s39338" xml:space="preserve"> Patet ergo totum, quod proponebatur.</s> <s xml:id="echoid-s39339" xml:space="preserve"/> </p> <div xml:id="echoid-div1583" type="float" level="0" n="0"> <figure xlink:label="fig-0603-01" xlink:href="fig-0603-01a"> <variables xml:id="echoid-variables687" xml:space="preserve">t i y n g z x q m b c l f h s a d p e k o u</variables> </figure> </div> </div> <div xml:id="echoid-div1585" type="section" level="0" n="0"> <head xml:id="echoid-head1199" xml:space="preserve" style="it">52. Superficie lineæ rectæuiſæ ſuperficiem, in quaest axis ſpeculi columnaris conuexi, ortho-<lb/>gonaliter ſecante, centró uiſus exiſtente in utra ſuperficie: à circumferentia circuli (quiest <lb/>communis ſectio dict arum ſuperſicierum & ſpeculi) fiet reflexio: lineǽ rectæ uiſæ imago erit <lb/>curua. Alhazen 28 n 6.</head> <p> <s xml:id="echoid-s39340" xml:space="preserve">Eſto linea th in ſuperficie plana orthogonaliter ſecante ſuperficiem, in qua ſunt centrum uiſus <lb/>e, & axis dati ſpeculi columnaris, qui ſit d ſ:</s> <s xml:id="echoid-s39341" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s39342" xml:space="preserve"> punctum e in eadẽ <lb/> <anchor type="figure" xlink:label="fig-0604-01a" xlink:href="fig-0604-01"/> ſuperficie cum linea t h:</s> <s xml:id="echoid-s39343" xml:space="preserve"> erit ergo punctum e in linea, in qua illæ duæ <lb/>ſuperficies ſe interſecant:</s> <s xml:id="echoid-s39344" xml:space="preserve"> quod neceſſe eſt eſſe per 19 th.</s> <s xml:id="echoid-s39345" xml:space="preserve"> 1 huius, & <lb/>per 1 p 11.</s> <s xml:id="echoid-s39346" xml:space="preserve"> Dico quòd formæ totius lineæ t h à circumſerentia circuli <lb/>(qui eſt communis ſectio ſuperficiei t h e, & ſuperficiei colũnæ ipſi-<lb/>us ſpeculi) quæ ſit g b, fiet reflexio ad uiſum.</s> <s xml:id="echoid-s39347" xml:space="preserve"> Aut enim centrum ui-<lb/>ſus (quod eſte) erit retro lineam t h:</s> <s xml:id="echoid-s39348" xml:space="preserve"> & tunc, cúm illa linea ſit corpo <lb/>ralis, & non diaphana, eius denſitas occultabit uiſui ſpeculum, & nõ <lb/>fiet reflexio, niſi fortè ſolæ ſormæ capitum lineę, quę ſunt t & h, ap-<lb/>pareant & reſlectantur ad uiſum à circulo ſpeculi, qui eſt b g:</s> <s xml:id="echoid-s39349" xml:space="preserve"> & erit <lb/>formarum horum capitum imago tendens ad curuitatem, ſicut per <lb/>56 th.</s> <s xml:id="echoid-s39350" xml:space="preserve"> 6 huius patuit de ſpeculis ſphæricis conuexis.</s> <s xml:id="echoid-s39351" xml:space="preserve"> Siuerò fuerit li <lb/>nea th diaphana groſſæ diaphanitatis, ut cryſtallus:</s> <s xml:id="echoid-s39352" xml:space="preserve"> de hoc ſermo al-<lb/>ter erit in decimo libro huius ſcientiæ.</s> <s xml:id="echoid-s39353" xml:space="preserve"> Sed ſi linea t h ſiue exiſtente <lb/>diaphana ſiue non, ſuerit uiſus ſub illa inter ipſam ſcilicet & ſpecu-<lb/>lum:</s> <s xml:id="echoid-s39354" xml:space="preserve"> tunc occultabitur pars lineæ t h propter interpoſitionem capi-<lb/>tis, in quo eſt uiſus:</s> <s xml:id="echoid-s39355" xml:space="preserve"> pars autem illa lineæ t h, quę uideri poteſt, non <lb/>obſtante capitis impedimẽto, reflectetur à circulo b g ad uiſum, eo-<lb/>dem penitus modo, quem de ſpeculis ſphæricis conuexis oſtendi-<lb/>mus ſuo loco.</s> <s xml:id="echoid-s39356" xml:space="preserve"> Eſt ergo imago lineæ rectæ t h taliter uiſę ſemper cur-<lb/>ua.</s> <s xml:id="echoid-s39357" xml:space="preserve"> Quòd ſi centrum uiſus e fuerit extra terminos lineę th in eadem <lb/>ſuperficie, ut prius, & fiat reflexio formæ lineę t h ad uiſum:</s> <s xml:id="echoid-s39358" xml:space="preserve"> uidebitur imago lineę t h tota curua, ut <lb/>patet ſecundum præmiſſa.</s> <s xml:id="echoid-s39359" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s39360" xml:space="preserve"/> </p> <div xml:id="echoid-div1585" type="float" level="0" n="0"> <figure xlink:label="fig-0604-01" xlink:href="fig-0604-01a"> <variables xml:id="echoid-variables688" xml:space="preserve">f d b g t e h e</variables> </figure> </div> </div> <div xml:id="echoid-div1587" type="section" level="0" n="0"> <head xml:id="echoid-head1200" xml:space="preserve" style="it">53. Lineæ recæ uiſæ ſuperſicie orthogonaliter axem ſpeculi columnaris conuexi ſecante, cen-<lb/>tró uiſus non exiſtente in eadem ſuperficie, factá reflexione aduiſum æqualiter diſtãtem ab <lb/>extremis illius lineæ: eius imago uidetur maximæ curuitatis. Alhazen 29 n 6.</head> <p> <s xml:id="echoid-s39361" xml:space="preserve">Sit ſuperficies plana, in qua eſt linea t h, orthogonaliter ſecans ſuperficiem, in qua ſunt centrum <lb/>uiſus e, & axis ſpeculi columnaris conuexi, quod ſit b k g:</s> <s xml:id="echoid-s39362" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s39363" xml:space="preserve"> cẽtrum uiſus e non in eadem ſuper-<lb/>ficie cum linea t h:</s> <s xml:id="echoid-s39364" xml:space="preserve"> cuius extrema t & h, ſicut proponitur, æqualiter diſtent à centro uiſus e:</s> <s xml:id="echoid-s39365" xml:space="preserve"> palamq́;</s> <s xml:id="echoid-s39366" xml:space="preserve"> <lb/>per 10 huius quoniam communes ſectiones omnium ſuperficierum reflexionis & ſpeculi, erũt oxy <lb/>goniæ.</s> <s xml:id="echoid-s39367" xml:space="preserve"> Et quoniam ex hypotheſi forma puncti h reflectitur a d uiſum e ab aliquo puncto ſpeculi ꝓ-<lb/>poſiti:</s> <s xml:id="echoid-s39368" xml:space="preserve"> ſit ergo, ut hoc fiat à puncto b per 29 huius.</s> <s xml:id="echoid-s39369" xml:space="preserve"> Et quia punctus t eiuſdem eſt diſtantiæ à puncto <lb/>e (quod eſt centrum uiſus) cuius eſt punctum h:</s> <s xml:id="echoid-s39370" xml:space="preserve"> patet quòd form a puncti t reflectitur ad uiſum c <lb/>ab aliquo puncto ſpeculi:</s> <s xml:id="echoid-s39371" xml:space="preserve"> ſit illud punctum g.</s> <s xml:id="echoid-s39372" xml:space="preserve"> Et cum extrem a puncta lineę h t ſint eiuſdem ſitus & <lb/>longitudinis à centro uiſus e:</s> <s xml:id="echoid-s39373" xml:space="preserve"> erunt etiam puncta reflexionum ſormarum illarum punctorum (quę <lb/>ſunt b & g) eiuſdem diſtantiæ & ſitus à puncto e centro uiſus.</s> <s xml:id="echoid-s39374" xml:space="preserve"> Igitur duo puncta b & g erunt in cir-<lb/>culo æquidiſtante baſibus ſpeculi, qui cadet ſemper inter lineam h t & inter ſuperficiem tranſeun-<lb/>tem centrum uiſus e, & ſecãtem ſpeculum æquidiſtanter baſibus ipſius ſpeculi:</s> <s xml:id="echoid-s39375" xml:space="preserve"> quod ideo accidit, <lb/>quia puncta reflexionum, quę ſunt b & g, plus declinant ad centrum uiſus, ad quod fit reflexio, <lb/>quàm ipſa puncta h & t, quorum formę reflectuntur.</s> <s xml:id="echoid-s39376" xml:space="preserve"> Sit ergo ille ciruclus b z g, cuius centrum ſit <lb/>d:</s> <s xml:id="echoid-s39377" xml:space="preserve"> ducantur itaq;</s> <s xml:id="echoid-s39378" xml:space="preserve"> lineæ incidentiæ, quę ſunt h b & t g:</s> <s xml:id="echoid-s39379" xml:space="preserve"> & lineæ reflexionum, quæ ſunt b e & g e:</s> <s xml:id="echoid-s39380" xml:space="preserve"> & à <lb/>centro d ducantur perpendiculares ſuper lineas circulum b z g contingentes in punctis b & g, quæ <lb/>ſint d g & d b o.</s> <s xml:id="echoid-s39381" xml:space="preserve"> Palamq́;</s> <s xml:id="echoid-s39382" xml:space="preserve"> per 21 huius, quoniam illarum perpendicularium partes, quæ ſunt g d & <lb/>d b ſunt ſemidiametri circuli b z g:</s> <s xml:id="echoid-s39383" xml:space="preserve"> & ducatur linea à puncto d centro circuli ad centrum uiſus, quę <lb/>ſit e d:</s> <s xml:id="echoid-s39384" xml:space="preserve"> & producantur lineæ incidentię, quę ſunt h b & t g, donec concurrant cum linea e d.</s> <s xml:id="echoid-s39385" xml:space="preserve"> Cũ aũt <lb/>puncta h & t ſint eiuſdẽ ſitus & diſtantiæ, reſpectu pũcti e, & reſpectu centri d:</s> <s xml:id="echoid-s39386" xml:space="preserve"> palã quòd lineæ h b <lb/> <pb o="303" file="0605" n="605" rhead="LIBER SEPTIMVS."/> & t g habebunt eundem ſitum, reſpectu lineæ e d:</s> <s xml:id="echoid-s39387" xml:space="preserve"> concurrent ergo in idem punctum illius lineæ <lb/>e t:</s> <s xml:id="echoid-s39388" xml:space="preserve"> eſto quòd concurrant in punctum l:</s> <s xml:id="echoid-s39389" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s39390" xml:space="preserve"> <lb/>linea longitudinis columnæ ſpeculi, in qua ſit pũ-<lb/> <anchor type="figure" xlink:label="fig-0605-01a" xlink:href="fig-0605-01"/> ctus z:</s> <s xml:id="echoid-s39391" xml:space="preserve"> & ſit hęc linea in ſuperſicie plana, in qua eſt <lb/>centrum uiſus & axis ſpeculi:</s> <s xml:id="echoid-s39392" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s39393" xml:space="preserve"> linea a z:</s> <s xml:id="echoid-s39394" xml:space="preserve"> & du-<lb/>cantur lineę l z n & d z c.</s> <s xml:id="echoid-s39395" xml:space="preserve"> Et quoniam ſuperficies, <lb/>in qua ſunt centrum uiſus & axis ſpeculi, interſe-<lb/>cat ſuperficiem, in qua eſt linea t h, ſit punctus li-<lb/>neæ t h, in quo fit h æ c ſectio, punctus q:</s> <s xml:id="echoid-s39396" xml:space="preserve"> & a pun-<lb/>cto q ducatur linea æquidiſtans lineæ d z c:</s> <s xml:id="echoid-s39397" xml:space="preserve"> cadet <lb/>quidem h æc linea per 2 th.</s> <s xml:id="echoid-s39398" xml:space="preserve"> 1 huius ſuper axem ſpe <lb/>culi exuna parte, & ſuper lineam l z n exalia:</s> <s xml:id="echoid-s39399" xml:space="preserve"> ca-<lb/>dat ergo in punctum n lineæ l z n.</s> <s xml:id="echoid-s39400" xml:space="preserve"> Palàm autem <lb/>per 20 th.</s> <s xml:id="echoid-s39401" xml:space="preserve"> 5 huius quoniam angulus h b o (qui <lb/>eſt angulus incidentiæ formæ puncti h) eſt ęqua-<lb/>lis angulo o b e, qui eſt angulus reflexionis:</s> <s xml:id="echoid-s39402" xml:space="preserve"> ſed <lb/>angulus h b o per 15 p 1 eſt æqualis angulo l b d, <lb/>quoniam eſt ei contrapoſitus :</s> <s xml:id="echoid-s39403" xml:space="preserve"> & angulus o b e æ-<lb/>qualis eſt duobus angulis b e d, & b d e per 32 p 1:</s> <s xml:id="echoid-s39404" xml:space="preserve"> <lb/>cum in triangulo e b d ipſe ſit extrinſecus:</s> <s xml:id="echoid-s39405" xml:space="preserve"> angu-<lb/>lus ergo l b d æqualis eſt eiſdem ducobus angulis, <lb/>ſcilicet b e d, & b d e.</s> <s xml:id="echoid-s39406" xml:space="preserve"> Seceturitaq, exangulo l b d <lb/>angulus, qui ſit m b d, æqualis angulo b d e per <lb/>27 th.</s> <s xml:id="echoid-s39407" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39408" xml:space="preserve"> remanet ergo angulus m b l æqua-<lb/>lis angulo b e d.</s> <s xml:id="echoid-s39409" xml:space="preserve"> Quia ergo in triangulo e b m an-<lb/>gulus b e m eſt æqualis angulo m b l trianguli b <lb/>m l, & angulus b m e communis utriq;</s> <s xml:id="echoid-s39410" xml:space="preserve"> illorum tri <lb/>gonorum:</s> <s xml:id="echoid-s39411" xml:space="preserve"> erit per 32 p 1 angulus m b e trigoni <lb/>maioris æqualis angulo m l b trigoni minoris:</s> <s xml:id="echoid-s39412" xml:space="preserve"> eſt <lb/>ergo per 4 p 6 proportio lineæ e m ad b m, ſicut <lb/>lineæ b m ad m l:</s> <s xml:id="echoid-s39413" xml:space="preserve">ergo per 17 p 6 illud, quod fit ex <lb/>ductu lineæ e m in m l æquale eſt quadrato lineę <lb/>b m.</s> <s xml:id="echoid-s39414" xml:space="preserve"> Ducatur quoq;</s> <s xml:id="echoid-s39415" xml:space="preserve"> linea m z.</s> <s xml:id="echoid-s39416" xml:space="preserve"> Et quoniam an-<lb/>gulus b d m maior eſt angulo z d m (quia enim <lb/>angulus s d e eſt æqualis angulo o d e propter <lb/>identitatem ſitus punctorum reflexionum, quæ ſunt b & g à centro uiſus e, quæ cauſſatur, ut <lb/>præoſtenſum eſt, exidentitate ſitus punctorum uiſorum, qui ſunt h & t, reſpectu uiſus e:</s> <s xml:id="echoid-s39417" xml:space="preserve"> angu-<lb/>lus uerò s d e maior angulo z d m, ut totum ſua parte:</s> <s xml:id="echoid-s39418" xml:space="preserve"> ergo & angulus b d m eſt maior angulo <lb/>z d m) ſed & duo latera z d & d m ſunt æqualia duobus lateribus b d & d m:</s> <s xml:id="echoid-s39419" xml:space="preserve"> quoniam d b & <lb/>z d ſunt ex centro ad circumferentiam, & latus d m eſt commune:</s> <s xml:id="echoid-s39420" xml:space="preserve"> erit ergo per 24 p 1 latus m b <lb/>maius latere m z:</s> <s xml:id="echoid-s39421" xml:space="preserve"> illud ergo, quod fit ex ductu lineæ e m in l m, maius eſt quadrato lineę z m:</s> <s xml:id="echoid-s39422" xml:space="preserve"> ſit <lb/>ergo ductus lineę e m in lineam m i, (quę minor eſt quàm ſit linea m l) æqualis quadrato lineæ <lb/>m z:</s> <s xml:id="echoid-s39423" xml:space="preserve"> & ducantur lineæ i b, i z, e z.</s> <s xml:id="echoid-s39424" xml:space="preserve"> Et quia trianguli e z m, & z i m (quorũ cõmunis angulus eſt z m i) <lb/>per 6 p 6 ſunt æquianguli, propter laterum ſuorum proportionalitatem ex 17 p 6, quę continent <lb/>illum communem angulum:</s> <s xml:id="echoid-s39425" xml:space="preserve"> erit ergo angulus m z i æqualis angulo z e i:</s> <s xml:id="echoid-s39426" xml:space="preserve"> eſt ergo angulus m z l <lb/>(qui eſt maior angulo m z i) maior angulo z e d:</s> <s xml:id="echoid-s39427" xml:space="preserve"> ſed quoniam angulus m b d conſtitutus eſt æqua-<lb/>lis angulo b d m:</s> <s xml:id="echoid-s39428" xml:space="preserve"> eritlinea m d æqualis lineæ m b per 6 p 1:</s> <s xml:id="echoid-s39429" xml:space="preserve"> ſed linea m b eſt maior quàm linea m z, <lb/>ut patet expręmiſsis:</s> <s xml:id="echoid-s39430" xml:space="preserve"> ergo linea m d eſt maior quàm linea m z:</s> <s xml:id="echoid-s39431" xml:space="preserve"> ergo per 18 p 1 erit angulus m z d ma <lb/>ior angulo m d z:</s> <s xml:id="echoid-s39432" xml:space="preserve"> igitur angulus d z l maior eſt duobus angulis e d z, & z e d.</s> <s xml:id="echoid-s39433" xml:space="preserve"> Angulus enim d z l <lb/>continet angulum m z l maiorem angulo z e d:</s> <s xml:id="echoid-s39434" xml:space="preserve"> quoniam angulus m zi qui eſt pars anguli m zl, æ-<lb/>qualis eſt angulo z e d, utſuprà patuit.</s> <s xml:id="echoid-s39435" xml:space="preserve"> Item pręter angulum m z l, continet angulus d z l & angulũ <lb/>d z m maiorem angulo m d z:</s> <s xml:id="echoid-s39436" xml:space="preserve"> angulus uerò n z c eſt æqualis anguolo d z l per 15 p 1, & angulus e z c <lb/>per 32 p 1 æqualis eſt duobus angulis z d e & z e d:</s> <s xml:id="echoid-s39437" xml:space="preserve"> eſt ergo angulus n z c maior angulo e z c.</s> <s xml:id="echoid-s39438" xml:space="preserve"> Sece-<lb/>tur ergo ex angulo n z c per 27th.</s> <s xml:id="echoid-s39439" xml:space="preserve"> 1 huius angulus æqualis angulo e z c, qui ſit f z c, ducta linea z f:</s> <s xml:id="echoid-s39440" xml:space="preserve"> <lb/>quę quidem concurret cum linea n q per 2 th.</s> <s xml:id="echoid-s39441" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39442" xml:space="preserve"> quoniam concurrit in puncto z cum linea <lb/>e d æ quidiſtante lineę n q:</s> <s xml:id="echoid-s39443" xml:space="preserve"> concurrat ergo ſuper punctum f.</s> <s xml:id="echoid-s39444" xml:space="preserve"> Cum ergo angulus f z c ſit æqualis an-<lb/>gulo e z c:</s> <s xml:id="echoid-s39445" xml:space="preserve"> palàm per 20 th.</s> <s xml:id="echoid-s39446" xml:space="preserve"> 5 huius quoniam reflectetur forma puncti fad uiſum e à puncto ſpecu-<lb/>li z:</s> <s xml:id="echoid-s39447" xml:space="preserve"> ſed forma puncti q reflectitur ad uiſum ab aliquo puncto lineę longitudinis ſpeculi tranſeun-<lb/>tis per punctum z:</s> <s xml:id="echoid-s39448" xml:space="preserve"> reflectitur ergo à puncto, quod eſt ultra punctum z.</s> <s xml:id="echoid-s39449" xml:space="preserve"> Quia ſi detur, utrefle ctatur <lb/>à puncto, quod ſit citra punctum z, propinquius puncto e, quàm ſit punctum z:</s> <s xml:id="echoid-s39450" xml:space="preserve"> tunc linea ducta à <lb/>puncto q ad illum punctum reflexionis ſecabit lineam f z:</s> <s xml:id="echoid-s39451" xml:space="preserve"> ille ergo punctus ſectionis reflectetur ad <lb/>uiſum e à duobus punctis lineę longitudinis ſpeculi, quæ eſt z a, ſcilicet à puncto z, & ab alio pun-<lb/>cto dato:</s> <s xml:id="echoid-s39452" xml:space="preserve"> quod eſt impoſsibile per 26 huius.</s> <s xml:id="echoid-s39453" xml:space="preserve"> Sumatur ergo punctus reflexionis formę puncti q <lb/>ultra punctum z:</s> <s xml:id="echoid-s39454" xml:space="preserve"> & ſit punctus k:</s> <s xml:id="echoid-s39455" xml:space="preserve"> à quo reflectatur forma puncti q ad uiſum e:</s> <s xml:id="echoid-s39456" xml:space="preserve"> & ducatur linea in-<lb/> <pb o="304" file="0606" n="606" rhead="VITELLONIS OPTICAE"/> cidentię, quę ſit q k, & linea reflexionis, quę e k:</s> <s xml:id="echoid-s39457" xml:space="preserve"> & producatur linea e k, donec concurrat cum linea <lb/>n q:</s> <s xml:id="echoid-s39458" xml:space="preserve"> concurret autem linea e k cum linea n q per 2 th.</s> <s xml:id="echoid-s39459" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39460" xml:space="preserve"> quia concurrit cum linea d c æquidi-<lb/>ſtante lineæ n q:</s> <s xml:id="echoid-s39461" xml:space="preserve"> hęc enim in eadem ſuperſicie eſt inter puncta e & k:</s> <s xml:id="echoid-s39462" xml:space="preserve"> concurrunt itaque lineę e k & <lb/>n q:</s> <s xml:id="echoid-s39463" xml:space="preserve"> & ſit punctus concurſus p:</s> <s xml:id="echoid-s39464" xml:space="preserve"> erit ergo per 37 th.</s> <s xml:id="echoid-s39465" xml:space="preserve"> 5 huius pũctus p locus imaginis formæ puncti q:</s> <s xml:id="echoid-s39466" xml:space="preserve"> <lb/>ſed punctus h reflectitur ad uiſum e à puncto ſectionis oxygonię, cum non ſit in eadem ſuperficie <lb/>cum uiſu e.</s> <s xml:id="echoid-s39467" xml:space="preserve"> Si ergo à puncto h ducatur cathetus incidentiæ formæ punctih, quæ erit linea perpen-<lb/>dicularis ſuper lineam rectam contingẽtem ſectionem oxygoniam in aliquo puncto ipſius ſectio-<lb/>nis:</s> <s xml:id="echoid-s39468" xml:space="preserve"> palàm quia cathetus illa concurret cum perpendiculari o b d ſub axe per 44 huius:</s> <s xml:id="echoid-s39469" xml:space="preserve"> concurrat <lb/>ergo in puncto aliquo.</s> <s xml:id="echoid-s39470" xml:space="preserve"> Similiter à puncto t eſt ducere unam cathetum incidentiæ, lineam ſcilicet <lb/>perpendicularem ſuper ſectionem oxygoniam, à cuius ſectionis puncto reflectitur ſorma punctit <lb/>ad uiſum e, quæ, ſicut prius, concurret cum perpendiculari s g d ſub axe.</s> <s xml:id="echoid-s39471" xml:space="preserve"> Et quoniam ſemidiametri <lb/>b d & g d non poſſunt eſſe linea una, ut patet per 78th.</s> <s xml:id="echoid-s39472" xml:space="preserve"> 4 huius:</s> <s xml:id="echoid-s39473" xml:space="preserve"> palàm per 112 th.</s> <s xml:id="echoid-s39474" xml:space="preserve"> 1 huius quoniam re <lb/>flexio formarum punctorum h & t fit ex hypotheſi, & per 23 nuius à duobus punctis duarum ſectio <lb/>num columnarium ſecundum lineam c d productam trans ſpeculum ſe interſecantium per 24 hu-<lb/>ius, & per 1 p 11, & 19 th.</s> <s xml:id="echoid-s39475" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s39476" xml:space="preserve"> Et quoniam puncta h & t lineæ h t ſunt eiuſdem ſitus, reſpectu lineæ <lb/>e d:</s> <s xml:id="echoid-s39477" xml:space="preserve"> ideo enim quòd illa pũcta h & t ſunt eiuſdem ſitus, reſpectu uiſus e ex hypotheſi, linea uerò e d, <lb/>quæ diameter uiſualis, eſt in eadem ſuperficie cum axe ſpeculi & centro uiſus:</s> <s xml:id="echoid-s39478" xml:space="preserve"> habentergo puncta <lb/>h & t eundem ſitum, reſpectu lineæ e d, & puncta ſectionis ſimiliter, per quæ tranſeunt catheti inci <lb/>dentiæ ductæ à punctis h & t:</s> <s xml:id="echoid-s39479" xml:space="preserve"> & hæc omnia accidunt propter identitatem ſitus punctorum h & t, <lb/>reſpectu uiſus e, & reſpectu lineę e d.</s> <s xml:id="echoid-s39480" xml:space="preserve"> Palàm ergo quòd illæ duę catheti à punctis h & t ductæ ſuper <lb/>illas ſectiones, quarum, ut patet ex pręmiſsis, quælibet concurrit cum linea e d, ambæ cõcurrent in <lb/>eodem puncto lineæ e d:</s> <s xml:id="echoid-s39481" xml:space="preserve"> concurrantergo in puncto u.</s> <s xml:id="echoid-s39482" xml:space="preserve"> Et quia linea e b producta concurret cum li-<lb/>nea h u:</s> <s xml:id="echoid-s39483" xml:space="preserve"> ſit punctus concurſus r:</s> <s xml:id="echoid-s39484" xml:space="preserve"> concurratq́;</s> <s xml:id="echoid-s39485" xml:space="preserve"> linea e g cum linea t u in puncto y:</s> <s xml:id="echoid-s39486" xml:space="preserve"> & ducatur linea r y.</s> <s xml:id="echoid-s39487" xml:space="preserve"> <lb/>Palàm ergo per 37 th.</s> <s xml:id="echoid-s39488" xml:space="preserve"> 5 huius quia pũctum r eſt imago formę punctih, & punctum y eſt imago for-<lb/>mæ punctit.</s> <s xml:id="echoid-s39489" xml:space="preserve"> Habemus quoq;</s> <s xml:id="echoid-s39490" xml:space="preserve"> triangulum e r y, & extra ſuperficiem huius trianguli eſt pũctum z:</s> <s xml:id="echoid-s39491" xml:space="preserve"> ſu <lb/>perficies ergo huius trianguli altior eſt quàm linea e p, ſi cẽtrum uiſus fuerit altius quàm linea h t, <lb/>& eſt baſsior, ſi cẽtrum uiſus fuerit baſsius quàm linea h t:</s> <s xml:id="echoid-s39492" xml:space="preserve"> eſt ergo pũctus p ſemper extra illã ſuper-<lb/>ficiem.</s> <s xml:id="echoid-s39493" xml:space="preserve"> Linea ergo r p y eſt ſemper curua per 1 p 11, ſed ipſa eſt imago lineę th, ut patet per 37 th.</s> <s xml:id="echoid-s39494" xml:space="preserve"> 5.</s> <s xml:id="echoid-s39495" xml:space="preserve"> Eſt <lb/>ergo imago lineę h t modo propoſito ſituatæ, reſpectu centri uiſus & ſpeculi columnaris conuexi, <lb/>ſemper curusa curuitate non modica.</s> <s xml:id="echoid-s39496" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s39497" xml:space="preserve"/> </p> <div xml:id="echoid-div1587" type="float" level="0" n="0"> <figure xlink:label="fig-0605-01" xlink:href="fig-0605-01a"> <variables xml:id="echoid-variables689" xml:space="preserve">c c s o l i g m k z b d t q h p n y r u a x</variables> </figure> </div> </div> <div xml:id="echoid-div1589" type="section" level="0" n="0"> <head xml:id="echoid-head1201" xml:space="preserve" style="it">54. Lineæ rectæ uiſæ non æquidiſt antis axi ſpeculi columnaris conuexi, cuius ſuperficies obli-<lb/>què ſecat axem: imago uidctur curua diuerſæ curuitatis ſecundum diuerſitatem ſuiſitus.</head> <p> <s xml:id="echoid-s39498" xml:space="preserve">Quia enim per 51 huius patet quòd linea recta æquidiſtãs axi ſpeculi columnaris conuexi imagi <lb/>nẽ habet nõ rectam ſed curuam, licet modicę curuitatis:</s> <s xml:id="echoid-s39499" xml:space="preserve"> lineę uerò (cuius ſuperficies orthogona-<lb/>liter ſecat axem ſpeculi, uiſu non exiſtẽte in eadem ſuperficie cũ linea uiſa) imago ſemper uidetur <lb/>curua per proximam pręmiſſam:</s> <s xml:id="echoid-s39500" xml:space="preserve"> palàm per eandẽ quoniã lineęinter has duas ſitę, quę magis acce-<lb/>dunt ad ſitũ lineæ æquidiſtantis lineæ lõgitudinis colũnę, habebũt imagines plus accedẽtes recti-<lb/>tudini:</s> <s xml:id="echoid-s39501" xml:space="preserve"> lineę uerò, quæ plus appropinquãt lineis, quarũ ſuperficies orthogonaliter ſecãt axem, plus <lb/>accedunt in ſuis imaginibus ad curuitatem:</s> <s xml:id="echoid-s39502" xml:space="preserve"> & augmẽtatur uel minuitur curuitas imaginum ſecun <lb/>dum acceſſum uel receſſum linearum ad alterum iſtorum ſituum.</s> <s xml:id="echoid-s39503" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s39504" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1590" type="section" level="0" n="0"> <head xml:id="echoid-head1202" xml:space="preserve" style="it">55. Forma omnis lineæ rectæ incidentis uertici ſpeculi pyramidalis cõuexi obliquè ſuper axẽ, <lb/>reflectitur ad centrum uiſus intra illam & ſuperficiem ſpeculi conſtitutum à linea longitudinis <lb/>ſpeculi: imagó ipſius uidetur curua modicæ curuitatis, cuius conuexitas est ad uiſum. Alha-<lb/>zen 32 n 6.</head> <p> <s xml:id="echoid-s39505" xml:space="preserve">Sit ſpeculum pyramidale conuexum a b c, cuius uertex ſit a, & cuius axis ſit a d:</s> <s xml:id="echoid-s39506" xml:space="preserve"> ſigneturq́;</s> <s xml:id="echoid-s39507" xml:space="preserve"> in ſu-<lb/>perficie conica eius linea longitudinis, utcũq;</s> <s xml:id="echoid-s39508" xml:space="preserve">; cõtingit, quę ſit a z, per 101 th.</s> <s xml:id="echoid-s39509" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39510" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s39511" xml:space="preserve"> per <lb/>punctum z ſuperficies æ quidiſtans baſi pyramidis:</s> <s xml:id="echoid-s39512" xml:space="preserve"> hęc ergo per 100 th.</s> <s xml:id="echoid-s39513" xml:space="preserve"> 1 huius ſecabit pyramidem <lb/>ſpeculi ſecũdum circulum, qui ſit z u:</s> <s xml:id="echoid-s39514" xml:space="preserve"> & ducatur per 11 p 1 à pũcto z perpẽdicularis ſuper lineam lon <lb/>gitudinis z a:</s> <s xml:id="echoid-s39515" xml:space="preserve"> quę producta ad axem ſpeculi, qui eſt a d, cadat in pũctum h:</s> <s xml:id="echoid-s39516" xml:space="preserve"> cõcurret autem cum axe <lb/>per 96 th.</s> <s xml:id="echoid-s39517" xml:space="preserve"> 1 huius, uel per 14 th.</s> <s xml:id="echoid-s39518" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39519" xml:space="preserve"> ideo quia angulus d a z eſt acutus.</s> <s xml:id="echoid-s39520" xml:space="preserve"> E t à pũcto z ducatur linea <lb/>contingens circulum z u per 17 p 3, quę ſit z m:</s> <s xml:id="echoid-s39521" xml:space="preserve"> & ducatur à pũcto a linea cõtinens cum utraq;</s> <s xml:id="echoid-s39522" xml:space="preserve"> linea-<lb/>rum a z & a h, angulum acutum:</s> <s xml:id="echoid-s39523" xml:space="preserve"> quę ſit extra ſuperficiẽ contingẽtem pyramidem ſuper lineam a z:</s> <s xml:id="echoid-s39524" xml:space="preserve"> <lb/>hoc enim eſt poſsibile, cũ angulus h a z ſit acutus.</s> <s xml:id="echoid-s39525" xml:space="preserve"> Sit ergo illa linea a n:</s> <s xml:id="echoid-s39526" xml:space="preserve"> & in ſuperficie, in qua ſunt <lb/>lineę a n & a h, ducatur à puncto h linea continens cum linea a h angulum æqualem angulo z h a <lb/>per 23 p 1:</s> <s xml:id="echoid-s39527" xml:space="preserve"> hęc ergo linea concurret cum linea a n per 14 th.</s> <s xml:id="echoid-s39528" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39529" xml:space="preserve"> ideo quòd, ut patet ex pręmiſsis, <lb/>duo anguli n a h & a h z ſunt acuti.</s> <s xml:id="echoid-s39530" xml:space="preserve"> Sit ergo pũctus concurſus o:</s> <s xml:id="echoid-s39531" xml:space="preserve"> linea itaq;</s> <s xml:id="echoid-s39532" xml:space="preserve"> h o ſecabit circũferentiã <lb/>circuli z u:</s> <s xml:id="echoid-s39533" xml:space="preserve"> ideo enim quod angulus a h o eſt æqualis angulo a h z, oportet quòd lineæ z h & o h ſint <lb/>in eadem ſuperficie.</s> <s xml:id="echoid-s39534" xml:space="preserve"> Secet ergo linea h o peripheriam circuli in puncto u:</s> <s xml:id="echoid-s39535" xml:space="preserve"> & ducatur linea lon-<lb/>gitudinis ſpeculi, quę a u:</s> <s xml:id="echoid-s39536" xml:space="preserve"> & extrahatur linea perpendicularis h z extra ſpeculum ad punctum t:</s> <s xml:id="echoid-s39537" xml:space="preserve"> & <lb/>ducatur linea o z:</s> <s xml:id="echoid-s39538" xml:space="preserve"> & producatur in continuum & directum:</s> <s xml:id="echoid-s39539" xml:space="preserve"> & ſit o z f:</s> <s xml:id="echoid-s39540" xml:space="preserve"> & producatur linea a z ad <lb/>punctum e.</s> <s xml:id="echoid-s39541" xml:space="preserve"> Angulus ergo f z h erit acutus per 15 p 1:</s> <s xml:id="echoid-s39542" xml:space="preserve"> quia linea o z cum linea t z continet angulum <lb/> <pb o="305" file="0607" n="607" rhead="LIBER SEPTIMVS."/> acutum:</s> <s xml:id="echoid-s39543" xml:space="preserve"> eſt enim angulus a z trectus.</s> <s xml:id="echoid-s39544" xml:space="preserve"> Et quia linea o z ſecat ſuperficiem contingentem ſpeculũ ſu-<lb/>per lineam a z, ſuper quam erecta eſt linea h z, ut patet ex pręmiſsis:</s> <s xml:id="echoid-s39545" xml:space="preserve"> angulo itaq;</s> <s xml:id="echoid-s39546" xml:space="preserve"> a z h exiſtente re-<lb/>cto, angulus o z a eſt acutus:</s> <s xml:id="echoid-s39547" xml:space="preserve"> ergo per 15 p 1 relin quitur ut angulus e z f ſit acutus.</s> <s xml:id="echoid-s39548" xml:space="preserve"> A puncto ergo f <lb/>ducatur perpendicularis ſuper lineam a e per 12 p 1:</s> <s xml:id="echoid-s39549" xml:space="preserve"> & producatur in continuum & directũ, donec <lb/>concurrat cum linea a o in puncto n:</s> <s xml:id="echoid-s39550" xml:space="preserve"> concurret autem linea f e cum linea a o per 14 th.</s> <s xml:id="echoid-s39551" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39552" xml:space="preserve"> ideo <lb/>quia angulus e a o eſt acutus, & <lb/>angulus a e n rectus:</s> <s xml:id="echoid-s39553" xml:space="preserve"> & ducatur <lb/> <anchor type="figure" xlink:label="fig-0607-01a" xlink:href="fig-0607-01"/> à puncto e linea e d æquidiſtans <lb/>lineę z h:</s> <s xml:id="echoid-s39554" xml:space="preserve"> erit ergo ք 8 p 11 linea <lb/>e d perp ẽ dicularis ſuper ſuper-<lb/>ficiem cõtingẽtem pyramidẽ ſe-<lb/>cundũ lineam a e:</s> <s xml:id="echoid-s39555" xml:space="preserve"> cũ linea z h ſit <lb/>perpẽdicularis ſuper eandem ſu <lb/>perficiẽ:</s> <s xml:id="echoid-s39556" xml:space="preserve"> & ducatur à pũcto e li-<lb/>nea e l æquidiſtãs lineæ z m:</s> <s xml:id="echoid-s39557" xml:space="preserve"> & i-<lb/>maginetur ſuքficies, in qua ſint <lb/>lineę e l & e d, ſecare pyramidẽ:</s> <s xml:id="echoid-s39558" xml:space="preserve"> <lb/>erit quo q;</s> <s xml:id="echoid-s39559" xml:space="preserve"> cõmunis ſectio huius <lb/>ſuperficiei & ſuperficiei conicæ <lb/>ipſius ſpeculi ſectio oxygonia ք <lb/>103 th.</s> <s xml:id="echoid-s39560" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39561" xml:space="preserve"> quoniã illa ſuperfi <lb/>cies l e d eſt obliqua ſuper axem <lb/>a d.</s> <s xml:id="echoid-s39562" xml:space="preserve"> Sit ergo illa ſectio d e c.</s> <s xml:id="echoid-s39563" xml:space="preserve"> Li-<lb/>nea uerò m z, quę eſt cõtingens <lb/>circulũ z u, eſt perpẽ dicularis ſuper lineã a e per 22 th.</s> <s xml:id="echoid-s39564" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39565" xml:space="preserve"> ideo quia axis a h erectus eſt ſuper ſu <lb/>perficiẽ illius circuli per 89 th.</s> <s xml:id="echoid-s39566" xml:space="preserve"> 1 huius, & linea z m eſt perpẽdicularis ſuper illius circuli ſemidiame <lb/>trũ per 18 p 3:</s> <s xml:id="echoid-s39567" xml:space="preserve"> eſt ergo linea z merecta ſuper ſuperficiẽ a z h, ut patuit in 41 huius:</s> <s xml:id="echoid-s39568" xml:space="preserve"> quoniã ſuperficies <lb/>circuli, & ſuperficies a z h ſunt a dinuicẽ rectę:</s> <s xml:id="echoid-s39569" xml:space="preserve"> ergo linea l e æquidiſtãs lineę z m, per 8 p 11 eſt per-<lb/>pẽdicularis ſuper ſuperficiem a d e:</s> <s xml:id="echoid-s39570" xml:space="preserve"> ergo angulus a e l eſtrectus:</s> <s xml:id="echoid-s39571" xml:space="preserve"> quod tamẽ facilius patet per 29 p 1.</s> <s xml:id="echoid-s39572" xml:space="preserve"> <lb/>Quia enim angulus a z m eſt rectus, erit & angulus a e l rectus:</s> <s xml:id="echoid-s39573" xml:space="preserve"> ſed & angulus a e n eſt rectus:</s> <s xml:id="echoid-s39574" xml:space="preserve"> & ſimi <lb/>liter angulus a e d eſt rectus ք 29 p 1:</s> <s xml:id="echoid-s39575" xml:space="preserve"> ideo quia angulus a z h eſt rectus, & linea e d ęquidiſtat lineę <lb/>z h:</s> <s xml:id="echoid-s39576" xml:space="preserve"> ergo per 5 p 11 lineæ n e, l e, d e ſunt in eadẽ ſuperficie ſectionis:</s> <s xml:id="echoid-s39577" xml:space="preserve"> & linea a e eſt erecta ſuper ſuper-<lb/>ficiẽ illius ſectionis:</s> <s xml:id="echoid-s39578" xml:space="preserve"> cũ oẽs illę lineę cũ linea a e cõcurrant ad angulos æquales & rectos:</s> <s xml:id="echoid-s39579" xml:space="preserve"> ergo linea <lb/>fn eſt in ſuperficie ſectionis.</s> <s xml:id="echoid-s39580" xml:space="preserve"> Protrahatur itaq;</s> <s xml:id="echoid-s39581" xml:space="preserve"> linea d e in continuũ& directũ uſq;</s> <s xml:id="echoid-s39582" xml:space="preserve"> ad pũctum q:</s> <s xml:id="echoid-s39583" xml:space="preserve"> & <lb/>extrahatur à pũcto flinea æ quidiſtãs lineæ d e q, quę ſit f p:</s> <s xml:id="echoid-s39584" xml:space="preserve"> hęc ergo linea æ quidiſtabit lineę h z per <lb/>30 p 1:</s> <s xml:id="echoid-s39585" xml:space="preserve"> & producatur à pũcto z in ſuperficie o z h linea recta continẽs cũ linea z t angulũ æqualẽ an-<lb/>gulo o z t, qui eſt acutus per 13 p 1:</s> <s xml:id="echoid-s39586" xml:space="preserve"> ideo quia, ut ſuprà patuit, angulus o z h eſt obtuſus:</s> <s xml:id="echoid-s39587" xml:space="preserve"> hæc ergo li-<lb/>nea cõcurret cũ linea f p per 2 th.</s> <s xml:id="echoid-s39588" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39589" xml:space="preserve"> quia ſecabit lineã z h æ quidiſtãtẽ lineę f p, & eſt in ſuperfi-<lb/>cie eius:</s> <s xml:id="echoid-s39590" xml:space="preserve"> ꝗ a linea z f eſt in ſuperficie eius:</s> <s xml:id="echoid-s39591" xml:space="preserve"> oẽs aũt lineę æquidiſtãtes ſunt in eadẽ ſuperficie per 1 th.</s> <s xml:id="echoid-s39592" xml:space="preserve"> 1 <lb/>huius:</s> <s xml:id="echoid-s39593" xml:space="preserve"> cõcurrat ergo in pũcto p:</s> <s xml:id="echoid-s39594" xml:space="preserve"> & ſit angulus p z t æqualis angulo o z t.</s> <s xml:id="echoid-s39595" xml:space="preserve"> Et quia angulus o z t eſt æ-<lb/>qualis angulo z f p per 29 p 1, quia eſt extrinſecus illi, & angulus t z p æqualis eſt angulo ſibi coalter <lb/>no, qui eſt angulus z p f:</s> <s xml:id="echoid-s39596" xml:space="preserve"> palã quòd angulus z f p eſt æ qualis angulo z p f:</s> <s xml:id="echoid-s39597" xml:space="preserve"> ergo per 6 p 1 lineæ z f & z <lb/>p ſunt æquales.</s> <s xml:id="echoid-s39598" xml:space="preserve"> Et quia linea f e n eſt in ſuperficie ſectionis, & linea fp eſt æquidiſtãs lineæ e d, quæ <lb/>eſt in ſuperficie ſectionis:</s> <s xml:id="echoid-s39599" xml:space="preserve"> eſt ergo per 2 th.</s> <s xml:id="echoid-s39600" xml:space="preserve"> 1 huius & per 7 p 11 linea f p in ſuperficie illius ſectionis.</s> <s xml:id="echoid-s39601" xml:space="preserve"> <lb/>Producatur quoq;</s> <s xml:id="echoid-s39602" xml:space="preserve"> linea p e:</s> <s xml:id="echoid-s39603" xml:space="preserve"> erit ergo linea p e ſimiliter in ſuperficie ſectionis per 7 p 11.</s> <s xml:id="echoid-s39604" xml:space="preserve"> Et quoniã <lb/>ſuperius declaratũ eſt quòd linea lõgitudinis ſpeculi, quę eſt e a, eſt perpẽdicularis ſuper ſuperficiẽ <lb/>ſectionis:</s> <s xml:id="echoid-s39605" xml:space="preserve"> uterq;</s> <s xml:id="echoid-s39606" xml:space="preserve"> ergo angulus a e p & a e f eſt rectus ք definitionẽ lineę ſuper ſuperficiẽ erectę:</s> <s xml:id="echoid-s39607" xml:space="preserve"> qua-<lb/>dratũ ergo lineę f z ualet duo quadrata linearũ z e & f e ք 4 7 p 1:</s> <s xml:id="echoid-s39608" xml:space="preserve"> ſimiliter quadratũ lineæ z p ualet <lb/>duo quadrata linearũ z e & e p:</s> <s xml:id="echoid-s39609" xml:space="preserve"> ſed quadratũ lineę z f eſt æquale quadrato lineę z p:</s> <s xml:id="echoid-s39610" xml:space="preserve"> quia & linea li-<lb/>neę eſt æqualis ex pręmiſsis:</s> <s xml:id="echoid-s39611" xml:space="preserve"> eſt aũt amborũ cõmune quadratũ lineę z e:</s> <s xml:id="echoid-s39612" xml:space="preserve"> relinquitur ergo quadra-<lb/>tũ lineę f e æquale quadrato lineę e p:</s> <s xml:id="echoid-s39613" xml:space="preserve"> erit ergo linea f e æqualis lineę p e:</s> <s xml:id="echoid-s39614" xml:space="preserve"> ergo per 5 p 1 duo anguli <lb/>e p f & e f p ſunt ęquales.</s> <s xml:id="echoid-s39615" xml:space="preserve"> Sed angulus n e q eſt ęqualis angulo e f p per 29 p 1:</s> <s xml:id="echoid-s39616" xml:space="preserve"> quia eſt ei extrinſecus, <lb/>& angul{us} q e p eſt ęqualis angulo e p f:</s> <s xml:id="echoid-s39617" xml:space="preserve"> quia eſt ei coalternus:</s> <s xml:id="echoid-s39618" xml:space="preserve"> ſunt ergo angulin e q & q e p ęquales.</s> <s xml:id="echoid-s39619" xml:space="preserve"> <lb/>Ergo ք 20 th.</s> <s xml:id="echoid-s39620" xml:space="preserve"> 5 huius form a pũctin reflectetur a d uiſum exiſtẽtẽ in pũcto p à pũcto ſpeculi e:</s> <s xml:id="echoid-s39621" xml:space="preserve"> & for-<lb/>ma pũcti o reflectetur ad uiſum exiſtẽtẽ in pũcto p à pũcto ſpeculi z.</s> <s xml:id="echoid-s39622" xml:space="preserve"> Et omnis linea producta à pũ-<lb/>cto f ad aliquod pũctũ lineę o n, ſecabit lineã z e.</s> <s xml:id="echoid-s39623" xml:space="preserve"> Patet quoq;</s> <s xml:id="echoid-s39624" xml:space="preserve"> ſecũdũ pręmiſſa quòd illa linea erit ę-<lb/>qualis lineę ꝓ ductę à pũcto p ad illud idẽ pũctum:</s> <s xml:id="echoid-s39625" xml:space="preserve"> quia linea a e eſt perpẽdicularis ſuper ſuperficiẽ, <lb/>in qua ſunt lineę p e & f e, quę eſt ſuperficies ſectionis:</s> <s xml:id="echoid-s39626" xml:space="preserve"> & duę lineę f e & p e ſunt æquales:</s> <s xml:id="echoid-s39627" xml:space="preserve"> oẽs ergo <lb/>lineę extractę à pũctis f & p ad aliquod unũ pũctum lineę z e, ſunt ęquales, iterãdo modũ ꝓbandi, <lb/>quo uſi ſumus prius.</s> <s xml:id="echoid-s39628" xml:space="preserve"> Patet ergo quòd forma omnis pũcti, qui eſt in linea o n, reflectetur ad uiſum <lb/>exiſtentẽ in pũcto p exillo pũcto ſpeculi, quod ſecatur in linea z e.</s> <s xml:id="echoid-s39629" xml:space="preserve"> Omnis quoq;</s> <s xml:id="echoid-s39630" xml:space="preserve"> linea extracta ex <lb/>uertice pyramidis, qui eſt a, cadensq́;</s> <s xml:id="echoid-s39631" xml:space="preserve"> obliquè ſuper axem pyramidis ſpeculi, qui eſt a d, itaut angu <lb/>los acutos contineat cũ axe a d, & cũ linea longitudinis, quę eſt a z, uel alia quacũq;</s> <s xml:id="echoid-s39632" xml:space="preserve">;, pręmiſſo mo-<lb/>do demonſtrari poteſt, quia aliqua parsipſius reflectitur ad uiſum tunc diſpoſitum, reſpectu illius <lb/> <pb o="306" file="0608" n="608" rhead="VITELLONIS OPTICAE"/> uiſibilis, ut nunc eſt diſpoſitus punctus p, reſpectu lineæ o n.</s> <s xml:id="echoid-s39633" xml:space="preserve"> Similiterq́;</s> <s xml:id="echoid-s39634" xml:space="preserve"> patet, quòd in hac diſpoſi-<lb/>tione formæ punctorum totius lineæ a o n reflectentur ad uiſum in puncto p exiſtentem.</s> <s xml:id="echoid-s39635" xml:space="preserve"> Et ſi pun-<lb/>ctus fulterius producatur in maiori diſtantia à puncto z:</s> <s xml:id="echoid-s39636" xml:space="preserve"> augmentabitur quantitas lineæ a o n ſe-<lb/>cundum illud.</s> <s xml:id="echoid-s39637" xml:space="preserve"> Et huius quidem ſimile demonſtratum eſt per 41 huius:</s> <s xml:id="echoid-s39638" xml:space="preserve"> nunc uerò hoc præmiſimus <lb/>in hoc propoſito theoremate, ut ſtudioſus in dagator ea, quæ ſequuntur, facilius acceptet.</s> <s xml:id="echoid-s39639" xml:space="preserve"> Omni-<lb/>bus itaq;</s> <s xml:id="echoid-s39640" xml:space="preserve"> his ſuo modo diſpoſitis cõtinuetur linea n d:</s> <s xml:id="echoid-s39641" xml:space="preserve"> ſecabit ergo linea n d circumferentiã ſectio-<lb/>nis:</s> <s xml:id="echoid-s39642" xml:space="preserve"> nam duo puncta d & n ſunt in eadem ſuperficie ſectionis, & punctum n eſt extra circumferen-<lb/>tiam ſectionis, d uerò eſt intra illam:</s> <s xml:id="echoid-s39643" xml:space="preserve"> ſecet ergo linea n d circumferentiam ſectionis in puncto c:</s> <s xml:id="echoid-s39644" xml:space="preserve"> & <lb/>quia triangulus a h o eſt totus in eadem ſuperficie per 2 p 11:</s> <s xml:id="echoid-s39645" xml:space="preserve"> palàm quoniam linea n d erit in ſuperfi <lb/>cie trianguli a o h per 1 p 11:</s> <s xml:id="echoid-s39646" xml:space="preserve"> puncta enim d & n ſunt in lineis a o & a h:</s> <s xml:id="echoid-s39647" xml:space="preserve"> ergo & linea n d eſt in ſuperfi-<lb/>cie eadem cum illis:</s> <s xml:id="echoid-s39648" xml:space="preserve"> erit ergo pun <lb/>ctus c in ſuperficie trianguli a o h.</s> <s xml:id="echoid-s39649" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0608-01a" xlink:href="fig-0608-01"/> Similiter etiam duo puncta a & u <lb/>ſunt in ſuperficie huιus trianguli <lb/>a o h, ut patet ex præmiſsis:</s> <s xml:id="echoid-s39650" xml:space="preserve"> quo-<lb/>niam linea h o ſecabat peripheriá <lb/>circuli z u in pũcto u:</s> <s xml:id="echoid-s39651" xml:space="preserve"> ſic enim uo-<lb/>cauimus pũctum illud.</s> <s xml:id="echoid-s39652" xml:space="preserve"> Tria ergo <lb/>puncta, quę ſunt a & u & c ſunt in <lb/>ſuperficie huius trianguli a o h:</s> <s xml:id="echoid-s39653" xml:space="preserve"> <lb/>ſed puncta a, b, c ſunt omnia in ſu <lb/>perficie ſpeculi:</s> <s xml:id="echoid-s39654" xml:space="preserve"> ergo tria pũcta a, <lb/>u, c ſunt in linea communi ſuper-<lb/>ficiei ſpeculi & ſuperficiei a n d:</s> <s xml:id="echoid-s39655" xml:space="preserve"> <lb/>ſed hæc linea communis eſt linea <lb/>recta per 90 th.</s> <s xml:id="echoid-s39656" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39657" xml:space="preserve"> fit enim <lb/>ſectio ſecũdum axẽ ſpeculi:</s> <s xml:id="echoid-s39658" xml:space="preserve"> ergo <lb/>puncta a, u, c ſunt in linea recta.</s> <s xml:id="echoid-s39659" xml:space="preserve"> <lb/>Protrahatur ergo linea a u rectè <lb/>ad punctum c:</s> <s xml:id="echoid-s39660" xml:space="preserve"> & producatur linea r z ultra punctum z:</s> <s xml:id="echoid-s39661" xml:space="preserve"> quæ ſecabit lineã o h per 29 th.</s> <s xml:id="echoid-s39662" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39663" xml:space="preserve"> ideo <lb/>quia lineæ r z & h o ſunt in eadem ſuperficie, & linea r z, quæ ſecat angulum f z t, ſecat angulum eius <lb/>contrapoſitum, qui eſt h z o:</s> <s xml:id="echoid-s39664" xml:space="preserve"> ergo & baſim illi ſubtenſam, quæ eſt h d, neceſſariò ſecabit:</s> <s xml:id="echoid-s39665" xml:space="preserve"> ſecet ergo <lb/>ipſam in puncto p.</s> <s xml:id="echoid-s39666" xml:space="preserve"> Eſt ergo punctus p in ſuperficie trianguli a o h.</s> <s xml:id="echoid-s39667" xml:space="preserve"> Producatur quoq;</s> <s xml:id="echoid-s39668" xml:space="preserve"> linea a p:</s> <s xml:id="echoid-s39669" xml:space="preserve"> & <lb/>protrahatur ultra p:</s> <s xml:id="echoid-s39670" xml:space="preserve"> ſecabit ergo lineá d n per 29 th.</s> <s xml:id="echoid-s39671" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39672" xml:space="preserve"> quoniã ſecat angulũ d a n:</s> <s xml:id="echoid-s39673" xml:space="preserve"> ſecet quoq;</s> <s xml:id="echoid-s39674" xml:space="preserve"> <lb/>ipſum in puncto g.</s> <s xml:id="echoid-s39675" xml:space="preserve"> Et quia punctus f nõ eſt in ſuperficie contingête pyramidem ſpeculi tranſeunte <lb/>per lineá a z e, ſed obliquè incidit eidẽ, ut patet ex pręmiſsis:</s> <s xml:id="echoid-s39676" xml:space="preserve"> eſt aũt in ſuperficie ſectiõis:</s> <s xml:id="echoid-s39677" xml:space="preserve"> & quoniã <lb/>ſuperficies ſectionis nõ eſt erecta ſuper ſuperficiẽ a d e per 103 th.</s> <s xml:id="echoid-s39678" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39679" xml:space="preserve"> patet per 4 p 11 quia neceſ <lb/>ſariò erit angulus f e d acutus, quoniã angulus a e f eſt rectus:</s> <s xml:id="echoid-s39680" xml:space="preserve"> angulus ergo d e n per 13 p 1 eſt obtu-<lb/>ſus:</s> <s xml:id="echoid-s39681" xml:space="preserve"> ergo angulus e d n eſt acutus per 32 p 1:</s> <s xml:id="echoid-s39682" xml:space="preserve"> cadit ergo in triangulo amblygonio, qui eſt d e n.</s> <s xml:id="echoid-s39683" xml:space="preserve"> Et ſit li <lb/>nea c x contingens ſectionem in puncto c.</s> <s xml:id="echoid-s39684" xml:space="preserve"> Per e a ergo, quæ pręmiſſa ſunt in demõſtratione 45 th.</s> <s xml:id="echoid-s39685" xml:space="preserve"> 5 <lb/>huius, & etiã ex eo quoniã angulus d c x eſt obtuſus:</s> <s xml:id="echoid-s39686" xml:space="preserve"> palã quòd per pẽdicularis extracta ex pũcto c <lb/>ſuper lineã c x cõtingétẽ ſectionẽ, ſecat angulũ d c x:</s> <s xml:id="echoid-s39687" xml:space="preserve"> & qđ cõcurret cũ linea e d ſub pũcto d:</s> <s xml:id="echoid-s39688" xml:space="preserve"> hęc er <lb/>go perpẽdicularis ſecet lineã e d producta m ultra pũctum d in pũcto s:</s> <s xml:id="echoid-s39689" xml:space="preserve"> perpẽdicularis ergo extra-<lb/>cta ex pũcto n ſuper lineã cõtingẽtem ſectionẽ, ſecabit lineã e d ultra pũctum s remotius à pũcto d, <lb/>quàm ſit pũctus s:</s> <s xml:id="echoid-s39690" xml:space="preserve"> ſiue iſtæ perpẽdiculares cũ linea e d cõcurrant ultra circũſerentiam ſectionis, uel <lb/>intra illã.</s> <s xml:id="echoid-s39691" xml:space="preserve"> Perpẽdicularis enim extracta à pũcto n ſuper lineã cõun gentẽ ſectionẽ nó ſecabit angulũ <lb/>d c x, ſicut linea perpẽdicularis ducta à pũcto c ſecat angulũ illũ.</s> <s xml:id="echoid-s39692" xml:space="preserve"> Vt enim patet per 46 huius, & per <lb/>113 th.</s> <s xml:id="echoid-s39693" xml:space="preserve"> 1 erit illa perpendicularis remotior à linea n e, ꝗ̃ ſit linea n d:</s> <s xml:id="echoid-s39694" xml:space="preserve"> hęc ergo perpendiculariter ſecat <lb/>axẽ ſpeculi, qui eſt a d, in pũcto altiori ꝗ̃ ſit pũctum d:</s> <s xml:id="echoid-s39695" xml:space="preserve"> ſit ergo per pẽdicularis extracta à pũcto n ſu-<lb/>per lineam contingẽtem ſectionem in puncto ſuæ incidentiæ linea n q:</s> <s xml:id="echoid-s39696" xml:space="preserve"> & linea r e ſecat lineam n e <lb/>in puncto e, qui eſt pũctus circũferentiæ ſectionis, & eſt in ipſius ſuperficie:</s> <s xml:id="echoid-s39697" xml:space="preserve"> & ſimiliter linea n q eſt <lb/>in ſuperficie ſectionis.</s> <s xml:id="echoid-s39698" xml:space="preserve"> Si ergo linea r e, quæ eſt linea reflexionis, extrahatur in cõtinuum & directũ:</s> <s xml:id="echoid-s39699" xml:space="preserve"> <lb/>palàm quòd ipſa ſecabit lineã n q per 29 th.</s> <s xml:id="echoid-s39700" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39701" xml:space="preserve"> quoniã ipſa protracta ſecat angulũ q e n:</s> <s xml:id="echoid-s39702" xml:space="preserve"> ſecabit <lb/>ergo baſim q n in trigono n e q:</s> <s xml:id="echoid-s39703" xml:space="preserve"> ſit ergo, ut ſecet ipſam in pũcto y.</s> <s xml:id="echoid-s39704" xml:space="preserve"> Itẽ quia pũctũ e (qđ eſt in ſuper-<lb/>ficie ſectionis) eſt extra ſuperficiẽ trigonia n d, patet qđ trigonũ a n d ſecabit ſuքficiẽ ſectiõis:</s> <s xml:id="echoid-s39705" xml:space="preserve"> quia <lb/>ſuperficies a n d nó eſt ſuperficies ſectionis:</s> <s xml:id="echoid-s39706" xml:space="preserve"> cũ, ſicut patet ex pręmiſsis, pũctus a ſit extra ſuperficiẽ <lb/>ſectionis, & linea a e ſit perpẽdicularis ſuper ſuperficiẽ ſectiõis, & pũctus e eſt in circũferẽtia ipſius <lb/>ſectionis:</s> <s xml:id="echoid-s39707" xml:space="preserve"> eſt aũt linea n c d cõmunis ambabus illis ſuperficiebus, trigoni ſcilicet a n d & ſectionis.</s> <s xml:id="echoid-s39708" xml:space="preserve"> er <lb/>go ք 19 th.</s> <s xml:id="echoid-s39709" xml:space="preserve"> 1 huius linea n c d eſt cõmunis ſectio illarũ ſuperficierũ, ſcilicet trigoni a n d & ſectióis, & <lb/>linea n q cõ currit cũ ipſa ſectiõe ultra pũctũ c, ut ſuprà declaratũ eſt:</s> <s xml:id="echoid-s39710" xml:space="preserve"> ergo linea n q eſt ultra ſuperfi-<lb/>ciẽ trigoni a n d:</s> <s xml:id="echoid-s39711" xml:space="preserve"> ſed linea a p g eſt in ipſa ſuperficie trigoni a n d:</s> <s xml:id="echoid-s39712" xml:space="preserve"> pũctus ergo y (qui ք 37 th.</s> <s xml:id="echoid-s39713" xml:space="preserve"> 5 huius <lb/>eſt locus imaginis formæ pũctin, cũ ipſe ſit cõmunis ſectio lineæ reflexionis, q̃ eſt r e, & catheti inci <lb/>dẽtiæ formę pũcti n, q̃ eſt linea n q) erit ultra lineã a p g.</s> <s xml:id="echoid-s39714" xml:space="preserve"> Viſu itaq;</s> <s xml:id="echoid-s39715" xml:space="preserve"> exiſtẽte in pũct o r, & forma alicu <lb/>ius rei uiſę reflexa a d cẽtrũ uiſus in pũcto r à linea lõgitudinis ſpeculi, quæ eſt z e (ut nũc in præce-<lb/> <pb o="307" file="0609" n="609" rhead="LIBER SEPTIMVS."/> dentibus oſtenſum eſt, quòd forma pũcti o reflectitur ad uiſum exiſtentem in punctor à pũcto ſpe-<lb/>culi z:</s> <s xml:id="echoid-s39716" xml:space="preserve"> & forma puncti n à puncto ſpeculi e) tunc punctus p erit locus imaginis formæ puncti o per <lb/>37 th.</s> <s xml:id="echoid-s39717" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s39718" xml:space="preserve"> quoniam ipſe punctus p eſt cõmunis ſectio line æ reflexionis, quæ eſt z r, & catheti in-<lb/>cidentiæ formæ puncti o, qui eſt lιnea o h:</s> <s xml:id="echoid-s39719" xml:space="preserve"> & punctus y eſt locus imaginis formæ punctin:</s> <s xml:id="echoid-s39720" xml:space="preserve"> forma ue <lb/>rò puncti a uidebitur in ſuo loco proprio:</s> <s xml:id="echoid-s39721" xml:space="preserve"> quia eſt in uertice pyramidis:</s> <s xml:id="echoid-s39722" xml:space="preserve"> & erit imago lineæ a o n li-<lb/>nea tranſiens per puncta a, p, y.</s> <s xml:id="echoid-s39723" xml:space="preserve"> Sed hæc linea eſt conuexa, quia punctum y eſt ultra lineam a p g:</s> <s xml:id="echoid-s39724" xml:space="preserve"> ſit <lb/>ergo illa linea imaginis curua, quæ eſt linea a p y.</s> <s xml:id="echoid-s39725" xml:space="preserve"> Iam autem patuit quòd formæ omnium punctorũ <lb/>lineæ a n reflectantur ad uiſum exiſtentem in puncto r à linea lõgitudinis ſpeculi, quæ eſt a e.</s> <s xml:id="echoid-s39726" xml:space="preserve"> Lineę <lb/>ergo reflexionum, per quas reflectuntur illæ formæ, ſunt omnes in ſuperficie trianguli r a e:</s> <s xml:id="echoid-s39727" xml:space="preserve"> omnes <lb/>ergo imagines punctorum lineę a n ſunt in hac ſuperficie:</s> <s xml:id="echoid-s39728" xml:space="preserve"> ergo linea a p y, quæ eſt cõuexa, eſt in hac <lb/>ſuperficie:</s> <s xml:id="echoid-s39729" xml:space="preserve"> & punctus p, qui eſt locus imaginis formæ puncti o, eſt propior centro uiſus, qui eſt pun-<lb/>ctus r, quã ſit punctus y, qui eſt locus imaginis formę puncti n:</s> <s xml:id="echoid-s39730" xml:space="preserve"> propter quod erit conuexitas huius <lb/>imaginis reſpiciens centrum uiſus:</s> <s xml:id="echoid-s39731" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s39732" xml:space="preserve"> conuexitas parua.</s> <s xml:id="echoid-s39733" xml:space="preserve"> Et diameter huius imaginis (quæ dia-<lb/>meter eſt linea a y) erit minor, quã ſit linea a n, cuius imaginis eſt ipſa diameter:</s> <s xml:id="echoid-s39734" xml:space="preserve"> erit aũt illius diuer-<lb/>ſitatis exceſſus in modica quantitate.</s> <s xml:id="echoid-s39735" xml:space="preserve"> Imagines ergo linearum, quæ extrahuntur ex uerticib.</s> <s xml:id="echoid-s39736" xml:space="preserve"> pyra-<lb/>midaliũ ſpeculorum conuexorũ obliquè ſuper axem ſpeculi, cõprehendũtur à uiſu in talib.</s> <s xml:id="echoid-s39737" xml:space="preserve"> ſpeculis <lb/>ſecundum lineam longitudinis ſuæ reflexę:</s> <s xml:id="echoid-s39738" xml:space="preserve"> & apparent conuexę.</s> <s xml:id="echoid-s39739" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s39740" xml:space="preserve"/> </p> <div xml:id="echoid-div1590" type="float" level="0" n="0"> <figure xlink:label="fig-0607-01" xlink:href="fig-0607-01a"> <variables xml:id="echoid-variables690" xml:space="preserve">a o u m h z t b s n c l d q e f p</variables> </figure> <figure xlink:label="fig-0608-01" xlink:href="fig-0608-01a"> <variables xml:id="echoid-variables691" xml:space="preserve">a o l u p m h z t x b q y c n s d g c k f r</variables> </figure> </div> </div> <div xml:id="echoid-div1592" type="section" level="0" n="0"> <head xml:id="echoid-head1203" xml:space="preserve" style="it">56. Omnis forma lineæ rectæ æquidiſt antis latitudini ſpeculi pyramidalis conuexi, uiſu exi-<lb/>ſtente extra eius ſuperficiem ſpeculum æquidiſt anter baſi ſecantem, reflectitur ad uiſum ſecun-<lb/>dum oxygonias ſectiones, imagó ipſius uidetur curua maximæ curuit atis, cuius cõuexit as eſt <lb/>ad uiſum. Alhazen 33 n 6.</head> <p> <s xml:id="echoid-s39741" xml:space="preserve">Eſto ſpeculum pyramidale conuexum:</s> <s xml:id="echoid-s39742" xml:space="preserve"> cuius uertex ſit a:</s> <s xml:id="echoid-s39743" xml:space="preserve"> diameter baſis b c:</s> <s xml:id="echoid-s39744" xml:space="preserve"> eſt ergo ipſius latitu <lb/>do trigonũ a b c:</s> <s xml:id="echoid-s39745" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s39746" xml:space="preserve"> centrum uiſus d, & linea recta uiſa ſit e f æqui-<lb/>diſtans ſuperficiei trigoni a b c, ſitq́;</s> <s xml:id="echoid-s39747" xml:space="preserve"> centrum uiſus d extra ſuperfi-<lb/> <anchor type="figure" xlink:label="fig-0609-01a" xlink:href="fig-0609-01"/> ciem, in qua linea e f exiftente per ipſam ſecaretur ſpeculum ęquidi <lb/>ſtanter ſuæ baſi.</s> <s xml:id="echoid-s39748" xml:space="preserve"> Dico quòd forma lineę e freflectitur ad uiſum d ſe-<lb/>cundum oxygonias ſectiones ſpeculi ſuperficiem ſecantes.</s> <s xml:id="echoid-s39749" xml:space="preserve"> Non e-<lb/>nim poteſt reflecti ſecundum lineam longitudinis ſpeculi:</s> <s xml:id="echoid-s39750" xml:space="preserve"> quoniã <lb/>tunc oporteret, ut cõcurreret cum axe ſpeculi uerſus uerticem per <lb/>41 huius, & quòd obliquè incideret eidem, cuius oppoſitũ dicit hy-<lb/>potheſis:</s> <s xml:id="echoid-s39751" xml:space="preserve"> à ſuperficie uerò iſtorum ſpeculorum ſecũdum circulum <lb/>non fit reflexio per 12 huius.</s> <s xml:id="echoid-s39752" xml:space="preserve"> Oportet ergo de neceſsitate, ut harum <lb/>linearum reflexio cũ fit ad uiſum, ſiat ſecundũ oxygonias ſectiones.</s> <s xml:id="echoid-s39753" xml:space="preserve"> <lb/>Et quoniam catheti incidentię, quæ ſunt perpendiculares ſuper il-<lb/>las oxygonias ſectiones, (quoniã ſunt perpẽdiculares ſuper lineas <lb/>illas ſectiones contingẽtes) cũ lineis reflexionum concurrunt non <lb/>in eadẽ linea ęquidiſtante lineę uiſæ, ſed in lineis diuerſis:</s> <s xml:id="echoid-s39754" xml:space="preserve"> ideo ima-<lb/>gines talium linearum ſic diſpoſitarum reſpectu ſuperficierum iſto-<lb/>rum ſpeculorum uidẽtur curæ:</s> <s xml:id="echoid-s39755" xml:space="preserve"> ſicut de ſpeculis columnarib.</s> <s xml:id="echoid-s39756" xml:space="preserve"> oſtẽ-<lb/>dimus in 53 huius.</s> <s xml:id="echoid-s39757" xml:space="preserve"> Sunt aũt imagines harũ linearum multũ curuæ, <lb/>ita ut ipſarum curuitas ſit maniſeſta ſenſui:</s> <s xml:id="echoid-s39758" xml:space="preserve"> fitq́;</s> <s xml:id="echoid-s39759" xml:space="preserve"> centrum illarũ ima-<lb/>ginum extra ſuperficies, in quibus eſt conuexitas formarum harum linearum:</s> <s xml:id="echoid-s39760" xml:space="preserve"> fiuntq́;</s> <s xml:id="echoid-s39761" xml:space="preserve"> diametriima <lb/>ginum harum linearum multò minores ipſis lineis:</s> <s xml:id="echoid-s39762" xml:space="preserve"> quod accidit propter augmentum ſuę curuita-<lb/>tis.</s> <s xml:id="echoid-s39763" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s39764" xml:space="preserve"/> </p> <div xml:id="echoid-div1592" type="float" level="0" n="0"> <figure xlink:label="fig-0609-01" xlink:href="fig-0609-01a"> <variables xml:id="echoid-variables692" xml:space="preserve">a e f d b c</variables> </figure> </div> </div> <div xml:id="echoid-div1594" type="section" level="0" n="0"> <head xml:id="echoid-head1204" xml:space="preserve" style="it">57. Linearum rectarum ſuperficiebus ſpeculorum pyr amidalium conuexorum non ſecundũ <lb/>concurſum cum uertice axis, ne æquιdiſt anter latitudini ſpeculi, ſed inter hæc obliquè inciden <lb/>tium imagines ſunt curuæ, diuerſæ curuit atis ſecundum modum, quo plus participant ſitib. ex-<lb/>tremis. Alhazen 34 n 6.</head> <p> <s xml:id="echoid-s39765" xml:space="preserve">Quod hic proponitur, ſatis euidentẽ habet cauſſam.</s> <s xml:id="echoid-s39766" xml:space="preserve"> Lineę enim rectę applicatæ his ſpeculis neq;</s> <s xml:id="echoid-s39767" xml:space="preserve"> <lb/>ſecondum lineam longitudinis, ut in 41 & 55 huius, neq;</s> <s xml:id="echoid-s39768" xml:space="preserve"> ęquidiſtanter latitudini ſpeculi, ut in præ-<lb/>miſſa:</s> <s xml:id="echoid-s39769" xml:space="preserve"> medio modo, ſecundum quod plus approximant uni ſitui uel alteri, participant modos curui <lb/>tatis.</s> <s xml:id="echoid-s39770" xml:space="preserve"> Vnde illæ, quę plus approximant in ſuo ſitu lineis exiſtentibus in longitudine ſpeculi, habent <lb/>formas minus conuexas, quę uerò plus approximant lineis ęquidiſtantibus latitudini ſpeculorum, <lb/>habent formas magis manifeſtè conuexas:</s> <s xml:id="echoid-s39771" xml:space="preserve"> ſed tortuosè tamen:</s> <s xml:id="echoid-s39772" xml:space="preserve"> quia, quę appropinquant plus uerti <lb/>ci ſpeculi, habent formas ſtrictiores & conuexiores, quę uerò appropin quant plus baſi ſpeculi, ha-<lb/>bent ſormas ampliores:</s> <s xml:id="echoid-s39773" xml:space="preserve"> ueruntamen omnium illarum imaginum conuexitas erit manifeſta.</s> <s xml:id="echoid-s39774" xml:space="preserve"> Patet <lb/>ergo propoſitum.</s> <s xml:id="echoid-s39775" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1595" type="section" level="0" n="0"> <head xml:id="echoid-head1205" xml:space="preserve" style="it">58. Omnis forma rei uiſæ in ſpeculis pyramidalib. conuexis uidetur pyramidalis, ſimilis ſpecu <lb/>lipyramidalitati. Alhazen 35 n 6.</head> <p> <s xml:id="echoid-s39776" xml:space="preserve">Quod hic proponitur, patet per 40 th.</s> <s xml:id="echoid-s39777" xml:space="preserve"> 6 huius:</s> <s xml:id="echoid-s39778" xml:space="preserve"> quoniam ibidem monſtratum eſt in ſpeculis ſphę <lb/>ricis conuexis, quòd quantò minus fuerit illud ſpeculum, tantò minores erunt circuli cadentes in <lb/>ſuperficie ipſιus:</s> <s xml:id="echoid-s39779" xml:space="preserve"> & ſic imagines erunt propinquiores centro, & ideo erũt minores.</s> <s xml:id="echoid-s39780" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s39781" xml:space="preserve"> <lb/> <pb o="308" file="0610" n="610" rhead="VITELLONIS OPTICAE"/> ſectiones cadentes in aliquo ſpeculo pyramidali:</s> <s xml:id="echoid-s39782" xml:space="preserve"> illę, quæ ſunt propinquiores uertici, ſunt minores <lb/>& ſtrictiores:</s> <s xml:id="echoid-s39783" xml:space="preserve"> & ſic locus imaginis erit propin quior puncto, in quo cum axe ſpeculi cõcurrunt per-<lb/>pendiculares ductæ ſuper ſuperficies, contingentes ipſa ſpecula in punctis reflexionum oxygonia-<lb/>rum ſectionum, à quarum punctis fit reflexio ad uiſum:</s> <s xml:id="echoid-s39784" xml:space="preserve"> erunt ergo illæ imagines minores.</s> <s xml:id="echoid-s39785" xml:space="preserve"> Sectio-<lb/>nes uerò oxygoniæ, quæ ſunt propinquiores baſi, habent contrariam diſpoſitionem alijs ſuperiori <lb/>bus, quoniam ipſæ ſunt ampliores, ut patet per 116 th.</s> <s xml:id="echoid-s39786" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39787" xml:space="preserve"> unde loca imaginum fiunt remotiora <lb/>à puncto, in quo concurrunt prædictæ perpendiculares, ductæ ſuper ſuperficies contingentes ipſa <lb/>ſpecula in punctis reflexionũ:</s> <s xml:id="echoid-s39788" xml:space="preserve"> fiunt ergo imagines maiores.</s> <s xml:id="echoid-s39789" xml:space="preserve"> Et propter hoc accidit, quòd imagines <lb/>formarum uiſarum in ſpeculis pyramidalib.</s> <s xml:id="echoid-s39790" xml:space="preserve"> conuexis fiunt pyramidales, ſimiles pyramidalitati ſpe <lb/>culorum.</s> <s xml:id="echoid-s39791" xml:space="preserve"> Quod enim ex formis fuerit propinquius uertici ſpeculi, erit ſtrictius:</s> <s xml:id="echoid-s39792" xml:space="preserve"> & quod fuerit pro-<lb/>pinquius baſi, erit latius.</s> <s xml:id="echoid-s39793" xml:space="preserve"> Omnino enim forma rei uiſæ, quæ comprehenditur per reſlexionem ab <lb/>aliquo ſpeculorum facta, aſsimilabitur ſuperficiei ſpeculi, à qua reflectitur illa forma, ut patet per <lb/>38 th.</s> <s xml:id="echoid-s39794" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s39795" xml:space="preserve"> Reliquæ uerò omnes fallaciæ, quæ accidunt uiſui ex ſpeculis columnarib.</s> <s xml:id="echoid-s39796" xml:space="preserve"> conuexis, <lb/>accidunt etiam ex iſtis ſpeculis pyramidalib.</s> <s xml:id="echoid-s39797" xml:space="preserve"> conuexis:</s> <s xml:id="echoid-s39798" xml:space="preserve"> unde non eſt hic reiterationi talium immo-<lb/>randum.</s> <s xml:id="echoid-s39799" xml:space="preserve"> Econuerſo etiam quæcunq;</s> <s xml:id="echoid-s39800" xml:space="preserve"> fallacię accidunt in ſpeculis his pyramidalibus, accidunt etiã <lb/>in ipſis columnaribus, excepta pyramidatione imaginum:</s> <s xml:id="echoid-s39801" xml:space="preserve"> quoniã oxygonię ſectiones columnariũ <lb/>ſpeculorũ, quę ſunt eiuſdem decliuitatis ſuper axẽ colũnę, oẽs ſunt ęquales:</s> <s xml:id="echoid-s39802" xml:space="preserve"> & pars omnis talis ſe-<lb/>ctionis cacumen ſpeculi reſpicientis eſt ſimilis parti ſibi ęqualι in eodẽ ſitu reſpicienti baſim ſpecu <lb/>li, quod non eſt in ſectionib.</s> <s xml:id="echoid-s39803" xml:space="preserve"> oxygonijs pyramidum, quę, ut oſtenſum eſt ք 116 th.</s> <s xml:id="echoid-s39804" xml:space="preserve"> 1 huius, omnes ad <lb/>partem baſis pyramidum dilatantur, ſecundum quod circuli ipſas æquidiſtanter baſibus ſecantes <lb/>ſunt maiores, qui circuli omnes in columnis ſunt æquales.</s> <s xml:id="echoid-s39805" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s39806" xml:space="preserve"> propoſitum.</s> <s xml:id="echoid-s39807" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1596" type="section" level="0" n="0"> <head xml:id="echoid-head1206" xml:space="preserve" style="it">59. In ſpeculis columnaribus uel pyramidalιbus conuexis maioribus maior a uidentur idola: <lb/>reí uiſæ propιnquioris imago uidetur maior. Alhazen 36 n 6.</head> <p> <s xml:id="echoid-s39808" xml:space="preserve">Propoſitæ paſsiones aliæq́;</s> <s xml:id="echoid-s39809" xml:space="preserve"> quá plures cõmunes ſunt his ſpeculis columnaribus & pyramidalib.</s> <s xml:id="echoid-s39810" xml:space="preserve"> <lb/>& ſpeculis ſphæricis conuexis:</s> <s xml:id="echoid-s39811" xml:space="preserve"> unde iſtarum paſsionum, ſicut & aliarum communium, idem hinc <lb/>inde demonſtrandi eſt modus.</s> <s xml:id="echoid-s39812" xml:space="preserve"> Verùm ſi in propoſitis his ſpeculis fiat communis ſectio ſuperficiei <lb/>reflexionis & ſpeculi ſectio oxygonia, quæ non accidit in ſpeculis ſphæricis, cum in illis ſolùm ſint <lb/>circuli:</s> <s xml:id="echoid-s39813" xml:space="preserve"> tunc ex his, quæ in hoc noſtro lιbro præmiſimus, hic erit in ipſis ſectionibus, utillic in circu <lb/>lis, demonſtrandum:</s> <s xml:id="echoid-s39814" xml:space="preserve"> patebitq́;</s> <s xml:id="echoid-s39815" xml:space="preserve"> propoſitum ingenio diligenti.</s> <s xml:id="echoid-s39816" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1597" type="section" level="0" n="0"> <head xml:id="echoid-head1207" xml:space="preserve" style="it">60. Poßbile eſt ſpeculum columnare uel pyramidale cõuexum taliter ſiſti, ut intuens uideat <lb/>in aere extra ſpeculum imaginem rei alterius non uiſæ.</head> <p> <s xml:id="echoid-s39817" xml:space="preserve">Sit ſpeculum columnare cõuexum:</s> <s xml:id="echoid-s39818" xml:space="preserve"> cuius linea longitudinis ſit a b c:</s> <s xml:id="echoid-s39819" xml:space="preserve"> quod erigatur ſuք baſim ſuá <lb/>in loco aliquo domus conuenienter amplæ, ita ut linea a c, cuius medius punctus ſit b, ſit erecta ſu-<lb/>per pauimentum domus:</s> <s xml:id="echoid-s39820" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s39821" xml:space="preserve"> linea contingens ſpeculum in puncto b perpendiculariter ſuք <lb/>lineam a b:</s> <s xml:id="echoid-s39822" xml:space="preserve"> quæ ſit d b e, quæ ſecundum puncta d & e tangat parietes domus:</s> <s xml:id="echoid-s39823" xml:space="preserve"> & illa puncta ſignen-<lb/>tur in ipſis domus parietibus.</s> <s xml:id="echoid-s39824" xml:space="preserve"> Superficies itaq;</s> <s xml:id="echoid-s39825" xml:space="preserve"> in qua eſt linea d b e (quę eſt orthogonalis ſuք axẽ <lb/>ſpeculi) palã quoniã ſecat ſpeculũ ſecundum circulũ ք 100 th.</s> <s xml:id="echoid-s39826" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s39827" xml:space="preserve"> Super punctũ itaq;</s> <s xml:id="echoid-s39828" xml:space="preserve"> d parietis <lb/>domus ſignato puncto f, ut propinquius cõuenienter poſsit fieri:</s> <s xml:id="echoid-s39829" xml:space="preserve"> ducatur à pũcto f linea ę quidiftãs <lb/>lineę ſpeculi, quę eſt a b c, cuiuſcũq;</s> <s xml:id="echoid-s39830" xml:space="preserve"> quãtitatis placuerit:</s> <s xml:id="echoid-s39831" xml:space="preserve"> quę ſit g f h:</s> <s xml:id="echoid-s39832" xml:space="preserve"> & eius medius punctus ſit f:</s> <s xml:id="echoid-s39833" xml:space="preserve"> co <lb/>puleturq́;</s> <s xml:id="echoid-s39834" xml:space="preserve"> linea f b:</s> <s xml:id="echoid-s39835" xml:space="preserve"> quę producatur ultra punctũ f trãs murum in punctũ k:</s> <s xml:id="echoid-s39836" xml:space="preserve"> & perforetur paries ſe-<lb/>cũdum li <lb/> <anchor type="figure" xlink:label="fig-0610-01a" xlink:href="fig-0610-01"/> neã g f h, <lb/>ita quòd <lb/>ex alia <lb/>parte ſu-<lb/>perficiei <lb/>muri ma <lb/>ior fiat <lb/>exciſio ri <lb/>mę parie <lb/>tis ꝗ̃ uer <lb/>ſus ſpe-<lb/>culũ, ſi-<lb/>cut con-<lb/>ſueuit fie <lb/>ri in fene <lb/>ſtris do <lb/>morũ:</s> <s xml:id="echoid-s39837" xml:space="preserve"> fi-<lb/>atq́;</s> <s xml:id="echoid-s39838" xml:space="preserve"> totalis illa exciſio rimę ſecũdũ extẽſionẽ lineę b f k:</s> <s xml:id="echoid-s39839" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s39840" xml:space="preserve"> illa rima f k l.</s> <s xml:id="echoid-s39841" xml:space="preserve"> Et à pũcto ſpeculi, qđ eſt <lb/>b, ducatur linea erecta ſuք ſuperficiẽ ſpeculi:</s> <s xml:id="echoid-s39842" xml:space="preserve"> quę erit քpẽdicularis ſuք lineã d b e:</s> <s xml:id="echoid-s39843" xml:space="preserve"> quę educta extra <lb/>ſpeculũ ſit b m.</s> <s xml:id="echoid-s39844" xml:space="preserve"> Angulo quo q;</s> <s xml:id="echoid-s39845" xml:space="preserve"> k b m fiat ſuք punctũ b terminũ lineę m b angulus ęqualis, ꝗ ſit m b n, <lb/>ducta linea b n.</s> <s xml:id="echoid-s39846" xml:space="preserve"> A pũctis quoq;</s> <s xml:id="echoid-s39847" xml:space="preserve"> g & h (quę ſunt extrema pũcta lineę g f h) ducãtur lineę ad ſpeculũ, <lb/>quę ſint g a & h c:</s> <s xml:id="echoid-s39848" xml:space="preserve"> quę productę cócurrant in puncto o ſuքficiei circuli ſecantis ſpeculũ in puncto b:</s> <s xml:id="echoid-s39849" xml:space="preserve"> <lb/> <pb o="309" file="0611" n="611" rhead="LIBER OCTAVVS."/> ducaturq́;</s> <s xml:id="echoid-s39850" xml:space="preserve"> linea b o:</s> <s xml:id="echoid-s39851" xml:space="preserve"> ſacta quo que talireſectione lineæ b n per 3 p 1 ut ipſa fiat æqualis lineæ b o.</s> <s xml:id="echoid-s39852" xml:space="preserve"> Di-<lb/>so quòd ſi in puncto n ponatur centrum uiſus, quòd ad ipſum reflectetur forma lιneæ g f h à linea <lb/>longitudinis ſpeculi, quæ a b c.</s> <s xml:id="echoid-s39853" xml:space="preserve"> Hoc autem patet per 30 huius.</s> <s xml:id="echoid-s39854" xml:space="preserve"> forma quoq;</s> <s xml:id="echoid-s39855" xml:space="preserve"> totius lineæ g f h uide-<lb/>bitur extra ſpeculum ſcilicet inter ſpeculum & inter lineam g f h, ſcilicet citra punctum d lineæ d e <lb/>contingentis ſpeculum in puncto b, ut patet per 49 huius.</s> <s xml:id="echoid-s39856" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s39857" xml:space="preserve"> lineę o g & o h producantur trãs <lb/>murum in puncta p & q, & copuletur linea una, quæ ſit p k q, in quam tabula alιqua depicta ordme-<lb/>tur ultra murum, ita ut media linea ſormæ in illa tabula depictæ ſituetur ſuper lineam p k q, taliterq́;</s> <s xml:id="echoid-s39858" xml:space="preserve"> <lb/>diſponatur, quòd per uiſum exiſtentem in puncto n, uel citra illum uιderi non poſsit forma depιcta <lb/>in tabula:</s> <s xml:id="echoid-s39859" xml:space="preserve"> uidebitur tamẽ uiſu ſic dιſpoſito imago illius formæ in aere reflexa à ſpeculi ſuperficie co <lb/>lumnaris.</s> <s xml:id="echoid-s39860" xml:space="preserve"> Simili quoq;</s> <s xml:id="echoid-s39861" xml:space="preserve"> modo dιligens intuitor poteſt ſiſtere ſpeculum pyramidale conuexũ & cen <lb/>trum uiſus per 41 & 49 huius.</s> <s xml:id="echoid-s39862" xml:space="preserve"> A‘ ſpeculis uerò ſphęricis conuexis adeò regularis reflexio non fiet, <lb/>ut à propoſitis ſpeculis:</s> <s xml:id="echoid-s39863" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s39864" xml:space="preserve"> Secundũ huncitaq;</s> <s xml:id="echoid-s39865" xml:space="preserve"> modum ſtudioſus percontator <lb/>inuigilet:</s> <s xml:id="echoid-s39866" xml:space="preserve"> quoniam hoc, quod hιc pręmiſimus in pręſenti theoremate, exempli cauſſa fecimus, ut ex <lb/>huius librι 7 dιffuſione, uia perquiſitionis diuerſi artificij pateat animę diligenti.</s> <s xml:id="echoid-s39867" xml:space="preserve"/> </p> <div xml:id="echoid-div1597" type="float" level="0" n="0"> <figure xlink:label="fig-0610-01" xlink:href="fig-0610-01a"> <variables xml:id="echoid-variables693" xml:space="preserve">p l e g a k f m b o n q h c d</variables> </figure> </div> </div> <div xml:id="echoid-div1599" type="section" level="0" n="0"> <head xml:id="echoid-head1208" xml:space="preserve">VITELLONIS FI-<lb/>LII THVRINGORVM ET PO-<lb/>LONORVM OPTICAE LIBER OCTAVVS.</head> <p style="it"> <s xml:id="echoid-s39868" xml:space="preserve"><emph style="sc">Notificatis</emph> aliqualiter paßionibus ſpeculorum planorum & cõ-<lb/>uexorum regularium, ut ſphæricorum, columnarium & pyramidalium:</s> <s xml:id="echoid-s39869" xml:space="preserve"> <lb/>ſuperſt nunc, ut de ſpeculorum concauorum proprietatib.</s> <s xml:id="echoid-s39870" xml:space="preserve"> aliqua conſcri-<lb/>bamus, ſicut de illis, in quibus plus reſultat reflexionum diuerſitas & mi-<lb/>rabilis diffuſio naturalium formarum, uiſuum que aſpicientium deceptio multiformis.</s> <s xml:id="echoid-s39871" xml:space="preserve"> <lb/>Specula uerò concaua regularia (prout in 5 huius ſcientiæ libro th.</s> <s xml:id="echoid-s39872" xml:space="preserve"> 8 declar auimus) ſunt <lb/>tantùm tria, ſcilicet ſphæricum, columnare & pyramidale:</s> <s xml:id="echoid-s39873" xml:space="preserve"> inter quæ primò de ſphæri-<lb/>cis concauis in præſentis libro tractabimus, utpote de illis, quorum paßiones ueluti ſimpli <lb/>ciores alijs, in reliqua concaua ſpecula deſcendunt.</s> <s xml:id="echoid-s39874" xml:space="preserve"> Et quoniam principia communia his <lb/>ſpeculis ſphæricis concauis & ſphæricis conuexis, in principio 6 libri ſciẽtiæ huius præmi-<lb/>ſimus:</s> <s xml:id="echoid-s39875" xml:space="preserve"> ideo ipſa, ut ex præmißs ſuppoſita, hic non reiter amus:</s> <s xml:id="echoid-s39876" xml:space="preserve"> ea tamen, quæ propria ſunt <lb/>his ſpeculis, duximus explicanda.</s> <s xml:id="echoid-s39877" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1600" type="section" level="0" n="0"> <head xml:id="echoid-head1209" xml:space="preserve">DEFINITIO.</head> <p> <s xml:id="echoid-s39878" xml:space="preserve">Imaginem conuerſam dicimus, quæ totalem ſitum rei uiſæ uariat:</s> <s xml:id="echoid-s39879" xml:space="preserve"> ut ſi caput <lb/>intuentis, quod eſt ſurſum, uideatur deorſum:</s> <s xml:id="echoid-s39880" xml:space="preserve"> & ſecũdum hoctotus ſitus partium <lb/>imaginis, reſpectu ſitus partium rei uiſæ uarietur.</s> <s xml:id="echoid-s39881" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1601" type="section" level="0" n="0"> <head xml:id="echoid-head1210" xml:space="preserve">THEOREMATA</head> <head xml:id="echoid-head1211" xml:space="preserve" style="it">1. Oppoſito uiſui ſpeculo ſphærico concauo: communis ſectio baſis pyramidis uiſionis & ſuperfi <lb/>ciei concauæ ſpeculi erit circulus ſphæræ, quando magnus, quando minor illo.</head> <p> <s xml:id="echoid-s39882" xml:space="preserve">Quandoq;</s> <s xml:id="echoid-s39883" xml:space="preserve"> enim tota ſphæræ concauæ ſuperficies uidetur, quandoque pars eius maior, quãdo-<lb/>que minor, ut patet per 72 th.</s> <s xml:id="echoid-s39884" xml:space="preserve"> 4 huius.</s> <s xml:id="echoid-s39885" xml:space="preserve"> Secundum hoc ergo illa communis ſectio baſis pyramidis ui <lb/>ſionis & ſuperficiei ſpeculi uariatur.</s> <s xml:id="echoid-s39886" xml:space="preserve"> Cum autem ſuperficies baſis pyramidis ſit ſuperficies plana, & <lb/>ſuperficies concauorum ſpeculorum ſit ſphærica:</s> <s xml:id="echoid-s39887" xml:space="preserve"> patet per 110 th.</s> <s xml:id="echoid-s39888" xml:space="preserve"> 1 huius quod ipſorum communis <lb/>ſectio ſemper eſt circulus.</s> <s xml:id="echoid-s39889" xml:space="preserve"> Hic ergo quando que eſt circulus magnus, ut quando tranſit cẽtrum ſpe-<lb/>culi:</s> <s xml:id="echoid-s39890" xml:space="preserve"> quando que minor circulo magno, ut cum non tranſit centrum ſpeculi, ſed cadit extra illud.</s> <s xml:id="echoid-s39891" xml:space="preserve"> Pa <lb/>tet ergo propoſitum.</s> <s xml:id="echoid-s39892" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1602" type="section" level="0" n="0"> <head xml:id="echoid-head1212" xml:space="preserve" style="it">2. Communem ſectionem ſuperficiei reflexionis & ſuperficici ſpeculi ſphærici concauineceſſe <lb/>eſt circulum magnum uel arcum circuli magni ſuæ ſphæræ eſſe: ex quo patet, quòd omnis ſuperfi-<lb/>cies reflexionis ſecat ſphæram ſpeculi concaui per æqualia.</head> <p> <s xml:id="echoid-s39893" xml:space="preserve">Huius propoſiti theorematis non eſt alia demonſtratio, quàm quæ ſacta eſt ſuprà in 1 th.</s> <s xml:id="echoid-s39894" xml:space="preserve"> 6 huius:</s> <s xml:id="echoid-s39895" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0611-01a" xlink:href="fig-0611-01"/> <pb o="310" file="0612" n="612" rhead="VITELLONIS OPTICAE"/> ubi idem proponitur de ſpeculis ſphęricis conuexis.</s> <s xml:id="echoid-s39896" xml:space="preserve"> Et quia ſphæræ concauitas ſic reſpicit centrũ, <lb/>ſicut & ipſius conuexitas, & ſuperficies reflexionis eſt ſemper ſuքfi <lb/>cies plana erecta ſuper ſuperficiem ſpeculi per 25 th.</s> <s xml:id="echoid-s39897" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s39898" xml:space="preserve"> patet <lb/> <anchor type="figure" xlink:label="fig-0612-01a" xlink:href="fig-0612-01"/> propoſitum:</s> <s xml:id="echoid-s39899" xml:space="preserve"> quoniam idem erit modus demonſtrandi hic, qui ſu-<lb/>prà.</s> <s xml:id="echoid-s39900" xml:space="preserve"> Eſto enim ſpeculum ſphæricum concauum a b c:</s> <s xml:id="echoid-s39901" xml:space="preserve"> cuius centrũ <lb/>d:</s> <s xml:id="echoid-s39902" xml:space="preserve"> & ſit centrum uiſus g:</s> <s xml:id="echoid-s39903" xml:space="preserve"> reflectaturq́ue forma puncti e ad uiſum g à <lb/>puncto ſpeculi b.</s> <s xml:id="echoid-s39904" xml:space="preserve"> Dico quòd ſuperficiei reflexionis, quę e ſt e b g, & <lb/>ſuperficiei ſpeculi communis ſectio eſt circulus a b c.</s> <s xml:id="echoid-s39905" xml:space="preserve"> Sit enim ſu-<lb/>perſicies plana contingens ſph ęram in puncto b, à quo puncto eri-<lb/>gatur linea f b ſuper ſuperficiem ſpeculum in illo puncto b contin-<lb/>gentem per 12 p 11:</s> <s xml:id="echoid-s39906" xml:space="preserve"> hæc ergo cadet neceſſariò in ipſam ſuperficiem <lb/>reflexionis per 27 th.</s> <s xml:id="echoid-s39907" xml:space="preserve"> 5 huius, & eadem linea f b producta ultra pun <lb/>ctũ b neceſſariò tranſibit centrũ ſphæræ per 72 th.</s> <s xml:id="echoid-s39908" xml:space="preserve"> 1 huius, qđ eſt d:</s> <s xml:id="echoid-s39909" xml:space="preserve"> <lb/>ꝓducta quoq;</s> <s xml:id="echoid-s39910" xml:space="preserve"> fit diameter ſphærę:</s> <s xml:id="echoid-s39911" xml:space="preserve"> ergo & circuli magni illius ſphę <lb/>rę.</s> <s xml:id="echoid-s39912" xml:space="preserve"> Et quoniam hęc diameter communis eſt 4uperficiei reflexionis <lb/>& ipſi ſphærę:</s> <s xml:id="echoid-s39913" xml:space="preserve"> palàm ergo propoſitum.</s> <s xml:id="echoid-s39914" xml:space="preserve"/> </p> <div xml:id="echoid-div1602" type="float" level="0" n="0"> <figure xlink:label="fig-0611-01" xlink:href="fig-0611-01a"> <image file="0611-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0611-01"/> </figure> <figure xlink:label="fig-0612-01" xlink:href="fig-0612-01a"> <variables xml:id="echoid-variables694" xml:space="preserve">f b a d c g e</variables> </figure> </div> </div> <div xml:id="echoid-div1604" type="section" level="0" n="0"> <head xml:id="echoid-head1213" xml:space="preserve" style="it">3. In omni ſuperficie reflexionis à ſpeculis ſphæricis concauis, <lb/>centrum uiſus: centrum ſpeculi: punctum reflexionis: punctum <lb/>uiſum: terminuḿ diametri uiſualis à centro uiſus per cẽtrum <lb/>ſphæræ ductæ ad ſphæræ ſuperficiem, conſistere eſt neceſſe. Alha-<lb/>zen 23. 45 n 4.</head> <p> <s xml:id="echoid-s39915" xml:space="preserve">Cum ſuperficies reflexionis contineat lineam incidentiæ & reflexionis:</s> <s xml:id="echoid-s39916" xml:space="preserve"> palàm quoniam conti-<lb/>net punctum rei uiſæ, cuius forma reflectitur, & punctum reflexionis, à quo reflectitur, & centrum <lb/>uiſus, ad quod reflectitur.</s> <s xml:id="echoid-s39917" xml:space="preserve"> Et quoniam communis ſectio ſuperficiei reflexionis & ſuperficiei ſpeculi <lb/>ſphęrici concaui eſt circulus magnus per æqualia diuidens ſphęram per pręmiſſam:</s> <s xml:id="echoid-s39918" xml:space="preserve"> palàm quia in <lb/>qualibet ſuperficie reflexionis eſt centrum ſpeculi:</s> <s xml:id="echoid-s39919" xml:space="preserve"> quia quælibet ipſarũ tranſit cẽtrum ſphærę ipſi-<lb/>us ſpeculi, cum quęlibet illarum ſuperficierum ſit erecta ſuper ſuperficiem planam ſpeculum in pũ-<lb/>cto reflexionis contingentem per 25 th.</s> <s xml:id="echoid-s39920" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s39921" xml:space="preserve"> ſed per 1 p 11 producta diametro uiſuali per centrũ <lb/>uiſus & centrum ſphęrę, terminus illius diametri neceſſariò erit in eadem ſuperficie cum alijs duo-<lb/>bus ſuis punctis.</s> <s xml:id="echoid-s39922" xml:space="preserve"> Parædicta ergo quinq;</s> <s xml:id="echoid-s39923" xml:space="preserve"> puncta neceſſariò ſunt in omni ſuperficie reflexionis, quę ſit <lb/>à propoſitis ſpeculis.</s> <s xml:id="echoid-s39924" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s39925" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1605" type="section" level="0" n="0"> <head xml:id="echoid-head1214" xml:space="preserve" style="it">4. Centro uiſus uel puncto rei uiſæ in centro ſpeculi ſphærici concaui exiſtente: à quolibet pun <lb/>cto fiet reflexio in ſe ipſum. Ex quo patet quòd in hoc ſitu uiſus non comprehendet niſi ſetantùm: <lb/>& quòd punctus rei uiſæ exiſtens in centro ſpeculi nõ reflectitur aliqualiter ad uiſum. Euclides <lb/>24 th. catoptr. Ptolemæus 1 th. 2 catoptr. Alhazen 44 n 4. Item 62 n 5.</head> <p> <s xml:id="echoid-s39926" xml:space="preserve">Eſto ſpeculum ſphæricum concauum, cuius centrum ſit a:</s> <s xml:id="echoid-s39927" xml:space="preserve"> & ſignetur in ipſo aliquis ſuorum ma-<lb/>gnorum circulorum, qui b c d e:</s> <s xml:id="echoid-s39928" xml:space="preserve"> & centrũ uiſus ſit in centro ſpeculi, quod eſt punctum a.</s> <s xml:id="echoid-s39929" xml:space="preserve"> Dico quòd <lb/>à quocunque puncto ſiet reflexio ad uiſum:</s> <s xml:id="echoid-s39930" xml:space="preserve"> ſemper oportet ut reflectatur radius in ſeipſum.</s> <s xml:id="echoid-s39931" xml:space="preserve"> Dato <lb/>enim quòd à puncto b fiat reflexio ad centrum ſpeculi a, in quo eſt cẽtrum uiſus:</s> <s xml:id="echoid-s39932" xml:space="preserve"> palàm ergo per 72 <lb/>th.</s> <s xml:id="echoid-s39933" xml:space="preserve"> 1 huius quoniam linea b a, quę eſt linea reflexionis, eſt per pendicularis ſuper ſuperficiem cõtin-<lb/>gentem ſpeculum in puncto b:</s> <s xml:id="echoid-s39934" xml:space="preserve"> ſed omnis perpen-<lb/>dicularis in ſe ipſam ſem per reflectitur per 21 th.</s> <s xml:id="echoid-s39935" xml:space="preserve"> 5 <lb/> <anchor type="figure" xlink:label="fig-0612-02a" xlink:href="fig-0612-02"/> huius.</s> <s xml:id="echoid-s39936" xml:space="preserve"> Si ergo linea b a eſt per pẽdicularis ſuper ſu <lb/>perficiem ſpeculi:</s> <s xml:id="echoid-s39937" xml:space="preserve"> palàm quia linea incidens fuit <lb/>perpendicularis, & eadem cũ linea b a.</s> <s xml:id="echoid-s39938" xml:space="preserve"> Dato enim <lb/>oppoſito, ſequitur angulũ incidentię inę qualẽ eſſe <lb/>angulo reflexionis:</s> <s xml:id="echoid-s39939" xml:space="preserve"> quod eſt contra 20 th.</s> <s xml:id="echoid-s39940" xml:space="preserve"> 5 huius, <lb/>& impoſsibile.</s> <s xml:id="echoid-s39941" xml:space="preserve"> Linea itaq;</s> <s xml:id="echoid-s39942" xml:space="preserve"> a b reflectitur in ſeipſã, <lb/>ut ipſa eſt ſacta linea b a.</s> <s xml:id="echoid-s39943" xml:space="preserve"> Et quo niam in hoc ſitu ui <lb/>ſus, omnes lineę incidẽtes ſuperficiei ſpeculi, ſunt <lb/>ſemidiametri ipſius:</s> <s xml:id="echoid-s39944" xml:space="preserve"> palàm quoniam omnes angu <lb/>li incidẽtię ſunt inter ſe ęquales per 43 th.</s> <s xml:id="echoid-s39945" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s39946" xml:space="preserve"> <lb/>quia ſunt anguli ſemicirculorum.</s> <s xml:id="echoid-s39947" xml:space="preserve"> Reflectuntur er-<lb/>go neceſſariò in ſeipſos:</s> <s xml:id="echoid-s39948" xml:space="preserve"> uidebiturq́;</s> <s xml:id="echoid-s39949" xml:space="preserve"> in tota ſuper-<lb/>ficie ſpeculi forma aſpicientis oculi una forma, & <lb/>apud ſuperficiem ſpeculi apparebit:</s> <s xml:id="echoid-s39950" xml:space="preserve"> & nulla alia <lb/>forma tunc uidebitur reflecti ad uiſum.</s> <s xml:id="echoid-s39951" xml:space="preserve"> Et ex hoc <lb/>patet, cũ uiſus ſuerit in centro a, quòd ipſe uidebit <lb/>ſe à quolibet pũcto ſpeculi dati քpẽdiculariter:</s> <s xml:id="echoid-s39952" xml:space="preserve"> & qđ nihil aluid uidebit ք reflexionẽ à ſuքficie ſpe-<lb/>ouli:</s> <s xml:id="echoid-s39953" xml:space="preserve"> quoniã ab uno pũcto ſpeculi ad cẽtrum plures perpendiculares duci nõ eſt poſsibile, ut patet <lb/>per 20 th.</s> <s xml:id="echoid-s39954" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s39955" xml:space="preserve"> Similiter neq;</s> <s xml:id="echoid-s39956" xml:space="preserve"> punctus rei uiſę exiſtens in centro uiſus reflectitur ad uiſum, ſed ſo-<lb/>lùm in ſe ipſum:</s> <s xml:id="echoid-s39957" xml:space="preserve"> quoniã omnes lineę incidentię ſunt perpendiculares ſuper ſuperficiẽ ſpeculi:</s> <s xml:id="echoid-s39958" xml:space="preserve"> unde <lb/> <pb o="311" file="0613" n="613" rhead="LIBER OCTAVVS."/> non reflectentur niſiin ſeipſas.</s> <s xml:id="echoid-s39959" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s39960" xml:space="preserve"> Ethæc quidem dicta ſunt, non præſtante im <lb/>pedimentum uiſui capitis denſitate.</s> <s xml:id="echoid-s39961" xml:space="preserve"> Siergo centrum uiſus hominis uidentis conſtitutum fueritin <lb/>diametro ſphæræ ſpeculi concaui, & in centro eius (cum quęlibet linea à uiſu ad ſuperficiem ſpecu <lb/>li ducta ſit perpendιcularis ſuperipſam) tunc, ut prius demonſtratum eſt, comprehendetuiſus ſe <lb/>ipſum, & non comprehendetur forma alιcuius puncti ſpeculi, niſi puncti portionis circuli interia-<lb/>centis lineas longitudinis pyramidis uiſualis, quę à centro ſpeculi intelligitur protendi:</s> <s xml:id="echoid-s39962" xml:space="preserve"> quoniam <lb/>forma cuiuslibet alterius puncti cadetin ſpeculum ſuper lineam à uiſu declinatam, & neceſſariò re <lb/>fle ctetur ſuper illam lineam declinatam.</s> <s xml:id="echoid-s39963" xml:space="preserve"> Quare linea reflexionis non tranſibit per centrum ſpeculi:</s> <s xml:id="echoid-s39964" xml:space="preserve"> <lb/>& ita non pertinget ad centrum uiſus.</s> <s xml:id="echoid-s39965" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s39966" xml:space="preserve"/> </p> <div xml:id="echoid-div1605" type="float" level="0" n="0"> <figure xlink:label="fig-0612-02" xlink:href="fig-0612-02a"> <variables xml:id="echoid-variables695" xml:space="preserve">b c a e d</variables> </figure> </div> </div> <div xml:id="echoid-div1607" type="section" level="0" n="0"> <head xml:id="echoid-head1215" xml:space="preserve" style="it">5. Centro uiſus exiſtente in aliqua ſemidiametro ſpeculi ſphærici concaui extra centrum ſpe-<lb/>culi: impoßibile eſt ad uiſum reflecti formam alicuius punctorũ illius ſemidiametriobliquè ſpe-<lb/>culo incidentem: reliquæ uerò ſemidiametri eſt poßibile. Alhazen 63 n 5.</head> <p> <s xml:id="echoid-s39967" xml:space="preserve">Hoc, quod hic proponitur, euidenter declaratur.</s> <s xml:id="echoid-s39968" xml:space="preserve"> Sienim cẽtrum uiſus fueritin in ſemidiametro a-<lb/>liqua propoſiti ſpeculi, ſed non in centro:</s> <s xml:id="echoid-s39969" xml:space="preserve"> non comprehendet uiſus formá alicuius puncti ſemidia-<lb/>metri, in qua eſt, obliquè ſpeculo incidentem:</s> <s xml:id="echoid-s39970" xml:space="preserve"> quoniam angulus, quem efficient duę lineæ, quarum <lb/>una ducitur à puncto ſumpto in illa ſemidiametro, & alia à centro uiſus in idem ſpeculi punctũ, nó <lb/>poterit diuidi per lineam perpendicularem ab illo puncto ſpeculi ductam:</s> <s xml:id="echoid-s39971" xml:space="preserve"> cum illa perpendicularis <lb/>tẽdat ad centrũ ſpeculi, formam uerò alicuius puncti alterius ſemidiametri coniunctæ ſemidiame-<lb/>tro, in qua eſt centrum uiſus, ad complendam diametrum ſpeculi, in qua conſtitutus eſt uiſus, obli-<lb/>què ſpeculo incidentem percipere poteſt uiſus:</s> <s xml:id="echoid-s39972" xml:space="preserve"> utpote formam illius puncti, à quo ducta linea in-<lb/>cidentię ad aliquod punctum ſpeculi, ab eodem puncto ſpeculi ducta linea reflexionis ad uifum, an <lb/>gulus ab illis lineis contentus diuiditur per æ qualia per lineam ab illo puncto reflexionis ad cen-<lb/>trum ſpeculi productam.</s> <s xml:id="echoid-s39973" xml:space="preserve"> Hæc enim eſt proprietas reflexionis in omnibus ſpeculis, ut angulum à li-<lb/>nea incidentię & linea reflexionis contentum diuidat perpendicularis à puncto reflexionis ducta <lb/>per æqualia per 26th.</s> <s xml:id="echoid-s39974" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s39975" xml:space="preserve"> Ille ergo punctus poteritin ſpeculo uideri:</s> <s xml:id="echoid-s39976" xml:space="preserve"> & non eſt niſi unicus talis <lb/>punctus in quibuſcunq;</s> <s xml:id="echoid-s39977" xml:space="preserve"> diametris ſpeculi conſiſtens, qui ab uno circulo ſpeculi ad uiſum reflecti <lb/>poſsit.</s> <s xml:id="echoid-s39978" xml:space="preserve"> Quoniam centro ſpeculi, ad quod terminatur perpendicularis ducta à puncto reflexionis, <lb/>& centro oculi exiſtentibus fixis, erit punctus ab uno circulo ſpeculi reflexus ſemper unus:</s> <s xml:id="echoid-s39979" xml:space="preserve"> à diuer <lb/>ſis uerò circulis ſpeculi diuerſa puncta diametri poſsibile eſt reflecti.</s> <s xml:id="echoid-s39980" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s39981" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1608" type="section" level="0" n="0"> <head xml:id="echoid-head1216" xml:space="preserve" style="it">6. Poſito uiſu extra centrum ſpeculiſphærici concauι: à quolibet puncto ſpeculi poteſt fierifor-<lb/>mæ alterius reflexio ad uiſum, niſiſolùm ab illo puncto, cuiincidit diameter uiſualis. Alha-<lb/>zen 45 n 4.</head> <p> <s xml:id="echoid-s39982" xml:space="preserve">Eſto per 2 huius communis ſectio ſuperficiei reflexionis, & ſuperficiei ſpeculi ſphærici concaui <lb/>circulus magnus, qui fit g d c:</s> <s xml:id="echoid-s39983" xml:space="preserve"> cuius centrum ſit b:</s> <s xml:id="echoid-s39984" xml:space="preserve"> & centrum uiſus ſit a:</s> <s xml:id="echoid-s39985" xml:space="preserve"> & ducatur à cẽtro uiſus per <lb/>b centrum ſpeculi diameter uiſualis:</s> <s xml:id="echoid-s39986" xml:space="preserve"> quę ſit a b d, incidens ſuperficiei ſpeculi in puncto d.</s> <s xml:id="echoid-s39987" xml:space="preserve"> Dico <lb/>quòd à quolibet puncto ſpeculi dati poteſt fieri reflexio formę puncti alterius rei uiſibilis ad uiſum <lb/>a, niſi à ſolo puncto d.</s> <s xml:id="echoid-s39988" xml:space="preserve"> Sit enim datus alius punctus, qui ſit g:</s> <s xml:id="echoid-s39989" xml:space="preserve"> ducaturq́e ad ipſum ſemidiameter <lb/>b g:</s> <s xml:id="echoid-s39990" xml:space="preserve"> & continuetur linea reflexionis, quæ ſit g a:</s> <s xml:id="echoid-s39991" xml:space="preserve"> & ducatur linea f g l contingens circulum magnum <lb/>ſpeculi tranſeuntem pũcta g, d, c.</s> <s xml:id="echoid-s39992" xml:space="preserve"> Palàm per 18 p 3 quia anguli b g f & b g l ſunt recti.</s> <s xml:id="echoid-s39993" xml:space="preserve"> Patet etiam per <lb/> <anchor type="figure" xlink:label="fig-0613-01a" xlink:href="fig-0613-01"/> 42 th.</s> <s xml:id="echoid-s39994" xml:space="preserve"> 1 huius quoniam angulus b g a erit acutus:</s> <s xml:id="echoid-s39995" xml:space="preserve"> ca-<lb/>dit enim linea a g inter diametrum & lineam contin <lb/>gentem f g l, quæ eſt extra ſpeculum, ubicunq;</s> <s xml:id="echoid-s39996" xml:space="preserve"> pona <lb/>tur eſſe cẽtrum uiſus, ſiue intra ſiue extra circulum <lb/>g c d.</s> <s xml:id="echoid-s39997" xml:space="preserve"> Conſtituatur quoque per 23 p 1 in eiuſdem cir-<lb/>culi ſuperficie ſuper lineam l g ad punctum g angu-<lb/>lus æ qualis angulo f g a:</s> <s xml:id="echoid-s39998" xml:space="preserve"> qui ſit h g l:</s> <s xml:id="echoid-s39999" xml:space="preserve"> erit ergo angu-<lb/>lus h g b æ qualis angulo b g a.</s> <s xml:id="echoid-s40000" xml:space="preserve"> Et quoniam angulus <lb/>contingentiæ eſt minimus angulorum per 16 p 3:</s> <s xml:id="echoid-s40001" xml:space="preserve"> pa <lb/>làm quòd ab angulo b g l recto abſciſſo quocunque <lb/>angulo acuto rectilineo, ſemper linea illum acutum <lb/>angulum continẽs cadetintra circulum g c d:</s> <s xml:id="echoid-s40002" xml:space="preserve"> quo-<lb/>niam ſolus angulus contingentiæ cadit extra circu <lb/>lum.</s> <s xml:id="echoid-s40003" xml:space="preserve"> Poſito itaque quocunque puncto uiſibili in li-<lb/>nea h g:</s> <s xml:id="echoid-s40004" xml:space="preserve"> ſemper fiet reflexio formæ alicuius ſui pun-<lb/>cti ad uiſum a.</s> <s xml:id="echoid-s40005" xml:space="preserve"> Et eodem modo de quolibet alio ſpe-<lb/>culi puncto extra punctum d dato dem onſtrãdum.</s> <s xml:id="echoid-s40006" xml:space="preserve"> <lb/>Sed & à puncto d fit reflexio.</s> <s xml:id="echoid-s40007" xml:space="preserve"> Cum enim linea a d ſit <lb/>perpendicularis ſuper ſuperficiem contingentem <lb/>ſpeculum in puncto d:</s> <s xml:id="echoid-s40008" xml:space="preserve"> palàm quia linea a d reflectitur in ſeipſam per 21 th.</s> <s xml:id="echoid-s40009" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s40010" xml:space="preserve"> Siergo aliquid <lb/>interponatur non diaphanum inter centrum uiſus, quod eſt a, & punctum ſpeculi d, nulla fiet refle-<lb/>xio ad uiſum impediẽte medio:</s> <s xml:id="echoid-s40011" xml:space="preserve"> ſi uerò nullum tale interponatur, ſolius puncti ſuperficiei oculi for-<lb/>ma uidebitur ab eodem oculo, nihilq́;</s> <s xml:id="echoid-s40012" xml:space="preserve"> aliud, Et hoc eſt propoſitum.</s> <s xml:id="echoid-s40013" xml:space="preserve"/> </p> <div xml:id="echoid-div1608" type="float" level="0" n="0"> <figure xlink:label="fig-0613-01" xlink:href="fig-0613-01a"> <variables xml:id="echoid-variables696" xml:space="preserve">f g l c a b d h</variables> </figure> </div> <pb o="312" file="0614" n="614" rhead="VITELLONIS OPTICAE"/> </div> <div xml:id="echoid-div1610" type="section" level="0" n="0"> <head xml:id="echoid-head1217" xml:space="preserve" style="it">7. In ſpeculis ſphæricis concauis ſi ſupraperipheriam uelextra ponatur centrum uiſus: oculus <lb/>non uidetur, niſi per diametrum ſpeculireflectatur. Euclides 25 th catoptr.</head> <p> <s xml:id="echoid-s40014" xml:space="preserve">Sit ſpeculi concaui ſphęrici circulus magnus a b g:</s> <s xml:id="echoid-s40015" xml:space="preserve"> ſitq́ue centrum uiſus in puncto b ſuper ſpecu <lb/>li peripheriam:</s> <s xml:id="echoid-s40016" xml:space="preserve"> & ducantur lineę b a & b g non per centrum.</s> <s xml:id="echoid-s40017" xml:space="preserve"> Et quoniam angulus maioris portio-<lb/> <anchor type="figure" xlink:label="fig-0614-01a" xlink:href="fig-0614-01"/> <anchor type="figure" xlink:label="fig-0614-02a" xlink:href="fig-0614-02"/> nis, ut patet per 43th.</s> <s xml:id="echoid-s40018" xml:space="preserve"> 1 huius, eſt maior:</s> <s xml:id="echoid-s40019" xml:space="preserve"> angulus uerò reflexionis ſem-<lb/>per debet eſſe æ qualis angulo incidentiæ, ut patet per 20 th.</s> <s xml:id="echoid-s40020" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s40021" xml:space="preserve"> <lb/>palàm quia non fiet reflexιo ſecundum lineam a b, ſed fiet ad partem <lb/>maioris anguli:</s> <s xml:id="echoid-s40022" xml:space="preserve"> & ſimiliter eſt de puncto g:</s> <s xml:id="echoid-s40023" xml:space="preserve"> quoniam non fiet reflexio <lb/>ſecundum lineam b g, ſed ad partem anguli maioris per 33 th.</s> <s xml:id="echoid-s40024" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s40025" xml:space="preserve"> <lb/>Si enim forma puncti b à punctis a & g reflecteretur in ſeipſam:</s> <s xml:id="echoid-s40026" xml:space="preserve"> tunc <lb/>anguli portιonum ad punctum a & ad punctũ g eſſent æquales:</s> <s xml:id="echoid-s40027" xml:space="preserve"> quod <lb/>eſt impoſsibile, & contra 43 th.</s> <s xml:id="echoid-s40028" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s40029" xml:space="preserve"> per diametrum tamen cuiuſ-<lb/>cunq;</s> <s xml:id="echoid-s40030" xml:space="preserve"> circuli magni totius ſpeculi ſphærici concaui poteſt uiſus inci-<lb/>dens reflectι in ſeipſum:</s> <s xml:id="echoid-s40031" xml:space="preserve"> quoniam omnium ſemicirculorũ eiuſdem cir <lb/>culi anguli ſunt æ quales per idem 43 th.</s> <s xml:id="echoid-s40032" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s40033" xml:space="preserve"> ſed tunc non fiet <lb/>reflexio niſi unius puncti ſuperficiei ſpeculi diametraliter incidentιs, <lb/>ut ſecundum lineam b c:</s> <s xml:id="echoid-s40034" xml:space="preserve"> qui non percipιtur, quia indiuiſibilis eſt:</s> <s xml:id="echoid-s40035" xml:space="preserve"> & <lb/>omne, quod uidetur, diuiſibile eſt, quia ſub angulo uidetur per 18th.</s> <s xml:id="echoid-s40036" xml:space="preserve"> 3 <lb/>huius:</s> <s xml:id="echoid-s40037" xml:space="preserve"> alιj uerò puncti incιdẽtes obliquè reflectuntur ad partẽ anguli <lb/>maioris, & non perueniunt ad uiſum, niſi illi, quorum reflexionis li-<lb/>neę incidunt ſuperficiei uifus, & figuranturin illo partium rei uiſæ ſi-<lb/>tibus permutatis:</s> <s xml:id="echoid-s40038" xml:space="preserve"> quod autem non ſic reflectitur, non uidetur.</s> <s xml:id="echoid-s40039" xml:space="preserve"> In his <lb/>itaq;</s> <s xml:id="echoid-s40040" xml:space="preserve"> ſpeculis ſphæricis concauis, ſiſupra peripheriam ſpeculi pona-<lb/>tur centrum uiſus, non uidetur oculus, niſi per diametrum ſpeculire-<lb/>flectatur.</s> <s xml:id="echoid-s40041" xml:space="preserve"> Idem enim accidit, ſi extra peripheriam ſpeculi propoſiti o-<lb/>culus ponarur:</s> <s xml:id="echoid-s40042" xml:space="preserve"> & eodem modo dem onſtrandum:</s> <s xml:id="echoid-s40043" xml:space="preserve"> quoniam linearum inæqualitas naturam reflexio <lb/>nis non immutat per 20 th.</s> <s xml:id="echoid-s40044" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s40045" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s40046" xml:space="preserve"/> </p> <div xml:id="echoid-div1610" type="float" level="0" n="0"> <figure xlink:label="fig-0614-01" xlink:href="fig-0614-01a"> <variables xml:id="echoid-variables697" xml:space="preserve">b a g c</variables> </figure> <figure xlink:label="fig-0614-02" xlink:href="fig-0614-02a"> <variables xml:id="echoid-variables698" xml:space="preserve">a b g c</variables> </figure> </div> </div> <div xml:id="echoid-div1612" type="section" level="0" n="0"> <head xml:id="echoid-head1218" xml:space="preserve" style="it">8. Ab alter a parte productæ diametri extra circulum ſpeculi ſphærici concaui uiſu poſito, ſi-<lb/>ue in tranſuerſali diametro, ſiue extraillam, ſiue citr a illam: nihilrerum in illa parte diſpoſit a-<lb/>rum poßibile eſt uideri. Euclides 26th. catoptr.</head> <p> <s xml:id="echoid-s40047" xml:space="preserve">Eſto communis ſectio ſuperficiei reflexionis & ſpeculi ſphærici concaui circulus a g d:</s> <s xml:id="echoid-s40048" xml:space="preserve"> cuius cen <lb/>trum ſit z:</s> <s xml:id="echoid-s40049" xml:space="preserve"> & producatur ſemidιameter z g extra peculum ad punctum h ducaturq́;</s> <s xml:id="echoid-s40050" xml:space="preserve"> à centro z per <lb/>11 p 1 alia diameter perpendiculariter ſuper lincam h g, quæ a z d:</s> <s xml:id="echoid-s40051" xml:space="preserve"> & ſit centrum uiſus in puncto b <lb/>ab altera parte diametri h g:</s> <s xml:id="echoid-s40052" xml:space="preserve"> & à puncto b ducatur linea ęquidiſtans lineæ h g per 31 p 1, quę ſit linea <lb/>b e, incidens ſuperſiciei ſpeculi m puncto e.</s> <s xml:id="echoid-s40053" xml:space="preserve"> Dico quòd nulla rerum uiſibilium poſitarum ab illa par <lb/>te diametri h g, & lineę b e, in qua ſcilicet eſt uiſus, poteſt uideri, Detur enim, ſi ſit poſsibile, ut pun-<lb/> <anchor type="figure" xlink:label="fig-0614-03a" xlink:href="fig-0614-03"/> ctus q ab illa parte poſitus ad uiſum exiſtentem in pũ-<lb/>cto b reflexus ualeat uideri:</s> <s xml:id="echoid-s40054" xml:space="preserve"> incidatq́ forma puncti q <lb/>ad punctum ſpeculi, quod eſt e, producta linea inci-<lb/>dentiæ, quæ ſit q e:</s> <s xml:id="echoid-s40055" xml:space="preserve"> & à puncto e contingens circulum <lb/>per 17 p 3:</s> <s xml:id="echoid-s40056" xml:space="preserve"> quæ ſit p e o:</s> <s xml:id="echoid-s40057" xml:space="preserve"> & ducatur linea e z.</s> <s xml:id="echoid-s40058" xml:space="preserve"> Si ergo for <lb/>ma puncti q à pũcto ſpeculi e reflectatur ad uiſum exi-<lb/>ſtentem in puncto b:</s> <s xml:id="echoid-s40059" xml:space="preserve"> palàm per 20 th.</s> <s xml:id="echoid-s40060" xml:space="preserve"> 5 huius quoni-<lb/>am angulus q e o erit æqualis angulo b e p:</s> <s xml:id="echoid-s40061" xml:space="preserve"> ſed angu <lb/>lus b e p eſt maior angulo recto:</s> <s xml:id="echoid-s40062" xml:space="preserve"> quia per 18 p 3 eſt an-<lb/>gulus z e p rectus:</s> <s xml:id="echoid-s40063" xml:space="preserve"> ergo & angulus q e o eſt maior re-<lb/>cto:</s> <s xml:id="echoid-s40064" xml:space="preserve"> quod eſt contra 13 p 1.</s> <s xml:id="echoid-s40065" xml:space="preserve"> Palàm ergo quòd forma pũ-<lb/>cti q non reflectitur à puncto e ad uiſum b.</s> <s xml:id="echoid-s40066" xml:space="preserve"> Sed neque <lb/>ab aliquo alio puncto arcus e d:</s> <s xml:id="echoid-s40067" xml:space="preserve"> quoniam idem accidit <lb/>impoſsibile.</s> <s xml:id="echoid-s40068" xml:space="preserve"> Sed ſuper terminum lineæ z e per 23 p 1 <lb/>conſtituto angulo æquali angulo b e z, poſsibile erit <lb/>punctorum lineę productæ, quæ ſit r e, formas à pun-<lb/>cto e reflecti ad uiſum exiſtentem in puncto b.</s> <s xml:id="echoid-s40069" xml:space="preserve"> Idem <lb/>quoque patet uiſu poſito in puncto t citra diametrum <lb/>a d, producta linea t k:</s> <s xml:id="echoid-s40070" xml:space="preserve"> uel poſito ipſo in puncto m <lb/>diametri a d, ducta linea m n:</s> <s xml:id="echoid-s40071" xml:space="preserve"> copulatis quoque lineis z k, z n, & facta deductione ut prius.</s> <s xml:id="echoid-s40072" xml:space="preserve"> Patet <lb/>ergo propoſitum.</s> <s xml:id="echoid-s40073" xml:space="preserve"/> </p> <div xml:id="echoid-div1612" type="float" level="0" n="0"> <figure xlink:label="fig-0614-03" xlink:href="fig-0614-03a"> <variables xml:id="echoid-variables699" xml:space="preserve">o p o p o p d n f m t z a q q q b h r</variables> </figure> </div> </div> <div xml:id="echoid-div1614" type="section" level="0" n="0"> <head xml:id="echoid-head1219" xml:space="preserve" style="it">9. In concauis ſpeculιs ſphæricis ſi inter centrum ſpeculi & peripheriam fuerit punctũ reiui-<lb/>ſæ: poßibile eſt, ut quando in centro unιus uiſus à diuerſis punctιs ſpeculi lineæ reflexionis con-<lb/>currant. Euclides 6 th. catoptr.</head> <p> <s xml:id="echoid-s40074" xml:space="preserve">Sitſpeculum ſphæricum concauum, cuius maior circulus ſit a g:</s> <s xml:id="echoid-s40075" xml:space="preserve"> centrum quoq:</s> <s xml:id="echoid-s40076" xml:space="preserve"> ſit punctus d:</s> <s xml:id="echoid-s40077" xml:space="preserve"> & <lb/> <pb o="313" file="0615" n="615" rhead="LIBER OCTAVVS."/> ſit punctus rei uiſæ b conſtitutus inter centrum d & peripheriam circuli a g:</s> <s xml:id="echoid-s40078" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s40079" xml:space="preserve"> reflexio formæ <lb/>puncti b à puncto ſpeculi, quod ſit a ι & à puncto ſpeculi, quod eſt g.</s> <s xml:id="echoid-s40080" xml:space="preserve"> Dico quòd lineæ inciden-<lb/>tiæ, quæ ſunt b a & b g, poſſunt reflecti ad centrum unius uiſus in puncto uno exiſtentis.</s> <s xml:id="echoid-s40081" xml:space="preserve"> Sit enim <lb/>primo, ut linea b g reflectatur ad uiſum exiſtentem in puncto p:</s> <s xml:id="echoid-s40082" xml:space="preserve"> producantur quoq;</s> <s xml:id="echoid-s40083" xml:space="preserve"> lineæ inciden-<lb/>tiæ à punctis a & g ad aliam partem peripheriæ, quæ ſint lineæ a c & g h.</s> <s xml:id="echoid-s40084" xml:space="preserve"> Hæ ergo lineæ aut ſunt <lb/>æquales, aut inæquales.</s> <s xml:id="echoid-s40085" xml:space="preserve"> Sint primò æquales:</s> <s xml:id="echoid-s40086" xml:space="preserve"> erit ergo arcus a g c per 28 p 3 æqualis arcui g c h:</s> <s xml:id="echoid-s40087" xml:space="preserve"> <lb/>erit ergo per 43 th.</s> <s xml:id="echoid-s40088" xml:space="preserve"> 1 huius angulus portionis (qui eſt c a g) æqualis angulo portionis, qui eſt b g c:</s> <s xml:id="echoid-s40089" xml:space="preserve"> <lb/>ſed & angulus h g c eſt æqualis angulo p g a per hypotheſim & per 20 th.</s> <s xml:id="echoid-s40090" xml:space="preserve"> 5 huius, quoniam angulus <lb/>incidentiæ eſt æqualis angulo reflexionis:</s> <s xml:id="echoid-s40091" xml:space="preserve"> & angulus c a g ſit æqualis angulo l a i:</s> <s xml:id="echoid-s40092" xml:space="preserve"> relinquitur ergo <lb/>æqualibus angulis hinc & inde ablatis, ut angulus h g p ſit æqualis angulo c a l.</s> <s xml:id="echoid-s40093" xml:space="preserve"> Sit autem punctus, <lb/> <anchor type="figure" xlink:label="fig-0615-01a" xlink:href="fig-0615-01"/> <anchor type="figure" xlink:label="fig-0615-02a" xlink:href="fig-0615-02"/> in quo linea p g ſecat lineam <lb/>c a, punctus r:</s> <s xml:id="echoid-s40094" xml:space="preserve"> angulus ergo <lb/>p r c per 16 p 1 maior eſt an-<lb/>gulo p g h:</s> <s xml:id="echoid-s40095" xml:space="preserve"> ergo & angulo l a <lb/>c.</s> <s xml:id="echoid-s40096" xml:space="preserve"> Quia ergo angulus p r a cũ <lb/>angulo p r c eſt ęqualis duo-<lb/>bus rectis per 13 p 1:</s> <s xml:id="echoid-s40097" xml:space="preserve"> patet <lb/>quòd angulus p r a cum an-<lb/>gulo r a l minor eſt duobus <lb/>rectis:</s> <s xml:id="echoid-s40098" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s40099" xml:space="preserve"> 1 huius <lb/>lineæ g p & a l concurrent:</s> <s xml:id="echoid-s40100" xml:space="preserve"> <lb/>ſit concurſus punctus p.</s> <s xml:id="echoid-s40101" xml:space="preserve"> Si <lb/>itaq;</s> <s xml:id="echoid-s40102" xml:space="preserve"> in pũcto p ponatur cen <lb/>trum uiſus:</s> <s xml:id="echoid-s40103" xml:space="preserve"> palàm quòd ipſe uidebitformã puncti b reflexam à duobus punctis ſpeculi, quæ ſunt a <lb/>& g:</s> <s xml:id="echoid-s40104" xml:space="preserve"> ſimiliterq́ demonſtrandum ſi lineæ a c & g h fuerint in æquales:</s> <s xml:id="echoid-s40105" xml:space="preserve"> ut ſi linea a c ſit maior quàm li-<lb/>nea g h:</s> <s xml:id="echoid-s40106" xml:space="preserve"> tũc enim per 43 th.</s> <s xml:id="echoid-s40107" xml:space="preserve"> 1 huius angulus portionis, qui eſt c a g, erit maior angulo portionis, qui <lb/>eſt h g c:</s> <s xml:id="echoid-s40108" xml:space="preserve"> remanetq́;</s> <s xml:id="echoid-s40109" xml:space="preserve"> per modum, quo proceſsimus prius, angulus h g p maior angulo c a l:</s> <s xml:id="echoid-s40110" xml:space="preserve"> fietq́ an-<lb/>gulus p r b maior angulo h g p & maior angulo l a r:</s> <s xml:id="echoid-s40111" xml:space="preserve"> ergo, ut prius, lineæ g p & a l concurrent:</s> <s xml:id="echoid-s40112" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s40113" xml:space="preserve"> <lb/>cõcurſus punctus p:</s> <s xml:id="echoid-s40114" xml:space="preserve"> & eſt idem, quod prius.</s> <s xml:id="echoid-s40115" xml:space="preserve"> Quòd ſi linea a c fuerit minor quàm linea g h:</s> <s xml:id="echoid-s40116" xml:space="preserve"> tunc per <lb/>modum, quo uſi ſumus prius, erit angulus l a c maior angulo p g h:</s> <s xml:id="echoid-s40117" xml:space="preserve"> ſed & angulus p r b maior eſt an-<lb/>gulo p g h.</s> <s xml:id="echoid-s40118" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s40119" xml:space="preserve"> angulus l a c ſit minor angulo p r b, concurſus fiet, ut prius, linearum a l & g p ad <lb/>punctum p per 14 th.</s> <s xml:id="echoid-s40120" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s40121" xml:space="preserve"> Si uerò angulus l a c ſit maior angulo p r b:</s> <s xml:id="echoid-s40122" xml:space="preserve"> fiet idem per 14 th.</s> <s xml:id="echoid-s40123" xml:space="preserve"> 1 huius <lb/>concurſus illarum linearum ultra arcum a g, qui impeditur per corpulentiam ſpeculi:</s> <s xml:id="echoid-s40124" xml:space="preserve"> unde tunc <lb/>non fiet reflexio ad uiſum.</s> <s xml:id="echoid-s40125" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s40126" xml:space="preserve"> ſi angulus l a c fuerit æqualis angulo p r b:</s> <s xml:id="echoid-s40127" xml:space="preserve"> tunc per 28 p 1 <lb/>lineę a l & g p æquidiſtabunt.</s> <s xml:id="echoid-s40128" xml:space="preserve"> In nullo ergo puncto concurrent.</s> <s xml:id="echoid-s40129" xml:space="preserve"> Nunquam ergo fiet formæ unius <lb/>puncti, quod eſt b, reflexio ad unum centrum uiſus à duobus punctis ſpeculi ſphærici concaui.</s> <s xml:id="echoid-s40130" xml:space="preserve"> Pa-<lb/>tet ergo propoſitum.</s> <s xml:id="echoid-s40131" xml:space="preserve"/> </p> <div xml:id="echoid-div1614" type="float" level="0" n="0"> <figure xlink:label="fig-0615-01" xlink:href="fig-0615-01a"> <variables xml:id="echoid-variables700" xml:space="preserve">g a r b c i l p d h</variables> </figure> <figure xlink:label="fig-0615-02" xlink:href="fig-0615-02a"> <variables xml:id="echoid-variables701" xml:space="preserve">a g l r p d b i c h</variables> </figure> </div> </div> <div xml:id="echoid-div1616" type="section" level="0" n="0"> <head xml:id="echoid-head1220" xml:space="preserve" style="it">10. Lineæ reflexionis à ſpeculis ſphæricis concauis (puncto rei uiſæ exiſtente in peripheria ſpe-<lb/>culi uel extr a illam) nõnunquam in uno centro uiſus à diuerſis punctis ſpeculi concurrunt. Eu-<lb/>clides 5 th. catoptr. Ptolemæus 2 th. 2 catoptr.</head> <p> <s xml:id="echoid-s40132" xml:space="preserve">Sit ſpeculum ſphæricũ concauum g a b s:</s> <s xml:id="echoid-s40133" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s40134" xml:space="preserve"> punctũ rei uiſæ g:</s> <s xml:id="echoid-s40135" xml:space="preserve"> quod ſit conſtitutũ in aliquo cir <lb/>cunferentiæ puncto, quod ſit etiá punctũ g:</s> <s xml:id="echoid-s40136" xml:space="preserve"> ſit q́;</s> <s xml:id="echoid-s40137" xml:space="preserve"> ut g punctum rei uiſę refle ctatur à duobus punctis <lb/>arcus g a b, quæ ſint puncta a & b:</s> <s xml:id="echoid-s40138" xml:space="preserve"> fiatq́ reflexio formæ puncti g à puncto ſpeculi b ad punctũ e:</s> <s xml:id="echoid-s40139" xml:space="preserve"> & à <lb/>pun cto a ad punctũ l.</s> <s xml:id="echoid-s40140" xml:space="preserve"> Dico quod lineas reflexionũ, quæ ſunt b e & a l, poſsibile eſt concurrere.</s> <s xml:id="echoid-s40141" xml:space="preserve"> Du-<lb/>cantur itaq;</s> <s xml:id="echoid-s40142" xml:space="preserve"> lineæ contingentes ſpeculum in punctis a & b:</s> <s xml:id="echoid-s40143" xml:space="preserve"> contingatq́;</s> <s xml:id="echoid-s40144" xml:space="preserve"> ipſum linea k a p in puncto <lb/>a:</s> <s xml:id="echoid-s40145" xml:space="preserve"> & linea k b f in puncto b:</s> <s xml:id="echoid-s40146" xml:space="preserve"> & ducantur lineæ e b & b g & l a & a g.</s> <s xml:id="echoid-s40147" xml:space="preserve"> Sit quoq;</s> <s xml:id="echoid-s40148" xml:space="preserve">, ut lineæ a l & g b ſecent <lb/> <anchor type="figure" xlink:label="fig-0615-03a" xlink:href="fig-0615-03"/> ſe in puncto h.</s> <s xml:id="echoid-s40149" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s40150" xml:space="preserve"> omnes anguli conſtituti <lb/>ſuper punctũ b ſunt æ quales omnibus angulis con-<lb/>ſtitutis ſuper punctum a per 13 p 1, & per 20 th.</s> <s xml:id="echoid-s40151" xml:space="preserve"> 5 hu-<lb/>ius angulus e b f eſt æ qualis angulo k b g, & angu-<lb/>lus l a k æ qualis eſt angulo p a g, & anguli cõtingen-<lb/>tiæ omnes ſunt æ quales per 16 p 3:</s> <s xml:id="echoid-s40152" xml:space="preserve"> angulus uerò g a <lb/>b maioris portionis circuli, maior eſt angulo g b s <lb/>minoris portionis per 43 th.</s> <s xml:id="echoid-s40153" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s40154" xml:space="preserve"> ergo angulus k <lb/>b h maior eſt angulo p a g:</s> <s xml:id="echoid-s40155" xml:space="preserve"> ergo angulus e b f maior <lb/>eſt angulo k a h propter æqualitatem angulorum <lb/>hincinde per 20 th.</s> <s xml:id="echoid-s40156" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s40157" xml:space="preserve"> palàm ergo quia angu-<lb/>lus e b g minor eſt angulo l a g:</s> <s xml:id="echoid-s40158" xml:space="preserve"> ſed angulus l a g eſt <lb/>minor angulo g h l per 16 p 1:</s> <s xml:id="echoid-s40159" xml:space="preserve"> angulus ergo g h l eſt <lb/>maior angulo g b e:</s> <s xml:id="echoid-s40160" xml:space="preserve"> ſed angulus l h g cum angulo b <lb/>h l ualet duos rectos per 13 p 1:</s> <s xml:id="echoid-s40161" xml:space="preserve"> ergo anguli g b e & b h l ſunt minores duobus rectis:</s> <s xml:id="echoid-s40162" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s40163" xml:space="preserve"> 1 <lb/>huius lineæ a l & b e concurrent:</s> <s xml:id="echoid-s40164" xml:space="preserve"> ſit concurſus punctus e.</s> <s xml:id="echoid-s40165" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s40166" xml:space="preserve"> centrum uiſus fuerit in puncto e:</s> <s xml:id="echoid-s40167" xml:space="preserve"> <lb/>patet quòd à duobus punctis ſpeculi fiet ad ipſum formæ puncti g reflexio.</s> <s xml:id="echoid-s40168" xml:space="preserve"> Quòd ſi extra periphe-<lb/> <pb o="314" file="0616" n="616" rhead="VITELLONIS OPTICAE"/> riam ponatur punctus g, accidet hocidem:</s> <s xml:id="echoid-s40169" xml:space="preserve"> & eadem eſt demonſtratio.</s> <s xml:id="echoid-s40170" xml:space="preserve"> Non eſt tamen hoc uniuer-<lb/>ſale:</s> <s xml:id="echoid-s40171" xml:space="preserve"> quia poſsibile eſt non concurrere:</s> <s xml:id="echoid-s40172" xml:space="preserve"> utſi anguli g b e & g h l ſint æquales uel maiores duobus re-<lb/>ctis:</s> <s xml:id="echoid-s40173" xml:space="preserve"> tunc enim lineæ b e & a l non concurrent:</s> <s xml:id="echoid-s40174" xml:space="preserve"> uel ſi concurrant hoc erit retro ſpeculum, ubi uiſus <lb/>conſtitutus retro ſpeculum formas reflexas non poterit uidere.</s> <s xml:id="echoid-s40175" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s40176" xml:space="preserve"/> </p> <div xml:id="echoid-div1616" type="float" level="0" n="0"> <figure xlink:label="fig-0615-03" xlink:href="fig-0615-03a"> <variables xml:id="echoid-variables702" xml:space="preserve">p a k g h b l e f s</variables> </figure> </div> </div> <div xml:id="echoid-div1618" type="section" level="0" n="0"> <head xml:id="echoid-head1221" xml:space="preserve" style="it">11. Locus imaginum formarum à ſpeculis ſphæricis concauis reflexarum quando eſt in ipſo <lb/>puncto reflexionis: quando eſt ultr a ſpeculum: quando inter uiſum & ſpeculum: quando in <lb/>ſuperficie ipſius uiſus: quando retro uiſum. Alhazen 60 n 5.</head> <p> <s xml:id="echoid-s40177" xml:space="preserve">Quando enim forma puncti rei uiſæ uidetur ſecundũ cathetum ſuæ incidentiæ:</s> <s xml:id="echoid-s40178" xml:space="preserve"> tunc neceſſariò <lb/>imago uidetur in ipſa ſuperficie ſpeculi, in puncto ſcilicet ſuæ reflexionis:</s> <s xml:id="echoid-s40179" xml:space="preserve"> quando uero formæ obli <lb/>què incidunt ſuperficiebus propoſitorũ ſpeculorum:</s> <s xml:id="echoid-s40180" xml:space="preserve"> tunc diuerſificantur loca imaginũ, ut propo-<lb/>nitur.</s> <s xml:id="echoid-s40181" xml:space="preserve"> Ad quod declarandum ſit a centrum uiſus:</s> <s xml:id="echoid-s40182" xml:space="preserve"> & punctus d centrum ſpeculi ſphærici concaui:</s> <s xml:id="echoid-s40183" xml:space="preserve"> & <lb/>ducatur ſuperficies plana per hæc duo puncta, quæ erit ſuperficies reflexionis:</s> <s xml:id="echoid-s40184" xml:space="preserve"> quoniã ipſa eſt or-<lb/>thogonalis ſuper quamlibet ſuperficiem, contingentem ſpeculum ſecundum punctũ illum ſuperfi <lb/>ciei ſpeculi, cui incidit diameter uiſualis.</s> <s xml:id="echoid-s40185" xml:space="preserve"> Secabit ergo ſuperficiem ſpeculi dati:</s> <s xml:id="echoid-s40186" xml:space="preserve"> & erit communis <lb/>ſectio illarum ſuperficierum circulus magnus per 2 th.</s> <s xml:id="echoid-s40187" xml:space="preserve"> huius.</s> <s xml:id="echoid-s40188" xml:space="preserve"> Sit ergo ille circulus h b f g:</s> <s xml:id="echoid-s40189" xml:space="preserve"> & duca-<lb/>tur linea à centro uiſus ad centrum ſpeculi:</s> <s xml:id="echoid-s40190" xml:space="preserve"> quę ſit a d:</s> <s xml:id="echoid-s40191" xml:space="preserve"> & à puncto a ducatur ad circuli peripheriam <lb/>linea maior quàm linea a d, quæ ſit a e:</s> <s xml:id="echoid-s40192" xml:space="preserve"> & à puncto d ducatur ad circulum linea æ quidiſtans lineæ a <lb/>e:</s> <s xml:id="echoid-s40193" xml:space="preserve"> quæ ſit d h:</s> <s xml:id="echoid-s40194" xml:space="preserve"> & producatur linea a d ex utraq;</s> <s xml:id="echoid-s40195" xml:space="preserve"> parte ſui ad circumferentiam in puncta i & b, taliter <lb/>ut compleatur diameter i a d b:</s> <s xml:id="echoid-s40196" xml:space="preserve"> & ducatur linea d e.</s> <s xml:id="echoid-s40197" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s40198" xml:space="preserve"> linea a e eſt maior quàm linea a d:</s> <s xml:id="echoid-s40199" xml:space="preserve"> pa <lb/>làm per 18 p 1 quoniam angulus a e d eſt minor angulo a d e:</s> <s xml:id="echoid-s40200" xml:space="preserve"> eſt ergo per 32 p 1 angulus a e d minor <lb/>angulo recto, ſiue angulus a d e fuerit rectus uel obtuſus, uel acutus:</s> <s xml:id="echoid-s40201" xml:space="preserve"> ſed per 29 p 1 angulus e d h eſt <lb/>æ qualis angulo a e d:</s> <s xml:id="echoid-s40202" xml:space="preserve"> quia ſunt coalterni:</s> <s xml:id="echoid-s40203" xml:space="preserve"> eſt ergo angulus e d h minor recto.</s> <s xml:id="echoid-s40204" xml:space="preserve"> Super punctum quoq;</s> <s xml:id="echoid-s40205" xml:space="preserve"> <lb/>e lineæ d e fiat per 23 p 1 angulus æ qualis angulo a e d, qui ſit d e t.</s> <s xml:id="echoid-s40206" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s40207" xml:space="preserve"> quoniam linea e t ca-<lb/> <anchor type="figure" xlink:label="fig-0616-01a" xlink:href="fig-0616-01"/> dit intra circulum.</s> <s xml:id="echoid-s40208" xml:space="preserve"> Quoniam ſi caderet extra circulum, fieret ille <lb/>angulus aut rectus, ſi linea producta circulum cõtingeret, aut ob-<lb/>tuſus, ſi ſecaret:</s> <s xml:id="echoid-s40209" xml:space="preserve"> quod totũ patet ducta linea contingente circulum <lb/>in puncto e per 17 p 3:</s> <s xml:id="echoid-s40210" xml:space="preserve"> & quia hoc eſt impoſsibile, ut patet ex præ-<lb/>miſsis:</s> <s xml:id="echoid-s40211" xml:space="preserve"> palàm quia linea t e cadet intra circulum, ſecabitq́;</s> <s xml:id="echoid-s40212" xml:space="preserve"> lineam <lb/>d h:</s> <s xml:id="echoid-s40213" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s40214" xml:space="preserve"> punctus ſectiõis t:</s> <s xml:id="echoid-s40215" xml:space="preserve"> & erit linea e t ęqualis lineę d t per 6 p 1:</s> <s xml:id="echoid-s40216" xml:space="preserve"> <lb/>ſunt enim anguli e d t & d e t æ quales.</s> <s xml:id="echoid-s40217" xml:space="preserve"> Et quoniã angulus a d e ma-<lb/>ior eſt angulo a e d per 18 p 1:</s> <s xml:id="echoid-s40218" xml:space="preserve"> palàm quia angulus a d e maior eſt an <lb/>gulo d e t:</s> <s xml:id="echoid-s40219" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s40220" xml:space="preserve"> 1 huius linea e t non æquidiſtat lineæ a b:</s> <s xml:id="echoid-s40221" xml:space="preserve"> <lb/>concurrent ergo:</s> <s xml:id="echoid-s40222" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s40223" xml:space="preserve"> punctus concurſus z.</s> <s xml:id="echoid-s40224" xml:space="preserve"> Deinde à puncto a du <lb/>catur ad arcum e h linea an:</s> <s xml:id="echoid-s40225" xml:space="preserve"> quæ cum concurrat cum linea a e in <lb/>puncto a, & inter ipſam lineam d h ſibi ęquidiſtantem producatur, <lb/>palàm per 2 th.</s> <s xml:id="echoid-s40226" xml:space="preserve"> 1 huius quia concurret cum linea d h.</s> <s xml:id="echoid-s40227" xml:space="preserve"> Sit ergo pun-<lb/>ctus concurſus l:</s> <s xml:id="echoid-s40228" xml:space="preserve"> & ducatur linea d n:</s> <s xml:id="echoid-s40229" xml:space="preserve"> & ſuper punctum n lineæ d n <lb/>fiat angulus æ qualis angulo d n a per lineam m n, qui ſit m n d.</s> <s xml:id="echoid-s40230" xml:space="preserve"> Et <lb/>quia angulus d n a eſt acutus per 42 th.</s> <s xml:id="echoid-s40231" xml:space="preserve"> 1 huius, erit etiam angulus <lb/>d n m acutus.</s> <s xml:id="echoid-s40232" xml:space="preserve"> Ideo enim, quia angulus in ſemicirculo eſt rectus per <lb/>31 p 3, omnis angulus cõtentus à quacũq;</s> <s xml:id="echoid-s40233" xml:space="preserve"> linea & termino diametri <lb/>palàm quòd eſt acutus:</s> <s xml:id="echoid-s40234" xml:space="preserve"> cõcurret ergo linea n m cũ linea d h:</s> <s xml:id="echoid-s40235" xml:space="preserve"> ſit con <lb/>curſus in puncto m.</s> <s xml:id="echoid-s40236" xml:space="preserve"> Ducatur etiam à puncto a linea ad arcum e i f, <lb/>quæ ſit a g:</s> <s xml:id="echoid-s40237" xml:space="preserve"> & ducatur linea d g fiatq́;</s> <s xml:id="echoid-s40238" xml:space="preserve"> angulus q g d æqualis angulo <lb/>d g a.</s> <s xml:id="echoid-s40239" xml:space="preserve"> Et quoniam, ut prius, angulus d g a eſt acutus per 42 th.</s> <s xml:id="echoid-s40240" xml:space="preserve"> 1 hu-<lb/>ius, erit etiã angulus q g d acutus:</s> <s xml:id="echoid-s40241" xml:space="preserve"> cõcurret ergo linea g q cũ linea <lb/>d h:</s> <s xml:id="echoid-s40242" xml:space="preserve"> ſit concurſus in puncto q.</s> <s xml:id="echoid-s40243" xml:space="preserve"> Palàm quoq;</s> <s xml:id="echoid-s40244" xml:space="preserve">, cum linea g a concur-<lb/>rat cum linea a e, quoniam per 2 th.</s> <s xml:id="echoid-s40245" xml:space="preserve"> 1 huius concurret cum linea d h <lb/>illius æ quidiſtante:</s> <s xml:id="echoid-s40246" xml:space="preserve"> ſit concurſus punctus o ex parte puncti f:</s> <s xml:id="echoid-s40247" xml:space="preserve"> angu <lb/>lus enim g a d eſt maior angulo e a d:</s> <s xml:id="echoid-s40248" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s40249" xml:space="preserve"> 1 huius ad par-<lb/>tem minorum angulorum fiet concurſus:</s> <s xml:id="echoid-s40250" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s40251" xml:space="preserve"> linea g o periphe <lb/>riam circuli in puncto y:</s> <s xml:id="echoid-s40252" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s40253" xml:space="preserve"> arcus g y maior arcu g h.</s> <s xml:id="echoid-s40254" xml:space="preserve"> Quòd autem linea g q cadat inter puncta d <lb/>& h, palàm ſatis eſt ex præmiſsis:</s> <s xml:id="echoid-s40255" xml:space="preserve"> ſed & idem patere poteſt ex hoc.</s> <s xml:id="echoid-s40256" xml:space="preserve"> Quia cum arcus, quẽ ſecat linea <lb/>g o ex circulo h b f g, (qui eſt arcus g y) ſit maior arcu g h:</s> <s xml:id="echoid-s40257" xml:space="preserve"> producatur linea g d ad peripheriam cir-<lb/>culi in punctum p:</s> <s xml:id="echoid-s40258" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s40259" xml:space="preserve"> arcus h p maior arcu y p:</s> <s xml:id="echoid-s40260" xml:space="preserve"> ergo per 33 p 6 erit angulus h g d maior angulo a <lb/>g d:</s> <s xml:id="echoid-s40261" xml:space="preserve"> ſed angulus q g d eſt æ qualis angulo a g d, ut patet ex præmiſsis:</s> <s xml:id="echoid-s40262" xml:space="preserve"> ergo angulus h g d eſt maior <lb/>angulo q g d:</s> <s xml:id="echoid-s40263" xml:space="preserve"> linea ergo g q diuidit angulum h g d:</s> <s xml:id="echoid-s40264" xml:space="preserve"> ergo per 29 th.</s> <s xml:id="echoid-s40265" xml:space="preserve"> 1 huius diuidit & baſim d h:</s> <s xml:id="echoid-s40266" xml:space="preserve"> cadet <lb/>ergo punctum q inter puncta d & h.</s> <s xml:id="echoid-s40267" xml:space="preserve"> Item à puncto a ducatur ad arcum f b linea a k ſecans lineam d <lb/>f in puncto s, ita utſit linea k s maior quàm pars diametri, quæ eſt s d.</s> <s xml:id="echoid-s40268" xml:space="preserve"> Hoc autem facile per 7 p 3:</s> <s xml:id="echoid-s40269" xml:space="preserve"> ut <lb/>ſi linea d f diuidatur per æ qualia in puncto aliquo, & linea a k ducatur per illum punctum, aut per <lb/>punctum alium uerſus punctũ d.</s> <s xml:id="echoid-s40270" xml:space="preserve"> Hac itaq;</s> <s xml:id="echoid-s40271" xml:space="preserve"> linea a k ſic ducta, ducatur linea d k.</s> <s xml:id="echoid-s40272" xml:space="preserve"> Palàm ergo per 42 <lb/>th.</s> <s xml:id="echoid-s40273" xml:space="preserve"> 1 huius quòd angulus d k a eſt acutus.</s> <s xml:id="echoid-s40274" xml:space="preserve"> Fiat ergo ſuper punctum k terminũ lineæ d k angulo d k a <lb/> <pb o="315" file="0617" n="617" rhead="LIBER OCTAVVS."/> angulus æ qualis, qui ſit d k u.</s> <s xml:id="echoid-s40275" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s40276" xml:space="preserve"> per 18 p 1 angulus k d s ſit maior angulo d k s:</s> <s xml:id="echoid-s40277" xml:space="preserve"> ideo quia li-<lb/>nea s k eſt maior quàm linea d s:</s> <s xml:id="echoid-s40278" xml:space="preserve"> erit ergo angulus k d s maior angulo d k u:</s> <s xml:id="echoid-s40279" xml:space="preserve"> palàm ergo per 14 th.</s> <s xml:id="echoid-s40280" xml:space="preserve"> 1 <lb/>huius, quia linea u k concurret cũ linea d h:</s> <s xml:id="echoid-s40281" xml:space="preserve"> ſit ergo concurſus in puncto u.</s> <s xml:id="echoid-s40282" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s40283" xml:space="preserve"> per 20 th.</s> <s xml:id="echoid-s40284" xml:space="preserve"> 5 <lb/>huius & ſecundum prædicta, quòd forma puncti t à puncto ſpeculi e reflectitur ad uiſum, qui eſt in <lb/>puncto a:</s> <s xml:id="echoid-s40285" xml:space="preserve"> cathetus quoq;</s> <s xml:id="echoid-s40286" xml:space="preserve"> incidentiæ formæ puncti t eſt linea t d, quæ per 72 th.</s> <s xml:id="echoid-s40287" xml:space="preserve"> 1 huius eſt perpendi <lb/>cularis ſuper ſuperficiem contingentem ſpeculum, cum ſit tranſiens per eius centrum, & ipſa eſt <lb/>æquidiſtans lineæ reflexionis, quæ eſt a e:</s> <s xml:id="echoid-s40288" xml:space="preserve"> nunquam ergo concurret cum illa.</s> <s xml:id="echoid-s40289" xml:space="preserve"> Apparebit ergo ima-<lb/>go formæ puncti t in ipſo puncto refle xionis, quod eſt e.</s> <s xml:id="echoid-s40290" xml:space="preserve"> Forma uerò puncti z reflectitur ſimiliter à <lb/>puncto e ad uiſum exiſtentem in puncto a:</s> <s xml:id="echoid-s40291" xml:space="preserve"> cathetus quoq;</s> <s xml:id="echoid-s40292" xml:space="preserve"> ſuæ incidentiæ, quę eſt b z d ducta à pun <lb/>cto z per centrum ſpeculi concurrit cum linea refle xionis, quæ eſt a e in puncto a:</s> <s xml:id="echoid-s40293" xml:space="preserve"> locus itaq;</s> <s xml:id="echoid-s40294" xml:space="preserve"> ima-<lb/>ginis formæ puncti z per 37 th.</s> <s xml:id="echoid-s40295" xml:space="preserve"> 5 huius erit centrum uiſus, quod eſt a.</s> <s xml:id="echoid-s40296" xml:space="preserve"> Forma uerò puncti m à pun-<lb/>cto ſpeculi, quod eſtn, reflectitur ad uiſum a:</s> <s xml:id="echoid-s40297" xml:space="preserve"> & perpendicularis ducta à puncto m, quæ eſt cathe-<lb/>tus incidentiæ, quæ m d, concurrit cum a n linea reflexionis in puncto 1, quod eſt ultra ſpeculum:</s> <s xml:id="echoid-s40298" xml:space="preserve"> <lb/>& forma puncti m habet locum imaginis in puncto l ſub ſpeculo.</s> <s xml:id="echoid-s40299" xml:space="preserve"> Forma uerò puncti q peruenit ad <lb/>punctum ſpeculi, quod eſt g, & ex puncto g reflectitur ad uiſum a:</s> <s xml:id="echoid-s40300" xml:space="preserve"> & locus imaginis ſuæ eſt in pun-<lb/>cto o, quod eſt ultra uiſum.</s> <s xml:id="echoid-s40301" xml:space="preserve"> Et forma puncti u peruenit ad punctum ſpeculi, quod eſt k, & reflecti-<lb/>tur ad uiſum in puncto a:</s> <s xml:id="echoid-s40302" xml:space="preserve"> & cathetus ſuæ incidentiæ, quæ eſt perpendicularis ab eo ducta trans <lb/>centrum ſpeculi d, eſt linea u d, concurrens cum linea a k linea reflexionis in puncto s:</s> <s xml:id="echoid-s40303" xml:space="preserve"> locus itaq;</s> <s xml:id="echoid-s40304" xml:space="preserve"> <lb/>imaginis ſuæ eſt punctum s, quod eſt inter uiſum & ſpeculũ.</s> <s xml:id="echoid-s40305" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s40306" xml:space="preserve"> ex prædictis quòdimagi-<lb/>num à ſpeculis ſphæricis concauis reflexarum quædam uidentur in ſuperficie ipſius ſpeculi, ut in <lb/>ipſo puncto reflexionis:</s> <s xml:id="echoid-s40307" xml:space="preserve"> quædam uidentur ultra ſpeculum:</s> <s xml:id="echoid-s40308" xml:space="preserve"> quædam inter uiſum & ſpeculum:</s> <s xml:id="echoid-s40309" xml:space="preserve"> <lb/>quædam in ſuperficie ipſius uiſus:</s> <s xml:id="echoid-s40310" xml:space="preserve"> quædam citra uiſum.</s> <s xml:id="echoid-s40311" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s40312" xml:space="preserve"> Et ſi centrum uiſus <lb/>ſit extra circulum ſpeculi uel in circumferentia ipſius, idem accidit, & eodem modo eſt dem on-<lb/>ſtrandum:</s> <s xml:id="echoid-s40313" xml:space="preserve"> quoniam ſemper linea a e ſit maior quàm linea a d:</s> <s xml:id="echoid-s40314" xml:space="preserve"> & accidunt omnia, ut prius.</s> <s xml:id="echoid-s40315" xml:space="preserve"> Patet er-<lb/>go, quod proponebatur.</s> <s xml:id="echoid-s40316" xml:space="preserve"/> </p> <div xml:id="echoid-div1618" type="float" level="0" n="0"> <figure xlink:label="fig-0616-01" xlink:href="fig-0616-01a"> <variables xml:id="echoid-variables703" xml:space="preserve">l g e n h m t q u i a d z b s k y f p o</variables> </figure> </div> </div> <div xml:id="echoid-div1620" type="section" level="0" n="0"> <head xml:id="echoid-head1222" xml:space="preserve" style="it">12. Imaginum reflexarum à ſpeculis ſphæricis concauis diuerſafit à uiſu comprehenſio, ſecun <lb/>dum ſuorum locorum propr iam diuerſit atem. Alhazen 61 n 5.</head> <p> <s xml:id="echoid-s40317" xml:space="preserve">Remaneat diſpoſitio præcedentis in tota forma figurationis.</s> <s xml:id="echoid-s40318" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s40319" xml:space="preserve"> locus imaginis fuerit ul-<lb/>tra ſpeculum, ut in puncto l:</s> <s xml:id="echoid-s40320" xml:space="preserve"> aut inter uiſum & ſpeculum, utin puncto s:</s> <s xml:id="echoid-s40321" xml:space="preserve"> tunc, quia formas ſibi op-<lb/>poſitas ſemper perfectius acquirit uiſus, comprehenditur ueritas illius imaginis.</s> <s xml:id="echoid-s40322" xml:space="preserve"> Cum uerò locus <lb/>imaginis fuerit in puncto reflexionis:</s> <s xml:id="echoid-s40323" xml:space="preserve"> ut cum perpendicularis ducta à puncto rei uiſę quidiſtat li-<lb/>neæ reflexionis:</s> <s xml:id="echoid-s40324" xml:space="preserve"> tunc enim locus imaginis eſt in puncto e:</s> <s xml:id="echoid-s40325" xml:space="preserve"> quia cum punctus e per 3 th.</s> <s xml:id="echoid-s40326" xml:space="preserve"> 2 huius ſit <lb/>punctus naturalis diuiſibilis, ſenſibilis, utpote capax imaginis formę rei ſenſibilis, quæ eſt diuiſibi-<lb/>lis, cum ſit naturalis (ſumpto in ſui medio puncto intellectuali) erit imago cuiuſcunq;</s> <s xml:id="echoid-s40327" xml:space="preserve"> illius pun-<lb/>cti ſenſibilis pars, quæ fuerit ultra medium punctum ſumptum, apparens ultra ſpeculum, & ima-<lb/>go partis alterius, quæ fuerit citra punctum medium, apparebit inter uiſum & ſpeculum.</s> <s xml:id="echoid-s40328" xml:space="preserve"> Et cum <lb/>totalis forma ſecundum partes ſui ulteriores ſuperficie ſpeculi & citeriores uerſus uiſum ſem-<lb/>per uideatur una & continua, neceſſariò forma illius puncti ſenſibilis pro ximi puncto intellectuali <lb/>uidebitur in ipſius ſpeculi ſuperficie, in puncto ſcilicet reflexionis:</s> <s xml:id="echoid-s40329" xml:space="preserve"> aliæ quoq;</s> <s xml:id="echoid-s40330" xml:space="preserve"> partes formæ ſenſibi <lb/>lis circumiacentis illud punctum uidebuntur ab illo puncto declinare modo dicto:</s> <s xml:id="echoid-s40331" xml:space="preserve"> quæ dam ad ui-<lb/>ſum & intra ſpeculum:</s> <s xml:id="echoid-s40332" xml:space="preserve"> quædam ultra ſpeculum.</s> <s xml:id="echoid-s40333" xml:space="preserve"> Verùm in imaginibus, quarum locus eſt punctus <lb/>a, quod eſt centrum uiſus, ueritas ipſarum non comprehenditur:</s> <s xml:id="echoid-s40334" xml:space="preserve"> unde ſæpius accidit error uiſui in <lb/>formis ſic uiſis.</s> <s xml:id="echoid-s40335" xml:space="preserve"> Ad huius autem maiorem euidentiam, ut non ſolùm demonſtratio, ſed etiam expe-<lb/>rientia doceat, quod præmiſimus:</s> <s xml:id="echoid-s40336" xml:space="preserve"> erigatur ſuper ſuperficiem ſpeculi ſphærici cõcaui ſtipes ligneus <lb/>uel ferreus perpendiculariter, qui ſit minor medietate ſemidiametri ſpeculi:</s> <s xml:id="echoid-s40337" xml:space="preserve"> & circa caput huius ſti <lb/>pitis ponatur centrum uiſus:</s> <s xml:id="echoid-s40338" xml:space="preserve"> & dirigatur uiſualis radius ad punctum ſpeculi, cuius diſtantia à ſtipi <lb/>te ſit maior quàm diſtantia centri uiſus à diametro per ſtipitem tranſeuntem:</s> <s xml:id="echoid-s40339" xml:space="preserve"> apparebit quoq;</s> <s xml:id="echoid-s40340" xml:space="preserve"> ima-<lb/>go illius ſtipitis ultra uiſum:</s> <s xml:id="echoid-s40341" xml:space="preserve"> nec erit certa comprehenſio formæ ipſius:</s> <s xml:id="echoid-s40342" xml:space="preserve"> imò apparebit quaſi curua, <lb/>cum tamen ſtipes ſit formæ lineæ rectæ.</s> <s xml:id="echoid-s40343" xml:space="preserve"> Ex quo patet quòd in his ſpeculis non comprehenditur ue <lb/>ritas imaginis, niſi cuius locus fuerit ultra ſpeculum, aut inter uiſum & ſpeculum, ut hæc patere <lb/>poſſunt per experientiá ſitum ſtipitis & uiſus uariè diuerſificanti:</s> <s xml:id="echoid-s40344" xml:space="preserve"> & accidit eidem, quòd, cum cen-<lb/>trum uiſus fueritin perpendiculari per lignum tranſeunte, nó plenè comprehendet formam illius <lb/>ligni.</s> <s xml:id="echoid-s40345" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s40346" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1621" type="section" level="0" n="0"> <head xml:id="echoid-head1223" xml:space="preserve" style="it">13. In ſpeculo ſphærico concauo eſt proportio catheti incidentiæ ad rectam à centro ſpeculi ad <lb/>locum imaginis productam, ſicut lineæ à puncto rei uiſæ ad finem cõtingentiæ ductæ, adline am <lb/>à fine contingentiæ ad locum imaginis productam. Alhazen 64 n 5.</head> <p> <s xml:id="echoid-s40347" xml:space="preserve">Eſto ſpeculum ſphæricum concauum, cuius centrum ſite:</s> <s xml:id="echoid-s40348" xml:space="preserve"> & ſit b punctus rei uiſæ:</s> <s xml:id="echoid-s40349" xml:space="preserve"> & ſit a cen-<lb/>trum uiſus:</s> <s xml:id="echoid-s40350" xml:space="preserve"> & ſit g punctus reflexionis:</s> <s xml:id="echoid-s40351" xml:space="preserve"> & contingat linea z g circulũ, qui eſt cõmunis ſectio ſuperfi <lb/>ciei reflexionis & ſpeculi, in puncto g:</s> <s xml:id="echoid-s40352" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s40353" xml:space="preserve"> linea e g à puncto reflexionis ad centrũ ſpeculi, & <lb/>linea incidentiæ, quę b g, & cathetus incidentiæ:</s> <s xml:id="echoid-s40354" xml:space="preserve"> quæ ſit linea e b, quæ producta concurrat cum li-<lb/>nea z g in puncto t:</s> <s xml:id="echoid-s40355" xml:space="preserve"> concurrẽt autem per 14 th.</s> <s xml:id="echoid-s40356" xml:space="preserve"> 1 huius, cum ſintin eadem ſuperficie reflexionis per <lb/>3 huius & per 1 p 11, & cum per 18 p 3 angulus e g z ſit rectus, angulus uerò g e b ſit acutus.</s> <s xml:id="echoid-s40357" xml:space="preserve"> Sit ergo <pb o="316" file="0618" n="618" rhead="VITELLONIS OPTICAE"/> punctus t finis contingentiæ, ut patet è 6 defin.</s> <s xml:id="echoid-s40358" xml:space="preserve"> 6 huius.</s> <s xml:id="echoid-s40359" xml:space="preserve"> Educatur quoq;</s> <s xml:id="echoid-s40360" xml:space="preserve"> extra circulum linea refle <lb/>xionis, quæ ſit a g.</s> <s xml:id="echoid-s40361" xml:space="preserve"> Cathetus itaq;</s> <s xml:id="echoid-s40362" xml:space="preserve"> e b concurret cum a g linea reflexionis extra punctum g, qui eſt <lb/>punctus reflexionis:</s> <s xml:id="echoid-s40363" xml:space="preserve"> & hoc ideo, quia lineæ e b & a g ſunt duæ lineæ rectę, quarum a g ſecat lineam <lb/>z g in puncto g, & fit angulus a g t obtuſus, quoniam angulus e g t eſt rectus:</s> <s xml:id="echoid-s40364" xml:space="preserve"> linea uerò e b ſecat li-<lb/>neam z g in puncto t, & fit angulus e t g acutus per 32 p 1.</s> <s xml:id="echoid-s40365" xml:space="preserve"> Non ergo concurrũt lineæ e b & a g in pun <lb/>cto g.</s> <s xml:id="echoid-s40366" xml:space="preserve"> Autigitur lineæ a g & e b (cum non ſunt æ quidiſtantes, ut patet ex hypotheſi) concurrent ul-<lb/>tra punctum g:</s> <s xml:id="echoid-s40367" xml:space="preserve"> autinter puncta g & a.</s> <s xml:id="echoid-s40368" xml:space="preserve"> Sit ergo, ut <lb/> <anchor type="figure" xlink:label="fig-0618-01a" xlink:href="fig-0618-01"/> concurrant ultra punctum g:</s> <s xml:id="echoid-s40369" xml:space="preserve"> & ſit concurfus in <lb/>puncto h, qui erit locus imaginis per 37 th.</s> <s xml:id="echoid-s40370" xml:space="preserve"> 5 hu-<lb/>ius.</s> <s xml:id="echoid-s40371" xml:space="preserve"> Dico quòd eſt eadẽ proportio catheti e b ad <lb/>lineam e h (interiacentem centrũ ſpeculi, & pun <lb/>ctum concurſus lineæ reflexionis & catheti inci-<lb/>dentiæ, qui eſt locus imaginis) quę eſt proportio <lb/>lineæ b t (interiacentis punctũ rei uiſæ, & finem <lb/>contingentiæ) ad lineam t h, quæ interiacet finẽ <lb/>contingentiæ, & punctum concurſus lineæ refle <lb/>xionis cum incidentiæ catheto, qui eft locus ima <lb/>ginis formę puncti b, qui eſt pũctus rei uiſæ.</s> <s xml:id="echoid-s40372" xml:space="preserve"> Pro-<lb/>ducatur enim perpendicularis, quæ e g, ultra ſpe <lb/>culum:</s> <s xml:id="echoid-s40373" xml:space="preserve"> & à puncto h, qui eſt locus imaginis for-<lb/>mæ puncti b, ducatur linea æ quidiſtãs lineę inci <lb/>dentiæ, quæ b g, per 31 p 1:</s> <s xml:id="echoid-s40374" xml:space="preserve"> quę neceſſariò per 2 th.</s> <s xml:id="echoid-s40375" xml:space="preserve"> <lb/>1 huius concurret cum producta linea e g:</s> <s xml:id="echoid-s40376" xml:space="preserve"> cum <lb/>ſua æ quidiſtans, quę b g, concurrat cum eadẽ.</s> <s xml:id="echoid-s40377" xml:space="preserve"> Sit <lb/>punctus concurſus l:</s> <s xml:id="echoid-s40378" xml:space="preserve"> & à puncto b ducatur linea <lb/>æquidiſtans lineæ g h, quæ, ut prius, neceſſariò <lb/>concurret cum linea z t per 2 th.</s> <s xml:id="echoid-s40379" xml:space="preserve"> 1 huius, cum linea g h cõcurrat cum eadem:</s> <s xml:id="echoid-s40380" xml:space="preserve"> ſit cõcurſus punctus q.</s> <s xml:id="echoid-s40381" xml:space="preserve"> <lb/>Et quoniam angulus b g e eſt æqualis angulo a g e per 20 th.</s> <s xml:id="echoid-s40382" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s40383" xml:space="preserve"> ſed angulus b g e eſt æqualis <lb/>angulo g l h per 29 p 1, & angulus a g e æqualis eſt angulo l g h per 15 p 1:</s> <s xml:id="echoid-s40384" xml:space="preserve"> erit ergo angulus g l h æqua <lb/>lis angulo h g l:</s> <s xml:id="echoid-s40385" xml:space="preserve"> ergo per 6 p 1 erit linea l h æqualis lineæ g h.</s> <s xml:id="echoid-s40386" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s40387" xml:space="preserve"> angulus b g q æ qualis <lb/>eſt angulo a g z:</s> <s xml:id="echoid-s40388" xml:space="preserve"> quia cũ anguli e g z & e g q ſint æquales, quia recti, & angulib g e & e g a ſint æ qua-<lb/>les:</s> <s xml:id="echoid-s40389" xml:space="preserve"> remanent anguli reſidui æquales:</s> <s xml:id="echoid-s40390" xml:space="preserve"> ſed & angulus a g z æ qualis eſt angulo b q g per 29 p 1:</s> <s xml:id="echoid-s40391" xml:space="preserve"> angu-<lb/>lus ergo b g q æ qualis eſt angulo b q g:</s> <s xml:id="echoid-s40392" xml:space="preserve"> ergo per 6 p 1 linea b g eſt æ qualis lineæ b q.</s> <s xml:id="echoid-s40393" xml:space="preserve"> Proportio itaq;</s> <s xml:id="echoid-s40394" xml:space="preserve"> <lb/>lineæ b g ad lineam h l eſt, ſicut lineę b q ad lineam h g per 7 p 5:</s> <s xml:id="echoid-s40395" xml:space="preserve"> ſunt enim antecedentia æ qualia in-<lb/>ter ſe & conſequentia æ qualia inter ſe.</s> <s xml:id="echoid-s40396" xml:space="preserve"> Quia uerò angulus g h t æqualis eſt angulo t b q per 29 p 1:</s> <s xml:id="echoid-s40397" xml:space="preserve"> <lb/>ſunt enim illi anguli coalterni inter lineas æquidiſtantes:</s> <s xml:id="echoid-s40398" xml:space="preserve"> & angulus q t b æ qualis eſt angulo h t g <lb/>per 15 p 1:</s> <s xml:id="echoid-s40399" xml:space="preserve"> ſed & angulus h g t æ qualis eſt angulo t q b per 29 p 1:</s> <s xml:id="echoid-s40400" xml:space="preserve"> ergo triangulit q b & g t h ſunt æ qui <lb/>anguli:</s> <s xml:id="echoid-s40401" xml:space="preserve"> ergo per 4 p 6 eſt proportio lineę q b ad lineam h g, ſicu lineæ b t ad lineam t h:</s> <s xml:id="echoid-s40402" xml:space="preserve"> ſed linea b q <lb/>æ qualis eſt lineæ b g:</s> <s xml:id="echoid-s40403" xml:space="preserve"> ergo per 7 p 5 eſt proportio lineæ b g ad lineam h g, ſicut lineæ b t ad lineã th:</s> <s xml:id="echoid-s40404" xml:space="preserve"> <lb/>ergo per 11 p 5 eſt proportio lineæ b t ad lineã t <lb/>h, ſicut lineæ b g ad lineã h l.</s> <s xml:id="echoid-s40405" xml:space="preserve"> Quia uerò ք 29 p 1 <lb/>trianguli h e l & b e g ſunt æ quianguli:</s> <s xml:id="echoid-s40406" xml:space="preserve"> erit per <lb/> <anchor type="figure" xlink:label="fig-0618-02a" xlink:href="fig-0618-02"/> <anchor type="figure" xlink:label="fig-0618-03a" xlink:href="fig-0618-03"/> <anchor type="figure" xlink:label="fig-0618-04a" xlink:href="fig-0618-04"/> 4 p 6 proportio lineę e b ad lineam e h, ſicut li-<lb/>neæ b g ad lineam h l:</s> <s xml:id="echoid-s40407" xml:space="preserve"> ergo, ut prius, erit proportio lineę e b ad lineã e h, ſicut lineæ b t ad lineam t h:</s> <s xml:id="echoid-s40408" xml:space="preserve"> <lb/>quod eſt propoſitũ.</s> <s xml:id="echoid-s40409" xml:space="preserve"> Eadem quoq;</s> <s xml:id="echoid-s40410" xml:space="preserve"> eſt demonſtratio, ſi locus imaginis fuerit inter a centrum uiſus, <lb/>& g punctum reflexionis:</s> <s xml:id="echoid-s40411" xml:space="preserve"> aut ſi fuerit in puncto a:</s> <s xml:id="echoid-s40412" xml:space="preserve"> aut ultra illũ.</s> <s xml:id="echoid-s40413" xml:space="preserve"> Si uerò linea in puncto reflexionis <lb/>ſpeculũ contingens, quæ eſt z g, non cõcurrat cum catheto incidentiæ, quæ eſt b e h, ſed ſit ei æ qui-<lb/> <pb o="317" file="0619" n="619" rhead="LIBER OCTAVVS."/> diſtans:</s> <s xml:id="echoid-s40414" xml:space="preserve"> ducatur à puncto contingentiæ, quod eſt g, linea perpendicularis, quæ ſit g e, ſuper lineam <lb/>b h per 12 p 1:</s> <s xml:id="echoid-s40415" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s40416" xml:space="preserve"> per 29 p 1 linea e g perpendicularis ſuper lineam z g.</s> <s xml:id="echoid-s40417" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s40418" xml:space="preserve"> angulus b e g eſt <lb/>æ qualis angulo h e g:</s> <s xml:id="echoid-s40419" xml:space="preserve"> qula uterq;</s> <s xml:id="echoid-s40420" xml:space="preserve"> eſt rectus:</s> <s xml:id="echoid-s40421" xml:space="preserve"> & angulus b g e æ qualis eſt angulo h g e per 20 th.</s> <s xml:id="echoid-s40422" xml:space="preserve"> 5 hu-<lb/>ius:</s> <s xml:id="echoid-s40423" xml:space="preserve"> palàm per 32 p 1 quoniã triangulus b g e æ quiangulus eſt triangulo h g e:</s> <s xml:id="echoid-s40424" xml:space="preserve"> ergo per 4 p 6 eſt pro-<lb/>portio lineę b e ad lineã e h, ſicut lineæ b g ad lineam g h:</s> <s xml:id="echoid-s40425" xml:space="preserve"> quod eſt propoſitũ, ut prius.</s> <s xml:id="echoid-s40426" xml:space="preserve"> Non enim ta-<lb/>li facta diſpoſitione eſt alius punctus finis contingentiæ quàm punctus g, qui eſt punctus contin-<lb/>gentię.</s> <s xml:id="echoid-s40427" xml:space="preserve"> Similiterq́;</s> <s xml:id="echoid-s40428" xml:space="preserve"> demonſtrandum, ſi locus <lb/>imaginis fuerit in ipſo centro uiſus:</s> <s xml:id="echoid-s40429" xml:space="preserve"> tunc e-<lb/> <anchor type="figure" xlink:label="fig-0619-01a" xlink:href="fig-0619-01"/> nim punctum h, qui eſt concurſus lineæ re-<lb/>flexionis & catheti incidentiæ, & eſt locus <lb/>imaginis, fit idẽ cum puncto a:</s> <s xml:id="echoid-s40430" xml:space="preserve"> qui eſt centrũ <lb/>uiſus:</s> <s xml:id="echoid-s40431" xml:space="preserve"> nec oportet in illius demonſtratione <lb/>aliud adijci, niſi quia per 3 p 6 eſt proportio <lb/>catheti b e ad lineam e a ductã à centro ſpe-<lb/>culi a d locum imaginis, ſicut lineæ b g ad li-<lb/>neam g a:</s> <s xml:id="echoid-s40432" xml:space="preserve"> quoniam linea g e diuidit angulum a g b per ęqualia per 20 th.</s> <s xml:id="echoid-s40433" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s40434" xml:space="preserve"> Erit ergo, ut prius, <lb/>proportio lineæ b t a d lineam t a, ſicut lineæ b e ad lineam e a:</s> <s xml:id="echoid-s40435" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s40436" xml:space="preserve"> Et hoc eſt uni-<lb/>uerſale ad omnes modos imaginum ubicunq;</s> <s xml:id="echoid-s40437" xml:space="preserve"> uiſui occurrentium.</s> <s xml:id="echoid-s40438" xml:space="preserve"> Pater ergo propoſitum.</s> <s xml:id="echoid-s40439" xml:space="preserve"/> </p> <div xml:id="echoid-div1621" type="float" level="0" n="0"> <figure xlink:label="fig-0618-01" xlink:href="fig-0618-01a"> <variables xml:id="echoid-variables704" xml:space="preserve">l h z g c q a b e</variables> </figure> <figure xlink:label="fig-0618-02" xlink:href="fig-0618-02a"> <variables xml:id="echoid-variables705" xml:space="preserve">q ſ g c a e b</variables> </figure> <figure xlink:label="fig-0618-03" xlink:href="fig-0618-03a"> <variables xml:id="echoid-variables706" xml:space="preserve">b a ſ e h q g t z</variables> </figure> <figure xlink:label="fig-0618-04" xlink:href="fig-0618-04a"> <variables xml:id="echoid-variables707" xml:space="preserve">l t b e a q g z</variables> </figure> <figure xlink:label="fig-0619-01" xlink:href="fig-0619-01a"> <variables xml:id="echoid-variables708" xml:space="preserve">z g q h e b</variables> </figure> </div> </div> <div xml:id="echoid-div1623" type="section" level="0" n="0"> <head xml:id="echoid-head1224" xml:space="preserve" style="it">14. In ſpeculis ſphæricis concauis poßibile eſt quando reflexionem fieri ſecundum totam pe-<lb/>ripheriam unius circuli. Alhazen 65 n 5.</head> <p> <s xml:id="echoid-s40440" xml:space="preserve">Sit circulus magnus ſpeculi ſphærici concaui, qui a b g d:</s> <s xml:id="echoid-s40441" xml:space="preserve"> cuius diameter eſt b e d, & centrũ e:</s> <s xml:id="echoid-s40442" xml:space="preserve"> ſi-<lb/>gnenturq́;</s> <s xml:id="echoid-s40443" xml:space="preserve"> ſuper diametrũ b e d duo puncta exutraq;</s> <s xml:id="echoid-s40444" xml:space="preserve"> parte centri e:</s> <s xml:id="echoid-s40445" xml:space="preserve"> quæ ſint h & z æ qualiter diſtan <lb/>tia à centro e:</s> <s xml:id="echoid-s40446" xml:space="preserve"> erunt ergo lineæ h e & z e æ quales.</s> <s xml:id="echoid-s40447" xml:space="preserve"> Ducatur quoq;</s> <s xml:id="echoid-s40448" xml:space="preserve"> à centro per 11 p 1 diameter g e a <lb/>perpen diculariter ſuper diametrum b d:</s> <s xml:id="echoid-s40449" xml:space="preserve"> & copulentur <lb/> <anchor type="figure" xlink:label="fig-0619-02a" xlink:href="fig-0619-02"/> lineæ h a & z a.</s> <s xml:id="echoid-s40450" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s40451" xml:space="preserve"> in trigonis h e a & z e a duo la <lb/>tera h e & z e ſunt æ qualia exhypotheſi, & linea e a com <lb/>munis, & utriuſq;</s> <s xml:id="echoid-s40452" xml:space="preserve"> trigonorum anguli h e a & z e a ſunt <lb/>æ quales, quia recti:</s> <s xml:id="echoid-s40453" xml:space="preserve"> palàm per 4 p 1 quoniam angulus h <lb/>a e eſt æ qualis angulo z a e:</s> <s xml:id="echoid-s40454" xml:space="preserve"> ergo per 20 th.</s> <s xml:id="echoid-s40455" xml:space="preserve"> 5 huius pun-<lb/>cta h & z ad ſeinuicẽ mutuò reflectuntur à puncto ſpe-<lb/>culi, quod eſt a.</s> <s xml:id="echoid-s40456" xml:space="preserve"> Idem quoq;</s> <s xml:id="echoid-s40457" xml:space="preserve"> patet ductis lineis h g & z g:</s> <s xml:id="echoid-s40458" xml:space="preserve"> <lb/>quoniam iſtorũ punctorũ mutua reflexio fiet à puncto <lb/>g.</s> <s xml:id="echoid-s40459" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s40460" xml:space="preserve"> fixa diametro b d, imaginemur reuolui trigo-<lb/>num a h z circa diametrũ b d, linea trigoni (quæ eſt h z) <lb/>manente fixa:</s> <s xml:id="echoid-s40461" xml:space="preserve"> tũc punctum a motũ peruenit in punctũ <lb/>g, & exinde reuertetur ad locum ſuum primum, motuq́;</s> <s xml:id="echoid-s40462" xml:space="preserve"> <lb/>ſuo deſcribet in concauitate ſpeculi circulum, à quo to-<lb/>tali fiet formarũ punctorum h & z ad ſeinuicem mutua <lb/>reflexio:</s> <s xml:id="echoid-s40463" xml:space="preserve"> quoniã ad quemcunq;</s> <s xml:id="echoid-s40464" xml:space="preserve"> punctum illius circuli <lb/>ducantur lineæ à punctis h & z, ſemper ducta ſemidia-<lb/>metro à centro ad illud punctum, anguli ad punctum illius circuli erunt æ quales:</s> <s xml:id="echoid-s40465" xml:space="preserve"> & ita ab illo pun-<lb/>cto fiet reflexio per 20 th.</s> <s xml:id="echoid-s40466" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s40467" xml:space="preserve"> Si ergo centrũ uiſus fuerit in puncto h, reflectetur adipſum forma <lb/>puncti z à tota peripheria illius circuli.</s> <s xml:id="echoid-s40468" xml:space="preserve"> Si tamen puncta h & zinæqualiter diſtent à centro e, non <lb/>fiet reflexio à circulo illo, ſed fortè fiet ab alio circulo, quem deſcribit motu ſuo punctus refl exio-<lb/>nis.</s> <s xml:id="echoid-s40469" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s40470" xml:space="preserve"/> </p> <div xml:id="echoid-div1623" type="float" level="0" n="0"> <figure xlink:label="fig-0619-02" xlink:href="fig-0619-02a"> <variables xml:id="echoid-variables709" xml:space="preserve">b z a e g h d</variables> </figure> </div> </div> <div xml:id="echoid-div1625" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables710" xml:space="preserve">b p q h a e g t k z d</variables> </figure> <head xml:id="echoid-head1225" xml:space="preserve" style="it">15. Duobus pũctis in una diametrorũ ſpeculi ſphæ-<lb/> rici concaui ſe orthogonaliter ſecantiũ exiſtentibus, ſub inæquali diſtantia à centro: impoßibile eſt ab ali- quo punctorũ peripheriæ ſemicirculi, in quo eſt pun- ctus à centro remotior illorum punctorũ adinuicem fieri reflexionem: à reliqui uerò ſemicirculi duobus punctis eſt poßibile.</head> <p> <s xml:id="echoid-s40471" xml:space="preserve">Sit ſpeculi ſphærici concaui circulus magnus, qui a <lb/>b g d:</s> <s xml:id="echoid-s40472" xml:space="preserve"> cuius centrũ e:</s> <s xml:id="echoid-s40473" xml:space="preserve"> ſecentq́;</s> <s xml:id="echoid-s40474" xml:space="preserve"> ſe in ipſo duę diametri or <lb/>thogonaliter, quæ ſint a g & b d:</s> <s xml:id="echoid-s40475" xml:space="preserve"> in quarũ una, quæ b d, <lb/>ſint duo pũcta h & z inæ qualiter diſtãtia à cẽtro e:</s> <s xml:id="echoid-s40476" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s40477" xml:space="preserve"> <lb/>h ꝓpinquius centro e, & z remotius:</s> <s xml:id="echoid-s40478" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s40479" xml:space="preserve"> punctus h in <lb/>ſemicirculo a b g, & pũctus z in ſemicirculo a d g.</s> <s xml:id="echoid-s40480" xml:space="preserve"> Dico <lb/>qđ ab aliquo punctorũ ſemicirculi a d g non poteſt fie <lb/>ri iſtorũ punctorũ adinuicem reflexio.</s> <s xml:id="echoid-s40481" xml:space="preserve"> Sit enim, ſi poſsi <lb/>bile eſt, ut fiat à puncto a:</s> <s xml:id="echoid-s40482" xml:space="preserve"> & ducatur linea h a:</s> <s xml:id="echoid-s40483" xml:space="preserve"> abſcinda <lb/>turq́;</s> <s xml:id="echoid-s40484" xml:space="preserve"> à linea e z linea æ qualis lineæ h e ք 3 p 1, quæ ſit e t:</s> <s xml:id="echoid-s40485" xml:space="preserve"> & ducatur linea t a.</s> <s xml:id="echoid-s40486" xml:space="preserve"> Palá ergo per 4 p 1 quia <lb/> <pb o="318" file="0620" n="620" rhead="VITELLONIS OPTICAE"/> angulus h a e eſt ęqualis angulo t a e:</s> <s xml:id="echoid-s40487" xml:space="preserve"> ſed angulus e a t ք 29 th.</s> <s xml:id="echoid-s40488" xml:space="preserve"> 1 huius eſt minor angulo e a z:</s> <s xml:id="echoid-s40489" xml:space="preserve"> angulus <lb/>ergo h a e eſt minor angulo z a e.</s> <s xml:id="echoid-s40490" xml:space="preserve"> Non ergo fiet punctorũ h & z mutua reflexio à pũcto ſpeculi a per <lb/>20 th.</s> <s xml:id="echoid-s40491" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s40492" xml:space="preserve"> Sed neq;</s> <s xml:id="echoid-s40493" xml:space="preserve"> ab aliquo alio puncto arcus a d g.</s> <s xml:id="echoid-s40494" xml:space="preserve"> Sit enim, ſi poſsibile eſt, ut fiat iſtorũ pun-<lb/>ctorũ reflexio à puncto k peripheriæ ſemicirculi, qui a d g:</s> <s xml:id="echoid-s40495" xml:space="preserve"> & ducátur lineę h k, e k, & z k.</s> <s xml:id="echoid-s40496" xml:space="preserve"> Eruntitaq-<lb/>ք 20 th.</s> <s xml:id="echoid-s40497" xml:space="preserve"> 5 huius anguli h k e, & z k e ę quales:</s> <s xml:id="echoid-s40498" xml:space="preserve"> linea ergo k e diuidit angulũ h k e ք ę qualia:</s> <s xml:id="echoid-s40499" xml:space="preserve"> ergo ք 3 p 6 <lb/>erit ꝓportio lineæ h k ad lineã k z, ſicut lineę h e ad lineã e z:</s> <s xml:id="echoid-s40500" xml:space="preserve"> ſed linea e h eſt minor ꝗ̃ linea e z, ut pa <lb/>tet ex hypotheſi:</s> <s xml:id="echoid-s40501" xml:space="preserve"> ergo linea h k eſt minor ꝗ̃ linea h z:</s> <s xml:id="echoid-s40502" xml:space="preserve"> eſt aũt linea h k maior ꝗ̃ linea k z:</s> <s xml:id="echoid-s40503" xml:space="preserve"> quoniam eſt <lb/>maior ꝗ̃ linea e k per 19 p 1.</s> <s xml:id="echoid-s40504" xml:space="preserve"> ut enim patet, angulus h e k eſt obtuſus maior angulo h e a recto:</s> <s xml:id="echoid-s40505" xml:space="preserve"> ſed li-<lb/>nea e k eſt æ qualis lineę e a, quę eſt maior ꝗ̃ linea k z, ut patet.</s> <s xml:id="echoid-s40506" xml:space="preserve"> Eſt ergo linea h k maior ꝗ̃ linea z k:</s> <s xml:id="echoid-s40507" xml:space="preserve"> <lb/>& ſequitur ex datis ipſam eſſe minorem:</s> <s xml:id="echoid-s40508" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s40509" xml:space="preserve"> Non ergo fiet reflexio form æ puncti <lb/>had punctũ z, uel econuerſo ab aliquo punctorũ arcus a k g.</s> <s xml:id="echoid-s40510" xml:space="preserve"> Ab aliquibus uerò punctis peripheriæ <lb/>ſemicirculi a b g mutuam reflexionem iſtorum punctorum fieri eſt poſsibile:</s> <s xml:id="echoid-s40511" xml:space="preserve"> quoniam eſt poſsibile <lb/>eſſe aliquod punctum arcus a b, utpote p, ad quod ductis lineis h p, e p, z p, fiat proportio lineæ z p <lb/>ad lineam h p, ſicut lineæ z e ad lineam e h:</s> <s xml:id="echoid-s40512" xml:space="preserve"> ergo per 3 p 6 angulus h p z diuidetur per æ qualia per li <lb/>neam e p:</s> <s xml:id="echoid-s40513" xml:space="preserve"> & ſimiliter poteſt fieri in arcu b g.</s> <s xml:id="echoid-s40514" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s40515" xml:space="preserve"> quod proponebatur:</s> <s xml:id="echoid-s40516" xml:space="preserve"> quoniã ab aliquo pun-<lb/>cto arcus b g, ut à puncto q, ſimiliter poteſt fieri reflexio ductis lineis h q, e q, z q.</s> <s xml:id="echoid-s40517" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1626" type="section" level="0" n="0"> <head xml:id="echoid-head1226" xml:space="preserve" style="it">16. Duobus punctis in una diametro ſpeculi ſphærici ſuperficiei concaui exiſtentibus, ſub inæ-<lb/>quali diſt antia à centro ſpeculi, ſi exceſſus diſtantiarũ ad minorẽ diſt antiã proportionẽ habeat, <lb/>quã pars diametri interiacẽtis ambo puncta, ad partẽ interiacentẽ punctũ cẽtro propinquius <lb/>& ſpeculum: impoßibile eſt à circulo illius diametriillorum punctorũ fieri mutuã reflexionem.</head> <p> <s xml:id="echoid-s40518" xml:space="preserve">Sit ſpeculi ſphærici concaui magnus circulus a b g d:</s> <s xml:id="echoid-s40519" xml:space="preserve"> cuius centrũ e:</s> <s xml:id="echoid-s40520" xml:space="preserve"> & diameter b d:</s> <s xml:id="echoid-s40521" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s40522" xml:space="preserve"> duo <lb/>puncta z & h conſtituta ſuper illam diametrum b d:</s> <s xml:id="echoid-s40523" xml:space="preserve"> quorum remotior â centro e ſit punctus z, & <lb/>propinquior punctus h:</s> <s xml:id="echoid-s40524" xml:space="preserve"> erit ergo linea z e maior quàm linea h e:</s> <s xml:id="echoid-s40525" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s40526" xml:space="preserve"> ipſarũ exceſſus linea z t.</s> <s xml:id="echoid-s40527" xml:space="preserve"> Dico <lb/>quòd ſi proportio lineę z t ad lineã t e, uel ad lineã h e fuerit, ſicut lineæ z h ad lineã h b, qđ impoſsi <lb/>bile eſt reflexionẽ fieri ab aliquo punctorũ circuli a b g d.</s> <s xml:id="echoid-s40528" xml:space="preserve"> Patet enim per præmiſſam quod non po-<lb/>teſt fierl reflexio ab aliquo pũctorũ ſemicirculi a d g:</s> <s xml:id="echoid-s40529" xml:space="preserve"> ſed neq;</s> <s xml:id="echoid-s40530" xml:space="preserve"> ab aliquo punctorũ ſemicirculi a b g.</s> <s xml:id="echoid-s40531" xml:space="preserve"> <lb/>Detur enim, ſi ſit poſsibile, ut fiat à puncto l arcus a b:</s> <s xml:id="echoid-s40532" xml:space="preserve"> & ducatur linea l b, & ipſi æ quidiſtãs ducatur <lb/>à centro ſpeculi per 31 p 1, quę ſit linea m e n:</s> <s xml:id="echoid-s40533" xml:space="preserve"> & ducan <lb/>tur lineæ l z, l e, & l h:</s> <s xml:id="echoid-s40534" xml:space="preserve"> ſecabit itaq;</s> <s xml:id="echoid-s40535" xml:space="preserve"> per 2 th.</s> <s xml:id="echoid-s40536" xml:space="preserve"> 1 huius li-<lb/> <anchor type="figure" xlink:label="fig-0620-01a" xlink:href="fig-0620-01"/> nea l z lineam n m:</s> <s xml:id="echoid-s40537" xml:space="preserve"> ſit punctus ſectionis m.</s> <s xml:id="echoid-s40538" xml:space="preserve"> Produca-<lb/>tur quoq;</s> <s xml:id="echoid-s40539" xml:space="preserve"> linea l h ultra punctũ h, quæ ſimiliter per 2 <lb/>th.</s> <s xml:id="echoid-s40540" xml:space="preserve"> 1 huius ſecabit lineam m n:</s> <s xml:id="echoid-s40541" xml:space="preserve"> ſit punctus ſectionis n.</s> <s xml:id="echoid-s40542" xml:space="preserve"> <lb/>Quia itaq;</s> <s xml:id="echoid-s40543" xml:space="preserve"> ex hypotheſi eſt proportio lineæ z t ad li-<lb/>neam t e, ſicut lineæ z h ad lineam b h:</s> <s xml:id="echoid-s40544" xml:space="preserve"> erit ergo per 18 <lb/>p 5 coniunctim proportio lineæ z e ad lineam t e, uel <lb/>per 7 p 5 ad lineam h e, ſicut lineæ z b ad linea b h:</s> <s xml:id="echoid-s40545" xml:space="preserve"> er-<lb/>go per 16 p 5 erit permutatim proportio lineæ z e ad <lb/>lineam z b, ſicut lineæ h e ad lineam b h.</s> <s xml:id="echoid-s40546" xml:space="preserve"> Quia uerò li-<lb/>neæ b l & n e æ quidiſtant:</s> <s xml:id="echoid-s40547" xml:space="preserve"> patet per 15 & 29 p 1 quia <lb/>trigona b l h & n h e ſunt æ quiangula:</s> <s xml:id="echoid-s40548" xml:space="preserve"> ergo per 4 p 6 <lb/>eſt proportio lineę e n ad lineã b l, ſicut lineæ e h ad li <lb/>neam b h.</s> <s xml:id="echoid-s40549" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s40550" xml:space="preserve"> trigona b l z & e m z ſunt <lb/>æ quian gula per 29 p 1, quia lineæ b l & e m æ quidi-<lb/>ſtant:</s> <s xml:id="echoid-s40551" xml:space="preserve"> erit ergo proportio lineę e m ad lineam b l, ſicut <lb/>lineæ z e ad lineam z b:</s> <s xml:id="echoid-s40552" xml:space="preserve"> ſed eadẽ eſt proportio lineæ e <lb/>h ad lineam h b, quæ lineæ z e ad lineam z b:</s> <s xml:id="echoid-s40553" xml:space="preserve"> eadem ergo eſt proportio lineæ e n ad lineam b l, quę li <lb/>neæ e m ad eandem lineam b l.</s> <s xml:id="echoid-s40554" xml:space="preserve"> Quia ergo linearũ n e & m e ad lineam b l eadem eſt proportio:</s> <s xml:id="echoid-s40555" xml:space="preserve"> ergo <lb/>per 9 p 5 lineę n e & m e ſunt ęquales.</s> <s xml:id="echoid-s40556" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s40557" xml:space="preserve"> angulus n l m diuiditur per ęqualia per lineã le, ut <lb/>patet per 20 th.</s> <s xml:id="echoid-s40558" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s40559" xml:space="preserve"> (fit enim reflexio punctoru h & z à puncto l) erit per 3 p 6 proportio lineæ l <lb/>n ad lineam l m, ſicut lineæ n e ad lineã e m:</s> <s xml:id="echoid-s40560" xml:space="preserve"> eſt ergo linea l n æ qualis lineæ l m:</s> <s xml:id="echoid-s40561" xml:space="preserve"> linea uerò l e eſt com <lb/>munis ambobus trigonis l e n & l e m:</s> <s xml:id="echoid-s40562" xml:space="preserve"> ergo per 8 p 1 anguli l e m & ſunt æ quales:</s> <s xml:id="echoid-s40563" xml:space="preserve"> ſunt ergo recti <lb/>per definitionem angulorum rectorũ:</s> <s xml:id="echoid-s40564" xml:space="preserve"> ergo per 29 p 1 angulus b l e erit rectus:</s> <s xml:id="echoid-s40565" xml:space="preserve"> linea ergo blcontin-<lb/>git circulũ, & oadit extra circulum per 16 p 3:</s> <s xml:id="echoid-s40566" xml:space="preserve"> quod eſt impoſsibile:</s> <s xml:id="echoid-s40567" xml:space="preserve"> eſt enim ducta ſecás circulũ per <lb/>2 p 3.</s> <s xml:id="echoid-s40568" xml:space="preserve"> Non ergo fiet reflexio à puncto l.</s> <s xml:id="echoid-s40569" xml:space="preserve"> Sequitur autem magis impoſsibile, ſi ſit proportio lineæ zt <lb/>ad lineam t e, ſicut lineæ z h ad aliquam lineam minorem linea h b.</s> <s xml:id="echoid-s40570" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s40571" xml:space="preserve"> quoniã <lb/>de quolibet dato puncto eſt penitus eodem modo deducendum.</s> <s xml:id="echoid-s40572" xml:space="preserve"/> </p> <div xml:id="echoid-div1626" type="float" level="0" n="0"> <figure xlink:label="fig-0620-01" xlink:href="fig-0620-01a"> <variables xml:id="echoid-variables711" xml:space="preserve">a l m b h e t z d n g</variables> </figure> </div> </div> <div xml:id="echoid-div1628" type="section" level="0" n="0"> <head xml:id="echoid-head1227" xml:space="preserve" style="it">17. Centro uiſus & puncto rei uiſæ exiſtentibus in una diametro ſpeculi ſphærici concaui, & <lb/>inæqualiter diſtantibus à centro, ſi exceſſus diſt antiarum ad minorem diſt antiam proportionẽ <lb/>habeat, quam pars diametri interiacentis puncta data adlineam maiorem parte diametri in-<lb/>teriacente punctum centro propinquius & peripheriam, fiet reflexio: poßibilé eſt punctum re <lb/>flexionis inueniri.</head> <p> <s xml:id="echoid-s40573" xml:space="preserve">Sit ſpeculi ſphęrici concaui maior circulus a b g d:</s> <s xml:id="echoid-s40574" xml:space="preserve"> cuius centrum e:</s> <s xml:id="echoid-s40575" xml:space="preserve"> & diameter ſit b d:</s> <s xml:id="echoid-s40576" xml:space="preserve"> in qua ſit <lb/> <pb o="319" file="0621" n="621" rhead="LIBER OCTAVVS."/> centrum uiſus, quod ſit z:</s> <s xml:id="echoid-s40577" xml:space="preserve"> & punctus rei uiſę, qui ſith:</s> <s xml:id="echoid-s40578" xml:space="preserve"> diſtetq́ centrum uiſus z plus à centro ſpeculi, <lb/>quod eſt e, quàm punctus rei uiſæ, qui eſt h:</s> <s xml:id="echoid-s40579" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s40580" xml:space="preserve"> proportio exceſſus diſtantiæ maioris, quæ eſt z e, <lb/>ad minorem, quę eſt h e, ſicut partis diametri inter puncta data cadentis, quæ eſt z h, ad lineam ma-<lb/>iorem parte diametri, quæ eſt inter pũctum h & peripheriam, quę eſt h b.</s> <s xml:id="echoid-s40581" xml:space="preserve"> Dico quòd in hoc ſitu fiet <lb/>reflexio:</s> <s xml:id="echoid-s40582" xml:space="preserve"> & quòd eſt poſsibile punctum reflexionis inueniri.</s> <s xml:id="echoid-s40583" xml:space="preserve"> Ducatur enim diameter a g orthogo-<lb/>naliter ſuper diametrum b d.</s> <s xml:id="echoid-s40584" xml:space="preserve"> Et quia linea z e eſt maior quàm linea h e:</s> <s xml:id="echoid-s40585" xml:space="preserve"> ſit linea e t æqualis lineæ h e <lb/>per 3 p 1:</s> <s xml:id="echoid-s40586" xml:space="preserve"> erit ergo linea z t exceſſus lineæ z e ſuper lineam h e:</s> <s xml:id="echoid-s40587" xml:space="preserve"> quæ ergo eſt proportio lineæ z t ad li-<lb/>neam he, eadem ſit per 3 th.</s> <s xml:id="echoid-s40588" xml:space="preserve"> 1 huius pro <lb/> <anchor type="figure" xlink:label="fig-0621-01a" xlink:href="fig-0621-01"/> portio lineæ z h ad aliam lineam, quæ <lb/>ſit h k:</s> <s xml:id="echoid-s40589" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s40590" xml:space="preserve"> ex hypotheſi linea h k ma-<lb/>ior quàm linea h b:</s> <s xml:id="echoid-s40591" xml:space="preserve"> cader ergo pũctum <lb/>k extra peripheriam circuli.</s> <s xml:id="echoid-s40592" xml:space="preserve"> A puncto <lb/>itaq;</s> <s xml:id="echoid-s40593" xml:space="preserve"> k ducatur linea contingens circu-<lb/>lum a b g d per 17 p 3, quæ ſit k l, contin-<lb/>gens circulum in puncto l:</s> <s xml:id="echoid-s40594" xml:space="preserve"> & copulen-<lb/>tur lineę lz, & l h, & l e:</s> <s xml:id="echoid-s40595" xml:space="preserve"> & à puncto e per <lb/>31 p 1 ducatur linea æ quidiſtans lineæ k <lb/>l:</s> <s xml:id="echoid-s40596" xml:space="preserve"> quæ ſit n e m ſecans lineam l zin pun-<lb/>cto m:</s> <s xml:id="echoid-s40597" xml:space="preserve"> & linea l h producatur:</s> <s xml:id="echoid-s40598" xml:space="preserve"> hæc ergo <lb/>per 2 th.</s> <s xml:id="echoid-s40599" xml:space="preserve"> 1 huius concurret cum linea m <lb/>e n:</s> <s xml:id="echoid-s40600" xml:space="preserve"> quia concurrit cũ eius æ quidiſtan-<lb/>te, quę eſt linea l k:</s> <s xml:id="echoid-s40601" xml:space="preserve"> ſit punctus concur-<lb/>ſus n.</s> <s xml:id="echoid-s40602" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s40603" xml:space="preserve"> eſt proportio lineæ z h <lb/>ad lineam h k, ſicut lineæ z t ad lineam e h, uel ad eius æqualem lineam, ſcilicette per 7 p 5:</s> <s xml:id="echoid-s40604" xml:space="preserve"> erit per <lb/>18 p 5 coniunctim proportio lineæ z k ad lineam h k, ſicut lineæ z e ad lineam t e:</s> <s xml:id="echoid-s40605" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s40606" xml:space="preserve"> permutatim <lb/>per 16 p 5 proportion lineæ z k ad lineam z e, ſicut lineę h k ad lineam t e, uel ad eius æ qualem lineam <lb/>h e.</s> <s xml:id="echoid-s40607" xml:space="preserve"> Eſt autem proportio lineę k h ad lineam e h, ſicut lineæ k l ad lineam e n per 4 p 6:</s> <s xml:id="echoid-s40608" xml:space="preserve"> quoniam tri-<lb/>gona h l k & h n e ſunt æ quiangula per 29 p 1:</s> <s xml:id="echoid-s40609" xml:space="preserve"> ideo quia lineæ k l & n e ſunt æ quidiſtantes.</s> <s xml:id="echoid-s40610" xml:space="preserve"> Propor-<lb/>tio uerò lineę k z ad lineam e z, eſt ſicut proportio lineæ k l ad lineam e m per 4 p 6:</s> <s xml:id="echoid-s40611" xml:space="preserve"> quoniam trigo-<lb/>na k l z & e m z ſuntæ quiangula per 29 p 1, quia linea e m æ quidiſtat lineę k l.</s> <s xml:id="echoid-s40612" xml:space="preserve"> Lineæ itaq;</s> <s xml:id="echoid-s40613" xml:space="preserve"> n e & m e <lb/>ad lineam k l ean dem habent proportionẽ:</s> <s xml:id="echoid-s40614" xml:space="preserve"> quoniam ex hypotheſi eſt proportio lineę k z ad lineam <lb/>z e, ſicut lineæ k h ad lineam h e:</s> <s xml:id="echoid-s40615" xml:space="preserve"> ergo per 9 p 5 lineę en & e m ſunt æ quales:</s> <s xml:id="echoid-s40616" xml:space="preserve"> linea ueròl e eſt cõmu-<lb/>nis duobus trigonis l e n & l e m, & anguli l e n & l e m ſunt æ quales, quia ſunt recti per 29 p 1, angu-<lb/>lus enim k l e eſt rectus per 18 p 3:</s> <s xml:id="echoid-s40617" xml:space="preserve"> ergo per 4 p 1 duo anguli z l e & e l h ſunt ęquales.</s> <s xml:id="echoid-s40618" xml:space="preserve"> Ergo per 20 th.</s> <s xml:id="echoid-s40619" xml:space="preserve"> 5 <lb/>huius forma puncti h reflectitur ad punctum z, uel econuerſo, à puncto ſpeculi, quod eſtl.</s> <s xml:id="echoid-s40620" xml:space="preserve"> Pater er-<lb/>go propoſitum.</s> <s xml:id="echoid-s40621" xml:space="preserve"> Oſtenſum eſt enim quia fit reflexio mutua datorum punctorũ in hoc ſitu:</s> <s xml:id="echoid-s40622" xml:space="preserve"> & inuen-<lb/>tus eſt punctus reflexionis:</s> <s xml:id="echoid-s40623" xml:space="preserve"> quód proponebatur.</s> <s xml:id="echoid-s40624" xml:space="preserve"> Ex his itaq;</s> <s xml:id="echoid-s40625" xml:space="preserve"> manifeſtum eſt, quòd ſi linea e z fuerit <lb/>maior quàm linea e h, & ſit proportio lineæ k z ad li-<lb/> <anchor type="figure" xlink:label="fig-0621-02a" xlink:href="fig-0621-02"/> neam z e, ſicut lineæ k h ad lineã e h;</s> <s xml:id="echoid-s40626" xml:space="preserve"> quòdin omnibus <lb/>ſpeculis ſphęricis concauis conſtitutis ſuper centrum <lb/>e, quorum ſemidiameter fuerit maior quàm linea e h, <lb/>& minor quàm linea e k, fiet mutua reflexio punctorũ <lb/>h & z adinuicem à duobus punctis cõmunis ſectionis <lb/>circuli ſpeculi & circuli, cuius diameter eſt linea e k.</s> <s xml:id="echoid-s40627" xml:space="preserve"> <lb/>Sit enim in linea k h punctus, qui ſit b:</s> <s xml:id="echoid-s40628" xml:space="preserve"> & ſuper cẽtrum <lb/>e deſcribatur circulus ad quantitatem ſemidiametri e <lb/>b, qui ſit a b g d:</s> <s xml:id="echoid-s40629" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s40630" xml:space="preserve"> in ſpeculo ſphærico concauo:</s> <s xml:id="echoid-s40631" xml:space="preserve"> & <lb/>diuidatur linea e k per æqualia in puncto f per 10 p 1:</s> <s xml:id="echoid-s40632" xml:space="preserve"> <lb/>fiatq́;</s> <s xml:id="echoid-s40633" xml:space="preserve"> ſuper centrum ſcirculus, cuius diameter ſit e k:</s> <s xml:id="echoid-s40634" xml:space="preserve"> <lb/>hic ergo ſecabit circulum a b g d in duobus pũctis per <lb/>10 p 3, quę ſint puncta l & p.</s> <s xml:id="echoid-s40635" xml:space="preserve"> Dico quòd punctorum h <lb/>& z mutua reflexio fiet à punctis l & p.</s> <s xml:id="echoid-s40636" xml:space="preserve"> Ducátur enim <lb/>lineę k l, k p, el, e p:</s> <s xml:id="echoid-s40637" xml:space="preserve"> erit ergo angulus k l e rectus per 31 <lb/>p 3:</s> <s xml:id="echoid-s40638" xml:space="preserve"> ergo per 16 p 3 linea k l contingit circulum a b g d, <lb/>cum ſit perpen dicularis ſuper ſemidiametrum ipſius, <lb/>quæ eſt e l.</s> <s xml:id="echoid-s40639" xml:space="preserve"> Ducta itaq;</s> <s xml:id="echoid-s40640" xml:space="preserve"> à puncto e linea n e m ęquidi-<lb/>ſtanter lineę k l, dem onſtrabitur ut prius, quoniá pun-<lb/>cta h & z mutuò reflectentur adinuicem à puncto l.</s> <s xml:id="echoid-s40641" xml:space="preserve"> <lb/>Similiter quoque ductis lineis z p & h p, & linea q e s <lb/>ęquidiſtante lineę k p:</s> <s xml:id="echoid-s40642" xml:space="preserve"> nam eadẽ eſt demõſtratio hinc <lb/>inde.</s> <s xml:id="echoid-s40643" xml:space="preserve"> Semper enim anguli incidentię & reflexionis ad <lb/>puncta l & p fiunt ęquales.</s> <s xml:id="echoid-s40644" xml:space="preserve"> Patet etiam expręmiſsis quòd ſi linea in cidentię uel reflexionis, quę eſt <lb/>hl, ſit perpen dicularis ſuper lineam e k, quoniá lineá z l neceſſariò circulum cõtingit, cuius diame-<lb/>ter eſt linea e k:</s> <s xml:id="echoid-s40645" xml:space="preserve"> efficiturq́;</s> <s xml:id="echoid-s40646" xml:space="preserve"> tunc angulus z l h maximus illorũ angulorum, ſecundũ quos in hoc ſitu <lb/> <pb o="320" file="0622" n="622" rhead="VITELLONIS OPTICAE"/> poteſt fieri reflexio.</s> <s xml:id="echoid-s40647" xml:space="preserve"> Ducatur enim à puncto f, quod eſt centrum circuli k l e p, linea fl:</s> <s xml:id="echoid-s40648" xml:space="preserve"> erit per 5 p 1 <lb/>angulus fle ęqualis angulo fel:</s> <s xml:id="echoid-s40649" xml:space="preserve"> ſed angulus fel eſt ęqua-<lb/> <anchor type="figure" xlink:label="fig-0622-01a" xlink:href="fig-0622-01"/> lis duobus angulis e z l, & e l z per 32 p 1, cũ ſit illis extrin-<lb/>ſecus in trigono z e l:</s> <s xml:id="echoid-s40650" xml:space="preserve"> angulus ergo fl e eſt æ qualis duo-<lb/>bus angulis e z l, & e l z:</s> <s xml:id="echoid-s40651" xml:space="preserve"> ſed angulus e l z eſt ęqualis angu <lb/>lo e l h ex præmiſsis:</s> <s xml:id="echoid-s40652" xml:space="preserve"> remanet ergo angulus fl h æqualis <lb/>angulo e z l.</s> <s xml:id="echoid-s40653" xml:space="preserve"> Sit quoq;</s> <s xml:id="echoid-s40654" xml:space="preserve"> angulus h l z communiter additus <lb/>utrobiq;</s> <s xml:id="echoid-s40655" xml:space="preserve">: erit ergo angulus flz ęqualis duobus angulis e <lb/>z l & h l z:</s> <s xml:id="echoid-s40656" xml:space="preserve"> ſed quia angulus l h z ex hypotheſi eſt rectus:</s> <s xml:id="echoid-s40657" xml:space="preserve"> <lb/>patet per 32 p 1 quòd illi duo anguli (qui ſunth l z & h zl) <lb/>ſunt æ quales uni recto.</s> <s xml:id="echoid-s40658" xml:space="preserve"> Angulus ergo flz eſt rectus.</s> <s xml:id="echoid-s40659" xml:space="preserve"> Li-<lb/>nea ergo l z contingit circulum k l e p per 16 p 3.</s> <s xml:id="echoid-s40660" xml:space="preserve"> Sequitur <lb/>ergo idem, quod prius.</s> <s xml:id="echoid-s40661" xml:space="preserve"> Et hoc eſt notandum, quòd in hac <lb/>diſpoſitione cẽtri uiſus & ipſorum uiſibilium ſemper lo-<lb/>cus imaginis eſt in centro uiſus per 37 th.</s> <s xml:id="echoid-s40662" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s40663" xml:space="preserve"> quo-<lb/>niam, ut patet, ibi concurrit cathetus incidentiæ cum li-<lb/>nea reflexionis:</s> <s xml:id="echoid-s40664" xml:space="preserve"> patetq́;</s> <s xml:id="echoid-s40665" xml:space="preserve"> ex præmiſsis, quomodo in hac <lb/>diſpoſitione de facili inuenitur punctus reflexionis:</s> <s xml:id="echoid-s40666" xml:space="preserve"> imò <lb/>puncta duo, quæ ſuntinter ſectiones duorum circulorũ.</s> <s xml:id="echoid-s40667" xml:space="preserve"> <lb/>Pater ergo propoſitum.</s> <s xml:id="echoid-s40668" xml:space="preserve"/> </p> <div xml:id="echoid-div1628" type="float" level="0" n="0"> <figure xlink:label="fig-0621-01" xlink:href="fig-0621-01a"> <variables xml:id="echoid-variables712" xml:space="preserve">l a m k b h e t z d n g</variables> </figure> <figure xlink:label="fig-0621-02" xlink:href="fig-0621-02a"> <variables xml:id="echoid-variables713" xml:space="preserve">k b f l n h s p a e g q m z d</variables> </figure> <figure xlink:label="fig-0622-01" xlink:href="fig-0622-01a"> <variables xml:id="echoid-variables714" xml:space="preserve">k f b l n h s p a e g q m z d</variables> </figure> </div> </div> <div xml:id="echoid-div1630" type="section" level="0" n="0"> <head xml:id="echoid-head1228" xml:space="preserve" style="it">18. Duorum punctorum in eadem diametro ſpeculi <lb/>ſphærici concaui exiſtentium formis ex aliquo puncto <lb/>ſpeculi adinuicem reflexis: eaſdem ab aliquo pũcto alio <lb/>eiuſdem quartæ illius circuli impoßibile eſt reflecti.</head> <p> <s xml:id="echoid-s40669" xml:space="preserve">Sit diſpoſitio, quæ in figuris proximis:</s> <s xml:id="echoid-s40670" xml:space="preserve"> reflectaturq́;</s> <s xml:id="echoid-s40671" xml:space="preserve"> forma puncti h ad punctum z à puncto ſpe <lb/>culi l.</s> <s xml:id="echoid-s40672" xml:space="preserve"> Dico quòd impoſsibile eſt, ut formarum illorum punctorum reflexio fiat ad inuicem ab ali.</s> <s xml:id="echoid-s40673" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0622-02a" xlink:href="fig-0622-02"/> quo alio puncto illius eiuſdem quartæ circuli, quæ <lb/>eſt b a, quàm à puncto l.</s> <s xml:id="echoid-s40674" xml:space="preserve"> Sit enim, ſi poſsibile eſt, ut <lb/>fiat à puncto s eiuſdem quartæ:</s> <s xml:id="echoid-s40675" xml:space="preserve"> & ducantur lineæ z <lb/>l, h l, z s, h s, e l, e s.</s> <s xml:id="echoid-s40676" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s40677" xml:space="preserve"> angulus z l h diuiſus eſt <lb/>peræ qualia per lineá e l:</s> <s xml:id="echoid-s40678" xml:space="preserve"> pater per 3 p 6 quia eſt pro-<lb/>portio lineæ z l ad lineá l h, ſicut lineæ z e ad lineam <lb/>e h.</s> <s xml:id="echoid-s40679" xml:space="preserve"> Similiter quia angulus z s h diuiſus eſt peræ qua <lb/>lia per lineam e s:</s> <s xml:id="echoid-s40680" xml:space="preserve"> erit per 3 p 6 proportio lineæ z s <lb/>ad lineam s h, ſicut lineæ z e ad lineam e h:</s> <s xml:id="echoid-s40681" xml:space="preserve"> ergo per <lb/>11 p 5 erit proportio lineæ z s ad lineam s h, ſicut li-<lb/>neæ z l ad lineam l h:</s> <s xml:id="echoid-s40682" xml:space="preserve"> ergo per 16 p 5 erit permutatim <lb/>proportio lineæ z s ad lineam z l, ſicut lineæ s h ad <lb/>lineam l h:</s> <s xml:id="echoid-s40683" xml:space="preserve"> ſed linea z s eſt minor quàm linea z l per <lb/>7 p 3:</s> <s xml:id="echoid-s40684" xml:space="preserve"> ergo linea s h eſt minor quàm linea h l:</s> <s xml:id="echoid-s40685" xml:space="preserve"> quod <lb/>eſt contra eandem 7 p 3:</s> <s xml:id="echoid-s40686" xml:space="preserve"> quoniam eſt linea s h pro-<lb/>pinquior centro ſpeculi, quod eſt e, quàm linea h l.</s> <s xml:id="echoid-s40687" xml:space="preserve"> <lb/>Et quoniam de quolibet puncto arcus a b poteſt ea-<lb/>dem fieri deductio:</s> <s xml:id="echoid-s40688" xml:space="preserve"> patet ergo quòd nõ poteſt fieri reflexio ab aliquo puncto quartæ circuli ab alio <lb/>quàm à puncto l.</s> <s xml:id="echoid-s40689" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s40690" xml:space="preserve"> demonſtrandum eſt in quarta circuli, quę eſt b g.</s> <s xml:id="echoid-s40691" xml:space="preserve"> ſi ab illius aliquo <lb/>puncto fiat reflexio.</s> <s xml:id="echoid-s40692" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s40693" xml:space="preserve"/> </p> <div xml:id="echoid-div1630" type="float" level="0" n="0"> <figure xlink:label="fig-0622-02" xlink:href="fig-0622-02a"> <variables xml:id="echoid-variables715" xml:space="preserve">b l s h a e g z d</variables> </figure> </div> </div> <div xml:id="echoid-div1632" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables716" xml:space="preserve">h e k z a d g x b</variables> </figure> <head xml:id="echoid-head1229" xml:space="preserve" style="it">19. Centro ſpeculi ſphærici concaui exiſtẽte extra <lb/>lineam connectentem cẽtrum uiſus, & punctum rei <lb/> uiſæ in diametris diuerſis exiſtentia, & æqualiter di- ſtantia à centro ſpeculi: ab uno tantùm puncto ſemi- circuli, in cuius ſemidiametris illa puncta non conſi- ſtunt, fit reflexio ad uiſum.</head> <p> <s xml:id="echoid-s40694" xml:space="preserve">Sit ſpeculi ſphærici concaui circulus a b g:</s> <s xml:id="echoid-s40695" xml:space="preserve"> cuius cen <lb/>trum ſit d:</s> <s xml:id="echoid-s40696" xml:space="preserve"> diameter a g:</s> <s xml:id="echoid-s40697" xml:space="preserve"> & ſemidiameter d b orthogona <lb/>liter erigatur ſuper diametrum a g:</s> <s xml:id="echoid-s40698" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s40699" xml:space="preserve"> centrum uiſus <lb/>punctum z:</s> <s xml:id="echoid-s40700" xml:space="preserve"> & punctus rei uiſæ ſit h:</s> <s xml:id="echoid-s40701" xml:space="preserve"> & ducatur linea z u <lb/>ſecans productã ſemidiametrũ d b in pũcto e, ita quòd <lb/>centrũ ſpeculi d ſit inter lineã z h & ſuperficiẽ ſpeculi, à <lb/>qua fit reflexio:</s> <s xml:id="echoid-s40702" xml:space="preserve"> diſtentq́;</s> <s xml:id="echoid-s40703" xml:space="preserve"> pũcta z & h æ qualiter à pũcto <lb/>d, qđ eſt centrũ ſpeculi:</s> <s xml:id="echoid-s40704" xml:space="preserve"> ꝓpter qđ erit linea bd e քpen-<lb/>dicularis ſuք lineã z h ք 8 p & 10 defi.</s> <s xml:id="echoid-s40705" xml:space="preserve"> 1.</s> <s xml:id="echoid-s40706" xml:space="preserve"> Dico quòd forma pũcti h reflectitur ad uiſũ z ab uno tãtùm <pb o="321" file="0623" n="623" rhead="LIBER OCTAVVS."/> puncto ſemicirculi a b g, quod eſt b.</s> <s xml:id="echoid-s40707" xml:space="preserve"> Ducantur enim lineæ d z, d h, z b, & b h.</s> <s xml:id="echoid-s40708" xml:space="preserve"> Et quia per 3 p 3 lineæ <lb/>b d e diuidit lineam h z per æqualia:</s> <s xml:id="echoid-s40709" xml:space="preserve"> patet quòd duo latera b e & e h ſunt æqualia duobus lateri-<lb/>bus b e & z e, & anguli b e h & b e z ſunt æquales, quia recti:</s> <s xml:id="echoid-s40710" xml:space="preserve"> ergo per 4 p 1 patet quoniam an-<lb/>goli z b e & h b e ſunt æquales.</s> <s xml:id="echoid-s40711" xml:space="preserve"> Fit ergo per 20 th.</s> <s xml:id="echoid-s40712" xml:space="preserve"> 5 huius reflexio form æ puncti h à puncto ſpe-<lb/>culi b ad centrum uiſus, quod eſt z.</s> <s xml:id="echoid-s40713" xml:space="preserve"> Dico itaque quòd non poteſt ab aliquo alio puncto ſpeculi fie-<lb/>ri hæc reflexio.</s> <s xml:id="echoid-s40714" xml:space="preserve"> Si enim detur, quòd fiat à puncto t:</s> <s xml:id="echoid-s40715" xml:space="preserve"> ducantur lineæ z t & t h:</s> <s xml:id="echoid-s40716" xml:space="preserve"> & à centro d du-<lb/>catur ad punctum refle xionis t linea d t:</s> <s xml:id="echoid-s40717" xml:space="preserve"> quæ producta ad lineam z h ſecet ipſam in puncto k.</s> <s xml:id="echoid-s40718" xml:space="preserve"> <lb/>Quia ita que per 20 th.</s> <s xml:id="echoid-s40719" xml:space="preserve"> 5 huius linea k t diuidit angulum z t h per æqualia:</s> <s xml:id="echoid-s40720" xml:space="preserve"> patet per 3 p 6 quo-<lb/>niam eſt proportio lineæ z t ad lineam t h, ſicut lineæ z k ad lineam k h:</s> <s xml:id="echoid-s40721" xml:space="preserve"> ſed linea z k eſt minor <lb/>quàm linea z e:</s> <s xml:id="echoid-s40722" xml:space="preserve"> ergo & minor quàm linea k h.</s> <s xml:id="echoid-s40723" xml:space="preserve"> Erit ergo linea z t minor quàm linea t h:</s> <s xml:id="echoid-s40724" xml:space="preserve"> ſed per <lb/>7 p 3 linea z t eſt maior quàm linea z b, & linea h b maior quàm linea h t:</s> <s xml:id="echoid-s40725" xml:space="preserve"> erit ergo z b mi-<lb/>nor quàm linea h b, quod eſt contra præmiſſa & contra 4 p 1.</s> <s xml:id="echoid-s40726" xml:space="preserve"> Non ergo reflectetur forma puncti h <lb/>ad centrum uiſus exiſtens in puncto z à puncto ſpeculi t.</s> <s xml:id="echoid-s40727" xml:space="preserve"> Similiter quoque demonſtrandum eſt de <lb/>quolibet puncto ſemicirculi a b g.</s> <s xml:id="echoid-s40728" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s40729" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1633" type="section" level="0" n="0"> <head xml:id="echoid-head1230" xml:space="preserve" style="it">20. Centro niſus & puncto reiuiſæ exiſtentibus in diametris diuerſis circuli magni ſpeculi <lb/>ſphærici cõcaui: poßibile eſt reflexionem fieri ab aliquo pũcto arcuũ interiacentiũ diametros cir <lb/>culitranſeuntes per illa puncta: non autem ab aliquo puncto arcuum aliorũ. Alhaz. 66 n 5.</head> <p> <s xml:id="echoid-s40730" xml:space="preserve">Circulus (qui eſt communis ſectio ſuperficiei reflexionis & ſpeculi ſphærici cõcaui) ſit d g t q:</s> <s xml:id="echoid-s40731" xml:space="preserve"> <lb/>& ſit a centrum uiſus intra ſpeculum ſphæricum concauum:</s> <s xml:id="echoid-s40732" xml:space="preserve"> & ſit e centrum ſpeculi:</s> <s xml:id="echoid-s40733" xml:space="preserve"> & ſit b pũctus <lb/>rei uiſæ:</s> <s xml:id="echoid-s40734" xml:space="preserve"> & ducatur diameter d a g per centrum uiſus a:</s> <s xml:id="echoid-s40735" xml:space="preserve"> & ducatur diameter t q, ut contingit.</s> <s xml:id="echoid-s40736" xml:space="preserve"> Dico <lb/>quòd ſi fuerit b punctus rei uiſæ in ſemidiametro e t, poteſt fieri reflexio formæ eius ad uiſum a ab <lb/>aliquo puncto ſemicirculi d t g, & ab aliquo pũcto ſemicirculi ſibi oppoſiti, qui eſt d q g.</s> <s xml:id="echoid-s40737" xml:space="preserve"> Ducatur e-<lb/>nim à puncto b rei uiſæ ad aliquẽ punctum ſemicirculi g t d arcus quartæ t d, qui ſit pũctus m, linea <lb/>incidentiæ, quæ ſit b m:</s> <s xml:id="echoid-s40738" xml:space="preserve"> & ducantur lineæ b a & m a:</s> <s xml:id="echoid-s40739" xml:space="preserve"> & ducatur ſemidiameter e m, quæ quia diuidit <lb/>b aſim a b trigonia m b, diuidit ergo angulum b m a <lb/>ք 29 th.</s> <s xml:id="echoid-s40740" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s40741" xml:space="preserve"> Producatur ergo ſemidiameter m e <lb/>ad partẽ circũferentiæ, quæ opponitur puncto m, in <lb/> <anchor type="figure" xlink:label="fig-0623-01a" xlink:href="fig-0623-01"/> punctũ, qui ſit punctus h, arcus g q:</s> <s xml:id="echoid-s40742" xml:space="preserve"> & ducátur lineę <lb/>b h & a h:</s> <s xml:id="echoid-s40743" xml:space="preserve"> ſecabit quoq;</s> <s xml:id="echoid-s40744" xml:space="preserve"> linea a h diametrũ t q.</s> <s xml:id="echoid-s40745" xml:space="preserve"> Sit.</s> <s xml:id="echoid-s40746" xml:space="preserve"> ut <lb/>ſecet ipſam in pũcto c:</s> <s xml:id="echoid-s40747" xml:space="preserve"> & linea h b ſecabit eandẽ dia-<lb/>metrũ t q in pũcto b.</s> <s xml:id="echoid-s40748" xml:space="preserve"> Sunt quoq;</s> <s xml:id="echoid-s40749" xml:space="preserve"> pũcta b & c ex di-<lb/>uerſis partibus cẽtri e:</s> <s xml:id="echoid-s40750" xml:space="preserve"> linea ergo e h diuidet angulũ <lb/>a h b per 29 th.</s> <s xml:id="echoid-s40751" xml:space="preserve"> 1 huius, quoniá diuidit baſim ei ſub-<lb/>tenſam, quæ eſt b c.</s> <s xml:id="echoid-s40752" xml:space="preserve"> Dico itaq;</s> <s xml:id="echoid-s40753" xml:space="preserve"> quòd forma puncti b <lb/>poreſt reflecti ad uiſum a ab aliquo pũcto arcus in-<lb/>teriacentis ſemidiametros et & e d, in quibus ſunt <lb/>puncta a & b, qui eſt arcus t d:</s> <s xml:id="echoid-s40754" xml:space="preserve"> & ſimiliter ab aliquo <lb/>puncto arcus illi arcui oppoſiti, interiacentis alias <lb/>ſemidiametros illis conterminales, quæ ſunt e g & <lb/>e q, utpote ab aliquo puncto arcus, qui eſt q g:</s> <s xml:id="echoid-s40755" xml:space="preserve"> & qđ <lb/>non poteſt reflecti ab aliquo puncto arcus g t.</s> <s xml:id="echoid-s40756" xml:space="preserve"> Si enim hoc dicatur eſſe poſsibile:</s> <s xml:id="echoid-s40757" xml:space="preserve"> ſumatur tunc ali-<lb/>quis pũctus arcus g t, qui ſit k, propinquior pũcto t:</s> <s xml:id="echoid-s40758" xml:space="preserve"> & ducãtur lineæ a k & k b:</s> <s xml:id="echoid-s40759" xml:space="preserve"> & producatur linea <lb/> <anchor type="figure" xlink:label="fig-0623-02a" xlink:href="fig-0623-02"/> k b, donec cadat ſuք diametrũ d g in punctũ o:</s> <s xml:id="echoid-s40760" xml:space="preserve"> cadet <lb/>autem per 14 th.</s> <s xml:id="echoid-s40761" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s40762" xml:space="preserve"> ideo quia angulus b e d <lb/>eſt rectus, & angulus k b t eſt acutus:</s> <s xml:id="echoid-s40763" xml:space="preserve"> & oẽs illę lineę <lb/>ſunt in eadẽ ſuperficie.</s> <s xml:id="echoid-s40764" xml:space="preserve"> Quoniam ergo puncta o & a <lb/>ſunt in eadẽ parte centri circuli, qđ eſt e, patet quòd <lb/>perpendicularis ducta à puncto k ad centrũ e, nõ di <lb/>uidit angulũ o k a:</s> <s xml:id="echoid-s40765" xml:space="preserve"> & ita forma pũcti b nõ poteſt re-<lb/>flecti ad uiſum a à puncto ſpeculi, quod eſt k.</s> <s xml:id="echoid-s40766" xml:space="preserve"> Simili-<lb/>ter ſumpto alio puncto, quod ſit f, ita ut linea b f ſit <lb/>ęquidiſt ans diametro d g, uel quòd angulus f b t fiat <lb/>obtuſus.</s> <s xml:id="echoid-s40767" xml:space="preserve"> Semper enim tunc patebit quoniam per-<lb/>pendicularis e f non diuidet angulum b f a per <lb/>29 th.</s> <s xml:id="echoid-s40768" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s40769" xml:space="preserve"> quoniam cadet extra a b baſim tri-<lb/>goni a b f.</s> <s xml:id="echoid-s40770" xml:space="preserve"> Non ergo poteſt reflecti forma puncti b <lb/>ad uiſum a à pũcto ſpeculi f:</s> <s xml:id="echoid-s40771" xml:space="preserve"> ergo neq;</s> <s xml:id="echoid-s40772" xml:space="preserve"> ab aliquo pũ-<lb/>cto arcus oppoſiti arcui g t, qui eſt arcus d q.</s> <s xml:id="echoid-s40773" xml:space="preserve"> Eodem <lb/>quoq;</s> <s xml:id="echoid-s40774" xml:space="preserve"> modo demonſtrandum, ſi b punctus rei uiſæ <lb/>fuerit in ſuperficie ſpeculi aut extra ſpeculum:</s> <s xml:id="echoid-s40775" xml:space="preserve"> dum <lb/>tamen punctum a, quod eſt centrum uiſus, ſit intra <lb/>ſpeculum:</s> <s xml:id="echoid-s40776" xml:space="preserve"> & idem erit modus probãdi.</s> <s xml:id="echoid-s40777" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s40778" xml:space="preserve"> ſi punctus a centrum uiſus fueritin ſuper-<lb/> <pb o="322" file="0624" n="624" rhead="VITELLONIS OPTICAE"/> ficie ſpeculi, & punctus b fuerit interius uel exterius, idem eſt probandi modus.</s> <s xml:id="echoid-s40779" xml:space="preserve"> Si etiam centrnm <lb/>uiſus a ſuerit extra ſpeculum, & punctus b rei uiſæ fuerit intra ſpeculum:</s> <s xml:id="echoid-s40780" xml:space="preserve"> patet idem, quod propoſi-<lb/>tum eſt.</s> <s xml:id="echoid-s40781" xml:space="preserve"> Ducantur enim à puncto a centro uiſus lineæ contingẽtes circulum g t d per 17 p 3, quæ ſint <lb/>lineæ a h & a z:</s> <s xml:id="echoid-s40782" xml:space="preserve"> & ducantur duæ diametri, una uiſualis, quæ ſit a e g, & alia, quæ ſit t e q:</s> <s xml:id="echoid-s40783" xml:space="preserve"> & ſit b pun-<lb/>ctus rei uiſæ in diametro t e q.</s> <s xml:id="echoid-s40784" xml:space="preserve"> Palàm ita q;</s> <s xml:id="echoid-s40785" xml:space="preserve"> ex præmiſsis, quia reflectitur forma puncti b ad uiſum a <lb/>ab aliquo puncto arcus t d.</s> <s xml:id="echoid-s40786" xml:space="preserve"> Igitur ab aliquo puncto arcus t z:</s> <s xml:id="echoid-s40787" xml:space="preserve"> quia impoſsibile eſt, utreflectatur ab <lb/>aliquo puncto arcus z d:</s> <s xml:id="echoid-s40788" xml:space="preserve"> quoniam ille arcus cadit ſub puncto contingentiæ, & etιam propter inæ-<lb/>qualitatem angulorum:</s> <s xml:id="echoid-s40789" xml:space="preserve"> quoniam per 18 p 3 angulus e z a eſt rectus, & angulus b z e per 42 th.</s> <s xml:id="echoid-s40790" xml:space="preserve"> 1 hu-<lb/>ius eſt minorrecto, cui fiunt inæquales omnes anguli conſtituti ſuper lineam z a.</s> <s xml:id="echoid-s40791" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s40792" xml:space="preserve"> <lb/>ab aliquo puncto arcus q g (qui eſt oppoſitus arcui t d) poteſt fieri reflexio formæ puncti b ad ui-<lb/>ſum exiſtentem in puncto a ſed ab arcu t g uel d q nulla fiet reflexio propter ſupradicta:</s> <s xml:id="echoid-s40793" xml:space="preserve"> ſimiliterq́;</s> <s xml:id="echoid-s40794" xml:space="preserve"> <lb/>permutato puncto b in aliam diametrum, quę ſit eadem diameter t q, idem accidet, quod prius.</s> <s xml:id="echoid-s40795" xml:space="preserve"> Pa-<lb/>tet ergo propoſitum.</s> <s xml:id="echoid-s40796" xml:space="preserve"/> </p> <div xml:id="echoid-div1633" type="float" level="0" n="0"> <figure xlink:label="fig-0623-01" xlink:href="fig-0623-01a"> <variables xml:id="echoid-variables717" xml:space="preserve">c k m b f d a o e g c q h</variables> </figure> <figure xlink:label="fig-0623-02" xlink:href="fig-0623-02a"> <variables xml:id="echoid-variables718" xml:space="preserve">a m d z h t b t l e q q g</variables> </figure> </div> </div> <div xml:id="echoid-div1635" type="section" level="0" n="0"> <head xml:id="echoid-head1231" xml:space="preserve" style="it">21. Centro uiſus & puncto rei uiſæ exiſtentibus in diuerſis diametris circuli magni ſpeculi <lb/>ſphærici concaui, ſi à centro uiſus ducatur linea æquidiſtans diametro, in qua eſt punctum rei <lb/>uiſæ ſecans circulum: erũt omnia loca imaginum punctorum reflexorum ab arcu ſpeculi inter-<lb/>iacente terminum diametri rei uιſæ & illam æquidiſtantẽ, extra ſpeculum: & loca imaginum <lb/>refle xarum à reliquo arcus interiacente diametros, erunt ultra uiſum: oppoſiti uerò arcus loca <lb/>imaginum erunt inter centrum uiſus & ſpeculum. Alhazen 67 n 5.</head> <figure> <variables xml:id="echoid-variables719" xml:space="preserve">l p m t n b d a e g x s q u</variables> </figure> <p> <s xml:id="echoid-s40797" xml:space="preserve">Sit diſpoſitio, quæ prius:</s> <s xml:id="echoid-s40798" xml:space="preserve"> & ducatur à puncto a linea æquidiſtans <lb/>ſemidiametro t e, quæ ſit a p.</s> <s xml:id="echoid-s40799" xml:space="preserve"> Dico quòd loca imaginum reflexarum <lb/>à punctis arcus t p erunt extra ſpeculum:</s> <s xml:id="echoid-s40800" xml:space="preserve"> loca uerò imaginum ar-<lb/>cus p d erunt ultra centrum uiſus, quod eſt a:</s> <s xml:id="echoid-s40801" xml:space="preserve"> loca uerò imaginum <lb/>arcus q g erunt inter centrum uiſus & ſpeculi ſuperficiem.</s> <s xml:id="echoid-s40802" xml:space="preserve"> Dato <lb/>enim quòd form a puncti b exiſtens in ſemidiametro t e, reflectatur <lb/>ad uiſum a exiſtentem in ſemidiametro d e ab aliquo puncto arcus <lb/>p t, quiſit m:</s> <s xml:id="echoid-s40803" xml:space="preserve"> palàm per 14 th.</s> <s xml:id="echoid-s40804" xml:space="preserve"> 1 huius quòd lineæ a m & b e concur-<lb/>rentultra puncta m & b extra ſpeculum.</s> <s xml:id="echoid-s40805" xml:space="preserve"> Sit quoque punctus con-<lb/>curſusl, qui per 37 th.</s> <s xml:id="echoid-s40806" xml:space="preserve"> 5 huius erit locus imaginis formæ puncti b.</s> <s xml:id="echoid-s40807" xml:space="preserve"> <lb/>Quòd ſi à puncto n arcus d p fiat reflexio:</s> <s xml:id="echoid-s40808" xml:space="preserve"> patet per eandem 14 th.</s> <s xml:id="echoid-s40809" xml:space="preserve"> 1 <lb/>huius quoniam lineæ a n & b e concurrent ultra puncta a & e:</s> <s xml:id="echoid-s40810" xml:space="preserve"> ſit <lb/>concurſus in puncto s:</s> <s xml:id="echoid-s40811" xml:space="preserve"> eritq́ue punctum s locus imaginis formæ <lb/>puncti b retro uiſum.</s> <s xml:id="echoid-s40812" xml:space="preserve"> Si uerò forma puncti b reflectatur ad uiſum <lb/>a ab aliquo puncto arcus q g:</s> <s xml:id="echoid-s40813" xml:space="preserve"> quoniam in præmiſſa oſtenſum eſt hoc <lb/>eſſe poſsibile:</s> <s xml:id="echoid-s40814" xml:space="preserve"> ſit, ut illa reflexio fiat à puncto arcus q g, quod ſit u.</s> <s xml:id="echoid-s40815" xml:space="preserve"> <lb/>Palàm itaque quoniam linea b e producta diuidit angulum a e u:</s> <s xml:id="echoid-s40816" xml:space="preserve"> <lb/>ergo per 29 th.</s> <s xml:id="echoid-s40817" xml:space="preserve"> 1 huius patet quòdipſa ſecat baſim a u:</s> <s xml:id="echoid-s40818" xml:space="preserve"> ſit, ut ſecet <lb/>ipſam in puncto x.</s> <s xml:id="echoid-s40819" xml:space="preserve"> Linea itaque a u, quæ eſt linea reflexionis, & ca-<lb/>thetus incidentiæ ſormæ puncti b, quæ eſt b e, ſecant ſe in puncto x:</s> <s xml:id="echoid-s40820" xml:space="preserve"> eſt ergo per 37 th.</s> <s xml:id="echoid-s40821" xml:space="preserve"> 5 huius <lb/>punctum x locus imaginis formæ puncti b:</s> <s xml:id="echoid-s40822" xml:space="preserve"> & ipſe eſtinter uiſum & ſpeculum.</s> <s xml:id="echoid-s40823" xml:space="preserve"> Secundum hæc <lb/>itaq;</s> <s xml:id="echoid-s40824" xml:space="preserve"> loca imaginum diuerſantur, ut etiam declaratum eſt in 11 huius.</s> <s xml:id="echoid-s40825" xml:space="preserve"> Nunquam autem eſt poſsibile <lb/>locum imaginis eſſe in centro uiſus, niſi cum punctus rei uiſæ & centrum uiſus in eadem ſunt dia-<lb/>metro.</s> <s xml:id="echoid-s40826" xml:space="preserve"> Tunc enim facta reflexione, utcunq;</s> <s xml:id="echoid-s40827" xml:space="preserve"> ſit poſsibile, ſemper patet quòd linea reflexionis & ca-<lb/>thetus incidentiæ concurrunt in centro uiſus:</s> <s xml:id="echoid-s40828" xml:space="preserve"> quoniam ſolus ille punctus ambabus illis lineis eſt <lb/>communis.</s> <s xml:id="echoid-s40829" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s40830" xml:space="preserve">, quod proponebatur.</s> <s xml:id="echoid-s40831" xml:space="preserve"> Semper enim eodem modo eſt demonſtrandum pro-<lb/>poſitum, ſiue punctum a centrum uiſus ſit intra ſpeculum:</s> <s xml:id="echoid-s40832" xml:space="preserve"> ſiue in ſuperficie ipſius ſpeculi:</s> <s xml:id="echoid-s40833" xml:space="preserve"> ſiue extra <lb/>ſpeculum:</s> <s xml:id="echoid-s40834" xml:space="preserve"> dum tamen linea à puncto a ducta æ quidiſtanter diametro, in qua eſt punctus rei uiſæ, <lb/>ſecet circulum ſpeculi, & non contingat ipſum.</s> <s xml:id="echoid-s40835" xml:space="preserve"> Forma uerò reflexa à puncto p ſecundum lineam p <lb/>a (ſi punctus, cuius forma reflectitur, fuerit in ſemidiametro t e, cui æquidiſtat linea a p) poteſt ui-<lb/>deri in ipſa ſpeculi ſuperficie, ut oſtendimus in 11 & 12 th.</s> <s xml:id="echoid-s40836" xml:space="preserve"> huius.</s> <s xml:id="echoid-s40837" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1636" type="section" level="0" n="0"> <head xml:id="echoid-head1232" xml:space="preserve" style="it">22. Quilibet punctus diametri circuli magniſpeculi ſphærici concaui poteſt eſſe locus imagi-<lb/>num, quantumcun producatur. Alhazen 68 n 5.</head> <p> <s xml:id="echoid-s40838" xml:space="preserve">Sit a g diameter circuli ſpeculi ſphærici concaui, qui ſit a t m g:</s> <s xml:id="echoid-s40839" xml:space="preserve"> cuius circuli cẽtrum ſit d:</s> <s xml:id="echoid-s40840" xml:space="preserve"> produ-<lb/>caturq́;</s> <s xml:id="echoid-s40841" xml:space="preserve"> extra circulum:</s> <s xml:id="echoid-s40842" xml:space="preserve"> & ſignetur in ipſa punctum z:</s> <s xml:id="echoid-s40843" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s40844" xml:space="preserve"> punctus e centrum uiſus intra circulum <lb/>in ſemidiametro m d.</s> <s xml:id="echoid-s40845" xml:space="preserve"> Dico quòd punctus z poteſt eſſe locus imaginis.</s> <s xml:id="echoid-s40846" xml:space="preserve"> Ducatur enim linea e t z <lb/>per t punctum circumferentiæ circuli:</s> <s xml:id="echoid-s40847" xml:space="preserve"> & ducatur linea d t:</s> <s xml:id="echoid-s40848" xml:space="preserve"> eritq́ue angulus e t d acutus per <lb/>42 th.</s> <s xml:id="echoid-s40849" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s40850" xml:space="preserve"> Fiat itaque angulus d t l ſuper terminum lineæ d t æqualis angulo e t d per 23 <lb/>p 1:</s> <s xml:id="echoid-s40851" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s40852" xml:space="preserve"> linea tl diametrum d a in puncto l.</s> <s xml:id="echoid-s40853" xml:space="preserve"> Palàm itaque per 20 th.</s> <s xml:id="echoid-s40854" xml:space="preserve"> 5 huius quoniam forma <lb/>puncti l reflectitur ad uiſum exiſtentem in puncto e à puncto ſpeculi quod eſt t:</s> <s xml:id="echoid-s40855" xml:space="preserve"> & eius ima-<lb/>ginis locus eſt in puncto z per 37 th.</s> <s xml:id="echoid-s40856" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s40857" xml:space="preserve"> quoniam in illo puncto concurrit cathetus <lb/> <pb o="323" file="0625" n="625" rhead="LIBER OCTAVVS."/> incidentiæ, qui eſt d l z, cum linea reflexionis, quæ eſt c e.</s> <s xml:id="echoid-s40858" xml:space="preserve"> Et ſi ſumatur punctus diametri a gintra <lb/>circulum, qui debet oſtendi poſſe eſſe locus imaginis, utſi ille punctus ſit l:</s> <s xml:id="echoid-s40859" xml:space="preserve"> palàm quia & ipſe erit <lb/>locus imaginis alicuius formæ.</s> <s xml:id="echoid-s40860" xml:space="preserve"> Dueatur enim linea e l, & produca-<lb/>tur uſque ad punctum circumferentiæ, quod ſit b, & ducatur linea <lb/> <anchor type="figure" xlink:label="fig-0625-01a" xlink:href="fig-0625-01"/> d b:</s> <s xml:id="echoid-s40861" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s40862" xml:space="preserve"> angulus d b e acutus per 42 th.</s> <s xml:id="echoid-s40863" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s40864" xml:space="preserve"> Fiat ergo æqualis <lb/>ſibi, qui ſit d b p.</s> <s xml:id="echoid-s40865" xml:space="preserve"> Palàm itaque per 20 th.</s> <s xml:id="echoid-s40866" xml:space="preserve"> 5 huius quoniam reflecti-<lb/>tur forma puncti p ad uiſum e à puncto ſpeculi b:</s> <s xml:id="echoid-s40867" xml:space="preserve"> & locus imaginis <lb/>formæ puncti p eſt punctus l per 37 th.</s> <s xml:id="echoid-s40868" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s40869" xml:space="preserve"> Sumpto quoq;</s> <s xml:id="echoid-s40870" xml:space="preserve"> quoli-<lb/>bet puncto alio eadem eſt probatio.</s> <s xml:id="echoid-s40871" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s40872" xml:space="preserve"/> </p> <div xml:id="echoid-div1636" type="float" level="0" n="0"> <figure xlink:label="fig-0625-01" xlink:href="fig-0625-01a"> <variables xml:id="echoid-variables720" xml:space="preserve">z t a l m e d b p g</variables> </figure> </div> </div> <div xml:id="echoid-div1638" type="section" level="0" n="0"> <head xml:id="echoid-head1233" xml:space="preserve" style="it">23. Centro uiſus & puncto rei uiſæ in eadẽ circuli magni dia-<lb/>metro exιſtentibus: punctorũ reflexorum à ſpeculis ſphæricis con-<lb/>cauis, quibus eſt locus imaginis centrum uiſus, poßibile eſt, ut ab <lb/>uno tantùm ſemicirculi puncto fiat reflexio ad uiſum: uel tantùm <lb/>à quolibet unius alterius circuli determinati puncto. Alha-<lb/>zen 69 n 5.</head> <p> <s xml:id="echoid-s40873" xml:space="preserve">Eſto circulus ſpeculi ſphærici concaui g z b a:</s> <s xml:id="echoid-s40874" xml:space="preserve"> cuius centrum ſit d:</s> <s xml:id="echoid-s40875" xml:space="preserve"> <lb/>& interſecent ſe in ipſo duæ diametri z a & g b orthogonaliter:</s> <s xml:id="echoid-s40876" xml:space="preserve"> & ſit <lb/>in diametro z a punctus e:</s> <s xml:id="echoid-s40877" xml:space="preserve"> qui ſit centrum uiſus:</s> <s xml:id="echoid-s40878" xml:space="preserve"> & h, qui ſit punctus <lb/>rei uiſæ, ſit in eadem diametro z a:</s> <s xml:id="echoid-s40879" xml:space="preserve"> quoniam ubicunque fuerint cen-<lb/>trũ uiſus & pũctus rei uiſę in una illius circuli diametro, ſemper poſ-<lb/>ſunt dictæ diametri taliter produci, ut ſe orthogonaliter interſecent, diametro z a per puncta e & h <lb/>tranſeunte.</s> <s xml:id="echoid-s40880" xml:space="preserve"> Aut ergo linea e d interiacens cẽtra ui-<lb/>ſus & ſpeculi eſt æqualis lineæ d h:</s> <s xml:id="echoid-s40881" xml:space="preserve"> aut non.</s> <s xml:id="echoid-s40882" xml:space="preserve"> Si ſit <lb/> <anchor type="figure" xlink:label="fig-0625-02a" xlink:href="fig-0625-02"/> æqualis, ita quòd illa pũcta æqualiter diſtent à cen-<lb/>tro ſpeculi:</s> <s xml:id="echoid-s40883" xml:space="preserve"> ducantur lineæ h g, h b, e g, e b.</s> <s xml:id="echoid-s40884" xml:space="preserve"> Palàm <lb/>itaq;</s> <s xml:id="echoid-s40885" xml:space="preserve"> per 4 p 1 quoniam triangulus h g d eſt æqualis <lb/>triangulo g d e, & æqualis triangulo h d b & trian-<lb/>gulo e d b, & ipſorum anguli reſpicientes æqualia <lb/>latera ſunt æquales.</s> <s xml:id="echoid-s40886" xml:space="preserve"> Et quoniam angulus h g d eſt <lb/>ęqualis angulo d g e:</s> <s xml:id="echoid-s40887" xml:space="preserve"> palàm quía angulus h g e diui-<lb/>ditur per æqualia per lineam g d:</s> <s xml:id="echoid-s40888" xml:space="preserve"> poteſt ergo per 20 <lb/>th.</s> <s xml:id="echoid-s40889" xml:space="preserve"> 5 huius forma pũcti h à puncto ſpeculi g reflecti <lb/>ad uiſum in punctum e:</s> <s xml:id="echoid-s40890" xml:space="preserve"> & erit per 37 th.</s> <s xml:id="echoid-s40891" xml:space="preserve"> 5 huius lo-<lb/>cus imaginis punctus e, qui eſt cẽtrum uiſus.</s> <s xml:id="echoid-s40892" xml:space="preserve"> Simi-<lb/>literq́;</s> <s xml:id="echoid-s40893" xml:space="preserve"> poteſt forma puncti h à puncto ſpeculi b re-<lb/>flecti ad uiſum in punctum e:</s> <s xml:id="echoid-s40894" xml:space="preserve"> & erit iterum locus <lb/>imaginis punctum e per eadem, quæ prius.</s> <s xml:id="echoid-s40895" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s40896" xml:space="preserve"> <lb/>diametro z a manente immobili, ſemicirculus z g a <lb/>imaginetur moueri per ſphæram ſpeculi, aut etiam <lb/>ſolus triangulus h g e moueatur fixo manente latere e h:</s> <s xml:id="echoid-s40897" xml:space="preserve"> palàm quia punctus g motu ſuo deſcribit <lb/>circulum, & à quolibet puncto illius circuli reflecti poteſt forma puncti h ad uiſum e:</s> <s xml:id="echoid-s40898" xml:space="preserve"> & locus ima-<lb/>ginis erit ſemper punctus e, quod eſt centrum uiſus.</s> <s xml:id="echoid-s40899" xml:space="preserve"> Quòd autem ab alio puncto ſpeculi quàm ab <lb/>aliquo puncto illius circuli non poſsit forma puncti h reflecti ad uiſum e, manifeſtum eſt.</s> <s xml:id="echoid-s40900" xml:space="preserve"> Si enim <lb/>reflecteretur ab alio circulo quàm ab illo, quem motu ſuo cauſſat punctum g uel punctum b:</s> <s xml:id="echoid-s40901" xml:space="preserve"> tunc <lb/>reflecteretur ab alio puncto illius ſemicirculi a g z.</s> <s xml:id="echoid-s40902" xml:space="preserve"> Sit ergo, ut reflectatur à puncto illius, qui ſit c:</s> <s xml:id="echoid-s40903" xml:space="preserve"> & <lb/>hoc erit extra illum circulum imaginatum in ſuperficie ſpeculi.</s> <s xml:id="echoid-s40904" xml:space="preserve"> Ducantur quoque lineæ h c & e c:</s> <s xml:id="echoid-s40905" xml:space="preserve"> <lb/>eritq́;</s> <s xml:id="echoid-s40906" xml:space="preserve"> linea e c maior quàm linea e g per 7 p 3, & erit linea h c minor quàm h g per eandem 7 p 3:</s> <s xml:id="echoid-s40907" xml:space="preserve"> non <lb/>ergo erit proportio lineæ e c ad lineam h c, ſicut lineæ e dad lineam b d, quæ ſunt æquales:</s> <s xml:id="echoid-s40908" xml:space="preserve"> ergo <lb/>per 3 p 6 angulus e ch non diuiditur in duo æqualia per lineam d c.</s> <s xml:id="echoid-s40909" xml:space="preserve"> Non ergo reflectitur forma <lb/>puncti h ad uiſum e à puncto ſpeculi c.</s> <s xml:id="echoid-s40910" xml:space="preserve"> Et eadem eſt deductio, ſi ſumatur punctus c inter puncta <lb/>g & z in arcu z g.</s> <s xml:id="echoid-s40911" xml:space="preserve"> Palàm itaque quoniam centro uiſus, quod eſt e, & puncto rei uiſæ, qui eſt h, exi-<lb/>ſtentibus in eadem diametro, & æqualiter diftantibus à centro ſpeculi, ſemper fit reflexio for-<lb/>mæ puncti uiſi ad uiſum modo propoſito.</s> <s xml:id="echoid-s40912" xml:space="preserve"> Quòd ſi puncta dicta in eadem diametro exiſtentia inæ-<lb/>qualiter diſtent à centro ſpeculi, puncto d, utpote ſi linea e d fuerit maior quàm linea d h, addatur <lb/>lineæ d h linea h q per 126 th.</s> <s xml:id="echoid-s40913" xml:space="preserve"> 1 huius, taliter, utillud, quod fit ex ductu lineæ e q in q h ſit æqua-<lb/>le quadrato lineæ d q:</s> <s xml:id="echoid-s40914" xml:space="preserve"> erit ergo per 17 p 6 proportio lineæ e q ad lineam d q, ſicut lineæ d q ad li-<lb/>neam h q.</s> <s xml:id="echoid-s40915" xml:space="preserve"> Fiat ergo circulus ad quantitatem ſemidiametri d q, cuius centrum fit q.</s> <s xml:id="echoid-s40916" xml:space="preserve"> Et quoniam <lb/>ille circulus interſecat circulum g z b a in duobus locis per 10 p 3:</s> <s xml:id="echoid-s40917" xml:space="preserve"> ſint illa loca ſectionis puncta <lb/>g & b:</s> <s xml:id="echoid-s40918" xml:space="preserve"> & ducantur lineæ e g, e b, q g, q b, d g, d b, h g, h b.</s> <s xml:id="echoid-s40919" xml:space="preserve"> Et quia linea q g eſt æqualis lineæ q d <lb/>per definitionem circuli:</s> <s xml:id="echoid-s40920" xml:space="preserve"> palàm per 7 p 5 quoniam eadem eſt proportio lineæ e q ad lineam q g <lb/>& ad lineam q d:</s> <s xml:id="echoid-s40921" xml:space="preserve"> eſt ergo proportio lineæ e q ad lineam q g, ſicut lineæ g q ad lineam q h:</s> <s xml:id="echoid-s40922" xml:space="preserve"> an-<lb/>gulus uerò g q h communis eſt utrique triangulorum, qui ſunt e q g & h q g:</s> <s xml:id="echoid-s40923" xml:space="preserve"> ergo per 6 p 6 illi duo <lb/> <pb o="324" file="0626" n="626" rhead="VITELLONIS OPTICAE"/> trianguli ſunt æquianguli:</s> <s xml:id="echoid-s40924" xml:space="preserve"> erunt quoq;</s> <s xml:id="echoid-s40925" xml:space="preserve"> eorum latera proportionalia per 4 p 6:</s> <s xml:id="echoid-s40926" xml:space="preserve"> erit ergo proportio <lb/>lineæ e q ad lineam q g, ſicut li-<lb/> <anchor type="figure" xlink:label="fig-0626-01a" xlink:href="fig-0626-01"/> neæ e g ad lineam g h:</s> <s xml:id="echoid-s40927" xml:space="preserve"> erit quoq;</s> <s xml:id="echoid-s40928" xml:space="preserve"> <lb/>per 19 p 5 proportio lineæ e d ad <lb/>lineam d h, ſicut lineæ e q ad li-<lb/>neam d q:</s> <s xml:id="echoid-s40929" xml:space="preserve"> ergo per 11 p 5 erit pro <lb/>portio lineæ e d ad lineam d h, <lb/>ſicut lineæ e g ad lineam g h:</s> <s xml:id="echoid-s40930" xml:space="preserve"> er-<lb/>go per 3 p 6 linea d g diuidit an-<lb/>gulum h g e per æqualia.</s> <s xml:id="echoid-s40931" xml:space="preserve"> Igitur <lb/>per 20 th.</s> <s xml:id="echoid-s40932" xml:space="preserve"> 5 huius forma puncti <lb/>h à pũcto ſpeculi g reflectitur ad <lb/>punctum e, qui eſt centrum ui-<lb/>ſus:</s> <s xml:id="echoid-s40933" xml:space="preserve"> & eſt punctus e locus ima-<lb/>ginis ſuæ.</s> <s xml:id="echoid-s40934" xml:space="preserve"> Et ſimiliter forma pũ-<lb/>cti h à puncto ſpeculi b reflecti-<lb/>tur ad punctum e, qui eſt centrũ <lb/>uiſus:</s> <s xml:id="echoid-s40935" xml:space="preserve"> & eſt punctũ e locus ima-<lb/>ginis ſuæ.</s> <s xml:id="echoid-s40936" xml:space="preserve"> Si ergo imaginetur <lb/>moueri triangulus e g h trans ſphæram ſpeculi (linea h e remanente immota) tunc punctus g de-<lb/>ſcribet circulum in ſuperficie concaua ſpeculi, à cuius quolibet puncto reflectetur forma puncti h <lb/>ad uiſum exiſtentem in puncto e:</s> <s xml:id="echoid-s40937" xml:space="preserve"> & ſemper erit locus imaginis punctus e.</s> <s xml:id="echoid-s40938" xml:space="preserve"> Quòd uerò ab alio pun-<lb/>cto quàm ab aliquo punctorum illius circuli, non poſsit forma puncti h reflecti ad uiſum e, patet, ut <lb/>prius.</s> <s xml:id="echoid-s40939" xml:space="preserve"> Si enim ſumatur punctus c inter puncta g & a:</s> <s xml:id="echoid-s40940" xml:space="preserve"> erit per 7 p 3 linea e c maior quàm linea e g, & <lb/>linea h c minor quàm linea h g:</s> <s xml:id="echoid-s40941" xml:space="preserve"> non erit igitur proportio lineæ e c ad c h, ſicut e d ad d h per 8 p 5:</s> <s xml:id="echoid-s40942" xml:space="preserve"> <lb/>ergo per 3 p 6 linea c d non diuidit angulum e c h per æqualia.</s> <s xml:id="echoid-s40943" xml:space="preserve"> Non ergo reflectetur forma puncti h <lb/>ad uiſum e à pũcto ſpeculic.</s> <s xml:id="echoid-s40944" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s40945" xml:space="preserve"> ſi punctus c, à quo debeat fieri reflexio, cadat in arcum <lb/>g z, idem ſequitur impoſsibile.</s> <s xml:id="echoid-s40946" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s40947" xml:space="preserve"> Sicut autem hæc de punctis & circulis ma-<lb/>thematicis demonſtrata ſunt:</s> <s xml:id="echoid-s40948" xml:space="preserve"> ſic de punctis medijs naturalium imaginum reflexarum intelligenda <lb/>ſunt.</s> <s xml:id="echoid-s40949" xml:space="preserve"> Forma enim puncti h continua uidetur formis aliorum punctorum:</s> <s xml:id="echoid-s40950" xml:space="preserve"> & eſt media intelligenda <lb/>in tota imagine naturali reflexa:</s> <s xml:id="echoid-s40951" xml:space="preserve"> & punctus medius totius illius formæ erit in puncto e, quod eſt <lb/>centrum uiſus, & reflectetur tota forma à loco circulari ſpeculi habẽte ſenſibilem latitudinem, cu-<lb/>ius medium mathematicum eſt circulus prædictus:</s> <s xml:id="echoid-s40952" xml:space="preserve"> & ſunt puncta e & h poli illius circuli.</s> <s xml:id="echoid-s40953" xml:space="preserve"> Cum au-<lb/>tem linea e d fuerit maior quàm linea d h, in tantum poterit eſſe maior, quòd non reflectetur forma <lb/>puncti h ad uiſum e à puncto ſpeculi g, prout oſtendimus per 17 huius:</s> <s xml:id="echoid-s40954" xml:space="preserve"> niſi enim fuerit proportio <lb/>exceſſus lineæ e d ſupra lineam d h ad lineam h d maior quàm lineæ e h ad lineam a h, non poterit <lb/>forma pũcti h reflecti ad uiſum e per 16 huius:</s> <s xml:id="echoid-s40955" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s40956" xml:space="preserve"> proportio lineæ e a ad lineam a h, maior quàm <lb/>lineæ e d ad lineam d h:</s> <s xml:id="echoid-s40957" xml:space="preserve"> aliàs enim non poterit reflecti forma puncti h ad uiſum in punctum e.</s> <s xml:id="echoid-s40958" xml:space="preserve"> Quia <lb/>ſi detur, quòd poſsit reflecti:</s> <s xml:id="echoid-s40959" xml:space="preserve"> ſit, ut reflectatur à pũcto g.</s> <s xml:id="echoid-s40960" xml:space="preserve"> Dico itaq;</s> <s xml:id="echoid-s40961" xml:space="preserve"> quòd neceſſariò ſequitur, ut ma-<lb/>ior ſit proportio lineæ e a ad lineam h a, quàm lineæ e d ad lineam d h:</s> <s xml:id="echoid-s40962" xml:space="preserve"> erit enim ex 42 th.</s> <s xml:id="echoid-s40963" xml:space="preserve"> 1 huius an-<lb/>gulus h d g acutus:</s> <s xml:id="echoid-s40964" xml:space="preserve"> erit quoq;</s> <s xml:id="echoid-s40965" xml:space="preserve"> peridẽ 42 th.</s> <s xml:id="echoid-s40966" xml:space="preserve"> 1 huius angulus d g h minor recto.</s> <s xml:id="echoid-s40967" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s40968" xml:space="preserve"> à pun-<lb/>cto g linea contingens circulum a g z b, quæ ſit g f:</s> <s xml:id="echoid-s40969" xml:space="preserve"> hæc ergo neceſſariò concurret cum linea e h per <lb/>14 th.</s> <s xml:id="echoid-s40970" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s40971" xml:space="preserve"> cum angulus h d g ſit a cutus, & angulus d g f rectus per 18 p 3:</s> <s xml:id="echoid-s40972" xml:space="preserve"> ſit concurſus punctus f:</s> <s xml:id="echoid-s40973" xml:space="preserve"> <lb/>erit ergo per 13 huius proportio catheti incidentiæ, quæ eſt h d, ad lineam d e ductam à centro ſpe-<lb/>culi ad locum imaginis, ſicut lineæ h f ductæ à puncto rei uiſæ ad finem contingentiæ, ad lineam f e <lb/>ducta m à fine contingentiæ ad locum imaginis.</s> <s xml:id="echoid-s40974" xml:space="preserve"> Ergo per 5 th.</s> <s xml:id="echoid-s40975" xml:space="preserve"> 1 huius erit econuerſo proportio li-<lb/>neæ e f ad lineam fh, ſicut lineæ e d ad lineam d h:</s> <s xml:id="echoid-s40976" xml:space="preserve"> ſed maior eſt proportio lineæ e a ad lineam a h, <lb/>quàm ſit lineæ e f ad lineam fh per 4 th.</s> <s xml:id="echoid-s40977" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s40978" xml:space="preserve"> quoniam æquali linea (quę eſt f a) addita utrobiq;</s> <s xml:id="echoid-s40979" xml:space="preserve">, <lb/>ininuitur proportio:</s> <s xml:id="echoid-s40980" xml:space="preserve"> igitur maior eſt proportio lineæ e a ad lineam h a, quàm ſit lineæ e d ad lineam <lb/>d h.</s> <s xml:id="echoid-s40981" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s40982" xml:space="preserve"> forma puncti h reflectitur ad uiſum e:</s> <s xml:id="echoid-s40983" xml:space="preserve"> neceſſarium eſt, ut proportio lineæ e a ad lineam <lb/>h a ſit maior quàm lineæ e d ad lineam d h.</s> <s xml:id="echoid-s40984" xml:space="preserve"> Hoc itaque cum fuerit, erit in hac diſpoſitione centri ui-<lb/>ſus & puncti rei uiſæ, ſicut prius, demonſtrandum.</s> <s xml:id="echoid-s40985" xml:space="preserve"> Palàm ergo ſunt omnia, quæ propoſita ſunt;</s> <s xml:id="echoid-s40986" xml:space="preserve"> <lb/>cum centrum uiſus & punctus rei uiſæ ſuerint in eadem diametro circuli propoſiti ſpeculi.</s> <s xml:id="echoid-s40987" xml:space="preserve"> Patet <lb/>ergo propoſitum.</s> <s xml:id="echoid-s40988" xml:space="preserve"/> </p> <div xml:id="echoid-div1638" type="float" level="0" n="0"> <figure xlink:label="fig-0625-02" xlink:href="fig-0625-02a"> <variables xml:id="echoid-variables721" xml:space="preserve">g c z e d h a b</variables> </figure> <figure xlink:label="fig-0626-01" xlink:href="fig-0626-01a"> <variables xml:id="echoid-variables722" xml:space="preserve">g c f q a h d e z b</variables> </figure> </div> </div> <div xml:id="echoid-div1640" type="section" level="0" n="0"> <head xml:id="echoid-head1234" xml:space="preserve" style="it">24. Puncto rei uiſæ & centro uiſus exiſtentibus extra ſpeculum ſphæricum concauum non <lb/>in eadem diametro circuli (qui eſt communis ſectio ſuperficiei reflexionis & ſpeculi) non eſt poſ-<lb/>ſibile, ut fiat ad uiſum reflexio niſi ab uno tantùm puncto: & unicus tantùm imaginis erit lo-<lb/>cus. Alhazen 70 n 5.</head> <p> <s xml:id="echoid-s40989" xml:space="preserve">Eſto t punctus rei uiſæ:</s> <s xml:id="echoid-s40990" xml:space="preserve"> & h centrum uiſus:</s> <s xml:id="echoid-s40991" xml:space="preserve"> & ſit d centrum ſpeculi:</s> <s xml:id="echoid-s40992" xml:space="preserve"> & ducantur lineæ h d, t d, h t:</s> <s xml:id="echoid-s40993" xml:space="preserve"> <lb/>ſuperficies itaq;</s> <s xml:id="echoid-s40994" xml:space="preserve"> reflexionis, quæ per 3 huius eſt ſuperficies h d t, ſecat ſuperficiem ſpeculi per 2 hu-<lb/>ius ſuper circulum, qui ſit e b q g.</s> <s xml:id="echoid-s40995" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s40996" xml:space="preserve"> quòd forma puncti t non reflectitur ad uiſum h, niſi ab <lb/>aliquo puncto huius circuli:</s> <s xml:id="echoid-s40997" xml:space="preserve"> non enim fit aliqua reflexio extra ſuperficiem reflexionis.</s> <s xml:id="echoid-s40998" xml:space="preserve"> Producatur <lb/> <pb o="325" file="0627" n="627" rhead="LIBER OCTAVVS."/> itaq;</s> <s xml:id="echoid-s40999" xml:space="preserve"> linea h d ultra centrum d, donec ſecet circum ſerẽtiam circuli:</s> <s xml:id="echoid-s41000" xml:space="preserve"> & ſit punctus ſectionis a:</s> <s xml:id="echoid-s41001" xml:space="preserve"> & pro-<lb/>ducatur linea t d ultra punctum d, ſecans circulum in puncto q:</s> <s xml:id="echoid-s41002" xml:space="preserve"> incidatq́;</s> <s xml:id="echoid-s41003" xml:space="preserve"> linea h d circul o in pũcto <lb/>g, & linea t d in puncto b.</s> <s xml:id="echoid-s41004" xml:space="preserve"> Palàm ergo (cum per 20 huius ſolùm ſit poſsibilis reflexio ab arcubus in-<lb/>teriacentibus dιametros, in quibus ſunt centrum uiſus, & pũctus rei uiſæ) quòd forma puncti t ad <lb/>uiſum exiſtentem in puncto h non reflectitur ab aliquo puncto arcus q g uel arcus b a:</s> <s xml:id="echoid-s41005" xml:space="preserve"> refle ctitur <lb/>itaq;</s> <s xml:id="echoid-s41006" xml:space="preserve"> aut ab aliquo puncto arcus g b, aut ab aliquo puncto arcus q a.</s> <s xml:id="echoid-s41007" xml:space="preserve"> <lb/>Diuidatur itaq;</s> <s xml:id="echoid-s41008" xml:space="preserve"> angulus t d h per æqualia per 9 p 1:</s> <s xml:id="echoid-s41009" xml:space="preserve"> diuidatq́;</s> <s xml:id="echoid-s41010" xml:space="preserve"> ipſum <lb/>linea d e l, ſecans circuli peripheriam in puncto e, & lineá h t in pun-<lb/>cto l:</s> <s xml:id="echoid-s41011" xml:space="preserve"> & à puncto e ducatur linea contingens circulum per 17 p 3:</s> <s xml:id="echoid-s41012" xml:space="preserve"> quæ <lb/>ſit k e f.</s> <s xml:id="echoid-s41013" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s41014" xml:space="preserve"> puncti t & h fuerint ſuper illam lineam contingen-<lb/>tem, ubicunq;</s> <s xml:id="echoid-s41015" xml:space="preserve"> conſiſtant:</s> <s xml:id="echoid-s41016" xml:space="preserve"> palàm quòd non eſt poſsibile reflecti for-<lb/>mam puncti t ad uiſum h ab aliquo puncto arcus b g.</s> <s xml:id="echoid-s41017" xml:space="preserve"> Si enim à pun-<lb/>cto t ducatur linea ad aliquem interiorem punctum huius arcus, li-<lb/>nea à puncto h ad idem punctum ducta cadet ſuper eundem arcum <lb/>exterius & nó interius, cum punctum ſit extra ſpeculum:</s> <s xml:id="echoid-s41018" xml:space="preserve"> & ita non <lb/>erit reſlexio à parte interiori concauitatis ſcilicet ſpeculi, ipſo cor-<lb/>pore ſpeculi impediente.</s> <s xml:id="echoid-s41019" xml:space="preserve"> Ab arcu uerò a q poſsibile eſt ut fiat refle-<lb/>xio:</s> <s xml:id="echoid-s41020" xml:space="preserve"> quoniam lineas ductas à puncto t & à puncto h, concauitati il-<lb/>lius arcus poſsibile eſt incidere.</s> <s xml:id="echoid-s41021" xml:space="preserve"> Producatur itaq;</s> <s xml:id="echoid-s41022" xml:space="preserve"> linea l d donec ſe-<lb/>cet arcum a q:</s> <s xml:id="echoid-s41023" xml:space="preserve"> & ſit punctus ſectionis z.</s> <s xml:id="echoid-s41024" xml:space="preserve"> Dico quòd à puncto z refle-<lb/>ctetur forma puncti t ad h centrum uiſus.</s> <s xml:id="echoid-s41025" xml:space="preserve"> Ducantur enim lineæ t z, <lb/>h z:</s> <s xml:id="echoid-s41026" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s41027" xml:space="preserve"> linea h z cathetum incidentiæ, quæ eſt t d q, in puncto p.</s> <s xml:id="echoid-s41028" xml:space="preserve"> <lb/>Cum itaq;</s> <s xml:id="echoid-s41029" xml:space="preserve"> angulus t d h ſit diuiſus per æqualia:</s> <s xml:id="echoid-s41030" xml:space="preserve"> patet quòd angulus <lb/>t d z eſt æqualis angulo h d z per 13 p 1.</s> <s xml:id="echoid-s41031" xml:space="preserve"> Lineæ itaq;</s> <s xml:id="echoid-s41032" xml:space="preserve"> t d & h d aut ſunt <lb/>æquales, aut non.</s> <s xml:id="echoid-s41033" xml:space="preserve"> Si ſunt æquales, & linea d z eſt communis:</s> <s xml:id="echoid-s41034" xml:space="preserve"> erit per <lb/>4 p 1 triangulus t z d æqualis triangulo h z d:</s> <s xml:id="echoid-s41035" xml:space="preserve"> & angulus t z h eſt diuiſus per æqualia per lineam d z:</s> <s xml:id="echoid-s41036" xml:space="preserve"> <lb/>ergo per 20 th.</s> <s xml:id="echoid-s41037" xml:space="preserve"> 5 huius forma puncti t reflectetur ad uiſum in pũctum h à puncto ſpeculi z.</s> <s xml:id="echoid-s41038" xml:space="preserve"> Sed neq;</s> <s xml:id="echoid-s41039" xml:space="preserve"> <lb/>eſt poſsibile à puncto alio arcus a q reflecti formam puncti t ad h.</s> <s xml:id="echoid-s41040" xml:space="preserve"> Sit enim, ſi eſt poſsibile, quòd re-<lb/>flectatur à puncto o, & ducantur lineæ t o & h o:</s> <s xml:id="echoid-s41041" xml:space="preserve"> linea quoq;</s> <s xml:id="echoid-s41042" xml:space="preserve"> o d m ducta per centrum ſpeculi diui-<lb/>dat angulum t o h per æqualia:</s> <s xml:id="echoid-s41043" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s41044" xml:space="preserve"> lineam h t in puncto m.</s> <s xml:id="echoid-s41045" xml:space="preserve"> Palàm ergo per 8 p 3 quoniam linea t <lb/>z eſt minor quàm linea t o, & linea h o eſt minor quàm linea h z:</s> <s xml:id="echoid-s41046" xml:space="preserve"> eſt autem per 3 p 6 (cum angulus t z <lb/>h ſit diuiſus per ęqualia) proportio lineæ t z ad lineam h z, ſicut lineę tl ad lineam h l:</s> <s xml:id="echoid-s41047" xml:space="preserve"> proportio ue-<lb/>rò lineæ t o ad lineam h o per eandem 3 p 6 eſt, ſicut lineę t m ad lineam m h:</s> <s xml:id="echoid-s41048" xml:space="preserve"> ſed per 9 th.</s> <s xml:id="echoid-s41049" xml:space="preserve"> 1 huius ma-<lb/>ior eſt proportio lineæ h z ad lineam t z, quàm lineæ h o ad lineam t o:</s> <s xml:id="echoid-s41050" xml:space="preserve"> ergo per 11 p 5 maior eſt pro-<lb/>portio lineæ h l ad lineam l t, quàm lineæ h m maioris, quàm ſit linea h l, ad lineam m t minorem, <lb/>quàm ſit linea l t:</s> <s xml:id="echoid-s41051" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s41052" xml:space="preserve"> Semper enim eſt minor proportio quantitatis minoris ad <lb/>maiorem quàm maioris ad minorem:</s> <s xml:id="echoid-s41053" xml:space="preserve"> quod faciliter patet per 9 th.</s> <s xml:id="echoid-s41054" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s41055" xml:space="preserve"> Non ergo fiet reflexio <lb/>form æ puncti t ad uiſum h à puncto ſpeculi o.</s> <s xml:id="echoid-s41056" xml:space="preserve"> Similiter etiam demonſtrandum quod à nullo alio <lb/>niſi à ſolo puncto z:</s> <s xml:id="echoid-s41057" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s41058" xml:space="preserve"> Quòd ſi lineæ t d & h d ſint inæquales, fiat reſectio maio-<lb/>ris ad æqualitatem minoris per 3 p 1:</s> <s xml:id="echoid-s41059" xml:space="preserve"> & ordinetur demonſtratio, ut prius.</s> <s xml:id="echoid-s41060" xml:space="preserve"> Et quoniam forma puncti <lb/>cuiuſcunq;</s> <s xml:id="echoid-s41061" xml:space="preserve"> rei uiſæ in eadem linea exiſtentis ſemper reflectitur ab eodem puncto cuiuſcunq;</s> <s xml:id="echoid-s41062" xml:space="preserve"> ſpe-<lb/>culi ad uiſum in quocunq;</s> <s xml:id="echoid-s41063" xml:space="preserve"> puncto eiuſdem lineæ exiſtentis (quoniam linearum inæqualitas natu-<lb/>ram reflexionis non immutat, ut patet per 20 th.</s> <s xml:id="echoid-s41064" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s41065" xml:space="preserve"> ſemper enim angulus incidẽtiæ eſt æqua-<lb/>lis angulo reflexionis) patet quòd quæcunq;</s> <s xml:id="echoid-s41066" xml:space="preserve"> iſtarum linearum fuerit maior quàm alia, quòd non <lb/>impedietur propter hæc reflexio:</s> <s xml:id="echoid-s41067" xml:space="preserve"> & quòd tantùm ab uno puncto ſpeculi fiet reflexio:</s> <s xml:id="echoid-s41068" xml:space="preserve"> & hoc per di-<lb/>ligentiam perquirentis ſecundum modum præmiſſum poterit declarari.</s> <s xml:id="echoid-s41069" xml:space="preserve"> Et quia in tali diſpoſitio-<lb/>ne centri uiſus & puncti rei uiſæ ab uno tantùm puncto ſpeculi fit reflexio ad uiſum:</s> <s xml:id="echoid-s41070" xml:space="preserve"> patet quòd <lb/>unica eſt linea reflexionis, quæ h z:</s> <s xml:id="echoid-s41071" xml:space="preserve"> unicus eſt ergo locus imaginis, ſcilicet pũctus p, in quo linea re-<lb/>flexionis (quæ eſt h z) ſecat cathetum incidentiæ, quæ eſt t d q.</s> <s xml:id="echoid-s41072" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s41073" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1641" type="section" level="0" n="0"> <head xml:id="echoid-head1235" xml:space="preserve" style="it">25. Si angulum à duabus diametris circuli magni ſpeculi ſphærici concaui contentum diui-<lb/>dat tertia diameter per æqualia, & à puncto ſectionis circumferentiæ & diametri medio du-<lb/>cantur perpendiculares ſuper alias duas diametros: punct a diametrorum, in quæ cadunt per-<lb/>pendiculares, ad ſe inuicem reflectuntur tantùm ab illo puncto circumferẽtiæ, & à puncto ſibi <lb/>oppoſito: & quodlibet punctum diametri interiacens illa puncta, & centrum ſpeculi reflectitur <lb/>ad punctum alterius diametri æqualiter ei condiſtans à centro, ab eiſdem duobus punct is: & <lb/>loca imaginum erunt tantùm duo. Alhazen 71 n 5.</head> <p> <s xml:id="echoid-s41074" xml:space="preserve">Sint circuli (qui eſt communis ſectio ſuperficiei reflexionis & ſpeculi ſphærici concaui) cuius <lb/>centrum d, duæ diametri a g & b q:</s> <s xml:id="echoid-s41075" xml:space="preserve"> & diameter e d z diuidat angulum b d g per æqualia per 9 p 1:</s> <s xml:id="echoid-s41076" xml:space="preserve"> & <lb/>à puncto ſpeculi, cuì incidit diameter z d e, ducantur duæ perpendiculares ſuper duas ſemidiame-<lb/>tros b d & d g per 12 p 1, quæ ſint e t & e h.</s> <s xml:id="echoid-s41077" xml:space="preserve"> Palàm ergo per 26 p 1 quòd trianguli e t d & e h d ſunt <lb/>æquales & æquianguli.</s> <s xml:id="echoid-s41078" xml:space="preserve"> Quoniam enim angulus b d g diuiſus eſt per æqualia per lineam d e, & an-<lb/>guli e t d & e h d ſunt recti, & linea e d eſt ambobus illis trigonis communis:</s> <s xml:id="echoid-s41079" xml:space="preserve"> patet ergo quòd an- <pb o="326" file="0628" n="628" rhead="VITELLONIS OPTICAE"/> gulus t e d eſt æ qualis angulo d e h:</s> <s xml:id="echoid-s41080" xml:space="preserve"> ergo per 20 th.</s> <s xml:id="echoid-s41081" xml:space="preserve"> 5 huius forma puncti t refle ctitur ad uiſum exi-<lb/>ftentem in puncto h à puncto ſpeculi, quod eſt e:</s> <s xml:id="echoid-s41082" xml:space="preserve"> & eodem modo forma puncti h reflectitur ad ui-<lb/>ſum exiſtentem in puncto t à puncto ſpeculi e.</s> <s xml:id="echoid-s41083" xml:space="preserve"> Similiterq́;</s> <s xml:id="echoid-s41084" xml:space="preserve"> fiet reflexio à puncto z ductis lineis t z & <lb/>h z.</s> <s xml:id="echoid-s41085" xml:space="preserve"> Cum enim ex præmiſsis lineæ t d & h d ſint æ quales, & per 13 p 1 anguli h d z & t d z ſint æqua <lb/>les:</s> <s xml:id="echoid-s41086" xml:space="preserve"> erunt per 4 p 1 anguli t z d & d z h æquales:</s> <s xml:id="echoid-s41087" xml:space="preserve"> fiet ergo mutua reflexio pũctorum t & h ad inuicem <lb/>à puncto ſpeculi, quod eſt z.</s> <s xml:id="echoid-s41088" xml:space="preserve"> Patet autem per 20 huius quòd nõ reflectetur forma puncti t ad uiſum <lb/>exiſtentem in puncto h ab aliquo puncto arcus a b, uel ab aliquo puncto arcus g q:</s> <s xml:id="echoid-s41089" xml:space="preserve"> nec ab aliquo <lb/>puncto arcus a q, niſi à puncto z per 19 huius, & quòd idem accidet impoſsibile contra 9 th.</s> <s xml:id="echoid-s41090" xml:space="preserve"> 1 huius, <lb/>quod in proxima pręmiſſa, ducta prius linea th.</s> <s xml:id="echoid-s41091" xml:space="preserve"> Quòd uerò ab aliquo puncto arcus b g alio quàm à <lb/>puncto e, non poſsit fieri reflexio formæ puncti t a d uiſum h, ſic patebit.</s> <s xml:id="echoid-s41092" xml:space="preserve"> Detur enim quòd illa refle-<lb/>xio poſsit fieri à puncto o:</s> <s xml:id="echoid-s41093" xml:space="preserve"> & ducanturline æ t o & h o, d o:</s> <s xml:id="echoid-s41094" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s41095" xml:space="preserve"> circulus ſecundum quantitatem <lb/>diametri d e.</s> <s xml:id="echoid-s41096" xml:space="preserve"> Palàm ergo cũ anguli e t d & e h d ſint <lb/>recti, quoniá ille circulus tranſibit per quatuor pun-<lb/> <anchor type="figure" xlink:label="fig-0628-01a" xlink:href="fig-0628-01"/> cta, quæ ſunt t, d, h, e per 31 p 3.</s> <s xml:id="echoid-s41097" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s41098" xml:space="preserve"> pũctus e ſit <lb/>communis utriq;</s> <s xml:id="echoid-s41099" xml:space="preserve"> illorum circulorũ, & ſit ſuper ean-<lb/>dem diametrum e d, continget circulus maior mino <lb/>rem tantùm in puncto e per 13 p 3, & non in alio.</s> <s xml:id="echoid-s41100" xml:space="preserve"> Cir-<lb/>culus ita que minor, qui eſt e t d h, ſecabit lineam d o <lb/>productam in maiori circulo.</s> <s xml:id="echoid-s41101" xml:space="preserve"> Quoniam ſi non ſeca-<lb/>ret, tunc contingeret in pũcto o circulum maiorem, <lb/>& ſic ipſum cótingeret in duobus punctis, quod eſt <lb/>impoſsibile.</s> <s xml:id="echoid-s41102" xml:space="preserve"> Sit, ut ſecet ipſam in puncto l:</s> <s xml:id="echoid-s41103" xml:space="preserve"> & ducan-<lb/>tur lineæ t l & h l.</s> <s xml:id="echoid-s41104" xml:space="preserve"> Quia uerò, ut patet ex præmiſsis, <lb/>linea t d eſt æqualis lineæ d h:</s> <s xml:id="echoid-s41105" xml:space="preserve"> erit arcus d h circuli <lb/>minoris æqualis arcui d t per 28 p 3:</s> <s xml:id="echoid-s41106" xml:space="preserve"> ergo per 27 p 3 <lb/>angulus tl d eſt æqualis angulo d l h:</s> <s xml:id="echoid-s41107" xml:space="preserve"> ergo per 13 p 1 <lb/>angulus t l o eſt æqualis angulo h l o:</s> <s xml:id="echoid-s41108" xml:space="preserve"> ſed angulus l o <lb/>t eſt æqualis angulo l o h per 20 th.</s> <s xml:id="echoid-s41109" xml:space="preserve"> 5 huius, & ex hy-<lb/>potheſi, & latus o l eſt cómune ambobus trigonis t o l & h o l:</s> <s xml:id="echoid-s41110" xml:space="preserve"> ergo per 26 p 1 illi trigoni ſunt æqua-<lb/>les & æquianguli:</s> <s xml:id="echoid-s41111" xml:space="preserve"> erit ergo linea t o æqualis lineæ h o, quod eſt impoſsibile:</s> <s xml:id="echoid-s41112" xml:space="preserve"> quoniam per 7 p 3 li-<lb/>nea h o eſt maior quàm linea h e, & linea t o eſt minor quàm linea t e per eãdem 7 p 3:</s> <s xml:id="echoid-s41113" xml:space="preserve"> linea uero t e, <lb/>ut præmiſſum eſt, æqualis eſt lineæ h e:</s> <s xml:id="echoid-s41114" xml:space="preserve"> eſt ergo linea h o maior quàm linea t o.</s> <s xml:id="echoid-s41115" xml:space="preserve"> Non ergo reflecte-<lb/>tur forma puncti t ad uiſum exiſtentem in puncto h à puncto ſpeculi o:</s> <s xml:id="echoid-s41116" xml:space="preserve"> ſed neq;</s> <s xml:id="echoid-s41117" xml:space="preserve"> ab aliquo alio pun-<lb/>cto arcus e b.</s> <s xml:id="echoid-s41118" xml:space="preserve"> Similiterq́;</s> <s xml:id="echoid-s41119" xml:space="preserve"> eſt de ducendum, ſi punctus o, à quo ſupponitur fieri reflexio, cadat in ali-<lb/>quem punctum arcus e g inter puncta e & g.</s> <s xml:id="echoid-s41120" xml:space="preserve"> Reſtat ergo, ut forma puncti t non reflectatur ad uiſum <lb/>h ab aliquo puncto arcus b g, niſi à ſolo puncto e:</s> <s xml:id="echoid-s41121" xml:space="preserve"> nec ab aliquo puncto arcus a q, niſi à ſolo puncto <lb/>z.</s> <s xml:id="echoid-s41122" xml:space="preserve"> Item à puncto e ducatur, ut contingit, linea e m ſuper partem diametri b q, quæ eſt t d:</s> <s xml:id="echoid-s41123" xml:space="preserve"> & ſecetur <lb/>â linea h d pars æqualis lineæ d m per 3 p 1:</s> <s xml:id="echoid-s41124" xml:space="preserve"> quæ ſit d n:</s> <s xml:id="echoid-s41125" xml:space="preserve"> & ducatur linea e n.</s> <s xml:id="echoid-s41126" xml:space="preserve"> Palàm per 16 p 1 quòd <lb/>angulus e m d eſt obtuſus, cum angulus e t d ſit rectus.</s> <s xml:id="echoid-s41127" xml:space="preserve"> Ab angulo itaq;</s> <s xml:id="echoid-s41128" xml:space="preserve"> e m d per 27 th 1 huius reſe-<lb/>cetur angulus rectus, qui ſit d m p:</s> <s xml:id="echoid-s41129" xml:space="preserve"> & ducatur linea m p.</s> <s xml:id="echoid-s41130" xml:space="preserve"> Hæc ergo erit æquidiſtans lineæ e t per 28 <lb/>p 1:</s> <s xml:id="echoid-s41131" xml:space="preserve"> concurret ergo linea m p per 2 th.</s> <s xml:id="echoid-s41132" xml:space="preserve"> 1 huius cum linea e d;</s> <s xml:id="echoid-s41133" xml:space="preserve"> cum qua concurrit ſua æ quidiſtans, quæ <lb/>eſt e t:</s> <s xml:id="echoid-s41134" xml:space="preserve"> ſit concurſus punctus p:</s> <s xml:id="echoid-s41135" xml:space="preserve"> & ducatur linea n p:</s> <s xml:id="echoid-s41136" xml:space="preserve"> & fiat circulus ſecundum quantitatem diametri <lb/>d p:</s> <s xml:id="echoid-s41137" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s41138" xml:space="preserve"> per 31 p 3 ille circulus tranſiens per quatuor puncta m, d, n, p.</s> <s xml:id="echoid-s41139" xml:space="preserve"> Quia cum angulus p m d ſit <lb/>rectus, & angulus m d p æ qualis angulo p d n, & latus p d commune, erit per 4 p 1 angulus p n d re-<lb/>ctus.</s> <s xml:id="echoid-s41140" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s41141" xml:space="preserve"> arcus d n ſit æqualis arcui d m per 28 <lb/>p 3, erit angulus d p n æqualis angulo d p m per 27 p 3, <lb/> <anchor type="figure" xlink:label="fig-0628-02a" xlink:href="fig-0628-02"/> eruntq́;</s> <s xml:id="echoid-s41142" xml:space="preserve"> trianguli d m p & d n p æquianguli per 32 p 1.</s> <s xml:id="echoid-s41143" xml:space="preserve"> <lb/>Et quia linea n d eſt æqualis lineæ d m:</s> <s xml:id="echoid-s41144" xml:space="preserve"> erit per 4 p 6 <lb/>linea m p æqualis lineæ n p.</s> <s xml:id="echoid-s41145" xml:space="preserve"> Et quia angulus m p d eſt <lb/>æqualis angulo n p d:</s> <s xml:id="echoid-s41146" xml:space="preserve"> erit ergo per 13 p 1 angulus m p e <lb/>æqualis angulo n p e:</s> <s xml:id="echoid-s41147" xml:space="preserve"> ergo per 4 p 1 linea e p exiſtente <lb/>communi triangulo n e p, & triangulo m e p, erit an-<lb/>gulus n e p æqualis angulo m e p.</s> <s xml:id="echoid-s41148" xml:space="preserve"> Palàm ergo quòd <lb/>forma puncti m reflectitur ad uiſum exiſtenté in pun-<lb/>cto n à pũcto ſpeculi, quod eſt e:</s> <s xml:id="echoid-s41149" xml:space="preserve"> & eorum adinuicem <lb/>fiet mutua reflexio:</s> <s xml:id="echoid-s41150" xml:space="preserve"> ſimiliter à puncto z:</s> <s xml:id="echoid-s41151" xml:space="preserve"> & non ab ali-<lb/>quo puncto arcus b a, uel arcus g q per 20 huius:</s> <s xml:id="echoid-s41152" xml:space="preserve"> neq;</s> <s xml:id="echoid-s41153" xml:space="preserve"> <lb/>ab alio puncto arcus b g, quàm à puncto e:</s> <s xml:id="echoid-s41154" xml:space="preserve"> nec ab alio <lb/>puncto arcus q a, quàm à puncto z.</s> <s xml:id="echoid-s41155" xml:space="preserve"> In his enim eſt ea-<lb/>dem deductio, quæ prius.</s> <s xml:id="echoid-s41156" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s41157" xml:space="preserve"> ſecundum mo-<lb/>dum prædictum:</s> <s xml:id="echoid-s41158" xml:space="preserve"> quia ſumpto puncto lineæ m d, & <lb/>ductis lineis ad punctum illud à punctis t d h, & ſum-<lb/>pto puncto ultimo, in quo circulus minor ſecabit diametrum, & à puncto ſectionis ductis lineis ad <lb/>punctat & h:</s> <s xml:id="echoid-s41159" xml:space="preserve"> ſemper formæ illius puncti erit reflexio ad punctum ſibi ſimile lineæ d n, tantundem <lb/> <pb o="327" file="0629" n="629" rhead="LIBER OCTAVVS."/> diſtans à centro ſpeculi, quod eſt d:</s> <s xml:id="echoid-s41160" xml:space="preserve"> fietq́;</s> <s xml:id="echoid-s41161" xml:space="preserve"> illa reflexio à puncto ſpecûli e, & à pũcto illi oppoſito dia-<lb/>metraliter, qui eſt punctus z:</s> <s xml:id="echoid-s41162" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s41163" xml:space="preserve"> loca imaginum tantùm duo, in quibus duæ lineæ reflexionis, <lb/>quæ ſunt e h & z h, concurruot cum catheto incidẽtiæ, quæ t d.</s> <s xml:id="echoid-s41164" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s41165" xml:space="preserve"> Hoc tamen <lb/>eſt magis euidens, ſi diametri b q & a g ſecent ſe ad angulos non rectos:</s> <s xml:id="echoid-s41166" xml:space="preserve"> quoniam tunc loca imagi-<lb/>num cadunt aut retro uiſum;</s> <s xml:id="echoid-s41167" xml:space="preserve"> aut inter uiſum & ſpeculum.</s> <s xml:id="echoid-s41168" xml:space="preserve"> Si uerò illæ diametri ſecuerint ſe ad an-<lb/>gulos rectos:</s> <s xml:id="echoid-s41169" xml:space="preserve"> tunc a d huc loca imaginum erunt tantùm duo:</s> <s xml:id="echoid-s41170" xml:space="preserve"> quoniam tunc, ut patet per 28 p 1;</s> <s xml:id="echoid-s41171" xml:space="preserve"> linea <lb/>reflexionis, quæ e h, eſt æquidiſtans catheto incidentiæ, quæ eſt t d:</s> <s xml:id="echoid-s41172" xml:space="preserve"> & uidebitur una imago formæ <lb/>puncti tin pũcto reflexionis, quod eſt e, per 11 huius:</s> <s xml:id="echoid-s41173" xml:space="preserve"> reliqua uerò uidebitur in pũcto x, qui ſit com-<lb/>munis ſectio lineæ reflexionis, quæ eſt z h, & catheti incidẽtiæ, quæ eſt td.</s> <s xml:id="echoid-s41174" xml:space="preserve"> Et ſic loca imaginum di-<lb/>uerſantur ſecundum quantitates angulorum à diametris contentorum.</s> <s xml:id="echoid-s41175" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s41176" xml:space="preserve"/> </p> <div xml:id="echoid-div1641" type="float" level="0" n="0"> <figure xlink:label="fig-0628-01" xlink:href="fig-0628-01a"> <variables xml:id="echoid-variables723" xml:space="preserve">e o p l g b h n d m t q a z</variables> </figure> <figure xlink:label="fig-0628-02" xlink:href="fig-0628-02a"> <variables xml:id="echoid-variables724" xml:space="preserve">e o g a z l y p f h n m t d x</variables> </figure> </div> </div> <div xml:id="echoid-div1643" type="section" level="0" n="0"> <head xml:id="echoid-head1236" xml:space="preserve" style="it">26. Si angulum à duabus diametris magni circuli ſpeculi ſphærici concaui contentum diui-<lb/>dat tertia diameter per æqualia, & à puncto ſectionis cir cumferentiæ & diametri medio du-<lb/>cantur perpendiculares ſuper alias duas diametros: quilibet punctus unius diametrorum ſe-<lb/>ctarum interiacens perpendiculares & circumferẽtiam, reflectitur ad punctum alterius dia-<lb/>metri æqualiter ei condiſtans à centro, à quatuor tantùm circumferentiæ punctis: & ſecũdum <lb/>hæc loca imaginum numer antur. Alhazen 72 n 5.</head> <p> <s xml:id="echoid-s41177" xml:space="preserve">Sint, ut in proxima, circuli (qui eſt communis ſectio ſpeculi ſphærici cócaui, & ſuperficiei refle-<lb/>xionis) duæ diametri b q & a g ſecantes ſe ſuper punctum d centrum ſpeculi ſphærici concaui:</s> <s xml:id="echoid-s41178" xml:space="preserve"> & <lb/>diameter e z diuidat an gulum b d g ab eis in centro contentum per æ qualia:</s> <s xml:id="echoid-s41179" xml:space="preserve"> & ſumatur in ſemidia-<lb/>metro b d punctus t ſupra punctum, in quem cadit perpẽ dicularis ducta à puncto e ſuper ſemidia-<lb/>metrum b d:</s> <s xml:id="echoid-s41180" xml:space="preserve"> & in linea d g ſumatur eius pars (quæ ſit d h) æqualis lineæ d t per 3 p 1, & ducantur li-<lb/>neæ t e & h e.</s> <s xml:id="echoid-s41181" xml:space="preserve"> Dico quòd forma punctit refle ctitur ad uiſum exiſtentem in puncto h à puncto ſpe-<lb/>culi, quod eſte, & à puncto z ſibi diametraliter oppoſito:</s> <s xml:id="echoid-s41182" xml:space="preserve"> non autem reflectitur ab aliquo puncto <lb/>arcus b a, uel arcus g q.</s> <s xml:id="echoid-s41183" xml:space="preserve"> Eſt autem neceſſarium formam puncti t refle cti ad uiſum exiſtétem in pun-<lb/>cto h ab aliquo puncto arcus e g, & ab aliquo puncto arcus e b.</s> <s xml:id="echoid-s41184" xml:space="preserve"> Extrahatur enim à puncto t perpen-<lb/>dicularis ſuper lineam t d per 11 p 1, quæ ſit t o.</s> <s xml:id="echoid-s41185" xml:space="preserve"> Et quia linea t o eſt æquidiſtans perpendiculari du-<lb/>ctæ à puncto e ſuper ſemidiametrum d b per 28 p 1:</s> <s xml:id="echoid-s41186" xml:space="preserve"> palàm quia linea t o producta cadet extra circu-<lb/>lum ſpeculi, non ſecans punctum e.</s> <s xml:id="echoid-s41187" xml:space="preserve"> Producatur er-<lb/> <anchor type="figure" xlink:label="fig-0629-01a" xlink:href="fig-0629-01"/> go linea d e ultra punctum e.</s> <s xml:id="echoid-s41188" xml:space="preserve"> Et quia angulus b de <lb/>eſt acutus:</s> <s xml:id="echoid-s41189" xml:space="preserve"> ideo quia ſemidiameter d e diuidit an-<lb/>gulum b d g per æqualia:</s> <s xml:id="echoid-s41190" xml:space="preserve"> propter quod uterq;</s> <s xml:id="echoid-s41191" xml:space="preserve"> ipſo-<lb/>rũ eſt minor recto:</s> <s xml:id="echoid-s41192" xml:space="preserve"> palàm quòd linea t o per 14 th.</s> <s xml:id="echoid-s41193" xml:space="preserve"> 1 <lb/>huius concurret cum linea d e:</s> <s xml:id="echoid-s41194" xml:space="preserve"> concurrant ergo in <lb/>puncto o:</s> <s xml:id="echoid-s41195" xml:space="preserve"> & ducatur linea h o.</s> <s xml:id="echoid-s41196" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s41197" xml:space="preserve"> per 4 p 1 <lb/>cum angulus d t o ſit rectus, quòd etiam angulus d <lb/>h o eſt rectus.</s> <s xml:id="echoid-s41198" xml:space="preserve"> Fiat ita q;</s> <s xml:id="echoid-s41199" xml:space="preserve"> per 5 p 4 circulus tranſiens <lb/>per tria puncta t, d, h, qui per 31 p 1 neceſſariò tranſi-<lb/>bit per punctum o:</s> <s xml:id="echoid-s41200" xml:space="preserve"> & erit linea d o diameter eius:</s> <s xml:id="echoid-s41201" xml:space="preserve"> & <lb/>ducarur per 17 p 3 linea contingens circulum b a z g <lb/>in puncto e, quæ ſit k e.</s> <s xml:id="echoid-s41202" xml:space="preserve"> Et quoniam circulus t d h o <lb/>ſecat circulũ b a z g:</s> <s xml:id="echoid-s41203" xml:space="preserve"> neceſſe eſt ipſum ſecari in duo-<lb/>bus punctis per 10 p 3:</s> <s xml:id="echoid-s41204" xml:space="preserve"> ſint illa duo puncta l & m:</s> <s xml:id="echoid-s41205" xml:space="preserve"> & <lb/>ducantur lineæ t l, h l, d l, t m, h m, d m.</s> <s xml:id="echoid-s41206" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s41207" xml:space="preserve"> li-<lb/>nea recta, quæ eſt t d, ſit æqualis lineæ h d, ut patet <lb/>expræmiſsis, erit arcus t d æqualis arcui d h per 28 <lb/>p 3 erit ergo per 27 p 3 angulus t l d æqualis angulo d l h:</s> <s xml:id="echoid-s41208" xml:space="preserve"> & ita forma puncti treflectitur ad uiſum h <lb/>à puncto l.</s> <s xml:id="echoid-s41209" xml:space="preserve"> Et ſimiliter angulus t m d eſt æqualis angulo d m h per 27 p 3:</s> <s xml:id="echoid-s41210" xml:space="preserve"> ergo forma puncti trefle-<lb/>ctitur ad uiſum h à puncto m.</s> <s xml:id="echoid-s41211" xml:space="preserve"> Palàm igitur quòd forma punctit reflectitur ad uiſum h à quatuor <lb/>punctis e, z, l, m.</s> <s xml:id="echoid-s41212" xml:space="preserve"> Et quoniam lineæ reflexionis ſunt quatuor, ſcilicet h e, h l, h m, h z:</s> <s xml:id="echoid-s41213" xml:space="preserve"> patet quòd in <lb/>communi ſectione uniuſcuiuſcunq;</s> <s xml:id="echoid-s41214" xml:space="preserve"> ipſarum & catheti incidentiæ, quæ eſt t d, ſit locus imaginis.</s> <s xml:id="echoid-s41215" xml:space="preserve"> <lb/>Et ſi aliqua illarum linearum fuerit æquidiſtans catheto t d:</s> <s xml:id="echoid-s41216" xml:space="preserve"> erit locus imaginis in puncto reflexio-<lb/>nis per 11 & 12 huius.</s> <s xml:id="echoid-s41217" xml:space="preserve"> Loca ergo imaginum ſunt quatuor, ſecũdum numerum locorum reflexionis.</s> <s xml:id="echoid-s41218" xml:space="preserve"> <lb/>Non poteſt autem forma puncti treflecti ad uiſum hab alijs punctis præter hęc.</s> <s xml:id="echoid-s41219" xml:space="preserve"> Detur enim, ſi poſ-<lb/>ſibile eſt, ut fiat reflexio formæ puncti t ad uiſum h à puncto alio ſpeculi præter hæc quatuor, quod <lb/>ſit punctum f:</s> <s xml:id="echoid-s41220" xml:space="preserve"> & ducantur lineæ t f, h f, d f:</s> <s xml:id="echoid-s41221" xml:space="preserve"> & producatur d f, quouſq;</s> <s xml:id="echoid-s41222" xml:space="preserve"> concurrat cum linea contin-<lb/>gente circulum b a z q in puncto e:</s> <s xml:id="echoid-s41223" xml:space="preserve"> & ſit, exempli cauſſa, punctus concurſus k, qui ſit communis ſe-<lb/>ctio lineæ e k, & peripheriæ circuli t d h o:</s> <s xml:id="echoid-s41224" xml:space="preserve"> concurrent autem lineæ d f & e k per 14 th.</s> <s xml:id="echoid-s41225" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s41226" xml:space="preserve"> & du-<lb/>cantur lineæ t k & h k:</s> <s xml:id="echoid-s41227" xml:space="preserve"> erit itaq;</s> <s xml:id="echoid-s41228" xml:space="preserve"> ex hypotheſi, & per 20 th 5 huius angulus t f d æqualis angulo d fh:</s> <s xml:id="echoid-s41229" xml:space="preserve"> <lb/>ergo per 13 p 1 erit angulus t f k æqualis angulo h f k:</s> <s xml:id="echoid-s41230" xml:space="preserve"> ſed angulus t k f eſt æqualis angulo f k h per 27 <lb/>p 3:</s> <s xml:id="echoid-s41231" xml:space="preserve"> arcus enim in quos ad peripheriam cadunt illi anguli, ſcilicet arcus citculi t d h o (qui ſunt d h <lb/>& d t) ſunt æquales, & linea f k eſt communis:</s> <s xml:id="echoid-s41232" xml:space="preserve"> erunt ergo per 26 p 1 trianguli t k f & h k f æquian-<lb/>guli:</s> <s xml:id="echoid-s41233" xml:space="preserve"> eſt ergo per 4 p 6 linea t k æqualis lineæ h k:</s> <s xml:id="echoid-s41234" xml:space="preserve"> quod eſt impoſsibile:</s> <s xml:id="echoid-s41235" xml:space="preserve"> quoniam, ut patet per 7 p 2, <lb/> <pb o="328" file="0630" n="630" rhead="VITELLONIS OPTICAE"/> linea h k eſt maior quàm linea h o, & linea t k minor eſt quàm linea t o:</s> <s xml:id="echoid-s41236" xml:space="preserve"> linea uerò t o eſt æqualis li-<lb/>neæ h o, per præmiſſa, Et eodem modo deducendũ, ſi in <lb/>arcu m g ſit datus pũctus f:</s> <s xml:id="echoid-s41237" xml:space="preserve"> quoniam idem ſequitur im-<lb/>poſsibile dato puncto fin arcu g b ubicunq;</s> <s xml:id="echoid-s41238" xml:space="preserve"> extra tria <lb/> <anchor type="figure" xlink:label="fig-0630-01a" xlink:href="fig-0630-01"/> puncta m, e, l.</s> <s xml:id="echoid-s41239" xml:space="preserve"> Quia ſi punctus k, qui eſt pũctus lineę con <lb/>tingentis, cadat extra peripheriam circuli m d t o, copu-<lb/>latis lineis à punctis ſectionis lineæ e k ad peripheriam <lb/>circuli minoris, præmiſſo modo erit deducendum.</s> <s xml:id="echoid-s41240" xml:space="preserve"> Pa-<lb/>làm ergo quòd non reflectitur forma puncti t ad uiſum <lb/>h ab aliquo alio puncto quàm ab his quatuor punctis.</s> <s xml:id="echoid-s41241" xml:space="preserve"> <lb/>Sienim circulus fiat habẽs centrum in linea d z ad mo-<lb/>dum circuli t d h o habentis centrum in linea d o:</s> <s xml:id="echoid-s41242" xml:space="preserve"> palàm <lb/>per modum 24 huius, ducta linea t h, quoniam lineæ à <lb/>punctis t & h ad punctum z terminum diametri d z du-<lb/>ctæ, ſi ad partem aliam ultra puncta t & h fuerint pro-<lb/>ductæ:</s> <s xml:id="echoid-s41243" xml:space="preserve"> arcus interiacẽtes earum alteram & diametrum <lb/>e d z, qui ſunt p e & s e æquales reſecant:</s> <s xml:id="echoid-s41244" xml:space="preserve"> ergo æquales <lb/>angulos cum diametro in puncto z conſtituunt:</s> <s xml:id="echoid-s41245" xml:space="preserve"> & eſt <lb/>poſsibilis reflexio, quæ fit à puncto z.</s> <s xml:id="echoid-s41246" xml:space="preserve"> Ad alia uerò pun <lb/>cta arcuum uicinorum productæ à punctis t & h lineæ <lb/>ſemper arcus inæ quales reſecant:</s> <s xml:id="echoid-s41247" xml:space="preserve"> & ob hoc inæquales <lb/>angulos cóſtituunt ſuper circumferentiam circuli ma-<lb/>ioris:</s> <s xml:id="echoid-s41248" xml:space="preserve"> & per modum, quo uſi ſumus in 24 huius, ſequi-<lb/>tur impoſsibile contra 9 th.</s> <s xml:id="echoid-s41249" xml:space="preserve"> 1 huius, ut manifeſtũ eſt per <lb/>ea, quę præmiſſa ſunt.</s> <s xml:id="echoid-s41250" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s41251" xml:space="preserve"> quoniam <lb/>tantùm à quatuor punctis fit reflexio tali exiſtente diſpoſitione:</s> <s xml:id="echoid-s41252" xml:space="preserve"> & tantùm ſunt quatuor loca ima-<lb/>ginum.</s> <s xml:id="echoid-s41253" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s41254" xml:space="preserve"/> </p> <div xml:id="echoid-div1643" type="float" level="0" n="0"> <figure xlink:label="fig-0629-01" xlink:href="fig-0629-01a"> <variables xml:id="echoid-variables725" xml:space="preserve">o e k m f l g h b t d q a z</variables> </figure> <figure xlink:label="fig-0630-01" xlink:href="fig-0630-01a"> <variables xml:id="echoid-variables726" xml:space="preserve">o e k s m p f l g b h d t q a z</variables> </figure> </div> </div> <div xml:id="echoid-div1645" type="section" level="0" n="0"> <head xml:id="echoid-head1237" xml:space="preserve" style="it">27. Puncto reiuiſæ & centro uiſus in eadem ſuperficie circuli magni ſpeculi ſphærici cocaui, <lb/>diuerſis tamen diametris, & ſub inæquali diſtantia à centro ſpeculi exiſtentibus, in arcu illius <lb/>circuli interiacente reliquas ſemidiametros, in quibus illa puncta non conſiſtunt, punctum re-<lb/>flexionis inuenire: ex quo patet, quòd ab unot antùm puncto illius arcus fit reflexio in hoc ſitu. <lb/>Alhazen 73 n 5.</head> <p> <s xml:id="echoid-s41255" xml:space="preserve">Sit, ut prius, circulus (qui eſt communis ſectio ſuperficiei reflexionis & ſuperficiei ſpeculi ſphæ-<lb/>rici concaui) a b g q, cuius centrum d:</s> <s xml:id="echoid-s41256" xml:space="preserve"> & ducantur duæ diametri a d g & b d q:</s> <s xml:id="echoid-s41257" xml:space="preserve"> & diameter e d z di-<lb/>uidat angulum ab alijs duabus diametris contentum per æqualia:</s> <s xml:id="echoid-s41258" xml:space="preserve"> ſitq;</s> <s xml:id="echoid-s41259" xml:space="preserve"> t punctus rei uiſæ poſitus in <lb/>ſemidiametro b d propinquior centro ſpeculid, quàm ſit punctus h, qui ſit centrum uiſus pofitus <lb/>in ſemidiametro g d.</s> <s xml:id="echoid-s41260" xml:space="preserve"> Dico quòd in hac diſpoſitione punctorum t & h, poſsibile eſt in arcu a q pun-<lb/>ctum reflexionis inueniri:</s> <s xml:id="echoid-s41261" xml:space="preserve"> & quòd in illo arcu unicus huius reflexionis eſt punctus.</s> <s xml:id="echoid-s41262" xml:space="preserve"> Sumatur enim <lb/>extra circulum linea l y:</s> <s xml:id="echoid-s41263" xml:space="preserve"> & diuidatur per 119 th.</s> <s xml:id="echoid-s41264" xml:space="preserve"> 1 huius in pũcto m taliter, ut ſit proportio lineæ y m <lb/>ad lineam m l, ſicut lineæ h d ad lineam d t:</s> <s xml:id="echoid-s41265" xml:space="preserve"> & diuidatur item linea y l per æqualia in puncto n per 10 <lb/>p 1:</s> <s xml:id="echoid-s41266" xml:space="preserve"> & à puncto n educatur perpendicularis n k ſuper lineam y m per 11 p 1:</s> <s xml:id="echoid-s41267" xml:space="preserve"> & ſuper punctum l termi-<lb/>num lineæ y l fiat per 23 p 1 angulus æqualis medietati anguli a d t per lineam fl:</s> <s xml:id="echoid-s41268" xml:space="preserve"> erit itaq;</s> <s xml:id="echoid-s41269" xml:space="preserve"> angulus <lb/>f l y acutus, ſiue angulus a d t fuerit acutus ſiue rectus, uel etiam obtuſus:</s> <s xml:id="echoid-s41270" xml:space="preserve"> ſed an gulus f n l eſt rectus:</s> <s xml:id="echoid-s41271" xml:space="preserve"> <lb/>ergo per 14 th.</s> <s xml:id="echoid-s41272" xml:space="preserve"> 1 huius linea f l concurret cum linea n k:</s> <s xml:id="echoid-s41273" xml:space="preserve"> concurrant ergo in puncto f:</s> <s xml:id="echoid-s41274" xml:space="preserve"> & per 134 th.</s> <s xml:id="echoid-s41275" xml:space="preserve"> 1 <lb/>huius à puncto m ducatur linea ad baſim fl, concurrens cum latere n k in puncto k:</s> <s xml:id="echoid-s41276" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s41277" xml:space="preserve"> lineam <lb/>l fin puncto c taliter, ut ſit proportio lineæ k c ad lineam c l, ſicut lineę h d ad lineam b d.</s> <s xml:id="echoid-s41278" xml:space="preserve"> Deinde ſu-<lb/>per punctum d terminum lineæ a d fiat angulus æqualis angulo l c m, qui ſit i d a:</s> <s xml:id="echoid-s41279" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s41280" xml:space="preserve"> punctus cir-<lb/>cumferentiæ, qui eſt a, ſupra punctum z, uel infra illum:</s> <s xml:id="echoid-s41281" xml:space="preserve"> & ſuper punctum i terminum lineæ d i fiat <lb/>angulus æqualis angulo c l m, qui ſit o i d, ducta linea o i ſecante lineam d a in puncto o:</s> <s xml:id="echoid-s41282" xml:space="preserve"> quæ pro-<lb/>ducatur ultra punctum o:</s> <s xml:id="echoid-s41283" xml:space="preserve"> & ſuper lineam o i ducatur perpendicularis à puncto h per 12 p 1, quæ ſit <lb/>h r:</s> <s xml:id="echoid-s41284" xml:space="preserve"> & producatur linea r x, quouſq;</s> <s xml:id="echoid-s41285" xml:space="preserve"> ipſa æqualis ſit lineæ r i:</s> <s xml:id="echoid-s41286" xml:space="preserve"> & ducantur lineæ h x & h i.</s> <s xml:id="echoid-s41287" xml:space="preserve"> Palàm au-<lb/>tem per 120 th.</s> <s xml:id="echoid-s41288" xml:space="preserve"> 1 huius quoniam à puncto m impoſsibile eſt duci aliam lineam ſuper lineam fl, diui-<lb/>dentem eam ſecũdum proportionẽ, qua diuiſit ipſam linea m c k.</s> <s xml:id="echoid-s41289" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s41290" xml:space="preserve"> angulus o d i ſit æqua-<lb/>lis angulo l c m, & angulus o i d æ qualis angulo c l m:</s> <s xml:id="echoid-s41291" xml:space="preserve"> erit per 32 p 1 angulus i o d æqualis angulo l m <lb/>c:</s> <s xml:id="echoid-s41292" xml:space="preserve"> erit ergo per 13 p 1 angulus r o h æqualis angulo k m n:</s> <s xml:id="echoid-s41293" xml:space="preserve"> & angulus h r o eſt æqualis angulo k n m:</s> <s xml:id="echoid-s41294" xml:space="preserve"> <lb/>quia uterq;</s> <s xml:id="echoid-s41295" xml:space="preserve"> eſt rectus:</s> <s xml:id="echoid-s41296" xml:space="preserve"> ergo per 32 p 1 angulus n k m eſt æqualis angulo r h o.</s> <s xml:id="echoid-s41297" xml:space="preserve"> Trigona itaq;</s> <s xml:id="echoid-s41298" xml:space="preserve"> n k m & r <lb/>h o ſunt æquiangula:</s> <s xml:id="echoid-s41299" xml:space="preserve"> ergo per 4 p 6 latera ipſorum æquos angulos reſpicientia ſunt proportiona-<lb/>lia.</s> <s xml:id="echoid-s41300" xml:space="preserve"> Producatur itaq;</s> <s xml:id="echoid-s41301" xml:space="preserve"> linea i d ultra punctum d, donec concurrat cum linea h r:</s> <s xml:id="echoid-s41302" xml:space="preserve"> concurret autem per <lb/>14 th.</s> <s xml:id="echoid-s41303" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s41304" xml:space="preserve"> angulus enim h r i eſt rectus, & angulus r i d eſt acutus:</s> <s xml:id="echoid-s41305" xml:space="preserve"> concurſus autem punctum ſit <lb/>s:</s> <s xml:id="echoid-s41306" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s41307" xml:space="preserve"> angulus s d h æqualis angulo k c f per 15 p 1.</s> <s xml:id="echoid-s41308" xml:space="preserve"> Erunt ergo trigona f c k & s d h æquiangula per <lb/>32 p 1:</s> <s xml:id="echoid-s41309" xml:space="preserve"> ergo per 4 p 6 erit proportio lineæ s d ad lineam d h, ſicut lineæ f c ad lineam k c:</s> <s xml:id="echoid-s41310" xml:space="preserve"> ſed lineæ h d <lb/>ad lineam d i per 7 p 5 eſt proportio ſicut lineæ h d ad lineam d b:</s> <s xml:id="echoid-s41311" xml:space="preserve"> quoniam per definitionem circuli <lb/> <pb o="329" file="0631" n="631" rhead="LIBER OCTAVVS."/> lineæ d i & d b ſunt æquales:</s> <s xml:id="echoid-s41312" xml:space="preserve"> eſt ergo proportio lineæ h d ad lineam d i, ſicut lineæ k c ad lineam c l.</s> <s xml:id="echoid-s41313" xml:space="preserve"> <lb/>Ex pręmiſsis enim eſt pro-<lb/> <anchor type="figure" xlink:label="fig-0631-01a" xlink:href="fig-0631-01"/> portio lineæ k c ad lineam <lb/>c l, ſicut lineę h d ad lineam <lb/>b d:</s> <s xml:id="echoid-s41314" xml:space="preserve"> eſt ergo per 22 p 5 per <lb/>æquam ſcilicet proportio-<lb/>nem proportio lineę s d ad <lb/>lineam d i, ſicut lineę f c ad <lb/>lineã c l:</s> <s xml:id="echoid-s41315" xml:space="preserve"> ergo per 18 p 5 erit <lb/>cõiunctim proportio lineę <lb/>si ad lineam d i, ſicut lineę <lb/>flad lineá l c.</s> <s xml:id="echoid-s41316" xml:space="preserve"> Sed cũ trian-<lb/>gulus d i o ſit æquiangulus <lb/>triãgulo cl m, ut ſuprà pa-<lb/>tuit:</s> <s xml:id="echoid-s41317" xml:space="preserve"> palàm per 4 p 6 quo-<lb/>niam eſt proportio lineæ d <lb/>i ad lineam i o, ſicut lineæ c <lb/>l ad lineam l m:</s> <s xml:id="echoid-s41318" xml:space="preserve"> eſt igitur per 22 p 5 proportio lineæ s i ad lineam i o, ſicut lineæ fl ad lineá l m:</s> <s xml:id="echoid-s41319" xml:space="preserve"> ergo <lb/>per 5th.</s> <s xml:id="echoid-s41320" xml:space="preserve"> 1.</s> <s xml:id="echoid-s41321" xml:space="preserve"> huius erit econtrariò proportio lineæ i o ad lineam s i, ſicut lineæ l m ad lineam fl:</s> <s xml:id="echoid-s41322" xml:space="preserve"> ſed eſt <lb/>proportio lineæ s i ad lineam i r, ſicut lineæ fl ad lineam l n per 4 p 6:</s> <s xml:id="echoid-s41323" xml:space="preserve"> quoniam triangulus ris eſt <lb/>ſimilis triangulo fl n per 32 p 1.</s> <s xml:id="echoid-s41324" xml:space="preserve"> Cum enim anguli s r i & f n l ſint æquales, quia recti, & anguli r i s & <lb/>n l f ſunt æquales ex præmiſsis:</s> <s xml:id="echoid-s41325" xml:space="preserve"> erit angulus r s i æqualis angulon fl:</s> <s xml:id="echoid-s41326" xml:space="preserve"> igitur per 22 p 5 erit proportio <lb/>lineæ i o ad lineam i r, ſicut lineæ l m ad lineam l n:</s> <s xml:id="echoid-s41327" xml:space="preserve"> erit ergo econtrariò per 5th.</s> <s xml:id="echoid-s41328" xml:space="preserve"> 1 huius proportio <lb/>lineæ i r ad lineam i o, ſicut lineæ l n ad lineam l m.</s> <s xml:id="echoid-s41329" xml:space="preserve"> Et quoniam linea x i eſt dupla lineæ i r, & line <lb/>y l eſt dupla lineæ l n:</s> <s xml:id="echoid-s41330" xml:space="preserve"> erit per 15 p 5 eadem proportio lineæ x i ad lineam i o, ſicut lineæ y l ad lineam <lb/>l m:</s> <s xml:id="echoid-s41331" xml:space="preserve"> ergo per 17 p 5 erit diuiſim proportio lineæ y m ad lineam m l, ſicut lineæ x o ad lineam i o.</s> <s xml:id="echoid-s41332" xml:space="preserve"> Du-<lb/>catur itaq;</s> <s xml:id="echoid-s41333" xml:space="preserve"> à puncto i linea æquidiſtans lineæ h x per 31 p 1, quæ ſit iu.</s> <s xml:id="echoid-s41334" xml:space="preserve"> Producatur quoq;</s> <s xml:id="echoid-s41335" xml:space="preserve"> linea da, <lb/>donec concurrat cum linea i u:</s> <s xml:id="echoid-s41336" xml:space="preserve"> concurret autem per 2th.</s> <s xml:id="echoid-s41337" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s41338" xml:space="preserve"> quia concurrit cum eius æquidi-<lb/>ſtante, quæ eſt h x:</s> <s xml:id="echoid-s41339" xml:space="preserve"> ſit itaq;</s> <s xml:id="echoid-s41340" xml:space="preserve"> concurſus punctus u:</s> <s xml:id="echoid-s41341" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s41342" xml:space="preserve"> triangulus o u i per 15 & 29 p 1 æquiangulus <lb/>triangulo h o x:</s> <s xml:id="echoid-s41343" xml:space="preserve"> ergo per 4 p 6 eſt proportio lineæ h o ad lineam o u, ſicut lineæ x o ad lineam o i:</s> <s xml:id="echoid-s41344" xml:space="preserve"> eſt <lb/>autem, ut patuit ex præmiſsis, proportio lineæ x o ad lineam o i, ſicut lineæ y m ad lineam m l:</s> <s xml:id="echoid-s41345" xml:space="preserve"> ergo <lb/>per 11 p 5 erit proportio lineæ h o ad lineam o u, ſicut lineæ y m ad lineam l m:</s> <s xml:id="echoid-s41346" xml:space="preserve"> eſt ergo per eandem <lb/>11 p 5 proportio lineæ h o ad lineam o u, ſicut lineæ h d ad lineam d t.</s> <s xml:id="echoid-s41347" xml:space="preserve"> Sed quoniam triangulus h r i <lb/>æquiangulus eſt triangulo h r x per 4 p 1, quoniam ex hypotheſi linea x r eſt æqualis lineæ r i, & li-<lb/>nea h r eſt perpendicularis ſuper lineam x i:</s> <s xml:id="echoid-s41348" xml:space="preserve"> palàm quia angulus h x r eſt æqualis angulo r i h:</s> <s xml:id="echoid-s41349" xml:space="preserve"> ergo <lb/>angulus r i h eſt æqualis angulo u i o:</s> <s xml:id="echoid-s41350" xml:space="preserve"> quia per 29 p 1 anguli h x i & u i o ſunt æquales:</s> <s xml:id="echoid-s41351" xml:space="preserve"> cum ſint coal-<lb/>terni inter lineas x h & u i æquidiſtantes:</s> <s xml:id="echoid-s41352" xml:space="preserve"> ergo per 3 p 6 erit proportio lineæ h o ad lineam o u, ſicut <lb/>lineæ h i ad lineam i u:</s> <s xml:id="echoid-s41353" xml:space="preserve"> eſt ergo proportio lineæ h i ad lineam iuper 11 p 5 ſicut lineæ h d ad lineam <lb/>d t.</s> <s xml:id="echoid-s41354" xml:space="preserve"> Verùm angulus u i d, ut patet per præmiſſa, maior eſt angulo d i h:</s> <s xml:id="echoid-s41355" xml:space="preserve"> ſecetur ergo ab angulo u i d <lb/>angulus æqualis d i h angulo per 27 th.</s> <s xml:id="echoid-s41356" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s41357" xml:space="preserve"> & ſit angulus p i d:</s> <s xml:id="echoid-s41358" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s41359" xml:space="preserve"> punctus p in diametro d a:</s> <s xml:id="echoid-s41360" xml:space="preserve"> & <lb/>ducatur linea p t.</s> <s xml:id="echoid-s41361" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s41362" xml:space="preserve"> per 13 th.</s> <s xml:id="echoid-s41363" xml:space="preserve"> 1 huius quòd proportio lineę h i ad lineam iu conſtat ex pro-<lb/>portione lineæ h i ad lineam p i, & ex proportione lineæ p i ad lineam i u:</s> <s xml:id="echoid-s41364" xml:space="preserve"> ſed per 3 p 6 proportio eſt <lb/>lineæ h i ad lineam i p, ſicut lineæ h d ad lineam d p:</s> <s xml:id="echoid-s41365" xml:space="preserve"> quoniam angulus p i h diuiſus eſt per æqualia <lb/>per lineam d i:</s> <s xml:id="echoid-s41366" xml:space="preserve"> igitur proportio lineæ h i ad lineam i u, quæ eſt proportio lineæ h d ad lineam d t, <lb/>conſtat ex proportione lineæ h d ad d p, & lineæ p i ad i u:</s> <s xml:id="echoid-s41367" xml:space="preserve"> & proportio lineæ h d ad d t conſtat ex <lb/>proportione lineę h d ad lineam d p, & exproportione lineæ d p ad lineam d t:</s> <s xml:id="echoid-s41368" xml:space="preserve"> eſt igitur per 13 th.</s> <s xml:id="echoid-s41369" xml:space="preserve"> 1 <lb/>huius proportio lineæ d p ad lineam d t, ſicut lineæ p i ad lineam ui.</s> <s xml:id="echoid-s41370" xml:space="preserve"> Verùm, ut ſuprà patuit, angu-<lb/>lus ri u eſt medietas anguli u i h:</s> <s xml:id="echoid-s41371" xml:space="preserve"> quoniam angulus uir eſt æqualis angulo h xi per 29 p 1, & angu-<lb/>lus h x i eſt æqualis r i h per 4 p 1:</s> <s xml:id="echoid-s41372" xml:space="preserve"> eſt ergo angulus r i h medietas anguli u i h:</s> <s xml:id="echoid-s41373" xml:space="preserve"> & angulus d i h eſt me-<lb/>dietas anguli p i h:</s> <s xml:id="echoid-s41374" xml:space="preserve"> reſtat ergo, ut angulus d i o ſit medietas anguli p i u:</s> <s xml:id="echoid-s41375" xml:space="preserve"> ſed angulus d i o, cũ ſit æqua <lb/>lis angulo fly, eſt medietas anguli p d t:</s> <s xml:id="echoid-s41376" xml:space="preserve"> igitur angulus p i u eſt æqualis angulo p d t.</s> <s xml:id="echoid-s41377" xml:space="preserve"> Eſt autem, ut <lb/>patet per præmiſſa, proportio lineæ d p ad lineam d t, ſicut lineæ p i ad lineam i u:</s> <s xml:id="echoid-s41378" xml:space="preserve"> igitur per 6 p 6 <lb/>trianguli p i u & d p t ſuntę quianguli:</s> <s xml:id="echoid-s41379" xml:space="preserve"> igitur per 4 p 6 illi trigoni ſunt ſimiles:</s> <s xml:id="echoid-s41380" xml:space="preserve"> & angulus u p i ęqua <lb/>lis eſt angulo d p t:</s> <s xml:id="echoid-s41381" xml:space="preserve"> ergo per 14 p 1 linea t p i eſt linea una recta:</s> <s xml:id="echoid-s41382" xml:space="preserve"> cum angulo enim o p t uterq;</s> <s xml:id="echoid-s41383" xml:space="preserve"> illorum <lb/>angulorum æqualium, qui ſunt u p i & t p d, ualet duos angulos rectos per 13 p 1.</s> <s xml:id="echoid-s41384" xml:space="preserve"> Quoniam ergo li-<lb/>neat p i eſt linea una recta:</s> <s xml:id="echoid-s41385" xml:space="preserve"> erit ipſa linea incidentiæ formę puncti t:</s> <s xml:id="echoid-s41386" xml:space="preserve"> & anguli ti d & d i h ſunt ęqua-<lb/>les, ut patet ex præmiſsis.</s> <s xml:id="echoid-s41387" xml:space="preserve"> Palàm ergo per 20 th.</s> <s xml:id="echoid-s41388" xml:space="preserve"> 5 huius quòd forma pũcti treflectitur ad uiſum exi-<lb/>ſtentem in puncto h à puncto ſpeculi, quod eſti:</s> <s xml:id="echoid-s41389" xml:space="preserve"> ſem perq́;</s> <s xml:id="echoid-s41390" xml:space="preserve"> eadem eſt probatio, ſiue punctus rei ui-<lb/>ſæ, qui eſt t, ſit extra circulum ſpeculi, ſiue intra:</s> <s xml:id="echoid-s41391" xml:space="preserve"> ſimiliter ſiue punctum, quod eſt centrum uiſus, <lb/>ſit extra circulum ſpeculi, ſiue intra:</s> <s xml:id="echoid-s41392" xml:space="preserve"> dum tamen diſtent inęqualiter à centro ſpeculi.</s> <s xml:id="echoid-s41393" xml:space="preserve"> Patet ergo pro <lb/>poſitum.</s> <s xml:id="echoid-s41394" xml:space="preserve"> Fit enim reflexio ab uno tantùm puncto arcus a q interiacéte illas ſemidiametros, in qui-<lb/>bus puncta h & t non conſiſtunt.</s> <s xml:id="echoid-s41395" xml:space="preserve"> Et quoniam à puncto m impoſsibile eſt duci aliam lineam ſuper <lb/>lineam fl, diuidentem ipſam ſecũdum proportionem, qua diuiſit ipſam linea m c k, ut patet per 120 <lb/> <pb o="330" file="0632" n="632" rhead="VITELLONIS OPTICAE"/> th 1 huius:</s> <s xml:id="echoid-s41396" xml:space="preserve"> manifeſtum eſt quia non eſt poſsibile in propoſito arcu inueniri aliud punctum pręmiſ-<lb/>ſæ reflexionis.</s> <s xml:id="echoid-s41397" xml:space="preserve"> Patet ergo, quod proponebatur.</s> <s xml:id="echoid-s41398" xml:space="preserve"/> </p> <div xml:id="echoid-div1645" type="float" level="0" n="0"> <figure xlink:label="fig-0631-01" xlink:href="fig-0631-01a"> <variables xml:id="echoid-variables727" xml:space="preserve">b u i a x r o i m t p c e d z k p q o f n s h g q y</variables> </figure> </div> </div> <div xml:id="echoid-div1647" type="section" level="0" n="0"> <head xml:id="echoid-head1238" xml:space="preserve" style="it">28. Si angulum à duabus diametris circuli magni ſpeculi ſphærici concaui contentum diui-<lb/>dat alia diameter per æqualia: ab omni puncto arcus interiacentis ſemidiametros primas, in <lb/>quibus punct a reflexanõ conſiſtunt (præter punctum, cui incidit diameter angulum diuidens) <lb/>infinit a punctorum paria inæqualiter à centro circuli diſtantiũ reflectuntur. Alhaz. 74 n 5.</head> <p> <s xml:id="echoid-s41399" xml:space="preserve">Sit diſpoſitio figuræ præ cedentis:</s> <s xml:id="echoid-s41400" xml:space="preserve"> ſecentq́;</s> <s xml:id="echoid-s41401" xml:space="preserve"> circulum (qui eſt communis ſectio ſuperficiei refle-<lb/>xionis & ſuperficiei ſpeculi ſphærici concaui) duæ diametri, quæ ſint b q & a g, ſuper centrum d:</s> <s xml:id="echoid-s41402" xml:space="preserve"> <lb/>diuidatq́;</s> <s xml:id="echoid-s41403" xml:space="preserve"> diameter e d z angulum b d g per æqualia.</s> <s xml:id="echoid-s41404" xml:space="preserve"> Dico quòd quicunq;</s> <s xml:id="echoid-s41405" xml:space="preserve"> punctus ſumatur in arcu <lb/>a q, pręter punctum z, ab illo poſſunt reflecti infinita paria punctorum inæqualiter à centro diſtan-<lb/>tium.</s> <s xml:id="echoid-s41406" xml:space="preserve"> Sumatur enim in arcu a q punctus h:</s> <s xml:id="echoid-s41407" xml:space="preserve"> & ſumatur in ſemidiametro d g punctus l:</s> <s xml:id="echoid-s41408" xml:space="preserve"> & à ſemidia-<lb/>metro b d ſecetur linea m d æqualis lineæ l d:</s> <s xml:id="echoid-s41409" xml:space="preserve"> & ducantur lineæ l m, l h, m h, d h:</s> <s xml:id="echoid-s41410" xml:space="preserve"> ſecabitq́;</s> <s xml:id="echoid-s41411" xml:space="preserve"> diameter <lb/>e z lineam m l per 29 th.</s> <s xml:id="echoid-s41412" xml:space="preserve"> 1 huius, quia ſecat angulum b d g, cui ſubtenditur linea l m:</s> <s xml:id="echoid-s41413" xml:space="preserve"> ſit ergo punctus <lb/>ſectionis f:</s> <s xml:id="echoid-s41414" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s41415" xml:space="preserve"> per 4 p 1 & ex hypotheſi linea m f æqualis lineæ fl.</s> <s xml:id="echoid-s41416" xml:space="preserve"> Producatur quoq;</s> <s xml:id="echoid-s41417" xml:space="preserve"> linea h d, <lb/>quouſq;</s> <s xml:id="echoid-s41418" xml:space="preserve"> cadat ſuper lineam m l:</s> <s xml:id="echoid-s41419" xml:space="preserve"> cadet autem per 29 th.</s> <s xml:id="echoid-s41420" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s41421" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s41422" xml:space="preserve"> punctus ſectionis n:</s> <s xml:id="echoid-s41423" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s41424" xml:space="preserve"> linea <lb/>l n minor quàm linea n m:</s> <s xml:id="echoid-s41425" xml:space="preserve"> ideo, quia linea d n ſecat angulum f d l:</s> <s xml:id="echoid-s41426" xml:space="preserve"> quia angulus h d z (qui per 15 p 1 <lb/>eſt æqualis angulo n d f) minor eſt angulo a d z.</s> <s xml:id="echoid-s41427" xml:space="preserve"> Ve-<lb/> <anchor type="figure" xlink:label="fig-0632-01a" xlink:href="fig-0632-01"/> rùm eum angulus f d m ſit æqualis angulo ſ d l ex hy-<lb/>potheſi, & angulo q d z per 15 p 1, & angulus m d a ſit <lb/>æqualis angulo l d q:</s> <s xml:id="echoid-s41428" xml:space="preserve"> & angulus a d h æqualis angulo <lb/>n d l:</s> <s xml:id="echoid-s41429" xml:space="preserve"> angulus uerò m d n eſt maior angulo n d l, & <lb/>angulus h d q eſt maior angulo a d h:</s> <s xml:id="echoid-s41430" xml:space="preserve"> ergo totus an-<lb/>gulus l d h eſt maior toto angulo m d h:</s> <s xml:id="echoid-s41431" xml:space="preserve"> igitur per 24 <lb/>p 1 linea l h eſt maior quàm linea h m, cum linea m d <lb/>ſit æqualis lineæ d l, & linea d h communis ambobus <lb/>trigonis m d h & l d h.</s> <s xml:id="echoid-s41432" xml:space="preserve"> Erit ergo angulus d h l minor <lb/>angulo d h m.</s> <s xml:id="echoid-s41433" xml:space="preserve"> Quoniã ſi detur, quòd ſit æqualis:</s> <s xml:id="echoid-s41434" xml:space="preserve"> tunc <lb/>erit proportio lineæ l h ad lineam m h, ſicut lineæ l n <lb/>ad lineam n m per 3 p 6:</s> <s xml:id="echoid-s41435" xml:space="preserve"> quod eſt impoſsibile per 8 p <lb/>5.</s> <s xml:id="echoid-s41436" xml:space="preserve"> Si uerò detur quòd angulus d h l ſit maior angulo <lb/>d h m:</s> <s xml:id="echoid-s41437" xml:space="preserve"> ergo per 27 th.</s> <s xml:id="echoid-s41438" xml:space="preserve"> 1 huius ſecetur ex angulo d h l <lb/>angulus æqualis angulo d h m:</s> <s xml:id="echoid-s41439" xml:space="preserve"> & ſequetur impoſsi-<lb/>bile, ut prius, producta illa linea ſecante, ad lineam l <lb/>n per 29 th.</s> <s xml:id="echoid-s41440" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s41441" xml:space="preserve"> Eſt igitur angulus d h l minor an-<lb/>gulo d h m.</s> <s xml:id="echoid-s41442" xml:space="preserve"> Secetur igitur ab angulo m h d angulus æqualis angulo d h l, qui ſit angulus t h d.</s> <s xml:id="echoid-s41443" xml:space="preserve"> Ergo <lb/>forma puncti t per 20 th.</s> <s xml:id="echoid-s41444" xml:space="preserve"> 5 huius refle ctetur ad uiſum exiſtentem in puncto l à puncto ſpeculi, quod <lb/>eſt h:</s> <s xml:id="echoid-s41445" xml:space="preserve"> & linea t d eſt minor quàm linea l d:</s> <s xml:id="echoid-s41446" xml:space="preserve"> quoniam eſt minor quàm linea d m.</s> <s xml:id="echoid-s41447" xml:space="preserve"> Similiter ſi ſumantur <lb/>in ſemidiam etris b g & g d alia pũcta quàm l & m, æqualiter diſtantia à punctis l & m:</s> <s xml:id="echoid-s41448" xml:space="preserve"> ſimiliter pro-<lb/>babitur quòd à puncto h fit reflexio punctorum in æqualiter diſtantium à centro adinuicem:</s> <s xml:id="echoid-s41449" xml:space="preserve"> & ita <lb/>de infinitis punctis in his diametris ſumptis ſemper ſimilis erit probatio:</s> <s xml:id="echoid-s41450" xml:space="preserve"> & à quocunq;</s> <s xml:id="echoid-s41451" xml:space="preserve"> puncto ar-<lb/>cus a q, præter quàm à puncto z, eadem eſt demonſtratio.</s> <s xml:id="echoid-s41452" xml:space="preserve"> A puncto uero z non eſt poſsibilis refle-<lb/>xio propter angulorum t z d & d z linæqualitatem:</s> <s xml:id="echoid-s41453" xml:space="preserve"> quæ patet per 4 p 1, reſecta per 3 p 1 linea l d in <lb/>puncto p ad æqualitatem lineæ d t, & copulata linea p z.</s> <s xml:id="echoid-s41454" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s41455" xml:space="preserve"/> </p> <div xml:id="echoid-div1647" type="float" level="0" n="0"> <figure xlink:label="fig-0632-01" xlink:href="fig-0632-01a"> <variables xml:id="echoid-variables728" xml:space="preserve">b a m h t e f d z p n l g q</variables> </figure> </div> </div> <div xml:id="echoid-div1649" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables729" xml:space="preserve">b a t h e p d z n l k g q</variables> </figure> <head xml:id="echoid-head1239" xml:space="preserve" style="it">29. Puncto rei uiſæ & cẽtro uiſus intra ſpeculum in diuerſis diametris circuli magni ſpeculi <lb/>ſphærici concaui exiſtẽtibus, inæqualiteŕ dιſtan-<lb/>tibus à centro: ſi ab aliquo puncto ſpeculi arcus ſci-<lb/> licet interiacentis ſemidiametros, in quibus illa punct a non conſiſtunt, fiat reflexio formarũ eiuſ- dem puncti ad eundem uiſum: ab alio puncto eiuſ- dem arcus eſt impoßibile reflecti. Alhaz. 75 n 5.</head> <p> <s xml:id="echoid-s41456" xml:space="preserve">Remaneat omnimoda diſpoſitio theorematis prę-<lb/>cedentis:</s> <s xml:id="echoid-s41457" xml:space="preserve"> & ſit, ut pũctus rei uiſæ, (qui eſt t) in ſemi-<lb/>diametro circuli d b à puncto arcus a q, (quod ſit h) <lb/>refle ctatur ad uiſum exiſtentẽ in pũcto l ſemidiame-<lb/>tri d g plus diſtantẽ à cẽtro ſpeculi, qđ eſt d, quã pũ-<lb/>ctus rei uiſæ, qui eſt t:</s> <s xml:id="echoid-s41458" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s41459" xml:space="preserve"> puncta t & l ambo intra <lb/>ſpeculũ.</s> <s xml:id="echoid-s41460" xml:space="preserve"> Dico quòd formá pũcti t ad uiſum limpoſ-<lb/>ſibile eſt reflecti ab alio pũcto arc{us} a q, quàm à pũcto <lb/>h.</s> <s xml:id="echoid-s41461" xml:space="preserve"> Si enim ſit ipſum poſsibile ab alio puncto reflecti <lb/>ad uiſum l:</s> <s xml:id="echoid-s41462" xml:space="preserve"> ſit illud punctũ k:</s> <s xml:id="echoid-s41463" xml:space="preserve"> & ducátur lineæ t k, l k, <lb/>d k, l t, t h, l h:</s> <s xml:id="echoid-s41464" xml:space="preserve"> & linea n d h:</s> <s xml:id="echoid-s41465" xml:space="preserve"> & producatur linea k d, <lb/>quouſq;</s> <s xml:id="echoid-s41466" xml:space="preserve"> cadatin lineá l t in punctũ p:</s> <s xml:id="echoid-s41467" xml:space="preserve"> cadet autẽ per 29 th.</s> <s xml:id="echoid-s41468" xml:space="preserve"> 1 huius, ut in præmiſſa oſtendimus.</s> <s xml:id="echoid-s41469" xml:space="preserve"> Quia <pb o="331" file="0633" n="633" rhead="LIBER OCTAVVS."/> itaq;</s> <s xml:id="echoid-s41470" xml:space="preserve">, ut patet ex hypotheſi, forma puncti t refle ctitur ad uiſum exiſtentem in puncto l à puncto ſpe <lb/>culi h:</s> <s xml:id="echoid-s41471" xml:space="preserve"> palàm per 20 th.</s> <s xml:id="echoid-s41472" xml:space="preserve"> 1 huius quoniam angulus th l diuiditur per æ qualia per lineam n d h.</s> <s xml:id="echoid-s41473" xml:space="preserve"> Ergo <lb/>per 3 p 6 patet quoniam eſt proportio lineæ l h ad lineam t h, ſicut lineæ l n ad lineam n t.</s> <s xml:id="echoid-s41474" xml:space="preserve"> Et ſimili-<lb/>ter cum angulus t k p ſit æqualis angulo l k p ex hypotheſi:</s> <s xml:id="echoid-s41475" xml:space="preserve"> erit per eandem 3 p 6 proportio lineæ <lb/>l k ad lineam t k, ſicut lineæ l p ad p t:</s> <s xml:id="echoid-s41476" xml:space="preserve"> ſed linea lh eſt maior quàm linea l k per 7 p 3, & linea t h eſt <lb/>minor quàm linea t k:</s> <s xml:id="echoid-s41477" xml:space="preserve"> igitur per 9 th.</s> <s xml:id="echoid-s41478" xml:space="preserve"> 1 huius maior eſt proportio lineæ l h ad lineam t h, quàm li-<lb/>neæ l k ad lineam t k:</s> <s xml:id="echoid-s41479" xml:space="preserve"> maior ergo erit proportio lineæ l n ad lineam n t, quàm lineæ l p ad lineam p <lb/>t:</s> <s xml:id="echoid-s41480" xml:space="preserve"> quod eſt impoſsibile, & contra idem 9 th.</s> <s xml:id="echoid-s41481" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s41482" xml:space="preserve"> Quocunq;</s> <s xml:id="echoid-s41483" xml:space="preserve"> uerò alio puncto dicti arcus h q <lb/>dato, idem accidit impoſsibile.</s> <s xml:id="echoid-s41484" xml:space="preserve"> Palàm ergo quoniam ab alio puncto arcus a q, quàm à puncto h, eſt <lb/>impoſsibile formam puncti t ad l centrum uiſus reflecti.</s> <s xml:id="echoid-s41485" xml:space="preserve"> Ergo nec aliquem punctorum æqualiter <lb/>diſtantiũ à puncto t, & à puncto l, poſsibile eſt ab alio puncto arcus a q, quàm à puncto h reflecti.</s> <s xml:id="echoid-s41486" xml:space="preserve"> Et <lb/>hoc eſt ꝓpoſitũ.</s> <s xml:id="echoid-s41487" xml:space="preserve"> Ex his itaq;</s> <s xml:id="echoid-s41488" xml:space="preserve"> duob.</s> <s xml:id="echoid-s41489" xml:space="preserve"> theorematibus patet uniuerſalis paſsio, quę accidit uiſibilibus, <lb/>& uiſui ſic diſpoſito, reſpectu centri ſpeculi ab omnibus punctis arcus a q:</s> <s xml:id="echoid-s41490" xml:space="preserve"> quoniam à nullo puncto <lb/>aliorum arcuũ eſt poſsibilis reflexio punctorũ taliter diſpoſitorũ, ut etiam hoc patet per 27 huius.</s> <s xml:id="echoid-s41491" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1650" type="section" level="0" n="0"> <head xml:id="echoid-head1240" xml:space="preserve" style="it">30. Centro uiſus intra circulum (qui eſt cõmunis ſectio ſuperficiei reflexionis & ſpeculi ſphæ <lb/>rici concaui) in eius diametro existente: à quolibet puncto illius ſemicirculi reflectuntur ad ui-<lb/>ſum formæ punctorum æqualis uelinæqualis diſtantiæ à centro ſpeculi cum ipſo centro uiſus. <lb/>Alhazen 76 n 5.</head> <p> <s xml:id="echoid-s41492" xml:space="preserve">Sit a centrum uiſus:</s> <s xml:id="echoid-s41493" xml:space="preserve"> centrum uerò ſpeculi ſphærici concaui ſit b:</s> <s xml:id="echoid-s41494" xml:space="preserve"> & ſit a intra ſpeculum:</s> <s xml:id="echoid-s41495" xml:space="preserve"> duca-<lb/>turq́;</s> <s xml:id="echoid-s41496" xml:space="preserve"> una diameter, quæ ſit d a b g:</s> <s xml:id="echoid-s41497" xml:space="preserve"> & imaginetur ſuperficies plana, in quaſunt puncta a & b quo-<lb/>cunq;</s> <s xml:id="echoid-s41498" xml:space="preserve"> modo extenſa:</s> <s xml:id="echoid-s41499" xml:space="preserve"> hæc ergo per 69 th.</s> <s xml:id="echoid-s41500" xml:space="preserve"> 1 huius ſecabit ſphæram ſpeculi ſecundum circulum, qui <lb/>ſit d l g.</s> <s xml:id="echoid-s41501" xml:space="preserve"> Dico quòd à quolibet puncto alterius iſto-<lb/>rum ſemicirculorum reflectuntur ad uiſum a for-<lb/> <anchor type="figure" xlink:label="fig-0633-01a" xlink:href="fig-0633-01"/> mæ punctorum inæ qualiter diſtantiũ à centro ſpe-<lb/>culi cũ ipſo puncto a.</s> <s xml:id="echoid-s41502" xml:space="preserve"> Sumatur enim in alicuius ſe-<lb/>micirculorum illorum peripheria punctus e:</s> <s xml:id="echoid-s41503" xml:space="preserve"> & du-<lb/>cantur lineæ e a & e b.</s> <s xml:id="echoid-s41504" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s41505" xml:space="preserve"> quoniam angu-<lb/>lus a e b erit acutus per 42 th.</s> <s xml:id="echoid-s41506" xml:space="preserve"> 1 huius, & quia cadit <lb/>in minorem arcum ſemicirculo.</s> <s xml:id="echoid-s41507" xml:space="preserve"> Super punctum <lb/>itaq;</s> <s xml:id="echoid-s41508" xml:space="preserve"> e terminum lineæ b e fiat per 23 p 1 angulus æ-<lb/>qualis angulo a e b, qui ſit p e b:</s> <s xml:id="echoid-s41509" xml:space="preserve"> & producatur linea <lb/>p e quántum placet.</s> <s xml:id="echoid-s41510" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s41511" xml:space="preserve"> per 20 th.</s> <s xml:id="echoid-s41512" xml:space="preserve"> 5 huius <lb/>quoniam quodlibet punctum illius lineæ reflecti-<lb/>tur ad uiſum a à puncto ſpeculi, quod eſt e.</s> <s xml:id="echoid-s41513" xml:space="preserve"> Ducta <lb/>quoq;</s> <s xml:id="echoid-s41514" xml:space="preserve"> à centro ſpeculi, quod eſt b, ad lineá p e per-<lb/>pendiculari per 12 p 1:</s> <s xml:id="echoid-s41515" xml:space="preserve"> aut illa perpendicularis erit <lb/>æqualis lineæ b a, ſecundum quam diſtat centrum <lb/>uiſus à centro ſpeculi:</s> <s xml:id="echoid-s41516" xml:space="preserve"> aut maior:</s> <s xml:id="echoid-s41517" xml:space="preserve"> aut minor.</s> <s xml:id="echoid-s41518" xml:space="preserve"> Si fue-<lb/>rit æ qualis:</s> <s xml:id="echoid-s41519" xml:space="preserve"> tunc, cum omnes lineæ ductæ à centro <lb/>b ad lineam p e, præter illam perpendicularẽ, ſint maiores illa perpendiculari per 19 p 1, quoniam <lb/>opponuntur angulo recto in illo triangulo:</s> <s xml:id="echoid-s41520" xml:space="preserve"> palàm quòd omnes lineæ erunt maiores quàm linea b <lb/>a:</s> <s xml:id="echoid-s41521" xml:space="preserve"> & ita quodlibet punctum lineæ p e, excepto puncto unico, in quod cadit perpendicularis ducta <lb/>à centro b ſuper lineæ p e, inæ qualiter diſtabit à <lb/>centro b cum puncto a centro uiſus.</s> <s xml:id="echoid-s41522" xml:space="preserve"> Siuerò per-<lb/>pendicularis fuerit maior quàm linea b a:</s> <s xml:id="echoid-s41523" xml:space="preserve"> tũc pa-<lb/>tet ſecundũ præmiſſa, quòd omnia puncta lineæ <lb/>p e plus diſtabunt à centro b, quàm punctus a.</s> <s xml:id="echoid-s41524" xml:space="preserve"> Si <lb/>aũt illa perpendicularis fuerit minor quàm linea <lb/>b a:</s> <s xml:id="echoid-s41525" xml:space="preserve"> tũc poſsibile eſt duci à puncto b duas lineas <lb/>ex diuerſis partibus perpendicularis æ quales li-<lb/>neæ b a:</s> <s xml:id="echoid-s41526" xml:space="preserve"> quod fiet ſubtenſis illis angulis, rectis ex <lb/>utraq;</s> <s xml:id="echoid-s41527" xml:space="preserve"> parte lineis, æ qualibus lineę a b per 26 th.</s> <s xml:id="echoid-s41528" xml:space="preserve"> <lb/>1 huius:</s> <s xml:id="echoid-s41529" xml:space="preserve"> & omnes lineę aliæ ductæ à centro b ad <lb/>lineã p e aut ſunt minores, aut maiores, quàm li-<lb/>nea b a.</s> <s xml:id="echoid-s41530" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s41531" xml:space="preserve"> per 28 huius quoniã à pun-<lb/>cto e reflectũtur omnia pũcta lineę p e ad a cẽtrũ <lb/>uiſus, quorũ diſtantia à centro ſpeculi in æqualis <lb/>eſt diſtantiæ centri uiſus, quod eſt a, ab eodẽ cen <lb/>tro ſpeculi.</s> <s xml:id="echoid-s41532" xml:space="preserve"> Sed, ut patet ex præmiſsis, inter hæc <lb/>ſunt puncta æqualiter diſtantia à centro ſpeculi <lb/>cũ puncto a.</s> <s xml:id="echoid-s41533" xml:space="preserve"> Sumpto quoq;</s> <s xml:id="echoid-s41534" xml:space="preserve"> quocũq;</s> <s xml:id="echoid-s41535" xml:space="preserve"> puncto in toto ſemicirculo illo, in quo ſumptũ eſt punctum e, <lb/>ſemper eſt eodem modo demonſtran dum.</s> <s xml:id="echoid-s41536" xml:space="preserve"> Eodem quoq;</s> <s xml:id="echoid-s41537" xml:space="preserve"> modo poteſt in alio ſemicirculo circuli d <lb/>l g demonſtratio fermari.</s> <s xml:id="echoid-s41538" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s41539" xml:space="preserve"/> </p> <div xml:id="echoid-div1650" type="float" level="0" n="0"> <figure xlink:label="fig-0633-01" xlink:href="fig-0633-01a"> <variables xml:id="echoid-variables730" xml:space="preserve">l e p d a b g</variables> </figure> </div> <pb o="332" file="0634" n="634" rhead="VITELLONIS OPTICAE"/> </div> <div xml:id="echoid-div1652" type="section" level="0" n="0"> <head xml:id="echoid-head1241" xml:space="preserve" style="it">31. Centro uiſus extra circulũ (qui ect cõmunis ſectio ſuperficiei reflexiõis & ſpeculi ſphærici <lb/>concaui) exiſtente, ſi à uiſu ducantur duæ lineæ circulum contingentes, & diameter circuli: à <lb/>quolibet puncto arcus interiacentis terminum ultιmum diametri & punctum contingentiæ <lb/>(præter quàm ab illis punctis) poteſt fieri reflexio ad uiſum punctorum inæqualiter diſtantium <lb/>à centro circuli cum centro uiſus. Alhazen 77 n 5.</head> <figure> <variables xml:id="echoid-variables731" xml:space="preserve">h d t b q g</variables> </figure> <p> <s xml:id="echoid-s41540" xml:space="preserve">Huius demonſtratio euidens eſt per præmiſſa.</s> <s xml:id="echoid-s41541" xml:space="preserve"> Sit enim centrum <lb/>uiſus h extra circulum d t g q, cuius centrum eſt b:</s> <s xml:id="echoid-s41542" xml:space="preserve"> & ducatur dia-<lb/>meter h d b g:</s> <s xml:id="echoid-s41543" xml:space="preserve"> patetq́;</s> <s xml:id="echoid-s41544" xml:space="preserve"> per 6 huius quòd à puncto g non fit aliqua <lb/>reflexio ad uiſum:</s> <s xml:id="echoid-s41545" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s41546" xml:space="preserve"> à puncto h (quod eſt centrum uiſus) <lb/>duæ lineæ contingentes circulum d t g q per 17 p 3 quæ ſint lineæ h <lb/>t & h q:</s> <s xml:id="echoid-s41547" xml:space="preserve"> palamq́;</s> <s xml:id="echoid-s41548" xml:space="preserve"> eſt per ea, quæ dicta ſunt in 24 huius, quoniam ab <lb/>arcu q d t nulla fit reflexio ad uiſum exiſtentem in puncto h:</s> <s xml:id="echoid-s41549" xml:space="preserve"> ſed nec <lb/>ab aliquo punctorum contingentiæ, quæ ſunt q & t, poteſt fieri refle <lb/>xio ad uiſum exiſtentem in puncto h:</s> <s xml:id="echoid-s41550" xml:space="preserve"> quoniam angulus contingen-<lb/>tiæ eſt indiuiſibilis:</s> <s xml:id="echoid-s41551" xml:space="preserve"> & lineæ q h & t h ſunt circulum contingentes, & <lb/>ut patet per 42 th.</s> <s xml:id="echoid-s41552" xml:space="preserve"> 1 huius omnis angulus contentus ſub termino <lb/>chordę & diametri eſt acutus:</s> <s xml:id="echoid-s41553" xml:space="preserve"> angulus uerò b q h eſt rectus.</s> <s xml:id="echoid-s41554" xml:space="preserve"> Non er-<lb/>go fiet ab illis punctis reflexio alicuius formæ ad uiſum in punctum <lb/>h:</s> <s xml:id="echoid-s41555" xml:space="preserve"> à reliquis uerò punctis arcus q g t, excepto puncto g, poteſt fieri <lb/>reflexio, demonſtratione 6 & 24 huius repetita.</s> <s xml:id="echoid-s41556" xml:space="preserve"> Patet ergo propoſi-<lb/>tum, ſeruata hypotheſi præmiſſa.</s> <s xml:id="echoid-s41557" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1653" type="section" level="0" n="0"> <head xml:id="echoid-head1242" xml:space="preserve" style="it">32. Centro uiſus intra circulum (qui eſt cõmunis ſectio ſuperfi-<lb/>cieireflexionis & ſpeculi ſphærici concaum) exiſtente, factá refle-<lb/>xione ab aliquo puncto circumferentiæ formæ alιcuius punctorum inæqualiter diſtantiũ à cen <lb/>tro ſpeculi cum centro uiſus: diameter circuli, in qua eſt punctus reflexus, cum diametro, in qua <lb/>eſt centrum uiſus facit angulum extrinſecum angulo reflexionis quando maiorẽ: quando <lb/>minorem angulo conctante ex angulis incidentiæ & reflexionis. Alhazen 78 n 5.</head> <p> <s xml:id="echoid-s41558" xml:space="preserve">Stante priori diſpoſitione 30 huius, ducatur à centro ſpeculi, quod eſt b, linea b f perpendicula-<lb/>ris ſuper lineam e p.</s> <s xml:id="echoid-s41559" xml:space="preserve"> Aut ergo linea b a eſt perpendicularis ſuper lineam e a:</s> <s xml:id="echoid-s41560" xml:space="preserve"> aut non.</s> <s xml:id="echoid-s41561" xml:space="preserve"> Sit primò <lb/>perpendicularis:</s> <s xml:id="echoid-s41562" xml:space="preserve"> & erũt duo anguli f b a & f e a æquales duobus rectis per 32 p 1, ideo quòd in qua-<lb/>drilatero f b a e alij duo anguli ſuntrecti ex hypotheſi.</s> <s xml:id="echoid-s41563" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s41564" xml:space="preserve"> linea b o ſuper lineam e f:</s> <s xml:id="echoid-s41565" xml:space="preserve"> & <lb/>erunt duo anguli o b a & o e a minores duobus rectis:</s> <s xml:id="echoid-s41566" xml:space="preserve"> ideo quòd angulus b o e eſt obtuſus, & angu <lb/>lus b a e rectus:</s> <s xml:id="echoid-s41567" xml:space="preserve"> erit ergo angulus o b g (qui per 13 p 1 <lb/> <anchor type="figure" xlink:label="fig-0634-02a" xlink:href="fig-0634-02"/> cũ angulo o b a ualet duos rectos) maior angulo o <lb/>e a, qui eſt angulus cõſtans exangulo reflexionis & <lb/>incidentiæ.</s> <s xml:id="echoid-s41568" xml:space="preserve"> Et cum triangulus e b f ſit æ qualis trian <lb/>gulo e b a:</s> <s xml:id="echoid-s41569" xml:space="preserve"> quia cum angulus b f e ſit æ qualis angu-<lb/>lo b a e (quoniam uterq;</s> <s xml:id="echoid-s41570" xml:space="preserve"> rectus) & angulus b e f eſt <lb/>æ qualis angulo b e a per 20 th.</s> <s xml:id="echoid-s41571" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s41572" xml:space="preserve"> erit per 26 <lb/>p 1 angulus e b a æ qualis angulo e b f:</s> <s xml:id="echoid-s41573" xml:space="preserve"> eſt enim b e la <lb/>tus utriq;</s> <s xml:id="echoid-s41574" xml:space="preserve"> illorum trigonorum cõmune:</s> <s xml:id="echoid-s41575" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s41576" xml:space="preserve"> per <lb/>4 p 6 latus f b æ quale lateri b a:</s> <s xml:id="echoid-s41577" xml:space="preserve"> quoniam ipſa reſpi <lb/>ciunt angulos æ quales.</s> <s xml:id="echoid-s41578" xml:space="preserve"> Sed latus o b per 19 p 1 eſt <lb/>maius latere b f:</s> <s xml:id="echoid-s41579" xml:space="preserve"> ergo & ipſum eſt maius latere b a.</s> <s xml:id="echoid-s41580" xml:space="preserve"> <lb/>Ducta uerò linea b n ſuper aliquod punctum lineæ <lb/>f p:</s> <s xml:id="echoid-s41581" xml:space="preserve"> erũt per præmiſſa duo anguli n b a & n e a maio-<lb/>res duobus rectis:</s> <s xml:id="echoid-s41582" xml:space="preserve"> ſed per 13 p 1 duo anguli n b a & n <lb/>b g ualent duos rectos:</s> <s xml:id="echoid-s41583" xml:space="preserve"> ergo angulus n b g min or <lb/>eſt angulo n e a:</s> <s xml:id="echoid-s41584" xml:space="preserve"> & linea n b cũ ſit per 19 p 1 maior ꝗ̃ <lb/>linea b f, erit ipſa maior quàm linea b a.</s> <s xml:id="echoid-s41585" xml:space="preserve"> Itaq;</s> <s xml:id="echoid-s41586" xml:space="preserve"> forma <lb/>puncti n reflectitur ad uiſum exiſtentem in puncto a, à puncto ſpeculi, quod eſt e:</s> <s xml:id="echoid-s41587" xml:space="preserve"> & in æ qualiter di-<lb/>ſtat à centro ſpeculi, quod eſt b, cum centro uiſus, quod eſt a:</s> <s xml:id="echoid-s41588" xml:space="preserve"> & diameter b n, in qua eſt punctus rei <lb/>uiſæ, quod eſt n, cum diametro a b g, in qua eſt centrum uiſus, quod eſt a, facit angulum n b g mino-<lb/>rem angulo n e a, qui eſt angulus conſtans ex angulis incidentiæ & reflexionis:</s> <s xml:id="echoid-s41589" xml:space="preserve"> diameter uerò o b <lb/>cum diametro a b g continet angulum o b g maiorem angulo o e a.</s> <s xml:id="echoid-s41590" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s41591" xml:space="preserve"> Siuerò <lb/>linea b a non fuerit perpendicularis ſuper lineam e a:</s> <s xml:id="echoid-s41592" xml:space="preserve"> tunc per 12 p 1 à puncto b ſuper productam li-<lb/>neam e a ducatur perpendicularis, quę ſit b k:</s> <s xml:id="echoid-s41593" xml:space="preserve"> quę quidẽ ſiue cadatultra lineam a b, uel citra uerſus <lb/>punctũ e, ſemper eadẽ probatio.</s> <s xml:id="echoid-s41594" xml:space="preserve"> Sit enim linea b f perpendicularis ſuper lineã e p:</s> <s xml:id="echoid-s41595" xml:space="preserve"> & ſit linea f t æ-<lb/>qualis lineę a k:</s> <s xml:id="echoid-s41596" xml:space="preserve"> & ducatur linea t b.</s> <s xml:id="echoid-s41597" xml:space="preserve"> Palãitaq;</s> <s xml:id="echoid-s41598" xml:space="preserve"> quoniã in trigonok e b angulus e k b eſt rectus æqua-<lb/> <pb o="333" file="0635" n="635" rhead="LIBER OCTAVVS."/> lis angulo f b e trigoni fe b:</s> <s xml:id="echoid-s41599" xml:space="preserve"> & angulus k e b per 20 th.</s> <s xml:id="echoid-s41600" xml:space="preserve"> 5 huius eſt æqualis angulo f e b, linea uerò <lb/>e b eſt latus commune:</s> <s xml:id="echoid-s41601" xml:space="preserve"> ergo per 26 p 1 illa trigona f b e & k b e ſunt æ qualia:</s> <s xml:id="echoid-s41602" xml:space="preserve"> & erit linea b f æ qualis <lb/>lineæ k b:</s> <s xml:id="echoid-s41603" xml:space="preserve"> ſed linea a k æ qualis eſt lineæ f t ex hypotheſi:</s> <s xml:id="echoid-s41604" xml:space="preserve"> ergo per 4 p 1 in trigonis b t f & b k a erit <lb/>linea b t æ qualis lineæ b a:</s> <s xml:id="echoid-s41605" xml:space="preserve"> & angulus a b k æqualis angulo f b t:</s> <s xml:id="echoid-s41606" xml:space="preserve"> addito ergo utrobiq;</s> <s xml:id="echoid-s41607" xml:space="preserve"> communi an-<lb/>gulo f b a, erit angulus k b f æ qualis angulo a b t:</s> <s xml:id="echoid-s41608" xml:space="preserve"> ſed duo anguli k b f & fe a ualent duos rectos per <lb/>32 p 1, quia in quadrilatero k b ſ e alij duo anguli (qui ſunt b f e & b k e) ſunt recti:</s> <s xml:id="echoid-s41609" xml:space="preserve"> ergo duo angulit <lb/>b a & t e a ualent duos rectos:</s> <s xml:id="echoid-s41610" xml:space="preserve"> ſed per 13 p 1 angulus t b g cum angulo t b a ualet duos rectos:</s> <s xml:id="echoid-s41611" xml:space="preserve"> ergo <lb/>angulus t b g æqualis eſt angulo t e a, quieſt angulus conſtans ex angulo incidentiæ & angulo refle <lb/>xionis.</s> <s xml:id="echoid-s41612" xml:space="preserve"> Si igitur à centro ſpeculi, quod eſt b, ad lineam t e ducatur linea ultra punctum t, faciet an-<lb/> <anchor type="figure" xlink:label="fig-0635-01a" xlink:href="fig-0635-01"/> <anchor type="figure" xlink:label="fig-0635-02a" xlink:href="fig-0635-02"/> gulum cum diametro b g ex parte puncti g minorem angulo t e a:</s> <s xml:id="echoid-s41613" xml:space="preserve"> quoniam faciet minorem angulo <lb/>t b g, qui eſt æ qualis angulo t e a:</s> <s xml:id="echoid-s41614" xml:space="preserve"> & erit illa linea maior quàm linea a b:</s> <s xml:id="echoid-s41615" xml:space="preserve"> quia erit per 19 p 1 maior <lb/>quàm linea b t, quæ eſt æ qualis lineæ a b.</s> <s xml:id="echoid-s41616" xml:space="preserve"> Quælibet uerò linea ducta ab aliquo puncto lineæ t e ad <lb/>centrum ſpeculi, quod eſt b, faciet angulum cum diametro b g maiorem angulo t b g:</s> <s xml:id="echoid-s41617" xml:space="preserve"> ergo & maio-<lb/>rem angulo t e a:</s> <s xml:id="echoid-s41618" xml:space="preserve"> & erit quælibet illarum linearum minor quàm linea b t:</s> <s xml:id="echoid-s41619" xml:space="preserve"> ergo erit minor quàm li-<lb/>nea b a.</s> <s xml:id="echoid-s41620" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s41621" xml:space="preserve"/> </p> <div xml:id="echoid-div1653" type="float" level="0" n="0"> <figure xlink:label="fig-0634-02" xlink:href="fig-0634-02a"> <variables xml:id="echoid-variables732" xml:space="preserve">e o f n p d a b g</variables> </figure> <figure xlink:label="fig-0635-01" xlink:href="fig-0635-01a"> <variables xml:id="echoid-variables733" xml:space="preserve">e o f t p d a b g k</variables> </figure> <figure xlink:label="fig-0635-02" xlink:href="fig-0635-02a"> <variables xml:id="echoid-variables734" xml:space="preserve">e o f l p k d a b g</variables> </figure> </div> </div> <div xml:id="echoid-div1655" type="section" level="0" n="0"> <head xml:id="echoid-head1243" xml:space="preserve" style="it">33. Centro uiſus & puncto rei uiſæ in diuerſis diametris circuli (qui eſt communis ſectio ſu-<lb/>perficiei reflexionis & ſpeculi ſphærici concaui) exiſtentibus, & inæqualiter diſtantibus à cen-<lb/>tro ſpeculi: ſi ab aliquo puncto circumferentiæ circuli fiat reflexio, impoßibile eſt diametrum, in <lb/>qua eſt punctus rei uiſæ cumdiametro, in qua eſt centrum uiſus, angulum extrinſecum æqua-<lb/>lem conſtituere angulo conſtanti ex angulis incidentiæ & reflexionis. Alhazen 79 n 5.</head> <p> <s xml:id="echoid-s41622" xml:space="preserve">Sit b centrum uiſus:</s> <s xml:id="echoid-s41623" xml:space="preserve"> & centrum ſpeculi ſphærici concaui ſit g:</s> <s xml:id="echoid-s41624" xml:space="preserve"> & ducatur diameter per pun-<lb/>cta b & g:</s> <s xml:id="echoid-s41625" xml:space="preserve"> quæ ſit z d:</s> <s xml:id="echoid-s41626" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s41627" xml:space="preserve"> a punctus rei uiſæ:</s> <s xml:id="echoid-s41628" xml:space="preserve"> & eſto, ut aliqua ſuperficies plana ſecet ſphæram ſpecu <lb/>li ſuper circulum z e d per 69 th.</s> <s xml:id="echoid-s41629" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s41630" xml:space="preserve"> Dico (ſi forma puncti a exiſtentis in diametro h g e reflecti <lb/>tur ad uiſum exiſtentem in puncto b ab aliquo puncto cir-<lb/>culi z e d:</s> <s xml:id="echoid-s41631" xml:space="preserve"> & ſi in-<lb/> <anchor type="figure" xlink:label="fig-0635-03a" xlink:href="fig-0635-03"/> <anchor type="figure" xlink:label="fig-0635-04a" xlink:href="fig-0635-04"/> æqualis eſt di-<lb/>ſtantia puncto-<lb/>rum a & b à cen <lb/>tro ſpeculi, qđ <lb/>eſt g) quòd dia <lb/>meter a g cũ dia <lb/>metro b g d ex <lb/>parte pũcti d fa-<lb/>eiet angulum a <lb/>g d, quem im-<lb/>poſsibile eſt eſſe <lb/>æqualem angu-<lb/>lo conſtanti ex <lb/>angulis inciden <lb/>tiæ & reflexiõis.</s> <s xml:id="echoid-s41632" xml:space="preserve"> <lb/>Si uerò hoc ſit <lb/>poſsibile, ponatur eſſe:</s> <s xml:id="echoid-s41633" xml:space="preserve"> & ſit punctus reflexionis t:</s> <s xml:id="echoid-s41634" xml:space="preserve"> ſit q́;</s> <s xml:id="echoid-s41635" xml:space="preserve"> linea à g inæqualis lineæ b g:</s> <s xml:id="echoid-s41636" xml:space="preserve"> & ducantur <lb/>lineæ t a, t b, t g, b a:</s> <s xml:id="echoid-s41637" xml:space="preserve"> & fiat circulus tranſiens pertria puncta a g b trigoni a b g per 5 p 4:</s> <s xml:id="echoid-s41638" xml:space="preserve"> trãſibit ergo <lb/> <pb o="334" file="0636" n="636" rhead="VITELLONIS OPTICAE"/> ille circulus neceſſariò per punctum t.</s> <s xml:id="echoid-s41639" xml:space="preserve"> Si enim trãſeat extra punctum t:</s> <s xml:id="echoid-s41640" xml:space="preserve"> tunc ductis lineis à punctis <lb/>a & b ad aliquod punctum unum illius circuli extra punctum t, & ducta linea b a:</s> <s xml:id="echoid-s41641" xml:space="preserve"> erit angulus con-<lb/>tentus per lineas ductas ad illud punctum circumferentię minoris circuli per 21 p 1 minor angulo a <lb/>t b, ſed accidit ipſum eſſe æ qualem angulo a t b.</s> <s xml:id="echoid-s41642" xml:space="preserve"> Palàm enim per 22 p 3 quoniã ille angulus cum an-<lb/>gulo a g b ualet duos rectos:</s> <s xml:id="echoid-s41643" xml:space="preserve"> quoniam omnes duo anguli quadrilateri inſcripti circulo ex aduerſo <lb/>collocati ualent duos rectos:</s> <s xml:id="echoid-s41644" xml:space="preserve"> ſed angulus a g b cum angulo a g d per 13 p 1 ualet duos rectos:</s> <s xml:id="echoid-s41645" xml:space="preserve"> angu-<lb/>lus uerò a g d æqualis eſt angulo a t b ex hypotheſi:</s> <s xml:id="echoid-s41646" xml:space="preserve"> ergo angulus a g b cum angulo a t b ualet duos <lb/>rectos:</s> <s xml:id="echoid-s41647" xml:space="preserve"> erit ergo ille angulus conſtitutus ſuper arcum minoris circuli æqualis angulo a t b, quod eſt <lb/>contra 21 p 1.</s> <s xml:id="echoid-s41648" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s41649" xml:space="preserve"> accidit idem impoſsibile, ſi circulus ille tranſiens puncta illa tria, quæ <lb/>ſunt a, g, b, non ceciderit in punctum t, ſed citra illud, & erit eadem deductio, quęprius.</s> <s xml:id="echoid-s41650" xml:space="preserve"> Reſtat ergo <lb/>ut circulus tranſiens per puncta a, g, b tranſeat etiam per punctum t.</s> <s xml:id="echoid-s41651" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s41652" xml:space="preserve"> angulus a t g ſit per <lb/>20 th.</s> <s xml:id="echoid-s41653" xml:space="preserve"> 5 huius æ qualis angulo b t g:</s> <s xml:id="echoid-s41654" xml:space="preserve"> erit arcus a g æqualis arcui b g per 26 p 3:</s> <s xml:id="echoid-s41655" xml:space="preserve"> ergo per 29 p 3 erit li-<lb/>nea b g æqualis lineæ g a:</s> <s xml:id="echoid-s41656" xml:space="preserve"> poſita autem eſt eſſe mæqualis:</s> <s xml:id="echoid-s41657" xml:space="preserve"> hoc ergo eſt impoſsibile.</s> <s xml:id="echoid-s41658" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s41659" xml:space="preserve"> pro-<lb/>poſitum, quoniã angulus a t b conſtans ex angulis incidentiæ & reflexionis formæ puncti a ad cen <lb/>trũ uiſus exiſtẽs in puncto b, ſemper eſt inæ qualis angulo cõtento à diametris, in quibus ſunt pun <lb/>ctus rei uiſæ, & centrũ uiſus, extrinſeco ad illũ angulũ incidentiæ & reflexionis.</s> <s xml:id="echoid-s41660" xml:space="preserve"> Quod eſt ꝓpoſitũ.</s> <s xml:id="echoid-s41661" xml:space="preserve"/> </p> <div xml:id="echoid-div1655" type="float" level="0" n="0"> <figure xlink:label="fig-0635-03" xlink:href="fig-0635-03a"> <variables xml:id="echoid-variables735" xml:space="preserve">t z e b a g h d</variables> </figure> <figure xlink:label="fig-0635-04" xlink:href="fig-0635-04a"> <variables xml:id="echoid-variables736" xml:space="preserve">t z c b a g h d</variables> </figure> </div> </div> <div xml:id="echoid-div1657" type="section" level="0" n="0"> <head xml:id="echoid-head1244" xml:space="preserve" style="it">34. Centro uiſus & puncto rei uiſæ in diuerſis diametris circuli (qui eſt communis ſectio ſu-<lb/>perficiei reflexionis & ſpeculi ſphærici cõcaui) exiſtentibus, & inæqualiter diſtantibus à centro <lb/>ſpeculi: ſi à duobus punctis arcus interiacentis diametrum, in qua eſt centrum uiſus, & aliam, <lb/>in qua eſt punctus rei uiſæ, fiat reflexio: non erit uter angulus conſtãs ex angulo incidentiæ & <lb/>reflexionis minor angulo, extrinſeco ad angulum cadentem in eundem arcum, à dictis diame-<lb/>tris contento. Alhazen 80 n 5.</head> <p> <s xml:id="echoid-s41662" xml:space="preserve">Sit, ut in præmiſſa proxima, centrum uiſus b:</s> <s xml:id="echoid-s41663" xml:space="preserve"> & punctus rei uiſæ a:</s> <s xml:id="echoid-s41664" xml:space="preserve"> centrum ſpeculi ſphærici con <lb/>caui ſit g:</s> <s xml:id="echoid-s41665" xml:space="preserve"> & ducatur diameter per puncta b & g, quæ ſit z d:</s> <s xml:id="echoid-s41666" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s41667" xml:space="preserve"> ſuperficies plana ſpeculũ ſecun-<lb/>dum diametrum z d:</s> <s xml:id="echoid-s41668" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s41669" xml:space="preserve"> per 69 th.</s> <s xml:id="echoid-s41670" xml:space="preserve"> 1 huius ſectio communis circulus, qui ſit e d h z:</s> <s xml:id="echoid-s41671" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s41672" xml:space="preserve"> dia <lb/>meter e h, in qua ſit punctus rei uiſæ, qui eſt a:</s> <s xml:id="echoid-s41673" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s41674" xml:space="preserve"> linea b g, quæ eſt diſtantia centri uiſus à centro <lb/>ſpeculi, maior quàm linea a g.</s> <s xml:id="echoid-s41675" xml:space="preserve"> Dico quòd ſi forma puncti a reflectitur ad uiſum exiſtentem in pun-<lb/>cto b à duobus punctis arcus e z, non erit uterq;</s> <s xml:id="echoid-s41676" xml:space="preserve"> angulus conſtans ex angulis incidentiæ & refle-<lb/>xionis minor angulo a g d.</s> <s xml:id="echoid-s41677" xml:space="preserve"> Sint enim duo puncta, à quibus fit reflexio formæ puncti a ad uiſum exi <lb/>ſtentem in puncto b, quæ ſunt puncta t & q:</s> <s xml:id="echoid-s41678" xml:space="preserve"> & ducantur line æ b t, g t, a t, b q, g q, a q.</s> <s xml:id="echoid-s41679" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s41680" xml:space="preserve"> angulus <lb/>b t a cõſtans ex angulo incidentiæ, qui eſt a t g, & exangulo reflexionis, qui eſt g t b, ſit minorangu <lb/>lo a g d, qui eſt angulus extrinſecus angulo cadenti in arcum e z:</s> <s xml:id="echoid-s41681" xml:space="preserve"> & eſt ipſe angulus a g d cadens in <lb/>arcum e d.</s> <s xml:id="echoid-s41682" xml:space="preserve"> Dico quò d angulus a q b, qui conſtat <lb/> <anchor type="figure" xlink:label="fig-0636-01a" xlink:href="fig-0636-01"/> exangulo incidentiæ a q g, & angulo reflexionis g <lb/>q b, non erit minorangulo a g d.</s> <s xml:id="echoid-s41683" xml:space="preserve"> Dato enim quòd <lb/>ſit minor:</s> <s xml:id="echoid-s41684" xml:space="preserve"> ducatur linea g n diuidens angulum e g <lb/>z per æ qualia per 9 p 1:</s> <s xml:id="echoid-s41685" xml:space="preserve"> & ducatur linea a b conti-<lb/>nuans punctum rei uiſæ, quod eſt a, cum centro ui <lb/>ſus, quod eſt b.</s> <s xml:id="echoid-s41686" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s41687" xml:space="preserve"> per 29 th.</s> <s xml:id="echoid-s41688" xml:space="preserve"> 1 huius, cum <lb/>linea g n ſecet angulum b g a, cui ſubtenditur linea <lb/>a b, quòd linea g n etiã ſecabit lineã a b:</s> <s xml:id="echoid-s41689" xml:space="preserve"> ſit punctus <lb/>ſectiõis f:</s> <s xml:id="echoid-s41690" xml:space="preserve"> erit ergo per 3 p 6 proportio lineæ b g a d <lb/>lineam g a, ſicut lineæ b f ad lineam f a:</s> <s xml:id="echoid-s41691" xml:space="preserve"> ſed linea b g <lb/>ex hypotheſi eſt maior quàm linea g a:</s> <s xml:id="echoid-s41692" xml:space="preserve"> eſt ergo li-<lb/>nea b f maior quàm linea f a.</s> <s xml:id="echoid-s41693" xml:space="preserve"> Diuidatur itaq;</s> <s xml:id="echoid-s41694" xml:space="preserve"> linea <lb/>a b per æ qualia in puncto k per 10 p 1:</s> <s xml:id="echoid-s41695" xml:space="preserve"> & fiat per 5 p <lb/>4 circulus tranſiens per tria puncta, quę ſunt a, b, t:</s> <s xml:id="echoid-s41696" xml:space="preserve"> <lb/>qui circulus non tranſibit per punctum g, ſed citra <lb/>illud uerſus puncta a & b.</s> <s xml:id="echoid-s41697" xml:space="preserve"> Dato enim quòd circu-<lb/>lus ille tranſeat centrum g, ſequeretur per 22 p 3 an <lb/>gulum a g b cum angulo a t b ęqualẽ eſſe duobus rectis:</s> <s xml:id="echoid-s41698" xml:space="preserve"> quoniã illi duo anguli erunt ex aduerſo col <lb/>locati in quadrilatero inſcripto illi minori circulo:</s> <s xml:id="echoid-s41699" xml:space="preserve"> ſuntautẽ illi duo anguli minores duobus rectis, <lb/>quod patet ex hypotheſi, cum angulus b t a ſit minor angulo a g d, qui per 13 p 1 cum angulo a g b ua <lb/>let duos rectos.</s> <s xml:id="echoid-s41700" xml:space="preserve"> Igitur ille minor circulus non tranſibit per centrũ maioris circuli, quod eſt g.</s> <s xml:id="echoid-s41701" xml:space="preserve"> Simi-<lb/>liter quoq;</s> <s xml:id="echoid-s41702" xml:space="preserve"> dico quòd non tranſibit ille circulus minor punctũ reflexionis ſecundũ, quod eſt q.</s> <s xml:id="echoid-s41703" xml:space="preserve"> Da <lb/>to enim quòd tranſeat punctũ q, cũ non tranſeat centrum g:</s> <s xml:id="echoid-s41704" xml:space="preserve"> ſit punctus, in quo linea g q ſecat peri-<lb/>pheriã illius circuli, punctus m.</s> <s xml:id="echoid-s41705" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s41706" xml:space="preserve"> anguli a q m & m q b ſunt ęquales per 20 th 5 huius, quo-<lb/>niã angulus incidentiæ eſt æqualis angulo reflexionis:</s> <s xml:id="echoid-s41707" xml:space="preserve"> & ſunt cõſtituti ſuper illius circuli circum-<lb/>ferentiam:</s> <s xml:id="echoid-s41708" xml:space="preserve"> palàm per 26 p 3 quoniã arcus a m æ qualis erit arcui m b:</s> <s xml:id="echoid-s41709" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s41710" xml:space="preserve"> Sitenim <lb/>punctus in quo linea g t ſecat circulũ, punctus o:</s> <s xml:id="echoid-s41711" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s41712" xml:space="preserve"> palàm ք 20 th.</s> <s xml:id="echoid-s41713" xml:space="preserve"> 5 huius & 26 p 3 quoniã arcus <lb/>a o eſt ęqualis arcui o b:</s> <s xml:id="echoid-s41714" xml:space="preserve"> eſt aũt arcus a o maior arcu a m:</s> <s xml:id="echoid-s41715" xml:space="preserve"> fiet ergo arcus o b maior arcu m b, pars ſuo <lb/>toto:</s> <s xml:id="echoid-s41716" xml:space="preserve"> qđ eſt impoſsibile.</s> <s xml:id="echoid-s41717" xml:space="preserve"> Nõ ergo trãſibit ille circulus per punctũ q:</s> <s xml:id="echoid-s41718" xml:space="preserve"> reſtat ergo, ut ille circulus tran-<lb/> <pb o="335" file="0637" n="637" rhead="LIBER OCTAVVS."/> ſeat ultra punctum q:</s> <s xml:id="echoid-s41719" xml:space="preserve"> ſi enim citra punctum q tranſeat, eadem penitus erit improbatio, quęprius.</s> <s xml:id="echoid-s41720" xml:space="preserve"> <lb/>Ducatur item linea à puncto o ad punctum k, quę ſit o k:</s> <s xml:id="echoid-s41721" xml:space="preserve"> hæc ergo diuidit chordam b a per æ qualia, <lb/>& ſimiliter arcum b a, ut patet ex præmiſsis.</s> <s xml:id="echoid-s41722" xml:space="preserve"> Ductis ergo chordis b o & a o, quæ erunt æquales per <lb/>29 p 3, patet per 8 p 1 quod linea o k perpendicularis erit ſuper lineam b a.</s> <s xml:id="echoid-s41723" xml:space="preserve"> Sed per 18 p 1 angulus b <lb/>a g maior eſt angulo a b g:</s> <s xml:id="echoid-s41724" xml:space="preserve"> eſt enim linea b g maior quàm linea g a ex hypotheſi, & per 32 p 1 angulus <lb/>b f g ualet duos angulos f a g & f g a, & per eandem 32 p 1 angulus a f g ualet duos angulos f b g & f <lb/>g b:</s> <s xml:id="echoid-s41725" xml:space="preserve"> ſed ex præ miſsis angulus a g f eſt æ qualis angulo t g b, & angulus f a g maior eſt angulo f b g:</s> <s xml:id="echoid-s41726" xml:space="preserve"> er-<lb/>go angulus a f g minor eſt angulo g f b:</s> <s xml:id="echoid-s41727" xml:space="preserve"> eſt ergo angulus g f a acutus, & angulus g f b obtuſus per 13 <lb/>p 1:</s> <s xml:id="echoid-s41728" xml:space="preserve"> ergo angulus n f k eſt acutus per eandem 13 p 1:</s> <s xml:id="echoid-s41729" xml:space="preserve"> ſed angulus o k b eſt rectus, ut patet ex præmiſ-<lb/>ſis:</s> <s xml:id="echoid-s41730" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s41731" xml:space="preserve"> 1 huius linea o k producta concurret cum linea g n ultra lineam b f, non autem <lb/>ſub illa:</s> <s xml:id="echoid-s41732" xml:space="preserve"> ideo, quia ſi concurreret cum linea g f in puncto k:</s> <s xml:id="echoid-s41733" xml:space="preserve"> fierent per 1 p 6 trigona a g k & b g k æ-<lb/>qualia:</s> <s xml:id="echoid-s41734" xml:space="preserve"> cum ipſa ſint eiuſdem altitudinis, & eorum baſes, quæ ſunt b k & a k, ſint æ quales:</s> <s xml:id="echoid-s41735" xml:space="preserve"> ſed & eo-<lb/>rum anguli, qui ſunt b g k & a g k ſunt æ quales:</s> <s xml:id="echoid-s41736" xml:space="preserve"> angulus enim a g b diuiſus eſt per æqualia per li-<lb/>neam g f, in quam cadit punctum k:</s> <s xml:id="echoid-s41737" xml:space="preserve"> ergo per 15 p 6 ſequitur latus b g fieri ęquale lateri a g:</s> <s xml:id="echoid-s41738" xml:space="preserve"> quod eſt <lb/>contra hypotheſim:</s> <s xml:id="echoid-s41739" xml:space="preserve"> uel ſequitur per 3 p 6 lineam b k fieri maiorem quàm fuit linea a k:</s> <s xml:id="echoid-s41740" xml:space="preserve"> quod item <lb/>eſt contra præmiſſa.</s> <s xml:id="echoid-s41741" xml:space="preserve"> Idem quoq;</s> <s xml:id="echoid-s41742" xml:space="preserve"> accidit impoſsibile, ſi punctus f cadat inter puncta b & k:</s> <s xml:id="echoid-s41743" xml:space="preserve"> fiet enim <lb/>linea b k maior quàm linea b f:</s> <s xml:id="echoid-s41744" xml:space="preserve"> eſt aũt linea b f per 3 p 6 maior quàm linea f a:</s> <s xml:id="echoid-s41745" xml:space="preserve"> & ita eſt linea b f maior <lb/>quàm linea k a:</s> <s xml:id="echoid-s41746" xml:space="preserve"> quod totum eſt impoſsibile:</s> <s xml:id="echoid-s41747" xml:space="preserve"> cadet ergo punctus f inter puncta k & a.</s> <s xml:id="echoid-s41748" xml:space="preserve"> Fiet ergo li-<lb/>nearum o k & g n concurſus ultra lineam b f.</s> <s xml:id="echoid-s41749" xml:space="preserve"> Facto item circulo tranſeunte per tria puncta, quæ <lb/>ſunt a, q, b:</s> <s xml:id="echoid-s41750" xml:space="preserve"> tranſibit ille circulus citra punctum g:</s> <s xml:id="echoid-s41751" xml:space="preserve"> quoniam, ut prius oſtenſum eſt, ſi tranſiret per <lb/>punctum g, fieret per 22 p 3 angulus a q b æ qualis angulo a <lb/>g d per 13 p 1, quod eſt contra pręmiſſam proximam:</s> <s xml:id="echoid-s41752" xml:space="preserve"> tranſi <lb/> <anchor type="figure" xlink:label="fig-0637-01a" xlink:href="fig-0637-01"/> bit ergo ille circulus citra punctum g, & per 20 th.</s> <s xml:id="echoid-s41753" xml:space="preserve"> 5 huius <lb/>& per 26 p 3 linea g q diuidet arcum illius circuli, qui eſt a <lb/>b, per æqualia in puncto, qui ſit o:</s> <s xml:id="echoid-s41754" xml:space="preserve"> quoniam ipſa diuidit <lb/>angulum b q a per æqualia.</s> <s xml:id="echoid-s41755" xml:space="preserve"> Ducatur quoq;</s> <s xml:id="echoid-s41756" xml:space="preserve"> linea k o, quæ, <lb/>ut patet ex pręmiſsis, diuidit chordam b a per æqualia:</s> <s xml:id="echoid-s41757" xml:space="preserve"> er-<lb/>go linea k o concurret cum linea g n infra lineam f b, & ul-<lb/>tra punctum o Quia enim, ut ſuprà oſtenſum eſt, linea o k <lb/>eſt perpendicularis ſuper lineam b a, punctumq́;</s> <s xml:id="echoid-s41758" xml:space="preserve"> o cadit <lb/>in peripheriam cireuli minoris, qui eſt a q b:</s> <s xml:id="echoid-s41759" xml:space="preserve"> à punctis er-<lb/>go a & b copulentur, ut prius, chordæ b o & a o, patetq́;</s> <s xml:id="echoid-s41760" xml:space="preserve"> <lb/>per 4 p 1 quoniam chordæ b o & a o ſunt & quales:</s> <s xml:id="echoid-s41761" xml:space="preserve"> ergo <lb/>per 28 p 3 arcus a o eſt æ qualis arcui b o:</s> <s xml:id="echoid-s41762" xml:space="preserve"> arcus enim b a di <lb/>uiſus eſt per æ qualia in puncto o per lineam g q:</s> <s xml:id="echoid-s41763" xml:space="preserve"> lineæ er-<lb/>go o k & g n concurrunt in puncto aliquo citra lineam b <lb/>f, & ultra punctum o:</s> <s xml:id="echoid-s41764" xml:space="preserve"> quoniam linea g n diuidens per æ <lb/>qualia angulum a g b, caditinter puncta k & o, ut ſuprà patuit.</s> <s xml:id="echoid-s41765" xml:space="preserve"> Linea ergo k o cõcurrens cum linea <lb/>b a, de neceſsitate prius concurret cum linea g n:</s> <s xml:id="echoid-s41766" xml:space="preserve"> concurret ergo cum linea g n ſub linea b f:</s> <s xml:id="echoid-s41767" xml:space="preserve"> cuius <lb/>contrarium iam patuit in præmiſsis:</s> <s xml:id="echoid-s41768" xml:space="preserve"> oſtenſum enim fuit, quia concurrebat cum linea g n ultra li-<lb/>neam b f:</s> <s xml:id="echoid-s41769" xml:space="preserve"> & ita ſequeretur duas rectas lineas includere ſuperficiem:</s> <s xml:id="echoid-s41770" xml:space="preserve"> quod eſt manifeſtum impoſsi-<lb/>bile.</s> <s xml:id="echoid-s41771" xml:space="preserve"> Reſtat ergo ut angulus a q b non ſit minor angulo a g d:</s> <s xml:id="echoid-s41772" xml:space="preserve"> aut quòd forma puncti a non reflecta-<lb/>tur ad uiſum in punctum b à puncto q:</s> <s xml:id="echoid-s41773" xml:space="preserve"> quod eſt contra hypotheſim & impoſsibile.</s> <s xml:id="echoid-s41774" xml:space="preserve"> Eſt ergo angu-<lb/>lus a q b non minor angulo a g d:</s> <s xml:id="echoid-s41775" xml:space="preserve"> ex quo ſequitur propoſitum, quòd in hac diſpoſitione non erit u-<lb/>terq;</s> <s xml:id="echoid-s41776" xml:space="preserve"> angulorum conſtantium ex angulis incidentię & reflexionis minor angulo extrinſeco, ad an-<lb/>gulum cadentem in arcum contentum à duabus diametris circuli, in quarum una eſt centrũ uiſus, <lb/>& in altera punctus rei uiſę.</s> <s xml:id="echoid-s41777" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s41778" xml:space="preserve"> quoniã ſemper ſimilis erit improbatio, ſumpto <lb/>quocunq;</s> <s xml:id="echoid-s41779" xml:space="preserve"> alio puncto arcus e n.</s> <s xml:id="echoid-s41780" xml:space="preserve"> Sed neq;</s> <s xml:id="echoid-s41781" xml:space="preserve"> ab aliquo puncto arcus z n poſsibile eſt fieri reflexionem <lb/>formæ puncti a rei uiſæ ad uiſum exiſtentẽ in puncto b, ita ut angulus conſtans ex angulis inciden-<lb/>tiæ & refle xionis factæ à puncto t, & ab illo alio puncto arcus n z ſit uterq;</s> <s xml:id="echoid-s41782" xml:space="preserve"> minor angulo a g d.</s> <s xml:id="echoid-s41783" xml:space="preserve"> Re-<lb/>manente enim diſpoſitione figuræ prioris, ſit, ut à puncto arcus n z fiat reflexio formæ puncti a ad <lb/>uiſum b.</s> <s xml:id="echoid-s41784" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s41785" xml:space="preserve"> quòd angulus conſtans ex angulo incidentię & reflexionis, qui ſit ſuper punctum <lb/>p, ſit minor angulo a g d, ſicut & angulus conſtans ex angulo incidentiæ & reflexionis, qui eſt ſupra <lb/>punctum t, minor eſt eodem angulo a g d.</s> <s xml:id="echoid-s41786" xml:space="preserve"> Ducanturitaq;</s> <s xml:id="echoid-s41787" xml:space="preserve"> lineæ a p, b p, g p:</s> <s xml:id="echoid-s41788" xml:space="preserve"> ſecabit ergo linea g p li-<lb/>neam k o:</s> <s xml:id="echoid-s41789" xml:space="preserve"> quoniam, ut præmiſſum eſt, linea g t diuidit arcum a b minoris circuli per æ qualia in pun <lb/>cto o per 26 p 3:</s> <s xml:id="echoid-s41790" xml:space="preserve"> eſt enim per 20 th.</s> <s xml:id="echoid-s41791" xml:space="preserve"> 5 huius angulus a t g æ qualis angulo g t b, & eundem arcum diui <lb/>dit linea k o per æ qualia.</s> <s xml:id="echoid-s41792" xml:space="preserve"> Et quoniã, ut præoſtenſum eſt, patet quod linea k o concurrit cum linea <lb/>g n, linea g p ſecat angulũ n g t, cui ſubtenditur linea k o, concurrens cum linea n g ultra lineam b f:</s> <s xml:id="echoid-s41793" xml:space="preserve"> <lb/>ergo per 29 th.</s> <s xml:id="echoid-s41794" xml:space="preserve"> 1 huius linea g p ſecabit lineam k o.</s> <s xml:id="echoid-s41795" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s41796" xml:space="preserve"> punctus ſectionis linearum g p & k <lb/>o punctus l:</s> <s xml:id="echoid-s41797" xml:space="preserve"> & ducatur linea t p.</s> <s xml:id="echoid-s41798" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s41799" xml:space="preserve"> duæ lineæ g t & g p ſint æquales:</s> <s xml:id="echoid-s41800" xml:space="preserve"> quia ſunt ſemidia-<lb/>metri eiuſd m citculi:</s> <s xml:id="echoid-s41801" xml:space="preserve"> erit per 5 p 1 angulus g t p æqualis angulo g p t, & uterq;</s> <s xml:id="echoid-s41802" xml:space="preserve"> acutus per 32 p 1.</s> <s xml:id="echoid-s41803" xml:space="preserve"> <lb/>Ducta ergo linea perpendiculari à puncto t ſuper lineam g t, erit illa perpendicularis per 16 p <lb/>3 contingens ſpeculi circulum, qui eſt e d h z:</s> <s xml:id="echoid-s41804" xml:space="preserve"> & producta cadet ſuper terminum diametri mi-<lb/>noris circuli per 31 p 3:</s> <s xml:id="echoid-s41805" xml:space="preserve"> cum angulus, quem efficit illa perpendicularis cum linea t g, reſpiciat <lb/> <pb o="336" file="0638" n="638" rhead="VITELLONIS OPTICAE"/> ſemicirculũ minoris:</s> <s xml:id="echoid-s41806" xml:space="preserve"> linea enim t o cadit ſuper lineã k o, fitq́;</s> <s xml:id="echoid-s41807" xml:space="preserve"> angulus t o k minor recto per 42 th.</s> <s xml:id="echoid-s41808" xml:space="preserve"> 1 <lb/>huius:</s> <s xml:id="echoid-s41809" xml:space="preserve"> linea enim o k eſt pars diametri circuli minoris, propter hoc quòd angulus o k b eſt rectus:</s> <s xml:id="echoid-s41810" xml:space="preserve"> <lb/>& linea k o producta ſecat circulum minorẽ, tranſiens per eius centrũ per 1 p 3:</s> <s xml:id="echoid-s41811" xml:space="preserve"> ideo quòd ipſa ſecãs <lb/>lineam b a orthogonaliter, & per æqualia ſecat ipſam neceſſariò:</s> <s xml:id="echoid-s41812" xml:space="preserve"> ergo illa perpendicularis produ-<lb/>cta concurret cum linea k o per 14 th.</s> <s xml:id="echoid-s41813" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s41814" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s41815" xml:space="preserve"> punctus concurſus in puncto termini diametri <lb/>circuli minoris per 31 p 3:</s> <s xml:id="echoid-s41816" xml:space="preserve"> cum ille angulus in ſemicirculo ſit rectus, qui fit ſuper punctum t terminũ <lb/>lineæ g t:</s> <s xml:id="echoid-s41817" xml:space="preserve"> ſed linea t p eſt inferior illa perpendiculari ex parte puncti n.</s> <s xml:id="echoid-s41818" xml:space="preserve"> Igitur quæcunq;</s> <s xml:id="echoid-s41819" xml:space="preserve"> linea duca-<lb/>tur à puncto g centro ſpeculi ad lineam t p, ſecans diametrum o k:</s> <s xml:id="echoid-s41820" xml:space="preserve"> illa cadet neceſſariò in aliquod <lb/>punctum lineæ t p citra perpendicularem.</s> <s xml:id="echoid-s41821" xml:space="preserve"> Cum igitur linea g p cadat in punctum p, & ſecet lineam <lb/>o k:</s> <s xml:id="echoid-s41822" xml:space="preserve"> erit punctus p citra illam perpendicularem, & infra arcum minoris circuli, cui ſubtenditur illa <lb/>perpendicularis.</s> <s xml:id="echoid-s41823" xml:space="preserve"> Facto igitur circulo trãſeunte per tria puncta, quę ſunt a, b, p, tranſibit quidem ille <lb/>circulus per punctum l:</s> <s xml:id="echoid-s41824" xml:space="preserve"> quoniam linea p l ſecabit illum circulum, ſicuti priorem circulum a b t ſeca <lb/>bat linea t o.</s> <s xml:id="echoid-s41825" xml:space="preserve"> Circulus itaq;</s> <s xml:id="echoid-s41826" xml:space="preserve"> a b p ſecabit circulum a b t in duobus punctis a & b:</s> <s xml:id="echoid-s41827" xml:space="preserve"> & cum exeat à pun-<lb/>cto b, & iterum redeat in punctum p inferiorem puncto t (cum ſit citra illum circulum uerſus pun-<lb/>ctum t) neceſſariò ſecabit illum circulum in tertio puncto, quod eſt contra 10 p 3 & impoſsibile.</s> <s xml:id="echoid-s41828" xml:space="preserve"> Re <lb/>ſtat igitur, ut forma puncti rei uiſæ, qui eſt a, non reflectatur ad uiſum exiſtentẽ in puncto b à duo-<lb/>bus punctis arcus z n:</s> <s xml:id="echoid-s41829" xml:space="preserve"> ita ut quilibet angulorum illorum ſit minor angulo a g d.</s> <s xml:id="echoid-s41830" xml:space="preserve"> Palàm ergo, quòd <lb/>impoſsibile eſt, ut forma puncti a reflectatur ad uiſum b à duobus punctis arcus interiacentis eo-<lb/>rum diametros, qui eſt e z, ita ut uter q;</s> <s xml:id="echoid-s41831" xml:space="preserve"> angulorum conſtantium ex angulis incidentiæ & reflexio-<lb/>nis ſit minor angulo a g d.</s> <s xml:id="echoid-s41832" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s41833" xml:space="preserve"/> </p> <div xml:id="echoid-div1657" type="float" level="0" n="0"> <figure xlink:label="fig-0636-01" xlink:href="fig-0636-01a"> <variables xml:id="echoid-variables737" xml:space="preserve">n q t p z b k f a l m g h d</variables> </figure> <figure xlink:label="fig-0637-01" xlink:href="fig-0637-01a"> <variables xml:id="echoid-variables738" xml:space="preserve">n q t z e b k f a g h d</variables> </figure> </div> </div> <div xml:id="echoid-div1659" type="section" level="0" n="0"> <head xml:id="echoid-head1245" xml:space="preserve" style="it">35. In ſpeculis ſphæricis cõcauis duos pũctos, qui in diuerſis diametris, & inæqualis diſtantiæ <lb/>à centro ſpeculi exiſtentes à duobus punctis ſpeculi arcus ſcilicet interiacentis ſemidiametros, <lb/>in quibus illi puncti conſiſtunt, ad ſe mutuò reflectantur, poßibile eſt inueniri. Alhazen 81 n 5.</head> <p> <s xml:id="echoid-s41834" xml:space="preserve">Sit circulus (qui eſt communis ſectio ſuperficiei reflexionis, & ſuperficiei ſpeculi ſphærici con-<lb/>caui) cuius centrum d:</s> <s xml:id="echoid-s41835" xml:space="preserve"> & ſumantur in ipſo duæ diametri, quæ ſint g a & b h, ſecantes ſe in centro d:</s> <s xml:id="echoid-s41836" xml:space="preserve"> <lb/>dico quòd poſsibile eſt fieri, quod proponitur.</s> <s xml:id="echoid-s41837" xml:space="preserve"> Diuidatur enim angulus g d b per æqualia per ſemi <lb/>diametrum d e:</s> <s xml:id="echoid-s41838" xml:space="preserve"> & in ſemidiametro b d ſumatur punctus m ultra punctũ, in quem cadit perpendicu <lb/>laris ducta à puncto e ſuper diametrũ b d:</s> <s xml:id="echoid-s41839" xml:space="preserve"> & ſumatur linea n d in diametro d g æqualis lineæ m d:</s> <s xml:id="echoid-s41840" xml:space="preserve"> & <lb/>fiat per 5 p 4 circulus tranſiens per tria puncta m, d, n:</s> <s xml:id="echoid-s41841" xml:space="preserve"> hic ergo neceſſariò tranſibit ultra punctum e.</s> <s xml:id="echoid-s41842" xml:space="preserve"> <lb/>Si enim detur, quòd ille circulus tranſeat punctum e, ducantur lineæ m e & n e:</s> <s xml:id="echoid-s41843" xml:space="preserve"> fietq́;</s> <s xml:id="echoid-s41844" xml:space="preserve"> quadrangulũ <lb/>d m e n intra circulũ:</s> <s xml:id="echoid-s41845" xml:space="preserve"> ergo per 22 p 3 duo anguli iſtius quadranguli ex aduerſo collocati, ut qui ſunt <lb/>ad punctos m & n, ſunt æquales duobus rectis:</s> <s xml:id="echoid-s41846" xml:space="preserve"> quod eſt impoſsibile:</s> <s xml:id="echoid-s41847" xml:space="preserve"> quoniam duo anguli e m d & <lb/>e n d ambo ſunt acuti, minores duobus rectis:</s> <s xml:id="echoid-s41848" xml:space="preserve"> ideo quòd lineæ e m & e n cadunt ultra perpendicu-<lb/>lares ductas à pũcto e ſuper ſemidiametros b d & g d.</s> <s xml:id="echoid-s41849" xml:space="preserve"> Similis quoq;</s> <s xml:id="echoid-s41850" xml:space="preserve"> fiet deductio, ſi circulus trãſeat <lb/>citra punctum e:</s> <s xml:id="echoid-s41851" xml:space="preserve"> tunc enim anguli illius quadranguli cadentes ſuper punctum m & n, erũt iterum <lb/>minores duobus rectis.</s> <s xml:id="echoid-s41852" xml:space="preserve"> Tranſit igitur circulus d m n extra punctum e:</s> <s xml:id="echoid-s41853" xml:space="preserve"> ſecabit ergo circulum pro-<lb/>poſitum ipſius ſpeculi in duo bus punctis per 10 p 3:</s> <s xml:id="echoid-s41854" xml:space="preserve"> ſint illa duo puncta t & l:</s> <s xml:id="echoid-s41855" xml:space="preserve"> & ducantur lineæ n t, <lb/>m t, n l, d l, m l:</s> <s xml:id="echoid-s41856" xml:space="preserve"> & ducatur linea m n ſecans lineã t d in puncto f, & lineam e d in puncto p.</s> <s xml:id="echoid-s41857" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s41858" xml:space="preserve">, <lb/>ut patet ex præmiſsis, linea m d ſit æqualis lineæ n d, & li-<lb/> <anchor type="figure" xlink:label="fig-0638-01a" xlink:href="fig-0638-01"/> nea p d cómunis ambobus trigonis p d m & p d n, & an-<lb/>gulus p d m æqualis angulo p d n:</s> <s xml:id="echoid-s41859" xml:space="preserve"> palàm per 4 p 1 quoniã <lb/>triangulus p m d æqualis eſt triangulo p n d:</s> <s xml:id="echoid-s41860" xml:space="preserve"> erit quoq;</s> <s xml:id="echoid-s41861" xml:space="preserve"> an <lb/>gulus f p d æqualis angulo n p d, & uterq;</s> <s xml:id="echoid-s41862" xml:space="preserve"> rectus:</s> <s xml:id="echoid-s41863" xml:space="preserve"> angulus <lb/>itaq;</s> <s xml:id="echoid-s41864" xml:space="preserve"> p f d eſt acutus per 32 p 1.</s> <s xml:id="echoid-s41865" xml:space="preserve"> Ducatur ergo à puncto ſ li-<lb/>nea perpendicularis ſuper lineam d t per 11 p 1, quæ produ <lb/>cta ad circunferentiam minoris circuli ſit linea f k.</s> <s xml:id="echoid-s41866" xml:space="preserve"> Hæc <lb/>itaq;</s> <s xml:id="echoid-s41867" xml:space="preserve"> ſecabit lineam l n:</s> <s xml:id="echoid-s41868" xml:space="preserve"> uel non ſecabit.</s> <s xml:id="echoid-s41869" xml:space="preserve"> Si non ſecet:</s> <s xml:id="echoid-s41870" xml:space="preserve"> erit <lb/>quilibet punctus lineę l n propinquior puncto n, quàm <lb/>punctus k.</s> <s xml:id="echoid-s41871" xml:space="preserve"> Si ſecet:</s> <s xml:id="echoid-s41872" xml:space="preserve"> palàm itaq;</s> <s xml:id="echoid-s41873" xml:space="preserve"> quoniam aliquis punctus <lb/>lineæ l n erit inſerior puncto k, plus approximans ad pun <lb/>ctum n quàm punctũ k:</s> <s xml:id="echoid-s41874" xml:space="preserve"> ſit ille punctus z:</s> <s xml:id="echoid-s41875" xml:space="preserve"> & ducatur linea <lb/>t z:</s> <s xml:id="echoid-s41876" xml:space="preserve"> quæ producatur uſq;</s> <s xml:id="echoid-s41877" xml:space="preserve"> ad circunferentiam circuli mino <lb/>ris, cadatq́;</s> <s xml:id="echoid-s41878" xml:space="preserve"> in punctum o.</s> <s xml:id="echoid-s41879" xml:space="preserve"> Arcus itaq;</s> <s xml:id="echoid-s41880" xml:space="preserve"> n o aut eſt minor ar <lb/>cu t l:</s> <s xml:id="echoid-s41881" xml:space="preserve"> aut non.</s> <s xml:id="echoid-s41882" xml:space="preserve"> si non fuerit minor, abſcindatur ex eo ar-<lb/>cus minor arcul t, & ducatur ad terminum illius arcus li-<lb/>nea à puncto t, & erit idem, ſicuti ſi arcus n o ſit min or ar-<lb/>cu l t.</s> <s xml:id="echoid-s41883" xml:space="preserve"> Sit ergo arcus n o minor quàm ſit arcus t l:</s> <s xml:id="echoid-s41884" xml:space="preserve"> ergo per 33 p 6 angulus t n l eſt maior angulo o t n.</s> <s xml:id="echoid-s41885" xml:space="preserve"> <lb/>Secetur ergo ex angulo t n l angulus æqualis angulo o t n, qui ſit i n z:</s> <s xml:id="echoid-s41886" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s41887" xml:space="preserve"> punctum i in lineam <lb/>t z per 29 th.</s> <s xml:id="echoid-s41888" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s41889" xml:space="preserve"> & ſuper punctum t lineæ m t per 23 p 1 fiat angulus æqualis angulo o t n, qui ſit <lb/>angulus q t m.</s> <s xml:id="echoid-s41890" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s41891" xml:space="preserve"> angulus t m l ſit maior angulo m t q:</s> <s xml:id="echoid-s41892" xml:space="preserve"> quia arcus t l eſt maior arcu n o, ut pa-<lb/>tet ex præmiſsis:</s> <s xml:id="echoid-s41893" xml:space="preserve"> arcus uerò n o determinat quantitatem anguli m t q, qui eſt æqualis angulo o t n:</s> <s xml:id="echoid-s41894" xml:space="preserve"> <lb/>palàm ergo per 14 th.</s> <s xml:id="echoid-s41895" xml:space="preserve"> 1 huius quoniá concurret linea t q cum linea l m:</s> <s xml:id="echoid-s41896" xml:space="preserve"> ſit itaq;</s> <s xml:id="echoid-s41897" xml:space="preserve"> concurſus in puncto <lb/> <pb o="337" file="0639" n="639" rhead="LIBER OCTAVVS."/> q.</s> <s xml:id="echoid-s41898" xml:space="preserve"> Cum igitur angulus l m t ſit ęqualis duobus angulis m q t & m t q per 32 p 1, & angulus l n t ſit æ-<lb/>qualis angulo l m t per 27 p 3, ſunt enim conſtituti ſuper eundem arcum:</s> <s xml:id="echoid-s41899" xml:space="preserve"> qui eſt l t:</s> <s xml:id="echoid-s41900" xml:space="preserve"> & cum angulus <lb/>i n z ex præmiſsis ſit ęqualis angulo m t q:</s> <s xml:id="echoid-s41901" xml:space="preserve"> erit angulus i n t, ęqualis angulo m q t:</s> <s xml:id="echoid-s41902" xml:space="preserve"> eſt ergo per 32 p 1 <lb/>triangulus m t q æquiangulus triangulo i n t, cum angulus o t n ſit ęqualis angulo m t q:</s> <s xml:id="echoid-s41903" xml:space="preserve"> & ſimiliter <lb/>triangulus i n z eſt per 32 p 1 æquiangulus triangulo t n z, cum angulus t z n amb obus illis triangu-<lb/>lis ſit cõmunis, & angulus i n t ſit ęqualis angulo o t n.</s> <s xml:id="echoid-s41904" xml:space="preserve"> Eſt ergo per 4 p 6 proportio lineę n t ad lineã <lb/>t q, ſicut lineę n i ad lineá m q:</s> <s xml:id="echoid-s41905" xml:space="preserve"> & ſimiliter eſt proportio lineę t n ad lineá t z, ſicut lineę n i ad lineá n z:</s> <s xml:id="echoid-s41906" xml:space="preserve"> <lb/>ſed linea t z eſt maior quàm linea t q:</s> <s xml:id="echoid-s41907" xml:space="preserve"> quod patet per hoc.</s> <s xml:id="echoid-s41908" xml:space="preserve"> Sit enim r punctus, in quo linea t z ſecat li-<lb/>neam k f:</s> <s xml:id="echoid-s41909" xml:space="preserve"> angulus itaque t f r eſt rectus, cum linea f k ſit perpen dicularis ſuper lineam t d:</s> <s xml:id="echoid-s41910" xml:space="preserve"> ergo per 32 <lb/>p 1 angulus f t r eſt acutus.</s> <s xml:id="echoid-s41911" xml:space="preserve"> Quia uerò linea d m, ut patet ex præmiſsis, eſt ęqualis lineæ d n:</s> <s xml:id="echoid-s41912" xml:space="preserve"> erit per <lb/>28 p 3 arcus d m æqualis arcuid n:</s> <s xml:id="echoid-s41913" xml:space="preserve"> ergo per 27 p 3 angulus m t d eſt ęqualis angulo d t n:</s> <s xml:id="echoid-s41914" xml:space="preserve"> ſed angulus <lb/>q t m eſt æqualis angulo o t n ex præmiſsis:</s> <s xml:id="echoid-s41915" xml:space="preserve"> fit ergo angulus q t f æqualis angulo f t r:</s> <s xml:id="echoid-s41916" xml:space="preserve"> quia ex æquali <lb/>bus angulis conſtat:</s> <s xml:id="echoid-s41917" xml:space="preserve"> angulus ergo q t f eſt acutus, & linea k f eſt perpendicularis ſuper lineam t d:</s> <s xml:id="echoid-s41918" xml:space="preserve"> an <lb/>gulus quo quet f k eſt rectus:</s> <s xml:id="echoid-s41919" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s41920" xml:space="preserve"> 1 huius linea k f producta concurret cum linea t q:</s> <s xml:id="echoid-s41921" xml:space="preserve"> ſit pũ <lb/>ctum concurſus s:</s> <s xml:id="echoid-s41922" xml:space="preserve"> & linea producta à puncto t uſque ad punctum s, quod eſt punctum concurſus, <lb/>cuius pars eſt linea t q, eſt ęqualis lineæ t r.</s> <s xml:id="echoid-s41923" xml:space="preserve"> Quoniam enim illorum trigonorum anguli ad punctum <lb/>f fiunt recti, & ad punctum t ex præmiſsis ſunt ęquales:</s> <s xml:id="echoid-s41924" xml:space="preserve"> patet per 32 p 1 quoniam illi trigoni t f s & t f r <lb/>ſunt æquian guli, & linea t f communis:</s> <s xml:id="echoid-s41925" xml:space="preserve"> reliqua ergo latera, quæ ſunt t s & t r ſunt æqualia per 26 p 1:</s> <s xml:id="echoid-s41926" xml:space="preserve"> <lb/>ſed linea t s eſt maior quàm linea t q, & linea t z eſt maior quã linea t r:</s> <s xml:id="echoid-s41927" xml:space="preserve"> linea ergo t q eſt minor quàm <lb/>linea t z.</s> <s xml:id="echoid-s41928" xml:space="preserve"> Eſt ergo per 8 p 5 maior proportio lineæ n t ad lineam t q, quàm lineæ n t ad lineam t z:</s> <s xml:id="echoid-s41929" xml:space="preserve"> igi-<lb/>tur maior eſt proportio lineæ i n ad m q, quàm lineę i n ad lineã n z:</s> <s xml:id="echoid-s41930" xml:space="preserve"> quare per 10 p 5 linea m q eſt mi-<lb/>nor quàm linea n z.</s> <s xml:id="echoid-s41931" xml:space="preserve"> Secetur ergo ex linea n z linea æqualis lineę m q, quę ſit n x:</s> <s xml:id="echoid-s41932" xml:space="preserve"> & ducatur linea d x.</s> <s xml:id="echoid-s41933" xml:space="preserve"> <lb/>Et quoniam per 22 p 3 angulus l n d cum angulo l m d ualet duos rectos, & angulus q m d cum angu <lb/>lo l m d per 13 p 1 ualet duos rectos:</s> <s xml:id="echoid-s41934" xml:space="preserve"> erit angulus l n d ęqualis angulo q m d:</s> <s xml:id="echoid-s41935" xml:space="preserve"> ergo per 4 p 1 triangulus <lb/>x n d eſt ęqualis triangulo d m q:</s> <s xml:id="echoid-s41936" xml:space="preserve"> & linea d x ęqualis lineæ d q:</s> <s xml:id="echoid-s41937" xml:space="preserve"> & angulus x d n eſt ęqualis angulo q <lb/>d m:</s> <s xml:id="echoid-s41938" xml:space="preserve"> & angulus d x n æqualis angulo d q m.</s> <s xml:id="echoid-s41939" xml:space="preserve"> Sed angulus d x z eſt maior recto, cum ſit maior angulo <lb/>d n x per 16 p 1:</s> <s xml:id="echoid-s41940" xml:space="preserve"> & angulus d n x eſt maior recto per 31 p 3:</s> <s xml:id="echoid-s41941" xml:space="preserve"> quoniam cadit in portionem minorem ſe-<lb/>micirculo, quę eſt d n l:</s> <s xml:id="echoid-s41942" xml:space="preserve"> & etiam patet hoc per 22 p 3.</s> <s xml:id="echoid-s41943" xml:space="preserve"> Quoniam enim angulus l m d eſt acutus:</s> <s xml:id="echoid-s41944" xml:space="preserve"> quia <lb/>angulus e m d eſt acutus ex præmiſsis:</s> <s xml:id="echoid-s41945" xml:space="preserve"> patet quod angulus d n l eſt obtuſus:</s> <s xml:id="echoid-s41946" xml:space="preserve"> ergo per 19 p 1 linea d z <lb/>eſt maior quàm linea d x:</s> <s xml:id="echoid-s41947" xml:space="preserve"> ergo linea z d eſt maior quàm linea q d.</s> <s xml:id="echoid-s41948" xml:space="preserve"> Forma ergo pũcti q poteſt reflecti <lb/>ad punctum z à duobus punctis ſpeculi, quę ſunt t & l:</s> <s xml:id="echoid-s41949" xml:space="preserve"> & puncta q & z ſunt inęqualis diſtantię à cen <lb/>tro & in diuerſis diametris:</s> <s xml:id="echoid-s41950" xml:space="preserve"> quod patetideo quòd angulus x d n eſt ęqualis angulo q d m:</s> <s xml:id="echoid-s41951" xml:space="preserve"> addito er <lb/>go cõmuniangulo x d m, erit angulus n d m æqualis angulo q d x:</s> <s xml:id="echoid-s41952" xml:space="preserve"> ſed angulus n d m eſt minor duo-<lb/>bus rectis:</s> <s xml:id="echoid-s41953" xml:space="preserve"> ergo & angulus q d x:</s> <s xml:id="echoid-s41954" xml:space="preserve"> ergo magis angulus q d z eſt minor duobus rectis.</s> <s xml:id="echoid-s41955" xml:space="preserve"> Ergo duo pun-<lb/>cta q & z non ſunt in eadem diametro, ſed in diuerſis.</s> <s xml:id="echoid-s41956" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s41957" xml:space="preserve"/> </p> <div xml:id="echoid-div1659" type="float" level="0" n="0"> <figure xlink:label="fig-0638-01" xlink:href="fig-0638-01a"> <variables xml:id="echoid-variables739" xml:space="preserve">k e l t r o z i g x b n p f m q d s n a</variables> </figure> </div> </div> <div xml:id="echoid-div1661" type="section" level="0" n="0"> <head xml:id="echoid-head1246" xml:space="preserve" style="it">36. A ſpeculis ſphæricis concauis duobus punctis inæqualiter diſtantib. à centro & in diuer <lb/>ſis diametris exiſtentibus, ad ſe inuicem reflexis à duobus punctis arcus interiacentis illas ſemi-<lb/>diametros, in quibus illa puncta conſiſtunt: impoßibile eſt ipſa à puncto alio illius arcus ad ſe in-<lb/>uicem reflecti. Alhazen 82. n t.</head> <p> <s xml:id="echoid-s41958" xml:space="preserve">Sit circulus ſpeculi ſphærici concaui a g b, cuius centrum ſit d:</s> <s xml:id="echoid-s41959" xml:space="preserve"> & ſint duo pũcta k & o ad ſe inui-<lb/>cem reflexa à duobus punctis arcus b g:</s> <s xml:id="echoid-s41960" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s41961" xml:space="preserve"> punctum k remotius à cẽtro ſpeculi, quod eſt d, quàm <lb/>punctus o:</s> <s xml:id="echoid-s41962" xml:space="preserve"> & ſint lineæ g d a & b d m duę diametri, in quibus ſunt puncta illa k & o:</s> <s xml:id="echoid-s41963" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s41964" xml:space="preserve"> punctum k <lb/>in ſemidiametro g d, & punctus o in ſemιdiametro b d:</s> <s xml:id="echoid-s41965" xml:space="preserve"> reflectanturq́ue formæ iſtorum punctorum <lb/>ad inuicem à duobus punctis arcus g b, ut oſtenditur per præcedentem:</s> <s xml:id="echoid-s41966" xml:space="preserve"> & ſit t unus punctus arcus <lb/>b g, à quo fit reflexio.</s> <s xml:id="echoid-s41967" xml:space="preserve"> Palàm ergo ex 34 huius quòd non uterque duorum angulorum conſtantium <lb/>exangulis incidentię & reflexionis erit minor angulo o d a:</s> <s xml:id="echoid-s41968" xml:space="preserve"> neque eſt aliquis illorum angulorum ę-<lb/>qualis angulo o d a, ut patet per ea, quæ declarata ſunt in 33 huius:</s> <s xml:id="echoid-s41969" xml:space="preserve"> alter ergo illorum erit maior an-<lb/>gulo o d a:</s> <s xml:id="echoid-s41970" xml:space="preserve"> ſit itaque angulus, qui eſt ſuper punctum t, maior angulo o d a:</s> <s xml:id="echoid-s41971" xml:space="preserve"> & ducantur lineę o t, d t, k <lb/>t:</s> <s xml:id="echoid-s41972" xml:space="preserve"> & ex angulo o t k ſecetur per 27 th.</s> <s xml:id="echoid-s41973" xml:space="preserve"> 1 huius angulus æqualis angulo o d a, qui ſit o t f, ducta linea t f <lb/>ſuper diametrum g d:</s> <s xml:id="echoid-s41974" xml:space="preserve"> & diuidatur angulus f t k per æqualia per 9 p 1, ducta linea t e ſuper lineam <lb/>k f:</s> <s xml:id="echoid-s41975" xml:space="preserve"> & à puncto k ducatur linea ęquidiſtans lineæ t f per 31 p 1:</s> <s xml:id="echoid-s41976" xml:space="preserve"> quæ ſit k z.</s> <s xml:id="echoid-s41977" xml:space="preserve"> Et quoniam linea t f ęqui-<lb/>diſtans lineę k z, concurrit cum linea t e in puncto t:</s> <s xml:id="echoid-s41978" xml:space="preserve"> patet quòd linea k z concurret cum linea t e ꝓ-<lb/>ducta per 2 th.</s> <s xml:id="echoid-s41979" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s41980" xml:space="preserve"> ſit ergo linea k z concurrens cum linea t e in puncto z:</s> <s xml:id="echoid-s41981" xml:space="preserve"> & ducatur linea o k:</s> <s xml:id="echoid-s41982" xml:space="preserve"> & <lb/>per 9 p 1 diuidatur angulus o d k per æqualia per lineam d u ſecantem lineam k o in puncto p.</s> <s xml:id="echoid-s41983" xml:space="preserve"> Cum <lb/>ergo ſit linea k d maior quàm linea o d, ut patet ex hypotheſi:</s> <s xml:id="echoid-s41984" xml:space="preserve"> & quia per 3 p 6 eſt proportio lineæ <lb/>k d ad lineam d o, ſicut lineæ k p ad lineam p o:</s> <s xml:id="echoid-s41985" xml:space="preserve"> erit linea k p maior quàm linea p o.</s> <s xml:id="echoid-s41986" xml:space="preserve"> Item ſit, ut linea <lb/>d t ſecet lineam k o in puncto n:</s> <s xml:id="echoid-s41987" xml:space="preserve"> palàm quòd linea d p u cadet inter duo puncta k & n:</s> <s xml:id="echoid-s41988" xml:space="preserve"> non autem <lb/>inter duo puncta n & o.</s> <s xml:id="echoid-s41989" xml:space="preserve"> Quia enim angulus k p d ualet duos angulos p o d & p d o, & angulus o p d <lb/>ualet duos angulos p d k & p k d per 32 p 1:</s> <s xml:id="echoid-s41990" xml:space="preserve"> ſed angulus p d o eſt æqualis angulo p d k:</s> <s xml:id="echoid-s41991" xml:space="preserve"> & angulus k o <lb/>d maior eſt angulo o k d per 19 p 1:</s> <s xml:id="echoid-s41992" xml:space="preserve"> ergo angulus k p d maior eſt angulo o p d:</s> <s xml:id="echoid-s41993" xml:space="preserve"> eſt ergo angulus k p d <lb/>maior recto per 13 p 1:</s> <s xml:id="echoid-s41994" xml:space="preserve"> & angulus o p d eſt acutus.</s> <s xml:id="echoid-s41995" xml:space="preserve"> Sed & angulus k n d eſt acutus:</s> <s xml:id="echoid-s41996" xml:space="preserve"> quod patet:</s> <s xml:id="echoid-s41997" xml:space="preserve"> ſi fiat <lb/>circulus tranſiens per tria puncta o, t, k per 5 p 4:</s> <s xml:id="echoid-s41998" xml:space="preserve"> hic enim tranſibit infra punctum d, quod eſt cen-<lb/> <pb o="338" file="0640" n="640" rhead="VITELLONIS OPTICAE"/> trum circuli maioris.</s> <s xml:id="echoid-s41999" xml:space="preserve"> Quoniam cum angulus o t k ſit maior angulo o d a ex hypotheſi:</s> <s xml:id="echoid-s42000" xml:space="preserve"> erunt duo an <lb/>guli o d k & o t k maiores duobus rectis:</s> <s xml:id="echoid-s42001" xml:space="preserve"> quod eſſet <lb/> <anchor type="figure" xlink:label="fig-0640-01a" xlink:href="fig-0640-01"/> impoſsibile per 22 p 3, ſi circulus ille tranſiret punctũ <lb/>d:</s> <s xml:id="echoid-s42002" xml:space="preserve"> uel ſupra punctum d:</s> <s xml:id="echoid-s42003" xml:space="preserve"> quoniam eadem eſt demon-<lb/>ſtratio.</s> <s xml:id="echoid-s42004" xml:space="preserve"> Linea uerò n d diuidet k o arcum illius circu-<lb/>li per æqualia per 26 p 3:</s> <s xml:id="echoid-s42005" xml:space="preserve"> quoniam diuidit angulum o <lb/>t k per æqualia ex hypotheſi:</s> <s xml:id="echoid-s42006" xml:space="preserve"> fiet autem illa diuiſio ar <lb/>cus k o infra punctum d.</s> <s xml:id="echoid-s42007" xml:space="preserve"> Si uerò ab illo puncto diui-<lb/>ſionis arcus o k ducatur linea ad medium punctũ li-<lb/>neę o k (quę eſt chorda illius arcus o k) erit linea illa <lb/>perpendicularis ſuper lineam o k per 8 p 1, & cadet il-<lb/>la perpendicularis inter puncta p & k, cũ linea k p ſit <lb/>maior quàm linea p o ex præmiſsis:</s> <s xml:id="echoid-s42008" xml:space="preserve"> & angulus ſuper <lb/>punctum n ex parte illius perpẽdicularis erit acutus:</s> <s xml:id="echoid-s42009" xml:space="preserve"> <lb/>ergo & ex parte p erit acutus:</s> <s xml:id="echoid-s42010" xml:space="preserve"> & angulus ſuper pũctũ <lb/>p ex parte o erit acutus:</s> <s xml:id="echoid-s42011" xml:space="preserve"> hoc enim oſtenſum eſt ſupe-<lb/>rius.</s> <s xml:id="echoid-s42012" xml:space="preserve"> Si ergo detur quòd punctus p cadat inter duo <lb/>puncta n & o:</s> <s xml:id="echoid-s42013" xml:space="preserve"> impoſsibile erit perpẽdicularem illam <lb/>cadere inter puncta n & p:</s> <s xml:id="echoid-s42014" xml:space="preserve"> quia tunc ſecaret lineam <lb/>d p:</s> <s xml:id="echoid-s42015" xml:space="preserve"> & fieret triangulus:</s> <s xml:id="echoid-s42016" xml:space="preserve"> cuius unus angulus eſſet rectus, & alius obtuſus:</s> <s xml:id="echoid-s42017" xml:space="preserve"> quod cum ſit impoſsibile, <lb/>neceſſe eſt angulum k n d eſſe acutum:</s> <s xml:id="echoid-s42018" xml:space="preserve"> ergo per 13 p 1 angulus o n d eſt obtuſus:</s> <s xml:id="echoid-s42019" xml:space="preserve"> punctum ergo p nõ <lb/>cadet inter puncta n & o.</s> <s xml:id="echoid-s42020" xml:space="preserve"> Quoniam cum angulus o n d ſit obtuſus, & ut patet ex pręmiſsis, angulus <lb/>d p k eſt obtuſus:</s> <s xml:id="echoid-s42021" xml:space="preserve"> ſequeretur ergo in trigono d n p duos eſſe angulos obtuſos:</s> <s xml:id="echoid-s42022" xml:space="preserve"> quod cum ſit impoſsi <lb/>bile per 32 p 1:</s> <s xml:id="echoid-s42023" xml:space="preserve"> palàm quia punctus p non cadet inter puncta n & o:</s> <s xml:id="echoid-s42024" xml:space="preserve"> nea cadit etiam in punctum n, ut <lb/>eſt euidẽs.</s> <s xml:id="echoid-s42025" xml:space="preserve"> Cadet ergo inter puncta k & n.</s> <s xml:id="echoid-s42026" xml:space="preserve"> Quia ergo, ut patet ex præmiſsis, angulus k t d eſt medie-<lb/>tas anguli k t o:</s> <s xml:id="echoid-s42027" xml:space="preserve"> ſed & angulus k t e eſt medietas anguli k t f:</s> <s xml:id="echoid-s42028" xml:space="preserve"> angulus uerò k t o maior eſt angulo f t o, <lb/>in angulo k t f:</s> <s xml:id="echoid-s42029" xml:space="preserve"> reſtat ergo ut angulus e t d ſit medietas anguli f t o:</s> <s xml:id="echoid-s42030" xml:space="preserve"> ſed angulus f t o eſt æqualis angu <lb/>lo o d a:</s> <s xml:id="echoid-s42031" xml:space="preserve"> igitur angulus e t d eſt medietas anguli o d a:</s> <s xml:id="echoid-s42032" xml:space="preserve"> ſed angulus o d a cum angulo o d f ualet duos <lb/>rectos per 13 p 1, & tres anguli trianguli e t d ualent duos rectos per 32 p 1:</s> <s xml:id="echoid-s42033" xml:space="preserve"> tres ergo anguli trigoni e t <lb/>d ſunt æqualis duobus angulis o d a & o d f:</s> <s xml:id="echoid-s42034" xml:space="preserve"> ablato ergo angulo e d t hinc inde illis angulis comma <lb/>ni, & ablato angulo e t d, qui eſt medietas anguli o d a:</s> <s xml:id="echoid-s42035" xml:space="preserve"> reſtat ut angulus t e d æqualis ſit medietati <lb/>anguli o d a, & toti angulo o d n:</s> <s xml:id="echoid-s42036" xml:space="preserve"> ſed angulus o d p, qui eſt medietas anguli o d k cum medietate an-<lb/>guli o d a eſt rectus:</s> <s xml:id="echoid-s42037" xml:space="preserve"> eſt autem angulus o d p maior angulo o d n, quod patet per 29 th.</s> <s xml:id="echoid-s42038" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s42039" xml:space="preserve"> cum, <lb/>ficut patet ex præmiſsis, punctum n lineæ d n cadat inter puncta p & o:</s> <s xml:id="echoid-s42040" xml:space="preserve"> eſt ergo angulus o d p cum <lb/>medietate anguli o d a maior angulo t e d cum medietate anguli o d a.</s> <s xml:id="echoid-s42041" xml:space="preserve"> Patet ergo cum angulus o d k <lb/>cum medietate anguli o d a ſit rectus, quoniam angulus t e d eſt acutus:</s> <s xml:id="echoid-s42042" xml:space="preserve"> quare per 15 p 1 ei contrà po <lb/>ſitus, qui eſt angulus k e z, eſt acutus.</s> <s xml:id="echoid-s42043" xml:space="preserve"> Igitur ſi per 12 p 1 à puncto k ducatur perpendicularis ſuper li-<lb/>neam t z, illa cadetinter puncta e & z:</s> <s xml:id="echoid-s42044" xml:space="preserve"> quia, ut patet ex præmiſsis, linea k e non eſt perpen dicularis <lb/>ſuper lineam t e z.</s> <s xml:id="echoid-s42045" xml:space="preserve"> Si uerò dicatur quòd illa perpendicularis cadat ultra pũctum e ſuper lineam t e:</s> <s xml:id="echoid-s42046" xml:space="preserve"> <lb/>tunc cum angulus t e k per pręmiſſa & 13 p 1 ſit obtuſus:</s> <s xml:id="echoid-s42047" xml:space="preserve"> accidet triãgulum habere duos angulos u-<lb/>num rectum & alium obtuſum:</s> <s xml:id="echoid-s42048" xml:space="preserve"> quod eſt impoſsibile per 32 p 1.</s> <s xml:id="echoid-s42049" xml:space="preserve"> Cadet itaque perpendicularis illa <lb/>inter puncta e & z:</s> <s xml:id="echoid-s42050" xml:space="preserve"> quæ ſit linea k q.</s> <s xml:id="echoid-s42051" xml:space="preserve"> Hoc autem ſeruato, nunc quidem neceſſarium interponimus:</s> <s xml:id="echoid-s42052" xml:space="preserve"> <lb/>ſcilicet quòd linea k t ſe habet ad lineam t f, ſicut linea k d ad lineam d o.</s> <s xml:id="echoid-s42053" xml:space="preserve"> Eſt enim linea t o aut æqui-<lb/>diſtans lineę k d, aut concurrens cum illa.</s> <s xml:id="echoid-s42054" xml:space="preserve"> Sit primũ ęquidiſtans:</s> <s xml:id="echoid-s42055" xml:space="preserve"> erit ergo per 29 p 1, angulus o d a ę-<lb/>qualis angulo t o d:</s> <s xml:id="echoid-s42056" xml:space="preserve"> eſt ergo angulus t o d ęqualis angulo o t f, quoniam, ut patet ex pręmiſsis, angu <lb/> <anchor type="figure" xlink:label="fig-0640-02a" xlink:href="fig-0640-02"/> <anchor type="figure" xlink:label="fig-0640-03a" xlink:href="fig-0640-03"/> li o t f & o d a ſunt æquales.</s> <s xml:id="echoid-s42057" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s42058" xml:space="preserve"> lineæ o d & t faut ęquidiſtabunt, aut concurrent.</s> <s xml:id="echoid-s42059" xml:space="preserve"> Si æ-<lb/>quidiſtent, cum illæ cadant inter lineas k d & t o æquidiſtantes:</s> <s xml:id="echoid-s42060" xml:space="preserve"> palàm per 34 p 1 quoniam ipfę erũt <lb/> <pb o="339" file="0641" n="641" rhead="LIBER OCTAVVS."/> æquales.</s> <s xml:id="echoid-s42061" xml:space="preserve"> Si uerò lineæ o d & t f concurrunt, facient triangulum, cuius duo latera erunt æqualia per <lb/>6 p 1:</s> <s xml:id="echoid-s42062" xml:space="preserve"> quoniam duo eius anguli, qui ſunt f t o & d o t, ſunt ęquales.</s> <s xml:id="echoid-s42063" xml:space="preserve"> Linea uerò f d ſecat illa duo latera <lb/>æqualia æquidiſtanter baſi d o:</s> <s xml:id="echoid-s42064" xml:space="preserve"> erit ergo per 2 p 6 & 18 p 5 proportio unius illorum laterum ad lineá <lb/>d o, ſicut alterius ad lineam f t:</s> <s xml:id="echoid-s42065" xml:space="preserve"> eſt ergo linea t f ęqualis lineæ o d per 9 p 5.</s> <s xml:id="echoid-s42066" xml:space="preserve"> Fit autem hęc deductio <lb/>cum lineę illæ concurrunt ſub linea k d.</s> <s xml:id="echoid-s42067" xml:space="preserve"> Quòd ſi cõcurrãt ſub linea t o, erit eadẽ probatio:</s> <s xml:id="echoid-s42068" xml:space="preserve"> quia fiet <lb/>triãgulus, cuius unũ latus eſt linea t o, & alia duo latera æqualia ք 6 p 1, ut prius:</s> <s xml:id="echoid-s42069" xml:space="preserve"> ꝗ a linea t o eſt ęqui <lb/>diſtãs lineæ d f, erit per 2 p 6 & 18 p 5 proportio unius illorũ duorũ laterũ ad lineã t o, ſicut alterius <lb/>ad lineã t f, eruntq́;</s> <s xml:id="echoid-s42070" xml:space="preserve"> ut prius, per 9 p 5, lineę t f & d o ęquales.</s> <s xml:id="echoid-s42071" xml:space="preserve"> Itẽ patet quòd angulus t d k eſt æqualis <lb/>angulo d t o per 29 p 1:</s> <s xml:id="echoid-s42072" xml:space="preserve"> ideo quòd linea t o data eſt æquidiſtans eſſe lineæ k d:</s> <s xml:id="echoid-s42073" xml:space="preserve"> ergo angulus t d f eſt <lb/>æqualis angulo d t k, cum anguli d t o & d t k ſint æquales ex hypotheſi & per 20 th.</s> <s xml:id="echoid-s42074" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s42075" xml:space="preserve"> ergo per <lb/>6 p 1 lineę d k & t k ſunt æquales:</s> <s xml:id="echoid-s42076" xml:space="preserve"> eſt ergo per 7 p 5 proportio lineę t k ad lineam t f, ſicut lineæ k d ad <lb/>lineam d o:</s> <s xml:id="echoid-s42077" xml:space="preserve"> ideo quòd antecedentia & conſequentia ſunt hinc & inde æqualia.</s> <s xml:id="echoid-s42078" xml:space="preserve"> Si uerò linea t o non <lb/>ęquidiſtat, ſed cõcurrit cũ linea k d:</s> <s xml:id="echoid-s42079" xml:space="preserve"> aut hoc eſt ad par <lb/> <anchor type="figure" xlink:label="fig-0641-01a" xlink:href="fig-0641-01"/> tẽ pũcti a:</s> <s xml:id="echoid-s42080" xml:space="preserve"> aut ad partẽ pũcti g diametri a g.</s> <s xml:id="echoid-s42081" xml:space="preserve"> Si fiat cõ <lb/>curſus ex parte a:</s> <s xml:id="echoid-s42082" xml:space="preserve"> ſit hoc in pũcto l.</s> <s xml:id="echoid-s42083" xml:space="preserve"> Manifeſtũ ergo ք <lb/>13 th.</s> <s xml:id="echoid-s42084" xml:space="preserve"> 1 huius quoniã ꝓportio lineę t k ad lineã t f cõpo <lb/>nitur ex proportione lineę t k ad lineã t l, & ex pro-<lb/>portione lineę t l ad lineam t f:</s> <s xml:id="echoid-s42085" xml:space="preserve"> ſed proportio lineę k t <lb/>ad lineam t l eſt, ſicut proportio lineę k d ad lineá d l <lb/>per 3 p 6:</s> <s xml:id="echoid-s42086" xml:space="preserve"> linea enim d t diuidit angulum k t o per æ-<lb/>qualia ex hypotheſi.</s> <s xml:id="echoid-s42087" xml:space="preserve"> Quia uerò angulus o d l per prę-<lb/>miſſa eſt æqualis angulo l t f, & angulus ad punctum l <lb/>communis eſt am bobus trigonis t l f & o d l:</s> <s xml:id="echoid-s42088" xml:space="preserve"> patet ք <lb/>32 p 1 quòd tertius angulus eſt tertio ęqualis:</s> <s xml:id="echoid-s42089" xml:space="preserve"> erit er-<lb/>go per 4 p 6 proportio lineę t l ad lineá t f, ſicut lineæ <lb/>d l ad lineam d o.</s> <s xml:id="echoid-s42090" xml:space="preserve"> Proportio itaque lineę t k ad lineá <lb/><gap/>f conſtat ex proportione lineę k d ad lineam d l, & li <lb/>neę d l ad lineam d o:</s> <s xml:id="echoid-s42091" xml:space="preserve"> ſed proportio lineę k d ad lineá <lb/>d o cóſtat ex eiſdem proportionibus, poſita linea d l <lb/>media per 13 th.</s> <s xml:id="echoid-s42092" xml:space="preserve"> 1 hurus:</s> <s xml:id="echoid-s42093" xml:space="preserve"> ergo proportio lineę k t ad li-<lb/>neam t f eſt, ſicut proportio lineę k d ad lineam d o.</s> <s xml:id="echoid-s42094" xml:space="preserve"> Siautem linea t o cócurrat cum linea k d ex par <lb/>te g:</s> <s xml:id="echoid-s42095" xml:space="preserve"> ſit concurſus in puncto s:</s> <s xml:id="echoid-s42096" xml:space="preserve"> & à puncto d ducatur linea ęquidiſtans lineę k t, quæ ſit d r, concur-<lb/>rens cum linea t o producta ultra punctum o in pũcto r:</s> <s xml:id="echoid-s42097" xml:space="preserve"> <lb/>igitur angulus k t d æqualis eſt angulo t d r per 29 p 1:</s> <s xml:id="echoid-s42098" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0641-02a" xlink:href="fig-0641-02"/> ſed & angulus k t d ex hypotheſi æqualis eſt angulo d <lb/>t o:</s> <s xml:id="echoid-s42099" xml:space="preserve"> ergo anguli d t r & d r t ſunt æquales:</s> <s xml:id="echoid-s42100" xml:space="preserve"> ergo per 6 p <lb/>1 linea d r eſt æqualis lineæ t r.</s> <s xml:id="echoid-s42101" xml:space="preserve"> Sed quoniam triangu-<lb/>lus s t k æquiangulus eſt triangulo s r d per 29 p 1, & <lb/>ꝓpter angulũ a s d cõmunẽ:</s> <s xml:id="echoid-s42102" xml:space="preserve"> erit ergo per 4 p 6 propor-<lb/>tio lineæ d r ad lineã s r, ſicut lineę k t ad lineã t s:</s> <s xml:id="echoid-s42103" xml:space="preserve"> ſed li-<lb/>nea d r eſt ęqualis lineę r t:</s> <s xml:id="echoid-s42104" xml:space="preserve"> eſt ergo ք 7 p 5 proportio li-<lb/>neę r t ad lineã r s, ſicut lineæ k t ad lineã t s:</s> <s xml:id="echoid-s42105" xml:space="preserve"> ſed propor-<lb/>tio lineę r t ad lineam r s eſt, ſicut proportio lineæ d k <lb/>ad lineam d s per 2 p 6 & per 18 p 5:</s> <s xml:id="echoid-s42106" xml:space="preserve"> igitur per 11 p 5 eſt <lb/>proportio lineæ k t ad lineam t s, ſicut lineæ k d ad d s.</s> <s xml:id="echoid-s42107" xml:space="preserve"> <lb/>Sed quoniam angulus f t o æqualis eſt angulo o d a:</s> <s xml:id="echoid-s42108" xml:space="preserve"> e-<lb/>rit angulus o d s æqualis angulo f t s per 13 p 1, & angu-<lb/>lus ad punctum s eſt communis:</s> <s xml:id="echoid-s42109" xml:space="preserve"> erit ergo triangulus o <lb/>d s æquiangulus triangulo f t s per 32 p 1:</s> <s xml:id="echoid-s42110" xml:space="preserve"> ergo per 4 p 6 <lb/>eſt proportio lineæ t s ad lineam t f, ſicut lineæ d s ad li-<lb/>neam d o:</s> <s xml:id="echoid-s42111" xml:space="preserve"> eſt autem proportio lineæ k t ad lineam t s, <lb/>ficut lineæ k d ad lineam d s:</s> <s xml:id="echoid-s42112" xml:space="preserve"> ergo per 22 p 5 erit pro-<lb/>portio lineæ k t ad lineam t f, ſicut lineę k d ad lineam <lb/>d o.</s> <s xml:id="echoid-s42113" xml:space="preserve"> Quia uerò linea k z æquidiſtat lineæ t f, ut patet ex <lb/>præmiſsis, erit per 29 p 1 angulus k z e æqualis angulo <lb/>e t f:</s> <s xml:id="echoid-s42114" xml:space="preserve"> ſed angulus k e z eſt æqualis angulo t e f per 15 p 1:</s> <s xml:id="echoid-s42115" xml:space="preserve"> <lb/>ergo trigoni k z e & e t f ſunt æquianguli per 32 p 1:</s> <s xml:id="echoid-s42116" xml:space="preserve"> <lb/>ergo per 4 p 6 erit proportio lineæ k e ad lineam e f, <lb/>ſicut lineæ k z ad lineam t f:</s> <s xml:id="echoid-s42117" xml:space="preserve"> ſed proportio lineę k e ad li <lb/>neam e f eſt, ſicut lineę k t ad lineam t f per 3 p 6:</s> <s xml:id="echoid-s42118" xml:space="preserve"> quia <lb/>angulus k e f diuiſus eſt per æqualia per lineam t e:</s> <s xml:id="echoid-s42119" xml:space="preserve"> li-<lb/>neæ ergo k z & k t ad eandem lineam t f eandem habent proportionem:</s> <s xml:id="echoid-s42120" xml:space="preserve"> ergo per 9 p 5 lineam k z <lb/>eſt æqualis lineę k t:</s> <s xml:id="echoid-s42121" xml:space="preserve"> ſed ex pręmiſsis patet, quòd eſt proportio lineæ z k ad lineam t f, ſicut lineæ <lb/>z e ad lineam e t:</s> <s xml:id="echoid-s42122" xml:space="preserve"> eſt ergo per 11 p 5 proportio lineę z e ad lineam e t, ſicut lineæ k d ad lineam <lb/> <pb o="340" file="0642" n="642" rhead="VITELLONIS OPTICAE"/> d o:</s> <s xml:id="echoid-s42123" xml:space="preserve"> ſed linea k d ex hypotheſi eſt maior quàm linea d o:</s> <s xml:id="echoid-s42124" xml:space="preserve"> linea ergo z e eſt maior quàm linea e t:</s> <s xml:id="echoid-s42125" xml:space="preserve"> <lb/>& hoc quidem pro alijs reſeruantes nunc ad propoſitum redeamus.</s> <s xml:id="echoid-s42126" xml:space="preserve"> Quia uerò, (ut ſuprà patuit) <lb/>linea k q eſt perpendicularis ſuper lineam e z:</s> <s xml:id="echoid-s42127" xml:space="preserve"> erunt omnes anguli circa punctum q recti:</s> <s xml:id="echoid-s42128" xml:space="preserve"> ſed angu <lb/>lus e t d eſt acutus, quoniam eſt medietas anguli f t o, ut ſuperius oſtenſum eſt:</s> <s xml:id="echoid-s42129" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s42130" xml:space="preserve"> 1 <lb/>huius linea k q concurret cum linea t d:</s> <s xml:id="echoid-s42131" xml:space="preserve"> ſit punctus concurſus h:</s> <s xml:id="echoid-s42132" xml:space="preserve"> & ducatur linea e h:</s> <s xml:id="echoid-s42133" xml:space="preserve"> & à puncto <lb/>e ducatur linea ęquidiſtans lineę k h producta uſq;</s> <s xml:id="echoid-s42134" xml:space="preserve"> ad lineã d h, quę ſit e x, ſecans h d lineã in pun-<lb/>cto x:</s> <s xml:id="echoid-s42135" xml:space="preserve"> fiatq́ue per 5 p 4 circulus tranſiens per tria puncta, quę ſunt e, t, x:</s> <s xml:id="echoid-s42136" xml:space="preserve"> & immutetur figura (ſi <lb/>placet) propter diuerſam intricationem linearum.</s> <s xml:id="echoid-s42137" xml:space="preserve"> Quia itaque angulus t q h eſt rectus, ut patet <lb/>ex præmiſsis:</s> <s xml:id="echoid-s42138" xml:space="preserve"> erit per 29 p 1 angulus t e x rectus:</s> <s xml:id="echoid-s42139" xml:space="preserve"> er <lb/> <anchor type="figure" xlink:label="fig-0642-01a" xlink:href="fig-0642-01"/> go per 31 p 3 linea x t erit diameter illius circuli, qui <lb/>eſt e t x:</s> <s xml:id="echoid-s42140" xml:space="preserve"> & producatur linea k e pertriangulum or <lb/>thogonium t e x, & trans circulum, cadens in pun-<lb/>ctum m circumferentiæ circuli t e x:</s> <s xml:id="echoid-s42141" xml:space="preserve"> & ducatur li-<lb/>nea m t:</s> <s xml:id="echoid-s42142" xml:space="preserve"> & erit angulus t m e æqualis angulo t x e <lb/>per 27 p 3:</s> <s xml:id="echoid-s42143" xml:space="preserve"> cadunt enim ambo illi anguli in eundem <lb/>arcum, qui eſt e t:</s> <s xml:id="echoid-s42144" xml:space="preserve"> ſed angulus t x e æqualis eſt an-<lb/>gulo t h k per 29 p 1:</s> <s xml:id="echoid-s42145" xml:space="preserve"> quoniam lineæ e x & k h du-<lb/>ctæ ſunt æquidiſtantes:</s> <s xml:id="echoid-s42146" xml:space="preserve"> erit ergo angulus t m e æ-<lb/>qualis angulo t h k:</s> <s xml:id="echoid-s42147" xml:space="preserve"> ſed angulus t h k maior eſt an <lb/>gulo d h e:</s> <s xml:id="echoid-s42148" xml:space="preserve"> quod patet per 29 th.</s> <s xml:id="echoid-s42149" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s42150" xml:space="preserve"> ſecat enim <lb/>linea h e baſim k d:</s> <s xml:id="echoid-s42151" xml:space="preserve"> ergo angulus t m e maior eſt <lb/>eodem angulo d h e.</s> <s xml:id="echoid-s42152" xml:space="preserve"> Reſecetur ergo ab angulo t <lb/>m e angulus æqualis angulo d h e ք 27 th.</s> <s xml:id="echoid-s42153" xml:space="preserve"> 1 huius, <lb/>qui ſit angulus f m d ducta linea f m:</s> <s xml:id="echoid-s42154" xml:space="preserve"> & punctus, <lb/>in quo linea f m ſecat lineam t x, ſit i.</s> <s xml:id="echoid-s42155" xml:space="preserve"> Palàm ergo, <lb/>cum ex præmiſsis angulus i m d ſit æqualis angu-<lb/>lo d h e, & per 15 p 1 angulus i d m ſit æqualis angu-<lb/>lo e d h:</s> <s xml:id="echoid-s42156" xml:space="preserve"> quoniam per 32 p 1 triangulus i m d eſt æquiangulus triangulo d h e:</s> <s xml:id="echoid-s42157" xml:space="preserve"> ergo per 4 p 6 eſt <lb/>proportio lineę h d ad lineam d m, ſicut lineę e h ad lineam i m.</s> <s xml:id="echoid-s42158" xml:space="preserve"> Et ſimiliter triangulus t m d fit ſi-<lb/>milis triangulo k h d:</s> <s xml:id="echoid-s42159" xml:space="preserve"> cum, ſicut patet ex præmiſsis, angulus d h k ſit æqualis angulo t m d, & per 15 <lb/>p 1 angulus t d m ſit æqualis angulo k d b, & tertius tertio per 32 p 1:</s> <s xml:id="echoid-s42160" xml:space="preserve"> erit ergo proportio lineę k d ad <lb/>lineam d t, ſicut lineę h d ad lineam d m:</s> <s xml:id="echoid-s42161" xml:space="preserve"> eſt autem proportio lineæ h d ad lineam d m, ſicut lineę e h <lb/>ad lineam i m:</s> <s xml:id="echoid-s42162" xml:space="preserve"> eſt ergo per 11 p 5 proportio lineę k d ad lineam d t, ſicut lineę e h ad lineam i m:</s> <s xml:id="echoid-s42163" xml:space="preserve"> ſed <lb/>proportio lineæ k d ad lineam d t eſt nota:</s> <s xml:id="echoid-s42164" xml:space="preserve"> quoniam ſemper una & eadem permanet, quicunque pũ <lb/>ctus reflexionis ſit t in arcu b g:</s> <s xml:id="echoid-s42165" xml:space="preserve"> quia ſemper linea d t, quæ eſt ſemidiameter, eſt una, & linea k d ſi-<lb/>militer eſt ſemper una:</s> <s xml:id="echoid-s42166" xml:space="preserve"> quoniam ipſa eſt diſtantia alterius punctorum reflexorum à centro ſpeculi.</s> <s xml:id="echoid-s42167" xml:space="preserve"> <lb/>Linea etiam e h una permanet in quacun que reflexione, & non mutatur eius quantitas:</s> <s xml:id="echoid-s42168" xml:space="preserve"> quoniam <lb/>non mutatur quantitas anguli e d h, qui eſt medietas anguli o d a, qui non mutatur.</s> <s xml:id="echoid-s42169" xml:space="preserve"> Quare linea i m <lb/>ſemper erit una & æ qualis:</s> <s xml:id="echoid-s42170" xml:space="preserve"> erit ergo punctus circumferentiæ, in quem cadit linea i m producta ul <lb/>tra punctum i, qui eſt punctus f, ſemper notus & determinatus.</s> <s xml:id="echoid-s42171" xml:space="preserve"> Si ergo à tribus punctis arcus b g <lb/>poſsit fieri reflexio:</s> <s xml:id="echoid-s42172" xml:space="preserve"> continget ducere à puncto f ad circulum t x e tres lineas, quarum cuiuslibet <lb/>pars interiacens diametrum t x & peripheriã circuli ſit æqualis lineæ i m per 9 p 5:</s> <s xml:id="echoid-s42173" xml:space="preserve"> quia ſemper erit <lb/>proportio lineę k d ad lineam d t, ſicut lineę e h ad quamlibet illarum linearum:</s> <s xml:id="echoid-s42174" xml:space="preserve"> patet autẽ hoc eſſe <lb/>impoſsibile per 133 th.</s> <s xml:id="echoid-s42175" xml:space="preserve"> 1 huius, quòd ab eodem puncto dato in circũſerentia circuli extra diametrũ <lb/>per ipſam diametrum ad circumferentiam (ita ut pars lineæ interiacentis diametrum ad reliquam <lb/>partem circunferentiæ ſit æqualis datæ lineæ) non niſi duæ lineę ęquales duci poſſunt.</s> <s xml:id="echoid-s42176" xml:space="preserve"> Quare à <lb/>duobus tantùm punctis illius propoſiti arcus fiet reflexio.</s> <s xml:id="echoid-s42177" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s42178" xml:space="preserve"/> </p> <div xml:id="echoid-div1661" type="float" level="0" n="0"> <figure xlink:label="fig-0640-01" xlink:href="fig-0640-01a"> <variables xml:id="echoid-variables740" xml:space="preserve">b k o u n p g k e f d a q z m</variables> </figure> <figure xlink:label="fig-0640-02" xlink:href="fig-0640-02a"> <variables xml:id="echoid-variables741" xml:space="preserve">b t o u p n g k e f d a q z m</variables> </figure> <figure xlink:label="fig-0640-03" xlink:href="fig-0640-03a"> <variables xml:id="echoid-variables742" xml:space="preserve">b t o u p n g k e f d a q v m</variables> </figure> <figure xlink:label="fig-0641-01" xlink:href="fig-0641-01a"> <variables xml:id="echoid-variables743" xml:space="preserve">u t b o n p g k f d l a q m z</variables> </figure> <figure xlink:label="fig-0641-02" xlink:href="fig-0641-02a"> <variables xml:id="echoid-variables744" xml:space="preserve">s g t z k e f d o b r a</variables> </figure> <figure xlink:label="fig-0642-01" xlink:href="fig-0642-01a"> <variables xml:id="echoid-variables745" xml:space="preserve">t f k e d m z q x h</variables> </figure> </div> </div> <div xml:id="echoid-div1663" type="section" level="0" n="0"> <head xml:id="echoid-head1247" xml:space="preserve" style="it">37. Secundum modũ datæ lineæ à dato puncto ſpeculi ſphærici concaui ductæ: poßibile ect duo <lb/>punct a reperiri, quæ in diuerſis diametris inæqualiter à centro ſpeculi diſtantia, ab eodem dato <lb/>puncto ſpeculi, & uno tantùm alio eiuſdem arcus interiacentis ſemidiametros, in quibus illa pũ <lb/>cta conſiſtunt, ad ſe mutuò reflectantur. Alhazen 83 n 5.</head> <p> <s xml:id="echoid-s42179" xml:space="preserve">Remaneat diſpoſitio proximæ:</s> <s xml:id="echoid-s42180" xml:space="preserve"> ſitq́ue datus quicunque punctus ſpeculi:</s> <s xml:id="echoid-s42181" xml:space="preserve"> qui ſit t:</s> <s xml:id="echoid-s42182" xml:space="preserve"> proponitur no <lb/>bis, ut inueniãtur duo puncta, quæ in diuerſis diametris ſpeculi exiſtentia ab illo dato puncto ſu-<lb/>perficiei ſpeculi, & uno tantùm alio propoſiti arcus puncto ad ſe mutuò reflectantur.</s> <s xml:id="echoid-s42183" xml:space="preserve"> Sit enim, ut, <lb/>quantacunque placuerit, ſumatur linea z t:</s> <s xml:id="echoid-s42184" xml:space="preserve"> quæ per 119 th.</s> <s xml:id="echoid-s42185" xml:space="preserve"> 1 huius diuidatur taliter in puncto e, ut <lb/>ſit proportio lineę z e ad lineam e t, ſicut in præcedẽte propoſitione prima ſcilicet eius figuratione, <lb/>eſt proportio lineę k d ad lineam d o.</s> <s xml:id="echoid-s42186" xml:space="preserve"> Et quoniam ex hypotheſi illius linea k d eſt maior quàm linea <lb/>d o:</s> <s xml:id="echoid-s42187" xml:space="preserve"> erit linea z e maior quàm linea e t:</s> <s xml:id="echoid-s42188" xml:space="preserve"> diuidaturq́ue linea z t per æqualia in puncto q per 10 p 1:</s> <s xml:id="echoid-s42189" xml:space="preserve"> & à <lb/>puncto q ducatur perpendicularis ſuper lineam z t per 11 p 1:</s> <s xml:id="echoid-s42190" xml:space="preserve"> & fiat angulus e t d æqualis medietati <lb/>anguli o d a per 23 p 1:</s> <s xml:id="echoid-s42191" xml:space="preserve"> erit quidem ille angulus e t d acutus:</s> <s xml:id="echoid-s42192" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s42193" xml:space="preserve"> 1 huius linea t d concur-<lb/>ret cum perpendiculari ducta à puncto q ſuper lineam z t:</s> <s xml:id="echoid-s42194" xml:space="preserve"> ſit concurſus in puncto h.</s> <s xml:id="echoid-s42195" xml:space="preserve"> Completum <lb/>eſt ergo trigonum orthogonium, quod eſt t q h, in cuius altero laterum rectum angulum t q h con-<lb/> <pb o="341" file="0643" n="643" rhead="LIBER OCTAVVS."/> tinentium, quod eſt t q, datus eſt punctus e:</s> <s xml:id="echoid-s42196" xml:space="preserve"> poſs ibile eſt ergo à puncto e per 137 th.</s> <s xml:id="echoid-s42197" xml:space="preserve"> 1 huius duci li-<lb/> <anchor type="figure" xlink:label="fig-0643-01a" xlink:href="fig-0643-01"/> neam ad baſim trigoni t q h, quę eſt t h, ex alia ſui <lb/>parte concurrentem cum altero laterum rectum <lb/>angulum continentium, quod eſt q h, producto ul <lb/>tra punctum q, ita ut tota producta linea ſe habe-<lb/>at ad partem abſciſſam baſis, ſicut linea data ad li-<lb/>neam datam.</s> <s xml:id="echoid-s42198" xml:space="preserve"> Sit à puncto e taliter producta linea <lb/>d e k, ita ut ſit proportio totius lineæ k d ad lineam <lb/>d t, ſicut lineæ k d ad ſemidiametrum ſphæræ ſpe-<lb/>culi:</s> <s xml:id="echoid-s42199" xml:space="preserve"> ergo per 9 p 5 linea d t erit æqualis ſemidia-<lb/>metro.</s> <s xml:id="echoid-s42200" xml:space="preserve"> Punctum ergo d eſt centrum ſpeculi.</s> <s xml:id="echoid-s42201" xml:space="preserve"> Et an <lb/>gulo k t d fiat per 23 p 1 ſuper punctum t terminum <lb/>lineę d t æqualis angulus, quiſit o t d.</s> <s xml:id="echoid-s42202" xml:space="preserve"> Dico quo-<lb/>niam punctus ſpeculi (qui eſt t) eſt punctus refle-<lb/>xionis formæ puncti o ad uiſum exiſtentem in pũ <lb/>cto k:</s> <s xml:id="echoid-s42203" xml:space="preserve"> uel econuerſo formæ puncti k ad punctum <lb/>o:</s> <s xml:id="echoid-s42204" xml:space="preserve"> & quòd ab illo dato puncto t & ab uno tantùm <lb/>alio propoſiti arcus puncto fit illorum pũctorum <lb/>mutua reflexio.</s> <s xml:id="echoid-s42205" xml:space="preserve"> Et hęc omnia faciliter patent repe <lb/>tita priori demonſtratione theorematis præcedentis, prout huic propoſito eſt neceſſe.</s> <s xml:id="echoid-s42206" xml:space="preserve"> Patet er-<lb/>go propoſitum.</s> <s xml:id="echoid-s42207" xml:space="preserve"/> </p> <div xml:id="echoid-div1663" type="float" level="0" n="0"> <figure xlink:label="fig-0643-01" xlink:href="fig-0643-01a"> <variables xml:id="echoid-variables746" xml:space="preserve">c t k e d b q h k d o z</variables> </figure> </div> </div> <div xml:id="echoid-div1665" type="section" level="0" n="0"> <head xml:id="echoid-head1248" xml:space="preserve" style="it">38. Duobus punctis in diuerſis diametris circuli ſpeculi ſphærici concaui exiſtentibus, ambo-<lb/>bus extra circulum, uel uno intra circulum, & alio extra illum, & inæqualiter dictantibus à <lb/>centro, reſpicientibus arcum ſpeculi, à quo fit reflexio: ſireflectantur ab aliquo puncto arcus op-<lb/>poſiti illis diametris, non eſt ea poßibile reflecti ab alio puncto eiuſdem arcus. Alhazen 84 n 5.</head> <p> <s xml:id="echoid-s42208" xml:space="preserve">Sint duo puncta a & b in diuerſis diametris extra circulum (qui eſt communis ſectio ſuperficiei <lb/>reflexionis & ſpeculi ſphærici concaui) cuius centrum fit g:</s> <s xml:id="echoid-s42209" xml:space="preserve"> ſintq́ue illę diametri a e & b d:</s> <s xml:id="echoid-s42210" xml:space="preserve"> & fit pũ <lb/> <anchor type="figure" xlink:label="fig-0643-02a" xlink:href="fig-0643-02"/> ctus reflexionis t:</s> <s xml:id="echoid-s42211" xml:space="preserve"> & ducantur lineę b t, a t, g t.</s> <s xml:id="echoid-s42212" xml:space="preserve"> Linea itaque b t ſecabit <lb/>arcum circuli:</s> <s xml:id="echoid-s42213" xml:space="preserve"> ſit punctus ſectionis q:</s> <s xml:id="echoid-s42214" xml:space="preserve"> ſed & linea a t ſecabit periphe-<lb/>riam eiuſdem circuli:</s> <s xml:id="echoid-s42215" xml:space="preserve"> ſit punctus ſectionis m.</s> <s xml:id="echoid-s42216" xml:space="preserve"> Et quoniam angulus b <lb/>t g æqualis eſt angulo a t g:</s> <s xml:id="echoid-s42217" xml:space="preserve"> palàm per 26 p 3 quoniam cadunt in ar-<lb/>cus æquales.</s> <s xml:id="echoid-s42218" xml:space="preserve"> Producatur ergo diameter t g ad aliam partem periphe <lb/>riæ in punctum p:</s> <s xml:id="echoid-s42219" xml:space="preserve"> & erit arcus q p arcui m p æqualis.</s> <s xml:id="echoid-s42220" xml:space="preserve"> Si igitur for-<lb/>ma puncti b reflectitur ad uiſum exiſtentem in puncto a ab aliquo <lb/>alio puncto ſpeculi arcus eiuſdem:</s> <s xml:id="echoid-s42221" xml:space="preserve"> ſit illud aliud punctum h:</s> <s xml:id="echoid-s42222" xml:space="preserve"> & du-<lb/>cantur lineę a h, b h, g h:</s> <s xml:id="echoid-s42223" xml:space="preserve"> & ſecet linea b h circulum in pũcto l, & linea <lb/>a h in puncto n:</s> <s xml:id="echoid-s42224" xml:space="preserve"> producaturq́;</s> <s xml:id="echoid-s42225" xml:space="preserve"> ſemidiameter h g in punctum circum-<lb/>ferentiæ, qui ſit k.</s> <s xml:id="echoid-s42226" xml:space="preserve"> Secundum prædicta itaque erit arcus l k æqualis <lb/>arcui n k:</s> <s xml:id="echoid-s42227" xml:space="preserve"> ſed habitum eſt prius, quòd arcus q p eſt æqualis p m:</s> <s xml:id="echoid-s42228" xml:space="preserve"> ſed <lb/>arcus q p maior eſt arcu l k, & arcus k n maior arcu m p:</s> <s xml:id="echoid-s42229" xml:space="preserve"> accidit igitur <lb/>impoſsibile, ſcilicet minus eſſe maiori æquale.</s> <s xml:id="echoid-s42230" xml:space="preserve"> Quocunque uero alio <lb/>pũcto illius arcus d t e dato, idem accidit impoſsibile.</s> <s xml:id="echoid-s42231" xml:space="preserve"> Reſtat ergo ut <lb/>forma puncti b non reflectatur ad uiſum a à puncto h, uel ab alio pun <lb/>cto arcus d t e, oppoſiti diametris, in quibus ſunt puncta a & b, præ-<lb/>terquàm à puncto t.</s> <s xml:id="echoid-s42232" xml:space="preserve"> Idem quoque accidit impoſsibile, & eodem mo <lb/>do deducendum, ſi unum dato rum punctorũ ſit in circulo, reliquum <lb/>uerò extra circulum.</s> <s xml:id="echoid-s42233" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s42234" xml:space="preserve"/> </p> <div xml:id="echoid-div1665" type="float" level="0" n="0"> <figure xlink:label="fig-0643-02" xlink:href="fig-0643-02a"> <variables xml:id="echoid-variables747" xml:space="preserve">a b n m p k l q g d h c e</variables> </figure> </div> </div> <div xml:id="echoid-div1667" type="section" level="0" n="0"> <head xml:id="echoid-head1249" xml:space="preserve" style="it">39. Duobus punctis in diuerſis diametris circuli ſpeculi ſphærici concaui exiſtentibus ambo-<lb/>bus extra circulum: ſi linea continuans illa punct a conting at illum circulum, aut tota ſit extra <lb/>circulum: non eſt poßibile unum illorum punctorum ad alterum reflecti, niſi ab uno tantùm il-<lb/>lius ſpeculi puncto. Alhazen 85 n 5.</head> <p> <s xml:id="echoid-s42235" xml:space="preserve">Sint, ut in pręcedente theoremate, duo puncta a & b in diuerſis diametris extra circulum (qui eſt <lb/>communis ſectio ſuperficiei reflexionis & ſpeculi ſphærici concaui) cuius centrum ſit g:</s> <s xml:id="echoid-s42236" xml:space="preserve"> ſintq́ue il-<lb/>læ diametri l d & m n:</s> <s xml:id="echoid-s42237" xml:space="preserve"> ſitq́ue punctus a in ſemidiametro l g, & punctus b in ſemidiametro m g:</s> <s xml:id="echoid-s42238" xml:space="preserve"> & du <lb/>catur linea continuans puncta a & b, quæ ſit a b:</s> <s xml:id="echoid-s42239" xml:space="preserve"> & hæc contingat circulum illum, à quo per 2 hu-<lb/>ius poteſt fieri reflexio:</s> <s xml:id="echoid-s42240" xml:space="preserve"> ſitq́ue ille contactus in arcu circuli, qui ſit arcus l m:</s> <s xml:id="echoid-s42241" xml:space="preserve"> aut ſi linea illa ſit tota <lb/>extra ſpeculum:</s> <s xml:id="echoid-s42242" xml:space="preserve"> dico quò d à nullo puncto arcus l m interiacentis diametros, in quibus ſunt illa <lb/>puncta, fit reflexio formæ unius punctorum a uel b ad punctum reliquum.</s> <s xml:id="echoid-s42243" xml:space="preserve"> Sumpto enim quocun-<lb/>que puncto in arculm, ut puncto t, ductisq́ue lineis a t & b t, ſi linea a t cadat intra ſpeculum, li-<lb/>nea b t neceſſariò cadet extra ſpeculum:</s> <s xml:id="echoid-s42244" xml:space="preserve"> quoniam hoc requirit talis ſitus ſpeculi:</s> <s xml:id="echoid-s42245" xml:space="preserve"> & econuerſo ſi li-<lb/> <pb o="342" file="0644" n="644" rhead="VITELLONIS OPTICAE"/> nea b t cadat in ſpeculo, linea a t cadet extra:</s> <s xml:id="echoid-s42246" xml:space="preserve"> ſemper enim altera linearum ab illis duobus punctis <lb/> <anchor type="figure" xlink:label="fig-0644-01a" xlink:href="fig-0644-01"/> a & b ad illud punctum ſpeculi ductarum tota e <lb/>rit extra ſpeculum:</s> <s xml:id="echoid-s42247" xml:space="preserve"> & ſic item neuter illorum pun <lb/>ctorum ad alterum reflectetur ab aliquo puncto <lb/>illius arcus l m.</s> <s xml:id="echoid-s42248" xml:space="preserve"> Similiter quoque patet idem, ſili <lb/>nea tota ſit extra ſpeculũ, non contingens ipſum, <lb/>reſpiciat tamen arcum l m:</s> <s xml:id="echoid-s42249" xml:space="preserve"> quia neque tunc am-<lb/>bæ lineę at & b t cadent intra ſpeculum:</s> <s xml:id="echoid-s42250" xml:space="preserve"> ſed ſi u-<lb/>na erit intra ſpeculum, reliqua erit tota extra ſpe-<lb/>culum:</s> <s xml:id="echoid-s42251" xml:space="preserve"> unde non ſiet reflexio ſecundum illam:</s> <s xml:id="echoid-s42252" xml:space="preserve"> ab <lb/>aliquo tamen puncto arcus d n poteſt fieri re-<lb/>flexio per 27 huius:</s> <s xml:id="echoid-s42253" xml:space="preserve"> & ab uno tantùm puncto illi-<lb/>us arcus, ut patet per præcedentem:</s> <s xml:id="echoid-s42254" xml:space="preserve"> & ita forma-<lb/>rum illorum punctorum reflexio ad inuicem <lb/>non fiet niſi ab uno ſolo puncto ſpeculi.</s> <s xml:id="echoid-s42255" xml:space="preserve"> Quod eſt <lb/>propoſitum.</s> <s xml:id="echoid-s42256" xml:space="preserve"/> </p> <div xml:id="echoid-div1667" type="float" level="0" n="0"> <figure xlink:label="fig-0644-01" xlink:href="fig-0644-01a"> <variables xml:id="echoid-variables748" xml:space="preserve">b a b a m t l g d n</variables> </figure> </div> </div> <div xml:id="echoid-div1669" type="section" level="0" n="0"> <head xml:id="echoid-head1250" xml:space="preserve" style="it">40. Exiſtẽtib. duobus pũctis in diuerſis diame <lb/>tris circuli ſpeculi ſphærici concaui, inæqualiter <lb/>dictantibus à centro: ſi linea continuans illa <lb/>puncta product a ſecet circulum, unum illorum punctorum ad alterum ab uno tantùm puncto <lb/>ſpeculi, uel à duobus, aut à tribus, aut à quatuor poßible eſt reflecti: & ſecundum hæc locaima-<lb/>ginum numer antur. Alhazen 86 n 5.</head> <p> <s xml:id="echoid-s42257" xml:space="preserve">Sint, ut ſuprà, duo puncta a & b in diuerſis diametris circuli ſpeculi ſphærici concaui, ita ut pun-<lb/>ctus a ſit in diametro l d, & pũctus b in diametro m n:</s> <s xml:id="echoid-s42258" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0644-02a" xlink:href="fig-0644-02"/> ſintq́ue illa puncta inæ qualiter diſtantia à centro ſpe <lb/>culi, quod eſt g:</s> <s xml:id="echoid-s42259" xml:space="preserve"> & linea a b ducta ab uno illorũ pun-<lb/>ctorum ad alterum producta ſecet circulum:</s> <s xml:id="echoid-s42260" xml:space="preserve"> dico <lb/>quòd uerum eſt, quod proponitur.</s> <s xml:id="echoid-s42261" xml:space="preserve"> Fiat enim circu-<lb/>lus pertranſiens per centrum ſpeculi, quod eſt g, & <lb/>per illa duo puncta a & b per 5 p 4:</s> <s xml:id="echoid-s42262" xml:space="preserve"> circulus itaque il-<lb/>le a b g aut totus erit intra circulum ſpeculi:</s> <s xml:id="echoid-s42263" xml:space="preserve"> aut con-<lb/>tinget ipſum intrin ſecus:</s> <s xml:id="echoid-s42264" xml:space="preserve"> aut ſecabit ipſum.</s> <s xml:id="echoid-s42265" xml:space="preserve"> Sitotus <lb/>circulus a b g fuerit intra ſpeculi circulum, palàm per <lb/>6 huius quòd unum illorum punctorum reflectetur <lb/>ad alterum ab aliquo puncto ſpeculi & propoſiti cir-<lb/>culi, ut patet per 2 huius, & per 20 th.</s> <s xml:id="echoid-s42266" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s42267" xml:space="preserve"> Sit ergo <lb/>punctus reflexionis t:</s> <s xml:id="echoid-s42268" xml:space="preserve"> palamq́;</s> <s xml:id="echoid-s42269" xml:space="preserve"> per 20 huius quòd pũ <lb/>ctus t eſt in arcu interiacente diametros, in quibus <lb/>funt puncta a & b:</s> <s xml:id="echoid-s42270" xml:space="preserve"> qui ſit arcus l m:</s> <s xml:id="echoid-s42271" xml:space="preserve"> & ducantur lineæ <lb/>at, bt, gt:</s> <s xml:id="echoid-s42272" xml:space="preserve"> erit quoque angulus a t b minor angulo b <lb/>g d.</s> <s xml:id="echoid-s42273" xml:space="preserve"> Sit enim, ut ſemidiameter g t ſecet circulum a b g <lb/>in puncto ſ:</s> <s xml:id="echoid-s42274" xml:space="preserve"> & ducantur lineæ a f & b f:</s> <s xml:id="echoid-s42275" xml:space="preserve"> fientq́ue duo trigona a t b & a f b ſuper unam baſim, quæ eſt <lb/>a b:</s> <s xml:id="echoid-s42276" xml:space="preserve"> palàm ergo per 21 p 1 quoniam angulus a f b eſt maior angulo a t b:</s> <s xml:id="echoid-s42277" xml:space="preserve"> ſed per 22 p 3 angulus a f b cũ <lb/> <anchor type="figure" xlink:label="fig-0644-03a" xlink:href="fig-0644-03"/> angulo a g b ualet duos rectos:</s> <s xml:id="echoid-s42278" xml:space="preserve"> ergo per 13 p 1 angu-<lb/>lus a f b eſt ęqualis angulo b g d:</s> <s xml:id="echoid-s42279" xml:space="preserve"> angulusergo a t b eſt <lb/>minor angulo b g d.</s> <s xml:id="echoid-s42280" xml:space="preserve"> Quilibet quoq;</s> <s xml:id="echoid-s42281" xml:space="preserve"> angulus ſic fa-<lb/>ctus ſuper arcum l m, ut ſuper punctũ h, erit minor <lb/>angulo b g d.</s> <s xml:id="echoid-s42282" xml:space="preserve"> Ab arcu itaq;</s> <s xml:id="echoid-s42283" xml:space="preserve"> ſpeculi, qui eſt l m, nõ fiet <lb/>reflexio niſi ab uno tantũ pũcto ſpeculi:</s> <s xml:id="echoid-s42284" xml:space="preserve"> quoniã iam <lb/>oſtenſum eſt per 34 huius, quia non eſt in huiuſmodi <lb/>pũctorũ reflexorũ diſpoſitione poſsibile reflexionẽ <lb/>fieri à duobus punctis ſpeculi, ita ut uterq;</s> <s xml:id="echoid-s42285" xml:space="preserve"> angulorũ <lb/>conſtans ex angulo incidentiæ & reflexionis, ſit mi-<lb/>nor angulo b g d.</s> <s xml:id="echoid-s42286" xml:space="preserve"> In hac ergo diſpoſitione ab uno tã-<lb/>tùm puncto ſpeculi fiet reflexio:</s> <s xml:id="echoid-s42287" xml:space="preserve"> quod eſt unũ pro-<lb/>poſitorũ.</s> <s xml:id="echoid-s42288" xml:space="preserve"> Siuerò circulus a b g ſit intrinſecus contin <lb/>gens circulũ ſpeculi:</s> <s xml:id="echoid-s42289" xml:space="preserve"> ſit punctũ contactus h:</s> <s xml:id="echoid-s42290" xml:space="preserve"> & ducã <lb/>tur lineę a h, b h, g h.</s> <s xml:id="echoid-s42291" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s42292" xml:space="preserve"> angulus a h b ք 22 p 3 <lb/>cum angulo a g b ualet duos rectos:</s> <s xml:id="echoid-s42293" xml:space="preserve"> patet per 13 p 1 <lb/>quòd angulus a h b eſt æqualis angulo b g d.</s> <s xml:id="echoid-s42294" xml:space="preserve"> Quare <lb/>ab illo puncto contactus non fiet reflexio per 33 hu-<lb/>ius.</s> <s xml:id="echoid-s42295" xml:space="preserve"> Angulus quoque factus ſuper quodcun que aliud punctum arcus circuli ſpeculi, erit minor illo <lb/> <pb o="343" file="0645" n="645" rhead="LIBER OCTAVVS."/> angulo per modum, quoiam ſuperius præoſtenſum eſt.</s> <s xml:id="echoid-s42296" xml:space="preserve"> Quare à duobus punctis illius arcus non <lb/>fiet reflexio per 34 huius, ſed ſolùm ab uno puncto.</s> <s xml:id="echoid-s42297" xml:space="preserve"> Si uerò circulus a b g ſecet circulum ſpeculi:</s> <s xml:id="echoid-s42298" xml:space="preserve"> pa-<lb/>tet quòd tantùm in duobus punctis ſecare neceſſe eſt per 10 p 3:</s> <s xml:id="echoid-s42299" xml:space="preserve"> & illa duo puncta a & b aut ambo <lb/>erunt extra circulum ſpeculi:</s> <s xml:id="echoid-s42300" xml:space="preserve"> aut ambo intra:</s> <s xml:id="echoid-s42301" xml:space="preserve"> aut unum extra circulum, aliud intra illum:</s> <s xml:id="echoid-s42302" xml:space="preserve"> aut unum <lb/>illorum punctorum in circumferentia circuli, & aliud extra illum uel intra illum.</s> <s xml:id="echoid-s42303" xml:space="preserve"> Si fuerint ambo <lb/> <anchor type="figure" xlink:label="fig-0645-01a" xlink:href="fig-0645-01"/> extra circulum ſpeculi:</s> <s xml:id="echoid-s42304" xml:space="preserve"> tunc patet quòd linea a b non ſecabit circulũ <lb/>ſpeculi:</s> <s xml:id="echoid-s42305" xml:space="preserve"> fietq́ reflexio ab uno tantũ ſpeculi puncto, ut patet per præ-<lb/>cedentem.</s> <s xml:id="echoid-s42306" xml:space="preserve"> Tunc enim manifeſtè patet, quòd circulus a b g non ſeca-<lb/>bit circulum ſpeculi ſecundum arcum l m:</s> <s xml:id="echoid-s42307" xml:space="preserve"> quoniam ille arcus inter-<lb/>iacet lineas a g & b g, & arcus b g a cadit extra illas lineas in alia pun <lb/>cta peripheriæ circuli ipſius ſpeculi, cum ambo puncta a & b ſunt <lb/>extra circulum ſpeculi.</s> <s xml:id="echoid-s42308" xml:space="preserve"> Si uerò punctus b ſit in peripheria circuli ſpe <lb/>culi uel intra, puncto a conſtituto extra:</s> <s xml:id="echoid-s42309" xml:space="preserve"> patet tunc quòd arcus l m in <lb/>duobus punctis non ſecabitur, ſed arcus b g tranſibit punctum ali <lb/>quod arcus l m, quod ſit t:</s> <s xml:id="echoid-s42310" xml:space="preserve"> ergo angulus factus ſuper arcũ l m erit ma <lb/>ior angulo b g d:</s> <s xml:id="echoid-s42311" xml:space="preserve"> quoniam ductis lineis l t, b t & a t, patet ſecundum <lb/>præmiſſa per 22 p 3 quoniam angulus l t b eſt æqualis angulo b g d:</s> <s xml:id="echoid-s42312" xml:space="preserve"> <lb/>angulus uerò a t b eſt maior illo.</s> <s xml:id="echoid-s42313" xml:space="preserve"> Patet ergo per 24 huius quoniam in <lb/>hac diſpoſitione ab unico puncto, uel à duobus pũctis arcus l m fiet <lb/>formarum illorum punctorum adinuicem reflexio.</s> <s xml:id="echoid-s42314" xml:space="preserve"> Si uerò duo pun <lb/>cta a & b fuerint intra circulum ſpeculi, & circulus a b g ſecet circulũ <lb/>ſpeculi:</s> <s xml:id="echoid-s42315" xml:space="preserve"> tunc patet quòd circulus a b g ſecabit arcum l m in duobus <lb/>punctis:</s> <s xml:id="echoid-s42316" xml:space="preserve"> quoniam duæ ſemidiametri circuli maioris, quæ ſunt g l & g m, ſecant circulũ a b g in pun-<lb/> <anchor type="figure" xlink:label="fig-0645-02a" xlink:href="fig-0645-02"/> ctis a & b, & tranſeuntes reſecant ex circulo ſpeculi arcum l m:</s> <s xml:id="echoid-s42317" xml:space="preserve"> ſecet <lb/>ergo circulus a b g arcum l m in duobus punctis, quę ſint t & h:</s> <s xml:id="echoid-s42318" xml:space="preserve"> & re-<lb/>ſtabũt ex ipſo arcu l m duo arcus in diuerſis partibus ipſius, qui ſunt <lb/>arcus l t & h m:</s> <s xml:id="echoid-s42319" xml:space="preserve"> omnisq́;</s> <s xml:id="echoid-s42320" xml:space="preserve"> angulus conſtitutus ſuper arcũ circuli ſpe-<lb/>culi, qui eſt th, erit maior angulo b d:</s> <s xml:id="echoid-s42321" xml:space="preserve"> quod patet, ſi ſuper peripheriã <lb/>ſpeculi fiat angulus a e b:</s> <s xml:id="echoid-s42322" xml:space="preserve"> ille enim eſt maior angulo b g d.</s> <s xml:id="echoid-s42323" xml:space="preserve"> Produ-<lb/>cta enim linea b e ad peripheriam circuli a b g in punctum f, ſi co-<lb/>puletur linea a f, erit per 22 p 3 & per 13 p 1 angulus a f b æqualis an <lb/>gulo b g d:</s> <s xml:id="echoid-s42324" xml:space="preserve"> ſed per 21 uel per 16 p 1 angulus a e b eſt maior angulo <lb/>a f b:</s> <s xml:id="echoid-s42325" xml:space="preserve"> ergo & angulo b g d.</s> <s xml:id="echoid-s42326" xml:space="preserve"> Et ſimiliter erit de quolibet alio puncto <lb/>arcus t e h demonſtrandum.</s> <s xml:id="echoid-s42327" xml:space="preserve"> Ab hoc itaque arcu t e h, ut patet per <lb/>34 huius, poterit fieri reflexio, forſan a b uno tantùm puncto, & for-<lb/>ſan à duobus.</s> <s xml:id="echoid-s42328" xml:space="preserve"> Quòd ſi fiat reflexio à duobus arcubus l t & h m, qui <lb/>reſtant ſuper arcum t e h exarcu l m & ex diuerſis partibus ipſius <lb/>circuli a b g:</s> <s xml:id="echoid-s42329" xml:space="preserve"> tunc ſecundum præmiſſa omnes anguli ſuper illos ar-<lb/>cus conſiſtentes contenti ſub lineis à punctis a & b productis, e <lb/>runt minores angulo b g d.</s> <s xml:id="echoid-s42330" xml:space="preserve"> Fiat enim angulus b k a ſuper punctum <lb/>arcus l t.</s> <s xml:id="echoid-s42331" xml:space="preserve"> Et quoniã arcus at circuli a b g eſt intra circulũ ſpeculi ſub <lb/>arcult, ſecet linea b karcũ a t in puncto o:</s> <s xml:id="echoid-s42332" xml:space="preserve"> & ducatur linea a o:</s> <s xml:id="echoid-s42333" xml:space="preserve"> patet <lb/>ergo per 22 p 3 & per 13 p 1 quòd angulus a o b eſt æqualis angulo b g d:</s> <s xml:id="echoid-s42334" xml:space="preserve"> ſed angulus a o b eſt maior <lb/>angulo a k b per 16 p 1:</s> <s xml:id="echoid-s42335" xml:space="preserve"> patet ergo quòd angulus a k b eſt minor angulo b g d.</s> <s xml:id="echoid-s42336" xml:space="preserve"> Et ſimiliter de quo-<lb/>libet puncto arcuum l t & h m eſt demonſtrandum.</s> <s xml:id="echoid-s42337" xml:space="preserve"> Ergo per 34 huius ab uno tantùm illorum ar-<lb/>cuum puncto fiet reflexio.</s> <s xml:id="echoid-s42338" xml:space="preserve"> In hoc itaque ſitu fiet reflexio à duobus punctis arcus l m interiacentis <lb/> <anchor type="figure" xlink:label="fig-0645-03a" xlink:href="fig-0645-03"/> diametros, aut forſan à tribus:</s> <s xml:id="echoid-s42339" xml:space="preserve"> palàm uerò per 27 & 29 huius quòd <lb/>ab uno tantùm puncto arcus n d fiet reflexio:</s> <s xml:id="echoid-s42340" xml:space="preserve"> & ita in hoc ſitu ali-<lb/>quando à tribus punctis ſpeculi, aliquando uerò à quatuor pun-<lb/>ctis fiet reflexio.</s> <s xml:id="echoid-s42341" xml:space="preserve"> Si uerò unus punctorum a uel b fuerit in peri-<lb/>pheria circuli, alius uerò intra circulum, & circulus a b g ſecet <lb/>circulum ſpeculi:</s> <s xml:id="echoid-s42342" xml:space="preserve"> tunc ſecabit arcum l m in uno tantùm pun-<lb/>cto, qui ſit t:</s> <s xml:id="echoid-s42343" xml:space="preserve"> quoniam in loco alterius punctorum l uel m erit <lb/>punctum a uel b:</s> <s xml:id="echoid-s42344" xml:space="preserve"> exiſtens enim in altera diametrorum n m uel <lb/>l d, & in ipſa circuli peripheria, erit in puncto, quod eſt commu <lb/>nis ſectio illarum:</s> <s xml:id="echoid-s42345" xml:space="preserve"> & ſic puncto b exiſtente in puncto m, & pun-<lb/>cto a intra ſpeculum:</s> <s xml:id="echoid-s42346" xml:space="preserve"> reſtabιt unicus tantùm arcus totius arcus l m:</s> <s xml:id="echoid-s42347" xml:space="preserve"> <lb/>qui ſitlt.</s> <s xml:id="echoid-s42348" xml:space="preserve"> Patet itaque ſecundum præmiſſa ductis, ut prius, lineis <lb/>a f & b f ſuper arcum circuli a b g, & lineis a e & b e ſuper ali-<lb/>quod punctum arcus l m, quod ſit e:</s> <s xml:id="echoid-s42349" xml:space="preserve"> quoniam per 21 p 1 omnes <lb/>anguli conſiſtentes ſuper arcum t b ſunt maiores angulo b g d:</s> <s xml:id="echoid-s42350" xml:space="preserve"> er-<lb/>go per 34 huius poteſt fieri reflexio à duobus punctis illius ar-<lb/>cus, uel ab uno.</s> <s xml:id="echoid-s42351" xml:space="preserve"> Omnes uerò anguli arcus l t erunt minores <lb/>angulo b g d, ut præoſtenſum eſt prius:</s> <s xml:id="echoid-s42352" xml:space="preserve"> & ita per 34 huius ab uno tantùm puncto arcus l t <lb/> <pb o="344" file="0646" n="646" rhead="VITELLONIS OPTICAE"/> fiet reflexio:</s> <s xml:id="echoid-s42353" xml:space="preserve"> ſed & per 27 uel 29 huius ab uno tantũ puncto arcus n d fiet reflexio.</s> <s xml:id="echoid-s42354" xml:space="preserve"> Fiet itaq:</s> <s xml:id="echoid-s42355" xml:space="preserve"> in hoc <lb/>ſitu reflexio quádoq;</s> <s xml:id="echoid-s42356" xml:space="preserve"> à tribus punctis:</s> <s xml:id="echoid-s42357" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s42358" xml:space="preserve"> à quatuor, & non â pluribus.</s> <s xml:id="echoid-s42359" xml:space="preserve"> Quòd ſi pun cto b <lb/>exiſtente in peripheria circuli ſpeculi, punctus a ſit extra illum circulum:</s> <s xml:id="echoid-s42360" xml:space="preserve"> tunc patet quòd circulus <lb/>a b g nunquam ſecabit circulum ſpeculi ſecundum arcum l m:</s> <s xml:id="echoid-s42361" xml:space="preserve"> quoniam ſemidiameter g m, & peri-<lb/>pheriæ circuli communis ſectio eſt punctus m, in quo eſt punctus b:</s> <s xml:id="echoid-s42362" xml:space="preserve"> ſemidiameter uerò glproce-<lb/>dens ad punctum a extra circulum ſecat arcum t b.</s> <s xml:id="echoid-s42363" xml:space="preserve"> Omnes itaq;</s> <s xml:id="echoid-s42364" xml:space="preserve"> anguli arcus l m ſunt maiores an-<lb/>gulo b g d, ut patet ex præmiſsis:</s> <s xml:id="echoid-s42365" xml:space="preserve"> ergo per 34 huius ab uno tantùm pũcto uel forſan â duobus pun-<lb/>ctis arcus l m poteſt fieri reflexio punctorum a & b adinuicem:</s> <s xml:id="echoid-s42366" xml:space="preserve"> & ſimiliter ab uno pũcto arcus n d.</s> <s xml:id="echoid-s42367" xml:space="preserve"> <lb/>Fiet itaq;</s> <s xml:id="echoid-s42368" xml:space="preserve"> in hoc ſitu reflexio à duob.</s> <s xml:id="echoid-s42369" xml:space="preserve"> aut à tribus pũctis ſpeculi, & nõ à pluribus.</s> <s xml:id="echoid-s42370" xml:space="preserve"> Palàm ergo quòd <lb/>puncta inæqualiter diſtantia à centro ſpeculi aliquando ab uno tantùm puncto ſpeculi:</s> <s xml:id="echoid-s42371" xml:space="preserve"> aliquan-<lb/>do à duobus:</s> <s xml:id="echoid-s42372" xml:space="preserve"> aliquando à tribus:</s> <s xml:id="echoid-s42373" xml:space="preserve"> aliquando à quatuor:</s> <s xml:id="echoid-s42374" xml:space="preserve"> nunquam à pluribus reflectuntur:</s> <s xml:id="echoid-s42375" xml:space="preserve"> ſecun-<lb/>dum hęc quoq;</s> <s xml:id="echoid-s42376" xml:space="preserve"> loca imaginum numerantur, quemadmodum patuit iam pluries in præmiſsis.</s> <s xml:id="echoid-s42377" xml:space="preserve"> Et <lb/>hoc eſt, quod ꝓponebatur declarandum.</s> <s xml:id="echoid-s42378" xml:space="preserve"/> </p> <div xml:id="echoid-div1669" type="float" level="0" n="0"> <figure xlink:label="fig-0644-02" xlink:href="fig-0644-02a"> <variables xml:id="echoid-variables749" xml:space="preserve">m t h l b f p a g d n</variables> </figure> <figure xlink:label="fig-0644-03" xlink:href="fig-0644-03a"> <variables xml:id="echoid-variables750" xml:space="preserve">m t h l b a g d n</variables> </figure> <figure xlink:label="fig-0645-01" xlink:href="fig-0645-01a"> <variables xml:id="echoid-variables751" xml:space="preserve">a b l m l t a b m g n d n d</variables> </figure> <figure xlink:label="fig-0645-02" xlink:href="fig-0645-02a"> <variables xml:id="echoid-variables752" xml:space="preserve">g e t h h o l m l a g n d</variables> </figure> <figure xlink:label="fig-0645-03" xlink:href="fig-0645-03a"> <variables xml:id="echoid-variables753" xml:space="preserve">f e b m a f l d g n</variables> </figure> </div> </div> <div xml:id="echoid-div1671" type="section" level="0" n="0"> <head xml:id="echoid-head1251" xml:space="preserve" style="it">41. Exiſtentibus duobus punctis in diuerſis diametris circuli ſpeculi ſphærici concaui, & <lb/>æqualiter diſtantibus à centro, ſi linea coutinuans illa puncta ſecet circulum: poßibile eſt <lb/>unũ illorum punctorum ad alterum reflecti ab uno tantumpuncto ſpeculi: uelà duobus: aut à <lb/>quatuor: ſed impoßibile ect à tribus: & ſecundum hæc loca imaginum numerantur. Alha-<lb/>zen 87 n 5.</head> <p> <s xml:id="echoid-s42379" xml:space="preserve">Sint, ut in præmiſſa, duo puncta a & b in diuerſis diametris circuli ſpeculi ſphærici concaui, quæ <lb/>ſintl d & m n, ita ut pũctus a ſit in diametro l d, & pũctus b in diametro m n:</s> <s xml:id="echoid-s42380" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s42381" xml:space="preserve"> pũcta a & b æqua <lb/>liter diſtantia à centro ſpeculi, & linea a b ſit ducta ab uno illorũ pũctorũ ad alterũ ſecundũ circulũ <lb/>(qui eſt cõmunis ſectio ſuperficiei reflexionis & ſpeculi) cuius centrũ ſit g:</s> <s xml:id="echoid-s42382" xml:space="preserve"> dico quòd uerũ eſt qđ <lb/> <anchor type="figure" xlink:label="fig-0646-01a" xlink:href="fig-0646-01"/> proponitur.</s> <s xml:id="echoid-s42383" xml:space="preserve"> Quòd enim ab uno tantũ puncto ſpecu-<lb/>li quandoq;</s> <s xml:id="echoid-s42384" xml:space="preserve"> fiat illorũ pũctorum adinuicẽ mutua re-<lb/>flexio, patet per 19 huius:</s> <s xml:id="echoid-s42385" xml:space="preserve"> & etiã idẽ oſtẽdi poteſt per <lb/>modũ 24 huius:</s> <s xml:id="echoid-s42386" xml:space="preserve"> linearũ enim inæ qualitas in illo ſitu <lb/>naturã reflexionis nõ immutat, ut declaratum eſt in <lb/>20 th.</s> <s xml:id="echoid-s42387" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s42388" xml:space="preserve"> Quãdoq;</s> <s xml:id="echoid-s42389" xml:space="preserve"> uerò fit mutua reflexio iſto-<lb/>rum pũctorum a & b à duobus tantũ pũctis ſpeculi, <lb/>ut patet per 25 huius.</s> <s xml:id="echoid-s42390" xml:space="preserve"> Quandoq;</s> <s xml:id="echoid-s42391" xml:space="preserve"> uerò fit reflexio mu <lb/>tua propoſitorũ pũctorum, quę ſunt a & b, à quatuor <lb/>pũctis circũferentiæ ipſius ſpeculi, ut patet per 26 hu <lb/>ius.</s> <s xml:id="echoid-s42392" xml:space="preserve"> A tribus uerò tantũ pũctis iſtorum ſpeculorum <lb/>formas pũctorum æqualiter diſtantium à centro ſpe <lb/>culi ad ſe mutuò reflecti eſt impoſsibile.</s> <s xml:id="echoid-s42393" xml:space="preserve"> Si enim ab <lb/>aliquibus duobus pũctis unius arcus fiat iſta mutua <lb/>reflexio, diuiſo arcu interiacente illa pũcta per æqua <lb/>lia, & ductis ad illud pũctum lineis, patet per 27 p 3 & <lb/>propter ęqualitatem laterum g a & g b, quoniam an-<lb/>guli conſtituti ſuper illud punctum fiunt æquales:</s> <s xml:id="echoid-s42394" xml:space="preserve"> ab <lb/>illo ergo pũcto fiet reflexio per 20 th.</s> <s xml:id="echoid-s42395" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s42396" xml:space="preserve"> ſed & <lb/>fiet ab aliquo pũcto arcus oppoſiti illi arcui.</s> <s xml:id="echoid-s42397" xml:space="preserve"> Palã ergo quòd à quatuor pũctis ſpeculi fiet reflexio, <lb/>& non à tribus.</s> <s xml:id="echoid-s42398" xml:space="preserve"> Et quoniam, ut patet per præmiſſam & ex plur bus propoſitionibus huius libri, <lb/>nunquam fit à tribus punctis ſpeculi reflexio aliquorum duorum punctorum adinuicem, niſi fiat à <lb/>duobus punctis unius arcus, & ab aliquo puncto arcus oppoſiti interiacente illas diametros:</s> <s xml:id="echoid-s42399" xml:space="preserve"> patet <lb/>ergo quòd in hac diſpoſitione reflexio fiet ſemper à quatuor punctis ſpeculi propoſiti, & nunquam <lb/>à tribus.</s> <s xml:id="echoid-s42400" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s42401" xml:space="preserve"> Et quoniam hęc duo præmiſſa theoremata diſpoſuimus ſecundum <lb/>modum epilogi plurimorum præmiſſorum theorematum, ęſtimamus ipſa memorię cõmendanda.</s> <s xml:id="echoid-s42402" xml:space="preserve"/> </p> <div xml:id="echoid-div1671" type="float" level="0" n="0"> <figure xlink:label="fig-0646-01" xlink:href="fig-0646-01a"> <variables xml:id="echoid-variables754" xml:space="preserve">l m a b g n d</variables> </figure> </div> </div> <div xml:id="echoid-div1673" type="section" level="0" n="0"> <head xml:id="echoid-head1252" xml:space="preserve" style="it">42. Siab uno puncto arcus circuli ſpeculi ſphærici concaui formæ unius termini lineæ totali-<lb/>ter uiſæ, ab alio quo puncto eiuſdem arcus formæ alterius termini eiuſdem lineæ fiat reflexio: <lb/>neceſſe eſt omnia punct a media lineæ uiſæ abillius arcus punctis medijs reflecti: ex quo patet <lb/>quòd loca imaginum punct orum mediorum cadunt inter imagines punctorum extremorum. <lb/>Alhazen 45 n 6.</head> <p> <s xml:id="echoid-s42403" xml:space="preserve">Quod hic proponitur ſpecialiter, quantùm ad primam ſui partem, uniuerſaliter eſt pręmiſſum in <lb/>24 th.</s> <s xml:id="echoid-s42404" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s42405" xml:space="preserve"> Eſto ergo arcus circuli ſpeculi ſphærici concaui a f h:</s> <s xml:id="echoid-s42406" xml:space="preserve"> cuius centrum e:</s> <s xml:id="echoid-s42407" xml:space="preserve"> & ſit z centrũ <lb/>uiſus:</s> <s xml:id="echoid-s42408" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s42409" xml:space="preserve"> g r linea uiſa:</s> <s xml:id="echoid-s42410" xml:space="preserve"> cuius unus terminus (quig) reflectatur à puncto ſpeculi, qui ſit f:</s> <s xml:id="echoid-s42411" xml:space="preserve"> & ille ſit <lb/>aliquis punctus arcus dati, qui eſt a f h:</s> <s xml:id="echoid-s42412" xml:space="preserve"> & alter terminus lineę (qui eſt r) reflectatur à puncto h ar-<lb/>cus a f h.</s> <s xml:id="echoid-s42413" xml:space="preserve"> Dico quòd omnia puncta media lineæ g r reflectentur à punctis medijs arcus h f.</s> <s xml:id="echoid-s42414" xml:space="preserve"> Coapte-<lb/>tur enim linea g r (exempli cauſſa) diametro ſpeculi, q̃ ſit o a cadatq́;</s> <s xml:id="echoid-s42415" xml:space="preserve"> in ſemidiametrũ o e:</s> <s xml:id="echoid-s42416" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s42417" xml:space="preserve"> pũct{us} <lb/>z, ꝗ eſt cẽtrũ uiſus, in alia diametro eiuſdẽ circuli, quæ ſit d b, cadẽs in ſemidiametrũ e b:</s> <s xml:id="echoid-s42418" xml:space="preserve"> & ducãtur <lb/> <pb o="345" file="0647" n="647" rhead="LIBER OCTAVVS."/> lineæ g f, e f, z f, r h, e h, z h:</s> <s xml:id="echoid-s42419" xml:space="preserve"> & copuletur linea g z:</s> <s xml:id="echoid-s42420" xml:space="preserve"> producaturq́;</s> <s xml:id="echoid-s42421" xml:space="preserve"> linea ſe ultra punctum e ad lineam <lb/>g z in in punctum m:</s> <s xml:id="echoid-s42422" xml:space="preserve"> & ſignetur in linea grpunctus c.</s> <s xml:id="echoid-s42423" xml:space="preserve"> Dico quòd forma puncti c reflectetur ab aliquo <lb/>puncto arcus fh.</s> <s xml:id="echoid-s42424" xml:space="preserve"> Quòd enim reflectatur forma puncti c ad uiſum exiſtentem in puncto z palàm, cũ <lb/>extrema lineæ, quæ ſunt g & r, reflectantur ad uiſum exiſtentem in puncto z:</s> <s xml:id="echoid-s42425" xml:space="preserve"> fiet ergo reflexio ab a-<lb/>liquo puncto arcus a d, & non ab alio.</s> <s xml:id="echoid-s42426" xml:space="preserve"> Oſtenſum enim eſt per 20 huius quòd in hoc ſitu à duobus <lb/>arcubus a b & d o nõ poteſt fieri reflexio formæ pũcti c ad uiſum exiſtentẽ in pũcto z:</s> <s xml:id="echoid-s42427" xml:space="preserve"> oportet ergo <lb/>quòd fiat reflexio ab aliquo pũcto arcus a d:</s> <s xml:id="echoid-s42428" xml:space="preserve"> quoniam patet ſolùm offerri uiſui arcũ ſpeculi b a d o <lb/> <anchor type="figure" xlink:label="fig-0647-01a" xlink:href="fig-0647-01"/> per 72 th.</s> <s xml:id="echoid-s42429" xml:space="preserve"> 4 huius:</s> <s xml:id="echoid-s42430" xml:space="preserve"> ideo quòd cẽtrum uiſus eſt in pun-<lb/>cto z diametri d b.</s> <s xml:id="echoid-s42431" xml:space="preserve"> Oſtenſum etiam eſt per eandem 20 <lb/>huius quòd forma cuiuſcunq;</s> <s xml:id="echoid-s42432" xml:space="preserve"> pũcti ſemidiametri e o <lb/>reflectitur ab aliquo pũcto arcus a d:</s> <s xml:id="echoid-s42433" xml:space="preserve"> fit autem per 27 <lb/>huius formę cuiuslibet puncti lineæ g r reflexio ad ui <lb/>ſum ab uno tantùm pũcto arcus a d cadente inter ſe-<lb/>midiametros, in quibus non conſiſtut puncta reflexa <lb/>& ipſum centrum uiſus.</s> <s xml:id="echoid-s42434" xml:space="preserve"> Forma ergo pũcti c reflecte-<lb/>tur ab uno tantùm puncto arcus a d ad uiſum exiſten <lb/>tem in puncto z.</s> <s xml:id="echoid-s42435" xml:space="preserve"> Si ergo illud punctum ſit in arcu f h:</s> <s xml:id="echoid-s42436" xml:space="preserve"> <lb/>habemus propoſitum.</s> <s xml:id="echoid-s42437" xml:space="preserve"> Si nõ:</s> <s xml:id="echoid-s42438" xml:space="preserve"> eſto primò quòd ipſum <lb/>ſit in aliquo puncto arcus a f:</s> <s xml:id="echoid-s42439" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s42440" xml:space="preserve"> punctum u:</s> <s xml:id="echoid-s42441" xml:space="preserve"> & du-<lb/>cantur lineæ z u, c u, e u, g u:</s> <s xml:id="echoid-s42442" xml:space="preserve"> eſt ergo per 7 p 3 linea g u <lb/>maior quàm linea g f:</s> <s xml:id="echoid-s42443" xml:space="preserve"> ſed per eandẽ 7 p 3 linea z u eſt <lb/>minor quàm linea z f:</s> <s xml:id="echoid-s42444" xml:space="preserve"> ergo per 9 th.</s> <s xml:id="echoid-s42445" xml:space="preserve"> 1 huius proportio <lb/>lineæ gu ad lineam z u eſt maior proportione lineæ <lb/>g f ad lιneam f z:</s> <s xml:id="echoid-s42446" xml:space="preserve"> ſed per 3 p 6 & ex hypotheſi propor-<lb/>tio lineæ g f ad lineã f z eſt, ſicut proportio lineæ g m <lb/>ad lineam m z:</s> <s xml:id="echoid-s42447" xml:space="preserve"> proportio ergo lineæ g u ad lineam z u eſt maior quàm proportio lineæ g m ad linea <lb/>m z:</s> <s xml:id="echoid-s42448" xml:space="preserve"> linea ergo, quæ diuidit angulum g u z per æqualia, ſecat lineam z m:</s> <s xml:id="echoid-s42449" xml:space="preserve"> ſecat ergo lineam z e per 32 <lb/>th.</s> <s xml:id="echoid-s42450" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s42451" xml:space="preserve"> angulus ergo g u e eſt minor angulo e u z:</s> <s xml:id="echoid-s42452" xml:space="preserve"> ergo angulus c u e eſt multò minor angulo e <lb/>u z.</s> <s xml:id="echoid-s42453" xml:space="preserve"> Non ergo fiet reflexio formę puncti c ad uiſum z à puncto ſpeculiu, ut patet per 20 th.</s> <s xml:id="echoid-s42454" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s42455" xml:space="preserve"> <lb/>Similiter quoq;</s> <s xml:id="echoid-s42456" xml:space="preserve"> poteſt fieri deductio de quolibet puncto arcus a f.</s> <s xml:id="echoid-s42457" xml:space="preserve"> Forma ergo puncti c non reflecti <lb/>tur ad uiſum exiſtentem in puncto z ab aliquo puncto arcus a f.</s> <s xml:id="echoid-s42458" xml:space="preserve"> Sed neque ab aliquo puncto arcus <lb/>h d.</s> <s xml:id="echoid-s42459" xml:space="preserve"> Sit enim, ſi poſsibile eſt, ut reflectatur ab aliquo puncto arcus h d, & reflectatur à puncto eius, <lb/>quod ſit q:</s> <s xml:id="echoid-s42460" xml:space="preserve"> & ducantur lineæ z q, e q, c q, r q, z r:</s> <s xml:id="echoid-s42461" xml:space="preserve"> & producatur linea e h ultra punctũ e ad lineam r z:</s> <s xml:id="echoid-s42462" xml:space="preserve"> <lb/>incidatq́;</s> <s xml:id="echoid-s42463" xml:space="preserve"> in punctũ n:</s> <s xml:id="echoid-s42464" xml:space="preserve"> ergo per 7 p 3 linea z q eſt maior quàm linea z h, & linea q r eſt minor ꝗ̃ linea <lb/>rh:</s> <s xml:id="echoid-s42465" xml:space="preserve"> eſt ergo per 9 th.</s> <s xml:id="echoid-s42466" xml:space="preserve"> 1 huius proportio lineæ z q ad lineam q r maior proportione lineæ z h ad line-<lb/>am hr:</s> <s xml:id="echoid-s42467" xml:space="preserve"> ſed per 3 p 6 & ex hypotheſi, quæ eſt proportio lineæ z h ad lineam h r, eadem eſt lineę z n ad <lb/>lineam n r:</s> <s xml:id="echoid-s42468" xml:space="preserve"> eſt ergo proportio lineę z q ad lineam q r maior proportione lineæ z n ad lineam n r:</s> <s xml:id="echoid-s42469" xml:space="preserve"> li-<lb/>nea ergo diuidens angulum z q r per æqualia, ſecat lineam n r:</s> <s xml:id="echoid-s42470" xml:space="preserve"> ergo per 32 th.</s> <s xml:id="echoid-s42471" xml:space="preserve"> 1 huius ſecat lineã r e:</s> <s xml:id="echoid-s42472" xml:space="preserve"> <lb/>angulus ergo r q e eſt maior angulo e q z:</s> <s xml:id="echoid-s42473" xml:space="preserve"> angulus ergo c q e eſt multo maior angulo e q z.</s> <s xml:id="echoid-s42474" xml:space="preserve"> Non ergo <lb/>fiet reflexio formę puncti c ad uiſum in punctum z à puncto ſpeculi, quod eſt q, arcus h d.</s> <s xml:id="echoid-s42475" xml:space="preserve"> Eodemq́;</s> <s xml:id="echoid-s42476" xml:space="preserve"> <lb/>modo deducendum quocunq;</s> <s xml:id="echoid-s42477" xml:space="preserve"> puncto arcus h d dato.</s> <s xml:id="echoid-s42478" xml:space="preserve"> Forma ergo puncti c nõ reflectitur ad uiſum <lb/>exiſtentem in puncto z ex arcu h d:</s> <s xml:id="echoid-s42479" xml:space="preserve"> ſed neq;</s> <s xml:id="echoid-s42480" xml:space="preserve"> ex arcu a f, neq;</s> <s xml:id="echoid-s42481" xml:space="preserve"> ab aliquo punctorum h uel f, ut per 29 <lb/>th.</s> <s xml:id="echoid-s42482" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s42483" xml:space="preserve"> Omnia ergo puncta media lineæ g r reflectuntur à punctis medijs arcus h f:</s> <s xml:id="echoid-s42484" xml:space="preserve"> nec poſſunt <lb/>à punctis alijs reflecti, niſi fortè ab alio arcu reflectantur puncta g & r.</s> <s xml:id="echoid-s42485" xml:space="preserve"> Etex hoc patet quia tã lineæ <lb/>reflexionum punctorum mediorum, quàm catheti ſuarum incidentiarum concurrunt inter locai-<lb/>maginum punctorum extremorum.</s> <s xml:id="echoid-s42486" xml:space="preserve"> Et quia illarum linearum communis ſectio eſt locus imaginis <lb/>per 37 th.</s> <s xml:id="echoid-s42487" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s42488" xml:space="preserve"> patet ergo quòd loca imaginum punctorum mediorum cadunt inter loca imagi-<lb/>num pũctorum extrem orum.</s> <s xml:id="echoid-s42489" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s42490" xml:space="preserve"> Idem enim accidit, etiamſi res uiſa uel centrũ <lb/>uiſus extra illas ſpeculi diametros collocentur:</s> <s xml:id="echoid-s42491" xml:space="preserve"> quoniam ſemper trans illa puncta diam etri aliæ du <lb/>cipoſſunt.</s> <s xml:id="echoid-s42492" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s42493" xml:space="preserve"/> </p> <div xml:id="echoid-div1673" type="float" level="0" n="0"> <figure xlink:label="fig-0647-01" xlink:href="fig-0647-01a"> <variables xml:id="echoid-variables755" xml:space="preserve">o d q h f u a b y c r e n m z</variables> </figure> </div> </div> <div xml:id="echoid-div1675" type="section" level="0" n="0"> <head xml:id="echoid-head1253" xml:space="preserve" style="it">43. Siduorum punctorum in ſpeculo ſphærico concauo à duobus punctis ad unum uiſum fiat <lb/>reflexio, ſic quòd loca imaginum ſint in eadem ſpeculi diametro: maior erit proportio lineæ in-<lb/>teriacentis centrum ſpeculi & locum imaginis remotiorem, ad lineam interiacentem idem cen <lb/>trum & punctum reflexum à centro ſpeculi remotiorem, quàm lineæ interiacẽtis idem centrũ <lb/>& locum imaginis propinquiorem, ad lineam ductam à centro ad punctum reflexum centro <lb/>ſpeculi propinquiorem. Alhaz. 48 n 6.</head> <p> <s xml:id="echoid-s42494" xml:space="preserve">Sit ſpeculum ſphæricum concauum, per cuius centrum tranſeat ſuperficies plana:</s> <s xml:id="echoid-s42495" xml:space="preserve"> ſecabit ergo il <lb/>la ſuperficιem ſpeculi ſecundũ circulum magnum illius ſphærę per 69 th.</s> <s xml:id="echoid-s42496" xml:space="preserve"> 1 huius, qui a b g:</s> <s xml:id="echoid-s42497" xml:space="preserve"> & eius <lb/>centrũ ſit d:</s> <s xml:id="echoid-s42498" xml:space="preserve"> & extrahatur à centro d linea quocunq;</s> <s xml:id="echoid-s42499" xml:space="preserve"> modo placuerit, quę ſit d g:</s> <s xml:id="echoid-s42500" xml:space="preserve"> & tranſeat à centro <lb/>ad circũferentiã in pũctũ g:</s> <s xml:id="echoid-s42501" xml:space="preserve"> & ducatur à cẽtro d in ſuperficie illius circuli linea perpendicularis ſu-<lb/>per lineã d g, q̃ ſit d a:</s> <s xml:id="echoid-s42502" xml:space="preserve"> & abſcindatur ab angulo a d g recto parua particula quocũq;</s> <s xml:id="echoid-s42503" xml:space="preserve"> modo cõtingat:</s> <s xml:id="echoid-s42504" xml:space="preserve"> <lb/>& ſit angulus g d e, ita qđ inter angulũ rectũ, qui eſt a d g, & inter angulũ a d e ſit proportio multi-<lb/> <pb o="346" file="0648" n="648" rhead="VITELLONIS OPTICAE"/> plicitatis relatę and angulũ e d g.</s> <s xml:id="echoid-s42505" xml:space="preserve"> Hoc autem poteſt fieri, ſi angulus rectus, qui eſt a d g, diuidatur per <lb/>æqualia, & item eius medietas per æqualia, & ſic deinceps quouſq;</s> <s xml:id="echoid-s42506" xml:space="preserve"> fiat angulus a d e multiplex an-<lb/>guli e d g:</s> <s xml:id="echoid-s42507" xml:space="preserve"> ut ſi angulus a d e ſit ſeptuplus angulo e d g, erit rectus a d g ſeſquiſeptuplus angulo a d e:</s> <s xml:id="echoid-s42508" xml:space="preserve"> <lb/>& diuidatur angulus a d e in duo æqualia per lineam d b per 9 p 1.</s> <s xml:id="echoid-s42509" xml:space="preserve"> A puncto quoq;</s> <s xml:id="echoid-s42510" xml:space="preserve"> d centro ſpecu-<lb/>li extrahatur linea continens cum linea b d angulum rectum per 23 p 1, qui ſit angulus b d x:</s> <s xml:id="echoid-s42511" xml:space="preserve"> & extra <lb/>hatur linea a d ultra punctum d ad peripheriam, ut compleat diametrum:</s> <s xml:id="echoid-s42512" xml:space="preserve"> & ſit linea d k:</s> <s xml:id="echoid-s42513" xml:space="preserve"> & à puncto <lb/>d ducatur linea d z continens cum linea a d angulum æqualem angulo e d g, qui ſit angulus a d z:</s> <s xml:id="echoid-s42514" xml:space="preserve"> & <lb/>à puncto z ducatur linea ſuper lineam d z conſtituens angulum æqualem angulo k d x:</s> <s xml:id="echoid-s42515" xml:space="preserve"> qui ſit h z d <lb/>ducta linea z h ad diametrum h d k:</s> <s xml:id="echoid-s42516" xml:space="preserve"> hoc autem eſt poſsibile.</s> <s xml:id="echoid-s42517" xml:space="preserve"> Quia enim anguli k d x & a d z ſunt mi-<lb/>nores duobus rectis:</s> <s xml:id="echoid-s42518" xml:space="preserve"> erũt quoq;</s> <s xml:id="echoid-s42519" xml:space="preserve"> anguli a d z & h z d ęquales k d x, minores duobus rectis:</s> <s xml:id="echoid-s42520" xml:space="preserve"> ergo con <lb/>currẽt illæ lineæ, quæ ſunt a d & z h per 14 th.</s> <s xml:id="echoid-s42521" xml:space="preserve"> 1 hu-<lb/> <anchor type="figure" xlink:label="fig-0648-01a" xlink:href="fig-0648-01"/> ius:</s> <s xml:id="echoid-s42522" xml:space="preserve"> ſit concurſus punctus h.</s> <s xml:id="echoid-s42523" xml:space="preserve"> Et quia anguli trian-<lb/>guli ualent duos rectos per 32 p 1, & anguli a d z & <lb/>z d x & x d k ualent duos rectos per 13 p 1:</s> <s xml:id="echoid-s42524" xml:space="preserve"> angulus <lb/>uerò h z d eſt æqualis angulo x d k, & angulus <lb/>a d z communis:</s> <s xml:id="echoid-s42525" xml:space="preserve"> relinquitur angulus z h d ęqua-<lb/>lis angulo z d x.</s> <s xml:id="echoid-s42526" xml:space="preserve"> Et extrahatur à puncto z linea zl, <lb/>per 23 p 1 continens cum linea z h angulum ęqua-<lb/>lem angulo b d k obtuſo, qui ſit angulus h z l.</s> <s xml:id="echoid-s42527" xml:space="preserve"> Duo <lb/>ergo anguli l z d & b d z ſunt minores duobus re-<lb/>ctis:</s> <s xml:id="echoid-s42528" xml:space="preserve"> deficiunt enim à duob.</s> <s xml:id="echoid-s42529" xml:space="preserve"> rectis in angulo z d a:</s> <s xml:id="echoid-s42530" xml:space="preserve"> <lb/>linea ergo z l per 14 th.</s> <s xml:id="echoid-s42531" xml:space="preserve"> 1 huius cõcurret cum linea <lb/>d b:</s> <s xml:id="echoid-s42532" xml:space="preserve"> ſit concurſus punctus l:</s> <s xml:id="echoid-s42533" xml:space="preserve"> & ducatur linea lh:</s> <s xml:id="echoid-s42534" xml:space="preserve"> & <lb/>triangulo h ld circumſcribatur circulus per 5 p 4, <lb/>qui ſit circulus d h l.</s> <s xml:id="echoid-s42535" xml:space="preserve"> Trãſibit ergo ille circulus per <lb/>punctum z per 22 p 3:</s> <s xml:id="echoid-s42536" xml:space="preserve"> quia duo anguli l z h & l d h <lb/>ſunt æquales duobus rectis:</s> <s xml:id="echoid-s42537" xml:space="preserve"> ſunt autem illi anguli <lb/>in quadrilatero d h z l:</s> <s xml:id="echoid-s42538" xml:space="preserve"> eſt ergo illud quadrilaterũ <lb/>in circulo.</s> <s xml:id="echoid-s42539" xml:space="preserve"> Anguli ergo l h z & l d z ſunt æquales ք <lb/>27 p 3, cadunt enim in arcum eundem circuli d h l, <lb/>qui eſt arcus z l:</s> <s xml:id="echoid-s42540" xml:space="preserve"> ſed, ut ſuprà oſtendimus, angulus <lb/>z h d eſt æqualis angulo z d x:</s> <s xml:id="echoid-s42541" xml:space="preserve"> æqualibus ergo an-<lb/>gulis, qui ſunt l h z & l d z hinc inde ablatis, rema-<lb/>net angulus l h d æqualis angulo l d x:</s> <s xml:id="echoid-s42542" xml:space="preserve"> ſed angulus <lb/>l d x eſt rectus:</s> <s xml:id="echoid-s42543" xml:space="preserve"> angulus ergo l h d eſt rectus.</s> <s xml:id="echoid-s42544" xml:space="preserve"> Ab-<lb/>ſcindatur quoq;</s> <s xml:id="echoid-s42545" xml:space="preserve"> ex linea d e linea d m æqualis li-<lb/>neæ d h:</s> <s xml:id="echoid-s42546" xml:space="preserve"> & ducatur linea l m.</s> <s xml:id="echoid-s42547" xml:space="preserve"> Angulus ergo l m d <lb/>eſt rectus.</s> <s xml:id="echoid-s42548" xml:space="preserve"> Quia enim angulus b d e eſt ęqualis an-<lb/>gulo b d h:</s> <s xml:id="echoid-s42549" xml:space="preserve"> quoniam angulus a d e diuiſus fuit per <lb/>æqualia per lineam d b:</s> <s xml:id="echoid-s42550" xml:space="preserve"> linea quoq;</s> <s xml:id="echoid-s42551" xml:space="preserve"> d m eſt æqua-<lb/>lis lineæ d h:</s> <s xml:id="echoid-s42552" xml:space="preserve"> ſed latus l d eſt commune ambobus <lb/>trigonis l h d & l m d:</s> <s xml:id="echoid-s42553" xml:space="preserve"> ergo per 4 p 1 linea h l eſt æ-<lb/>qualis lineæ l m:</s> <s xml:id="echoid-s42554" xml:space="preserve"> & angulus l m d eſt ęqualis angu-<lb/>lo l h d:</s> <s xml:id="echoid-s42555" xml:space="preserve"> ſed angulus l h d oſtenſus eſt rectus eſſe:</s> <s xml:id="echoid-s42556" xml:space="preserve"> er <lb/>go angulus l m d eſt rectus.</s> <s xml:id="echoid-s42557" xml:space="preserve"> Ergo per 22 p 3 circu-<lb/>lus l h d tran ſit per punctum m:</s> <s xml:id="echoid-s42558" xml:space="preserve"> & ſecat arcum b e circuli a b g in puncto compari puncto z:</s> <s xml:id="echoid-s42559" xml:space="preserve"> qui ſit <lb/>punctus f:</s> <s xml:id="echoid-s42560" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s42561" xml:space="preserve"> linea l d diameter circuli l h d per 31 p 3:</s> <s xml:id="echoid-s42562" xml:space="preserve"> & ducatur linea d f.</s> <s xml:id="echoid-s42563" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s42564" xml:space="preserve"> circuli l h d <lb/>arcus d m eſt ęqualis arcui d h per 28 p 3.</s> <s xml:id="echoid-s42565" xml:space="preserve"> quoniam lineæ d m & d h ſunt æquales:</s> <s xml:id="echoid-s42566" xml:space="preserve"> ſed & arcus d f eſt <lb/>æqualis arcui d z per 64 th.</s> <s xml:id="echoid-s42567" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s42568" xml:space="preserve"> relin quitur ergo arcus m f æqualis arcui h z:</s> <s xml:id="echoid-s42569" xml:space="preserve"> & arcus l z ęqualis <lb/>arcui l f:</s> <s xml:id="echoid-s42570" xml:space="preserve"> ergo per 27 p 3 angulus l d ferit æqualis angulo l d z.</s> <s xml:id="echoid-s42571" xml:space="preserve"> Ducantur ergo lineæ h b, h f, z f, m f, <lb/>b m, b f.</s> <s xml:id="echoid-s42572" xml:space="preserve"> Et quia angulus l h d eſt rectus:</s> <s xml:id="echoid-s42573" xml:space="preserve"> patet quòd angulus b h d eſt a cutus:</s> <s xml:id="echoid-s42574" xml:space="preserve"> & angulus g d h eſt re <lb/>ctus:</s> <s xml:id="echoid-s42575" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s42576" xml:space="preserve"> 1 huius linea h b concurret cum linea d g extra circulũ a b g:</s> <s xml:id="echoid-s42577" xml:space="preserve"> concurrant ergo in <lb/>pũcto q.</s> <s xml:id="echoid-s42578" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s42579" xml:space="preserve"> per 14 th.</s> <s xml:id="echoid-s42580" xml:space="preserve"> 1 huius linea h f concurret cum linea d g extra circulum:</s> <s xml:id="echoid-s42581" xml:space="preserve"> ſit con-<lb/>curſus punctus n:</s> <s xml:id="echoid-s42582" xml:space="preserve"> & producatur linea f b ultra punctum b, quouſq;</s> <s xml:id="echoid-s42583" xml:space="preserve"> ſecet arcũ l z:</s> <s xml:id="echoid-s42584" xml:space="preserve"> ſecet ergo ipſum <lb/>in puncto r:</s> <s xml:id="echoid-s42585" xml:space="preserve"> & ducatur linea r m.</s> <s xml:id="echoid-s42586" xml:space="preserve"> Angulus ergo f r m (qui eſt in circumferentia) reſpicit ar-<lb/>cum f m, & angulus f b m eſt maior angulo f r m per 16 p 1:</s> <s xml:id="echoid-s42587" xml:space="preserve"> eſt enim extrinſecus in triangulo r b m:</s> <s xml:id="echoid-s42588" xml:space="preserve"> <lb/>& angulus f b m eſt in circumferentia circuli a b g:</s> <s xml:id="echoid-s42589" xml:space="preserve"> ergo ſi linea b m protrahatur ex parte puncti m, <lb/>ab ſcindet de circulo a b g arcũ maiorẽ quodã arcui, ſimili arcu f m circuli l h d ք 33 p 6:</s> <s xml:id="echoid-s42590" xml:space="preserve"> ſed arcus f m <lb/>in ſuo circulo l h d eſt ſimilis duplo arcus f e in circulo a b g:</s> <s xml:id="echoid-s42591" xml:space="preserve"> quoniã duplũ arcus f e correſpõdet du-<lb/>plo anguli f d e ſuper peripheriã ſui circuli conſtituti per 33 p 6, & per 20 p 3:</s> <s xml:id="echoid-s42592" xml:space="preserve"> eſt aũt arcus fe æqualis <lb/>arcui e g per 26 p 3:</s> <s xml:id="echoid-s42593" xml:space="preserve"> ideo quò d angulus e d g eſt æqualis angulo f d e:</s> <s xml:id="echoid-s42594" xml:space="preserve"> cũ uterq;</s> <s xml:id="echoid-s42595" xml:space="preserve"> ipſorũ ſit æqualis an-<lb/>gulo a d z, ut patet ex pręmiſsis:</s> <s xml:id="echoid-s42596" xml:space="preserve"> arcus ergo g f eſt duplus arcui f e:</s> <s xml:id="echoid-s42597" xml:space="preserve"> eſt ergo arcus f g in circulo a b g <lb/>ſimilis arcui f m in circulo l h d.</s> <s xml:id="echoid-s42598" xml:space="preserve"> Si ergo linea b m extrahatur rectè in partem m, abſcindet de circulo <lb/>a b g arcum ultra punctum g maiorem arcu f g.</s> <s xml:id="echoid-s42599" xml:space="preserve"> Si enim caderet in punctum g, fieret angulus f b g <lb/> <pb o="347" file="0649" n="649" rhead="LIBER OCTAVVS."/> æqualis angulo f r g, extrinſecus intrinſeco:</s> <s xml:id="echoid-s42600" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s42601" xml:space="preserve"> Linea ergo b m non cadetin pun <lb/>ctum g, ſed ſecabit lineam d g inter duo puncta g & d:</s> <s xml:id="echoid-s42602" xml:space="preserve"> ſecet ergo in puncto o.</s> <s xml:id="echoid-s42603" xml:space="preserve"> Producatur quoq;</s> <s xml:id="echoid-s42604" xml:space="preserve"> li-<lb/>nea f m ultra punctum m:</s> <s xml:id="echoid-s42605" xml:space="preserve"> hæc ergo, quia ſecat angulum d m o, patet per 29 th.</s> <s xml:id="echoid-s42606" xml:space="preserve"> 1 huius quia ſecabit li <lb/>neam d o:</s> <s xml:id="echoid-s42607" xml:space="preserve"> ſecet illã in pũcto u:</s> <s xml:id="echoid-s42608" xml:space="preserve"> & producatur linea m b ultra punctũ b:</s> <s xml:id="echoid-s42609" xml:space="preserve"> ſecabitq́ arcũ l r:</s> <s xml:id="echoid-s42610" xml:space="preserve"> ſecet ipſum <lb/>in puncto c:</s> <s xml:id="echoid-s42611" xml:space="preserve"> & ducatur linea c d à puncto c ad centrũ ſpeculi.</s> <s xml:id="echoid-s42612" xml:space="preserve"> Quia ergo angulus b f z eſt in circum-<lb/>ferentia circuli a b g, erit angulus b f z medietas anguli b d z ք zo p 3:</s> <s xml:id="echoid-s42613" xml:space="preserve"> ſed angulus b d z eſt multiplus <lb/>anguli z d a:</s> <s xml:id="echoid-s42614" xml:space="preserve"> ergo angulus b f z multiplus angulo z d h:</s> <s xml:id="echoid-s42615" xml:space="preserve"> ergo & angulus r f z eſt multiplus eidẽ:</s> <s xml:id="echoid-s42616" xml:space="preserve"> ergo <lb/>per 33 p 6 arcus r z eſt multiplus arcui z h:</s> <s xml:id="echoid-s42617" xml:space="preserve"> arcus uerò c z eſt maior arcu r z, ut totũ ſua parte:</s> <s xml:id="echoid-s42618" xml:space="preserve"> ergo ar-<lb/>cus c z eſt multiplus arcus z h, uel maior multiplo.</s> <s xml:id="echoid-s42619" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s42620" xml:space="preserve"> linea c h:</s> <s xml:id="echoid-s42621" xml:space="preserve"> angulus ergo c h d & an-<lb/>gulus c m d ſunt æquales duobus rectis per 22 p 3:</s> <s xml:id="echoid-s42622" xml:space="preserve"> ſed angulus b m d cũ angulo b m e ualet duos re-<lb/>ctos per 13 p 1:</s> <s xml:id="echoid-s42623" xml:space="preserve"> relin quitur ergo ut angulus c h d ſit æqualis angulo b m e:</s> <s xml:id="echoid-s42624" xml:space="preserve"> ſed angulus z h d addit ſu-<lb/>per angulum c h d angulũ c h z, qui eſt per 27 p 3 æqualis angulo c d z:</s> <s xml:id="echoid-s42625" xml:space="preserve"> & angulus c d z eſt multiplus <lb/>anguli z d a per 33 p 6:</s> <s xml:id="echoid-s42626" xml:space="preserve"> quoniã, ut ſuprà patuit, arcus c z eſt multiplus arcui z h:</s> <s xml:id="echoid-s42627" xml:space="preserve"> ergo angulus c h z eſt <lb/>multiplus anguli e d g:</s> <s xml:id="echoid-s42628" xml:space="preserve"> angulus ergo d h z excedit angulum c h d in multiplo anguli e d g.</s> <s xml:id="echoid-s42629" xml:space="preserve"> Et quia ar <lb/>cus f m d eſt æqualis arcuι z h d per 64 th.</s> <s xml:id="echoid-s42630" xml:space="preserve"> 1 huius, remanet arcus f z d æqualis arcui z f d:</s> <s xml:id="echoid-s42631" xml:space="preserve"> ergo erit <lb/>per 27 p 3 angulus f m d æqualis angulo z h d:</s> <s xml:id="echoid-s42632" xml:space="preserve"> ſed angulus c h d eſt æqualis b m e:</s> <s xml:id="echoid-s42633" xml:space="preserve"> ergo angulus f m d <lb/>excedit angulum b m e in multiplo anguli e g d:</s> <s xml:id="echoid-s42634" xml:space="preserve"> ſed angulus o m d eſt æqualis angulo b m e per 15 <lb/>p 1:</s> <s xml:id="echoid-s42635" xml:space="preserve"> ergo angulus f m d excedit angulum o m d in multiplo anguli e d g.</s> <s xml:id="echoid-s42636" xml:space="preserve"> Et quia angulus g o m ualet <lb/>angulum o m d, & angulũ o d m per 32 p 1:</s> <s xml:id="echoid-s42637" xml:space="preserve"> palàm quia angulus f m d excedit angulum m o g in mul-<lb/>tiplo anguli e d g:</s> <s xml:id="echoid-s42638" xml:space="preserve"> ſed angulus f m d per 32 p 1 excedit angulum m u d in ſolo angulo e d g:</s> <s xml:id="echoid-s42639" xml:space="preserve"> eſt ergo <lb/>angulus m u d maior angulo m o g:</s> <s xml:id="echoid-s42640" xml:space="preserve"> ergo angulus m o u eſt maior angulo m u o per 13 p 1 bιs ſumptá:</s> <s xml:id="echoid-s42641" xml:space="preserve"> <lb/>ergo per 19 p 1 linea m u eſt maior quàm linea m o.</s> <s xml:id="echoid-s42642" xml:space="preserve"> Et quia arcus h d eſt æqualis arcuι m d per præ-<lb/>miſſa, erũt duo anguli h f d & m f d æquales per 27 p 3.</s> <s xml:id="echoid-s42643" xml:space="preserve"> Formæ ergo punctorum duarum linearum h <lb/>f & f u ad ſeinuicem reflectuntur:</s> <s xml:id="echoid-s42644" xml:space="preserve"> & ſimiliter formæ punctorum linearum h b & b o ad ſe inuicẽ re-<lb/>flectuntur:</s> <s xml:id="echoid-s42645" xml:space="preserve"> quoniã per præmiſſa angulus d b h eſt æqualis angulo d b m per 4 p 1 & per hypotheſes <lb/>præmiſſas.</s> <s xml:id="echoid-s42646" xml:space="preserve"> Duo ergo puncta, quæ ſunt o & u ad uiſum exiſtentem in puncto h reflectũtur à duobus <lb/>punctis ſpeculi, quæ ſunt b & f.</s> <s xml:id="echoid-s42647" xml:space="preserve"> Eſt ergo per 37 th.</s> <s xml:id="echoid-s42648" xml:space="preserve"> 5 huius punctus q imago puncti o, & punctus n <lb/>imago puncti u.</s> <s xml:id="echoid-s42649" xml:space="preserve"> Ducatur ergo expũcto m linea æquidiſtans lineæ h q per 31 p 1:</s> <s xml:id="echoid-s42650" xml:space="preserve"> quæ ſit linea m s:</s> <s xml:id="echoid-s42651" xml:space="preserve"> & <lb/>linea ęquidiſtans lineę h n, quę ſit m p.</s> <s xml:id="echoid-s42652" xml:space="preserve"> Quia ergo angulus h n d eſt maior angulo h q d per 16 p 1, erit <lb/>angulus m p o, qui per 29 p 1 eſt æqualis angulo h n d, maior angulo m s o, qui per 29 p 1 eſt æqualis <lb/>h q d:</s> <s xml:id="echoid-s42653" xml:space="preserve"> erit ergo punctum p inter duo puncta s & u per conuerſam 21 p 1.</s> <s xml:id="echoid-s42654" xml:space="preserve"> Et quia angulus h d n eſt re-<lb/>ctus:</s> <s xml:id="echoid-s42655" xml:space="preserve"> erit per 32 p 1 angulus h n d acutus:</s> <s xml:id="echoid-s42656" xml:space="preserve"> ergo angulus m p d eſt acutus:</s> <s xml:id="echoid-s42657" xml:space="preserve"> angulus ergo m p s eſt obtu-<lb/>ſus per 13 p 1:</s> <s xml:id="echoid-s42658" xml:space="preserve"> ergo linea m s eſt maior quàm linea m p per 19 p 1.</s> <s xml:id="echoid-s42659" xml:space="preserve"> Sed ex pręmiſsis linea m u eſt maior <lb/>quàm linea m o:</s> <s xml:id="echoid-s42660" xml:space="preserve"> ergo per 9 th.</s> <s xml:id="echoid-s42661" xml:space="preserve"> 1 huius maior eſt proportio lineæ m s ad lineam m o quàm lineæ m p <lb/>ad lineam m u:</s> <s xml:id="echoid-s42662" xml:space="preserve"> ſed proportio lineæ s m ad lineam m o eſt, ſicut proportio lineæ q b ad b o per 4 p 6:</s> <s xml:id="echoid-s42663" xml:space="preserve"> <lb/>trigoni enim q b o & s m o ſunt æquianguli per 29 p 1:</s> <s xml:id="echoid-s42664" xml:space="preserve"> cum lineam s ſit æquidiſtans lineæ q b, & an-<lb/>gulus q o b ſit communis illis ambobus trigonis.</s> <s xml:id="echoid-s42665" xml:space="preserve"> Et ſimiliter proportio lineæ p m ad lineã m u eſt, <lb/>ſicut proportio lineæ n f ad lineam f u:</s> <s xml:id="echoid-s42666" xml:space="preserve"> per eadem ergo, quæ prius, & per 11 p 5 erit proportio lineæ <lb/>q b ad lineam b o maior proportione lineæ n f ad lineam f u:</s> <s xml:id="echoid-s42667" xml:space="preserve"> ſed proportio lineæ q b ad lineam b o <lb/>eſt, ſicut lineæ q d ad lineam d o:</s> <s xml:id="echoid-s42668" xml:space="preserve"> & proportio lineæ n f ad f u eſt, ſicut lineæ n d ad d u per ea, quæ <lb/>ſunt oſtenſa in 13 huius, quorum declarationem, cum manifeſta ſit, hic omittimus propter figuratio <lb/>nis multitudinem.</s> <s xml:id="echoid-s42669" xml:space="preserve"> Palàm ergo quòd proportio lineę q d ad lineam d o eſt maior proportione lineę <lb/>n d ad lineam d u.</s> <s xml:id="echoid-s42670" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s42671" xml:space="preserve"/> </p> <div xml:id="echoid-div1675" type="float" level="0" n="0"> <figure xlink:label="fig-0648-01" xlink:href="fig-0648-01a"> <variables xml:id="echoid-variables756" xml:space="preserve">q n p e f o g x u m l b c r z k d h a</variables> </figure> </div> </div> <div xml:id="echoid-div1677" type="section" level="0" n="0"> <head xml:id="echoid-head1254" xml:space="preserve" style="it">44. In ſpeculis ſphæricis concauis imagine retro ſpecu-<lb/>lum occurrente: maior erit diſtantia imaginis à ſpeculo quàm <lb/>reiuiſæ.</head> <figure> <variables xml:id="echoid-variables757" xml:space="preserve">n t l m s h s b k d e z a</variables> </figure> <p> <s xml:id="echoid-s42672" xml:space="preserve">Eſto ſpeculi ſphærici concaui circulus, qui a b g d:</s> <s xml:id="echoid-s42673" xml:space="preserve"> cuius cẽtrum <lb/>ſit e:</s> <s xml:id="echoid-s42674" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s42675" xml:space="preserve"> centrum uiſus z:</s> <s xml:id="echoid-s42676" xml:space="preserve"> & punctus rei uiſæ h:</s> <s xml:id="echoid-s42677" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s42678" xml:space="preserve"> reflexio for <lb/>mæ puncti h ad uiſum z à puncto ſpeculi b, appareatq́;</s> <s xml:id="echoid-s42679" xml:space="preserve"> imago retro <lb/>ſpeculum:</s> <s xml:id="echoid-s42680" xml:space="preserve"> dico quòd maior erit diſtantia imaginis à ſpeculi ſuperfi <lb/>cie quàm ipſius rei uiſæ.</s> <s xml:id="echoid-s42681" xml:space="preserve"> Ducantur enim lineæ h b incidentię, & z b <lb/>reflexionis:</s> <s xml:id="echoid-s42682" xml:space="preserve"> & ducatur cathetus incidentiæ, quæ ſit e h g t:</s> <s xml:id="echoid-s42683" xml:space="preserve"> produ-<lb/>catur quoque linea reflexionis, quæ z b, donec lineæ e h & z b <lb/>concurrant in puncto t:</s> <s xml:id="echoid-s42684" xml:space="preserve"> erit ergo per 37 th.</s> <s xml:id="echoid-s42685" xml:space="preserve"> 5 huius punctum t lo-<lb/>cus imaginis.</s> <s xml:id="echoid-s42686" xml:space="preserve"> Dico quòd linea t b (quæ eſt diſtantia imaginis à <lb/>ſpeculo) eſt maior quàm linea b h, quæ eſt diſtantia rei uiſæ à pun-<lb/>cto reflexionis.</s> <s xml:id="echoid-s42687" xml:space="preserve"> Et ſimiliter linea h g eſt minor quàm linea g t.</s> <s xml:id="echoid-s42688" xml:space="preserve"> Du-<lb/>catur enim linea e b:</s> <s xml:id="echoid-s42689" xml:space="preserve"> & à puncto b ducatur linea contingens cir-<lb/>culum in puncto b per 17 p 3:</s> <s xml:id="echoid-s42690" xml:space="preserve"> quæ ſit l b k.</s> <s xml:id="echoid-s42691" xml:space="preserve"> Quia itaque anguli <lb/>cõtingentiæ, qui ſunt a b k & g b l, ſunt æquales per 16 p 3:</s> <s xml:id="echoid-s42692" xml:space="preserve"> & anguli <lb/>z b a & h b g æquales per 20 th.</s> <s xml:id="echoid-s42693" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s42694" xml:space="preserve"> fit ergo angulus k b z æqualis angulo l b h:</s> <s xml:id="echoid-s42695" xml:space="preserve"> ſed angulus t b l <lb/> <pb o="348" file="0650" n="650" rhead="VITELLONIS OPTICAE"/> eſt æqualis angulo k b z per 15 p 1:</s> <s xml:id="echoid-s42696" xml:space="preserve"> angulus ergo t b l eſt æqualis angulo l b h:</s> <s xml:id="echoid-s42697" xml:space="preserve"> ſed angulus l b h eſt a-<lb/>cutus:</s> <s xml:id="echoid-s42698" xml:space="preserve"> quoniã angulus l b e eſt rectus:</s> <s xml:id="echoid-s42699" xml:space="preserve"> ergo & angulus t b l eſt acutus.</s> <s xml:id="echoid-s42700" xml:space="preserve"> Sed angulus e l b eſt acutus:</s> <s xml:id="echoid-s42701" xml:space="preserve"> <lb/>quoniã in trigono e b l angulus e b l eſt rectus:</s> <s xml:id="echoid-s42702" xml:space="preserve"> ergo per 13 p 1 angulus b l t eſt obtuſus:</s> <s xml:id="echoid-s42703" xml:space="preserve"> angulus itaq;</s> <s xml:id="echoid-s42704" xml:space="preserve"> <lb/>t b l eſt minor angulo b l t.</s> <s xml:id="echoid-s42705" xml:space="preserve"> Reſecetur quoq;</s> <s xml:id="echoid-s42706" xml:space="preserve"> ab angulo b l t angulus æqualis angulo b l h per 27 th.</s> <s xml:id="echoid-s42707" xml:space="preserve"> 1 <lb/>huius, qui ſit b l m.</s> <s xml:id="echoid-s42708" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s42709" xml:space="preserve"> angulus m b l eſt ęqualis angulo l b h, & angul{us} b l m ęqualis angulo b <lb/>l h:</s> <s xml:id="echoid-s42710" xml:space="preserve"> erũt per 32 p 1 trigona l b m & l b h æquiãgula:</s> <s xml:id="echoid-s42711" xml:space="preserve"> ergo ք 4 p 6 latera ipſorũ ſunt proportionalia:</s> <s xml:id="echoid-s42712" xml:space="preserve"> ſed <lb/>latus l b (cũ ſit commune ambobus) eſt æquale ſibijpſi:</s> <s xml:id="echoid-s42713" xml:space="preserve"> ergo latus m b eſt æquale lateri b h:</s> <s xml:id="echoid-s42714" xml:space="preserve"> ſed li-<lb/>nea m b eſt mιnor quàm linea b t:</s> <s xml:id="echoid-s42715" xml:space="preserve"> ergo linea h b eſt minor quàm linea b t.</s> <s xml:id="echoid-s42716" xml:space="preserve"> Et ꝗa linea l b diuidit an-<lb/>gulum t b h per æqualia:</s> <s xml:id="echoid-s42717" xml:space="preserve"> patet per 3 p 6 quoniã eſt proportio lineæ h l ad lineam l t, ſicut lineæ b h <lb/>ad lineam b t:</s> <s xml:id="echoid-s42718" xml:space="preserve"> ſed linea b h eſt minor quàm linea b t, ut patet ex præmiſsis:</s> <s xml:id="echoid-s42719" xml:space="preserve"> ergo & linea h l eſt minor <lb/>ꝗ̃ linea l t:</s> <s xml:id="echoid-s42720" xml:space="preserve"> linea ergo g h eſt multò minor ꝗ̃ linea g t.</s> <s xml:id="echoid-s42721" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s42722" xml:space="preserve"> Et ex his patet quòd rerũ, <lb/>quarum diſtãtia ab eodem uiſu maior eſt, uel augetur:</s> <s xml:id="echoid-s42723" xml:space="preserve"> etiã diſtantia imaginũ retro ſpeculũ retro ſpeculũ uiſarum <lb/>maior eſt uel augetur.</s> <s xml:id="echoid-s42724" xml:space="preserve"> Si enim protrahatur linea b h ultra punctum h ad punctum s, & producatur <lb/>cathetus e s, quouſq;</s> <s xml:id="echoid-s42725" xml:space="preserve"> concurrat cum linea reflexionis z b in puncto n:</s> <s xml:id="echoid-s42726" xml:space="preserve">erit punctum n locus imagi-<lb/>nis formæ puncti s:</s> <s xml:id="echoid-s42727" xml:space="preserve"> & erit linea b n maior quàm linea b s, ut prius patuit:</s> <s xml:id="echoid-s42728" xml:space="preserve"> & erunt lineæ b s & b n <lb/>maiores quàm lineæ b h & b t.</s> <s xml:id="echoid-s42729" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1678" type="section" level="0" n="0"> <head xml:id="echoid-head1255" xml:space="preserve" style="it">45. In concauis ſpeculis ſphæricis inter uiſum & ſpeculum imagine occurrente: nonnunquã <lb/>minor erit diſtantia imaginis à uiſu, quàm ſit ipſius rei uiſæ: à ſuperficie uerò ſperculi quando <lb/>erit minor: quando maior: quando æqualis.</head> <p> <s xml:id="echoid-s42730" xml:space="preserve">Eſto in ſpeculo ſphærico concauo circulus magnus a b g:</s> <s xml:id="echoid-s42731" xml:space="preserve"> cuius centrũ ſit d:</s> <s xml:id="echoid-s42732" xml:space="preserve"> & ſit ſemidiameter <lb/>d b:</s> <s xml:id="echoid-s42733" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s42734" xml:space="preserve"> centrum uiſus in puncto e:</s> <s xml:id="echoid-s42735" xml:space="preserve"> & linea rei uiſę ſit z t m:</s> <s xml:id="echoid-s42736" xml:space="preserve"> quæ reflectatur ad uiſum à pũcto ſpecu <lb/>li b:</s> <s xml:id="echoid-s42737" xml:space="preserve"> ſitq;</s> <s xml:id="echoid-s42738" xml:space="preserve"> linea incidentiæ z b, & linea reflexionis b e.</s> <s xml:id="echoid-s42739" xml:space="preserve"> Dico quòd uerum eſt, quod proponitur.</s> <s xml:id="echoid-s42740" xml:space="preserve"> Duca <lb/>tur enim per centrum d ad lineam reflexionis e b linea, quæ ſit t d h:</s> <s xml:id="echoid-s42741" xml:space="preserve"> & eſto ut ipſa ſit perpendicula-<lb/>ris ſuper ſemidiametrũ d b.</s> <s xml:id="echoid-s42742" xml:space="preserve"> Ducatur quoq;</s> <s xml:id="echoid-s42743" xml:space="preserve"> ſimiliter à pũcto rei uiſæ, qđ eſt z, linea z d:</s> <s xml:id="echoid-s42744" xml:space="preserve"> quę ꝓducta <lb/>ultra punctum d ad lineam reflexionis, quæ eſt e b, ſecet ipſam in puncto k:</s> <s xml:id="echoid-s42745" xml:space="preserve"> & ſimiliter à puncto ui-<lb/>ſo, quod eſt m, ducatur linea m d:</s> <s xml:id="echoid-s42746" xml:space="preserve"> quæ producta ad lineã reflexionis, quæ eſt e b, ſecet ipſam in pun <lb/>cto l.</s> <s xml:id="echoid-s42747" xml:space="preserve"> Eſt ergo per 37 th.</s> <s xml:id="echoid-s42748" xml:space="preserve"> 5 huius punctus k locus imaginis formæ puncti z:</s> <s xml:id="echoid-s42749" xml:space="preserve"> & punctus h locus ima-<lb/>ginis punctι t:</s> <s xml:id="echoid-s42750" xml:space="preserve"> & pũctus l locus imaginis pũcti m.</s> <s xml:id="echoid-s42751" xml:space="preserve"> Et palàm quia puncta k & h cadunt inter puncta e <lb/>& b:</s> <s xml:id="echoid-s42752" xml:space="preserve"> palamq́;</s> <s xml:id="echoid-s42753" xml:space="preserve"> cum loca imaginũ approximẽt uiſui, qui <lb/> <anchor type="figure" xlink:label="fig-0650-01a" xlink:href="fig-0650-01"/> eſt in pũcto e, ꝗ a multò minor erit diſtantia ipſarũima <lb/>ginum à uiſu, quàm ſit ipſius rei uiſæ.</s> <s xml:id="echoid-s42754" xml:space="preserve"> Quoniam enim <lb/>linea d b ſemper diuidit angulũ omnis reflexionis per <lb/>æqualia:</s> <s xml:id="echoid-s42755" xml:space="preserve"> patet quòd centrum uiſus & punctum rei uiſę <lb/>ſemper collocantur ex diuerſis partib.</s> <s xml:id="echoid-s42756" xml:space="preserve"> centri.</s> <s xml:id="echoid-s42757" xml:space="preserve"> Ducatur <lb/>quoq;</s> <s xml:id="echoid-s42758" xml:space="preserve"> linea e z:</s> <s xml:id="echoid-s42759" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s42760" xml:space="preserve"> in trigono k e z angulus e k z nõ-<lb/>nun quam maior angulo k z e:</s> <s xml:id="echoid-s42761" xml:space="preserve"> ergo per 19 p 1 erit tunc <lb/>linea e z (quæ eſt diſtantia rei uiſæ à centro uiſus) ma <lb/>ior quàm linea e k, quæ eſt diſtantia imaginis k à cẽtro <lb/>uiſus:</s> <s xml:id="echoid-s42762" xml:space="preserve"> minus aũt diſtant à uiſu loca imaginũ, quæ ſunt <lb/>h & l.</s> <s xml:id="echoid-s42763" xml:space="preserve"> Quia uerò in trigonis b d t & b d h duo anguli, ꝗ <lb/>ſunt b d t & b d h ſunt ęquales:</s> <s xml:id="echoid-s42764" xml:space="preserve"> quia recti ex hypotheſi:</s> <s xml:id="echoid-s42765" xml:space="preserve"> <lb/>& duo anguli h b d & t b d ſunt æquales per 20 th.</s> <s xml:id="echoid-s42766" xml:space="preserve"> 5 hu <lb/>ius, cum ſint anguli incidentiæ & reflexionis:</s> <s xml:id="echoid-s42767" xml:space="preserve"> erũt per <lb/>32 p 1 illi trigoni æquianguli:</s> <s xml:id="echoid-s42768" xml:space="preserve"> ergo per 4 p 6 cum linea <lb/>b d ſit æqualis ſibijpſi, erit linea b t æqualis lineæ b h.</s> <s xml:id="echoid-s42769" xml:space="preserve"> <lb/>Æqualiter ergo diſtabũtimago & res uiſa à ſuperficie ſpeculi.</s> <s xml:id="echoid-s42770" xml:space="preserve"> Sed linea b k eſt minor quàm linea <lb/>b h & linea b z eſt maior ꝗ̃ linea b k:</s> <s xml:id="echoid-s42771" xml:space="preserve"> erit ergo tũc locus imaginis <lb/>& imago ꝓ pinquior ſuperficiei ſpeculi ꝗ̃ res uiſa, cuius illa eſt imago.</s> <s xml:id="echoid-s42772" xml:space="preserve"> Et quia linea b m eſt minor ꝗ̃ <lb/>linea b l:</s> <s xml:id="echoid-s42773" xml:space="preserve"> eſt aũt pũctus l locus imaginis pũcti m:</s> <s xml:id="echoid-s42774" xml:space="preserve"> pater quòd res uiſa propinquior eſt ſpeculo ꝗ̃ eius <lb/>imago.</s> <s xml:id="echoid-s42775" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s42776" xml:space="preserve"> ꝓpoſitũ.</s> <s xml:id="echoid-s42777" xml:space="preserve"> Et ex his patet, quoniã rerũ q̃ magis elõgatę ſunt à ſpeculis, & quarũ for <lb/>mę reflectũtur ad uiſum, ita quò loca imaginũ ſint inter uiſum & ſpeculi ſuperficiẽ;</s> <s xml:id="echoid-s42778" xml:space="preserve">, fiunt imagines <lb/>ipſarũ propin quiores ſuperficiei ſpeculi, & elongatæ plus à cẽtro uiſus.</s> <s xml:id="echoid-s42779" xml:space="preserve"> Rerũ quo q̃ ſunt propin-<lb/>quiores ſpeculis, & quarũ formę reflectũtur ad uiſum, & loca imaginũ ſunt inter ſpeculũ & uiſum, <lb/>imagines plus elongantur à ſuperficie ſpeculi, & fiunt propinquiores ad uiſum.</s> <s xml:id="echoid-s42780" xml:space="preserve"/> </p> <div xml:id="echoid-div1678" type="float" level="0" n="0"> <figure xlink:label="fig-0650-01" xlink:href="fig-0650-01a"> <variables xml:id="echoid-variables758" xml:space="preserve">b m k t d h g l a z e</variables> </figure> </div> </div> <div xml:id="echoid-div1680" type="section" level="0" n="0"> <head xml:id="echoid-head1256" xml:space="preserve" style="it">46. Centro uiſus & re uiſa exiſtentibus intra ſpeculum ſphæricum cõcauum, in eadem linea <lb/>recta æqualiter à centro ſpeculi ſecundum ſui extrema diſtante: imago rei uiſæ uidebitur ultra <lb/>ſpeculum, maior re uiſa. Alhazen 39 n 6.</head> <p> <s xml:id="echoid-s42781" xml:space="preserve">Sit ſpeculũ ſphæricũ concauũ, cuius centrũ ſit a:</s> <s xml:id="echoid-s42782" xml:space="preserve"> dico qđ ſi centrũ uiſus fuerit intra ſpeculũ, & ſi-<lb/>militer linea uiſa:</s> <s xml:id="echoid-s42783" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s42784" xml:space="preserve"> illorũ diſp oſitio modo quo proponitur, uerũ eſſe qđ proponitur.</s> <s xml:id="echoid-s42785" xml:space="preserve"> Secetur e-<lb/>nim ſpeculũ ք ſuperficiẽ planã tranſeuntẽ ք cẽtrũ ſpeculi:</s> <s xml:id="echoid-s42786" xml:space="preserve"> erit ergo ք 69 th.</s> <s xml:id="echoid-s42787" xml:space="preserve"> 1 huius cõmunis ſectio <lb/>illius ſuքficiei planæ & ſuperficiei ſpeculi circulus, qui ſit b g:</s> <s xml:id="echoid-s42788" xml:space="preserve"> & ducatur in hoc circulo linea à cen-<lb/> <pb o="349" file="0651" n="651" rhead="LIBER OCTAVVS."/> tro ſpeculi ad circumferentiã, quocunq;</s> <s xml:id="echoid-s42789" xml:space="preserve"> modo contingat:</s> <s xml:id="echoid-s42790" xml:space="preserve"> & ſit linea a u, quæ diuidatur per æqualia <lb/>in puncto o:</s> <s xml:id="echoid-s42791" xml:space="preserve"> & à centro a ſecundum quãtitatem lineæ a o deſcribatur circulus, qui ſit e z:</s> <s xml:id="echoid-s42792" xml:space="preserve"> & in linea <lb/>o u ſignetur pũctus t, utcunq;</s> <s xml:id="echoid-s42793" xml:space="preserve"> contingat:</s> <s xml:id="echoid-s42794" xml:space="preserve"> & à puncto t ducantur lineę t n & t m perpẽdiculariter ſu-<lb/>per lineã a u per 11 p 1:</s> <s xml:id="echoid-s42795" xml:space="preserve"> & ducãtur à pũcto t lineæ t e & t z contingentes circulum e z per 17 p 3:</s> <s xml:id="echoid-s42796" xml:space="preserve"> & ſint <lb/>pũcta contactuũ e & z.</s> <s xml:id="echoid-s42797" xml:space="preserve"> Ducãtur quoq;</s> <s xml:id="echoid-s42798" xml:space="preserve"> à cẽtro ſpeculi pũcto a ad pũcta cõtactuũ lineæ a e & a z:</s> <s xml:id="echoid-s42799" xml:space="preserve"> quę <lb/>productę ſecent ſpeculũ in punctis b & g.</s> <s xml:id="echoid-s42800" xml:space="preserve"> Copulentur quoq;</s> <s xml:id="echoid-s42801" xml:space="preserve"> lineę t b & t g à pũcto t, & ducatur li-<lb/>nea b m æquidiſtans lineæ a u per 31 p 1:</s> <s xml:id="echoid-s42802" xml:space="preserve"> & linea g n ducatur æquidiſtãs eiſdẽ lineis a u & b m:</s> <s xml:id="echoid-s42803" xml:space="preserve"> & du-<lb/>cantur à centro ſpeculi ad puncta m & n lineæ a m & a n:</s> <s xml:id="echoid-s42804" xml:space="preserve"> quæ producantur ulterius extra circulum <lb/> <anchor type="figure" xlink:label="fig-0651-01a" xlink:href="fig-0651-01"/> g b.</s> <s xml:id="echoid-s42805" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s42806" xml:space="preserve"> linea a o eſt æqualis lineę o u:</s> <s xml:id="echoid-s42807" xml:space="preserve"> palàm quo-<lb/>niam linea a e eſt æqualis lineæ e b, & linea a z æqualis li-<lb/>neæ z g:</s> <s xml:id="echoid-s42808" xml:space="preserve"> oẽs enim diametri circuli e z ſunt medietates dia-<lb/>metrorũ circuli b g:</s> <s xml:id="echoid-s42809" xml:space="preserve"> ergo linea, quę interiacet circulos exi-<lb/>ens à cẽtro a, eſt æqualis ſemidiametro circuli e z.</s> <s xml:id="echoid-s42810" xml:space="preserve"> Et quia <lb/>linea t e cõtingit circulũ minorẽ, qui eſt e z:</s> <s xml:id="echoid-s42811" xml:space="preserve"> erit per 18 p 3 li <lb/>nea t e perpẽdicularis ſuper lineã b a:</s> <s xml:id="echoid-s42812" xml:space="preserve"> & ſimiliter erit linea <lb/>t z per pẽdicularis ſuper lineã g a:</s> <s xml:id="echoid-s42813" xml:space="preserve"> ergo per 4 p 1 linea t e exi <lb/>ſtente cõmuni ambobus trigonis b e t & t e a, erit linea b t <lb/>æqualis lineæ t a:</s> <s xml:id="echoid-s42814" xml:space="preserve"> & ſimiliter erit linea g t æqualis lineæ t a:</s> <s xml:id="echoid-s42815" xml:space="preserve"> <lb/>ergo per 5 p 1 in trigono t b a erit angulus t a b æqualis an-<lb/>gulo t b a:</s> <s xml:id="echoid-s42816" xml:space="preserve"> & in trigono t g a erit angulus t g a æqualis angu <lb/>lo t a g.</s> <s xml:id="echoid-s42817" xml:space="preserve"> Et quia linea b m eſt æquidiſtans lineæ a t:</s> <s xml:id="echoid-s42818" xml:space="preserve"> erit ք 29 <lb/>p 1 angulus m b a æqualis angulo t a b:</s> <s xml:id="echoid-s42819" xml:space="preserve"> quoniã ſunt coalter <lb/>ni:</s> <s xml:id="echoid-s42820" xml:space="preserve"> angulus ergo m b a æqualis eſt angulo a b t:</s> <s xml:id="echoid-s42821" xml:space="preserve"> & ſimiliter <lb/>angulus n g a æqualis eſt angulo a g t.</s> <s xml:id="echoid-s42822" xml:space="preserve"> Cũ ergo uiſus fuerit in pũcto t, & in linea m b fuerit aliquod <lb/>uiſibile (ut pũctũm) tunc forma pũcti m à pũcto ſpeculi, quod eſt b, reflectetur ad uiſum exiſtentẽ <lb/>in pũcto t:</s> <s xml:id="echoid-s42823" xml:space="preserve"> & forma pũcti n reflectetur à pũcto ſpeculi g ad uiſum exiſtentẽ in pũcto t.</s> <s xml:id="echoid-s42824" xml:space="preserve"> Viſus itaq;</s> <s xml:id="echoid-s42825" xml:space="preserve"> exi <lb/>ſtẽs in pũcto t cõprehẽdet formas pũctorum n & m reflexas ad ſe à pũctis ſpeculi g & b.</s> <s xml:id="echoid-s42826" xml:space="preserve"> Cõprehen-<lb/>det ergo eadẽ ratione & totã lineã n m reflexam ad ſe extoto arcu g b, ut patet per 42 huius.</s> <s xml:id="echoid-s42827" xml:space="preserve"> Etꝗa <lb/>linea m t eſt perpendicularis ſuper lineã a t:</s> <s xml:id="echoid-s42828" xml:space="preserve"> erit angulus m t b acutus.</s> <s xml:id="echoid-s42829" xml:space="preserve"> Quia enim angulus m t u eſt <lb/>rectus:</s> <s xml:id="echoid-s42830" xml:space="preserve"> ergo per 29 p 1 angulus b m t eſt rectus:</s> <s xml:id="echoid-s42831" xml:space="preserve"> ergo angulus m t b eſt acutus ք 32 p 1:</s> <s xml:id="echoid-s42832" xml:space="preserve"> ergo per 19 p 1 <lb/>erit linea t b maior ꝗ̃ linea b m.</s> <s xml:id="echoid-s42833" xml:space="preserve"> Sed, ut pręmiſſum eſt, linea t b eſt æqualis lineæ at:</s> <s xml:id="echoid-s42834" xml:space="preserve"> ergo linea at eſt <lb/>maior ꝗ̃ linea b m:</s> <s xml:id="echoid-s42835" xml:space="preserve"> ſed lineæ a t & b m ſunt ęquidiſtantes:</s> <s xml:id="echoid-s42836" xml:space="preserve"> ergo per 16 th.</s> <s xml:id="echoid-s42837" xml:space="preserve"> 1 huius linea t b cõcurret cũ <lb/>linea a m:</s> <s xml:id="echoid-s42838" xml:space="preserve"> concurrant ergo in puncto f:</s> <s xml:id="echoid-s42839" xml:space="preserve"> eſt itaq;</s> <s xml:id="echoid-s42840" xml:space="preserve"> ք 37 th.</s> <s xml:id="echoid-s42841" xml:space="preserve"> 5 huius pũctus flocus imaginis formæ pun-<lb/>ctim.</s> <s xml:id="echoid-s42842" xml:space="preserve"> Eodẽ quoq;</s> <s xml:id="echoid-s42843" xml:space="preserve"> modo linea t g cõcurret cũ linea a n in pũcto, qui ſit q:</s> <s xml:id="echoid-s42844" xml:space="preserve"> & erit punctus q locus ima <lb/>ginis formæ pũcti n:</s> <s xml:id="echoid-s42845" xml:space="preserve"> quoniã cathetus incidentiæ formæ pũcti m eſt linea a m, & cathetus inciden-<lb/>tię formę pũcti n eſt linea a n:</s> <s xml:id="echoid-s42846" xml:space="preserve"> lineæ quoq;</s> <s xml:id="echoid-s42847" xml:space="preserve"> reflexionis ſunt lineę t b & t g.</s> <s xml:id="echoid-s42848" xml:space="preserve"> Cõtinuẽtur itaq;</s> <s xml:id="echoid-s42849" xml:space="preserve"> pũcta f & <lb/>q per lineã f q:</s> <s xml:id="echoid-s42850" xml:space="preserve"> & erit linea f q diameter imaginis formę totius lineę n m.</s> <s xml:id="echoid-s42851" xml:space="preserve"> Et quia lineæ t e & t z ſunt <lb/>æquales per 58 th.</s> <s xml:id="echoid-s42852" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s42853" xml:space="preserve"> erũt anguli t a e & t a z æquales.</s> <s xml:id="echoid-s42854" xml:space="preserve"> Anguli enim t z a & t e a ſuntrecti per 18 <lb/>p 3, & lineæ z a & e a ſunt æquales, quia ſemidiametri eiuſdẽ circuli:</s> <s xml:id="echoid-s42855" xml:space="preserve"> linea quoq;</s> <s xml:id="echoid-s42856" xml:space="preserve"> t a eſt cõmunis am-<lb/>bobus trigonis t z a & t e a:</s> <s xml:id="echoid-s42857" xml:space="preserve"> ergo ք 8 p 1 anguli z t a & e t a ſunt æquales:</s> <s xml:id="echoid-s42858" xml:space="preserve"> & ſimiliter anguli t a e & t a z.</s> <s xml:id="echoid-s42859" xml:space="preserve"> <lb/>ſunt æquales:</s> <s xml:id="echoid-s42860" xml:space="preserve"> ergo & angulus t a b æqualis angulo t a g:</s> <s xml:id="echoid-s42861" xml:space="preserve"> ergo ք 4 p 1 erũt lineæ t b & t g æquales.</s> <s xml:id="echoid-s42862" xml:space="preserve"> Et <lb/>quia angulus e t a eſt æqualis angulo z t a:</s> <s xml:id="echoid-s42863" xml:space="preserve"> erit angulus u t b æqualis angulo u t g:</s> <s xml:id="echoid-s42864" xml:space="preserve"> relin quitur ergo <lb/>angulus b t m æqualis angulo g t n:</s> <s xml:id="echoid-s42865" xml:space="preserve"> quoniã anguli u t m & u t n ſunt æquales, quia recti:</s> <s xml:id="echoid-s42866" xml:space="preserve"> ſed & angu <lb/>li b m t & g n t ſunt recti:</s> <s xml:id="echoid-s42867" xml:space="preserve"> ergo trigona g t n & b t m ſunt ք 32 p 1 æquiangula.</s> <s xml:id="echoid-s42868" xml:space="preserve"> Ergo ք 4 p 6 cũ linea t g <lb/>ſit æqualis lineæ t b:</s> <s xml:id="echoid-s42869" xml:space="preserve"> erũt lineę b m & g n æquales, & linea t m æqualis lineę t n:</s> <s xml:id="echoid-s42870" xml:space="preserve">ergo ք 4 p 1 cũ angu <lb/>lin t a & m t a ſint recti & æquales, erũt lineæ a m & a n æquales:</s> <s xml:id="echoid-s42871" xml:space="preserve"> & ſic pũcta m & n ęqualiter diſta-<lb/>bunt à cẽtro ſpeculi, qđ eſt a:</s> <s xml:id="echoid-s42872" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s42873" xml:space="preserve"> ք 29 p 1 & 4 p 6 ꝓportio lineę a f ad lineã f m, ſicut lineę a t ad li-<lb/>neã b m:</s> <s xml:id="echoid-s42874" xml:space="preserve"> & erit ꝓportio lineæ a q ad lineã q n, ſicut lineę a t ad lineã g n:</s> <s xml:id="echoid-s42875" xml:space="preserve"> ſed ք 7 p 5 eadẽ eſt ꝓportio <lb/>lineæ a t ad lineam b m, & ad lineam g n:</s> <s xml:id="echoid-s42876" xml:space="preserve"> quoniã illę duę ſunt æquales:</s> <s xml:id="echoid-s42877" xml:space="preserve"> eadẽ ergo eſt proportio lineę <lb/>a f ad lineã f m, quę eſt lineę a q ad lineã q n:</s> <s xml:id="echoid-s42878" xml:space="preserve"> ergo ք 7 th.</s> <s xml:id="echoid-s42879" xml:space="preserve"> 1 huius erit euerſim eadẽ proportio lineæ a f <lb/>ad lineam a m, quæ eſt lineę a q ad lineã a n:</s> <s xml:id="echoid-s42880" xml:space="preserve"> ergo ք 16 p & corollariũ 4 p 5 erit permutatim ꝓportio <lb/>lineæ a q ad lineã a f, ſicut lineę a n ad lineã a m:</s> <s xml:id="echoid-s42881" xml:space="preserve"> ſed linea a m eſt æqualis lineę a n:</s> <s xml:id="echoid-s42882" xml:space="preserve"> ergo linea a f eſt ę-<lb/>qualis lineæ a q.</s> <s xml:id="echoid-s42883" xml:space="preserve"> Linea itaq;</s> <s xml:id="echoid-s42884" xml:space="preserve"> f q æquidiſtat lineę n m ք 2 p 6.</s> <s xml:id="echoid-s42885" xml:space="preserve"> Ergo linea f q eſt maior quàm linea n m.</s> <s xml:id="echoid-s42886" xml:space="preserve"> <lb/>Si itaq;</s> <s xml:id="echoid-s42887" xml:space="preserve"> centrũ uiſus fuerit in puncto t, & in linea n m fuerit aliquod uiſibile:</s> <s xml:id="echoid-s42888" xml:space="preserve"> tũc uiſus cõprehendet <lb/>imaginẽ illius uiſibilis maiorẽ ꝗ̃ ſit ſecũdum ueritatẽ.</s> <s xml:id="echoid-s42889" xml:space="preserve"> Et hoc eſt propoſitũ.</s> <s xml:id="echoid-s42890" xml:space="preserve"> Et ſi arcus cuiuſcũq;</s> <s xml:id="echoid-s42891" xml:space="preserve"> cir-<lb/>culi copulentur ad has chordas n m & q f:</s> <s xml:id="echoid-s42892" xml:space="preserve"> patet idem de arcubus, quod de lineis rectis.</s> <s xml:id="echoid-s42893" xml:space="preserve"/> </p> <div xml:id="echoid-div1680" type="float" level="0" n="0"> <figure xlink:label="fig-0651-01" xlink:href="fig-0651-01a"> <variables xml:id="echoid-variables759" xml:space="preserve">f u q b s m l n c o z q</variables> </figure> </div> </div> <div xml:id="echoid-div1682" type="section" level="0" n="0"> <head xml:id="echoid-head1257" xml:space="preserve" style="it">47. Centro uiſus & re uiſa oppoſitis ſpeculo ſphærico concauo taliter, ut uiſus ſit altior re uiſa <lb/>ſecundum ſui extrema æqualiter diſtante à centro ſpeculi: imago lineæ uiſæ uidebitur ultra ſpe-<lb/>culum, maior re uiſa. Alhazen 40 n 6.</head> <p> <s xml:id="echoid-s42894" xml:space="preserve">Sit circulus ſpeculi ſphærici cõcaui, ſicut in pręmiſſa, qui eſt b g:</s> <s xml:id="echoid-s42895" xml:space="preserve"> cuius centrũ a:</s> <s xml:id="echoid-s42896" xml:space="preserve"> & ducantur lineę <lb/>à centro circuli a ad peripheriã, quę ſint a b, a g, a u:</s> <s xml:id="echoid-s42897" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s42898" xml:space="preserve"> linea a u diuidens per ęqualia arcũ g b:</s> <s xml:id="echoid-s42899" xml:space="preserve"> quę <lb/>diuidatur, ut in pręcedẽte, ſecũdũ punctũ tultra ſui mediũ uerſus circũferẽtiã g b:</s> <s xml:id="echoid-s42900" xml:space="preserve"> & ducãtur lineæ <lb/> <pb o="350" file="0652" n="652" rhead="VITELLONIS OPTICAE"/> g t & t b:</s> <s xml:id="echoid-s42901" xml:space="preserve"> & erigatur à puncto t linea perpẽdiculairs ſuper ſuperficiẽ circuli ք 12 p 11, quæ ſit linea t k:</s> <s xml:id="echoid-s42902" xml:space="preserve"> <lb/>& ducantur lineæ a k, b k & g k.</s> <s xml:id="echoid-s42903" xml:space="preserve"> Superficies itaq;</s> <s xml:id="echoid-s42904" xml:space="preserve"> trigonorum k b a, k g a ſunt ſecantes ſphærã ſpecu-<lb/>li ſuper cẽtrum a:</s> <s xml:id="echoid-s42905" xml:space="preserve"> & ſunt erectæ ſuper ſuperficiẽ circuli b g per 18 p 11 & ſuper oẽs ſuperficies cõtin-<lb/>gentes ſphærã in punctis b & g, uel quibuſcũq;</s> <s xml:id="echoid-s42906" xml:space="preserve"> pũctis alijs circulorũ, qui ſunt cõmunis ſectio illarũ <lb/>ſuperficierũ & ſpeculi ք 2 huius.</s> <s xml:id="echoid-s42907" xml:space="preserve"> Quoniã enim cõmunes ſectiones circuli b g & ſuperficierũ illorũ <lb/>trigonorũ ſunt ſemidiametri a b & a g, qui ſunt erecti ſuperficies in illis pũctis b & g ſpeculũ <lb/>cõtingẽtes:</s> <s xml:id="echoid-s42908" xml:space="preserve"> patet quòd illæ ſuperficies per 18 p 11 ſunt erectæ ſuper ſuperficies in illis punctis cõtin <lb/>gentes.</s> <s xml:id="echoid-s42909" xml:space="preserve"> Et ſimiliter patet hoc de alijs ſuperficiebus ſecũdum puncta illorum circulorum contingen <lb/>tibus.</s> <s xml:id="echoid-s42910" xml:space="preserve"> In illis itaq;</s> <s xml:id="echoid-s42911" xml:space="preserve"> ſuperficiebus fit reflexio à punctis circũferentiæ circulorũ cõmunium eis & ſpe-<lb/>culo.</s> <s xml:id="echoid-s42912" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s42913" xml:space="preserve"> linea b m in ſuperficie b k a æquidiſtanter lineę a k:</s> <s xml:id="echoid-s42914" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s42915" xml:space="preserve"> linea b m minor ꝗ̃ linea <lb/>a k:</s> <s xml:id="echoid-s42916" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s42917" xml:space="preserve"> taliter, ut linea b m tota penetret ſuperficiẽ circuli b g ad partẽ aliã, ꝗ̃ linea t k, ita ut lineæ <lb/>t k & b m ſint in diuerſis partibus ſpeculi reſectis ք ſuperficiem circuli b g.</s> <s xml:id="echoid-s42918" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s42919" xml:space="preserve"> linea a m:</s> <s xml:id="echoid-s42920" xml:space="preserve"> <lb/>& extrahantur lineę b k & a m, donec cõcurrant in puncto f:</s> <s xml:id="echoid-s42921" xml:space="preserve"> cõcurrent aũt per 16 th.</s> <s xml:id="echoid-s42922" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s42923" xml:space="preserve"> cum li-<lb/>nea b m ſit minor ꝗ̃ ſua æquidiſtãs linea a k:</s> <s xml:id="echoid-s42924" xml:space="preserve"> & in ſuperficie g k a ducatur linea g n æquidiſtans lineę <lb/>a k:</s> <s xml:id="echoid-s42925" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s42926" xml:space="preserve"> linea g n æqualis lineæ b m, & ad eandẽ partẽ ſuperficiei circuli producta:</s> <s xml:id="echoid-s42927" xml:space="preserve"> & ducatur linea <lb/>a n:</s> <s xml:id="echoid-s42928" xml:space="preserve"> producanturq́;</s> <s xml:id="echoid-s42929" xml:space="preserve"> lineæ a n & k g, donec per 16 th.</s> <s xml:id="echoid-s42930" xml:space="preserve"> 1 huius concurrant in puncto q:</s> <s xml:id="echoid-s42931" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s42932" xml:space="preserve"> li-<lb/>nea f q, & linea m n.</s> <s xml:id="echoid-s42933" xml:space="preserve"> Quia ergo (ut in pręcedente proxi-<lb/> <anchor type="figure" xlink:label="fig-0652-01a" xlink:href="fig-0652-01"/> ma oſtẽdimus) linea b t eſt æqualis lineæ t a, & linea t k <lb/>eſt communis duobus trigonis b k t & a k t, & anguli ad <lb/>pũctũ t ſunt recti per definitionẽ lineę ſuper ſuperficiẽ <lb/>erectæ:</s> <s xml:id="echoid-s42934" xml:space="preserve"> palàm ք 4 p 1 quia linea b k eſt æqualis lineæ k a:</s> <s xml:id="echoid-s42935" xml:space="preserve"> <lb/>& ք eadem erit linea g k æqualis lineę a k:</s> <s xml:id="echoid-s42936" xml:space="preserve"> ergo per 5 p 1 <lb/>anguli k a b & k b a ſunt æqules:</s> <s xml:id="echoid-s42937" xml:space="preserve"> & ſimiliter ſunt anguli <lb/>k a g & k g a æqualis.</s> <s xml:id="echoid-s42938" xml:space="preserve"> Itẽ quia linea g k eſt æqualis lineæ <lb/>a k:</s> <s xml:id="echoid-s42939" xml:space="preserve"> igitur linea g k ęqualis eſt lineę b k:</s> <s xml:id="echoid-s42940" xml:space="preserve"> ſed & linea a g eſt <lb/>æqualis lineę a b:</s> <s xml:id="echoid-s42941" xml:space="preserve"> quia ſunt ſemidiametri eiuſdem circu <lb/>li:</s> <s xml:id="echoid-s42942" xml:space="preserve"> & linea a k eſt cõmunis:</s> <s xml:id="echoid-s42943" xml:space="preserve"> trigona itaq;</s> <s xml:id="echoid-s42944" xml:space="preserve"> a k b & a k g ſunt <lb/>æquilatera:</s> <s xml:id="echoid-s42945" xml:space="preserve"> ergo per 8 p 1 angulus k b a eſt æqualis angu <lb/>l o k g a, & angulus k a b ęqualis angulo k a g.</s> <s xml:id="echoid-s42946" xml:space="preserve"> Et quoniã <lb/>per 29 p 1 angulus a b m eſt æqualis angulo k a b:</s> <s xml:id="echoid-s42947" xml:space="preserve"> ergo & <lb/>angulo k b a:</s> <s xml:id="echoid-s42948" xml:space="preserve"> quia lineæ a k & b m æquidiſtant, & iſti an-<lb/>guliſunt coalterni.</s> <s xml:id="echoid-s42949" xml:space="preserve"> Et ſimiliter angulus a g n eſt propter <lb/>eadem ęqualis angulo k a g:</s> <s xml:id="echoid-s42950" xml:space="preserve"> quoniã etiã lineę a k & g n <lb/>æquidiſtant:</s> <s xml:id="echoid-s42951" xml:space="preserve"> ergo & angulo k g a.</s> <s xml:id="echoid-s42952" xml:space="preserve"> Et quonιam anguli k <lb/>a g & k a b ſunt æquales, ut pręoſtenſum eſt:</s> <s xml:id="echoid-s42953" xml:space="preserve"> erit ergo angulus a b m æqualis angulo a g n, & linea <lb/>b m ex hypotheſi eſt æqualis lineę g n:</s> <s xml:id="echoid-s42954" xml:space="preserve"> ergo per 4 p 1 linea a m eſt æqualis lineæ a n:</s> <s xml:id="echoid-s42955" xml:space="preserve"> ergo, utin præ-<lb/>miſſa, linea a f erit æqualis lineæ a q:</s> <s xml:id="echoid-s42956" xml:space="preserve"> ergo per 2 p 6 linea q f æquidiſtat lineę m n:</s> <s xml:id="echoid-s42957" xml:space="preserve"> & linea f q eſt ma-<lb/>ior quàm linea m n.</s> <s xml:id="echoid-s42958" xml:space="preserve"> Cum itaque uiſus fuerit in puncto k uel ſuper punctum k in linea t k:</s> <s xml:id="echoid-s42959" xml:space="preserve"> & fuerit li <lb/>nea m n in aliquo uiſibili inferiore ipſo uiſu:</s> <s xml:id="echoid-s42960" xml:space="preserve"> tunc forma puncti m incidet ſpeculo ſecũdum lineam <lb/>m b, & reflectetur à puncto ſpeculi b ad uiſum ſecundum lineam b k in ſuperficie circuli tranſeun-<lb/>tis per puncta b, a, k:</s> <s xml:id="echoid-s42961" xml:space="preserve"> & forma puncti n incidet ſpeculo ſecũdum lineam n g, & à puncto ſpeculi g re-<lb/>flectetur ad uiſum ſecundum lineam g k inſuperficie circuli tranſeuntis per puncta g, a, k:</s> <s xml:id="echoid-s42962" xml:space="preserve"> & erit per <lb/>37 th.</s> <s xml:id="echoid-s42963" xml:space="preserve"> 5 huius imago puncti m punctum f:</s> <s xml:id="echoid-s42964" xml:space="preserve"> & imago puncti n punctum q:</s> <s xml:id="echoid-s42965" xml:space="preserve"> & erit linea q f diameter <lb/>imaginis lineæ n m:</s> <s xml:id="echoid-s42966" xml:space="preserve"> & linea f q erit maior quàm linea m n.</s> <s xml:id="echoid-s42967" xml:space="preserve"> Imago itaque rei uiſæ apparebit maior <lb/>ipſa re uiſa, & ultra ſpeculum.</s> <s xml:id="echoid-s42968" xml:space="preserve"> In hoc ergo ſitu uiſus & uiſibilis patet propoſitum.</s> <s xml:id="echoid-s42969" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s42970" xml:space="preserve"> reuolua-<lb/>tur tota figura in circuitu lineę a u, ipſa linea a u permanente immobili:</s> <s xml:id="echoid-s42971" xml:space="preserve"> tunc punctum k deſcribet <lb/>motu ſuo quendam circulum, ſuper quem erecta eſt linea a u tranſiens ad utramq;</s> <s xml:id="echoid-s42972" xml:space="preserve"> partem ſuperfi-<lb/>ciei illius circuli:</s> <s xml:id="echoid-s42973" xml:space="preserve"> & omne punctum illius circuli habebit ſitum reſpectu lineæ comparis lineæ m n.</s> <s xml:id="echoid-s42974" xml:space="preserve"> <lb/>Si itaque uiſus fuerit in aliquo puncto circumferentię huius circuli, & linea compar lineę m n fue-<lb/>ritin ſuperficie alicuius rei uiſę, reſpicientis centrum uiſus ſecundum illum ſitum, ut res uiſa (in <lb/>qua eſt linea m n) reſpiciebat uiſum exiſtentem in puncto k:</s> <s xml:id="echoid-s42975" xml:space="preserve"> tunc uiſus comprehendet formam il-<lb/>lius lineę maiorem ſua propria quantitate.</s> <s xml:id="echoid-s42976" xml:space="preserve"> Et ſimiliter ſi extrahatur linea t k in continuum & dire-<lb/>ctum:</s> <s xml:id="echoid-s42977" xml:space="preserve"> & ſignetur in ea punctum aliud pręter punctum k, ut punctum p:</s> <s xml:id="echoid-s42978" xml:space="preserve"> & ducantur lineę ad illud <lb/>punctum p, ſicut ad punctum k ſunt prius ductę:</s> <s xml:id="echoid-s42979" xml:space="preserve"> erit idem eueniens, quod prius accidit in puncto <lb/>k.</s> <s xml:id="echoid-s42980" xml:space="preserve"> Pluries itaq;</s> <s xml:id="echoid-s42981" xml:space="preserve">, ut patet per pręſens theorema, & per proximè præmiſſum, in ſpeculis ſphæricis con <lb/>cauis uidetur imago rei uiſę maior ipſa re uiſa:</s> <s xml:id="echoid-s42982" xml:space="preserve"> quod eſt notandum.</s> <s xml:id="echoid-s42983" xml:space="preserve"/> </p> <div xml:id="echoid-div1682" type="float" level="0" n="0"> <figure xlink:label="fig-0652-01" xlink:href="fig-0652-01a"> <variables xml:id="echoid-variables760" xml:space="preserve">f q b u g m l n k p a</variables> </figure> </div> </div> <div xml:id="echoid-div1684" type="section" level="0" n="0"> <head xml:id="echoid-head1258" xml:space="preserve" style="it">48. In ſpeculis ſphæricis conauis quando comprehendituringago æqualis ipſirei uiſæ: quæ <lb/>occurrens inter uiſum & ſpeculum, conuerſum: retro uiſum uerò conformem habet ſitum rei <lb/>uiſæ. Alhazen 41 n 6.</head> <p> <s xml:id="echoid-s42984" xml:space="preserve">Sit ſpeculum ſphęricum concauũ a b:</s> <s xml:id="echoid-s42985" xml:space="preserve"> cuius centrũ ſit e:</s> <s xml:id="echoid-s42986" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s42987" xml:space="preserve"> ipſum ſuperficies plana tranſiens <lb/>centrũ e, cuius cõmunis ſectio & ſuperficiei ſpeculi erit circulus per 69 th.</s> <s xml:id="echoid-s42988" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s42989" xml:space="preserve"> ꝗ ſit a b:</s> <s xml:id="echoid-s42990" xml:space="preserve"> & duca-<lb/>tur à centro linea e z, utcũq;</s> <s xml:id="echoid-s42991" xml:space="preserve"> contingit, nõ in ipſa ſuperficie circuli a b, ſed obliquè ſuper illam, ſicut <lb/> <pb o="351" file="0653" n="653" rhead="LIBER OCTAVVS."/> placet:</s> <s xml:id="echoid-s42992" xml:space="preserve"> quę producatur ultra circuli peripheriã ad pũctum g:</s> <s xml:id="echoid-s42993" xml:space="preserve"> & à pũcto g extrahatur linea perpen di <lb/>cularis ſuper ſuperficiẽ circuli a b per 12 p 11:</s> <s xml:id="echoid-s42994" xml:space="preserve"> & in illa perpendiculari ſignetur pũctum d:</s> <s xml:id="echoid-s42995" xml:space="preserve"> & ducatur <lb/>linea d e:</s> <s xml:id="echoid-s42996" xml:space="preserve"> quę protrahatur ultra centrũ e ad pũctum o:</s> <s xml:id="echoid-s42997" xml:space="preserve"> & ducatur linea e b cõtinens cũ linea d e an-<lb/>gulũ obtuſum:</s> <s xml:id="echoid-s42998" xml:space="preserve"> & ducatur linea e a cõtinens cũ linea e d angulũ obtuſum æqualẽ angulo d e b per 23 <lb/>p 1:</s> <s xml:id="echoid-s42999" xml:space="preserve"> & ducãtur lineę d a, d b:</s> <s xml:id="echoid-s43000" xml:space="preserve"> erũtq́;</s> <s xml:id="echoid-s43001" xml:space="preserve"> per 4 p 1 trigona d e a & d e b æ quiangula.</s> <s xml:id="echoid-s43002" xml:space="preserve"> Superficies itaq;</s> <s xml:id="echoid-s43003" xml:space="preserve"> duo-<lb/>rũ trigonorũ d e a & d e b ſecãt ſe ſuper lineam d e:</s> <s xml:id="echoid-s43004" xml:space="preserve"> & duo anguli d b e & d a e ſunt acuti & æ quales <lb/>per 4 p 1:</s> <s xml:id="echoid-s43005" xml:space="preserve"> linea enim e b eſt ęqualis lineę e a, & linea d e eſt cõmunis ambobus trigonis d e a & d e b:</s> <s xml:id="echoid-s43006" xml:space="preserve"> <lb/>& anguli d e b & d e a ſunt æquales.</s> <s xml:id="echoid-s43007" xml:space="preserve"> A pũcto quoq;</s> <s xml:id="echoid-s43008" xml:space="preserve"> b in ſuperfi <lb/> <anchor type="figure" xlink:label="fig-0653-01a" xlink:href="fig-0653-01"/> cie trianguli d e b ducatur per 23 p 1 linea continens cũ linea e b <lb/>angulũ æqualem angulo d b e:</s> <s xml:id="echoid-s43009" xml:space="preserve"> quę ſit linea b o:</s> <s xml:id="echoid-s43010" xml:space="preserve"> hæc igitur linea <lb/>cõcurret cũ linea d e per 14 th.</s> <s xml:id="echoid-s43011" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43012" xml:space="preserve"> ideo quòd angulus b e d <lb/>eſt obtuſus, & angulus e b o, qui eſt apud pũctum b, eſt acutus, <lb/>nõ ualẽs cũ angulo d e b duos rectos:</s> <s xml:id="echoid-s43013" xml:space="preserve"> cũ angulus o b e ſit æqua-<lb/>lis angulo d b e, qui cũ angulo b e d & angulo b d e ualet duos <lb/>rectos ք 32 p 1.</s> <s xml:id="echoid-s43014" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s43015" xml:space="preserve"> linearũ d e & b o cõcurſus in pũcto o:</s> <s xml:id="echoid-s43016" xml:space="preserve"> & <lb/>à pũcto a ducatur linea in ſuperficie trianguli d e a cõtinens cũ <lb/>linea a e angulũ ęqualẽ angulo d a e:</s> <s xml:id="echoid-s43017" xml:space="preserve"> cõcurret ergo illa, ut prius, <lb/>cũ linea e o in pũcto o:</s> <s xml:id="echoid-s43018" xml:space="preserve"> quoniá anguli a e o & b e o ք 13 p 1 & ex <lb/>pręmiſsis ſunt ęquales:</s> <s xml:id="echoid-s43019" xml:space="preserve"> & anguli e b o & e a o ex pręmiſsis inter <lb/>ſe ſunt æquales:</s> <s xml:id="echoid-s43020" xml:space="preserve"> ergo ք 32 p 1 anguli reliqui, ꝗ ſunt e o b & e o a, <lb/>ſunt æquales:</s> <s xml:id="echoid-s43021" xml:space="preserve"> ergo per 4 p 6 latera ipſorũ ſunt proportionalia:</s> <s xml:id="echoid-s43022" xml:space="preserve"> <lb/>ſed linea e a eſt æqualis lineæ e b:</s> <s xml:id="echoid-s43023" xml:space="preserve"> ergo linea e o eſt æqualis ſibi-<lb/>ipſi:</s> <s xml:id="echoid-s43024" xml:space="preserve"> cadũt ergo lineę b o & a o in unũ punctũ lineę d e ꝓductæ, <lb/>qui eſt o.</s> <s xml:id="echoid-s43025" xml:space="preserve"> Ducatur etiam linea e t a d lineá b d, ita quòd cõtineat <lb/>cũ linea e b angulũ rectũ per 11 p 1:</s> <s xml:id="echoid-s43026" xml:space="preserve"> & protrahatur linea t e ultra <lb/>pũctum e, & linea b o ultra pũctum o:</s> <s xml:id="echoid-s43027" xml:space="preserve"> cõcurrẽtq́;</s> <s xml:id="echoid-s43028" xml:space="preserve"> lineæ t e & b o <lb/>per 14 th.</s> <s xml:id="echoid-s43029" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43030" xml:space="preserve"> quia cũ angulus b e t ſit rectus, angulus e b o <lb/>eſt acutus:</s> <s xml:id="echoid-s43031" xml:space="preserve"> ſit ergo cõcurſus pũctus h:</s> <s xml:id="echoid-s43032" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s43033" xml:space="preserve"> linea t e æqualis li-<lb/>neæ e h, & linea t b æqualis lineę b h ք 4 p 6:</s> <s xml:id="echoid-s43034" xml:space="preserve"> trigona enim t e b <lb/>& b e h per 26 p 1 & ex pręmiſsis ſunt æquiangula, quibus latus <lb/>e b eſt cõmune.</s> <s xml:id="echoid-s43035" xml:space="preserve"> Et ſimiliter producatur linea e k ad lineã a d, ita <lb/>quòd cõtineat cũlinea e a angulũ rectũ per 11 p 1, & producatur <lb/>ultra pũctum e:</s> <s xml:id="echoid-s43036" xml:space="preserve"> & producatur linea a o ultra pũctum o:</s> <s xml:id="echoid-s43037" xml:space="preserve"> concur <lb/>rentq́;</s> <s xml:id="echoid-s43038" xml:space="preserve"> lineę k e & a o ք 14 th.</s> <s xml:id="echoid-s43039" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43040" xml:space="preserve"> quia cũ angulus k e a ſit rectus, angulus e a o eſt acutus:</s> <s xml:id="echoid-s43041" xml:space="preserve"> ſit con-<lb/>curſus punctus l:</s> <s xml:id="echoid-s43042" xml:space="preserve"> & erit linea k e æqualis lineæ e l:</s> <s xml:id="echoid-s43043" xml:space="preserve"> quia cũ angulus k e a ſit rectus, erit angulus e a l <lb/>rectus:</s> <s xml:id="echoid-s43044" xml:space="preserve"> ſed & angulus e a l eſt æqualis angulo k a e, ut patet ex pręmiſsis:</s> <s xml:id="echoid-s43045" xml:space="preserve"> ergo per 32 p 1 triguona k e a <lb/>& e a l ſunt æquiangula:</s> <s xml:id="echoid-s43046" xml:space="preserve"> ergo per 4 p 6 cũ linea e a ſit ambobus illis trigonis cómunis, erit linea k a <lb/>æqualis lineę a l, & linea k e æqualis lineę e l.</s> <s xml:id="echoid-s43047" xml:space="preserve"> Et hoc etiam prteſt concludi per 3 p 6.</s> <s xml:id="echoid-s43048" xml:space="preserve"> Et per eundem <lb/>modũ oſtenſæ ſunt lineæ t e & e h adinuicẽ, & lineæ t b & b h adinuicẽ æquales.</s> <s xml:id="echoid-s43049" xml:space="preserve"> Ducátur ergo li-<lb/>neæ t k & l h.</s> <s xml:id="echoid-s43050" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s43051" xml:space="preserve"> duo latera t e & k e ſunt æqualia duobus lateribus e h & e l, & per 15 p 1 an-<lb/>gulus t e k eſt æqualis angulo l e h:</s> <s xml:id="echoid-s43052" xml:space="preserve"> patet per 4 p 1 quoniam lineę t k & l h erunt æquales inter ſe.</s> <s xml:id="echoid-s43053" xml:space="preserve"> Si <lb/>ergo uiſus fuerit in pũcto d, & linea l h fuerit in aliquo uiſibili:</s> <s xml:id="echoid-s43054" xml:space="preserve"> tũc uiſus exiſtens in puncto d cõpre-<lb/>hendet formá pũcti h in ſpeculo a b reflexam à pũcto b:</s> <s xml:id="echoid-s43055" xml:space="preserve"> & erit formę pũcti h imago pũctum t per 37 <lb/>th.</s> <s xml:id="echoid-s43056" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s43057" xml:space="preserve"> quoniã cathetus ſuę incidẽtię, quę eſt linea h e, cõcurrit cũ linea reflexionis, quę eſt d b.</s> <s xml:id="echoid-s43058" xml:space="preserve"> <lb/>in pũcto t:</s> <s xml:id="echoid-s43059" xml:space="preserve"> ſimiliterq́;</s> <s xml:id="echoid-s43060" xml:space="preserve"> forma pũcti l reflectetur ad uiſum in punctum d à pũcto ſpeculi, quod eſt a:</s> <s xml:id="echoid-s43061" xml:space="preserve"> & <lb/>quia cathetus ſuę incidentiæ, que eſt l e, concurrit cũ linea reflexionis, quæ eſt d a, in puncto k:</s> <s xml:id="echoid-s43062" xml:space="preserve"> erit <lb/>per 37 th.</s> <s xml:id="echoid-s43063" xml:space="preserve"> 5 huius pũctum k imago formę pũcti l:</s> <s xml:id="echoid-s43064" xml:space="preserve"> & erit linea t k diameter imaginis lineę l h:</s> <s xml:id="echoid-s43065" xml:space="preserve"> & erit ei <lb/>æqualis.</s> <s xml:id="echoid-s43066" xml:space="preserve"> Si ergo reuoluatur tota figura ſpeculi, & linearũ productarũ, linea h l immobili exiſtente:</s> <s xml:id="echoid-s43067" xml:space="preserve"> <lb/>tunc pũctus d deſcribet circulũ, in cuius circũferẽtię pũcto aliquo cẽtro uiſus exiſtẽte poterit com <lb/>prehendere aliquod uiſibile cõparem habens ſitũ ad uiſum, ſicut nũc habet linea l h ad uiſum d:</s> <s xml:id="echoid-s43068" xml:space="preserve"> & <lb/>erit imago illius uiſibilis æqualis ei.</s> <s xml:id="echoid-s43069" xml:space="preserve"> Et ſimiliter ſi uiſus fuerit intra circulũ ſpeculi in pũcto o, & res <lb/>uiſa fuerit diſpoſita ſecũdum lineam t k:</s> <s xml:id="echoid-s43070" xml:space="preserve"> erit imago lineæ t k linea l h, æqualis rei uiſæ.</s> <s xml:id="echoid-s43071" xml:space="preserve"> Sed tamen re <lb/>uiſa exiſtente in linea l h, & uiſu exiſtẽte in pũcto d, cũ imago rei uiſę fuerit linea t k:</s> <s xml:id="echoid-s43072" xml:space="preserve"> erit forma ima <lb/>ginis cõuerſa reſpectu ſitus rei.</s> <s xml:id="echoid-s43073" xml:space="preserve"> Si enim pũctus h fuerit in dextra, erit punctus t in ſiniſtra:</s> <s xml:id="echoid-s43074" xml:space="preserve"> & ſi pun-<lb/>ctus h fuerit ſupra lineá aliquá eleuatus, erit punctus t infra illã lineã depreſſus & inclinatus:</s> <s xml:id="echoid-s43075" xml:space="preserve"> & ſimi <lb/>liter eſt de pũcto l reſpectu pũcti k.</s> <s xml:id="echoid-s43076" xml:space="preserve"> Sed cũ res uiſa fuerit in linea t k, & uiſus fuerit in pũcto o, & ima <lb/>go lineę t k fuerit linea l h:</s> <s xml:id="echoid-s43077" xml:space="preserve"> erit forma nõ cõuerſa ſed directa.</s> <s xml:id="echoid-s43078" xml:space="preserve"> Ná imago, quę eſt linea l h, erit retro ui <lb/>ſum, ut oſtẽſum eſt in 11 hui{us}:</s> <s xml:id="echoid-s43079" xml:space="preserve"> & uiſus cõprehẽdet pũctũ h, q đ eſt imago pũcti t, retro ſe in linea h o, <lb/>& pũctum l, quod eſt imago pũcti k, in linea l o retro ſe:</s> <s xml:id="echoid-s43080" xml:space="preserve"> & pars formę uiſibilis, quæ reflectitur ad ui <lb/>ſum, erit reſpiciens uiſum in ipſa imagine, ſicut & in ipſa ſuperficie rei uiſę.</s> <s xml:id="echoid-s43081" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s43082" xml:space="preserve"/> </p> <div xml:id="echoid-div1684" type="float" level="0" n="0"> <figure xlink:label="fig-0653-01" xlink:href="fig-0653-01a"> <variables xml:id="echoid-variables761" xml:space="preserve">d g t k z b e a l h</variables> </figure> </div> </div> <div xml:id="echoid-div1686" type="section" level="0" n="0"> <head xml:id="echoid-head1259" xml:space="preserve" style="it">49. In ſpeculis ſphæricis concauis imago quando cõprehenditur minor re uiſa: quæ occurrẽs <lb/>inter uiſum & ſpeculum conuerſum habet ſitum rei uiſæ: quando uerò uidetur maior re uiſa: <lb/>quæ occurrens retro uiſum conformem habet ſitum rei uiſæ. Alhazen 42 n 6.</head> <p> <s xml:id="echoid-s43083" xml:space="preserve">Sit diſpoſitio totius figurę omnino eadem, quę in pręcedente theoremate:</s> <s xml:id="echoid-s43084" xml:space="preserve"> & producatur linea <lb/> <pb o="352" file="0654" n="654" rhead="VITELLONIS OPTICAE"/> b h in continuum & directũ:</s> <s xml:id="echoid-s43085" xml:space="preserve"> & in ipſa ſignetur punctus r:</s> <s xml:id="echoid-s43086" xml:space="preserve"> & ducatur linea r e ad centrũ ſpeculi.</s> <s xml:id="echoid-s43087" xml:space="preserve"> Et <lb/>quoniá angulus t e b eſt rectus, patet per 13 p 1 quòd angulus h e b eſt rectus:</s> <s xml:id="echoid-s43088" xml:space="preserve"> palàm ergo quia angu <lb/>lus r e b erit obtuſus:</s> <s xml:id="echoid-s43089" xml:space="preserve"> producaturq́;</s> <s xml:id="echoid-s43090" xml:space="preserve"> linea r e ultra punctum e ad lineã b d:</s> <s xml:id="echoid-s43091" xml:space="preserve"> incidatq́;</s> <s xml:id="echoid-s43092" xml:space="preserve"> in pũctum n:</s> <s xml:id="echoid-s43093" xml:space="preserve"> ca-<lb/>d etq́;</s> <s xml:id="echoid-s43094" xml:space="preserve"> punctũ n inter pũcta t & b.</s> <s xml:id="echoid-s43095" xml:space="preserve"> Cum enim angulus b e r ſit obtuſus:</s> <s xml:id="echoid-s43096" xml:space="preserve"> patet per 13 p 1 quòd angulus <lb/>b e n eſt a cutus:</s> <s xml:id="echoid-s43097" xml:space="preserve"> linea itaq;</s> <s xml:id="echoid-s43098" xml:space="preserve"> e n diuidit angulũ t e b, qui eſt rectus:</s> <s xml:id="echoid-s43099" xml:space="preserve"> ergo per 29 th.</s> <s xml:id="echoid-s43100" xml:space="preserve"> 1 huius ipſa ſecabit <lb/>baſim t b:</s> <s xml:id="echoid-s43101" xml:space="preserve"> erit ergo linea n b minor ꝗ̃ linea t b:</s> <s xml:id="echoid-s43102" xml:space="preserve"> ſed linea t b, ut patuit in pręcedẽte, eſt æqualis lineæ <lb/>b h, & linea b r eſt maior quàm linea b h:</s> <s xml:id="echoid-s43103" xml:space="preserve"> erit ergo linea r b maior ꝗ̃ linea b n.</s> <s xml:id="echoid-s43104" xml:space="preserve"> Et quia, ut patet ex prę <lb/>miſsis in proxima pręcedente, angulus n b e eſt æqualis angulo e b r:</s> <s xml:id="echoid-s43105" xml:space="preserve"> palàm quod linea e b diuidit <lb/>angulum n b r per æqualia.</s> <s xml:id="echoid-s43106" xml:space="preserve"> Erit ergo per 3 p 6 proportio lineæ r b ad lineam b n, ſicut proportio li-<lb/>neæ r e ad lineam e n:</s> <s xml:id="echoid-s43107" xml:space="preserve"> ſed linea r b eſt maior quàm linea b n:</s> <s xml:id="echoid-s43108" xml:space="preserve"> ergo linea r e eſt maior quàm linea e n.</s> <s xml:id="echoid-s43109" xml:space="preserve"> <lb/>Producatur quoq;</s> <s xml:id="echoid-s43110" xml:space="preserve"> ſimiliter linea a l in continuum & directum, donec ſit linea a m ęqualis lineę b r:</s> <s xml:id="echoid-s43111" xml:space="preserve"> <lb/>& ducatur linea m e, quę producta concurrat cũ linea d a in puncto u:</s> <s xml:id="echoid-s43112" xml:space="preserve"> cõcurret autem, ut prius de-<lb/>monſtratũ eſt per 29 th.</s> <s xml:id="echoid-s43113" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s43114" xml:space="preserve"> Et quia duo anguli e a m & o b r ſunt æquales, ut patet in cõmento <lb/>pręmiſſæ propoſitionis, & duo latera e a & a m trigoni e a m <lb/> <anchor type="figure" xlink:label="fig-0654-01a" xlink:href="fig-0654-01"/> ſunt æqualia duobus laterib.</s> <s xml:id="echoid-s43115" xml:space="preserve"> trigoni b e r, quę ſunt b e & b r:</s> <s xml:id="echoid-s43116" xml:space="preserve"> <lb/>erit per 4 p 1 linea m e æ qualis lineæ r e:</s> <s xml:id="echoid-s43117" xml:space="preserve"> & angulus m e æ-<lb/>qualis angulo r e b:</s> <s xml:id="echoid-s43118" xml:space="preserve"> ſed angulus r e b maior eſt angulo recto <lb/>& obtuſus:</s> <s xml:id="echoid-s43119" xml:space="preserve"> erit ergo angulus m e a obtuſus:</s> <s xml:id="echoid-s43120" xml:space="preserve"> ergo ք 13 p 1 angu <lb/>lus u e a eſt acutus.</s> <s xml:id="echoid-s43121" xml:space="preserve"> Quia ergo in trigono a e u angulus u a e <lb/>eſt æqualis angulo e a m trigoni m e a, & angulus u e a eſt mi <lb/>nor angulo m e a:</s> <s xml:id="echoid-s43122" xml:space="preserve"> erit angulus e u a maior angulo a m e ք 32 <lb/>p 1:</s> <s xml:id="echoid-s43123" xml:space="preserve"> ergo in trigono m a u latus m a eſt maius latere u a:</s> <s xml:id="echoid-s43124" xml:space="preserve"> ſed li-<lb/>nea a e diuidit angulũ u a m ք æqualia.</s> <s xml:id="echoid-s43125" xml:space="preserve"> Ergo ք 3 p 6 linea m e <lb/>eſt maior ꝗ̃ linea e u:</s> <s xml:id="echoid-s43126" xml:space="preserve"> & ſimiliter eſt linea r e maior ꝗ̃ linea e n.</s> <s xml:id="echoid-s43127" xml:space="preserve"> <lb/>Ducátur itaq;</s> <s xml:id="echoid-s43128" xml:space="preserve"> lineę n u & m r.</s> <s xml:id="echoid-s43129" xml:space="preserve"> Et quia per 26 p 1 linea n e eſt <lb/>æqualis lineę e u:</s> <s xml:id="echoid-s43130" xml:space="preserve"> quoniam ex pręmiſsis angulus u a e eſt æ-<lb/>qualis angulo n b e, & angulus a e u eſt æqualis angulo b e n, <lb/>cũ uterq;</s> <s xml:id="echoid-s43131" xml:space="preserve"> ipſorũ ſuper angulũ æqualẽ obtuſum ſit cõplemen <lb/>tum duorũ rectorũ per 13 p 1, & latus a e eſt æquale lateri b e.</s> <s xml:id="echoid-s43132" xml:space="preserve"> <lb/>Sunt igitur per 15 p 1 & per 7 p 5 & per 6 p 6 trigoni m e r & <lb/>n e u æquianguli:</s> <s xml:id="echoid-s43133" xml:space="preserve"> ergo per 4 p 6 erit ꝓportio lineę m e ad li-<lb/>neã e u, ſicut lineę m r ad lineã n u:</s> <s xml:id="echoid-s43134" xml:space="preserve"> ſed, ut patet ex pręmiſsis, <lb/>linea m e eſt maior ꝗ̃ linea e u:</s> <s xml:id="echoid-s43135" xml:space="preserve"> ergo linea m r eſt maior ꝗ̃ li-<lb/>nea n u.</s> <s xml:id="echoid-s43136" xml:space="preserve"> Si ergo linea m r fuerit in aliquo uiſibili, & uiſus fue <lb/>rit in puncto d:</s> <s xml:id="echoid-s43137" xml:space="preserve"> erit linea n u diameter imaginis lineę m r:</s> <s xml:id="echoid-s43138" xml:space="preserve"> & <lb/>eſt minor ꝗ̃ linea r m.</s> <s xml:id="echoid-s43139" xml:space="preserve"> Et ſi uiſus fuerit in pũcto o, & linea n u <lb/>fuerit in aliquo uiſibili:</s> <s xml:id="echoid-s43140" xml:space="preserve"> erit linea m r imago lineę n u:</s> <s xml:id="echoid-s43141" xml:space="preserve"> & eſt <lb/>maior ꝗ̃ linea n u.</s> <s xml:id="echoid-s43142" xml:space="preserve"> Sed cũm in linea m r fuerit aliquod uiſibile, <lb/>& uiſus in pũcto d:</s> <s xml:id="echoid-s43143" xml:space="preserve"> imago n u eritinter uiſum & ſpeculũ:</s> <s xml:id="echoid-s43144" xml:space="preserve"> & <lb/>uidebitur imago reuerſa, habens ſitũ alium ꝗ̃ res uiſa, prout <lb/>declarauimus in the oremate pręcedente.</s> <s xml:id="echoid-s43145" xml:space="preserve"> Cum uerò res uiſa fuerit in linea n u, & uiſus in pũcto o:</s> <s xml:id="echoid-s43146" xml:space="preserve"> <lb/>imago m r uidebitur retro uiſum, & erit eius forma conformis ſitui rei uiſę, ut in pręmiſſa patuit.</s> <s xml:id="echoid-s43147" xml:space="preserve"> <lb/>Nã imago ſi fuerit ultra uiſum uidebitur anterius ipſius, & omne punctum imaginis uidebitur in li <lb/>nea ſuę reflexionis.</s> <s xml:id="echoid-s43148" xml:space="preserve"> Patet ergo manifeſtè totum, quod proponebatur.</s> <s xml:id="echoid-s43149" xml:space="preserve"/> </p> <div xml:id="echoid-div1686" type="float" level="0" n="0"> <figure xlink:label="fig-0654-01" xlink:href="fig-0654-01a"> <variables xml:id="echoid-variables762" xml:space="preserve">d g t z k n u b e a f o h m v</variables> </figure> </div> </div> <div xml:id="echoid-div1688" type="section" level="0" n="0"> <head xml:id="echoid-head1260" xml:space="preserve" style="it">50. In ſpeculis ſph æricis concauis imago quando comprehenditur maior re uiſa, & conuer-<lb/>ſa ſecundum ſitum formæ rei uiſæ, ipſa imagine inter uiſum & ſpeculũ occurrente: retro uiſum <lb/>non uidetur minor, ſedhabens ſitum conformem rei uiſæ. Alhazen 43 n 6.</head> <p> <s xml:id="echoid-s43150" xml:space="preserve">Remaneat diſpoſitio, quę prius in 48 huius:</s> <s xml:id="echoid-s43151" xml:space="preserve"> & ſignetur in linea o h punctum q:</s> <s xml:id="echoid-s43152" xml:space="preserve"> & ducatur linea <lb/>e q:</s> <s xml:id="echoid-s43153" xml:space="preserve"> & producta ultra cẽtrũ <lb/> <anchor type="figure" xlink:label="fig-0654-02a" xlink:href="fig-0654-02"/> e tranſeat ad punctum p li-<lb/>neę d b:</s> <s xml:id="echoid-s43154" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s43155" xml:space="preserve">, ut à linea o l <lb/>abſcindatur linea o f æ qua-<lb/>lis lineę o q per 3 p 1:</s> <s xml:id="echoid-s43156" xml:space="preserve"> & duca <lb/>turlinea f e:</s> <s xml:id="echoid-s43157" xml:space="preserve"> quę produca-<lb/>tur ultra punctũ e ad lineá <lb/>d a in punctũ i:</s> <s xml:id="echoid-s43158" xml:space="preserve"> erunt itaq;</s> <s xml:id="echoid-s43159" xml:space="preserve"> <lb/>ſecundum prędictũ in prę-<lb/>miſsis proban di modũ duę <lb/>lineę p e & i e maiores dua-<lb/>bus lineis e f & e q.</s> <s xml:id="echoid-s43160" xml:space="preserve"> Quia e-<lb/>nim linea l e eſt maior ꝗ̃ li-<lb/>nea f e, ut patet ex pręmiſ-<lb/>ſis duobus theorematibus, & linea e h eſt maior quàm linea e q:</s> <s xml:id="echoid-s43161" xml:space="preserve"> lianea uerò p e eſt maior ꝗ̃ linea t e, <lb/> <pb o="353" file="0655" n="655" rhead="LIBER OCTAVVS."/> & linea i e maior quàm linea e k:</s> <s xml:id="echoid-s43162" xml:space="preserve"> linea uerò l e eſt æqualis lineę k e, & linea h e eſt æqualis lineę e t:</s> <s xml:id="echoid-s43163" xml:space="preserve"> <lb/>patet quòd duę lineę e p & e i ſunt maiores duabus lineis f e & e q.</s> <s xml:id="echoid-s43164" xml:space="preserve"> Et quia ex præmiſsis in pręce-<lb/>dentibus duobus theorematibus anguli e h q & e l f ſunt æquales, & lineę e h & e l æquales:</s> <s xml:id="echoid-s43165" xml:space="preserve"> nũc au-<lb/>tem lineę h q & l f acceptę ſunt ęquales:</s> <s xml:id="echoid-s43166" xml:space="preserve"> ergo per 4 p 1 lineę f e & q e ſunt æquales:</s> <s xml:id="echoid-s43167" xml:space="preserve"> & angulus f e o ę-<lb/>qualis angulo q e o:</s> <s xml:id="echoid-s43168" xml:space="preserve"> ergo per 15 p 1 angulus p e d eſt ęqualis angulo d e i:</s> <s xml:id="echoid-s43169" xml:space="preserve"> relinquitur ergo ex pręmiſ <lb/>ſis angulus p e b æqualis angulo i e a:</s> <s xml:id="echoid-s43170" xml:space="preserve"> ergo per 32 p 1 trigona p e b & i e a ſunt ęquiangula:</s> <s xml:id="echoid-s43171" xml:space="preserve"> ergo per 4 <lb/>p 6 cum linea e b ſit æ qualis lineę e a:</s> <s xml:id="echoid-s43172" xml:space="preserve"> erit linea p e æqualis lineę e i.</s> <s xml:id="echoid-s43173" xml:space="preserve"> Ducantur ergo lineę p i & f q:</s> <s xml:id="echoid-s43174" xml:space="preserve"> e-<lb/>rit per 15 p 1 & per 7 p 5 & per 6 & 4 p 6 linea p i maior quàm linea f q.</s> <s xml:id="echoid-s43175" xml:space="preserve"> Si ergo uiſus fuerit in puncto <lb/>o, & linea p i ſit in aliquo uiſibili:</s> <s xml:id="echoid-s43176" xml:space="preserve"> erit linea f q imago lineę p i:</s> <s xml:id="echoid-s43177" xml:space="preserve"> & eſt linea f q minor quàm linea p i:</s> <s xml:id="echoid-s43178" xml:space="preserve"> & <lb/>imago f q uidebitur ſuper duas lineas reflexionis, quæ ſunt a o & b o:</s> <s xml:id="echoid-s43179" xml:space="preserve"> erit ergo forma imaginis re-<lb/>tro uiſum minor quàm res uiſa:</s> <s xml:id="echoid-s43180" xml:space="preserve"> & erit directa habens ſitum conformem ſitui rei uiſę.</s> <s xml:id="echoid-s43181" xml:space="preserve"> Si uerò uiſus <lb/>fuerit in puncto d, & linea f q fuerit in aliquo uiſibili:</s> <s xml:id="echoid-s43182" xml:space="preserve"> tunc erit linea p i imago lineę f q, & erit maio-<lb/>ris quantitatis quàm linea f q:</s> <s xml:id="echoid-s43183" xml:space="preserve"> & erit forma ante uiſum, conuerſum & contrarium habens ſitum, re-<lb/>ſpectu ſitus formę rei uiſę.</s> <s xml:id="echoid-s43184" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s43185" xml:space="preserve"/> </p> <div xml:id="echoid-div1688" type="float" level="0" n="0"> <figure xlink:label="fig-0654-02" xlink:href="fig-0654-02a"> <variables xml:id="echoid-variables763" xml:space="preserve">y b f j a q t k p l d g</variables> </figure> </div> </div> <div xml:id="echoid-div1690" type="section" level="0" n="0"> <head xml:id="echoid-head1261" xml:space="preserve" style="it">51. Centro uiſus exiſtẽte in aliquo pũcto, inter quod & ſuperficiẽ ſpeculi ſphærici cõcaui fuerit <lb/>cẽtrũ ſpeculi: formæ uiſæ exiſtẽtis ultr a cẽtrũ ſpeculi imago cõuerſa uidetur, & minor forma rei <lb/>uiſæ. In hoc quo ſitu uiſus cõprehendet propriãimaginẽ minorẽ & cõuerſam. Alhaz. 44 n 6.</head> <p> <s xml:id="echoid-s43186" xml:space="preserve">Sit ſpeculum ſphęricum concauũ a b d:</s> <s xml:id="echoid-s43187" xml:space="preserve"> cuius cẽtrum g:</s> <s xml:id="echoid-s43188" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s43189" xml:space="preserve"> ipſum ſuperficies plana per cen-<lb/>trum g:</s> <s xml:id="echoid-s43190" xml:space="preserve"> erit ergo per 69 th.</s> <s xml:id="echoid-s43191" xml:space="preserve"> 1 huius cõmunis ſectio circulus:</s> <s xml:id="echoid-s43192" xml:space="preserve"> qui ſit a b d:</s> <s xml:id="echoid-s43193" xml:space="preserve"> & ducatur linea g d, utcũq;</s> <s xml:id="echoid-s43194" xml:space="preserve"> <lb/>contingit:</s> <s xml:id="echoid-s43195" xml:space="preserve"> & producatur linea g d ultra punctũ g ad pũctum e:</s> <s xml:id="echoid-s43196" xml:space="preserve"> in quo ſit cẽtrum uiſus in ſuperficie <lb/>circuli a b d:</s> <s xml:id="echoid-s43197" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s43198" xml:space="preserve"> pũctus t in eadẽ linea e d ultra cẽtrum ſpeculi, quod eſt pũctum g:</s> <s xml:id="echoid-s43199" xml:space="preserve"> & ducatur linea <lb/>t h per 11 p 1 perpẽdiculariter ſuper lineá e d:</s> <s xml:id="echoid-s43200" xml:space="preserve"> & producatur linea h t ultra púctum t ad púctum z, do <lb/>nec ſit linea z t æqualis lineę t h:</s> <s xml:id="echoid-s43201" xml:space="preserve"> comprehẽdatq́;</s> <s xml:id="echoid-s43202" xml:space="preserve"> uiſus exiſtẽs in pũcto e formã pũcti h ք reflexionẽ <lb/>factá à puncto ſpeculi, quod ſit a.</s> <s xml:id="echoid-s43203" xml:space="preserve"> Erũt itaq;</s> <s xml:id="echoid-s43204" xml:space="preserve"> duo pũcta a & h à duobus lateribus pũcti g:</s> <s xml:id="echoid-s43205" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s43206" xml:space="preserve"> ita, ut ſi <lb/>linea g h producatur ad peripheriam circuli in punctum p, fiat arcus a p maior quarta circuli:</s> <s xml:id="echoid-s43207" xml:space="preserve"> & erit <lb/>angulus a g p obtuſus per 33 p 6.</s> <s xml:id="echoid-s43208" xml:space="preserve"> Non eſt autẽ poſsibile, ut pũcta a & h conſiſtãt in eodẽ latere pũcti <lb/>g, inter diametros g d & g q, producta ſemidiametro g p in pũctum q.</s> <s xml:id="echoid-s43209" xml:space="preserve"> Nõ enim poſſet fieri reflexio, <lb/>ut patet per 20 huius, niſi linea producta à pũcto g cẽtro ſpeculi ad pũctum a diuideret angulũ h a e <lb/>per ęqualia.</s> <s xml:id="echoid-s43210" xml:space="preserve"> Ducantur itaq;</s> <s xml:id="echoid-s43211" xml:space="preserve"> lineę e a & a h:</s> <s xml:id="echoid-s43212" xml:space="preserve"> & producta linea h g ad lineã a e, incidat ipſa in pũctũ k.</s> <s xml:id="echoid-s43213" xml:space="preserve"> <lb/>Angulus itaq;</s> <s xml:id="echoid-s43214" xml:space="preserve"> h a g eſt ęqualis angulo g a e per 20 th.</s> <s xml:id="echoid-s43215" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43216" xml:space="preserve"> & eſt punctus k locus imaginis pũcti h <lb/>per 37 th.</s> <s xml:id="echoid-s43217" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s43218" xml:space="preserve"> Sit quoq;</s> <s xml:id="echoid-s43219" xml:space="preserve"> arcus b d æqualis arcui d a:</s> <s xml:id="echoid-s43220" xml:space="preserve"> quod fiet ք 26 p 3:</s> <s xml:id="echoid-s43221" xml:space="preserve"> ſi angulus d g b fiat ęqua-<lb/>lis angulo d g a:</s> <s xml:id="echoid-s43222" xml:space="preserve"> & ducátur lineę e b, z b, g b:</s> <s xml:id="echoid-s43223" xml:space="preserve"> & producatur linea z g ad lineá b e, incidatq́;</s> <s xml:id="echoid-s43224" xml:space="preserve"> in pũctũ l:</s> <s xml:id="echoid-s43225" xml:space="preserve"> <lb/>ſecetq́;</s> <s xml:id="echoid-s43226" xml:space="preserve"> linea z b ſemidiametrũ d g in pũcto f.</s> <s xml:id="echoid-s43227" xml:space="preserve"> Et quia, ut patet ex pręmiſsis, duę lineę z t & t h ſunt <lb/>ęquales, & pũcta z & h ęqualẽ habẽt diſpoſitionẽ, reſpectu cẽtri, & reſpectu peripherię circuli:</s> <s xml:id="echoid-s43228" xml:space="preserve"> patet <lb/>quòd lineę h a & z b interſecabũt ſemidiametrũ d g in eodẽ pũcto f.</s> <s xml:id="echoid-s43229" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s43230" xml:space="preserve"> in trigonis t z f & h t <lb/>f duo latera h t & t z ſunt ęqualia, & latus t f eſt cõmune, & anguli a d t recti:</s> <s xml:id="echoid-s43231" xml:space="preserve"> palàm ք 4 p 1 quoniã li-<lb/>nea z f eſt ęqualis lineę h f.</s> <s xml:id="echoid-s43232" xml:space="preserve"> Sed & in trigonis a g f & b g f accidet ք candẽ 4 p 1 angulũ f a g æqualem <lb/>eſſe angulo f b g, & lineá a f æqualẽ fieri lineę f b.</s> <s xml:id="echoid-s43233" xml:space="preserve"> Eſt enim ex pręmiſsis angulus a f g ęqualis angulo <lb/>b g f, & lineę a g & b g ſunt ſemidiametri, cõmunis uerò ambobus trigonis a f g & b f g eſt linea f g:</s> <s xml:id="echoid-s43234" xml:space="preserve"> <lb/>ergo ք 4 p 1 angulus f a g ęqualis eſt angulo f b g:</s> <s xml:id="echoid-s43235" xml:space="preserve"> ſimiliterq́;</s> <s xml:id="echoid-s43236" xml:space="preserve"> per 13 p 1 & ք eandẽ 4 p 1 linea e a ęqua-<lb/>lis fit lineę e b, & angulus g b e ęqualis angulo g a e:</s> <s xml:id="echoid-s43237" xml:space="preserve"> ſed anguli f a g & g a e ſunt ęquales:</s> <s xml:id="echoid-s43238" xml:space="preserve"> ergo & an-<lb/>guli f b g & g b e ſunt ęquales:</s> <s xml:id="echoid-s43239" xml:space="preserve"> ergo angulus z b g ęqualis eſt angulo e b g.</s> <s xml:id="echoid-s43240" xml:space="preserve"> Ergo per 20 th.</s> <s xml:id="echoid-s43241" xml:space="preserve"> 5 huius for <lb/>ma pũcti z reflectetur à pũcto ſpeculi, quod eſt b, ad uiſum exiſtẽtem in puncto e:</s> <s xml:id="echoid-s43242" xml:space="preserve"> & erit pũctus l lo-<lb/>cus imaginis formę pũcti z.</s> <s xml:id="echoid-s43243" xml:space="preserve"> Ducatur quo q;</s> <s xml:id="echoid-s43244" xml:space="preserve"> linea k l:</s> <s xml:id="echoid-s43245" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0655-01a" xlink:href="fig-0655-01"/> quę erit diameter imaginis lineę z h.</s> <s xml:id="echoid-s43246" xml:space="preserve"> Et ꝗa linea z h <lb/>eſt perpẽdicularis ſuք lineã d e, & linea z t eſt ęqua-<lb/>lis lineę t h ex hypotheſi, & quia, ut patet ex pręmiſ-<lb/>ſis, duę lineę z f & h f ſunt ęquales, & duę lineę a f & <lb/>b f ſunt ęquales:</s> <s xml:id="echoid-s43247" xml:space="preserve"> tota ergo linea z b eſt ęqualis toti li <lb/>neę h a:</s> <s xml:id="echoid-s43248" xml:space="preserve"> ſed & duę lineę a e & e b ſunt ęquales:</s> <s xml:id="echoid-s43249" xml:space="preserve"> ducan <lb/>tur quoq;</s> <s xml:id="echoid-s43250" xml:space="preserve"> lineę e h & e z.</s> <s xml:id="echoid-s43251" xml:space="preserve"> In trigonis itaq;</s> <s xml:id="echoid-s43252" xml:space="preserve"> e a h & e z <lb/>b duo latera unius, quę ſunt e a & h a, ſunt æqualia <lb/>duobus alterius lateribus, q̃ ſunt e b & b z:</s> <s xml:id="echoid-s43253" xml:space="preserve"> & angu-<lb/>lus h a e eſt ęqualis angulo z b e:</s> <s xml:id="echoid-s43254" xml:space="preserve"> ergo ք 4 p 1 baſis z e <lb/>eſt ęqualis baſi h e:</s> <s xml:id="echoid-s43255" xml:space="preserve"> ſimiliterq́;</s> <s xml:id="echoid-s43256" xml:space="preserve"> in trigonis z t g & h t g <lb/>duo anguli ad pũctum t ſunt recti, & latus z t ęquale <lb/>lateri h t, latus quoq;</s> <s xml:id="echoid-s43257" xml:space="preserve"> t g eſt cõmune:</s> <s xml:id="echoid-s43258" xml:space="preserve"> ergo per 4 p 1 li <lb/>nea g h eſt æqualis lineę z g:</s> <s xml:id="echoid-s43259" xml:space="preserve"> lineę uerò a g & g b ſunt <lb/>ſemidiametri circuli a b d & ęquales:</s> <s xml:id="echoid-s43260" xml:space="preserve"> ergo duę lineę <lb/>a g & g h ſunt æquales duabus lineis b g & g z, & ba-<lb/>ſis a h eſt æqualis baſi b z:</s> <s xml:id="echoid-s43261" xml:space="preserve"> ergo per 8 p 1 erit angulus <lb/>a h k æqualis angulo b z l, & angulus h a k æqualis angulo z b l:</s> <s xml:id="echoid-s43262" xml:space="preserve"> erit ergo ք 32 p 1 angulus h k´a æqua <lb/>lis angulo z l b.</s> <s xml:id="echoid-s43263" xml:space="preserve"> Trigona itaq;</s> <s xml:id="echoid-s43264" xml:space="preserve"> h a k & z b l ſunt ęquiangula:</s> <s xml:id="echoid-s43265" xml:space="preserve"> ergo ք 4 p 6 erit proportio line e h k ad <lb/> <pb o="354" file="0656" n="656" rhead="VITELLONIS OPTICAE"/> lineam z l, ſicut lineæ z b ad lineã h a:</s> <s xml:id="echoid-s43266" xml:space="preserve"> ſed linea z b eſt æqualis lineæ h a, ut patet ex pręmiſsis:</s> <s xml:id="echoid-s43267" xml:space="preserve"> ergo li <lb/>nea h k eſt æqualis lineæ z l:</s> <s xml:id="echoid-s43268" xml:space="preserve"> ſed & linea h g eſt æqualis lineæ z g, ut ſuprà patuit:</s> <s xml:id="echoid-s43269" xml:space="preserve"> erit ergo reliquum <lb/>æquale reliquo:</s> <s xml:id="echoid-s43270" xml:space="preserve"> ergo linea g k eſt æqualis lineæ g l.</s> <s xml:id="echoid-s43271" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s43272" xml:space="preserve"> duæ lineæ z g & h g inter ſe ſunt ęqua <lb/>les, & duæ lineæ g k & k l inter ſe ſunt æquales:</s> <s xml:id="echoid-s43273" xml:space="preserve"> patet ք 7 p 5 quoniá eſt proportio lineæ z g ad lineã <lb/>g l, ſicut lineæ h g ad lineá g k:</s> <s xml:id="echoid-s43274" xml:space="preserve"> ſed angulus z g h & k g l ſunt ęquales ք 15 p 1:</s> <s xml:id="echoid-s43275" xml:space="preserve"> ergo ք 6 p 6 erũt trigona <lb/>z g h & k g h æquiangula:</s> <s xml:id="echoid-s43276" xml:space="preserve"> angulus ergo z h k eſt æ qualis angulo l k h:</s> <s xml:id="echoid-s43277" xml:space="preserve"> ergo ք 27 p 1 lineę z h & k l ſunt <lb/>æquidiſtátes:</s> <s xml:id="echoid-s43278" xml:space="preserve"> quod etiá patere poteſt ք 14 th.</s> <s xml:id="echoid-s43279" xml:space="preserve"> 1 huius Itẽ angulus h g a, ut patet ex p̃miſsis, eſt obtu-<lb/>ſus:</s> <s xml:id="echoid-s43280" xml:space="preserve"> ergo ք 13 p 1 angulus a g k eſt acutus:</s> <s xml:id="echoid-s43281" xml:space="preserve"> duo uerò anguli h a g & g a k ſunt æquales:</s> <s xml:id="echoid-s43282" xml:space="preserve"> relin quitur er-<lb/>go ք 32 p 1 angulus a k g maior angulo a h g:</s> <s xml:id="echoid-s43283" xml:space="preserve"> ergo per 19 p 1 in trigono a h k latus a h eſt maius latere <lb/>a k, & duo anguli apud a ſunt æquales:</s> <s xml:id="echoid-s43284" xml:space="preserve"> ergo per 3 p 6 linea h g eſt maior ꝗ̃ linea g k:</s> <s xml:id="echoid-s43285" xml:space="preserve"> & ſimiliter linea <lb/>z g eſt maior ꝗ̃ linea g l.</s> <s xml:id="echoid-s43286" xml:space="preserve"> Ergo linea z h eſt maior ꝗ̃ linea k l per 4 p 6:</s> <s xml:id="echoid-s43287" xml:space="preserve"> ſed linea k l eſt diameter imagi-<lb/>nιs lineæ z h:</s> <s xml:id="echoid-s43288" xml:space="preserve"> linea ergo z h uidebitur minor ꝗ̃ ſit ſecũdũ ueritatẽ.</s> <s xml:id="echoid-s43289" xml:space="preserve"> Si ergo reuoluerim{us} circulũ a b d, <lb/>linea e d immobili exiſtẽte:</s> <s xml:id="echoid-s43290" xml:space="preserve"> ex duob.</s> <s xml:id="echoid-s43291" xml:space="preserve"> pũctis a & b deſcribetur circulus in ſuperficie ſpeculi:</s> <s xml:id="echoid-s43292" xml:space="preserve"> & ſicut <lb/>ſe habet uiſus exiſtens in pũcto e ad rẽ uiſam, in qua eſt linea z h:</s> <s xml:id="echoid-s43293" xml:space="preserve"> ſic ſe habebit reſpectu cuiuslibet <lb/>cõparis lineæ cadẽtis intra illũ circulũ, quẽ ſignant pũcta z & h reflexa ex arcu cõpari arcui a b, ex <lb/>protione ſpeculi, quam diuidit circulus, quẽ ſignát duo pũcta a & b.</s> <s xml:id="echoid-s43294" xml:space="preserve"> Et ſimiliter poteſt declarari, ſi <lb/>linea z h ponatur maior uel minor, ꝗ̃ nũc eſt poſita.</s> <s xml:id="echoid-s43295" xml:space="preserve"> Vniuerſaliter enim in hoc ſitu diameter imagi-<lb/>nis uel faciei aſpιcientis cõprehenditur in ſpeculo ſphærico concauo minor ꝗ̃ ſit:</s> <s xml:id="echoid-s43296" xml:space="preserve"> ſed etiã imago ui-<lb/>detur conuerſa.</s> <s xml:id="echoid-s43297" xml:space="preserve"> Si enim uiſus ſuerit in pũcto e:</s> <s xml:id="echoid-s43298" xml:space="preserve"> tũc aſpiciens cõprehendet formá ſuá in tali ſpeculo <lb/>minorẽ ꝗ̃ ſit.</s> <s xml:id="echoid-s43299" xml:space="preserve"> Et ꝗ a pũctus k eſt imago pũcti h, & pũctus l eſt imago pũcti z:</s> <s xml:id="echoid-s43300" xml:space="preserve"> erit imago cõuerſa:</s> <s xml:id="echoid-s43301" xml:space="preserve"> quo-<lb/>niá pars dextra uidebitur ſiniſtra, & ſiniſtra dextra:</s> <s xml:id="echoid-s43302" xml:space="preserve"> & ſimiliter ſuperior uidebitur inferior, & infe-<lb/>rior ſuperior.</s> <s xml:id="echoid-s43303" xml:space="preserve"> Et ſimiliter etiá uiſus cõprehendet ſuá formá:</s> <s xml:id="echoid-s43304" xml:space="preserve"> quia illud, quod eſt in dextro, cõprehen <lb/>det in ſiniſtro, & econuerſo:</s> <s xml:id="echoid-s43305" xml:space="preserve"> & quod deorſum eſt, cõprehendet ſurſum, & ecõuerſo.</s> <s xml:id="echoid-s43306" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s43307" xml:space="preserve"> <lb/>ſi uiſus fuerit in quolibet qũcto, ιnter quod & ſuperficié ſpeculi fuerit centrũ ſpeculi:</s> <s xml:id="echoid-s43308" xml:space="preserve"> ſemper cópre <lb/>hendet ſuá formã cóuerſam.</s> <s xml:id="echoid-s43309" xml:space="preserve"> Et hoc eſt propoſitũ.</s> <s xml:id="echoid-s43310" xml:space="preserve"> Ex his itaq;</s> <s xml:id="echoid-s43311" xml:space="preserve"> pręmiſsis quatuor theorematibus pa <lb/>tet, quòd in ſpeculis ſphæricis concauis imago rei uiſæ cõprehenditur à uiſæ quãdoq;</s> <s xml:id="echoid-s43312" xml:space="preserve"> maior:</s> <s xml:id="echoid-s43313" xml:space="preserve"> quan-<lb/>doq;</s> <s xml:id="echoid-s43314" xml:space="preserve"> minor:</s> <s xml:id="echoid-s43315" xml:space="preserve"> quãdoq;</s> <s xml:id="echoid-s43316" xml:space="preserve"> æ qualis rei uiſæ:</s> <s xml:id="echoid-s43317" xml:space="preserve"> & nũc conformẽ habens ſitum ipſi rei uiſæ, & nũc cõuerſum-<lb/>Et quoniam, ſicut oſten dimus per 40 huius, quan doq;</s> <s xml:id="echoid-s43318" xml:space="preserve"> unius rei una uidetur imago:</s> <s xml:id="echoid-s43319" xml:space="preserve"> quãdoq;</s> <s xml:id="echoid-s43320" xml:space="preserve"> duæ:</s> <s xml:id="echoid-s43321" xml:space="preserve"> <lb/>quandoq;</s> <s xml:id="echoid-s43322" xml:space="preserve"> tres:</s> <s xml:id="echoid-s43323" xml:space="preserve"> & quandoq;</s> <s xml:id="echoid-s43324" xml:space="preserve"> quatuor:</s> <s xml:id="echoid-s43325" xml:space="preserve"> illud ergo, quod habet unã imaginẽ maiorẽ ſe, forſan habebit <lb/>alias minores uel æqualies:</s> <s xml:id="echoid-s43326" xml:space="preserve"> & quod habet unam imaginem ſe minorẽ, forſan habebit alias maiores <lb/>uel æquales:</s> <s xml:id="echoid-s43327" xml:space="preserve"> & quod habet unã æqualẽ, forſan habebit alias maiores uel minores:</s> <s xml:id="echoid-s43328" xml:space="preserve"> & quod habet u-<lb/>nam, cuius ſitus eſt directus compar rei uiſæ, forſan uidebitur ſub alijs imaginibus habentibus con <lb/>uerſum ſitum in contrarium rei uiſæ.</s> <s xml:id="echoid-s43329" xml:space="preserve"> Et hęc omnia ex diuerſitate ſitus rei uiſæ, & ipſius uiſus, reſpe <lb/>ctu punctorum reflexionis patere poſſunt ex pręmiſsis.</s> <s xml:id="echoid-s43330" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s43331" xml:space="preserve"/> </p> <div xml:id="echoid-div1690" type="float" level="0" n="0"> <figure xlink:label="fig-0655-01" xlink:href="fig-0655-01a"> <variables xml:id="echoid-variables764" xml:space="preserve">p d h t z f b g a l e k q</variables> </figure> </div> </div> <div xml:id="echoid-div1692" type="section" level="0" n="0"> <head xml:id="echoid-head1262" xml:space="preserve" style="it">52. Lineis incidentiæ ſe interſecantibus in ſpeculis ſphæricis concauis: altitudines & proſun-<lb/>ditates erect æ ſuper ſuperficiem ſpeculi citra punctum ſectionis exiſtẽtes: reuerſæ: quæ uerò ſunt <lb/>in eiſdem lineis ultra ſectionem, quemadmodum ſunt, ſic apparent. Eucl. 11 th. catoptr.</head> <p> <s xml:id="echoid-s43332" xml:space="preserve">Eſto ſpeculũ ſphęricũ concauũ a g:</s> <s xml:id="echoid-s43333" xml:space="preserve"> cuius centrũ q:</s> <s xml:id="echoid-s43334" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s43335" xml:space="preserve"> duæ altitudines d e & h n erectę ſuper <lb/>ſuperficiẽ ſpeculi:</s> <s xml:id="echoid-s43336" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s43337" xml:space="preserve"> cõmunis ſectio ſuperficiei re-<lb/> <anchor type="figure" xlink:label="fig-0656-01a" xlink:href="fig-0656-01"/> flexionis & ſpeculi circulus a g:</s> <s xml:id="echoid-s43338" xml:space="preserve"> reflectaturq́;</s> <s xml:id="echoid-s43339" xml:space="preserve"> forma <lb/>puncti e ad uiſum (cuius centrũ ſit b) à pũcto ſpecu <lb/>li, quod ſit a:</s> <s xml:id="echoid-s43340" xml:space="preserve"> & forma pũcti d à pũcto g:</s> <s xml:id="echoid-s43341" xml:space="preserve"> interſecẽtq́;</s> <s xml:id="echoid-s43342" xml:space="preserve"> <lb/>ſe lineę incidentię d g & e a in pũcto z:</s> <s xml:id="echoid-s43343" xml:space="preserve"> citra quẽ pun <lb/>ctũ ſectionis ſit altitudo h n:</s> <s xml:id="echoid-s43344" xml:space="preserve"> cuius pũctum h ſit in li-<lb/>nea e a, & eius punctũn ſit in linea d g.</s> <s xml:id="echoid-s43345" xml:space="preserve"> Cum ergo o-<lb/>mnia puncta lineæ e a reflectantur ad uiſum b à pun <lb/>cto ſpeculi a:</s> <s xml:id="echoid-s43346" xml:space="preserve"> & omnia puncta lineæ d g à puncto ſpe <lb/>culi g:</s> <s xml:id="echoid-s43347" xml:space="preserve"> palàm quòd forma puncti h reflectetur à pun-<lb/>cto ſpeculi a:</s> <s xml:id="echoid-s43348" xml:space="preserve"> & forma puncti n à puncto ſpeculi g.</s> <s xml:id="echoid-s43349" xml:space="preserve"> <lb/>Quia uerò lineæ h n & d e ſunt erectæ ſuper ſuperfi-<lb/>ciem ſpeculi:</s> <s xml:id="echoid-s43350" xml:space="preserve"> patet per 72 th.</s> <s xml:id="echoid-s43351" xml:space="preserve"> 1 huius quoniam quęli-<lb/>bet ipſarum tranſit punctum q cẽtrum ſpeculi.</s> <s xml:id="echoid-s43352" xml:space="preserve"> Pro-<lb/>ducatur ergo à centro ſpeculi (quod eſt q) per li-<lb/>neam h n linea q n h:</s> <s xml:id="echoid-s43353" xml:space="preserve"> producaturq́;</s> <s xml:id="echoid-s43354" xml:space="preserve"> ab eodem cen-<lb/>tro q per lineam e d, linea, quæ producatur extra ſpe <lb/>culum.</s> <s xml:id="echoid-s43355" xml:space="preserve"> Et quia linea q e d eſt perpendicularis ſuper <lb/>ſuperficiem ſpeculi, & linea b g obliqua:</s> <s xml:id="echoid-s43356" xml:space="preserve"> patet per <lb/>14 th.</s> <s xml:id="echoid-s43357" xml:space="preserve"> 1 huius quòd lineæ e d & b g concurrent ultra <lb/>ſpeculum:</s> <s xml:id="echoid-s43358" xml:space="preserve"> ſit concurſus punctus i.</s> <s xml:id="echoid-s43359" xml:space="preserve"> Palàm etiam per <lb/>eandem 14 th.</s> <s xml:id="echoid-s43360" xml:space="preserve"> 1 huius quoniam linea q n h produ-<lb/>cta concurret cum linea b g i:</s> <s xml:id="echoid-s43361" xml:space="preserve"> ſit concurſus pun-<lb/>ctus p:</s> <s xml:id="echoid-s43362" xml:space="preserve"> & linea b a concurrat cum linea q h in puncto l:</s> <s xml:id="echoid-s43363" xml:space="preserve"> & cum linea q i in puncto c.</s> <s xml:id="echoid-s43364" xml:space="preserve"> Manifeſtum <lb/> <pb o="355" file="0657" n="657" rhead="LIBER OCTAVVS."/> autem per 37 th.</s> <s xml:id="echoid-s43365" xml:space="preserve"> 5 huius quoniam locus imaginis formæ puncti h erit in puncto l:</s> <s xml:id="echoid-s43366" xml:space="preserve"> & locus imaginig <lb/>formæ puncti n erit in puncto p.</s> <s xml:id="echoid-s43367" xml:space="preserve"> Erit ergo linea l p imago totius lineæ h n.</s> <s xml:id="echoid-s43368" xml:space="preserve"> Habet autem imago l p ſi-<lb/>tum reuerſum, reſpectu ſitus lineę h n:</s> <s xml:id="echoid-s43369" xml:space="preserve"> quoniam pun-<lb/> <anchor type="figure" xlink:label="fig-0657-01a" xlink:href="fig-0657-01"/> ctus h eſt altior puncto n, & punctus l, qui eſt imago <lb/>puncti h, eſt baſsior puncto p, qui eſt imago puncti n.</s> <s xml:id="echoid-s43370" xml:space="preserve"> <lb/>Punctus uerò i eſt locus imaginis puncti d:</s> <s xml:id="echoid-s43371" xml:space="preserve"> & punctus <lb/>c eſt locus imaginis puncti e.</s> <s xml:id="echoid-s43372" xml:space="preserve"> Et quia punctus i eſt al-<lb/>tior puncto c, ſicut punctus d eſt altior ipſo puncto e:</s> <s xml:id="echoid-s43373" xml:space="preserve"> <lb/>palàm quoniam imago lineę d e (quæ eſt linea i c) con <lb/>formem ſituationem habet ipſi lineæ d e, cuius ipſa eſt <lb/>imago:</s> <s xml:id="echoid-s43374" xml:space="preserve"> quoniam imago ſituata apparet, ſicut ſe habet <lb/>ipſa res uiſa.</s> <s xml:id="echoid-s43375" xml:space="preserve"> Et hoc eſt propoſitũ de altitudinibus.</s> <s xml:id="echoid-s43376" xml:space="preserve"> De <lb/>profunditatibus uerò idem patet:</s> <s xml:id="echoid-s43377" xml:space="preserve"> ut ſi lineæ h n & d e <lb/>quędam profunditates ponantur eſſe:</s> <s xml:id="echoid-s43378" xml:space="preserve"> tunc enim eadẽ <lb/>eſt demõſtratio.</s> <s xml:id="echoid-s43379" xml:space="preserve"> Apparet enim profunditas h n reuer-<lb/>ſa, & profunditas d e quemadmodum eſt diſpoſita, ſic <lb/>apparet.</s> <s xml:id="echoid-s43380" xml:space="preserve"> Hoc itaq;</s> <s xml:id="echoid-s43381" xml:space="preserve"> eſt propoſitum.</s> <s xml:id="echoid-s43382" xml:space="preserve"> Si uerò ambę lineę <lb/>d e & h n eſſent ex una quacunq;</s> <s xml:id="echoid-s43383" xml:space="preserve"> parte ſectionis linea <lb/>rum incidentiæ, fieret ſuarum imaginum conformis <lb/>ſituatio:</s> <s xml:id="echoid-s43384" xml:space="preserve"> ut patet per præmiſſa.</s> <s xml:id="echoid-s43385" xml:space="preserve"/> </p> <div xml:id="echoid-div1692" type="float" level="0" n="0"> <figure xlink:label="fig-0656-01" xlink:href="fig-0656-01a"> <variables xml:id="echoid-variables765" xml:space="preserve">b q a h z d n g e l p e i c</variables> </figure> <figure xlink:label="fig-0657-01" xlink:href="fig-0657-01a"> <variables xml:id="echoid-variables766" xml:space="preserve">i c d l p e h a z n g q b</variables> </figure> </div> </div> <div xml:id="echoid-div1694" type="section" level="0" n="0"> <head xml:id="echoid-head1263" xml:space="preserve" style="it">53. Lineis incidentiæ ſe interſecantib. in ſpeculis <lb/>ſphæricis concauis: obliquæ longitudinis citr a pun-<lb/>ctum ſectionis exiſtentes, quem admodum ſunt, ſic <lb/>apparent: earum uerò, quæ ſunt ultra ſectionem in <lb/>eiſdem lineis, uidentur imagines reuerſæ. Euclides 12 th. catoptr.</head> <p> <s xml:id="echoid-s43386" xml:space="preserve">Sit ſpeculum ſphæricum concauum a g:</s> <s xml:id="echoid-s43387" xml:space="preserve"> cuius centrum m:</s> <s xml:id="echoid-s43388" xml:space="preserve"> & ſit centrum uiſus b:</s> <s xml:id="echoid-s43389" xml:space="preserve"> & ſit linea d e <lb/>obliqua ſuper ſuperficiem ſpeculi:</s> <s xml:id="echoid-s43390" xml:space="preserve"> cuius puncti d forma reflectatur ad uiſum b à puncto ſpeculi, <lb/>quod eſt a:</s> <s xml:id="echoid-s43391" xml:space="preserve"> formaq́;</s> <s xml:id="echoid-s43392" xml:space="preserve"> puncti e à puncto g:</s> <s xml:id="echoid-s43393" xml:space="preserve"> & lineæ incidentiæ (quæ ſunt d a & e g) interſecẽt ſe in pũ-<lb/>cto i:</s> <s xml:id="echoid-s43394" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s43395" xml:space="preserve"> citra punctum i linea obliquè incidẽs ſuperficiei <lb/> <anchor type="figure" xlink:label="fig-0657-02a" xlink:href="fig-0657-02"/> ſpeculi, quęſit k c, cuius punctus k reflectatur à puncto ſpe <lb/>culi g, & punctus c à puncto ſpeculi a.</s> <s xml:id="echoid-s43396" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s43397" xml:space="preserve"> linea <lb/>d m à puncto d ad centrum ſpeculi, quę (propter obliqui <lb/>tatẽ lineę b a ſuper ſuperficiẽ ſpeculi, cũ linea d m ſit perpẽ <lb/>dicularis ſuper eandem ſpeculi ſuperficiem per 72 th.</s> <s xml:id="echoid-s43398" xml:space="preserve"> 1 hu-<lb/>ius:</s> <s xml:id="echoid-s43399" xml:space="preserve"> ideo quia tranſit centrum ſpeculi, quod eſt m) concur <lb/>ret cum linea b a obliquè ſuperficiei ſpeculi incidente, ut <lb/>patere poteſt per 14 th.</s> <s xml:id="echoid-s43400" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43401" xml:space="preserve"> ſit concurſus in puncto l.</s> <s xml:id="echoid-s43402" xml:space="preserve"> <lb/>Similiter quoq;</s> <s xml:id="echoid-s43403" xml:space="preserve"> linea e m concurret cum linea b g:</s> <s xml:id="echoid-s43404" xml:space="preserve"> ſit pun <lb/>ctum concurſus n.</s> <s xml:id="echoid-s43405" xml:space="preserve"> Palàm ergo per 37 th.</s> <s xml:id="echoid-s43406" xml:space="preserve"> 5 huius quoniam <lb/>in puncto l eſt imago formæ puncti d, & in puncto n ima-<lb/>go formæ puncti e:</s> <s xml:id="echoid-s43407" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s43408" xml:space="preserve"> linea n l:</s> <s xml:id="echoid-s43409" xml:space="preserve"> quæ erit imago to-<lb/>tius lineæ d e.</s> <s xml:id="echoid-s43410" xml:space="preserve"> Habet quoque imago n l reuerſè ſe ad ſitum <lb/>lineę d e:</s> <s xml:id="echoid-s43411" xml:space="preserve"> quoniã punctus n, qui eſt imago puncti e baſsio-<lb/>ris, eſt altior puncto l, qui eſt imago puncti d altioris.</s> <s xml:id="echoid-s43412" xml:space="preserve"> Pro-<lb/>ducatur quo que linea m k donec concurrat cum linea b g <lb/>producta:</s> <s xml:id="echoid-s43413" xml:space="preserve"> concurret autem propter obliquitatẽ lineæ b g <lb/>ſuper ſuperficiem ſpeculi, & propter perpendicularitatẽ li <lb/>neę m k:</s> <s xml:id="echoid-s43414" xml:space="preserve"> ſit concurſus punctus f:</s> <s xml:id="echoid-s43415" xml:space="preserve"> & producatur linea m c <lb/>donec cõcurrat cũ linea b a producta:</s> <s xml:id="echoid-s43416" xml:space="preserve"> & ſit punctus cõcur <lb/>ſus s:</s> <s xml:id="echoid-s43417" xml:space="preserve"> copuleturq́;</s> <s xml:id="echoid-s43418" xml:space="preserve"> linea f s:</s> <s xml:id="echoid-s43419" xml:space="preserve"> erit ergo linea f s imago lineæ <lb/>k c:</s> <s xml:id="echoid-s43420" xml:space="preserve"> & ſicut punctũ k eſt altius puncto c, ſic erit punctũ f al-<lb/>tius puncto s.</s> <s xml:id="echoid-s43421" xml:space="preserve"> Eſt ita q;</s> <s xml:id="echoid-s43422" xml:space="preserve"> imago f s cõformẽ habens ſitũ ipſi <lb/>rei uiſæ, quę eſt k c, occurrens ſpeculo citra punctum ſe-<lb/>ctionis linearũ incidentię, qui eſt i.</s> <s xml:id="echoid-s43423" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s43424" xml:space="preserve"/> </p> <div xml:id="echoid-div1694" type="float" level="0" n="0"> <figure xlink:label="fig-0657-02" xlink:href="fig-0657-02a"> <variables xml:id="echoid-variables767" xml:space="preserve">d e b m z l n c k a g f s</variables> </figure> </div> </div> <div xml:id="echoid-div1696" type="section" level="0" n="0"> <head xml:id="echoid-head1264" xml:space="preserve" style="it">54. In ſpeculis ſphæricis concauis uiſus in quibuſdam ſitibus cõprehendit lineæ rect æ uiſæ ima <lb/>ginem plenè rectam. Alhaz. 45 n 6.</head> <p> <s xml:id="echoid-s43425" xml:space="preserve">Sit ſpeculũ ſphęricũ concauũ a b:</s> <s xml:id="echoid-s43426" xml:space="preserve"> cuius cẽtrũ e:</s> <s xml:id="echoid-s43427" xml:space="preserve"> ſecetruq́;</s> <s xml:id="echoid-s43428" xml:space="preserve"> ք ſuperficiẽ planã ք centrũ:</s> <s xml:id="echoid-s43429" xml:space="preserve"> erit ergo ք.</s> <s xml:id="echoid-s43430" xml:space="preserve"> <lb/>69 th.</s> <s xml:id="echoid-s43431" xml:space="preserve"> 1 huius cõmunis ſectio circulus magnus, qui ſit a b:</s> <s xml:id="echoid-s43432" xml:space="preserve"> & eius centrũ e:</s> <s xml:id="echoid-s43433" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s43434" xml:space="preserve"> duę diametri <lb/>huius circuli, quę ſunt a e o & b e d:</s> <s xml:id="echoid-s43435" xml:space="preserve"> & ſpeculũ non excedat arcũ b a d o:</s> <s xml:id="echoid-s43436" xml:space="preserve"> aſſumaturq́;</s> <s xml:id="echoid-s43437" xml:space="preserve"> in ſemidiame-<lb/>tro b e, quicũq;</s> <s xml:id="echoid-s43438" xml:space="preserve"> pũctus placuerit:</s> <s xml:id="echoid-s43439" xml:space="preserve"> & ſit z:</s> <s xml:id="echoid-s43440" xml:space="preserve"> in quo ponatur cẽtrũ uiſus:</s> <s xml:id="echoid-s43441" xml:space="preserve"> & ſumatur in ſemidiametro a e <lb/>punctus k, taliter, ut linea a k ſit maior ꝗ̃ linea k e:</s> <s xml:id="echoid-s43442" xml:space="preserve"> & ducatur linea z k:</s> <s xml:id="echoid-s43443" xml:space="preserve"> & ꝓtrahatur ad circũferentiã, <lb/>incidatq́;</s> <s xml:id="echoid-s43444" xml:space="preserve"> in punctũ f:</s> <s xml:id="echoid-s43445" xml:space="preserve"> & ducatur linea e f:</s> <s xml:id="echoid-s43446" xml:space="preserve"> & ſuք f terminũ lineę e f conſtituatur angulus æ qualis an-<lb/> <pb o="356" file="0658" n="658" rhead="VITELLONIS OPTICAE"/> gulo z f e ք 23 p 1, qui ſit angulus g f e, ducta linea g f:</s> <s xml:id="echoid-s43447" xml:space="preserve"> cuius pũctus g cadetin ſemidiametrũ o e.</s> <s xml:id="echoid-s43448" xml:space="preserve"> Quia <lb/>enim linea f k eſt maior ꝗ̃ linea k a per 7 p 3, & linea k a eſt maior ꝗ̃ k e exhypotheſi:</s> <s xml:id="echoid-s43449" xml:space="preserve"> erit linea f k ma-<lb/>ior ꝗ̃ linea k e:</s> <s xml:id="echoid-s43450" xml:space="preserve"> ergo per 18 p 1 angulus f e k maior eſt angulo e f k:</s> <s xml:id="echoid-s43451" xml:space="preserve"> eſt ergo angulus f e k maior angulo <lb/>e f g:</s> <s xml:id="echoid-s43452" xml:space="preserve"> linea ergo f g per 14 th.</s> <s xml:id="echoid-s43453" xml:space="preserve"> 1 huius concurret cũ linea g e:</s> <s xml:id="echoid-s43454" xml:space="preserve"> cõcurrat ergo in puncto g.</s> <s xml:id="echoid-s43455" xml:space="preserve"> Duarum ergo <lb/>linearũ z ſ & ſ g puncta reflectuntur ad ſe inuicem à puncto ſpeculi, quod eſt f, propter angulorũ æ-<lb/>qualitatem per 20 th.</s> <s xml:id="echoid-s43456" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s43457" xml:space="preserve"> Eſt ergo punctus k imago puncti g centro uiſus exiſtente in puncto z.</s> <s xml:id="echoid-s43458" xml:space="preserve"> <lb/>Ducatur itaq;</s> <s xml:id="echoid-s43459" xml:space="preserve"> linea z h ſecans diametrũ o a in pun-<lb/> <anchor type="figure" xlink:label="fig-0658-01a" xlink:href="fig-0658-01"/> cto l, & peripheriã circuli in pũcto h, utcunq;</s> <s xml:id="echoid-s43460" xml:space="preserve"> cõtin <lb/>git:</s> <s xml:id="echoid-s43461" xml:space="preserve"> ducáturq́;</s> <s xml:id="echoid-s43462" xml:space="preserve"> lineę e h, h g, z g:</s> <s xml:id="echoid-s43463" xml:space="preserve"> & protrahatur linea <lb/>f e ſuք lineã z g:</s> <s xml:id="echoid-s43464" xml:space="preserve"> incidatq́;</s> <s xml:id="echoid-s43465" xml:space="preserve"> in punctũ m:</s> <s xml:id="echoid-s43466" xml:space="preserve"> ergo ք 3 p 6 <lb/>erit proportio lineę z m ad lineã m g, ſicut lineę z f <lb/>ad lineã f g:</s> <s xml:id="echoid-s43467" xml:space="preserve"> ſed ք 7 p 3 linea z h eſt maior ꝗ̃ linea z f, <lb/>& linea g h eſt min or ꝗ̃ linea g ſ per eandẽ 7 p 3:</s> <s xml:id="echoid-s43468" xml:space="preserve"> er-<lb/>go per 9 th.</s> <s xml:id="echoid-s43469" xml:space="preserve"> 1 huius maior eſt proportio lineę z h ad <lb/>lineam g h, quã lineæ z f ad f g:</s> <s xml:id="echoid-s43470" xml:space="preserve"> eſt ergo ꝓportio li-<lb/>neæ z h ad lineã g h maior quã lineę z m ad lineam <lb/>m g.</s> <s xml:id="echoid-s43471" xml:space="preserve"> Ergo ք 3 p 6 linea, quæ diuidit angulum z h g <lb/>per ęqualia, ſecat lineam m g:</s> <s xml:id="echoid-s43472" xml:space="preserve"> ſecat ergo prius lineã <lb/>e g per 32 th.</s> <s xml:id="echoid-s43473" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43474" xml:space="preserve"> quoniã linea e g eſt uicinior ad <lb/>punctũ h quã linea m g:</s> <s xml:id="echoid-s43475" xml:space="preserve"> maior erit ergo angulus g <lb/>h e angulo e h z, argumẽto 29 th.</s> <s xml:id="echoid-s43476" xml:space="preserve"> 1 huius, & ex præ-<lb/>miſsis.</s> <s xml:id="echoid-s43477" xml:space="preserve"> Ponamus ergo angulũ e h r ęqualẽ angulo e <lb/>h z:</s> <s xml:id="echoid-s43478" xml:space="preserve"> linea ergo h r ſecat lineá g f, & ſecat lineã g e ք 29 th.</s> <s xml:id="echoid-s43479" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43480" xml:space="preserve"> ſecet ergo g e in pũcto r:</s> <s xml:id="echoid-s43481" xml:space="preserve"> & ſecet linea <lb/>h z ſemidiametrũ e a in puncto l.</s> <s xml:id="echoid-s43482" xml:space="preserve"> Pũcta ergo duarũ linearũ z h & h r refle ctũtur adinuicẽ propter ę-<lb/>qualitatẽ angulorũ r h e, e h z:</s> <s xml:id="echoid-s43483" xml:space="preserve"> fietq́;</s> <s xml:id="echoid-s43484" xml:space="preserve"> reflexio à puncto ſpeculi, quod eſt h, ք 20 th.</s> <s xml:id="echoid-s43485" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s43486" xml:space="preserve"> & erit l pũ-<lb/>ctus imago puncti r.</s> <s xml:id="echoid-s43487" xml:space="preserve"> Palã uerò quoniã forma cuiuslibet puncti lineę g r reflectitur ad uiſum in pun-<lb/>ctum z, ex aliquo puncto arcus f h, & nõ ex alio ք 42 huius.</s> <s xml:id="echoid-s43488" xml:space="preserve"> Sumatur itaq;</s> <s xml:id="echoid-s43489" xml:space="preserve"> aliꝗs punctus lineę g r, ꝗ <lb/>ſit c, & hic refle ctatur ab aliquo puncto arcus f h, qui ſit t:</s> <s xml:id="echoid-s43490" xml:space="preserve"> & ducãtur lineę c t & z t.</s> <s xml:id="echoid-s43491" xml:space="preserve"> Quia ergo pũctus <lb/>t eſt inter duo puncta f & h arcus f h:</s> <s xml:id="echoid-s43492" xml:space="preserve"> palã quia linea z t cadet inter duas lineas z f & z h:</s> <s xml:id="echoid-s43493" xml:space="preserve"> linea ergo z t <lb/>ք 29 th.</s> <s xml:id="echoid-s43494" xml:space="preserve"> 1 huius ſecat lineã k l:</s> <s xml:id="echoid-s43495" xml:space="preserve"> ſecet ergo in puncto i.</s> <s xml:id="echoid-s43496" xml:space="preserve"> Eſt ergo per 37 th.</s> <s xml:id="echoid-s43497" xml:space="preserve"> 5 huius punctus i imago for-<lb/>mæ puncti c:</s> <s xml:id="echoid-s43498" xml:space="preserve"> & pũctus c nõ habet aliã imaginẽ niſi punctũ i, quoniã tãtũ ab uno puncto arcus f h fit <lb/>reflexio formę pũcti c ad uiſum exiſtentẽ in pũcto z, ut patet ք 19 uel ք 27 huius.</s> <s xml:id="echoid-s43499" xml:space="preserve"> Imago itaq;</s> <s xml:id="echoid-s43500" xml:space="preserve"> cuiusli <lb/>bet puncti lineę gr erit in aliquo puncto lineę k l:</s> <s xml:id="echoid-s43501" xml:space="preserve"> eſt ergo tota linea k l imago formę totius lineę g r:</s> <s xml:id="echoid-s43502" xml:space="preserve"> <lb/>& eſt recta:</s> <s xml:id="echoid-s43503" xml:space="preserve"> quia eſt pars ſemidiametri circuli a e.</s> <s xml:id="echoid-s43504" xml:space="preserve"> Viſus ergo exiſtẽs in puncto z cõprehẽdit formã li <lb/>neę rectę, quę eſt g r, imaginẽ l k rectã exiſtentẽ in ſpeculo ſphęrico cõcauo a b.</s> <s xml:id="echoid-s43505" xml:space="preserve"> Et hoc eſt propoſitũ.</s> <s xml:id="echoid-s43506" xml:space="preserve"/> </p> <div xml:id="echoid-div1696" type="float" level="0" n="0"> <figure xlink:label="fig-0658-01" xlink:href="fig-0658-01a"> <variables xml:id="echoid-variables768" xml:space="preserve">h t f d l i k a r e z b c m o g</variables> </figure> </div> </div> <div xml:id="echoid-div1698" type="section" level="0" n="0"> <head xml:id="echoid-head1265" xml:space="preserve" style="it">55. In ſpeculis ſphœricis cõcauis cõprebendet uiſus ex quibuſdã ſitib. imaginẽ lineœ cõuexœ cõue-<lb/>xam, & cõcauœ concauã: erit́ lineæ, cuius cõuexitas reſpicit ſpeculũ, imago cõuexa reſpiciẽs ui-<lb/>ſum: & lineœ, cuius cõcauit as reſpicit ſpeculũ, imago concauareſpiciẽs uiſum. Alhazen 46 n 6.</head> <p> <s xml:id="echoid-s43507" xml:space="preserve">Sit diſpoſitio, quæ in proxima præcedente:</s> <s xml:id="echoid-s43508" xml:space="preserve"> conſtituanturq́;</s> <s xml:id="echoid-s43509" xml:space="preserve"> ſuper lineam g r à duobus ſuis late-<lb/>ribus duo arcus, utcun q;</s> <s xml:id="echoid-s43510" xml:space="preserve"> contigerit:</s> <s xml:id="echoid-s43511" xml:space="preserve"> qui ſint g n r & g q r:</s> <s xml:id="echoid-s43512" xml:space="preserve"> & ſit arcus g n r non ſecás lineam g h:</s> <s xml:id="echoid-s43513" xml:space="preserve"> & po <lb/>natur in linea recta g r punctũ m, quomodocunq;</s> <s xml:id="echoid-s43514" xml:space="preserve"> ſit <lb/> <anchor type="figure" xlink:label="fig-0658-02a" xlink:href="fig-0658-02"/> illud.</s> <s xml:id="echoid-s43515" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s43516" xml:space="preserve"> puncti m reflectitur ad uiſum z ex <lb/>aliquo puncto arcus f h per 42 huius:</s> <s xml:id="echoid-s43517" xml:space="preserve"> ſit itaq;</s> <s xml:id="echoid-s43518" xml:space="preserve">, ut refle <lb/>ctatur ex puncto t & ducantur lineę z t, e t, m t:</s> <s xml:id="echoid-s43519" xml:space="preserve"> duo <lb/>itaq;</s> <s xml:id="echoid-s43520" xml:space="preserve"> anguli z t e & e t m ſunt ęquales per 20 th.</s> <s xml:id="echoid-s43521" xml:space="preserve"> 5 hu <lb/>ius.</s> <s xml:id="echoid-s43522" xml:space="preserve"> Linea ergo m t ſecabit arcum g n r:</s> <s xml:id="echoid-s43523" xml:space="preserve"> ſit, ut ſeceti-<lb/>pſum in puncto n:</s> <s xml:id="echoid-s43524" xml:space="preserve"> & producatur linea t m uerſus ar-<lb/>cũ g q r:</s> <s xml:id="echoid-s43525" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s43526" xml:space="preserve"> illũ in puncto q:</s> <s xml:id="echoid-s43527" xml:space="preserve"> & ducatur linea n e:</s> <s xml:id="echoid-s43528" xml:space="preserve"> <lb/>producaturq́;</s> <s xml:id="echoid-s43529" xml:space="preserve"> ultra punctũ e:</s> <s xml:id="echoid-s43530" xml:space="preserve"> ſecabit ergo lineam z t <lb/>ſub linea k l ք 29 th.</s> <s xml:id="echoid-s43531" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43532" xml:space="preserve"> quoniam ſecat angulum <lb/>k e z, cui ſubtẽditur pars lineę t z:</s> <s xml:id="echoid-s43533" xml:space="preserve"> ſecet ergo illam in <lb/>puncto i.</s> <s xml:id="echoid-s43534" xml:space="preserve"> Quia ergo duo anguli z t e & n t e ſunt æ-<lb/>quales:</s> <s xml:id="echoid-s43535" xml:space="preserve"> patet per 20 th.</s> <s xml:id="echoid-s43536" xml:space="preserve"> 5 huius quòd forma puncti n <lb/>reflectitur ad uiſum z à puncto ſpeculi t.</s> <s xml:id="echoid-s43537" xml:space="preserve"> Eſt ergo pa-<lb/>làm per 37 th.</s> <s xml:id="echoid-s43538" xml:space="preserve"> 5 huius quoniam punctus i eſt locusi-<lb/>maginis formę puncti n:</s> <s xml:id="echoid-s43539" xml:space="preserve"> & duo puncta k & l ſunt imagines duorũ punctorũ g & r, ut patuit per præ <lb/>miſſam.</s> <s xml:id="echoid-s43540" xml:space="preserve"> Imago ergo arcus g n r eſt linea tranſiens per puncta k i l:</s> <s xml:id="echoid-s43541" xml:space="preserve"> ſed linea k i l eſt conuexa ex parte <lb/>uiſus z:</s> <s xml:id="echoid-s43542" xml:space="preserve"> & arcus g n r eſt conuexus ex parte ſpeculi.</s> <s xml:id="echoid-s43543" xml:space="preserve"> Viſus itaque exiſtens in puncto z comprehen-<lb/>det formam lineę g n r conuexæ conuexam lineam.</s> <s xml:id="echoid-s43544" xml:space="preserve"> Ducatur quoque linea q e:</s> <s xml:id="echoid-s43545" xml:space="preserve"> & producatur ul-<lb/>tra punctum e:</s> <s xml:id="echoid-s43546" xml:space="preserve"> ſecabit quoque lineam z t ultra lineam l k per 29 th.</s> <s xml:id="echoid-s43547" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43548" xml:space="preserve"> quoniam ſecat angulum <lb/>t e k:</s> <s xml:id="echoid-s43549" xml:space="preserve"> ſecet ergo in puncto p.</s> <s xml:id="echoid-s43550" xml:space="preserve"> Et quia anguli p t e & q t e ſunt æ quales:</s> <s xml:id="echoid-s43551" xml:space="preserve"> patet per 20 th.</s> <s xml:id="echoid-s43552" xml:space="preserve"> 5 huius quoniã <lb/>à puncto ſpeculi, quod eſt t, reflectetur ſorma puncti q ad uiſum z:</s> <s xml:id="echoid-s43553" xml:space="preserve"> & locus imaginis formæ puncti q <lb/>eſt punctus p:</s> <s xml:id="echoid-s43554" xml:space="preserve"> & erit, ut ſuperà, linea l p q ex parte uiſus concaua:</s> <s xml:id="echoid-s43555" xml:space="preserve"> & ipſa eſt imago arcus g q r <lb/> <pb o="357" file="0659" n="659" rhead="LIBER OCTAVVS."/> concaui ex parte ſpeculi.</s> <s xml:id="echoid-s43556" xml:space="preserve"> Comprehendet ergo uiſus in puncto z exiſtens formam arcus g q r conca <lb/>ui lineam concauam.</s> <s xml:id="echoid-s43557" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s43558" xml:space="preserve"/> </p> <div xml:id="echoid-div1698" type="float" level="0" n="0"> <figure xlink:label="fig-0658-02" xlink:href="fig-0658-02a"> <variables xml:id="echoid-variables769" xml:space="preserve">o d h t f a b p k l z e r n m g q</variables> </figure> </div> </div> <div xml:id="echoid-div1700" type="section" level="0" n="0"> <head xml:id="echoid-head1266" xml:space="preserve" style="it">56. In ſpeculis ſphœricis concauis comprehendet uiſus ex quibuſdam ſitibus lineœ rectœ imagi-<lb/>nes quatuor curuas: lineœ́ curuœ, cuius conuexitas ect ad ſpeculum, imaginem comprehendit <lb/>curuam: omniuḿ harum imaginum concauit as reſpiciens eſt ad uiſum. Alhazen 47 n 6.</head> <p> <s xml:id="echoid-s43559" xml:space="preserve">Sit ſpeculum ſphęricum concauum, in quo ſit circulus maximus, qui a b d:</s> <s xml:id="echoid-s43560" xml:space="preserve"> cuius centrum g:</s> <s xml:id="echoid-s43561" xml:space="preserve"> & <lb/>extrahatur à centro g ſemidiameter g b, utcunq;</s> <s xml:id="echoid-s43562" xml:space="preserve"> contingit:</s> <s xml:id="echoid-s43563" xml:space="preserve"> quę diuidatur per inęqualia in puncto t <lb/>taliter, ut linea g t ſit maior medietate lineę b g:</s> <s xml:id="echoid-s43564" xml:space="preserve"> & à puncto t ducatur linea t z perpendiculariter ſuք <lb/>lineam g b per 11 p 1:</s> <s xml:id="echoid-s43565" xml:space="preserve"> & producatur linea z t ultra pũctũ t ad punctũ e:</s> <s xml:id="echoid-s43566" xml:space="preserve"> fiantq́;</s> <s xml:id="echoid-s43567" xml:space="preserve"> lineę z t & e t utręq;</s> <s xml:id="echoid-s43568" xml:space="preserve"> æ-<lb/>quales lineę t g per 3 p 1:</s> <s xml:id="echoid-s43569" xml:space="preserve"> & ducantur lineæ g e & g z:</s> <s xml:id="echoid-s43570" xml:space="preserve"> & trigono e g z circumſcribatur circulus per 5 <lb/>p 4:</s> <s xml:id="echoid-s43571" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s43572" xml:space="preserve"> centrum illius circuli punctus t per 9 p 3.</s> <s xml:id="echoid-s43573" xml:space="preserve"> Et quia linea t g maior eſt quã linea t b:</s> <s xml:id="echoid-s43574" xml:space="preserve"> palàm <lb/>quoniam ille circulus ſecabit circulum a b d:</s> <s xml:id="echoid-s43575" xml:space="preserve"> in duobus ergo pũctis illum ſecabit per 10 p 3:</s> <s xml:id="echoid-s43576" xml:space="preserve"> ſint itaq;</s> <s xml:id="echoid-s43577" xml:space="preserve"> <lb/>illa duo puncta a & d.</s> <s xml:id="echoid-s43578" xml:space="preserve"> Ducantur quoque lineæ g a, g d, e a, e b, e d, z a, z b, z d.</s> <s xml:id="echoid-s43579" xml:space="preserve"> Quia ergo duæ lineæ e t <lb/>& t z ſunt æ quales, & anguli ad punctum t ſunt recti, & linea t g communis:</s> <s xml:id="echoid-s43580" xml:space="preserve"> erunt per 4 p 1 duę li-<lb/>neę e g & z g æ quales:</s> <s xml:id="echoid-s43581" xml:space="preserve"> & ſimiliter per eandem 4 p 1 duæ lineæ e b & z b ſunt æ quales:</s> <s xml:id="echoid-s43582" xml:space="preserve"> ergo per 28 p <lb/>3 duo arcus e g & g z ſunt æ quales:</s> <s xml:id="echoid-s43583" xml:space="preserve"> ergo per 27 p 3 angulus e a g eſt æ qualis angulo g a z:</s> <s xml:id="echoid-s43584" xml:space="preserve"> & angulus <lb/>e d g æqualis eſt angulo g d z:</s> <s xml:id="echoid-s43585" xml:space="preserve"> & angulus e b g æqualis angulo g b z:</s> <s xml:id="echoid-s43586" xml:space="preserve"> quoniam omnes illi anguli ca-<lb/>duntin eoſdem arcus.</s> <s xml:id="echoid-s43587" xml:space="preserve"> Forma ergo puncti z reflectitur ad punctum e à punctis ſpeculi a & d & b, uel <lb/>econuerſo per 20 th.</s> <s xml:id="echoid-s43588" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s43589" xml:space="preserve"> Et quia linea g t eſt maior quàm linea t b:</s> <s xml:id="echoid-s43590" xml:space="preserve"> duæ uerò lineæ e b & z e ad <lb/>inuicem, & duę lineę e g & z g adinuicem ſunt æquales per 4 p 11:</s> <s xml:id="echoid-s43591" xml:space="preserve"> palàm per 47 p 1 quoniam linea <lb/>g e eſt maior quàm linea b e.</s> <s xml:id="echoid-s43592" xml:space="preserve"> Quadratum enim lineæ g e ualet ambo quadrata linearum g t & t e, & <lb/>quadratum lineæ e b ualet ambo quadrata linearum e t & t b:</s> <s xml:id="echoid-s43593" xml:space="preserve"> ablato ergo quadrato lineæ t e com-<lb/>muni, relinquitur quadratum lineę g e maius quadrato lineæ e b:</s> <s xml:id="echoid-s43594" xml:space="preserve"> quoniam linea g t eſt maior quàm <lb/>linea t b.</s> <s xml:id="echoid-s43595" xml:space="preserve"> Ergo linea g e eſt maior quàm linea e b in trigono g e b, &, ut patet per 18 p 1, angulus g b <lb/>e eſt maior angulo e g b:</s> <s xml:id="echoid-s43596" xml:space="preserve"> ſed angulus e g b eſt medietas unius recti per 5 & per 32 p 1:</s> <s xml:id="echoid-s43597" xml:space="preserve"> duo ergo an-<lb/>guli, qui b g e & e b g, ſimul ſum pti ſunt maiores recto:</s> <s xml:id="echoid-s43598" xml:space="preserve"> ergo angulus b e g eſt minor recto per 32 p 1:</s> <s xml:id="echoid-s43599" xml:space="preserve"> <lb/>ſed angulus e g z eſt rectus per 31 p 3:</s> <s xml:id="echoid-s43600" xml:space="preserve"> & ideo quoniam anguli e g t & t g z ſunt duę medietates unius <lb/>recti:</s> <s xml:id="echoid-s43601" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s43602" xml:space="preserve"> 1 huius duæ lineæ e b & g z <lb/> <anchor type="figure" xlink:label="fig-0659-01a" xlink:href="fig-0659-01"/> productę concurrent extra circulum:</s> <s xml:id="echoid-s43603" xml:space="preserve"> ſit earũ con <lb/>curſus pũctus l.</s> <s xml:id="echoid-s43604" xml:space="preserve"> Et quia linea e d eſt intra triangu <lb/>lum l e g:</s> <s xml:id="echoid-s43605" xml:space="preserve"> palàm quoniam i pſa producta concur-<lb/>ret cum linea g l per 29 th.</s> <s xml:id="echoid-s43606" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43607" xml:space="preserve"> concurrant er-<lb/>go in puncto m.</s> <s xml:id="echoid-s43608" xml:space="preserve"> Et quia linea g b tranſit per pun-<lb/>ctum t, quod eſt centrum circuli e g z:</s> <s xml:id="echoid-s43609" xml:space="preserve"> linea uerò <lb/>a g ducitur extra illam à centro ad peripheriã:</s> <s xml:id="echoid-s43610" xml:space="preserve"> pa-<lb/>lã quia portio a e g eſt minor ſemicirculo:</s> <s xml:id="echoid-s43611" xml:space="preserve"> ergo ք <lb/>31 p 3 angulus a e g eſt obtuſus, & angulus e g z eſt <lb/>rectus:</s> <s xml:id="echoid-s43612" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s43613" xml:space="preserve"> 1 huius illæ duę lineæ a e & <lb/>z g concurrent in partem lineæ e g:</s> <s xml:id="echoid-s43614" xml:space="preserve"> concurrant er <lb/>go in puncto f.</s> <s xml:id="echoid-s43615" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s43616" xml:space="preserve"> uiſus fuerit in puncto e, & <lb/>punctus z in aliquo uiſibili:</s> <s xml:id="echoid-s43617" xml:space="preserve"> tunc tria puncta m, l, f <lb/>erunt imagines puncti z.</s> <s xml:id="echoid-s43618" xml:space="preserve"> Sic ergo punctus z com <lb/>prehenditur in tribus locis:</s> <s xml:id="echoid-s43619" xml:space="preserve"> quoniam à tribus pũ-<lb/>ctis ſpeculi, quæ ſunt a, b, d fit reflexio formæ pun <lb/>cti ipſius z ad uiſum e.</s> <s xml:id="echoid-s43620" xml:space="preserve"> Item protrahatur à pũcto e <lb/>linea ſuper arcum d z, utcunq;</s> <s xml:id="echoid-s43621" xml:space="preserve"> contingat, quæ ſit <lb/>linea e k:</s> <s xml:id="echoid-s43622" xml:space="preserve"> & ducatur linea g k, quæ ſecet arcum d z <lb/>in puncto k:</s> <s xml:id="echoid-s43623" xml:space="preserve"> & ducatur linea z k.</s> <s xml:id="echoid-s43624" xml:space="preserve"> Quia ergo arcus <lb/>e g & g z ſunt æ quales:</s> <s xml:id="echoid-s43625" xml:space="preserve"> erunt duo anguli e k g & g <lb/>k z æ quales per 27 p 3:</s> <s xml:id="echoid-s43626" xml:space="preserve"> producaturq́;</s> <s xml:id="echoid-s43627" xml:space="preserve"> linea g k ad <lb/>circumferentiam circuli a b d:</s> <s xml:id="echoid-s43628" xml:space="preserve"> incidatq́;</s> <s xml:id="echoid-s43629" xml:space="preserve"> in punctũ <lb/>r:</s> <s xml:id="echoid-s43630" xml:space="preserve"> & ducantur lineę er & z r.</s> <s xml:id="echoid-s43631" xml:space="preserve"> Et quoniam angulus <lb/>e k g eſt æ qualis angulo g k z, erit angulus e k r æ-<lb/>qualis angulo z k r per 13 p 1:</s> <s xml:id="echoid-s43632" xml:space="preserve"> erit ergo angulus e r <lb/>k maior angulo k r z.</s> <s xml:id="echoid-s43633" xml:space="preserve"> Si enim ſit æ qualis:</s> <s xml:id="echoid-s43634" xml:space="preserve"> tunc per <lb/>32 p 1 & 4 p 6 ſequitur lineam e k æ qualem eſſe li-<lb/>neæ z k, & arcũ z k ęqualẽ eſſe arcui e a k:</s> <s xml:id="echoid-s43635" xml:space="preserve"> quod eſt <lb/>contra præmiſſa:</s> <s xml:id="echoid-s43636" xml:space="preserve"> eſt enim arcus ea ęqualis arcui <lb/>d z.</s> <s xml:id="echoid-s43637" xml:space="preserve"> Quòd ſi angulus e r k ſit minor angulo z r k:</s> <s xml:id="echoid-s43638" xml:space="preserve"> e-<lb/>rit ergo ex pręmiſsis angulus r e k maior angulo k <lb/>z r:</s> <s xml:id="echoid-s43639" xml:space="preserve"> reſecetur ergo angulus r e k ad ęqualitatẽ an-<lb/>guli r z k ք 27 th.</s> <s xml:id="echoid-s43640" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43641" xml:space="preserve"> & ſequetur idẽ impoſsibile qđ prius, ꝓducta illa linea ad lineã r k:</s> <s xml:id="echoid-s43642" xml:space="preserve"> reſtat er <lb/>go ut angulus e r g ſit maior angulo g r z.</s> <s xml:id="echoid-s43643" xml:space="preserve"> Fiat ergo ք 23 p 1 ſuք pũctũ r terminũ lineę g r angulus g r n <lb/> <pb o="358" file="0660" n="660" rhead="VITELLONIS OPTICAE"/> æqualis angulo e r g:</s> <s xml:id="echoid-s43644" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s43645" xml:space="preserve"> punctus n in lineam z m per 29 th.</s> <s xml:id="echoid-s43646" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43647" xml:space="preserve"> duæ ergo lineę e r & r n à pũ <lb/>cto ſpeculi (quod eſtr) reflectentur ad ſe inuicem per 20 th.</s> <s xml:id="echoid-s43648" xml:space="preserve"> 5 huius, propter æqualitatem angulorũ <lb/>ad punctum r.</s> <s xml:id="echoid-s43649" xml:space="preserve"> Producatur quoq;</s> <s xml:id="echoid-s43650" xml:space="preserve"> linea e r ad lineam g m:</s> <s xml:id="echoid-s43651" xml:space="preserve"> concurret autem cum illa per 14 th.</s> <s xml:id="echoid-s43652" xml:space="preserve"> 1 hu-<lb/>ius:</s> <s xml:id="echoid-s43653" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s43654" xml:space="preserve"> punctus concurſus q:</s> <s xml:id="echoid-s43655" xml:space="preserve"> erit ergo punctus q imago formæ puncti n, reſpectu uiſus e.</s> <s xml:id="echoid-s43656" xml:space="preserve"> Imagine-<lb/>mur ergo ſuperficiem exeuntem à linea m g f, quæ ſit perpendiculariter erecta ſuper ſuperficiem cir <lb/>culi a b d:</s> <s xml:id="echoid-s43657" xml:space="preserve"> & extrahatur à puncto z linea in hac ſuperficie, quæ ſit perpendicularis ſuper lineam g z:</s> <s xml:id="echoid-s43658" xml:space="preserve"> <lb/>& tranſeat in utranque partem ſuperficiei circuli a b d:</s> <s xml:id="echoid-s43659" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s43660" xml:space="preserve"> linea c z p.</s> <s xml:id="echoid-s43661" xml:space="preserve"> Poſito itaque puncto g cen-<lb/>tro circuli fiat arcus circuli ſecundum quantitatem lineæ g n, qui ſit c n p:</s> <s xml:id="echoid-s43662" xml:space="preserve"> ſecans lineam c z p in duo <lb/>bus punctis c & p:</s> <s xml:id="echoid-s43663" xml:space="preserve"> & producantur lineę g c & g p:</s> <s xml:id="echoid-s43664" xml:space="preserve"> erunt ergo iſtę lineæ in ſuperficie perpendicula-<lb/>ri ſuper ſuperficiem a b d per 2 p 11.</s> <s xml:id="echoid-s43665" xml:space="preserve"> Producantur item lineę g c & g p ultra puncta c & p extra ſpecu-<lb/>lum:</s> <s xml:id="echoid-s43666" xml:space="preserve"> & ſuper centrum g ſecundum longitudinem lineę g q in ſuperficie tranſeunte lineam m g f, ſe-<lb/>cante circulum, in qua ſunt lineæ g c & g p, fiat arcus circuli.</s> <s xml:id="echoid-s43667" xml:space="preserve"> Hic ergo iterum ſecabit duas lineas g c <lb/>& g p productas:</s> <s xml:id="echoid-s43668" xml:space="preserve"> ſecet ergo lineam g c in puncto s, & lineam g p in puncto o.</s> <s xml:id="echoid-s43669" xml:space="preserve"> Quia ergo ſuperficies <lb/>circuli a b d eſt perpendiculariter erecta ſuper ſuperficiem duarum linearũ g c & g p:</s> <s xml:id="echoid-s43670" xml:space="preserve"> palàm per defi <lb/>nitionem quoniam duo anguli e g s & e g o erunt recti:</s> <s xml:id="echoid-s43671" xml:space="preserve"> linea ergo e g erit erecta ſuper ſuperficiẽ g c <lb/>p:</s> <s xml:id="echoid-s43672" xml:space="preserve"> ergo per 18 p 11 erit utraq;</s> <s xml:id="echoid-s43673" xml:space="preserve"> ſuperficierum, quæ ſunt e g s & e g o, perpendicularis ſuper ſuperficiẽ <lb/>s g o, & utra que iſtarum ſuperficierum facit in ſpeculo circulum magnum comparem circulo a b d <lb/>per 69 th.</s> <s xml:id="echoid-s43674" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43675" xml:space="preserve"> punctum ergo circuli, quem facit ſuperficies e g s, quod eſt compar puncto circu-<lb/>li a b d ſcilicet puncto r, eundem habet ſitum reſpectu centri ipſius ſpeculi, quod eſt g, & reſpectu ui <lb/>ſus, qui eſt in puncto e, quem habet punctum r.</s> <s xml:id="echoid-s43676" xml:space="preserve"> Concurrunt ergo exipſo ſecundum angulos ęqua-<lb/>les duę lineę inter duo puncta e & c:</s> <s xml:id="echoid-s43677" xml:space="preserve"> & ſimiliter accidit inter duo pũcta e & p:</s> <s xml:id="echoid-s43678" xml:space="preserve"> & lineæ g c & g p ſunt <lb/>æ quales per definitionem circuli:</s> <s xml:id="echoid-s43679" xml:space="preserve"> & ſimiliter lineæ g s, g q, g o ſunt æ quales per definitionem cir-<lb/>culi:</s> <s xml:id="echoid-s43680" xml:space="preserve"> & punctus q eſt imago puncti n:</s> <s xml:id="echoid-s43681" xml:space="preserve"> & punctus s eſt imago puncti c:</s> <s xml:id="echoid-s43682" xml:space="preserve"> & punctus o eſt imago puncti <lb/>p.</s> <s xml:id="echoid-s43683" xml:space="preserve"> Imago ergo arcus c n p conuexi ex parte ſpeculi eſt arcus s q o concauus ex parte uiſus:</s> <s xml:id="echoid-s43684" xml:space="preserve"> & pun-<lb/>ctus l eſt imago formæ puncti z:</s> <s xml:id="echoid-s43685" xml:space="preserve"> & duo puncta s & o ſunt imagines formarum duorum punctorum <lb/>c & p:</s> <s xml:id="echoid-s43686" xml:space="preserve"> imago ergo lineę rectæ, quę eſt c z p, eſt linea curua, tranſiens pertria puncta s, l, o:</s> <s xml:id="echoid-s43687" xml:space="preserve"> hęc autem <lb/>linea s l o eſt concaua ex parte uiſus.</s> <s xml:id="echoid-s43688" xml:space="preserve"> Ducatur itaque linea tranſiens per puncta s, l, o:</s> <s xml:id="echoid-s43689" xml:space="preserve"> & extrahatur <lb/>linea e g ad circumferentiam circuli a b d in punctum h.</s> <s xml:id="echoid-s43690" xml:space="preserve"> Si ergo ſpeculum non peruenit ad duo pun <lb/>cta b & h, ſed alter duorum ſuorum terminorum fueritinter duo puncta b & d, & reliquus fuerit in-<lb/>fra punctum h, & uiſus fuerit in puncto e, & duę lineę p z crecta, & p n c conuexa ex parte ſpenculi, <lb/>fuerint in aliquo uiſibili:</s> <s xml:id="echoid-s43691" xml:space="preserve"> tunc forma lineæ p z c rectę apparebit concaua, ſcilicet s l o:</s> <s xml:id="echoid-s43692" xml:space="preserve"> & forma lineę <lb/>p n c conuexæ reſpectu ſpeculi erit concaua uiſui occurrẽs, ſcilicet s q o.</s> <s xml:id="echoid-s43693" xml:space="preserve"> Et forma lineę p z c unã tan <lb/>tũ habebit imaginẽ:</s> <s xml:id="echoid-s43694" xml:space="preserve"> & arcus p n c tantũ unã.</s> <s xml:id="echoid-s43695" xml:space="preserve"> Item producatur linea b g ultra punctũ g ad aliã partẽ <lb/>peripherię circuli ad punctũ i:</s> <s xml:id="echoid-s43696" xml:space="preserve"> & producantur lineę e i & e z:</s> <s xml:id="echoid-s43697" xml:space="preserve"> erit ergo ex pręmiſsis, & per 4 p 1 angu <lb/>lus b i e ęqualis angulo b i z:</s> <s xml:id="echoid-s43698" xml:space="preserve"> ergo ք 20 th.</s> <s xml:id="echoid-s43699" xml:space="preserve"> 5 huius reflectetur forma pũcti z ad uiſum in punctum e à <lb/>puncto ſpeculi, quod eſt i;</s> <s xml:id="echoid-s43700" xml:space="preserve"> & linea e i ſecabit lineã f g:</s> <s xml:id="echoid-s43701" xml:space="preserve"> ſecet ergo in puncto u:</s> <s xml:id="echoid-s43702" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s43703" xml:space="preserve"> punctus u imago <lb/>formę pũcti z reflexæ à pũcto ſpeculi, quod eſt i.</s> <s xml:id="echoid-s43704" xml:space="preserve"> Pũcta ergo quatuor, quę ſunt m, l, u, f ſunt loca ima <lb/>ginũ formæ puncti z.</s> <s xml:id="echoid-s43705" xml:space="preserve"> Et ſi ſpeculũ exceſſerit duo puncta a & d, & uiſus fuerit in pũcto e, & dorſum <lb/>aſpicientis ſuerit ex parte arcus a i, & uiſus cõprehenderit totum arcũ i d a:</s> <s xml:id="echoid-s43706" xml:space="preserve"> tunc punctũ z uidebitur <lb/>in quatuor locis, ſcilicet in punctis m, l, u, f:</s> <s xml:id="echoid-s43707" xml:space="preserve"> & uidebuntur duo puncta lineę rectæ p z c uel arcus p c <lb/>in duobus punctis s & o:</s> <s xml:id="echoid-s43708" xml:space="preserve"> & ſiclinea recta p z c habebit quatuor imagines concauas:</s> <s xml:id="echoid-s43709" xml:space="preserve"> & una tranſit ք <lb/>puncta s, m, o:</s> <s xml:id="echoid-s43710" xml:space="preserve"> & ſecunda pertranſit puncta s, l, o:</s> <s xml:id="echoid-s43711" xml:space="preserve"> tertia pertrãſit pũcta s, u, o:</s> <s xml:id="echoid-s43712" xml:space="preserve"> & quarta pertrãſit pun-<lb/>cta s, f, o ſcilicet lineam s f o.</s> <s xml:id="echoid-s43713" xml:space="preserve"> In his tamen omnib.</s> <s xml:id="echoid-s43714" xml:space="preserve"> imaginibus ſemper cõcauitas imaginis reſpicit ui <lb/>ſum.</s> <s xml:id="echoid-s43715" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s43716" xml:space="preserve"> Patet quoq;</s> <s xml:id="echoid-s43717" xml:space="preserve"> quòd imagines eiuſdem lineę rectę, ut patet nunc in linea <lb/>p z c, fiunt diuerſę curuitatis maioris & minoris:</s> <s xml:id="echoid-s43718" xml:space="preserve"> & fit principium formæ monſtruoſæ.</s> <s xml:id="echoid-s43719" xml:space="preserve"/> </p> <div xml:id="echoid-div1700" type="float" level="0" n="0"> <figure xlink:label="fig-0659-01" xlink:href="fig-0659-01a"> <variables xml:id="echoid-variables770" xml:space="preserve">l m s q o d r b n p t h c g u l f</variables> </figure> </div> </div> <div xml:id="echoid-div1702" type="section" level="0" n="0"> <head xml:id="echoid-head1267" xml:space="preserve" style="it">57. In ſpeculis ſphœricis concauis uiſus in quibuſdam ſitibus comprehendet lineœ rectœ imagi-<lb/>nem conuexam, conuexitate uiſum reſpiciente. Alhazen 49 n 6.</head> <p> <s xml:id="echoid-s43720" xml:space="preserve">Sit circulus magnus ſpeculi ſphærici concaui, qui a b g:</s> <s xml:id="echoid-s43721" xml:space="preserve"> cuius centrum d:</s> <s xml:id="echoid-s43722" xml:space="preserve"> & ducatur ſemidiame-<lb/>ter d g, ut contingit:</s> <s xml:id="echoid-s43723" xml:space="preserve"> in qua ſituetur linea recta, quæ ſit o u:</s> <s xml:id="echoid-s43724" xml:space="preserve"> & ſit punctum o remotius à centro ſpecu <lb/>li d, & u propinquius illi:</s> <s xml:id="echoid-s43725" xml:space="preserve"> & ſuper hanc ſemidiametrum d g ducatur perpen diculariter linea, quæ ſit <lb/>d h:</s> <s xml:id="echoid-s43726" xml:space="preserve"> in cuius puncto h ſit centrũ uiſus:</s> <s xml:id="echoid-s43727" xml:space="preserve"> & ſit linea h d erecta ſuper ſuperficiem circuli a b g:</s> <s xml:id="echoid-s43728" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s43729" xml:space="preserve"> linea <lb/>h d minor ſemidiametro ſpeculi ſecun dũ diſpoſitionẽ lineę h d, quę aſſumpta fuit in 43 huius, ad cu <lb/>ius modum & cętera referuntur:</s> <s xml:id="echoid-s43730" xml:space="preserve"> reflectaturq́;</s> <s xml:id="echoid-s43731" xml:space="preserve"> forma puncti o, quod eſt rem otius à cẽtro ſpeculi, ad <lb/>uiſum in punctũ h à puncto ſpeculi b:</s> <s xml:id="echoid-s43732" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s43733" xml:space="preserve"> locus imaginis pũctus q:</s> <s xml:id="echoid-s43734" xml:space="preserve"> & producatur ſemidiameter d g <lb/>in punctum q, ut ſit linea d q:</s> <s xml:id="echoid-s43735" xml:space="preserve"> refle ctaturq́;</s> <s xml:id="echoid-s43736" xml:space="preserve"> forma puncti u ad uiſum exiſtentẽ in puncto h à puncto <lb/>ſpeculi, quod eſt f:</s> <s xml:id="echoid-s43737" xml:space="preserve"> & locus imaginis eius ſit punctum n.</s> <s xml:id="echoid-s43738" xml:space="preserve"> Et quia puncta o & u ſunt in ſemidiametro <lb/>d g:</s> <s xml:id="echoid-s43739" xml:space="preserve"> erũt loca imaginũ, quę ſint puncta q & n, in eadem ſemidiametro producta, quæ erit linea d u o <lb/>n q:</s> <s xml:id="echoid-s43740" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s43741" xml:space="preserve"> quantitas linearũ d q, d u, d n, d o illis omnino ęqualis, quę ſunt aſſumptę in 43 huius:</s> <s xml:id="echoid-s43742" xml:space="preserve"> & erit <lb/>linea h d perpendicularis ſuper lineam d q, ut patet ex præmiſsis:</s> <s xml:id="echoid-s43743" xml:space="preserve"> eſt enim ipſa perpendicularis ſu-<lb/>per ſuperficiem circuli:</s> <s xml:id="echoid-s43744" xml:space="preserve"> eſtq́;</s> <s xml:id="echoid-s43745" xml:space="preserve"> linea d h æqualis illi lineæ d h, quæ in figura 43 huius.</s> <s xml:id="echoid-s43746" xml:space="preserve"> Angulus ergo h <lb/>d q eſt rectus:</s> <s xml:id="echoid-s43747" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s43748" xml:space="preserve"> communis ſectio ſuperficiei planæ, in qua ſunt lineę h d & d q, & ſuperficiei ſpe <lb/>culi circulus, cuius arcus interiacens lineas d h & d q per 20 huius eſt arcus, ex quo fit reflexio for-<lb/>marum, quarum imagines ſunt in punctis q & n:</s> <s xml:id="echoid-s43749" xml:space="preserve"> & erit arcus ille æ qualis arcui a g aſſumpto in 43 <lb/> <pb o="359" file="0661" n="661" rhead="LIBER OCTAVVS."/> huius:</s> <s xml:id="echoid-s43750" xml:space="preserve"> & ex duobus punctis illius arcus ſimilibus duobus punctis b & f in 43 huius fit ab hoc arcu <lb/>illa reflexio ſormarum duorum punctorum, quæ ſunt u & o:</s> <s xml:id="echoid-s43751" xml:space="preserve"> erit ergo q imago puncti o, & n imago <lb/>puncti u.</s> <s xml:id="echoid-s43752" xml:space="preserve"> Ducatur ergo à puncto u in ſuperficie cir <lb/> <anchor type="figure" xlink:label="fig-0661-01a" xlink:href="fig-0661-01"/> culi a b g recta perpendicularis ſuper lineam d u:</s> <s xml:id="echoid-s43753" xml:space="preserve"> <lb/>quæ ſit z u e:</s> <s xml:id="echoid-s43754" xml:space="preserve"> & à centro d ſecundum longitudinẽ <lb/>ſemidiametri d o fiat circulus:</s> <s xml:id="echoid-s43755" xml:space="preserve"> hic ergo circulus ſe <lb/>cabit lineã z u e in duobus pũctis per 2 p 3:</s> <s xml:id="echoid-s43756" xml:space="preserve"> ſecet er <lb/>go in punctis z & e:</s> <s xml:id="echoid-s43757" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s43758" xml:space="preserve"> arcus circuli ſecundum <lb/>quantitatem lineæ d q à centro d:</s> <s xml:id="echoid-s43759" xml:space="preserve"> & ducantur à cẽ <lb/>tro ſpeculi d lineæ d z, d e:</s> <s xml:id="echoid-s43760" xml:space="preserve"> & producãtur extra ſpe <lb/>culum ad arcum circuli deſcripti à centro d ſecun <lb/>dum quantitatem ſemidiametri d q:</s> <s xml:id="echoid-s43761" xml:space="preserve"> & ſint d t, d k:</s> <s xml:id="echoid-s43762" xml:space="preserve"> <lb/>& ducatur linea t k:</s> <s xml:id="echoid-s43763" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s43764" xml:space="preserve"> lineam d q in puncto l.</s> <s xml:id="echoid-s43765" xml:space="preserve"> <lb/>Quia ergo linea h d eſt perpendicularis ſuper ſu-<lb/>perficiem circuli, palàm per definitionem lineæ e-<lb/>rectæ quoniam uterque angulus h d t, h d k eſt re-<lb/>ctus:</s> <s xml:id="echoid-s43766" xml:space="preserve"> & utraq;</s> <s xml:id="echoid-s43767" xml:space="preserve"> ſuperficies h d t & h d k in ſuperficie <lb/>ſpeculi continet arcũ interiacentẽ lineas h d & d t, <lb/>& lineas h d & d k per 69 th.</s> <s xml:id="echoid-s43768" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43769" xml:space="preserve"> quorũ arcuũ <lb/>quilibet eſt ęqualis arcui, qui eſt inter duas lineas <lb/>h d & d q:</s> <s xml:id="echoid-s43770" xml:space="preserve"> & utraq;</s> <s xml:id="echoid-s43771" xml:space="preserve"> linearum d z & d e eſt ęqualis <lb/>lineæ d o:</s> <s xml:id="echoid-s43772" xml:space="preserve"> quoniã omnes ſunt ſemidiametri eiuſdẽ <lb/>circuli.</s> <s xml:id="echoid-s43773" xml:space="preserve"> Illi ergo duo arcus ſunt huiuſmodi, quòd <lb/>exillis poſsibile eſt fieri reflexionẽ formarũ duo-<lb/>rum punctorum, quæ ſunt z & e, ab aliquibus pun <lb/>ctis illorum arcuum, ut patet per 20 huius.</s> <s xml:id="echoid-s43774" xml:space="preserve"> Interia <lb/>cent enim illi arcus ſemidiametros ſpeculi, in qui-<lb/>bus conſiſtunt centrum uiſus, quod eſt in puncto <lb/>h, & puncta, quorum formæ reflectũtur, quæ ſunt <lb/>e & z:</s> <s xml:id="echoid-s43775" xml:space="preserve"> incidentq́;</s> <s xml:id="echoid-s43776" xml:space="preserve"> formæ eorum illis punctis illorũ <lb/>arcuum, & reflectentur ad uiſum in punctum h ſe-<lb/>cundum angulos ęquales à duobus punctis ſpecu <lb/>li:</s> <s xml:id="echoid-s43777" xml:space="preserve"> & duę lineæ d t & d k ſunt æ quales lineę d q:</s> <s xml:id="echoid-s43778" xml:space="preserve"> er-<lb/>go punctum t eſt locus imaginis puncti z, & pun-<lb/>ctum k eſt locus imaginis puncti e.</s> <s xml:id="echoid-s43779" xml:space="preserve"> Et quia lineæ <lb/>d t, d q, d k ſunt æ quales, & lineę d z, d o, d e æ quales, erit per 7 p 5 proportio lineæ d t ad lineã d z, <lb/>ſicut lineæ d q ad lineam d o, & ſicut lineæ k d ad lineam d e:</s> <s xml:id="echoid-s43780" xml:space="preserve"> ſed per 43 huius proportio lineę d q a d <lb/>lineam d o eſt maior proportione lineę d n ad lineam d u:</s> <s xml:id="echoid-s43781" xml:space="preserve"> ergo ſimiliter proportio lineę k d ad lineã <lb/>d e eſt maior proportione lineę n d ad lineã d u:</s> <s xml:id="echoid-s43782" xml:space="preserve"> & ſimiliter proportio lineę d t ad lineam d z eſt ma-<lb/>ior proportione lineæ d n a d lineam d u.</s> <s xml:id="echoid-s43783" xml:space="preserve"> Et quia duę lineę d e & z d ſunt æ quales, & duæ lineæ d t <lb/>& d k ſunt æquales:</s> <s xml:id="echoid-s43784" xml:space="preserve"> erit per 7 p 5 proportio lineę d t a d lineã d z, ſicut lineę d k ad lineã d e:</s> <s xml:id="echoid-s43785" xml:space="preserve"> ergo per <lb/>17 p 5 erit proportio lineæ t z ad lineam z d, ſicut lineæ k e ad lineam d e:</s> <s xml:id="echoid-s43786" xml:space="preserve"> ergo per 2 p 6 linea t k eſt ę-<lb/>quidiſtans lineę ez:</s> <s xml:id="echoid-s43787" xml:space="preserve"> erit ergo per eandem 2 p 6 & per 18 p 5 proportio lineę l d ad lineam d u, ſicut li-<lb/>neę d k ad lineam d e, & ſicut lineę d t ad lineam d z:</s> <s xml:id="echoid-s43788" xml:space="preserve"> proportio ergo lineę l d ad lineẽ d u eſt maior <lb/>proportione lineę n d ad lineam d u:</s> <s xml:id="echoid-s43789" xml:space="preserve"> ergo per 10 p 5 linea l d eſt maior quàm linea n d:</s> <s xml:id="echoid-s43790" xml:space="preserve"> ergo punctus <lb/>n eſt inter puncta l & u:</s> <s xml:id="echoid-s43791" xml:space="preserve"> ſed punctus n eſt imago puncti u:</s> <s xml:id="echoid-s43792" xml:space="preserve"> & duo puncta t & k ſunt imagines duorũ <lb/>punctorũ z & e:</s> <s xml:id="echoid-s43793" xml:space="preserve"> ergo imago lineę z u e rectæ eſt linea tranſiens per tria puncta t, n, k:</s> <s xml:id="echoid-s43794" xml:space="preserve"> linea uerò per-<lb/>tranſiens hæc puncta eſt conuexa.</s> <s xml:id="echoid-s43795" xml:space="preserve"> Patet ergo quòd imago lineę z e rectę uidebitur in hoc ſitu con-<lb/>uexa.</s> <s xml:id="echoid-s43796" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s43797" xml:space="preserve"/> </p> <div xml:id="echoid-div1702" type="float" level="0" n="0"> <figure xlink:label="fig-0661-01" xlink:href="fig-0661-01a"> <variables xml:id="echoid-variables771" xml:space="preserve">k q t l n f g b o l u z d h a</variables> </figure> </div> </div> <div xml:id="echoid-div1704" type="section" level="0" n="0"> <head xml:id="echoid-head1268" xml:space="preserve" style="it">58. In quibuſdam ſitibus reflexione facta à ſpeculis ſphœricis concauis, uiſus comprehendet i-<lb/>maginem concauam reflexam ex linea concaua uel conuexa. Alhazen 50 n 6.</head> <p> <s xml:id="echoid-s43798" xml:space="preserve">Sit diſpoſitio omnino, quę in pręcedente.</s> <s xml:id="echoid-s43799" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s43800" xml:space="preserve">, ut patet in præmiſſa, imago formæ puncti o <lb/>eſt punctum q, & imago formę puncti z eſt punctum t, & imago ſormæ puncti e eſt punctũ k:</s> <s xml:id="echoid-s43801" xml:space="preserve"> erit er-<lb/>go linea concaua reſpectu uiſus, quæ eſt t q k, imago lineę curuę reſpectu uiſus, conuexę tamen re-<lb/>ſpectu ſpeculi, quæ eſt linea z o e.</s> <s xml:id="echoid-s43802" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s43803" xml:space="preserve">, ſi in linea z u ſignetur punctum m, qualitercunq;</s> <s xml:id="echoid-s43804" xml:space="preserve"> <lb/>hoc contingat:</s> <s xml:id="echoid-s43805" xml:space="preserve"> & circa centrum m ſecundum longitudinẽ ſemidiametri m u deſcribatur arcus par-<lb/>ui circuli, qui ſit r u ſ:</s> <s xml:id="echoid-s43806" xml:space="preserve"> hic ergo arcus ſecabit circulum z o e in duobus punctis per 10 p 3:</s> <s xml:id="echoid-s43807" xml:space="preserve"> ſint illa duo <lb/>puncta f & r:</s> <s xml:id="echoid-s43808" xml:space="preserve"> & ducantur lineæ d r & d f:</s> <s xml:id="echoid-s43809" xml:space="preserve"> quę protrahantur uſque ad arcum t q k eductum:</s> <s xml:id="echoid-s43810" xml:space="preserve"> incidatq́;</s> <s xml:id="echoid-s43811" xml:space="preserve"> <lb/>linea d f in punctum i, & linea d r in punctum p.</s> <s xml:id="echoid-s43812" xml:space="preserve"> Superficies ergo duarum linearũ h d & d p ſecabit <lb/>ſpeculum ſecundum circulum, à cuius circum ſerentię puncto aliquo duci poterunt ſecundũ angu-<lb/>los ęquales & ęqualiter ſe habentes lineę ad punctum h, in quo eſt centrũ uiſus, & a d punctũ r, qui <lb/>eſt punctus lineę uiſæ.</s> <s xml:id="echoid-s43813" xml:space="preserve"> Et ſimiliter ſuperficies duarum linearum h d & d i ſaciet in ſpeculo circulũ, à <lb/>cuius circumſerentia reflectetur ad uiſum ſorma puncti ſarcus r u f.</s> <s xml:id="echoid-s43814" xml:space="preserve"> Eſt ergo punctus p imago for-<lb/> <pb o="360" file="0662" n="662" rhead="VITELLONIS OPTICAE"/> mæ puncti r:</s> <s xml:id="echoid-s43815" xml:space="preserve"> & punctus i imago formæ puncti f:</s> <s xml:id="echoid-s43816" xml:space="preserve"> & punctus n eſt imago formæ puncti u.</s> <s xml:id="echoid-s43817" xml:space="preserve"> Imago ita-<lb/>que arcus r u f eſt linea tranſiens per puncta i p n:</s> <s xml:id="echoid-s43818" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0662-01a" xlink:href="fig-0662-01"/> ſed hæc linea i p n eſt concaua reſpectu uiſus, & ar <lb/>cus r u f eſt concauus ex parte ſuperficiei ſpeculi, <lb/>& cõuexus ex parte uiſus.</s> <s xml:id="echoid-s43819" xml:space="preserve"> Cum ergo uiſus fuerit <lb/>in puncto h, & linea r u f conuexa cum fuerit in ali <lb/>quo uiſibili:</s> <s xml:id="echoid-s43820" xml:space="preserve"> cõprehendetur imago eius concaua:</s> <s xml:id="echoid-s43821" xml:space="preserve"> <lb/>& linea z o e cõuexa cõprehenditur ſimiliter ima-<lb/>ginis cõcauę.</s> <s xml:id="echoid-s43822" xml:space="preserve"> Si ergo unaquęq;</s> <s xml:id="echoid-s43823" xml:space="preserve"> duarum linearũ, <lb/>quę ſunt z o e & r u f, habuerit unã imaginem:</s> <s xml:id="echoid-s43824" xml:space="preserve"> erit <lb/>forma illarum imaginum ſecundũ modũ declara-<lb/>tum.</s> <s xml:id="echoid-s43825" xml:space="preserve"> Et ſi aliqua ipſarum plures habuerit imagi-<lb/>nes:</s> <s xml:id="echoid-s43826" xml:space="preserve"> fortè accidet diuerſitas ſitus in illis imaginib.</s> <s xml:id="echoid-s43827" xml:space="preserve"> <lb/>ut ſuprà diximus.</s> <s xml:id="echoid-s43828" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s43829" xml:space="preserve"> Palàm <lb/>itaq;</s> <s xml:id="echoid-s43830" xml:space="preserve"> ex his præmiſsis quinq;</s> <s xml:id="echoid-s43831" xml:space="preserve"> theorematib.</s> <s xml:id="echoid-s43832" xml:space="preserve"> quòd <lb/>lineæ rectę imago in ſpeculis ſphęricis concauis <lb/>quandoq;</s> <s xml:id="echoid-s43833" xml:space="preserve"> cõprehenditur recta:</s> <s xml:id="echoid-s43834" xml:space="preserve"> quãdoq;</s> <s xml:id="echoid-s43835" xml:space="preserve"> cõuexa:</s> <s xml:id="echoid-s43836" xml:space="preserve"> <lb/>& quandoq;</s> <s xml:id="echoid-s43837" xml:space="preserve"> cõcaua:</s> <s xml:id="echoid-s43838" xml:space="preserve"> & imago lineæ cõuexę quan <lb/>doq;</s> <s xml:id="echoid-s43839" xml:space="preserve"> uidetur cõuexa:</s> <s xml:id="echoid-s43840" xml:space="preserve"> quãdoq;</s> <s xml:id="echoid-s43841" xml:space="preserve"> concaua:</s> <s xml:id="echoid-s43842" xml:space="preserve"> & lineæ <lb/>concauæ imago quandoq;</s> <s xml:id="echoid-s43843" xml:space="preserve"> uidetur cõuexa:</s> <s xml:id="echoid-s43844" xml:space="preserve"> quan-<lb/>doq;</s> <s xml:id="echoid-s43845" xml:space="preserve"> cõcaua.</s> <s xml:id="echoid-s43846" xml:space="preserve"> Formę ergo ſuperficierum uiſibiliũ <lb/>cõprehenduntur aliter ꝗ̃ ſint in his ſpeculis:</s> <s xml:id="echoid-s43847" xml:space="preserve"> nam <lb/>lineę rectę non ſunt, niſi in ſuperficieb.</s> <s xml:id="echoid-s43848" xml:space="preserve"> planis.</s> <s xml:id="echoid-s43849" xml:space="preserve"> Cũ <lb/>ergo lineæ rectæ cõprehen duntur cõuexę uel cõ-<lb/>cauę:</s> <s xml:id="echoid-s43850" xml:space="preserve"> tunc ſuperficies plana cõprehẽditur cõuexa <lb/>uel concaua.</s> <s xml:id="echoid-s43851" xml:space="preserve"> Cũ ita q;</s> <s xml:id="echoid-s43852" xml:space="preserve"> uiſus cõprehen dit lineas re <lb/>ctas cõuexas uel cõcauas aliter, ꝗ̃ ſint:</s> <s xml:id="echoid-s43853" xml:space="preserve"> cõprehen-<lb/>dit ſuperficies, in quibus ſunt illę lineę aliter, quã <lb/>ſint:</s> <s xml:id="echoid-s43854" xml:space="preserve"> & ſimiliter eſt de lineis cõuexis & cõcauis re-<lb/>ſpectu illarum ſuperficierum.</s> <s xml:id="echoid-s43855" xml:space="preserve"> Et per hoc patet ra-<lb/>tio & cauſſa aliorum multorum errorũ, qui ex mo <lb/>dis talium uiſibilium accidunt in uiſu.</s> <s xml:id="echoid-s43856" xml:space="preserve"/> </p> <div xml:id="echoid-div1704" type="float" level="0" n="0"> <figure xlink:label="fig-0662-01" xlink:href="fig-0662-01a"> <variables xml:id="echoid-variables772" xml:space="preserve">k q p i t l n g b e o r f u m z d h a</variables> </figure> </div> </div> <div xml:id="echoid-div1706" type="section" level="0" n="0"> <head xml:id="echoid-head1269" xml:space="preserve" style="it">59. In concauis ſphœricis ſpeculis à duobus ui-<lb/>dentibus ſecundum aliquem ſitum res una uiſa unum habebit idolum, ſecundum alium ue-<lb/>rò plura.</head> <p> <s xml:id="echoid-s43857" xml:space="preserve">Sit ſpeculum ſphæricum concauum:</s> <s xml:id="echoid-s43858" xml:space="preserve"> cuius cõmunis ſectio cum ſuperficie reflexionis ſit circulus <lb/>e b h:</s> <s xml:id="echoid-s43859" xml:space="preserve"> cuius diameter ſit e h:</s> <s xml:id="echoid-s43860" xml:space="preserve"> centrum uerò p:</s> <s xml:id="echoid-s43861" xml:space="preserve"> & ducatur linea a b perpendiculariter ſuper ſuperficiẽ <lb/>ſpeculi.</s> <s xml:id="echoid-s43862" xml:space="preserve"> Palàm ergo per 72 th.</s> <s xml:id="echoid-s43863" xml:space="preserve"> 1 huius quoniam ipſa tranſit per centrum ſpeculi, quod eſt punctum <lb/>p:</s> <s xml:id="echoid-s43864" xml:space="preserve"> & producatur ultra ſpeculum:</s> <s xml:id="echoid-s43865" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s43866" xml:space="preserve"> a b l, ſecans diametrum e h perpendiculariter in centro p:</s> <s xml:id="echoid-s43867" xml:space="preserve"> & <lb/>in diametro e h ſignentur duo puncta æqualiter diſtantia à cẽtro p:</s> <s xml:id="echoid-s43868" xml:space="preserve"> quæ ſintg & f:</s> <s xml:id="echoid-s43869" xml:space="preserve"> erit ergo linea g p <lb/>æqualis lineæ p f.</s> <s xml:id="echoid-s43870" xml:space="preserve"> Et à punctis g & f ducãtur duę lineę <lb/> <anchor type="figure" xlink:label="fig-0662-02a" xlink:href="fig-0662-02"/> ad circumferentiam æquales, quę angulos acutos cõ <lb/>tineant cum diametro e h, reſpectu centri p, & lineæ <lb/>a p b:</s> <s xml:id="echoid-s43871" xml:space="preserve"> quod fiet auxilio 33 p 3, ſi ex utraq;</s> <s xml:id="echoid-s43872" xml:space="preserve"> parte pũcti b <lb/>arcus æ quales a b ſcindantur parui, quorũ chordę ſint <lb/>minores quàm lineæ g p & p f, qui ſint arcus d b, t b:</s> <s xml:id="echoid-s43873" xml:space="preserve"> <lb/>& ad puncta t & d ducantur lineę, quę ſint g d & ft.</s> <s xml:id="echoid-s43874" xml:space="preserve"> Et <lb/>quia arcus b t & b d ſunt æquales, & arcus b h & b e <lb/>ęquales, remanent arcus t h & d e ęquales:</s> <s xml:id="echoid-s43875" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s43876" xml:space="preserve"> an <lb/>guli portionis, qui ſunt g d e & f t h, inter ſe æquales <lb/>per 43 th.</s> <s xml:id="echoid-s43877" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s43878" xml:space="preserve"> Et à puncto d ducatur linea contin-<lb/>gens circulum per 17 p 3, quæ ſit d q:</s> <s xml:id="echoid-s43879" xml:space="preserve"> & ſimiliter à pun <lb/>cto t ducatur linea circulum contingens, quæ ſit t m:</s> <s xml:id="echoid-s43880" xml:space="preserve"> <lb/>producanturq́;</s> <s xml:id="echoid-s43881" xml:space="preserve"> illę lineę contingentes ad diametrum <lb/>a l, & concurrent in puncto uno per 59 th.</s> <s xml:id="echoid-s43882" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43883" xml:space="preserve"> ſit <lb/>concurſus punctus r.</s> <s xml:id="echoid-s43884" xml:space="preserve"> Et quoniam per 16 p 3 anguli <lb/>contingentiæ, qui ſunt q d e & m t h ſunt æquales, & <lb/>anguli portionis, qui ſunt g d e & f t h, ſunt æquales:</s> <s xml:id="echoid-s43885" xml:space="preserve"> <lb/>erit totus angulus q d g æqualis toti angulo m t f.</s> <s xml:id="echoid-s43886" xml:space="preserve"> Su-<lb/>per punctum itaq;</s> <s xml:id="echoid-s43887" xml:space="preserve"> d terminum lineę r d conſtituatur <lb/>angulus æqualis angulo q d g per 23 p 1, qui ſit r d k:</s> <s xml:id="echoid-s43888" xml:space="preserve"> <lb/>linea quoque d k producta concurret cum linea a b <lb/>per 14 th.</s> <s xml:id="echoid-s43889" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43890" xml:space="preserve"> ſit concurſus punctus k:</s> <s xml:id="echoid-s43891" xml:space="preserve"> & ſuper punctum t terminum lineæ r t conſtituatur an-<lb/> <pb o="361" file="0663" n="663" rhead="LIBER OCTAVVS."/> gulus æqualis angulo r d k, qui ſit r t k:</s> <s xml:id="echoid-s43892" xml:space="preserve"> concurrent enim illæ lineæ ambæ in uno puncto diametri, <lb/>quod eſt k:</s> <s xml:id="echoid-s43893" xml:space="preserve"> quia cum angulus r t k ſit æqualis angulo r d k per præmiſſa, & angulus k r t ſit æqualis <lb/>angulo k r d per 59 th.</s> <s xml:id="echoid-s43894" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s43895" xml:space="preserve"> trigoni ergo d k r & t k r ſunt æquianguli per 32 p 1:</s> <s xml:id="echoid-s43896" xml:space="preserve"> ergo per 4 p 6 late <lb/>ra illorum trigonorum ſunt proportionalia.</s> <s xml:id="echoid-s43897" xml:space="preserve"> Sed linea r t æqualis eſt lineæ d r per 58 th.</s> <s xml:id="echoid-s43898" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43899" xml:space="preserve"> <lb/>erit ergo linea k r æqualis ſibijpſi:</s> <s xml:id="echoid-s43900" xml:space="preserve"> concurrent ergo lineæ d k & t k in puncto uno diametri b p, quod <lb/>eſt k.</s> <s xml:id="echoid-s43901" xml:space="preserve"> Poſitis itaq;</s> <s xml:id="echoid-s43902" xml:space="preserve"> duobus oculis diuerſorũ uιdentiũ in punctis g & f, & puncto rei uiſę in puncto k:</s> <s xml:id="echoid-s43903" xml:space="preserve"> <lb/>tũc forma pũcti k uidebitur ab utroq;</s> <s xml:id="echoid-s43904" xml:space="preserve"> uiſuũ reflexa à duobus pũctis ſpeculi d & t.</s> <s xml:id="echoid-s43905" xml:space="preserve"> Sed & idolũ eius <lb/>uιdebitur unũ & in eodẽloco.</s> <s xml:id="echoid-s43906" xml:space="preserve"> Producátur enim lineę g d & f t extra circulũ:</s> <s xml:id="echoid-s43907" xml:space="preserve"> cõcurrẽt itaq;</s> <s xml:id="echoid-s43908" xml:space="preserve"> ambę cũ <lb/>diametro a b ꝓducta ք 14 th.</s> <s xml:id="echoid-s43909" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43910" xml:space="preserve"> quonia anguli g p b & f p b ſunt recti, & anguli p g d & p ſ tacu <lb/>ti, ut patet expręmiſsis:</s> <s xml:id="echoid-s43911" xml:space="preserve"> cõcurrat ergo linea g d cũ linea a b in puncto l.</s> <s xml:id="echoid-s43912" xml:space="preserve"> Dico quod linea ft cócurret <lb/>cũ eadem lιnea a b in eodẽ puncto l.</s> <s xml:id="echoid-s43913" xml:space="preserve"> Cum enim angulus q d g ſit æqualis angulo f t m, ut ſuprà pa-<lb/>tuit, & angulus r d l ſit æqualis angulo g d q per 15 p 1, & angulus r t l æqualis angulo f t m:</s> <s xml:id="echoid-s43914" xml:space="preserve"> erit angu-<lb/>lus r d l æqualis angulo r t l:</s> <s xml:id="echoid-s43915" xml:space="preserve"> ſed & angulus t r b eſt æqualis angulo b r d per 59 th.</s> <s xml:id="echoid-s43916" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43917" xml:space="preserve"> ergo per <lb/>13 p 1 angulus t r l eſt æqualis angulo d r l:</s> <s xml:id="echoid-s43918" xml:space="preserve"> ergo per 32 p 1 trigoni t r l & d r l ſunt æquianguli.</s> <s xml:id="echoid-s43919" xml:space="preserve"> Ergo cũ <lb/>linea tr ſit æqualis lineæ r d per 58th.</s> <s xml:id="echoid-s43920" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s43921" xml:space="preserve"> erit per 4 p 6 linea r l æqualis ſibijpſi, & lιnea t l æqua <lb/>lis lineæ d l.</s> <s xml:id="echoid-s43922" xml:space="preserve"> In uno ergo puncto diametri a b l concurrent lineæ t l & d l:</s> <s xml:id="echoid-s43923" xml:space="preserve"> & hoc eſt punctum l.</s> <s xml:id="echoid-s43924" xml:space="preserve"> Patet <lb/>ergo cum per 37 th.</s> <s xml:id="echoid-s43925" xml:space="preserve"> 5 huius punctus l ſit locus imagιnis formæ puncti rei uiſæ, qui eſt k, quòd am bo <lb/>bus uiſibus uni exiſtenti in puncto g, & alij in puncto f unica tantùm occurrit imago:</s> <s xml:id="echoid-s43926" xml:space="preserve"> uiſibus uerò <lb/>permutatis ab hoc ſitu, plures occurrunt imagines.</s> <s xml:id="echoid-s43927" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s43928" xml:space="preserve"> Quandocunq;</s> <s xml:id="echoid-s43929" xml:space="preserve"> tamen <lb/>aliquid in his ſpeculis percipitur duplici uiſu, ſi linea reflexionis æquidiftãs fuerit catheto inciden <lb/>tiæ:</s> <s xml:id="echoid-s43930" xml:space="preserve"> erit locus imaginis ipſe punctus reflexionis per 11 huius:</s> <s xml:id="echoid-s43931" xml:space="preserve"> & cum diſtant à ſe puncta reflexionis, <lb/>reſpectu amborũ uiſuum:</s> <s xml:id="echoid-s43932" xml:space="preserve"> apparebunt uiſibus duæ imagines eiuſdem puncti, & locus cuiuſq;</s> <s xml:id="echoid-s43933" xml:space="preserve"> ima-<lb/>ginis eſt in ipſo puncto ſuæ reflexionis.</s> <s xml:id="echoid-s43934" xml:space="preserve"> Si uerò linea reflexionis non ſit æquidiſtãs catheto inciden <lb/>tiæ, & punctus rei uiſæ tantùm diſtet ab uno uiſu, quantùm ab altero:</s> <s xml:id="echoid-s43935" xml:space="preserve"> uel ſit modica differentia di-<lb/>ſtantιæ, ſi locus imaginis fuerit in ipſa ſuperficie uiſus:</s> <s xml:id="echoid-s43936" xml:space="preserve"> duæ adhuc imagines uidebuntur:</s> <s xml:id="echoid-s43937" xml:space="preserve"> aliàs autẽ <lb/>ut plurimùm locus imaginis reſpectu utriuſq;</s> <s xml:id="echoid-s43938" xml:space="preserve"> uiſus eritidem, aut modicùm diſtans:</s> <s xml:id="echoid-s43939" xml:space="preserve"> unde aut tan-<lb/>tùm una uidebitur imago, aut penè una.</s> <s xml:id="echoid-s43940" xml:space="preserve"/> </p> <div xml:id="echoid-div1706" type="float" level="0" n="0"> <figure xlink:label="fig-0662-02" xlink:href="fig-0662-02a"> <variables xml:id="echoid-variables773" xml:space="preserve">l r b d k t q m e g p f h a</variables> </figure> </div> </div> <div xml:id="echoid-div1708" type="section" level="0" n="0"> <head xml:id="echoid-head1270" xml:space="preserve" style="it">60. In una diametro ſpeculi ſphærici concaui poſitis ambobus oculis æqualiter à centro ſpeculi <lb/>diſtantibus, neuter uidebitur oculorum. Euclides 27th. catoptr.</head> <p> <s xml:id="echoid-s43941" xml:space="preserve">Sit ſpeculum concauum ſphæricum a t g d:</s> <s xml:id="echoid-s43942" xml:space="preserve"> cuius centrum z:</s> <s xml:id="echoid-s43943" xml:space="preserve"> & diameter a d:</s> <s xml:id="echoid-s43944" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s43945" xml:space="preserve"> duo oculi b <lb/>& e conſtituti in diametro a d, æqualiter diſtantes à centro z.</s> <s xml:id="echoid-s43946" xml:space="preserve"> Dico quòd neuter oculorum uidebi-<lb/>tur.</s> <s xml:id="echoid-s43947" xml:space="preserve"> Ducatur enim ſemidiameter z g perpendiculariter ſuper diametrum a d:</s> <s xml:id="echoid-s43948" xml:space="preserve"> & ducantur lineæ b g <lb/> <anchor type="figure" xlink:label="fig-0663-01a" xlink:href="fig-0663-01"/> & e g.</s> <s xml:id="echoid-s43949" xml:space="preserve"> Quia ergo in trigonis e z g & b z g la-<lb/>tus e z eſt ęquale lateri z b ex hypotheſi, & la <lb/>tus z g commune:</s> <s xml:id="echoid-s43950" xml:space="preserve"> anguli quoq;</s> <s xml:id="echoid-s43951" xml:space="preserve"> e z g & b z g <lb/>ſunt æquales, quia ſunt ambo recti:</s> <s xml:id="echoid-s43952" xml:space="preserve"> erit per <lb/>4 p 1 angulus b g z æqualis angulo e g z.</s> <s xml:id="echoid-s43953" xml:space="preserve"> For-<lb/>ma ergo puncti b reflectitur ad punctum e à <lb/>puncto g ſpeculi, & econuerſo per 20 th.</s> <s xml:id="echoid-s43954" xml:space="preserve"> 5 <lb/>huius.</s> <s xml:id="echoid-s43955" xml:space="preserve"> Sed neq;</s> <s xml:id="echoid-s43956" xml:space="preserve"> poſsibile eſt ab alio puncto <lb/>ſpeculi ſormam puncti b ad punctum e refle <lb/>cti.</s> <s xml:id="echoid-s43957" xml:space="preserve"> Sit enim, ut fuerit id datũ eſſe poſsibile, <lb/>ut ſorma puncti b reflectatur ad punctum e <lb/>â puncto alio ſpeculi, quod ſit t:</s> <s xml:id="echoid-s43958" xml:space="preserve"> & ducantur <lb/>lineæ b t, t e, t z:</s> <s xml:id="echoid-s43959" xml:space="preserve"> linea ergo t z diuidit angulum b t e per duo æqualia per 20 th.</s> <s xml:id="echoid-s43960" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s43961" xml:space="preserve"> erit ergo per <lb/>3 p 6 proportio lineæ b t ad lineam t e, ſicut lineæ b z ad lineam e z:</s> <s xml:id="echoid-s43962" xml:space="preserve"> ſed linea b t eſt maior quàm <lb/>linea b g per 7 p 3:</s> <s xml:id="echoid-s43963" xml:space="preserve"> linea uero b g eſt æqualis lineæ e g, ut patuit ſuperius:</s> <s xml:id="echoid-s43964" xml:space="preserve"> linea uerò e g eſt maior <lb/>quàm linea t e per 7 p 3:</s> <s xml:id="echoid-s43965" xml:space="preserve"> erit ergo linea b t maior quàm linea e t:</s> <s xml:id="echoid-s43966" xml:space="preserve"> ergo linea b z maior erit quàm linea <lb/>e z:</s> <s xml:id="echoid-s43967" xml:space="preserve"> quod eſt contra hypotheſim & impoſsibile.</s> <s xml:id="echoid-s43968" xml:space="preserve"> Et eodem modo de quolibet puncto ſemicirculi a g <lb/>d poteſt demonſtrari.</s> <s xml:id="echoid-s43969" xml:space="preserve"> Non ergo reflectitur forma puncti b ad punctum e ab alio ſpeculi puncto <lb/>quàm à puncto g.</s> <s xml:id="echoid-s43970" xml:space="preserve"> Non ergo uidebit oculus b oculum e:</s> <s xml:id="echoid-s43971" xml:space="preserve"> ideo quia linea reflexionis, quæ eſt b g, non <lb/>concurrit cum catheto e z ducta à puncto e per centrum ſpeculi z, niſi in puncto b:</s> <s xml:id="echoid-s43972" xml:space="preserve"> & linea re-<lb/>flexionis, quæ eſt e g, non concurrit cum catheto b z, niſi in puncto e.</s> <s xml:id="echoid-s43973" xml:space="preserve"> Locus itaq;</s> <s xml:id="echoid-s43974" xml:space="preserve"> imaginis e eſt <lb/>punctus b:</s> <s xml:id="echoid-s43975" xml:space="preserve"> ſed b eſt ſimile ipſi e in forma, & e ipſi b.</s> <s xml:id="echoid-s43976" xml:space="preserve"> Non ergo cóprehenditur aliqua diſtantia, quæ <lb/>ſit cauſſa diuerſitatis inter illos uiſus.</s> <s xml:id="echoid-s43977" xml:space="preserve"> Non ergo unus uiſus percipiet formam alterius in ſeipſo exi-<lb/>ſtentis, ſed æſtimabit formam propriam ſe uidere.</s> <s xml:id="echoid-s43978" xml:space="preserve"> Non ergo unus oculus taliter diſpoſitus uiſibus <lb/>alium oculum uidebit:</s> <s xml:id="echoid-s43979" xml:space="preserve"> & hoc eſt propoſitum.</s> <s xml:id="echoid-s43980" xml:space="preserve"> Aliæ tamen partes corporis circumſtantes centrum <lb/>uiſus poterunt uideri:</s> <s xml:id="echoid-s43981" xml:space="preserve"> quarum catheti incidentiæ cum lineis ſuarum reflexionum concurrunt:</s> <s xml:id="echoid-s43982" xml:space="preserve"> ſiue <lb/>ille concurſus ſit in ſuperficie uiſus uel in alijs punctis quibuſcunq;</s> <s xml:id="echoid-s43983" xml:space="preserve">: & circa hæc multa diuerſi-<lb/>tas uiſibus occurrit.</s> <s xml:id="echoid-s43984" xml:space="preserve"/> </p> <div xml:id="echoid-div1708" type="float" level="0" n="0"> <figure xlink:label="fig-0663-01" xlink:href="fig-0663-01a"> <variables xml:id="echoid-variables774" xml:space="preserve">g t d b z e a</variables> </figure> </div> </div> <div xml:id="echoid-div1710" type="section" level="0" n="0"> <head xml:id="echoid-head1271" xml:space="preserve" style="it">61. Si in linea à puncto medio ſemidiametri ſuper diametrũ ſpeculi ſphærici concaui perpen-<lb/>diculariter erectæ duct a æquidiſtanter diametro ambo ponantur oculi, æqualiter diſtates à cen <lb/>tro ſpeculi: imago una tantùm oculi apparebit in puncto reflexionis. Euclides 28 th catoptr.</head> <pb o="362" file="0664" n="664" rhead="VITELLONIS OPTICAE"/> <p> <s xml:id="echoid-s43985" xml:space="preserve">Sit ſpeculum ſphæricum concauum a g d:</s> <s xml:id="echoid-s43986" xml:space="preserve"> cuius centrum k:</s> <s xml:id="echoid-s43987" xml:space="preserve"> & diameter a d:</s> <s xml:id="echoid-s43988" xml:space="preserve"> dueaturq́;</s> <s xml:id="echoid-s43989" xml:space="preserve"> ſemidia-<lb/>meter k g perpendiculariter ſuper diametrum a d:</s> <s xml:id="echoid-s43990" xml:space="preserve"> & à medio puncto ſemidiametri k g ducatur li-<lb/>n e a æquidiſtans diametro a d:</s> <s xml:id="echoid-s43991" xml:space="preserve"> & in hac poſiti ſint uiſus ambo æqualiter diſtantes à centro k.</s> <s xml:id="echoid-s43992" xml:space="preserve"> Dico <lb/>quòd amborum oculorũ una tantùm imago, in uno ſcilicet puncto reflexionis uidebitur.</s> <s xml:id="echoid-s43993" xml:space="preserve"> Sit enim <lb/>ut à puncto p (quod ſit medius punctus lineę k g per 10 p 1) ducatur linea æ quidiſtans diametro a d <lb/>per 31 p 1, quæ ſit e z:</s> <s xml:id="echoid-s43994" xml:space="preserve"> & ſint in illa perpẽdiculari e z poſiti ambo oculi, quiſint b & t, æ qualiter diſtan-<lb/>tes à centro k, & à linea k g:</s> <s xml:id="echoid-s43995" xml:space="preserve"> erunt ergo lineæ b p & t p æquales:</s> <s xml:id="echoid-s43996" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s43997" xml:space="preserve"> lineæ b g, t g, b k, t k:</s> <s xml:id="echoid-s43998" xml:space="preserve"> er-<lb/> <anchor type="figure" xlink:label="fig-0664-01a" xlink:href="fig-0664-01"/> go per 4 p 1 linea p g exiſtente communiam <lb/>bobus trigonis b p g & t p g, cum anguli b p <lb/>g & t p g ſint recti:</s> <s xml:id="echoid-s43999" xml:space="preserve"> erit angulus b g p ęqualis <lb/>angulo t g p.</s> <s xml:id="echoid-s44000" xml:space="preserve"> Reflectetur ergo forma puncti <lb/>b ad punctũ t à puncto ſpeculi g:</s> <s xml:id="echoid-s44001" xml:space="preserve"> & econuer-<lb/>ſo.</s> <s xml:id="echoid-s44002" xml:space="preserve"> Et quia linea k p eſt ęqualis lineę p g, quo <lb/>niam punctus p eſt medius punctus lineæ k <lb/>g, & lineæ b p & t p ſunt æquales:</s> <s xml:id="echoid-s44003" xml:space="preserve"> angulus <lb/>quoq;</s> <s xml:id="echoid-s44004" xml:space="preserve"> k p t eſt æqualis angulo b p g per 15 p <lb/>1:</s> <s xml:id="echoid-s44005" xml:space="preserve"> ergo per 4 p 1 angulus t k p eſt æqualis an-<lb/>gulo b g p:</s> <s xml:id="echoid-s44006" xml:space="preserve"> ergo per 27 p 1 linea t k æquidiſtat <lb/>lineæ b g:</s> <s xml:id="echoid-s44007" xml:space="preserve"> ſed linea t k eſt cathetus puncti t, <lb/>& linea b g eſt linea reflexionis:</s> <s xml:id="echoid-s44008" xml:space="preserve"> nunquam ergo concurrent:</s> <s xml:id="echoid-s44009" xml:space="preserve"> ergo per 11 huius non uidebitur forma <lb/>puncti t, qui eſt unus oculorum ab alio oculo, qui eſt b:</s> <s xml:id="echoid-s44010" xml:space="preserve"> neq;</s> <s xml:id="echoid-s44011" xml:space="preserve"> econuerſo per eandem rationem, niſi <lb/>in puncto g, qui eſt punctus reflexionis.</s> <s xml:id="echoid-s44012" xml:space="preserve"> Linea enim b g, quæ eſt linea reflexionis formę puncti t ad <lb/>uiſum b, non concurrit cum catheto incidentiæ formę puncti t, quę eſt linea t k.</s> <s xml:id="echoid-s44013" xml:space="preserve"> Quilibet ergo ocu-<lb/>lorum uidebit alterum in uno tantùm puncto reflexionis.</s> <s xml:id="echoid-s44014" xml:space="preserve"> Imago ergo amborum oculorũ erit tan-<lb/>tùm una:</s> <s xml:id="echoid-s44015" xml:space="preserve"> & ſic unus tantùm oculus apparebit.</s> <s xml:id="echoid-s44016" xml:space="preserve"> Et quoniam reliqua pars faciei uidentis offertur am <lb/>bobus uiſibus retro uiſus) quia ad illam partem catheti incidentię cum lineis reflexionum concur-<lb/>runt, ut patet intuenti.</s> <s xml:id="echoid-s44017" xml:space="preserve"> Si enim lineę b k & t g caderent inter lineas concurrentes:</s> <s xml:id="echoid-s44018" xml:space="preserve"> tunc & ipſæ con-<lb/>currerent:</s> <s xml:id="echoid-s44019" xml:space="preserve"> quod eſt impoſsibile:</s> <s xml:id="echoid-s44020" xml:space="preserve"> cum ſint æ quidiſtantes:</s> <s xml:id="echoid-s44021" xml:space="preserve"> concurrent ergo retro ambos uiſus illę li-<lb/>neę) ergo per 37 th.</s> <s xml:id="echoid-s44022" xml:space="preserve"> 5 huius apparebit tunc facies uidentis monocula ad modum picturę cyclopis, <lb/>eritq́;</s> <s xml:id="echoid-s44023" xml:space="preserve"> oculus ultra faciem prominens:</s> <s xml:id="echoid-s44024" xml:space="preserve"> quoniam nõ uidetur, niſi in puncto reflexionis per 11 huius.</s> <s xml:id="echoid-s44025" xml:space="preserve"> <lb/>Patet ergo propoſitum.</s> <s xml:id="echoid-s44026" xml:space="preserve"/> </p> <div xml:id="echoid-div1710" type="float" level="0" n="0"> <figure xlink:label="fig-0664-01" xlink:href="fig-0664-01a"> <variables xml:id="echoid-variables775" xml:space="preserve">g z t p b e d k a</variables> </figure> </div> </div> <div xml:id="echoid-div1712" type="section" level="0" n="0"> <head xml:id="echoid-head1272" xml:space="preserve" style="it">62. Si à puncto propinquiori diametro ſpeculi ſphærici concaui quàm medius punctus ſemi-<lb/>diametri ſuper illam diametrum orthogonaliter productæ linea æquidiſtãs diametro producd-<lb/>tur: in illa uiſus, in æquidiſtantia à centro ſpeculi poſitiretro ſe apparebũt: dextra pars dextra, <lb/>& ſiniſtra ſiniſtra: idolum maius facie: & imago plus diſtabit à uiſu quàm facies uidentis à ſu-<lb/>perficie ſpeculi. Euclides 29 th. catoptr.</head> <p> <s xml:id="echoid-s44027" xml:space="preserve">Sit communis ſectio ſuperficiei reflexionis & ſpeculi ſphærici concaui circulus a g d:</s> <s xml:id="echoid-s44028" xml:space="preserve"> cuius dia-<lb/>meter ſit a d:</s> <s xml:id="echoid-s44029" xml:space="preserve"> & ducatur ſemidiameter k g perpendiculariter ſuper diametrũ a d:</s> <s xml:id="echoid-s44030" xml:space="preserve"> cuius femidiame-<lb/>tri k g medius punctus ſit p:</s> <s xml:id="echoid-s44031" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s44032" xml:space="preserve"> centra amborum uiſuum puncta b & t.</s> <s xml:id="echoid-s44033" xml:space="preserve"> Si ergo ab aliquo puncto <lb/>lineę p k (qui ſit n) ducatur linea æquidiſtanter diametro a d:</s> <s xml:id="echoid-s44034" xml:space="preserve"> quę ſit l m:</s> <s xml:id="echoid-s44035" xml:space="preserve"> & uiſus b & t poſiti in li-<lb/>neal m, æqualiter diſtent à puncto n, uel à centro ſpeculi, quod eſt k:</s> <s xml:id="echoid-s44036" xml:space="preserve"> dico quod accidet, ut proponi <lb/>tur.</s> <s xml:id="echoid-s44037" xml:space="preserve"> Ducantur enim lineę b g, t g, b k, t k:</s> <s xml:id="echoid-s44038" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s44039" xml:space="preserve"> ex hypotheſi & per 4 p 1 anguli b g n & t g n æqua-<lb/>les:</s> <s xml:id="echoid-s44040" xml:space="preserve"> ergo à puncto g reflectentur uiſus adinuicem mutuò per 20 th.</s> <s xml:id="echoid-s44041" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s44042" xml:space="preserve"> ſed linea n g eſt maior <lb/>quàm linea n k:</s> <s xml:id="echoid-s44043" xml:space="preserve"> reſecetur ergo per 3 p 1 linea n g ad æqualitatem lineę n k in puncto q:</s> <s xml:id="echoid-s44044" xml:space="preserve"> & ducatur li-<lb/> <anchor type="figure" xlink:label="fig-0664-02a" xlink:href="fig-0664-02"/> nea b q:</s> <s xml:id="echoid-s44045" xml:space="preserve"> erit ergo per 15 & 4 p 1 angulus b q n æqua-<lb/>lis angulo t k n:</s> <s xml:id="echoid-s44046" xml:space="preserve"> ſed angulus b q n eſt maior angulo <lb/>b g q per 16 p 1:</s> <s xml:id="echoid-s44047" xml:space="preserve"> ergo angulus t k n eſt maior angulo <lb/>b g q:</s> <s xml:id="echoid-s44048" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s44049" xml:space="preserve"> 1 huius lineę t k & g b concur-<lb/>rentretro uiſum b:</s> <s xml:id="echoid-s44050" xml:space="preserve"> cõcurrant ergo in puncto s.</s> <s xml:id="echoid-s44051" xml:space="preserve"> Eſt <lb/>aũt linea t k cathetus puncti t, & linea g b linea re-<lb/>flexionis.</s> <s xml:id="echoid-s44052" xml:space="preserve"> Videbitur ergo forma puncti g retro ui-<lb/>fum b.</s> <s xml:id="echoid-s44053" xml:space="preserve"> Et ſimiliter per eadem penitus uidebitur for <lb/>ma puncti b retro uiſum t:</s> <s xml:id="echoid-s44054" xml:space="preserve"> quia lineæ b k & g t con-<lb/>current, ut præ oſtenſum eſt per 14 th.</s> <s xml:id="echoid-s44055" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s44056" xml:space="preserve"> ſit, ut <lb/>concurrant in puncto x:</s> <s xml:id="echoid-s44057" xml:space="preserve"> & ducatur linea s x.</s> <s xml:id="echoid-s44058" xml:space="preserve"> Et <lb/>quoniam linea s x eſt maior quàm linea b t:</s> <s xml:id="echoid-s44059" xml:space="preserve"> ideo <lb/>quòd in triangulo s g t angulus s t g, ut patet ex <lb/>præmiſsis, eſt æqualis angulo x b g trigoni x g b, & <lb/>angulus s g x cõmunis:</s> <s xml:id="echoid-s44060" xml:space="preserve"> erunt ergo per 32 p 1 triangu <lb/>li illi s t g & x b g ęquiáguli:</s> <s xml:id="echoid-s44061" xml:space="preserve"> eſt ergo ք 4 p 6 propor <lb/>tío lineę x g e ad lineá g s, ſicutlineæ b g ad lineã g t:</s> <s xml:id="echoid-s44062" xml:space="preserve"> ſed linea b g eſt æqualis lineę g t:</s> <s xml:id="echoid-s44063" xml:space="preserve"> ergo linea x g <lb/> <pb o="363" file="0665" n="665" rhead="LIBER OCTAVVS."/> eſt æqualis lineæ g s, & linea x b æqualis lineæ s t:</s> <s xml:id="echoid-s44064" xml:space="preserve"> ergo per 7 p 5 erit proportio lineæ x g ad lineam <lb/>g t, ſicut lineæ s g ad lineam g b:</s> <s xml:id="echoid-s44065" xml:space="preserve"> ergo per 17 p 5 erit proportio lineæ x t ad lineam t g, ſicut lineæ s b <lb/>ad lineam b g.</s> <s xml:id="echoid-s44066" xml:space="preserve"> In trigono ergo s g x per 2 p 6 linea b t æquidiſtat lineæ s x:</s> <s xml:id="echoid-s44067" xml:space="preserve"> eſt igitur per 4 p 6 pro-<lb/>portio lineæ s x ad lineam b t, ſicut lineæ x g ad lineam g t:</s> <s xml:id="echoid-s44068" xml:space="preserve"> ſed linea x g maior eſt quàm linea g t:</s> <s xml:id="echoid-s44069" xml:space="preserve"> er-<lb/>go linea s x maior eſt quàm linea b c.</s> <s xml:id="echoid-s44070" xml:space="preserve"> Imago ergo erit facie maior:</s> <s xml:id="echoid-s44071" xml:space="preserve"> quoniam linea s x eſt diameter <lb/>imaginis, & linea b t pars diametri faciei:</s> <s xml:id="echoid-s44072" xml:space="preserve"> ſcilicet linea continenes diſtantiam oculorum.</s> <s xml:id="echoid-s44073" xml:space="preserve"> Item linea <lb/>g k producta ſecat lineam s x per 29 th.</s> <s xml:id="echoid-s44074" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s44075" xml:space="preserve"> ſecat enim angulum s g x:</s> <s xml:id="echoid-s44076" xml:space="preserve"> ſecet ergo in puncto u.</s> <s xml:id="echoid-s44077" xml:space="preserve"> <lb/>Quia itaq;</s> <s xml:id="echoid-s44078" xml:space="preserve"> in trigono s u g linea b n æquidiſtat baſi s u:</s> <s xml:id="echoid-s44079" xml:space="preserve"> patet per 4 p 6 & 17 p 5 quia eſt propor-<lb/>tio lineæ u n ad lineam n g, ſicut lineæ s u ad lineam b n:</s> <s xml:id="echoid-s44080" xml:space="preserve"> ſed linea s u eſt maior quàm linea b n per <lb/>4 p 6:</s> <s xml:id="echoid-s44081" xml:space="preserve"> quoniam linea u g eſt maior quàm linea n g:</s> <s xml:id="echoid-s44082" xml:space="preserve"> erit ergo linea u n maior quàm linea n g:</s> <s xml:id="echoid-s44083" xml:space="preserve"> ſed li-<lb/>nea u n eſt diſtantia imaginis à uiſu, & linea n g eſt diſtantia uiſus à ſpeculi ſuperficie.</s> <s xml:id="echoid-s44084" xml:space="preserve"> Patet er-<lb/>go propoſitum.</s> <s xml:id="echoid-s44085" xml:space="preserve"/> </p> <div xml:id="echoid-div1712" type="float" level="0" n="0"> <figure xlink:label="fig-0664-02" xlink:href="fig-0664-02a"> <variables xml:id="echoid-variables776" xml:space="preserve">g q p l b n t m a k d s u a</variables> </figure> </div> </div> <div xml:id="echoid-div1714" type="section" level="0" n="0"> <head xml:id="echoid-head1273" xml:space="preserve" style="it">63. Si à puncto remotiori à diametro ſpeculi ſphærici concaui quàm medius punctus ſemidia <lb/>metri orthogonaliter ſuper illam ſemidiametrum productæ, linea æquidiſtans diametro produ <lb/>catur, uiſibus æquidiſtanter à centro ſpeculi in linea illa poſitis, dextra apparent ſinictra, & ſi-<lb/>niſtra dextra: & imago uidentis maior facie: maioŕ erit diſtantia imaginis à ſpeculo quàm fa <lb/>ciei uidentis. Euclides alter a parte 28 th. catoptr.</head> <p> <s xml:id="echoid-s44086" xml:space="preserve">Eſto ſpeculum ſphæricum concauum, cuius ſuperficiei & ſuperficiei reflexionis communis ſe-<lb/>ctio ſit circulus a k f:</s> <s xml:id="echoid-s44087" xml:space="preserve"> cuius centrum z:</s> <s xml:id="echoid-s44088" xml:space="preserve"> & diameter a f:</s> <s xml:id="echoid-s44089" xml:space="preserve"> & à centro z ducatur perpendicularis ſuper <lb/>diametrum a f, ſemidiameter z k:</s> <s xml:id="echoid-s44090" xml:space="preserve"> quæ diuidatur per æqualia in puncto e:</s> <s xml:id="echoid-s44091" xml:space="preserve"> & à puncto e ducatur æ-<lb/>quidiſtans diametro a f, linea c d:</s> <s xml:id="echoid-s44092" xml:space="preserve"> diuidatur quoq;</s> <s xml:id="echoid-s44093" xml:space="preserve"> linea e k per æqualia in puncto n:</s> <s xml:id="echoid-s44094" xml:space="preserve"> & à puncto n li <lb/>neæ e k ducatur linea æquidiſtans lineæ a f, quæ ſit l m:</s> <s xml:id="echoid-s44095" xml:space="preserve"> in hac itaq;</s> <s xml:id="echoid-s44096" xml:space="preserve"> linea l m ponantur uiſus æqua-<lb/>liter diſtantes à centro z:</s> <s xml:id="echoid-s44097" xml:space="preserve"> dico quòd uerum eſt, quod proponitur.</s> <s xml:id="echoid-s44098" xml:space="preserve"> Sint enim uiſus b & g diſpoſiti in <lb/>linea l m, ut proponitur:</s> <s xml:id="echoid-s44099" xml:space="preserve"> erunt ergo, ut in præmiſſa propoſitione, anguli b k n & g k n æquales per 4 <lb/>p 1:</s> <s xml:id="echoid-s44100" xml:space="preserve"> reflectentur ergo uiſus b & g ad ſe inuicem mutuò à puncto k.</s> <s xml:id="echoid-s44101" xml:space="preserve"> Ducantur quoq;</s> <s xml:id="echoid-s44102" xml:space="preserve"> lineæ b e & g e:</s> <s xml:id="echoid-s44103" xml:space="preserve"> <lb/>& erit per 4 p 1 angulus b e n æqualis angulo b k n:</s> <s xml:id="echoid-s44104" xml:space="preserve"> ſed angulus b e n per 16 p 1 eſt maior angulo b <lb/>z e:</s> <s xml:id="echoid-s44105" xml:space="preserve"> ergo angulus b k z maior eſt angulo b z k:</s> <s xml:id="echoid-s44106" xml:space="preserve"> ergo & maior eſt angulo k z g:</s> <s xml:id="echoid-s44107" xml:space="preserve"> ergo per 14 th.</s> <s xml:id="echoid-s44108" xml:space="preserve"> 1 huius <lb/>lineæ b k & z g concurrent:</s> <s xml:id="echoid-s44109" xml:space="preserve"> ſit concurſus punctus q:</s> <s xml:id="echoid-s44110" xml:space="preserve"> ſed & per eandem lineæ g k & z b concurrent:</s> <s xml:id="echoid-s44111" xml:space="preserve"> <lb/>ſit concurſus punctus p.</s> <s xml:id="echoid-s44112" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s44113" xml:space="preserve"> linea g k ſit linea reflexionis formæ puncti b à puncto ſpeculi k, <lb/> <anchor type="figure" xlink:label="fig-0665-01a" xlink:href="fig-0665-01"/> & linea z b ſit cathetus incidentiæ:</s> <s xml:id="echoid-s44114" xml:space="preserve"> erit ergo per 37 <lb/>th.</s> <s xml:id="echoid-s44115" xml:space="preserve"> 5 huius punctus p imago formæ puncti b:</s> <s xml:id="echoid-s44116" xml:space="preserve"> & ſimi-<lb/>liter erit punctus q imago formæ puncti g.</s> <s xml:id="echoid-s44117" xml:space="preserve"> Ducatur <lb/>ergo linea p q:</s> <s xml:id="echoid-s44118" xml:space="preserve"> & hæc erit imago lineæ b g.</s> <s xml:id="echoid-s44119" xml:space="preserve"> Videbi-<lb/>tur ergo dextrum ſiniſtrum, & ſiniſtrum dextrum, <lb/>propter interſectionẽ linearũ reflexionis b q & g p, <lb/>ut patet per 53 huius.</s> <s xml:id="echoid-s44120" xml:space="preserve"> Itẽ per 4 p 1 linea z b eſt æqua-<lb/>lis lineæ z g:</s> <s xml:id="echoid-s44121" xml:space="preserve"> ergo per 5 p 1 angulus z b n eſt æqualis <lb/>angulo z g n, & angulus p b g eſt æqualis angulo q g <lb/>b:</s> <s xml:id="echoid-s44122" xml:space="preserve"> ſed angulus n b k æqualis eſt angulo n g k:</s> <s xml:id="echoid-s44123" xml:space="preserve"> relin-<lb/>quitur ergo angulus k b p æqualis angulo k g q:</s> <s xml:id="echoid-s44124" xml:space="preserve"> ſed <lb/>angulus b k p eſt æqualis angulo g k q per 15 p 1:</s> <s xml:id="echoid-s44125" xml:space="preserve"> er-<lb/>go per 32 p 1 trigoni b k p & g k q ſunt æquianguli:</s> <s xml:id="echoid-s44126" xml:space="preserve"> <lb/>ſunt ergo anguli b p k & g q k æquales.</s> <s xml:id="echoid-s44127" xml:space="preserve"> Et quia angu <lb/>li p b g & q g b;</s> <s xml:id="echoid-s44128" xml:space="preserve"> ut patet ex præmiſsis, ſunt æquales, <lb/>ergo per 32 p 1 trigona p b g & q g b ſunt æquiangula:</s> <s xml:id="echoid-s44129" xml:space="preserve"> ergo per 4 p 6 erit proportio lineæ b p ad li-<lb/>neam g q, ſicut lineæ b g ad ſeipſam:</s> <s xml:id="echoid-s44130" xml:space="preserve"> erit ergo linea b p æqualis lineæ g q:</s> <s xml:id="echoid-s44131" xml:space="preserve"> erit ergo linea z p æqualis <lb/>lineæ z q:</s> <s xml:id="echoid-s44132" xml:space="preserve"> quæ eſt ergo proportio lineæ p z ad lineam z b, eadem eſt lineæ q z ad lineam z g:</s> <s xml:id="echoid-s44133" xml:space="preserve"> ergo <lb/>per 17 p 5 & per 2 p 6 linea b g æquidiſtat lineæ p q:</s> <s xml:id="echoid-s44134" xml:space="preserve"> ergo per 29 p 1 trigoni p z q & b z g ſunt æquian <lb/>guli:</s> <s xml:id="echoid-s44135" xml:space="preserve"> erit ergo per 4 p 6 proportio lineæ p z ad lineam z b, ſicut lineæ p q ad lineam b g:</s> <s xml:id="echoid-s44136" xml:space="preserve"> ſed linea p <lb/>z eſt maior quàm linea b z:</s> <s xml:id="echoid-s44137" xml:space="preserve"> ergo linea p q eſt maior quàm linea b g:</s> <s xml:id="echoid-s44138" xml:space="preserve"> eſt ergo idolum maius re uiſa.</s> <s xml:id="echoid-s44139" xml:space="preserve"> <lb/>Item linea z k producta ſecat lineam p q per 29 th.</s> <s xml:id="echoid-s44140" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s44141" xml:space="preserve"> ſecat enim angulum p z q:</s> <s xml:id="echoid-s44142" xml:space="preserve"> ſecet ergo i-<lb/>pſum in puncto o:</s> <s xml:id="echoid-s44143" xml:space="preserve"> erit ergo per præmiſſa;</s> <s xml:id="echoid-s44144" xml:space="preserve"> & per 29 p 1 angulus p o k trigoni k p o æqualis angulo g <lb/>n k trigoni k g n:</s> <s xml:id="echoid-s44145" xml:space="preserve"> ſed & angulus p k o æqualis eſt angulo g k n per 15 p 1:</s> <s xml:id="echoid-s44146" xml:space="preserve"> ergo per 32 p 1 trigona p k o <lb/>& g n k ſunt æquiangula:</s> <s xml:id="echoid-s44147" xml:space="preserve"> ergo per 4 p 6 quæ eſt proportio lineę p o ad lineam g n, eadem eſt lineæ o <lb/>k ad lineam k n:</s> <s xml:id="echoid-s44148" xml:space="preserve"> eſt autem, ut patet ex præmiſsis, linea b n æqualis lineę g n:</s> <s xml:id="echoid-s44149" xml:space="preserve"> ſed linea p o eſt maior <lb/>quàm linea b n:</s> <s xml:id="echoid-s44150" xml:space="preserve"> ideo quòd tota linea p q eſt maior quàm tota linea b g:</s> <s xml:id="echoid-s44151" xml:space="preserve"> & p o eſt medietas li-<lb/>neę p q, ſicut linea b n medietas lineæ b g.</s> <s xml:id="echoid-s44152" xml:space="preserve"> Cum enim lineę b q & g p ſint ęquales, & lineę b k & g k <lb/>ęquales:</s> <s xml:id="echoid-s44153" xml:space="preserve"> erit linea k q ęqualis lineę k p, & anguli p k o & q k o ſunt equales per 15 p 1, & per præmiſ-<lb/>ſa:</s> <s xml:id="echoid-s44154" xml:space="preserve"> erit ergo linea p o ęqualis lineę q o.</s> <s xml:id="echoid-s44155" xml:space="preserve"> Si ergo linea p o eſt maior quàm linea b n:</s> <s xml:id="echoid-s44156" xml:space="preserve"> patet quod linea o <lb/>k eſt maior quàm linea k n:</s> <s xml:id="echoid-s44157" xml:space="preserve"> & linea o k eſt diſtantia imaginis ſub ſpeculo, & linea n k eſt diſtantia rei <lb/>reflexę à ſuperficie ſpeculi.</s> <s xml:id="echoid-s44158" xml:space="preserve"> Palàm ergo propoſitum.</s> <s xml:id="echoid-s44159" xml:space="preserve"/> </p> <div xml:id="echoid-div1714" type="float" level="0" n="0"> <figure xlink:label="fig-0665-01" xlink:href="fig-0665-01a"> <variables xml:id="echoid-variables777" xml:space="preserve">p o q k l b n g m c e d a z f</variables> </figure> </div> </div> <div xml:id="echoid-div1716" type="section" level="0" n="0"> <head xml:id="echoid-head1274" xml:space="preserve" style="it">64. Circa diametrum ſpeculi ſphærici concaui extr a ſpeculum product æ ambobus poſitis ocu-<lb/> <pb o="364" file="0666" n="666" rhead="VITELLONIS OPTICAE"/> lis ſecundum æqualem diſtantiam à diametro, & centro ſpeculi: dextra apparent ſiniſtra, & ſi-<lb/>niſtra dextra: & imago minor facie apparet inter uiſus & ſuperficiem ſpeculi.</head> <p> <s xml:id="echoid-s44160" xml:space="preserve">Sit communis ſectio ſuperficiei reflexionis, & ſuperficiei ſpeculi ſphærici concaui circulus d b <lb/>kιcuius centrum o:</s> <s xml:id="echoid-s44161" xml:space="preserve"> & diameter d k:</s> <s xml:id="echoid-s44162" xml:space="preserve"> & orthogonaliter ſuper diametrum d k producatur diameter b <lb/>o a extra ſpeculum:</s> <s xml:id="echoid-s44163" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s44164" xml:space="preserve"> duo oculi in punctis e & c lineæ c e perpendicularis ſuper lineam b a:</s> <s xml:id="echoid-s44165" xml:space="preserve"> & <lb/>ſint ambo oculi æqualiter diſtantes ab ipſa diametro b a, & à puncto a:</s> <s xml:id="echoid-s44166" xml:space="preserve"> erit ergo linea e a æqualis li <lb/>neæ a c:</s> <s xml:id="echoid-s44167" xml:space="preserve"> & ducantur lineæ e b & c b:</s> <s xml:id="echoid-s44168" xml:space="preserve"> erit ergo per 4 p 1 angulus e b a æqualis angulo a b c:</s> <s xml:id="echoid-s44169" xml:space="preserve"> ergo per <lb/>20 th.</s> <s xml:id="echoid-s44170" xml:space="preserve"> 1 huius uiſus ambo e & c ad ſe inuicem reflectuntur à puncto b.</s> <s xml:id="echoid-s44171" xml:space="preserve"> Producatur itaq;</s> <s xml:id="echoid-s44172" xml:space="preserve"> linea à pun-<lb/>cto e ad centrum o:</s> <s xml:id="echoid-s44173" xml:space="preserve"> hæc ergo producta concurret cum linea c b per 29 th.</s> <s xml:id="echoid-s44174" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s44175" xml:space="preserve"> ſit concurſus pun <lb/> <anchor type="figure" xlink:label="fig-0666-01a" xlink:href="fig-0666-01"/> ctus ſ:</s> <s xml:id="echoid-s44176" xml:space="preserve"> & ſimiliter à puncto c ducatur linea per cen-<lb/>trum o concurrens cumlinea e b in puncto g.</s> <s xml:id="echoid-s44177" xml:space="preserve"> Appa-<lb/>ret ergo per 37 th.</s> <s xml:id="echoid-s44178" xml:space="preserve"> 5 huius imago formæ puncti e in <lb/>puncto f:</s> <s xml:id="echoid-s44179" xml:space="preserve"> & imago formæ puncti c in puncto g:</s> <s xml:id="echoid-s44180" xml:space="preserve"> appa-<lb/>rẽt ergo dextra ſiniſtra, & ſiniſtra dextra.</s> <s xml:id="echoid-s44181" xml:space="preserve"> Sed & per 5 <lb/>p 1 angulus b e c eſt æqualis angulo b c e:</s> <s xml:id="echoid-s44182" xml:space="preserve"> quoniam li-<lb/>neæ b e & b c ſunt æquales.</s> <s xml:id="echoid-s44183" xml:space="preserve"> Sed cum trigonorum e a <lb/>o & c a o duo latera e a & c a ſint æqualia, & latus a o <lb/>commune, anguliq́;</s> <s xml:id="echoid-s44184" xml:space="preserve"> c a o & e a o ſint æquales, quia re-<lb/>cti:</s> <s xml:id="echoid-s44185" xml:space="preserve"> erit per 4 p 1 angulus f e a æqualis angulo g c a:</s> <s xml:id="echoid-s44186" xml:space="preserve"> tri-<lb/>anguli ergo e f c & c e g ſunt æquianguli per præmiſ-<lb/>fa & 32 p 1:</s> <s xml:id="echoid-s44187" xml:space="preserve"> ergo per 4 p 6 eſt proportio lineæ e g ad <lb/>lineam c f, & lineæ e f ad lineam c g, ſicut lineæ e c ad <lb/>ſeipſam:</s> <s xml:id="echoid-s44188" xml:space="preserve"> ſunt ergo lineæ e g & c f æquales, & lineæ e f <lb/>& c g æquales.</s> <s xml:id="echoid-s44189" xml:space="preserve"> Sed totalis linea b e eſt æqualis totali <lb/>lineæ b c:</s> <s xml:id="echoid-s44190" xml:space="preserve"> ergo relinquitur linea b g æqualis lineæ b f:</s> <s xml:id="echoid-s44191" xml:space="preserve"> <lb/>ergo per 5 p 1 angulus b g f ęqualis eſt angulo b f g:</s> <s xml:id="echoid-s44192" xml:space="preserve"> ſed <lb/>illi anguli cũ angulo g b ſ ualẽt duos rectos per 32 p 1:</s> <s xml:id="echoid-s44193" xml:space="preserve"> <lb/>ſunt ergo illi duo anguli æquales duobus angulis b e <lb/>c, b c e:</s> <s xml:id="echoid-s44194" xml:space="preserve"> illi ergo trigoni e b c & g b f ſunt æquianguli:</s> <s xml:id="echoid-s44195" xml:space="preserve"> ergo per 4 p 6 quæ eſt proportio lineę b g ad <lb/>lineam b e, eadem eſt proportio lineæ g f ad lineam e c:</s> <s xml:id="echoid-s44196" xml:space="preserve"> ſed linea b g eſt minor quàm linea b e:</s> <s xml:id="echoid-s44197" xml:space="preserve"> ergo <lb/>linea g f eſt minor quàm linea e c.</s> <s xml:id="echoid-s44198" xml:space="preserve"> Imago ergo faciei uidentis eſt minor facie conſpecta.</s> <s xml:id="echoid-s44199" xml:space="preserve"> Apparet au-<lb/>tem inter oculos & ſpeculi ſuperficiem:</s> <s xml:id="echoid-s44200" xml:space="preserve"> quoniam linea g f (quæ eſt diameter imaginis) cadit inter <lb/>lineam e c, in qua ſunt ambo uiſus, & inter ſuperficiem ſpeculi.</s> <s xml:id="echoid-s44201" xml:space="preserve"> Palàm ergo propoſitum.</s> <s xml:id="echoid-s44202" xml:space="preserve"/> </p> <div xml:id="echoid-div1716" type="float" level="0" n="0"> <figure xlink:label="fig-0666-01" xlink:href="fig-0666-01a"> <variables xml:id="echoid-variables778" xml:space="preserve">b g f d o l e a c</variables> </figure> </div> </div> <div xml:id="echoid-div1718" type="section" level="0" n="0"> <head xml:id="echoid-head1275" xml:space="preserve" style="it">65. Imagines rerum retro ſpecula ſphærica concaua apparentes, motis rebus, quarũ ſunt ima <lb/>gines, ad eandem partem moueri uidentur.</head> <p> <s xml:id="echoid-s44203" xml:space="preserve">Sit in ſpeculo ſphærico concauo circulus a g b:</s> <s xml:id="echoid-s44204" xml:space="preserve"> cuius centrum ſit d:</s> <s xml:id="echoid-s44205" xml:space="preserve"> & ſit centrum uiſus pun-<lb/>ctum e:</s> <s xml:id="echoid-s44206" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s44207" xml:space="preserve"> duo puncta rei uiſæ ex utraq;</s> <s xml:id="echoid-s44208" xml:space="preserve"> parte puncti e:</s> <s xml:id="echoid-s44209" xml:space="preserve"> quæ ſint z & h:</s> <s xml:id="echoid-s44210" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s44211" xml:space="preserve"> duæ catheti <lb/>incidentiæ, quæ ſint d z c & d h k:</s> <s xml:id="echoid-s44212" xml:space="preserve"> reflectaturq́;</s> <s xml:id="echoid-s44213" xml:space="preserve"> forma puncti z ad uiſum e à puncto ſpeculi a:</s> <s xml:id="echoid-s44214" xml:space="preserve"> & for-<lb/>ma puncti h à puncto ſpeculi b:</s> <s xml:id="echoid-s44215" xml:space="preserve"> & ducantur reflexionum lineæ, quę ſint a e & b e:</s> <s xml:id="echoid-s44216" xml:space="preserve"> concurratq́;</s> <s xml:id="echoid-s44217" xml:space="preserve"> linea <lb/> <anchor type="figure" xlink:label="fig-0666-02a" xlink:href="fig-0666-02"/> a e cum catheto d z in puncto c, & linea e b cum ca-<lb/>theto d h in puncto k:</s> <s xml:id="echoid-s44218" xml:space="preserve"> erunt ergo per 37 th.</s> <s xml:id="echoid-s44219" xml:space="preserve"> 5 huius <lb/>pũcta c & k loca imaginũ ultra ſpeculum:</s> <s xml:id="echoid-s44220" xml:space="preserve"> ita quòd <lb/>punctum c ſit locus imaginis formæ puncti z, & <lb/>punctum k locus imaginis formę puncti h:</s> <s xml:id="echoid-s44221" xml:space="preserve"> & erunt <lb/>loca imaginum in partibus illis, in quibus ſentiun-<lb/>tur & res, quarum ſunt illę imagines.</s> <s xml:id="echoid-s44222" xml:space="preserve"> Transferatur <lb/>itaq;</s> <s xml:id="echoid-s44223" xml:space="preserve"> punctus rei uiſæ, qui eſt h ad punctum l:</s> <s xml:id="echoid-s44224" xml:space="preserve"> & re-<lb/>flectatur ad uiſum e à puncto g:</s> <s xml:id="echoid-s44225" xml:space="preserve"> & ducatur cathe-<lb/>tus d l concurrens cum linea reflexionis, quæ eſt e <lb/>g in puncto m:</s> <s xml:id="echoid-s44226" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s44227" xml:space="preserve"> locus imaginis ſormę pun-<lb/>cti h in puncto m translata ad ipſum à puncto k, <lb/>quilocus m erit in illa parte, ad quam translata eſt <lb/>ipſa res, cuius in puncto m eſt imago.</s> <s xml:id="echoid-s44228" xml:space="preserve"> Quòd ſi pun-<lb/>cta rei uiſę fuerint h & l, & ſint ſupra uiſum:</s> <s xml:id="echoid-s44229" xml:space="preserve"> erunt <lb/>loca imaginum, quæ ſunt k & m, ſupra uiſum:</s> <s xml:id="echoid-s44230" xml:space="preserve"> & ap-<lb/>parebunt ſupra res, quarum ſunt formæ.</s> <s xml:id="echoid-s44231" xml:space="preserve"> Et ſi pun-<lb/>cta h & l ſuerint à dextris ipſius uiſus:</s> <s xml:id="echoid-s44232" xml:space="preserve"> & loca i-<lb/>maginum ſuarum, quę ſunt k & m, erunt à dextris:</s> <s xml:id="echoid-s44233" xml:space="preserve"> ſed non putabuntur eſſe dextra, ut patuit <lb/>ſuprà per 51 huius:</s> <s xml:id="echoid-s44234" xml:space="preserve"> quoniam propter reuerberationem dextra apparent ſiniſtra, & ſiniſtra dextra.</s> <s xml:id="echoid-s44235" xml:space="preserve"> <lb/>Patet itaq;</s> <s xml:id="echoid-s44236" xml:space="preserve"> propoſitum.</s> <s xml:id="echoid-s44237" xml:space="preserve"/> </p> <div xml:id="echoid-div1718" type="float" level="0" n="0"> <figure xlink:label="fig-0666-02" xlink:href="fig-0666-02a"> <variables xml:id="echoid-variables779" xml:space="preserve">k m c b g a h l e z d</variables> </figure> </div> <pb o="365" file="0667" n="667" rhead="LIBER OCTAVVS."/> </div> <div xml:id="echoid-div1720" type="section" level="0" n="0"> <head xml:id="echoid-head1276" xml:space="preserve" style="it">66. Imagines rerum inter ſpecula ſphærica concaua & uiſus apparentes, motis rebus, uiden-<lb/>tur ad partem contrariam moueri.</head> <p> <s xml:id="echoid-s44238" xml:space="preserve">Sit ſpeculi ſphærici concaui circulus a b g:</s> <s xml:id="echoid-s44239" xml:space="preserve"> cuius centrum ſit punctus d:</s> <s xml:id="echoid-s44240" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s44241" xml:space="preserve"> centrum uiſus e ci-<lb/>tra centrũ ſpeculi, quod eſt d:</s> <s xml:id="echoid-s44242" xml:space="preserve"> & ex lateribus aſpicientis ſint duo puncta rei uiſæ:</s> <s xml:id="echoid-s44243" xml:space="preserve"> quę ſint z & h:</s> <s xml:id="echoid-s44244" xml:space="preserve"> quæ <lb/> <anchor type="figure" xlink:label="fig-0667-01a" xlink:href="fig-0667-01"/> reflectantur ad uiſum à duobus punctis a & b:</s> <s xml:id="echoid-s44245" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s44246" xml:space="preserve"> li <lb/>neæ reflexionum e a puncti z, & e b puncti h:</s> <s xml:id="echoid-s44247" xml:space="preserve"> ducan-<lb/>turq́;</s> <s xml:id="echoid-s44248" xml:space="preserve"> catheti incidentiæ z d c & h d k ſecantes lineas <lb/>reflexionum in punctis c & k:</s> <s xml:id="echoid-s44249" xml:space="preserve"> erunt ergo per 37 th.</s> <s xml:id="echoid-s44250" xml:space="preserve"> 5 <lb/>huius puncta c & k loca imaginũ:</s> <s xml:id="echoid-s44251" xml:space="preserve"> c puncti z, & k pun-<lb/>cti h.</s> <s xml:id="echoid-s44252" xml:space="preserve"> Videbuntur itaq;</s> <s xml:id="echoid-s44253" xml:space="preserve"> formæ illorum punctorum in <lb/>diuerſis partibus alijs, quàm ſint res ipſæ per 49 hu-<lb/>ius.</s> <s xml:id="echoid-s44254" xml:space="preserve"> Quòd ſi punctus h rei uiſæ transferatur ad pun-<lb/>ctum l:</s> <s xml:id="echoid-s44255" xml:space="preserve"> & reflectatur à puncto ſpeculi g ad uiſum e:</s> <s xml:id="echoid-s44256" xml:space="preserve"> du <lb/>caturq́;</s> <s xml:id="echoid-s44257" xml:space="preserve"> linea reflexionis, quę ſit e g:</s> <s xml:id="echoid-s44258" xml:space="preserve"> & cathetus l d <lb/>m, ſecans lineam reflexionis, quæ eſt e g, in puncto m:</s> <s xml:id="echoid-s44259" xml:space="preserve"> <lb/>erit per 37 th.</s> <s xml:id="echoid-s44260" xml:space="preserve"> 5 huius punctus m locus imaginis for-<lb/>mæ puncti l.</s> <s xml:id="echoid-s44261" xml:space="preserve"> Imago itaq;</s> <s xml:id="echoid-s44262" xml:space="preserve"> puncti h, quę eſt k, erit tranſ-<lb/>lata ad partem diuerſam illi, ad quam res uera tranſ-<lb/>lata eſt.</s> <s xml:id="echoid-s44263" xml:space="preserve"> Et ſi puncta h & l fuerint ſurſum mota ſupra <lb/>uiſum:</s> <s xml:id="echoid-s44264" xml:space="preserve"> tunc imagines ipſorum, quæ ſunt k & m, uide-<lb/>buntur moueri deorſum.</s> <s xml:id="echoid-s44265" xml:space="preserve"> Et ſi puncta h & l ſuerint mo <lb/>ta ad dextram partem uiſus:</s> <s xml:id="echoid-s44266" xml:space="preserve"> formæ imaginum uide-<lb/>buntur moueri ad ſiniſtram:</s> <s xml:id="echoid-s44267" xml:space="preserve"> & ita ſemper mouentur imagines ad partem contrariam rebus.</s> <s xml:id="echoid-s44268" xml:space="preserve"> Patet <lb/>ergo propoſitum.</s> <s xml:id="echoid-s44269" xml:space="preserve"/> </p> <div xml:id="echoid-div1720" type="float" level="0" n="0"> <figure xlink:label="fig-0667-01" xlink:href="fig-0667-01a"> <variables xml:id="echoid-variables780" xml:space="preserve">a b g l c d k m h c z</variables> </figure> </div> </div> <div xml:id="echoid-div1722" type="section" level="0" n="0"> <head xml:id="echoid-head1277" xml:space="preserve" style="it">67. Per ſpecula ſphærica concaua, quot libuerit, poßibile eſt formæ eiuſdem puncti imaginem <lb/>uideri. Euclides 15 th. catoptr. Ptolemæus 8 th. 2 catoptr.</head> <p> <s xml:id="echoid-s44270" xml:space="preserve">Fiat diſpoſitio, quæ in planis & conuexis ſphæricis ſpeculis:</s> <s xml:id="echoid-s44271" xml:space="preserve"> & ſit centrum uiſus a:</s> <s xml:id="echoid-s44272" xml:space="preserve"> & punctus <lb/>rei uiſæ ſit b:</s> <s xml:id="echoid-s44273" xml:space="preserve"> & ſecundum diſtantiam centri uiſus (quod eſt a) à puncto rei uiſæ, (quod eſt b) deſcri <lb/>batur polygonium æquilaterum & æquiangulum, quotcunq;</s> <s xml:id="echoid-s44274" xml:space="preserve"> angulorum placuerit:</s> <s xml:id="echoid-s44275" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s44276" xml:space="preserve">, exempli <lb/> <anchor type="figure" xlink:label="fig-0667-02a" xlink:href="fig-0667-02"/> cauſſa, pentagonum:</s> <s xml:id="echoid-s44277" xml:space="preserve"> quod ſit a b g d e:</s> <s xml:id="echoid-s44278" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s44279" xml:space="preserve"> circu-<lb/>lus circumſcribens illud polygoniũ pentagonum <lb/>per 12 p 4:</s> <s xml:id="echoid-s44280" xml:space="preserve"> & ſuper illius pentagoni angulos ortho <lb/>gonaliter ſuper lineas à centro circuli circumſcri-<lb/>bentis polygoniũ productas ad circumferentiam <lb/>ſecundum ipſorum puncta media ſtatuantur ſpe-<lb/>cula ſphærica concaua, quæ ſint partes eiuſdem <lb/>ſphæræ & æquales portiones.</s> <s xml:id="echoid-s44281" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s44282" xml:space="preserve"> quoniam <lb/>ſuperficies plana pentagoni a b g d e ſecabit quod-<lb/>libet ſpeculorum ſecundum circulum per 69 th.</s> <s xml:id="echoid-s44283" xml:space="preserve"> 1 <lb/>huius.</s> <s xml:id="echoid-s44284" xml:space="preserve"> Vnus itaq;</s> <s xml:id="echoid-s44285" xml:space="preserve"> arcus unius illorum circulorum <lb/>ſit z g c:</s> <s xml:id="echoid-s44286" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s44287" xml:space="preserve"> lineæ contingentes quemlibet <lb/>illorum arcuum in punctis g, d, e:</s> <s xml:id="echoid-s44288" xml:space="preserve"> contingatq́;</s> <s xml:id="echoid-s44289" xml:space="preserve"> ar-<lb/>cum z g c in puncto g linea l k.</s> <s xml:id="echoid-s44290" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s44291" xml:space="preserve"> per 43 th.</s> <s xml:id="echoid-s44292" xml:space="preserve"> <lb/>1 huius angulus portionis, qui eſt b g z, eſt æqualis <lb/>angulo d g c:</s> <s xml:id="echoid-s44293" xml:space="preserve"> anguli quoq;</s> <s xml:id="echoid-s44294" xml:space="preserve"> contingentiæ, qui ſunt <lb/>k g z & l g c ſunt æquales:</s> <s xml:id="echoid-s44295" xml:space="preserve"> palàm ergo per 20 th.</s> <s xml:id="echoid-s44296" xml:space="preserve"> 5 <lb/>huius quoniam fit reflexio formæ puncti b à puncto ſpeculi g ad punctũ ſpeculi alterius, quod eſt <lb/>d.</s> <s xml:id="echoid-s44297" xml:space="preserve"> Et ſimiliter per eandem demonſtrationem fiet reflexio à puncto d ad punctum ſpeculi alterius, <lb/>quod eſt e, & à puncto e ad centrum uiſus, quod eſt a.</s> <s xml:id="echoid-s44298" xml:space="preserve"> Palàm ergo propoſitum.</s> <s xml:id="echoid-s44299" xml:space="preserve"> Et ſic quotcunq;</s> <s xml:id="echoid-s44300" xml:space="preserve"> fue <lb/>rint anguli polygonij, tot aſſumantur ſpecula, & ſemper accidet illud, quod præmiſſum eſt.</s> <s xml:id="echoid-s44301" xml:space="preserve"/> </p> <div xml:id="echoid-div1722" type="float" level="0" n="0"> <figure xlink:label="fig-0667-02" xlink:href="fig-0667-02a"> <variables xml:id="echoid-variables781" xml:space="preserve">d l c e g z k a b</variables> </figure> </div> </div> <div xml:id="echoid-div1724" type="section" level="0" n="0"> <head xml:id="echoid-head1278" xml:space="preserve" style="it">68. A ſpeculis ſphæricis concauis ſoli oppoſitis ignem poßibile est accendi. Euclides 31 <lb/>th. catoptr.</head> <p> <s xml:id="echoid-s44302" xml:space="preserve">Eſto ſpeculum ſphæricum concauum ſoli oppoſitum:</s> <s xml:id="echoid-s44303" xml:space="preserve"> in quo ſignetur circulus k a b g x, cuius <lb/>centrum ſit c:</s> <s xml:id="echoid-s44304" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s44305" xml:space="preserve">, ut ſuperficies plana ſecans ſpeculum ſecundum hunc circulum ſecet etiam cor-<lb/>pus ſolis trans centrum:</s> <s xml:id="echoid-s44306" xml:space="preserve"> ergo per 69 th.</s> <s xml:id="echoid-s44307" xml:space="preserve"> 1 huius communis ſectio illius ſuperficiei planæ & ſolis e-<lb/>rit circulus magnus, qui ſit d e z:</s> <s xml:id="echoid-s44308" xml:space="preserve"> & ab aliquo puncto illius circuli ſolaris, ut à puncto d, ducatur li-<lb/>nea, ſecundum quam procedens radius ad centrum ſpeculi, quod eſt c, incidat in punctum ſpeculi, <lb/>quod ſit g:</s> <s xml:id="echoid-s44309" xml:space="preserve"> & à puncto circuli ſolis, quod ſit e, procedens radius ad centrum ſpeculi, quod eſt c, inci-<lb/>dat in punctum ſpeculi b:</s> <s xml:id="echoid-s44310" xml:space="preserve"> & à puncto ſolis, quod ſit z, incidens radius per centrum ſpeculi c, cadat <lb/>in punctum ſpeculi a.</s> <s xml:id="echoid-s44311" xml:space="preserve"> Quia ergo omnes radij tranſeuntes per centrum c ſunt perpendiculares ſu-<lb/> <pb o="366" file="0668" n="668" rhead="VITELLONIS OPTICAE"/> per ſuperficiem ſpeculi a b g per 72 th.</s> <s xml:id="echoid-s44312" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s44313" xml:space="preserve"> patet per 21 th 5 huius quoniam omnes reflectuntur <lb/>in ſeipſos:</s> <s xml:id="echoid-s44314" xml:space="preserve"> concurrent ergo tam incidentes quàm reflexi omnes in puncto c, quod eſt centrum ſpe-<lb/>culi:</s> <s xml:id="echoid-s44315" xml:space="preserve"> omnes enim illi radij ſunt diametri ipſius ſpeculi, & omnes <lb/> <anchor type="figure" xlink:label="fig-0668-01a" xlink:href="fig-0668-01"/> anguli ſemicirculi ſunt æquales per 43 th.</s> <s xml:id="echoid-s44316" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s44317" xml:space="preserve"> reflexio autem <lb/>omnis fit ſecundum angulos æquales, ut patet per 20 th.</s> <s xml:id="echoid-s44318" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s44319" xml:space="preserve"> <lb/>Quicunq;</s> <s xml:id="echoid-s44320" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s44321" xml:space="preserve"> radiorum ſolarium pertranſierint per centrum <lb/>ſpeculi, quod eſt c, & peruenerint ad quęcunq;</s> <s xml:id="echoid-s44322" xml:space="preserve"> puncta ſuperficiei <lb/>ſpeculi:</s> <s xml:id="echoid-s44323" xml:space="preserve"> illi omnes reflectentur in ſeipſos, & concurrent in centro <lb/>ipſius:</s> <s xml:id="echoid-s44324" xml:space="preserve"> radij uerò æquidiſtantes illis radijs non concurrunt.</s> <s xml:id="echoid-s44325" xml:space="preserve"> Sit e <lb/>nim radius perpẽdicularis ſuper ſuperficiẽ ſpeculi, qui eſt e b:</s> <s xml:id="echoid-s44326" xml:space="preserve"> hic <lb/>ergo, ut præmiſſum eſt, tranſibit centrum ſpeculi, quod eſt c:</s> <s xml:id="echoid-s44327" xml:space="preserve"> & <lb/>reflectetur in ſeipſum.</s> <s xml:id="echoid-s44328" xml:space="preserve"> Huic ergo ducatur per 31 p 1 aliquis radius <lb/>æquidiſtans:</s> <s xml:id="echoid-s44329" xml:space="preserve"> qui ſit l n:</s> <s xml:id="echoid-s44330" xml:space="preserve"> & alius, qui o s:</s> <s xml:id="echoid-s44331" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s44332" xml:space="preserve"> arcus n b inæqualis <lb/>arcui b s:</s> <s xml:id="echoid-s44333" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s44334" xml:space="preserve"> linea l n circulum a b g in puncto y:</s> <s xml:id="echoid-s44335" xml:space="preserve"> & in arcu y <lb/>n ſignetur punctum k:</s> <s xml:id="echoid-s44336" xml:space="preserve"> & ducatur linea c n.</s> <s xml:id="echoid-s44337" xml:space="preserve"> Quia itaq, angulus l n <lb/>k eſt minor angulo c n k, ut pars ſuo toto:</s> <s xml:id="echoid-s44338" xml:space="preserve"> patet quòd angulus l n <lb/>k eſt minor angulo c n b:</s> <s xml:id="echoid-s44339" xml:space="preserve"> quoniam anguli c n b & c n k ſunt æqua-<lb/>les per 43 th.</s> <s xml:id="echoid-s44340" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s44341" xml:space="preserve"> Patet ergo per 20 th.</s> <s xml:id="echoid-s44342" xml:space="preserve"> 5 huius quòd radius l n <lb/>non reflectetur in punctum c.</s> <s xml:id="echoid-s44343" xml:space="preserve"> Fiat itaq;</s> <s xml:id="echoid-s44344" xml:space="preserve"> angulus b n f æqualis l <lb/>n k:</s> <s xml:id="echoid-s44345" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s44346" xml:space="preserve"> punctum f citra punctum c in punctum aliquod ſe-<lb/>midiametri c b:</s> <s xml:id="echoid-s44347" xml:space="preserve"> & in corpore ſolari continuetur linea e l.</s> <s xml:id="echoid-s44348" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s44349" xml:space="preserve"> <lb/>quadrangulum n f e l, (fixo permanẽte ſuo latere e f) imaginetur <lb/>moueri, quouſq;</s> <s xml:id="echoid-s44350" xml:space="preserve"> linea l n incidat ad locum, unde exiuit:</s> <s xml:id="echoid-s44351" xml:space="preserve"> tunc pun <lb/>ctus n motu ſuo deſcribet quendam circulum in ſuperficie ſpe-<lb/>culi:</s> <s xml:id="echoid-s44352" xml:space="preserve"> & in tota peripheria illius circuli angulus l n f remanet æ-<lb/>qualis:</s> <s xml:id="echoid-s44353" xml:space="preserve"> ergo angulus l n k eſt æqualis angulo b n f.</s> <s xml:id="echoid-s44354" xml:space="preserve"> Fiet ergo per <lb/>20 th.</s> <s xml:id="echoid-s44355" xml:space="preserve"> 5 huius à tota peripheria illius circuli reflexio omnium ra-<lb/>diorum incidentium ad punctum f.</s> <s xml:id="echoid-s44356" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s44357" xml:space="preserve"> ſi à puncto ſo <lb/>lis, quod eſt o, ducatur per 31 p 1 radius æquidiſtans radio perpen <lb/>diculari, qui eſt e b:</s> <s xml:id="echoid-s44358" xml:space="preserve"> & ſit ille radius æquidiſtans o s, ſecans circu-<lb/>lum a b g in puncto x:</s> <s xml:id="echoid-s44359" xml:space="preserve"> & in arcu x s ſignetur punctum q in linea <lb/>n f producta, ſitq́;</s> <s xml:id="echoid-s44360" xml:space="preserve">, ut perpendicularis e b ſecet circulum a b g in <lb/>puncto p:</s> <s xml:id="echoid-s44361" xml:space="preserve"> & ſit arcus b s mioor arcu n b:</s> <s xml:id="echoid-s44362" xml:space="preserve"> ergo & arcus x p (qui eſt æqualis arcui b s per 53 th.</s> <s xml:id="echoid-s44363" xml:space="preserve"> 1 huius) <lb/>minor eſt arcu p y æquali b n:</s> <s xml:id="echoid-s44364" xml:space="preserve"> ergo arcus x q s remanet maior arcu y k n:</s> <s xml:id="echoid-s44365" xml:space="preserve"> ergo per 43 th.</s> <s xml:id="echoid-s44366" xml:space="preserve"> 1 huius an-<lb/>gulus x s q eſt maior angulo y n k.</s> <s xml:id="echoid-s44367" xml:space="preserve"> Radius ergo o s non reflectitur ad punctum f, ſed ad aliquod pun <lb/>ctum lineæ f c, quod ſit h.</s> <s xml:id="echoid-s44368" xml:space="preserve"> Portio enim circuli y k n, quæ eſt æqualis portioni n b q, eſt minor por-<lb/>tione x-q s, quæ eſt æqualis portioni s b k:</s> <s xml:id="echoid-s44369" xml:space="preserve"> copuletur quoq;</s> <s xml:id="echoid-s44370" xml:space="preserve"> linea o e.</s> <s xml:id="echoid-s44371" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s44372" xml:space="preserve"> fixo latere e h, quadran <lb/>gulum o e h s intelligatur moueri, quouſq;</s> <s xml:id="echoid-s44373" xml:space="preserve"> linea o s redeat ad locum, unde exiuit:</s> <s xml:id="echoid-s44374" xml:space="preserve"> tunc punctum s <lb/>motu ſuo deſcribet in ſuperficie ſpeculi circulum, à cuius totali peripheria fiet reflexio ad pun-<lb/>ctum diametri ſpeculi, qui eſt h.</s> <s xml:id="echoid-s44375" xml:space="preserve"> Et ſimiliter eſt de quibuſcunq;</s> <s xml:id="echoid-s44376" xml:space="preserve"> alijs radijs incidentibus ſuperficiei <lb/>ſpeculi æquidiſtanter radio e b.</s> <s xml:id="echoid-s44377" xml:space="preserve"> Semper enim fiet reflexio omnium ſibi ſimilium radiorum à peri-<lb/>pheria unius circuli totius ſpeculi ad unum punctum diametri ipſius ſpeculi:</s> <s xml:id="echoid-s44378" xml:space="preserve"> & lineæ radiales pro-<lb/>phinquiores diametro, reflectuntur ad punctum propinquius centro c:</s> <s xml:id="echoid-s44379" xml:space="preserve"> & lineæ radiales remotiores <lb/>à diametro, & æquidiſtantes illi, reflectuntur ad punctum remotius à centro, quod eſt c.</s> <s xml:id="echoid-s44380" xml:space="preserve"> In quo-<lb/>cunq;</s> <s xml:id="echoid-s44381" xml:space="preserve"> autem illorum punctorum ponatur aliquod corpus combuſtibile, per radios reflexos in-<lb/>cendetur.</s> <s xml:id="echoid-s44382" xml:space="preserve"> Sed quia radij ſunt pauci & debiles, oportet ut combuſtile diutius in puncto collectio-<lb/>nis radiorum moram trahat.</s> <s xml:id="echoid-s44383" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s44384" xml:space="preserve"> Et hoc ſpeculum, quantùm ad a ctum combu-<lb/>ſtionis, efficacius eſt ſpeculo compoſito explanis ſpeculis, de quo locuti ſumus in fine quinti li-<lb/>bri huius ſcientiæ.</s> <s xml:id="echoid-s44385" xml:space="preserve"> Poſſet quoq;</s> <s xml:id="echoid-s44386" xml:space="preserve"> per diligentiam artificis aliquod ſpeculum ex pluribus hu-<lb/>iuſmodi ſpeculis componi, quod eſſet maioris efficacię ad comburendum:</s> <s xml:id="echoid-s44387" xml:space="preserve"> hoc <lb/>autem relinquimus induſtriæ perquirentis:</s> <s xml:id="echoid-s44388" xml:space="preserve"> quia ſufficit nobis <lb/>ut propoſitum ſit hoc modo demon.</s> <s xml:id="echoid-s44389" xml:space="preserve"> <lb/>ſtratum.</s> <s xml:id="echoid-s44390" xml:space="preserve"/> </p> <div xml:id="echoid-div1724" type="float" level="0" n="0"> <figure xlink:label="fig-0668-01" xlink:href="fig-0668-01a"> <variables xml:id="echoid-variables782" xml:space="preserve">d z l e o y p x k c h f q a b g s</variables> </figure> </div> <pb o="367" file="0669" n="669"/> </div> <div xml:id="echoid-div1726" type="section" level="0" n="0"> <head xml:id="echoid-head1279" xml:space="preserve">VITELLONIS FI-<lb/>LII THVRINGORVM ET PO-<lb/>LONORVM OPTICAE LIBER NONVS.</head> <p style="it"> <s xml:id="echoid-s44391" xml:space="preserve">IN præmiſſo libro paßiones ſpeculorum ſphæricorum cõcauorum pro noſtro <lb/>poſſe pertractauimus:</s> <s xml:id="echoid-s44392" xml:space="preserve"> ſupereſt nũc, ut ſpeculorum columnarium & pyra-<lb/>midalium concauorũ proprietates aliquas demonſtremus.</s> <s xml:id="echoid-s44393" xml:space="preserve"> In his enim ſpe-<lb/>culis quaſi omnium præmiſſorum ſpeculorum proprietutes concurrunt:</s> <s xml:id="echoid-s44394" xml:space="preserve"> pla-<lb/>norum quidem, cum in illis à linea longitudinis ſpeculi fit reflexio.</s> <s xml:id="echoid-s44395" xml:space="preserve"> Columnarium quo <lb/>& pyramidalium conuexorum plurimæ paßiones in hæc concaua ſpecula deſcẽdunt:</s> <s xml:id="echoid-s44396" xml:space="preserve"> <lb/>quoniam iſtorum & illorum cõformis eſt generatio ſecundũ figur as, à quibus in utriſ <lb/>prouenit quædam conformitas paßionum:</s> <s xml:id="echoid-s44397" xml:space="preserve"> niſi quòd hinc & inde ſecundum naturam <lb/>conuexi & concaui illæ paßiones quodammodo ſecundũ ſitum contrariè diſponuntur.</s> <s xml:id="echoid-s44398" xml:space="preserve"> <lb/>Ex quo accidit, ut quandoq;</s> <s xml:id="echoid-s44399" xml:space="preserve"> lιneæ reflexæ in cõuexis ſpeculis fiat locus imaginis in con-<lb/>cauis, & econuerſo:</s> <s xml:id="echoid-s44400" xml:space="preserve"> & ob hæc eadem principia in his ſpeculis & in illis ſunt (præmißis <lb/>figuris) conformiter aſſumenda.</s> <s xml:id="echoid-s44401" xml:space="preserve"> Sic ita omnium ſpeculorum regularium pro noſtra-<lb/>rum uirium & experientiæ poßibilitate paßionibus aliqualiter pertractatis, ad aliqua <lb/>ſpecula figurarum irregularium & compoſitarum mentem conuertimus:</s> <s xml:id="echoid-s44402" xml:space="preserve"> uidentesq́;</s> <s xml:id="echoid-s44403" xml:space="preserve"> <lb/>quòd antiquorum geometrarum diligentia & ſolicitudo circa ſpecula comburentia, <lb/>(à quorum totali ſuperficie ad unum punctum natur alem uel mathematicum fit re-<lb/>flexio luminis & formarum incidentium) plurimum eſt uerſata:</s> <s xml:id="echoid-s44404" xml:space="preserve"> ut circa rem ſcientiæ <lb/>geometriæ plurimam ſubtilitatem rebus naturalibus applicantem:</s> <s xml:id="echoid-s44405" xml:space="preserve"> actionem quoq;</s> <s xml:id="echoid-s44406" xml:space="preserve"> na-<lb/>turalium formarum accelerantem in productione effectuum mirandorum:</s> <s xml:id="echoid-s44407" xml:space="preserve"> huic nego-<lb/>tio curam conſequenter in hoc libro dedimus, ut rei, ad quam, ſicut ad finem nobilißi-<lb/>mum, omne, quod de natura quorumlibet ſpeculorum præmiſimus, aliqualιter ordina-<lb/>tur.</s> <s xml:id="echoid-s44408" xml:space="preserve"> Ex præmißis uerò libris ſatis patet, quòd figur a talium ſpeculorum comburentium <lb/>in una ſuperficierum planarum, ut patet per 65 th.</s> <s xml:id="echoid-s44409" xml:space="preserve"> 5 huius, non eſt poßibilis:</s> <s xml:id="echoid-s44410" xml:space="preserve"> ſicut nec ab <lb/>aliqua una ſuperficierum conuexarum quacunque, ſiue illa couexa ſuperficies fuerit <lb/>ſphærica, ut patet per 65 th.</s> <s xml:id="echoid-s44411" xml:space="preserve"> 6 huius, ſiue fuerit columnaris uelpyramidalis, ut patet <lb/>per 59 th.</s> <s xml:id="echoid-s44412" xml:space="preserve"> 7 huius, poßibile eſt radios aliquos aggregari ad punctum unum mathema-<lb/>ticum uel etiam naturalem.</s> <s xml:id="echoid-s44413" xml:space="preserve"> A‘ concauis quoſpeculis ſphæricis non fit ad unum axis <lb/>punctum mathematicum reflexio, niſi à peripheria unius tantùm circuli, & à tota ſu-<lb/>perficie unius hemiſphærij ad totam ſemidiametrum ſiue axem ſpeculi, ut oſtenſum eſt <lb/>per 68 th.</s> <s xml:id="echoid-s44414" xml:space="preserve"> 8 huius.</s> <s xml:id="echoid-s44415" xml:space="preserve"> Non fit autem omnium radiorum, æquidiſtanter axi ſpeculi ſuperfi-<lb/>ciei talis ſpeculi incidentium reflexio ad punctum unum.</s> <s xml:id="echoid-s44416" xml:space="preserve"> Sed ne ab aliqua ſuperficie-<lb/>rum ſpeculorum columnariũ uelpyramidalium concauorũ eſt hoc poßibile fieri:</s> <s xml:id="echoid-s44417" xml:space="preserve"> prout <lb/>infrà in præſenti libro demonſtrabimus.</s> <s xml:id="echoid-s44418" xml:space="preserve"> Reſtat ergo, ut ſuperficies alias huic noſtro <lb/>propoſito competentes cum demonſtrationis diligentia perquiramus:</s> <s xml:id="echoid-s44419" xml:space="preserve"> quoniam illud, <lb/>quod ex plurium ſpeculorum regularium compoſitione ad hũc effectum poßibile prius <lb/>fore diximus, unius ſuperficiei (à qua totali ad unum punctum fiat reflexio) certitu-<lb/>dinem non attingit:</s> <s xml:id="echoid-s44420" xml:space="preserve"> ne ad illorum peruenit commoditatem:</s> <s xml:id="echoid-s44421" xml:space="preserve"> ne in illis adeò relucet <lb/>humani bonitas ingenij & utilitas figurarum.</s> <s xml:id="echoid-s44422" xml:space="preserve"> In his ita columnaribus & pyrami-<lb/>dalibus, & alijs irregularibus quibuſcunque ſpeculis, & etiam in ιpſis comburentibus <lb/> <pb o="368" file="0670" n="670" rhead="VITELLONIS OPTICAE"/> ſpeculis ſupponimus principia, quæ in libris præcedentibus ſunt præmiſſa, ut patet in 5.</s> <s xml:id="echoid-s44423" xml:space="preserve"> 6 <lb/>& præcipuè 7 & 8 libris huius ſcientiæ:</s> <s xml:id="echoid-s44424" xml:space="preserve"> quæ uerò ex præſuppoſitis principijs & cõclu-<lb/>ſionibus demonſtranda de his ſpeculis prænominatis uidimus, ſunt iſta.</s> <s xml:id="echoid-s44425" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1727" type="section" level="0" n="0"> <head xml:id="echoid-head1280" xml:space="preserve">THEOREMATA</head> <head xml:id="echoid-head1281" xml:space="preserve" style="it">1. In ſpeculis column aribus concauis communis ſectio ſuperficiei reflexionis & ſpeculi quan-<lb/>do eſt linea longitudιnis ſpeculi: quando circulus: quandó oxygonia ſectio. Alhaz. 89 n 5.</head> <p> <s xml:id="echoid-s44426" xml:space="preserve">Quod hic proponitur, patet ex præmiſsis in libro 7 huius de ſpeculis columnaribus conuexis-<lb/>Et quia ſpeculum columnare concauum non minus participat formam & proprietatem columnæ <lb/>quàm conuexum:</s> <s xml:id="echoid-s44427" xml:space="preserve"> patet quòd propoſita paſsio eodem penitùs modo demonſtráda eſt de ſpeculis.</s> <s xml:id="echoid-s44428" xml:space="preserve"> <lb/>columnaribus concauis, ut de columnaribus conuexis.</s> <s xml:id="echoid-s44429" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s44430" xml:space="preserve"> nec enim neceſſa-<lb/>rium talibus amplius immorari.</s> <s xml:id="echoid-s44431" xml:space="preserve"> Et quando fuerit communis illa ſectio linea longitudinis ſpeculi:</s> <s xml:id="echoid-s44432" xml:space="preserve"> <lb/>erunt modi reflexionum & loca imaginum ſicut in ſpeculis planis:</s> <s xml:id="echoid-s44433" xml:space="preserve"> quãdo uerò illa ſectio commu-<lb/>nis fuerit circulus:</s> <s xml:id="echoid-s44434" xml:space="preserve"> erunt modi reflexionis & loca reflexionum, ſicut in ſpeculis ſphæricis cõcauis.</s> <s xml:id="echoid-s44435" xml:space="preserve"> <lb/>Eruntq́;</s> <s xml:id="echoid-s44436" xml:space="preserve"> loca imaginum quandoq;</s> <s xml:id="echoid-s44437" xml:space="preserve"> ultra ſpeculum:</s> <s xml:id="echoid-s44438" xml:space="preserve"> quãdoq;</s> <s xml:id="echoid-s44439" xml:space="preserve"> in ipſa ſuperficie ſpeculi:</s> <s xml:id="echoid-s44440" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s44441" xml:space="preserve"> in-<lb/>ter uiſum & ſpeculum:</s> <s xml:id="echoid-s44442" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s44443" xml:space="preserve"> in ipſa ſuperficie uiſus:</s> <s xml:id="echoid-s44444" xml:space="preserve"> & omnium iſtorum idem eſt demonſtran-<lb/>di modus, qui in illis ſphærieis concauis ſpeculis patuit per 11 th.</s> <s xml:id="echoid-s44445" xml:space="preserve"> 8 huius.</s> <s xml:id="echoid-s44446" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1728" type="section" level="0" n="0"> <head xml:id="echoid-head1282" xml:space="preserve" style="it">2. In ſpeculis pyramidalibus concauis communem ſectionem ſuperficiei reflexionis & ſpecu-<lb/>li lineam longitudinis ſpeculi aut ſectionem oxygoniam poßibile eſt eſſe: circulum uerò impoßi-<lb/>bile. Alhazen 97 n 5.</head> <p> <s xml:id="echoid-s44447" xml:space="preserve">Paſsiones propoſitæ de præſentibus ſpeculis eodem penitùs modo demonſtrabiles ſunt, quo & <lb/>de ſpeculis pyramidalibus cõuexis ſunt oſtẽſæ per diuerias propoſitiones 7 huius.</s> <s xml:id="echoid-s44448" xml:space="preserve"> Patet ergo pro-<lb/>poſitum.</s> <s xml:id="echoid-s44449" xml:space="preserve"> Et quando cõmunis ſectio ſuperficiei reflexionis & ſpeculi fuerit linea longitudinis:</s> <s xml:id="echoid-s44450" xml:space="preserve"> erũt <lb/>modi reflexionum & loca imaginum, quæ & in ſpeculis planis oſtenſa ſunt per 49 th.</s> <s xml:id="echoid-s44451" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s44452" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1729" type="section" level="0" n="0"> <head xml:id="echoid-head1283" xml:space="preserve" style="it">3. In omni ſuperficie reflexionis à ſpeculis columnaribus uel pyramidalibus concauis, centrũ <lb/>uiſus: & punctum rei uiſæ: punctum reflexionis: & punctum axis, (in quem cadit perpendicu-<lb/>laris duct a à puncto reflexionis ſuper ſuperficiem ſpeculum in puncto reflexionis contingẽtem) <lb/>conſiſtere eſt neceſſe. Alhazen 46 n 4.</head> <p> <s xml:id="echoid-s44453" xml:space="preserve">Sit ſpeculum columnare concauum:</s> <s xml:id="echoid-s44454" xml:space="preserve"> cuius axis ſit a b:</s> <s xml:id="echoid-s44455" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s44456" xml:space="preserve"> centrum uiſus t:</s> <s xml:id="echoid-s44457" xml:space="preserve"> & punctum rei uiſæ <lb/>d:</s> <s xml:id="echoid-s44458" xml:space="preserve"> reflectaturq́;</s> <s xml:id="echoid-s44459" xml:space="preserve"> forma puncti rei uiſæ, quod eſt d, ad uiſum t à puncto ſpeculi e:</s> <s xml:id="echoid-s44460" xml:space="preserve"> & in puncto e con-<lb/>tingat ſuperficiem ſpeculi ſuperficies plana:</s> <s xml:id="echoid-s44461" xml:space="preserve"> ſuper quam ſuperficiem à puncto e ducatur linea per-<lb/>pendicularis per 12 p 11:</s> <s xml:id="echoid-s44462" xml:space="preserve"> quæ ſecetlineã a b axem ſpeculi in puncto f:</s> <s xml:id="echoid-s44463" xml:space="preserve"> & ſit linea e f.</s> <s xml:id="echoid-s44464" xml:space="preserve"> Dico quòd pun-<lb/>cta t, d, e, f neceſſariò erunt ſemper in ea dẽ ſuperficie reflexionis.</s> <s xml:id="echoid-s44465" xml:space="preserve"> Aut enim hæc ſuperficies reflexio-<lb/>nis æ quidiſta bit baſibus columnæ, aut nõ.</s> <s xml:id="echoid-s44466" xml:space="preserve"> Si ſic:</s> <s xml:id="echoid-s44467" xml:space="preserve"> patet per 100 th.</s> <s xml:id="echoid-s44468" xml:space="preserve"> 1 huius quòd cominunis ſectio ſu-<lb/>perficiei reflexionis & ſuperficiei ſpeculi erit circulus æquidiſtans baſibus columnę:</s> <s xml:id="echoid-s44469" xml:space="preserve"> & linea ducta <lb/>à puncto reflexionis, quod eſt e, tranſiens per centrũ illius circuli, eſt perpendicularis ſuper ſuper-<lb/>ficiem columnæ, ut patet per 96 & per 100 th.</s> <s xml:id="echoid-s44470" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s44471" xml:space="preserve"> Et ſi centrum uiſus, quod eſt t, & punctum rei <lb/>uiſæ quod eſt d, fuerint in illa linea:</s> <s xml:id="echoid-s44472" xml:space="preserve"> fiet reflexio formarum punctorum uiſorum tantùm ſecundum <lb/>illam lineam per 21 th.</s> <s xml:id="echoid-s44473" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s44474" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s44475" xml:space="preserve"> ιlla quatuor puncta, (quæ ſunt t, <lb/>d, e, f) omnia in ſuperficie reflexionis.</s> <s xml:id="echoid-s44476" xml:space="preserve"> Quòd ſi centrum uiſus uel pun-<lb/>ctum rei uiſæ non fuerit in hac linea perpendiculari:</s> <s xml:id="echoid-s44477" xml:space="preserve"> ſem per tamẽ linea <lb/>e f perpẽdiculariter à puncto e ducta, cadet in axem a b per 96 th.</s> <s xml:id="echoid-s44478" xml:space="preserve"> 1 hu-<lb/>ius, & linea reflexionis continebit cum illa perpendiculari angulum <lb/>acutum:</s> <s xml:id="echoid-s44479" xml:space="preserve"> quoniam cadet inter perpendicularem e f, & inter lineam, cir-<lb/>culum (qui eſt communis ſectio ſuperficiei reflexionis & ſpeculi) in <lb/>puncto e contingentem.</s> <s xml:id="echoid-s44480" xml:space="preserve"> Et quoniam hæc linea reflexionis cadit ſem-<lb/>perintra ſpeculum:</s> <s xml:id="echoid-s44481" xml:space="preserve"> quia ſecundum ſui partem, qua incidit ſpeculo, ne-<lb/>ceſſariò cadet inter ſuperficies planas per cẽtrum uiſus ductas, portio-<lb/>nem apparentẽ ſpeculi contingẽtes:</s> <s xml:id="echoid-s44482" xml:space="preserve"> & quoniam per 20 th.</s> <s xml:id="echoid-s44483" xml:space="preserve"> 5 huius ſem-<lb/>per angulus incidentiæ eſt æqualis angulo reflexionis:</s> <s xml:id="echoid-s44484" xml:space="preserve"> patet quòd ſi <lb/>unus illorum punctorum eſt in ſuperficie reflexionis, quod & reliquus.</s> <s xml:id="echoid-s44485" xml:space="preserve"> <lb/>Quia enim angulus d e f erit æqualis angulo ſ e t, cadent hi anguli ex di-<lb/>uerſis partibus perpendicularis lineæ, quæ eſt e f, intra ſpeculũ.</s> <s xml:id="echoid-s44486" xml:space="preserve"> In eadẽ <lb/>itaq;</s> <s xml:id="echoid-s44487" xml:space="preserve"> ſuperficie cadent omnia puncta t, d, e, f.</s> <s xml:id="echoid-s44488" xml:space="preserve"> Et eodem modo demon-<lb/>ſtrandũ eſt, à quocunq;</s> <s xml:id="echoid-s44489" xml:space="preserve"> pũcto circuli, (qui eſt cõmunis ſectio ſuperficiei <lb/>reflexionis & ſpeculi) fiat reflexio:</s> <s xml:id="echoid-s44490" xml:space="preserve"> ſem per enim illa quatuor pũcta erũt <lb/>in ſuperficie reflexionis.</s> <s xml:id="echoid-s44491" xml:space="preserve"> Quòd ſi cõmunis ſectio ſuperficiei reflexionis <lb/>& ſuperficiei ſpeculi ſit linea lõgitudinis ſpeculi:</s> <s xml:id="echoid-s44492" xml:space="preserve"> tũc iterũ à quocũq;</s> <s xml:id="echoid-s44493" xml:space="preserve"> pũcto illius lineæ flat reflexio:</s> <s xml:id="echoid-s44494" xml:space="preserve"> <lb/> <pb o="369" file="0671" n="671" rhead="LIBER NONVS."/> ſem per propoſita quatuor puncta erũt in ſuperficie reflexionis, ut patet per 27 th.</s> <s xml:id="echoid-s44495" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s44496" xml:space="preserve"> Similiter <lb/>quoq;</s> <s xml:id="echoid-s44497" xml:space="preserve"> patet idem, ſi cõmunis ſectio ſuperficiei reflexionis & horum ſpeculorum fuerit ſectio oxy-<lb/>gonia:</s> <s xml:id="echoid-s44498" xml:space="preserve"> quoniam illa ſectio ſecabit ſpeculũ trans axem per 103 th.</s> <s xml:id="echoid-s44499" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s44500" xml:space="preserve"> & linea à puncto reflexio-<lb/>nis perpendiculariter ducta ſuperſuperficiem, ſpeculum in puncto reflexionis contingẽtem, ſem-<lb/>per cadet in axe, ut hæc in ſpeculis columnaribus & pyramidalibus conuexis ſunt amplius decla-<lb/>rata.</s> <s xml:id="echoid-s44501" xml:space="preserve"> Et ille modus demonſtrãdi eſt uniuocus & iſtis ſpeculis.</s> <s xml:id="echoid-s44502" xml:space="preserve"> Quòd ſi ſpeculum propoſitum fuerit <lb/>pyramidale concauum:</s> <s xml:id="echoid-s44503" xml:space="preserve"> tunc (ut ſuprà oſtenſum eſt per præmiſſam) impoſsibile eſt communem <lb/>ſectionem ſuperficiei reflexionis & ſuperficiei ſpeculi circulum eſſe:</s> <s xml:id="echoid-s44504" xml:space="preserve"> quę ſectio ſi fuerit linea longi-<lb/>tudinis uel ſectio oxygonia:</s> <s xml:id="echoid-s44505" xml:space="preserve"> tunc eadem erit declaratio quòd quatuor prædicta puncta t, d, e, f con-<lb/>ſiſtunt in ſuperficie reflexionis, quæ prius in ſpeculis columnaribus concauis.</s> <s xml:id="echoid-s44506" xml:space="preserve"> Patet ergo illud, <lb/>quod proponebatur.</s> <s xml:id="echoid-s44507" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1730" type="section" level="0" n="0"> <head xml:id="echoid-head1284" xml:space="preserve" style="it">4. Centro uiſus exiſtente intra ſpeculum columnare uel pyramidale concauum: à quolibet <lb/>puncto ſpeculi fiet reflexio ad uiſum. Alhazen 49 n 4.</head> <p> <s xml:id="echoid-s44508" xml:space="preserve">Sit ſpeculum columnare concauum:</s> <s xml:id="echoid-s44509" xml:space="preserve"> cuius axis ſit a b:</s> <s xml:id="echoid-s44510" xml:space="preserve"> & ſit centrum uiſus t, ſitq́;</s> <s xml:id="echoid-s44511" xml:space="preserve"> punctum t intra <lb/>ſpeculum:</s> <s xml:id="echoid-s44512" xml:space="preserve"> dico quòd ab omni puncto ſuperficiei ſpeculi fiet reflexio ad uiſum.</s> <s xml:id="echoid-s44513" xml:space="preserve"> Siue enim cõmunis <lb/>ſectio ſu perficiei reflexionis & huius ſpeculi ſuerit linea longitudinis columnæ ſpeculi:</s> <s xml:id="echoid-s44514" xml:space="preserve"> ut cum ſu-<lb/>perſicies reflexionis ſecat ſuperficiem ſpeculi ſecũdum axis longitudinem, ut patet per 93 th.</s> <s xml:id="echoid-s44515" xml:space="preserve"> 1 hu-<lb/>ius:</s> <s xml:id="echoid-s44516" xml:space="preserve"> ſiue fuerit circulus æquidiſtans baſibus colũnæ ipſius ſpeculi:</s> <s xml:id="echoid-s44517" xml:space="preserve"> ſiue fuerit ſectio oxygonia:</s> <s xml:id="echoid-s44518" xml:space="preserve"> ſem-<lb/>per patet per præmiſſam quòd punctus reflexionis & centrum circuli ſiue punctus axis, in quem <lb/>cadit perpendicularis ducta à puncto reflexionis ſuper ſuperficiem ſpeculi, ſunt in eadẽ ſuperficie.</s> <s xml:id="echoid-s44519" xml:space="preserve"> <lb/>Eſt ergo ſemper poſsibile, ut ab illo puncto flat reflexio ad uiſum:</s> <s xml:id="echoid-s44520" xml:space="preserve"> quoniam in concauitate talium <lb/>ſpeculorum non eſt corpus aliquod denſum reſiſtens multiplicationi formarum per medium.</s> <s xml:id="echoid-s44521" xml:space="preserve"> A <lb/>quolibet ergo puncto ſuperficiei talium ſpeculorum fiet formarum reflexio ad uiſum.</s> <s xml:id="echoid-s44522" xml:space="preserve"> Idem quoq;</s> <s xml:id="echoid-s44523" xml:space="preserve"> <lb/>patet in ſpeculis pyramidalibus concauis.</s> <s xml:id="echoid-s44524" xml:space="preserve"> Quoniam enim centrum uiſus ſemper eſt intra talia ſpe-<lb/>cula, non refert à quocunq;</s> <s xml:id="echoid-s44525" xml:space="preserve"> puncto ſuperficiei ſpeculi flat reflexio:</s> <s xml:id="echoid-s44526" xml:space="preserve"> quoniam ſemper poſsibile erit <lb/>formam ad uiſum peruenire, niſi fortè denſitas occipitis in quibuſdã ſitibus impediat reflexionẽ-<lb/>Patet ergo propoſitum, reſumpta figuratione pręmiſſæ, poſitoq́;</s> <s xml:id="echoid-s44527" xml:space="preserve"> puncto tintra ſuperficiẽ ſpeculi in <lb/>linea t e.</s> <s xml:id="echoid-s44528" xml:space="preserve"> Quicunq;</s> <s xml:id="echoid-s44529" xml:space="preserve"> enim punctus in utroq;</s> <s xml:id="echoid-s44530" xml:space="preserve"> ſpeculorũ fuerit datus:</s> <s xml:id="echoid-s44531" xml:space="preserve"> ſit ille punctus e:</s> <s xml:id="echoid-s44532" xml:space="preserve"> & ab eo extra-<lb/>hatur perpendicularis ſuper ſuperficiem planam in illo puncto ſpeculum contingentem per 12 p 11.</s> <s xml:id="echoid-s44533" xml:space="preserve"> <lb/>Et quoniam illa cadet in axem ſpeculi per 96 th.</s> <s xml:id="echoid-s44534" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s44535" xml:space="preserve"> ſit, ut cadat in punctum f:</s> <s xml:id="echoid-s44536" xml:space="preserve"> & ſuper pũctum <lb/>e terminũ lineæ e f flat per 23 p 1 angulus æ qualis angulo t e f, qui f e d.</s> <s xml:id="echoid-s44537" xml:space="preserve"> Palàm ergo quòd forma pun-<lb/>cti d reflectetur ad uiſum in puncto t exiſtentem per 20 th.</s> <s xml:id="echoid-s44538" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s44539" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s44540" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1731" type="section" level="0" n="0"> <head xml:id="echoid-head1285" xml:space="preserve" style="it">5. Centro uiſus exiſtente extra ſpeculũ columnare uelpyramidale concauum nõ integrum, <lb/>à maiore parte ſuperficiei ſpeculi fiet reflexio ad uiſum. Alhazen 49 n 4.</head> <p> <s xml:id="echoid-s44541" xml:space="preserve">Eſto ſpeculum columnare uel pyramidale concauum:</s> <s xml:id="echoid-s44542" xml:space="preserve"> cuius axis ſit a b:</s> <s xml:id="echoid-s44543" xml:space="preserve"> & ſit centrum uiſus pun-<lb/>ctum t:</s> <s xml:id="echoid-s44544" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s44545" xml:space="preserve"> extra ſpeculum:</s> <s xml:id="echoid-s44546" xml:space="preserve"> dico quòd à maiore parte ſuperficiei concauæ ſpeculi flet reflexio ad <lb/>uiſum.</s> <s xml:id="echoid-s44547" xml:space="preserve"> Imaginentur enim ſuperficies contin gentes columnam uel pyramidem à uiſu productæ ad <lb/>ſpeculum:</s> <s xml:id="echoid-s44548" xml:space="preserve"> palamq́;</s> <s xml:id="echoid-s44549" xml:space="preserve"> per 1 p 7 huius quoniam ſolùm pars ſuperficiei ſpeculi interiacens illas ſuperfi-<lb/>cies contingentes eſt illa, à qua, ſpeculo exiſtente conuexo, fit reflexio ad uiſum.</s> <s xml:id="echoid-s44550" xml:space="preserve"> Eſt autem illa pars <lb/>minor pars ſuperficiei ſpeculi, ut patet de ſpeculis columnaribus per 78 th.</s> <s xml:id="echoid-s44551" xml:space="preserve"> 4 huius, & de pyrami-<lb/>dalibus per 84 th.</s> <s xml:id="echoid-s44552" xml:space="preserve"> 4 huius:</s> <s xml:id="echoid-s44553" xml:space="preserve"> ablata itaq;</s> <s xml:id="echoid-s44554" xml:space="preserve"> illa parte remanet maior pars ſuperficiei ſpeculi.</s> <s xml:id="echoid-s44555" xml:space="preserve"> Fit autem <lb/>à tota illa ſuperficie reflexio ad uiſum:</s> <s xml:id="echoid-s44556" xml:space="preserve"> quoniam omnis linea ducta ſub lineis contingentibus ſpe-<lb/>culum in aliqua illarum ſuperficierum, producta ſecat ſuperficiem ſpeculi per 4 th.</s> <s xml:id="echoid-s44557" xml:space="preserve"> 7 huius:</s> <s xml:id="echoid-s44558" xml:space="preserve"> ſecun-<lb/>dum illam ergo poteſt fieri reflexio ad uiſum.</s> <s xml:id="echoid-s44559" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s44560" xml:space="preserve"/> </p> <figure> <variables xml:id="echoid-variables783" xml:space="preserve">a d e f b c</variables> </figure> </div> <div xml:id="echoid-div1732" type="section" level="0" n="0"> <head xml:id="echoid-head1286" xml:space="preserve" style="it">6. Speculo pyramidali concauo integro exiſtẽte, oppoſitó ipſo <lb/>uiſui ex parte ſuæ baſis exiſtenti: nullius puncti forma uidebi-<lb/>tur, niſi intra ſpeculum exiſtentis. Alhazen 50 n 4.</head> <p> <s xml:id="echoid-s44561" xml:space="preserve">Eſto ſpeculum pyramidale concauum:</s> <s xml:id="echoid-s44562" xml:space="preserve"> cuius axis ſit a b:</s> <s xml:id="echoid-s44563" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s44564" xml:space="preserve"> eius <lb/>conica ſuperficies tota integra:</s> <s xml:id="echoid-s44565" xml:space="preserve"> baſis uerò eius, quæ eſt ſuperficies <lb/>plana, ſit ſubmota ab ipſo ſpeculo:</s> <s xml:id="echoid-s44566" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s44567" xml:space="preserve"> centrum uiſus c ex parte ba-<lb/>ſis ſubmotæ:</s> <s xml:id="echoid-s44568" xml:space="preserve"> dico quòd uiſus non percipiet formam alicuius puncti <lb/>rei uiſæ, niſi illius, quę fuerit intra ipſum ſpeculum.</s> <s xml:id="echoid-s44569" xml:space="preserve"> Si enim centrum <lb/>uiſus c in aliqua cõſiſtat linea longitudinis ſpeculi, fiatq́;</s> <s xml:id="echoid-s44570" xml:space="preserve"> reflexio ab <lb/>illa linea longitudinis ad uiſum:</s> <s xml:id="echoid-s44571" xml:space="preserve"> tunc patet quia punctum rei uiſæ <lb/>oportebit conſiſtere intra ſpeculum:</s> <s xml:id="echoid-s44572" xml:space="preserve"> quoniam ex hypotheſi cen-<lb/>trum uiſus eſt ex parte baſis ſpeculi:</s> <s xml:id="echoid-s44573" xml:space="preserve"> oportebitq́ue punctum rei ui-<lb/>ſæ in eadem linea longitudinis exiſtere:</s> <s xml:id="echoid-s44574" xml:space="preserve"> aliàs enim non fieret re-<lb/>flexio propter in æqualitatem angulorum.</s> <s xml:id="echoid-s44575" xml:space="preserve"> Quòd ſi centrum uiſus <lb/>c ſit ſub aliqua linearum longitudinis ſpeculi:</s> <s xml:id="echoid-s44576" xml:space="preserve"> tunc adhuc patet <lb/>propoſitum.</s> <s xml:id="echoid-s44577" xml:space="preserve"> Quoniam enim omnis perpendicularis ducta à quo-<lb/>cunque puncto reflexionis, quæ fieri poſsit ad uiſum c in hoc ſitu, <lb/> <pb o="370" file="0672" n="672" rhead="VITELLONIS OPTICAE"/> tenet angulum acutum cum linea reflexionis:</s> <s xml:id="echoid-s44578" xml:space="preserve"> patet per 33 th.</s> <s xml:id="echoid-s44579" xml:space="preserve"> 5 huius cũ ſemper fiat reflexio ex par-<lb/>te anguli maioris, quòd ſem per fiet reflexio ex parte acuminis pyramidis ſpeculi.</s> <s xml:id="echoid-s44580" xml:space="preserve"> Oportet ergo de <lb/>neceſsitate, ut puncta rei uiſæ, quorum formæ reflectuntur ad uiſum à quibuſcunq;</s> <s xml:id="echoid-s44581" xml:space="preserve"> punctis ſuper-<lb/>ficiei totius ſpeculi, ſemper ſint intra ipſum ſpeculum.</s> <s xml:id="echoid-s44582" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s44583" xml:space="preserve"> Si uerò auferatur à <lb/>ſpeculo tali portio aliqua ſecundum longitudinem ſpeculi:</s> <s xml:id="echoid-s44584" xml:space="preserve"> tunc poterunt comprehendi exteriora, <lb/>quæ ſunt extra ſpeculum:</s> <s xml:id="echoid-s44585" xml:space="preserve"> quoniam patebunt liberi introitus lineis incidentiæ formarum extrinſe-<lb/>carum, quę reflectentur ad uiſum.</s> <s xml:id="echoid-s44586" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s44587" xml:space="preserve"> accidit, ſi ſecetur pyramis ſpeculi ad modum an-<lb/>nuli ſecundum aliquem circulum æquidiſtantem baſi, uel etiam ſecundum oxygoniam ſectionem, <lb/>taliter, ut auferatur uertex pyramidis ſpeculi:</s> <s xml:id="echoid-s44588" xml:space="preserve"> tunc enim lineæ incidentiæ liberum habebũt ingreſ-<lb/>ſum:</s> <s xml:id="echoid-s44589" xml:space="preserve"> plures tamen formæ reflectentur ad uiſum ſi centrum uiſus fuerit ex parte ſuperficiei conca-<lb/>uitatis ſpeculi, quàm ſi fuerit ex parte ſuæ baſis:</s> <s xml:id="echoid-s44590" xml:space="preserve"> quia tunc lineis incidentibus latior uia patet.</s> <s xml:id="echoid-s44591" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1733" type="section" level="0" n="0"> <head xml:id="echoid-head1287" xml:space="preserve" style="it">7. A quocun puncto ſpeculi columnaris uelpyramidalis concaui non eſt poßibile niſifor-<lb/>mam unius puncti ad eundem uiſum reflecti. Alhazen 51 n 4.</head> <p> <s xml:id="echoid-s44592" xml:space="preserve">Eſto, ut in præmiſſa, ſpeculum columnare uel pyramidale concauũ, cuius axis a b:</s> <s xml:id="echoid-s44593" xml:space="preserve"> ab eius quoq, <lb/>puncto e reflectatur ad uiſum c forma pũcti d:</s> <s xml:id="echoid-s44594" xml:space="preserve"> dico quòd ab eodẽ puncto e formam alterius puncti, <lb/>quàm d, ad uiſum exiſtentem in puncto c impoſsibile eſt reflecti.</s> <s xml:id="echoid-s44595" xml:space="preserve"> Ducatur enim à puncto reflexio-<lb/>nis, qui eſt e, linea perpendicularis ſuper ſuperficiem ſpeculum in puncto e contingentem:</s> <s xml:id="echoid-s44596" xml:space="preserve"> quæ ſe-<lb/>cabit axem ſpeculi per 96 th.</s> <s xml:id="echoid-s44597" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s44598" xml:space="preserve"> ſecet ergo in puncto f.</s> <s xml:id="echoid-s44599" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s44600" xml:space="preserve"> per 3 huius quoniam pun-<lb/>cta c, d, e, f ſunt in eadem ſuperficie.</s> <s xml:id="echoid-s44601" xml:space="preserve"> Et quoniam una ſola linea recta à centro uiſus, quod eſt e, duci-<lb/>bilis eſt ad punctum reflexionis, quod eſt e:</s> <s xml:id="echoid-s44602" xml:space="preserve"> patet quòd angulus c e f non poteſt uariari:</s> <s xml:id="echoid-s44603" xml:space="preserve"> ergo nec <lb/>angulus d e f, qui per 20 th.</s> <s xml:id="echoid-s44604" xml:space="preserve"> 5 huius eſt æqualis angulo c e f.</s> <s xml:id="echoid-s44605" xml:space="preserve"> Linea ergo e d eſt tãtùm unica linea, cu-<lb/>ius alicuius puncti forma poteſt reflecti ad uiſum c:</s> <s xml:id="echoid-s44606" xml:space="preserve"> ſed ex hypotheſi forma puncti d reflectitur ad <lb/>uiſum:</s> <s xml:id="echoid-s44607" xml:space="preserve"> nullius ergo alterius puncti forma ad ipſum reflectetur.</s> <s xml:id="echoid-s44608" xml:space="preserve"> Cum enim aliqua linea incidentiæ <lb/>peruenit ad aliquod punctum corporis:</s> <s xml:id="echoid-s44609" xml:space="preserve"> non poteſt forma alterius puncti per illam lineam incidere <lb/>ſpeculo:</s> <s xml:id="echoid-s44610" xml:space="preserve"> quoniam punctus altior occultat poſteriorem, nec præſtat tranſitum formæ illius.</s> <s xml:id="echoid-s44611" xml:space="preserve"> Patet <lb/>ergo propoſitum:</s> <s xml:id="echoid-s44612" xml:space="preserve"> quoniam in his ſpeculis à quocunq;</s> <s xml:id="echoid-s44613" xml:space="preserve"> puncto facta reflexione formę unius puncti, <lb/>non poteſt ab eodem puncto ſpeculi forma alterius puncti reflecti ad eundem uiſum.</s> <s xml:id="echoid-s44614" xml:space="preserve"> Sed à duo-<lb/>bus uiſibus poſſunt in eodem puncto ſpeculi duorum punctorum formæ comprehẽdi, ſicut à plu-<lb/>ribus uiſibus plures formæ diuerſorum punctorum:</s> <s xml:id="echoid-s44615" xml:space="preserve"> quoniam, ut patet per 18 th.</s> <s xml:id="echoid-s44616" xml:space="preserve"> 7 huius, infinitæ <lb/>poſſunt ſumi ſuperficies ſuper perpendicularem e f ſe ſecantes, in quarum qualibet ex utraq;</s> <s xml:id="echoid-s44617" xml:space="preserve"> parte <lb/>perpendicularis e f ſumi poſſunt duo ariguli acuti æquales.</s> <s xml:id="echoid-s44618" xml:space="preserve"> Licet autem illud, quod hic proponi-<lb/>tur, ſatis patuerit per 29 th.</s> <s xml:id="echoid-s44619" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s44620" xml:space="preserve"> hic tamen idem declarauimus:</s> <s xml:id="echoid-s44621" xml:space="preserve"> ideo quia oppoſitum in his ſpe-<lb/>culis plus ueriſimile uidebatur.</s> <s xml:id="echoid-s44622" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1734" type="section" level="0" n="0"> <head xml:id="echoid-head1288" xml:space="preserve" style="it">8. Linea longitudinis ſpeculi columnaris uelpyramidalis concaui exiſtente communi ſectio-<lb/>ne ſuperficiei reflexιonis & ſpeculi: unus eſt tantùm punctus reflexionis, & unius punctirei ui-<lb/>ſæ ad unius uiſus centrum: & uidetur unica imago.</head> <p> <s xml:id="echoid-s44623" xml:space="preserve">Non oportet huic propoſitioni declaran dæ aliter inſiſti, niſi ſicut idem oſtenſum eſt in ſpeculis <lb/>planis, quòd ab uno tantùm puncto fit reflexio, & una tantùm occurrit uiſui imago, ut patet per 46 <lb/>& 48 th.</s> <s xml:id="echoid-s44624" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s44625" xml:space="preserve"> Linea enim recta eſt communis ſectio ſuperficiei reflexionis & ſuperficiei ſpeculi <lb/>hinc inde:</s> <s xml:id="echoid-s44626" xml:space="preserve"> unicus ergo tãtùm eſt punctus reflexionis:</s> <s xml:id="echoid-s44627" xml:space="preserve"> unica tátùm ergo uidebitur imago ſub ſuper-<lb/>ficie ſpeculi ſemper apparens, ut in planis ſpeculis:</s> <s xml:id="echoid-s44628" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s44629" xml:space="preserve"> per 49 th.</s> <s xml:id="echoid-s44630" xml:space="preserve"> 5 huius diſtantia imaginis ſub <lb/>ſpeculo æqualis diſtantiæ rei uiſæ ſupra ſpeculum.</s> <s xml:id="echoid-s44631" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s44632" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1735" type="section" level="0" n="0"> <head xml:id="echoid-head1289" xml:space="preserve" style="it">9. Communi ſectione ſuperficiei reflexionis & ſpeculi columnaris uel pyramidalis concaui <lb/>oxygonia exiſtente: à pluribus punctis illius ſectionis poteſt fieri reflexio formæ eiuſdem puncti <lb/>reiuiſæ adidem centrum uiſus. Alhazen 48 n 4. Item 93 n 5.</head> <p> <s xml:id="echoid-s44633" xml:space="preserve">Sit ſpeculum columnare uel pyramidale concauum, cuius axis a b:</s> <s xml:id="echoid-s44634" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s44635" xml:space="preserve"> centrũ uiſus c:</s> <s xml:id="echoid-s44636" xml:space="preserve"> & punctũ <lb/>rei uiſæ ſit d, ut patet in figura 6 huius.</s> <s xml:id="echoid-s44637" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s44638" xml:space="preserve"> cõmunis ſectio ſuperficiei reflexionis & ſpeculi fue-<lb/>rit ſectio oxygonia:</s> <s xml:id="echoid-s44639" xml:space="preserve"> dico quòd forma puncti d ad centrum uiſus c à pluribus punctis illius ſectionis <lb/>reflecti poteſt.</s> <s xml:id="echoid-s44640" xml:space="preserve"> Iam enim oſtendimus ſuprà per 22 th.</s> <s xml:id="echoid-s44641" xml:space="preserve"> 7 huius quòd à ſpeculis columnaribus conue-<lb/>xis ab uno tantùm pũcto ſectionis oxygoniæ fit formæ eiuſdem puncti reflexio ad uiſum eundem:</s> <s xml:id="echoid-s44642" xml:space="preserve"> <lb/>& diximus quòd ſi diameter column æ fuerit æqualis diſtantiæ oculorum, quòd à duobus punctis <lb/>ſectionis oxygoniæ poteſt fieri reflexio ad uiſum:</s> <s xml:id="echoid-s44643" xml:space="preserve"> aliàs enim latebunt uiſum puncta reflexionis ſe <lb/>reſpicientia, ſcilicet illa, per quæ tranſit circulus columnæ, ductus per punctum reflexionis æ qui-<lb/>diſtanter baſibus:</s> <s xml:id="echoid-s44644" xml:space="preserve"> unde uiſo uno illorum punctorum alius pũctus latebit propter minoris portio-<lb/>nis columnæ ipſius apparentiam.</s> <s xml:id="echoid-s44645" xml:space="preserve"> In his uerò ſpeculis columnaribus concauis apparet uiſui maior-<lb/>portio columnæ, ut patet per 5 huius:</s> <s xml:id="echoid-s44646" xml:space="preserve"> unde ab unico uiſu poſſunt percipl ambo puncta, quæ ſunt <lb/>extremitates diametri circuli æquidiſtãtis baſibus columnæ.</s> <s xml:id="echoid-s44647" xml:space="preserve"> Et eodem modo penitùs de ſpeculis <lb/>pyramidalibus cõcauis declaran dũ eius enim ſuperficiei plus medietate uni uiſui occurrit:</s> <s xml:id="echoid-s44648" xml:space="preserve"> & duo <lb/>pũcta per diametrũ circuli æ quidiſtãtis baſi pyramidis oppoſita uideri poſsũt.</s> <s xml:id="echoid-s44649" xml:space="preserve"> Patet ergo ꝓpoſitũ.</s> <s xml:id="echoid-s44650" xml:space="preserve"/> </p> <pb o="371" file="0673" n="673" rhead="LIBER NONVS."/> </div> <div xml:id="echoid-div1736" type="section" level="0" n="0"> <head xml:id="echoid-head1290" xml:space="preserve" style="it">10. Communi ſectione ſuperficiei reflexionis & ſpeculi columnaris uelpyramidalis concaui <lb/>oxygonia exiſtente: erit locus imaginis quando ultra ſpeculum: quando citra uiſum: quan-<lb/>doque in centro uiſus: quandoque in ſuperficie ſpeculi: quandoque inter uiſum & ſpeculum. <lb/>Alhazen 90 n 5.</head> <p> <s xml:id="echoid-s44651" xml:space="preserve">Eſto ſpeculum columnare concauum:</s> <s xml:id="echoid-s44652" xml:space="preserve"> cuius pars axis ſit d k:</s> <s xml:id="echoid-s44653" xml:space="preserve"> & eius ſuperficiei columnaris & ſu-<lb/>perficiei reflexionis communis ſectio ſit oxygonia:</s> <s xml:id="echoid-s44654" xml:space="preserve"> quæ a b g:</s> <s xml:id="echoid-s44655" xml:space="preserve"> dico quòd poſsibile eſt totum;</s> <s xml:id="echoid-s44656" xml:space="preserve"> quod <lb/>hic proponitur.</s> <s xml:id="echoid-s44657" xml:space="preserve"> Ducatur enim in hac ſectione perpendicularis ſuper ſuperficiem ſpeculum cõtin-<lb/>gentem in pũcto reflexionis, quæ ſit d g:</s> <s xml:id="echoid-s44658" xml:space="preserve"> hæc itaq;</s> <s xml:id="echoid-s44659" xml:space="preserve"> per 112 & per 104 th.</s> <s xml:id="echoid-s44660" xml:space="preserve"> 1 huius erit ſemidiameter cu-<lb/>iuſdam circuli ſecundum illum punctum ſecantιs columnam ſpeculi æquidiſtanter baſibus:</s> <s xml:id="echoid-s44661" xml:space="preserve"> ſeca-<lb/>bitq́;</s> <s xml:id="echoid-s44662" xml:space="preserve"> axẽ ſpeculi, qui eſt k d:</s> <s xml:id="echoid-s44663" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s44664" xml:space="preserve">, ut ſecet ipſum in puncto d:</s> <s xml:id="echoid-s44665" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s44666" xml:space="preserve"> illa perpendicularis tantùm una:</s> <s xml:id="echoid-s44667" xml:space="preserve"> <lb/>cum à nullo alio puncto ſectionis a b g poſsit duci linea perpendicularis ſuper ſuperficiem contin-<lb/>gentem ſpeculum in puncto reflexionis quàm ab uno puncto reflexionis:</s> <s xml:id="echoid-s44668" xml:space="preserve"> cum omnes aliæ lineæ à <lb/>quibuſcunq;</s> <s xml:id="echoid-s44669" xml:space="preserve"> punctis ſectionis a b g ductæ ad axem d h, ſint obliquæ ſuper ſuperficiem illam ſpecu-<lb/>lum contingentem, ut patet per pręnominatas propoſitiones 1 huius.</s> <s xml:id="echoid-s44670" xml:space="preserve"> Sumatur item alιus punctus <lb/>ſectionis a b g, qui ſit b:</s> <s xml:id="echoid-s44671" xml:space="preserve"> & ducatur ab illo puncto b linea <lb/> <anchor type="figure" xlink:label="fig-0673-01a" xlink:href="fig-0673-01"/> perpendicularis ſuper lineam rectam contingẽtem ſectio-<lb/>nem a b g in puncto b:</s> <s xml:id="echoid-s44672" xml:space="preserve"> & hæc quidem linea per 114 th.</s> <s xml:id="echoid-s44673" xml:space="preserve"> 1 hu-<lb/>ius neceſſariò concurret cum perpendiculari g d.</s> <s xml:id="echoid-s44674" xml:space="preserve"> Sit ergo, <lb/>exempli cauſſa, cócurſus in puncto d:</s> <s xml:id="echoid-s44675" xml:space="preserve"> quoniam ſi concur-<lb/>rant ſub puncto d, eadem eſt demonſtratio:</s> <s xml:id="echoid-s44676" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s44677" xml:space="preserve"> punctus <lb/>b taliter ſumptus in ſectione a b g circa punctum g, ut an-<lb/>gulus b d g ſit acutus.</s> <s xml:id="echoid-s44678" xml:space="preserve"> Deinde à puncto g ducatur in ſuper-<lb/>ficie ſectionis a b g linea æquidiſtás lineæ b d per 31 p 1:</s> <s xml:id="echoid-s44679" xml:space="preserve"> quę <lb/>ſit g h:</s> <s xml:id="echoid-s44680" xml:space="preserve"> & hæc linea cadet intra pyramidalẽ ſectionem:</s> <s xml:id="echoid-s44681" xml:space="preserve"> ideo <lb/>quia cum angulus g d b ſit acutus ex hypotheſi, erit ſuus <lb/>coalternus (qui eſt angulus h g d) ſimiliter acutus per 29 <lb/>p 1:</s> <s xml:id="echoid-s44682" xml:space="preserve"> cum lineæ b g & g h adinuicem æquidiſtent.</s> <s xml:id="echoid-s44683" xml:space="preserve"> Item inter <lb/>puncta d & h ducatur à puncto g linea in ſuperficie ſectio-<lb/>nis, quæ per 2 th.</s> <s xml:id="echoid-s44684" xml:space="preserve"> 1 huius neceſſariò concurret cum linea b <lb/>d:</s> <s xml:id="echoid-s44685" xml:space="preserve"> quoniam ipſa cõcurrit cum linea h g æquidiſtante lineæ <lb/>b d:</s> <s xml:id="echoid-s44686" xml:space="preserve"> ſit ergo punctus concurſus n:</s> <s xml:id="echoid-s44687" xml:space="preserve"> cadet itaq;</s> <s xml:id="echoid-s44688" xml:space="preserve"> linea g n inter <lb/>lineas g h & b n.</s> <s xml:id="echoid-s44689" xml:space="preserve"> In hacitaq;</s> <s xml:id="echoid-s44690" xml:space="preserve"> linea g n ſumatur pũctus qui-<lb/>cunq;</s> <s xml:id="echoid-s44691" xml:space="preserve">, qui ſit o, inter duo puncta g & n, & ultra punctum n <lb/>ſumatur punctus tin linea g n.</s> <s xml:id="echoid-s44692" xml:space="preserve"> Item à puncto g ducatur <lb/>extra ambas lineas g h & b d, alia linea intra ſectionem a b <lb/>g, quæ ſit g z.</s> <s xml:id="echoid-s44693" xml:space="preserve"> Hæc itaq;</s> <s xml:id="echoid-s44694" xml:space="preserve"> linea g z, quia concurrit cum linea <lb/>h g in puncto g, neceſſariò concurret cum linea d b produ-<lb/>cta ultra punctum b per 2 th.</s> <s xml:id="echoid-s44695" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s44696" xml:space="preserve"> ſit concurſus in pun-<lb/>cto e:</s> <s xml:id="echoid-s44697" xml:space="preserve"> & ſuper g terminum lineæ g d fiat angulus æqualis <lb/>angulo z g d per 23 p 1, qui ſit angulus d g q:</s> <s xml:id="echoid-s44698" xml:space="preserve"> cadatq́;</s> <s xml:id="echoid-s44699" xml:space="preserve"> punctũ <lb/>q in linea b d.</s> <s xml:id="echoid-s44700" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s44701" xml:space="preserve"> fiat angulus l g d æqualis angulo h g d:</s> <s xml:id="echoid-s44702" xml:space="preserve"> & fiat angulus m g d æqualis <lb/>angulo n g d:</s> <s xml:id="echoid-s44703" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s44704" xml:space="preserve"> omnia puncta q, l, & m in linea b d.</s> <s xml:id="echoid-s44705" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s44706" xml:space="preserve"> per 20 th.</s> <s xml:id="echoid-s44707" xml:space="preserve"> 5 huius, quòd ſi centrũ <lb/>uiſus fuerit in puncto z, reflectetur ad ipſum forma puncti q à pũcto ſpeculi g:</s> <s xml:id="echoid-s44708" xml:space="preserve"> & erit per 37 th.</s> <s xml:id="echoid-s44709" xml:space="preserve"> 5 hu-<lb/>ius locus imaginis punctum e:</s> <s xml:id="echoid-s44710" xml:space="preserve"> & ſi fuerit centrũ uiſus in puncto h, reflectetur ad ipſum forma pun-<lb/>cti l à puncto ſpeculi g.</s> <s xml:id="echoid-s44711" xml:space="preserve"> Et quoniá cathetus incidentiæ, quæ eſt l d, æ quidiſtat lineæ reflexionis, quæ <lb/>eſt g h:</s> <s xml:id="echoid-s44712" xml:space="preserve"> palàm quòd lineæ l d & g h nunquam concurrent.</s> <s xml:id="echoid-s44713" xml:space="preserve"> Erit ergo locus imaginis in puncto ſuper-<lb/>ficiei ſpeculi, à quo fit reflexio (quod eſt punctum g) qui locus eſt primus & proprius ipſius imagi-<lb/>nis propter continuitatem totius formæ reflexæ, prout diximus in 12 th.</s> <s xml:id="echoid-s44714" xml:space="preserve"> 8 huius.</s> <s xml:id="echoid-s44715" xml:space="preserve"> Si uerò centrum <lb/>uiſus fuerit in puncto o, reflectetur ad ipſum forma puncti m à puncto ſpeculi, quod eſt g:</s> <s xml:id="echoid-s44716" xml:space="preserve"> & locus <lb/>imaginis erit punctum n.</s> <s xml:id="echoid-s44717" xml:space="preserve"> Si uerò centrum uiſus fuerit in puncto n:</s> <s xml:id="echoid-s44718" xml:space="preserve"> erit locus imaginis formæ pun-<lb/>cti m in ipſo centro uiſus, quod eſt in puncto n.</s> <s xml:id="echoid-s44719" xml:space="preserve"> Quòd ſi cẽtrum uiſus fuerit in puncto t:</s> <s xml:id="echoid-s44720" xml:space="preserve"> erit iterum <lb/>locus imaginis formæ puncti m in puncto n, quod erit tunc inter uiſum & ſuperficiem ſpeculi.</s> <s xml:id="echoid-s44721" xml:space="preserve"> Pa-<lb/>tet ergo propoſitum:</s> <s xml:id="echoid-s44722" xml:space="preserve"> quoniã in ſpeculis pyramidalibus concauis poterit ſecundum præmiſſa, coo-<lb/>perante 113 huius, demonſtratio faciliter coaptari.</s> <s xml:id="echoid-s44723" xml:space="preserve"> Hoc itaq;</s> <s xml:id="echoid-s44724" xml:space="preserve"> proponebatur.</s> <s xml:id="echoid-s44725" xml:space="preserve"/> </p> <div xml:id="echoid-div1736" type="float" level="0" n="0"> <figure xlink:label="fig-0673-01" xlink:href="fig-0673-01a"> <variables xml:id="echoid-variables784" xml:space="preserve">e b g c q l m d z o a f n t h k</variables> </figure> </div> </div> <div xml:id="echoid-div1738" type="section" level="0" n="0"> <head xml:id="echoid-head1291" xml:space="preserve" style="it">11. Centro uiſus & puncto rei uiſœ exiſtentibus in eadem linea perpendiculari ſuper ſuperfi-<lb/>ciem ſpeculi columnaris uel pyramidalis concaui: quando ab unopuncto ſpeculi: quando à <lb/>duobus fit reflexio: & locus imaginis ſemper erit centrum uiſus. Alhazen 91 n 5.</head> <p> <s xml:id="echoid-s44726" xml:space="preserve">Sit ſpeculum columnare concauum:</s> <s xml:id="echoid-s44727" xml:space="preserve"> cuius a xis ſit a b:</s> <s xml:id="echoid-s44728" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s44729" xml:space="preserve"> centrum uiſus c:</s> <s xml:id="echoid-s44730" xml:space="preserve"> & punctum rei uiſæ <lb/>d:</s> <s xml:id="echoid-s44731" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s44732" xml:space="preserve"> puncta c & d in una linea perpendiculari ſuper ſuperficiẽ ſpeculi, quæ ſit e f, uel in alia linea <lb/>perpendiculari ſuper lineá e f:</s> <s xml:id="echoid-s44733" xml:space="preserve"> quæ ſit h p, ita quòd punctus e ſit pũctus ſuperficiei ſpeculi, & pũctus <lb/>f ſit pũctus axis a b:</s> <s xml:id="echoid-s44734" xml:space="preserve"> & producatur linea e f ad aliá partem ſpeculi in punctũ g.</s> <s xml:id="echoid-s44735" xml:space="preserve"> Dico quòd quandoq;</s> <s xml:id="echoid-s44736" xml:space="preserve"> <lb/>ab uno puncto ſpeculi, ut à puncto e:</s> <s xml:id="echoid-s44737" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s44738" xml:space="preserve"> à duobus, ut à punctis e & g, poteſt forma puncti d <lb/> <pb o="372" file="0674" n="674" rhead="VITELLONIS OPTICAE"/> reflecti ad uiſum c.</s> <s xml:id="echoid-s44739" xml:space="preserve"> Palàm enim per 21 th.</s> <s xml:id="echoid-s44740" xml:space="preserve"> 5 huius quòd linea c e, in qua eſt punctus rei uiſæ, qui eſt d, <lb/>reflectitur in ſeipſam:</s> <s xml:id="echoid-s44741" xml:space="preserve"> tunc enim infinitæ poſſunt intelligi ſuperficies ſecantes ſe ſuper lineam e f, <lb/>quarum quælibet eſt erecta ſuper ſuperficiem contingẽtem ſpeculum per 18 p 11:</s> <s xml:id="echoid-s44742" xml:space="preserve"> cum linea e f, quæ <lb/>eſt communis ſectio illarum ſuperficierum, ſit erecta ſuper ſuperficiem ſpeculum in puncto e con-<lb/>tingentem.</s> <s xml:id="echoid-s44743" xml:space="preserve"> Quádo ergo quarundam illarum ſuperficierum & ſuperficiei ipſius ſpeculi communis <lb/>ſectio eſt linea recta, quę eſt linea longitudinis ſpeculi æquidiſtans axi a b:</s> <s xml:id="echoid-s44744" xml:space="preserve"> tunc, ſicut per 21 th.</s> <s xml:id="echoid-s44745" xml:space="preserve"> 5 hu-<lb/>ius in ſpeculis quibuſcunq;</s> <s xml:id="echoid-s44746" xml:space="preserve"> oſtendimus, non ſiet reflexio, niſi ſuper eandem lineam perpendicula-<lb/>rem, quæ eſt e c:</s> <s xml:id="echoid-s44747" xml:space="preserve"> & (ut patet per 32 & 36.</s> <s xml:id="echoid-s44748" xml:space="preserve"> th.</s> <s xml:id="echoid-s44749" xml:space="preserve"> 5 huius) locus imaginis eſt centrum uiſus, qui eſt pun-<lb/>ctus c:</s> <s xml:id="echoid-s44750" xml:space="preserve"> nec uidebitur aliquis pũctus rei uiſæ, niſi ſolus ille, qui fuerit in ſuperficie ipſius uiſus.</s> <s xml:id="echoid-s44751" xml:space="preserve"> Quã-<lb/>do uerò aliqua illarum ſuperficierum perpendicu-<lb/> <anchor type="figure" xlink:label="fig-0674-01a" xlink:href="fig-0674-01"/> larium ſuper ſuperficiem ſpeculum in puncto e con <lb/>tingentem, ſecat ſuperficiem concauam ipſius ſpe-<lb/>culi, ita quòd communis ſectio illarum ſuperficierũ <lb/>eſt circulus æquidiſtans baſibus colũnæ, cuius cen-<lb/>trum eſt f punctum axis:</s> <s xml:id="echoid-s44752" xml:space="preserve"> & tunc ſi punctum f fuerit <lb/>in diametro p h inter punctum c, quod eſt centrum <lb/>uiſus, & punctum d, quod eſt punctum rei uiſæ, ita <lb/>quòd æqualiter diſtet ab utroq;</s> <s xml:id="echoid-s44753" xml:space="preserve">: ſitq́;</s> <s xml:id="echoid-s44754" xml:space="preserve"> linea c f æqua-<lb/>lis lineæ f d:</s> <s xml:id="echoid-s44755" xml:space="preserve"> poterit forma puncti d ad uiſum c refle-<lb/>cti à duobus punctis ſpeculi, quæ ſunt e & g:</s> <s xml:id="echoid-s44756" xml:space="preserve"> & ſunt <lb/>puncta terminantia diametrũ illius circuli.</s> <s xml:id="echoid-s44757" xml:space="preserve"> A quo-<lb/>libet enim illorũ punctorum fit reflexio formę pun-<lb/>cti d ad uiſum c:</s> <s xml:id="echoid-s44758" xml:space="preserve"> ideo quòd angulus d e f eſt æqualis <lb/>angulo f e c:</s> <s xml:id="echoid-s44759" xml:space="preserve"> & ſimilιter angulus d g f æqualis angu-<lb/>lo f g c per 4 p 1.</s> <s xml:id="echoid-s44760" xml:space="preserve"> Duorum enim trigonorum d f e & f <lb/>e c duo latera d f & f c ſunt æqualia ex hypotheſi, & <lb/>latus f e eſt commune, angulusq́;</s> <s xml:id="echoid-s44761" xml:space="preserve"> d f e eſt æqualis angulo c f e:</s> <s xml:id="echoid-s44762" xml:space="preserve"> quia uterq;</s> <s xml:id="echoid-s44763" xml:space="preserve"> eſt rectus:</s> <s xml:id="echoid-s44764" xml:space="preserve"> & ſimiliter eſt <lb/>in trigonis d f g & c f e.</s> <s xml:id="echoid-s44765" xml:space="preserve"> Angulum itaq;</s> <s xml:id="echoid-s44766" xml:space="preserve"> d e c per æqualia diuidit perpendicularis e f:</s> <s xml:id="echoid-s44767" xml:space="preserve"> & angulum d g <lb/>c per æqualia diuidit perpendicularis f g ducta à puncto reflexionis ad centrũ illius circuli.</s> <s xml:id="echoid-s44768" xml:space="preserve"> Et quo-<lb/>niam cathetus incidentiæ, quæ eſt d f, cum linea reflexionis e c uel g c non concurrit niſi in centro <lb/>uiſus, quod eſt c:</s> <s xml:id="echoid-s44769" xml:space="preserve"> patet per 37 th.</s> <s xml:id="echoid-s44770" xml:space="preserve"> 5 huius quoniam centrum uiſus eſt locus imaginis formæ puncti <lb/>d.</s> <s xml:id="echoid-s44771" xml:space="preserve"> Alia uerò puncta lineæ perpendicularis, quæ eſt c d h, non reflectuntur ad uiſum c à puncto ſpe-<lb/>culi h, niſi ſolus ille punctus, qui eſt in ſuperficie ipſius uiſus, ut ſuprà patuit:</s> <s xml:id="echoid-s44772" xml:space="preserve"> ideo quòd nõ reflecti-<lb/>tur niſi per eandem perpendicularem.</s> <s xml:id="echoid-s44773" xml:space="preserve"> Cum uerò alicuius illarum ſuperficierum perpẽdicularium <lb/>ſuper ſuperficiem ſpeculum propoſitum in puncto e contingẽtem & ſuperficiei ſpeculi fuerit oxy-<lb/>gonia ſectio:</s> <s xml:id="echoid-s44774" xml:space="preserve"> non poterunt puncta lineæ reflexionis reflecti ad uiſum ab aliquibus alijs punctis ſe-<lb/>ctionis:</s> <s xml:id="echoid-s44775" xml:space="preserve"> cũ (ſicut patet per 112 th.</s> <s xml:id="echoid-s44776" xml:space="preserve"> 1 huius) duæ lineæ perpendiculares ſuper ſuperficiẽ ſpeculi in ſu-<lb/>perficie ſectionis ſe interſecare nó poſsint, ſicut in ſuperficie circuli æquidiſtantis baſibus ſpeculi <lb/>ſe tales duæ diametri ſecant ſupercentrum f, ut iam patuit, quę ſunt p h & e g.</s> <s xml:id="echoid-s44777" xml:space="preserve"> Non enim eſt diame-<lb/>ter ſectionis (quæ eſt p h) perpendicularis ſuper ſuperficiem contingẽtem ſpeculum in puncto h, <lb/>ſed obliquè incidit ſuper illam, quando diameter e g perpendicularis eſt ſuper ſuperficiem ſpeculi:</s> <s xml:id="echoid-s44778" xml:space="preserve"> <lb/>& hoc accidit propter obliquationem ſectionis oxygoniæ ſuper axem columnæ ſpeculi.</s> <s xml:id="echoid-s44779" xml:space="preserve"> Non ergo <lb/>reflectetur forma puncti d ad uiſum c per lineam c d h.</s> <s xml:id="echoid-s44780" xml:space="preserve"> Sed ſi puncta d & c æqualiter diſtent à pun-<lb/>cto f, ita ut linea d f ſit æqualis lineæ f c:</s> <s xml:id="echoid-s44781" xml:space="preserve"> tunc à punctis ſpeculi e & g, quæ ſunt termini lineę perpen-<lb/>dicularis ſuper ſuperficiem ſpeculi, quæ eſt linea e f g, poteſt fieri reflexio formæ puncti d ad uiſum <lb/>c per 20 th.</s> <s xml:id="echoid-s44782" xml:space="preserve"> 5 huius, & per 4 p 1, ut ſuprà patuit:</s> <s xml:id="echoid-s44783" xml:space="preserve"> quoniam anguli d e f, & f e c ſunt æquales:</s> <s xml:id="echoid-s44784" xml:space="preserve"> & itẽ an-<lb/>guli d g f & f g c ſunt æquales, & punctum rei uiſæ, quod eſt d, & cẽtrum uiſus, quod eſt c, ſunt cum <lb/>ambobus punctis reflexionis, qui ſunt e & g, & cum pũcto axis f, cui incidit linea e f g, quæ eſt per-<lb/>pendicularis ſuper ſuperficies contingentes ſpeculum in punctis e & g in eadem ſuperficie ipſius <lb/>ſectionis.</s> <s xml:id="echoid-s44785" xml:space="preserve"> Patet ergo quòd fiet ab illis duobus pũctis reflexio formæ puncti d ad uiſum c:</s> <s xml:id="echoid-s44786" xml:space="preserve"> & erit l o-<lb/>cus imaginis in utriſq;</s> <s xml:id="echoid-s44787" xml:space="preserve"> centrum uiſus, quod eſt c.</s> <s xml:id="echoid-s44788" xml:space="preserve"> Sed ſi puncta d & c fuerint in perpendiculari e f:</s> <s xml:id="echoid-s44789" xml:space="preserve"> <lb/>tunc non fiet reflexio ab aliquo puncto ſectionis oxygoniæ, niſi ſolùm à puncto e:</s> <s xml:id="echoid-s44790" xml:space="preserve"> quoniam forma <lb/>incidens ſuperficiei ſpeculi ſecundum lineam perpendicularem, reflectitur ſecũdum eandem per-<lb/>pendicularem:</s> <s xml:id="echoid-s44791" xml:space="preserve"> & in ſectione oxygonia eſt unica linea perpendicularis ſuper ſuperficiem ſpeculum <lb/>contingentem.</s> <s xml:id="echoid-s44792" xml:space="preserve"> Quare, ut prius dictum eſt, per illam ſolam fit reflexio ſolius puncti lineæ perpendi-<lb/>cularis, qui eſt in ſuperficie uiſus:</s> <s xml:id="echoid-s44793" xml:space="preserve"> & ſicut prius, erit locus imaginis in centro uiſus.</s> <s xml:id="echoid-s44794" xml:space="preserve"> Eodem quoque <lb/>modo deducendo, patet idem propoſitũ in ſpeculis pyramidalibus concauis.</s> <s xml:id="echoid-s44795" xml:space="preserve"> Ducta enim à centro <lb/>uiſus ad ſuperficiem contingẽtem ſpeculum pyramidale linea recta perpendiculari ſuper illam ſu-<lb/>perficiem:</s> <s xml:id="echoid-s44796" xml:space="preserve"> ſi in illa perpẽdiculari ſumatur punctus corporeus inter uiſum & ſpeculum:</s> <s xml:id="echoid-s44797" xml:space="preserve"> patet quòd <lb/>non reflectetur ſorma eius ad uiſum ſecundum illam perpendicularem:</s> <s xml:id="echoid-s44798" xml:space="preserve"> quoniam punctus ille oc-<lb/>cultabit terminum perpendicularis, & non reflectetur ab ipſo.</s> <s xml:id="echoid-s44799" xml:space="preserve"> Si autem nullus punctus corporeus <lb/>fuerit in illa perpendiculari:</s> <s xml:id="echoid-s44800" xml:space="preserve"> reflectetur ad uiſum ſecundum hác perpendicularẽ forma ſolius pun-<lb/>cti ſuperficiei uiſus, quod punctum ex illa ſuperficie uiſus ſecat ipſa perpendicularis.</s> <s xml:id="echoid-s44801" xml:space="preserve"> Si communis <lb/>ſectio ſuperficiei reflexionis & ſpeculi fuerit linea longitudinis ſpeculi:</s> <s xml:id="echoid-s44802" xml:space="preserve"> ab uno tantùm puncto ſpe-<lb/> <pb o="373" file="0675" n="675" rhead="LIBER NONVS."/> culi ſit reflexio, ſicut & in alio ſpeculo columnari præ oſtenſum eſt.</s> <s xml:id="echoid-s44803" xml:space="preserve"> Quòd ſi ſectio fuerit oxygonia, <lb/>quandoq;</s> <s xml:id="echoid-s44804" xml:space="preserve"> ab uno puncto:</s> <s xml:id="echoid-s44805" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s44806" xml:space="preserve"> à duobus poteſt fieri reflexio ſecundũ diuerſitatem ſitus pun-<lb/>cti rei uiſæ & centri uiſus:</s> <s xml:id="echoid-s44807" xml:space="preserve"> quoniam punctis c & d exiſtentibus in linea f p, fiet reflexio à puncto h:</s> <s xml:id="echoid-s44808" xml:space="preserve"> <lb/>& ſi puncto c exiſtente in linea f g, punctus d ſit in linea f e:</s> <s xml:id="echoid-s44809" xml:space="preserve"> fiet reflexio ſortè à punctis h & p:</s> <s xml:id="echoid-s44810" xml:space="preserve"> & ſem-<lb/>per locus imaginis eſt centrum uiſus.</s> <s xml:id="echoid-s44811" xml:space="preserve"> Vniuerſaliter enim tam in ſpeculis pyramidalibus quàm co-<lb/>lumnaribus concauis, exiſtẽte axe ſpeculi inter uiſum & ſpeculum, non fiet reflexio per lineam ad <lb/>uiſum perpendicularem, niſi ab uno tantùm puncto ſpeculi, quẽ ſecat illa perpẽdicularis:</s> <s xml:id="echoid-s44812" xml:space="preserve"> & ſolùm <lb/>illius puncti ſuperficiei uiſus, quem ſecatilla perpendicularis ducta à cẽtro uiſus.</s> <s xml:id="echoid-s44813" xml:space="preserve"> Hoc quoq;</s> <s xml:id="echoid-s44814" xml:space="preserve"> quod <lb/>præmiſimus, tunc demum uerùm eſt, ſi linea f h ſuerit perpendicularis ſuper lineam longitudinis <lb/>ſpeculi:</s> <s xml:id="echoid-s44815" xml:space="preserve"> quod eſt poſsibile fieri in ſpeculis pyramidalibus, non autẽ in ſpeculis columnaribus:</s> <s xml:id="echoid-s44816" xml:space="preserve"> quia <lb/>tunc ſemper ſectio eſt obliqua ſuper ſuperficiem ſpeculi:</s> <s xml:id="echoid-s44817" xml:space="preserve"> & ſimiliter eſt de linea f p.</s> <s xml:id="echoid-s44818" xml:space="preserve"> Patet ergo pro-<lb/>poſitum:</s> <s xml:id="echoid-s44819" xml:space="preserve"> quoniam ſectionem pyramidalem poſsibile eſt ſic diſponi, ut linea p h ſit perpendicularis <lb/>ſuper ſpeculi ſuperficiem, & ut ordinetur reflexio ſecun dum illud.</s> <s xml:id="echoid-s44820" xml:space="preserve"/> </p> <div xml:id="echoid-div1738" type="float" level="0" n="0"> <figure xlink:label="fig-0674-01" xlink:href="fig-0674-01a"> <variables xml:id="echoid-variables785" xml:space="preserve">a g c p c f d h d e b</variables> </figure> </div> </div> <div xml:id="echoid-div1740" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables786" xml:space="preserve">a b b</variables> </figure> <head xml:id="echoid-head1292" xml:space="preserve" style="it">12. Centro uiſus exiſtente in centro baſis ſpeculi columnaris cõcaui, <lb/>aut circuli æquidiſtantis baſi: fiet reflexio formæ ipſius oculi ab arcu cir-<lb/> culi ſpeculi ſimili arcui circuli magni, qui eſt in ſuperficie oculi: erit́ locus imaginis centrum uiſus. Alhazen 92 n 5.</head> <p> <s xml:id="echoid-s44821" xml:space="preserve">Sit ſpeculum columnare concauum:</s> <s xml:id="echoid-s44822" xml:space="preserve"> cuius axis ſit a b:</s> <s xml:id="echoid-s44823" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s44824" xml:space="preserve"> centrum ui-<lb/>ſus in puncto b:</s> <s xml:id="echoid-s44825" xml:space="preserve"> quod per 92 th.</s> <s xml:id="echoid-s44826" xml:space="preserve"> 1 huius eſt cẽtrum circuli, qui eſt baſis ſpe-<lb/>culi:</s> <s xml:id="echoid-s44827" xml:space="preserve"> dico quòd forma ipſius oculi uidentis reflectetur ad ipſum uiſum ab <lb/>arcu circuli baſis ſpeculi, ſimili arcui circuli magni, qui eſt totius ſphæræ <lb/>oculi, tranſiens per centrum foraminis uueæ & per centrum oculi:</s> <s xml:id="echoid-s44828" xml:space="preserve"> hoc eſt <lb/>arcui, qui interiacet extremas perpendiculares, quæ à centro uiſus ſecan-<lb/>tes peripheriam foraminis uueæ duci poſſunt ad peripheriam circuli ſpe-<lb/>culi.</s> <s xml:id="echoid-s44829" xml:space="preserve"> Imaginentur enim illæ lineæ à centro oculi per centrum foraminis <lb/>uueæ & per totam peripheriam cuiuſdã arcus circuli magni ſphęræ ipſius <lb/>oculi, ſecantis portionem ſphæræ oculi, cui correſpondet foramen uueæ, <lb/>per æqualia.</s> <s xml:id="echoid-s44830" xml:space="preserve"> Illæ ergo lineæ omnes erunt perpendiculares ſuper ſuperfi-<lb/>ciem ſphæræ oculi per 72 th.</s> <s xml:id="echoid-s44831" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s44832" xml:space="preserve"> quoniam ducuntur à centro:</s> <s xml:id="echoid-s44833" xml:space="preserve"> ſed eæ-<lb/>dem lineæ ad peripheriam circulι baſis ſpeculi productæ ſunt perpendi-<lb/>culares ſuper ſuperficiem ſpeculi per eandem rationem:</s> <s xml:id="echoid-s44834" xml:space="preserve"> quoniam exeunt <lb/>à centro illius circuli, quod eſt b.</s> <s xml:id="echoid-s44835" xml:space="preserve"> Iſtæ ergo lineæ ſunt perpendiculares ſu-<lb/>per utraſq;</s> <s xml:id="echoid-s44836" xml:space="preserve"> iſtas ſuperficies:</s> <s xml:id="echoid-s44837" xml:space="preserve"> ergo per 21 th.</s> <s xml:id="echoid-s44838" xml:space="preserve"> 5 huius ipſæ reflectuntur in ſeipſas.</s> <s xml:id="echoid-s44839" xml:space="preserve"> Formæ ergo puncto-<lb/>rum ſuperficiei oculi in illis perpendicularibus cadentes, reflectũtur ad uiſum per eaſdem.</s> <s xml:id="echoid-s44840" xml:space="preserve"> Et quo-<lb/>niam circulus ſphæræ oculi & circulus baſis ſpeculi (cum idẽ centrum habeant) ſunt circuli æqui-<lb/>diſtantes:</s> <s xml:id="echoid-s44841" xml:space="preserve"> patet per defin itionẽ ſimilium arcuum, quòd arcus quaſq;</s> <s xml:id="echoid-s44842" xml:space="preserve"> duas ipſarum ſemidiametros <lb/>interiacentes ſunt ſimiles.</s> <s xml:id="echoid-s44843" xml:space="preserve"> Arcus itaque circuli ſpeculi, à <lb/> <anchor type="figure" xlink:label="fig-0675-02a" xlink:href="fig-0675-02"/> quo ſit reflexio, eſt ſimilis arcui oculi, qui reflectitur.</s> <s xml:id="echoid-s44844" xml:space="preserve"> Et <lb/>fortè ille arcus hinc inde eſt quadrans circuli:</s> <s xml:id="echoid-s44845" xml:space="preserve"> quia ſicut <lb/>in 4 th.</s> <s xml:id="echoid-s44846" xml:space="preserve"> 3 huius diximus, latus rectum ſubtenſum arcui <lb/>circuli magni, & ſphæræ ipſius oculi tranſeunti per cen-<lb/>trum uueæ & trans totum foramẽ uueæ, eſt quaſi æqua-<lb/>le lateri quadrati inſcriptibilis ipſi ſphæræ oculi:</s> <s xml:id="echoid-s44847" xml:space="preserve"> illi au-<lb/>tẽ correſpondet in centro angulus rectus, & in ſuperficie <lb/>ipſius ſphæræ quadrans circuli per 33 p 6.</s> <s xml:id="echoid-s44848" xml:space="preserve"> Locus autem <lb/>imaginis omnium pũctorum ſuperficiei oculi taliter re-<lb/>flexorum eſt in centro ipſius uiſus, ut patet per præmiſ-<lb/>ſam.</s> <s xml:id="echoid-s44849" xml:space="preserve"> Et quoniã de quocunq;</s> <s xml:id="echoid-s44850" xml:space="preserve"> circulo ſpeculi æquidiſtan-<lb/>te baſi eſt eadem demonſtratio:</s> <s xml:id="echoid-s44851" xml:space="preserve"> patet ergo propoſitum.</s> <s xml:id="echoid-s44852" xml:space="preserve"/> </p> <div xml:id="echoid-div1740" type="float" level="0" n="0"> <figure xlink:label="fig-0675-02" xlink:href="fig-0675-02a"> <variables xml:id="echoid-variables787" xml:space="preserve">d z b t m q l i p h k f g e a</variables> </figure> </div> </div> <div xml:id="echoid-div1742" type="section" level="0" n="0"> <head xml:id="echoid-head1293" xml:space="preserve" style="it">13. In ſpeculis columnaribus concauis ſumptis duo-<lb/>bus punct is in axe ſpeculi: poßibile eſt unum reflecti ad <lb/>alterum à toto uno circulo ſpeculi: locuś imaginis erit <lb/>quidã circulus extra ſuperficiẽ ſpeculi. Alhaz. 94 n 5.</head> <p> <s xml:id="echoid-s44853" xml:space="preserve">Eſto ſpeculum columnare concauum:</s> <s xml:id="echoid-s44854" xml:space="preserve"> cuius axis ſit e <lb/>z, ſintq́;</s> <s xml:id="echoid-s44855" xml:space="preserve"> t & h duo pũcta ſignata in axe:</s> <s xml:id="echoid-s44856" xml:space="preserve"> dico quòd eſt poſ-<lb/>ſibile unum illorum punctorum reflecti ad alterum, ut <lb/>proponitur.</s> <s xml:id="echoid-s44857" xml:space="preserve"> Sint enim circuli a g & b d baſes ſpeculi:</s> <s xml:id="echoid-s44858" xml:space="preserve"> & <lb/>diuidatur linea th per æqualia in puncto q per 10 p 1:</s> <s xml:id="echoid-s44859" xml:space="preserve"> & <lb/>ſuper cẽtrum q deſcribatur circulus in ſuperficie ſpeculi <lb/>æquidiſtás baſibus ſpeculi per 102 th.</s> <s xml:id="echoid-s44860" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s44861" xml:space="preserve"> cuius dia-<lb/>meter ſit linea l q m:</s> <s xml:id="echoid-s44862" xml:space="preserve"> ducantur quoq;</s> <s xml:id="echoid-s44863" xml:space="preserve"> lineæ longitudinis <lb/>ſpeculi per 101 th.</s> <s xml:id="echoid-s44864" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s44865" xml:space="preserve"> quæ ſint b l a, & d m g:</s> <s xml:id="echoid-s44866" xml:space="preserve"> fiat quoq;</s> <s xml:id="echoid-s44867" xml:space="preserve"> circa centrum h circulus:</s> <s xml:id="echoid-s44868" xml:space="preserve"> cuius diameter <lb/> <pb o="374" file="0676" n="676" rhead="VITELLONIS OPTICAE"/> ſit linea k h p:</s> <s xml:id="echoid-s44869" xml:space="preserve"> & ducantur lineæ t l, t m, h l, h m.</s> <s xml:id="echoid-s44870" xml:space="preserve"> Et quia axis ſpeculi, qui eſt e z, per 92 th.</s> <s xml:id="echoid-s44871" xml:space="preserve"> 1 huius ere-<lb/>ctus eſt ſuք ſuperficiẽ circuli l m:</s> <s xml:id="echoid-s44872" xml:space="preserve"> patet quia anguli t q l & t q m & h q l & h q m ſunt recti:</s> <s xml:id="echoid-s44873" xml:space="preserve"> ſed & lιnea <lb/>t q eſt æqualis lineæ q h exhypotheſi:</s> <s xml:id="echoid-s44874" xml:space="preserve"> & lineæ q m & q l ſunt æquales per definitionem circuli:</s> <s xml:id="echoid-s44875" xml:space="preserve"> ergo <lb/>per 4 p 1 trigona quatuor, quæ ſunt t q m & h q m & t q l & h q l ſunt æquiangula:</s> <s xml:id="echoid-s44876" xml:space="preserve"> angulus itaq;</s> <s xml:id="echoid-s44877" xml:space="preserve"> t l q <lb/>eſt æqualis angulo q l h:</s> <s xml:id="echoid-s44878" xml:space="preserve"> & angulus t m q æqualis angulo q m h.</s> <s xml:id="echoid-s44879" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s44880" xml:space="preserve"> centrum uiſus fuerit in pun-<lb/>cto t, & alicuius rei uiſæ punctus fuerith:</s> <s xml:id="echoid-s44881" xml:space="preserve"> reflectetur forma puncti h ad uiſum exiſtẽtem in puncto <lb/>t, à puncto ſpeculi, quod eſt l:</s> <s xml:id="echoid-s44882" xml:space="preserve"> & ſimiliter à puncto m.</s> <s xml:id="echoid-s44883" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s44884" xml:space="preserve"> triangulus t l h, fixo manente latere t h, <lb/>quod eſt pars axis ſpeculi, imaginetur moueri quouſq;</s> <s xml:id="echoid-s44885" xml:space="preserve"> redeat ad locum, unde ſumpſit motus prin-<lb/>cipium:</s> <s xml:id="echoid-s44886" xml:space="preserve"> tunc punctus l motu ſuo deſcribet circulum:</s> <s xml:id="echoid-s44887" xml:space="preserve"> & ſemper duo anguli t l q & q l h manebunt <lb/>æquales:</s> <s xml:id="echoid-s44888" xml:space="preserve"> & ſemper in hoc motu reflectetur ſorma puncti h ad uiſum exiſtentem in puncto t.</s> <s xml:id="echoid-s44889" xml:space="preserve"> Quia <lb/>uerò diameter p h k eſt perpendicularis ſuper ſuperficiem ſpeculi:</s> <s xml:id="echoid-s44890" xml:space="preserve"> palàm quia ipſe eſt cathetus in-<lb/>cidentiæ formæ puncti h.</s> <s xml:id="echoid-s44891" xml:space="preserve"> Producatur itaq;</s> <s xml:id="echoid-s44892" xml:space="preserve"> eadem cathetus p h k ultra pũctum k extra ſuperficiem <lb/>ſpeculi, donec concurrat cum linea reflexionis, quæ t l, producta:</s> <s xml:id="echoid-s44893" xml:space="preserve"> cõcurret autem per 14 th.</s> <s xml:id="echoid-s44894" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s44895" xml:space="preserve"> <lb/>quoniam cũ angulus t h k ſit rectus, angulus h t l eſt acutus:</s> <s xml:id="echoid-s44896" xml:space="preserve"> ſit punctus concurſus f.</s> <s xml:id="echoid-s44897" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s44898" xml:space="preserve"> <lb/>producta catheto h p ultra punctum p:</s> <s xml:id="echoid-s44899" xml:space="preserve"> cõcurret ipſe cum linea reflexionis, quæ eſt t m:</s> <s xml:id="echoid-s44900" xml:space="preserve"> ſit punctus <lb/>concurſus r:</s> <s xml:id="echoid-s44901" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s44902" xml:space="preserve"> per 37 th.</s> <s xml:id="echoid-s44903" xml:space="preserve"> 5 huius puncta f & r loca imaginũ ſormæ puncti h:</s> <s xml:id="echoid-s44904" xml:space="preserve"> motoq́;</s> <s xml:id="echoid-s44905" xml:space="preserve"> triangulo <lb/>t l h, mouebitur ſimul cum illo triangulus t f h:</s> <s xml:id="echoid-s44906" xml:space="preserve"> & in hoc motu punctus f deſcribet circulum extra <lb/>columnam ſpeculi:</s> <s xml:id="echoid-s44907" xml:space="preserve"> totusq́;</s> <s xml:id="echoid-s44908" xml:space="preserve"> ille circulus erit locus imaginis.</s> <s xml:id="echoid-s44909" xml:space="preserve"> Et idem erit probandi modus ſumptis <lb/>quibuſcunq;</s> <s xml:id="echoid-s44910" xml:space="preserve"> duobus pũctis in axe ſpeculi.</s> <s xml:id="echoid-s44911" xml:space="preserve"> Oportebit taméhoc modo uiſum taliter ſiſti, ut centrũ <lb/>eius ſit directè in axe ſpeculi, & punctus rei uiſæ ſit in aliquo cẽtro circuli ſpeculi, aut circuli baſis, <lb/>aut æquidiſtantis ei:</s> <s xml:id="echoid-s44912" xml:space="preserve"> aliàs enim locus imaginis nó occurret uiſui extra ſpeculũ.</s> <s xml:id="echoid-s44913" xml:space="preserve"> Patet ergo ꝓpoſitũ.</s> <s xml:id="echoid-s44914" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1743" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables788" xml:space="preserve">a d e g b h e f</variables> </figure> <head xml:id="echoid-head1294" xml:space="preserve" style="it">14. Communi ſectione ſuperficiei reflexionis & ſpeculi columnaris concaui exiſtẽte circulo: <lb/>quando unum: quando duo: quando tria: quando quatuor <lb/>erunt puncta reflexionis & non plura: & ſecũdum hæc loca ima-<lb/> ginum numer antur. Alhazen 95 n 5.</head> <p> <s xml:id="echoid-s44915" xml:space="preserve">Eſto ſpeculum columnare concauum, cuius axis a b:</s> <s xml:id="echoid-s44916" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s44917" xml:space="preserve"> com-<lb/>munis ſectio ſuperficiei reflexionis & ſpeculi circulus, quic d e f:</s> <s xml:id="echoid-s44918" xml:space="preserve"> cu-<lb/>ius centrum ſit b:</s> <s xml:id="echoid-s44919" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s44920" xml:space="preserve"> centrum uiſus g:</s> <s xml:id="echoid-s44921" xml:space="preserve"> & punctũ rei uiſæ h:</s> <s xml:id="echoid-s44922" xml:space="preserve"> quæ ſint <lb/>intra illum circulum æqualiter uel inæqualiter diſtantia à centro b:</s> <s xml:id="echoid-s44923" xml:space="preserve"> <lb/>ſintq́;</s> <s xml:id="echoid-s44924" xml:space="preserve"> ambo ab una parte centri b.</s> <s xml:id="echoid-s44925" xml:space="preserve"> Dico quòd uerum, quod propo-<lb/>nitur.</s> <s xml:id="echoid-s44926" xml:space="preserve"> Ducantur enim diametri g b & h b:</s> <s xml:id="echoid-s44927" xml:space="preserve"> quæ producantur ad peri <lb/>pheriam circuli:</s> <s xml:id="echoid-s44928" xml:space="preserve"> patetq́;</s> <s xml:id="echoid-s44929" xml:space="preserve"> per 40 th.</s> <s xml:id="echoid-s44930" xml:space="preserve"> 8 huius quoniá poſsibile eſt quá-<lb/>doq;</s> <s xml:id="echoid-s44931" xml:space="preserve"> formam puncti h reflecti ad uiſum exiſtentem in puncto g ab <lb/>uno tantùm puncto circuli c d e f:</s> <s xml:id="echoid-s44932" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s44933" xml:space="preserve"> à duobus:</s> <s xml:id="echoid-s44934" xml:space="preserve"> quandoque <lb/>uerò a tribus:</s> <s xml:id="echoid-s44935" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s44936" xml:space="preserve"> uerò à quatuor:</s> <s xml:id="echoid-s44937" xml:space="preserve"> non autem à pluribus.</s> <s xml:id="echoid-s44938" xml:space="preserve"> Et <lb/>quoniam in propoſito, cum reflexio fiat à circulo ſpeculi, nõ eſt ali-<lb/>qua differentia quo ad illud:</s> <s xml:id="echoid-s44939" xml:space="preserve"> patet ergo primum propoſitum.</s> <s xml:id="echoid-s44940" xml:space="preserve"> Patet <lb/>etiã, prout oſtenſum eſt in 11 th.</s> <s xml:id="echoid-s44941" xml:space="preserve"> 8 huius, ſiue catheti incidentiæ con-<lb/>currant cum lineis reflexionis ſiue æquidiſtent, quòd ſecũdum nu-<lb/>merum linearum reflexionis imagines numerantur.</s> <s xml:id="echoid-s44942" xml:space="preserve"> Et hoc eſt to-<lb/>tum, quod proponebatur.</s> <s xml:id="echoid-s44943" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1744" type="section" level="0" n="0"> <head xml:id="echoid-head1295" xml:space="preserve" style="it">15. In columnaribus cõcauis ſpeculis communi ſectione ſuperficiei reflexionis & ſpeculi exi-<lb/>ſtente oxygònia: formarum punctorum rei uiſœ quarundam fit ab uno tantùm puncto ſpeculi <lb/>reflexio ad uiſum: quarundam à duobus: quarundam à tribus: quarundam à quatuor: non au-<lb/>tem à pluribus: & ſecundum hœc loca imaginum numer antur. Alhazen 95 n 5.</head> <p> <s xml:id="echoid-s44944" xml:space="preserve">Eſto ſpeculum columnare concauum:</s> <s xml:id="echoid-s44945" xml:space="preserve"> cuius axis ſit linea x h:</s> <s xml:id="echoid-s44946" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s44947" xml:space="preserve"> punctus rei uiſæ obliquè in-<lb/>cidens ſpeculo, ita quòd non ſit in aliqua linearum perpẽdicularium ſuper ſuperficiem ſpeculi:</s> <s xml:id="echoid-s44948" xml:space="preserve"> qui <lb/>ſit punctus a:</s> <s xml:id="echoid-s44949" xml:space="preserve"> taliter ut communis ſectio ſuperficiei reflexionis & ſpeculi ſit ſectio oxygonia.</s> <s xml:id="echoid-s44950" xml:space="preserve"> Dico <lb/>quòd poſsibile eſt, ut ab uno puncto, uel à duobus, uel à tribus, uel à quatuor punctis alicuius oxy-<lb/>goniæ ſectionis fiat reflexio ad uiſum:</s> <s xml:id="echoid-s44951" xml:space="preserve"> & quandoq;</s> <s xml:id="echoid-s44952" xml:space="preserve"> unica appareat imago, quãdoq;</s> <s xml:id="echoid-s44953" xml:space="preserve"> duæ, quandoq;</s> <s xml:id="echoid-s44954" xml:space="preserve"> <lb/>tres, quandoq;</s> <s xml:id="echoid-s44955" xml:space="preserve"> quatuor & non plures imagines:</s> <s xml:id="echoid-s44956" xml:space="preserve"> quoniam totidem ſunt puncta reflexionis tantùm <lb/>poſsibilia.</s> <s xml:id="echoid-s44957" xml:space="preserve"> Imaginetur itaq;</s> <s xml:id="echoid-s44958" xml:space="preserve"> ſuperficies plana tranſiens per punctum a æquidiſtans baſibus ſpeculi <lb/>propoſiti:</s> <s xml:id="echoid-s44959" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s44960" xml:space="preserve"> cómunis ſectio huius ſuperficiei & ſuperficiei ſpeculi circulus per 100 th.</s> <s xml:id="echoid-s44961" xml:space="preserve"> 1 huius, <lb/>cuius circuli centrum ſit h:</s> <s xml:id="echoid-s44962" xml:space="preserve"> ſumaturq́;</s> <s xml:id="echoid-s44963" xml:space="preserve"> in ſuperficie illius circuli aliud punctum, quod ſit b, inæqua-<lb/>liter diſtans à centro h cum puncto a:</s> <s xml:id="echoid-s44964" xml:space="preserve"> & ducantur à punctis a & b ad centrum circuli h lineæ a h & <lb/>b h:</s> <s xml:id="echoid-s44965" xml:space="preserve"> & compleantur diametri illius circuli eiſdẽ lineis ad peripheriam circuli hinc inde productis.</s> <s xml:id="echoid-s44966" xml:space="preserve"> <lb/>Palàm ergo per ea, quæ dicta ſunt in theoremate pręcedente, & in 40 th.</s> <s xml:id="echoid-s44967" xml:space="preserve"> 8 huius, quòd ab uno pun-<lb/>cto arcus interiacentis duas ſemidiametros a h & b h poteſt forma puncti a reflecti ad uiſum exi-<lb/>ſtentem in puncto b:</s> <s xml:id="echoid-s44968" xml:space="preserve"> uel forſitan à duobus uel à tribus:</s> <s xml:id="echoid-s44969" xml:space="preserve"> ſed nõ à pluribus:</s> <s xml:id="echoid-s44970" xml:space="preserve"> ab arcu uerò oppoſito iſti <lb/>arcui (utpote ab illo arcu, qui cadit inter eaſdem ſemidiametros productas ad aliam partem peri-<lb/>pheriæ circuli) non poteſt fieri reflexio formæ pũcti a ad uiſum b, niſi ab uno tantùm puncto.</s> <s xml:id="echoid-s44971" xml:space="preserve"> Eſto <lb/>itaq;</s> <s xml:id="echoid-s44972" xml:space="preserve"> quòd forma pũcti a reſlectatur ad uiſum b à tribus pũctis ſpeculi propoſiti arcus, ſcilicet unius <lb/> <pb o="375" file="0677" n="677" rhead="LIBER NONVS."/> inter iacentis ſemidiametros a h & b h:</s> <s xml:id="echoid-s44973" xml:space="preserve"> quæ ſint puncta g, d, e:</s> <s xml:id="echoid-s44974" xml:space="preserve"> & ducantur lineæ a g, h g, b g, a d, h d, <lb/>b d, a e, h e, b e:</s> <s xml:id="echoid-s44975" xml:space="preserve"> & à puncto a rei uiſæ ducantur in eadem ſuperficie tres lineæ æquidiſtantes tribus <lb/>ſemidiametris, quæ ſunt h g, h d, h e:</s> <s xml:id="echoid-s44976" xml:space="preserve"> quæ lineæ æquidiſtantes ſint a k, a f, an:</s> <s xml:id="echoid-s44977" xml:space="preserve"> ita quòd linea a k ſit <lb/>æquidiſtás ſemidiametro h g:</s> <s xml:id="echoid-s44978" xml:space="preserve"> & linea a f ſemidiametro h d:</s> <s xml:id="echoid-s44979" xml:space="preserve"> & linea a n ſemidiametro h e.</s> <s xml:id="echoid-s44980" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s44981" xml:space="preserve"> <lb/>linea a k ſit æquidiſtãs ſemidiametro h g, & linea <lb/> <anchor type="figure" xlink:label="fig-0677-01a" xlink:href="fig-0677-01"/> b g concurrat cum eadem ſemidi a metro in pũcto <lb/>g:</s> <s xml:id="echoid-s44982" xml:space="preserve"> palàm per 2 th.</s> <s xml:id="echoid-s44983" xml:space="preserve"> 1 huius quoniá linea b g concur-<lb/>ret cum linea a k:</s> <s xml:id="echoid-s44984" xml:space="preserve"> ſit ergo punctus cócurſus k.</s> <s xml:id="echoid-s44985" xml:space="preserve"> Si-<lb/>militer quoq;</s> <s xml:id="echoid-s44986" xml:space="preserve"> per eandem ration em linea b d con <lb/>curret cum linea a f:</s> <s xml:id="echoid-s44987" xml:space="preserve"> ſit concurſus punctus f:</s> <s xml:id="echoid-s44988" xml:space="preserve"> ſimi-<lb/>liter quoque linea b e concurret cum linea a n:</s> <s xml:id="echoid-s44989" xml:space="preserve"> ſit <lb/>punctus cõcurſus n.</s> <s xml:id="echoid-s44990" xml:space="preserve"> Deinde à puncto b erigatur <lb/>perpendicularis ſuper ſuperficiem circuli, cuius <lb/>centrum h, per 12 p 11:</s> <s xml:id="echoid-s44991" xml:space="preserve"> quæſit b t:</s> <s xml:id="echoid-s44992" xml:space="preserve"> & quoniam axis <lb/>x h eſt perpẽdicularis ſuper ſuperficiem illius cir-<lb/>culi:</s> <s xml:id="echoid-s44993" xml:space="preserve"> erit per 6 p 11 linea b t æquidiſtãs axi x h.</s> <s xml:id="echoid-s44994" xml:space="preserve"> Su-<lb/>matur quoq;</s> <s xml:id="echoid-s44995" xml:space="preserve"> in linea b t punctũ quodcũq;</s> <s xml:id="echoid-s44996" xml:space="preserve">, quod <lb/>ſit t:</s> <s xml:id="echoid-s44997" xml:space="preserve"> & ab illo ducantur tres lineę ad tria puncta k, <lb/>f, n, quæ ſint lineæ t k, t f, t n:</s> <s xml:id="echoid-s44998" xml:space="preserve"> & à tribus punctis g, <lb/>d, e erigantur per 12 p 11 tres perpẽdiculares ſuper <lb/>ſuperficiem circuli, cuius cẽtrum h:</s> <s xml:id="echoid-s44999" xml:space="preserve"> quæ ſint g m, <lb/>d l, e q:</s> <s xml:id="echoid-s45000" xml:space="preserve"> erunt ergo per 6 p 11 lineæ b t & e q æ qui-<lb/>diſtantes.</s> <s xml:id="echoid-s45001" xml:space="preserve"> Et quoniam, ut patet per 1 th.</s> <s xml:id="echoid-s45002" xml:space="preserve"> 1 huius, o-<lb/>mnes lineę æquidiſtantes ſunt in eadẽ ſuperficie:</s> <s xml:id="echoid-s45003" xml:space="preserve"> <lb/>palàm per 1 p 11 quoniã lineæ b t & e q ſunt in ſu-<lb/>perficie trianguli b t n:</s> <s xml:id="echoid-s45004" xml:space="preserve"> igitur linea e q ſecabit li-<lb/>neam t n:</s> <s xml:id="echoid-s45005" xml:space="preserve"> ſit ut ſecet ipſam in puncto q:</s> <s xml:id="echoid-s45006" xml:space="preserve"> & penitus <lb/>per eundem modum ſit, ut linea d l ſecet lineam t <lb/>fin puncto l:</s> <s xml:id="echoid-s45007" xml:space="preserve"> & linea g m ſecet lineam t k in pũcto <lb/>m:</s> <s xml:id="echoid-s45008" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s45009" xml:space="preserve"> per 92th.</s> <s xml:id="echoid-s45010" xml:space="preserve"> 1 huius hæ tres lineæ ſcilicet <lb/>e q & d l & g m partes linearum longitudinis ſpe-<lb/>culi:</s> <s xml:id="echoid-s45011" xml:space="preserve"> cum ſint in ſuperficie columnæ ſpeculi per-<lb/>pendiculariter productæ ſuper ſuper ficiẽ circuli, <lb/>cuius centrum h:</s> <s xml:id="echoid-s45012" xml:space="preserve"> & per conſequẽs ſint erectæ ſu-<lb/>per baſes ſpeculi per 23 th.</s> <s xml:id="echoid-s45013" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s45014" xml:space="preserve"> Et à puncto q <lb/>ducatur per 31 p 1 linea æ quidiſtans lineæ n a:</s> <s xml:id="echoid-s45015" xml:space="preserve"> quæ <lb/>ſit linea q u:</s> <s xml:id="echoid-s45016" xml:space="preserve"> hęc itaq;</s> <s xml:id="echoid-s45017" xml:space="preserve"> per 30 p 1 erit æquidiſtans li-<lb/>neæ h e:</s> <s xml:id="echoid-s45018" xml:space="preserve"> quoniam ipſa h e æquidiſtat lineæ a n, ut patet ex præmiſsis.</s> <s xml:id="echoid-s45019" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s45020" xml:space="preserve"> axis x h concurrit <lb/>cum linea h e in puncto h:</s> <s xml:id="echoid-s45021" xml:space="preserve"> palàm per 2 th.</s> <s xml:id="echoid-s45022" xml:space="preserve"> 1 huius quoniam ipſe axis cõcurret cum eius æquidiſtan-<lb/>te ducta à puncto q:</s> <s xml:id="echoid-s45023" xml:space="preserve"> ſit cõcurſus in puncto u:</s> <s xml:id="echoid-s45024" xml:space="preserve"> & ſit illa æquidiſtans linea q u:</s> <s xml:id="echoid-s45025" xml:space="preserve"> & ducatur linea t a:</s> <s xml:id="echoid-s45026" xml:space="preserve"> hæc <lb/>itaq;</s> <s xml:id="echoid-s45027" xml:space="preserve"> ſecabit lineam q u:</s> <s xml:id="echoid-s45028" xml:space="preserve"> quoniã linea q u ducitur à latere trianguli t b n, & à termino lineę e q æqui-<lb/>diſtantis baſi t b, & omnes illæ lineæ ſunt in eadem ſuperficie, lineaq́;</s> <s xml:id="echoid-s45029" xml:space="preserve"> t a producta eſt inter lineam <lb/>t u æquidiſtantem axi h u, & inter ipſum axem:</s> <s xml:id="echoid-s45030" xml:space="preserve"> patet quòd linea t a ſecabit lineã q u:</s> <s xml:id="echoid-s45031" xml:space="preserve"> ſunt enim am-<lb/>bæ in eadem ſuperficie:</s> <s xml:id="echoid-s45032" xml:space="preserve"> ſit itaq;</s> <s xml:id="echoid-s45033" xml:space="preserve"> linearum t a & q u pũctus ſectionis i:</s> <s xml:id="echoid-s45034" xml:space="preserve"> & ducatur linea q a.</s> <s xml:id="echoid-s45035" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s45036" xml:space="preserve"> <lb/>lineæ h e & a n ſunt æquidiſtátes, ut ſuprà patuit:</s> <s xml:id="echoid-s45037" xml:space="preserve"> palàm per 29 p 1 quia angulus b e h extrinſecus eſt <lb/>æqualis angulo e n a intrinſeco, & anguli h e a & e a n ſunt æquales, quia coalterni:</s> <s xml:id="echoid-s45038" xml:space="preserve"> ſed & angulus <lb/>reflexionis, qui eſt h e b, eſt æqualis angulo incidentiæ, qui eſt a e h, per 20 th.</s> <s xml:id="echoid-s45039" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s45040" xml:space="preserve"> Erit ergo an-<lb/>gulus e a n æqualis angulo a n e:</s> <s xml:id="echoid-s45041" xml:space="preserve"> ergo per 6 p 1 in trigono e a n duo latera e a & e n ſunt æqualia:</s> <s xml:id="echoid-s45042" xml:space="preserve"> ſed <lb/>linea e q eſt perpendicularis ſuper ſuperficiem trigoni a e n:</s> <s xml:id="echoid-s45043" xml:space="preserve"> quia & ſuper ſuperficiem circuli, cuius <lb/>cẽtrum eſt h, eſt erecta, ut ſuprà patuit.</s> <s xml:id="echoid-s45044" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s45045" xml:space="preserve"> linea q e ſit communis duobus trigonis q e a & q <lb/>e n:</s> <s xml:id="echoid-s45046" xml:space="preserve"> patet per 4 p 1 quoniam illa trigona ſunt æqualia:</s> <s xml:id="echoid-s45047" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s45048" xml:space="preserve"> linea q n æqualis lineę q a:</s> <s xml:id="echoid-s45049" xml:space="preserve"> ergo per 5 p 1 <lb/>quia trigoni q a n duo latera q a & q n ſunt æqualia, erit angulus q a n æqualis angulo q n a.</s> <s xml:id="echoid-s45050" xml:space="preserve"> Quia <lb/>itaq;</s> <s xml:id="echoid-s45051" xml:space="preserve"> linea q i æquidiſtat lineæ a n:</s> <s xml:id="echoid-s45052" xml:space="preserve"> patet per 29 p 1 quoniam angulus t q i extrinſecus ęqualis eſt an-<lb/>gulo t n a intrinſeco:</s> <s xml:id="echoid-s45053" xml:space="preserve"> & angulus i q a æqualis eſt angulo q a n, quia ſunt coalterni:</s> <s xml:id="echoid-s45054" xml:space="preserve"> erit ergo angulus <lb/>i q t æqualis angulo i q a.</s> <s xml:id="echoid-s45055" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s45056" xml:space="preserve"> puncti a per 20 th.</s> <s xml:id="echoid-s45057" xml:space="preserve"> 5 huius reflectetur ad uiſum exiſtentem in <lb/>puncto t à puncto ſpeculi, quod eſt q.</s> <s xml:id="echoid-s45058" xml:space="preserve"> Et eodem modo demonſtrandum quòd forma puncti a refle-<lb/>ctitur ad uiſum exiſtentem in puncto t ab alijs duobus punctis ſpeculi ſimilibus puncto q, quę ſunt <lb/>puncta l & m.</s> <s xml:id="echoid-s45059" xml:space="preserve"> Sic ergo formæ puncti a ad uiſum in punctum t fiet reflexio à tribus punctis ſpeculi <lb/>columnaris concaui, quę ſunt q, l, m, & ex eadem parte colũnæ ſpeculi:</s> <s xml:id="echoid-s45060" xml:space="preserve"> nec eſt poſsibile, ut fiat eiuſ-<lb/>modi reflexio à pluribus punctis ſpeculi exilla parte.</s> <s xml:id="echoid-s45061" xml:space="preserve"> Si enim detur quodcunq;</s> <s xml:id="echoid-s45062" xml:space="preserve"> pũctum ſuperficiei <lb/>ſpeculi columnaris concaui aliud ab iſtis tribus, à quo dicatur poſſe fieri reflexio formę puncti a ad <lb/>uiſum in punctum t:</s> <s xml:id="echoid-s45063" xml:space="preserve"> ducatur ab illo puncto dato linea longitudinis ſpeculi ſuper circulum, cuius <lb/>centrum h:</s> <s xml:id="echoid-s45064" xml:space="preserve"> & oſten detur modo præmiſſo, quòd à puncto peripheriæ illius circuli, cui incidit illa li-<lb/>nea longitudinis, poteſt forma puncti a reflecti ad uiſum exiſtentẽ in pũcto b:</s> <s xml:id="echoid-s45065" xml:space="preserve"> & ſic à quatuor pun-<lb/> <pb o="376" file="0678" n="678" rhead="VITELLONIS OPTICAE"/> ctis arcus interiacentis diametros circuli, in quibus ſunt centrum uiſus & punctum rei uiſæ, fiet re-<lb/>flexio ad uiſum, ſcilicet à tribus punctis g, d, e, & à quarto dato:</s> <s xml:id="echoid-s45066" xml:space="preserve"> quod eſt contra 40 th.</s> <s xml:id="echoid-s45067" xml:space="preserve"> 8 huius, & im-<lb/>poſsibile.</s> <s xml:id="echoid-s45068" xml:space="preserve"> Non ergo ſiet reflexio formæ puncti a ad uiſum exiſtentẽ in puncto t, niſi à tribus punctis <lb/>ſpeculi columnaris concaui:</s> <s xml:id="echoid-s45069" xml:space="preserve"> quę ſunt, q, l, m ex una parte ipſius ſpeculi.</s> <s xml:id="echoid-s45070" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s45071" xml:space="preserve"> alia pars columnaris <lb/>ſpeculi abſciſſa fuerit:</s> <s xml:id="echoid-s45072" xml:space="preserve"> patet quòd tantùm fiet reflexio à tribus pũctis ſpeculi:</s> <s xml:id="echoid-s45073" xml:space="preserve"> quò d ſi totum ſpecu-<lb/>lum integrũ fuerit, poſsibile eſt fieri reflexionẽ à pũctis quatuor.</s> <s xml:id="echoid-s45074" xml:space="preserve"> Iam enim patuit per 27 th.</s> <s xml:id="echoid-s45075" xml:space="preserve"> 8 huius, <lb/>quòd ex arcu circuli, cuius centrũ h, oppoſito arcui g d e, poteſt forma pũcti a reflecti ad uiſum exi-<lb/>ftentem in puncto b ab uno tantùm puncto.</s> <s xml:id="echoid-s45076" xml:space="preserve"> Sit ergo illud punctũ z:</s> <s xml:id="echoid-s45077" xml:space="preserve"> & ducatur ſemidiameter h z:</s> <s xml:id="echoid-s45078" xml:space="preserve"> & <lb/>à puncto a per 31 p 1 ducatur linea ei æ quidiſtans:</s> <s xml:id="echoid-s45079" xml:space="preserve"> quæ ſit a s:</s> <s xml:id="echoid-s45080" xml:space="preserve"> & ducatur linea reflexionis, quæ ſit b z <lb/>concurrens cum linea a s in puncto s:</s> <s xml:id="echoid-s45081" xml:space="preserve"> concurret autẽ per 2 th.</s> <s xml:id="echoid-s45082" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s45083" xml:space="preserve"> quoniã concurrit cum linea h <lb/>z æ quidiſtante ipſi a s:</s> <s xml:id="echoid-s45084" xml:space="preserve"> & à puncto z erigatur ſuper ſuperficiẽ circuli, cuius centrũ h, linea z o perpẽ-<lb/>diculariter per 12 p 11:</s> <s xml:id="echoid-s45085" xml:space="preserve"> hæc ergo per 6 p 11 æ quidiſtabit lineæ b t.</s> <s xml:id="echoid-s45086" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s45087" xml:space="preserve"> linea t s, quæ, ſicut <lb/>prius in alijs declarauimus, ſeca bit lineã z o:</s> <s xml:id="echoid-s45088" xml:space="preserve"> quoniá ſunt in eadẽ ſuperficie:</s> <s xml:id="echoid-s45089" xml:space="preserve"> ſit ergo punctus ſectio-<lb/>nis o:</s> <s xml:id="echoid-s45090" xml:space="preserve"> patebitq́ ſecundũ pręmiſſo prius modos, quoniã forma punctis s reflectitur ad uiſum exiſtẽ-<lb/>tem in puncto t à puncto ſpeculi, quod eſt o:</s> <s xml:id="echoid-s45091" xml:space="preserve"> nec erit poſsibilis reflexio ab aliquo puncto ſuperficiei <lb/>ſpeculi ex illa parte, præter quàm à puncto o.</s> <s xml:id="echoid-s45092" xml:space="preserve"> Si enim detur, quòd ab aliquo alio pũcto hoc ſit poſ-<lb/>fibile:</s> <s xml:id="echoid-s45093" xml:space="preserve"> ſequetur, ut prius deduximus, quòd ſimiliter ab alio puncto illius arcus circuli, cuius centrũ <lb/>h, quàm à puncto z, poſsit forma puncti a reflecti ad uiſum exiſtentẽ in puncto b, quod eſt impoſsi-<lb/>bile, & contra 29 th.</s> <s xml:id="echoid-s45094" xml:space="preserve"> 8 huius.</s> <s xml:id="echoid-s45095" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s45096" xml:space="preserve"> forma puncti a ab uno pũcto circuli, cuius centrũ h, reflectitur <lb/>ad uiſum exiſtentem in puncto b:</s> <s xml:id="echoid-s45097" xml:space="preserve"> reflectetur eadẽ forma puncti a ex eadẽ parte ſpeculi columnaris <lb/>concaui ad uiſum exiſtentẽ in puncto t ab uno tantùm ſpeculi puncto:</s> <s xml:id="echoid-s45098" xml:space="preserve"> & ſi à duobus pũctis ſpeculi <lb/>fiat reflexio formæ puncti a ad b:</s> <s xml:id="echoid-s45099" xml:space="preserve"> & à duobus punctis ſpeculi reflectetur a ad t.</s> <s xml:id="echoid-s45100" xml:space="preserve"> Si uerò una harum <lb/>reflexionum à tribus fiat punctis:</s> <s xml:id="echoid-s45101" xml:space="preserve"> ſiet etiá refliqua à tribus:</s> <s xml:id="echoid-s45102" xml:space="preserve"> & ab illa parte circuli uel ſpeculi non eſt <lb/>poſsibile fieri plures reflexiones.</s> <s xml:id="echoid-s45103" xml:space="preserve"> Sicut autẽ ab uno tantùm puncto arcus oppoſiti in circulo fit re-<lb/>flexio formæ puncti a ad punctum b:</s> <s xml:id="echoid-s45104" xml:space="preserve"> ſic etiá ex illa parte ſpeculi ab uno tantum puncto fit reflexio <lb/>formę puncti a ad uiſum exiſtentẽ in puncto t.</s> <s xml:id="echoid-s45105" xml:space="preserve"> Item linea t b æ quidiſtat axi x h:</s> <s xml:id="echoid-s45106" xml:space="preserve"> ſunt ergo in eadẽ ſu-<lb/>perficie per 1 th.</s> <s xml:id="echoid-s45107" xml:space="preserve"> 1 huius, quę eſt ſuperficies t b h u:</s> <s xml:id="echoid-s45108" xml:space="preserve"> nec enim poteſt alia ſumi plana ſuperficies, in qua <lb/>fint illæ lineæ t b & h x per 1 p 11.</s> <s xml:id="echoid-s45109" xml:space="preserve"> Itẽ n ec poteſt ſumi aliqua plana ſuperſicies, in qua ſit punctus a, & <lb/>axis x h, pręter ſuperficiẽ a u h, quę per 18 p 11 eſt electa perpẽdiculariter ſuper ſuper ficiẽ circuli, cu-<lb/>ius centrũ eſt punctũ h:</s> <s xml:id="echoid-s45110" xml:space="preserve"> cum per 92 th.</s> <s xml:id="echoid-s45111" xml:space="preserve"> 1 huius axis h u ſit perpẽ dicularis ſuperipſam.</s> <s xml:id="echoid-s45112" xml:space="preserve"> Punctus ergo <lb/>tnó eſt in eadẽ ſuperficie cũ pũcto a erecta ſuper ſuperficiẽ dicti circuli:</s> <s xml:id="echoid-s45113" xml:space="preserve"> ſed neq;</s> <s xml:id="echoid-s45114" xml:space="preserve"> illa pũcta t & a ſunt <lb/>in eodẽ circulo:</s> <s xml:id="echoid-s45115" xml:space="preserve"> ſed neq;</s> <s xml:id="echoid-s45116" xml:space="preserve"> ſunt in axe ſpeculi:</s> <s xml:id="echoid-s45117" xml:space="preserve"> quoniá linea b t eſt æ quidiſtans axi ſpeculi, qui eſt x h.</s> <s xml:id="echoid-s45118" xml:space="preserve"> <lb/>Superficies ergo, in qua forma pũcti a refle ctitur ad uiſum exiſtentẽ in pũcto t, eſt oxygonia ſectio.</s> <s xml:id="echoid-s45119" xml:space="preserve"> <lb/>Verùm ꝓducta linea t a ex utraq;</s> <s xml:id="echoid-s45120" xml:space="preserve"> parte ultra pũcta t & a, ut fiat linea p r:</s> <s xml:id="echoid-s45121" xml:space="preserve"> cũ quatuor ſint ſuperficies <lb/>reflexionis:</s> <s xml:id="echoid-s45122" xml:space="preserve"> quia à quatuor pũctis fit reflexio, qu æ ſunt q, l, m, o, & in qualibetillarũ quatuor ſuper-<lb/>ficierum neceſſe eſt eſſe duo puncta, quæ ſunt a & t:</s> <s xml:id="echoid-s45123" xml:space="preserve"> patet quòd linea p reſt communis illis quatuor <lb/>ſuperficiebus per 1 p 11:</s> <s xml:id="echoid-s45124" xml:space="preserve"> quoniam in linea p r ſunt cẽtrum uiſus, quod eſt punctum t:</s> <s xml:id="echoid-s45125" xml:space="preserve"> & punctum rei <lb/>uiſæ, quod eſt punctum a:</s> <s xml:id="echoid-s45126" xml:space="preserve"> quæ neceſſe eſt eſſe in omni ſuperficie refle xionis ſactæ ab his ſpeculis, ut <lb/>patet per 3 huius.</s> <s xml:id="echoid-s45127" xml:space="preserve"> Quælibet autem illarum ſupetſicierum ſecat ſpeculum ſuper ſuperficiem contin-<lb/>gentem ſpeculum in puncto ſuæ reflexionis:</s> <s xml:id="echoid-s45128" xml:space="preserve"> & cuilibet iſtarum ſuperficierum reflexionis & ſuper-<lb/>ficiei in illo puncto ſpeculum contingentis communis ſectio eſt linea recta per 3 p 11.</s> <s xml:id="echoid-s45129" xml:space="preserve"> Et ſicut pũcta <lb/>reflexionis non ſunt eadem:</s> <s xml:id="echoid-s45130" xml:space="preserve"> ſic neq;</s> <s xml:id="echoid-s45131" xml:space="preserve"> lineæ communes illarum ſectionum ſunt eæ dem:</s> <s xml:id="echoid-s45132" xml:space="preserve"> linea itaq;</s> <s xml:id="echoid-s45133" xml:space="preserve"> p <lb/>reſt perpendicularis ſuper unam tantùm illarum quatuor communium linearum, non ſuper duas.</s> <s xml:id="echoid-s45134" xml:space="preserve"> <lb/>Quoniam ſi eſſet perpendicularis ſuper duas illarum linearum:</s> <s xml:id="echoid-s45135" xml:space="preserve"> eſſet perpẽdicularis ſuper duas ſu-<lb/>perſicies ſpeculum ſecun dum puncta illarum linearum cõtingentes:</s> <s xml:id="echoid-s45136" xml:space="preserve"> linea itaq;</s> <s xml:id="echoid-s45137" xml:space="preserve"> prneceſſariò tran-<lb/>ſiret axem:</s> <s xml:id="echoid-s45138" xml:space="preserve"> cum tam en oſten ſum ſit prius, quòd linea t a, (quæ eſt pars lineæ t p r) cadat citra axem <lb/>ſpeculi, qui eſt x h.</s> <s xml:id="echoid-s45139" xml:space="preserve"> Neceſſariò ergo oportet duci quatuor diuerſas lineas perpendiculares ad illas <lb/>quatuor lineas communes à puncto rei uiſæ, quod eſt a:</s> <s xml:id="echoid-s45140" xml:space="preserve"> quæ erunt quatuor catheti incidentiæ per-<lb/>pendiculares ſuper oxygonias ſectiones, cõmunes illis ſuperſiciebus reflexionũ & ſpeculi.</s> <s xml:id="echoid-s45141" xml:space="preserve"> Quæ-<lb/>libet itaq;</s> <s xml:id="echoid-s45142" xml:space="preserve"> iſtanrũ perpendiculariũ aut erit æ quidiſtans lineæ refle xionis:</s> <s xml:id="echoid-s45143" xml:space="preserve"> aut cõcurret cũ illa ſiue in-<lb/>tra ſpeculũ ſiue extra.</s> <s xml:id="echoid-s45144" xml:space="preserve"> Si fuerit æ quidiſtans:</s> <s xml:id="echoid-s45145" xml:space="preserve"> erit locus imaginis ipſe pũctus reflexionis, ut ſuperà pa-<lb/>tuit in 11 huius.</s> <s xml:id="echoid-s45146" xml:space="preserve"> Et cũ quatuor ſint huiuſmodi քpẽdiculares:</s> <s xml:id="echoid-s45147" xml:space="preserve"> erũt quatuor loca imaginũ, & quatuor <lb/>imagines:</s> <s xml:id="echoid-s45148" xml:space="preserve"> ideo quòd quatuor ſunt loca reflexionũ.</s> <s xml:id="echoid-s45149" xml:space="preserve"> Si uerò oẽs illę quatuor perpẽdiculares cõcur-<lb/>runt cũ lin eis ſuarũ reflexionũ:</s> <s xml:id="echoid-s45150" xml:space="preserve"> erũt itẽ quatuor imagines:</s> <s xml:id="echoid-s45151" xml:space="preserve"> quia quatuor ſunt cócurſus illarũ linea-<lb/>rum.</s> <s xml:id="echoid-s45152" xml:space="preserve"> Sic ergo loca imaginũ num erátur ſecundũ num erũ punctorũ reflexionis.</s> <s xml:id="echoid-s45153" xml:space="preserve"> Et hoc eſt ꝓpoſitũ.</s> <s xml:id="echoid-s45154" xml:space="preserve"/> </p> <div xml:id="echoid-div1744" type="float" level="0" n="0"> <figure xlink:label="fig-0677-01" xlink:href="fig-0677-01a"> <variables xml:id="echoid-variables789" xml:space="preserve">s z o r x a h k g m b d e i t f q p f n</variables> </figure> </div> </div> <div xml:id="echoid-div1746" type="section" level="0" n="0"> <head xml:id="echoid-head1296" xml:space="preserve" style="it">16. In ſpeculis columnaribus concauis dato centro uiſus & pũcto rei uiſæ, punctum reflexio-<lb/>nis inuenire. Alhazen 96 n 5.</head> <p> <s xml:id="echoid-s45155" xml:space="preserve">Sit ſpeculum columnare concauum, cuius axis ſit x h:</s> <s xml:id="echoid-s45156" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s45157" xml:space="preserve"> punctum rei uiſæ a:</s> <s xml:id="echoid-s45158" xml:space="preserve"> & centrum uiſus <lb/>b:</s> <s xml:id="echoid-s45159" xml:space="preserve"> quæ ſint in locis datis.</s> <s xml:id="echoid-s45160" xml:space="preserve"> Dico quòd eſt poſsibile punctum reflexionis inueniri.</s> <s xml:id="echoid-s45161" xml:space="preserve"> Si enim pũctum rei <lb/>uiſæ (quod eſt a) & centrum uiſus (quod eſt b) fuerint in un a plana ſuperficie ſpeculũ trans axem <lb/>ſecante:</s> <s xml:id="echoid-s45162" xml:space="preserve"> tũc patet per 93 th.</s> <s xml:id="echoid-s45163" xml:space="preserve"> 1 huius, quia communis ſection ſuperficiei reflexionis & ſpecnli eſt linea <lb/>longitudinis.</s> <s xml:id="echoid-s45164" xml:space="preserve"> Poteſt itaq;</s> <s xml:id="echoid-s45165" xml:space="preserve"> inueniri punctum reflexionis, ſicut in ſpeculis planis per 46 th.</s> <s xml:id="echoid-s45166" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s45167" xml:space="preserve"> <lb/>Quod ſi puncta a & b non ſuerint in tali ſuperficie:</s> <s xml:id="echoid-s45168" xml:space="preserve"> imaginetur ſuperſicies tranſiens per punctum a, <lb/>ſecans ſpeculum æ quidiftanter bafibus:</s> <s xml:id="echoid-s45169" xml:space="preserve"> erit ergo per 100 th.</s> <s xml:id="echoid-s45170" xml:space="preserve"> 1 communis ſectio ſuperficiti illius & <lb/> <pb o="377" file="0679" n="679" rhead="LIBER NONVS."/> ſuperficiei ſpeculi circulus.</s> <s xml:id="echoid-s45171" xml:space="preserve"> Centrum itaq;</s> <s xml:id="echoid-s45172" xml:space="preserve"> uiſus, quod eſt punctum b, aut eſt in ſuperſicie illius cir-<lb/>culi, aut non.</s> <s xml:id="echoid-s45173" xml:space="preserve"> Si ſic:</s> <s xml:id="echoid-s45174" xml:space="preserve"> poteſt reflexionis punctum inuenlri in peripheria illius circuli, ſicut ſuprà in 27 <lb/>th.</s> <s xml:id="echoid-s45175" xml:space="preserve"> 8.</s> <s xml:id="echoid-s45176" xml:space="preserve"> huius d dcuimus in ſpeculis ſphæricis cõcuais.</s> <s xml:id="echoid-s45177" xml:space="preserve"> Si uerò centrum uiſus b non ſuerit in ſuperficie <lb/>illius circuli:</s> <s xml:id="echoid-s45178" xml:space="preserve"> tũc cũ punctũ rei uiſæ, & centrũ uiſus ſemper ſint in ſuperficie reflexionis per;</s> <s xml:id="echoid-s45179" xml:space="preserve"> 3 huius.</s> <s xml:id="echoid-s45180" xml:space="preserve"> <lb/>pater quòd cõmunis ſectio ſuperficiei reflexionis & ſpeculi in hoc ſitu eſt ſectio oxygonia.</s> <s xml:id="echoid-s45181" xml:space="preserve"> Duca-<lb/>tur ergo à puncto b cẽtro uiſus perpẽdicularis ſuper ſuperficiẽ illius circuli per 11 p 11:</s> <s xml:id="echoid-s45182" xml:space="preserve"> & replicetur <lb/>tota ꝓbatio proximæ præ cedẽcedẽtis:</s> <s xml:id="echoid-s45183" xml:space="preserve"> & palàm, quia inuenietur pũctus reflexionis.</s> <s xml:id="echoid-s45184" xml:space="preserve"> Quod eſt ꝓpoſitũ.</s> <s xml:id="echoid-s45185" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1747" type="section" level="0" n="0"> <head xml:id="echoid-head1297" xml:space="preserve" style="it">17. Centro uiſus exiſtente in puncto, qui eſt communis ſectio axis & lineæ perpendicularis <lb/>ſuper ſuperficiem, contingentem ſpeculum pyramidale concauum: fiet reflexio formæ ipſius ocu-<lb/>li ab una totali peripheria circuli ſpeculi æquidiſtantis baſi: & ſolùm per line as perpẽdiculares: <lb/>locuś imaginis erit in centro uiſus. Alhazen 98 n 5.</head> <p> <s xml:id="echoid-s45186" xml:space="preserve">Eſto ſpeculum pyramidale concauum, cuius axis ſit a h:</s> <s xml:id="echoid-s45187" xml:space="preserve"> & ducatur à puncto h linea perpendicu-<lb/>laris ſuper ſupeficiem, contingentem ſpeculum in puncto b:</s> <s xml:id="echoid-s45188" xml:space="preserve"> erit itaq;</s> <s xml:id="echoid-s45189" xml:space="preserve"> punctus h communis ſectio <lb/> <anchor type="figure" xlink:label="fig-0679-01a" xlink:href="fig-0679-01"/> axis a h & lineæ perpendicularis, quæ eſt h b.</s> <s xml:id="echoid-s45190" xml:space="preserve"> Dico quòd ſi centrum <lb/>uiſus poſitum fuerit in puncto h:</s> <s xml:id="echoid-s45191" xml:space="preserve"> fiet reflexio formæ oculi uidentis a <lb/>tota peripheria unius circuli ſpeculi æquidiftantis baſi, cuius polus erit <lb/>punctus h.</s> <s xml:id="echoid-s45192" xml:space="preserve"> Sit enlm punctus a uertex ſpeculi:</s> <s xml:id="echoid-s45193" xml:space="preserve"> & ducatur linea a b:</s> <s xml:id="echoid-s45194" xml:space="preserve"> ut ergo <lb/>pater per 95 th.</s> <s xml:id="echoid-s45195" xml:space="preserve"> 1 huius, erit linea a b pars lineæ lõgitudinis ſpeculi:</s> <s xml:id="echoid-s45196" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s45197" xml:space="preserve"> <lb/>trigonum h b a orth ogonium:</s> <s xml:id="echoid-s45198" xml:space="preserve"> quoniam angulus a b h erit rectus propter <lb/>perpendicularitatem lineæ h b ſuper lineam a b.</s> <s xml:id="echoid-s45199" xml:space="preserve"> Imaginẽtur ergo à pun-<lb/>cto h plurimæ duci perpendiculares ſuper lineas longitudinis ſpeculi, <lb/>ſicut eſt linea h b perpendicularis ſuper lineam longitu dinis, quæ eſt a b:</s> <s xml:id="echoid-s45200" xml:space="preserve"> <lb/>uel remanente fixo a h latere trignoi a b h, & circum ducto trigono, quo-<lb/>uſq;</s> <s xml:id="echoid-s45201" xml:space="preserve"> ad locum, unde exiuit, redeat:</s> <s xml:id="echoid-s45202" xml:space="preserve"> deſcribet punctũ b circulum in con-<lb/>cauitate ſpeculi, à cuius quolibet peripheriæ pũcto fiet reflexio ad uiſum <lb/>exiſtentem in puncto h ſecundum lineas perpendiculares, ſimiles lineæ <lb/>h b:</s> <s xml:id="echoid-s45203" xml:space="preserve"> hoc eſt ſecun dum lineas, quas motu ſuo determinabit linea h b.</s> <s xml:id="echoid-s45204" xml:space="preserve"> Fiet <lb/>autem reflexio ſolùm ſuperficiei ipſius uiſus per 21 th.</s> <s xml:id="echoid-s45205" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s45206" xml:space="preserve"> & ſolùm <lb/>partis ſuperficiei uiſus, quam ſecant duę lineæ perpendiculares à centro <lb/>oculi exeuntes, & maiorem angulum, qui eſt ibi poſsibilis, continentes.</s> <s xml:id="echoid-s45207" xml:space="preserve"> <lb/>Erit autem in omnibus his reflexionibus ſemper locus imaginis in cen-<lb/>tro uiſus:</s> <s xml:id="echoid-s45208" xml:space="preserve"> quoniam non ſit reflexio niſi ſecundum lineas perpẽdiculares.</s> <s xml:id="echoid-s45209" xml:space="preserve"> <lb/>Patet itaq;</s> <s xml:id="echoid-s45210" xml:space="preserve"> propoſitum:</s> <s xml:id="echoid-s45211" xml:space="preserve"> ita tamen quòd inter centrum uiſus & ſpeculi <lb/>ſuperficiem non ſit aliquod corpus ſolidum, quod obſiſtat.</s> <s xml:id="echoid-s45212" xml:space="preserve"/> </p> <div xml:id="echoid-div1747" type="float" level="0" n="0"> <figure xlink:label="fig-0679-01" xlink:href="fig-0679-01a"> <variables xml:id="echoid-variables790" xml:space="preserve">a b h</variables> </figure> </div> </div> <div xml:id="echoid-div1749" type="section" level="0" n="0"> <head xml:id="echoid-head1298" xml:space="preserve" style="it">18. Exiſtentibus centro uiſus punctó rei uiſæ in axe ſpeculi pyramidalis concaui: poßibile <lb/>eſt reflexionem fieri à toto uno circulo ſuperficiei reflexionis ſpeculi: locuś imaginis erit quidũ <lb/>circulus extra ſpeculum. Alhazen 99 n 5.</head> <p> <s xml:id="echoid-s45213" xml:space="preserve">Eſto ſpeculum pyramidale concauum:</s> <s xml:id="echoid-s45214" xml:space="preserve"> cuius axis ſit linea a h:</s> <s xml:id="echoid-s45215" xml:space="preserve"> & uertex a:</s> <s xml:id="echoid-s45216" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s45217" xml:space="preserve"> centrum uiſus in <lb/> <anchor type="figure" xlink:label="fig-0679-02a" xlink:href="fig-0679-02"/> puncto h:</s> <s xml:id="echoid-s45218" xml:space="preserve"> & ſit punctus rei uiſæ in puncto axis:</s> <s xml:id="echoid-s45219" xml:space="preserve"> qui ſit t:</s> <s xml:id="echoid-s45220" xml:space="preserve"> ima-<lb/>gineturq́ ſuperficies plana ſecans pyramidem ſpeculi ſecũ-<lb/>dum axis longitudinem, quæ ſit a b h g.</s> <s xml:id="echoid-s45221" xml:space="preserve"> Et quoniam linea a h <lb/>eſt axis ſpeculi:</s> <s xml:id="echoid-s45222" xml:space="preserve"> erunt lineæ a b & a g lineæ longitudinis ſpe-<lb/>culi per 90 th.</s> <s xml:id="echoid-s45223" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s45224" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s45225" xml:space="preserve"> â puncto rei uiſæ (quod <lb/>eſt t) linea perpẽdicularis ſuper lineam a b:</s> <s xml:id="echoid-s45226" xml:space="preserve"> quę ſit t q:</s> <s xml:id="echoid-s45227" xml:space="preserve"> & pro-<lb/>ducatur ultra punctum q extra ſpeculum ad pũctum l, donec <lb/>linea q l ſit æ qualis lineæ t q:</s> <s xml:id="echoid-s45228" xml:space="preserve"> & à puncto h ducatut linea ad <lb/>punctum l, quæ ſit h l.</s> <s xml:id="echoid-s45229" xml:space="preserve"> Hęc itaq:</s> <s xml:id="echoid-s45230" xml:space="preserve"> neceſſariò ſecabit lineam a b:</s> <s xml:id="echoid-s45231" xml:space="preserve"> <lb/>quoniam eſt cũ illa in eadẽ ſuperſicie:</s> <s xml:id="echoid-s45232" xml:space="preserve"> ſit ergo, ut ſecet ipſam <lb/>in puncto b:</s> <s xml:id="echoid-s45233" xml:space="preserve"> & à puncto b ducatur linea æ quidiſtans lineæ t <lb/>q per 31 p 1:</s> <s xml:id="echoid-s45234" xml:space="preserve"> quæ producta ad axem ſpeculi, ſit linea b d, ſecans <lb/>axem a h in puncto d:</s> <s xml:id="echoid-s45235" xml:space="preserve"> & copuletur linea t b.</s> <s xml:id="echoid-s45236" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s45237" xml:space="preserve">, cum <lb/>linea t q ſit perpendicularis ſuper lineam a b, & æ qualis lineę <lb/>q l:</s> <s xml:id="echoid-s45238" xml:space="preserve"> erit per 4 p 1 triangulus t b q ęqualis triangulo q b l:</s> <s xml:id="echoid-s45239" xml:space="preserve"> & an-<lb/>gulus q l b æqualis angulo q t b:</s> <s xml:id="echoid-s45240" xml:space="preserve"> ſed angulus q t b æqualis eſt <lb/>angulo t b d per 29 p 1 quia ſunt coalterni:</s> <s xml:id="echoid-s45241" xml:space="preserve"> & angulus d b h <lb/>extrinſecus eſt æqualis angulo q l bintrinſeco.</s> <s xml:id="echoid-s45242" xml:space="preserve"> Eſt ergo angulus t b d ęqualis angulo d b h:</s> <s xml:id="echoid-s45243" xml:space="preserve"> ergo per <lb/>20 th.</s> <s xml:id="echoid-s45244" xml:space="preserve"> 5 huius forma punctit reflectitur à puncto ſpeculi, quod eſt b, ad centrum uiſus exiſtens in <lb/>puncto h.</s> <s xml:id="echoid-s45245" xml:space="preserve"> Et quoniam linea t q eſt perpẽdicularis ſuper ſuperficiem ſpeculi:</s> <s xml:id="echoid-s45246" xml:space="preserve"> pater per definitionem <lb/>quoniam ipſa eſt cathetus incidentiæ formę puncti t:</s> <s xml:id="echoid-s45247" xml:space="preserve"> concurrit autem cathetus t q cum linea refle-<lb/>xionis, quæ eſt h b, in puncto l:</s> <s xml:id="echoid-s45248" xml:space="preserve"> eſt ergo punctus l locus imaginis formæ puncti t per 37 th.</s> <s xml:id="echoid-s45249" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s45250" xml:space="preserve"> <lb/>Si itaq;</s> <s xml:id="echoid-s45251" xml:space="preserve"> fixo latere t h imaginetur trigonus t l h moueri, quouſq;</s> <s xml:id="echoid-s45252" xml:space="preserve"> redeat ad locum, unde incepit:</s> <s xml:id="echoid-s45253" xml:space="preserve"> tũc <lb/> <pb o="378" file="0680" n="680" rhead="VITELLONIS OPTICAE"/> punctus b motu ſuo deſcribet circulum in ſuperficie cõcaua ſpeculi:</s> <s xml:id="echoid-s45254" xml:space="preserve"> & à quolibet puncto periphe-<lb/>riæ illius circuli reflectetur forma puncti t ad uiſum exiſtentem in puncto h.</s> <s xml:id="echoid-s45255" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s45256" xml:space="preserve"> l motu <lb/>ſuo deſcribet circulum extra ſpdculum, in cuius totali peripheria erit locus imaginis formæ puncti <lb/>t:</s> <s xml:id="echoid-s45257" xml:space="preserve"> quoniam in tota illius circuli peripheria catheti incidentiæ formæ puncti t, & lineæ reflexionum <lb/>formæ puncti t ad uiſum h, concurrent.</s> <s xml:id="echoid-s45258" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s45259" xml:space="preserve"> propoſitum.</s> <s xml:id="echoid-s45260" xml:space="preserve"/> </p> <div xml:id="echoid-div1749" type="float" level="0" n="0"> <figure xlink:label="fig-0679-02" xlink:href="fig-0679-02a"> <variables xml:id="echoid-variables791" xml:space="preserve">a j t q s d b h</variables> </figure> </div> </div> <div xml:id="echoid-div1751" type="section" level="0" n="0"> <head xml:id="echoid-head1299" xml:space="preserve" style="it">19. In pyramidalibus concauis ſpeculis cõmuni ſectione ſuperficiei reflexionis & ſpeculi oxy-<lb/>gonia exiſtente, & centro uiſus, punctó rei uiſæ exiſtentibus in eadem ſuperficie baſis ſpeculi, <lb/>aut ei æquidiſtantis, ne ſit ipſorum aliquod in axe ſpeculi: formarum punctorum rei uiſæ qua-<lb/>rundam fit ab uno tantùm pũcto ſpeculi reflexio: quarundã à quarundã à tribus: qua-<lb/>rundã à quatuor: nõ aũt à pluribus: & ſecundũ hæc loca imaginũ numerãtur. Alhaz. 100 n 5.</head> <p> <s xml:id="echoid-s45261" xml:space="preserve">Eſto ſpeculum pyramidale concauum a g u:</s> <s xml:id="echoid-s45262" xml:space="preserve"> cuius axis ſit a d & uertex a:</s> <s xml:id="echoid-s45263" xml:space="preserve"> ſitq́ punctus e centrum <lb/>uifus:</s> <s xml:id="echoid-s45264" xml:space="preserve"> & ſit z punctus rei uiſæ obliquè incidens ſpeculo:</s> <s xml:id="echoid-s45265" xml:space="preserve"> ita quòd nõ ſit in aliqua linearum perpen-<lb/>dicularium ſuper ſuperficiem uiſus:</s> <s xml:id="echoid-s45266" xml:space="preserve"> neq;</s> <s xml:id="echoid-s45267" xml:space="preserve"> ſit in axe ſpeculi, qui eſt a d:</s> <s xml:id="echoid-s45268" xml:space="preserve"> neq;</s> <s xml:id="echoid-s45269" xml:space="preserve"> fiat reflexio ab aliqual li-<lb/>nearum longitudinis ſpeculi:</s> <s xml:id="echoid-s45270" xml:space="preserve"> fiat tamẽ reflexio formæ puncti z ad uiſum e ab aliquo puncto ſuper-<lb/>ficiei propoſiti ſpeculi.</s> <s xml:id="echoid-s45271" xml:space="preserve"> Erit ergo neceſſariò cõmunis ſectio ſuperficiei reflexionis & ſpeculi ſectio <lb/>oxygonia per 2 th.</s> <s xml:id="echoid-s45272" xml:space="preserve"> huius:</s> <s xml:id="echoid-s45273" xml:space="preserve"> & ſint puncta e & z in eadẽ ſuperficie circuli baſis ſpeculi, aut æ quidiſtan-<lb/>tis ei.</s> <s xml:id="echoid-s45274" xml:space="preserve"> Dico quòd eſt poſsibile, ut ab uno tantùm pũcto ſpeculi:</s> <s xml:id="echoid-s45275" xml:space="preserve"> uel duobus:</s> <s xml:id="echoid-s45276" xml:space="preserve"> uel tribus:</s> <s xml:id="echoid-s45277" xml:space="preserve"> uel quatuor:</s> <s xml:id="echoid-s45278" xml:space="preserve"> <lb/>& non à pluribus fiat reflexio ad uiſum:</s> <s xml:id="echoid-s45279" xml:space="preserve"> & quandoq;</s> <s xml:id="echoid-s45280" xml:space="preserve"> unica apparebit imago:</s> <s xml:id="echoid-s45281" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s45282" xml:space="preserve"> duæ:</s> <s xml:id="echoid-s45283" xml:space="preserve"> quan-<lb/>doq;</s> <s xml:id="echoid-s45284" xml:space="preserve"> tres:</s> <s xml:id="echoid-s45285" xml:space="preserve"> quandoque quatuor:</s> <s xml:id="echoid-s45286" xml:space="preserve"> nec eſt poſsibile uideri plures imagines:</s> <s xml:id="echoid-s45287" xml:space="preserve"> quoniam totidem tantùm <lb/>ſunt puncta reflexionis poſsibilia.</s> <s xml:id="echoid-s45288" xml:space="preserve"> Imaginetur itaq;</s> <s xml:id="echoid-s45289" xml:space="preserve"> ſuperficies plana tranſiens per pũctum z æqui-<lb/>diſtans baſi ſpeculi:</s> <s xml:id="echoid-s45290" xml:space="preserve"> hæc itaq;</s> <s xml:id="echoid-s45291" xml:space="preserve"> ſuperficies per 100 th.</s> <s xml:id="echoid-s45292" xml:space="preserve"> 1 huius ſecabit ſpeculum ſecundum circulum:</s> <s xml:id="echoid-s45293" xml:space="preserve"> <lb/>centrum itaq;</s> <s xml:id="echoid-s45294" xml:space="preserve"> uiſus (quod eſt punctum e, ut patet ex hypotheſi) erit in ſuperficie illius circuli, cu-<lb/>ius cẽtrum ſit t:</s> <s xml:id="echoid-s45295" xml:space="preserve"> & ducatur linea e z, quæ producta ſecet illum circulum.</s> <s xml:id="echoid-s45296" xml:space="preserve"> Palàm ergo per ea, quæ de-<lb/>monſtrata ſunt in ſpeculis ſphæricis cõcauis per 40 th.</s> <s xml:id="echoid-s45297" xml:space="preserve"> 8 huius, quoniam in tali diſpoſitione forma <lb/>puncti z reflectitur ad uiſum exiſtentem in puncto e à peripheria illius circuli ex una parte, ſcilicet <lb/>ab arcu interiacente ſemidiametros, in quibus puncta z & e conſiſtunt, aut ab uno puncto ſpeculi:</s> <s xml:id="echoid-s45298" xml:space="preserve"> <lb/>aut à duobus:</s> <s xml:id="echoid-s45299" xml:space="preserve"> aut à tribus:</s> <s xml:id="echoid-s45300" xml:space="preserve"> & ex alia parte ab arcu ſcilicet interiacẽte illas ſemidiametros reliquas, <lb/>in quibus puncta z & e non conſiſtunt, ab uno tantùm puncto.</s> <s xml:id="echoid-s45301" xml:space="preserve"> Sumatur itaq;</s> <s xml:id="echoid-s45302" xml:space="preserve"> aliquis punctus cir-<lb/>culi, à quo fiat hæc reflexio, quod ſit h:</s> <s xml:id="echoid-s45303" xml:space="preserve"> & ducantur lineæ z h & e h:</s> <s xml:id="echoid-s45304" xml:space="preserve"> & ſemidiameter th.</s> <s xml:id="echoid-s45305" xml:space="preserve"> Pater itaq;</s> <s xml:id="echoid-s45306" xml:space="preserve"> <lb/>per 18 p 3 quoniam linea th eſt perpendicularis ſuper lineam circulum in puncto h contingentem:</s> <s xml:id="echoid-s45307" xml:space="preserve"> <lb/>& per 20 th.</s> <s xml:id="echoid-s45308" xml:space="preserve"> 5 huius palàm eſt quoniam linea t h diuidit angulum z h e per æqualia:</s> <s xml:id="echoid-s45309" xml:space="preserve"> ergo per 29 th.</s> <s xml:id="echoid-s45310" xml:space="preserve"> 1 <lb/> <anchor type="figure" xlink:label="fig-0680-01a" xlink:href="fig-0680-01"/> huius linea th ſeca bit lineam e z:</s> <s xml:id="echoid-s45311" xml:space="preserve"> ſit ergo punctus ſectionis q:</s> <s xml:id="echoid-s45312" xml:space="preserve"> du-<lb/>catuŕ per 101 th.</s> <s xml:id="echoid-s45313" xml:space="preserve"> 1 huius linea longitudinis ſpeculi:</s> <s xml:id="echoid-s45314" xml:space="preserve"> quæ ſit a h:</s> <s xml:id="echoid-s45315" xml:space="preserve"> & <lb/>â puncto q ducatur linea cadens perpendiculariter ſuper lineam <lb/>a h per 12 p 1:</s> <s xml:id="echoid-s45316" xml:space="preserve"> quæ ſit q m, ſecans lineam a h in puncto m:</s> <s xml:id="echoid-s45317" xml:space="preserve"> & produ-<lb/>cta ultra punctum q fecet a xem ſpeculi, qui eſt a d, in puncto d:</s> <s xml:id="echoid-s45318" xml:space="preserve"> & <lb/>ducantur lineæ z m & e m:</s> <s xml:id="echoid-s45319" xml:space="preserve"> & à puncto z, quod eſt pũctum rei ui-<lb/>fæ ducatur in ſuperficle illius circuli linea æquidiſtans lineæ q h:</s> <s xml:id="echoid-s45320" xml:space="preserve"> <lb/>quæ ſit z l.</s> <s xml:id="echoid-s45321" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s45322" xml:space="preserve"> linea e h concurrit cum linea q h in puncto <lb/>h:</s> <s xml:id="echoid-s45323" xml:space="preserve"> patet per 2 th.</s> <s xml:id="echoid-s45324" xml:space="preserve"> 1 huius quoniã linea e h producta ultra punctum <lb/>h concutret cum linea z l:</s> <s xml:id="echoid-s45325" xml:space="preserve"> ſit concurſus punctus l:</s> <s xml:id="echoid-s45326" xml:space="preserve"> & à puncto h <lb/>ducatur linea perpendicularis ſuper lineam l z, quæ ſit h p:</s> <s xml:id="echoid-s45327" xml:space="preserve"> dein-<lb/>de in ſuperficie e m z ducatur à puncto z linea æ quidiſtans lineæ <lb/>q m:</s> <s xml:id="echoid-s45328" xml:space="preserve"> quæ ſit linea z o.</s> <s xml:id="echoid-s45329" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s45330" xml:space="preserve"> linea e m concurrit eum linea m <lb/>q:</s> <s xml:id="echoid-s45331" xml:space="preserve"> patet per 2 th.</s> <s xml:id="echoid-s45332" xml:space="preserve"> 1 huius quòd ipſa concurret cum linea z o ipſius <lb/>æ quidiſtante:</s> <s xml:id="echoid-s45333" xml:space="preserve"> ſit ergo concurſus in puncto o:</s> <s xml:id="echoid-s45334" xml:space="preserve"> & ducatur linea lo:</s> <s xml:id="echoid-s45335" xml:space="preserve"> <lb/>& à puncto p ducatur linea æ quidiſtãs lineæ l o;</s> <s xml:id="echoid-s45336" xml:space="preserve"> quæ ſit linea p n, <lb/>ſecans lineam z o in puncto n:</s> <s xml:id="echoid-s45337" xml:space="preserve"> & ducatur linea m n.</s> <s xml:id="echoid-s45338" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s45339" xml:space="preserve"> <lb/>ex præ miſsis, & per 20 th.</s> <s xml:id="echoid-s45340" xml:space="preserve"> 5 huius quòd angulus e h q eſt æqualis <lb/>angulo q h z:</s> <s xml:id="echoid-s45341" xml:space="preserve"> ſed quia lineæ th & l z æquidiſtant:</s> <s xml:id="echoid-s45342" xml:space="preserve"> pater per 29 p 1 <lb/>quòd anguli q h z & h z l ſunt æqualies:</s> <s xml:id="echoid-s45343" xml:space="preserve"> quia conalterni:</s> <s xml:id="echoid-s45344" xml:space="preserve"> ſed & an-<lb/>gulus q h e extrinſecus eſt æ qualis angulo h l z intrinſeco:</s> <s xml:id="echoid-s45345" xml:space="preserve"> anguli <lb/>ergo h l z & h z l ſunt æ qualies:</s> <s xml:id="echoid-s45346" xml:space="preserve"> ergo per 6 p 1 latera h l & h z ſunt <lb/>æ qualia:</s> <s xml:id="echoid-s45347" xml:space="preserve"> ſed lineæ h p eſt perpẽdicularis ſuper lineã l z baſim iſo-<lb/>ſcelis h l z:</s> <s xml:id="echoid-s45348" xml:space="preserve"> erũt ergo per 31 th.</s> <s xml:id="echoid-s45349" xml:space="preserve"> 1 huius trigona h l p & h p z ſimilia:</s> <s xml:id="echoid-s45350" xml:space="preserve"> <lb/>ergo per 4 p 6, cum linea h p ſit ambobus illis trigonis cõmunis:</s> <s xml:id="echoid-s45351" xml:space="preserve"> <lb/>erit linea l p æqualis lineæ p z:</s> <s xml:id="echoid-s45352" xml:space="preserve"> ſed in trigono l o z linea p n eſt <lb/>æquidiſtans lineæ l o:</s> <s xml:id="echoid-s45353" xml:space="preserve"> ergo per 2 p 6 erit proportio lineæ z n ad lineam on, ſicut lineæ z p ad lineam <lb/>p l.</s> <s xml:id="echoid-s45354" xml:space="preserve"> Eſt ergo linea z n æ qualis lineæ n o.</s> <s xml:id="echoid-s45355" xml:space="preserve"> Item cum, ſicut patet ex præmiſisi, linea o z ſit æ quidiſtans <lb/>lineæ q m & linea h q ſit æ quidiſtans lineæ l z:</s> <s xml:id="echoid-s45356" xml:space="preserve"> ergo p 15 p 11 erit ſuperficies z l o æ quidiſtans ſuper-<lb/>ficiei q m h:</s> <s xml:id="echoid-s45357" xml:space="preserve"> & ſuperficies e o l ſecat illas duas ſuperficies:</s> <s xml:id="echoid-s45358" xml:space="preserve"> ſuperficiẽ quidẽ q h m ſecundũ lineã h m, <lb/>& ſuperficiẽ l o z ſecundũ lineã l o:</s> <s xml:id="echoid-s45359" xml:space="preserve"> ergo per 16 p 11 cõmunes ſectiones ſuperficiei e ol cum illis dua-<lb/>bus ſuperficiebus æ quidiſtantibus ſunt æ quidiſtantes.</s> <s xml:id="echoid-s45360" xml:space="preserve"> linea ergo h m æ quidiſtabit lineæ l o:</s> <s xml:id="echoid-s45361" xml:space="preserve"> ſed <lb/> <pb o="379" file="0681" n="681" rhead="LIBER NONVS."/> linea p n æ quidiſtat lineæ l o:</s> <s xml:id="echoid-s45362" xml:space="preserve"> ergo per 30 p 1 lineæ h m & p n æ quidiſtant.</s> <s xml:id="echoid-s45363" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s45364" xml:space="preserve"> linea h p cadit <lb/>inter lineas h t & l z æ quidiſtantes:</s> <s xml:id="echoid-s45365" xml:space="preserve"> patet per 29 p 1 quia anguli h p l & p h t ſunt æ qualies:</s> <s xml:id="echoid-s45366" xml:space="preserve"> quia coal-<lb/>terni:</s> <s xml:id="echoid-s45367" xml:space="preserve"> ſed angulus h pleſt rectus:</s> <s xml:id="echoid-s45368" xml:space="preserve"> ergo angulus p h t eſt rectus:</s> <s xml:id="echoid-s45369" xml:space="preserve"> ergo per 16 p 3 linea p h contingit <lb/>circulum:</s> <s xml:id="echoid-s45370" xml:space="preserve"> igitur ſuperficies a h p eſt contingens pyramidem ſpeculi:</s> <s xml:id="echoid-s45371" xml:space="preserve"> ergo per 95 th.</s> <s xml:id="echoid-s45372" xml:space="preserve"> 1 huius contin-<lb/>git illam ſecun dum lineam longitudinis, quæ eſt a h:</s> <s xml:id="echoid-s45373" xml:space="preserve"> & in hac ſuperficie erunt ambæ lineæ p n & n <lb/>m:</s> <s xml:id="echoid-s45374" xml:space="preserve"> linea quidem m h, quoniam eſt part lineæ longitudinis, quæ eſt a h:</s> <s xml:id="echoid-s45375" xml:space="preserve"> linea uerò p n per 1 th.</s> <s xml:id="echoid-s45376" xml:space="preserve"> 1 hu-<lb/>ius:</s> <s xml:id="echoid-s45377" xml:space="preserve"> omnes enim lineę æquidiſtantes neceſſariò ſunt in eadem ſuperficie, & linea p n & h m æquidi <lb/>ſtant:</s> <s xml:id="echoid-s45378" xml:space="preserve"> linea uerò n m eſt in eadem ſuperficie per 1 p 11, quoniam puncta n & m ſunt in illa ſuperficie:</s> <s xml:id="echoid-s45379" xml:space="preserve"> <lb/>eſt autem linea d m perpendicularis ſuper ſuperficiem a h p ſpeculum contingentem:</s> <s xml:id="echoid-s45380" xml:space="preserve"> ergo linea d <lb/>m eſt perpendicularis ſuper lineam n m per definitionem lineę perpendicularis ſuper ſuperficiem.</s> <s xml:id="echoid-s45381" xml:space="preserve"> <lb/>Sed lineæ d m & o z æ quidiſtant, ut prius patuit:</s> <s xml:id="echoid-s45382" xml:space="preserve"> ergo per 29 p 1 linean m, quæ eſt perpendicularis <lb/>ſuper lineam d m, erit perpendicularis ſuper eius æquidiſtantem, quæ eſt z o:</s> <s xml:id="echoid-s45383" xml:space="preserve"> ſed linea o n eſt æ qua <lb/>lis z n:</s> <s xml:id="echoid-s45384" xml:space="preserve"> ergo per 4 p 1 erit linea m o æqualis m z:</s> <s xml:id="echoid-s45385" xml:space="preserve"> ergo per 7 p 5 erit proportio lineæ e m ad lineam m <lb/>o, ſicut eiuſdem ad lineam m z:</s> <s xml:id="echoid-s45386" xml:space="preserve"> eſt autem proportio lineæ e m ad lineam m o, ſicut lineæ e q ad li-<lb/>neam q z per 2 p 6:</s> <s xml:id="echoid-s45387" xml:space="preserve"> cum lineæ m q & o z ſint æ quidiſtantes in trigono o z e:</s> <s xml:id="echoid-s45388" xml:space="preserve"> uel ſic:</s> <s xml:id="echoid-s45389" xml:space="preserve"> eſt autem pro-<lb/>portio lineæ e m ad lineam m o, ſicut lineæ e h ad lineam h l:</s> <s xml:id="echoid-s45390" xml:space="preserve"> ſed lineæ l h & h z ſunt æqualies per <lb/>præmiſſa:</s> <s xml:id="echoid-s45391" xml:space="preserve"> ergo per 7 p 5 eſt proportio lineæ e h ad lineam h z, ſicut ad lineam h l:</s> <s xml:id="echoid-s45392" xml:space="preserve"> eſt autem per 3 p 6 <lb/>cum linea h q diuidat angulum e h z per æqualia, proportio lineæ e h ad h z, ſicut e q ad q z.</s> <s xml:id="echoid-s45393" xml:space="preserve"> Eſt er-<lb/>go per 11 p 5 proportio lineæ e m ad lineam m z, ſicut lineæ e q ad lineam q z:</s> <s xml:id="echoid-s45394" xml:space="preserve"> ergo linea m q diuidit <lb/>angulum e m z per æqualia per 3 p 6.</s> <s xml:id="echoid-s45395" xml:space="preserve"> Eſt ergo angulus e m q æqualis angulo q m z:</s> <s xml:id="echoid-s45396" xml:space="preserve"> ergo per 20 th.</s> <s xml:id="echoid-s45397" xml:space="preserve"> <lb/>5 huius forma puncti z reflectitur ad uiſum exiſtentem in puncto e à puncto ſpeculi, quod eſt m.</s> <s xml:id="echoid-s45398" xml:space="preserve"> <lb/>Sicut itaq;</s> <s xml:id="echoid-s45399" xml:space="preserve"> forma puncti z reflectitur ad uiſum exiſtentem in puncto e à ſolo puncto circuli, quod <lb/>eſt h:</s> <s xml:id="echoid-s45400" xml:space="preserve"> ita ſimiliter reflectetur eadem forma puncti z ad uiſum e à ſolo puncto ſpeculi, quod eſt m.</s> <s xml:id="echoid-s45401" xml:space="preserve"> <lb/>Quòd ſi fiat in hoc ſitu reflexio à duobus punctis circuli:</s> <s xml:id="echoid-s45402" xml:space="preserve"> erit etiam reflexio à duobus punctis ſpe-<lb/>culi:</s> <s xml:id="echoid-s45403" xml:space="preserve"> & per eadem demonſtrandum:</s> <s xml:id="echoid-s45404" xml:space="preserve"> & ſi à tribus punctis circuli fiat reflexio:</s> <s xml:id="echoid-s45405" xml:space="preserve"> fiet etiam à tribus pun <lb/>ctis ſpeculi:</s> <s xml:id="echoid-s45406" xml:space="preserve"> & ſi fiat à quatuor punctis unius, fiet etiam à quatuor punctis alterius:</s> <s xml:id="echoid-s45407" xml:space="preserve"> & ſicut ab alia <lb/>parte círculi fiet reflexio ab uno pũcto circuli, ita fiet etiam ab uno puncto ſpeculi ex eadem parte.</s> <s xml:id="echoid-s45408" xml:space="preserve"> <lb/>Patet ergo propoſitum.</s> <s xml:id="echoid-s45409" xml:space="preserve"/> </p> <div xml:id="echoid-div1751" type="float" level="0" n="0"> <figure xlink:label="fig-0680-01" xlink:href="fig-0680-01a"> <variables xml:id="echoid-variables792" xml:space="preserve">a g e u t m q d o n z h p f</variables> </figure> </div> </div> <div xml:id="echoid-div1753" type="section" level="0" n="0"> <head xml:id="echoid-head1300" xml:space="preserve" style="it">20. In ſpeculis pyramidalibus concauis, communi ſectione ſuperficiei reflexionis & ſpeculio-<lb/>xygonia exiſtente, & centro uiſus, punctó rei uiſæ exiſtentibus intra ſpeculum, non in axe, <lb/>nec in eadem ſuperficie baſis ſpeculi, aut ei æquidiſt ante: formarum punctorum rei uiſæ quarun <lb/>dam reflexio fit ab uno tantùm puncto ſpeculi: quarundam à duobus: quarundam à tribus: <lb/>quarundam à quatuor: non autem à pluribus: & ſecundum hæc loca imaginum numerantur. <lb/>Alhazen 101 n 5.</head> <p> <s xml:id="echoid-s45410" xml:space="preserve">Sit, ut in propoſitione præcedente, ſpeculi pyramidalis concaui (quod ſit a g u) uertex a:</s> <s xml:id="echoid-s45411" xml:space="preserve"> & axis <lb/>a d:</s> <s xml:id="echoid-s45412" xml:space="preserve"> ſitq́ punctus rei uiſæ z:</s> <s xml:id="echoid-s45413" xml:space="preserve"> & centrum uiſus e:</s> <s xml:id="echoid-s45414" xml:space="preserve"> ductaq́ per punctum z ſperficie ſecante ſpeculum <lb/>æquidiſtanter baſi ſpeculi, non ſit punctúm e in illa ſuperficie, ſed ſub illa, uel ſupera illam.</s> <s xml:id="echoid-s45415" xml:space="preserve"> Sit autem <lb/>nunc, exempli cauſſa, ſupera illam:</s> <s xml:id="echoid-s45416" xml:space="preserve"> quia ſi ponatur eſſe ſub illa, eadem erit demonſtratio:</s> <s xml:id="echoid-s45417" xml:space="preserve"> dico itaq;</s> <s xml:id="echoid-s45418" xml:space="preserve"> <lb/>quòd uerum eſt id, quod proponitur.</s> <s xml:id="echoid-s45419" xml:space="preserve"> Quia enim, ut patet per 100 th.</s> <s xml:id="echoid-s45420" xml:space="preserve"> 1 huius, cõmunis ſectio illius <lb/>ſuperficiei & ſpeculi eſt circulus:</s> <s xml:id="echoid-s45421" xml:space="preserve"> ducatur à ueritice ſpeculi, quod eſt a, linea per centrum uiſus e, ſe-<lb/>cans ſuperficiem præmiſsi circuli extra ipſius centrum in puncto h, quę ſit a e h:</s> <s xml:id="echoid-s45422" xml:space="preserve"> hoc autem eſt poſ-<lb/>ſibile:</s> <s xml:id="echoid-s45423" xml:space="preserve"> ideo quia centrum uiſus, quod eſt punctum e, ut patet ex hypotheſi, eſt intra ſpeculum, non <lb/>in axe:</s> <s xml:id="echoid-s45424" xml:space="preserve"> ſiq́t;</s> <s xml:id="echoid-s45425" xml:space="preserve"> centrum illius circuli punctum q.</s> <s xml:id="echoid-s45426" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s45427" xml:space="preserve"> per 20 th.</s> <s xml:id="echoid-s45428" xml:space="preserve"> 8 huius quia forma puncti z po-<lb/>teſt reflecti ad uiſum exiſtentem in puncto h ab aliquo puncto circuli:</s> <s xml:id="echoid-s45429" xml:space="preserve"> ſit illud punctum t:</s> <s xml:id="echoid-s45430" xml:space="preserve"> & ducan-<lb/>tur lineæ h t & z t & h z, & ſemidiameter q t:</s> <s xml:id="echoid-s45431" xml:space="preserve"> quæcum ſit perpendicularis ſuperlineam contingen-<lb/>tem circulum in puncto t per 18 p 3, ergo per 26 th.</s> <s xml:id="echoid-s45432" xml:space="preserve"> 5 huius palàm quòd linea q t diuidit angulum h <lb/>t z per æqualia:</s> <s xml:id="echoid-s45433" xml:space="preserve"> ergo per 29 th.</s> <s xml:id="echoid-s45434" xml:space="preserve"> 1 huius pater quòd linea q t ſecabit lineã h z:</s> <s xml:id="echoid-s45435" xml:space="preserve"> ſit punctus ſectionis n:</s> <s xml:id="echoid-s45436" xml:space="preserve"> <lb/>& ducatur linea z e à puncto rei uiſæ ad centrum uiſus in punctum e, & linea longitudinis ſpeculi:</s> <s xml:id="echoid-s45437" xml:space="preserve"> <lb/>quæ ſit a t.</s> <s xml:id="echoid-s45438" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s45439" xml:space="preserve"> ex præmiſsis, cum punctus z ſit ex illa parte diametri q t, & ex una parte e-<lb/>iuſdem ſit punctum e, quod eſt centrum uiſus, quoniam punctum h, quod eſt in linea a e, eſt in ea-<lb/>dem parte ſemidiametri q t, in qua eſt & punctume.</s> <s xml:id="echoid-s45440" xml:space="preserve"> Pater ergo quòd linea e z ſecabit ſuperficiem <lb/>a q t:</s> <s xml:id="echoid-s45441" xml:space="preserve"> ſit, ut ſecet ipſam in puncto o:</s> <s xml:id="echoid-s45442" xml:space="preserve"> & ab illo puncto o per 12 p 1 ducatur perpendicularis ſuper <lb/>lineam at, ſcilicet lineam longitudinis ſpeculi:</s> <s xml:id="echoid-s45443" xml:space="preserve"> quæ perpendicularis ſit o p:</s> <s xml:id="echoid-s45444" xml:space="preserve"> hæc itaq;</s> <s xml:id="echoid-s45445" xml:space="preserve"> producta ul-<lb/>tra punctum o neceſſariò cadet ſuper axem ſpeculi, qui eſt a d, ut patet per 96 th.</s> <s xml:id="echoid-s45446" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s45447" xml:space="preserve"> ſit, ut ca-<lb/>dat in punctum d:</s> <s xml:id="echoid-s45448" xml:space="preserve"> & ducantur lineæ e p & z p.</s> <s xml:id="echoid-s45449" xml:space="preserve"> Dico quòd forma puncti z reflectitur ad uiſum exi-<lb/>ftentem in puncto e à puncto ſpeculi, quod eſt p.</s> <s xml:id="echoid-s45450" xml:space="preserve"> Ducatur enim à puncto z linea æ quidiſtans ſemi-<lb/>diametro q t per 31 p 1, quæ ſit z f.</s> <s xml:id="echoid-s45451" xml:space="preserve"> Et quoniam linea h t concurrit cum linea q t in puncto t:</s> <s xml:id="echoid-s45452" xml:space="preserve"> palàm <lb/>per 2 th.</s> <s xml:id="echoid-s45453" xml:space="preserve"> 1 huius quoniam ipſa concurret cum eius æquidiſtante, ſcilicet cum linea z f:</s> <s xml:id="echoid-s45454" xml:space="preserve"> ſit pun-<lb/>ctus concurſus f.</s> <s xml:id="echoid-s45455" xml:space="preserve"> Item à puncto z ducatur linea æquidiſtans lineæ o p:</s> <s xml:id="echoid-s45456" xml:space="preserve"> quæ ſit z k.</s> <s xml:id="echoid-s45457" xml:space="preserve"> Et quoniam <lb/>linea e p concurrit cum linea o p:</s> <s xml:id="echoid-s45458" xml:space="preserve"> patet quòd ipſa producta ultra punctum p concurret cum il-<lb/>la z k:</s> <s xml:id="echoid-s45459" xml:space="preserve"> ſit punctus concurſus k:</s> <s xml:id="echoid-s45460" xml:space="preserve"> & ducantur lineæ k f & k h.</s> <s xml:id="echoid-s45461" xml:space="preserve"> Et quia, ut patet ex præmiſsis, angu-<lb/>lus o pt eſt rectus, angulus uerò p t q minor recto per 89 th.</s> <s xml:id="echoid-s45462" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s45463" xml:space="preserve"> quoniam ipſe eſt angulus, <lb/> <pb o="380" file="0682" n="682" rhead="VITELLONIS OPTICAE"/> quem continet linea longitudinis cum ſemidiam etro baſis:</s> <s xml:id="echoid-s45464" xml:space="preserve"> patet ergo per 14 th.</s> <s xml:id="echoid-s45465" xml:space="preserve"> 1 huius quoniam <lb/>lineæ o p & q t concurrunt in aliquo puncto productæ ultra puncta d & q.</s> <s xml:id="echoid-s45466" xml:space="preserve"> Cum ita linea z f ſit <lb/>æquidiſtans lineæ q t, & linea z k æ quidiſtãs lineæ o p, & lineæ z f & z k concurrãtin puncto z:</s> <s xml:id="echoid-s45467" xml:space="preserve"> lineę <lb/>quoq;</s> <s xml:id="echoid-s45468" xml:space="preserve"> d p & q t ſimiliter concurrant in aliquo puncto, ut præ oſtenſum eſt:</s> <s xml:id="echoid-s45469" xml:space="preserve"> patet quod ſuperficies <lb/>f k z, & ſuperficies o p q t, quæ eſt ſuperficies a q t, ſunt æ quidiſtantes per 15 p 11.</s> <s xml:id="echoid-s45470" xml:space="preserve"> Quod qutem ſuper-<lb/>ficies o p q t ſit pars ſuperficiei a q t, patet ex his.</s> <s xml:id="echoid-s45471" xml:space="preserve"> Quoniam enim linea p o producta cadit in pun-<lb/>ctum axis, quod eſt d, patet per 1 p 11 quòd linea p o eſt in ſuperficie a q t:</s> <s xml:id="echoid-s45472" xml:space="preserve"> ſed & linea q t eſt in illa ſu-<lb/> <anchor type="figure" xlink:label="fig-0682-01a" xlink:href="fig-0682-01"/> perſicie:</s> <s xml:id="echoid-s45473" xml:space="preserve"> tota ergo ſuperficies o p q t eſt pars ſuperficiei a q k.</s> <s xml:id="echoid-s45474" xml:space="preserve"> <lb/>Et quia ſuperficies h k f ſecat duas ſuperficies z k f & a q t ſuք <lb/>duas lineast p & k f:</s> <s xml:id="echoid-s45475" xml:space="preserve"> patet quòd illæ duæ lineæ t p & k f ſunt <lb/>ęquidiſtantes per 16 p 11.</s> <s xml:id="echoid-s45476" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s45477" xml:space="preserve"> à pũcto t linea perpén <lb/>dicularis ſuք lineã z f per 12 p 1:</s> <s xml:id="echoid-s45478" xml:space="preserve"> quæ ſit linea t s:</s> <s xml:id="echoid-s45479" xml:space="preserve"> erit ergo angu <lb/>lus t s frectus:</s> <s xml:id="echoid-s45480" xml:space="preserve"> ergo ք29 p 1 angulus s t q eſt rectus:</s> <s xml:id="echoid-s45481" xml:space="preserve"> quoniã li-<lb/>neæ z f & t q æ quidiſtãt:</s> <s xml:id="echoid-s45482" xml:space="preserve"> ergo ք16 p 3 linea t s cõtingit in pun-<lb/>cto t circuliũ, cuius centrum eſt punctũ q.</s> <s xml:id="echoid-s45483" xml:space="preserve"> Superficies itaq;</s> <s xml:id="echoid-s45484" xml:space="preserve"> a <lb/>t s eſt contingens pyramidem ſpeculi:</s> <s xml:id="echoid-s45485" xml:space="preserve"> continget ergo illam <lb/>per 95 th.</s> <s xml:id="echoid-s45486" xml:space="preserve"> 1 huius ſecundum lineam longitudinis, quæ eſt a t.</s> <s xml:id="echoid-s45487" xml:space="preserve"> <lb/>Sed linea o p eſt perpendicularis ſuper lineam a t:</s> <s xml:id="echoid-s45488" xml:space="preserve"> eſt ergo li-<lb/>nea o p eracta ſuper ſuperficiem a t s contingentem pyrami-<lb/>dem:</s> <s xml:id="echoid-s45489" xml:space="preserve"> quoniam linea o p eſt in ſuperficie a q t, tranſeunte per <lb/>axem a d, & perlineam longitudinis a t:</s> <s xml:id="echoid-s45490" xml:space="preserve"> talis autem ſuperfi-<lb/>cies, ut patet per 97 th.</s> <s xml:id="echoid-s45491" xml:space="preserve"> 1 huius erecta eſt ſuper ſuperficiem <lb/>contingentem ſpeculum in linea longitudinis, quæ eſt a t.</s> <s xml:id="echoid-s45492" xml:space="preserve"> <lb/>Quia ergo ſuperficies a t s ſecat duas ſuperficies o p q t & z k <lb/>f, quæ ſunt æ quidiſtantes:</s> <s xml:id="echoid-s45493" xml:space="preserve"> patet per 16 p 11 quoniam duæ li-<lb/>neæ, quæ ſunt illarum ſuperficierum communes ſectiones, <lb/>ſunt æ quidiſtantes:</s> <s xml:id="echoid-s45494" xml:space="preserve"> quarum linearum una eſt linea p t, & alte-<lb/>ra ſit linea si ſecans lineam z k in puncto i.</s> <s xml:id="echoid-s45495" xml:space="preserve"> Patet quoq;</s> <s xml:id="echoid-s45496" xml:space="preserve"> quia <lb/>punctus i cadit inter puncta k & z:</s> <s xml:id="echoid-s45497" xml:space="preserve"> lineæ itaq;</s> <s xml:id="echoid-s45498" xml:space="preserve"> p t & siæqui-<lb/>diſtant:</s> <s xml:id="echoid-s45499" xml:space="preserve"> ſed lineæ p t & f k æquidiftant ad inuicem:</s> <s xml:id="echoid-s45500" xml:space="preserve"> quoniam <lb/>ſunt in ſuperficiebus æ quidiſtantibus:</s> <s xml:id="echoid-s45501" xml:space="preserve"> ergo per 30 p 1 lineæ si <lb/>& f k ſunt æ quidiſtantes.</s> <s xml:id="echoid-s45502" xml:space="preserve"> Et quoniam lineæ q t & z f æquidi-<lb/>ſtant:</s> <s xml:id="echoid-s45503" xml:space="preserve"> patet per 29 p 1 quòd angulus n t z eſt æqualis angulo <lb/>t z f:</s> <s xml:id="echoid-s45504" xml:space="preserve"> quia ſunt coalterni:</s> <s xml:id="echoid-s45505" xml:space="preserve"> & angulus h t n extrinſecus eſt æqua <lb/>lis angulo t f z intrinſeco:</s> <s xml:id="echoid-s45506" xml:space="preserve"> ſed anguli h t n & n t z ſunt æquales:</s> <s xml:id="echoid-s45507" xml:space="preserve"> ergo anguli t f z & t z f ſunt æquales:</s> <s xml:id="echoid-s45508" xml:space="preserve"> <lb/>ergo per 6 p 1 lineæ t f & t z ſunt æquales:</s> <s xml:id="echoid-s45509" xml:space="preserve"> & linea t s eſt perpendicularis ſuper baſim iſoſcelis t f z:</s> <s xml:id="echoid-s45510" xml:space="preserve"> <lb/>trigona itaq;</s> <s xml:id="echoid-s45511" xml:space="preserve"> partialia (quæ ſunt t s f & t s z) ſunt ſimilia per 31 th.</s> <s xml:id="echoid-s45512" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s45513" xml:space="preserve"> ergo per 4 p 6 cum linea <lb/>ts ambobus illis trigonis ſit communis:</s> <s xml:id="echoid-s45514" xml:space="preserve"> erit linea s f æ qualis lineæ s z.</s> <s xml:id="echoid-s45515" xml:space="preserve"> Sed cum linea s i æ quidiſtet <lb/>lineæ f k in trigono f k z:</s> <s xml:id="echoid-s45516" xml:space="preserve"> erit per 2 p 6 proportio lineæ f s ad lineam s z, ſicut lineæ ki ad lineam i z:</s> <s xml:id="echoid-s45517" xml:space="preserve"> <lb/>erit ergo linea k i æ qualis lineæ i z:</s> <s xml:id="echoid-s45518" xml:space="preserve"> ducaturq́ linea p i.</s> <s xml:id="echoid-s45519" xml:space="preserve"> Cum ergo ſuperficies a t s i, in qua ducta eſt <lb/>linea p i, ſit erecta ſuper ſuperficiem z k f, in qua cadit linea z k:</s> <s xml:id="echoid-s45520" xml:space="preserve"> erit per definitionem ſuperficiei ſu-<lb/>per ſuperficiem erectæ linea p i erecta ſuper lineam z k:</s> <s xml:id="echoid-s45521" xml:space="preserve"> ergo per 4 p 1 cum linea ki ſit æqualis li-<lb/>neæ i z, lineaq́ p i ſit communis, & anguli ad punctum i ſint æ qualies, quia recti:</s> <s xml:id="echoid-s45522" xml:space="preserve"> erit angulus p k z <lb/>æqualis angulo p z k:</s> <s xml:id="echoid-s45523" xml:space="preserve"> ſed per 29 p 1 angulus e p o extrinſecus æqualis eſt angulo p k z intrinſeco:</s> <s xml:id="echoid-s45524" xml:space="preserve"> <lb/>quoniam lineæ o p & z k æquidiſtant:</s> <s xml:id="echoid-s45525" xml:space="preserve"> & angulus o p z eſt æqualis angulo p z k:</s> <s xml:id="echoid-s45526" xml:space="preserve"> quia ſunt coaleterni:</s> <s xml:id="echoid-s45527" xml:space="preserve"> <lb/>anguli ergo e p d & d p z ſunt æquales:</s> <s xml:id="echoid-s45528" xml:space="preserve"> cum anguli p k z & p z k ſunt æ qualies.</s> <s xml:id="echoid-s45529" xml:space="preserve"> Ergo per 20 th.</s> <s xml:id="echoid-s45530" xml:space="preserve"> 5 hu-<lb/>ius forma puncti z reflectitur ad uiſum exiſtentem in puncto e à puncto ſuperficieie ſpeculi, quod <lb/>eſt p:</s> <s xml:id="echoid-s45531" xml:space="preserve"> quod eſt unũ propoſitorũ.</s> <s xml:id="echoid-s45532" xml:space="preserve"> si aũt ſumatur aliud punctũ in circulo, cuius centrum eſt punctum <lb/>q, à quo forma puncti z reflectatur ad uiſum exiſtentẽ in puncto h:</s> <s xml:id="echoid-s45533" xml:space="preserve"> præ miſſo modo poteſt declarari <lb/>quòd ab alio puncto ſpeculi reflectetur forma puncti z ad uiſum exiſtentẽ in puncto e ab alio pũcto <lb/>ꝗ̃ à puncto p.</s> <s xml:id="echoid-s45534" xml:space="preserve"> similiter quoq;</s> <s xml:id="echoid-s45535" xml:space="preserve"> ſi forma puncti z reflectatur ad uiſum exiſtentẽ in puncto h à tribus <lb/>punctis circuli:</s> <s xml:id="echoid-s45536" xml:space="preserve"> reflectetur forma puncti z ad uiſum e à tribus punctis ſpeculi:</s> <s xml:id="echoid-s45537" xml:space="preserve"> & ſi à quatuor pun-<lb/>ctis reflexio fiat in circulo:</s> <s xml:id="echoid-s45538" xml:space="preserve"> & à quatuor punctis reflexio erit in ſpeculo:</s> <s xml:id="echoid-s45539" xml:space="preserve"> & ſecundũ hæc loca ima-<lb/>ginum numerantur.</s> <s xml:id="echoid-s45540" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s45541" xml:space="preserve"> Quòd ſi dicatur quòd à pluribus punctis ſpeculi <lb/>quàm à quatuor poſsit fieri reflexio formæ puncti z ad uiſum exiſtentem in puncto e:</s> <s xml:id="echoid-s45542" xml:space="preserve"> ducta ab illo <lb/>puncto linea longitudinis ſuper peripheriam circuli, cuius centrum eſt punctum q:</s> <s xml:id="echoid-s45543" xml:space="preserve"> poterit per con <lb/>uerſionem præmiſſæ demonſtratioris oſtendi, quòd forma puncti z reflectetur ad uiſum exiſten-<lb/>tem in puncto h à pluribus punctis circuli quàm à quatuor:</s> <s xml:id="echoid-s45544" xml:space="preserve"> quod eſt impoſsibile, & contra 40 th.</s> <s xml:id="echoid-s45545" xml:space="preserve"> 8 <lb/>huius.</s> <s xml:id="echoid-s45546" xml:space="preserve"> Semper enim;</s> <s xml:id="echoid-s45547" xml:space="preserve"> ut patuit ex præmiſsis, à quotcunq;</s> <s xml:id="echoid-s45548" xml:space="preserve"> punctis circuli reflectitur forma puncti <lb/>z ad punctum h:</s> <s xml:id="echoid-s45549" xml:space="preserve"> à totidem punctis ſpeculi reflectetur eadem forma puncti z ad punctum e:</s> <s xml:id="echoid-s45550" xml:space="preserve"> & econ <lb/>uerſo:</s> <s xml:id="echoid-s45551" xml:space="preserve"> & dicenti contrarium accidit impoſsibile modo prædicto.</s> <s xml:id="echoid-s45552" xml:space="preserve"> Pater itaq;</s> <s xml:id="echoid-s45553" xml:space="preserve"> quòd punctorum rei <lb/>uiſæ in his ſpeculis quædam habent unicam imaginẽ quæ dam duas:</s> <s xml:id="echoid-s45554" xml:space="preserve"> quæ dam tres:</s> <s xml:id="echoid-s45555" xml:space="preserve"> quæ dam qua-<lb/>tuor:</s> <s xml:id="echoid-s45556" xml:space="preserve"> & quòd nõ eſt poſsibile cauſſari plures imagines in ſpeculis colũnaribus uel pyramidalibus <lb/>concauis:</s> <s xml:id="echoid-s45557" xml:space="preserve"> ſicut neq;</s> <s xml:id="echoid-s45558" xml:space="preserve"> in ſphæricis concauis:</s> <s xml:id="echoid-s45559" xml:space="preserve"> quod eſt notandum.</s> <s xml:id="echoid-s45560" xml:space="preserve"/> </p> <div xml:id="echoid-div1753" type="float" level="0" n="0"> <figure xlink:label="fig-0682-01" xlink:href="fig-0682-01a"> <variables xml:id="echoid-variables793" xml:space="preserve">a u e g d o p h q n z i k t s f</variables> </figure> </div> <pb o="381" file="0683" n="683" rhead="LIBER NONVS."/> </div> <div xml:id="echoid-div1755" type="section" level="0" n="0"> <head xml:id="echoid-head1301" xml:space="preserve" style="it">21. Dato centro uiſus & puncto rei uiſæ in ſpeculis pyramidalibus concauis, punctum refle-<lb/>xionis inuenire. Alhazen 102 n 5.</head> <p> <s xml:id="echoid-s45561" xml:space="preserve">Sit ſpeculum pyramidale concauum, cuius axis ſit linea a d:</s> <s xml:id="echoid-s45562" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s45563" xml:space="preserve"> punctus rei uiſæ z:</s> <s xml:id="echoid-s45564" xml:space="preserve"> & centrum <lb/>uiſus ſit punctum e, quæ ſint in locis datis:</s> <s xml:id="echoid-s45565" xml:space="preserve"> dico quòd eſt poſsibile punctum reflexionis inueniri.</s> <s xml:id="echoid-s45566" xml:space="preserve"> <lb/>Si enim punctum rei uiſę, quod eſt z, & centrum uiſus, quod eſt e, fuerint in una plana ſuperficie ſpe <lb/>culum trans axem ſecante:</s> <s xml:id="echoid-s45567" xml:space="preserve"> tunc patet per 90 th.</s> <s xml:id="echoid-s45568" xml:space="preserve"> 1 huius quia communis ſectio ſuperficiei reflexio-<lb/>nis & ſpeculi eſt linea longitudinis pyramidis ſpeculi:</s> <s xml:id="echoid-s45569" xml:space="preserve"> poteſt itaq;</s> <s xml:id="echoid-s45570" xml:space="preserve"> punctum reflexionis inueniri ſi-<lb/>cuti in ſpeculis planis per 46 th.</s> <s xml:id="echoid-s45571" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s45572" xml:space="preserve"> Quòd ſi puncta z & b non fuerint in illa totali ſuperficie:</s> <s xml:id="echoid-s45573" xml:space="preserve"> <lb/>imaginetur ſuperficies tran ſiens per punctum z, ſecans ſpeculum æquidiſtanter ſuæ baſi:</s> <s xml:id="echoid-s45574" xml:space="preserve"> erit ergo <lb/>per 100 th.</s> <s xml:id="echoid-s45575" xml:space="preserve"> 1 huius communis ſectio illius ſuperficiei & ſpeculi circulus.</s> <s xml:id="echoid-s45576" xml:space="preserve"> Centrum itaq;</s> <s xml:id="echoid-s45577" xml:space="preserve"> uiſus, quod <lb/>eſt punctum e, aut erit in illa ſuperficie circuli, aut non.</s> <s xml:id="echoid-s45578" xml:space="preserve"> Quomodocunq;</s> <s xml:id="echoid-s45579" xml:space="preserve"> autem ſit:</s> <s xml:id="echoid-s45580" xml:space="preserve"> qura, ut pater <lb/>per 12 th 7 huius impoſsibile eſt communem ſectionem ſuperficiei reflexionis & huius ſpeculi cir-<lb/>culum eſſe:</s> <s xml:id="echoid-s45581" xml:space="preserve"> ergo erit ſemper tunc illa cõmunis ſectio oxygonia:</s> <s xml:id="echoid-s45582" xml:space="preserve"> replicata ergo demonſtratione 19 <lb/>huius, uel proximæ præmiſſæ:</s> <s xml:id="echoid-s45583" xml:space="preserve"> patebit ſaciliter inuentio puncti reflexionis.</s> <s xml:id="echoid-s45584" xml:space="preserve"> Forma enim puncti z re <lb/>flectetur ad uiſum exiſtentem in puncto h ab aliquo puncto circumferentiæ circuli, cuius centrum <lb/>eſt q:</s> <s xml:id="echoid-s45585" xml:space="preserve"> uel fortè à duobus:</s> <s xml:id="echoid-s45586" xml:space="preserve"> uel à tribus:</s> <s xml:id="echoid-s45587" xml:space="preserve"> uel à quatuor:</s> <s xml:id="echoid-s45588" xml:space="preserve"> &quotcunq;</s> <s xml:id="echoid-s45589" xml:space="preserve"> fuerint, ſemper modo præmiſſo <lb/>inuenietur punctum reflexionis illi puncto circuli correſpondens, inuento puncto reflexionis il-<lb/>lorum punctorum in peripheria circuli, per ea, quæ declarauimus in diuerſis propoſitionibus octa <lb/>ui huius.</s> <s xml:id="echoid-s45590" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s45591" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1756" type="section" level="0" n="0"> <head xml:id="echoid-head1302" xml:space="preserve" style="it">22. Ambobus uiſibus à ſpeculis columnaribus uel pyramidalibus concanis quaſi unica oc-<lb/>currit imago.</head> <p> <s xml:id="echoid-s45592" xml:space="preserve">In his enim ſpeculis puncta reflexionis formæ eiuſdẽpunctirei uiſæ, ad diuerſos uiſus einſdem <lb/>uidentis non habent multam diuerſitatem diſtantiæ, propter uiſuum approximationem ad ſe inui <lb/>cem.</s> <s xml:id="echoid-s45593" xml:space="preserve"> Vnde etſi puncti unius formæ imago ſit aliqualiter ambobus uiſibus occurrẽs duplicata:</s> <s xml:id="echoid-s45594" xml:space="preserve"> ſunt <lb/>tamen illæ imagines contiguæ & admixtæ:</s> <s xml:id="echoid-s45595" xml:space="preserve"> unde uidebuntur quaſi unica imago.</s> <s xml:id="echoid-s45596" xml:space="preserve"> Diuerſitas enim <lb/>locorum illarum imaginum propter ſui imperceptibilitatem non inducit aliquam diſtantiam in ui <lb/>ſu, nec aliquẽ efficit errorem.</s> <s xml:id="echoid-s45597" xml:space="preserve"> Videtur ergo imago quſi una.</s> <s xml:id="echoid-s45598" xml:space="preserve"> Et ſimiliter per modum, quo in 59 th.</s> <s xml:id="echoid-s45599" xml:space="preserve"> <lb/>8 huius oſtendimus, poſsibile eſt quòd diuerſorum uidentium uiſibus diſtantibus & diuerſis, uni-<lb/>ca quandoq;</s> <s xml:id="echoid-s45600" xml:space="preserve"> in his ſpeculis, ſicut & in alijs, occurrat imago:</s> <s xml:id="echoid-s45601" xml:space="preserve"> cui propter identitatem illius ſitus hic <lb/>non duxim us immorandum.</s> <s xml:id="echoid-s45602" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s45603" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1757" type="section" level="0" n="0"> <head xml:id="echoid-head1303" xml:space="preserve" style="it">23. Lineæ rectæ æquidiſtantis axi ſpeculi columnaris cõcaui (centro uiſus exiſtente in eadem <lb/>ſuperficie uel in alia) reflexio fit à linea longitudinis ſpeculis ad uiſum.</head> <p> <s xml:id="echoid-s45604" xml:space="preserve">Eſto axis ſpeculi columnaris concaui linea, quæ z k:</s> <s xml:id="echoid-s45605" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s45606" xml:space="preserve"> linea uiſa axi ſpeculi æquidiſtans t q h, <lb/>ſitq́;</s> <s xml:id="echoid-s45607" xml:space="preserve"> centrum uiſus punctum e:</s> <s xml:id="echoid-s45608" xml:space="preserve"> dico quòd forma lineæ t q h reflectitur ad uiſum e à linea longitudi-<lb/>dinis ſpeculi a b g, quæ eſt communis ſectio ſuperficiei t h z k, & ſuperficiei ſpeculi:</s> <s xml:id="echoid-s45609" xml:space="preserve"> & hoc quidem <lb/>ſi centrum uiſus (quod eſt e) non ſuerit in ſu <lb/> <anchor type="figure" xlink:label="fig-0683-01a" xlink:href="fig-0683-01"/> perficie t h z k, demonſtrari poteſt omnimo-<lb/>dè ſicut in 30 th.</s> <s xml:id="echoid-s45610" xml:space="preserve"> 7 huius.</s> <s xml:id="echoid-s45611" xml:space="preserve"> Si uerò centrum ui-<lb/>ſus fuerit in eadem ſuperficie, demonſtrabi-<lb/>tur idem propoſitum, ſicut in 51 th.</s> <s xml:id="echoid-s45612" xml:space="preserve"> 7 huius:</s> <s xml:id="echoid-s45613" xml:space="preserve"> <lb/>reflecteturq́;</s> <s xml:id="echoid-s45614" xml:space="preserve"> forma puncti t à puncto ſpecu-<lb/>li g:</s> <s xml:id="echoid-s45615" xml:space="preserve"> & forma puncti q à puncto ſpeculi b:</s> <s xml:id="echoid-s45616" xml:space="preserve"> & <lb/>forma puncti h à puncto ſpeculi a.</s> <s xml:id="echoid-s45617" xml:space="preserve"> Erit itaq;</s> <s xml:id="echoid-s45618" xml:space="preserve"> <lb/>angulus t g n æqualis angulo n g e:</s> <s xml:id="echoid-s45619" xml:space="preserve"> & angu-<lb/>lus q b m æ qualis angulo m b e:</s> <s xml:id="echoid-s45620" xml:space="preserve"> & angulus h <lb/>a r æ qualis angulo r a e.</s> <s xml:id="echoid-s45621" xml:space="preserve"> Patet etiam per 30 <lb/>th.</s> <s xml:id="echoid-s45622" xml:space="preserve"> 7 huius quòd lineæ e k, h a, q b, t g concur <lb/>runt in puncto o.</s> <s xml:id="echoid-s45623" xml:space="preserve"> Patet etiam ibidem quòd li <lb/>nea a b g eſt linea recta extenſa in longitudi-<lb/>n e ſpeculi:</s> <s xml:id="echoid-s45624" xml:space="preserve"> & quòd lineæ g z, b l & a d ſunt <lb/>perpendiculares ſuper ſuperficiem, contingentem ſpecualum, quà cõtingit ipſum ſecundũ lineam <lb/>a b g:</s> <s xml:id="echoid-s45625" xml:space="preserve"> & quòd linea a b g eſt perpendicularis ſuper ſuperficiẽ, in qua eſt triangulus e b o:</s> <s xml:id="echoid-s45626" xml:space="preserve"> & quòd li-<lb/>nea t q eſt æ qualis lineæ q h:</s> <s xml:id="echoid-s45627" xml:space="preserve"> & linea a b æ qualis lineæ b g.</s> <s xml:id="echoid-s45628" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s45629" xml:space="preserve">, cum in his & in illis ſpeculis <lb/>hinc inde eadem ſit demonſtratio:</s> <s xml:id="echoid-s45630" xml:space="preserve"> quoniam forma lineæ t q h reflectitur a b his ſpeculis à linea lon-<lb/>gitudinis ipſorum.</s> <s xml:id="echoid-s45631" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s45632" xml:space="preserve"> quoniam ſiue linea longitudinis, quæ eſt a b g, ſit in con <lb/>uexo uel in concauo ipſius ſpeculi, quantùm ad hoc, nulla eſt diuerſit as in propoſito.</s> <s xml:id="echoid-s45633" xml:space="preserve"/> </p> <div xml:id="echoid-div1757" type="float" level="0" n="0"> <figure xlink:label="fig-0683-01" xlink:href="fig-0683-01a"> <variables xml:id="echoid-variables794" xml:space="preserve">t n q g z m b l f h r a d c e o</variables> </figure> </div> </div> <div xml:id="echoid-div1759" type="section" level="0" n="0"> <head xml:id="echoid-head1304" xml:space="preserve" style="it">24. Imago lineæ æquidiſtantis axi ſpeculi columnaris concaui (centro uiſus exiſtente in ea-<lb/>dem ſuperficie) uidebitur recta, æqualis & conformis rei uiſæ.</head> <p> <s xml:id="echoid-s45634" xml:space="preserve">Sit enim diſpoſitio, quę in pręcedente:</s> <s xml:id="echoid-s45635" xml:space="preserve"> reflectaturq́;</s> <s xml:id="echoid-s45636" xml:space="preserve"> forma lineę t q h à ſuperficie ſpeculi ſecundũ <lb/>lineã lõgitudinis, quę eſt a g:</s> <s xml:id="echoid-s45637" xml:space="preserve"> & ſit centrũ uiſus e in ipſa ſupficie t h z k.</s> <s xml:id="echoid-s45638" xml:space="preserve"> Dico quòd imago lineę t q h <lb/> <pb o="382" file="0684" n="684" rhead="VITELLONIS OPTICAE"/> uidebitur recta, æqualis ipſi lineæ t q h.</s> <s xml:id="echoid-s45639" xml:space="preserve"> Quælibet enim perpendicularis ducta ab aliquo punctorũ <lb/>lineęt q h erit ſemper in eadem ſuperficie cũ centro uiſus & axe:</s> <s xml:id="echoid-s45640" xml:space="preserve"> & probabuntur loca imaginũ pun <lb/>ctorum lineę t q h ſituari ſecundum lineam rectam, ſicut in ſpeculis planis per 52 th.</s> <s xml:id="echoid-s45641" xml:space="preserve"> 5 huius oſten-<lb/>ſum eſt de lineis rectis uiſis.</s> <s xml:id="echoid-s45642" xml:space="preserve"> Vt ſi aliqua linea recta rei uiſę imaginetur in his ſpeculis collocari in lo <lb/>co imaginis, & uiſus ſituetur proportionaliter ad illam, ſicut nunc ſituatus eſt ad lineam t h:</s> <s xml:id="echoid-s45643" xml:space="preserve"> erit lo-<lb/>cus imaginis illius lineæ linea t h, & apparebit recta & æ qualis rei uiſæ.</s> <s xml:id="echoid-s45644" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s45645" xml:space="preserve"> illud, quod <lb/>eſt in linea rei uiſæ ſuperius, erit in imagine ſuperius:</s> <s xml:id="echoid-s45646" xml:space="preserve"> & quod in re uiſa eſt inferius, erit in imagine <lb/>inferius.</s> <s xml:id="echoid-s45647" xml:space="preserve"> Erit itaq;</s> <s xml:id="echoid-s45648" xml:space="preserve"> imago conformis rei uiſæ.</s> <s xml:id="echoid-s45649" xml:space="preserve"> Latitudo uerò talium uiſorum erit maior quàm latitu <lb/>do ſuarum imaginum:</s> <s xml:id="echoid-s45650" xml:space="preserve"> quoniam imagines ſecundum latitudinem conſtringuntur propter puncta <lb/>reflexionum, quæ anguſtantur, & puncta latitudinis diuerſantur:</s> <s xml:id="echoid-s45651" xml:space="preserve"> quoniam ſiniſtrũ rei ſit dextrum <lb/>imaginis, & dextrum rei fit imaginis ſiniſtrum.</s> <s xml:id="echoid-s45652" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s45653" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1760" type="section" level="0" n="0"> <head xml:id="echoid-head1305" xml:space="preserve" style="it">25. Lineæ rectæ æquidiſt antis axi ſpeculi columnaris concaui (centro uiſus non exiſtente in <lb/>eadem ſuperficie) imago quando uidebitur recta maior re uiſa: quando concaua: quando <lb/>conuexa: quando unica: quando plures. Alhazen 51 n 6.</head> <p> <s xml:id="echoid-s45654" xml:space="preserve">Remaneat diſpoſitio præcedentis, niſi quòd centrum uiſus (quod eſt e) non ſit in ſuperfioie t h <lb/>z k:</s> <s xml:id="echoid-s45655" xml:space="preserve"> dico quòd erit, ut proponitur.</s> <s xml:id="echoid-s45656" xml:space="preserve"> Repetita enim demonſtratione 51 th.</s> <s xml:id="echoid-s45657" xml:space="preserve"> 7 huius, patebit quòd in ſpe <lb/>culis columnari bus cõuexis locus imaginis formæ puncti h lineę t q h eſt in puncto s:</s> <s xml:id="echoid-s45658" xml:space="preserve"> & locus ima-<lb/>ginis formæ q eſt in puncto c:</s> <s xml:id="echoid-s45659" xml:space="preserve"> & locus imaginis formæ puncti t eſt in puncto i.</s> <s xml:id="echoid-s45660" xml:space="preserve"> Sic ergo in linea s c i <lb/>ſunt imagines formarum omnium punctorum lineæ h q t:</s> <s xml:id="echoid-s45661" xml:space="preserve"> & patet quòd punctus c eſt propinquior <lb/>centro uiſus quod eſt e, quàm linea recta s i:</s> <s xml:id="echoid-s45662" xml:space="preserve"> & quòd linea s i eſt in ſuperficie trigoni u h t:</s> <s xml:id="echoid-s45663" xml:space="preserve"> & quòd <lb/>duæ lineæ u h & u t ſunt æquales:</s> <s xml:id="echoid-s45664" xml:space="preserve"> & quòd duæ lineæ u s & u i ſunt æquales:</s> <s xml:id="echoid-s45665" xml:space="preserve"> relin quitur ergo, ut duę <lb/>lineæ t i & h s ſint æquales:</s> <s xml:id="echoid-s45666" xml:space="preserve"> eſt ergo proportio lineę t i ad lineam i u, ſicut lineæ h s ad lineã s u:</s> <s xml:id="echoid-s45667" xml:space="preserve"> ergo <lb/>per 2 p 6 linea s i æquidiſtat lineæ t h.</s> <s xml:id="echoid-s45668" xml:space="preserve"> Patet etiam ex eodem 51 th.</s> <s xml:id="echoid-s45669" xml:space="preserve"> 7 quia duæ s e & e i ſunt æ-<lb/>quales.</s> <s xml:id="echoid-s45670" xml:space="preserve"> Ducatur ergo linea q u, quę ſecet lineã s i in puncto æ:</s> <s xml:id="echoid-s45671" xml:space="preserve"> diuidet ergo ipſam per æqualia:</s> <s xml:id="echoid-s45672" xml:space="preserve"> nam <lb/>linea t h diuiſa eſt in duo æqualia in puncto q:</s> <s xml:id="echoid-s45673" xml:space="preserve"> & erit linea c u in ſuperficie trigoni q u e:</s> <s xml:id="echoid-s45674" xml:space="preserve"> quæ eſt ſu-<lb/>perficies circuli b f æ quidiſtãs baſibus ſpeculi.</s> <s xml:id="echoid-s45675" xml:space="preserve"> Punctus itaq;</s> <s xml:id="echoid-s45676" xml:space="preserve"> c erit in ſuperficie trigoni c u e:</s> <s xml:id="echoid-s45677" xml:space="preserve"> & fimi <lb/>liter erit punctũ t in ſuperficie trigoni c e i:</s> <s xml:id="echoid-s45678" xml:space="preserve"> eſt ergo punctũ c in linea, quę eſt cõmunis ſectio illarum <lb/>duarũ ſuperficierũ, ſcilicet trigonorũ q u e & c e i:</s> <s xml:id="echoid-s45679" xml:space="preserve"> ſed hæc cõmunis ſectio eſt linea e b per 19 th.</s> <s xml:id="echoid-s45680" xml:space="preserve"> 1 hu-<lb/>ius.</s> <s xml:id="echoid-s45681" xml:space="preserve"> Punctus ergo c cadit in rectitudinẽ lineæ e b:</s> <s xml:id="echoid-s45682" xml:space="preserve"> linea ergo q c ſecat lineã e b in rectitudine ipſius:</s> <s xml:id="echoid-s45683" xml:space="preserve"> <lb/>& duæ lineę h u & t u ſunt ſub duobus punctis d & z:</s> <s xml:id="echoid-s45684" xml:space="preserve"> nam duæ lineę h u & t u ſunt duæ catheti inci <lb/>dentię, ſcilicet duę lineę perpendiculares exeuntes à duobus terminis lineę t h ſuper duas lineas, <lb/>cõtingentes duas portiones duarũ ſectionum columnariũ ſpeculi, in quarũ circũferentia ſunt duo <lb/>puncta a & g, à quibus ſit reflexio punctorũt & h a d uiſum in punctũ e.</s> <s xml:id="echoid-s45685" xml:space="preserve"> Superficies ergo trianguliu <lb/>h t eſt ſub axe ſpeculi, qui eſt z k.</s> <s xml:id="echoid-s45686" xml:space="preserve"> Sed nullũ punctũ ipſius axis, etſi ꝓtrahaturin infinitũ, erit unquã <lb/>in ſuperficie trianguli u h t.</s> <s xml:id="echoid-s45687" xml:space="preserve"> Nam ſi hoc eſſet poſsibile:</s> <s xml:id="echoid-s45688" xml:space="preserve"> tunc ſi axis k z continuaretur cũ aliquo pun-<lb/>cto lineæ h t ſecundũ lineam rectã:</s> <s xml:id="echoid-s45689" xml:space="preserve"> tunc illa ſuperficies, in qua eſſet illa linea recta, & linea u h t eſſet <lb/>ſuperficies trianguli u h t:</s> <s xml:id="echoid-s45690" xml:space="preserve"> & illa ſuperficies eſſet illa, in qua ſunt duæ lineæ æ quidiſtantes, quę ſunt <lb/>h t, & axis z k:</s> <s xml:id="echoid-s45691" xml:space="preserve"> & ſic ſuperficies, in qua ſunt duæ lineæ h t & k z, eſſet ſuperficies trianguli h u t:</s> <s xml:id="echoid-s45692" xml:space="preserve"> & ſic <lb/>totus axis z k erit in ſuperficie trianguli h u t:</s> <s xml:id="echoid-s45693" xml:space="preserve"> ſed ex hypotheſi axis eſt æ quidiſtãs lineæ h t:</s> <s xml:id="echoid-s45694" xml:space="preserve"> & ſic <lb/>dum iſtum modũ accideret quòd axis k z ſecaret duas lineas h u & t u.</s> <s xml:id="echoid-s45695" xml:space="preserve"> Sed & linea t h ſecundũ eius <lb/>punctũ h eſt in ſuperficie trianguli u e h, quæ eſt ſuperficies reflexionis:</s> <s xml:id="echoid-s45696" xml:space="preserve"> & ſectio cõmmunis huic ſu-<lb/>perficiei & ſuperficiei colũnaris ſpeculi eſt ſectio oxygonia:</s> <s xml:id="echoid-s45697" xml:space="preserve"> ſuperficies ergo e u h ſecat axẽ colũna-<lb/>ris ſpeculi in uno puncto, ſcilicet in puncto d, ut to tũ præ oſtenſum eſt in cõmento 51 th.</s> <s xml:id="echoid-s45698" xml:space="preserve"> 7 huius.</s> <s xml:id="echoid-s45699" xml:space="preserve"> Si <lb/>ergo axis k z <lb/> <anchor type="figure" xlink:label="fig-0684-01a" xlink:href="fig-0684-01"/> ſecet lineã h <lb/>u:</s> <s xml:id="echoid-s45700" xml:space="preserve"> pũctus ſe-<lb/>ctionis cũ li-<lb/>nea h u erit in <lb/>ſuperficie tri-<lb/>anguli u e h:</s> <s xml:id="echoid-s45701" xml:space="preserve"> <lb/>ſed in hac ſu-<lb/>perficie non <lb/>eſt punctũ, ք <lb/>qđ axis tráſ-<lb/>eat, niſi pun-<lb/>ctũ d:</s> <s xml:id="echoid-s45702" xml:space="preserve"> ſecabit <lb/>ergo axis k z <lb/>lineam h u in <lb/>puncto d:</s> <s xml:id="echoid-s45703" xml:space="preserve"> ſed <lb/>per 114 th.</s> <s xml:id="echoid-s45704" xml:space="preserve"> 1 huius, uel per 44 th.</s> <s xml:id="echoid-s45705" xml:space="preserve"> 7 huius oſtenſum eſt, quòd linea h u ſecat axem ſub puncto d:</s> <s xml:id="echoid-s45706" xml:space="preserve"> in <lb/>duobus ergo punctis ſecabit linea h u axem k z:</s> <s xml:id="echoid-s45707" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s45708" xml:space="preserve"> Axis ergo k z totus eſt extra <lb/>ſuperficiem h u t:</s> <s xml:id="echoid-s45709" xml:space="preserve"> & propin quior uiſui exiſtenti in pũcto e, quàm ſuperficies h u t.</s> <s xml:id="echoid-s45710" xml:space="preserve"> Superficies ergo, <lb/>in qua ſunt lineę h t, & axis k z, propinquior eſt centro uiſus puncto e ꝗ̃ ſuperficies u h t:</s> <s xml:id="echoid-s45711" xml:space="preserve"> & punctũ c <lb/>eſt in ſuperficie, in qua ſunt linea h t, & axis k z:</s> <s xml:id="echoid-s45712" xml:space="preserve"> quia punctus c eſt in linea q l:</s> <s xml:id="echoid-s45713" xml:space="preserve"> & q l ք 7 p 11 eſt in eadẽ <lb/> <pb o="383" file="0685" n="685" rhead="LIBER NONVS."/> ſuperficic cum lineis æ quidiſtantibus, quas copulat, quæ ſunt h t & z k.</s> <s xml:id="echoid-s45714" xml:space="preserve"> Punctum ergo c eſt propin <lb/>quius puncto e centro uiſus;</s> <s xml:id="echoid-s45715" xml:space="preserve"> quàm ſit linea s i:</s> <s xml:id="echoid-s45716" xml:space="preserve"> ſed punctum c cum ſit communis ſectio linearum e <lb/>b & q l, ut in 51 th.</s> <s xml:id="echoid-s45717" xml:space="preserve"> 7 huius præ oſtendimus:</s> <s xml:id="echoid-s45718" xml:space="preserve"> palàm quòd eſt in rectitudine lineæ e b.</s> <s xml:id="echoid-s45719" xml:space="preserve"> Si ergo linea e b <lb/>educatur ultra punctum b, inſa perueniet ad punctum c:</s> <s xml:id="echoid-s45720" xml:space="preserve"> ſupponatur itaq;</s> <s xml:id="echoid-s45721" xml:space="preserve"> perueniſſe ad punctũ c.</s> <s xml:id="echoid-s45722" xml:space="preserve"> <lb/>His itaq;</s> <s xml:id="echoid-s45723" xml:space="preserve"> ſic præ miſsis, patet quòd ſi linea s i (quę oſtenſa eſt per 51 th.</s> <s xml:id="echoid-s45724" xml:space="preserve"> 7 huius in ſpeculis columna.</s> <s xml:id="echoid-s45725" xml:space="preserve"> <lb/>ribus conuexis eſſe imago lineæ t h, & eſſe æ quidiſtans lineæ t h, & axi z k) ſit in aliquo corpore <lb/>uiſibili, & uiſus fuerit in puncto o ex parte concauitatis ſpeculi columnarisitunc forma lineæ s i re-<lb/>fle ctetur a d uiſum in puncto o à linea longitu dinis ſpeculi, quę eſt a b g:</s> <s xml:id="echoid-s45726" xml:space="preserve"> & diuerſabuntur imagines <lb/>eius ſecundum diuerſitatẽ diſtantiæ ſuæ a b axe ſpeculi, qui eſt z k.</s> <s xml:id="echoid-s45727" xml:space="preserve"> Quia enim angulus e b m eſt acu <lb/>tus:</s> <s xml:id="echoid-s45728" xml:space="preserve"> ergo per 15 p 1 angulus l b c eſt acutus:</s> <s xml:id="echoid-s45729" xml:space="preserve"> & linea e b c eſt in ſuperficie circuli b f:</s> <s xml:id="echoid-s45730" xml:space="preserve"> & linea l b eſt ſe-<lb/>mιdιameter illius circuli per 21 th.</s> <s xml:id="echoid-s45731" xml:space="preserve"> 7 huius:</s> <s xml:id="echoid-s45732" xml:space="preserve"> linea ergo e b c ſecat circulum, & eius pars, quæ eſt b c, <lb/>eſt intra circulum & intra concauitatem ſpeculi.</s> <s xml:id="echoid-s45733" xml:space="preserve"> Et ſimiliter eſt de linea o b, quoniam ipſa cadit <lb/>intra concauitatem ſpeculi:</s> <s xml:id="echoid-s45734" xml:space="preserve"> ideo quòd angulus o b l eſt acutus:</s> <s xml:id="echoid-s45735" xml:space="preserve"> & duo anguli o b l & c b l ſunt æqua <lb/>les:</s> <s xml:id="echoid-s45736" xml:space="preserve"> quonia n ipſi per 15 p 1 ſunt æquales duobus angulis q b m & m b e æ qualibus:</s> <s xml:id="echoid-s45737" xml:space="preserve"> & ſemidiameter <lb/>l b eſt per pendicularis ſuper ſperficiem contingentem columnam ſpeculi ſecundum lineam lon-<lb/>gitudιnis ſpeculi, tranſeuntem per punctum b.</s> <s xml:id="echoid-s45738" xml:space="preserve"> forma itaq;</s> <s xml:id="echoid-s45739" xml:space="preserve"> puncti c incidit ſpeculo per lineam <lb/>c b:</s> <s xml:id="echoid-s45740" xml:space="preserve"> & à puncto ſpeculi b reflectitur per lineam b o, & comprehen ditur à uiſu exiſtente in puncto <lb/>o.</s> <s xml:id="echoid-s45741" xml:space="preserve"> Item patet per 51 th.</s> <s xml:id="echoid-s45742" xml:space="preserve"> 7 huius, & ibι declaratum eſt, quòd ſuperficies contingens ſpeculum colu-<lb/>mnare in puncto g, eſt ſub puncto e centro uiſus:</s> <s xml:id="echoid-s45743" xml:space="preserve"> linea ergo e g ſecat illam ſuperficiem contingen-<lb/>tem.</s> <s xml:id="echoid-s45744" xml:space="preserve"> Secat ergo in puncto g (qui eſt punctus reflexionis) lineam in eodem puncto g contingen-<lb/>tem peripheriam ſectionis columnaris;</s> <s xml:id="echoid-s45745" xml:space="preserve"> quæ eſt communis ſectio ſuperficiei reflexi onis formę pun <lb/>cti t lineæ t h, & ſpeculi columnaris conuexi.</s> <s xml:id="echoid-s45746" xml:space="preserve"> Et quia ſecat illam lineam contin gentem in puncto i-<lb/>pſius ſpeculi, quod eſt g:</s> <s xml:id="echoid-s45747" xml:space="preserve"> ſecat ergo ſectionem oxygoniam, & cadit intra ipſam:</s> <s xml:id="echoid-s45748" xml:space="preserve"> cadit ergo intra con <lb/>cauitate in ſpeculi:</s> <s xml:id="echoid-s45749" xml:space="preserve"> & eſt linea g i:</s> <s xml:id="echoid-s45750" xml:space="preserve"> duæ ergo lineæ o g & g i cadunt intra concauitatem ſpeculi:</s> <s xml:id="echoid-s45751" xml:space="preserve"> & li-<lb/>nea z g eſt perpendicularis ſuper ſuperficiem contingentem columnam ſpeculi per 96 th.</s> <s xml:id="echoid-s45752" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s45753" xml:space="preserve"> <lb/>quoniam ducitur ab axe perpendiculariter ſuper lineam longitudinis ſpeculi, tranſeuntem per <lb/>punctum g:</s> <s xml:id="echoid-s45754" xml:space="preserve"> & duo anguli o g z & z g i ſunt æ quales per 15 p 1 ut prius.</s> <s xml:id="echoid-s45755" xml:space="preserve"> Forma ergo puncti i incidit <lb/>ſuperficιei concauæ ipſius ſpeculi ſecundum lineam i g:</s> <s xml:id="echoid-s45756" xml:space="preserve"> & à puncto ſpeculi greflectitur ad uiſum <lb/>exiſtentem in puncto o ſecundum lineam reflexionis, quæ eſt g o.</s> <s xml:id="echoid-s45757" xml:space="preserve"> Et eodem modo patet quòd for-<lb/>mæ puncti sincidit ſpeculo ſecundum lineam s a, & reflexctitur à puncto ſpeculi a ad uiſum exiſten-<lb/>tem in puncto o ſecundũ lineã reflexionis, quę eſt a o.</s> <s xml:id="echoid-s45758" xml:space="preserve"> Et etiã patuit in cõmento 51 th.</s> <s xml:id="echoid-s45759" xml:space="preserve"> 7 huius quo-<lb/>niam duæ lineæ h u & tu ſunt perpendiculares ſuper duas lineas contingentes ſectiones oxygo-<lb/>nias, tranſeuntes per duo puncta a & g.</s> <s xml:id="echoid-s45760" xml:space="preserve"> Imago ergo formæ puncti s eſt in linea h u per 36 th.</s> <s xml:id="echoid-s45761" xml:space="preserve"> 5 hu-<lb/>ius:</s> <s xml:id="echoid-s45762" xml:space="preserve"> ſed linea a o eſt linea reflexionis formæ punctis s:</s> <s xml:id="echoid-s45763" xml:space="preserve"> quoniam à puncto reflexionis, quod eſt a, pro-<lb/>ducitur ad uiſum exiſtentem in puncto o.</s> <s xml:id="echoid-s45764" xml:space="preserve"> Imago itaq;</s> <s xml:id="echoid-s45765" xml:space="preserve"> formę puncti s eſt in linea s o, per 37 th.</s> <s xml:id="echoid-s45766" xml:space="preserve"> 5 hu-<lb/>ius:</s> <s xml:id="echoid-s45767" xml:space="preserve"> punctum ergo h, quod eſt communis ſectio linearum h u & o a, eſt locus imaginis formæ pun-<lb/>cti s.</s> <s xml:id="echoid-s45768" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s45769" xml:space="preserve"> patet quòd punctum t eſt locus imaginis formæ puncti i.</s> <s xml:id="echoid-s45770" xml:space="preserve"> Ducatur quoq;</s> <s xml:id="echoid-s45771" xml:space="preserve"> li-<lb/>nea c l à puncto c ad punctum, centrum circuli b:</s> <s xml:id="echoid-s45772" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s45773" xml:space="preserve"> linea c l producta ultra punctum c perpen.</s> <s xml:id="echoid-s45774" xml:space="preserve"> <lb/>dicularis ſuper lineam contingentem circulum per 18 p 3:</s> <s xml:id="echoid-s45775" xml:space="preserve"> eſt ergo linea c l cathetus incidentiæ for-<lb/>mæ puncti c per definitionem illius catheti.</s> <s xml:id="echoid-s45776" xml:space="preserve"> Quia ergo forma puncti c reflectitur a d uiſum in pun-<lb/>ctũ o à puncto ſpeculi b:</s> <s xml:id="echoid-s45777" xml:space="preserve"> erit imago formæ puncti c in linea q c l, quæ eſt cathetus ſuæ incidentiæ:</s> <s xml:id="echoid-s45778" xml:space="preserve"> <lb/>ſed & in linea reflexionis, quæ eſt b o, neceſſe eſt eſſe eandem imaginem per 37 th.</s> <s xml:id="echoid-s45779" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s45780" xml:space="preserve"> Im ago <lb/>itaq;</s> <s xml:id="echoid-s45781" xml:space="preserve"> formæ puncti c neceſſariò erit in puncto, quod eſt communis ſectio linearum l c q & o b:</s> <s xml:id="echoid-s45782" xml:space="preserve"> hoc <lb/>autem poteſt eſſe in partibus diuerſis.</s> <s xml:id="echoid-s45783" xml:space="preserve"> Patuit enim per 11 th.</s> <s xml:id="echoid-s45784" xml:space="preserve"> 8 huius quòd imago formæ puncti, <lb/>quę reflectitur à cõcauitate circuli ſpeculi, quandoq;</s> <s xml:id="echoid-s45785" xml:space="preserve"> occurrit uiſui inter uiſum & ſpeculum:</s> <s xml:id="echoid-s45786" xml:space="preserve"> quan-<lb/>doq;</s> <s xml:id="echoid-s45787" xml:space="preserve"> ultra ſpeculum:</s> <s xml:id="echoid-s45788" xml:space="preserve"> quand oq;</s> <s xml:id="echoid-s45789" xml:space="preserve"> in centro uiſus:</s> <s xml:id="echoid-s45790" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s45791" xml:space="preserve"> ultra uiſum:</s> <s xml:id="echoid-s45792" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s45793" xml:space="preserve"> in ipſa ſuperficie <lb/>ſpeculi:</s> <s xml:id="echoid-s45794" xml:space="preserve"> & (ut patet per 40 th.</s> <s xml:id="echoid-s45795" xml:space="preserve"> 8 huius) quãdoq;</s> <s xml:id="echoid-s45796" xml:space="preserve"> apparet una imago:</s> <s xml:id="echoid-s45797" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s45798" xml:space="preserve"> duæ:</s> <s xml:id="echoid-s45799" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s45800" xml:space="preserve"> tres:</s> <s xml:id="echoid-s45801" xml:space="preserve"> <lb/>quand oq;</s> <s xml:id="echoid-s45802" xml:space="preserve"> quatuor.</s> <s xml:id="echoid-s45803" xml:space="preserve"> Imago ergo puncti c, cum formæ ipſius reflexio fiat à puncto peripheriæ circu-<lb/>li æquidiſtantis baſibus ſpeculi, erit ſorte in linea h q ultra ſpeculum:</s> <s xml:id="echoid-s45804" xml:space="preserve"> & ſortè erit ultra lineam b q:</s> <s xml:id="echoid-s45805" xml:space="preserve"> <lb/>& fortè ultra lineam b o retro uiſum:</s> <s xml:id="echoid-s45806" xml:space="preserve"> & fortè erit in linea b o inter uiſum & ſpeculum:</s> <s xml:id="echoid-s45807" xml:space="preserve"> & fortè erit <lb/>in puncto o, ſcilicet in ipſo centro uiſus:</s> <s xml:id="echoid-s45808" xml:space="preserve"> & fortè erit unica imago, forté duæ:</s> <s xml:id="echoid-s45809" xml:space="preserve"> fortè tres:</s> <s xml:id="echoid-s45810" xml:space="preserve"> fortè qua-<lb/>tuor.</s> <s xml:id="echoid-s45811" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s45812" xml:space="preserve"> locus imaginis formæ puncti c, uel alicuius puncti formæ lineæ s i (utpote illius, ſe-<lb/>cundum quem b c producta ultra punctum c ſecat lineam i s:</s> <s xml:id="echoid-s45813" xml:space="preserve"> quia & illud punctum reflectitur à <lb/>puncto ſpeculi columnaris coricaui, quod eſt b, ad uiſum exiſtentem in puncto o per 20 th.</s> <s xml:id="echoid-s45814" xml:space="preserve"> 5 hu-<lb/>ius) ſuerit punctum q:</s> <s xml:id="echoid-s45815" xml:space="preserve"> tunc linea h q t erit diameter imaginis formæ lineæ i s.</s> <s xml:id="echoid-s45816" xml:space="preserve"> Si ergo omnes imagi <lb/>nes omnium punctorum lineæ s i fuerint in linea h q t:</s> <s xml:id="echoid-s45817" xml:space="preserve"> tunc imago eius erit linea recta:</s> <s xml:id="echoid-s45818" xml:space="preserve"> nam mediũ <lb/>eius punctum, quod eſt punctum q, eſt in rectitudine duarum ſuarum extremitatum, quæ ſunth & <lb/>t.</s> <s xml:id="echoid-s45819" xml:space="preserve"> Quòd ſi locus imaginis formæ puncti c fuerit ultra punctum q:</s> <s xml:id="echoid-s45820" xml:space="preserve"> tunc imago lineæ rectæ, quæ eſt <lb/>s i, erit concaua:</s> <s xml:id="echoid-s45821" xml:space="preserve"> eiusq́;</s> <s xml:id="echoid-s45822" xml:space="preserve"> concauitas reſpiciet uiſum.</s> <s xml:id="echoid-s45823" xml:space="preserve"> Et ſi imago formæ puncti c fuerit in linea b o:</s> <s xml:id="echoid-s45824" xml:space="preserve"> <lb/>uel in puncto o centro uiſus:</s> <s xml:id="echoid-s45825" xml:space="preserve"> aut inter ſpeculum & uiſum:</s> <s xml:id="echoid-s45826" xml:space="preserve"> tunc uidebitur imago lineæ s i conue-<lb/>xa, cuius conuexitas reſpiciet uiſum.</s> <s xml:id="echoid-s45827" xml:space="preserve"> Et ſi fuerit imago formæ puncti c in linea b o retro ui-<lb/>ſum:</s> <s xml:id="echoid-s45828" xml:space="preserve"> tunc iterum uidebitur imago concaua, in cuius concauitate ſituabitur centrum uiſus.</s> <s xml:id="echoid-s45829" xml:space="preserve"> <lb/>Quòd ſi punctum c plures habuerit imagines:</s> <s xml:id="echoid-s45830" xml:space="preserve"> tunc linea s i plures habebit imagines:</s> <s xml:id="echoid-s45831" xml:space="preserve"> quarum <lb/>omnium extremitates cõiungchtur in punctis h & t, & media ipſorũ erunt diſtincta & ſeparata:</s> <s xml:id="echoid-s45832" xml:space="preserve"> & <lb/> <pb o="384" file="0686" n="686" rhead="VITELLONIS OPTICAE"/> linea h terit communis diameter omnium illiarum imaginum, quotcunq;</s> <s xml:id="echoid-s45833" xml:space="preserve"> ſuerint imagines:</s> <s xml:id="echoid-s45834" xml:space="preserve"> & ſor-<lb/>tè linea h t, quæ eſt diameter imaginis, erit maior quàm linea rei uiſæ, quæ s i, in modica quantita-<lb/>te.</s> <s xml:id="echoid-s45835" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s45836" xml:space="preserve"/> </p> <div xml:id="echoid-div1760" type="float" level="0" n="0"> <figure xlink:label="fig-0684-01" xlink:href="fig-0684-01a"> <variables xml:id="echoid-variables795" xml:space="preserve">t n g i l z x y m b c a l f h r a c d p e k o u</variables> </figure> </div> </div> <div xml:id="echoid-div1762" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables796" xml:space="preserve">f b d g t e h e</variables> </figure> <head xml:id="echoid-head1306" xml:space="preserve" style="it">26. Superſicie lineæ rect æ uel curuæ cureæ uiſæ ſuperficiem (in qua eſt axis ſpeculi columnaris conca <lb/>ui) or thogonaliter ſecante, centró uiſus exiſtente in utra ſuperficie: à circumferentia circuli <lb/>(qui eſt communis ſectio dictæ ſuperficiei & ſpeculi) fiet reſlexio: <lb/> imagó lineæ uiſæ quando erit rect a: uel aliquando conuexa.</head> <p> <s xml:id="echoid-s45837" xml:space="preserve">Eſto, ſicut in 52 th.</s> <s xml:id="echoid-s45838" xml:space="preserve"> 7 huius proponitur, linea t h in ſuperficie pla <lb/>na orthogonaliter ſecarite ſuperficiem, in qua ſunt centrum uiſus <lb/>e, & axis dati ſpeculi columnaris, qui ſit d f:</s> <s xml:id="echoid-s45839" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s45840" xml:space="preserve"> centrũ uiſus (quod <lb/>ſit e) in eadem ſuperficie lineæ t h.</s> <s xml:id="echoid-s45841" xml:space="preserve"> Facta quoq;</s> <s xml:id="echoid-s45842" xml:space="preserve"> figuratione 52 th 7 <lb/>huius, compleatur demonſtratio, ut in illa propoſitione:</s> <s xml:id="echoid-s45843" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s45844" xml:space="preserve"> ima <lb/>go lineæ rectæ, quæ eſt t h, curua.</s> <s xml:id="echoid-s45845" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s45846" xml:space="preserve"> ſpeculum idem, quod ibi <lb/>conuexum accipitur, aſſumatur concauum:</s> <s xml:id="echoid-s45847" xml:space="preserve"> & in locum imaginis <lb/>collocata intelligatur linea curua, ſecundum cuius terminos extre <lb/>mos ducatur etiam linea recta, quæ ſit in ſuperficie rei uiſæ:</s> <s xml:id="echoid-s45848" xml:space="preserve"> & cen-<lb/>trum uiſus diſponatur proportionaliter circa illam lineam in ea-<lb/>dem ſuperficie:</s> <s xml:id="echoid-s45849" xml:space="preserve"> tunc locus imaginis lineæ curuæ uel rectæ uiſæ e-<lb/>rit linea t h recta.</s> <s xml:id="echoid-s45850" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s45851" xml:space="preserve"> & fortè linea imaginis e-<lb/>rit æqualis recta uel fortè conuexa:</s> <s xml:id="echoid-s45852" xml:space="preserve"> ſicut oſtenſum eſt in 57 th.</s> <s xml:id="echoid-s45853" xml:space="preserve"> 8 hu <lb/>ius:</s> <s xml:id="echoid-s45854" xml:space="preserve"> & hic eodem modo eſt deducendum.</s> <s xml:id="echoid-s45855" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1763" type="section" level="0" n="0"> <head xml:id="echoid-head1307" xml:space="preserve" style="it">27. Superficie lineæ rect æ uiſæ, orthogonaliter axem ſpeculi co <lb/>lumnaris concaui ſecante, centro uiſus nõ exiſtente in eadem ſu-<lb/>perficie, reflexioné facta aduiſum æqualiter diſtantẽ ab ex-<lb/>tremis illius lineæ: eius imago uideditur cõcauitatis magnæuiſum reſpiciẽtis. Alhazen 52 n 6.</head> <p> <s xml:id="echoid-s45856" xml:space="preserve">Fiat omnimoda diſpoſitio figuræ, quæ in 53 th.</s> <s xml:id="echoid-s45857" xml:space="preserve"> 7 huius:</s> <s xml:id="echoid-s45858" xml:space="preserve"> dico quòd uerum eſt, quod proponi-<lb/>tur.</s> <s xml:id="echoid-s45859" xml:space="preserve"> Patet enim per ea, quæ in commento illius dicta ſunt, quòd puncta t & h (quia æ qualiter di-<lb/>ſtant à centro uiſus, puncto ſcilicet e) reflectuntur ad uiſum à duobus punctis oxygoniarum ſectio <lb/>num, cadentibus tamen in quodam circulo æ quidiſtante baſibus ſpeculi, qui circulus erit medius <lb/>inter lineam h t, & inter ſuperficiem tranſeuntem centrum uiſus e, ſecantem ſpeculum æ quidiſtan <lb/>ter baſibus ipſius ſpeculi.</s> <s xml:id="echoid-s45860" xml:space="preserve"> Sit ergo, ut forma puncti h reflectatur in punctum e à puncto ſpeculi b, <lb/>qui eſt punctus peripheriæ cuiuſdam ſectionis oxygoniæ (quæ eſt communis ſuperficiei reflexio-<lb/>nis & ſuperficiei ſpeculi) cadens in circulo b g:</s> <s xml:id="echoid-s45861" xml:space="preserve"> lineæ ergo h b & b e continẽt angulos æquales cum <lb/>linea contingente illum circulũ in puncto b.</s> <s xml:id="echoid-s45862" xml:space="preserve"> Et ſimiliter forma punctit reflectitur ad uiſum e à pun <lb/>cto ſpeculi g:</s> <s xml:id="echoid-s45863" xml:space="preserve"> & lineæ t g & g e continent angulos æ quales cum linea contingente circulum ſpeeuli <lb/>in puncto g.</s> <s xml:id="echoid-s45864" xml:space="preserve"> Lineæ quoq;</s> <s xml:id="echoid-s45865" xml:space="preserve"> h b & t g concurrunt in puncto l:</s> <s xml:id="echoid-s45866" xml:space="preserve"> & linea h b continet cum linea perpen-<lb/>diculari, quæ eſt b o, angulum acutum:</s> <s xml:id="echoid-s45867" xml:space="preserve"> linea ergo h b ſecat ſuperficiem contingentem ſuperficiem <lb/>columnæ in linea longitudinis, in qua eſt punctum b:</s> <s xml:id="echoid-s45868" xml:space="preserve"> linea itaq;</s> <s xml:id="echoid-s45869" xml:space="preserve"> b l cadit intra concauitatem colu-<lb/>mnæ:</s> <s xml:id="echoid-s45870" xml:space="preserve"> & ſimiliter linea g l.</s> <s xml:id="echoid-s45871" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s45872" xml:space="preserve"> duæ lineæ l f & g y cadũt intra concauitatẽ colũuę:</s> <s xml:id="echoid-s45873" xml:space="preserve"> & per <lb/>15 p 1 duo anguli l b d & b b r ſunt æ quales:</s> <s xml:id="echoid-s45874" xml:space="preserve"> cum ipſorũ contrapoſiti, qui ſunt e b o & o b h ſint æ qua <lb/>les per 20 th.</s> <s xml:id="echoid-s45875" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s45876" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s45877" xml:space="preserve"> duo anguli l g d & d g y ſunt æquales.</s> <s xml:id="echoid-s45878" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s45879" xml:space="preserve"> linea r y (quæ in <lb/>ſpeculo columnari conuexo eſt imago lineę t h) ſuerit nuncin aliquo uiſibili oppoſita ſpeculo co-<lb/>lumnari concauo, & centrum uiſus fuerit in puncto l:</s> <s xml:id="echoid-s45880" xml:space="preserve"> tunc form a puncti r incider in ſpeculo ſecun-<lb/>dum lineam r b, & reflectetur ad uiſum in punctum l à puncto ſpeculi b:</s> <s xml:id="echoid-s45881" xml:space="preserve"> & linea h u eſt perpendicu-<lb/>laris ſuper lineam contingentem ſectionem, in cuius peripheria eſt punctum b, à quo ſit reflexio:</s> <s xml:id="echoid-s45882" xml:space="preserve"> <lb/>imago ergo formæ puncti r erit in cathero r h per 36 th.</s> <s xml:id="echoid-s45883" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s45884" xml:space="preserve"> ſed & eadem imago neceſſario eſt <lb/>in linea reflexionis, quæ eſt b l.</s> <s xml:id="echoid-s45885" xml:space="preserve"> Erit ergo in communi illarum ſectione in puncto h.</s> <s xml:id="echoid-s45886" xml:space="preserve"> eft ergo pun-<lb/>ctum h imago punctir, ut hæc omnia patent per 37 th.</s> <s xml:id="echoid-s45887" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s45888" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s45889" xml:space="preserve"> declarabitur, quòd <lb/>forma puncti y incidet ſpeculo per lineam y g:</s> <s xml:id="echoid-s45890" xml:space="preserve"> & reflectetur per lineam g l à puncto ſpeculi g:</s> <s xml:id="echoid-s45891" xml:space="preserve"> & e-<lb/>ius imago uidebitur in puncto t.</s> <s xml:id="echoid-s45892" xml:space="preserve"> Et ducatur linea q u:</s> <s xml:id="echoid-s45893" xml:space="preserve"> hæc ergo ſecabit lineam r y:</s> <s xml:id="echoid-s45894" xml:space="preserve"> quoniam punctũ <lb/>ueſt ultra lineamillam r y, quæ eſt inter duo puncta q & u:</s> <s xml:id="echoid-s45895" xml:space="preserve"> puncta quoq;</s> <s xml:id="echoid-s45896" xml:space="preserve"> h, q, t, u ſunt omnia in ſu-<lb/>perficie circuli b g, ut patet ex præ miſsis:</s> <s xml:id="echoid-s45897" xml:space="preserve"> ſecet ergo linea q u lineam r y in punctom:</s> <s xml:id="echoid-s45898" xml:space="preserve"> punctum itaq;</s> <s xml:id="echoid-s45899" xml:space="preserve"> <lb/>m erit in ſuperficie tranſeunte per axem ſpeculi, & per centrum uiſus punctum l.</s> <s xml:id="echoid-s45900" xml:space="preserve"> Nam, ut in com-<lb/>mento præaſſumptæ 53 propoſitionis 7 huius patuit, punctal & q ſuntin illa ſuperficie:</s> <s xml:id="echoid-s45901" xml:space="preserve"> nam, ut ibi <lb/>acceptum eſt, patet quòd in illa ſuperficie, in qua erat centrum uiſus e, & axis ſpeculi, in eadem erat <lb/>linea e l d:</s> <s xml:id="echoid-s45902" xml:space="preserve"> ſed & illa ſuperficies ſecabat lineam h t in puncto q, & in linea e d cadebat punctum u:</s> <s xml:id="echoid-s45903" xml:space="preserve"> <lb/>ergo per 1 p 11 linea q u eſt in illa ſuperficie:</s> <s xml:id="echoid-s45904" xml:space="preserve"> ergo & punctum m.</s> <s xml:id="echoid-s45905" xml:space="preserve"> Et quia duo puncta m & l ſunt in ſu-<lb/>perficie tranſeunte per axem columnæ:</s> <s xml:id="echoid-s45906" xml:space="preserve"> ideo forma puncti m poteſt reflecti ad uiſum in pun-<lb/>ctum l in illa ſuperficie, & linea a z eſt communis ſectio ſuperficiei columnæ ſpeculi & ſuperficiei <lb/>tranſeunti per ſuum axem;</s> <s xml:id="echoid-s45907" xml:space="preserve"> & per punctum l, quod eſt centrum diſus.</s> <s xml:id="echoid-s45908" xml:space="preserve"> forma ergo puncti m reflecte-<lb/>tur ad uiſum in punctum l (quod eſt centrum uiſus) ab aliquo puncoto ſpeculi lineæ a z.</s> <s xml:id="echoid-s45909" xml:space="preserve"> Et ducatur <lb/> <pb o="385" file="0687" n="687" rhead="LIBER NONVS."/> linea e m, quæ erit in illa ſuperficie:</s> <s xml:id="echoid-s45910" xml:space="preserve"> & linea e l etiam erit in illa ſuperficie:</s> <s xml:id="echoid-s45911" xml:space="preserve"> & punctum e, ut ſuperà pa-<lb/>tuit, eſt elongatum à ſuperficie contingente columnam ſpeculi in linea a z, ut patet per 53 th.</s> <s xml:id="echoid-s45912" xml:space="preserve"> 7 hu-<lb/>ius, Si ergo linea a z ducatur in continuum & dire <lb/> <anchor type="figure" xlink:label="fig-0687-01a" xlink:href="fig-0687-01"/> ctum ultra punctum z, concurret cum duabus li-<lb/>neis e m & e l, quę ſunt in una ſuperficie cum linea <lb/>a z:</s> <s xml:id="echoid-s45913" xml:space="preserve"> concurrat ergo cum linea e m in puncto i, & <lb/>cum linea e l in puncto n.</s> <s xml:id="echoid-s45914" xml:space="preserve"> Punctum itaque n cadet <lb/>inter duo puncta e & l:</s> <s xml:id="echoid-s45915" xml:space="preserve"> quia punctum l eſt intra cõ <lb/>cauitatem columnę, & pũctum n eſt extra, in ipſi-<lb/>us conuexitate in ſuperficie columnæ:</s> <s xml:id="echoid-s45916" xml:space="preserve"> quoniam <lb/>eſt in linea longitudinis columnæ, quæ eſt a z:</s> <s xml:id="echoid-s45917" xml:space="preserve"> pun <lb/>ctum uerò e, quod in ſpeculis columnaribus con-<lb/>uexis ſuppoſitum fuit eſſe centrum uiſus, eſt elon-<lb/>gatum à ſuperficie columnaris ſpeculi.</s> <s xml:id="echoid-s45918" xml:space="preserve"> Patuit quo <lb/>que in demonſtratione 53 th.</s> <s xml:id="echoid-s45919" xml:space="preserve"> 7 huius quòd circu-<lb/>lus b g eſt medius inter lineam h t, & inter ſuperfi <lb/>ciem exeuntem à puncto e ęquidiſtantem baſibus <lb/>columnæ ſpeculi:</s> <s xml:id="echoid-s45920" xml:space="preserve"> & linea perpendicularis exiens <lb/>â puncto e ſuper lineam a z, eſt in ſuperficie tran-<lb/>ſeunte punctum e, & ſecante ſpeculum ęquidiſtan <lb/>ter baſibus columnæ:</s> <s xml:id="echoid-s45921" xml:space="preserve"> ergo linea perpendicularis <lb/>exiens à puncto e ſuper lineam a z n cadit extra <lb/>triangulum e i n, & uerſus partem puncti n:</s> <s xml:id="echoid-s45922" xml:space="preserve"> quo-<lb/>niam linea e n l d u eſt communis ſectio ſuperfi-<lb/>cierum reflexionis, ſecundum quas reflect untur <lb/>formæ punctorum h & t:</s> <s xml:id="echoid-s45923" xml:space="preserve"> quæ cum ſint oxyg oniæ <lb/>ſectiones, patet per 103 th.</s> <s xml:id="echoid-s45924" xml:space="preserve"> 1 huius quoniam i p ſæ <lb/>ſunt obliquè ſecantes axem ſpeculi:</s> <s xml:id="echoid-s45925" xml:space="preserve"> ergo & ipſarũ <lb/>communis ſectio obliquè incidit illi axi ſpeculi:</s> <s xml:id="echoid-s45926" xml:space="preserve"> ergo per 32 p 1 angulus e in eſt acutus:</s> <s xml:id="echoid-s45927" xml:space="preserve"> ergo per 15 <lb/>p 1 angulus m i a eſt acutus:</s> <s xml:id="echoid-s45928" xml:space="preserve"> & angulus m in erit obtuſus per 13 p 1.</s> <s xml:id="echoid-s45929" xml:space="preserve"> Educatur ergo per 12 p 1 à puncto <lb/>m linea perpendicularis ſuper lineam ai:</s> <s xml:id="echoid-s45930" xml:space="preserve"> quæ ſit m k, ſecans lineam a i in puncto k:</s> <s xml:id="echoid-s45931" xml:space="preserve"> punctum ergo k <lb/>erit inter puncta i & a.</s> <s xml:id="echoid-s45932" xml:space="preserve"> Quoniã ſi caderet inter pũcta i & n, fieret unius trigoni unus angulus rectus <lb/>& alter obtuſus, qui eſt m i n:</s> <s xml:id="echoid-s45933" xml:space="preserve"> q đ eſt impoſsibile:</s> <s xml:id="echoid-s45934" xml:space="preserve"> cadet ergo punctũ k inter pũcta i & a.</s> <s xml:id="echoid-s45935" xml:space="preserve"> Producatur <lb/>itaq;</s> <s xml:id="echoid-s45936" xml:space="preserve"> linea m k ultra punctũ k ad punctũ s, donec linea k s fiat æ qualis lineæ m k:</s> <s xml:id="echoid-s45937" xml:space="preserve"> erit ergo punctus s <lb/>extra ſuperficiem ſpecul, & ultra concauitatem eius:</s> <s xml:id="echoid-s45938" xml:space="preserve"> & punctus l, in quo eſt centrum uiſus, erit in-<lb/>tra ipſius ſpeculi concauitatem.</s> <s xml:id="echoid-s45939" xml:space="preserve"> ducatur itaque linea s l, quæ ſecabit lineam n k.</s> <s xml:id="echoid-s45940" xml:space="preserve"> Quoniam cum li-<lb/>nea n k ſit pars lineæ longitudinis ſpeculi:</s> <s xml:id="echoid-s45941" xml:space="preserve"> patet quòd ipſa eſt cadens inter puncta s & l:</s> <s xml:id="echoid-s45942" xml:space="preserve"> ſecet ergo <lb/>ipſam in puncto f:</s> <s xml:id="echoid-s45943" xml:space="preserve"> & à puncto f ducatur per 31 p 1 linea ęquidiſtans lineę k m, quæ producta ad axem <lb/>ſpeculi ſecet ipſum in puncto x:</s> <s xml:id="echoid-s45944" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s45945" xml:space="preserve"> linea f x.</s> <s xml:id="echoid-s45946" xml:space="preserve"> Erit ergo per 29 p 1 linea f x perpendicularis ſuper li-<lb/>neam longitudinis ſpeculi, quæ eſt a n:</s> <s xml:id="echoid-s45947" xml:space="preserve"> quoniam linea m k æ quidiſtans lineæ f x, eſt perpendicula-<lb/>ris ſuper ipſam a n:</s> <s xml:id="echoid-s45948" xml:space="preserve"> eritq́ue linea f x in ſuperficie tranſeunte per axem ſpeculi, & per punctum l.</s> <s xml:id="echoid-s45949" xml:space="preserve"> Eſt <lb/>ergo linea f x ſemidiameter circuli tranſeuntis per punctum f æ quidiſtanter baſibus columnę per <lb/>21 th.</s> <s xml:id="echoid-s45950" xml:space="preserve"> 7 huius:</s> <s xml:id="echoid-s45951" xml:space="preserve"> linea ergo f x eſt perpendicularis ſuper ſuperficiem contingentem columnam ſpeculi <lb/>ſecundum lineam longitudinis, quæ eſt a z.</s> <s xml:id="echoid-s45952" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s45953" xml:space="preserve"> linea m f.</s> <s xml:id="echoid-s45954" xml:space="preserve"> Quia ergo duorum trigonorum <lb/>m k f & f k s duo latera m k & k s ſunt æ qualia ex hypotheſi, & latus k f commune ambobus illis tri-<lb/>gonis, anguliq́ue ad punctum k ſunt recti:</s> <s xml:id="echoid-s45955" xml:space="preserve"> ergo per 4 p 1 latus m f eſt æ quale lateri f s:</s> <s xml:id="echoid-s45956" xml:space="preserve"> ergo per 5 p 1 <lb/>angulus f m s æqualis erit angulo f s m:</s> <s xml:id="echoid-s45957" xml:space="preserve"> linea uerò f x æquidiſtat lineæ s m:</s> <s xml:id="echoid-s45958" xml:space="preserve"> ergo per 29 p 1 angulus <lb/>x f l extrinſecus æ qualis eſt angulo f s m intrin ſeco, &anguli x f m & f m s ſunt æ quales, quia coal-<lb/>terni:</s> <s xml:id="echoid-s45959" xml:space="preserve"> angulus ergo x f m eſt æ qualis angulo x f l.</s> <s xml:id="echoid-s45960" xml:space="preserve"> Forma ergo puncti m incidens ſpeculo ſecundum <lb/>lineam m f, ſecun dum lineam reflexionis, quæ eſt f l, reflectitur ad uiſum exiſtẽtem in puncto l à pũ-<lb/>cto ſpeculi f, per 20 th.</s> <s xml:id="echoid-s45961" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s45962" xml:space="preserve"> & linea x f eſt perpendicularis ſuper ſuperficiem contingentem ſpe-<lb/>culum in puncto f.</s> <s xml:id="echoid-s45963" xml:space="preserve"> Et quoniam linea m k eſt perpẽdicularis ſuper ſuperficiem ſpeculi:</s> <s xml:id="echoid-s45964" xml:space="preserve"> quia eſt per-<lb/>pendicularis ſuper lineam longitudinis, quæ eſt a z:</s> <s xml:id="echoid-s45965" xml:space="preserve"> patet quòd linea m k eſt cathetus incidentiæ <lb/>formæ puncti m:</s> <s xml:id="echoid-s45966" xml:space="preserve"> in ipſa ergo erit locus imaginis formę puncti m per 36 th.</s> <s xml:id="echoid-s45967" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s45968" xml:space="preserve"> ſed & idem locus <lb/>eſt in linea reflexionis, quæ eſt l f.</s> <s xml:id="echoid-s45969" xml:space="preserve"> In illarum ergo linearum communi ſectione, quæ eſt punctus s, <lb/>eſt locus imaginis formæ puncti m per 37 th.</s> <s xml:id="echoid-s45970" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s45971" xml:space="preserve"> Et quia duæ lineę r y & h t ſunt æquidiſtantes <lb/>& perpendiculares ſuper ſuperficiem tranſeuntem per axem ſpeculi & per centrum uiſus, quod eſt <lb/>nunc punctũ l:</s> <s xml:id="echoid-s45972" xml:space="preserve"> quoniam linea h t taliter fuit diſpoſita in 53 th.</s> <s xml:id="echoid-s45973" xml:space="preserve"> 7 huius:</s> <s xml:id="echoid-s45974" xml:space="preserve"> duę igitur ſuperficies unifor-<lb/>miter exeuntes à duabus lineis h t & r y erunt æ quidiſtantes & perpendiculares ſuper ſuperficiem <lb/>tranſeuntem per axem per 18 p 11.</s> <s xml:id="echoid-s45975" xml:space="preserve"> Et quia linea r y eſt perpendicularis ſuper ſuperficiem tranſeuntẽ <lb/>per axem & per punctum l:</s> <s xml:id="echoid-s45976" xml:space="preserve"> ideo per 18 p 11 ſuperficies duarum linearum, quæ ſuntr my & m s, erit <lb/>perpendicularis ſuper ſuperficiem tranſeuntem per axem, & per punctum l:</s> <s xml:id="echoid-s45977" xml:space="preserve"> & erit per 19 th.</s> <s xml:id="echoid-s45978" xml:space="preserve"> 1 huius <lb/>linea m s communis ſectio illarum duarum ſuperficierum.</s> <s xml:id="echoid-s45979" xml:space="preserve"> Et quia linea a k, cum ſit pars lineę longi <lb/>tudinis ſpeculi, quæ eſt a z, eſt in ſuperficie tranſeunte per axem:</s> <s xml:id="echoid-s45980" xml:space="preserve"> quia omnis ſuperficies ſecans co-<lb/> <pb o="386" file="0688" n="688" rhead="VITELLONIS OPTICAE"/> lumnam ſecundum lineam longitudinis per ęqualia, trãfit per axem illius columnę, ut patet per 93 <lb/>th.</s> <s xml:id="echoid-s45981" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s45982" xml:space="preserve"> Sed & linea a k eſt perpendicularis ſuper lineam m s, quæ eſt communis ſectio inter ſuքfi <lb/>ciem tranſeuntem per axem, & inter ſuperficiem duarum linearum, quæ ſuntrm & m s:</s> <s xml:id="echoid-s45983" xml:space="preserve"> ergo linea <lb/>a k n eſt erecta ſuper ſuperficiem r m s:</s> <s xml:id="echoid-s45984" xml:space="preserve"> & linea a n eſt æquidiſtans axi ſpeculi:</s> <s xml:id="echoid-s45985" xml:space="preserve">ergo per 8 p 11 erit axis <lb/>ſpeculi perpendicularis ſuper ſuperficiem, in qua ſunt duæ lineæ r m & m s.</s> <s xml:id="echoid-s45986" xml:space="preserve"> Illa ergo ſuperficies eſt <lb/>perpendicularis ſuper axem columnæ.</s> <s xml:id="echoid-s45987" xml:space="preserve"> Punctum itaque s eſt in ſuperficie exeunte ex linea r y per-<lb/>pendiculariter ſuper axem columnæ ſpeculi:</s> <s xml:id="echoid-s45988" xml:space="preserve">ſed linea h t eſt in ſuperficie perpendiculari ſuper axẽ <lb/>ſpeculi, æquidiſtanti ſuperficiei exeunti exlinea r y:</s> <s xml:id="echoid-s45989" xml:space="preserve"> punctum ergo s eſt extra lineam h t, & propin <lb/>quius puncto l centro uiſus, quàm ſint duo puncta h & t:</s> <s xml:id="echoid-s45990" xml:space="preserve"> & duo puncta h & t ſunt imagines forma-<lb/>rum duorum punctorum r & y:</s> <s xml:id="echoid-s45991" xml:space="preserve"> & punctum s eſt imago formæ punctim.</s> <s xml:id="echoid-s45992" xml:space="preserve"> Palàm ergo quia imago for <lb/>mæ lineę r m y eſt linea tranſiens per puncta h, s, t:</s> <s xml:id="echoid-s45993" xml:space="preserve"> ſed talis linea eſt arcualis:</s> <s xml:id="echoid-s45994" xml:space="preserve"> quia punctum s eſt ex-<lb/>tra rectitudinem lineę h t.</s> <s xml:id="echoid-s45995" xml:space="preserve"> Tranſeat itaque per puncta h, s, tlinea arcualis, quæ ſit h s t.</s> <s xml:id="echoid-s45996" xml:space="preserve"> Et quia linea <lb/>h t ſecundum hypotheſim 53 th.</s> <s xml:id="echoid-s45997" xml:space="preserve"> 7 huius fuit elongata à conuexo columnæ:</s> <s xml:id="echoid-s45998" xml:space="preserve"> erit linea h t ultra ſuper-<lb/>ficiem ſpeculi, reſpectu punctil, quod eſt nunc centrum uiſus.</s> <s xml:id="echoid-s45999" xml:space="preserve"> Etiam ſuprà oſtenſum eſt quòd pun-<lb/>ctum s eſt extra concauitatem ſpeculi, reſpectu punctil, & punctum l eſt intra concauitatem ſpecu-<lb/>li:</s> <s xml:id="echoid-s46000" xml:space="preserve"> punctum ergo l, quod eſt centrum uiſus, eſt extra ſuperficiem, in qua eſt linea h s t:</s> <s xml:id="echoid-s46001" xml:space="preserve"> arcualitas er-<lb/>go lineæ h s tapparebit uiſui manifeſtè.</s> <s xml:id="echoid-s46002" xml:space="preserve"> Et quia punctum f eſt in ſuperficie columnæ ſpeculi extra <lb/>ſuperficiem circuli b g, & linea t h eſt ultra ſpeculum in ſuperficie circuli b g:</s> <s xml:id="echoid-s46003" xml:space="preserve"> quoniam eſt in ſuperfi-<lb/>cie trigonil h t:</s> <s xml:id="echoid-s46004" xml:space="preserve"> erit linea l f s altior quàm ſuperficies trigoni l h t:</s> <s xml:id="echoid-s46005" xml:space="preserve"> linea ergo l s erit altior duabus li-<lb/>neis l h & l t, reſpectu uiſus l:</s> <s xml:id="echoid-s46006" xml:space="preserve"> punctum ergo s eſt altius quàm duo puncta h & t.</s> <s xml:id="echoid-s46007" xml:space="preserve"> Linea ergo h s t ap-<lb/>parebit uiſui exiſtenti in puncto l concaua, concauitate uiſum reſpiciente.</s> <s xml:id="echoid-s46008" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s46009" xml:space="preserve"/> </p> <div xml:id="echoid-div1763" type="float" level="0" n="0"> <figure xlink:label="fig-0687-01" xlink:href="fig-0687-01a"> <variables xml:id="echoid-variables797" xml:space="preserve">u r h d x b y m l o n g f y k q z t c c s a</variables> </figure> </div> </div> <div xml:id="echoid-div1765" type="section" level="0" n="0"> <head xml:id="echoid-head1308" xml:space="preserve" style="it">28. Superficie incidentiæ lineærectæuiſæ, obliquè ſecantis axem ſpeculi columnaris concaui, <lb/>centro uiſus exiſtente in eadem ſuperficie: imago uidetur concaua reſpectu uiſus & conuerſa ſe-<lb/>cundum ſitum. Alhazen 53 n 6.</head> <p> <s xml:id="echoid-s46010" xml:space="preserve">Eſto ſpeculum columnare concauum, quod ſecetur per ſuperficiem obliquã ſuper axem:</s> <s xml:id="echoid-s46011" xml:space="preserve"> erit er-<lb/>go communis ſectio illius ſuperficiei & ſuperficiei ſpeculi ſectio oxygonia per 103 th.</s> <s xml:id="echoid-s46012" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s46013" xml:space="preserve"> ſit illa <lb/>fectio a b g:</s> <s xml:id="echoid-s46014" xml:space="preserve"> ſed in 11 huius oſtenſum eſt, quòd quandoque in ſuperficie oxygonię ſectionis à puncto <lb/>reflexionis erit linea perpendicularis ſuper ſuperficiem contingentem ſpeculum columnare, ex cu <lb/>ius duobus terminis ſcilicet ex duabus communibus ſectionibus ſui & ſuperficiei ipſius ſpeculi fit <lb/>reflexio formarum ad uiſum.</s> <s xml:id="echoid-s46015" xml:space="preserve"> Sit ergo in ſectione a b g huiuſmodi perpendicularis, quæ ſit g a:</s> <s xml:id="echoid-s46016" xml:space="preserve"> & ſit <lb/>linea b e k perpendicularis ſuper lineam contingentem peripheriam ſectionis in puncto b:</s> <s xml:id="echoid-s46017" xml:space="preserve"> & ſit pũ-<lb/>ctum b prope punctum g, ita quòd linea ducta à puncto b cum linea perpendiculari ducta ſuper ſu-<lb/>perficiem ſpeculi à puncto reflexionis (qui ſit g) contineat ſuper axem ſpeculiangulum acutum.</s> <s xml:id="echoid-s46018" xml:space="preserve"> Pa <lb/>tet ergo per 44th.</s> <s xml:id="echoid-s46019" xml:space="preserve"> 7 huius quoniam linea b e k ſecabit lineam perpendicularem, quæ eſt g a, ſub axe <lb/>ſpeculi, & continebit cum ipſa angulum acutum:</s> <s xml:id="echoid-s46020" xml:space="preserve"> fiat ergo illarum linearum ſectio in puncto e.</s> <s xml:id="echoid-s46021" xml:space="preserve"> An-<lb/>gulus ergo b e g erit acutus per 32 p 1, ut patet:</s> <s xml:id="echoid-s46022" xml:space="preserve"> cadatq́;</s> <s xml:id="echoid-s46023" xml:space="preserve"> punctum k in peripheriam ſectionis:</s> <s xml:id="echoid-s46024" xml:space="preserve"> & à pun <lb/>cto g ducatur per 31 p 1 linea æquidiſtans line ę b k:</s> <s xml:id="echoid-s46025" xml:space="preserve"> quę ſit linea g d:</s> <s xml:id="echoid-s46026" xml:space="preserve"> erit ergo angulus d g e per 29 p 1 <lb/>æqualis angulo b e g:</s> <s xml:id="echoid-s46027" xml:space="preserve"> ergo uterque eſt acutus:</s> <s xml:id="echoid-s46028" xml:space="preserve"> linea ergo g d erit intra concauitatem ſpeculi:</s> <s xml:id="echoid-s46029" xml:space="preserve"> quoniã <lb/>linea à puncto g termino perpendicularis, quæ eſt a g, extra ſectionem ducta continget ſectionem, <lb/>& continebit angulum rectum cum linea a g:</s> <s xml:id="echoid-s46030" xml:space="preserve"> aut non continget, & continebit angulum obtuſum-<lb/> <anchor type="figure" xlink:label="fig-0688-01a" xlink:href="fig-0688-01"/> Fiatitaque per 23 p 1 ſuper punctum g terminum lineæ e g an-<lb/>gulus ęqualis angulo e g d:</s> <s xml:id="echoid-s46031" xml:space="preserve"> quiſit e g l linea ergo g l concurret <lb/>cum linea b e k per 14 th.</s> <s xml:id="echoid-s46032" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s46033" xml:space="preserve"> ideo quòd anguli g e l & lge <lb/>ambo ſunt acuti:</s> <s xml:id="echoid-s46034" xml:space="preserve"> ſit concurſus in puncto l:</s> <s xml:id="echoid-s46035" xml:space="preserve"> qui ſit punctus li-<lb/>neæ b k:</s> <s xml:id="echoid-s46036" xml:space="preserve"> & in linea le, ut contigerit, ſignetur punctũ m:</s> <s xml:id="echoid-s46037" xml:space="preserve"> & du-<lb/>catur linea a m:</s> <s xml:id="echoid-s46038" xml:space="preserve"> erit ergo angulus m a g acutus ք 32 p 1:</s> <s xml:id="echoid-s46039" xml:space="preserve"> ideo, <lb/>ut prius oſtendimus, quia angulus m e g, qui eſt maior angu-<lb/>lo m a g, cum ſit ei extrinſecus, eſt acutus, ut patet ex pręmiſ <lb/>ſis:</s> <s xml:id="echoid-s46040" xml:space="preserve"> linea ergo m a cadit intra ſectionẽ.</s> <s xml:id="echoid-s46041" xml:space="preserve"> Fiat quoque ſuper pũ-<lb/>ctum a terminũ lineæ a g angulus æqualis angulo g a m, qui <lb/>ſit angulus g a d:</s> <s xml:id="echoid-s46042" xml:space="preserve"> linea enim a d concurret cum linea g d per <lb/>14th.</s> <s xml:id="echoid-s46043" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s46044" xml:space="preserve"> ideo quia anguli d g a & d a g ſunt acuti:</s> <s xml:id="echoid-s46045" xml:space="preserve"> ſit er-<lb/>go concurſus in puncto d.</s> <s xml:id="echoid-s46046" xml:space="preserve"> Linea itaque a d ſecabit lineam <lb/>b k, concurrens cum ipſa per 2th.</s> <s xml:id="echoid-s46047" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s46048" xml:space="preserve"> quoniam concur-<lb/>rit cum eius ęquidiſtante, quæ eſt d g:</s> <s xml:id="echoid-s46049" xml:space="preserve"> ſecet ergo ipſam b k in puncto t.</s> <s xml:id="echoid-s46050" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s46051" xml:space="preserve"> linealk fuerit in <lb/>aliquo corpore uiſibili, & centrum uiſus fuerit in puncto d:</s> <s xml:id="echoid-s46052" xml:space="preserve">tunc forma puncti l uidebitur in puncto <lb/>ſpeculi g, quod eſt punctum reflexionis:</s> <s xml:id="echoid-s46053" xml:space="preserve"> & hoc accidit per 10 huius:</s> <s xml:id="echoid-s46054" xml:space="preserve"> ideo quia forma punctil reflecti <lb/>tur ad uiſum exiſtentem in puncto d à puncto ſpeculig, & linea k l b, quæ eſt cathetus incidentię for <lb/>mæ punctil, æquidiſtat lineæ g d, quæ eſt linea reflexionis.</s> <s xml:id="echoid-s46055" xml:space="preserve"> Nunquam ergo concurrent:</s> <s xml:id="echoid-s46056" xml:space="preserve"> & ſic locus <lb/>imaginis formæ punctil erit in puncto reflexionis, quod eſt g.</s> <s xml:id="echoid-s46057" xml:space="preserve"> Similiter quoque forma puncti m re-<lb/>flectitur ad uiſum exiſtentem in puncto d à puncto ſpeculi, quod eſt a:</s> <s xml:id="echoid-s46058" xml:space="preserve"> & cathetus incidentiæ, quæ <lb/>eſt linea b m k, ſecat lineam reflexionis, quæ eſt a d, in puncto t:</s> <s xml:id="echoid-s46059" xml:space="preserve"> ergo punctũ t eſt locus imaginis for <lb/> <pb o="387" file="0689" n="689" rhead="LIBER NONVS."/> mæ punctim per 37 th.</s> <s xml:id="echoid-s46060" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s46061" xml:space="preserve"> Tranſeatitaque per punctum d, quod eſt centrum uiſus, ſuperficies <lb/>plana æ quidiſtans baſibus columnæ:</s> <s xml:id="echoid-s46062" xml:space="preserve"> hæc ergo ſuperficies ſecabit columnam ſpeculi ſecundum cir <lb/>culum per 100 th.</s> <s xml:id="echoid-s46063" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s46064" xml:space="preserve"> qui circulus ſit p o r.</s> <s xml:id="echoid-s46065" xml:space="preserve"> Et quoniam centrum uiſus deſt in ſuperficie ſectio-<lb/>nis a b g:</s> <s xml:id="echoid-s46066" xml:space="preserve"> palàm quòd ille circulus p o r ſecabιt ſectionẽ oxygoniam a b g in duobus punctis per 104 <lb/>th.</s> <s xml:id="echoid-s46067" xml:space="preserve">1 huιus:</s> <s xml:id="echoid-s46068" xml:space="preserve"> ſuperficies ergo illius circuli ſecabit lineam b k:</s> <s xml:id="echoid-s46069" xml:space="preserve"> quoniam ſecat lineam g d æquidiſtantẽ <lb/>lineę b k:</s> <s xml:id="echoid-s46070" xml:space="preserve"> ducitur enim per punctum d:</s> <s xml:id="echoid-s46071" xml:space="preserve">ſit ergo, ut ſecet lineam b k in puncto k:</s> <s xml:id="echoid-s46072" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s46073" xml:space="preserve"> centrum circuli <lb/>p o r punctum h:</s> <s xml:id="echoid-s46074" xml:space="preserve"> & ducatur linea k h, quæ ducta per circulum ſecet ipſius peripheriam in puncto p:</s> <s xml:id="echoid-s46075" xml:space="preserve"> <lb/>& ducatur linea d h:</s> <s xml:id="echoid-s46076" xml:space="preserve"> quæ producta ad peripheriam circuli incidat ipſi in punctor.</s> <s xml:id="echoid-s46077" xml:space="preserve"> Forma ergo pun <lb/>cti k reflectitur ad uiſum exiſtentem in puncto d ab aliquo puncto arcusr p, ut patet per 27 th.</s> <s xml:id="echoid-s46078" xml:space="preserve"> 8 hu-<lb/>ius, ubi hoc oſtenſum eſt de reflexione formæ uiſibilis ad uiſum ſecundum talem ſitum ab aliquo <lb/>puncto peripheriæ circuli.</s> <s xml:id="echoid-s46079" xml:space="preserve"> Sit ergo ut fiat illa reflexio àpuncto ſpeculi ſcilicetarcus p r, quod ſit pũ <lb/>ctum o:</s> <s xml:id="echoid-s46080" xml:space="preserve"> & ducantur lineę k o, d o, h o:</s> <s xml:id="echoid-s46081" xml:space="preserve"> ergo angulus k o h eſt æqualis angulo h o d per 20 th.</s> <s xml:id="echoid-s46082" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s46083" xml:space="preserve"> <lb/>Et quoniam linea reflexionis, quę eſt d o, ſecat ſemidiametrum h p:</s> <s xml:id="echoid-s46084" xml:space="preserve"> ideo quia linea d h r tranſit per <lb/>centrum circuli, citra quam reſpectu puncti o ducitur linea d o:</s> <s xml:id="echoid-s46085" xml:space="preserve"> hęc ergo ſecat ſemidiametrum h p:</s> <s xml:id="echoid-s46086" xml:space="preserve"> <lb/>ſit, ut ſecetipſam in puncto n.</s> <s xml:id="echoid-s46087" xml:space="preserve"> Eſt aũt linea k h p cathetus incidentię formę puncti k:</s> <s xml:id="echoid-s46088" xml:space="preserve"> ergo per 37 th.</s> <s xml:id="echoid-s46089" xml:space="preserve"> 5 <lb/>huius punctum n eſt locus imaginis formæ puncti k.</s> <s xml:id="echoid-s46090" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s46091" xml:space="preserve"> linea k d:</s> <s xml:id="echoid-s46092" xml:space="preserve"> quæ per 19 th.</s> <s xml:id="echoid-s46093" xml:space="preserve"> 1 huius <lb/>erit cõmunis ſectio ſuperficiei circuli p o r & ſectiõis a b g, uel pars illius cõmunis ſectionis:</s> <s xml:id="echoid-s46094" xml:space="preserve"> nã duo <lb/>puncta k & d ſunt in utraq;</s> <s xml:id="echoid-s46095" xml:space="preserve"> illarũ ſuperficierũ, & nihil deſuքficie ſectionis oxygoniæ (quę eſt a b g) <lb/>eſt in ſuperficie circuli r p, niſi linea k d, uellinea, cuius pars eſt linea k d:</s> <s xml:id="echoid-s46096" xml:space="preserve"> punctũ ergo g eſt extra cir-<lb/>culũ & ſimiliter pũctũ b:</s> <s xml:id="echoid-s46097" xml:space="preserve"> & ſunt in ſuperficie ſectionis:</s> <s xml:id="echoid-s46098" xml:space="preserve"> & punctũn eſt in ſuperficie circuli r o:</s> <s xml:id="echoid-s46099" xml:space="preserve"> & for-<lb/>ma imaginis lineę l m k tranſit per puncta g, t, n:</s> <s xml:id="echoid-s46100" xml:space="preserve"> linea uerò pertranſiens hęc puncta eſt arcualis:</s> <s xml:id="echoid-s46101" xml:space="preserve"> quia <lb/>ſuperſicies ſectionis eſt decliuis ſuper ſuperficiem columnę per 103 th.</s> <s xml:id="echoid-s46102" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s46103" xml:space="preserve"> lõgior ergo diameter <lb/>ipſius ſectionis non tranſit per totum axem colũnæ, neq;</s> <s xml:id="echoid-s46104" xml:space="preserve"> eſt ſuperficies ſectionis æ quidiſtans baſi <lb/>columnæ:</s> <s xml:id="echoid-s46105" xml:space="preserve"> linea ergo t n g, quæ eſt imago lineæ rectę k m l, cuius ſuperficies ſecat axem ſpeculi obli <lb/>què, eſt curua maximæ curuitatis:</s> <s xml:id="echoid-s46106" xml:space="preserve"> & eius concauitas reſp cit uiſum exiſtẽtem in puncto d.</s> <s xml:id="echoid-s46107" xml:space="preserve"> Et quia <lb/>punctum t eſt imago formæ punctim:</s> <s xml:id="echoid-s46108" xml:space="preserve"> & punctum n imago formę puncti k:</s> <s xml:id="echoid-s46109" xml:space="preserve"> & punctum g eſt imago <lb/>formę punctil:</s> <s xml:id="echoid-s46110" xml:space="preserve"> patet quòd imago lineæ l m k eſt cõuerſa, ita quòd ſuperior punctus imaginis, reſpe-<lb/>ctu uiſus, quieſt g, correſpondet infimo puncto lineæuiſę, quieſt l, & infimus punctus imaginis, qui <lb/>eſt n, correſpondet ſupremo puncto lineæ uiſæ, qui eſt k.</s> <s xml:id="echoid-s46111" xml:space="preserve"> Sic ergo ſitus partium imaginis non eſt cõ <lb/>formis ſitui partium rei uiſæ, ſed conuerſus & difformis.</s> <s xml:id="echoid-s46112" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s46113" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s46114" xml:space="preserve"> exhac <lb/>propoſitione & duabus præmiſsis, quòd lineę rectę ęquidiſtátes axi ſpeculi columnaris concaui, & <lb/>ęquidiſtantes baſi eius, & etiá illæ, quę.</s> <s xml:id="echoid-s46115" xml:space="preserve"> ſunt obliquę ſuper ſuperficiem eius, quandoq;</s> <s xml:id="echoid-s46116" xml:space="preserve"> uidebũtur ar <lb/>cuales:</s> <s xml:id="echoid-s46117" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s46118" xml:space="preserve"> rectę:</s> <s xml:id="echoid-s46119" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s46120" xml:space="preserve"> cõuerſę.</s> <s xml:id="echoid-s46121" xml:space="preserve"> Forma ergo eorũ, quę cõprehenduntur in ſpeculis colũ-<lb/>narib, concauis, quandoq;</s> <s xml:id="echoid-s46122" xml:space="preserve"> erit directa, cõformis in ſuo ſitu ſitui partiũ rei uiſæ:</s> <s xml:id="echoid-s46123" xml:space="preserve"> & quandoq;</s> <s xml:id="echoid-s46124" xml:space="preserve"> erit dif-<lb/>formis, cõuerſum habẽs ſitũ ſuarum partiũ, reſpectu uiſus partiũ rei uiſæ, & in reſpectu ad uiſum.</s> <s xml:id="echoid-s46125" xml:space="preserve"/> </p> <div xml:id="echoid-div1765" type="float" level="0" n="0"> <figure xlink:label="fig-0688-01" xlink:href="fig-0688-01a"> <variables xml:id="echoid-variables798" xml:space="preserve">p b g o n m d r h e t a k</variables> </figure> </div> <figure> <variables xml:id="echoid-variables799" xml:space="preserve">d s p i t k n u b e a f q l h m r</variables> </figure> </div> <div xml:id="echoid-div1767" type="section" level="0" n="0"> <head xml:id="echoid-head1309" xml:space="preserve" style="it">29. Imago lineæ rectæ exicttentis in ſuperficie ſpeculum co-<lb/>lumnare concauumtrans axem orthogonaliter ſecante, cen-<lb/>tró uiſus exiſtente in eadem ſuperficie, uidebitur recta: quan <lb/>do maior: quando æqualis: quando minor reuiſa: ſed ſem-<lb/>per conuerſum habens ſitum: & quando una: quando plu-<lb/>res imagines uiſui occurrent. Alhazen 54 n 6.</head> <p> <s xml:id="echoid-s46126" xml:space="preserve">Sit ſecundum diſpoſitionem 48th.</s> <s xml:id="echoid-s46127" xml:space="preserve"> 8 huius circulus a b z in ſu <lb/>perficie ſpeculi columnaris concaui æquidiſtans baſibus ſpecu-<lb/>li:</s> <s xml:id="echoid-s46128" xml:space="preserve">cuius centrum e:</s> <s xml:id="echoid-s46129" xml:space="preserve"> & ſit centrum uiſus in puncto d:</s> <s xml:id="echoid-s46130" xml:space="preserve"> erit ergo li-<lb/>nea d g, ut in prædicta 48 præmiſſum eſt, perpendiculariter ere-<lb/>cta ſuper ſuperficiem circuli:</s> <s xml:id="echoid-s46131" xml:space="preserve"> & ſint duæ lineę e a & e b perpendi <lb/>culares ſuper ſuperficies cõtingentes ſuperficiem columnę ſpe <lb/>culi:</s> <s xml:id="echoid-s46132" xml:space="preserve"> & erit ſuքficies triangulι d e g քpendiculariter erecta ſuper <lb/>ſuperficiẽ circuli a b z per 18 p 11:</s> <s xml:id="echoid-s46133" xml:space="preserve"> quia linea g d eſt perpendicula-<lb/>ris ſuper ſuperficiem circuli:</s> <s xml:id="echoid-s46134" xml:space="preserve"> hoc eſt ſuper eam ſuperficiem, cu-<lb/>ius ſectio efficit circulum a b z.</s> <s xml:id="echoid-s46135" xml:space="preserve"> Superficies ergo trigoni d e g, ut <lb/>patet per 19 p 11 & per 92 th.</s> <s xml:id="echoid-s46136" xml:space="preserve"> 1 huius, tranſit per totum axem ſpe-<lb/>culi, & per centrum uiſus, quod eſt punctum d:</s> <s xml:id="echoid-s46137" xml:space="preserve"> & neutra ſuper-<lb/>ficies earum, quæſunt d b o & d a o, quæ ſecant ſe in linea d o, ut <lb/>patet per 19 th.</s> <s xml:id="echoid-s46138" xml:space="preserve"> 1 huius, tranſit per totum axẽ:</s> <s xml:id="echoid-s46139" xml:space="preserve"> & in neutra illarũ <lb/>ſuperficierum eſt aliquid de axe, niſipunctum e, quod eſt cen-<lb/>trum circuli a b z.</s> <s xml:id="echoid-s46140" xml:space="preserve"> Vtraque ergo ſuperficies, quęſunt d b o & d a <lb/>o, ſecat ſuperficiem columnarem ſpeculi ſecũdum oxygoniam <lb/>ſectionem:</s> <s xml:id="echoid-s46141" xml:space="preserve"> & fit reflexio formarum ad uiſum à duobus punctis <lb/>illarum ſectionũ quæ ſunt a & b, ut patet per præmiſſa in 48 th.</s> <s xml:id="echoid-s46142" xml:space="preserve"> <lb/>8 huius.</s> <s xml:id="echoid-s46143" xml:space="preserve"> Forma ergo puncti r reflectur ad uiſum exiſtentem in puncto d à puncto ſpeculi, quod eft <lb/>b:</s> <s xml:id="echoid-s46144" xml:space="preserve"> & forma puncti m reflectetur ad uiſum in punctum d à puncto ſpeculi, quod eſt a.</s> <s xml:id="echoid-s46145" xml:space="preserve"> Et quoniam <lb/> <pb o="388" file="0690" n="690" rhead="VITELLONIS OPTICAE"/> cathetus incidentiæ formæ puncti r eſt linear e n, ſecans lineam b d, quæ eſt linea reflexionis, in pũ <lb/>cto n:</s> <s xml:id="echoid-s46146" xml:space="preserve"> & cathetus incidentiæ formę puncti m eſt linea m e u, ſecans lineam reflexionis, quæ eſt a d in <lb/>puncto u:</s> <s xml:id="echoid-s46147" xml:space="preserve"> patet quòd puncta n & u ſunt loca imaginum formarum pũctorum r & m:</s> <s xml:id="echoid-s46148" xml:space="preserve"> & erit linea n u <lb/>diameter imaginis formæ lineę m r:</s> <s xml:id="echoid-s46149" xml:space="preserve"> & eſt minor quá linea m r, ut patuit in 49 th.</s> <s xml:id="echoid-s46150" xml:space="preserve"> 8 huius.</s> <s xml:id="echoid-s46151" xml:space="preserve"> Et ſimiliter <lb/>formę duorum punctorum h & l reflectentur ad uiſum in punctum d à duobus punctis ſpeculi, quæ <lb/>ſunt a & b:</s> <s xml:id="echoid-s46152" xml:space="preserve"> & erit per modum prius dictum, linea t k diameter imaginis formę lineę l h:</s> <s xml:id="echoid-s46153" xml:space="preserve"> & ſecundum.</s> <s xml:id="echoid-s46154" xml:space="preserve"> <lb/>pręmiſſa in 48th.</s> <s xml:id="echoid-s46155" xml:space="preserve"> 8 huius erit diameter imaginis t k æqualis diametro rei uiſæ, quę eſt linea l h.</s> <s xml:id="echoid-s46156" xml:space="preserve"> Simi <lb/>liter quoque linea p i erit diameter imaginis formę lineę f q:</s> <s xml:id="echoid-s46157" xml:space="preserve"> & eſt maior quã diameter rei uiſæ, quæ <lb/>eſt linea f q:</s> <s xml:id="echoid-s46158" xml:space="preserve"> & omnes iſtæ imaginies erunt cóuerſę, ut oſtẽſum eſt in 50 th.</s> <s xml:id="echoid-s46159" xml:space="preserve"> 8 huius.</s> <s xml:id="echoid-s46160" xml:space="preserve"> Siuerò centrũ ui-<lb/>ſus fuerit in puncto o, & formæ linearum, quæ ſunt p i, t k & n u, reflectantur ad uiſum in puncto o à <lb/>punctis ſpeculi, quæ ſunt a & b:</s> <s xml:id="echoid-s46161" xml:space="preserve"> tunc erit ecõuerſo.</s> <s xml:id="echoid-s46162" xml:space="preserve"> Erit enim diameter imaginis lineę p i, quę eſt li-<lb/>nea f q, minor diametro t k rei uiſæ & erit linea l h diameter imaginis lineę t k, & ęqualis ei:</s> <s xml:id="echoid-s46163" xml:space="preserve"> & erit li-<lb/>nea m r diameter imaginis lineæ n u, & maior quáilla:</s> <s xml:id="echoid-s46164" xml:space="preserve"> omnesq́;</s> <s xml:id="echoid-s46165" xml:space="preserve"> imagines linearum iſtarum recta-<lb/>rũ erunt rectæ ſed conuerſę ſecundum ſitum & ordinem partiũ, quem habent ipſæres.</s> <s xml:id="echoid-s46166" xml:space="preserve"> Nam dextrũ <lb/>rei fit ſiniſtrum imaginis, & ſiniſtrum reifit dextrum imaginis:</s> <s xml:id="echoid-s46167" xml:space="preserve"> & fimiliter eſt de partib.</s> <s xml:id="echoid-s46168" xml:space="preserve"> quę ſunt ſur <lb/>ſum & deorſum.</s> <s xml:id="echoid-s46169" xml:space="preserve"> Item cũ utraq;</s> <s xml:id="echoid-s46170" xml:space="preserve"> extremitatum harum linearum unicã habuerit imaginẽ, & aliquod <lb/>aliud punctũ in medio plures habuerit imagines:</s> <s xml:id="echoid-s46171" xml:space="preserve"> tunc forma illius lineę tot habebit imagines, quot <lb/>punctum medium ipſius:</s> <s xml:id="echoid-s46172" xml:space="preserve"> & omnes iſtę imagines copulabuntur ad puncta extrema illius imaginis:</s> <s xml:id="echoid-s46173" xml:space="preserve"> <lb/>& erit illa linea unica diameter omnium illarum imaginum.</s> <s xml:id="echoid-s46174" xml:space="preserve"> Et ſi utraq;</s> <s xml:id="echoid-s46175" xml:space="preserve"> extremitas illius lineę uel <lb/>altera ipſarum plures habuerit imagines, punctum uerò medium habuerit tantùm unam:</s> <s xml:id="echoid-s46176" xml:space="preserve"> iterũ tota <lb/>illa linea tot habebit imagines, quot eius puncta extrema ambo, uel ſaltem alterũ ſuum punctũ ex-<lb/>tremum.</s> <s xml:id="echoid-s46177" xml:space="preserve"> Et ſiutraq;</s> <s xml:id="echoid-s46178" xml:space="preserve"> extremitas uel altera plures habuerit imaginies, & ſimiliter punctum medium <lb/>multas habuerit imaginies:</s> <s xml:id="echoid-s46179" xml:space="preserve"> tunc tota linea habebit imagines ſecundum numerum maiorem:</s> <s xml:id="echoid-s46180" xml:space="preserve"> & hoc <lb/>patebit, ſicut patuit ſuprà de imaginibus ſpeculorum ſphæricorum concauorum.</s> <s xml:id="echoid-s46181" xml:space="preserve"> In ſpeculis enim <lb/>columnaribus concauis accidit fallacia in omnibus, quæ in eis comprehenduntur, ſicut accidit in <lb/>ſpeculis ſphæricis concauis:</s> <s xml:id="echoid-s46182" xml:space="preserve"> ſcilicet de formis ſpecierum uiſibilium:</s> <s xml:id="echoid-s46183" xml:space="preserve"> & de quantitatib, & de numero <lb/>ſuarum imaginum:</s> <s xml:id="echoid-s46184" xml:space="preserve"> & de conformitate ipſarum ad res, quarum ipſę ſunt imagines:</s> <s xml:id="echoid-s46185" xml:space="preserve"> & de difformita-<lb/>te ſitus ipſarum ſecundum cõuerſionem formarum partialium, cum omnib.</s> <s xml:id="echoid-s46186" xml:space="preserve"> fallacijs, quę appropriã <lb/>tur conuerſioni:</s> <s xml:id="echoid-s46187" xml:space="preserve"> & omnes fallaciæ ſunt in his, ut in ſpeculis prędictis ſphęricis concauis.</s> <s xml:id="echoid-s46188" xml:space="preserve"> Patet ergo <lb/>illud, quod proponebatur.</s> <s xml:id="echoid-s46189" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1768" type="section" level="0" n="0"> <head xml:id="echoid-head1310" xml:space="preserve" style="it">30. Line æ rectæuiſæ, non æquidiſtantis axi ſpeculi columnaris concaui, cuius ſuperſicies inci-<lb/>dentiæ ſecat axem obliquè, centro uiſus non exiſtente in eadem ſuperficie, uidetur imago curua, <lb/>diuerſæ curuitatis ſecundum diuer ſitatem ſui ſitus: & conuerſa.</head> <p> <s xml:id="echoid-s46190" xml:space="preserve">Fiat in iſto propoſito theoremate diſpoſitio totalis, quæ in 28 huius:</s> <s xml:id="echoid-s46191" xml:space="preserve"> apparebitq́;</s> <s xml:id="echoid-s46192" xml:space="preserve"> totum, quodibi <lb/>proponitur in his ſpeculis columnaribus concauis.</s> <s xml:id="echoid-s46193" xml:space="preserve"> Poſito itaq;</s> <s xml:id="echoid-s46194" xml:space="preserve">, ut aliqua linea recta non ęquidiſtet <lb/>axi ſpeculi columnaris concaui, cuius ſuperficies incidentiæ obliquè ſecat illum axem, ſicentrũ ui-<lb/>ſus fuerit in illa ſuperficie:</s> <s xml:id="echoid-s46195" xml:space="preserve"> tũc patet per 28 huius quòd imago illius lineę uidetur curua, reſpectu ui <lb/>ſus, & conuerſa ſecundum ſitum ipſius rei uiſæ.</s> <s xml:id="echoid-s46196" xml:space="preserve"> Quòd ſi centrum uiſus fuerit extra illam ſuperficiẽ <lb/>in linea erecta ſuper illam ſuperficiem à puncto d, in quo eſt illic centrum uiſus:</s> <s xml:id="echoid-s46197" xml:space="preserve"> tunc ſi à pũctis a, g, <lb/>o(à quibus fit ibi reflexio) erigãtur lineæ longitudinis ſpeculiper 101th.</s> <s xml:id="echoid-s46198" xml:space="preserve"> 1 huius, & inueniantur pũ <lb/>cta reflexionum formarũ punctorũ m, b, k:</s> <s xml:id="echoid-s46199" xml:space="preserve"> patebit ſecundũ modũ plurium præmiſſarum, quòd for-<lb/>mę punctorum k, m, b reflectentur ad uiſum ſecundum diſpoſitionẽ ſuo ſitui diuerſam:</s> <s xml:id="echoid-s46200" xml:space="preserve"> & ſecundũ <lb/>hoc diſponetur curuitas imaginũ & conuerſio figuræ.</s> <s xml:id="echoid-s46201" xml:space="preserve"> Quòd ſi centrum uiſus nõ fuerit in linea per-<lb/>pendiculariter erecta ſuper illã ſuperficiẽ à pũcto d:</s> <s xml:id="echoid-s46202" xml:space="preserve"> tũc à centro uiſus ducatur perpendicularis ſu-<lb/>per illam ſuperficiem per 11 p 11, & inuentis punctis reflexionum formarum punctorum b, m, k:</s> <s xml:id="echoid-s46203" xml:space="preserve"> pate <lb/>bit propoſitum ut prius.</s> <s xml:id="echoid-s46204" xml:space="preserve"> Ethoc proponebatur.</s> <s xml:id="echoid-s46205" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1769" type="section" level="0" n="0"> <head xml:id="echoid-head1311" xml:space="preserve" style="it">31. Forma alicuius line æ curuæ incidẽtis uertici ſpeculi pyramidalis cõcaui obliquè ſuper axẽ, <lb/>reflectitur ad centrũ uiſus inter illãlineã & ſuperficiẽ ſpeculi conſtitutũ, à linea logitudinis ſpe-<lb/>culi: imagó ipſius uidetur recta: & ſi illa linea incidẽs fuerit rect a: eius imago uidebitur cur-<lb/>ua, modicæ curuitatis, cuius conuexitas uel concauitas eſt ad uiſum. Alhazen 55 n 6.</head> <p> <s xml:id="echoid-s46206" xml:space="preserve">Fiat diſpoſitio omnimoda, quę in 55 th.</s> <s xml:id="echoid-s46207" xml:space="preserve"> 7 huius:</s> <s xml:id="echoid-s46208" xml:space="preserve"> inuenieturq́;</s> <s xml:id="echoid-s46209" xml:space="preserve"> in ſpeculis pyramidalib.</s> <s xml:id="echoid-s46210" xml:space="preserve"> conuexis li <lb/>neęrectę, quę eſt a n, propoſito modo illud ſpeculum reſpicientis, imago curua intra concauitatem <lb/>ſpeculi, quę eſt a p y.</s> <s xml:id="echoid-s46211" xml:space="preserve"> Punctum quoq;</s> <s xml:id="echoid-s46212" xml:space="preserve">, quod eſt ſub ſuperficie ſpeculum cótingente ſecundum lineá <lb/>lógitudinis ſpeculi, quę eſt a e, à qua fit reflexio formę lineę rectę uiſę, quę eſt a n, ad uiſum exiſtentẽ <lb/>in puncto r, erat illic punctum f:</s> <s xml:id="echoid-s46213" xml:space="preserve"> in quo puncto f ſi fuerit centrum uiſus, erunt omnia pũcta, quęſunt <lb/>in illa curua imagine, uel quę ſunt in linea recta, ſcilicet in diametro imaginis, reflexa ad punctũ f:</s> <s xml:id="echoid-s46214" xml:space="preserve"> & <lb/>imago lineę curuæ, quæ a p y, erit linea recta, quæ eſt a n:</s> <s xml:id="echoid-s46215" xml:space="preserve">uel imagines duarum extremitatum lineę <lb/>a p y erunt in linea a n, & in extremitatibus illius:</s> <s xml:id="echoid-s46216" xml:space="preserve"> & loca imaginis punctip, quod eſt in medio lineę <lb/>a y, diuerſabuntur.</s> <s xml:id="echoid-s46217" xml:space="preserve"> Et hoc poteſt eodem modo declarari, ſicut ſibi ſimile declaratum eſt in 55 th.</s> <s xml:id="echoid-s46218" xml:space="preserve"> 7 <lb/>huius.</s> <s xml:id="echoid-s46219" xml:space="preserve"> Quoniam enim, ut ibi declaratum eſt, angulus z r f eſt ęqualis angulo z f r per 29 p 1:</s> <s xml:id="echoid-s46220" xml:space="preserve"> fed per eandem <lb/>29 p 1 angulus h z f eſt ęqualis angulo z f r:</s> <s xml:id="echoid-s46221" xml:space="preserve"> eſt ergo angulus p z h æqualis angulo h z f.</s> <s xml:id="echoid-s46222" xml:space="preserve"> Palàm ergo <lb/> <pb o="389" file="0691" n="691" rhead="LIBER NONVS."/> per 20 th.</s> <s xml:id="echoid-s46223" xml:space="preserve"> 5 huius quoniam fiet reflexio formę puncti p ad uiſum exiſtentem in puncto f à pũcto ſpe <lb/>culi pyramidalis concaui, quod eſt z.</s> <s xml:id="echoid-s46224" xml:space="preserve"> Et quoniam linea h p o eſt cathetus incidentiæ formæ puncti <lb/> <anchor type="figure" xlink:label="fig-0691-01a" xlink:href="fig-0691-01"/> p:</s> <s xml:id="echoid-s46225" xml:space="preserve"> & linea f z o eſt linea ſuę reflexio <lb/>nis ad uiſum exiſtẽtem in puncto <lb/>f:</s> <s xml:id="echoid-s46226" xml:space="preserve"> patet per 37 th.</s> <s xml:id="echoid-s46227" xml:space="preserve"> 5 huius quoniam <lb/>punctum o eſt locus imaginis for-<lb/>mæ puncti p.</s> <s xml:id="echoid-s46228" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s46229" xml:space="preserve"> an-<lb/>gulus y e d eſt ęqualis angulo k e r <lb/>per 15 p 1, qui per 29 p 1 eſt ęqualis <lb/>angulo er ſ:</s> <s xml:id="echoid-s46230" xml:space="preserve"> & per eandem 29 p 1 <lb/>angulus d e f eſt æqualis angulo e <lb/>fr:</s> <s xml:id="echoid-s46231" xml:space="preserve"> ſed, ut in commento 55th.</s> <s xml:id="echoid-s46232" xml:space="preserve"> 7 hu <lb/>ius oftenſum eſt, angulus e f r eſt <lb/>æqualis angulo e r f:</s> <s xml:id="echoid-s46233" xml:space="preserve"> eſtigitur an-<lb/>gulus y e d æqualis angulo d e f:</s> <s xml:id="echoid-s46234" xml:space="preserve"> <lb/>ergo per 20 th.</s> <s xml:id="echoid-s46235" xml:space="preserve"> 5 huius forma pun <lb/>cti y reflectitur ad uiſum exiſtentẽ <lb/>in puncto f à puncto ſpeculi con-<lb/>caui, quod eſt e.</s> <s xml:id="echoid-s46236" xml:space="preserve"> Et quoniam linea <lb/>y n eſt cathetus incidentię formæ <lb/>puncti y:</s> <s xml:id="echoid-s46237" xml:space="preserve"> & linea f e n eſt linea ſuę <lb/>reflexionis:</s> <s xml:id="echoid-s46238" xml:space="preserve"> patet per 37 th.</s> <s xml:id="echoid-s46239" xml:space="preserve"> 5 huius quòd locus imaginis formę puncti y eſt pũctũ n:</s> <s xml:id="echoid-s46240" xml:space="preserve"> & pũctũ a, ſicut <lb/>reflectitur à uertice ſpeculi, ſic locus imaginis ſuę eſt ibidem per ea, quæ dicta ſunt in 11 & 12 th.</s> <s xml:id="echoid-s46241" xml:space="preserve"> 8 hu <lb/>ius, & in 10 huius.</s> <s xml:id="echoid-s46242" xml:space="preserve"> Erit ergo imago totius lineæ a p y curuæ linea a o n recta:</s> <s xml:id="echoid-s46243" xml:space="preserve"> quoniá de alijs punctis <lb/>eſt eodem modo demonſtrandum.</s> <s xml:id="echoid-s46244" xml:space="preserve"> Quòd ſi aliquod uiſibile ſtatuatur in loco lineæ rectæ a y, quę eſt <lb/>diameter illius curuæ imaginis lineę a p y:</s> <s xml:id="echoid-s46245" xml:space="preserve"> tunc duæ extremitates lineę a y (quæ ſunt a & y) habe-<lb/>bunt, ut prius, loca ſuarum imaginum in punctis a & n.</s> <s xml:id="echoid-s46246" xml:space="preserve"> Loca uerò imaginis puncti medij correſpon <lb/>dentis puncto p, qui cadit in producta linea z p, & aliorum punctorum mediorum diuerſabuntur:</s> <s xml:id="echoid-s46247" xml:space="preserve"> <lb/>& ſecundum diuerſitatem concurſus cathetorum in cidentiæ formarum illorum punctorum cum <lb/>lineis ſuarum reflexionum, ſecundum quas à punctis lineę longitudinis, quę eſt a z e, ſpeculi propo <lb/>ſiti concaui reflectuntur ad uiſum exiſtentem in puncto f, uel ultra lineam a o n uel citra illam loca <lb/>imaginum illorum punctorum diuerſabuntur, quandoque ad concauitatem:</s> <s xml:id="echoid-s46248" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s46249" xml:space="preserve"> ad conuexi <lb/>tatem, reſpicientem centrum uiſus.</s> <s xml:id="echoid-s46250" xml:space="preserve"> Erit tamen illa curuitas modica:</s> <s xml:id="echoid-s46251" xml:space="preserve"> quoniam prędictorum locorũ <lb/>imaginum (reſpectu lineæ a o n) modicus eſt exceſſus.</s> <s xml:id="echoid-s46252" xml:space="preserve"> Palàm itaque ex pręmiſsis quòd ſi linea re-<lb/>cta, quæ eſt diameter imaginis curuæ, quæ eſt a p y, fuerit in aliquo uiſibili, & centrum uiſus fuerit <lb/>in puncto f:</s> <s xml:id="echoid-s46253" xml:space="preserve"> tunc imago lineę rectę pręmiſſo modo diſpoſitę fortè uidebitur cõuexa, & fortè uidebi <lb/>tur concaua.</s> <s xml:id="echoid-s46254" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s46255" xml:space="preserve"/> </p> <div xml:id="echoid-div1769" type="float" level="0" n="0"> <figure xlink:label="fig-0691-01" xlink:href="fig-0691-01a"> <variables xml:id="echoid-variables800" xml:space="preserve">a l h p u o z m t x b g c n q s d g e l k f r</variables> </figure> </div> </div> <div xml:id="echoid-div1771" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables801" xml:space="preserve">u r h d b o y m x l n f i g t q k z e c s a</variables> </figure> <head xml:id="echoid-head1312" xml:space="preserve" style="it">32. Lineæ rectæ uiſæ ſuperficie incidentiæ axẽ <lb/>ſpeculi pyramidalis concaui orthogonaliter ſecã <lb/> te, centró uiſus non exiſtente in eadem ſuperfi- cie, imago uidebitur concaua, mir abilis concaui tatis, uiſum reſpicientis.</head> <p> <s xml:id="echoid-s46256" xml:space="preserve">Sit, ut in 27 huius libri, centrum uiſus punctum <lb/>l:</s> <s xml:id="echoid-s46257" xml:space="preserve"> & linea uiſa r m y:</s> <s xml:id="echoid-s46258" xml:space="preserve"> cuius extrema puncta, quæ <lb/>ſunt r & y, æ qualiter diſtent à centro uiſus l:</s> <s xml:id="echoid-s46259" xml:space="preserve"> ſitq́ue <lb/>centrum uiſus extra ſuperficiem lineę r y:</s> <s xml:id="echoid-s46260" xml:space="preserve"> quę pro <lb/>ducta ſecat ſpeculum pyramidale cõcauum æ qui-<lb/>diſtanter baſi ſecundũ circulum, qui ſit b g:</s> <s xml:id="echoid-s46261" xml:space="preserve"> cuius <lb/>centrũ ſit d:</s> <s xml:id="echoid-s46262" xml:space="preserve"> reflectaturq̃;</s> <s xml:id="echoid-s46263" xml:space="preserve"> forma puncti r ad uiſum l <lb/>à puncto ſpeculi b, & forma pũcti y refle ctatur ad <lb/>uiſum l à puncto ſpeculi g:</s> <s xml:id="echoid-s46264" xml:space="preserve"> erũtq́;</s> <s xml:id="echoid-s46265" xml:space="preserve"> pũcta b & g, quã-<lb/>uis ſint in circulo, ut tñ ſunt puncta reflexionum, <lb/>erunt in duabus oxygonijs ſectionib.</s> <s xml:id="echoid-s46266" xml:space="preserve"> ſecantib.</s> <s xml:id="echoid-s46267" xml:space="preserve"> ſe <lb/>ſecundum lineam d l:</s> <s xml:id="echoid-s46268" xml:space="preserve">ut patent hæc per 7 th.</s> <s xml:id="echoid-s46269" xml:space="preserve"> 7 hu-<lb/>ius, & per 19 th.</s> <s xml:id="echoid-s46270" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s46271" xml:space="preserve"> Et quoniam quantùm ad <lb/>propoſitum demonſtrandũ non eſt aliqua diuerſi <lb/>tas inter ſpecula columnaria & pyramidalia con-<lb/>caua:</s> <s xml:id="echoid-s46272" xml:space="preserve"> tunc patet quòd reiterata demõſtratione 27 <lb/>huius, erit locus imaginis formę pũcti r in pũcto <lb/>h:</s> <s xml:id="echoid-s46273" xml:space="preserve"> & locus imaginis formę pũcti y erit in puncto t:</s> <s xml:id="echoid-s46274" xml:space="preserve"> <lb/>locus uerò imaginis formæ pũcti m erit punctũ s, <lb/>quod eſt extra rectitudinẽ lineę t h.</s> <s xml:id="echoid-s46275" xml:space="preserve"> Imago itaque <lb/>lineę r m y eſt in quadã linea trãſeũte ք pũcta h, s, t:</s> <s xml:id="echoid-s46276" xml:space="preserve"> ſed talis linea eſt curua, Eſt ergo lineę rectę, quæ <lb/> <pb o="390" file="0692" n="692" rhead="VITELLONIS OPTICAE"/> eſt r m y, imago curua.</s> <s xml:id="echoid-s46277" xml:space="preserve"> Et quoniam punctum s eſt ultra concauitatem ſpeculi, reſpectu punctil <lb/>centri uiſus, & punctum l eſt intra illam cõcauitatem:</s> <s xml:id="echoid-s46278" xml:space="preserve"> palàm quòd punctum l eſt extra ſuperficiem, <lb/>in qua eſt linea h s t:</s> <s xml:id="echoid-s46279" xml:space="preserve"> curuitas ergo lineę h s t apparebit uiſui maniſeſtè.</s> <s xml:id="echoid-s46280" xml:space="preserve"> Et quia punctũ f cadit in ipſa <lb/>ſuperficie ſpeculi pyramidalis concaui extra ſuperficiem circuli b g, & linea t h eſt ultra ſpeculum in <lb/>ſuperficie circuli b g:</s> <s xml:id="echoid-s46281" xml:space="preserve"> erit linea l f s altior quàm ſuperficies trigoni l h t:</s> <s xml:id="echoid-s46282" xml:space="preserve"> linea ergo l s erit altior dua-<lb/>bus lineis l h & h t:</s> <s xml:id="echoid-s46283" xml:space="preserve"> punctũ ergo s, reſpectu uiſus l, eſt altius ꝗ̃ duo puncta h & t.</s> <s xml:id="echoid-s46284" xml:space="preserve"> Linea ergo h s t appa <lb/>rebit uiſui exiſtẽti in pũcto l cõcaua maxima cõcauitate uiſum reſpiciente.</s> <s xml:id="echoid-s46285" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s46286" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1772" type="section" level="0" n="0"> <head xml:id="echoid-head1313" xml:space="preserve" style="it">33. Line æ rect æ uiſæ non æquidiſtantis axi ſpeculi pyramidalis concaui, cuius ſuperficies inci-<lb/>dentiæ ſecat axem ſpeculi obliquè, imago uidetur curua, diuerſæ curuitatis ſecundum diuer ſita-<lb/>tem ſuiſitus.</head> <p> <s xml:id="echoid-s46287" xml:space="preserve">Quoniam enim, ut in 31 huius oſtenſum eſt, forma lineæ rectæ incidentis uertici huius ſpeculi <lb/>propoſiti obliquè ſuper axem, imaginiem curuam uiſui, ad quem fit reflexio, repræſentat:</s> <s xml:id="echoid-s46288" xml:space="preserve"> & per præ <lb/>miſſam proximam patet, quòd linea recta, cuius ſuperficies incidentiæ ſecat axem ſpeculi orthogo <lb/>naliter, uidetur mirabilis concauitatis uiſum reſpicientis.</s> <s xml:id="echoid-s46289" xml:space="preserve"> Si ergo inter has diſpoſitiones ſituetur li-<lb/>nea recta, cuius ſuperficies incidentiæ, ut hic proponitur, obliquè ſecet axem ſpeculi:</s> <s xml:id="echoid-s46290" xml:space="preserve"> patet quòdi-<lb/>mago illius lineę diuerſificabitur ſecundum modos diuerſæ curuitatis:</s> <s xml:id="echoid-s46291" xml:space="preserve"> qui accidunt hinc & inde <lb/>lineis ſecundum ambos præmiſſos modos ſituatis, cuius conformis eſt demonſtratio cum præmiſ-<lb/>ſis.</s> <s xml:id="echoid-s46292" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s46293" xml:space="preserve"> nec enim dignum uidimus talibus immorandum, quę ex prædemon-<lb/>ſtratis concluſtionibus ſuæ certitudinis ſubſiſtentiam lucidè accipiunt:</s> <s xml:id="echoid-s46294" xml:space="preserve"> unde talia relinquimus ani-<lb/>mæ perquirenti.</s> <s xml:id="echoid-s46295" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1773" type="section" level="0" n="0"> <head xml:id="echoid-head1314" xml:space="preserve" style="it">34. Imago line æ; rectæ exiſtentis in ſuperficie ſpeculũ pyramidale trãs axem ſecante, centró <lb/>uiſus existente in communi ſectione eiuſdem ſuperficiei, & ſuperficiei ſpeculum ſecũdum axem <lb/>ſecantis, uidebitur rect a: quando maior: quando æqualis: quando minor reuiſa: ſed ſemper <lb/>conuerſum habens ſitum: & quando una: quandoque plures imagines uiſui occurrent. Al-<lb/>hazen 56 n 6.</head> <p> <s xml:id="echoid-s46296" xml:space="preserve">Fiatitem (utin 29 huius) eadem diſpoſitio figuræ, quæ facta eſt in 48 th.</s> <s xml:id="echoid-s46297" xml:space="preserve"> 8 huius.</s> <s xml:id="echoid-s46298" xml:space="preserve"> Siergo aliquod <lb/>punctum commune ambabus ſuperficiebus d a o & d b o, fuerit in axe pyramidis, ut punctum o:</s> <s xml:id="echoid-s46299" xml:space="preserve"> & <lb/> <anchor type="figure" xlink:label="fig-0692-01a" xlink:href="fig-0692-01"/> ſi duæ lineæ a e & b e fuerint perpendiculares ſuper ſuperficies <lb/>contingentes pyramidem ſpeculi:</s> <s xml:id="echoid-s46300" xml:space="preserve"> hoc autem eſt poſsibile, quia li <lb/>neę a e & b e ſunt æquales:</s> <s xml:id="echoid-s46301" xml:space="preserve"> poſſunt enim cum axe continere duos <lb/>angulos acutos æquales.</s> <s xml:id="echoid-s46302" xml:space="preserve"> Cum ergo hæ duæ lineę fuerint perpen <lb/>diculares ſuper illas ſuperficies, & uiſus fuerit in puncto d:</s> <s xml:id="echoid-s46303" xml:space="preserve"> tune <lb/>ſuperficies trigoni d e g, in qua ſunt lineæ g e & d e, tranſibit per <lb/>totum axem & per centrum uiſus:</s> <s xml:id="echoid-s46304" xml:space="preserve"> & utraque ſuperficies d a o & <lb/>d b o erit decliuis ſuper axem ſpeculi:</s> <s xml:id="echoid-s46305" xml:space="preserve"> & communes ipſarum ſe-<lb/>ctiones cum ſuperficie conica ſpeculi erunt duę ſectiones oxygo <lb/>niæ & formæ trium punctorum, quæ ſunt r, h, q, reflectentur ad <lb/>uiſum exiſtentem in pũcto d à puncto ſpeculi, quod eſt b.</s> <s xml:id="echoid-s46306" xml:space="preserve"> Formę <lb/>quoque trium punctorum, quæ ſunt m, l, f, refle ctentur ad uiſum <lb/>in punctum d à puncto ſpeculi a.</s> <s xml:id="echoid-s46307" xml:space="preserve"> Cum ergo lineæ m l f & r h q fue <lb/>rint in aliqua ſuperficie corporis uiſibilis, & uiſus fuerit in pun-<lb/>cto d:</s> <s xml:id="echoid-s46308" xml:space="preserve"> tunc, ut ſuprà in 29 huius patuit, linea n u erit imago lineæ <lb/>m r:</s> <s xml:id="echoid-s46309" xml:space="preserve"> & linea t k erit imago lineę l h:</s> <s xml:id="echoid-s46310" xml:space="preserve"> & linea p i erit imago lineæ <lb/>f q:</s> <s xml:id="echoid-s46311" xml:space="preserve"> erit itaque imago lineę m r, quæ eſt linea n u, minor quàm <lb/>linea m r:</s> <s xml:id="echoid-s46312" xml:space="preserve"> & imago lineę f q, quæ eſt p i, erit maior quàm linea f q:</s> <s xml:id="echoid-s46313" xml:space="preserve"> <lb/>& imago lineę l h, quę eſt t k, erit æqualis ipſilineæ l h.</s> <s xml:id="echoid-s46314" xml:space="preserve"> Omnes <lb/>quoque iſtę imagines conuerſum habebunt ſitum, reſpectu rerũ, <lb/>quarum ipſæ ſunt imagines, uiſu exiſtente in puncto d.</s> <s xml:id="echoid-s46315" xml:space="preserve"> Quòd ſi <lb/>uiſus fuerit in puncto o, & lineę n u, t k, & p i, quæ ſunt imagines <lb/>linearum m r, l h & f q, uiſu exiſtente in puncto o, fuerint in ſu-<lb/>perficiebus corporum uiſibilium:</s> <s xml:id="echoid-s46316" xml:space="preserve"> tunc per eandem præmiſſam <lb/>rationem in 29 huius imagines illarum linearum n u, t k, & p i <lb/>erunt lineæ m r, l h & f q:</s> <s xml:id="echoid-s46317" xml:space="preserve"> eritq́ue imago lineæ p i, quæ eſt li-<lb/>nea f q, minor quàm linea p i:</s> <s xml:id="echoid-s46318" xml:space="preserve"> & imago lineæ t k, quæ eſt linea <lb/>l h, erit æqualis ſuæ:</s> <s xml:id="echoid-s46319" xml:space="preserve"> & imago lineæ n u, quæ eſt linea m r, <lb/>erit maior ipſa linea n u:</s> <s xml:id="echoid-s46320" xml:space="preserve"> & iſtæ imagines omnes erunt lineæ rectæ:</s> <s xml:id="echoid-s46321" xml:space="preserve"> & apparebunt ultra centrum <lb/>uiſus, quod eſt in puncto o.</s> <s xml:id="echoid-s46322" xml:space="preserve"> Et ſi imaginentur continuari capita illarum linearum per lineas n t p & <lb/>u k i:</s> <s xml:id="echoid-s46323" xml:space="preserve"> eruntloca imaginum illarum linearum lineæ m l f & r h q.</s> <s xml:id="echoid-s46324" xml:space="preserve"> Puncta itaque iſtarum imagi-<lb/>num, quæ ſunt m, l, f, comprehenduntur ſuper eandem lineam reflexionis, quę eſt a o:</s> <s xml:id="echoid-s46325" xml:space="preserve"> & puncta <lb/>r, h, q comprehenduntur ſuper eandem lineam reflexionis, quę eſt b o:</s> <s xml:id="echoid-s46326" xml:space="preserve"> & imago puncti remotio-<lb/>ris à uiſu, erit propinquior uiſui, & imago puncti propinquioris uiſui, erit remotior à uiſu.</s> <s xml:id="echoid-s46327" xml:space="preserve"> Con-<lb/> <pb o="391" file="0693" n="693" rhead="LIBER NONVS."/> uerſum ltaq;</s> <s xml:id="echoid-s46328" xml:space="preserve"> habebunt ſitum omnes iſtæ imagines.</s> <s xml:id="echoid-s46329" xml:space="preserve"> Quod eſt propoſitũ.</s> <s xml:id="echoid-s46330" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s46331" xml:space="preserve"> exhis quatuor <lb/>propoſitionibus, quòd lineæ rectæ quandoq;</s> <s xml:id="echoid-s46332" xml:space="preserve"> in his ſpeculis pyramidalibus concauis uidẽtur con-<lb/>uexæ:</s> <s xml:id="echoid-s46333" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s46334" xml:space="preserve"> concauæ:</s> <s xml:id="echoid-s46335" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s46336" xml:space="preserve"> rectæ:</s> <s xml:id="echoid-s46337" xml:space="preserve"> & quandoq;</s> <s xml:id="echoid-s46338" xml:space="preserve"> maiores:</s> <s xml:id="echoid-s46339" xml:space="preserve"> & quandoq;</s> <s xml:id="echoid-s46340" xml:space="preserve"> minores:</s> <s xml:id="echoid-s46341" xml:space="preserve"> & quãdo-<lb/>que æquales rebus uiſis:</s> <s xml:id="echoid-s46342" xml:space="preserve"> & ſunt omnes rectæ imagines difformem ſitum habentes, reſpectu ſitus <lb/>rerum, quarũ ſunt imagines.</s> <s xml:id="echoid-s46343" xml:space="preserve"> Et accidit in his ſpeculis, ſicut in alijs ſpeculis, numerari imagines ſe-<lb/>cundum numerum punctorum reflexionis:</s> <s xml:id="echoid-s46344" xml:space="preserve"> & fortè imagines eiuſdem rei diuerſarum erunt forma <lb/>rum ſecundum diuerſum ſitum ſuarum partium:</s> <s xml:id="echoid-s46345" xml:space="preserve"> quæ omnia ex pręmiſsis principijs poſſunt facili-<lb/>ter declarari.</s> <s xml:id="echoid-s46346" xml:space="preserve"> Hæc itaq;</s> <s xml:id="echoid-s46347" xml:space="preserve"> de regularibus ſpeculis ſufficiant ad præſens.</s> <s xml:id="echoid-s46348" xml:space="preserve"> Deinceps uerò in ſequẽtibus <lb/>huius libri ad tractatum quorũdam irregularium ſpeculorum & cõburentiũ ingeniũ cõuertemus.</s> <s xml:id="echoid-s46349" xml:space="preserve"/> </p> <div xml:id="echoid-div1773" type="float" level="0" n="0"> <figure xlink:label="fig-0692-01" xlink:href="fig-0692-01a"> <variables xml:id="echoid-variables802" xml:space="preserve">d g p i t k n z u b e a m l f o q h r</variables> </figure> </div> </div> <div xml:id="echoid-div1775" type="section" level="0" n="0"> <head xml:id="echoid-head1315" xml:space="preserve" style="it">35. Poßibile eſt ſpeculum ex conuexo & concauo compoſitum fieri, in quo dextra apparent <lb/>dextra, & ſiniſtra ſiniſtra, & multa diuerſitas imagιnum occurrit. Euclides 30 th. catoptr. Pto <lb/>lemæus 3 th. 2 catoptr.</head> <p> <s xml:id="echoid-s46350" xml:space="preserve">Aſſumatur in illa magnitudine, qua quis conſtruere uoluerit tale ſpeculum, circulus, qui ſit a b g:</s> <s xml:id="echoid-s46351" xml:space="preserve"> <lb/>& inſcribatur ei latus pentagoni inſcriptibilis eidem circulo per 11 p 4:</s> <s xml:id="echoid-s46352" xml:space="preserve"> quod ſit a b:</s> <s xml:id="echoid-s46353" xml:space="preserve"> & ſimiliter in-<lb/> <anchor type="figure" xlink:label="fig-0693-01a" xlink:href="fig-0693-01"/> ſcribatur eidẽ circulo latus hexagoni per 15 p 4, <lb/>quod ſit b g:</s> <s xml:id="echoid-s46354" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s46355" xml:space="preserve"> per eádem 15 p 4 linea b g ę-<lb/>qualis ſemidiametro circuli.</s> <s xml:id="echoid-s46356" xml:space="preserve"> Et abſcindatur ab <lb/>illo circulo portio a e b, cuius arcus a b ք 28 p 3 <lb/>eſt æqualis quintæ parti peripheriæ circuli.</s> <s xml:id="echoid-s46357" xml:space="preserve"> Et <lb/>ſimiliter abſcindatur ab eodem circulo portio <lb/>g z b, cuius arcus b g eſt æqualis ſextæ parti <lb/>circuli.</s> <s xml:id="echoid-s46358" xml:space="preserve"> Fiant quoq;</s> <s xml:id="echoid-s46359" xml:space="preserve"> formæ regulares ad quanti-<lb/>tatem illarum duarum portionum:</s> <s xml:id="echoid-s46360" xml:space="preserve"> quarum una <lb/>fiat ſecundum quantitatem portionis a e b, quæ <lb/>ſit concaua, ut eſt figura, quam deſcripſimus, <lb/>z h t f k m l:</s> <s xml:id="echoid-s46361" xml:space="preserve"> altera uerò facta ad quantitatem <lb/>portionis, quę eſt g z b, ſit conuexa, ut eſt fi-<lb/>gura x o p.</s> <s xml:id="echoid-s46362" xml:space="preserve"> Et aſſumatur petia uel pars ferri re-<lb/>ctangula, cuius longitudo ſit maior quàm am-<lb/>bę chordę a b, & b g, latitudo quoque ſit maior <lb/>quàm chorda b g:</s> <s xml:id="echoid-s46363" xml:space="preserve"> & incuruetur ferrũ taliter, ut <lb/>eius longitudo ſit conuexitatis portionis a e b, <lb/>ita ut ſuperficies cõcaua, quæ eſt k f t, ſibi extrin <lb/>ſecus applicetur:</s> <s xml:id="echoid-s46364" xml:space="preserve"> & eius latitudo ſit in parte lon <lb/>gitudinis reſiduæ concauitatis portionis g z b, <lb/>ita ut cóuexitas ſuperficiei x o p ſibi intrinſecus <lb/>applicetur.</s> <s xml:id="echoid-s46365" xml:space="preserve"> Taliter uerò fiat, ne forma cóuexita <lb/>tis impedimentum accipiat ex forma concaui-<lb/>tatis, ſed in eadem ſuperficie ſpeculi ipſarũ quę-<lb/>libet imprimatur:</s> <s xml:id="echoid-s46366" xml:space="preserve"> poliaturq́;</s> <s xml:id="echoid-s46367" xml:space="preserve"> ſpeculum ex parti-<lb/>bus ambabus:</s> <s xml:id="echoid-s46368" xml:space="preserve"> propter quod oportet ut lamina <lb/>ſpeculanda ſit conuenienter ſpiſſa, ut ex utra-<lb/>que parte ſalua diſpoſitione reliqua ualeat poli-<lb/>ri.</s> <s xml:id="echoid-s46369" xml:space="preserve"> Hoc itaque ſpeculum ſi ſuper ſedem uolubi-<lb/>lem ad hoc præparatam componatur, & ſuper <lb/>ipſam uoluatur, ita quòd nunc conuexa, nunc <lb/>concaua ſuperficies uiſui ſe offerat:</s> <s xml:id="echoid-s46370" xml:space="preserve"> tunc appa-<lb/>rebunt dextra dextra, & ſiniſtra ſiniſtra:</s> <s xml:id="echoid-s46371" xml:space="preserve"> & di-<lb/>ſtanti quaſi duobus cubitis apparet imago commenſurata & ſimilis ueræ formę:</s> <s xml:id="echoid-s46372" xml:space="preserve"> magis uerò diſtan <lb/>ti protenditur imago in anterius:</s> <s xml:id="echoid-s46373" xml:space="preserve"> propius uerò accedenti ad conuexam ſuperficiem ſpeculi, fit i-<lb/>mago penitus informis:</s> <s xml:id="echoid-s46374" xml:space="preserve"> & magis accedenti informitas plus augetur, & contraria ei, quod uidetur, <lb/>fit imago, magisq́;</s> <s xml:id="echoid-s46375" xml:space="preserve"> accedẽti prolixior apparet, & fit facies uidentis cõſimilis formæ equi:</s> <s xml:id="echoid-s46376" xml:space="preserve"> & ſemper <lb/>magis inclinato ſpeculo, imago apparet plus inclinata.</s> <s xml:id="echoid-s46377" xml:space="preserve"> Permutato quoque ſpeculo, imago quan-<lb/>doque habet caput ſurſum & pedes deorſum:</s> <s xml:id="echoid-s46378" xml:space="preserve"> & quandoque pedes ſurſum & caput deorſum:</s> <s xml:id="echoid-s46379" xml:space="preserve"> & <lb/>plus experientia, quàm ſcriptura, docebit imaginum diuerſitates:</s> <s xml:id="echoid-s46380" xml:space="preserve"> quia ſi connectantur duo ſpe-<lb/>cula ſphærica, quorum unum ſit concauum, reliquum conuexum, non moto etiam ſpeculo, ua-<lb/>riatur diſpoſitio imaginum.</s> <s xml:id="echoid-s46381" xml:space="preserve"> Propter reuerberationem enim formæ reflexæ ab uno ſpeculo in al-<lb/>terum, dextra apparebunt dextra, & ſiniſtra ſiniſtra:</s> <s xml:id="echoid-s46382" xml:space="preserve"> & in parte conuexa non mutabitur ſitus i-<lb/>maginis ſecundum ſurſum & deorſum:</s> <s xml:id="echoid-s46383" xml:space="preserve"> ſed in parte concaua uidebitur imago ſupercapitalis, ue-<lb/>lut antipodes.</s> <s xml:id="echoid-s46384" xml:space="preserve"> Cauſſa uerò omnium horum in ſimplicibus ſpeculis dicta eſt per præmiſſa:</s> <s xml:id="echoid-s46385" xml:space="preserve"> mo-<lb/>do quoq;</s> <s xml:id="echoid-s46386" xml:space="preserve"> tali in præmiſſo ſpeculo permiſcentur imaginies.</s> <s xml:id="echoid-s46387" xml:space="preserve"> Et ſi in eadem continuitate ſit ſpeculum <lb/>planum ipſis ſpeculis ſphęricis conuexis & cõcauis interpoſitum:</s> <s xml:id="echoid-s46388" xml:space="preserve"> uaria bitur imaginum quátitas:</s> <s xml:id="echoid-s46389" xml:space="preserve"> <lb/> <pb o="392" file="0694" n="694" rhead="VITELLONIS OPTICAE"/> quia in planis eſt imago æqualis rei uiſę ք 52 th.</s> <s xml:id="echoid-s46390" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s46391" xml:space="preserve"> in cõuexis uerò eſt minor ք 39 th.</s> <s xml:id="echoid-s46392" xml:space="preserve"> 6 huius:</s> <s xml:id="echoid-s46393" xml:space="preserve"> <lb/>in cõcauis uerò quádoq;</s> <s xml:id="echoid-s46394" xml:space="preserve"> æqualis:</s> <s xml:id="echoid-s46395" xml:space="preserve"> quádoq;</s> <s xml:id="echoid-s46396" xml:space="preserve"> maior:</s> <s xml:id="echoid-s46397" xml:space="preserve"> & quádoq;</s> <s xml:id="echoid-s46398" xml:space="preserve"> minor, ut patet ք 45.</s> <s xml:id="echoid-s46399" xml:space="preserve"> 47.</s> <s xml:id="echoid-s46400" xml:space="preserve"> 49 th.</s> <s xml:id="echoid-s46401" xml:space="preserve"> 8 hu-<lb/>ius:</s> <s xml:id="echoid-s46402" xml:space="preserve"> & tale ſpeculũ poteſt taliter cõponi.</s> <s xml:id="echoid-s46403" xml:space="preserve"> Sit ſuperficies aliqua plana, quę a b:</s> <s xml:id="echoid-s46404" xml:space="preserve"> & fiant in ipſa ſpecula <lb/>cõuexa, quę ſint a c g & t r k:</s> <s xml:id="echoid-s46405" xml:space="preserve"> & ſimiliter fiant in ipſa ſpecula cõcaua, quæ ſint g d e & z i t:</s> <s xml:id="echoid-s46406" xml:space="preserve"> & fiant ſpe-<lb/>cula plana, quę ſint e z & k b:</s> <s xml:id="echoid-s46407" xml:space="preserve"> ponaturq́;</s> <s xml:id="echoid-s46408" xml:space="preserve"> res uiſa in puncto m, quæ à ſpeculis illis ad uiſum reflecta-<lb/>tur.</s> <s xml:id="echoid-s46409" xml:space="preserve"> A planis itaq;</s> <s xml:id="echoid-s46410" xml:space="preserve"> ſpeculis apparent æqualia idola & æqualiter diſtãtia:</s> <s xml:id="echoid-s46411" xml:space="preserve"> & à cõuexis minora & mi-<lb/> <anchor type="figure" xlink:label="fig-0694-01a" xlink:href="fig-0694-01"/> nus diſtátia:</s> <s xml:id="echoid-s46412" xml:space="preserve"> à cócauis <lb/>uerò diuerſa & diuerſi <lb/>modè uiſui occurren-<lb/>tia, ſicut in alijs pręde-<lb/>monſtratũ eſt.</s> <s xml:id="echoid-s46413" xml:space="preserve"> Ingeniũ <lb/>uerò modernorû & fu <lb/>turorũ addat, quod li-<lb/>buerit:</s> <s xml:id="echoid-s46414" xml:space="preserve"> quia ſufficiẽter <lb/>dedimus cogitantibus <lb/>principia multarum talium adinuentionum:</s> <s xml:id="echoid-s46415" xml:space="preserve"> & nos, quætalia digna memoria inuenerimus, pe-<lb/>ſterius conſcribemus.</s> <s xml:id="echoid-s46416" xml:space="preserve"/> </p> <div xml:id="echoid-div1775" type="float" level="0" n="0"> <figure xlink:label="fig-0693-01" xlink:href="fig-0693-01a"> <variables xml:id="echoid-variables803" xml:space="preserve">b e z a g z m h t f k l x o p a b g</variables> </figure> <figure xlink:label="fig-0694-01" xlink:href="fig-0694-01a"> <variables xml:id="echoid-variables804" xml:space="preserve">b k t i z e d g a r m</variables> </figure> </div> </div> <div xml:id="echoid-div1777" type="section" level="0" n="0"> <head xml:id="echoid-head1316" xml:space="preserve" style="it">36. A ſpeculis columnaribus uel pyramidalibus concauis ignem difficile eſt accendi.</head> <p> <s xml:id="echoid-s46417" xml:space="preserve">Si enim in ſpeculis colũnaribus cõcauis ſuperficiei reflexionis & ſpeculi cõmunis ſectio ſit linea <lb/>longitudinis, nó eſt neceſſariũ ignẽ ab ipſis accẽdi, ſicut neq;</s> <s xml:id="echoid-s46418" xml:space="preserve"> à ſpeculis planis, etiã ſi ſuperficies re-<lb/>flexιonis oẽs ſe in axe colũnę interſecẽt:</s> <s xml:id="echoid-s46419" xml:space="preserve"> radij enim æquidiſtáter ſuperficiei ſpeculi incidẽtes, æqui <lb/>diſtáter utiq;</s> <s xml:id="echoid-s46420" xml:space="preserve"> reflectẽtur:</s> <s xml:id="echoid-s46421" xml:space="preserve"> perpẽdiculares quidẽ in ſe ipſos ad diuerſa pũcta ſpeculi colũnaris ſecun-<lb/>dum quę, cũ ipſi ſpeculo incidebãt, axẽ ſecabãt & ita nunquã in pũcto concurrent, ſed in tota linea <lb/>axis diſtendẽtur:</s> <s xml:id="echoid-s46422" xml:space="preserve"> nõ perpẽdiculares uerò radij, obliquè ſcilicet ſuperficiei ſpeculi incidentes, quo-<lb/>niã ſecundũ angulos, quos faciũt cũ perpendiculari ducta ab axe ad lineã lõgitudinis, quę eſt com-<lb/>munis ſectio ſuperficiei reflexionis & ſuperficiei cõtingentis columnã, ad partẽ aliã in eadẽ ſuper-<lb/>ficie à dicta perpendiculari reflectuntur.</s> <s xml:id="echoid-s46423" xml:space="preserve"> Patet ergo quia ſecundũ quod ęquidiſtantes ad inuicẽ in-<lb/>cidunt, ſic quaſi æquidiſtãtes ad inuicẽ reflectuntur, & nõ in puncto, ſed in linea cõcurrent ք 29 p 1.</s> <s xml:id="echoid-s46424" xml:space="preserve"> <lb/>Quòd ſi dicatur, quòd aliquæ ſuperficies reflexionis ſe in axe colũnę nõ interſecent, ſed ſint æqui-<lb/>diſtantes (quod eſt impoſsibile, ut patet per 7 th.</s> <s xml:id="echoid-s46425" xml:space="preserve"> 7 huius) palàm tamẽ eſt quòd in eis reflexi radij <lb/>nunquã cõcurrẽt.</s> <s xml:id="echoid-s46426" xml:space="preserve"> Siuerò ſectio cõmunis ſuperficiei reflexionis & ſuperficiei columnę ſit circulus:</s> <s xml:id="echoid-s46427" xml:space="preserve"> <lb/>tunc patet quòd ք eius centrũ tranſeuntes radij (quoniá oẽs ſunt perpendiculares ſuper ſuperfi-<lb/>cies cõtingentes ιn punctis ſuæ incidentiæ, ut per 21 th.</s> <s xml:id="echoid-s46428" xml:space="preserve"> 7 huius oſtenſum eſt) oẽs reflectuntur in <lb/>ſeipſos, & cõcurrentin centro circuli illius ſiue ſit baſis columnæ ſpeculi, ſiue ſit circulus baſi æqui <lb/>diſtans.</s> <s xml:id="echoid-s46429" xml:space="preserve"> Hoc aũt centrũ erit ſemper in axe:</s> <s xml:id="echoid-s46430" xml:space="preserve"> & ſunt tot cẽtra talium circulorũ in axe, quot ſunt circu-<lb/>li in columna:</s> <s xml:id="echoid-s46431" xml:space="preserve"> ad unũ ergo punctũ non reflectuntur radij totius ſuperficiei ſpeculi columnaris, ſed <lb/>ad totã axis lineam.</s> <s xml:id="echoid-s46432" xml:space="preserve"> Quod ſi radij reflexi ſecundũ circulum nõ tranſeunt centrũ circuli:</s> <s xml:id="echoid-s46433" xml:space="preserve"> tunc ſecun-<lb/>dũ angulorũ incidentiæ diuerſitatẽ fiet diuerſitas reflexionis ad ſemidiametrũ circuli:</s> <s xml:id="echoid-s46434" xml:space="preserve"> nec fiet con-<lb/>curſus in centro circuli radiorũ, ſed in tota ſemidiametro:</s> <s xml:id="echoid-s46435" xml:space="preserve"> & ſic ignis difficiliter accendi poterit, ſi-<lb/>cut etiã prius dictum eſt in ſpeculo ſphærico concauo, ut patet per 68 th.</s> <s xml:id="echoid-s46436" xml:space="preserve"> 8 huius.</s> <s xml:id="echoid-s46437" xml:space="preserve"> Quòd ſi cõmunis <lb/>ſectio dictarũ duarum ſuperficierũ ſit ſectio columnaris:</s> <s xml:id="echoid-s46438" xml:space="preserve"> tunc radij pauciſsimi cõcurrent.</s> <s xml:id="echoid-s46439" xml:space="preserve"> Patet er-<lb/>go quòd nõ eſt poſsibile oẽs radios ſuperficiei ſpeculi columnaris concaui in unũ locũ uel etiá in <lb/>unã lineam aggregari:</s> <s xml:id="echoid-s46440" xml:space="preserve"> & ob hoc pauci antiquorum tali ſpeculo pro cõbuſtionibus ſunt uſi.</s> <s xml:id="echoid-s46441" xml:space="preserve"> Ex ſpe-<lb/>culis etiá pyramidalibus lumen aggregatũ ignẽ accendere non eſt neceſſariũ, quáuis ad hæc multa <lb/>rum acclinetur imaginatio:</s> <s xml:id="echoid-s46442" xml:space="preserve"> cuius cauſſa eſt, quia in talibus ſpeculis communis ſectio ſuperficiei re-<lb/>flexionis & ſuperficiei ſpeculi non poteſt eſſe circulus aliquis nec baſis, nec æquidiſtans baſi:</s> <s xml:id="echoid-s46443" xml:space="preserve"> pro-<lb/>pter hoc, quod prius dictum eſt, & petet per 2 th.</s> <s xml:id="echoid-s46444" xml:space="preserve"> huius.</s> <s xml:id="echoid-s46445" xml:space="preserve"> In nullo ergo euentu poſſunt radij à peri-<lb/>pheria circuli in centro concurrere:</s> <s xml:id="echoid-s46446" xml:space="preserve"> ſicut aliqualiter accidit in ſpeculo colũnari.</s> <s xml:id="echoid-s46447" xml:space="preserve"> Quòd ſi ſectio com <lb/>munis ſuperſicierum dictarũ ſit linea lõgitudinis ſpeculi:</s> <s xml:id="echoid-s46448" xml:space="preserve"> tũc, quoniã ſuperficies ſpeculũ contingẽs <lb/>contingit in linea longitudinis, accidet in his ſpeculis, ſicut prius dictũ eſt in planis & colũnaribus <lb/>ſpeculis.</s> <s xml:id="echoid-s46449" xml:space="preserve"> Radij enim incidentes quoſcunq;</s> <s xml:id="echoid-s46450" xml:space="preserve"> angulos fecerint cũlinea longitudinis, eoſdẽ facient cũ <lb/>eadẽ reflexi:</s> <s xml:id="echoid-s46451" xml:space="preserve"> & ſic radij incidẽtes ęquidiſtát, & ęquidiſtáter reflectũtur.</s> <s xml:id="echoid-s46452" xml:space="preserve"> Nõ ergo cõ currẽt, etiá ſi ſint <lb/>in eadẽ ſuperficie reflexionis:</s> <s xml:id="echoid-s46453" xml:space="preserve"> & ſi in diuerſis ſint ſuperficiebus, patet qđ nó cõcurrent niſi in axe:</s> <s xml:id="echoid-s46454" xml:space="preserve"> <lb/>quia ſuքficies reflexionis ſe ſuper axẽ pyramidis interſecát:</s> <s xml:id="echoid-s46455" xml:space="preserve"> & tũc cõcurſus radiorũ fiet in linea, nõ <lb/>in pũcto.</s> <s xml:id="echoid-s46456" xml:space="preserve"> Si cõmunis ſectio ſuperficierũ dictarũ ſit ſectio pyramidalis:</s> <s xml:id="echoid-s46457" xml:space="preserve"> nec adhuc oẽs uel plures ra-<lb/>dij eiuſdẽ ſuperficiei uel diuerſarũ aliqualiter concurrẽt.</s> <s xml:id="echoid-s46458" xml:space="preserve"> Nullo ergo modo radij incidentes pyra-<lb/>mιdali ſpeculo omnes, uel plures ipſorum, uel etiam pauci in puncto uno poſſunt concurrere, ut a-<lb/>liquid ignitioni reſiſtens ualeant ignire:</s> <s xml:id="echoid-s46459" xml:space="preserve"> nec etiam pluralitas coniun ctorum ſpeculorũ aliquid ua-<lb/>lidum reſpectu laboris ſuperadditi apportabit.</s> <s xml:id="echoid-s46460" xml:space="preserve"> Patet ergo illud, quod prop onebatur.</s> <s xml:id="echoid-s46461" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1778" type="section" level="0" n="0"> <head xml:id="echoid-head1317" xml:space="preserve" style="it">37. Ex plurium ſpeculorum ſphæricorum concaurum interſectione ſpeculum comburens <lb/>conſtitui eſt poßibile.</head> <p> <s xml:id="echoid-s46462" xml:space="preserve">Verbi gratia, ſit circulus alicuius ſpeculi ſphęrici concaui, qui a b c d:</s> <s xml:id="echoid-s46463" xml:space="preserve"> & eius centrum ſit e:</s> <s xml:id="echoid-s46464" xml:space="preserve"> inter-<lb/> <pb o="393" file="0695" n="695" rhead="LIBER NONVS."/> ſecentq́;</s> <s xml:id="echoid-s46465" xml:space="preserve"> fe in ipſo duę diametri a c & b d orthogonaliter:</s> <s xml:id="echoid-s46466" xml:space="preserve"> incidantq́;</s> <s xml:id="echoid-s46467" xml:space="preserve"> radij ſolares illi circulo:</s> <s xml:id="echoid-s46468" xml:space="preserve"> palàm <lb/>itaq;</s> <s xml:id="echoid-s46469" xml:space="preserve"> per ea, quæ in 68 th.</s> <s xml:id="echoid-s46470" xml:space="preserve"> 8 huius dicta ſunt, quoniam radius incidens circulo ſecundum aliquá dia <lb/>metrorum (uerbi gratia, ſecundum diametrum a c) reflectitur in ſeipſum trans centrum:</s> <s xml:id="echoid-s46471" xml:space="preserve"> radiorũ <lb/>uerò æ quidiſtantium illi diametro a c, is, qui contingit circulum, palàm quia incidit in punctum <lb/>b per 29 p 1:</s> <s xml:id="echoid-s46472" xml:space="preserve"> angulus enim, quem linea contingens cõtinet cũ diametro, eſt rectus per 18 p 3, & an-<lb/>gulus b e a eſt rectus ex hypotheſi.</s> <s xml:id="echoid-s46473" xml:space="preserve"> Ille ergo radιus contingens circulum non reflectitur:</s> <s xml:id="echoid-s46474" xml:space="preserve"> quia nihil <lb/>inuenit reſiſtens:</s> <s xml:id="echoid-s46475" xml:space="preserve"> procedit ergo in continuum & directum.</s> <s xml:id="echoid-s46476" xml:space="preserve"> Alius uerò radιus æquidiſtãs diametro <lb/>a c cum linea in puncto ſuæ incidentię ſpeculum contingente continet angulum rectilineum acu-<lb/> <anchor type="figure" xlink:label="fig-0695-01a" xlink:href="fig-0695-01"/> tiſsimum, & modicam abſcin dit portionem circuli in-<lb/>cidens, & modicùm ſe reflectẽs, ſed æqualiter.</s> <s xml:id="echoid-s46477" xml:space="preserve"> Sic itaq;</s> <s xml:id="echoid-s46478" xml:space="preserve"> <lb/>omnes radij æquidiſtantes diametro a c incidentes cir <lb/>culo ſpeculi æquales abſcindũt circuli portiones:</s> <s xml:id="echoid-s46479" xml:space="preserve"> ſem-<lb/>per enim angulus reflexionis eſt æqualis angulo inci-<lb/>dentię:</s> <s xml:id="echoid-s46480" xml:space="preserve"> illi autem anguli æquales ſemper æquales ab-<lb/>ſcindunt portiones per 43 th.</s> <s xml:id="echoid-s46481" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s46482" xml:space="preserve"> ſolus autem radi-<lb/>us incidens circulo ęquidiſtanter diametro a c, abſcin-<lb/>dens portionem, cuius arcus eſt ſexta pars peripherię <lb/>circuli, & cuius chorda eſt æqualis lateri hexagoni in-<lb/>ſcriptibilis eidem circulo, reflectitur ad punctum c ter-<lb/>minum diametri c a:</s> <s xml:id="echoid-s46483" xml:space="preserve"> eſt enim diameter a c æquidiſtans <lb/>medio lateri hexagoni ſuo circulo inſcripti, quẽ hexa-<lb/>gonũ diuidit illa diameter per æqualia, ut patet per 63 <lb/>th.</s> <s xml:id="echoid-s46484" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s46485" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s46486" xml:space="preserve">, ut talis radius incidat circulo ιn pũcto <lb/>f.</s> <s xml:id="echoid-s46487" xml:space="preserve"> O ẽs quoq;</s> <s xml:id="echoid-s46488" xml:space="preserve"> rad j ę quidiſtantes ſemidiametro a c, inci-<lb/>dentes reliquo arcui quartę circuli, cuius chorda eſt æ-<lb/>qualιs reſiduo lateri hexagoni, & eſt arcus f c, reflectũ-<lb/>tur ad illá partem cιrculi portiones æquales abſcιdentes:</s> <s xml:id="echoid-s46489" xml:space="preserve"> & omnes illi radij tranſeunt per aliquod <lb/>punctum ſemidιametri c e:</s> <s xml:id="echoid-s46490" xml:space="preserve"> & quodcunq;</s> <s xml:id="echoid-s46491" xml:space="preserve"> punctum reflexionis imaginetur moueri circa axem a c, <lb/>quouſq;</s> <s xml:id="echoid-s46492" xml:space="preserve"> redeat ad locũ à quo exiuit:</s> <s xml:id="echoid-s46493" xml:space="preserve"> ιllud pũctũ motu ſuo deſcrιbet circulũ, cuius polus erit pun-<lb/>ctũ c:</s> <s xml:id="echoid-s46494" xml:space="preserve"> & à tota illius circuli peripheria fiet reflexio ad idẽ punctum ſemιdiametri ſpeculi, quę eſt c e:</s> <s xml:id="echoid-s46495" xml:space="preserve"> <lb/>fietq́;</s> <s xml:id="echoid-s46496" xml:space="preserve"> in illis punctιs diametri combuſtio, oppoſita alιqua materia cõbuſtibili, ſed debιlis & cũ mo-<lb/>ra tẽporis.</s> <s xml:id="echoid-s46497" xml:space="preserve"> Quòd ſi fieri poſsit, ut loca plura cóbuſtionis uel omnia in unũ punctũ congregẽtur, fiet <lb/>fortior cóbuιtio:</s> <s xml:id="echoid-s46498" xml:space="preserve"> hoc aũt uiſum eſt poſsibile fierι ք interſectionẽ ſphęricã plurium ſpeculorũ ſphęri <lb/>corũ cõcauorũ:</s> <s xml:id="echoid-s46499" xml:space="preserve"> nõ aũt inęqualiũ:</s> <s xml:id="echoid-s46500" xml:space="preserve"> quia in illis nõ cõueniẽter uniformis poteſt inueniri ꝓportio.</s> <s xml:id="echoid-s46501" xml:space="preserve"> Re <lb/> <anchor type="figure" xlink:label="fig-0695-02a" xlink:href="fig-0695-02"/> linꝗtur er-<lb/>go quòd ę-<lb/>qualiũ ſpe <lb/>culorum <lb/>ſphęrico-<lb/>rũ ſit illa in <lb/>terſectio:</s> <s xml:id="echoid-s46502" xml:space="preserve"> <lb/>ita, ut illud <lb/>qđ uariat <lb/>in locis cõ <lb/>buſtionũ <lb/>diuerſitas <lb/>diſtátiæ ra <lb/>diorũ ęqui <lb/>diſtãtium <lb/>axi ſpecu-<lb/>li, & ad i-<lb/>pſum axẽ <lb/>reflexorũ, <lb/>cõformet <lb/>diuerſifica <lb/>tio centro <lb/>rũ:</s> <s xml:id="echoid-s46503" xml:space="preserve"> utſi cẽ-<lb/>tra ſphęra-<lb/>rũ ſpeculo <lb/>rũ ſe inter <lb/>ſecantium <lb/>ſecundum omnia puncta unius ſemidiametri ſphæræ uarientur:</s> <s xml:id="echoid-s46504" xml:space="preserve"> tunc enim puncta combuſtionis <lb/>aut omnia, aut plurima in unũ pũctũ colligẽtur:</s> <s xml:id="echoid-s46505" xml:space="preserve"> & fortificabitur cóbuſtio ſecũdum illud.</s> <s xml:id="echoid-s46506" xml:space="preserve"> Huius aũt <lb/>rei mechanicũ artificiũ tradẽdũ cogitauimus illis, ꝗք manualẽ fabricã intẽdere uoluerint p̃miſsis, <lb/> <pb o="394" file="0696" n="696" rhead="VITELLONIS OPTICAE"/> cuius forma talis eſt.</s> <s xml:id="echoid-s46507" xml:space="preserve"> Aſſumatur regula lignea uel ænea quadrágula planarũ ſuperficierũ, quáta pla <lb/>cet:</s> <s xml:id="echoid-s46508" xml:space="preserve"> & ſit eius latitudo tripla ſuæ ſpiſsitudini uel circa illud:</s> <s xml:id="echoid-s46509" xml:space="preserve"> deinde in medio ſuæ latitudinis caue-<lb/>tur ſecundũ lineã rectã, & planetur ſoramẽ, & or dinetur taliter, ut intra ipſam decurrere poſsit na-<lb/>uicula ad modũ artificij tornatorũ, in qua nauicula uncus ferreus in figatur.</s> <s xml:id="echoid-s46510" xml:space="preserve"> & hęc regula ſic cõcaua <lb/>ta & diſpoſita, taliter ſituetur, ut eius cauata ſuperficies ſit erecta ſuper ſuperficiem horizontis, & li <lb/>neæ profunditatis ſuæ concauitatis ſint perpendiculares ſuper ſuperficiem horizontis:</s> <s xml:id="echoid-s46511" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s46512" xml:space="preserve"> linea, <lb/>quá motu ſuo deſcribet uncus motæ nauiculę, æqualis ſemidiametro propoſiti circuli, quæ eſt e d, <lb/>ita quòd pũctũ e cadat in intrinſeca ſuperficie ipſius unci ferrei, qui motu nauiculæ, cui infixus eſt, <lb/>mouetur.</s> <s xml:id="echoid-s46513" xml:space="preserve"> Deinde aſſumatur alia regula lignea uel ænea ſimiliter quadrangula, ut prima, & planarũ <lb/>ſuperficierũ:</s> <s xml:id="echoid-s46514" xml:space="preserve"> & hæc ſimiliter in ſui ſuperficie latiori cauetur ſubtiliter ſecũdum lineas rectas, & pla-<lb/>nentur ſuperficies cócauitatis, ita ut ſine impedimẽto ք illã cõcauitatẽ poſsit alia ſubtilis regula uel <lb/>funiculus moueri:</s> <s xml:id="echoid-s46515" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s46516" xml:space="preserve"> cõcauitas illius regulæ duplalιneæ e d, hoc eſt ut ſit æqualis diametro circu <lb/>li, quæ eſt a c:</s> <s xml:id="echoid-s46517" xml:space="preserve"> & hæc regula cũ priori regula taliter adaptetur, ut eius ſuperficies nõ cõcauata æquι-<lb/>diſtet horizonti, & eius ſuperficies cauata reſpiciat cauaturã regulæ prioris:</s> <s xml:id="echoid-s46518" xml:space="preserve"> & ordinetur orthogo-<lb/>naliter ſuper illá, ita ut angulus d e c ſit rectus:</s> <s xml:id="echoid-s46519" xml:space="preserve"> & ſit medius pũctus lógitudinis ſuæ cócauitatis cor-<lb/>reſpondens puncto e, qui eſt punctus unci ipſius nauiculæ:</s> <s xml:id="echoid-s46520" xml:space="preserve"> & ſint omnia hæc in eadẽ ſuperficie æ-<lb/>quidiſtante ſuperficiei horizontis.</s> <s xml:id="echoid-s46521" xml:space="preserve"> Fiat quoq;</s> <s xml:id="echoid-s46522" xml:space="preserve"> tertia regula ęnea longa quadrangularũ ſuperficierú <lb/>planarú & rectarũ linearũ, quę ſit e f g:</s> <s xml:id="echoid-s46523" xml:space="preserve"> ſit q́;</s> <s xml:id="echoid-s46524" xml:space="preserve"> eius pars e f æqualis ſemidiametro circuli, quæ eſt e c:</s> <s xml:id="echoid-s46525" xml:space="preserve"> <lb/>ſitq́;</s> <s xml:id="echoid-s46526" xml:space="preserve"> taliter diſpoſita, ut per aliquã armillã uel foramẽ applicetur unco nauiculę ſecundum pũctum <lb/>e, & ut ipſa moueri poſsit per cócauitaté lineæ a c:</s> <s xml:id="echoid-s46527" xml:space="preserve"> ſit q́;</s> <s xml:id="echoid-s46528" xml:space="preserve"> in puncto f nodus, cuius diameter ſit maior <lb/>diametro concauitatis regulæ a c:</s> <s xml:id="echoid-s46529" xml:space="preserve"> fiat quoq;</s> <s xml:id="echoid-s46530" xml:space="preserve"> reliqua pars lineæ e f g, quę eſt f g, longitudinis placitæ <lb/>cuiuſcunq;</s> <s xml:id="echoid-s46531" xml:space="preserve">: & in puncto g adhibeatur clauus acutus in fine acuitatis, qui ſit illius quátitatis, ut mo-<lb/>ta linea e f g, attingere poſsit pauimentum uel illi aliam ſuperficiem ſubſtratam.</s> <s xml:id="echoid-s46532" xml:space="preserve"> His itaq;</s> <s xml:id="echoid-s46533" xml:space="preserve"> omnibus <lb/>ſic diſpoſitis immittatur regula e f g ſecundum foramen puncti e in uncum nauiculę, & trahatur na <lb/>uicula planè per cochleam uel modo alio, ut uidebitur, plano tamen & æquali tractu:</s> <s xml:id="echoid-s46534" xml:space="preserve"> & ſequetur re <lb/>gula e f g tractum nauiculę, decurretq́;</s> <s xml:id="echoid-s46535" xml:space="preserve"> punctus f in ſuperficie regulę a c:</s> <s xml:id="echoid-s46536" xml:space="preserve"> & ſemper mutabιtur cen-<lb/>trum circuli, cuius diameter eſt linea e f.</s> <s xml:id="echoid-s46537" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s46538" xml:space="preserve"> punctus e peruenerit in punctũ d:</s> <s xml:id="echoid-s46539" xml:space="preserve"> tũc punctus f <lb/>erit in medio puncto lineæ a c, quod eſt centrum circuli præmiſsi:</s> <s xml:id="echoid-s46540" xml:space="preserve"> omniumq́ punctorum reflexio-<lb/>nis luminis uel quarum cunq;</s> <s xml:id="echoid-s46541" xml:space="preserve"> formarum à quarta circuli, quæ eſt c b, concurſus radiorum uel diffu <lb/>ſæ uirtutis erit in centro circuli, quod eſt e:</s> <s xml:id="echoid-s46542" xml:space="preserve"> quoniã omnia puncta combuſtionum concurrentia in <lb/>axe e b, reducta ſunt ad punctum e, quod eſt centrum circuli, utpote omnium radiorum incidentiũ <lb/>circulo ſpeculi æquidiſtáter diametro a c.</s> <s xml:id="echoid-s46543" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s46544" xml:space="preserve">, ſi placet, fiat in alia quarta circuli deſcen <lb/>dente planè ipſa nauicula, reducendo punctum f ad pũctum a:</s> <s xml:id="echoid-s46545" xml:space="preserve"> tunc enim punctum g lineę f g motu <lb/>ſuo deſcribet quandam lineam, quę per clauum ſibi affixum in pauimento figurabitur:</s> <s xml:id="echoid-s46546" xml:space="preserve"> & hác lineá <lb/>dicimus lineam eccentralem:</s> <s xml:id="echoid-s46547" xml:space="preserve"> quoniam eſt interſectio infinitorum circulorum.</s> <s xml:id="echoid-s46548" xml:space="preserve"> Quilibet enim pun <lb/>ctus illius lineæ (exceptis punctis extremis correſpondentibus punctis a & c, ipſius diametri a c, <lb/>& quibuslibet duobus punctis æqualiter diſtantibus à puncto medio totius lineæ eccentralis) di-<lb/>uerſo correſpondet centro, ſicut & quęlibet duo puncta æqualiter diſtantia à puncto ſui medio, re-<lb/>ſpiciunt idem centrum:</s> <s xml:id="echoid-s46549" xml:space="preserve"> & ſunt puncta unius circuli alterum circulum ſecantis.</s> <s xml:id="echoid-s46550" xml:space="preserve"> Hac ergo linea ad <lb/>conſtitutionem propoſiti ſpeculi utemur ſecundum ipſam aliquam ſpecularem ſuperficiem conca <lb/>uantes, ſicut per modum demonſtrationis & artificij inſerius dicetur.</s> <s xml:id="echoid-s46551" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s46552" xml:space="preserve"/> </p> <div xml:id="echoid-div1778" type="float" level="0" n="0"> <figure xlink:label="fig-0695-01" xlink:href="fig-0695-01a"> <variables xml:id="echoid-variables805" xml:space="preserve">b f a e c d</variables> </figure> <figure xlink:label="fig-0695-02" xlink:href="fig-0695-02a"> <variables xml:id="echoid-variables806" xml:space="preserve">a c a c e c f g</variables> </figure> </div> </div> <div xml:id="echoid-div1780" type="section" level="0" n="0"> <head xml:id="echoid-head1318" xml:space="preserve" style="it">38. Ex interſectione plurium ſpeculorum pyramidalium concauorum ignem est poßibi-<lb/>le accendi.</head> <p> <s xml:id="echoid-s46553" xml:space="preserve">Quod hic proponimus, primum fuit, quod nobis harum rerum ſcientiam perquirentibus occur <lb/>rit, & in cuius rei inuentione primò animus noſter conquieuit.</s> <s xml:id="echoid-s46554" xml:space="preserve"> Quia etſi non ad unum punctum <lb/>mathematicum, ad unum tamen punctum naturalem modicam & quaſi inſenſibilem latitudinem <lb/>habentem radij unius totalis ſuperficiei poſſunt faciliter aggregari:</s> <s xml:id="echoid-s46555" xml:space="preserve"> quę nobis uerò poſtea occur-<lb/>rerunt, ualidiora ſunt.</s> <s xml:id="echoid-s46556" xml:space="preserve"> Nihil tamen iſtorum duximus prętermittendum, ut poſteriorum animi in al <lb/>tius excreſcát.</s> <s xml:id="echoid-s46557" xml:space="preserve"> Pręſenti itaq;</s> <s xml:id="echoid-s46558" xml:space="preserve"> demõſtrationi opus ipſum mechanicũ duximus aliqualiter immiſcen <lb/>dũ, nihil tamẽ de demõſtrationis ſubſtantia omittentes.</s> <s xml:id="echoid-s46559" xml:space="preserve"> Aſſumatur ergo quæcũq;</s> <s xml:id="echoid-s46560" xml:space="preserve"> pyramis, quæ ſit <lb/>a b c d:</s> <s xml:id="echoid-s46561" xml:space="preserve"> cuius uertex ſit punctũ a:</s> <s xml:id="echoid-s46562" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s46563" xml:space="preserve"> lineę lõgitudinis illius pyramidis a b & a c:</s> <s xml:id="echoid-s46564" xml:space="preserve"> & ſit axis ipſius li <lb/>nea a d:</s> <s xml:id="echoid-s46565" xml:space="preserve"> quæ ſit, exempli cauſſa, partes 18, ſecundũ quod diameter circuli ſuę baſis, quę eſt f b e c, eſt <lb/>partes 6:</s> <s xml:id="echoid-s46566" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s46567" xml:space="preserve"> per 89 th.</s> <s xml:id="echoid-s46568" xml:space="preserve"> 1 huius punctũ d cẽtrũ circuli, qui eſt baſis ipſius pyramidis:</s> <s xml:id="echoid-s46569" xml:space="preserve"> inſcribaturq́;</s> <s xml:id="echoid-s46570" xml:space="preserve"> <lb/>circulo baſis linea æqualis ſemidiametro ipſius per 1 p 4:</s> <s xml:id="echoid-s46571" xml:space="preserve"> quę ſit f e:</s> <s xml:id="echoid-s46572" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s46573" xml:space="preserve"> aliqua diameter in circulo <lb/>æquidiſtans inſcriptæ lineæ:</s> <s xml:id="echoid-s46574" xml:space="preserve"> quoniã diuiſa linea f e per æqualia ex 10 p 1, ꝓducatur à pũcto diuiſio <lb/>nis (ꝗ ſit g) perpendicularis ſuper illam lineam ex 11 p 1:</s> <s xml:id="echoid-s46575" xml:space="preserve"> hęc quoq;</s> <s xml:id="echoid-s46576" xml:space="preserve"> tranſibit per centrũ circuli per <lb/>1 p 3:</s> <s xml:id="echoid-s46577" xml:space="preserve"> producaturq́;</s> <s xml:id="echoid-s46578" xml:space="preserve"> linea illa ad utranq;</s> <s xml:id="echoid-s46579" xml:space="preserve"> partẽ circũferentiæ:</s> <s xml:id="echoid-s46580" xml:space="preserve"> & ſit b c:</s> <s xml:id="echoid-s46581" xml:space="preserve"> extrahatur ergo perpẽdicula-<lb/>ris à centro circulι baſis, quod eſt d, ſuper diametrum b c:</s> <s xml:id="echoid-s46582" xml:space="preserve"> quæ ſit d h:</s> <s xml:id="echoid-s46583" xml:space="preserve"> & producatur ad partem aliã <lb/>circuli:</s> <s xml:id="echoid-s46584" xml:space="preserve"> fietq́;</s> <s xml:id="echoid-s46585" xml:space="preserve"> diameter, quę ſit h k, æquidiſtans lineæ e f per 28 p 1:</s> <s xml:id="echoid-s46586" xml:space="preserve"> producanturq́;</s> <s xml:id="echoid-s46587" xml:space="preserve"> à punctis h & k <lb/>duę lineæ longitudinis pyramidis ad uerticem, quę ſint h a & k a.</s> <s xml:id="echoid-s46588" xml:space="preserve"> Producatur quoq;</s> <s xml:id="echoid-s46589" xml:space="preserve"> à puncto e li-<lb/>nea æquidiſtans lineę k a, & à puncto f æquidiſtans linea h a ex 31 p 1:</s> <s xml:id="echoid-s46590" xml:space="preserve"> & concurrant productę lineæ <lb/>in puncto x:</s> <s xml:id="echoid-s46591" xml:space="preserve"> concurrent autem ideo, quia ipſarum æquidiſtantes, quę ſunt k a & h a, concurrunt in <lb/>puncto a.</s> <s xml:id="echoid-s46592" xml:space="preserve"> Inter duas ergo lineas e x & f x continuata plana ſuperficies & terminata ad lineam f e <lb/> <pb o="395" file="0697" n="697" rhead="LIBER NONVS."/> (quæ ſit trigonum f e x) palàm quoniam interſecabit pyramidem:</s> <s xml:id="echoid-s46593" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s46594" xml:space="preserve"> triangulus x f e propter <lb/>æquidiſtantiam laterum ęquidiſtans triangulo magno in pyramide, qui eſt a h k:</s> <s xml:id="echoid-s46595" xml:space="preserve"> & ſicut triangulus <lb/> <anchor type="figure" xlink:label="fig-0697-01a" xlink:href="fig-0697-01"/> a h k diuidit pyramidem per æqualia, eò quòd ſit duabus lineis lon <lb/>gitudinis & diametro baſis contentus:</s> <s xml:id="echoid-s46596" xml:space="preserve"> ſic etiam triangulus x f e ali <lb/>quam pyramidis reſecat portionem.</s> <s xml:id="echoid-s46597" xml:space="preserve"> Abſcindatur ergo hæc portio <lb/>â tota pyramide, quæ ſit l f b e g:</s> <s xml:id="echoid-s46598" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s46599" xml:space="preserve"> lineę rectæ, quæ ſint l e <lb/>& l f:</s> <s xml:id="echoid-s46600" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s46601" xml:space="preserve"> lineę l f & l e per 89 th.</s> <s xml:id="echoid-s46602" xml:space="preserve"> 1 huius partes æquales unius <lb/>ſectióis conicæ, quę eſt e l f, diuiſa per æqualia in ſui ſupremo pun-<lb/>cto, qui eſt l.</s> <s xml:id="echoid-s46603" xml:space="preserve"> Linea uerò l b, quę eſt pars lineæ longitudinis pyrami <lb/>dis, erit minoris quantitatis qualibet linearum l e & l f:</s> <s xml:id="echoid-s46604" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s46605" xml:space="preserve"> linea <lb/>b g linea proſun ditatis huius portionis:</s> <s xml:id="echoid-s46606" xml:space="preserve"> linea uerò f e linea latitu-<lb/>dinis:</s> <s xml:id="echoid-s46607" xml:space="preserve"> & linea l g latus portionis erectum, æquidiſtans lineæ d a, <lb/>quę eſt axis pyramidis.</s> <s xml:id="echoid-s46608" xml:space="preserve"> Expedit ergo ut operi mechanico conſu-<lb/>lentes notitiam harum linearum omniũ per quiramus, ſupponen-<lb/>tes ea, quę in chordis & arcubus ſunt probata.</s> <s xml:id="echoid-s46609" xml:space="preserve"> Palàm autẽ ex præ-<lb/>miſsis quoniã linea f e, quæ inſcripta circulo, quia eſt æqualis eius <lb/>ſemidiametro, eſt partes 60, ſecũdũ quod diameter circuli eſt 120:</s> <s xml:id="echoid-s46610" xml:space="preserve"> <lb/>arcus ergo f e ſimiliter eſt 60:</s> <s xml:id="echoid-s46611" xml:space="preserve"> ſecũdũ qđ circulus eſt 360.</s> <s xml:id="echoid-s46612" xml:space="preserve"> Ducátur <lb/>quoq;</s> <s xml:id="echoid-s46613" xml:space="preserve"> lineę b f & b e.</s> <s xml:id="echoid-s46614" xml:space="preserve"> Et quoniã diameter b c diuidit chordá f e per <lb/>æqualia & orthogona liter:</s> <s xml:id="echoid-s46615" xml:space="preserve"> patet quoniã line ærectę f b & b e ęqua-<lb/>les ſunt ք 4 p 1:</s> <s xml:id="echoid-s46616" xml:space="preserve"> ergo arcus f b & b e ſunt ęquales per 28 p 3:</s> <s xml:id="echoid-s46617" xml:space="preserve"> arcus <lb/>itaq;</s> <s xml:id="echoid-s46618" xml:space="preserve"> f e diuiſus eſt per æqualia in pũcto b:</s> <s xml:id="echoid-s46619" xml:space="preserve"> ergo arcus f b eſt partes <lb/>30:</s> <s xml:id="echoid-s46620" xml:space="preserve"> chorda ergo f b eſt 31 partes, 3 minuta, & 30 ſecunda:</s> <s xml:id="echoid-s46621" xml:space="preserve"> ſed quoniá <lb/>m linea f g eſt medietas lineę f e, quę fuit 60:</s> <s xml:id="echoid-s46622" xml:space="preserve"> patet qđ linea f g eſt <lb/>30:</s> <s xml:id="echoid-s46623" xml:space="preserve"> quadrentur ergo ex 46 p 1 linea f b, & ſimiliter linea f g.</s> <s xml:id="echoid-s46624" xml:space="preserve"> Et quia <lb/>quadratũ lineæ f b in triangulo f b g ſubtenditur angulo recto:</s> <s xml:id="echoid-s46625" xml:space="preserve"> palá <lb/>ex 47 p 1 quia quadratũ lineæ f b ualet ambo quadrata linearũ f b <lb/>& b g:</s> <s xml:id="echoid-s46626" xml:space="preserve"> ablato ergo ex quadrato f b quadrato f g, remanet quadratũ <lb/>b g.</s> <s xml:id="echoid-s46627" xml:space="preserve"> Extrahatur ergo radix quadrata illius reſidui, & ipſa eſt quanti <lb/>tas lineæ b g:</s> <s xml:id="echoid-s46628" xml:space="preserve"> & ſecundũ quod linea f g eſt 30 partes, erit ipſa 8 par-<lb/>tes, 2 minuta, 29 ſecunda:</s> <s xml:id="echoid-s46629" xml:space="preserve"> ſecundũ uerò quòd diameter b c eſt par-<lb/>tes 6, & ſemidiameter f e partes 3, & linea f g partes 8 & 30 minuta:</s> <s xml:id="echoid-s46630" xml:space="preserve"> <lb/>erit linea b g 24 minuta & 6 ſecunda, prout ex tribus notis quartũ <lb/>ignotum perquirens auxilio 19 p 7 diligens inquiſitor facile pote-<lb/>ritinuenire.</s> <s xml:id="echoid-s46631" xml:space="preserve"> Quoniam uerò linea g l erecta ęquidiſtans eſt axi py-<lb/>ramidis, quę eſt d a, patet ex 29 p 1 quoniam trianguli d a b & g l b ſunt ęquianguli:</s> <s xml:id="echoid-s46632" xml:space="preserve"> ergo per <lb/>4 p 6 erit proportio lineę d a ad lineam g l, ſicut lineę d b ad lineam g b:</s> <s xml:id="echoid-s46633" xml:space="preserve"> ergo per 16 p 5 erit per-<lb/>mutatim proportio lineę d a ad lineam d b, ſicut lineę l g ad lineam g b:</s> <s xml:id="echoid-s46634" xml:space="preserve"> ſed linea d a ſextupla eſt <lb/>ad lineam d b ex hypotheſi:</s> <s xml:id="echoid-s46635" xml:space="preserve"> erit ergo linea l g ſextupla lineę b g:</s> <s xml:id="echoid-s46636" xml:space="preserve"> patet ergo quoniam linea l g e-<lb/>rit 2 partes, 24 minuta, 36 ſecunda, ſecundum quod linea d a eſt partes 18.</s> <s xml:id="echoid-s46637" xml:space="preserve"> Sed quia in triangulo <lb/>l b g angulus l g b eſt rectus:</s> <s xml:id="echoid-s46638" xml:space="preserve"> quia latus g l, quemadmodum linea d a, orthogonaliter erectum eſt <lb/>ſuper ſuperficiem circuli baſis pyramidis per 89 th.</s> <s xml:id="echoid-s46639" xml:space="preserve"> 1 huius, & per 8 p 11:</s> <s xml:id="echoid-s46640" xml:space="preserve"> patet ergo quia quadratum <lb/>lineę l b ualet quadrata ambarum linearum l g & b g ex 47 p 1:</s> <s xml:id="echoid-s46641" xml:space="preserve"> componantur ergo quadrata, & <lb/>aggregati radix quadrata extrahatur, & ipſa eſt quantitas lineę l b:</s> <s xml:id="echoid-s46642" xml:space="preserve"> quę ſecundum propoſitum nu-<lb/>merum quo ſemidiameter baſis eſt 3 partes, erit 2 partes, 26 minuta, 35 ſecunda.</s> <s xml:id="echoid-s46643" xml:space="preserve"> Et quia li-<lb/>nea l g erecta eſt ſuper ſuperficiem baſis pyramidis:</s> <s xml:id="echoid-s46644" xml:space="preserve"> palàm ex definitione lineę erectę ſuper ſu-<lb/>perficiem, quoniam ipſa cum lineis g f & g e angulos rectos facit, ſicut etiam cum omnibus li-<lb/>neis in dicta ſuperficie productis.</s> <s xml:id="echoid-s46645" xml:space="preserve"> Quadratum ergo lineę e l rectę, quę in triangulo rectilineo <lb/>(qui eſt e g l) angulo recto opponitur, ualet quadratum lineæ l g & lineæ g e:</s> <s xml:id="echoid-s46646" xml:space="preserve"> coniunctis ergo il-<lb/>lis quadratis, ipſius aggregati extrahatur radix:</s> <s xml:id="echoid-s46647" xml:space="preserve"> & patet quòd linea recta, quæ eſt l e, eſt 2 partes, 50 <lb/>minuta, 19 ſecunda.</s> <s xml:id="echoid-s46648" xml:space="preserve"> Et quia per eadem quadratum lineæ rectæ, quę eſt fl, ualet quadratũ lineæ f g, <lb/>quæ eſt æqualis lineæ g e, & quadratum lineæ l g:</s> <s xml:id="echoid-s46649" xml:space="preserve"> patet quia linea l f eſt æqualis lineæ e l:</s> <s xml:id="echoid-s46650" xml:space="preserve"> erit ergo li <lb/>nea f l 2 partes, 50 minuta, 19 ſecunda.</s> <s xml:id="echoid-s46651" xml:space="preserve"> Habetur itaq;</s> <s xml:id="echoid-s46652" xml:space="preserve"> notitia omnium linearũ portionis pyramidis <lb/>aſſumptæ, neceſſariæ operi præſenti.</s> <s xml:id="echoid-s46653" xml:space="preserve"> Cum autẽ difficile ſit aſſumi pyramidẽ propoſito cõpetẽtem, <lb/>(quoniã oporteret, ut ipſa tota eſſet concaua ſolidi corporis dẽſi & polibilis pro factura ſpeculi, ut <lb/>prius dictum eſt, & ab illa difficilis fieret abſciſio) ſufficiat ipſam habere mathematicam in imagi-<lb/>natione.</s> <s xml:id="echoid-s46654" xml:space="preserve"> Cum ergo ad opus ſpeculi libeat procedere:</s> <s xml:id="echoid-s46655" xml:space="preserve"> fiat de corpore polibili albo, utpote argenteo <lb/>uel ferreo bono portio pyramidis concaua, ſic ut baſis illius ſectionis ſit portio circuli, qui eſt baſis <lb/>imaginatæ pyramidis, cuius chorda ſit medietas diametri imaginati circuli, & eſt linea f e:</s> <s xml:id="echoid-s46656" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s46657" xml:space="preserve"> par <lb/>tes 3:</s> <s xml:id="echoid-s46658" xml:space="preserve"> ſinus uerò uerſus, qui g b, ſit ſecundum illam quantitatẽ 24 minuta, 6 ſecunda, quę eſt linea ꝓ-<lb/>funditatis acceptæ ſectionis:</s> <s xml:id="echoid-s46659" xml:space="preserve"> & fortè, quádo protrahitur, aſsimilatur ſagittæ, ſecundum quod illę li <lb/>neæ chordæ & arcui aſsimilátur:</s> <s xml:id="echoid-s46660" xml:space="preserve"> & erunt lineæ e l & f l rectæ æquales:</s> <s xml:id="echoid-s46661" xml:space="preserve"> & ipſarum quęlibet eſt 2 par-<lb/>tes, 50 minuta, 19 ſecunda:</s> <s xml:id="echoid-s46662" xml:space="preserve"> & erit linea l b 2 partes, 26 minuta, 35 ſecunda, ſecundum dictã quantita-<lb/>tẽ:</s> <s xml:id="echoid-s46663" xml:space="preserve"> quę omnia ſi bene menſurata ſuerint:</s> <s xml:id="echoid-s46664" xml:space="preserve"> patet qđ habetur portio pyramidis, cuius circuli baſis dia <lb/> <pb o="396" file="0698" n="698" rhead="VITELLONIS OPTICAE"/> meter eſt partes 6, & axis pyramidis partes 18:</s> <s xml:id="echoid-s46665" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s46666" xml:space="preserve"> tale ſpeculũ latius ꝗ̃ ſit lõgum, & in breue ſpa-<lb/>tium radios plurimos cõgregabit.</s> <s xml:id="echoid-s46667" xml:space="preserve"> Quòd ſi axẽ pyramidis imaginatus ſueris 24 partes, ſecun dũ qđ <lb/>diameter eſt partes 6:</s> <s xml:id="echoid-s46668" xml:space="preserve"> tunc erit linea l g 4 partes, & longius radij protendentur:</s> <s xml:id="echoid-s46669" xml:space="preserve"> eruntq́ exharum li <lb/>nearum notitia, & ex notitia linearũ e g & g f (quarum notitia ſupponitur, eò quòd ſunt medietas <lb/>ſemidiametri) oẽs aliæ lineæ notę cõponenti quadrata linearũ notarum, & radicẽ lateris oppoſiti <lb/>recto angulo extrahenti:</s> <s xml:id="echoid-s46670" xml:space="preserve"> & numerorũ taliũ eſt infinitas, eò quòd ſecun dum omnẽ numerum axem <lb/>pyramidis accipi eſt poſsibile, diametro tamẽ circuli baſis nõ mutata ſecundũ numerũ, & ſi mute-<lb/>tur ſecundum quãtitatẽ partium numeratarum.</s> <s xml:id="echoid-s46671" xml:space="preserve"> Certitudo ergo numerorum operationi indagato <lb/>ris ſoliciti relin quatur:</s> <s xml:id="echoid-s46672" xml:space="preserve"> ſinus enim uerſus & medietas ſemidiametri, circulo in ſcriptæ, ſecundũ quã <lb/>fit baſis portionis abſciſsio, nõ poterũt uariari:</s> <s xml:id="echoid-s46673" xml:space="preserve"> ex quorum notitia ad aliarum linearum notitiã po-<lb/>terit procedi.</s> <s xml:id="echoid-s46674" xml:space="preserve"> Quòd ſi radios ad longã diſtantiã aggregari placuerit (ex quo tamẽ uirtutẽ ipſorum <lb/>debilitari patulum eſt, niſi quãtitas aggregationis quãtitatẽ uincat diſtãtiæ) illud erit in exceſſu la <lb/>teris erecti ipſius ſcilicet axis pyramidis, reſpectu ſemidiametri baſis, & ſemidiametri baſis, reſpe-<lb/>ctu ſinus uerſi.</s> <s xml:id="echoid-s46675" xml:space="preserve"> Poteſt ergo, ſiplacet, circulo baſis inſcribi medietas ſemidiametri:</s> <s xml:id="echoid-s46676" xml:space="preserve"> hæc aũt cũ ſit par-<lb/>tes 30 ſecundum quod tota diameter eſt partes 120, ſi ex notis notum extrahatur:</s> <s xml:id="echoid-s46677" xml:space="preserve"> inuenietur arcus <lb/>ſibi correſpódens in circulo 28 partium, 57 minutorum, 21 ſecundorum, qui ex 30 p 3 ſi per æqualia <lb/>diuidatur, erit medietas ipſius 14 partes, 28 minuta, 40 ſecunda, 30 tertia, ſecundum quod circulus <lb/>eſt 360, cuius arcus chordá operás inueniet 15 partes, 7 minuta, 13 ſecunda, 20 tertia, ſecundum qđ <lb/>diameter eſt 120:</s> <s xml:id="echoid-s46678" xml:space="preserve"> ſemidiameter quoq;</s> <s xml:id="echoid-s46679" xml:space="preserve"> partes 60.</s> <s xml:id="echoid-s46680" xml:space="preserve"> Sed ſecundũ quod ſemidiameter eſt partes 3:</s> <s xml:id="echoid-s46681" xml:space="preserve"> erit <lb/>prædicta chorda 45 minuta 21 ſecunda, 40 tertia:</s> <s xml:id="echoid-s46682" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s46683" xml:space="preserve"> latus f b.</s> <s xml:id="echoid-s46684" xml:space="preserve"> Sed linea f e inſcripta circulo ęqua-<lb/>lis medietati ſemidiametri, per diametrum orthogonaliter ſuperſtátem ei, ex 3 p 3 diuiditur per æ-<lb/>qualia in puncto g:</s> <s xml:id="echoid-s46685" xml:space="preserve"> ergo linea f g eſt medietas lineæ f e (quę eſt pars & 30 minuta) linea ergo f g eſt <lb/>45 minuta.</s> <s xml:id="echoid-s46686" xml:space="preserve"> Quadratum itaq;</s> <s xml:id="echoid-s46687" xml:space="preserve"> f g auferatur ex quadrato f b, & reſidui extrahatur radix quadrata, & <lb/>erit linea g b (quę eſt ſinus uerſus ipſius arcus f e) 5 minuta, 42 ſecunda, 44 tertia:</s> <s xml:id="echoid-s46688" xml:space="preserve"> cuius immutabi <lb/>li hic poſita quãtitate numerali, axe pyramidis quomodocunq;</s> <s xml:id="echoid-s46689" xml:space="preserve"> in numero & quátitate uariato, dia <lb/>metro baſis 6 partium cuiuſcunq;</s> <s xml:id="echoid-s46690" xml:space="preserve"> quãtitatis exiſtẽte, oés lineæ abſciſſæ ſectionis, ut prius, operãti <lb/>poſſunt faciliter inueniri.</s> <s xml:id="echoid-s46691" xml:space="preserve"> Fabricata itaq;</s> <s xml:id="echoid-s46692" xml:space="preserve"> ſectione pyramidis, ſi placet, ex ferro cõpetẽtis ſpiſsitudi <lb/>nis, menſurationeq́;</s> <s xml:id="echoid-s46693" xml:space="preserve"> facta linearum pręmiſſarum in illa, ſecundum proportionẽ axis imagιnatæ py <lb/>ramidis, & ſecundum diuerſitatẽ lineæ baſi inſcriptæ, quá fieri poſſe diximus ſecundũ quátitatem <lb/>ſemidiametri uel medietatẽ ipſius, ut ſecundum hęc quátitas ſinus uerſi & tota proportio uarietur, <lb/>planetur ſpeculum intrinſecus ne partes partibus multum pręmineant, quátùm eſt poſsibile.</s> <s xml:id="echoid-s46694" xml:space="preserve"> Quia <lb/>uerò, & ſi hoc ſpeculum ſecũdū ultimum poſsibilitatis poliretur:</s> <s xml:id="echoid-s46695" xml:space="preserve"> tamẽ quia eſt pars pyramidis, oẽs <lb/>radios ipſius uel plures ad unum punctũ aggregari eſſet impoſsibile, ut patet per 36 huius:</s> <s xml:id="echoid-s46696" xml:space="preserve"> oportet <lb/>ergo ante politionẽ completã aliã ſibi adhibere medelã, ſcilicet, ut in eo fiant diuerſarũ interſectio-<lb/>nes pyramidum:</s> <s xml:id="echoid-s46697" xml:space="preserve"> qđ ք tale artificium poterit cõpleri.</s> <s xml:id="echoid-s46698" xml:space="preserve"> Quoniam enim in aſſumpta pyramidis por-<lb/>tione, triangulus l b g, qui continetur à lineis intra ſectionem aſſumptis, eſt notorum laterũ:</s> <s xml:id="echoid-s46699" xml:space="preserve"> æqua-<lb/>lis ei triangulus in aliquo plano deſcribatur, qui ſit item l b g:</s> <s xml:id="echoid-s46700" xml:space="preserve"> qui ſi duplatus fuerit, protracto late-<lb/>relg, quouſq;</s> <s xml:id="echoid-s46701" xml:space="preserve"> linea g m ſit æqualis lineæ g l, & compleatur triangulus l b m:</s> <s xml:id="echoid-s46702" xml:space="preserve"> palàm quòd ſiue ſit or-<lb/> <anchor type="figure" xlink:label="fig-0698-01a" xlink:href="fig-0698-01"/> thogonius, ſiue amblygonius, ſiue oxygonius, quia ex doctrina 5 p 4 circulus ſibi poteſt circũſcri-<lb/>bi:</s> <s xml:id="echoid-s46703" xml:space="preserve"> circũſcribatur ergo:</s> <s xml:id="echoid-s46704" xml:space="preserve"> quod ut facilius fiat, aſſumatur prior diſpoſitio, ſcilicet, ut linea b g ſit 24 mi <lb/>nutorũ, 6 ſecun dorũ, & linea l g 2 partiũ, 24 minutorũ, 36 ſecũ dorũ:</s> <s xml:id="echoid-s46705" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s46706" xml:space="preserve"> l g ſextupla lineę b g.</s> <s xml:id="echoid-s46707" xml:space="preserve"> Pro-<lb/>ducatur ergo linea b g in continuum & directum ad punctum p, donec linea g p ſit ſextupla lineæ <lb/>l g:</s> <s xml:id="echoid-s46708" xml:space="preserve"> erit ergo proportio lineæ p g ad lineam g l, ſicut lineæ g l ad lineam g b:</s> <s xml:id="echoid-s46709" xml:space="preserve"> ergo per 17 p 6 illud, qđ <lb/>fit ex ductu lineæ g p in lineá b g, erit æquale quadrato lineæ g l:</s> <s xml:id="echoid-s46710" xml:space="preserve"> ſed quadratum lineæ g l æquale eſt <lb/>ei, quod fit ex ductu lineæ g l in lineam g m, quia linea l g eſt æqualis lineæ g m.</s> <s xml:id="echoid-s46711" xml:space="preserve"> Illud ergo, quod fit <lb/>ex ductu lineę p g in lineam g b, eſt æquale ei, quod fit ex ductu lineę l g in lineá g m:</s> <s xml:id="echoid-s46712" xml:space="preserve"> ergo lineæ p g <lb/>& l m in circulo aliquo ſeinterſecant ex conuerſa 35 p 3:</s> <s xml:id="echoid-s46713" xml:space="preserve"> ſed linea p b ſecat lineam l m per æqualia, & <lb/>orthogonaliter ei ſuperſtat exprius datis:</s> <s xml:id="echoid-s46714" xml:space="preserve"> tráſit ergo linea b p per cẽtrũ circuli ex 1 p 3:</s> <s xml:id="echoid-s46715" xml:space="preserve"> quę diuida-<lb/>tur per 10 p 1 per ęqualia, & erit in puncto diuiſionis centrum circuli circumſcriptibilis triangulo <lb/>l g b:</s> <s xml:id="echoid-s46716" xml:space="preserve"> & erit diameter circuli, quę eſt linea b p, 14 partes, 51 minuta, 42 ſecunda:</s> <s xml:id="echoid-s46717" xml:space="preserve"> cuius medietas eſt <lb/>7 partes, 25 minuta, 51 ſecunda:</s> <s xml:id="echoid-s46718" xml:space="preserve"> & eſt punctus ille poſt completam fabricam locus aggregationis <lb/> <pb o="397" file="0699" n="699" rhead="LIBER NONVS."/> radiorum ſpeculi ſecundum dictam diſpoſitionis quantitatem, præter quàm modicum, quod per-<lb/>ditur in limando.</s> <s xml:id="echoid-s46719" xml:space="preserve"> Quòd ſi baſi eiuſdem pyramidis inſcribatur medietas ſemidiametri axe pyrami-<lb/>dis exiſtente 18:</s> <s xml:id="echoid-s46720" xml:space="preserve"> erit linea b g 5 minuta, 42 ſecunda, 44 tertia, cuius ſextuplum eſt latus l g, quod e-<lb/>rit 34 minuta, 16 ſecunda, 24 tertia:</s> <s xml:id="echoid-s46721" xml:space="preserve"> cuius item ſextuplum erit linea g p:</s> <s xml:id="echoid-s46722" xml:space="preserve"> & ipſa erit 3 partes, 25 minu-<lb/>ta, 38 ſecunda, 24 tertia:</s> <s xml:id="echoid-s46723" xml:space="preserve"> adiecta ergo linea b g, erit linea b p 3 partes, 31 minuta, 21 ſecunda, 8 tertia:</s> <s xml:id="echoid-s46724" xml:space="preserve"> <lb/>cuius medietas eſt pars una 45 minuta, 40 ſecunda, 34 tertia:</s> <s xml:id="echoid-s46725" xml:space="preserve"> & eſt punctus ille locus aggregatio-<lb/>nis radiorũ ſpeculi ſecundũ talẽ quátitatẽ diſpoſiti, pręter illud, qđ deperditur in limádo.</s> <s xml:id="echoid-s46726" xml:space="preserve"> Similiter <lb/>etiã eſt in reliquis formis ſpeculorũ ſecúdũ quátitates uarias acceptorũ & ſemper ſecũdũ ꝓportio <lb/>nẽ axis pyramidis, reſpectu diametri baſis, & ſemidiametri, reſpectu ſinus uerſi, fit diuerſitas elóga <lb/>tionis pũcti aggregationis radiorũ à ſpeculo, qui ſecundũ eundem modum eſt in omnibus perqui-<lb/>rendus.</s> <s xml:id="echoid-s46727" xml:space="preserve"> Aſſumatur ergo pars circuli circum ſcribentis triangulum l m b, & reſecetur ſecundum li-<lb/>neam b p, quæ eſt diameter:</s> <s xml:id="echoid-s46728" xml:space="preserve"> & deinde ducatur à centro illius circuli (quod ſit q) linea q l:</s> <s xml:id="echoid-s46729" xml:space="preserve"> & re-<lb/> <anchor type="figure" xlink:label="fig-0699-01a" xlink:href="fig-0699-01"/> ſecetur circulus ſecundum illam, remaneatq́;</s> <s xml:id="echoid-s46730" xml:space="preserve"> q l b ſector:</s> <s xml:id="echoid-s46731" xml:space="preserve"> in quo poſtea fiant interſectiones trian-<lb/>gulorum diuerſarum pyramidum hoc modo.</s> <s xml:id="echoid-s46732" xml:space="preserve"> Quoniam enim angulus l b g eſt angulus ſemicircu-<lb/>li:</s> <s xml:id="echoid-s46733" xml:space="preserve"> patet ex 16 p 3 quoniam ipſe eſt maximus omnium angulorum acutorum:</s> <s xml:id="echoid-s46734" xml:space="preserve"> ergo eſt maior quo-<lb/>libet angulo trianguli cuiuslibet pyramidis.</s> <s xml:id="echoid-s46735" xml:space="preserve"> Reſecetur ergo ab ipſo angulo alicuius trianguli, cu-<lb/>ius latus tertium à centro circuli puncto q productum rectum angulum contineat cum linea b q, <lb/>quæ eſt ſemidiameter circuli:</s> <s xml:id="echoid-s46736" xml:space="preserve"> producaturq́;</s> <s xml:id="echoid-s46737" xml:space="preserve"> à puncto b linea ſecans arcum b l, prout uicinius <lb/>poſsit puncto b:</s> <s xml:id="echoid-s46738" xml:space="preserve"> & ſit arcus reſectus b t.</s> <s xml:id="echoid-s46739" xml:space="preserve"> Deinde adhuc à puncto b ducantur latera aliorum tri-<lb/>angulorum interſecantia arcum b l:</s> <s xml:id="echoid-s46740" xml:space="preserve"> & ſint loca interſectionum t, d, e, f, l:</s> <s xml:id="echoid-s46741" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s46742" xml:space="preserve"> lineæ productæ, <lb/>quoniam angulum acutum continent cum linea b q, omnes concurrentes cum linea à puncto q <lb/>orthogonaliter imaginata erigi, quæ ſit q s, ut patet per 14 th.</s> <s xml:id="echoid-s46743" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s46744" xml:space="preserve"> facientq́;</s> <s xml:id="echoid-s46745" xml:space="preserve"> triangulos inclu-<lb/>dentes ſemper altiores ipſis triangulis incluſis ex 21 p 1:</s> <s xml:id="echoid-s46746" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s46747" xml:space="preserve"> omnium illorum trigonorum ſupe-<lb/>riora puncta ſignata per notam s:</s> <s xml:id="echoid-s46748" xml:space="preserve"> quorum triangulorum quilibet ſi <lb/>moueatur, latere erecto fixo manente, deſcribet pyramidem rotun-<lb/>dam:</s> <s xml:id="echoid-s46749" xml:space="preserve"> & pars motus partem pyramidis efficiet axi copulatam, & pars <lb/> <anchor type="figure" xlink:label="fig-0699-02a" xlink:href="fig-0699-02"/> triangulι reſecta cauſſabit partem pyramidis habentem proportio-<lb/>nem ad totam pyramidem, ſicut pars trianguli ad totum triangulum, <lb/>& ſicut partialis motus ad totum motum.</s> <s xml:id="echoid-s46750" xml:space="preserve"> Quoniam uerò patet per 2 <lb/>huius quòd in ſpeculo pyramidali concauo ſecundum lιneas longi-<lb/>tudinis pyramidis fit reflexio, ita quòd angulus, quem facit radius <lb/>incidens cum linea longitudinis ſpeculi, eſt æqualis angulo reflexio-<lb/>nis, ſcilicet ei, quem facit radius reflexus cum eadem linea longitu-<lb/>dinis ſpeculi (ut ſi ſuper lineam longitudinis pyramidis alicuius ſpe-<lb/>culi, quæ ſit a b, reflectatur radius e c, æquidiſtanter ſemidiametro <lb/>baſi incidens, quæ ſit b d:</s> <s xml:id="echoid-s46751" xml:space="preserve"> patet quia angulus e c a æqualis eſt angu-<lb/>lo d c b:</s> <s xml:id="echoid-s46752" xml:space="preserve"> quoniam, ut patet per 20 th.</s> <s xml:id="echoid-s46753" xml:space="preserve"> 5 huιus, quoſcunque angulos <lb/>facit radius incidens cum perpendiculari erecta ſuper ſuperficiem <lb/>contingentem ſpeculum in puncto incidentiæ, eoſdem facit radius <lb/>reflexus cum eadem perpendiculari:</s> <s xml:id="echoid-s46754" xml:space="preserve"> uniuerſaliter enim angulus in-<lb/>cidentiæ eſt æqualis angulo reflexionis.</s> <s xml:id="echoid-s46755" xml:space="preserve">) Reſumatur ergo q l b ſe-<lb/>ctor, & eius trianguli:</s> <s xml:id="echoid-s46756" xml:space="preserve"> quia quod demonſtratum eſt in pyramidibus, <lb/>uerum etiam eſt in triangulis cauſſantibus pyramides.</s> <s xml:id="echoid-s46757" xml:space="preserve"> Incidat ergo ipſi ſectori in puncto e radius <lb/> <pb o="398" file="0700" n="700" rhead="VITELLONIS OPTICAE"/> æquidiſtans lineæ q b, qui ſſt h t.</s> <s xml:id="echoid-s46758" xml:space="preserve"> Erit ergo angulus incidentiæ, qui eſt h t s, ęqualis angulo reflexio-<lb/>nis:</s> <s xml:id="echoid-s46759" xml:space="preserve"> ſed angulus h t s æqualis eſt angulo q b t per 29 p 1, & angulus q b t eſt per 5 p 1 æqualis angulo <lb/>q t b:</s> <s xml:id="echoid-s46760" xml:space="preserve"> ideo quòd latera q b & q t ſunt æqualia per definitionem circuli:</s> <s xml:id="echoid-s46761" xml:space="preserve"> erit ergo angulus refle-<lb/>xionis æqualis angulo q b t:</s> <s xml:id="echoid-s46762" xml:space="preserve"> ergo linea reflexionis æqualis erit lineæ q b per 6 p 1:</s> <s xml:id="echoid-s46763" xml:space="preserve"> ſecundum li-<lb/>neam ergo q t fit reflexio.</s> <s xml:id="echoid-s46764" xml:space="preserve"> Incidens ergo radius in punctum b, & reflexus à puncto t, concurrunt <lb/>in puncto q:</s> <s xml:id="echoid-s46765" xml:space="preserve"> quia à puncto t aliam lineam æqualem lineæ q b, continentem cum linea b t angu-<lb/>lum æqualem angulo q b t duci eſt impoſsibile.</s> <s xml:id="echoid-s46766" xml:space="preserve"> Similiter etiam angulus incidentiæ, qui eſt k d s, <lb/>æqualis eſt angulo reflexionis:</s> <s xml:id="echoid-s46767" xml:space="preserve"> ſed & idem eſt æqualis angulo q b d ſecundum præmiſſum mo-<lb/>dum deducendo ex 29 p 1:</s> <s xml:id="echoid-s46768" xml:space="preserve"> ergo angulus q b d & angulus reflexionis radij k d incidentis ſunt æ-<lb/>quales:</s> <s xml:id="echoid-s46769" xml:space="preserve"> ergo ſecundum lineam q d fit reflexio.</s> <s xml:id="echoid-s46770" xml:space="preserve"> Similiter etiam eſt & in alijs demonftrandum.</s> <s xml:id="echoid-s46771" xml:space="preserve"> Pa-<lb/>tet ergo quòd omnes radij incidentes in puncta ſectionum factarum per latera triangulorum pro-<lb/>ductorum à puncto b uerſus axem q s reflectuntur ad punctum unum, qui eſt centrum accepti <lb/>circuli.</s> <s xml:id="echoid-s46772" xml:space="preserve"> Et quia ſectiones illæ fieri poſſunt quaſi infinitæ ab una linea ſic ordinata in ſectore, ad u-<lb/>num punctum mathematicum fiunt aggregationes radiorum quaſi infinitæ.</s> <s xml:id="echoid-s46773" xml:space="preserve"> Hoc ergo demonſtra-<lb/>to patet quòd omnes radij incidentes punctis b, t, d, e, f, l reflectuntur ad unum punctum, qui eſt <lb/>q.</s> <s xml:id="echoid-s46774" xml:space="preserve"> Et ſi portiunculæ præeminentes auferantur, regulabunt termini t d & e f interiacentes lineas, <lb/>ita quòd reflexio ab illis facta, non multùm diſtabιt à puncto reflexionis, qui eſt q:</s> <s xml:id="echoid-s46775" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s46776" xml:space="preserve"> aggrega-<lb/>tio omnium radiorum totali lineæ b l incidentium ad unum punctum ſenſibilem naturalem, in <lb/>circuitu puncti q.</s> <s xml:id="echoid-s46777" xml:space="preserve"> Hæc ergo linea b l motu ſuo ſuperficiem ſectionis præaſſumptæ ſuperius pyra-<lb/>midis limando & cauando producet:</s> <s xml:id="echoid-s46778" xml:space="preserve"> à qua tota fiet reflexio ad punctum unum naturalem, ut in-<lb/>ferius docebitur.</s> <s xml:id="echoid-s46779" xml:space="preserve"> Patet ergo propoſitum:</s> <s xml:id="echoid-s46780" xml:space="preserve"> faciunt enim iſti trianguli motu ſuo pyramides ſe in-<lb/>terſecantes.</s> <s xml:id="echoid-s46781" xml:space="preserve"/> </p> <div xml:id="echoid-div1780" type="float" level="0" n="0"> <figure xlink:label="fig-0697-01" xlink:href="fig-0697-01a"> <variables xml:id="echoid-variables807" xml:space="preserve">x a j b g f h d e k c</variables> </figure> <figure xlink:label="fig-0698-01" xlink:href="fig-0698-01a"> <variables xml:id="echoid-variables808" xml:space="preserve">l p g b m</variables> </figure> <figure xlink:label="fig-0699-01" xlink:href="fig-0699-01a"> <variables xml:id="echoid-variables809" xml:space="preserve">s u l s n f s m e s k d s h t q g b</variables> </figure> <figure xlink:label="fig-0699-02" xlink:href="fig-0699-02a"> <variables xml:id="echoid-variables810" xml:space="preserve">a e c f d b</variables> </figure> </div> </div> <div xml:id="echoid-div1782" type="section" level="0" n="0"> <head xml:id="echoid-head1319" xml:space="preserve" style="it">39. Siſectionem parabolam linea recta contingat, & à puncto contactus ducatur recta <lb/>perpendiculariter ſuper diametrum ſectionis productam ad concurſum cum contingente: erit <lb/>pars diametri interiacens perpendicularem & peripheriam ſectionis æqualis parti interiacen <lb/>tiſectionem & contingentem.</head> <p> <s xml:id="echoid-s46782" xml:space="preserve">Sit ſectio parabola, cuius nomen prius libro primo in commento propoſitionis 98 expoſuimus:</s> <s xml:id="echoid-s46783" xml:space="preserve"> <lb/>quæ ſit l a g, cuius latus rectum ſit l g:</s> <s xml:id="echoid-s46784" xml:space="preserve"> & dia <lb/>meter a d:</s> <s xml:id="echoid-s46785" xml:space="preserve"> contingatq́;</s> <s xml:id="echoid-s46786" xml:space="preserve"> hác ſectionẽ in pun <lb/>cto b linea recta:</s> <s xml:id="echoid-s46787" xml:space="preserve"> quæ ſit h b k:</s> <s xml:id="echoid-s46788" xml:space="preserve"> concur-<lb/>ratq́;</s> <s xml:id="echoid-s46789" xml:space="preserve"> diamcter:</s> <s xml:id="echoid-s46790" xml:space="preserve"> quæ ſit d a, producta ex-<lb/>tra ſectionem cum linea contingente, quæ <lb/>eſt h b k, in puncto h:</s> <s xml:id="echoid-s46791" xml:space="preserve"> & à puncto contin-<lb/> <anchor type="figure" xlink:label="fig-0700-01a" xlink:href="fig-0700-01"/> gentiæ, quod eſt b, ducatur per 12 p 1 linea <lb/>perpendicularis ſuper diametrum a d, ſe-<lb/>cans ipſam in puncto z:</s> <s xml:id="echoid-s46792" xml:space="preserve"> & ſit b z.</s> <s xml:id="echoid-s46793" xml:space="preserve"> Dico <lb/>quòd linea z a pars diametri interiacens <lb/>punctum ſectionis perpendicularis b z, & <lb/>peripheriam ſectionis, quæ eſt l a g, eſt <lb/>æqualis lineæ a h, parti eductæ diametri, <lb/>quæ interiacet punctum h, quod eſt pun-<lb/>ctum concurſus diametri cum linea con-<lb/>tingente, quæ eſt h b k, & pũctum a, quod <lb/>eſt terminus diametri, cadens in ipſam pe-<lb/>ripheriam ſectionis.</s> <s xml:id="echoid-s46794" xml:space="preserve"> Et hoc uniuerſale eſt:</s> <s xml:id="echoid-s46795" xml:space="preserve"> <lb/>etiam ſi linea recta ſectionem contingat in puncto g.</s> <s xml:id="echoid-s46796" xml:space="preserve"> Hoc autem demonſtratum eſt ab Apollonio <lb/>Pergæo in libro de Conicis elementis:</s> <s xml:id="echoid-s46797" xml:space="preserve"> & hic utemur ipſo, ut demonſtrato.</s> <s xml:id="echoid-s46798" xml:space="preserve"/> </p> <div xml:id="echoid-div1782" type="float" level="0" n="0"> <figure xlink:label="fig-0700-01" xlink:href="fig-0700-01a"> <variables xml:id="echoid-variables811" xml:space="preserve">h o b z k g d l</variables> </figure> </div> </div> <div xml:id="echoid-div1784" type="section" level="0" n="0"> <head xml:id="echoid-head1320" xml:space="preserve" style="it">40. Omne quadr atum lineæ perpendicularis ductæ ab aliquo puncto ſectionis parabolæ ſu-<lb/>per diametrum ſectionis, est æquale rectangulo contento ſub parte diametri interiacente illam <lb/>perpendicularem & peripheriam ſectionis, & ſub latere recto ipſius ſectionis.</head> <p> <s xml:id="echoid-s46799" xml:space="preserve">Verbi gratia:</s> <s xml:id="echoid-s46800" xml:space="preserve"> ſit, ut in præmiſſa, ſectio parabola, quæſit l a g:</s> <s xml:id="echoid-s46801" xml:space="preserve"> cuius latus rectum ſit l g, & eius <lb/>diameter ſit a d:</s> <s xml:id="echoid-s46802" xml:space="preserve"> & à puncto aliquo ſectionis, quod ſit b, ducatur ſuper diametrum ſectionis, quæ <lb/>eſt a d, perpendicularis b z.</s> <s xml:id="echoid-s46803" xml:space="preserve"> Dico quòd quadratum lineæ perpendicularis, quæ b z, eſt æquale <lb/>ei rectangulo, quod fit ex ductu lineæ z a, quæ eſt pars diametri a d, interiacens ipſam perpen di-<lb/>cularem b z, & peripheriam ſectionis, in lineam l g, quæ eſt latus rectum ipſius ſectionis.</s> <s xml:id="echoid-s46804" xml:space="preserve"> Eſt er-<lb/>go per 17 p 6 proportio lineæ l g ad lineam z b, ſicut ipſius z b ad lineam z a.</s> <s xml:id="echoid-s46805" xml:space="preserve"> Hoc autem ſimili-<lb/>ter demonſtratum eſt ab Apollonio Pergæo in libro de Conicis elementis:</s> <s xml:id="echoid-s46806" xml:space="preserve"> & nos ipſoutemur, ut <lb/>demõſtrato.</s> <s xml:id="echoid-s46807" xml:space="preserve"> Hæc uerò duo theoremata cum alijs Apollonij theorematibus in principio librinon <lb/>connumerauimus:</s> <s xml:id="echoid-s46808" xml:space="preserve"> quia ſolùm illis indigemus ad theorema ſubſequens explicandum, & in nul-<lb/>lo aliorum theorematum totius huius libri.</s> <s xml:id="echoid-s46809" xml:space="preserve"/> </p> <pb o="399" file="0701" n="701" rhead="LIBER NONVS."/> </div> <div xml:id="echoid-div1785" type="section" level="0" n="0"> <head xml:id="echoid-head1321" xml:space="preserve" style="it">41. Si in ſectione parabola ab extremitate diametri ex parte peripheriæ ſectionis reſece-<lb/>tur æquale quartæ parti lateris recti ipſius ſectionis: omnis linea æquidiſt anter diametro inci-<lb/>dens alicui puncto ſectionis, & linea ab eodem puncto ſectionis ad punctum abſcißionis dia-<lb/>metri producta, cum linea contingente ſectionem ſuper illud punctum, continent angu-<lb/>los æquales.</head> <p> <s xml:id="echoid-s46810" xml:space="preserve">Sit, ut ſuperius, ſectio parabola, quæ l a b g:</s> <s xml:id="echoid-s46811" xml:space="preserve"> cuius diameter ſit a d:</s> <s xml:id="echoid-s46812" xml:space="preserve"> & eius latus rectum ſit l g:</s> <s xml:id="echoid-s46813" xml:space="preserve"> ab <lb/>extremitate quoque diametri a d ex parte peripheriæ ſectionis, hoc eſt à parte puncti a reſecetur <lb/>per 3 p 1 linea a e æqualis quartæ parti lateris recti ipſius ſectionis, quod eſt l g:</s> <s xml:id="echoid-s46814" xml:space="preserve"> incidatq́;</s> <s xml:id="echoid-s46815" xml:space="preserve"> linea t b <lb/>puncto ſectionis, quod eſt b, æquidiſtanter diametro a d:</s> <s xml:id="echoid-s46816" xml:space="preserve"> & continuetur linea à puncto b ad pun-<lb/>ctum e, quod ſeparat à diametro a d lineam a e æqualem quartæ parti lineæ l g:</s> <s xml:id="echoid-s46817" xml:space="preserve"> & ducatur à <lb/>puncto b linea contingens ſectionem:</s> <s xml:id="echoid-s46818" xml:space="preserve"> quæ ſit h b k.</s> <s xml:id="echoid-s46819" xml:space="preserve"> Dico quòd duælineæ t b & b e cum linea <lb/>ſectionem contingente, quæ eſt h b k, in puncto b continent angulos æquales:</s> <s xml:id="echoid-s46820" xml:space="preserve"> ita quòd angulus <lb/> <anchor type="figure" xlink:label="fig-0701-01a" xlink:href="fig-0701-01"/> t b k eſt æqualis angulo e b h.</s> <s xml:id="echoid-s46821" xml:space="preserve"> Angulus <lb/>enim b e h non poteſt euadere unam tri-<lb/>um conditionum.</s> <s xml:id="echoid-s46822" xml:space="preserve"> Aut enim erit acutus:</s> <s xml:id="echoid-s46823" xml:space="preserve"> <lb/>aut rectus:</s> <s xml:id="echoid-s46824" xml:space="preserve"> aut obtuſus.</s> <s xml:id="echoid-s46825" xml:space="preserve"> Sit primò acutus:</s> <s xml:id="echoid-s46826" xml:space="preserve"> <lb/>& à puncto b ducatur per 12 p 1 ſuper dia-<lb/>metrum a d perpendicularis b z:</s> <s xml:id="echoid-s46827" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s46828" xml:space="preserve"> <lb/>per 32 p 1 punctum z inter duo puncta a <lb/>& e:</s> <s xml:id="echoid-s46829" xml:space="preserve"> & producatur diameter a d ultra pun <lb/>ctum a, donec per 2 th.</s> <s xml:id="echoid-s46830" xml:space="preserve"> 1 huius concurrat <lb/>cum linea contingente ſectionem, quę eſt <lb/>k b h:</s> <s xml:id="echoid-s46831" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s46832" xml:space="preserve"> concurſus in puncto h:</s> <s xml:id="echoid-s46833" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s46834" xml:space="preserve"> <lb/>angulus a h b acutus:</s> <s xml:id="echoid-s46835" xml:space="preserve"> cadet ergo perpen-<lb/>dicularis b z inter puncta h & e, & erit <lb/>per 39 huius linea a z æqualis lineæ a h.</s> <s xml:id="echoid-s46836" xml:space="preserve"> <lb/>Quia itaque linea a e eſt diuiſa in pun-<lb/>cto z, & ei eſt æqualis uni parti diuiden-<lb/>tium adiecta, quæ eſt a h:</s> <s xml:id="echoid-s46837" xml:space="preserve"> erit ergo per 8 <lb/>p 2 quadratum lineæ e h æquale ei, quod <lb/>fit ex ductu lineæ e a in lineam h a, uel <lb/>in lineam a z quater, & quadrato lineæ <lb/>z e:</s> <s xml:id="echoid-s46838" xml:space="preserve"> ſed linea e a eſt quarta pars lineæ l g ex hypotheſi:</s> <s xml:id="echoid-s46839" xml:space="preserve"> ergo per 1 p 2 uel per 1 p 6 illud, quod fit ex <lb/>ductu lineæ a z in lineam a e quater, eſt æquale ei, quod fit ex ductu lineæ a z in lineam l g ſemel.</s> <s xml:id="echoid-s46840" xml:space="preserve"> Il-<lb/>lud ergo, quod fit ex ductu lineæ a z in lineam l g cum quadrato lineæ z e eſt æquale quadrato li-<lb/>neæ e h:</s> <s xml:id="echoid-s46841" xml:space="preserve"> ſed per præmiſſam patet, quòd illud, quod fit ex ductu lineæ a z in lineam l g, eſt æquale <lb/>quadrato lineæ b z:</s> <s xml:id="echoid-s46842" xml:space="preserve"> quoniam linea b z eſt perpendicularis ſuper diametrum a d:</s> <s xml:id="echoid-s46843" xml:space="preserve"> duo uerò quadra <lb/>ta b z & z e ſunt per 47 p 1 æqualia quadrato lineæ b e:</s> <s xml:id="echoid-s46844" xml:space="preserve"> quadrata ergo linearum e h & e b ſunt <lb/>æqualia:</s> <s xml:id="echoid-s46845" xml:space="preserve"> ergo linea e b eſt æqualis lineæ e h:</s> <s xml:id="echoid-s46846" xml:space="preserve"> ergo per 5 p 1 in trigono e b h angulus e h b eſt æ-<lb/>qualis angulo e b h:</s> <s xml:id="echoid-s46847" xml:space="preserve"> ſed linea t b & d a ſunt æquidiſtantes:</s> <s xml:id="echoid-s46848" xml:space="preserve"> ergo per 29 p 1 angulus t b k extrin-<lb/>ſecus eſt ęqualis d h b intrinſeco:</s> <s xml:id="echoid-s46849" xml:space="preserve"> angulus ergo e b h eſt æqualis angulo t b k.</s> <s xml:id="echoid-s46850" xml:space="preserve"> Eodem quo-<lb/>que modo dem onſtrandum de qualibet linea æquidiſtante diametro a d & b e linea copula-<lb/>ta ad punctum e, quando illa linea ſuper pun-<lb/>ctum e cum diametro a d angulum continet <lb/> <anchor type="figure" xlink:label="fig-0701-02a" xlink:href="fig-0701-02"/> acutum.</s> <s xml:id="echoid-s46851" xml:space="preserve"> Patet ergo propoſitum ſecundum <lb/>hunc modum.</s> <s xml:id="echoid-s46852" xml:space="preserve"> Quòd ſi angulus b e h fuerit re-<lb/>ctus, adhuc patet propoſitum, Quòd angulus t <lb/>b k eſt æqualis angulo e b h.</s> <s xml:id="echoid-s46853" xml:space="preserve"> Quoniam enim an-<lb/>gulus b e h eſt rectus:</s> <s xml:id="echoid-s46854" xml:space="preserve"> patet quòd linea b e eſt <lb/>perpendicularis ſuper diametrum a d:</s> <s xml:id="echoid-s46855" xml:space="preserve"> ergo li-<lb/>nea e a per 39 huius eſt æqualis lineę a h:</s> <s xml:id="echoid-s46856" xml:space="preserve"> ſed li-<lb/>nea e a ex hypotheſi eſt quarta pars lineæ l g:</s> <s xml:id="echoid-s46857" xml:space="preserve"> <lb/>ergo linea h e, quæ eſt dupla lineæ a e, eſt me-<lb/>dietas lineæ l g:</s> <s xml:id="echoid-s46858" xml:space="preserve"> ergo per 4 p 2 quadratum lineę <lb/>e h eſt quarta pars quadrati lineæ l g.</s> <s xml:id="echoid-s46859" xml:space="preserve"> Id quoq;</s> <s xml:id="echoid-s46860" xml:space="preserve">, <lb/>quod fit ex ductu lineæ e a in lineam l g eſt æ-<lb/>quale quartæ parti quadrati lineæ l g per 1 p 6:</s> <s xml:id="echoid-s46861" xml:space="preserve"> <lb/>quoniam linea e a eſt ex hypotheſi quarta pars <lb/>lineæ l g.</s> <s xml:id="echoid-s46862" xml:space="preserve"> Illud ergo, quod fit ex ductu lineæ <lb/>e a in lineam l g eſt æquale quadrato lineæ e h:</s> <s xml:id="echoid-s46863" xml:space="preserve"> <lb/>ſed id, quod fit ex ductu lineæ e a in lineam l g, <lb/>eſt æquale quadrato lineæ e b per præmiſſam:</s> <s xml:id="echoid-s46864" xml:space="preserve"> <lb/>quoniam linea e b eſt perpendicularis ſuper diametrum a d:</s> <s xml:id="echoid-s46865" xml:space="preserve"> quadratum ergo lineæ e h eſt æquale <lb/> <pb o="400" file="0702" n="702" rhead="VITELLONIS OPTICAE"/> quadrato lineæ e b:</s> <s xml:id="echoid-s46866" xml:space="preserve"> ergo linea e h eſt æqualis lineæ b e:</s> <s xml:id="echoid-s46867" xml:space="preserve"> ergo, ut prius per 5 p 1 anguli e b h & e h b <lb/>ſunt æquales.</s> <s xml:id="echoid-s46868" xml:space="preserve"> Et quoniam linea t b æquidiſtat lineæ a d:</s> <s xml:id="echoid-s46869" xml:space="preserve"> patet per 29 p 1 quoniam angulus t b k <lb/>eſt æqualis angulo e b h.</s> <s xml:id="echoid-s46870" xml:space="preserve"> Et ſimiliter demonſtrandum de omni linea incidente ipſi ſectioni, cum <lb/>angulus b e h eſt rectus:</s> <s xml:id="echoid-s46871" xml:space="preserve"> & illud eſt, quod proponebatur.</s> <s xml:id="echoid-s46872" xml:space="preserve"> Siuerò angulus b e h ſit obtuſus:</s> <s xml:id="echoid-s46873" xml:space="preserve"> dico <lb/>quod adhuc angulus t b k eſt æqualis angu-<lb/> <anchor type="figure" xlink:label="fig-0702-01a" xlink:href="fig-0702-01"/> lo e b h.</s> <s xml:id="echoid-s46874" xml:space="preserve"> Ducatur enim linea perpendicula-<lb/>ris, quæ ſit b z à puncto b ipſius ſectionis, cui <lb/>incidit linea æquidiſtans diametro a d, quæ <lb/>eſt t b:</s> <s xml:id="echoid-s46875" xml:space="preserve"> illa quoque perpendicularis ſuper dia <lb/>metrum a d ſit b z:</s> <s xml:id="echoid-s46876" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s46877" xml:space="preserve"> hæc perpendicu-<lb/>laris b z inter puncta diametri, quæ ſunt d & <lb/>e:</s> <s xml:id="echoid-s46878" xml:space="preserve"> aliàs enim duo anguli unius trigoni b e z <lb/>fierent maiores duobus rectis:</s> <s xml:id="echoid-s46879" xml:space="preserve"> quoniam uno <lb/>exiſtente recto, qui b z e, angulus b e z eſſet <lb/>obtuſus:</s> <s xml:id="echoid-s46880" xml:space="preserve"> quod eſt impoſsibile:</s> <s xml:id="echoid-s46881" xml:space="preserve"> cadit ergo pun <lb/>ctum zinter puncta e & d:</s> <s xml:id="echoid-s46882" xml:space="preserve"> linea ergo a z eſt <lb/>maior quàm linea a e.</s> <s xml:id="echoid-s46883" xml:space="preserve"> Et quoniam linea h b k <lb/>contingit ſectionem, & linea b z eſt perpen-<lb/>dicularis ſuper diametrum a d:</s> <s xml:id="echoid-s46884" xml:space="preserve"> erit per 39 hu <lb/>ius linea a z æqualis lineæ a h:</s> <s xml:id="echoid-s46885" xml:space="preserve"> ergo linea h a <lb/>eſt maior quàm linea a e:</s> <s xml:id="echoid-s46886" xml:space="preserve"> ſiat per 3 p 1 linea a m <lb/>æqualis lineæ a e:</s> <s xml:id="echoid-s46887" xml:space="preserve"> remanet ergo linea h m æ-<lb/>qualis lineę z e:</s> <s xml:id="echoid-s46888" xml:space="preserve"> linea ergo e m addita utrobi-<lb/>que, erit linea z m æqualis lineæ h e:</s> <s xml:id="echoid-s46889" xml:space="preserve"> quadra-<lb/>tum ergo lineę z m eſt æquale quadrato lineę <lb/>e h.</s> <s xml:id="echoid-s46890" xml:space="preserve"> Quia itaque linea z a eſt diuiſa in puncto <lb/>e, & ei eſt adiecta æqualis uni diuidentium, quę eſt m a, æqualis ipſi a e:</s> <s xml:id="echoid-s46891" xml:space="preserve"> patet per 8 p 2 quòd illud, <lb/>quod fit ex ductu lineę z a in lineam a m, uel in eius æqualem lineam a e quater, cum quadrato li-<lb/>neæ z e, eſt æquale quadrato lineę z m, uel lineę e h, quę ſunt æquales:</s> <s xml:id="echoid-s46892" xml:space="preserve"> ſed illud, quod fit ex ductu <lb/>lineę z a in lineam a e quater, ut patet ex pręmiſsis, eſt æquale ei, quod fit ex ductu lineę a z in li-<lb/>neam l g per 1 p 2 uel per 1 p 6:</s> <s xml:id="echoid-s46893" xml:space="preserve"> quoniam linea a e eſt æqualis quartę parti lineę l g ex hypotheſi.</s> <s xml:id="echoid-s46894" xml:space="preserve"> Il-<lb/>lud ergo, quod fit ex ductu lineę a z in lineam l g, cum quadrato lineę z e, eſt æquale quadrato li-<lb/>neę e h:</s> <s xml:id="echoid-s46895" xml:space="preserve"> ſed illud, quod fit ex ductu lineę z a in lineam l g eſt æquale quadrato lineę b z per præ-<lb/>cedentem:</s> <s xml:id="echoid-s46896" xml:space="preserve"> quoniam linea b z eſt perpendicularis ſuper diametrum a d:</s> <s xml:id="echoid-s46897" xml:space="preserve"> quadratum uerò lineę b e <lb/>per 47 p 1 eſt ęquale quadratis ambarum linearum b z & e z.</s> <s xml:id="echoid-s46898" xml:space="preserve"> Patet ergo quòd quadratum lineę b e <lb/>eſt ęquale quadrato lineę e h:</s> <s xml:id="echoid-s46899" xml:space="preserve"> ergo linea e b eſt ęqualis lineæ e h:</s> <s xml:id="echoid-s46900" xml:space="preserve"> ergo per 5 p 1 anguli e b h & a h b <lb/>ſunt ęquales:</s> <s xml:id="echoid-s46901" xml:space="preserve"> ſed, ut prius, lineę t b & d h ſunt æquidiſtantes:</s> <s xml:id="echoid-s46902" xml:space="preserve"> angulus ergo t b k per 29 p 1 eſt æqua-<lb/>lis angulo d h b:</s> <s xml:id="echoid-s46903" xml:space="preserve"> ergo & angulus e b h.</s> <s xml:id="echoid-s46904" xml:space="preserve"> Et ſimiliter demonſtrandum in omnilinea incidente ſectio-<lb/>ni æquidiſtanter diametro a d, cum angulus b e h eſt obtuſus.</s> <s xml:id="echoid-s46905" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s46906" xml:space="preserve"> generaliter propoſitum.</s> <s xml:id="echoid-s46907" xml:space="preserve"> <lb/>Nam omnis linea incidens peripherię ſectionis ęquidiſtanter diametro, & alia linea, quę ab illo eo-<lb/>dem puncto ducitur ad punctum abſcindens à diametro ex parte peripherię ſectionis partẽ, æqua-<lb/>lem quartæ partilateris recti ipſius ſectionis, cum linea ſectionem in illo puncto contingente con-<lb/>tinent angulos ęquales.</s> <s xml:id="echoid-s46908" xml:space="preserve"> Ethoc proponebatur.</s> <s xml:id="echoid-s46909" xml:space="preserve"/> </p> <div xml:id="echoid-div1785" type="float" level="0" n="0"> <figure xlink:label="fig-0701-01" xlink:href="fig-0701-01a"> <variables xml:id="echoid-variables812" xml:space="preserve">h a b z e k g d l t</variables> </figure> <figure xlink:label="fig-0701-02" xlink:href="fig-0701-02a"> <variables xml:id="echoid-variables813" xml:space="preserve">h a b e k g d l t</variables> </figure> <figure xlink:label="fig-0702-01" xlink:href="fig-0702-01a"> <variables xml:id="echoid-variables814" xml:space="preserve">h m a b e z k g d l t</variables> </figure> </div> </div> <div xml:id="echoid-div1787" type="section" level="0" n="0"> <figure><variables xml:id="echoid-variables815" xml:space="preserve">l a b h k e g t d z</variables> </figure> <head xml:id="echoid-head1322" xml:space="preserve" style="it">42. In omni ſuperficie concaua conca-<lb/> uitatis ſectionis parabolæ, ſiab extremi- tate axis contingentis ſectionem abſcin- datur pars æqualis quartæ lateris recti ipſius parabolæ: omnis linea æquidiſtan- ter axi incidens illi ſuperficiei, & linea à puncto incidentiæ ad punctum ſignatũ in axe producta, cũ linea in illo pũcto ſu- քficiẽ cõtingẽte cõtinẽt angulos æquales.</head> <p> <s xml:id="echoid-s46910" xml:space="preserve">Sit ſuperficies concaua concauitate ſe-<lb/>ctionis parabolę, cuius uertex ſit pũctũ a:</s> <s xml:id="echoid-s46911" xml:space="preserve"> <lb/>& hęc eſt ſuperficies illa, quã motu ſuo cir-<lb/>ca axẽ fixũ efficit ipſa parabola per 117 th.</s> <s xml:id="echoid-s46912" xml:space="preserve"> 1 <lb/>huius.</s> <s xml:id="echoid-s46913" xml:space="preserve"> Et quoniã, utibidẽ patuit, huius ſu-<lb/>perficiei baſis eſt circulus, quẽ circa pũctũ <lb/>d motu ſuo deſcribit linea g d:</s> <s xml:id="echoid-s46914" xml:space="preserve"> ſit ille circu-<lb/>lus g e z:</s> <s xml:id="echoid-s46915" xml:space="preserve"> & ſit huius ſuքficiei concauę axis <lb/>linea a d, quę fuit prius diameter ſectionis <lb/>parabolę:</s> <s xml:id="echoid-s46916" xml:space="preserve"> & ab extremitate axis à puncto ſcilicet a abſcin datur ab axe linea a h ęqualis quartę partl <lb/> <pb o="401" file="0703" n="703" rhead="LIBER NONVS."/> lateris recti ipſius ſe ctionis, quę ſit linea g z, cuius quartę parti ęqualis ſit linea a h:</s> <s xml:id="echoid-s46917" xml:space="preserve"> & ducatur à pun <lb/>cto ſuperficiei b linea b t ęquidiſtanter axi a d per 31 p 1:</s> <s xml:id="echoid-s46918" xml:space="preserve"> & ducatur linea b h.</s> <s xml:id="echoid-s46919" xml:space="preserve"> Dico quòd duę lineę t b <lb/>& h b continent cum linea continegnte ſuperficiẽ concauam propoſitam in pũcto b duos angulos <lb/>æ quales.</s> <s xml:id="echoid-s46920" xml:space="preserve"> Quoniam enim linea a d & b t ſunt ęquidiſtantes:</s> <s xml:id="echoid-s46921" xml:space="preserve"> patet quòd ipſę ſunt in eadé ſuperficie <lb/>per 1 th.</s> <s xml:id="echoid-s46922" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s46923" xml:space="preserve"> ſed linea b h cadit inter illas:</s> <s xml:id="echoid-s46924" xml:space="preserve"> ergo per 7 p 11 ipſa eſt in eadẽ ſuperficie cum illis:</s> <s xml:id="echoid-s46925" xml:space="preserve"> lineę <lb/>ergo t b & b h & a d ſunt in una ſuperficie.</s> <s xml:id="echoid-s46926" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s46927" xml:space="preserve">, ut aliqua ſuperficies plana contingat ſuperficiẽ <lb/>propoſitam ſuper punctum b:</s> <s xml:id="echoid-s46928" xml:space="preserve"> ſuperficies itaq;</s> <s xml:id="echoid-s46929" xml:space="preserve"> b t d a ſecabit ſuperficiem concauã:</s> <s xml:id="echoid-s46930" xml:space="preserve"> & erit per 19 th.</s> <s xml:id="echoid-s46931" xml:space="preserve"> 1 <lb/>huius communis ſectio ipſarum parabola:</s> <s xml:id="echoid-s46932" xml:space="preserve"> quę ſit a b g:</s> <s xml:id="echoid-s46933" xml:space="preserve"> cuius diameter erit linea a d:</s> <s xml:id="echoid-s46934" xml:space="preserve"> & erit commu-<lb/>nis ſectio ſuperficiei b t d a & ſuperficiei planę contingentis iſtam ſuperficiem concauam linea con <lb/>tingens ſectionem a b g in puncto b:</s> <s xml:id="echoid-s46935" xml:space="preserve"> quę ſit linea l b k.</s> <s xml:id="echoid-s46936" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s46937" xml:space="preserve"> linea l b k contingit ſectionẽ a b g <lb/>in puncto b, & linea a h eſt quarta pars lateris recti, & linea t b ęquidiſtat lineę a d:</s> <s xml:id="echoid-s46938" xml:space="preserve"> patet per pręmiſ-<lb/>ſam, quoniam duę lineę t b & b h continent angulos æquales cum linea l b k contingente ſectionẽ <lb/>in puncto b:</s> <s xml:id="echoid-s46939" xml:space="preserve"> & quòd imaginata moueri ſuperficie b t d a circa axem fixũ, qui eſt a d:</s> <s xml:id="echoid-s46940" xml:space="preserve"> patet quòd pun <lb/>ctum b motu ſuo eſficit circulum in ſuperficie cõcaua, à cuius totali peripheria lineę ductę ad pun-<lb/>ctum h continent angulos ęquales.</s> <s xml:id="echoid-s46941" xml:space="preserve"> Et idem accidit in quacunque parte ſectionis parabolę, quę eſt <lb/>a b g, cadat punctus b:</s> <s xml:id="echoid-s46942" xml:space="preserve"> ſiue angulus b h a fiat acutus, rectus uel obtuſus.</s> <s xml:id="echoid-s46943" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s46944" xml:space="preserve"> quòd omnis li-<lb/>nea ęquidiſtans axi a d & incidens ſuperficiei concauę propoſitę, & linea ab illo puncto ad punctũ <lb/>h ducta continent angulos ęquales cum linea in illo pũcto ſuperficiẽ cõtingẽte.</s> <s xml:id="echoid-s46945" xml:space="preserve"> Et hoc eſt ꝓpoſitũ.</s> <s xml:id="echoid-s46946" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1788" type="section" level="0" n="0"> <head xml:id="echoid-head1323" xml:space="preserve" style="it">43. Speculo concauo concauitatis ſectionis parabolæ ſoli oppoſito, ita ut axis ipſius ſit in di-<lb/>recto corporis ſolaris: omnes radij incidentes ſpeculo æquidiſtanter axi, reflectuntur ad punctũ <lb/>unum axis, distantem à ſuperficie ſpeculi ſecundum quart am lateris recti ipſius ſectionis para-<lb/>bole, ſpeculi ſuperficiem cauſſantis. Ex quo patet quòd à ſuperficie talium ſpeculorum ignem est <lb/>poßibile accendi.</head> <p> <s xml:id="echoid-s46947" xml:space="preserve">Sit ſpeculũ concauum concauitate ſectionis parabolæ:</s> <s xml:id="echoid-s46948" xml:space="preserve"> cuius uertex ſit pũctum a:</s> <s xml:id="echoid-s46949" xml:space="preserve"> & baſis ipſius <lb/>ſit circulus q e z:</s> <s xml:id="echoid-s46950" xml:space="preserve"> & eius axis a d:</s> <s xml:id="echoid-s46951" xml:space="preserve"> & diſtantia puncti axis (quod ſith) à puncto uerticis ſpeculi (qđ <lb/>eſt a) ſit æ qualis quartæ parti lineę q z, ſcilicet lateris recti ſectionis parabolæ a g q, cauſſantis mo-<lb/>tu ſuo ſuper axem a d ſuperficiem ipſius ſpeculi concaui:</s> <s xml:id="echoid-s46952" xml:space="preserve"> quod ſoli opponatur ſecundum eius axẽ <lb/>ad.</s> <s xml:id="echoid-s46953" xml:space="preserve"> Sit enim corporis ſolaris centrum k:</s> <s xml:id="echoid-s46954" xml:space="preserve"> ſitueturq́;</s> <s xml:id="echoid-s46955" xml:space="preserve"> ſpeculũ taliter, ut eius axis a d ſic productus per <lb/>ueniat ad centrum ſolis in punctum k.</s> <s xml:id="echoid-s46956" xml:space="preserve"> Dico quòd <lb/> <anchor type="figure" xlink:label="fig-0703-01a" xlink:href="fig-0703-01"/> omnes radij ſolares æquidiſtanter radio k a ſuperfi-<lb/>ciei ſpeculi propoſi ti incidétes, reflectuntur ad pun <lb/>ctum h lineę a d, quę eſt axis ſpeculi.</s> <s xml:id="echoid-s46957" xml:space="preserve"> Quoniá enim <lb/>omnes radij egredientes â quocunq;</s> <s xml:id="echoid-s46958" xml:space="preserve"> puncto corpo <lb/>ris ſolaris ſuper aliquod punctum ſuperficiei ſpecu <lb/>li, egrediuntur ſecundum lineas rectas, ut patet per <lb/>1 th.</s> <s xml:id="echoid-s46959" xml:space="preserve"> 2 huius:</s> <s xml:id="echoid-s46960" xml:space="preserve"> tunc palàm eſt quia linea k a eſt linea re <lb/>cta.</s> <s xml:id="echoid-s46961" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s46962" xml:space="preserve"> ſuper peripheriam alicuius ſectionis pa <lb/>rabolæ ipſius ſpeculi (quę ſit g a z q) pũctũ g ſigna <lb/>tum, utcunq;</s> <s xml:id="echoid-s46963" xml:space="preserve"> contingit:</s> <s xml:id="echoid-s46964" xml:space="preserve"> & à puncto ſpeculi g per 31 <lb/>p 1 ad aliquod punctum corporis ſolaris (quod ſit <lb/>t) ducatur linea g t æquidiſtans radio a k, qui inci-<lb/>dit ſuperficiei ſpeculi ſecũdum axem a d.</s> <s xml:id="echoid-s46965" xml:space="preserve"> Eſt autem <lb/>neceſſarium omnem lineam à quocunq;</s> <s xml:id="echoid-s46966" xml:space="preserve"> puncto ſpe <lb/>culi æquidiſtanter radio a k productam ad ſuperfi-<lb/>ciem corporis ſolis incidere:</s> <s xml:id="echoid-s46967" xml:space="preserve"> quoniá ſuperficiei ſpe-<lb/>culi ad ſuperficiẽ ſolaris corporis aut nulla, aut mo-<lb/>dica eſt proportio:</s> <s xml:id="echoid-s46968" xml:space="preserve"> ſit ergo punctum t, quod eſt ter-<lb/>minus lineæ g t, in ipſa ſuperficie corporis ſolaris.</s> <s xml:id="echoid-s46969" xml:space="preserve"> <lb/>Omnes itaque lineæ, quæ poſſunt duci à ſuperficie <lb/>ipſius ſpeculi ęquidiſtanter ſuo axi a d, incidunt cor <lb/>pori ſolari, & ſecũdum illas lineas fit incidentia ſu-<lb/>perficiei ſpeculi, reſpectu radij, qui incidit ſecũdum <lb/>axem omnium æquidiſtantium axi radiorum:</s> <s xml:id="echoid-s46970" xml:space="preserve"> hoc <lb/>autem eſt omnium radiorum cuicunque puncto ſu-<lb/>perficiei totius ſpeculi incidẽtium:</s> <s xml:id="echoid-s46971" xml:space="preserve"> quoniam per 31 <lb/>p 1 à quolibet pũcto propè uel remotè dato ſcimus <lb/>cuilibet datæ lineæ, utin propoſito eſt axis a d, duce <lb/>re lineam æquidiſtantem.</s> <s xml:id="echoid-s46972" xml:space="preserve"> Dico itaq;</s> <s xml:id="echoid-s46973" xml:space="preserve"> quòd omnes <lb/>illi radij reflectuntur à totali ſuperficie ſpeculi ad u-<lb/>num punctum axis ſpeculi, quod eſt punctum h.</s> <s xml:id="echoid-s46974" xml:space="preserve"> O-<lb/>mnes enim illi radij cum ſint lineæ rectæ:</s> <s xml:id="echoid-s46975" xml:space="preserve"> patet per <lb/>præmiſſam, quòd cum lineis ab omnibus pũctis ſuarum incidentiarũ ad punctũ h ductis continent <lb/> <pb o="402" file="0704" n="704" rhead="VITELLONIS OPTICAE"/> angulos æquales:</s> <s xml:id="echoid-s46976" xml:space="preserve"> ergo per 20 th.</s> <s xml:id="echoid-s46977" xml:space="preserve"> 5 huius omnes illi radij reflectuntur ſecundum illas lineas tranſe-<lb/>untes punctum h.</s> <s xml:id="echoid-s46978" xml:space="preserve"> Et ex hoc pater, quòd omnes radij incidentes peripheriæ ſectionis æquidiſtan-<lb/>ter radio incidenti ſecundum lineam, quę eſt diameter ipſius ſectionis, reflectuntur ad punctũ dia-<lb/>metri, qui abſcindit ex capite diametri à parte peripherię ſectionis partem æ qualem quarιę parti la <lb/>teris recti ipſius ſectionis g a z q:</s> <s xml:id="echoid-s46979" xml:space="preserve"> quoniam omnis reflexio à quolibet corporum politorum regula-<lb/>rium fit ſecundum æ qualitatem angulorum, quos continent linea incidens & reflexa cum linea in <lb/>illo puncto ſuperficiem ſpeculi, à qua fit reflexio, contingente.</s> <s xml:id="echoid-s46980" xml:space="preserve"> Et quoniam omnes illę lineę ſecant <lb/>ſe in puncto h:</s> <s xml:id="echoid-s46981" xml:space="preserve"> patet quòd in puncto h eſt concurſus omnium illorũ radiorum.</s> <s xml:id="echoid-s46982" xml:space="preserve"> In illo ergo puncto <lb/>aggregatur omnis uirtus omnium radiorum totali ſuperficiei ſpeculi incidentium.</s> <s xml:id="echoid-s46983" xml:space="preserve"> Et quoniã qui-<lb/>libet radiolus defert ſecũ aliquid uirtutis actiuę corporis ſolaris:</s> <s xml:id="echoid-s46984" xml:space="preserve"> patet quòd in illo puncto tota uir <lb/>tus eſt concurrẽs, omnium ſcilicet radiorũ ſuperficiei ſpeculi ęquidiſtanter ipſi axi a d incidentιũ.</s> <s xml:id="echoid-s46985" xml:space="preserve"> <lb/>Ex quo patet quòd in illo puncto h poſito aliquo combuſtibili ignem eſt poſsibile accendi.</s> <s xml:id="echoid-s46986" xml:space="preserve"> Et hæc <lb/>eſt melior & fortior figura omniũ figurarũ radios ſolares ad unũ pũctũ aggregãtiũ:</s> <s xml:id="echoid-s46987" xml:space="preserve"> quoniá à tota e-<lb/>ius ſuքficie, & à quolibet pũcto ipſius radij ſolares in unũ pũctũ aggregãtur.</s> <s xml:id="echoid-s46988" xml:space="preserve"> Patet ergo ꝓpoſitum.</s> <s xml:id="echoid-s46989" xml:space="preserve"/> </p> <div xml:id="echoid-div1788" type="float" level="0" n="0"> <figure xlink:label="fig-0703-01" xlink:href="fig-0703-01a"> <variables xml:id="echoid-variables816" xml:space="preserve">k t q d z e h y a</variables> </figure> </div> </div> <div xml:id="echoid-div1790" type="section" level="0" n="0"> <head xml:id="echoid-head1324" xml:space="preserve" style="it">44. Speculum ſecundũ formã ſectionis parabolæ, uel lineæ eccentr alis, uel interſectionis pyra <lb/>midalis, uel cuiuſcun alterius regularis uel irregularis datæ lineæ artificialiter conſtituere.</head> <p> <s xml:id="echoid-s46990" xml:space="preserve">Lineam, quá dicimus peripheriá ſectionis, inueniat induſtria operantis:</s> <s xml:id="echoid-s46991" xml:space="preserve"> quę & apud nos multis <lb/>conatibus artificialiter eſt inuenta:</s> <s xml:id="echoid-s46992" xml:space="preserve"> faciliter tamen eſt imaginabilis:</s> <s xml:id="echoid-s46993" xml:space="preserve"> quoniam, ut in 98 th.</s> <s xml:id="echoid-s46994" xml:space="preserve"> 1 huius di-<lb/>ximus, ipſa eſt linea, quę eſt cõmunis ſectio ſuperficiei conicę cuiuſcunq;</s> <s xml:id="echoid-s46995" xml:space="preserve"> pyramidis, maximè uerò <lb/>rectangulę & ſuperficiei pyramidem per diametrum baſis ſecanti æquidiſtanter alicui lineę longi-<lb/>tudinis illius pyramidis:</s> <s xml:id="echoid-s46996" xml:space="preserve"> utpote ei, cuius & axis pyramidis cõmunis ſuperficies eſt erecta ſuper pla <lb/>nam ſuperficιem dicto modo pyramidẽ ſecantem.</s> <s xml:id="echoid-s46997" xml:space="preserve"> Talis itaq;</s> <s xml:id="echoid-s46998" xml:space="preserve"> ſectio parabola ſic artificialiter inuen <lb/>ta ſit a e g:</s> <s xml:id="echoid-s46999" xml:space="preserve"> & aſſumatur lamina ferri boni uel chalybis, menſuræ & quantitatis cuius placuerit:</s> <s xml:id="echoid-s47000" xml:space="preserve"> quę <lb/>ſit a b g d:</s> <s xml:id="echoid-s47001" xml:space="preserve"> & protrahatur in ipſa ſectio parabola, quę <lb/> <anchor type="figure" xlink:label="fig-0704-01a" xlink:href="fig-0704-01"/> ſit æqualis & ſimilis ſectioni a e g:</s> <s xml:id="echoid-s47002" xml:space="preserve"> & abſcindatur la-<lb/>mina ſecũdũ illam ſectionẽ a e g, uel ſecundũ aliquá <lb/>partem ipſius:</s> <s xml:id="echoid-s47003" xml:space="preserve"> ſiue placeat à parte uerticis, qui eſt a:</s> <s xml:id="echoid-s47004" xml:space="preserve"> <lb/>ſiue ex parte unius ſui capitis, quod eſt g:</s> <s xml:id="echoid-s47005" xml:space="preserve"> ſiue ex par <lb/>te alterius ſui capitis, quod eſt in latere eius recto op <lb/>poſitum puncto g:</s> <s xml:id="echoid-s47006" xml:space="preserve"> fit enim magna diuerſitas proie-<lb/>ctionis radiorum ſecũdum illá partium ſectionis di-<lb/>uerſitatem.</s> <s xml:id="echoid-s47007" xml:space="preserve"> Reſecta itaq;</s> <s xml:id="echoid-s47008" xml:space="preserve"> lamina a b d g ſecundũ for-<lb/>mam & figurá ſectionis a e g:</s> <s xml:id="echoid-s47009" xml:space="preserve"> acuatur extremitas la-<lb/>minę, quę eſt ſecũdum formam ſectionis, acuitione <lb/>bona, ſcilicet ut radere ualeat totum illud, ſuper qđ <lb/>mouetur.</s> <s xml:id="echoid-s47010" xml:space="preserve"> Et aſſumatur item alia lamina de chalybe <lb/>forti alicuius competentis ſpiſsitudinis:</s> <s xml:id="echoid-s47011" xml:space="preserve"> quę incida-<lb/>tur iterum ſecundũ formá pręaſſumptę partis illius <lb/>ſectionis:</s> <s xml:id="echoid-s47012" xml:space="preserve"> & illa ſuperficies ſimilis parabolę ſecetur <lb/>cótiguè multis ſectionibus ad modum limę, ita ut ք <lb/>ipſam poſsit limari ferrũ.</s> <s xml:id="echoid-s47013" xml:space="preserve"> Deinde fiat corpus ferreũ <lb/>cõueniens illi figurę, cuius ſuperficiẽ ſecundũ formã <lb/>intentã ꝓponimus cõcauare & polire ad modũ ſpe-<lb/>culi:</s> <s xml:id="echoid-s47014" xml:space="preserve"> ſiue illud ſit ſecundũ ſormã partis ſectionis ad-<lb/>iacentẽ uertici ſectionis parabolę, ſiue capitis.</s> <s xml:id="echoid-s47015" xml:space="preserve"> In his <lb/>enim eſt multa diuerſitas formę uel figurę ſpeculi.</s> <s xml:id="echoid-s47016" xml:space="preserve"> <lb/>Forma enim figurę ſpeculi cõcauati ſecundũ partes <lb/>adiacẽtes uertici ſectionis, ęqualiter hinc inde diſtá-<lb/>tes à pũcto uerticis, eſt figurę quaſi annularis:</s> <s xml:id="echoid-s47017" xml:space="preserve"> & for <lb/>ma ſpeculi cõcauati ſecundũ partes adiacentes capi <lb/>tibus ſectionis, eſt figurę quaſi oualis, hoc eſt ad mo-<lb/>dũ longitudinis oui.</s> <s xml:id="echoid-s47018" xml:space="preserve"> Limeturitaq;</s> <s xml:id="echoid-s47019" xml:space="preserve"> ſpeculũ cuiuſcun <lb/>que figurę fieri debuerit per limam ſibi ſimilẽ in figu <lb/>ra, taliter ut ſuperficies limę, quę eſt ſecta ad limádũ, <lb/>occurrat toti ſuperficiei ipſius ſpeculi.</s> <s xml:id="echoid-s47020" xml:space="preserve"> Si ergo ſpecu <lb/>lum limatũ fuerit ſecundũ figurã oualẽ:</s> <s xml:id="echoid-s47021" xml:space="preserve"> tunc ordine <lb/>tur in loco fixo, ita ut eius cõcaua ſuperficies, quãtũ ad lineã peripherię ſuę baſis, ſit in peripheria il <lb/>lius circuli baſis:</s> <s xml:id="echoid-s47022" xml:space="preserve"> uel ſi fuerit figurę annularis, ad peripheriá circuli ęquidiſtátis baſi:</s> <s xml:id="echoid-s47023" xml:space="preserve"> & in loco axis <lb/>figatur lamina limę ſuperficiẽ radendã planãtis, moueaturq́;</s> <s xml:id="echoid-s47024" xml:space="preserve"> ad cõcauandũ ſpeculũ:</s> <s xml:id="echoid-s47025" xml:space="preserve"> & tornetur, ſi-<lb/>cut tornantur alia inſtrumẽta, donec peripheria acutę laminę occurrat toti ſuperficiei ſpeculi, & e-<lb/>uacuetur omnis aſperitas ipſius:</s> <s xml:id="echoid-s47026" xml:space="preserve"> planetur quoq;</s> <s xml:id="echoid-s47027" xml:space="preserve">, quãtũ eſt poſsibile:</s> <s xml:id="echoid-s47028" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s47029" xml:space="preserve"> tũc ſuperficies illius ſpe <lb/>culi ſecundũ ſe totá habens figurá ſectionis parabolę:</s> <s xml:id="echoid-s47030" xml:space="preserve"> & fiet ab omnibus punctis ſuę ſuperficiei re-<lb/>flexio in punctũ unũ.</s> <s xml:id="echoid-s47031" xml:space="preserve"> Simili modo faciat ingenioſus artifex in alijs lineis quibuſcunq;</s> <s xml:id="echoid-s47032" xml:space="preserve">, utin illis li-<lb/> <pb o="403" file="0705" n="705" rhead="LIBER DECIMVS."/> neis, quas per 37 & 38 huius docuimus inuenire:</s> <s xml:id="echoid-s47033" xml:space="preserve"> quoniam in omnibus his idem eſt operan-<lb/>di modus:</s> <s xml:id="echoid-s47034" xml:space="preserve"> ut ſecundum fixam diametrum a c in 37 huius:</s> <s xml:id="echoid-s47035" xml:space="preserve"> uel ſecundum fixum punctum q in 38 hu-<lb/>ius fiat dictarum linearũ reuolutio ſuper ſubiectas ſibi proportionales corporis ferrei ſuperficies:</s> <s xml:id="echoid-s47036" xml:space="preserve"> <lb/>proueniẽtq́;</s> <s xml:id="echoid-s47037" xml:space="preserve"> figuræ ſimiles illis lineis, à quarum ſuperficiebus reflexi radij omnes ad unum punctũ <lb/>naturalem uel mathematicum concurrent.</s> <s xml:id="echoid-s47038" xml:space="preserve"> Patet itaque propoſιtum.</s> <s xml:id="echoid-s47039" xml:space="preserve"/> </p> <div xml:id="echoid-div1790" type="float" level="0" n="0"> <figure xlink:label="fig-0704-01" xlink:href="fig-0704-01a"> <variables xml:id="echoid-variables817" xml:space="preserve">a g b e d</variables> </figure> </div> </div> <div xml:id="echoid-div1792" type="section" level="0" n="0"> <head xml:id="echoid-head1325" xml:space="preserve">VITELLONIS FI-<lb/>LII THVRINGORVM ET PO-<lb/>LONORVM OPTICAE LIBER DECIMVS.</head> <p style="it"> <s xml:id="echoid-s47040" xml:space="preserve"><emph style="sc">Svperivs</emph> duos modos uiſionis, ſcilicet eum, qui fit directè per unum <lb/>medium diaphanum:</s> <s xml:id="echoid-s47041" xml:space="preserve"> & eum, qui fit per reflexionem à politis corporibus, <lb/>tractauimus:</s> <s xml:id="echoid-s47042" xml:space="preserve"> ſupereſt nunc, ut tertium uidendi modum, qui fit per refra-<lb/>ctionem, factamà pluribus diaphanis corporibus medijs inter uiſum & <lb/>rem uiſam proſequamur:</s> <s xml:id="echoid-s47043" xml:space="preserve"> quoniam & ſecundum hunc modum diuerſimodè uariatur <lb/>actio naturalium formarum & modus actionis.</s> <s xml:id="echoid-s47044" xml:space="preserve"> Virtutes enim formarum natura-<lb/>lium aggregatæ per refractionem fortius agunt, & plus actionis formæ corporibus ſuſce-<lb/>ptibilibus imprimunt:</s> <s xml:id="echoid-s47045" xml:space="preserve"> unde etiam accenditur ignis ex radijs ſolis ſub corpore ſphærico <lb/>diaphano denſiore aere uelaqua, ut ſub glacie uel cryſtallo.</s> <s xml:id="echoid-s47046" xml:space="preserve"> Vniuerſaliter uerò aggrega-<lb/>tio uirtutis radiorum ſtellarum uel aliarum formarum in eodem puncto naturali uel <lb/>circa illud fit fortioris actionis:</s> <s xml:id="echoid-s47047" xml:space="preserve"> diſperſio uerò uirtutum naturalium formarum debili-<lb/>tat actiones naturales:</s> <s xml:id="echoid-s47048" xml:space="preserve"> diſgregata enim uirtus debilius & minus agit.</s> <s xml:id="echoid-s47049" xml:space="preserve"> In his autem o-<lb/>mnibus, ſicut & in alijs modis uidendi ſuperius diximus, uiſiua cognitio ſignum eſt, nõ <lb/>cauſſa.</s> <s xml:id="echoid-s47050" xml:space="preserve"> Non enim, quia uiſus ſic uidet, ideo ſic accidit in formis rerum agentium:</s> <s xml:id="echoid-s47051" xml:space="preserve"> ſed <lb/>quia ſic agunt formæ naturales, ideo ipſas ſic agentes uidet uiſus, niſi fortè in quibuſdam <lb/>deceptionibus, quæ uiſui accidunt per ſeipſum.</s> <s xml:id="echoid-s47052" xml:space="preserve"> Omnis autem paßio ſecundum modos cu <lb/>iuſcunque refractionis naturæ accidens uel uiſui, fit ſemper propter diuerſitatem diapha <lb/>nitatis mediorum corporum inter agens & paſſum, uel inter uiſum & rem uiſam.</s> <s xml:id="echoid-s47053" xml:space="preserve"> Cor <lb/>pora uerò diaphana nobis aſſueta, ſunt aer, qui ect rarioris diaphanitatis omnibus alijs <lb/>diaphanis corporibus, (excepto corpore cœli) quod eſt rarius aere, ut poſtmodum demõ <lb/>ſtrabimus in progreſſu.</s> <s xml:id="echoid-s47054" xml:space="preserve"> Hic autem & in toto ſequente tractatu nomine aeris & ignem <lb/>accipimus:</s> <s xml:id="echoid-s47055" xml:space="preserve"> quia licet inter hæc ſit differentia ſpecifica formalis & dιuerſa raritas in diſ-<lb/>poſitionibus materiæ:</s> <s xml:id="echoid-s47056" xml:space="preserve"> non tamen ex hac diuerſitate aliqua accidit diuerſitas ſenſibilis <lb/>in formarum refractione:</s> <s xml:id="echoid-s47057" xml:space="preserve"> quoniam ignis, qui apud nos eſt hic inferius, eſt in materia <lb/>groſſa terrea uel aquea uel aerea, & ſecundum hoc ſequitur paßiones corporum alio-<lb/>rum:</s> <s xml:id="echoid-s47058" xml:space="preserve"> ignis uerò in ſphæra ſua eſt ſecundum ſui formalem diſtinctionem aeri contiguus, <lb/>& ſecundum naturam diaphanitatis continuus, non habens diſtinctam ſuperficiem ab <lb/>aere, in qua ſit poßibile refractionem ſenſibilem fieri.</s> <s xml:id="echoid-s47059" xml:space="preserve"> Aer enim quantò propinquior <lb/>eſt cœlo, tantò fit rarioris diaphanitatis:</s> <s xml:id="echoid-s47060" xml:space="preserve"> ſimiliter & ignis, ita quòd infimum ignis & <lb/>ſupremum aeris eſt diaphanitas quaſi una, in qua refractio ſenſibilis fieri non poteſt:</s> <s xml:id="echoid-s47061" xml:space="preserve"> & <lb/>ita quòd ſuperficies concaua ignis non eſt diuerſæ diaphanitatis & ſenſibiliter determi-<lb/>natæ à ſuperficie conuexa aeris:</s> <s xml:id="echoid-s47062" xml:space="preserve"> ideo non fit refractio inter illa:</s> <s xml:id="echoid-s47063" xml:space="preserve"> & ſic ignem in hoc tra-<lb/>ctatu ſub nomine aeris implicamus.</s> <s xml:id="echoid-s47064" xml:space="preserve"> Ect tamen aliqualis refractionum diuerſitas in <lb/>aere denſiori & rariori, quando illa diuerſitas denſitatis fit ſenſibilis:</s> <s xml:id="echoid-s47065" xml:space="preserve"> ſicut plurimum <lb/> <pb o="404" file="0706" n="706" rhead="VITELLONIS OPTICAE"/> accidit in aere condenſato prope terram:</s> <s xml:id="echoid-s47066" xml:space="preserve"> & maximè in crepuſculis ſerotinis & matu-<lb/>tinis.</s> <s xml:id="echoid-s47067" xml:space="preserve"> Diaphanum uerò aliud diuerſum ab iſtis eſt aqua continens etiam in ſe diuerſita-<lb/>tem refractionis ſecundum rarius & denſius, quod eſt in illo ſuo genere:</s> <s xml:id="echoid-s47068" xml:space="preserve"> uno tamen no-<lb/>mine nuncupatur.</s> <s xml:id="echoid-s47069" xml:space="preserve"> Sunt enim aquæ calidæ ſulphureæ & aquæ ſalſæ, ut maris, großioris <lb/>diaphanitatis, quàm aliæ aquæ frigidæ, claræ, dulces.</s> <s xml:id="echoid-s47070" xml:space="preserve"> Alia uerò corpora diaphana no-<lb/>bis aſſueta ſunt quidam lapides, ut crystallus, beryllus, & ſimiles, ut ſunt uitra.</s> <s xml:id="echoid-s47071" xml:space="preserve"> Dicitur <lb/>etiam de quibuſdã corporib.</s> <s xml:id="echoid-s47072" xml:space="preserve"> animatis, quòd ſint diaphana, ut de istis, quæ colorantur co <lb/>loribus corporum, quibus ſuperſtant:</s> <s xml:id="echoid-s47073" xml:space="preserve"> quorum animatorum corporum paßiones nõ pro-<lb/>ſequimur, quia ſunt figuræ irregularis.</s> <s xml:id="echoid-s47074" xml:space="preserve"> Superficies itaque cœli, quæ occurrit uiſui, eſt <lb/>ſphærica concaua:</s> <s xml:id="echoid-s47075" xml:space="preserve"> quæ ſi ſecetur ab aliqua plana ſuperficie:</s> <s xml:id="echoid-s47076" xml:space="preserve"> erit communis ſectio illa-<lb/>rum ſuperficierum linea circularis, cuius concauũ eſt ex parte uiſus, ut patet per 69 th.</s> <s xml:id="echoid-s47077" xml:space="preserve"> 1 <lb/>huius:</s> <s xml:id="echoid-s47078" xml:space="preserve"> & ſuperficies aeris, quæ tangit illam, eſt ſphærica conuexa:</s> <s xml:id="echoid-s47079" xml:space="preserve"> quæ ſiſecetur à pla-<lb/>na ſuperficie:</s> <s xml:id="echoid-s47080" xml:space="preserve"> communis ſectio erit linea circularis:</s> <s xml:id="echoid-s47081" xml:space="preserve"> cuius conuexum eſt ex parte cœli.</s> <s xml:id="echoid-s47082" xml:space="preserve"> <lb/>Superficies uerò aquæ ex parte uiſus ſuperctantis aquæ eſt ſphærica conuexa:</s> <s xml:id="echoid-s47083" xml:space="preserve"> quæ ſi ſe-<lb/>cetur à plana ſuperficie:</s> <s xml:id="echoid-s47084" xml:space="preserve"> erit communis ſectio linea cιrcularis:</s> <s xml:id="echoid-s47085" xml:space="preserve"> cuius cõuexum eſt ex par-<lb/>te illius uiſus.</s> <s xml:id="echoid-s47086" xml:space="preserve"> Vitrorum uerò & lapidum diaphanorum figuræ ſunt rotũdæ:</s> <s xml:id="echoid-s47087" xml:space="preserve"> aut planæ:</s> <s xml:id="echoid-s47088" xml:space="preserve"> <lb/>aut irregulares:</s> <s xml:id="echoid-s47089" xml:space="preserve"> unde ſi ſecentur à planis ſuperficiebus, fient in illis communes ſectiones <lb/>aut circuli:</s> <s xml:id="echoid-s47090" xml:space="preserve"> aut lineæ rectæ:</s> <s xml:id="echoid-s47091" xml:space="preserve"> aut irregulares, ſecundum quarum linearum & ſuperficie-<lb/>rum diuerſitatem uariatur diuerſitas paßionum, quæ uiſibus occurrunt.</s> <s xml:id="echoid-s47092" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1793" type="section" level="0" n="0"> <head xml:id="echoid-head1326" xml:space="preserve">DEFINITIONES.</head> <p> <s xml:id="echoid-s47093" xml:space="preserve">1.</s> <s xml:id="echoid-s47094" xml:space="preserve"> Linea incidentiæ dicitur linea, ſecundum quam forma directè diffun ditur per <lb/>medium unius diaphani.</s> <s xml:id="echoid-s47095" xml:space="preserve"> Et eadem dicitur linea extenſionis formæ.</s> <s xml:id="echoid-s47096" xml:space="preserve"> 2.</s> <s xml:id="echoid-s47097" xml:space="preserve"> Refractio <lb/>dicitur incuruatio eiuſdem lineæ ad angulum cõtinendum:</s> <s xml:id="echoid-s47098" xml:space="preserve"> ut cum lineæ, per quas <lb/>una forma rei uiſæ peruenit ad uiſum, non rectè prodeunt, ſed franguntur in ſuper-<lb/>ficie alterius corporis diaphani.</s> <s xml:id="echoid-s47099" xml:space="preserve"> 3.</s> <s xml:id="echoid-s47100" xml:space="preserve"> Punctus refractionis eſt punctus ſuperficiei <lb/>corporis diaphani, à quo fit lineæ incidentiæ uel lineæ extẽſionis formæ refractio <lb/>ad uiſum.</s> <s xml:id="echoid-s47101" xml:space="preserve"> 4.</s> <s xml:id="echoid-s47102" xml:space="preserve"> Linea refractionis dicitur linea à puncto refractionis ad centrum <lb/>uiſus extenſa.</s> <s xml:id="echoid-s47103" xml:space="preserve"> 5.</s> <s xml:id="echoid-s47104" xml:space="preserve"> Linea perpendicularis hic nunc dicitur linea, quæ à puncto re-<lb/>fractionis erigitur ſuper ſuperficiem corporis, à qua fit refractio.</s> <s xml:id="echoid-s47105" xml:space="preserve"> 6.</s> <s xml:id="echoid-s47106" xml:space="preserve"> Cathetus <lb/>incidentiæ dicitur linea à puncto rei uiſæ ſuper ſuperficiem corporis, in quo eſt res <lb/>uiſa, & à qua fit refractio, perpendiculariter producta.</s> <s xml:id="echoid-s47107" xml:space="preserve"> 7.</s> <s xml:id="echoid-s47108" xml:space="preserve"> Superficies refractio-<lb/>nis dicitur ſuperficies, in qua contin entur lineæ incidentiæ & refractionis.</s> <s xml:id="echoid-s47109" xml:space="preserve"> 8.</s> <s xml:id="echoid-s47110" xml:space="preserve"> An-<lb/>gulus incidentiæ dicitur minor angulus, quem continet linea in cidentiæ cum li-<lb/>nea perpendiculari, ducta à puncto refractionis ſuper ſuperficiem corporis, à qua <lb/>fit illa refractio.</s> <s xml:id="echoid-s47111" xml:space="preserve"> 9.</s> <s xml:id="echoid-s47112" xml:space="preserve"> Angulus refractus dicitur angulus minor, quem continet li-<lb/>nea refracta cum dicta perpendiculari.</s> <s xml:id="echoid-s47113" xml:space="preserve"> 10.</s> <s xml:id="echoid-s47114" xml:space="preserve"> Angulus refractionis dicitur angu-<lb/>lus, quem continet linea refractionis cum linea incidētiæ trans corpus diaphanũ, <lb/>à cuius ſuperficie fit refractio, in continuum protracta.</s> <s xml:id="echoid-s47115" xml:space="preserve"> 11.</s> <s xml:id="echoid-s47116" xml:space="preserve"> Directè uideri dicitur, <lb/>ſicut & ſuperius 1 defin.</s> <s xml:id="echoid-s47117" xml:space="preserve"> 4 huius definitum eſt, quando forma rei uiſæ ſine refractio <lb/>ne peruenit ad uiſum.</s> <s xml:id="echoid-s47118" xml:space="preserve"> 12.</s> <s xml:id="echoid-s47119" xml:space="preserve"> Obliquè dicitur uideri, cum forma rei uiſæ ad uiſum <lb/>peruenit refractè.</s> <s xml:id="echoid-s47120" xml:space="preserve"> 13.</s> <s xml:id="echoid-s47121" xml:space="preserve"> Imago refracta dicitur forma rei uiſæ obliquè perueni-<lb/>ens ad uiſum.</s> <s xml:id="echoid-s47122" xml:space="preserve"> 14.</s> <s xml:id="echoid-s47123" xml:space="preserve"> Locus imaginis refractæ, dicitur locus, in quo imago refra-<lb/>cta uiſibus occurrit.</s> <s xml:id="echoid-s47124" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1794" type="section" level="0" n="0"> <head xml:id="echoid-head1327" xml:space="preserve">PETITIONES.</head> <p> <s xml:id="echoid-s47125" xml:space="preserve">Supponimus autem hæc.</s> <s xml:id="echoid-s47126" xml:space="preserve"> 1.</s> <s xml:id="echoid-s47127" xml:space="preserve"> Lumen Solis aliqualiter in matutinis & ſerotinis <lb/>crepuſculis uideri.</s> <s xml:id="echoid-s47128" xml:space="preserve"> 2.</s> <s xml:id="echoid-s47129" xml:space="preserve"> Item iridem ſecundum ſiguram rotundam <lb/>& colores uarios uideri.</s> <s xml:id="echoid-s47130" xml:space="preserve"/> </p> <pb o="405" file="0707" n="707" rhead="LIBER DECIMVS."/> </div> <div xml:id="echoid-div1795" type="section" level="0" n="0"> <head xml:id="echoid-head1328" xml:space="preserve">THEOREMATA</head> <head xml:id="echoid-head1329" xml:space="preserve" style="it">1. In omni ſuperficie refractionis neceſſariò ſunt punctum, cuius forma refringitur: & pun-<lb/>ctum refractionis: & centrum ipſius uiſus: & perpendicularis ducta à puncto refractionis ſu-<lb/>per ſuperficiem, à qua fit refractio. Ex quo patet quòd unius refractionis unica tantùm eſt <lb/>ſuperficies.</head> <p> <s xml:id="echoid-s47131" xml:space="preserve">Sit ſuperficies ſecundi diaphani denſioris uel rarioris primo diaphano, in qua ſit linea a b c:</s> <s xml:id="echoid-s47132" xml:space="preserve"> & ſit <lb/>punctũ, cuius forma refringitur, punctum d:</s> <s xml:id="echoid-s47133" xml:space="preserve"> ſitq́ue centrum uiſus e:</s> <s xml:id="echoid-s47134" xml:space="preserve"> fiatq́ue refractio in puncto ſu-<lb/>perficiei ſecundi diaphani, quod eſt b:</s> <s xml:id="echoid-s47135" xml:space="preserve"> & à puncto b ſuper ſuperficiem a b c ducatur perpendicula-<lb/>ris b f.</s> <s xml:id="echoid-s47136" xml:space="preserve"> Dico quòd puncta d, e, b, & linea b f ſunt ſemper in eadem ſuperficie refractionis.</s> <s xml:id="echoid-s47137" xml:space="preserve"> Quoniam <lb/>eni m, ut patet per definitionem præmiſſam in principijs libri huius, & per 46 th.</s> <s xml:id="echoid-s47138" xml:space="preserve"> 2 huius linea radia <lb/>lis incidens (quæ eſt d b) & refracta (quæ eſt b e) ſunt in eadem ſuperficie refractionis:</s> <s xml:id="echoid-s47139" xml:space="preserve"> punctum er <lb/>go d, cuius forma incidit & refringitur, & punctum refractionis, ſcilicet pũctum, à quo fit refra ctio, <lb/>(quod eſt b) & centrum uiſus (quod eſt e) ſunt in eadem ſuperficie per 1 p 11:</s> <s xml:id="echoid-s47140" xml:space="preserve"> ſed & per 2 p 11 linea <lb/>b f, quæ eſt perpendicularis ſuper ſuperficiem a b c, <lb/>eſt in eadem ſuperficie cum linea b c:</s> <s xml:id="echoid-s47141" xml:space="preserve"> ergo & cum <lb/> <anchor type="figure" xlink:label="fig-0707-01a" xlink:href="fig-0707-01"/> lineis d b & b e:</s> <s xml:id="echoid-s47142" xml:space="preserve"> quoniam linea b f eſt perpendi-<lb/>cularis ſuper lineam a b c, & cum illa in eadem ſu-<lb/>perficie.</s> <s xml:id="echoid-s47143" xml:space="preserve"> Similiter protracta linea d b ultra punctum <lb/>b ad punctum g, eſt in eadem ſuperficie.</s> <s xml:id="echoid-s47144" xml:space="preserve"> Puncta ita-<lb/>que d, b, e & linea b f ſunt in eadem ſuperficie per 1 <lb/>& 2 p 11.</s> <s xml:id="echoid-s47145" xml:space="preserve"> Omnis enim refractio aut fit ad ipſam per-<lb/>pendicularem b f, aut ab ipſa:</s> <s xml:id="echoid-s47146" xml:space="preserve"> & ſemper in eadem <lb/>ſuperficie, in qua fiebat incidentia formę refringen <lb/>dæ.</s> <s xml:id="echoid-s47147" xml:space="preserve"> Quoniam enim omnis refractio fit ad omnem <lb/>differentiam poſitionis (quia qua ratione fit ad u-<lb/>nam partem, eadem ratiõe fit ad quamlibet aliam) <lb/>determinatio ergo refractionis ad certam differen <lb/>tiam poſitionis fit tantùm per uiſum:</s> <s xml:id="echoid-s47148" xml:space="preserve"> quia in qua-<lb/>cunque ſuperficie centrum uiſus fuerit, in illa tan-<lb/>tùm percipitur fieri refractio.</s> <s xml:id="echoid-s47149" xml:space="preserve"> Patet ergo propoſi-<lb/>tum.</s> <s xml:id="echoid-s47150" xml:space="preserve"> Et ex hoc patet, cum iſta puncta refractionis omnia ſcilicet d, e, b & linea b f ſuperficiem refra-<lb/>ctionis conſtituant, quòd horum aliquo deficiente non eſt ſuperficies refractionis:</s> <s xml:id="echoid-s47151" xml:space="preserve"> & quòd unius <lb/>refractionis unica tantùm eſt ſuperficies refractionis:</s> <s xml:id="echoid-s47152" xml:space="preserve"> quoniam hæc omnia puncta in unica tantùm <lb/>ſuperficie ſimili concurrere eſt poſsibile, & non in pluribus.</s> <s xml:id="echoid-s47153" xml:space="preserve"> Et hoc eſt, quod proponebatur.</s> <s xml:id="echoid-s47154" xml:space="preserve"/> </p> <div xml:id="echoid-div1795" type="float" level="0" n="0"> <figure xlink:label="fig-0707-01" xlink:href="fig-0707-01a"> <variables xml:id="echoid-variables818" xml:space="preserve">e g f a b c d</variables> </figure> </div> </div> <div xml:id="echoid-div1797" type="section" level="0" n="0"> <head xml:id="echoid-head1330" xml:space="preserve" style="it">2. Neceſſe eſt omnem ſuperficiem refractionis ſuper ſuperficiem corporis, à qua fit refractio <lb/>(ſiue illa ſuperficies ſit plana conuexa uel concaua) erectam eſſe. Alhazen 9 n 7.</head> <p> <s xml:id="echoid-s47155" xml:space="preserve">Hoc, quod hic proponitur, patet per præmiſſam.</s> <s xml:id="echoid-s47156" xml:space="preserve"> Quoniam enim in omni ſuperficie refractionis <lb/>neceſſariò ſunt:</s> <s xml:id="echoid-s47157" xml:space="preserve"> punctum, cuius forma refringitur:</s> <s xml:id="echoid-s47158" xml:space="preserve"> & punctum ſuperficiei corporis, à quo fit refra-<lb/>ctio:</s> <s xml:id="echoid-s47159" xml:space="preserve"> & centrum uiſus & perpendicularis ducta à puncto refractionis ſuper ſuperficiem corporis il <lb/>lius, à qua fit refractio:</s> <s xml:id="echoid-s47160" xml:space="preserve"> ergo per 18 p 11 patet quòd omnis ſuperficies refractionis eſt perpendicula-<lb/>ris ſuper ſuperficiem corporis, à qua fit refractio.</s> <s xml:id="echoid-s47161" xml:space="preserve"> Si enim illa ſuperficies fuerit plana:</s> <s xml:id="echoid-s47162" xml:space="preserve"> tunc euiden-<lb/>ter patet propoſitum per 18 p 11, ut præ miſſum eſt.</s> <s xml:id="echoid-s47163" xml:space="preserve"> Si uerò fuerit illa ſuperficies conuexa uel con-<lb/>caua ſphærica:</s> <s xml:id="echoid-s47164" xml:space="preserve"> tunc patet per 72 th.</s> <s xml:id="echoid-s47165" xml:space="preserve"> 1 huius quoniam perpendicularis ducta à puncto refractionis ſu <lb/>per ipſam ſuperficiem corporis, à qua fit refractio, ſemper tranſit centrũ illius corporis:</s> <s xml:id="echoid-s47166" xml:space="preserve"> & eſt perpẽ-<lb/>dicularis ſuper ſuperficiem illud corpus in puncto refractionis contingentem:</s> <s xml:id="echoid-s47167" xml:space="preserve"> ergo itẽ per 18 p 11 <lb/>ſuperficies refractionis eſt erecta ſuper illã ſuperficiẽ contingentẽ:</s> <s xml:id="echoid-s47168" xml:space="preserve"> ergo & ſuper ipſam corporis ſu-<lb/>perficiem.</s> <s xml:id="echoid-s47169" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s47170" xml:space="preserve"> demonſtrandum, ſiue figura corporis, à qua fit refractio, fuerit columna <lb/>ris ſiue pyramidalis ſiue alterius figuræ cuiuſcunq;</s> <s xml:id="echoid-s47171" xml:space="preserve">: ſemper enim ſuperficies refractionis erit erecta <lb/>ſuper ſuperficiem corporis, à qua fit refractio.</s> <s xml:id="echoid-s47172" xml:space="preserve"> Et ſi accidat, ut illa ſuperficies corporis, à qua fit refra <lb/>ctio, fuerit æquidiſtans horizonti:</s> <s xml:id="echoid-s47173" xml:space="preserve"> tunc perpendicularis ducta à puncto refractionis ſuper ſuperfi-<lb/>ciem corporis, à qua fit reſractio, eſt ctiam perpendicularis ſuper ſuperficiem horizontis per 23 th.</s> <s xml:id="echoid-s47174" xml:space="preserve"> 1 <lb/>huius:</s> <s xml:id="echoid-s47175" xml:space="preserve"> ergo & per 18 p 11 ſuperficies refractionis eſt perpendicularis, & erecta ſuper ſuperficiem ho-<lb/>rizontis.</s> <s xml:id="echoid-s47176" xml:space="preserve"> Sed & hoc patet per declarationem, quæ fit in inſtrumento, quod in 1 th.</s> <s xml:id="echoid-s47177" xml:space="preserve"> 2 huius præmiſi-<lb/>mus.</s> <s xml:id="echoid-s47178" xml:space="preserve"> Quoniam enim linea radialis incidens & refracta ab aliqua ſuperficie unius corporis diapha-<lb/>ni ad aliud corpus diaphanum, ut patet per 46 th.</s> <s xml:id="echoid-s47179" xml:space="preserve"> 2 huius, ſemper ſunt in una plana ſuperficie, quæ <lb/>eſt medius circulus illorũ triũ circulorũ ſignatorũ in interiori parte oræ inſtrumenti, æquidiſtans <lb/>ſuperficiei interioris laminæ inſtrumẽti:</s> <s xml:id="echoid-s47180" xml:space="preserve"> ſed illa ſuperficies laminæ æ quidiſtat ſuքficiei dorſi inſtru <lb/>mẽti, cui extrinſecus ſuքponitur ſuperficies regulæ cubitalis tenentis inſtrumentũ.</s> <s xml:id="echoid-s47181" xml:space="preserve"> Suքficies itaq;</s> <s xml:id="echoid-s47182" xml:space="preserve"> <lb/>medij circuli ęquidiſtat ſuքficiei regulę lógę quadrãgulę ſuքpoſitę dorſo laminę ք 24.</s> <s xml:id="echoid-s47183" xml:space="preserve"> th.</s> <s xml:id="echoid-s47184" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s47185" xml:space="preserve"> ſed <lb/>illa ſuքficies քpẽdicularis eſt ſuք ſuքficies laterũ lógitudinis regulę erectas ſuք oras inſtrumẽti.</s> <s xml:id="echoid-s47186" xml:space="preserve"> Su <lb/>perficies itaq;</s> <s xml:id="echoid-s47187" xml:space="preserve"> medij circuli eſt ք cõuerſam 14 p 11 քpendicularis ſuper ſuքficies lõgitudinis regulæ <lb/> <pb o="406" file="0708" n="708" rhead="VITELLONIS OPTICAE"/> erectas ſuք oras inſtruméti:</s> <s xml:id="echoid-s47188" xml:space="preserve"> ſed illę duę ſuperficies regulę ſunt ęquidiſtátes horizonti tẽpore expe-<lb/>rimẽtationis ք inſtrumentũ poſitum in uaſe, ut cõſueuit.</s> <s xml:id="echoid-s47189" xml:space="preserve"> Superficies itaq;</s> <s xml:id="echoid-s47190" xml:space="preserve"> medij circuli eſt perpédi-<lb/>cularis ſuք ſuperficiẽ horizótis.</s> <s xml:id="echoid-s47191" xml:space="preserve"> Et quia ſuperficies medij circuli eſt ſuքficies refractιõis, patet pro-<lb/>poſitũ.</s> <s xml:id="echoid-s47192" xml:space="preserve"> Idem quoq;</s> <s xml:id="echoid-s47193" xml:space="preserve"> poteſt oſtẽdi producta per imaginationẽ linea à centro medij circuli ad centrú <lb/>mundi.</s> <s xml:id="echoid-s47194" xml:space="preserve"> Hæc enim linea, cum ſit ſemidiameter mundi, perpendicularis eſt ſuper ſuperficiem aquæ, <lb/>quę eſt in uaſe:</s> <s xml:id="echoid-s47195" xml:space="preserve"> eſt autem illa linea in ſuperficie medij circuli, quæ eſt ſuperficies refractionis.</s> <s xml:id="echoid-s47196" xml:space="preserve"> Eſt er-<lb/>go per 18 p 11 illa ſuperficies perpendicularis ſuper ſuperficiem horizontis.</s> <s xml:id="echoid-s47197" xml:space="preserve"> cum enim lux refringi-<lb/>tur ab aere ad aquam:</s> <s xml:id="echoid-s47198" xml:space="preserve"> erit refractionis linea cadens inter primam lineam, per quá extenditur in ae-<lb/>re, quæ eſt linea in cidentiæ ſuę, & inter perpendicularem exeuntem à centro medij circuli ſuper ſu <lb/>perficiem aquæ:</s> <s xml:id="echoid-s47199" xml:space="preserve"> & centrum lucis intra aquam ſemper procedit à centro medij circuli.</s> <s xml:id="echoid-s47200" xml:space="preserve"> Palàm ergo <lb/>quòd lux, quæ refringitur ab aere ad aquam, reſringitur in ſuperficie perpendiculari ſuper ſuperfi-<lb/>ciem aquæ:</s> <s xml:id="echoid-s47201" xml:space="preserve"> ergo & ſuper ſuperficiem horizontis.</s> <s xml:id="echoid-s47202" xml:space="preserve"> Idem quoq;</s> <s xml:id="echoid-s47203" xml:space="preserve"> accidit cum ab aere ad uitrum fit refra <lb/>ctio.</s> <s xml:id="echoid-s47204" xml:space="preserve"> Patet ergo ſiue ſuperficies corporis, à qua fit refractio, ſit plana conuexa uel, cócaua, quòd ſem-<lb/>per ſuperficies refractionis eſt erecta ſuper illam.</s> <s xml:id="echoid-s47205" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s47206" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1798" type="section" level="0" n="0"> <head xml:id="echoid-head1331" xml:space="preserve" style="it">3. Centro uiſus exiſtente ultra medium ſecundi diaphani: omnes formæ obliquè incidẽtes ſu-<lb/>perficiei ſecundi diaphani, reſpectu uiſus, refractè uiſuioccurrunt: perpendiculariter uerò inci-<lb/>dentes uidentur directè. Alhazen 13 n 7.</head> <p> <s xml:id="echoid-s47207" xml:space="preserve">Quoniam enim lux pertranſit corpora diaphana, quibus incidit, aut directè, ut cũ radius incidés <lb/>eſt perpendicularis ſuper ſuperficiem corporis ſibi oppoſiti:</s> <s xml:id="echoid-s47208" xml:space="preserve"> aut obliquè, ut cum radius incidit ob-<lb/>liquè:</s> <s xml:id="echoid-s47209" xml:space="preserve"> & ab uno puncto corporis luminoſi ſecundum omnem lineam ab illo puncto ducibilem fit lu <lb/>minis diffuſio, ut patet per 20 th.</s> <s xml:id="echoid-s47210" xml:space="preserve"> 2 huius:</s> <s xml:id="echoid-s47211" xml:space="preserve"> & quia forma coloris ſemper diffundit ſe cum lumine:</s> <s xml:id="echoid-s47212" xml:space="preserve"> pa-<lb/>tet quòd cuiuslibet puncti cuiuſcun que corporis luminoſi colorati uel lucidi exiſtentis in aliquo <lb/>corpore diaphano, forma lucis & coloris extenditur in uniuerſo corpore diaphano ſibi proximo, & <lb/>peruenit ad ſuperficiem corporis diaphani ſibi oppoſiti.</s> <s xml:id="echoid-s47213" xml:space="preserve"> Et ſi fuerit aliud corpus diaphanum cótin-<lb/>gens illud ſecundum corpus diaphanum, quod ſit alterius diaphanitatis ab illo:</s> <s xml:id="echoid-s47214" xml:space="preserve"> tunc forma diffuſa <lb/>penetrat illud, & omnes lineæ radiales, ſecundum quas illis corporib diaphanis obliquè lumen uel <lb/>color incidit, refringentur, præter quàm linea incidens perpendiculariter:</s> <s xml:id="echoid-s47215" xml:space="preserve"> ſola enim illa extenditur <lb/>ſecundum rectitudinem in corpore diaphano proximo ſibi, & in corpore alio diaphano proximum <lb/>corpus diaphanum contingente:</s> <s xml:id="echoid-s47216" xml:space="preserve"> dum tamen perpendiculariter incidat utriq;</s> <s xml:id="echoid-s47217" xml:space="preserve">. Et ſi fortè aliqua li-<lb/>nearum radialium perpendiculariter inciderit puncto ſuperficiei continuæ cum ſuperficie corpo-<lb/>ris diaphani proximi:</s> <s xml:id="echoid-s47218" xml:space="preserve"> nec ſit illius ſuperficiei ſecundæ corpus diaphanum:</s> <s xml:id="echoid-s47219" xml:space="preserve"> uel ſi fuerit diaphanum, <lb/>non ſit tamen eius ſuperficies prioris diaphani ſuperficiei ęquidiſtans:</s> <s xml:id="echoid-s47220" xml:space="preserve"> tunc à puncto incidentiæ li-<lb/>neæ radialis ſuper ſuperficiem ſecundi corporis alia perpendicularis duci poteſt:</s> <s xml:id="echoid-s47221" xml:space="preserve"> ergo tunc illa for <lb/>ma, quę ſuperficiei prioris corporis ſecundum perpendicularem incidebat, delebitur:</s> <s xml:id="echoid-s47222" xml:space="preserve"> quoniam ab <lb/>uno puncto ad unam ſuperficiẽ duas lineas perpendiculares duci eſt impoſsibile per 13 p 11.</s> <s xml:id="echoid-s47223" xml:space="preserve"> Omnes <lb/>ergo formæ illius puncti tranſeuntes in corpus diaphanum contingens proximum illi pũcto aliud <lb/>corpus diaphanum, erunt reſractæ.</s> <s xml:id="echoid-s47224" xml:space="preserve"> Et quoniam à quolibet pũcto cuiuslibet corporis luminoſi uel <lb/>colorati extenditur lumen & color penetrans totum corpus diaphanum obiectum, & refringitur à <lb/>ſuperficie alterius corporis diuerſæ diaphanitatis illi ſuccedentis per 47 th.</s> <s xml:id="echoid-s47225" xml:space="preserve"> 2 huius:</s> <s xml:id="echoid-s47226" xml:space="preserve"> patet quòd ſor <lb/>ma lucis & coloris erit una forma continua, coniuncta:</s> <s xml:id="echoid-s47227" xml:space="preserve"> & refringitur tota continua & coniuncta, ſu <lb/>perficie corporis diaphani exiſtente continua, & cum forma refracta fuerit continua.</s> <s xml:id="echoid-s47228" xml:space="preserve"> Si ergo cor-<lb/>pus denſioris diaphanitatis quàm ſit primum diaphanum, illi formę occurrerit:</s> <s xml:id="echoid-s47229" xml:space="preserve"> tunc forma conti-<lb/>nua magis aggregata & unita perueniet ad illud corpus:</s> <s xml:id="echoid-s47230" xml:space="preserve"> & occurrente item corpore diaphano ra-<lb/>riore:</s> <s xml:id="echoid-s47231" xml:space="preserve"> tunc quilibet punctus corporis diaphani, per quod extenditur forma puncti, quod eſt in pri-<lb/>mo corpore luminoſo uel colorato, tranſmittet formam lucis & coloris ad quodlibet punctũ ipſius <lb/>ſecundi uel tertij corporis diaphani per omnem lineam rectam, quæ poteſt extendi ab illo puncto.</s> <s xml:id="echoid-s47232" xml:space="preserve"> <lb/>Si itaq;</s> <s xml:id="echoid-s47233" xml:space="preserve"> aliquis fuerit imaginatus pyramides rectilineas, exeuntes à quolibet puncto aeris ad ſuper-<lb/>ficiem corporis diaphanitatis alterius pertingentes:</s> <s xml:id="echoid-s47234" xml:space="preserve"> & ſi in ſuperficie huius ſecundi corporis dia-<lb/>phani lineę obliquè in cidentes refringi imaginentur (perpendiculari linea, quę eſt axis illius pyra-<lb/>midis imaginatæ, ſine refractione tranſeunte) tunc adhuc fit unum corpus continuum in refractio <lb/>ne, ſicut & una eſt forma corporis incidens ſuperficiei illius ſecundi corporis diaphani.</s> <s xml:id="echoid-s47235" xml:space="preserve"> Si ergo in <lb/>loco imaginatæ pyramidis ſiſtatur ſecundum ueritatem in aere pyramis ſenſibilis, cuius corpus ſit <lb/>coloratũ uel luminoſum dẽſum:</s> <s xml:id="echoid-s47236" xml:space="preserve"> miſcebitur lux uel color illius pyramidis cum luce uel colore cor-<lb/>poris, à quo fit refractio:</s> <s xml:id="echoid-s47237" xml:space="preserve"> & fiet ipſorum multiplicatio per omnem lineam rectam, quæ poterit ex-<lb/>tendi ab illo puncto, cui incidit:</s> <s xml:id="echoid-s47238" xml:space="preserve"> & forma puncti incidens alicui puncto corporis denſi, extendetur <lb/>per quamlibet linearum refractarum ad illum punctum corporis, in quo fit refractio, ſibi correſpon <lb/>dentem.</s> <s xml:id="echoid-s47239" xml:space="preserve"> Et ſi uiſus fuerit ex parte altera illius diaphani:</s> <s xml:id="echoid-s47240" xml:space="preserve"> tunc illæ formæ perueniunt ad uiſum:</s> <s xml:id="echoid-s47241" xml:space="preserve"> ſed <lb/>perpendicularis (quia non reſringitur) peruenit perpẽdiculariter ad centrum uiſus:</s> <s xml:id="echoid-s47242" xml:space="preserve"> & formę per li <lb/>neas obliquas incidentes, refractè & obliquè perueniunt ad uiſum.</s> <s xml:id="echoid-s47243" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s47244" xml:space="preserve"> lineę, ſecundũ quas <lb/>forma refringitur, ſe in aere per omne corpus medium diffundant, quando coniunguntur apud u-<lb/>num punctum aeris:</s> <s xml:id="echoid-s47245" xml:space="preserve"> ideo quòd ipſarum multa fit interſectio propter æqualitatẽ diffuſionis forma-<lb/>rum illarum ad omnem differentiam poſitionis:</s> <s xml:id="echoid-s47246" xml:space="preserve"> tunc ſi centrum uiſus poſitũ ſit in illo puncto, com <lb/>prehendet uiſus illud uiſum ſecundũ refractionem (excepto unico puncto perpendiculariter inci-<lb/> <pb o="407" file="0709" n="709" rhead="LIBER DECIMVS."/> dente) quoniam ille non refringitur, ut in 47 th.</s> <s xml:id="echoid-s47247" xml:space="preserve"> 2 huius oſtenſum eſt.</s> <s xml:id="echoid-s47248" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s47249" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1799" type="section" level="0" n="0"> <head xml:id="echoid-head1332" xml:space="preserve" style="it">4. Omnis formæ per refractionem uiſæ ſi fiat refractio à medio ſecundi diaphani denſioris pri <lb/>mo ad uiſum, uidetur fieri ad partem perpendicularis, ductæ à puncto refractionis ſuper ſuperfi <lb/>ciem, à qua fit refractio. Si uerò fiat à diaphano rariori, uidetur fieri ad partem contrariam il-<lb/>lius perpendicularis. Alhazen 14 n 7.</head> <p> <s xml:id="echoid-s47250" xml:space="preserve">Quod hic proponitur, poteſt inſtrumentaliter demonſtrari, ita ut demonſtratio auxilio inſtru-<lb/>menti ſenſibiliter exprimatur.</s> <s xml:id="echoid-s47251" xml:space="preserve"> Accipiatur itaq;</s> <s xml:id="echoid-s47252" xml:space="preserve"> prædictum inſtrumentum, quo in præcedentib.</s> <s xml:id="echoid-s47253" xml:space="preserve"> uſi <lb/>ſumus:</s> <s xml:id="echoid-s47254" xml:space="preserve"> cuius diametrũ, quã ibi ſignauimus per literas f, g, nunc dicimus b q g, ita ut punctũ q ſit cẽ-<lb/>trum laminæ baſis inſtrumenti.</s> <s xml:id="echoid-s47255" xml:space="preserve"> Hoc itaque inſtrumentum ponatur in uaſe æquidiſtáter ſuperficiei <lb/>horizontis ſituato, & infundatur aqua uſque ad centrum laminæ, quod eſt q:</s> <s xml:id="echoid-s47256" xml:space="preserve"> oppilentur quoq;</s> <s xml:id="echoid-s47257" xml:space="preserve"> fora <lb/>mina inſtrumenti cum cera uel alio modo, ita quòd modicùm remaneat de foraminibus circa me-<lb/>dium ipſorum, quod in ambobus foraminibus ſit æquale:</s> <s xml:id="echoid-s47258" xml:space="preserve"> & hoc poteſt æquali colum na illis forami <lb/>nibus immiſſa menſurari.</s> <s xml:id="echoid-s47259" xml:space="preserve"> Dein de moueatur inſtrumentum, donec diameter b q g ſit perpendicula <lb/>ris ſuper ſuperficiem aquæ.</s> <s xml:id="echoid-s47260" xml:space="preserve"> Immittatur quoque ſtilus albus ſubtilis in ipſum uas, ita quòd eius ex-<lb/>tremitas cadat in punctum z, quod eſt extremitas diametri circuli medij, quæ ſit k f z:</s> <s xml:id="echoid-s47261" xml:space="preserve"> ponaturq́;</s> <s xml:id="echoid-s47262" xml:space="preserve"> u-<lb/>nus uiſuum ſuper ſuperius foramen in punctum k, & claudatur reliquus:</s> <s xml:id="echoid-s47263" xml:space="preserve"> tunc enim uidebitur extre <lb/>mitas ſtili ſecundum rectitudinem perpendicularis exeuntis ab extremitate ſtili ſuper ſuperficiem <lb/>aquæ:</s> <s xml:id="echoid-s47264" xml:space="preserve"> nam centrum uiſus & extremitas ſtili tunc ſunt in linea k f z perpendiculari ſuper ſuperficiẽ <lb/>aquę, ſecundum quam fit uiſio.</s> <s xml:id="echoid-s47265" xml:space="preserve"> Eſt enim linea k f z perpendicularis ſuper ſuperficiẽ aquæ per 8 p 11:</s> <s xml:id="echoid-s47266" xml:space="preserve"> <lb/>ideo quòd ipſa æquidiſtat lineæ b q g, quæ ex hypothe <lb/>ſi eſt perpendicularis ſuper eandem ſuperficiem aquę.</s> <s xml:id="echoid-s47267" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0709-01a" xlink:href="fig-0709-01"/> Deinde declinetur inſtrum entum, donec linea b q g <lb/>obliquetur ſuper ſuperficiem aquæ:</s> <s xml:id="echoid-s47268" xml:space="preserve"> ponaturq́ue <lb/>uiſus ſuper ſuperius foramen:</s> <s xml:id="echoid-s47269" xml:space="preserve"> & non uidebitur ex-<lb/>tremitas ſtili.</s> <s xml:id="echoid-s47270" xml:space="preserve"> Moueatur itaque extremitas ſtili in <lb/>circumferentia medij circuli paulatim ad partem op-<lb/>poſitam uiſui, donec uideatur illa extremitas, & figa-<lb/>tur in illo pũcto circuli medij, in quo apparet.</s> <s xml:id="echoid-s47271" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s47272" xml:space="preserve"> <lb/>tunc ponatur aliquod corpuſculum denſum in ſuper-<lb/>ficie aquæ in centro medij circuli, quod eſt f:</s> <s xml:id="echoid-s47273" xml:space="preserve"> tunc nó <lb/>uidebitur illa extremitas ſtili:</s> <s xml:id="echoid-s47274" xml:space="preserve"> ablato uerò illo corpu-<lb/>ſculo, uidebitur illa extremitas ſtili.</s> <s xml:id="echoid-s47275" xml:space="preserve"> Quòd ſi cóſidere-<lb/>tur in numero graduũ medij circuli diſtátia extremita <lb/>tis ſtili à pũcto z:</s> <s xml:id="echoid-s47276" xml:space="preserve"> inuenietur diſtantia ſenſibilis.</s> <s xml:id="echoid-s47277" xml:space="preserve"> Poteſt <lb/>aũt punctus z, qui eſt extremitas diametri medij cir-<lb/>culi, tranſeuntis per centrum duorum foraminum ſic <lb/>inueniri:</s> <s xml:id="echoid-s47278" xml:space="preserve"> ſcilicet ut regulæ ſubtilis latior extremitas ponatur ſuper centrum laminæ, & media linea <lb/>ipſius protendatur ſecundum diametrum laminæ:</s> <s xml:id="echoid-s47279" xml:space="preserve"> tunc enim acumen regulæ cadit ſuper punctũ z, <lb/>ut præmiſſum eſt prius in propoſitionibus 2 huius.</s> <s xml:id="echoid-s47280" xml:space="preserve"> Quòd ſi aſſumpto uitro, quod ſit pars alicuius <lb/>ſphæræ, ut in illis propoſitionibus aliquib.</s> <s xml:id="echoid-s47281" xml:space="preserve"> aſſumptũ eſt, cuius uitri ſuperficies aliqua ſit plana & ali <lb/>qua cõuexa ſphęrica:</s> <s xml:id="echoid-s47282" xml:space="preserve"> & illud uitrũ applicetur laminę, ita ut eius plana ſuperficies ſit ex parte ſorami <lb/>num, lineaq́;</s> <s xml:id="echoid-s47283" xml:space="preserve"> (quę eſt ſuarũ ſuperficierum planarũ cõmunis differentia) ſit ſuper lineam o d, ſecantẽ <lb/>b q ſemidiametrum laminæ perpendiculariter:</s> <s xml:id="echoid-s47284" xml:space="preserve"> ſic ergo erit diameter k f z perpẽdicularis ſuper pla-<lb/>ná ſuperficiem uitri & ſuper conuexá.</s> <s xml:id="echoid-s47285" xml:space="preserve"> Deinde ponatur inſtrumentũ in aqua, ponaturq́;</s> <s xml:id="echoid-s47286" xml:space="preserve"> extremitas <lb/>ſtili ſuper punctum z, & centrũ uiſus ſuper ſuperius foramen:</s> <s xml:id="echoid-s47287" xml:space="preserve"> uidebiturq́;</s> <s xml:id="echoid-s47288" xml:space="preserve"> extremitas ſtili, quæ in a-<lb/>lio puncto circuli medij non poterat uideri.</s> <s xml:id="echoid-s47289" xml:space="preserve"> Ex quo patet quoniam extremitas ſtili, quando eſt in li <lb/>nea perpendiculari ſuper ſuperficiem corporis, à qua fit refractio, (ut nunc eſt linea k f z perpẽdicu-<lb/>laris ſuper ſuperficiem uitri) forma ipſius uidetur non per refractionẽ, ſed rectè.</s> <s xml:id="echoid-s47290" xml:space="preserve"> Ex quo patet quò d <lb/>forma perpendiculariter incidens non refringitur.</s> <s xml:id="echoid-s47291" xml:space="preserve"> Quòd ſi conuexum uitri ponatur ex parte ſecũ-<lb/>da foraminum, & differentia communis duarum ſuperficierum planarum uitri ponatur ſuper pri-<lb/>mum locum, ſcilicet lineę o d:</s> <s xml:id="echoid-s47292" xml:space="preserve"> quoniam & tunc linea k f z eſt perpendicularis ſuper utraſque ſuperfi <lb/>cies uitri:</s> <s xml:id="echoid-s47293" xml:space="preserve"> uidebitur ergo tunc, ut prius, extremitas ſtili in puncto z.</s> <s xml:id="echoid-s47294" xml:space="preserve"> Quòd ſi à ſuperficie laminę in-<lb/>ſtrumenti euulſo uitro à centro laminæ, quod eſt q, in ſuperficie laminę ducatur ſemidiameter q r, <lb/>continens cum ſemidiametro b q angulum obtuſum:</s> <s xml:id="echoid-s47295" xml:space="preserve"> deinde ducatur ſemidiameter q u, continens <lb/>cũ linea q r angulũ rectum:</s> <s xml:id="echoid-s47296" xml:space="preserve"> & protrahatur ad aliã oram inſtrumenti:</s> <s xml:id="echoid-s47297" xml:space="preserve"> erit ergo angulus b q u acutus, <lb/>& erit ſemidiameter b q obliqua ſuք lineã q u.</s> <s xml:id="echoid-s47298" xml:space="preserve"> Deinde linea, quę eſt cómunis differẽtia ſuperficierũ <lb/>planarum uitri, ponatur ſuper lineá q u, & ſit plana uitri ſuperficies ex parte foraminum, & ſit me-<lb/>dium differentiæ communis planarũ ſuperficierum ipſius uitri ſuper centrum q.</s> <s xml:id="echoid-s47299" xml:space="preserve"> Erit itaq;</s> <s xml:id="echoid-s47300" xml:space="preserve"> tunc cen <lb/>trum uitri ſuper centrum medij circuli, ut pręoſtenſum eſt in alijs, & linea k f tranſit per centrũ uitri <lb/>& eſt obliqua ſuք ſuperficiem ipſius planá:</s> <s xml:id="echoid-s47301" xml:space="preserve"> quoniã diameter b q ęquidiſtans illi lineę, quę eſt k f, ob-<lb/>liquè cadit ſuper lineam q u:</s> <s xml:id="echoid-s47302" xml:space="preserve"> & quoniã linea k f tranſit per centrũ uitri:</s> <s xml:id="echoid-s47303" xml:space="preserve"> palàm quoniam ipſa eſt per-<lb/>pendicularis ſuper conuexam ſuperficiem uitri.</s> <s xml:id="echoid-s47304" xml:space="preserve"> Deinde à puncto r ſuper lineam q r ducatur per-<lb/> <pb o="408" file="0710" n="710" rhead="VITELLONIS OPTICAE"/> pendicularis in ora inſtrumenti uſq;</s> <s xml:id="echoid-s47305" xml:space="preserve"> ad circúſerentiam medij circuli, quę ſit r e:</s> <s xml:id="echoid-s47306" xml:space="preserve"> & fiat nigra utraque <lb/>illarum linearum q r & r e, ut melius per uiſum ualeant notari:</s> <s xml:id="echoid-s47307" xml:space="preserve"> & imaginetur duci linea e f:</s> <s xml:id="echoid-s47308" xml:space="preserve"> hęc itaq;</s> <s xml:id="echoid-s47309" xml:space="preserve"> <lb/>per 72 th.</s> <s xml:id="echoid-s47310" xml:space="preserve"> 1 huius erit perpendicularis ſuper conuexam ſuperficiem uitri:</s> <s xml:id="echoid-s47311" xml:space="preserve"> quoniam tranſit per eius <lb/>centrum:</s> <s xml:id="echoid-s47312" xml:space="preserve"> & eſt perpendicularis ſuper planam uitri ſuperficiem:</s> <s xml:id="echoid-s47313" xml:space="preserve"> quoniam eſt æquidiſtans lineę q r <lb/>perpendiculari ſuper lineam q u, cui ſuperpoſita eſt illa communis ſectio planarum ſuperficierum <lb/>ipſius uitri.</s> <s xml:id="echoid-s47314" xml:space="preserve"> Punctus ita que e eſt punctus medij circuli, in quem cadit perpen dicularis, exiens à cen <lb/>tro uitri ſuper planam ſuperficiem ipſius.</s> <s xml:id="echoid-s47315" xml:space="preserve"> Ponatur itaque inſtrumentum ſie diſpoſitum in uas, & po <lb/>natur extremitas ſtili albi, ut prius, in puncto z:</s> <s xml:id="echoid-s47316" xml:space="preserve"> & ponatur uiſus ſuper foramen ſuperius in puncto <lb/>k:</s> <s xml:id="echoid-s47317" xml:space="preserve"> tunc non uidebitur extremitas ſtili.</s> <s xml:id="echoid-s47318" xml:space="preserve"> Moueatur itaq;</s> <s xml:id="echoid-s47319" xml:space="preserve"> ſtilus in circum ferentia medij circuli ad par-<lb/>tem contrariam puncto e, nec tunc uidebitur extremitas ſtili:</s> <s xml:id="echoid-s47320" xml:space="preserve"> moueatur autem ad partem puncti e <lb/>paulatim, & uidebitur extremitas ſtili.</s> <s xml:id="echoid-s47321" xml:space="preserve"> Quòd ſi tunc punctum f, quod eſt centrum medij circuli, co-<lb/>operiatur aliquo corpuſculo:</s> <s xml:id="echoid-s47322" xml:space="preserve"> non uidebitur extremitas ſtili, ſed illo corpuſculo remoto, iterum ui-<lb/>debitur illa extremitas ſtili.</s> <s xml:id="echoid-s47323" xml:space="preserve"> Ex hoc itaq;</s> <s xml:id="echoid-s47324" xml:space="preserve"> patet, quòd formæ illius extremitatis ſtili comprehenſio, <lb/>quæ ſit a, eſt ſecundum reſractionem factam à centro uitri:</s> <s xml:id="echoid-s47325" xml:space="preserve"> & quòd forma refracta eſt in ſuperficie <lb/>circuli medij, quæ eſt perpendicularis ſuper ſuperficiem planam uitri:</s> <s xml:id="echoid-s47326" xml:space="preserve"> & inuenietur locus formæ <lb/>extremitatis ſtili, quæ eſt a, inter puncta e & z.</s> <s xml:id="echoid-s47327" xml:space="preserve"> Et quoniam reſractio fit à centro uitri, linea ducta à <lb/>centro uitri ad extremitatem uitri, quæ media eſt inter lineas f z & f e, & ſit a f:</s> <s xml:id="echoid-s47328" xml:space="preserve"> palàm quia eſt perpé-<lb/>dicularis ſuper conuexam ſuperficiem uitri, & peruenit eius forma ad uiſum per lineam k f, per cen <lb/>tra amborum foraminum tranſeuntem, quæ magis diſtat à linea perpendiculari ſuper ſuperficiem <lb/>planam uitri, quæ eſt linea f e æquidiſtans lineę q r, quàm linea, per quam incidit ipſi uitro forma pũ <lb/>ctia.</s> <s xml:id="echoid-s47329" xml:space="preserve"> cum itaque forma puncti a inciderit uitro per lineam a f, & tranſiuerit per totum corpus uitri <lb/>perpendiculariter:</s> <s xml:id="echoid-s47330" xml:space="preserve"> quoniam ipſa linea q f, cum tranſeat centrum uitri, eſt perpendicularis ſuper ſu-<lb/>perficiem uitri:</s> <s xml:id="echoid-s47331" xml:space="preserve"> cumq́ue pertranſito corpore uitri peruenit ad aerem, cuius corpus eſt rarioris dia-<lb/>phanitatis, quàm ſit corpus uitri, & peruenit ad centrum uiſus:</s> <s xml:id="echoid-s47332" xml:space="preserve"> patet quòd eſt refracta à ſuo primo <lb/>progreſſu lineæ a f, & peruenit ad progreſſum lineę z f k.</s> <s xml:id="echoid-s47333" xml:space="preserve"> Et quoniam linea z f eſt remotior à perpẽ-<lb/>diculari, ducta à puncto refractionis ſuper planam ſuperficiem uitri, quæ eſt linea e f, quàm ſit linea <lb/>a f:</s> <s xml:id="echoid-s47334" xml:space="preserve"> quoniam punctum a cadit in ſuperficie medij circuli inter puncta e & z:</s> <s xml:id="echoid-s47335" xml:space="preserve"> patet quòd hęc refractio <lb/>erit ad partem contrariam perpendicularis e f, ductæ à puncto refractionis ſuper ſuperficiem aeris <lb/>contingentis planam ſuperficiem uitri.</s> <s xml:id="echoid-s47336" xml:space="preserve"> Nam linea f z pertranſiens centra amborum ſoraminum, <lb/>magis diſtat ab illa perpendiculari e f, quàm linea exiens ab extremitate ſtili ad centrum uitri, quæ <lb/>eſt a f, quę producta in continuum & directum caderetinter perpendicularem e f productam & in-<lb/>ter lineam ſ k.</s> <s xml:id="echoid-s47337" xml:space="preserve"> Quia itaque peruenit ad punctum k, quoniam in illo uidetur:</s> <s xml:id="echoid-s47338" xml:space="preserve"> palàm quia fit refractio <lb/>ad partem contrariam ipſius perpendicularis, quæ eſt e f.</s> <s xml:id="echoid-s47339" xml:space="preserve"> Et quoniam hæc forma refringitur ex ui-<lb/>tro ad aerem, qui ſubtilior eſt uitro:</s> <s xml:id="echoid-s47340" xml:space="preserve"> patet quòd ſimili modo fit refractio ab aqua ad aerem:</s> <s xml:id="echoid-s47341" xml:space="preserve"> quoniá <lb/>etiam aer eſt ſubtilior quàm aqua.</s> <s xml:id="echoid-s47342" xml:space="preserve"> Quòd ſi conuexum uitri ponatur ex parte ſecunda foraminum:</s> <s xml:id="echoid-s47343" xml:space="preserve"> <lb/>& communis differentia ſuarum planarum ſuperficierum ponatur ſuper lineam q u:</s> <s xml:id="echoid-s47344" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s47345" xml:space="preserve"> medium <lb/>punctum illius communis differentiæ ſuper centrum laminæ, quod eſt q:</s> <s xml:id="echoid-s47346" xml:space="preserve"> palàm quia linea k f erit <lb/>obliqua ſuper planam uitri ſuperficiem, & perpendicularis ſuper eius ſuperficiem conuexam:</s> <s xml:id="echoid-s47347" xml:space="preserve"> e-<lb/>ritq́;</s> <s xml:id="echoid-s47348" xml:space="preserve"> linea r q perpendicularis ſuper planam ſuperficiem uitri:</s> <s xml:id="echoid-s47349" xml:space="preserve"> quoniam eſt perpendicularis ſuper <lb/>lineam u q:</s> <s xml:id="echoid-s47350" xml:space="preserve"> & erit linea e f perpendicularis ſuper conuexam ſuperficiem uitri per 72 th.</s> <s xml:id="echoid-s47351" xml:space="preserve"> 1 huius, & <lb/>ſuper eius planam ſuperficiem per 8 p 11:</s> <s xml:id="echoid-s47352" xml:space="preserve"> quoniam lineæ e f & r q æquidiſtant.</s> <s xml:id="echoid-s47353" xml:space="preserve"> Ponatur quoque ex-<lb/>tremitas ſtili albi, quæ ſit a, ſuper punctum z, ut prius:</s> <s xml:id="echoid-s47354" xml:space="preserve"> ſtatuaturq́ue uiſus ſuper ſuperius foramẽ in-<lb/>ſtrumenti in puncto k:</s> <s xml:id="echoid-s47355" xml:space="preserve"> & tunc non uidebitur extremitas ſtili, quæ eſt a.</s> <s xml:id="echoid-s47356" xml:space="preserve"> Moueatur itaque ſtilus ad <lb/>partem puncti e per circumferentiam medij circuli:</s> <s xml:id="echoid-s47357" xml:space="preserve"> & tunc etiá non uidebitur extremitas ſtili.</s> <s xml:id="echoid-s47358" xml:space="preserve"> De-<lb/>inde moueatur ad partem contrariam puncti e:</s> <s xml:id="echoid-s47359" xml:space="preserve"> & tunc uidebitur extremitas ſtili:</s> <s xml:id="echoid-s47360" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s47361" xml:space="preserve"> linea f z <lb/>inter lineam a f rectam, exeuntem ab extremitate ſtili ad centrum uitri, ſecundum quam extẽditur <lb/>illa forma puncti a, & inter perpendicularem f e:</s> <s xml:id="echoid-s47362" xml:space="preserve"> reſringeturq́ue forma puncti a extremitatis ſtili à <lb/>centro uitri ad uiſum per lineam f k tranſeuntem centra amborum foraminum:</s> <s xml:id="echoid-s47363" xml:space="preserve"> propterea quòd li-<lb/>nea a f obliquè incidit ſuperficiei uitri planæ, à qua fit refractio.</s> <s xml:id="echoid-s47364" xml:space="preserve"> Erit quoq;</s> <s xml:id="echoid-s47365" xml:space="preserve"> illa refractio ad partem <lb/>perpendicularis lineę, ſcilicet f e exeuntis à loco refractionis ſuper planam ſuperficiem uitri:</s> <s xml:id="echoid-s47366" xml:space="preserve"> & hęc <lb/>forma exit ab aere, & refringitur in uitro, quod eſt groſsius aere.</s> <s xml:id="echoid-s47367" xml:space="preserve"> Formę itaque, quæ refringuntur à <lb/>groſsiori corpore ad ſubtilius, declinant ad partem contrariam illi parti, in qua eſt perpendicularis, <lb/>exiens à loco refractionis ſuper ſuperficiem corporis diaphani, à qua fit refractio:</s> <s xml:id="echoid-s47368" xml:space="preserve"> & formę reflexæ <lb/>à corpore ſubtiliore ad groſsius, declinant ad partem, in qua eſt perpendicularis producta.</s> <s xml:id="echoid-s47369" xml:space="preserve"> Et hoc <lb/>eſt propoſitum.</s> <s xml:id="echoid-s47370" xml:space="preserve"/> </p> <div xml:id="echoid-div1799" type="float" level="0" n="0"> <figure xlink:label="fig-0709-01" xlink:href="fig-0709-01a"> <variables xml:id="echoid-variables819" xml:space="preserve">k b d o f q u g z r e a</variables> </figure> </div> </div> <div xml:id="echoid-div1801" type="section" level="0" n="0"> <head xml:id="echoid-head1333" xml:space="preserve" style="it">5. Quantitates angulorum refractionis ex aere ad aquam experimẽtaliter declarare. Al-<lb/>hazen 10 n 7.</head> <p> <s xml:id="echoid-s47371" xml:space="preserve">Differentia angulorum refractionis eſt ſecundum quantitates angulorum incidentiæ contento <lb/>rum ſub linea incidentiæ uel extenſionis radij in primo corpore, & ſub perpendiculari exeunte à <lb/>puncto refractionis ſuper ſuperficiem corporis ſecundi.</s> <s xml:id="echoid-s47372" xml:space="preserve"> Anguli enim refractionũ creſcũt & decre-<lb/>ſcunt ſecundum diſpoſitiones illorũ angulorũ incidẽtiæ in corporib.</s> <s xml:id="echoid-s47373" xml:space="preserve"> & ſitib.</s> <s xml:id="echoid-s47374" xml:space="preserve"> diuerſis.</s> <s xml:id="echoid-s47375" xml:space="preserve"> Et quia à cor <lb/>pore ſubtilioris diaphani ad corpus groſsius fit refractio ad perpendicularẽ productá à pũcto re fra <lb/>ctionis ſuper ſuperficiẽ ſecũdi corporis:</s> <s xml:id="echoid-s47376" xml:space="preserve"> & à corpore groſsioris diaphani ad ſubtilius fit refractio ad <lb/> <pb o="409" file="0711" n="711" rhead="LIBER DECIMVS."/> partem contrariam perpendicularis ſic ductæ, ut patuit per præmiſſam:</s> <s xml:id="echoid-s47377" xml:space="preserve"> tunc patet quia differunt <lb/>etiam illi anguli ſecundum diuerſitatẽ diaphanitatis ſecundi corporis.</s> <s xml:id="echoid-s47378" xml:space="preserve"> Et ut hæc differentia angu-<lb/>lorũ experimentaliter probetur:</s> <s xml:id="echoid-s47379" xml:space="preserve"> diuidatur à circulo medio, qui eſt in peripheria inſtrumenti ex par <lb/>te centri foraminis, quod eſt in circumferentia inſtrumenti circa punctum k, arcus 10.</s> <s xml:id="echoid-s47380" xml:space="preserve"> partiũ ex illis <lb/>partibus, quibus tota peripheria medij circuli diuiſa eſt in 360 partes:</s> <s xml:id="echoid-s47381" xml:space="preserve"> quì arcus ſit k n:</s> <s xml:id="echoid-s47382" xml:space="preserve"> & à puncto n <lb/>ducatur in ora inſtrumenti linea perpendicularis ſuper ſuperficiem laminæ:</s> <s xml:id="echoid-s47383" xml:space="preserve"> quę ſit n l:</s> <s xml:id="echoid-s47384" xml:space="preserve"> cadatq́;</s> <s xml:id="echoid-s47385" xml:space="preserve"> pun <lb/>ctus l in ſuperficie laminæ:</s> <s xml:id="echoid-s47386" xml:space="preserve"> ducatur quoq;</s> <s xml:id="echoid-s47387" xml:space="preserve"> ab hoc pũcto l ad centrum laminæ inſtrumenti, quod eſt <lb/>q, linea l q:</s> <s xml:id="echoid-s47388" xml:space="preserve"> & à centro medij circuli, quod eſt f, ducatur linea ad punctum n:</s> <s xml:id="echoid-s47389" xml:space="preserve"> quæ ſit f n, ſitq́;</s> <s xml:id="echoid-s47390" xml:space="preserve"> diame-<lb/>ter medij circuli ducta à puncto k per centrum f linea k f z, tranſiens per centra amborum forami-<lb/>num, quæ ſunt k & y, & per centrum medij circuli.</s> <s xml:id="echoid-s47391" xml:space="preserve"> Deinde in circumferentia medij circuli à puncto <lb/>n ſeparetur arcus 90 partium, ſequens arcum k n:</s> <s xml:id="echoid-s47392" xml:space="preserve"> <lb/>qui ſit arcus n s:</s> <s xml:id="echoid-s47393" xml:space="preserve"> & à centro medij circuli, quod eſt f, <lb/> <anchor type="figure" xlink:label="fig-0711-01a" xlink:href="fig-0711-01"/> ad punctum s ducatur linea, quæ ſit f s:</s> <s xml:id="echoid-s47394" xml:space="preserve"> quæ erit per-<lb/>pendicularis ſuper lineam f n per 33 p 6:</s> <s xml:id="echoid-s47395" xml:space="preserve"> ideo quia il-<lb/>læ duæ lineæ continent quartam partem circuli:</s> <s xml:id="echoid-s47396" xml:space="preserve"> re-<lb/>manebitq́;</s> <s xml:id="echoid-s47397" xml:space="preserve"> arcus reſiduus ex medio circulo, qui eſt <lb/>s z partes 80.</s> <s xml:id="echoid-s47398" xml:space="preserve"> Deinde ponatur inſtrumentum in ua-<lb/>ſe:</s> <s xml:id="echoid-s47399" xml:space="preserve"> & ſituetur uas æquidiſtanter horizonti:</s> <s xml:id="echoid-s47400" xml:space="preserve"> & infun-<lb/>datur aqua clara uſq;</s> <s xml:id="echoid-s47401" xml:space="preserve"> ad punctum q centrum lami-<lb/>næ:</s> <s xml:id="echoid-s47402" xml:space="preserve"> & in ortu ſolis in mane moueatur inſtrumentũ, <lb/>donec linea l q contingat ſuperficiem aquæ.</s> <s xml:id="echoid-s47403" xml:space="preserve"> In hoc <lb/>ergo ſitu diameter medij circuli, quæ eſt æquidiſtás <lb/>lineæ l q, ſignatæ in ſuperficie laminæ ſimiliter con-<lb/>tinget ſuperficiem aquæ:</s> <s xml:id="echoid-s47404" xml:space="preserve"> locus enim iſtarum dua-<lb/>rum linearum non differt in reſpectu ſuperficiei a-<lb/>quæ, quo ad ſenſum:</s> <s xml:id="echoid-s47405" xml:space="preserve"> & linea n f continet cum linea <lb/>f s angulum rectum, ut ſuprà patuit:</s> <s xml:id="echoid-s47406" xml:space="preserve"> eſt ergo linea f <lb/>s perpendicularis ſuper ſuperficiẽ aquæ:</s> <s xml:id="echoid-s47407" xml:space="preserve"> & ſemidia-<lb/>meter f z continet cum linea f s angulum, cuius quantitas per 33 p 6 eſt 80 partium:</s> <s xml:id="echoid-s47408" xml:space="preserve"> quoniá illi an-<lb/>gulo ſubtenditur arcus partium 80:</s> <s xml:id="echoid-s47409" xml:space="preserve"> qui eſt arcus s z:</s> <s xml:id="echoid-s47410" xml:space="preserve"> arcus uerò interiacens puncta k & n, ſub-<lb/>tendit angulum declinationis puncti k à puncto n, & à ſuperficie ipſius aquæ.</s> <s xml:id="echoid-s47411" xml:space="preserve"> Deinde mutetur in-<lb/>ſtrumentum in præmiſſo modo diſpoſitum cum toto uaſe, donec eleuato ſole ſuper horizonta ſe-<lb/>cundum altitudinem arcus k n, lux tranſeat per duo foramina:</s> <s xml:id="echoid-s47412" xml:space="preserve"> & ſignetur centrum lucis in ora in-<lb/>ſtrumenti, quæ eſt intra aquam:</s> <s xml:id="echoid-s47413" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s47414" xml:space="preserve"> ſuper centrum lucis ſignum aliquod per aliquam puncturá:</s> <s xml:id="echoid-s47415" xml:space="preserve"> <lb/>eritq́;</s> <s xml:id="echoid-s47416" xml:space="preserve"> ſignum illud, quod ſit h, in circumferentia medij circuli.</s> <s xml:id="echoid-s47417" xml:space="preserve"> Auferatur itaq;</s> <s xml:id="echoid-s47418" xml:space="preserve"> inſtrumentum, & re-<lb/>ſpiciatur punctum h:</s> <s xml:id="echoid-s47419" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s47420" xml:space="preserve"> ipſum inter punctum z, quod eſt extremitas diametri medij circuli, <lb/>tranſeuntis per centra duorum foraminum, & inter punctum s, quod eſt extremitas perpendicula-<lb/>ris, exeuntis à centro medij circuli erectæ ſuper ſuperficiem aquę, ut patet per præmiſſam.</s> <s xml:id="echoid-s47421" xml:space="preserve"> Patet er-<lb/>go tunc quod angulus refractionis eſt ille, quem ſubtendit arcus z h interiacẽs punctum h, & pun-<lb/>ctũ z:</s> <s xml:id="echoid-s47422" xml:space="preserve"> & ex numero partiũ huius arcus patebit quátitas anguli refracti & anguli refractionis:</s> <s xml:id="echoid-s47423" xml:space="preserve"> & pro-<lb/>portio anguli refractionis ad 80 partes, quæ ſunt tunc quantitas anguli incidentiæ.</s> <s xml:id="echoid-s47424" xml:space="preserve"> Deinde ſigne-<lb/>tur in circumferentia medij circuli arcus k m, pertranſiens punctum n:</s> <s xml:id="echoid-s47425" xml:space="preserve"> qui ſit partium 20:</s> <s xml:id="echoid-s47426" xml:space="preserve"> & duca-<lb/>tur linea m p in ora inſtrumenti perpendiculariter ſuper ſuperficiem laminę:</s> <s xml:id="echoid-s47427" xml:space="preserve"> & ducatur linea p q in <lb/>ſuperficie laminæ ad centrum q:</s> <s xml:id="echoid-s47428" xml:space="preserve"> & ab arcu m z reſecetur arcus m t partium 90:</s> <s xml:id="echoid-s47429" xml:space="preserve"> & ducatur linea t f à <lb/>puncto t ad centrum circuli medij, quod eſt f:</s> <s xml:id="echoid-s47430" xml:space="preserve"> relinquetur ergo arcus t z partium 70.</s> <s xml:id="echoid-s47431" xml:space="preserve"> Deinde pona <lb/>tur inſtrumẽtum in uas, & reuoluatur quouſq;</s> <s xml:id="echoid-s47432" xml:space="preserve"> linea p q tangat ſuperficiem aquæ:</s> <s xml:id="echoid-s47433" xml:space="preserve"> erit ergo linea t q <lb/>perpendicularis ſuper ſuperficiem aquæ:</s> <s xml:id="echoid-s47434" xml:space="preserve"> & linea k f z tranſiens per centra amborum foraminum <lb/>continet cum linea t f angulum 70 partium.</s> <s xml:id="echoid-s47435" xml:space="preserve"> Deinde conſideretur altitudo ſolis, & moueatur inſtru <lb/>mentum, quouſq;</s> <s xml:id="echoid-s47436" xml:space="preserve"> lux tranſeat per ambo foramina:</s> <s xml:id="echoid-s47437" xml:space="preserve"> & ſignetur ſuper centrum lucis cadentis intra a-<lb/>quam ſignum u.</s> <s xml:id="echoid-s47438" xml:space="preserve"> Deinde conſideretur arcus u z.</s> <s xml:id="echoid-s47439" xml:space="preserve"> Et quia ipſe ſubtenditur angulo refractionis:</s> <s xml:id="echoid-s47440" xml:space="preserve"> patet <lb/>quantitas illius anguli per computationem partium arcus:</s> <s xml:id="echoid-s47441" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s47442" xml:space="preserve"> nota proportio anguli z f u ad an-<lb/>gulum incidentię, qui eſt z f t, quem continet diameter tranſiens per centra amborum ſoraminum, <lb/>cum perpendiculari f t:</s> <s xml:id="echoid-s47443" xml:space="preserve"> qui angulus incidentiæ eſt partium 70.</s> <s xml:id="echoid-s47444" xml:space="preserve"> Similiterq́;</s> <s xml:id="echoid-s47445" xml:space="preserve"> procedatur ſignando ar <lb/>cum k x, qui ſit partium 30:</s> <s xml:id="echoid-s47446" xml:space="preserve"> & eſt eadem experimentatio.</s> <s xml:id="echoid-s47447" xml:space="preserve"> Deinde ſumatur arcus partium 40:</s> <s xml:id="echoid-s47448" xml:space="preserve"> dein-<lb/>de 50:</s> <s xml:id="echoid-s47449" xml:space="preserve"> deinde 60:</s> <s xml:id="echoid-s47450" xml:space="preserve"> deinde 70:</s> <s xml:id="echoid-s47451" xml:space="preserve"> deinde 80:</s> <s xml:id="echoid-s47452" xml:space="preserve"> & ſemper per computationem partium arcus circuli medij <lb/>interiacentis punctum z, & centrum lucis, erunt anguli refractionis noti:</s> <s xml:id="echoid-s47453" xml:space="preserve"> & ipſorum proportio ad <lb/>angulos incidentiæ contentos ſub perpendicularibus & diametris tranſeuntibus centra foraminũ <lb/>ſemper erit nota.</s> <s xml:id="echoid-s47454" xml:space="preserve"> Non ſolùm autem per 10, ſed etiam per alios quoſcunq;</s> <s xml:id="echoid-s47455" xml:space="preserve"> numeros integros uel fra <lb/>ctos præmiſſa arcuũ diuiſio poteſt procedere:</s> <s xml:id="echoid-s47456" xml:space="preserve"> quia ſemper eſt idem modus declarandi.</s> <s xml:id="echoid-s47457" xml:space="preserve"> Et, ut ſum-<lb/>mariè horum angulorum quátitates & proportiones perſtringamus.</s> <s xml:id="echoid-s47458" xml:space="preserve"> Quandocunq;</s> <s xml:id="echoid-s47459" xml:space="preserve"> alicuius radij <lb/>tranſeuntis per corpus aeris ſuæ debitę diſpoſitionis exiſtens fuerit in ſuperficie aquæ facta refra-<lb/>ctio:</s> <s xml:id="echoid-s47460" xml:space="preserve"> fueritq́;</s> <s xml:id="echoid-s47461" xml:space="preserve"> aqua ſuæ propriæ diſpoſitionis in diaphanitate cópetenti formę aquę, ſi angulus inci-<lb/>dentię cótentus in centro f ſub ſemidiametro k f, & linea radij incidentis fuerit 10 partiũ:</s> <s xml:id="echoid-s47462" xml:space="preserve"> erit angu <lb/>lus cótentus in centro f ſub ſemidiametro f z & ſub linea radiali refracta quaſi 2 partiũ & 5 minuto-<lb/> <pb o="410" file="0712" n="712" rhead="VITELLONIS OPTICAE"/> rum:</s> <s xml:id="echoid-s47463" xml:space="preserve"> & ſic conſequẽter ſecundũ formã tabulæ, quam inferius ſubiungemus.</s> <s xml:id="echoid-s47464" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s47465" xml:space="preserve"/> </p> <div xml:id="echoid-div1801" type="float" level="0" n="0"> <figure xlink:label="fig-0711-01" xlink:href="fig-0711-01a"> <variables xml:id="echoid-variables820" xml:space="preserve">x m n k b l p s t f h z u</variables> </figure> </div> </div> <div xml:id="echoid-div1803" type="section" level="0" n="0"> <head xml:id="echoid-head1334" xml:space="preserve" style="it">6. Quantitates angulorum refractionis ex aere uel aqua ad uιtrum planum uel cõuexum, <lb/>& econuerſo experimentaliter declarare. Alhazen 11 n 7.</head> <p> <s xml:id="echoid-s47466" xml:space="preserve">Diuidatur arcus medij circuli inſtrumenti modo illo, ut in præmiſſa:</s> <s xml:id="echoid-s47467" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s47468" xml:space="preserve"> arcus k n 10 partium:</s> <s xml:id="echoid-s47469" xml:space="preserve"> <lb/>& ducatur linea n l perpendicularis ſuper ſuperficiẽ laminæ:</s> <s xml:id="echoid-s47470" xml:space="preserve"> copuletur quoq;</s> <s xml:id="echoid-s47471" xml:space="preserve"> linea l q:</s> <s xml:id="echoid-s47472" xml:space="preserve"> & ſuperpo-<lb/>natur uitrum formatum cubicè ſuperficiei ipſius tabulæ, ita ut cómunis ſectio duarum ſuperficie-<lb/>rum planarum, quæ eſt linea recta (ut patet per 3 p 11) ſuperponatur lineæ l q, taliter ut ſecundum <lb/>ſui punctum medium ſuperponatur lineæ ſignatæ in ſuperficie tabulæ perpẽdiculari ſuper lineam <lb/>l q, quæ eſt æ quidiſtans lineæ s f ductæ in ſuperficie medij circuli:</s> <s xml:id="echoid-s47473" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s47474" xml:space="preserve"> medium pũctum illius lineæ <lb/>uitri ſuper punctum q centrum laminæ:</s> <s xml:id="echoid-s47475" xml:space="preserve"> ponaturq́;</s> <s xml:id="echoid-s47476" xml:space="preserve"> ſuperficies uitri plana ex parte foraminum:</s> <s xml:id="echoid-s47477" xml:space="preserve"> & <lb/>applicetur benè uitrum laminæ:</s> <s xml:id="echoid-s47478" xml:space="preserve"> & inſtrumẽtum poſitum in uaſe moueatur, donec lux tranſeat per <lb/>ambo ſoramina:</s> <s xml:id="echoid-s47479" xml:space="preserve"> ſigneturq́;</s> <s xml:id="echoid-s47480" xml:space="preserve"> ſuper centrum lucis ſignum:</s> <s xml:id="echoid-s47481" xml:space="preserve"> & conſiderentur quantitates angulorum <lb/>refractionis ex aere ad uitrum per quantitates arcuum, ut in præcedente.</s> <s xml:id="echoid-s47482" xml:space="preserve"> Quòd ſi aliquis perſcru-<lb/>tari uoluerit angulos refractionis ex uitro ad aerem <lb/>uel aquam:</s> <s xml:id="echoid-s47483" xml:space="preserve"> accipiat uitrum, quod eſt pars ſphæræ, ut <lb/> <anchor type="figure" xlink:label="fig-0712-01a" xlink:href="fig-0712-01"/> ipſi ſuperius uſi ſumus in propoſitionibus 2 libri hu-<lb/>ius ſcientiæ, & in 4 th.</s> <s xml:id="echoid-s47484" xml:space="preserve"> huius:</s> <s xml:id="echoid-s47485" xml:space="preserve"> & ponatur conuexum <lb/>uitri ex parte centrorum duorum foraminum:</s> <s xml:id="echoid-s47486" xml:space="preserve"> pona-<lb/>turq́;</s> <s xml:id="echoid-s47487" xml:space="preserve"> medium lineæ, quæ eſt differentia communis <lb/>ſuperficierum planarum, ſuper centrum laminæ, ita <lb/>quòd illa communis differentia ſit ſuper lineam l q.</s> <s xml:id="echoid-s47488" xml:space="preserve"> <lb/>Tunc ergo lux, quæ tranſit centra duorum ſorami-<lb/>num, peruenit rectè ad centrum uitri, & reflectitur <lb/>apud illud de uitro ad aerem:</s> <s xml:id="echoid-s47489" xml:space="preserve"> diuidanturq́;</s> <s xml:id="echoid-s47490" xml:space="preserve"> poſtmo-<lb/>dum arcus ſucceſsiuè, ut in præmiſſa, & mutetur ui-<lb/>tri poſitio, ita ut illa communis planarum ſuperficie-<lb/>rum ipſius uitri ſectio ſit ſuper lineã p q:</s> <s xml:id="echoid-s47491" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s47492" xml:space="preserve"> iterum <lb/>medius punctus illius lineæ uitri ſuper punctum q <lb/>centrum laminæ:</s> <s xml:id="echoid-s47493" xml:space="preserve"> & ſic factis ulterioribus diuiſioni-<lb/>bus circuli medij, ductisq́;</s> <s xml:id="echoid-s47494" xml:space="preserve"> lineis, ut prius, & mutato <lb/>uitro ſecundum illas:</s> <s xml:id="echoid-s47495" xml:space="preserve"> habebuntur anguli refractio-<lb/>num particulares, & ipſorum proportio ad angulum incidentiæ, quem continet diameter pertran-<lb/>ſiens centra foraminum cum perpẽdiculari producta à loco refractionis ſuper ſuperficiem planam <lb/>ipſam ſuperficiem uitri conuexam contingentem.</s> <s xml:id="echoid-s47496" xml:space="preserve"> In his enim diſpoſitionibus uitri, reſpectu lami-<lb/>næ inſtrumenti, ſemper erit cêtrum uitreæ ſphæræ in puncto f:</s> <s xml:id="echoid-s47497" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s47498" xml:space="preserve"> per 72 th.</s> <s xml:id="echoid-s47499" xml:space="preserve"> 1 huius linea s f ſimi-<lb/>lis illi, perpendicularis ſuper ſuperficiem conuexam uitri, & ſuper ſuperficiem planam ipſius, à cu-<lb/>ius punctorum aliquo ſit refractio:</s> <s xml:id="echoid-s47500" xml:space="preserve"> quoniam quælibet illarum linearum eſt perpendicularis ſuper <lb/>lineas æquidiſtátes lineis l q & p q, & ſimilibus illis quibuſcunq;</s> <s xml:id="echoid-s47501" xml:space="preserve">. Scieturq́;</s> <s xml:id="echoid-s47502" xml:space="preserve">, ut prius, reiterata ope-<lb/>ratione cum extremitate ſtipitis totius refractionis modus, & anguli refractionis à uitro ad centrú <lb/>uiſus exiſtens in puncto k centro foraminis ſuperioris.</s> <s xml:id="echoid-s47503" xml:space="preserve"> Et in his duobus ſitibus, cum refractio fit ab <lb/>aere ad uitrum, uel à uitro ad aerem, ſemper inuenientur quantitates angulorũ refractionis de aere <lb/>ad uitrum, & de uitro ad aerem æquales:</s> <s xml:id="echoid-s47504" xml:space="preserve"> quando angulus contentus à linea, per quam extenditur <lb/>lux ad locum reſractionis, & à linea perpendiculari ducta à puncto refractionis, cum fit refractio de <lb/>aere ad uitrum, æqualis fuerit angulo contento à linea, per quam extenditur lux, & à perpendicu-<lb/>lari ducta à loco refractionis, cum refringitur de uitro ad aerem, ut patet inſtrumentaliter operan-<lb/>ti.</s> <s xml:id="echoid-s47505" xml:space="preserve"> Si uerò uoluerit aliquis experiri quantitates angulorum refractionis à conuexo uitri ad aerem:</s> <s xml:id="echoid-s47506" xml:space="preserve"> <lb/>diuidat, ut prius, de circumferentia medij circuli ex parte puncti k centri ſoraminis, quod eſt in ora <lb/>inſtrumenti, arcum 10 partium, qui ſit k n:</s> <s xml:id="echoid-s47507" xml:space="preserve"> & ducantur, ut prius, linea n l, & linea l q:</s> <s xml:id="echoid-s47508" xml:space="preserve"> & à linea l q, quę <lb/>eſt ſemidiameter laminæ ex parte centri q, abſcindatur linea æqualis ſemidiametro ſphæræ ipſius <lb/>uitri, quæ ſit q o:</s> <s xml:id="echoid-s47509" xml:space="preserve"> & à puncto o ducatur perpendicularis ſuper diametrum laminæ b q g:</s> <s xml:id="echoid-s47510" xml:space="preserve"> quæ pro-<lb/>tracta ultra diametrum ſit o d, ſecans diametrum b q g in puncto d.</s> <s xml:id="echoid-s47511" xml:space="preserve"> Deinde ſuperponatur com mu-<lb/>nis ſectio planarum ſuperficierum uitri huic perpendiculari o d:</s> <s xml:id="echoid-s47512" xml:space="preserve"> ita quòd punctum medium illius <lb/>ſectionis ſit ſuper punctum o.</s> <s xml:id="echoid-s47513" xml:space="preserve"> Erit itaq;</s> <s xml:id="echoid-s47514" xml:space="preserve"> centrum uitri in ſuperficie medij circuli:</s> <s xml:id="echoid-s47515" xml:space="preserve"> & eiuſdem circuli <lb/>diameter, quæ eſt k f z, erit perpendicularis uper ſuperficiem uitri planam per 8 p 11:</s> <s xml:id="echoid-s47516" xml:space="preserve"> quoniam eſt <lb/>æquidiftans diametro laminæ b q g, quæ eſt perpendicularis ſuper illam ſuperficiem, & ſuper illam <lb/>differentiam communem illarum duarum planarum ſuperficierum uitri.</s> <s xml:id="echoid-s47517" xml:space="preserve"> Erit quoq;</s> <s xml:id="echoid-s47518" xml:space="preserve"> centrũ circuli <lb/>medij in ſuperficie cóuexa uitri:</s> <s xml:id="echoid-s47519" xml:space="preserve"> ideo quia linea f q exiens à centro medij circuli, quod eſt f, ad cen-<lb/>trum laminæ, quod eſt q, eſt æqualis lineæ productæ à centro uitri ad medium lineæ, quæ eſt diffe-<lb/>rentia communis ſuperficierum planarum uitri, ut patet ex his, quæ præmiſſa ſunt in figuratione <lb/>huius figuræ uitreæ in 45 th.</s> <s xml:id="echoid-s47520" xml:space="preserve"> 2 huius:</s> <s xml:id="echoid-s47521" xml:space="preserve"> & utraq;</s> <s xml:id="echoid-s47522" xml:space="preserve"> iſtarum linearum eſt perpendicularis ſuper ſuperfi-<lb/>ciem laminæ:</s> <s xml:id="echoid-s47523" xml:space="preserve"> ergo per 25 th.</s> <s xml:id="echoid-s47524" xml:space="preserve"> 1 huius, illæ duæ lineæ ſunt æquales & æquidiſtantes:</s> <s xml:id="echoid-s47525" xml:space="preserve"> ergo per 33 p 1 li-<lb/>nea copulans centrum uitri, quod eſt in aliquo puncto planæ ſuperficiei ipſius uitri, cum centro <lb/>medij circuli, eſt æqualis lineæ q o copulanti centrum laminæ, quod eſt q, cum medio puncto dif-<lb/> <pb o="411" file="0713" n="713" rhead="LIBER DECIMVS."/> ferentiæ cõmunis duarum planarũ ſuperficierum ipſius uitri, quod eſt punctum o:</s> <s xml:id="echoid-s47526" xml:space="preserve"> ſed linea q o po-<lb/>ſita eſt æ qualis ſemidiametro uitri:</s> <s xml:id="echoid-s47527" xml:space="preserve"> ergo & linea æ quidiſtans ei eſt æqualis ſemidiametro uitri.</s> <s xml:id="echoid-s47528" xml:space="preserve"> Cen <lb/>trum ergo medij circuli eſt in conuexo uitri:</s> <s xml:id="echoid-s47529" xml:space="preserve"> linea ergo k f, quæ eſt ſemidiameter medij circuli, cum <lb/>nõ tranſeat centrũ ſphæræ uitreæ:</s> <s xml:id="echoid-s47530" xml:space="preserve"> patet quia eſt obliquè incidens ſuper eius conuexã ſuperficiem:</s> <s xml:id="echoid-s47531" xml:space="preserve"> <lb/>ergo per 47 th.</s> <s xml:id="echoid-s47532" xml:space="preserve"> 2 huius, cum eadẽ diameter obliquè incidat ſuperficiei aeris contingentis, refringe-<lb/>tur ipſa à perpendiculari ducta à puncto refractiõ is ſuper ipſam ſuperficiem aeris.</s> <s xml:id="echoid-s47533" xml:space="preserve"> Imaginetur itaq;</s> <s xml:id="echoid-s47534" xml:space="preserve"> <lb/>ſemidiameter uitri produci ex utraq;</s> <s xml:id="echoid-s47535" xml:space="preserve"> parte ad circumſferentiã circuli medij, quę fiat linea n f u, ſe-<lb/>cans diamctrũ circuli medij, quæ eſt k f z in puncto f.</s> <s xml:id="echoid-s47536" xml:space="preserve"> Erit itaq;</s> <s xml:id="echoid-s47537" xml:space="preserve"> per 15 p 1 angulus k f n æ qualis an-<lb/>gulo z f u:</s> <s xml:id="echoid-s47538" xml:space="preserve"> & erit per 26 p 3 arcus u z æ qualis arcui k n, qui eſt poſitus eſſe 10 partium.</s> <s xml:id="echoid-s47539" xml:space="preserve"> Eſt ergo arcus <lb/>u z 10 partium notus:</s> <s xml:id="echoid-s47540" xml:space="preserve"> ergo & angulus u f z eſt notus.</s> <s xml:id="echoid-s47541" xml:space="preserve"> Intueatur itaq;</s> <s xml:id="echoid-s47542" xml:space="preserve"> aliquis centrum lucis refractæ:</s> <s xml:id="echoid-s47543" xml:space="preserve"> <lb/>& inuenietur remotius à puncto z, quod eſt extremitas lineæ tranſeuntis per centra duorum fora-<lb/>minum, quàm ſit punctum u, quod eſt extremitas lineæ tranſeuntis per centrum uitri ab eodem <lb/>puncto z, qui eſt extremitas diametri circuli medij.</s> <s xml:id="echoid-s47544" xml:space="preserve"> Hæc ergo refractio ſacta eſt ad partem contra-<lb/>riam diametri productæ à loco refractionis, quæ tranſit centrum uitri:</s> <s xml:id="echoid-s47545" xml:space="preserve"> & arcus medij circuli interia <lb/>cens punctum z & centrum lucis ſignatum, eſt quantitas anguli refractionis:</s> <s xml:id="echoid-s47546" xml:space="preserve"> angulus enim refra-<lb/>ctionis eſt apud centrũ circuli medij:</s> <s xml:id="echoid-s47547" xml:space="preserve"> quoniam, ut patuit per 44 th.</s> <s xml:id="echoid-s47548" xml:space="preserve"> 2 huius lux extenditur ſuper li-<lb/>neam tranſeuntem per centra duorum foraminum rectè, donec perueniat ad conuexum uitri.</s> <s xml:id="echoid-s47549" xml:space="preserve"> Et <lb/>cum eſt angulus incidentiæ 10 partium, fit angulus reſractus quaſi 13 partium, & angulus refractio-<lb/>nis quaſi partiũ triũ:</s> <s xml:id="echoid-s47550" xml:space="preserve"> factisq́;</s> <s xml:id="echoid-s47551" xml:space="preserve">, ut in præcedentibus, diuiſionibus arcuũ à puncto k:</s> <s xml:id="echoid-s47552" xml:space="preserve"> inuenietur diuer <lb/>ſitas angulorum refractionis per inſtrumentum.</s> <s xml:id="echoid-s47553" xml:space="preserve"> Et ſi infundatur aqua uaſi:</s> <s xml:id="echoid-s47554" xml:space="preserve"> tunc erit aqua loco ae-<lb/>ris:</s> <s xml:id="echoid-s47555" xml:space="preserve"> & præmiſſo modoinuenietur diuerſitas angulorum refractionis à uitro ad aquam:</s> <s xml:id="echoid-s47556" xml:space="preserve"> & differen-<lb/>tia ſecundum quod illi refractioni eſt propria:</s> <s xml:id="echoid-s47557" xml:space="preserve"> & quantitas angulorum refractorum & angulorum <lb/>refractionis, reſpectu eorum, qui ſunt in aere.</s> <s xml:id="echoid-s47558" xml:space="preserve"> Quòd ſi à puncto z ducere placuerit extremitatem ſti <lb/>li, ut prius:</s> <s xml:id="echoid-s47559" xml:space="preserve"> tunc ſecundum illud facta diſpoſitione ſitus uitri, occurret eadem quantitas angulorũ, <lb/>quæ prius.</s> <s xml:id="echoid-s47560" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s47561" xml:space="preserve"/> </p> <div xml:id="echoid-div1803" type="float" level="0" n="0"> <figure xlink:label="fig-0712-01" xlink:href="fig-0712-01a"> <variables xml:id="echoid-variables821" xml:space="preserve">k n m b l d p o q f g u z</variables> </figure> </div> </div> <div xml:id="echoid-div1805" type="section" level="0" n="0"> <head xml:id="echoid-head1335" xml:space="preserve" style="it">7. Zuantitates angulorum refractionis ex aere uel aqua ad uitrum concauum, uel econuer-<lb/>ſo experimentaliter inuenire. Alhazen 12 n 7.</head> <p> <s xml:id="echoid-s47562" xml:space="preserve">Accipiatur uitrum clarum mundum, æquidiſtantium ſuperficierum omnium:</s> <s xml:id="echoid-s47563" xml:space="preserve"> cuius longitudo <lb/>ſit maior in uno grano hordei, quàm diameter uitri ſphærici conuexi, quo ſuperius uſi ſumus:</s> <s xml:id="echoid-s47564" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s47565" xml:space="preserve"> <lb/>latitudo eius æqualis longitudini:</s> <s xml:id="echoid-s47566" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s47567" xml:space="preserve"> ſpiſsitudo eius dupla diametro foraminis, quod eſt in ora <lb/>inſtrumenti:</s> <s xml:id="echoid-s47568" xml:space="preserve"> & fiat in uno ſuorum laterum quadratorum concauitas rotunda ſemicolumnaris :</s> <s xml:id="echoid-s47569" xml:space="preserve"> ita <lb/>quòd ſemidiameter baſis columnæ cõcauę ſit in quantitate ſemidiametri uitri ſphærici:</s> <s xml:id="echoid-s47570" xml:space="preserve"> & ſint com <lb/>inunes ſectiones planarum ſuperficierum huius uitri lineæ rectiſsimæ.</s> <s xml:id="echoid-s47571" xml:space="preserve"> Poteſt autem hæc forma ui <lb/>tri ſic fieri per artificium, ita quòd fiat talis forma ex ære uel lapide, & uitrum liquefactum ſundatur <lb/>ſuper ipſam, & poliatur.</s> <s xml:id="echoid-s47572" xml:space="preserve"> Diuidatur itaq;</s> <s xml:id="echoid-s47573" xml:space="preserve"> à centro foraminis oræ inſtrumenti, quod eſt k, in circum <lb/>ferentia medij circuli arcus, cuius quantitas ſit illa, ſecundum quam quis uult experiri quantitates <lb/>angulorum, qui ſit arcus k n:</s> <s xml:id="echoid-s47574" xml:space="preserve"> & à puncto n ducatur <lb/>in ora inſtrumẽti linea n l perpendiculariter ſuper <lb/> <anchor type="figure" xlink:label="fig-0713-01a" xlink:href="fig-0713-01"/> ſuperficiem laminæ:</s> <s xml:id="echoid-s47575" xml:space="preserve"> & ducatur linea l q in ſu-<lb/>perficie laminæ ad centrum eius, quod eſt q:</s> <s xml:id="echoid-s47576" xml:space="preserve"> & à <lb/>ſemidiametro l q reſecetur ex parte centri q linea <lb/>q o, æ qualis ſemidiametro baſis concauitatis co-<lb/>lumnę:</s> <s xml:id="echoid-s47577" xml:space="preserve"> & à puncto o extrahatur per 12 p 1 perpen-<lb/>dicularis ſuper diametrum laminæ b q, & protra-<lb/>hatur in utramq;</s> <s xml:id="echoid-s47578" xml:space="preserve"> partem:</s> <s xml:id="echoid-s47579" xml:space="preserve"> & ſit o e, ſecans diame-<lb/>trum b q g in puncto e:</s> <s xml:id="echoid-s47580" xml:space="preserve"> & ſuperponatur uitrum la-<lb/>minæ, ita quòd dorſum concauitatis, hoc eſt ſu-<lb/>perficies plana cõcauitati ſuperpoſita ſit ex parte <lb/>duorum ſoraminum:</s> <s xml:id="echoid-s47581" xml:space="preserve"> & quòd concauitate reſpi-<lb/>ciente foramina, duę ſuperfluitates rectilineę, quę <lb/>ſuperfluunt ſuper diametrum columnæ, ſint dire-<lb/>ctæ & fixè ſuperpoſitæ iſti lineæ perpendiculari o <lb/>e:</s> <s xml:id="echoid-s47582" xml:space="preserve"> & præ ſeruetur hoc, ut diſtantiæ duarum extre-<lb/>mitatum diametri baſis concauitatis columnaris <lb/>diſtent æ qualiter à puncto o, à quo exeunt directè perpendiculares.</s> <s xml:id="echoid-s47583" xml:space="preserve"> Erit ergo tunc centrum baſis <lb/>concauitatis columnaris ſuper punctum o, à quo exiuit lines o e perpendicularis ſuper lineam q b, <lb/>& ſuper punctum, cuius diſtantia à centro laminę, quod eſt q, eſt æqualis ſemidiametro concauita-<lb/>tis columnaris.</s> <s xml:id="echoid-s47584" xml:space="preserve"> Secundum hanc ergo diſpoſitionem applicetur uitrum firmiter ſuperficiei laminę:</s> <s xml:id="echoid-s47585" xml:space="preserve"> <lb/>& erit ſuperficies medij circuli ſecans concauitatem columnarem & æquidiſtans baſi eius:</s> <s xml:id="echoid-s47586" xml:space="preserve"> quoniá <lb/>baſis eius in hac diſpoſitione eſt in ſuperficie laminæ inſtrumenti.</s> <s xml:id="echoid-s47587" xml:space="preserve"> Superficies ergo medij circuli <lb/>per 100 th.</s> <s xml:id="echoid-s47588" xml:space="preserve"> 1 huius ſecat ſuperficiem columnarem concauam ſecundum circulum, cuius ſemidiame <lb/>ter æquidiſtat ſemidiametro baſis concauitatis ipſius columnæ:</s> <s xml:id="echoid-s47589" xml:space="preserve"> & linea continuans centra iſtorũ <lb/>duorũ ſemicirculorũ, ſcilicet baſis, & alterius ſibi æquidiſtantis erit perpendicularis ſuper ſuperfi-<lb/>ciem laminę incidẽs ad punctũ o:</s> <s xml:id="echoid-s47590" xml:space="preserve"> quoniã ipſa per 25 th.</s> <s xml:id="echoid-s47591" xml:space="preserve"> 1 huius eſt æ qualis lineæ perpendiculari f q <lb/> <pb o="412" file="0714" n="714" rhead="VITELLONIS OPTICAE"/> cexunti à centro medij circuli, quod eſt f, ſuper centrũ laminæ, quod eſt q:</s> <s xml:id="echoid-s47592" xml:space="preserve"> ſed & linea o q eſt æqua <lb/>lis ſemidiametro baſis columnæ ex hypotheſi:</s> <s xml:id="echoid-s47593" xml:space="preserve"> ergo per 33 p 1 linea, quæ.</s> <s xml:id="echoid-s47594" xml:space="preserve"> exità centro medij circuli <lb/>(quod eſt f) ad centrum ſemicirculi, qui fit in ſuperficie columnæ concauæ æquidiſtans baſi, eſt æ-<lb/>qualis ſemidiametro baſis concauitatis concauæ columnæ.</s> <s xml:id="echoid-s47595" xml:space="preserve"> Centrum itaq;</s> <s xml:id="echoid-s47596" xml:space="preserve"> medij circuli, quod eſt <lb/>f, eſt in circumſerentia ſemicirculi in columna uitrea facti.</s> <s xml:id="echoid-s47597" xml:space="preserve"> Eſt ergo centrum f in concaua ſuperficie <lb/>columnæ.</s> <s xml:id="echoid-s47598" xml:space="preserve"> Et quia terminus planus uitri ſuperp onitur lineæ perpendiculari, productæ à puncto o <lb/>ſuper b q diametrum laminæ:</s> <s xml:id="echoid-s47599" xml:space="preserve"> palàm quia diameter laminæ, quæ eſt q b, eſt perpendicularis ſuper <lb/>planam uitri ſuperficiem:</s> <s xml:id="echoid-s47600" xml:space="preserve"> quia etiã planæ ſuperficies ſunt ſuper ſe inuicem perpendiculariter ere-<lb/>ctæ.</s> <s xml:id="echoid-s47601" xml:space="preserve"> Erit ergo linea k f z pertranſiẽs centra amborum foraminũ, perpendicularis ſuper ſuperficiem <lb/>planam, quæ eſt in parte conuexa uitri per 8 p 11:</s> <s xml:id="echoid-s47602" xml:space="preserve"> quia illa linea k f z eſt æ quidiſtans diametro lami-<lb/>næ b q g:</s> <s xml:id="echoid-s47603" xml:space="preserve"> quæ eſt perpendicularis ſuper illam ſuperficiem, ut patet ex præmiſsis:</s> <s xml:id="echoid-s47604" xml:space="preserve"> & hæc ſuperficies <lb/>plana uitri eſt ex parte foraminum.</s> <s xml:id="echoid-s47605" xml:space="preserve"> In hoc ergo ſitu lux, quæ extenditur per lineam tranſeuntẽ cen <lb/>tra duorum foraminum, extenditur in corpore uitri rectè, donec perueniat ad concauum uitri:</s> <s xml:id="echoid-s47606" xml:space="preserve"> & <lb/>tunc reflectitur apud concauam ſuperficiem uitri.</s> <s xml:id="echoid-s47607" xml:space="preserve"> Cum enim nõ tranſeat per centrũ circuli, qui eſt <lb/>in concaua ſuperficie uitri:</s> <s xml:id="echoid-s47608" xml:space="preserve"> patet per 72 th.</s> <s xml:id="echoid-s47609" xml:space="preserve"> 1 huius quoniã ipſa non eſt perpendicularis ſuper cõ ca-<lb/>uam ſuperficiem uitri:</s> <s xml:id="echoid-s47610" xml:space="preserve"> refringetur ergo in concaua ſuperficie uitri:</s> <s xml:id="echoid-s47611" xml:space="preserve"> & cõmunis ſectio illius lineæ & <lb/>concauitatis uitri eſt centrum circuli medij:</s> <s xml:id="echoid-s47612" xml:space="preserve"> & in hoc puncto fit refractio ex aere ad uitrum.</s> <s xml:id="echoid-s47613" xml:space="preserve"> Arcus <lb/>itaq;</s> <s xml:id="echoid-s47614" xml:space="preserve"> cadens inter centrum lucis & punctũ z, qui eſt terminus diametri, tranſeuntis per centra am-<lb/>borum foraminum, ſubtenditur angulo refractionis.</s> <s xml:id="echoid-s47615" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s47616" xml:space="preserve"> patet in cuiuslibet aliorũ ar-<lb/>cuum refractione à puncto k:</s> <s xml:id="echoid-s47617" xml:space="preserve"> & poteſt oſtendi quantitas omnium angulorum refractionis à con-<lb/>caua uitri ſuperficie.</s> <s xml:id="echoid-s47618" xml:space="preserve"> Quòd ſi uitrum ſic diſponatur, ut communi ſectione ſuarum planarum ſuper-<lb/>ficierum poſita ſuper lineam o e, conuexitas uitri reſpiciat centra foraminum:</s> <s xml:id="echoid-s47619" xml:space="preserve"> tunc quia linea k f z <lb/>pertranſiens uitrũ, peruenit ad concauũ uitri irreſracta, cum ſit perpendicularis ſuper planam ſu-<lb/>perficiem ipſius, obliqua uerò ſuper concauã eius ſuperficiem:</s> <s xml:id="echoid-s47620" xml:space="preserve"> ergo & ſuper cõuexam ſuperficiem <lb/>aeris contingentis uitrum:</s> <s xml:id="echoid-s47621" xml:space="preserve"> refringetur ergo àconcaua uitri ſuperficie:</s> <s xml:id="echoid-s47622" xml:space="preserve"> & hæc refractio eſt à conca-<lb/>uo uitri ad aerem:</s> <s xml:id="echoid-s47623" xml:space="preserve"> & anguli, qui ſiunt ex aere ad uitrum in concauo uitri, ſunt ijdem iſtis:</s> <s xml:id="echoid-s47624" xml:space="preserve"> quoniam <lb/>ſemper anguli refractionis à uitro ad aerem, & ab aere ad uitrum ſunt ijdem:</s> <s xml:id="echoid-s47625" xml:space="preserve"> cum angulus, quẽ con <lb/>tinet linea, per quam primò extenditur lux, & perpendicularis exiens à loco refractionis, ſit idem <lb/>angulus.</s> <s xml:id="echoid-s47626" xml:space="preserve"> Et eodem modo poſſunt ſciri anguli refractionis de aqua ad uitrum & de uitro ad aquam <lb/>in ſuperficie uitri cõcaua, uel in ſuperficie alia quacunq;</s> <s xml:id="echoid-s47627" xml:space="preserve">. Quòd ſi extremitas ſtili ducatur à puncto <lb/>z in peripheria medij circuli, ut prius:</s> <s xml:id="echoid-s47628" xml:space="preserve"> tunc facta diſpoſitione ſitus uitri ſecundũ exigentiã illius re <lb/>fractionis, occurret notitia angulorũ huius refractiõ is ad uiſum, ſicut prius.</s> <s xml:id="echoid-s47629" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s47630" xml:space="preserve"/> </p> <div xml:id="echoid-div1805" type="float" level="0" n="0"> <figure xlink:label="fig-0713-01" xlink:href="fig-0713-01a"> <variables xml:id="echoid-variables822" xml:space="preserve">k n l b o e q f g u z</variables> </figure> </div> </div> <div xml:id="echoid-div1807" type="section" level="0" n="0"> <head xml:id="echoid-head1336" xml:space="preserve" style="it">8. Anguli omnium refractionum per tabulas declar antur. Alhazen 12 n 7.</head> <p> <s xml:id="echoid-s47631" xml:space="preserve">Acceptis inſtrumẽtaliter, prout potuimus propinquius, angulis omnium refractionũ à quibuſ-<lb/>cunq;</s> <s xml:id="echoid-s47632" xml:space="preserve"> diaphanis notis adinuicem (ut ab aere ad aquam & uitrum, & ab aqua ad uitrum:</s> <s xml:id="echoid-s47633" xml:space="preserve"> & econuer <lb/>ſo ab aqua & uitro ad aerem, & à uitro ad aquam) inuenimus quòd ſemper ijdem ſunt anguli refra-<lb/>ctionum à quocunq;</s> <s xml:id="echoid-s47634" xml:space="preserve"> raro diaphano ad diaphanum denſius illo, & ab eodem denſo ad idem rarum:</s> <s xml:id="echoid-s47635" xml:space="preserve"> <lb/>ſecundum hoc ſecimus has tabulas, quarum hæc eſt forma.</s> <s xml:id="echoid-s47636" xml:space="preserve"> Et præmittimus angulos incidentiæ <anchor type="table" xlink:label="tab-0708-01a" xlink:href="tab-0708-01"/> <anchor type="table" xlink:label="tab-0708-02a" xlink:href="tab-0708-02"/><lb/> <pb o="413" file="0715" n="715" rhead="LIBER DECIMVS."/> in primis:</s> <s xml:id="echoid-s47637" xml:space="preserve"> deinde alios angulos ſubiungimus ſecundum modos ſuorum circulorũ, quos præmitti-<lb/>mus in capitibus ſuarum linearum.</s> <s xml:id="echoid-s47638" xml:space="preserve"> Poteſt itaq;</s> <s xml:id="echoid-s47639" xml:space="preserve"> ſecundum has tabulas experimentaliter inuen-<lb/>tas per inſtrumentum præmiſſum, diligens inquiſitor ſcire omnes angulos refractionum à medijs <lb/>diuerſæ diaphanitatis quibuſcunq;</s> <s xml:id="echoid-s47640" xml:space="preserve">. Et patet ex eis quoniam anguli incidentię formæ eiuſdem pun <lb/>cti propinquiores radio, à puncto rei uiſæ ſuperficiei corporis diaphani (à qua fit refractio) perpen <lb/>diculariter incidenti, ſunt minores:</s> <s xml:id="echoid-s47641" xml:space="preserve"> & remotiores ab illo ſunt maiores:</s> <s xml:id="echoid-s47642" xml:space="preserve"> Ablato enim angulo maio-<lb/>re à ſuo recto, qui relinquitur, fit minor alio angulo, quando à recto aufertur angulus minor:</s> <s xml:id="echoid-s47643" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s47644" xml:space="preserve"> <lb/>in eodem diaphano denſiore primo angulus refractionis ab angulo incidentiæ maiore, maior an-<lb/>gulo refractionis ab angulo incidentiæ minore:</s> <s xml:id="echoid-s47645" xml:space="preserve"> exceſſus quoq;</s> <s xml:id="echoid-s47646" xml:space="preserve"> anguli refractionis maioris ſupra <lb/>angulum refractionis minorem, erit minor exceſſu anguli incidentiæ maioris ſupra minorem:</s> <s xml:id="echoid-s47647" xml:space="preserve"> & <lb/>proportio anguli refractionis ab angulo incidentiæ maiore ad illum angulum maiorem, erit ma-<lb/>ior proportione anguli refractionis ab angulo incidentiæ minore ad illum minorem:</s> <s xml:id="echoid-s47648" xml:space="preserve"> & angulus re <lb/>fractus, ſcilicet ille, quem addit angulus incidentiæ maior ſupra angulum ſuæ refractionis, eſt <lb/>maior angulo refracto, quem addit angulus incidentię minor ſupra angulum ſuæ refractionis.</s> <s xml:id="echoid-s47649" xml:space="preserve"> Sem <lb/>per itaq;</s> <s xml:id="echoid-s47650" xml:space="preserve"> in medio ſecundi diaphani denſiore primo erit angulus refractus minor angulo inciden-<lb/>tiæ:</s> <s xml:id="echoid-s47651" xml:space="preserve"> & proportio iſtorum angulorum refractorum ad æquales angulos incidentię diuerſiſicatur ſe-<lb/>cundum diuerſitatem denſitatis ipſorum mediorum.</s> <s xml:id="echoid-s47652" xml:space="preserve"> Cum enim per aerem eundem & ſecundum <lb/>æ qualitatem anguli incidentiæ fit refractio in aqua & uitro, acutiores fiunt anguli refracti in uitro <lb/>quàm in aqua:</s> <s xml:id="echoid-s47653" xml:space="preserve"> & ſic ſecundum diuerſitatem diaphanitatis anguli uariantur.</s> <s xml:id="echoid-s47654" xml:space="preserve"> Si uerò medium ſecun <lb/>di diaphani fuerit rarius:</s> <s xml:id="echoid-s47655" xml:space="preserve"> tunc ſemper angulus reſractus erit maior angulo incidentiæ:</s> <s xml:id="echoid-s47656" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s47657" xml:space="preserve"> iſtorũ <lb/>angulorum habitudo ad alios angulos reuerſè ſe habens angulis præ miſsis, ac ſi præmiſſæ tabulæ <lb/>modo reuerſo ordinentur.</s> <s xml:id="echoid-s47658" xml:space="preserve"> Et iſtorũ angulorum refractorũ & refractionis ſecundũ maiorẽ & mino <lb/>rem raritatẽ diaphanitatis ſecundi medij ad eundẽ angulum incidentiæ proportio uariatur.</s> <s xml:id="echoid-s47659" xml:space="preserve"> Quan <lb/>do enim à uitro ad aquam uel ad aerem fit refractio:</s> <s xml:id="echoid-s47660" xml:space="preserve"> tunc anguli, qui ſiunt in aere, ſunt maiores an-<lb/>gulis, qui fiunt in aqua:</s> <s xml:id="echoid-s47661" xml:space="preserve"> & ſecundum hoc angulorum reſractionis ad angulos incidentiæ proportio <lb/>uariatur.</s> <s xml:id="echoid-s47662" xml:space="preserve"> Hæc itaq;</s> <s xml:id="echoid-s47663" xml:space="preserve"> ſunt, quæ accidũt lucibus & coloribus, & uniuerſaliter omnibus formis in diffu <lb/>ſione ſui in corporibus diaphanis, & in refractione, quæ accidit in illis omnibus tam ſecundum ſe <lb/>quàm in reſpectu ad uiſus.</s> <s xml:id="echoid-s47664" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s47665" xml:space="preserve"> quod quærebatur.</s> </p> <div xml:id="echoid-div1807" type="float" level="0" n="0"> <xhtml:table xlink:label="tab-0707-01" xlink:href="tab-0707-01a" xml:id="t1"> <xhtml:tr> <xhtml:th><s xml:id="echoid-s47666" xml:space="preserve">Tabula quãtitatis angu <lb/>lorũ incidentiæ omnibus <lb/>ſequentibus communis.</s></xhtml:th> <xhtml:th colspan="2"><s xml:id="echoid-s47667" xml:space="preserve">Anguli refra- <lb/>cti ab aere ad <lb/>aquam.</s></xhtml:th> <xhtml:th colspan="2"><s xml:id="echoid-s47668" xml:space="preserve">Anguli refra- <lb/>ctionis eiuſ- <lb/>dem.</s></xhtml:th> <xhtml:th colspan="2"><s xml:id="echoid-s47669" xml:space="preserve">Anguli refra- <lb/>cti ab aere ad <lb/>uitrum.</s></xhtml:th> <xhtml:th colspan="2"><s xml:id="echoid-s47670" xml:space="preserve">Anguli refra- <lb/>ctionis eiuſ- <lb/>dem.</s></xhtml:th> <xhtml:th colspan="2"><s xml:id="echoid-s47671" xml:space="preserve">Anguli refra- <lb/>cti ab aqua ad <lb/>uitrum.</s></xhtml:th> <xhtml:th colspan="2"><s xml:id="echoid-s47672" xml:space="preserve">Anguli refra- <lb/>ctionis eiuſ- <lb/>dem.</s></xhtml:th> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47673" xml:space="preserve"/></xhtml:td> <xhtml:td><s xml:id="echoid-s47674" xml:space="preserve">par.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47675" xml:space="preserve">minut.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47676" xml:space="preserve">par.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47677" xml:space="preserve">minut.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47678" xml:space="preserve">par.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47679" xml:space="preserve">minut.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47680" xml:space="preserve">par.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47681" xml:space="preserve">minut.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47682" xml:space="preserve">par.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47683" xml:space="preserve">minut.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47684" xml:space="preserve">par.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47685" xml:space="preserve">minuta.</s></xhtml:td> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47686" xml:space="preserve">10</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47687" xml:space="preserve">7</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47688" xml:space="preserve">45</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47689" xml:space="preserve">2</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47690" xml:space="preserve">5</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47691" xml:space="preserve">7</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47692" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47693" xml:space="preserve">3</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47694" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47695" xml:space="preserve">9</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47696" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47697" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47698" xml:space="preserve">30</s></xhtml:td> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47699" xml:space="preserve">20</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47700" xml:space="preserve">15</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47701" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47702" xml:space="preserve">4</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47703" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47704" xml:space="preserve">13</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47705" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47706" xml:space="preserve">6</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47707" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47708" xml:space="preserve">18</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47709" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47710" xml:space="preserve">1</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47711" xml:space="preserve">30</s></xhtml:td> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47712" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47713" xml:space="preserve">22</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47714" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47715" xml:space="preserve">7</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47716" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47717" xml:space="preserve">19</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47718" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47719" xml:space="preserve">10</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47720" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47721" xml:space="preserve">27</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47722" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47723" xml:space="preserve">3</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47724" xml:space="preserve">0</s></xhtml:td> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47725" xml:space="preserve">40</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47726" xml:space="preserve">29</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47727" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47728" xml:space="preserve">11</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47729" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47730" xml:space="preserve">25</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47731" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47732" xml:space="preserve">15</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47733" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47734" xml:space="preserve">35</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47735" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47736" xml:space="preserve">5</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47737" xml:space="preserve">0</s></xhtml:td> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47738" xml:space="preserve">50</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47739" xml:space="preserve">35</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47740" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47741" xml:space="preserve">15</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47742" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47743" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47744" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47745" xml:space="preserve">20</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47746" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47747" xml:space="preserve">42</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47748" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47749" xml:space="preserve">7</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47750" xml:space="preserve">30</s></xhtml:td> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47751" xml:space="preserve">60</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47752" xml:space="preserve">40</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47753" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47754" xml:space="preserve">19</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47755" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47756" xml:space="preserve">34</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47757" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47758" xml:space="preserve">25</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47759" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47760" xml:space="preserve">49</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47761" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47762" xml:space="preserve">10</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47763" xml:space="preserve">30</s></xhtml:td> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47764" xml:space="preserve">70</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47765" xml:space="preserve">45</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47766" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47767" xml:space="preserve">24</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47768" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47769" xml:space="preserve">38</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47770" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47771" xml:space="preserve">31</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47772" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47773" xml:space="preserve">56</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47774" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47775" xml:space="preserve">14</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47776" xml:space="preserve">0</s></xhtml:td> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47777" xml:space="preserve">80</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47778" xml:space="preserve">50</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47779" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47780" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47781" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47782" xml:space="preserve">42</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47783" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47784" xml:space="preserve">38</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47785" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47786" xml:space="preserve">62</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47787" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47788" xml:space="preserve">18</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47789" xml:space="preserve">0</s></xhtml:td> </xhtml:tr> </xhtml:table> <xhtml:table xlink:label="tab-0707-02" xlink:href="tab-0707-02a" xml:id="t2"> <xhtml:tr> <xhtml:th><s xml:id="echoid-s47790" xml:space="preserve">Tabula quantitatis angu <lb/>lorũ incidentiæ omnibus <lb/>ſequentibus communis.</s></xhtml:th> <xhtml:th colspan="2"><s xml:id="echoid-s47791" xml:space="preserve">Anguli refra- <lb/>cti ab aqua <lb/>ad aerem.</s></xhtml:th> <xhtml:th colspan="2"><s xml:id="echoid-s47792" xml:space="preserve">Anguli refra- <lb/>ctionis eiuſ- <lb/>dem.</s></xhtml:th> <xhtml:th colspan="2"><s xml:id="echoid-s47793" xml:space="preserve">Anguli refra- <lb/>cti à uitro ad <lb/>aerem.</s></xhtml:th> <xhtml:th colspan="2"><s xml:id="echoid-s47794" xml:space="preserve">Anguli refra- <lb/>ctionis eiuſ- <lb/>dem.</s></xhtml:th> <xhtml:th colspan="2"><s xml:id="echoid-s47795" xml:space="preserve">Anguli refra- <lb/>cti à uitro ad <lb/>aquam.</s></xhtml:th> <xhtml:th colspan="2"><s xml:id="echoid-s47796" xml:space="preserve">Anguli refra- <lb/>ctionis eiuſ- <lb/>dem.</s></xhtml:th> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47797" xml:space="preserve"/></xhtml:td> <xhtml:td><s xml:id="echoid-s47798" xml:space="preserve">par.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47799" xml:space="preserve">minut.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47800" xml:space="preserve">par.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47801" xml:space="preserve">mιnut.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47802" xml:space="preserve">par.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47803" xml:space="preserve">minut.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47804" xml:space="preserve">par.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47805" xml:space="preserve">mιnut.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47806" xml:space="preserve">par.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47807" xml:space="preserve">minut.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47808" xml:space="preserve">par.</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47809" xml:space="preserve">minut.</s></xhtml:td> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47810" xml:space="preserve">10</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47811" xml:space="preserve">12</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47812" xml:space="preserve">5</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47813" xml:space="preserve">2</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47814" xml:space="preserve">5</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47815" xml:space="preserve">13</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47816" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47817" xml:space="preserve">3</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47818" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47819" xml:space="preserve">10</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47820" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47821" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47822" xml:space="preserve">30</s></xhtml:td> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47823" xml:space="preserve">20</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47824" xml:space="preserve">24</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47825" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47826" xml:space="preserve">4</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47827" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47828" xml:space="preserve">26</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47829" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47830" xml:space="preserve">6</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47831" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47832" xml:space="preserve">21</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47833" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47834" xml:space="preserve">1</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47835" xml:space="preserve">30</s></xhtml:td> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47836" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47837" xml:space="preserve">37</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47838" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47839" xml:space="preserve">7</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47840" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47841" xml:space="preserve">40</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47842" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47843" xml:space="preserve">10</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47844" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47845" xml:space="preserve">33</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47846" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47847" xml:space="preserve">3</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47848" xml:space="preserve">0</s></xhtml:td> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47849" xml:space="preserve">40</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47850" xml:space="preserve">51</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47851" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47852" xml:space="preserve">11</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47853" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47854" xml:space="preserve">55</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47855" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47856" xml:space="preserve">15</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47857" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47858" xml:space="preserve">45</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47859" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47860" xml:space="preserve">5</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47861" xml:space="preserve">0</s></xhtml:td> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47862" xml:space="preserve">50</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47863" xml:space="preserve">65</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47864" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47865" xml:space="preserve">15</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47866" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47867" xml:space="preserve">70</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47868" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47869" xml:space="preserve">20</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47870" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47871" xml:space="preserve">57</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47872" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47873" xml:space="preserve">7</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47874" xml:space="preserve">30</s></xhtml:td> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47875" xml:space="preserve">60</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47876" xml:space="preserve">79</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47877" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47878" xml:space="preserve">19</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47879" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47880" xml:space="preserve">85</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47881" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47882" xml:space="preserve">25</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47883" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47884" xml:space="preserve">70</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47885" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47886" xml:space="preserve">10</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47887" xml:space="preserve">30</s></xhtml:td> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47888" xml:space="preserve">70</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47889" xml:space="preserve">94</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47890" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47891" xml:space="preserve">24</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47892" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47893" xml:space="preserve">101</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47894" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47895" xml:space="preserve">31</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47896" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47897" xml:space="preserve">84</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47898" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47899" xml:space="preserve">14</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47900" xml:space="preserve">0</s></xhtml:td> </xhtml:tr> <xhtml:tr> <xhtml:td><s xml:id="echoid-s47901" xml:space="preserve">80</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47902" xml:space="preserve">110</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47903" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47904" xml:space="preserve">30</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47905" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47906" xml:space="preserve">118</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47907" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47908" xml:space="preserve">38</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47909" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47910" xml:space="preserve">98</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47911" xml:space="preserve">0</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47912" xml:space="preserve">18</s></xhtml:td> <xhtml:td><s xml:id="echoid-s47913" xml:space="preserve">0</s></xhtml:td> </xhtml:tr> </xhtml:table> </div> </div> <div xml:id="echoid-div1809" type="section" level="0" n="0"> <head xml:id="echoid-head1337" xml:space="preserve" style="it">9. Centro uiſus & puncto reiper refractionem uiſæ in diuerſis diaphanis loca propria permu <lb/>tantibus, eædem lineæ incidentie & refractionis nomina permutant. Alhazen 34 n 7.</head> <p> <s xml:id="echoid-s47914" xml:space="preserve">Satis iam patuit ex præmiſsis huius 10 tractatus propoſitionibus, quòd formę uiſæ per refractio <lb/>nem extenduntur directè per lineam rectã, donec perueniant ad ſuperficiem alterius corporis dia-<lb/>phani, in quo eſt uiſus:</s> <s xml:id="echoid-s47915" xml:space="preserve"> deinde refringuntur in illo alio corpore diaphano per aliã lineam rectã, quæ <lb/>continet cum linea incidentiæ angulum.</s> <s xml:id="echoid-s47916" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s47917" xml:space="preserve"> centrum uiſus a:</s> <s xml:id="echoid-s47918" xml:space="preserve"> & punctũ rei uiſæ b.</s> <s xml:id="echoid-s47919" xml:space="preserve"> Sitq̃;</s> <s xml:id="echoid-s47920" xml:space="preserve"> ſuper-<lb/>ficies corporis, in quo eſt punctũ b, ſuperficies c d e, & refringatur forma puncti b ad uiſum exiſten <lb/>tem in puncto a à ſuperficie corporis c d e, puncto <lb/>d:</s> <s xml:id="echoid-s47921" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s47922" xml:space="preserve"> linea incidẽtię, quæ b d:</s> <s xml:id="echoid-s47923" xml:space="preserve"> & linea refractionis, <lb/> <anchor type="figure" xlink:label="fig-0715-01a" xlink:href="fig-0715-01"/> quę d a.</s> <s xml:id="echoid-s47924" xml:space="preserve"> Dico quòd ſi centrũ uiſus & punctũ rei uiſę <lb/>permutent loca, ita ut centrum uiſus poſitum ſit in <lb/>puncto b, & punctum rei uiſæ in puncto a:</s> <s xml:id="echoid-s47925" xml:space="preserve"> tunc ad-<lb/>huc fiet refractio ab eodem puncto corporis, qui eſt <lb/>d:</s> <s xml:id="echoid-s47926" xml:space="preserve"> & linea a d erit linea incidentiæ, & linea d b erit li <lb/>nea refractionis:</s> <s xml:id="echoid-s47927" xml:space="preserve"> & ſic tantùm linearũ nomina per-<lb/>mutantur, manentibus eiſdem lineis & eodem ari-<lb/>gulo.</s> <s xml:id="echoid-s47928" xml:space="preserve"> Hoc autem patet per experientiã.</s> <s xml:id="echoid-s47929" xml:space="preserve"> Cum enim <lb/>aliquis exiſtens in aere inſpexerit aliquod corpus <lb/>contentum ſub alio corpore, quod eſt diapha-<lb/>num, differens in ſua diaphanitate ab aeris dia-<lb/>phanitate:</s> <s xml:id="echoid-s47930" xml:space="preserve"> tunc uiſus comprehendet omnia, quæ <lb/>ſunt ultra illud corpus, quæcunque opponuntur <lb/>uiſui:</s> <s xml:id="echoid-s47931" xml:space="preserve"> & ſi cooperuerit alterum uiſuum, & aſpexe-<lb/>rit cum reliquo:</s> <s xml:id="echoid-s47932" xml:space="preserve"> uidebit illa eadem, quę prius, ſiue <lb/>illud medium ſit aer uel aqua uel uitrum uel cryſtallus:</s> <s xml:id="echoid-s47933" xml:space="preserve"> Quòd ſi uiſus ponatur intra aquam aut ſub <lb/>uitro uel cryſtallo:</s> <s xml:id="echoid-s47934" xml:space="preserve"> uidebit omnia corpora uiſibilia, quę ſunt ultra illud aliud corpus diaphanum in <lb/>ipſo aere.</s> <s xml:id="echoid-s47935" xml:space="preserve"> Siue ergo uiſus ſuerit in aere uel in uitro, ſemper comprehendet omnia eade, quę prius.</s> <s xml:id="echoid-s47936" xml:space="preserve"> <lb/>Patuit aũt per 4 huius quòd uiſus per mediũ diaphani diuerſi non cõprehenditres, quę non ſunt <lb/>in perpẽdiculari ducta à centro uiſus ſuper ſuperficiẽ diaphani corporis, niſi per refractionẽ:</s> <s xml:id="echoid-s47937" xml:space="preserve"> omne <lb/>ergo punctum comprehen ſum à uiſu, præter illud punctum, quod eſt in prædicta perpendiculari, <lb/>comprehenditur per refractionem.</s> <s xml:id="echoid-s47938" xml:space="preserve"> Et quoniam formæ omnium punctorum, quæ ſunt in omnibus <lb/>uiſibilibus exiſtentibus ultra corpus diaphanum, refringuntur in eodem tempore ad centrum u-<lb/>nius uiſus:</s> <s xml:id="echoid-s47939" xml:space="preserve"> patet quòd ſi alicuius rei uiſæ punctum eſſet in puncto, in quo tunc eſt centrum uiſus, <lb/>refringeretur forma illius puncti ad omnia puncta, quæ ſunt in omnibus uiſibilibus exiſtentibus <lb/> <pb o="414" file="0716" n="716" rhead="VITELLONIS OPTICAE"/> ultra illud corpus diaphanum, oppoſitum uiſui in illo tempore:</s> <s xml:id="echoid-s47940" xml:space="preserve"> fieretq́;</s> <s xml:id="echoid-s47941" xml:space="preserve"> illa refractio eodem modo.</s> <s xml:id="echoid-s47942" xml:space="preserve"> <lb/>Et ſimiliter eſt de quolibet puncto propinquo illi puncto, in quo eſt centrum uiſus:</s> <s xml:id="echoid-s47943" xml:space="preserve"> quoniam ſi cen <lb/>tro uiſus in eodem puncto remanente moueatur oculus ad omnem differentiam poſitionis, com-<lb/>prehendet omnia illa uiſibilia.</s> <s xml:id="echoid-s47944" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s47945" xml:space="preserve"> cuiuslibet puncti cuiuſcunq;</s> <s xml:id="echoid-s47946" xml:space="preserve"> rei uiſæ cum fuerit ultra <lb/>aliquod corpus diaphanum, extend tur ad ſuperficiem corporis diaphani, ultra quod eſt, & re-<lb/>fringitur ad uniuerſum eius, quod opponitur ei ex corpore a eris uel alterius diaphani:</s> <s xml:id="echoid-s47947" xml:space="preserve"> & illa forma <lb/>erit apud quodlibet punctum illius ſecundi corporis diaphani:</s> <s xml:id="echoid-s47948" xml:space="preserve"> & ob hoc forma totius rei uiſæ <lb/>coniungitur apud quodlibet punctum aeris uel alterius corporis diaphani:</s> <s xml:id="echoid-s47949" xml:space="preserve"> forma enim cuiuslibet <lb/>punctorum rei uiſæ diffundit ſe per lineam rectam ad unum quodq;</s> <s xml:id="echoid-s47950" xml:space="preserve"> punctum corporis diaphani.</s> <s xml:id="echoid-s47951" xml:space="preserve"> <lb/>Vnde ſi tot fuerint centra uiſuum in aere, quot ſunt puncta aeris:</s> <s xml:id="echoid-s47952" xml:space="preserve"> quilibet illorum uiſuum uidebit <lb/>totalem formam rei uiſibilis, quæ eſt ſub altero diaphano.</s> <s xml:id="echoid-s47953" xml:space="preserve"> Nam ſemper forma rei uiſæ tunc erit <lb/>apud punctum, apud quem erit & centrum uiſus:</s> <s xml:id="echoid-s47954" xml:space="preserve"> unde etiam uiſus motus de loco ad locum ſuper <lb/>idem diaphanum, ſemper eandem uidet ſormam, quamdiu forma illa ſecundum lineas rectas po-<lb/>teſt pertingere ad uiſum.</s> <s xml:id="echoid-s47955" xml:space="preserve"> Et ſimiliter plures aſpicientes comprehendunt unam rem in cœlo & in <lb/>aqua uno & eodem tempore.</s> <s xml:id="echoid-s47956" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s47957" xml:space="preserve"> cuiuslibet puncti rei uiſæ extenditur ad quodlibet pun-<lb/>ctum corporis diaphani, in quo eſt illa res uiſa:</s> <s xml:id="echoid-s47958" xml:space="preserve"> & formæ omnium punctorum rei uiſæ congregan-<lb/>tur apud quodlibet punctum cuiuslibet corporis diaphani, in quo exiſtit, & apud quodlibet pun-<lb/>ctum corporis diaphani diuerſi ab illo corpore diaphano, in quo exiſtit res uiſa.</s> <s xml:id="echoid-s47959" xml:space="preserve"> Inter quodlibet e-<lb/>nim punctum aeris, & quamlibet rem uiſibilem exiſtentem in aliquo corpore diaphano, diuerſo <lb/>ab aere fit pyramis, cuius uertex eſt in aliquo puncto aeris, & baſis in ſuperficie rei uiſæ:</s> <s xml:id="echoid-s47960" xml:space="preserve"> fiuntq́;</s> <s xml:id="echoid-s47961" xml:space="preserve"> tot <lb/>pyramides, quot ſunt puncta aeris, uel alterius corporis diaphani, in quo fit diffuſio formarum.</s> <s xml:id="echoid-s47962" xml:space="preserve"> <lb/>Quia itaque totum medium eſt plenum formis rerum:</s> <s xml:id="echoid-s47963" xml:space="preserve"> anguli uerò refractionis, qui fiunt ab ae-<lb/>re ad aquam, ſunt ijdem cum angulis refractionum, qui fiunt ab aqua ad aerem, ut patet per præ-<lb/>miſſam in tabulis:</s> <s xml:id="echoid-s47964" xml:space="preserve"> ijdem uerò anguli ſemper per eaſdem lineas continentur.</s> <s xml:id="echoid-s47965" xml:space="preserve"> Patet ergo quia locus <lb/>centri uiſus & punctum rei uiſæ de uno diaphano ad alterum permutatis, ſemper quidem fit for.</s> <s xml:id="echoid-s47966" xml:space="preserve"> <lb/>marum uniuerſalis diffuſio:</s> <s xml:id="echoid-s47967" xml:space="preserve"> non tamen percipitur quælibet forma à quolibet uiſu in quolibet pun <lb/>cto, ſed ſolùm in illo, à quo fit directio refractæ lineæ ad illum uiſum.</s> <s xml:id="echoid-s47968" xml:space="preserve"> Patet itaque quia illæ lineæ <lb/>manent eædem ſecundum ſubſtantiam, nominibus tantùm hinc inde permutatis:</s> <s xml:id="echoid-s47969" xml:space="preserve"> ut quæ prius <lb/>fuit linea incidentiæ uel extenſionis ipſius formæ, poſtea fiat linea refractionis, & econuerſo.</s> <s xml:id="echoid-s47970" xml:space="preserve"> <lb/>Patet ergo propoſitum.</s> <s xml:id="echoid-s47971" xml:space="preserve"/> </p> <div xml:id="echoid-div1809" type="float" level="0" n="0"> <figure xlink:label="fig-0715-01" xlink:href="fig-0715-01a"> <variables xml:id="echoid-variables823" xml:space="preserve">a c d o l</variables> </figure> </div> </div> <div xml:id="echoid-div1811" type="section" level="0" n="0"> <head xml:id="echoid-head1338" xml:space="preserve" style="it">10. Omnis refractio formam lucis & coloris, que ſunt in re uiſa, debilius uiſui repræſentat. <lb/>Alhazen 38 n 7.</head> <p> <s xml:id="echoid-s47972" xml:space="preserve">Hoc patet per experientiam.</s> <s xml:id="echoid-s47973" xml:space="preserve"> Cum enim aliquid uiſum eſt in medio ſecundi diaphani, utpote <lb/>per aerem in aqua, & uiſus fuerit ualde obliquus à perpendicularibus exeuntibus à punctis rei ui-<lb/>ſæ ſuper ſuperficiem a quæ:</s> <s xml:id="echoid-s47974" xml:space="preserve"> & deinde uiſus moueatur, donec fiat poſitus in perpendiculari aliqua, <lb/>exeunte à re uiſa ſuper ſuperficiem aquæ:</s> <s xml:id="echoid-s47975" xml:space="preserve"> tunc lux & color rei uiſæ fiunt manifeſtiora quàm eſſent, <lb/>cum a ſpiciebantur obliquè.</s> <s xml:id="echoid-s47976" xml:space="preserve"> Tunc enim figura exiens ad uiſum ſecundum lineas obliquas eſt reſra <lb/>cta, & multùm obliqua:</s> <s xml:id="echoid-s47977" xml:space="preserve"> in perpendiculari uerò forma tota exit rectè:</s> <s xml:id="echoid-s47978" xml:space="preserve"> & quædam partes eius obli-<lb/>què aut ferè rectè, ſecundum quod plus uel minus diſtant à perpendiculari.</s> <s xml:id="echoid-s47979" xml:space="preserve"> Patet ergo ex hoc, <lb/>quoniam refractio debilitat in formis refractis luces & colores, quas formæ rerum uiſarum per <lb/>quodcunq;</s> <s xml:id="echoid-s47980" xml:space="preserve"> corpus diaphanum ſecum deferunt ad uiſum:</s> <s xml:id="echoid-s47981" xml:space="preserve"> nec enim eſt aliqua alia differentia illa-<lb/>rum formarum in eſſe ſuo:</s> <s xml:id="echoid-s47982" xml:space="preserve"> ergo nec quo ad uiſum, niſi ſola obliquitas inducens refractionem, & <lb/>perpendicularitas adiuuans directioné uiſionis:</s> <s xml:id="echoid-s47983" xml:space="preserve"> & ſecundum illa uiſus iudicat formas lucis & co-<lb/>loris debiles uel fortes.</s> <s xml:id="echoid-s47984" xml:space="preserve"> Accidit itaq;</s> <s xml:id="echoid-s47985" xml:space="preserve"> in corporibus uiſis per medium ſecundi diaphani propter re-<lb/>fractionem fallacia, quæ non accideret in illis, ſi uiderentur rectè:</s> <s xml:id="echoid-s47986" xml:space="preserve"> quia etiam, ut patet per 33 th.</s> <s xml:id="echoid-s47987" xml:space="preserve"> 4 hu <lb/>ius, omnis linea uel ſuperficies rei uiſæ directè uiſibus oppoſita perfectius uidetur quàm obliqua-<lb/>ta:</s> <s xml:id="echoid-s47988" xml:space="preserve"> & ſecundum quantitatem obliquationis fit imperfectio uiſionis.</s> <s xml:id="echoid-s47989" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s47990" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1812" type="section" level="0" n="0"> <head xml:id="echoid-head1339" xml:space="preserve" style="it">11. Imago refracta rei uiſibilis nunquam occurrit uiſui in loco rei uiſæ, ſed ſemper extra ſuum <lb/>locum. Euclides 7 hypotheſicatoptr. Alhazen 17 n 7.</head> <p> <s xml:id="echoid-s47991" xml:space="preserve">Quod hic proponitur, patet ratione & experientia.</s> <s xml:id="echoid-s47992" xml:space="preserve"> Ratio autem eſt hæc.</s> <s xml:id="echoid-s47993" xml:space="preserve"> Nam forma compre-<lb/>henſa à uiſu in corpore diaphano alio ab aere, non eſt ipſa res uiſa:</s> <s xml:id="echoid-s47994" xml:space="preserve"> quoniam uiſus non compre-<lb/>hendit rem tunc in ſua forma uel in figura, ſed in alijs diſpoſitionibus & alio modo:</s> <s xml:id="echoid-s47995" xml:space="preserve"> comprehendit <lb/>enim imaginem refractam in ſua oppoſitione:</s> <s xml:id="echoid-s47996" xml:space="preserve"> cum tamen res non ſit directè uiſui oppoſita.</s> <s xml:id="echoid-s47997" xml:space="preserve"> Et <lb/>quia comprehendit rem refractè:</s> <s xml:id="echoid-s47998" xml:space="preserve"> ideo quia uiſus eſt decliuis à perpendicularibus exeuntibus à re <lb/>uiſa ſuper ſuperficiem corporis diaphani:</s> <s xml:id="echoid-s47999" xml:space="preserve"> comprehendit ergo ipſam ut extra ſuum locum, non in <lb/>ſuo loco.</s> <s xml:id="echoid-s48000" xml:space="preserve"> Per experientiá quoq;</s> <s xml:id="echoid-s48001" xml:space="preserve"> idem patet.</s> <s xml:id="echoid-s48002" xml:space="preserve"> Aſſumatur uas habẽs oras erectas ſuper baſim eius:</s> <s xml:id="echoid-s48003" xml:space="preserve"> <lb/>& in medio fundi uaſis ponatur denarius argenteus:</s> <s xml:id="echoid-s48004" xml:space="preserve"> & elonget ſe experimentans, quouſq;</s> <s xml:id="echoid-s48005" xml:space="preserve"> uideat <lb/>illũ denariũ in fundo uaſis:</s> <s xml:id="echoid-s48006" xml:space="preserve"> deinde elonget ſe paulatim ulterius, quouſq;</s> <s xml:id="echoid-s48007" xml:space="preserve"> nõ uideat ipſum, & in prin <lb/>cipio occultatiõ is ſtet in ſuo loco, uiſu immoto:</s> <s xml:id="echoid-s48008" xml:space="preserve"> & præcipiat inſundi aquã in uas, ita ut denarius nõ <lb/>mutet locum:</s> <s xml:id="echoid-s48009" xml:space="preserve"> & tunc uidebit denarium in eius oppoſitione ipſo nõ exiſtente in eius oppoſitione.</s> <s xml:id="echoid-s48010" xml:space="preserve"> <lb/>Ex quo patet quòd forma, quam experimentans uidet in aqua, non eſt in loco rei uiſæ.</s> <s xml:id="echoid-s48011" xml:space="preserve"> Nam ſi for-<lb/> <pb o="415" file="0717" n="717" rhead="LIBER DECIMVS."/> ma eſſet in loco rei uiſæ:</s> <s xml:id="echoid-s48012" xml:space="preserve"> tunc etiam res uiſa comprehendi poſſet ſine inſuſione aquæ in uas:</s> <s xml:id="echoid-s48013" xml:space="preserve"> quod <lb/>non accidit in tanta diſtantia, ut patuit.</s> <s xml:id="echoid-s48014" xml:space="preserve"> Imago itaq;</s> <s xml:id="echoid-s48015" xml:space="preserve"> rei uiſæ per refractionem non uidetur in loco <lb/>ipſius rei.</s> <s xml:id="echoid-s48016" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s48017" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1813" type="section" level="0" n="0"> <head xml:id="echoid-head1340" xml:space="preserve" style="it">12. Omnis forma punctiper refractionem uiſi comprehenditur in rectitudine linea, per <lb/>quam à puncto refractionis forma extenditur ad uiſum. Alhazen 19 n 7.</head> <p> <s xml:id="echoid-s48018" xml:space="preserve">Sit enim punctus per refractionem uiſus, qui eſt a:</s> <s xml:id="echoid-s48019" xml:space="preserve"> cuius forma refringatur ad uiſum ab aliquo <lb/>puncto ſuperficiei corporis laterius diaphani, qui ſit b:</s> <s xml:id="echoid-s48020" xml:space="preserve"> & <lb/>ſit centrũ uiſus d:</s> <s xml:id="echoid-s48021" xml:space="preserve"> dico quòd ſorma puncti a comprehen-<lb/> <anchor type="figure" xlink:label="fig-0717-01a" xlink:href="fig-0717-01"/> ditur à uiſu ſecundum rectitudinem lineæ d b.</s> <s xml:id="echoid-s48022" xml:space="preserve"> Hoc au-<lb/>tem inſtrumentaliter declarandum.</s> <s xml:id="echoid-s48023" xml:space="preserve"> Accipiatur itaq;</s> <s xml:id="echoid-s48024" xml:space="preserve"> in-<lb/>ſtrumentum primum, & ponatur in uaſe impleto aqua, <lb/>ut prius:</s> <s xml:id="echoid-s48025" xml:space="preserve"> & ſignetur aliquod uidendum per refractionem <lb/>in ora in ſtrumenti in oppoſitione uiſus:</s> <s xml:id="echoid-s48026" xml:space="preserve"> & intueatur ex-<lb/>perimentans per ambo foramina, ita ut uideat illud per <lb/>refractionem:</s> <s xml:id="echoid-s48027" xml:space="preserve"> deinde claudatur ſecundum foramen in-<lb/>ſtrumenti:</s> <s xml:id="echoid-s48028" xml:space="preserve"> & tunc non comprehẽdetur res uiſa:</s> <s xml:id="echoid-s48029" xml:space="preserve"> & ſi clau-<lb/>datur primum foramen, ſimiliter nihil uidebitur:</s> <s xml:id="echoid-s48030" xml:space="preserve"> quoniã <lb/>abſciſſa eſt linea recta imaginabiliter exiens à cẽtro uiſus ad locum refractionis.</s> <s xml:id="echoid-s48031" xml:space="preserve"> Forma enim pun-<lb/>cti uiſi per refractionem extẽditur in corpore diaphano, in quo eſt res uiſa, & refringitur in corpo-<lb/>re diaphano, quod eſt inter ipſum & centrum uiſus, peruenitq́;</s> <s xml:id="echoid-s48032" xml:space="preserve"> ad uiſum per lineam rectam, exeun-<lb/>tem à centro uiſus ad punctum refractionis:</s> <s xml:id="echoid-s48033" xml:space="preserve"> & uiſus non comprehendit aliquid, niſi in rectitudine <lb/>linearum radialium, per quas forma uiſibilis mouetur ad uiſum.</s> <s xml:id="echoid-s48034" xml:space="preserve"> Et ſi fiat operatio per interpoſitio-<lb/>nem alicuius uitri uiſui & rei uiſæ, ut ſuprà:</s> <s xml:id="echoid-s48035" xml:space="preserve"> eodem modo penitùs operando, partebit idem.</s> <s xml:id="echoid-s48036" xml:space="preserve"> Et hoc <lb/>eſt propoſitum.</s> <s xml:id="echoid-s48037" xml:space="preserve"> Viſus enim nihil comprehendit niſi in rectitudine linearum radialium:</s> <s xml:id="echoid-s48038" xml:space="preserve"> non enim <lb/>patitur niſi in progrſsione iſtarum linearum à punctis rerum uiſibilium ad uiſum:</s> <s xml:id="echoid-s48039" xml:space="preserve"> quoniam non <lb/>uidet niſi res ſibi oppoſitas, quarum formę ſecundum lineas rectas multiplicant ſe ad uiſum, ut pa-<lb/>tuit per 2 th.</s> <s xml:id="echoid-s48040" xml:space="preserve"> 3 huius, & per multa ſimilia.</s> <s xml:id="echoid-s48041" xml:space="preserve"> Patet ergo, quod proponebatur.</s> <s xml:id="echoid-s48042" xml:space="preserve"/> </p> <div xml:id="echoid-div1813" type="float" level="0" n="0"> <figure xlink:label="fig-0717-01" xlink:href="fig-0717-01a"> <variables xml:id="echoid-variables824" xml:space="preserve">d b a</variables> </figure> </div> </div> <div xml:id="echoid-div1815" type="section" level="0" n="0"> <head xml:id="echoid-head1341" xml:space="preserve" style="it">13. Omnis forma uiſa per refractionem comprehenditur in linea perpendiculari, ducta à <lb/>puncto rei uiſæ ſuper ſuperficiem corporis, à qua fit refractio. Alhazen 19 n 7.</head> <p> <s xml:id="echoid-s48043" xml:space="preserve">Quod hic proponitur, patet ideo:</s> <s xml:id="echoid-s48044" xml:space="preserve"> quia lux extenditur in corpore diaphano trãſitu uelociſsimo, <lb/>intelligendo illam uelocitatem modo prius expoſito:</s> <s xml:id="echoid-s48045" xml:space="preserve"> & iam patuit ex his, quæ dicta ſunt in 47 th.</s> <s xml:id="echoid-s48046" xml:space="preserve"> 2 <lb/>huius, quia trãſitus lucis in corpore diaphano ſuper lineam decliuem ſuper ſuperficiem illius cor-<lb/>poris, eſt compoſitus ex motu ſuper lineam perpendicularem, exeuntem à puncto, à quo extendi-<lb/>tur lux ſuper ſuperficiem illius corporis diaphani, & ex motu ſuper lineam, quæ eſt perpendicula-<lb/>ris ſuper hanc lineam perpendicularem.</s> <s xml:id="echoid-s48047" xml:space="preserve"> Forma uerò, quę extenditur à puncto rei per refractionem <lb/>uiſæ ad ipſum punctum refractionis, quæ eſt forma lucis exiſtentis in puncto rei uiſæ mixta cum <lb/>forma coloris, ſemper extenditur ſuper lineam decliuem ſuper ſuperficiem corporis diaphani.</s> <s xml:id="echoid-s48048" xml:space="preserve"> Hęc <lb/>ergo forma extenditur ad locũ ſuæ refractionis motu compoſito ex motu ſuper perpendicularem, <lb/>exeuntem à puncto ipſo uiſo ſuper ſuperficiem corporis diaphani, & ex motu ſuper lineam, quę eſt <lb/>perpendicularis ſuper hanc perpendicularem.</s> <s xml:id="echoid-s48049" xml:space="preserve"> Eſt ergo motus formæ, quæ mouetur ad uiſum, aut <lb/>ſuper perpendicularem ductam ab ipſo pũcto, cuius ipſa eſt ſorma, ſuper ſuperficíem corporis dia-<lb/>phani:</s> <s xml:id="echoid-s48050" xml:space="preserve"> quamuis poſtmodum translata ſit ab hac perpendiculari alio motu:</s> <s xml:id="echoid-s48051" xml:space="preserve"> aut motus eius eſt ſuper <lb/>perpendicularem, ductam ſuper illam priorem perpendicularem, & translata eſt poſt motum eius <lb/>ſuper primam perpendicularem, ductam à puncto rei formæ motę ſuper ſuperficiem corporis dia-<lb/>phani:</s> <s xml:id="echoid-s48052" xml:space="preserve"> fitq́;</s> <s xml:id="echoid-s48053" xml:space="preserve"> hęc translatio propter compoſitionẽ ex prædictis duobus motibus.</s> <s xml:id="echoid-s48054" xml:space="preserve"> Forma ergo exiens <lb/>à loco refractionis peruenit ad ipſum uiſum per motum formæ, quæ mouetur ſuper lineã perpen-<lb/>dicularem ductam à puncto rei uiſæ ſuper ſuperficiem corporis diaphani:</s> <s xml:id="echoid-s48055" xml:space="preserve"> deinde translata eſt ab <lb/>hac perpendiculari per motum in rectitudine lineæ, per quam forma ad uiſum.</s> <s xml:id="echoid-s48056" xml:space="preserve"> Palàm eſt etiã quod <lb/>proponitur per hoc.</s> <s xml:id="echoid-s48057" xml:space="preserve"> Quia ſi punctum ſuperficiei corporis diaphani, cui incidit perpẽdicularis du-<lb/>cta à puncto rei uiſæ, contingat abſcondi à uiſu, utpote propter interpoſitionem alicuius corporis <lb/>opaci:</s> <s xml:id="echoid-s48058" xml:space="preserve"> non fiet uiſio illius puncti rei uiſæ.</s> <s xml:id="echoid-s48059" xml:space="preserve"> Forma ergo rei uiſæ comprehenditur in perpendiculari <lb/>ducta à puncto rei uiſæ ſuper ſuperficiem corporis, à qua fit refractio.</s> <s xml:id="echoid-s48060" xml:space="preserve"> Patet ergo propoſitum, quod <lb/>& maniſeſtius poſtmodum inſtrumentaliter ſtudebimus declarare.</s> <s xml:id="echoid-s48061" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1816" type="section" level="0" n="0"> <head xml:id="echoid-head1342" xml:space="preserve" style="it">14. Omnium formarum punctorum rei uiſæ plus diſtantium à linea perpendiculari, ducta <lb/>à centro uiſus ſuper ſuperficiem corporis diaphani, à qua fit refractio, maior eſt refractio quàm <lb/>punctorum minus diſtantium ab illa.</head> <p> <s xml:id="echoid-s48062" xml:space="preserve">Eſto centrum uiſus a:</s> <s xml:id="echoid-s48063" xml:space="preserve"> & linea uiſa per refractionem ſit b c d e:</s> <s xml:id="echoid-s48064" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s48065" xml:space="preserve"> communis ſectio ſuperficiei <lb/>refractionis & corporis, à cuius ſuperficie fit refractio, linea f g h i:</s> <s xml:id="echoid-s48066" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s48067" xml:space="preserve"> perpẽdicularis ducta à cen-<lb/>tro uiſus ſuper ſuperficiem illius corporis linea a f:</s> <s xml:id="echoid-s48068" xml:space="preserve"> quæ incidat in punctum b rei uiſæ:</s> <s xml:id="echoid-s48069" xml:space="preserve"> & ſit a f b:</s> <s xml:id="echoid-s48070" xml:space="preserve"> <lb/> <pb o="416" file="0718" n="718" rhead="VITELLONIS OPTICAE"/> diſtetq́;</s> <s xml:id="echoid-s48071" xml:space="preserve"> à puncto b & à perpendiculari a f b plus punctum d quàm punctum c, & plus punctum e <lb/>quàm punctum d.</s> <s xml:id="echoid-s48072" xml:space="preserve"> Dico quòd maior erit refractio puncti e quàm puncti d:</s> <s xml:id="echoid-s48073" xml:space="preserve"> & maior puncti d quàm <lb/>puncti c.</s> <s xml:id="echoid-s48074" xml:space="preserve"> Forma enim puncti a cum ſit in ipſa linea perpendiculari:</s> <s xml:id="echoid-s48075" xml:space="preserve"> patet per 3 th.</s> <s xml:id="echoid-s48076" xml:space="preserve"> huius quia non re-<lb/>fringitur.</s> <s xml:id="echoid-s48077" xml:space="preserve"> Formæ uerò aliorum punctorum, quæ ſunt c, d, e, patet quòd refringuntur per 4 huius.</s> <s xml:id="echoid-s48078" xml:space="preserve"> <lb/>Et quoniam, ut patet per 49 th.</s> <s xml:id="echoid-s48079" xml:space="preserve"> 2 huius, nulla refractio tranſmutat ſitum partium formæ refractæ, <lb/>ſed ſolùm auget uel minuit figuram:</s> <s xml:id="echoid-s48080" xml:space="preserve"> patet quòd de neceſsitate diuerſitas formarum pũctorum rei <lb/>uiſæ refringitur à diuerſis punctis ſuperficiei ipſius rei uiſæ:</s> <s xml:id="echoid-s48081" xml:space="preserve"> ita quòd forma puncti remotioris à ui-<lb/>ſu refringitur à puncto ſuperficiei remotiori à centro uiſus:</s> <s xml:id="echoid-s48082" xml:space="preserve"> aliàs enim fieret tranſmutatio forma-<lb/>rum uiſarum per refractionẽ.</s> <s xml:id="echoid-s48083" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0718-01a" xlink:href="fig-0718-01"/> Sit ergo, ut forma puncti c re-<lb/>fringatur à puncto g:</s> <s xml:id="echoid-s48084" xml:space="preserve"> & for-<lb/>ma puncti d à pũcto h:</s> <s xml:id="echoid-s48085" xml:space="preserve"> & for-<lb/>ma puncti e à pũcto i:</s> <s xml:id="echoid-s48086" xml:space="preserve"> & edu-<lb/>cantur à puncto g linea gl:</s> <s xml:id="echoid-s48087" xml:space="preserve"> & <lb/>â pũcto h linea h m:</s> <s xml:id="echoid-s48088" xml:space="preserve"> & à pun-<lb/>cto i linea i n perpendicula-<lb/>res ſuper ſuperficiem corpo-<lb/>ris diaphani per 12 p 11:</s> <s xml:id="echoid-s48089" xml:space="preserve"> & pro-<lb/>ducantur lineę incidẽtiæ for-<lb/>marum ultra ſuperficiẽ cor-<lb/>poris, linea c g in punctum o:</s> <s xml:id="echoid-s48090" xml:space="preserve"> <lb/>& linea d h in punctum q:</s> <s xml:id="echoid-s48091" xml:space="preserve"> & co-<lb/>linea e i in punctum q:</s> <s xml:id="echoid-s48092" xml:space="preserve"> & co-<lb/>pulẽtur lineæ refractæ à pun-<lb/>ctis g, h, i ad uiſum, quæ ſunt g a, h a, i a.</s> <s xml:id="echoid-s48093" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s48094" xml:space="preserve"> in trigono a fi ductæ ſunt lineæ a g & a h:</s> <s xml:id="echoid-s48095" xml:space="preserve"> patet <lb/>per 21 p 1 quoniam angulus a g f eſt maior angulo a h f.</s> <s xml:id="echoid-s48096" xml:space="preserve"> Quia ergo anguli l g f & m h f ſunt recti & <lb/>æquales:</s> <s xml:id="echoid-s48097" xml:space="preserve"> relinquitur angulus a g l minor angulo a h m:</s> <s xml:id="echoid-s48098" xml:space="preserve"> ſed angulus o g l & p h m ſunt æquales:</s> <s xml:id="echoid-s48099" xml:space="preserve"> quæ-<lb/>libet enim linea incidentiæ cum ſua perpendiculari continet angulos æquales propter æqualem <lb/>diſtantiam punctorum b, c, d, e ab inuicem, & à ſuperficie diaphani, à qua fit refractio.</s> <s xml:id="echoid-s48100" xml:space="preserve"> Eſt ergo an-<lb/>gulus p h a maior angulo o g a:</s> <s xml:id="echoid-s48101" xml:space="preserve"> & angulus q i a maior angulo p h a:</s> <s xml:id="echoid-s48102" xml:space="preserve"> eſt autem eadem diſpoſitio me-<lb/>dij, in quo ſit refractio formarum punctorum c & d à punctis g & h.</s> <s xml:id="echoid-s48103" xml:space="preserve"> Patet ergo quòd maior fit refra-<lb/>ctio à pũcto h remotiore à uiſu a, quàm à puncto g propinquiore uiſui illo pũcto h.</s> <s xml:id="echoid-s48104" xml:space="preserve"> Similiter quoq.</s> <s xml:id="echoid-s48105" xml:space="preserve"> <lb/>patet per eundem modum de puncto i, reſpectu puncti h:</s> <s xml:id="echoid-s48106" xml:space="preserve"> fit enim ſecundum pręmiſſa angulus a i q <lb/>maior angulo a h p:</s> <s xml:id="echoid-s48107" xml:space="preserve"> eſt ergo maior refractio puncti i quàm puncti h:</s> <s xml:id="echoid-s48108" xml:space="preserve"> ergo & maior quàm puncti g.</s> <s xml:id="echoid-s48109" xml:space="preserve"> <lb/>Patet ergo uniuerſaliter, quod proponebatur.</s> <s xml:id="echoid-s48110" xml:space="preserve"> In omnibus enim punctis & ſuperficiebus, à quibus <lb/>fit refractio, eſt eadem demonſtratio.</s> <s xml:id="echoid-s48111" xml:space="preserve"/> </p> <div xml:id="echoid-div1816" type="float" level="0" n="0"> <figure xlink:label="fig-0718-01" xlink:href="fig-0718-01a"> <variables xml:id="echoid-variables825" xml:space="preserve">a o l j p m q n f g y i b c d e</variables> </figure> </div> </div> <div xml:id="echoid-div1818" type="section" level="0" n="0"> <head xml:id="echoid-head1343" xml:space="preserve" style="it">15. Locus imaginis refract æ cuiuslibet punctirei per refr actionem uiſæ eſt in cõmuni ſectio-<lb/>ne lineæ refractionis, per quam peruenit forma ad uiſum, & catheti incidẽtiæ, exeuntis ab illo <lb/>puncto rei uiſæ ſuper ſuperficiem corporis diaphani uiſum contingentis. Ex quo patet quòd lo-<lb/>cus imaginis formæ punctirei uiſæ exiſtentis in medio ſecundi diaphani denſioris primo appro-<lb/>ximat uiſui: in rariore uerò elongatur. Alhazen 18 n 7.</head> <p> <s xml:id="echoid-s48112" xml:space="preserve">Verbi gratia:</s> <s xml:id="echoid-s48113" xml:space="preserve"> ſit punctus rei uiſæ per medium ſecundi diaphania:</s> <s xml:id="echoid-s48114" xml:space="preserve"> & ſuperficies ſecundi diapha.</s> <s xml:id="echoid-s48115" xml:space="preserve"> <lb/>ni ſit, in qua eſt linea b c:</s> <s xml:id="echoid-s48116" xml:space="preserve"> & ſit b punctus refractionis:</s> <s xml:id="echoid-s48117" xml:space="preserve"> & centrum ui-<lb/> <anchor type="figure" xlink:label="fig-0718-02a" xlink:href="fig-0718-02"/> ſus ſit d:</s> <s xml:id="echoid-s48118" xml:space="preserve"> perueniatq́;</s> <s xml:id="echoid-s48119" xml:space="preserve"> forma punctia ad uiſum d ſecundum lineam <lb/>refractionis:</s> <s xml:id="echoid-s48120" xml:space="preserve"> quæ ſit b d.</s> <s xml:id="echoid-s48121" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s48122" xml:space="preserve"> à puncto a perpendicularis <lb/>ſuper ſuperficiem b c:</s> <s xml:id="echoid-s48123" xml:space="preserve"> quæ ſit a e.</s> <s xml:id="echoid-s48124" xml:space="preserve"> Dico quod in puncto, qui eſt com-<lb/>munis ſectio lineæ perpẽdicularis a e.</s> <s xml:id="echoid-s48125" xml:space="preserve"> & productę d b eſt locus ima-<lb/>ginis refractæ.</s> <s xml:id="echoid-s48126" xml:space="preserve"> Hoc autem patet.</s> <s xml:id="echoid-s48127" xml:space="preserve"> Quoniá enim per 12 huius forma <lb/>refracta occurrit uiſui in linea d b, & per 13 huius occurrit in linea <lb/>perpendiculari:</s> <s xml:id="echoid-s48128" xml:space="preserve"> quæ eſt a e:</s> <s xml:id="echoid-s48129" xml:space="preserve"> occurrit ergo in communi ipſarum ſe-<lb/>ctione, quæ ſit punctum x.</s> <s xml:id="echoid-s48130" xml:space="preserve"> Hoc autem fortius inſtrumentaliter de-<lb/>monſtrandum.</s> <s xml:id="echoid-s48131" xml:space="preserve"> Accipiatur columna rotunda lignea, cuius baſis dia-<lb/>meter ſit unius cubiti, & altitudo modica, utpote duorum uel trium <lb/>digitorum:</s> <s xml:id="echoid-s48132" xml:space="preserve"> & planentur ſuperficies baſium eius:</s> <s xml:id="echoid-s48133" xml:space="preserve"> & in una baſium <lb/>ſuarum in uento per 1 p 3 cẽtro (quod ſite) ducantur diametri quæ-<lb/>cunq;</s> <s xml:id="echoid-s48134" xml:space="preserve"> placuerint:</s> <s xml:id="echoid-s48135" xml:space="preserve"> & ſint duæ, quæ g h & i k, obliquè ſe fecantes:</s> <s xml:id="echoid-s48136" xml:space="preserve"> quæ <lb/>profundentur ferro, ut appareãt uiſui:</s> <s xml:id="echoid-s48137" xml:space="preserve"> & impleantur profunditates <lb/>ipſarum ceruſa diſtemperata cum lacte uel cũ alio albo liquore aut <lb/>albo alio colore quocunq;</s> <s xml:id="echoid-s48138" xml:space="preserve">. punctum uerò centri, quod eſte, ſit ni-<lb/>grum.</s> <s xml:id="echoid-s48139" xml:space="preserve"> Deinde accipiatur uas magnum profundum habẽs oras ere-<lb/>ctas, & ponatur in loco luminoſo:</s> <s xml:id="echoid-s48140" xml:space="preserve"> infundaturq́ in uas aqua tanta, <lb/>quòd cum immiffa fuerit columna in a quam erecta taliter, ut eius ſuperficies planæ perpendicula-<lb/> <pb o="417" file="0719" n="719" rhead="LIBER DECIMVS."/> res ſint ſuper fundum uaſis:</s> <s xml:id="echoid-s48141" xml:space="preserve"> tuncipſa aqua excedat punctum e centrum circuli bafis columnæ ad <lb/>aliquot digitos:</s> <s xml:id="echoid-s48142" xml:space="preserve"> expecteturq́;</s> <s xml:id="echoid-s48143" xml:space="preserve"> donec aqua quieſcat in ipſo uaſe.</s> <s xml:id="echoid-s48144" xml:space="preserve"> Moueaturitaq;</s> <s xml:id="echoid-s48145" xml:space="preserve"> columna, donec g h <lb/>diameter baſis ſit perpendicularis ſuper ſuperficiem aquæ:</s> <s xml:id="echoid-s48146" xml:space="preserve"> declinetur quoq;</s> <s xml:id="echoid-s48147" xml:space="preserve"> uiſus extra oras uafis, <lb/>quouſq;</s> <s xml:id="echoid-s48148" xml:space="preserve"> appropinquet æquidiſtantiæ ſuperficiei aquæ in tantùm, ut poſsit uideri punctum e cen-<lb/>trum circuli & diameter g h:</s> <s xml:id="echoid-s48149" xml:space="preserve"> & inuenietur centrum circuli e in rectitudine illius diametri.</s> <s xml:id="echoid-s48150" xml:space="preserve"> Deinde <lb/>intueatur uiſus diametrum i k decliuem ſuper ſuperficiem aquæ:</s> <s xml:id="echoid-s48151" xml:space="preserve"> & inuenietur incuruari & frangi <lb/>apud ſuperficiem aquæ:</s> <s xml:id="echoid-s48152" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s48153" xml:space="preserve"> pars eius intra aquam cum parte eius extra aquam continens angu-<lb/>lum obtuſum reſpectu uiſus:</s> <s xml:id="echoid-s48154" xml:space="preserve"> cum tamen diameter g h extra aquam & intra aquam remaneat linea <lb/>una recta ſine refractione uel continentia anguli.</s> <s xml:id="echoid-s48155" xml:space="preserve"> Ex quo patet quòd forma puncti centralis, quod <lb/>eſt e, quam uiſus comprehendit, non eſt apud centrum circuli baſis:</s> <s xml:id="echoid-s48156" xml:space="preserve"> quia tunc eſſet etiam in recti-<lb/>tudine diametri decliuis, quæ eſt i k:</s> <s xml:id="echoid-s48157" xml:space="preserve"> quia ſecundũ ueritatem ille eſt eius ſitus.</s> <s xml:id="echoid-s48158" xml:space="preserve"> Cũ ergo uiſus com-<lb/>prehẽdit illud punctũ extra rectitudinem diame-<lb/> <anchor type="figure" xlink:label="fig-0719-01a" xlink:href="fig-0719-01"/> tri decliuis, quæ eſti k, & angulus, quem continẽt <lb/>partes diametri decliuis i k, ſequũtur perpendicu-<lb/>larem g h:</s> <s xml:id="echoid-s48159" xml:space="preserve"> patet quòd punctus, in quo uidetur for-<lb/>ma centri e, eſt eleuatus à centro baſis columnæ.</s> <s xml:id="echoid-s48160" xml:space="preserve"> <lb/>Et quia uiſus hoc punctum comprehẽdit in recti-<lb/>tudine diametri g h:</s> <s xml:id="echoid-s48161" xml:space="preserve"> patet quòd forma centri e eſt <lb/>eleuata à uero loco cẽtri ſecundũ rectitudinẽ dia-<lb/>metri perpendicularis, quæ eſt g h.</s> <s xml:id="echoid-s48162" xml:space="preserve"> Patet etiam ex <lb/>diametri decliuis i k incuruatione apud ſuperficiẽ <lb/>aquæ, & ex rectitudine & continuitate partis ſuæ <lb/>intra aquam, quòd omne pũctum partis diametri <lb/>i k, quod eſt intra aquam, eſt eleuatum à ſuo loco.</s> <s xml:id="echoid-s48163" xml:space="preserve"> <lb/>Deinde reuoluatur circulus baſis columnæ, quo-<lb/>uſq;</s> <s xml:id="echoid-s48164" xml:space="preserve"> diameter i k fiat perpendicularis ſuper ſuper-<lb/>ficiem aquæ:</s> <s xml:id="echoid-s48165" xml:space="preserve"> erit ergo tunc g h diameter decliuis <lb/>ſuper ſuperficiem aquæ:</s> <s xml:id="echoid-s48166" xml:space="preserve"> & tunc uidebitur forma <lb/>centri e in rectitudine diametri i k, & extra rectitudinem diametri g h:</s> <s xml:id="echoid-s48167" xml:space="preserve"> quoniá illa uidebitur frangi <lb/>& incuruari ſuper ſuperficiem aquæ:</s> <s xml:id="echoid-s48168" xml:space="preserve"> & angulus incuruationis obtuſus erit reſpiciens uiſum & dia-<lb/>metrum i k perpendicularẽ ſuper aquæ ſuperficiem.</s> <s xml:id="echoid-s48169" xml:space="preserve"> Idẽ quoq;</s> <s xml:id="echoid-s48170" xml:space="preserve"> accidet ſi plures ſint diametri ſigna-<lb/>tæ in ſuperficie baſis colũnæ:</s> <s xml:id="echoid-s48171" xml:space="preserve"> ſemper enim forma cẽtri e uidebitur in rectitudine diametri perpen-<lb/>dicularis:</s> <s xml:id="echoid-s48172" xml:space="preserve"> & diameter decliuis uidebitur incuruari apud ſuperficiẽ aquæ, & cõtinere angulũ obtu-<lb/>ſum cũ parte ſui, quæ eſt intra aquã:</s> <s xml:id="echoid-s48173" xml:space="preserve"> quæ pars intra aquã ſemper uidebitur cõtinua & recta.</s> <s xml:id="echoid-s48174" xml:space="preserve"> Exhoc <lb/>itaq;</s> <s xml:id="echoid-s48175" xml:space="preserve"> patet quòd forma pũcti a uiſi in corpore diaphanitatis groſsioris, quàm ſit a eris diaphanitas, <lb/>uidetur extra locũ ſuũ eleuata in rectitudine perpẽdicularis, exeuntis ab illo pũcto ſuperficiei cor-<lb/>poris diaphani:</s> <s xml:id="echoid-s48176" xml:space="preserve"> cũ linea d b cõtinuans d centrũ uiſus cũ puncto refractionis b, nõ fuerit perpẽdicu-<lb/>laris;</s> <s xml:id="echoid-s48177" xml:space="preserve"> ſuper ſuperficiẽ corporis diaphani.</s> <s xml:id="echoid-s48178" xml:space="preserve"> Et quia (ſicut inſtrumentaliter & per rationẽ oſtenſum eſt <lb/>per 11 huius) omne punctum comprehẽditur ã uiſu in ipſius uiſus oppoſitione & in rectitudine li-<lb/>neæ, per quam extenditur forma ad uiſum:</s> <s xml:id="echoid-s48179" xml:space="preserve"> puncta ergo, quæ uiſus comprehẽdit per refractionem, <lb/>quia ſunt in oppoſitione uiſus ſecundũ lineam rectam, in cõmuni ſectione perpẽdicularis a e & li-<lb/>neæ d a productæ ad perpendicularẽ neceſſariõ uidẽtur.</s> <s xml:id="echoid-s48180" xml:space="preserve"> Eſt ergo punctus ille, in quo illę lineæ duæ <lb/>ſecant ſe, locus imaginis refractæ.</s> <s xml:id="echoid-s48181" xml:space="preserve"> Quòd ſi fiat refractio formæ puncti uiſi à corpore diaphano ſub-<lb/>tiliori ad groſsius, adhuc idẽ accidit quod in præmiſsis:</s> <s xml:id="echoid-s48182" xml:space="preserve"> quoniã adhuc locus imaginis refractæ erit <lb/>in cómuni ſectione lineæ refractionis, per quam forma peruenit ad uiſum, & lineæ perpẽdicularis, <lb/>ductæ à puncto rei uiſæ ſuper ſuperficiẽ corporis, à qua fit refractio.</s> <s xml:id="echoid-s48183" xml:space="preserve"> Aſſumatur enim uitrum ſuper-<lb/>ficierum planarum & æ quidiſtantium, cuius longitudo ſit octo digitorum, latitudo & ſpiſsitudo ſit <lb/>æqualis:</s> <s xml:id="echoid-s48184" xml:space="preserve"> quælibet quatuor digitorũ.</s> <s xml:id="echoid-s48185" xml:space="preserve"> Deinde baſi columnæ ligneæ prędictæ prius inſcribatur linea <lb/>decem digitorũ per 1 p 4:</s> <s xml:id="echoid-s48186" xml:space="preserve"> quæ ſit l m:</s> <s xml:id="echoid-s48187" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s48188" xml:space="preserve"> medietas lineæ l m quinq;</s> <s xml:id="echoid-s48189" xml:space="preserve"> digitorũ:</s> <s xml:id="echoid-s48190" xml:space="preserve"> diuidaturq́;</s> <s xml:id="echoid-s48191" xml:space="preserve"> in duo <lb/>ęqualia in pũcto n:</s> <s xml:id="echoid-s48192" xml:space="preserve"> & à cẽtro baſis, quod eſt e, ducatur linea e n:</s> <s xml:id="echoid-s48193" xml:space="preserve"> & ꝓducatur illa linea ex utraq;</s> <s xml:id="echoid-s48194" xml:space="preserve"> par-<lb/>te ad peripheriá ut fiat diameter o n e p.</s> <s xml:id="echoid-s48195" xml:space="preserve"> Erit itaq;</s> <s xml:id="echoid-s48196" xml:space="preserve"> per 3 p 3 linea e n perpendicularis ſuper lineã l m:</s> <s xml:id="echoid-s48197" xml:space="preserve"> <lb/>& ducatur linea e l:</s> <s xml:id="echoid-s48198" xml:space="preserve"> & compleatur diameter l q:</s> <s xml:id="echoid-s48199" xml:space="preserve"> hæitaq;</s> <s xml:id="echoid-s48200" xml:space="preserve"> duæ diametri o p & l q profundẽtur cultro:</s> <s xml:id="echoid-s48201" xml:space="preserve"> <lb/>& impleatur diametri p o concauitas colore albo, & diametri l q cõcauitas colore alio.</s> <s xml:id="echoid-s48202" xml:space="preserve"> Deinde po-<lb/>natur uitrum ſuper baſim columnæ, taliter, ut altera extremitas longitudinis ſuperponatur medie-<lb/>tati lineæ l m, quæ eſt n l.</s> <s xml:id="echoid-s48203" xml:space="preserve"> Et quia uitrum eſt in longitudine octo digitorũ, & linea l n quinq;</s> <s xml:id="echoid-s48204" xml:space="preserve"> digito-<lb/>rum:</s> <s xml:id="echoid-s48205" xml:space="preserve"> patet quòd longitudo uitri excedit quantitatẽ lineæ l n in tribus digitis:</s> <s xml:id="echoid-s48206" xml:space="preserve"> & diſtinguátur deui-<lb/>tro tres digiti, de quibus duo erũt ex parte diametri l q decliuis extra circulũ:</s> <s xml:id="echoid-s48207" xml:space="preserve"> & remanebit de lon-<lb/>gitudine uitri unus digitus ultra diametrũ p o perpendicularẽ ſuper lineã l m:</s> <s xml:id="echoid-s48208" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s48209" xml:space="preserve"> corpus uitri ex <lb/>parte centri e, ſcilicet inter lineã l m & centrũ e:</s> <s xml:id="echoid-s48210" xml:space="preserve"> & ſic applicetur uitrum tabulæ per glutinum:</s> <s xml:id="echoid-s48211" xml:space="preserve"> erit <lb/>itaq;</s> <s xml:id="echoid-s48212" xml:space="preserve"> perpendicularis p o erecta ſuper extremitates uitri, quæ ſunt ſuperficies duæ æ quidiftantes:</s> <s xml:id="echoid-s48213" xml:space="preserve"> <lb/>& diameter l q erit obliqua ſuper illas duas ſuperficies.</s> <s xml:id="echoid-s48214" xml:space="preserve"> Ponatur itaq;</s> <s xml:id="echoid-s48215" xml:space="preserve"> peripheria circuli, cui ſupere-<lb/>minet extremitas uitri, ex parte uiſus experimentantis:</s> <s xml:id="echoid-s48216" xml:space="preserve"> & ponatur alter uiſuũ in differẽtia cõmuni <lb/>circumferentiæ baſis & extremitatis uitri:</s> <s xml:id="echoid-s48217" xml:space="preserve"> hoc eſt in pũcto l, quod eſt extremitas diametri decliuis, <lb/>quę eſt l q:</s> <s xml:id="echoid-s48218" xml:space="preserve"> & applicetur taliter uitro, ita qđ nihil uideatur cũillo oculo, niſi ſolus pũctus l:</s> <s xml:id="echoid-s48219" xml:space="preserve"> reliquus <lb/> <pb o="418" file="0720" n="720" rhead="VITELLONIS OPTICAE"/> uerò uiſus ſit in parte, in qua eſt uitrũ & circulus:</s> <s xml:id="echoid-s48220" xml:space="preserve"> & cooperiatur illud, quod opponitur ei exſuper-<lb/>ficie uitri cum panno linteo uel bombace, applicata taliter ſuperficiei columnæ, utnon uideatur, <lb/>niſi ſola diameter decliuis l q per unum uiſum contingentem uitrum:</s> <s xml:id="echoid-s48221" xml:space="preserve"> diameter uerò p o perpendi-<lb/>cularis alba uideatur utroq;</s> <s xml:id="echoid-s48222" xml:space="preserve"> uiſu.</s> <s xml:id="echoid-s48223" xml:space="preserve"> Sic itaq;</s> <s xml:id="echoid-s48224" xml:space="preserve"> diſpoſito uiſu & inſtrumento:</s> <s xml:id="echoid-s48225" xml:space="preserve"> centrum circuli e inuenie-<lb/>tur in rectitudine diametri p o albæ, quæ eſt erecta ſuper ſuperſiciem uitri:</s> <s xml:id="echoid-s48226" xml:space="preserve"> & inuenietur diameter <lb/>decliuis, quæ eſt l q, incuruata in ſuperficie uitri, quæ eſt ex parte centri e:</s> <s xml:id="echoid-s48227" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s48228" xml:space="preserve"> angulus incurua.</s> <s xml:id="echoid-s48229" xml:space="preserve"> <lb/>tionis ex parte circumferentiæ:</s> <s xml:id="echoid-s48230" xml:space="preserve"> ſed uiſus comprehendet partem diametri l q, quæ eſt ſub uitro, in <lb/>rectitudine.</s> <s xml:id="echoid-s48231" xml:space="preserve"> Et quoniam uiſus tangit ſuperficiẽ uitri, & diametri perpendicularis (quæ eſt p o) ali-<lb/>qua pars eſt ſub uitro, & alia extra uitrum ex parte cẽtri e, & alia extra uitrum ex parte extremitatis <lb/>diametri, ut eſt eius pars, quæ o n:</s> <s xml:id="echoid-s48232" xml:space="preserve"> pars illa, quę eſt ſub uitro, comprehen ditur à uiſu exiſtente extra <lb/>uitrum ſecundũ refractionem:</s> <s xml:id="echoid-s48233" xml:space="preserve"> & parson, quæ eſt ex parte extremitatis diametri, comprehenditur <lb/>à uiſu extra uitrum exiſtente rectè & ſine refractione:</s> <s xml:id="echoid-s48234" xml:space="preserve"> pars autem, quæ eſt ex parte centri, compre-<lb/>henditur ab utroq;</s> <s xml:id="echoid-s48235" xml:space="preserve"> uiſu per refractionem.</s> <s xml:id="echoid-s48236" xml:space="preserve"> Nam lineæ exeuntes à centro uiſus contingẽtis uitrum, <lb/>& extenſæ in corpore uitri peruenientes ad ſuperficiem uitri, quæ eſt ex parte centri, omnes fiunt <lb/>decliues ſuper ſuperficiem uitri.</s> <s xml:id="echoid-s48237" xml:space="preserve"> Pars ergo perpendicularis diametri p o, illa, quæ eſt ex parte cẽtri, <lb/>comprehenditur à uiſu contingente uitrum per refractionem:</s> <s xml:id="echoid-s48238" xml:space="preserve"> lineæ uerò exeuntes à reliquo uiſu <lb/>ad ſuperiorem uitri ſuperficiem erunt decliues ſuper ſuperiorem uitri ſuperficiem.</s> <s xml:id="echoid-s48239" xml:space="preserve"> Cum ergo ex-<lb/>tenduntur ad ſuperficiem uitri reliquam, quę eſt ex parte centrie, erunt etiam decliues ſuper illam, <lb/>ut patet per 23 th.</s> <s xml:id="echoid-s48240" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s48241" xml:space="preserve"> illæ enim ſuperficies uitri ſunt æ quidiſtantes ex hypotheſi.</s> <s xml:id="echoid-s48242" xml:space="preserve"> Viſus itaq;</s> <s xml:id="echoid-s48243" xml:space="preserve"> ille <lb/>comprehendet etiam partem diametri p o, quæ eſt uerſus cẽtrum e, duabus refractionibus:</s> <s xml:id="echoid-s48244" xml:space="preserve"> partem <lb/>uerò, quæ eſt ſub uitro, una ſola refractione:</s> <s xml:id="echoid-s48245" xml:space="preserve"> partem uerò ſuperiorem, quæ eſt p o, comprehendet <lb/>abſq;</s> <s xml:id="echoid-s48246" xml:space="preserve"> refractione:</s> <s xml:id="echoid-s48247" xml:space="preserve"> uterq, tamen uiſuum comprehendit hãc diametrum p o rectam.</s> <s xml:id="echoid-s48248" xml:space="preserve"> Et ſi experimen-<lb/>tator cooperto altero uiſu aſpiciat ſolum per uiſum, qui poſitus eſt ſuper uitrum:</s> <s xml:id="echoid-s48249" xml:space="preserve"> comprehendet <lb/>perpẽdicularem p o rectam:</s> <s xml:id="echoid-s48250" xml:space="preserve"> & ſi eleuauerit uiſum à ſuperficie uitri, & intueatur diametrum p o ul-<lb/>tra uitrum:</s> <s xml:id="echoid-s48251" xml:space="preserve"> comprehendet tamen ipſam lineam rectam, quamuis comprehendat ipſam ſecundum <lb/>refractionem:</s> <s xml:id="echoid-s48252" xml:space="preserve"> quoniam quilibet punctus diametri p o, & ſi non comprehendatur à uiſu in ſuo loco, <lb/>comprehenditur tamen in rectitudine perpendicularis, quæ exità puncto illo ſuper ſuperficiem <lb/>uitri:</s> <s xml:id="echoid-s48253" xml:space="preserve"> hæc autem eſt ſola ipſa linea p o per 20 th.</s> <s xml:id="echoid-s48254" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s48255" xml:space="preserve"> quoniam ab uno puncto ſuper quam cunq;</s> <s xml:id="echoid-s48256" xml:space="preserve"> <lb/>ſuperficiem unam tantùm perpendicularem duci eſt poſsibile:</s> <s xml:id="echoid-s48257" xml:space="preserve"> hæc autem linea, quæ eſt p o, à quo-<lb/>libet ſui puncto procedit perpendiculariter ſuper ſuperficiem uitri.</s> <s xml:id="echoid-s48258" xml:space="preserve"> Omnis ergo refractio ſuorum <lb/>punctorum fit ſuper ipſam eandem.</s> <s xml:id="echoid-s48259" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s48260" xml:space="preserve"> centri e, quando uiſus tangit uitrum, comprehen-<lb/>ditur in rectitudine diametri p o, exeuntis perpendiculariter à centro e ſuper ſuperficiem uitri:</s> <s xml:id="echoid-s48261" xml:space="preserve"> & <lb/>diametri decliuis l q pars extra uitrum exiſtens uerſus centrum e comprehẽditur non in ſuo loco:</s> <s xml:id="echoid-s48262" xml:space="preserve"> <lb/>ideo quia punctus centri e non comprehenditur à uiſu, niſi præter ſuum locum:</s> <s xml:id="echoid-s48263" xml:space="preserve"> & cum angulus in-<lb/>curuationis ſuerit ex parte circumferentiæ:</s> <s xml:id="echoid-s48264" xml:space="preserve"> tunc forma centri e uidetur ſub centro baſis columnæ.</s> <s xml:id="echoid-s48265" xml:space="preserve"> <lb/>Quia ergo forma cuiuslibet puncti comprehenſi à uiſu in ſecũdo medio rarioris diaphani illo dia-<lb/>phano, in quo eſt uiſus, eſt in rectitudine perpendicularis, productæ ab illo puncto ſuper ſuperficiẽ <lb/>corporis diaphani, quod eſt contingens uiſum, & eſt remotior à ſuperficie eiuſdem diaphani quàm <lb/>ipſum punctum, cuius uidetur forma:</s> <s xml:id="echoid-s48266" xml:space="preserve"> & quoniam omne punctum comprehenſum à uiſu per 12 hu-<lb/>ius eſt in rectitudine lineæ, per quam forma peruenit ad uiſum:</s> <s xml:id="echoid-s48267" xml:space="preserve"> patet quòd forma cuiuslibet pun-<lb/>cti in quibuſcunq;</s> <s xml:id="echoid-s48268" xml:space="preserve"> diaphanis taliter ſituatis comprehenditur in puncto, qui eſt communis ſectio li-<lb/>neæ, per quam forma peruenit ad uiſum, & lineæ perpendicularis, exeuntis à puncto rei uiſæ ſuper <lb/>ſuperficiem corporis diaphani, quod eſt contingens uiſum.</s> <s xml:id="echoid-s48269" xml:space="preserve"> Et patet ex præmiſsis corollarium.</s> <s xml:id="echoid-s48270" xml:space="preserve"> Lo-<lb/>cus enim formæ puncti rei uiſæ per refractionem, quãdo fit illa refractio in medio ſecundi diapha-<lb/>ni denſiore primo:</s> <s xml:id="echoid-s48271" xml:space="preserve"> tunc locus imaginis approximatipſi uiſui, ut patet in experimentatione prima <lb/>de centro e, cum ipſum uidetur ſub aqua:</s> <s xml:id="echoid-s48272" xml:space="preserve"> cum uerò fit reſractio à ſuperficie alterius diaphani rario-<lb/>ris primo diaphano contingente uiſum:</s> <s xml:id="echoid-s48273" xml:space="preserve"> tunc locus imaginis elongatur à uiſu, ut patet in experi-<lb/>mentatione ſecunda de centro e uiſo ſub uitro approximato uiſibus, cuius forma per medium ra-<lb/>rius uitro, quod eſt aer, diffunditur ad uitri ſuperficiẽ, & per uitrum refringitur ad uiſum:</s> <s xml:id="echoid-s48274" xml:space="preserve"> ut etiam <lb/>exemplariter patet in prima figura præſentis propoſitionis:</s> <s xml:id="echoid-s48275" xml:space="preserve"> punctum enim x propinquius eſt uiſui <lb/>exiſtenti in puncto d, quàm punctum z.</s> <s xml:id="echoid-s48276" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s48277" xml:space="preserve"> propoſitum.</s> <s xml:id="echoid-s48278" xml:space="preserve"/> </p> <div xml:id="echoid-div1818" type="float" level="0" n="0"> <figure xlink:label="fig-0718-02" xlink:href="fig-0718-02a"> <variables xml:id="echoid-variables826" xml:space="preserve">e d d c b r a z</variables> </figure> <figure xlink:label="fig-0719-01" xlink:href="fig-0719-01a"> <variables xml:id="echoid-variables827" xml:space="preserve">h m k o n q e j p i g</variables> </figure> </div> </div> <div xml:id="echoid-div1820" type="section" level="0" n="0"> <head xml:id="echoid-head1344" xml:space="preserve" style="it">16. Formæ puncti rei uiſæ per refr actionem, exiſtentis in medio ſecundi diaphani, locus ima-<lb/>ginis quando eſt in ipſo ſecundo corpore diaphano: quando in eius ſuperficie ut in ipſo puncto <lb/>refractionis: quando eſt inter uiſum & illud corpus diaphanum: quando retro uiſum: quan-<lb/>do in ipſa ſuperficie uiſus.</head> <p> <s xml:id="echoid-s48279" xml:space="preserve">Quia enim oſtenſum eſt per præmiſſam, quòd locus imaginis refractæ cuiuslibet puncti rei per <lb/>refractionem uiſæ eſt in communi ſectione lineæ, per quam forma peruenit ad uiſum, & lineæ per-<lb/>pendicularis, exeuntis ab illo puncto rei uiſæ ſuper ſuperficiem corporis diaphani uiſum contin-<lb/>gentis:</s> <s xml:id="echoid-s48280" xml:space="preserve"> cum itaq;</s> <s xml:id="echoid-s48281" xml:space="preserve"> illæ lineæ neceſſariò concurrant:</s> <s xml:id="echoid-s48282" xml:space="preserve"> aut æ quidiſtent:</s> <s xml:id="echoid-s48283" xml:space="preserve"> patet quòd ſi concurrunt, ubi-<lb/>cunq;</s> <s xml:id="echoid-s48284" xml:space="preserve"> illæ lineæ ſe interſecuerint, ſiue hoc ſit intra corpus diaphanũ, in quo eſt pũctus rei uiſæ:</s> <s xml:id="echoid-s48285" xml:space="preserve"> ſiue <lb/>fuerit extra illud corpus inter uiſum & ſuperficiẽ illius corporis:</s> <s xml:id="echoid-s48286" xml:space="preserve"> ſiue hoc fuerit in centro uiſus, ſiue <lb/>retro uiſum:</s> <s xml:id="echoid-s48287" xml:space="preserve"> ibi ſemper erit locus imaginis formę puncti rei uiſæ.</s> <s xml:id="echoid-s48288" xml:space="preserve"> Si uerò illa linea, per quam forma <lb/>peruenit ad uiſum, fuerit æquidiſtans illi perpẽdiculari:</s> <s xml:id="echoid-s48289" xml:space="preserve"> tuncnon erit aliqua certitudo propria loci <lb/> <pb o="419" file="0721" n="721" rhead="LIBER DECIMVS."/> illius imaginis, niſi ſolum ipſum punctum refractionis.</s> <s xml:id="echoid-s48290" xml:space="preserve"> In illo ergo uidebitur imago illius formæ <lb/>ſicut etiam acciditidẽ, quando linea refractionis & dicta perpen dicularis in ipſo puncto refractio-<lb/>nis ſe interſecant:</s> <s xml:id="echoid-s48291" xml:space="preserve"> nec indigent hæc alia demonſtratione, niſi illa quam in 11 th.</s> <s xml:id="echoid-s48292" xml:space="preserve"> 8 huius in ſpeculis <lb/>ſphæricis cõcauis poſuimus:</s> <s xml:id="echoid-s48293" xml:space="preserve"> hæc enim refractio, ut patet per 7 huius, quandoq;</s> <s xml:id="echoid-s48294" xml:space="preserve"> fit à ſuperficie con-<lb/>caua corporis diaphani, quod corpus eſt ex parte uiſus contingens conuexum corporis diaphani, <lb/>quod eſt ex parte rei uiſæ:</s> <s xml:id="echoid-s48295" xml:space="preserve"> unde eſt omnimoda demonſtrationis ſimilitudo faciendæ hinc & inde.</s> <s xml:id="echoid-s48296" xml:space="preserve"> <lb/>Patet ergo propoſitum:</s> <s xml:id="echoid-s48297" xml:space="preserve"> diuerſantur enim illæ perpendiculares ſecundum diuerſitatem ſuperficie-<lb/>rum corporum, à quibus fit refractio.</s> <s xml:id="echoid-s48298" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1821" type="section" level="0" n="0"> <head xml:id="echoid-head1345" xml:space="preserve" style="it">17. In refractione formarum à ſuperficiebus corporũ alterius diaphanitatis ad uiſum, ſem-<lb/>per fit deceptio in ſitu.</head> <p> <s xml:id="echoid-s48299" xml:space="preserve">Quoniam enim ſecundum omnes lineas, per quas forma extenditur ad uiſum, ſemper fit refra-<lb/>ctio in ſuperficie corporis alterius diaphanitatis, ut linea, per quam forma extenditur in medio <lb/>unius diaphani, angulum contineat cum linea illa, per quam in ſecundo diàphano forma peruenit <lb/>ad uiſum:</s> <s xml:id="echoid-s48300" xml:space="preserve"> ſola uero perpendicularis ducta à puncto uiſo ſuper ſuperficiem corporis diaphani non <lb/>refringitur:</s> <s xml:id="echoid-s48301" xml:space="preserve"> & omnis imaginis refractæ locus eſt in communi ſectione lineæ ſecũdæ, per quam for-<lb/>ma refracta extenditur ad uiſum, & lineæ perpendicularis, exeuntis à puncto rei uiſæ ſuper ſuperfi-<lb/>ciem corporis diaphani uiſum contingentis per 15 th.</s> <s xml:id="echoid-s48302" xml:space="preserve"> huius:</s> <s xml:id="echoid-s48303" xml:space="preserve"> hæc autem ſectio ſemper eſt extra lo-<lb/>cum uerum puncti uiſi:</s> <s xml:id="echoid-s48304" xml:space="preserve"> quoniam ſola linea incidentiæ concurrit cũ illa perpendiculari in ipſo pun-<lb/>cto rei uiſæ, à quo ambæ illæ lineæ producũtur.</s> <s xml:id="echoid-s48305" xml:space="preserve"> Palàm ergo quia uiſus nunquam uidet formam rel <lb/>uiſæ per refractionem niſi in alio loco & ſitu, quàm ſit ipſa res uiſa:</s> <s xml:id="echoid-s48306" xml:space="preserve"> erit ltaq;</s> <s xml:id="echoid-s48307" xml:space="preserve"> poſitio formæ compre-<lb/>henſæ à uiſu alia à poſitione rei uiſæ.</s> <s xml:id="echoid-s48308" xml:space="preserve"> Et ſimiliter eſt de remotione:</s> <s xml:id="echoid-s48309" xml:space="preserve"> hæc autem ſunt quidam ſitus.</s> <s xml:id="echoid-s48310" xml:space="preserve"> <lb/>Punctus enim communis ſectionis dictarum linearum faciens locum imaginis, in refractione ex <lb/>diaphano denſiore ad ſubtilius ſe eleuat approximando uiſui, & in refractione ex diaphano rariori <lb/>ad denſius ſe deprimit, remouendo ſe à centro uiſus, ut patuit per corollarium 15 huius.</s> <s xml:id="echoid-s48311" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s48312" xml:space="preserve"> <lb/>quòd locus imaginis ſemper ſe uariat:</s> <s xml:id="echoid-s48313" xml:space="preserve"> & ſecundum hoc decipitur uiſus ſecundum ſitum imaginis;</s> <s xml:id="echoid-s48314" xml:space="preserve"> <lb/>alium locum rei uiſæ & ſituationem aliam accipiens ſecundum illud.</s> <s xml:id="echoid-s48315" xml:space="preserve"> Pater ergo propoſitum.</s> <s xml:id="echoid-s48316" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1822" type="section" level="0" n="0"> <head xml:id="echoid-head1346" xml:space="preserve" style="it">18. Omnis forma rei uiſæ per refractionem comprehenditur, acſi res illius formæ ſit in loco <lb/>imaginis conſtituta. Alhazen 19 n 7.</head> <p> <s xml:id="echoid-s48317" xml:space="preserve">Sicut enim in 13 th.</s> <s xml:id="echoid-s48318" xml:space="preserve"> huius dictum eſt, forma exiſtens in puncto refractionis peruenit ad ipſum ui-<lb/>ſum per motum formæ, quæ mouetur ſuper lmeam perpendicularem ſuper ſuperficiem corporis <lb/>diaphani, ductam à puncto rei uiſæ:</s> <s xml:id="echoid-s48319" xml:space="preserve"> deinde transfertur ab hac perpendiculari per motum in recti-<lb/>tudine lineæ, per quam forma peruenit ad uiſum.</s> <s xml:id="echoid-s48320" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s48321" xml:space="preserve">, quæ eſt ſuper lineam perpendicula-<lb/>riter incidentem ſuperficiei corporis diaphani, & deinde mouetur in rectitudine lineæ, per quam <lb/>forma extenditur ad uiſum, eſt forma, quę extenditur à pũcto uiſo in rectitudine perpendicularis, <lb/>exeuntis exipſo ſuper ſuperficiem corporis diaphani, donec perueniat ad punctum ſectionis inter <lb/>hanc perpendicularem & lineam, per quam forma extenditur ad uiſum.</s> <s xml:id="echoid-s48322" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s48323" xml:space="preserve">, quam uiſus <lb/>comprehendit refractam ultra corpus diaphanum, eſt per motum formæ, quæ peruenit ad uiſum a <lb/>loco imaginis:</s> <s xml:id="echoid-s48324" xml:space="preserve"> comprehendit autem uiſus hanc formam in loco imaginis ſicut alia, quæ in ſuo loco <lb/>comprehendit ſine refractione per medium unius diaphani & directè.</s> <s xml:id="echoid-s48325" xml:space="preserve"> Videturitaq;</s> <s xml:id="echoid-s48326" xml:space="preserve"> res diſtãs tan-<lb/>tùm à centro uiſus, quantùm punctus imaginis diſtat ab eodem centro uiſus:</s> <s xml:id="echoid-s48327" xml:space="preserve"> quoniam ſitus loci <lb/>imaginis in reſpectu uiſus, eſt ſitus formæ, quæ eſt in loco imaginis:</s> <s xml:id="echoid-s48328" xml:space="preserve"> unde propter refractionẽ for-<lb/>marei uiſæ comprehenditur in loco imaginis.</s> <s xml:id="echoid-s48329" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s48330" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1823" type="section" level="0" n="0"> <head xml:id="echoid-head1347" xml:space="preserve" style="it">19. Communi ſectione ſuperficieirefr actionis & ſuperficiei corporis diaphani, à qua fit re-<lb/>fractio, exiſtente linea recta, punctó rei uiſæ exiſtente in perpendiculari ducta à centro uiſus <lb/>ſuper ſuperficiem corporis diaphani qualiſcun: à nullo puncto illius ſuperficiei fiet refractio: & <lb/>una tantùm imago uiſui occurret. Alhazen 21 n 7.</head> <p> <s xml:id="echoid-s48331" xml:space="preserve">Eſto centrum uiſus punctus a:</s> <s xml:id="echoid-s48332" xml:space="preserve"> & punctus rei uiſæ b:</s> <s xml:id="echoid-s48333" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s48334" xml:space="preserve"> g aliquod punctum ſuperficiei corpo-<lb/>ris, à qua fit refractio, quod ſit groſsioris uel rarioris diaphanitatis quàm corpus, quod eſt contin-<lb/>gens uiſum:</s> <s xml:id="echoid-s48335" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s48336" xml:space="preserve"> à puncto a cẽtro uiſus linea a g c:</s> <s xml:id="echoid-s48337" xml:space="preserve"> quæ ſit perpendicularis ſuper ſuperficiem <lb/>corporis ſecundi diaphani per 11 p 11:</s> <s xml:id="echoid-s48338" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s48339" xml:space="preserve"> punctus rei uiſæ, qui eſt b, in linea g c.</s> <s xml:id="echoid-s48340" xml:space="preserve"> Palàm ergo per 3 th.</s> <s xml:id="echoid-s48341" xml:space="preserve"> <lb/>huius quoniam uiſus a comprehendet ſormam puncti b rectè ſine omni refractione.</s> <s xml:id="echoid-s48342" xml:space="preserve"> Quia enim <lb/>forma puncti b in rectitudine extenditur per lineam b g ad ſuperficiem corporis diaphani, quod eſt <lb/>contingens uiſum in puncto a:</s> <s xml:id="echoid-s48343" xml:space="preserve"> & quia linea b g eſt perpendicularis ſuper ſuperficiem corporis dia-<lb/>phani contingentis uiſum:</s> <s xml:id="echoid-s48344" xml:space="preserve"> comprehendet ergo uiſus a punctum b in ſuo loco ſecundum recti-<lb/>tudinem lineæ a g b.</s> <s xml:id="echoid-s48345" xml:space="preserve"> Non eſt itaque poſsibile, ut punctum b extra lineam b g a refringatur ad ui-<lb/>ſum a.</s> <s xml:id="echoid-s48346" xml:space="preserve"> Siautem detur hoc eſſe poſsibile:</s> <s xml:id="echoid-s48347" xml:space="preserve"> ſit ſuperſiciei illius diaphani, in qua eſt punctus refractio-<lb/>nis g, alter punctus refractionis, qui ſit p, extra lineam a g b:</s> <s xml:id="echoid-s48348" xml:space="preserve"> & refringatur forma puncti b ad a <lb/>centrum uiſus à puncto p.</s> <s xml:id="echoid-s48349" xml:space="preserve"> Imaginemur itaque ſuperficiem refractionis, in qua fit linea perpen-<lb/>dicularis, quæ a g b, tranſire perpunctum p:</s> <s xml:id="echoid-s48350" xml:space="preserve"> & ſit communis ſectio huius ſuperficiei & ſuper-<lb/>ficiei corporis diaphani, in qua fit refractio, linea recta, quæ eſt g p d per 3 p 11:</s> <s xml:id="echoid-s48351" xml:space="preserve"> & à puncto p ex-<lb/>trahatur perpendicularis ſuper lineam g d per 11 p 1:</s> <s xml:id="echoid-s48352" xml:space="preserve"> quæ ſit k p l:</s> <s xml:id="echoid-s48353" xml:space="preserve"> & ſit linea k p l producta <lb/> <pb o="420" file="0722" n="722" rhead="VITELLONIS OPTICAE"/> ſecans ipſum corpus diaphanum, à cuius ſuperficie fit refractio formæ pũcti b ad uiſum a.</s> <s xml:id="echoid-s48354" xml:space="preserve"> Eſt erge <lb/>linea k p l perpendicularis ſuper ſuper ſuperficiem illius corporis diapha-<lb/> <anchor type="figure" xlink:label="fig-0722-01a" xlink:href="fig-0722-01"/> ni:</s> <s xml:id="echoid-s48355" xml:space="preserve"> ducatur itaq;</s> <s xml:id="echoid-s48356" xml:space="preserve"> linea b p, & producatur ultra corpus diaphanũ uſq;</s> <s xml:id="echoid-s48357" xml:space="preserve"> <lb/>ad punctũ h.</s> <s xml:id="echoid-s48358" xml:space="preserve"> Erit ergo angulus k p h contentus â linea p h, per quam <lb/>extenditur forma, & à linea k p perpendiculari, exeunte à puncto re-<lb/>fractionis, quod eſt p, ſuper ſuperficiẽ corporis diaphani.</s> <s xml:id="echoid-s48359" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s48360" xml:space="preserve"> <lb/>corpus diaphanum, quod eſt ex parte uiſus a, eſt ſubtilius illo, quod <lb/>eſt ex parte ipſius b puncti rei uiſæ:</s> <s xml:id="echoid-s48361" xml:space="preserve"> tunc, cum forma puncti b per-<lb/>uenerit ad p punctum refractionis:</s> <s xml:id="echoid-s48362" xml:space="preserve"> palàm per 4 huius quia refringe-<lb/>tur ad partem cótrariam illi parti, in qua eſt perpendicularis k l:</s> <s xml:id="echoid-s48363" xml:space="preserve"> non <lb/>ergo perueniet forma refracta ad lineam a b:</s> <s xml:id="echoid-s48364" xml:space="preserve"> ergo neq;</s> <s xml:id="echoid-s48365" xml:space="preserve"> ad punctum <lb/>a, quod eſt centrum uiſus:</s> <s xml:id="echoid-s48366" xml:space="preserve"> ſed datum eſt ipſum refringi à puncto p ad <lb/>punctum a:</s> <s xml:id="echoid-s48367" xml:space="preserve"> accidit igitur impoſsibile cõtra hypotheſim:</s> <s xml:id="echoid-s48368" xml:space="preserve"> & quocũq;</s> <s xml:id="echoid-s48369" xml:space="preserve"> <lb/>alio puncto dato idem accidit impoſsibile.</s> <s xml:id="echoid-s48370" xml:space="preserve"> Non ergo refringitur for-<lb/>ma puncti b ad uiſum a ex aliquo puncto ſuperficiei illius corporis <lb/>diaphani dato extra lineam a g b:</s> <s xml:id="echoid-s48371" xml:space="preserve"> ſed ſolùm forma illa pũcti b ſecun-<lb/>dum rectitudinem peruenit ad uiſum a.</s> <s xml:id="echoid-s48372" xml:space="preserve"> Quòd ſi corpus diaphanum <lb/>contingẽs ſuperficiem uiſus ſit denſius illo corpore diaphano, quod <lb/>eſt continens punctum rei uiſæ:</s> <s xml:id="echoid-s48373" xml:space="preserve"> tunc eadem linea p h refringetur ad <lb/>partem perpendicularis p k;</s> <s xml:id="echoid-s48374" xml:space="preserve"> propter denſitatem diaphani ſecundi:</s> <s xml:id="echoid-s48375" xml:space="preserve"> <lb/>nec tamen concurret unquam cum perpendiculari p k:</s> <s xml:id="echoid-s48376" xml:space="preserve"> ergo neque <lb/>cum linea a b æ quidiſtante ipſi p k per 6 p 11:</s> <s xml:id="echoid-s48377" xml:space="preserve"> quoniam am b æ lineæ a b & k l ſunt erectę ſuper ſuper-<lb/>ficiem corporis diaphani, in qua eſt linea g p d.</s> <s xml:id="echoid-s48378" xml:space="preserve"> Qualecunq;</s> <s xml:id="echoid-s48379" xml:space="preserve"> ergo fuerit diaphanum ſecundum, ſci-<lb/>licet rarius uel denſius primo diaphano, ſemper puncto rei uiſæ ſic diſpoſito, à nullo puncto illius <lb/>ſuperficiei diaphani fiet refractio ad uiſum:</s> <s xml:id="echoid-s48380" xml:space="preserve"> ſed uidebitur res in ipſa linea perpendiculari ducta à <lb/>centro uiſus ad punctum rei uiſæ, ſecante ſuperficiem corporis ſecundi diaphani in uno tantùm <lb/>puncto g.</s> <s xml:id="echoid-s48381" xml:space="preserve"> Forma ergo illius puncti non comprehẽditur niſi ex uno tantùm puncto ſuperſiciei illius <lb/>corporis diaphani:</s> <s xml:id="echoid-s48382" xml:space="preserve"> habet ergo tantùm unicam imaginem non refractam.</s> <s xml:id="echoid-s48383" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s48384" xml:space="preserve"/> </p> <div xml:id="echoid-div1823" type="float" level="0" n="0"> <figure xlink:label="fig-0722-01" xlink:href="fig-0722-01a"> <variables xml:id="echoid-variables828" xml:space="preserve">a k h g d p b c j</variables> </figure> </div> </div> <div xml:id="echoid-div1825" type="section" level="0" n="0"> <head xml:id="echoid-head1348" xml:space="preserve" style="it">20. Comuni ſectione ſuperficieirefractionis & ſuperficiei corporis diaphani, à qua fit refra-<lb/>ctio, exiſtente linea recta, punctó uiſo exiſtente extr a perpendicularem duct am à centro ui-<lb/>ſus ſuper ſuperficiem corporis diaphani denſioris diaphano uiſum contingente: ab uno tantùm <lb/>puncto fiet refractio: & uidebitur unica imago. Alhazen 22 n 7.</head> <p> <s xml:id="echoid-s48385" xml:space="preserve">Remaneat diſpoſitio, quæ in proxima præcedente:</s> <s xml:id="echoid-s48386" xml:space="preserve"> & ſit punctus b extra lineam perpendicula-<lb/>rem ductam à centro uiſus a ſuper ſuperficiem ſecundi diaphani, quæ eſt a g c.</s> <s xml:id="echoid-s48387" xml:space="preserve"> Educatur quoq;</s> <s xml:id="echoid-s48388" xml:space="preserve"> ſu-<lb/>perficies plana per lineam a g c & per punctum b:</s> <s xml:id="echoid-s48389" xml:space="preserve"> hæc itaq;</s> <s xml:id="echoid-s48390" xml:space="preserve"> erit perpendicularis ſuper ſuperficiem <lb/>ſecundi corporis diaphani per 18 p 11:</s> <s xml:id="echoid-s48391" xml:space="preserve"> & ſecabit ſuperficiem corporis diaphani ſecũdum lineam re-<lb/>ctam per 3 p 11:</s> <s xml:id="echoid-s48392" xml:space="preserve"> quæ ſit g d.</s> <s xml:id="echoid-s48393" xml:space="preserve"> Non ergo refringetur per 2 th.</s> <s xml:id="echoid-s48394" xml:space="preserve"> huius forma pũcti b ad uiſum a, niſi a b ali-<lb/>quo puncto ſuperficiei, in qua eſt linea g d:</s> <s xml:id="echoid-s48395" xml:space="preserve"> non enim tranſit per duo puncta a & b ſuperficies per-<lb/>pendicularis ſuper ſuperſiciem ſecundi corporis diaphani, niſi ſolùm ſuperficies tranſiens per per-<lb/>pendicularem a c:</s> <s xml:id="echoid-s48396" xml:space="preserve"> ſed per perpendicularem a c, & per punctũ b non tranſit aliqua ſuperficies plana <lb/>niſi una ſola tantûm.</s> <s xml:id="echoid-s48397" xml:space="preserve"> Forma ergo puncti b reſringitur a d punctum a centrum uiſus ab aliquo pun-<lb/>cto lineæ g d:</s> <s xml:id="echoid-s48398" xml:space="preserve"> qui ſit e:</s> <s xml:id="echoid-s48399" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s48400" xml:space="preserve"> duæ lineæ b e & e a:</s> <s xml:id="echoid-s48401" xml:space="preserve"> & extrahatur à puncto e linea perpendicula-<lb/>ris ſuper ſuperficiem g e d per 12 p 11:</s> <s xml:id="echoid-s48402" xml:space="preserve"> quæ ſit h e z:</s> <s xml:id="echoid-s48403" xml:space="preserve"> quæ per 1 th.</s> <s xml:id="echoid-s48404" xml:space="preserve"> huius erit in illa ſuperficie refractio-<lb/> <anchor type="figure" xlink:label="fig-0722-02a" xlink:href="fig-0722-02"/> nis:</s> <s xml:id="echoid-s48405" xml:space="preserve"> erit ergo linea h e z perpen-<lb/>dicularis ſuper duas ſuperficies <lb/>illorum duorum corporum dia-<lb/>phanorum:</s> <s xml:id="echoid-s48406" xml:space="preserve"> quia ducta eſt perpẽ-<lb/>diculariter in ſuperſicie erecta <lb/>ſuper illas ambas ſuքficies.</s> <s xml:id="echoid-s48407" xml:space="preserve"> Pro-<lb/>ducatur itaque linea b e in con-<lb/>tinuum & directum:</s> <s xml:id="echoid-s48408" xml:space="preserve"> & ſit linea b <lb/>e p:</s> <s xml:id="echoid-s48409" xml:space="preserve"> erit ergo linea e p cadens in-<lb/>ter duas lineas e h & e a per 41 th.</s> <s xml:id="echoid-s48410" xml:space="preserve"> <lb/>huius.</s> <s xml:id="echoid-s48411" xml:space="preserve"> Nam corpus diaphanum, <lb/>quod eſt ex parte a centri uiſus, <lb/>eſt ſubtilius corpore diaphano, <lb/>quod eſt ex parte b:</s> <s xml:id="echoid-s48412" xml:space="preserve"> ergo peridẽ <lb/>4 th.</s> <s xml:id="echoid-s48413" xml:space="preserve"> huius forma puncti b, quæ <lb/>extenditur per lineam b e, cum <lb/>perueniet ad e punctum datum <lb/>refractionis, refringetur ad par-<lb/>tem contrariam parti perpendicularis, quæ eft z e h:</s> <s xml:id="echoid-s48414" xml:space="preserve"> erit ergo linea e p inter duas lineas e h & e a.</s> <s xml:id="echoid-s48415" xml:space="preserve"> <lb/> <pb o="421" file="0723" n="723" rhead="LIBER DECIMVS."/> Ducaturitaq;</s> <s xml:id="echoid-s48416" xml:space="preserve"> à puncto uiſo b linea perpendicularis ſuper lineam g d per 12 p 1:</s> <s xml:id="echoid-s48417" xml:space="preserve"> quæ ſit b k:</s> <s xml:id="echoid-s48418" xml:space="preserve"> erit ergo <lb/>linea b k perpendicularis ſuper ſuperſiciem corporis diaphani, quod eſt ex parte b per conuerſam <lb/>4 definitionis 11:</s> <s xml:id="echoid-s48419" xml:space="preserve"> quia ducta eſt perpendiculariter in ſuperficie a b g erecta ſuper illã.</s> <s xml:id="echoid-s48420" xml:space="preserve"> Educatur itaq;</s> <s xml:id="echoid-s48421" xml:space="preserve"> <lb/>linea a e in continuum:</s> <s xml:id="echoid-s48422" xml:space="preserve"> hęc itaq;</s> <s xml:id="echoid-s48423" xml:space="preserve"> reſecabit ab angulo b e k angulum æqualem angulo p e a per 15 p 1:</s> <s xml:id="echoid-s48424" xml:space="preserve"> <lb/>ſecabit ergo per 29 th.</s> <s xml:id="echoid-s48425" xml:space="preserve"> 1 huius & lineam b k illi angulo ſubtenſam.</s> <s xml:id="echoid-s48426" xml:space="preserve"> Secetipſaitaq;</s> <s xml:id="echoid-s48427" xml:space="preserve"> lineam b k in pun-<lb/>ctom.</s> <s xml:id="echoid-s48428" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s48429" xml:space="preserve"> per 15 th.</s> <s xml:id="echoid-s48430" xml:space="preserve"> huius quoniam punctus meſt locus imaginis formæ puncti b:</s> <s xml:id="echoid-s48431" xml:space="preserve"> & angu-<lb/>lus p e a eſt angulus reſractionis.</s> <s xml:id="echoid-s48432" xml:space="preserve"> Dico itaq;</s> <s xml:id="echoid-s48433" xml:space="preserve"> quòd punctus b non habebit aliam imaginem præter <lb/>quàm illam, quæ eſt in punctom:</s> <s xml:id="echoid-s48434" xml:space="preserve"> nec forma eius refringetur ad uiſum in punctum a ab alio puncto <lb/>ſuperficiei corporis diaphani, quàm à puncto e.</s> <s xml:id="echoid-s48435" xml:space="preserve"> Necenim poteſt forma puncti b comprehendi à ui-<lb/>ſu, niſi ſecundum perpendicularem b k per 13 th.</s> <s xml:id="echoid-s48436" xml:space="preserve"> huius.</s> <s xml:id="echoid-s48437" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s48438" xml:space="preserve"> puncaus b aliam habuerit imaginem <lb/>quàm in puncto m:</s> <s xml:id="echoid-s48439" xml:space="preserve"> erit ille punctus in linea b k, & inter duo puncta b & k per 15 th.</s> <s xml:id="echoid-s48440" xml:space="preserve"> huius:</s> <s xml:id="echoid-s48441" xml:space="preserve"> quia cor-<lb/>pus, quod eſt ex parte b puncti uiſi, eſt groſsioris diaphanitatis illo corpore, quod eſt ex parte uiſus <lb/>a.</s> <s xml:id="echoid-s48442" xml:space="preserve"> Sit ita q;</s> <s xml:id="echoid-s48443" xml:space="preserve"> ſi poſsibile eſt illa alia imago formæ puncti b in puncto lineæ b k, quod ſit n.</s> <s xml:id="echoid-s48444" xml:space="preserve"> Erit itaque <lb/>punctus n aut inter duo puncta m, k:</s> <s xml:id="echoid-s48445" xml:space="preserve"> aut inter duo puncta m.</s> <s xml:id="echoid-s48446" xml:space="preserve"> b.</s> <s xml:id="echoid-s48447" xml:space="preserve"> Ducatur quoq;</s> <s xml:id="echoid-s48448" xml:space="preserve"> linea a n à centro ui-<lb/>ſus ad punctum n:</s> <s xml:id="echoid-s48449" xml:space="preserve"> hæc itaq;</s> <s xml:id="echoid-s48450" xml:space="preserve"> ſecabit lineam g d:</s> <s xml:id="echoid-s48451" xml:space="preserve"> ſunt enim puncta a, b, kin eadem ſuperficie cum li-<lb/>nea g d, ut patet ex præmiſsis, Secet ergo linea a n lineam g d in puncto o:</s> <s xml:id="echoid-s48452" xml:space="preserve"> ducaturá;</s> <s xml:id="echoid-s48453" xml:space="preserve"> linea b o :</s> <s xml:id="echoid-s48454" xml:space="preserve"> quæ <lb/>producta ultra puncturm o ſignetur ad punctum 1:</s> <s xml:id="echoid-s48455" xml:space="preserve"> erit itaq;</s> <s xml:id="echoid-s48456" xml:space="preserve"> pũctum o punctum refractionis formæ <lb/>puncti b ad uiſum in punctum a:</s> <s xml:id="echoid-s48457" xml:space="preserve"> quia b o l eſt linea, per qua m extenditur forma:</s> <s xml:id="echoid-s48458" xml:space="preserve"> & eſt angulus l o a <lb/>angulus refractionis.</s> <s xml:id="echoid-s48459" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s48460" xml:space="preserve"> à puncto o linea perpẽdicularis ſuper lineam g d per 11 p 1, quæ <lb/>ſit linea f o q:</s> <s xml:id="echoid-s48461" xml:space="preserve"> erit itaq;</s> <s xml:id="echoid-s48462" xml:space="preserve"> linea f o q perpendicularis ſuper ſuperficiem corporis diaphani per 28 p 1 & <lb/>per 8 p 11:</s> <s xml:id="echoid-s48463" xml:space="preserve"> & erit angulus l o f æqualis angulo o b k contento à perpendiculari k b & à linea b o, per <lb/>quam extenditur forma ad locum refractionis per 29 p 1:</s> <s xml:id="echoid-s48464" xml:space="preserve"> quoniam, ut patet per 6 p 11, lineæ b k & f o <lb/>q ſunt æ quidiſtantes.</s> <s xml:id="echoid-s48465" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s48466" xml:space="preserve"> punctus n fuerit inter duo puncta m & k:</s> <s xml:id="echoid-s48467" xml:space="preserve"> tũc punctus o erit inter duo <lb/>puncta e & k, ſecans lineam e k per 32 th.</s> <s xml:id="echoid-s48468" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s48469" xml:space="preserve"> erit ita q;</s> <s xml:id="echoid-s48470" xml:space="preserve"> angulus e b k maior angulo o b k per 29 <lb/>th.</s> <s xml:id="echoid-s48471" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s48472" xml:space="preserve"> quia omne totum eſt maius ſua parte.</s> <s xml:id="echoid-s48473" xml:space="preserve"> Et quia angulus p e h eſt æ qualis angulo e b k per <lb/>29 p 1, & angulus l o f æ qualis angulo l b k per eandem 29 p 1:</s> <s xml:id="echoid-s48474" xml:space="preserve"> quoniam lineæ h z & f q & b k ſuntin-<lb/>ter ſe æ quidiſtátes:</s> <s xml:id="echoid-s48475" xml:space="preserve"> erit ergo angulus p e h maior angulo l o f:</s> <s xml:id="echoid-s48476" xml:space="preserve"> & angulus p e a eſt angulus refractio-<lb/>nis ex angulo incidentiæ, qui eſt p e h:</s> <s xml:id="echoid-s48477" xml:space="preserve"> & angulus l o a eſt angulus refractionis ex angulo inciden-<lb/>tiæ, qui eſt l o f:</s> <s xml:id="echoid-s48478" xml:space="preserve"> angulus ergo p e a eſt maior angulo l o a per 8 huius.</s> <s xml:id="echoid-s48479" xml:space="preserve"> Oſtenſum eſt enium in corolla-<lb/>rio, quod ſequitur, tabulas ibi poſitas, cuius ueritas patet ex præcedente experimentatione:</s> <s xml:id="echoid-s48480" xml:space="preserve"> quo-<lb/>niam anguli refractionum in medio ſecundi dia phani groſsioris, quibus differunt anguli inciden-<lb/>tiæ ab angulis refractis contentis ſub linea perẽdiculari, ducta à puncto refractionis ſuper ſuper-<lb/>ficiem diaphani, & à lineis refractis ad uiſum, in maioribus angulis incidentiæ ſunt maiores, & in <lb/>minoribus ſunt minores:</s> <s xml:id="echoid-s48481" xml:space="preserve"> ergo angulus a e h eſt minor angulo a o f:</s> <s xml:id="echoid-s48482" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s48483" xml:space="preserve"> Quoniam <lb/>enim per 21 p 1 angulus a e g eſt maior angulo a o g, & anguli h e g & f o g ſunt æ quales per 29 p 1, & <lb/>quia ſunt recti:</s> <s xml:id="echoid-s48484" xml:space="preserve"> patet ergo quoniam angulus a o f eſt maior angulo a e h.</s> <s xml:id="echoid-s48485" xml:space="preserve"> Cum ergo ſequatur impoſ-<lb/>ſibile ex datis:</s> <s xml:id="echoid-s48486" xml:space="preserve"> patet quòd punctum n non cadit inter puncta m & k.</s> <s xml:id="echoid-s48487" xml:space="preserve"> Similiter quoque ſequitur ex <lb/>illis datis, ut angulus a e b ſit minor angulo a o b:</s> <s xml:id="echoid-s48488" xml:space="preserve"> quod eſt impoſsibile, & contra 21 p 1 producta li-<lb/>nea a b, quæ ambobus illis angulis ſubtenditur, & à cuius punctis terminalibus illæ lineæ produ-<lb/>cuntur.</s> <s xml:id="echoid-s48489" xml:space="preserve"> Si enim angulus p e a ſit maior angulo l o a:</s> <s xml:id="echoid-s48490" xml:space="preserve"> ergo per 13 p 1 angulus a e b eſt maior angulo a <lb/>o b:</s> <s xml:id="echoid-s48491" xml:space="preserve"> eſt enim uterq;</s> <s xml:id="echoid-s48492" xml:space="preserve"> illorum ſuper angulum ſuæ refractionis reſiduum duorum rectorum.</s> <s xml:id="echoid-s48493" xml:space="preserve"> Quòd ſi <lb/>punctus n, qui datus eſt eſſe locus ſecundæ imaginis formæ puncti b, fuerit inter duo puncta m & <lb/>b lineæ b k:</s> <s xml:id="echoid-s48494" xml:space="preserve"> tunc punctus e erit <lb/> <anchor type="figure" xlink:label="fig-0723-01a" xlink:href="fig-0723-01"/> inter duo puncta o & k per 32 th.</s> <s xml:id="echoid-s48495" xml:space="preserve"> <lb/>1 huius:</s> <s xml:id="echoid-s48496" xml:space="preserve"> quod poteſt oſtendi, ut <lb/>purius:</s> <s xml:id="echoid-s48497" xml:space="preserve"> & erit angulus e b k minor <lb/>angulo o b k:</s> <s xml:id="echoid-s48498" xml:space="preserve"> erit ergo, ut prius, <lb/>angulus p e h minor angulo l o f:</s> <s xml:id="echoid-s48499" xml:space="preserve"> <lb/>& erit angulus p e a, qui eſt an-<lb/>gulus refractionis, minor angulo <lb/>l o a, qui eſt etiam angulus refra-<lb/>ctionis:</s> <s xml:id="echoid-s48500" xml:space="preserve"> angulus ergo a e b eſt <lb/>maior angulo a o b:</s> <s xml:id="echoid-s48501" xml:space="preserve"> quod eſtim-<lb/>poſsibile, ut prius per 21 p 1 ducta <lb/>linea a b.</s> <s xml:id="echoid-s48502" xml:space="preserve"> Impoſsibile eſt ergo <lb/>quòd punctus n ſit locus imagi-<lb/>nis formæ puncti b:</s> <s xml:id="echoid-s48503" xml:space="preserve"> ergo neque <lb/>aliquod aliud punctum lineæ b <lb/>k, præter punctum m.</s> <s xml:id="echoid-s48504" xml:space="preserve"> Punctus <lb/>itaq;</s> <s xml:id="echoid-s48505" xml:space="preserve"> b exiſtens in propoſito ſitu <lb/>non habebit alium locum imaginis, reſpectu uiſus a, niſi ſolum punctum m:</s> <s xml:id="echoid-s48506" xml:space="preserve"> nec refringetur ab alio <lb/>puncto ſuperficiei corporis diaphani ad uiſum a, niſi à ſolo puncto e.</s> <s xml:id="echoid-s48507" xml:space="preserve"> Quod eſt propoſitum.</s> <s xml:id="echoid-s48508" xml:space="preserve"/> </p> <div xml:id="echoid-div1825" type="float" level="0" n="0"> <figure xlink:label="fig-0722-02" xlink:href="fig-0722-02a"> <variables xml:id="echoid-variables829" xml:space="preserve">a p h j f g e o k d n c z q g m</variables> </figure> <figure xlink:label="fig-0723-01" xlink:href="fig-0723-01a"> <variables xml:id="echoid-variables830" xml:space="preserve">a l f h p g o e k d m n c q z b</variables> </figure> </div> </div> <div xml:id="echoid-div1827" type="section" level="0" n="0"> <head xml:id="echoid-head1349" xml:space="preserve" style="it">21. Communi ſectione ſuperficiei refractionis & ſuperficiei corporis diapbani, à quo fit re-<lb/> <pb o="422" file="0724" n="724" rhead="VITELLONIS OPTICAE"/> fractio, exiſtente linearecta, punctó uiſo exiſtente extra perpendicularem ductam à centro <lb/>uiſas ſuper ſuperficiem corporis diaphani rarioris corpore diapbano uiſum contingente: ab uno <lb/>tantùm puncto fiet refr actio: & unica uidebitur imago. Alhazen 23 n 7.</head> <p> <s xml:id="echoid-s48509" xml:space="preserve">Remaneat omnis diſpofιtio, ut in præcedentibus, niſi quòd corpus diaphanum, in cuius ſuper-<lb/>ficie eſt linea g d & perpendicularis g c, quod eſt ex parte uiſus a, ſit groſioris diaphanitatis illo <lb/>corpore, quod eſt ex parte b puncti rei uiſæ:</s> <s xml:id="echoid-s48510" xml:space="preserve"> & illud, quod eſt ex parte puncti b ſit rarius:</s> <s xml:id="echoid-s48511" xml:space="preserve"> & ſit linea <lb/>b k ducta à puncto rei uiſæ per 11 p 11 perpendicularis ſuper ſuperficiem corporis diapliani:</s> <s xml:id="echoid-s48512" xml:space="preserve"> fiatq́ <lb/>refractio formæ puncti b ad uiſiam a ex puncto ſuperficiei illius corporis, quod ſit e:</s> <s xml:id="echoid-s48513" xml:space="preserve"> & ducantur li-<lb/>neæ b e & e a:</s> <s xml:id="echoid-s48514" xml:space="preserve"> protrahaturq́;</s> <s xml:id="echoid-s48515" xml:space="preserve"> linea b e uſq;</s> <s xml:id="echoid-s48516" xml:space="preserve"> ad punctum p ultra ſuperficiem corporis, in qua eſt linea <lb/>g d, & à puncto refractionis, quod eſte, ducatur linea h e z perpendiculariter ſuper lineam g k:</s> <s xml:id="echoid-s48517" xml:space="preserve"> ca-<lb/>det ergo linea a e media inter duas lineas e p & e h.</s> <s xml:id="echoid-s48518" xml:space="preserve"> Nam prima linea, per quam extenditur forma <lb/>ad locum refractionis, eſt linea b e p:</s> <s xml:id="echoid-s48519" xml:space="preserve"> fit autem refractio ad partem perpendicularis e h per 4 huius:</s> <s xml:id="echoid-s48520" xml:space="preserve"> <lb/>nam corpus, quod eſt ex parte uiſus a, eſt groſsioris diaphanitatis corpore, quod eſt ad partem rei <lb/>uiſæ b, ut patet ex hypotheſi.</s> <s xml:id="echoid-s48521" xml:space="preserve"> Protrahatur itaque linea a e ultra punctum e, quoufq;</s> <s xml:id="echoid-s48522" xml:space="preserve"> concurrat cum <lb/>linea k b:</s> <s xml:id="echoid-s48523" xml:space="preserve"> concurret autem cum illa per 2 th 1 huius:</s> <s xml:id="echoid-s48524" xml:space="preserve"> ſecat enim eius æquidiſtantem h e z:</s> <s xml:id="echoid-s48525" xml:space="preserve"> ſecet <lb/>ergo lineam k b in puncto m.</s> <s xml:id="echoid-s48526" xml:space="preserve"> Eſtitaque per 15 th.</s> <s xml:id="echoid-s48527" xml:space="preserve"> huius punctus m locus imaginis formæ puncti <lb/>b:</s> <s xml:id="echoid-s48528" xml:space="preserve"> & profundabitur ſub puncto b ultra ſitum rei uiſæ, cuius ipſum habet formam.</s> <s xml:id="echoid-s48529" xml:space="preserve"> Nam corpus.</s> <s xml:id="echoid-s48530" xml:space="preserve"> <lb/>quod eſt ex parte b, eſt ſubtilius il-<lb/> <anchor type="figure" xlink:label="fig-0724-01a" xlink:href="fig-0724-01"/> lo corpore, quod eſt ex parte uiſus <lb/>a.</s> <s xml:id="echoid-s48531" xml:space="preserve"> Dico itaque quòd forma puncti <lb/>b non refringitur ad uiſum a, niſi à <lb/>ſolo puncto e:</s> <s xml:id="echoid-s48532" xml:space="preserve"> & quod non habet <lb/>imaginem, niſi in ſolo puncto m.</s> <s xml:id="echoid-s48533" xml:space="preserve"> <lb/>Si enim hoc ſit poſsibile.</s> <s xml:id="echoid-s48534" xml:space="preserve"> ut plu-<lb/>res habeat imagines quàm illam, <lb/>quæ eſt in puncto m:</s> <s xml:id="echoid-s48535" xml:space="preserve"> ſit, ut habeat <lb/>imaginem in puncto alio, quod ſit <lb/>n:</s> <s xml:id="echoid-s48536" xml:space="preserve"> erit itaque punctus n in linea per <lb/>pendiculari b k per 13 huius:</s> <s xml:id="echoid-s48537" xml:space="preserve"> & in-<lb/>fra punctum b per 15 th.</s> <s xml:id="echoid-s48538" xml:space="preserve"> huius:</s> <s xml:id="echoid-s48539" xml:space="preserve"> pro <lb/>pter corporum diaphanorum me-<lb/>diorum propoſitam diuerſitatem.</s> <s xml:id="echoid-s48540" xml:space="preserve"> <lb/>Autigitur erit punctus ninter duo <lb/>punctam & b:</s> <s xml:id="echoid-s48541" xml:space="preserve"> aut ſub puncto m.</s> <s xml:id="echoid-s48542" xml:space="preserve"> <lb/>Sit primò inter duo puncta b & m:</s> <s xml:id="echoid-s48543" xml:space="preserve"> <lb/>ducaturq́, linea a n:</s> <s xml:id="echoid-s48544" xml:space="preserve"> quæ ſecabit lineam e k per 32 th.</s> <s xml:id="echoid-s48545" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s48546" xml:space="preserve"> quia ipſa producta à puncto lateris m <lb/>e ſecatlatus k m trigoni e k m remotius à puncto a, quàm eſt latus k e:</s> <s xml:id="echoid-s48547" xml:space="preserve"> & etiam ideo, quia puncta <lb/>a & b ſunt in eadem ſuperficie, & linea e d eſt iacens inter illa puncta.</s> <s xml:id="echoid-s48548" xml:space="preserve"> Secet ergo ipſam in puncto <lb/>o:</s> <s xml:id="echoid-s48549" xml:space="preserve"> eſt itaque o punctus refractionis:</s> <s xml:id="echoid-s48550" xml:space="preserve"> & ducatur linea b o:</s> <s xml:id="echoid-s48551" xml:space="preserve"> quæ tranſeat uſq;</s> <s xml:id="echoid-s48552" xml:space="preserve"> ad punctuml:</s> <s xml:id="echoid-s48553" xml:space="preserve"> & ex pun-<lb/>cto o extrahatur linea f o q perpendiculariter ſuper lineam g o d per 11 p 1.</s> <s xml:id="echoid-s48554" xml:space="preserve"> Linea ita que b o eſt illa <lb/>linea, per quam forma puncti b extenditur ad punctum refractionis, quod eſto.</s> <s xml:id="echoid-s48555" xml:space="preserve"> Linea quoque o a <lb/>eritinter duas lineas o 1 & o f:</s> <s xml:id="echoid-s48556" xml:space="preserve"> quoniam in tali diſpoſitione mediorum diaphanorum ſemper fit re-<lb/>fractio ad perpendicularem per 4 th.</s> <s xml:id="echoid-s48557" xml:space="preserve"> huius.</s> <s xml:id="echoid-s48558" xml:space="preserve"> Si itaque punctus n fuerit inter duo punxta m & b:</s> <s xml:id="echoid-s48559" xml:space="preserve"> <lb/>erit per 32 th.</s> <s xml:id="echoid-s48560" xml:space="preserve"> 1 huius punctum o inter duo puncta e & k:</s> <s xml:id="echoid-s48561" xml:space="preserve"> ergo, ut in præmiſſa per 29 th.</s> <s xml:id="echoid-s48562" xml:space="preserve"> 1 huius, an-<lb/>gulus o b k erit minor angulo e b <lb/> <anchor type="figure" xlink:label="fig-0724-02a" xlink:href="fig-0724-02"/> k:</s> <s xml:id="echoid-s48563" xml:space="preserve"> quoniam pars eſt minor ſuo to-<lb/>to:</s> <s xml:id="echoid-s48564" xml:space="preserve"> ſed per 29 p 1 angulus 1 o f eſt <lb/>æ qualis angulo o b k, & angulus <lb/>p e h eſt æ qualis angulo e b k:</s> <s xml:id="echoid-s48565" xml:space="preserve"> ideo <lb/>quòd lineæ h e & f o & k b ſunt <lb/>æ quidiſtantes:</s> <s xml:id="echoid-s48566" xml:space="preserve"> eſt ergo angulus 1 <lb/>o f minor angulo p e h:</s> <s xml:id="echoid-s48567" xml:space="preserve"> angulus <lb/>itaque l o a, qui eſt angulus refra-<lb/>ctionis, per corollarium 8 huius <lb/>eſt minor angulo p e a, qui eſt etiá <lb/>angulus refractionis:</s> <s xml:id="echoid-s48568" xml:space="preserve"> ergo angu-<lb/>lus a o f, qui remanet de angulo 1 <lb/>o f ſuper angulum refractionis, <lb/>qui eſt l o a, eſt minor angulo a e h, <lb/>qui remanet de angulo p e h ſuper <lb/>angulum refractionis, qui eſt p e a <lb/>per eandem 8 huius:</s> <s xml:id="echoid-s48569" xml:space="preserve"> ſed angulus <lb/>a o f eſt æqualis angulo a n k per 29 p 1:</s> <s xml:id="echoid-s48570" xml:space="preserve"> & angulus a e h eſt æ qualis an gulo a m k per eandem 29 p 1:</s> <s xml:id="echoid-s48571" xml:space="preserve"> <lb/> <pb o="423" file="0725" n="725" rhead="LIBER DECIMVS."/> angulus itaque a n k eſt minor angulo a m k:</s> <s xml:id="echoid-s48572" xml:space="preserve"> quod eſt impoſsibile, & contra 16 p 1.</s> <s xml:id="echoid-s48573" xml:space="preserve"> Si autem pun-<lb/>ctus n fuerit infra punctum m:</s> <s xml:id="echoid-s48574" xml:space="preserve"> tunc, ut prius in proxima huius, deductione facta punctus e cadet <lb/>inter puncta o & k:</s> <s xml:id="echoid-s48575" xml:space="preserve"> & erit angulus o b k maior angulo e b k per 29 th.</s> <s xml:id="echoid-s48576" xml:space="preserve"> 1 huius, & quia totum eſt ma-<lb/>ius parte:</s> <s xml:id="echoid-s48577" xml:space="preserve"> angulus ergo l o ferit maior angulo p e h per 29 p 1:</s> <s xml:id="echoid-s48578" xml:space="preserve"> ergo angulus l o a eſt maior angulo p <lb/>e a :</s> <s xml:id="echoid-s48579" xml:space="preserve"> & angulus a o f eſt maior angulo a e h per 8 huius, ut prius:</s> <s xml:id="echoid-s48580" xml:space="preserve"> ergo angulus a n k per 29 p 1 eſt ma-<lb/>ior angulo a m k:</s> <s xml:id="echoid-s48581" xml:space="preserve"> quod eſt impoſsibile, & contra 16 p 1.</s> <s xml:id="echoid-s48582" xml:space="preserve"> Non eſt ergo imago form æ puncti b in pun-<lb/>cto n, nec in aliquo alio puncto lineæ m b k, præter quàm in puncto m:</s> <s xml:id="echoid-s48583" xml:space="preserve"> quoniam idem impoſsibile <lb/>accidit in omnibus datis punctis.</s> <s xml:id="echoid-s48584" xml:space="preserve"> Ab unico ergo puncto in hac diſpoſitione fiet refraction:</s> <s xml:id="echoid-s48585" xml:space="preserve"> & unica <lb/>uiſui occurret imago.</s> <s xml:id="echoid-s48586" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s48587" xml:space="preserve"/> </p> <div xml:id="echoid-div1827" type="float" level="0" n="0"> <figure xlink:label="fig-0724-01" xlink:href="fig-0724-01a"> <variables xml:id="echoid-variables831" xml:space="preserve">a h f p e o k d n m g z q</variables> </figure> <figure xlink:label="fig-0724-02" xlink:href="fig-0724-02a"> <variables xml:id="echoid-variables832" xml:space="preserve">a f h p l g o a k d b</variables> </figure> </div> </div> <div xml:id="echoid-div1829" type="section" level="0" n="0"> <head xml:id="echoid-head1350" xml:space="preserve" style="it">22. Communi ſectione ſuperficieirefractionis & ſuperficiei corporis diaphani, à quo fit re-<lb/>fractio, exiſtente circulo, punctó quſo exiſtente in perpendiculari, duct a à centro uiſus ſuper <lb/>conuexam ſuperficiem corporis diaphani: formæreruiſæ à nullo puncto fiet refractio: & una <lb/>tantùm uidebitur imago. Alhazen 26 n 7.</head> <p> <s xml:id="echoid-s48588" xml:space="preserve">Sit centrum uiſus punctum a:</s> <s xml:id="echoid-s48589" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s48590" xml:space="preserve"> b punctus rei uiſæ ultra corpus diaphanum groſsius illo cor-<lb/>pore diaphano, quod eſt circa uiſum:</s> <s xml:id="echoid-s48591" xml:space="preserve"> & ſit ſuperficies illius corporis diaphani, quod eſt ex parte b, <lb/>ſuperficies conuexa, illa, quæ eſt ex parte uiſus a:</s> <s xml:id="echoid-s48592" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s48593" xml:space="preserve"> communis fectio ſuperficiei refractionis & <lb/>ſuperficiei illius corporis diaphani per 69 th.</s> <s xml:id="echoid-s48594" xml:space="preserve"> 1 huius circulus c d e:</s> <s xml:id="echoid-s48595" xml:space="preserve"> cuius centrum ſit punctus z:</s> <s xml:id="echoid-s48596" xml:space="preserve"> & <lb/>ducatur linea a c z d, quæ neceſſariò erit perpendicularis ſuper ſuperficiem corporis diaphani per <lb/>72 th 1 huius:</s> <s xml:id="echoid-s48597" xml:space="preserve"> quoniam tranſit per punctum z centrum eius:</s> <s xml:id="echoid-s48598" xml:space="preserve"> ſtiq́;</s> <s xml:id="echoid-s48599" xml:space="preserve"> b punctus rei uiſæ in perpen dicu-<lb/>lari linea, quæ eſt a d.</s> <s xml:id="echoid-s48600" xml:space="preserve"> Tunc itaque uiſus a comprehendet formam puncti b ſine aliqua refractio-<lb/>ne.</s> <s xml:id="echoid-s48601" xml:space="preserve"> Nam forma, quæ extenditur ſecundum lineam d a, extenditur rectè in corpore diaphano, quod <lb/>eſt ex parte uiſus a per 3 huius:</s> <s xml:id="echoid-s48602" xml:space="preserve"> ideo quòd linea d a eſt ex parte uiſus:</s> <s xml:id="echoid-s48603" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0725-01a" xlink:href="fig-0725-01"/> laris ſuper ſuperficiem corporis diaphani, quod eſt ex parte uiſus:</s> <s xml:id="echoid-s48604" xml:space="preserve"> <lb/>comprehendet itaque uiſus a formam puncti b in ſuo loco, & rectè:</s> <s xml:id="echoid-s48605" xml:space="preserve"> <lb/>ſed & in hac diſpoſitione forma puncti b nunquam refringitur ad <lb/>a uiſum.</s> <s xml:id="echoid-s48606" xml:space="preserve"> Aut enim punctus rei uiſæ, qui eſt b, erit in centro corpo-<lb/>ris diaphani, quod eſt z:</s> <s xml:id="echoid-s48607" xml:space="preserve"> aut extra illud, Si fuerit in cenrro z:</s> <s xml:id="echoid-s48608" xml:space="preserve"> tunc <lb/>nulla linea, per quam extenditur forma punctib ad circumferẽtiam <lb/>circuli c d e, refringitur ad uiſum a:</s> <s xml:id="echoid-s48609" xml:space="preserve"> quoniam omnes illæ ſunt ſemi-<lb/>diametri, perpendiculares ſuper ſuperſiciem conuexam corporis <lb/>diaphani.</s> <s xml:id="echoid-s48610" xml:space="preserve"> Et quia ſola linea z a exit à centro circuli c d e ad uiſum:</s> <s xml:id="echoid-s48611" xml:space="preserve"> <lb/>patet quò d forma puncti b non refringitur ad uiſum a, cum punctus <lb/>b fuerit in centro z.</s> <s xml:id="echoid-s48612" xml:space="preserve"> Quòd ſi punctus b fuerit in linea c d extra cen-<lb/>trum z:</s> <s xml:id="echoid-s48613" xml:space="preserve"> aut igitur erit in linea d z:</s> <s xml:id="echoid-s48614" xml:space="preserve"> aut in linea z c.</s> <s xml:id="echoid-s48615" xml:space="preserve"> Si ſit in linea z c, ad-<lb/>huc nulla ſui fiet refractio ad uiſum a.</s> <s xml:id="echoid-s48616" xml:space="preserve"> Quod ſi fuerit poſsibile, eſto <lb/>quòd refringatur ex puncto e:</s> <s xml:id="echoid-s48617" xml:space="preserve"> & ducatur linea b e:</s> <s xml:id="echoid-s48618" xml:space="preserve"> & protrahatur <lb/>extra circulum ad punctum h:</s> <s xml:id="echoid-s48619" xml:space="preserve"> & protrahatur linea z e extra circu-<lb/>lum ad punctum p:</s> <s xml:id="echoid-s48620" xml:space="preserve"> erit itaque linea z p perpendicularis ſuper ſu-<lb/>perficiem corporis diaphani, quod eſt ex parte uiſus.</s> <s xml:id="echoid-s48621" xml:space="preserve"> Cum itaque <lb/>corpus diaphanum, quod eſt circa uiſum, fuerit rarius corpore dia-<lb/>phano, quod eſt circa rem uiſam, & circa punctum b:</s> <s xml:id="echoid-s48622" xml:space="preserve"> patet per 4 hu-<lb/>ius quòd forma puncti b, quádo exten ditur per lineam b e, refringitur in puncto e ad partem con-<lb/>trariam illi parti, in qua eſt perpendicularis z p:</s> <s xml:id="echoid-s48623" xml:space="preserve"> non ergo refringitur tunc forma puncti b ad uiſum <lb/>a.</s> <s xml:id="echoid-s48624" xml:space="preserve"> Quòd ſi punctum b ſit in linea d z, adhuc non refringitur forma puncti b ad uiſum a.</s> <s xml:id="echoid-s48625" xml:space="preserve"> Si enim hoc <lb/>eſt poſsibile, ſit, ut refringatur ex puncto e:</s> <s xml:id="echoid-s48626" xml:space="preserve"> & producatur linea b e ad punctum r:</s> <s xml:id="echoid-s48627" xml:space="preserve"> & protrahatur li-<lb/>nea z e ad punctum p:</s> <s xml:id="echoid-s48628" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s48629" xml:space="preserve">, ut forma puncti b refrin gatur ad uiſum a ex puncto e per lineam e a:</s> <s xml:id="echoid-s48630" xml:space="preserve"> pa-<lb/>làm itaq;</s> <s xml:id="echoid-s48631" xml:space="preserve"> quoniam angulus r e a eſt angulus refractionis:</s> <s xml:id="echoid-s48632" xml:space="preserve"> & angulus r e p eſt contentus à linea b e r, <lb/>per quam extenditur forma puncti b, & à perpendiculari exeunte ab e puncto refractionis ſuper-<lb/>ſuperficiem corporis diaphani, à qua fit refractio:</s> <s xml:id="echoid-s48633" xml:space="preserve"> ergo per corollarium 8 huius angulus refractio-<lb/>nis, qui eſt r e a, eſt minor angulo incidentiæ, qui eſt r e p:</s> <s xml:id="echoid-s48634" xml:space="preserve"> & linea b z aut eſt minor quàm linea z e:</s> <s xml:id="echoid-s48635" xml:space="preserve"> <lb/>aut æ qualis ei:</s> <s xml:id="echoid-s48636" xml:space="preserve"> quia punctus b aut eſt inter duo puncta d & z, aut in puncto d:</s> <s xml:id="echoid-s48637" xml:space="preserve"> eſt itaque per 18 & <lb/>per 5 p 1 angulus e b z aut maior angulo b e z, auto æqualis ei:</s> <s xml:id="echoid-s48638" xml:space="preserve"> ſed angulus a er per 16 p 1 maior eſt <lb/>angulo e b z:</s> <s xml:id="echoid-s48639" xml:space="preserve"> ergo & angulo b e z:</s> <s xml:id="echoid-s48640" xml:space="preserve"> & angulus r e p per 15 p 1 eſt æ qualis angulo b e z.</s> <s xml:id="echoid-s48641" xml:space="preserve"> Erit ergo angu-<lb/>lus a e r maior angulo r e p:</s> <s xml:id="echoid-s48642" xml:space="preserve"> quod eſt contra præoſtenſa & impoſsibile.</s> <s xml:id="echoid-s48643" xml:space="preserve"> Forma ergo puncti b non re-<lb/>fringitur ad uiſum a ex puncto e:</s> <s xml:id="echoid-s48644" xml:space="preserve"> ſed nec ex alio puncto circuli c d e:</s> <s xml:id="echoid-s48645" xml:space="preserve"> nex ex alia circumferentia ali-<lb/>cuius circulorum in ſuperficie corporis diaphani, in quo eſt punctum b, exiſtentium, ut patet per 1 <lb/>huius.</s> <s xml:id="echoid-s48646" xml:space="preserve"> Palàm ergo quoniam exiſtente puncto b in linea g d, non comprehenditur forma eius à uiſu <lb/>a per refractionem ex aliquo puncto ſuperficiei corporis denſioris:</s> <s xml:id="echoid-s48647" xml:space="preserve"> & non comprehenditur niſi ſo-<lb/>lum unum punctum:</s> <s xml:id="echoid-s48648" xml:space="preserve"> quoniam linea perpendicularis ſuper ſuperficiem corporis diaphani denſio-<lb/>ris non ſecat illius corporis ſuperficiem, niſi in un o tantùm pũcto:</s> <s xml:id="echoid-s48649" xml:space="preserve"> unica ergo tantùm uidetur ima-<lb/>go.</s> <s xml:id="echoid-s48650" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s48651" xml:space="preserve"> demonſtrandum ſi corpus diaphanum, quod eſt circa centrum uiſus punctum <lb/>a, fuerit denſius corpore diaphano, quod eſt circa punctum rei uiſæ, quod eſt b.</s> <s xml:id="echoid-s48652" xml:space="preserve"> Tunc enim ſemper <lb/> <pb o="424" file="0726" n="726" rhead="VITELLONIS OPTICAE"/> fiet refractio ad perpendicularem ductam à dato puncto refractionis, & nunquam fiet ad centrum <lb/>uiſus punctum a:</s> <s xml:id="echoid-s48653" xml:space="preserve"> ſiue punctum rei uiſæ fuerit in linea c z:</s> <s xml:id="echoid-s48654" xml:space="preserve"> uel in linea z d:</s> <s xml:id="echoid-s48655" xml:space="preserve"> & ſequuntur maiora im-<lb/>poſsibilia quàm prius.</s> <s xml:id="echoid-s48656" xml:space="preserve"> Et ſi fuerit in centro z:</s> <s xml:id="echoid-s48657" xml:space="preserve"> patet quòd non refringitur, ſed uidetur directè forma <lb/>cius:</s> <s xml:id="echoid-s48658" xml:space="preserve"> & unica eſt eius imago.</s> <s xml:id="echoid-s48659" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s48660" xml:space="preserve"> propoſitum ſecundum omnes eius modos.</s> <s xml:id="echoid-s48661" xml:space="preserve"/> </p> <div xml:id="echoid-div1829" type="float" level="0" n="0"> <figure xlink:label="fig-0725-01" xlink:href="fig-0725-01a"> <variables xml:id="echoid-variables833" xml:space="preserve">a r c e u p b h z b</variables> </figure> </div> </div> <div xml:id="echoid-div1831" type="section" level="0" n="0"> <head xml:id="echoid-head1351" xml:space="preserve" style="it">23. Cõmuni ſectione ſuperficiei refractionis & ſuperficiei corporis diaphani, à quo fit refra-<lb/>ctio, exiſtente circulo, punctóuiſo iacente extr a perpendicularem, duct am à centro uiſus ſu-<lb/>per ſuperficiem conuexam corporis diaphani großioris corpore diaphano uiſumcõtingente: ab <lb/>uno tantùm puncto fiet refracgtio: & unica uidebitur imago: loco tamen imaginis diuerſificato <lb/>ſecundum diuerſitatem loci puncti uiſi uel centri uiſus. Alhazen 27 n 7.</head> <p> <s xml:id="echoid-s48662" xml:space="preserve">Eſto diſpoſitio, quæ in proxima præmiſſa, niſi quòd punctus rei uiſæ, qui eſt b, ſit extra lineam a <lb/>c d, tamen intra circulum c d e.</s> <s xml:id="echoid-s48663" xml:space="preserve"> Et quia forma puncti b non refringitur ad uiſum a, niſi à circumfe-<lb/>rentia circuli c d e:</s> <s xml:id="echoid-s48664" xml:space="preserve"> quæ eſt in ſuperficie refractionis, ut patet per 1 huius, & ex hypotheſit:</s> <s xml:id="echoid-s48665" xml:space="preserve"> fit q́;</s> <s xml:id="echoid-s48666" xml:space="preserve"> illa re-<lb/>fractio à concauitate corporis diaphani, quod eſt ex parte uiſus contingens conuexum corporis <lb/>diaphani ex parte rei uiſæ:</s> <s xml:id="echoid-s48667" xml:space="preserve"> ſit, ut refringatur ad uiſum a ex puncto e circuli c d e:</s> <s xml:id="echoid-s48668" xml:space="preserve"> dico quòd non po-<lb/>teſt ex alio puncto ſuperficiei corporis illius refringi ad uiſum.</s> <s xml:id="echoid-s48669" xml:space="preserve"> Sit enim, ſi poſsibile eſt, ut refringa-<lb/>tur ex puncto alio circuli c d e, quàm ex puncto e:</s> <s xml:id="echoid-s48670" xml:space="preserve"> qui ſit punctus in :</s> <s xml:id="echoid-s48671" xml:space="preserve"> & ducantur lineæ b e, a e, b m, <lb/>a m, z e, z m:</s> <s xml:id="echoid-s48672" xml:space="preserve"> ſit quoq;</s> <s xml:id="echoid-s48673" xml:space="preserve"> ut lineæ z e & b m, cum ſint in eadem ſuperficie circuli c d e, ſecent ſe in pun-<lb/>cto, quod ſit g:</s> <s xml:id="echoid-s48674" xml:space="preserve"> & producatur linea b e extra circulum uſq;</s> <s xml:id="echoid-s48675" xml:space="preserve"> ad punctum h:</s> <s xml:id="echoid-s48676" xml:space="preserve"> & linea b m uſq;</s> <s xml:id="echoid-s48677" xml:space="preserve"> ad pun-<lb/>ctum n:</s> <s xml:id="echoid-s48678" xml:space="preserve"> & linea z e uſq;</s> <s xml:id="echoid-s48679" xml:space="preserve"> ad punctum p:</s> <s xml:id="echoid-s48680" xml:space="preserve"> & linea z m uſq;</s> <s xml:id="echoid-s48681" xml:space="preserve"> ad punctum l.</s> <s xml:id="echoid-s48682" xml:space="preserve"> Erit itaq;</s> <s xml:id="echoid-s48683" xml:space="preserve"> angulus h e p per 15 <lb/>p 1 æqualis angulo in cidentiæ:</s> <s xml:id="echoid-s48684" xml:space="preserve"> quoniam uterque illorum eſt contentus ſub linea e b, per quam ex-<lb/>ten ditur forma, & ſub perpendiculari e p, exeunte à loco refractionis, qui eſt e, ſuper ſuperficiem <lb/>corporis, à quo ſit refractio:</s> <s xml:id="echoid-s48685" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s48686" xml:space="preserve"> angulus h e a angulus refractionis:</s> <s xml:id="echoid-s48687" xml:space="preserve"> & erit angulus l m n per 15 p 1 <lb/>æqualis angulo incidentiæ contentus ſub linea n m, per quam extenditur forma, & ſub perpendi-<lb/>culari l m, exeunte à loco refractionis, qui eſt m:</s> <s xml:id="echoid-s48688" xml:space="preserve"> & angulus n m a eſt angulus refractionis, Erit itaq;</s> <s xml:id="echoid-s48689" xml:space="preserve"> <lb/>angulus h e p aut æ qualis angulo n m l:</s> <s xml:id="echoid-s48690" xml:space="preserve"> aut maior:</s> <s xml:id="echoid-s48691" xml:space="preserve"> aut minor.</s> <s xml:id="echoid-s48692" xml:space="preserve"> Sit ſit æqualis:</s> <s xml:id="echoid-s48693" xml:space="preserve"> tunc per 8 huius erit an-<lb/>gulus h e a refractionis æqualis angulo n m a, qui eſt ſimiliter angulus refractionis.</s> <s xml:id="echoid-s48694" xml:space="preserve"> Et quoniam <lb/>uterque ipſorum cum ſuo compari ualet duos rectos per 13 p 1:</s> <s xml:id="echoid-s48695" xml:space="preserve"> erit tunc angulus a m b æ qualis an-<lb/>gulo a e b:</s> <s xml:id="echoid-s48696" xml:space="preserve"> quòd producta linea a b patet eſſe impoſsibi-<lb/> <anchor type="figure" xlink:label="fig-0726-01a" xlink:href="fig-0726-01"/> le, & contra 21 p 1.</s> <s xml:id="echoid-s48697" xml:space="preserve"> Si autem angulus h e p ſit minor an-<lb/>gulo l m n:</s> <s xml:id="echoid-s48698" xml:space="preserve"> erit angulus h e a minor angulo n m a per 8 <lb/>huius:</s> <s xml:id="echoid-s48699" xml:space="preserve"> erit ergo per 13 p 1 angulus a m b minor angulo a <lb/>e b:</s> <s xml:id="echoid-s48700" xml:space="preserve"> quod iterum eſt contra 21 p 1 & impoſsibile.</s> <s xml:id="echoid-s48701" xml:space="preserve"> Si uerò <lb/>angulus h e p ſit maior angulo l m n:</s> <s xml:id="echoid-s48702" xml:space="preserve"> extrahatur linea e b <lb/>in partem puncti b ad punctum circumferentiæ, qui ſit <lb/>f:</s> <s xml:id="echoid-s48703" xml:space="preserve"> & extrahatur linea m b ultra punctum b ad punctum <lb/>circumferentiæ, qui ſit o.</s> <s xml:id="echoid-s48704" xml:space="preserve"> Angulus itaque e b m erit per <lb/>54 th.</s> <s xml:id="echoid-s48705" xml:space="preserve"> 1 huius æqualis angulo, qui eſt apud circumferen-<lb/>tiam, cadens in arcum æqualem duobus arcubus e m & <lb/>f o.</s> <s xml:id="echoid-s48706" xml:space="preserve"> Et cum angulus h e p ex hypotheſi ſit maior angulo <lb/>n m l:</s> <s xml:id="echoid-s48707" xml:space="preserve"> erit angulus z e b per 15 p 1 maior angulo n m l:</s> <s xml:id="echoid-s48708" xml:space="preserve"> ergo <lb/>& angulo b m z per eandem 15.</s> <s xml:id="echoid-s48709" xml:space="preserve"> Cum ergo angulus z e b <lb/>ſit maior angulo b m z:</s> <s xml:id="echoid-s48710" xml:space="preserve"> erit exceſſus anguli m z e ſupra <lb/>angulum e b m, æqualis exceſſui anguli z e b ſupra angu-<lb/>lum b m z per 32 p 1.</s> <s xml:id="echoid-s48711" xml:space="preserve"> Cum enim in trigonis e b g & m g z <lb/>anguli interſectionia ad punctum g ſint æquales, ut pa-<lb/>tet per 15 p 1, & quilibet reliquorum duorum angulorum <lb/>cum ſuo tertio ualeat duos rectos:</s> <s xml:id="echoid-s48712" xml:space="preserve"> patet quòd duo an-<lb/>guli reliqui unius trigoni ſunt æquales duobus reliquis <lb/>angulis alterius trigoni.</s> <s xml:id="echoid-s48713" xml:space="preserve"> In quanto ergo angulus z e b eſt <lb/>maior angulo b m z, in tanto angulus m z e eſt maioran-<lb/>gulo e b m.</s> <s xml:id="echoid-s48714" xml:space="preserve"> Arcus uerò reſpiciens an gulum m z e, cum fuerit apud circumferentiam, erit duplus ad <lb/>arcum m e per 20 p 3 & 33 p 6.</s> <s xml:id="echoid-s48715" xml:space="preserve"> Si ergo angulus m z e fuerit maior angulo m b e:</s> <s xml:id="echoid-s48716" xml:space="preserve"> tunc arcus m e du-<lb/>plicatus erit maior duobus arcubus m e & f o:</s> <s xml:id="echoid-s48717" xml:space="preserve"> & erit exceſſus arcus m e duplicati ſupra duos arcus <lb/>m e & f o, æqualis exceſſui arcus m e ſupra arcum f o:</s> <s xml:id="echoid-s48718" xml:space="preserve"> quoniam arcus m e utrique eſt communis, <lb/>quo ablato remanet idem exceſſus:</s> <s xml:id="echoid-s48719" xml:space="preserve"> & ſi uarietur proportio geometrica, non tamen uariatur pro-<lb/>portio arithmetica.</s> <s xml:id="echoid-s48720" xml:space="preserve"> Exceſſus ergo anguli m z e ſupra angulum e b m, eſt ille, qui reſpicit apud cir-<lb/>cumferentiam exceſſum arcus m e ſupr a arcum f o:</s> <s xml:id="echoid-s48721" xml:space="preserve"> ſed exceſſus arcus m e ſupra arcum f o eſt mi-<lb/>nor duobus arcubus m e & f o:</s> <s xml:id="echoid-s48722" xml:space="preserve"> quonitam eſt pars arcus m e:</s> <s xml:id="echoid-s48723" xml:space="preserve"> ergo exceſſus anguli m z e ſupra angu-<lb/>lum m b e eſt minor angulo m b e per 33 p 6, & ut patet ex præmiſsis.</s> <s xml:id="echoid-s48724" xml:space="preserve"> Exceſſus itaque anguli z e b <lb/>ſupra angulum z m b eſt minor angulo m b e:</s> <s xml:id="echoid-s48725" xml:space="preserve"> ergo, ut ſuprà patuit per 15 p 1 exceſſus anguli h e p <lb/>fupra angulum n m l eſt minor angulo m b e:</s> <s xml:id="echoid-s48726" xml:space="preserve"> ergo exceſſus anguli refractionis, qui eſt h e a, ſupra <lb/>angulum refractionis, qui eſt n m a, eſt multo min or angulo m b e per 8 huius:</s> <s xml:id="echoid-s48727" xml:space="preserve"> ſed exceſſus an-<lb/> <pb o="425" file="0727" n="727" rhead="LIBER DECIMVS."/> guli h e a ſupra angulum n m a eſt exceſſus anguli a m b ſupra angulum a e b per 13 p 1:</s> <s xml:id="echoid-s48728" xml:space="preserve"> exceſſus ita-<lb/>que anguli a m b ſupra angulum a e b eſt minor angulo m b e:</s> <s xml:id="echoid-s48729" xml:space="preserve"> exceſſus uerò anguli a m b ſupra an-<lb/>gulum a e b eſt duo anguli m a e & m b e:</s> <s xml:id="echoid-s48730" xml:space="preserve"> quod patet per 33 th.</s> <s xml:id="echoid-s48731" xml:space="preserve"> 1 huius producta linea a b.</s> <s xml:id="echoid-s48732" xml:space="preserve"> Duo ita-<lb/>que anguli m a e & m b e ſunt minores angulo m b e, totum ſua parte:</s> <s xml:id="echoid-s48733" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s48734" xml:space="preserve"> Forma <lb/>itaque puncti b non refringitur ad uiſum a ex alio puncto circuli c d e, quàm ex puncto e:</s> <s xml:id="echoid-s48735" xml:space="preserve"> unicam <lb/>ergo habebit imaginem.</s> <s xml:id="echoid-s48736" xml:space="preserve"> Et hoc eſt propoſitum primum.</s> <s xml:id="echoid-s48737" xml:space="preserve"> Sed & locus imaginis diuerſatur ſecun-<lb/>dum diuerſitatem loci, in quo eſt propoſitum primum.</s> <s xml:id="echoid-s48738" xml:space="preserve"> Sed & locus imaginis diuerſatur ſecun-<lb/>cta b & z ad utram que partem trans cir culum c d e:</s> <s xml:id="echoid-s48739" xml:space="preserve"> quæ aut concurret cum linea e a:</s> <s xml:id="echoid-s48740" xml:space="preserve"> aut erit æqui-<lb/>diſtans ei.</s> <s xml:id="echoid-s48741" xml:space="preserve"> Si concurrat:</s> <s xml:id="echoid-s48742" xml:space="preserve"> tunc concurſus aut erit ad partem diametri, ad quam eſt b, propinquior pe-<lb/>ripheriæ, ut in puncto k:</s> <s xml:id="echoid-s48743" xml:space="preserve"> aut concurrent in puncto aliquo alio ad partem uiſus, ut in puncto r.</s> <s xml:id="echoid-s48744" xml:space="preserve"> Si <lb/>itaque concurſus fuerit in puncto k:</s> <s xml:id="echoid-s48745" xml:space="preserve"> tunc per 15 th.</s> <s xml:id="echoid-s48746" xml:space="preserve"> huius erit imago ante uiſum:</s> <s xml:id="echoid-s48747" xml:space="preserve"> & erit forma ma-<lb/>nifeſtè comprehenſa à uiſu:</s> <s xml:id="echoid-s48748" xml:space="preserve"> quoniam eſt in perpendiculari z k producta à centro corporis diapha-<lb/>ni ſuper ſuperficiem corporis diaphani.</s> <s xml:id="echoid-s48749" xml:space="preserve"> Quòd ſi concurſus fuerit in puncto r:</s> <s xml:id="echoid-s48750" xml:space="preserve"> erit imago in puncto <lb/>r:</s> <s xml:id="echoid-s48751" xml:space="preserve"> & tunc forma comprehenditur à uiſu in eius oppoſitione:</s> <s xml:id="echoid-s48752" xml:space="preserve"> ſed non manifeſtè, quia comprehen-<lb/>ditur à uiſu extra ſuum locum, ſcilicet extra ſuperficiem corporis diaphani inter uiſum & illam ſu-<lb/>perficiem.</s> <s xml:id="echoid-s48753" xml:space="preserve"> Siuerò linea b z fuerit æquidiſtans lineæ e a:</s> <s xml:id="echoid-s48754" xml:space="preserve"> tunc erit linea b z media inter duas lineas <lb/>k b z & b z r:</s> <s xml:id="echoid-s48755" xml:space="preserve"> & tuncimago uidebitur indeterminata:</s> <s xml:id="echoid-s48756" xml:space="preserve"> & forma comprehendetur in loco refractio-<lb/>nis, ut patet per 15 huius.</s> <s xml:id="echoid-s48757" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s48758" xml:space="preserve"> Ex his itaque patete, quòd re, cuius forma compre-<lb/>henditur à uiſu, exiſtente ultra corpus diaphanum groſsius corpore diaphano, quod eſt ex parte <lb/>uiſus, non ſit refractio niſi ab uno tantùm ſuperficiei illius corporis puncto:</s> <s xml:id="echoid-s48759" xml:space="preserve"> & res illa non habet, <lb/>niſi imaginem unicam:</s> <s xml:id="echoid-s48760" xml:space="preserve"> neque comprehenditur, niſui unum tantùm.</s> <s xml:id="echoid-s48761" xml:space="preserve"> hæc enim refractio eſt à con-<lb/>cauitate corporis diaphani, quod eſt ex parte uiſus contingentis conuexum corporis diaphani, <lb/>quod eſt ex parte rei uiſæ.</s> <s xml:id="echoid-s48762" xml:space="preserve"> Patet etiam, quod ſecundum diuerſitatem ſituationis puncti a, qui eſt <lb/>centrum uiſus, fit diuerſitas locorum imaginum formę puncti b non tranſmutati ſecundum ſitum:</s> <s xml:id="echoid-s48763" xml:space="preserve"> <lb/>quoniam eadem eſt huius cum præmiſſo modo alio declaratio, niſi quòd tunc puncta refractio-<lb/>num diuerſificantur.</s> <s xml:id="echoid-s48764" xml:space="preserve"/> </p> <div xml:id="echoid-div1831" type="float" level="0" n="0"> <figure xlink:label="fig-0726-01" xlink:href="fig-0726-01a"> <variables xml:id="echoid-variables834" xml:space="preserve">a n r l c m e h p g z b s d o k</variables> </figure> </div> </div> <div xml:id="echoid-div1833" type="section" level="0" n="0"> <head xml:id="echoid-head1352" xml:space="preserve" style="it">24. Communi ſectione ſuperficiei refractionis & ſuperficiei corporis diaphani, à quo fit re-<lb/>fraction, exiſtente circulo, punctó uiſo iacente extr a perpendicularem ductam à centro uiſus <lb/>ſuper ſuperficiem corporis diaphani rarioris diaphano uiſum contingente: ab uno tantùm pun-<lb/>cto fiet refractio: & unica refracta uidebitur imago, loco tamę imaginis diuerſificato ſecundum <lb/>diuerſitatem loci puncti uiſi uel centri uiſus. Alhazen 28 n 7.</head> <p> <s xml:id="echoid-s48765" xml:space="preserve">Eſto omnis diſpofitio, ut in præcedente, niſi quòd punctum b nunc ponimus eſſe cẽtrum uiſus, <lb/>& punctum a punctum rei uiſæ.</s> <s xml:id="echoid-s48766" xml:space="preserve"> R efringatur itaq;</s> <s xml:id="echoid-s48767" xml:space="preserve"> forma puncti a ad uiſum b à puncto e:</s> <s xml:id="echoid-s48768" xml:space="preserve"> & erit linea <lb/>refractionis e b.</s> <s xml:id="echoid-s48769" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s48770" xml:space="preserve"> extenſa per lineam a e refringitur per lineam e b, ſicut in præcedente <lb/>propoſitione forma extenſa per lineam b e refringitur per lineam e a.</s> <s xml:id="echoid-s48771" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s48772" xml:space="preserve"> forma puncti a refrin-<lb/>gitur ad uiſum b ex alio puncto circuli c d e, quàm ex puncto è:</s> <s xml:id="echoid-s48773" xml:space="preserve"> tunc utiq;</s> <s xml:id="echoid-s48774" xml:space="preserve"> forma puncti b refringe-<lb/>retur ad uiſum a ex eodem puncto, ut oſtenſum eſt in 9 huius:</s> <s xml:id="echoid-s48775" xml:space="preserve"> ſed iam in præcedente declaratum <lb/>eſt hoc eſſe impoſsibile.</s> <s xml:id="echoid-s48776" xml:space="preserve"> Forma enim extẽſa per lineam b e, & refracta per lineam e a, per præceden-<lb/>tem proximam non poteſt refringi ad uiſum exiſtẽtem in puncto a ab alio puncto circuli c d e, neq;</s> <s xml:id="echoid-s48777" xml:space="preserve"> <lb/>ex alio puncto ſuperficiei corporis diaphani:</s> <s xml:id="echoid-s48778" xml:space="preserve"> quoniam in ſuperficie refractoinis ſolus cadit ille cir-<lb/>culus.</s> <s xml:id="echoid-s48779" xml:space="preserve"> Non ergo refringetur forma puncti a ad uiſum exiaſtentem in puncto b ex alio puncto circuli <lb/>c d e, niſi ex puncto e:</s> <s xml:id="echoid-s48780" xml:space="preserve"> & unica tantùm uidebitur imago, De diuerſitate quoq;</s> <s xml:id="echoid-s48781" xml:space="preserve"> locorum imaginum <lb/>eſtitem, ſicut in præmiſſa, declarandum.</s> <s xml:id="echoid-s48782" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s48783" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1834" type="section" level="0" n="0"> <head xml:id="echoid-head1353" xml:space="preserve" style="it">25. Cum ſuperficies ſphærica conuexa corporis diaphani denſioris aere fuerit oppoſita uiſui <lb/>exiſtenti extra circulum cõmunis ſextionis ſuperficiei refractionis & corporis ſphærici diapha-<lb/>nidenſioris: proßibile eſt lineam rectam taliter ſiſti, ut aliquis ipſius punctus directè, & diuerſa <lb/>puncta eiuſdem lineæ uideãtur refractè: totá forma illius line æ refringatur à protione ſuper-<lb/>ficiei corporis illius terminata circulo non magno: & locus imaginis ſuæ ſit in centro uiſus. <lb/>Alhazen 29 n 7.</head> <p> <s xml:id="echoid-s48784" xml:space="preserve">Eſto communis ſectio ſuperficiei refractionis & corporis ſphærici conuexi denſioris diaphani <lb/>quàm eſt aer, circulus g e d, cuius centrum ſit z:</s> <s xml:id="echoid-s48785" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s48786" xml:space="preserve"> ſemidiameter z e:</s> <s xml:id="echoid-s48787" xml:space="preserve"> ſuper cuius terminum <lb/>e fiat per 23 p 1 angulus z e k æqualis maximo angulo incidentiæ, quem continet linea extenſionis <lb/>formæ puncti rei exiſtentis ſub illo diaphano, ad uiſum exiſtentem extra illud diaphanum in aere <lb/>uel in alio diaphano rariori, cum linea perpẽdiculari ducta à puncto e ſuper ſuperficiem illius cor-<lb/>poris, à qua fit refractio:</s> <s xml:id="echoid-s48788" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s48789" xml:space="preserve"> angulus k e t per eandem 23 p 1 æ qualis medietati maximi angulire-<lb/>fractionis, qui poteſt fieri inter corpora diaphana quæcunq;</s> <s xml:id="echoid-s48790" xml:space="preserve"> data, ut inter aquam & aerẽ, uel econ-<lb/>uerſo:</s> <s xml:id="echoid-s48791" xml:space="preserve"> hoc autem eſt poſsibile:</s> <s xml:id="echoid-s48792" xml:space="preserve"> quoniam omnes iſti anguli per 8 huius ſuntnoti.</s> <s xml:id="echoid-s48793" xml:space="preserve"> Et à puncto z cen-<lb/>tro corporis groſsioris ducatur linea æquidiſtans lineæ e t per 31 p 1:</s> <s xml:id="echoid-s48794" xml:space="preserve"> quæ producta ex utraque par-<lb/>te ad circumferentiam ſit g z d:</s> <s xml:id="echoid-s48795" xml:space="preserve"> & linea e z ex parte puncti e protrahatur extra corpus illud uſq;</s> <s xml:id="echoid-s48796" xml:space="preserve"> ad <lb/>h punctum.</s> <s xml:id="echoid-s48797" xml:space="preserve"> Cum ita que, ut patet ex præmiſsis, proportio anguli z e k ad duplum anguli k e t ſit ma- <pb o="426" file="0728" n="728" rhead="VITELLONIS OPTICAE"/> xima proportio, quam angulus incidentiæ, quem continet linea, per quam extenditur forma pun-<lb/>cti rei uiſæ ad ſuperficiem corporis, à qua refringitur, cum linea per <lb/> <anchor type="figure" xlink:label="fig-0728-01a" xlink:href="fig-0728-01"/> pendiculari à puncto refractionis ſuper ſuperficiem illius corporis <lb/>educta, poſsit habere ad angulum reſractionis, quem exigit ille an-<lb/>gulus incidẽtiæ quo ad ſenſum (anguli enim refractionis, qui ſiunt <lb/>inter duo corpora diuerſæ diaphanitatis, à luce tranſeunte perilla <lb/>corpora diuerſantur, quorũ diuerſitas quo ad ſenſum, habet finem, <lb/>quem ſi angulus exceſſerit:</s> <s xml:id="echoid-s48798" xml:space="preserve"> tunc ſenſus non comprehendet quan-<lb/>titatem refractionis:</s> <s xml:id="echoid-s48799" xml:space="preserve"> cõprehendet enim directè cẽtrum lucis tran-<lb/>ſeuntis per illa duo corpora in rectitudine lineæ, per quam exten-<lb/>ditur:</s> <s xml:id="echoid-s48800" xml:space="preserve"> & hoc plenius experiri poteſt per inſtrumentum, quo ſupe-<lb/>rius uſi ſumus) & cum, ut patet expræmiſsis, angulus e z d ſit ma-<lb/>ior angulo k et:</s> <s xml:id="echoid-s48801" xml:space="preserve"> ponatur ergo angulus d z t æ qualis angulo k e t per <lb/>27th.</s> <s xml:id="echoid-s48802" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s48803" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s48804" xml:space="preserve"> linea e k concurrit cum linea e t:</s> <s xml:id="echoid-s48805" xml:space="preserve"> patet per <lb/>2 th.</s> <s xml:id="echoid-s48806" xml:space="preserve"> 1 huius quia concurrit cũ linea a d eius æ quidiſtante:</s> <s xml:id="echoid-s48807" xml:space="preserve"> ſit ut con-<lb/>currat in puncto b.</s> <s xml:id="echoid-s48808" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s48809" xml:space="preserve"> linea z t cõcurret cum linea e t:</s> <s xml:id="echoid-s48810" xml:space="preserve"> <lb/>ſit, ut concurrat in puncto t.</s> <s xml:id="echoid-s48811" xml:space="preserve"> Et quia lineæ e b & z e ſunt inter duas <lb/>lineas æ quidiſtantes, & in eadem ſuperficie:</s> <s xml:id="echoid-s48812" xml:space="preserve"> patet quòd ipſæ ſe in-<lb/>terſecant:</s> <s xml:id="echoid-s48813" xml:space="preserve"> ſit pũctus ſectionis k:</s> <s xml:id="echoid-s48814" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s48815" xml:space="preserve"> per 32 p 1 angulus z k e æqua-<lb/>lis duobus angulis k z b & k b z:</s> <s xml:id="echoid-s48816" xml:space="preserve"> ſed angulus k b z eſt per 29 p 1 <lb/>æ qualis angulo k e t:</s> <s xml:id="echoid-s48817" xml:space="preserve"> angulus ergo z k e eſt æ qualis duplo anguli k <lb/>e t:</s> <s xml:id="echoid-s48818" xml:space="preserve"> ergo per 7 p 5 erit proportio anguli z e k ad angulem z k e ma-<lb/>xima proportio, quę eſt poſsibilis inueniri inter angulum inciden-<lb/>tiæ (quem continet linea, per quam extenditur forma, & perpen-<lb/>dicularis exiens à loco refractionis) & inter angulum refractionis, <lb/>quem exigit ille angulus incidentiæ.</s> <s xml:id="echoid-s48819" xml:space="preserve"> Item à puncto e per 31 p 1 du-<lb/>catur linea æ quidiſtans lineæ t z, quæ per 2 th.</s> <s xml:id="echoid-s48820" xml:space="preserve"> 1 huius cõcurret cum <lb/>linea z g uerſus punctum g:</s> <s xml:id="echoid-s48821" xml:space="preserve"> ſit itaq;</s> <s xml:id="echoid-s48822" xml:space="preserve"> punctus concurſus a:</s> <s xml:id="echoid-s48823" xml:space="preserve"> & extra-<lb/>hatur linea b e extra circulum g e d uſq;</s> <s xml:id="echoid-s48824" xml:space="preserve"> ad punctum li erit ergo an-<lb/>gulus l e a æ qualis angulo z k e per 29 p 1:</s> <s xml:id="echoid-s48825" xml:space="preserve"> & angulus l e h æ qualis eſt angulo z e k per 15 p 1.</s> <s xml:id="echoid-s48826" xml:space="preserve"> Erir <lb/>ergo, ut patet ex præ miſsis, angulus l e a angulus ille refractionis, quem exigit angulus l e h:</s> <s xml:id="echoid-s48827" xml:space="preserve"> quo-<lb/>niam per 15 p 1 angulus l e h eſt æ qualis angulo z e k, qui acceptus eſt talis, ut proponitur.</s> <s xml:id="echoid-s48828" xml:space="preserve"> Si itaque <lb/>centrum uiſus fuerit in puncto a, aliquo ſcilicet puncto aeris, & corpus diaphanum denſius aere <lb/>(cuius conuexum eſt ex parte uiſus a) ſuerit continuatum uſque ad punctum b, & non fuerit di-<lb/>ſtinctum apud circulum g d e e x parte b, ita ut diuerſitas alterius diaphani non impediat naturam <lb/>refractionis:</s> <s xml:id="echoid-s48829" xml:space="preserve"> tunc forma punctio b extenditur per lineam b e, & refringitur per lineam e a:</s> <s xml:id="echoid-s48830" xml:space="preserve"> & com-<lb/>prehenditur à uiſu in puncto a per lineam e a.</s> <s xml:id="echoid-s48831" xml:space="preserve"> Et quoniam angulus refractionis, qui eſt a el, poteſt <lb/>diuidi pluribus portionibus earum, quæ poſſunt eſſe inter angulos refractionis & angulos inci-<lb/>dentiæ, quos continent dictæ perpendiculares cum lineis, per quas incidunt formæ corporibus <lb/>diaphanis, à quarum ſuperficie refringuntur:</s> <s xml:id="echoid-s48832" xml:space="preserve"> in linea itaque d b erunt plura puncta, quoram for-<lb/>mæ extenduntur ad arcum g e, & refringuntur ab illo ad uiſum a:</s> <s xml:id="echoid-s48833" xml:space="preserve"> & forma totius lineæ d b, in qua <lb/>ſunt omnia illa puncta, refringuntur ad uiſum a e x arcu g e.</s> <s xml:id="echoid-s48834" xml:space="preserve"> Si itaque figatur linea a g b, & reuolua-<lb/>tur trigonum a e b in cirucuitu lineæ a b fixæ, & pars ſuperficiei corporis diaphani, quæ eſt ex par-<lb/>te rei uiſæ, fuerit ſphærica:</s> <s xml:id="echoid-s48835" xml:space="preserve"> tunc punctum e, quod eſt punctum refractionis, ſignabit motu ſuo in <lb/>ſuperficie corporis ſphærica conuexa circulum exparte uiſus a, à quo tota refringetur forma pun-<lb/>cti b ad uiſum a:</s> <s xml:id="echoid-s48836" xml:space="preserve"> ſed locus imaginis in tota peripheria circuli refractionis erit unus:</s> <s xml:id="echoid-s48837" xml:space="preserve"> quoniam, ut pa-<lb/>tet per 15 huius, locus imaginis eſt centrum uiſus, in quo concurrit linea extenſionis formæ, quę eſt <lb/>e a, & perpẽdicularis b z a.</s> <s xml:id="echoid-s48838" xml:space="preserve"> Similiterq̃;</s> <s xml:id="echoid-s48839" xml:space="preserve"> formæ omnium punctorum lineæ d b, excepto puncto d, re-<lb/>fringuntur ab aliquo puncto arcus e g, ſecundum quod præmiſſum eſt:</s> <s xml:id="echoid-s48840" xml:space="preserve"> & locus imaginis omnium <lb/>illorum punctorum ſemper erit in centro uiſus:</s> <s xml:id="echoid-s48841" xml:space="preserve"> & ſic tota imago illius rei uiſæ eſt una.</s> <s xml:id="echoid-s48842" xml:space="preserve"> Compre-<lb/>henditur itaq;</s> <s xml:id="echoid-s48843" xml:space="preserve"> forma huius rei uiſæ ab ipſo uiſu formæ circularis apud ciculũ refractionis:</s> <s xml:id="echoid-s48844" xml:space="preserve"> & uni-<lb/>cus eius punctus ſuperior circa punctum d uidetur in rectitudine perpendicularis, tranſeuntis per <lb/>centrum uiſus & rem uiſam.</s> <s xml:id="echoid-s48845" xml:space="preserve"> Cum ergo centrum uiſus fuerit in uno corpore diaphano, & res uiſa <lb/>fuerit in alio diaphano denſiori:</s> <s xml:id="echoid-s48846" xml:space="preserve"> & ſuperficies corporis diaphani denſioris, quæ eſt ex parte uiſus, <lb/>fuerit ſphærica conuexa:</s> <s xml:id="echoid-s48847" xml:space="preserve"> ſueritq́;</s> <s xml:id="echoid-s48848" xml:space="preserve"> uiſus extra circulum, cuius conuexum eſt ex parte uiſus;</s> <s xml:id="echoid-s48849" xml:space="preserve"> fueritq́;</s> <s xml:id="echoid-s48850" xml:space="preserve"> <lb/>ille circulus remotior à uiſu quàm pũctum remotius formæ (cuius fit refractio, ut eſt in propoſito <lb/>punctum b) diſtans fuerit à duo bus punctis ſectionis factæ inter perpendiculares & circumferen-<lb/>tiam:</s> <s xml:id="echoid-s48851" xml:space="preserve"> & cũ corpus diaphanũ denſius, quod eſt à parte rei uiſæ, fuerit totũ continuũ uſq;</s> <s xml:id="echoid-s48852" xml:space="preserve"> ad locũ, in <lb/>quo eſt res uiſa, nec fuerit in aliquo puncto mediũ interciſum:</s> <s xml:id="echoid-s48853" xml:space="preserve"> tunc uiſus cõprehendet formã illius <lb/>rei uiſæ & uerè & refractè:</s> <s xml:id="echoid-s48854" xml:space="preserve"> & locus imaginis illius rei uiſæ erit in cẽtro uiſus:</s> <s xml:id="echoid-s48855" xml:space="preserve"> uidebitur aũtin in ſuper-<lb/>ficie uiſus.</s> <s xml:id="echoid-s48856" xml:space="preserve"> Quod eſt propoſitũ.</s> <s xml:id="echoid-s48857" xml:space="preserve"> Si uerò ſic accidat, ut perpũdicularis ducta à re uiſa ſuper ſuperficiẽ <lb/>corporis, à qua fit refractio, æ quidiſtet alicui illarum linearũ, per quas forma peruenit ad uiſum, & <lb/>alicui non:</s> <s xml:id="echoid-s48858" xml:space="preserve"> poſsibile erit, ut forma rei uideatur partim in ſuperficie corporis, à quo fit refractio, & <lb/> <pb o="427" file="0729" n="729" rhead="LIBER DECIMVS."/> partim in ſuperficie uiſus:</s> <s xml:id="echoid-s48859" xml:space="preserve"> & hoc erit ut monſtruoſum.</s> <s xml:id="echoid-s48860" xml:space="preserve"> Huiuſmodi quoq;</s> <s xml:id="echoid-s48861" xml:space="preserve"> inſinita accidunt ſecun-<lb/>dum diuerſitatem lineæ perpendicularis, reſpectu lineæ extenſionis ipſius formæ.</s> <s xml:id="echoid-s48862" xml:space="preserve"> Eodem quoq;</s> <s xml:id="echoid-s48863" xml:space="preserve"> <lb/>modo demonſtrandum eſt, ſi punctus rei uiſæ fuerit in diaphano rariori, & centrum uiſus in dia-<lb/>phano denſiori, diſpoſita figura ſecundum diſpoſitionem illorum angulorum, qui tali perti-<lb/>nent refractioni.</s> <s xml:id="echoid-s48864" xml:space="preserve"/> </p> <div xml:id="echoid-div1834" type="float" level="0" n="0"> <figure xlink:label="fig-0728-01" xlink:href="fig-0728-01a"> <variables xml:id="echoid-variables835" xml:space="preserve">a l g h e z d k b t</variables> </figure> </div> </div> <div xml:id="echoid-div1836" type="section" level="0" n="0"> <head xml:id="echoid-head1354" xml:space="preserve" style="it">26. Communi ſectione ſuperficiei refractionis & ſuperficiei corporis diaphani, à quo fit refra <lb/>ctio, exiſtente circulo, punctó rei uiſæ exiſtente in perpendiculari ducta à centro uiſus ſuper <lb/>concauam ſuperficiem corporis diaphani oppoſit am uiſui: forma reiuiſæ rectè occurret uiſui, & <lb/>à nullo puncto fiet refractio: una quo tantùm uidebitur imago. Alhazen 30 n 7.</head> <p> <s xml:id="echoid-s48865" xml:space="preserve">Sit a centrum uiſus:</s> <s xml:id="echoid-s48866" xml:space="preserve"> & ſit b punctus rei uiſæ ultra corpus diaphanum, quod ſit, exempli cauſſa, <lb/>groſsius illo, in quo eſt centrũ uiſus a:</s> <s xml:id="echoid-s48867" xml:space="preserve"> ſit quoq;</s> <s xml:id="echoid-s48868" xml:space="preserve"> corporis groſsioris ſuperficies, quæ eſt ex parte ui-<lb/>ſus ſphærica cõcaua:</s> <s xml:id="echoid-s48869" xml:space="preserve"> cuius ſit centrũ g.</s> <s xml:id="echoid-s48870" xml:space="preserve"> Dico quòd pũctis a & b exiſten <lb/>tibus in una linea perpendiculari ſuper ſuperficiẽ illius corporis con-<lb/> <anchor type="figure" xlink:label="fig-0729-01a" xlink:href="fig-0729-01"/> caui:</s> <s xml:id="echoid-s48871" xml:space="preserve"> tunc b punctus rei uiſæ unam ſolam habebit imaginem, & unam <lb/>tantùm formam apud centrum uiſus a.</s> <s xml:id="echoid-s48872" xml:space="preserve"> Ducatur enim linea a g:</s> <s xml:id="echoid-s48873" xml:space="preserve"> & ex-<lb/>trahatur rectè uſq;</s> <s xml:id="echoid-s48874" xml:space="preserve"> ad punctum z.</s> <s xml:id="echoid-s48875" xml:space="preserve"> Erit ergo per 72th.</s> <s xml:id="echoid-s48876" xml:space="preserve"> 1 huius linea a z <lb/>perpendicularis ſuper ſuperficiem concauam corporis diaphani.</s> <s xml:id="echoid-s48877" xml:space="preserve"> Sit <lb/>quoq;</s> <s xml:id="echoid-s48878" xml:space="preserve"> punctus b in linea a z:</s> <s xml:id="echoid-s48879" xml:space="preserve"> uiſus itaq;</s> <s xml:id="echoid-s48880" xml:space="preserve"> a comprehendet formam pun-<lb/>cti b in rectitudine lineæ a b:</s> <s xml:id="echoid-s48881" xml:space="preserve"> quoniam linea a b eſt perpendicularis ſu-<lb/>per concauam ſuperficiem illius corporis, quod eſt diaphanum groſ-<lb/>ſius:</s> <s xml:id="echoid-s48882" xml:space="preserve"> neq;</s> <s xml:id="echoid-s48883" xml:space="preserve"> ab aliquo puncto ipſam poterit comprehendere refractam, <lb/>Cuius contrarium ſi detur eſſe poſsibile:</s> <s xml:id="echoid-s48884" xml:space="preserve"> eſto, ut forma puncti b re-<lb/>fringatur ad a uiſum à puncto corporis e:</s> <s xml:id="echoid-s48885" xml:space="preserve"> & ducantur lineæ b e & g e:</s> <s xml:id="echoid-s48886" xml:space="preserve"> <lb/>eritq́;</s> <s xml:id="echoid-s48887" xml:space="preserve"> linea g e perpendicularis ſuper ſuperficiem corporis, à qua fit re-<lb/>fractio:</s> <s xml:id="echoid-s48888" xml:space="preserve"> & extrahatur linea b e uſq;</s> <s xml:id="echoid-s48889" xml:space="preserve"> ad punctum t:</s> <s xml:id="echoid-s48890" xml:space="preserve"> angulus itaq;</s> <s xml:id="echoid-s48891" xml:space="preserve"> t e g eſt <lb/>angulus incidentiæ contentus à linea, per quam extenditur forma, & <lb/>à linea perpendiculari exeunte à loco refractionis ſuper ſuperficiem <lb/>corporis, à qua fit refractio.</s> <s xml:id="echoid-s48892" xml:space="preserve"> Et quia corpus, quod eſt ex parte uiſus a, <lb/>ſubtilius eſt illo, quod eſt ex parte rei uiſæ, in qua eſt punctum b:</s> <s xml:id="echoid-s48893" xml:space="preserve"> pa-<lb/>làm per 4 huius quoniam erit refractio ad partem contrariam illi par-<lb/>ti, in qua eſt perpendicularis, quæ e g:</s> <s xml:id="echoid-s48894" xml:space="preserve"> & linea e t non concurrit cum li-<lb/>nea b a aliquo modo.</s> <s xml:id="echoid-s48895" xml:space="preserve"> Forma ergo puncti b non refrin gitur ad uiſum a.</s> <s xml:id="echoid-s48896" xml:space="preserve"> <lb/>Non ergo cõprehendet uiſus ipſam refractè, ſed ſolùm rectè:</s> <s xml:id="echoid-s48897" xml:space="preserve"> non ergo <lb/>habebit apud uiſum a punctum b niſi unam folam formam & unam <lb/>imaginem.</s> <s xml:id="echoid-s48898" xml:space="preserve"> Si uerò corpus, in quo eſt res uiſa, ſuerit rarius corpore, in <lb/>quo eſt centrũ uiſus, adhuc eadem eſt demonſtratio:</s> <s xml:id="echoid-s48899" xml:space="preserve"> nec enim adhue <lb/>perueniet refractio ad centrum uiſus.</s> <s xml:id="echoid-s48900" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s48901" xml:space="preserve"/> </p> <div xml:id="echoid-div1836" type="float" level="0" n="0"> <figure xlink:label="fig-0729-01" xlink:href="fig-0729-01a"> <variables xml:id="echoid-variables836" xml:space="preserve">t a e y z b</variables> </figure> </div> </div> <div xml:id="echoid-div1838" type="section" level="0" n="0"> <head xml:id="echoid-head1355" xml:space="preserve" style="it">27. Communi ſectione ſuperficiei refractionis & ſuperficiei corpo <lb/>ris diaphani, à quo ſit refractio, exiſtente circulo, puncto uiſoiacen <lb/>te extra perpendicularem ductam à centro uiſus ſuper ſuperficiem <lb/>concauam oppoſitam uiſui corporis großioris diaphano contingente uiſum: ab unot tantùm pun <lb/>cto ſiet refractio: & unica refract a uidebitur imago: loco tamen imaginis diuerſiſicato ſecundũ <lb/>diuerſitatem loci punctiuiſi. Alhazen 31 n 7.</head> <p> <s xml:id="echoid-s48902" xml:space="preserve">Eſto diſpoſitio, quæ in præcedente:</s> <s xml:id="echoid-s48903" xml:space="preserve"> & ſit punctus b extra lineam a z.</s> <s xml:id="echoid-s48904" xml:space="preserve"> Et quoniam, ut patet per 2 <lb/>th.</s> <s xml:id="echoid-s48905" xml:space="preserve"> huius, omnis ſuperficies refractionis perpendicularis eſt ſuper ſuperficiem corporis, à quo fit re <lb/>fractio, ſit per 69 th.</s> <s xml:id="echoid-s48906" xml:space="preserve"> 1 huius communis ſectio ſuperficiei refractionis, & ſuperficxiei concauæ cor-<lb/>poris diaphani, à quo fit refractio, circulus h d k, cuius centrum ſit g:</s> <s xml:id="echoid-s48907" xml:space="preserve"> & ſit punctus refractionis for-<lb/>mæ puncti b ad uiſum a punctum h.</s> <s xml:id="echoid-s48908" xml:space="preserve"> Dico quòd nõ fiet refractio formæ puncti b ad uiſum a ex alio <lb/>puncto circuli h d k, quàm ex puncto h.</s> <s xml:id="echoid-s48909" xml:space="preserve"> Si enim hoc ſit poſsibile, ſit illud aliud punctum refractio-<lb/>nis m:</s> <s xml:id="echoid-s48910" xml:space="preserve"> & ducãtur lineæ a h, b h, g h, a m, b m, g m:</s> <s xml:id="echoid-s48911" xml:space="preserve"> ſecetq̃;</s> <s xml:id="echoid-s48912" xml:space="preserve"> linea h a lineã m g in pũcto f:</s> <s xml:id="echoid-s48913" xml:space="preserve"> & protrahatur <lb/>linea b h intra corpus diaphanum reliquum ad punctum c:</s> <s xml:id="echoid-s48914" xml:space="preserve"> & linea b m ad punctum n:</s> <s xml:id="echoid-s48915" xml:space="preserve"> & linea g h <lb/>ad punctũ l:</s> <s xml:id="echoid-s48916" xml:space="preserve"> & linea g m ad punctũ p:</s> <s xml:id="echoid-s48917" xml:space="preserve"> ſecet quoq;</s> <s xml:id="echoid-s48918" xml:space="preserve"> linea a g protracta ultra punctũ g circumferentiã <lb/>circuli in puncto k.</s> <s xml:id="echoid-s48919" xml:space="preserve"> Aut igitur centrum uiſus a erit in linea k d, quę eſt diameter circuli:</s> <s xml:id="echoid-s48920" xml:space="preserve"> aut extra il-<lb/>lam ultra punctum k.</s> <s xml:id="echoid-s48921" xml:space="preserve"> Si uiſus a fuerit in linea k d:</s> <s xml:id="echoid-s48922" xml:space="preserve"> tunc aut erit in centro g:</s> <s xml:id="echoid-s48923" xml:space="preserve"> auto in altera duarum li-<lb/>nearum g k uel g d.</s> <s xml:id="echoid-s48924" xml:space="preserve"> Si ergo fuerit a centrũ uiſus in centro g:</s> <s xml:id="echoid-s48925" xml:space="preserve"> tunc forma puncti b non refringetur ad <lb/>uiſum a per pręmiſſam proximã propoſitionem:</s> <s xml:id="echoid-s48926" xml:space="preserve"> lineæ enim continuantes corpus diaphanũ ſphæ-<lb/>ricũ cũ centro g, per 72.</s> <s xml:id="echoid-s48927" xml:space="preserve"> th.</s> <s xml:id="echoid-s48928" xml:space="preserve"> 1 huius ſunt perpendiculares ſuper ſuperficiẽ corporis, quod eſt ex par-<lb/>re uiſus:</s> <s xml:id="echoid-s48929" xml:space="preserve"> non fit autem aliqua refractio formarum incidentium ſecundum lineas perpendiculares, <lb/> <pb o="428" file="0730" n="730" rhead="VITELLONIS OPTICAE"/> ut ibi oſtenſum eſt.</s> <s xml:id="echoid-s48930" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s48931" xml:space="preserve"> puncti b non refringetur ad uiſum a in centro corporis diapha-<lb/>ni exiſtentem.</s> <s xml:id="echoid-s48932" xml:space="preserve"> Quòd ſi uiſus a fuerit in linea g d:</s> <s xml:id="echoid-s48933" xml:space="preserve"> tunc linea h cerit inter duas lineas h a & h g:</s> <s xml:id="echoid-s48934" xml:space="preserve"> & ſi-<lb/>militer linea n m erit inter duas lineas m a & m g:</s> <s xml:id="echoid-s48935" xml:space="preserve"> quoniam per 4 huius & ex hypotheſi refractio fit <lb/>ad partem contrariã parti ambarũ perpendicularium, quæ ſunt h g & m g:</s> <s xml:id="echoid-s48936" xml:space="preserve"> corpus enim diaphanũ, <lb/>quod eſt parte uiſus a, eſt ſubtilius illo corpore diaphano, quod eſt ex parte rei uiſæ.</s> <s xml:id="echoid-s48937" xml:space="preserve"> Si autem linea <lb/>h c fuerit inter duas lineas h a & h g, & a centrum uiſus <lb/>fuerit in linea g d:</s> <s xml:id="echoid-s48938" xml:space="preserve"> tũe angulus b h a erit ex parte puncti <lb/> <anchor type="figure" xlink:label="fig-0730-01a" xlink:href="fig-0730-01"/> d, ſeilicet reſpiciens punctũ d:</s> <s xml:id="echoid-s48939" xml:space="preserve"> & ſimiliter angulus b m a <lb/>erit ex parte puncti d:</s> <s xml:id="echoid-s48940" xml:space="preserve"> & erit punctum b ultra lineam g h <lb/>l uerſus punctũ k:</s> <s xml:id="echoid-s48941" xml:space="preserve"> quod patet per 15 p 1.</s> <s xml:id="echoid-s48942" xml:space="preserve"> Si enim linea h c <lb/>cadit inter lineas h a & h g:</s> <s xml:id="echoid-s48943" xml:space="preserve"> tunc oportet quòd linea h b <lb/>cadat inter lineas h l & g k:</s> <s xml:id="echoid-s48944" xml:space="preserve"> & erit angulus c h g angulus <lb/>incidentiæ contentus à linea, per quam extenditur for-<lb/>ma, & à perpendiculari g h:</s> <s xml:id="echoid-s48945" xml:space="preserve"> & ſittiliter erit angulus n m <lb/>gangulus incidentię:</s> <s xml:id="echoid-s48946" xml:space="preserve"> & erit angulus c h a angulus refra <lb/>ctionis:</s> <s xml:id="echoid-s48947" xml:space="preserve"> & ſimiliter angulus n m a.</s> <s xml:id="echoid-s48948" xml:space="preserve"> Angulus uerò n m g <lb/>aut erit ęqualis angulo c h g:</s> <s xml:id="echoid-s48949" xml:space="preserve"> aut maior:</s> <s xml:id="echoid-s48950" xml:space="preserve"> aut minor.</s> <s xml:id="echoid-s48951" xml:space="preserve"> Si æ-<lb/>qualis:</s> <s xml:id="echoid-s48952" xml:space="preserve"> ergo & angulus n m a erit æ qualis angulo c h a ք <lb/>8 huius:</s> <s xml:id="echoid-s48953" xml:space="preserve"> & angulus b m a erit æ qualis angulo b h a ք 13 p <lb/>1:</s> <s xml:id="echoid-s48954" xml:space="preserve"> hoc aũt impoſsibile & cótra 33 th.</s> <s xml:id="echoid-s48955" xml:space="preserve"> 1 huius, & 21 p 1, ut pa <lb/>tet ducta linea b a.</s> <s xml:id="echoid-s48956" xml:space="preserve"> Si aũtangulus n m g ſit maior angulo <lb/>c h g:</s> <s xml:id="echoid-s48957" xml:space="preserve"> erit quoq:</s> <s xml:id="echoid-s48958" xml:space="preserve"> per 8 huius angulus n m a maior angulo <lb/>c h a:</s> <s xml:id="echoid-s48959" xml:space="preserve"> & ſic angulus b m a erit minor angulo b h a:</s> <s xml:id="echoid-s48960" xml:space="preserve"> quod <lb/>eſt item impoſsibile, ut prius.</s> <s xml:id="echoid-s48961" xml:space="preserve"> Quòd ſiangulus n m g ſit <lb/>minor angulo c h g:</s> <s xml:id="echoid-s48962" xml:space="preserve"> tunc angulus n m a per 8 huius erit <lb/>minor angulo c h a:</s> <s xml:id="echoid-s48963" xml:space="preserve"> & ſic totus angulus refractus, qui eſt <lb/>a m g, erit minor toto angulo refracto, qui eſt a h g:</s> <s xml:id="echoid-s48964" xml:space="preserve"> & e-<lb/>rit diminutio anguli refractionis, qui eſt n m a, ab angu-<lb/>l o refractionis, qui eſt c h a, minor quàm diminutio an-<lb/>guli a m g ab angulo a h g, qui ambo ſunt anguli refracti:</s> <s xml:id="echoid-s48965" xml:space="preserve"> <lb/>(& ſi quandoq;</s> <s xml:id="echoid-s48966" xml:space="preserve"> in eadem proportione plus excedit an-<lb/>gulus refractus maior minorem, quàm illorum angulo-<lb/>rum refractionis maior minorem, ut pater per 8 huius, <lb/>& ex tabulis) ſed diminutio anguli a m g ab angulo a h g eſt æ qualis diminutioni anguli h g m ab <lb/>angulo h a m:</s> <s xml:id="echoid-s48967" xml:space="preserve"> ideo quia duo anguli contrapoſiti, qui ſunt ad punctum f, punctũ ſcilicet ſectionis li-<lb/>nearum h a & m g, ſunt æqualies per 15 p 1, & reliqui duo anguli trigonorum g f h & a f m cuiuslibet <lb/>cum ſuo tertio ualent duos rectos per 32 p 1.</s> <s xml:id="echoid-s48968" xml:space="preserve"> Diminutio itaq;</s> <s xml:id="echoid-s48969" xml:space="preserve"> anguli refractionis, qui n m a, ab angu <lb/>lo refractionis a h c eſt minor quàm diminutio anguli h g m ab angulo h a m.</s> <s xml:id="echoid-s48970" xml:space="preserve"> Educantur itaq;</s> <s xml:id="echoid-s48971" xml:space="preserve"> duæ <lb/>lineę h a & m a ad circumferentiã circuli:</s> <s xml:id="echoid-s48972" xml:space="preserve"> & incidat linea a h puncto e:</s> <s xml:id="echoid-s48973" xml:space="preserve"> & linea m a puncto o:</s> <s xml:id="echoid-s48974" xml:space="preserve"> erit er-<lb/>go angulus h a m ille angulus, quem reſpiciunt in circumferentia circuli h d k duo arcus h m & o e <lb/>per 54 th.</s> <s xml:id="echoid-s48975" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s48976" xml:space="preserve"> & an gulũ h g m reſpicit in circumferentia arcus h m duplicatus per 20 p 3.</s> <s xml:id="echoid-s48977" xml:space="preserve"> Et quo-<lb/>niam angulus h g m eſt minor angulo h a m:</s> <s xml:id="echoid-s48978" xml:space="preserve"> ideo quia, ut patet ex præmiſsis, angulus a h g eſt ma-<lb/>ior angulo a m g:</s> <s xml:id="echoid-s48979" xml:space="preserve"> patet per 33 p.</s> <s xml:id="echoid-s48980" xml:space="preserve"> 6 quia arcus duplicatus h m eſt minor duobus arcubus h m & e o:</s> <s xml:id="echoid-s48981" xml:space="preserve"> <lb/>& erit diminutio arcus duplicati h m à duobus arcubus h m & e o, diminutio arcus h m ab ar-<lb/>cu e o:</s> <s xml:id="echoid-s48982" xml:space="preserve"> quoniam arcus h m utrobiq;</s> <s xml:id="echoid-s48983" xml:space="preserve"> eſt communis.</s> <s xml:id="echoid-s48984" xml:space="preserve"> Ergo diminutio anguli n m a ab angulo cha <lb/>erit minor angulo, quem reſpicit apud circumferentiam diminutio arcus h m ab arcu e o:</s> <s xml:id="echoid-s48985" xml:space="preserve"> ſed an-<lb/>gulus, quem reſpicit apud circumſerentiam diminutio arcus h m ab arcu e o eſt minor angulo h <lb/>a m, ut patet ex præmiſsis:</s> <s xml:id="echoid-s48986" xml:space="preserve"> ergo diminutio anguli n m a ab angulo c h a erit minor angulo h a <lb/>m:</s> <s xml:id="echoid-s48987" xml:space="preserve"> ergo per 13 p 1 exceſſus anguli b m a ſupera angulum b h a eſt minor angulo h a m:</s> <s xml:id="echoid-s48988" xml:space="preserve"> ſed exceſſus an-<lb/>guli b m a ſupera angulum b h a per 33 th.</s> <s xml:id="echoid-s48989" xml:space="preserve"> 1 huius ſunt duo anguli h a m & h b m:</s> <s xml:id="echoid-s48990" xml:space="preserve"> ergo illi duo angu-<lb/>li ſunt minores an gulo h a m, totum ſua parte:</s> <s xml:id="echoid-s48991" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s48992" xml:space="preserve"> Quòd ſi centrum uiſus a, fuerit <lb/>in linea g k:</s> <s xml:id="echoid-s48993" xml:space="preserve"> tunc, ſicut prius oſtenſum eſt, linea h c erit inter duas lineas h g & h a:</s> <s xml:id="echoid-s48994" xml:space="preserve"> & linea m n erit <lb/>inter duas lineas m g & m a:</s> <s xml:id="echoid-s48995" xml:space="preserve"> erit ergo angulus b h a ex parte pucti k:</s> <s xml:id="echoid-s48996" xml:space="preserve"> & ſimiliter angulus b m a e-<lb/>rit ex parte puncti k:</s> <s xml:id="echoid-s48997" xml:space="preserve"> & erit punctũ rei uiſæ, quod eſt b, infra lineam g m p ex parte d:</s> <s xml:id="echoid-s48998" xml:space="preserve"> & itẽ, ut prius, <lb/>anguli c h g & n m g ſunt anguli incidentię contenti à lineis, per quas extenditur forma, & à perpen <lb/>dicularibus exeuntibus à punctis refractiõis:</s> <s xml:id="echoid-s48999" xml:space="preserve"> & anguli c h a & n m a ſunt anguli refractiõis.</s> <s xml:id="echoid-s49000" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s49001" xml:space="preserve"> <lb/>angulus c h g fuerit æ qualis angulo n m g:</s> <s xml:id="echoid-s49002" xml:space="preserve"> tũc erit, ut prius, per 8 huius angulus c h a æ qualis angu-<lb/>lo n m a:</s> <s xml:id="echoid-s49003" xml:space="preserve"> & ſic item per 13 p 1 angulus b h a erit æ qualis angulo b m a:</s> <s xml:id="echoid-s49004" xml:space="preserve"> quod eſt impoſsibile & contra <lb/>21 p 1 ducta linea b a, ut ſuperà.</s> <s xml:id="echoid-s49005" xml:space="preserve"> Si uerò angulus c h g eſt maior angulo n m g:</s> <s xml:id="echoid-s49006" xml:space="preserve"> tũc per 8 huius angulus <lb/>c h a erit maior angulo n m a:</s> <s xml:id="echoid-s49007" xml:space="preserve"> & ſic itẽ angulus b h a erit minor angulo b m a:</s> <s xml:id="echoid-s49008" xml:space="preserve"> quod eſt impoſsibile, <lb/>ut ſuprà.</s> <s xml:id="echoid-s49009" xml:space="preserve"> Quòd ſi angulus c h g fuerit minor angulo n m g:</s> <s xml:id="echoid-s49010" xml:space="preserve"> tunc angulus c h a eſt minor angu-<lb/>lo n m a:</s> <s xml:id="echoid-s49011" xml:space="preserve"> & ſic totus angulus g h a erit minor totali angulo g m a:</s> <s xml:id="echoid-s49012" xml:space="preserve"> eritá;</s> <s xml:id="echoid-s49013" xml:space="preserve"> tũc modo pręoſtenſo angu <lb/>lus h g m minor angulo h a m.</s> <s xml:id="echoid-s49014" xml:space="preserve"> Ergo diminutio anguli h g m ab angulo h a m erit minor ꝗ̃ angulus <lb/>g m a:</s> <s xml:id="echoid-s49015" xml:space="preserve"> & diminutio anguli c h a ab angulo n m a eſt minor ꝗ̃ diminutio anguli g h a ab angulo g m a:</s> <s xml:id="echoid-s49016" xml:space="preserve"> <lb/> <pb o="429" file="0731" n="731" rhead="LIBER DECIMVS."/> eſt ergo minor quàm diminutio anguli h g m ab angulo h a m:</s> <s xml:id="echoid-s49017" xml:space="preserve"> ergo diminutio anguli c h a ab an-<lb/>gulo n m a eſt minor quàm angulus g m a.</s> <s xml:id="echoid-s49018" xml:space="preserve"> Sed diminutio anguli c h a ab angulo n m a eſt exceſſus <lb/> <anchor type="figure" xlink:label="fig-0731-01a" xlink:href="fig-0731-01"/> <anchor type="figure" xlink:label="fig-0731-02a" xlink:href="fig-0731-02"/> anguli b h a ſupera angulũ b m a ք 13 p 1:</s> <s xml:id="echoid-s49019" xml:space="preserve"> exceſſus uerò anguli b h a ſupera angulum b m a ſunt duo an-<lb/>guli h a m & h b m per 33 th.</s> <s xml:id="echoid-s49020" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s49021" xml:space="preserve"> ergo iſti duo anguli ſimul ſuperti ſunt minores angulo h a m, <lb/>totum ſua parte:</s> <s xml:id="echoid-s49022" xml:space="preserve"> quod eſt impoſsibile.</s> <s xml:id="echoid-s49023" xml:space="preserve"> Si uerò centrum uiſus a fuerit extra diametrum k d:</s> <s xml:id="echoid-s49024" xml:space="preserve"> hoc e-<lb/>rit ad partem k, quæ reſpicit partem concauam ſuperficiei ſphæræ diaphanæ:</s> <s xml:id="echoid-s49025" xml:space="preserve"> quoniam ad par-<lb/>tem z eſt conuexitas ſphæræ corporis diaphani, à cuius ſuperficie fit refractio.</s> <s xml:id="echoid-s49026" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s49027" xml:space="preserve"> tunc cor-<lb/>pus diaphanum, in quo eſt centrum uiſus a, fuerit continuum ad uiſum a, ducantur duæ lineæ a <lb/>h & a m.</s> <s xml:id="echoid-s49028" xml:space="preserve"> et quoniam illæ lineæ non ſunt contingentes circulum d m k:</s> <s xml:id="echoid-s49029" xml:space="preserve"> palàm per 57th.</s> <s xml:id="echoid-s49030" xml:space="preserve"> 1 huius <lb/>quoniam circulum ſecabunt:</s> <s xml:id="echoid-s49031" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s49032" xml:space="preserve"> ipſum linea a h in puncto q:</s> <s xml:id="echoid-s49033" xml:space="preserve"> & linea a m in puncto r:</s> <s xml:id="echoid-s49034" xml:space="preserve"> & produ <lb/>cantur aliæ lineæ, ut prius.</s> <s xml:id="echoid-s49035" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s49036" xml:space="preserve"> angulus c h g fuerit æqualis angulo n m g:</s> <s xml:id="echoid-s49037" xml:space="preserve"> tunc angulus b h a <lb/>eſt æqualis angulo b m a:</s> <s xml:id="echoid-s49038" xml:space="preserve"> quod eſt impoſsibile, ut prius.</s> <s xml:id="echoid-s49039" xml:space="preserve"> Et ſiangulus c h g fuerit maior angulo n <lb/>m g, & angulus c h a erit maior angulo n m a:</s> <s xml:id="echoid-s49040" xml:space="preserve"> erit ergo per 13 p 1 angulus b h a minor angulo b m a:</s> <s xml:id="echoid-s49041" xml:space="preserve"> <lb/>quod item eſt impoſsibile, ut ſuprà.</s> <s xml:id="echoid-s49042" xml:space="preserve"> Si uerò angulus c h g fuerit minor anguluo n m g:</s> <s xml:id="echoid-s49043" xml:space="preserve"> erit angulus <lb/>c h a minor angulo n m a:</s> <s xml:id="echoid-s49044" xml:space="preserve"> & totus angulus g h a minor toto anguluo g m a:</s> <s xml:id="echoid-s49045" xml:space="preserve"> ergo, ut prius, erit angu-<lb/>lus h g m minor angulo h a m:</s> <s xml:id="echoid-s49046" xml:space="preserve"> ſed angulus h g m eſt ille, quẽ apud circumferentiã reſpicit arcus h m <lb/>duplicatus:</s> <s xml:id="echoid-s49047" xml:space="preserve"> & angulus h a m eſt ille angulus, quẽ reſpicitin circum ferẽtia exceſſus arcus h m ſupra <lb/>arcũ r q, ut patet ք 55 th.</s> <s xml:id="echoid-s49048" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s49049" xml:space="preserve"> ergo arcus h m duplicatus eſt minor exceſſu arcus h m ſupra arcũ r <lb/>q:</s> <s xml:id="echoid-s49050" xml:space="preserve"> qđ eſt impoſsibile:</s> <s xml:id="echoid-s49051" xml:space="preserve"> quoniam ſic ſequitur totum eſſe minus ſuna parte.</s> <s xml:id="echoid-s49052" xml:space="preserve"> Vbicunq;</s> <s xml:id="echoid-s49053" xml:space="preserve"> ergo ſecundum <lb/>hypotheſim præmiſſam ſit punctum rei uiſibilis, quod eſt b, extra perpendicularem ductam à cen-<lb/>tro uiſus a ſuper ſuperficiem corporis diaphani oppoſiti uiſui:</s> <s xml:id="echoid-s49054" xml:space="preserve"> patet quia imago formæ puncti <lb/>b non refringitur ad uiſum a, niſi ab uno tantùm puncto:</s> <s xml:id="echoid-s49055" xml:space="preserve"> & erit una tantùm imago reſracta.</s> <s xml:id="echoid-s49056" xml:space="preserve"> Diuer-<lb/>ſificabitur quoq;</s> <s xml:id="echoid-s49057" xml:space="preserve"> ſemper locus imaginis ſecundum diuerſitatẽ concurſus perpendicularis ductæ <lb/>à puncto b rei uiſæ ſuper ſuperficiem corporis diaphani, à quo fit refractio, cum linea, per quam <lb/>extenditur forma ad centrum uiſus a:</s> <s xml:id="echoid-s49058" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s49059" xml:space="preserve"> locus imaginis quandoq;</s> <s xml:id="echoid-s49060" xml:space="preserve"> retro uiſum:</s> <s xml:id="echoid-s49061" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s49062" xml:space="preserve"> an-<lb/>te uiſum:</s> <s xml:id="echoid-s49063" xml:space="preserve"> quandoque in centro uiſus.</s> <s xml:id="echoid-s49064" xml:space="preserve"> Et ſi illas lineas contingat fieri æquidiſtantes, ut non con-<lb/>currant:</s> <s xml:id="echoid-s49065" xml:space="preserve"> erit locus imaginis in puncto refractionis:</s> <s xml:id="echoid-s49066" xml:space="preserve"> ſcilicet in ſuperficie corporis, à qua fit refra-<lb/>ctio, ut hæc omnia declarata ſunt per 16 huius.</s> <s xml:id="echoid-s49067" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s49068" xml:space="preserve"/> </p> <div xml:id="echoid-div1838" type="float" level="0" n="0"> <figure xlink:label="fig-0730-01" xlink:href="fig-0730-01a"> <variables xml:id="echoid-variables837" xml:space="preserve">k g o e a d z c n s h m</variables> </figure> <figure xlink:label="fig-0731-01" xlink:href="fig-0731-01a"> <variables xml:id="echoid-variables838" xml:space="preserve">e o a c n g d z k h m l p b</variables> </figure> <figure xlink:label="fig-0731-02" xlink:href="fig-0731-02a"> <variables xml:id="echoid-variables839" xml:space="preserve">a k r q c n h l m d p e b</variables> </figure> </div> </div> <div xml:id="echoid-div1840" type="section" level="0" n="0"> <head xml:id="echoid-head1356" xml:space="preserve" style="it">28. Communi ſectione ſuperficiei refractionis & ſuperficiei corporis diaphani, à quo fit re-<lb/>fractio, exiſtente circulo, punctó rei uiſæ iacente extra perpendicularem ductam à centro ui-<lb/>ſus ſuper concauam ſuperficiem, oppoſitam uiſui corporis rarioris diaphano cõtingente uiſum: <lb/>ab uno tantùm puncto fiet refr actio: & unica refracta uidebitur imago. Alhazen 32 n 7.</head> <p> <s xml:id="echoid-s49069" xml:space="preserve">Remaneat omnis diſpoſitio proximæ præcedentis, niſi quòd punctum b ſit centrum uiſus, & a <lb/>ſit punctũ rei uiſę.</s> <s xml:id="echoid-s49070" xml:space="preserve"> Refringatur itaq;</s> <s xml:id="echoid-s49071" xml:space="preserve"> forma puncti a à puncto ſuperficiei corporis diaphani, quod eſt <lb/>h:</s> <s xml:id="echoid-s49072" xml:space="preserve"> & erit linea refracta, quę a h b:</s> <s xml:id="echoid-s49073" xml:space="preserve"> forma itaq;</s> <s xml:id="echoid-s49074" xml:space="preserve"> extenſa per lineã a h, refringitur քlineã h b:</s> <s xml:id="echoid-s49075" xml:space="preserve"> ſicut in prę-<lb/>cedẽte figuratiõe forma extenſa ք lineã b h, refringitur ք lineã h a.</s> <s xml:id="echoid-s49076" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s49077" xml:space="preserve"> forma pũcti a refringitur <lb/>ad uiſum b ex alio pũcto circuli h d k, ꝗ̃ ex puncto h:</s> <s xml:id="echoid-s49078" xml:space="preserve"> tũc utiq;</s> <s xml:id="echoid-s49079" xml:space="preserve"> forma pũcti b refringitur ad uiſum exi <lb/> <pb o="430" file="0732" n="732" rhead="VITELLONIS OPTICAE"/> ſtentem in puncto a ex eodem puncto, ut patet per 9 huius:</s> <s xml:id="echoid-s49080" xml:space="preserve"> ſed iam in pręcedente declaratum eſt <lb/>hoc eſſe impoſsibile.</s> <s xml:id="echoid-s49081" xml:space="preserve"> Forma enim extenſa per lineam b h & refracta per lineam h a, nõ poteſt refrin-<lb/>gi ad uiſum in punctum h ab alio puncto circuli h d k, quàm ex puncto h:</s> <s xml:id="echoid-s49082" xml:space="preserve"> neq;</s> <s xml:id="echoid-s49083" xml:space="preserve"> exaliquo alio puncto <lb/>ſuperficiei corporis diaphani:</s> <s xml:id="echoid-s49084" xml:space="preserve"> quoniam in ſuperficie refractionis ſolus cadit ille circulus.</s> <s xml:id="echoid-s49085" xml:space="preserve"> Non ergo <lb/>refringitur forma puncti a ad uiſum exiſtentem in puncto b ex alio puncto circuli h d k, niſi ex pun-<lb/>cto h:</s> <s xml:id="echoid-s49086" xml:space="preserve"> & unica tantùm uidebitur imago.</s> <s xml:id="echoid-s49087" xml:space="preserve"> Et hoc eſt propoſitum.</s> <s xml:id="echoid-s49088" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1841" type="section" level="0" n="0"> <head xml:id="echoid-head1357" xml:space="preserve" style="it">29. Concaua ſuperficic corporis diaphani denſioris aere uiſui oppoſita: poßibile eſt lineam re-<lb/>ctam taliter ſiſti, ut aliquis eius punctus directè, & diuerſa puct a eiuſdem lineæ uide antur re <lb/>fractè: totá forma illius lineæ refringatur à portione ſuperficiei illius corporis: & locus imagi-<lb/>nis ſuæ ſit in centro uiſus. Alhazen 29 n 7.</head> <p> <s xml:id="echoid-s49089" xml:space="preserve">Eſto per modum 25 huius, Communis ſectio ſuperficiei refractionis, & corporis ſphærici conca-<lb/>ui denſioris aere (ut uitri uel cryſtalli) circulus g e d, cuius centrum ſit punctum z:</s> <s xml:id="echoid-s49090" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s49091" xml:space="preserve"> ſemi-<lb/>diameter z e:</s> <s xml:id="echoid-s49092" xml:space="preserve"> ſuper cuius terminum, punctum e fiat per 23 p 1 angu-<lb/>lus z e k æqualis maximo angulo incidentiæ, quem continet linea <lb/>extenſionis formæ puncti rei exiſtentis ſub illo diaphano, ad uniſum <lb/> <anchor type="figure" xlink:label="fig-0732-01a" xlink:href="fig-0732-01"/> exiſtentem extra illud diaphanum in aere uel in alio diaphano ra-<lb/>riori, cum linea perpendiculari ducta à puncto e ſuper ſuperficiem <lb/>illius corporis, à qua fit refractio:</s> <s xml:id="echoid-s49093" xml:space="preserve"> fiatq́;</s> <s xml:id="echoid-s49094" xml:space="preserve"> angulus k e t per eandẽ 23 p 1 <lb/>æ qualis medietati maximi anguli refractionis, qui poteſt fieri inter <lb/>illa corpora diaphana quæcuq;</s> <s xml:id="echoid-s49095" xml:space="preserve"> data, ut, exempli cauſſa, inter ui-<lb/>trum concauum & aerem (hoc autem eſt poſsibile:</s> <s xml:id="echoid-s49096" xml:space="preserve"> quoniam o-<lb/>mnes iſti anguli per 8 huius ſunt noti) & à puncto z centri corporis <lb/>concaui, ut uitri uel cryſtalli, ducatur linea æquidiſtans lineę e t per <lb/>31 p 1, quæ producta ex utraq;</s> <s xml:id="echoid-s49097" xml:space="preserve"> parte ad circumferenctiam ſit g z d:</s> <s xml:id="echoid-s49098" xml:space="preserve"> <lb/>& linea e z ex parte puncti e protrahatur extra corpus illud uſ-<lb/>que ad punctum h.</s> <s xml:id="echoid-s49099" xml:space="preserve"> Atque ſic completa totali figuratione & de-<lb/>monſtratione 25 huius:</s> <s xml:id="echoid-s49100" xml:space="preserve"> patet quòd concaua ſuperficie corporis <lb/>diaphani denſioris aere uiſui oppoſita, poſsibile eſt lineam rectam <lb/>taliter ſiſti, ut aliquis eius punctus uideatur directè, & diuerſa pun-<lb/>cta eiuſdem lineæ uideantur refractè:</s> <s xml:id="echoid-s49101" xml:space="preserve"> totaq́;</s> <s xml:id="echoid-s49102" xml:space="preserve"> forma illius lineæ re-<lb/>fringatur ab una portione ſuperficiei illius corporis concaui uitrei <lb/>uel cryſtallini, terminata ad circulum non magnum illius ſphæræ:</s> <s xml:id="echoid-s49103" xml:space="preserve"> <lb/>& quòd punctus d uidetur ſecundum perpendicularem a d ſine re-<lb/>fractione:</s> <s xml:id="echoid-s49104" xml:space="preserve"> omnium uerò aliorum punctorum lineæ d b formæ re-<lb/>fringuntur.</s> <s xml:id="echoid-s49105" xml:space="preserve"> Perpendiculares quoq;</s> <s xml:id="echoid-s49106" xml:space="preserve"> omnium illorum punctorum <lb/>ſuntin linea b a concurrentes cum lineis, per quas ueniunt formæ <lb/>ad uiſum in ipſo centro uiſus puncto a.</s> <s xml:id="echoid-s49107" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s49108" xml:space="preserve"> propoſitum per <lb/>15 th.</s> <s xml:id="echoid-s49109" xml:space="preserve"> huius.</s> <s xml:id="echoid-s49110" xml:space="preserve"> Expræmiſsis itaq;</s> <s xml:id="echoid-s49111" xml:space="preserve"> octo theorematibus patent paſsio-<lb/>nes occurrentes uiſui propter medium ſecundi diaphani, in quo <lb/>res eſt uiſa, cuius figuara eſt ſphærica, ſiue ſit conuexa, ſiue concaua.</s> <s xml:id="echoid-s49112" xml:space="preserve"> <lb/>Et quando corpore ſecundi diaphani exiſtente figuræ columna-<lb/>ris uel pyramidalis, communis ſectio ſuperficiei refractionis eſt li-<lb/>nea recta:</s> <s xml:id="echoid-s49113" xml:space="preserve"> tunc omniono uniformis paſsio accidit uiſui per illa, ſicut <lb/>accidit per corpora alia diaphana planarum ſuperficierum.</s> <s xml:id="echoid-s49114" xml:space="preserve"> quarum communis ſectio & ſuperficiei <lb/>refractionis eſt linea recta, & eodem modo demonſtrandum.</s> <s xml:id="echoid-s49115" xml:space="preserve"> Quando uerò illa communis ſectio <lb/>eſt circulus:</s> <s xml:id="echoid-s49116" xml:space="preserve"> tunc accidunt eadem in corporibus columnaribus diaphanis, quæ accidunt in corpo-<lb/>ribus ſphæricis concauis uel conuexis:</s> <s xml:id="echoid-s49117" xml:space="preserve"> præter hæc quòd à circumterentia unius circuli ſuperficiei <lb/>corporis ſecũdi diaphani nõ poteſt in talibus corporibus fieri refractio ad uiſum, ſicut oſtendimus <lb/>in 23 huius à corporibus ſphæricis conuexis fieri, In corporibus uerò pyramidalibus diaphanis <lb/>concauis uel conuexis non poteſt communis ſectio ſuperficiei refractionis & ſuperficiei illius cor-<lb/>poris eſſe circulus, ſicut oſtenſum eſt in ſuperficie bus reflexionum per 12 th.</s> <s xml:id="echoid-s49118" xml:space="preserve"> 7 & 2th.</s> <s xml:id="echoid-s49119" xml:space="preserve"> 9 huius:</s> <s xml:id="echoid-s49120" xml:space="preserve"> quo-<lb/>niam etiam omnes ſuperficies refractionum erectæ ſunt ſuper ſuperficies corporum, à quibus fit <lb/>refractio, ut patet per 2 huius:</s> <s xml:id="echoid-s49121" xml:space="preserve"> unde iſtæ paſsiones non pertinent ad illa.</s> <s xml:id="echoid-s49122" xml:space="preserve"> Quòd ſi communis ſectio <lb/>ſuperficiei corporis diaphani & ſuperficiei refractionis in corporibus columnaribus uel pyrami-<lb/>dalibus diaphanis fuerit ſectio oxygonia:</s> <s xml:id="echoid-s49123" xml:space="preserve"> ab uno tantùm puncto fiet refractio, ſicut nunc oſtendi-<lb/>mus in circulis uel conuexis uel concauis.</s> <s xml:id="echoid-s49124" xml:space="preserve"> Et imago formæ rei uiſæ quandoq;</s> <s xml:id="echoid-s49125" xml:space="preserve"> uid ebitur intra cor-<lb/>pus dlaphanum:</s> <s xml:id="echoid-s49126" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s49127" xml:space="preserve"> inter uiſum & corpus diaphanum:</s> <s xml:id="echoid-s49128" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s49129" xml:space="preserve"> in ſuperficie ipſius cor-<lb/>poris diaphani:</s> <s xml:id="echoid-s49130" xml:space="preserve"> quando que in ſuperficie ipſius uiſus, ſicut acciderit lineam perpendicularem du-<lb/>ctam à puncto rei uiſæ ſuper ſuperficiem corporis diaphani concurrere uel æ quidiſtare lineæ ex-<lb/>tenſionis ipſius formæ, per quam forma peruerit ad uiſum:</s> <s xml:id="echoid-s49131" xml:space="preserve"> unde non duximus talibus am-<lb/>plius immorandum.</s> <s xml:id="echoid-s49132" xml:space="preserve"/> </p> <div xml:id="echoid-div1841" type="float" level="0" n="0"> <figure xlink:label="fig-0732-01" xlink:href="fig-0732-01a"> <variables xml:id="echoid-variables840" xml:space="preserve">s f g h z e d k b t</variables> </figure> </div> </div> <div xml:id="echoid-div1843" type="section" level="0" n="0"> <head xml:id="echoid-head1358" xml:space="preserve" style="it">30. Superficiebus corporum diaphanorum oppoſitorum uiſui diuerſarum figurarum, uelipſis <lb/> <pb o="431" file="0733" n="733" rhead="LIBER DECIMVS."/> eorporibus diuerſæ diaphanitatis exiſtentibus: loca imaginum form arum trans illacorpor aui <lb/>ſarum diuer ſantur: & occurunt uiſui forme monſtruoſæ & ιmagines numeratæ numero pun-<lb/>ctorum refractionis. Alhazen 33 n 7.</head> <p> <s xml:id="echoid-s49133" xml:space="preserve">Expræmiſsis enlm patet quòd in corporibus diaphanis, quæ ſunt unius figuræ & ſubſtantiæ, u-<lb/>na tantùm occurrιt uiſui imago omnium corporum, quorum formæ trans illa corpora diaphana ſe <lb/>m ιltiplicantad uiſum.</s> <s xml:id="echoid-s49134" xml:space="preserve"> Siuero corpus diaphanum, per quod fit uiſio, fuerit ſuperficiei compoſitæ <lb/>ex diuerſis ſiguris:</s> <s xml:id="echoid-s49135" xml:space="preserve"> ut ſortè ex plana & ſphærica, uel ex ſphærica & columnariuúc (cum ſuperſicies <lb/>oppoſita uiſui fuerit diuerſa ex diuerſis figuris compoſita, & natura perpendicularium & linearum <lb/>extenſionis formarum ſecundum diuerſitatem ſigurarum ipſarum ſuperficierum diuerſificetur) <lb/>pater per 15 huius quod loca imaginum formarum uiſarum diuerſantur:</s> <s xml:id="echoid-s49136" xml:space="preserve"> & fortaſſe diuerſa erunt <lb/>puncta reſractionum ſormæ eiuſdem puncti rei uiſæ ad eundem uiſum, & diuerſæ lineæ extenſio-<lb/>nis ſormarum, & diuerſæ perpen diculares:</s> <s xml:id="echoid-s49137" xml:space="preserve"> propter quod plures uidebuntur imagines eiuſdem rei <lb/>uiſæ reſractæ à ſuperſiciebus talium corporum.</s> <s xml:id="echoid-s49138" xml:space="preserve"> Vnde ſi quis aſpexerit aliquod uiſibile exiſtens ul-<lb/>tra corpus diaphanum, cuius ſuperficies oppoſita uiſui, ſit ſiguræ compoſitæ ex ſuperficie ſphæræ <lb/>magnæ & paruæ, ut ſæpe acciditin cryſtallis uel alijs lapidibus diaphanis & uitris:</s> <s xml:id="echoid-s49139" xml:space="preserve"> patet quòd cen-<lb/>tra illarum ſphærarum ſunt diuerſa per 81 th.</s> <s xml:id="echoid-s49140" xml:space="preserve">1 huius:</s> <s xml:id="echoid-s49141" xml:space="preserve"> illæ enim ſphærę ſe interſecant.</s> <s xml:id="echoid-s49142" xml:space="preserve"> Erunt ergo <lb/>perpendiculares illę ductę ab uno puncto rei uiſę ſuper ſuperſiciem illius corporis magnam haben <lb/>tes diuerſitatem.</s> <s xml:id="echoid-s49143" xml:space="preserve"> Et ſi figura ſuperficiei illorum corporum ſuerit compoſita ex ſuperficie ſphærica <lb/>& columnari:</s> <s xml:id="echoid-s49144" xml:space="preserve"> patet quod maior eſt diuerſitas & punctorum reſractionis & perpendicularium <lb/>ductarum.</s> <s xml:id="echoid-s49145" xml:space="preserve"> Diſſormabitur ergo diſpoſitio imaginũ trans hæc corpora diaphana:</s> <s xml:id="echoid-s49146" xml:space="preserve"> & fortè illa ſorma <lb/>uidebitur monſtruoſa, propter confluxum diuerſarum imaginum ad conſtitutionem unius formę, <lb/>cum puncta refractionum ſuerint adinuicem propinqua, & interſectiones perpendicularium & li <lb/>nearum extenſionis ſormarum ſuerint adinuicem propinquæ.</s> <s xml:id="echoid-s49147" xml:space="preserve"> Si uerò puncta reſractionum uel <lb/>prædictarum ſectionum ſuerint ad inuicem ſenſibiliter diſtantia:</s> <s xml:id="echoid-s49148" xml:space="preserve"> tunc uidentur plures imagines e-<lb/>iuſdem rei uiſæ:</s> <s xml:id="echoid-s49149" xml:space="preserve"> quoniam illarum reſractio non eſt una, neq;</s> <s xml:id="echoid-s49150" xml:space="preserve"> unitur, ſed remanet diuerſa.</s> <s xml:id="echoid-s49151" xml:space="preserve"> Forma e-<lb/>nim rei uiſæ extenditur ab ipſa re ad ſuperficies ſphæricas uel columnares uel alterius figurę ipſius <lb/>corporis diaphani, & reſringitur abillis apud concauitatem aeris contingentis illud corpus dia-<lb/>phanum:</s> <s xml:id="echoid-s49152" xml:space="preserve"> & ita ſit comprehenſio ſormarum eiuſdem rei ex diuerſis reſractionibus:</s> <s xml:id="echoid-s49153" xml:space="preserve"> unde imagines <lb/>diuerſæ fiunt numeratæ numero punctorum reſractionis:</s> <s xml:id="echoid-s49154" xml:space="preserve"> Idem quoq;</s> <s xml:id="echoid-s49155" xml:space="preserve"> accidit ſi corpus diaphanũ <lb/>uniſorme in ſuperſicie, ſuerit diuerſæ diaphanitatis:</s> <s xml:id="echoid-s49156" xml:space="preserve"> ſcilicet in una ſui parte denſius, & in alia parte <lb/>rarius:</s> <s xml:id="echoid-s49157" xml:space="preserve"> tunc enim ſecundum unam ſui partem fit refractio ad partem perpendicularis, & in alia ſui <lb/>parte ad partem contrariam:</s> <s xml:id="echoid-s49158" xml:space="preserve"> & ſic iterum aut ſormæ fiunt monſtruoſæ:</s> <s xml:id="echoid-s49159" xml:space="preserve"> aut ſortè aliter diuerſæ & <lb/>numero diſſerentes.</s> <s xml:id="echoid-s49160" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s49161" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1844" type="section" level="0" n="0"> <head xml:id="echoid-head1359" xml:space="preserve" style="it">31 Cõmuni ſectione ſuper ficiei refractionis & ſuperficiei corporis, à quo fit refr actio exiſtente <lb/>linea rect a: uiſu quo exiſtente in perpendiculari exeunte à medio puncto lineæ uiſæ ſuper pla-<lb/>nam ſuper ſiciem corpor is diaphani, à qua forma illius lineæ refringitur ad uiſum, ſi linea uiſa <lb/>æ quidiſtans fuerit ſuperficiei corporis diaphani cuiuſcũ ſiue denſioris ſiue rarιoris primo: ima <lb/>go refract a rei uiſæ comprehenditur maior re uiſa. Alhazen 39 n 7.</head> <p> <s xml:id="echoid-s49162" xml:space="preserve">Eſto punctus a centrum uiſus:</s> <s xml:id="echoid-s49163" xml:space="preserve"> & ſit linea uiſa in medio ſecundi diaphani, quæ b c:</s> <s xml:id="echoid-s49164" xml:space="preserve"> cuius medius <lb/>punctus ſit z:</s> <s xml:id="echoid-s49165" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s49166" xml:space="preserve"> cõmunis ſectio ſuperficiei refractionis & planæ ſuperficiei corporis diaphani li-<lb/>nea d e:</s> <s xml:id="echoid-s49167" xml:space="preserve"> ducaturq́:</s> <s xml:id="echoid-s49168" xml:space="preserve"> à pucto z, quod eſt medius punctus lineæ b c, linea perpendicularis ſuper li-<lb/>neam d e per 12 p 1:</s> <s xml:id="echoid-s49169" xml:space="preserve"> quæ producatur ultra punctum m.</s> <s xml:id="echoid-s49170" xml:space="preserve"> Erit itaq;</s> <s xml:id="echoid-s49171" xml:space="preserve"> linea z m perpendicu <lb/>lariter erecta ſuper ſuperficiem corporis planam, in qua eſt linea d e:</s> <s xml:id="echoid-s49172" xml:space="preserve"> quoniam ſuperficies refractid <lb/>nis, in qua producitur linea z m, & in qua eſt linea c d, erecta eſt ſuper illam ſuperficiem corporis <lb/>diaphani per 2 th.</s> <s xml:id="echoid-s49173" xml:space="preserve"> huius:</s> <s xml:id="echoid-s49174" xml:space="preserve"> ſitq́ linea b c æ quidiſtans lineæ d e.</s> <s xml:id="echoid-s49175" xml:space="preserve"> Exiſtente itaq;</s> <s xml:id="echoid-s49176" xml:space="preserve"> centro uiſus a in linea <lb/>z m:</s> <s xml:id="echoid-s49177" xml:space="preserve"> dico quòd linea b c uidetur maior quàm ſit ſecundum ueritatem.</s> <s xml:id="echoid-s49178" xml:space="preserve"> Nec enim tranſit per centru <lb/>uiſus, quod eſt a & per aliquod punctum lineæ b c, præter punctum z, ſup erficies, quæ ſit erecta ſu-<lb/>per ſuperficiem corporis diaphani, niſi ſola ſuperficies refractionis, in qua ſunt lineę a z & b c.</s> <s xml:id="echoid-s49179" xml:space="preserve"> Non <lb/>enim tranſit per a ſuperficies erecta ſuper ſuperficiem corporis diaphani, niſi illa, quæ tranſit per li <lb/>neam a z, quæ eſt linea perpendicularis ſuper ſuperficiẽ corporis diaphani:</s> <s xml:id="echoid-s49180" xml:space="preserve"> nec exit a puncto a per-<lb/>pendicularis ſuper ſuperficiem corporis diaphani, niſi linea a z per 20 th.</s> <s xml:id="echoid-s49181" xml:space="preserve">1 huius.</s> <s xml:id="echoid-s49182" xml:space="preserve"> Non ergo tranſit <lb/>per punctũ a aliqua ſuperficies perpen dicularis ſuper ſuperficiem corporis diaphani, niſi ſolũ illa, <lb/>quæ tranſit per lineam a z:</s> <s xml:id="echoid-s49183" xml:space="preserve"> & non tranſit aliqua ſuperſicies per aliquod punctum lineę b c, aliud à <lb/>puncto z, & per lineam a z, niſi ſolùm illa ſuperſicies, in qua ſunt duæ lineę a z & b c.</s> <s xml:id="echoid-s49184" xml:space="preserve"> Non tranſit er-<lb/>go per uiſum a & per aliquod punctũ lineæ b c, præter punctũ z, ſuperficies aliqua perpendicularis <lb/>ſuper ſuperficiem corporis diaphani, niſi ſolùm illa, in qua ſunt lineę a z & b c.</s> <s xml:id="echoid-s49185" xml:space="preserve"> Non ergo refringitur <lb/>forma alicuius punctorũ, quę ſunt in linea b c, niſi ex aliquo punctorũ lineę d e.</s> <s xml:id="echoid-s49186" xml:space="preserve"> Ducantur itaq;</s> <s xml:id="echoid-s49187" xml:space="preserve"> per <lb/>11 p 1 ex prædictis punctis b & c duæ perpendiculares ſuper lineam d e:</s> <s xml:id="echoid-s49188" xml:space="preserve"> quæ, ut patet ex præmiſsis, <lb/>neceſſariò caduntin illá:</s> <s xml:id="echoid-s49189" xml:space="preserve"> & ſint lineæ b d & c e.</s> <s xml:id="echoid-s49190" xml:space="preserve"> Et quoniã lineę b c & d e ſunt ęquidiſtantes ex hypo <lb/>theſi, & lineę b d & c e ſunt æ quidiſtantes per 28 p I:</s> <s xml:id="echoid-s49191" xml:space="preserve"> patet quia quæ libet illarum linearum, que ſunt <lb/>b d & c e, æquidiſtant lineæ a z per eandem 28 p I.</s> <s xml:id="echoid-s49192" xml:space="preserve"> Et patet quòd non reſringetur forma punctib <lb/> <pb o="432" file="0734" n="734" rhead="VITELLONIS OPTICAE"/> ad uiſum a ex puncto d per 3 huius:</s> <s xml:id="echoid-s49193" xml:space="preserve"> neq;</s> <s xml:id="echoid-s49194" xml:space="preserve"> forma puncti cà puncto e:</s> <s xml:id="echoid-s49195" xml:space="preserve"> quoniã lineæ c e & d b ſunt per-<lb/>pendiculares ſuper ſuperficiem corporis diaphani:</s> <s xml:id="echoid-s49196" xml:space="preserve"> nulla aũt perpendicularis refringitur in aliquo <lb/>corpore medio.</s> <s xml:id="echoid-s49197" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s49198" xml:space="preserve">, ut forma puncti b refringatur ad uiſum a ex puncto p, & forma puncti c ex <lb/>puncto h:</s> <s xml:id="echoid-s49199" xml:space="preserve"> & ducantur lineæ b p, p a, c h, h a:</s> <s xml:id="echoid-s49200" xml:space="preserve"> & protrahatur <lb/> <anchor type="figure" xlink:label="fig-0734-01a" xlink:href="fig-0734-01"/> linea a p ultra punctum p ad perpendicularem b d.</s> <s xml:id="echoid-s49201" xml:space="preserve"> Et quo-<lb/>niam linea p a cõcurrit cum linea z a:</s> <s xml:id="echoid-s49202" xml:space="preserve"> patet per 2 th.</s> <s xml:id="echoid-s49203" xml:space="preserve">1 huius <lb/>quoniam ipſa concurret cum eius æquidiſtante, ſcilicet li-<lb/>nea b d:</s> <s xml:id="echoid-s49204" xml:space="preserve"> ſit ergo concurſus in puncto l.</s> <s xml:id="echoid-s49205" xml:space="preserve"> Et eadẽ ratione con <lb/>curret linea a h cũ linea e c in puncto k:</s> <s xml:id="echoid-s49206" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s49207" xml:space="preserve"> per 15 th.</s> <s xml:id="echoid-s49208" xml:space="preserve"> hu-<lb/>ius hoc punctum limago ſormæ puncti b:</s> <s xml:id="echoid-s49209" xml:space="preserve"> & punctũ k ima-<lb/>go formæ puncti c.</s> <s xml:id="echoid-s49210" xml:space="preserve"> Quia uerò linea a z eſt perpendicularis <lb/>ſuper lineam b c, erit per ſ p 1 linea c a æ qualis lineæ b a:</s> <s xml:id="echoid-s49211" xml:space="preserve"> æ-<lb/>qualiter ergo diſtant puncta b & c à puncto a.</s> <s xml:id="echoid-s49212" xml:space="preserve"> Puncta itaq;</s> <s xml:id="echoid-s49213" xml:space="preserve"> <lb/>refractionis, quæ ſunt p & h, æ qualiter diſtabunt æ puncto <lb/>a:</s> <s xml:id="echoid-s49214" xml:space="preserve"> quoniam medium, per quod ſit illorum punctorum for-<lb/>marum diffuſio, eſt uniforme, & linea e d æquidiſtat lineæ <lb/>b c.</s> <s xml:id="echoid-s49215" xml:space="preserve"> Linea itaq;</s> <s xml:id="echoid-s49216" xml:space="preserve"> a p eſt æ qualis lineę a h:</s> <s xml:id="echoid-s49217" xml:space="preserve"> ergo per 5 p 1 an-<lb/>gulus a p h eſt æ qualis angulo a h p:</s> <s xml:id="echoid-s49218" xml:space="preserve"> ergo per 15 p 1 erit an-<lb/>gulus d p l æ qualis angulo e h k:</s> <s xml:id="echoid-s49219" xml:space="preserve"> ſed duo anguli p d l & h e <lb/>k ſunt recti:</s> <s xml:id="echoid-s49220" xml:space="preserve"> ergo angulus p l d per 32 p 1 eſt æ qualis angulo <lb/>h k e:</s> <s xml:id="echoid-s49221" xml:space="preserve"> ergo per 4 p 6 latera iſtorũ trigonorum ſunt propor <lb/>tionalia, quæ æquos angulos reſpiciunt.</s> <s xml:id="echoid-s49222" xml:space="preserve"> Sed linea p d eſt <lb/>æqualis lineæ e h:</s> <s xml:id="echoid-s49223" xml:space="preserve"> quia linea p m eſt æ qualis lineæ h m per <lb/>26 p 1:</s> <s xml:id="echoid-s49224" xml:space="preserve"> trigonorum enim a m p & a m h anguli ad m ſuntre-<lb/>cti, & anguli a h p & a p h ſunt æ quales, & latus a m commu <lb/>ne, æquale ſibijpſi.</s> <s xml:id="echoid-s49225" xml:space="preserve"> Eſt ergo linea p m æqualis lineæ m h.</s> <s xml:id="echoid-s49226" xml:space="preserve"> <lb/>Hocetiã patet per 31 th.</s> <s xml:id="echoid-s49227" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s49228" xml:space="preserve"> iſoſcelis enim eſt trigonus <lb/>h a p, & perpẽdicularis eſt linea a m:</s> <s xml:id="echoid-s49229" xml:space="preserve"> trigona ergo partialia <lb/>ſunt æquiangula:</s> <s xml:id="echoid-s49230" xml:space="preserve"> ergo per 4 p 6 (quia latus a m æquale eſt <lb/>ſibijpſi) erit linea p m æqualis lineæ h m.</s> <s xml:id="echoid-s49231" xml:space="preserve"> Eſt ergo linea e h <lb/>æqualis lineę p d:</s> <s xml:id="echoid-s49232" xml:space="preserve"> patet ergo quoniã linea d l eſt æ qualis lineę e k.</s> <s xml:id="echoid-s49233" xml:space="preserve"> Ducatur itaq;</s> <s xml:id="echoid-s49234" xml:space="preserve"> linea l k:</s> <s xml:id="echoid-s49235" xml:space="preserve"> erit ergo ք <lb/>33 p 1 linea k l æ qualis & æquidiſtãs lineæ b c.</s> <s xml:id="echoid-s49236" xml:space="preserve"> Angulus itaq;</s> <s xml:id="echoid-s49237" xml:space="preserve"> k a l eſt maior angulo b a c ք 34 th.</s> <s xml:id="echoid-s49238" xml:space="preserve">1 hu-<lb/>ius:</s> <s xml:id="echoid-s49239" xml:space="preserve"> & linea l k eſt diameter imaginis lineę b c:</s> <s xml:id="echoid-s49240" xml:space="preserve"> nam omne punctũ lineę b c refrin gitur ad uiſum a ab <lb/>aliquo puncto lineæ p h.</s> <s xml:id="echoid-s49241" xml:space="preserve"> Sicut enim forma puncti b refringitur à puncto p, & punctũ z perpendicu <lb/>lariter ſine reſractione tranſiens punctum m, peruenit ad uiſum a:</s> <s xml:id="echoid-s49242" xml:space="preserve"> ſic punctum, quod eſt inter b & <lb/>z, refringitur ab aliquo puncto lineæ p m, quod eſt inter puncta p & m:</s> <s xml:id="echoid-s49243" xml:space="preserve"> & ſicut forma puncti c reſrin <lb/>gitur ad uiſum a à puncto lineæ e m, quod eſt h:</s> <s xml:id="echoid-s49244" xml:space="preserve"> ſic omne punctum lineæ c z reſringitur ab aliquo <lb/>puncto lineæ h m:</s> <s xml:id="echoid-s49245" xml:space="preserve"> & omne punctum lineæ b z ab aliquo puncto lineæ p m:</s> <s xml:id="echoid-s49246" xml:space="preserve"> ut ſi ſuper lineam b z ſit <lb/>punctum n.</s> <s xml:id="echoid-s49247" xml:space="preserve"> Sι itaq;</s> <s xml:id="echoid-s49248" xml:space="preserve"> dicatur quòd forma puncti n reſringatur ab aliquo puncto lineę m d extra lineã <lb/>ιn p ex parte d, ut à puncto g:</s> <s xml:id="echoid-s49249" xml:space="preserve"> ducatur linea n g.</s> <s xml:id="echoid-s49250" xml:space="preserve"> Palàm itaq:</s> <s xml:id="echoid-s49251" xml:space="preserve"> quoniam linea n g ſecabit lineam b p:</s> <s xml:id="echoid-s49252" xml:space="preserve"> & <lb/>ſit punctus ſectionis q.</s> <s xml:id="echoid-s49253" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s49254" xml:space="preserve"> puncti q perueniet ad uiſum a ex duobus punctis reſractionis, <lb/>ſcilicet p & g:</s> <s xml:id="echoid-s49255" xml:space="preserve"> quod eſt contra 20 uel 21 huius, & impoſsibile.</s> <s xml:id="echoid-s49256" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s49257" xml:space="preserve"> puncti n non reſringetur <lb/>ad uiſum a, niſi ex aliquo puncto lineæ p m, quod eſt inter puncta p & m.</s> <s xml:id="echoid-s49258" xml:space="preserve"> Idẽ quoq;</s> <s xml:id="echoid-s49259" xml:space="preserve"> eſt de omni pun <lb/>cto linę z c, quod eſt inter puncta z & c:</s> <s xml:id="echoid-s49260" xml:space="preserve"> nullum enim illorum refringitur ad uiſum a, niſi ex aliquo <lb/>puncto lineæ h m, quod eſt inter puncta h & m.</s> <s xml:id="echoid-s49261" xml:space="preserve"> Et quia in linea l k omnes perpendiculares ductæ à <lb/>punctis lineæ b & c cum lineis refractionis protractis ſe interſecant:</s> <s xml:id="echoid-s49262" xml:space="preserve"> patet quia linea l k eſt diame-<lb/>terimaginis lineæ b c.</s> <s xml:id="echoid-s49263" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s49264" xml:space="preserve"> lineę b c uidetur in linea l k maior quàm ſecundum ueritatem ſit <lb/>linea b c per 20 th.</s> <s xml:id="echoid-s49265" xml:space="preserve"> 4 huius.</s> <s xml:id="echoid-s49266" xml:space="preserve"> Sub maiori enim angulo uidetur:</s> <s xml:id="echoid-s49267" xml:space="preserve"> quia angulus k a l eſt maior angulo b a <lb/>c per 34 th.</s> <s xml:id="echoid-s49268" xml:space="preserve">1 huius:</s> <s xml:id="echoid-s49269" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s49270" xml:space="preserve"> Et huiuſmodi deceptio accιdit uiſui propter debilitatem <lb/>formæ reſractæ, ut patet per 10 huius:</s> <s xml:id="echoid-s49271" xml:space="preserve"> propter quod aſsimilat ipſam uiſus ſormæ rei, quæ uidetur <lb/>à maiori remotione:</s> <s xml:id="echoid-s49272" xml:space="preserve"> mai or enim diſtantia debilitat formam.</s> <s xml:id="echoid-s49273" xml:space="preserve"> Comprehendit itaq;</s> <s xml:id="echoid-s49274" xml:space="preserve"> uiſus formam li-<lb/>neæ b c reftactiuè ex comparatione anguli k a l maioris angulo b a c, ad diſtantiam maiorem quàm <lb/>ſit diſtantia lineæ b c, & ad poſitionem æqualem poſitioni b c.</s> <s xml:id="echoid-s49275" xml:space="preserve"> Sic itaq;</s> <s xml:id="echoid-s49276" xml:space="preserve"> quantitas lineæ b c compre <lb/>henditur refractè maior propter magnitudinem anguli, quem ſacit propinquitas ad uiſum, & pro-<lb/>pter ſormæ debilitatem, quę cauſſatur propter reſractionem.</s> <s xml:id="echoid-s49277" xml:space="preserve"> Et ſic uniuerſaliter cauſſa, quare linea <lb/>b c apparet maior, eſt reſractio formæ ſuæ in medio ſecũdi diaphani ad uiſum:</s> <s xml:id="echoid-s49278" xml:space="preserve"> & eſt ſemper demon <lb/>ſtratio eadem, ſiue ſiat reſractio in ſuperſicie ſecundi diaphani denſioris ſiue rarioris primo, in quo <lb/>eſt linea b c:</s> <s xml:id="echoid-s49279" xml:space="preserve"> nec enim eſt aliqua diſſerentia quo a d illud:</s> <s xml:id="echoid-s49280" xml:space="preserve"> ſi tamen ſuerit poſsibile inueniri corpora <lb/>diaphana taliter collocata, ut ſuperficies plana poſsit eſſe in corpore rariore corpore contingente <lb/>ipſum uiſum:</s> <s xml:id="echoid-s49281" xml:space="preserve"> ſicut accidit cum uitrum planum contingit uιſum, ita quòd centrum foraminis uueę <lb/>in uitri plana ſuperficie collocatur.</s> <s xml:id="echoid-s49282" xml:space="preserve"/> </p> <div xml:id="echoid-div1844" type="float" level="0" n="0"> <figure xlink:label="fig-0734-01" xlink:href="fig-0734-01a"> <variables xml:id="echoid-variables841" xml:space="preserve">a d g p m h e l k q b n z c</variables> </figure> </div> </div> <div xml:id="echoid-div1846" type="section" level="0" n="0"> <head xml:id="echoid-head1360" xml:space="preserve" style="it">32. Cõmuni ſectione ſuper ſiciei refr actiõis & corporis, à quo ſit reſractio exiſtẽte linea recta: <lb/>uiſu quo exiſtente in perpendiculari, exeunte à medio puncto lineæ uiſæ ſuper planã ſuperficiẽ <lb/> <pb o="433" file="0735" n="735" rhead="LIBER DECIMVS."/> corporis diaphani, à qua forma eius refringitur ad uiſum, ſi linea uiſa non fuerit æquidiſtans ſu <lb/>perficiei corporis diaphani: imago eius comprehẽditur maior ipſa: & maior quàm ſi eſſet ſuper-<lb/>ficiei corporis diaphani æquidiſtans. Alhazen 40 n 7.</head> <p> <s xml:id="echoid-s49283" xml:space="preserve">Sit diſpoſitio eadem, quæ in præcedente, niſi quòd linea b c non ſit æquidiſtans lineæ d e, ſed ſit <lb/>punctus c remotior à puncto a, quàm ſit punctus b:</s> <s xml:id="echoid-s49284" xml:space="preserve"> & à puncto c ducatur linea æ quidiſtans & ęqua <lb/>lis lineæ d e per 31 p 1:</s> <s xml:id="echoid-s49285" xml:space="preserve"> quæ ſit linea c q:</s> <s xml:id="echoid-s49286" xml:space="preserve"> cuius medius punctus ſit o:</s> <s xml:id="echoid-s49287" xml:space="preserve"> & à puncto o per 11 p 11 protraha-<lb/>tur linea perpendicularis ſuper ſuperſiciem corporis diaphani, ſecans lineam d e in puncto m, & li-<lb/>neam b c in puncto z:</s> <s xml:id="echoid-s49288" xml:space="preserve"> & ſit centrum uiſus, quod eſta, in illa perpendiculari, quæ eſt o m:</s> <s xml:id="echoid-s49289" xml:space="preserve"> eritq́ue pũ-<lb/>ctus z in medio puncto lineæ c b.</s> <s xml:id="echoid-s49290" xml:space="preserve"> Quia enim linea b q eſt ęquidiſtans lineæ z o:</s> <s xml:id="echoid-s49291" xml:space="preserve"> erit per 2 p 6 propor <lb/>tio lineæ q o ad o c, ſicut b z ad z c:</s> <s xml:id="echoid-s49292" xml:space="preserve"> ſed linea q o, ut patet ex pręmiſsis, eſt æ qualis lineæ o c:</s> <s xml:id="echoid-s49293" xml:space="preserve"> erit ergo <lb/>linea b z æqualis lineæ z c:</s> <s xml:id="echoid-s49294" xml:space="preserve"> eſt ergo punctus z in medio lineæ c b.</s> <s xml:id="echoid-s49295" xml:space="preserve"> Punctus ita que lineæ d e, à quo for <lb/>ma puncti q reſrin gitur ad uiſum a, ſit p:</s> <s xml:id="echoid-s49296" xml:space="preserve"> & punctus, à quo reſringitur forma punctic, ſit h:</s> <s xml:id="echoid-s49297" xml:space="preserve"> ducãturq́;</s> <s xml:id="echoid-s49298" xml:space="preserve"> <lb/>lineæ a h & a p:</s> <s xml:id="echoid-s49299" xml:space="preserve"> & protrahatur linea a p ad l punctum lineæ d b:</s> <s xml:id="echoid-s49300" xml:space="preserve"> & linea a h ad k punctum lineæ e c:</s> <s xml:id="echoid-s49301" xml:space="preserve"> <lb/>concurrent autem illæ lineæ per 2 th.</s> <s xml:id="echoid-s49302" xml:space="preserve">1 huius, ut oſtendi-<lb/>mus in præmiſſa:</s> <s xml:id="echoid-s49303" xml:space="preserve"> eritq́ue punctum k locus imaginis for <lb/> <anchor type="figure" xlink:label="fig-0735-01a" xlink:href="fig-0735-01"/> mæ puncti c, & punctum l forma puncti q, per 15 th.</s> <s xml:id="echoid-s49304" xml:space="preserve"> hu-<lb/>ius:</s> <s xml:id="echoid-s49305" xml:space="preserve"> ducaturq́ue linea l k:</s> <s xml:id="echoid-s49306" xml:space="preserve"> quæ erit diameter imaginis li-<lb/>neæ q c:</s> <s xml:id="echoid-s49307" xml:space="preserve"> & ducantur lineę a q & a c:</s> <s xml:id="echoid-s49308" xml:space="preserve"> erit itaque, ut in prę-<lb/>cedente, angulus k a l maior angulo c a q per 34 th.</s> <s xml:id="echoid-s49309" xml:space="preserve"> 1 hu-<lb/>ius:</s> <s xml:id="echoid-s49310" xml:space="preserve"> uiſus ergo comprehendetimaginem lineę q c maio-<lb/>rem quàm ſit linea q c, ut patet per pręcedentem.</s> <s xml:id="echoid-s49311" xml:space="preserve"> Et quia <lb/>linea q p ſecat lineam b c:</s> <s xml:id="echoid-s49312" xml:space="preserve"> ſit punctus ſectionis r:</s> <s xml:id="echoid-s49313" xml:space="preserve"> palàm <lb/>itaque cum punctus r ſit in linea q p, quoniam ipſe reſrin <lb/>getur ad uiſum a ex puncto p.</s> <s xml:id="echoid-s49314" xml:space="preserve"> Forma itaque puncti bre-<lb/>fringetur ad uiſum a ex aliquo puncto lineę p d, quod ſit <lb/>inter puncta p & d.</s> <s xml:id="echoid-s49315" xml:space="preserve"> Nam ſi daretur refringi ex aliquo pũ-<lb/>cto inter p & m:</s> <s xml:id="echoid-s49316" xml:space="preserve"> ſequeretur propter interſectionẽ lineæ <lb/>incidentiæ formæ puncti b & lineæ r p unius puncti for <lb/>mam reſringi ad uiſum à duobus punctis lineę d e, quod <lb/>eſt contra 20 uel 21 huius, & impoſsibile.</s> <s xml:id="echoid-s49317" xml:space="preserve"> Reſringaturi-<lb/>taque forma puncti b ad uiſum a ex ſ puncto lineæ p d:</s> <s xml:id="echoid-s49318" xml:space="preserve"> <lb/>& ducatur linea a f:</s> <s xml:id="echoid-s49319" xml:space="preserve"> quæ protracta ad lineam d e, ſecabit <lb/>illam per 14 th.</s> <s xml:id="echoid-s49320" xml:space="preserve">1 huius:</s> <s xml:id="echoid-s49321" xml:space="preserve"> ſecet ergo in puncto i:</s> <s xml:id="echoid-s49322" xml:space="preserve"> eritq́ue per <lb/>15 huius pũctus i locus imaginis formæ puncti b:</s> <s xml:id="echoid-s49323" xml:space="preserve"> & duca <lb/>tur linea i k:</s> <s xml:id="echoid-s49324" xml:space="preserve"> quæ erit diameter imaginis lineæ b c.</s> <s xml:id="echoid-s49325" xml:space="preserve"> Erit <lb/>quoque ſit us lineæ i k reſpectu ſitus a ſimilis ſitui lineæ <lb/>b c.</s> <s xml:id="echoid-s49326" xml:space="preserve"> Quia linea i k aut erit æquidiſtans lineæ b c:</s> <s xml:id="echoid-s49327" xml:space="preserve"> aut <lb/>non erit inter ip ſarum æquidiſtantiam diuerſitas ſenſibi <lb/>lis, mutans ſitum ipſarum reſpectu uiſus a:</s> <s xml:id="echoid-s49328" xml:space="preserve"> quia non eſt <lb/>inter æquidiſtantiam lineæ i k & æquidiſtantiam lineæ <lb/>b c à uiſu grã dis diuerſitas:</s> <s xml:id="echoid-s49329" xml:space="preserve"> declinatio enim lineæ i k à li-<lb/>nea æ quidiſtante lineæ b c, quæ exit à puncto k, erit ualde parua:</s> <s xml:id="echoid-s49330" xml:space="preserve"> angulus ita que i a k eſt maior angu <lb/>lolak per 29 th.</s> <s xml:id="echoid-s49331" xml:space="preserve">1 huius:</s> <s xml:id="echoid-s49332" xml:space="preserve"> & ſimiliter angulus i a k eſt maior angulo b a c per 34 th.</s> <s xml:id="echoid-s49333" xml:space="preserve">1 huius.</s> <s xml:id="echoid-s49334" xml:space="preserve"> Videturi-<lb/>taque linea i k maior quàm linea b c:</s> <s xml:id="echoid-s49335" xml:space="preserve"> & ſitus imaginis lineæ i k <lb/>comprehenditur quaſi remotior propter debilitatem formæ.</s> <s xml:id="echoid-s49336" xml:space="preserve"> Quia itaque linea i k eſt imago formæ <lb/>lineæ b c:</s> <s xml:id="echoid-s49337" xml:space="preserve"> palàm quòd in hoc ſitu linea b c uidetur maior quàm ſit ſecundum ueritatem:</s> <s xml:id="echoid-s49338" xml:space="preserve"> & uidetur <lb/>linea c q minor quàm linea b c:</s> <s xml:id="echoid-s49339" xml:space="preserve"> quia, ut præoſtenſum eſt, angulus i a k eſt maior angulo l a k, <lb/>ſecundum quem uidetur imago lineæ q c.</s> <s xml:id="echoid-s49340" xml:space="preserve"> Et hoc eſt propoſitum:</s> <s xml:id="echoid-s49341" xml:space="preserve"> nec eſt diuerſitas ſitus diuerſo-<lb/>rum diaphanorum attendenda.</s> <s xml:id="echoid-s49342" xml:space="preserve"/> </p> <div xml:id="echoid-div1846" type="float" level="0" n="0"> <figure xlink:label="fig-0735-01" xlink:href="fig-0735-01a"> <variables xml:id="echoid-variables842" xml:space="preserve">a d c i f p m h l k b z q o c</variables> </figure> </div> </div> <div xml:id="echoid-div1848" type="section" level="0" n="0"> <head xml:id="echoid-head1361" xml:space="preserve" style="it">33. Centro uiſus exiſtente extra ſuperſiciem perpendicularium à punctis rei uiſæ ſub medio <lb/>ſecundi diaphani plan am habente ſuperficiem ſuper eandem ſuperficiem productarum, lineá <lb/>uiſa ſuperficiei eiuſdem corporis æquidiſtante: ιmago lineæ uiſæ cõprehenditur maior ipſa. Al-<lb/>hazen 41 n 7.</head> <p> <s xml:id="echoid-s49343" xml:space="preserve">Sit, utſuprà, punctus a centrum uiſus:</s> <s xml:id="echoid-s49344" xml:space="preserve"> & linea b c res uiſa:</s> <s xml:id="echoid-s49345" xml:space="preserve"> & ſuper ſuperficiem corporis, à qua fit <lb/>refractio, educantur perpendiculares b d & c e:</s> <s xml:id="echoid-s49346" xml:space="preserve"> & continuetur linea d e in ſuperficie ipſius corporis <lb/>diaphani, per quod fit uiſio refracta:</s> <s xml:id="echoid-s49347" xml:space="preserve"> ſitq́ue linea b c æquidiſtans lineæ d e:</s> <s xml:id="echoid-s49348" xml:space="preserve"> & ſit a centrum uiſus ex-<lb/>tra ſuperficiem, in qua ſunt lineæ b c & d e:</s> <s xml:id="echoid-s49349" xml:space="preserve"> & diuidatur linea b c in duo æ qualia in puncto z:</s> <s xml:id="echoid-s49350" xml:space="preserve"> & du-<lb/>catur linea z m perpen diculariter ſuper lineam b c:</s> <s xml:id="echoid-s49351" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s49352" xml:space="preserve"> lineam d e in puncto m:</s> <s xml:id="echoid-s49353" xml:space="preserve"> & à centro uiſus <lb/>a ducatur perpendicularis ſuper ſuperficiem b c d e per 11 p 11:</s> <s xml:id="echoid-s49354" xml:space="preserve"> quæ ſit a h, ita ut punctus h imagine-<lb/>tur cadere in lineam m z:</s> <s xml:id="echoid-s49355" xml:space="preserve"> producaturq́ue linea a z:</s> <s xml:id="echoid-s49356" xml:space="preserve"> quæ per 22 th.</s> <s xml:id="echoid-s49357" xml:space="preserve">1 huius, & expræmiſsis erit perpẽ-<lb/>dicularis ſuper lineam b c.</s> <s xml:id="echoid-s49358" xml:space="preserve"> Situatio ιtaque puncti b uerſus a centrum uiſus eſt ſimilis ſituationi <lb/> <pb o="434" file="0736" n="736" rhead="VITELLONIS OPTICAE"/> puncti c reſpectu a:</s> <s xml:id="echoid-s49359" xml:space="preserve"> & diſtantia puncti c à uiſu a eſt æqualis diſtantiæ puncti b ab a.</s> <s xml:id="echoid-s49360" xml:space="preserve"> Refringatur ita-<lb/>que forma puncti b ad uiſum a ex puncto p:</s> <s xml:id="echoid-s49361" xml:space="preserve"> & forma puncti c ex puncto k:</s> <s xml:id="echoid-s49362" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s49363" xml:space="preserve"> puncta p & k extra <lb/>lineam d e æ quidiſtantem lineæ b c in ſuperficie corporis <lb/> <anchor type="figure" xlink:label="fig-0736-01a" xlink:href="fig-0736-01"/> diaphani:</s> <s xml:id="echoid-s49364" xml:space="preserve"> ſituatio itaque & diſtantia puncti p ad a uiſum <lb/>eſt, ſicut ſituatio & diſtantia puncti h ad a uiſum.</s> <s xml:id="echoid-s49365" xml:space="preserve"> Ducan-<lb/>tur itaque lineę b p, p a, c k, k a.</s> <s xml:id="echoid-s49366" xml:space="preserve"> Eſt ergo ſuperficies, in qua <lb/>ſunt duę lineę a p & b p, perpendicularis ſuper ſuperficiẽ <lb/>corporis diaphani per 2 huius, cũ ſit ſuperficies refractio-<lb/>nis:</s> <s xml:id="echoid-s49367" xml:space="preserve"> ergo & linea b d, quæ eſt perpẽdicularis ſuper ſuperfi <lb/>ciem corporis diaphani ducta à puncto b, erit in hac ſuքfi <lb/>cie.</s> <s xml:id="echoid-s49368" xml:space="preserve"> Et ſimiliter ſuperficies, in qua ſunt lineæ a k & c k, eſt <lb/>perpendicularis ſuper ſuperficiem corporis diaphani:</s> <s xml:id="echoid-s49369" xml:space="preserve"> er-<lb/>go & in illa ſuperficie eſt linea c e, quæ eſt perpendicula-<lb/>ris ſuper eandem ſuperficiem corporis ducta à puncto c.</s> <s xml:id="echoid-s49370" xml:space="preserve"> <lb/>Protrahatur itaque linea a p ultra p punctum:</s> <s xml:id="echoid-s49371" xml:space="preserve"> & palàm ք <lb/>iam dicta & per 2 th.</s> <s xml:id="echoid-s49372" xml:space="preserve"> 1 huius quoniam ipſa ſecabit lineam <lb/>b d:</s> <s xml:id="echoid-s49373" xml:space="preserve"> quia, ut patet per 28 p 1, lineæ a z & b d ęquidiſtant.</s> <s xml:id="echoid-s49374" xml:space="preserve"> <lb/>Quia ergo linea a p ſecat lineam b d, ſecet ipſam in pun-<lb/>cto l:</s> <s xml:id="echoid-s49375" xml:space="preserve"> ſecetq́ue propter eadem linea k d protracta ultra pũ <lb/>ctum k lineam c e in puncto o.</s> <s xml:id="echoid-s49376" xml:space="preserve"> Eſt ergo per 15 huius pun-<lb/>ctum l locus imaginis formæ puncti b:</s> <s xml:id="echoid-s49377" xml:space="preserve"> & punctum o lo-<lb/>cus imaginis ſormæ puncti c.</s> <s xml:id="echoid-s49378" xml:space="preserve"> Erit quoque ſituatio lineæ <lb/>a l ſicut lineę a o:</s> <s xml:id="echoid-s49379" xml:space="preserve"> & lineę b l ſicut lineæ c o.</s> <s xml:id="echoid-s49380" xml:space="preserve"> Ducatur etiam <lb/>linea l o:</s> <s xml:id="echoid-s49381" xml:space="preserve"> hęc itaq;</s> <s xml:id="echoid-s49382" xml:space="preserve"> erit diameter imaginis lineę b c, & ęqua <lb/>lis eidem b c per 33 p 1.</s> <s xml:id="echoid-s49383" xml:space="preserve"> Ducantur itaque lineæ a b & a c:</s> <s xml:id="echoid-s49384" xml:space="preserve"> u-<lb/>traque ergo ſuperſicies a l b & a o c eſt erecta ſuper ſuper-<lb/>ficiem corporis diaphani per 2 huius.</s> <s xml:id="echoid-s49385" xml:space="preserve"> Tres ita que ſuperfi <lb/>cies ſunt erectę ſuper ſuperficiem corpoſis diaphani, quę <lb/>ſunt a l b, ao c, a m z:</s> <s xml:id="echoid-s49386" xml:space="preserve"> & hæ ſuperficies neceſſario ſecant ſe <lb/>ſuper lineam perpendicularem, quæ eſt a h, exeuntem à pũcto a ſuper ſuperficiem corporis diapha-<lb/>niper 19 p 11:</s> <s xml:id="echoid-s49387" xml:space="preserve"> quoniam communis ſectio illarum neceſſariò eſt perpendicularis ſuper ſuperficiem, <lb/>cui ſuperſtant:</s> <s xml:id="echoid-s49388" xml:space="preserve"> & ab uno puncto una tantùm perpendicularis ſuper ſuperficiem planam duci poteſt <lb/>per 20 th.</s> <s xml:id="echoid-s49389" xml:space="preserve">1 huius.</s> <s xml:id="echoid-s49390" xml:space="preserve"> Erititaq;</s> <s xml:id="echoid-s49391" xml:space="preserve"> angulus b p l per 15 p 1 æqualis angulo reſractionis, & linea b l d eſt perpẽ <lb/>dicularis ſuper ſuperficiem corporis, à qua fit refractio:</s> <s xml:id="echoid-s49392" xml:space="preserve"> ergo linea a l eſt obliqua ſuper ipſam per 13 <lb/>p 11:</s> <s xml:id="echoid-s49393" xml:space="preserve"> linea ergo a p continet cum perpendiculari ſuper ean dem ſuperficiẽ exeuntẽ à puncto p.</s> <s xml:id="echoid-s49394" xml:space="preserve"> (quæ <lb/>ſit p g) angulum acutum, qui eſt l p g:</s> <s xml:id="echoid-s49395" xml:space="preserve"> & erit perpendicularis p g æquidiſtãs lineæ d l per 6 p 11, quo-<lb/>niam ambę lineæ p g & d l ſunt erctę ſuper unam ſuperficiem:</s> <s xml:id="echoid-s49396" xml:space="preserve"> ergo per 29 p 1 angulus p l d eſt acu-<lb/>tus:</s> <s xml:id="echoid-s49397" xml:space="preserve"> ergo per 13 p 1 angulus a l b eſt obtuſus:</s> <s xml:id="echoid-s49398" xml:space="preserve"> ergo per 19 p 1 linea a b eſt longior quã linea a l.</s> <s xml:id="echoid-s49399" xml:space="preserve"> Et ſimili <lb/>ter patere poteſt quòd linea a o minor eſt quã linea a c:</s> <s xml:id="echoid-s49400" xml:space="preserve"> ſed lineę a l & a o ſunt æquales:</s> <s xml:id="echoid-s49401" xml:space="preserve"> & lineę a b & <lb/>a c ſunt æquales:</s> <s xml:id="echoid-s49402" xml:space="preserve"> & linea l o eſt æ qualis lineę b c:</s> <s xml:id="echoid-s49403" xml:space="preserve"> ergo per 34 th.</s> <s xml:id="echoid-s49404" xml:space="preserve">1 huius angulus l a o eſt maior angu <lb/>lo b a c:</s> <s xml:id="echoid-s49405" xml:space="preserve"> & ſitus lineæ l o eſt ſimilis ſitui lineę b c:</s> <s xml:id="echoid-s49406" xml:space="preserve"> quia linea exiens à puncto a ad medium lineę l o, eſt <lb/>perpendicularis ſuper lineam l o per 22th.</s> <s xml:id="echoid-s49407" xml:space="preserve">1 huius cum per 29 p 1 linea l o ſit æ quidiſtans lineę b c:</s> <s xml:id="echoid-s49408" xml:space="preserve"> <lb/>& etiam, quia linea b c eſt perpendicularis ſuper ſuperficiem, in qua ſunt lineę a z & m z, ſuper quá <lb/>ſimiliter per 8 p 11 perpendicularis eſt linea l o:</s> <s xml:id="echoid-s49409" xml:space="preserve"> ergo linea l o eſt perpendicularis ſuper ſuperſiciem <lb/>continuantem centrum uiſus, quod eſt punctum a, cum medio puncto lineę l o.</s> <s xml:id="echoid-s49410" xml:space="preserve"> Situs ergo lineæ l o <lb/>reſpectu uiſus a eſt, ſicut lineæ b creſpectu eiuſdem uiſus a:</s> <s xml:id="echoid-s49411" xml:space="preserve"> ſed & linea l o comprehenditur remo-<lb/>tior propter debilitatem formæ:</s> <s xml:id="echoid-s49412" xml:space="preserve"> linea itaque l o uidetur maior quàm linea b c:</s> <s xml:id="echoid-s49413" xml:space="preserve"> ſed linea l o eſt ima-<lb/>go lineæ b c.</s> <s xml:id="echoid-s49414" xml:space="preserve"> Palàm itaque quia linea b c uidetur maior quàm ſit eius uera quantitas.</s> <s xml:id="echoid-s49415" xml:space="preserve"> Et hoc eſt <lb/>propoſitum:</s> <s xml:id="echoid-s49416" xml:space="preserve"> nec ad iſtud aliquid coadiuuat in diuerſitatem ipſa diuerſa ſituatio mediorum plus uel <lb/>minus diaphanorum.</s> <s xml:id="echoid-s49417" xml:space="preserve"/> </p> <div xml:id="echoid-div1848" type="float" level="0" n="0"> <figure xlink:label="fig-0736-01" xlink:href="fig-0736-01a"> <variables xml:id="echoid-variables843" xml:space="preserve">a p k d m e l o g h b z c</variables> </figure> </div> </div> <div xml:id="echoid-div1850" type="section" level="0" n="0"> <head xml:id="echoid-head1362" xml:space="preserve" style="it">34. Centro uiſus exiſtente extra ſuperficiem perpendicularium à punctis rei uiſæ ſub medio <lb/>ſecundi diaphani planam habente ſuperficiem ſuper eandem ſuperficiem productarum, lineá <lb/>uiſa ſuperficiei eiuſdem corporis non æquidiſtante: imago rei comprehenditur maior re uiſa: ma <lb/>ior quo quàm ſi eſſet ſuperficiei corpori æquidistans. Alhazen 42 n 7.</head> <p> <s xml:id="echoid-s49418" xml:space="preserve">Remaneat diſpoſitio, quæ in præcedente, niſi quòd linea b c non ſit æquidiſtans lineę d e, quę eſt <lb/>in ſuperficie corporis diaphani:</s> <s xml:id="echoid-s49419" xml:space="preserve"> & educatur à puncto c linea c f ęquidiſtans lineę d e:</s> <s xml:id="echoid-s49420" xml:space="preserve"> & continuetur <lb/>linea f b protrahen do lineam d b perpendiculariter ſuper lineam c f:</s> <s xml:id="echoid-s49421" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s49422" xml:space="preserve">, prout in præmiſſa oſten-<lb/>ſum eſt, p punctum refractionis formæ puncti fad uiſum a:</s> <s xml:id="echoid-s49423" xml:space="preserve"> & punctum refractionis formæ puncti b <lb/>ad uiſum a ſit punctum q:</s> <s xml:id="echoid-s49424" xml:space="preserve"> & ducatur linea a q:</s> <s xml:id="echoid-s49425" xml:space="preserve"> & protrahatur a d lineam d b:</s> <s xml:id="echoid-s49426" xml:space="preserve"> concurret autem cum <lb/>illa, ut in proxima oſtenſum eſt.</s> <s xml:id="echoid-s49427" xml:space="preserve"> Sit ergo punctus concurſus g, qui eſt altior quã punctus l:</s> <s xml:id="echoid-s49428" xml:space="preserve"> nam pun <lb/>ctus b eſt ultra lineam a f:</s> <s xml:id="echoid-s49429" xml:space="preserve"> linea ita que a g neceſſariò erit ultra lineam a l:</s> <s xml:id="echoid-s49430" xml:space="preserve"> punctus ergo g eſt altior pũ <lb/>cto l:</s> <s xml:id="echoid-s49431" xml:space="preserve"> & ducatur linea g o.</s> <s xml:id="echoid-s49432" xml:space="preserve"> Erit ergo ſecundum pręmiſſa linea g o diameter imaginis lineę b c:</s> <s xml:id="echoid-s49433" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s49434" xml:space="preserve"> li <lb/> <pb o="435" file="0737" n="737" rhead="LIBER DECIMVS."/> nea g o maior quàm linea l o per 19 p 1, quoniam angulus g l o eſt rectus:</s> <s xml:id="echoid-s49435" xml:space="preserve"> & linea a g minor quàm li-<lb/>nea a l per eãdem 19 p 1 quoniam angulus a g l eſt obtuſus, <lb/> <anchor type="figure" xlink:label="fig-0737-01a" xlink:href="fig-0737-01"/> ut ſuprà patuit:</s> <s xml:id="echoid-s49436" xml:space="preserve"> & duę lineę a g & a o ſunt in duabus ſuperfi <lb/>ciebus ſecantibus ſe, ſcilicet a g b & a o c:</s> <s xml:id="echoid-s49437" xml:space="preserve"> & differentia cõ-<lb/>munis iſtarum duarum ſuperficierum tranſit per a centrũ <lb/>uiſus per 1 huius:</s> <s xml:id="echoid-s49438" xml:space="preserve"> quia ambæ illæ ſuperficies ſunt ſuperfici-<lb/>es refractionis:</s> <s xml:id="echoid-s49439" xml:space="preserve"> & centrum uiſus ſemper oportet quòd ſit <lb/>in ſuperficie refractionis.</s> <s xml:id="echoid-s49440" xml:space="preserve"> Et quoniam, ut patet per 2 huius, <lb/>illæ ambæ ſuperficies ſunt erectæ ſuper ſuperficiem corpo <lb/>ris diaphani, à quo fit refractio:</s> <s xml:id="echoid-s49441" xml:space="preserve"> patet per 19 p 11 quoniam li <lb/>nea recta, quę eſt communis ipſarum differentia, eſt erecta <lb/>ſuper illam ſuperficiem:</s> <s xml:id="echoid-s49442" xml:space="preserve"> ergo duæ lineę exeuntes à puncto <lb/>a non perpendiculariter ſuper illam corporis diaphani ſu-<lb/>perficiem, ſunt extra hanc communem differentiam in his <lb/>duabus ſuperficiebus:</s> <s xml:id="echoid-s49443" xml:space="preserve"> quæ lineę ſunt a b & a c:</s> <s xml:id="echoid-s49444" xml:space="preserve"> ſuntq́ altio <lb/>res duabus lineis a g & a o:</s> <s xml:id="echoid-s49445" xml:space="preserve"> cadunt enim ultra illas lineas.</s> <s xml:id="echoid-s49446" xml:space="preserve"> <lb/>Angulus itaq;</s> <s xml:id="echoid-s49447" xml:space="preserve"> g a o eſt maior angulo b a c ք 34 th.</s> <s xml:id="echoid-s49448" xml:space="preserve">1 huius:</s> <s xml:id="echoid-s49449" xml:space="preserve"> <lb/>diuerſitas enim ſituum linearum g o & b c à uiſu a non eſt <lb/>magna:</s> <s xml:id="echoid-s49450" xml:space="preserve"> quia linea g o aut eſt ęquidiſtans lineæ a c, aut non <lb/>eſt in hoc differentia ſenſibilis.</s> <s xml:id="echoid-s49451" xml:space="preserve"> Eſt ergo ſitus lineæ g o re-<lb/>ſpectu uiſus a, ſicut linea b c reſpectu eiuſdem uiſus a.</s> <s xml:id="echoid-s49452" xml:space="preserve"> Vi-<lb/>debitur itaque per 20th.</s> <s xml:id="echoid-s49453" xml:space="preserve"> 4 huius linea g o maior quã linea <lb/>b c:</s> <s xml:id="echoid-s49454" xml:space="preserve"> ſed linea g o eſt imago lineę b c.</s> <s xml:id="echoid-s49455" xml:space="preserve"> Palàm ergo quia linea <lb/>b c uidebitur maior quàm ipſa ſit ſecundum ueritatem.</s> <s xml:id="echoid-s49456" xml:space="preserve"> Et <lb/>quia, ſicut in præmiſsis patuit, angulus o a g eſt maior an-<lb/>gulo o a l, uidebitur imago o g maior imagine o l, quę eſt i-<lb/>mago lineæ c fæ quidiſtãtis lineę e d, quæ eſt in ſuperficie <lb/>corporis, à qua fit refractio.</s> <s xml:id="echoid-s49457" xml:space="preserve"> Et hoc proponebatur.</s> <s xml:id="echoid-s49458" xml:space="preserve"/> </p> <div xml:id="echoid-div1850" type="float" level="0" n="0"> <figure xlink:label="fig-0737-01" xlink:href="fig-0737-01a"> <variables xml:id="echoid-variables844" xml:space="preserve">a q p k d m e g l o b f z o</variables> </figure> </div> </div> <div xml:id="echoid-div1852" type="section" level="0" n="0"> <head xml:id="echoid-head1363" xml:space="preserve" style="it">35. In omnibus refractionibus factis à planis ſuperficiebus corporum diap hanorũ aduiſum: <lb/>imagine apparente maiore ipſa re uiſa, & pars imaginis uidebitur maior parte rei uiſæ ſibi pro-<lb/>portionalι. Alhazen 43 n 7.</head> <p> <s xml:id="echoid-s49459" xml:space="preserve">Sit diſpoſitio omnimoda, quæ prius in 31 huius:</s> <s xml:id="echoid-s49460" xml:space="preserve"> & ſit linea a m z ſecans perpendiculariter lineam <lb/>k l in puncto o:</s> <s xml:id="echoid-s49461" xml:space="preserve"> erit itaq;</s> <s xml:id="echoid-s49462" xml:space="preserve"> linea l o medietas lineæ l k:</s> <s xml:id="echoid-s49463" xml:space="preserve"> & for-<lb/>ma punctiz uidebitur in puncto o:</s> <s xml:id="echoid-s49464" xml:space="preserve"> quia uidetur in perpen-<lb/> <anchor type="figure" xlink:label="fig-0737-02a" xlink:href="fig-0737-02"/> diculari z o:</s> <s xml:id="echoid-s49465" xml:space="preserve"> tota quoq;</s> <s xml:id="echoid-s49466" xml:space="preserve"> linea b c uidebitur in linea l k:</s> <s xml:id="echoid-s49467" xml:space="preserve"> & li-<lb/>nea b z eſt medietas lineæ b c:</s> <s xml:id="echoid-s49468" xml:space="preserve"> & linea l o eſt medietas lineę <lb/>l k:</s> <s xml:id="echoid-s49469" xml:space="preserve"> & linea l k uidebitur maior quàm linea b c:</s> <s xml:id="echoid-s49470" xml:space="preserve"> ergo & linea <lb/>l o uidebitur maior quã linea b z:</s> <s xml:id="echoid-s49471" xml:space="preserve"> & erit utriuſq;</s> <s xml:id="echoid-s49472" xml:space="preserve"> iſtorũ cauſ-<lb/>ſa refractio.</s> <s xml:id="echoid-s49473" xml:space="preserve"> Et quia centrum uiſus a eſt in perpendiculari <lb/>a z exeunte à puncto z, qui eſt extremitas lineę b z, ſuper ſu <lb/>perficiem corporis diaphani, aut ſuper ſuperficiem tranſeũ <lb/>tem per extremitatem medietatis perpendicularis ſuper ſu <lb/>perficiem corporis diaphani æquidiſtanter ſuperficiei cor-<lb/>poris diaphani per 23 th.</s> <s xml:id="echoid-s49474" xml:space="preserve">1 huius:</s> <s xml:id="echoid-s49475" xml:space="preserve"> uiſus itaq;</s> <s xml:id="echoid-s49476" xml:space="preserve"> comprehendit <lb/>medietates uiſibilium maiores quàm ſint.</s> <s xml:id="echoid-s49477" xml:space="preserve"> Nam punctus o, <lb/>qui eſt medium imaginis k l, eſt in perpendiculari exeunte <lb/>à medio rei uiſæ, ſiue res uiſa ſit ęquidiſtans ſuperficiei cor-<lb/>poris diaphani, ſiue non.</s> <s xml:id="echoid-s49478" xml:space="preserve"> Sit item linea b n pars aliqua lineę <lb/>b z:</s> <s xml:id="echoid-s49479" xml:space="preserve"> & à puncto n educatur linea n g perpẽdiculariter ſuper <lb/>lineam b z:</s> <s xml:id="echoid-s49480" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s49481" xml:space="preserve"> lineam l o in puncto g:</s> <s xml:id="echoid-s49482" xml:space="preserve"> erit ergo ſecundũ <lb/>præmiſſa linea l g imago lineæ b n.</s> <s xml:id="echoid-s49483" xml:space="preserve"> Sit ita que punctus g ima <lb/>go punctin.</s> <s xml:id="echoid-s49484" xml:space="preserve"> Aut ergo punctus g erit in linea l g, aut prope:</s> <s xml:id="echoid-s49485" xml:space="preserve"> <lb/>quocunque uerò iſtorum exiſtente erit linea l g ęqualis li-<lb/>neę b n, aut ferè.</s> <s xml:id="echoid-s49486" xml:space="preserve"> Et quia formarum plus diſtantium à per-<lb/>pendiculari a z maior eſt refractio quàm minus diſtantium <lb/>per 14 th.</s> <s xml:id="echoid-s49487" xml:space="preserve"> huius:</s> <s xml:id="echoid-s49488" xml:space="preserve"> erit refractio ſormæ lineę b n ad uiſum a <lb/>maior quàm refractio lineę z n ad uiſum a.</s> <s xml:id="echoid-s49489" xml:space="preserve"> Siergo minor re <lb/>fractio facit totam l o imaginem lineę b z apparere uiſui ma <lb/>iorem quã ſit linea b z:</s> <s xml:id="echoid-s49490" xml:space="preserve"> ergo maior reſractio faciet lineã l g <lb/>imaginẽ lineę b n uideri maiorẽ quã ſit ipſa linea b n:</s> <s xml:id="echoid-s49491" xml:space="preserve"> cũ maiorem efficaciã habeat refractio maiorre <lb/>ſpectu minoris.</s> <s xml:id="echoid-s49492" xml:space="preserve"> Linea ergo l g, quæ eſt imago lineę b n, cõprehenditur maior ꝗ̃ ſit ipſa linea b n.</s> <s xml:id="echoid-s49493" xml:space="preserve"> Et ſi <lb/>uiſus non cõprehenderet lineã l g imaginẽ lineę b n maiorem ipſa linea b n, nõ cõprehenderet ima-<lb/>gínes partiũ lineę b n, quę ſunt propinquiores ad punctũ z, maiores ipſis partib.</s> <s xml:id="echoid-s49494" xml:space="preserve"> quia formę illarum <lb/> <pb o="436" file="0738" n="738" rhead="VITELLONIS OPTICAE"/> partium ſunt minoris refractionis per 14 th.</s> <s xml:id="echoid-s49495" xml:space="preserve"> huius quã remotiores à puncto z:</s> <s xml:id="echoid-s49496" xml:space="preserve">ſed refractio eft cauf-<lb/>ſa magnitudinis imaginis.</s> <s xml:id="echoid-s49497" xml:space="preserve"> Viſus ergo a ſi nõ cõprehendet imaginẽ lineæ l g maiorẽ quã ſit linea b n, <lb/>nec comprehendet imaginẽ lineę l o maiorẽ ipſa linea b z, nec totã linea m l k maiorẽ tota linea b c, <lb/>quod eſt impoſsibile, & contra 31 huius.</s> <s xml:id="echoid-s49498" xml:space="preserve"> Viſus ergo cõprehendet lineam l g, quę eſt imago lineę b n, <lb/>maiorem ipſa linea b n:</s> <s xml:id="echoid-s49499" xml:space="preserve"> & ita comprehendet lineã b n maiorem quã ſit ſecundum ueritatẽ.</s> <s xml:id="echoid-s49500" xml:space="preserve"> Eodem <lb/>quoque modo poteſtidẽ in alijs refractionib.</s> <s xml:id="echoid-s49501" xml:space="preserve"> declarari:</s> <s xml:id="echoid-s49502" xml:space="preserve">ut cũ per modum 33 huius fuerit centrum ui <lb/>ſus extra ſuperficiẽ perpendiculariũ illarum productarũ:</s> <s xml:id="echoid-s49503" xml:space="preserve">quoniã idem accidit in omnib.</s> <s xml:id="echoid-s49504" xml:space="preserve"> illis modis, <lb/>quibus imago rei uidetur maior ipſa re uiſa:</s> <s xml:id="echoid-s49505" xml:space="preserve"> ſemper enim pars imaginis uidebitur maior parte rei ui <lb/>ſę ſibi correſpõdente:</s> <s xml:id="echoid-s49506" xml:space="preserve"> quod eſt propoſitũ.</s> <s xml:id="echoid-s49507" xml:space="preserve"> Et quia cõmunis ſectio ſuperficiei refractionis & ſuperfi-<lb/>ciei corporis diaphani, ut plurimum & per ſe eſt linea recta, quando illud corpus diaphanum fuerit <lb/>groſsius aere:</s> <s xml:id="echoid-s49508" xml:space="preserve"> per accidens uerò accidit quandoq;</s> <s xml:id="echoid-s49509" xml:space="preserve"> contrarium propter uoluntariam ſituationẽ cor-<lb/>poris dẽſioris plani iuxta uiſum, ut diximus in fine cõmenti 31 huius:</s> <s xml:id="echoid-s49510" xml:space="preserve"> patet euidenter quòd 5 proxi <lb/>mè præmiſſa theoremata per ſe intelligenda ſunt, quando à ſuperficie corporis diaphani groſsioris <lb/>aere fit refractio ad uiſum in aere exiſtentem:</s> <s xml:id="echoid-s49511" xml:space="preserve"> & per accidens econuerſo.</s> <s xml:id="echoid-s49512" xml:space="preserve"/> </p> <div xml:id="echoid-div1852" type="float" level="0" n="0"> <figure xlink:label="fig-0737-02" xlink:href="fig-0737-02a"> <variables xml:id="echoid-variables845" xml:space="preserve">a d p m h e l g o f b n z c</variables> </figure> </div> </div> <div xml:id="echoid-div1854" type="section" level="0" n="0"> <head xml:id="echoid-head1364" xml:space="preserve" style="it">36. Communi ſectione ſuperficiei refractionis & corporis ſphærici diaphani denſioris aere, à <lb/>quo fit refr actio, exιtente circulo, centró uiſus in eadem ſuperficie extra circulum in lineæ <lb/>per pendiculari ſuper illius corporis ſuperficiem, & re uiſa inter centrum corporis & uiſus exi-<lb/>ſtentibus, ita quòd extrema rei uiſæ æqualiter diſtent à centro corporis: imago uidebitur maior <lb/>re uiſa. Alhazen 44 n 7.</head> <p> <s xml:id="echoid-s49513" xml:space="preserve">Sit ſuperficies ſphærica corporis diaphani groſsioris aere:</s> <s xml:id="echoid-s49514" xml:space="preserve"> cuius conuexum ſit ex parte uiſus, cu <lb/>ius centrum ſit a:</s> <s xml:id="echoid-s49515" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s49516" xml:space="preserve"> res uiſa b c:</s> <s xml:id="echoid-s49517" xml:space="preserve"> ſitq̃;</s> <s xml:id="echoid-s49518" xml:space="preserve"> centrum corporis ſphærici punctum d:</s> <s xml:id="echoid-s49519" xml:space="preserve"> quod ſit ultra lineam <lb/>b c reſpectu uiſus a:</s> <s xml:id="echoid-s49520" xml:space="preserve">ſitq́;</s> <s xml:id="echoid-s49521" xml:space="preserve"> punctus z medius punctus lineæ b c:</s> <s xml:id="echoid-s49522" xml:space="preserve"> & ducantur lineæ d b, d z, d c:</s> <s xml:id="echoid-s49523" xml:space="preserve"> & pro-<lb/>trahantur quouſq;</s> <s xml:id="echoid-s49524" xml:space="preserve"> concurrant cum ſuperficie corporis diaphani ſphærici:</s> <s xml:id="echoid-s49525" xml:space="preserve"> linea d b in puncto e:</s> <s xml:id="echoid-s49526" xml:space="preserve"> & <lb/>linea d z in puncto m:</s> <s xml:id="echoid-s49527" xml:space="preserve"> & linea d c in puncto n:</s> <s xml:id="echoid-s49528" xml:space="preserve"> & ſit uiſus a in linea z m, quæ eſt perpendicularis ſu-<lb/>per ſuperficiem illius diaphani corporis per 72 th.</s> <s xml:id="echoid-s49529" xml:space="preserve">1 huius.</s> <s xml:id="echoid-s49530" xml:space="preserve"> Erit itaq;</s> <s xml:id="echoid-s49531" xml:space="preserve"> a m z linea recta.</s> <s xml:id="echoid-s49532" xml:space="preserve"> Et quoniam li-<lb/>nea b z eſt æqualis lineę z c, & quia puncta b & c (quę ſunt extrema rei uiſæ) æqualiter diſtant à cẽ-<lb/>tro d ex hypotheſi:</s> <s xml:id="echoid-s49533" xml:space="preserve">erit etiam linea d b æqualis lineę d c:</s> <s xml:id="echoid-s49534" xml:space="preserve">erunt ergo trigona b d z & c d z æ quilatera:</s> <s xml:id="echoid-s49535" xml:space="preserve"> <lb/>quoniam linea z d eſt communis ambobus illis trigonis:</s> <s xml:id="echoid-s49536" xml:space="preserve"> ergo per 8 p 1 erunt anguli ad punctum d <lb/>ęquales, qui ſunt anguli z d b & z d c:</s> <s xml:id="echoid-s49537" xml:space="preserve"> & ſimiliter erunt an <lb/>guli ad punctũ z ęquales:</s> <s xml:id="echoid-s49538" xml:space="preserve"> ſunt ergo recti:</s> <s xml:id="echoid-s49539" xml:space="preserve"> eſt ergo per deſi <lb/> <anchor type="figure" xlink:label="fig-0738-01a" xlink:href="fig-0738-01"/> nitionem perpendicularis, linea a z perpendicularis ſu-<lb/>per lineam b c.</s> <s xml:id="echoid-s49540" xml:space="preserve"> Ducantur quoque lineę a b & a c:</s> <s xml:id="echoid-s49541" xml:space="preserve"> ergo ք <lb/>4 p 1 erunt trigona a z b & a z c æqualia:</s> <s xml:id="echoid-s49542" xml:space="preserve"> linea ergo a c eſt <lb/>æqualis lineæ a b:</s> <s xml:id="echoid-s49543" xml:space="preserve"> puncta ergo b & c ęqualiter diſtant à cẽ <lb/>tro uiſus a:</s> <s xml:id="echoid-s49544" xml:space="preserve">habebunt itaque puncta b & c ęqualem reſpe-<lb/>ctum ad uiſum a.</s> <s xml:id="echoid-s49545" xml:space="preserve"> Extrahatur quoque ſuperſicies plana, <lb/>in qua ſunt lineę d e & d n & d m:</s> <s xml:id="echoid-s49546" xml:space="preserve"> hæc itaq;</s> <s xml:id="echoid-s49547" xml:space="preserve"> ſuperficies ſe-<lb/>cabit ſuperficiem corporis ſphærici ſecundum circulum <lb/>magnum per 69 th.</s> <s xml:id="echoid-s49548" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s49549" xml:space="preserve"> cuius arcus oppoſitus uiſui <lb/>ſit n m e:</s> <s xml:id="echoid-s49550" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s49551" xml:space="preserve"> in illa ſuperficie centrum uiſus a, & linea <lb/>uiſa, quę eſt b c:</s> <s xml:id="echoid-s49552" xml:space="preserve"> erit ergo per 1 huius illa ſuperficies ſuper-<lb/>ficies refractionis, quæ eſt perpendicularis ſuper ſuperſi-<lb/>ciem ſphęricam per 2 th huius:</s> <s xml:id="echoid-s49553" xml:space="preserve"> nec fit refractio formæ li-<lb/>neæ b c ad uiſum a extra illam ſuperficiem:</s> <s xml:id="echoid-s49554" xml:space="preserve"> & linea a z eſt <lb/>perpendicularis ſuper ſuperficiem ſphæricam corporis.</s> <s xml:id="echoid-s49555" xml:space="preserve"> <lb/>Dico ita que quòd imago lineę b c in hac diſpoſitione ui-<lb/>debitur maior ipſa linea b c.</s> <s xml:id="echoid-s49556" xml:space="preserve"> Quia enim, ut patet ex prę-<lb/>miſsis, forma cuiuſcunq;</s> <s xml:id="echoid-s49557" xml:space="preserve"> partis lineę b c non refringitur <lb/>ad uiſum a, niſi ex aliquo puncto arcus e m n:</s> <s xml:id="echoid-s49558" xml:space="preserve"> ſit ergo, ut <lb/>forma puncti b refringatur ad uiſum a ex puncto circuli <lb/>h:</s> <s xml:id="echoid-s49559" xml:space="preserve"> & forma puncti c ex puncto g.</s> <s xml:id="echoid-s49560" xml:space="preserve"> Quia itaque puncta b & <lb/>c æqualiter diſtant à puncto a centro uiſus:</s> <s xml:id="echoid-s49561" xml:space="preserve"> patet quòd i-<lb/>pſorũ erit uniformis refractio ad uiſum per 14th.</s> <s xml:id="echoid-s49562" xml:space="preserve"> huius:</s> <s xml:id="echoid-s49563" xml:space="preserve"> <lb/>puncta ergo h & g æqualiter diſtabunt à puncto m.</s> <s xml:id="echoid-s49564" xml:space="preserve"> Arcus <lb/>autem e m & m n ſunt æquales per 26 p 3:</s> <s xml:id="echoid-s49565" xml:space="preserve"> ideo quia angu-<lb/>li m d e & m d n ſunt æquales, quod patet ex præmiſsis:</s> <s xml:id="echoid-s49566" xml:space="preserve"> tantùm ergo diſta bit punctus reſractionis, <lb/>qui eſt h, à puncto e, quantùm punctus g à puncto n:</s> <s xml:id="echoid-s49567" xml:space="preserve"> & erit punctorum iſtorum ſitus & reſpectus <lb/>æqualis.</s> <s xml:id="echoid-s49568" xml:space="preserve"> Ducantur itaque lineæ b h, a h, c g, a g:</s> <s xml:id="echoid-s49569" xml:space="preserve"> & producatur linea a h ad lineã d e:</s> <s xml:id="echoid-s49570" xml:space="preserve"> ſit q̃ punctus ſe-<lb/>ctionis k:</s> <s xml:id="echoid-s49571" xml:space="preserve"> & ſimiliter producatur linea a g ad lιneam d n in punctum l:</s> <s xml:id="echoid-s49572" xml:space="preserve">ducaturq́;</s> <s xml:id="echoid-s49573" xml:space="preserve"> linea k l.</s> <s xml:id="echoid-s49574" xml:space="preserve"> Quia itaqs <lb/>in trigonis d a k & d a l anguli a d k & a d l ſunt æquales, ut patuit ſuprà:</s> <s xml:id="echoid-s49575" xml:space="preserve"> anguli quoque l a d & k a d <lb/>ſunt æ quales (quod patet ductis lineis d h & d g:</s> <s xml:id="echoid-s49576" xml:space="preserve"> tunc enim, cum arcus m g & m h ſint æquales ex <lb/>pręmiſsis, erunt per 27 p 3 angulia d g & a d h æquales:</s> <s xml:id="echoid-s49577" xml:space="preserve"> ergo per 4 p 1 anguli l a d & k a d ſunt ęqua-<lb/> <pb o="437" file="0739" n="739" rhead="LIBER DECIMVS."/> les) ergo per.</s> <s xml:id="echoid-s49578" xml:space="preserve"> 32 p 1 trigona d a k & d a l ſunt æquiangula:</s> <s xml:id="echoid-s49579" xml:space="preserve"> ergo per 4 p 6 cum linea a d ſit æqualis ſi-<lb/>bijpſi, erit linea d l ęqualis lineę d k, & linea a k ęqualis lineę a l:</s> <s xml:id="echoid-s49580" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s49581" xml:space="preserve"> linea l k, ut patet per 15 huius, <lb/>imago lineę b c, & erit linea l k ęquidiſtans lineę b c:</s> <s xml:id="echoid-s49582" xml:space="preserve"> uide biturq́ue per 20 th.</s> <s xml:id="echoid-s49583" xml:space="preserve">4 huius maïor quàm ſit <lb/>linea b c:</s> <s xml:id="echoid-s49584" xml:space="preserve"> quoniam angulus k a l, ſecundum quem uidetur linea l k, eſt maior angulo b a c.</s> <s xml:id="echoid-s49585" xml:space="preserve"> Et quia po <lb/>ſitio & ſitus lineę k l eſt conſimilis poſitioni & ſitui b c lineę:</s> <s xml:id="echoid-s49586" xml:space="preserve"> quod patet ex hoc:</s> <s xml:id="echoid-s49587" xml:space="preserve"> quòd cum linea d l <lb/>ſit ęqualis lineę d k, & linea c d ęqualis lineę b d, erit linea l cęqualis lineę k b:</s> <s xml:id="echoid-s49588" xml:space="preserve"> ergo per 7 p 5 & 2 p 6 <lb/>lineę b c & l k ſunt æquidiſtantes:</s> <s xml:id="echoid-s49589" xml:space="preserve"> ipſarum ergo ſitus & poſitio reſpectu uiſus a eſt conſimιlis.</s> <s xml:id="echoid-s49590" xml:space="preserve"> Inter <lb/>lineas ergo k l & b c non eſt differentia in diſtantia, quę ſit ſenſibilis.</s> <s xml:id="echoid-s49591" xml:space="preserve"> Palàm ergo quia linea k l uide-<lb/>bitur maior quàm ſit:</s> <s xml:id="echoid-s49592" xml:space="preserve"> quia imago eius eſt maior ipſa:</s> <s xml:id="echoid-s49593" xml:space="preserve"> & hoc accidit etiam ideo:</s> <s xml:id="echoid-s49594" xml:space="preserve"> quia forma eius refra <lb/>cta eſt debilior quã uera forma, ut patet per 10 huius.</s> <s xml:id="echoid-s49595" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s49596" xml:space="preserve"/> </p> <div xml:id="echoid-div1854" type="float" level="0" n="0"> <figure xlink:label="fig-0738-01" xlink:href="fig-0738-01a"> <variables xml:id="echoid-variables846" xml:space="preserve">a h m g e k b d c l n</variables> </figure> </div> </div> <div xml:id="echoid-div1856" type="section" level="0" n="0"> <head xml:id="echoid-head1365" xml:space="preserve" style="it">37. Communi ſectione ſuperficiei refractionis & corporis ſphærici diaphani denſioris aere, à <lb/>quo fit refr actio, exiſtente circulo, uiſú exiſtente in eadem ſuperficie extra circulum in linea <lb/>perpendiculari ſuper illius corporis ſuperficiem, & re uiſa inter centrum corporis & uiſus exi-<lb/>ſtentibus, ita quòd extremæ rei uiſæ inæqualiter diſtent à centro: imago uidetur maior re uiſa. <lb/>Alhazen 45 n 7.</head> <p> <s xml:id="echoid-s49597" xml:space="preserve">Remaneat diſpoſitio præcedentis, niſi quò alterum extrem orũ lineę b c punctum c ſit propin-<lb/>quius puncto d centro corporis diaphanì, & punctũ b rem otius ab illo.</s> <s xml:id="echoid-s49598" xml:space="preserve"> Dico quòd ahuc imago li-<lb/>neæ b c uidebitur maior ipſa linea b c.</s> <s xml:id="echoid-s49599" xml:space="preserve"> Ducatur enim à puncto c linea c q, cuius extrem a æqualiter <lb/>diſtent à puncto d:</s> <s xml:id="echoid-s49600" xml:space="preserve"> quod poteſt fieri, ſi à linea d e abſcindatur per 3 p 1 linea æqualis lineæ d c:</s> <s xml:id="echoid-s49601" xml:space="preserve">quæ ſit <lb/>d q.</s> <s xml:id="echoid-s49602" xml:space="preserve"> Palàm itaq;</s> <s xml:id="echoid-s49603" xml:space="preserve"> per ea, quæ in demonſtratione præceden-<lb/>tis oſtenſa ſunt, quoniam imago lineæ c q uidetur maior i-<lb/> <anchor type="figure" xlink:label="fig-0739-01a" xlink:href="fig-0739-01"/> pſa linea c q:</s> <s xml:id="echoid-s49604" xml:space="preserve"> ſit ita que illa imago linea l p.</s> <s xml:id="echoid-s49605" xml:space="preserve"> Et palàm per 13 <lb/>huius quòd punctum p illius imaginis, quod eſt imago pũ <lb/>cti q, neceſſariò cadet in linea perpendiculari ducta à pun-<lb/>cto q ſuper ſuperficiẽ corporis diaphani, quæ eſt linea d e, <lb/>inter puncta d & e:</s> <s xml:id="echoid-s49606" xml:space="preserve"> & quòd punctum l, quod eſt imago pun <lb/>cti c, erit in linea perpendiculari ducta à puncto c ſuper ſu-<lb/>perficiem corporis diaphani, quæ eſt d n.</s> <s xml:id="echoid-s49607" xml:space="preserve"> Et quia forma pũ <lb/>cti c refringitur ad uiſum a ex puncto circuli g:</s> <s xml:id="echoid-s49608" xml:space="preserve"> ſit ut forma <lb/>puncti q refringatur ad eundem uiſum ex puncto h.</s> <s xml:id="echoid-s49609" xml:space="preserve"> Patet <lb/>itaque per hypotheſim, & per præcedentem quoniam pun <lb/>cta g & h ęqualiter diſtabunt à puncto m.</s> <s xml:id="echoid-s49610" xml:space="preserve"> Et quia punctum <lb/>b eſt remotius à centro corporis d, quàm punctum q:</s> <s xml:id="echoid-s49611" xml:space="preserve"> erit <lb/>per ea, quæ oſtendimus in 14 th.</s> <s xml:id="echoid-s49612" xml:space="preserve"> huius, punctum ſuæ refra-<lb/>ctionis remotius à puncto m, quàm punctum h:</s> <s xml:id="echoid-s49613" xml:space="preserve"> ſit itaque <lb/>pũctum illud ſ:</s> <s xml:id="echoid-s49614" xml:space="preserve"> & ducatur linea a f:</s> <s xml:id="echoid-s49615" xml:space="preserve"> quę cadet extra lineam <lb/>a h:</s> <s xml:id="echoid-s49616" xml:space="preserve"> & hęc producta ad perpendicularem d e, ſecet ipſam <lb/>in puncto k:</s> <s xml:id="echoid-s49617" xml:space="preserve"> cadetq́ue punctum k in linea p e inter puncta <lb/>p & e.</s> <s xml:id="echoid-s49618" xml:space="preserve"> Si enim caderet in punctum e, eſſet linea a k contin-<lb/>gens circulum in puncto e, & ſecans in puncto f:</s> <s xml:id="echoid-s49619" xml:space="preserve"> quod eſt <lb/>impoſsibile:</s> <s xml:id="echoid-s49620" xml:space="preserve"> & ſi caderet in punctum p uel citra illum:</s> <s xml:id="echoid-s49621" xml:space="preserve"> tũc <lb/>linea a k ſecaret lineam a p, & punctus p uel alter punctus <lb/>illius ſectionis refringeretur ad uiſum a ex duobus pũctis <lb/>h & f:</s> <s xml:id="echoid-s49622" xml:space="preserve"> quod eſt impoſsibile & contra 23 uel 24th.</s> <s xml:id="echoid-s49623" xml:space="preserve"> huius:</s> <s xml:id="echoid-s49624" xml:space="preserve"> ca <lb/>det ita que punctum k inter duo puncta p & e:</s> <s xml:id="echoid-s49625" xml:space="preserve"> eritq́ue per <lb/>15 huius punctum k imago formæ puncti b.</s> <s xml:id="echoid-s49626" xml:space="preserve"> Ducaturita-<lb/>que linea l k, quæ erit diameter imaginis formę lineę b c.</s> <s xml:id="echoid-s49627" xml:space="preserve"> Quia itaque linea l k uidetur ſub angulo <lb/>l a k, & linea b c ſub angulo b a c:</s> <s xml:id="echoid-s49628" xml:space="preserve"> eſt autem angulus l a k maior angulo b a c, ut manifeſtum eſt:</s> <s xml:id="echoid-s49629" xml:space="preserve"> quia <lb/>totum eſt maius ſua parte.</s> <s xml:id="echoid-s49630" xml:space="preserve"> Patet ergo per 20 th.</s> <s xml:id="echoid-s49631" xml:space="preserve"> 4 huius quia linea l k uidetur maior quàm linea b c:</s> <s xml:id="echoid-s49632" xml:space="preserve"> <lb/>quod enim ſub maiori angulo uidetur, maius uidetur.</s> <s xml:id="echoid-s49633" xml:space="preserve"> Et etiam quia ſitus & poſitio lineæ l k re-<lb/>ſpectu uiſus a eſt conſimilis ſitui & poſitioni lineę b c reſpectu eiuſdem uiſus a:</s> <s xml:id="echoid-s49634" xml:space="preserve"> patet quia lineę b c <lb/>& k l aut ſunt æ quidiſtantes ſimpliciter:</s> <s xml:id="echoid-s49635" xml:space="preserve"> aut inter illarum ęquidiſtantiam non eſt diuerſitas ſenſibi-<lb/>lis:</s> <s xml:id="echoid-s49636" xml:space="preserve"> ergo per 29 p 1 & per 4 p 6, linea k l eſt maior quàm linea b c.</s> <s xml:id="echoid-s49637" xml:space="preserve"> Et quia illarum linearum l k & b c <lb/>ab ipſo uiſu non eſt diſtantia ſenſibilis diuerſitatis in remotione:</s> <s xml:id="echoid-s49638" xml:space="preserve"> uidetur ergo linea l k maior quàm <lb/>linea b c, quia eſt maior:</s> <s xml:id="echoid-s49639" xml:space="preserve"> ſed linea k l eſt imago formæ lineę b c.</s> <s xml:id="echoid-s49640" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s49641" xml:space="preserve"> Comprehen <lb/>ditur etiam linea l k quaſi maior à uiſu quàm linea b c propter debilitatem formę reſractæ:</s> <s xml:id="echoid-s49642" xml:space="preserve"> quoniã, <lb/>ut patet per 10 huius, refractio debilitat omnes formas lucis & coloris.</s> <s xml:id="echoid-s49643" xml:space="preserve"/> </p> <div xml:id="echoid-div1856" type="float" level="0" n="0"> <figure xlink:label="fig-0739-01" xlink:href="fig-0739-01a"> <variables xml:id="echoid-variables847" xml:space="preserve">a f h m g e k b p q d c l n</variables> </figure> </div> </div> <div xml:id="echoid-div1858" type="section" level="0" n="0"> <head xml:id="echoid-head1366" xml:space="preserve" style="it">38. Centro uiſus exiſtente extra ſuperficiem linearum perpendicularium, à punctis rei uiſæ <lb/>ſub corpore ſphærico diaphano denſiore aere, ſuper eius conuexam ſuperficiem oppoſitã uiſui pro-<lb/>ductarum, lineá uiſa ſecundum ſui extrema à centro corporis æquidiſtante: imago lineæ ui-<lb/> <pb o="438" file="0740" n="740" rhead="VITELLONIS OPTICAE"/> ſæ comprehenditur maior ipſa linea uiſa. Alhazen 46 n 7.</head> <p> <s xml:id="echoid-s49644" xml:space="preserve">Eſto centrum uiſus punctum a:</s> <s xml:id="echoid-s49645" xml:space="preserve"> & linea uiſa per refra ctionem ſit b c:</s> <s xml:id="echoid-s49646" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s49647" xml:space="preserve"> punctus d centrum cor-<lb/>poris diaphani denſioris aere:</s> <s xml:id="echoid-s49648" xml:space="preserve"> ſitq́ue ita, ut linea b c ſit intra illud corpus ſecundum ſui extrema b & <lb/>c æqualiter diſtans à centro d:</s> <s xml:id="echoid-s49649" xml:space="preserve">à medio quoq;</s> <s xml:id="echoid-s49650" xml:space="preserve"> puncto lineę b c, quod ſit z, & à duobus extremis eius <lb/>punctis ducantur in eadem ſuperficie lineę perpendiculares ſuper ſuperficiem corporis:</s> <s xml:id="echoid-s49651" xml:space="preserve"> quæ pro-<lb/>ductæ ad peripheriam circuli, ſint b e, z m, & c n:</s> <s xml:id="echoid-s49652" xml:space="preserve"> hę itaque omnes per 72 th.</s> <s xml:id="echoid-s49653" xml:space="preserve">1 huius ſecabunt ſe in cõ <lb/>tro d.</s> <s xml:id="echoid-s49654" xml:space="preserve"> Erit ergo arcus n m e in ſuperficie illius corporis <lb/>diaphani, reſpiciens centrum d:</s> <s xml:id="echoid-s49655" xml:space="preserve"> non ſit autem centrum <lb/> <anchor type="figure" xlink:label="fig-0740-01a" xlink:href="fig-0740-01"/> uiſus in aliqua iſtarum linearum:</s> <s xml:id="echoid-s49656" xml:space="preserve"> ſed ſit extra ſuperficiẽ, <lb/>in qua ſunt illę lineę.</s> <s xml:id="echoid-s49657" xml:space="preserve"> Dico quòd imago lineę b c uidebi-<lb/>tur maior quàm ipſa linea b c.</s> <s xml:id="echoid-s49658" xml:space="preserve"> Ducatur enim linea a z:</s> <s xml:id="echoid-s49659" xml:space="preserve"> & <lb/>â centro uiſus puncto a ducatur perpendicularis linea ſu <lb/>per ſuperficiẽ circuli n m e per 11 p 11, quę ſit a x.</s> <s xml:id="echoid-s49660" xml:space="preserve"> Et quia, <lb/>ut patet ex præmiſsis, & per 22 th.</s> <s xml:id="echoid-s49661" xml:space="preserve">1 huius eſt linea a z per-<lb/>pendicularis ſuper lineam b c:</s> <s xml:id="echoid-s49662" xml:space="preserve"> ſituatio itaque puncti b <lb/>uerſus uiſum a eſt per 4 p 1 & ex præmiſsis conſimilis ſi-<lb/>tuationi puncti c uerſus eundem uiſum a, & illorum pun <lb/>ctorum à uiſu a diſtantia eſt ęqualis.</s> <s xml:id="echoid-s49663" xml:space="preserve"> Sit itaque, ut forma <lb/>puncti b refringatur ad uiſum a à puncto corporis dia-<lb/>phani, quod ſit h:</s> <s xml:id="echoid-s49664" xml:space="preserve"> & forma puncti c à puncto g:</s> <s xml:id="echoid-s49665" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s49666" xml:space="preserve"> pũ-<lb/>cta g & h extra ſuperficiem circuli n m e:</s> <s xml:id="echoid-s49667" xml:space="preserve"> eritq́ue illorum <lb/>punctorum h & g à uiſu a diſtantia æqualis.</s> <s xml:id="echoid-s49668" xml:space="preserve"> Ducantur <lb/>itaque lineæ b h, a h, c g, a g:</s> <s xml:id="echoid-s49669" xml:space="preserve"> eritq́ue ſuperficies, in qua <lb/>ſunt duę lineę a h & b h, erecta ſuper ſuperficiem corpo-<lb/>ris diaphani per 2 huius:</s> <s xml:id="echoid-s49670" xml:space="preserve"> quoniam ipſa eſt ſuperficies re-<lb/>fractionis:</s> <s xml:id="echoid-s49671" xml:space="preserve"> ergo & linea b e (quę eſt perpendicularis ſu-<lb/>per ſuperficiem corporis diaphani ducta à puncto b) e-<lb/>rit in illa ſuperficie per 1 huius.</s> <s xml:id="echoid-s49672" xml:space="preserve"> Similiter quoque ſuperfi <lb/>cies, in qua ſunt lineę c g & a g, cum ſit ſuperficies refra-<lb/>ctionis:</s> <s xml:id="echoid-s49673" xml:space="preserve"> patetper 2 huius quoniam ipſa eſt erecta ſuper <lb/>ſuperficiem corporis diaphani:</s> <s xml:id="echoid-s49674" xml:space="preserve"> ergo & in illa ſuperficie <lb/>eſt linea c n, quæ eſt perpendicularis ſuper eandem cor-<lb/>poris ſuperficiem ducta à puncto c.</s> <s xml:id="echoid-s49675" xml:space="preserve"> Protrahatur itaque <lb/>linea a h ultra punctum h:</s> <s xml:id="echoid-s49676" xml:space="preserve"> & palàm per præmiſſa & per 14.</s> <s xml:id="echoid-s49677" xml:space="preserve"> th.</s> <s xml:id="echoid-s49678" xml:space="preserve"> huius quòd ipſa ſecabit lineam b e:</s> <s xml:id="echoid-s49679" xml:space="preserve"> ſit <lb/>ergo ut ſecet ipſam in puncto k.</s> <s xml:id="echoid-s49680" xml:space="preserve"> Similiter quoque linea a g producta ultra punctum g ſecet lineam <lb/>d n in puncto l:</s> <s xml:id="echoid-s49681" xml:space="preserve"> eritq́ ſituatio lineę a k, reſpectu uiſus a, ſicut lineę a l:</s> <s xml:id="echoid-s49682" xml:space="preserve"> unde linea a k & a l erunt ę-<lb/>quales:</s> <s xml:id="echoid-s49683" xml:space="preserve"> & ſimiliter erit linea d k ęqualis lineę d l, quę omnia oſten dipoſſunt ſecũdum modum, quo <lb/>proceſsimus in pręmiſſa 34 huius.</s> <s xml:id="echoid-s49684" xml:space="preserve"> Copuletur ergo linea l k:</s> <s xml:id="echoid-s49685" xml:space="preserve"> hęc itaque erit diameter imaginis lineę <lb/>b c.</s> <s xml:id="echoid-s49686" xml:space="preserve"> Quia itaque linea b d eſt æqualis lineæ d c ex hypotheſi, & linea d k æqualis lineę d l:</s> <s xml:id="echoid-s49687" xml:space="preserve"> erit li-<lb/>nea k b ęqualis lineæ l c:</s> <s xml:id="echoid-s49688" xml:space="preserve"> ergo per 7 p 5 & per 2 p 6 lineæ l k & b c ęquidiſtant:</s> <s xml:id="echoid-s49689" xml:space="preserve"> ergo per 29 p 1 & <lb/>per 4 p 6 linea l k eſt maior quàm linea b c:</s> <s xml:id="echoid-s49690" xml:space="preserve"> & quia ſub maiori angulo uidetur, apparet maior.</s> <s xml:id="echoid-s49691" xml:space="preserve"> Et <lb/>hoc eſt propoſitum.</s> <s xml:id="echoid-s49692" xml:space="preserve"/> </p> <div xml:id="echoid-div1858" type="float" level="0" n="0"> <figure xlink:label="fig-0740-01" xlink:href="fig-0740-01a"> <variables xml:id="echoid-variables848" xml:space="preserve">a h g m x e k b z d c l n</variables> </figure> </div> <figure> <variables xml:id="echoid-variables849" xml:space="preserve">a f h g m r e k b p q d c l a</variables> </figure> </div> <div xml:id="echoid-div1860" type="section" level="0" n="0"> <head xml:id="echoid-head1367" xml:space="preserve" style="it">39. Centro uiſus exiſtente extra ſuperficiem perpen-<lb/>dicularium à puncto rei uiſæ ſub corpore ſphærico dia-<lb/>phano denſiore aere, ſuper eius conuexam ſuperficiem <lb/>oppoſitam uiſui productarum, lineǽ uiſæ extremis cẽ-<lb/>tro corporis inæqualiter approximatis: imago lineæ ui-<lb/>ſæ comprehenditur maior ipſa linea uiſa. Alha-<lb/>zen 46 n 7.</head> <p> <s xml:id="echoid-s49693" xml:space="preserve">Remaneat omnis diſpoſitio proximę pręmiſſę, niſi qđ <lb/>extrema lineę b c inęqualiter diſtent à cẽtro corporis dia-<lb/>phani, quod eſt d:</s> <s xml:id="echoid-s49694" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s49695" xml:space="preserve"> linea d b maior quàm linea d c.</s> <s xml:id="echoid-s49696" xml:space="preserve"> Se-<lb/>cetur ergo ex linea d b per 3 p 1 linea d q ęqualis lineę d c:</s> <s xml:id="echoid-s49697" xml:space="preserve"> <lb/>& copuletur linea c q:</s> <s xml:id="echoid-s49698" xml:space="preserve"> cuius extrema ęqualiter diſtabunt <lb/>à centro d:</s> <s xml:id="echoid-s49699" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s49700" xml:space="preserve"> per pręmiſſam imago lineę c q, quę ſit l p, <lb/>maior quàm linea c q.</s> <s xml:id="echoid-s49701" xml:space="preserve"> Et quia puncta q & b ſunt in ea dẽ <lb/>linea perpendiculari ſuper ſuperficiem corporis diapha-<lb/>ni, quę eſt d e:</s> <s xml:id="echoid-s49702" xml:space="preserve"> patet quòd ipſa ambo ſunt in eadem ſuperfi <lb/>cie refractionis, quę eſt a d e:</s> <s xml:id="echoid-s49703" xml:space="preserve"> & refringũtur ad uiſum a ex <lb/>eodẽ arcu circuli, qui eſt cõmunis ſectio illius ſuքficiei & <lb/>ſuքficiei corporis diaphani.</s> <s xml:id="echoid-s49704" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s49705" xml:space="preserve">, ut forma pũcti q re-<lb/>fringatur à puncto illius arcus, qui eſt h, cõformiter ſe ha-<lb/> <pb o="439" file="0741" n="741" rhead="LIBER DECIMVS."/> bente ad uiſum a cum puncto g, à quo refringitur forma puncti c:</s> <s xml:id="echoid-s49706" xml:space="preserve"> patet per 14 huius quò d punctũ, <lb/>à quo refringitur forma puncti b, quod ſit f, erit baſsius puncto h:</s> <s xml:id="echoid-s49707" xml:space="preserve"> producta quoq;</s> <s xml:id="echoid-s49708" xml:space="preserve"> linea a fintra cor <lb/>pus diaphanum ad diametrum d e in punctum k:</s> <s xml:id="echoid-s49709" xml:space="preserve">patet quoq;</s> <s xml:id="echoid-s49710" xml:space="preserve">, ut in 37 huius, quia punctum k cadet <lb/>inter puncta p & e:</s> <s xml:id="echoid-s49711" xml:space="preserve"> copulata quoq;</s> <s xml:id="echoid-s49712" xml:space="preserve"> linea l k, erit ipſa quaſi æquidiſtans lineæ b c, & in eadem ſuper-<lb/>ficie cum illa.</s> <s xml:id="echoid-s49713" xml:space="preserve"> Erit ergo maior per 29 p 1 & 4 p 6:</s> <s xml:id="echoid-s49714" xml:space="preserve"> & etiam quia ſub maiori angulo uidetur, maior ui-<lb/>detur.</s> <s xml:id="echoid-s49715" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s49716" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1861" type="section" level="0" n="0"> <head xml:id="echoid-head1368" xml:space="preserve" style="it">40. Lineæ refractè uiſæ, tranſeuntis per centrum corporis diaphani ſphærici denſioris aere, <lb/>non exiſtẽtis in perpendiculari ducta à centro uiſus ſuper illius corporis ſuperficiem, imago ſem <lb/>per uidetur maior ipſa linea.</head> <p> <s xml:id="echoid-s49717" xml:space="preserve">Sit a centrum uiſus extra corpus diaphanum groſsius aere:</s> <s xml:id="echoid-s49718" xml:space="preserve"> cuius centrũ ſit d:</s> <s xml:id="echoid-s49719" xml:space="preserve"> ſitq̃ linea uiſa b c <lb/>pertranſiens centrum d:</s> <s xml:id="echoid-s49720" xml:space="preserve"> ita tamen quòd centrum uiſus non ſit in illa linea b c utcunq;</s> <s xml:id="echoid-s49721" xml:space="preserve"> protracta:</s> <s xml:id="echoid-s49722" xml:space="preserve"> <lb/>dico quòd eius imago ſemper uidetur maior ipſa linea.</s> <s xml:id="echoid-s49723" xml:space="preserve"> Quoniã enim per-<lb/>pendiculares ſuper ſuperficiem corporis à quibuſcunq;</s> <s xml:id="echoid-s49724" xml:space="preserve"> punctis lineæ b c <lb/> <anchor type="figure" xlink:label="fig-0741-01a" xlink:href="fig-0741-01"/> productæ, omnes continent lineam b c:</s> <s xml:id="echoid-s49725" xml:space="preserve"> uiſu quoq;</s> <s xml:id="echoid-s49726" xml:space="preserve"> in aere exiſtente fit re-<lb/>fractio ſemper ad contrariam partem perpendicularis ductæ à puncto re-<lb/>fractionis ſuper ſuperficiem corporis, ut patet per 4 huius.</s> <s xml:id="echoid-s49727" xml:space="preserve"> Ergo ſecun-<lb/>dum præmiſſas demonſtrationes patet, quòd lineæ extenſionis forma-<lb/>rum punctorum extremorum lineæ b c, quæ ſunt b & c, productæ intra <lb/>corpus diaphanum, à cuius ſuperficie fit refractio, interſecabunt perpen-<lb/>diculares punctorum b & c:</s> <s xml:id="echoid-s49728" xml:space="preserve"> maior ergo ſemper uidebitur imago lineæ <lb/>b c, quàm ipſa linea:</s> <s xml:id="echoid-s49729" xml:space="preserve"> quæ tunc fit pars ſuæ propriæ imaginis ſecundum <lb/>ueritatem.</s> <s xml:id="echoid-s49730" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s49731" xml:space="preserve"> Poſſet quoq;</s> <s xml:id="echoid-s49732" xml:space="preserve"> ampliari modus iſte de-<lb/>monſtrandi ad alios ſitus lineæ uiſæ, qui poſſent eſſe ultra centrum cor-<lb/>poris diaphani denſioris aere, uiſu exiſtente extra illud corpus in aere, & <lb/>conuexitate corporis reſpiciente uiſum.</s> <s xml:id="echoid-s49733" xml:space="preserve"> Videtur enim & tunc imago <lb/>quandoq;</s> <s xml:id="echoid-s49734" xml:space="preserve"> maior re uiſa præmiſſo modo in alijs ſitibus an<gap/> centrum:</s> <s xml:id="echoid-s49735" xml:space="preserve"> ut <lb/>cum linea uiſa fuerit propinqua centro corporis diaphani.</s> <s xml:id="echoid-s49736" xml:space="preserve"> Et ſi linea uiſa <lb/>b c fuerit perpendicularis ſuper lineam a d z à centro uiſus per centrum <lb/>corporis productam:</s> <s xml:id="echoid-s49737" xml:space="preserve"> & lineæ extenſionis formarum extremorum pun-<lb/>ctorum lineæ b c ſecent corporis ſphærici diaphani ſuperficiem, & ſe-<lb/>cent lineas perpendiculares ductas à pũctis b & c ſuper ſuperficiẽ corpo-<lb/>ris diaphani intra corpus:</s> <s xml:id="echoid-s49738" xml:space="preserve"> tunc imago uidebitur minor re uiſa.</s> <s xml:id="echoid-s49739" xml:space="preserve"> Si uerò li-<lb/>neæ extenſionis formarum punctorum b & c fuerint contingentes circulum corporis diaphani <lb/>in term inis perpendicularium ductarum à punctis c & b ſuper ſuperficiem corporis, uel ſecan-<lb/>tes circulum in eiſdem terminis:</s> <s xml:id="echoid-s49740" xml:space="preserve"> tunc ſemper imago erit æqualis rei uiſæ per 15 p 1 & per 26 & 28 <lb/>p 3:</s> <s xml:id="echoid-s49741" xml:space="preserve"> & uidebitur imago lineæ b c ſicut quædam chorda arcus illius circuli.</s> <s xml:id="echoid-s49742" xml:space="preserve"> Et ſi lineas extenſionis <lb/>formarum accideret contingere circulum corporis diaphani in duobus punctis medijs illius ar-<lb/>cus:</s> <s xml:id="echoid-s49743" xml:space="preserve"> ut ſi uiſus ſit ualde propinquus ſuperficiei corporis diaphani:</s> <s xml:id="echoid-s49744" xml:space="preserve"> tunc illæ lineæ concurrent cum <lb/>perpendicularibus extra corporis ſuperficiem:</s> <s xml:id="echoid-s49745" xml:space="preserve"> uidebiturq;</s> <s xml:id="echoid-s49746" xml:space="preserve"> imago lineæ b c maior ipſa linea:</s> <s xml:id="echoid-s49747" xml:space="preserve"> & <lb/>extra ſuperficiem corporis ſecundum ſui extrema extenſa.</s> <s xml:id="echoid-s49748" xml:space="preserve"> Quòd ſi linea uiſa b c ſit extra corpus <lb/>diaphanum, contingens ipſum, uel diſtans ab ipſo, non exiſtens tamen pars lineæ a d:</s> <s xml:id="echoid-s49749" xml:space="preserve"> tunc ima-<lb/>go eius uidebitur minor re uiſa, quando occurrit inter ipſum corpus diaphanum, uel ultra illud, <lb/>inter rem uiſam & ſuperficiem corporis.</s> <s xml:id="echoid-s49750" xml:space="preserve"> Sed in aſſuetis uiſibilibus non eſt aliquid tale, niſi fortè <lb/>fuerit aliquod corpus diaphanum uitreum aut lapideum, & fuerit totum corpus ſolidum, & res ui-<lb/>ſa fuerit intra ipſum:</s> <s xml:id="echoid-s49751" xml:space="preserve"> uel ſi res uiſa fuerit ultra ſphæram cryſtallinam aut uitream.</s> <s xml:id="echoid-s49752" xml:space="preserve"> Horum autem ſi-<lb/>tuum diuerſitatem ex præhabitis principijs demonſtran dam relinquimus ingenio perquirentis.</s> <s xml:id="echoid-s49753" xml:space="preserve"/> </p> <div xml:id="echoid-div1861" type="float" level="0" n="0"> <figure xlink:label="fig-0741-01" xlink:href="fig-0741-01a"> <variables xml:id="echoid-variables850" xml:space="preserve">a b d c b c z</variables> </figure> </div> </div> <div xml:id="echoid-div1863" type="section" level="0" n="0"> <head xml:id="echoid-head1369" xml:space="preserve" style="it">41. In omnibus refractionibus factis à ſuperficiebus ſphæricis corporum diaphanorum ad <lb/>uiſum, imagine apparente maiore re uiſa: pars imaginis uidebitur maior parte rei uiſæ ſibi pro-<lb/>portinoali. Alhazen 47 n 7.</head> <p> <s xml:id="echoid-s49754" xml:space="preserve">Fiat diſpoſitio, quæ in 35 huius:</s> <s xml:id="echoid-s49755" xml:space="preserve"> & ſit, ut linea d m ſecet lineam k l, quæ eſt diameter imaginis, <lb/>in puncto o:</s> <s xml:id="echoid-s49756" xml:space="preserve"> erit ergo linea k o imago lineæ b z:</s> <s xml:id="echoid-s49757" xml:space="preserve"> quoniam punctum z uidetur ſecundum perpendi-<lb/>cularem a z per 3 huius:</s> <s xml:id="echoid-s49758" xml:space="preserve"> & erit angulus k a o maior angulo b a z:</s> <s xml:id="echoid-s49759" xml:space="preserve"> & ſitus lineæ k o reſpectu uiſus a <lb/>eſt ſimilis poſitioni lineæ b z reſpectu eiuſdem uiſus:</s> <s xml:id="echoid-s49760" xml:space="preserve"> & ambæ illæ lineæ æqualiter diſtant à cen-<lb/>tro uiſus:</s> <s xml:id="echoid-s49761" xml:space="preserve"> uel ſi in hoc ſit aliqua differentia, illa non erit ſenſibilis, reſpectu uiſus.</s> <s xml:id="echoid-s49762" xml:space="preserve"> Imago itaque k o <lb/>uidetur maior quàm linea b z:</s> <s xml:id="echoid-s49763" xml:space="preserve"> & earum puncta z & o cadunt in linea z a, quæ eſt ducta à <lb/>centro uiſus, & cuius pars eſt linea z m, exiens ab extremitate lineæ b z perpendiculariter ſu-<lb/>per ſuperficiem corporis diaphani, cadens in punctum m.</s> <s xml:id="echoid-s49764" xml:space="preserve"> Quòd ſi aſſumatur alia pars lineæ <lb/>b z:</s> <s xml:id="echoid-s49765" xml:space="preserve"> quæ ſit b f:</s> <s xml:id="echoid-s49766" xml:space="preserve"> & ſit locus imaginis formæ puncti f in puncto r lineæ k o:</s> <s xml:id="echoid-s49767" xml:space="preserve">tunc erit linea kr <lb/>imago lineæ b f:</s> <s xml:id="echoid-s49768" xml:space="preserve"> & ſicut ſuprà, oſtenſum eſt, patet quòd linea k r uidebitur maior quàm li-<lb/>nea b f:</s> <s xml:id="echoid-s49769" xml:space="preserve"> quoniam plus refractionis accidit lineæ b f, quàm lineæ f z per 14 th.</s> <s xml:id="echoid-s49770" xml:space="preserve"> huius:</s> <s xml:id="echoid-s49771" xml:space="preserve"> maior <lb/> <pb o="440" file="0742" n="742" rhead="VITELLONIS OPTICAE"/> ergo ei debetur exceſſus imaginis quàm lineæ f z.</s> <s xml:id="echoid-s49772" xml:space="preserve"> Si uerò pũctum a centrum uiſus ſit extra ſuper-<lb/>ficiem, in qua ſunt omnes perpendiculares, exeuntes ex <lb/>punctis lineæ b c ſuper ſuperficiẽ corporis diaphani, à qua <lb/> <anchor type="figure" xlink:label="fig-0742-01a" xlink:href="fig-0742-01"/> fit reſractio (nam linea a z, quæ exit à puncto a perpendicu <lb/>lariter ſuper medium pũctũ lineę b c, quod eſt z, nõ propter <lb/>hoc eſt perpendicularis ſuper ſuperficiem corporis, in qua <lb/>eſt linea b c) idẽ patebit.</s> <s xml:id="echoid-s49773" xml:space="preserve"> Nã quoniam lineæ b c & k l ſunt e-<lb/>rectæ ſuper lineam a z d, & linea k o eſt imago lineæ b z, & li <lb/>nea l o eſt imago lineę z c, & angulus, quẽ reſpicit linea k o <lb/>apud centrum uiſus a, qui eſt angulus k a z, eſt maior angu-<lb/>lo b a z, quem reſpicit linea b z apud centrum uiſus a:</s> <s xml:id="echoid-s49774" xml:space="preserve"> linea <lb/>ergo k o per 20 th.</s> <s xml:id="echoid-s49775" xml:space="preserve"> 4 huius uidebitur maior quàm linea b z:</s> <s xml:id="echoid-s49776" xml:space="preserve"> <lb/>& ſimiliter linea k r uidebitur maior quàm linea b f.</s> <s xml:id="echoid-s49777" xml:space="preserve"> Et o-<lb/>mnia hæc patent exillis, quę pręmiſſa ſunt in 33 huius.</s> <s xml:id="echoid-s49778" xml:space="preserve"> Siue <lb/>ergo ſuperficies corporum diaphanorũ oppoſitæ uiſui fue-<lb/>rint planę, ſiue ſphæricę:</s> <s xml:id="echoid-s49779" xml:space="preserve"> accidit imaginem rei uiſę <lb/>uideri maiorem ipſa re uiſa.</s> <s xml:id="echoid-s49780" xml:space="preserve"> In hoc tamẽ eſt differentia, quia <lb/>in corporib.</s> <s xml:id="echoid-s49781" xml:space="preserve"> diaphanis planarũ ſuperficierũ exceſſus magni <lb/>tudinis imaginis ſuper rẽ uiſam eſt ſolũ in apparentia uiſus, <lb/>propter exceſſum angulorũ, ſecũdum quos uidetur & ima-<lb/>go & res ipſa uiſa:</s> <s xml:id="echoid-s49782" xml:space="preserve"> aliàs enim imagines ſecũdũ ueritatẽ ſunt <lb/>æquales ip ſis rebus uiſis:</s> <s xml:id="echoid-s49783" xml:space="preserve"> ſed in refractione facta à corpori-<lb/>bus conuexis ſphæricis imago eſt ſecũdum ueritatẽ maior <lb/>ipſa re uiſa:</s> <s xml:id="echoid-s49784" xml:space="preserve"> & etiam ſecũdum apparentiam in uiſu propter <lb/>angulorum exceſſum uidetur maior:</s> <s xml:id="echoid-s49785" xml:space="preserve"> quoniam in hoc ſitu i-<lb/>mago reſpicit maiorem angulum apud cẽtrum uiſus quàm <lb/>reſpiciat ipſa res uiſa:</s> <s xml:id="echoid-s49786" xml:space="preserve"> & ſunt utroq;</s> <s xml:id="echoid-s49787" xml:space="preserve"> modo partes imaginũ <lb/>maiores partibus rerum uiſarum ſibi proportionalium.</s> <s xml:id="echoid-s49788" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s49789" xml:space="preserve"/> </p> <div xml:id="echoid-div1863" type="float" level="0" n="0"> <figure xlink:label="fig-0742-01" xlink:href="fig-0742-01a"> <variables xml:id="echoid-variables851" xml:space="preserve">a m h g e k r o <gap/> b f z c l d</variables> </figure> </div> </div> <div xml:id="echoid-div1865" type="section" level="0" n="0"> <head xml:id="echoid-head1370" xml:space="preserve" style="it">42. Omne corpus uiſum in aqua, comprehenditur maius quàm ſit ſecundum ueritatem. <lb/>Alhazen 48 n 7.</head> <p> <s xml:id="echoid-s49790" xml:space="preserve">Quod hic proponitur, patet ſatis ex præmiſsis:</s> <s xml:id="echoid-s49791" xml:space="preserve"> ſed & idẽ placuit experimentaliter declarare, & <lb/>uniuerſalẽ cauſſam particulariter exemplare.</s> <s xml:id="echoid-s49792" xml:space="preserve"> Aſſumatur itaq;</s> <s xml:id="echoid-s49793" xml:space="preserve"> corpus colũnare longitudinis unius <lb/>cubiti, & ali quãtę gro ſsiciei:</s> <s xml:id="echoid-s49794" xml:space="preserve"> & ſit albũ, ut manifeſtius in aqua poſsit diſtingui:</s> <s xml:id="echoid-s49795" xml:space="preserve"> ſintq́;</s> <s xml:id="echoid-s49796" xml:space="preserve"> ſuքficies eius <lb/>baſis planæ, ita quod perſe ſuper illas poſsit ſtare æqualiter ſuper ſuperficiem horizontis uel terræ <lb/>uel uaſis.</s> <s xml:id="echoid-s49797" xml:space="preserve"> Deinde infundatur aqua clara in uas aliquod, cuius fuperficies baſis ſit plana:</s> <s xml:id="echoid-s49798" xml:space="preserve"> ita quòd a-<lb/>qua non immergat totam corporis longitudinem:</s> <s xml:id="echoid-s49799" xml:space="preserve"> & erigatur corpus ſuper mediam baſim uaſis in <lb/>aqua.</s> <s xml:id="echoid-s49800" xml:space="preserve"> Remanebit ergo aliqua pars eius extra aquã:</s> <s xml:id="echoid-s49801" xml:space="preserve"> quia profunditas a quæ eſt minor corporis lon-<lb/>gitudine.</s> <s xml:id="echoid-s49802" xml:space="preserve"> Cũ itaq;</s> <s xml:id="echoid-s49803" xml:space="preserve"> quieuerit a qua:</s> <s xml:id="echoid-s49804" xml:space="preserve"> uidebitur pars corporis intra a quam groſsior, quàm illa, quæ eſt <lb/>extra a quam.</s> <s xml:id="echoid-s49805" xml:space="preserve"> Patet ergo propoſitũ per experimentũ.</s> <s xml:id="echoid-s49806" xml:space="preserve"> Sed & idẽ patet aliter.</s> <s xml:id="echoid-s49807" xml:space="preserve"> Quoniã enim conuexũ <lb/>ſuperficiei a quę eſt figuræ ſphæricæ, & opponitur uiſui:</s> <s xml:id="echoid-s49808" xml:space="preserve"> & centrũ ſuperficiei aquæ, quod eſt centrũ <lb/>uniuerſi (ut aliàs oſtẽdimus) ſemper eſt ultra omnia illa uiſibilia, quę cõprehendunturin aqua, & <lb/>aqua eſt groſsior aere:</s> <s xml:id="echoid-s49809" xml:space="preserve"> ſiue extremitas rei uiſæ fuerit æqualiter diſtans à cẽtro aquę, ſiueinæquali-<lb/>ter:</s> <s xml:id="echoid-s49810" xml:space="preserve"> & ſiue uiſus fuerit in aliqua linearũ perpendiculariũ exeuntiũ ab aliquo pũctorũ rei uiſæ ſuper <lb/>ſuperficiẽ aquę, ſiue oẽs extra illas perpendiculares:</s> <s xml:id="echoid-s49811" xml:space="preserve"> ſemper eſt neceſſariũ, ut patet expręmiſsis ſex <lb/>propoſitionibus proximis, ſormã rei uiſę uideri maiorẽ ipſa re uiſa exiſtẽte intra corpus aquę.</s> <s xml:id="echoid-s49812" xml:space="preserve"> Sed <lb/>fortè ſi a qua fuerit clara ualde, & pauca:</s> <s xml:id="echoid-s49813" xml:space="preserve"> quales aquas in loco ſubterraneo in concauitate montis, <lb/>qui eſt inter ciuitates Paduã & Vincentiã (qui locus dicitur Cubalus) nos uidimus lucidas, quaſi <lb/>ut aerem:</s> <s xml:id="echoid-s49814" xml:space="preserve"> tũc fortè non cõprehẽdetur imago formę rei uiſæ ſub aqua tali eſſe maior quàm ſi in aere <lb/>uideretur:</s> <s xml:id="echoid-s49815" xml:space="preserve"> quia tũc non eſt differentia in quantitate iſtorũ quo ad ſenſum:</s> <s xml:id="echoid-s49816" xml:space="preserve"> quoniam denſitas a quæ <lb/>modicũ addit ſuper aeris denſitatẽ:</s> <s xml:id="echoid-s49817" xml:space="preserve"> & ideo ſenſus tũc non diſtinguet quantitatis additionẽ:</s> <s xml:id="echoid-s49818" xml:space="preserve"> ſemper <lb/>tamen ſecũdum ueritatẽ imago fit maior ipſare uiſa:</s> <s xml:id="echoid-s49819" xml:space="preserve"> licet illud quan doque lateat ſenſum.</s> <s xml:id="echoid-s49820" xml:space="preserve"> Patet er-<lb/>go propoſitũ:</s> <s xml:id="echoid-s49821" xml:space="preserve"> magis tamen eſt hoc euidens in aquis groſsioribus, ut ſulphureis calidis:</s> <s xml:id="echoid-s49822" xml:space="preserve"> in quarum <lb/>intuitu & mirabilii tran ſmutatione formarum primùm nos amor huius ſtudij allexit.</s> <s xml:id="echoid-s49823" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1866" type="section" level="0" n="0"> <head xml:id="echoid-head1371" xml:space="preserve" style="it">43. Re uiſa ultra corpus diaphanum ſphæricum großius aere exiſtente, itaquòd centrum ui <lb/>ſus & res uiſa & centrum corporis ſphærici ſint in eadem linea recta: comprehenditur imago <lb/>rei uiſæ figuræ armillaris, multò maior re uiſa. Alhazen 49 n 7.</head> <p> <s xml:id="echoid-s49824" xml:space="preserve">Sit centrum uiſus a:</s> <s xml:id="echoid-s49825" xml:space="preserve"> & corpus ſphæricum diaphanum ſit b d z g:</s> <s xml:id="echoid-s49826" xml:space="preserve"> cuius centrum ſit e:</s> <s xml:id="echoid-s49827" xml:space="preserve"> & ducatur li <lb/>nea a e:</s> <s xml:id="echoid-s49828" xml:space="preserve"> quę protracta ſecet ſuperficiæ ſphærę diaphanę in duobus pũctis b & d:</s> <s xml:id="echoid-s49829" xml:space="preserve"> protrahatur quoq;</s> <s xml:id="echoid-s49830" xml:space="preserve"> <lb/>ultra punctum d uſq;</s> <s xml:id="echoid-s49831" xml:space="preserve"> ad punctum h:</s> <s xml:id="echoid-s49832" xml:space="preserve"> tran ſeatq́;</s> <s xml:id="echoid-s49833" xml:space="preserve"> per lineam a b d h ſuperficies plana ſecans ſphęram:</s> <s xml:id="echoid-s49834" xml:space="preserve"> <lb/>& ſit communis ſectio illius ſuperficiei planæ, & ſuperficiei ſphærę diaphanæ per 69 th.</s> <s xml:id="echoid-s49835" xml:space="preserve">1 huius cir-<lb/>culus b d z g.</s> <s xml:id="echoid-s49836" xml:space="preserve"> Iam autem oſtenſum eſt in 25 huius quòd in linea d h ſunt plura puncta, quorum <lb/>formæ refringuntur ad uiſum a ex circumferentia circuli b d z g:</s> <s xml:id="echoid-s49837" xml:space="preserve"> & quòd forma totius illius <lb/> <pb o="441" file="0743" n="743" rhead="LIBER DECIMVS."/> lineæ refringitur ad uiſum a, ſi arcus b g z d fuerit continuus, unius ſcilicet diaphanitatis continen-<lb/>tis lineam d h l.</s> <s xml:id="echoid-s49838" xml:space="preserve"> Et ſi forma puncti h re fringatur ad uiſum a ex puncto corporis g:</s> <s xml:id="echoid-s49839" xml:space="preserve"> & forma punctil <lb/>refringatur ad uiſum a ex pũcto corporis p:</s> <s xml:id="echoid-s49840" xml:space="preserve"> manifeſtum eſt quod forma totius lineæ refringetur ad <lb/>a uiſum ex arcu g p:</s> <s xml:id="echoid-s49841" xml:space="preserve"> & ducantur lineæ g h, p l, g a, p a:</s> <s xml:id="echoid-s49842" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s49843" xml:space="preserve"> linea g h circũferentiam circuli in pun-<lb/>cto m, & linea p lin pũcto z.</s> <s xml:id="echoid-s49844" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s49845" xml:space="preserve"> pũcti h extenditur per lineam h g, & refringitur per lineam <lb/>g a:</s> <s xml:id="echoid-s49846" xml:space="preserve"> & forma puctil extenditurper lineã l p, & refringitur per line-<lb/> <anchor type="figure" xlink:label="fig-0743-01a" xlink:href="fig-0743-01"/> am p a:</s> <s xml:id="echoid-s49847" xml:space="preserve"> & ducantur lineæ e m & e z:</s> <s xml:id="echoid-s49848" xml:space="preserve"> & extrahatur linea e m ad pun-<lb/>ctum c:</s> <s xml:id="echoid-s49849" xml:space="preserve"> & linea e z ad punctum f.</s> <s xml:id="echoid-s49850" xml:space="preserve"> Forma ergo, quę exten ditur per <lb/>lineam a g (quoniam peruenit ad punctum g) refringitur per li-<lb/>neam g h ad punctum h:</s> <s xml:id="echoid-s49851" xml:space="preserve"> & forma, quæ extenditur per lineam a p <lb/>perueniens ad punctum p, per lineam p l refringitur & peruenit <lb/>ad punctum l:</s> <s xml:id="echoid-s49852" xml:space="preserve"> & hoc ſi corpus diaphanum fuerit continuum & u-<lb/>num uſq;</s> <s xml:id="echoid-s49853" xml:space="preserve"> ad punctum l.</s> <s xml:id="echoid-s49854" xml:space="preserve"> Si uerò corpus ſphæricum fuérit ſignatum <lb/>& terminatum apud ſuperficiem ſphæricam citra lineam h l:</s> <s xml:id="echoid-s49855" xml:space="preserve"> tunc <lb/>forma, quæ extenditur per lineam a g, refringitur per lineam g m <lb/>in partem perpendicularis e h:</s> <s xml:id="echoid-s49856" xml:space="preserve"> & cum forma peruenerit ad pun-<lb/>ctum m, refringetur ſecundò in partem contrariam perpendicula-<lb/>ris, quæ eſt e m c, & concurret cum perpendiculari e l:</s> <s xml:id="echoid-s49857" xml:space="preserve"> refringa-<lb/>tur ergo in punctum k perpendicularis e l.</s> <s xml:id="echoid-s49858" xml:space="preserve"> Etſimiliter forma, quæ <lb/>extenditur per lineam a p, refringetur per lineam p z:</s> <s xml:id="echoid-s49859" xml:space="preserve"> & cum per-<lb/>uenerit ad punctum z, refringetur ſecundò ad partem contrariam <lb/>perpendicularis e z f in partem perpendicularis e h, & concurret <lb/>cum illa perpendiculari h e:</s> <s xml:id="echoid-s49860" xml:space="preserve"> ſit punctum concurſus o.</s> <s xml:id="echoid-s49861" xml:space="preserve"> Sic ergo re-<lb/>fractio ſormæ quæ eſt à puncto p, peruenit ad punctum z:</s> <s xml:id="echoid-s49862" xml:space="preserve"> abillo <lb/>puncto zrefringitur ad diametrum e l per lineam z o.</s> <s xml:id="echoid-s49863" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s49864" xml:space="preserve"> <lb/>puncti k per 9 huius extenditur per lineam k m, & à puncto m re-<lb/>fringitur per lineam m g in punctum g:</s> <s xml:id="echoid-s49865" xml:space="preserve"> deinde ſecundò refringitur <lb/>à puncto g per lineam g a ad uiſum a.</s> <s xml:id="echoid-s49866" xml:space="preserve"> Etſimiliter forma puncti o <lb/>extenditur perlineam o z:</s> <s xml:id="echoid-s49867" xml:space="preserve"> & à puncto z refringitur perlineam z p <lb/>in punctum p:</s> <s xml:id="echoid-s49868" xml:space="preserve"> deinde refringitur ab illo puncto p per lineam p a ad <lb/>uiſum a.</s> <s xml:id="echoid-s49869" xml:space="preserve"> Forma ergo totius lineæ k o refringitur ad uiſum a ex arcu <lb/>g p.</s> <s xml:id="echoid-s49870" xml:space="preserve"> Et ſi linea a k o fuerit fixa, & imaginati fuerimus figuram k a g p <lb/>circumuolui eirca lineam a k o fixam:</s> <s xml:id="echoid-s49871" xml:space="preserve"> tunc arcus g p deſcribet fi-<lb/>guram circularem, utpote armillam, à cuius totali ſuperficie refrin-<lb/>getur forma lineæ k o ad uiſum a:</s> <s xml:id="echoid-s49872" xml:space="preserve"> & erit centrum uniſus a locus ima-<lb/>ginis per 15 th.</s> <s xml:id="echoid-s49873" xml:space="preserve"> huius.</s> <s xml:id="echoid-s49874" xml:space="preserve"> Forma ergo lineæ k o uidebitur in tota ſuperficie circulari, quæ eſt locus res <lb/>fractiõis:</s> <s xml:id="echoid-s49875" xml:space="preserve"> & eſt armillaris in ſuperficie ſphæræ.</s> <s xml:id="echoid-s49876" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s49877" xml:space="preserve"> lineæ k o uidebitur multò maior ſeipſa:</s> <s xml:id="echoid-s49878" xml:space="preserve"> <lb/>& erit figura formæ diuerſa à figura k o.</s> <s xml:id="echoid-s49879" xml:space="preserve"> Hoc autem poteſt ſic experimento declarari.</s> <s xml:id="echoid-s49880" xml:space="preserve"> Accipiatur <lb/>ſphæra cryſta llina aut uitrea perſectę rotunditatis:</s> <s xml:id="echoid-s49881" xml:space="preserve"> & accipiatur corpuſculum paruum, ut cera ni-<lb/>gra ſphærica, quæ ponatur in capite acus:</s> <s xml:id="echoid-s49882" xml:space="preserve"> ponaturq́;</s> <s xml:id="echoid-s49883" xml:space="preserve"> ſphæra cryſtallina in oppoſitione alterius ui-<lb/>ſuum, & claudatur reliquus:</s> <s xml:id="echoid-s49884" xml:space="preserve"> eleueturq́;</s> <s xml:id="echoid-s49885" xml:space="preserve"> acus ultra ſphæram:</s> <s xml:id="echoid-s49886" xml:space="preserve"> & aſpiciatur medium ſphæræ:</s> <s xml:id="echoid-s49887" xml:space="preserve"> & ſit ce-<lb/>ra oppoſita medio ſphæræ in linea recta:</s> <s xml:id="echoid-s49888" xml:space="preserve"> uide biturq́;</s> <s xml:id="echoid-s49889" xml:space="preserve"> in ſuperficie ſphæræ nigredo rotunda in fi-<lb/>gura armillæ.</s> <s xml:id="echoid-s49890" xml:space="preserve"> Quòd ſi non uideatur talis figura:</s> <s xml:id="echoid-s49891" xml:space="preserve"> moueatur cera antè & retro, donec uideatur talis <lb/>rotunditas:</s> <s xml:id="echoid-s49892" xml:space="preserve"> & tunc auferatur cera, & recedet nigredo:</s> <s xml:id="echoid-s49893" xml:space="preserve"> quòd ſi ceram reduxerit quis ad locum & <lb/>ſitum priorem, reuertetur ſtatim nigredo rotunda armillaris.</s> <s xml:id="echoid-s49894" xml:space="preserve"> Sed & in his multa eſt diuerſitas, <lb/>quam relin quimus ſtudio perquirentis.</s> <s xml:id="echoid-s49895" xml:space="preserve"/> </p> <div xml:id="echoid-div1866" type="float" level="0" n="0"> <figure xlink:label="fig-0743-01" xlink:href="fig-0743-01a"> <variables xml:id="echoid-variables852" xml:space="preserve">a b g p e d b m z o h f l c</variables> </figure> </div> </div> <div xml:id="echoid-div1868" type="section" level="0" n="0"> <head xml:id="echoid-head1372" xml:space="preserve" style="it">44. Reuiſatrans corpus diaphanum columnare denſius aere, it a quòd centrum uiſus, & cen <lb/>trum alicuius circuli corporis æquidiſtantis b aſibus columnæ, & res uiſa ſint in eadem linea re-<lb/>cta: imago reiuidebitur duplicata. Alhazen 50 n7.</head> <p> <s xml:id="echoid-s49896" xml:space="preserve">Sitin corpore columnari groſsioris diaphanitatis quàm ſit aer, circulus b g d z:</s> <s xml:id="echoid-s49897" xml:space="preserve"> & ſit centrum ui <lb/>fus a:</s> <s xml:id="echoid-s49898" xml:space="preserve"> & cætera, ut prius in præcedente:</s> <s xml:id="echoid-s49899" xml:space="preserve"> dico quòd forma lineæ k o uidebitur duplicata:</s> <s xml:id="echoid-s49900" xml:space="preserve"> quoniam <lb/>ipſa uidebitur apud arcum g p, & apud arcum ſibi æqualem & ſibi correſpondentem exarcu b d <lb/>in alia parte ſemicylindri.</s> <s xml:id="echoid-s49901" xml:space="preserve"> Sed hæc forma non erit circularis:</s> <s xml:id="echoid-s49902" xml:space="preserve"> quia figura a h p g cum fuerit circũ-<lb/>noluta circa a k lineam immotam atq;</s> <s xml:id="echoid-s49903" xml:space="preserve"> fixam, non tranſibit perillam lineã arcus g p per totã ſuperſi-<lb/>ciem columnarẽ:</s> <s xml:id="echoid-s49904" xml:space="preserve"> ſed reſringetur forma ex aliquibus portionibus colũnæ, & erit cõtinua in una par <lb/><gap/>e, & ſimiliter in alia, Ná ſuperficies, in qua ſunt pũcta l, k, tranſiẽs per axẽ colũnę, facit in ſuperficie <lb/>colũnę, quę eſt ex parte uiſus a, lineam rectã tranſeuntẽ per pũctũ b, & extẽſam in lõgitudine colũ-<lb/>næ:</s> <s xml:id="echoid-s49905" xml:space="preserve"> & non refringetur ſorma lineæ k o ex illa linea recta:</s> <s xml:id="echoid-s49906" xml:space="preserve"> nam linea k h erit perpendicularis ſuper il <lb/>lam lineam rectam.</s> <s xml:id="echoid-s49907" xml:space="preserve"> Non ergo erit forma rotũda corpore diaphano exiſtente colũnari:</s> <s xml:id="echoid-s49908" xml:space="preserve"> ſed erũt duæ <lb/>formę, quarũ altera refringetur ſuper alteram.</s> <s xml:id="echoid-s49909" xml:space="preserve"> Videbitur ergo linea k o habẽs imagines duas, qua-<lb/>rum utraq;</s> <s xml:id="echoid-s49910" xml:space="preserve"> eſt maior quàm linea k o:</s> <s xml:id="echoid-s49911" xml:space="preserve"> & erũt illæ duæ formæ eædem apud pũctum a, quod eſt cen-<lb/>trum uiſus:</s> <s xml:id="echoid-s49912" xml:space="preserve"> quoniam in illo pũcto a eſt locus ambarum illarum imaginum, ut patet per 15 th.</s> <s xml:id="echoid-s49913" xml:space="preserve"> huius.</s> <s xml:id="echoid-s49914" xml:space="preserve"> <lb/>Patet ergo propoſitũ.</s> <s xml:id="echoid-s49915" xml:space="preserve"> Non poteſt autem fieri huiuſmodi refractio à ſuperficie corporum pyrami-<lb/> <pb o="442" file="0744" n="744" rhead="VITELLONIS OPTICAE"/> dalium:</s> <s xml:id="echoid-s49916" xml:space="preserve"> quoniam linea k a non eſt perpendiculariter erecta ſuper ſuperficiem conicam talium cor-<lb/>porum:</s> <s xml:id="echoid-s49917" xml:space="preserve"> neque poteſt eſſe, ut ſuperficies refractionis ſecet huiuſmodi corpora ſecundum circulum, <lb/>quemadmodum etiam de ſuperficiebus reflexionũ & de ſpeculis pyramidalibus conuexis & con-<lb/>cauis oſtenſum eſt in præmiſsis libris.</s> <s xml:id="echoid-s49918" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1869" type="section" level="0" n="0"> <head xml:id="echoid-head1373" xml:space="preserve" style="it">45. Centro uiſus exiſtente in diametro corporis diaphani ſphærici concaui denſior is aere, & <lb/>reuiſa reſpiciente conuexum illius corporis: imago uidebitur quando minor re uiſa: quando <lb/>maior, ut cum fit figuræ armillaris.</head> <p> <s xml:id="echoid-s49919" xml:space="preserve">Sit centrum uiſus a:</s> <s xml:id="echoid-s49920" xml:space="preserve"> lineaq́;</s> <s xml:id="echoid-s49921" xml:space="preserve"> uiſa ſit b c:</s> <s xml:id="echoid-s49922" xml:space="preserve"> & ſit corpus ſphæricum concauũ denſioris diaphanitatis, <lb/>quàm ſit aer, cuius centrum ſit d:</s> <s xml:id="echoid-s49923" xml:space="preserve"> & diameter e d f:</s> <s xml:id="echoid-s49924" xml:space="preserve"> ſitq́ linea b c extra conuexum illius corporis:</s> <s xml:id="echoid-s49925" xml:space="preserve"> & <lb/>centrũ uiſus a ſit in diametro illius intra corpus cõcauũ:</s> <s xml:id="echoid-s49926" xml:space="preserve">dico quòd ſemper imago rei uiſæ lineæ b c <lb/>erit minor ipſa re uiſa.</s> <s xml:id="echoid-s49927" xml:space="preserve"> Si enim centrũ uiſus a fuerit in centro corpo <lb/>ris puncto d:</s> <s xml:id="echoid-s49928" xml:space="preserve">palàm per 72th.</s> <s xml:id="echoid-s49929" xml:space="preserve">1 huius quoniam omnes lineæ exten-<lb/> <anchor type="figure" xlink:label="fig-0744-01a" xlink:href="fig-0744-01"/> ſionis formarũ pũctorum lineæ b c ad uiſum a, erũt perpendicula-<lb/>res ſuper ſuperficiem corporis:</s> <s xml:id="echoid-s49930" xml:space="preserve"> quoniam tranſeunt centrũ eius:</s> <s xml:id="echoid-s49931" xml:space="preserve"> lo-<lb/>cus ergo imaginis per 15 huius erit ipſe arcus refractionis, uidebi-<lb/>turq́;</s> <s xml:id="echoid-s49932" xml:space="preserve"> imago curua minor re uiſa.</s> <s xml:id="echoid-s49933" xml:space="preserve"> Quòd ſi a centrum uiſus fuerit in <lb/>aliquo punctorũ ſemidiametri e d propinquioris rei uiſę, uel in ali-<lb/>quo punctorum ſemidiametri d ſ remotioris:</s> <s xml:id="echoid-s49934" xml:space="preserve"> adhuc ſemper lineæ <lb/>extenſionis ſormarum ad uiſum ſecabũt perpendiculares ductas à <lb/>pũctis rei uiſæ ſuper ſuperficiem corporis diaphani, à qua fit refra-<lb/>ctio, in ipſis pũctis refractionũ:</s> <s xml:id="echoid-s49935" xml:space="preserve"> hoc eſt in pũctis arcus, à quo fit re-<lb/>fractio, uel circa illa pũcta intra corpus diaphanum uel extra illud.</s> <s xml:id="echoid-s49936" xml:space="preserve"> <lb/>Videbitur ergo imago quãdoq;</s> <s xml:id="echoid-s49937" xml:space="preserve"> curua:</s> <s xml:id="echoid-s49938" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s49939" xml:space="preserve"> recta:</s> <s xml:id="echoid-s49940" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s49941" xml:space="preserve"> <lb/>irregularis:</s> <s xml:id="echoid-s49942" xml:space="preserve"> ſed ſemper minor re uiſa:</s> <s xml:id="echoid-s49943" xml:space="preserve"> quoniam, ut patet, chorda uel <lb/>alia diameter imaginis eſt minor re uiſa:</s> <s xml:id="echoid-s49944" xml:space="preserve"> & omnis linea cadens in-<lb/>ter centrũ uiſus pũctum a & inter lineam b c, eſt minor quàm linea <lb/>b c, cũ ceciderit inter lineas a b & a c:</s> <s xml:id="echoid-s49945" xml:space="preserve"> ot hæc patere poſſunt per 29 <lb/>p 1, uel per 4 p 6.</s> <s xml:id="echoid-s49946" xml:space="preserve"> Eſt itaq;</s> <s xml:id="echoid-s49947" xml:space="preserve"> in tali diſpoſitione ſemper imago minor <lb/>ipſa re uiſa:</s> <s xml:id="echoid-s49948" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s49949" xml:space="preserve"> eius imago quandoq;</s> <s xml:id="echoid-s49950" xml:space="preserve"> maior, ut cum fit figuræ ar-<lb/>millaris.</s> <s xml:id="echoid-s49951" xml:space="preserve"> Si enim linea b c ſituetur in diametro f d e:</s> <s xml:id="echoid-s49952" xml:space="preserve"> tũc formarum <lb/>punctorũ b & c fiet refractio ab aliquibus duobus pũctis unius arcus circuli corporis, & pũctorum <lb/>mediorum lineæ b c fiet refractio à pũctis medijs illius arcus.</s> <s xml:id="echoid-s49953" xml:space="preserve"> Et ſi linea a b c remanente fixa, imagi-<lb/>netur illa figura circũuolui, quouſq;</s> <s xml:id="echoid-s49954" xml:space="preserve"> redeat ad locũ, unde motus accepit principiũ:</s> <s xml:id="echoid-s49955" xml:space="preserve"> deſcri betur per <lb/>arcũ refractionis quędam ſuperficies armillaris in tota ſphęrica ſuperficie corporis, à qua totali fier <lb/>refractio ad uiſum:</s> <s xml:id="echoid-s49956" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s49957" xml:space="preserve"> locus imaginis in centro uiſus:</s> <s xml:id="echoid-s49958" xml:space="preserve"> qui applicans formã uiſam ipſi ſuperficiei <lb/>refractionis, rem iudicat figurę armillaris:</s> <s xml:id="echoid-s49959" xml:space="preserve"> ut hæc amplius omnia declarauimus in 43 huius.</s> <s xml:id="echoid-s49960" xml:space="preserve"> Patet <lb/>ergo propoſitum.</s> <s xml:id="echoid-s49961" xml:space="preserve"> Sed in uiſibilibus nobis aſſuetis nihil comprehẽditur à uiſu ultra corpus diapha-<lb/>num ſphæricum denſius aere, cuius cõcauitas ſit ex parte uiſus, niſi fortè tale corpus fiat artificiali-<lb/>ter ex uitro, uel cryſtallo, uel glacie, aut aliquo illis ſimili:</s> <s xml:id="echoid-s49962" xml:space="preserve"> refractio tamen, quæ fit ad uiſum à ſuperfi <lb/>cie concaua cœli ſimilis eſt iſti, niſi quòd ſecundum illam non fit refractio niſi formarum ſphærica-<lb/>rum, quarum naturam & modum inferius duximus perſequendum.</s> <s xml:id="echoid-s49963" xml:space="preserve"/> </p> <div xml:id="echoid-div1869" type="float" level="0" n="0"> <figure xlink:label="fig-0744-01" xlink:href="fig-0744-01a"> <variables xml:id="echoid-variables853" xml:space="preserve">b c e a a d a f</variables> </figure> </div> </div> <div xml:id="echoid-div1871" type="section" level="0" n="0"> <head xml:id="echoid-head1374" xml:space="preserve" style="it">46. Imago formæ cuiuslibet rei uiſæ figuratur diuer ſimodè ſecundum figuram ſuperficiei cor <lb/>poris, à qua fit refractio ad uiſum. Alhazen 35 n 7.</head> <p> <s xml:id="echoid-s49964" xml:space="preserve">Quoniam enim locus imaginis refractæ eſt ſemperin communi ſectione catheti incidẽtiæ, quæ <lb/>eſt perpendiculariter à puncto rei uiſæ producta ſuper ſuperficiem corporis diaphani, in quo eſt <lb/>res uiſa, & lineę, per quam ſorma peruenit ad uiſum, ut patet per 15 th.</s> <s xml:id="echoid-s49965" xml:space="preserve"> huius.</s> <s xml:id="echoid-s49966" xml:space="preserve"> Si ergo imaginati fue-<lb/>rimus quòd ab unoquoq;</s> <s xml:id="echoid-s49967" xml:space="preserve"> puncto rei uiſæ exeat cathetus incidentiæ, quę eſt perpẽdicularis ſuper <lb/>ſuperficiem corporis, in quo eſt res uiſa:</s> <s xml:id="echoid-s49968" xml:space="preserve"> tũc habebimus quãdã figurã columnarẽ uel corporalem, <lb/>exeuntẽ à ſuperficie totius uiſi corporis ad ſuperficiẽ corporis diaphani:</s> <s xml:id="echoid-s49969" xml:space="preserve"> & hęc figura ſecat pyrami <lb/>dẽ radialẽ, ſecundũ quã fit uiſio refracta, cuius uertex eſt in cẽtro uiſus:</s> <s xml:id="echoid-s49970" xml:space="preserve"> & iſtarũ duarũ figurarũ cor <lb/>poralium, columnaris ſcilicet & pyramidalis communis ſectio eſt locus imaginis formæ rei uiſæ.</s> <s xml:id="echoid-s49971" xml:space="preserve"> <lb/>Si itaq;</s> <s xml:id="echoid-s49972" xml:space="preserve"> ſuperficies corporis, à qua fit refractio formæ rei uiſæ, fuerit plana:</s> <s xml:id="echoid-s49973" xml:space="preserve"> tunc corpus imagina-<lb/>tum continens omnes perpendiculares erit ſimiliter planæ ſuperficiei:</s> <s xml:id="echoid-s49974" xml:space="preserve"> quare illa imago erit æqua-<lb/>lis, uel modicò maior quàm ſit forma rei uiſæ:</s> <s xml:id="echoid-s49975" xml:space="preserve"> uidebitur tamen ſemper multò maior re uiſa.</s> <s xml:id="echoid-s49976" xml:space="preserve"> Quòd <lb/>fi corpus, à quo fit refractio, fuerit ſphæricum, & conuexum eius ſit ex parte uiſus, fueritq́;</s> <s xml:id="echoid-s49977" xml:space="preserve"> res uiſa <lb/>in centro ipſius corporis diaphani, uel inter illud centrum & uiſum:</s> <s xml:id="echoid-s49978" xml:space="preserve"> tunc imago rei uiſæ erit figu-<lb/>ræ pyramidalis, quoniam omnes perpendiculares, quæ ſunt catheti incidentię, concurrunt in cen-<lb/>tro corporis diaphani per 72 th.</s> <s xml:id="echoid-s49979" xml:space="preserve">1 huius:</s> <s xml:id="echoid-s49980" xml:space="preserve"> & hęc imago quantò magis extenditur uerſus ſuperficiem <lb/>conuexam corporis diaphani, tantò magis amplificatur:</s> <s xml:id="echoid-s49981" xml:space="preserve"> & ubicunq;</s> <s xml:id="echoid-s49982" xml:space="preserve"> locus imaginis fuerit inter <lb/>rem uiſam & ſuperficiem corporis ſphæricam:</s> <s xml:id="echoid-s49983" xml:space="preserve"> ſemper imago erit amplior re uiſa.</s> <s xml:id="echoid-s49984" xml:space="preserve"> Si autem lo-<lb/>cus imaginis fuerit ultra rem uiſam :</s> <s xml:id="echoid-s49985" xml:space="preserve"> tunc imago erit ſtrictior re uiſa.</s> <s xml:id="echoid-s49986" xml:space="preserve"> Si uerſo4;</s> <s xml:id="echoid-s49987" xml:space="preserve"> res uiſa fuerit ultra <lb/> <pb o="443" file="0745" n="745" rhead="LIBER DECIMVS."/> ſuperficiem ſphæricam corporis diaphani uel ultra centrum eius:</s> <s xml:id="echoid-s49988" xml:space="preserve"> tunc, (cum omnes catheti inci-<lb/>dentię ſecẽt ſe in centro corporis) erit corpus imaginatũ duæ pyramides oppoſitæ, quarũ uertices <lb/>coniunguntur in centro corporis diaphani:</s> <s xml:id="echoid-s49989" xml:space="preserve"> & loca imaginum tunc poſſunt eſſe diuerſa:</s> <s xml:id="echoid-s49990" xml:space="preserve"> & fortè ac <lb/>cidet quandoq;</s> <s xml:id="echoid-s49991" xml:space="preserve"> imaginem uideri maiorem re uiſa:</s> <s xml:id="echoid-s49992" xml:space="preserve"> quando q;</s> <s xml:id="echoid-s49993" xml:space="preserve"> æqualem:</s> <s xml:id="echoid-s49994" xml:space="preserve"> & quandoq;</s> <s xml:id="echoid-s49995" xml:space="preserve"> minorẽ.</s> <s xml:id="echoid-s49996" xml:space="preserve"> Quòd <lb/>ſi corporis diaphani ſphærici concauitas ſuerit à parte uiſus, & conuexitas ex parte rei uiſæ:</s> <s xml:id="echoid-s49997" xml:space="preserve"> tunc <lb/>per eandem rationem, per quam prius, corpus imaginatum erit pyramis, cuius uertex erit in cẽtro <lb/>corporis diaphani.</s> <s xml:id="echoid-s49998" xml:space="preserve"> Quantò ergo magis hoc corpus imaginatum extendιtur uerſus centrum corpo <lb/>ris diaphani, tantò magis conſtringitur:</s> <s xml:id="echoid-s49999" xml:space="preserve"> & quantò magis extenditur ad partem illam, tantò magis <lb/>dilatatur & amplificatur ſuperficies:</s> <s xml:id="echoid-s50000" xml:space="preserve"> unde ſecundum hoc locis imaginum diuerſiſicatis, diuerſifica <lb/>tur & quantitas imaginum formarum.</s> <s xml:id="echoid-s50001" xml:space="preserve"> Quia ſi locus imaginis fuerit propinquior centro corporis <lb/>diaphani concaui, quàm ipſa res uiſa:</s> <s xml:id="echoid-s50002" xml:space="preserve"> erit imago minor ipſa re uiſa:</s> <s xml:id="echoid-s50003" xml:space="preserve"> & ſi ſuerit locus imaginis remo-<lb/>tior à centro corporis quàm res uiſa:</s> <s xml:id="echoid-s50004" xml:space="preserve"> erit imago maior ipſa re uiſa.</s> <s xml:id="echoid-s50005" xml:space="preserve"> Et quomodo hoc exemplificaui-<lb/>mus in corporibus diaphanis ſphæricis conuexis & concauis:</s> <s xml:id="echoid-s50006" xml:space="preserve"> eodem modo in corporibus colu-<lb/>mnaribus & pyramidalibus conuexis & concauis poteſt intelligi.</s> <s xml:id="echoid-s50007" xml:space="preserve"> Vniuerſaliter autẽ quando locus <lb/>imaginis eſt ſuperficies corporis diaphani, à qua fit refractio:</s> <s xml:id="echoid-s50008" xml:space="preserve"> tunc ſemper imago induit figuram ſu <lb/>perficiei, à qua fit refractio.</s> <s xml:id="echoid-s50009" xml:space="preserve"> Vnde in conuexis ſuperficiebus fit conuexa:</s> <s xml:id="echoid-s50010" xml:space="preserve"> in cõcauis concaua:</s> <s xml:id="echoid-s50011" xml:space="preserve"> in co-<lb/>lumnaribus corporibus fit oblonga columnaris:</s> <s xml:id="echoid-s50012" xml:space="preserve"> & in pyramidalibus corporibus pyramidalis.</s> <s xml:id="echoid-s50013" xml:space="preserve"> Di-<lb/>uerſificantur etiam figuræ imaginum in eodem diaphano ſecũdum diuerſum ſitum eiuſdem rei ui-<lb/>ſæ reſpectu uiſus.</s> <s xml:id="echoid-s50014" xml:space="preserve"> Vnde forma eiuſdem rei, ut pedis uel manus, quandoq;</s> <s xml:id="echoid-s50015" xml:space="preserve"> uidetur ſtricta & curta:</s> <s xml:id="echoid-s50016" xml:space="preserve"> <lb/>quandoq;</s> <s xml:id="echoid-s50017" xml:space="preserve"> arcta & longa, ſecũdum quod perpendiculares à punctis illius rei ad ſuperficiem corpo-<lb/>ris diaphani productæ illi ſuperficiei incidunt diuerſimodè:</s> <s xml:id="echoid-s50018" xml:space="preserve"> ſic enim uariè à lineis extenſionis for-<lb/>marum interſecantur:</s> <s xml:id="echoid-s50019" xml:space="preserve"> & uariatur multiformiter imago, ut patet per 15 & 16 huius.</s> <s xml:id="echoid-s50020" xml:space="preserve"> Horum quoq;</s> <s xml:id="echoid-s50021" xml:space="preserve"> o-<lb/>mnium cauſſa ſufficienter patet ex præmiſsis.</s> <s xml:id="echoid-s50022" xml:space="preserve"> Palàm ergo eſt id, quod proponebatur.</s> <s xml:id="echoid-s50023" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1872" type="section" level="0" n="0"> <head xml:id="echoid-head1375" xml:space="preserve" style="it">47. Vna imago refr acta occurrit eiuſdem uidentis uiſibus ambobus. Alhazen 36 n 7.</head> <p> <s xml:id="echoid-s50024" xml:space="preserve">Quoniam enim forma eiuſdẽ rei uiſæ refracta ab aliqua ſuperficie corporis diaphani, in quo eſt <lb/>illa res, ſe offert ambobus uiſibus eiuſdem uidentis:</s> <s xml:id="echoid-s50025" xml:space="preserve"> tunc in ipſius uiſione non fit quantùm ad actũ <lb/>uidendi, differentia à ſimplici uiſione, quam pertractauimus in tertio & quarto libro huius ſciẽtiæ:</s> <s xml:id="echoid-s50026" xml:space="preserve"> <lb/>ubi diximus quòd res ſecundum pyramidem uidetur, cuius uertex eſt in centro uiſus, & baſis in ſu <lb/>perficie rei uiſæ:</s> <s xml:id="echoid-s50027" xml:space="preserve"> & oſtendimus quòd tunc ab ambobus uiſibus uidetur una forma:</s> <s xml:id="echoid-s50028" xml:space="preserve"> unde illud hic <lb/>fupponimus in ſormis refractis, ut in formis directè uiſis.</s> <s xml:id="echoid-s50029" xml:space="preserve"> Si enim homo comprehenderit aliquod <lb/>uiſibile in cœlo aut in aqua, aut ſub uitro uel cryſtallo ambobus uiſibus, & claudat unũ uiſuũ:</s> <s xml:id="echoid-s50030" xml:space="preserve"> nihi-<lb/>lominus comprehendet illud uiſibile.</s> <s xml:id="echoid-s50031" xml:space="preserve"> Ambobus ergo uiſibus & uno tantũ uiſu cõprehenditur ea-<lb/>dem forma.</s> <s xml:id="echoid-s50032" xml:space="preserve"> Et hoc eſt propoſitũ:</s> <s xml:id="echoid-s50033" xml:space="preserve"> non enim uidimus in talibus aliquid ulteriori mora dignum.</s> <s xml:id="echoid-s50034" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1873" type="section" level="0" n="0"> <head xml:id="echoid-head1376" xml:space="preserve" style="it">48. Cryſtallo ſphærica ſoli oppoſita ignem poſsibile eſt accendi <lb/>in re combuſtibili, quæ est post illam.</head> <figure> <variables xml:id="echoid-variables854" xml:space="preserve">a d c g b e f</variables> </figure> <p> <s xml:id="echoid-s50035" xml:space="preserve">Sit centrum ſolis punctum a:</s> <s xml:id="echoid-s50036" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s50037" xml:space="preserve"> cryſtallus ſibi oppoſita, cu-<lb/>ius centrum b:</s> <s xml:id="echoid-s50038" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s50039" xml:space="preserve">, ut ſuperficies plana centra amborum, quæ <lb/>ſunt a & b, pertranſiens ſecet ipſam cryſtallum ſphæricam ſecun-<lb/>dum circulum per 69 th.</s> <s xml:id="echoid-s50040" xml:space="preserve">1 huius:</s> <s xml:id="echoid-s50041" xml:space="preserve"> qui ſit c d e f g.</s> <s xml:id="echoid-s50042" xml:space="preserve"> Dico quòd ſi ali-<lb/>quod combuſtibile ponatur poſt hanc cryſtallum:</s> <s xml:id="echoid-s50043" xml:space="preserve"> ita quòd cry-<lb/>ſtallus ſit media inter ſolem & rem combuſtibilem, ut ſtupam uel <lb/>aliquid conſimile:</s> <s xml:id="echoid-s50044" xml:space="preserve"> poſsibile eſt, ut ignis in illo corpore accen-<lb/>datur.</s> <s xml:id="echoid-s50045" xml:space="preserve"> Imaginetur enim à centro ſolis a uſque ad centrum cryſtal <lb/>li, quod eſt b, diffundi radium, qui ſit a b.</s> <s xml:id="echoid-s50046" xml:space="preserve"> Cum itaque radius iſte ſit <lb/>perpendicularis ſuper corpus ſolis & ſuper corpus cryſtalli per 72 <lb/>th.</s> <s xml:id="echoid-s50047" xml:space="preserve"> 1 huius, quoniam tranſit per amborum centra:</s> <s xml:id="echoid-s50048" xml:space="preserve"> palàm per 47 <lb/>th.</s> <s xml:id="echoid-s50049" xml:space="preserve"> 2 huius, quia non refringitur, ſed tranſit corpus cryſtalli irrefra-<lb/>ctus:</s> <s xml:id="echoid-s50050" xml:space="preserve"> omnesq́;</s> <s xml:id="echoid-s50051" xml:space="preserve"> radij ſolis ſuperſiciei ſphæricæ cryſtalli ęquidiſtan-<lb/>ter radio a b incidentes, palàm quoniam incidunt obliquè:</s> <s xml:id="echoid-s50052" xml:space="preserve"> ergo <lb/>per 47 th.</s> <s xml:id="echoid-s50053" xml:space="preserve"> 2 huius patet quoniam omnes illi radij refringuntur ad <lb/>perpendicularem a b:</s> <s xml:id="echoid-s50054" xml:space="preserve"> quoniam quilibet illorum radiorum refrin <lb/>gitur ad perpendicularem à puncto refractionis ſuper ſuperficiem <lb/>cryſtalli:</s> <s xml:id="echoid-s50055" xml:space="preserve"> quæ perpendiculares omnes concurrunt cum diametro <lb/>a b in centro ſphæræ cryſtalli:</s> <s xml:id="echoid-s50056" xml:space="preserve"> fit autem ad illas perpendiculares <lb/>refractio:</s> <s xml:id="echoid-s50057" xml:space="preserve"> ideo quòd corpus cryſtalli denſius eſt corpore aeris, per <lb/>quod tranſeunt radij inter corpus ſolis & corpus cryſtalli inciden-<lb/>tes.</s> <s xml:id="echoid-s50058" xml:space="preserve">Et quoniam in diſtantia æquali à radio a b, alij radij à corpo-<lb/>re ſolis procedentes, corpori cryſtalli incidunt ſecundum angulos <lb/>æquales per 43 th.</s> <s xml:id="echoid-s50059" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s50060" xml:space="preserve"> palàm per 8 huius quoniam ſecun-<lb/>dum æquales angulos refringuntur.</s> <s xml:id="echoid-s50061" xml:space="preserve"> Imaginetur itaque radius a b <lb/>produci ultra corpus cryſtalli:</s> <s xml:id="echoid-s50062" xml:space="preserve"> & patet quoniam à quolibet circu-<lb/>lo cryſtalli totius ſuperficiei ſolis oppoſitæ refringuntur radij ad unum pu n ctum perpendicula-<lb/> <pb o="444" file="0746" n="746" rhead="VITELLONIS OPTICAE"/> ris a b, ſicut & omnes perpendiculares concurrunt in centro b.</s> <s xml:id="echoid-s50063" xml:space="preserve"> In aliquo itaque illorum puncto-<lb/>rum perpendicularis a b retro corpus cryſtalli poſito combuſtibili, ignis accendetur in illo, ſi mo-<lb/>ram duxerit.</s> <s xml:id="echoid-s50064" xml:space="preserve"> Omnes enim anguli refractionis ex aere ad ſuperficiem ſuperiorem cryſtalli unius <lb/>circuli (cuius polus eſt punctus, ſecundum quem linea a b ſecat ſuperficiem cryſtalli) ſunt æ-<lb/>quales:</s> <s xml:id="echoid-s50065" xml:space="preserve"> & eorum radiorum anguli refractionis à ſuperficie.</s> <s xml:id="echoid-s50066" xml:space="preserve"> Et quo-<lb/>niam quilibet illorum radiorum refringitur à linea perpendiculari à puncto ſuæ refractionis ſu-<lb/>per ſuperficiem cryſtalli producta:</s> <s xml:id="echoid-s50067" xml:space="preserve"> patet quòd omnes illi radij æqualiter refracti, concurruntin <lb/>uno pũcto lineæ a b productæ ultra ſuperficiem cryſtalli.</s> <s xml:id="echoid-s50068" xml:space="preserve">Et quia illa pũcta naturalia latitudinẽ ha-<lb/>bent:</s> <s xml:id="echoid-s50069" xml:space="preserve"> patet quòd in ipſis radij plurimi concurrũt:</s> <s xml:id="echoid-s50070" xml:space="preserve"> poſſunt ergo rem combuſtibilem ibi poſitam in-<lb/>flam mare:</s> <s xml:id="echoid-s50071" xml:space="preserve"> quod eſt propoſitũ.</s> <s xml:id="echoid-s50072" xml:space="preserve"> Fortè tamen portio ſphæræ cryſtallinæ minor hemiſphærio fortius <lb/>inflammaret in loco centri ſui poſita re inflammabili:</s> <s xml:id="echoid-s50073" xml:space="preserve"> quoniã omnes radij totali illi ſuper ficiei ſphæ <lb/>ricæ perpendiculariter incidentes concurrerent in centro per 72 th.</s> <s xml:id="echoid-s50074" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s50075" xml:space="preserve"> Sed & in horum expe-<lb/>rimentatione eſt maxima latitudo, quam relinquimus ad talia curioſis.</s> <s xml:id="echoid-s50076" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1874" type="section" level="0" n="0"> <head xml:id="echoid-head1377" xml:space="preserve" style="it">49. Stellas cæli & lunam ſecundum refr actionem à uiſibus comprehendi inſtrument ali-<lb/>ter declaratur. Alhazen 15 n 7.</head> <p> <s xml:id="echoid-s50077" xml:space="preserve">Inſtrumentum armillarum ponatur in loco eminente:</s> <s xml:id="echoid-s50078" xml:space="preserve"> unde appareat horizontis pars orientalis, <lb/>ita quòd armilla, quę eſt in loco circuli meridiei, ſit poſita in ſuperficie circuli meridiei:</s> <s xml:id="echoid-s50079" xml:space="preserve"> & polus e-<lb/>ius ſit exaltatus à ſuperficie terræ ſecundum eleuationem poli mundi ſuper illius habitabilis hori-<lb/>zonta:</s> <s xml:id="echoid-s50080" xml:space="preserve"> & in nocte obſeruetur aliqua ſtellarum fixarum magnarum, quæ cum peruenit ad circulum <lb/>meridianum, ſit tranſiens per centrum capitis experimentantis aut prope:</s> <s xml:id="echoid-s50081" xml:space="preserve"> & cõſideretur illa in or-<lb/>tu ſuo, dum eleuatur ſuper ſuperficiem horizontis:</s> <s xml:id="echoid-s50082" xml:space="preserve"> & tunc reuoluatur armilla reuolubilis in circui-<lb/>tu poli mundi, qui eſt polus æquinoctialis, donec ſiat æquidiſtans circulo magno cœli tranſeunti <lb/>per polos æquinoctialis, & per centrum corporis illius ſtellæ:</s> <s xml:id="echoid-s50083" xml:space="preserve"> & certificetur locus ſtellæ ex armilla, <lb/>ita ut habeatur diſtantia ſtellæ à polo mundi.</s> <s xml:id="echoid-s50084" xml:space="preserve"> Deinde obſeruetur ſtella, donec ueniat ad circulum <lb/>meridiei:</s> <s xml:id="echoid-s50085" xml:space="preserve"> moueaturq́;</s> <s xml:id="echoid-s50086" xml:space="preserve"> armilla mobilis, donec fiat æquidiſtans circulo ſtellæ, ut prius:</s> <s xml:id="echoid-s50087" xml:space="preserve"> & ſit in ſu-<lb/>perficie circuli meridiani:</s> <s xml:id="echoid-s50088" xml:space="preserve"> & tunc iterum habebitur diſtantia ſtellæ à polo mundi, cum ſtella ſuerit <lb/>in zenith capitis aut prope:</s> <s xml:id="echoid-s50089" xml:space="preserve"> inuenieturq́;</s> <s xml:id="echoid-s50090" xml:space="preserve"> diſtantia ſtellæ à polo mundi in tempore ortus & eleua-<lb/>tionis ſtellæ minor ipſius diſtantia ab eodem polo, tempore, quo eſt in zenith capitis uel prope.</s> <s xml:id="echoid-s50091" xml:space="preserve"> Pa <lb/>tet itaque ex iſtis quia uiſus comprehendit formas ſtellarum orientium reſractè, & non rectè:</s> <s xml:id="echoid-s50092" xml:space="preserve"> quo-<lb/>niam quælibet ſtellarum fixarum ſemper mouetur per eundem circulum ex circulis æquidiſtanti-<lb/>bus æquinoctiali, niſi fortè ſecundum motum latitudinis uarietur parum in tempore lõgo:</s> <s xml:id="echoid-s50093" xml:space="preserve"> de quo <lb/>alibi plenius dicemus.</s> <s xml:id="echoid-s50094" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s50095" xml:space="preserve"> uiſus comprehenderet ſtellas rectè, non refractè:</s> <s xml:id="echoid-s50096" xml:space="preserve">tunc uiſus compre <lb/>henderet quamlibet ſtellarum in ſuo loco:</s> <s xml:id="echoid-s50097" xml:space="preserve"> & eſſet omni hora noctis eiuſdem ſtellæ à polo mundi <lb/>eadem diſtantia in uiſu:</s> <s xml:id="echoid-s50098" xml:space="preserve"> cuius contrarium accidit uiſui per inſtrumentum.</s> <s xml:id="echoid-s50099" xml:space="preserve"> Similiter quoque acci-<lb/>dit in luna.</s> <s xml:id="echoid-s50100" xml:space="preserve"> Si enim aliquis per tabulas æquauerit locum lunæ in aliqua hora prope ortum eius:</s> <s xml:id="echoid-s50101" xml:space="preserve"> & <lb/>habeat latitudinem eius & diſtantiam à polo mundi notam:</s> <s xml:id="echoid-s50102" xml:space="preserve"> & item æquet ipſam pro tempore me-<lb/>diæ noctis:</s> <s xml:id="echoid-s50103" xml:space="preserve"> & ſciat latitudinem eius & diftantiam à polo mundi.</s> <s xml:id="echoid-s50104" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s50105" xml:space="preserve"> inueniatur locus lunæ per <lb/>armillas tempore ortus ſui:</s> <s xml:id="echoid-s50106" xml:space="preserve"> non accidet diuerſitas inter computationem per tabulas & experimen <lb/>tationem per inſtrumentum.</s> <s xml:id="echoid-s50107" xml:space="preserve"> Inuento uerò loco lunæ per armillas, dum eſt in meridiano circulo:</s> <s xml:id="echoid-s50108" xml:space="preserve"> <lb/>erit diſtantia lunæ à zenith capitis inuenta per inſtrumentum, cum latitudo lunæ eſt meridiana, <lb/>maior, & cum eſt ſeptentrionalis, minor uera diſtantia eius à zenith capitis inuenta per computa-<lb/>tionem tabularum.</s> <s xml:id="echoid-s50109" xml:space="preserve"> Patet ergo quòd lux lunæ non peruenit ad uiſum rectè, ſed refringitur in ali-<lb/>quo medio corpore ſecundi diaphani:</s> <s xml:id="echoid-s50110" xml:space="preserve"> quia niſi refringeretur, eadem eius eſſet diſtantia à zenith ca <lb/>pitis per inſtrumentum & per tabularum computationem, ut accidit cum eſt in horizonte:</s> <s xml:id="echoid-s50111" xml:space="preserve"> nunc <lb/>autem differt.</s> <s xml:id="echoid-s50112" xml:space="preserve"> Palàm eſt ergo propoſitum, quòd omnes ſtellæ uidentur per refractionem.</s> <s xml:id="echoid-s50113" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1875" type="section" level="0" n="0"> <head xml:id="echoid-head1378" xml:space="preserve" style="it">50. Diaphanitas corporis cæleſtis rarior est aeris & ignis diaphanitate. Alhazen 16 n 7.</head> <p> <s xml:id="echoid-s50114" xml:space="preserve">Diſpoſito enim inſtrumento armillarum, ut ſuprà, inuenienda eſt diſtantia alicuius ſtellarum à <lb/>zenith capitis:</s> <s xml:id="echoid-s50115" xml:space="preserve"> & in loco experimentationis ſit circulus meridiei a b g:</s> <s xml:id="echoid-s50116" xml:space="preserve"> & ſit zenith capitis punctũ <lb/>b:</s> <s xml:id="echoid-s50117" xml:space="preserve"> & polus mundi ſit punctum d:</s> <s xml:id="echoid-s50118" xml:space="preserve"> centrum quoque mundi ſit punctus e:</s> <s xml:id="echoid-s50119" xml:space="preserve"> & ducatur ſemidiameter <lb/>meridiani circuli:</s> <s xml:id="echoid-s50120" xml:space="preserve"> quæ ſit e b, pertranſiens centrum uiſus experimentantis, qui ſit punctus z:</s> <s xml:id="echoid-s50121" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s50122" xml:space="preserve"> <lb/>circulus h t æquidiſtans circulo æquinoctiali & polo ipſius, qui eſt d:</s> <s xml:id="echoid-s50123" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s50124" xml:space="preserve"> polus illius circuli h t <lb/>punctus d per 68 th.</s> <s xml:id="echoid-s50125" xml:space="preserve"> 1 huius, propter æquidiſtantiam illorum circulorum:</s> <s xml:id="echoid-s50126" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s50127" xml:space="preserve"> circuli h t diſtantia à <lb/>puncto d polo mundi illa, in qua inuenitur ſtella in hora certificationis diſtantiæ primæ, quæ eſt in <lb/>ipſo puncto ſui ortus:</s> <s xml:id="echoid-s50128" xml:space="preserve"> & ſit locus ſtellæ in illa hora punctus h:</s> <s xml:id="echoid-s50129" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s50130" xml:space="preserve"> circulus alter, qui k b g, æquidi-<lb/>ſtans æquinoctiali circulo, & etiam circulo h t:</s> <s xml:id="echoid-s50131" xml:space="preserve"> cuius diftantia à polo mundi, qui eſt d, ſit illa, in qua <lb/>inuenitur ſtella in ſecunda hora conſiderationis, quæ ſit ſtella exiſtente iuxta zenith capitis in cir-<lb/>culo meridiano, qui eſt a b g:</s> <s xml:id="echoid-s50132" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s50133" xml:space="preserve"> circulus k b g æquidiſtans polo mundi, qui eſt d, & ualde pro-<lb/>pinquus ip ſi zenith capitis, aut tranſiens per punctum b, quod eſt zenith capitis.</s> <s xml:id="echoid-s50134" xml:space="preserve"> Ille ergo circulus <lb/>k b g eſt, in quo ceſſat obliquitas refractionis.</s> <s xml:id="echoid-s50135" xml:space="preserve"> Nam cum ſtella fuerit in zenith capitis in pũcto b, aut <lb/>ualde prope:</s> <s xml:id="echoid-s50136" xml:space="preserve"> tũc uiſus comprehendet eius formã rectè.</s> <s xml:id="echoid-s50137" xml:space="preserve"> Nã linea e z b à centro mũdi e per centrũ ui <lb/>ſus z ad zenith capitis b pertingẽs, eſt perpẽdicularis ſuper cõcauũ ſphęrę cœleſtis, & ſuper cõuexũ <lb/> <pb o="445" file="0747" n="747" rhead="LIBER DECIMVS."/> ſphæræ aeris per 72 th.</s> <s xml:id="echoid-s50138" xml:space="preserve">1 huius:</s> <s xml:id="echoid-s50139" xml:space="preserve"> quoniam tranſit per centrum utriuſq;</s> <s xml:id="echoid-s50140" xml:space="preserve"> illarum ſphærarũ.</s> <s xml:id="echoid-s50141" xml:space="preserve"> Viſus itaq;</s> <s xml:id="echoid-s50142" xml:space="preserve"> <lb/>propter perpendicularitatem lineæ z b ſuper ſphęras aeris & cœli, comprehendet ſtellam exiſten-<lb/>tem ſu per hanc lineam rectè, ſiue corpus cœli & aeris ſint eiuſdem diaphanitatis, ſiue diuerſæ:</s> <s xml:id="echoid-s50143" xml:space="preserve"> quo-<lb/>niam, ut ſuprà oſtenſum eſt per 3 th.</s> <s xml:id="echoid-s50144" xml:space="preserve"> huius, perpendicularis linea radialis non refringitur in medio <lb/>ſecundi diaphani.</s> <s xml:id="echoid-s50145" xml:space="preserve"> Forma itaq;</s> <s xml:id="echoid-s50146" xml:space="preserve"> ſtellę apparentis in <lb/> <anchor type="figure" xlink:label="fig-0747-01a" xlink:href="fig-0747-01"/> puncto b ſine omni refractione peruenit ad uiſum <lb/>per medium corpus cœleſte & ignis & aeris (quo-<lb/>rum in hoc loco acceptio eſt uniſormis, quanquã <lb/>ignis plus diaphanus eſt aere:</s> <s xml:id="echoid-s50147" xml:space="preserve"> & ex lucibus cœle-<lb/>ſtibus nihil ad nos peruenit uel ad noſtros uiſus, <lb/>niſi per medias ſphæras ignis & aeris, quæ quantũ <lb/>ad illud, ſunt ſphæra quaſi una:</s> <s xml:id="echoid-s50148" xml:space="preserve">) Stellam itaq;</s> <s xml:id="echoid-s50149" xml:space="preserve"> exi <lb/>ſtẽtem in zenith capitis aut prope illud, compre-<lb/>hendet uiſus in ſuo uero circulo æquidiſtante cir-<lb/>culo æquinoctiali, ſuper quem mouebatur ab ini-<lb/>tio noctis, quouſq;</s> <s xml:id="echoid-s50150" xml:space="preserve"> peruenit ad circulum meridia-<lb/>num.</s> <s xml:id="echoid-s50151" xml:space="preserve"> In circulo itaq;</s> <s xml:id="echoid-s50152" xml:space="preserve"> k b g fuit ſtella in prima expe <lb/>rimentatione ſecundum ueritatem.</s> <s xml:id="echoid-s50153" xml:space="preserve"> Sit autem cir-<lb/>culus altitudinis tranſiens per ſtellam in prima ho <lb/>ra experimentationis circulus b h k:</s> <s xml:id="echoid-s50154" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s50155" xml:space="preserve"> iſte circulus circulum k b g in ambobus punctis:</s> <s xml:id="echoid-s50156" xml:space="preserve"> ſcili-<lb/>cet in puncto k, qui eſt in parte orientis, & in puncto g illi directè oppoſito:</s> <s xml:id="echoid-s50157" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s50158" xml:space="preserve"> circulum h tin <lb/>puncto h, in quo corpus ſtellæ uidetur eſſe in tempore primę conſiderationis.</s> <s xml:id="echoid-s50159" xml:space="preserve"> Et quia diſtantia ſtel <lb/>læ ſecundum uiſum à polo mundi ſuit in prima experimentatione minor, quàm in ſecunda:</s> <s xml:id="echoid-s50160" xml:space="preserve"> patet <lb/>quòd circulus h t eſt propinquior polo d, quàm circulus k b g:</s> <s xml:id="echoid-s50161" xml:space="preserve"> pũctus itaq;</s> <s xml:id="echoid-s50162" xml:space="preserve"> h circuli altitudinis, qui <lb/>eſt b h k, propinquior eſt ipſi zenith capitis b quàm punctus k.</s> <s xml:id="echoid-s50163" xml:space="preserve"> Ducantur itaq;</s> <s xml:id="echoid-s50164" xml:space="preserve"> duæ lineæ h z & k z <lb/>ad centrũ uiſus z.</s> <s xml:id="echoid-s50165" xml:space="preserve"> Quia ergo ſtella comprehenditur à uiſu in prima hora experimentationis in pun <lb/>cto circuli h t:</s> <s xml:id="echoid-s50166" xml:space="preserve"> & tunc erat in ſuperficie circuli b h k:</s> <s xml:id="echoid-s50167" xml:space="preserve"> & tamen ſtella erat in illa hora ſecundum ueri-<lb/>tatem in circumferentia circuli k b g:</s> <s xml:id="echoid-s50168" xml:space="preserve"> oportet neceſſariò, uoſtella in illa hora fuerit ſecundum ueri-<lb/>tatem in puncto communi illis duobus circulis, qui ſunt k b g & b h k, qui eſt punctus k ſupra terrã:</s> <s xml:id="echoid-s50169" xml:space="preserve"> <lb/>comprehenditur autem à uiſu in puncto h per lineam z h:</s> <s xml:id="echoid-s50170" xml:space="preserve"> quia forma ſtellæ peruenit ad uiſum in <lb/>rectitudine lineæ h z:</s> <s xml:id="echoid-s50171" xml:space="preserve"> & linea, quę eſt inter ſtellam & uiſum ſecũdum ueritatem, eſt linea k z.</s> <s xml:id="echoid-s50172" xml:space="preserve"> Palàm <lb/>ergo quòd uiſus non comprehendit ſtellam, quę eſt in puncto k, rectè:</s> <s xml:id="echoid-s50173" xml:space="preserve"> comprehẽdit ergo ipſam re-<lb/>fractè.</s> <s xml:id="echoid-s50174" xml:space="preserve"> Et quia in corpore cœleſti propter homogeneitatem ſuæ diaphanitatis non poteſt fieri re-<lb/>fractio:</s> <s xml:id="echoid-s50175" xml:space="preserve"> fiet ergo illa in aliquo puncto corporis illi propinqui.</s> <s xml:id="echoid-s50176" xml:space="preserve"> Sit itaque locus refractionis factæ in <lb/>medio ſecundi diaphani (quod eſt aer uel ignis) punctus m:</s> <s xml:id="echoid-s50177" xml:space="preserve"> & ducatur linea k m:</s> <s xml:id="echoid-s50178" xml:space="preserve"> & protrahatur <lb/>à puncto m linea recta uſque ad punctum z centrum uiſus.</s> <s xml:id="echoid-s50179" xml:space="preserve"> Quia ergo forma ſtellæ extenditur à ſtel <lb/>la per lineam k m, & refringitur ad uiſum per lineam k m z:</s> <s xml:id="echoid-s50180" xml:space="preserve"> formæ uerò non refringuntur niſi occur <lb/>rerit corpus diuerſæ diaphanitatis, ut oſten dimus in ſecundo libro huius, & in præmiſsis huius li-<lb/>bri propoſitionibus.</s> <s xml:id="echoid-s50181" xml:space="preserve"> Ergo corpus cœleſte, in quo eſt ſtella, eſt differentis diaphanitatis ab aeris uel <lb/>ignis diaphanitate.</s> <s xml:id="echoid-s50182" xml:space="preserve"> Et quia locus refractionis eſt apud ſuperficiem tranſeuntem inter duo corpora <lb/>differentia in diaphanitate, ut patet per 4 huius:</s> <s xml:id="echoid-s50183" xml:space="preserve"> punctus itaque m eſt in cõcauitate cœli.</s> <s xml:id="echoid-s50184" xml:space="preserve"> Et ſi pro-<lb/>ducatur linea e m:</s> <s xml:id="echoid-s50185" xml:space="preserve"> hæc ſecundum ueritatem erit ſemidiameter ſphæræ cœli, cuius concauum attin <lb/>git conuexum ipſius ignis.</s> <s xml:id="echoid-s50186" xml:space="preserve"> Eſt ergo perpendicularis ſuper ſuperficiem cœli concauam, contingen-<lb/>tem aerem uel ignem, & ſuper ſuperficiem aeris uel ignis conuexam.</s> <s xml:id="echoid-s50187" xml:space="preserve"> Et quia forma ſtellę extẽſa in <lb/>corpore cœleſti per lineam k m, refringitur in aere ad uiſum per lineam m z:</s> <s xml:id="echoid-s50188" xml:space="preserve"> linea uerò k m protra-<lb/>cta ultra punctum m ſecaret lineam z m, elongans ſe à puncto e centro mundi:</s> <s xml:id="echoid-s50189" xml:space="preserve"> ideo quia obliquè <lb/>incidit concauæ ſuperficiei ipſius cœli:</s> <s xml:id="echoid-s50190" xml:space="preserve"> palàm quia illa refractio eſt ad partem, in qua eſt perpendi-<lb/>cularis e m, tranſiens per punctum refractionis perpendiculariter ſuper conuexam ſuperficiem ae-<lb/>ris.</s> <s xml:id="echoid-s50191" xml:space="preserve"> Et quoniam neque in cœlo, neque in aere eſt aliquod corpus denſum politum, à quo poſsit fie-<lb/>ri reflexio, ut à ſpeculo:</s> <s xml:id="echoid-s50192" xml:space="preserve"> patet quia illa diuerſitas accidit propter refractionem formæ in medio ſe-<lb/>cũdi diaphani.</s> <s xml:id="echoid-s50193" xml:space="preserve"> Corpus itaq;</s> <s xml:id="echoid-s50194" xml:space="preserve"> aeris eſt groſsius corpore cœli, ut patet ք 4 huius.</s> <s xml:id="echoid-s50195" xml:space="preserve"> Et hoc eſt ꝓpoſitũ.</s> <s xml:id="echoid-s50196" xml:space="preserve"/> </p> <div xml:id="echoid-div1875" type="float" level="0" n="0"> <figure xlink:label="fig-0747-01" xlink:href="fig-0747-01a"> <variables xml:id="echoid-variables855" xml:space="preserve">k h b t d m z e a g</variables> </figure> </div> </div> <div xml:id="echoid-div1877" type="section" level="0" n="0"> <head xml:id="echoid-head1379" xml:space="preserve" style="it">51. Diametri omnium ſtellarum & lineæ determinantes distantias quarumlibet duarum <lb/>ſtellarum in zenith capitis uel circa exiſtentium, minores comprehendũtur per refr actionem, <lb/>quàm ſi directè uiderentur. Alhazen 52 n 7.</head> <p> <s xml:id="echoid-s50197" xml:space="preserve">Sit circulus meridianus in aliquo horizonte b f k:</s> <s xml:id="echoid-s50198" xml:space="preserve"> & communis ſectio ſuperficiei huius circuli <lb/>& ſuperficiei conuexitatis ſphæræ cœli inſimi per 69 th.</s> <s xml:id="echoid-s50199" xml:space="preserve"> 1 huius ſit circulus m e z:</s> <s xml:id="echoid-s50200" xml:space="preserve"> erunt ergo iſti <lb/>duo circuli in eadem ſuperficie & concentrici.</s> <s xml:id="echoid-s50201" xml:space="preserve"> Sit ergo centrum ipſorum (quod eſt centrum mun-<lb/>di) punctum g:</s> <s xml:id="echoid-s50202" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s50203" xml:space="preserve"> centrum uiſus punctum t:</s> <s xml:id="echoid-s50204" xml:space="preserve"> & ducatur à centro mundi g ad centrum uiſus t li-<lb/>nea g t:</s> <s xml:id="echoid-s50205" xml:space="preserve"> & extrahatur linea g t in partem t e, donec occurrat circulo meridiei in puncto b:</s> <s xml:id="echoid-s50206" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s50207" xml:space="preserve"> cir <lb/>culum, qui eſt in ſuperficie cœli concaua, in puncto e:</s> <s xml:id="echoid-s50208" xml:space="preserve"> erit itaq;</s> <s xml:id="echoid-s50209" xml:space="preserve"> punctus b zenith capitis, quo ad ui-<lb/>ſum:</s> <s xml:id="echoid-s50210" xml:space="preserve"> ſit itaq;</s> <s xml:id="echoid-s50211" xml:space="preserve"> k l àrcus, cuius chorda kl ſit diameter alicuius ſtellæ aut diſtantia inter aliquas duas <lb/>ſtellas:</s> <s xml:id="echoid-s50212" xml:space="preserve"> & linea t b tranſeat per medium arcum k l ad punctum b:</s> <s xml:id="echoid-s50213" xml:space="preserve"> & ſecet chordam l k in puncto c:</s> <s xml:id="echoid-s50214" xml:space="preserve"> <lb/> <pb o="446" file="0748" n="748" rhead="VITELLONIS OPTICAE"/> arcus itaq;</s> <s xml:id="echoid-s50215" xml:space="preserve"> k b eſt æqualis arcui b l:</s> <s xml:id="echoid-s50216" xml:space="preserve"> & ducantur duæ t k & t l.</s> <s xml:id="echoid-s50217" xml:space="preserve"> Erit ergo angulus k t l quidam <lb/>angulus, ſecundum quem uiſust comprehendit arcum k l, quando ipſum rectè comprehendit.</s> <s xml:id="echoid-s50218" xml:space="preserve"> Sit <lb/>ita q;</s> <s xml:id="echoid-s50219" xml:space="preserve">, utforma puncti k refringatur ad uiſum t à puncto m circuli m e z, qui eſt ſignatus in concaua <lb/>ſuperficie ipſius cœli infimi, ut præaſſumptum eſt:</s> <s xml:id="echoid-s50220" xml:space="preserve"> & forma punctil refringatur ad uiſum t ex pun-<lb/>cto z, & ducantur lineæ g m & g z à centro mundi ad loca refractionum:</s> <s xml:id="echoid-s50221" xml:space="preserve"> ducantur quoq:</s> <s xml:id="echoid-s50222" xml:space="preserve"> lineæ k m:</s> <s xml:id="echoid-s50223" xml:space="preserve"> <lb/>m t, l z, z t.</s> <s xml:id="echoid-s50224" xml:space="preserve"> Formaitaq;</s> <s xml:id="echoid-s50225" xml:space="preserve"> puncti k extenditur per lineam k m, & refringitur ad uiſum t per linea m m t.</s> <s xml:id="echoid-s50226" xml:space="preserve"> <lb/>Et quoniam linea g m exit à centro ad circumferentiam:</s> <s xml:id="echoid-s50227" xml:space="preserve"> palàm per 72 th.</s> <s xml:id="echoid-s50228" xml:space="preserve"> 1 huius quòd ipſa eſt per-<lb/>pendicularis ſuper ſuperficiem ſphæræ cœli incidens puncto m, quod eſt punctum refractio nis.</s> <s xml:id="echoid-s50229" xml:space="preserve"> Et <lb/>eodem modo oſtendi poteſt, quòd g z eſt perpendicularis ſuper ſuperficiem cœli, incidens in pun-<lb/>cto z.</s> <s xml:id="echoid-s50230" xml:space="preserve"> Et quia per pręmiſſam corpus cœli, quod eſt z m, eſt rarioris diaphanitatis quàm corpus ae-<lb/>ris, in quo eſt uiſus t:</s> <s xml:id="echoid-s50231" xml:space="preserve"> palàm per 4 huius quia refractio, quæ fit ſecundum lineam m t, erit ad partem <lb/>perpendicularis lineæ, quę eſt m g.</s> <s xml:id="echoid-s50232" xml:space="preserve"> Erit itaq;</s> <s xml:id="echoid-s50233" xml:space="preserve"> punctum m <lb/>inter duas lineas t b & t k.</s> <s xml:id="echoid-s50234" xml:space="preserve"> Quia ſi punctus m eſſet ultra li <lb/> <anchor type="figure" xlink:label="fig-0748-01a" xlink:href="fig-0748-01"/> neam t k:</s> <s xml:id="echoid-s50235" xml:space="preserve"> tunc perpendicularis exiens à puncto gad pun <lb/>ctum m, eſſet etiam ultra punctum k:</s> <s xml:id="echoid-s50236" xml:space="preserve"> & ita cũ forma pun-<lb/>cti k refringeretur ad uiſum à puncto m, refringeretur ad <lb/>partẽ քpendicularis m g, & nõ քueniret ad քpendicularẽ <lb/>g e:</s> <s xml:id="echoid-s50237" xml:space="preserve"> ergo nõ քueniret ad uiſum t.</s> <s xml:id="echoid-s50238" xml:space="preserve"> Palã itaq;</s> <s xml:id="echoid-s50239" xml:space="preserve"> quoniã pũctus <lb/>m eſt inter duas lineas t k & tb:</s> <s xml:id="echoid-s50240" xml:space="preserve"> & eodem modo declarari <lb/>poteſt quia pũctũ z eſt inter duas lineas t b & t l.</s> <s xml:id="echoid-s50241" xml:space="preserve"> Extraha <lb/>turitaq;</s> <s xml:id="echoid-s50242" xml:space="preserve"> linea t m ad q pũctum circuli meridiani, & linea <lb/>t z ad punctũ reiuſdem circuli meridiani.</s> <s xml:id="echoid-s50243" xml:space="preserve"> Erit itaq;</s> <s xml:id="echoid-s50244" xml:space="preserve"> arcus <lb/>q k ęqualis arcui l r, & angulus q t r erit minor angulo k t <lb/>l:</s> <s xml:id="echoid-s50245" xml:space="preserve"> quoniã eſt pars eius.</s> <s xml:id="echoid-s50246" xml:space="preserve"> Sed angul{us} q t r eſt angulus, ք quẽ <lb/>uiſus t cõprehendit arcum k l refractè:</s> <s xml:id="echoid-s50247" xml:space="preserve"> & angulus k tl eſt <lb/>angulus, per quem uiſus t comprehẽdetarcum k l rectè:</s> <s xml:id="echoid-s50248" xml:space="preserve"> <lb/>ſi ipſum rectè poſſet comprehendere:</s> <s xml:id="echoid-s50249" xml:space="preserve"> ſed remotio arcus <lb/>k l à uiſu eſt maxima:</s> <s xml:id="echoid-s50250" xml:space="preserve"> quapropter quantitas eius non cer <lb/>tificatur.</s> <s xml:id="echoid-s50251" xml:space="preserve"> Viſus itaq;</s> <s xml:id="echoid-s50252" xml:space="preserve"> per exiſtimationem non per certitu-<lb/>dinem accipitremotionem arcus k l:</s> <s xml:id="echoid-s50253" xml:space="preserve"> ſed exiſtimatio uiſus quando comprehendit refractè, nõ dif-<lb/>fert ab exiſtimatione eius, quando comprehendit rectè, niſi in hoc ſolùm, quòd putat ſe rectè com <lb/>prehendere, quando comprehenditrefractè.</s> <s xml:id="echoid-s50254" xml:space="preserve"> Viſus itaq;</s> <s xml:id="echoid-s50255" xml:space="preserve"> t comprehendit arcum k l refractè ex an-<lb/>gulo minori, quàm ille angulus, quo ipſum comprehendit rectè, & ſecundum comparationem ad <lb/>illam eandem remotionem, ad quam comparat, ſi ipſam rectè comprehenderet.</s> <s xml:id="echoid-s50256" xml:space="preserve"> Sed uiſus t compre <lb/>hendit magnitudinem ex quantitate anguli reſpectu remotionis puncti t (quod eſt cẽtrum uiſus) <lb/>à ſuperficie rei uiſæ per 27 th.</s> <s xml:id="echoid-s50257" xml:space="preserve"> 4 huius:</s> <s xml:id="echoid-s50258" xml:space="preserve"> ergo comprehẽdit quantitatem arcus k l refractè min orem, <lb/>quàm ſi comprehẽderetillam rectè.</s> <s xml:id="echoid-s50259" xml:space="preserve"> Et ſi figura, in qua ſunt pũcta k, l, t, b imaginetur circumuolui <lb/>liǹea t b exiſtente immobili:</s> <s xml:id="echoid-s50260" xml:space="preserve"> deſcribetur circulus ſecans meridianum circulum in duobus punctis, <lb/>cuius circuli polus erit pũctum b zenith capitis:</s> <s xml:id="echoid-s50261" xml:space="preserve"> & erũt omnes anguli, qui ſunt apud uiſum t cõten-<lb/>ti duabus lineis ſimilibus lineis t k & tl, inter ſe quilibet ſuo compari æqualis.</s> <s xml:id="echoid-s50262" xml:space="preserve"> Viſus ergo t compre <lb/>hendet formam arcus k l refractè in omni ſitu in reſpectu circuli meridiei, cum fuerit in uertice ca-<lb/>pitis, minorem, quàm ſi comprehenderet ipſam rectè.</s> <s xml:id="echoid-s50263" xml:space="preserve"> Et ſilinea t b ſecuerit arcum k l in duo æqua-<lb/>lia:</s> <s xml:id="echoid-s50264" xml:space="preserve"> tũc duo pũcta q & r erũt inter duo puncta k & l:</s> <s xml:id="echoid-s50265" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s50266" xml:space="preserve"> angulus q t r minor angulo k tl:</s> <s xml:id="echoid-s50267" xml:space="preserve"> & erit o-<lb/>mnis angulus æqualis angulo q t r, exiens à pũcto t ſecans ſtellam:</s> <s xml:id="echoid-s50268" xml:space="preserve"> & linea exiens à cẽtro uiſus t in <lb/>ſuperficie illius circuli ſecabit circulum maiorem ipſius ſtellæ, & comprehenditur quantitas eius <lb/>minor quàm ſit:</s> <s xml:id="echoid-s50269" xml:space="preserve"> & ſic tota ſtella uidebitur minor quàm ſit.</s> <s xml:id="echoid-s50270" xml:space="preserve"> Omnis ergo ſtella uidetur minor, cũ eſt <lb/>in zenith capitis, quàm ſi uideretur directè.</s> <s xml:id="echoid-s50271" xml:space="preserve"> Et ſimiliter eſt de omni diſtantia inter quaslibet duas <lb/>ſtellas, cum zenith capitis fuerit inter duas extremitates illius diſtantiæ:</s> <s xml:id="echoid-s50272" xml:space="preserve"> comprehẽdetur enim in <lb/>omnibus ſuis poſitionibus minor, quàm ſi directè comprehenderetur ſine refractione.</s> <s xml:id="echoid-s50273" xml:space="preserve"> Omnis ita-<lb/>que ſtella in uertice capitis aſpicientis exiſtens uidetur minor quàm in alio loco cœli:</s> <s xml:id="echoid-s50274" xml:space="preserve"> & quãtò ma-<lb/>gis remouetur à uertice capitis, tantò ſemper apparet maior:</s> <s xml:id="echoid-s50275" xml:space="preserve"> itaque in horizonte apparet maior <lb/>quàm in alio loco.</s> <s xml:id="echoid-s50276" xml:space="preserve"> Et hoc eſt commune omnibus ſtellis, planetis ſcilicet & fixis, quòd in zenith ca-<lb/>pitis uel prope illud ſemper ſunt minores.</s> <s xml:id="echoid-s50277" xml:space="preserve"> Et hoc ſimiliter apparetin lineis determinantibus ſtel-<lb/>larum diſtantias:</s> <s xml:id="echoid-s50278" xml:space="preserve"> hoc eſt in ipſis ſtellarum diſtantijs, ut ſpatiorum cœli, quæ ſunt inter ſtellas, ma-<lb/>gis quàm in quantitatibus ſtellarum:</s> <s xml:id="echoid-s50279" xml:space="preserve"> nam quantitas ſtellæ, quo ad uiſum, eſt res parua, & exceſſus <lb/>ſuæ quantitatis res parua, ſed magis comprehenditur diuerſitas & exceſſus diſtantiarum.</s> <s xml:id="echoid-s50280" xml:space="preserve"> Patet <lb/>ergo propoſitum.</s> <s xml:id="echoid-s50281" xml:space="preserve"/> </p> <div xml:id="echoid-div1877" type="float" level="0" n="0"> <figure xlink:label="fig-0748-01" xlink:href="fig-0748-01a"> <variables xml:id="echoid-variables856" xml:space="preserve">q f h o r k c p m e z t g</variables> </figure> </div> </div> <div xml:id="echoid-div1879" type="section" level="0" n="0"> <head xml:id="echoid-head1380" xml:space="preserve" style="it">52. Diametri ſtellarum, uel lineæ ſtellarum diſt antiã determinantes, exiſtentes in horizon-<lb/>te, aut inter horizonta & circulum meridiei, taliter ut æquidiſtent horizonti: uidebuntur pro-<lb/>pter refractionem minores, quàm ſi directè uiderentur. Alhazen 53 n 7.</head> <p> <s xml:id="echoid-s50282" xml:space="preserve">Sit item circulus meridianus, qui b p:</s> <s xml:id="echoid-s50283" xml:space="preserve"> cuius centrũ, quod eſt centrũ mũdi, ſit pũctus m:</s> <s xml:id="echoid-s50284" xml:space="preserve"> & ſit cen-<lb/>trũ uiſus a:</s> <s xml:id="echoid-s50285" xml:space="preserve"> & zenith capitis pũctum b:</s> <s xml:id="echoid-s50286" xml:space="preserve"> & ducatur linea a b:</s> <s xml:id="echoid-s50287" xml:space="preserve"> & ſit diameter ſtellẽ aut diſtãtia inter ali <lb/>quas duas ſtellas linea d e æ quidiſtans horizonti:</s> <s xml:id="echoid-s50288" xml:space="preserve"> & ſit circulus altitudinis tranſiens per unã extre-<lb/> <pb o="447" file="0749" n="749" rhead="LIBER DECIMVS."/> mitatem diametri ſtellę aut diſtãtiæ inter duas ſtellas, circulus b d:</s> <s xml:id="echoid-s50289" xml:space="preserve"> & alius circulus altitudinis tran <lb/>ſiens per alteram extremitatem diametri ſtellæ aut diſtantię ſit circulus b e:</s> <s xml:id="echoid-s50290" xml:space="preserve"> cõmunes quoq;</s> <s xml:id="echoid-s50291" xml:space="preserve"> ſectio-<lb/>nes ſuperficierũ iſtorũ duorũ circulorũ & ſuperficiei concauę cœli inſimi ſint duo circuli g h & g z.</s> <s xml:id="echoid-s50292" xml:space="preserve"> <lb/>Forma itaq;</s> <s xml:id="echoid-s50293" xml:space="preserve"> pũcti d refringitur ad uiſum a in ſuperficie circuli g h:</s> <s xml:id="echoid-s50294" xml:space="preserve"> eſto, ut hoc fiat in pũcto h:</s> <s xml:id="echoid-s50295" xml:space="preserve"> & for-<lb/>ma pũcti e refringitur ad uiſum a in ſuperficie circu-<lb/>li g z:</s> <s xml:id="echoid-s50296" xml:space="preserve"> & ſit in pũcto z:</s> <s xml:id="echoid-s50297" xml:space="preserve"> ducãtur itaq;</s> <s xml:id="echoid-s50298" xml:space="preserve"> lineæ a d, a e, a h, <lb/> <anchor type="figure" xlink:label="fig-0749-01a" xlink:href="fig-0749-01"/> a z, m z, m h:</s> <s xml:id="echoid-s50299" xml:space="preserve"> & producatur linea m z ad arcum b e in <lb/>pũctũ n:</s> <s xml:id="echoid-s50300" xml:space="preserve"> & linea m h producatur ad arcũ b d in pun-<lb/>ctum f.</s> <s xml:id="echoid-s50301" xml:space="preserve"> Et quoniã linea d e æquidiſtat horizonti, cũ <lb/>ſit quędam pars circuli æquidiſtantis circulo hori-<lb/>zontis, ut alicuius illorũ circulorum, qui arabicè di-<lb/>cũtur almicantarah:</s> <s xml:id="echoid-s50302" xml:space="preserve"> palàm per 68 th.</s> <s xml:id="echoid-s50303" xml:space="preserve"> 1 huius quoniã <lb/>zenith capitis, quod eſt punctus b, eſt polus circuli <lb/>d e:</s> <s xml:id="echoid-s50304" xml:space="preserve"> quoniã ipſe eſt polus horizontis.</s> <s xml:id="echoid-s50305" xml:space="preserve"> Arcus itaq;</s> <s xml:id="echoid-s50306" xml:space="preserve"> b d <lb/>eſt æqualis arcui b e per 28 p 3:</s> <s xml:id="echoid-s50307" xml:space="preserve"> chordæ enim illorum <lb/>arcuũ ſunt æquales per 65 th.</s> <s xml:id="echoid-s50308" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s50309" xml:space="preserve"> linea itaq;</s> <s xml:id="echoid-s50310" xml:space="preserve"> m h <lb/>eſt perpẽdicularis ſuper ſuperficiẽ corporis diapha-<lb/>ni cœleſtis per 72 th.</s> <s xml:id="echoid-s50311" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s50312" xml:space="preserve"> quoniam exit à centro <lb/>mũdi.</s> <s xml:id="echoid-s50313" xml:space="preserve"> Linea itaq;</s> <s xml:id="echoid-s50314" xml:space="preserve"> h a refringitur à pũcto h ad uiſum <lb/>a:</s> <s xml:id="echoid-s50315" xml:space="preserve"> & erit eius refractio ad partem diametri h m per 4 <lb/>huius:</s> <s xml:id="echoid-s50316" xml:space="preserve"> aer enim eſt denſior corpore cœleſti, ut patet <lb/>per 50 huius:</s> <s xml:id="echoid-s50317" xml:space="preserve"> refringetur ergo ad partẽ contrariam <lb/>illi parti, in qua eſt pars reliqua perpendicularis, quę h f:</s> <s xml:id="echoid-s50318" xml:space="preserve"> ergo h pũctum refractionis eſt altius quàm <lb/>linea a d.</s> <s xml:id="echoid-s50319" xml:space="preserve"> Et ſimiliter declarabitur quòd z pũctus refractionis eſt altior ꝗ̃ linea a e.</s> <s xml:id="echoid-s50320" xml:space="preserve"> Duo ergo pũcta f <lb/>& n, quę ſunt termini duarũ linearũ perpendiculariũ m f & m n, ſunt inter duo puncta d & e, & ze-<lb/>nith capitis, quod eſt b:</s> <s xml:id="echoid-s50321" xml:space="preserve"> ita quòd pũctum f eſt inter duo pũcta d & b, & pũctum n inter duo pũcta o <lb/>& b:</s> <s xml:id="echoid-s50322" xml:space="preserve"> & angulus refractionis, qui eſt apud pũctum h, eſt æqualis angulo refractiõis, qui eſt apud pun <lb/>ctũ z per 14 th.</s> <s xml:id="echoid-s50323" xml:space="preserve"> huius:</s> <s xml:id="echoid-s50324" xml:space="preserve"> quoniã ſitus duorũ pũctorum d & e reſpectu uiſus a, eſt cõſimilis exhypothe-<lb/>ſi.</s> <s xml:id="echoid-s50325" xml:space="preserve"> Tantùm ergo diſtat pũctus f à pũcto d, quantũ pũctus n à puncto e.</s> <s xml:id="echoid-s50326" xml:space="preserve"> Extrahatur itaq:</s> <s xml:id="echoid-s50327" xml:space="preserve"> linea a h ad <lb/>pũctum t:</s> <s xml:id="echoid-s50328" xml:space="preserve"> & linea a z ad pũctum k:</s> <s xml:id="echoid-s50329" xml:space="preserve"> diſtabit itaq;</s> <s xml:id="echoid-s50330" xml:space="preserve"> pũctus t à pũcto d tantũ, quantũ pũctus k à pũcto e:</s> <s xml:id="echoid-s50331" xml:space="preserve"> <lb/>& ducatur linea t k, quæ neceſſariò erit æquidiſtans lineæ d e per 88 th.</s> <s xml:id="echoid-s50332" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s50333" xml:space="preserve"> quoniã arcus e k eſt <lb/>æqualis arcui d t:</s> <s xml:id="echoid-s50334" xml:space="preserve"> eſt ergo linea t k minor quàm linea d e per idẽ 88th.</s> <s xml:id="echoid-s50335" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s50336" xml:space="preserve"> & lineæ a t, a k, a d, a e <lb/>ſunt æquales:</s> <s xml:id="echoid-s50337" xml:space="preserve"> quia pũctum a centrũ uiſus, eſt quaſi centrũ mũdi, & omniũ arcuũ ſignatorũ, ut b d & <lb/>b e.</s> <s xml:id="echoid-s50338" xml:space="preserve"> Duæ itaq;</s> <s xml:id="echoid-s50339" xml:space="preserve"> lineæ a t & a k ſunt æquales duabus lineis a d & a e, & baſis t k trigonia t k eſt minor <lb/>quàm baſis d e trigoni a d e:</s> <s xml:id="echoid-s50340" xml:space="preserve"> ergo per 25 p 1 erit angulus t a k minor angulo d a e:</s> <s xml:id="echoid-s50341" xml:space="preserve"> ſed angulus t a k <lb/>eſt angulus, ſecũdum quẽ linea d e comprehẽditur refractè:</s> <s xml:id="echoid-s50342" xml:space="preserve"> & angulus d a e eſt angulus, ſecũdũ quẽ <lb/>linea d e comprehẽditur rectè.</s> <s xml:id="echoid-s50343" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s50344" xml:space="preserve"> illud, quod proponebatur, ſiue linea d e ſit diameter ali-<lb/>cuius ſtellarum, ſiue ipſa ſit linea determinans diſtantiam inter ſtellas.</s> <s xml:id="echoid-s50345" xml:space="preserve"/> </p> <div xml:id="echoid-div1879" type="float" level="0" n="0"> <figure xlink:label="fig-0749-01" xlink:href="fig-0749-01a"> <variables xml:id="echoid-variables857" xml:space="preserve">h g f r n h d k z a m e p</variables> </figure> </div> </div> <div xml:id="echoid-div1881" type="section" level="0" n="0"> <head xml:id="echoid-head1381" xml:space="preserve" style="it">53. Diametri ſtellarum aut lineæ determinantes diſtantiam ſtellarum in aliquo circulo alti-<lb/>tudinis ſuper horizonta erectæ, per refractionem uidentur minores, quàm ſi directè uideren-<lb/>tur. Alhazen 54 n 7.</head> <p> <s xml:id="echoid-s50346" xml:space="preserve">Remaneat diſpoſitio, quæ ſuprà, & ſit diameter alicuius ſtellarũ uel diſtantia aliquarũ duarũ ſtel <lb/>larum linea d e:</s> <s xml:id="echoid-s50347" xml:space="preserve"> quæ ſit erecta in aliquo circulo altitudinis tranſeunte per zenith capitis, quod eſt <lb/>pũctum b, qui circulus altitudinis ſit b d e:</s> <s xml:id="echoid-s50348" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s50349" xml:space="preserve"> cõmunis ſectio ſuperficiei circuli b d e, & ſuperficiei <lb/>concauitatis ſphæræ infimæ c œleſtis circulus g h z per 69 th.</s> <s xml:id="echoid-s50350" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s50351" xml:space="preserve"> & ducantur lineæ a d & a e:</s> <s xml:id="echoid-s50352" xml:space="preserve"> & <lb/>refringatur forma pũcti d ad uiſum a ex pũcto h:</s> <s xml:id="echoid-s50353" xml:space="preserve"> & forma pũcti e ex pũcto z.</s> <s xml:id="echoid-s50354" xml:space="preserve"> Copulentur quoq;</s> <s xml:id="echoid-s50355" xml:space="preserve"> li-<lb/>neæ, d h, quę producatur ultra pũctum h in pũctum n:</s> <s xml:id="echoid-s50356" xml:space="preserve"> & e z, quę producatur ultra pũctum z in pun-<lb/>ctum o.</s> <s xml:id="echoid-s50357" xml:space="preserve"> Patet ergo, ut in præcedente proxima, quòd pũctum h eſt altius ꝗ̃ linea a d:</s> <s xml:id="echoid-s50358" xml:space="preserve"> & qđ punctũ z <lb/>eſt altius ꝗ̃ linea a e.</s> <s xml:id="echoid-s50359" xml:space="preserve"> Ducantur itaq;</s> <s xml:id="echoid-s50360" xml:space="preserve"> lineæ a h, h d, a z, z e, m h, m z:</s> <s xml:id="echoid-s50361" xml:space="preserve"> & protrahatur linea m h ultra pun <lb/>ctum h ad circulũ altitudinis in pũctum t:</s> <s xml:id="echoid-s50362" xml:space="preserve"> & linea m z ultra pũctum z in pũctum k:</s> <s xml:id="echoid-s50363" xml:space="preserve"> erit ergo angu-<lb/>lus refractus, qui fit ex refractione formæ pũcti e ad uiſum a (qui eſt angulus a z m) ualde paruus <lb/>(quoniã linea a m, quę eſt ſemidiameter terræ, reſpectu tantæ diſtantiæ, non eſt alicuius ſenſibilis <lb/>quãtitatis, ut aliàs declarauimus in ſcientia motuũ cœleſtiũ) & angulus refractionis eius erit par-<lb/>uus, ſequens modũ illius anguli a z m:</s> <s xml:id="echoid-s50364" xml:space="preserve"> quoniã cũ aer ſit denſior corpore cœleſti, ut patet per 50 hu-<lb/>ius:</s> <s xml:id="echoid-s50365" xml:space="preserve"> palã per 4 huius quoniã fit refractio ad perpendicularẽ, quæ eſt z m.</s> <s xml:id="echoid-s50366" xml:space="preserve"> Erit ergo per 8 huius angu <lb/>lus e z k acutus:</s> <s xml:id="echoid-s50367" xml:space="preserve"> & ſimiliter erit angulus d h t acutus:</s> <s xml:id="echoid-s50368" xml:space="preserve"> ergo angulorũ a h d & a z e uterq;</s> <s xml:id="echoid-s50369" xml:space="preserve"> erit obtuſus <lb/>per 13 p 1.</s> <s xml:id="echoid-s50370" xml:space="preserve"> Pũctum itaq;</s> <s xml:id="echoid-s50371" xml:space="preserve"> z aut erit in ſuperficie horizontis, aut altius.</s> <s xml:id="echoid-s50372" xml:space="preserve"> Si erit in ſuperficie horizontis:</s> <s xml:id="echoid-s50373" xml:space="preserve"> <lb/>erit ergo in extremitate perpendicularis exeuntis à centro uiſus, quod eſt a, ſuper lineã b a perpen-<lb/>diculariter ſuperficiei horizontis inſiſtentẽ, quę perpendicularis imaginatur eſſe ducta in ſuperfi-<lb/>cie horizontis:</s> <s xml:id="echoid-s50374" xml:space="preserve"> aut ſi fuerit altius horizonte, erit altius illa linea perpendiculari:</s> <s xml:id="echoid-s50375" xml:space="preserve"> & punctum h erit <lb/>ſemper altius puncto z.</s> <s xml:id="echoid-s50376" xml:space="preserve"> Angulus ergo a h m eſt minor angulo a z m:</s> <s xml:id="echoid-s50377" xml:space="preserve"> quod patet, ſi ſuper pũctum m <lb/>terminum lineę a m fiat per 23 p 1 angulus ęqualis angulo a m z, qui ſit a m p, ducta linea m p ad pe-<lb/>ripheriam circuli g h z:</s> <s xml:id="echoid-s50378" xml:space="preserve"> facto quoq;</s> <s xml:id="echoid-s50379" xml:space="preserve"> angulo q a g ęquali angulo h a g:</s> <s xml:id="echoid-s50380" xml:space="preserve"> ita ut per 7 p 3 linea a q ſit ęqua, <lb/> <pb o="448" file="0750" n="750" rhead="VITELLONIS OPTICAE"/> lis lineæ a h:</s> <s xml:id="echoid-s50381" xml:space="preserve"> & eopuletur linea h p.</s> <s xml:id="echoid-s50382" xml:space="preserve"> In trigono ergo h m p duo anguli m p h ſunt ęquales per <lb/>5 p 1:</s> <s xml:id="echoid-s50383" xml:space="preserve"> ſed in trigono h a p latus a p eſt maius latere <lb/>a h:</s> <s xml:id="echoid-s50384" xml:space="preserve"> quia eſt maius latere a q per 7 p 3:</s> <s xml:id="echoid-s50385" xml:space="preserve"> eſt ergo per <lb/> <anchor type="figure" xlink:label="fig-0750-01a" xlink:href="fig-0750-01"/> 19 p 1 angulus a h p maior angulo h p a:</s> <s xml:id="echoid-s50386" xml:space="preserve"> relinqui-<lb/>tur ergo angulus a p m maior angulo a h m.</s> <s xml:id="echoid-s50387" xml:space="preserve"> Eſt au-<lb/>tem per 4 p 1 angulus a p m ęqualis angulo a z m:</s> <s xml:id="echoid-s50388" xml:space="preserve"> <lb/>eſt ergo angulus a h m minor angulo a z m:</s> <s xml:id="echoid-s50389" xml:space="preserve"> ergo ք <lb/>8 huius angulus formę incidẽtiæ puncti d, qui eſt <lb/>angulus d h t, eſt minor angulo incidentiæ formæ <lb/>pũcti e, qui eſt angulus e z k:</s> <s xml:id="echoid-s50390" xml:space="preserve"> ergo angulus a h d eſt <lb/>maior angulo a z e per 13 p 1:</s> <s xml:id="echoid-s50391" xml:space="preserve"> quia per 8 huius mino <lb/>res anguli incidentiæ minores habent angulos re.</s> <s xml:id="echoid-s50392" xml:space="preserve"> <lb/>fractionum:</s> <s xml:id="echoid-s50393" xml:space="preserve"> & ita angulus n h a eſt minor angulo <lb/>o z a:</s> <s xml:id="echoid-s50394" xml:space="preserve"> relin quitur ergo angulus a h d maior angulo <lb/>a z e:</s> <s xml:id="echoid-s50395" xml:space="preserve"> & duæ lineæ m t & m k ſunt ſemidiametri cir-<lb/>culi b d e:</s> <s xml:id="echoid-s50396" xml:space="preserve"> & duæ lineę m h & m z ſunt ſemidiame-<lb/>tri circuli g h z.</s> <s xml:id="echoid-s50397" xml:space="preserve"> Linea itaque m t eſt ęqualis lineæ <lb/>m k, & linea m h eſt æ qualis lineæ m z per defini-<lb/>tionem circuli:</s> <s xml:id="echoid-s50398" xml:space="preserve"> linea itaq;</s> <s xml:id="echoid-s50399" xml:space="preserve"> h t eſt æ qualis lineę z k:</s> <s xml:id="echoid-s50400" xml:space="preserve"> <lb/>quoniam ſunt remanentiæ linearũ æqualiũ ablatis æqualibus:</s> <s xml:id="echoid-s50401" xml:space="preserve"> & angulus d h t eſt minor angulo e <lb/>z k:</s> <s xml:id="echoid-s50402" xml:space="preserve"> ergo linea h d eſt minor quâm linea e z:</s> <s xml:id="echoid-s50403" xml:space="preserve"> quoniam linea continens cũ linea t h angulũ æ qualem <lb/>angulo k z e (qui eſt maior angulo d h t) erit maior ꝗ̃ linea d h per 7 p 3:</s> <s xml:id="echoid-s50404" xml:space="preserve"> linea ergo d h eſt minor ꝗ̃ li <lb/>nea e z:</s> <s xml:id="echoid-s50405" xml:space="preserve"> & duæ lineæ a d & a e ſunt æ quales:</s> <s xml:id="echoid-s50406" xml:space="preserve"> ſimiliter duæ lineæ a h & a z ſunt æquales:</s> <s xml:id="echoid-s50407" xml:space="preserve"> quia punctũ <lb/>a centrum uiſus eſt quaſi centrũ circulorũ b d e & g h z:</s> <s xml:id="echoid-s50408" xml:space="preserve"> triangulus ergo a h d eſt minor triangulo a <lb/>z e:</s> <s xml:id="echoid-s50409" xml:space="preserve"> quoniam illorũ duorum trigonorũ duobus lateribus exiſtentibus æqualibus, tertium eſt inæ-<lb/>quale.</s> <s xml:id="echoid-s50410" xml:space="preserve"> Ergo circulus continens trigonũ a h d eſt maior circulo cõtinente trigonũ a z e:</s> <s xml:id="echoid-s50411" xml:space="preserve"> quia angu-<lb/>lus a h d eſt maior angulo a z e, & linea h d eſt minor quàm linea z e.</s> <s xml:id="echoid-s50412" xml:space="preserve"> Linea itaq;</s> <s xml:id="echoid-s50413" xml:space="preserve"> h d diſtinguit de cir <lb/>culo maiore, continente triangulũ a h d, arcum min orẽ arcu ſimili illi arcui, quẽ reſecat linea z e ex <lb/>circulo minore cõtinente triangulũ a e z:</s> <s xml:id="echoid-s50414" xml:space="preserve"> angulus ergo h a d eſt minor angulo z a e:</s> <s xml:id="echoid-s50415" xml:space="preserve"> ſit ergo angulus <lb/>za d cõmunis illis ambobus angulis:</s> <s xml:id="echoid-s50416" xml:space="preserve"> erit ergo angulus h a z minor angulo d a e:</s> <s xml:id="echoid-s50417" xml:space="preserve"> angulus uerò h a z <lb/>eſt angulus, ſecũdum quẽ uiſus a comprehẽdit lineam d e per refractionẽ:</s> <s xml:id="echoid-s50418" xml:space="preserve"> & angulus d a e eſt angu-<lb/>lus, ſecũdum quẽ comprehẽderetur forma lineæ d e rectè (ſi hoc poſſet fieri.</s> <s xml:id="echoid-s50419" xml:space="preserve">) Viſus itaq;</s> <s xml:id="echoid-s50420" xml:space="preserve"> a compre <lb/>hendit lineam d e refractè minorem quàm rectè per 20 th.</s> <s xml:id="echoid-s50421" xml:space="preserve"> 4 huius:</s> <s xml:id="echoid-s50422" xml:space="preserve"> quoniã ſub maiori angulo com-<lb/>prehenditip ſam refractè quàm rectè.</s> <s xml:id="echoid-s50423" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s50424" xml:space="preserve"/> </p> <div xml:id="echoid-div1881" type="float" level="0" n="0"> <figure xlink:label="fig-0750-01" xlink:href="fig-0750-01a"> <variables xml:id="echoid-variables858" xml:space="preserve">b t d g h q n o k z a p e m</variables> </figure> </div> </div> <div xml:id="echoid-div1883" type="section" level="0" n="0"> <head xml:id="echoid-head1382" xml:space="preserve" style="it">54. Omnes ſtellæ uidentur rotundæ: maiores in horiz onte quàm in medio cœli: niſi quando <lb/>contrarium accidat propter interpoſitos uapores uiſibus & ſtellis. Alhazen 55 n 7.</head> <p> <s xml:id="echoid-s50425" xml:space="preserve">Omnes ſtellę cõprehendũtur rotũdæ:</s> <s xml:id="echoid-s50426" xml:space="preserve"> quoniam utraq;</s> <s xml:id="echoid-s50427" xml:space="preserve"> diametrorum ſuarũ ſcilicet lõgitudinis & <lb/>latitudinis comprehẽditur æqualiter minor, quàm ſi comprehẽderetur rectè:</s> <s xml:id="echoid-s50428" xml:space="preserve"> quælibet ergo ſuarũ <lb/>diametrorũ decliuiũ comprehẽditur æqualiter minor per refractionẽ, quàm ſi comprehenderetur <lb/>rectè.</s> <s xml:id="echoid-s50429" xml:space="preserve"> Stella ergo comprehẽditur rotũda in omni ſuo ſitu.</s> <s xml:id="echoid-s50430" xml:space="preserve"> Omnes quoq;</s> <s xml:id="echoid-s50431" xml:space="preserve"> ſtellæ comprehẽdũtur mi-<lb/>nores per refractionẽ, ꝗ̃ ſi directè uiderẽtur:</s> <s xml:id="echoid-s50432" xml:space="preserve"> quoniã ipſarũ diametri comprehẽdũtur minores, ut pa <lb/>tet ex propoſitionibus pręmiſsis.</s> <s xml:id="echoid-s50433" xml:space="preserve"> Et hoc uerũ eſt, quãtũ eſt à parte refractionis, q̃ fit in medio ſecũ-<lb/>di diaphani, quod eſt aer, qui eſt dẽſior cœlo per 50 huius.</s> <s xml:id="echoid-s50434" xml:space="preserve"> In cœleſti itaq;</s> <s xml:id="echoid-s50435" xml:space="preserve"> concaua ſuperficie fit re.</s> <s xml:id="echoid-s50436" xml:space="preserve"> <lb/>fractio ad perpẽdicularem exeuntẽ à pũcto refractionis ſuper illã ſuperficiẽ:</s> <s xml:id="echoid-s50437" xml:space="preserve"> hoc eſt ad lineã (q̃ eſt <lb/>ſemidiameter mũdi) per 4 huius.</s> <s xml:id="echoid-s50438" xml:space="preserve"> Diuerſitas uerò refractionis, quę fit ſecũdũ diſtãtiã ſtellarũ à po-<lb/>lo mũdi, inuenitur parua:</s> <s xml:id="echoid-s50439" xml:space="preserve"> quoniam illi anguli refractionis ſunt parui.</s> <s xml:id="echoid-s50440" xml:space="preserve"> Vnde ſecũdum ipſos non di-<lb/>uerſificatur ſenſibiliter quãtitas ſtellarũ.</s> <s xml:id="echoid-s50441" xml:space="preserve"> Sed magnitudo ſtellarum & quantitas diſtãtię ipſarum ab <lb/>inuicem, multum differũt, cum ſunt in horizonte, & cum ſunt iuxta zenith capitis, uel in medio cœ <lb/>li, propter ſenſibilem diuerſitatẽ ſuę refractionis.</s> <s xml:id="echoid-s50442" xml:space="preserve"> Et hic eſt error perpetuus, quia cauſſa eius eſt per-<lb/>petua, ſcilicet uictoria raritatis corporis cœleſtis ſuper aeris raritatem.</s> <s xml:id="echoid-s50443" xml:space="preserve"> Accidit tamẽ quãdoq;</s> <s xml:id="echoid-s50444" xml:space="preserve"> uideri <lb/>ſtellas maiores una uice ꝗ̃ alia:</s> <s xml:id="echoid-s50445" xml:space="preserve"> ut ſi uapor groſſus ſit inter uiſum & ſtellas:</s> <s xml:id="echoid-s50446" xml:space="preserve"> tũc enim propter refra-<lb/>ctionẽ linearũ extẽſionis formarũ ſtellarũ in illo uapore ad perpẽdicularẽ, & propter refractionem <lb/>â ſuperficie illius uaporis factã iterũ ad aerẽ, in quo eſt uiſus, quæ refractio fit ab illa perpẽdiculari, <lb/>diſperſior occurrit forma uiſui, & ſub angulis maioribus uidẽtur formę ſtellarũ:</s> <s xml:id="echoid-s50447" xml:space="preserve"> ſicut etiam accidit <lb/>de denario ſub a qua uiſo, qui uidetur maior quàm ſi in aere uideretur.</s> <s xml:id="echoid-s50448" xml:space="preserve"> Huiuſmodiaũt quantitas ui-<lb/>ſionis ſtellarũ maximè accidit cum ſtellæ ſunt in horizonte, aut prope illum:</s> <s xml:id="echoid-s50449" xml:space="preserve"> & ſic duę refractiones <lb/>ſubſequentes primam (quę fit in concaua ſuperficie ipſius c œli, & fit ſemper in omni ſtellarum ui-<lb/>ſione) faciunt nouas immutationes circa ſtellarum uiſionem.</s> <s xml:id="echoid-s50450" xml:space="preserve"> Vapor enim ille groſſus cum fuerit <lb/>in horizonte aut prope, & non fuerit continuus uſq;</s> <s xml:id="echoid-s50451" xml:space="preserve"> ad medium c œli, erit portio cuiuſdam ſphærę <lb/>concentricæ mundo, & erit ſuperficies eius, quę eſt ex parte uiſus, plana:</s> <s xml:id="echoid-s50452" xml:space="preserve"> propter quod formæ aut <lb/>diſtantię ſtellarũ, quæ ſunt ultra illum uaporẽ, uidebuntur maiores, quàm ſi ſine illo uapore uide-<lb/>rentur.</s> <s xml:id="echoid-s50453" xml:space="preserve"> In illo enim loco cõcauitatis cœli, ex quo refringitur forma ſtellæ ad uiſum, eſt forma ſtellę, <lb/>& exipſo extenditur ad uiſum, ſi non interuenerit uapor groſſus, Quòd ſi uapor groſſus uiſibus & <lb/> <pb o="449" file="0751" n="751" rhead="LIBER DECIMVS."/> ſtellis interuen erit:</s> <s xml:id="echoid-s50454" xml:space="preserve"> tũc extenditur forma ſtellę ad ſuperficiem uaporis ſupremã, & refringitur in il-<lb/>la ad perpẽdicularem:</s> <s xml:id="echoid-s50455" xml:space="preserve"> deinde extẽditur ad ſuperficiẽ infimã uaporis, & refringitur ab illa ad aerem <lb/>purũ continentẽ uiſum:</s> <s xml:id="echoid-s50456" xml:space="preserve"> & ſit illa refractio ad partẽ contrariam perpẽdicularis, exeuntis à pũcto re-<lb/>fractionis ſuper planam ſuperficiem uaporis.</s> <s xml:id="echoid-s50457" xml:space="preserve"> Sic ergo forma ſtellæ & earũ diſtantia uidetur maior, <lb/>quàm ſi uideretur poſt refractionem factam in concauo c œli à ſupremo corporis elementaris, nul-<lb/>la facta refractione in ſuperficie uaporis ad aerem, qui eſt ſub uapore, ut ſub dẽſiore corpore rarior <lb/>conſiſtens & continens ipſum uiſum.</s> <s xml:id="echoid-s50458" xml:space="preserve"> Cauſſa uero, propter quam omniuapore medio excluſo, ui-<lb/>dẽtur ſtellę & ſtellarũ diſtantiæ maiores in horizonte ꝗ̃ in medio c œli aut prope, coadiuuatur plu-<lb/>rimũ per exiſtimationẽ uidentis:</s> <s xml:id="echoid-s50459" xml:space="preserve"> quoniã exiſtimat ſtellas plus diſtare à uiſu in horizõte ꝗ̃ in medio <lb/>cœli:</s> <s xml:id="echoid-s50460" xml:space="preserve"> exiſtimãs ipſam partẽ cœli, quę eſt iuxta zenith capitis propinquiorẽ ſibi, ꝗ̃ eã, quæ eſt in hori-<lb/>zonte, ut oſten dimus per 13 th.</s> <s xml:id="echoid-s50461" xml:space="preserve"> 4 huius.</s> <s xml:id="echoid-s50462" xml:space="preserve"> Comprehendit ergo uiſus quantitatem ſtellę, & quantitatẽ <lb/>diſtantię, quę eſt inter ſtellas, cum fuerint in horizonte aut prope, ex comparatione anguli, ſub quo <lb/>fit uiſio, ad diſtantiam remotam:</s> <s xml:id="echoid-s50463" xml:space="preserve"> & cũ fuerint in medio cœli aut prope illud, comprehendit ipſarũ <lb/>quantitatem ex comparatione anguli æqualis primo aut ferè, ad diſtãtiam propinquam, inter quã <lb/>& diſtantiam horizontis uidetur diuerſitas maxima.</s> <s xml:id="echoid-s50464" xml:space="preserve"> Et ſic iudicat ſtellarum quantitatem ſecundũ <lb/>modũ, quo dijudicat quantitatem uiſibilium conſuetorũ.</s> <s xml:id="echoid-s50465" xml:space="preserve"> Quæ enim à rem otiori ſub eodẽ angulo <lb/>uidentur, quo alia propin quiora:</s> <s xml:id="echoid-s50466" xml:space="preserve"> illa remotiora iudicãtur à uidentibus eſſe maiora, ut oſtendimus <lb/>hoc 4 libro huius.</s> <s xml:id="echoid-s50467" xml:space="preserve"> Hęc enim cauſſa uiſionis ſtellarum eſt perpetua & immutabilis, omnibus uidẽti-<lb/>bus cõmunis.</s> <s xml:id="echoid-s50468" xml:space="preserve"> Et eodẽ modo accidit uidentibus in cõprehenſione diſtantiarũ ipſarũ ſtellarũ:</s> <s xml:id="echoid-s50469" xml:space="preserve"> nam <lb/>formæ harum diſtantiarum non diuerſantur apud uiſum in diuerſis temporibus, ſed ſunt ſemper <lb/>eodem modo ſe habentes, & uiſus aſsimilat ipſas diſtantijs rerum aſſuetarum, quæ maximæ diſtant <lb/>à uiſu ſuper ſuperficiem terræ ipſius.</s> <s xml:id="echoid-s50470" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s50471" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1884" type="section" level="0" n="0"> <head xml:id="echoid-head1383" xml:space="preserve" style="it">55. Scintillatio accidit ſemper omnibus ſtellis fixis propter diuaricationẽ formæ in loco ima-<lb/>ginis ex motu ſubiecti corporis accidentem.</head> <p> <s xml:id="echoid-s50472" xml:space="preserve">Quoniam enim, ut patet ex præmiſsis quinq;</s> <s xml:id="echoid-s50473" xml:space="preserve"> theorematibus, locus imaginis formæ cuiuslibet <lb/>ſtellarum eritin conuexo aeris uel ignis ſub concauo cœli infimi ignem continentis:</s> <s xml:id="echoid-s50474" xml:space="preserve"> horũ aũt ele-<lb/>mentorũ quodlibet mobile eſt per ſemotu recto, utpote ſurſum propter leuitatẽ, quę eſt in illis:</s> <s xml:id="echoid-s50475" xml:space="preserve"> mo <lb/>uetur aũt per accidens motu circulari unà cũ motu diurno cœli, propter quod formã ſtellarũ ipſis <lb/>incidentẽ neceſſe eſt diuaricari & diſtrahi, ſicut & ipſa forma uidetur aliqualiter locũ mutare pro-<lb/>pter motum corporis, in quo uidetur:</s> <s xml:id="echoid-s50476" xml:space="preserve"> nec eſt diuerſitas in iſto, ſiue lumen ſtellarũ per ſe ipſum dif-<lb/>fundatur, ſiue fiat hoc propter reflexionẽ luminis ſolaris à ſtellis.</s> <s xml:id="echoid-s50477" xml:space="preserve"> Semper enim tã lumen per ſe dif-<lb/>fuſum à corpore luminoſo, quàm lumen ab alijs corporibus diffuſum (quando per refractionẽ ui-<lb/>detur) fit debilius per 10 huius.</s> <s xml:id="echoid-s50478" xml:space="preserve"> Vnde cum habet locum imaginis in corpore mobili diuerſis moti-<lb/>bus, aut uno motu forti:</s> <s xml:id="echoid-s50479" xml:space="preserve"> neceſſe eſt formam illã debilitatã diuaricatã & diſtractã uideri, propter mo <lb/>tũ corporis ſubiecti, in quo uidetur:</s> <s xml:id="echoid-s50480" xml:space="preserve"> unde in his talis reflexio luminis nõ eſt cauſſa.</s> <s xml:id="echoid-s50481" xml:space="preserve"> Et huius ſimile <lb/>eſt in aqua uelociter currente, à cuius ſuperficie formæ ſtellarum reflexæ uidentur plus ſcintillare <lb/>quàm in ipſo loco ſuæ imaginis refractè per aerẽ uideantur:</s> <s xml:id="echoid-s50482" xml:space="preserve"> quoniã propter motũ aqę diſtrahitur <lb/>forma reflexa, & mutatur locus imaginis reflexę:</s> <s xml:id="echoid-s50483" xml:space="preserve"> propter quod & ſtellarũ formę plus moueriuiden <lb/>tur:</s> <s xml:id="echoid-s50484" xml:space="preserve"> & ideo apparent amplius ſcintillantes.</s> <s xml:id="echoid-s50485" xml:space="preserve"> Similiter quoq;</s> <s xml:id="echoid-s50486" xml:space="preserve"> formę ſtellarũ in loco ſuę imaginis tẽpo <lb/>re uentorũ propter maiorẽ motũ corporis medij plus ſcintillãt.</s> <s xml:id="echoid-s50487" xml:space="preserve"> In planetis uerò nõ ſemper accidit <lb/>ſcintillatio:</s> <s xml:id="echoid-s50488" xml:space="preserve"> quoniã licet plus ſcintillẽt, & in eis ſit idẽ locus imaginis, & ipſorũ formæ propter refra <lb/>ctionẽ debilitẽtur:</s> <s xml:id="echoid-s50489" xml:space="preserve"> tamẽ propter ipſorũ ꝓpinquitatẽ ad nos uidẽtes nõ accidit eis multa debilitas:</s> <s xml:id="echoid-s50490" xml:space="preserve"> <lb/>quia minor fit in eis refractio per 14 th.</s> <s xml:id="echoid-s50491" xml:space="preserve"> huius.</s> <s xml:id="echoid-s50492" xml:space="preserve"> Perueniũt ergo formę ipſorũ fortes ad uiſum:</s> <s xml:id="echoid-s50493" xml:space="preserve"> unde <lb/>& locũ imaginis ſuæ (quãuis corpus ſubiectũ moueatur) penetrãt immotè & ſine omni diuarica-<lb/>tione:</s> <s xml:id="echoid-s50494" xml:space="preserve"> niſi fortè aliquod corpus groſsius aere uiſibus & planetarũ formis interponatur:</s> <s xml:id="echoid-s50495" xml:space="preserve"> utpote ua-<lb/>por a quaticus groſſus:</s> <s xml:id="echoid-s50496" xml:space="preserve"> tũc etenim {por}pter incertitudinẽ motus illius uaporis (pręſertim cũ à uentis <lb/>agitatur) formę planetarũ quaſi ſcintillãtes քueniũt ad uiſum.</s> <s xml:id="echoid-s50497" xml:space="preserve"> Et ex hac cauſſa aliquãdo & ipſum <lb/>ſolẽ uidemus ſcintillãtẽ in mane, cũ fuerit in ortu ſuo uiſibilis ſecũdũ ſpirituũ uiſibiliũ reſolutionẽ, <lb/>propter quorũ reſolutionẽ & motũ, ſol ſemper aliquãdiu aſpectus uidetur ſcintillare & moueri for <lb/>ma eius:</s> <s xml:id="echoid-s50498" xml:space="preserve"> quoniã recipitur in ſpiritib.</s> <s xml:id="echoid-s50499" xml:space="preserve"> motis, qui propter uictoriã luminis cũ fuerint in fine ſuę cor-<lb/>ruptionis ab actu uiſiõis, rarificãtur ſuper ſuę naturę cõſiſtẽtiã:</s> <s xml:id="echoid-s50500" xml:space="preserve"> unde mouẽtur motu ſibi impro por <lb/>tio nato & inſolito, fiuntq́;</s> <s xml:id="echoid-s50501" xml:space="preserve"> cauſſa motus formę uiſę:</s> <s xml:id="echoid-s50502" xml:space="preserve"> & tũc uidetur forma rei uiſæ ſcintillare:</s> <s xml:id="echoid-s50503" xml:space="preserve"> ſicut e-<lb/>tiã accidit cũ à corporibus politis fit fortis reflexio luminis ad uiſum:</s> <s xml:id="echoid-s50504" xml:space="preserve"> tũc enim ꝓpter improportio <lb/>nẽ illius luminis ad ſpiritus uiſibiles fit motus illorũ ſpirituũ, & uidẽtur formę illorũ corporũ ſcin-<lb/>tillãtes & motę, ꝗ a recipiũtur in corpore cõmoto.</s> <s xml:id="echoid-s50505" xml:space="preserve"> Sic itaq;</s> <s xml:id="echoid-s50506" xml:space="preserve"> ſcintillatio ſemper accidit omnib.</s> <s xml:id="echoid-s50507" xml:space="preserve"> ſtellis <lb/>fixis:</s> <s xml:id="echoid-s50508" xml:space="preserve"> quoniã cauſſa illius eſt քpetua, ſcilicet diuaricatio formę ſuę in loco imaginis, accidẽs ex mo-<lb/>tu ſubiecti corporis.</s> <s xml:id="echoid-s50509" xml:space="preserve"> In planetis uerò ſcintillatio accidit ut rarò:</s> <s xml:id="echoid-s50510" xml:space="preserve"> ꝗa cauſſa eius eſt eueniẽs ut rarò.</s> <s xml:id="echoid-s50511" xml:space="preserve"> In <lb/>alijs uerò corporũ formis, quarũ excellẽtia corrũpit ſenſum, nõ eſt propriè ſcintillatio, ſiue illa cor-<lb/>ruptio fiat per ſimplicẽ luminis immiſsionẽ, uel per reflexionẽ à corporibus politis:</s> <s xml:id="echoid-s50512" xml:space="preserve"> quia illa ſcintil <lb/>latio nõ accidit ſenſui, ut eſt ſuæ ꝓpriæ diſpoſitionis, ſed ut eſt in fine ſuę corruptionis.</s> <s xml:id="echoid-s50513" xml:space="preserve"> Etenim ſi ha <lb/>bẽtibus in oculis formã rei motæ, aut etiã mouẽtibus, omnia moueri uideantur ꝓpter motũ ſpiri-<lb/>tuũ ſine regimine animæ diſcurrentiũ:</s> <s xml:id="echoid-s50514" xml:space="preserve"> nõ propter hoc dicũtur formę rerũ omniũ ſcintillare.</s> <s xml:id="echoid-s50515" xml:space="preserve"> Patet <lb/>ergo ꝓpoſitũ.</s> <s xml:id="echoid-s50516" xml:space="preserve"> Et quia ſecũdũ pręmiſſos refractionũ modos paſsiones uiſibiliũ infimorũ & ſupre-<lb/> <pb o="450" file="0752" n="752" rhead="VITELLONIS OPTICAE"/> morum tranſcurrimus:</s> <s xml:id="echoid-s50517" xml:space="preserve"> reſtat, ut refractiones, quæ in medijs accidunt corporibus, aliqualiter per-<lb/>tractemus, utpote illas, quę in uaporibus medijs occurrunt.</s> <s xml:id="echoid-s50518" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1885" type="section" level="0" n="0"> <head xml:id="echoid-head1384" xml:space="preserve" style="it">56. Non aggregatis radijs corporis luminoſi in corpore non luminoſo plus, quàm in medio: lu-<lb/>men ſenſibilius fieriest impoßibile.</head> <p> <s xml:id="echoid-s50519" xml:space="preserve">Quod hic proponitur, patet:</s> <s xml:id="echoid-s50520" xml:space="preserve"> quia lato lumine per aliquã partẽ medij, uniformis erit extenſio ra-<lb/>dij ſecũdũ lineã rectã per 1 th.</s> <s xml:id="echoid-s50521" xml:space="preserve"> 2 huius:</s> <s xml:id="echoid-s50522" xml:space="preserve"> unde ſi nõ aggregẽtur radij in corpore aliquo occurrẽre ipſis <lb/>radijs luminis, nõ erit plus ſenſibile lumẽ in illo corpore ꝗ̃ fuerit in alia parte medij:</s> <s xml:id="echoid-s50523" xml:space="preserve"> per quã fereba <lb/>tur ſecũdũ extẽſionẽ ad modũ linearũ rectarũ.</s> <s xml:id="echoid-s50524" xml:space="preserve"> Lumine enim æ qualiter lato ք unũ corpus, & aliud, <lb/>niſi fiat aliqua diuerſitas ipſius luminis:</s> <s xml:id="echoid-s50525" xml:space="preserve"> nõ magis in uno ꝗ̃ in alio corpore ſentietur (alijs circũſtan <lb/>rijs in uiſu & remotione exiſtentibus æqualibus.</s> <s xml:id="echoid-s50526" xml:space="preserve">) Quòd ſi fiat diuerſitas luminis in radijs, reſpectu <lb/>diuerſorum corporum, ut patet per 4 huius:</s> <s xml:id="echoid-s50527" xml:space="preserve"> tunc in eo corpore, in quo magis radij diſgregãtur, mi <lb/>nus luminis apparet.</s> <s xml:id="echoid-s50528" xml:space="preserve"> Si ergo in aliquo corpore plus luminis apparebit:</s> <s xml:id="echoid-s50529" xml:space="preserve"> neceſſe eſt in illo corpore <lb/>radios plus aggregari.</s> <s xml:id="echoid-s50530" xml:space="preserve"> Patet ergo quòd nõ aggregatis radijs corporis luminoſi in corpore nõ lumi-<lb/>noſo plus, quàm in medio, lumẽ ſenſibilius fieri in alio corpore, quàm ſit in medio unius diaphani, <lb/>impoſsibile eſt.</s> <s xml:id="echoid-s50531" xml:space="preserve"> Ex quo patet, quòd ſi radij in aliquo corpore plus aggregẽtur quàm in medio, quòd <lb/>in illo corpore lumen ſenſibilius quàm in medio apparebit:</s> <s xml:id="echoid-s50532" xml:space="preserve"> & ſecundum quantitatem aggregatio, <lb/>nis radiorum lumen uidebitur intendi.</s> <s xml:id="echoid-s50533" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1886" type="section" level="0" n="0"> <head xml:id="echoid-head1385" xml:space="preserve" style="it">57. Radios corporis luminoſi per reflexionem uel refractionem aggregari palàm eſt.</head> <p> <s xml:id="echoid-s50534" xml:space="preserve">Iſtud patet per hoc.</s> <s xml:id="echoid-s50535" xml:space="preserve"> Quoniã cum radius reuerberatur uel reflectitur ab aliquo corpore:</s> <s xml:id="echoid-s50536" xml:space="preserve"> tũc quia <lb/>ք 20 th.</s> <s xml:id="echoid-s50537" xml:space="preserve"> 5 huius angulus incidẽtiæ eſt æqualis angulo reflexionis, & radius incidẽs & reflexus ſunt <lb/>in eadẽ ſuperficie, ut patet ք 27 th.</s> <s xml:id="echoid-s50538" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s50539" xml:space="preserve"> in ſuperficie ergo eadẽ radij duo ad ęquales angulos inci <lb/>dentes reflectuntur & uniuntur ſic, ut fiant unum:</s> <s xml:id="echoid-s50540" xml:space="preserve"> aggregantur ergo, quia duo obtinẽt unum locũ:</s> <s xml:id="echoid-s50541" xml:space="preserve"> <lb/>imò uerius fiunt unũ.</s> <s xml:id="echoid-s50542" xml:space="preserve"> Verbi gratia, ſit, ut in ſuperficie una reflexionis, quæ ſit a b c, incidãt duo radij <lb/>à diuerſis partibus diametri corporis luminoſi, ſcilicet a & c ad unum punctum corporis, à quo fit <lb/>reflexio:</s> <s xml:id="echoid-s50543" xml:space="preserve"> quod ſit b:</s> <s xml:id="echoid-s50544" xml:space="preserve"> & ſint anguli incidẽtiæ æquales.</s> <s xml:id="echoid-s50545" xml:space="preserve"> Producta ergo à puncto b linea in dicta ſuperfi <lb/>cie ad utramq;</s> <s xml:id="echoid-s50546" xml:space="preserve"> partẽ, ſcilicetea, quæ eſt cõmunis ſectio ſuperficiei reflexionis & ſuperficiei corpo-<lb/>ris, à quo fit reflexio, quæ ſit d b e:</s> <s xml:id="echoid-s50547" xml:space="preserve"> erit angulus incidentiæ, qui eſt a b d, æ qualis angulo reflexionis, <lb/>qui eſt c b e, per 20 th.</s> <s xml:id="echoid-s50548" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s50549" xml:space="preserve"> ſed & ſecundum angulum incidẽtię, qui eſt c b e fit reflexio radij c b:</s> <s xml:id="echoid-s50550" xml:space="preserve"> <lb/>ergo radius b a reflexus & radius b incidens efficiuntur <lb/>unus radius:</s> <s xml:id="echoid-s50551" xml:space="preserve"> & radius b c reflexus, radius quoq;</s> <s xml:id="echoid-s50552" xml:space="preserve"> a b inci-<lb/> <anchor type="figure" xlink:label="fig-0752-01a" xlink:href="fig-0752-01"/> dens efficiuntur unus.</s> <s xml:id="echoid-s50553" xml:space="preserve"> Sic aũt eſt de alijs omnibus, qui <lb/>incidunt ſecundum pyramidem, cuius conus eſt in ali-<lb/>quo puncto corporis, à quo fit reflexio, & baſis in corpo <lb/>reluminoſo.</s> <s xml:id="echoid-s50554" xml:space="preserve"> Patet ergo quòd ad minus omnes illi radij <lb/>in ſe duplicãtur.</s> <s xml:id="echoid-s50555" xml:space="preserve"> Vnde cum ipſi ſint infiniti, quoniã ſolũ <lb/>ſunt entes in potẽtia in cõtinuo, & tales pyramides ſunt <lb/>tot, quot ſunt puncta in corpore, à quo fit reflexio:</s> <s xml:id="echoid-s50556" xml:space="preserve"> patet <lb/>quòd ipſi ք reflexionẽ aggregãtur.</s> <s xml:id="echoid-s50557" xml:space="preserve"> Sed & ք refractionẽ in <lb/>medio ſecũdi diaphani lumẽ aggregari per experiẽtiã ſenſibiliter ad hibitã patere poteſt.</s> <s xml:id="echoid-s50558" xml:space="preserve"> Cũenim <lb/>oſtẽſum ſit qđ in medio ſecũdi diaphani dẽſioris aere à parte oppoſita ſuperficiei incidẽtię ſemper <lb/>fit radiorum aggregatio, imò cõcurſus in punctum unum, & ibi lumẽ & calorẽ generant:</s> <s xml:id="echoid-s50559" xml:space="preserve"> imò quòd <lb/>ignitionẽ efficiunt in corpore inflammabili, cui immorãtur, ut patet per 48 huius.</s> <s xml:id="echoid-s50560" xml:space="preserve"> Refractio itaque <lb/>lumen generat, quoniã adunat radios.</s> <s xml:id="echoid-s50561" xml:space="preserve"> Sed & in ſuperficie, à qua fit refractio, in proſundum corpo-<lb/>ris denſioris diaphani radius incidens & refractus (qui in medio unius diaphani producti, eſſent li <lb/>nea una) angulum refractionis conſtituunt:</s> <s xml:id="echoid-s50562" xml:space="preserve"> ſuntq́;</s> <s xml:id="echoid-s50563" xml:space="preserve"> per 46 th.</s> <s xml:id="echoid-s50564" xml:space="preserve"> 2 huius in una ſuperficie, quæ dicitur <lb/>ſuperficies refractionis, & eſt ſemք orthogonalis ſuper ſuperficiẽ corporis, à quo fit refractio per 2 <lb/>huius:</s> <s xml:id="echoid-s50565" xml:space="preserve"> unde tales radij omnes, ſic ſibijpſis incidentes, quando ſunt refracti, uicinantur & aggregan <lb/>tur, ſecundum diaphani ſecundi diſpoſitionẽ angulo refractionis ad angulum incidẽtiæ ſuæ uaria-<lb/>to.</s> <s xml:id="echoid-s50566" xml:space="preserve"> In groſsiori enim uel denſiori diaphano radius non perpendicularis magis debilitatur:</s> <s xml:id="echoid-s50567" xml:space="preserve"> unde ad <lb/>perpẽdicularẽ uehemẽtius refringitur, & in uiciniorẽ punctum axis cadit:</s> <s xml:id="echoid-s50568" xml:space="preserve"> angulus ergo fit acutior <lb/>angulo incidẽtiæ ſuæ, reſpectu eius, ſi ſecundum idẽ pũctum radius ſubtiliori diaphano incidiſſet.</s> <s xml:id="echoid-s50569" xml:space="preserve"> <lb/>Et ob hoc (quoniã angulus ex omnibus refractis radijs cum linea, quę eſt cõmunis ſectio ſuperfi-<lb/>ciei refractionis, & ſuperficiei corporis, à quo fit refractio, eſt minor in corporibus dẽſioris diapha-<lb/>ni quàm minus dẽſi) patet quòd in corporibus dẽſioribus & radij plus aggregãtur ꝗ̃ in minus den <lb/>ſis per 8 huius.</s> <s xml:id="echoid-s50570" xml:space="preserve"> Fit itaq;</s> <s xml:id="echoid-s50571" xml:space="preserve"> illorum radiorum aggregatio quandoq;</s> <s xml:id="echoid-s50572" xml:space="preserve"> propter lucis reflexionẽ ad punctũ <lb/>unum mathematicum uel naturalẽ, ut in nono libro huius ſcientiæ per ſpecula comburẽtia oſten-<lb/>dimus fieri aggregationem radiorum, & in alijs libris ubi de talibus ſermo fuit.</s> <s xml:id="echoid-s50573" xml:space="preserve"> Fit etiam hæc aggre <lb/>gatio quandoq;</s> <s xml:id="echoid-s50574" xml:space="preserve"> per refractionem:</s> <s xml:id="echoid-s50575" xml:space="preserve"> quoniam radij ſecundum æquales angulos incidentes, per 8 hu-<lb/>ius ſecundum æquales angulos refringuntur:</s> <s xml:id="echoid-s50576" xml:space="preserve"> & quandoque concurrunt in puncto uno, ut patet <lb/>per 48 th.</s> <s xml:id="echoid-s50577" xml:space="preserve"> huius.</s> <s xml:id="echoid-s50578" xml:space="preserve"> Semper autem in talibus & radij reflexi & refracti quodammodo in eadem parte <lb/>medij ſe duplicant:</s> <s xml:id="echoid-s50579" xml:space="preserve"> unde faciunt maius lumen.</s> <s xml:id="echoid-s50580" xml:space="preserve"> Aggregatis autem per refractionem radijs, ut pa-<lb/>tet ex præmiſsis:</s> <s xml:id="echoid-s50581" xml:space="preserve"> tunc in uiſu exiſtente in loco aggregationis lumen generatur.</s> <s xml:id="echoid-s50582" xml:space="preserve"> Et quoniam <lb/> <pb o="451" file="0753" n="753" rhead="LIBER DECIMVS."/> in corporibus diaphanis ſuperficiem lenem habentibus, denſioribus aere propter lenitatem ſuper-<lb/>ficiei lumen incidens ab ipſis reflectitur, ut oſtendimus per 1 th.</s> <s xml:id="echoid-s50583" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s50584" xml:space="preserve"> tunc patet quòd propterre <lb/>flexionem lumen aggregatur:</s> <s xml:id="echoid-s50585" xml:space="preserve"> & item quia in illis corporibus propter diuerſitatem dẽſioris diapha <lb/>ni fit luminis refractio ad perpendicularem intra corpus, ut patet per 4 huius:</s> <s xml:id="echoid-s50586" xml:space="preserve"> tunc in peripheria cu <lb/>iuslibet ſuperficiei refraction is propter acutum angulum refractionis ipſis adinuicem radijs uici-<lb/>natis fortificatur ſenſibilitas luminis.</s> <s xml:id="echoid-s50587" xml:space="preserve"> Quando ergo ſuperficies talium corporum ſunt lenes, ut po-<lb/>litæ per naturam:</s> <s xml:id="echoid-s50588" xml:space="preserve"> tunc licet in ipſis fiat refractio:</s> <s xml:id="echoid-s50589" xml:space="preserve"> ab eorum tamen ſuperficie fit etiam reflexio radio <lb/>rum, licet debiliter.</s> <s xml:id="echoid-s50590" xml:space="preserve"> Et propter hoc duabus his cauſsis concurrentibus, in ſuperficie corporũ taliũ <lb/>lumẽ aggregatur, & apparẽt corpora plurimũ luminoſa:</s> <s xml:id="echoid-s50591" xml:space="preserve"> quáuis magis dẽſa magis appareát lumino <lb/>ſa.</s> <s xml:id="echoid-s50592" xml:space="preserve"> Non ſunt aũt modi alij aggregationis radiorum, quá reflexio & refractio:</s> <s xml:id="echoid-s50593" xml:space="preserve"> ad hos enim, ut ad pri-<lb/>mos, ſi qui alij modi apparuerint, radicaliter reducuntur.</s> <s xml:id="echoid-s50594" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s50595" xml:space="preserve"/> </p> <div xml:id="echoid-div1886" type="float" level="0" n="0"> <figure xlink:label="fig-0752-01" xlink:href="fig-0752-01a"> <variables xml:id="echoid-variables859" xml:space="preserve">a c d b e</variables> </figure> </div> </div> <div xml:id="echoid-div1888" type="section" level="0" n="0"> <head xml:id="echoid-head1386" xml:space="preserve" style="it">58. Sine oppoſitione corporis denſioris, quàm ſit medium proximum radijs corporis lumino-<lb/>ſi: ipſorum radiorum reflexionem uel refractionem uel maiorem ſenſibιlitatem impoßibile <lb/>eſt fieri.</head> <p> <s xml:id="echoid-s50596" xml:space="preserve">Iſtud patet per hoc.</s> <s xml:id="echoid-s50597" xml:space="preserve"> Quoniá enim radij cuiuslibet corporis radioſi ſunt in ſe ſemper luminoſi & <lb/>uniformes:</s> <s xml:id="echoid-s50598" xml:space="preserve"> ſi ergo medium, per quod feruntur, ſit uniforme:</s> <s xml:id="echoid-s50599" xml:space="preserve"> nunquá reflectentur uel refringentur, <lb/>ſed ſemper ferentur in continuum & directum, ut patet per 1 th.</s> <s xml:id="echoid-s50600" xml:space="preserve"> 2 huius:</s> <s xml:id="echoid-s50601" xml:space="preserve"> nec lumen propter eorum <lb/>diſperſionem aggregabitur, ut uincat lumen, quod ex æquali diffuſione luminis receptum eſt in o-<lb/>culo uidentis.</s> <s xml:id="echoid-s50602" xml:space="preserve"> Nec etiam ad uiſum fiet reflexio, nec refractio in partem oppoſitam ad axem pyrami <lb/>dis uiſualis:</s> <s xml:id="echoid-s50603" xml:space="preserve"> nec lumen uel ſenſibilitas luminis maior efficietur.</s> <s xml:id="echoid-s50604" xml:space="preserve"> Patet ergo propoſitum, quòd ſine <lb/>oppoſitione corporis denſioris, quá ſit primũ medium, per quod fertur radius corporis luminoſi, i-<lb/>pſorum radiorum reflexionem uel refractionem fieri nõ eſt poſsibile:</s> <s xml:id="echoid-s50605" xml:space="preserve"> quoniam omnis reflexio uel <lb/>refractio ſemper fit ab aliquo talium corporum, ut eſt habitum expræmiſsis.</s> <s xml:id="echoid-s50606" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1889" type="section" level="0" n="0"> <head xml:id="echoid-head1387" xml:space="preserve" style="it">59. Quantitatem arcus circuli magniterræ, ſecundum quem illuminatur à ſole, poßibile est <lb/>declarari. Alhazen 5 n libride crepuſculis.</head> <p> <s xml:id="echoid-s50607" xml:space="preserve">Suppoſito ex his, quę alibi declarata ſunt per antiquos & nos, quòd corpus ſolis ſit maius corpo <lb/>re terræ:</s> <s xml:id="echoid-s50608" xml:space="preserve"> palàm per 27 th.</s> <s xml:id="echoid-s50609" xml:space="preserve"> 2 huius quoniam ſol <lb/>aſpicit terram ſecundum ſuperficiẽ terræ maio-<lb/>rem medietate ſuperficiei ipſius terrę.</s> <s xml:id="echoid-s50610" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s50611" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0753-01a" xlink:href="fig-0753-01"/> circulus, ſecundum quem terra illuminatur à ſo <lb/>le, qui b c d e, cuius centrum ſit a:</s> <s xml:id="echoid-s50612" xml:space="preserve"> & ſit circulus <lb/>maior ſolaris corporis, qui g h:</s> <s xml:id="echoid-s50613" xml:space="preserve"> cuius centrum <lb/>ſit f:</s> <s xml:id="echoid-s50614" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s50615" xml:space="preserve"> lineæ contingentes utrunq;</s> <s xml:id="echoid-s50616" xml:space="preserve"> ho <lb/>rum circulorum:</s> <s xml:id="echoid-s50617" xml:space="preserve"> quę ſint b h & e g.</s> <s xml:id="echoid-s50618" xml:space="preserve"> Portio itaq;</s> <s xml:id="echoid-s50619" xml:space="preserve"> <lb/>b c d e terræ eſt illuminata à ſole, quæ eſt maior <lb/>hemiſphærio.</s> <s xml:id="echoid-s50620" xml:space="preserve"> Ducantur itaq;</s> <s xml:id="echoid-s50621" xml:space="preserve"> lineæ a b & f h:</s> <s xml:id="echoid-s50622" xml:space="preserve"> <lb/>quæ erunt ęquidiſtantes per 28 p 1:</s> <s xml:id="echoid-s50623" xml:space="preserve"> quoniam u-<lb/>traq;</s> <s xml:id="echoid-s50624" xml:space="preserve"> ipſarum eſt perpendicularis ſuper lineam <lb/>b h utroſque circulos contingentem per 18 p 3.</s> <s xml:id="echoid-s50625" xml:space="preserve"> <lb/>Et quoniá linea h f eſt maior quàm linea b a (ut <lb/>patet ex ſuppoſitis) reſecetur à linea f h ęqualis <lb/>lineæ a b per 3 p 1:</s> <s xml:id="echoid-s50626" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s50627" xml:space="preserve"> h k æqualis ipſi a b:</s> <s xml:id="echoid-s50628" xml:space="preserve"> & du <lb/>catur linea a k:</s> <s xml:id="echoid-s50629" xml:space="preserve"> eritq́ue per 33 p 1 linea a k æqui-<lb/>diſtans lineæ h b:</s> <s xml:id="echoid-s50630" xml:space="preserve">ergo linea a k eſt perpendicu-<lb/>laris ſuper lineam f h.</s> <s xml:id="echoid-s50631" xml:space="preserve"> Et quia linea f h eſt 5 par-<lb/>tes & medietas partis ferè, ſecundũ quod linea <lb/>a b eſt pars una, ut demonſtratum eſt in Aſtro-<lb/>nomicis:</s> <s xml:id="echoid-s50632" xml:space="preserve"> remanet linea k f 4 partes & media.</s> <s xml:id="echoid-s50633" xml:space="preserve"> <lb/>Per eandem quoq;</s> <s xml:id="echoid-s50634" xml:space="preserve"> uiam aſtronomicam oſten-<lb/>ſum eſt, quòd ſecundum quantitatem, qua ſemi <lb/>diameter terræ eſt pars una, linea a f eſt partes <lb/>12 10:</s> <s xml:id="echoid-s50635" xml:space="preserve"> cum ſit diftantia ſolis à terra in medijs lon <lb/>gitudinibus eius.</s> <s xml:id="echoid-s50636" xml:space="preserve"> Si ergo ſecundum quantita-<lb/>tem, qua linea a f eſt 12 10 partes, linea f k eſt 4 <lb/>partes, & medietas partis:</s> <s xml:id="echoid-s50637" xml:space="preserve"> erit ſecundum quan-<lb/>titatem, qua linea a f eſt 120 partes, linea f k 29 <lb/>minuta, 12 ſecunda:</s> <s xml:id="echoid-s50638" xml:space="preserve"> & ſecundũ quantitatẽ qua <lb/>linea a f eſt 60 partes, linea f k eſt 14 minuta, <lb/>& 36 ſecunda.</s> <s xml:id="echoid-s50639" xml:space="preserve"> Circumſcripto ergo circulo illi <lb/>trigono orthogonio, qui eſt f k a, per 5 p 4:</s> <s xml:id="echoid-s50640" xml:space="preserve"> erit <lb/>arcus, quem ſubtendit chorda f k quaſi 13 minuta, & 56 ſecunda:</s> <s xml:id="echoid-s50641" xml:space="preserve"> ergo per 33 p 6 erit angulus k a f 13 <lb/>minuta, & 56 ſecunda, ſecundum quòd angulus rectus eſt 90 partes:</s> <s xml:id="echoid-s50642" xml:space="preserve"> arcus ergo c d erit 13 minuta, <lb/> <pb o="452" file="0754" n="754" rhead="VITELLONIS OPTICAE"/> & 56 ſecunda, ſecundum quod arcus b c eſt partes 90 per 33 p 6:</s> <s xml:id="echoid-s50643" xml:space="preserve"> quoniam angulus b a c eſt rectus <lb/>per 34 p 1:</s> <s xml:id="echoid-s50644" xml:space="preserve"> angulus enim k h b eſt rectus:</s> <s xml:id="echoid-s50645" xml:space="preserve"> totus ergo arcus b d erit 90 partes, 13 minuta, & 56 ſecun-<lb/>da:</s> <s xml:id="echoid-s50646" xml:space="preserve"> ſed arcus d e eſt æqualis arcui d b:</s> <s xml:id="echoid-s50647" xml:space="preserve"> totus ergo arcus b c d e eſt 180 partes, 27 minuta, & 52 ſecun-<lb/>da.</s> <s xml:id="echoid-s50648" xml:space="preserve"> Quod quærebamus.</s> <s xml:id="echoid-s50649" xml:space="preserve"/> </p> <div xml:id="echoid-div1889" type="float" level="0" n="0"> <figure xlink:label="fig-0753-01" xlink:href="fig-0753-01a"> <variables xml:id="echoid-variables860" xml:space="preserve">g f k h d c e a b</variables> </figure> </div> </div> <div xml:id="echoid-div1891" type="section" level="0" n="0"> <head xml:id="echoid-head1388" xml:space="preserve" style="it">60. Summorum uaporum conſiſtentiam ad quantum poßint eleuati pertingere, poßibile eſt <lb/>inueniri. Alhazen 6 n libri de crepuſculis.</head> <p> <s xml:id="echoid-s50650" xml:space="preserve">Ad hoc, quod hic proponitur, demonſtrandum, utemur conſuetis in ſcientia aſtrorum, ut in <lb/>præcedente.</s> <s xml:id="echoid-s50651" xml:space="preserve"> Sit itaq;</s> <s xml:id="echoid-s50652" xml:space="preserve"> per 69 th.</s> <s xml:id="echoid-s50653" xml:space="preserve"> 1 huius circulus, ſecundum quem ſuperficies plana tranſiens centrũ <lb/>ſolis & terræ, ſecat terram, circulus a b c:</s> <s xml:id="echoid-s50654" xml:space="preserve"> & ſit locus uiſus a:</s> <s xml:id="echoid-s50655" xml:space="preserve"> & ſit linea d a e contingens circulum.</s> <s xml:id="echoid-s50656" xml:space="preserve"> Et <lb/>quoniam angulus contingentiæ eſt indiuiſibilis, quia eſt minimus acutorum per 16 p 3:</s> <s xml:id="echoid-s50657" xml:space="preserve"> tunc patet <lb/>quòd uiſus non cadet ſub linea d a e, ſed tantùm ſupra illam.</s> <s xml:id="echoid-s50658" xml:space="preserve"> Et quoniam, ut patet per 27 th.</s> <s xml:id="echoid-s50659" xml:space="preserve"> 2 huius, <lb/>umbra terræ eſt pyramidalis:</s> <s xml:id="echoid-s50660" xml:space="preserve"> ſit illa pyramis umbrę terræ ante crepuſculum matutinum, quando <lb/>primò uidetur aer albeſcere in mane, c f e g:</s> <s xml:id="echoid-s50661" xml:space="preserve"> cuius uertex ſit f.</s> <s xml:id="echoid-s50662" xml:space="preserve"> Aer itaque cadens intra hanc pyrami-<lb/>dem non illuminatur à ſole, ſed radius ſolaris cadit ſuper omnem <lb/>aerem, qui eſt extra hanc pyramidem, quoniam ille nõ impeditur <lb/>per obſtaculum terræ.</s> <s xml:id="echoid-s50663" xml:space="preserve"> Non tamen uidetur uiſui illuminatum hoc, <lb/> <anchor type="figure" xlink:label="fig-0754-01a" xlink:href="fig-0754-01"/> quod eſt extra hãc pyramidem:</s> <s xml:id="echoid-s50664" xml:space="preserve"> quoniam (ut patet per 56 & 58 th.</s> <s xml:id="echoid-s50665" xml:space="preserve"> <lb/>huius) non fit luminis reflexio ab aere puro & ſubtili.</s> <s xml:id="echoid-s50666" xml:space="preserve"> Tria ſunt er <lb/>go, quæ in hac diſpoſitione res faciunt non uideri:</s> <s xml:id="echoid-s50667" xml:space="preserve"> ut ſi cadant ſub <lb/>linea contingente, & per uiſum tranſeunte:</s> <s xml:id="echoid-s50668" xml:space="preserve"> uel ſi cadant intra ſu-<lb/>perficiem conicam pyramidis umbræ terræ:</s> <s xml:id="echoid-s50669" xml:space="preserve"> uel ſi tanta ſit ſubtili-<lb/>tas materiæ corporum mediorum, ut ab ipſis non fiat reflexio ad <lb/>uiſum.</s> <s xml:id="echoid-s50670" xml:space="preserve"> Sit quoq;</s> <s xml:id="echoid-s50671" xml:space="preserve">, ut linea e a d contingens terram in puncto a cen <lb/>tro uiſus, ſecet ſuperficiem pyramidis illius umbrę in pũcto extra <lb/>pyramidem, quod ſit punctũe, ut propinquum umbræ.</s> <s xml:id="echoid-s50672" xml:space="preserve"> Aer ergo, <lb/>qui eſt apud punctum e, eſt inuiſibilis:</s> <s xml:id="echoid-s50673" xml:space="preserve"> non quòd cadat ſub linea <lb/>terram contingente:</s> <s xml:id="echoid-s50674" xml:space="preserve"> quoniam ille aer eſt in ſuperficie horizontis:</s> <s xml:id="echoid-s50675" xml:space="preserve"> <lb/>nec quòd cadat intra ſuperficiem pyramidis umbræ terrę:</s> <s xml:id="echoid-s50676" xml:space="preserve"> quoniã <lb/>eſt extra illam:</s> <s xml:id="echoid-s50677" xml:space="preserve"> ſed manet inuiſibilis propter ſubtilitatem materię <lb/>ſuę, quia non habet admixtionem uaporis denſioris aere, à quo re <lb/>flectatur lumen ſolis ad uiſum, ut patet per 56 huius.</s> <s xml:id="echoid-s50678" xml:space="preserve"> Imaginemur <lb/>ergo moueri ſolem uſq;</s> <s xml:id="echoid-s50679" xml:space="preserve"> ad principiũ crepuſculi matutini.</s> <s xml:id="echoid-s50680" xml:space="preserve"> Et quo-<lb/>niam uertex pyramidis umbræ terræ ad locum nadir ſolis ſemper <lb/>procedit, ut patet per 27 th.</s> <s xml:id="echoid-s50681" xml:space="preserve"> 2 huius, & ex cauſſa eclipſium lunariũ:</s> <s xml:id="echoid-s50682" xml:space="preserve"> <lb/>patet quòd illa pyramis omne corpus medium habet neceſſariò <lb/>tranſire.</s> <s xml:id="echoid-s50683" xml:space="preserve"> Sit ergo tunc pyramis umbræ terræ h i k:</s> <s xml:id="echoid-s50684" xml:space="preserve"> cuius uertex ſit <lb/>h:</s> <s xml:id="echoid-s50685" xml:space="preserve"> quæ interſecet lineam e d (quæ eſt diameter horizontis) in pũ <lb/>cto m.</s> <s xml:id="echoid-s50686" xml:space="preserve"> In hoc itaque puncto m, exſignificato ipſius nominis cre-<lb/>puſculi, primò uidebitur reflexum lumẽ ſolis, ut fiat ſenſibile.</s> <s xml:id="echoid-s50687" xml:space="preserve"> Hoc <lb/>autem neceſſe eſt accidere ex denſitate aeris inſpiſſati per natu-<lb/>ram uaporum:</s> <s xml:id="echoid-s50688" xml:space="preserve"> quia ab aere ſimplici non fit reflexio, ut patet ex <lb/>præmiſsis huius libri propoſitionibus:</s> <s xml:id="echoid-s50689" xml:space="preserve"> punctum ergo m eſt pun-<lb/>ctum altiſsim um, in quo conſiſtit eleuatio uaporum aerem ínſpiſ-<lb/>ſantium.</s> <s xml:id="echoid-s50690" xml:space="preserve"> Deſcribatur quoque conſequenter circulus alitudinis pertranſiens centrum ſolis in ho-<lb/>ra dicti crepuſculi:</s> <s xml:id="echoid-s50691" xml:space="preserve"> qui ſit a b c d:</s> <s xml:id="echoid-s50692" xml:space="preserve"> qui per 69 th.</s> <s xml:id="echoid-s50693" xml:space="preserve"> 1 huius ſecabit ſphæram terræ ſecundum circulum:</s> <s xml:id="echoid-s50694" xml:space="preserve"> <lb/>qui ſit e f g h, cuius centrum ſit k:</s> <s xml:id="echoid-s50695" xml:space="preserve"> ſitq́ue linea à centro terræ ad zenith capitis ducta, quæ ſit a e k:</s> <s xml:id="echoid-s50696" xml:space="preserve"> <lb/>ſitq́ue linea b k d perpendicularis ſuper lineam a k ſemidiam etrum circuli altitudinis:</s> <s xml:id="echoid-s50697" xml:space="preserve"> eritq́ue linea <lb/>b k d diameter cuiuſdam circuli, cuius ſuperficies per 18 p 11 erit erecta ſuper ſuperficiem circuli al-<lb/>titudinis ſecans ſphæram terræ in duo hæmiſphæria:</s> <s xml:id="echoid-s50698" xml:space="preserve"> nec eſt differentia ſenſibilis ſuperficiei huius <lb/>circuli à ſuperficie circuli horizontis.</s> <s xml:id="echoid-s50699" xml:space="preserve"> Sit itaque corporis ſolis centrum in puncto c:</s> <s xml:id="echoid-s50700" xml:space="preserve"> eritq́ue per <lb/>acceptionem aſtronomicam, ſcilicet inſtrumentalem armillarum uel aſtrolabij, uel tabularum to-<lb/>talis arcus b c, quo diſtat centrum ſolis ab ipſa ſuperficie horizontis ferè 19 partes, ſecundum <lb/>quod circulus altitudinis eſt 360.</s> <s xml:id="echoid-s50701" xml:space="preserve"> Et quoniam diameter ſolis eſt quintupla diametro terræ, & <lb/>eius continens medietatem:</s> <s xml:id="echoid-s50702" xml:space="preserve"> fiat circa centrum c circulus l m ſecundum diametrum quintu-<lb/>plam & continentem medietatem lineæ e k, quæ eſt ſemidiameter terræ.</s> <s xml:id="echoid-s50703" xml:space="preserve"> Erit quoque, ut pa-<lb/>tet ex præmiſsis, circulus l m maximus circulorum corporis ſolaris:</s> <s xml:id="echoid-s50704" xml:space="preserve"> producaturq́ue linea c k à <lb/>centro ſolis ad centrum terræ, ſecans ſuperficiem terræ in puncto g.</s> <s xml:id="echoid-s50705" xml:space="preserve"> Et quoniam longior radius <lb/>à corpore ſolis exiens, & ad terram pertingens quaſi linea contingens eſt per 16 th.</s> <s xml:id="echoid-s50706" xml:space="preserve"> 2 huius:</s> <s xml:id="echoid-s50707" xml:space="preserve"> du-<lb/>cantur duæ lineæ contingentes ambos circulos, ſolis ſcilicet & terræ, quæ ſint l f n & m h n, ſecun-<lb/>dum quas lineas per 27 th.</s> <s xml:id="echoid-s50708" xml:space="preserve"> 2 huius, continetur illuminatio ſolis & umbra terræ.</s> <s xml:id="echoid-s50709" xml:space="preserve"> Producatur quo-<lb/>que linea contingens circulum terræ in puncto e, quæ ſit p o:</s> <s xml:id="echoid-s50710" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s50711" xml:space="preserve"> linea m h n lineam p o, in pũ-<lb/>cto q:</s> <s xml:id="echoid-s50712" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s50713" xml:space="preserve"> punctum q locus luminoſus in tempore crepuſculi.</s> <s xml:id="echoid-s50714" xml:space="preserve"> Et quoniã punctus n, qui eſt uertex <lb/> <pb o="453" file="0755" n="755" rhead="LIBER DECIMVS."/> pyramidis umbrę, (quia ſemper eſt in nadir ſolis) ſecũdũ motũ ſolís declinat:</s> <s xml:id="echoid-s50715" xml:space="preserve"> patet qđ primũ, in qđ <lb/>radius ſolis cadit extra pyramidẽ, eſt ſummitas <lb/>uaporũ eleuatorũ à terra & aqua.</s> <s xml:id="echoid-s50716" xml:space="preserve"> Producatur <lb/>ergo linea k r q à cẽtro terræ ad ſummitatẽ ua-<lb/> <anchor type="figure" xlink:label="fig-0755-01a" xlink:href="fig-0755-01"/> porum, ſigneturq́;</s> <s xml:id="echoid-s50717" xml:space="preserve"> punctus r in ſuperficie terrę:</s> <s xml:id="echoid-s50718" xml:space="preserve"> <lb/>& ducantur lineę k f, k h.</s> <s xml:id="echoid-s50719" xml:space="preserve"> Eritq́;</s> <s xml:id="echoid-s50720" xml:space="preserve"> arcus f g h pars <lb/>terræ illuminata:</s> <s xml:id="echoid-s50721" xml:space="preserve"> cuius quantitas (ut patet per <lb/>præmiſſam) eſt 180 partium, 27 minutorum & <lb/>52 ſecundorum, ſecũdum quod totus circulus <lb/>e f g h eſt 360 partes:</s> <s xml:id="echoid-s50722" xml:space="preserve"> eritq́ue medietas ipſius, <lb/>quæ eſt f g, partes 90, & 13 minuta, & 56 ſecun-<lb/>da.</s> <s xml:id="echoid-s50723" xml:space="preserve"> Hæc eſt ergo quantitas anguli f k g, ſecundũ <lb/>quod 4 recti ſunt 360 partes:</s> <s xml:id="echoid-s50724" xml:space="preserve"> ſed angulus b k c <lb/>ex præmiſsis & per 33 p 6 eſt 19 partes:</s> <s xml:id="echoid-s50725" xml:space="preserve"> quoniã <lb/>eſt angulus crepuſcularis:</s> <s xml:id="echoid-s50726" xml:space="preserve"> remanet ergo angu-<lb/>lus h k b 71 partes, 13 minuta, & 56 ſecunda:</s> <s xml:id="echoid-s50727" xml:space="preserve"> ſed <lb/>angulus e k b eſt 90 partes, quoniam eſt rectus:</s> <s xml:id="echoid-s50728" xml:space="preserve"> <lb/>remanet ergo angulus e k h 18 partes, 46 minu <lb/>ta, 4 ſecunda.</s> <s xml:id="echoid-s50729" xml:space="preserve"> Et quoniá linea q e eſt æqualis li-<lb/>neæ q h per 58 th.</s> <s xml:id="echoid-s50730" xml:space="preserve"> 1 huius (quoniam ab uno pun <lb/>cto ducuntur eundem circulum contingẽtes) <lb/>erit per 8 p 1 angulus q k e ęqualis angulo q k h:</s> <s xml:id="echoid-s50731" xml:space="preserve"> <lb/>erit ergo angulus q k e 9 partes, 23 minuta, & 2 <lb/>ſecunda.</s> <s xml:id="echoid-s50732" xml:space="preserve"> Et quoniam angulus q e k eſt rectus ք <lb/>18 p 3:</s> <s xml:id="echoid-s50733" xml:space="preserve"> erit angulus k q e per 32 p 1 cóplementum <lb/>unius recti, hoc eſt 80 partes, 36 minuta, & 58 ſe <lb/>cunda, prout 4 recti ualent 360 partes:</s> <s xml:id="echoid-s50734" xml:space="preserve"> & ſecun <lb/>dũ quod duo recti ualent 360 partes, erit angu-<lb/>lus k q e 161 partes, 13 minuta, & 56 ſecunda.</s> <s xml:id="echoid-s50735" xml:space="preserve"> Cir <lb/>cumſcripto ergo circulo ipſi trigono q e k:</s> <s xml:id="echoid-s50736" xml:space="preserve"> erit <lb/>arcus, quem ſubtendit linea k e 161 partes, 13 mi <lb/>nuta, & 56 ſecunda:</s> <s xml:id="echoid-s50737" xml:space="preserve"> chorda ergo eius, quæ eſt li <lb/>nea k e, erit 118 partes, 23 minuta, & 20 ſecunda, <lb/>18 tertia, ſecundum quantitatem, qua diameter <lb/>q k eſt 120 partes:</s> <s xml:id="echoid-s50738" xml:space="preserve"> & ſecundũ quantitatem, qua <lb/>diameter q k eſt 60, erit chorda k e 59 partes, 11 <lb/>minuta, 40 ſecunda, 9 tertia:</s> <s xml:id="echoid-s50739" xml:space="preserve"> ergo ſecundum quantitatẽ, qua linea k e eſt 60, erit linea k q 60 partes, <lb/>& 48 minuta, & 50 ſecunda.</s> <s xml:id="echoid-s50740" xml:space="preserve"> Ablatis itaq;</s> <s xml:id="echoid-s50741" xml:space="preserve"> à linea k q partibus 60, quę eſt quantitas lineę k r ſemidia <lb/>metri terræ:</s> <s xml:id="echoid-s50742" xml:space="preserve"> remanet linea r q (quæ eſt ſumma uaporum eleuatio) 48 minuta, & 50 ſecunda, ſecun <lb/>dum illam quantitatem, qua diameter terræ eſt 120 partes.</s> <s xml:id="echoid-s50743" xml:space="preserve"> Et quoniam ſecundum coſmographos <lb/>maximus circulus terræ ſecundum milliaria eſt notus:</s> <s xml:id="echoid-s50744" xml:space="preserve"> ergo ſecundum illum quantitas diametri eſt <lb/>nota:</s> <s xml:id="echoid-s50745" xml:space="preserve"> ergo & linea r q eſt nota.</s> <s xml:id="echoid-s50746" xml:space="preserve"> Ethoc eſt propoſitum.</s> <s xml:id="echoid-s50747" xml:space="preserve"> Eſt aũt ſecundum cóputationem Abbomadi <lb/>ex milliarib.</s> <s xml:id="echoid-s50748" xml:space="preserve"> (quibus terrę circumferentia eſt 24 000 milliaria) linea r q 51 milliaria, 47 minuta, & 34 <lb/>ſecunda, & 31 tertia ferè.</s> <s xml:id="echoid-s50749" xml:space="preserve"> Summum ergo, ad quod eleuantur uapores ſecundum ipſorum conſiſten-<lb/>tiam, eſt minus quã 52000 paſſuum, ut patere poteſt perquirenti.</s> <s xml:id="echoid-s50750" xml:space="preserve"/> </p> <div xml:id="echoid-div1891" type="float" level="0" n="0"> <figure xlink:label="fig-0754-01" xlink:href="fig-0754-01a"> <variables xml:id="echoid-variables861" xml:space="preserve">h f d a m e c i k y b</variables> </figure> <figure xlink:label="fig-0755-01" xlink:href="fig-0755-01a"> <variables xml:id="echoid-variables862" xml:space="preserve">n a p e q o d r k h f g b r c m</variables> </figure> </div> </div> <div xml:id="echoid-div1893" type="section" level="0" n="0"> <head xml:id="echoid-head1389" xml:space="preserve" style="it">61. Ab aqua & aere denſo & uapore rorido reflexionem radiorum corporis luminoſi fieri <lb/>manifeſtum eſt.</head> <p> <s xml:id="echoid-s50751" xml:space="preserve">Iſtud in politis corporib.</s> <s xml:id="echoid-s50752" xml:space="preserve"> (ut in ſpeculis & ſimilibus) ſenſus comperit, nosq́;</s> <s xml:id="echoid-s50753" xml:space="preserve"> in pluribus pręmiſ-<lb/>ſis huius ſcientiæ libris iſtud ſumus eum amplitudine ſtudij perſequuti.</s> <s xml:id="echoid-s50754" xml:space="preserve"> In aqua uerò ſoli expoſita <lb/>idẽ patet:</s> <s xml:id="echoid-s50755" xml:space="preserve"> quia radius in parte ſoli oppoſita uidetur, & maximè ſi locus oppoſitus ſit obſcurus:</s> <s xml:id="echoid-s50756" xml:space="preserve"> hoc <lb/>aũt fit per reflexionẽ.</s> <s xml:id="echoid-s50757" xml:space="preserve"> In aere etiam aliqualiter dẽſiore idem euenit:</s> <s xml:id="echoid-s50758" xml:space="preserve"> ut quando inſpiſſatus eſt & con <lb/>ſiſtens quaſi in nubem:</s> <s xml:id="echoid-s50759" xml:space="preserve"> tunc enim ab ipſo fit luminis reflexio, ut apparet in crepuſculis ſerotinis & <lb/>matutinis.</s> <s xml:id="echoid-s50760" xml:space="preserve"> Huic etiam atteſtatur quòd tẽpore pluuiali radij ſolis ſępe in aere diſpergũtur, & uix te-<lb/>nuiter ad terrã pertingunt propter humiditatẽ & groſsiciẽ aeris contrapoſiti ipſi ſoli.</s> <s xml:id="echoid-s50761" xml:space="preserve"> Hoc etiã pa-<lb/>tet:</s> <s xml:id="echoid-s50762" xml:space="preserve"> quoniam in aere modicę denſitatis in hyeme, maximè flãte auftro circa lucernas frequenter ui-<lb/>detur lumen reflecti ſecundum formam circularem:</s> <s xml:id="echoid-s50763" xml:space="preserve"> & maximè uiſibus humidis, ad quos de facili fit <lb/>luminis reflexio & formarum, cum uirtus uiſiua propter debilitatem organi debilitatur, ſic quòd <lb/>non poteſt denſitatem modicam aeris penetrare, ſed ad ipſum forma rei uiſæ reflectitur ab aere mo <lb/>dicę denſitatis:</s> <s xml:id="echoid-s50764" xml:space="preserve"> ſicut ad uiſus fortes reflectitur ſolũ ab aliquo ſolido peruietatem non habente.</s> <s xml:id="echoid-s50765" xml:space="preserve"> Vn-<lb/>de etiam in uiſu aliquis debilitatus & non acutè uidẽs, propter ophthalmiã uel propter aliud, uidet <lb/>quandoq;</s> <s xml:id="echoid-s50766" xml:space="preserve"> imaginem ſuã in aere groſſo ante ſe, ſicut in ſpeculo, ſtantem contra ſe, & ambulantẽ cum <lb/>ipſo, quando ipſe ambulat, & reſpicientem ad ipſum.</s> <s xml:id="echoid-s50767" xml:space="preserve"> Et ſic quidã notus meus poſt plurium noctiũ <lb/>uigilias cum cõpulſus nocte ſequente equitaret, formã ſuam, hoc eſt uirũ alium ſecum equitantem <lb/> <pb o="454" file="0756" n="756" rhead="VITELLONIS OPTICAE"/> uidit, cum tranſiret quandá aquã, circa quam groſſus fuit aer, & cũ ſtaret, ſtetit & ille alius, & omnia <lb/>opera ipſius faciebat:</s> <s xml:id="echoid-s50768" xml:space="preserve"> cum autem ad aerẽ ſerenum uenit ille notus meus:</s> <s xml:id="echoid-s50769" xml:space="preserve"> tunc ſocius eius diſparuit, <lb/>quia non fuerat niſi forma ſua.</s> <s xml:id="echoid-s50770" xml:space="preserve"> Et ſic uiſui debili error accidit:</s> <s xml:id="echoid-s50771" xml:space="preserve"> nec mirum:</s> <s xml:id="echoid-s50772" xml:space="preserve"> quia & quandoq;</s> <s xml:id="echoid-s50773" xml:space="preserve"> ſanis ui <lb/>ſibus hoc accidit ab aere ſpiſſo & longè diſtante:</s> <s xml:id="echoid-s50774" xml:space="preserve"> ſicut etiam auxilio ſpeculorum (ut in 60 th.</s> <s xml:id="echoid-s50775" xml:space="preserve"> 7 hu-<lb/>ius oſtendimus) poſſet fieri, quòd aliquis imaginem propriam uel aliam non in ſpeculo, ſed extra <lb/>ſpeculum uideret in aere, in loco imaginis, qui per induſtriam poſſet ad locum certum uariari.</s> <s xml:id="echoid-s50776" xml:space="preserve"> In ua <lb/>pore etiã rorido fit reuerberatio luminis, quando incipit uapor aqueus diſſolui in guttas:</s> <s xml:id="echoid-s50777" xml:space="preserve"> quia quę-<lb/>libet ſuarum partium fit quaſi ſpeculum:</s> <s xml:id="echoid-s50778" xml:space="preserve"> & ob hoc lumẽ reflectitur ab ipſo:</s> <s xml:id="echoid-s50779" xml:space="preserve"> & iſtud apparet in aqua <lb/>guttatim ſparſa:</s> <s xml:id="echoid-s50780" xml:space="preserve"> quoniam ab illa lumen etiam ad partem oppoſitam reflectitur:</s> <s xml:id="echoid-s50781" xml:space="preserve"> quamuis poſt refle-<lb/>xionem coloretur.</s> <s xml:id="echoid-s50782" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s50783" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1894" type="section" level="0" n="0"> <head xml:id="echoid-head1390" xml:space="preserve" style="it">62. A ſuperficie aquæ & aeris denſi, & uaporis roridi, & ſimilibus refractionem fieri ad <lb/>perpendicularem patens eſt.</head> <p> <s xml:id="echoid-s50784" xml:space="preserve">Quod hic declarandum proponitur, patet ք 4 huius:</s> <s xml:id="echoid-s50785" xml:space="preserve"> ſed etiã experimẽtis cóprobatur:</s> <s xml:id="echoid-s50786" xml:space="preserve"> & hoc eſt <lb/>uniuerſale.</s> <s xml:id="echoid-s50787" xml:space="preserve"> Quando forma rei uel radius per mediũ rarius ad dẽſius diaphanum procedit:</s> <s xml:id="echoid-s50788" xml:space="preserve"> tũc ſemք <lb/>in medio ſecũdi diaphani fit refractio ad perpendicularẽ.</s> <s xml:id="echoid-s50789" xml:space="preserve"> Verbi gratia, expoſita a qua in uaſe ſoli, in <lb/>fundo uaſis uidebuntur radij aggregari.</s> <s xml:id="echoid-s50790" xml:space="preserve"> Luceſcente etiã ſole ſuper aerẽ denſum uiſui & ſoli interpo <lb/>ſitũ quãdoq;</s> <s xml:id="echoid-s50791" xml:space="preserve"> lux aggregatur, & maior calor peruenit in nobis, quãuis multa pars luminis ſuperius <lb/>ad nubes uicinas reflectitur:</s> <s xml:id="echoid-s50792" xml:space="preserve"> & hoc fit maximè in tẽpore præcedente tẽpus pluuiarũ:</s> <s xml:id="echoid-s50793" xml:space="preserve"> unde poſt talẽ <lb/>improportionatũ tempori calorem & lumen inſolitum ſæpius pluuia deſcendit.</s> <s xml:id="echoid-s50794" xml:space="preserve"> Ex quo patet, quia <lb/>nube in uaporẽ roridũ reſoluta, refractio fit radiorũ in ipſo uapore rorido, & ad nos perueniuntra-<lb/>dij ſolis aggregati per refractionem.</s> <s xml:id="echoid-s50795" xml:space="preserve"> Patet ergo quòd in aqua & aere denſo & uapore rorido, qñ for <lb/>ma uel lumẽ eſt in rariore diaphano, & incidit illis diaphanis denſiorib.</s> <s xml:id="echoid-s50796" xml:space="preserve"> diaphanũ quoq;</s> <s xml:id="echoid-s50797" xml:space="preserve">, in quo eſt <lb/>uiſus, multũ differt à diaphano, à quo fit refractio:</s> <s xml:id="echoid-s50798" xml:space="preserve"> tũc fiet refractio ſenſibilis ad perpẽdicularẽ.</s> <s xml:id="echoid-s50799" xml:space="preserve"> Q đ <lb/>ſi forma uel lumen ſit in denſiore diaphano, uel ultra denſius diaphanũ uideatur:</s> <s xml:id="echoid-s50800" xml:space="preserve"> tunc fiet refractio <lb/>à perpendiculari:</s> <s xml:id="echoid-s50801" xml:space="preserve"> & ob hoc omnia talia uiſui apparent maiora ſua certa quantitate, ut patet per 42 <lb/>huius.</s> <s xml:id="echoid-s50802" xml:space="preserve"> Et ob hoc accidit quòd ſummitates rerum in mari uiſarum refractè uidentur:</s> <s xml:id="echoid-s50803" xml:space="preserve"> eò quòd forma <lb/>ipſarũ diſpergitur à perpendiculari in ſecundo diaphano ſubtiliori, ſcilicet in aere, & uidẽtur formę <lb/>illorũ in cõcurſu lineę refractionis cũ perpẽdiculari ducta àre uiſa ad ſuperficiẽ aquę, ut patet ք 15 <lb/>th.</s> <s xml:id="echoid-s50804" xml:space="preserve"> huius:</s> <s xml:id="echoid-s50805" xml:space="preserve"> & denarius uidetur poſitus in uaſe ſub aqua in ea diſtãtia, in qua uiſus propter altitudinẽ <lb/>peripherię uaſis ſine aqua ipſum denariũ directè non uideret:</s> <s xml:id="echoid-s50806" xml:space="preserve"> & tunc uidetur etiã maior, quoniã ſub <lb/>maiori angulo uidetur.</s> <s xml:id="echoid-s50807" xml:space="preserve"> In aere etiã dẽſo, utpote qñ euri flãt, & aer humidus fit & ingroſſatur, omniũ <lb/>rerũ uidentur magnitudines maiores.</s> <s xml:id="echoid-s50808" xml:space="preserve"> Sol quoq;</s> <s xml:id="echoid-s50809" xml:space="preserve"> & omnia aſtra orientia & occidentia propter cali-<lb/>ginẽ & aerẽ uaporib.</s> <s xml:id="echoid-s50810" xml:space="preserve"> terræ ingroſſatum illis uiſibus interpoſitum, uidentur maiora, quàm in medio <lb/>cœli exiſtentia, ut patet per 54 huius:</s> <s xml:id="echoid-s50811" xml:space="preserve"> & hęc eſt cauſſa temporalis:</s> <s xml:id="echoid-s50812" xml:space="preserve"> alia uerò eſt perpetua, quam dixi-<lb/>musibidẽ.</s> <s xml:id="echoid-s50813" xml:space="preserve"> Ex hoc etiã prouenit quòd ſi in loco imaginis, uel inter imaginẽ & uiſum ponatur uitrũ <lb/>clarũ uel cryſtallus, ita utimago reflexa à ſpeculo ad certũ locum aeris uideatur per uitrũ:</s> <s xml:id="echoid-s50814" xml:space="preserve"> tũc enim <lb/>imago maior uidebitur, & ſecundũ qđ media diaphana multiplicata à dẽſiore in rarius fuerint, for <lb/>ma ſe uiſibus ita uicináte, qđ ultimò ipſa ք aerẽ uideatur:</s> <s xml:id="echoid-s50815" xml:space="preserve"> tunc forma maxima uidebitur:</s> <s xml:id="echoid-s50816" xml:space="preserve"> cuius ratio <lb/>patet ex præmiſsis pluribus theorematib.</s> <s xml:id="echoid-s50817" xml:space="preserve"> huius libri.</s> <s xml:id="echoid-s50818" xml:space="preserve"> In iſtis ergo corporib.</s> <s xml:id="echoid-s50819" xml:space="preserve"> medijs omnib.</s> <s xml:id="echoid-s50820" xml:space="preserve"> ſic diſpo <lb/>ſitis fit refractio à perpendiculari, ducta à cẽtro rei uiſæ ad ſuperficiẽ corporis diaphani rẽipſam uel <lb/>formã refractã continẽtis.</s> <s xml:id="echoid-s50821" xml:space="preserve"> His ergo modis fit in propoſitis corporib.</s> <s xml:id="echoid-s50822" xml:space="preserve"> uel ſimilib.</s> <s xml:id="echoid-s50823" xml:space="preserve"> ſibi ad uiſum refra-<lb/>ctio:</s> <s xml:id="echoid-s50824" xml:space="preserve"> inter hęc uerò maximè fit in aqua:</s> <s xml:id="echoid-s50825" xml:space="preserve"> magis aũt fit in uapore rorido incipiẽte aqua fieri, ꝗ̃ fiat ab ae <lb/>re:</s> <s xml:id="echoid-s50826" xml:space="preserve"> nec mirum:</s> <s xml:id="echoid-s50827" xml:space="preserve"> quia uapor roridus (qui fit tẽpore trãſmutationis nubiũ ex uapore cõtinuo in gutta <lb/>tim ſparſam aquã) eſt groſsior aere:</s> <s xml:id="echoid-s50828" xml:space="preserve"> unde in ipſa facta refractio plus ſentitur.</s> <s xml:id="echoid-s50829" xml:space="preserve"> Non poteſt aũt tunc fi-<lb/>gura rei uiſæ, cuius forma refringitur, diſtin ctè ad uiſum peruenire, propter refractionũ multitudi-<lb/>nẽ:</s> <s xml:id="echoid-s50830" xml:space="preserve"> ſed peruenit uiſui tantũ aliqua forma rei:</s> <s xml:id="echoid-s50831" xml:space="preserve"> ſicut patet etiã quòd in ſpeculis paruarũ partiũ uel ſu-<lb/>perficierum fractarũ alterius ſuper alterã eleuatarum, & ſi modicę pręeminentię ſint, ita tamẽ quòd <lb/>ſuperficies ipſorum ſpeculorum non ſint in eadem linea recta uel curua:</s> <s xml:id="echoid-s50832" xml:space="preserve"> tunc non apparet rei pro-<lb/>pria quantitas uel figura, ſed apparet tantũ color ipſius rei uiſæ, cuius forma reflectitur ab ipſis.</s> <s xml:id="echoid-s50833" xml:space="preserve"> Per <lb/>quod manifeſtè patet quòd forma corporis luminoſi, quæ ab aqua uel aere groſſo integrè, ſcilicet <lb/>quo ad figuram & lucem uel colorem reflectitur ad uiſum, à uapore rorido reflectitur, ſine figura & <lb/>quantitate certa, ſed tantũ cum ſuo colore uel lumine.</s> <s xml:id="echoid-s50834" xml:space="preserve"> Et ita, cum à uapore rorido fit reflexio ad ui-<lb/>ſum luminis ſolaris uel ſtellarum, non uidentur formarum reflexarum figuræ propriæ, ſed tantùm <lb/>ſormæ luminis reflexi.</s> <s xml:id="echoid-s50835" xml:space="preserve"> Patet ergo propoſitum</s> </p> </div> <div xml:id="echoid-div1895" type="section" level="0" n="0"> <head xml:id="echoid-head1391" xml:space="preserve" style="it">63. Omnis corporis ſphærici luminoſi irradiationem in corpore, (cuius ſuperficies æquidiſtat <lb/>ſuperficiei contingenti corpus luminoſum ſphæricum in puncto, ubi perpendicular is ducta à cen <lb/>tro corporis ſphærici ſuper ſuperficiem corporis illumin andi ſecat ſuperficiem corporis ſphærici) <lb/>poßibile eſt fieri ſecundum pyramidem rotundam, cuius baſis eſt in corpore irradiato, uertex ue <lb/>rò in centro corporis luminoſi. Ex quo patet omnem huiuſmodi irradiationem fieri ſecundũ an-<lb/>gulos incidentiæ æquales.</head> <p> <s xml:id="echoid-s50836" xml:space="preserve">Sit corpus luminoſum ſphæricum, in quo ſit circulus magnus, qui b c d:</s> <s xml:id="echoid-s50837" xml:space="preserve"> & eius centrũ ſit pun-<lb/> <pb o="455" file="0757" n="757" rhead="LIBER DECIMVS."/> ctum a:</s> <s xml:id="echoid-s50838" xml:space="preserve"> contingatq́;</s> <s xml:id="echoid-s50839" xml:space="preserve"> ipſum ſuperficies plana, quę ſit s p in puncto c:</s> <s xml:id="echoid-s50840" xml:space="preserve"> & ſit ſuperficies corporis illumi-<lb/>nandi à corpore ſphærico, ſuperficies g, quæ eſt ex hypotheſi æquidiſtãs ſuperficiei s p:</s> <s xml:id="echoid-s50841" xml:space="preserve"> & ſit linea a <lb/>c g ducta à centro corporis ſphærici perpendicularis ſuper dicti corporis ſuperficiem:</s> <s xml:id="echoid-s50842" xml:space="preserve"> dico quòd ir <lb/>radiationem illius corporis poſsibile eſt fieri ſecundum pyramidem rotundam, cuius baſis eſt in ſu-<lb/>perficie corporis g, uertex uerò in puncto a centro corporis luminoſi.</s> <s xml:id="echoid-s50843" xml:space="preserve"> Si enim perpendicularis a g <lb/>in centrum uel in medium ſuperficiei g non ceciderit:</s> <s xml:id="echoid-s50844" xml:space="preserve"> ducatur ad ipſius ſuperficiei g breuius extre-<lb/>mum linea a f:</s> <s xml:id="echoid-s50845" xml:space="preserve"> ſuper cuius terminũ in puncto a conſtitua-<lb/>tur angulus ex 23 p 1 æqualis angulo g a f, qui ſit g a h:</s> <s xml:id="echoid-s50846" xml:space="preserve"> pro-<lb/> <anchor type="figure" xlink:label="fig-0757-01a" xlink:href="fig-0757-01"/> ducaturq́;</s> <s xml:id="echoid-s50847" xml:space="preserve"> linea a h ad ſuperficiem g:</s> <s xml:id="echoid-s50848" xml:space="preserve"> & producantur in ſu-<lb/>perficie g lineę g f, & g h.</s> <s xml:id="echoid-s50849" xml:space="preserve"> Et quoniam duorum triangulo-<lb/>rum a g f & a g h anguli a g f & a g h, qui ſunt ad baſim, ſunt <lb/>ęquales ex definitione lineæ erectę ſuper ſuperficiẽ, & an-<lb/>guli g a f & g a h ſunt ęquales, & latus a g commune:</s> <s xml:id="echoid-s50850" xml:space="preserve"> patet <lb/>ex 26 p 1 quia latus a f erit æquale lateri a h, & f g æquale <lb/>g h.</s> <s xml:id="echoid-s50851" xml:space="preserve"> Similiter etiam facto alio angulo æquali g a f & g a h <lb/>angulis triãgulorum a g f & a g h, qui ſit g a k:</s> <s xml:id="echoid-s50852" xml:space="preserve"> productisq́;</s> <s xml:id="echoid-s50853" xml:space="preserve"> <lb/>lineis a k & g k:</s> <s xml:id="echoid-s50854" xml:space="preserve"> erit, ſicut in præcedentibus, linea a k ęqua <lb/>lis lineę a f uel a h, & erit linea g k æqualis lineę g f uel g h.</s> <s xml:id="echoid-s50855" xml:space="preserve"> <lb/>Cum ergo ex puncto g exeant tres lineæ ęquales & in ea-<lb/>dem ſuperficie:</s> <s xml:id="echoid-s50856" xml:space="preserve"> patet ex 9 p 3 lineam f h k ſecundum quan <lb/>titatem lineæ g f à puncto g productam eſſe circularem.</s> <s xml:id="echoid-s50857" xml:space="preserve"> <lb/>Quia ita que irradiatio fit ſecundum has lineas, ſcilicet a f, <lb/>a h, a k, & ſecundum alias omnes ducibiles, angulos æqua <lb/>les cum linea a g prædictorum triangulorum angulis, qui <lb/>ſunt ad punctum a, continentes, ut eſt linea a l, & aliæ:</s> <s xml:id="echoid-s50858" xml:space="preserve"> pa-<lb/>tet ex definitione pyramidis rotundæ, quoniam fit irradia <lb/>tio ſecundum pyramidem rotundam.</s> <s xml:id="echoid-s50859" xml:space="preserve"> Fit enim ſecundum <lb/>figuram, quæ deſcribi poſsit per triangulum a g f orthogo <lb/>nium latere a g fixo manente, & a f & g f lateribus reuolu-<lb/>tis ad locum, unde inceperant moueri.</s> <s xml:id="echoid-s50860" xml:space="preserve"> Et ex pręmiſsis pa-<lb/>tet quoniam huius irradiatio ſemper fit ſecundum angu-<lb/>los incidentiæ æquales.</s> <s xml:id="echoid-s50861" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s50862" xml:space="preserve"> Si dicatur <lb/>quòd etiam fit irradiatio extra hanc pyramidem:</s> <s xml:id="echoid-s50863" xml:space="preserve"> hoc eſt <lb/>uerum:</s> <s xml:id="echoid-s50864" xml:space="preserve"> ſed quia natura lucis eſt ſemper æqualiter diffundi, ut patet per 20 th.</s> <s xml:id="echoid-s50865" xml:space="preserve"> 2 huius:</s> <s xml:id="echoid-s50866" xml:space="preserve"> tunc fiet ad <lb/>omnem partem ſuperficiei g ſecundum pyramidem uel ſecundum partem pyramidis in ipſa rece-<lb/>ptam irradiatio (parte alia pyramidis ad ſuperficiem corporis non illuminabilis protenſa.</s> <s xml:id="echoid-s50867" xml:space="preserve">) Vnde ſi <lb/>pars illuminata extra ſignatam pyramidem modica fuerit:</s> <s xml:id="echoid-s50868" xml:space="preserve"> nó fiet in ea ſenſibilis irradiatio propter <lb/>radiorum paucitatem:</s> <s xml:id="echoid-s50869" xml:space="preserve"> quæ ſi magna fuerit, cum in ipſa ad ęquales angulos multi radij conueniant:</s> <s xml:id="echoid-s50870" xml:space="preserve"> <lb/>tunc irradiatio ſenſibilis erit propter multorum radiorum concurſum & æqualitatem angulorum.</s> <s xml:id="echoid-s50871" xml:space="preserve"> <lb/>Et ſic eſt poſsibile propter lucis unigenitatem irradiationem fieri ſecũdum lineam circularem, quę <lb/>ſit terminus baſis pyramidis uel partis baſis.</s> <s xml:id="echoid-s50872" xml:space="preserve"> Eodem autem modo demonſtrandum, ſi ſuperficies g <lb/>æquidiſtet ſuperficiei s p contingenti corpus lumiaoſum in b, d punctis, uel in alijs punctis ſigna-<lb/>tis.</s> <s xml:id="echoid-s50873" xml:space="preserve"> Vniuerſaliter autem corporum, quæ ſplendorem ſenſibilem à corpore aliquo luminoſo acci-<lb/>piunt, oportet quòd ſit talis aſpectus ad corpus luminoſum, ut theorema ſupponit:</s> <s xml:id="echoid-s50874" xml:space="preserve"> ſcilicet æqui-<lb/>diſtantia ad ſuperficiem planam contingentem corpus lumino ſum in puncto, ubi perpendicularis <lb/>ducta à centro corporis luminoſi ad ſuperficiem corporis illuminandi ſecat ſuperficiem corporis <lb/>luminoſi.</s> <s xml:id="echoid-s50875" xml:space="preserve"> Et huius ſignum eſt irradiatio lunæ, quæ nunquam, niſi in parte ſoli oppoſita illumina-<lb/>tur:</s> <s xml:id="echoid-s50876" xml:space="preserve"> & ſemper medietas illius, ea ſcilicet, quæ ſolem reſpicit, eſt illuminata neceſſariò propter natu-<lb/>ram præmiſsi aſpectus:</s> <s xml:id="echoid-s50877" xml:space="preserve"> aliam uerò partem irradiatio ſolis, niſi fortè per refractionem, nullatenus <lb/>attingit.</s> <s xml:id="echoid-s50878" xml:space="preserve"> Et quoniam pyramides uerticem habentes in centro corporis luminoſi, ad infinitas ba-<lb/>ſes in corpore irradiando una baſi alteri inſcripta applicantur:</s> <s xml:id="echoid-s50879" xml:space="preserve"> ideo tota ſuperficies irradiati corpo-<lb/>ris corpus luminoſum aſpiciens multiformiter irradiatur, & augmentatur irradiatio:</s> <s xml:id="echoid-s50880" xml:space="preserve"> quoniam o-<lb/>portet ut tale corpus ſit denſius medio, per quod lumen uenit ad ipſum:</s> <s xml:id="echoid-s50881" xml:space="preserve"> oportet enim quòd tale <lb/>corpus habeat aliquid denſitatis.</s> <s xml:id="echoid-s50882" xml:space="preserve"> Vnde ſi lumen nihil haberet reſiſtentiæ, trãſiret, nec corpus per-<lb/>tranſitum irradiaret:</s> <s xml:id="echoid-s50883" xml:space="preserve"> aliter etiam in ipſo non fieret reflexio uel refractio per 58 huius.</s> <s xml:id="echoid-s50884" xml:space="preserve"> Et quoniam <lb/>per reflexionem radij aggregantur, & ſimiliter per refractionem ex 57 huius:</s> <s xml:id="echoid-s50885" xml:space="preserve"> tunc per 56 hu-<lb/>ius radijs non aggregatis plus ſenſibilis non fieret irradiatio quàm in medio:</s> <s xml:id="echoid-s50886" xml:space="preserve"> nunc autem irradia-<lb/>tio in theoremate ſupponitur:</s> <s xml:id="echoid-s50887" xml:space="preserve"> patet ergo quòd oportet corpus irradiandum eſſe denſius quàm ſit <lb/>corpus propinquum corpori luminoſo.</s> <s xml:id="echoid-s50888" xml:space="preserve"> Exemplariter uerò id declarari poteſt per hoc, quod <lb/>in 37 th.</s> <s xml:id="echoid-s50889" xml:space="preserve"> 2 huius oſtendimus.</s> <s xml:id="echoid-s50890" xml:space="preserve"> Quia ſi per foramen rotundum penetret radius ſolis:</s> <s xml:id="echoid-s50891" xml:space="preserve"> ſtatim in cor-<lb/>pore oppoſito ad baſim applicatur, & in formam pyramidis lumen figuratur.</s> <s xml:id="echoid-s50892" xml:space="preserve"> Signum ergo eſt quòd <lb/>in quolibet radio corporis luminoſi idem fiat, qui cum ſint naturæ homogeneæ, eadem eſt natura <lb/>in toto & in parte:</s> <s xml:id="echoid-s50893" xml:space="preserve"> & ad minus, ſi illud non ſit neceſſarium ſemper fieri:</s> <s xml:id="echoid-s50894" xml:space="preserve"> eſt tamen poſsibile fieri, ut <lb/>proponitur.</s> <s xml:id="echoid-s50895" xml:space="preserve"> Patet ergo intentum.</s> <s xml:id="echoid-s50896" xml:space="preserve"/> </p> <div xml:id="echoid-div1895" type="float" level="0" n="0"> <figure xlink:label="fig-0757-01" xlink:href="fig-0757-01a"> <variables xml:id="echoid-variables863" xml:space="preserve">d a b p c s k h y f l</variables> </figure> </div> <pb o="456" file="0758" n="758" rhead="VITELLONIS OPTICAE"/> </div> <div xml:id="echoid-div1897" type="section" level="0" n="0"> <head xml:id="echoid-head1392" xml:space="preserve" style="it">64. Si ad idem cẽtrum uiſus ab aliqua ſuperficie fiat luminis refractio uel reflexio: neceſſe eſt <lb/>extremum illius luminis ſuperficiei uiſus circulariter ſecundum rotundam pyramidem incide-<lb/>re. Ex quo patet tunc centrum corporis irr adiantis, & centrum uiſus, centrṹ circuli baſis py-<lb/>ramidis irradiationis refractæ uel reflexæ in eadem recta linea conſistere oportere.</head> <p> <s xml:id="echoid-s50897" xml:space="preserve">Suppoſito quòd aliquod corpus irradiatum ſit inter uiſum & inter corpus luminoſum irradiãs:</s> <s xml:id="echoid-s50898" xml:space="preserve"> <lb/>& ſit illud medium corpus diaphanum, ita quòd radij refracti in centro uiſus ualeant aggregari:</s> <s xml:id="echoid-s50899" xml:space="preserve"> ali-<lb/>ter enim non uideretur irradiatio.</s> <s xml:id="echoid-s50900" xml:space="preserve"> Sit quo que centrum corporis irradiantis a:</s> <s xml:id="echoid-s50901" xml:space="preserve"> ſuperficiesq́;</s> <s xml:id="echoid-s50902" xml:space="preserve"> corpo-<lb/>ris irradiati ſit f h i k:</s> <s xml:id="echoid-s50903" xml:space="preserve"> perpendicularis ducta à centro corporis luminoſi ſuper illam ſuperficiem ſit <lb/>a g:</s> <s xml:id="echoid-s50904" xml:space="preserve"> & ducantur lineæ a f, a h, a i, a k:</s> <s xml:id="echoid-s50905" xml:space="preserve"> & lineę g f, g h, g i, g k:</s> <s xml:id="echoid-s50906" xml:space="preserve"> & ſit centrum uiſus b:</s> <s xml:id="echoid-s50907" xml:space="preserve"> ducanturq́;</s> <s xml:id="echoid-s50908" xml:space="preserve"> lineæ b f, <lb/>b h, b i, b k, b g.</s> <s xml:id="echoid-s50909" xml:space="preserve"> Quoniã itaque, ut patet ex hypotheſi, lumẽ corporis irradiantis per refractionem ui <lb/>detur in puncto b:</s> <s xml:id="echoid-s50910" xml:space="preserve"> & per 3 huius perpendicularis non refringitur, ſed trãſit ad angulos rectos, ut in-<lb/>cidebat ad lineas f g, h g, i g, k g, & in uno puncto, ut in centro oculi, concurrunt plures radij refra-<lb/>cti, qui obliquè incidunt illi ſuperficiei ex hypotheſi:</s> <s xml:id="echoid-s50911" xml:space="preserve"> qua autẽ ratione aliquis radius refractus per-<lb/>uenit ad centrum uiſus, eadem ratione omnes radij incidentes ſuperficiei corporis f h i k, ſecundũ <lb/>circulum (cuius centrum eſt punctum g) refracti perueniunt ad centrum uiſus, ut patuit in 48 hu-<lb/>ius:</s> <s xml:id="echoid-s50912" xml:space="preserve"> ſunt enim illi anguli incidentiæ omnes æquales, ut patet per præmiſſam:</s> <s xml:id="echoid-s50913" xml:space="preserve"> ergo & anguli refra-<lb/>ctionis omnes erunt æquales per 8 huius.</s> <s xml:id="echoid-s50914" xml:space="preserve"> In centro ergo unius ui-<lb/> <anchor type="figure" xlink:label="fig-0758-01a" xlink:href="fig-0758-01"/> ſus nulli radij extremi concurrunt, niſi qui refringuntur ſecundum <lb/>angulos æquales.</s> <s xml:id="echoid-s50915" xml:space="preserve"> Sit ergo, ut ſit illa refractio ſecundum aliquos an <lb/>gulos extremos, qui ſint b f g, b h g, b k g, b i g:</s> <s xml:id="echoid-s50916" xml:space="preserve"> erunt ergo illi anguli <lb/>æquales:</s> <s xml:id="echoid-s50917" xml:space="preserve"> ſed & anguli ad punctum g ſub linea b g & ſub lineis f g, <lb/>h g, k g, i g, ſuntæ quales:</s> <s xml:id="echoid-s50918" xml:space="preserve"> quia ſunt recti.</s> <s xml:id="echoid-s50919" xml:space="preserve"> Sunt ergo trigona b g f, b g <lb/>h, b g k, b g i æquiãgula per 32 p1:</s> <s xml:id="echoid-s50920" xml:space="preserve"> ergo per 4 p 6 ipſorum latera ſunt <lb/>proportionalia:</s> <s xml:id="echoid-s50921" xml:space="preserve"> ſed latus b g eſt æ quale ſibijpſi, cum omnib.</s> <s xml:id="echoid-s50922" xml:space="preserve"> ſit illis <lb/>trigonis commune:</s> <s xml:id="echoid-s50923" xml:space="preserve"> latera ergo b f, b h, b k, b i ſunt æqualia inter ſe, <lb/>& latera g f, g h, g k, g i ſunt inter ſe æqualia.</s> <s xml:id="echoid-s50924" xml:space="preserve"> Ergo per 9 p 3 linea h f <lb/>i k eſt perpheria circuli, cuius centrum eſt punctum g:</s> <s xml:id="echoid-s50925" xml:space="preserve"> & ſic deſcri-<lb/>bitur in oculi ſuperficie.</s> <s xml:id="echoid-s50926" xml:space="preserve"> Fit ergo pyramis refracta, cuius uertex eſt <lb/>in puncto b centro uiſus, & eius baſis eſt in illuminata ſuperficie:</s> <s xml:id="echoid-s50927" xml:space="preserve"> <lb/>eſtq́;</s> <s xml:id="echoid-s50928" xml:space="preserve"> alia pyramis illuminationis, cuius uertex eſt in puncto a cen-<lb/>tro luminoſi, & eius baſis eſt etiam circulus f h i k.</s> <s xml:id="echoid-s50929" xml:space="preserve"> Patet ergo quòd <lb/>iſtarum duarum pyramidum lineæ g f, g h, g i, g k ſunt in eadem ſu-<lb/>perficie, ut prius:</s> <s xml:id="echoid-s50930" xml:space="preserve"> quoniam ab eiſdem lineis, in quas radius incidit, <lb/>etiam refringitur.</s> <s xml:id="echoid-s50931" xml:space="preserve"> Vna eſt ergo ſuperficies communis terminans i-<lb/>ſtas duas pyramides, quæ eſt circulus f h i k:</s> <s xml:id="echoid-s50932" xml:space="preserve"> & eſt baſis ambarum il <lb/>larum pyramidum.</s> <s xml:id="echoid-s50933" xml:space="preserve"> Patet etiam hoc ex 5 p 11:</s> <s xml:id="echoid-s50934" xml:space="preserve"> quia illæ lineæ ſecun-<lb/>dum unum punctum, qui eſt g, cum linea b a angulos rectos faciũt.</s> <s xml:id="echoid-s50935" xml:space="preserve"> <lb/>Angulus enim f g b eſt æqualis angulo f g a:</s> <s xml:id="echoid-s50936" xml:space="preserve"> quoniam uterque ipſo-<lb/>rum eſt rectus, ex eo quòd ſuppoſitum eſt angulum a g f eſſe rectũ:</s> <s xml:id="echoid-s50937" xml:space="preserve"> <lb/>eritq́ue ſuperficies, in qua ſunt lineę f g, h g, i g, k g orthogonalis ſu-<lb/>per ſuperficies omnes refractionis.</s> <s xml:id="echoid-s50938" xml:space="preserve"> Patet ergo unum propoſitorũ.</s> <s xml:id="echoid-s50939" xml:space="preserve"> <lb/>Quòd ſi centrum uiſus fuerit inter corpus irradiatum, & corpus ir-<lb/>radians conſtitutum:</s> <s xml:id="echoid-s50940" xml:space="preserve"> tunc eadem diſpoſitione manente, niſi ſolo <lb/>puncto b inter a & g puncta conſtituto, patet propoſitum ex eo:</s> <s xml:id="echoid-s50941" xml:space="preserve"> <lb/>quòd tunc corpus irradiatum non uidetur niſi per reflexionem lu-<lb/>minis recepti à corpore luminoſo:</s> <s xml:id="echoid-s50942" xml:space="preserve"> & ſemper angulus incidentiæ <lb/>erit æqualis angulo reflexionis per 20th.</s> <s xml:id="echoid-s50943" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s50944" xml:space="preserve"> quia angulus extrinſecus angulo a g fin triangu-<lb/>lo a g f pyramidis illuminationis erit æqualis angulo b f g, qui fit ad baſim trianguli b f g pyrami-<lb/>dis reflexionis:</s> <s xml:id="echoid-s50945" xml:space="preserve"> nec erit poſsibilis uiſio irradiationis niſi in puncto axis pyramidis illuminationis:</s> <s xml:id="echoid-s50946" xml:space="preserve"> <lb/>ubi ſecundum æquales angulos reflexi radij à tota ſuperficie illuminati corporis concurrunt:</s> <s xml:id="echoid-s50947" xml:space="preserve"> e-<lb/>runtq́ue omnes anguli triangulorum pyramidis reflexionis, qui ſunt ad baſim, æquales inter ſe per <lb/>20th.</s> <s xml:id="echoid-s50948" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s50949" xml:space="preserve"> quoniam anguli extrinſeci pyramidis irradiationis, qui ſunt anguli incidentiæ, o-<lb/>mnes ſunt æquales inter ſe.</s> <s xml:id="echoid-s50950" xml:space="preserve"> Omnes ita que radij ad uiſum reflexi, qui ſunt in eadem ſuperficie, per <lb/>6 p 1 erunt æquales.</s> <s xml:id="echoid-s50951" xml:space="preserve"> Et quoniam lineæ f g, h g, i g, i g, k g ſunt æquales:</s> <s xml:id="echoid-s50952" xml:space="preserve"> patet per 9 p 3 lineam f h i k eſſe <lb/>peripheriam circuli:</s> <s xml:id="echoid-s50953" xml:space="preserve"> quod eſt ſecundum propoſitum.</s> <s xml:id="echoid-s50954" xml:space="preserve"> Et quoniam linea b g, quæ eſt perpendicula-<lb/>ris ſuper illam ſuperficiem, omnibus illis trigonis eſt communis, & angulus cuiuslibet triangulo-<lb/>rum, qui ſunt ad baſim, æqualis eſt alteri ſibi correſpondenti per 4 p 1, cum lineæ f g, h g, i g, k g ſint <lb/>adinuicem æquales, ut declaratum eſt prius, & ab ipſis fiat reflexio ad uiſum:</s> <s xml:id="echoid-s50955" xml:space="preserve"> patet per 106 th.</s> <s xml:id="echoid-s50956" xml:space="preserve"> 1 hu-<lb/>ius quia erit per radios ab ipſis reflexos pyramis inſcripta pyramidi ad eandem baſim, ſed diuerſæ <lb/>altitudinis:</s> <s xml:id="echoid-s50957" xml:space="preserve"> quoniam punctus b, qui eſt centrum uiſus, poſitus eſt eſſe inter corpus irradians & <lb/>corpus irradiatum:</s> <s xml:id="echoid-s50958" xml:space="preserve"> & erit illa baſis communis duabus pyramidibus, ſcilicet pyramidi irradiatio-<lb/>nis & pyramidi reflexionis orthogonalis ſuper omnes ſuperficies reflexionis.</s> <s xml:id="echoid-s50959" xml:space="preserve"> Patet ergo, quod co-<lb/>rollario proponebatur per 107 th.</s> <s xml:id="echoid-s50960" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s50961" xml:space="preserve"> Viſum eſt etiam quibuſdam ad propoſitam uiſorum <lb/> <pb o="457" file="0759" n="759" rhead="LIBER DECIMVS."/> circulationem coadunare circulationem foraminis uueæ, ac ſi ad peripheriam foraminis ſolùm ra-<lb/>dij incidant:</s> <s xml:id="echoid-s50962" xml:space="preserve"> & ſic in ſuperficie uiſus rotundentur.</s> <s xml:id="echoid-s50963" xml:space="preserve"> Quòd etſi ſit aliquando poſsibile, non tamen eſt <lb/>uniuerſaliter neceſſarium:</s> <s xml:id="echoid-s50964" xml:space="preserve"> quia etiam cuicunq;</s> <s xml:id="echoid-s50965" xml:space="preserve"> parti ſuperficiei uiſus radij incidant ſecundum an-<lb/>gulos æquales:</s> <s xml:id="echoid-s50966" xml:space="preserve"> ſemper accidet neceſſario figuram uideri circula-<lb/>rem.</s> <s xml:id="echoid-s50967" xml:space="preserve"> Ex iſtis itaq, manifeſtè patet, quia etſi tota ſuperficies alicu-<lb/> <anchor type="figure" xlink:label="fig-0759-01a" xlink:href="fig-0759-01"/> ius corporis irregularis uel regularis, rectilinea uel circularis ſit <lb/>irradiata:</s> <s xml:id="echoid-s50968" xml:space="preserve"> non tamen uidebitur niſi circularis pars eius irradiata, <lb/>quando per reflexionem uel refractionem uidetur.</s> <s xml:id="echoid-s50969" xml:space="preserve"> Quia oportet <lb/>ad hoc, quòd uiſus ipſum iudicet irradiatum, radios plures in cen <lb/>tro oculi aggregari:</s> <s xml:id="echoid-s50970" xml:space="preserve"> non autem concurrunt niſi illi, qui inci dentes <lb/>ad ſuperficiem corporis irradiati & reflexi ad centrũ oculi omnes <lb/>æquales angulos conſtituunt:</s> <s xml:id="echoid-s50971" xml:space="preserve"> tales autem in cidunt ſecundum cir <lb/>culum:</s> <s xml:id="echoid-s50972" xml:space="preserve"> faciunt enim pyramidem, ut patet ex præmiſſa, & refle-<lb/>ctuntur uel refringuntur neceſſariò ſecundũ circulum eundem.</s> <s xml:id="echoid-s50973" xml:space="preserve"> <lb/>Ergo ſuperficies illius corporis ſemper uidebitur circulariter irra <lb/>diata:</s> <s xml:id="echoid-s50974" xml:space="preserve"> nec uidebit uiſus illam irradiationem, niſi fuerit in puncto <lb/>concurſus linearum taliter reflexarũ conſtitutus.</s> <s xml:id="echoid-s50975" xml:space="preserve"> Et propter hoc <lb/>in eadem ſuperficie irràdiati corporis diuerſis uiſibus diuerſi ap-<lb/>parebunt circuli:</s> <s xml:id="echoid-s50976" xml:space="preserve"> quia eæ dem lineæ in diuerſis punctis non con-<lb/>currunt, ſed in uno tantùm:</s> <s xml:id="echoid-s50977" xml:space="preserve"> & remotioribus maiores apparebunt <lb/>circuli:</s> <s xml:id="echoid-s50978" xml:space="preserve"> ſcllicet illi, quibus ad maiores angulos incidebant radij, & <lb/>ad maiores reflectuntur uel refringuntur:</s> <s xml:id="echoid-s50979" xml:space="preserve"> & ſunt exteriores in pe <lb/>ripheria baſis.</s> <s xml:id="echoid-s50980" xml:space="preserve"> Sic ergo pyramis interior, ſcilicet reflexionis uel re <lb/>fractionis inſcribitur pyramidi alteri reflexionis uel refractionis <lb/>minorẽ exterius ambienti:</s> <s xml:id="echoid-s50981" xml:space="preserve"> centrumq́;</s> <s xml:id="echoid-s50982" xml:space="preserve"> uiſus propinquius ſuperfi-<lb/>ciei irradiatæ minorẽ uidebit circulũ, ꝗ̃ uiſus remotior:</s> <s xml:id="echoid-s50983" xml:space="preserve"> quoniã ra <lb/>dij in minori circulo ſecundũ angulos minores incidunt, & ſecun <lb/>dum angulos minores reflectuntur per 20th.</s> <s xml:id="echoid-s50984" xml:space="preserve"> 5 huius, uel ſecundũ <lb/>minores angulos refringuntur per 8 huius.</s> <s xml:id="echoid-s50985" xml:space="preserve"> Patet aũt ք 106 th.</s> <s xml:id="echoid-s50986" xml:space="preserve"> 1 hu <lb/>ius quia ſecundũ quòd angulus refractionis uel reflexionis plus <lb/>minuitur, ſecundum hoc angulus in uiſu contentus augmenta-<lb/>tur.</s> <s xml:id="echoid-s50987" xml:space="preserve"> Et quia angulus refractionis uel reflexionis ſemper eſt acutus <lb/>rectilineus diuiſibilis:</s> <s xml:id="echoid-s50988" xml:space="preserve"> propter hoc angulus in oculo ſemper eſt acutus, nec ad rectum poteſt excre <lb/>ſcere, ut quartã partẽ circuli altitudinis ſibi faciat reſpõdere:</s> <s xml:id="echoid-s50989" xml:space="preserve"> quoniã inter angulos cauſſantes pyra <lb/>midem ille angulus in oculo & angulus reflexionis uel refractionis ualent unũ rectũ:</s> <s xml:id="echoid-s50990" xml:space="preserve"> cum angulus <lb/>ad axẽ ſemper ſit rectus per 89 primi huius.</s> <s xml:id="echoid-s50991" xml:space="preserve"> Ex præmiſsis quoq;</s> <s xml:id="echoid-s50992" xml:space="preserve"> patet corollariũ perpulchrũ auxi-<lb/>lio 12 huius.</s> <s xml:id="echoid-s50993" xml:space="preserve"> Quoniam enim in pyramide orthogonia centrum circuli baſis & conus ſemper ſunt in <lb/>eadem linea (ut in axe) in propoſito erunt a & g in axe a g:</s> <s xml:id="echoid-s50994" xml:space="preserve"> ſed eadem ratione erunt b & g in eadem <lb/>linea:</s> <s xml:id="echoid-s50995" xml:space="preserve"> lineæ uerò b g & g a coniunctæ ſunt linea una:</s> <s xml:id="echoid-s50996" xml:space="preserve"> eò quòd f g à termino ipſarum exiens cum am <lb/>babus facit angulos rectos.</s> <s xml:id="echoid-s50997" xml:space="preserve"> Quo modocunq;</s> <s xml:id="echoid-s50998" xml:space="preserve"> ergo ſe habeat uiſus ad corpus irradiatum, dummo-<lb/>do ad ipſum fiat reflexio uel refractio:</s> <s xml:id="echoid-s50999" xml:space="preserve"> patet propoſitum quoniã ſemper centrum corporis irradian <lb/>tis & centrum oculi & centrũ circuli baſis utriuſq;</s> <s xml:id="echoid-s51000" xml:space="preserve"> pyramidis, irradiationis ſcilicet & uiſionis ſunt <lb/>in eadem linea, ſcilicet axe pyramidis irradiationis:</s> <s xml:id="echoid-s51001" xml:space="preserve"> nec aliter eſt poſsibile uideri irradiationem.</s> <s xml:id="echoid-s51002" xml:space="preserve"/> </p> <div xml:id="echoid-div1897" type="float" level="0" n="0"> <figure xlink:label="fig-0758-01" xlink:href="fig-0758-01a"> <variables xml:id="echoid-variables864" xml:space="preserve">a f h g k i b</variables> </figure> <figure xlink:label="fig-0759-01" xlink:href="fig-0759-01a"> <variables xml:id="echoid-variables865" xml:space="preserve">a g f y h k i</variables> </figure> </div> </div> <div xml:id="echoid-div1899" type="section" level="0" n="0"> <head xml:id="echoid-head1393" xml:space="preserve" style="it">65. Iridem ex reflexione & refractione radiorum corporis luminoſi uideri neceſſe ect.</head> <p> <s xml:id="echoid-s51003" xml:space="preserve">Locuturi de iride, de illa principaliter intendimus, quæ interſecans horizontem ad diuerſas par <lb/>tes mundi protenditur:</s> <s xml:id="echoid-s51004" xml:space="preserve"> quamuis etiam de alijs, quæ illi iridi ſimiles uidentur, intentionem non <lb/>principaliter facturi ſimus.</s> <s xml:id="echoid-s51005" xml:space="preserve"> Quòd uerò iris fiat ex multitudine luminis corporis luminoſi in uiſu re <lb/>cepti, hoc patet ſenſui:</s> <s xml:id="echoid-s51006" xml:space="preserve"> quòd autem (non aggregatis radijs corporis luminoſi) lumen ſenſibilius <lb/>poſsit fieri in corpore non luminoſo, quàm in medio, per quod prius lumen ferebatur, oſtenſum <lb/>eſt per 56 huius impoſsibile eſſe.</s> <s xml:id="echoid-s51007" xml:space="preserve"> Vnde patet ex hoc quòd lumẽ uigoratur ex aggregatione radio-<lb/>rum corporis luminoſi, ut ſenſibilius fiat in aliquo corpore quàm in medio.</s> <s xml:id="echoid-s51008" xml:space="preserve"> Quod uerò aggregatio <lb/>radiorum corporis luminoſi fiat per reflexionem uel per refractionem, quæ fit in corpore denſio-<lb/>ris diaphani quàm medium, per quod antea ferebatur, declaratum eſt per 57 huius.</s> <s xml:id="echoid-s51009" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s51010" xml:space="preserve"> ge-<lb/>neraliter quòd luminis maior ſenſibilitas per reflexionẽ uel per refractionem in omnibus uiſibili-<lb/>bus cauſſatur.</s> <s xml:id="echoid-s51011" xml:space="preserve"> Quòd uerò iris ſpecialiter ex reflexione fiat:</s> <s xml:id="echoid-s51012" xml:space="preserve"> patet per hoc:</s> <s xml:id="echoid-s51013" xml:space="preserve"> quia lumen eius ſenſibile <lb/>peruenit ad uiſum, ut ſuppoſitũ eſt in 2 petitione libri huius.</s> <s xml:id="echoid-s51014" xml:space="preserve"> Oſtenſum eſt quoq;</s> <s xml:id="echoid-s51015" xml:space="preserve"> per 20 th.</s> <s xml:id="echoid-s51016" xml:space="preserve"> 5 huius <lb/>quòd omne, quod uidetur per reflexionem, ſic uidetur, quòd angulus, ſecundum quẽ forina ſpecu <lb/>lo uel alteri corpori polito incidit, fit æqualis angulo, ſecundũ quẽ illa forma reflectitur ad uiſum:</s> <s xml:id="echoid-s51017" xml:space="preserve"> <lb/>quod etiam patet per 26 th.</s> <s xml:id="echoid-s51018" xml:space="preserve"> 5 huius ducta perpendiculari à puncto incidentiæ ſuper ſuperficiem <lb/>corporis politi, ad quam reflexionis anguli referuntur:</s> <s xml:id="echoid-s51019" xml:space="preserve"> continet enim radius incidens & radius re-<lb/>flexus cum eadem perpendiculari angulos æquales.</s> <s xml:id="echoid-s51020" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s51021" xml:space="preserve"> forma iridis fiat in uiſu:</s> <s xml:id="echoid-s51022" xml:space="preserve"> patet iri-<lb/>dem per reflexionẽ radiorũ corporis luminoſi ad uiſum cauſſari.</s> <s xml:id="echoid-s51023" xml:space="preserve"> Quòd uerò iris per refractionem <lb/> <pb o="458" file="0760" n="760" rhead="VITELLONIS OPTICAE"/> ctiam radiorum corporis lumino ſi fiat:</s> <s xml:id="echoid-s51024" xml:space="preserve"> patet per hoc quia non generatur iris, niſi in aliqua diapha-<lb/>na materia exiſtente in medio, & prohibente tranſitum luminis.</s> <s xml:id="echoid-s51025" xml:space="preserve"> Iam quoq;</s> <s xml:id="echoid-s51026" xml:space="preserve"> dictum eſt in 4 huius <lb/>quod in corporibus diaphanis denſioribus primo diaphano, & ſi ab ipſorum ſuperficie fiat refle-<lb/>xio:</s> <s xml:id="echoid-s51027" xml:space="preserve"> ſemper tamen fit refractio ad perpendicularem:</s> <s xml:id="echoid-s51028" xml:space="preserve"> & ſic lumen talium corporum ſuperficiebus <lb/>obliquè incidens quaſi ſecundum unam lineam ad duas partes oppoſitas diuiſum protenditur.</s> <s xml:id="echoid-s51029" xml:space="preserve"> Fit <lb/>itaq;</s> <s xml:id="echoid-s51030" xml:space="preserve"> per refractionem in talibus corporibus luminis aggregatio, quæ uiſui offertur, ſicut & quodli <lb/>bet aliud uiſibile:</s> <s xml:id="echoid-s51031" xml:space="preserve"> & ſicut nubes alba, & lumen ab illorum corporum ſuperficie ad uiſum reflexum <lb/>coadiuuat, ut actũ maioris ſenſibilitatis faciat in uiſu:</s> <s xml:id="echoid-s51032" xml:space="preserve"> ſicut uidemus quòd à corporibus albis, quæ <lb/>plus habent luminis, ſenſibilior fit reflexio quàm à corporibus medio colore coloratis.</s> <s xml:id="echoid-s51033" xml:space="preserve"> Hoc etiam <lb/>patet per luminis profundationem in iridis generatione.</s> <s xml:id="echoid-s51034" xml:space="preserve"> Cum enim ea, quæ ſolùm reflexionem lu-<lb/>minis habent, tantùm in ſuperficie irradientur, materia iridis ſenſibiliter inuenitur in profundo ir-<lb/>radiata:</s> <s xml:id="echoid-s51035" xml:space="preserve"> & ob hoc (ut comperit Philippus ſodalis Platonis, & ut quotidie quoq;</s> <s xml:id="echoid-s51036" xml:space="preserve"> circa iridem deam-<lb/>bulantibus cõtingit, & nos ipſi experimento hoc didicimus) iris mutatur ſecundum mutationem <lb/>uidentis.</s> <s xml:id="echoid-s51037" xml:space="preserve"> Sequitur enim fugientem ab ea, & illum, qui progreditur ad eam, fugiens antecedit.</s> <s xml:id="echoid-s51038" xml:space="preserve"> Et ſi <lb/>quis ad dextrum uel ſiniſtrum latus progreſſus fuerit:</s> <s xml:id="echoid-s51039" xml:space="preserve"> iris ad idem latus uidebitur moueri.</s> <s xml:id="echoid-s51040" xml:space="preserve"> Sed ſe-<lb/>cundum reflexionem ſolùm uiſa fugiunt fugientem, & occurrunt accedenti:</s> <s xml:id="echoid-s51041" xml:space="preserve"> uidentur enim talia <lb/>ſemper in concurſu lineæ reflexionis ad uiſum progredientis, cum perpendiculari ducta à puncto <lb/>rei uiſæ ſuper ſuperficiem corporis, à qua fit reflexio formæ uiſæ, ut patet per 37 th.</s> <s xml:id="echoid-s51042" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s51043" xml:space="preserve"> Iris er-<lb/>go non ſolùm uidetur per reflexionẽ, ſed etiam per refractionẽ luminis intra corpus, à quo reflecti-<lb/>tur:</s> <s xml:id="echoid-s51044" xml:space="preserve"> quamuis accedenti ad iridem, uel ab ipſa elongato ab alijs & alijs ſuperficiebus corporum lu-<lb/>mini obuiantium fiat reflexio luminis ad uiſum:</s> <s xml:id="echoid-s51045" xml:space="preserve"> quoniam fuga iridis à progrediente ad eam, & ſe-<lb/>cutio fugientis ab ea, accidit propter diuerſas reflexiones, quę fiunt ad uiſum à diuerſis partibus <lb/>materiæ iridis:</s> <s xml:id="echoid-s51046" xml:space="preserve"> ſcilicet ſecundũ quod uiſus mutat puncta, in quibus ab angulis baſis unius pyrami-<lb/>dis omnes radij in centro ipſius oculi concurrũt.</s> <s xml:id="echoid-s51047" xml:space="preserve"> Et quia tales baſes ſunt infinitæ, & puncta, in qui-<lb/>bus earum radij reflexi, in axe colliguntur, ſunt infinita:</s> <s xml:id="echoid-s51048" xml:space="preserve"> patet etiam quòd per reflexionem multifa <lb/>riam uidentur irides infinitæ ſecundum infinitatem punctorum in axe pyramidis occurrentium <lb/>accedenti uel recedenti ſecundũ lineam eiuſdem axis, uel etiam à latere eunti ſecundum mutatio-<lb/>nem axis à centro corporis lumin oſi per alium punctum ſuæ ſuperficlei exeuntis, quàm per illum, <lb/>quo primus axis exibat.</s> <s xml:id="echoid-s51049" xml:space="preserve"> Fit enim uiſum ad latera ſic mutanti noua pyramis & noua baſis:</s> <s xml:id="echoid-s51050" xml:space="preserve"> aliudq́ <lb/>eſt punctum ſuperficiei corporis luminoſi, per quod uenit radius perpendicularis ad ſuperficiem <lb/>materiæ iridis, qui (in ipſum cadente centro oculi) fit axis pyramidis utriuſq;</s> <s xml:id="echoid-s51051" xml:space="preserve">. Videntur itaq;</s> <s xml:id="echoid-s51052" xml:space="preserve"> hoc <lb/>modo irides infinitę ad quamcunq;</s> <s xml:id="echoid-s51053" xml:space="preserve"> differentiam poſitionis quis uidentium motus fuerit:</s> <s xml:id="echoid-s51054" xml:space="preserve"> dum mo <lb/>dò contra corpus luminoſum non moueatur.</s> <s xml:id="echoid-s51055" xml:space="preserve"> Quod etiam ſi uerum ſit per reflexionis naturã poſſe <lb/>fieri:</s> <s xml:id="echoid-s51056" xml:space="preserve"> refractio tamen radiorum corporis luminoſi ſemper augmentat lumen, ut uideri ualeat ſenſi-<lb/>bilius à uiſu.</s> <s xml:id="echoid-s51057" xml:space="preserve"> Patet enim quòd refractio radiorum corporis luminoſi aggregat lumen, ut fiat magis <lb/>uiſibile:</s> <s xml:id="echoid-s51058" xml:space="preserve"> quoniam propter ipſam refractionem radiorum circa eandem partem medij radius dupli-<lb/>catur:</s> <s xml:id="echoid-s51059" xml:space="preserve"> ſimiliterq́;</s> <s xml:id="echoid-s51060" xml:space="preserve"> ipſorum radiorum reflexio lumen aggregat & ad uiſum ſenſibiliter reducit:</s> <s xml:id="echoid-s51061" xml:space="preserve"> iris ue <lb/>rò non fit, niſi ex aggregato lumine, nec fit ex illo, niſi occurrat uiſui.</s> <s xml:id="echoid-s51062" xml:space="preserve"> Ergo ad generationem iridis <lb/>refractio radiorum corporis luminoſi & reflexio eorundem neceſſariæ exiſtunt.</s> <s xml:id="echoid-s51063" xml:space="preserve"> Et hoc eſt, quod <lb/>in præſente theoremate perquirere uolebamus.</s> <s xml:id="echoid-s51064" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1900" type="section" level="0" n="0"> <head xml:id="echoid-head1394" xml:space="preserve" style="it">66. In uapore rorido iridem gener ari neceſſarium eſt.</head> <p> <s xml:id="echoid-s51065" xml:space="preserve">Quod hic ꝓponitur, patet.</s> <s xml:id="echoid-s51066" xml:space="preserve"> Quia cũ iris non fiat ſinelumine, imò luminis multitudine:</s> <s xml:id="echoid-s51067" xml:space="preserve"> lumẽ aũt <lb/>non aggregetur niſi ex reflexione aut refractione radiorum corporis luminoſi, ut patet per 57 hu-<lb/>ius:</s> <s xml:id="echoid-s51068" xml:space="preserve"> hæc autẽ non fiant, niſi lumini fiat obiectio corporis denſioris aere puro per 56 huius.</s> <s xml:id="echoid-s51069" xml:space="preserve"> Ergo in <lb/>loco generationis iridis non erit ipſius generatio ſine corpore irradiabili, à cuius ſuperficie poſsit <lb/>fieri reflexio & refractio luminis incidentis.</s> <s xml:id="echoid-s51070" xml:space="preserve"> Aliquod uerò ſolidorum planorum ibi eſſe eſt impoſsi <lb/>bile.</s> <s xml:id="echoid-s51071" xml:space="preserve"> Sed neq;</s> <s xml:id="echoid-s51072" xml:space="preserve"> aquam:</s> <s xml:id="echoid-s51073" xml:space="preserve"> quoniam hæc curreret ſubitò ad inferiorem locorum ſibi poſsibilem:</s> <s xml:id="echoid-s51074" xml:space="preserve"> iris ue-<lb/>rò aliquo tempore manet, non eadem, ſed ſemper diuerſa propter continuum deſcenſum roratio-<lb/>nis:</s> <s xml:id="echoid-s51075" xml:space="preserve"> nec tamen poſſet in aqua continua figura iridis generari:</s> <s xml:id="echoid-s51076" xml:space="preserve"> quoniam lumen integrum reflectere-<lb/>tur à ſuperficie aquæ propter continuitatem ipſius aquæ.</s> <s xml:id="echoid-s51077" xml:space="preserve"> Iris enim, quæ fit in aqua diffuſa per re-<lb/>mos, fit proter aquæ diſperſionem:</s> <s xml:id="echoid-s51078" xml:space="preserve"> quia tunc temone pro manu utitur nauta aquam rorans:</s> <s xml:id="echoid-s51079" xml:space="preserve"> & ob <lb/>hoc cum aqua ſic fuerit fuſa, in ipſa colores iridis apparent.</s> <s xml:id="echoid-s51080" xml:space="preserve"> Non etiam poteſt eſſe quòd ſit aer groſ-<lb/>ſus, in quo iris generatur:</s> <s xml:id="echoid-s51081" xml:space="preserve"> quoniam impreſsio luminis in aere non efficeret colores iridis, ſed face-<lb/>ret quandam albedinem, ut apparet in crepuſculis matutinis in ipſarum principijs & etiam termi-<lb/>nis crepuſculorum ſerotinorum:</s> <s xml:id="echoid-s51082" xml:space="preserve"> & uniuerſaliter in ſimilibus quibuſcunq;</s> <s xml:id="echoid-s51083" xml:space="preserve">. Non etiam poteſt eſſe <lb/>uapor continuus, ſiue ſit eleuatus ad generationem nubis, ſiue ſit in nubem cõdenſatus.</s> <s xml:id="echoid-s51084" xml:space="preserve"> Eſto enim <lb/>quòd ſit poſsibile à uapore continuo iridem generari.</s> <s xml:id="echoid-s51085" xml:space="preserve"> Ponatur ergo corpus radioſum (cuius cen-<lb/>trum ſit a) in circulo horizontis:</s> <s xml:id="echoid-s51086" xml:space="preserve"> ſecetq́;</s> <s xml:id="echoid-s51087" xml:space="preserve"> ipſum ſuperficies ortho gonaliter erecta ſuper ſuperficiem <lb/>horizontis per centrum ipſius corporis:</s> <s xml:id="echoid-s51088" xml:space="preserve"> & ducatur in illa ſuperficie ſecante per centrum corporis <lb/>luminoſi linea h g.</s> <s xml:id="echoid-s51089" xml:space="preserve"> Huic itaq;</s> <s xml:id="echoid-s51090" xml:space="preserve"> ſuperficiei ſecanti aut æ quidiſtat uapor continuus irradiabilis:</s> <s xml:id="echoid-s51091" xml:space="preserve"> aut <lb/>non.</s> <s xml:id="echoid-s51092" xml:space="preserve"> Si æ quidiſtat:</s> <s xml:id="echoid-s51093" xml:space="preserve"> ſit linea in eius ſuperficie b c d æ quidiſtans lineæ h g:</s> <s xml:id="echoid-s51094" xml:space="preserve"> incidantq́;</s> <s xml:id="echoid-s51095" xml:space="preserve"> ſibi radij a b, a c, <lb/>a d:</s> <s xml:id="echoid-s51096" xml:space="preserve"> & ſit linea a b perpendicularis ſuper ſuperficiẽ uaporis, quæ in ſe reflectetur per 21th.</s> <s xml:id="echoid-s51097" xml:space="preserve"> 5 huius:</s> <s xml:id="echoid-s51098" xml:space="preserve"> <lb/>& reflectentur etiam lineæ a c, a d:</s> <s xml:id="echoid-s51099" xml:space="preserve"> quia non ſunt perpendiculares.</s> <s xml:id="echoid-s51100" xml:space="preserve"> Quoniam autem angulus a c b <lb/>eſt acutus per 32 p 1, cum an angulus a b c ſit rectus:</s> <s xml:id="echoid-s51101" xml:space="preserve"> patet per 13 p 1 quòd angulus d c a eſt obtuſus:</s> <s xml:id="echoid-s51102" xml:space="preserve"> per-<lb/> <pb o="459" file="0761" n="761" rhead="LIBER DECIMVS."/> pendicularis ergo extracta à puncto c non concurret cum axe a b:</s> <s xml:id="echoid-s51103" xml:space="preserve"> ergo nec radius reflexus.</s> <s xml:id="echoid-s51104" xml:space="preserve"> Cum <lb/>ergo centrum uiſus ex 64 huius neceſſariò ſit ſitum in linea a b, quæ eſt in ſuperficie horizontis, & <lb/>centrum uiſus ſit centrum horizontis, quod ſit pun-<lb/> <anchor type="figure" xlink:label="fig-0761-01a" xlink:href="fig-0761-01"/> ctus f:</s> <s xml:id="echoid-s51105" xml:space="preserve"> patet quòd lumen ſic reflexum centrum uiſus <lb/>nullatenus attinget, niſi fortè radius ille reflexus ſu-<lb/>perficiei alterius corporis plani incidens reflectere-<lb/>tur ad uiſum.</s> <s xml:id="echoid-s51106" xml:space="preserve"> Ergo uapore taliter diſpoſito iris non <lb/>uidebitur.</s> <s xml:id="echoid-s51107" xml:space="preserve"> Quòd ſi uaporis continui ſuperficies ſu-<lb/>perficiei ſecanti corpus luminoſum non æquidiſtet, <lb/>ſed cum ipſa cõcurrat:</s> <s xml:id="echoid-s51108" xml:space="preserve"> ſi illę ſuperficies ſub horizon-<lb/>te cõcurrant, idem accidit impoſsibile, & eodem mo <lb/>do deducendũ.</s> <s xml:id="echoid-s51109" xml:space="preserve"> Quia & ſi hoc modo radios aliquos <lb/>ſub horizonte ad uiſum reflecti ſit poſsibile:</s> <s xml:id="echoid-s51110" xml:space="preserve"> non <lb/>tamen uiſus illorum paſsionem aliquam iudicabit:</s> <s xml:id="echoid-s51111" xml:space="preserve"> <lb/>non enim uidentur ea, quæ ſunt ſub horizonte:</s> <s xml:id="echoid-s51112" xml:space="preserve"> cum <lb/>horizon ſit circulus, qui eſt terminator uiſus.</s> <s xml:id="echoid-s51113" xml:space="preserve"> Et cum <lb/>ſuperficies horizontis ſit obliqua ſuper ſuperficiem <lb/>uaporis:</s> <s xml:id="echoid-s51114" xml:space="preserve"> patet quòd radius à centro corporis lumi-<lb/>noſi perpendiculariter incidens ſuperficiei uaporis <lb/>cadit ſub horizonte:</s> <s xml:id="echoid-s51115" xml:space="preserve"> omnesq́;</s> <s xml:id="echoid-s51116" xml:space="preserve"> radij non perpendiculariter ſuperficiei uaporis ultra ſuperficiem ho <lb/>rizontis incidentes, reflectuntur ad partem contrariã centro uiſus in centro horizontis conſtituti.</s> <s xml:id="echoid-s51117" xml:space="preserve"> <lb/>Non ergo uidebitur iris, centro uiſus & ſuperficie illius uap oris taliter ad inuicẽ diſpoſitis.</s> <s xml:id="echoid-s51118" xml:space="preserve"> Quòd ſi <lb/>nõ ſub horizonte, ſed ſupra horizontẽ cõcurrãt illę <lb/> <anchor type="figure" xlink:label="fig-0761-02a" xlink:href="fig-0761-02"/> duæ ſuperficies, una uaporis & alia ſecans lumino-<lb/>ſum corpus:</s> <s xml:id="echoid-s51119" xml:space="preserve"> tunc iterũ lumen ad uiſum reflecti non <lb/>eſt poſsibile, ex cauſsis prius dictis.</s> <s xml:id="echoid-s51120" xml:space="preserve"> Sẽper enim an-<lb/>gulus a c d, cũ ſit extrinſecus angulo a b c, in triãgu-<lb/>lo orthogonio a b c, erit maior recto per 16 p 1:</s> <s xml:id="echoid-s51121" xml:space="preserve"> ergo <lb/>reflexio nunꝗ̃ fiet ad uiſum, ꝗ eſt in cẽtro horizõtis.</s> <s xml:id="echoid-s51122" xml:space="preserve"> <lb/>Sed etiã dato qđ in aliqua præ miſſarũ diſpoſitionũ <lb/>fiat reflexio ad uiſum (qđ tamẽ eſt impoſsibile) nõ <lb/>ꝓpter hoc iris uidebitur:</s> <s xml:id="echoid-s51123" xml:space="preserve"> quoniã propter uaporis <lb/>cõtinuitatẽ fiet luminis multa in ſuperficie uaporis <lb/>generatio:</s> <s xml:id="echoid-s51124" xml:space="preserve"> & erit lumẽ continuũ, qđ ad uiſum refle <lb/>xũ ipſum debilitabit, nec in ꝓfundũ uaporis ipſum <lb/>քmittet inſpicere.</s> <s xml:id="echoid-s51125" xml:space="preserve"> Et dicit uulgus qđ tale lumẽ eſt <lb/>ſol aqueus:</s> <s xml:id="echoid-s51126" xml:space="preserve"> nec habet diſtinctionẽ aliquam colorũ.</s> <s xml:id="echoid-s51127" xml:space="preserve"> <lb/>Et etiã ſi dictæ ſuperficies ſupra horizontẽ cõcurre <lb/>rẽt:</s> <s xml:id="echoid-s51128" xml:space="preserve"> tunc iris deflexa uideretur à zenith capitis ſenſi <lb/>biliter ſecundũ gibbũ circuli, quo uidetur:</s> <s xml:id="echoid-s51129" xml:space="preserve"> qđ totũ <lb/>ſenſui eſt cõtrariũ, nec apparet uiſui.</s> <s xml:id="echoid-s51130" xml:space="preserve"> In tali ergo ua <lb/>pore nõ eſt conueniẽs iridẽ cauſſari.</s> <s xml:id="echoid-s51131" xml:space="preserve"> Sed inter uapo <lb/>rem aqueũ cõtinuũ, & inter aquã depluentẽ à nubi <lb/>bus eſt quoddã mediũ, quod dicitur uapor roridus.</s> <s xml:id="echoid-s51132" xml:space="preserve"> <lb/>Et fit quando frigus condenſans incipit uaporẽ a-<lb/>queũ in formã propriam ſcilicet a quę reducere:</s> <s xml:id="echoid-s51133" xml:space="preserve"> tũc <lb/>enim cõdenſantur raræ partes uaporis, & fit partιũ uaporis diſtãtia, quę rotundari incipiũt:</s> <s xml:id="echoid-s51134" xml:space="preserve"> nondũ <lb/>tamen ꝓpter debilitatẽ agentis reducuntur ad formã propriã, quę ſibi det motũ ad inferius:</s> <s xml:id="echoid-s51135" xml:space="preserve"> & tũc <lb/>illę uaporis particulę ſunt quaſi quædã parua ſpecula, in quibus ſolũ apparet color corporis radioſi <lb/>ſine quantitate & figura, ut diximus in 62 th.</s> <s xml:id="echoid-s51136" xml:space="preserve"> huius.</s> <s xml:id="echoid-s51137" xml:space="preserve"> Si ergo ad talia corpuſcula incipiẽtia rotũdari <lb/>ꝓpter ęqualẽ ex om ni parte uirtutis cõdenſantis actionẽ, quouſq;</s> <s xml:id="echoid-s51138" xml:space="preserve"> materiã cõdenſet, incidat lumen <lb/>corporis luminoſi:</s> <s xml:id="echoid-s51139" xml:space="preserve"> refringitur ad poſterius ipſius quilibet radiorũ ſibi incidentiũ ad lineã perpen-<lb/>dicularẽ, à pũcto incidẽtię ſuք ſuperficiẽ illius corporis productã ք 4 huius.</s> <s xml:id="echoid-s51140" xml:space="preserve"> Et quoniã ք 72 th.</s> <s xml:id="echoid-s51141" xml:space="preserve"> 1 hu-<lb/>ius illa perpendicularis tranſit centrũ illius corporis ſphærici:</s> <s xml:id="echoid-s51142" xml:space="preserve"> patet quòd radius refractus obliquè <lb/>cadet ſuք ſuperficiẽ illius corporis oppoſitã corpori luminoſo, & aggregabitur lumẽ in profundo <lb/>totius cõſiſtentię iſtorũ corpuſculorũ, propter refractionẽ factã in quolibet ipſorũ:</s> <s xml:id="echoid-s51143" xml:space="preserve"> ſicut uidemus <lb/>in cryſtallo rotunda:</s> <s xml:id="echoid-s51144" xml:space="preserve"> quoniã ultra ſuperficiẽ illius poſteriorẽ fit aggregatio radiorũ in aere ad pun-<lb/>ctum unũ, ut patet ք 48 huius.</s> <s xml:id="echoid-s51145" xml:space="preserve"> In quolibet aũt iſtorũ corpuſculorũ (ſiue ipſa ſint maiora guttis ex <lb/>ipſis poſtmodũ uia cõdenſationis generatis, ut quãdoq;</s> <s xml:id="echoid-s51146" xml:space="preserve"> poſsibile eſt fieri:</s> <s xml:id="echoid-s51147" xml:space="preserve"> ſiue ք modũ aggregatio-<lb/>nis ex pluribus corpuſculis fiat gutta:</s> <s xml:id="echoid-s51148" xml:space="preserve"> in hoc enim, quo ad iridis generationẽ, nõ eſt diuerſitas quo-<lb/>niã ſemper incidũt radij infiniti, qui etiã reflectũtur à ſuperficie ipſorũ corpuſculorũ ſecundũ angu <lb/>los incidentię ſuę, quos faciũt cũ lineis maiores circulos dictorũ corpuſculorũ in puncto ſuę inci-<lb/>dentię cõtingentibus, qui anguli diuerſi ſunt:</s> <s xml:id="echoid-s51149" xml:space="preserve"> & ob hoc anguli reflexionis efficiuntur diuerſi, ut pa-<lb/>tet per totum 6 librũ huius ſcientię:</s> <s xml:id="echoid-s51150" xml:space="preserve"> & radij faciẽtes angulos cum lineis cõtingentibus corpuſcula <lb/>prædicta & cũ lineis ſignatis in ſuperficie corpus luminoſum ſecante concurrentibus ſupra hori-<lb/> <pb o="460" file="0762" n="762" rhead="VITELLONIS OPTICAE"/> zontem, & interſecantibus axẽ pyramidis illuminationis ultra punctũ b remotius à corpore lumi-<lb/>noſo (ut in puncto m) quia anguli tales intra pyramidẽ obtuſi ſunt:</s> <s xml:id="echoid-s51151" xml:space="preserve"> ideo per 33 th.</s> <s xml:id="echoid-s51152" xml:space="preserve"> 5 huius illi radij <lb/>ſic incidentes ad uiſum reflectuntur:</s> <s xml:id="echoid-s51153" xml:space="preserve"> & in puncto, ubi talium radiorum plurimorũ fit concurfus in <lb/>axe, inter corpus luminoſum & uaporẽ uiſu poſito uidetur lumẽ.</s> <s xml:id="echoid-s51154" xml:space="preserve"> Et quoniã iſtorum corpuſculorũ <lb/>quædã ſunt, in quæ ſecundũ æquales angulos, ut dictũ eſt, radij incidũt à centro corporis lumin oſi:</s> <s xml:id="echoid-s51155" xml:space="preserve"> <lb/>tales aũtradij ex omni parte nubis diſperſi ſunt infiniti (cũ enim tota conſiſtentla uaporis ſit plena <lb/>talibus corpuſculis, infiniti ſunt tales radij in ſuperficie nubis uel uaporis roridi cõcurrente, uel e-<lb/>tiam æquidiſtante ſuperficiei ſecanti corpus luminoſum, ſecundum quod reſpicit uaporis coſiſten <lb/>tiam:</s> <s xml:id="echoid-s51156" xml:space="preserve"> & in illorũ irradiatione pyramis figuratur, cuius uertex eſt in centro corporis luminoſi, baſis <lb/>uerò in conſiſtentia uaporis roridi, & lineæ longitudinis illius pyramidis termin antur ad diuerſas <lb/>partes diuerſorũ corpuſculorum:</s> <s xml:id="echoid-s51157" xml:space="preserve"> quę cum ſecundũ ſimiles angulos ſuæ incidentię refle ctuntur ad <lb/>uiſum, aliã faciunt pyramidẽ, cuius uertex eſt in centro uiſus, baſis uerò eadẽ cum baſi pyramidis <lb/>prioris:</s> <s xml:id="echoid-s51158" xml:space="preserve"> & eſt circulus, ut oſtenſum eſt uniuerſaliter in 64 huius) uidebitur illud lumen reflexum <lb/>continuũ propter uicinitatẽ partium uaporis, & eorum diſtantiæ inſenſibilitatẽ à uiſu, qui proten-<lb/>ſus ab illis fallitur propter ſui debilitatẽ:</s> <s xml:id="echoid-s51159" xml:space="preserve"> & ob hoc uiſus aggregatũ ab omnibus illis corpuſculis re-<lb/>flexum lumen ſine cognitione uel perceptione diſtantiæ partium recipit, & iudicat tanquã unum.</s> <s xml:id="echoid-s51160" xml:space="preserve"> <lb/>Patet itaq;</s> <s xml:id="echoid-s51161" xml:space="preserve"> ex præmiſsis, quòd licet tota conſiſtentia uaporis ſit radioſa, & fortè tota irradiata ſu-<lb/>perficies ſit multilatera:</s> <s xml:id="echoid-s51162" xml:space="preserve"> tamen ſemper uidetur circularis:</s> <s xml:id="echoid-s51163" xml:space="preserve"> cuius ratio eſt:</s> <s xml:id="echoid-s51164" xml:space="preserve"> quia nõ uidetur, niſi quod <lb/>de ipſo ſecundum æquales angulos ad unum punctũ axis pyramidis radialis eſt reflexum.</s> <s xml:id="echoid-s51165" xml:space="preserve"> Quando <lb/>uerò anguli ad baſim ſunt æquales:</s> <s xml:id="echoid-s51166" xml:space="preserve"> latera æquos angulos continentia ſunt æqualia per 6 p 1:</s> <s xml:id="echoid-s51167" xml:space="preserve"> ergo <lb/>per 65 th.</s> <s xml:id="echoid-s51168" xml:space="preserve"> 1 huius centrũ uiſus eſt polus, & ſuperficies, ad quã illæ ęquales lineę terminantur, eſt cir-<lb/>culus:</s> <s xml:id="echoid-s51169" xml:space="preserve"> & ita uidetur iris circularis.</s> <s xml:id="echoid-s51170" xml:space="preserve"> Poteſt etiam (exempli cauſſa) idem aliter declarari:</s> <s xml:id="echoid-s51171" xml:space="preserve"> ſcilicet du-<lb/>ctis tribus lineis uel pluribus à punctis reflexionis orthogonaliter ſuper lineam ipſi totali cõſiſten <lb/>tiæ uaporis à centro luminoſi corporis perpendiculariter incidentẽ:</s> <s xml:id="echoid-s51172" xml:space="preserve"> illę enim erunt in eadẽ ſuperfi <lb/>cie ex 5 p 11:</s> <s xml:id="echoid-s51173" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s51174" xml:space="preserve"> æ quales ex 32 & 26 p 1:</s> <s xml:id="echoid-s51175" xml:space="preserve"> ergo in puncto cõcurſus earum in axe eſt centrum circu <lb/>li ex 9 p 3.</s> <s xml:id="echoid-s51176" xml:space="preserve"> Et quia totius baſis radij non ad æquales angulos reflectuntur:</s> <s xml:id="echoid-s51177" xml:space="preserve"> nõ uidetur totus circulus <lb/>radioſus, quãuis in tota nubis cõſiſtentia ubiq;</s> <s xml:id="echoid-s51178" xml:space="preserve"> lumen exiſtat.</s> <s xml:id="echoid-s51179" xml:space="preserve"> Radij enim, qui ad maiores angulos <lb/>reflectuntur, quàm ſint anguli radiorũ ad uiſum reflexorũ, ultra punctũ uιſus ad alium locum axis <lb/>reflectuntur:</s> <s xml:id="echoid-s51180" xml:space="preserve"> radij aũt, qui ad minores angulos eis, qui ad uiſum perueniũt, reflectuntur ad locum <lb/>alium axis infra centrum uiſus concurrunt:</s> <s xml:id="echoid-s51181" xml:space="preserve"> & ſic neurri uidentur, niſi fortè ab alijs uiſibus in locis <lb/>ſuorũ concurſuũ exiſtentibus.</s> <s xml:id="echoid-s51182" xml:space="preserve"> Et propter hoc accidit moto homine in antè uel retro, aliã & aliã iri-<lb/>dem uideri:</s> <s xml:id="echoid-s51183" xml:space="preserve"> quoniã ſemper uiſus progredientis uel recedentis incidit in puncta aggregationis di-<lb/>uerſorum radiorũ:</s> <s xml:id="echoid-s51184" xml:space="preserve"> ſicut etiã accidit in hominibus diuerſis magis uel minus à centro ſolis ſecundũ <lb/>diuerſam zenith capitis elongationẽ diſpoſitis, ſub eodẽ tamẽ exiſtentibus circulo meridiano uel <lb/>alio circulo altitudinis.</s> <s xml:id="echoid-s51185" xml:space="preserve"> Iris itaq;</s> <s xml:id="echoid-s51186" xml:space="preserve"> propter has cauſſas uidetur circularis cõcaua:</s> <s xml:id="echoid-s51187" xml:space="preserve"> quia nec exteriores <lb/>nec interiores radij incidentes ſuperficiei totius conſiſtentiæ roridæ, in eodem puncto cõcurrunt <lb/>ad uiſum:</s> <s xml:id="echoid-s51188" xml:space="preserve"> unde uiſus partes uaporis alias iudicat lumine priuatas.</s> <s xml:id="echoid-s51189" xml:space="preserve"> Et ſignũ huius eſt, quod accidit <lb/>in ſuperficie plana aquæ, in qua in quolibet puncto eſt forma ſolis uellunæ, uel ſtellarũ:</s> <s xml:id="echoid-s51190" xml:space="preserve"> non tamen <lb/>uidetur, niſi in puncto uel loco uno, à quo eſt poſsibilis reuerberatio ad uiſum:</s> <s xml:id="echoid-s51191" xml:space="preserve"> & mutato uidente <lb/>ulterius, alia iterũ forma corporis luminoſi uidetur in loco alio, à quo eſt ad uiſum poſsibilis refle-<lb/>xio.</s> <s xml:id="echoid-s51192" xml:space="preserve"> Et idẽ uidetur de cãdela uel lumine aliquo diſtincto in cultello nouo uel ferro polito, uel alio:</s> <s xml:id="echoid-s51193" xml:space="preserve"> <lb/>quia ſemք re immobili exiſtente mutatur ſorma uiſa, uiſu mutato ſecundũ modũ, quo poſsibile eſt <lb/>ipſam ad oculum reflecti:</s> <s xml:id="echoid-s51194" xml:space="preserve"> & in puncto alio non uidetur.</s> <s xml:id="echoid-s51195" xml:space="preserve"> Aliud etiã ſignum huius eſt:</s> <s xml:id="echoid-s51196" xml:space="preserve"> quia ſi aliquo <lb/>exiſtente in radio ſolis, per aliũ, qui eſt extra radium, tranſuerſaliter ſpargatur ore uel aliquo alio at <lb/>tificio aqua roratim in radiũ:</s> <s xml:id="echoid-s51197" xml:space="preserve"> uiſus eius, qui eſt in radio, fortè non uidebit niſi colorẽ album:</s> <s xml:id="echoid-s51198" xml:space="preserve"> cum ta <lb/>men ſpargens, cui opponitur uapor directus, uideat lumen & colores iridis, ſed cõſuſos, niſi diſpo-<lb/>fitio corpuſculorũ roridorum ſic diſponatur, ut poſsit fieri certa reflexio ad uiſum in medio radij <lb/>exiſtentẽ.</s> <s xml:id="echoid-s51199" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s51200" xml:space="preserve"> ex præmiſsis quoniã iris in uapore rorido generatur.</s> <s xml:id="echoid-s51201" xml:space="preserve"> Signũ aut illius eſt:</s> <s xml:id="echoid-s51202" xml:space="preserve"> quia <lb/>modicùm ſtat iris:</s> <s xml:id="echoid-s51203" xml:space="preserve"> eò quòd uapor talis, cum ſit ex materia graui iam ad formã grauis accedẽte, ſta-<lb/>re nõ poteſt ſuper ſuperficiẽ horizontis, niſi moueatur ad centrum grauium, qđ eſt centrum mun-<lb/>di, ſecundum quod ei eſt poſsibile.</s> <s xml:id="echoid-s51204" xml:space="preserve"> Et ob hoc etiã poſt apparitionẽ iridis quan do operatione agen-<lb/>tis cõdenſatur materia, & reducitur ad formã potentẽ mouere, fit pluuia, & ex corpuſculorum quo <lb/>libet in uapore prius ſeparatorũ ſit per condenſationẽ materiæ gutta aquea deſcendens.</s> <s xml:id="echoid-s51205" xml:space="preserve"> Signũ etiã <lb/>eius eſt, qđ dictũ eſt prius:</s> <s xml:id="echoid-s51206" xml:space="preserve"> quoniã aqua uaporoſè ſparſa ore, manu, uel remo, ut apud nautas:</s> <s xml:id="echoid-s51207" xml:space="preserve"> in ra-<lb/>dio ſolari apparet iris, & iridis colores, & diuerſi aſpicientes uident illud:</s> <s xml:id="echoid-s51208" xml:space="preserve"> quia radij incidentes gur-<lb/>tulis diuerſimodè reflectuntur.</s> <s xml:id="echoid-s51209" xml:space="preserve"> Patet ergo propoſitũ:</s> <s xml:id="echoid-s51210" xml:space="preserve"> quod eſt:</s> <s xml:id="echoid-s51211" xml:space="preserve"> iridẽ in uapore rorido generari.</s> <s xml:id="echoid-s51212" xml:space="preserve"> Si <lb/>aũt dicatur, quòd partes corpuſculorũ in materia iridis nõ ſunt omnes omnino ſphæricæ, non eſt <lb/>uim faciens inſtantia:</s> <s xml:id="echoid-s51213" xml:space="preserve"> quia idem accidit omnino in non ſphæricis, quod nunc dictum eſt de ſphæri <lb/>cis:</s> <s xml:id="echoid-s51214" xml:space="preserve"> nun quam enim fiet iris, niſi multi congregati radij ad uiſum uniformiter reflectantur.</s> <s xml:id="echoid-s51215" xml:space="preserve"/> </p> <div xml:id="echoid-div1900" type="float" level="0" n="0"> <figure xlink:label="fig-0761-01" xlink:href="fig-0761-01a"> <variables xml:id="echoid-variables866" xml:space="preserve">d c g c d f h a g</variables> </figure> <figure xlink:label="fig-0761-02" xlink:href="fig-0761-02a"> <variables xml:id="echoid-variables867" xml:space="preserve">h d c m b f a g</variables> </figure> </div> </div> <div xml:id="echoid-div1902" type="section" level="0" n="0"> <head xml:id="echoid-head1395" xml:space="preserve" style="it">67. Tricolor eſt omnis iris.</head> <p> <s xml:id="echoid-s51216" xml:space="preserve">Dubitatum propter ſui difficultatẽ ab antiquis hoc theorema proponitur.</s> <s xml:id="echoid-s51217" xml:space="preserve"> Multis enim mathe-<lb/>maticorũ patuit figura & quãtitas iridis:</s> <s xml:id="echoid-s51218" xml:space="preserve"> & ſunt hæc ab ipſis naturalis philoſophiæ inquiſitoribus <lb/>ſuppoſita:</s> <s xml:id="echoid-s51219" xml:space="preserve"> color tamen, quẽ uidemus, nondum conuenienter ab aliquo eſt pertractatus, niſi per di-<lb/>ſtinctionẽ materiæ iridis ſecundũ aduſti, indigeſti & opaci naturã:</s> <s xml:id="echoid-s51220" xml:space="preserve"> quòd ſi hoc motum & poſsibili.</s> <s xml:id="echoid-s51221" xml:space="preserve"> <lb/>tatẽ rerum naturalium ſeruet & ſeruare ualeat, intellectui eorũ, qui ſcripſerũt talia, duximus relin- <pb o="461" file="0763" n="763" rhead="LIBER DECIMVS."/> quendum.</s> <s xml:id="echoid-s51222" xml:space="preserve"> Colores autẽ iridis ſecundũ uerum, quod ſe nobis poſt multos cogitatus & experiẽtias <lb/>obtulit, ſic poſſunt declarari.</s> <s xml:id="echoid-s51223" xml:space="preserve"> Quia enim totus uaporroridus (qui eſt materia iridis) in ſuperficie & <lb/>profundo eſt irradiatus, & ipſius eſt multa profunditas:</s> <s xml:id="echoid-s51224" xml:space="preserve"> patet quia ipſe in aſpectu ſui ad lolem ſere-<lb/>nius & immixtius habet lumẽ, mixtum tamẽ cum colore uaporis, qui niger eſt, ut in aquoſis uapo-<lb/>ribus euidẽs eſt (ſunt enim omnes nigri) natura autẽ lucis eſt immiſcere ſe coloribus rerũ, ad quas <lb/>reflectitur:</s> <s xml:id="echoid-s51225" xml:space="preserve"> eſt enim in principio 2 huius 7 petitione ſuppoſitũ, lucem res coloratas tranſeuntẽ illa-<lb/>rum coloribus colorari:</s> <s xml:id="echoid-s51226" xml:space="preserve"> hoc enim patet ſenſui:</s> <s xml:id="echoid-s51227" xml:space="preserve"> unde etiá lumẽ reflexum ſecum defert colorem rei, <lb/>à qua reflectitur ad uiſum, ſicut patet in radio tranſeunte per uitrum coloratum.</s> <s xml:id="echoid-s51228" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s51229" xml:space="preserve"> lumen <lb/>de natura ſua fulgidum ſit (ut patet) & recipiatur in generatione iridis in uapore nigro aqueo:</s> <s xml:id="echoid-s51230" xml:space="preserve"> ne-<lb/>ceſſe eſt ipſum per 157 th.</s> <s xml:id="echoid-s51231" xml:space="preserve"> 4 huius uiſui colorẽ pręſentare puniceum:</s> <s xml:id="echoid-s51232" xml:space="preserve"> & iridẽ in parte illa ſecundum <lb/>uiſum colorẽ habere puniceum, propter fortitudinẽ uiſus & plurimam ad ipſum exloco uicino re-<lb/>flexionem fortiorum radiorú, propter uicinitatem corporis luminoſi, à quo fit impreſsio lucis re-<lb/>flexæ ſecundum lineam breuiorem.</s> <s xml:id="echoid-s51233" xml:space="preserve"> Et quo <lb/> <anchor type="figure" xlink:label="fig-0763-01a" xlink:href="fig-0763-01"/> niam tota nubes eſt luminoſa, & lumen ſem-<lb/>per ſecũ dum æ quales angulos reflexum à di-<lb/>uerſis ſuperficiebus in profundo nubis æ qui-<lb/>diſtantibus baſi pyramidis primæ illumina-<lb/>tionis, ad cundẽ reflectitur uiſum per ſuperfi.</s> <s xml:id="echoid-s51234" xml:space="preserve"> <lb/>ciem prioris pyramidis uicinioris uiſui (quo-<lb/>niam, ut patet per 68 th.</s> <s xml:id="echoid-s51235" xml:space="preserve"> 1 huius, circuli æ qui-<lb/>diſtantes in codem axè ſuos habent polos:</s> <s xml:id="echoid-s51236" xml:space="preserve"> & <lb/>idem pũctus eſt polus diuerſorũ circulorum) <lb/>patet quia etíam lumen, quod eſt in profundo nubis, uidetur.</s> <s xml:id="echoid-s51237" xml:space="preserve"> Quoniã uerò illud lumen, eſt lumen <lb/>refractum, debile, multo colori nubis, qui niger eſt, admixtum:</s> <s xml:id="echoid-s51238" xml:space="preserve"> & quoniá uidetur per pyramidẽ ui-<lb/>ſualem, inſcriptam ab eodem uertice (utpote à centro oculi) ipſi primæ pyramidi uiſuali, ſecũdum <lb/>quam uiciniores radij, qui punicei apparent, ad uiſum reflectuntur:</s> <s xml:id="echoid-s51239" xml:space="preserve"> patet per 106 th.</s> <s xml:id="echoid-s51240" xml:space="preserve"> 1 huius quoniá <lb/>anguli, qui ad baſim inſcriptę pyramidis fiunt, maiores erunt angulis, qui fiunt ad baſim primæ py-<lb/>ramidis.</s> <s xml:id="echoid-s51241" xml:space="preserve"> Lumẽ ergo ab illo loco in radijs ſub maiori angulo ad uiſum reflectitur:</s> <s xml:id="echoid-s51242" xml:space="preserve"> unde radij minus <lb/>lumini uniti ſunt, & debilius uiſui offerũtur.</s> <s xml:id="echoid-s51243" xml:space="preserve"> Anguli etiá, quos in cẽtro uiſus faciunt, ſunt minores, <lb/>ut patet per idem 106 th.</s> <s xml:id="echoid-s51244" xml:space="preserve"> 1 huius, quàm anguli, qui fiunt per radios primæ pyramidis in cẽtro uiſus.</s> <s xml:id="echoid-s51245" xml:space="preserve"> <lb/>Sub minori ergo angulo uidetur lumen in corpore nubis, quàm in ſuperficie:</s> <s xml:id="echoid-s51246" xml:space="preserve"> quod autem fub mi-<lb/>nori angulo uidetur, minus uidetur, ut patet per 20 th.</s> <s xml:id="echoid-s51247" xml:space="preserve"> 4 huius.</s> <s xml:id="echoid-s51248" xml:space="preserve"> Hoc autem patet experimentanti <lb/>in lumine ſtellæ uel can delæ.</s> <s xml:id="echoid-s51249" xml:space="preserve"> Quod enim prius uiſum eſt aperto oculo fulgidum, claudendo planè <lb/>oculum amittit fulgorem, & incipit nigreſcere.</s> <s xml:id="echoid-s51250" xml:space="preserve"> Item quoniam à remotiori uidetur tale lumen:</s> <s xml:id="echoid-s51251" xml:space="preserve"> ideo <lb/>debilius uidetur:</s> <s xml:id="echoid-s51252" xml:space="preserve"> remotio enim ſiue protẽſio uiſibilis à uiſu eſt cauſſa debilitatis uiſus, ut patèt per <lb/>158 th.</s> <s xml:id="echoid-s51253" xml:space="preserve"> 4 huius.</s> <s xml:id="echoid-s51254" xml:space="preserve"> Item quia uapor remotior à corpore luminoſo groſsior eſt & nigrior, & magis <lb/>aqueus:</s> <s xml:id="echoid-s51255" xml:space="preserve"> unde nigredo uaporis Iumini incorporata plus denigrat, & magis ipſum uiſui obfuſcatum <lb/>præſentat.</s> <s xml:id="echoid-s51256" xml:space="preserve"> Et hæc quidẽ in coloribus iridis aliquam cauſſalitatẽ habent.</s> <s xml:id="echoid-s51257" xml:space="preserve"> T otalis uerò cauſſa omni-<lb/>bus huius coloribus uniuerſalis eſt immixtio umbrarum ipſi fulgori luminis.</s> <s xml:id="echoid-s51258" xml:space="preserve"> Quoniam enim (ut <lb/>patet per præmiſſam) uaporroridus eſt materia iridis, à cuius corpuſculis fit reflexio luminis ad <lb/>uiſum, & per 11 th.</s> <s xml:id="echoid-s51259" xml:space="preserve"> 2 huius omnia corpora denſa in par̀tẽ luminoſo corpori aduerſam umbram pro-<lb/>ijciunt:</s> <s xml:id="echoid-s51260" xml:space="preserve"> patet quòd radij reflexi à remotiorum corpuſculorum ſuperficiebus, umbrarũ anteriorum <lb/>corpuſculorum nigredini ſe immiſcent:</s> <s xml:id="echoid-s51261" xml:space="preserve"> & ſic permixti colore nigro umbrarum perueniunt refle xi <lb/>ad uiſum:</s> <s xml:id="echoid-s51262" xml:space="preserve"> & ſecundum quod plus uel minus umbrarum nigredine permiſcentur, ſecũdum hoc di-<lb/>uerſificant actú ſuæ luminoſitatis in uarios colores.</s> <s xml:id="echoid-s51263" xml:space="preserve"> Et huius rei ſignum eſt in coloribus ſimilibus <lb/>iridi, qui obducto uiſu ipſa manu uel alio umbroſo de ſub manu in feneſtrarũ peripherijs uidẽtur.</s> <s xml:id="echoid-s51264" xml:space="preserve"> <lb/>Signum quoq;</s> <s xml:id="echoid-s51265" xml:space="preserve"> huius eſt nigredo maris, quæ propter umbrarum multiplicationem accidit in mari-<lb/>bus aquarum limpidarum, in quas lumẽ ſe profundat, cum exturbulentis aquis marium, quas lux <lb/>non penetrat, ut umbras efficiat, ipſis maribus non nigredo ſed uiriditas accedat:</s> <s xml:id="echoid-s51266" xml:space="preserve"> & obductis pal-<lb/>pebris, uiſui reſpectu luminis ex umbris pilorum ipſarũ palpebrarum colores iridis uidentur.</s> <s xml:id="echoid-s51267" xml:space="preserve"> Sin-<lb/>gula quoq;</s> <s xml:id="echoid-s51268" xml:space="preserve"> particularia, in quibus colores iridis apparent, ad hanc umbrarũ cauſſam, ut ad quod dã <lb/>uniuocum reducuntur:</s> <s xml:id="echoid-s51269" xml:space="preserve"> ut patet in collis anatum & pauonum, quæ ſecundũ diuerſam diſp oſitionẽ <lb/>diuerſimodè colorantur.</s> <s xml:id="echoid-s51270" xml:space="preserve"> Criſpitudo enim ſuarũ pennarum alias hinc & inde proijcit umbras, quæ <lb/>permixtæ lumini diuerſos hinc & inde procreant colores, ut patet intuenti.</s> <s xml:id="echoid-s51271" xml:space="preserve"> Nec enim alias pręmiſ-<lb/>ſorũ cauſſas noſtro potuimus indagare ingenio.</s> <s xml:id="echoid-s51272" xml:space="preserve"> Exiſtētibus enim tantùm 22 uiſibilibus, nullũ alio-<lb/>rum uiſibiliũ, præter umbrá, & lumẽ horũ colorũ apparentiũ uiſui uidetur eſſe cauſſa:</s> <s xml:id="echoid-s51273" xml:space="preserve"> unde & hanc <lb/>colorũ iridis æſtim amus proximã eſſe cauſſam:</s> <s xml:id="echoid-s51274" xml:space="preserve"> nullũ tamẽ uidimus, quẽ intellectus ſuus in hoc mo <lb/>dicũ intelligibile direxerit:</s> <s xml:id="echoid-s51275" xml:space="preserve"> ſed huius rei facilis omnes alij difficiles uiſi ſunt dare cauſſas.</s> <s xml:id="echoid-s51276" xml:space="preserve"> Nos tamẽ <lb/>hac cauſſa ut uniuoca & cõuertibili erimus cõtenti, alia, quæ præmifimus, ponentes, ut quædã ad-<lb/>miniculantia huic cauſſæ.</s> <s xml:id="echoid-s51277" xml:space="preserve"> Iſtis itaq;</s> <s xml:id="echoid-s51278" xml:space="preserve"> præmiſsis cauſsis uel omnibus, uel pluribus, uel aliquot ſenſi-<lb/>biliter cõcurrentibus interſectione pyramidũ reflexionis baſiũ æquidiſtantiũ:</s> <s xml:id="echoid-s51279" xml:space="preserve"> tunc deficit iudiciũ <lb/>uiſus, & lumẽ magis mixtũ uaporis nigredini, minusq́;</s> <s xml:id="echoid-s51280" xml:space="preserve"> refractũ, ſub maiori quoq;</s> <s xml:id="echoid-s51281" xml:space="preserve"> angulo reflexũ, & <lb/>ſub minori angulo uiſum, & in maiori diſtátia à ſe ipſo poſitũ, & in materia groſsiori radiatũ, & um-<lb/>bris pluribus permixtũ uiſus íudicat magis ab albo recedere quàm puniceum:</s> <s xml:id="echoid-s51282" xml:space="preserve"> uideturq́;</s> <s xml:id="echoid-s51283" xml:space="preserve"> illud lu-<lb/>men reflexũ ſibi uiride ſeu praſsinum.</s> <s xml:id="echoid-s51284" xml:space="preserve"> Et poſt hunc colorem praſsinum, plurium pyramidum facta <lb/> <pb o="462" file="0764" n="764" rhead="VITELLONIS OPTICAE"/> reflexione, cum dictæ conditiones ſenſibiliter à prius entibus cõditionibus uariantur:</s> <s xml:id="echoid-s51285" xml:space="preserve"> uidetur lu-<lb/>mẽ plus nigro accedere, & fit uiſui color alurgus ſiue lazulius, qui uaporis nigredini umbrisq́;</s> <s xml:id="echoid-s51286" xml:space="preserve"> plu-<lb/>ribus magis permixtus eſt quàm praſsinus.</s> <s xml:id="echoid-s51287" xml:space="preserve"> Et demum cum ſecundũ hunc colorem alurgum plu-<lb/>rium pyramidum uiſis circumferentijs baſium, ſenſibiliter incipiunt prędictæ conditiones uariari, <lb/>& cum lumen amplius ad uiſum ſic diſpoſitũ non reflectitur:</s> <s xml:id="echoid-s51288" xml:space="preserve"> fit nigrum, quod amplius permixtum <lb/>lumini non uidetur.</s> <s xml:id="echoid-s51289" xml:space="preserve"> Signum uerò prædictorũ eſt:</s> <s xml:id="echoid-s51290" xml:space="preserve"> quia cũ aliquis poſtquam ſolem uel aliquod cor-<lb/>pus fulgidum aſpexerit, claudit oculos ſubitò & fortiter:</s> <s xml:id="echoid-s51291" xml:space="preserve"> primò quidem obducto oculo pelle, quod <lb/>prius uidit fulgidum, uidebit puniceum:</s> <s xml:id="echoid-s51292" xml:space="preserve"> deinde praſsinum:</s> <s xml:id="echoid-s51293" xml:space="preserve"> deinde purpureum:</s> <s xml:id="echoid-s51294" xml:space="preserve"> pòſt in nigrum co-<lb/>lorem forma lucis decidens exterminatur:</s> <s xml:id="echoid-s51295" xml:space="preserve"> & ſic facto motu in uiſu ab albo ipſo paulatim extermi-<lb/>nato, ſemper in propin quius nigro fit reſolutio.</s> <s xml:id="echoid-s51296" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s51297" xml:space="preserve"> ex præmiſsis quòd iris ſit tricolor:</s> <s xml:id="echoid-s51298" xml:space="preserve"> quo-<lb/>rum colorũ ſupremus eſt puniceus:</s> <s xml:id="echoid-s51299" xml:space="preserve"> & color uiridis ſub puniceo continetur, quoniá color circum-<lb/>ferentiarũ baſium uiridiũ ſub colore baſium circumferentiarũ punicearum fertur ad uiſum:</s> <s xml:id="echoid-s51300" xml:space="preserve"> & ſimi <lb/>liter color alurgus ſub uiridi cõtinetur eadẽ ratione:</s> <s xml:id="echoid-s51301" xml:space="preserve"> & ſic uidetur unus arcus coloratus ſub alio ar-<lb/>cu cõtinuo colorato.</s> <s xml:id="echoid-s51302" xml:space="preserve"> Color uerò xanthus, qui inter colorẽ uiridẽ & colorẽ puniceũ uidetur, in iride <lb/>non eſt color diſtinctus ab alijs, ſed ex cómixtione uiridis & rubei uiſibus occurrit.</s> <s xml:id="echoid-s51303" xml:space="preserve"> Puniceus enim <lb/>color iuxta praſsinũ uiſus albus uidetur:</s> <s xml:id="echoid-s51304" xml:space="preserve"> quia & purpureus coloriuxta nigrũ albus uidetur:</s> <s xml:id="echoid-s51305" xml:space="preserve"> uiride <lb/>etiá permixtũ eſt albo:</s> <s xml:id="echoid-s51306" xml:space="preserve"> & ob hoc color xanthus, quia ppinquior eſt nigro quàm puniceus, inter pu-<lb/>niceũ & uiridẽ uidetur.</s> <s xml:id="echoid-s51307" xml:space="preserve"> Vnde etiá facta iride in nube nigerrima, color ſuperior nó eſt puniceus, ſed <lb/>xanthus uidetur, propter multá nigredinis uaporis cũ lumine permixtionẽ, & reſoluta nube, quod <lb/>prius uidebatur puniceum, demũ albũ uidetur:</s> <s xml:id="echoid-s51308" xml:space="preserve"> praſsinus quoq;</s> <s xml:id="echoid-s51309" xml:space="preserve"> uidetur tendere ad xanthum colo-<lb/>rem, & alurgus ad uiridem.</s> <s xml:id="echoid-s51310" xml:space="preserve"> Et iam uidit quidá uir experientiæ iridẽ totã albã:</s> <s xml:id="echoid-s51311" xml:space="preserve"> quod accidit propter <lb/>materiæ raritatẽ & luminis claritatẽ:</s> <s xml:id="echoid-s51312" xml:space="preserve"> & uiſus optimã diſpoſitionẽ in ſe, & in diſtãtia proportionata <lb/>ad rẽ uiſam:</s> <s xml:id="echoid-s51313" xml:space="preserve"> uel fortè propter uaporis plurimã groſsiciem & denſitatẽ, in quo nõ potuit lumẽ pene-<lb/>trare in profundũ:</s> <s xml:id="echoid-s51314" xml:space="preserve"> ſed fiebat à ſuperficie uaporis reflexio:</s> <s xml:id="echoid-s51315" xml:space="preserve"> & propter hoclumen nõ receperat colo-<lb/>rem à colore corporis ſibi cõmixto, nec miſcebatur nigredini umbrarũ:</s> <s xml:id="echoid-s51316" xml:space="preserve"> unde reflexio faciens iridẽ.</s> <s xml:id="echoid-s51317" xml:space="preserve"> <lb/>in forma luminis reflectebatur ſine admixtione nigredinis & umbrarũ.</s> <s xml:id="echoid-s51318" xml:space="preserve"> Signũ uerò diuerſæ appa-<lb/>ritionis colorũ eſt, quod uidetur in texturis purpurarum:</s> <s xml:id="echoid-s51319" xml:space="preserve"> in quibus colores iuxta alios poſiti pluri-<lb/>mam faciũt differentiá & mixtionẽin unu.</s> <s xml:id="echoid-s51320" xml:space="preserve"> Non enim idẽ uidetur purpureũ iuxtà poſitũ albo & ni-<lb/>gro, aut alicui alteri colori.</s> <s xml:id="echoid-s51321" xml:space="preserve"> Et ex hoc propter claritatẽ aliqualẽ, quam color accipit à uicino ſibi co-<lb/>lore, aliæ phantaſiæ colorũ in uiſibus oriũtur.</s> <s xml:id="echoid-s51322" xml:space="preserve"> Sicut etiã accidit operãtibus ad lucerná decipi in co-<lb/>loribus, propter admixtionẽ impuri luminis:</s> <s xml:id="echoid-s51323" xml:space="preserve"> & accidit eos peccare, & alios colores pro alijs acci-<lb/>pere, colorũ alietate eximmixtione impuri luminis generata.</s> <s xml:id="echoid-s51324" xml:space="preserve"> Et ſic nõ inconuenienter dici poſsit, <lb/>quòd medij colores iridis à medijs pyramidibus ſecundũ dictas circũſtantias & diuerſarũ umbra-<lb/>rum permixtionẽ cũ lumine generẽtur.</s> <s xml:id="echoid-s51325" xml:space="preserve"> Numerum autẽ colorũ iridis ſecundũ antiquos in ternario <lb/>decreuimus:</s> <s xml:id="echoid-s51326" xml:space="preserve"> extendunt enim in tantum colorũ nomina, ut color medius illius extremi coloris no-<lb/>men habeat, cũ quo magis participat in natura.</s> <s xml:id="echoid-s51327" xml:space="preserve"> Et ſic iridem tantũ tricolorẽ eſſe neceſſariò cópro-<lb/>batur:</s> <s xml:id="echoid-s51328" xml:space="preserve">nec poſſunt pictores tales colores plenariè ſimulare.</s> <s xml:id="echoid-s51329" xml:space="preserve"> De coloribus etiã, qui apparent in iride <lb/>generata in uapore aqueo ſparſo ore uel alio ſubtili artificio, manu, uel remo, tota cauſſa dicta eſt.</s> <s xml:id="echoid-s51330" xml:space="preserve"> <lb/>Cũ enim lumen ad talia corpuſcula incidit, & ab eis reflectitur ad uiſum in radio poſitũ, uel in ſene-<lb/>ſtra, per quam incidit radius, uerſo occipite directè ad centrũ ſolis:</s> <s xml:id="echoid-s51331" xml:space="preserve"> tunc lumẽ propin quius reflexũ <lb/>tanti eſt luminis, quod remotius reflexũ lumẽ, propter admixtionẽ umbrarum ſuperiorum corpu-<lb/>ſculorum propinquiorũ uiſibus, & corpori luminoſo, magis & magis obtenebratur ſecũdum mo-<lb/>dos prius dictos:</s> <s xml:id="echoid-s51332" xml:space="preserve"> uidebiturq́;</s> <s xml:id="echoid-s51333" xml:space="preserve"> ſic cõſtituto uiſu iris ex cauſsis prius dictis rotundata.</s> <s xml:id="echoid-s51334" xml:space="preserve"> Aliter autẽ ui-<lb/>ſu diſpoſito ad radium:</s> <s xml:id="echoid-s51335" xml:space="preserve"> uidebũtur propter inordinatam reflexionem ad uiſum colores iridis inor-<lb/>dinati:</s> <s xml:id="echoid-s51336" xml:space="preserve"> quoniam illa reflexio cum non fiat ſecundum angulos æquales ad figuram iridis rotundam <lb/>non pertingit:</s> <s xml:id="echoid-s51337" xml:space="preserve"> & ſecundum quod lumen corpuſcula rorida contingit:</s> <s xml:id="echoid-s51338" xml:space="preserve"> ſic ſecundum aliquam refle-<lb/>xionem perceptam lumen colores uarios uiſui inducit.</s> <s xml:id="echoid-s51339" xml:space="preserve"> Sed quantò remotiores ſunt radij à princi-<lb/>pio ſuæ aggregationis in feneſtra:</s> <s xml:id="echoid-s51340" xml:space="preserve"> tantò colores magis efficiũt opacos propter plurium umbrarum <lb/>immixtionem ipſi lumini reflexo.</s> <s xml:id="echoid-s51341" xml:space="preserve"> Inuenimus & nos diebus æſtiuis circa horá ueſpertinam uel mo-<lb/>dicùm antè circa Viterbium in quodam præcipitio apud balneum, (quod dicitur ſcopuli) aquam <lb/>uehementer præcipitari:</s> <s xml:id="echoid-s51342" xml:space="preserve"> deſcendentesq;</s> <s xml:id="echoid-s51343" xml:space="preserve"> ad uidendum, quid in ipſo poſſet accidere ſoli ſibi oppo-<lb/>ſito:</s> <s xml:id="echoid-s51344" xml:space="preserve">uidimus iridem perpetuam ſole circa aſpectum illi debitum exiftente:</s> <s xml:id="echoid-s51345" xml:space="preserve"> & multas ex proprieta-<lb/>tibus iridis notauimus.</s> <s xml:id="echoid-s51346" xml:space="preserve"> Vnde, quia ea, quæ prius ſcripta de iride fuerant, nobis non per omnia ſuf-<lb/>ficere uidebantur (excepto eo, quod inuolutè ſcripſerat Ariſtoteles) illud nobis principium cogi-<lb/>tationis fuit, ut præſenti negotio ſtudium applicaremus.</s> <s xml:id="echoid-s51347" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s51348" xml:space="preserve"> propoſitum.</s> <s xml:id="echoid-s51349" xml:space="preserve"/> </p> <div xml:id="echoid-div1902" type="float" level="0" n="0"> <figure xlink:label="fig-0763-01" xlink:href="fig-0763-01a"> <caption xml:id="echoid-caption2" xml:space="preserve">PVNICEVS <lb/>XAN THVS VIRIDIS VELINDICVS <lb/>ALVRGVS</caption> </figure> </div> </div> <div xml:id="echoid-div1904" type="section" level="0" n="0"> <head xml:id="echoid-head1396" xml:space="preserve" style="it">68. Corona fit ex refractione luminis ſolis, uellunæ, uel ſtellarum primæ magnitudinis à ua-<lb/>pore humido circulariter ad uiſum.</head> <p> <s xml:id="echoid-s51350" xml:space="preserve">Impreſsio (quæ græcè dicitur, ά<unsure/>λως & arabicè alileti) latinè dicitur corona.</s> <s xml:id="echoid-s51351" xml:space="preserve"> Fit autẽ hæc impreſ-<lb/>ſio in uiſu ex incorporatione luminis in aliqua cõſiſtentia uaporis.</s> <s xml:id="echoid-s51352" xml:space="preserve"> Cũ enim, ut patet per 56 huius, <lb/>non aggregatis radijs corporis luminoſi in corpore non luminoſo plus, quàm in medio lumen ſen-<lb/>ſibilius fieri ſit impoſsibile:</s> <s xml:id="echoid-s51353" xml:space="preserve"> patet quòd ad generationẽ halonis neceſſarium eſt aliquem uaporem <lb/>corpori luminoſo & uiſibus interponi.</s> <s xml:id="echoid-s51354" xml:space="preserve"> Cum ergo aliquis uapor humidus continuus interponitur <lb/>uiſibus, & corpori luminoſo nõ potẽte illũ uaporem citò diſſoluere uel diſgregare:</s> <s xml:id="echoid-s51355" xml:space="preserve"> túc fit ad uiſum <lb/>refractio luminis ſecundũ circulũ ք 64 th.</s> <s xml:id="echoid-s51356" xml:space="preserve"> huius.</s> <s xml:id="echoid-s51357" xml:space="preserve"> Lumẽ enim ſecundũ æquales angulos illi uapori <lb/>per ignem & aerem incidens, ſecundũ æ quales angulos refringitur ad uiſum per 8 huius:</s> <s xml:id="echoid-s51358" xml:space="preserve"> uidetur <lb/> <pb o="463" file="0765" n="765" rhead="LIBER DECIMVS."/> itaq lumẽ circulare propter æ qualem refractionem luminis aggregati ad uiſum:</s> <s xml:id="echoid-s51359" xml:space="preserve"> quoniam propter <lb/>reſractionem luminis, ut patet per 57 th.</s> <s xml:id="echoid-s51360" xml:space="preserve"> huius, aggregantur radij in profundo uaporis.</s> <s xml:id="echoid-s51361" xml:space="preserve"> Cum enim <lb/>lineæ radiales franguntur ad angulos:</s> <s xml:id="echoid-s51362" xml:space="preserve"> tunc lumen uιſui quaſi duplicatur, & peruenit uehementius <lb/>ad uiſum.</s> <s xml:id="echoid-s51363" xml:space="preserve"> Et ſi fortè uaporille ſit roridus, diſtinctus per corpuſcula:</s> <s xml:id="echoid-s51364" xml:space="preserve"> tunc plures fiunt refractiones, <lb/>& àugetur lumen.</s> <s xml:id="echoid-s51365" xml:space="preserve"> Et quoniam idem radius incidens ſuperficiei uaporis, in corpore uaporis refrin-<lb/>gitur ad perpendicularem, à puncto ſuæ incidentiæ ſuper ſuperficiem corporis, à quo refringitur <lb/>productam, & ſecundum extenſionem lineæ incidentiæ umbra protenditur per 11 th.</s> <s xml:id="echoid-s51366" xml:space="preserve"> 2 huius:</s> <s xml:id="echoid-s51367" xml:space="preserve"> & <lb/>quoniam radius incidens & refractus non ſunt linea una, ſed angulum continent:</s> <s xml:id="echoid-s51368" xml:space="preserve"> ideo patet quia <lb/>radius refractus refugit umbram proiectam à corpore, cui incidebat, quæ tamen eſt modica:</s> <s xml:id="echoid-s51369" xml:space="preserve"> quia <lb/>ut plurimum corona uidetur in uapore raro, leuiter condenſato.</s> <s xml:id="echoid-s51370" xml:space="preserve"> Veruntamen quia retro uaporis <lb/>illius conſiſtentiam fit noua refractio in aere medio inter uaporem & uiſum, quæ fit à perpendicu-<lb/>lari per 4 huius:</s> <s xml:id="echoid-s51371" xml:space="preserve"> patet quòd lumen refractum perueniens ad centrum uiſus non eſt umbrarum ni-<lb/>gredine permixtum, ſed liberum ab illis:</s> <s xml:id="echoid-s51372" xml:space="preserve"> & propter hoc ſemper uidetur album, uel fortè modico & <lb/>indiſtincto colore aliqualiter rubeo ſecundum ſe totum coloratum.</s> <s xml:id="echoid-s51373" xml:space="preserve"> Iris uerò quia fit per reflexio-<lb/>nem radiorum umbras proiectas penetrantium:</s> <s xml:id="echoid-s51374" xml:space="preserve"> ideo illi radij ſub actu coloris perueniũt ad uiſum:</s> <s xml:id="echoid-s51375" xml:space="preserve"> <lb/>fitq́;</s> <s xml:id="echoid-s51376" xml:space="preserve"> diſtinctio colorum ſecundum modum diuerſitatis luminis & umbrarum.</s> <s xml:id="echoid-s51377" xml:space="preserve"> Videtur itaq;</s> <s xml:id="echoid-s51378" xml:space="preserve"> coro-<lb/>na ex refractione luminis quandoq;</s> <s xml:id="echoid-s51379" xml:space="preserve"> ſolaris:</s> <s xml:id="echoid-s51380" xml:space="preserve"> ſed rarò accidit hoc, propter fortitudinẽ & uehemen-<lb/>tiam illius luminis, uaporem (qui eſt materia coronæ) ſubitò diſſoluentis.</s> <s xml:id="echoid-s51381" xml:space="preserve"> Sæpe tamen accidit hoc <lb/>exlumine lunæ & ſtellarum primæ magnitudinis, quarum lumen illam conſiſtentíam uaporis diſ-<lb/>ſoluere non poteſt.</s> <s xml:id="echoid-s51382" xml:space="preserve"> A minoribus uerò ſtellis non accidit halo propter ſui luminis debilitatẽ, quod <lb/>tantum effectum imprimere non poteſt.</s> <s xml:id="echoid-s51383" xml:space="preserve"> In circuitu quoq;</s> <s xml:id="echoid-s51384" xml:space="preserve"> luminis candelarum quandoq;</s> <s xml:id="echoid-s51385" xml:space="preserve"> accidit <lb/>uideri coronam in aere groſſo, ut plurimum flante euro:</s> <s xml:id="echoid-s51386" xml:space="preserve"> & tũc quandoq;</s> <s xml:id="echoid-s51387" xml:space="preserve"> p̀ropter denſitatem aeris <lb/>proijcientis umbram partium ſuperiorum ſuper infimas, accidit uiſibus colorem purpureum à tali <lb/>refracto uel reflexo lumine præſentari.</s> <s xml:id="echoid-s51388" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s51389" xml:space="preserve"> propoſitum.</s> <s xml:id="echoid-s51390" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1905" type="section" level="0" n="0"> <head xml:id="echoid-head1397" xml:space="preserve" style="it">69. Iridem in parte mundi meridionali à ſeptentrionalibus uiſibus non eſt poßibile uideri.</head> <p> <s xml:id="echoid-s51391" xml:space="preserve">Quod per 107 th.</s> <s xml:id="echoid-s51392" xml:space="preserve"> 1 huius patet in pyramidibus purè mathematicis ſibi ad inuicẽ inſcriptis:</s> <s xml:id="echoid-s51393" xml:space="preserve"> idem <lb/>patet per 64 huius de pyramidibus reflexis, iridem cauſſantibus, quæ naturam mathematicarum <lb/>pyramidum conſequuntur.</s> <s xml:id="echoid-s51394" xml:space="preserve"> Semper enim oportet, ut centrum uiſus ſit inter centrum corporis lu-<lb/>minoſi & centrum iridis, ad hoc ut illa impreſsio uideatur, quam propriè iridem nominamus:</s> <s xml:id="echoid-s51395" xml:space="preserve"> licet <lb/>aliæ impreſsiones, colores iridis ſimulantes, quandoq;</s> <s xml:id="echoid-s51396" xml:space="preserve"> per modos alios uideri ualeant, ut inferius <lb/>patebit.</s> <s xml:id="echoid-s51397" xml:space="preserve"> Quòd autem iris meridiana à uiſibus ſeptentrionalibus uideri nõ ualeat, ſatis patet ex his, <lb/>quæ diximus in generatione colorum iridis:</s> <s xml:id="echoid-s51398" xml:space="preserve"> qui propter reflexionem luminis & umbrarum lumi-<lb/>ni admixtionem perſe cauſſantur.</s> <s xml:id="echoid-s51399" xml:space="preserve"> Poteſt etiam occaſionaliter id patere per hoc:</s> <s xml:id="echoid-s51400" xml:space="preserve"> quòd materia iri-<lb/>dis in approximatione corporis luminoſi de facili reſoluitur in aquam, uel ſubtiliatur in aerem <lb/>lucidum, à cuius ſuperficie non poſſunt fieri reflexiones:</s> <s xml:id="echoid-s51401" xml:space="preserve"> quæ etſi fierent, tamen tenderent in par-<lb/>tem, in qua eſt ſol, nec ad uiſum peruenirent.</s> <s xml:id="echoid-s51402" xml:space="preserve"> Et etiam quia colores iridis, qui fiunt propter debi-<lb/>litationem reflexæ lucis, non poſſunt in taliloco cauſſari:</s> <s xml:id="echoid-s51403" xml:space="preserve"> quia circa corpus luminoſum cum ſem-<lb/>per plus ſit luminis, radij reflexi non debilitantur, ſed magis uiſibiles efficiuntur.</s> <s xml:id="echoid-s51404" xml:space="preserve"> In talibus tamen <lb/>locis facta radiorum refractione ad uiſum per uaporem uel aerem denſum, aliquod lumen aggre-<lb/>gatum uideri poteſtin uapore uel aere condenſato, ut diximus in præmiſſa de generatione coro-<lb/>næ, quæ fit ex refractione luminis ſolis quandoque:</s> <s xml:id="echoid-s51405" xml:space="preserve"> & tamen rarò, propter luminis illius fortitu-<lb/>dinem:</s> <s xml:id="echoid-s51406" xml:space="preserve"> ſæpe uerò exlumine lunæ & ſtellarum primæ & principalis magnitudinis generatur.</s> <s xml:id="echoid-s51407" xml:space="preserve"> Iris <lb/>ergo quando debet generari, oportet quòd radij ad oculum reflectantur, & quòd retro uaporem <lb/>roridum (qui eſt materia iridis per 66 huius) non ſit lumen aliud irradians.</s> <s xml:id="echoid-s51408" xml:space="preserve"> Vnde etiam co-<lb/>rona groſſa apparente uiſui, ſcilicet in groſſa materia & ſpiſſa ſiue denſa, à forti lumine cauſſata, <lb/>eſt poſsibile, ut in ipſa aliqui colores iridis appareant, uiſu poſito inter corpus luminoſum & ua-<lb/>porem.</s> <s xml:id="echoid-s51409" xml:space="preserve"> Tunc enim omnes conditiones & cauſſæ colorum iridis in loco tali concurrent, & ma-<lb/>teria ſubeſt.</s> <s xml:id="echoid-s51410" xml:space="preserve"> Iris ergo ſic poterit apparere.</s> <s xml:id="echoid-s51411" xml:space="preserve"> Fortè ergo accidit quòd materia, in qua plus meridio-<lb/>nalibus à uapore rorido iris uidetur reflexa:</s> <s xml:id="echoid-s51412" xml:space="preserve"> tunc hominibus plus ſeptentrionalibus ab eodem ua-<lb/>pore (ita quòd uaporidem eodem tempore utriſq;</s> <s xml:id="echoid-s51413" xml:space="preserve"> habitatoribus appareat, & ſecundum eundem <lb/>circulum altitudinis) uideatur corona propter luminis refractionem:</s> <s xml:id="echoid-s51414" xml:space="preserve"> & idem erit in quolibet cir-<lb/>culo altitudinis prædicto modo quibuslibet uidentibus conſtitutis.</s> <s xml:id="echoid-s51415" xml:space="preserve"> Exhis quoque, quæ dicta <lb/>ſunt, patere poteſt, quòd quandoque ex fortibus ſolis radijs reflexis à nube aquoſa integra ad lo-<lb/>cum, in quo eſt uapor roridus, à latere ſolis aliquo poſſunt colores iridis generari in plenis circu-<lb/>lis uel circulorum portionibus incompletis:</s> <s xml:id="echoid-s51416" xml:space="preserve"> ut quando corpori ſolis nubes ſolida aquoſa diame-<lb/>traliter opponitur, & in ipſam incidens radius reflectitur, & reflexo radio nubes rorida obſiſtit, <lb/>in qua fit radiorum refractio & reflexio perueniens ad uiſum.</s> <s xml:id="echoid-s51417" xml:space="preserve"> Tunc enim colores iridis apparent <lb/>uiſui recti, ut cum uapor non recte opponitur uiſui:</s> <s xml:id="echoid-s51418" xml:space="preserve"> & tales colores ſunt in uapore raro aqueo per-<lb/>mixto:</s> <s xml:id="echoid-s51419" xml:space="preserve"> quandoq;</s> <s xml:id="echoid-s51420" xml:space="preserve"> uerò apparent circulares:</s> <s xml:id="echoid-s51421" xml:space="preserve"> & fiunt quaſi irides.</s> <s xml:id="echoid-s51422" xml:space="preserve"> Oportet autem ad hoc, ut talis iris <lb/>uideatur, quòd nubes, ad quam fit radiorum ſolis reflexio ad oppoſitum uaporem, & uapor rori-<lb/>dus, ad quem, & à quo ad uiſum fit luminis reflexio, & uiſus, ad quem fit reflexio, in eadem recta <lb/>linea conſiſtant:</s> <s xml:id="echoid-s51423" xml:space="preserve"> & quòd ſuperficies nubis, à qua fit reflexio, & ſuperficies uaporis, à qua, & ad quam <lb/>fit reflexio, productæ ſupra horizontem quaſi in ſuperiori hemiſphærio concurrant.</s> <s xml:id="echoid-s51424" xml:space="preserve"> Aliter enim <lb/> <pb o="464" file="0766" n="766" rhead="VITELLONIS OPTICAE"/> uix fieret ſenſibilis reflexio ad uiſum poſteriorem nube, à qua fit reflexio:</s> <s xml:id="echoid-s51425" xml:space="preserve"> fierct autem modica pro-<lb/>pter naturam reflexionis à corpuſculis paruis, de quibus ſermo fuit in 62 th.</s> <s xml:id="echoid-s51426" xml:space="preserve"> huius.</s> <s xml:id="echoid-s51427" xml:space="preserve"> Nos autem per <lb/>hunc concurſum ſuperficierum, intelligimus concurſum linearum contingentium corpuſcula ua-<lb/>poris roridi in ipſo puncto reflexionis.</s> <s xml:id="echoid-s51428" xml:space="preserve"> Oportet etiam quòd nubes aquea reuerberans lumen, uici-<lb/>na ſit circa ſolem, ubi radij ſolares fortes exiſtunt:</s> <s xml:id="echoid-s51429" xml:space="preserve"> & talem iridem non unam, nec duas tantùm, ſed <lb/>etiam quatuor ſimul oidimus Paduæ ſole iam ad ueſperam declinante, & nõ erant irides in diſtan-<lb/>tia 10 graduum à ſole:</s> <s xml:id="echoid-s51430" xml:space="preserve"> & omnes circulorum completorum, & in ſuperficiebus diuerſis:</s> <s xml:id="echoid-s51431" xml:space="preserve"> & erát quæ-<lb/>dam quaſi ſe extrinſecus contingẽtes.</s> <s xml:id="echoid-s51432" xml:space="preserve"> Eas autem irides, quæ fiunt ex radijs corporis luminoſi non <lb/>ab alia nube reflexis ad uaporem, ſed ab ipſo uapore ad uiſum reflexis non eſt poſsibile fieri, nιſi in <lb/>oppoſιtione corporis luminoſi ad uaporem, uiſu in medio exiſtente.</s> <s xml:id="echoid-s51433" xml:space="preserve"> Vnde in noſtra habitabili non <lb/>poteſt uideri iris ad meridiem:</s> <s xml:id="echoid-s51434" xml:space="preserve"> quia non interponituribi uiſui uapor & corpori luminoſo.</s> <s xml:id="echoid-s51435" xml:space="preserve"> Curſus <lb/>enim ſtellarum erraticarum terminantur ſecũdum partem, qua extremitas zodiaci terminatur, qui <lb/>in noſtra habitabili ſeptentrionali fieri non poteſt.</s> <s xml:id="echoid-s51436" xml:space="preserve"> Et hoc eſt, quod proponebatur.</s> <s xml:id="echoid-s51437" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1906" type="section" level="0" n="0"> <head xml:id="echoid-head1398" xml:space="preserve" style="it">70. Exradijs ſolaribus & lunaribus tantùm irides generantur.</head> <p> <s xml:id="echoid-s51438" xml:space="preserve">Quoniam tantùm horum duorum corporum radij ſecũdum mundi diametrum ſenſibiliter ex-<lb/>tenduntur:</s> <s xml:id="echoid-s51439" xml:space="preserve"> ſolis utpote, quia eſt corpus maximum quantitate omnium luminoſorum corporum & <lb/>puriſsimæ ſubſtantiæ:</s> <s xml:id="echoid-s51440" xml:space="preserve"> lunæ uerò, quia ipſa terræ eſt uicinior:</s> <s xml:id="echoid-s51441" xml:space="preserve"> unde eius radij uiſui ſenſibilius offe-<lb/>runtur.</s> <s xml:id="echoid-s51442" xml:space="preserve"> Ab aliorum uerò corporum luminis ſenſibilitate excuſat uiſum paruitas ipſorum corporũ, <lb/>reſpectu ſolis, & magna à nobis diſtantia, reſpectu lunæ.</s> <s xml:id="echoid-s51443" xml:space="preserve"> A ſole autem iridem fieri cognitũ eſt ſen-<lb/>ſui.</s> <s xml:id="echoid-s51444" xml:space="preserve"> Ex radijs etiam lunæ iridem fieri eſt poſsibile:</s> <s xml:id="echoid-s51445" xml:space="preserve"> & hoc eſt ſæpe uiſum:</s> <s xml:id="echoid-s51446" xml:space="preserve"> maximè apud plus ſepten-<lb/>trionales, quibus ſæpe offertur materia.</s> <s xml:id="echoid-s51447" xml:space="preserve"> Vnde uiderunt lunæ iridem obſeruatores nocturni in Ale-<lb/>mania bis in uno anno:</s> <s xml:id="echoid-s51448" xml:space="preserve"> & fortè pluries uideretur, ſecundũ quod ſe offerũt agens & materia.</s> <s xml:id="echoid-s51449" xml:space="preserve"> Apud <lb/>meridionales uerò rarius uidetur:</s> <s xml:id="echoid-s51450" xml:space="preserve"> quia non offert ſe toties materia, & ſi agens ſemper ſit diſpoſi-<lb/>tum ad diffuſionem luminis, ut in omni plenilunio uel circa illud.</s> <s xml:id="echoid-s51451" xml:space="preserve"> Vnde Ariſtoteles non conſide-<lb/>rauit fieri iridem lunæ in loco ſuæ habitationis, niſi bis in 50 annis.</s> <s xml:id="echoid-s51452" xml:space="preserve"> Fiunt autem irides lunæ plures <lb/>in crepuſculis luna plena uel gibberoſa, magna exiſtéte, poſita circa orientem ſuper horizonta ſic, <lb/>ne radij ſolis uideantur.</s> <s xml:id="echoid-s51453" xml:space="preserve"> Fiunt etiam in nocte, ſemper tamẽ in oppoſito lunæ:</s> <s xml:id="echoid-s51454" xml:space="preserve"> habetq́;</s> <s xml:id="echoid-s51455" xml:space="preserve"> iris lunæ for-<lb/>mam & materiam, quam & iris ſolis:</s> <s xml:id="echoid-s51456" xml:space="preserve"> ſimiliter & colorum diſtinctiones:</s> <s xml:id="echoid-s51457" xml:space="preserve"> qui tamen ſunt albiores co-<lb/>loribus iridis ſolis:</s> <s xml:id="echoid-s51458" xml:space="preserve"> cuius cauſſa eſt, quoniam in nube nigra & in nocte fit iridis lunæ apparitio:</s> <s xml:id="echoid-s51459" xml:space="preserve"> un-<lb/>de duplicato nigro;</s> <s xml:id="echoid-s51460" xml:space="preserve"> ſcilicet noctis & nubis, album, quod fit ex radijs lunæ, magis uidetur album.</s> <s xml:id="echoid-s51461" xml:space="preserve"> Et <lb/>quia puniceum eſt debiliter album:</s> <s xml:id="echoid-s51462" xml:space="preserve"> ideo puniceum magis album tũc uidebitur comparatione plus <lb/>nigri.</s> <s xml:id="echoid-s51463" xml:space="preserve"> Et ſimiliter eſt de unoquoque aliorum colorum:</s> <s xml:id="echoid-s51464" xml:space="preserve"> quilibet enim illorum colorum albior uide-<lb/>tur.</s> <s xml:id="echoid-s51465" xml:space="preserve"> Et ſic tota iris lunæ albior uidetur, quàm iris ſolis.</s> <s xml:id="echoid-s51466" xml:space="preserve"> Vmbræ enim radijs lunæ accidentes non <lb/>ſunt tam nigræ ut umbræſolis:</s> <s xml:id="echoid-s51467" xml:space="preserve"> & huius cauſſæ ſunt diuerſæ, ut dictum eſt.</s> <s xml:id="echoid-s51468" xml:space="preserve"> Lumen enim lunæ eſt <lb/>pallidius lumine ſolis:</s> <s xml:id="echoid-s51469" xml:space="preserve"> unde colores ex cómixtione ſui informati inficiuntur, nec accedunt ad ſum-<lb/>mum formæ ſibi propriæ:</s> <s xml:id="echoid-s51470" xml:space="preserve"> ſicut etiam accidit propter pallorem luminis candelæ uariari plurimos <lb/>colores, & alios pro alijs accipi per ſenſum.</s> <s xml:id="echoid-s51471" xml:space="preserve"> Sic ergo patet à quorum corporum radijs irides gene-<lb/>rantur:</s> <s xml:id="echoid-s51472" xml:space="preserve"> quoniam ex radijs ſolis & lunæ tantùm, non autem ex aliarum ſtellarum radijs quarum-<lb/>cunque:</s> <s xml:id="echoid-s51473" xml:space="preserve"> quod eſt propoſitum.</s> <s xml:id="echoid-s51474" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1907" type="section" level="0" n="0"> <head xml:id="echoid-head1399" xml:space="preserve" style="it">71. Non plures duabus iridibus, ſitu colorum differentibus, poßibile eſt uideri.</head> <p> <s xml:id="echoid-s51475" xml:space="preserve">Verbi gratia.</s> <s xml:id="echoid-s51476" xml:space="preserve"> Cum enim non ſint, niſi tres colores ιridis, ut patet per 67 th.</s> <s xml:id="echoid-s51477" xml:space="preserve"> huius:</s> <s xml:id="echoid-s51478" xml:space="preserve"> non eſt poſsi-<lb/>bile diuerſificari colores iridis in ſitu, niſi ſecundum extremorum colorum, ſcilicet punicei & alur-<lb/>gi localem tranſpoſitionem:</s> <s xml:id="echoid-s51479" xml:space="preserve"> quia ſemper medius manet in cauſſalitate media inter iſtos.</s> <s xml:id="echoid-s51480" xml:space="preserve"> Et ob hoc <lb/>patet quòd plures, quàm duæ irides ſitu colorum differentes fieri non poſſunt:</s> <s xml:id="echoid-s51481" xml:space="preserve"> quia color medius <lb/>non poteſt habere cauſſam generationis alijs coloribus manentibus in forma propria, quamuis <lb/>ſint tranſpoſiti in ſitu.</s> <s xml:id="echoid-s51482" xml:space="preserve"> Quòd autem quandoque plures irides eiuſdem ſitus in coloribus uidentur, <lb/>una ſub alia, ut primò rubeú:</s> <s xml:id="echoid-s51483" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0766-01a" xlink:href="fig-0766-01"/> deinde uiride:</s> <s xml:id="echoid-s51484" xml:space="preserve"> & deinde a-<lb/>lurgum:</s> <s xml:id="echoid-s51485" xml:space="preserve"> & iterum rubeum:</s> <s xml:id="echoid-s51486" xml:space="preserve"> & <lb/>iterum uiride:</s> <s xml:id="echoid-s51487" xml:space="preserve"> & demum a-<lb/>lurgum:</s> <s xml:id="echoid-s51488" xml:space="preserve"> hoc accidit propter <lb/>diuerſitatẽ materiæ in diuer-<lb/>ſis ſuperficiebus, quarũ una <lb/>eſt ante aliam, & quas accidit <lb/>ſub uno angulo uideri:</s> <s xml:id="echoid-s51489" xml:space="preserve"> unde <lb/>uidentur quaſi ſint habitę uel <lb/>contiguæ.</s> <s xml:id="echoid-s51490" xml:space="preserve"> Quòd ſi in angulo <lb/>ſit diuerſitas, ut quando linea <lb/>exiens à uiſu & tranſiens per <lb/>gibbum iridis unius, ſcilicet <lb/>inferioris, non tranſit per gib-<lb/>bum ſuperioris:</s> <s xml:id="echoid-s51491" xml:space="preserve"> tunc uidebuntur conſequenter entes, & inter alurgũ ſuperioris & puniceum infe, <lb/>rioris erit notabilis differẽtia, ſcilicet alba:</s> <s xml:id="echoid-s51492" xml:space="preserve"> quoniã ab illa parte nubis remotioris uel propinquioris <lb/> <pb o="465" file="0767" n="767" rhead="LIBER DECIMVS."/> ipſi uiſui, quàm naturæ reflexionis ad uiſum illũ conueniat, non fit reflexio luminis ad uiſum:</s> <s xml:id="echoid-s51493" xml:space="preserve"> quod <lb/>non accidit quando ſub eodem angulo uidentur.</s> <s xml:id="echoid-s51494" xml:space="preserve"> Sunt tamen huiuſmodi irides ſemper in diuerſis <lb/>ſuperficiebus, & ab una pyramide reflexi luminis cauſſantur:</s> <s xml:id="echoid-s51495" xml:space="preserve"> & ob hoc ipſorum eſt quaſi centrum <lb/>unum, quod eſt centrum pyramidis irradiationis, & uidentur æ quidiſtantes in uiſu ipſorum peri-<lb/>pheriæ.</s> <s xml:id="echoid-s51496" xml:space="preserve"> Et poſsibile eſt (licet non ſæpe eueniat) quòd plures tales irides, una uidelicet intra aliam <lb/>uiſui offerantur.</s> <s xml:id="echoid-s51497" xml:space="preserve"> Et iſtud poterit probari duobus aquam in radio ſpargentibus, uno ſcilicet ſub reli-<lb/>quo:</s> <s xml:id="echoid-s51498" xml:space="preserve"> tunc enim iris ſub iride poterit uideri.</s> <s xml:id="echoid-s51499" xml:space="preserve"> Sed idem erit ordo in ſitu colorum iridis utriuſq;</s> <s xml:id="echoid-s51500" xml:space="preserve">: neu-<lb/>ter tamen alterius iridem uidebit, ſed unicuiq;</s> <s xml:id="echoid-s51501" xml:space="preserve"> ſua in eodem tempore uiſui occurret.</s> <s xml:id="echoid-s51502" xml:space="preserve"> Impoſsibile <lb/>autem eſt quòd id fiat in eadem ſuperficie:</s> <s xml:id="echoid-s51503" xml:space="preserve"> ſcilicet quòd plures irides eiuſdem ſitus in coloribus <lb/>appareant:</s> <s xml:id="echoid-s51504" xml:space="preserve"> quoniam ab illa ſola parte ſuperficiei fit reflexio, ubi ſecundum æ quales angulos radij <lb/>incidunt, & non ab alijs partibus eiuſdem ſuperficiei ſuperioribus uel inferioribus peripheria prę-<lb/>dicta, ut patet per 66 th.</s> <s xml:id="echoid-s51505" xml:space="preserve"> huius.</s> <s xml:id="echoid-s51506" xml:space="preserve"> Colores autem iridis exterioris coloribus iridis interioris ſemper <lb/>debiliores apparent:</s> <s xml:id="echoid-s51507" xml:space="preserve"> quoniam fiunt à radijs magis diſtantibus à perpendiculari & remotioribus <lb/>à uiſu:</s> <s xml:id="echoid-s51508" xml:space="preserve"> unde lumen per eos reflexum debilius uidetur, reſpectu eius, quod ex interioribus ra-<lb/>dijs cauſſatur.</s> <s xml:id="echoid-s51509" xml:space="preserve"/> </p> <div xml:id="echoid-div1907" type="float" level="0" n="0"> <figure xlink:label="fig-0766-01" xlink:href="fig-0766-01a"> <caption xml:id="echoid-caption3" xml:space="preserve">ALVRGVS <lb/>VIRIDIS <lb/>PVNICEVS <lb/>PVNICEVS <lb/>VIRIDIS <lb/>ALVRGVS</caption> </figure> </div> </div> <div xml:id="echoid-div1909" type="section" level="0" n="0"> <head xml:id="echoid-head1400" xml:space="preserve" style="it">72. In iride exteriori quando colores interioris iridis contr apoſiti & debiliores uidentur.</head> <p> <s xml:id="echoid-s51510" xml:space="preserve">Colores iridis contrapoſitos dicimus, quando ſicut iridis interioris color eſt puniceus, qui eſt <lb/>in exteriori circumferentia ipſius, ſic exterioris iridis color eſt puniceus, qui eſt in interiori peri-<lb/>pheria ipſius, mediusq́;</s> <s xml:id="echoid-s51511" xml:space="preserve"> utriufq;</s> <s xml:id="echoid-s51512" xml:space="preserve"> iridis color eſt praſsinus:</s> <s xml:id="echoid-s51513" xml:space="preserve"> interiorq́;</s> <s xml:id="echoid-s51514" xml:space="preserve"> color interioris iridis eſt alur-<lb/>gus, ſicut exterior color iridis exterioris.</s> <s xml:id="echoid-s51515" xml:space="preserve"> Sic autem diſpoſitis duabus iridibus:</s> <s xml:id="echoid-s51516" xml:space="preserve"> tunc omnes colo-<lb/>res exterioris iridis ſunt debiliores quàm interioris iridis colores.</s> <s xml:id="echoid-s51517" xml:space="preserve"> Huius quoque cauſſa aliqua eſſe <lb/>poſſet, ſi illi colores o-<lb/>mnes in una nubis ſuper <lb/>ficie uiderẽtur:</s> <s xml:id="echoid-s51518" xml:space="preserve"> quia tunc <lb/>colores exterioris iridis <lb/>per magnam diſtantiam <lb/>uiſui apparerent, ſicut & <lb/>interiores peripherię iri-<lb/>dis interioris.</s> <s xml:id="echoid-s51519" xml:space="preserve"> Ad quod in <lb/>telligẽdum ponamus ex-<lb/>empli cauſſa ſolem ſupra <lb/>horizonta 20 gradibus <lb/>eleuatum.</s> <s xml:id="echoid-s51520" xml:space="preserve"> Et quoniã pa-<lb/>tuit prius in 64 th.</s> <s xml:id="echoid-s51521" xml:space="preserve"> huius <lb/>quòd centrum baſis py-<lb/>ramidis irradiationis & <lb/>centrum uiſus, & cẽtrum corporis radioſi, quod eſt ſol, ſunt ſemper in eadem linea:</s> <s xml:id="echoid-s51522" xml:space="preserve"> centrumq́;</s> <s xml:id="echoid-s51523" xml:space="preserve"> ba-<lb/>ſis pyramidis irradiationis & pyramidis uiſionis eſt unum punctum centro ſolis diametraliter op-<lb/>poſitum:</s> <s xml:id="echoid-s51524" xml:space="preserve"> unde ipſum eſt nadir ſolis, & mouetur ſemper ſecundum motum ſolis:</s> <s xml:id="echoid-s51525" xml:space="preserve"> motuq́;</s> <s xml:id="echoid-s51526" xml:space="preserve"> ſuo ſimi-<lb/>lem circulum deſcribit circulo motus ſolis, ſcilicet ei parallelo, quem ſol motu diurno deſcribit ſu-<lb/>pra horizonta:</s> <s xml:id="echoid-s51527" xml:space="preserve"> talem enim dictum centrum iridis deſcribit, quod eſt centrum baſis pyramidis illu-<lb/>minationis ſub horizonte.</s> <s xml:id="echoid-s51528" xml:space="preserve"> Et ſicut cum ſol fuerit in puncto horizontis orientali, centrum fit in par-<lb/>te horizontis occidentali:</s> <s xml:id="echoid-s51529" xml:space="preserve"> ſic cum ſol fit in puncto horizontis occidentali, centrum illud fit in parte <lb/>orientali.</s> <s xml:id="echoid-s51530" xml:space="preserve"> Et quoniam lineæ ductæ à centro ſolis ad circumferentiam baſis pyramidis illuminatio-<lb/>nis, ſunt æ quales per 89 th.</s> <s xml:id="echoid-s51531" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s51532" xml:space="preserve"> palàm quòd ſuperficies baſis prædictæ pyramidis ſic horizonta <lb/>interſecat, quòd ipſa cũ ſuperficie ſecante ſolem, orthogonaliter inſiſtente horizonti concurret ſub <lb/>horizonte:</s> <s xml:id="echoid-s51533" xml:space="preserve"> ergo facit angulum ſuper horizontem obtuſum reſpectu uiſus.</s> <s xml:id="echoid-s51534" xml:space="preserve"> Nec mirum quoniam <lb/>horizon cum tranſeat per unum polorum circuli baſis, ut per centrum uiſus, qui eſt polus illius cir-<lb/>culi per 65 th.</s> <s xml:id="echoid-s51535" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s51536" xml:space="preserve"> patet quòd per polum alterum illius circuli non tranſit.</s> <s xml:id="echoid-s51537" xml:space="preserve"> Quælibet ergo pars <lb/>ſuperficiei uaporis, in qua fit iris exterior, illa pars, quæ eſt ſuper circulũ iridis in parte altiori, plus <lb/>à uiſu elongatur:</s> <s xml:id="echoid-s51538" xml:space="preserve"> & ſi ab ipſa reflecti accidat radios ad uiſum, neceſſe eſt ſuperiores nigriores uiſui <lb/>apparere, reſpectu eorum radiorum, qui à partibus eiuſdem ſuperficiei inferioribus illis ad uiſum <lb/>reflectuntur, ut patet per 158 & 159 th.</s> <s xml:id="echoid-s51539" xml:space="preserve"> 4 huius.</s> <s xml:id="echoid-s51540" xml:space="preserve"> Et ſic ſuperioris iridis inferioris peripheriæ, quæ ui-<lb/>cinior eſt uiſui, colores puniceos, mediæ uerò praſsinos, ſupremæ uerò alurgos neceſſe eſt uideri:</s> <s xml:id="echoid-s51541" xml:space="preserve"> <lb/>& uincit quantitas diſtantiæ in magnitudine exceſſus elongationis quantitatem angulorum refle-<lb/>xionis & quantitatem anguli uiſionis.</s> <s xml:id="echoid-s51542" xml:space="preserve"> Et ob hoc colores iridis ſuperioris contrapoſiti quandoque <lb/>uidentur coloribus iridis interioris, in qua ſuperior peripheria ſem per uidetur punicea.</s> <s xml:id="echoid-s51543" xml:space="preserve"> Quoniam <lb/>quando ad uiſum ab illa parte ſuperficiei fit reflexio improportionata reflexionibus diſtantia:</s> <s xml:id="echoid-s51544" xml:space="preserve"> tunc <lb/>radij inferiores eiuſdem ſuperficiei in eadẽ diſtantia ad uiſum reflecti non poſſunt, eò quòd in pro-<lb/>ximitate debitam diſtantiam excedunt:</s> <s xml:id="echoid-s51545" xml:space="preserve"> ſunt enim tali uiſui proportionata reflexioni diſtantia ui-<lb/>ciniores.</s> <s xml:id="echoid-s51546" xml:space="preserve"> Quod ergo uiſui de proximo uapore irradiatum apparere poteſt, punicèum apparet pro-<lb/>pter uicinitatẽ & alias cauſsas in 67 huius prius dictas.</s> <s xml:id="echoid-s51547" xml:space="preserve"> Viſui uerò profundato ulterius in uapore, <lb/>ſecundum modũ diſtantiæ ſulgor luminis umbrarum nigredine permiſcetur, & uariantur colores <lb/> <pb o="466" file="0768" n="768" rhead="VITELLONIS OPTICAE"/> ſecundum prius dicta.</s> <s xml:id="echoid-s51548" xml:space="preserve"> Sic ergo in uapore irradiato fit quædam gibboſitas, quo ad uiſum.</s> <s xml:id="echoid-s51549" xml:space="preserve"> Et ob hoc <lb/>fortè dictum eſt à quibuſdam, nubem fore cõcauam, in qua iris generatur:</s> <s xml:id="echoid-s51550" xml:space="preserve"> quamuis ea, quæ uiden-<lb/>tur, nubis concauitati non oporteat adſcribi:</s> <s xml:id="echoid-s51551" xml:space="preserve"> quia uapor (quo ad conſiſtentiam ſui totius) eſt in-<lb/>teger, plenus corpuſculis diſtinctis, ſicut uidẽtur atomi totum ſolis radium implere:</s> <s xml:id="echoid-s51552" xml:space="preserve"> & eſt talis ua-<lb/>por à parte poſteriori à ſole groſsior quàm à parte anteriori ſolem aſpiciente.</s> <s xml:id="echoid-s51553" xml:space="preserve"> Quòd ſi cẽtrum ſolis <lb/>in periheria horizontis poſitum fuerit, ſic ut baſis pyramidis illuminationis ſit orthogonaliter ho-<lb/>rizonti inſiſtens:</s> <s xml:id="echoid-s51554" xml:space="preserve"> adhuc radij exteriores ad uiſum reflexi ſunt longiores, reſpectu eorum, qui ab in-<lb/>terioribus peripherijs refle ctuntur per 19 p 1:</s> <s xml:id="echoid-s51555" xml:space="preserve"> in eodem enim triangulo ad uiſum terminato maiori <lb/>angulo opponuntur.</s> <s xml:id="echoid-s51556" xml:space="preserve"> Sic ergo patet, quòd corpore ſolis ubicunq;</s> <s xml:id="echoid-s51557" xml:space="preserve"> poſito exterioris iridis colores, <lb/>reſpectu colorum iridis interioris, poſsibile eſt contrapoſitos apparere.</s> <s xml:id="echoid-s51558" xml:space="preserve"> Omnes autem colores ſe-<lb/>cundæ iridis ſunt debiliores neceſſariò coloribus primæ iridis:</s> <s xml:id="echoid-s51559" xml:space="preserve"> quoniá fiunt à radijs magis diſtan-<lb/>tibus à perpendiculari, & ſecũdum maiores angulos ad uiſum reflexis:</s> <s xml:id="echoid-s51560" xml:space="preserve"> propter quod iſti radij cum <lb/>radijs incidentibus minus aggregatur:</s> <s xml:id="echoid-s51561" xml:space="preserve"> unde minus eſſciunt luminis & coloris.</s> <s xml:id="echoid-s51562" xml:space="preserve"> Nos autẽ eo, quod <lb/>nunc præmiſimus, utimur pro principio ad propoſitum declarandum diſponente (& ſi ipſum non <lb/>ſit certa cauſſa.</s> <s xml:id="echoid-s51563" xml:space="preserve">) Manifeſtum eſt enim quòd illi radij (cum ſint extra peripheriam proportionatam <lb/>reflexioni ad illum uiſum, ſcilicet ultra puniceam interioris iridis) non reflectentur ad uiſum cum <lb/>lumine, niſi propter reflexos radios ab interiori prima iride ad reflexionem diſponantur, & niſi lu-<lb/>men eorum in actum uiſibilitatis per aggregationem luminis illorum radiorum cũ ipſis ad uiſum <lb/>reflexorum perducatur.</s> <s xml:id="echoid-s51564" xml:space="preserve"> Et huius ſignũ eſt albedo, quæ circulariter apparet in nube inter periphe-<lb/>riam ſuperiorem iridis inferioris puniceam, & inferiorem iridis ſuperioris puniceam:</s> <s xml:id="echoid-s51565" xml:space="preserve"> quia hæc al-<lb/>bedo fit per lumen nubem irradians ad uiſum nõ reflexum.</s> <s xml:id="echoid-s51566" xml:space="preserve"> Cum enim radiorum ab eadem ſuper-<lb/>ficie reflexibilium, qui ad uiſum in aliquo uno loco diſpoſitum reflecti poſſunt, ſint hi, qui ab ulti-<lb/>ma peripheria inferioris iridis reflectũtur:</s> <s xml:id="echoid-s51567" xml:space="preserve"> nullus ſuperiorum radiorum reflectetur ad illum uiſum, <lb/>ſed nubes alba ex commixtione luminis non reflexi per modum uiſionis ſimplicis illi uiſioni oc-<lb/>curret.</s> <s xml:id="echoid-s51568" xml:space="preserve"> Ex peripheria uerò punicea inferioris iridis & ſi plurimi radij, pręter eos, qui ad illum uiſum <lb/>refle ctuntur, ad partes uicinas uaporis roridi ſe diffundant:</s> <s xml:id="echoid-s51569" xml:space="preserve"> lumen tamen ad illum uiſum ex eorum <lb/>incidentia, à uicino uapore reflecti non poteſt:</s> <s xml:id="echoid-s51570" xml:space="preserve"> quoniam cadunt illi radij in ſuperficiem uaporis, à <lb/>qua, ſicut à ſuperficie improportionata adhuc uiſui, non eſt conueniens diſtantia reflexioni.</s> <s xml:id="echoid-s51571" xml:space="preserve"> Hoc <lb/>enim in principio peripheriæ puniceæ incipit, ubi ſecundum angulos in illa pyramide acutiſsimos <lb/>radij incidunt ipſi nubi:</s> <s xml:id="echoid-s51572" xml:space="preserve"> alij uerò ra dij poſteriores his radijs in punicea peripheria inferioris iridis <lb/>ad maiores angulos incidunt, quo ad uiſum (cũ ſint in profundiore ſuperficie à uiſu) & ad illam ſu-<lb/>perficiem uaporis, in qua eſt inferior ſuperioris iridis peripheria punicea, reflectũtur:</s> <s xml:id="echoid-s51573" xml:space="preserve"> & ibi aggre-<lb/>gati cum radijs illi parti uaporis incidentibus à ſole, illam partem ſuperficiei ex aggre gatione ma-<lb/>ioris luminis uiſibilem faciunt, radijs ad uiſum reflexis, qui prius propter luminis debilitatem ſen-<lb/>ſibiliter non poterãt reflecti.</s> <s xml:id="echoid-s51574" xml:space="preserve"> Et quoniam radij ab inferiori parte ſurſum ad alias partes uaporis ro-<lb/>ridi reflexi (ſiue uapor, ad quem fit reflexio in eadem ſuperficie cum prima iride, ſiue in alia ſuper-<lb/>ficie ſit conſiſtens) cum radijs ab eadem peripheria ad uiſum reflexis in generatione primæ iridis, <lb/>ut declaratum eſt in 66 huius, angulos conſtituunt:</s> <s xml:id="echoid-s51575" xml:space="preserve"> fiunt trianguli, quorum anguli ſunt in centro <lb/>uiſus, baſes uerò ſunt lineæ interiacentes puniceam peripheriam inferioris iridis, & puniceam ſu-<lb/>perioris:</s> <s xml:id="echoid-s51576" xml:space="preserve"> & quia ab illis baſibus nulla fit uiſui ſenſibilis reflexio;</s> <s xml:id="echoid-s51577" xml:space="preserve"> tota ipſarum ſuperficies uidetur <lb/>alba, non reflexo ab ipſa aliquo lumine ad uiſum.</s> <s xml:id="echoid-s51578" xml:space="preserve"> Simili quoq;</s> <s xml:id="echoid-s51579" xml:space="preserve"> modo fit reflexio ab alijs coloribus <lb/>inferioris iridis ad iridem ſupremam.</s> <s xml:id="echoid-s51580" xml:space="preserve"> Et quoniam anguli incidentiæ radiorum illas partes iridis <lb/>cauſſantium, ſunt maiores, ut ſuprà patuit per 106 th.</s> <s xml:id="echoid-s51581" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s51582" xml:space="preserve"> ideo per 20 th.</s> <s xml:id="echoid-s51583" xml:space="preserve"> 5 huius & anguli refle-<lb/>xionum ſunt maiores.</s> <s xml:id="echoid-s51584" xml:space="preserve"> Altius ergo in uaporem ſuperiorẽ illi radij pertingunt, proceantes ſibi ſimi-<lb/>les colores:</s> <s xml:id="echoid-s51585" xml:space="preserve"> quoniam illi radij propter admixtionem umbrarum aliorum corpuſculorum colorem <lb/>participant, qui ad corpus oppoſitum mixtum cum lumine tranſinittitur per 2 th.</s> <s xml:id="echoid-s51586" xml:space="preserve"> 5 huius.</s> <s xml:id="echoid-s51587" xml:space="preserve"> Et ſicut <lb/>oſtenſum eſt per 55th.</s> <s xml:id="echoid-s51588" xml:space="preserve"> 5 huius, quòd propter refle xionem dextra apparét ſiniſtra, & ſiniſtra dextra:</s> <s xml:id="echoid-s51589" xml:space="preserve"> <lb/>ſic etiam accidit in iſta reflexione colores iſtarum iridum contrapoſitos uideri.</s> <s xml:id="echoid-s51590" xml:space="preserve"> Colores quoq;</s> <s xml:id="echoid-s51591" xml:space="preserve"> ſe-<lb/>cundæ iridis debiliores uidentur quàm primæ iridis, ſcilicet inferioris:</s> <s xml:id="echoid-s51592" xml:space="preserve"> quoniã radij remoti ab axe <lb/>pyramidis irradiationis nubi incidentes ſunt debiles, & uiſui propter diſtantiam magnam inſenſi-<lb/>biles, ut patet per 158 th.</s> <s xml:id="echoid-s51593" xml:space="preserve"> 4 huius:</s> <s xml:id="echoid-s51594" xml:space="preserve"> & etiam radij reflexi à primæ iridis refractis radijs ſunt debiles, ut <lb/>patet per 3 th.</s> <s xml:id="echoid-s51595" xml:space="preserve"> 5 huius, & per 10 th.</s> <s xml:id="echoid-s51596" xml:space="preserve"> huius.</s> <s xml:id="echoid-s51597" xml:space="preserve"> Sequitur ergo neceſſariò eorum reflexionem ad uiſum fie-<lb/>ri debilem:</s> <s xml:id="echoid-s51598" xml:space="preserve"> & ſic omnes ſecundæ iridis colores ſunt debiles, magisq́;</s> <s xml:id="echoid-s51599" xml:space="preserve"> nigredine umbrarum permi-<lb/>ſcentur.</s> <s xml:id="echoid-s51600" xml:space="preserve"> Neceſſariò ergo primæ iridis coloribus ſecundæ iridis colores debiliores apparent:</s> <s xml:id="echoid-s51601" xml:space="preserve"> nec fit <lb/>aliqua ulterior reflexio ab illis ad partes ſuperiores roridi uaporis, propter illorum radiorum de-<lb/>bilitatem.</s> <s xml:id="echoid-s51602" xml:space="preserve"> Et fortè ob hoc dixit Ariſtoteles quòd plures duabus iridibus non poſſunt uideri:</s> <s xml:id="echoid-s51603" xml:space="preserve"> quo-<lb/>niam tantùm duæ ſunt, quæ ſitu colorum formaliter diſtinguuntur:</s> <s xml:id="echoid-s51604" xml:space="preserve"> quamuis plures quandoq;</s> <s xml:id="echoid-s51605" xml:space="preserve"> ui-<lb/>deantur, ut in præmiſſa declaratur.</s> <s xml:id="echoid-s51606" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s51607" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1910" type="section" level="0" n="0"> <head xml:id="echoid-head1401" xml:space="preserve" style="it">73. Omnem arcum ſenſibilem iridis per circulum ſuæ altitudinis in duo &qualia diuidi eſt <lb/>neceſſe. Vnde manifeſtum eſt quemlibet uidentem propriam iridem uidere.</head> <p> <s xml:id="echoid-s51608" xml:space="preserve">Cum enim, ut ex præcedẽtibus patet, ſuperficies horizontis interſecet ſuperficiem circuli iridis:</s> <s xml:id="echoid-s51609" xml:space="preserve"> <lb/>tunc eorum cõmunis ſectio ex 3 p 11 eſt linea recta.</s> <s xml:id="echoid-s51610" xml:space="preserve"> Sed quia circulus altitudinis iridis ſemper tran-<lb/>ſit per zenith capitis:</s> <s xml:id="echoid-s51611" xml:space="preserve"> quoniam (ut patet per 64 th.</s> <s xml:id="echoid-s51612" xml:space="preserve"> huius, & declaratum eſt in præhabitis) cen-<lb/>trum uiſus eſt polus iridis:</s> <s xml:id="echoid-s51613" xml:space="preserve"> illius uero circuli altitudinis centrum eſt centrum mundi & horizontis:</s> <s xml:id="echoid-s51614" xml:space="preserve"> <lb/> <pb o="467" file="0769" n="769" rhead="LIBER DECIMVS."/> ergo ipſe tranſit per polos horizontis:</s> <s xml:id="echoid-s51615" xml:space="preserve"> zenith enim capitis eſt polus ipſius horizontis:</s> <s xml:id="echoid-s51616" xml:space="preserve"> linea uerò à <lb/>polo ad cẽtrum horizontis deducta, eſt erecta ſuper ſuperficiem horizontis ex principio primi hu-<lb/>ius.</s> <s xml:id="echoid-s51617" xml:space="preserve"> Ergo per 18 p 11 circulus ille altitudinis iridis eſt erectus ſuper ſuperficiem horizontis:</s> <s xml:id="echoid-s51618" xml:space="preserve"> & ipſe <lb/>tranſit eius centrum:</s> <s xml:id="echoid-s51619" xml:space="preserve"> quoniã cum ipſi ambo ſint circuli magni ſphæræ mundi, patet quoniam ipſo-<lb/>rum eſt idem centrum, quod eſt cẽtrum mundi.</s> <s xml:id="echoid-s51620" xml:space="preserve"> Ille ergo circulus altitudinis ſecat horizontem per <lb/>æqualia & orthogonaliter.</s> <s xml:id="echoid-s51621" xml:space="preserve"> Similiter autẽ & idem circulus altitudinis cum per centrum uiſus tran-<lb/>ſeat, & per centrum circuli iridis, & per centrum ſolis, (hęc enim ſunt in eadem linea per 64 th.</s> <s xml:id="echoid-s51622" xml:space="preserve"> hu-<lb/>ius) tranſit ergo per polos circuli iridis:</s> <s xml:id="echoid-s51623" xml:space="preserve"> & ſecundum præmiſſa ſecat eum per æqualia & orthogo-<lb/>naliter.</s> <s xml:id="echoid-s51624" xml:space="preserve"> Sed ſi horizonta & circulum iridis circulus altitudinis iridis per æqualia ſecat & orthogo-<lb/>naliter:</s> <s xml:id="echoid-s51625" xml:space="preserve"> ergo illorum ſectionem per æqualia ſecabit & orthogonaliter per 19 p 11.</s> <s xml:id="echoid-s51626" xml:space="preserve"> Sit ergo illa com-<lb/>munis ſectio linea a b, quam productus circulus altitudinis diuidat per æqualia in puncto c:</s> <s xml:id="echoid-s51627" xml:space="preserve"> duca-<lb/>turq́;</s> <s xml:id="echoid-s51628" xml:space="preserve"> ſurſum in ſuperficie circuli altitudinis à puncto clinea c d:</s> <s xml:id="echoid-s51629" xml:space="preserve"> quæ ſit communis ſectio ſuperfi-<lb/> <anchor type="figure" xlink:label="fig-0769-01a" xlink:href="fig-0769-01"/> cierum illius circuli & iridis:</s> <s xml:id="echoid-s51630" xml:space="preserve"> & hęc linea c d erit perpendicu-<lb/>laris ſuper lineam a b per 19 p 11:</s> <s xml:id="echoid-s51631" xml:space="preserve"> eò quòd circulus altitudinis <lb/>erectus eſt ſuper ſuperficiem cuiuſq;</s> <s xml:id="echoid-s51632" xml:space="preserve"> duorum illorum circu-<lb/>lorum, quorum eſt communis ſectio linea a b:</s> <s xml:id="echoid-s51633" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s51634" xml:space="preserve"> commu-<lb/>nis ſectio peripheriarũ circuli altitudinis & iridis punctus d:</s> <s xml:id="echoid-s51635" xml:space="preserve"> <lb/>angulus ergo d c a eſt rectus, & ſimiliter angulus d c b:</s> <s xml:id="echoid-s51636" xml:space="preserve"> ſubten <lb/>dantur ergo illis angulis lineæ a d & b d:</s> <s xml:id="echoid-s51637" xml:space="preserve"> & patet ex 4 p 1 & ex <lb/>pręmiſsis quòd ipſæ ſunt ęquales:</s> <s xml:id="echoid-s51638" xml:space="preserve"> ergo per 28 p 3 arcus iridis, <lb/>qui eſt a d, eſt æ qualis ipſius arcui b d.</s> <s xml:id="echoid-s51639" xml:space="preserve"> Pars ergo peripheriæ <lb/>iridis, quæ eſt ſupra horizontem (quoniam illa ſola eſt ſenſi-<lb/>bilis) per circulũ altitudinis per æqualia eſt diuiſa.</s> <s xml:id="echoid-s51640" xml:space="preserve"> Quod eſt <lb/>propoſitum.</s> <s xml:id="echoid-s51641" xml:space="preserve"> Vnde manifeſtũ eſt corollarium perpulchrum:</s> <s xml:id="echoid-s51642" xml:space="preserve"> <lb/>ſcilicet quemlibet uidentem iridem propriam uidere, ex eo, <lb/>quòd moto aliquo uidente ſecundum locum ſemper zenith <lb/>capitis uariatur:</s> <s xml:id="echoid-s51643" xml:space="preserve"> patet enim quòd diuerſorũ diuerſa ſunt ze-<lb/>nith, & diuerſi horizõtes:</s> <s xml:id="echoid-s51644" xml:space="preserve"> nec eſt poſsibile aliquos duos eun-<lb/>dem habere horizonta:</s> <s xml:id="echoid-s51645" xml:space="preserve"> quoniam ſemper oculus uidentis eſt <lb/>centrum horizontis.</s> <s xml:id="echoid-s51646" xml:space="preserve"> Si ergo aliquorum diuerſitas ſit ſecundum diſtãtiam latitudinis uniuerſi tan-<lb/>tùm:</s> <s xml:id="echoid-s51647" xml:space="preserve"> tunc ad eorundem oculos diuerſimodè radij reflexi à corpore nubis ſecundũ diuerſa puncta <lb/>aggregationis concurrent:</s> <s xml:id="echoid-s51648" xml:space="preserve"> & remotior ipſorũ à uapore rorido maiorẽ iridem uidebit, propinquior <lb/>minorem, ſi in eadem ſuperficie appareant irides:</s> <s xml:id="echoid-s51649" xml:space="preserve"> quæ ſi appareant in ſuperficiebus diuerſis æqui-<lb/>diſtantibus:</s> <s xml:id="echoid-s51650" xml:space="preserve"> tunc ſecundũ æquales circulos iris uideri poterit:</s> <s xml:id="echoid-s51651" xml:space="preserve"> & ſequetur iris fugientem, & fugiet <lb/>ſequentem, ut diximus in 65 huius:</s> <s xml:id="echoid-s51652" xml:space="preserve"> eſt tamen eis idem circulus altitudinis, ſed nõ eodem modo ſe <lb/>habens.</s> <s xml:id="echoid-s51653" xml:space="preserve"> Quòd ſit diuerſitas aliquorum ſit ſecundũ longitudinem uniuerſi tantùm:</s> <s xml:id="echoid-s51654" xml:space="preserve"> tũc erunt diuerſi <lb/>circuli altitudinis, & quilibet illorum circulorũ diuidit per præmiſſa arcum iridis, qui eſt ſupra ho-<lb/>rizonta, in duo æqualia:</s> <s xml:id="echoid-s51655" xml:space="preserve">ergo ipſa diuiſa, ſicut & ipſa diuidẽtia, ſunt diuerſa:</s> <s xml:id="echoid-s51656" xml:space="preserve"> quilibet ergo propriam <lb/>iridem uidebit.</s> <s xml:id="echoid-s51657" xml:space="preserve"> Quòd ſi latitudo & longitudo uidentium differant:</s> <s xml:id="echoid-s51658" xml:space="preserve"> tunc per præmiſſa patet, quòd <lb/>nullo modo eandem iridem uidebunt.</s> <s xml:id="echoid-s51659" xml:space="preserve"> Patet ergo quod intendebamus.</s> <s xml:id="echoid-s51660" xml:space="preserve"> Et ſignum huius eſt:</s> <s xml:id="echoid-s51661" xml:space="preserve"> quòd <lb/>ſi aliquis ſtans in radio ſolis auerſa ſoli facie aquã ore ſpargat:</s> <s xml:id="echoid-s51662" xml:space="preserve"> uidebit cũ ambobus oculis ante fron <lb/>tem ſuam colores iridis, & arcũ æqualiter ab utroq;</s> <s xml:id="echoid-s51663" xml:space="preserve"> oculo diſtãtem.</s> <s xml:id="echoid-s51664" xml:space="preserve"> Quòd ſi aquam ſecundò ſpar-<lb/>ſerit, & oculum dextrum clauſerit uel manu cooperiat:</s> <s xml:id="echoid-s51665" xml:space="preserve"> uidebit arcum æqualiter diſtantem à cẽtro <lb/>ſiniſtri oculi, arcumq́ue iridis dextrum oculũ ſecantem:</s> <s xml:id="echoid-s51666" xml:space="preserve"> & econuerſo erit, ſi oculũ ſiniſtrum clau-<lb/>ſerit:</s> <s xml:id="echoid-s51667" xml:space="preserve"> tunc enim iterum uidebit arcum æquidiſtantem à centro dextri oculi, ſiniſtrumq́;</s> <s xml:id="echoid-s51668" xml:space="preserve"> oculum ſe-<lb/>cantem.</s> <s xml:id="echoid-s51669" xml:space="preserve"> Ex quo manifeſtè patere poteſt, quòd color iridis eſt paſsio uiſus:</s> <s xml:id="echoid-s51670" xml:space="preserve"> & quòd mutatur iris ſe-<lb/>cundum uidentium mutationem:</s> <s xml:id="echoid-s51671" xml:space="preserve"> & quòd materia ſua eſt uapor roridus:</s> <s xml:id="echoid-s51672" xml:space="preserve"> & quòd diſtinctio colo-<lb/>rum non eſt ex qualitate materiæ, ſed ex reflexione luminis ad uiſum, cui color eſſentialiter adue-<lb/>nit ex commixtione nigredinis umbrarum.</s> <s xml:id="echoid-s51673" xml:space="preserve"/> </p> <div xml:id="echoid-div1910" type="float" level="0" n="0"> <figure xlink:label="fig-0769-01" xlink:href="fig-0769-01a"> <variables xml:id="echoid-variables868" xml:space="preserve">d a c b</variables> </figure> </div> </div> <div xml:id="echoid-div1912" type="section" level="0" n="0"> <head xml:id="echoid-head1402" xml:space="preserve" style="it">74. In aliquo puncto horizontis exiſtente centro corporis luminoſi, neceſſe eſt tantùm ſemi-<lb/>circulum ab eo cauſſatæ iridis uideri.</head> <p> <s xml:id="echoid-s51674" xml:space="preserve">Quoniam enim non eſt poſsibile ſolis uellunæ (quorum ſolummodò corporum, ut 70 th.</s> <s xml:id="echoid-s51675" xml:space="preserve"> huius <lb/>diximus, radij iridem faciunt) centra in horizonte exiſtere, niſi in oriente uel occidente, in noſtra <lb/>terra, ſcilicet Poloniæ, habitabili, quæ eſt circa latitudinem 50 graduum:</s> <s xml:id="echoid-s51676" xml:space="preserve"> (quamuis in regionibus <lb/>maximæ latitudinis, ſole exiſtente in capite capricorni, ut in his, quæ ſunt 66 graduũ & 9 minuto-<lb/>rum ſol in meridiano exiſtens circulo, uideatur in peripheria horizontis:</s> <s xml:id="echoid-s51677" xml:space="preserve"> & in alijs regionibus di-<lb/>uerſificata latitudine regionis & declinatione ſolis in diuerſis circulis altitudinis quandoq;</s> <s xml:id="echoid-s51678" xml:space="preserve"> ſol ui-<lb/>deatur in horizonte.</s> <s xml:id="echoid-s51679" xml:space="preserve">) Ponamus itaq;</s> <s xml:id="echoid-s51680" xml:space="preserve"> ſolem in oriente, cuius cẽtrum ſit a:</s> <s xml:id="echoid-s51681" xml:space="preserve"> fiatq́ iris in parte ſibi op-<lb/>poſita, uiſu intermedio exiſtente:</s> <s xml:id="echoid-s51682" xml:space="preserve"> & erit illa iris ad occidentem per 67 huius:</s> <s xml:id="echoid-s51683" xml:space="preserve"> & ſit centrum iridis <lb/>punctum b:</s> <s xml:id="echoid-s51684" xml:space="preserve"> ducaturq́;</s> <s xml:id="echoid-s51685" xml:space="preserve"> diameter circuli iridis trans ſuperficiem horizontis per centrũ b, quod cen-<lb/>trum tunc neceſſariò erit in ſuperficie horizontis:</s> <s xml:id="echoid-s51686" xml:space="preserve"> quoniã per 64 th.</s> <s xml:id="echoid-s51687" xml:space="preserve"> huius oſten ſum eſt, quòd cen-<lb/>trum ſolis, & centrum uiſus, & centrum iridis neceſſe eſt in eadem linea eſſe.</s> <s xml:id="echoid-s51688" xml:space="preserve"> Eiuſdẽ uerò lineę par-<lb/>tem in ſubiecta ſuperficie, partẽ in ſublimi eſſe eſt impoſsibile per 1 p 11:</s> <s xml:id="echoid-s51689" xml:space="preserve"> in ſuperficie uerò horizon-<lb/>tis eſt ex hypotheſi centrum ſolis, & centrum uiſus eſt centrum horizontis:</s> <s xml:id="echoid-s51690" xml:space="preserve"> ergo & linea copulans <lb/> <pb o="468" file="0770" n="770" rhead="VITELLONIS OPTICAE"/> illa cẽtra erit in ſupficie horizõtis:</s> <s xml:id="echoid-s51691" xml:space="preserve"> & ſit diameter illa iridis, quæ e d:</s> <s xml:id="echoid-s51692" xml:space="preserve"> & coniungãtur lineę a b, a c, a d:</s> <s xml:id="echoid-s51693" xml:space="preserve"> <lb/> <anchor type="figure" xlink:label="fig-0770-01a" xlink:href="fig-0770-01"/> fientq́, duo trianguli a c b & a d b.</s> <s xml:id="echoid-s51694" xml:space="preserve"> Et quoniam in his <lb/>triãgulis latus a c eſt æ quale lateri a d ք 89 t 1 huius:</s> <s xml:id="echoid-s51695" xml:space="preserve"> <lb/>quoniá ſunt lineæ lõgitudinis unius & eiuſdẽ pyra-<lb/>midis:</s> <s xml:id="echoid-s51696" xml:space="preserve"> & latus c b æquale eſt lateri d b, ꝗ a ſunt ſemi-<lb/>diametri circuli iridis:</s> <s xml:id="echoid-s51697" xml:space="preserve"> latus uerò a b cõmune eſt am <lb/>bobus illis triãgulis:</s> <s xml:id="echoid-s51698" xml:space="preserve"> patet ergo ք 8 p 1 quia angulus <lb/>c b a eſt æ qualis angulo d b a:</s> <s xml:id="echoid-s51699" xml:space="preserve"> uterq;</s> <s xml:id="echoid-s51700" xml:space="preserve"> itaq;</s> <s xml:id="echoid-s51701" xml:space="preserve"> eſt rectus.</s> <s xml:id="echoid-s51702" xml:space="preserve"> <lb/>Ergo per 18 p 11 erit ſuperficies horizontis erecta ſu-<lb/>per ſuperficiẽ circuli iridis:</s> <s xml:id="echoid-s51703" xml:space="preserve"> tranſit autẽ per centrum <lb/>iridis.</s> <s xml:id="echoid-s51704" xml:space="preserve"> Palàm ergo quoniã circulus horizontis diui-<lb/>dit circulũ iridis per æqualia:</s> <s xml:id="echoid-s51705" xml:space="preserve"> cõmunis enim ſectio <lb/>illorũ circulorũ non poteſt eſſe, niſi diameter circuli <lb/>iridis, quæ ſemper ſuũ circulũ diuidit ք æqualia per <lb/>diametri definitionẽ:</s> <s xml:id="echoid-s51706" xml:space="preserve"> quod autẽ de circulo iridis eſt <lb/>ſupra horizonta, hoc uidetur.</s> <s xml:id="echoid-s51707" xml:space="preserve"> Sic ergo poſito cẽtro <lb/>ſolis uel lunæ in pũcto horizõtis, ſemicirculus iridis uidetur:</s> <s xml:id="echoid-s51708" xml:space="preserve"> niſi fortè tantò minus, quantũ eſt dif-<lb/>ferẽtiæ, ꝓpter hoc, quòd centrũ uiſus nõ eſt uerum centrũ uniuerſi.</s> <s xml:id="echoid-s51709" xml:space="preserve"> In hoc aũt nõ eſt ſenſibilis dif-<lb/>ferentia:</s> <s xml:id="echoid-s51710" xml:space="preserve"> & ſi ſit, nõ eſt in generatione iridis, ſed in uiſione ipſius.</s> <s xml:id="echoid-s51711" xml:space="preserve"> Et hoc eſt, quod hic ꝓponitur de-<lb/>monſtrand ũ.</s> <s xml:id="echoid-s51712" xml:space="preserve"> Po teſt & idẽ aliter demõſtrari.</s> <s xml:id="echoid-s51713" xml:space="preserve"> Sit ergo ſecundũ diſpoſitionẽ priorẽ centrũ ſolis in ali-<lb/>quo pũcto horizõtis, quod ſit punctũ h:</s> <s xml:id="echoid-s51714" xml:space="preserve"> & ſit k centrũ uiſus, quod eſt centrũ horizontis:</s> <s xml:id="echoid-s51715" xml:space="preserve"> & ſit hori-<lb/>zontis diameter linea h g.</s> <s xml:id="echoid-s51716" xml:space="preserve"> Erigatur ergo ſemicirculus unus altitudinis ſuք horizontẽ orthogonali-<lb/>ter ex cẽtro k, qui ſit h m g:</s> <s xml:id="echoid-s51717" xml:space="preserve"> hũc ergo ſemicirculũ altitudinis arcus iridis generatæ in oppoſito ſolis <lb/>(interpoſito cẽtro uiſus) ſecet in puncto m:</s> <s xml:id="echoid-s51718" xml:space="preserve"> & producatur linea k m.</s> <s xml:id="echoid-s51719" xml:space="preserve"> Et quoniã lineæ h k, k m & k g <lb/>omnes ſunt ex cẽtro circuli altitudinis, omnes ergo ſunt æ quales & omnes notæ:</s> <s xml:id="echoid-s51720" xml:space="preserve"> quoniam mundi <lb/>ſemidiameter eſt nota, ut ſi ipſa ſupponatur eſſe 60 partiũ.</s> <s xml:id="echoid-s51721" xml:space="preserve"> Producatur itaq;</s> <s xml:id="echoid-s51722" xml:space="preserve"> linea h m:</s> <s xml:id="echoid-s51723" xml:space="preserve"> & ſi notus eſt <lb/>angulus h k m:</s> <s xml:id="echoid-s51724" xml:space="preserve"> tũc linea h m erit nota.</s> <s xml:id="echoid-s51725" xml:space="preserve"> Sc<gap/><gap/>i aũt poteſt angulus h k m ք hoc, ut ſciatur arcus m g, qui <lb/>eſt arcus altitudinis, qui ſciri poteſt per inſtrumentũ, ut per armillam uel per aſtrolabium uel qua-<lb/>drantem:</s> <s xml:id="echoid-s51726" xml:space="preserve"> quo ſcito, ſcietur angulus m k g:</s> <s xml:id="echoid-s51727" xml:space="preserve"> qui ſi auferatur de duobus rectis, ſeietur angulus h k m:</s> <s xml:id="echoid-s51728" xml:space="preserve"> <lb/>& ſic ſcietur linea h m, reſpectu ſemidiametri k m, operatione illa, qua utimur in ſciẽtia aſtrorũ.</s> <s xml:id="echoid-s51729" xml:space="preserve"> Li-<lb/>nea uerò h m cũ ſit linea lõgitudinis pyramidis illuminationis, & per 89 th.</s> <s xml:id="echoid-s51730" xml:space="preserve"> 1 huius omnes lineæ ló-<lb/>gitudinis unius pyramidis ſint æquales:</s> <s xml:id="echoid-s51731" xml:space="preserve"> erũt tunc omnes lineæ lógitudinis illius pyramidis notæ.</s> <s xml:id="echoid-s51732" xml:space="preserve"> <lb/>Circumducatur itaq;</s> <s xml:id="echoid-s51733" xml:space="preserve"> circulus iridis ſuper ſuperficiem horizontis, eam interſecãs, quæ (ut patet ex <lb/>præmiſsis) tranſibit punctũ m circuli altitudinis:</s> <s xml:id="echoid-s51734" xml:space="preserve"> ſit ergo, ut ipſe circulus iridis ſecet horizontẽ in <lb/>puncto n.</s> <s xml:id="echoid-s51735" xml:space="preserve"> Duos itaq;</s> <s xml:id="echoid-s51736" xml:space="preserve"> circulos contingẽt lineæ k m & h m in puncto m, ſecundũ eorũ communẽ ſci-<lb/>licet ſectionẽ.</s> <s xml:id="echoid-s51737" xml:space="preserve"> Quoniã uerò punctũ m in circulo altitudinis datũ eſt, & lineæ h m & k m ſunt notæ:</s> <s xml:id="echoid-s51738" xml:space="preserve"> <lb/>erit proportio lineæ h m ad lineã k <lb/> <anchor type="figure" xlink:label="fig-0770-02a" xlink:href="fig-0770-02"/> m nota.</s> <s xml:id="echoid-s51739" xml:space="preserve"> Et quoniã quæ eſt ꝓportio <lb/>alicuius lineæ primę ad aliquam ſe-<lb/>cundam, eadẽ eſt cuiuslibet tertiæ <lb/>ad aliquã quartá:</s> <s xml:id="echoid-s51740" xml:space="preserve"> tũc per 3 th.</s> <s xml:id="echoid-s51741" xml:space="preserve"> 1 huius <lb/>eſto, ut ſit proportio lineæ rectæ a b <lb/>ad rectá b c, ſicut lineæ h m ad lineá <lb/>k m.</s> <s xml:id="echoid-s51742" xml:space="preserve"> Et quoniá linea h m eſt maior <lb/>quàm linea k m per 19 p 1, eò quòd <lb/>maiori angulo opponitur in trian-<lb/>gulo h m k:</s> <s xml:id="echoid-s51743" xml:space="preserve"> patet ergo quòd linea a <lb/>b eſt maior quàm linea b c.</s> <s xml:id="echoid-s51744" xml:space="preserve"> Produca <lb/>tur ergo linea b c ad punctũ d in tan <lb/>tùm, ut ſit proportio lineæ b d ad li-<lb/>neam a b, ſicut lineę a b ad lineã b c.</s> <s xml:id="echoid-s51745" xml:space="preserve"> <lb/>Et quia quæ eſt proportio lineę h m <lb/>adlineã k m, eadé eſt lineæ a b ad b <lb/>c:</s> <s xml:id="echoid-s51746" xml:space="preserve"> erit ergo per 11 p 5 proportio lineæ h m ad lineá m k, ſicut lineæ b d ad lineã a b.</s> <s xml:id="echoid-s51747" xml:space="preserve"> Et quia proportio <lb/>lineæ h m ad lineã k m, uel ad lineã h k æqualẽ per 7 p 5 ex præmiſsis eſt nota:</s> <s xml:id="echoid-s51748" xml:space="preserve"> ꝓportio ergo lineæ a <lb/>b ad lineã b c erit nota:</s> <s xml:id="echoid-s51749" xml:space="preserve"> ergo ipſarũ utraq;</s> <s xml:id="echoid-s51750" xml:space="preserve"> eſt nota ſecundũ aliquam quantitatẽ ſuppoſitam in altera <lb/>ipſarum.</s> <s xml:id="echoid-s51751" xml:space="preserve"> Sed & proportio lineæ b d ad lineã a b eſt nota:</s> <s xml:id="echoid-s51752" xml:space="preserve"> ergo & linea a b eſt nota, & linea b d eſt no-<lb/>ta:</s> <s xml:id="echoid-s51753" xml:space="preserve"> ſed linea b c fuit nota:</s> <s xml:id="echoid-s51754" xml:space="preserve"> ergo relin quitur, ut linea c d ſit nota.</s> <s xml:id="echoid-s51755" xml:space="preserve"> Sed linea h k eſt nota:</s> <s xml:id="echoid-s51756" xml:space="preserve"> quia cũ ipſa ſit <lb/>ſemidiameter horizontis, erit ipſa partiũ 60:</s> <s xml:id="echoid-s51757" xml:space="preserve"> ergo proportio lineæ c d ad h k erit nota.</s> <s xml:id="echoid-s51758" xml:space="preserve"> Quæ eſt ergo <lb/>proportio lineæ c d ad lineã h k, eadẽ erit lineæ b c notæ ad aliquã aliam per 3 th.</s> <s xml:id="echoid-s51759" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s51760" xml:space="preserve"> Quia uerò <lb/>eſt proportio a b ad b c, ſicut b d ad a b, & ab eſt maior quàm b c, ut patet ex præmiſsis:</s> <s xml:id="echoid-s51761" xml:space="preserve"> erit ergo b d <lb/>maior quã a b:</s> <s xml:id="echoid-s51762" xml:space="preserve"> relin queturq́;</s> <s xml:id="echoid-s51763" xml:space="preserve"> c d maior ꝗ̃ b c (hoc aũt patet in numeris taliter diſpoſitis quibuſcũq;</s> <s xml:id="echoid-s51764" xml:space="preserve">.) <lb/>Linea ergo proportionalis lineæ b c, ſicut linea h k eſt lineę c d, illa erit minor ꝗ̃ linea h k uel ꝗ̃ linea <lb/>k g.</s> <s xml:id="echoid-s51765" xml:space="preserve"> Abſcindatur ergo à ſemidiametro k g per 3 p 1 æqualis illi lineę:</s> <s xml:id="echoid-s51766" xml:space="preserve"> & ſit linea k p:</s> <s xml:id="echoid-s51767" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s51768" xml:space="preserve"> linea k p ſe-<lb/> <pb o="469" file="0771" n="771" rhead="LIBER DECIMVS."/> cundum præmiſſa nota.</s> <s xml:id="echoid-s51769" xml:space="preserve"> Copuletur itaq;</s> <s xml:id="echoid-s51770" xml:space="preserve"> à puncto p ad punctũ m linea in ſuperficie circuli altitudi-<lb/>nis, quæ ſit p m:</s> <s xml:id="echoid-s51771" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s51772" xml:space="preserve"> neceſſariò, ut quæ eſt ꝓportio lineæ c d ad h k, uel lineæ b c ad k p, eadẽ ſit ꝓ-<lb/>portio lineæ a b ad lineã p m.</s> <s xml:id="echoid-s51773" xml:space="preserve"> Quòd ſi dicatur hoc nõ eſſe poſsibile:</s> <s xml:id="echoid-s51774" xml:space="preserve"> quę eſt ergo proportio lineæ c d <lb/>ad h k, uel b c ad k p:</s> <s xml:id="echoid-s51775" xml:space="preserve"> eadẽ erit lineæ a b ad aliquã aliã lineam maiorẽ uel minorẽ linea p m, per 3 th.</s> <s xml:id="echoid-s51776" xml:space="preserve"> 1 <lb/>huius.</s> <s xml:id="echoid-s51777" xml:space="preserve"> Sit ergo nũc illa proportio lineæ a b ad quandã minorem linea m p, quæ ſit p r.</s> <s xml:id="echoid-s51778" xml:space="preserve"> Quæ eſt ergo <lb/>proportio lineæ c d ad lineã h k, uel b c ad lineã k p, eadẽ eſt lineæ a b ad lineã p r:</s> <s xml:id="echoid-s51779" xml:space="preserve"> quæ autẽ eſt pro-<lb/>portio lineæ c d ad lineã h k, eadẽ eſt lineæ b c ad lineã k p:</s> <s xml:id="echoid-s51780" xml:space="preserve"> ergo per 16 p 5 quæ eſt proportio lineæ <lb/>c d ad b c, eadẽ eſt h k ad k p:</s> <s xml:id="echoid-s51781" xml:space="preserve"> & quæ eſt proportio lineæ b c ad k p, eadẽ eſt lineæ a b ad lineã p r:</s> <s xml:id="echoid-s51782" xml:space="preserve"> ergo <lb/>itẽ per 16 p 5 quæ eſt proportio lineæ b c ad a b, eadẽ eſt lineæ k p ad p r:</s> <s xml:id="echoid-s51783" xml:space="preserve"> & ſic lineæ c d, b c, a b pro-<lb/>portionales erunt lineis h k, k p, p r:</s> <s xml:id="echoid-s51784" xml:space="preserve"> ſed quæ eſt proportio lineæ a b ad b c, eadẽ eſt lineæ b d ad a b:</s> <s xml:id="echoid-s51785" xml:space="preserve"> <lb/>ergo & in ipſarũ comproportionalibus ſic erit, quòd ſicut ſe habet linea r p ad p k, ſic coniunctim ſe <lb/>habebit tota p h ad lineã p r.</s> <s xml:id="echoid-s51786" xml:space="preserve"> Ducãtur ergo lineæ h r & k r:</s> <s xml:id="echoid-s51787" xml:space="preserve">fientq́;</s> <s xml:id="echoid-s51788" xml:space="preserve"> duo triãguli, qui h r p & k r p, quo-<lb/>rum cõmunis eſt angulus r p h, & latera dictũ angulũ continẽtia reſpectu diuerſorũ trigonorũ ſunt <lb/>proportionalia:</s> <s xml:id="echoid-s51789" xml:space="preserve"> quæ enim eſt ꝓportio lineæ p r lateris maioris trianguli ad lineã p k latus minoris <lb/>trianguli:</s> <s xml:id="echoid-s51790" xml:space="preserve"> eadẽ ꝓportio lineę h p lateris maioris trigoni ad lineã p r latus trigoni p r k minoris:</s> <s xml:id="echoid-s51791" xml:space="preserve"> ergo <lb/>per 6 p 6 illi trianguli ſunt æ quianguli:</s> <s xml:id="echoid-s51792" xml:space="preserve"> ergo per 4 p 6 latera ipſorũ æ quos angulos reſpiciẽtia ſunt <lb/>proportionalia.</s> <s xml:id="echoid-s51793" xml:space="preserve"> Eſt ergo ꝓportio lineæ h p ad lineã p r, & lineæ p r ad lineã p k, ſicut lineæ h r ad li-<lb/>neã k r:</s> <s xml:id="echoid-s51794" xml:space="preserve">ſed quam proportionẽ habet linea h p ad lineã p r, hanc habet linea b d ad lineã a b:</s> <s xml:id="echoid-s51795" xml:space="preserve"> & quam <lb/>habet linea b d ad a b, hãc habet linea a b ad b c:</s> <s xml:id="echoid-s51796" xml:space="preserve"> & quam habet a b ad b c, hãc habet linea h m ad k m <lb/>ex hypotheſi:</s> <s xml:id="echoid-s51797" xml:space="preserve"> per 11 ergo p 5 patet quòd quam proportionẽ habet linea h r ad lineã k r, hãc habet li-<lb/>nea h m ad lineã k m:</s> <s xml:id="echoid-s51798" xml:space="preserve"> hoc aũt eſt impoſsibile, & cõtra 56 th.</s> <s xml:id="echoid-s51799" xml:space="preserve"> 1 huius:</s> <s xml:id="echoid-s51800" xml:space="preserve"> quoniã in ſemicirculo quocunq;</s> <s xml:id="echoid-s51801" xml:space="preserve"> <lb/>duab.</s> <s xml:id="echoid-s51802" xml:space="preserve"> lineis ductis ad quẽcũq;</s> <s xml:id="echoid-s51803" xml:space="preserve"> pũctũ քipherię, ſcilicet una à termino diametri & alia à cẽtro, ut ſunt <lb/>in ꝓpoſito lineę h m & k m, duas alias lineas ab eiſdẽ pũctis ad ali udpũctũ circũſerentiæ quodcũq;</s> <s xml:id="echoid-s51804" xml:space="preserve"> <lb/>duabus prioribus ꝓportionales ducere eſt impoſsibile.</s> <s xml:id="echoid-s51805" xml:space="preserve"> Eſt ergo impoſsibile lineã a b ad aliá mino-<lb/>rem lineá quam linea p m, eandẽ habere ꝓportionẽ quam linea b d ad lineã h p, uel quam linea c d <lb/>ad h k, uel quã linea b c ad k p.</s> <s xml:id="echoid-s51806" xml:space="preserve"> Sed neq;</s> <s xml:id="echoid-s51807" xml:space="preserve"> poteſt linea a b habere illã proportionẽ ad aliquá lineá ma-<lb/>iorẽ linea p m:</s> <s xml:id="echoid-s51808" xml:space="preserve"> quoniã eadẽ eſt ratio, & eodẽ modo deducitur ad impoſsibile.</s> <s xml:id="echoid-s51809" xml:space="preserve"> Ergo quę eſt ꝓportio <lb/>c d ad lineã h k, uel lineæ b c ad k p:</s> <s xml:id="echoid-s51810" xml:space="preserve"> eadę erit lineæ a b ad p m:</s> <s xml:id="echoid-s51811" xml:space="preserve"> & ſequetur repetita priori demõſtra-<lb/>tione, quæ ducebat ad impoſsibile, ſcilicet, ut quæ eſt ꝓportio lineæ h p ad p m, & lineæ m p ad p k, <lb/>eadẽ ſit lineæ h m ad k m.</s> <s xml:id="echoid-s51812" xml:space="preserve"> Ductis itaq;</s> <s xml:id="echoid-s51813" xml:space="preserve"> pluribus ſemicirculis altitudinis circa centrũ k ſub horizõte, <lb/>proportionales lineæ prędictis lineis h m & k m ducãtur ſecundũ modũ 56 th.</s> <s xml:id="echoid-s51814" xml:space="preserve"> 1 huius.</s> <s xml:id="echoid-s51815" xml:space="preserve"> Si ergo linea <lb/>m p ſit perpẽdiculariter inſiſtẽs diametro h g:</s> <s xml:id="echoid-s51816" xml:space="preserve"> tũc poſito cẽtro p ſecundũ ſemidiametrũ p m deſcri-<lb/>batur circulus:</s> <s xml:id="echoid-s51817" xml:space="preserve"> quòd ſi linea p m nõ ſit perpẽdicularis ſuper diametrũ h g:</s> <s xml:id="echoid-s51818" xml:space="preserve"> polo itaq;</s> <s xml:id="echoid-s51819" xml:space="preserve"> exiſtẽte pũcto <lb/>p per 65 th.</s> <s xml:id="echoid-s51820" xml:space="preserve"> 1 huius (quoniã ille punctus æqualiter diſtabit ab omnibus in illis ſemicirculis ſignatis <lb/>pũctis, ſimilibus pũcto m) ducatur circulus ſecundũ diſtantiã lineæ p m:</s> <s xml:id="echoid-s51821" xml:space="preserve"> qui attinget omnia dicta <lb/>pũcta ſemicirculorũ altitudinis, in quæ cadũt prædictæ proportionales lineæ, ſiue anguli reflexio-<lb/>num iridẽ cauſſantes.</s> <s xml:id="echoid-s51822" xml:space="preserve"> Si enim dicatur quòd nõ attingat:</s> <s xml:id="echoid-s51823" xml:space="preserve"> accidet ſecundũ pręmiſſa contrariũ 56 th.</s> <s xml:id="echoid-s51824" xml:space="preserve"> 1 <lb/>huius, quod eſt impoſsibile.</s> <s xml:id="echoid-s51825" xml:space="preserve"> Poteſt etiá ſic fieri, ut ſemicirculus h m g ſit medietas horizõtis, & facta <lb/>diuiſione in pũcto m, intelligatur circũduci idẽ ſemicirculus:</s> <s xml:id="echoid-s51826" xml:space="preserve"> nihil enim refert ſemicirculos diuer-<lb/>ſos deſcribere uel unũ circũducere:</s> <s xml:id="echoid-s51827" xml:space="preserve"> punctusq́;</s> <s xml:id="echoid-s51828" xml:space="preserve"> m circumductus deſcribet circulũ iridis, qui eſt n m, <lb/>circa centrũ uel polũ p ſecundũ diſtantiã lineæ p m:</s> <s xml:id="echoid-s51829" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s51830" xml:space="preserve"> anguli à termino diametri, ſcilicet pũ-<lb/>cto h & à centro k ductarum linearũ ad circulũ n m, omnes æquales in qualibet ſuperficie reflexio-<lb/>nis:</s> <s xml:id="echoid-s51831" xml:space="preserve"> quia triangulus h m k in tota circum ductione ſimiles ſibi triangulos cauſſat in qualibet ſuper-<lb/>ficie reflexionis:</s> <s xml:id="echoid-s51832" xml:space="preserve"> & ſimiliter triangulus h m p motu ſuo deſcribet ſimiles triangulos:</s> <s xml:id="echoid-s51833" xml:space="preserve"> & triangulus k <lb/>m p ſimiliter ſimiles triangulos deſcribet.</s> <s xml:id="echoid-s51834" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s51835" xml:space="preserve"> linea m p non ſit perpẽdicularis ſuper diametrũ h <lb/>g:</s> <s xml:id="echoid-s51836" xml:space="preserve"> ducatur ergo perpẽdicularis à pũcto m per 12 p 1 ſuper diametrũ h g:</s> <s xml:id="echoid-s51837" xml:space="preserve"> cadetq́;</s> <s xml:id="echoid-s51838" xml:space="preserve"> illa perpendicularis <lb/>per 29 th.</s> <s xml:id="echoid-s51839" xml:space="preserve"> 1 huius inter pũcta k & p, uel inter pũcta p & g:</s> <s xml:id="echoid-s51840" xml:space="preserve"> quoniã linea m p cũ diametro h g ex aliqua <lb/>ſui parte angulũ acutũ continet, ut patet ex pręmiſsis:</s> <s xml:id="echoid-s51841" xml:space="preserve"> & ſimiliter linea m k;</s> <s xml:id="echoid-s51842" xml:space="preserve"> quia iris nõ apparet niſi <lb/>ultra mediũ diametri horizontis, ut prius patuit:</s> <s xml:id="echoid-s51843" xml:space="preserve"> cadat ergo illa perpẽdicularis in punctũ o.</s> <s xml:id="echoid-s51844" xml:space="preserve"> Simili-<lb/>ter quoq;</s> <s xml:id="echoid-s51845" xml:space="preserve"> ad idem punctũ diametri neceſſariò cadent ab omnibus aliorũ ſemicirculorum angulis <lb/>lineæ perpẽdiculares:</s> <s xml:id="echoid-s51846" xml:space="preserve"> uel angulus k o m motu ſuo in omnibus ſuքficiebus reflexionũ æquales an-<lb/>gulos cauſſabit.</s> <s xml:id="echoid-s51847" xml:space="preserve"> Punctũ ergo o eſt centrũ circuli reflexionis factę ad uiſum.</s> <s xml:id="echoid-s51848" xml:space="preserve"> Cũ ergo centrũ iridis ſit <lb/>in horizontis diametro:</s> <s xml:id="echoid-s51849" xml:space="preserve"> medietas eius erit ſupra horizontẽ, quæ eſt n m, & medietas ſub horizõte:</s> <s xml:id="echoid-s51850" xml:space="preserve"> <lb/>quoniã tũc cõmunis ſectio ſuքficierũ horizontis & iridis eſt diameter iridis.</s> <s xml:id="echoid-s51851" xml:space="preserve"> Idẽq́;</s> <s xml:id="echoid-s51852" xml:space="preserve"> accideret ſi linea <lb/>m p eſſet քpẽdicularis ſuք diametrũ.</s> <s xml:id="echoid-s51853" xml:space="preserve"> Et hic eſt modus, quo Ariſtoteles ꝓpoſitũ cõcluſit.</s> <s xml:id="echoid-s51854" xml:space="preserve"> Sed tamen <lb/>nõ eſt nobis uiſa fore neceſſaria notitia linearũ, quia ſine illa idem & eodẽ modo declarari poteſt.</s> <s xml:id="echoid-s51855" xml:space="preserve"/> </p> <div xml:id="echoid-div1912" type="float" level="0" n="0"> <figure xlink:label="fig-0770-01" xlink:href="fig-0770-01a"> <variables xml:id="echoid-variables869" xml:space="preserve">d b a c</variables> </figure> <figure xlink:label="fig-0770-02" xlink:href="fig-0770-02a"> <variables xml:id="echoid-variables870" xml:space="preserve">m r n h k o p g a b c d</variables> </figure> </div> </div> <div xml:id="echoid-div1914" type="section" level="0" n="0"> <head xml:id="echoid-head1403" xml:space="preserve" style="it">75. In aliquo circulo altitudinis ſuper horizontem exiſtente centro corporis luminoſi, ſecun-<lb/>dum eius eleuationem centrum circuli iridis ſub horizonte deprimitur: & portio iridis minor <lb/>ſemicirculo uidetur.</head> <p> <s xml:id="echoid-s51856" xml:space="preserve">Eſto ſecundum diſpoſitionem proximæ, ſcilicet ut ſit horizon circulus h m g:</s> <s xml:id="echoid-s51857" xml:space="preserve"> cuius diameter ſit <lb/>linea m h.</s> <s xml:id="echoid-s51858" xml:space="preserve"> & centrum k:</s> <s xml:id="echoid-s51859" xml:space="preserve"> ſitq́;</s> <s xml:id="echoid-s51860" xml:space="preserve"> circulus altitudinis tranſiens per zenith capitis & per centrum corpo-<lb/>ris luminoſi:</s> <s xml:id="echoid-s51861" xml:space="preserve"> qui eſt l m n h:</s> <s xml:id="echoid-s51862" xml:space="preserve"> & ſit centrum ſolis eleuatum ſupra horizontem in circulo altitudinis in <lb/>puncto n.</s> <s xml:id="echoid-s51863" xml:space="preserve"> Et quoniam per 64 th.</s> <s xml:id="echoid-s51864" xml:space="preserve"> huius centrum corporis luminoſi, & cẽtrum oculi, & centrũ baſis <lb/> <pb o="470" file="0772" n="772" rhead="VITELLONIS OPTICAE"/> pyramidis irradiationis ſemper ſunt in eadem linea, & cum centrum uiſus ſit centrum circuli alti-<lb/>tudinis:</s> <s xml:id="echoid-s51865" xml:space="preserve"> ſi ducatur linea à centro luminoſi corporis <lb/> <anchor type="figure" xlink:label="fig-0772-01a" xlink:href="fig-0772-01"/> per centrum uiſus, illa neceſſariò erit diameter cir-<lb/>culi altitudinis:</s> <s xml:id="echoid-s51866" xml:space="preserve"> erit ergo illa linea à pũcto n produ-<lb/>cta per centrum k neceſſariò cadens in aliqué pun-<lb/>ctum circuli altitudinis, qui ſit l:</s> <s xml:id="echoid-s51867" xml:space="preserve"> & erit ſemicirculus <lb/>altitudinis eleuatus ſupra circulum horizontis, qui <lb/>eſt h n m, ęqualis ſemicirculo n m l:</s> <s xml:id="echoid-s51868" xml:space="preserve"> quoniã ſunt me-<lb/>dietates eiuſdem circuli:</s> <s xml:id="echoid-s51869" xml:space="preserve"> ablato ergo cõmuni arcu, <lb/>qui eſt n m:</s> <s xml:id="echoid-s51870" xml:space="preserve"> erit arcus h n æ qualis arcui m l:</s> <s xml:id="echoid-s51871" xml:space="preserve">ſed pun-<lb/>ctum l eſt locus centri circuli irradiationis:</s> <s xml:id="echoid-s51872" xml:space="preserve"> & pun-<lb/>ctum n eſt locus centri ſolis.</s> <s xml:id="echoid-s51873" xml:space="preserve"> Patet ergo quòd quan-<lb/>tùm cẽtrum ſolis eleuatur ſupra horizonta, tantùm <lb/>cẽtrum circuli baſis pyramidis irradiationis depri-<lb/>mitur ſub horizóta.</s> <s xml:id="echoid-s51874" xml:space="preserve"> Ethoc eſt primum propoſitum.</s> <s xml:id="echoid-s51875" xml:space="preserve"> <lb/>Cum autem erit cẽtrorum utrunq;</s> <s xml:id="echoid-s51876" xml:space="preserve"> in circulo hori-<lb/>zontis, medietas circuli iridis uide tur, ut in præce-<lb/>denti theoremate eſt oſtenſum:</s> <s xml:id="echoid-s51877" xml:space="preserve"> ergo cum centrum <lb/>ſolis eleuatur, & centrum circuli deprimitur, minus ſemicirculo uidebitur.</s> <s xml:id="echoid-s51878" xml:space="preserve"> Et hoc eſt, quod ſecun-<lb/>dò proponebatur.</s> <s xml:id="echoid-s51879" xml:space="preserve"> Quod autem nunc diximus exponentes propoſitum, ſole exiſtente in oriente, <lb/>idem eſt ſi ſit in horizontis parte occidẽtali, uel in quacunq;</s> <s xml:id="echoid-s51880" xml:space="preserve"> parte ſit horizontis:</s> <s xml:id="echoid-s51881" xml:space="preserve"> ut eſt his, quorum <lb/>latitudo eſt 66 graduum & 9 minutorum:</s> <s xml:id="echoid-s51882" xml:space="preserve"> his enim eſt ſol in meridie in puncto tropici hiemalis in <lb/>horizonte.</s> <s xml:id="echoid-s51883" xml:space="preserve"> Et ſic ſecundum regiones diuerſas uniuerſale ſemper eſt propoſitum theorema.</s> <s xml:id="echoid-s51884" xml:space="preserve"/> </p> <div xml:id="echoid-div1914" type="float" level="0" n="0"> <figure xlink:label="fig-0772-01" xlink:href="fig-0772-01a"> <variables xml:id="echoid-variables871" xml:space="preserve">n h k m g l</variables> </figure> </div> </div> <div xml:id="echoid-div1916" type="section" level="0" n="0"> <head xml:id="echoid-head1404" xml:space="preserve" style="it">76. Iridis nunquam uideri poſſe completum circulum manifeſtum eſt.</head> <p> <s xml:id="echoid-s51885" xml:space="preserve">Quoniam enim ſi ſol eſt in horizonte, ſemicirculus tantùm uidetur, ut patet ex 74 th.</s> <s xml:id="echoid-s51886" xml:space="preserve"> huius:</s> <s xml:id="echoid-s51887" xml:space="preserve"> & ſi <lb/>ſit ſupra horizonta in aliquo circulo altitudinis, patet per pręmiſſam quòd quantùm centrum ſolis <lb/>uel lunæ eleuatur ſupra horizonta, tantùm cẽtrum iridis deprimitur ſub horizonte.</s> <s xml:id="echoid-s51888" xml:space="preserve"> Vnde tune ſu-<lb/>pra horizontem ſemper pars iridis minor ſemicirculo uidetur, ſicut patet in alijs parallelis in ſphę-<lb/>ra, per quorum centrum non tranſit horizon.</s> <s xml:id="echoid-s51889" xml:space="preserve"> Hi enim in portiones inæquales ſub horizonte & ſu-<lb/>pra horizontem ſecantur.</s> <s xml:id="echoid-s51890" xml:space="preserve"> Patet ergo cõ corpus luminoſum in tempore uiſionis iridis ſit aut in ho-<lb/>rizonte aut ſupra horizonta, quòd nunquam completus circulus iridis poterit uideri:</s> <s xml:id="echoid-s51891" xml:space="preserve"> niſi fortè fiat <lb/>exreuerberatione luminis ſolis à nube forti ad terram uel ad aliam nubem, ubi ſit uapor roridus in <lb/>medio, & uiſus inter uaporem & nubem, à qua fit reuerberatio, uel in eadẽ linea, ſic quòd ad ipſum <lb/>poſsit fieri reflexio:</s> <s xml:id="echoid-s51892" xml:space="preserve"> tunc enim poſsibile eſt integras irides uideri:</s> <s xml:id="echoid-s51893" xml:space="preserve"> ſed de talibus ſermo propoſitus <lb/>non intendit:</s> <s xml:id="echoid-s51894" xml:space="preserve"> diximus enim de talibus iridibus in 67 th.</s> <s xml:id="echoid-s51895" xml:space="preserve"> huius.</s> <s xml:id="echoid-s51896" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s51897" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1917" type="section" level="0" n="0"> <head xml:id="echoid-head1405" xml:space="preserve" style="it">77. Datæ iridis ſemidiametrum inuenire.</head> <p> <s xml:id="echoid-s51898" xml:space="preserve">Ad quantum enim ſummorum uaporum conſiſtentia eleuari poſsit iam oſtẽdimus in 60 th.</s> <s xml:id="echoid-s51899" xml:space="preserve"> hu-<lb/>ius:</s> <s xml:id="echoid-s51900" xml:space="preserve">ſed non ſecundum totam eleuationem illorũ poſsibile eſt iridem eleuari:</s> <s xml:id="echoid-s51901" xml:space="preserve"> quoniam materia iri-<lb/>dis eſt uapor roridus per 66 huius, qui non adeò eleuatur, ut uapor ſiccus.</s> <s xml:id="echoid-s51902" xml:space="preserve"> Si ergo datæ iridis ſemi-<lb/>diametrum uolumus inuenire, & data iris ſit ſemicircularis, faciliter habetur propoſitum.</s> <s xml:id="echoid-s51903" xml:space="preserve"> Accipia-<lb/>tur enim altitudo ſua per inſtrumentum:</s> <s xml:id="echoid-s51904" xml:space="preserve"> circuliq̀;</s> <s xml:id="echoid-s51905" xml:space="preserve"> altitudinis ſuæ portio ſiue arcus interiacens ho-<lb/>rizonta & gibbum iridis duplicetur, & cum arcu duplicato intrentur tabulæ chordarum & arcuum <lb/>prima dictione almageſti poſitarum, & extrahatur chorda arte conſueta:</s> <s xml:id="echoid-s51906" xml:space="preserve"> eritq́;</s> <s xml:id="echoid-s51907" xml:space="preserve"> chorda inuenta dia-<lb/>meter totius iridis:</s> <s xml:id="echoid-s51908" xml:space="preserve"> & ea diuiſa per æqualia medietas ipſius erit ſemidiameter iridis:</s> <s xml:id="echoid-s51909" xml:space="preserve"> & ita ſinus cir-<lb/>culi altitudinis erit ſemidiameter iridis, quæ ſub hoc ſitu in tali altitudine uidetur.</s> <s xml:id="echoid-s51910" xml:space="preserve"> Si dicatur quòd <lb/>illa linea non eſt ſemidiameter iridis, ſed cuiuſdam alterius circuli æquidiſtantis iridi, ſed maioris <lb/>iride:</s> <s xml:id="echoid-s51911" xml:space="preserve"> hoc non obſtat:</s> <s xml:id="echoid-s51912" xml:space="preserve"> quia illi duo circuli in eundem angulum ſolidum cadunt apud cẽtrum mun-<lb/>di, quod tunc eſt cẽtrum uiſus:</s> <s xml:id="echoid-s51913" xml:space="preserve"> unde quod de uno dicitur, de reliquo poteſt intelligi, quo ad quan-<lb/>titatem.</s> <s xml:id="echoid-s51914" xml:space="preserve"> Et quia per talium diametrorum proportiones habetur completa proportio iridis ad iri-<lb/>dem:</s> <s xml:id="echoid-s51915" xml:space="preserve"> ideo talem diametrum iridis diametrũ appellamus.</s> <s xml:id="echoid-s51916" xml:space="preserve"> Si uerò iris ſit portio minor ſemicircirculo:</s> <s xml:id="echoid-s51917" xml:space="preserve"> <lb/>accipiatur ipſius altitudo.</s> <s xml:id="echoid-s51918" xml:space="preserve"> Et quia, ut patet per 75 huius, tunc ſol eſt ſupra horizonta in eodẽ circu-<lb/>lo, accipiatur altitudo ſolis.</s> <s xml:id="echoid-s51919" xml:space="preserve"> Quia ergo, ut in illa declaratum eſt, diſtantia centri iridis ſub horizonte <lb/>eſt æqualis eleuationi ſolis ſupra horizontem:</s> <s xml:id="echoid-s51920" xml:space="preserve"> coniungãtur iſti duo arcus altitudinis, iridis ſcilicet <lb/>& ſolis, prouenietq́;</s> <s xml:id="echoid-s51921" xml:space="preserve"> arcus interiacens punctum circuli altitudinis, in quo incidit diameter ducta à <lb/>centro corpotis ſolis per centrum uiſus & per cẽtrum iridis ad ipſum circulum altitudinis (& hoc <lb/>eſt nadir ſolis) & punctum ſuperiorẽ circuli altitudis iridis:</s> <s xml:id="echoid-s51922" xml:space="preserve"> duplicetur ergo ille arcus, & extra-<lb/>hatur chorda ut prius, diuidaturq́;</s> <s xml:id="echoid-s51923" xml:space="preserve"> per æqualia:</s> <s xml:id="echoid-s51924" xml:space="preserve"> & habebitur intentum.</s> <s xml:id="echoid-s51925" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s51926" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1918" type="section" level="0" n="0"> <head xml:id="echoid-head1406" xml:space="preserve" style="it">78. Iridis ſemicirculus uiſus eſt medietas circuli minoris: portio uerò minor ſemicirculo uiſa, <lb/>eſt portio circuli maioris.</head> <p> <s xml:id="echoid-s51927" xml:space="preserve">Huius propoſitæ rei cauſſa pater ſecundum præmiſſa huius libri.</s> <s xml:id="echoid-s51928" xml:space="preserve"> Quoniam enim, ut patet per <lb/>64 huius, centrum ſolis & uiſus & iridis ſemper in eadem linea conſiſtunt, quæ eſt axis pyrami-<lb/>dis illuminationis uaporis roridi:</s> <s xml:id="echoid-s51929" xml:space="preserve"> propter quod pater quia in omni reflexione, ex qua apparet <lb/>iris, ſemper centrum uiſus eſt polus circuli iridis:</s> <s xml:id="echoid-s51930" xml:space="preserve"> palàm ergo quòd nullam facit diuerſitatem in <lb/> <pb o="471" file="0773" n="773" rhead="LIBER DECIMVS."/> uilu erectio uel obliquatio ſuperficiel iridis ſupra ſuperficiem horizontis.</s> <s xml:id="echoid-s51931" xml:space="preserve"> Quoniam ſemper li-<lb/>nea pertranſiens centrum ſolis & uiſus eſt erecta ſuper ſuperficiem iridis:</s> <s xml:id="echoid-s51932" xml:space="preserve"> & ſic peripheria iridis <lb/>ſemper ſe habet uniformiter ad uiſum, quantũ eſt de ſe, ut patet per 65 th.</s> <s xml:id="echoid-s51933" xml:space="preserve">1 huius.</s> <s xml:id="echoid-s51934" xml:space="preserve"> Quod tamen hic <lb/>proponitur, cauſſam habet non ex reflexione, ſed ex refractione:</s> <s xml:id="echoid-s51935" xml:space="preserve"> quia ut in 8 huius declarauimus, <lb/>diuerſitas angulorum refractionis cauſſatur ex diuerſitate diaphanitatis corporũ diaphanorum e-<lb/>tiam eiuſdem ſpeciei:</s> <s xml:id="echoid-s51936" xml:space="preserve"> maior enim fit refractio ad lineam perpendicularem in aqua groſsiori quàm <lb/>in aqua ſubtiliori.</s> <s xml:id="echoid-s51937" xml:space="preserve"> Quia itaq;</s> <s xml:id="echoid-s51938" xml:space="preserve"> ſole exiſtente in peripheria horizontis, aer eſt groſsior ſeip ſo, poſtmo-<lb/>dum per luminis ſolaris præſentiam ſubtiliato:</s> <s xml:id="echoid-s51939" xml:space="preserve"> palàm quòd in groſsiori illo aere minor fit refractio <lb/>â perpendiculari:</s> <s xml:id="echoid-s51940" xml:space="preserve"> radij itaq;</s> <s xml:id="echoid-s51941" xml:space="preserve"> tunc refracti magis approximant perpendiculari quàm poſtmodum <lb/>aere ſubtiliato.</s> <s xml:id="echoid-s51942" xml:space="preserve"> Ad propinquiorem ergo locum ſuperficie iridis fit aggregatio radiorum inciden-<lb/>tium ſuperficiebus uiſuum ibi exiſtentium, quàm fiat aere rariori exiſtente.</s> <s xml:id="echoid-s51943" xml:space="preserve"> Subtiliato uerò aere, fit <lb/>ad eoſdem uiſus à partibus remotioribus ipſius uaporis refractio:</s> <s xml:id="echoid-s51944" xml:space="preserve"> non enim fit à partibus propin-<lb/>pin quioribus:</s> <s xml:id="echoid-s51945" xml:space="preserve"> quoniam ab illis neq;</s> <s xml:id="echoid-s51946" xml:space="preserve"> prius fiebat.</s> <s xml:id="echoid-s51947" xml:space="preserve"> Sed neq;</s> <s xml:id="echoid-s51948" xml:space="preserve"> fit illa refractio à partibus uaporis, à qui-<lb/>bus fiebat prius:</s> <s xml:id="echoid-s51949" xml:space="preserve"> quoniam medio immutato eſt ipſa refractio immutata per 8 huius:</s> <s xml:id="echoid-s51950" xml:space="preserve"> fit ergo neceſſa <lb/>riò refractio à partibus uaporis remotioribus, quàm prius.</s> <s xml:id="echoid-s51951" xml:space="preserve"> Radij ergo refracti ſunt longiores his, <lb/>qui prius refringebantur:</s> <s xml:id="echoid-s51952" xml:space="preserve"> pyramis ergo illuminationis eſt maior:</s> <s xml:id="echoid-s51953" xml:space="preserve"> ergo & baſis eius (quæ, ut patet <lb/>expræhabitis, eſt peripheria iridis) erit maior.</s> <s xml:id="echoid-s51954" xml:space="preserve"> Exiſtente uerò ſole in peripheria horizontis, tunc <lb/>tantùm cauſſatæ iridis ſemicirculus uidetur, ut patet per 74 huius:</s> <s xml:id="echoid-s51955" xml:space="preserve"> eleuato uerò ſole ſupra horizon <lb/>ta:</s> <s xml:id="echoid-s51956" xml:space="preserve"> tunc portio iridis minor ſemicirculo uidetur, ut patet per 75 huius.</s> <s xml:id="echoid-s51957" xml:space="preserve"> Maniſeſtum eſt ergo propo-<lb/>ſitum.</s> <s xml:id="echoid-s51958" xml:space="preserve"> Eſt autem quorundam experientia, quòd altitudo iridis, & altitudo ſolis coniunctæ ſemper <lb/>faciunt gradus 42:</s> <s xml:id="echoid-s51959" xml:space="preserve"> quod per præſens theorema impoſsibile eſſe oſten ditur.</s> <s xml:id="echoid-s51960" xml:space="preserve"> Si enim ſemidiameter <lb/>circuli iridis ſit quan doq;</s> <s xml:id="echoid-s51961" xml:space="preserve"> minor, quandoq;</s> <s xml:id="echoid-s51962" xml:space="preserve"> maior ſecun dum mediorum diaphanorum & ſuarum <lb/>reſractionum diuerſitatẽ, ut præoſtenſum eſt:</s> <s xml:id="echoid-s51963" xml:space="preserve"> tunc non poterit rationabiliter uideri alicui, quòd o-<lb/>mnes aliorum circulorũ diuerſarum iridum ſemidiametri ſint æquales:</s> <s xml:id="echoid-s51964" xml:space="preserve"> poſſet tamen eſſe modica <lb/>differentia, quæ fortè per in ſtrumentum modicum improportionale circulo altitudinis non poſ-<lb/>fit aliqualiter perpendi.</s> <s xml:id="echoid-s51965" xml:space="preserve"> Et etiam eorum experientia eſt in portionibus iridum min oribus ſemicir-<lb/>culo, quod patet per altitudinem ſolis, quam tales uerſo inſtrumento uel mutato uiſu, fixo inſtru-<lb/>mento accipiunt, quæ nulla eſt ſole exiſtente in peripheria horizontis.</s> <s xml:id="echoid-s51966" xml:space="preserve"> Et fortè talium portionum <lb/>uel ſuarum diametrorum non eſt ſenſibilis differentia:</s> <s xml:id="echoid-s51967" xml:space="preserve"> quia etiam Ariſtoteles de illa nihil ſcripſit:</s> <s xml:id="echoid-s51968" xml:space="preserve"> <lb/>cum tamen de præſente theoremate magnam fecerit mentionem:</s> <s xml:id="echoid-s51969" xml:space="preserve"> quamuis nec ipſe nec alius, cu-<lb/>ius ſcripta uiderimus, ſuper hoc attulerit declarationem.</s> <s xml:id="echoid-s51970" xml:space="preserve"> De differentia uerò climatum nullus ex-<lb/>cuſationem afferat:</s> <s xml:id="echoid-s51971" xml:space="preserve"> quia quod in uno climate accidit, in omnibus climatibus euenire neceſſe eſt in <lb/>iridis generatione.</s> <s xml:id="echoid-s51972" xml:space="preserve"> Semper enim centra ſolis, uiſus, & circuli iridis in eadem linea conſiſtunt:</s> <s xml:id="echoid-s51973" xml:space="preserve"> & ar-<lb/>cus altitu dinis ſub horizonte centri circuli irldis, ſolis altitudini in omnibus climatibus eſt æ qua-<lb/>lis:</s> <s xml:id="echoid-s51974" xml:space="preserve"> nec in hoc aliquis differentiam perpendet.</s> <s xml:id="echoid-s51975" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1919" type="section" level="0" n="0"> <head xml:id="echoid-head1407" xml:space="preserve" style="it">79. In quibuſdam regionibus ſole exiſtente in meridie, iris ſenſibilis non apparet.</head> <p> <s xml:id="echoid-s51976" xml:space="preserve">Ad oſten den dum propoſitum ponatur primò centrum ſolis in aliqua regione in meridie in ze-<lb/>nith capitis:</s> <s xml:id="echoid-s51977" xml:space="preserve"> & palàm ex præmiſsis, quòd tunc baſis pyramidis irradiationis erit ſub horizonte æ-<lb/>quidiſtans horizonti.</s> <s xml:id="echoid-s51978" xml:space="preserve"> Et quoniam tunc altitudo ſolis erit partium 90:</s> <s xml:id="echoid-s51979" xml:space="preserve"> ſole deſcen dente (ſiue hoe <lb/>ſit propter ip ſum motum ſolis, ſiue propter altitudinem regionum diſtantium plus ab æquino-<lb/>ctiali, quàm regio, in qua ſol fuit perpendicularis in meridie, ut ab ea, quæ eſt directè ſub capite can <lb/>cri) nunquam fiet iris in meridie, quandiu ſinus circuli altitudinis ſolis in meridie fuerit maior <lb/>diametro iridis, quam per 77 huius diligens perquiſitor poterit inuenire.</s> <s xml:id="echoid-s51980" xml:space="preserve"> Quantùm autem ſinus <lb/>circuli altitudinis ſolis in meridie minuetur à diametro iridis:</s> <s xml:id="echoid-s51981" xml:space="preserve"> tantùm apparebit uiſui in meridie <lb/>de diametro iridis & de iride.</s> <s xml:id="echoid-s51982" xml:space="preserve"> Et ob hoc in diebus æſtiualibus ab æquinoctio uernali ad autumna-<lb/>le in conſuetis nobis regionibus, quæ ſunt ultra clima quartum uſq;</s> <s xml:id="echoid-s51983" xml:space="preserve"> ad finem notorum ſeptem <lb/>climatum in meridie iris non apparet:</s> <s xml:id="echoid-s51984" xml:space="preserve"> & ſi in alia parte anni appareat quandoq;</s> <s xml:id="echoid-s51985" xml:space="preserve">. Totum autem <lb/>hoc diximus propter regiones, quæ ſunt extra climata, in quibus præmiſſa regula doctrinæ gene-<lb/>rali poterit committi.</s> <s xml:id="echoid-s51986" xml:space="preserve"> In omnibus autem regionibus ſole exiſtente ſupra horizontem, in qualibet <lb/>hora diei iris poterit apparere, præter quàm in meridie:</s> <s xml:id="echoid-s51987" xml:space="preserve"> in illis tamen horis, in quibus ſinus circuli <lb/>altitudinis ſolis maior eſt iridis diametro.</s> <s xml:id="echoid-s51988" xml:space="preserve"> Et hæc ſufficiant pro iridis intento:</s> <s xml:id="echoid-s51989" xml:space="preserve"> quia irim de cœlo <lb/>miſit Saturnia Iuno.</s> <s xml:id="echoid-s51990" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1920" type="section" level="0" n="0"> <head xml:id="echoid-head1408" xml:space="preserve" style="it">80. Nubium apparens color fit ſecundũ diſpoſitionem materia & luminis incorpor ationem.</head> <p> <s xml:id="echoid-s51991" xml:space="preserve">Quoniá enim nubiũ conſiſtentia ex duobus fit uaporibus, ſicco ſcilicet & humido, ut declaratũ <lb/>eſt in philoſophia naturali:</s> <s xml:id="echoid-s51992" xml:space="preserve"> tũc quando ſol agendo ex ſicco penitus extrahit humidũ, aduritur ſiccũ <lb/>rerreſtre, ita quòd lumen in ipſum penetrare non poteſt:</s> <s xml:id="echoid-s51993" xml:space="preserve"> & ideo fit tunc nubes nigra multæ nigre-<lb/>dinis:</s> <s xml:id="echoid-s51994" xml:space="preserve"> & ſunt tales nubes materia uentorum.</s> <s xml:id="echoid-s51995" xml:space="preserve"> In uapore uerò aqueo generatur nigredo ex conden-<lb/>ſatione frigoris, propter quam in ipſum penetrare non poteſt radius ſolaris uel ſtellarum:</s> <s xml:id="echoid-s51996" xml:space="preserve"> & ideo re <lb/>manet nubes humida multũ nigra.</s> <s xml:id="echoid-s51997" xml:space="preserve"> Ex uapore uerò quocunq;</s> <s xml:id="echoid-s51998" xml:space="preserve"> diſgregato ſubtili, recipiente ingreſ-<lb/>ſum luminis ſolaris fit nubes alba:</s> <s xml:id="echoid-s51999" xml:space="preserve"> unde etiam aliquando uidetur nebula alba.</s> <s xml:id="echoid-s52000" xml:space="preserve"> Quan do autẽ nubes <lb/>habet in ſe humidũ fumo ſum admixtum aliquantulũ terreſtri aduſto:</s> <s xml:id="echoid-s52001" xml:space="preserve"> tunc in ipſo recepto lumine <lb/>fit nubes rubea, & aliquando purpurea:</s> <s xml:id="echoid-s52002" xml:space="preserve"> ut cum radij terminantur ad inferiorẽ partem nubis humi-<lb/> <pb o="472" file="0774" n="774" rhead="VITELLONIS OPTICAE"/> dę in mane uel in ſero:</s> <s xml:id="echoid-s52003" xml:space="preserve"> & hæc ſignificant pluuiã futuram.</s> <s xml:id="echoid-s52004" xml:space="preserve"> Et ſi quidem ſit in oriente, defertur pluuia <lb/>ſuper homines illius habitabilis:</s> <s xml:id="echoid-s52005" xml:space="preserve"> ſi uerò ſit in occaſu, tunc defertur pluuia ad mundi inferius hemi-<lb/>ſphęrium ſub homines uidentes:</s> <s xml:id="echoid-s52006" xml:space="preserve"> & erit ibi pluuia in nocte:</s> <s xml:id="echoid-s52007" xml:space="preserve"> & redibit illa pars cœli fortè ſpoliata nu <lb/>bibus in mane:</s> <s xml:id="echoid-s52008" xml:space="preserve"> & ſic ſignificat rubor nubium in ſero ſerenitatem in die ſequente.</s> <s xml:id="echoid-s52009" xml:space="preserve"> Quando uerò nu <lb/>bes depreſſa habet ſuperius reſperſam purpureitatẽ obſcuram ualde:</s> <s xml:id="echoid-s52010" xml:space="preserve"> tunc illa rubedo eſt ex parti-<lb/>bus terreis aduſtis, quæ iam incipiunt inflammari in uentre nubis:</s> <s xml:id="echoid-s52011" xml:space="preserve"> & ſunt nubes tales periculoſæ <lb/>continentes materiam tonitru & ſimilium.</s> <s xml:id="echoid-s52012" xml:space="preserve"> Quòd ſi nubes ſit rorans & in fine ſuæ reſolutionis:</s> <s xml:id="echoid-s52013" xml:space="preserve"> <lb/>tunc illa nubes in ſe recepto lumine, quandoq;</s> <s xml:id="echoid-s52014" xml:space="preserve"> iridis acquirit colorẽ:</s> <s xml:id="echoid-s52015" xml:space="preserve"> & ſecundum ſui uarias diſpo-<lb/>ſitiones fit multa uarietas colorum lumine nubibus præſente:</s> <s xml:id="echoid-s52016" xml:space="preserve"> ſiue lumẽ nubi incidens refringatur <lb/>ad uiſum propter denſitatem ſecundi diaphani, ſiue reflectatur ad uiſum à ſuperficie ipſius nubis.</s> <s xml:id="echoid-s52017" xml:space="preserve"> <lb/>Sed in his coloribus medijs nubium non modicum effectum habet admixtio umbrarum, cum nu-<lb/>bes ſuperior per nubem ſubtilem umbroſam uiſibus occurrit.</s> <s xml:id="echoid-s52018" xml:space="preserve"> Tunc enim uario colore coloratur <lb/>nubes uiſa fecundum illarum umbrarum admixtionem.</s> <s xml:id="echoid-s52019" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s52020" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1921" type="section" level="0" n="0"> <head xml:id="echoid-head1409" xml:space="preserve" style="it">81. Virgæ fiunt ex refr actione radiorum ſolarium ad uiſum ab aliqua conſiſtentia nuboſa, ra <lb/>ritate & ſpißitudine inæqualiter dictincta.</head> <p> <s xml:id="echoid-s52021" xml:space="preserve">Virgæ dicuntur extenſiones radiorũ per nubes, quæ uulgo dicuntur funes tentorij.</s> <s xml:id="echoid-s52022" xml:space="preserve"> Interpoſita <lb/>enim nube aliqua aquoſa inter ſolem & uiſus noſtros fit refractio radiorũ ſolariũ ad uiſum:</s> <s xml:id="echoid-s52023" xml:space="preserve"> & hoc <lb/>accidit in medio ſecundi diaphani.</s> <s xml:id="echoid-s52024" xml:space="preserve"> Et ob hoc quandoq;</s> <s xml:id="echoid-s52025" xml:space="preserve"> ibi uidentur iridis colores ſecundũ quaſdã <lb/>lineas rectas protenſæ, eò quòd habeant quandam ſubtiliorẽ & quandam groſsiorẽ conſiſtentiam, <lb/>in quibus permixtũ ſolis lumen phantaſiam coloris in ipſis facit.</s> <s xml:id="echoid-s52026" xml:space="preserve"> Potior tamen in his cauſſa eſt ad-<lb/>mixtio umbrarũ, quæ diuerſimodè immixtę lumini colores diuerſos uiſibus repræſentant.</s> <s xml:id="echoid-s52027" xml:space="preserve"> Et quia <lb/>radius ſolis perpen dicularis ſuper ſuperficiẽ nubis penetrat nubẽ, & ad uiſus non reflectitur:</s> <s xml:id="echoid-s52028" xml:space="preserve"> ideo <lb/>nubes in medio alba & incolorata uidetur:</s> <s xml:id="echoid-s52029" xml:space="preserve"> & ſol per illã uiſus uidetur ſine figura, ſed in colore puni <lb/>ceus aut colorẽ aliũ habens:</s> <s xml:id="echoid-s52030" xml:space="preserve"> ſol enim per conſiſtentiã nubis groſsiorẽ & caliginoſam aliũ & alium <lb/>præſentat uiſibus colorẽ.</s> <s xml:id="echoid-s52031" xml:space="preserve"> Non eſt aũt in hoc differentia ſiue ſol uideatur per nubẽ, ſic qđ fiat ſuorũ <lb/>radiorũ ad uiſus refractio, ſiue radij ſolis reflectantur ad uiſum.</s> <s xml:id="echoid-s52032" xml:space="preserve"> Aſpicienti uerò ad ſolis latera uide <lb/>tur quandoq;</s> <s xml:id="echoid-s52033" xml:space="preserve"> iridis color uirgatus, ut præ miſimus, quãdo nubes ſecundũ aliquid eſt ſpiſſa, & ſecun <lb/>dum aliquid rara, & ſecundũ aliquã ſui partẽ plus aquoſa, & ſecundũ aliquã minus:</s> <s xml:id="echoid-s52034" xml:space="preserve"> & quandoq;</s> <s xml:id="echoid-s52035" xml:space="preserve"> ui-<lb/>detur aliqua pars punicea, alia uerò uiridis aut flaua.</s> <s xml:id="echoid-s52036" xml:space="preserve"> Virgæ itaq;</s> <s xml:id="echoid-s52037" xml:space="preserve"> fiũt propter irregularitatẽ diuerſi <lb/>ſitus & qualitatis ſpeculorũ, nõ propter figurę anomaliã.</s> <s xml:id="echoid-s52038" xml:space="preserve"> Sunt enim quędã ſpecula, quę propter ſui <lb/>anomaliã figuras anomalas & permutatas uiſibus oſtendunt formarũ uiſarũ per ipſa, de quibus in <lb/>nono libro ſcientiæ huius aliquis ſermo fuit.</s> <s xml:id="echoid-s52039" xml:space="preserve"> Vnde & nubes figurã ſolis non oſtendit:</s> <s xml:id="echoid-s52040" xml:space="preserve"> quia ſpecula <lb/>nubis non ſunt propriè oſtendentia figuram propter ſpeculorũ paruitatẽ, ſed oſten dunt colorem, <lb/>quod conuenit diaphanitati ſpeculorũ & nubis totius:</s> <s xml:id="echoid-s52041" xml:space="preserve"> & diſtinguuntur illi colores ſecundum diſ-<lb/>poſitionẽ materiæ, cui lux incorporatur, & ſecundũ umbrarum immixtionẽ.</s> <s xml:id="echoid-s52042" xml:space="preserve"> Patet ergo propoſitũ.</s> <s xml:id="echoid-s52043" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1922" type="section" level="0" n="0"> <head xml:id="echoid-head1410" xml:space="preserve" style="it">82. Pareliæ fiunt ex reflexione radiorum ſolarium ad uiſum ab æquali conſiſtentia nuboſa.</head> <p> <s xml:id="echoid-s52044" xml:space="preserve">Pareliæ dicuntur quaſi paria ſoli, {κ~'}λιος enim græcè, ſol dicitur latinè, & ſignificant ſoles aqueos, <lb/>qui in nube uidentur.</s> <s xml:id="echoid-s52045" xml:space="preserve"> Nube enim interpoſita ſoli & uiſibus, exiſtente æ quali ſecundũ ſua ſpecula, <lb/>neq;</s> <s xml:id="echoid-s52046" xml:space="preserve"> denſiore neq;</s> <s xml:id="echoid-s52047" xml:space="preserve"> rariore, neq;</s> <s xml:id="echoid-s52048" xml:space="preserve"> plus aquoſa, neq;</s> <s xml:id="echoid-s52049" xml:space="preserve"> minus ſecundũ ſuas partes:</s> <s xml:id="echoid-s52050" xml:space="preserve"> tunc radio ſolis illis <lb/>incidente, propter ſimilitudinẽ & æ qualitatẽ ſpeculorum, & ipſorum regularitatẽ unius coloris fit <lb/>phan taſia:</s> <s xml:id="echoid-s52051" xml:space="preserve"> albi aũt uidetur coloris propter ſpiſsitudinem conſiſtentiæ & regularitatẽ ipſius nubis.</s> <s xml:id="echoid-s52052" xml:space="preserve"> <lb/>Radij enim ad ipſam nubẽ ſic diſpoſitũ incidentes, & ab ipſa reflexi ad uiſus (maximè nube illa non <lb/>exiſtente aquoſa neq;</s> <s xml:id="echoid-s52053" xml:space="preserve"> nigra, uicina tamen aquæ) ſine admixtione alicuius umbræ reflectuntur ad <lb/>uiſum:</s> <s xml:id="echoid-s52054" xml:space="preserve"> propter quod proprium ſolis colorem, qui luminoſus & albus eſt, in tota nubis conſiften-<lb/>tia apparere faciunt uiſibus:</s> <s xml:id="echoid-s52055" xml:space="preserve"> fiuntq́ue pareliæ albæ, ſicut etiam ab omni corpore polito reflectitur <lb/>lumen ſolis ad uiſum propter ſpiſsitudinẽ conſiſtentiæ:</s> <s xml:id="echoid-s52056" xml:space="preserve"> ut oſtenſum eſt per 1 th.</s> <s xml:id="echoid-s52057" xml:space="preserve">5 huius.</s> <s xml:id="echoid-s52058" xml:space="preserve"> Sunt autẽ <lb/>parelię magis ſignum pluuiæ quàm uirgæ:</s> <s xml:id="echoid-s52059" xml:space="preserve"> quia æ qualis nubium conſiſtentia, quæ eſt materia pare <lb/>lijs, ſignum eſt quòd aer idoneè habet ſe ad permutationẽ & ad generationem aquæ.</s> <s xml:id="echoid-s52060" xml:space="preserve"> Et quia auſtra <lb/>lis aer facilius in aquam permutatur propter ſui facilitatem in patlendo, quâm aer borealis, qui ſic-<lb/>cior eſt propter frigoris conſtrictionẽ:</s> <s xml:id="echoid-s52061" xml:space="preserve"> ideo pareliæ auſtrales magis ſunt ſignum pluuiæ quàm bo-<lb/>reales.</s> <s xml:id="echoid-s52062" xml:space="preserve"> Fiunt aũt pareliæ ſicut & uirgæ magis ſole exiſtente in oriente uel occidentę quàm in meri-<lb/>die:</s> <s xml:id="echoid-s52063" xml:space="preserve"> quoniã ſol exiſtẽs in medio cœli ſoluit tales nubium conſiſtentias, & plurimũ ſegregat illas:</s> <s xml:id="echoid-s52064" xml:space="preserve"> & <lb/>neq;</s> <s xml:id="echoid-s52065" xml:space="preserve"> fiunt deſuper ſolẽ neq;</s> <s xml:id="echoid-s52066" xml:space="preserve"> deſubtus, ſed à lateribus ſolis obliquis, quæ ſunt ſecundum polos mũ-<lb/>di:</s> <s xml:id="echoid-s52067" xml:space="preserve"> & neq;</s> <s xml:id="echoid-s52068" xml:space="preserve"> fiunt multũ prope ſolem:</s> <s xml:id="echoid-s52069" xml:space="preserve"> quia à propinquo citò diſſoluitur nubiũ conſiſtentia:</s> <s xml:id="echoid-s52070" xml:space="preserve"> neq;</s> <s xml:id="echoid-s52071" xml:space="preserve"> fiunt <lb/>multùm longè à ſole:</s> <s xml:id="echoid-s52072" xml:space="preserve"> quia nõ eſt inde poſsibile reflexionẽ fieri ad uiſus:</s> <s xml:id="echoid-s52073" xml:space="preserve"> reflexio enim facta à paruo <lb/>ſpeculo ſubtilis eſt:</s> <s xml:id="echoid-s52074" xml:space="preserve"> unde longè protenſa debilitatur & euaneſcit, antequã perueniat ad uiſus.</s> <s xml:id="echoid-s52075" xml:space="preserve"> Et ex <lb/>eiſdẽ cauſsis nõ fiunt hæ pareliæ ſupra ſolem, neq;</s> <s xml:id="echoid-s52076" xml:space="preserve"> ſub ſole, quia prope ſolẽ exiſtentes conſiſtentiæ <lb/>nubium ſoluuntur, remotè uerò diſtantes non perueniunt ſecundũ ipſorũ reflexionẽ ad uiſum:</s> <s xml:id="echoid-s52077" xml:space="preserve"> ſe-<lb/>cundum lateralẽ rerò ſolis ſitum eſt inuenire mediocrẽ diſtantiam, in qua cõſiſtentia non diffolui-<lb/>tur, & tamẽ ſit reflexio ad uiſum:</s> <s xml:id="echoid-s52078" xml:space="preserve"> ut cum non eſt nimis propè ad terrã deſcendens illa nubis conſi-<lb/>ſtentia.</s> <s xml:id="echoid-s52079" xml:space="preserve"> Quando enim nubes ſunt nimis propinquæ horizonti:</s> <s xml:id="echoid-s52080" xml:space="preserve"> tũc ab ipſis nubibus reflexi radij nõ <lb/>pertingunt ad uiſus propter diſtantiã minorem improportionatã reflexioni luminis:</s> <s xml:id="echoid-s52081" xml:space="preserve"> quoniã enim <lb/>uiſus funt apud terrã, patet quòd tunc luminis reflexio à nube non concurrit cum uiſibus.</s> <s xml:id="echoid-s52082" xml:space="preserve"> Sub ſole <lb/>etiam nõ poteſt fieri parelia:</s> <s xml:id="echoid-s52083" xml:space="preserve"> quia & tunc nubes uicina terræ perpendicularem ſolis radium reſpi-<lb/> <pb o="473" file="0775" n="775" rhead="LIBER DECIMVS."/> ciens diſſoluitur à radio ſolari, remota uerò nubes à uiſu nullam cauſſat reflexionem uel refractio-<lb/>nem ad uiſum, propter longitudinem diſtantię:</s> <s xml:id="echoid-s52084" xml:space="preserve"> quia ſi etiam à latere ſolis eſſet cõſiſtentia nubis ni-<lb/>mis alta, non accideret reflexionem luminis fieri ad uiſum:</s> <s xml:id="echoid-s52085" xml:space="preserve"> nec tunc apparerent pareliæ ipſis uiſi-<lb/>bus.</s> <s xml:id="echoid-s52086" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s52087" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1923" type="section" level="0" n="0"> <head xml:id="echoid-head1411" xml:space="preserve" style="it">83. Ex cryſt allo hexagona ſoli oppoſita colores iridis generantur.</head> <p> <s xml:id="echoid-s52088" xml:space="preserve">Huiuſinodi enim colores generantur ex debilitatione luminis propter refractionem ad perpen <lb/>dicularem, ductam à centro corporis ſolis ad ſuperficiem unius parallelogrãmi ex lateribus cryſtal <lb/>li.</s> <s xml:id="echoid-s52089" xml:space="preserve"> Et quoniã (ut declarauimus in 27 th.</s> <s xml:id="echoid-s52090" xml:space="preserve">2 huius ſciẽtiæ) manifeſtũ eſt quòd à ſole illuminatur magis <lb/>me dietate cylindri ſibi oppoſiti, ſi rotun dus ſit cylindrus:</s> <s xml:id="echoid-s52091" xml:space="preserve"> hoc autem in cylin dro angulato eſſe non <lb/>poteſt (angulis ueniẽtibus in diametrũ corporis baſim per æqualia diuidentẽ) tunc enim ſola me-<lb/>dietas illuminatur propter radiorũ incidentiã, ut diximus ibidẽ.</s> <s xml:id="echoid-s52092" xml:space="preserve"> Sed ſi corpus illud columnare dia-<lb/>phanũ ſuerit:</s> <s xml:id="echoid-s52093" xml:space="preserve"> tunc alia medietas illius corporis illuminatur propter radiorum refractionẽ.</s> <s xml:id="echoid-s52094" xml:space="preserve"> Si itaq;</s> <s xml:id="echoid-s52095" xml:space="preserve"> <lb/>ſuperficies corporis diaphani ſoli oppoſita unica fuerit, ut in corporibus quadrangulis:</s> <s xml:id="echoid-s52096" xml:space="preserve"> tunc una <lb/>fit luminis refractio fortis:</s> <s xml:id="echoid-s52097" xml:space="preserve"> & lumen ſub forma luminis tranſibit ad partem oppoſitam corporis, & <lb/>aggregabitur extra corpus ſub forma luminis:</s> <s xml:id="echoid-s52098" xml:space="preserve"> ſicut etiã hoc fortius euenit in corpore ſphærico dia <lb/>phano nõ cõcauo:</s> <s xml:id="echoid-s52099" xml:space="preserve"> eò quòd à ſuperficie maioris partis totius illius corporis ſphærici fit refractio ad <lb/>radiũ, qui perpendiculariter incidit ſuper ſuperficiem corpus ſphæricum contingentẽ, æquidiſtan <lb/>tem ſuperficiei ſecanti corpus ſolis per centrũ ſecundum aſpectũ, quo ab ipſo reſpicitur corpus illu <lb/>minandũ, ut oſten dimus in 48 huius.</s> <s xml:id="echoid-s52100" xml:space="preserve"> Ex tantorũ ergo & tot radiorũ aggregatione, & ſi nõ ad pun-<lb/>ctum unũ (quo niã hoc eſt impoſsibile propter diuerſitatẽ ſuperficierum incidentiæ) ad locũ tamẽ <lb/>naturalem paruũ fit luminis aggre gatio, ipſo lumine abſq;</s> <s xml:id="echoid-s52101" xml:space="preserve"> coloratione ſub forma luminis manẽte;</s> <s xml:id="echoid-s52102" xml:space="preserve"> <lb/>& illud lumen aggregatũ calefacit corpus oppoſitũ, & incendit ex mora corpus inflãmabile ſubitò, <lb/>ut ſtupam uel aliud aliquid potentiã actiuã in ſe habentẽ ad inflam mationẽ.</s> <s xml:id="echoid-s52103" xml:space="preserve"> Si uerò corpus diapha-<lb/>num ſoli oppoſitũ ſit plurium ſuperficierũ ꝗ̃ unius planæ uel circularis, ſecundũ eam ſcilicet partẽ, <lb/>quæ ſoli opponitur:</s> <s xml:id="echoid-s52104" xml:space="preserve"> utpote ſi corpus quadrangulũ ſecundum unũ ſuorum angulorũ ſoli oppona-<lb/>tur:</s> <s xml:id="echoid-s52105" xml:space="preserve"> tunc fiet refractio radiorũ incidentium uni ſuperficiei ad ambas ſuքficies oppoſitas, & ſimiliter <lb/>radiorũ incidentium alteri ſuperficiei.</s> <s xml:id="echoid-s52106" xml:space="preserve"> Et cum ex parte oppoſita lumini refracto aer (qui eſt corpus <lb/>rarioris diaphani) occurrerit:</s> <s xml:id="echoid-s52107" xml:space="preserve"> refringentur radij ab utraq;</s> <s xml:id="echoid-s52108" xml:space="preserve"> ſuperficie ab illa perpendiculari, quæ ab <lb/>angulo ad angulũ ducta in corpore baſim ipſius per ęqualia duideret, uel alia ei æquidiſtante, & in <lb/>alio corpore denſo illi corpori diaphano ſubiecto, ut terra uel alio corpore quocunq;</s> <s xml:id="echoid-s52109" xml:space="preserve">: tunc quan-<lb/>doq;</s> <s xml:id="echoid-s52110" xml:space="preserve"> apparebunt duo lumina clara, aliquando uerò colorata:</s> <s xml:id="echoid-s52111" xml:space="preserve"> ut ſi corpus diaphanũ æqualium fue-<lb/>rit angulorũ & ſuperficierũ:</s> <s xml:id="echoid-s52112" xml:space="preserve"> & hoc patet experimentanti:</s> <s xml:id="echoid-s52113" xml:space="preserve"> eruntq́;</s> <s xml:id="echoid-s52114" xml:space="preserve"> tunc ibi duo colores confuſi, non <lb/>plures, color ſcilicet rubeus, & alius mixtus, quaſi uiridis, qui ſecundum cryſtalli uel alterius parui <lb/>corporis diſpoſitionẽ magis ſunt intenſi uel remiſsi.</s> <s xml:id="echoid-s52115" xml:space="preserve"> Quòd ſi ſuperficies corporis (quo ad partẽ ſoli, <lb/>oppoſitã) fuerint tres, ut ſunt in cryſtallo hexagona:</s> <s xml:id="echoid-s52116" xml:space="preserve"> tunc à qualibet ſuperficierũ oppoſitarum ſoli, <lb/>quæ ſunt tres, receptũ lumen cuiuslibet ſuperiorũ trium ſuperficierũ red ditur corpori oppoſito, ut <lb/>terrę uel alteri corpori cuicunq;</s> <s xml:id="echoid-s52117" xml:space="preserve">. Atq;</s> <s xml:id="echoid-s52118" xml:space="preserve"> horũ trium luminũ medium manet in ipſa perpen diculari co <lb/>lumnę cryſtallinæ baſim ſuam per æ qualia diuidente, uel ipſi diuidenti æ quidiſtante:</s> <s xml:id="echoid-s52119" xml:space="preserve"> & fit uiſibile <lb/>lumen illud, niſi lumẽ ſolis impediat:</s> <s xml:id="echoid-s52120" xml:space="preserve"> alia uerò duo refringuntur à dicta perpendiculari propter na-<lb/>turã ſecundi diaphani rarioris, ſcilicet aeris (dictũ enim eſt in 4 th.</s> <s xml:id="echoid-s52121" xml:space="preserve"> huius quòd medio ſecundi dia-<lb/>phani rariore exiſtente refractio ſit à perpendiculari) & eſt quaſi quædã diſperſio radiorũ.</s> <s xml:id="echoid-s52122" xml:space="preserve"> Apparẽt <lb/>aũt colores in iſtis luminibus ſic reflexis & refractis propter mixtionẽ nigredinis coloris cryſtalli-<lb/>ni cum lumine penetrante:</s> <s xml:id="echoid-s52123" xml:space="preserve"> & propter admixtiones umbrarum partium ipſius cryſtalli prominen-<lb/>tium ſecundũ acumen ſuorum angulorũ, quæ per 11 th.</s> <s xml:id="echoid-s52124" xml:space="preserve">2 huius proijciuntur ad partem oppoſitã in-<lb/>cidentiæ radiorũ in partem aduerſam corpori luminoſo:</s> <s xml:id="echoid-s52125" xml:space="preserve"> quarum umbrarum numerus facit diuer-<lb/>ſitatem colorum, quando lumini permiſcentur.</s> <s xml:id="echoid-s52126" xml:space="preserve"> Quoniam ubi radio luminis perpendiculari magis, <lb/>quo ad ſuperficiem incidentię (circa quã in uiciniori multorum radiorum fit aggregatio) color cry <lb/>ſtalli & umbræ cõmix us refle ctitur (quia ille radius magis eſt lumin oſus) tunc fit color rubeus.</s> <s xml:id="echoid-s52127" xml:space="preserve"> In <lb/>alijs uerò radijs ſecundum ſui debilitatẽ & coloris corporis & umbrarum plurium cõmixtionem <lb/>alij colores medij generantur.</s> <s xml:id="echoid-s52128" xml:space="preserve"> Fiunt aũt tres colores:</s> <s xml:id="echoid-s52129" xml:space="preserve"> quoniã ex tribus ſuperficie bus ſuperioribus <lb/>radij colliguntur ad quamlibet inferiorũ ſuperficierum:</s> <s xml:id="echoid-s52130" xml:space="preserve"> & color rubeus ſemper ab illa parte uide-<lb/>bitur, ubi radius perpendicularis ſuper ſuperficiem cryſtalli in contrario ſitu generatæ iridis oppo <lb/>ſitam ſoli aggregatis omnibus radijs, ſuæ ſuperficiei incidit, poſt refractionem factam ex aeris in-<lb/>terpoſiti diaphanitate.</s> <s xml:id="echoid-s52131" xml:space="preserve"> Et tunc quandoq;</s> <s xml:id="echoid-s52132" xml:space="preserve"> tres irides generantur, propter triplicem naturam refra-<lb/>ctionis in medio ſecundi diaphani rarioris, ut præmiſſum eſt:</s> <s xml:id="echoid-s52133" xml:space="preserve"> & quia ter tria ſaciũt quadratũ, qui eſt <lb/>9:</s> <s xml:id="echoid-s52134" xml:space="preserve"> erunt tunc 9 colorum indiuidua numero multiplicitatis trium ſuperficierum ſuperiorum, in nu-<lb/>merum trium in feriorum.</s> <s xml:id="echoid-s52135" xml:space="preserve"> Tres uerò erunt ſpecificæ differentiæ colorum:</s> <s xml:id="echoid-s52136" xml:space="preserve"> & fit iſtorum colorũ per <lb/>angulos corporis nulla ſenſibilis diſtinctro:</s> <s xml:id="echoid-s52137" xml:space="preserve"> quoniam & à linea angulorũ, quæ a ctu eſt indiuiſibilis, <lb/>reflexi uel refracti radij in diuiſibiles, nihil ſenſibile producunt.</s> <s xml:id="echoid-s52138" xml:space="preserve"> Non autem fiunt iſti colores iridis <lb/>per cryſtallam penitus per naturam colorum ueræ iridis, quorum diſtinctio formaliter eſt tantùm <lb/>in uiſu:</s> <s xml:id="echoid-s52139" xml:space="preserve"> ſed ſiunt per naturã lucis reflexæ à figura dicti corporis.</s> <s xml:id="echoid-s52140" xml:space="preserve"> unde etiam cauſſa ipſorum non eſt <lb/>ad uiſum facta reflexio:</s> <s xml:id="echoid-s52141" xml:space="preserve"> non enim uidentur per modum reflexionis, ſed per modum ſimplicis uiſio-<lb/>nis ut alia uiſibilia, quæ uiſui offerũtur, & à quolibet in eodẽ loco uidẽtur.</s> <s xml:id="echoid-s52142" xml:space="preserve"> Fit itaq;</s> <s xml:id="echoid-s52143" xml:space="preserve"> colorũ diftinctio <lb/>à figura corporis:</s> <s xml:id="echoid-s52144" xml:space="preserve"> quoniã à qualibet alia cryſtallo uel corpore peruio alterius figuræ colores uarij <lb/> <pb o="474" file="0776" n="776" rhead="VITELLONIS OPTICAE LIBER X."/> apparent, qui ſecundũ ſitũ colorũ iridis nõ ſunt diſtincti.</s> <s xml:id="echoid-s52145" xml:space="preserve"> Et iſtius ſignũ eſt:</s> <s xml:id="echoid-s52146" xml:space="preserve"> quòd ſi accipiatur cry-<lb/>ſtallus hexagona, & duę eius ſuperſicies cera rubea uel alia tegantur, ſic quòd inter illas duas tertia <lb/>ſuperficies maneat nõ opaca:</s> <s xml:id="echoid-s52147" xml:space="preserve"> tunc tribus alijs ſoli tranſeunti per ſoramen non magnum oppoſitis, <lb/>ſi locus operationis nõ ſit aliàs ualde luminoſus, & aliquod nigrũ ſupponatur:</s> <s xml:id="echoid-s52148" xml:space="preserve"> tunc uidebitur etiã <lb/>ex cryſtallo modica iris maxima & pulcherrima & coloris clariſsimi:</s> <s xml:id="echoid-s52149" xml:space="preserve"> quod fit propter aggregatio-<lb/>nem radiorum totius luminis ab omnibus ſuperficiebus ſuperioribus ad inferiores incidentis, <lb/>qui ad locum uicinũ unicum aggregantur.</s> <s xml:id="echoid-s52150" xml:space="preserve"> Si uerò illæ ſuperficies tres, quę nunc ſoli fuerunt oppo <lb/>ſitæ, inferiores fiant, & econuerſo aliæ tres ſuperiores:</s> <s xml:id="echoid-s52151" xml:space="preserve"> tunc iris quandoq;</s> <s xml:id="echoid-s52152" xml:space="preserve"> una, quandoq;</s> <s xml:id="echoid-s52153" xml:space="preserve"> nulla ap-<lb/>parebit.</s> <s xml:id="echoid-s52154" xml:space="preserve"> Et qui ludum iſtum iocoſum reuoluerit:</s> <s xml:id="echoid-s52155" xml:space="preserve"> inueniet, quę hic ſcripſimus, & etiam plura, quàm <lb/>per nos in tali ſolatio ſunt inuenta.</s> <s xml:id="echoid-s52156" xml:space="preserve"> Et ſi unã ex ſex ſuperficiebus dictis experimentans opacauerit:</s> <s xml:id="echoid-s52157" xml:space="preserve"> <lb/>ille ſimilia per reuolutionẽ cryſtalli ad diuerſos ſitus inueniet.</s> <s xml:id="echoid-s52158" xml:space="preserve"> Et ſi cryſtallum oculo oppoſuerit, ſic <lb/>ut tres non opacatæ ſuperficies ad oculũ uertantur:</s> <s xml:id="echoid-s52159" xml:space="preserve"> per omnes tres oculo oppoſitas illam cerã ru-<lb/>beam uidebit.</s> <s xml:id="echoid-s52160" xml:space="preserve"> Et ſi reuoluerit cryſtallum coram oculo, plures occurrent diuerſitates, quas genera-<lb/>tionibus colorum applicare quis poterit:</s> <s xml:id="echoid-s52161" xml:space="preserve"> ſemper conſiderans umbrarum immixtionem:</s> <s xml:id="echoid-s52162" xml:space="preserve"> quoniam <lb/>eadem eſt natura reflexionis formarum ad uiſum, & luminis, ad ea, quibus incidit.</s> <s xml:id="echoid-s52163" xml:space="preserve"> Non enim defer <lb/>tur color uel forma uiſibilis ad uiſum, niſi per naturam lucis, quæ eſt in ipſo:</s> <s xml:id="echoid-s52164" xml:space="preserve"> poteritq́;</s> <s xml:id="echoid-s52165" xml:space="preserve"> per experien <lb/>tiam his dictis multa addere diligens inquiſitor.</s> <s xml:id="echoid-s52166" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s52167" xml:space="preserve"> propoſitum.</s> <s xml:id="echoid-s52168" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1924" type="section" level="0" n="0"> <head xml:id="echoid-head1412" xml:space="preserve" style="it">84. Sub uaſe uitreo roiundo, pleno aqua, ſoli expoſito: colores ſimiles iridis coloribus uidẽtur.</head> <p> <s xml:id="echoid-s52169" xml:space="preserve">Sit, ut exponatur ſoli uas uitreũ rotundum ad modum urinalis, plenũ aqua pura:</s> <s xml:id="echoid-s52170" xml:space="preserve"> dico quòd ue-<lb/>rum eſt, quod proponitur.</s> <s xml:id="echoid-s52171" xml:space="preserve"> Videntur enim in ſuperficie corporis ſuppoſiti illi corpori, ut in terræ ſu <lb/>perſicie uel in alio corpore colores ſimiles iridis coloribus:</s> <s xml:id="echoid-s52172" xml:space="preserve"> quorũ generatio eſt propter uarias lu-<lb/>minis ſolis refractiones.</s> <s xml:id="echoid-s52173" xml:space="preserve"> Vt enim patet per 4 th.</s> <s xml:id="echoid-s52174" xml:space="preserve"> huius, fit una refractio ab aere ad uitrũ, alia quoq;</s> <s xml:id="echoid-s52175" xml:space="preserve"> <lb/>à uitro ad aquam:</s> <s xml:id="echoid-s52176" xml:space="preserve"> & item alia ab aqua ad uitrum, & alia à uitro ad aerem ſubiectum:</s> <s xml:id="echoid-s52177" xml:space="preserve"> quarum refra-<lb/>ctionum anguli ſunt diuerſi, ut patet per 8 huius.</s> <s xml:id="echoid-s52178" xml:space="preserve"> Secundum hos itaq;</s> <s xml:id="echoid-s52179" xml:space="preserve"> refractionũ modos cum ad-<lb/>mixtione coloris ipſorum corporum diaphanorum & umbrarum proiectarum à corporibus, lumẽ <lb/>penetrat, & circulariter diffuſum uel fortè irre gulariter ſecundum corporum diaphanorũ conue-<lb/>xas ſuperficies, uarios uiſui præſentat colores diſtinctos ſecundum præmiſſas cauſſas.</s> <s xml:id="echoid-s52180" xml:space="preserve"> Quòd ſi uas <lb/>illud extrin ſecus aqua perfuſum fuerit:</s> <s xml:id="echoid-s52181" xml:space="preserve"> pulchriores colores uiſui præſentabit:</s> <s xml:id="echoid-s52182" xml:space="preserve"> quoniã tunc nume-<lb/>rus refractionum aliqualiter augetur, & ſimiliter numerus umbrarum.</s> <s xml:id="echoid-s52183" xml:space="preserve"> Non ſunt autem hi colores <lb/>uerè colores iridis:</s> <s xml:id="echoid-s52184" xml:space="preserve"> quoniam numerantur alio colorum numero quàm colores iridis, & non perue-<lb/>niunt ad uiſum per reflexionem quemadmodum colores iridis, ſed uidentur directè, ſicut & ipſum <lb/>lumen & alij colores.</s> <s xml:id="echoid-s52185" xml:space="preserve"> Patet itaq;</s> <s xml:id="echoid-s52186" xml:space="preserve"> propoſitum.</s> <s xml:id="echoid-s52187" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1925" type="section" level="0" n="0"> <head xml:id="echoid-head1413" xml:space="preserve" style="it">85. Speculo quocun ſub aqua ſoli expoſito: figura ſolis uidebitur quaſi duplicata.</head> <p> <s xml:id="echoid-s52188" xml:space="preserve">In ſpeculo enim receptum lumen radiorum ſuper ſuperficiem aquæ perpendicularium, ſuperfi-<lb/>ciei uero ſpeculi obliquè incidentium, reflectitur à ſuperficie ſpeculi ad uiſum in loco reflexionis <lb/>exiſtentem:</s> <s xml:id="echoid-s52189" xml:space="preserve"> & ſic offert uiſui figuram ſolis.</s> <s xml:id="echoid-s52190" xml:space="preserve"> Lumen uero radiorum obliquè ſuperficiei aquę inciden <lb/>tium refringitur in ſuperficie aquæ ad perpendicularem, ductam à puncto incidentiæ ad ſuperficiẽ <lb/>aquæ per 4 th.</s> <s xml:id="echoid-s52191" xml:space="preserve"> huius.</s> <s xml:id="echoid-s52192" xml:space="preserve"> Cum itaq;</s> <s xml:id="echoid-s52193" xml:space="preserve"> illa forma refracta peruenit ad ſpeculi ſuperficiem:</s> <s xml:id="echoid-s52194" xml:space="preserve"> tunc ab illa ſu-<lb/>perficie, cui obliquè incidit, reflectitur iterũ ad uiſum:</s> <s xml:id="echoid-s52195" xml:space="preserve"> apparentq́;</s> <s xml:id="echoid-s52196" xml:space="preserve"> duæ figuræ 4olis:</s> <s xml:id="echoid-s52197" xml:space="preserve"> una maior pro-<lb/>pter ſimplicem reflexionem:</s> <s xml:id="echoid-s52198" xml:space="preserve"> alia quoq;</s> <s xml:id="echoid-s52199" xml:space="preserve"> minor propter refractionem, quę in medio denſiori minuit <lb/>figuram poſtmodum reflexam:</s> <s xml:id="echoid-s52200" xml:space="preserve"> uideturq́;</s> <s xml:id="echoid-s52201" xml:space="preserve"> illa ſecunda figura ſolis, quaſi ſit forma ſtellæ ſequentis <lb/>corpus ſolis.</s> <s xml:id="echoid-s52202" xml:space="preserve"> Eſt autem & ipſa forma ſolis:</s> <s xml:id="echoid-s52203" xml:space="preserve"> quod patet:</s> <s xml:id="echoid-s52204" xml:space="preserve"> quoniam extra radium ſolis cum figura ſolis <lb/>à ſuperficie ſpeculi per ſe non reflectitu.</s> <s xml:id="echoid-s52205" xml:space="preserve"> Et hanc refractam formam accidit uideri.</s> <s xml:id="echoid-s52206" xml:space="preserve"> Et ſi planè ſpe-<lb/>culum ſub aqua deducatur in ſolis radium:</s> <s xml:id="echoid-s52207" xml:space="preserve"> tunc eadem numero forma, quæ prius ſub minori lumi-<lb/>ne fuit uiſa, uidebitur amplius, quàm prius, luminoſa:</s> <s xml:id="echoid-s52208" xml:space="preserve"> & ſecundum motum aquæ uidebitur moueri <lb/>circa reflexam figuram ſolis.</s> <s xml:id="echoid-s52209" xml:space="preserve"> Patet ergo propoſitum.</s> <s xml:id="echoid-s52210" xml:space="preserve"> Et quoniam nos diuinæ gratiæ ſuffragante <lb/>præſidio, tres propoſitos uidendi modos ſecundum omnem ipſorum, quatenus potuimus, diuerſi <lb/>tatem tranſcurrimus, nec condignum aliquid tantæ munificentiæ diuinæ bonitati red dere poſsi-<lb/>bile nobis eſt:</s> <s xml:id="echoid-s52211" xml:space="preserve"> ad illas tamen, quas poſſumus, gratiarum actiones conſurgimus ei, qui uerè trinus & <lb/>unus eſt:</s> <s xml:id="echoid-s52212" xml:space="preserve"> ſoli nihil in rebus entibus conforme, nihil coæternum, nihil æ quebonum æſtimantes:</s> <s xml:id="echoid-s52213" xml:space="preserve"> cui <lb/>ſit honor & gloria per infinita ſecula.</s> <s xml:id="echoid-s52214" xml:space="preserve"> Amen.</s> <s xml:id="echoid-s52215" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div1926" type="section" level="0" n="0"> <head xml:id="echoid-head1414" xml:space="preserve">VITELLONIS FILII THVRINGORVM ET POLO-<lb/>norum opticæ finis.</head> <head xml:id="echoid-head1415" xml:space="preserve">BASILE AE, <lb/>EX OFFICINA EPISCOPIANA, PER EVSEBIVM <lb/>Epiſcopium, & Nicolai F hæredes. Anno M. D. LXXII. <lb/>Menſe Auguſto.</head> <pb file="0777" n="777"/> <pb file="0778" n="778"/> <figure> <caption xml:id="echoid-caption4" xml:space="preserve">EPISCOP.</caption> </figure> </div></text> </echo>